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  <row Id="1" PostTypeId="1" AcceptedAnswerId="132" CreationDate="2014-10-28T20:17:22.573" Score="15" ViewCount="516" Body="&lt;p&gt;A well known &lt;a href=&quot;http://math.berkeley.edu/~robin/Cauchy/&quot; rel=&quot;nofollow&quot;&gt;urban legend&lt;/a&gt; states that &lt;a href=&quot;http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy&quot; rel=&quot;nofollow&quot;&gt;Cauchy's&lt;/a&gt; last words to the Academy where: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;C'est ce que j'expliquerai plus au long dans un prochain mémoire.&lt;/em&gt; (&quot;I will explain it in greater detail in my next memoire.&quot;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've always wondered, never managing to figure it out, what was to be explained in his next memoire. Is there any guess about what the new theory to be unveiled consisted off?&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="26" LastEditDate="2015-06-08T08:47:00.643" LastActivityDate="2015-06-08T08:47:00.643" Title="Cauchy's undead theory" Tags="|mathematics|biographical-details|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2" PostTypeId="1" CreationDate="2014-10-28T21:05:00.990" Score="25" ViewCount="5007" Body="&lt;p&gt;We know that Pythagoreans in Ancient Greece discovered that the square root of two is an irrational number. Why was that discovery historically significant? What value was that knowledge to the Ancient Greeks?&lt;/p&gt;&#xA;" OwnerUserId="11" LastEditorUserId="26" LastEditDate="2014-10-28T22:05:54.117" LastActivityDate="2015-09-17T17:15:38.030" Title="Irrationality of the square root of 2" Tags="|mathematics|ancient-greece|number-theory|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3" PostTypeId="1" AcceptedAnswerId="53" CreationDate="2014-10-28T21:11:41.460" Score="75" ViewCount="14985" Body="&lt;p&gt;Aside from the fact that Fermat was a genius, is it probable that he actually did have a proof?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some specifics that I think would point one way or another:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Would the mathematics of his day allow him to prove it similarly to&#xA;how &lt;a href=&quot;http://en.wikipedia.org/wiki/Andrew_Wiles&quot;&gt;Andrew Wiles&lt;/a&gt; did?&lt;/li&gt;&#xA;&lt;li&gt;Have we lost any large portion of his works?&lt;/li&gt;&#xA;&lt;li&gt;Is this something he did frequently?  That is, say that he had a&#xA;proof on one paper and prove it later in another paper?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="19" LastEditorUserId="4251" LastEditDate="2021-02-27T07:42:06.850" LastActivityDate="2021-02-27T07:42:06.850" Title="What evidence is there that Fermat had a proof for his Last Theorem?" Tags="|mathematics|mathematicians|number-theory|17th-century|fermat|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4" PostTypeId="1" AcceptedAnswerId="155" CreationDate="2014-10-28T21:13:19.517" Score="23" ViewCount="10613" Body="&lt;p&gt;It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using Roman numerals was notably clumsy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the ancient Chinese did not really have place value notation either, and did not have a symbol for zero. Yet that culture discovered many pretty advanced arithmetic results (for example, the Chinese Remainder Theorem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what exactly was it about the invention of zero in Western culture that was so useful?&lt;/p&gt;&#xA;" OwnerUserId="17" LastEditorUserId="62" LastEditDate="2014-11-06T17:31:46.897" LastActivityDate="2020-01-22T07:28:53.990" Title="Ancient Chinese numbering system" Tags="|mathematics|number-theory|ancient-china|zero|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5" PostTypeId="2" ParentId="3" CreationDate="2014-10-28T21:17:07.257" Score="33" Body="&lt;p&gt;There is no way that Fermat could have had anything approaching the now commonly-accepted proof. Almost none of the concepts in that proof were known in any form in Fermat's time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, Fermat is known for publishing very few of his proofs; almost none survive today, and even in the 1800s there was significant doubt in the mathematical community that he had proofs for much of what he stated as fact. (This is not to diminish his results; in fact, it makes it all the more impressive that his intuition led him to so many results later proved true).&lt;/p&gt;&#xA;" OwnerUserId="17" LastActivityDate="2014-10-28T21:17:07.257" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6" PostTypeId="1" AcceptedAnswerId="86" CreationDate="2014-10-28T21:19:02.733" Score="25" ViewCount="1036" Body="&lt;p&gt;Many scientific units are named after scientists, for example,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Tesla for magnetic flux&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Farad for capacitance&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Newton for force.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;When did the tradition of naming scientific units begin?&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="4251" LastEditorDisplayName="user22" LastEditDate="2020-07-13T00:33:04.600" LastActivityDate="2020-07-13T07:58:15.013" Title="When did the names of scientists first become the names of scientific units?" Tags="|terminology|discoveries|units|scientists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7" PostTypeId="2" ParentId="2" CreationDate="2014-10-28T21:30:09.017" Score="15" Body="&lt;p&gt;According to &lt;a href=&quot;http://mathworld.wolfram.com/PythagorassConstant.html&quot; rel=&quot;noreferrer&quot;&gt;this link&lt;/a&gt;, Legend has it that &lt;a href=&quot;http://en.wikipedia.org/wiki/Hippasus&quot; rel=&quot;noreferrer&quot;&gt;Hippasus&lt;/a&gt; first discovered the irrationality of $\sqrt{2}$. The second link in fact mentions a legend that held that supporters of Pythagoras murdered Hippasus -- who allegedly discovered the irrationality of $\sqrt{2}$ on a boat in the middle of the sea -- by throwing him overboard immediately after he informed them of his discovery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Pythagoreans had the belief that all numbers could be constructed as the ratio of 2 numbers.  (That they were rational)  So basically it was a big deal because it flew in the face of knowledge.  All of their work was based on the premise of rational numbers being all the numbers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any new evidence that completely overturns a fundamental truth has often been met with derision.  Even in (relatively) modern times, &lt;a href=&quot;http://en.wikipedia.org/wiki/Imaginary_number&quot; rel=&quot;noreferrer&quot;&gt;Imaginary Numbers&lt;/a&gt; were considered &quot;fictitious or useless, much as zero and the negative numbers once were.&quot;&lt;/p&gt;&#xA;" OwnerUserId="19" LastEditorUserId="26" LastEditDate="2014-12-03T10:14:43.643" LastActivityDate="2014-12-03T10:14:43.643" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="8" PostTypeId="1" AcceptedAnswerId="41" CreationDate="2014-10-28T21:35:24.573" Score="9" ViewCount="732" Body="&lt;p&gt;Currently, the &lt;a href=&quot;http://en.wikipedia.org/wiki/Wave-particle_duality&quot; rel=&quot;nofollow&quot;&gt;wave-particle duality&lt;/a&gt; model for light is the accepted model. From &lt;a href=&quot;http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html&quot; rel=&quot;nofollow&quot;&gt;HyperPhysics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The evidence for the description of light as waves was well established at the turn of the century when the photoelectric effect introduced firm evidence of a particle nature as well. On the other hand, the particle properties of electrons was well documented when the DeBroglie hypothesis and the subsequent experiments by Davisson and Germer established the wave nature of the electron.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What theories preceded both the the wave and particle theories of light?&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="62" LastEditDate="2015-04-12T16:31:41.250" LastActivityDate="2015-04-12T16:31:41.250" Title="What theories preceded the wave, particles and duality models of light?" Tags="|particle-physics|quantum-mechanics|light|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="9" PostTypeId="1" CreationDate="2014-10-28T21:35:58.003" Score="74" ViewCount="14696" Body="&lt;p&gt;It is well known that Évariste Galois died a young man. I have heard that he died in a duel. What was the duel about? More rather what is the back story behind his death and did he really write down most of his great ideas days before his death?&lt;/p&gt;&#xA;" OwnerUserId="27" LastEditorUserId="4251" LastEditDate="2021-06-23T00:49:49.637" LastActivityDate="2021-10-10T21:16:23.760" Title="Why was Évariste Galois killed?" Tags="|mathematicians|biographical-details|19th-century|galois|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="10" PostTypeId="1" AcceptedAnswerId="60" CreationDate="2014-10-28T21:43:09.557" Score="29" ViewCount="2227" Body="&lt;p&gt;In theoretical physics, the modern perspective on &lt;a href=&quot;http://en.wikipedia.org/wiki/Gauge_theory&quot; rel=&quot;noreferrer&quot;&gt;gauge theory&lt;/a&gt; is that it is most elegantly described in the 'language' of &lt;a href=&quot;http://en.wikipedia.org/wiki/Differential_geometry&quot; rel=&quot;noreferrer&quot;&gt;differential geometry&lt;/a&gt;. I am interested in the history behind these ideas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, it appears (e.g. &lt;a href=&quot;http://universe-review.ca/R15-21-YangPauli.htm&quot; rel=&quot;noreferrer&quot;&gt;this anecdote by C.N. Yang&lt;/a&gt;) that the 'fathers' of (physical) gauge theory were unaware of the deep connection to geometry. In fact, I am not sure &lt;a href=&quot;http://en.wikipedia.org/wiki/Gauge_theory#Mathematical_formalism&quot; rel=&quot;noreferrer&quot;&gt;the rather advanced mathematical framework&lt;/a&gt; that one needs to understand gauge theories was even in place (i.e. understood as a fully abstract theory) at all at the time. I also &lt;em&gt;think&lt;/em&gt;, but do not know for sure, that the mathematical theory of principal bundles and other related mathematical objects and/or structures was first developed fully abstractly, before it was realized how useful it was in describing physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess what I'm asking for is a (cursory) account of the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) What were the key insights that allowed physicists and mathematicians to understand gauge theories in this light?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) When (and by whom?) were these essential steps first made?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For those interested, &lt;a href=&quot;https://physics.stackexchange.com/questions/122946/em-and-geometry-a-historical-perspective&quot;&gt;this&lt;/a&gt; is a (tangentially) related question of mine, from physics.SE&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2017-04-21T21:27:52.117" Title="When and how was the geometric understanding of gauge theories developed?" Tags="|mathematics|theoretical-physics|mathematical-physics|differential-geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="11" PostTypeId="1" CreationDate="2014-10-28T21:47:27.107" Score="10" ViewCount="93" Body="&lt;p&gt;&lt;em&gt;Chow, Tommy WS, and John Sum. &quot;&lt;a href=&quot;http://link.springer.com/article/10.1007/s00521-010-0427-z&quot;&gt;Guest editorial: special issue on the emerging applications of neural networks.&lt;/a&gt;&quot; Neural Computing &amp;amp; Applications 20.7 (2011): 923-924.&lt;/em&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The take off of neural computing research was probably due to the&#xA;  announcement that Japan had resumed the research of neural networks.&#xA;  The US feared that they would be overtaken by the Japanese. This&#xA;  resulted in Ira Skurnick, a program funding manager in the US defense&#xA;  Science Office, decided to fund neural computing research again.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How true is it that the take off of &lt;a href=&quot;http://en.wikipedia.org/wiki/Artificial_neural_network&quot;&gt;neural computing research&lt;/a&gt; in the 80's was due to Japan?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-10-31T16:10:49.113" Title="Was the take off of neural computing research in the 80's due to Japan?" Tags="|computer-science|japan|machine-learning|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="12" PostTypeId="1" AcceptedAnswerId="104" CreationDate="2014-10-28T21:47:39.240" Score="17" ViewCount="309" Body="&lt;p&gt;There are many claimants for the first telescope. Amongst these are the claims placed at the doors of Bacon and Digges. The Bacon claim is very sketchy, boiling down to one sentence and is easy to dismiss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Thomas Digges claim seems more accomplished. From what I gather the main objection is that the claimed performance is much greater than what was possible with the technology of the day combined with the theoretical refraction limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But could this be just Elizabethan hyperbole - figure of speech that was typical during the era? It made things appear closer, but not to the extent that a literal interpretation of the descriptions would suggest?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="26" LastEditDate="2014-10-28T22:03:05.573" LastActivityDate="2014-11-07T21:39:14.367" Title="Did the Digges Telescope actually exist?" Tags="|telescope|physics|experimental-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="14" PostTypeId="2" ParentId="4" CreationDate="2014-10-28T21:56:23.303" Score="13" Body="&lt;p&gt;The successful numbering system did not have 0, even as a place holder, due to them not needing it, according to the University of St. Andrew's website &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Chinese_numerals.html&quot; rel=&quot;noreferrer&quot;&gt;Chinese numerals&lt;/a&gt;, due to them developing a system that had symbols for the larger values, i.e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/4qbed.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a second form of Chinese numerals began to be used from the 4th century BC when counting boards came into use. A counting board consisted of a checker board with rows and columns.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/nHi2x.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Both images are from the link above).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To represent a '0' place holder, a gap was left in its place. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Xiahou Yang's Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) written in the 5th century AD notes that to multiply a number by 10, 100, 1000, or 10000 all that needs to be done is that the rods on the counting board are moved to the left by 1, 2, 3, or 4 squares. Similarly to divide by 10, 100, 1000, or 10000 the rods are moved to the right by 1, 2, 3, or 4 squares. What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The significance to Western ideas, according to the YaleGlobal web article &lt;a href=&quot;http://yaleglobal.yale.edu/about/zero.jsp&quot; rel=&quot;noreferrer&quot;&gt;The History of Zero&lt;/a&gt; (Wallin, 2002), originally, zero was a placeholder (in the sense of being used for tens, hundreds, thousands etc).  According to the article, it was in India where the notion of zero started to have meaning by itself in about the 7th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The significance of zero in Western Culture when it arrived in Europe in the mid-12th century through Spain and via Fibonacci in the early 13th century, gaining merchants a more efficient way to balance their books, with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fibonacci's developments quickly gained notice by Italian merchants and German bankers, especially the use of zero. Accountants knew their books were balanced when the positive and negative amounts of their assets and liabilities equaled zero. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This actually got the number outlawed, so was named 'cipher' in merchant based encrypted messages to overcome the ban.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rene Descarte was able to progress the use of zero in what we know as Cartesian coordinate system.  With the origin of the coordinate system occurring at (0,0).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From there, arguably the greatest innovation of the notion of zero was from the dilemma of dividing by zero, which (from the article):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the 1600's, Newton and Leibniz solved this problem independently and opened the world to tremendous possibilities. By working with numbers as they approach zero, calculus was born without which we wouldn't have physics, engineering, and many aspects of economics and finance.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastEditorDisplayName="user22" LastEditDate="2014-10-29T00:27:42.867" LastActivityDate="2014-10-29T00:27:42.867" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="15" PostTypeId="5" CreationDate="2014-10-28T22:03:34.060" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-10-28T22:03:34.060" LastActivityDate="2014-10-28T22:03:34.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="16" PostTypeId="4" CreationDate="2014-10-28T22:03:34.060" Score="0" Body="An island country in the western Pacific, located east of the Asian mainland." OwnerUserId="10" LastEditorUserId="10" LastEditDate="2014-11-02T21:26:31.517" LastActivityDate="2014-11-02T21:26:31.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="17" PostTypeId="1" AcceptedAnswerId="145" CreationDate="2014-10-28T22:14:15.787" Score="20" ViewCount="779" Body="&lt;p&gt;The so-called &amp;quot;Hilbert space&amp;quot; is named after mathematician &lt;a href=&quot;http://en.wikipedia.org/wiki/David_Hilbert&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Hilbert&lt;/a&gt;. Later, this was generalized into &amp;quot;Banach spaces&amp;quot; by &lt;a href=&quot;http://en.wikipedia.org/wiki/Stefan_Banach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stefan Banach&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that Hilbert was German and Banach was Polish, and there did not appear to be any &amp;quot;major&amp;quot; connection between them (that is, no more than between two &amp;quot;random&amp;quot; European mathematicians, although this was a very small circle at the time). Yet there is a fairly strong connection between Hilbert's work and Banach's work.&lt;/p&gt;&#xA;&lt;p&gt;How did Banach manage to take off on Hilbert's work without knowing him well? (E.g. Banach seems to have been much closer to Hugo Steinhaus of the Banach-Steinhaus Theorem.) Or did the two work together/know each other better than I have given them credit for?&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="3434" LastEditDate="2020-07-12T02:40:42.143" LastActivityDate="2022-03-05T16:26:58.430" Title="What was the connection between David Hilbert and Stefan Banach?" Tags="|mathematics|mathematicians|biographical-details|hilbert|banach|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="18" PostTypeId="1" CreationDate="2014-10-28T22:14:40.623" Score="10" ViewCount="359" Body="&lt;p&gt;I read on &#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Academic_publishing&quot;&gt;http://en.wikipedia.org/wiki/Academic_publishing&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One of the earliest research journals is the Philosophical&#xA;  Transactions of the Royal Society, created in the 17th century. At&#xA;  that time, the act of publishing academic inquiry was controversial,&#xA;  and widely ridiculed. It was not at all unusual for a new discovery to&#xA;  be announced as an anagram, reserving priority for the discoverer, but&#xA;  indecipherable for anyone not in on the secret: both Isaac Newton and&#xA;  Leibniz used this approach. However, this method did not work well.&#xA;  Robert K. Merton, a sociologist, found that &lt;strong&gt;92% of cases of&#xA;  simultaneous discovery in the 17th century ended in dispute&lt;/strong&gt;. The&#xA;  number of disputes dropped to 72% in the 18th century, 59% by the&#xA;  latter half of the 19th century, and 33% by the first half of the 20th&#xA;  century. The decline in contested claims for priority in research&#xA;  discoveries can be credited to the increasing acceptance of the&#xA;  publication of papers in modern academic journals, with estimates&#xA;  suggesting that around 50 million journal articles (see Reference 1) have been&#xA;  published since the first appearance of the Philosophical&#xA;  Transactions.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Reference 1: Jinha, A. E. (2010). &quot;Article 50 million: An estimate of the number of scholarly articles in existence&quot;. Learned Publishing 23 (3): 258–263. doi:10.1087/20100308. Archived from the original on 2012-05-23&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How comes 92% of cases of simultaneous discovery in the 17th century ended in dispute, whereas authors had often &quot;encrypted&quot; their results to be able to claim paternity later on?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="4251" LastEditDate="2020-07-08T21:35:07.750" LastActivityDate="2020-07-08T21:35:07.750" Title="Why did 92% of cases of simultaneous discovery in the 17th century end in dispute?" Tags="|discoveries|publishing|academia|17th-century|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="19" PostTypeId="1" AcceptedAnswerId="25" CreationDate="2014-10-28T22:23:46.030" Score="17" ViewCount="678" Body="&lt;p&gt;Kekulé, who discovered the chemical formula for benzene in 1865, claimed that he had guessed the ring formula with the alternating single and double bonds in a day-dream. Are there similar examples of significant scientific progress claimed to be caused by dreaming?&lt;/p&gt;&#xA;" OwnerUserId="37" LastEditorUserId="13" LastEditDate="2014-11-04T16:38:08.567" LastActivityDate="2019-05-06T17:57:44.707" Title="Scientific progress claimed to be caused by dreaming" Tags="|discoveries|chemistry|examples|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="20" PostTypeId="1" CreationDate="2014-10-28T22:24:29.163" Score="11" ViewCount="1204" Body="&lt;p&gt;Nowadays, science is aware of the divisions of the Earth's atmosphere, particularly the division between the troposphere and the stratosphere, as shown in the diagram below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Yt0bD.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://essayweb.net/geology/weather.shtml&quot; rel=&quot;noreferrer&quot;&gt;Image source&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When and how was the division between the troposphere and stratosphere determined?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorDisplayName="user22" LastEditDate="2014-10-28T22:43:58.280" LastActivityDate="2014-10-30T20:16:22.490" Title="When was the division between the troposphere and stratosphere determined?" Tags="|experimental-physics|atmosphere|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="21" PostTypeId="1" CreationDate="2014-10-28T22:33:58.553" Score="14" ViewCount="142" Body="&lt;p&gt;Nowadays researchers are generally not paid to review research articles. Has it always been the case? If not, how come it has changed?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="25" LastEditDate="2014-10-28T22:38:51.820" LastActivityDate="2014-10-29T01:46:24.777" Title="Have reviewers never been paid?" Tags="|academia|publishing|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="22" PostTypeId="1" AcceptedAnswerId="43" CreationDate="2014-10-28T22:35:17.300" Score="12" ViewCount="3290" Body="&lt;p&gt;Related to my previous question &lt;a href=&quot;https://hsm.stackexchange.com/questions/20/when-was-the-division-between-the-troposphere-and-stratosphere-determined&quot;&gt;When was the division between the troposphere and stratosphere determined?&lt;/a&gt;, an important atmospheric feature at this boundary is the Ozone Layer, as shown in the image below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/BNQcW.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nc-climate.ncsu.edu/edu/k12/.ozonelayer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Image source&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that the Ozone Layer concentration is very low, according to &lt;a href=&quot;http://ozonewatch.gsfc.nasa.gov/facts/ozone.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;NASA&lt;/a&gt;,  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The peak concentration of ozone occurs at an altitude of roughly 32 kilometers (20 miles) above the surface of the Earth. At that altitude, ozone concentration can be as high as 15 parts per million (0.0015 percent).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How was the Ozone Layer discovered?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="-1" LastEditorDisplayName="user22" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-09-05T07:53:49.437" Title="How was the Ozone Layer discovered?" Tags="|discoveries|atmosphere|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="23" PostTypeId="1" AcceptedAnswerId="183" CreationDate="2014-10-28T22:37:51.037" Score="20" ViewCount="3838" Body="&lt;p&gt;The year before he died, mathematician &lt;a href=&quot;http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes&quot;&gt;Rene Descartes&lt;/a&gt; accepted an invitation to tutor the brilliant 19-year old Queen Christina of Sweden (some thirty years younger). He apparently died from the Swedish cold and long hours on the job. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it surprising that he would accept such an offer, since as a young military man, Descartes had served as an officer in the army of Maximilian of Bavaria, an enemy of Christina's father, Gustavus Adolphus. (On the other hand, his native France was an ally of Sweden).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is believed by some (as cited in the Wikipedia article), that he was &quot;very close&quot; to Queen Christina, e.g. perhaps converting her to Catholicism, and that this may have contributed to his death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any records or published contemporary accounts about how he felt about Queen Christina, Sweden, or other factors that would have motivated him to take a job that ultimately killed him?&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="4251" LastEditDate="2020-07-07T22:51:52.073" LastActivityDate="2020-07-07T22:51:52.073" Title="Why did Rene Descartes go to Sweden?" Tags="|mathematics|biographical-details|reference-request|descartes|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="24" PostTypeId="1" AcceptedAnswerId="28" CreationDate="2014-10-28T22:44:19.353" Score="13" ViewCount="1939" Body="&lt;p&gt;While I have heard speculative answers to this question, I do not know one which can be supported.  Is there any information explaining why Nobel did not chose to include this topic? Has there even been a serious attempt within the committee to add one or an explanation why they will not?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="26" LastEditDate="2014-10-28T22:46:21.820" LastActivityDate="2014-10-29T12:58:36.273" Title="Why isn't there a Nobel prize in Mathematics?" Tags="|mathematics|academia|academic-awards|nobel-prize|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="25" PostTypeId="2" ParentId="19" CreationDate="2014-10-28T22:44:58.147" Score="8" Body="&lt;p&gt;Indian mathematician  &lt;a href=&quot;http://en.wikipedia.org/wiki/Srinivasa_Ramanujan&quot;&gt;Srinivasa Ramanujan&lt;/a&gt; claimed and is often said to have come up with theorems and questions in his dreams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: &lt;a href=&quot;http://www.huffingtonpost.com/2012/12/27/ramanujans-mock-modular-forms_n_2371680.html&quot;&gt;Ramanujan's Mock Modular Forms: Indian Mathematician's Dream Conjecture Finally Proven&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While on his death bed, the brilliant Indian mathematician Srinivasa&#xA;  Ramanujan cryptically wrote down functions he said came to him in&#xA;  dreams, with a hunch about how they behaved. Now 100 years later,&#xA;  researchers say they've proved he was right.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;We've solved the problems from his last mysterious letters. For&#xA;  people who work in this area of math, the problem has been open for 90&#xA;  years,&quot; Emory University mathematician Ken Ono said.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, it is difficult to verify such claims. But regardless I have always found Srinivasa Ramanujan's achievements amazing.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-10-28T22:44:58.147" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="26" PostTypeId="1" AcceptedAnswerId="220" CreationDate="2014-10-28T22:56:10.307" Score="49" ViewCount="5623" Body="&lt;p&gt;In brief, I am looking for an example where Occam’s razor favoured a theory A over another theory B, but theory B turned out to be a better description of reality later. But let me formulate some criteria of what I mean by that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First of all, as our modern perspective may be biased – e.g., due to didactical advances in the prevailing theory or new insights into historic experiments –, consider my criteria on qualities of theories to refer to historical scientific opinions and statements, as long as they can be considered to be based on reason (instead of, e.g., being strongly influenced by a religious bias).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At a given time, two theories (A and B) were comparably good at describing the same aspect of reality as it was observable at that time. They need not have been perfect descriptions of available observations, but they should not have been so far-off that they were applicable only to special cases or none at all.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Occam’s razor was reasonably invoked in a scientific dispute in favor of theory A. This invocation needs neither to have happened by name nor in a peer-reviewed publication (if such existed at that time at all). I am also interested in, but do not prefer, cases in which adherents of both theories invoked Occam’s razor (or similar) to argue against the respective other theory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;At a later time, theory B or a reasonably small modification thereof turned out to be a better description of reality than theory A. Alternatively, theory B is still used today for some aspects, while theory A isn’t. Theory B does not need to be the prevailing theory today.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am asking out of curiosity. I am very well aware that the existence of such an example does not invalidate Occam’s razor.&lt;/p&gt;&#xA;" OwnerUserId="13" LastEditorUserId="473" LastEditDate="2014-12-09T00:17:41.047" LastActivityDate="2020-08-13T18:12:17.263" Title="Was Occam’s razor ever wrong?" Tags="|examples|scientific-method|philosophy-of-science|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="27" PostTypeId="2" ParentId="24" CreationDate="2014-10-28T22:56:30.233" Score="2" Body="&lt;p&gt;This &quot;oversight&quot; of Mr. Nobel's was rectified in 1980 by the &lt;a href=&quot;http://en.wikipedia.org/wiki/Crafoord_Prize&quot; rel=&quot;nofollow&quot;&gt;Crafoord Prize&lt;/a&gt; in mathematics. Now that there is such a prize, it's unlikely that the Nobel Committee will create another one for mathematics, even though the Crafoord Prize is much less well known,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mr. Nobel not only created the prizes, but selected the Swedish academies to award them (except for the Peace Prize, which is awarded by the Norwegians, whom he wanted to placate).&#xA;Perhaps he lacked confidence in the Swedish mathematics academy. He also didn't create a prize in economics, but the Swedish Academy of Science took it upon itself to rectify this.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-10-28T22:56:30.233" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="28" PostTypeId="2" ParentId="24" CreationDate="2014-10-28T23:00:48.443" Score="11" Body="&lt;p&gt;Let's first mention that the popular myth that Nobel decided not to fund a prize for mathematicians because his wife was cheating on him with a mathematician (often said to be &lt;a href=&quot;http://en.wikipedia.org/wiki/G%C3%B6sta_Mittag-Leffler&quot;&gt;Gösta Mittag-Leffler&lt;/a&gt;) is (predictably) not true. In fact, it's trivially false since Nobel was never married! Furthermore, in the correspondence between him and his lover, there is no sign of anything like an affair. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that the reason there is no Nobel Prize in Mathematics is much more dull: Nobel probably simply wasn't &lt;em&gt;that&lt;/em&gt; interested in pure mathematics. The other categories are more natural, considering the context: Nobel was personally very much involved in physics and chemistry, so prizes for that were beyond questioning. It was also clear to him that medicine enormously benefits mankind (this is almost the &lt;em&gt;definition&lt;/em&gt; of medicine, when you think about it!) It appears that the peace prize was suggested by his secretary and old lover, who would go on to win the prize in 1905. Finally, the prize in literature seems to simply have come from the fact that Nobel was very interested in literature. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an alternative theory, however. At the time, the same Mittag-Leffler that is often accused of stealing Nobel's wife had just recently ensured for King Oscar II to create an 'endowment prize for various mathematicians throughout Europe'. Perhaps, this convinced Nobel that no additional prize for mathematicians was needed.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources: &lt;a href=&quot;http://en.wikipedia.org/wiki/Nobel_Prize&quot;&gt;Wikipedia article on the Nobel Prizes&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Alfred_Nobel&quot;&gt;Wikipedia article on Alfred Nobel&lt;/a&gt;, and &lt;a href=&quot;http://www.snopes.com/science/nobel.asp&quot;&gt;other&lt;/a&gt; &lt;a href=&quot;http://www.todayifoundout.com/index.php/2010/10/the-reason-there-is-no-nobel-prize-for-mathematics-had-nothing-to-do-with-any-wifemistress-of-alfred-nobel/&quot;&gt;websites&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2014-10-28T23:00:48.443" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="29" PostTypeId="1" AcceptedAnswerId="33" CreationDate="2014-10-28T23:03:07.440" Score="23" ViewCount="3336" Body="&lt;p&gt;Is it known how Hilbert initially reacted to Gödel's incompleteness theorems upon their announcement at the Königsberg conference in 1930, or their publication in 1931?&lt;/p&gt;&#xA;" OwnerUserId="37" LastEditorUserId="26" LastEditDate="2020-07-11T19:08:03.317" LastActivityDate="2020-07-11T19:08:03.317" Title="Hilbert's reaction to Gödel's incompleteness theorems" Tags="|mathematics|mathematicians|biographical-details|mathematical-logic|hilbert|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="30" PostTypeId="1" CreationDate="2014-10-28T23:12:17.827" Score="21" ViewCount="1694" Body="&lt;p&gt;From a young age I have been taught that the stones used to create the great pyramids were mined.  Concrete and cements were largely invented by the Romans/Greeks especially for use as mortar.  More recently, I've heard other explanations.&lt;/p&gt;&#xA;&lt;p&gt;Prof. Joseph Davidovits has been making claims that the Egyptians used geopolymer chemistry in order to create the stones used in the pyramids, rather than mining them. This would explain a lot about how the pyramids were constructed and about the developmental history of cement, one of the most important man-made materials and chemical reactions. Researching this I also learned that gypsum was used by the Egyptians to make simple cements.&lt;/p&gt;&#xA;&lt;p&gt;What do we currently know about the chemistry knowledge the ancient Egyptians had with regard to cementitious materials?  Would they have had the ability to manufacture artificial stones on the scale needed to make the pyramids?&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="7205" LastEditDate="2022-09-28T13:36:10.440" LastActivityDate="2022-09-28T13:36:10.440" Title="To what degree did ancient Egyptians use Cement Chemistry" Tags="|chemistry|ancient-egypt|architectural-engineering|materials-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="31" PostTypeId="1" CreationDate="2014-10-28T23:13:33.977" Score="17" ViewCount="1144" Body="&lt;p&gt;On Nov 12, 1946 the Americans organized a &lt;a href=&quot;http://www.wired.com/2009/11/1112abacus-beats-calculator/&quot; rel=&quot;noreferrer&quot;&gt;contest&lt;/a&gt; in Japan to compare the Japanese Abacus with the American Electric Calculator. The &lt;a href=&quot;http://www.ee.ryerson.ca/~elf/abacus/abacus-contest.html&quot; rel=&quot;noreferrer&quot;&gt;Abacus won&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Civilization, on the threshold of the atomic age, tottered Monday&#xA;  afternoon as the 2,000-year-old abacus beat the electric calculating&#xA;  machine in adding, subtracting, dividing and a problem including all&#xA;  three with multiplication thrown in, according to UP. Only in&#xA;  multiplication alone did the machine triumph...&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/sQuAi.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How comes the abacus lost the multiplication sub-contest, and won the 3 others (addition, subtraction and division)? From what I recall, but I cannot find the source anymore, this was at least partly due to the fact that an officer has to pronounce the numbers: unlike the  calculator operator, the abacus operator could start computing before hearing the last figure of the numbers to be computed. Is that true and if so would giving the numbers to be completed written down on some papers have made any difference?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2015-03-01T13:36:30.677" LastActivityDate="2022-09-19T08:11:59.907" Title="The Abacus vs. the Electric Calculator (Nov 12, 1946): Why did the latter lose?" Tags="|mathematics|computer-science|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="32" PostTypeId="2" ParentId="18" CreationDate="2014-10-28T23:14:44.910" Score="10" Body="&lt;p&gt;Looking at a brief &lt;a href=&quot;http://www.historyworld.net/wrldhis/PlainTextHistories.asp?ParagraphID=fly&quot;&gt;history of communication&lt;/a&gt; and your post it seems that there were disputes because who originally published the discovery was easily disputable.   The fact that there was a steady decline in such disputes was due to improving communication technology AND standard publishing practices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the point of using an anagram while publishing.  Many scientific discoveries have been stolen.  &lt;a href=&quot;http://www.the-scientist.com/?articles.view/articleNo/32287/title/All-s-Not-Fair-in-Science-and-Publishing/&quot;&gt;This article&lt;/a&gt; states that the discovery of the structure of DNA and the AIDS virus was controversial, and both of those topic were quite significant.  Its fairly well known that people steal ideas, even Family Guy pokes fun at Einstein for working at a patent office before becoming a significant person in his field.  (The joke being that someone else was patenting The Theory of Relativity and he stole it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'de conjecture that intelligent people were (unintentionally?) using a &lt;a href=&quot;http://en.wikipedia.org/wiki/Shared_secret&quot;&gt;shared secret&lt;/a&gt; to communicate their ideas in order to prevent such an occurrence.  This way by the time thieves would be able to decipher the information, there would already be widespread knowledge of who originally created the idea.  &lt;/p&gt;&#xA;" OwnerUserId="19" LastEditorUserId="19" LastEditDate="2014-10-28T23:19:49.527" LastActivityDate="2014-10-28T23:19:49.527" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="33" PostTypeId="2" ParentId="29" CreationDate="2014-10-28T23:33:29.117" Score="18" Body="&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/2273764?seq=2&quot;&gt;The paper 'Some facts about Kurt Gödel' by Wang (1981)&lt;/a&gt; (regrettably paywalled) contains a section that suggests Hilbert was &lt;em&gt;not&lt;/em&gt; present when Gödel originally announced his sketch of the First Incompleteness Theorem at Königsberg, on the 7th of September, 1930. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notable mathematicians that &lt;em&gt;were&lt;/em&gt; present include Carnap, Heyting and most importantly von Neumann, who shortly thereafter managed to independently prove the Second Incompleteness Theorem, but decided to leave the credit for its discovery to Gödel, after the latter informed von Neumann that a paper containing both important theorems was already in the process of being published. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;http://books.google.de/books?id=UXk2AAAAQBAJ&amp;amp;pg=PA188&amp;amp;lpg=PA188&amp;amp;dq=hilbert+reaction+g%C3%B6del+incompleteness+theorem&amp;amp;source=bl&amp;amp;ots=_4o3gA7qyJ&amp;amp;sig=CSWFPWd5yLS6N-yv6uUEd5xhDNM&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ei=0SNQVJ_GDKPcywP-hYK4BA&amp;amp;redir_esc=y#v=onepage&amp;amp;q=hilbert%20reaction%20g%C3%B6del%20incompleteness%20theorem&amp;amp;f=false&quot;&gt;link provided by Philipp&lt;/a&gt; in the comments, it is clear that Hilbert reacted &lt;em&gt;angrily&lt;/em&gt; when the paper by Gödel was published, since it meant the failure of his program. However, being a mathematician, he could not argue with the validity of the proof and therefore resigned himself to the truth eventually.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-10-29T07:40:58.010" LastActivityDate="2014-10-29T07:40:58.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="34" PostTypeId="1" AcceptedAnswerId="37" CreationDate="2014-10-28T23:42:13.643" Score="19" ViewCount="1966" Body="&lt;p&gt;In a Wiki article on &lt;a href=&quot;http://en.wikipedia.org/wiki/Imaginary_number&quot;&gt;imaginary numbers&lt;/a&gt; it was asserted that &quot;the use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What motivated Euler's and Gauss's contributions to the theory of imaginary numbers? For instance, I know that Euler produced the formula that later led to DeMoivre's theorem, but don't quite understand why. And their lives barely overlapped, so why did no one &quot;in between&quot; pick up the &quot;baton&quot; from Euler to Gauss?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Ironically, Rene Descartes, who derided imaginary numbers, founded the &quot;Cartesian&quot; (2x2) co-ordinate system, which parallels the plane on which imaginary numbers are also graphed. This may have been a case of an &quot;accidental&quot; contribution.)&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="26" LastEditDate="2014-10-29T13:20:09.337" LastActivityDate="2019-04-23T14:24:16.497" Title="What mathematical developments/discoveries caused imaginary numbers to gain acceptance at the time (18th century) they did?" Tags="|mathematics|biographical-details|elementary-algebra|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="35" PostTypeId="1" AcceptedAnswerId="40" CreationDate="2014-10-28T23:45:03.670" Score="19" ViewCount="543" Body="&lt;p&gt;It's well known that Kepler worked out his laws by fitting curves to Tycho Brahe's data on the trajectories of planets through the sky. What &lt;em&gt;was&lt;/em&gt; this data? How does one record the trajectory of a planet through the sky? A pair of angles measured from some reference point? What sort of reference point - a steeple on the horizon? Also, how did Tycho Brahe keep accurate enough time to know he was recording the position of the planet at equal intervals of time?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="4251" LastEditDate="2021-09-11T10:16:51.980" LastActivityDate="2021-09-12T00:56:20.217" Title="What data did Kepler work out his laws from?" Tags="|physics|astronomy|celestial-mechanics|17th-century|kepler|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="36" PostTypeId="2" ParentId="9" CreationDate="2014-10-28T23:57:31.737" Score="46" Body="&lt;p&gt;According to the webpage &lt;a href=&quot;http://www.math.rutgers.edu/%7Ekellenm/deaths.html&quot; rel=&quot;noreferrer&quot;&gt;Mathematics - Means to an End&lt;/a&gt;, which lists the reasons for deaths of famous mathematicians, there is a considerable amount of uncertainty about Galois' death.  However, they speculate&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;that the duel was motivated either by a matter related to his involvement with the radical Républicain movement, or by conflict arising from a romantic entanglement. The duel occurred only a month after his release from a six-month incarceration stemming from his disruptive political activities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The site also states that, having predicted his own demise, he wrote down his work the night before.&lt;/p&gt;&#xA;&lt;p&gt;A different theory according to the &lt;a href=&quot;http://www.galois-group.net/&quot; rel=&quot;noreferrer&quot;&gt;The Evariste Galois Archive&lt;/a&gt;, particularly in their page regarding &lt;a href=&quot;http://www.galois-group.net/g/EN/duel.html&quot; rel=&quot;noreferrer&quot;&gt;the duel&lt;/a&gt;, about the death of Galois is the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most likely reason is: He was weary of life, because of his unhappy love affair, his fruitless efforts for gaining recognition for his mathematical work, his financial and work situation and he felt finished up a blind alley in politics as well. So his duel was like a staged suicide.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is some further evidence of Galois' malaise presented in the National University of Singapore's paper &lt;a href=&quot;http://sms.math.nus.edu.sg/smsmedley/Vol-22-1/Evariste%20Galois%20(Ng%20LM).pdf&quot; rel=&quot;noreferrer&quot;&gt;Evariste Galois&lt;/a&gt;, describing that he endured several disappointments academically and romantically, and, according to the author, had become bitter, and the conclusion that the author makes is quite salient:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If he had known that his work would be&#xA;of great significance to mathematics, life would have been more meaningful for him. As it&#xA;was, he died a very disappointed man.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="12330" LastEditDate="2021-10-10T21:16:23.760" LastActivityDate="2021-10-10T21:16:23.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="37" PostTypeId="2" ParentId="34" CreationDate="2014-10-29T00:11:05.143" Score="20" Body="&lt;p&gt;The first serious use of complex numbers is in finding the roots of quadratic, cubic, and quartic polynomials. Cardano, in his &lt;em&gt;Ars Magna&lt;/em&gt; (1545), first showed that quadratic equations could have (formally) complex roots, although he didn't call them that; he said they were &quot;as subtle as [they are] useless&quot;. In Bombelli's algebra text (1572), he developed the rules of complex arithmetic, and showed that Cardano's formula for the cubic could lead to real solutions even though intermediate results were imaginary. By the way, I've been told on multiple occassions that the notation &lt;span class=&quot;math-container&quot;&gt;$i= \sqrt{-1}$&lt;/span&gt; was only developed to guard against the common mistake of 'proving' &lt;span class=&quot;math-container&quot;&gt;$$(\sqrt{-1})^2=\sqrt{(-1)^2}=\sqrt{1}=1$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A key insight that was achieved in the early 18th century is the deep connection between complex numbers and geometry. It was observed that &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; can be used to simplify many trigonometric identities, and in 1748 Euler discovered his famous and beautiful formula &lt;span class=&quot;math-container&quot;&gt;$$e^{it}=\cos t+i \sin t$$&lt;/span&gt;&#xA;(The derivation was rather different from the one usually presented in today's textbooks; see &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2007-08.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this entry&lt;/a&gt; in the series &lt;em&gt;How Euler Did It&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conception of a complex number as a point in the plane is another discovery worth noting. This construction was already &lt;a href=&quot;http://books.google.de/books?id=idM6nvbz9xgC&amp;amp;printsec=frontcover&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;used by Wessel in 1799&lt;/a&gt;, and was independently rediscovered by Argand, but it really gained popularity when Gauss published his treatise on complex numbers. This book also established much of the modern notation and terminology that is used in complex analysis.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2019-04-23T14:24:16.497" LastActivityDate="2019-04-23T14:24:16.497" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="38" PostTypeId="1" CreationDate="2014-10-29T00:19:24.037" Score="15" ViewCount="469" Body="&lt;ul&gt;&#xA;&lt;li&gt;1954: The Georgetown experiment in 1954 involved fully automatic translation of more than sixty Russian sentences into English. The authors claimed that &lt;em&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_natural_language_processing&quot;&gt;within&lt;/a&gt; three or five years, machine translation would be a solved problem&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;1956 -  &lt;a href=&quot;http://en.wikipedia.org/wiki/Herbert_A._Simon&quot;&gt;Herbert Simon&lt;/a&gt;: &quot;&lt;em&gt;machines will be capable, within twenty years, of doing any work a man can do&lt;/em&gt;&quot;&lt;/li&gt;&#xA;&lt;li&gt;1967 - &lt;a href=&quot;http://newsoffice.mit.edu/2013/center-for-brains-minds-and-machines-0909&quot;&gt;Marvin Minsky&lt;/a&gt;: “&lt;em&gt;within a generation ... the problem of creating ‘artificial intelligence’ will substantially be solved.&lt;/em&gt;”&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Why were so many AI founders so optimistic in the early stages of AI?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="29" LastEditDate="2014-11-07T22:53:20.720" LastActivityDate="2016-03-06T21:28:36.717" Title="Why were so many artificial intelligence founders so optimistic?" Tags="|computer-science|artificial-intelligence|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="39" PostTypeId="1" CreationDate="2014-10-29T00:20:28.080" Score="23" ViewCount="758" Body="&lt;p&gt;I was rewriting the Wikipedia article for metamathematics, and it was very difficult to find any references after the 1930s. The most important works seem to have been Gödel's completeness and incompleteness theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a field of mathematics today that is the spiritual successor to metamathematics as studied by Gödel, Hilbert, and the authors of Principia Mathematica?&lt;/p&gt;&#xA;" OwnerUserId="56" LastEditorUserId="2442" LastEditDate="2016-07-25T23:29:35.533" LastActivityDate="2022-09-01T02:50:18.363" Title="In what form does the field of metamathematics exist today?" Tags="|mathematics|mathematical-logic|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="40" PostTypeId="2" ParentId="35" CreationDate="2014-10-29T00:37:21.603" Score="12" Body="&lt;p&gt;Today, sky coordinates are measured as &quot;Right Ascension&quot; (RA) and declination. These are similar to the angular coordinates we use for the Earth's surface but are measured on the celestial sphere relative to the celestial equator and pole. By using the current sidereal time, it is possible to map the local sky coordinates (ie. a horizontal bearing relative to true north, and an inclination above the horizon) to RA &amp;amp; Declination.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brahe used devices such as Quadrants to measure this inclination. There's a engraving of one mounted to the side of a building (presumably at Uraniborg) at &lt;a href=&quot;http://en.wikipedia.org/wiki/Uraniborg&quot;&gt;http://en.wikipedia.org/wiki/Uraniborg&lt;/a&gt; (Brahe did most of his observations at &lt;a href=&quot;http://en.wikipedia.org/wiki/Stjerneborg&quot;&gt;Stjerneborg&lt;/a&gt; due to problems with the Uraniborg instrument mountings).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for time, the clocks of the day were accurate enough because the escarpment had been invented a couple of centuries earlier. A sidereal clock can be reset every night (using a known star and a meridian) so it only needs to be accurate for 24 hours. Temperature changes will be relatively limited in this period (and minimized further by resetting the clock twice a night), and there will be no motion.&#xA;Yes the clocks available to Brahe would be have been inaccurate compared to Harrison's chronometers of a few centuries later but Harrison was designing clocks to last months without adjustment and to survive violent motion and significant temperature changes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definitions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Meridian&lt;/em&gt;: A great circle line that goes through the location (eg. Uraniborg Observatory) and the poles. It would be a line due south from Uraniborg. A star crosses the meridian when it appears to be due south of the celestial pole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Sidereal Clock&lt;/em&gt;: A sidereal clock is one that is synchronized with the stars and not the Sun. The Earth actually rotates in one sidereal day (about 4mins short of a normal solar day) but because it has moved a little bit in its orbit, the usual point of reference (the Sun) has moved slightly, hence the solar day is slightly longer than a sidereal day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern solution would be to use a solar clock and then apply a correction based on the calendar (trivial with a microprocessor). But if you're dealing with mechanical clocks, it is just as easy to set the pendulum to run slightly fast.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-10-30T01:24:11.253" LastActivityDate="2014-10-30T01:24:11.253" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="41" PostTypeId="2" ParentId="8" CreationDate="2014-10-29T00:38:17.387" Score="11" Body="&lt;p&gt;The wave and the particle (or corpuscular) theories of light go back to the 17-th century and are often associated with Huygens and Newton, respectively, as their founders. What preceeded them was called geometric optics, where light consisted of rays connecting an eye to an object. There were two theories in geometric optics too. According to the emission theory rays of light came out of our eyes and &quot;sensed&quot; objects in a way similar to touch, Hero and Ptolemy supported this theory. According to the intromission theory it was objects that emanated the light, Aristotle and Galen favored this version. Mathematically both theories are equivalent for optical phenomena, and essentially reduce optics to geometry except where the refraction law is concerned. More historical details are given &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_optics#The_beginnings_of_geometrical_optics&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;particles&quot; of light are photons rather than electrons, and although &quot;wave-particle duality&quot; is a popular way to explain their nature it is rather inaccurate, the real duality would have produced a &lt;a href=&quot;http://en.wikipedia.org/wiki/Pilot_wave&quot; rel=&quot;nofollow&quot;&gt;pilot wave theory&lt;/a&gt;. The accepted model is &lt;a href=&quot;http://en.wikipedia.org/wiki/Quantum_electrodynamics&quot; rel=&quot;nofollow&quot;&gt;quantum electrodynamics&lt;/a&gt; developed in late 1940-s, that describes also other forms of electromagnetism, and in the first approximation quantum mechanics. While photons do have some properties of waves and some of particles they do not have all, and they also have some properties (such as entanglement) that neither of the two has. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-10-30T00:27:42.270" LastActivityDate="2014-10-30T00:27:42.270" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="42" PostTypeId="1" AcceptedAnswerId="47" CreationDate="2014-10-29T00:50:38.260" Score="53" ViewCount="6248" Body="&lt;p&gt;The natural logarithm function ($\ln x$) and the base of the natural logarithm function ($e$) are both extremely useful. They're also both closely related: $\ln (e^x)=x$, and $e^{\ln x}=x$. But which came first? I would think it's likely they were developed together, but each could have been developed separately. For example, $\int 1/x \,dx=\ln x$, and the $\cosh$ function can be described in terms of $e$. So which came first: the natural logarithm function or the base of the natural logarithm function?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="48" LastEditDate="2014-11-18T21:52:50.147" LastActivityDate="2019-10-02T17:38:13.457" Title="Which came first, the natural logarithm or the base of the natural logarithm?" Tags="|mathematics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="43" PostTypeId="2" ParentId="22" CreationDate="2014-10-29T01:11:26.977" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ozone_layer&quot;&gt;Wikipedia's&lt;/a&gt; article isn't quite up to snuff. It's rather short and largely skips over the early history of research regarding the ozone layer. Here's what it has to say about it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ozone layer was discovered in 1913 by the French physicists Charles Fabry and Henri Buisson.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not a lot of great information. &lt;a href=&quot;http://www.physics.rutgers.edu/ugrad/387/Mulligan98.pdf&quot;&gt;Here&lt;/a&gt;, though, it is stated that Fabry worked closely with &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfred_Perot&quot;&gt;Alfred Perot&lt;/a&gt; to develop the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer&quot;&gt;Fabry-Perot interferometer&lt;/a&gt;. Its initial applications were in the field of astrophysics, but later&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;in 1913 they [Fabry and Buisson] were the first to demonstrate&#xA;  that the ultraviolet absorption in the Earth’s upper&#xA;  atmosphere was due to ozone.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So Fabry took his prior work regarding interferometry and used it to collaborate with Buisson. That said, &lt;a href=&quot;https://en.wikipedia.org/wiki/G._M._B._Dobson&quot;&gt;G. M. B. Dobson&lt;/a&gt; was one of the first people to truly explore the ozone layer's properties in detail, using a &lt;a href=&quot;https://en.wikipedia.org/wiki/Dobson_ozone_spectrophotometer&quot;&gt;spectrophotometer&lt;/a&gt; to monitor ozone concentrations.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-29T01:11:26.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="44" PostTypeId="1" AcceptedAnswerId="46" CreationDate="2014-10-29T01:41:02.587" Score="22" ViewCount="3073" Body="&lt;p&gt;It is well known that famous names such as Gauss, Euler and Newton were &lt;a href=&quot;http://en.wikipedia.org/wiki/Polymath&quot;&gt;polymaths&lt;/a&gt; as well as their main fields of study and contributed from optics to ship building. Why was this the case in the past? It is known, as far as I am aware, to exist since the Greeks. Why are there so few modern polymaths?&lt;/p&gt;&#xA;" OwnerUserId="27" LastActivityDate="2014-11-07T01:27:33.370" Title="Why were so many pre-18th century Mathematicians polymaths?" Tags="|mathematics|polymath|mathematicians|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="45" PostTypeId="2" ParentId="21" CreationDate="2014-10-29T01:46:24.777" Score="7" Body="&lt;p&gt;You probably mean referees, those who referee papers for the journals.&#xA;(&quot;Reviewers&quot; is the usual name for those who write the reviews for Math reviews or for Zentralblatt Math. They are payed small amounts per review.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some journals do pay to the referees. For example, Indiana University Math journal&#xA;used to offer $50 per report, I don't know whether this continues. (Also notice that this always&#xA;was one of the cheapest journals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course this is an exception. Traditionally refereeing is considered a service to&#xA;the community, and mathematicians do this voluntary. I suppose other scientists do this voluntary as well. As far as I know this always was the case.&#xA;In general, fundamental science was not considered a commercial enterprise. Scientists&#xA;are usually not payed for their published papers (though they may collect royalties for books). So it was assumed that can do the refereeing jobs on the same basis.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-10-29T01:46:24.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="46" PostTypeId="2" ParentId="44" CreationDate="2014-10-29T01:51:19.197" Score="17" Body="&lt;p&gt;Really, it's because that was the social protocol at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://en.wikipedia.org/wiki/Polymath&quot;&gt;the Wikipedia article&lt;/a&gt; on polymaths,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many notable polymaths lived during the Renaissance period, a cultural movement that spanned roughly the 14th through to the 17th century and that began in Italy in the late Middle Ages and later spread to the rest of Europe. These polymaths had a rounded approach to education that reflected the ideals of the humanists of the time. A gentleman or courtier of that era was expected to speak several languages, play a musical instrument, write poetry, and so on, thus fulfilling the Renaissance ideal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So you could say it was one of the primary tenets of &lt;a href=&quot;http://en.wikipedia.org/wiki/Renaissance_humanism&quot;&gt;Renaissance humanism&lt;/a&gt;. This approach stressed a person being proficient in many subjects, specifically the humanities. The philosophy was laid down in a book, &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Book_of_the_Courtier&quot;&gt;The Book of the Courtier&lt;/a&gt;, written by &lt;a href=&quot;http://en.wikipedia.org/wiki/Baldassare_Castiglione&quot;&gt;Baldassare Castiglione&lt;/a&gt;. It laid down the ideas that the optimal person (symbolized by the main characters, a group of courtiers) should be extremely well-rounded.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why are there so few modern polymaths?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suspect a combination of apathy and the fact that society no longer values people who have a wide range of talents (unless we count colleges!). Today, we typically only major in one subject in college (although we can focus on a minor, as well). The major is in a specific field, which the student hopes to go into upon leaving college. Our education is utilitarian, but in a different sense than in the Renaissance: we don't &lt;em&gt;need&lt;/em&gt; to learn shipbuilding if we're going to work at, say, an art museum, and society no longer expects us too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This bit is a wee bit dubious, but hopefully my logic makes sense. The various fields of study today, especially in the sciences, are much broader than they were in the time of the Renaissance. It was &lt;em&gt;much&lt;/em&gt; easier to learn physics back in pre-Newtonian times (and during Newtonian times, too!) because learning physics didn't entail learning everything from Lagrangian mechanics to tensor calculus. True, the equivalent then of a &quot;physicist&quot; would have to have a broad knowledge of philosophy and metaphysics (as well as possibly alchemy), but probably not as much as a physicist needs to know today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, it takes &lt;em&gt;time&lt;/em&gt; nowadays to become an expert in something. Here's how to become a physicist - in only a decade or two:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Work hard for 4 years in high school and get good grades; show an interest in science, especially physics, to attract colleges.&lt;/li&gt;&#xA;&lt;li&gt;Spend 4 years in college; majoring in physics, with a possible minor, often in a related field.&lt;/li&gt;&#xA;&lt;li&gt;Spend 4 to 8 years to get a &lt;a href=&quot;https://en.wikipedia.org/wiki/Doctor_of_Philosophy#Time&quot;&gt;PhD&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Work as a postdoc at a university for 5 or so years.&lt;/li&gt;&#xA;&lt;li&gt;Become an assistant professor; work at a university for 5 more years.&lt;/li&gt;&#xA;&lt;li&gt;Look for a job as a physicist.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-02T16:08:36.997" LastActivityDate="2014-11-02T16:08:36.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="47" PostTypeId="2" ParentId="42" CreationDate="2014-10-29T02:02:53.907" Score="60" Body="&lt;p&gt;It may seem strange but logarithms were invented much earlier.&#xA;Napier used the base &lt;span class=&quot;math-container&quot;&gt;$(1-10^{-7})^{10^7}$&lt;/span&gt; which is very close to 1/&lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; (within 0.00000002 of 1/&lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;).&#xA;Number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; (as a limit) was formally defined by Euler approximately 100 years after&#xA;Napier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Napier's &lt;a href=&quot;http://www.17centurymaths.com/contents/napier/constructiobookone.pdf&quot; rel=&quot;noreferrer&quot;&gt;MIRIFICI LOGARITHMORUM CANONIS CONSTRUCTIO&lt;/a&gt; (English translation by Ian Bruce) contains tables of logarithms, and explanations of the construction of the tables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Natural logarithms, and the formula &lt;span class=&quot;math-container&quot;&gt;$\ln x=\int_1^xdt/t$&lt;/span&gt; defining them, were known long before Euler. Modern texts usually define them as the inverse function of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;, but historically this was not the case: &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; is much later invention than logarithms.&#xA;According to Wikipedia, this definition using &quot;the area under the hyperbola&quot; is due to&#xA;Alphonse Antonio de Sarasa (1649), that is a century before Euler.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="5659" LastEditDate="2019-10-02T17:38:13.457" LastActivityDate="2019-10-02T17:38:13.457" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="48" PostTypeId="1" AcceptedAnswerId="50" CreationDate="2014-10-29T02:07:47.330" Score="11" ViewCount="708" Body="&lt;p&gt;We have fermions (named after &lt;a href=&quot;http://en.wikipedia.org/wiki/Enrico_Fermi&quot; rel=&quot;nofollow&quot;&gt;Fermi&lt;/a&gt;) and bosons (named after &lt;a href=&quot;http://en.wikipedia.org/wiki/Satyendra_Nath_Bose&quot; rel=&quot;nofollow&quot;&gt;Bose&lt;/a&gt;). Why don't we name the particle corresponding to the Higgs field a &quot;higgson&quot;? The superpartner particle (sparticle) of the Higgs boson is the higgsino, which makes the name &quot;higgson&quot; more reasonable (the sparticle of the photon is the photino).&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="62" LastEditDate="2015-05-12T15:15:01.583" LastActivityDate="2022-07-02T19:49:38.520" Title="Why don't we name the Higgs boson a &quot;higgson&quot;?" Tags="|terminology|naming-conventions|particle-physics|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="49" PostTypeId="2" ParentId="34" CreationDate="2014-10-29T02:12:57.663" Score="6" Body="&lt;p&gt;Just to complement Danu's answer. Some people used complex numbers since the 16th century, however, WIDE acceptance came later (at the end of the 18th century) when several people (Argand, Vessel, Gauss) discovered the geometric interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was apparently a crucial step. Still, they were not universally recognized. They say that even Chebyshev never used them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another event which could be significant: in the early 19th century, physicists started to use them (Fresnel).   &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorDisplayName="user22" LastEditDate="2014-10-29T18:15:06.220" LastActivityDate="2014-10-29T18:15:06.220" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="50" PostTypeId="2" ParentId="48" CreationDate="2014-10-29T02:19:39.830" Score="9" Body="&lt;p&gt;There are a number of reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The thing that differentiates fermions and bosons from the Higgs boson specifically is that they are general classes of particles (based on spin), while the Higgs boson is a specific particle (or particle type, if you will).&lt;/li&gt;&#xA;&lt;li&gt;The names &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermion&quot;&gt;&quot;fermions&quot;&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Boson&quot;&gt;&quot;bosons&quot;&lt;/a&gt; were coined by &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Dirac&quot;&gt;Paul Dirac&lt;/a&gt;, who was no longer in prominence when the idea behind the Higgs field was proposed. In general, particles aren't named after people (the Higgs boson is an exception).&lt;/li&gt;&#xA;&lt;li&gt;Just ask &lt;a href=&quot;https://en.wikipedia.org/wiki/Benjamin_W._Lee&quot;&gt;Benjamin Lee&lt;/a&gt; (unfortunately deceased) who first used the term.&lt;/li&gt;&#xA;&lt;li&gt;The idea behind the higgsino, &lt;a href=&quot;https://en.wikipedia.org/wiki/Supersymmetry&quot;&gt;supersymmetry&lt;/a&gt;, didn't come around until the early 1970s, and didn't rise to prominence until much later; by then, the term &quot;Higgs boson&quot; was well established, and &quot;higgson&quot; wouldn't have been catchy enough.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-29T02:19:39.830" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="51" PostTypeId="2" ParentId="44" CreationDate="2014-10-29T02:22:45.917" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Why are there so few modern polymaths?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Because it is pretty much impossible nowadays to master many fields at a level to be able to make significant contributions due to the incredible size of knowledge we have now reached. E.g. David Hilbert was probably one of the last universal mathematicians. The time investment of becoming an expert in just one narrow field is such that one does not have the time to become expert at many.&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-10-29T02:22:45.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="52" PostTypeId="1" CreationDate="2014-10-29T03:06:44.707" Score="27" ViewCount="982" Body="&lt;p&gt;In modern physics textbooks, we teach Newton's laws of motion, then Newton's law of Universal Gravitation, and then &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion&quot;&gt;Kepler's laws of planetary motion&lt;/a&gt;. Specifically, from the $1/r^2$ form of the gravitational force, and some other parts of Newton's laws, we can derive Kepler's 3rd law, that the period of motion of a planet is proportional to the $3/2$ power of its distance from the sun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But historically, Kepler developed his laws before Newton wrote the &lt;em&gt;Principia&lt;/em&gt;. Newton formulated his laws in the Principia, then (also in the &lt;em&gt;Principia&lt;/em&gt;) derived the specific $1/r^2$ form of his gravitational law from the $3/2$ power form of Kepler's 3rd law. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: when did physics texts and/or courses switch from the historical order of these two laws to the more recent (and possibly more pedagogical)? Was there a reason given at the time? The historical order was more inductive in its reasoning, while the modern presentation is more deductive in its reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possibility I can think of is that we derive the $1/r^2$ form of &lt;a href=&quot;http://en.wikipedia.org/wiki/Coulomb%27s_law&quot;&gt;Coulomb's law&lt;/a&gt; using &lt;a href=&quot;http://en.wikipedia.org/wiki/Gauss%27s_law&quot;&gt;Gauss's law&lt;/a&gt; and the fact that (macroscopic) space is 3-dimensional. That derivation carries over word-for-word to gravity. That becomes a very logical reason to say gravity should have the $1/r^2$ form once you know vector calculus. That might be a fruitful time period to look at.&lt;/p&gt;&#xA;" OwnerUserId="24" LastEditorUserId="2442" LastEditDate="2015-07-18T20:22:52.330" LastActivityDate="2015-07-18T20:22:52.330" Title="When did physics texts start to teach Kepler's $3/2$'s power law as a result of Newton's $1/r^2$ law of gravitation, rather than the other way around?" Tags="|physics|teaching|gravity|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="53" PostTypeId="2" ParentId="3" CreationDate="2014-10-29T03:25:08.223" Score="83" Body="&lt;p&gt;The other answer is correct. In addition, there is significant evidence that Fermat did not have a proof of the theorem now known as Fermat's Last Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, we should note that Fermat was not a professional mathematician, only an amateur. He never published any mathematics himself. With just that, it would not seem strange that he did not publish his proof. However, his son Samuel decided to collect Fermat's writings and letters. The famous margin which was too small for the proof was the margin of Fermat's copy of Diophantus's &lt;a href=&quot;http://en.wikipedia.org/wiki/Arithmetica&quot;&gt;&lt;em&gt;Arithmetica&lt;/em&gt;&lt;/a&gt;. Fermat likely wrote this note around 1630 when he first began studying this text. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From his writings and letters, we see a common trend. Almost all of the problems Fermat mentioned having solved were included in his work more than once, typically being restated as challenge problems which he then sent to various mathematicians with whom he was in correspondence. However, FLT appears only once, in this margin. It is never again mentioned in any writings that Samuel was able to find. Assuming he truly had a (long) proof which was not incorrect, he would have likely written this down or discussed it with other mathematicians, as he did with essentially every other result he found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, in his writings, we do find later references to special cases of the theorem (see &lt;a href=&quot;http://en.wikipedia.org/w/index.php?title=Proof_of_Fermat%27s_Last_Theorem_for_specific_exponents&amp;amp;oldid=622173251&quot;&gt;Wikipedia, Proofs of Fermat's Last Theorem for specific exponents&lt;/a&gt;). Both the $n=3$ and $n=4$ cases are found in his later writings. It's not clear whether he knew a proof for the $n=3$ case, but if he did it was unknown at the time of his death and none was found until Euler in 1760. However, he did send multiple letters, in 1636, 1640, and 1657, containing this case as a problem. The single surviving proof by Fermat is equivalent to a proof of the $n=4$ case. It would seem very strange for him to state the problem in 1630 for all $n$, and then in much later writings, to specialize to two cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this in mind, it seems there are three possibilities.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Fermat never meant to claim that he knew a proof of FLT for all $n&gt;2$, and only wanted to state FLT as a conjecture. He may have intended to specialize to $n=3$ and/or $n=4$. This was, after all, a private writing by an amateur mathematician who was just learning number theory. It was never intended to be communicated to others, and in his communications we can find no indication of such a claim. It's not clear what he would have meant by the note in the margin.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fermat believed he had a proof of this for all $n&gt;2$ at the time. However, he was wrong, and very likely discovered this himself, possibly while trying to write down the proof. This would explain his later specializations to the $n=3$ and $n=4$ cases, themselves not trivial, and his decision not to communicate the result to any of the mathematicians he was in contact with. Exactly what this proof might have been is not clear. Many people since have failed to prove FLT in many ways. Being very generous, he could have made an assumption similar or somehow equivalent to one made by Lamé in his failed attempt from 1847, i.e. that $\mathbb Z[\zeta_n]$ (where $\zeta_n$ is a primitive $n$-th root of unity) has unique factorization for all $n$. Even that would have been far ahead of his time. But his error could have been something more mundane as well.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fermat (who was just an amateur, albeit an extremely gifted one) found a correct elementary proof of FLT which has since evaded thousands of mathematicians with more sophisticated technology and more complete understanding of number theory over a period of over 350 years. He never wrote this down or communicated this result to any mathematicians, preferring to discuss only two specific cases. This can not be technically ruled out, but it seems highly unlikely, and the only evidence supporting it is a private note scribbled in the margin of a text by a man who was learning number theory for the first time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The first two possibilities both seem reasonable, while the third is almost completely absurd. This would not be the only case in which Fermat believed he had a result which was only completely proven later. The &lt;a href=&quot;http://en.wikipedia.org/wiki/Fermat_polygonal_number_theorem&quot;&gt;polygonal number theorem&lt;/a&gt; is another major case, which was only proven by Legendre for squares in 1770, Gauss for triangles in 1796, and Cauchy in general in 1812. Gauss in particular cast some serious doubts as to whether Fermat had a proof of this. The most popular guess is possibility 2, that Fermat had some sort of argument which was flawed but perhaps worked for some small exponents. Not enough of his writings have survived to guess what that method was, and the proof he gave for $n=4$ doesn't generalize in any obvious way to other exponents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is simply not possible that Fermat discovered a proof which is equivalent to Wiles' proof. That would have been impossible; the concepts required to even understand Wiles' proof were not developed until the 20th century.&lt;/p&gt;&#xA;" OwnerUserId="58" LastActivityDate="2014-10-29T03:25:08.223" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="54" PostTypeId="1" CreationDate="2014-10-29T05:21:41.490" Score="36" ViewCount="2696" Body="&lt;p&gt;If you ask a mathematician today, many will tell you that mathematics is not a science. Many physicists, chemists, and scientists in other disciplines would say something similar. Mathematicians will emphasize several differences between mathematics and empirical science, from aesthetic differences to the unfalsifiability of mathematics. The biggest distinction seems to be that Mathematicians don't accept the Baconian method of induction, having different standards of proof, and as such, can't reasonably be classified as an empirical science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the original definition of the term &quot;science&quot;, as per &lt;a href=&quot;http://www.etymonline.com/index.php?term=science&quot; rel=&quot;noreferrer&quot;&gt;its etymology&lt;/a&gt;, more or less just means &quot;knowledge&quot;. The term began to mean &quot;empirical science&quot; (in the sense of Bacon) later, perhaps in the 17th and 18th centuries, but it seems to me that Mathematics was still implicitly included by use of the phrase &quot;the sciences&quot; until significantly later. This was a source of &lt;a href=&quot;https://area51.meta.stackexchange.com/questions/14650/suggest-a-url-for-this-site#comment24852_14654&quot;&gt;some debate&lt;/a&gt; on our Area 51 proposal, and as Conifold suggested, I'm moving it here so that we can get an authoritative answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When did use of the term &quot;the sciences&quot; by mathematicians and scientists stop implicitly including mathematics?&lt;/strong&gt; I am particularly looking for specific quotes/statements from scientists or mathematicians (preferably in English) addressing the question as to whether Mathematics is one of the sciences, though other forms of evidence are also welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some evidence, Gauss famously referred to mathematics as &quot;the Queen of the sciences&quot;, attributed to him by his biographer &lt;a href=&quot;http://en.wikipedia.org/wiki/Wolfgang_Sartorius_von_Waltershausen&quot; rel=&quot;noreferrer&quot;&gt;Sartorius von Waltershausen&lt;/a&gt; (but see the comments below questioning the accuracy of this interpretation and translation). On the other hand, by the 20th century, people like Einstein were saying things like &quot;As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality.&quot; In addition, people like Hardy were emphasizing the aesthetic aspects of mathematics, e.g. in &lt;a href=&quot;http://en.wikipedia.org/wiki/A_Mathematician&amp;#39;s_Apology&quot; rel=&quot;noreferrer&quot;&gt;his Apology&lt;/a&gt;. So it seems to me like the biggest shift happened sometime in the 19th or 20th century, but it's hard to pinpoint exactly when this change began or what sparked it.&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="-1" LastEditDate="2017-05-03T08:41:04.823" LastActivityDate="2016-06-19T17:50:38.930" Title="When did Mathematics stop being one of &quot;the Sciences&quot;?" Tags="|mathematics|philosophy-of-science|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="55" PostTypeId="1" AcceptedAnswerId="95" CreationDate="2014-10-29T05:25:55.803" Score="40" ViewCount="17762" Body="&lt;p&gt;Are there any differences between the study of Calculus done by Newton as compared to that done by Leibniz? If yes, please mention point by point.&lt;/p&gt;&#xA;" OwnerUserId="76" LastEditorUserId="10106" LastEditDate="2020-05-14T03:16:49.843" LastActivityDate="2022-10-26T10:05:34.357" Title="What is the difference between Calculus of Newton and that of Leibniz?" Tags="|mathematics|biographical-details|calculus|newton|leibniz|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="56" PostTypeId="1" CreationDate="2014-10-29T05:41:25.197" Score="16" ViewCount="1118" Body="&lt;p&gt;François Viète's &lt;em&gt;On the Numerical Resolution of Powers by Exegetics&lt;/em&gt; published&#xA;in 1610 (Viete, 2006, pp. 311-370) introduced one way of numerically solving polynomial equations, a special case of which is the equation where the square of an unknown is equal to a given number.  Thomas Harriot's &lt;em&gt;The Practice of the Analytic Art&lt;/em&gt; (Harriot, 1631) built upon Viète's work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that Viète's and Harriot's procedure is the basis of the manual decimal digit-by-digit method of obtaining square roots (as seen in Wikipedia, 2014).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was taught this procedure when I was in elementary school.  My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When was the manual decimal digit-by-digit method of obtaining square roots first taught to school children?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm looking for evidence such as a textbook containing the procedure and explicitly stating that children (say, those below 18 years of age) are the intended audience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Harriot, T. (1631). &lt;em&gt;Artis analyticae praxis, ad aequationes algebraicas noua, expedia, &amp;amp; generali methodo, resoluendas : tractatus&lt;/em&gt;. Londini: Barker. Retrieved January 29, 2014, from &lt;a href=&quot;http://echo.mpiwg-berlin.mpg.de/MPIWG:DBZ6XPZN&quot;&gt;http://echo.mpiwg-berlin.mpg.de/MPIWG:DBZ6XPZN&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Viète, F. (2006). &lt;em&gt;The anayltic art: Nine studies in algebra, geometry and trigonometry from the&lt;/em&gt; opus restitutae mathematicae analyseos, seu algebrâ novâ (T. R. Witmer, Trans.).&#xA;Mineola, New York: Dover Publications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia. (2014). Methods of computing square roots---Wikipedia, the free encyclopedia. Retrieved January 29, 2014, from &lt;a href=&quot;http://en.wikipedia.org/w/index.php?title=Methods_of_computing_square_roots&amp;amp;oldid=589727868&quot;&gt;http://en.wikipedia.org/w/index.php?title=Methods_of_computing_square_roots&amp;amp;oldid=589727868&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="72" LastEditorUserId="8" LastEditDate="2015-05-18T16:37:37.010" LastActivityDate="2015-05-25T14:35:34.863" Title="When was the method of getting square roots (invented by Viète in 1610 and developed by Harriot in 1631) first taught to school children?" Tags="|mathematics|elementary-algebra|teaching|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="57" PostTypeId="1" AcceptedAnswerId="65" CreationDate="2014-10-29T06:08:57.467" Score="14" ViewCount="1291" Body="&lt;p&gt;I read the book &quot;A Brief History of Time&quot; By Stephan Hawking. It states that Einstein helped scientists like Pauli etc. in the development of the quantum theory and even shared the Nobel Prize with them for his contributions, but to his dying day, didn't agree with the theory. He even made the quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I, for one, do not believe that He plays dice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If he was so much against the theory, when then he helped in its development?&lt;/p&gt;&#xA;" OwnerUserId="78" LastEditorUserId="26" LastEditDate="2014-10-29T08:14:45.823" LastActivityDate="2016-08-05T22:58:50.670" Title="Why did Einstein help in the development of the Quantum theory if he didn't agree with it?" Tags="|biographical-details|quantum-mechanics|theoretical-physics|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="58" PostTypeId="1" AcceptedAnswerId="77" CreationDate="2014-10-29T06:13:14.940" Score="40" ViewCount="3330" Body="&lt;p&gt;I can't imagine mathematics without sets, but the question &quot;what was mathematics like before there were sets&quot; is not answerable, I think. Instead, a good answer to the title question should cover a certain aspect of the more general question.&lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="4251" LastEditDate="2020-06-06T13:00:31.033" LastActivityDate="2020-06-06T13:00:31.033" Title="What motivated Cantor to invent set theory?" Tags="|mathematics|mathematicians|biographical-details|set-theory|cantor|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="59" PostTypeId="1" CreationDate="2014-10-29T07:13:52.747" Score="34" ViewCount="1187" Body="&lt;p&gt;As asked in the title: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there any written sources (from the 19th century) explicitly stating the belief that any function satisfying the intermediate value property is continuous?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I do not believe it makes sense to search for earlier sources, since the notion of continuity itself was not made rigorous until the 19th century. This question originated in an &lt;a href=&quot;https://math.stackexchange.com/a/622083/462&quot;&gt;answer&lt;/a&gt; I gave at &lt;a href=&quot;https://math.stackexchange.com/&quot;&gt;Math.Stackexchange&lt;/a&gt;. What follows borrows heavily from that answer.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I is an interval, and f : I → ℝ, we say that f has the &lt;em&gt;intermediate value property&lt;/em&gt; if and only if whenever a≠b are points of I, if c is between f(a) and f(b), then there is a d &lt;em&gt;between&lt;/em&gt; a and b with f(d)=c. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bolzano published in 1817 his paper &lt;em&gt;Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege&lt;/em&gt; (&lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086080900361&quot; rel=&quot;noreferrer&quot;&gt;Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation&lt;/a&gt;). There, he proves that continuous functions satisfy the intermediate value property. As he indicates in the paper, the proposition was widely believed to be true, and several &quot;geometric&quot; arguments had been given trying to justify it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, now we know that the intermediate value property is far weaker than continuity. A nice survey containing detailed examples of functions that are discontinuous and yet have the intermediate value property is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I. Halperin, &lt;a href=&quot;http://cms.math.ca/10.4153/CMB-1959-016-1&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Discontinuous functions with the Darboux property&lt;/em&gt;&lt;/a&gt;, Can. Math. Bull., &lt;strong&gt;2 (2)&lt;/strong&gt;, (May 1959), 111-118. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Halperin's paper we find the amusing quote &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Until the work of Darboux in 1875 some mathematicians believed that [the intermediate value] property actually implied continuity of f(x). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This claim is repeated in (many) other places. For example, &lt;a href=&quot;http://www.encyclopediaofmath.org/index.php/Darboux_property&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; one reads &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the 19th century some mathematicians believed that [the intermediate value] property is equivalent to continuity. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is very similar to what we find in A. Bruckner, &lt;strong&gt;Differentiation of real functions&lt;/strong&gt;, AMS, 1994. In page 5 we read&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This property was believed, by some 19th century mathematicians, to be equivalent to the property of continuity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Intermediate_value_theorem#Converse_is_false&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions [citation needed].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have been unable to find a direct source expressing this belief. That this was indeed the case is perhaps supported by the following two quotes from Gaston Darboux's &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1875_2_4_/ASENS_1875_2_4__57_0/ASENS_1875_2_4__57_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mémoire sur les fonctions discontinues&lt;/em&gt;&lt;/a&gt;, Ann. Sci. Scuola Norm. Sup., &lt;strong&gt;4&lt;/strong&gt;, (1875), 161–248. First, on pp. 58-59 we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Au risque d'être trop long, j'ai tenu avant tout, sans y réussir peutêtre, à être rigoureux. Bien des points, qu'on regarderait  à bon droit comme évidents ou que l'on accorderait dans les applications de la science aux fonctions usuelles, doivent être soumis à une critique rigoureuse dans l'exposé des propositions relatives aux fonctions les plus générales. Par exemple, on verra qu'il existe des fonctions continues qui ne sont ni croissantes ni décroissantes dans aucun intervalle, qu'il y a des fonctions discontinues qui ne peuvent varier d'une valeur à une autre sans passer par toutes les valeurs intermédiaires.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Darboux's paper proves that derivatives have the intermediate value property, and that there are discontinuous derivatives, thus first verifying that the two notions are not equivalent. (For this reason, the intermediate value property is sometimes called the &lt;a href=&quot;http://www.encyclopediaofmath.org/index.php/Darboux_property&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Darboux property&lt;/em&gt;&lt;/a&gt; or, even, one says that a function with this property is &lt;em&gt;Darboux continuous&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof that derivatives have the intermediate value property starts on page 109, where we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;En partant de la remarque précédente, nous allons montrer qu'il existe des fonctions discontinues qui jouissent d'une propriété que l'on regarde quelquefois comme le caractère distinctif des fonctions continues, celle de ne pouvoir varier d'une valeur à une autre sans passer par toutes les valeurs intermediaires.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wikipedia &lt;a href=&quot;http://en.wikipedia.org/wiki/Intermediate_value_theorem#History&quot; rel=&quot;noreferrer&quot;&gt;mentions&lt;/a&gt; the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function. Proponents include Louis Arbogast, who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The article cites &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Arbogast.html&quot; rel=&quot;noreferrer&quot;&gt;this site&lt;/a&gt;, though I have been unable to verify this from Arbogast's writings (or from the linked site). Indeed, Arbogast seems to have a notion of function that is significantly more restrictive that our modern notion of continuity, and therefore the intermediate value theorem holds there. I do not see that he directly addresses the intermediate value property, or indicates that it implies continuity. (Given his understanding of what a function is, I am not even certain that this would have been meaningful.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, let me ask: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If it is not actually the case that the belief in the equivalence of these two notions was explicitly stated in the literature, where does the false claim originate? (Is it in Halperin's paper?)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="18" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2019-06-13T20:34:34.063" Title="Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity?" Tags="|mathematics|real-analysis|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="60" PostTypeId="2" ParentId="10" CreationDate="2014-10-29T07:27:22.787" Score="17" Body="&lt;p&gt;I'll focus on the geometry of Yang-Mills theories specifically, but as &lt;a href=&quot;https://hsm.stackexchange.com/a/147&quot;&gt;Conifold's answer&lt;/a&gt; points out, gauge theories were studied geometrically long before the work of Yang and Mills. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The foreward to &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198532792&quot; rel=&quot;nofollow noreferrer&quot;&gt;volume 5 of Atiyah's collected works&lt;/a&gt; (on gauge theories) contains some historical comments on this from the mathematics side. You can read it &lt;a href=&quot;http://celebratio.org/Atiyah_MF/article/41/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. This is probably somewhat later than what you're looking for, since it has more to do with the later study of the deep geometric properties of Yang-Mills theory than the (comparatively) simple interpretation of Yang-Mills as an action on the space of connections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Atiyah says his own interests in the subject began in 1977, when he says interest in gauge theories among mathematicians was just beginning to start (citing Yang's influence in mathematical circles). That matches pretty well his writings. The first writing included in his collection is [1]. This paper he cites with bringing many of the ideas in gauge theory to the mathematics community. In it, he shows that the multi-instanton problem can be reduced to constructing suitable vector bundles on 3-dimensional projective space. The construction was completed in [2]. He wrote several more papers that year and in the following years on the geometry and topology of Yang-Mills fields. His (and other) papers in the late 70s are the earliest that I know of by mathematicians on the geometry of Yang-Mills theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the early 80s, several other people began publishing on this topic. Some of the big names are Donaldson, Hitchin, and Witten. In particular, Donaldson's study of 4-manifolds via instantons in [3] proved to be of great interest. By that point, it had become clear that the Yang-Mills equations could be used to great effect for more than just physics. It's fair to say that the interest in them continued well through the 80s and in some cases to the present day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earlier developments before this were almost completely taken on by physicists. I know less of the story here because physicists seem less inclined to write detailed accounts of the chronology of events. It's clear that by 1977, it was already known by physicists that Yang-Mills could be viewed in terms of an action functional on the space of connections, though the geometric consequences had not been explored. (Of course, physicists had bigger problems to deal with before that, like understanding how to give gauge bosons mass and proving the renormalizability of quantum Yang-Mills.) The earliest source I know for this is by Popov in [4] in 1975. In this, he shows that the now-standard geometrical interpretation of Yang-Mills via principal bundles and connections yields the Yang-Mills equations. However, it's quite possible that some of the ideas there originated earlier, though I can't see anything in the citations to indicate such. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] M. F. Atiyah and R. S. Ward: “In­stan­tons and al­geb­ra­ic geo­metry,” Comm. Math. Phys. 55 : 2 (1977), pp. 117–​124.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] M. F. Atiyah, N. J. Hitchin, V. G. Drin­fel’d, and Yu. I. Man­in: “Con­struc­tion of in­stan­tons,” Phys. Lett. A 65 : 3 (1978), pp. 185–​187.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[3]  S. K. Donaldson, &quot;An application of gauge theory to four-dimensional topology&quot;, Jour. Differential Geometry 18 (1983), 279-315.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[4] Popov, D. A., &quot;Theory of Yang-Mills Fields&quot;, 1975, Teor. Mat. Fiz. 24, 347.&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-10-30T23:59:02.800" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="62" PostTypeId="1" AcceptedAnswerId="457" CreationDate="2014-10-29T07:41:46.787" Score="15" ViewCount="338" Body="&lt;p&gt;Van Leuwenhoek &lt;em&gt;observed&lt;/em&gt; the first protist in 1674, but his description of it was somewhat superficial and fragmentary:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Passing lately over the Sea at a time, when it blew a fresh gale of wind, and observing the water as above described, I took up some of it in a Glass-vessel, which having view'd the next day, I found moving in it several Earthy particles, and some green streaks, spirally ranged, after the manner of the copper or Tin-worms, used by distillers to cool their distilled waters; and the whole compass of each of these streaks was about the thickness of a man's-hair on his head: Other particles had but the beginning of the said streak; all consisting of small green globuls interspersed; among all which there crawled abundance of little animals, some of which were roundish; those that were somewhat bigger than others were of an oval figure: On these latter I saw two legs near the head, and two little fins on the other end of the body: Others were somewhat larger than an oval, and there were very flow in their motion, and few in number. These &lt;em&gt;animalcula&lt;/em&gt; had divers colours, some being whitish, others pellucid: others had green and very shining little scales; others again were green in the middle, and before and behind white, others grayish. And the motion of most of them in the water was so swift, and so various, upwards, downwards, and round about, that I confess I could not but wonder at it. I judge, that some of these little creatures were above a thousand times smaller than the smallest ones, which I have hitherto seen in chees, wheaten flower, mould, and the like.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;Extract from:&lt;br&gt;&#xA;van Leuwenhoek, A. 1674. More observations from Mr Leewenhook in a letter of Sept. 7, 1674 sent to the publisher. &lt;em&gt;Philosophical transactions of the Royal Society of London,&lt;/em&gt; 9: 178-182.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did he later redescribed them more thoroughly?&lt;br&gt;&#xA;If not, when and by whom was an eukaryotic cell properly described for the first time?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What strikes me the most in this first description is that he doesn't seem to acknowledge what's inside the cell (apart from &lt;em&gt;&quot;others again were green in the middle, and before and behind white, others grayish&quot;&lt;/em&gt;), despite the fact that some organelles (nucleus, chloroplasts, etc.) should be visible with an optical microscope.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-11-13T12:16:53.820" LastActivityDate="2014-11-14T08:17:44.453" Title="When was an eukaryotic cell described for the first time?" Tags="|biology|cell-biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="63" PostTypeId="2" ParentId="57" CreationDate="2014-10-29T07:56:45.110" Score="11" Body="&lt;p&gt;It's not true that Einstein rejected quantum mechanics completely. He acknowledged that it gave numerically accurate predictions in a wide variety of cases, as did any competent physicist by 1935. In that year, he introduced the &lt;a href=&quot;http://en.wikipedia.org/wiki/EPR_paradox&quot;&gt;EPR paradox&lt;/a&gt;, which shows that quantum mechanics doesn't respect locality in special relativity. In particular, if one considers an entangled state of two spins at spacial separation, and measures the spin of one of them, the state of the other must immediately change in order to accommodate the measurement of the first. He considered this sort of faster-than-light action at a distance to be at odds with any &quot;reasonable definition of the nature of reality&quot;. This was his best objection to quantum mechanics as it was interpreted at the time. Of course, Einstein also did some early work on quantum mechanics before these issues were apparent to him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, Einstein chose to reject not quantum mechanics outright, but it's conventional interpretation. He favored a theory in which all physical measurements were determined, but could not all be measured. This would be a so-called &lt;a href=&quot;http://en.wikipedia.org/wiki/Hidden_variable_theory&quot;&gt;hidden-variable theory&lt;/a&gt;, arguing that quantum mechanics was not complete and that additional local degrees of freedom exist which would give a theory that was essentially classical. However, these additional &quot;hidden variables&quot; couldn't hope to be measured in practice, and so quantum mechanics is what we end up seeing. This philosophical position is sometimes known as &quot;local realism&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Local realism, while not necessary to explain any physical measurement, was still on good experimental standing until 1964. Up until that point, the prevailing view was that any quantum mechanical theory could be made into a local hidden variable theory, though no one knew exactly how. In that year, Bell derived his now-famous &lt;a href=&quot;http://en.wikipedia.org/wiki/Bell%27s_theorem&quot;&gt;inequalities&lt;/a&gt; showing that quantum mechanics predicts smaller correlations between certain measurements than could ever be accommodated by any classical hidden variable theory. This gave rise to actual measurements which definitively showed that local hidden variables aren't what we have in nature. By that point, one had to either settle for non-local hidden variables, which wouldn't have satisfied Einstein very much, or just quantum mechanics. Einstein didn't live long enough to have to make that decision though, as he died in 1955.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, it wasn't so much that Einstein believed quantum mechanics was incorrect. Rather, he thought it was incomplete. His later frustrations with it were more that no one had managed to figure out how to realize it with hidden variables (and few people were even trying). When he said things like &quot;God doesn't play dice&quot; and the like, he wasn't saying that quantum mechanics was wrong, so much as incomplete, and was upset that no one was doing what he thought was necessary to make it complete. Through the lens of history, we can see that he was wrong, but at the time this was a seemingly reasonable stance to take.&lt;/p&gt;&#xA;" OwnerUserId="58" LastActivityDate="2014-10-29T07:56:45.110" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="64" PostTypeId="1" CreationDate="2014-10-29T07:56:50.200" Score="10" ViewCount="922" Body="&lt;p&gt;How come do we attribute general theory of relativity to Einstein when David Hilbert published first?&lt;/p&gt;&#xA;" OwnerUserId="88" LastEditorUserId="26" LastEditDate="2016-05-26T17:40:57.287" LastActivityDate="2016-06-21T19:11:01.043" Title="How come we attribute the general theory of relativity to Einstein?" Tags="|mathematics|theoretical-physics|mathematical-physics|differential-geometry|relativity-theory|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="65" PostTypeId="2" ParentId="57" CreationDate="2014-10-29T08:02:54.690" Score="15" Body="&lt;p&gt;Einstein made a number of contributions of momentous importance to quantum theory in the 'early days'. In 1905, his famous &lt;em&gt;annus mirabilis&lt;/em&gt;, he published a paper on the photo-electric effect that laid the basis for the modern understanding of photons (i.e. quantized wavepackets). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was twenty years before the foundations of quantum mechanics were properly formulated by Heisenberg, Schrödinger, Dirac, Born, etc. At that time, the implications of Einstein's work were not quite clear to him, or anyone else for that matter. Another important contribution came in 1924, when Einstein ensured that Bose's work on what later became known as Bose-Einstein statistics were published in a mainstream journal. By that time, however, Einstein already worried much about the foundations of quantum mechanics and the lack of complete determinacy it presents us with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, Einstein did not do much &lt;em&gt;constructive&lt;/em&gt; work on quantum mechanics, but his continuous criticism was important in forcing the proponents of quantum mechanics to give shape to their ideas, and consider how they apply in complicated situations. The most famous instance of this occurring is the Fifth Solvay Conference in 1927, when Einstein went head-to-head with Niels Bohr, proposing a number of 'inconsistencies' of the Heisenberg principle, with Bohr coming up with a refutation time and again. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will not give a serious account of Einstein's later work on the EPR paradox; I think the answer by Logan Maingi already addresses this sufficiently. In conclusion, I would like to point out that most of Einstein's &lt;em&gt;constructive&lt;/em&gt; work on quantum theory was done before the theory was well-understood, but his importance in sharpening the minds of the inventors of quantum mechanics cannot be underestimated. I don't think it's  fair to say that Einstein was &lt;em&gt;against&lt;/em&gt; quantum theory: He simply thought it was not the final theory.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2014-10-29T08:02:54.690" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="66" PostTypeId="2" ParentId="64" CreationDate="2014-10-29T08:07:46.850" Score="15" Body="&lt;p&gt;This seems a bit of a naive question to me. Einstein had been working on this problem for several years (starting as early as 1907), and had developed much of the physics by 1912. He greatly struggled to find the correct mathematical formulation of his theory, finally discovering the necessary tools from differential geometry during a collaboration with the mathematician Grossmann in 1912-1913. However, Einstein's ideas on e.g. the equivalence principle were already known in the community by the time he and Hilbert finally found the correct formulation in terms of the Einstein Field Equations or, alterntively, the Einstein-Hilbert action. It is therefore only natural that the physical theory is regarded as &lt;em&gt;Einstein&lt;/em&gt;'s.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2016-05-26T17:39:56.680" LastActivityDate="2016-05-26T17:39:56.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="67" PostTypeId="2" ParentId="57" CreationDate="2014-10-29T08:21:25.327" Score="3" Body="&lt;p&gt;While the answers provided so far are both good, their authors forget to mention what is the starting point of Einstein's disagreement with quantum mechanics - the Copenhagen interpretation. This (up to date) most spread interpretation of quantum mechanics states that observable quantities do not have a specific value prior to a measurement, after which the quantum state of the system randomly collapses into one of possible measurement eigenstates. It is this inherent randomness which bothered Einstein (and not only him, another famous example is Schroedinger's cat that is dead and alive at the same time until one measures its state). This also gives rise to the super-luminal effects in EPR paradox and led to the quote about god not playing dice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's major contribution to quantum physics - the explanation of the photoelectric effect is about 20 years older than the Copenhagen interpretation and was formulated in the very early years of quantum mechanics. The EPR paradox, on the other hand is older than the Copenhagen interpretation and was designed primarily to show its apparent inconsistency. There is thus no inconsistency in Einstein helping develop a theory he did not agree with.&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="90" LastEditDate="2014-10-29T19:49:03.617" LastActivityDate="2014-10-29T19:49:03.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="68" PostTypeId="1" AcceptedAnswerId="69" CreationDate="2014-10-29T08:38:00.533" Score="15" ViewCount="1142" Body="&lt;p&gt;Compasses are used for various reasons since the 2nd century BC, and for navigation in particular since the 11th century.&lt;br&gt;&#xA;When did people realize that the reason it indicated North have something to do with an intrinsic property of Earth?&lt;br&gt;&#xA;In short, when was it discovered that Earth possess a magnetic field?&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="26" LastEditDate="2014-10-29T09:18:44.163" LastActivityDate="2018-06-30T09:33:53.940" Title="Discovery of Earth's magnetic field" Tags="|experimental-physics|earth-sciences|magnetism|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="69" PostTypeId="2" ParentId="68" CreationDate="2014-10-29T09:03:41.133" Score="12" Body="&lt;p&gt;As it turns out, it was already known by some philosophers from the 1200s that certain types of rocks naturally tend to rotate to point north. However, they had no idea why: For instance, it was generally thought that compas needles were attracted by the Pole Star. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some more serious work on compasses (e.g. discovering the '&lt;a href=&quot;http://en.wikipedia.org/wiki/Magnetic_dip&quot; rel=&quot;noreferrer&quot;&gt;magnetic dip&lt;/a&gt;') was first done by Norman, near the end of the 16th century. He, however, also didn't understand that compasses rely on the magnetic field of the Earth itself. The first to grasp this was &lt;a href=&quot;http://en.wikipedia.org/wiki/William_Gilbert_(astronomer)&quot; rel=&quot;noreferrer&quot;&gt;William Gilbert&lt;/a&gt;, also known as the father of magnetic science. In his primary academic work, published under the name &lt;em&gt;De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure&lt;/em&gt; or, more briefly, &lt;em&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/De_Magnete&quot; rel=&quot;noreferrer&quot;&gt;De Magnete&lt;/a&gt;&lt;/em&gt; he concluded - based on experiments carried out on a model planet - that the Earth itself can be regarded as a giant magnet (as is reflected in the title itself),  correctly attributing it to the iron core of our planet. Thus, William Gilbert was the first to realize that Earth possesses an intrinsic magnetic field. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, it is interesting to note that Gauss was in fact the first to accurately measure Earth's magnetic field and fit it to &lt;a href=&quot;http://en.wikipedia.org/wiki/Spherical_harmonics&quot; rel=&quot;noreferrer&quot;&gt;spherical harmonics&lt;/a&gt; in order to understand its variations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.magnet.fsu.edu/education/tutorials/pioneers/gilbert.html&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt; (aside from Wikipedia).&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-10-30T19:38:39.700" LastActivityDate="2014-10-30T19:38:39.700" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="70" PostTypeId="2" ParentId="68" CreationDate="2014-10-29T09:09:04.870" Score="5" Body="&lt;p&gt;To add to Danu's answer, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to New Zealand's GNS webpage &lt;a href=&quot;http://www.gns.cri.nz/Home/Our-Science/Earth-Science/Earth-s-Magnetic-Field/Discovery-of-the-Earth-s-magnetic-field&quot;&gt;Discovery of the Earth’s magnetic field&lt;/a&gt;, from William Gilbert's work concluding that the planet acted as a giant magnet, it took several centuries to determine the actual mechanism as to why the Earth behaves this way, due to it being found that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It gradually became apparent that the obvious theory, that the earth is composed of magnetic rock, was incorrect, as rocks lose their magnetism at the temperatures found at any significant depth within the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It was in the 20th century the widel accepted intrinsic property for the magnetic field became clearer, with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Larmor suggested in 1919 that a self-exciting dynamo could explain the magnetic field of the earth, as well as that of the sun and other stars, but it was Elsasser and Bullard in the 1940s who showed how motion in the liquid core of the earth might produce a self-sustaining magnetic field.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastEditorDisplayName="user22" LastEditDate="2014-10-29T09:15:39.637" LastActivityDate="2014-10-29T09:15:39.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="71" PostTypeId="1" AcceptedAnswerId="76" CreationDate="2014-10-29T09:22:23.357" Score="10" ViewCount="168" Body="&lt;p&gt;Inspired by the question &lt;a href=&quot;https://hsm.stackexchange.com/questions/68/discovery-of-earths-magnetic-field/70#70&quot;&gt;Discovery of Earth's magnetic field&lt;/a&gt;, the article from New Zealand's webpage &lt;a href=&quot;http://www.gns.cri.nz/Home/Our-Science/Earth-Science/Earth-s-Magnetic-Field/Discovery-of-the-Earth-s-magnetic-field&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discovery of the Earth’s magnetic field&lt;/a&gt;, also states that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The earth's dynamo is unstable, as is shown by magnetic reversals, when the polarity of the whole magnetic field changes over. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When was the fact that Earth's magnetic field reversed discovered?&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-10-29T15:00:02.540" Title="Discovery of magnetic field reversals?" Tags="|experimental-physics|earth-sciences|magnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="72" PostTypeId="5" CreationDate="2014-10-29T09:34:51.690" Score="0" Body="&lt;p&gt;If the concerned question is about a more specific area of Chemistry (such as &lt;a href=&quot;/questions/tagged/atomic-theory&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;atomic-theory&amp;#39;&quot; rel=&quot;tag&quot;&gt;atomic-theory&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/peridic-table&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;peridic-table&amp;#39;&quot; rel=&quot;tag&quot;&gt;peridic-table&lt;/a&gt; etc.), this tag should be used along with the tag of that specific area of Chemistry.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-20T13:46:15.383" LastActivityDate="2022-03-20T13:46:15.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="73" PostTypeId="4" CreationDate="2014-10-29T09:34:51.690" Score="0" Body="For questions about the study of chemical reactions, dynamics and related phenomena" OwnerDisplayName="user22" LastEditorUserId="10106" LastEditorDisplayName="user22" LastEditDate="2022-06-04T02:35:01.560" LastActivityDate="2022-06-04T02:35:01.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="74" PostTypeId="5" CreationDate="2014-10-29T09:38:04.993" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-10-29T09:38:04.993" LastActivityDate="2014-10-29T09:38:04.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="75" PostTypeId="4" CreationDate="2014-10-29T09:38:04.993" Score="0" Body="Questions related to the chemistry and dynamics of the varying layers of the atmosphere." OwnerDisplayName="user22" LastEditorDisplayName="user22" LastEditDate="2014-11-02T21:26:36.347" LastActivityDate="2014-11-02T21:26:36.347" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="76" PostTypeId="2" ParentId="71" CreationDate="2014-10-29T10:31:45.660" Score="8" Body="&lt;p&gt;&lt;strong&gt;Cox et al. 1964&lt;/strong&gt; gives a good account on the different steps of how this discovery was made.&lt;br&gt;&#xA;The most important first step was to acknowledge the fact that some rocks (namely magmatic rocks) acquired a remanent magnetism based on the magnetic field in which they were &quot;cooked&quot; (&lt;strong&gt;Brunhes 1906&lt;/strong&gt;).&lt;br&gt;&#xA;Followed observations of different directions in remanent magnetism of some lavas by &lt;strong&gt;Chevallier 1925&lt;/strong&gt;, thus implying that the surrounding magnetic field changed direction.&lt;br&gt;&#xA;&lt;strong&gt;Matuyama 1929&lt;/strong&gt; then managed to place the latest reversal of Earth's magnetic field in a stratigraphic context: it is now known to have occured at 0.781 Ma. The event later took his name, &quot;Matuyama reversed polarity interval&quot;, now known more formally as chron C1r-1r (the current chron, C1n, is by the way nicknamed &quot;Brunhes normal polarity interval&quot;).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, interestingly, the discovery of reversed interval of magnetic polarity actually predates the discovery of the mechanism explaining earth magnetic field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;References:&lt;br&gt;&#xA;Brunhes, 1906. &lt;a href=&quot;https://hal.archives-ouvertes.fr/file/index/docid/241149/filename/ajp-jphystap_1906_5_705_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;Recherches sur la direction d'aimantation des roches volcaniques&lt;/a&gt;. Journal de Physique Théorique et Appliquée, 5: 705-724.&lt;br&gt;&#xA;Chevallier, 1925. &lt;a href=&quot;http://dx.doi.org/10.1007/BF02719509&quot; rel=&quot;noreferrer&quot;&gt;L'aimantation des laves de l'Etna et l'orientation du champ terrestre en Sicile du XIIe au XVIIe siècle&lt;/a&gt;. Bulletin Volcanique, 2(2): 234-244.&lt;br&gt;&#xA;Cox, Doell &amp;amp; Dalrymple, 1964. &lt;a href=&quot;https://courses.seas.harvard.edu/climate/eli/Courses/EPS281r/Sources/Plate-tectonics/Sea-floor-spreading-and-magnetic-reversals/more/2_Cox_Doell_1964_Science.pdf&quot; rel=&quot;noreferrer&quot;&gt;Reversals of the Earth's magnetic field&lt;/a&gt;. Science, 144 (3626): 1537-1543.&lt;br&gt;&#xA;Matuyama, 1929. &lt;a href=&quot;https://www.jstage.jst.go.jp/article/pjab1912/5/5/5_5_203/_article&quot; rel=&quot;noreferrer&quot;&gt;On the Direction of Magnetisation of Basalt in Japan, Tyôsen and Manchuria&lt;/a&gt;. Proceedings of the Imperial Academy, 5: 203-205.&#xA;&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-10-29T15:00:02.540" LastActivityDate="2014-10-29T15:00:02.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="77" PostTypeId="2" ParentId="58" CreationDate="2014-10-29T10:41:32.383" Score="44" Body="&lt;p&gt;An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set,  the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to consider first the intersection of all these sets, and then the derived set of &lt;em&gt;that&lt;/em&gt; set and so on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So he came to consider transfinite ordinals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is discussed in various places, including &lt;a href=&quot;http://www.math.caltech.edu/~kechris/papers/uniqueness.pdf&quot;&gt;&quot;Set Theory and Uniqueness of Trogonometric Series&quot; by Kechris&lt;/a&gt; or &quot;&lt;a href=&quot;http://dx.doi.org/10.1007/BF01886630&quot;&gt;Uniqueness of trigonometric series and descriptive set theory, 1870–1985&lt;/a&gt;&quot; by Roger Cooke (Archive for History of Exact Sciences, 1993)   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original paper is (I think) &lt;a href=&quot;http://resolver.sub.uni-goettingen.de/purl?GDZPPN00224179X&quot;&gt;&quot;Ueber die Ausdehnung eines Satzes ais der Theorie der trigonometrischen Reihen (Math. Annalen, 1872)&quot;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another motivation was his earlier work on number theory. Using what is now called a diagonalisation argument he was able to prove results on the existence of transcendental numbers. This is in his 1874 paper &quot;Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen&quot; (&quot;On a Property of the Collection of All Real Algebraic Numbers&quot;)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief, the original motivation was to have better tools for making progress on existing problems. &lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="205" LastEditDate="2015-05-08T12:01:34.207" LastActivityDate="2015-05-08T12:01:34.207" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="79" PostTypeId="2" ParentId="54" CreationDate="2014-10-29T12:00:19.160" Score="4" Body="&lt;p&gt;By the etymology of science, any pursuit of knowledge may be wrongly attributed as science. But science in reality is much more than the just knowledge, it means systematizing the existing knowledge and then suing that knowledge to gain more knowledge and so on ad infinitum. But how do we gain this knowledge, we do that by using what is now called the &lt;i&gt;scientific method&lt;/i&gt;, although at times it becomes very difficult to really tell where the boundaries of this method ends.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is slightly different, to be called a scientific method we must have some empirical and measurable evidence of what we are discussing. At times, this becomes very hard to achieve. Case in point are statements for which we know we can never give a definite answer. Another instance is the construction of the standard topological set $S_\Omega$, we know that it exists. but we do not know what it is. This in some sense is against the ethics of what we call science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This demarcation of math and science has become more prominent in the 20th century after many abstract mathematical concepts came into being, which at first sight seemed to be counter intuitive or which lacked any evidence whatsoever about their existence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But still, we still call it the mathematical sciences when we include fields which are inspired by mathematics, but maybe whose specialists we do not call mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="102" LastActivityDate="2014-10-29T12:00:19.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="80" PostTypeId="2" ParentId="2" CreationDate="2014-10-29T12:13:44.190" Score="8" Body="&lt;p&gt;The fact that $\sqrt{2}$ existed and is irrational was a blow to the ancient Greeks who only believed in numbers that they could calculate to a certain degree of precision whenever required. Or in other words, they were familiar with rational numbers. The fact that others numbers existed would have carried the same sort of feelings in them as and when we first encounter topics such as countability and uncountability and the continuum hypothesis in set theory. At first it may seem to be all some sort of circular and wrong argument but given some time we get used to it. And perhaps so did the Greeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as practicality is concerned, it would not have been much practical to them as they would not have been able to measure these new numbers to a degree of precision that they were used too. But the whole point of knowledge gathering is not where to put that knowledge to use, but why should that knowledge exist in the first place.&lt;/p&gt;&#xA;" OwnerUserId="102" LastEditorUserId="26" LastEditDate="2014-12-03T10:13:16.500" LastActivityDate="2014-12-03T10:13:16.500" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="81" PostTypeId="2" ParentId="44" CreationDate="2014-10-29T12:24:12.300" Score="6" Body="&lt;p&gt;The last great polymaths were John von Neumann and David Hilbert. After that we perhaps do not see examples in their class. Some may say that Terence Tao can be considered one considering the fact that he has contributed to so many different mathematical fields, but I don't think his diversity can rival say Gauss' or Euler's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main reason is that the length and breadth of human knowledge has expanded many folds now than in the time of the great polymaths we know about. Nowadays to get a PhD we work on a sub field of a sub field of a subfield of a field, and many a times we cannot even consider ourselves to be a expert in that particular sub-sub-sub field. In practice thus, it would be nearly impossible now to be a polymath, but one can always try.&lt;/p&gt;&#xA;" OwnerUserId="102" LastActivityDate="2014-10-29T12:24:12.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="82" PostTypeId="1" CreationDate="2014-10-29T12:44:22.510" Score="13" ViewCount="551" Body="&lt;p&gt;Today the language of topological spaces via open sets is fundamental in many different areas of mathematics, and it is a bit mysterious that the same formalism successfully captures such a wide variety of behavior.  I can think of several independent reasons to invent the definition of a topology, all of which would have been on mathematicians' radar screens around the time the definition was first being mulled over in the early 20th century:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To provide a foundation for Klein's Erlangen program and Poincare's work on Betti numbers and the fundamental group&lt;/li&gt;&#xA;&lt;li&gt;To clarify the foundations of calculus, e.g. the role of compactness in the extreme value theorem&lt;/li&gt;&#xA;&lt;li&gt;To distinguish among various notions of convergence of functions (leading to functional analysis)&lt;/li&gt;&#xA;&lt;li&gt;To give meaning to arguments involving &quot;generic&quot; configurations in algebraic geometry&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My understanding is that it took quite some time for the modern formalism of topological spaces to emerge, so I'm wondering what specific results or examples were most influential in its development?  And which modern applications of the theory were only realized after it matured?&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="58" LastEditDate="2014-10-30T10:36:37.990" LastActivityDate="2018-09-13T17:37:11.597" Title="What examples led to the modern definition of a topological space?" Tags="|mathematics|topology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="83" PostTypeId="1" AcceptedAnswerId="217" CreationDate="2014-10-29T12:55:21.747" Score="29" ViewCount="1660" Body="&lt;p&gt;From what I have read, the atomic theory of matter was cemented by a 1905 paper by Einstein in which he explained the erratic motion of a bit of pollen suspended in water using the assumption that water is made of tiny molecules.  The question of whether matter was atomic was subsequently laid to rest with the rise of atomic physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: what were the most persuasive non-atomic theories of matter?  Did they enjoy any scientific advantages over atomism?&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="2172" LastEditDate="2019-03-26T13:58:19.263" LastActivityDate="2019-03-26T13:58:19.263" Title="What were the dominant non-atomic theories of matter in the 19th century?" Tags="|physics|thermodynamics|atomic-theory|statistical-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="84" PostTypeId="2" ParentId="24" CreationDate="2014-10-29T12:58:36.273" Score="1" Body="&lt;p&gt;One more slightly different claim is also that Nobel was not in good terms with Mittag Lefler and when he asked someone if a Nobel in math could ever go to Lefler, the person said yes and so Nobel striked off math from his list (which again some claimed contained math as one of the subjects).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the most logical explanation that has ever been put forward was that Nobel was not much interested in math as he could not see any tangible applications which served mankind as the others subjects did. And also there existing a good prize for math at that time was probably another reason for him giving math a miss.&lt;/p&gt;&#xA;" OwnerUserId="102" LastActivityDate="2014-10-29T12:58:36.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="85" PostTypeId="1" CreationDate="2014-10-29T13:04:11.690" Score="3" ViewCount="129" Body="&lt;p&gt;Until the Abel prize came along, the Fields medals were considered to be the Nobel for math. But when did this tradition of calling it the Nobel for math start and why? Because the prize money is nowhere as much as the Nobels and also the rules governing the two prizes are entirely different.&lt;/p&gt;&#xA;" OwnerUserId="102" LastActivityDate="2014-10-29T13:17:01.633" Title="When did the Fields Medal get considered as the Nobel for math?" Tags="|nobel-prize|academic-awards|" AnswerCount="0" CommentCount="2" ClosedDate="2014-10-29T20:29:48.710" ContentLicense="CC BY-SA 3.0" />
  <row Id="86" PostTypeId="2" ParentId="6" CreationDate="2014-10-29T13:06:16.970" Score="19" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_metric_system#The_first_structured_metric_system:_CGS&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;History of the metric system - Wikipedia&quot;&gt;History of the metric system - Wikipedia&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1861, Charles Bright and Latimer Clark proposed the names of ohm, volt, and farad in honour of Georg Ohm, Alessandro Volta and Michael Faraday respectively for the practical units based on the centimetre-gramme-second absolute system. This was supported by Thomson (Lord Kelvin). These names were later scaled for use in the Practical System. The concept of naming units of measure after noteworthy scientists was subsequently used for other units.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="102" LastEditorUserId="65" LastEditDate="2020-07-13T07:58:15.013" LastActivityDate="2020-07-13T07:58:15.013" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="87" PostTypeId="5" CreationDate="2014-10-29T13:07:56.120" Score="0" Body="&lt;p&gt;The Fields Medal, formally called the 'International Medal for Outstanding Discoveries in Mathematics' and sometimes informally referred to as the 'mathematicians' Nobel Prize', is one of the most prestigious prizes that a mathematician may receive. It was awarded for the first time in 1936, and continues to be awarded every four years to between two and four mathematicians below the age of 40.&lt;/p&gt;&#xA;" OwnerUserId="-1" LastEditorUserId="26" LastEditDate="2015-03-01T13:45:03.857" LastActivityDate="2015-03-01T13:45:03.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="88" PostTypeId="4" CreationDate="2014-10-29T13:07:56.120" Score="0" Body="The Fields Medals is a prestigious prize awarded every four year at the International Congress of Mathematicians to between two and four mathematicians below the age of 40." OwnerUserId="102" LastEditorUserId="26" LastEditDate="2015-03-01T13:41:47.263" LastActivityDate="2015-03-01T13:41:47.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="90" PostTypeId="2" ParentId="55" CreationDate="2014-10-29T13:26:55.317" Score="3" Body="&lt;p&gt;From a practical point of view, the notation was vastly different.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A particular sore point for me is that the Leibniz notation lets you &lt;a href=&quot;http://en.wikipedia.org/wiki/Abuse_of_notation#Derivative&quot; rel=&quot;nofollow&quot;&gt;incorrectly&lt;/a&gt; work with derivatives as though they were a mathematical fraction.  Unfortunately this 'works out' a lot of the time so its still used, even in college courses, today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think there is anything wrong with shortcuts, up to the point that they don't interfere with understanding.  In this case, I do believe it creates a misunderstanding of the subject matter.  This alone I think puts Newtons notation above Leibniz's. &lt;/p&gt;&#xA;" OwnerUserId="19" LastActivityDate="2014-10-29T13:26:55.317" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="91" PostTypeId="2" ParentId="9" CreationDate="2014-10-29T13:34:25.353" Score="11" Body="&lt;p&gt;There have been many accounts of the life of Galois and the duel. Most sources suggest that the duel was due to a failed love affair. It seems Galois fell in love with the physician's daughter when he was in jail, but as happens most of the time, the girl was not in love with Galois but someone else. Galois challenged him to a duel and the rest is now known. It was very common back then to challenge one another to a duel to resolve matters, but what was sad was that Galois had to lie there after the duel for many hours before help reached him. Perhaps timely help would have saved him. But that is just speculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the matter of whether he wrote most of his math before he died, that would most certainly be false because he must have had the idea beforehand. He just wrote it down before the night of his duel for safety and sent it to Chavelier. Hermann Weyl says about this letter : &quot;This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an easy to read description about him, you can check &lt;a href=&quot;http://gonitsora.com/evariste-galois/&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;. Or read an interesting version &lt;a href=&quot;http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Rothman84-106.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="102" LastEditorUserId="1878" LastEditDate="2015-06-14T14:26:31.520" LastActivityDate="2015-06-14T14:26:31.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="92" PostTypeId="2" ParentId="82" CreationDate="2014-10-29T13:36:28.480" Score="3" Body="&lt;p&gt;Topological spaces appear to have their roots in the &lt;em&gt;nineteenth&lt;/em&gt; century. It started, indirectly, with the theory of limits and delta-epsilon proofs. A major breakthrough occurred with the development of set theory &lt;a href=&quot;http://en.wikipedia.org/wiki/De_Morgan&amp;#39;s_laws&quot; rel=&quot;nofollow&quot;&gt;(e.g. DeMorgan's Laws)&lt;/a&gt; in the middle to latter part of the century. This led to the &quot;generalization&quot; of limit, convergence, and accumulation point axioms using the theory of open and closed sets. Topology is sometimes referred to as &quot;point set&quot; theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The applications you cite came &quot;later,&quot; that is, in the twentieth century. So did the so-called &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_the_separation_axioms&quot; rel=&quot;nofollow&quot;&gt;Separation Axioms&lt;/a&gt;, beginning with Hausdorff spaces, in 1914, and extended in the middle of the century. But the foundations for these applications were laid in the previous century.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-10-29T13:36:28.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="93" PostTypeId="2" ParentId="38" CreationDate="2014-10-29T13:57:01.510" Score="6" Body="&lt;p&gt;According to &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0688119123&quot; rel=&quot;noreferrer&quot;&gt;Generations&lt;/a&gt;, by William Strauss and Neil Howe (S&amp;amp;H), the so-called World War II, or &quot;Greatest&quot; Generation, born 1901-1924 in America, were a particularly optimistic generation, in large part because of their life cycle. They won  World War II as young men, landed a man on the moon in their middle age, and enjoyed the Reagan (economic) boom as oldsters. Simon (born 1914), Minsky (born 1919) and Reagan himself (born 1911) were all members of this generation. &quot;Artificial intelligence&quot; emerged during the period of these men's middle age, and that's why their optimism is sometimes associated with their science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They had been shaped this way as children by parents of the two preceding generations: FDR's &#xA;&quot;Rendezvous With Destiny&quot; Generation (or what S&amp;amp;H called &quot;Missionaries,&quot;) and the Lost Generation of FitzGerald and Hemingway. Mr. Howe (and Mr. Strauss before he passed away) believe that today's young people, the so-called Millennials, born 1982-2001, are another optimistic generation, raised by the Boomers (the modern &quot;Rendezvous&quot;) and Generation X (the &quot;new Lost&quot;). Hopefully without a &quot;World War&quot; attached to their designation. This would be particularly true of Americans, somewhat true of others in a &quot;global&quot; economy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These prognosticators made one fundamental mistake; that of assuming that succeeding generations will be like them and continue their trajectory. S&amp;amp;H postulated four different generations with different priorities, with the fourth remotest (the Millennials), taking up the &quot;baton&quot; of the World War II generation. So the vision of these World War II generation men will be realized in the middle age of the Millennials, in the 2020s-2030s, not during the middle age of the so-called Silent generation (of Alan Greenspan and Neil Armstrong), (1965-1985).&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="31" LastEditDate="2014-10-29T14:10:31.660" LastActivityDate="2014-10-29T14:10:31.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="94" PostTypeId="1" AcceptedAnswerId="105" CreationDate="2014-10-29T14:16:47.293" Score="13" ViewCount="1451" Body="&lt;p&gt;One of the first distinctions encountered in science education is that substances can assume different states of matter: for example, water can be found as a liquid, as a solid (ice), or as a gas (water vapor).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This concept gradually becomes more and more sophisticated: The state reflects the macroscopic variables of the problem (e.g. pressure, temperature), different substances have different phase diagrams, different phases can coexist under certain conditions, etc. It's thus not surprising that the phase concept is central to modern science and in particular our attempts to discover/create/understand novel forms of matter. (Most discussions of high-temperature superconductivity, for example, revolve around trying to obtain a theoretical understanding of the observed phase diagrams of various materials.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the purpose of this question, what I'm specifically interested in is the early history of the phase concept, and what references are available on the subject. For example, roughly when did human beings recognize that liquid water, solid ice, and water vapor could all be understood as different states of the same substance?&lt;/p&gt;&#xA;" OwnerUserId="107" LastEditorUserId="107" LastEditDate="2014-10-29T14:34:52.623" LastActivityDate="2024-01-04T14:18:49.347" Title="Early history of the phase concept in the physical sciences" Tags="|reference-request|materials-science|thermodynamics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="95" PostTypeId="2" ParentId="55" CreationDate="2014-10-29T15:56:39.987" Score="30" Body="&lt;p&gt;Newton's notation, Leibniz's notation and Lagrange's notation are all in use today to some extent. They are, respectively:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dot{f} = \frac{df}{dt}=f'(t)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\ddot{f} = \frac{d^2f}{dt^2}=f''(t)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can find more notation examples on &lt;a href=&quot;http://en.wikipedia.org/wiki/Notation_for_differentiation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The standard integral(&lt;span class=&quot;math-container&quot;&gt;$\displaystyle\int_0^\infty f dt$&lt;/span&gt;) notation was developed by Leibniz as well.  Newton did not have a standard notation for integration.&lt;/p&gt;&#xA;&lt;p&gt;I have read from &amp;quot;The Information&amp;quot; by James Gleick the following: According to Babbage, who eventually took the Lucasian Professorship at Cambridge which Newton held, Newton's notation crippled mathematical development.  He worked as an undergraduate to institute Leibniz's notation as it is used today at Cambridge, despite the distaste the university still had because of the Newton/Leibniz conflict.  This notation is a lot more useful that Newton's for most cases.  It does, however, imply that it can be treated as a simple fraction which is incorrect.&lt;/p&gt;&#xA;" OwnerUserId="42" LastEditorUserId="7205" LastEditDate="2022-10-26T10:05:34.357" LastActivityDate="2022-10-26T10:05:34.357" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="96" PostTypeId="1" AcceptedAnswerId="231" CreationDate="2014-10-29T16:16:22.420" Score="12" ViewCount="915" Body="&lt;p&gt;I've always been curious about how great forgotten ideas are rediscovered.  This question: &lt;a href=&quot;https://hsm.stackexchange.com/questions/59/are-there-written-19th-century-sources-expressing-the-belief-that-the-intermed&quot;&gt;Are there written (19th century) sources expressing the belief that the intermediate value property is equivalent to continuity?&lt;/a&gt; led me to the following paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Schubring, Gert: &quot;&lt;a href=&quot;http://dx.doi.org/10.1006/hmat.1993.1005&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernard Bolzano--Not as unknown to His contemporaries as is commonly believed?&lt;/a&gt;&quot; &lt;em&gt;Historia Mathematica&lt;/em&gt;, 20(1):45-53, 1993.  (Elsevier's paywall, sorry I couldn't find a freed version.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which says that &quot;Herman Hankel is credited with having been the first to bring Bolzano to  the general attention of the mathematical community in 1871.&quot;  (and Schubring goes on to argue that in fact, Bolzano's work had probably been known to Crelle's circle in Berlin around 1825 before being forgotten.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did Hankel come upon Bolzano's work and recognize that Bolzano had priority over Weierstrass?&lt;/p&gt;&#xA;" OwnerUserId="16" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-11-03T04:14:21.420" Title="How and when was Bolzano's proof of the Bolzano-Weierstrass theorem rediscovered?" Tags="|mathematics|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="97" PostTypeId="1" AcceptedAnswerId="103" CreationDate="2014-10-29T16:30:45.593" Score="11" ViewCount="119" Body="&lt;p&gt;Adam Sedgwick was a master structural geologist and was one of the primary participants in the early days of stratigraphy - working out how rocks could be distinguished, extrapolated, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although he did not consider himself a biostratigraphic expert (he would usually send his fossils to the relevant experts for precise identification), he would have definitely been aware of the &quot;succession&quot; of species: That the species in rocks of different ages are different (so much so that they could be used for system identification), and they tend to be fewer and simpler in his own Cambrian, but much more complex and diverse higher up (e.g. in the Secondary and Tertiary rocks).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was probably also aware of some of the similarities between fossils in rocks of neighboring ages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite retaining a good friendship with one of his former students, Charles Darwin, he never accepted evolution. Presumably this is related to his position in the more conservative side of the Church of England.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what was Sedgwick's explanation for these patterns seen in the stratigraphic distribution of fossils?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="26" LastEditDate="2015-03-01T13:39:41.913" LastActivityDate="2015-03-01T13:39:41.913" Title="How did Sedgwick explain the succession of species?" Tags="|earth-sciences|evolution|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="98" PostTypeId="1" AcceptedAnswerId="249" CreationDate="2014-10-29T17:18:34.950" Score="29" ViewCount="1406" Body="&lt;p&gt;It's common knowledge that Darwin is considered the father of evolution, but even humans have been breeding animals (and probably themselves) selectively for specific traits for thousands of years. &lt;strong&gt;How much of the theory of evolution actually existed before Darwin caused big waves?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="113" LastEditorUserId="26" LastEditDate="2014-10-29T18:39:12.050" LastActivityDate="2015-12-15T12:19:00.140" Title="Was evolution ever discussed (perhaps using different terms) prior to Darwin?" Tags="|evolution|biology|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="99" PostTypeId="2" ParentId="98" CreationDate="2014-10-29T17:26:05.867" Score="14" Body="&lt;p&gt;The first known discussion concerning evolution are, to my knowledge, pre-Socratic philosophers. A more well known ancient publication evoking the idea may be &lt;em&gt;De Rerum Natura&lt;/em&gt;, by Roman philosopher Lucretius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answers your title question, I don't have sufficient knowledge to satisfactorily address the question in the body of the text. It might be worth it to look into &lt;a href=&quot;http://en.wikipedia.org/wiki/Pierre_Louis_Maupertuis&quot;&gt;Maupertuis&lt;/a&gt;'s work on &quot;natural modifications&quot;, as he called them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Kirk, Geoffrey; Raven, John; Schofield, John (1984a). The Presocratic Philosophers: A Critical History with a Selection of Texts (3rd ed.). Chicago: The University of Chicago Press. pp. 100–321)&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="26" LastEditDate="2014-10-29T19:05:26.390" LastActivityDate="2014-10-29T19:05:26.390" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="100" PostTypeId="2" ParentId="29" CreationDate="2014-10-29T17:44:37.613" Score="13" Body="&lt;p&gt;Constance Reid's biography &lt;em&gt;Hilbert&lt;/em&gt; contains a brief discussion at the beginning of chapter 23. Hilbert was &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;somewhat angry... but then began to try to deal constructively with the problem... Broadened methods would permit the loosening of the requirements of formalizing. Hilbert himself now took a step in this direction. This was the replacing of the schema of complete induction by a stronger rule called &quot;transfinite induction&quot;. In 1931 two papers in the new direction appeared.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think the last sentence may refer to the consistency proof for Peano arithmetic due to Gerhard Gentzen, who was Hilbert's assistant. Gödel himself gave a consistency proof for Peano arithmetic using so-called functionals of higher types (see Shoenfeld's &lt;em&gt;Mathematical Logic&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-10-30T20:05:23.370" LastActivityDate="2014-10-30T20:05:23.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="101" PostTypeId="2" ParentId="98" CreationDate="2014-10-29T17:48:18.383" Score="20" Body="&lt;p&gt;Some more pre-Charles Darwin discussions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Erasmus Darwin (Charles' grandfather): &lt;a href=&quot;http://www.ucmp.berkeley.edu/history/Edarwin.html&quot;&gt;http://www.ucmp.berkeley.edu/history/Edarwin.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Jean-Batiste Lamarck: &lt;a href=&quot;http://en.wikipedia.org/wiki/Lamarckism&quot;&gt;http://en.wikipedia.org/wiki/Lamarckism&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Robert Chambers &quot;Vestiges of the Natural History of Creation&quot; (heavily criticized at the time and with political undercurrents)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;James Hutton also wrote on the topic and his ideas sound suspiciously Darwinian (it should be noted that Hutton also had his own farm and was aware of selective breeding). However, Hutton was noted for his obtuse writings (his geological ideas would have been ignored if it were not for John Playfair), and more importantly, his evolutionary writings were unpublished and only discovered in the 20th century.&lt;/p&gt;&#xA;" OwnerUserId="25" LastActivityDate="2014-10-29T17:48:18.383" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="102" PostTypeId="2" ParentId="42" CreationDate="2014-10-29T18:05:41.670" Score="1" Body="&lt;p&gt;Logarithm tables have been used since at least the middle ages by merchants in order to perform big multiplications. Guess that makes them come first, although the formal definition came latter, as shown by Alexandre's answer.&lt;/p&gt;&#xA;" OwnerUserId="8" LastActivityDate="2014-10-29T18:05:41.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="103" PostTypeId="2" ParentId="97" CreationDate="2014-10-29T18:13:45.650" Score="9" Body="&lt;p&gt;Based on what I've read in Stephen Jay Gould, &lt;em&gt;The Structure of Evolutionary Theory&lt;/em&gt;, Cambridge MA: Belknap Press of Harvard University Press, 2002, I would assume that, in fact, the intellectual conflict was the opposite of what you propose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Darwin's variant of natural selection is founded in Lyell's theory of &lt;a href=&quot;http://en.wikipedia.org/wiki/Gradualism&quot;&gt;&lt;em&gt;gradualism&lt;/em&gt;&lt;/a&gt;.  That is, it assumed that infinitesimally small mutations accumulate over geological time until there are so many of them that you have achieved a new species.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lyell's (and pretty much everyone elses) explanation for the sudden changes in species in the geological record were simply due to missing data.  Lyell (and Darwin) held that everything in the geological record could be explained in terms of processes (like erosion) that are observed every day.  This was opposed to &lt;em&gt;cataclysmic&lt;/em&gt; theories that claimed that some geological strata were actually caused by sudden massive, global events (like meteor strikes).  According to the Wikipedia article on Sedgwick, for much of his career he was an opponent of gradualism and uniformatarianism and favored explanations of certain gravel deposits as evidence for the biblical &quot;Great Flood.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The recognition that the sudden species jumps in the geological record represent &lt;em&gt;evidence&lt;/em&gt; of very fast (non gradual) evolution, rather than &lt;em&gt;missing&lt;/em&gt; data, was popularized in the 1970s and 1980s by Gould.&lt;/p&gt;&#xA;" OwnerUserId="16" LastActivityDate="2014-10-29T18:13:45.650" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="104" PostTypeId="2" ParentId="12" CreationDate="2014-10-29T18:19:29.500" Score="12" Body="&lt;p&gt;It all depends on your definition of a telescope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Leonard_Digges_(scientist)&quot; rel=&quot;noreferrer&quot;&gt;Digges&lt;/a&gt; absolutely built some sort of device that was capable of magnifying objects. It seems that is agreed upon. But the divide is really about &lt;em&gt;whether or not this was a telescope&lt;/em&gt;. Some historians shrug it off as little more than a powerful spyglass; others herald it as the first telescope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several accounts that attest to the device's power. &lt;a href=&quot;http://space.about.com/od/astronomerbiographies/a/Leonard_Digges.htm&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt;, Digges' son is quoted as saying&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;..... my father by his continual pain-full practices [practical experiments], assisted with Demonstrations Mathematicall, was able and sundrie Times hath by proportionall Glasses duly situate in convenient angles, not onely discovered things farre off, read letters, numbered peeces of money with the very coyne and superscription thereof, cast by some of his freends of purpose uppon Downes in open fields, but also at seven miles declared what had been doon at that instant in private places.....&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it could see things from seven miles off. &lt;a href=&quot;http://www.chocky.demon.co.uk/oas/diggeshistory.html&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt;, though, the same quote is discussed and considered mere exaggeration. It does analyze another testimony, from William Bourne:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Yet I am assured that the Glasse that ys grounde, beeynge of very cleare stuffe, and of good largeness, and placed so that the beame dothe come thorowe, and so reseaved into a very large concave lookinge Glasse, that yt will shewe the thinge of marvelous largenes, in a manner uncredable to bee believed of the common people.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is from a report, commissioned by Queen Elizabeth I, by an independent, outside observer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Leonard_Digges_(scientist)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; page on Digges ends with this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, the construction of lenses to the required optical precision would have been very difficult in the 16th century, and the construction of an adequate mirror would have been much harder still. It is doubtful that Digges built a successful instrument, and the optical performance required to see the details of coins lying about in fields, or private activities seven miles away was far beyond the technology of the time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it is likely that Digges built a device capable of magnifying objects at a distance (although probably not seven miles), but it would have been fairly rudimentary, and not much like a telescope today.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;By comparison, &lt;a href=&quot;http://www.universetoday.com/15763/galileos-telescope/&quot; rel=&quot;noreferrer&quot;&gt;Galileo's telescope&lt;/a&gt; was much more powerful. His first one could only magnify an object 3x, but his final version could magnify an object 30x. That's fairly powerful! A piece of writing 7 one inch tall that was seven miles away would appear to be 30 inches tall. When you think about it, that's a huge improvement. And, after all, &lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_Galilei#Engineering&quot; rel=&quot;noreferrer&quot;&gt;Galileo&lt;/a&gt; used it to discover some of Jupiter's moons. I highly doubt Digges' device was capable of that!&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-07T21:39:14.367" LastActivityDate="2014-11-07T21:39:14.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="105" PostTypeId="2" ParentId="94" CreationDate="2014-10-29T18:44:13.120" Score="7" Body="&lt;p&gt;Presumably the recognition that water, ice and vapor are different states of the same substance goes back to prehistoric times: e.g. boiling water over a fire or observing the melting of ice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The early history of the phase diagram in physics is connected to Gibbs, Maxwell and van der Waals: the van der Waals equation (1873) implies that there are coexisting liquid and gas phases below the critical temperature in real gases. Van der Waals received the 1910 Nobel Prize in physics for this work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell, based on the work of Gibbs, constructed a three-dimensional &lt;a href=&quot;http://en.wikipedia.org/wiki/Maxwell%27s_thermodynamic_surface&quot;&gt;plot&lt;/a&gt; of the variables volume, entropy and energy for a given pressure and temperature (1874). He did not exactly construct a phase diagram of water but &#xA; &quot;… modelled a fictitious substance, in which the volume is greater when solid than when liquid; and in which, as in water, the saturated vapour becomes superheated by compression.&quot;&lt;/p&gt;&#xA;" OwnerUserId="37" LastActivityDate="2014-10-29T18:44:13.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="106" PostTypeId="1" AcceptedAnswerId="133" CreationDate="2014-10-29T19:10:29.007" Score="15" ViewCount="3362" Body="&lt;p&gt;Base ten makes a lot of sense as a numbering system, given the number of digits humans typically have on their hands. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, some older money systems weren't based on the number of fingers we have, e.g. pounds, shillings, and pence. Similarly, a day is divided into 24 hours, not ten. Both of these suggest perhaps some cultures may have used alternate numbering systems in the past.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Have any non decimal numbering systems been used frequently in the past? If so, what were they?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="113" LastEditorUserId="113" LastEditDate="2014-10-29T20:14:03.167" LastActivityDate="2022-08-11T14:10:43.377" Title="Have numbering systems other than base ten ever been used or popular?" Tags="|mathematics|number-theory|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="107" PostTypeId="2" ParentId="64" CreationDate="2014-10-29T19:11:00.137" Score="9" Body="&lt;p&gt;First, to echo and elaborate on previous answers, Hilbert only appeared on the scene during the last few minutes of the last act, so to speak.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein made his first step towards the General Theory of Relativity (GR) in a 1907 paper &quot;Relativitätsprinzip und die aus demselben gezogenen Folgerungen&quot; (On the Relativity Principle and the Conclusions Drawn from It), in which he first introduced the Equivalence Principle; he later called that &quot;The happiest thought of my life&quot;. In 1911 he returned to it in the paper &quot;Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes&quot; (On the Influence of Gravitation on the Propagation of Light) (This paper is available in English translation in the collection &lt;em&gt;The Principle of Relativity&lt;/em&gt;.) In that paper he calculated the bending of a light beam passing close to the sun, getting half the correct value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both these papers were a response to the clear incompatibility of Newton's theory of gravity with Special Relativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein continued to work on the theory after 1911, eventually realizing the need for the mathematics of differential geometry. There followed a long series of papers where Einstein (with help from Grossmann) overcame various misconceptions. Pais's biography &lt;em&gt;Subtle Is The Lord...&lt;/em&gt; gives a detailed technical discussion of the path Einstein traveled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In June and July 1915, Einstein spent some time in Göttingen, giving lectures on his work and having prolonged discussions with Felix Klein and David Hilbert. In Oct-Nov 1915, Einstein surmounted the final obstacles and obtained the field equations of GR.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Independently Hilbert obtained the field equations, via a somewhat different approach. (As Pais points out, Hilbert had to overcome his own, somewhat different, misconceptions.) During the crucial month Nov 1915, Einstein and Hilbert exchanged several letters. The publication by Hilbert and by Einstein of the field equations was nearly simultaneous (5 days apart).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Einstein had made a whole series of fundamental conceptual advances, before Hilbert ever became engaged with the topic. Hilbert had the benefit of Einstein's personal discussions. Incidentally, Hilbert also needed a crucial assist from Emmy Noether with the math (the famous Noether's theorem on symmetries came out of this).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Felix Klein wrote, regarding the field equations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;there can be no question of priority, since both authors pursued entirely different trains of thought to such an extent that the compatibility of the results did not at once seem assured&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and Pais's final verdict:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I do believe that Einstein was the sole creator of the physical theory of general relativity and that both he and Hilbert should be credited for the discovery of the fundamental equation [i.e., the field equations]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, I should mention the &quot;priority dispute&quot;; the Wikipedia entries &lt;a href=&quot;http://en.wikipedia.org/wiki/Relativity_priority_dispute#Did_Hilbert_claim_priority_for_parts_of_General_Relativity.3F&quot;&gt;&quot;Did Hilbert claim priority for parts of General Relativity?&quot;&lt;/a&gt; and &lt;a href=&quot;http://en.wikipedia.org/wiki/Relativity_priority_dispute#Did_Einstein_develop_the_field_equations_independently.3F&quot;&gt;Did Einstein develop the field equations independently?&lt;/a&gt; contain a detailed discussion.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-10-29T19:11:00.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="108" PostTypeId="2" ParentId="106" CreationDate="2014-10-29T19:18:13.560" Score="13" Body="&lt;p&gt;Old Babylonians used base 60 for calculations. (This is also where 60 seconds in a minute and 60 minutes in an hour come from.) According to &lt;a href=&quot;http://en.wikipedia.org/wiki/Babylonian_mathematics&quot;&gt;Wikipedia&lt;/a&gt;, the main advantage of this was that it made practical calculations rather easy due to the number 60 having many divisors. Their mathematics was generally developed for the time but that would make up for a completely different question and answer.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-10-29T19:18:13.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="109" PostTypeId="2" ParentId="106" CreationDate="2014-10-29T19:29:29.687" Score="11" Body="&lt;p&gt;The Celts (ie. Iron Age Britons and their descendents) are reputed to have used a base 20 system. This is meant to be the origins of the &lt;strong&gt;&lt;em&gt;Yan Tan Tethera&lt;/em&gt;&lt;/strong&gt; counting system used by many upland sheep farmers well into the 19th Century and surviving today in a few places such as Swaledale. See &lt;a href=&quot;http://en.wikipedia.org/wiki/Yan_tan_tethera&quot;&gt;http://en.wikipedia.org/wiki/Yan_tan_tethera&lt;/a&gt; for a list of the variations.&lt;/p&gt;&#xA;" OwnerUserId="25" LastActivityDate="2014-10-29T19:29:29.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="110" PostTypeId="2" ParentId="82" CreationDate="2014-10-29T19:42:23.687" Score="8" Body="&lt;p&gt;I believe that our modern definition of a topological space came primarily from Hausdorff's book &lt;em&gt;Grundzüge der Mengenlehre&lt;/em&gt; (Foundations of Set Theory), first published in 1914, 2nd ed. 1927.  Hausdorff started with metric spaces, but then generalized them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the background to Hausdorff's work was the 19th work on continuity, and the so-called &quot;arithmetization of analysis&quot; --- the attempt to put calculus on a firm logical foundation. The biggest names here are Cauchy, Weierstrass, Dedekind, Bolzano, and Cantor. But the axiomatization of general topology in terms of open or closed sets is due to Hausdorff.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-10-29T19:42:23.687" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="111" PostTypeId="2" ParentId="106" CreationDate="2014-10-29T19:51:53.917" Score="11" Body="&lt;p&gt;The French language still has the residue of a base-twenty system.  Their word for 80 is 'quatre-vingts', which literally translates to 'four twenties'.&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="26" LastEditDate="2014-10-29T21:26:35.697" LastActivityDate="2014-10-29T21:26:35.697" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="112" PostTypeId="2" ParentId="106" CreationDate="2014-10-29T19:58:22.640" Score="6" Body="&lt;p&gt;The binary system is widely used today, and hexadecimal and octals numbers have gained a lot of attention due to their proximity to the binaries. This is due to computers working mainly with boolean values, which makes it natural to count in base 2 when dealing with such machines.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="8" LastEditDate="2014-10-29T20:57:02.500" LastActivityDate="2014-10-29T20:57:02.500" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="113" PostTypeId="5" CreationDate="2014-10-29T20:26:46.730" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-10-29T20:26:46.730" LastActivityDate="2014-10-29T20:26:46.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="114" PostTypeId="4" CreationDate="2014-10-29T20:26:46.730" Score="0" Body="A polymath is a person who is specialized in many areas of study." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-02T21:26:18.303" LastActivityDate="2014-11-02T21:26:18.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="115" PostTypeId="5" CreationDate="2014-10-29T20:27:26.520" Score="0" Body="&lt;p&gt;If the concerned question is about a more specific area of Mathematics (such as &lt;a href=&quot;/questions/tagged/calculus&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;calculus&amp;#39;&quot; rel=&quot;tag&quot;&gt;calculus&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/number-theory&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;number-theory&amp;#39;&quot; rel=&quot;tag&quot;&gt;number-theory&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/real-analysis&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;real-analysis&amp;#39;&quot; rel=&quot;tag&quot;&gt;real-analysis&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/euclidean-geometry&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;euclidean-geometry&amp;#39;&quot; rel=&quot;tag&quot;&gt;euclidean-geometry&lt;/a&gt; etc.), this tag should be used along with the tag of that specific area of Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-17T08:02:45.707" LastActivityDate="2022-03-17T08:02:45.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="116" PostTypeId="4" CreationDate="2014-10-29T20:27:26.520" Score="0" Body="For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns" OwnerUserId="27" LastEditorUserId="10106" LastEditDate="2022-02-02T23:56:09.663" LastActivityDate="2022-02-02T23:56:09.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="117" PostTypeId="2" ParentId="17" CreationDate="2014-10-29T20:27:58.550" Score="9" Body="&lt;p&gt;As far as I know there is no particular connection between Hilbert and Banach. Of course, Hilbert being one of the most dominant mathematicians of the time his influence was wide spread. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would however also be wrong to consider first Hilbert then Banach as a direct succession. There were various influences and contributors in the development of what are now Banach spaces. [Indeed, the notion was almost in parallel introduced by others, too, Wiener in particular. (Banach was the one that made most out of it and rightly got the &quot;name credit&quot;)]&#xA;Other names one could mention besides Hilbert include Fredholm, Riesz, Fischer, Fréchet, Lebesgue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To wit, the chronology in Pietsch's History of Banach spaces and linear operators has 12 entries (starting 1902) before Banachs thesis in 1920. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this context it might be also  noteworthy that Banach visited Paris in 1924-25.&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-10-29T20:27:58.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="118" PostTypeId="5" CreationDate="2014-10-29T20:29:24.400" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-10-29T20:29:24.400" LastActivityDate="2014-10-29T20:29:24.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="119" PostTypeId="4" CreationDate="2014-10-29T20:29:24.400" Score="0" Body="A field of mathematics studying numbers, their properties and structures that arise from them." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-02T21:26:33.783" LastActivityDate="2014-11-02T21:26:33.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="120" PostTypeId="5" CreationDate="2014-10-29T20:30:10.280" Score="0" Body="&lt;p&gt;Mathematicians study one or more particular fields or subfields of Mathematics. This tag should be used if the concerned question is about the lives or works of historical Mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;/questions/tagged/biographical-details&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;biographical-details&amp;#39;&quot; rel=&quot;tag&quot;&gt;biographical-details&lt;/a&gt; tag should be used along with this tag if the question is about the lives of the concerned Mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;If the Mathematicians are well-renowned (such as &lt;a href=&quot;/questions/tagged/euler&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;euler&amp;#39;&quot; rel=&quot;tag&quot;&gt;euler&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/gauss&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;gauss&amp;#39;&quot; rel=&quot;tag&quot;&gt;gauss&lt;/a&gt; etc.), this tag should be used along with the tags of those well-renowned Mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-08-14T18:52:01.933" LastActivityDate="2022-08-14T18:52:01.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="121" PostTypeId="4" CreationDate="2014-10-29T20:30:10.280" Score="0" Body="For questions about those who did mathematics" OwnerUserId="27" LastEditorUserId="10106" LastEditDate="2022-08-21T13:21:04.853" LastActivityDate="2022-08-21T13:21:04.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="122" PostTypeId="2" ParentId="106" CreationDate="2014-10-29T20:31:06.670" Score="8" Body="&lt;p&gt;Maya used a &lt;a href=&quot;https://en.wikipedia.org/wiki/Maya_numerals&quot;&gt;vigesimal&lt;/a&gt; system too. Note that many decimal systems use a quinary subsystem: think at the Latin numerals, but also at the Japanese abacus. &lt;/p&gt;&#xA;" OwnerUserId="81" LastActivityDate="2014-10-29T20:31:06.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="123" PostTypeId="1" AcceptedAnswerId="125" CreationDate="2014-10-29T20:49:13.633" Score="21" ViewCount="582" Body="&lt;p&gt;Who was the first person to conjecture that our Sun is just another star?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Relatedly, what was the first evidence that this was true?&lt;/p&gt;&#xA;" OwnerUserId="34" LastEditorUserId="26" LastEditDate="2014-10-31T22:25:04.877" LastActivityDate="2014-10-31T22:25:04.877" Title="When &amp; how was it known that our Sun is the same thing as the night time stars?" Tags="|experimental-physics|discoveries|astronomy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="124" PostTypeId="1" AcceptedAnswerId="129" CreationDate="2014-10-29T21:27:32.927" Score="28" ViewCount="593" Body="&lt;p&gt;A lot of chemical terms such as alcohol, aldehyde, sugar/azucar, amalgam etc. are of Arabic origin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Arabic chemistry in medieval times achieve any scientific insights still valid today (such as 'alcohol can be transformed into aldehyde by oxidation', although I am aware that oxidation is a more recent term)? Or was it rather the practice of alchemy?&lt;/p&gt;&#xA;" OwnerUserId="37" LastEditorUserId="8" LastEditDate="2015-05-18T19:03:13.800" LastActivityDate="2015-05-18T19:03:13.800" Title="Contributions to chemistry from medieval Arabia" Tags="|terminology|chemistry|middle-ages|islamic-science|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="125" PostTypeId="2" ParentId="123" CreationDate="2014-10-29T21:31:26.410" Score="17" Body="&lt;p&gt;As quoted from &lt;a href=&quot;http://solar-center.stanford.edu/FAQ/Qsunasstar.html&quot;&gt;this article&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many people's work was needed to prove that the Sun is a star. The first person we know of to suggest that the Sun is a star up close (or, conversely, that stars are Suns far away) was Anaxagoras, around 450 BC. It was again suggested by Aristarchus of Samos, but this idea did not catch on. About 1800 years later, around AD 1590, Giordano Bruno suggested the same thing, and was burnt at the stake for it. Through the work of Galileo, Kepler, and Copernicus during the 16th and 17th centuries the nature of the solar system and the Sun's place in it became clear, and finally in the 19th century the distances to stars and other things about them could be measured by various people. Only then was it proved that the Sun is a star.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the &lt;em&gt;first conjecture&lt;/em&gt; would date back to the Greeks, however there was much controversy surrounding the idea because to it's resemblance of the gods. It seems that it is a relatively &lt;em&gt;recent&lt;/em&gt; fact.&lt;/p&gt;&#xA;" OwnerUserId="27" LastActivityDate="2014-10-29T21:31:26.410" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="126" PostTypeId="2" ParentId="54" CreationDate="2014-10-29T22:40:11.180" Score="18" Body="&lt;p&gt;Let's start with everyone's first go-to source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics&quot;&gt;Wikipedia&lt;/a&gt;. Right in the long introduction at the top lies the passage&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Carl Friedrich Gauss (1777–1855) referred to mathematics as &quot;the Queen of the Sciences&quot;. Benjamin Peirce (1809–1880) called mathematics &quot;the science that draws necessary conclusions&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's a good start. &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics#Mathematics_as_science&quot;&gt;Later on&lt;/a&gt;, we find that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ouch. That's a valid point, though. But if we continue on to the next sentence,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that &quot;most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And in the concluding paragraph of that section,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The opinions of mathematicians on this matter are varied.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This sums up the sense one gets from reading the section: people are pretty divided.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can deduce that, given the varying time periods during which these mathematicians and scientists worked, that there has been a lot of debate on the matter for centuries, and that debate endures today. Nobody can seem to agree, which could render the issue moot.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Next up is &lt;a href=&quot;http://euclid.trentu.ca/math/sb/misc/mathsci.html&quot;&gt;this&lt;/a&gt; rather interesting essay, which starts off with, as its abstract,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is not a science, but there are grey areas at the fringes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Interesting. Let's delve in deeper . . . only to find that it is merely opinionated, citing sources but not anyone famous. It does make some interesting points, though:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;In mathematics, however, the ultimate arbiter of correctness is proof rather than empirical evidence.&quot; This alone seems to separate it from the sciences, which require absolute evidence (or as close to it as possible) for an idea to be accepted. Also (a point of my own), you can never prove a scientific theory; this is clearly not the case in mathematics, although there are some basic axioms that can never be proved.&lt;/li&gt;&#xA;&lt;li&gt;&quot;The basic problem is that one can be confident of a fact derived by mathematical methods only to the extent that the mathematical object being considered is an accurate model of the relevant parts of the universe.&quot; In other words, many conclusions derived from mathematical models can be proven to be true, though the models are only as accurate as the data they are given.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This essay, unfortunately, merely gives arguments, instead of citing famous mathematicians, and so we will push it aside. I'd also suggest &lt;a href=&quot;http://undsci.berkeley.edu/article/mathematics&quot;&gt;this page&lt;/a&gt; to look at criteria for determining what a science is.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This last bit is a personal opinion, so feel free to ignore it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I get the feeling that mathematics began to separate from the sciences when it became more abstract. With the rise of &lt;a href=&quot;https://en.wikipedia.org/wiki/Pure_mathematics&quot;&gt;pure mathematics&lt;/a&gt;, many mathematicians began to venture into the discipline entirely for the sake of mathematics, without any thought for its applications to physical theories. We could put a finger on this point at some time during the career of &lt;a href=&quot;https://en.wikipedia.org/wiki/David_Hilbert&quot;&gt;David Hilbert&lt;/a&gt;, who, while making extraordinary contributions to applied mathematics, also made numerous advances in pure mathematics. In fact, &lt;a href=&quot;https://en.wikipedia.org/wiki/Pure_mathematics&quot;&gt;Wikipedia&lt;/a&gt; credits him as heavily influencing the field of pure mathematics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I love getting &lt;a href=&quot;https://en.wikipedia.org/wiki/Bertrand_Russell&quot;&gt;Bertrand Russell&lt;/a&gt; in there, but I would argue that Hilbert's devotion solely to mathematics (as opposed to Russell, who could be viewed as a jack-of-all-trades) puts him at the top of the list of those leading the charge into pure mathematics, thus taking it away from the sciences.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The debate about whether or not mathematics is a science still goes on today. The center of the debate lies on the empirical predictions (or lack thereof) or purely mathematical ideas, and whether or not mathematical ideas that can be proved to be true can be true in the real world.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-29T22:40:11.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="127" PostTypeId="5" CreationDate="2014-10-29T22:48:57.067" Score="0" Body="&lt;p&gt;For questions about the history of calculus and its theoretical foundations, including topics such as different constructions and axiomatizations of the reals, continuity, differentiability, infinite series, Lebesgue measurability, and the Baire property. Related topics include questions on the history of the theory of differential equations, functional analysis, measure theory, and some aspects of general topology and classical descriptive set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider also the &lt;a href=&quot;/questions/tagged/calculus&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged 'calculus'&quot; rel=&quot;tag&quot;&gt;calculus&lt;/a&gt; tag.&lt;/p&gt;&#xA;" OwnerUserId="18" LastEditorUserId="62" LastEditDate="2015-04-12T16:37:43.250" LastActivityDate="2015-04-12T16:37:43.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="128" PostTypeId="4" CreationDate="2014-10-29T22:48:57.067" Score="0" Body="For questions about the history of calculus and its theoretical foundations, including topics such as continuity, differentiability, and infinite series. Related topics include questions on the history of measure theory, and some aspects of general topology and classical descriptive set theory." OwnerUserId="18" LastEditorUserId="62" LastEditDate="2015-04-12T16:37:43.250" LastActivityDate="2015-04-12T16:37:43.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="129" PostTypeId="2" ParentId="124" CreationDate="2014-10-29T22:55:22.367" Score="26" Body="&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/J%C4%81bir_ibn_Hayy%C4%81n&quot;&gt;Jabir ibn Hayyan&lt;/a&gt;&lt;/strong&gt; was the first to describe processes such as liquefaction, crystallisation, distillation, purification, oxidisation, evaporation and filtration. He also did an early classification of chemical elements around their properties which seems pertinent, and noted that &quot;a certain quantity of acid is necessary in order to neutralize a given amount of base.&quot; It is noteworthy that he valued experiments a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jabir probably set the base for medieval Arabic chemistry and for modern chemistry, and his results are significant, but on the other hand, he was practising alchemy, and worked only towards this goal. I believe that putting a barrier between what is chemistry and what is alchemy is a subjective, if not unanswerable question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Georges C. Anawati, &quot;Arabic alchemy&quot;, in R. Rashed (1996), The Encyclopaedia of the History of Arabic Science, Vol. 3, p. 853-902, through Wikipedia&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="520" LastEditDate="2015-04-13T15:30:32.733" LastActivityDate="2015-04-13T15:30:32.733" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="130" PostTypeId="1" CreationDate="2014-10-30T00:08:38.767" Score="30" ViewCount="812" Body="&lt;p&gt;In one of the lab courses I took as an undergraduate, I remember that the professor noted while discussing some statistical test (almost certainly chi-squared) that one could use it to show that a lot of early scientists (prior to the development of rigorous statistical analysis) had data that were too good to be realistic, and thus (with very high probability) were either deliberately falsified or the result of poor experimental design. He in particular cited Mendel as an example of a scientist whose data were far too good to be believable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia has &lt;a href=&quot;http://en.wikipedia.org/wiki/Gregor_Mendel#Controversy&quot;&gt;a section&lt;/a&gt; related to Mendel's case specifically, and some discussion as to the possibilities. I'm more interested in the general case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it true that a large fraction (relative to today) of prominent scientists before the 20th century presented data to support their conclusions which were statistically too good to be true? And if so, how did they avoid being wrong about a lot of their results?&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="8" LastEditDate="2015-06-05T14:29:33.910" LastActivityDate="2021-09-04T13:49:29.917" Title="Was fake/rigged data common prior to the 20th century?" Tags="|academia|statistics|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="131" PostTypeId="1" AcceptedAnswerId="509" CreationDate="2014-10-30T00:16:48.927" Score="10" ViewCount="483" Body="&lt;p&gt;George Cayley, aka &quot;The Father of Aerodynamics&quot; essentially, invented the field of heavier-than-air aerodynamics and built a number of model and man-carrying gliders, all in the first half of the 19th century. He defined the four forces acting on a plane, understood aerofoils and invented a device to test them. The layout of a modern plane can be traced back to him. He even invented the &quot;tension wheel&quot; to keep undercarriage weight down - an invention still found on virtually all bicycles today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However he never built a true plane. Two problems alluded him: stability/control, and power. Both problems were only truly cracked by the Wright Brothers, and it could be argued that solving these problems were essentially the Brothers' main achievement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cayley had no hope of cracking the power problem. Steam engines were far too heavy and internal combustion engines were decades away (even the Wright Brothers had to build their own engine).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However the problem of stability and control (they are two sides of the same coin) were definitely within his grasp. If you look at the tail on his man-carrying machine, he appears to be thinking along the right lines - his tail is more like that of most aircraft, rather than the split tail used by the Wright Brothers (which was probably the more practical considering the technology available).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how did he manage to miss this important step? He built a lot of models over an extended period of time (almost 50 years by some estimates) - how come he didn't come up with the idea of warping the wings or adding control surfaces? I think everyone who has built paper aeroplanes has accidentally warped their wings! Isn't Cayley likely to have made essentially the same mistake?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="26" LastEditDate="2014-10-30T12:26:19.833" LastActivityDate="2014-11-18T13:50:50.537" Title="Why did George Cayley never crack the stability problem?" Tags="|experimental-physics|aerodynamics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="132" PostTypeId="2" ParentId="1" CreationDate="2014-10-30T00:44:07.587" Score="14" Body="&lt;p&gt;Of course, the phrasing &amp;quot;This is what I shall explain...&amp;quot; implies that Cauchy has just stated what this theorem is, so it would seem that yes, we should have a very good chance at finding out what it was, provided we can find out where the statement is from.&lt;/p&gt;&#xA;&lt;p&gt;It appears to be from a note published on the 4th of May 1857 called &lt;em&gt;Sur l'utilisation des régulateurs en Astronomie&lt;/em&gt;. This is the last article in the last volume of &amp;quot;Série 1&amp;quot; of Cauchy's collected &lt;em&gt;Oeuvres&lt;/em&gt;, excluding what appears to be the minutes of Académie on the day Cauchy's death was announced, and assuming the date under the article's title is the publication date, it appeared just a few weeks before Cauchy's death. Thus it appears likely that this was indeed his last publication.&lt;/p&gt;&#xA;&lt;p&gt;It's article 589 in Tome XII, Série 1 of &lt;em&gt;Œuvres Complètes d'Augustin Cauchy&lt;/em&gt;, page 455, an online copy of which can be found &lt;a href=&quot;http://www.e-rara.ch/zut/content/structure/5705720&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The note seems to be about the use &amp;quot;Régulateurs&amp;quot; in astronomy, where I assume a &amp;quot;Régulateur&amp;quot; is an &lt;em&gt;astronomical regulator&lt;/em&gt;, an old type of pendulum clock used in observatories.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/faWyA.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Translation (my own, and although I'm fluent in french I'm less fluent in 19th century astronomy, and without knowing what exactly Cauchy is talking about I have no idea what the correct english  technical terms are - I've therefore simply translated them as literally as possible):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On the use of regulators in Astronomy&lt;/p&gt;&#xA;&lt;p&gt;I noted in the previous session the advantages of using regulators in mathematical Analysis. I will add that not only the variables, but indeed the parameters contained in the given equations, finite or differential, or even partial differential, can be supposed to be developped according to the ascending powers of a given regulator. In many problems, particularly in Astronomy, this observation will allow us to render monodrome or monogeneous the variations of the various orders of unknowns which are expanded as series following the increasing powers of a single given regulator. This solves the question raised in my previous Mémoire, on the possibility of developping the coordinates which determine the orbits of the planets about the Sun, or of satellites about the planets, according to the ascending and descending powers of the trigonometric exponentials whose arguments are the excentric or average anomalies, and thereby according to the ascending or descending powers powers of the keys of the orbits. This I will explain in more detail in a future Mémoire.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I frankly have no idea what any of the above means. The article is followed by &lt;a href=&quot;https://i.stack.imgur.com/9Pkt1.jpg&quot; rel=&quot;noreferrer&quot;&gt;the aforementioned bit of text about Cauchy's death&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-10-30T00:44:07.587" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="133" PostTypeId="2" ParentId="106" CreationDate="2014-10-30T01:32:48.920" Score="14" Body="&lt;p&gt;To this day many people use various systems besides the decimal one. &#xA;I was surprised to read that &quot;Old Babylonians used the system based on 60&quot;:-) &#xA;Do not we all&#xA;use their system today? Not only for time but for angle measurement?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the French revolutionary government introduced the decimal system as a standard, they also&#xA;tried to reform the angle and time measurement. &#xA;10 hours in a day and 10 in a night, 100 minutes&#xA;in an hour, 100 second in a minute. For the angles: 100 degrees in a right angle, 400 in a circle, each degree is 100 minutes of arc, etc. This is why kilometer was defined as&#xA;1/400000 of the meridian. It is nothing but the &quot;decimal minute&quot;.&#xA;(The nautical mile is one &quot;Babylonian minute&quot; of the meridian). Clocks, watches and angle measuring instruments with decimal scales can be found in museums.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this did not work. So we still use the Babylonian system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, how many inches are there in a foot? How many ounces in a pound ?&#xA;Not everyone uses decimal system today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting system is used for angle measuring in artillery. The circle is divided into&#xA;6000 parts. This is convenient for quick estimation of the distance to an object of known size, using the crude approximation pi=3. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was an attempt to introduce the system based on 8 for everyday use. (Charles XII of&#xA;Sweden). This did not work.   &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-10-30T01:32:48.920" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="135" PostTypeId="2" ParentId="54" CreationDate="2014-10-30T01:56:06.247" Score="9" Body="&lt;p&gt;Some people would still call mathematics a science. (V. I. Arnold is a notable example).&#xA;The distinction became commonly accepted in the first half of the 20-s century, but the process was slow and it was different in different cultures. For example, in the Soviet universities the degree in mathematics is still called &quot;Doctor of physical and mathematical sciences&quot;. In the middle of 20-s century&#xA;there were very few mathematics departments in Soviet universities. Most departments were called &quot;Department of mathematics and Mechanics&quot;, and earlier they were departments of&#xA;mathematics and physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you go further back to 19 century, you discover that many of the greatest mathematicians&#xA;worked in both mathematics and physics or astronomy. (Gauss, Riemann, Green, Kelvin, for example).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-10-30T01:56:06.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="136" PostTypeId="1" AcceptedAnswerId="137" CreationDate="2014-10-30T02:00:35.647" Score="13" ViewCount="2279" Body="&lt;p&gt;The steam engine was one of the main enabling technologies of the industrial revolution.&#xA;Many histories of the Industrial Revolution chart the steam engine's rise with James Watt. However, Thomas Newcomen is often credited with inventing the steam engine. So who did it?&lt;/p&gt;&#xA;" OwnerUserId="25" LastActivityDate="2023-06-16T14:30:32.737" Title="Who invented the steam engine: Newcomen or Watt?" Tags="|thermodynamics|steam-engines|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="137" PostTypeId="2" ParentId="136" CreationDate="2014-10-30T02:00:35.647" Score="12" Body="&lt;p&gt;Both Newcomen and Watt invented 'steam' engines. Both operated by injecting steam into a cylinder, moving a piston in the cylinder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newcomen's engine is often described as a steam engine but the steam pressure is not really performing the work.&#xA;Instead, the steam is condensed (by injecting a small amount of water) to produce a partial vacuum. This partial &#xA;vacuum pulls the piston (to perform work).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newcomen's design was actually an incremental improvement of a thermic syphon designed by Thomas Savery.&#xA;Newcomen added a piston into the vacuum container (/cylinder), and used a rocking beam to operate the pump mechanism (most Newcomen engines were used as water pumps for mines).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watt realized that the condensing arrangement was very inefficient because most of the heat was used heating the &#xA;cylinder which was then cooled during the condensing stage.&#xA;Although thermodynamics was at a very primitive stage, Watt realised the importance of latent heat during this condensing process.&#xA; He operated the cylinder at steam temperature with a&#xA;steam jacket, and moved the condensing stage out of the cylinder. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The piston was now operated by steam being admitted into &#xA;the cylinder. Although initial steam pressures were not much higher, this was significantly more efficient than the Newcomen arrangement.&#xA;As technology improved and safety valves were invented (and improved!), pressures increased for greater power and efficiency: early &#xA;steam locomotives (early 19th century) often operated at or below 50psi, whilst 200-250psi was common during the 20th &#xA;century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watt initially had a lot of trouble getting his designs to work. The primary problem was the creation of a large cylinder with a steam-tight piston. This was eventually solved by partnering with Boulton. Boulton also introduced the idea of converting the reciprocating beam motion into a rotating wheel motion - suitable for use in mills.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Further Information&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.animatedengines.com/newcomen.html&quot; rel=&quot;noreferrer&quot;&gt;A wonderful animated Newcomen engine and explanation&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://technology.niagarac.on.ca/people/mcsele/newcomen.htm&quot; rel=&quot;noreferrer&quot;&gt;Further information including survivors and Smeaton's famous Royal Society drawing of a Newcomen Engine&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.egr.msu.edu/~lira/supp/steam/wattengine.htm&quot; rel=&quot;noreferrer&quot;&gt;Description of a surviving Watt Engine at the Henry Ford Museum&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Watt_steam_engine&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia Page for the Watt Engine&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-10-30T18:32:29.980" LastActivityDate="2014-10-30T18:32:29.980" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="138" PostTypeId="1" CreationDate="2014-10-30T04:52:26.093" Score="24" ViewCount="385" Body="&lt;p&gt;Nuclear applications—be it in the form of electricity production, weapon creation, etc.—inevitably involves &lt;a href=&quot;http://en.wikipedia.org/wiki/Radioactive_waste&quot;&gt;radioactive waste.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When was it realized that there might be a need for a specialized waste disposal process? How does this date compare to landmarks in nuclear theory/experiment/refinement?&lt;/li&gt;&#xA;&lt;li&gt;What was the first such disposal facility?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="34" LastEditorUserId="26" LastEditDate="2015-03-01T14:06:59.387" LastActivityDate="2015-08-27T00:44:36.057" Title="How soon after or before the development of nuclear applications was waste disposal considered?" Tags="|physics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="139" PostTypeId="1" AcceptedAnswerId="6111" CreationDate="2014-10-30T08:44:21.350" Score="9" ViewCount="236" Body="&lt;p&gt;Prior to the theory of &lt;a href=&quot;http://en.wikipedia.org/wiki/Symbiogenesis&quot; rel=&quot;nofollow&quot;&gt;symbiogenesis&lt;/a&gt; (heralded especially by Lynn Margulis in the late 60s), how did people explained the presence of chlorophyll in various, separate groups such as cyanobacteria, land plants, chromists and a variety of small separate groups of eukaryots (namely, if I don't forget anyone: euglens, chlorarachniophyceae, cryptophytes, haptophytes and dinoflagellates)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did people think that chlorophyll-based photosynthesis appeared once and that all those groups were descendants of the same ancester?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or did they think it appeared multiple times, using basically the same structure, by convergence?&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorDisplayName="user466" LastEditDate="2017-07-05T21:22:18.250" LastActivityDate="2017-07-05T21:22:18.250" Title="Theories on the beginning of photosynthesis, prior to Lynn Margulis work on symbiogenesis" Tags="|biology|evolution|cell-biology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="140" PostTypeId="1" AcceptedAnswerId="5260" CreationDate="2014-10-30T12:30:46.013" Score="45" ViewCount="63949" Body="&lt;p&gt;In most of science, it is typical to have the independent variable on the horizontal axis and the dependent variable on the vertical axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in economics, this is often (traditionally?) flipped around. As &lt;a href=&quot;http://gregmankiw.blogspot.com/2006/09/who-invented-supply-and-demand.html&quot; rel=&quot;noreferrer&quot;&gt;an economist (Greg Mankiw) writes&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;given the way we now teach supply and demand, it makes more sense to&#xA;  have price on the horizontal axis. The price is viewed as the variable&#xA;  that determines quantity supplied and quantity demanded, and we&#xA;  usually put the dependent variable (which here is quantity) on the&#xA;  vertical axis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(To elaborate: Demand is typically taught as a function which takes as input prices and gives as output quantity demanded.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When drawing the supply and demand for loanable funds, the interest rate is on the vertical axis, while savings and investment are on the horizontal.&lt;/li&gt;&#xA;&lt;li&gt;When drawing the supply and demand for money, the interest rate is on the vertical axis and money supply and demand are on the horizontal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It seems that it was Alfred Marshall who popularized this convention, though possibly he was merely following Cournot. What precisely is the correct historical account for why price is on the vertical axis and quantity on the horizontal? Why does economics deviate from the rest of science with respect to this particular convention? When did it happen?&lt;/p&gt;&#xA;" OwnerDisplayName="user67" LastEditorDisplayName="user67" LastEditDate="2016-10-06T02:36:04.590" LastActivityDate="2021-06-14T20:58:50.753" Title="Why is price on the vertical axis and quantity on the horizontal axis?" Tags="|notation|economics|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="141" PostTypeId="2" ParentId="140" CreationDate="2014-10-30T13:10:04.143" Score="8" Body="&lt;p&gt;To your three points&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When drawing supply and demand, price (which one might more naturally&#xA;think of as being the independent variable) is on the vertical axis&#xA;and quantity on the horizontal axis.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For elastic &lt;code&gt;demand&lt;/code&gt; you are sometimes able to dictate the demand by the price.  Normally though, the demand dictates the price.  For (purely) inelastic &lt;code&gt;demand&lt;/code&gt;, the price is entirely set by &lt;code&gt;demand&lt;/code&gt;.  So price is the dependent variable.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When drawing the investment schedule, the interest rate is on the&#xA;vertical axis and investment on the horizontal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is is basically a &lt;code&gt;demand&lt;/code&gt; curve but its geared toward predicting rates rather than reporting rates at different investments.   &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When drawing the money market (money supply and demand), the interest&#xA;rate is on the vertical axis and money supply/demand on the&#xA;horizontal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As you said this is also a &lt;code&gt;demand&lt;/code&gt; curve.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I think the confusion is because you &lt;em&gt;usually&lt;/em&gt; see price as being the independent variable.  The price isn't the independent variable because its on the x axis.  The price is on the x axis because its the independent variable.  As far as (basic) supply and demand go, the price is going to be set by &lt;code&gt;demand&lt;/code&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, in my explanations I always said &lt;code&gt;demand&lt;/code&gt; to keep it clean.  You could switch it to Supply and it still makes sense (unless i made a mistake.)&lt;/p&gt;&#xA;" OwnerUserId="19" LastEditorUserId="19" LastEditDate="2014-10-30T13:49:43.113" LastActivityDate="2014-10-30T13:49:43.113" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="142" PostTypeId="1" AcceptedAnswerId="193" CreationDate="2014-10-30T15:44:08.230" Score="32" ViewCount="4507" Body="&lt;p&gt;The Americans and the French use a different notation for &lt;a href=&quot;http://mathworld.wolfram.com/OpenInterval.html&quot;&gt;open intervals&lt;/a&gt;: The Americans use (x, y) while the French use ]x, y[. How did this notational divergence appear?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="27" LastEditDate="2014-10-30T23:52:03.283" LastActivityDate="2017-10-17T03:17:11.013" Title="Why is American and French notation different for open intervals (x, y) vs. ]x, y[?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="143" PostTypeId="2" ParentId="2" CreationDate="2014-10-30T15:48:18.883" Score="25" Body="&lt;p&gt;I do not agree on some details of the interpretation regarding the discovery of the irrationality of $\sqrt{2}$ as a confutation of the &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pythagoreans [...] belief that all numbers could be constructed as the ratio of 2 numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My undestanding is that all &quot;archaic&quot; Greek mathematics shared the (implicit) assumption that, given two magnitudes, e.g. two segments of lenght $a,b$, it is always possible to find a segment of &quot;unit lenght&quot; $u$ such that it &lt;em&gt;measures&lt;/em&gt; both, i.e. such that [using modern algebraic formulae which are totally foreign to Greek math] :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ a=n\times u\ \text{and}\ b=m\times u\ \text{for suitable}\ n,m$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the above instance of the assumption, it follows that :&#xA;$$ \frac{a}{b}=\frac{n\times u}{m\times u}=\frac{n}{m}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The assumption amounts to saying that the ratio between two magnitudes is always a ratio between &lt;em&gt;integers&lt;/em&gt; (i.e. in modern terms: a &lt;em&gt;rational&lt;/em&gt; number).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But note that for Greek math the &lt;strong&gt;only&lt;/strong&gt; numbers are the &lt;em&gt;natural&lt;/em&gt; ones and they must be distinguished from &lt;em&gt;magnitudes&lt;/em&gt; : a segment, a square, ... which are &quot;measured by&quot; numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For ancient Greeks there are &lt;strong&gt;no&lt;/strong&gt; rational numbers; but only magnitudes measurable with multiples of a suitable unit one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery of the existence of irrational magnitudes, through the proof that the case where $a$ is the side of the square and $b$ its diagonal is not expressible as a ratio between (natural) numbers, leads Greek math to the withdrawal of the above (implicit) assumption, that we may call : &quot;commensurability assumption&quot; and to the axiomatization of geometry, i.e. the systematic effort to explicitly lists all the needed assumptions.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="26" LastEditDate="2014-12-03T10:17:15.320" LastActivityDate="2014-12-03T10:17:15.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="144" PostTypeId="2" ParentId="57" CreationDate="2014-10-30T15:59:20.103" Score="5" Body="&lt;p&gt;A bit more about Einstein's indirect contributions to quantum theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Heisenberg's autobiographical essay &quot;Theory, Criticism, and Philosophy&quot;, in the section &quot;Einstein on Theory and Observation&quot;, Heisenberg recounts a conversation he had with Einstein shortly after Heisenberg had proposed his version of QM (called matrix mechanics).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein asked me to come to his flat and discuss the matters with him. The first thing he asked me was: &quot;What was the philosophy underlying your kind of very strange theory? The theory looks quite nice, but what did you mean by only observable quantities?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here Einstein alludes to Heisenberg's claim that physics should deal only with observable quantities; this justified discarding the idea of electron trajectories. Heisenberg responded,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I felt that one should go back to those quantities which can really be observed and I also felt that this was just the kind of philosophy which he had used in relativity; because he also had abandoned absolute time... Well, he laughed at me and then he said, &quot;But you must realize that it is completely wrong.&quot; I answered: &quot;But why, is it not true that you have used this philosophy?&quot; &quot;Oh yes&quot;, he said, &quot;I may have used it, but still it is nonsense!&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Einstein explained to me that it is really the other way around. He said, &quot;Whether you can observe a thing or not depends on the theory which you use. It is the theory which decides what can be observed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Heisenberg explains that this conversation set him on the train of thought which culminated in his Uncertainty Principle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Turning to Schrödinger, in a footnote to his paper &quot;On the Relation of the Heisenberg-Born-Jordan Quantum Mechanics to Mine&quot;, he wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein (Berl. Ber. 1925, p.9ff)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I believe the cited paper is Einstein's second one on Bose-Einstein statistics.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-10-30T15:59:20.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="145" PostTypeId="2" ParentId="17" CreationDate="2014-10-30T16:36:23.993" Score="16" Body="&lt;p&gt;It is worth noting that the abstract definition of a Hilbert space (as a complete inner-product space) is not due to Hilbert. Weyl recounts the history in his memorial essay, &quot;David Hilbert and His Mathematical Work&quot; (Bull. Amer. Math. Soc. v.50 p.612--654). In his work on integral equations, Hilbert investigated only one particular Hilbert space: the space of square-summable infinite sequences. He did not use Lebesgue integration; only later did Riesz and Fischer show the equivalence with Lebesgue square-integrable functions. Weyl adds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I mention these details because the historic order of events may have fallen into oblivion with many of our younger mathematicians, for whom Hilbert space has assumed that abstract connotation which no longer distinguishes between the two realizations...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(There is also an apocryphal story that Hilbert attended a lecture, and came up at the end to ask the speaker, &quot;What is a Hilbert space?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Banach by contrast gave the abstract formulation of Banach spaces in his dissertation, along with his motivation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This present work has the object of establishing certain theorems &#xA;  that hold in several different branches of mathematics, which will be &#xA;  specified later. However, in order to avoid proving these theorems for &#xA;  each branch individually, which would be very wearisome, I have &#xA;  chosen a different way, which is this: I consider in a general way sets &#xA;  of elements for which I postulate certain properties. From these I &#xA;  deduce theorems and then I prove for each separate branch of mathematics that the postulates adopted are true of it. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words, Banach is seeking economy of proof via the axiomatic method. His motivation is thus entirely different from Hilbert's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To return to your original question: I have not been able to uncover any personal connection between Hilbert and Banach. The name &quot;Banach&quot; does not appear in the index of Constance Reid's biography &lt;em&gt;Hilbert&lt;/em&gt;; the MacTutor entry for Hilbert does not contain &quot;Banach&quot;, and the MacTutor entry for Banach contains only once occurrence of Hilbert, where it notes that Banach's work &quot;generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that one sentence is probably sufficient explanation. Hilbert did his work on integral equations in the early 1900s, and it was soon developed further by Riesz, Fischer, Schmidt, and others. Banach's dissertation was written in 1920. It is hardly surprising that entering this field, Banach would pay close attention to relevant published work by one of the foremost mathematicians of the day.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-02T05:27:18.130" LastActivityDate="2014-11-02T05:27:18.130" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="146" PostTypeId="1" AcceptedAnswerId="494" CreationDate="2014-10-30T18:37:56.553" Score="11" ViewCount="2453" Body="&lt;p&gt;As steam engine pressures increased, it was realized the exhaust steam still had useful pressure. The compound engine was invented to take advantage of this. The exhaust from a small high pressure cylinder fed into a larger low pressure cylinder to extract further work. The most famous example is probably the marine Triple Expansion Engine which greatly improved the efficiency of marine steam engines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compound engines were popular in marine and static applications, but their usage in railway locomotives was patchy at best. Although popular in a few locations (France comes to mind), the vast majority of 20th century steam locomotives were built with simple expansion engines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-10-30T18:43:10.227" LastActivityDate="2017-09-03T14:43:25.457" Title="Why did most steam locomotives use simple expansion?" Tags="|thermodynamics|steam-engines|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="147" PostTypeId="2" ParentId="10" CreationDate="2014-10-30T19:07:56.943" Score="17" Body="&lt;p&gt;I will focus on the history before the Yang-Mills paper. The first harbinger was the introduction of the scalar potential for the gravitational field by Lagrange in 1773. In 1864 Maxwell introduced vector potential for the magnetic field, which can be interpreted as a connection form, making magnetic theory the first gauge theory in hindsight. In the famous 1905 paper Einstein joined the scalar and the vector potentials into a 4-potential, which is a connection form on the four dimensional background. At the same time developments in differential geometry led to another type of connection, the Riemannian one, in the form of covariant derivative. Christoffel symbols were the first appearence of it, and Ricci and Levi-Civita developed the theory based on &quot;absolute differentiation&quot; (covariant derivatives) systematized in their 1900 book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this qualifies as more of a pre-history. The real history of gauge theory begins in Hermann Weyl's papers of 1918-1920, where he pulled together different strands that developed up to that point. Ironically, Weyl was also motivated by the theory of gravity, the general relativity. First, he noticed that one does not need a Riemannian metric to define parallel transport, what is now called affine connection suffices. He then realized that Riemannian geometry is not entirely local, lengths of vectors at different points, being numbers, can be compared in the absolute sense. To localize it entirely he switched to conformal metrics accompanied by a field of scales, in modern terminology sections of a principal bundle with the fiber the multiplicative group of positive real numbers $R^+$. Transporting scales (gauges) requires specifying a 1-form, a principal connection form, and changing them induces a transformation of this form, a gauge transformation. Weyl then formulated for the first time the principle of gauge invariance, the form of natural laws must be invariant under local changes of gauge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the formulas he obtained were identical to those of electromagnetism Weyl conjectured that the curvature of his scale connection was exactly the electromagnetic field. He even wrote coupled equations for electromagnetic and gravitational fields producing the first unified field theory (Kaluza proposed his five dimensional theory around the same time, he published in 1921). Unfortunately, it was unphysical, as Einstein pointed out shortly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weyl came back to the gauge idea in 1929, from the quantum mechanical perspective. This time instead of the field of scales he used the field of phases, that is replaced $R^+$ with U(1) as the fiber. The formulas are almost the same, except for the presence of i, he never considered non-Abelian principal bundles. In 1930 Dirac used non-trivial U(1) bundles to describe magnetic monopoles. These  developments are described in detail by Varadarajan in his &lt;a href=&quot;http://www.math.ucla.edu/~vsv/papers/paper.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;survey paper&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After Weyl the mathematical and the physical paths diverge once again. Elie Cartan used specialized connections to study Pfaffian systems in 1926, they lived on fiber bundles with fibers being homogeneous spaces (Kleinian geometries), his work solidified the view of connections as matrix-valued 1-forms. Two big developments came in 1950, &lt;a href=&quot;http://en.wikipedia.org/wiki/Covariant_derivative#Introduction_and_history&quot; rel=&quot;nofollow noreferrer&quot;&gt;Koszul&lt;/a&gt; gave a general algebraic description of connections on vector bundles as covariant derivatives, and eliminated the need for non-tensorial objects like Christoffel symbols. &lt;a href=&quot;http://en.wikipedia.org/wiki/Ehresmann_connection#Introduction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ehresmann&lt;/a&gt;, Cartan's student, finally gave a general definition of connection on a principal bundle, Abelian or not, and clarified the general relationship between connections on principal and associated bundles. His notion of connection was very abstract though, a horizontal distribution in the tangent bundle to the total space, which can be used to define the parallel transport directly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When Yang and Mills introduced in their paper the first non-Abelian gauge theory (with SU(2) as a fiber) in 1954 they weren't aware of these mathematical developments, the relationship between the principal connections and gauge fields was clarified in the years following their publication. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-04-21T21:27:52.117" LastActivityDate="2017-04-21T21:27:52.117" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="148" PostTypeId="2" ParentId="20" CreationDate="2014-10-30T20:16:22.490" Score="6" Body="&lt;p&gt;According to the webpage &lt;a href=&quot;http://birner.atmos.colostate.edu/tropopause.html&quot; rel=&quot;noreferrer&quot;&gt;the tropopause&lt;/a&gt; (Birner, 2008), ironically the discovery of the tropopause (boundary between the troposphere and stratosphere) was almost entirely accidental and was considered to be an error at first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery started out of a dilemma, from an earlier discovery in 1787 by Horae Benedict de Saussure that air temperature drops with higher altitude, (&lt;a href=&quot;http://empslocal.ex.ac.uk/people/staff/gv219/classics.d/Hoinka-tropo97.pdf&quot; rel=&quot;noreferrer&quot;&gt;The tropopause discovery and demarcation&lt;/a&gt;, Hoinka, 1997):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;200 years ago people knew from the temperature difference between mountain tops and valleys that temperature drops with height at roughly 7 K/km (this is about 7.8 F/mi). It is a simple exercise to realize that this would lead to 0 K (= -273.15 C or -459.67 F, i.e. absolute zero) at an altitude of 40 km (assuming a surface temperature of 280 K = 44 F).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So many researchers of the day (mainly in France and Germany) went for a balloon ride, and took temperature and pressure soundings at very high altitudes and noted that there was an area of constant temperature - this is what was originally noted as a measurement error initially.  But repeated soundings confirmed the existence of a 'pause' in the temperature drop in the troposphere and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It took Teisserence de Bort almost 10 years and more than 200 collected soundings to trust his results enough and have the heart to report his results to the French Acadamy of Sciences in April 1902 and announce the discovery of an isothermal layer (&quot;zone isotherme&quot; in french). As little as three days(!) later the German R. Assmann announced essentially the same discovery to the German Academy of Science referring to the isothermal layer as upper inversion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The boundary and the description of the lower stratosphere continued during what was known as &quot;International Aerological Days&quot; in the early years of the 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;isothermal layer&quot; detected by Teisserence was the lower stratosphere (the isotherm can be seen in the diagram below):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HpgHy.gif&quot; alt=&quot;enter image description here&quot;&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;http://www.physicalgeography.net/fundamentals/7b.html&quot; rel=&quot;noreferrer&quot;&gt;Image source&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-10-30T20:16:22.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="149" PostTypeId="1" CreationDate="2014-10-30T21:40:34.407" Score="13" ViewCount="684" Body="&lt;p&gt;Hipparchus applied Apollonius's epicycle/eccenter construction to the motion of the Sun and the Moon, but not to the planets. There are speculations about epicyclic gears for planets in the Antikythera mechanism, but apparently they did not survive. On the other hand, a few centuries later Ptolemy had well advanced models for all the planets, and even introduced equant to make them more accurate. What happened in between? Who first applied epicycles to planetary motions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lucio Russo even &lt;a href=&quot;http://www.academia.edu/2119678/The_astronomy_of_Hipparchus_and_his_time_A_study_based_on_pre-ptolemaic_sources&quot;&gt;speculated&lt;/a&gt; about dynamical ideas from this time period concerning celestial motions based on some passages from Plutarch and Strabo. What is the consensus on that?&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-07-07T17:22:07.467" Title="What happened in astronomy between Hipparchus and Ptolemy?" Tags="|astronomy|celestial-mechanics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="150" PostTypeId="2" ParentId="149" CreationDate="2014-10-30T23:33:43.553" Score="3" Body="&lt;p&gt;Actually, I've found some sources that suggest that &lt;a href=&quot;https://en.wikipedia.org/wiki/Apollonius_of_Perga&quot; rel=&quot;nofollow&quot;&gt;Apollonius&lt;/a&gt; applied epicycles to planetary motion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://jwilson.coe.uga.edu/EMAT6680/Greene/EMAT6000/greek%20geom/Apollonius/apollonius.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Apollonius also was an important person in founding Greek mathematical astronomy. He used geometrical models to explain planetary theory. He introduced systems of eccentric and epicyclic motion to explain the motion of the planets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Apollonius.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Apollonius was also an important founder of Greek mathematical astronomy, which used geometrical models to explain planetary theory. Ptolemy in his book Syntaxis says Apollonius introduced systems of eccentric and epicyclic motion to explain the apparent motion of the planets across the sky. This is not strictly true since the theory of epicycles certainly predates Apollonius. Nevertheless, Apollonius did make substantial contributions particularly using his great geometric skills. In particular, he made a study of the points where a planet appears stationary, namely the points where the forward motion changes to a retrograde motion or the converse.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://schoolworkhelper.net/apollonius-of-perga-biography-mathematician/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lastly, from references in Ptolemy’s Almagest, it is known that Apollonius introduced the systems of eccentric and epicyclic motion to explain planetary motion. Of particular interest was his determination of the points where a planet appears stationary.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And &lt;a href=&quot;https://en.wikipedia.org/wiki/Apollonius_of_Perga&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;, by the way, uses the slightly dubious phrasing:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The hypothesis of eccentric orbits, or equivalently, deferent and epicycles, to explain the apparent motion of the planets and the varying speed of the Moon, is also attributed to him.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This last passage, of course, does not state outright that &lt;em&gt;he&lt;/em&gt; applied the theory to planetary orbits.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-30T23:33:43.553" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="151" PostTypeId="1" AcceptedAnswerId="213" CreationDate="2014-10-31T00:48:07.580" Score="13" ViewCount="273" Body="&lt;p&gt;Many results in group theory were proven for permutation groups before the general definition of a group was established (for example: Lagrange's theorem, Sylow's theorems). However, permutation groups weren't the only groups being studied in the nineteenth century, there were also groups of geometric transformations and group arising from number theory (I can't really give more details because I don't frankly know the details).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were any general group theory results known for &lt;em&gt;several&lt;/em&gt; specific cases other than just permutation groups, before the general definition of a group had been formulated? I ask because I'm wondering if such &quot;coincidences&quot; could have motivated the general definition of a group. As an example, Lagrange's theorem was known in the 19th century both for permutation groups and for the multiplicative group of $\mathbb Z/n\mathbb Z$ (via &lt;a href=&quot;http://en.wikipedia.org/wiki/Euler%27s_theorem&quot;&gt;Euler&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-11-02T04:52:38.767" Title="What group theoretic results were known for several special cases before the general definition of a group was established?" Tags="|mathematics|group-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="152" PostTypeId="2" ParentId="138" CreationDate="2014-10-31T00:58:03.117" Score="5" Body="&lt;p&gt;Take a look at &lt;a href=&quot;http://berniesteam.com/industry-services/bt-dry-cask-storage-services/history-of-nuclear-waste-policy/&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt; handy-dandy webpage:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1957: A study by the National Academy of Science, determined that the federal government should build a permanent geologic repository for high-level nuclear waste.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To put this in perspective, &lt;a href=&quot;http://en.wikipedia.org/wiki/Nuclear_power#History&quot; rel=&quot;nofollow&quot;&gt;the first time&lt;/a&gt; a nuclear reactor successfully generated a substantial amount of electricity was in &lt;a href=&quot;http://en.wikipedia.org/wiki/Experimental_Breeder_Reactor_I&quot; rel=&quot;nofollow&quot;&gt;1951&lt;/a&gt; (okay, it wasn't much, but still, it was a start). The first power plant to bring electricity to the power grid was in &lt;a href=&quot;http://en.wikipedia.org/wiki/Obninsk_Nuclear_Power_Plant&quot; rel=&quot;nofollow&quot;&gt;1954&lt;/a&gt;, in Russia. Successful commercial reactors didn't open until the mid-1950s; considering this, the initial talk of waste disposal was rather early and on-time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, though, federal activities regarding nuclear power slowed down. Flash-forward 25 years:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1982: The &lt;a href=&quot;http://en.wikipedia.org/wiki/Nuclear_Waste_Policy_Act&quot; rel=&quot;nofollow&quot;&gt;Nuclear Waste Policy Act&lt;/a&gt; required the US Department of Energy to start taking utilities’ spent fuel by Jan. 31, 1998. It directed DOE to begin studying sites for permanent repositories and established a schedule for that process.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But it took a while for them to get around to it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1987: Congress amended the Nuclear Waste Policy Act to require DOE to focus its efforts on studying Yucca Mountain, Nevada as the permanent repository site. The amendments also created a federal Nuclear Waste Negotiator to find a volunteer host for an interim storage facility or a permanent repository.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But &lt;a href=&quot;http://en.wikipedia.org/wiki/Yucca_Mountain_nuclear_waste_repository&quot; rel=&quot;nofollow&quot;&gt;Yucca Mountain&lt;/a&gt; really hasn't been so successful. Note, though, that I don't believe that it was the first planned storage site.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-08-27T00:44:36.057" LastActivityDate="2015-08-27T00:44:36.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="153" PostTypeId="1" AcceptedAnswerId="540" CreationDate="2014-10-31T02:17:34.073" Score="11" ViewCount="1364" Body="&lt;p&gt;For many years, large buildings with impressive spans between the building columns were erected without precisely calculating what loads the columns, walls, and buttresses would have to support. Load on a building is a complex calculation taking in account the weight of the roof and potential snow, the size and spacing of columns, how building members are braced, and in modern times wind loads and seismic forces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read that the architects of Gothic cathedrals were not able to calculate with any accuracy the size and number of columns and flying buttresses they needed to support a given design. They worked from geometry, rules of thumb, and prior experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At what point in history did mathematics begin to make building design more predictable and foolproof? I suspect that at some point algebraic equations replaced geometry and that after the advent of calculus, the section modulus of various shapes began to be understood, but I cannot seem to find information as to where and when these things happened. A good answer would point out which mathematical concepts were first used rigorously in structural design and also what notable buildings were the result of this more formalized process.&lt;/p&gt;&#xA;" OwnerUserId="138" LastActivityDate="2014-11-21T22:36:36.000" Title="When did architects learn how to calculate load on building members accurately?" Tags="|mathematics|architectural-engineering|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="154" PostTypeId="2" ParentId="138" CreationDate="2014-10-31T02:32:12.263" Score="16" Body="&lt;p&gt;While it may have been known that a need for safe storage of nuclear waste existed before 1943-1944, it was certainly known by 1944. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Los Alamos National Laboratory  began research and development on the first nuclear weapon in 1943 &lt;a href=&quot;http://energy.gov/em/los-alamos-national-laboratory&quot;&gt;LANL&lt;/a&gt; under the &lt;a href=&quot;http://en.wikipedia.org/wiki/Manhattan_Project&quot;&gt;Manhattan Project&lt;/a&gt; which lasted from 1942 to 1946. The &lt;a href=&quot;http://web.archive.org/web/20100202080017/http://www.energy.gov/about/breactor.htm&quot;&gt;B Reactor&lt;/a&gt; at the &lt;a href=&quot;http://en.wikipedia.org/wiki/Hanford_Site&quot;&gt;Hanford&lt;/a&gt; site in Washington state was the first reactor to produce plutonium on a full scale, and that was the source of the first nuclear test bombs, as well as the actual bombs dropped on Japan. That reactor was built and in production by 1944.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The experiments, and this development, created a substantial amount of waste. This is the sort of waste that once created in and around 1943-1944 would immediately present itself as being quite hazardous and thus in immediate need of safe disposal.&#xA;While I do not see any sources telling us that droves of scientists knew that this material was toxic, nor do I see that droves of scientists had to die before we discovered that this material needed to be stored safely, we can deduce that certainly by 1943 there was a need for specialized waste disposal and storage by virtue of the fact that they were quite careful to store experimental waste underground in special containers at this time. Many of these early containers are causing trouble today, and that is old news.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At Hanford, waste was stored in underground tanks, and these are leaking today. I quote and source, &lt;a href=&quot;http://koin.com/2014/04/24/6-things-know-hanfords-nuclear-waste/&quot;&gt;&quot;Some underground storage tanks filled with radioactive sludge from as early as the 1940s are slowly giving out, with 67 known to be leaking into the soil.&quot;&lt;/a&gt;. The fact that they tried to store it implies they knew of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.lasg.org/waste.htm&quot;&gt;This source&lt;/a&gt; suggests in the first sentence that &quot;officially-designated material disposal areas&quot; started at Los Alamos in 1944.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generally it does not take days for scientists to recognize this sort of waste as being quite toxic and naturally the need for waste storage would occur as soon as the waste starts being created. While I say that the answer to your question is certainly by 1943, it appears that the Nazis buried what appears to be nuclear waste, and this has been a modern surprise, as historically no one thought the Germans had a program that was advanced enough to produce nuclear waste. Source for that: &lt;a href=&quot;http://www.dailymail.co.uk/news/article-2014146/Nazi-nuclear-waste-Hitlers-secret-A-bomb-programme-mine.html&quot;&gt;Nazi nuclear waste&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am convinced that certainly by 1943 the need was known, and your question asks for a date. I suspect however that, in light of the human knowledge gained since the time of &lt;a href=&quot;http://en.wikipedia.org/wiki/Marie_Curie&quot;&gt;Marie Curie&lt;/a&gt;, that the scientists whether German or American knew quite well that they were about to create hazardous waste material that would need to be stored safely before they even began doing their work. In that light, someone may have known sooner than the WW2 start of nuclear weapons development that the process (theoretical before then) would require waste storage.   &lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2014-10-31T02:37:18.820" LastActivityDate="2014-10-31T02:37:18.820" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="155" PostTypeId="2" ParentId="4" CreationDate="2014-10-31T03:37:27.177" Score="20" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using Roman numerals was notably clumsy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Zero is not necessary in order to have a place-value system.  Nor is zero necessary (or sufficient) in order to develop sophisticated mathematics.  The Mesopotamians had a place value system by, at the latest, 1800 BCE, but had no symbol for zero at that time.  Theirs was a sexagesimal (base-60) system.  Where we would put a zero, they left a blank space.  (But only between digits, and not as a trailing digit, which had the implication that a string of digits defined a number only up to a power of 60 (which could, effectively, be negative).  The magnitude of a number had to be inferred from context.)  By around 200 BCE, the Mesopotamians had introduced a place-holder zero, but there is no evidence the zero was treated as a stand-alone number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Greeks did not use a place value system (although Archimedes devised a special purpose place value system for dealing with very large numbers in &lt;em&gt;The Sand Reckoner&lt;/em&gt;) and did not recognize zero as a number.  Nevertheless, they accomplished great things in mathematics, including developing a sophisticated axiomatic geometry, proving the irrationality of the square root of 2, proving the infinitude of the primes, developing a sophisticated theory of ratios of incommensurable magnitudes, developing the method of exhaustion and using it to compute the volume of the sphere and the area of a parabolic segment, developing the theory of conic sections, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Maya, by 1000 CE (and probably well before), had a place value system that did use zero.  Theirs was a vigesimal (or base-20) system.  Furthermore, zero was treated as a number: counting started from zero rather than one in their calendar.  Their calendar was quite sophisticated and they also had a well-developed astronomy.  Nevertheless, it does not appear that their mathematics advanced very far (although it is hard to say for sure, since the majority of their artifacts were destroyed).  For example, there is no evidence that they had a decent multiplication algorithm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example of China has been addressed in the other answer.  The rod numeral/counting board system certainly was a place value system.  In that system, an empty place was indicated by leaving a gap rather than by an explicit zero.  Many of the problem solutions in classic Chinese mathematical texts were formulated as steps for carrying out a computation on the counting board.  To develop something like the Chinese remainder theorem doesn't require a zero.  The notion &quot;leaves remainder zero&quot; can always be formulated as &quot;divides evenly&quot;.  (A similar formulation would have been used in Greek mathematics.)  As already noted, the counting board technology had an implicit place-holder zero, but not a full-fledged zero.  The counting board technology was sufficiently powerful to enable many great achievements, such as a very accurate estimate of pi.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So what exactly was it about the invention of zero in Western culture that was so useful?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The modern zero was, in fact, invented in India.  Knowledge of the Indian system diffused, first to the Islamic world, and later to Europe.  Indians knew how to perform arithmetic with zero, and introduced the notion that $1/0$ produces an infinite result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it is clear that efficient calculation systems can be developed without the recognition of an explicit zero.  That is not to say that recognition of zero as a number was not extremely important.  To give but one example, the theory of roots of polynomials becomes much less cumbersome when zero (and negative numbers) are recognized.  Before that happened, the theory involved a complicated case-by-case analysis.  The algebraic solution of the cubic and quartic, discovered by del Ferro, Fontana, Cardano, and Ferrari during the Renaissance, as well as earlier work by Omar Khayyam, saw the equations $x^3 = a$, $x^3 + bx = a$, $x^3 = bx + a$, $x^3 + bx^2 + cx = d$, and so on, as different equations, each requiring an individual treatment.  The ability to treat such sets of related equations in a uniform manner certainly accelerated progress. &lt;/p&gt;&#xA;" OwnerUserId="139" LastEditorUserId="62" LastEditDate="2014-11-21T02:29:39.750" LastActivityDate="2014-11-21T02:29:39.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="156" PostTypeId="2" ParentId="153" CreationDate="2014-10-31T07:36:13.373" Score="2" Body="&lt;p&gt;&lt;em&gt;De Re Metallica&lt;/em&gt; was written by Georges Agricola and published in approximately 1550. It was written in Latin, and kept available in European churches. Builders (and readers in general) could physically access the book there. In addition, for those who were illiterate or could not read Latin, the priest or deacon would translate and read requested passage when needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Architects may have been able to accurately calculate building loads prior to 1550. After 1650, such matters became more standardized due to the existence and accessibility of &lt;em&gt;&lt;a href=&quot;https://archive.org/details/deremetallica50agri&quot; rel=&quot;nofollow&quot;&gt;De Re Metallica&lt;/a&gt;&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-10-31T07:36:13.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="159" PostTypeId="2" ParentId="26" CreationDate="2014-10-31T08:11:38.393" Score="11" Body="&lt;p&gt;In the 1960s the mathematical structure of &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Turing_degree&quot; rel=&quot;nofollow&quot;&gt;Turing degrees&lt;/a&gt;&lt;/em&gt; was conjectured to be rather simple and homogeneous. This was consistent with what was known at the time. It later turned out that the opposite is true in a sense: the Turing degrees are as complicated as can be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Details in Ambos-Spies and Fejer, &lt;em&gt;History of degree theory&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="144" LastEditorUserId="3983" LastEditDate="2016-04-14T22:12:30.100" LastActivityDate="2016-04-14T22:12:30.100" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="160" PostTypeId="1" AcceptedAnswerId="3075" CreationDate="2014-10-31T14:07:46.370" Score="10" ViewCount="133" Body="&lt;p&gt;Binomial nomenclature, and with it modern taxonomy, is said to have appeared with Carl Linnaeus' &lt;em&gt;Systema Naturae&lt;/em&gt; (1758, the year during which the 10th edition of the book was published, being Year 0 of nomenclature as far as the &lt;a href=&quot;http://iczn.org/code&quot;&gt;ICZN&lt;/a&gt; is concerned).&#xA;But how much of a leap was it from previous works in taxonomy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was taxonomy slowly working its way toward it, with Linnaeus' work being the very last step?&#xA;Or was it indeed a monumental leap forward?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generaly, what was the state of nomenclature before the tenth edition of &lt;em&gt;Systema naturae&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2015-11-15T17:11:07.780" Title="How much of a conceptual leap forward was 'Systema Naturae'?" Tags="|biology|taxonomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="161" PostTypeId="2" ParentId="39" CreationDate="2014-10-31T15:18:51.877" Score="9" Body="&lt;p&gt;There are various more recent works on subjects that can be considered as metamathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;http://en.wikipedia.org/wiki/Reverse_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reverse Mathematics&lt;/a&gt; was started by Harvey Friedman in the mid-seventies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently, there was quite a bit of excitement around &lt;a href=&quot;http://homotopytypetheory.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Homotopy Type Theory and Univalent Foundations&lt;/a&gt; not only but also as it ties in nicely with effort to have automatically checkable proof. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, needless to say, there is various other work in proof theory and other branches of mathematical logic. &#xA;The issue that you might be perceiving is what is expressed in an &lt;a href=&quot;https://mathoverflow.net/a/172848/&quot;&gt;answer on MathOverflow to a question around metamathematics&lt;/a&gt;; the problems are still studied but not anymore perceived as &lt;em&gt;meta&lt;/em&gt;-mathematics but rather &quot;just regular mathematics.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking your question in a somewhat different direction on might argue that efforts to make more and more of mathematics amenable to formal verification via &lt;a href=&quot;http://en.wikipedia.org/wiki/Proof_assistant&quot; rel=&quot;nofollow noreferrer&quot;&gt;proof assistants&lt;/a&gt; or even &lt;a href=&quot;http://en.wikipedia.org/wiki/Automated_theorem_proving&quot; rel=&quot;nofollow noreferrer&quot;&gt;automated theorem proving&lt;/a&gt; is as natural and current continuation of early effort to formalize mathematics. &lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2014-10-31T15:18:51.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="162" PostTypeId="1" CreationDate="2014-10-31T15:52:32.050" Score="47" ViewCount="9204" Body="&lt;p&gt;Recently, I read an interesting &lt;a href=&quot;http://www.bbc.com/news/magazine-29543708&quot;&gt;article&lt;/a&gt; about how English replaced German as the language in which scientists communicate. But how did German become the leading language in the first place? In the early years of what we could call modern science, around the time when Galileo worked, I would expect latin to dominate the scientific discussion. What happened that scientists switched to German?&lt;/p&gt;&#xA;" OwnerUserId="90" LastEditorUserId="13" LastEditDate="2014-11-04T16:20:51.733" LastActivityDate="2018-09-24T23:56:33.317" Title="How did German become the language of science?" Tags="|academia|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="163" PostTypeId="1" AcceptedAnswerId="309" CreationDate="2014-10-31T15:54:05.797" Score="7" ViewCount="177" Body="&lt;p&gt;If a &lt;a href=&quot;https://hsm.stackexchange.com/questions/12/did-the-digges-telescope-actually-exist&quot;&gt;working telescope was barely possible at the time of Digges&lt;/a&gt; (circa 1570s), what technical advances allowed Lippershey (patent 1608) and especially Galileo (1609 onwards) to create working telescopes? In Galileo's case, he could easily view Jupiter's moons, craters on the Moon, and probably even Neptune (Jupiter aligned with Neptune and Galileo appears to have dutifully recorded it as a fifth moon), so his telescope was &quot;pretty good&quot; for the era.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-11-08T12:16:07.120" Title="What advances allowed Lippershey and Galileo to make working telescopes?" Tags="|telescope|optics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="164" PostTypeId="1" AcceptedAnswerId="174" CreationDate="2014-10-31T16:03:16.397" Score="8" ViewCount="2197" Body="&lt;p&gt;The concept of the constant acceleration for different objects due to gravity (at the same height and ignoring atmospheric effects) is usually attributed to Galileo. In reality, Galileo merely popularized the concept with his Tower of Pisa Thought Experiment (What happens if you drop a cannon ball and a musket ball from the top of the Tower of Pisa?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;John Dee (Elizabethan England) was well aware of the concept and did not think it particularly note-worthy - implying it was a well known concept amongst his peers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So who was the first person to discover this non-intuitive concept?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-10-31T21:17:42.693" LastActivityDate="2018-02-24T17:10:37.380" Title="Who discovered that different objects accelerate at the same rate due to gravity?" Tags="|physics|gravity|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="165" PostTypeId="2" ParentId="44" CreationDate="2014-10-31T16:05:11.170" Score="1" Body="&lt;p&gt;In the &quot;old days&quot; (probably up to the eighteenth century), when the base of knowledge was narrow, the search for greater knowledge largely involved discovering hidden commonalities and &quot;synergies&quot; between mathematical ideas (e.g., the laws governing gravitation and electric fields are similar; imaginary numbers govern trigonometric calculations through DeMoivre's Theorem, etc.) In that kind of a world, &quot;getting the big picture&quot; or being an &quot;expert&quot; meant knowing a little bit about a lot of different math fields (and connecting them together).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays, the &quot;lowing hanging fruit&quot; has been picked, the basic knowledge (mostly) discovered, and the exploration into further connections goes &quot;deeper&quot; in &quot;narrower&quot; fields. Barring someone who is exceptional at 19th century-style &quot;horizontal&quot; thinking, the trend is for greater specialization, and fewer &quot;poly&quot;maths, or interdisciplinary people.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-10-31T16:05:11.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="166" PostTypeId="2" ParentId="11" CreationDate="2014-10-31T16:10:49.113" Score="4" Body="&lt;p&gt;It's not just &quot;neural computing research,&quot; but a number of other scientific fields such as &quot;high definition&quot; video that were spurred by the threat of competition from Japan, real or imagined, in the 1980s and 1990s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 1950s, it was the threat of &quot;Sputnik,&quot; and Russia's (temporary) lead in the space program that spurred a renewed American interest is science, particularly space and atomic science (and helped introduce &quot;new math&quot; to elementary schools).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite a few scientific discoveries/revolutions in history have been spurred by the threat of foreign competition.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-10-31T16:10:49.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="167" PostTypeId="1" AcceptedAnswerId="178" CreationDate="2014-10-31T16:23:55.640" Score="22" ViewCount="484" Body="&lt;p&gt;Everything is in the title but just to expand on the question, I am wondering if there was any publication process similar to the peer-review process in the past (and when). If not, does it mean that anybody could widely release its results without any control?&lt;/p&gt;&#xA;" OwnerUserId="146" LastActivityDate="2014-10-31T21:03:24.703" Title="How old is the peer-review process in scientific publishing?" Tags="|publishing|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="168" PostTypeId="1" AcceptedAnswerId="172" CreationDate="2014-10-31T16:30:31.127" Score="20" ViewCount="1926" Body="&lt;p&gt;I frequently hear and read that Wegener's theory was very controversial at the time it was suggested. Why was this so? Was it due to Wegener himself, to evidence that was erroneously interpreted at that time, or to something else?&lt;/p&gt;&#xA;" OwnerUserId="146" LastEditorUserId="25" LastEditDate="2014-11-04T16:42:06.043" LastActivityDate="2022-03-02T01:32:14.293" Title="Why was the continental drift theory of Alfred Wegener so controversial in the first part of the 20th century?" Tags="|earth-sciences|plate-tectonics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="169" PostTypeId="2" ParentId="162" CreationDate="2014-10-31T18:03:38.127" Score="25" Body="&lt;p&gt;One reason is geography: German is spoken in the middle of Europe and contains  - or is close to - many eminent academic cities. Germany was not a homogeneous country (like e.g. France) before 1871, but a collection of small dynastic states, which were culturally quite diverse. The protestant part of Germany had a strict work ethic and there was also a substantial Jewish population, all of which contributed to the development of mathematics and science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The age of Enlightenment ('Aufklärung') and the romantic period played an important role in Germany and lead to a surge of interest in topics such as nature, philosophy, and also science and math. Traces of this period are still visible in the classical curriculum of some high schools ('humanistisches Gymnasium'). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the German language is good at creating new composites: for example 'Eigenwert' (eigenvalue), 'Nullsummenspiel' (a game without favorable outcome), 'Gedankenexperiment' (thought experiment) and so on. This is a very convenient thing in scientific discourse, and many of these terms have been borrowed or translated into English.&lt;/p&gt;&#xA;" OwnerUserId="37" LastActivityDate="2014-10-31T18:03:38.127" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="170" PostTypeId="1" AcceptedAnswerId="173" CreationDate="2014-10-31T18:28:51.907" Score="23" ViewCount="3010" Body="&lt;p&gt;How did the physicists in the 1920s become aware of the importance of group theory in quantum mechanics? Was group theory already part of the physics curriculum at that time, perhaps in connection to crystallography?&lt;/p&gt;&#xA;" OwnerUserId="37" LastEditorUserId="26" LastEditDate="2014-11-09T12:42:05.140" LastActivityDate="2014-11-17T21:24:14.577" Title="How did group theory enter quantum mechanics?" Tags="|theoretical-physics|quantum-mechanics|group-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="171" PostTypeId="2" ParentId="162" CreationDate="2014-10-31T18:31:18.653" Score="22" Body="&lt;p&gt;The change from Latin to German wasn't instantaneous. True, Galileo was one of the first to break the mold and write in his native language, but he used Italian, not German. Newton, too, used English (though, admittedly, also a lot of Latin). There doesn't appear to be a sudden shift to German around this time period. The change was gradual. As &lt;a href=&quot;http://www.bbc.com/news/magazine-29543708&quot;&gt;the article&lt;/a&gt; you mentioned says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Then it [Latin] began to fracture. Latin became one of many languages in which science was done.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why the change? I found that &lt;a href=&quot;http://www.popsci.com/article/science/fyi-how-did-english-get-be-international-language-science&quot;&gt;this&lt;/a&gt; put it best:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But researchers began to move away from Latin in the 17th century. Galileo, Newton, and others started writing papers in their native tongues in part to make their work more accessible and in part as a reaction to the Protestant Reformation and the declining influence of the Catholic Church.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After all, how many people in the 17th century were fluent in Latin?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I'd say that German became the language of science sometime in the late 19th century, growing until the late part of the early half of the 20th century. Physics exemplifies this. Look at this list of physicists:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Max Planck&lt;/li&gt;&#xA;&lt;li&gt;Albert Einstein&lt;/li&gt;&#xA;&lt;li&gt;Werner Heisenberg&lt;/li&gt;&#xA;&lt;li&gt;Emmy Noether&lt;/li&gt;&#xA;&lt;li&gt;Heinrich Hertz&lt;/li&gt;&#xA;&lt;li&gt;Wilhelm Rontgen&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There were also many Austrians in physics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Erwin Schrodinger&lt;/li&gt;&#xA;&lt;li&gt;Wolfgang Pauli&lt;/li&gt;&#xA;&lt;li&gt;Ernst Mach&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, Germany was strong in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bernhard Riemann&lt;/li&gt;&#xA;&lt;li&gt;David Hilbert&lt;/li&gt;&#xA;&lt;li&gt;Felix Klein&lt;/li&gt;&#xA;&lt;li&gt;Hermann Weyl&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-31T18:31:18.653" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="172" PostTypeId="2" ParentId="168" CreationDate="2014-10-31T18:58:37.827" Score="18" Body="&lt;p&gt;Two primary reasons come to mind:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;At the time the crust appeared solid, and the mantle was not known to be fluid and convecting (although plastic deformation is implied by isostasy). Moving solid rock through solid rock was too strange a concept - especially without a fluid mantle.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Continental Drift&quot; had no mechanism. Why did the continents appear to move? Today we think of it more as a &lt;em&gt;symptom&lt;/em&gt; of plate tectonics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, Wegener was a meteorologist by training.&#xA;Proposing an extreme hypothesis is difficult at best, but proposing it in a field that is not your own only makes it easier to dismiss. Wegener was a distinguished meteorologist but he must have had the same problems as Agassiz and Glaciation Theory (Agassiz was a fossil fish expert not an alpine geologist!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ucmp.berkeley.edu/geology/techist.html&quot;&gt;http://www.ucmp.berkeley.edu/geology/techist.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="25" LastActivityDate="2014-10-31T18:58:37.827" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="173" PostTypeId="2" ParentId="170" CreationDate="2014-10-31T19:13:40.740" Score="16" Body="&lt;p&gt;From Pais's book &lt;em&gt;Inward Bound&lt;/em&gt;, chapter 13:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wigner had become interested in $n&amp;gt;2$ identical particle problem. He rapidly mastered the case $n=3$ (without spin). His methods were rather laborious; for example, he had to solve a (reducible) equation of degree six. It would be pretty awful to go on this way to higher $n$. So, Wigner told me, he went to consult his friend the mathematician Johnny von Neumann. Johnny thought a few moments then told him that he should read certain papers by Frobenius and by Schur which he promised to bring the next day. As a result Wigner's paper on the case of general $n$ (no spin), was ready soon and was submitted in November 1926. It contains an acknowledgement to von Neumann, and also the following phrase: &quot;There exists a well-developed mathematical theory which one can use here: the theory of transformation groups which are isomorphic with the symmetric group (the group of permutations)&quot;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Thus did group theory enter quantum mechanics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The introduction of group theory was not universally welcomed. In the preface to Sternberg's &lt;em&gt;Group Theory and Physics&lt;/em&gt;, he quotes from the autobiography of John Slater, head of the MIT physics department for many years (p.60-62):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their &quot;Gruppenpest&quot;: the pest of the group theory... The authors of the &quot;Gruppenpest&quot; wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practicle consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I had what I can only describe as a feeling of outrage at the turn which the subject had taken...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;As soon as this [Slater's] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as &quot;Slater has slain the 'Gruppenpest'&quot;. I believe that no other piece of work I have done was so universally popular.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sternberg comments that this was not an atypical reaction, but adds &quot;It is, however, amazing to consider that this autobiography was published in 1975, after the major triumphs of group theory in elementary particle physics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wigner writes in the preface to the 1959 English edition of his &lt;em&gt;Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When the original German version was first published, in 1931, there was a great reluctance among physicists toward accepting group theoretical arguments and the group theoretical point of view. It pleases the author that this reluctance has virtually vanished and that, in fact, the younger generation does not understand the causes and basis for this reluctance. Of the older generation it was probably M. von Laue who first recognized the significance of group theory as the natural tool with which to obtain a first orientation in problems of quantum mechanics.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...The initial stimulus for these articles [on which this book is based] was given by the investigations of Heisenberg and Dirac on the quantum theory of assemblies of identical particles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-17T21:24:14.577" LastActivityDate="2014-11-17T21:24:14.577" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="174" PostTypeId="2" ParentId="164" CreationDate="2014-10-31T20:21:25.457" Score="11" Body="&lt;p&gt;From the comments, it seems you are really asking about the fact that (neglecting air friction) different objects fall with the same acceleration. Before the concept of acceleraton was fully understood, scholars would phrase this differently: different objects take the same time to fall a given distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;John Philoponus, in the 6th century, is said to have performed the &quot;Tower of Pisa&quot; experiment (except not with that tower); the &lt;a href=&quot;http://plato.stanford.edu/entries/philoponus/&quot;&gt;Stanford Encyclopedia of Philosophy&lt;/a&gt; has an extensive article on him, and the &lt;a href=&quot;http://en.wikipedia.org/wiki/John_Philoponus&quot;&gt;Wikipedia entry&lt;/a&gt; has this quote from his works:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Marshall Clagett's &lt;em&gt;The Science of Mechanics in the Middle Ages&lt;/em&gt; is a classic reference on this topic; it contains many original sources (in translation), with extensive discussion. &lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-10-31T20:21:25.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="175" PostTypeId="1" AcceptedAnswerId="261" CreationDate="2014-10-31T20:39:52.643" Score="17" ViewCount="546" Body="&lt;p&gt;The &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Poincare.html&quot;&gt;McTutor&lt;/a&gt; biography on Poincaré says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Poincaré's &lt;em&gt;Analysis situs&lt;/em&gt;, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The fundamental group is credited to Poincaré, plus much of homology theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What specific problems stimulated his topological work? Was it an outgrowth of his work on celestial mechanics, or automorphic functions, or something else? The McTutor article mentions that he introduced the fundamental group to help classify two-dimensional surfaces, but why was he trying to do that?&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-05T03:49:49.093" Title="What specific problems motivated Poincaré's work on topology?" Tags="|mathematics|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="176" PostTypeId="2" ParentId="167" CreationDate="2014-10-31T20:43:46.803" Score="6" Body="&lt;p&gt;According to the article &lt;a href=&quot;http://jama.jamanetwork.com/article.aspx?articleid=380935&quot;&gt;Peer Review in 18th-Century Scientific Journalism&lt;/a&gt; (Kronick, 1990), state that peer-review had its antecedents in the 17th century, and that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the beginnings of &quot;peer review&quot; are frequently associated with the Royal Society of London when it took over official responsibility for the Philosophical Transactions in 1752&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, according to the article &lt;a href=&quot;http://advan.physiology.org/content/31/2/145.short&quot;&gt;The ups and downs of peer review&lt;/a&gt; (Benos et al. 2007) suggests that the scientific peer review system had its basis in Edinburgh in the early 18th century, from the article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was not until 1731 that the Royal Society of Edinburgh published Medical Essays and Observations, the first peer-reviewed collection of medical articles &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Where the editor of the journal&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;distributed the submitted essays for review to individuals he considered to be “most versed in these matters&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There was a disclaimer even then that the peer-reviewed articles did necessarily mean that the content of the article was truthful or accurate and stated that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As always, the submitting authors were ultimately responsible for the quality and veracity of their own research&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, an important point made by both authors is that there has been many cases of editors exercising different policies and methods of ensuring the veracity of scientific articles, with Benos et al. stating:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The development of peer review was gradual and somewhat haphazard. Different editors employed varying styles of peer review.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-10-31T20:43:46.803" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="177" PostTypeId="2" ParentId="153" CreationDate="2014-10-31T21:00:50.990" Score="3" Body="&lt;p&gt;The Roman architect, author and civil engineer, &lt;a href=&quot;http://en.wikipedia.org/wiki/Vitruvius&quot; rel=&quot;nofollow&quot;&gt;Vitruvius&lt;/a&gt; wrote 10 books of architecture, &lt;a href=&quot;http://en.wikipedia.org/wiki/De_architectura&quot; rel=&quot;nofollow&quot;&gt;De Architectura&lt;/a&gt;, the text is translated in the Project Gutenburg Ebook &lt;a href=&quot;http://www.gutenberg.org/files/20239/20239-h/29239-h.htm&quot; rel=&quot;nofollow&quot;&gt;Vitruvius: The Ten Books on Architecture&lt;/a&gt; - while not necessarily being the first to apply mathematical principles to architecture, e.g Egyptians, Mayans etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the Romans certainly had a consistency of how their structures (aqueducts, amphitheaters etc) were built and have withstood disasters, modernisation and time itself.  A pertinent quote in De Architectura is in the section &lt;a href=&quot;http://www.gutenberg.org/files/20239/20239-h/29239-h.htm#Page_13&quot; rel=&quot;nofollow&quot;&gt;The Fundamental Principals of Arctitecture&lt;/a&gt; include:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Architecture depends on Order (in Greek τἁξις), Arrangement (in Greek διἁθεσις), Eurythmy, Symmetry, Propriety, and Economy (in Greek οἱκονομἱα).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some particular points are:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Order gives due measure to the members of a work considered separately, and symmetrical agreement to the proportions of the whole.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-10-31T21:00:50.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="178" PostTypeId="2" ParentId="167" CreationDate="2014-10-31T21:03:24.703" Score="10" Body="&lt;p&gt;Wikipedia has a nice page about the &lt;a href=&quot;http://en.wikipedia.org/wiki/Peer_review#History&quot;&gt;peer-review process&lt;/a&gt;, which includes a section on its history:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first recorded editorial pre-publication peer-review process was&#xA;  at the Royal Society of London in 1665 by the founding editor of&#xA;  Philosophical Transactions of the Royal Society, Henry Oldenburg. In&#xA;  the 20th century, peer review became common for science funding&#xA;  allocations. This process appears to have developed independently from&#xA;  that of editorial peer review.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The first peer-reviewed publication might have been the Medical Essays&#xA;  and Observations published by the Royal Society of Edinburgh in 1731.&#xA;  The present-day peer-review system evolved from this 18th-century&#xA;  process.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A prototype professional peer-review process is recommended in the&#xA;  Ethics of the Physician written by Ishāq ibn ʻAlī al-Ruhāwī (854–931).&#xA;  His work states that a visiting physician must make duplicate notes of&#xA;  a patient's condition on every visit. When the patient was cured or&#xA;  had died, the notes of the physician were examined by a local medical&#xA;  council of other physicians, who would decide whether the treatment&#xA;  had met the required standards of medical care.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Peer review has long been a touchstone of the scientific method, but&#xA;  it had only been performed by editors in chief or editorial committees&#xA;  until the end of the 19th century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Wikipedia also points to some very interesting, thorough resources:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Spier, R. (2002). &quot;&lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0167779902019856&quot;&gt;The history of the peer review process&lt;/a&gt;&quot;. TRENDS in Biotechnology 20 (8): 357–358. doi:10.1016/S0167-7799(02)01985-6. PMID 12127284.&lt;/li&gt;&#xA;&lt;li&gt;Gould, T.P.H. (2012). &lt;a href=&quot;https://rowman.com/ISBN/9780810885745&quot;&gt;Do We Still Need Peer Review?&lt;/a&gt;. The Scarecrow Press.&lt;/li&gt;&#xA;&lt;li&gt;Biagioli, M. (2002). &quot;&lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/1045722022000003435#.VFP4mvnF8rk&quot;&gt;From book censorship to academic peer review&lt;/a&gt;&quot;. Emergences 12 (1): 11–45. doi:10.1080/1045722022000003435.&lt;/li&gt;&#xA;&lt;li&gt;Rip, A. (1985). &quot;&lt;a href=&quot;http://sth.sagepub.com/content/10/3/82&quot;&gt;Commentary: Peer Review is Alive and Well in the United States&lt;/a&gt;&quot;. Science, Technology, and Human Values 10 (3): 82–86. doi:10.1177/016224398501000310.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-10-31T21:03:24.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="179" PostTypeId="2" ParentId="39" CreationDate="2014-10-31T21:08:40.893" Score="16" Body="&lt;p&gt;Nowadays, metamathematics is a standard part of the landscape of mathematical logic.&lt;/p&gt;&#xA;&lt;p&gt;On the one hand, most work on foundations of mathematics should probably be considered metamathematical. The standard foundation is set-theoretic, with ZFC and its variants being the usual formalizations. But this is by far not the only option and, for instance, there is recent work on what we now call univalent foundations based on abstract homotopy theory. In a sense this is perhaps closer to Principia than ZFC, since type theory plays a serious role. On the other hand, the approach is really category theoretic, and categories were not really conceived at the time of Principia. Although this new approach is receiving a large amount of attention, the logicians community at large is only beginning to understand its scope and possibilities. A recent series of &lt;a href=&quot;http://www.cs.nyu.edu/pipermail/fom/2014-October/thread.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;threads&lt;/a&gt; at the FOM (foundations of mathematics) email list illustrates the current tension.&lt;/p&gt;&#xA;&lt;p&gt;A large fraction of research in standard areas of mathematical logic is driven by metamathematical considerations, even if not in the sense of revised foundations.&lt;/p&gt;&#xA;&lt;p&gt;For instance, reverse mathematics (also mentioned in another answer) studies the question of what set-existence axioms are actually needed for standard mathematical arguments. Typical results here argue that a standard theorem (such as the intermediate value theorem in classical analysis) is equivalent to or, at least, implies (over a reasonably weak background theory where the discussion takes place) an abstract &amp;quot;existence&amp;quot; axiom (e.g., every infinite binary tree has an infinite branch) or an instance of mathematical induction.&lt;/p&gt;&#xA;&lt;p&gt;Proof theory deals with theories as mathematical objects, and studies their strength, based on either the length of proofs (suitably defined) when compared to some standard options, or  in more subtle ways (such as considerations of the so-called proof-theoretic ordinals). For instance, within Peano arithmetic, the standard first-order system of axioms for number theory, we can easily define Turing machines, the usual formalization of &amp;quot;computer programs&amp;quot;. We can then state whether a binary relation &amp;lt;' on the natural numbers is recursive, meaning that there is an algorithm (a Turing machine) that can decide of any pair of numbers n,m, whether or not n&amp;lt;'m. Many recursive relations are actually well-orderings, and given such a relation R and a theory T (extending Peano arithmetic) we can ask whether T can prove that R is a well-ordering. In general, the length of the provable well-orderings is significantly small, when compared with the length of all recursive well-orderings. We can then compare theories by checking which ones can prove well-orderability of longer (recursive) well-orderings. Based on this description, this seems a bit eccentric, but this is closely tied up with how much transfinite induction the theory can formalize and prove, so these &lt;a href=&quot;https://mathoverflow.net/q/52926/6085&quot;&gt;proof-theoretic ordinals&lt;/a&gt; are actually very reasonable yardsticks of the power of expression and strength of theories.&lt;/p&gt;&#xA;&lt;p&gt;In set theory, one of the standard themes is the comparison of consistency strength of theories. From Gödel's work, we know that a reasonable theory T cannot prove its own consistency, so if a theory T manages to prove the consistency of a theory S, this gives us a natural way in which T is stronger than S. The resulting consistency strength hierarchy is a fascinating mathematical object. It turns out that for natural extensions T of ZFC, we tend to be able to identify a large cardinal axiom that when added to ZFC results in a theory equiconsistent with T. This gives us a large cardinal companion of T, and the purely mathematical study of large cardinals then reflects the study of strengths of theories. That there is such a thing at all is remarkable. Inner model theory is the area of set theory that most directly concerns itself with trying to explain this phenomenon. The actual identification of the companion of a theory, on the other hand, is nowadays a mostly combinatorial question, thanks to Cohen's development of the forcing method.&lt;/p&gt;&#xA;&lt;p&gt;References on univalent foundations can be found &lt;a href=&quot;http://homotopytypetheory.org/book/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://homotopytypetheory.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. On reverse mathematics, see for example &lt;a href=&quot;http://rmzoo.math.uconn.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; in addition to the link given at the other answer. On proof theory, see &lt;a href=&quot;http://math.ucsd.edu/%7Esbuss/ResearchWeb/handbookI/ChapterI.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. For the consistency strength hierarchy in set theory, see &lt;a href=&quot;http://math.berkeley.edu/%7Esteel/talks/stanford2013copy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, although many papers and talks by John Steel are also relevant. Also, many of my posts in MathOverflow and Math.Stackexchange are related to this topic. Let me single out &lt;a href=&quot;https://mathoverflow.net/a/44185/6085&quot;&gt;this one&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="18" LastEditorUserId="18" LastEditDate="2022-09-01T02:50:18.363" LastActivityDate="2022-09-01T02:50:18.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="180" PostTypeId="1" AcceptedAnswerId="182" CreationDate="2014-10-31T21:10:31.200" Score="9" ViewCount="3065" Body="&lt;p&gt;Nazi Germany tried to win World War II by creating &quot;wonder weapons.&quot; They actually produced the V-2 Missiles capable of reaching England, and tried, but failed, to develop the atomic bomb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the weapons on the drawing board was the &lt;a href=&quot;http://en.wikipedia.org/wiki/Sun_gun&quot;&gt;Sun gun&lt;/a&gt;. It was supposed to work as a &quot;space station&quot; from which the sun's rays could be concentrated, and fired onto enemy cities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguably, the rocket program was a prelude to a &quot;space program,&quot; not an Apollo (moon-walk) program, per se, but the creation of a space station from which &quot;bombs,&quot; either the sun's rays, or other space material, could be &quot;fired&quot; onto enemy cities on earth. The end goal was a weapon, like the atomic bomb, that was an order of magnitude more destructive than conventional bombs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How far along did the Germans get in creating such a &quot;space station,&quot; or at least the means to access outer space? Was the &quot;sun gun&quot; just on the drawing board of a few &quot;mad scientists,&quot; or something within the realm of technical possibility, given enough time?&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="26" LastEditDate="2014-11-01T09:16:51.083" LastActivityDate="2021-05-28T08:37:08.280" Title="How far did Nazi Germany get in researching the &quot;sun gun?&quot;" Tags="|thermodynamics|aerodynamics|nazi-germany|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="181" PostTypeId="2" ParentId="180" CreationDate="2014-10-31T21:54:53.560" Score="4" Body="&lt;p&gt;In the Oddly Historical web article &lt;a href=&quot;http://www.oddlyhistorical.com/2014/03/04/sun-gun-nazi-plan-build-orbital-death-ray/&quot; rel=&quot;nofollow&quot;&gt;The Sun Gun: The Nazi Plan to Build an Orbital Death Ray&lt;/a&gt;, the idea started by Dr. Hermann Oberth, a German engineer and rocketry pioneer who had already started developing plans for a space station in the 1920's.  The Nazi's liked this idea, and made plans to turn it into a weapon, using a sodium mirror.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plans for the logistics of this plan were even drawn up, from the article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nazi scientists working on the project envisioned using rockets to lift prefabricated sections into orbit. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The planning was very intricate, even to the point that the workers in the station would wear magnetic boots!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, it was the fact that a lot of the technology required did not exist.  Even with their advanced (for the day) rocketry&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Their best rocket, the infamous V-2, would not have been able to reach the 5100 mile orbit called for in the plan by itself, much less with enough payload to begin building the mirror. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, the scientists envisaged that it would take several decades to perfect their technology and techniques - the end of the war put an end to that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting twist, when analysed, the feasibility of such a weapon is not so good, with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It turns out that a quirk of optics mean that rather than destruction from on high, the sun gun would probably unleash nothing more horrifying than a pleasant spring day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-10-31T21:54:53.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="182" PostTypeId="2" ParentId="180" CreationDate="2014-10-31T22:03:24.917" Score="7" Body="&lt;p&gt;It never got off the drawing board (literally).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a few sources that suggest that the scientists never got beyond some basic technical drawings and schematics related to the Sun Gun. Admittedly, they aren't the &lt;em&gt;best&lt;/em&gt; sources I could hope for, but they agree, so I'm inclined to believe them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea was conceived by Hermann Oberth in 1923 (and published as part of a book in 1929). He was a rocket scientist, not a military man, and he originally wanted to use it for the purposes of capturing energy and heat transfer. He called his concept the &quot;space mirror&quot;, and it was envisioned  - well, exactly as it sounded: a giant mirror in space. It could have been part of a space station, with a manned crew. It was extremely implausible at the time, but later developments made it seem possible. In the 1940s, Germany was undergoing an incredible acceleration of scientific and engineering ideas, courtesy of the Nazis, and Oberth's idea was turned into a weapon. But, fortunately, it never got far.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.damninteresting.com/the-third-reichs-diabolical-orbiting-superweapon/&quot;&gt;here&lt;/a&gt;, we find out what the United States got:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In addition, they [the German scientists] handed over the schematics and calculations for their formidable Sun Gun concept.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.news.com.au/technology/science/death-star-nazis-had-plan-for-giant-sun-gun-to-fry-cities-from-outer-space-using-solar-power/story-fn5fsgyc-1226611613409&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1945, when the victorious Allies began sifting through captured war plans, it emerged that the Nazis had updated Oberth’s proposals and begun looking into the possibility of the Third Reich building a mirror weapon in orbit 22,236 miles above the Earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Interestingly enough, though, &lt;a href=&quot;http://www.huffingtonpost.com/2013/04/04/nazi-sun-gun-space-mirror_n_3015475.html&quot;&gt;this&lt;/a&gt; suggests that the plans are lost:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;No schematics survive to show how the Nazis might have built the mirror.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Read through these sites more and you'll come to the same conclusion the Allies did: The scientists made very elaborate calculations and blueprints, but there was no testing of any prototypes, or any other form of experimentation.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-10-31T22:03:24.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="183" PostTypeId="2" ParentId="23" CreationDate="2014-11-01T06:15:58.363" Score="9" Body="&lt;p&gt;According to the Oregon State University webpage &lt;a href=&quot;http://oregonstate.edu/instruct/phl302/philosophers/descartes.html&quot;&gt;RENE DESCARTES (1595-1650)&lt;/a&gt;, part of your question is explained by&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two months before the publication of the Passions Descartes set sail for Stockholm, Sweden, at the invitation of Queen Christina of Sweden. Descartes' death in Stockholm of pneumonia, has &lt;strong&gt;regularly been attributed to the rigours of the Swedish climate and the fact that Descartes (no early riser) was sometimes required to give the Queen lessons as early as five in the morning.&lt;/strong&gt; However unpleasant these conditions may have been, it seems plain that Descartes acquired his fatal malady as a result of nursing his friend the French ambassador (who had pneumonia) back to health.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similarly, according to the Washington and Lee University webpage &lt;a href=&quot;http://home.wlu.edu/~mahonj/Descartes.Bio.htm&quot;&gt;RENÉ DESCARTES (1596-1650)&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Because the Queen was often occupied with other business, and wanting to give full attention to philosophy, she arranged for her lessons to be at 5.00 a.m. René was staying at the French embassy opposite the Royal Palace, and had to take a carriage over the bridge to get to Palace each morning. He caught a cold, succumbed to a chill and died on the 11th February, 1650.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, why did Descartes accept the offer from the Queen of Sweden and move to Stockholm?  A possible answer is included in Stanford Encyclopedia of Philosophy webpage &lt;a href=&quot;http://plato.stanford.edu/entries/descartes-works/&quot;&gt;Descartes' Life and Works&lt;/a&gt; which suggests that the offer from Sweden came at a right to to allow him to get away from a situation that he felt was increasingly threatening his way of life.  From the section of the Stanford website about Descartes &lt;a href=&quot;http://plato.stanford.edu/entries/descartes-works/#Pas&quot;&gt;The Passion&lt;/a&gt;, it becomes clear that a very tense and very public academic challenge to Descartes work was happening.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For many reasons, which would certanily include those related to his concerns about Voetius, Descartes accepted the offer&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and went to Sweden.  A decision, according to the Stanford document linked above, he began to have reservations about.&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-01T06:15:58.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="184" PostTypeId="1" CreationDate="2014-11-01T14:57:23.507" Score="16" ViewCount="2150" Body="&lt;p&gt;I read on &lt;a href=&quot;http://en.wikipedia.org/wiki/Walter_Pitts&quot;&gt;the Wikipedia page on Walter Pitts&lt;/a&gt; that Norbert Wiener broke off relations with Walter Pitts, which &quot;sent Walter Pitts into 'cognitive suicide'&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did Norbert Wiener break off relations with Walter Pitts? The Wikipedia page argues that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wiener suddenly turned against McCulloch, on account of his wife&#xA;  Margaret Wiener who hated McCulloch, and broke off relations with&#xA;  anyone connected to him including Pitts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Was it the only reason and if so why did Margaret Wiener hated McCulloch?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="4251" LastEditDate="2020-07-11T12:33:05.460" LastActivityDate="2020-07-11T12:33:05.460" Title="Why did Norbert Wiener break off relations with Walter Pitts?" Tags="|computer-science|neuroscience|death|cybernetics|wiener|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="185" PostTypeId="1" CreationDate="2014-11-01T15:04:46.383" Score="13" ViewCount="163" Body="&lt;p&gt;I read on &lt;a href=&quot;http://en.wikipedia.org/wiki/Walter_Pitts&quot;&gt;the Wikipedia page on Walter Pitts&lt;/a&gt; that  :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pitts was also described as an eccentric, refusing to allow his name&#xA;  to be made publicly available. He refused all offers of advanced&#xA;  degrees or official positions at MIT as he would have to sign his&#xA;  name.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It came to me as a surprise as he seems to have use his name in his &lt;a href=&quot;http://scholar.google.com/scholar?hl=en&amp;amp;q=Walter%20Pitts&amp;amp;btnG=&amp;amp;as_sdt=1%2C22&amp;amp;as_sdtp=&quot;&gt;research articles&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Walter Pitts refuse to allow his name to be made publicly available, or that some myth?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2015-03-21T17:16:02.300" LastActivityDate="2015-03-21T17:16:02.300" Title="Did Walter Pitts refuse to allow his name to be made publicly available?" Tags="|biographical-details|computer-science|neuroscience|cybernetics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="186" PostTypeId="1" CreationDate="2014-11-01T18:13:13.733" Score="11" ViewCount="574" Body="&lt;p&gt;I read the Wikipedia page on &lt;a href=&quot;http://en.wikipedia.org/wiki/Software_bug&quot;&gt;Software bugs&lt;/a&gt;. It does have a section on its etymology. Albeit interesting it doesn't answer my question but merely notes that the term &lt;em&gt;bug&lt;/em&gt; was used in 1878:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Use of the term &quot;bug&quot; to describe inexplicable defects has been a part&#xA;  of engineering jargon for many decades and predates computers and&#xA;  computer software; it may have originally been used in hardware&#xA;  engineering to describe mechanical malfunctions. For instance, Thomas&#xA;  Edison wrote the following words in a letter to an associate in 1878:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It has been just so in all of my inventions. The first step is an&#xA;  intuition, and comes with a burst, then difficulties arise — this&#xA;  thing gives out and [it is] then that &quot;Bugs&quot; — as such little faults&#xA;  and difficulties are called — show themselves and months of intense&#xA;  watching, study and labor are requisite before commercial success or&#xA;  failure is certainly reached.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Baffle Ball, the first mechanical pinball game, was advertised as&#xA;  being &quot;free of bugs&quot; in 1931. Problems with military gear during&#xA;  World War II were referred to as bugs (or glitches).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A page from the Harvard Mark II electromechanical computer's log,&#xA;  featuring a dead moth that was removed from the device The term &quot;bug&quot;&#xA;  was used in an account by computer pioneer Grace Hopper, who&#xA;  publicized the cause of a malfunction in an early electromechanical&#xA;  computer. A typical version of the story is given by this quote:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;In 1946, when Hopper was released from active duty, she joined the&#xA;    Harvard Faculty at the Computation Laboratory where she continued her&#xA;    work on the Mark II and Mark III. Operators traced an error in the&#xA;    Mark II to a moth trapped in a relay, coining the term bug. This bug&#xA;    was carefully removed and taped to the log book. Stemming from the&#xA;    first bug, today we call errors or glitches in a program a bug.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So why was a bug called bug in the first place (e.g. before 1878)? Also, was nobody in Hopper's lab aware of the engineering meaning of bug?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2015-02-11T12:00:48.967" Title="What's the etymology of an engineering/software bug?" Tags="|computer-science|terminology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="187" PostTypeId="1" CreationDate="2014-11-01T18:43:28.607" Score="18" ViewCount="3775" Body="&lt;p&gt;A &lt;strong&gt;Kuhn loss&lt;/strong&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a success, empirical or theoretical, of a prior theory – or paradigm as Kuhn would have preferred – that does not carry over to the theory or paradigm that replaced it. [&lt;a href=&quot;http://arxiv.org/pdf/1205.0179.pdf&quot;&gt;Midwinter and Janssen&lt;/a&gt;, see below.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kuhn introduced the concept in &lt;em&gt;The Structure of Scientific Revolutions&lt;/em&gt; (p. 99–100, 3&lt;sup&gt;rd&lt;/sup&gt; ed.). The examples he gives there come from the phlogiston theory, which explained:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why metals are so much more alike than their ores: All metals contain phlogiston.&lt;/li&gt;&#xA;&lt;li&gt;Why burning something in a closed flask reduces the pressure of the air: The phlogiston released by combustion “spoils” the elasticity of the air, just as fire “spoils” the elasticity of a steel spring.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have also run into two examples from Cartesian vortex theory, which explained:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why all the planets revolve around the sun in the same direction: They are all carried by a common vortex.&lt;/li&gt;&#xA;&lt;li&gt;Why there is gravity: The vortex “pushes” bodies “downward”, due to differential impact of the vortex stream (see Aiton’s book &lt;em&gt;The Vortex Theory of Planetary Motions&lt;/em&gt; for more details).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, the &lt;a href=&quot;http://arxiv.org/pdf/1205.0179.pdf&quot;&gt;article by Midwinter and Janssen&lt;/a&gt;, &quot;Kuhn Losses Regained: Van Vleck from Spectra to Susceptibilities&quot; gives a lengthy account of a technical example in the theory of electric and magnetic susceptibilities. Briefly, there is a constant &lt;em&gt;C&lt;/em&gt; in that theory. According to classical physics, &lt;em&gt;C&lt;/em&gt; = ⅓. The old quantum theory (1913–1925) gave much larger values, the exact value depending on the model and the way quantum conditions were imposed. In the new quantum theory (i.e., modern quantum mechanics), the experimentally correct value of ⅓ is restored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since posting this, another near-example has occurred to me: Kepler's 1609 prediction of the sun's rotation, confirmed shortly by the discovery of sunspots. Kepler's prediction stemmed from his theory of a whirlpool-like force from the sun sweeping the planets around; in Newton's system, the rotation of the sun is just an unexplained fact.&#xA;But this example doesn't quite qualify, since Kepler's whirlpool force was never widely accepted; also, the prediction was not fully successful. (Kepler predicted the wrong period and the wrong axis of rotation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should mention that the concept of Kuhn loss remains controversial, but these examples seem reasonable at first glace. Note that a Kuhn loss can later be regained with a yet newer theory. Electron theory, for example, explains the similarities of metals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Taxonomy.&lt;/strong&gt; Kuhn loss comes in several varieties. Both explanations and predictions can be lost. Most examples concern lost explanations. Subsequent history can take several courses. (a) Changing standards of scientific explanation can turn the old issue into a non-problem. For example, Newton's gravity was eventually accepted as fact, not needing a mechanical cause. (b) A yet newer theory can provide a new explanation, as in the metal example. (Here, the electron theory of metals does not refute Lavoisier's &quot;chemical revolution&quot;; on the other hand, the new quantum theory replaced its immediate predecessor. So two versions of this case.) (c) The orphaned phenomenon (deprived of its old explanation) can be treated as an anomaly, something still requiring resolution. Example: geocentric theories explained the lack of stellar parallax, and for almost 300 years, detecting it was an outstanding problem. When Bessel finally detected it in 1838, heliocentricity had long been accepted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer a question in a comment, then:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Doesn't this also mean that they must have been &quot;rediscovered&quot; during a later theory/paradigm shift, since they are defined in terms of previous successes? Or do all things that were believed to be successes in the previous paradigm but later discarded count, even though they were due to e.g. biased observation methods or other misconceptions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;All these count as Kuhn loss. The C-constant example is of type (b); the stellar parallax example was of type (c) until 1838, transitioning then to type (b); the metal example transitioned from type (a) to type (b).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last clarification: you don't have to accept Kuhn's relativism to be interested in Kuhn loss. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What other examples of Kuhn loss are there?&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-20T20:26:12.257" LastActivityDate="2018-03-26T09:53:56.687" Title="Examples of Kuhn loss?" Tags="|examples|philosophy-of-science|kuhn|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="188" PostTypeId="1" AcceptedAnswerId="194" CreationDate="2014-11-01T19:05:43.450" Score="31" ViewCount="14573" Body="&lt;p&gt;The U.S. picked up a whole bunch of German scientists after World War II in Operation Paperclip. One of the most notable was Wernher von Braun, who jump-started the American space program. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Soviet Union also got some scientists, although not exactly top-tier guys. What happened to them? Did they play a role in the Space Race, working on the other side of the Iron Curtain?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="4251" LastEditDate="2020-07-18T21:41:12.527" LastActivityDate="2020-07-18T21:41:12.527" Title="What did the Soviets do with German scientists after WWII?" Tags="|scientists|nazi-germany|soviet-union|cold-war|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="189" PostTypeId="2" ParentId="186" CreationDate="2014-11-01T19:19:50.413" Score="7" Body="&lt;p&gt;This first sourcing I take directly from my paper copy of the &lt;strong&gt;The Oxford Dictionary of English Etymology&lt;/strong&gt; (&lt;em&gt;c. 1966, 1996 reprint&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here we have two etymologies for the word bug. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The less appropriate etymology is as &quot;insect, beetle (as still in U.S.)&quot; with reference to a 16th century occurrence &quot;in &lt;em&gt;turd-bug&lt;/em&gt; dung beetle&quot;. It is next noted &quot;Origin unascertained&quot; with indication that this is possibly an alteration of the Old English word &lt;em&gt;buddle&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more appropriate sense, and I think clearly the context from whence this entered into the use we are looking for (I expand the abbreviations to form at least a few complete sentences, but in general I quote with liberties):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;bug&lt;/strong&gt; object of dread. First seen in the English language with this meaning in the 14th century. The earliest of several words, mostly evidenced from the 16th century, of similar form and meaning ('goblin', 'spectre', 'bugbear', 'bogey'), the connexions of which are obscure; namely the obsolete dialectic words &lt;em&gt;'bog'&lt;/em&gt;, &lt;em&gt;'boggard'&lt;/em&gt;, &lt;em&gt;'bogle'&lt;/em&gt; (Dunbar), &lt;em&gt;'bogle-bo'&lt;/em&gt;, BUGABOO, BUGBEAR, and the more recent BOGEY. Comparison with Welsh &lt;em&gt;'bwg'&lt;/em&gt;, &lt;em&gt;'bwgan'&lt;/em&gt; meaning ghost, hobgoblin,....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This etymology continues, but that is the heart of the context we are looking for, and I suspect the earlier portion of the best known total path to the more recent 19th and 20th century etymological traces described in the cited Wikipedia article. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have two other good paper sourcings that cast even more references and details along this line:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;strong&gt;The Merriam-Webster New Book of Word Histories&lt;/strong&gt; (c. 1991) The authors point out that &quot;A Middle English &lt;em&gt;bugge&lt;/em&gt; was either a scarecrow or a hobgoblin.&quot; The author sources that 14th century OED reference as being from the 1382 Wycliffe bible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the &lt;strong&gt;Dictionary of Word Origins by John Ayto&lt;/strong&gt; (c. 1990), the author states, &quot;Originally the word &lt;em&gt;bug&lt;/em&gt; meant 'something frightening'&quot;. He goes on to say. &quot;It is one of a set of words (others are 'bogle' and perhaps 'bugaboo') for alarming or annoying phenomena, usually supernatural, whose interrelationship and ultimate source have never been explained.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That last sourcing sounds distinctly like a software bug to my ears in the sense of some alarming and annoying thing that might (initially) seem supernatural.&lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2014-11-01T19:48:11.220" LastActivityDate="2014-11-01T19:48:11.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="190" PostTypeId="2" ParentId="162" CreationDate="2014-11-01T19:46:45.147" Score="33" Body="&lt;p&gt;My impression is that German never was &quot;THE language of science&quot; in the same sense as English in now. After the switch from Latin to modern languages, there was no single dominating language of science. Up to the middle 20-s century there were at least 4 common languages&#xA;of science: French, German, English and Italian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me state more precisely what I mean: French scientists wrote in French (not in German),&#xA;English/American wrote in English (not in German, not in French). And so on.&#xA;Scientists of other nations had an option: to write in their native language or in one&#xA;of these 4 &quot;languages of science&quot;. For example Russians wrote in Russian, English, German and French.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This multilingual situation prevailed in Mathematics until 1970-s. Since the end of 70-s&#xA;very few Germans write mathematics in German. There are still many French mathematicians&#xA;writing in French, but a talk in French in an international conference is really rare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is incorrect to say that German (or French) ever was THE &quot;language of science&quot;.&#xA;Until 1970-s the Germans wrote in German, the English wrote in English and the French&#xA;wrote in French. These languages were &quot;one of the languages of science&quot; each.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. It seems to be true, however that in terms of VOLUME of published scientific works German was on the first place in the 19th and early 20th century. (I can confirm this by some statistics I made myself). But still English, French and Italian scientists wrote in their own languages. Nowadays the situation is very different: most scientists of any nation write in English.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-09-07T04:28:44.377" LastActivityDate="2018-09-07T04:28:44.377" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="191" PostTypeId="2" ParentId="149" CreationDate="2014-11-01T19:55:25.050" Score="4" Body="&lt;p&gt;You can look in Wikipedia for the list of astronomers between Hipparchus and Ptolemy,&#xA;or for more detail consult Neugebauer. (Or Ptolemy himself:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that 90% of our knowledge about astronomy before Ptolemy (except Babylonian&#xA;for which we have an independent source) is based&#xA;on Ptolemy books, and this is not an exaggeration. An evidence of this is&#xA;Neugebauer's History of Ancient Mathematical astronomy. The first volume is an analysis of Ptolemy, the second of EVERYTHING else (Including Hipparchus, Babylonians, Egyptians,&#xA;and everything in between), and the 3-d volume is Appendixes, tables plates etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Wikipedia lists the following astronomers in the period between Hipparchus and Ptolemy: Aglaonice, Agrippa, Andronicus of Cyrrhus, Hypcicles, Geminus, Menelaus and Theon of Smyrna (these two are contemporaries of Ptolemy), Posidonius, Seleucus of Seleucia,&#xA;Theodosius of Bythinia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of these, Ptolemy mentions: Agrippa, Theon of Smyrna and Menelaus. One book of Geminus still exists. All others are just mentioned somewhere in other sources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2. There is no consensus about Russo's speculations. The fact is that most of the Greek pre-Ptolemy astronomy is lost. (Including almost everything that Hipparchus wrote. Only one minor work of Hipparchus survives.) I am inclined to think that Russo is right in his general conclusions.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-04T00:30:20.913" LastActivityDate="2014-11-04T00:30:20.913" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="192" PostTypeId="2" ParentId="151" CreationDate="2014-11-01T20:01:48.407" Score="2" Body="&lt;p&gt;&quot;Almost everything&quot; was found before the general modern definition of a group was extablished:-) I am not sure who gave the first definition of the abstract group&#xA;(as a set with an operation satisfying such and such axioms). But probably this happened&#xA;in 20-s century (various people are credited with this).&#xA;For 19 century mathematicians a group was a group of transformations of a set into itself.&#xA;And the first deep results belong to Lagrange and Galois.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-01T20:01:48.407" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="193" PostTypeId="2" ParentId="142" CreationDate="2014-11-01T20:08:00.930" Score="25" Body="&lt;p&gt;Notation $()$ is traditional, and $].[$ was introduced by &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicolas_Bourbaki&quot; rel=&quot;noreferrer&quot;&gt;Bourbaki&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of the Bourbaki notations and terminology became standard, but English speaking people are the most conservative ones in this respect:-) &#xA;(Recall the history of the metric system:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example of the same is &quot;injection&quot;, &quot;surjection&quot;, &quot;bijection&quot;.&#xA;Many English authors still write &quot;one-to-one&quot;, &quot;onto&quot; and &quot;one-to-one and onto&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example: Bourbaki taught us that &quot;positive&quot; is $\geq 0$,&#xA;and &quot;strictly positive&quot; is $&amp;gt;0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But many people still prefer &quot;positive&quot; to mean $&amp;gt;0$ and &quot;non-negative&quot; for $\geq0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. I am educated in Ukraine in 1970-s, and I experienced a strong influence&#xA;of Bourbaki on education. But I still like $(,)$, perhaps just for aesthetic reasons.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-17T03:17:11.013" LastActivityDate="2017-10-17T03:17:11.013" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="194" PostTypeId="2" ParentId="188" CreationDate="2014-11-01T20:26:51.760" Score="38" Body="&lt;p&gt;The Russians picked up a similar number of &quot;rocket scientists&quot; as the West, but the lesser ones. These were taken to newly-constructed but isolated scientific facilities at places like &lt;a href=&quot;http://www.russianspaceweb.com/a4_team_moscow.html&quot;&gt;Gorodomiya Island&lt;/a&gt; on a lake northwest of Moscow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They were housed with Russian scientists in relatively comfortable (by Russian standards) facilities, near their place of work. Basically, the Germans' job was to write papers on rocket technology to educate their Russian counterparts, while they received very little knowledge in return, so that their technical expertise would fall behind the Russians'. This continued for more than five years, until Stalin's death. By this time, the German scientists had &quot;drained&quot; of their knowledge, and having been kept in isolation, no longer represented a threat. Between this fact and the more liberal atmosphere that prevailed after Stalin's death, it was possible to send them home to West Germany.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Germans in Russia did very little of the actual design work, but their theoretical knowledge was of some help to the Russians in understanding rocketry and designing missiles; to a lesser extent in designing rockets for the space program.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-11-01T20:26:51.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="195" PostTypeId="1" AcceptedAnswerId="242" CreationDate="2014-11-01T20:30:44.907" Score="14" ViewCount="1505" Body="&lt;p&gt;At first, my initiate question was: &lt;a href=&quot;http://timesofindia.indiatimes.com/home/stoi/What-is-the-difference-between-an-atomic-and-a-nuclear-bomb/articleshow/1345040.cms&quot;&gt;What is the difference between an atomic and a nuclear bomb?&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nuclear bombs are of two types — those that depend on fission, like atomic bombs, and those that depend on fusion, like hydrogen bombs&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regardless of fission or fusion, both of them are reactions of subatomic particles, no reaction of atoms. Therefore, &lt;em&gt;atomic bomb&lt;/em&gt; is a wrong name. Do you have any explanation for this? Why has &lt;em&gt;atomic bomb&lt;/em&gt; instead of another better term become the predominant term?&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="26" LastEditDate="2015-04-26T14:28:25.300" LastActivityDate="2015-04-26T14:28:25.300" Title="How did we come up with the name &quot;atomic bomb&quot;?" Tags="|physics|terminology|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="196" PostTypeId="2" ParentId="35" CreationDate="2014-11-01T20:36:13.390" Score="10" Body="&lt;p&gt;At the time of Brahe and Kepler they did not use the right ascention and declination to record the movement of planets. These coordinates are related to the Earth, and it is known since the times of Hipparchus and Ptolemy that one has to use the ecliptic coordinates, that is a system related to the Sun motion. (Ecliptic is the large circle in the sky on which the Sun moves. All planets move in the planes that are close to the ecliptic plane).The corresponding coordinates are called (celestial) longitude and latitude.&lt;/p&gt;&#xA;&lt;p&gt;Brahe's data for the planets consisted (roughly speaking) of positions of the planets&#xA;(their longitudes and latitudes) at some&#xA;known moments of time. Because the latitudes of all planets are small, one can consider the&#xA;motion in longitude independently of the motion in latitude, which simplifies the problem.&#xA;This motion is not uniform (and even not always in the same direction), so the problem was&#xA;to create a kinematic mechanism which would &amp;quot;explain&amp;quot; this motion. Such a mechanism is&#xA;known from the antiquity, but it gives a small disagreement with the observed data.&#xA;It is in his attempts to explain this disagreement (in the case of Mars) that Kepler made&#xA;his great discoveries (the first and second laws). Much later he also discovered the 3-d law. The most fundamental discovery was that the orbits are not combinations of uniform motions on circles, but a motion on an ellipse according to the &amp;quot;law of equal areas&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;This remarkable discovery was probably the greatest revolution in the history of astronomy.&lt;/p&gt;&#xA;&lt;p&gt;Here one can find a bit more explanation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.purdue.edu/%7Eeremenko/dvi/kepler.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.purdue.edu/~eremenko/dvi/kepler.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-12T00:56:20.217" LastActivityDate="2021-09-12T00:56:20.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="197" PostTypeId="2" ParentId="195" CreationDate="2014-11-01T20:45:27.930" Score="3" Body="&lt;p&gt;&lt;strong&gt;They really do involve atoms. . .&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two types of nuclear weapons: fission bombs and fusion bombs. Fission bombs depend on - surprise! - &lt;a href=&quot;https://en.wikipedia.org/wiki/Nuclear_fission&quot; rel=&quot;nofollow&quot;&gt;nuclear fission&lt;/a&gt;, while fusion bombs depend on - wait for it - &lt;a href=&quot;https://en.wikipedia.org/wiki/Nuclear_fusion&quot; rel=&quot;nofollow&quot;&gt;nuclear fusion&lt;/a&gt;. A good overview of the two can, of course, be found &lt;a href=&quot;https://en.wikipedia.org/wiki/Nuclear_weapon&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Nuclear_weapon_design&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. Just don't let the NSA catch you. . .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, nuclear bombs &lt;em&gt;do&lt;/em&gt; involve atoms. In fission bombs, these are generally uranium and plutonium, the scary elements you think about whenever you hear the word &quot;nuclear.&quot; Here's how the reaction goes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Stick a chunk of a material for the reaction (perhaps uranium, for simplicity).&lt;/li&gt;&#xA;&lt;li&gt;Fire a neutron at it.&lt;/li&gt;&#xA;&lt;li&gt;Let the (now unstable) uranium nucleus decay into other products, and emit free neutrons.&lt;/li&gt;&#xA;&lt;li&gt;Repeat. Watch as a lot of energy is released, in the form of photons.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;That's really how nuclear fission works. It &lt;em&gt;does&lt;/em&gt; involve atoms, because it involves nuclei - in the form of atoms.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-01T20:45:27.930" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="198" PostTypeId="1" CreationDate="2014-11-01T21:02:13.600" Score="45" ViewCount="5196" Body="&lt;p&gt;Let me tell what I know about this. It is well-known that Heisenberg invented matrix multiplication himself, in his great paper that is considered part of the foundation of quantum mechanics. This was in 1925, and the story is very well documented. Then very shortly after that Born and Jordan recognized that this is matrix multiplication,&#xA;BECAUSE one of them had a course on &quot;hypercomplex numbers&quot; as a student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The clear conclusion that I make of this is that in the first decade (when they were&#xA;all students) of 20th century matrix multiplication was not taught to students on a regular basis in the best European universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first edition of &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0471504475&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Courant-Hilbert&lt;/a&gt; was published in 1924. (I am not sure what was the standard course of mathematics for physicists before that, but probably &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1402186843&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomson-Tait&lt;/a&gt;, which has no matrices).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand at the present time matrices are taught to ALL (science) undergraduates; this is even more standard than Calculus (I judge from my experience in Soviet Union and USA, but&#xA;I suppose this is the case everywhere).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When did this dramatic transition in undergraduate curriculum happen?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;and even more interestingly:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Why did it happen?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;On the second question I have a conjecture: it is exactly because of the invention of&#xA;quantum mechanics. I have some supporting evidence and &quot;philosophical arguments&quot; in favor of this. But to investigate this matter, it is good first of all to find the answer on the first question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that matrix multiplication was probably introduced by Cayley, but it is a very long way for a new mathematical object to the undergraduate curriculum, and most of our inventions never make it this way :-) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar question is posted on &lt;a href=&quot;https://mathoverflow.net/questions/185954/when-exactly-and-why-did-matrix-multiplication-become-a-part-of-the-undergraduat&quot;&gt;MO&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-19T12:47:16.127" LastActivityDate="2020-05-19T12:47:16.127" Title="When exactly (and why) did matrices become a part of the undergraduate curriculum?" Tags="|mathematics|physics|education|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="199" PostTypeId="1" CreationDate="2014-11-01T22:18:58.163" Score="29" ViewCount="8363" Body="&lt;p&gt;I've encountered the claim that around the end of the 19th century, physicists believed that their understanding of the physical world was close to being complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example of this claim can be found in Wikipedia's article on &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_physics&quot;&gt;History of physics&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Was this really generally accepted? What sources support that?&lt;/p&gt;&#xA;" OwnerUserId="162" LastEditorUserId="24" LastEditDate="2014-11-10T22:32:02.277" LastActivityDate="2014-11-10T22:32:02.277" Title="Did physicists around 1900 really believe they were close to &quot;figuring it all out&quot;?" Tags="|physics|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="200" PostTypeId="2" ParentId="198" CreationDate="2014-11-01T22:35:46.343" Score="29" Body="&lt;p&gt;I will start by answering why matrix algebra became important, and then discuss approximately when.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Matrices&quot; underpin what is often called &lt;a href=&quot;http://en.wikipedia.org/wiki/Operations_research&quot; rel=&quot;nofollow noreferrer&quot;&gt;operations research&lt;/a&gt;. That is, the theory of decision making. They are particularly useful in computer science, which features strings, arrays, etc., with machines substituting for human beings in (mechanical) decision making.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Operations research took a giant step forward during World War II, when the quantity of men, materials, weaponry etc. were &quot;mind-boggling&quot; for their time. As my father, a retired engineering professor would say, numerous &quot;systems of equations&quot; needed to be solved. (His first job out engineering school was to design an airfield.) During the war, the British government had some 1000 people in their &quot;operational research&quot; department, and likewise for the U.S. Some ten members of the U.S. group went to Harvard Business School together, then &quot;parachuted&quot; into Ford Motor Company as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Whiz_Kids_(Ford)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;whiz kids.&quot;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &quot;matrices&quot; was introduced into the undergraduate curriculum not long after World War II. The subject was given a boost by the newly-developed technique of &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Linear_programming&quot; rel=&quot;nofollow noreferrer&quot;&gt;linear programming&lt;/a&gt;&quot; (1947), followed by other decision-making tools such as input-output tables, which &lt;a href=&quot;http://en.wikipedia.org/wiki/Wassily_Leontief&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wassily Leontief&lt;/a&gt; popularized in 1953. By the mid-1950s, &quot;matrices&quot; were taught at most of the better colleges, and by the late 1960s, they were finding their way into the high school curriculum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's true, as some commenters pointed out, that matrices are now taught earlier in secondary school in countries outside the United States than &quot;here.&quot; But that wasn't the question, which was about when (and where) matrices were taught earlier at the undergraduate level in &lt;strong&gt;history.&lt;/strong&gt; That would be the United States in the 1950s.&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="3235" LastEditDate="2020-03-23T17:00:49.023" LastActivityDate="2020-03-23T17:00:49.023" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="201" PostTypeId="2" ParentId="199" CreationDate="2014-11-01T22:52:04.323" Score="10" Body="&lt;p&gt;I'll add more later, but for now consider &lt;a href=&quot;http://www.todayinsci.com/K/Kelvin_Lord/KelvinLord-Quotations.htm&quot;&gt;this quote by Lord Kelvin&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I believe this is what the Wikipedia article you found is talking about. &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Thomson%2C_1st_Baron_Kelvin#Pronouncements_later_proven_to_be_false&quot;&gt;Wikipedia&lt;/a&gt; also references this. Note, though, that its accuracy is disputed. However, it still sums up the feelings of some at the time.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-01T22:52:04.323" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="202" PostTypeId="2" ParentId="199" CreationDate="2014-11-01T23:02:38.930" Score="17" Body="&lt;p&gt;According to (the late) William Strauss and Neil Howe in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0688119123&quot;&gt;Generations&lt;/a&gt;, there are periodic episodes of scientific exhaustion, at least in the United States. For instance, around 1910,  the head of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Charles_Holland_Duell&quot;&gt;U.S. patent office&lt;/a&gt; was (mis)quoted as saying: &quot;Everything that can be invented has been invented.&quot; In 1992, &lt;a href=&quot;http://en.wikipedia.org/wiki/Francis_Fukuyama&quot;&gt;Francis Fukuyama&lt;/a&gt; famously opined about &quot;the end of history.&quot;  These &quot;periods&quot; occur roughly eighty years, or four generations apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What really happened was that these remarks followed a recent burst of scientific achievement; the space program and &quot;star wars&quot; in Fukuyama's case, and the industrialization of America, in the earlier case. At one level, such people are &quot;right;&quot; the recent pace of scientific achievement can't be maintained. But they then wrongly conclude that scientific progress has &quot;come to an end.&quot;&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="31" LastEditDate="2014-11-01T23:09:27.547" LastActivityDate="2014-11-01T23:09:27.547" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="203" PostTypeId="2" ParentId="184" CreationDate="2014-11-01T23:54:41.883" Score="9" Body="&lt;p&gt;Another view of the split comes from the John Hopkins University document &lt;a href=&quot;http://mwbdvjh.muse.jhu.edu/journals/perspectives_in_biology_and_medicine/v043/43.2smalheiser.pdf&quot; rel=&quot;noreferrer&quot;&gt;Walter Pitts&lt;/a&gt; (Smalheiser, 2000), it seemed to be a compounded effect of Pitts and his friends McCulloch, Selfridge and Lettvin's activities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly starting with Wiener's concern about the progress Pitts was making with his PhD thesis&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wiener was worried about Pitts making adequate progress on his Ph.D.&#xA;  thesis (concerning the state functions of neuronal networks arbitrarily&#xA;  connected in two and three dimensions), and tried to separate him from&#xA;  the distracting and corrupting influence of his pals Lettvin and Selfridge,&#xA;  who were engaged in endless scientific and money-making schemes. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wiener almost broke ties when Pitts was supposed to return some documents back to him:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One comedy of errors nearly led to Wiener rupturing his connections&#xA;  with Pitts and the others. Pitts had been given a copy of a manuscript of&#xA;  Wiener’s to return to him, but he had checked it in the cloakroom in&#xA;  Grand Central Station in New York and had given the claim check to Jerry&#xA;  Lettvin—who, after a fortnight, gave it to Oliver Selfridge to pick up.&#xA;  Selfridge failed to find the case at the checkroom. Several months later,&#xA;  when Lettvin was next in New York, he inquired about the case, and eventually&#xA;  found that it had been sent to Chicago as unclaimed property,&#xA;  requiring several more months to be recovered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But, it appears that the final straw was&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Norbert Wiener made a sudden, irrevocable break with the others over an&#xA;  alleged slander involving one of the members of his family. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It appears that Norbet Wiener  had simply had enough, a decision that no doubt was influenced by his wife (and possibly other family members and other friend).&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="29" LastEditDate="2014-11-02T00:00:05.067" LastActivityDate="2014-11-02T00:00:05.067" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="204" PostTypeId="2" ParentId="184" CreationDate="2014-11-02T00:35:50.410" Score="3" Body="&lt;p&gt;I make this answer tentatively since I cannot find the document, an autobiographical chapter by Lettvin I read some years ago which discussed among other things this situation. Lettvin had a whole book available online, but searching I can't find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, Lettvin claimed that Pitts and McCulloch were not sufficiently sympathetic to socialism, so far Wiener was concerned. (Apparently Wiener, explains Lettvin, travelled with his wife to Mexico to visit Trotsky.) Lettvin wrote that the dislike Wiener's wife had for McCulloch was the last straw in a strained relationship, compounded by other misunderstandings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, McCulloch was not particularly bohemian in any way, so far as Alex Andrew, a younger member of the group Wiener split, recalls in an article in &lt;em&gt;Kybernetes&lt;/em&gt;. He and his wife were of very wealthy families, Andrew recalls, although their wealth was strongly decreased over time. Andrew does not say anything about any Bohemian behaviour of McCulloch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lettvin argued: Pitts was the only one in the group who was dependent on Wiener for funding and help in academia after Pitts and McCulloch left Nicholas Rashevsky at Chicago to work with Wiener. After Wiener broke off relations with the McCulloch group, Pitts was put under a lot of stress.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2014-11-02T00:35:50.410" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="205" PostTypeId="1" AcceptedAnswerId="206" CreationDate="2014-11-02T00:54:40.500" Score="48" ViewCount="3821" Body="&lt;p&gt;Göttingen was the place in which many important mathematicians such as Riemann worked. It was also one of the main locations for the development of quantum theory in the twenties (e.g. Heisenberg, Born worked there) What led to its decline, was it previously building up to it or was WW2 the only cause? Also when did it occur?&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="4251" LastEditDate="2021-09-13T14:25:55.503" LastActivityDate="2021-09-13T14:25:55.503" Title="What led to the fall of Göttingen?" Tags="|mathematics|theoretical-physics|nazi-germany|gottingen|germany|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="206" PostTypeId="2" ParentId="205" CreationDate="2014-11-02T01:18:44.717" Score="49" Body="&lt;p&gt;It's not exactly the &lt;em&gt;war&lt;/em&gt;, but the Nazi regime more generally that caused the decline of Göttingen. When the Nazis came to power in 1933, they started implementing antisemitic measures quite quickly. An important step in the Nazi policy was what is now dubbed &lt;a href=&quot;http://en.wikipedia.org/wiki/Law_for_the_Restoration_of_the_Professional_Civil_Service&quot; rel=&quot;noreferrer&quot;&gt;'the great purge of 1933'&lt;/a&gt;, which (basically) aimed to expel all Jews from positions in government or academics.&lt;/p&gt;&#xA;&lt;p&gt;The 'purge' caused a huge amount of influential scientists to leave Göttingen, mostly involuntarily (fired or fled the country), &lt;a href=&quot;http://www.mphpa.org/classic/HISTORY/H-02b.htm&quot; rel=&quot;noreferrer&quot;&gt;including&lt;/a&gt; Born, Noether, Wigner, von Neumann, Franck and many others. Born was even officially stripped of his PhD. degree. This turned out to be a decisive blow (something which applies more generally to science in Germany, which was really unrivaled in the early 20th century): The university never quite regained its former glory. One very telling example is given &lt;a href=&quot;http://books.google.de/books?id=9tUrarQYhKMC&amp;amp;pg=PA4540&amp;amp;lpg=PA4540&amp;amp;dq=gottingen+purge+1933&amp;amp;source=bl&amp;amp;ots=24AvXWdFdF&amp;amp;sig=UjmT-_RO-lIBh40mRHaZO12Pkwc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=sYVVVPvoLcH4OuSsgKgH&amp;amp;redir_esc=y#v=onepage&amp;amp;q=gottingen%20purge%201933&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;in this book&lt;/a&gt;. Between 1931 and 1939, the number of mathematics students in Germany declined from over 7000 to around 1250. The link that I provided to the Historical Encyclopedia of Natural and Mathematical Sciences also contains a very 'nice' anecdote, illustrating the damage that was done:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1934, David Hilbert was sitting next to the Nazis' newly appointed minister of education, at a banquet. When asked, &amp;quot;And how is mathematics in Göttingen, now that it has been freed of the Jewish influence?&amp;quot; Hilbert replied: &amp;quot;Mathematics in Göttingen, there is really no such thing any more.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="26" LastEditorUserId="643" LastEditDate="2021-03-09T22:16:53.223" LastActivityDate="2021-03-09T22:16:53.223" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="207" PostTypeId="1" CreationDate="2014-11-02T01:36:26.543" Score="12" ViewCount="560" Body="&lt;p&gt;If I recall correctly, Stephen Hawking mentioned in &lt;em&gt;A Brief History of Time&lt;/em&gt; that Newton remarked once that the famed apple actually &lt;strong&gt;distracted&lt;/strong&gt; him from his lucubrations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know where it was that Newton wrote (or said) the remark under consideration or, failing that, where it may have been that Hawking read (or heard) it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much for your learned replies...&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="26" LastEditDate="2014-12-17T12:26:32.280" LastActivityDate="2014-12-20T18:38:47.183" Title="Hawking on Newton and the famed apple" Tags="|biographical-details|debunking|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="208" PostTypeId="2" ParentId="207" CreationDate="2014-11-02T01:44:59.643" Score="14" Body="&lt;p&gt;Of course, the naive description as many heard it (which includes Newton having an apple fall on his head) is not true. There also does not exist any known source where Newton discusses anything about apples, and how they relate to his thoughts on gravitation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there are multiple secondary sources, providing accounts of a related &lt;a href=&quot;http://en.wikipedia.org/wiki/Isaac_Newton#Apple_incident&quot; rel=&quot;noreferrer&quot;&gt;'apple incident'&lt;/a&gt;, some event involving an apple falling down to Earth, which helped Newton in his thought process which would eventually lead him to the law of universal gravitation. It is, of course, difficult to investigate the veracity of such anecdotes, but it's the best I can do. The first account is by William Stukeley, who wrote the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;we went into the garden, &amp;amp; drank thea under the shade of some appletrees; only he, &amp;amp; my self. amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. &quot;why should that apple always descend perpendicularly to the ground,&quot; thought he to himself; occasion'd by the fall of an apple, as he sat in a contemplative mood. &quot;why should it not go sideways, or upwards? but constantly to the earths center? assuredly, the reason is, that the earth draws it. there must be a drawing power in matter. &amp;amp; the sum of the drawing power in the matter of the earth must be in the earths center, not in any side of the earth. therefore dos this apple fall perpendicularly, or toward the center. if matter thus draws matter; it must be in proportion of its quantity. therefore the apple draws the earth, as well as the earth draws the apple.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Newton's assistant John Conduitt also mentions an apple in a story about Newton:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the year 1666 he retired again from Cambridge to his mother in Lincolnshire. Whilst he was pensively meandering in a garden it came into his thought that the power of gravity (which brought an apple from a tree to the ground) was not limited to a certain distance from earth, but that this power must extend much further than was usually thought. Why not as high as the Moon said he to himself &amp;amp; if so, that must influence her motion &amp;amp; perhaps retain her in her orbit, whereupon he fell a calculating what would be the effect of that supposition but being absent from books &amp;amp; taking the common estimate in use among Geographers &amp;amp; our sea men before Norwood had measured the earth, that 60 English miles were contained in one degree of latitude his computation did not agree with his Theory &amp;amp; inclined him then to entertain a notion that together with the power of gravity there might be a mixture of that force which the moon would have if it was carried along in a vortex, but when the Tract of Picard of the measure of the earth came out shewing that a degree was about 69.5 English miles.  He began his calculation a new &amp;amp; found it perfectly agreable to his Theory&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt; 17-12-2014 in response to fdb's comment: The original sources for the quotes are found &lt;a href=&quot;http://books.google.co.uk/books?id=1xKFSqsDj0MC&amp;amp;pg=PT57#v=onepage&amp;amp;q=newton&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; (first quote, check out the page titled 'William Stukeley') and &lt;a href=&quot;http://www.newtonproject.sussex.ac.uk/view/texts/normalized/THEM00167&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; (second quote). Indeed, Wikipedia also reproduces these quotes, but that does not detract anything for my answer, in my opinion.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-17T16:01:09.553" LastActivityDate="2014-12-17T16:01:09.553" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="209" PostTypeId="2" ParentId="44" CreationDate="2014-11-02T01:59:35.807" Score="6" Body="&lt;p&gt;Polymaths today exist. For example, the (late) Clifford Truesdell, Roger Penrose, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fred Hoyle, Paul Dirac's student, wrote on everything from physics to science fiction, to economics, to astronomy. Eddington dabbled in philosophy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;William Clifford, although he died in his thirties, wrote on almost every subject. So did William Strutt, Lord Raleigh. So did James Hutton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Carl (von) Menger, the economist, father of the famous mathematician Karl Menger, had a library of over 30,000 books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Condillac wrote over forty volumes. So did Wolff. Cauchy was a master of everything and anything except economics and history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Waterston gave the modern kinetic theory of heat in a book about a neural network connectionist explanation of the brain (in 1840's!) prior to publishing in the Philosophical Journal and submitting his physics and thermodynamics work to other journals and presenting it to the royal society.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect the real issue is quite basic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today's physics requires substantial time to learn. But, then again, we have better tools. In ten pages, using fiber bundles and groups and modern methods of integration, one can discuss dynamical histories with more precision and detail than a thousand pages in the nineteenth century. It's not true that one cannot know, for example, physics if one is a (mathematical) biologist, or that a physicist cannot know biology and economics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We learn so much more, in depth and in breadth, and in terms of empirical knowledge, besides mathematical concepts. But our greater human capital makes the process much easier. We easily solve problems that would require months of correspondence and effort a hundred years earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare the enormous literature prior the 1930's, on special functions, made useless by advances in basic methods in analysis, including use of operator methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, although more is required to be known in each field, access to literature is much easier and faster than any time in the past, where one had to spend vast sums of money to obtain rare monographs several times a year, and this often by advance subscription or by chance purchase.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only about 60 copies of one of Euler's major calculus books was sold in his lifetime. Within fifty years, all continental mathematics was taught using his methods. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;No, the issue is elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) There is general lack of respect for a scientist, at least relative to the past in Western Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Truesdell wrote once, people who became scientists in the past gained tremendously in social &quot;rank&quot;, status, income, if they succeeded. This is no longer the case. Scientists were very rare, and interesting persons, with whom the nobility liked to meet. Recall how the king of England, George, invited and met with Lichtenberg, Gauss's teacher.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, several order of magnitude more people are scientists, engineers, and most of these are per se, as is statistically necessary, unexceptional individuals. So each one is less valuable to the public, unless the public can understand what exactly one can do that another cannot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) There are far more opportunities to do other things today than in the past, so FEWER people devote as MUCH of their time to study and writing exclusively, despite our population being much larger. In the past it was done partly to entertain oneself, today it is partly work, compared to other things one could be doing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider this: the foregone opportunities, the cost of spending AS MUCH TIME AS EULER on science, for example, is much greater today. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(So is the cost of having children by the way, since it reduces time that could be spent working or making use of all modern leisure goods, which is why people have three kids, not thirteen.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make their life interesting, the polymaths of the past, they sat around and read and read, and wrote, wrote, studied, studied, and corresponded, and sometimes, rarely, they also met. There was neither the television nor the internet nor rapid travel nor many stores nor even many restaurants, and social gatherings were in private homes or at the court. Few products existed. Few books were easily obtained. There were few industries willing to pay them to work on challenging problems with good pay. They filled their entire day in study. Of course they knew everything that was known, and were able to contribute something as well. They devoted their entire lives to knowledge for its own sake.Today, very few people are willing to do this, even inside one profession. It's too costly, unless you really like reading and writing.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2014-11-02T02:08:07.780" LastActivityDate="2014-11-02T02:08:07.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="210" PostTypeId="1" AcceptedAnswerId="234" CreationDate="2014-11-02T02:40:11.710" Score="13" ViewCount="475" Body="&lt;p&gt;In the note &quot;What St. Augustine didn't say about mathematicians&quot; (which appeared sometime in 1991 in the pages of &lt;a href=&quot;http://www.pme-math.org/journal/overview.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;the Pi Mu Epsilon Journal&lt;/em&gt;&lt;/a&gt;), R. P. Boas, Jr. mentioned, among other things, that in the days of Augustine of Hippo the word &quot;mathematicians&quot; meant what we nowadays call &quot;astrologers&quot;. Boas would even add the following: &quot;The old usage seems to have occurred occasionally as recently as the 1700's, although the modern meaning goes back to around 1400.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big, big question here is plain and simple to express:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How did R. P. Boas, Jr. find it all out?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among his principal references one can find Marvin J. Greenberg's book on &lt;em&gt;Euclidean and Non-Euclidean Geometries&lt;/em&gt; but, to be a hundred percent honest, I don't think that Mr. Greenberg touched upon the etymology of the word &quot;mathematicians&quot; at all in his book (of course, I'm always more than glad to be proven wrong).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, the said article also appears in the (hilarious) book &lt;em&gt;Lion hunting and other mathematical pursuits (A collection of mathematics, verse, and stories by Ralph P. Boas, Jr.)&lt;/em&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me thank you in advance for your insightful replies.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2014-11-03T18:43:13.380" LastActivityDate="2017-06-08T20:31:53.037" Title="Question on &quot;What St. Augustine didn't say about mathematicians&quot;" Tags="|terminology|debunking|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="211" PostTypeId="1" AcceptedAnswerId="337" CreationDate="2014-11-02T03:47:28.727" Score="10" ViewCount="1722" Body="&lt;p&gt;It is pretty common knowledge nowadays that prolonged exposure to solar ultraviolet radiation is dangerous to human skin, potentially resulting in cancers such as melanoma and photoaging.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A bit of background from the Skin Cancer Foundation's webpage &lt;a href=&quot;http://www.skincancer.org/prevention/uva-and-uvb/understanding-uva-and-uvb&quot; rel=&quot;nofollow noreferrer&quot;&gt;Understanding UVA and UVB&lt;/a&gt;, the UV radiation reaching the Earth's surface is generally considered to go from a wavelength of 290nm (UVB), and the majority is UVA, as per the diagram (from the link above):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/naowB.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both UVA and UVB affect the skin, with UVA penetrating further, as per the diagram below (from the link above):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/dVGpB.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There has also been research in the dangers of tanning booths that often use UVA, for example the CDC web-article &lt;a href=&quot;http://www.cdc.gov/cancer/skin/basic_info/indoor_tanning.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indoor Tanning Is Not Safe&lt;/a&gt; and the article &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.3322/caac.21248/full&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indoor tanning increases melanoma risk, even in the absence of a sunburn&lt;/a&gt; (Barton, 2014) among many other articles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When and how was it confirmed that ultraviolet radiation can be dangerous to human skin?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-09T16:18:55.910" Title="When and how was UV radiation found to be dangerous to human skin?" Tags="|physics|biology|medicine|radiation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="212" PostTypeId="1" CreationDate="2014-11-02T04:06:26.780" Score="5" ViewCount="176" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Hideyo_Noguchi#Death&quot; rel=&quot;nofollow&quot;&gt;On the Wikipedia page on Hideyo Noguchi&lt;/a&gt;, one can find the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Possibly his erratic and irresponsible behaviour was caused by the&#xA;  untreated syphilis with which he was diagnosed in 1913, and which may&#xA;  have progressed to neurosyphilis. In any case, he was careless not&#xA;  only of his own safety, but also of the safety of those around him,&#xA;  and despite repeated promises to Young, failed to keep infected&#xA;  mosquitoes in the specially designed secure housing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there any study confirming or refuting that Hideyo Noguchi had neurosyphilis?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2014-11-02T10:01:05.893" LastActivityDate="2016-07-29T10:13:57.470" Title="Is it true that Hideyo Noguchi had neurosyphilis when carrying out his last experiments in Africa?" Tags="|biographical-details|death|medicine|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="213" PostTypeId="2" ParentId="151" CreationDate="2014-11-02T04:27:13.500" Score="5" Body="&lt;p&gt;The article &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_groups.html&quot; rel=&quot;nofollow&quot;&gt;The Abstract Group Concept&lt;/a&gt;, from the McTutor archive, gives an accounting of the steps towards the modern abstract definition. In brief, Cayley made the first stumbling attempts (citing the associative law explicitly) in an 1854 paper, but not until 1895 did Weber give the modern definition, in his &lt;em&gt;Lehrbuch der Algebra&lt;/em&gt;. Weber included infinite groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the original question: other than the Lagrange theorem you mention, I'm not aware of any cases where the abstract definition unified previous separate results. The abstract definition does not seem to have been motivated by this desire. However, it is true that Lie groups were directly inspired by Galois's permutation groups, and Lie's desire to develop a theory for differential equations analogous to Galois theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also seems plausible that Weber, writing a comprehensive text on Algebra, saw the possibility of uniting disparate notions. But that's just a conjecture on my part. &lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-02T04:52:38.767" LastActivityDate="2014-11-02T04:52:38.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="216" PostTypeId="1" CreationDate="2014-11-02T13:59:38.800" Score="15" ViewCount="847" Body="&lt;p&gt;One famous quote I've heard of &lt;a href=&quot;https://en.wikipedia.org/wiki/Wolfgang_Pauli&quot;&gt;Wolfgang Pauli&lt;/a&gt; is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is not even wrong.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've also heard that Pauli had a blustery, often unpleasant personality, and I vaguely recall hearing &lt;a href=&quot;https://en.wikipedia.org/wiki/Not_even_wrong&quot;&gt;the phrase&lt;/a&gt; used in multiple instances. In which instances did Pauli use this phrase?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="26" LastEditDate="2014-11-09T12:41:26.327" LastActivityDate="2018-04-23T17:15:14.437" Title="In what instances did Wolfgang Pauli say that something was 'not even wrong'?" Tags="|biographical-details|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="217" PostTypeId="2" ParentId="83" CreationDate="2014-11-02T14:12:54.047" Score="21" Body="&lt;p&gt;All theories of matter, starting with the ancient Greek philosophers, can be classified as either continuous or discrete (i.e., particulate). This dichotomy is due to Aristotle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle held that matter was continuous: infinitely divisible. Aristotle believed that a vacuum was impossible (indeed, he claimed to prove this). Since a particulate theory requires a vacuum between the particles, that ruled out atomism for Aristotle and his followers. The Stanford Encyclopedia of Philosophy is a good starting point for Aristotle's philosophy of matter; here, the sections on &lt;a href=&quot;http://plato.stanford.edu/entries/aristotle/#Hyl&quot;&gt;Hylomorphism&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/aristotle/#Sub&quot;&gt;Substance&lt;/a&gt; are most relevant. Aristotle's thought is, as usual, complicated: he did allow that for some sorts of substance (e.g., blood, bone), there was a smallest amount of it that would retain the character of the bulk substance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Atomism goes back to the ancient Greeks, Leucippus and his student Democritus being given credit. These two currents of thought persist, as you say, right up to the 20th C. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo in a lengthy passage in the &lt;em&gt;Dialog on the Two Chief World Systems&lt;/em&gt; discusses material vs. abstract shapes, e.g., a mathematical sphere vs. a sphere made of bronze. Do the theorems of geometry hold for material objects? Galileo (through his mouthpiece Salviati) says yes. It is true that the bronze sphere is probably not perfect, but it does correspond perfectly to some mathematical shape. Moreover, there is no reason in principle why we couldn't have a bronze sphere that &lt;em&gt;did&lt;/em&gt; match the mathematical sphere perfectly. Clearly these opinions are incompatible with the atomic view of matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theories of bulk matter, like fluid dynamics or solid mechanics (jointly called continuum mechanics), use continuous models of matter. Nowadays these are regarded as convenient fictions. I am not sure of the views of the pioneers here (Hooke, Euler, Ricatti, Young). Galileo's work on how strength depends on size is well-known, and as noted he was not an atomist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Skipping to the 19th C.: the chapter &quot;The Reality of Molecules&quot; in Pais's Einstein biography &lt;em&gt;Subtle is the Lord...&lt;/em&gt; covers the ground succinctly. On the side of atomism, Dalton's law of multiple proportions and Gay-Lussac's law of combining volumes both seem like powerful arguments for atomism. Prout's hypothesis (that all atomic weights are multiples of the atomic weight of hydrogen) also seems firmly atomistic. But Pais writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Yet Prout did not consider his hypothesis as a hint for the reality of atoms. &quot;The light in which I have always been accustomed to consider it [the atomic hypothesis] has been ... as a conventional artifice, exceedingly convenient for many purposes but which does not represent nature.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To quote Pais again: &quot;The principle point of debate among chemists was whether atoms were real objects or only mnemonic devices for coding chemical regularities and laws.&quot; In other words, does the atomic hypothesis tell us anything new, beyond what we can already deduce directly from the laws of Dalton and Gay-Lussac?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among the physicists, the argument centered around the kinetic theory of gases. Mach and Ostwald were the most famous opponents of atomism. At an address Ostwald gave in 1895, he attacked atomism with an argument already made by Loschmidt twenty years earlier: at the microscopic level all known laws of physics are time-reversible, yet at the macroscopic level we have entropy and obvious irreversibility. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You ask about scientific advantages of continuum theories over atomism. We should careful not to import modern ideas of scientific evidence wholesale into early periods (a sin historians call presentism or whiggism). What we dismiss as metaphysical or maybe linguistic arguments held great weight throughout most of the history of science. Parmenides's argument against the vacuum -- to speak of a thing, one has to speak of a thing that exists -- was persuasive to many. While Aristotle rejected Parmenides's argument, he had his own philosophical &quot;proofs&quot; of the impossibility of a void, based on his theories of motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A metaphysical argument more palatable to modern ears is Occam's razor: &quot;entities should not be multiplied without necessity&quot;. In other words, don't hypothesize the existence of something unless there are testable consequences. This is the argument used in relativity theory against absolute space, and in quantum mechanics against classical trajectories for particles. Mach and many chemists felt that the atomic hypothesis provided nothing beyond the regularities (like the laws of Dalton and Gay-Lussac) used as arguments for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kuhn pointed out that highly articulated, unified theories always have an advantage over less developed theories with multiple variants. In the 19th C. the &quot;atomists&quot; by no means presented a unified front. Were atoms divisible or not? What exactly was the difference between an atom and a molecule? Chemists spoke of the distinction between chemical and physical molecules, with no consensus on what the difference was or if there even was one. (Nowadays we say no difference.) The work of Maxwell and Boltzmann on statistical mechanics seemed merely to reproduce, with great mathematical difficulty, results obtained easily with classical thermodynamics, a continuous theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, many physicists (like Planck for many years) regarded the second law of thermodynamics as absolutely and not merely statistically true. I've already noted Loschmidt's devastating argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will finish with a tongue-in-cheek remark. If string theory holds, does that perhaps mean that the continuum crowd was right all along?&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-02T14:12:54.047" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="218" PostTypeId="2" ParentId="199" CreationDate="2014-11-02T14:48:16.907" Score="22" Body="&lt;p&gt;On the other hand, consider this quote from &lt;em&gt;&lt;a href=&quot;http://www.feynmanlectures.caltech.edu/I_40.html#Ch40-S6&quot;&gt;The Feynman Lectures on Physics&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[In 1869], in a lecture, [Maxwell] said, “I have now put before you what I consider to be the greatest difficulty yet encountered by the molecular theory.” These words represent the first discovery that the laws of classical physics were wrong. This was the first indication that there was something fundamentally impossible, because a rigorously proved theorem did not agree with experiment. About 1905, Sir James Hopwood Jeans and Lord Rayleigh (John William Strutt) were to talk about this puzzle again. One often hears it said that physicists at the latter part of the nineteenth century thought they knew all the significant physical laws and that all they had to do was to calculate more decimal places. Someone may have said that once, and others copied it. But a thorough reading of the literature of the time shows they were all worrying about something. Jeans said about this puzzle that it is a very mysterious phenomenon, and it seems as though as the temperature falls, certain kinds of motions “freeze out.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-02T14:48:16.907" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="219" PostTypeId="1" CreationDate="2014-11-02T14:49:36.557" Score="16" ViewCount="404" Body="&lt;p&gt;&lt;em&gt;Cooper, S. M., and R. P. R. Dawber. &quot;&lt;a href=&quot;http://scholar.google.com/scholar?cluster=1180038322841065357&amp;amp;hl=en&amp;amp;as_sdt=0,22&quot;&gt;The history of cryosurgery.&lt;/a&gt;&quot; Journal of the royal society of medicine 94.4 (2001): 196-201.&lt;/em&gt; claims:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The benefits of cold have been appreciated for many thousands of&#xA;  years. The ancient Egyptians, and later Hippocrates, were aware of the&#xA;  analgesic and anti-inflammatory properties of cold.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What cold object/water/etc. did ancient Egyptians use for analgesic and anti-inflammatory purpose? I am surprised they could have objects that were cold enough for this use.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2014-11-02T15:15:24.037" LastActivityDate="2014-11-14T16:26:05.270" Title="How did ancient Egyptians become aware of the analgesic and anti-inflammatory properties of cold?" Tags="|medicine|ancient-egypt|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="220" PostTypeId="2" ParentId="26" CreationDate="2014-11-02T16:44:12.403" Score="30" Body="&lt;p&gt;By way of throat-clearing, what is Occam's razor? &lt;a href=&quot;http://math.ucr.edu/home/baez/physics/General/occam.html&quot; rel=&quot;noreferrer&quot;&gt;John Baez&lt;/a&gt; has a useful essay giving the history and some examples. William of Ockham's original formulation was&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Entities should not be multiplied unnecessarily.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words, don't assume the existence of something unless there is good evidence for it. Again quoting Baez, &quot;In physics we use the razor to shave away metaphysical concepts.&quot; The canonical example is discarding the ether. Newton's absolute time and space, mechanical explanations for gravity, and classical trajectories for particles all have felt the edge of the razor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Baez also mentions a famous failure of this version of Occam's razor:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mach and his followers claimed that molecules were metaphysical because they were too small to detect directly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mach's point is that the molecular hypothesis is just unnecessary decoration layered on top of empirical regularities (laws of Dalton and Gay-Lussac in chemistry, Boyle's law) that work just fine without the added ornaments. We have (or so Mach would claim) an analogy:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;ether : relativity = molecules : (chemistry+physics)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Occam's razor is often strengthened to the &lt;strong&gt;rule of simplicity&lt;/strong&gt;: in one formulation (taken from Baez's essay),&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The simplest explanation for some phenomenon is more likely to be accurate than more complicated explanations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Often people say Occam's razor when they really mean the rule of simplicity. The obvious problem with the rule of simplicity is its subjectivity. A sterling example is the heliocentric hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To Copernicans in the 16th C. (Galileo, Kepler, a few others) heliocentricity was clearly simpler. In this period, the contest was between true heliocentricity and so-called geoheliocentric hybrids: the planets revolve around the sun which revolves around the earth. (Tycho's geoheliocentric system was the most famous, but not the only one.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To modern eyes, heliocentricity looks obviously simpler. But advocates of geoheliocentricity deployed two powerful arguments from simplicity.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Heliocentricity was inconsistent with physics as then understood. Kepler responded by inventing his own celestial physics, with three different forces guiding each planet, &lt;em&gt;plus&lt;/em&gt; the force of gravity, which had nothing to do with the planet's orbit.&lt;/li&gt;&#xA;&lt;li&gt;The lack of detectable stellar parallax implied enormously greater distances to the fixed stars than with a geoheliocentric theory. The apparent sizes of stellar disks (an artifact of optics, not understood at the time) then implies that &lt;em&gt;every other star&lt;/em&gt; is &lt;em&gt;way&lt;/em&gt; bigger than the sun. Tycho first put forward this argument, quite convincing to many of his contemporaries. (See &lt;a href=&quot;http://arxiv.org/abs/1112.1988&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; by Chris Graney for more details.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Simplicity is not simple.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2016-04-24T11:53:18.113" LastActivityDate="2016-04-24T11:53:18.113" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="221" PostTypeId="1" AcceptedAnswerId="223" CreationDate="2014-11-02T16:47:33.120" Score="8" ViewCount="84" Body="&lt;p&gt;Ophiolites are distinctive sequences of rock, typically capped by sediments followed by basic rocks (pillow lavas, sheeted dykes, and gabbro) followed by ultramafics (eg. dunite).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the exact emplacement process has evolved over time, these are generally recognized as pieces of ocean crust which have been &quot;obducted&quot; onto continents. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How and when was it realized that ophiolites could be explained as pieces of ocean crust?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="84" LastEditDate="2014-11-26T11:13:30.823" LastActivityDate="2014-11-26T11:13:30.823" Title="How and when was it realized that ophiolites represented pieces of ocean crust?" Tags="|earth-sciences|plate-tectonics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="222" PostTypeId="1" AcceptedAnswerId="233" CreationDate="2014-11-02T17:09:59.343" Score="10" ViewCount="2527" Body="&lt;p&gt;The terms Primary, Secondary, Tertiary, and Quaternary were coined by early geologists as they tried to make sense of the rocks around them. As they began to understand what was going on, they found &quot;Primary&quot; and &quot;Secondary&quot; were unhelpful terms, but &quot;Tertiary&quot; and &quot;Quaternary&quot; were more useful and evolved into terms for geological periods within the Cenozoic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The International Commission on Stratigraphy has chosen to drop &quot;Tertiary&quot; and replace it with the Paleogene &amp;amp; Neogene; but keep the Quaternary. Why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a terrestrial point of view, I can see the value of the Quaternary - it represents a period of polar ice caps and clear climate variations according to Milankovitch cycles. But why was the Tertiary considered less useful? What was Paleogene&amp;amp;Neogene considered more meaningful as a complete replacement for the Tertiary (rather than in addition to)?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="84" LastEditDate="2014-11-26T11:13:53.370" LastActivityDate="2014-11-26T11:13:53.370" Title="Why was the Tertiary period dropped but the Quaternary retained?" Tags="|earth-sciences|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="223" PostTypeId="2" ParentId="221" CreationDate="2014-11-02T17:15:33.620" Score="6" Body="&lt;p&gt;The first place I looked, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ophiolite&quot;&gt;Wikipedia&lt;/a&gt;, gave a fairly good answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term dates back to the 19th century, but it first came it to use in the current context only about 50 years ago, when Frederick Vine and Drummond Matthews used it&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;to represent the formation of new crust at the oceanic ridge and its subsequent symmetric spreading away from that ridge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This was inspired by the discovery of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;magnetic anomaly stripes on the seafloor&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words, odd bits under the sea that shouldn't have been where they were, or with the same characteristics as they had, pertaining to magnetic fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More key work was done by Ian Graham Gass et. al., who found a piece of an ophiolite that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;must have formed by repetitive extension of crust and intrusion of magma, resulting in a formation consisting of 100% dikes with no older wall rocks preserved within the complex.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-02T17:15:33.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="224" PostTypeId="2" ParentId="222" CreationDate="2014-11-02T18:25:54.057" Score="4" Body="&lt;p&gt;It appears to be part of the reason the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tertiary&quot; rel=&quot;nofollow&quot;&gt;Tertiary Period&lt;/a&gt; was dropped. Looking at the Wikipedia page on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Paleogene&quot; rel=&quot;nofollow&quot;&gt;Paleogene&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By dividing the Tertiary Period into two periods instead of directly into five epochs, the periods are more closely comparable to the duration of 'periods' in the Mesozoic and Paleozoic Eras.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The page on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Neogene&quot; rel=&quot;nofollow&quot;&gt;Neogene&lt;/a&gt; says much the same thing:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By dividing the Cenozoic Era into three (arguably two) periods (Paleogene, Neogene, Quaternary) instead of 7 epochs, the periods are more closely comparable to the duration of periods in the Mesozoic and Paleozoic eras.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-02T18:25:54.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="225" PostTypeId="2" ParentId="59" CreationDate="2014-11-02T21:13:09.850" Score="10" Body="&lt;p&gt;One answer to your question could be that the separation actually came pretty late. Wikipedia claims that &quot;Earlier authors held the result to be intuitively obvious, and requiring no proof.&quot;, so until continuity was formalized by Bolzano and Cauchy, I believe it doesn't make sense to find evidence. So we need to look for people who read Bolzano or Cauchy, and believed that intermediate value property is equivalent to continuity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you already stated in your question, Darboux showed in 1875 you could verify the theorem without being continuous. This leaves a small window - 1817-1875 - to find for published absurdities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here comes &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1875_2_4_/ASENS_1875_2_4__57_0/ASENS_1875_2_4__57_0.pdf&quot;&gt;Darboux himself&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;La propriété précédente a souvent étée prise pour la définition des fonctions continues&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which translates :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The aforementioned proposition was often confused with the definition of continuous fonction&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So this answers your second question: if no prior evidence can be found, &lt;strong&gt;Darboux himself made the claim that the error was widespread&lt;/strong&gt; before his own work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the introduction of the same memoire, Darboux states that &lt;strong&gt;M. Hankel's work of 1870 regarding Riemann's memoire&lt;/strong&gt; where not beyond any reproach, but it's not clear whether he is talking about the existency of a derivative to all functions or the intermediate value theorem at this point. I believe one willing to find evidence of a confusion could look into M. Hankel's work, but I couldn't find the paper which Darboux describes.&lt;/p&gt;&#xA;" OwnerUserId="8" LastActivityDate="2014-11-02T21:13:09.850" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="226" PostTypeId="2" ParentId="216" CreationDate="2014-11-02T21:46:04.567" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;This isn’t right. This isn’t even wrong.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Response after reading a colleague’s paper, quoted in &lt;em&gt;The Successful Toastmaster: A Treasure Chest of Introductions, Epigrams, Humor, and Quotations&lt;/em&gt; (1966) by Herbert Victor Prochnow, p. 350, and in &lt;em&gt;Mathematical Apocrypha Redux: More Stories and Anecdotes of Mathematicians and the Mathematical&lt;/em&gt; (2005) by Steven George Krantz, p. 194.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This paper is so bad it is not even wrong.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As quoted in &lt;em&gt;Comic Sections: The Book of Mathematical Jokes, Humour, Wit, and Wisdom&lt;/em&gt; (1993) by Des MacHale&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taken from &lt;a href=&quot;http://en.wikiquote.org/wiki/Wolfgang_Pauli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikiquote page of Pauli&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="102" LastEditorUserId="65" LastEditDate="2018-04-23T09:39:34.653" LastActivityDate="2018-04-23T09:39:34.653" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="227" PostTypeId="5" CreationDate="2014-11-02T21:55:52.610" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-02T21:55:52.610" LastActivityDate="2014-11-02T21:55:52.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="228" PostTypeId="4" CreationDate="2014-11-02T21:55:52.610" Score="0" Body="Nobel Prizes in Sciences are awarded every year to at the most three distinguished scientists in the fields of Chemistry, Physics and Physiology each. There are also separate Nobel Prizes for Literature, Economics and Peace." OwnerUserId="102" LastEditorUserId="102" LastEditDate="2014-11-03T13:56:32.137" LastActivityDate="2014-11-03T13:56:32.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="229" PostTypeId="2" ParentId="210" CreationDate="2014-11-03T00:42:24.280" Score="4" Body="&lt;p&gt;I knew I'd already heard this story that astrologers used to be called mathematicians, and now I've found where. Morris Kline mentions it in his celebrated &lt;em&gt;Mathematical Thought from Ancient to Modern Times&lt;/em&gt;, the excerpt can be read &lt;a href=&quot;http://books.google.fr/books?id=Lei2wjJyWEAC&amp;amp;lpg=PA179&amp;amp;ots=nvc82q9MA6&amp;amp;dq=mathematics%20romans%20astrology&amp;amp;pg=PA179#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; on page 179. You might find more details in the (short) bibliography a few pages further down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm afraid I don't have any references for specific examples of this term being used by anyone, let alone as recently as the eighteenth century. All I can offer you is the above non-constructive existence proof.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2014-11-03T00:42:24.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="230" PostTypeId="2" ParentId="199" CreationDate="2014-11-03T02:44:39.513" Score="13" Body="&lt;p&gt;See, for instance, A. A. Michelson's claim in the second lecture of his 1899 (pub. 1903) lectures &lt;em&gt;Light Waves and Their Uses&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before entering into these details, however, it may be well&#xA;  to reply to the very natural question: What would be the&#xA;  use of such extreme refinement in the science of measurement? Very briefly and in general terms the answer would&#xA;  be that in this direction the greater part of all future discovery must lie. The more important fundamental laws and&#xA;  facts of physical science have all been discovered, and these&#xA;  are now so firmly established that the possibility of their ever&#xA;  being supplanted in consequence of new discoveries is exceedingly remote. Nevertheless, it has been found that there&#xA;  are apparent exceptions to most of these laws, and this is&#xA;  particularly true when the observations are pushed to a limit,&#xA;  i. e., whenever the circumstances of experiment are such that&#xA;  extreme cases can be examined. Such examination almost&#xA;  surely leads, not to the overthrow of the law, but to the discovery of other facts and laws whose action produces the&#xA;  apparent exceptions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is unclear to me whether this was merely an unpopular, optimistic bluster, or whether this quote represents an entire school of thought at the time. Especially troublesome to me is Michelson's involvement in the famous 1880s experiments on light and ether... surely the conflicting results gave him some pause?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, note his last two sentences above - Michelson is clearly not positing that all of physics is &quot;done&quot; per se. Instead, his view seems to be that &lt;strong&gt;&quot;the more important fundamental laws&quot; are correct, modulo certain small improvements that can still be made&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps it is this bolded statement that represented the general attitude of the time, and not the statement that physics was &quot;finished.&quot; Of course, while this quote does not conclusively answer your first question, it does provide another data point towards the second.&lt;/p&gt;&#xA;" OwnerUserId="23" LastEditorUserId="23" LastEditDate="2014-11-03T16:04:30.323" LastActivityDate="2014-11-03T16:04:30.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="231" PostTypeId="2" ParentId="96" CreationDate="2014-11-03T04:14:21.420" Score="8" Body="&lt;p&gt;Fair warning: this answer doesn't completely answer the question, but I think it may answer the question as well as it is possible to do.&lt;/p&gt;&#xA;&lt;p&gt;The article which Hankel wrote (published 1971) that is generally credited with &amp;quot;rediscovering&amp;quot; Bolzano's work was an article in section 1 Theil 90 (Gregorius - Grezin) of &lt;a href=&quot;http://en.wikipedia.org/wiki/Allgemeine_Encyclop%C3%A4die_der_Wissenschaften_und_K%C3%BCnste&quot; rel=&quot;noreferrer&quot;&gt;Allgemeine Encyclopädie der Wissenschaften und Künste&lt;/a&gt;, one of the largest encyclopedias ever written (spanning 167 volumes despite being uncompleted). You can read this section freely &lt;a href=&quot;http://books.google.com/books?id=SUpYAAAAYAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;here on Google Books&lt;/a&gt;. Hankel's article is on &amp;quot;Grenze&amp;quot; (&amp;quot;limits&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;The relevant paragraph concerning Bolzano is on pages 209-210, while discussing the history of analysis. I'll provide a rough translation in modern English of this paragraph. I should note that I know next-to no German, so the translation is a lot of guesswork, and I could be wrong in some places. Anyone who knows German should feel free to note errors.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Even worse was another contemporary who remained then and now almost entirely unknown among mathematicians: We have to reclaim the priority of the first rigorous development in the series of algebraic analysis in favor of the excellent Bernhard Bolzano. Bolzano's notions of the convergence of the series are quite clearly and correctly written, its operations with infinite series all strictly proved, and nothing is wrong with the development of those statements for real arguments, which he supposes everywhere. In the preface he gives an apt criticism of the previous derivations of the Binominal theorem and then of ordinary unrestricted use of infinite series. In short, this work was not only a French art, he should be placed in this respect on the same level with Cauchy, and stated his thoughts in a pleasant way. But Bolzano remained unknown and was soon forgotten; Cauchy was the lucky one, the one praised as a reformer of science and whose elegant writings in a short time found general dissemination.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this paragraph, Hankel basically credits Bolzano with developing much of the foundations of analysis independently of (and years before) Cauchy. However, Bolzano's work remained unknown, while Cauchy, who was well-connected in French math circles, found it easy to communicate his work. Hankel makes no mention of where or how he found Bolzano's work.&lt;/p&gt;&#xA;&lt;p&gt;Some historical commentary is in order here. 1871 is a significant year; specifically, it's the year in which the &lt;a href=&quot;http://en.wikipedia.org/wiki/Franco-Prussian_War&quot; rel=&quot;noreferrer&quot;&gt;Franco-Prussian war&lt;/a&gt;, a time of strong national pride in Germany and general dislike of all things French. The encyclopedia Hankel was writing in was intended to be something of an encyclopedia &amp;quot;for and by the German people&amp;quot;. Hankel surely would not have been happy having to give the credit of developing analysis to Cauchy, a Frenchman. It was far better to give it to Bolzano. Sure, Bolzano wasn't the ideal German mathematician, having spent most of his academic career in Austria, and being as much a philosopher and theologian as a mathematician (and a controversial one at that), but he did speak and write in German, and equally importantly, was not French. And Bolzano did really do (for the most part) the things Hankel attributed to him. To be clear, I'm not accusing Hankel of any wrongdoing by pointing this out, only saying that he had considerable interest in attributing as much as he could to Bolzano.&lt;/p&gt;&#xA;&lt;p&gt;However, there is something of an issue attributing the development of limits in analysis to Bolzano over Cauchy, though it is more philosophical than mathematical. Bolzano probably had a very different interpretation of his theorems than later readers. Indeed, in &amp;quot;The Mathematical Works of Bernard Bolzano&amp;quot;, Steve Russ argues that Bolzano would not have thought of his theorems in terms of limits at all, which he would have associated with the very infinities he was trying to do away with. From pages 146-147:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, the modern recognition of Bolzano's work raises a historical problem. From Hankel's article in 1871 to the extracts in Bitkhoff (1973) commentators have been inclined to give particular credit to Bolzano for matters which at the time he saw in a very different light from these later critics. We are thinking here of the arithmetic concept of limit and the concept of the convergence of infinite series that are commonly adopted today. These concepts had been used in some form for a long time, and judging from other examples in his writings, Bolzano would not have been too modest to claim them as new and original if he had regarded them as so. He does not do so. Undoubtedly he had great confidence in these definitions; they satisfied his conceptual requirements, he knew they would be fruitful and effective in the development of analysis, but never does he claim them to be his own...&lt;/p&gt;&#xA;&lt;p&gt;It is commonly assumed  that following the introduction of quantities labelled as ω, or Ω, possibly with subscripts, there is outlined in &lt;em&gt;BL&lt;/em&gt; §14 ff. a fairly standard theory of limits. The irony is that Bolzano, along with most of his contemporaries, would have associated limits with infinite processes (or infinitely small quantities). And so he would, at this time, have been horrified to be associated with such a theory. Similar remarks apply to his work on the convergence of series. He believed he was treating the binomial series for negative and rational exponents in a purely finite manner. The way in which he use his ω quantities—variable quantities that can become smaller than any given quantity, or that can become as small as we please, naturally appealed to an infinite range of values. We might call them 'arbitrarily small quantities' . Rusnock suggests that such a concept of a variable that can become as small as desired was common at the time. It is some sort of counterpart to a physical variable quantity. He suggests that Bolzano's ω's might be interpreted as &lt;em&gt;ranges&lt;/em&gt; of values containing zero...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is to say that Hankel's judgement of Bolzano as an independent discoverer of the rigorous theory of limits in analysis, while correct in terms of &lt;em&gt;mathematical&lt;/em&gt; content, is surely false if we take into account the philosophical aspects of his work. But of course even if Hankel realized this, he had little to gain by pointing it out explicitly in his article. In any case, neither Bolzano's methods nor his theorems were any less rigorous than Cauchy's; just his interpretation of the definitions and content of the theorems was different.&lt;/p&gt;&#xA;&lt;p&gt;In any case, you'll note that Hankel didn't mention Bolzano-Weierstrass specifically, nor the intermediate value theorem (which was Bolzano's ultimate goal, toward which Bolzano-Weierstrass was just a lemma). That isn't terribly surprising. While Hankel was likely aware of Weierstrass' result (they were well acquainted with each other, Hankel having worked with Weierstrass in Berlin in 1861 prior to his doctorate) it was probably too recent to appreciate its significance, especially in the context of this sort of publication. It isn't even clear that Hankel had read the parts of Bolzano's work related to the intermediate value theorem; the parts which he cites in the article are elsewhere. So it wasn't actually Hankel who established Bolzano's priority here.&lt;/p&gt;&#xA;&lt;p&gt;After Hankel's original citation, some mathematicians went back and read Bolzano's various works, reinterpreting them in more modern language. &lt;a href=&quot;http://en.wikipedia.org/wiki/Otto_Stolz&quot; rel=&quot;noreferrer&quot;&gt;Otto Stolz&lt;/a&gt; in particular is credited with rediscovering and republishing many of his mathematical works in 1881. This included the relevant article, &lt;em&gt;Bedeutung in der Geschichte der Infinitesimalrechnung&lt;/em&gt;, which predates Weierstrass and even Cauchy, establishing Bolzano's priority for the intermediate value theorem and the Bolzano-Weierstrass theorem. A number of other influential German mathematicians and philosophers also read Bolzano's works, and a few other interesting mathematical results were found. His legacy was likely cemented in the various historical notes in the highly influential (at least at Göttingen) &lt;a href=&quot;http://en.wikipedia.org/wiki/Klein%27s_encyclopedia&quot; rel=&quot;noreferrer&quot;&gt;Enzyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen&lt;/a&gt;, which mention his work several times as being far ahead of his time.&lt;/p&gt;&#xA;&lt;p&gt;That answers the question on Bolzano-Weierstrass, but there's still a lingering unanswered question, namely, how did Hankel even find Bolzano in the first place (which was a big part of your original question). I don't  know the answer to that, and to the best of my knowledge, no one does. Perhaps he had some concept that there was an Eastern European school of analytic philosophy which in the early 19th century dealt with issues related to infinities and was unhappy with Leibniz's informal approach to calculus. Or maybe, in writing the article, he got into a conversation with someone (possibly someone who was familiar with Bolzano's work from the period in the 1820s mentioned in the article you quoted, or possibly not even a mathematician) who suggested he look in that direction. Hankel did a decent amount of study on the history of mathematics (though his historical works typically had notable errors), also pointing out the importance of the work of &lt;a href=&quot;http://en.wikipedia.org/wiki/Hermann_Grassmann&quot; rel=&quot;noreferrer&quot;&gt;Hermann Grassmann&lt;/a&gt; in 1867 two decades after Grassman had essentially stopped doing mathematics, so he certainly had some broader understanding of the works of his predecessors than the average mathematician of his time. How exactly Hankel found Bolzano is anyone's guess, but once he did, it's pretty clear that he wasn't just going to ignore it in his article, regardless of what Bolzano did/didn't think about how to interpret his results. Hankel died in 1873, just 2 years after the publication of the article, and to the best of my knowledge he didn't ever comment on Bolzano's work again. While one might be able to track the movements of various mathematicians from 1817 to 1871 to try to figure out how the idea might have been transmitted to Hankel (a seemingly Herculean task, though not technically impossible), at best we'd end up with a guess, and the truth is quite possibly lost to history.&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-03T04:14:21.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="232" PostTypeId="1" AcceptedAnswerId="407" CreationDate="2014-11-03T06:46:22.303" Score="16" ViewCount="903" Body="&lt;p&gt;Asthma has been known since Ancient Egypt, according to the Medical News Today web-article &lt;a href=&quot;http://www.medicalnewstoday.com/info/asthma/asthma-history.php&quot;&gt;Asthma History - Through The Ages&lt;/a&gt;, with the first use of the word by Hippocrates in Ancient Greece.  Throughout most of history from Ancient Egypt and Greece, asthma was treated as a physical condition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in the 16th and 17th centuries, asthma was identified as being a pulmonary condition and that dust and exercise could be triggers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the beginning of the 20th century asthma was seen as a psychosomatic disease - an approach that probably undermined any medical breakthroughs at the time. During the 1930s to 1950s, asthma was known as one of the &lt;strong&gt;holy seven psychosomatic illnesses&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Treatment often involved &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;psychoanalysis and other 'talking cures'. A child's wheeze was seen as&#xA;  a suppressed cry for his or her mother. Psychoanalysts thought that&#xA;  patients with asthma should be treated for depression.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These ideas were eventually thrown out, but the question is &lt;strong&gt;why was asthma considered a psychosomatic condition during the first half of the 20th century?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-12T14:26:44.290" Title="Why was asthma considered a psychosomatic condition during the first half of the 20th century?" Tags="|medicine|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="233" PostTypeId="2" ParentId="222" CreationDate="2014-11-03T07:29:29.993" Score="8" Body="&lt;p&gt;What happened is that Tertiary/Quaternary were dropped first, together, by the International Commission on Stratigraphy (ICS) formally in 2004 (see &lt;strong&gt;&lt;a href=&quot;http://www.episodes.co.in/www/backissues/272/Time%20Scale.pdf&quot; rel=&quot;noreferrer&quot;&gt;Gradstein et al. 2004a&lt;/a&gt;&lt;/strong&gt; for instance, along with the 2004 edition of &quot;A geological timescale&quot;, &lt;strong&gt;Gradstein et al 2004b&lt;/strong&gt;) but then Quaternary was reinstated as a formal Period in 2009.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.stratigraphy.org/bak/geowhen/TQ.html&quot; rel=&quot;noreferrer&quot;&gt;Geowhen database&lt;/a&gt; has a nice page on the pre-2009 story (I'm not sure this database has been updated since, unfortunately). To summarize what they say: Giovanni Arduino (1759) separated geologic time in Primary/Secondary/Tertiary; in 1829, Jules Desnoyers added the term Quaternary; the term has been used since then to describe the recent period of glacial/interglacial cycles. But since then, the concepts of stratigraphy evolved and came the need to be able to pinpoint the boundaries between periods/systems/stages accurately and because they were different definitions on the Tertiary and Quaternary top and bottom boundaries in the literature, the ICS opted for a completely different system: Paleozoic/Mesozoic/Cenozoic, with the Cenozoic separated into a Paleogene and a Neogene.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in 2009, the Quaternary was reinstated (see &lt;strong&gt;&lt;a href=&quot;http://dx.doi.org/10.1002/jqs.1338&quot; rel=&quot;noreferrer&quot;&gt;Gibbard et al 2010&lt;/a&gt;&lt;/strong&gt;): the rationale, if I understand correctly, was that the US Geological Survey and the Geological Society of America were still using the Tertiary/Quaternary System and reinstating the Quaternary was somewhat of a compromise. Since the Paleogene and early Neogene had already many GSSPs (which are precisely defined boundaries, with a golden nail planted at the precise datum in the type section), reinstating the Tertiary as well would have been counterproductive at that stage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To add to the confusion, the &quot;new&quot; Quaternary, as defined from 2009 on, has his bottom boundary defined as the bottom boundary of the Gelasian (the &quot;old&quot; Quaternary bottom boundary corresponded roughly to the bottom of the Calabrian, thus 750kyr younger than the new one), making it overlap with part of the &quot;old&quot; Tertiary.&lt;br&gt;&#xA;According to the 2012 edition of &quot;A geological Timescale&quot; (thus &lt;strong&gt;Gradstein et al. 2012&lt;/strong&gt;), the rationale for the lowering of the base of the Quaternary to the base of the Gelasian is because of a now well-established cooling at 2.6/2.7Ma, which is considered to be the beginning of the first glacial/interglacial cycle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/8Oo7A.png&quot; alt=&quot;International Chronostratigraphic Chart v2014/02&quot;&gt;&#xA;&lt;sub&gt;Current ICS chart. © International Commission on Stratigraphy, 2014.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-11-03T08:00:33.610" LastActivityDate="2014-11-03T08:00:33.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="234" PostTypeId="2" ParentId="210" CreationDate="2014-11-03T19:02:44.863" Score="11" Body="&lt;p&gt;From the article &quot;Astrology&quot;, by Sheila J. Rabin, in &lt;em&gt;Encyclopedia of the Scientific Revolution from Copernicus to Newton&lt;/em&gt;, p.77:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In fact, astrology was part of the mathematics curriculum of every Western university&#xA;  from their founding in the twelfth century to the seventeenth century, and &lt;em&gt;mathematicus&lt;/em&gt;&#xA;  was a synonym for astrologer&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lynn Thorndike's magisterial &lt;em&gt;A History of Magic and Experimental Science&lt;/em&gt; (8 volumes; volume 2 alone is over 1000 pages!) has a wealth of material (of which I've read only a smidgen). For example, he mentions Bernard Silvester's 12th C.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;narrative poem whose plot hinges upon an astrologer's prediction and whose very title is &lt;em&gt;Mathematicus&lt;/em&gt; (v.2, p.101).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Other medieval scholars made a careful lexical distinction. Hugh of St. Victor (also 12th C.) distinguishes &lt;em&gt;mathematica&lt;/em&gt; (with an 'h' after the 't'), mathematics in our sense, and &lt;em&gt;matesis&lt;/em&gt; (no 'h'), astrology (&quot;that superstitious vanity which places the fate of man under the constellations&quot;) (v.2, p.11).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Michael Scot, a 13th C. astrologer at the court of Frederick II, in his &lt;em&gt;Introduction to Astrology&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;distinguishes between &lt;em&gt;mathesis&lt;/em&gt;, or knowledge, and &lt;em&gt;matesis&lt;/em&gt;, or divination, and between &lt;em&gt;mathematica&lt;/em&gt;, which may be taught freely and publicly, and &lt;em&gt;matematica&lt;/em&gt;, which is forbidden to Christians [this being the version of astrology which denied the existence of free will]. (v.2, p.319)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Scot often uses &lt;em&gt;astronomia&lt;/em&gt; to mean astrology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, after Scot's death the poet Henry of Avranches wrote a poem about him, in which he&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;explains how astrologers (&lt;em&gt;mathematici&lt;/em&gt;) &quot;reveal the secrets of things&quot;, by their art affecting numbers, by numbers affecting the procession of the stars, and by the stars moving the universe. (v.2, p.307-308)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-03T19:02:44.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="235" PostTypeId="1" CreationDate="2014-11-03T20:11:18.883" Score="7" ViewCount="1207" Body="&lt;p&gt;I have yet to read a math book that mentions ancient African mathematicians at any length or level of detail. There is scant coverage in academic settings in the United States, perhaps elsewhere too.  Most books have side notes that mention only European and east Asian mathematicians. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, the ancient Egyptians had a lot of knowledge about trigonometry, given that they built the pyramids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well-regarded ancient Greek intellectuals acknowledged the achievements of African mathematicians throughout their historic writings. Modern academicians consider these ancient Greek and Roman authors to be credible, yet that credibility is selective regarding African mathematicians. For specific examples, please see &lt;strong&gt;&lt;a href=&quot;http://www.math.buffalo.edu/mad/Ancient-Africa/&quot; rel=&quot;nofollow&quot;&gt;this link&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="175" LastEditorUserId="13" LastEditDate="2014-11-10T00:41:21.473" LastActivityDate="2016-05-31T23:56:50.250" Title="Are ancient African mathematicians acknowledged in the history of mathematics?" Tags="|academia|ancient-egypt|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="236" PostTypeId="2" ParentId="235" CreationDate="2014-11-03T21:19:42.650" Score="16" Body="&lt;p&gt;I do not think the following is true:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is it so hard for many white scholars and writers of today to&#xA;  acknowledge that Ancient African Mathematicians played a very&#xA;  important role in the development of mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mathematicians of north Africa, what is present day Egypt, Tunisia (old Carthage), Algeria and Morocco are widely and readily acknowledged by scholars* of all races for their achievements in number theory, mathematical logic, algebra, geometry and trigonometry. These mathematicians of antiquity had Arab, Jewish, Latin and Greek names, but they were all African, geographically. &quot;African mathematicians&quot; also include the mathematicians of ancient Syracuse, Alexandria and the land area just north of the Straits of Gibraltar (I am thinking too of the Moors, who were more contemporary than ancient... maybe).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the Great Pyramids at Giza, we still don't know how these structures were built, nor the names of those who built them. That makes any explicit acknowledgement difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Specific examples can be found in &lt;strong&gt;&lt;a href=&quot;https://math.stackexchange.com/search?q=Apostol&quot;&gt;mathematics textbooks by Apostol&lt;/a&gt;&lt;/strong&gt; specifically this textbook, &lt;strong&gt;&lt;a href=&quot;http://math-blog.com/mathematics-books/#calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus I&lt;/a&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2014-11-03T22:55:39.950" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="237" PostTypeId="5" CreationDate="2014-11-03T21:59:11.527" Score="0" Body="&lt;p&gt;If the concerned question is about a more specific area of Biology (such as &lt;a href=&quot;/questions/tagged/genetics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;genetics&amp;#39;&quot; rel=&quot;tag&quot;&gt;genetics&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/zoology&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;zoology&amp;#39;&quot; rel=&quot;tag&quot;&gt;zoology&lt;/a&gt; etc.), this tag should be used along with the tag of that specific area of Biology.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-20T13:46:40.000" LastActivityDate="2022-03-20T13:46:40.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="238" PostTypeId="4" CreationDate="2014-11-03T21:59:11.527" Score="0" Body="For questions relating to the study of life and life-related processes" OwnerDisplayName="user22" LastEditorUserId="10106" LastEditorDisplayName="user22" LastEditDate="2022-03-20T13:46:04.417" LastActivityDate="2022-03-20T13:46:04.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="239" PostTypeId="5" CreationDate="2014-11-03T22:00:00.717" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-03T22:00:00.717" LastActivityDate="2014-11-03T22:00:00.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="240" PostTypeId="4" CreationDate="2014-11-03T22:00:00.717" Score="0" Body="Questions related to the study of the structure, dynamics and composition of the Earth." OwnerDisplayName="user22" LastEditorDisplayName="user22" LastEditDate="2014-11-03T22:31:45.727" LastActivityDate="2014-11-03T22:31:45.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="241" PostTypeId="1" AcceptedAnswerId="255" CreationDate="2014-11-03T22:26:11.397" Score="9" ViewCount="153" Body="&lt;p&gt;In (theoretical) physics, it is customary to describe the system under consideration in terms of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Lagrangian&quot;&gt;Lagrangian&lt;/a&gt;. One of the major advantages of this approach is that it allows us to analyze the symmetries of the system in a systematic way. Arguably the most important result in this context is what is known as &lt;a href=&quot;http://en.wikipedia.org/wiki/Noether%27s_theorem&quot;&gt;Noether's theorem&lt;/a&gt;: For any continuous symmetry of the Lagrangian, we have what is known as a 'conserved current'. Mathematically speaking: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \delta \mathcal L=0 \Leftrightarrow \partial_\mu j^\mu=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $j^\mu$ is the conserved current associated with this symmetry of the Lagrangian, defined by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$j^\mu= \frac{\partial \mathcal L}{\partial (\partial_\mu \phi)}\delta\phi $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or some variation/generalization thereof. The zero-component $j^0$ is conventionally called the &lt;em&gt;charge&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Where did this terminology (charge, current) have its origins? Can something meaningful be said about this, beyond the fact that we can use the often-cited example of a complex scalar field to demonstrate that something similar to classical (electromagnetic) current and charge is conserved due to Noether's theorem? As an aside, any historically oriented reviews of the development of Noether's theorem would also be much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2014-11-04T22:44:12.840" Title="What is the reason for the 'electromagnetism terminology' when discussing the conserved quantities found through Noether's theorem?" Tags="|theoretical-physics|terminology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="242" PostTypeId="2" ParentId="195" CreationDate="2014-11-03T22:54:42.153" Score="14" Body="&lt;p&gt;I'm going to try to answer the question you end with, &quot;Why has &lt;em&gt;atomic bomb&lt;/em&gt; instead of another better term become the predominant term?&quot;, rather than the question in your title, because that's the historical question. (The other might be interesting to debate, but unlikely to produce a satisfying final answer.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's start with the Google ngram, showing the relative frequency of &lt;em&gt;atomic&lt;/em&gt; and &lt;em&gt;nuclear&lt;/em&gt; in the period 1900 to 1940:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/nNWEo.jpg&quot; alt=&quot;Google ngram of atomic vs. nuclear&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, &lt;em&gt;atomic&lt;/em&gt; was much more common than &lt;em&gt;nuclear&lt;/em&gt; throughout this period. Note that the Manhattan project didn't get started until 1942.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next let's look at &lt;em&gt;atomic bomb&lt;/em&gt; during the same period:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/KW6hf.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, the term was around for quite some time before the actual bomb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, we run out of hard data, but I can still offer some food for thought. First, the etymology of &lt;em&gt;nuclear&lt;/em&gt;: its use in physics derives (around 1914) from its biological meaning. By contrast, &lt;em&gt;atomic&lt;/em&gt; has had roughly the same meaning since the ancient Greeks (&lt;em&gt;very&lt;/em&gt; roughly, but close enough for this topic).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also searched through some of the novels of H.G. Wells. One stood out: &lt;em&gt;The World Set Free&lt;/em&gt;, published in 1914. Richard Rhodes recounts in &lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt; that Wells's novel made a great impression on Leo Szilard when he read it in 1932, a few years before he discovered the nuclear chain reaction. This novel describes (in Szilard's summary):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...the liberation of atomic energy on a large scale for industrial purposes, the development of atomic bombs, and a world war .... in this war the major cities of the world are all destroyed by atomic bombs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wells uses the terms &lt;em&gt;atomic energy&lt;/em&gt; and &lt;em&gt;atomic bomb&lt;/em&gt; throughout. Where did he get these terms? Wells inscribes the book&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;TO Frederick Soddy's &lt;em&gt;Interpretation Of Radium&lt;/em&gt; This Story, Which Owes Long Passages To the Eleventh Chapter Of That Book, Acknowledges And Inscribes Itself&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now, Soddy was Rutherford's student, and the two of them discovered that the thorium atom was &lt;em&gt;not&lt;/em&gt; unsplittable: a piece (the alpha particle) splintered off during radioactive decay, energy being released as well. (True also for radium.) This work was done in 1900-1903. Rutherford's nuclear model of the atom came &lt;em&gt;later&lt;/em&gt;: the Geiger-Marsden experiment in 1909, the model in 1911.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here's a conjectural history of the terms &lt;em&gt;atomic energy&lt;/em&gt; and &lt;em&gt;atomic bomb&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rutherford-Soddy make their discoveries and Soddy writes his book. The term &lt;em&gt;nuclear&lt;/em&gt; is not used, since its use in physics is still in the future.&lt;/li&gt;&#xA;&lt;li&gt;Wells reads Soddy's book and writes his own work of fiction, picking up on &lt;em&gt;atomic&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Szilard reads Wells's book, and when he discovers the basic mechanism of the atomic bomb, he naturally perpetuates the term &lt;em&gt;atomic&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As I said, conjecture. But we can be pretty sure that in the period 1910 to 1940 (roughly), &lt;em&gt;nuclear&lt;/em&gt; was a fairly technical term, not in general use --- like &lt;em&gt;atomic&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-03T22:54:42.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="243" PostTypeId="1" AcceptedAnswerId="248" CreationDate="2014-11-03T23:47:33.373" Score="14" ViewCount="872" Body="&lt;p&gt;A well-known and specific example is that Leibniz is less well regarded than Newton for his calculus, the &lt;strong&gt;&lt;a href=&quot;https://hsm.stackexchange.com/a/90/123&quot;&gt;reason being notation&lt;/a&gt;&lt;/strong&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Leibniz notation lets you incorrectly work with derivatives as though&#xA;  they were a mathematical fraction&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;although this is &lt;strong&gt;&lt;a href=&quot;http://www.uiowa.edu/~c22m025c/history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;debatable&lt;/a&gt;&lt;/strong&gt;. There are numerous, more philosophical critiques of Leibniz's shortcomings though, with respect to Spinoza and &lt;strong&gt;&lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;otherwise&lt;/a&gt;&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1700 the academy of Berlin was created on his advice, and he drew&#xA;  up the first body of statutes for it. On the accession in 1714 of his&#xA;  master, George I., to the throne of England, Leibnitz was thrown aside&#xA;  as a useless tool; he was forbidden to come to England; and the last&#xA;  two years of his life were spent in neglect and dishonour.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why?&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="4251" LastEditDate="2020-05-13T11:21:43.413" LastActivityDate="2020-05-13T11:21:43.413" Title="Why is Leibniz less well regarded?" Tags="|mathematics|calculus|leibniz|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="244" PostTypeId="5" CreationDate="2014-11-03T23:53:35.523" Score="0" Body="&lt;p&gt;Leibniz invented the differential and integral calculus in 1674, simultaneously but independently from Sir Thomas Newton. For a complete biography, see &lt;strong&gt;&lt;a href=&quot;http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Leibniz.html&quot; rel=&quot;nofollow&quot;&gt;Leibniz summary&lt;/a&gt;&lt;/strong&gt; (St. Andrews).&lt;/p&gt;&#xA;" OwnerUserId="123" LastEditorUserId="123" LastEditDate="2014-11-04T02:16:42.713" LastActivityDate="2014-11-04T02:16:42.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="245" PostTypeId="4" CreationDate="2014-11-03T23:53:35.523" Score="0" Body="Gottfried Wilhelm von Leibniz (1646 - 1716) was a German mathematician and philosopher; also spelled &quot;Leibnitz&quot;." OwnerUserId="123" LastEditorUserId="123" LastEditDate="2014-11-04T02:00:35.767" LastActivityDate="2014-11-04T02:00:35.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="246" PostTypeId="2" ParentId="243" CreationDate="2014-11-04T00:11:50.283" Score="8" Body="&lt;p&gt;I think I can answer the last bit. From &lt;a href=&quot;https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz#House_of_Hanover.2C_1676.E2.80.931716&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of the history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the Dowager Electress Sophia, died in 1714.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What &quot;history of the Brunswick family&quot;? Go up a little to find this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Elector Ernest Augustus commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared; the next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wow, that's a large chunk of text. Here's a quick summary and/or interpretation:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A while back, Leibniz is asked to write a big book on the history of the Brunswick family (including George I).&lt;/li&gt;&#xA;&lt;li&gt;Leibniz holds himself to high standards; combined with his attention to detail and the many other project on his plate, he never finishes the book.&lt;/li&gt;&#xA;&lt;li&gt;Years later, George I becomes king. He realizes that Leibniz never finished the book he was asked to write three decades prior.&lt;/li&gt;&#xA;&lt;li&gt;George I refuses to ask Leibniz to join his court in London. A second reason is that it would insult Isaac Newton, who had been recognized as the inventor of calculus.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Interestingly enough, Leibniz &lt;em&gt;had&lt;/em&gt; written a large portion of the book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They never knew that he had in fact carried out a fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-04T00:11:50.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="247" PostTypeId="1" CreationDate="2014-11-04T00:16:56.090" Score="21" ViewCount="690" Body="&lt;p&gt;Before Eudoxus's theory of proportion there was a theory of irrationals based on continued fraction expansions, which &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183544897&quot;&gt;Fowler&lt;/a&gt; calls anthyphairesis. Theaetetus is said to develop a classification of quadratic irrationals in this setting. Given positive $x$ and $y$, with $x/y$ being a square root of a rational number, $x+y$ is called a binomial, $x-y$ an apotome (if $x&amp;gt;y$), and $\sqrt{xy}$ a medial (if it is irrational). There are some interesting results &lt;a href=&quot;http://www.math.umt.edu/tmme/vol6no1and2/TMME_vol6nos1and2_article22_pp.277_294.pdf&quot;&gt;described&lt;/a&gt; in book X of Euclid's Elements: the classes are mutually exclusive, for any binomial or apotome the split into $x$ and $y$ is unique, etc. Further classes are considered, apotome of a medial, binomial of a medial, and so on.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Theaetetus seems to deal with positive elements of $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots)$, but I can not think of any algebraic or number theoretic classification scheme that would produce his classes. Are they related to something in modern algebra?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-03T01:58:44.250" LastActivityDate="2022-12-06T23:16:51.000" Title="What is the modern significance of Theaetetus's classification of quadratic irrationals?" Tags="|mathematics|number-theory|ancient-greece|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="248" PostTypeId="2" ParentId="243" CreationDate="2014-11-04T00:47:16.300" Score="19" Body="&lt;p&gt;You say: &quot;A well-known and specific example is that Leibniz is less well regarded than Newton for his calculus&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well known??&#xA;I think this is just incorrect. Leibniz version of calculus lead to an explosive development&#xA;of calculus in continental Europe. Think of l'Hopitale, Bernoulli's, Euler and many others.&#xA;While calculus in England experienced stagnation in the period after Newton.&#xA;Exactly for the reason that English mathematicians ignored Leibniz's version.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Name me ONE English mathematician of 18 century in the class of Bernoullis, Euler, Clairaut, d'Alembert etc. Only in 19-s century English mathematicians overcome Newton's dominance and accepted Leibniz notation. Which we are all using now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Of course I agree that Newton was one of the greatest mathematicians of all times.&#xA;But nevertheless he certainly had some negative influence on development of mathematics, mainly in England, but not only in England. On my opinion, this happened for three reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One is that he was so much superior to everyone else. Not surprisingly his cult was formed. No one could contradict him. (Think of the history of Optics, where his wrong opinion was dominating for almost 100 years).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;His domineering personality. He suppressed and ruined great people around him (Flamsteed, Hooke and others). He staged this shameful &quot;priority dispute with Leibniz&quot;.&#xA;He did everything to put Leibniz down, to destroy his reputation (like he did to Hook). And he partially succeeded. &#xA;And this discussion is another evidence of this.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally he was reluctant to publish his results, while Leibniz did really a lot to promote calculus. He created a journal. He had students, etc.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In his life time Newton was surrounded by really great people. And look how it changed after his death. Read the history of the Royal society in Wikipedia, to begin with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Newton wrote once: &quot;If I saw farther than others, this is because I stood on the shoulders of giants&quot;.&#xA;Recently I read a paraphrase of this: &quot;If I did not see as far as some others, this is because giants stood on my shoulders&quot; :-) I think Flamsteed or Hooke would be justified if they said this. They were the giants on whose shoulders Newton stood. And what did he do to them? &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-04T03:31:59.363" LastActivityDate="2014-11-04T03:31:59.363" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="249" PostTypeId="2" ParentId="98" CreationDate="2014-11-04T08:41:04.583" Score="13" Body="&lt;p&gt;To add to the two previous answers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bernard-Germain-Etienne de Lacepède&lt;/strong&gt; in &quot;Histoire naturelle des poissons&quot; (Natural History of Fishes) 1798 wrote &lt;a href=&quot;http://www.lamarck.cnrs.fr/ice/ice_book_detail.php?lang=fr&amp;amp;type=text&amp;amp;bdd=koyre_lamarck&amp;amp;table=cpo_lamarck&amp;amp;bookId=13&amp;amp;typeofbookId=1&amp;amp;num=0&quot;&gt;the following&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Une espèce peut s'éteindre de deux manières. Elle peut périr toute entière, et dans un temps très-court, lorsqu'une catastrophe violente bouleverse la portion de la surface du globe sur laquelle elle vivoit, [...] Mais, indépendamment de ces grands coups que la Nature frappe rarement et avec éclat, une espèce disparoît par une longue suite de nuances insensibles et d'altérations successives.[...] Troisièmement, l'espèce peut subir un si grand nombre de modifications dans ses formes et dans ses qualités, que, sans rien perdre de son aptitude au mouvement vital, elle se trouve, par sa dernière conformation et par ses dernières propriétés, plus éloignée de son premier état que d'une espèce étrangère : elle est alors métamorphosée en une espèce nouvelle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which translates roughly to the following &lt;em&gt;[Disclaimer: own translation, hence poor quality]&lt;/em&gt;:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A species can become extinct in two ways. It can die out entirely, and in a very short time, when a violent catastrophe shatters the portion of the earth's surface it lives on [...] But independantly to these big blows that Nature strike rarely and resoundingly, a species disappears by a long serie of imperceptible differences and successives alterations. [...] Thirdly, the species can undergo such a large number of modifications in its forms and qualities that, without losing its aptitude to the vital movement, it finds itself, in its last conformation and its last properties, further away from its first state than another species: it is then metamorphosed into a new species.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Erasmus Darwin&lt;/strong&gt; was already named by @winwaed so I won't expand but here is an &lt;a href=&quot;http://www.lamarck.cnrs.fr/ice/ice_book_detail.php?lang=fr&amp;amp;type=text&amp;amp;bdd=koyre_lamarck&amp;amp;table=cpo_lamarck&amp;amp;bookId=15&amp;amp;typeofbookId=7&amp;amp;num=0&quot;&gt;extract of his &lt;em&gt;Zoonomia&lt;/em&gt; (1801)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As air and water are supplied to animals in sufficient profusion, the three great objects of desire, which have changed the forms of many animals by their exertions to gratify them, are those of lust, hunger, and security.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This prefigures quite obviously the notion of natural selection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Jean-Baptiste Lamarck&lt;/strong&gt; was also already quoted by @winwaed, but to expand on him: he expressed the idea that species were transforming through time for the first time in 1801 in its &lt;a href=&quot;http://www.lamarck.cnrs.fr/ice/ice_book_detail.php?lang=fr&amp;amp;type=text&amp;amp;bdd=koyre_lamarck&amp;amp;table=ouvrages_lamarck&amp;amp;bookId=5&amp;amp;typeofbookId=2&amp;amp;num=0&quot;&gt;&quot;Discours d'Ouverture du cours de Zoologie, donné dans le Museum d'Histoire Naturelle l'an 8 de la République&quot;&lt;/a&gt; (litt.: &quot;Opening speech of the Zoology course, given in the Natural History Museum in the year 8 of the Republic&quot;). He thought in particular that species were transforming to adapt to their environment. He later expanded more on that in his books (in particular &lt;a href=&quot;http://www.lamarck.cnrs.fr/ice/ice_book_detail.php?lang=fr&amp;amp;type=text&amp;amp;bdd=koyre_lamarck&amp;amp;table=ouvrages_lamarck&amp;amp;bookId=29&amp;amp;typeofbookId=1&amp;amp;num=0&quot;&gt;'Philosophie zoologique'&lt;/a&gt; in 1809).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Johann Wolfgang von Goethe&lt;/strong&gt;, more famously known as a poet, was also a naturalist, and a &quot;transformist&quot; (as pre-Darwinian evolutionist were often called, by opposition to &quot;fixists&quot;). here is what he wrote in &quot;Story of My Botanical Studies&quot; (1831) &lt;em&gt;[extract found on his wikipedia page]&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ever-changing display of plant forms, which I have followed for so many years, awakens increasingly within me the notion: The plant forms which surround us were not all created at some given point in time and then locked into the given form, they have been given... a felicitous mobility and plasticity that allows them to grow and adapt themselves to many different conditions in many different places.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One of the main tool in recognizing that two species evolved from the same ancester is the homology, and there is quite a string of researchers that led to the refinement of this concept, among them: &lt;strong&gt;Michel Adanson&lt;/strong&gt; (1727-1806) established a classification of beings based on &quot;the greatest number of similarities&quot; and &lt;strong&gt;Etienne Geoffroy Saint-Hilaire&lt;/strong&gt; (1772-1844) who not only formalized the concept of homology (under the name &quot;analogy&quot; at the time) but also thought that species changed according to their living conditions (according to the &lt;a href=&quot;http://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Geoffroy_Saint-Hilaire,_%C3%89tienne&quot;&gt;Encyclopedia britannica&lt;/a&gt; entry on him).&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-11-04T10:23:35.023" LastActivityDate="2014-11-04T10:23:35.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="251" PostTypeId="2" ParentId="235" CreationDate="2014-11-04T14:55:29.320" Score="9" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Category:African_mathematicians&quot;&gt;African mathematicians&lt;/a&gt; are acknowledged in history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is true is that most of the African mathematicians that are acknowledged are North Africans. That's because Egypt, Libya, Tunisia, Algeria, etc. are on the &quot;southern rim&quot; of the Mediterranean, and was therefore part of the &quot;classical&quot; world of southern Europe and the Middle East.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also true that Sub Sahara mathematicians are not well recognized at least until modern times. Those places were remote, both by sheer distance, but especially by transportation from &quot;centers&quot; of civilization further north.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-11-04T14:55:29.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="252" PostTypeId="1" CreationDate="2014-11-04T15:52:25.947" Score="11" ViewCount="439" Body="&lt;p&gt;Enderton &lt;em&gt;Elements of Set Theory&lt;/em&gt;, p. 43 (1977, Academic Press), writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There was a reluctance to separate the concept of a function itself from the idea of a written formula defining the function. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is the basis for the above historical claim? And at around what point did the concept of a function itself from the idea of a formula become firmly separated?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems interesting that what is today regarded as an elementary mistake had a strong historical basis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fuller quote from Enderton: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/NAy0G.png&quot; alt=&quot;Enderton, p. 43&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question originally posted over at &lt;a href=&quot;https://math.stackexchange.com/questions/746879&quot;&gt;Math.SE&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user67" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2022-03-14T12:35:33.620" Title="Concept of a function and Idea of a formula as a function" Tags="|mathematics|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="253" PostTypeId="2" ParentId="252" CreationDate="2014-11-04T16:35:19.667" Score="9" Body="&lt;p&gt;You can see &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_the_function_concept&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of the function concept&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For Euler (1748) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e. a function was a “symbolic expression” that, received a value as “input” allows us to calculate a corresponding “output” value.&lt;/p&gt;&#xA;&lt;p&gt;It seems that it is in &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k99435r/f148.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichelet (1837, page 135)&lt;/a&gt;, that we can find the first explicit definition of the concept of function as “arbitrary correspondence”:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If now a unique finite &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; corresponding to each &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and moreover in such a way that when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; ranges continuously over the interval from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; also varies continuously, then &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is called a continuous function of x for this interval.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;It is not at all necessary here that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; be given in terms of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by one and the same law throughout the entire interval, and it is not necessary that it be regarded as a dependence expressed using mathematical operations&lt;/em&gt; [emphasis added].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2022-03-14T12:35:33.620" LastActivityDate="2022-03-14T12:35:33.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="254" PostTypeId="1" AcceptedAnswerId="262" CreationDate="2014-11-04T21:44:31.233" Score="16" ViewCount="2439" Body="&lt;p&gt;The discoverers of radioactivity and its potential uses are renowned, e.g. Madame Curie and Roentgen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was it discovered that human exposure to ionizing radiation including radioactive isotopes often caused leukemia, other cancers and highly malignant cell growth in general? What other aspects of this realization are documented, specifically, how it was discovered  and by whom?&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-11-05T07:41:57.117" Title="When and how was ionizing radiation found to be harmful?" Tags="|medicine|radiation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="255" PostTypeId="2" ParentId="241" CreationDate="2014-11-04T22:44:12.840" Score="4" Body="&lt;p&gt;At the time Emmy Noether formulated her theorems (1910-s) electromagnetism was the only fundamental field theory with well developed terminology. But even in the electromagnetism itself the word current was borrowed earlier from analogies with fluids, and probably would have spread into other theories for the same reason. After Einstein's work on relativity matter fields (elasticity, fluids) were far from the center of attention, and for practical reasons no one studied currents in classical gravity. So charge, rather than mass, was a natural choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noether herself was focused on conservation of energy and its apparent failure in general relativity discovered by Klein. Hilbert conjectured that there is a distinction between &quot;proper&quot; conservation laws as in electromagnetism, and &quot;improper&quot; ones as in general relativity. Noether not only proved this in great generality, but supplied a link to symmetry groups. Conservation is proper when the symmetry group has at most countably many independent infinitesimal generators, and &quot;improper&quot; when it has a continuum of them as in general relativity. Klein, by the way called energy current &quot;Hilbert's energy vector&quot;, but Noether's theorems were general so this wasn't a good name. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Gursey &quot;Before Noether's Theorem the principle of conservation of energy was shrouded in mystery, leading to the obscure physical systems of Mach and Ostwald. Noether's simple and profound mathematical formulation did much to demystify physics.&quot;&#xA;More details on historical circumstances of Noether's work can be found &lt;a href=&quot;http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2014-11-04T22:44:12.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="256" PostTypeId="1" CreationDate="2014-11-04T22:56:16.433" Score="18" ViewCount="735" Body="&lt;p&gt;In the comments on &lt;a href=&quot;https://physics.stackexchange.com/q/135764/36360&quot;&gt;this question on Physics&lt;/a&gt; about the usefulness of expensive experiments such as the CERN, the following short discussion happened:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Has there ever been a major basic science result that did not lead to practical applications within the next couple of hundred years?  &lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;Isn't there a strong selection bias there? If something hasn't lead to anything much in the following couple of hundred years, we've probably forgotten about it, regardless of how big a deal it seemed at the time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While the second argument is indeed sound, I wonder whether there is any good example for this. More precisely and with slight deviations from the inspiration, I am looking for the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A scientific result that can be considered basic science in the sense that it was not mainly about application to begin with.&lt;/li&gt;&#xA;&lt;li&gt;This result was considered a breakthrough at its time by notable sources (in particular not by people gaining advantage from exaggerating something as a breakthrough).&lt;/li&gt;&#xA;&lt;li&gt;Neither this result &lt;strong&gt;nor its successors&lt;/strong&gt; are considered relevant today. There is no relevant technological application (nor has there ever been) and it does not appear in modern textbooks of any discipline.&lt;/li&gt;&#xA;&lt;li&gt;The result was not a negative one, such as the falsification of the ether theory.&lt;/li&gt;&#xA;&lt;li&gt;The result must be real, e.g., it should not have turned out to be due to experimental errors.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2017-09-26T02:37:40.700" Title="Considered a breakthrough at its time – almost forgotten nowadays" Tags="|discoveries|examples|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="257" PostTypeId="1" CreationDate="2014-11-05T01:43:15.967" Score="14" ViewCount="354" Body="&lt;p&gt;Publication bias is a &lt;a href=&quot;https://academia.stackexchange.com/q/30839/452&quot;&gt;huge issue&lt;/a&gt; is some academic fields. What are the earliest accounts of publication bias?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As requested in the comments, publication bias intuitively means that if you read one article showing that drug X works to treat disease Y, you should wonder how many articles showing the same drug has no affect are unpublished due to the habits of publishing only positive results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;https://www.youtube.com/watch?v=RKmxL8VYy0M&quot; rel=&quot;nofollow noreferrer&quot;&gt;video explaining what publication bias is&lt;/a&gt; (15 minutes, worth watching if not familiar with publication bias):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Half of all clinical trials ever completed on the medical treatments&#xA;  currently in use have never been published in the medical literature.&#xA;  Trials with positive results for the test treatment are about twice as&#xA;  likely to be published, and this applies to both academic research and&#xA;  industry studies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Definition from the &lt;a href=&quot;http://en.wikipedia.org/wiki/Publication_bias&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page on publication bias&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Publication bias is a bias with regard to what is likely to be&#xA;  published, among what is available to be published. Not all bias is&#xA;  inherently problematic – for instance, a bias against publishing lies&#xA;  is often a desirable bias – but one problematic and much-discussed&#xA;  bias is the tendency of researchers, editors, and pharmaceutical&#xA;  companies to handle the reporting of experimental results that are&#xA;  positive (i.e. showing a significant finding) differently from results&#xA;  that are negative (i.e. supporting the null hypothesis) or&#xA;  inconclusive, leading to a misleading bias in the overall published&#xA;  literature.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.757" LastActivityDate="2014-12-24T00:34:00.920" Title="What are the earliest accounts of publication bias?" Tags="|academia|publishing|bias|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="258" PostTypeId="2" ParentId="252" CreationDate="2014-11-05T02:48:18.697" Score="7" Body="&lt;p&gt;I recommend an excellent account by Luzin in the Monthly: MR1615544, MR1613935&#xA;(American math Monthly 105 (1998), 1 59-67 and 3, 263-270.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is usually overlooked that there are actually several different notions of function in modern mathematics. One is the Dirichlet's definition which is usually cited (where two sets are given X and Y, and a rule which to each element of X puts into correspondence an element of Y). Notice that X is a part of the definition!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the problem of the type &quot;find the domain of $\log((x-1)(x-2))$ makes no sense from the point of this definition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 18 century, Euler understood a function as some analytic expression whose domain is not given in advance. This different notion (from Dirichlet's definition) is not &quot;out of date&quot;. It evolved into a modern definition of an &quot;analytic function&quot;. Roughly speaking, an&#xA;&quot;analytic expression&quot; has a &quot;natural domain of definition&quot;, which is not given in advance.&#xA;And problems of the type &quot;find the domain of definition&quot; of an analytic function make perfect sense in modern mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also other notions of functions in modern mathematics (generalized functions, or distributions), which also do not fit in the Dirichlet definition. Moreover, these generalized functions are in some sense closer to what physicists and engineers mean by a function than the Dirichlet definition.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-05T02:48:18.697" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="259" PostTypeId="2" ParentId="130" CreationDate="2014-11-05T03:05:37.287" Score="14" Body="&lt;p&gt;The notion of &amp;quot;rigged data&amp;quot; evolved with time. Some ancient scientists are accused (by modern scientists) in rigging of the data.&lt;/p&gt;&#xA;&lt;p&gt;One notable example is Ptolemy. I do not want to discuss here the accusation of Ptolemy by Robert Newton, but here is another well-known example.&lt;/p&gt;&#xA;&lt;p&gt;In his Optics, Ptolemy gives a table of refraction. It looks like he measured refraction&#xA;of the light ray passing from the air to other media (water, glass). He gives a little table,&#xA;in such a way that a modern reader may conclude that these are experimental data.&#xA;Some even consider this the oldest surviving example of an experiment in physics.&#xA;However they are most likely made up. The data are inconsistent with the Snell law.&#xA;If you plot the numbers they better fit a parabola rather than a sinusoid (as it should be). It is clear from the text that Ptolemy had some theory about refraction (incorrect theory), and his table was most likely computed according to this theory.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, he never states unambiguously that he actually performed this experiment, and does not give any detail.&lt;/p&gt;&#xA;&lt;p&gt;The most serious accusation against Ptolemy is that he forged his star catalog.&#xA;That is he took the 200 years old catalog of Hipparchus, and corrected the data for precession. And used the wrong constant of precession which was not known exactly at that time. There is a large controversy about this among historians of astronomy, and as I understand, there is no consensus.&lt;/p&gt;&#xA;&lt;p&gt;EDIT. Let me add that I am not aware of any famous stories of forgery and rigged data in physics and mathematics. None prior to 20 century and VERY few in 20 century. Unlike in humanities like archeology or &amp;quot;finding some ancient manuscripts&amp;quot; which was quite common in 18 and 19 centuries.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="13566" LastEditDate="2021-08-31T21:32:39.907" LastActivityDate="2021-08-31T21:32:39.907" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="260" PostTypeId="2" ParentId="52" CreationDate="2014-11-05T03:38:15.810" Score="7" Body="&lt;p&gt;I am not familiar with physics textbooks of 18 century, so I do not know the answer to the question. However I want to add a comment which is too long for the comment window.&#xA;(The system does not allow me to post a comment of this length.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Principia actually contains both derivations. Of the inverse square law from Kepler laws and of the Kepler laws from the inverse square law. Kepler laws are empirical laws. While the law of gravitation is a theory. At the time when Principia was written, the equivalence of the law of gravitation to the Kepler laws was the crucial proof of the law of gravitation. Only later in 18 century other consequences of the law of gravitation were tested (the shape of Earth, for example, and most notably the theory of the Moon motion). Until the triumph of the Moon theory in the second part of 18 century, there were doubts about the inverse square law. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why Kepler laws are taught nowadays as consequence of the Newton's law? &#xA;I am not sure that this is the case in the astronomy books for the beginners.&#xA;And elementary physics books do not contain actual derivation of the Kepler laws from the law of gravity. I was taught in high school both Kepler laws and the law of gravitation,&#xA;and I was TOLD that Kepler's laws are a consequence, but no actual derivation was given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not always convenient to follow the historical development of the subject in teaching. For example, we are not taught Ptolemy system, neither in astronomy nor in physics courses. And this is not because it is &quot;wrong&quot;:-) It is not wrong. If you look to the modern Nautical almanac, it conforms more to the Ptolemy system than to the heliocentric one.&#xA;And the actual computation of ephemerides uses their representation as trigonometric series, in complete conformity with &quot;epicycles&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-05T03:38:15.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="261" PostTypeId="2" ParentId="175" CreationDate="2014-11-05T03:41:50.380" Score="10" Body="&lt;p&gt;One motivation, perhaps the principal one, was his work in ordinary differential equations.&#xA;(And celestial mechanics, as an application of ordinary differential equations).&#xA;He introduced what is called &quot;qualitative methods&quot; which are based to a large extend on&#xA;topology. For example, the existence of periodic orbits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the introduction to his paper Analysis Situs (1895) Poincare mentions the following motivations: classification of algebraic surfaces, qualitative theory of differential equations, and applications to what is known now as Lie groups.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-05T03:49:49.093" LastActivityDate="2014-11-05T03:49:49.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="262" PostTypeId="2" ParentId="254" CreationDate="2014-11-05T07:41:57.117" Score="10" Body="&lt;p&gt;Clark University have published a very comprehensive document &lt;a href=&quot;http://www.clarku.edu/mtafund/prodlib/clark/round6/Ionizing_Radiation.pdf&quot;&gt;Health Risks of Ionizing Radiation: An Overview of Epidemiological Studies&lt;/a&gt;, (Russ et al. 2006).  In their introductory chapter they state that the harmful effects of ionising radiation, in this case, x-rays, were observed very quickly after their discovery, specifically,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;X-rays were first discovered in late 1895 &#xA;  and dangers associated with exposure became &#xA;  apparent very quickly. In 1896 the first injuries &#xA;  due to x-ray exposure were recorded and in 1904 &#xA;  Thomas Edison’s assistant Clarence Dally was the &#xA;  first person recorded to have died as a result of x-ray &#xA;  exposure. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;More detail of this is provided in the article &lt;a href=&quot;http://fas.org/sgp/othergov/doe/lanl/pubs/00326631.pdf&quot;&gt;A Brief History of Radiation Protection Standards&lt;/a&gt; (Inkret et al. 1995),&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In July 1896, only one month after the discovery&#xA;  of x rays, a severe case of x-ray-induced dermatitis&#xA;  was published, and in 1902, the first dose limit of&#xA;  about 10 rad per day (or 3000 rad per year), was recommended.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This limit was not based on any biological limit, but on the, then, limit of detection.  But, it was a start.  The link between radiation and cancer came about in 1903:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By 1903, animal studies had&#xA;  shown that x rays could produce cancer and kill living&#xA;  tissue and that the organs most vulnerable to radiation&#xA;  damage were the skin, the blood-forming organs,&#xA;  and the reproductive organs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and specifically for the element radium (from Russ et al. 2006):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Radium was discovered in 1898 and its use in &#xA;  medicine also spread very quickly. In the 1920s, &#xA;  sickness and death in watch dial painters, who &#xA;  ingested small amounts of radium in their work, &#xA;  taught scientists and doctors that internal exposure &#xA;  to radium could be harmful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Around the same time, (from Inkert et al. 1995), standards were being formalised and developed particularly in reference to 'tolerance doses', with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A&#xA;  tolerance dose was &quot;assumed to be a radiation dose to which the body can be subjected&#xA;  without production of harmful effects.” Mutscheller presented his recommendation&#xA;  in a paper entitled, “&lt;a href=&quot;http://pubs.rsna.org/doi/abs/10.1148/6.4.314?journalCode=radiology&quot;&gt;Physical Standards of Protection Against Roentgen&#xA;  Ray Dangers&lt;/a&gt;,” which was published in 1925. Quite fortuitously, F. M. Sievert arrived&#xA;  at about the same limits using a similar approach.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;http://en.wikipedia.org/wiki/Sievert&quot;&gt;Sievert&lt;/a&gt; is now a unit of ionising radiation dosage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leading to the first internationally recognised radiation safety guidelines being published in 1928, and the founding of the &lt;a href=&quot;http://www.icrp.org/&quot;&gt;International Commission on Radiological Protection&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-05T07:41:57.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="263" PostTypeId="1" AcceptedAnswerId="265" CreationDate="2014-11-05T17:54:38.943" Score="5" ViewCount="426" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Giuseppe_Peano&quot; rel=&quot;nofollow&quot;&gt;Wikipedia page on Giuseppe Peano&lt;/a&gt; claims the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the conference Peano met Bertrand Russell and gave him a copy of&#xA;  Formulario. Russell was so struck by Peano's innovative logical&#xA;  symbols that he left the conference and returned home to study Peano's&#xA;  text.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, the claim is unreferenced. Did Bertrand Russell leave the Second International Congress of Mathematicians to read Giuseppe Peano's &lt;a href=&quot;http://en.wikipedia.org/wiki/Formulario_mathematico&quot; rel=&quot;nofollow&quot;&gt;Formulario&lt;/a&gt;, or is that just some myth?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Ray Monk, ‘Russell, Bertrand Arthur William, third Earl Russell (1872–1970)’, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, May 2014 (&lt;a href=&quot;http://www.oxforddnb.com/view/article/35875&quot; rel=&quot;nofollow&quot;&gt;http://www.oxforddnb.com/view/article/35875&lt;/a&gt;, accessed 5 Nov 2014)&lt;/em&gt; does not mention this premature departure:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At Paris, Russell met the Italian mathematician Giuseppe Peano, the&#xA;  head of a movement whose ultimate goal was to construct a single&#xA;  axiomatic system upon which the whole of mathematics could be founded.&#xA;  In pursuit of this aim Peano had invented a special symbolism which he&#xA;  used to construct a system of mathematical logic, at the heart of&#xA;  which is the now-familiar notion of a ‘propositional function’. Using&#xA;  this system, Peano and his colleagues had shown that arithmetic could&#xA;  be founded on a single elegant formal theory which used only three&#xA;  basic ideas (zero, number, and successor) and five initial axioms.&#xA;  Inspired by his meeting with Peano and his study of Peano's work,&#xA;  Russell returned from Paris with an almost ecstatic conviction that he&#xA;  knew the way ahead: if he could show that all mathematical notions&#xA;  were fundamentally arithmetical and that Peano's system was&#xA;  fundamentally a system of logic, then he would have succeeded in his&#xA;  stated aim of demonstrating that mathematics was logic. For this the&#xA;  crucial step would be to show that Peano's axioms could be founded&#xA;  upon a system of logic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2014-11-05T22:29:19.337" LastActivityDate="2014-11-05T22:29:19.337" Title="Did Bertrand Russell leave the Second International Congress of Mathematicians to read Giuseppe Peano's Formulario?" Tags="|mathematics|biographical-details|mathematical-logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="264" PostTypeId="1" AcceptedAnswerId="5346" CreationDate="2014-11-05T19:07:24.517" Score="56" ViewCount="6667" Body="&lt;p&gt;Where can I find a paper or reference that describes the timeline of measurements of the magnitude of the electron's electric charge.&lt;/p&gt;&#xA;&lt;p&gt;For context, &lt;a href=&quot;http://en.wikipedia.org/wiki/Oil_drop_experiment&quot; rel=&quot;noreferrer&quot;&gt;Millikan's oil drop experiment&lt;/a&gt; in 1908 determined the charge of the electron to be 1.59*10^(-19) C, which is lower than today's accepted value. Richard Feynman is quoted as &lt;a href=&quot;http://en.wikipedia.org/wiki/Oil_drop_experiment#Millikan.27s_experiment_as_an_example_of_psychological_effects_in_scientific_methodology&quot; rel=&quot;noreferrer&quot;&gt;saying, in 1974,&lt;/a&gt; that subsequent experimental attempts to measure the charge were &lt;em&gt;biased&lt;/em&gt; by Millikan's initial value:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;It's interesting to look at the history of measurements of the charge of an electron, after Millikan. If you plot them as a function of time, you find that one is a little bit bigger than Millikan's, and the next one's a little bit bigger than that, and the next one's a little bit bigger than that, until finally they settle down to a number which is higher.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Why didn't they discover the new number was higher right away? It's a thing that scientists are ashamed of—this history—because it's apparent that people did things like this: When they got a number that was too high above Millikan's, they thought something must be wrong—and they would look for and find a reason why something might be wrong. When they got a number close to Millikan's value they didn't look so hard. And so they eliminated the numbers that were too far off, and did other things like that...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is there a reference that shows the temporal evolution of the electron's charge values? The most I've found is a statement that the values &lt;em&gt;did&lt;/em&gt; change, but not what those values were.&lt;/p&gt;&#xA;" OwnerUserId="34" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-11-03T16:42:28.627" Title="Timeline of measurements of the electron's charge" Tags="|experimental-physics|reference-request|bias|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="265" PostTypeId="2" ParentId="263" CreationDate="2014-11-05T19:47:51.150" Score="9" Body="&lt;p&gt;I've found in Hubert Kennedy, &lt;em&gt;Twelve Articles on Giuseppe Peano&lt;/em&gt; (2002) : &lt;strong&gt;&lt;a href=&quot;http://projecteuclid.org/euclid.ndjfl/1093891001&quot;&gt;What Russell learned from Peano&lt;/a&gt;&lt;/strong&gt;, from &lt;em&gt;Notre Dame Journal of Formal Logic&lt;/em&gt; (1983) the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The evidence comes from Russell himself in, among other places, the description of the International Congress of Philosophy in Paris, 1900, in his &lt;em&gt;Autobiography&lt;/em&gt; (p.217–219):&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;The Congress was a turning point in my intellectual life, because I there met Peano. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. [...] As the days went by, I decided that this must be owing to his mathematical logic. &lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;I therefore got him to give me all his works, and &lt;em&gt;as soon as the Congress was over&lt;/em&gt; I retired to Fernhurst to study quietly every word written by him and his disciples [emphasis added].&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="29" LastEditDate="2014-11-05T19:54:01.737" LastActivityDate="2014-11-05T19:54:01.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="266" PostTypeId="1" AcceptedAnswerId="274" CreationDate="2014-11-05T23:05:13.457" Score="11" ViewCount="1931" Body="&lt;p&gt;One of my favorite scientists in history is Hero of Alexandria, who invented a variety of clever mechanical devices in the first century AD, long before their economic potential was realized. Included among them are the steam engine, wind power, mechanical &quot;computer programs&quot;, and a sort of vending machine.  It seems to me that if his ideas were taken seriously they could have triggered an industrial revolution centuries early, and I'm wondering why this didn't happen.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One explanation that I've heard is that the economy at the time relied heavily on slave labor and hence there was little demand for efficient industrial processes, but somehow this seems unsatisfactory.  If nothing else, were military applications of this technology not explored?  This question may call for a bit more speculation than may be desired, but I would be interested if anyone can comment on aspects of the historical context which mitigated the impact of Hero's discoveries.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="98" LastEditDate="2014-11-05T23:56:14.507" LastActivityDate="2014-11-06T20:29:29.070" Title="Could the industrial revolution have happened in Hellenistic Alexandria?" Tags="|ancient-greece|steam-engines|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="267" PostTypeId="1" CreationDate="2014-11-06T00:06:56.760" Score="21" ViewCount="2330" Body="&lt;p&gt;I'm quite desperate to understand the historical motivation and origin of all of the &quot;geometrical&quot; concepts of linear algebra, namely:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The concept of thinking of elements of $\mathbb R^n$ or some other vector space as points in a &quot;space&quot;, and of subspaces as lines and planes.&lt;/li&gt;&#xA;&lt;li&gt;The notions of norm and inner product as generalizations of length and angle.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;More generally, I'm interested in any &lt;strong&gt;detailed histories of linear algebra&lt;/strong&gt;, although my main motivation is still to try and get over my intense phobia of norms and inner products. I found the book &lt;em&gt;The Genesis of the Abstract Group Concept&lt;/em&gt; very helpful with similar troubles about the motivations of group theory, but I can't find anything similar for linear algebra, and brief, superficial synopses on Wikipedia articles just aren't cutting it.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="62" LastEditDate="2014-11-06T01:10:11.887" LastActivityDate="2019-09-20T21:52:34.403" Title="Books on the history of linear algebra" Tags="|mathematics|reference-request|linear-algebra|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="268" PostTypeId="2" ParentId="256" CreationDate="2014-11-06T01:19:28.400" Score="10" Body="&lt;p&gt;I'll give it a try but strictly speaking your conditions exclude pretty much everything.&#xA;Breakthroughs considered such by competent people not prone to exaggeration probably were &quot;real&quot; in some sense, in hindsight perhaps for wrong reasons. What was once considered &quot;real&quot; is not anymore, old models that were seen as advances and made sense in their time are viewed as errors today, due to obliviousness or poor measurement capabilities. Anything that described some phenomenon once would have a modern &quot;successor&quot; that describes that phenomenon, and modern textbooks usually have historical sections describing little known tidbits of times long gone. So examples below may not be what you are looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eudoxian model of &lt;a href=&quot;http://en.wikipedia.org/wiki/Concentric_spheres&quot; rel=&quot;nofollow noreferrer&quot;&gt;homocentric spheres&lt;/a&gt;, first geometric model in astronomy that cleverly reconciled uniform circular motions (required by Pythagorians and Plato for heavenly bodies) with messy and retrograde motions of the planets. Was later supplanted by the epicyclic model of Apollonius that lasted until Copernicus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Tusi_couple&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tusi couple&lt;/a&gt; that solved the problem of representing latitudinal motion without a longitudinal component in epicyclic astronomy. When a circle rolls without slipping inside another circle twice its size all points on its circumference oscillate along straight lines, there is a &lt;a href=&quot;https://www.youtube.com/watch?v=pNe6fsaCVtI&quot; rel=&quot;nofollow noreferrer&quot;&gt;curious video&lt;/a&gt; presenting this as an &quot;optical illusion&quot;. Tusi couple influenced Copernicus, but of course fell into oblivion along with epicyclic astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stahl's &lt;a href=&quot;http://en.wikipedia.org/wiki/Phlogiston_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;phlogiston&lt;/a&gt; allowed to treat heat exchange and combustion quantitatively, but was eventually rejected when Lavoisier clarified the oxidation process. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cuvier's &lt;a href=&quot;http://en.wikipedia.org/wiki/Georges_Cuvier#Catastrophism&quot; rel=&quot;nofollow noreferrer&quot;&gt;catastrophism&lt;/a&gt;, a theory explaining apparent replacement of species in the fossil record prior to Darwin's evolution theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gordan's &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF00327448&quot; rel=&quot;nofollow noreferrer&quot;&gt;construction of invariants of binary forms&lt;/a&gt; at the end of 19-th century earned him the title &quot;the king of invariants&quot;. Unfortunately, his (constructive) methods could not be extended beyond binary forms. After (non-constructive) Hilbert's basis theorem the classical invariant theory, along with Gordan's result, fell into obscurity. &quot;This is not mathematics; this is theology&quot; is anecdotally attributed to Gordan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these examples have a common theme. A breakthrough is made in a framework later replaced by a more advanced one, into which it does not translate.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-06-18T17:40:58.400" LastActivityDate="2017-06-18T17:40:58.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="269" PostTypeId="1" AcceptedAnswerId="270" CreationDate="2014-11-06T06:58:45.483" Score="10" ViewCount="358" Body="&lt;p&gt;In the Science Daily press release &lt;a href=&quot;http://www.sciencedaily.com/releases/2013/05/130501112855.htm&quot;&gt;New evidence on how fluoride fights tooth decay&lt;/a&gt; (ACS, 2013), they report on recent research that may explain how fluoride strengthens tooth enamel.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The report describes new evidence that fluoride also works by impacting the adhesion force of bacteria that stick to the teeth and produce the acid that causes cavities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When and by who first discovered that fluoride can prevent tooth decay?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-06T08:17:42.267" Title="When and by who was fluoride found to prevent tooth decay?" Tags="|biology|medicine|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="270" PostTypeId="2" ParentId="269" CreationDate="2014-11-06T08:17:42.267" Score="13" Body="&lt;p&gt;This is an interesting case that might have never been discovered had it not been for another effect of high fluoride levels in water. Here's the (relatively well-known) history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1901, after graduating from dental school, Dr. Frederick McKay moved from the east coast USA to Colorado Springs, Colorado to start a dental practice. When he reached there, he noticed that most of the children had very darkened, ugly teeth. This condition is now understood to be &lt;a href=&quot;http://en.wikipedia.org/wiki/Dental_fluorosis&quot;&gt;dental fluorosis&lt;/a&gt;. The stains were permanent and appeared severe, but there was no mention of such condition in the dental literature. McKay eventually gathered enough interest in the condition, then called &quot;Colorado Brown Stain&quot;, to get the interest of a more senior researcher &lt;a href=&quot;http://en.wikipedia.org/wiki/Greene_Vardiman_Black&quot;&gt;Dr. G.V. Black&lt;/a&gt;, who was initially skeptical but quickly got interested after observing the condition and realizing that it affected about 90% of the children in the area. Black and McKay experimented on this until Black's death in 1915, after which McKay continued. One of the surprising results of these experiments was that the children with this condition were actually &lt;em&gt;less&lt;/em&gt; prone to cavities and tooth decay than others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1923, McKay was able to perform an experiment that demonstrated that something in the water was causing the condition. The town of Oakley, Idaho had just installed a new water pipeline, and the Colorado Brown Stain had begun to show up there afterwards. McKay already had a hunch the water might have something to do with it, so he convinced the town's leaders to use a different water source. Within a few years the condition had all-but disappeared. McKay still didn't know what the difference was in the water, but after he and Dr. Grover Kempf from the United States Public Health Service investigated a third case in Bauxite, Arkansas, they were sure it was in the water.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, H. V. Churchill, an industrial chemist from &lt;a href=&quot;http://en.wikipedia.org/wiki/Alcoa&quot;&gt;ALCOA&lt;/a&gt;, was worried that the results would hurt his business, so he decided to collect samples from Bauxite and run more sophisticated tests. The results were surprising: high levels of fluoride, which was not known to be present in the water supply at all at the time. After some further tests of other locations in collaboration with McKay, in 1931 they had reached the surprising conclusion that indeed this was caused by high levels of fluoride.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The application of this wasn't fully realized until the 1940s, when more precise measurements of fluoride levels in water showed that sufficiently low levels only very rarely caused fluorosis. &lt;a href=&quot;http://en.wikipedia.org/wiki/Henry_Trendley_Dean&quot;&gt;Dr. Henry Trendley Dean&lt;/a&gt; came to the realization that at these low levels, cavity reduction demonstrated by Black and McKay might still be possible. After performing large-scale tests in several cities (the first of which was Grand Rapids, Michigan), this was definitively demonstrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like I said, this story is fairly well-known and documented in several locations. &lt;a href=&quot;http://en.wikipedia.org/wiki/Water_fluoridation_in_the_United_States#History&quot;&gt;Wikipedia&lt;/a&gt; has an article on fluoridation with some history. More extensive histories are compiled by the &lt;a href=&quot;http://www.nidcr.nih.gov/oralhealth/topics/fluoride/thestoryoffluoridation.htm&quot;&gt;National Institute of Dental and Craniofacial Research&lt;/a&gt;, by the &lt;a href=&quot;http://www.cdc.gov/mmwr/preview/mmwrhtml/mm4841a1.htm&quot;&gt;Centers for Disease Control and Prevention&lt;/a&gt;, and by &lt;a href=&quot;http://www.nature.com/bdj/journal/v199/n7s/full/4812863a.html&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="58" LastActivityDate="2014-11-06T08:17:42.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="271" PostTypeId="2" ParentId="267" CreationDate="2014-11-06T11:51:19.980" Score="9" Body="&lt;p&gt;This is specifically about the history of linear algebra, &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Matrices_and_determinants.html&quot;&gt;history of Matrices and determinants&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2014-11-06T11:51:19.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="272" PostTypeId="2" ParentId="266" CreationDate="2014-11-06T12:25:55.117" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lucio_Russo&quot;&gt;Lucio Russo&lt;/a&gt; contends in his &lt;em&gt;The Forgotten Revolution&lt;/em&gt; that a revolution could have been happened in the &lt;strong&gt;early&lt;/strong&gt; Hellenistic period, that Romans destroyed this (seeing no obvious advantage: even the description of Archimedes' works in Syracuse was added later and seen more as a mythical history) and that the works of Hero were just a copy of what was made two centuries earlier. This would explain why his machines were mostly toys.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't buy all the theses of Russo, but if you are interested in that period of history it is worth a serious reading.&lt;/p&gt;&#xA;" OwnerUserId="81" LastActivityDate="2014-11-06T12:25:55.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="273" PostTypeId="2" ParentId="267" CreationDate="2014-11-06T12:59:27.683" Score="5" Body="&lt;p&gt;From your sentence &quot;my main motivation is still to try and get over my intense phobia of norms and inner products&quot; I conclude that you need first of all a good book in Linear algebra itself, rather then history of linear algebra. In English, I recommend the textbook of P. Lax. There is a nice book of Dieudonne Algèbre linéaire et géométrie élémentaire (there is an English translation) which gives an exposition of high school geometry from the linear algebra point of view. Essentially this is the book which does all linear algebra in dimensions 2 and 3. That is elementary geometry, only exposed in a modern way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the history of linear algebra there is another book of Dieudonne, Abrege d'histoire des mathematiques, vol. I which explains genesis of these notions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I have to repeat that genesis was quite complicated and convoluted, before the modern clarity and simplicity were reached. So in this particular case, I recommend NOT to follow the historical development if your problem is to understand the linear algebra itself. &#xA;Only AFTER you overcome your &quot;fobia of norms and inner products&quot; you may read some of this history with a profit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Another good book is MR1885576 &#xA;Givental, Alexander&#xA;Linear algebra and differential equations. &#xA;Berkeley Mathematics Lecture Notes, 11. American Mathematical Society, Providence, RI; Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 2001.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It teaches you linear algebra in dimension 2. That is the linear algebra part covers the SAME material as middle school geometry course. Only in the modern language. If you had a geometry course at school, there must be nothing unfamiliar to you in linear algebra in dimension 2.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-06T20:42:14.860" LastActivityDate="2014-11-06T20:42:14.860" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="274" PostTypeId="2" ParentId="266" CreationDate="2014-11-06T13:26:49.677" Score="11" Body="&lt;p&gt;This is a very interesting question which occupied me for a long time. I agree with L. Russo that a &quot;scientific revolution&quot; really happened in Hellenistic Greece. It actually happened&#xA;2 centuries before the time of Hero, at the time of the first Ptolemy's. Hero himself was probably not a great inventor, he is famous mostly because his book survived. Unlike the books of great scientists of the previous  epoch. The fact is that scientific revolution started and finished in the period of less than 200 years. At the time of Hero science in Alexandria was already in a deep decline (if you compare with the period 200 years before). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no clear explanation why this happened, but this story shows that for an industrial revolution, scientific revolution is not sufficient. One needs some social conditions for an industrial revolution, and these were not present at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scientific inventions (of the time BEFORE Hero, when the true scientific revolution happened) were actually used in technology, especially in the military technology. Advanced military engines, enormous ships were built in the Hellenistic era.&#xA;(See &quot;Helepolis&quot;, &quot;Leonrophoros&quot;, &quot;Greek and Roman artillery&quot; in Wikipedia, as some notable examples). But only scattered hints of this remain in the surviving literature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that decline of science and technology coincided with the time of the Roman conquest. But I am not sure that the Roman conquest was the real reason of this.&#xA;(If the Hellenistic society were an industrialized society, how would the Romans conquer it?) Another thing is that the Romans borrowed and adopted almost all Hellenistic culture,&#xA;why did not they adopt also the science and technology?&#xA;The Romans annexed Egypt shortly before the time when Hero lived. Science was already in decline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The school of Alexandria was not destroyed by the Romans, it existed for several centuries&#xA;after the Roman conquest, and in the second century AD it produced Ptolemy,&#xA;who lived in the times of Trajan, and in the 3-d century there was Diophantus. &#xA;Science in Alexandria completely disappeared only with the spread of Christianity. This shows that the Romans did not &quot;eliminate&quot; or prosecute Science, they tolerated it. They just did not care about it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is really strange. There were many well-educated Romans, some of them even wrote on science (Pliny, Strabo, Plutarchus), but they were not interested in doing it! You cannot find any true mathematicians or astronomers in the whole history of Rome, though there were certainly some engineers. (Of course, Ptolemy lived in the Roman empire, and probably was a Roman citizen, but nobody counts him as a &quot;Roman&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The short period of explosive development of technology ended a generation before Hero. &#xA;(And there was apparently a short revival about 100 years later, I mean Ptolemy).&#xA;And on my opinion, the reason why science did not lead to a real industrial revolution lies somewhere in the social organization of society. Whether this was slavery or not, I am not sure. But it looks like &quot;capitalism&quot; is a necessary prerequisite for a true&#xA;industrial revolution:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. On my opinion, another good example, that great scientific discoveries do not necessary lead to industrial revolutions, is the history of science in China. The ancient Chinese invented paper, and according to some sources, even the printing press. This on my opinion is a much greater invention than a steam engine. Printing was invented in Europe in 15 century,&#xA;and look at the consequences! And compare them to the consequences of these inventions in China. Same in the military area: the Chinese invented gunpowder and rockets, and probably guns. And what happened to all this artillery when the Europeans arrived to the Far east? Chinese had to adopt European technology. There are many examples like this. The military of Byzantine empire was probably very advanced. They had advanced ships and they had this mysterious &quot;Greek fire&quot;. But they were defeated by a less technologically advanced civilization, and all their inventions were forgotten. I think something similar happened to the Hellenistic civilization.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-06T20:29:29.070" LastActivityDate="2014-11-06T20:29:29.070" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="275" PostTypeId="1" CreationDate="2014-11-06T14:50:51.463" Score="6" ViewCount="86" Body="&lt;p&gt;How and when was the significance of partial melting understood to be a significant form of melt production and could explain the variability in melt (and hence igneous rock) composition?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="84" LastEditDate="2014-11-26T11:13:38.830" LastActivityDate="2022-04-19T16:24:44.820" Title="Significance of partial melting as a form of melt production" Tags="|earth-sciences|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="276" PostTypeId="1" AcceptedAnswerId="277" CreationDate="2014-11-06T17:17:19.373" Score="22" ViewCount="4704" Body="&lt;p&gt;Children learn counting things, naturally like, 1, 2, 3, ... and so on. Because it seems obvious to them. But, zero is something we need to teach them about.&#xA;As far as my understanding goes zero was introduced after quite some time since the birth of mathematics as a science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, not sure when and where?&lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorUserId="8151" LastEditDate="2018-11-27T08:41:03.987" LastActivityDate="2019-04-29T05:14:12.570" Title="When was zero actually introduced in mathematics?" Tags="|mathematics|zero|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="277" PostTypeId="2" ParentId="276" CreationDate="2014-11-06T17:46:50.727" Score="25" Body="&lt;p&gt;&lt;strong&gt;TL;DR: The modern zero was born in India, in the latter half of the first millennium.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll start with &lt;a href=&quot;https://en.wikipedia.org/wiki/0_(number)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, and then work with some better sources. But first, from the Wikipedia article,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1740 BCE, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But that's not really what zero is used for today. Let's move a little.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Babylonian placeholder was not a true zero because it was not used alone.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Moving on a millennium or two, we go to India.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The concept of zero as a number and not merely a symbol or an empty space for separation is attributed to India, where, by the 9th century AD, practical calculations were carried out using zero, which was treated like any other number, even in case of division. The Indian scholar Pingala (circa 5th–2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code. He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero or void.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But perhaps most importantly of all,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe), written in 628 AD.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note, however, that he used some concepts little differently:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In some instances, his rules differ from the modern standard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One key example of this is that for him, $\frac{0}{0}=0$ that is, $0$ divided by $0$ is $0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Wikipedia indicates it was sometime in the 800s that zero got its start.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I'd rather not &lt;a href=&quot;https://hsm.meta.stackexchange.com/questions/83/will-we-consider-wikipedia-an-acceptable-source&quot;&gt;use Wikipedia&lt;/a&gt; as my main source here, so I'll now go elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://yaleglobal.yale.edu/about/zero.jsp&quot; rel=&quot;noreferrer&quot;&gt;This site&lt;/a&gt; confirms that India was possibly the birthplace of the modern zero, and also confirms Brahmagupta's rules:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Brahmagupta, around 650 AD, was the first to formalize arithmetic operations using zero. He used dots underneath numbers to indicate a zero. These dots were alternately referred to as 'sunya', which means empty, or 'kha', which means place. Brahmagupta wrote standard rules for reaching zero through addition and subtraction as well as the results of operations with zero. The only error in his rules was division by zero, which would have to wait for Isaac Newton and G.W. Leibniz to tackle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It goes on to explain that the concept of zero was studied more extensively in Europe hundreds of years later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note: &#xA;&lt;a href=&quot;http://www.storyofmathematics.com/indian_brahmagupta.html&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt; is a good outline of Brahmagupta, his life, and his work.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.scientificamerican.com/article/history-of-zero/&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt;, we have the a fact listed that Indians used zero in the 5th century:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It began to take shape as a number, rather than a punctuation mark between numbers, in India in the fifth century A.D., says Robert Kaplan, author of The Nothing That Is: A Natural History of Zero (Oxford University Press, 2000). &quot;It isn't until then, and not even fully then, that zero gets full citizenship in the republic of numbers,&quot; Kaplan says.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And in the teaser for &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0195142373&quot; rel=&quot;noreferrer&quot;&gt;Kaplan's book&lt;/a&gt;, we find&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We follow the trail to the East where, a millennium or two ago, Indian mathematicians took another crucial step. By treating zero for the first time like any other number, instead of a unique symbol, they allowed huge new leaps forward in computation, and also in our understanding of how mathematics itself works.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The consensus of these three sources is that while ancient cultures used &lt;em&gt;some&lt;/em&gt; concept of &quot;nothing&quot; or a &quot;placeholder&quot;, the idea of 0 as a number can be traced back to India, sometime in the second half of the first millennium. I'd recommend sifting through &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, possibly the most comprehensive history of 0 I was able to find. It agrees with all the information on India presented above, and talks about one of the earliest inscriptions utilizing 0: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The usage here seems to be more of a placeholder, though.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;By the way, the answers to &lt;a href=&quot;https://hsm.stackexchange.com/questions/4/ancient-chinese-numbering-system&quot;&gt;this question&lt;/a&gt; agree with the India-was-first claim; one uses the Yale source I used here.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-12-31T17:24:46.483" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="278" PostTypeId="1" AcceptedAnswerId="280" CreationDate="2014-11-06T17:49:24.063" Score="10" ViewCount="1326" Body="&lt;p&gt;It was in lime light for more than a century (late 18th century to late 19th century). They say it was inspected and found fake. But, what was fake about that. There are stories that this machine won most of the matches it played. It didn't just won chess games, but it was able to complete the knight's tour(a famous chess puzzle) easily. Which most experienced chess players of that time struggled with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does that mean Kempelen (who created this) was himself a genius or sort of world champion chess player of that time ?&lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorUserId="62" LastEditDate="2014-11-06T18:08:09.577" LastActivityDate="2014-11-06T18:52:57.230" Title="What is the truth behind &quot;The Turk&quot; (the mechanical chess machine)?" Tags="|mathematics|chess|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="279" PostTypeId="1" CreationDate="2014-11-06T18:40:03.597" Score="9" ViewCount="149" Body="&lt;p&gt;One of the most disturbing trends in modern politics (at least in America - I can't comment on other countries) is public distrust of science and scientists.  There is by now an overwhelming scientific consensus behind the reality of anthropogenic climate change and biological evolution, but many non-scientists remain skeptical of these ideas - 30% of Americans in the case of anthropogenic climate change (according to a 2012 study by Yale/George Mason) and 33% of Americans in the case of evolution (according to a 2013 Pew survey).  In both cases the skepticism is probably based in part on the belief espoused by journalists (and &quot;journalists&quot;) that scientists' judgements are corrupted by money, compromised by personal bias, and co-opted by activist groups.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I believe that while isolated cases of corruption or bias - even by influential scientists - are not as rare as one might hope, typically the broader scientific community identifies and purges the scientific malpractice before it has the chance to grow into a consensus.  My question is: are there any historical counter-examples to this statement?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To be clear, I want to exclude the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cases in which a single scientist or small group of scientists illegitimately manipulated results but were rebuked before the results were widely accepted.&lt;/li&gt;&#xA;&lt;li&gt;Cases in which scientific consensus coalesced for legitimate reasons around an idea that was later proved to be false.  For instance, there was a broad scientific consensus behind Newton's theory of gravity even though it ultimately proved to be inaccurate on large scales.&lt;/li&gt;&#xA;&lt;li&gt;Cases in which &quot;elder statesmen&quot; in a particular field were slow to react to a breakthrough.  For instance, many prominent physicists were slow to accept quantum mechanics in the early 20th century for philosophical reasons, but I wouldn't count this as systematic corruption.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I've thought of a few possible examples, but I just don't know the history well enough to properly evaluate them.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Geocentrism (corruption by the Catholic church?)&lt;/li&gt;&#xA;&lt;li&gt;Eugenics (corruption by racial activist groups?)&lt;/li&gt;&#xA;&lt;li&gt;Health effects of smoking (corruption by tobacco companies?)&lt;/li&gt;&#xA;&lt;li&gt;Health effects of dietary fat (corruption by farming lobbies?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;With these examples I don't really know how much of a consensus there was or the extent to which the consensus took hold among scientists.  Of course, I would be interested in any other examples.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-11-06T18:51:45.633" LastActivityDate="2014-11-06T23:05:48.087" Title="Has any broad scientific consensus ever been based on corruption rather than evidence?" Tags="|bias|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="280" PostTypeId="2" ParentId="278" CreationDate="2014-11-06T18:52:57.230" Score="7" Body="&lt;p&gt;I vaguely remember learning about the Turk in a documentary about Gary Kasparov and his match with Deep Blue. It seemed really impressive, even if it wasn't actually an automated chess-playing robot.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Turk&quot; rel=&quot;noreferrer&quot;&gt;Here's&lt;/a&gt; the relevant Wikipedia article. Once again, I won't use it as a primary source, but I think that the following passage from Wikipedia gets to the heart of the matter:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Turk was in fact a mechanical illusion that allowed a human chess master hiding inside to operate the machine. With a skilled operator, the Turk won most of the games played during its demonstrations around Europe and the Americas for nearly 84 years, playing and defeating many challengers including statesmen such as Napoleon Bonaparte and Benjamin Franklin. The operator(s) within the mechanism during Kempelen's original tour remains a mystery.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'll quote a passage from &lt;a href=&quot;http://www.uh.edu/engines/epi2765.htm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, summarizing how the machine worked:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The secret of the Mechanical Turk was kept for over 50 years – the machine was an elaborate illusion, and contained an ingeniously hidden compartment that housed a human operator. This hidden chess master could observe the position on the chessboard above, and manipulate the movements of the Turk.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;http://chess.about.com/od/chesshistory/fl/The-Turk-A-Chess-Playing-Machine.htm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, we have a slightly more in-depth summary (although I'm not too happy about the source; you can disregard it if you want):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;How It Worked&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although many people had guessed that the machine actually had an operator working from the inside, few were able figure out exactly how it worked. There were guesses that the operator must be a child to fit inside the cabinet, or that someone was actually poised partially inside the figure that moved the pieces.&lt;/p&gt;&#xA;&lt;p&gt;In fact, the operator inside the cabinet sat on a sliding seat, allowing them to adjust their position as a presenter opened doors to show what was supposedly going on inside the machine. Over the years, a number of talented chess players were responsible for playing the games, which helps explain why the machine rarely lost except when up against elite competition. From inside the cabinet, the operator could see a magnet version of the board that reacted to the movement of the pieces above (each had a strong magnet in its base) so that they were able to keep track of the position. Levers allowed for the movement of the Turk's arm and the opening and closing of its hand, which made it possible to move the pieces on command.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://en.chessbase.com/post/von-kempelen-s-che-turk-recreated&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, we get a re-creation of the machine. Here (from the site) is part of the mechanism controlling the arm:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://en.chessbase.com/portals/4/files/images2/2004/turk15.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is a picture of the arm mechanisms:&#xA;&lt;img src=&quot;https://en.chessbase.com/portals/4/files/images2/2004/turk16.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, I go to &lt;a href=&quot;http://ed5015.tripod.com/SkepticsTurk104.htm&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;, where we find some key information:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The concealed chess expert was cramped, hot, had to operate various levers, and worked by candlelight. But he had two advantages over his opponents:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;He almost always had White and so made the first move;&lt;/li&gt;&#xA;&lt;li&gt;He could explore, although only briefly, alternative variations on his second chess set, in effect cheat.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-06T18:52:57.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="281" PostTypeId="2" ParentId="267" CreationDate="2014-11-06T21:44:37.453" Score="11" Body="&lt;p&gt;Are you familiar with Michael J. Crowe's book, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486679101&quot;&gt;A History of Vector Analysis&lt;/a&gt;?&#xA;While I haven't read the book &lt;a href=&quot;https://www.math.ucdavis.edu/~temple/MAT21D/SUPPLEMENTARY-ARTICLES/Crowe_History-of-Vectors.pdf&quot;&gt;this article&lt;/a&gt; is well worth a read, and it seems to be a good summary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, vector analysis is the precursor to linear algebra, so it won't directly address your question. Crowe does discuss briefly Grassmann's &lt;em&gt;Ausdehnungslehre&lt;/em&gt;, one of the roots of (n-dimensional) linear algebra, and the (somewhat convoluted) history of the inner product.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-06T21:44:37.453" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="282" PostTypeId="2" ParentId="279" CreationDate="2014-11-06T22:44:09.420" Score="2" Body="&lt;p&gt;To the cases you mention, I can add a wide spread belief in astrology, which spread in the Hellenistic world almost at the same time when a true scientific revolution happened there.&#xA;Later, there was a large public demand for astrology and none for the real science, so existing astronomers and mathematicians had to do astrology to make their living. The last great astronomer who did this was Kepler, and it does not matter for this discussion, whether believed he in it or not. This was a normal occupation of a mathematician at that time, giving him money, because the &quot;general public&quot; believed it. Actually it exists to this day, and many journals and newspapers publish horoscopes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Situation with creationism is similar. For most ordinary people,&#xA;religion is more important than science, and wherever they see any contradiction &#xA;between the two they tend to side with religion. But this was always so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heliocentrism is a different kind of controversy on my opinion. I do not think that general pubic cared much about astronomical theories. Here the controversy was triggered by Galileo&#xA;(who started to popularize science by writing his Dialogues in Italian, for the general public, instead of using the language of science that is Latin) and by the Church establishment on the other side. The Church did not interfere by the way, while the discussion was limited to professional astronomers. It found necessary to interfere when&#xA;all this was brought to the general public, in fear that its authority may be undermined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar controversies happened more recently in Soviet Union when the government condemned genetics and other things for ideological reasons, with very bad consequences for science and agriculture. And in South Africa, where the government denied that AIDS is caused by an HIV virus, with disastrous consequences. &#xA;This type of controversy has nothing to do with the degree of education of general public. (It was actually quite high in Soviet Union, and of course most people did not care whether the acquired features are inherited or not:-) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your final two examples are controversial in a different sense. Current widespread propaganda definitely exaggerates the effects of smoking (especially of the so-called &quot;secondary smoking&quot;). The reasons of this campaign are definitely outside of the scope of&#xA;this list which is dedicated to history of science. This is more a kind of current politics, rather than history of science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in general, all this should not be surprising. After all, scientists are payed either by the public (in a democracy) or by the rulers. So they must please somehow those who pay:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-06T22:49:42.293" LastActivityDate="2014-11-06T22:49:42.293" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="283" PostTypeId="2" ParentId="279" CreationDate="2014-11-06T23:05:48.087" Score="2" Body="&lt;p&gt;I cannot think of any major international scientific consensus around the some ideas stemming from corruption or bias in the recent history (&gt; 1950). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;STEEN, RG. &lt;a href=&quot;http://scholar.google.com/scholar?cluster=10514083804833765043&amp;amp;hl=en&amp;amp;as_sdt=0,22&quot; rel=&quot;nofollow&quot;&gt;Retractions in the scientific literature: is the incidence of research fraud increasing?&lt;/a&gt;. Journal Of Medical Ethics. England, 37, 4, 249-253, Apr. 2011. ISSN: 1473-4257.&lt;/em&gt; analyzes PubMed&#xA;database between 2000 and 2010 and points out that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It may be noteworthy that, among 742 retracted articles, only one&#xA;  retraction notice described biased viewpoint as the reason for&#xA;  retraction (“authors have confounded the elegant analysis with a&#xA;  political viewpoint representing only one side”).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The same article relates some complains regarding some industry-funded research:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is a fear that clinical trial results may be interpreted in a&#xA;  distorted manner by certain authorsdperhaps authors in the pay of a&#xA;  pharmaceutical companydand one study concluded that 40% of randomised&#xA;  clinical trials show evidence of ‘spin’ in the published text.15 These&#xA;  findings feed a fear that medical journals have become an extension of&#xA;  the marketing department of pharmaceutical companies. Yet bias is in&#xA;  the eye of the beholder, and the paper entirely free of bias has not&#xA;  yet been written. We note that one study concluded that only 4% of&#xA;  retractions for misconduct had declared pharmaceutical sponsorship,&#xA;  while another study reported that pharmaceutical sponsorship of&#xA;  obesity-related randomised clinical trials was actually associated&#xA;  with higher quality reporting of results.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In medicine there are some cases of misconducts that may have resulted in some temporary consensus among physicians, &lt;a href=&quot;http://articles.mercola.com/sites/articles/archive/2012/07/12/drug-companies-on-scientific-fraud.aspx&quot; rel=&quot;nofollow&quot;&gt;example&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Vioxx is perhaps one of the better examples of what can happen when a&#xA;  drug is manufactured and sold under false pretenses. It killed more&#xA;  than 60,000 people in just a few years time, before it was removed&#xA;  from the market. In the case of Vioxx, there are lingering questions&#xA;  about the soundness of the research backing the drug in the first&#xA;  place. Back in 2008, Dr. Joseph S. Ross of New York's Mount Sinai&#xA;  School of Medicine came across ghostwritten research studies for Vioxx&#xA;  while reviewing documents related to lawsuits filed against Merck.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;According to an April 16, 2008 article on MedHeadlinesiii:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;&quot;In about 96 journal publications, Ross and his colleagues discovered&#xA;  internal Merck documents and e-mail messages pertaining to clinical&#xA;  study reports and review articles, some of which were developed by the&#xA;  company's marketing department, not its scientific department. In&#xA;  others, there is little evidence that the authors recruited for the&#xA;  report made substantial contribution to the research itself. ... Some&#xA;  of the authors listed in the Merck study reports of concern...&#xA;  question the true nature of ghostwriting. One neurologist originally&#xA;  listed as &quot;External author?&quot; and then listed as Dr. Leon J. Thal, of&#xA;  the University of California, San Diego in the final draft, died a&#xA;  year ago in an airplane crash.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;An editorial published in the Journal of the American Medical&#xA;  Association (JAMA)iv that year by Drs. Psaty and Kronmal also&#xA;  questioned whether Merck might have deliberately manipulated dozens of&#xA;  academic documents published in the medical literature, in order to&#xA;  promote Vioxx under false pretenses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-11-06T23:05:48.087" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="284" PostTypeId="1" AcceptedAnswerId="288" CreationDate="2014-11-06T23:18:18.283" Score="22" ViewCount="2919" Body="&lt;p&gt;Gauss is probably the most famous mathematician today, his face even appears on one of European bills. But he did not have a charismatic personality, like Galois, nor was particularly interested in self-promotion, more of a recluse. Also Euler, Galois and Riemann had stronger influence on mathematics after them than Gauss did. So how did he come to be the &quot;face&quot; of mathematics to general public?&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2020-06-24T22:01:21.753" LastActivityDate="2020-06-24T22:01:21.753" Title="How did Gauss become the &quot;the prince of mathematicians&quot;?" Tags="|mathematicians|gauss|depiction-of-scientists|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="285" PostTypeId="2" ParentId="284" CreationDate="2014-11-07T01:08:27.073" Score="14" Body="&lt;p&gt;Gauss portrait was printed on German paper money until recently (10 Mark note), before the introduction of the Euro. However my experience shows that the &amp;quot;general public&amp;quot; in many European countries does not recognize the people whose portraits are printed on their money:-) You can still buy some of these notes here &lt;a href=&quot;http://www.kleinbottle.com/gauss.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.kleinbottle.com/gauss.htm&lt;/a&gt;.&#xA;(I understand that the rules of this site probably prohibit advertisement, but I think this is an exceptional case, anyway you may edit my message and remove the reference:-)&#xA;This note is very well designed, showing some of the main achievements of Gauss in applied math and geodesy. Look at it carefully.&lt;/p&gt;&#xA;&lt;p&gt;No other mathematician that I know (except Euler) was pictured on the money.&#xA;(Hypparchus, the &amp;quot;father of astronomy&amp;quot;, is depicted on some coins of his country, which on&#xA;my opinion is truly amazing, and shows that in the Hellenistic world they also valued science).&lt;/p&gt;&#xA;&lt;p&gt;Gauss is also known for his reluctance to publish. However at a very young age he pubished his book &amp;quot;Disquisitiones Arithmeticae&amp;quot;, and this book is of such quality that it is still read by working mathematicians, and one of the recent Fields medalists, said that his main discovery is inspired by reading of this book (this is what he said himself).&lt;/p&gt;&#xA;&lt;p&gt;Another achievement of young Gauss was a complete clarification of the question which regular polygons can be constructed by a ruler and a compass. This brought him immediate fame. Unlike the Disquisitiones, this question can be understood by any one with a primary school education. So he's got an unofficial title of Princeps Mathematicorum&#xA;(King of Math), and a very decent position.&lt;/p&gt;&#xA;&lt;p&gt;After that, he lived a long life, and published very little in Math (I mean little in comparison with Euler or Cauchy). His main work however was in astronomy and geodesy, and he computed the orbits of the newly discovered small planets.&#xA;Again an achievement which &amp;quot;general pubic&amp;quot; can understand.&lt;/p&gt;&#xA;&lt;p&gt;He also did a lot in Physics, invented the first working electric telegraph, etc.&#xA;Having a very long career, and very diversified contributions, besides mathematics,&#xA;attracted to him more public opinion than to some other mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, Euclid, Archimedes, Newton, Leibniz, Euler and Riemann perhaps had stronger influence on the development of mathematics, but these names are also reasonably well recognized by the (educated part of the) general public. I do not notice that Gauss's name is much more recognized than these 6.&lt;/p&gt;&#xA;&lt;p&gt;Correction about money. Quick Google search shows many modern banknotes with Abel, Cauchy, Descartes, Euler, Newton, Luca Pacioli and Turing, not even mentioning coins.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-24T01:10:34.027" LastActivityDate="2020-06-24T01:10:34.027" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="286" PostTypeId="2" ParentId="44" CreationDate="2014-11-07T01:27:33.370" Score="2" Body="&lt;p&gt;On my opinion, this is an example of usual aberration which often happens when statistics is applied without careful thinking. The percentage of polymath's is probably the same.&#xA;To the names already mentioned, let me add Terence Tao, the most famous modern case. But there are many others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for this aberration is the following. We remember now only few 18 century mathematicians. I doubt that an average modern mathematician will immediately list 20, not speaking of the &quot;general public&quot;. These are the best of the best. Most of the rest are not&#xA;remembered. Not surprisingly, percentage of polymaths among them is large.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern mathematicians are not that famous yet; their biographies are not written,&#xA;general public does not know them yet:-) So many polymath's among them are simply not&#xA;so well known to the general public. But I suspect that the percentage is the same.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-07T01:27:33.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="287" PostTypeId="1" AcceptedAnswerId="2021" CreationDate="2014-11-07T02:46:55.917" Score="12" ViewCount="503" Body="&lt;p&gt;It is well known that every construction that can be performed by a compass and a ruler can be also performed by a compass only. This is a good (and difficult) exercise in elementary geometry. My question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When did mathematicians start investigating this question ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The literature on the subject that I know is from the 19th and 20th centuries. However, I also know that there was a time when this question had serious practical applications. And this was much earlier (until the second half of the 18th century).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The practical application I am talking about is the following. Since the 16th century, astronomers&#xA;began to measure angles to high accuracy. For this they used divided circles made of metal (bronze, brass). The circles were divided to fractions of a minute. How did you do this?&#xA;(Those who never tried may think that this is a trivial task, but it is not). Late 18th century encyclopedias have long articles called &quot;Division of the circle&quot;, where they explain in great detail how this was done and the history of the business. There is also literature written by some of these master dividers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One feature of the task is that one cannot really use a ruler for very high accuracy constructions. One of these masters puts it clearly: &quot;You cannot really find an intersection of&#xA;two lines with a ruler&quot;. They used a compass which is a much more accurate instrument.&#xA;Even division of a straight ruler was performed with a compass of very large radius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second half of 18th century this noble trade suddenly came to the end: a dividing engine was invented which permitted to divide an instrument circle hundreds times faster than by hand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. It is another interesting question: to what extent could they to this in antiquity? There is ONE archeological find which shows that this business existed in Hellenistic world: it is the Antikythera mechanism.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT (after the answer of Uri Zarfaty). I learned about this problem from the writings of &#xA;Bird, a famous instrument maker, a &quot;master divider&quot; as they called him. It is he who explained that &quot;You cannot find the intersection of two lines with a ruler&quot;. He meant &quot;this is impossible in practice, with sufficiently high precision&quot;. Now we learn that this very same Bird is mentioned in the introduction of Masceroni's paper!&#xA;So my conjecture is correct! Bird lived long enough to see the invention of the dividing engine which made his noble art obsolete.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-01T18:32:56.130" LastActivityDate="2015-03-01T18:32:56.130" Title="Division of the circle and compass constructions" Tags="|mathematics|astronomy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="288" PostTypeId="2" ParentId="284" CreationDate="2014-11-07T03:08:35.383" Score="21" Body="&lt;p&gt;I was originally looking for the first documented reference of another contemporary mathematician calling Gauss the prince to no avail. What we do know is that he was considered, by at least many of his contemporaries (by some accounts all), to be the greatest among them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do see in &lt;em&gt;The Beginnings and Evolution of Algebra, Volume 19  I. G. Bashmakova, G. S. Smirnova&lt;/em&gt; a citation that &lt;em&gt;Disquisitiones Arithmeticae (1801)&lt;/em&gt; &quot;...elevated him to the level of a leading mathematician of his time.&quot; In that source, the author states &quot;&lt;a href=&quot;http://books.google.com/books?id=eBefKDTfmO8C&amp;amp;pg=PA110&amp;amp;lpg=PA110&amp;amp;dq=what+person+called+gauss+Princeps+mathematicorum?&amp;amp;source=bl&amp;amp;ots=P9dI3FSwD6&amp;amp;sig=3ttO24IFdH9rTaap165so12Cqmk&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=Ki9cVKv2Ao3IsASDroCAAg&amp;amp;ved=0CB0Q6AEwADgK#v=onepage&amp;amp;q=what%20person%20called%20gauss%20Princeps%20mathematicorum%3F&amp;amp;f=false&quot;&gt;He was soon to be referred as 'Princeps Mathematicorum'&lt;/a&gt;&quot; (Prince of Mathematicians) after this time. This, at least by this source, dates the introduction of this title or nickname for Gauss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It does not surprise me that his contemporaries thought of him so highly. For every great Gauss story that can be posted here, one more could probably be added. He was pretty active in a variety of fields and topics from the fundamental theorem of algebra, number theory, differential geometry, which according to Charles Boyer in &lt;em&gt;A History of Mathematics c 1991 [pg 505]&lt;/em&gt;, Gauss himself initiated, and the list goes on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For current lack of a contemporary mathematicians ascribing this name to him, in &lt;em&gt;Numbers - A Springer Publication c 1991&lt;/em&gt; it is written that &quot;&lt;a href=&quot;http://books.google.com/books?id=OKcKowxXwKkC&amp;amp;pg=PA111&amp;amp;lpg=PA111&amp;amp;dq=Princeps+mathematicorum&amp;amp;source=bl&amp;amp;ots=sk6rnWvFGw&amp;amp;sig=txwAJauLM-18zohvQMacsVWFEcU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=VC1cVK6CMtaOsQS2loHYDg&amp;amp;ved=0CEkQ6AEwBQ#v=onepage&amp;amp;q=Princeps%20mathematicorum&amp;amp;f=false&quot;&gt;After his death medals were struck in the Kingdom of Hanover at the initiative of the King on which he was described as 'Princeps Mathematicorum', a name by which he had already been called during his lifetime&lt;/a&gt;.&quot; Here are &lt;a href=&quot;http://gausschildren.org/genwiki/index.php?title=File:GaussMedal.jpg&quot;&gt;actual images of the coins&lt;/a&gt;. Note that on these coins I read, &quot;MATHEMATICORVM PRINCIPI&quot;. Either way, close enough, it means the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having your nickname printed on coins by a King at your death is one way to get a nickname to stick, although the coins may not have been necessary in the case of the great Gauss. He was so highly influential in his time, and he was never forgotten even to this very day. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He and Euler have been my two favorites since the earliest days of my personally being influenced by mathematics. It seems that social effervescence and a &quot;charismatic personality&quot; do not carry as much weight in mathematical popularity contests as it might in some other topic. Many of the greatest mathematicians (but not all) might be considered quite awkward individuals by common standards, yet they are the revered and commonly known individuals of our history.&lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2014-11-07T03:31:09.250" LastActivityDate="2014-11-07T03:31:09.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="289" PostTypeId="1" CreationDate="2014-11-07T05:41:53.923" Score="16" ViewCount="2131" Body="&lt;p&gt;In 1555, John Dee was arrested for &quot;calculating&quot;. According to his &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Dee.html&quot;&gt;MacTutor biography&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At this time mathematics in England was considered to be equivalent to the possession of magical powers and Aubrey writes that the authorities had &quot;... burned mathematical books for conjuring books.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In our society it seems astonishing that performing calculations could be taken to be magic. When and why did the association between mathematics and calculation begin, was it peculiar to England, and when and why did it end?&lt;/p&gt;&#xA;" OwnerUserId="165" LastEditorUserId="165" LastEditDate="2014-11-07T05:47:55.370" LastActivityDate="2022-01-23T19:46:28.947" Title="When and why were mathematics and magic considered synonymous in England" Tags="|mathematics|social-context|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="290" PostTypeId="2" ParentId="289" CreationDate="2014-11-07T08:35:14.697" Score="10" Body="&lt;p&gt;Surely it was not peculiar to England. Saint Augustine wrote &quot;The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell&quot; (&lt;em&gt;De Genesi ad Litteram&lt;/em&gt;, Book II, xviii, 37, according to &lt;a href=&quot;https://people.math.osu.edu/easwaran.1/augustine.html&quot;&gt;this source&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that from the IV century a.D. onwards mathematics - besides the bare minimum for everyday calculations - was especially associated with astrology, and therefore with magic. Johannes Kepler and Girolamo Cardano are the latest names which come into my mind, so probably it ended with the beginning of the XVII century.&lt;/p&gt;&#xA;" OwnerUserId="81" LastActivityDate="2014-11-07T08:35:14.697" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="291" PostTypeId="1" AcceptedAnswerId="292" CreationDate="2014-11-07T09:27:24.637" Score="8" ViewCount="237" Body="&lt;p&gt;Of course it is difficult, if not impossible, to know whether classical philosophers that talked about atomism thought that &quot;atoms&quot; could be manipulated by men. However, I was startled to know that even at the beginning the 20th century, there were scientists who disputed the existence of atoms, and that it took the explanation of Brownian effect by Einstein to convince them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My understanding - but I may be wrong - is that the early chemists at the end of the 18th century talked about &lt;em&gt;substances&lt;/em&gt;, non atoms: some chemical compounds were considered basic. So, when did atomism started to gain widespread, even if not global, acceptance?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="24" LastEditDate="2014-11-10T22:11:46.203" LastActivityDate="2014-11-10T22:11:46.203" Title="When was atomism first considered a scientific theory?" Tags="|physics|chemistry|atomic-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="292" PostTypeId="2" ParentId="291" CreationDate="2014-11-07T12:22:19.203" Score="6" Body="&lt;p&gt;We can see the &lt;em&gt;SEP&lt;/em&gt; entry : &lt;a href=&quot;http://plato.stanford.edu/entries/atomism-modern/&quot; rel=&quot;noreferrer&quot;&gt;Atomism from the 17th to the 20th Century&lt;/a&gt; by Alan Chalmers for a useful overview.&lt;/p&gt;&#xA;&lt;p&gt;See in particular the &lt;strong&gt;Concluding Remarks&lt;/strong&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we take atomism to involve the claim that the properties of macroscopic matter arise as a result of the combinations and motions of tiny particles, then it is a position confirmed by the time of the Solvay Conference in 1911 in a way that left little room for sensible doubt.&lt;/p&gt;&#xA;&lt;p&gt;But if we take atomism in a stronger sense, to mean a theory that explains all of the properties of macroscopic matter in terms of underlying particles with specified properties and governed by specified laws, then it must be denied that atomism had reached its objective in 1911.&lt;/p&gt;&#xA;&lt;p&gt;There were identifiable inadequacies and gaps in the specification of the properties of atoms and the electrons and protons that compose them and there were to an increasing extent problematic experimental results that were eventually to lead to a radical change in the laws that were presumed to govern the behaviour of atomic and sub-atomic particles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-07T12:22:19.203" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="293" PostTypeId="1" CreationDate="2014-11-07T12:40:09.293" Score="8" ViewCount="260" Body="&lt;p&gt;I'm asking this question as I've noticed that scientists like Gauss, Newton, Euler, Lagrange etc developed theories in many scientific fields(these ones that I know of were mostly interested in math and physics). When I asked my physics teacher, he told me scientists were &quot;the jack of all trades&quot; back then, they were doctors, mathematicians, physicists, chemists etc. But then I remember that most Ancient Greeks mostly spent their time researching math. So how did the lineage evolve? Also when did science categorize itself into fields and then subfields?&lt;/p&gt;&#xA;" OwnerUserId="191" LastEditorUserId="62" LastEditDate="2014-11-07T19:27:38.030" LastActivityDate="2021-03-18T07:42:47.393" Title="What was the definition of a scientist and how did it evolve? When was science categorized?" Tags="|terminology|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="294" PostTypeId="1" AcceptedAnswerId="302" CreationDate="2014-11-07T13:13:57.000" Score="5" ViewCount="79" Body="&lt;p&gt;It is widely known nowadays that air pollution attenuates incoming light, particularly sunlight through scattering and absorption of certain wavelengths (both in the UV and visible wavelengths).  This attenuation of light is significant to science as it can adversely affect the biosphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When was the optical effects of air pollution first measured, and by whom?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-07T17:40:18.450" Title="When was the optical effects of air pollution first measured, and by whom?" Tags="|atmosphere|optics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="295" PostTypeId="1" AcceptedAnswerId="296" CreationDate="2014-11-07T14:17:36.127" Score="17" ViewCount="611" Body="&lt;p&gt;Cancer is one of the most well known and widespread conditions in the modern world, it can be argued that everyone is affected either directly and indirectly throughout their lives.  According to &lt;a href=&quot;http://www.cancerresearchuk.org/cancer-info/cancerstats/keyfacts/Allcancerscombined/&quot;&gt;Cancer Research UK&lt;/a&gt;, there are over 200 types of cancer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The literature of modern research into cancer (and its treatments) is immense; however, the question here is, &lt;strong&gt;when was cancer (in any form) first recognised, and by whom?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="26" LastEditorDisplayName="user22" LastEditDate="2015-03-01T13:40:16.880" LastActivityDate="2015-03-01T13:40:16.880" Title="When was cancer first recognised, and by whom?" Tags="|medicine|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="296" PostTypeId="2" ParentId="295" CreationDate="2014-11-07T14:39:11.327" Score="9" Body="&lt;p&gt;Let's start with &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_cancer&quot;&gt;the Wikipedia page&lt;/a&gt; on the history of cancer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The earliest known descriptions of cancer appear in seven papyri, discovered and deciphered late in the 19th century. They provided the first direct knowledge of Egyptian medical practice. Two of them, known as the &quot;Edwin Smith&quot; and &quot;George Ebers&quot; papyri, contain descriptions of cancer written around 1600 B.C., and are believed to date from sources as early as 2500 B.C.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Several Greeks are also credited with knowing about cancer (centuries after the Egyptians, of course), namely, Hippocrates and Celsus. More practical developments continued in Europe from the 16th century onwards, when autopsies revealed more about tumors. This was eventually applied to surgeries removing tumors.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://cancer.about.com/od/historyofcancer/a/cancerhistory.htm&quot;&gt;about.com&lt;/a&gt; also mentions Hippocrates (who apparently gave the disease its name) and the ancient Egyptians, noting that the first documented case of cancer occurred in 1500 B.C. It also talks about how autopsies eventually revealed how tumors factored in to the development of cancers. Here's the passage about Egypt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The world's oldest documented case of cancer hails from ancient Egypt, in 1500 b.c. The details were recorded on a papyrus, documenting 8 cases of tumors occurring on the breast. It was treated by cauterization, a method to destroy tissue with a hot instrument called &quot;the fire drill.&quot; It was also recorded that there was no treatment for the disease, only palliative treatment. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.news-medical.net/health/Cancer-History.aspx&quot;&gt;here&lt;/a&gt;, we find more evidence that the disease can be traced back to Egypt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although the word cancer was not used, the oldest description of the disease is from Egypt and dates back to about 3000 BC. It is called the Edwin Smith Papyrus and is a copy of part of an ancient Egyptian textbook on trauma surgery. It describes 8 cases of tumors or ulcers of the breast that were treated by cauterization with a tool called the fire drill. The description adds that there is not treatment for the condition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note the similarity between this passage and the previous one, as well as the date of the papyrus listed here. 1500 years older. It is followed by another passage regarding Hippocrates:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The disease was first called cancer by Greek physician Hippocrates (460-370 BC). He is considered the “Father of Medicine.” Hippocrates used the terms carcinos and carcinoma to describe non-ulcer forming and ulcer-forming tumors. In Greek this means a crab. The description was names after the crab because the finger-like spreading projections from a cancer called to mind the shape of a crab.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, we come to &lt;a href=&quot;http://www.cancer.org/cancer/cancerbasics/thehistoryofcancer/the-history-of-cancer-what-is-cancer&quot;&gt;another very similar page&lt;/a&gt;, with similar text. I suspect some of these sources used each other as sources. Regarding the &quot;any form&quot; part of your recent edit, there's something interesting here:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some of the earliest evidence of cancer is found among fossilized bone tumors, human mummies in ancient Egypt, and ancient manuscripts. Growths suggestive of the bone cancer called osteosarcoma have been seen in mummies. Bony skull destruction as seen in cancer of the head and neck has been found, too.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Evidence suggests that osteosarcoma was one of the first recorded cancers. However, the &lt;em&gt;first&lt;/em&gt; appears to be breast cancer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Our oldest description of cancer (although the word cancer was not used) was discovered in Egypt and dates back to about 3000 BC. It’s called the Edwin Smith Papyrus and is a copy of part of an ancient Egyptian textbook on trauma surgery. It describes 8 cases of tumors or ulcers of the breast that were removed by cauterization with a tool called the fire drill. The writing says about the disease, “There is no treatment.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Funny how that's almost exactly how about.com put it. . . That's because it appears the article used &lt;em&gt;this&lt;/em&gt; page as a source.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-08T22:37:36.030" LastActivityDate="2014-11-08T22:37:36.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="297" PostTypeId="2" ParentId="293" CreationDate="2014-11-07T14:40:37.177" Score="6" Body="&lt;p&gt;Some of the statements in your question are questionable:-)&#xA;Among the &amp;quot;ancient Greeks&amp;quot;: Archimedes was a mathematician, physicist and engineer.&#xA;(Equally famous in all these areas).&#xA;Hypparchus was an astronomer and matematician, doing not only mathematics related to astronomy but also number theory, for example.&lt;/p&gt;&#xA;&lt;p&gt;The legendary Pythagoras, according to the legend, was the founder not only of mathematics but also of physics and music theory. It is he who discovered (according to the legend) the&#xA;relation between the length of the string and the musical tone which it produces: probably the earliest true experimental discovery in physics. He was also the founder of a quasi-religious philosophical sect of Pythagoreans.&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy was an astronomer, mathematician, geographer, physicist and astrologer. His output covers almost all sciences that existed at that time. Hero of Alexandria was a mathematician, physicist and&#xA;engineer. And this list can be continued.&lt;/p&gt;&#xA;&lt;p&gt;In the past, when the total amount of scientific knowledge was relatively small, scientists were not differentiated as in later times. Mathematics and astronomy differentiated at an earlier period (in the ancient times), but until the later part of the 19-th century many mathematicians were also astronomers. (The last great pure mathematician who worked as an astronomer for his living was Pierre Fatou who died in 1930).&lt;/p&gt;&#xA;&lt;p&gt;As scientific knowledge grows, various disciplines gradually separate from each other.&#xA;More it grows, more they separate. Separation of mathematics from physics was a slow process which occupied 18-th, 19-th centuries, and continued well into 20-th century.&lt;/p&gt;&#xA;&lt;p&gt;This process of specialization and separation continues. It is frequent nowadays that people in one area of mathematics do not talk to people in another area of mathematics.&#xA;In my life time &amp;quot;computer science&amp;quot; separated from mathematics, and &amp;quot;logic and set theory&amp;quot; become essentially a separate science.&lt;/p&gt;&#xA;&lt;p&gt;So one cannot say that &amp;quot;science categorized itself into fields and subfieleds&amp;quot; at some definite time. This is a continuous process which started almost at the birth of science and continues today.&lt;/p&gt;&#xA;&lt;p&gt;It is another thing that many mathematicians until 18-th century practiced medicine, law or&#xA;astrology, or had clerical positions.&#xA;One had to make a living somehow. There were VERY few positions for mathematicians. As late as in the second half of 19-th century, two of the most famous British mathematicians, Cayley and Silvester, had to practice law for substantial periods of their careers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-03-16T01:21:10.563" LastActivityDate="2021-03-16T01:21:10.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="298" PostTypeId="2" ParentId="295" CreationDate="2014-11-07T14:49:23.260" Score="3" Body="&lt;p&gt;The Egyptian medical Ebers papyrus (c. 1550 BC. ) mentions, or can be argued to mention what we now would call 'cancer'. When cancer became 'cancer' (I do not know) Here is a link to   &lt;a href=&quot;http://oilib.uchicago.edu/books/bryan_the_papyrus_ebers_1930.pdf&quot; rel=&quot;nofollow&quot;&gt;The Papyrus Ebers&lt;/a&gt; (1930) by Cyril. P Bryan. The plants prescribed as treatments for the illnesses is described in the 21 paragraphs of the last section of the Ebers Papyrus.&lt;/p&gt;&#xA;" OwnerUserId="187" LastActivityDate="2014-11-07T14:49:23.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="299" PostTypeId="2" ParentId="291" CreationDate="2014-11-07T14:52:53.493" Score="4" Body="&lt;p&gt;Atomism as a scientific theory has two origins: development of chemistry and development of statistical physics. In chemistry it was experimentally discovered that that there are elements and compunds, and elements probably consist of atoms which combine into moleculas.&#xA;Atomic weights were measured in the middle of 19-s century, and then came the periodic system. However in chemistry atoms were a kind of theoretical construction.&#xA;&quot;Nobody has seen them&quot;, as many people in 19-th sentury said.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From statistical physics comes the Avogadro number, &#xA;and statistical physics was developing during the whole&#xA;19-th century. However until the very end of the 19-th century there were serious physicists&#xA;which denied the existence of moleculas and atoms (as physics defines them, not chemistry). &#xA;Ernst Mach is a notable example. The crucial thing was the discovery of Brownian motion and developing of&#xA;its mathematical theory by Einstein and Smoluchowski. This theory permitted to measure&#xA;the Avogadro number experimentally, and the existence of atoms and moleculas became an&#xA;undeniable experimental fact. This happened around 1905.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-07T14:52:53.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="301" PostTypeId="2" ParentId="293" CreationDate="2014-11-07T15:44:58.463" Score="3" Body="&lt;p&gt;Aristotle (384–322 B.C.E.) refers to the branches of learning as “sciences” (epistêmai), he groups both empirical and nonempirical pursuits into what he calls &lt;em&gt;natural sciences&lt;/em&gt; such as physics. He also makes an distinction between theoretical science from more practically oriented studies. Theoretical sciences is what Aristotle calls &lt;em&gt;first philosophy&lt;/em&gt;, or Natural philosophy which includes studies such as biology, botany etc. Prior to Aristotle this distinction was not applied, and all studies was encompassed by the term &lt;em&gt;philosophy&lt;/em&gt; (love of wisdom). Also it might be good to look at the etymology of the word 'science', and it appears to be from late 14th century but with a slightly different connotation. Namely &quot;knowledge&quot;, &quot;collective knowledge&quot;, etc. The modern scientist, i.e. an individual who uses the scientific method. We may want to look into Karl Popper who is known for his rejection of the classical inductivist views on the scientific method, in favour of empirical falsification. Which is in use today.  &lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2014-11-07T16:14:23.357" LastActivityDate="2014-11-07T16:14:23.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="302" PostTypeId="2" ParentId="294" CreationDate="2014-11-07T17:33:42.947" Score="3" Body="&lt;p&gt;My preliminary research starts with the dating of the first scientific studies of air pollution in general. I first cite some dates in a work by E.C. Halliday titled and sourced: &lt;a href=&quot;http://whqlibdoc.who.int/monograph/WHO_MONO_46_(p9).pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;A Historical Review of Atmospheric Pollution - Halliday (1961)&lt;/em&gt;&lt;/a&gt;. In this paper, Halliday notes that, while pollution was known to be a social problem starting in the 14th century, &quot;...a scientific review of the history of air pollution cannot commence much before the year 1850.... (pp.9-10)&quot; There are a couple of charts in this paper that enumerate the number of scientific papers by year on the general topic of air pollution, and it appears that the 1950's mark a sudden explosion of scientific research (pp. 11-12).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that the earliest papers dealt mainly with smoke, and control of these sorts of emissions (p. 15). I would not quite call scientific studies into controlling smoke alone scientific studies of optical effects of air pollution, but this motivates me to start hunting after the 1940s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point, and I could be predated here, I have found some early research into the optical effects of air pollution by a meteorologist named &lt;a href=&quot;http://www2.ametsoc.org/stac/index.cfm/committees/committee-on-atmospheric-radiation/sean-twomey-obituary/&quot; rel=&quot;nofollow&quot;&gt;Sean Twomey&lt;/a&gt;. I found the &lt;a href=&quot;http://en.wikipedia.org/wiki/Twomey_effect&quot; rel=&quot;nofollow&quot;&gt;&quot;Twomey Effect&quot;&lt;/a&gt; referred to in recent articles. Twomey's two papers, &lt;a href=&quot;http://www.clidyn.ethz.ch/ese101/Papers/twomey74a.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Pollution and The Planetary Albedo (1974)&lt;/em&gt;&lt;/a&gt;, and &lt;a href=&quot;http://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281977%29034%3C1149%3ATIOPOT%3E2.0.CO%3B2&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;The Influence of Pollution on the Shortwave Albedo of Clouds (1976)&lt;/em&gt;&lt;/a&gt; would be the date to try and predate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Twomey's papers deal with the influence of pollution on the refractive/reflective properties of clouds and the earth's atmosphere. They are full of mathematical formulas and scientific measurements. Both of those papers fit securely into the topic question of the post, and so I conclude that, at the latest, the 1970's would be the date. I traced through citations in the Twomey papers, as well as papers that cite Twomey, but have yet to find optical studies and measurements on this level dealing specifically with air pollution. For future researchers, this at least gives a date to predate.  &lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2014-11-07T17:40:18.450" LastActivityDate="2014-11-07T17:40:18.450" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="303" PostTypeId="1" AcceptedAnswerId="304" CreationDate="2014-11-07T18:51:40.000" Score="13" ViewCount="878" Body="&lt;p&gt;That we see things seems unquestionably obvious. But, when I try to think about how, I find it as a mystery. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conventional answer nowadays to this question revolves around absorption and reflection of waves of some specific wavelength. And this is established long back. But, this theory seem not really natural from a more basic perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were there other theories explaining vision? What was the historical development of theories of vision? &lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorDisplayName="user22" LastEditDate="2014-11-07T20:33:17.613" LastActivityDate="2015-08-20T22:05:55.420" Title="What were theories for why and how we see things?" Tags="|optics|light|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="304" PostTypeId="2" ParentId="303" CreationDate="2014-11-07T20:18:23.493" Score="9" Body="&lt;p&gt;As per your comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Were there any other theories challenging them in the past?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This topic was briefly discussed on Neil deGrasse Tyson's reincarnation of the TV show &lt;a href=&quot;http://www.imdb.com/title/tt2395695/&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Cosmos&lt;/em&gt;&lt;/a&gt;. While I obviously can't insert the episode here (and can't recall which one it was, anyway), I can explain the basic premises of the early alternate theories, from other sources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As usual, I'll get &lt;a href=&quot;http://en.wikipedia.org/wiki/Visual_perception#Early_studies&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt; out of the way first. The idea I alluded to above is the &lt;a href=&quot;http://en.wikipedia.org/wiki/Emission_theory_(vision)&quot; rel=&quot;nofollow&quot;&gt;emission theory&lt;/a&gt;, which stated that people emit light from their eyes, which bounces off objects and back to their eyes. It was first proposed in ancient Greece, and became a leading theory. &lt;a href=&quot;http://en.wikipedia.org/wiki/Emission_theory_(vision)#Evidence_for_the_theory&quot; rel=&quot;nofollow&quot;&gt;The evidence for it&lt;/a&gt;, though, was extremely poor. Wikipedia also talks briefly about a second theory, which I'm actually unable to explain, because it doesn't have anything in its favor:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The second school advocated the so-called 'intro-mission' approach which sees vision as coming from something entering the eyes representative of the object.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again, though, there is virtually no evidence in favor of either idea, so that about wraps it up for the ancient Greeks. Oh, but as &lt;a href=&quot;http://en.wikipedia.org/wiki/Emission_theory_(vision)#Persistence_of_the_theory&quot; rel=&quot;nofollow&quot;&gt;a parting shot&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Winer et al. (2002) have found recent evidence that as many as 50% of American college students believe in emission theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which is from &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/12094435?dopt=Abstract&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. Honestly, sometimes humanity can be so . . . I don't even want to say it.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I found a second discussion of it &lt;a href=&quot;http://people.brandeis.edu/~sekuler/SensoryProcessesMaterial/Alhazen.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. It goes along lines similar to the Wikipedia articles:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A theory of vision must explain how information crosses the space between the perceiver and the object he/she is looking at. Early theories of sight offered three major opinions on this matter. One opinion held that the eye sends out rays to objects and that these rays give the viewer information about color and shape. Among the best known proponents of this view were the Pythagoreans, adherents to the religious and scientific philosophy developed by Pythagoras (he of the theorem). A second opinion held that sight depended upon an interaction between images that were ejected from the eye and the perceiver's own spirit, or soul. Socrates and Plato were among the best known of this theory's supporters. Finally, another opinion held that when people see, they actually make contact with the objects they see, or with replicas of those objects.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To summarize, there were three theories:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Light shoots out of a person's eyes to an object and back (see the emission theory discussed above).&lt;/li&gt;&#xA;&lt;li&gt;A person's soul &quot;interacts&quot; with the things you see.&lt;/li&gt;&#xA;&lt;li&gt;A person &quot;makes contact&quot; with things they see.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Not to be outdone, Democritus proposed a &lt;em&gt;fourth&lt;/em&gt; theory:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the fifth century B.C., the Greek philospher Democritos believed that every object contained numerous replicas of itself. These replicas were called eidola (singular is eidolon). Every object continuously emits eidola. If someone looks at some object, its replicas fly toward the viewer and enter his/her eyes. According to this scheme, you are able to see some object only because one of the object's eidola has winged its way into your eye. Several observations were taken as support of the eidolon theory. For instance, the fact that someone could see his/her reflection in water or in a mirror proved that the viewer is emitting eidola.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Like the others, though, it had flaws:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Here are two of the arguments that contributed to the theory's ultimate rejection (I've excerpted these from V Ronchi's book, The Nature of Light: An Historical Survey, Cambridge: Harvard University Press, 1970). &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Suppose that you are watching a parade. The eidolon theory suggests that you see the marchers because each marcher's eidolon reaches your eyes. What, though, governs the relative positions of all the eidola? How do the eidola maintain themselves in a formation that corresponds to the parade formation? &lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;According to eidolon theory, you can see your reflection because one of your own eidola struck the mirror and returned to your eye. But if each eidolon was flying away from you, it should fly with its &quot;back&quot; toward you; how does the eidolon contort itself so not only do you see its front but also so that in the mirror your right ear appears toward your right and your left ear toward your left? &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;These were the earliest theories about sight. Returning to your comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Which one/both of these historic theories truly explains the science behind &quot;seeing things&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Both do. These aren't theories of light; they're theories of &lt;em&gt;physics&lt;/em&gt;. I suggest you take a look at the Wikipedia article on &lt;a href=&quot;http://en.wikipedia.org/wiki/Wave-particle_duality&quot; rel=&quot;nofollow&quot;&gt;wave-particle duality&lt;/a&gt; for some more good information.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-08-20T22:05:55.420" LastActivityDate="2015-08-20T22:05:55.420" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="305" PostTypeId="2" ParentId="293" CreationDate="2014-11-07T22:01:41.370" Score="6" Body="&lt;p&gt;This post, &lt;a href=&quot;http://thonyc.wordpress.com/2014/07/10/the-history-of-scientist/&quot; rel=&quot;noreferrer&quot;&gt;The history of “scientist”&lt;/a&gt;, traces the history of the term &lt;em&gt;scientist&lt;/em&gt;, which was not introduced until 1833 (although &lt;em&gt;science&lt;/em&gt; is much older).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the current taxonomy of science (the division into subfields), that would require a very long discussion. For example, Michael Faraday's discoveries united what we would now call chemistry and physics. I don't believe Faraday himself saw a sharp distinction. Roughly speaking, as the sciences became more professionalized, the modern taxonomy evolved during the late 19th and early 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lavoisier listed &lt;em&gt;Lumière&lt;/em&gt; (light) and &lt;em&gt;Calorique&lt;/em&gt; (heat) as the first elements in his list of 23 elements. Nowadays we regard the study of heat and light as part of physics, not chemistry, although of course heat plays a major role in chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a beautiful and celebrated passage from his &lt;em&gt;Lectures on Physics&lt;/em&gt;, Feynman emphasized the artificial nature of the division:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A poet once said, “The whole universe is in a glass of wine.” We will probably never know in what sense he meant that, for poets do not write to be understood. But it is true that if we look at a glass of wine closely enough we see the entire universe. There are the things of physics: the twisting liquid which evaporates depending on the wind and weather, the reflections in the glass, and our imagination adds the atoms. The glass is a distillation of the earth’s rocks, and in its composition we see the secrets of the universe’s age, and the evolution of stars. What strange array of chemicals are in the wine? How did they come to be? There are the ferments, the enzymes, the substrates, and the products. There in wine is found the great generalization: all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the consciousness that watches it! If our small minds, for some convenience, divide this glass of wine, this universe, into parts—physics, biology, geology, astronomy, psychology, and so on—remember that nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-07T22:01:41.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="306" PostTypeId="1" CreationDate="2014-11-08T04:37:44.477" Score="20" ViewCount="1224" Body="&lt;p&gt;&lt;a href=&quot;http://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1964-077A&quot; rel=&quot;noreferrer&quot;&gt;Mariner 4&lt;/a&gt; was the first spacecraft to return photographs of Mars from near the planet, making its flyby of Mars in 1964.  Those photographs showed what appeared to be a barren, desert surface.  Mariner 4 also discovered that Mars had no magnetic field protecting it from ionizing solar radiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read many places that until Mariner 4, Mars was often assumed to host life of some kind, causing the seasonal color changes on the planet.  Wikipedia's &lt;a href=&quot;http://en.wikipedia.org/wiki/Mars#Martian_.22canals.22&quot; rel=&quot;noreferrer&quot;&gt;article on Mars&lt;/a&gt; sums it up well:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The seasonal changes (consisting of the diminishing of the polar caps and the dark areas formed during Martian summer) in combination with the canals lead to speculation about life on Mars, and it was a long-held belief that Mars contained vast seas and vegetation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Was the general view of the scientific community at the time that Mars had life?&lt;/strong&gt;  Or did scientists generally believe Mars to be a lifeless place?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h3&gt;A few statements about life on Mars before Mariner 4:&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;NASA's Summer Study, a group of eminent scientists gathered at Stanford University, said this in 1964:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given all the evidence presently available, we believe is entirely reasonable that Mars is inhabited with living organisms and that life independently originated there.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://goo.gl/bHByJc&quot; rel=&quot;noreferrer&quot;&gt;Life magazine, 1944&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The photographs of Mars...present evidence that points to the existence of life on Mars. ... Given this environment, it is logical to conclude that the vast regions on Mars that change from green to brown in seasonal cycles are covered by vegetation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From a letter published in &lt;a href=&quot;http://books.google.com/books?id=kQtS2lKz1zEC&amp;amp;pg=PA1317#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;New Scientist, 1959&lt;/a&gt; by a Cambridge professor:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At present there seems to be rather convincing evidence for life on Mars, and it is this that lends support to the assumption that life will come into existence where conditions are suitable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A biologist said this in &lt;a href=&quot;http://books.google.com/books?id=DSEDAAAAMBAJ&amp;amp;pg=PA104#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Popular Science, 1962&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We've said already that the color change fits neatly into a spring-summer vegetation cycle.  A second point in favor of life is that when the yellow clouds do cover the planet they don't permanently tinge the markings yellow.  Within a couple of weeks, whatever is there shakes it off or grows up through the cover.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've been able to find plenty of citations that argue for life on Mars, and plenty that argue against.  What I haven't been able to get is any clear picture of the dominant scientific consensus of the time.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-02-27T01:30:50.540" LastActivityDate="2015-02-27T01:30:50.540" Title="Before 1964, did scientists generally believe there was life on Mars?" Tags="|discoveries|biology|astronomy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="308" PostTypeId="1" CreationDate="2014-11-08T11:32:32.797" Score="11" ViewCount="288" Body="&lt;p&gt;The ancient Egyptians used examination, diagnosis, treatment and prognosis, to the treatment of diseases, which are all described in  Egyptian medical textbooks from around c. 1600 BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did the ancient Egyptians use some kind of scientific method, if not, what is the the earliest accounts of an scientific method for investigating phenomena, and acquiring new knowledge?&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="8" LastEditDate="2015-05-20T11:46:37.900" LastActivityDate="2015-05-20T11:46:37.900" Title="What are the earliest accounts of a scientific method?" Tags="|scientific-method|ancient-egypt|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="309" PostTypeId="2" ParentId="163" CreationDate="2014-11-08T12:16:07.120" Score="4" Body="&lt;p&gt;One of the main technological advancements that allowed Lippershey and Galileo to make their comparatively powerful telescopes was a change in how lenses were manufactured.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somewhat sadly, the exact date and progenitor of this new innovative method is not clearly known, as according to the &lt;a href=&quot;http://galileo.rice.edu/sci/instruments/telescope.html&quot; rel=&quot;nofollow&quot;&gt;Galileo Project's telescope webpage&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the telescope was not the invention of scientists; rather, it was the product of craftsmen. For that reason, much of its origin is inaccessible to us since craftsmen were by and large illiterate and therefore historically often invisible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The change, according to the American Society of Mechanical Engineers' webpage &lt;a href=&quot;https://www.asme.org/engineering-topics/articles/history-of-mechanical-engineering/only-as-good-as-their-telescopes&quot; rel=&quot;nofollow&quot;&gt;Only as Good as &#xA;Their Telescopes&lt;/a&gt;, up to a certain point in the late 16th century&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Shaping lenses involved increasingly sophisticated use of primitive tools, all of which were some variation on the basic lathe. The operator tried to form a lens contour by comparing it to a metal gauge that had an edge cut to match a compass-scribed arc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The change came about when this practice was replaced with a process that used&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a series of turning operations in which the tools, rather than the lens itself, were lathe-turned. Lenses were produced by grinding them against a metal tool called a &quot;lap,&quot; which had been produced using a lathe. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to the &lt;a href=&quot;http://galileo.rice.edu/sci/lipperhey.html&quot; rel=&quot;nofollow&quot;&gt;Galileo Project Lippershey webpage&lt;/a&gt;, these new lens crafting techniques were introduced to the Dutch (where Lippershey was) from the Italians in the 1590's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One name does stand out, &lt;a href=&quot;http://catalogue.museogalileo.it/biography/IppolitoFrancini.html&quot; rel=&quot;nofollow&quot;&gt;Ippolito Francini&lt;/a&gt; (1593-1653), was a lens maker who was one of the pioneers of this new method, in collaboration with Galileo, he participated &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;in the construction of lenses for telescopes, achieving substantial fame. He is credited with introducing the vertical optical lathe, a design still in use. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastActivityDate="2014-11-08T12:16:07.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="310" PostTypeId="1" AcceptedAnswerId="321" CreationDate="2014-11-08T12:58:51.220" Score="11" ViewCount="1133" Body="&lt;p&gt;There is an interesting web site called Mathematical Genealogy &lt;a href=&quot;http://genealogy.math.ndsu.nodak.edu/&quot;&gt;http://genealogy.math.ndsu.nodak.edu/&lt;/a&gt; which lists about 140000 mathematicians with their student-advisor relations. I think that this site can be useful for historian of sciences in many respects. Of course, extension of the modern student-advisor relation to the past is anachronistic, and many people listed there can be hardly called mathematicians, but nevertheless the site contains a lot of interesting information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to it, the genealogy of most modern mathematicians can be traced back to 14-th century, to several people who apparently moved to Western Europe from Constantinopole when it was conquered by the Turks. Their names are Manuel Bryennios, Demetrios Kydones,&#xA;Elissaeus Judaeus and few others. 119K (of the total 142K) of modern mathematicians have this Elissaeus&#xA;as their ancestor. I am trying to find out&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What exactly is known about these people ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And where the information comes from? Internet search did not give much, though they are frequently mentioned in mathematicians web sites with the reference on this Genealogy.&#xA;But where the data of this Genealogy for 14-th century come from, from what sources?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. On the answer of Mauro ALLEGRANZA. I understand that the &quot;sources&quot; he cites is Wikipedia. Of course I looked into Wikipedia BEFORE asking this question:-) A &quot;totally unknown&quot; teacher of Gemistos I did not find in Wikipedia. So I wonder: where the information about him comes from? On what grounds was it included to Math Genealogy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also understand all that these people were not mathematicians but rather clerics or philosophers. The real question is: what is the source of the information in Math Genealogy, and how reliable this source is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT2. I check my &quot;genealogy&quot; periodically (once a year in the average) and it expands:-) Now I see some ancestors which lived at the same time but look independent (according to the Math Genealogy) from those who came from Constantinopole. For example, Johannes Stoffler, who was born a year before the fall of Constantinopole, and has no known &quot;PhD adviser&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-11T20:31:15.170" LastActivityDate="2014-11-11T20:31:15.170" Title="Who was Elissaeus Judaeus?" Tags="|mathematics|middle-ages|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="311" PostTypeId="2" ParentId="58" CreationDate="2014-11-08T13:11:31.387" Score="20" Body="&lt;p&gt;Actually Cantor was working on a specific problem from the theory of &lt;a href=&quot;https://en.wikipedia.org/wiki/Trigonometric_series&quot; rel=&quot;noreferrer&quot;&gt;trigonometric series&lt;/a&gt;, the so-called uniqueness problem (I cannot be more specific until MathJax is introduced to this site). This problem led him to consideration of arbitrary sets on the real line. I mean more complicated sets than finite sets or finite union of intervals. At that time there was no tools and no terminology to study arbitrary sets, so all this had to be created.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the process of this study he created not only the set theory but also what is called now&#xA;General topology. (In is interesting to notice that the original problem about trigonometric series has no complete solution to this day:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original method of proof, the so-called &quot;diagonal procedure&quot; goes back to Cantor's predecessor, &lt;a href=&quot;http://en.wikipedia.org/wiki/Paul_du_Bois-Reymond&quot; rel=&quot;noreferrer&quot;&gt;Paul du Bois Reymond&lt;/a&gt;, who was also studying trigonometric series.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="205" LastEditDate="2016-07-12T08:07:06.657" LastActivityDate="2016-07-12T08:07:06.657" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="312" PostTypeId="2" ParentId="308" CreationDate="2014-11-08T13:31:08.893" Score="5" Body="&lt;p&gt;Of course, the exact definition of the scientific method is open to discussion (and the scientific method may be very different in different sciences, compare chemistry, astronomy, geology&#xA;and mathematics, for example). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest account that I know of an experimental discovery is physics is credited to Pythagoras. According to the legend, he once visited a blacksmith workshop and noticed that&#xA;similar metal rods of various length produce sounds of various pitch when struck with a hummer.&#xA;Apparently he experimented with various rods and strings, and derived the correct law relating the musical intervals with ratios of length of the strings that produce them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently this discovery impressed him so much that he concluded that &quot;numbers rule the world&quot;, which is essentially the main paradigm of exact scienses. In modern formulation: the laws of nature are expressed in mathematical form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is what the legend says about the origin of (&quot;Western European&quot;) exact science.&#xA;This happened some time in VI cent. BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somewhat earlier in the same century mathematical method was born: Thales of Miletus proved the first theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, all this information comes from much later secondary sources. So these are property called &quot;legends&quot;. But this is the best information we apparently possess. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Lucio Russo, with whom I agree, this led to the full development of the scientific method which reached its culmination in II BC and was still practiced in II AD. But then there was an interruption, and all exact science, together with the method were lost almost for a&#xA;millenium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It had to be reborn, rediscovered, and this happened in 16-th century,&#xA;when Francis Bacon&#xA;explicitly described the newly discovered scientific method. Fortunately, some of the Hellenistic science was preserved outside of Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I was asked to add sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the story with Pythagoras, see &quot;Pythagorean hummers&quot; on Wikipedia, it lists original sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Thales, his story is told in almost every history of Greek mathematics, I used van der Waerden, Science awakening.(English tranls. Ocford Univ press 1961). He also has a book on Pythagoreans, but I do not have it besides me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lucio Russo's book is called Forgotten revolution (engl. transl. Springer 2004). &#xA;There is a nice long review of this book in the Notices Amer. Math. Soc., which I believe is freely available. There is about a dozen other reviews online.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Francis Bacon's main work on scientific method is called New Organon (Wikipedia article &quot;Francis Bacon&quot; has a link to a free online version, and actually to cmplete works of Bacon).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-08T23:56:48.803" LastActivityDate="2014-11-08T23:56:48.803" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="313" PostTypeId="2" ParentId="306" CreationDate="2014-11-08T14:09:03.820" Score="7" Body="&lt;p&gt;I do not see how Mariner 4 mission could lead to any conclusions about the life on Mars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The crucial mission to my understanding was Viking 2 (1976) which landed on Mars and made&#xA;biochemical experiments trying to find any signs of life. At that time almost nobody seriously expected that there is life on Mars. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The presence and chemical composition of the atmosphere was found in 1926 by spectral analysis from the Earth. (The earlier attempts in 1894 did not show the presence of&#xA;a substantial atmosphere. All speculations about the life before that time were totally groundless, I suppose).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These results showed that the presence of life (at least in the form we know it) is very unlikely. Exact composition of the atmosphere was found with the first probe&#xA;that landed on Mars; this was the Soviet Mars 2 in 1969, if I remember correctly. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-09T00:07:54.610" LastActivityDate="2014-11-09T00:07:54.610" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="314" PostTypeId="1" CreationDate="2014-11-08T15:36:21.543" Score="10" ViewCount="559" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Erd%C5%91s_number&quot;&gt;Erdős number&lt;/a&gt; is used as a tool to study how mathematicians cooperate to find answers to unsolved problems. But what is the history behind the Erdős number?&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="109" LastEditDate="2014-11-09T08:31:58.057" LastActivityDate="2014-11-09T08:31:58.057" Title="What is the history behind the Erdős number?" Tags="|mathematics|academia|publishing|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="315" PostTypeId="2" ParentId="314" CreationDate="2014-11-08T16:06:30.990" Score="10" Body="&lt;p&gt;According to the USNA website &lt;a href=&quot;http://www.usna.edu/Users/math/meh/erdos.html&quot;&gt;Erdös Numbers&lt;/a&gt;, it is named after a most prolific and supremely talented mathematician,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Paul Erdös (1913-1996) was a phenomenally prolific mathematician, who combined extraordinary talent with near-total devotion to mathematical research. For most of his career he did not have a conventional position; he was a mathematical nomad who disdained property, power, and even family life as distractions from what he really wanted to do: prove theorems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;His work in number theory, set theory, various kinds of combinatorics (including graph theory), analysis, and probablility theory resulted in well over 1500 publications, with over 500 collaborators, giving rise to the Erdös Number, which was started by his friends (&lt;a href=&quot;http://www.iap.tuwien.ac.at/www/atomic/group/aumayr/erdos&quot;&gt;Vienna University of Technology)&lt;/a&gt;, and is described as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All those mathematicians who co-authored a paper or book with Erdös are said to have Erdös number 1; those who have co-authored with someone of Erdös number 1 (but not with Erdös himself) have Erdös number 2; and so on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to the USNA website, the number of people is almost 10,000.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oakland University have a dedicated &lt;a href=&quot;http://www.oakland.edu/enp&quot;&gt;The Erdös Number Project&lt;/a&gt; webpage, essentially exploring the 'six degrees of separation' in mathematics based research, there is even a facility to &lt;a href=&quot;http://www.ams.org/mathscinet/collaborationDistance.html&quot;&gt;find out your own Erdös Number&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of timeframes, according to Oakland's webpage &lt;a href=&quot;http://www.oakland.edu/enp/research/&quot;&gt;Research on Collaboration in Research&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The earliest discussion we could find on Erdös numbers are articles by Casper Goffman [And what is your Erdos number?, American Mathematical Monthly 76 (1969), 791] and Ron Graham (writing under the pseudonym Tom Odda) [On properties of a well-known graph or what is your Ramsey number?, Topics in Graph Theory (New York, 1977) 166-172]. According to MELVIN HENRIKSEN, &lt;strong&gt;the idea was suggested by John Isbell at Princeton University around 1957.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Incidently, John Isbell has an Erdös number of 2)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting part of the website is the correlation of Fields Medal and Nobel Prize recipients and their Erdös Number - &lt;a href=&quot;http://www.oakland.edu/enp/erdpaths/&quot;&gt;Some Famous People with Finite Erdös Numbers&lt;/a&gt;, that many have a low Erdös Number - read into that what you will.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing is clear though, the Erdös Number and the associated project is a fitting tribute, started by his friends (according to the Vienna University of Technology) to a great scholar and a way to show that in some way, we are all linked in our drive for knowledge.&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorDisplayName="user22" LastEditDate="2014-11-08T17:19:36.350" LastActivityDate="2014-11-08T17:19:36.350" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="317" PostTypeId="1" CreationDate="2014-11-08T17:57:44.583" Score="33" ViewCount="4578" Body="&lt;p&gt;I am looking for examples of scientific breakthroughs that have been made within the confines of a prison cell.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="2324" LastEditDate="2021-09-26T05:27:56.227" LastActivityDate="2022-09-08T05:59:48.340" Title="What are some scientific breakthroughs that have been done during jail time?" Tags="|discoveries|" AnswerCount="8" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="318" PostTypeId="2" ParentId="317" CreationDate="2014-11-08T19:30:49.270" Score="28" Body="&lt;p&gt;Andre Weil's work on zeta functions of algebraic curves over finite fields.&lt;/p&gt;&#xA;&lt;p&gt;Andre Weil was one of the most famous mathematicians of the last century. He also was a pacifist, and when WWII approached, he left France to visit....Soviet Union:-)&#xA;After Soviet Union, he visited Finland... just few days before Soviet Union attacked it.&#xA;He was arrested by the Finns, as a Soviet spy with the nickname &amp;quot;Bourbaki&amp;quot;.&#xA;It did not help his case that he had a full suitcase of papers signed &amp;quot;Nicolas Bourbaki&amp;quot;,&#xA;and written in some mysterious code. In view of the quick Russian offense they decided to execute him by shooting...&lt;/p&gt;&#xA;&lt;p&gt;Fortunately for Weil, the great Finnish mathematician Nevanlinna (who was socializing with highest Finnish officials) met the security chief in a party. The security chief told&#xA;Nevanlinna that they caught a Russian spy whose name is Andre Weil, and this guy says that he knows Professor Nevanlinna personally! Nevanlinna confirmed that Weil is a harmless mathematician. And recommended to deport him to France.&lt;/p&gt;&#xA;&lt;p&gt;So Weil was deported to his native country and put to prison as a deserter....&#xA;He spent his time in prison until the Germans defeated France and released him&#xA;(as a French deserter. Probably they did not notice that he was also a Jew).&lt;/p&gt;&#xA;&lt;p&gt;Anyway, after many adventures, Weil made his way to the US. In his memoirs he says that the work that he did in the French prison was his best. And that at a later age he even considered, instead of writing a grant application, to ask to put him to a prison... perhaps he would do a work of similar quality:-)&lt;/p&gt;&#xA;&lt;p&gt;All this is from Weil's memoirs &amp;quot;The apprenticeship of a mathematician&amp;quot;, Translated from the 1991 French original by Jennifer Gage. Birkhäuser Verlag, Basel, 1992.&lt;/p&gt;&#xA;&lt;p&gt;The work he did in prison was truly revolutionary.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2021-09-23T12:31:35.220" LastActivityDate="2021-09-23T12:31:35.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="319" PostTypeId="2" ParentId="317" CreationDate="2014-11-08T19:38:42.753" Score="15" Body="&lt;p&gt;Another famous and really unique case was &lt;a href=&quot;http://en.wikipedia.org/wiki/Andr%C3%A9_Bloch_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Andre Bloch&lt;/a&gt;, another great French mathematician of 20-s century. He was drafted to WW I when he was a student (or maybe just graduated). He was severely wounded at his head (he fell from an artillery observation tower when it was hit).&#xA;After a hospital and a honorable discharge he lived with his relatives. One morning he killed all of them during the breakfast (his uncle, aunt and several more relatives).&#xA;It was decided that a war hero should not stand a trial during the war and he was confined in a mental asylum for the rest of his life. He died in his asylum in 1948.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During his stay in the asylum he worked and published many papers which revolutionized the subject of Complex Analysis. He is still remembered for the &quot;Bloch's Principle&quot;, &quot;Bloch Constant,&quot; &quot;Bloch functions&quot; and &quot;Bloch's conjecture&quot; (see Wikipedia).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="29" LastEditDate="2014-11-08T21:51:06.187" LastActivityDate="2014-11-08T21:51:06.187" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="320" PostTypeId="2" ParentId="317" CreationDate="2014-11-08T19:40:55.920" Score="17" Body="&lt;p&gt;The French mathematician &lt;a href=&quot;http://en.wikipedia.org/wiki/Andr%C3%A9_Weil&quot;&gt;André Weil&lt;/a&gt; was imprisoned for failure to report for duty for a spell in 1940. It is said that &quot;While in jail for six months at Rouen, he proved the Riemann hypothesis for curves over finite fields.&quot; &lt;a href=&quot;http://www.robertnowlan.com/pdfs/Weil,%20Andre.pdf&quot;&gt;Nowlan&lt;/a&gt;. He is said to have sent a 14 page letter to his sister from prison with some groundbreaking ideas on &quot;Analogy in Mathematics&quot;. That letter is reprinted in the AMS article &lt;a href=&quot;http://www.ams.org/notices/200503/fea-weil.pdf&quot;&gt;A 1940 Letter of &#xA;André Weil on Analogy&#xA;in Mathematics (trans. Krieger)&lt;/a&gt; with some introduction. Some more discussion on the contents and importance of this letter are &lt;a href=&quot;http://blogs.scientificamerican.com/guest-blog/2013/05/21/an-unheralded-breakthrough-the-rosetta-stone-of-mathematics/&quot;&gt;here&lt;/a&gt;. Sources and stories abound online, and they largely seem to corroborate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That seems to be a real solid example. I found several lesser examples of great scientific works, or accounts of Mathematicians and Scientists either writing something great or continuing with some great work. While these may not fit the request as straightforward as the Weil story, I believe they are worth noting nonetheless:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It is said about Galois that &quot;During his imprisonment, he continued developing his mathematical ideas.&quot; &lt;a href=&quot;http://en.wikipedia.org/wiki/%C3%89variste_Galois&quot;&gt;Galois&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Galileo wrote what is said to be his greatest work &lt;em&gt;Two New Sciences&lt;/em&gt; &lt;a href=&quot;http://en.wikipedia.org/wiki/Galileo_Galilei&quot;&gt;Galileo&lt;/a&gt; while in prison.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bertrand Russell wrote &lt;em&gt;Introduction to Mathematical Philosophy&lt;/em&gt; while in prison &lt;a href=&quot;http://www.personal.kent.edu/~rmuhamma/Philosophy/bertrandRussell.html&quot;&gt;Russell&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jean_Leray&quot;&gt;Jean Leray&lt;/a&gt; is said to have done seminal work on topology while in prison.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2014-11-08T19:48:02.437" LastActivityDate="2014-11-08T19:48:02.437" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="321" PostTypeId="2" ParentId="310" CreationDate="2014-11-08T20:27:35.970" Score="6" Body="&lt;p&gt;From Polymnia Athanassiadi, &lt;strong&gt;Byzantine Commentators on the Chaldaean Oracles : Psellos and Plethon&lt;/strong&gt;, in Katerina Ierodiakonou (editor), &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0199269718&quot;&gt;Byzantine Philosophy and its Ancient Sources&lt;/a&gt; (2004), page 237-on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Page 248 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a letter to Theodora Palaeologina, Gennadios Scholarios offers the following information on [Georgius Gemistos] Plethon's spiritual grounding: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;the sum total of his apostasy was consummated by a certain Jew with whom he studied because he was an expert on Aristotle. He was a follower of Averroes and of the other Arab and Persian commentators of Aristotle's works, which have been translated by the Jews into their own language. He is also the man who acquainted him with Zoroaster and the rest. With this man, who was ostensibly a Jew but in reality a Hellene, he stayed for a long time not only as his pupil but also in his service being supported by him. He was one of the most influential men at the Court of these barbarians; his name was Elissaeus. [&lt;em&gt;footnote&lt;/em&gt; : George Scholarios Gennadios, &lt;em&gt;Oeuvres complètes&lt;/em&gt;, iv, ed. L.Petit, M.Jugie, and X.A.&#xA;    Siderides (Paris, 1935), 152; Bidez and Cumont (1938: O 115, II 260).]&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Page 249 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Elissaeus - a mysterious figure who seems to have been burnt at the stake - appears to have accepted George as his &lt;em&gt;murid&lt;/em&gt;, that is as a pupil who had to live with him and serve him on a daily basis, thus slowly progressing along the stations of the spiritual way not least through the virtue of obedience. His teaching would have been heterodox by the standards of any official dogma, since it was an eclectic synthesis borrowing elements and figures from all traditions and reorganizing them according to his own judgement.  &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Scholarios' information that Elissaeus was eventually burnt alive would tally with the overall picture: spiritual masters who proved too original were condemned by the Islamic establishment to exemplary deaths. Such was indeed the fate of Shihaboddin Yahya Sohrawardi (1155-91), the Iranian mystic, whose influence Henry Corbin detects behind Plethon's theories. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Sohrawardi was an Azerbaidjani, that is a native of the Zoroastrian holy land of Atropatene. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Page 250 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] I do not believe in a direct influence of Sohrawardi on Plethon, though an indirect knowledge of his writings through oral channels seems to me very &#xA;  probable. Sohrawardi's renown was vast and Elissaeus sounds exactly the type of man to be fascinated by the synthesis of Zarathustra and Plato attempted by the Iranian scholar. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In conclusion, it seems highly implausible that Elissaeus has been a mathematician at all ...&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-11-08T20:27:35.970" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="322" PostTypeId="1" AcceptedAnswerId="324" CreationDate="2014-11-08T21:47:05.253" Score="13" ViewCount="578" Body="&lt;p&gt;Reading the excellent thread &lt;a href=&quot;https://hsm.stackexchange.com/questions/317/what-are-some-scientific-breakthroughs-that-have-been-done-during-jail-time&quot;&gt;What are some scientific breakthroughs that have been done during jail time?&lt;/a&gt;, it stands to reason to ask what are some scientific breakthroughs made by interned prisoners of war,  when the author was confined in a POW camp or a concentration camp.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How did prisoners of war discover scientific breakthroughs while interned?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="-1" LastEditorDisplayName="user22" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-10-05T09:34:56.887" Title="How did prisoners of war discover scientific breakthroughs while interned?" Tags="|discoveries|" AnswerCount="5" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="323" PostTypeId="2" ParentId="322" CreationDate="2014-11-08T22:35:11.973" Score="10" Body="&lt;p&gt;Poncelet fought in the Napoleonic campaign, and was held as a prisoner of war in Russia. From &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Poncelet.html&quot; rel=&quot;nofollow&quot;&gt;the MacTutor bio&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He was held in the prison from March 1813 to June 1814 when he returned to France. During his imprisonment he recalled the fundamental principles of geometry but, forgetting the details of what he had learnt from Monge, Carnot and Brianchon, he went on to develop projective properties of conics. He called the notes that he made the 'Saratov notebook,' but it was only fifty years later that he incorporated much of what he had written in his treatise on analytic geometry Applications d'analyse et de géométrie (1862). His development of the pole and polar lines associated with conics led to the principle of duality but this, as we explain below, led to a priority dispute. He also discovered circular points at infinity. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastEditorUserId="62" LastEditDate="2014-12-31T23:36:26.597" LastActivityDate="2014-12-31T23:36:26.597" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="324" PostTypeId="2" ParentId="322" CreationDate="2014-11-09T01:14:44.857" Score="12" Body="&lt;p&gt;Another outstanding example is Jean Leray (who was mentioned in an answer to the question on&#xA;jail prisoners). He was an outstanding French mathematician with main interests in fluid dynamics. When confined to a German POW camp, where he stayed until the liberation he was afraid that Germans will force him to work in his area (which has important military applications). Unwilling to help the Germans in their war effort, he switched his mathematical research to the purest area of pure mathematics, topology. And made a seminal contribution to topology which completely changed the whole subject.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I understand from inmate recollections, this prison camp did not have very severe conditions. They even had an informal university in the camp where they lectured to each other. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-09T01:14:44.857" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="325" PostTypeId="1" AcceptedAnswerId="326" CreationDate="2014-11-09T05:20:28.990" Score="13" ViewCount="580" Body="&lt;p&gt;According to the folks of &lt;a href=&quot;http://en.wikipedia.org/wiki/Euclid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, this is the portrait of &lt;strong&gt;Euclid of Alexandria&lt;/strong&gt; (the &amp;quot;Father of Geometry&amp;quot;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xVkKo.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yet, I read several years ago on &lt;a href=&quot;https://mathoverflow.net/a/26666&quot;&gt;MathOverflow&lt;/a&gt; that the individual in the picture is actually a namesake of Euclid of Alexandria who was born in Megara like a hundred years before Euclid of Alexandria.&lt;/p&gt;&#xA;&lt;p&gt;Thus, I would like to know if it is known when it was that the people started using the portraits of the two Euclids interchangeably...&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="4251" LastEditDate="2020-07-20T11:49:31.020" LastActivityDate="2020-07-20T11:49:31.020" Title="When did the people start using the portraits of the two Euclids interchangeably?" Tags="|mathematicians|debunking|depiction-of-scientists|euclid|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="326" PostTypeId="2" ParentId="325" CreationDate="2014-11-09T08:17:12.150" Score="11" Body="&lt;p&gt;We have very few information regarding the author of the &lt;em&gt;Elements&lt;/em&gt;, called Euclid of Alexandria.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See Euclid, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486600882&quot; rel=&quot;nofollow&quot;&gt;The Thirteen Books of the Elements, Vol. 1 : Books 1-2&lt;/a&gt; (ed. T.Heath, Dover reprint), page 1-on :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As in the case of the other great mathematicians of Greece, so in Euclid's case, we have on eagre particulars of the life and personality of the man.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Most of what we have is contained in the passage of Proclus' summary relating to him [...]. This passage shows that even Proclus had no direct knowledge of Euclid's birthplace or of the date of his birth or death. He proceeds by inference.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This man lived in the time of the first Ptolemy. [...] Thus Euclid must have flourished ca.300 B.C., which date agrees well with the fact that Ptolemy reigned from 306 to 283 B.C.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, if we know quite nothing about it, it is vey difficult that someone can know his face...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See page 3 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the middle ages most translators and editors spoke of Euclid as &lt;em&gt;Euclid of Megara&lt;/em&gt;. This description arose out of a confusion between our Euclid and the philosopher Euclid of Megara who lived about 400 B.C. The first trace of this confusion appears in Valerius&#xA;  Maximus (in the time of Tiberius) who says that Plato, on being appealed to for a solution of the problem of doubling the cubical altar, sent the inquirers to &quot;Euclid the geometer.&quot; [...] if Valerius Maximus took Euclid the geometer for a contemporary of Plato, it could only be through confusing him with Euclid of Megara. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The first specific reference to Euclid as Euclid of Megara belongs to the 14th century, occurring in [...] Theodorus Metochita (d. 1332) who speaks of &quot;Euclid of Megara, the Socratic philosopher, contemporary of Plato,&quot; as the author of treatises on plane and solid geometry, data, optics etc.: and a Paris MS. of the 14th century has &quot;Euclidis philosophi&#xA;  Socratici liber elementorum.&quot; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The misunderstanding was general in the period from Campanus' translation (Venice (1482) to those of Tartaglia (Venice 1565) and Candalla (Paris 1566). But one Constantinus Lascaris (d. about 1493) had already made the proper distinction by saying of our Euclid that &quot;he was different from him of Megara of whom Laertius wrote, and who wrote dialogues &quot;; and&#xA;  to Commandinus belongs the credit of being the first translator to put the matter beyond ,doubt: &quot;Let us then free a number of people from the error by which they have been induced to believe that our Euclid is the same as the philosopher of Megara &quot; etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, it is quite clear that the portrait of Euclid by the Dutch painter &lt;a href=&quot;http://en.wikipedia.org/wiki/Justus_van_Gent&quot; rel=&quot;nofollow&quot;&gt;Justus van Gent&lt;/a&gt;, circa 1474, was &lt;strong&gt;intended&lt;/strong&gt; to be a portrait of the geometer, made in a time when he was still known by many as Euclid of Megara.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Added&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the overall issue regarding the re-discovery of Euclid during the Renaissance, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Goulding, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/9048135419&quot; rel=&quot;nofollow&quot;&gt;Defending Hypatia : Ramus, Savile, and the Renaissance Rediscovery of Mathematical History&lt;/a&gt; (2010).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In particular, see &lt;strong&gt;Ch.5 : The Puzzling Lives of Euclid&lt;/strong&gt;, page 117-on, devoted to &quot;the Renaissance unmasking of Euclid&quot; [page 118], from :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1505, the Venetian humanist Bartolomeo Zamberti published his monumental Latin translation of the works of Euclid, containing not only the &lt;em&gt;Elements&lt;/em&gt;, but also the &lt;em&gt;Phaenomena, Optics, Catoptrics&lt;/em&gt; and &lt;em&gt;Data&lt;/em&gt;. In his preface, Zamberti acknowledged that his readers would likely want to know something about the author of these works, styled “Euclid of Megara, the Platonic Philosopher” on the title page [page 118].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;to &quot;the final resolution&quot; [page 141]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1572, Federico Commandino published a great edition of the &lt;em&gt;Elements&lt;/em&gt; to which he prefaced a few biographical remarks. He noted that many had taken Diogenes Laertius’s life of Euclid of Megara to be that of the geometer, so that the author of the &lt;em&gt;Elements&lt;/em&gt; was thought to be one of the Socratic philosophers. But, he observed, that could not be the case, since Proclus stated quite clearly that Euclid lived in the time of Ptolemy I: it would be impossible for the two to be the same person.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-06-14T17:25:51.043" LastActivityDate="2015-06-14T17:25:51.043" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="327" PostTypeId="2" ParentId="325" CreationDate="2014-11-09T08:21:07.450" Score="12" Body="&lt;p&gt;This is almost certainly intended to be an image of Euclid of Alexandria, not of Euclid of Megara. The image was painted by &lt;a href=&quot;http://en.wikipedia.org/wiki/Justus_van_Gent&quot; rel=&quot;nofollow noreferrer&quot;&gt;Justus van Gent&lt;/a&gt;, who lived in the 15th century. Among his most famous works is a series of pictures of 28 famous people in history, the &lt;em&gt;uomini famosi&lt;/em&gt; (&quot;Famous Men&quot;) in approximately 1475. As you can guess, this image was among them. There is actually some debate regarding the attribution of some of these paintings, but that's not particularly relevant here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suffice it to say, most of the &quot;Famous Men&quot; really are quite famous, or at least were in the 15th century. They're all portrayed the person in some way doing something for which that person was famous. For instance, here's the picture of &lt;a href=&quot;http://en.wikipedia.org/wiki/Ptolemy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy&lt;/a&gt;; one can immediately tell it is an astronomer by the &lt;a href=&quot;http://en.wikipedia.org/wiki/Armillary_sphere&quot; rel=&quot;nofollow noreferrer&quot;&gt;spherical astrolabe&lt;/a&gt; (although to the best of my knowledge it's unclear whether Ptolemy ever developed an astrolabe of any kind).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/LLFCIm.jpg&quot; alt=&quot;Ptolemaeus&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the portrait of Euclid of Megara is holding a compass, something which would not have been associated to him based on his own works (which were more philosophical than mathematical). Indeed, the inclusion of Euclid of Megara among the other &quot;Famous Men&quot; would seem somewhat strange; his own writings were not nearly as important as the others in the series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, it seems easy to attribute this to confusion on the part of the painter. Such confusion between the two men was not uncommon; one can find other medieval accounts of the &lt;em&gt;Elements&lt;/em&gt; incorrectly attributing them to Euclid of Megara rather than Euclid of Alexandria (see &lt;a href=&quot;https://hsm.stackexchange.com/a/326&quot;&gt;Mauro ALLEGRANZA's answer&lt;/a&gt; for more details regarding this).&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-11-09T08:37:44.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="328" PostTypeId="2" ParentId="322" CreationDate="2014-11-09T08:27:43.150" Score="4" Body="&lt;p&gt;Mulutin Milankovic was a First World War prisoner of war at the time of this discovery.&lt;br&gt;&#xA;He eponymously discovered the Milankovich cycles which cause cyclic climate variation due to factors such as precession of the earths axis and the elliptical nature of its orbit.   &lt;/p&gt;&#xA;" OwnerUserId="196" LastActivityDate="2014-11-09T08:27:43.150" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="329" PostTypeId="5" CreationDate="2014-11-09T12:37:07.767" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-09T12:37:07.767" LastActivityDate="2014-11-09T12:37:07.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="330" PostTypeId="4" CreationDate="2014-11-09T12:37:07.767" Score="0" Body="DO NOT USE THIS TAG. It is obsolete; any question in this site should, by definition, be about history. Therefore, this tag is pointless, and it should not be used." OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-11-09T13:41:49.160" LastActivityDate="2014-11-09T13:41:49.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="331" PostTypeId="5" CreationDate="2014-11-09T12:40:31.673" Score="0" Body="&lt;p&gt;If the concerned question is about a more specific area of Physics (such as &lt;a href=&quot;/questions/tagged/particle-physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;particle-physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;particle-physics&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/quantum-mechanics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;quantum-mechanics&amp;#39;&quot; rel=&quot;tag&quot;&gt;quantum-mechanics&lt;/a&gt;,  &lt;a href=&quot;/questions/tagged/electromagnetism&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;electromagnetism&amp;#39;&quot; rel=&quot;tag&quot;&gt;electromagnetism&lt;/a&gt; etc.), this tag should be used along with the tag of that specific area of Physics.&#xA;Using this tag along with &lt;a href=&quot;/questions/tagged/theoretical-physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;theoretical-physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;theoretical-physics&lt;/a&gt; and/or &lt;a href=&quot;/questions/tagged/experimental-physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;experimental-physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;experimental-physics&lt;/a&gt; should also be considered.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-17T08:02:40.983" LastActivityDate="2022-03-17T08:02:40.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="332" PostTypeId="4" CreationDate="2014-11-09T12:40:31.673" Score="0" Body="For questions about the scientific discipline that concerns itself with analysing the laws of nature in full generality" OwnerUserId="26" LastEditorUserId="10106" LastEditDate="2022-06-04T02:34:51.117" LastActivityDate="2022-06-04T02:34:51.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="333" PostTypeId="1" AcceptedAnswerId="336" CreationDate="2014-11-09T14:00:21.920" Score="21" ViewCount="22685" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;At the last point of his life, he noted that he slept 15 minutes extra every night then he calculated that he would die on the day that the extra 15 minutes a night accumulated to 24 hours. That day was November 27, 1754, the actual day of his death. He died in London and was buried at St. Martin-in-the-Fields.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this claim true? Is there any more information on similar predictions by mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="191" LastEditorUserId="191" LastEditDate="2023-06-12T05:10:45.583" LastActivityDate="2023-06-12T05:10:45.583" Title="Did Abraham De Moivre really predict his own death?" Tags="|biographical-details|death|mathematicians|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="334" PostTypeId="2" ParentId="333" CreationDate="2014-11-09T14:29:49.317" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Abraham_de_Moivre&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abraham de Moivre&lt;/a&gt; was a great statistician, but one of his greatest achievements may very well have been predicting his own death.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;http://robertnowlan.com/pdfs/de%20Moivre,%20Abraham.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; pdf:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;W.W. Rouse Ball wrote of de Moivre’s final days:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Manner of de Moivre’s death has a certain interest for psychologists. Shortly before it, he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer&#xA;each day than the preceding one: the day after he had thus reached a total of something over&#xA;twenty-three hours he slept up to the limit of twenty-four hours, and then died in his sleep.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.omgfacts.com/lists/9200/Abraham-de-Moivre-a-French-mathematician-successfully-predicted-the-date-of-his-own-death-in-1754&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He noted that he was sleeping 15 minutes extra every night, and calculated that he would die on the day that the extra 15 minutes a night accumulated to 24 hours. That day was November 27, 1754, the actual day of his death. He died in London and was buried at St. Martin-in-the-Fields.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note, though, that the source for that is Wikipedia, so perhaps we should throw that source out.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Finally, from &lt;a href=&quot;https://erich-friedman.github.io/periodic/html/Md.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erich Friedman's Periodic Table of Mathematicians&lt;/a&gt; page:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;De Moivre, like Cardan, is famed for predicting the day of his own death. He found that he was sleeping 15 minutes longer each night and summing the arithmetic progression, calculated that he would die on the day that he slept for 24 hours. He was right!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This answers your last question&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Any more information on similar predictions by mathematicians?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;“Cardan” appears to be &lt;a href=&quot;http://en.wikipedia.org/wiki/Gerolamo_Cardano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerolamo Cardano&lt;/a&gt;, another famous mathematician.&lt;/p&gt;&#xA;&lt;p&gt;By the way, &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/De_Moivre.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; page has the exact same text.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="2258" LastEditDate="2020-12-07T14:52:34.477" LastActivityDate="2020-12-07T14:52:34.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="335" PostTypeId="2" ParentId="333" CreationDate="2014-11-09T15:15:35.137" Score="17" Body="&lt;p&gt;I could not find any direct sources to confirm and support  this claim. W.W. Rouse Ball (1850 – 1825), writes in  &lt;em&gt;A Short Account of the History of Mathematics&lt;/em&gt; first published in 1908, pages 383-384: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The manner of his death has a certain interest for psychologists.&#xA;  Shortly before it he declared that it was necessary for him to sleep&#xA;  some ten minutes or a quarter of an hour longer each day than the&#xA;  preceding one. The day after he had thus reached a total of something&#xA;  over twenty-three hours he slept up to the limit of twenty-four hours,&#xA;  and then died in his sleep.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, no source is added to support this claim made by Rouse Ball. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read through Matthew Maty's (1718 - 1776) biography on de Moivre published 1755 in &lt;em&gt;Journal britannique&lt;/em&gt;, in which nothing of the sort is mentioned as far as I could tell. What Maty mentions about de Moivre is that he required much sleep.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He suffered partial loss of sight and hearing; his body required more&#xA;  rest and his mind, greater respite. Although he came to need twenty&#xA;  hours sleep, he spent the remaining three or four hours taking his&#xA;  only meal of the day and talking with his friends.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a letter to Mr. De Mairan which he found the energy to dictate and&#xA;  sign, he expressed his commitment and gratitude with enthusiasm.&#xA;  However, he overestimated the time that he probably had left to live&#xA;  and underestimated the difficulty of recovering the manuscripts that&#xA;  he had lent when he promised to repay the honor bestowed upon him&#xA;  through some scholarly tribute. He was to enjoy this recognition for a&#xA;  few months only. His health grew steadily worse and he needed to sleep&#xA;  longer and often. After being confined to bed for seven or eight days,&#xA;  he died in his sleep on November 27, 1754.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="187" LastEditorUserId="187" LastEditDate="2014-11-09T16:22:03.123" LastActivityDate="2014-11-09T16:22:03.123" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="336" PostTypeId="2" ParentId="333" CreationDate="2014-11-09T15:22:30.687" Score="27" Body="&lt;p&gt;This is a common claim which has been repeated enough times that one can find many sources claiming it to be true. However, this doesn't seem to be corroborated by accounts of the time or serious biographies. Indeed, Helen M. Walker, in her 1934 biography of De Moivre in &lt;a href=&quot;http://en.wikipedia.org/wiki/Scripta_Mathematica&quot;&gt;&lt;em&gt;Scripta Mathematica&lt;/em&gt;&lt;/a&gt; Volume II, Number 4, August 1934, (reproduced &lt;a href=&quot;https://play.google.com/store/books/details?id=mIIZAQAAIAAJ&amp;amp;rdid=book-mIIZAQAAIAAJ&amp;amp;rdot=1&quot;&gt;here freely&lt;/a&gt; on Google Books as an afterward) explicitly states the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is often related that De Moivre, always interested in number series, had foretold that each day he should need 15 minutes more sleep than on the preceding day, and that his death would occur when the total reached 24 hours. The story, passed on as an interesting fiction by several writers, has some basis in fact, but the arithmetic progression and De Moivre's prophetic statement are purely apocryphal, and are not to be found in any of the accounts written by men living at the time of his death.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Therein, she cites a number of biographies of De Moivre, including by Fouchy (original available through &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3552s.image.langEN.r&quot;&gt;Gallica&lt;/a&gt;) and by Maty (available translated through &lt;a href=&quot;http://arxiv.org/abs/0708.3965&quot;&gt;Arxiv&lt;/a&gt;). These are the only two major biographies to have appeared soon after his death. They're quite similar (see the annotations to Maty's biography for more details). Neither has any mention of this story. I have not checked every other biography cited therein, only those two linked above, but Walker extensively documents her sources and it can be reasonably safely assumed that none of them have anything related to this story or she would have noted it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story isn't completely false. It is true that De Moivre was, towards the end of his life, in an increasingly frail state. He required in excess of 20 hours per day of sleep, and his condition only worsened. However, the story about predicting his own death has no reliable source, and can reasonably be assumed to be at least partly apocryphal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not traced this back to its origin. Wikipedia cites Cajori's 1893 text &lt;a href=&quot;http://www.gutenberg.org/ebooks/31061&quot;&gt;A History of Mathematics&lt;/a&gt;. This contains a slightly different version of the story without citation. It should be noted that this was intended as a popular presentation of an abbreviated history of mathematics, not an extensive scholarly work, so the lack of references is not surprising. There may be earlier sources for this claim (in fact, I suspect this is likely, as Cajori probably didn't just make it up), but none by any of De Moivre's biographers who knew him personally.&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="58" LastEditDate="2014-11-09T15:51:37.873" LastActivityDate="2014-11-09T15:51:37.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="337" PostTypeId="2" ParentId="211" CreationDate="2014-11-09T16:18:55.910" Score="3" Body="&lt;p&gt;I was able to trace some information back to &lt;em&gt;The Ultraviolet Disinfection Handbook&lt;/em&gt;, by James R. Bolton and Christine A. Cotton. The relevant passages can be found &lt;a href=&quot;http://books.google.com/books?id=1-Aw1BVoJVgC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; on Google Books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A timeline of ultraviolet disinfection begins on page two of the introduction. It notes Ritter's 1801 discovery of UV light, and then several other early experiments. However, the first study of the effects of UV light on organisms appears to be a test conducted in 1877 by Downes and Blunt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1877: Downes and Blunt (1877) observed that test tubes filled with a broth containing bacteria, when exposed to sunlight, eventually becomes sterile.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next year, the pair determined that light towards the blue side of the spectrum had the greatest effect on bacteria.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the book does not appear to properly address the dangers of UV light to human skin. The closest it gets is citing an experiment done around 1900 by Niels Finsen:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;~1900: the Danish physician Neils Finsen, considered the founder of modern phototherapy, discovered a UV treatment for &lt;em&gt;lupus vulgaris&lt;/em&gt;, a form of skin tuberculosis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, other sources seem to indicate that Finsen did research on he effects of UV light on skin. He eventually won a Nobel Prize. From &lt;a href=&quot;http://www.nobelprize.org/nobel_prizes/medicine/laureates/1903/finsen-bio.html&quot; rel=&quot;nofollow&quot;&gt;a page&lt;/a&gt; in the Nobel prize website,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In beautiful but simple experiments Finsen demonstrated that the most refractive rays from the sun («the chemical rays») or from an electric arc may have a stimulating effect on the tissues. If the irradiation is too strong, however, it may give rise to tissue damage, but this may to some extent be prevented by pigmentation of the skin as in the negro or in those much exposed to the sun.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I haven't been able to confirm that this was the first experiment addressing UV light and human skin, but it appears that it was, at the least, &lt;em&gt;one&lt;/em&gt; of the first.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-09T16:18:55.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="338" PostTypeId="2" ParentId="38" CreationDate="2014-11-09T20:49:50.387" Score="8" Body="&lt;p&gt;I suggest that the tacit assumption in the question---that all founders of A.I. were optimistic about the goal being reached very soon---is historically false. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some were optimistic, but just as many were not, although they hoped that their work was a step to the eventual realization of this goal far in the future.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure it has anything to do with the generation they were part of being particularly optimistic. Many scientists in the same generation, including many noted physiologists, were sceptical about such claims, yet worked to make contributions toward this goal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many researchers in the same generation denied that artificial intelligence in the sense of pure Turing Machines COULD EVER do ALL the work biological brains were doing. M. Lighthill, R. Gerard, N. Rashevsky (originator of inhibition in neural network models), R. Rosen, Popper, Penrose, Hayek, Lucas, Pribram, etc., etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E.g., in a paper (&lt;em&gt;Church’s thesis and its relation to the concept of realizability in biology and physics&lt;/em&gt;) in 1962, Rosen, argued that most arbitrary neural networks are not computable. Which does not mean artificial intelligence is impossible, but that it requires other methods of physically realizing computation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most suggested schemes are neural networks conceptually, but require specialized physical systems to be realized, due to the number of simultaneous interconnections. Indeed, one can couple Turing machines in a particular way to make a &quot;as-a-whole&quot; noncomputable (hypercomputable) system. Such discussions were present at the same time as much of A.I. was being developed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This has been discussed by J. Copeland in the 1990's---and the one catch is that Turing computation is the only computation that has a universal computer. Most hypercomputation concepts lack universal realizations, but depend on their construction on what they can and cannot do, are more specialized.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;McCulluch and Pitts were students of Rashevsky for instance, and they published on related subjects all the way through the 1950's.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2014-11-09T20:49:50.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="339" PostTypeId="1" CreationDate="2014-11-09T20:54:54.347" Score="13" ViewCount="310" Body="&lt;p&gt;It is well known that Galois, and other mathematicians around that time, considered Galois groups to be permutation groups and approached Galois theory in this manner. At some point the theory took a different angle and it was approached using newly developed field theory. The study of roots of the polynomials was approached with splitting fields etc. This gave a better and more concise statement of the fundamental theory of Galois theory. &lt;strong&gt;&lt;em&gt;At which point did this modern approach arise?&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="27" LastEditorUserId="26" LastEditDate="2015-03-01T13:53:00.700" LastActivityDate="2015-03-01T13:53:00.700" Title="When was the modern field theory approach to Galois theory developed?" Tags="|mathematics|group-theory|galois-theory|abstract-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="340" PostTypeId="1" AcceptedAnswerId="488" CreationDate="2014-11-09T22:25:21.780" Score="7" ViewCount="141" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Superconducting_Super_Collider&quot; rel=&quot;nofollow&quot;&gt;Superconducting Super Collider&lt;/a&gt; was the greatest experiment that was never built. It was cancelled in 1993 after years of planning and effort, during which there was some construction of facilities. &lt;a href=&quot;http://en.wikipedia.org/wiki/Superconducting_Super_Collider#Current_status_of_site&quot; rel=&quot;nofollow&quot;&gt;The site&lt;/a&gt; may be used in the future, but I haven't been able to find much information on plans for it. When the project was cancelled, were any plans made for the future of the site?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-05-12T15:17:16.820" LastActivityDate="2015-05-12T15:17:16.820" Title="When the Superconducting Super Collider was built, was the future of the site planned for?" Tags="|experimental-physics|particle-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="341" PostTypeId="2" ParentId="339" CreationDate="2014-11-09T22:34:55.280" Score="6" Body="&lt;p&gt;A first general treatment (that is, with an abstract notion of field, which is how I understand the question) of Galois theory was given by &lt;a href=&quot;http://en.wikipedia.org/wiki/Heinrich_Martin_Weber&quot;&gt;Heinrich Weber&lt;/a&gt; in &#xA;&lt;a href=&quot;http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002254670&quot;&gt;&quot;Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie&quot; Mathematische Annalen 43 (1893) 548 - 549&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-11-09T22:34:55.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="342" PostTypeId="1" CreationDate="2014-11-10T02:56:11.877" Score="8" ViewCount="251" Body="&lt;p&gt;While the long notation or long scale uses multiples of millions to denominate larger numbers (e.g., 10^12 being a billion, a million of millions, thus bi-llion), the short notation (the standard in the US and UK) uses multiples of thousands (e.g., 10^9 being a billion, a thousand millions, which does not match the bi-llion intuition). Most of continental Europe and Latin America uses the long scale whereas most English-speaking countries use the short scale. The difference between the short and long notation is especially bothersome when you are trying to translate between languages that use the long notation and English. Any ideas as to why we have the two scales, why the regional differences and why we ended up with the short scale, which at least to me seems less intuitive?&lt;/p&gt;&#xA;" OwnerUserId="148" LastEditorUserId="26" LastEditDate="2014-11-11T21:24:39.967" LastActivityDate="2014-11-11T21:24:39.967" Title="When was the short notation for large numbers first introduced and why?" Tags="|notation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="343" PostTypeId="1" AcceptedAnswerId="345" CreationDate="2014-11-10T04:48:02.527" Score="40" ViewCount="6647" Body="&lt;p&gt;In a letter to Robert Hooke in 1676, Newton wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I have seen further it is by standing on the shoulders of giants.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Do we know which giants Newton was referring to? And was he referring to a particular piece of work or in general to all his accomplishments thus far? &lt;/p&gt;&#xA;" OwnerUserId="165" LastEditorUserId="26" LastEditDate="2014-11-11T11:23:52.853" LastActivityDate="2024-01-08T08:33:13.430" Title="Whose shoulders did Newton stand on?" Tags="|mathematics|physics|biographical-details|" AnswerCount="6" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="344" PostTypeId="1" CreationDate="2014-11-10T06:44:10.857" Score="20" ViewCount="1129" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I am looking for the original reference discussing a specific, elementary example of a rearrangement of series converging to a value different from the original series. In what follows, I give some (meager) context and make explicit the example whose origin I am trying to locate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Apparently, Dirichlet was first to note that a conditionally convergent series may be rearranged so that the resulting series converges to a different value. In 1837, in his paper showing that any arithmetic progression &lt;span class=&quot;math-container&quot;&gt;$(a+bn)_{n\ge0}$&lt;/span&gt; contains infinitely many primes provided that &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; are relatively prime, he gives an example and also proves that rearranging an absolutely convergent series has no effect on its limit.&lt;/p&gt;&#xA;&lt;p&gt;In 1853, Riemann proved his rearrangement theorem, although it was not published until 1866, as part of his &lt;em&gt;Habilitationsschrift&lt;/em&gt; on representation of functions as trigonometric series. See &lt;a href=&quot;http://www.emis.de/classics/Riemann/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for Riemann's papers.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt; (Riemann). Given any α≤β in the extended reals, a conditionally convergent series of reals can be rearranged so that the liminf of the partial sums of the rearranged series is α, while the limsup is β. In particular, any real can be obtained as the sum of some rearrangement of the original series.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Prior to Riemann's theorem, Ohm had obtained interesting related result, that were later extended by Schlömilch, and Pringsheim, see &lt;a href=&quot;https://math.stackexchange.com/a/118723/462&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://eaton.math.rpi.edu/faculty/Kovacic/Classes/Math-4210/Papers/harmonic_rearrangements.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for statements of some of their results, and modern extensions. They are usually discussed in terms of rearrangements of the alternating harmonic series, although their results are more general.&lt;/p&gt;&#xA;&lt;p&gt;The alternating harmonic series is the series&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1 - 1/2 + 1/3 - 1/4 + 1/5 - ... = \ln(2)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Ohm's theorem predicts the value of the series obtained if we fix positive integers &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, and the terms of the series are rearranged so that first we add the first&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; positive terms, then the first &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; negative terms, then the next p positive terms, then the next q negative ones, etc. For instance, if &lt;span class=&quot;math-container&quot;&gt;$p=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q=2$&lt;/span&gt;, the rearrangement is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - \cdots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Ohm's theorem states that this series converges to &lt;span class=&quot;math-container&quot;&gt;$$\ln(2) + (1/2) \ln(p/q)$$&lt;/span&gt; In particular, for &lt;span class=&quot;math-container&quot;&gt;$p=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q=2$&lt;/span&gt;, the limit is &lt;span class=&quot;math-container&quot;&gt;$(1/2) \ln(2)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, when discussing rearrangements (say, in calculus textbooks), it is common to find this specific example, evaluated as follows:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - \cdots = (1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 + \cdots = (1/2)(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The proof of the general result is more intricate, but introductory discussions of rearrangements omit it, while the argument above is completely basic and intuitive. (The other example that tends to be found in introductory discussions is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + \cdots = (3/2) \ln(2)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;corresponding to &lt;span class=&quot;math-container&quot;&gt;$p=2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q=1$&lt;/span&gt; in Ohm's theorem. This example can be verified by a similar, only slightly more involved trick, see &lt;a href=&quot;https://math.stackexchange.com/a/412851/462&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;I am trying to locate the first place where this specific proof is mentioned. Manning's classical (1906) book &lt;strong&gt;Irrational numbers and their representation by sequences and series&lt;/strong&gt; mentions the example, in pages 97, 98, but the only source it mentions is &amp;quot;(Laurent)&amp;quot;, without further bibliographical details.&lt;/p&gt;&#xA;&lt;p&gt;Finally, although perhaps off topic for this list, if a similar elementary observation can be used to prove Ohm's theorem, or a large fragment of it, I would be grateful for details. The general case is not difficult, but the proofs I am aware of are not quite as suitable for discussion in an introductory course.&lt;/p&gt;&#xA;" OwnerUserId="18" LastEditorUserId="-1" LastEditDate="2022-02-07T04:36:42.963" LastActivityDate="2022-02-07T16:57:29.307" Title="Conditionally convergent series" Tags="|mathematics|real-analysis|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="345" PostTypeId="2" ParentId="343" CreationDate="2014-11-10T09:08:09.547" Score="31" Body="&lt;p&gt;We must locate this passage into a double context :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the general historical context&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the specific context of the &quot;rivalry&quot; between Hooke and Newton.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See Robert Purrington, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00408AWGA&quot; rel=&quot;noreferrer&quot;&gt;The First Professional Scientist : Robert Hooke and the Royal Society of London&lt;/a&gt; (2009), &lt;strong&gt;Ch.8 :&lt;/strong&gt; &lt;em&gt;And All Was Light&lt;/em&gt; : &lt;strong&gt;Hooke and Newton on Light and Color&lt;/strong&gt;, page 135-on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the general context, we have to take into account the diffusion of &lt;a href=&quot;http://plato.stanford.edu/entries/descartes-physics/&quot; rel=&quot;noreferrer&quot;&gt;Descartes' Physics&lt;/a&gt; and philosophy around the middle of the 17th Century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton made extensive study of Descartes' theories (regarding : motion of bodies, celestial physics and light) : see Isaac Newton, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521538483&quot; rel=&quot;noreferrer&quot;&gt;Philosophical Writings&lt;/a&gt; (editor Andew Janiak, 2004).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All Newtonian's theory was in contrast with Descartes' ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abou the rivalry with Hooke, [see Purrington, &lt;em&gt;passim&lt;/em&gt; and pages 141-42], we have to consider the&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;delivering [by Newton of] a long manuscript on light and color to the [Royal] Society. The paper, consisting of an “Hypothesis” and a “Discourse,” was read by Oldenburg in five parts between 9 December and 10 February 1675/6 [&lt;em&gt;footnote&lt;/em&gt; : Newton to Oldenburg, 7 December 1675; Corresp. I, 362–386].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Newton began by saying that he would not make any hypothesis about the aether and its role in the propagation of light, not thinking it necessary «to concern my selfe whether the properties of Light, discoverred by me, be explained by this or Mr Hook’s or any other Hypothesis capable of explaining them . . . » [&lt;em&gt;footnote&lt;/em&gt; : «An Hypothesis explaining the Properties of Light discoursed of in my severall Papers.» &lt;em&gt;Corresp&lt;/em&gt;. I, 362–386.] Further on he discussed Hooke’s discovery of diffraction, noting others who had observed it as well, but certainly giving Hooke credit. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Hooke was, however, unimpressed, and after hearing the first part of the “Discourse” on 9 and 16 December, noted «that the main of it was contained in his &lt;em&gt;Micrographia&lt;/em&gt;, which Mr. Newton had only carried farther in some particulars.» [&lt;em&gt;footnote&lt;/em&gt; : &lt;em&gt;Birch&lt;/em&gt;, 3, 269. Hooke made very much the same comment on Newton’s theory of planetary motion. Everything we know of Hooke suggests that he really believed these assertions.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The result of this new exchange, indirect though it may have been, with Oldenburg as intermediary, was to rekindle the old animosity of four years before.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The rift between Hooke and Oldenburg was growing at this time, and Hooke felt that the Secretary was trying to exacerbate the ill will between himself and Newton. Ironically, this alienation prompted Hooke to write a very conciliatory letter to Newton [&lt;em&gt;footnote&lt;/em&gt; : Hooke to Newton, 20 January 1675/6. &lt;em&gt;Corresp&lt;/em&gt;. I, pp. 412–3], hoping to heal the rupture that he felt the Secretary had fomented. He wrote Newton on the day the latter’s heated reply was read to the Society, 20 January, and the &lt;em&gt;Diary&lt;/em&gt; suggests that he wrote him immediately after the meeting, hoping to heal the rupture. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In Newton’s more than cordial reply to this letter [&lt;em&gt;footnote&lt;/em&gt; : &lt;em&gt;Corresp&lt;/em&gt;., I, 416–7], he mentioned&#xA;  having called at Hooke’s lodgings only to find him out. And it was in this letter that&#xA;  Newton penned the famous line: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;«If I have seen further it is by standing on ye shoulders of Giants,» &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;making it clear that Hooke was one of those giants. This almost friendly exchange of letters between brought a sort of peace and established a basis for Hooke’s correspondence with Newton after Oldenburg’s death in 1677.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In conclusion, I think that we have to read it in a &quot;multiple way&quot; : Newton's theories and discoveries was made possible also by a careful study of the works of previous &quot;Giants&quot;, like Descartes and Hooke, all of them surpassed by the ability of Newton to &quot;see further&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Added&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the on-line transcription of David Brewster, &lt;em&gt;Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton&lt;/em&gt;, vol. 1 (Edinburgh: 1855), in &lt;a href=&quot;http://www.newtonproject.sussex.ac.uk/view/texts/normalized/OTHE00101&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Newton Project&lt;/strong&gt;&lt;/a&gt; we can recover a longer quotation from the letter :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;you defer too much to my ability in searching into this subject. What Descartes did was a good step. &lt;em&gt;You have added much several ways, and especially in considering the colours of thin plates&lt;/em&gt; [emphasis added]. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&lt;em&gt;If I have seen farther, it is by standing on the shoulders of giants&lt;/em&gt;. &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;But I make no question you have divers very considerable experiments beside those you have published, and some, it's very probable, the same with some of those in my late papers. Two at least there are, which I know you have often observed, — the dilatation of the coloured rings by the obliquation of the eye, and the apparition of a black spot at the contact of two convex glasses, and at the top of a water-bubble; and it's probable there may be more, besides others which I have not made, so that I have reason to defer as much or more in this respect to you, as you would to me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;How much of this is flattery? How much is gratitude? Who knows...&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="2542" LastEditDate="2015-07-06T17:25:23.107" LastActivityDate="2015-07-06T17:25:23.107" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="346" PostTypeId="2" ParentId="342" CreationDate="2014-11-10T16:14:14.140" Score="2" Body="&lt;p&gt;I can answer the &quot;when&quot; part of the question at the moment; I'm not yet sure of the &quot;why&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://books.google.com/books?id=uTytJGnTf1kC&amp;amp;lpg=PA84&amp;amp;dq=origins+of+number+names+billion+-wikipedia&amp;amp;pg=PA84&amp;amp;hl=en#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;History of Mathematics, Volume 2&lt;/em&gt;&lt;/a&gt;, by David Eugene Smith (page 85, emphasis mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The French use of &lt;em&gt;milliard&lt;/em&gt;, for 10^9, or &lt;em&gt;billion&lt;/em&gt; as an alternative is relatively late. The word appears at least as early as the beginning of the 16th century as the equivalent of both 10^9 and of 10^12, the latter being the billion of England today. By the 17th century, however, it was used in Holland to mean 10^9, and no doubt it was about this time that the usage began to change in France.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This indicates that the term was first used in Holland in the 1600s. The word was used earlier, though, but only as part of the long scale, to refer to 10^12.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-10T16:14:14.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="347" PostTypeId="1" AcceptedAnswerId="351" CreationDate="2014-11-10T16:40:55.567" Score="15" ViewCount="2224" Body="&lt;p&gt;Apparently, Cardano had learned a formula for solving &lt;a href=&quot;https://www.google.com/search?q=cardano%27s+formula&amp;amp;oq=cardan%27s+&amp;amp;aqs=chrome.1.69i57j0l5.4815j0j4&amp;amp;sourceid=chrome&amp;amp;es_sm=119&amp;amp;ie=UTF-8&quot; rel=&quot;noreferrer&quot;&gt;cubic equations&lt;/a&gt; from Tartaglia, who had sworn him to secrecy, and in any event, not to publish it without giving Tartaglia due credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cardano published it under his own name, claiming to have learned it elsewhere from a man named Ferro.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Everyone&quot; knows these facts. So why do people keep referring to it as Cardano's formula, rather than Tartagia's formula, or even Ferro's formula? &lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="8" LastEditDate="2015-06-28T11:56:02.870" LastActivityDate="2015-06-28T11:56:02.870" Title="Why is &quot;Cardano's Formula&quot; (wrongly) attributed to him?" Tags="|mathematics|publishing|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="348" PostTypeId="1" CreationDate="2014-11-10T17:41:56.213" Score="12" ViewCount="641" Body="&lt;p&gt;Because the real numbers are uncountable and the real algebraic numbers are countable, there are uncountably many transcendental numbers. So there are far more transcendentals than rationals. With the language of set theory these facts are easy to see, but historically it was very difficult to construct transcendental numbers (or prove that certain numbers were transcendental). It is my impression that, historically, nowhere-differentiable functions were considered pathological and rare among the continuous functions. Was this similarly true for transcendental numbers among the rationals: were they considered pathological and rare?&lt;/p&gt;&#xA;" OwnerDisplayName="user106" LastEditorDisplayName="user106" LastEditDate="2014-11-10T17:46:37.860" LastActivityDate="2019-12-27T03:45:55.683" Title="Were transcendental numbers considered rare, pre-Cantor?" Tags="|mathematics|number-theory|set-theory|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="349" PostTypeId="2" ParentId="348" CreationDate="2014-11-10T20:37:15.057" Score="11" Body="&lt;p&gt;As I understand, people did not think much about comparing the size of infinite sets before Cantor, so for most of them the question you are asking had no meaning, and they did not think about it.&#xA;Perhaps few people felt intuitively that a countable set is &quot;smaller&quot; than a continuum,&#xA;but Cantor was the first to state and prove this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. But there were actually exceptions to this. Nicola Oresme in 1360 (!) really argued that there are much more irrational numbers than rational ones,&#xA;and even that a &quot;random number&quot;, whatever it could mean, is irrational. For more details on this see&#xA;the answers to this question &lt;a href=&quot;https://mathoverflow.net/questions/269893&quot;&gt;https://mathoverflow.net/questions/269893&lt;/a&gt;&#xA;and references there. Of course his arguments could not be rigorous at that time. And he was discussing irrational numbers. That transcendental numbers exist at all was only proved in 19 century, by Cantor and Liouville, with different proofs.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-12-27T03:45:55.683" LastActivityDate="2019-12-27T03:45:55.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="350" PostTypeId="2" ParentId="347" CreationDate="2014-11-10T20:48:16.977" Score="7" Body="&lt;p&gt;It is not &quot;attributed to him&quot;, because as you mention yourself, everyone knows the story.&#xA;It is only &quot;named&quot; after him. Very many things are named with other names than their inventors, or discoverers or first persons who proved a them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some random widely known examples are: Schwarz Lemma (he did not state it in full generality), Schwarz inequality (due to Cauchy and Bunyakovski), Schwarzian derivative&#xA;(considered by Lagrange), Alexandrov compactification, Riemann zeta-function, and many many other things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Name of a theorem or formula is just a label. It has has to be a) well recognizable, and b) commonly accepted. Who proved it first, is not so important (except for the author:-)&#xA;and in many cases several people contributed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;America&quot; is also not named after Columbus (or whoever &quot;discovered&quot; it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Let me also mention the well-known &quot;Arnold's Principle&quot;: If a thing is called after a person, this indicates that this particular person had nothing to do with it.&#xA;Sir Michael Berry replied on this with a Theorem: Arnold's Principle applies to itself. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-10T23:25:15.323" LastActivityDate="2014-11-10T23:25:15.323" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="351" PostTypeId="2" ParentId="347" CreationDate="2014-11-10T20:53:28.880" Score="19" Body="&lt;p&gt;In short, because only positive numbers could be used as coefficients at the time there were several cases of the cubic that had to be treated separately. Before Cardano only depressed cubics (with one of the powers missing) were solved. He introduced the substitution that reduces general cubic to a depressed one, and made formulas for depressed ones public for the first time, with his own derivations. Ironically, the formula under his name is not due to him, but he was the first one to &quot;publish&quot; it, while acknowledging del Ferro's and Tartaglia's contributions. Since the use of complex numbers was rudimentary at the time even Cardano's generalization did not resolve the case of a cubic with three real roots, the &lt;a href=&quot;http://en.wikipedia.org/wiki/Casus_irreducibilis#Relation_to_angle_trisection&quot; rel=&quot;noreferrer&quot;&gt;irreducible case&lt;/a&gt;, as Bombelli pointed out. It was later resolved by Viète, who related it to the angle trisection problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cardano's masterpiece, &lt;a href=&quot;http://en.wikipedia.org/wiki/Ars_Magna_%28Gerolamo_Cardano%29&quot; rel=&quot;noreferrer&quot;&gt;Ars Magna&lt;/a&gt; (1545), was the most comprehensive treatise of algebra until Viète's &lt;a href=&quot;http://en.wikipedia.org/wiki/New_algebra&quot; rel=&quot;noreferrer&quot;&gt;Isagoge&lt;/a&gt; (1591). Unlike del Ferro and Tartaglia who dealt with special cases only, Cardano provided systematic treatment based on substitutions of all thirteen cases of depressed cubics, and of the general case. It is no accident that it was his student, Ferrari, who solved the quartic following the same methods (his solution was also included into Ars Magna). Moreover, Cardano acknowledged the possibility of negative (&quot;fictitious&quot;) roots, and in one example considered complex (&quot;sophistic&quot;) roots for the first time. Ars Magna's influence is often compared to that of Copernicus' De revolutionibus, and it did a lot to promote new methods in algebra at the time. Most mathematicians learned the cubic solution formula from it, which explains the naming. This is similar to &quot;Pascal's&quot; triangle. It was known even in Europe long before Pascal (curiously, Tartaglia claimed it as his), but Pascal was the first to systematically study its properties. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery itself is a fascinating story. It was &lt;a href=&quot;http://en.wikipedia.org/wiki/Scipione_del_Ferro&quot; rel=&quot;noreferrer&quot;&gt;Scipione del Ferro&lt;/a&gt; who solved the first depressed cubic. At the time it was common for mathematicians to call each other to &quot;duels&quot;, where having a secret trick for solving equations no one else could solve was a big advantage. These duels often carried monetary prizes, and affected recognition and appointments in ways publications do today. So del Ferro kept it a secret, which he entrusted to his student Fiore. Tartaglia challenged Fiore and shortly before the contest solved del Ferro's case and one other (he admitted though that what helped him solve it was knowing that solution existed). Needless to say he won the duel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After futile attempts to solve cubics by himself Cardano convinced Tartaglia to give him the formula with a promise that he would keep it a secret until Tartaglia publishes it in a book he was writing at the time. However, Tartaglia took his time and eventually Cardano found out that del Ferro discovered the formula before Tartaglia. He no longer felt bound by the promise, and published the formula in Ars Magna. Tartaglia was understandably livid. The coda was Ferrari challenging Tartaglia to a duel after discovering a solution to quartics. That one Tartaglia lost.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-11-11T18:02:51.333" LastActivityDate="2014-11-11T18:02:51.333" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="352" PostTypeId="1" CreationDate="2014-11-10T21:53:48.870" Score="11" ViewCount="361" Body="&lt;p&gt;Recently, while discussing, a friend claimed that Leibniz was fond of proceeding axiomatically and from definitions, which I find hard to believe. My conception is that Leibniz was more interested in intuition and the process of construction rather than endproducts. This is exemplified by these two quotes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This quote I found in Polya's &lt;em&gt;How to Solve It&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;I am convinced that the unwritten knowledge scattered among men of different callings surpasses in quantity and in importance anything we find in books, and that the greater part of our wealth has yet to be recorded.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is from Wikiquote, alledgedly from &lt;a href=&quot;http://en.wikisource.org/wiki/fr:Discours_touchant_la_m%C3%A9thode_de_la_certitude_et_l%E2%80%99art_d%E2%80%99inventer&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Am I wrong?&lt;/p&gt;&#xA;" OwnerUserId="44" LastEditorUserId="4251" LastEditDate="2023-06-26T10:44:04.307" LastActivityDate="2023-06-28T10:56:51.610" Title="Leibniz's stand on axioms and definitions" Tags="|mathematicians|biographical-details|philosophy-of-science|leibniz|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="353" PostTypeId="1" AcceptedAnswerId="355" CreationDate="2014-11-10T22:26:45.813" Score="12" ViewCount="494" Body="&lt;p&gt;Initially our ancestors believed of Earth as a flat place. Then after scientific discussions and experiments over a period of time, it was established that our Earth is round.&#xA;But, there exists one more theory that stats that the earth is Hollow completely or at least at the both Pole(North and South) positions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is who and on what basis(scientific evidence) pitched in this theory ?&lt;/p&gt;&#xA;" OwnerUserId="188" LastActivityDate="2014-11-10T23:06:44.040" Title="Who pitched in the theory of Hollow Earth?" Tags="|discoveries|earth-sciences|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="354" PostTypeId="2" ParentId="347" CreationDate="2014-11-10T22:28:39.733" Score="1" Body="&lt;p&gt;One thing to take into account is that recognition of the discoveries goes to the first who publishes it. You may discover the secret of live, the universe and everything, but if you just hide the knowledge you cannot claim its prior discovery when someone other publishes them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Gauss did investigate the possibilities of non Euclidian geometry, but chose not to publish his findings. So, hyperbolic geometry is also known as Lobachevskian and not as Gaussian geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in this case, even if Cardano's disclosure was dishonest, the naming would just follow the usual conventions.&lt;/p&gt;&#xA;" OwnerUserId="206" LastActivityDate="2014-11-10T22:28:39.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="355" PostTypeId="2" ParentId="353" CreationDate="2014-11-10T23:06:44.040" Score="10" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hollow_Earth&quot;&gt;Wikipedia&lt;/a&gt; notes that many ancients believed that there were hollowed-out areas underground, where places such as Hell (or the underworld in general, hence the &lt;em&gt;under&lt;/em&gt;world) existed. But these weren't really &quot;Hollow Earth&quot; theories. We didn't get that far until a few hundred years ago.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.unmuseum.org/hollow.htm&quot;&gt;This&lt;/a&gt; attributes the modern Hollow-Earth hypothesis to Edmund Halley, the famed astronomer, circa 1692. He was a brilliant guy, but this was not exactly his most brilliant idea. He proposed the idea after noticing some strange properties of the Earth's magnetic field:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Halley was fascinated by the earth's magnetic field. He noticed the direction of the field varied slightly over time and the only way he could account for this was there existed not one, but several, magnetic fields. Halley came to believe that the Earth was hollow and within it was a second sphere with another field. In fact, to account for all the variations in the field, Halley finally proposed that the Earth was composed of some four spheres, each nestled inside another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This isn't &lt;em&gt;too&lt;/em&gt; bad, but the idea of proposing loads of different spheres to account for corrections in observations reeks of epicycles. But it gets worse:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Halley also suggested that the interior of the Earth was populated with life and lit by a luminous atmosphere. He thought the aurora borealis, or northern lights, was caused by the escape of this gas through a thin crust at the poles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The site goes on to discuss those who came later to develop the theory: John Symmes, Sir John Leslie, and even (gasp - don't look unless you want to be really disappointed)&lt;/p&gt;&#xA;&#xA;&lt;blockquote class=&quot;spoiler&quot;&gt;&#xA;  &lt;p&gt; Leonhard Euler!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some of the more recent proponents didn't have such good reputations. Take Cyrus Read Teed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Teed changed his name to Koresh and founded what might today be called a cult. After buying a 300 acre tract in Florida, Koresh declared himself the messiah of a new religion. He died in 1908 without proving his ideas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www2.gi.alaska.edu/ScienceForum/ASF7/718.html&quot;&gt;A second source&lt;/a&gt; gives a similar discussion on the subject. It attributes the modern-day theory to John Symmes (mentioned above):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The hollow earth theory actually seems to have been originated in the early 1800s by John Symmes, an earnest American who devoted the greater part of his later life to convincing the world that the earth was formed by a series of concentric shells.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.skepdic.com/hollowearth.html&quot;&gt;This&lt;/a&gt; page gives an interesting discussion of the idea. At first, I wasn't sure if it supported the theory or not, but it appears to go against it. It, too, says that Halley was the first to develop the idea, but it does use the two pages I listed above as sources, so this could be a duplicate.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;John Symmes appears to have been one of the proponents of the modern theory, but we can trace it back all the way to Edmund Halley.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-10T23:06:44.040" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="357" PostTypeId="1" AcceptedAnswerId="392" CreationDate="2014-11-11T02:09:08.773" Score="12" ViewCount="3313" Body="&lt;p&gt;A well-known TV series hinted some time ago that Isaac Newton burnt the only known picture of Robert Hooke.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it true?&lt;/p&gt;&#xA;" OwnerUserId="209" LastEditorUserId="26" LastEditDate="2014-11-11T10:23:38.690" LastActivityDate="2020-04-14T17:32:31.167" Title="Did Isaac Newton burn Robert Hooke's picture?" Tags="|biographical-details|debunking|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="358" PostTypeId="5" CreationDate="2014-11-11T02:14:23.257" Score="0" Body="&lt;p&gt;Use this tag for questions about the origin of scientific and/or mathematical words.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-11-16T21:27:15.307" LastActivityDate="2014-11-16T21:27:15.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="359" PostTypeId="4" CreationDate="2014-11-11T02:14:23.257" Score="0" Body="The origin of scientific / mathematical words" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-11-12T20:19:09.517" LastActivityDate="2014-11-12T20:19:09.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="360" PostTypeId="5" CreationDate="2014-11-11T02:30:18.660" Score="0" Body="&lt;p&gt;This tag should be used along with &lt;a href=&quot;/questions/tagged/physicists&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;physicists&amp;#39;&quot; rel=&quot;tag&quot;&gt;physicists&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/chemists&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;chemists&amp;#39;&quot; rel=&quot;tag&quot;&gt;chemists&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/biologists&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;biologists&amp;#39;&quot; rel=&quot;tag&quot;&gt;biologists&lt;/a&gt; or &lt;a href=&quot;/questions/tagged/mathematicians&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;mathematicians&amp;#39;&quot; rel=&quot;tag&quot;&gt;mathematicians&lt;/a&gt; tags as the case may be.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="10106" LastEditDate="2022-08-21T13:20:58.157" LastActivityDate="2022-08-21T13:20:58.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="361" PostTypeId="4" CreationDate="2014-11-11T02:30:18.660" Score="0" Body="For questions about the lives of historical Scientists or Mathematicians" OwnerUserId="25" LastEditorUserId="10106" LastEditDate="2022-08-20T17:12:54.050" LastActivityDate="2022-08-20T17:12:54.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="362" PostTypeId="5" CreationDate="2014-11-11T02:32:30.490" Score="0" Body="&lt;p&gt;Use this tag for questions about new scientific discoveries - in contrast to biographical questions, or how theories were developed.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-11-12T20:18:21.363" LastActivityDate="2014-11-12T20:18:21.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="363" PostTypeId="4" CreationDate="2014-11-11T02:32:30.490" Score="0" Body="For questions about the location (in space and time) of scientific discoveries and their historical context." OwnerUserId="25" LastEditorUserId="26" LastEditDate="2014-12-12T11:26:03.053" LastActivityDate="2014-12-12T11:26:03.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="364" PostTypeId="5" CreationDate="2014-11-11T02:33:53.990" Score="0" Body="&lt;p&gt;Use this tag for questions that are asking for literature references. For example, they might ask about a specific quote, or references where the reader can read about the history of a specific piece of scientific theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answers to questions using this tag should include the work(s) in question and a short description of them, preferably with a link to where the work(s) can be found.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="62" LastEditDate="2015-04-12T16:30:26.363" LastActivityDate="2015-04-12T16:30:26.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="365" PostTypeId="4" CreationDate="2014-11-11T02:33:53.990" Score="0" Body="For questions that are requesting specific literature references" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-11-16T21:25:27.423" LastActivityDate="2014-11-16T21:25:27.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="366" PostTypeId="5" CreationDate="2014-11-11T02:45:07.853" Score="0" Body="&lt;p&gt;Use this tag for questions about physics experiments and observations of physical phenomena. For example, one could ask about a particular historically significant experiment (e.g. Joule's experiments with heat &amp;amp; water, or Galileo's observations of Venus and Jupiter's Moons), or about some (type of) experiments that were conducted in the past and by whom.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="10106" LastEditDate="2022-02-23T08:33:02.727" LastActivityDate="2022-02-23T08:33:02.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="367" PostTypeId="4" CreationDate="2014-11-11T02:45:07.853" Score="0" Body="For questions about historical physics experiments, the history of experimental methodology, observations of physical phenomena and related topics" OwnerUserId="25" LastEditorUserId="10106" LastEditDate="2022-02-23T16:13:46.930" LastActivityDate="2022-02-23T16:13:46.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="368" PostTypeId="1" AcceptedAnswerId="371" CreationDate="2014-11-11T04:06:42.307" Score="23" ViewCount="5793" Body="&lt;p&gt;That is, when did astronomy figure out how to predict &lt;strong&gt;when and where&lt;/strong&gt; a solar eclipse will be visible?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems people noticed fairly early on that the Sun, Moon and Earth return to the same approximate relative positions &lt;a href=&quot;http://en.wikipedia.org/wiki/Saros_(astronomy)&quot; rel=&quot;noreferrer&quot;&gt;periodically (~18 years and 11 days and 8 hours)&lt;/a&gt;. This tells them how long it takes for an eclipse to (almost) repeat itself. However, the shadow of the moon on Earth is only ~150km wide during a solar eclipse, and so only a small part of the planet will actually see it happening. Thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... since the Saros cycle does not include an integral number of days ... the next eclipse in the cycle will occur one third of the way around the world from the previous eclipse in the cycle. So, the Saros cycle allowed ancient astronomers to predict when a solar eclipse could be expected, but it did not help them figure out where the eclipse would occur.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;&lt;sub&gt; - &lt;a href=&quot;http://ds9.ssl.berkeley.edu/solarweek/THURSDAY/history.html&quot; rel=&quot;noreferrer&quot;&gt;http://ds9.ssl.berkeley.edu/solarweek/THURSDAY/history.html&lt;/a&gt;&lt;/sub&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So when did astronomers understood the entire mechanics well enough to be able to predict both the time and place of a solar eclipse?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Inspired by &lt;a href=&quot;https://history.stackexchange.com/questions/16959/what-was-the-first-solar-eclipse-that-was-demonstrably-predicted-in-advance&quot;&gt;this question&lt;/a&gt; on &lt;a href=&quot;https://history.stackexchange.com/&quot;&gt;History.SE&lt;/a&gt;.&lt;/em&gt; Note that I'm not just asking for when &lt;em&gt;one&lt;/em&gt; eclipse was correctly predicted (by chance or otherwise), but rather when the physics of it is understood well enough for reliable predictions.&lt;/p&gt;&#xA;" OwnerUserId="211" LastEditorUserId="-1" LastEditDate="2017-04-13T12:47:53.943" LastActivityDate="2020-05-07T17:02:43.187" Title="When did it become possible to predict the time and place of solar eclipses?" Tags="|astronomy|experimental-physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="369" PostTypeId="2" ParentId="352" CreationDate="2014-11-11T10:33:56.337" Score="5" Body="&lt;p&gt;In a letter to Hermann Conring 1678, Leibniz writes in latin (my asterisk): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Axiomata ego non ut ais, αναπόδεικτα*, sed tamen plerumque non&#xA;  necessaria demonstratu esse arbitror. Demonstrabilia vero esse pro&#xA;  certo habeo. Unde enim constat nobis de eorum veritate? non, opinor ex&#xA;  inductione, ita enim omnes scientiae redderentur empiricae, ergo ex&#xA;  ipsismet, id est ex earum terminis: quod fit vel quando idem dicitur&#xA;  de seipso (ex. causa, A est A, unumquodque sibi ipsi aequale est, et&#xA;  similes identicae) vel quando sola terminorum significatione sive quod&#xA;  idem est definitione, intellecta statim apparet propositionis veritas.&#xA;  Omnes ergo propositiones certae demonstrari possunt praeter identicas&#xA;  et empiricas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My poor attempt at an translation. The &lt;em&gt;Axiomata ego non ut ais, αναπόδεικτα&lt;/em&gt; part:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The axioms is not as you say, αναπόδεικτα (unsubstantiated/proofless),&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The letter in its entirety can be found &lt;a href=&quot;http://www.uni-muenster.de/Leibniz/DatenII1/ii1c_tl2.pdf#page=242&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. 162. LEIBNIZ AN HERMANN CONRING (Latin).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed that the letter in Marcelo Dascal's &lt;em&gt;Leibniz: What Kind of Rationalist?&lt;/em&gt; is  similar in some regards, that translation is as follows (page 162): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I regard axioms not, as you say, as something apodictic but only as&#xA;  something that in most cases does not require a proof. Yet that they&#xA;  are demonstrable, I believe to be certain. Whence does it come that we&#xA;  are certain about their truth? As I believe, not from induction, since&#xA;  in that way all sciences would be rendered empirical; thus, from&#xA;  themselves, i.e. from their terms, which happens when the same is said&#xA;  of the same (e.g., A is A, everything is equal to itself, and similar&#xA;  identical propositions) or when only from the signification of terms&#xA;  or, what is the same, from the understood definition the truth of the&#xA;  proposition is apparent. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*I am not entirely sure this is the greek word. It looks a bit different in the text.&lt;/p&gt;&#xA;" OwnerUserId="187" LastEditorUserId="191" LastEditDate="2014-11-14T01:02:09.523" LastActivityDate="2014-11-14T01:02:09.523" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="370" PostTypeId="2" ParentId="352" CreationDate="2014-11-11T13:38:33.957" Score="5" Body="&lt;p&gt;&lt;em&gt;This is a long comment...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not able to give a complete answer &quot;in general&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if we consider specifically his &lt;em&gt;logical&lt;/em&gt; works, we can see in :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gottfried Wilhelm Leibniz, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198243065&quot; rel=&quot;noreferrer&quot;&gt;Logical Papers : A Selection&lt;/a&gt; (edited by G.H.R.Parkinson, 1966), at least :&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;A Specimen of the Universal Calculus&lt;/em&gt; (1679-86 ?), page 33-on&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Addenda to the Specimen of the Universal Calculus&lt;/em&gt; (1679-86 ?), page 40-on.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the last one [page 42] we can find :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Six] &lt;em&gt;Propositions true in themselves&lt;/em&gt; :&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;(1) &lt;em&gt;a&lt;/em&gt; is &lt;em&gt;a&lt;/em&gt;. An animal is an animal.&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;(2) &lt;em&gt;ab&lt;/em&gt; is &lt;em&gt;a&lt;/em&gt;. A rational animal is an animal.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;An inference true in itself&lt;/em&gt; : &lt;em&gt;a&lt;/em&gt; is &lt;em&gt;b&lt;/em&gt; and &lt;em&gt;b&lt;/em&gt; is &lt;em&gt;c&lt;/em&gt;, therefore &lt;em&gt;a&lt;/em&gt; is &lt;em&gt;c&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, he has a clear understanding of axioms and rules in logic.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If we instead consider his mathematical works, the &quot;failure&quot; to axiomatize the &lt;em&gt;infinitesimal calculus&lt;/em&gt; must not be considered a failure : we have to consider that it takes a couple of centuries only to attain a viable axiomatization of the real number system.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-11-11T13:38:33.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="371" PostTypeId="2" ParentId="368" CreationDate="2014-11-11T13:46:19.263" Score="16" Body="&lt;p&gt;The principle was known long ago, to the Babylonians and Hellenistic Greeks but the accuracy of prediction depends on the detail of the Lunar motion (the motion of the Sun is relatively simple). Without a precise Lunar theory, it was possible to predict that an eclipse is&#xA;LIKELY to happen on such and such date and time, but not with a 100% certainty, and the place where it will be visible and other features could not be predicted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there were many cases when an eclipse was predicted and really happened,&#xA;see, for example &lt;a href=&quot;http://en.wikipedia.org/wiki/Solar_eclipse_of_August_21,_1560&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Solar_eclipse_of_August_21,_1560&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another question is how far in advance one could predict. I suppose at the time of Brahe&#xA;it was possible to predict an eclipse several months in advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give an example in 1598 the almanac gave a solar eclipse on March 7 and a lunar&#xA;eclipse with the error of one hour. This led Kepler to the discovery of the 4-th inequality (see below). An almanac is supposed to make predictions for one year. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sufficiently precise Lunar theory&#xA;for accurate predictions for several years in advance,&#xA;with 100% certainty and where it will be visible, and other features of the eclipse,&#xA;was developed in the middle of 18 century, as a result of combined efforts of several greatest mathematicians of that time (Clairaut, Euler, T. Mayer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there are many records of more or less accurate predictions before that time, beginning from antiquity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 1. This is very different from the Lunar eclipses which do not require &#xA;knowledge of fine detail of the Moon motion. Prediction of the Lunar eclipses&#xA;was possible since antiquity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 2. The main motivation for these efforts was not the eclipse prediction but more important practical problem: determination of the longitude at sea by the &quot;Method of Lunar distances&quot;. But almost at the same time, chronometer was invented, and for about 1/2 of a century the two methods were competing. When chronometers became affordable (in the first half of 19-th century, the method of Lunar distances gradually was displaced by a simpler method, based on a chronometer). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Let me elaborate as much as possible without MathJax. First of all, what does it mean to &quot;predict&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If 10% of the Sun disc area is obscured, is this an eclipse or not?&#xA;This is what I mean by &quot;features&quot;, full or not full, perhaps circular. The same eclipse will be full in one place and partial in another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second important question: predict how much in advance? 2 days? 2 months? 2 years? or 1000 years? This makes a great difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now a brief account of what was involved. I assume that the Sun motion is known precisely&#xA;(it was known to sufficient accuracy to Hipparchus). So we only discuss the Moon. Both Sun and Moon have visible diameter about 1/2 degree. So to predict an eclipse at a given location and time we need to know the Moon motion, say to minutes of angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the time of Hipparchus to this time, the (geocentric) coordinates of the Moon as functions of time are described by a series of the form &lt;span class=&quot;math-container&quot;&gt;$$At+E(t)+E'(t)+E''(t)+ \cdots$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is time, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is the &quot;mean motion&quot; and &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; s are periodic terms depending on the&#xA;mutual position of Sun and Moon. These periodic terms are called &quot;inequalities&quot;.&#xA;The first inequality is due to Hipparchus and its maximal amplitude is about 6 degrees.&#xA;Second inequality (a. k. a. evection) was discovered by Ptolemy, its maximal&#xA;value is 2.5 degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the first inequality is due to ellipticity of the orbit (Kepler's law, as we know now), the second inequality is due to the influence of the Sun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was no much progress between Ptolemy and Tycho, except perhaps that some&#xA;numerical constants were determined more precisely. Astronomers before Tycho did not measure angles with sufficient precision. Their theories were based on Lunar eclipses,&#xA;not on the direct angle measurements. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was Tycho Brahe who discovered the third inequality (a. k. a. variation) which can be as large as 40' (that is greater than the diameter of the sun  or of the Moon).&#xA;Later Kepler and Brahe discovered the forth inequality (of amplitude 11').&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At this point the development stopped, because already the second inequality cannot be explained by the Kepler laws (Kepler laws solve the 2 body problem, and here we have 3 bodies: the influence of the Sun is not negligible, as we see already from the second inequality). For example, Jupiter obeys Kepler's laws to very high accuracy, because the influence of other planets is negligible.&#xA;But Earth-Moon-Sun system is a real 3 body system, and Moon does not obey Kepler's laws with sufficient accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With Newton's Universal law of gravitation, the first and most important problem was&#xA;whether it gives anything new besides the Kepler Laws. (After all &quot;explanation of the Kepler Laws was not such a great achievement because the Gravitation Law was derived from these very Kepler laws). The problem was to predict NEW phenomena, or at least (as a first step) to explain the third inequality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is known (nowadays) that the three body problem has no closed form solution.&#xA;This is a system of differential equations based on the law of gravity, but this system&#xA;cannot be exactly solved. So the major efforts of the best 18 century mathematicians were spent on finding a useful approximate solution. Some of them were stimulated with the practical of this problem which I mentioned above. After the great efforts&#xA;(the main contributors are Clairaut, Euler and Mayers, but also many others)&#xA;the theory permitted to compose lunar tables which predicted Moons position for several years in advance with the accuracy to fractions of a minute.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;THIS was the decisive test of the law of gravitation. Another test, some times earlier was the correct prediction of the shape of the Earth, confirmed by precise measurements.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mayers and Euler shared a part of the Longitude Prize given for the solution of the problem of&#xA;longitude by the British Parliament. (The main part of this prize was awarded to Harrison&#xA;who invented chronometer at approximately the same time). Nautical Almanac based on Mayers tables was published.&#xA;It gave predictions for one year.&#xA;I examined the early issues of the Almanac, and can say that the maximal error in Moon's&#xA;position was about 0.2'. But every year it had to be corrected.&#xA;Further mathematical achievements of 19-th century made possible very long term predictions&#xA;(say, 2000 years) with the accuracy within seconds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern formula contains several THOUSAND periodic terms, &quot;inequalities&quot;.&#xA;So we can easily tell that some eclipse happened on such and such date (say 1545 BC)&#xA;at such time, when such and such Babylonian king reigned, and the eclipse was 35% at &#xA;in a given city. This degree of accuracy of &quot;predictions&quot; was only made possible in 20-s century,&#xA;approximately since 1910. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The previous, very simplified discussion concerns only Moon's motion in longitude. The motion in latitude is even more important for the eclipse prediction, and it is described similarly. The key steps are due to Hipparchus (inclination of the orbit), Brahe (first inequality), Kepler (elliptic orbit), Newton (law of gravity), d'Alembert, Clairaut and Euler&#xA;(approximate solution of Newton's equations) and Mayer (making of the tables).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I should also mention that at the time of Hipparchus, there existed a different theory,&#xA;of Babylonian astronomers which had approximately the same accuracy and was based on different mathematics. This theory was practiced as late as 19-th century in India, by Tamil astronomers, and it gave good predictions of Lunar (not Solar!) eclipses.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-07T17:02:43.187" LastActivityDate="2020-05-07T17:02:43.187" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="372" PostTypeId="2" ParentId="348" CreationDate="2014-11-11T13:59:39.520" Score="5" Body="&lt;p&gt;A &lt;a href=&quot;http://en.wikipedia.org/wiki/Transcendental_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;transcendental number&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x*x − 2 = 0.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, with insight, now that we know Cantor's proof of the &lt;em&gt;uncountability&lt;/em&gt; of the real numbers, we can easily conclude that we have a lot more transcendental numbers than non-transcendental ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is the history made from the point of view of modern time; as explained by Alexandre in his answer, at the time of Liouville's proof of the existence of transcendental numbers (1844) there was no clear understanding of the possibility (if any) of comparing the &quot;size&quot; of infinite collections [you can see this &lt;a href=&quot;https://math.stackexchange.com/questions/1012453/was-there-anybody-before-cantor-who-conjectured-existence-of-infinities-of-diffe&quot;&gt;post&lt;/a&gt;].&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2014-11-12T07:34:42.573" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="373" PostTypeId="2" ParentId="357" CreationDate="2014-11-11T14:09:14.923" Score="5" Body="&lt;p&gt;I can partially confirm this by the following evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First. No portrait of Hooke survived. Unlike the portraits of many other members of the Royal Society of that day. (You can check this with Wikipedia).&#xA;Second. This story is told by V. Arnold in his very popular book: Huygens and Barrow, Newton and Hooke.&#xA;I have to say that Arnold is not always a reliable source of historical data, and he does not cite any references, and his other&#xA;statements (mainly about ancient science) were widely criticized. However I have never seen&#xA;this statement criticized. &#xA;Third. It is known that Newton refused to be the President of the Royal Society while Hooke&#xA;was a secretary. He accepted this position after Hooke's death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course, it would be nice to have a direct contemporary account of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another reference:&#xA;&lt;a href=&quot;http://melpor.hubpages.com/hub/The-Rivalry-Between-Isaac-Newton-and-Robert-Hooke&quot; rel=&quot;nofollow&quot;&gt;http://melpor.hubpages.com/hub/The-Rivalry-Between-Isaac-Newton-and-Robert-Hooke&lt;/a&gt; .&#xA;But again, it does not refer to primary sources.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-11T20:36:31.600" LastActivityDate="2014-11-11T20:36:31.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="374" PostTypeId="2" ParentId="357" CreationDate="2014-11-11T16:05:26.090" Score="5" Body="&lt;p&gt;I can only support &quot;indirectly&quot; Alexandre's answer; see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Purrington, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B00408AWGA&quot; rel=&quot;nofollow&quot;&gt;The First Professional Scientist : Robert Hooke and the Royal Society of London&lt;/a&gt; (2009), page 4 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;No portrait is known to exist of Hooke&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and the &lt;em&gt;footnote&lt;/em&gt; n°24 [page 10] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It has been suggested that in view of his unprepossessing appearance, Hooke may have been reluctant to sit for a portrait. He does not seem, however, to have had much vanity, and there is little doubt that the Society did have a portrait. When Zacharias von Uffenbach visited the Society in 1710, he reported seeing portraits of “Boyle and Hoock.” (‘Espinasse, 1956, p. 13; Chapman, 1996). Although Jardine ([Lisa Jardine, &lt;a href=&quot;http://www.amazon.it/The-Curious-Life-Robert-Hooke/dp/0060538988&quot; rel=&quot;nofollow&quot;&gt;The Curious Life of Robert Hooke : The Man Who Measured London&lt;/a&gt;,] 2004) has claimed the discovery of a portrait of Hooke in London’s Natural History Museum, previously thought to be of John Ray, it now seems that the portrait is of Jan Baptist van Helmont.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Comment&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's animosity against Hooke is well known. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, I've difficulty to imagine that we ever know if he lost or stole or burned Hooke's portrait owned by the Royal Society...&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On this issue, it seems to me that Wiki's entry dedicated to &lt;a href=&quot;http://en.wikipedia.org/wiki/Robert_Hooke&quot; rel=&quot;nofollow&quot;&gt;Robert Hooke&lt;/a&gt; is quite reliable (and detailed).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2014-11-12T11:27:39.243" LastActivityDate="2014-11-12T11:27:39.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="375" PostTypeId="1" AcceptedAnswerId="378" CreationDate="2014-11-11T17:19:52.943" Score="26" ViewCount="8829" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Names_of_the_days_of_the_week&quot; rel=&quot;noreferrer&quot;&gt;seven days of the week&lt;/a&gt; seems to be commonly named after celestial bodies. What I find curious is that all seven days apparently share the same names in both West and East.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sunday is, of course, Sun's Day. In East Asia, it is &lt;code&gt;日曜日&lt;/code&gt; - Day of the Sun.&lt;/li&gt;&#xA;&lt;li&gt;Monday comes from &quot;Moon's Day&quot;. In East Asia, it is &lt;code&gt;月曜日&lt;/code&gt; - Day of the Moon.&lt;/li&gt;&#xA;&lt;li&gt;Tuesday is &lt;a href=&quot;http://en.wikipedia.org/wiki/Mars_(mythology)&quot; rel=&quot;noreferrer&quot;&gt;Mars&lt;/a&gt; / &lt;a href=&quot;http://en.wikipedia.org/wiki/T%C3%BDr&quot; rel=&quot;noreferrer&quot;&gt;Týr's&lt;/a&gt; day. In East Asia, it is &lt;code&gt;火曜日&lt;/code&gt; - Day of &lt;a href=&quot;http://en.wikipedia.org/wiki/Mars&quot; rel=&quot;noreferrer&quot;&gt;Mars&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Wednesday is &lt;a href=&quot;http://en.wikipedia.org/wiki/Mercury_(mythology)&quot; rel=&quot;noreferrer&quot;&gt;Mercury&lt;/a&gt; / &lt;a href=&quot;http://en.wikipedia.org/wiki/Odin&quot; rel=&quot;noreferrer&quot;&gt;Odin&lt;/a&gt; / &lt;a href=&quot;http://en.wikipedia.org/wiki/W%C5%8Dden&quot; rel=&quot;noreferrer&quot;&gt;Wodin's&lt;/a&gt; Day. In East Asia, it is &lt;code&gt;水曜日&lt;/code&gt; - Day of &lt;a href=&quot;http://en.wikipedia.org/wiki/Mercury_(planet)&quot; rel=&quot;noreferrer&quot;&gt;Mercury&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Thursday is &lt;a href=&quot;http://en.wikipedia.org/wiki/Jupiter_(mythology)&quot; rel=&quot;noreferrer&quot;&gt;Jupiter&lt;/a&gt; / &lt;a href=&quot;http://en.wikipedia.org/wiki/Thor&quot; rel=&quot;noreferrer&quot;&gt;Thor's&lt;/a&gt; Day. In East Asia, it is &lt;code&gt;木曜日&lt;/code&gt; - Day of &lt;a href=&quot;http://en.wikipedia.org/wiki/Jupiter&quot; rel=&quot;noreferrer&quot;&gt;Jupiter&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Friday is &lt;a href=&quot;http://en.wikipedia.org/wiki/Venus_(mythology)&quot; rel=&quot;noreferrer&quot;&gt;Venus&lt;/a&gt; / &lt;a href=&quot;http://en.wikipedia.org/wiki/Frigg&quot; rel=&quot;noreferrer&quot;&gt;Frigg&lt;/a&gt; / &lt;a href=&quot;http://en.wikipedia.org/wiki/Freyja&quot; rel=&quot;noreferrer&quot;&gt;Freyja's&lt;/a&gt; Day. In East Asia, it is &lt;code&gt;金曜日&lt;/code&gt; - Day of &lt;a href=&quot;http://en.wikipedia.org/wiki/Venus&quot; rel=&quot;noreferrer&quot;&gt;Venus&lt;/a&gt;. &lt;/li&gt;&#xA;&lt;li&gt;Saturday is &lt;a href=&quot;http://en.wikipedia.org/wiki/Saturn_(mythology)&quot; rel=&quot;noreferrer&quot;&gt;Saturn's&lt;/a&gt; Day. In East Asia, it is &lt;code&gt;土曜日&lt;/code&gt; - Day of &lt;a href=&quot;http://en.wikipedia.org/wiki/Saturn&quot; rel=&quot;noreferrer&quot;&gt;Saturn&lt;/a&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The choice of planets probably reflects their higher visibility. But this doesn't explain why they are &lt;strong&gt;ordered&lt;/strong&gt; the aforementioned way. So, &lt;strong&gt;is there an astronomical reason for this specific &lt;em&gt;order&lt;/em&gt;&lt;/strong&gt;? And does it explain the apparent agreement between Europe's and the Sinitic cultural sphere in naming their weeks - or did it come about due to cultural diffusion / pure coincidence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(Note: I used Japanese for the Asian examples, but Korea shares the same system. Traditional Chinese did too, but Monday through Saturday was simplified to &quot;Star Day 1-6&quot; in Modern Chinese while Sunday remained the Sun's.)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;@LieRyan has raised some evidence&lt;sup&gt;&lt;strong&gt;[1]&lt;/strong&gt;&lt;/sup&gt; supporting the idea that the order became globalised due to cultural diffusion. So this just leaves the question of why the order is the way it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt; &lt;strong&gt;[1]:&lt;/strong&gt; Interestingly, one of the linked sources noted that the order of the five planets differed from classical Chinese ordering of elements. While true, it is actually the reverse of the (different, but more common) traditional order.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="211" LastEditorUserId="211" LastEditDate="2014-11-13T05:33:23.363" LastActivityDate="2023-07-28T00:55:52.920" Title="Is there an astronomical reason behind the order for the names of days of the week?" Tags="|astronomy|archaeoastronomy|calendar|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="376" PostTypeId="1" AcceptedAnswerId="389" CreationDate="2014-11-11T18:57:26.813" Score="25" ViewCount="17794" Body="&lt;p&gt;I know that Kronecker claimed it was God's doing, and that even prehistoric humans used some ways of counting. But I am curious where the idea of a sequence of numbers stretching out into infinity appears for the first time explicitly. I suppose another way to phrase it would be to ask who invented infinity. &quot;Ancient cultures had various ideas about the nature of infinity&quot; is what Wikipedia says.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pythagoreans must have understood it already, and Aristotle even discusses the difference between potential and actual infinities, so it had to be before that. Maybe they borrowed it from Egyptians? Babylonians had a positional system which allows recording arbitrarily large numbers in principle. But is that enough? What gives me pause is the &lt;a href=&quot;https://hsm.stackexchange.com/questions/276/when-was-zero-actually-introduced-in-mathematics&quot;&gt;history of zero&lt;/a&gt;. Babylonians and Alexandrian astronomers were using it as a placeholder for centuries before the concept of zero as representing nothing was formed, and not by them. And if it happened to zero it could happen to infinity. By the way, how ironic that infinity was discovered before zero. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2022-09-23T09:27:27.540" Title="Who invented the integers?" Tags="|mathematics|terminology|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="377" PostTypeId="2" ParentId="376" CreationDate="2014-11-11T19:22:45.147" Score="14" Body="&lt;p&gt;Actually when we say Integer today, we mean set of all positive whole numbers, negative whole numbers and zero.&#xA;But this complete set was not discovered/invented in a day. People were working with&#xA;integers from the very beginning. They might be using different names though(like Whole numbers, Natural numbers, ...). According to Wikipedia&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Negative numbers appeared for the first time in history in the &amp;quot;Nine Chapters on the Mathematical Art&amp;quot;, which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220), but may well contain much older material.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In an article i found that the word &amp;quot;integer&amp;quot; was first used of whole numbers in 1571 by Thomas Digges (&lt;a href=&quot;https://web.archive.org/web/20180505061302/http://mathforum.org/library/drmath/view/57212.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;refer this&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The same article further says that,&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The positive and negative numbers did&#xA;not actually become part of a single &amp;quot;number line&amp;quot; (today's &amp;quot;set of&#xA;integers&amp;quot;) until the 1700's or 1800's.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorUserId="2277" LastEditDate="2022-09-19T06:23:42.413" LastActivityDate="2022-09-19T06:23:42.413" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="378" PostTypeId="2" ParentId="375" CreationDate="2014-11-11T20:33:16.837" Score="12" Body="&lt;p&gt;&lt;a href=&quot;http://www.webexhibits.org/calendars/week.html&quot; rel=&quot;noreferrer&quot;&gt;One theory&lt;/a&gt; says, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;`If you order the &quot;planets&quot; according to either their presumed distance from Earth (assuming the Earth to be the center of the universe) or their period of revolution around the Earth, you arrive at this order: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now, assign (in reverse order) these planets to the hours of the day:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1=Saturn, 2=Jupiter, 3=Mars, 4=Sun, 5=Venus, 6=Mercury, 7=Moon, 8=Saturn, 9=Jupiter, etc., 23=Jupiter, 24=Mars&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Then next day will then continue where the old day left off:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1=Sun, 2=Venus, etc., 23=Venus, 24=Mercury&lt;br&gt;&#xA;  And the next day will go&lt;br&gt;&#xA;  1=Moon, 2=Saturn, etc.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If you look at the planet assigned to the first hour of each day, you will note that the planets come in this order:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Saturn, Sun, Moon, Mars, Mercury, Jupiter, Venus&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This is exactly the order of the associated week days. Coincidence? Maybe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.timeanddate.com/calendar/days/&quot; rel=&quot;noreferrer&quot;&gt;Another article&lt;/a&gt; says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ancient Mesopotamian astrologers linked a planet-god to each hour of the day and then arranged them to their correct cosmological order. They used a seven-sided figure to keep track of the proper names of the hours and days in relation to the planet gods where each vertex was marked with a planet’s name in the proper order.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the conclusion is that the order of the week-days is supposed to be governed by the planetary positions of the planets of our solar system.&lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorUserId="-1" LastEditDate="2014-11-12T13:27:39.073" LastActivityDate="2014-11-12T13:27:39.073" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="379" PostTypeId="1" CreationDate="2014-11-11T21:01:39.380" Score="8" ViewCount="52" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Staudinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Staudinger&lt;/a&gt; was awarded the 1953 Nobel Prize in Chemistry for his work demonstrating that &lt;a href=&quot;https://en.wikipedia.org/wiki/Polymer&quot; rel=&quot;nofollow noreferrer&quot;&gt;polymers&lt;/a&gt; are high molecular weight covalently bonded molecules. Prior to his work, the consensus among chemists was that polymers were loosely aggregated of smaller molecules, similar to what today we call a &lt;a href=&quot;https://en.wikipedia.org/wiki/Colloid&quot; rel=&quot;nofollow noreferrer&quot;&gt;colloid&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately for me, Staudinger published in German, and my German reading skills are not quite good enough to understand his papers. What I would like to know is: what evidence did Staudinger use to convince his contemporaries that polymers are high molecular weight molecules?&lt;/p&gt;&#xA;" OwnerUserId="24" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-03-01T13:53:25.030" Title="How did Staudinger demonstrate that polymers are high molecular weight covalently bonded molecules?" Tags="|chemistry|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="380" PostTypeId="2" ParentId="375" CreationDate="2014-11-11T21:38:23.250" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The choice of planets probably reflects their higher visibility.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nice catch, but it's not perfect. I checked the Wikipedia pages for &lt;a href=&quot;https://en.wikipedia.org/wiki/Mercury_(planet)&quot; rel=&quot;nofollow&quot;&gt;Mercury&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Jupiter&quot; rel=&quot;nofollow&quot;&gt;Jupiter&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Saturn&quot; rel=&quot;nofollow&quot;&gt;Saturn&lt;/a&gt;, and found that they're not in that order for apparent magnitude. You've also probably heard (or perhaps even observed for yourself) how bright Venus is - there's no reason for it to be that far down the list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There aren't any other astronomical patterns that I can see.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Size:&lt;/strong&gt; The ancients wouldn't have been able to properly figure out the sizes of the planets, given their limited technology. There's also no pattern in the week.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Distance from Earth:&lt;/strong&gt; Nope. Once again, Venus is far, far away, while the Sun is number 1. Jupiter and Saturn are farther off, which would make sense, but the pattern isn't right.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Importance of their namesakes vs. importance of the bodies:&lt;/strong&gt; This is a weird one, but hey, it's a shot. I thought that perhaps the ancients would have given their most important deities the first spots in the week, but that's clearly not the case. Look how far down Jupiter is. The same goes for Venus. The bodies do seem to be listed in their physical importance - after all, the Sun is a lot more important to us than Jupiter.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-11T21:38:23.250" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="381" PostTypeId="1" AcceptedAnswerId="385" CreationDate="2014-11-11T22:35:29.490" Score="19" ViewCount="3896" Body="&lt;p&gt;There were always some or he other means of measuring(estimating) the time.&#xA;But I always wonder that when and how the present time system (1 Hr.  = 60 Min., &#xA;1 Min. = 60 Sec.) evolved ?&lt;/p&gt;&#xA;" OwnerUserId="188" LastActivityDate="2024-01-20T06:40:08.660" Title="When and on what basis was it decided that an hour have 60 minutes and a minute have 60 seconds?" Tags="|time|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="382" PostTypeId="1" AcceptedAnswerId="388" CreationDate="2014-11-11T22:45:54.357" Score="6" ViewCount="648" Body="&lt;p&gt;&lt;em&gt;Note: this was original an &lt;a href=&quot;https://area51.stackexchange.com/proposals/65204/history-of-science-and-mathematics/65241#65241&quot;&gt;example question&lt;/a&gt;, posted by neizod. As I mentioned in &lt;a href=&quot;https://hsm.meta.stackexchange.com/questions/108/how-can-we-get-the-example-questions-asked-here&quot;&gt;meta&lt;/a&gt;, it would be nice if we could get the example questions asked on the site. neizod did not commit to the site and has not yet joined, so I will place it here. I can make this post community wiki if the community feels this is the right thing to do. I have also rephrased it, as per the suggestion on area51 of @J.W.Perry.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who invented the Haversine formula (or at least the oldest paper in which it was cited)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The relevant &lt;a href=&quot;https://en.wikipedia.org/wiki/Haversine_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; page.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-03-16T16:42:44.343" LastActivityDate="2014-11-12T00:50:11.497" Title="Who invented the Haversine formula (or at least the oldest paper in which it was cited)?" Tags="|mathematics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="383" PostTypeId="1" AcceptedAnswerId="408" CreationDate="2014-11-11T22:50:03.097" Score="8" ViewCount="482" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Somnium_%28novel%29&quot; rel=&quot;nofollow&quot;&gt;Kepler's &lt;em&gt;Somnium&lt;/em&gt; (&quot;The Dream&quot;)&lt;/a&gt; is a work of fiction that is sometimes regarded as the first example of science fiction (e.g. by Carl Sagan). In it, Kepler describes a journey to the Moon and various scientific observations, along with a number of autobiographical notes or similarities. The scientific observations cover things like what the solar system would look like from the Moon and could put it in the category of &quot;hard science fiction&quot;. But it has been suggested that Kepler wrote these ideas in a work of fiction because they might have been considered heretical. Although the Copernicean model would have been less heretical when it was published after his death, more heretical concepts in the book include the possibility of &quot;aliens&quot; on other worlds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the reaction to &lt;em&gt;Somnium&lt;/em&gt; when it was published? Was it considered a curiosity and nothing more - &quot;a badly written piece of fiction&quot; perhaps?  Or was the physics in it taken more seriously by natural philosophers and other learned people?&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="4251" LastEditDate="2021-09-11T11:38:23.193" LastActivityDate="2021-09-11T11:38:23.193" Title="What was the reaction to Kepler's *Somnium* when it was published?" Tags="|physics|biographical-details|17th-century|kepler|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="384" PostTypeId="1" AcceptedAnswerId="471" CreationDate="2014-11-11T22:57:19.287" Score="49" ViewCount="16079" Body="&lt;p&gt;I heard Gauss's primary school teacher gave some busy-work to his class: to add all the numbers between 1 and 100 up. Gauss immediately wrote 5050. His teacher was shocked, so she told him to add up all the numbers to 1000. And just as quickly he wrote 500500.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Gauss derive the $1+2+3+\ldots+(n-2)+(n-1)+n=\frac{n(n+1)}{2}$ formula in elementary school? If so, what biography of Gauss discusses it?&lt;hr&gt;His proof:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1+2+3+\ldots+(n-2)+(n-1)+n$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$+$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$n+(n-1)+(n-2)+\ldots+3+2+1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$=$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\underbrace{(n+1)+(n+1)+(n+1)+\ldots+(n+1)+(n+1)+(n+1)}_{n}=n(n+1)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then he divided by $2$ because of double-counting to get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\frac{n(n+1)}{2}$&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="4251" LastEditDate="2020-05-14T03:16:40.773" LastActivityDate="2020-05-14T03:16:40.773" Title="Did Gauss find the formula for $1+2+3+\ldots+(n-2)+(n-1)+n$ in elementary school?" Tags="|mathematics|mathematicians|gauss|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="385" PostTypeId="2" ParentId="381" CreationDate="2014-11-11T23:03:23.110" Score="21" Body="&lt;p&gt;In case anyone is curious, Wikipedia provided no help. The articles on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hour&quot;&gt;hour&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Minute&quot;&gt;minute&lt;/a&gt; have no in-line citations in their relevant sections, and I don't think they'll lead anywhere interesting. Fortunately (believe it or not), there are &lt;em&gt;other&lt;/em&gt; sources online besides Wikipedia. So I'll go to those.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.ubr.com/clocks/frequently-asked-questions-faq/clocks-and-time-faq-origin-of-hours-and-minutes.aspx&quot;&gt;this&lt;/a&gt; site:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Babylonians (about 300-100 B.C.E.) did their astronomical calculations in the sexagesimal (base-60) system. This was extremely convenient for simplifying division, since 60 is divisible by 2, 3, 4, 5, 6, and 10. The first fractional sexagesimal place we now call a minute, the second place, a second.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the Babylonians become interesting players in this temporal saga. Their decision to use a sexagesimal system was a good one. Nobody likes using fractions like $\frac{24}{5}$, right? Even a base-24 system is convenient.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=191&quot;&gt;This&lt;/a&gt; page had a bit more information (actually, a &lt;em&gt;lot&lt;/em&gt; more), but it wasn't terribly illuminating. It says that nobody really cared about seconds until after the Middle Ages:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Even the first clocks can measure periods less than an hour, but soon striking the quarter-hours seems insufficient. With the arrival of dials for the faces of clocks, in the 14th century, something like a minute is required. The Middle Ages, by a tortuous route from Babylon, inherit a scale of scientific measurement based on 60. In medieval Latin the unit of one sixtieth is pars minuta prima ('first very small part'), and a sixtieth of that is pars minute secunda ('second very small part'). Thus, on a principle 3000 years old, minutes and seconds find their way into time.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(Read more: &lt;a href=&quot;http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=191#ixzz3Int6qdGQ&quot;&gt;http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=191#ixzz3Int6qdGQ&lt;/a&gt;)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Minutes are mentioned from the 14th century, but clocks are not precise enough for anyone to bother about seconds until two centuries later.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(Read more: &lt;a href=&quot;http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=191#ixzz3IntMQEtH&quot;&gt;http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=191#ixzz3IntMQEtH&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Notes: These &quot;Read more&quot; links copied-and-pasted themselves in when I took the other bits of the text. They may prove illuminating.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, from &lt;a href=&quot;http://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/&quot;&gt;Scientific American&lt;/a&gt; comes a much more comprehensive slew of information. It also mentions the Babylonians:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Babylonians made astronomical calculations in the sexagesimal (base 60) system they inherited from the Sumerians, who developed it around 2000 B.C. Although it is unknown why 60 was chosen, it is notably convenient for expressing fractions, since 60 is the smallest number divisible by the first six counting numbers as well as by 10, 12, 15, 20 and 30.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but notes that Hipparchus, Eratosthenes and Ptolemy were also important, from a geometric point of view. Ptolemy used the equivalents of minutes and seconds to divide up a circle, like we do today in certain geometric applications (e.g. astronomy):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In his treatise Almagest (circa A.D. 150), Claudius Ptolemy explained and expanded on Hipparchus' work by subdividing each of the 360 degrees of latitude and longitude into smaller segments. Each degree was divided into 60 parts, each of which was again subdivided into 60 smaller parts. The first division, partes minutae primae, or first minute, became known simply as the &quot;minute.&quot; The second segmentation, partes minutae secundae, or &quot;second minute,&quot; became known as the second. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But we didn't use them to deal with time for a long time:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Minutes and seconds, however, were not used for everyday timekeeping until many centuries after the Almagest. Clock displays divided the hour into halves, thirds, quarters and sometimes even 12 parts, but never by 60. In fact, the hour was not commonly understood to be the duration of 60 minutes. It was not practical for the general public to consider minutes until the first mechanical clocks that displayed minutes appeared near the end of the 16th century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastEditorUserId="137" LastEditDate="2015-11-29T20:00:44.757" LastActivityDate="2015-11-29T20:00:44.757" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="386" PostTypeId="2" ParentId="375" CreationDate="2014-11-11T23:30:17.487" Score="6" Body="&lt;p&gt;As Amit mentions, the naming probably originated from hellenistic astrology in Egypt, wherein each day would be associated with the influence of a particular celestial object. The origins are lost, but the writings of Roman historian Dion Cassius (AD 150-235) have survived; he describes the scheme as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The celestial objects were ordered according to their orbital period:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Saturn 29 years&lt;/li&gt;&#xA;&lt;li&gt;Jupiter 12 years&lt;/li&gt;&#xA;&lt;li&gt;Mars 687 days&lt;/li&gt;&#xA;&lt;li&gt;Sun 365 days&lt;/li&gt;&#xA;&lt;li&gt;Venus 224 days&lt;/li&gt;&#xA;&lt;li&gt;Mercury 88 days&lt;/li&gt;&#xA;&lt;li&gt;Moon 29 days&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now assign the first hour of the first day to Saturn, the second to Jupiter, etc, and repeat the cycle. The 24th hour is then assigned to Mars, the 1st hour of the second day to the Sun, and so forth. The first hour of the 8th day will correspond with Saturn again, and the whole cycle is complete. Each day would then be named after the celestial object assigned to the first hour (which corresponds with 6 a.m.). In this way, we get the correct order&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Saturn, Sun, Moon, Mars, Mercury, Jupiter, Venus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The association between Saturday and the Sabbath may come from the Book of Genesis: the planets were created on the 4th day, which according to Jewish tradition starts on the first Tuesday, at 6 p.m. (13th hour). If we start our naming scheme at that point, with Saturn first, we indeed arrive at Mercury on Wednesday 6 a.m., Jupiter on Thursday 6 a.m., etc. Another theory states that Saturn is associated with bad luck, so one should refrain from working on this day. The Christian adoption of Sunday as the day of prayer was probably meant to stomp out competing religions such as the sun-worshiping Mithraism; also, Sunday was believed to be the day of Jesus' resurrection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the first century AD the astrological week had reached India, where the names of the seven days in Sanskrit follow the names of the same seven planets. From India, the weekdays spread to Tibet, Burma, Nepal, Thailand, and Ceylon; by the Song dynasty, at the end of the first millennium, it had reached China.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;source: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0192862057&quot; rel=&quot;noreferrer&quot;&gt;Mapping Time: The Calendar and Its History&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="220" LastActivityDate="2014-11-11T23:30:17.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="388" PostTypeId="2" ParentId="382" CreationDate="2014-11-12T00:50:11.497" Score="6" Body="&lt;p&gt;Don Josef de Mendoza y Rios, F.R.S. Recherches sur les principaux Problemes de l'Astronomie Nautique, Proc. Royal Soc., Dec 22, 1796.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He himself calls this sinus-verse and cosinus-verse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia article which you cite in your question contains a reference on Mendoza y Rios,&#xA;but on a slightly later paper of 1801.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-12T00:50:11.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="389" PostTypeId="2" ParentId="376" CreationDate="2014-11-12T01:22:30.270" Score="19" Body="&lt;p&gt;Very few (if any) mathematicians before Cantor thought of the SET of integers.&#xA;Certainly for Euclid it was completely evident that the &#xA;sequence of integers extends without limit.&#xA;(He actually has a famous theorem that the sequence of PRIMES extends without limit).&#xA;Who discovered this we will never know because very few mathematical sources before Euclid survived. Perhaps Pythagoreans, but maybe earlier. (What we know about Pythagoreans comes from much later secondary sources, Pythagoreans themselves were a secret society and did not publish their discoveries).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I said, most mathematicians imagined infinite sets as POTENTIALLY infinite, that is without limit. To every integer you can add 1 and obtain a larger integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in theological literature, beginning from early Middle Age, we encounter ACTUAL infinity (infinite things, infinite sets). I think this begins in the Neoplatonic school,&#xA;but Augustine (of Hyppo) certainly discusses (in V century) that in the City of God, and Cantor mentions him. These discussion continue in the medieval scholastic literature, but there is little mathematics or science in them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notion of actual infinity was revived by Cantor in his set theory, and nowadays this&#xA;is a common language of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Since the Hellenistic times, it is a popular opinion that the first mathematicians&#xA;(Pythagoras, Thales) &quot;learned something from Egyptians&quot;. Some modern authors tend to say that they learned everything from Egyptians. Serious research on the history of mathematics and astronomy does not confirm that. With enormous number of surviving texts, we know pretty much about ancient Egypt. There was NOTHING for people like Thales to learn there. Egyptian astronomy and mathematics was in very primitive state in comparison with &#xA;contemporary Babylonian and Greek sciences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best source: O. Neugebauer. Exact sciences in antiquity.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-12T01:22:30.270" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="390" PostTypeId="1" CreationDate="2014-11-12T01:28:28.083" Score="8" ViewCount="79" Body="&lt;p&gt;Nowadays,  &lt;a href=&quot;https://academia.stackexchange.com/q/29923/452&quot;&gt;many academic journals have some impressively high subscription costs&lt;/a&gt;. E.g. Harvard's expenditures for library resources in 2012 was $16,391,638.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did journal subscriptions evolve over time, and what factors accounted for it?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.757" LastActivityDate="2014-11-12T03:50:40.583" Title="How did journal subscriptions evolve over time, and what factors account for it?" Tags="|academia|publishing|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="391" PostTypeId="1" AcceptedAnswerId="399" CreationDate="2014-11-12T02:03:49.787" Score="19" ViewCount="1354" Body="&lt;p&gt;Maybe we need some replies on current scholarly thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Judging from some replies here, many of us are still using the myths current 100 years ago.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it true (as I have heard) that most, if not all, of the things attributed to Pythagoras and the Pythagoreans were, in fact, due instead to someone in Plato's school?  Someone who promulgated these ideas many years later, adding the attribution of &quot;Pythagoras&quot; (a famous personage of the remote past) in order to lend credence to them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, if so, what are some good things to read about this?&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="26" LastEditDate="2014-11-12T13:26:05.137" LastActivityDate="2020-04-14T09:19:51.530" Title="Pythagoras vs. the idea of Pythagoras" Tags="|mathematics|ancient-greece|debunking|depiction-of-scientists|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="392" PostTypeId="2" ParentId="357" CreationDate="2014-11-12T02:18:52.820" Score="12" Body="&lt;p&gt;The question is whether there ever existed a portrait of Hooke. There is very little evidence to support this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Hooke mentions a certain 'Mr Bonust' in his diary. In an entry for 16 October 1674 he wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;'At Garaways. Left off taking tobacco — Mr Bonust drew picture.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The name might refer to a certain Mr Bownest. However, if he did draw Hooke’s picture in 1674, this is the only reference Hooke made to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) The only person who ever mentioned a portrait was Zacharias von Uffenbach, who visited the Royal Society in 1710. He was probably mistaken though, as no-one else made any reference to a portrait. Perhaps von Uffenbach had misheard the name 'Haak':&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;James Yonge, who visited the Society in November 1702 and was elected FRS at the time, recorded seeing ‘divers original pictures’ in the Council Room. He listed eleven portraits, including those of Robert Boyle and Theodore Haak, but Hooke’s was not among them. It seems unlikely that Yonge would have overlooked Hooke’s picture if it was there. Hooke was Yonge’s first contact at the Society: the two men had corresponded for many years and Yonge referred to Hooke in his journal as ‘my old friend’.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Neither Hooke's first biographer, Richard Waller, nor his posthumous editor, William Derham, mentioned the existence of a portrait either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;source: &lt;a href=&quot;http://blogs.royalsociety.org/history-of-science/2010/12/03/hooke-newton-missing-portrait/&quot;&gt;Hooke, Newton, and the ‘missing’ portrait&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="220" LastActivityDate="2014-11-12T02:18:52.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="393" PostTypeId="5" CreationDate="2014-11-12T02:25:26.667" Score="0" Body="&lt;p&gt;This tag is for questions about the history of the science of healing. This includes diagnosis, treatment, prevention of disease, and other health care, as applied to humans.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="62" LastEditDate="2015-04-12T16:39:19.963" LastActivityDate="2015-04-12T16:39:19.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="394" PostTypeId="4" CreationDate="2014-11-12T02:25:26.667" Score="0" Body="For questions about all areas of health care for humans, including diagnosis, treatment, and disease prevention." OwnerUserId="25" LastEditorUserId="62" LastEditDate="2015-04-12T16:39:19.963" LastActivityDate="2015-04-12T16:39:19.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="395" PostTypeId="5" CreationDate="2014-11-12T02:29:25.580" Score="0" Body="&lt;p&gt;Astronomy is the study of celestial objects (stars, planets, moons, etc) and phenomena (eg. supernovae explosions) outside of the Earth's atmosphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many ancient civilizations (eg. Babylonians, Chinese, and Maya) observed the night sky and as such astronomy is claimed to be one of the oldest sciences. Ancient civilizations often used astronomical observations to define their calendars (eg. Ancient Egypt used Sirius to predict the seasonal Nile floods).&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-11-12T20:19:15.920" LastActivityDate="2014-11-12T20:19:15.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="396" PostTypeId="4" CreationDate="2014-11-12T02:29:25.580" Score="0" Body="The study of celestial objects and phenomena outside of the Earth's atmosphere." OwnerUserId="25" LastEditorUserId="25" LastEditDate="2014-11-12T20:18:46.090" LastActivityDate="2014-11-12T20:18:46.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="397" PostTypeId="5" CreationDate="2014-11-12T02:32:41.820" Score="0" Body="&lt;p&gt;Thermodynamics covers the subset of physics that covers heat and temperature, and their relationship to energy and work. Thermodynamics uses statistics to define macroscopic variables such as entropy and pressure.&lt;/p&gt;&#xA;" OwnerUserId="25" LastEditorUserId="62" LastEditDate="2015-04-12T16:38:03.787" LastActivityDate="2015-04-12T16:38:03.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="398" PostTypeId="4" CreationDate="2014-11-12T02:32:41.820" Score="0" Body="For questions about the branch of physics concerning heat and temperature and their macroscopic relationship to energy and work." OwnerUserId="25" LastEditorUserId="62" LastEditDate="2015-04-12T16:38:03.787" LastActivityDate="2015-04-12T16:38:03.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="399" PostTypeId="2" ParentId="391" CreationDate="2014-11-12T03:32:11.513" Score="19" Body="&lt;p&gt;Yes the stories of Pythagoras that were common a few decades ago have all been been disproved, largely by Walter Burkert in &lt;a href=&quot;http://books.google.it/books/about/Lore_and_Science_in_Ancient_Pythagoreani.html?id=0qqp4Vk1zG0C&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lore and Science in Ancient Pythagoreanism&lt;/a&gt; (1972).  In short, Pythagoras never thought about any of the mathematics attributed to him.  Consequently he gave no mathematical theory of music, never said all is number, and never contemplated a harmony of the planets.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The place to start reading about this is Burnyeat's concise review of later lighter books, including &lt;em&gt;Pythagoras and the Pythagoreans: A Brief History&lt;/em&gt; by Charles Kahn.  The review is linked by Gerald Edgar,  &lt;a href=&quot;http://www.lrb.co.uk/v29/n04/mf-burnyeat/other-lives&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.lrb.co.uk/v29/n04/mf-burnyeat/other-lives&lt;/a&gt;. Burnyeat bases his review on Burkert's book. Since then Leonid Zhmud has urged the opposite, saying Pythagoras did pursue mathematics.  But the review at &lt;a href=&quot;https://web.archive.org/web/20141018100000/http://bmcr.brynmawr.edu/2014/2014-08-30.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.archive.org/web/20141018100000/http://bmcr.brynmawr.edu/2014/2014-08-30.html&lt;/a&gt; shows Zhmud also attributes essentially all  of the specific traditionally Pythagorean mathematics to later Pythagoreans, not to Pythagoras himself, and the review shows problems within Zhmud's argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not entirely trust any current attempt to find how the mathematics was produced, but the other thing to read is David Fowler's book &lt;a href=&quot;http://www.amazon.it/The-Mathematics-Platos-Academy-Reconstruction/dp/0198502583&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mathematics of Plato's Academy&lt;/a&gt; (1987).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The books by Burkert and Fowler are no light reading.  &lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="55" LastEditDate="2020-04-14T09:19:51.530" LastActivityDate="2020-04-14T09:19:51.530" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="400" PostTypeId="2" ParentId="390" CreationDate="2014-11-12T03:50:40.583" Score="3" Body="&lt;p&gt;To my understanding, the main factor is that in 1970-1980-th the Universities, learned societies, and other non-profit organizations gave up on the journal business. Most existing journals were taken over by big businesses, and competition from non-profit organizations vanished. This permitted the publishers to raise prices. The business became profitable (it never was very profitable&#xA;while the journals were owned by universities and other non-profit organizations), and this&#xA;triggered fight between the publishing corporations: they started buying each other. Now a lion share of the market is owned&#xA;by two giants, Springer Business Media and Elsevier, which bought many smaller publishers, and they do with prices whatever they want.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scientific community tries to resist (see for example &quot;The Cost of Knowledge&quot; on Wikipedia, which describes a partially successful boycott of Elsevier), but resistance is too week and not well organized. Meanwhile the university libraries are forced to cancel more and more subscriptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was speaking on mathematics and physics above. In biology the situation is somewhat different, because the US government takes some protective measures. On the argument that&#xA;&quot;taxpayers pay for science and have right to have access to the results&quot;. So some government agencies like the NIH require that scientific results obtained with their support be freely available on Internet archives. This policies and their extension to&#xA;other sciences are hotly debated in the Congress. Publishers lobby resists.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-12T03:50:40.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="401" PostTypeId="1" AcceptedAnswerId="406" CreationDate="2014-11-12T04:44:38.710" Score="16" ViewCount="8954" Body="&lt;p&gt;To clear any misunderstandings, I am interested not in the gossipy aspect of the matter (i.e. conflicts that had mainly to do with character-clashes and idiosyncratic psychs) but rather in conflicts that &lt;em&gt;today&lt;/em&gt; are characterized by the following:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Both scientists (or groups of scientists) that were involved are widely recognized as important contributors to their science.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) The conflict lasted, and represented a deeper methodological or philosophical distance.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the &quot;final result&quot;: Either some sort of &lt;em&gt;synthesis&lt;/em&gt; has emerged, during their life-time or later, or, if not synthesized, their respective and apparently conflicting positions/approaches have come to be understood as &quot;complementary&quot;, in some sense. But it may also be the case that the one side eventually &quot;lost definitely the argument&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sensibly, I am looking to cases considered &quot;settled&quot; by the contemporary scientific community at large, and not for current scientific debates/controversies.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example I have to offer is from &lt;strong&gt;Statistics&lt;/strong&gt;: The &lt;strong&gt;[Fisher] / [Neyman-Pearson] conflict regarding Hypothesis Testing&lt;/strong&gt; (one can check &lt;a href=&quot;https://stats.stackexchange.com/questions/112769/is-the-hybrid-between-fisher-and-neyman-pearson-approaches-to-statistical-test&quot;&gt;this SE post on Cross Validated&lt;/a&gt; for stimulus). This controversy satisfies the two criteria above. The &quot;synthesis&quot; that has emerged is criticized by many, while there is another view that considers the two approaches as complementary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: I am not sure what tags would be appropriate for this question.&lt;/p&gt;&#xA;" OwnerUserId="244" LastEditorUserId="-1" LastEditDate="2017-04-13T12:45:01.060" LastActivityDate="2016-10-27T17:35:17.973" Title="Famous Scientific Conflicts and Controversies" Tags="|examples|scientific-method|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="402" PostTypeId="1" AcceptedAnswerId="411" CreationDate="2014-11-12T06:34:34.503" Score="26" ViewCount="89071" Body="&lt;p&gt;I have read at many places that zero was discovered by &lt;a href=&quot;http://en.wikipedia.org/wiki/Aryabhata&quot; rel=&quot;noreferrer&quot;&gt;Aryabhata&lt;/a&gt; but when i was discussing this with my mathematics teacher he told me that zero had already been discovered before Aryabhata and it was also used by &lt;a href=&quot;http://en.wikipedia.org/wiki/Euclid&quot; rel=&quot;noreferrer&quot;&gt;Euclid&lt;/a&gt;. Aryabhata just made Asians aware of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Did Aryabhata discover zero?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If no then who discovered Zero?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I have read this &lt;a href=&quot;https://hsm.stackexchange.com/questions/276/when-was-zero-actually-introduced-in-mathematics&quot;&gt;post&lt;/a&gt; but it doesn't tell which mathematician first discovered  that there is 0 in number system. &lt;/p&gt;&#xA;" OwnerUserId="247" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2019-05-06T08:18:47.820" Title="Discovery of zero" Tags="|mathematicians|zero|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="403" PostTypeId="2" ParentId="402" CreationDate="2014-11-12T12:47:19.897" Score="7" Body="&lt;p&gt;Like with many other inventions, this did not happen once, suddenly at a specific time.&#xA;The idea gradually evolved. I am not sure about Euclid but a &quot;zero&quot; (place holder) is occasionally found in Ptolemy. However the use is not systematic and it does not have the modern shape. Other people used space for the same purpose.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-12T12:47:19.897" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="404" PostTypeId="1" AcceptedAnswerId="405" CreationDate="2014-11-12T12:47:23.303" Score="17" ViewCount="3265" Body="&lt;p&gt;At the start of my chemistry course, my professor told a story:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He said that Alfred Nobel invented dynamite (which is true) and he accumulated his wealth by selling it to both parties A and B (&lt;em&gt;where A and B are countries that were at War&lt;/em&gt;) who were fighting each other. And he said that at the end he was too rich that he created the &quot;Noble Prize&quot; using this money he collected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To what extent is this story correct? Did Alfred Nobel actually help A and B kill each other? &lt;/p&gt;&#xA;" OwnerUserId="257" LastEditorUserId="4251" LastEditDate="2021-09-25T17:38:15.397" LastActivityDate="2021-09-25T17:38:15.397" Title="Did Alfred Nobel collect his wealth by selling dynamite?" Tags="|biographical-details|nobel-prize|chemists|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="405" PostTypeId="2" ParentId="404" CreationDate="2014-11-12T13:00:14.907" Score="20" Body="&lt;p&gt;I too was told this story, by my father as we drove through &lt;a href=&quot;http://en.wikipedia.org/wiki/Nobel,_Ontario&quot; rel=&quot;noreferrer&quot;&gt;Nobel, Ontario&lt;/a&gt;. While the main purpose of dynamite may never have been warfare, it most certainly was used for that purpose during his lifetime, and he didn't expect or like that very much. As well, he invented a number of other chemicals which were explicitly for use in war, but he didn't feel they defined him - he invented a lot of other stuff too. Some quotes from &lt;a href=&quot;http://en.wikipedia.org/wiki/Nobel_Prize&quot; rel=&quot;noreferrer&quot;&gt;the Wikipedia article on the Nobel Prize&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nobel amassed a fortune during his lifetime, with most of his wealth from his 355 inventions, of which dynamite is the most famous.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That answers your title question. Also,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1888, Nobel was astonished to read his own obituary, titled &lt;em&gt;The merchant of death is dead&lt;/em&gt;, in a French newspaper. As it was Alfred's brother Ludvig who had died, the obituary was eight years premature. The article disconcerted Nobel and made him apprehensive about how he would be remembered. This inspired him to change his will.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.brainpickings.org/2013/09/17/molly-oldfield-secret-museum-alfred-nobel-will/?utm_content=buffera3f2b&amp;amp;utm_medium=social&amp;amp;utm_source=twitter.com&amp;amp;utm_campaign=buffer&quot; rel=&quot;noreferrer&quot;&gt;A recent article&lt;/a&gt; includes pictures of the hand written will and more details on both how and when he wrote it, and how the mechanisms of the prize were put into place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At around the same time, &lt;a href=&quot;http://www.infobarrel.com/Alfred_Nobel&quot; rel=&quot;noreferrer&quot;&gt;he met a pacifist&lt;/a&gt; who inspired him to change his ways from &quot;living the high life in Paris&quot; to something more, well, noble. In a letter to her he &lt;a href=&quot;http://www.irwinabrams.com/articles/oddcouple.html&quot; rel=&quot;noreferrer&quot;&gt;wrote&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Perhaps my factories will put an end to war even sooner than your Congresses; on the day when two army corps will be able to annihilate each other in a second, all civilised nations will recoil with horror and disband their troops.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Yes, mutually assured destruction as a peace strategy, but dynamite and other explosives rather than nuclear bombs.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems your chemistry professor was close enough to the spirit of what happened and why the prize was created. Nobel was embarrassed by the way he'd been using his money, mostly as a result of his friendship with Bertha, and didn't like the idea of being a &quot;merchant of death&quot; because armies used dynamite (or other explosives he invented and sold, such as cordite) to kill people. Even though he thought that in theory this could reduce war, by the time he died he had set up the prizes in his will to help advance peace, science, literature and so on and thus leave the world a better place. The precise details about supplying dynamite to two specific countries in one specific war don't seem right, but the overall spirit of the story does.&lt;/p&gt;&#xA;" OwnerUserId="259" LastEditorUserId="259" LastEditDate="2015-06-09T19:51:24.033" LastActivityDate="2015-06-09T19:51:24.033" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="406" PostTypeId="2" ParentId="401" CreationDate="2014-11-12T13:09:50.713" Score="11" Body="&lt;p&gt;Probably the most famous conflict that satisfies your requirements is the conflict between&#xA;&quot;Ptolemy system&quot; and &quot;Copernic system&quot;, though few people understand this even today:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This conflict became so famous because of the intrusion of a non-scientific authority, that is catholic church. From the point of view of astronomy, there are two aspects:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) the choice of coordinate system. We are taught in elementary physics courses that this is indeed a matter of choice, of convenience. Some coordinate systems are good for one purpose, others for another purpose. Heliocentric system is by far more convenient when you want to write equations of motion. But geocentric system is more convenient in navigation, where they do not care about &quot;how planets really move&quot; but in &quot;how this looks from the Earth&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) the more important question whether the orbits are combinations of circles with uniform motion, or Kepler ellipses and the motion obeys Kepler laws. Of course, the discovery of Kepler's laws was a really great discovery (probably more important than the Heliocentric system), because they led to the discovery of the law of gravity, etc., and because they have much simpler mathematical formulation. However when people try to solve the equations of motion for more than 2 bodies, they obtain a solution in a form of&#xA;a trigonometric series of functions depending on uniformly changing angles, which is&#xA;essentially the same as Ptolemy did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you can think of the final result as a representation of the motion as a combination&#xA;of epicycles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in this case we really have a kind of synthesis after a long controversy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this conflict had also other significance beyond astronomy. &#xA;I don't want to discuss philosophical consequences.&#xA;The interference of the church into purely scientific matters undermined the church authority, so I would say it lost the cause in the long run. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Galileo is also not so innocent in triggering this conflict. (Copernicus and Kepler did not have any conflict with the church). And the main &quot;proof&quot; that Galileo presented for &quot;moving Earth&quot; was wrong. So it would be much more reasonable on the part of Galileo to hold Copernicus view that the difference between the two systems is mathematical, the choice of the coordinate system, which after all is closer to the truth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo's main &quot;proof&quot; was based on his totally incorrect theory of the tides which contradicted principles of mechanics he discovered himself in the young age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Some references. They are really abundant (any comprehensive history of astronomy). Some of my favorite in Latin aplphabet:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Simon Gindikin, Tales of mathematicians and physicists. Springer, New York, 2007.&#xA;(This is a collection of biographies written by professional mathematician for general public).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Stillman Drake, Galileo at work. A very comprehensive scholarly analysis of Galileo's work and scientific biography, including all controversies. Based on his work and correspondence.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jean Meeus, Astronomical algorithms, Willmann-Bell, Richmond VA, 1998. How ephemerides&#xA;are computed today, with complete formulas for the motion of Sun, Moon and planets, and other related calculations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John Louis Emil Dreyer, History of planetary systems from Thales to Kepler. Very good book for a beginner.  Newer editions are called &quot;History of astronomy from Thales to Kepler&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;5.Jean Baptiste Joseph Delambre, Histoire de l'astronomie ancienne, Histoire de l'astronomie du Moyen Age,&#xA;Histoire de l'astronomie moderne. These three books are for a serious student of history of astronomy.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-25T01:52:55.733" LastActivityDate="2014-11-25T01:52:55.733" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="407" PostTypeId="2" ParentId="232" CreationDate="2014-11-12T13:33:21.523" Score="8" Body="&lt;p&gt;I think the easier answer is that asthma was not really universally seen as a psychosomatic illness. To understand why, one must problematise the definition of &lt;i&gt;disease&lt;/i&gt; and put dominant versions of what counts as disease in context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side note, this is hardly exclusive to asthma - the debate over the definition chronic fatigue syndrome / myalgic encephalopathy, a debate over psychological vs physiological establishment of the chronic disease is ongoing &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2539637/&quot;&gt;[1]&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going back the case of asthma, why recruit psychosomatic definitions, then? That is most likely to be connected to the psychosomatic movement of the early ~1930s. The first half of the 20th century was a &lt;i&gt;golden age&lt;/i&gt; for psychoanalysis. Around 1900, Freud's proposed theories had gained popularity and a psychoanalysis society was formed (which further institutionalised the approach). Some of the prominent psychoanalysts lectured in universities around the world. At this time, psychoanalytic explanations had gravitas in a clinical context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much like in the other diseases in the holy 7, the diagnosis for asthma stumped contemporary physicians. So, when the biomedical explanation fell short, others were recruited to cover the gap. That is not to say, however, that the biomedical definition of the disease was thrown out of the window - again, there were competing definitions for what actually constituted asthma. By the turn of the century, William Osler offered a very biomedical definition for what constituted asthma &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2892047/&quot;&gt;[2]&lt;/a&gt;. In addition, the biomedical definition was very present in therapeutics with, for example, beta 2-agonists and theophylline having used in asthma therapy since the 1920s &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0954611101911485&quot;&gt;[3]&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also like to point out that the psychosomatic explanations were not an artifact or a bump on the road to the &lt;i&gt;right&lt;/i&gt; explanation. Psychosomatic dimenstions continue to be an area of interest in defining asthma &lt;a href=&quot;http://www.karger.com/Article/Abstract/287312&quot;&gt;[4]&lt;/a&gt;. Some of the ones identified in the early 20th century have been downplayed. Others, like stress, have been incorporated as triggers / exacerbating factors of the disease &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0889159107000761&quot;&gt;[5]&lt;/a&gt;. In fact, even within the biomedical model, a lot has changed as to what is perceived to constitute the disease since the 1950s. It went from being a smooth muscle disease, to an inflammatory disease, to an airway remodelling disease, to a combination of all of the above [2]. It is even now moving towards a molecular definition &lt;a href=&quot;http://www.nature.com/nm/journal/v18/n5/abs/nm.2678.html&quot;&gt;[6]&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="256" LastEditorUserId="26" LastEditDate="2014-11-12T14:26:44.290" LastActivityDate="2014-11-12T14:26:44.290" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="408" PostTypeId="2" ParentId="383" CreationDate="2014-11-12T13:39:10.453" Score="6" Body="&lt;p&gt;I think that the authoritative &lt;em&gt;Introduction&lt;/em&gt; by Edward Rosen to his translation with commentary of :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Johannes Kepler, &lt;a href=&quot;http://www.amazon.it/Keplers-Somnium-Dream-Posthumous-Astronomy/dp/0486432823/ref=sr_1_fkmr0_3?ie=UTF8&amp;amp;qid=1415799634&amp;amp;sr=8-3-fkmr0&amp;amp;keywords=Johannes+Kepler%2C+Kepler%27s+Somnium+The+Dream%2C+or+Posthumous+Work+on+Lunar+Astronomy+%281967+-+Dover+reprint%29+can+give+us+the+ne&quot; rel=&quot;nofollow&quot;&gt;Kepler's Somnium The Dream, or Posthumous Work on Lunar Astronomy&lt;/a&gt; (1967 - Dover reprint)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;can give us the necessary informations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems quite clear that the booklet was not intended to be a &lt;em&gt;sci-fi&lt;/em&gt; novel at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The origin is in the draft of Kepler's dissertation of 1593 on the motions of the moon [page xvii]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler refers to it as his &lt;em&gt;Lunar Astronomy&lt;/em&gt; [page xviii] and around 1609 he &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;constructed the framework of the &lt;em&gt;Dream&lt;/em&gt; [...] in order to introduce a supernatural agency for the purpose of transporting a professional astronomical observer to the moon [page xix].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1629 Kepler mentioned his &quot;Astronomy of the Moon or of the Celestial Phenomena as seen from the Moon&quot; as a possible choice for a &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;mathematical textbook suitable for use in the classroom [page xx-xxi].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The posthumous book was published in 1631 by Kepler's son [page xxi].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to reference into Rosen's introduction, the first reactions to the book judged it bizarre or satyrical [page xxiii].&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Browsing Kepler's notes, we can see [page 68] clear discussion of celestial phenomena, with reference to Aristotle as well as Kepler's own &lt;em&gt;Epitome of Copernican Astronomy&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler published during his life several books explicitly copernicans; thus, I do not think that the &quot;narrative form&quot; was due to the necessity of concealing his ideas [see &lt;em&gt;Addendum&lt;/em&gt;].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the idea of extra-terrestrial life does not seem to be a key fature of the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;narrative form&quot; of the book must be understood in the Renaissance and Humanists context, with reference to Erasmus, Thomas More and Campanella and their &lt;em&gt;Utopia&lt;/em&gt;'s as well as &lt;a href=&quot;http://plato.stanford.edu/entries/plutarch/&quot; rel=&quot;nofollow&quot;&gt;Plutarch&lt;/a&gt;'s treatise &quot;On the Face Which Appears in the Orb of the Moon&quot; (Περὶ τοῦ ἐμφαινομένου προσώπου τῷ κύκλῳ τῆς σελήνης - &lt;em&gt;De facie in orbe lunae&lt;/em&gt;), included into the collection of 78 essays and transcribed speeches named &lt;em&gt;Moralia&lt;/em&gt; (Ancient Greek : Ethika). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler's notes refer also to &lt;a href=&quot;http://en.wikipedia.org/wiki/True_History&quot; rel=&quot;nofollow&quot;&gt;True History&lt;/a&gt; (Ancient Greek: Ἀληθῆ διηγήματα) the a parody of travel tales by the Greek-speaking Syrian author Lucian of Samosata and to &lt;a href=&quot;http://en.wikipedia.org/wiki/Ignatius_His_Conclave&quot; rel=&quot;nofollow&quot;&gt;Ignatius His Conclave&lt;/a&gt; (Latin: &lt;em&gt;Conclave ignati&lt;/em&gt;) published anonymously in 1611 (and thus, presumibly, Kepler was unaware of the name of the author) by the metaphysical poet John Donne. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We must be aware that this &quot;strange&quot; book was drafted in a time when there was not yet a clearcut division between science, philosophy, humanities ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt; : for a book-lenght analysis of :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Johannes Kepler’s &lt;em&gt;Somnium&lt;/em&gt;, [as] a work heavily indebted to the Neoplatonist cosmological allegory as a literary form&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Dean Swinford, &lt;a href=&quot;https://books.google.it/books?id=E8lfpqfgYI4C&amp;amp;printsec=frontcover&amp;amp;dq=Dean+Swinford+Through+The+Daemon%27s+Gate&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ei=-RGIVLvwOcXCOeaegOgL&amp;amp;ved=0CCUQ6AEwAA#v=onepage&amp;amp;q=Dean%20Swinford%20Through%20The%20Daemon&amp;#39;s%20Gate&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Through The Daemon's Gate : Kepler's Somnium Medieval Dream Narratives and the Polysemy of Allegorical Motifs&lt;/a&gt; (2006).&#xA;&lt;hr&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Addendum&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some chronology can help :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Copernicus, &lt;em&gt;De revolutionisbis&lt;/em&gt; : published 1543; in March 1616, the Roman Catholic Church's Congregation of the Index issued a decree suspending &lt;em&gt;De revolutionibus&lt;/em&gt; until it could be &quot;corrected&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Kepler, &lt;em&gt;Epitome astronomiae Copernicanae&lt;/em&gt; : published in three parts from 1618–1621&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Kelper died : 1630&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Descartes, &lt;em&gt;Le Monde&lt;/em&gt; : written between 1629 and 1633&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Galileo, &lt;em&gt;Dialogo&lt;/em&gt; : published 1632&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Galileo found &quot;vehemently suspect of heresy&quot; and condemned : 1633&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Descartes' decision to not to release &lt;em&gt;Le Monde&lt;/em&gt; upon news of the conviction of Galileo  : 1633&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Kepler's &lt;em&gt;Somnium&lt;/em&gt; : published 1634&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My conclusion is that the chronology does not support the hypotheses that Kepler's decision not to publish the &lt;em&gt;Somnium&lt;/em&gt; was due to the condemnation of Copericanism.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2014-12-10T09:28:28.190" LastActivityDate="2014-12-10T09:28:28.190" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="409" PostTypeId="2" ParentId="401" CreationDate="2014-11-12T13:41:33.330" Score="10" Body="&lt;p&gt;Quite a famous example from modern physics deals with the inherent randomness of quantum theory. In the early years of quantum mechanics when the formulation in terms of wave functions was developed, the question of interpretation arose. &lt;a href=&quot;http://en.wikipedia.org/wiki/Copenhagen_interpretation&quot;&gt;Copenhagen interpretation&lt;/a&gt;, developed mainly by Niels Bohr and Werner Heisenberg became the most accepted of all the possibilities. It states that the wave function gives a full description of a system and an outcome of a measurement is random with probability distribution which can be determined by the wave function and form of measurement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Albert Einstein was one of the scientists that did not like the idea of non-deterministic universe and argued that this randomness comes from some unknown parameters that we are not aware of. This &lt;a href=&quot;http://en.wikipedia.org/wiki/Local_hidden_variable_theory&quot;&gt;local hidden variable theory&lt;/a&gt; would then be completely deterministic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was not until a few decades later, in 1964, when John Bell formulated &lt;a href=&quot;http://en.wikipedia.org/wiki/Bell%27s_theorem&quot;&gt;inequalities&lt;/a&gt; that have to be obeyed by every deterministic theory but can be violated by quantum theory in an inherently random world. Since then, the violation of these inequalities has been observed many times showing that quantum theory indeed is fundamental and no deterministic theory can explain the effects that we observe.&lt;/p&gt;&#xA;" OwnerUserId="90" LastActivityDate="2014-11-12T13:41:33.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="410" PostTypeId="2" ParentId="2" CreationDate="2014-11-12T14:05:52.193" Score="12" Body="&lt;p&gt;These legends do exist, and have for along time.  But few if any specialist historians of the subject believe Pythagoreans discovered irrationality of  $\sqrt{2}$.  See: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/391/pythagoras-vs-the-idea-of-pythagoras/399#399&quot;&gt;Pythagoras vs. the idea of Pythagoras&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is very hard to judge of Greek mathematics before Euclid, let alone before Plato, as there is so little evidence.  The most widely read single study of it today is probably D. H. Fowler's &lt;em&gt;Mathematics of Plato's Academy&lt;/em&gt;, and for what it is worth I think he may well be right.  In short he argues that incommensurability was a well known topic easily handled by Greek mathematicians as far back as our evidence can take us.  &lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-12-03T14:17:35.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="411" PostTypeId="2" ParentId="402" CreationDate="2014-11-12T15:23:55.233" Score="16" Body="&lt;p&gt;The first evidence of zero is from the Sumerian culture in Mesopotamia, some 5000 years ago (source: Robert Kaplan, &lt;em&gt;The Nothing that is: a natural History of Zero&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting to note that different cultures discovered the concept of &quot;zero&quot; independently. The Babylonians, the Mayans, the Chinese and the Hindus all introduced symbols for zero. They are e.g. shown in this &lt;a href=&quot;http://www.scientificamerican.com/article/what-is-the-origin-of-zer/&quot;&gt;article&lt;/a&gt; on the origin of zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our symbol for zero has its origin in India in the fifth century (where it was written as a dot), from where it spread to Cambodia at the end of the 7th century, and then to China and the Islamic countries in the 8th century. It reached western Europe in the 12th century. The open circle symbol for zero is originally from China.&lt;/p&gt;&#xA;" OwnerUserId="37" LastEditorUserId="37" LastEditDate="2014-11-12T15:45:59.573" LastActivityDate="2014-11-12T15:45:59.573" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="412" PostTypeId="2" ParentId="401" CreationDate="2014-11-12T16:23:34.997" Score="4" Body="&lt;p&gt;You could argue that there are two big issues that have been argued:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Tobacco and lung cancer:&lt;/strong&gt; This is really a sad one. For years, the tobacco companies insisted that there wasn't a &lt;a href=&quot;http://en.wikipedia.org/wiki/Health_effects_of_tobacco&quot; rel=&quot;nofollow&quot;&gt;connection&lt;/a&gt; between tobacco and all the medical issue smokers began to have. There were some good &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2038856/&quot; rel=&quot;nofollow&quot;&gt;early&lt;/a&gt; papers that found links, but they didn't gain much support until warnings began to be put on cigarette packs. &lt;a href=&quot;http://en.wikipedia.org/wiki/Smoking_and_Health:_Report_of_the_Advisory_Committee_to_the_Surgeon_General_of_the_United_States&quot; rel=&quot;nofollow&quot;&gt;A 1964 report&lt;/a&gt; established that there certainly &lt;em&gt;was&lt;/em&gt; a link (text available &lt;a href=&quot;http://profiles.nlm.nih.gov/ps/retrieve/ResourceMetadata/NNBBMQ&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;), and more &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC437139/&quot; rel=&quot;nofollow&quot;&gt;long-term studies&lt;/a&gt; have now been done.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Global warming:&lt;/strong&gt; This is another sad story. The idea of &lt;a href=&quot;http://en.wikipedia.org/wiki/Global_warming&quot; rel=&quot;nofollow&quot;&gt;global warming&lt;/a&gt; was opposed by those whom it hurt most - in this case, industry. Many scientists agree, but there are unfortunately large numbers who  don't. Some quotes &lt;a href=&quot;http://www.publications.parliament.uk/pa/ld200506/ldselect/ldeconaf/12/12we24.htm&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; are illuminating:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It has become fashionable in some parts of the UK media to portray the scientific evidence that has been collected about climate change and the impact of greenhouse gas emissions from human activities as an exaggeration. Some articles have claimed that scientists are ignoring uncertainties in our understanding of the climate and the factors that affect it. Some have questioned the motives of the scientists who have presented the most authoritative assessments of the science of climate change, claiming that they have a vested interest in &quot;playing up&quot; the potential effects that climate change is likely to have.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  Some assessments reports from the &lt;a href=&quot;http://www.ipcc.ch/publications_and_data/publications_and_data_reports.shtml&quot; rel=&quot;nofollow&quot;&gt;IPCC&lt;/a&gt; can be found at this site.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;More important than either of those, though, was the debate about how the universe itself formed. There were originally two competing theories: the &lt;a href=&quot;http://en.wikipedia.org/wiki/Big_Bang&quot; rel=&quot;nofollow&quot;&gt;Big Bang theory&lt;/a&gt; and the &lt;a href=&quot;http://en.wikipedia.org/wiki/Steady_State_theory&quot; rel=&quot;nofollow&quot;&gt;Steady State&lt;/a&gt; theory. The Big Bang theory was originally conceptualized by Georges Lemaitre, who did work on the expansion of the universe in the 1920s. Decades later, George Gamow reincarnated it to compete against the other prevailing theory, the Steady State theory. Fred Hoyle was the most famous advocate of the theory (he coined the term &quot;Big Bang&quot; as a derogatory name). However, the dispute was settled in the 1960s, when &lt;a href=&quot;http://adsabs.harvard.edu/abs/1965ApJ...142..419P&quot; rel=&quot;nofollow&quot;&gt;Penzias and Wilson&lt;/a&gt; discovered the cosmic microwave background, which provided strong evidence for the Big Bang theory.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-12T16:38:56.653" LastActivityDate="2014-11-12T16:38:56.653" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="413" PostTypeId="1" AcceptedAnswerId="417" CreationDate="2014-11-12T20:26:53.923" Score="6" ViewCount="830" Body="&lt;p&gt;Who first discovered that comets were physical objects with a Keplerian orbit?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was this accidentally discovered or was it discovered as part of an intentional effort by astronomers after they established facts about Sun, Earth, other planets, moon, and other celestial objects.&lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorUserId="26" LastEditDate="2015-03-01T13:35:33.633" LastActivityDate="2015-03-01T13:35:33.633" Title="Who discovered the &quot;Comet&quot; as a celestial object?" Tags="|discoveries|astronomy|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="414" PostTypeId="1" CreationDate="2014-11-12T20:48:29.807" Score="8" ViewCount="1347" Body="&lt;p&gt;Reading through Wikipedia says that Coriolis was the first to introduce the notion of work, described as &lt;em&gt;&quot;weight lifted through a height&quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our modern conception of work is of a force that realizes a displacement. This is general and valid for most forces (I don't know if we can apply this in general relativity or quantum mechanics). The book I'm studying for thermodynamics says that Joule's experiments clarified the relationship between heat and work, moving on to the first law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We now regard heat as energy and give it his name. I don't think they saw these concepts in the same light we do now. Energy is one of the fundamental principles all of physics is based on, and it has many faces: Vibrational, kinetic (translational and rotational), heat, potential, internal energy, etc. &lt;strong&gt;When did work and energy they take the form we use now?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book is &lt;em&gt;&quot;Mere Thermodynamics&quot;&lt;/em&gt;, by &lt;em&gt;Don Lemons&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="44" LastActivityDate="2014-11-16T13:24:01.543" Title="When were the modern notions of work and energy created" Tags="|physics|thermodynamics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="415" PostTypeId="2" ParentId="414" CreationDate="2014-11-12T20:57:37.190" Score="5" Body="&lt;p&gt;Mainly in the work of Hermann von Helmholtz in 1847. But of course (as always) there was a long development before and after.&#xA;&lt;a href=&quot;http://books.google.com/books?id=C1i4AAAAIAAJ&amp;amp;pg=PA114#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;http://books.google.com/books?id=C1i4AAAAIAAJ&amp;amp;pg=PA114#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that Helmholtz did not use the word &quot;energy&quot;. At the time of Newton (and even before) it was noticed that some quantities are preserved in the systems that obey Newton's laws (one of them is momentum, another energy, using modern terminology). But there was a long confusion about this, and discussions on what is really presevred: mv or mv^2 :-) Later, different forms of energy were discovered (energy of heat, for example). But it was Helmholtz who finally made an order in this and stated conservation of all kinds of energy together as a universal law of nature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. At the time of writing this paper, Helmholtz was a military doctor (physician).&#xA;When he published this paper, his commanders were very pleased that an officer does useful research&#xA;on &quot;preservation of force&quot; :-) &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-16T13:24:01.543" LastActivityDate="2014-11-16T13:24:01.543" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="417" PostTypeId="2" ParentId="413" CreationDate="2014-11-12T21:45:42.987" Score="3" Body="&lt;p&gt;I'm answering the question as per your approval of winwaed's comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Or do you mean when it was realized that they were physical objects with Keplerian orbits?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I've looked at a bunch of sources, and they seem to indicate that the honor of discovering the comets are celestial bodies goes to Tycho Brahe, Isaac Newton and Edmund Halley.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.space.com/53-comets-formation-discovery-and-exploration.html/&quot; rel=&quot;nofollow&quot;&gt;space.com&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For centuries, scientists thought comets traveled in the Earth's atmosphere, but in 1577, observations made by Danish astronomer Tycho Brahe revealed they actually traveled far beyond the moon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It was Isaac Newton who expanded upon this to study their orbits:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Isaac Newton later discovered that comets move in elliptical, oval-shaped orbits around the Sun, and correctly predicted that they could return again and again.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.britannica.com/EBchecked/topic/127524/comet&quot; rel=&quot;nofollow&quot;&gt;Encyclopedia Britannica&lt;/a&gt; tells a slightly different story (though with the same conclusion). Apparently, Aristotle first theorized about comets, claiming that they were just strange atmospheric phenomena. Some disagree - notably, the philosopher Seneca, who could lay claim to being the first person to identify comets as celestial bodies - but, of course, Aristotle prevailed. But Brahe came to the rescue:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Finally, during the 16th century the Danish nobleman Tycho Brahe established critical proof that comets are heavenly bodies. He compared the lack of diurnal parallax of the comet of 1577 with the well-known parallax of the Moon (the diurnal parallax is the apparent change of position in the sky relative to the distant stars due to the rotation of Earth). Tycho deduced that the comet was at least four times farther away than the Moon, establishing for the first time that comets were heavenly bodies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Go to the next page in the Encyclopedia Britannica article, though, and you'll find that Newton and Halley did important work on the orbits of comets.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://cse.ssl.berkeley.edu/segwayed/lessons/CometsTale/background.html&quot; rel=&quot;nofollow&quot;&gt;This&lt;/a&gt; page completely leaves out Brahe (!) and credits Newton and Halley with dominating the field of studying comets:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The next real progress was to show that comets obeyed the same natural laws as other bodies in the heavens, and this had to wait until someone--Sir Isaac Newton--determined what those laws were. Sir Edmund Halley, a close friend of  Newton, tamed the mystery of comets and broke the prediction barrier. The popular recent theory of Kepler was that comets travel in straight lines.  This seemed to allow for the way comets' tails changed orientation to point anti-sunward as they moved. But Newton's finding that heavenly bodies moved on closed, gravitationally bound orbits appealed to Halley.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And, later on,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Halley did some very difficult calculations, and discovered that a comet with an elliptical orbit and  period of 76 years could account in detail for the sightings of 1607, 1531, and 1456. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So perhaps you can credit this to Brahe, Newton, and Halley.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, &lt;a href=&quot;http://www.eso.org/public/events/astro-evt/hale-bopp/comet-history-1/&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt; page is exemplary, but it repeats a lot of the information stated above. That said, it's a fairly reliable source.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-12T22:15:38.333" LastActivityDate="2014-11-12T22:15:38.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="418" PostTypeId="1" AcceptedAnswerId="420" CreationDate="2014-11-13T00:32:46.567" Score="7" ViewCount="214" Body="&lt;p&gt;I was reading James Ladyman's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0415221579&quot;&gt;Understanding Philosophy of Science&lt;/a&gt;, and came across the following consequence of Fresnel's theory of light:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The phenomenon is known as &lt;em&gt;conical refraction&lt;/em&gt; and has now been observed many times. However, Fresnel knew nothing of it when he developed his theory, and indeed did not even derived the result itself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(italics in original)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ladyman briefly describes the phenomenon, but does not provide any further details about the history. I don't recall ever discussing this in a class, despite nearly having a PhD in physics, and working with a lot of classical optics. I also cannot find any mention of it in the indexes of the E&amp;amp;M and optics books I have easy access to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was the prediction of this phenomenon first derived from Fresnel's model, and when was it first observed? Did the derivation/prediction spur attempts to observe the effect?&lt;/p&gt;&#xA;" OwnerUserId="24" LastActivityDate="2014-11-13T00:54:03.163" Title="When was conical refraction predicted and observed?" Tags="|physics|optics|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="419" PostTypeId="1" AcceptedAnswerId="496" CreationDate="2014-11-13T00:35:13.920" Score="14" ViewCount="1493" Body="&lt;p&gt;The most obvious occurrence of numbers in nature are the counting (natural) numbers, as they describe how many of something there are. It took a long time for people to even realize that the numbers extended to zero and the non-integers, and even longer for negative numbers, complex numbers, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if I'm correct in my assumption that numbers arose from people's need to &lt;em&gt;count objects&lt;/em&gt;, what is the first recorded instance of numbers being used for any purpose other than counting objects? I'm guessing it was for establishing the geometrical concepts of length and area, but where and when did this first take place?&lt;/p&gt;&#xA;" OwnerUserId="273" LastActivityDate="2017-05-19T21:16:51.447" Title="When were numbers first used for anything other than counting?" Tags="|mathematics|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="420" PostTypeId="2" ParentId="418" CreationDate="2014-11-13T00:54:03.163" Score="5" Body="&lt;p&gt;I'm finding information that attributes this to Sir Humphrey Lloyd, who looked into it at the beckoning of none other than Sir William Rowan Hamilton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;http://www.britannica.com/EBchecked/topic/253431/Sir-William-Rowan-Hamilton#ref282842&quot;&gt;Encyclopedia Britannica&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In applying his methods in 1832 to the study of the propagation of light in anisotropic media, in which the speed of light is dependent on the direction and polarization of the ray, Hamilton was led to a remarkable prediction: if a single ray of light is incident at certain angles on a face of a biaxial crystal (such as aragonite), then the refracted light will form a hollow cone.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's the theoretical background. Lloyd then performed an experiment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hamilton’s colleague Humphrey Lloyd, professor of natural philosophy at Trinity College, sought to verify this prediction experimentally. Lloyd had difficulty obtaining a crystal of aragonite of sufficient size and purity, but eventually he was able to observe this phenomenon of conical refraction. This discovery excited considerable interest within the scientific community and established the reputations of both Hamilton and Lloyd.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/ConicalRefraction/&quot;&gt;This page&lt;/a&gt; also attributes the theory to Hamilton and the experiment to Lloyd:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hamilton described his prediction when he presented the concluding part of his Third Supplement to an Essay on the Theory of Systems of Rays to the Royal Irish Academy on the 22nd of October 1832. He asked Humphrey Lloyd, the Professor of Natural and Experimental Philosophy at Trinity College, Dublin, to try to verify the prediction experimentally. Lloyd had difficulty in obtaining a suitable crystal, but, after obtaining a good specimen of arragonite, succeeded in observing external conical refraction on the 14th of December. He was subsequently able to observe internal conical refraction. Humphrey Lloyd gave an account of his experiments in two papers published in the London and Edinburgh Philosophical Magazine in February and March, 1833. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, from &lt;a href=&quot;http://www.photonics.com/Article.aspx?AID=34819&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In two months, Humphrey Lloyd succeeded at observing the predicted light ring of conical refraction. In the early 1830s he used sunlight and a natural crystal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And a confirmation of Hamilton's part:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Therefore Hamilton asked Humphrey Lloyd for experimental approval, and two months later Lloyd observed the predicted light ring. This discovery was startling to the scientists and was a triumph for the wave theory. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-13T00:54:03.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="421" PostTypeId="5" CreationDate="2014-11-13T01:17:41.797" Score="0" Body="&lt;p&gt;Computer science can be applied to the natural sciences by creating simulations of natural processes to determine results too complex to be found by humans alone.&lt;/p&gt;&#xA;" OwnerUserId="-1" LastEditorUserId="62" LastEditDate="2015-04-12T16:38:49.233" LastActivityDate="2015-04-12T16:38:49.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="422" PostTypeId="4" CreationDate="2014-11-13T01:17:41.797" Score="0" Body="For questions about the scientific approach to computations and its applications." OwnerUserId="27" LastEditorUserId="62" LastEditDate="2015-04-12T16:38:49.233" LastActivityDate="2015-04-12T16:38:49.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="423" PostTypeId="5" CreationDate="2014-11-13T01:18:24.523" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:18:24.523" LastActivityDate="2014-11-13T01:18:24.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="424" PostTypeId="4" CreationDate="2014-11-13T01:18:24.523" Score="0" Body="History of scientific and mathematical terminology and its meaning." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-17T10:07:08.237" LastActivityDate="2014-11-17T10:07:08.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="425" PostTypeId="5" CreationDate="2014-11-13T01:19:39.163" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:19:39.163" LastActivityDate="2014-11-13T01:19:39.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="426" PostTypeId="4" CreationDate="2014-11-13T01:19:39.163" Score="0" Body="For questions about academic institutions, their members and other topics connected to the social process of scientific work and the interaction among students, scholars and other practitioners of science and mathematics." OwnerUserId="27" LastEditorUserId="62" LastEditDate="2015-04-12T16:32:44.177" LastActivityDate="2015-04-12T16:32:44.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="427" PostTypeId="5" CreationDate="2014-11-13T01:20:35.203" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:20:35.203" LastActivityDate="2014-11-13T01:20:35.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="428" PostTypeId="4" CreationDate="2014-11-13T01:20:35.203" Score="0" Body="The event of a person becoming deceased." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-17T10:07:22.097" LastActivityDate="2014-11-17T10:07:22.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="429" PostTypeId="5" CreationDate="2014-11-13T01:21:39.413" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:21:39.413" LastActivityDate="2014-11-13T01:21:39.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="430" PostTypeId="4" CreationDate="2014-11-13T01:21:39.413" Score="0" Body="The study of algebraic structures known as groups." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-16T21:25:36.017" LastActivityDate="2014-11-16T21:25:36.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="431" PostTypeId="5" CreationDate="2014-11-13T01:23:28.613" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:23:28.613" LastActivityDate="2014-11-13T01:23:28.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="432" PostTypeId="4" CreationDate="2014-11-13T01:23:28.613" Score="0" Body="The study of connections between field theory and group theory through the idea of Galois groups and further concepts." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-16T21:26:52.890" LastActivityDate="2014-11-16T21:26:52.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="433" PostTypeId="5" CreationDate="2014-11-13T01:26:55.653" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:26:55.653" LastActivityDate="2014-11-13T01:26:55.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="434" PostTypeId="4" CreationDate="2014-11-13T01:26:55.653" Score="0" Body="For questions about the origin and development of the number zero, formally the additive identity in mathematics." OwnerUserId="27" LastEditorUserId="54" LastEditDate="2016-10-10T06:10:54.320" LastActivityDate="2016-10-10T06:10:54.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="435" PostTypeId="5" CreationDate="2014-11-13T01:27:45.157" Score="0" Body="&lt;p&gt;Debunking involves finding flaws in theories, myths or other ideas based on falsehoods. Theories can be disproven merely by finding evidence that is contrary to one or more predictions made by a theory. The term &quot;debunking,&quot; though, is often used purely in a negative sense, against pseudoscience and related ideas.&lt;/p&gt;&#xA;" OwnerUserId="-1" LastEditorUserId="62" LastEditDate="2015-04-12T16:36:57.600" LastActivityDate="2015-04-12T16:36:57.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="436" PostTypeId="4" CreationDate="2014-11-13T01:27:45.157" Score="0" Body="For questions about finding the truth in claims that give good reason to be skeptical." OwnerUserId="27" LastEditorUserId="62" LastEditDate="2015-04-12T16:36:57.600" LastActivityDate="2015-04-12T16:36:57.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="437" PostTypeId="5" CreationDate="2014-11-13T01:29:35.163" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2014-11-13T01:29:35.163" LastActivityDate="2014-11-13T01:29:35.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="438" PostTypeId="4" CreationDate="2014-11-13T01:29:35.163" Score="0" Body="or sol, a yellow dwarf in the middle of the solar system. It is the main source of energy for life on earth." OwnerUserId="27" LastEditorUserId="27" LastEditDate="2014-11-16T21:24:49.730" LastActivityDate="2014-11-16T21:24:49.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="439" PostTypeId="5" CreationDate="2014-11-13T01:40:55.863" Score="0" Body="&lt;p&gt;Use this tag for questions about theories that theoretical physicists came up with, generally using mathematics, to describe the world. For example, one could ask how a physicist arrived at a particular conclusion in their physical theory.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="10106" LastEditDate="2022-02-23T16:12:30.380" LastActivityDate="2022-02-23T16:12:30.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="440" PostTypeId="4" CreationDate="2014-11-13T01:40:55.863" Score="0" Body="For questions regarding theories in the discipline of Physics" OwnerUserId="62" LastEditorUserId="10106" LastEditDate="2022-02-24T07:28:03.160" LastActivityDate="2022-02-24T07:28:03.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="441" PostTypeId="5" CreationDate="2014-11-13T01:54:04.277" Score="0" Body="&lt;p&gt;Quantum theory was born in 1900, when Max Planck came up with the idea of treating light as tiny bundles, or &quot;quanta&quot;. It entered its golden age within a few decades, and formed the basis for what became known as quantum field theory, which is the modern description of particle physics. It has thus far not been possible to formulate a quantum theory that describes gravity; it is thought that a theory of quantum gravity could lead to a theory of everything.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:24:06.210" LastActivityDate="2015-04-12T16:24:06.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="442" PostTypeId="4" CreationDate="2014-11-13T01:54:04.277" Score="0" Body="The branch of physics that relates to the behavior of objects, typically particles, on small scales. Probability is very important in quantum mechanics." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:24:06.210" LastActivityDate="2015-04-12T16:24:06.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="443" PostTypeId="5" CreationDate="2014-11-13T01:55:58.943" Score="0" Body="&lt;p&gt;Mathematics plays a huge role in physics to model phenomena. Most theories include a mathematical framework from which predictions can be made.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-12-08T21:16:20.800" LastActivityDate="2014-12-08T21:16:20.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="444" PostTypeId="4" CreationDate="2014-11-13T01:55:58.943" Score="0" Body="For questions regarding the mathematical aspects of physics." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-17T10:07:11.677" LastActivityDate="2014-11-17T10:07:11.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="445" PostTypeId="1" CreationDate="2014-11-13T05:04:42.480" Score="16" ViewCount="886" Body="&lt;p&gt;A benefit of Newton's concept of gravity as a force--despite the fact that it involved what was considered mysterious action at a distance--was that it explained objects falling to earth, and the motion of heavenly bodies, in a unified manner.  In contrast, an earlier view was that celestial objects have natural tendencies to move along circular paths, while objects fall to Earth because the Earth is the center of the universe, and objects composed of the earth element, for example, have a natural tendency to seek the center of the universe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question: Did Newton first develop (a) the conception of gravity as a force, and (b) the inverse square law, in order to explain acceleration of objects toward the earth, or did he work out these ideas for celestial motions, later applying them to objects falling to earth?  Or is there some other story that's more plausible?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://hsm.stackexchange.com/questions/207/hawking-on-newton-and-the-famed-apple&quot;&gt;This question&lt;/a&gt; talks about the apple story, which if true, might suggest that Newton started with objects falling to Earth, but I'm interested in answers inspired by more careful scholarly research.)&lt;/p&gt;&#xA;" OwnerUserId="277" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2022-06-07T12:23:59.203" Title="Did Newton develop the concept of gravity first for falling objects or for celestial motion?" Tags="|discoveries|gravity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="446" PostTypeId="2" ParentId="375" CreationDate="2014-11-13T05:34:54.833" Score="1" Body="&lt;p&gt;I came across a rather in-depth treatment of the order in &lt;em&gt;The Seven Day Circle: The History and Meaning of the Week&lt;/em&gt; by Eviatar Zerubave, which supports the explanations given in Pulsar and Amit Tyagi's answers are correct.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If we examine how the sequence of the seven days of the astrological week might have derived from the order in which the Hellenistic astronomers arranged the seven planets, we are immediately struck by one obvious overall pattern ... one must proceed by &quot;planetary leaps&quot; that consist of &lt;strong&gt;skipping two planets each time.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Unfortunately, a piece written around A.D. 100 by the Greek essayist Plutarch and bearing the suggestive title &quot;Why Are the Days Named After the Planets Reckoned in a Different Order form the Actual Order?&quot; has been lost forever, and our earliest source on this matter is the third-century Roman historian Dio Cassius. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book recounts two explanations given by &lt;a href=&quot;http://en.wikipedia.org/wiki/Cassius_Dio&quot; rel=&quot;nofollow&quot;&gt;Cassius Dio&lt;/a&gt; on why this is. Since he is much closer to the origins of the seven day week as we know it, presumably he had a more complete understanding of the motivations behind the order of the days. Nonetheless, the first theory appears rather batshit crazy:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;One explanation [Cassius Dio] claims to have heard relates to the tetrachord&lt;/strong&gt;, a diatonic series of four tones with an interval of a perfect fourth between the first and the last tone. The principle underlying the process of constructing intervals of perfect fourths happens to be identifiable to the one we have just discovered in the case of the days of the astrological week, namely, proceeding through the cycle of seven tones by skipping two tones each time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The second theory however is presented as much more sensible and likely to be correct:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dio Cassius claimed to have also head a second explanation of the order of the planetary days. This explanation is much more convincing, since it takes into account not only the traditional Hellenistic arrangement of the planets in a sequence corresponding to their orbital periods or assumed geocentric distances, but also the Hellenistic doctrine of chronocratories, which provides the astrological context within which the planetary seven-day week actually evolved.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;According to this explanation, one would astrologically &quot;assign&quot; the first hour of the first day to the most distant planet, Saturn, and then proceed to assign each following hour to the following planet in the traditional sequence ...The twenty-fifth hour of the first day would have been assigned to the sun, yet the daily cycle was divided into only twenty-four hours, so it would be the first hour of the second day. Since &lt;strong&gt;the controller of the first hour of each day was also supposed to dominate that entire day as a whole&lt;/strong&gt;, the entire first day came to be astrologically assigned to Saturn, the second one to the sun ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So in conclusion, it appears that Pulsar and Amit are correct.&lt;/strong&gt; Namely, that the order of the days comes from assigning the seven planets to the hours in sequence, and then taking first hour as the name of the day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I can't accept both answers, so I'll pick the older one.&lt;/p&gt;&#xA;" OwnerUserId="211" LastActivityDate="2014-11-13T05:34:54.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="447" PostTypeId="2" ParentId="419" CreationDate="2014-11-13T06:01:33.903" Score="-1" Body="&lt;p&gt;This is an interesting question and after researching this subject it gets even more interesting. First, as I expected the oldest mathematical artifact was found in Africa and you can read about this in detail at this link:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.tamu.edu/~dallen/masters/origins/origins.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.math.tamu.edu/~dallen/masters/origins/origins.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, I found some information that adds a twist to your question because their is proven research available that show the ability that animals possess to understand numbers. That being said, the origins of math could have possibly started with another species other than the human race. More information of this claim can be read here: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pierce.ctc.edu/staff/dlippman/mathinsociety/HistoricalCounting1.0.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.pierce.ctc.edu/staff/dlippman/mathinsociety/HistoricalCounting1.0.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, if we are just focusing on the human race, all evidence seems to point to Africa since the oldest bones and mathematical artifact were found there. As for the proof of establishing this research in relationship with geometrical concepts of length and area is debatable. Yet still, your question does hold some merit and notable points because the measurements of length and area are good examples of numbers being used for other purposes other than counting objects.&lt;/p&gt;&#xA;" OwnerUserId="175" LastEditorUserId="175" LastEditDate="2014-11-13T06:21:42.333" LastActivityDate="2014-11-13T06:21:42.333" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="448" PostTypeId="2" ParentId="322" CreationDate="2014-11-13T08:18:49.923" Score="2" Body="&lt;p&gt;&lt;strong&gt;Willi Hennig&lt;/strong&gt; is thought to have written his most significant work (introducing the concepts of what is now called &lt;em&gt;cladistic&lt;/em&gt;, i.e. monophyly-based biological classifications) &lt;em&gt;Grundzüge einer Theorie der Phylogenetischen Systematik&lt;/em&gt; (published later, in 1950) while prisoner of war at the end of WWII.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, though prisoner of war, he was not held in camp but put to work into mosquito control (being already a renowned entomologist at the time) by British forces.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;Grundzüge&lt;/em&gt; content was mainly theoretical, but grounded in his observations on insects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources:&lt;br&gt;&#xA;- &lt;a href=&quot;http://www.cladistics.org/willi.html&quot; rel=&quot;nofollow&quot;&gt;The Willi Hennig Society&lt;/a&gt;&lt;br&gt;&#xA;- His entry in the Complete Dictionary of Scientific biography (2008).&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2016-10-05T09:34:56.887" LastActivityDate="2016-10-05T09:34:56.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="449" PostTypeId="2" ParentId="445" CreationDate="2014-11-13T14:26:16.183" Score="10" Body="&lt;p&gt;It is a quite long story and not a &quot;one-shot&quot; discovery, as suggested by the &quot;apple tale&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two critical sources to be usefully read are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Alexandre Koyré, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0412423006&quot;&gt;Newtonian studies&lt;/a&gt;, Cambridge (Mass.) 1965 : Harvard University Press;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I.Bernard Cohen, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521273803&quot;&gt;The Newtonian revolution : With illustrations of the transformation&#xA;of scientific ideas&lt;/a&gt;, Cambridge 1980 : Cambridge University Press.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In the last one, we can see &lt;strong&gt;Ch.5 : Newton and Kepler's laws: stages of transformation leading toward universal gravitation&lt;/strong&gt;, page 222-on.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are two widely held opinions concerning the development of Newton's scientific ideas: that he found the law of universal &quot;gravitation&quot; in the 1660s and then refrained from publishing it for twenty years, and that he found this law by &quot;deducing&quot;&#xA;  it from Kepler's &quot;laws&quot; (or, possibly, from only one of Kepler's laws). The analysis presented in this chapter will show that according to any reasonable definition of universal &quot;gravitation&quot;, Newton did not find this law until some time after November&#xA;  1684, and before 1686, and then published it forthwith. [page 222]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In Newton's memorandum [written in 1718 in the draft of a letter to Des Maizeaux] and in his early manuscript of the 1660s, there is as yet no explicit suggestion that there is a solar &quot;force&quot; exerted on the planets that is identical to the terrestrial force exerted on the moon (which he alleges that he supposed to be ordinary terrestrial gravity). The &quot;endeavours to recede&quot;, in which he believed in the 1660s, are very different in concept from the simple centripetal forces continually deflecting planets and moons from their inertial (rectilinear) paths in which he came to believe some time later, in 1679-1680 or thereafter. Accordingly, there is no legitimate ground for saying that Newton had &quot;known''&#xA;  of the law (or a law) of universal gravitation in 1665 and had &quot;delayed&quot; announcing it for twenty years. In fact, far from believing in a force that is &quot;universal&quot;, Newton did not then even have an awareness of the possibility of a planetary action on the sun, a&#xA;  lunar action on the earth, much less an action of one planet on another.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the memorandum quoted above, Newton says that he 'began to think of gravity extending' to the moon. Presumably, he would have guessed - by analogy - that if the planetary &quot;endeavours&quot; to recede from the sun (in his mind they were not at that time &quot;forces&quot;)&#xA;  vary inversely as the square of the distance, then it must be the same for the moon. Hence, if gravity extends to the moon, and varies inversely as the square of the distance, the intensity of gravity at the moon's orbit should be 1/r*r times the intensity here on&#xA;  earth, r being the earth's radius. [page 232]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In an untitled MS document of the mid- or late 1660s (U.L.C. MS Add. 3958, sect. 5, fol. 87), Newton computes the 'endeavour of receding from the center' ('conatus a centro' or 'conatus recedendi a centro') by determining how far out along a tangent a body would move in some given time if it had that same 'endeavour' in a linear direction along the tangent and there were no impediment. In short, Newton is measuring the 'endeavour of receding'&#xA;  (not yet &quot;centrifugal force&quot;) by the acceleration, and the acceleration by the distance through which a body would move, according to Galileo's rule for uniform acceleration, freely along a straight line in a given time, 'the time of one revolution'. Newton&#xA;  then computes how far a body would descend if its 'endeavour of approaching toward the centre in virtue of its gravity' ('conatus accedendi ad centrum virtute gravitatis') were equal in magnitude to its 'endeavour of receding from the center' at the equator, as a result of the earth's daily rotation. [page 238]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the manuscript in which the foregoing calculations are described, Newton does derive an inverse-square law for the planets, by combining Kepler's third law for 'the primary planets' with their 'endeavours of receding' from the sun. This occurs in some&#xA;  brief or summary paragraphs that are added onto the rather detailed discussion and exposition of the moon's 'endeavour of receding' and the 'endeavour of receding' at the earth's surface. Newton neither says expressly nor in any way implies in this document&#xA;  either that the earth's gravity may extend as far out as the moon's orbit or that the moon's 'endeavour of receding' is in accord with an inverse-square law of the distance. The only application that he makes of his calculations concerning the moon's orbital 'endeavour&#xA;  to recede' is an attempt to account for the fact that the moon always 'turns the same face toward the earth'. [page 240]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Newton's interest in astronomy was a long-standing one, easily traceable to at least his student days at Cambridge in the 1660s. [page 241]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Newton's attention was forcibly drawn to astronomical problems three years later, in 1679, when Robert Hooke (recently appointed secretary of the Royal Society) wrote to Newton, expressing the hope that Newton would renew his former 'philosophicall' exchanges with the Society. To start things off, Hooke invited Newton to comment on a 'hypothesis or opinion of mine . . . of compounding the celestiall motions of the planetts [out] of a direct&#xA;  motion by the tangent &amp;amp; an attractive motion towards the centrall body'. In his reply, Newton declined to discuss Hooke's 'hypothesis'. Instead, he advanced a 'fansy' of his own: the effects of 'the Earth's diurnal motion' on the path of freely falling bodies (Newton&#xA;  to Hooke, 28 November 1679). Newton had erred, however, as Hooke was quick to discover.&#xA;  Newton had proposed a spiral path, usually accompanying the publication of Newton's letter to Hooke (28 November 1679) [...]. Hooke corrected Newton; he showed that the curve would not be (as Newton thought) 'a kind of spirall', which after a few revolutions would bring the falling object to 'the Centre of the Earth'. The curve would be 'rather a kind [of] Elleptueid'. [page 242]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Once Newton had acknowledged Hooke's correction, Hooke was emboldened to write to Newton of 'my supposition' concerning the attractive force that keeps the planets in their orbits; it was '... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall and Consequently that the Velocity will be in a sub-duplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance' (Hooke to Newton, 6 January 1679/80). Newton did not comment directly on this statement. His eventual opinion may be readily gathered from the fact that he proved that the velocity of a body (such as a planet or planetary satellite) moving in an elliptical orbit under the action of an inverse-square force is not 'as Kepler [and as Hooke] Supposes Reciprocall to the Distance', but is rather reciprocally as (or in inverse proportion to) the perpendicular distance from the center of force to the tangent to the planetary orbit. [page 243]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In various unpublished documents, Newton admitted that in 1679-1680 Hooke had provided the occasion for his study of planetary dynamics, although he would not admit that Hooke had&#xA;  made any substantive contribution to his thinking. [page 248]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;it is a plain fact that he [Newton] did undertake an analysis of planetary motion according to the Hookean mode of conceiving a centrally directed force to be acting on a planet that had a component of linear inertial motion, and he was able to do so even&#xA;  though he still had not fully given up his adherence to the concept of aethereal vortices and though he apparently did not really believe in centripetal forces acting at a distance. Whatever his beliefs, the Newtonian style enabled him to explore the properties of this kind of force and eventually to discover [...] that universal gravity is useful and even necessary, and that it 'really exists' (as he later declared in the concluding general scholium of the &lt;em&gt;Principia&lt;/em&gt; in 1713) and acts according to the laws he had set forth. [page 254]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-11-13T14:26:16.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="450" PostTypeId="2" ParentId="445" CreationDate="2014-11-13T15:07:26.570" Score="3" Body="&lt;p&gt;Let me give a shorter answer. The idea of &amp;quot;attraction proportional to the inverse square of the distance&amp;quot; was around for some time. Newton's crucial idea was to compare the acceleration of gravity of the Earth surface and acceleration of the Moon. His first computation did not give the good agreement with observation, because he used an incorrect value for the&#xA;Earth radius. When later he inserted more accurate value, there was an agreement.&lt;/p&gt;&#xA;&lt;p&gt;This was a crucial argument for the &amp;quot;universal gravity&amp;quot;.&#xA;Newton did not publish this observation and did not tell about it to anyone until much later time.&lt;/p&gt;&#xA;&lt;p&gt;Many years later Ch. Wren asked whether the law of attraction will imply Kepler orbits.&#xA;The question was passed to Newton by Halley.&#xA;A similar question was earlier asked by Hooke in a letter to Newton.&#xA;Newton's first answer to Hooke was incorrect.&#xA;But when Wren's question reached him, he quickly realized that the answer is &amp;quot;yes&amp;quot;,&#xA;and wrote his Principia.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-06-07T12:23:59.203" LastActivityDate="2022-06-07T12:23:59.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="451" PostTypeId="1" CreationDate="2014-11-13T20:07:00.413" Score="27" ViewCount="1068" Body="&lt;p&gt;To what extent was Cantor motivated by &lt;a href=&quot;http://en.wikipedia.org/wiki/Galileo&amp;#39;s_paradox&quot; rel=&quot;noreferrer&quot;&gt;Galileo's paradox&lt;/a&gt;? More generally, to what extent were late 19th century mathematicians motivated by, or even aware of, Galileo's paradox?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is an issue I've been wondering about for many years but I have never gotten around to looking into it until this morning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yesterday, I read Omen's answer [later deleted] to the History of Science and Math StackExchange question &lt;a href=&quot;https://hsm.stackexchange.com/questions/58/what-motivated-cantor-to-invent-set-theory&quot;&gt;What motivated Cantor to invent set theory&lt;/a&gt; and this led me to make the following comment: &lt;em&gt;It was my understanding that Cantor did not know about Galileo's writing on infinity. In fact, I've been under the impression (for maybe 20 years) that Galileo's thoughts on infinity were mostly unknown to mathematicians until&lt;/em&gt; &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183418475&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Edward Kasner&lt;/em&gt;&lt;/a&gt; &lt;em&gt;called attention to the connection in 1904, but I have not (yet) looked into this matter.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(NEXT DAY)&lt;/em&gt; mweiss suggested in a comment that I isolate my question from my partial attempts at an answer, which I think is a good structural idea for this site.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="4251" LastEditDate="2020-06-07T18:39:50.347" LastActivityDate="2020-06-07T18:39:50.347" Title="Did Galileo's writings on infinity influence Cantor?" Tags="|mathematics|set-theory|galileo|infinity|cantor|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="452" PostTypeId="2" ParentId="402" CreationDate="2014-11-13T20:15:09.597" Score="5" Body="&lt;p&gt;Answering your first question, it was not Aryabhata. It was rather &lt;a href=&quot;http://www.storyofmathematics.com/indian_brahmagupta.html&quot; rel=&quot;noreferrer&quot;&gt;Brahmagupta&lt;/a&gt; and that too for first time publishing the rules governing zero. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I see three aspects about the question,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Concept of '0'&lt;/li&gt;&#xA;&lt;li&gt;'0' as a Symbol/placeholder&lt;/li&gt;&#xA;&lt;li&gt;Word 'Zero' for the symbol '0'&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As already answered by @Felix the concept itself was discovered across different cultures including Babylonians, Mayans, Chinese, Hindus and many other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to using '0' as a symbol. &lt;a href=&quot;http://en.wikipedia.org/wiki/0_(number)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; states that,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In AD 976 the Persian encyclopedist Muhammad ibn Ahmad al-Khwarizmi, in his &quot;Keys of the Sciences&quot;, remarked that if, in a calculation, no number appears in the place of tens, then a little circle should be used &quot;to keep the rows&quot;. This circle was called صفر (ṣifr, &quot;empty&quot;) in Arabic language. That was the earliest mention of the name ṣifr that eventually became zero.[8]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The same Wikipedia article says that the Arabic word 'sifr' became 'zefiro' in Italian and then later 'zero' in Venetian.&lt;/p&gt;&#xA;" OwnerUserId="188" LastActivityDate="2014-11-13T20:15:09.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="453" PostTypeId="2" ParentId="322" CreationDate="2014-11-13T23:20:50.353" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Konrad_Lorenz&quot;&gt;Konrad Lorenz&lt;/a&gt; (Nobel Prize for Physiology or Medicine 1973) wrote (part of) his important book &quot;Behind the Mirror&quot; (original title &quot;Die Rückseite des Spiegels&quot;) as a POW in the Soviet Union (from 1944-1948). I quote from the Wikipedia page. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He was sent to the Russian front in 1944 where he quickly became a prisoner of war in the Soviet Union from 1944 to 1948. In captivity he continued to work as a medic and &quot;got quite friendly with some Russians, mostly doctors.&quot; When he was repatriated, he was allowed to keep the manuscript of a book he had been writing, and his pet starling. He arrived back in Altenberg (his family home, near Vienna) &quot;with manuscript and bird intact.&quot; The manuscript became his book Behind the Mirror. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding the question how this was done and the working conditions, it might be relevant to remark that he wrote on the paper of a cement bag. I had a vague recollection of that from a documentary on him I saw long ago, I think it was even him who said it there, I saw a long time ago. Searching I found some confirmation of that recollection in &lt;a href=&quot;http://slideplayer.de/slide/893971/&quot;&gt;some slides of a presentation, slide 10 specifically&lt;/a&gt; &quot;In Einsamkeit und Freiheit? Wissenschaftliche Praxis im historisch-biografischen Zusammenhang.&quot; [In loneliness and freedom? Scientific practice in historic-biographical context} (Sorry for the strange source but it seems the presentation is an academic one, so it is not as bad as it might seem on first glance.)&lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2014-11-13T23:20:50.353" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="454" PostTypeId="1" AcceptedAnswerId="456" CreationDate="2014-11-13T23:40:07.317" Score="11" ViewCount="1296" Body="&lt;p&gt;Sound is an oscillation of a medium (waves in the medium), and the frequency of thse oscillations we hear as the pitch.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When this was discovered and by what experiment?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suppose this was already known to Marin Mersenne (whose book was published in 1637)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know what simple experiment can prove this. It is the so-called &quot;syrene&quot;, but the earliest descriptions of this device that I now are of 19-th century.&#xA;Though there is nothing in it which Hero of Alexandia could not understand or invent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question splits into several warts: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any evidence that this relation&#xA;  between sound and oscillations was known in antiquity?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If not, when this was discovered, and what experiment was used.&#xA;The source that I have says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The interpretation of sound as a vibration also goes back to antiquity, although its more precise expression waited early-modern mechanical philosophy...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the seventeenth century, Galileo Galilei and Marin Mersenne popularized the correspondence between pitch and frequency that Giovanni Battista Benedetti,&#xA;  Galileo's father Vincenzo, and Isaac Beeckman had earlier articulated. Accordingly, a musical tone is a periodic succession of pulses transmitted by the air to the eardrum&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but it gives no reference on &quot;antiquity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Thanks to those who answered. I just found the following &quot;cite&quot; in an acoustic book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pythagoras (570–497 bce) observed that “air motion generated by a vibrating body&#xA;  sounding a single musical note is also vibratory and of the same frequency as the&#xA;  body;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But the author does NOT give the source of this sentence in quotation marks!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is correct, than the most ancient exact science is acoustics, not astronomy,&#xA;as we all thought:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-14T00:58:09.647" LastActivityDate="2014-11-14T00:58:09.647" Title="When and how it was discovered that sound is related to some kind of oscillations?" Tags="|experimental-physics|acoustics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="455" PostTypeId="2" ParentId="454" CreationDate="2014-11-14T00:20:29.740" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there any evidence that this relation between sound and oscillations was known in antiquity?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It doesn't appear that the ancients had any &lt;em&gt;proof&lt;/em&gt; that sound waves were oscillations in a medium, although Aristotle appears to have been the first to theorize about it. From &lt;a href=&quot;http://www.physics.umd.edu/courses/Phys273/williams06/Acoustics%20History.pdf&quot;&gt;this&lt;/a&gt; pdf (with a transcript available &lt;a href=&quot;http://asa.aip.org/pierce.html&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The wave interpretation was also consistent with Aristotle's (384-322 B.C.) statement to the effect that air motion is generated by a source, &quot;thrusting forward in like&#xA;  manner the adjoining air, to that the sound travels unaltered in quality as far as the disturbance of the air manages to reach.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This &quot;wave interpretation&quot; is a reference to later ideas by Vetruvius and Chrysippus. Earlier, however, Boethius proposed something similar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://electricalfacts.com/Neca/Science/sound/history.shtml&quot;&gt;This&lt;/a&gt; gives very similar text:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One of the first references of sound as a wave is found in a statement made by Aristotle where he indicated that air motion is generated by a source, trusting forward so that the sound travels unaltered in quality as far as the disturbance in the air manages to reach.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, it doesn't seem like there were any experiments done in antiquity to confirm or disprove these theories. For this, we have to go to Robert Boyle (quoted from the previous source listed):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Robert Boyle's classic 1660 experiment on the sound radiation by a ticking watch in a partially evacuated glass vessel provided evidence that air is necessary, either for the production or transmission of sound.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiquote.org/wiki/Vitruvius#Book_V&quot;&gt;Wikiquote&lt;/a&gt; gives this passage from Vitruvius's The Ten Books on Architecture:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Voice is a flowing breath of air, perceptible to the hearing by contact. It moves in an endless number of circular rounds, like the innumerably increasing circular waves which appear when a stone is thrown into smooth water, and which keep on spreading indefinitely from the centre unless interrupted by narrow limits, or by some obstruction which prevents such waves from reaching their end in due formation. When they are interrupted by obstructions, the first waves, flowing back, break up the formation of those which follow. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This text can be found on page 139 &lt;a href=&quot;http://books.google.com/books?id=La4aAAAAYAAJ&amp;amp;%7C&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-14T00:29:25.170" LastActivityDate="2014-11-14T00:29:25.170" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="456" PostTypeId="2" ParentId="454" CreationDate="2014-11-14T00:30:03.057" Score="6" Body="&lt;p&gt;The concept of oscillation in air conveying sound did not develop all at once, but was instead a natural development of ancient ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle, for example, may have had a correct understanding of sound, thinking of the vibration of (say) the string of a musical instrument, causing a similar vibration in the air, this being received with a similar vibration in the ear of the listener.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his work &lt;a href=&quot;http://classics.mit.edu/Aristotle/soul.2.ii.html&quot;&gt;&lt;em&gt;On the Soul&lt;/em&gt;&lt;/a&gt;, he writes about the nature of sound propagation.  Sound, he says is conveyed by &quot;the air originally struck by the impinging body and set in movement by it&quot;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is rightly said that an empty space plays the chief part in the production of hearing, for what people mean by 'the vacuum' is the air, which is what causes hearing, when that air is set in movement as one continuous mass; but owing to its friability it emits no sound, being dissipated by impinging upon any surface which is not smooth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is the &lt;em&gt;friability&lt;/em&gt; of air that he comes back to repeatedly, the ability of air to be &quot;broken down&quot; into smaller parts by motion or friction.  This, to me, suggests something like how a wave compresses and rarifies very small portions of the air.&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2014-11-14T00:30:03.057" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="457" PostTypeId="2" ParentId="62" CreationDate="2014-11-14T00:50:41.950" Score="2" Body="&lt;p&gt;Eukaryotic cells are defined by the presence of a nucleus (and any other organelles).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Van Leeuwenhoek himself (in &lt;em&gt;Opera Omnia&lt;/em&gt;, 1719) was the first to describe a cell nucleus, which he saw in &lt;a href=&quot;http://commons.wikimedia.org/wiki/File:Leeuwenhoek1719RedBloodCells.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;salmon blood cells&lt;/a&gt;:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/PQ8YK.jpg&quot; alt=&quot;salmon blood cells&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(By the way, while mammalian red blood cells don't have a nucleus, the red blood cells of &lt;a href=&quot;http://en.wikipedia.org/wiki/Red_blood_cell#Nucleus&quot; rel=&quot;nofollow noreferrer&quot;&gt;most other creatures do&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2014-11-14T08:17:44.453" LastActivityDate="2014-11-14T08:17:44.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="458" PostTypeId="1" AcceptedAnswerId="463" CreationDate="2014-11-14T00:59:35.817" Score="12" ViewCount="769" Body="&lt;p&gt;I occasionally encounter mentions suggesting that late middle ages weren't as barren intellectually as commonly believed. For example, Occam and Scotus are  credited with developing  modal and temporal logic. Occam apparently came up with a theory about assigning truth values to past statements that became influential in the 20th century. Cantor explicitly cited scholasts as a source for his ideas about transfinite sets. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What developments that are relevant today can be traced back to scholasts?&lt;/li&gt;&#xA;&lt;li&gt;Did they have ideas about different orders of infinity?&lt;/li&gt;&#xA;&lt;li&gt;What other mathematicians/scientists did they influence?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2020-06-06T17:20:59.097" LastActivityDate="2021-05-17T12:48:10.897" Title="How much did scholasticism contribute to logic and mathematics?" Tags="|mathematics|mathematical-logic|set-theory|logic|middle-ages|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="459" PostTypeId="1" CreationDate="2014-11-14T01:21:57.023" Score="6" ViewCount="413" Body="&lt;p&gt;For a very long time, people had no reason to believe that an airless void could exist in nature at all, whether above the earth or anywhere else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Descartes, in his 1633 work &lt;a href=&quot;http://www.princeton.edu/~hos/mike/texts/descartes/world/world.htm#four&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The World&lt;/em&gt;&lt;/a&gt;, the world is already full of matter, whether merely air or something more substantial:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By the same token, if you consider in this regard some of the experiments the philosophers have been wont to use in showing that there is no void in nature, you will easily recognize that all those spaces that people think to be empty, and where we feel only air, are at least as full, and as full of the same matter, as those where we sense other bodies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He further elaborates on this idea with an illustration of wine flowing out of a cask, that it cannot displace the air outside until the air has somewhere to go, namely, into the space left by the wine inside the cask.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did we get from Descartes' position to the one we hold now, that there is an airless void naturally occurring above the atmosphere?&lt;/p&gt;&#xA;" OwnerUserId="60" LastActivityDate="2017-06-03T21:16:50.870" Title="When was the airless void above the earth discovered?" Tags="|astronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="460" PostTypeId="1" AcceptedAnswerId="462" CreationDate="2014-11-14T01:23:23.893" Score="7" ViewCount="1738" Body="&lt;p&gt;There are references of Greeks discovering static electricity around 600 BC. Mediterranean's were also thought of having the understanding of rubbing amber rods to attract light objects  like feather. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is when was it first used productively for the first time, after having understood it's properties ?&lt;/p&gt;&#xA;" OwnerUserId="188" LastActivityDate="2018-10-17T19:20:37.767" Title="When and where was Electricity used for the first time?" Tags="|electricity|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="461" PostTypeId="1" CreationDate="2014-11-14T02:44:54.527" Score="5" ViewCount="64" Body="&lt;p&gt;Some recent studies have analyzed the percentage of research that is &lt;a href=&quot;http://en.wikipedia.org/wiki/Reproducibility&quot; rel=&quot;nofollow&quot;&gt;reproducible&lt;/a&gt;.&#xA;E.g. &lt;em&gt;Prinz, F.; Schlange, T.; Asadullah, K. (2011). &quot;&lt;a href=&quot;http://www.nature.com/nrd/journal/v10/n9/full/nrd3439-c1.html&quot; rel=&quot;nofollow&quot;&gt;Believe it or not: How much can we rely on published data on potential drug targets?&lt;/a&gt;&quot;. Nature Reviews Drug Discovery 10 (9): 712. doi:10.1038/nrd3439-c1. PMID 21892149.&lt;/em&gt; found that found that 65% of medical studies were inconsistent when re-tested, and only 6% were completely reproducible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder how research reproducibility in medicine has evolved over time. &lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2014-11-15T16:08:06.237" LastActivityDate="2014-11-15T16:08:06.237" Title="How did research reproducibility in medicine evolve over time?" Tags="|academia|publishing|medicine|scientific-method|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="462" PostTypeId="2" ParentId="460" CreationDate="2014-11-14T04:09:19.573" Score="1" Body="&lt;p&gt;I do not know of any practical use of static electricity except for many funny demonstration experiments which were popular in 18-th century. Only after the invention of a battery (by A. Volta, around 1800) serious applications became possible, and the earliest that comes to my mind is telegraph. Wikipedia mentions 1809 as the date of the earliest experiments with electric telegraphy. So I guess that telegraph was the first practical application of electricity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I agree with SJuan that lighting rods was the first use. I would say, important use,&#xA;because it saves lives and property.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-14T18:14:12.250" LastActivityDate="2014-11-14T18:14:12.250" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="463" PostTypeId="2" ParentId="458" CreationDate="2014-11-14T11:00:04.300" Score="11" Body="&lt;p&gt;Regarding &lt;em&gt;logic&lt;/em&gt;, the answer is a mixed one.&lt;/p&gt;&#xA;&lt;p&gt;The development of medieval logic is wide area of study; see at least &lt;em&gt;SEP&lt;/em&gt;'s entries :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/medieval-syllogism/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Theories of the Syllogism&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/consequence-medieval/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Theories of Consequence&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/modality-medieval/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Theories of Modality&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/medieval-terms/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Theories: Properties of Terms&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and more ..., as well as :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Dov Gabbay &amp;amp; John Woods (editors), &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0444516255&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Handbook of the History of Logic. Vol 2 : Mediaeval and Renaissance Logic&lt;/a&gt; (2008).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The Reanaissance rediscovery of ancient Greek mathematics and philosophy induced a general &amp;quot;devaluation&amp;quot; of &lt;em&gt;scholastic&lt;/em&gt; logic, seen as cumbersome and futile.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;champions&amp;quot; of &lt;em&gt;scientific revolution&lt;/em&gt;, like Bacon, Galileo and Descates was highly critical or quite silent about it; see :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Stephen Gaukroger, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0198248253&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cartesian Logic : An Essay on Descartes Conception of Inference&lt;/a&gt; (1989)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;but also :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;William Wallace, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/904814115X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo's Logic of Discovery and Proof : The Background, Content, and Use of His Appropriated Treatises on Aristotle's &lt;em&gt;Posterior Analytics&lt;/em&gt;&lt;/a&gt; (1992).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Among modern &amp;quot;pioneer&amp;quot; of logic, like Leibniz, Bolzano and Frege (all quite unrecognized as such during their time), only Leibniz has a clear knowledge of medieval logic.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/algebra-logic-tradition/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Algebra of Logic Tradition&lt;/a&gt; was &amp;quot;motivated&amp;quot; more by the development of algebra than by traditional logic.&lt;/p&gt;&#xA;&lt;p&gt;See on this :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Dov Gabbay &amp;amp; John Woods (editors), &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0444516115&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Handbook of the History of Logic. Vol 3 The Rise of Modern Logic From Leibniz to Frege&lt;/a&gt; (2004).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;But there are at least two considerations to be done :&lt;/p&gt;&#xA;&lt;p&gt;First, the humanists devaluation of scholastic logic is contemporary to the rediscovery of ancient Greek philosophy, included Aristotle's one and its logic.&lt;/p&gt;&#xA;&lt;p&gt;Thus, we have a &amp;quot;underground&amp;quot; continuity in &amp;quot;aristotelian&amp;quot; logic also during the Renaissance and Early Modern Era; see for example :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;E.J.Ashworth, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/9027704643&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Language and Logic in the Post-Medieval Period&lt;/a&gt; (1974) [covering the period : 1550-1650]&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Marco Sgarbi, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/9400749503&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Aristotelian Tradition and the Rise of British Empiricism : Logic and Epistemology in the British Isles (1570-1689)&lt;/a&gt; (2013); see &lt;strong&gt;Ch.9.3 Hobbes and Logic as Calculus&lt;/strong&gt;, page 184-on :&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hobbes is generally considered by the scholars as one of the first great early modern&#xA;thinkers to break with tradition and direct his work instead towards the new philosophical&#xA;and scientific developments. Not infrequently, passages have been considered out&#xA;of context, with scholars stating that Hobbes abandoned Aristotelian philosophy and&#xA;logic, because of his vitriolic attack upon Scholastic philosophy and theology, even&#xA;though a large part of his thought, and especially of his logic, had been decisively&#xA;in fluenced by the Aristotelian tradition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is well known that Hobbes never wrote a textbook of logic, nor taught logic in&#xA;the university. However, the general introduction to his &lt;em&gt;Elementa philosophiae&lt;/em&gt;&#xA;presents a dense treatment of logic, the result of 10 years’ thought on the topic; this&#xA;can be considered as a work in its own right. [...] Hobbes’ knowledge of the Aristotelians must have increased with his travel in Italy (1610–1613) [...] where he knew Fulgenzio Micanzio, a friend of Paolo Sarpi and Galileo. In this period Hobbes began to read Galileo and Euclid, and hatched the plan to establish a rigorous mechanical science of reality as a whole. Probably under the impulse of Galilei’s philosophy, which was full of Aristotelian ideas, Hobbes focused his interests on Paduan logic, whose legacy is quite evident in his works.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From the incipit of his &lt;em&gt;Logica&lt;/em&gt;, which could be considered as a work in its own right, Hobbes shares with his contemporaries the intention of establishing a scientific method. Such a method should take as its model the advancements and developments of geometry and should proceed with the same rigour.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Russell Wahl, &lt;em&gt;Port Royal : The Stirrings of Modernity&lt;/em&gt;, in Handbook of the History of Logic. Vol 3, page 667-on :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Logic or the Art of Thinking&lt;/em&gt;, popularly known as the Port-Royal &lt;em&gt;Logic&lt;/em&gt;, was probably&#xA;the most important logic text book from the time after the mediaeval period until the middle of the nineteenth century. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Like other authors of the seventeenth century, Arnauld and Nicole were very critical of the old logic, which for them included not only the “scholastic” work on syllogisms, but also the humanistic logic of Ramus. They saw their logic as new and they were particularly influenced by Descartes, who had also been critical of syllogistic logic, and they incorporated many of his doctrines into their work. This new seventeenth-century logic, with its rejection of much traditional logic and its concern with clarifying ideas and determining the truth of simple propositions, has often been criticized by more recent logicians of mixing psychology and epistemology with logic. Often it has been compared unfavorably with mediaeval logic. Despite the fact that most of its topics are closer to the mediaeval period, the Port-Royal Logic has a very modern feel to it, and covers several topics now included in more informal introductory logic courses, such as discussions of clarification of concepts, informal fallacies, causal reasoning, and probability, as well as more traditional accounts of propositions and syllogisms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Second, what &amp;quot;complicates&amp;quot; a reconstruction of the status of logic during the Early Modern Era are :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the difficulty of separating the influence of &amp;quot;old&amp;quot; medieval logic from that of &amp;quot;new&amp;quot; (rediscovered) aristotelian one&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the characteristic &amp;quot;mixture&amp;quot; in the logical discussions of this period of &lt;em&gt;formal logic&lt;/em&gt;, &lt;em&gt;analysis of language&lt;/em&gt; (both present in medieval tradition) and search for a &lt;em&gt;method&lt;/em&gt; for acquiring knowledge, that was clearly a &amp;quot;modern&amp;quot; issue, but again linked to the (rediscovered) aristotelian doctrines of the &lt;em&gt;Posterior Analytics&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2021-05-17T12:48:10.897" LastActivityDate="2021-05-17T12:48:10.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="464" PostTypeId="1" CreationDate="2014-11-14T11:28:26.190" Score="28" ViewCount="952" Body="&lt;p&gt;In &lt;a href=&quot;https://math.stackexchange.com/users/1421/jack-lee&quot;&gt;Lee&lt;/a&gt;'s book &lt;a href=&quot;http://www.math.washington.edu/~lee/Books/ISM/&quot; rel=&quot;noreferrer&quot;&gt;'Introduction to Smooth Manifolds'&lt;/a&gt;, there is an interesting discussion (near the end of chapter three) of several different ways of viewing/constructing the notion of a &lt;em&gt;tangent space&lt;/em&gt; to a point on a manifold. As it turns out, there are quite a few. Tangent vectors can be understood as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;derivations of the space of germs&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;equivalence classes of curves&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;tangent vectors to charts&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;equivalence classes of $n$-tuples&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Lee makes the remark that the last of these, which is based on the transformation rule under coordinate transformations&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\tilde v^j=\frac{\partial \tilde x^j}{\partial x^i}v^i$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is actually the oldest way to make sense of the concept 'tangent vector', and is still heavily used by physicists. As someone who identifies as a physicist, I can confirm this ;-). However, the remark that this is the &lt;em&gt;original&lt;/em&gt; way of understanding tangent spaces surprised me, and sparks the following question: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who introduced the different constructions of tangent spaces (&lt;em&gt;including&lt;/em&gt; the one in terms of the transformation law), when did they do this, and --- most importantly --- what was their motivation for taking the different approaches that we know now?&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2018-11-20T18:21:11.877" Title="The history of different constructions of tangent spaces" Tags="|mathematics|differential-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="465" PostTypeId="2" ParentId="460" CreationDate="2014-11-14T16:17:00.107" Score="0" Body="&lt;p&gt;Arguably, the &quot;founding father&quot; of electricity (as well as of the United States), was &lt;a href=&quot;http://en.wikipedia.org/wiki/Benjamin_Franklin&quot; rel=&quot;nofollow&quot;&gt;Benjamin Franklin&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was the first to discuss positive and negative &quot;electricity,&quot; (charges), as well as the electric charge conservation law analogous to Newton's conservation laws of mass and energy.&#xA;These represented &quot;keystones&quot; in the foundation of understanding electricity, and eventually led to productive uses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Franklin &quot;proved&quot; his ideas through his well-known experiment flying a kite in lightening.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2014-11-14T16:17:00.107" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="467" PostTypeId="2" ParentId="219" CreationDate="2014-11-14T16:26:05.270" Score="5" Body="&lt;p&gt;The claim that Egyptians were aware of the benefits of using cold objects appears to be from the &lt;a href=&quot;http://en.wikipedia.org/wiki/Edwin_Smith_Papyrus&quot;&gt;Edwin Smith Papyrus&lt;/a&gt;, a famous Egyptian scroll on ancient medicine. I found the abstract of a paper &lt;a href=&quot;http://www.journalacs.org/article/S1072-7515(06)01658-9/abstract&quot;&gt;here&lt;/a&gt; that goes into some detail:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As long ago as 300) BC, the use of cold compresses to treat compound skull fractures and infected wounds were mentioned in an Egyptian papyrus, identified by the historian Breasted as the Edwin Smith Papyrus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It goes on to quote from the papyrus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The portion of cryotherapy is translated by Breasted as follows:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Thou shalt make for him cool applications for drawing out the inflammation from the mouth of the wound.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it appears that the Egyptians would create a compress filled with something slightly cooler, such as water, and use it to treat the wound.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This isn't yet a full answer; I'll add more later.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-14T16:26:05.270" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="468" PostTypeId="2" ParentId="451" CreationDate="2014-11-14T18:37:08.467" Score="17" Body="&lt;p&gt;This morning [13 November 2014] I spent several hours going through many Cantor-related papers and books that I have, and I am now nearly convinced that Galileo probably had no influence on Cantor and Galileo had very little if any influence on other mathematicians. For instance, I believe there was no German translation of Galileo's &amp;quot;Two New Sciences&amp;quot; until 1890. (I am not very sure about this, however.)&lt;/p&gt;&#xA;&lt;p&gt;I have recorded the relevant things I found this morning as excerpts and comments associated with the bibliography that follows. The items in this bibliography are not (by a long way) the only papers and books I looked at. I selected these items for several reasons: for their relevance to the issue with Galileo (and with Bolzano), for their interest to people interested in the posted question &lt;a href=&quot;https://hsm.stackexchange.com/questions/58/what-motivated-cantor-to-invent-set-theory&quot;&gt;What motivated Cantor to invent set theory&lt;/a&gt;, and for their interest to people interested in 19th century infinity issues. For the record, the other papers and books I looked through were by Irving Henry Anellis, Roger Lee Cooke, José Ferreirós, Abraham Adolf [Adolph] Halevi Fraenkel, Michael F. Hallett, Thomas William Hawkins, Ernest William Hobson, Arie Hinkis, Phillip Eugene Johnson, Philip Edward Bertrand Jourdain, Akihiro Kanamori, Fyodor Andreyevich Medvedev, and many others.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Carl Benjamin Boyer, &lt;a href=&quot;http://catalog.hathitrust.org/Record/000165835&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Concepts of the Calculus. A Critical and Historical Discussion of the Derivative and the Integral&lt;/strong&gt;&lt;/a&gt;, Columbia University Press, 1939, vii + 346 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Reprinted by Hafner Publishing Company in 1949 (xii + 346 pages) and by &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486605094&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dover Publications&lt;/a&gt; in 1959 (title changed to &lt;strong&gt;The History of the Calculus and Its Conceptual Development&lt;/strong&gt;, xii + 346 pages). &lt;strong&gt;(from pp. 270-271)&lt;/strong&gt; &amp;quot;[Bolzano's] view in this respect resembles that of Galileo, to whom he referred in this connection. [footnote 11, omitted here] Although he denied the existence of infinitely large and infinitely small magnitudes, he maintained, with Galileo, the possibility of an actual infinity with respect to aggregation. He remarked, with respect to such assemblages, the paradox which Galileo had pointed out: that the part could in this case be put into one-to-one correspondence with the whole. [...] [Bolzano's] work remained largely unnoticed until rediscovered by Hermann Hankel more than a half century later.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Georg Ferdinand Ludwig Philip Cantor, &lt;a href=&quot;http://www.digizeitschriften.de/dms/toc/?PPN=PPN235181684_0021&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Ueber unendliche, lineare Punktmannichfaltigkeiten 5&lt;/em&gt;&lt;/a&gt; [On infinite, linear point sets 5], &lt;strong&gt;Mathematische Annalen&lt;/strong&gt; 21 (1883), 545-591.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Georg Ferdinand Ludwig Philip Cantor, &lt;strong&gt;Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen&lt;/strong&gt; [Foundations of a General Theory of Manifolds: A Mathematical-Philosophical Essay in the Theory of the Infinite], B. G. Teubner (Leipzig), 1883, 47 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a separate printing of Cantor's paper above (pp. 545-591) in which a half-page preface and 4 footnotes have been added (these are in addition to the endnotes of the Mathematische Annalen, which also appear here). These additions are omitted in the 1932 publication of Cantor's &amp;quot;Collected Works&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Louis [Ludovicus] Couturat, &lt;a href=&quot;https://archive.org/details/delinfinimathma02coutgoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;De L'Infini Mathématique&lt;/strong&gt;&lt;/a&gt; [On the Mathematical Infinite], Félix Alcan (Paris), 1896, xxiv + 667 + 1 (errata) pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Galileo&amp;quot; does not appear in this book according to a word search I did of the .pdf file. This work is one of two manuscripts that Couturat wrote and submitted for his Ph.D. at the Sorbonne (Paris). The other work is &lt;strong&gt;De Platonicis Mythis&lt;/strong&gt; (1896, v + 119 pages), a literary work written in Latin that deals with the writings of the Greek philosopher Plato (specifically, with trying to distinguish dogmatic passages from irony and allegory passages).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Joseph Warren Dauben, &lt;a href=&quot;http://www.jstor.org/stable/2689498&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;C. S. Peirce's philosophy of infinite sets&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Magazine&lt;/strong&gt; 50 #3 (May 1977), 123-135.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Joseph Warren Dauben, &lt;a href=&quot;http://www.jstor.org/stable/231091&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Georg Cantor: The personal matrix of his mathematics&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Isis&lt;/strong&gt; 69 #249 (December 1978), 534-550.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; William Bragg Ewald, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0198532717&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;From Kant to Hilbert: A Source Book in the Foundations of Mathematics&lt;/strong&gt;&lt;/a&gt;, two volumes, Clarendon Press, 1996, xviii + 1340 pages (both volumes).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A discussion of Cantor &lt;strong&gt;[2]&lt;/strong&gt; and &lt;strong&gt;[3]&lt;/strong&gt; is given on pp. 878-881 and an English translation (by Ewald) of Cantor &lt;strong&gt;[3]&lt;/strong&gt; is given on pp. 881-920.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from p. 889)&lt;/strong&gt; &amp;quot;If we look about in history, we find that similar opinions were often held; they are already to be found in Aristotle.&amp;quot; After discussing Aristotle, Cantor goes on to mention (pages refer to Ewald's translation; it is entirely possible that the page lists for each person named is not complete) Locke (p. 890), Descartes (p. 890), Spinoza (pp. 890-892), Leibniz (pp. 890-895), Kant (p. 892). Bolzano is mentioned on p. 895:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(top of p. 895)&lt;/strong&gt; Bolzano is perhaps the only one for whom the proper-infinite numbers are legitimate (at any rate, he speaks about them a great deal); but I absolutely do &lt;em&gt;not&lt;/em&gt; agree with the manner in which he handles them without being able to give a correct definition, and I regard, for example, §§29-33 of that book as unsupported and erroneous. The author lacks two things which are necessary for a genuine grasp of the concept of determinate-infinite numbers: both the general &lt;em&gt;concept of power&lt;/em&gt; and the precise &lt;em&gt;concept of Anzahl&lt;/em&gt;. To be sure, both appear in germ in isolated passages and as special cases. But he does &lt;em&gt;not&lt;/em&gt; work his way through to full clarity and exactness, which explains many inconsistencies and even many errors in this valuable book. &lt;em&gt;Without&lt;/em&gt; these two concepts, I am convinced, one can &lt;em&gt;not&lt;/em&gt; make further progress in the theory of manifolds. The same is true, I believe, for the fields that are a part of the theory of manifolds or that have the most intimate contact with it--for example, modern function theory on the one hand and logic and epistemology on the other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Ivor Grattan-Guinness, &lt;a href=&quot;https://eudml.org/doc/146602&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The correspondence between Georg Cantor and Philip Jourdain&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung&lt;/strong&gt; 73 #3 (20 September 1971), 111-130.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The following letter from Philip Edward Bertrand Jourdain (dated 3 January 1901) to Cantor appears on pp. 112-112.&lt;/p&gt;&#xA;&lt;p&gt;[All subsequent additions using square brackets are by Grattan-Guinness, and &amp;quot;connexion&amp;quot; and &amp;quot;emphasised&amp;quot; were in the original] Dear Sir, In some researches on the early history of Manifold-theory, I came across a paper by A. de Morgan &amp;quot;On &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; and on the sign of equality&amp;quot; [[32]] written in 1864 (quite independently of the earlier work of Bolzano), which appears to be of some importance in this connexion. de M. was an upholder of the &amp;quot;eigentlich Unendlich&amp;quot;; and showed that the substitution of &amp;quot;increase without limit&amp;quot; (&amp;quot;uneigentlich Un.&amp;quot;) is not always safe. He emphasised the existence of the notion of infinity like space and time forms in the Kantian sense, and the depend[ence] on it of the concept of finite [sic]. The most important is his argument for the conception of an inf[inite] 'multitude' and his clear distinction of 'conception' from 'image'. He often reminds one of Bolzano. He also notices, en passant, the correspondence betw[een] 2 inf[inite] manifolds but not so clearly or with such a full sense of its importance as Bolzano (§80 of Paradoxien [[2]]). His remarks on the history of inf[inity], especially on Aristotle, may, I think, interest you in connexion with your memoir in Math. Ann. Bd. XXI [[8]]. As I think you might like to see this paper, I should have great pleasure in sending you a separate copy of it on hearing from you. Yours sincerely, Philip E. B. Jourdain.&lt;/p&gt;&#xA;&lt;p&gt;After the above letter, Grattan-Guinness says: &amp;quot;Cantor replied at once, and his answer showed that he had not previously read de Morgan's paper&amp;quot;. Cantor's reply is in German, not translated. I don't have time to type it now, but I will do so on another day if anyone is interested.&lt;/p&gt;&#xA;&lt;p&gt;Part of a letter written by Cantor (29 March 1905) to Jourdain (2nd paragraph on p. 124): &amp;quot;With Mr. Weierstrass I had good relations and I possess a most interesting correspondence with him, which I will show to you. Of the conception of enumerability of which he heared from me at Berlin on Christmas holydays 1873 he became at first quite amazed, but one or two days passed over, it became his own and helped him to an unexpected development of his wonderful theory of functions.&amp;quot; [Note: I believe Cantor discovered the uncountability of the reals on 7 December 1873. In footnote 17 on p. 124, Grattan-Guinness says it is not clear which of the results of Weierstrass that Cantor was talking about.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; Ivor Grattan-Guinness, &lt;a href=&quot;https://eudml.org/doc/146637&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The rediscovery of the Cantor-Dedekind correspondence&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung&lt;/strong&gt; 76 #2-3 (30 December 1974), 104-139.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Grattan-Guinness makes the following remark about a letter by Cantor to Dedekind on p. 125: &amp;quot;The only point worth quoting is Cantor's discovery of a work by Bolzano (presumably his book [2] on the infinite), in a note of 7 October 1882.&amp;quot; Bolzano's book was published in 1851, but I believe it was mostly unknown to mathematicians until the 1870s, and even then it was not very widely known. In  footnote 18 on p. 125, Grattan-Guinness writes: &amp;quot;A year later Cantor mentioned Bolzano for the first time in his papers […]&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Edward Kasner, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183418475&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Galileo and the modern concept of infinity&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Bulletin of the American Mathematical Society&lt;/strong&gt; 11 #9 (June 1905), 499-501.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[11]&lt;/strong&gt; Cassius Jackson Keyser, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183416540&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Theorems concerning positive definitions of finite assemblage and infinite assemblage&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Bulletin of the American Mathematical Society&lt;/strong&gt; 7 #5 (February 1901), 218-226.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[12]&lt;/strong&gt; Cassius Jackson Keyser, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183417426&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Concerning the axiom of infinity and mathematical induction&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Bulletin of the American Mathematical Society&lt;/strong&gt; 9 #8 (May 1903), 424-434.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[13]&lt;/strong&gt; Cassius Jackson Keyser, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183424500&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The rôle of the concept of infinity in the work of Lucretius&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Bulletin of the American Mathematical Society&lt;/strong&gt; 24 #7 (April 1918), 321-327.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[14]&lt;/strong&gt; Gregory Harvey Moore, &lt;a href=&quot;http://www.worldcat.org/title/zermelos-axiom-of-choice-its-origins-development-and-influence/oclc/852789297&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Zermelo's Axiom of Choice: Its Origins, Development and Influence&lt;/strong&gt;&lt;/a&gt;, Studies in the History of Mathematics and Physical Sciences #8, Springer-Verlag, 1982, xiv + 410 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486488411&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reprinted&lt;/a&gt; by Dover Publications in 2013. Galileo is discussed on p. 23, along with Bolzano (discussed on pp. 23-24). There is no indication of what Cantor might have known or thought of Galileo, but Moore does say the following at the bottom of p. 23 to the top of p. 24: &amp;quot;It is uncertain to what extent Bolzano's views on the infinite influenced Cantor, who discussed the &lt;em&gt;Paradoxes of the Infinite&lt;/em&gt; only in 1883. On that occasion he praised Bolzano's book for asserting that the actual infinite exists but criticized it for failing to provide either a concept of infinite number or the concept of &amp;quot;power&amp;quot; based on equipollence. [footnote 5, omitted here] Although he may have adopted the terms &lt;em&gt;Menge&lt;/em&gt; (set) and &lt;em&gt;Vielheit&lt;/em&gt; (multitude or multiplicity) from Bolzano, Cantor's interest in the infinite had originated much earlier in his career.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[15]&lt;/strong&gt; Augustus De Morgan, &lt;em&gt;On infinity; and on the sign of equality&lt;/em&gt;, &lt;strong&gt;Transactions of the Cambridge Philosophical Society&lt;/strong&gt; 11 Part 1 (1871), 145-189.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://catalog.hathitrust.org/Record/000125921&quot; rel=&quot;noreferrer&quot;&gt;Separately published as a booklet&lt;/a&gt; by Cambridge University Press in 1865 (same title; i + 45 pages).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[16]&lt;/strong&gt; Jules Tannery, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/cb343784049/date.langEN&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;De l'infini mathématique&lt;/em&gt;&lt;/a&gt; [On mathematical infinity], &lt;strong&gt;Revue Générale des Sciences Pures et Appliquées&lt;/strong&gt; 8 (1897), 129-140.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[17]&lt;/strong&gt; Abel Étienne Louis Transon, &lt;a href=&quot;http://books.google.com/books?id=uiNTAAAAcAAJ&amp;amp;pg=PA367#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur l'emploi de l'infini en mathématiques&lt;/em&gt;&lt;/a&gt; [On the use of the infinite in mathematics], &lt;strong&gt;Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences&lt;/strong&gt; (Paris) 73 #6 (1871), 367-369.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paper associated with the session dated 7 August 1871.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[18]&lt;/strong&gt; Giulio Vivanti, &lt;a href=&quot;http://books.google.com/books?id=3cHuAAAAMAAJ&amp;amp;pg=PA9&amp;amp;f=false#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Notice historique sur la théorie des ensembles&lt;/em&gt;&lt;/a&gt; [Historical report on the theory of sets], &lt;strong&gt;Bibliotheca Mathematica&lt;/strong&gt; (2) 6 #1 (1892), 9-25.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The bibliography has 56 items.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[19]&lt;/strong&gt; Giulio Vivanti, &lt;a href=&quot;http://books.google.com/books?id=gD9VAAAAYAAJ&amp;amp;pg=PA160#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Lista bibliografica della teoria degli aggregati 1893-1899&lt;/em&gt;&lt;/a&gt; [Bibliographic list of the theory of sets 1893-1899], &lt;strong&gt;Bibliotheca Mathematica&lt;/strong&gt; (3) 1 (1900), 160-165.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The bibliography has 67 items, which are listed in chronological order.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[20]&lt;/strong&gt; William Charles Waterhouse, &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086079900296&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Gauss on infinity&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Historia Mathematica&lt;/strong&gt; 6 #4 (1979), 430-436.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from p. 435)&lt;/strong&gt; References to Gauss in connection with set theory apparently all go back to a comment published by Cantor himself. In an article of 1885 (published in 1886) he wrote: &lt;em&gt;It is just two years ago that Herr Rudolf Lipschitz in Bonn drew my attention to a certain place in the Gauss-Schumacher correspondence where the former speaks out against&lt;/em&gt; any &lt;em&gt;introduction of actual infinity into mathematics (Letter of 12 July 1831). I answered thoroughly, and&lt;/em&gt; on this point &lt;em&gt;did not accept the authority of Gauss, which I respect so highly in all other areas&lt;/em&gt; ... [Cantor 1932, 371]. This in isolation makes it seem that Cantor considered Gauss an opponent of his set theory. But as one follows his discussion it becomes clear that this disagreement is only with the words, not with Gauss' actual ideas. The crucial sentence is this: &lt;strong&gt;[omitted]&lt;/strong&gt; Thus Cantor objected not to Gauss' statement in context but to the meaning attributed to it by his own contemporaries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[21]&lt;/strong&gt; William Henry Young, &lt;a href=&quot;http://www.jstor.org/stable/3602706&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The introduction of the mathematical idea of infinity&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 4 #67 (December 1907), 147-159.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[22]&lt;/strong&gt; William Henry Young and Grace Chisholm Young, &lt;a href=&quot;https://archive.org/details/theoryofsetsofpo00youniala&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Theory of Sets of Points&lt;/strong&gt;&lt;/a&gt;, Cambridge University Press, 1906, xii + 316 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The 2nd edition was published by Chelsea Publishing Company in 1972 (xvi + 326 pages). The 2nd edition was prepared by Rosalind Cecilia Hildegard Tanner and Ivor Grattan-Guinness, and it corrects printing mistakes and simple errors in the original and it includes an appendix of supplementary notes prepared by Grace Chisholm Young. By my count the bibliography has 308 items. &amp;quot;Galileo&amp;quot; does not appear in the &lt;em&gt;Index of Proper Names&lt;/em&gt; on p. 320 of the 2nd edition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-20T19:10:38.490" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="469" PostTypeId="1" AcceptedAnswerId="470" CreationDate="2014-11-14T23:58:00.200" Score="19" ViewCount="4990" Body="&lt;p&gt;Who created topology, when did that happen, and how was it discovered?&lt;/p&gt;&#xA;" OwnerUserId="299" LastEditorUserId="4251" LastEditDate="2020-07-06T15:39:17.980" LastActivityDate="2022-09-26T18:50:45.277" Title="Who created topology, and when, and what problems led to this creation?" Tags="|mathematics|mathematicians|topology|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="470" PostTypeId="2" ParentId="469" CreationDate="2014-11-15T00:28:02.493" Score="9" Body="&lt;p&gt;Many sources, including &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Topology_in_mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;, credit the idea of topology (and its applications) to Leonhard Euler, to solve the puzzle of the Seven Bridges of Königsberg (or, rather to prove that there was no solution). Euler first used the concept in the first half of the 18th century:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler. In 1736 Euler published a paper on the solution of the Königsberg bridge problem entitled &lt;em&gt;Solutio problematis ad geometriam situs pertinentis&lt;/em&gt; which translates into English as The solution of a problem relating to the geometry of position. The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Euler went on to make subsequent developments in the fledgling field of topology. The page gives essentially a complete history of topology, from Riemann to Poincaré, which might answer any other questions you have. &lt;a href=&quot;https://web.archive.org/web/20150311081257/http://www.britannica.com/EBchecked/topic/599686/topology/258081/History-of-topology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Britannica&lt;/a&gt; confirms those facts regarding Euler.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20171223110033/http://mathforum.org/isaac/problems/bridges1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This page&lt;/a&gt; also credits Euler with creating the concept of topology, using it to solve the Seven Bridges problem.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of these areas is the topology of networks, first developed by  Leonhard Euler in 1735. His work in this field was inspired by the following problem: The Seven Bridges of Konigsberg&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.uchicago.edu/%7Echonoles/expository-notes/courses/2012/317/McCleary%20-%20A%20History%20of%20Algebraic%20Topology.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; we find an interesting PDF on the development of algebraic topology. McCleary credits Poincaré with some of the earliest work in the field, including applying topology to manifolds, although Hilbert later extended Poincaré's ideas.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="2277" LastEditDate="2022-09-22T08:29:50.527" LastActivityDate="2022-09-22T08:29:50.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="471" PostTypeId="2" ParentId="384" CreationDate="2014-11-15T01:12:27.293" Score="32" Body="&lt;p&gt;According to &lt;a href=&quot;http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning/2&quot; rel=&quot;noreferrer&quot;&gt;an American Scientist article&lt;/a&gt; (&lt;em&gt;Gauss' day of reckoning&lt;/em&gt; by Brian Hayes, Volume 94 p. 200) mentioned in the comments, the original source for this story, or at least a story very similar to it, was &lt;a href=&quot;http://en.wikipedia.org/wiki/Wolfgang_Sartorius_von_Waltershausen#Gauss_zum_Ged.C3.A4chtnis&quot; rel=&quot;noreferrer&quot;&gt;Gauss zum Gedächtnis&lt;/a&gt;, a memorial written very soon after Gauss' death by Wolfgang Sartorius, a colleague of his at Göttingen (however, I am not sure if Gauss and Sartorius personally knew each other). Sartorius claims that Gauss was fond of recounting the tale himself in his old age, and indeed specifically mentions that the problem was &quot;the summing of an arithmetic series&quot;, without mentioning which arithmetic series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the American Scientist article, the first appearance of the 1-100 summation in this anecdote is in a biography of Gauss by Ludwig Bieberbach in 1938, which simultaneously introduces the story about Gauss' rearrangement method for summing the numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it could be that Bieberbach, the person who Bieberbach learned the story from, or the person they learned the story from, or somebody else in the 80-year long chain between Sartorius and Bieberbach made up the story about the 1-100 sum, or somehow mistakenly inserted it into the anecdote. However, it could also be the case, if Gauss was indeed as fond of telling the story as Sartorius claims, that Sartorius simply wasn't aware of the 1-100 detail, or felt it wasn't important enough to include in his biography, but that the fact survived in oral tradition among other German mathematicians until Bieberbach wrote it down in 1938. It's impossible to know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it would seem very odd for a primary school teacher to set a problem involving the summing of an arithmetic series that was much stranger than &quot;the numbers from one to a hundred&quot;. The purpose of the exercise was clearly to test the students' ability at large-scale, repetitive computation. It seems like summing the numbers from 1-100 would be sufficient, and something more complicated like &quot;sum every 11th numbers from 344 to 700&quot; or whatever would seem unnecessary.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2019-10-03T06:43:36.183" LastActivityDate="2019-10-03T06:43:36.183" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="472" PostTypeId="2" ParentId="146" CreationDate="2014-11-15T01:18:38.393" Score="0" Body="&lt;p&gt;Each of the following references indicates that the &quot;Compound steam engines&quot; were fast because of the increased thermal efficiency. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, the high maintenance costs brought on by the more complex equipment tended to outweigh efficiency advantages.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://books.google.com/books?id=QxYx-iC0FFQC&amp;amp;pg=PA30&amp;amp;lpg=PA30&amp;amp;dq=why%20Compound%20engines%20were%20considered%20complex%20and%20high%20maintenance%20for%20locomotives&amp;amp;source=bl&amp;amp;ots=wvpZVNJe3s&amp;amp;sig=HvowZuH8WQnRtvcmnwoPmR-rPmg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=SaVmVPTxKKWQigK9zYHgDw&amp;amp;ved=0CDwQ6AEwBQ#v=onepage&amp;amp;q=why%20Compound%20engines%20were%20considered%20complex%20and%20high%20maintenance%20for%20locomotives&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link1&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Im4TW.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://cs.trains.com/trn/f/740/p/209944/2298889.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There's a big difference between a marine VTE engine, which operates at relatively low speeds, and a Vauclain or de Glehn compound, which are high speed designs.  Also, most of the early compound designs were withdrawn with the advent of superheating.The various D&amp;amp;H experimentals in the 1400 series were complex high-maintenance designs.  It was said that they could pull well but you had to send half of the shop force out with them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.locomotives.com.pl/Freight%20Steam%20Locomotives/Ty4.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link3&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ty4 was the most powerful locomotive in the PKP service and these engines operated mainly on the Coal Trunk Line between Upper Silesia and Gdynia on the Baltic coast. Superseded byTy246s and later also Ty51s, they disappeared from this line, but soldiered on elsewhere. Later many were used as heavy switchers, often with smaller tenders. Repairs of these engines were complex and time-consuming, as their three-cylinder layout made them untypical and suitable maintenance facilities were scarce. Availability rate was thus, at least initially, quite low.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know this doesn't seems like a complete answer but this is the gist of my research.&lt;/p&gt;&#xA;" OwnerUserId="188" LastEditorUserId="188" LastEditDate="2014-11-17T18:56:53.580" LastActivityDate="2014-11-17T18:56:53.580" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="473" PostTypeId="2" ParentId="469" CreationDate="2014-11-15T01:24:48.930" Score="16" Body="&lt;p&gt;Some topological problems were considered long ago, for example by Euler (see the previous answer).&#xA;Some ideas about topology were even earlier proposed by Leibniz.&#xA;The famous kindergarten problem about three houses and three wells belongs to this class.&#xA;One notable invention was Möbius band, another Gauss knot invariant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, one can agree with Dieudonné's opinion that topology as a separate area of mathematrics begins with Riemann. Originally it was known as Analysis Situs, the term which &#xA;existed until the beginning of 20th century. The term &quot;Analysis Situs&quot; was introduced by Leibniz(!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &lt;em&gt;topology&lt;/em&gt; was used by the Germans and since the beginning of the 20th century it is accepted. Depending on the point of view one may also consider this whole period&#xA;before Poincaré as &quot;pre-history&quot;, however some basic results about closed surfaces were proved&#xA;at that time, and Betti introduced &quot;Betti numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern subject was really defined by the work of Poincaré&#xA;in the very end of 19th century with his work called Analysis Situs in six parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References: Dieudonné, Abrege d'jistoire des mathematiques, 1700-1900, vol. 2.&#xA;Poincare, Analysis situs, J. Ecole Polyutechnique 1995, 1, 1-121.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this concerns what is called today &quot;Algebraic topology&quot;. A different part of mathematics, called &quot;General topology&quot;, has a different origin: in real analysis (Cantor).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="2442" LastEditDate="2016-02-15T15:48:56.600" LastActivityDate="2016-02-15T15:48:56.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="474" PostTypeId="1" AcceptedAnswerId="479" CreationDate="2014-11-15T01:44:27.753" Score="12" ViewCount="654" Body="&lt;p&gt;I'm having difficulty with section 6 of Lagrange's &lt;em&gt;&lt;a href=&quot;https://dl.dropboxusercontent.com/u/93726541/Lagrange%20-%20Reflexions%20sur%20la%20Resolution.pdf&quot;&gt;Réfléxions sur la résolution algébrique des équations&lt;/a&gt;&lt;/em&gt;. That's page 11 of the paper, 215 of his &lt;em&gt;Oeuvres&lt;/em&gt;. Annoyingly, this is one of the most critical points of the paper, since it's where he mentions the connection to permutations for the first time. This very sentence that I'm struggling with is essentially the birthplace of Galois theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange begins with a general equation of degree 3 having roots &lt;strong&gt;a, b&lt;/strong&gt;, and &lt;strong&gt;c&lt;/strong&gt;. He has already derived an auxiliary equation whose root &lt;strong&gt;y&lt;/strong&gt; can be used to express &lt;strong&gt;a, b&lt;/strong&gt; and &lt;strong&gt;c&lt;/strong&gt;, and now he inverts these expressions to obtain:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&lt;strong&gt;y = (a + α b + β c)/3&lt;/strong&gt; &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Where &lt;strong&gt;α&lt;/strong&gt; and &lt;strong&gt;β&lt;/strong&gt; are the primitive cube roots of &lt;strong&gt;1&lt;/strong&gt;. Firstly I don't understand why Lagrange, right before section 6, says that we can &quot;exchange &lt;strong&gt;α&lt;/strong&gt; and &lt;strong&gt;β&lt;/strong&gt;&quot; (I'm frankly not even sure what that assertion &lt;em&gt;means&lt;/em&gt;), but my bigger problem is section 6, where he now claims that (my translation)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As the auxiliary [réduite] equation does not depend immediately on the roots &lt;strong&gt;a, b, c&lt;/strong&gt; of the original equation, but only on its coefficeints &lt;strong&gt;m, n, p&lt;/strong&gt; into which the three roots enter equally [également - see below], it is clear that in the expression of &lt;strong&gt;y&lt;/strong&gt; we muts be able to exchange at will the quantities &lt;strong&gt;a, b&lt;/strong&gt; and &lt;strong&gt;c&lt;/strong&gt;, thus the quantity &lt;strong&gt;y&lt;/strong&gt; should have as many different values as can be formed by all of the permutations of &lt;strong&gt;a, b&lt;/strong&gt; and &lt;strong&gt;c&lt;/strong&gt;, which we know from the theory of combinations to be &lt;strong&gt;3 x 2 x 1&lt;/strong&gt;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What Lagrange seems to be claiming here is that if we permute the values &lt;strong&gt;a, b&lt;/strong&gt; and &lt;strong&gt;c&lt;/strong&gt; in the expression of &lt;strong&gt;y&lt;/strong&gt;, the resulting value will always be another root of the auxiliary equation. If this is what's being claimed, then I can't see how to prove it. Lagrange only mentions the fact that the auxiliary equation only depends on the coefficients of the original equation, into which the roots &lt;strong&gt;a, b, c&lt;/strong&gt; &quot;enter equally&quot;. I suspect by &quot;equally&quot;, he means &quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/Vieta%27s_formulas&quot;&gt;symmetrically&lt;/a&gt;&quot; , in which case he may be claiming that the coefficients of the auxiliary equation are symmetric functions of the values &lt;strong&gt;a, b&lt;/strong&gt; and &lt;strong&gt;c&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="131" LastEditDate="2014-11-15T10:47:13.047" LastActivityDate="2014-11-18T14:38:14.900" Title="What does Lagrange mean in this passage from Reflexions sur la résolution?" Tags="|mathematics|elementary-algebra|galois-theory|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="475" PostTypeId="1" AcceptedAnswerId="476" CreationDate="2014-11-15T12:42:13.990" Score="22" ViewCount="1475" Body="&lt;p&gt;It's often stated that Napier invented logarithms in order to simplify the task of engineers and scientists who had to crunch numbers by hand. Now, logarithms obviously do simplify calculations (as I learned myself when I turned up at a chemistry exam with no calculator - thankfully chemists give all data in logarithmic form!), in the sense that they reduce multiplication to addition. But in order to do this in practice, you need to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Look up your factors in the logarithm table.&lt;/li&gt;&#xA;&lt;li&gt;Add the factors, which could be quite large or have quite a few decmial places.&lt;/li&gt;&#xA;&lt;li&gt;Look up the result in the logarithm table.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Which I would have thought totally outweighted the time gained by not having to do multiplication, especially considering that you could make a mistake while looking something up, and that even if you don't the result is only going to be an approximation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone provide a concrete example of a task performed in the era of John Napier which was demonstrably made easier by the use of logarithms? How &lt;em&gt;much&lt;/em&gt; easier? Ideally I'd like to see a &quot;before and after&quot;, that is an example of the same task (like maybe compiling a trig table) being done with logarithms and without, and how long it took in each case.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="48" LastEditDate="2014-11-15T14:05:32.540" LastActivityDate="2014-11-16T03:08:22.033" Title="What sorts of calculations called for the invention of logarithm tables?" Tags="|mathematics|computation|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="476" PostTypeId="2" ParentId="475" CreationDate="2014-11-15T14:04:23.623" Score="19" Body="&lt;p&gt;From the &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Kepler.html&quot;&gt;MacTutor biography of Kepler&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Calculating tables, the normal business for an astronomer, always involved heavy arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms (published in 1614). However, Mästlin promptly told him first that it was unseemly for a serious mathematician to rejoice over a mere aid to calculation and second that it was unwise to trust logarithms because no-one understood how they worked.... Kepler's answer to the second objection was to publish a proof of how logarithms worked... Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that logarithms also allow the extraction of roots, as Napier noted in his &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Napier.html&quot;&gt;introduction&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you have do a series of multiplications, divisions, and/or root extractions (e.g., finding a geometric mean), you needn't find the anti-logarithms of the intermediate results, saving more time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not surprising that astronomers were among the first to embrace logarithms. First, astronomy was the most mathematical of all the sciences at that time, and astronomical calculations were quite involved and varied. However, practical problems such as surveying and navigation would also benefit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Turning to your objections: many of these calculations inherently involve approximations or data of limited accuracy, so you weren't losing anything by using logarithms (provided the tables offered sufficient precision). As for errors, note Napier's complaint about the old-fashioned methods being &quot;subject to many slippery errors&quot;. As for time, just try multiplying eight-figure numbers with and without logarithms; you will find it much faster with logs, even taking account of table lookup. (The modern analysis of algorithms tells us that addition, using the standard &quot;schoolbook&quot; method, has time complexity proportional to $n$; for multiplication, the &quot;schoolbook&quot; algorithm has time complexity proportional to $n^2$. Here $n$ is the number of places.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cs.xu.edu/~otero/math147/briggs/briggsintro.html&quot;&gt;This webpage&lt;/a&gt; goes into detail about one of the problems common to astronomy, navigation, and cartography: solving spherical triangles. Equation like $$\cos a = \cos b \cos c + \sin b \sin c \cos\alpha$$ and &#xA;$$\sin b \sin\alpha = \sin a \sin \beta$$ make frequent appearances. To quote the webpage, &quot;Preparation of ephemeral tables...demanded reams of tedious calculations like this&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem was so acute that Christoph Clavius, a leading astronomer, devised a clever method (called &lt;em&gt;prosthaphaeresis&lt;/em&gt;) for reducing multiplication to addition via trig formulas. The idea is to reverse-engineer the addition formulas for sine and cosine to express products of trig values as sums and differences of other trig values. (See the web page for details.) Prosthaphaeresis was used by Tycho Brahe, but was superceded by logarithms, which involved fewer operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Final point: it was common in those days to construct tables of compound functions, e.g., $\log(\sin(x))$. (Aside: this fact lead to the empirical discovery of the formula $\int\sec(\phi) d\phi = \log |\sec(\phi) + \tan(\phi)|$, used in Mercator's projection, before its proof via calculus. See Eli Maor's book &lt;em&gt;Trigonometric Delights&lt;/em&gt; for the full story.) Thus many uses of logs might have been indirect.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-16T03:08:22.033" LastActivityDate="2014-11-16T03:08:22.033" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="477" PostTypeId="1" AcceptedAnswerId="483" CreationDate="2014-11-15T15:31:49.300" Score="6" ViewCount="1064" Body="&lt;p&gt;Most people (including doctors) say and also newspapers states that an average person should drink 2 litres (approximately 8 glasses) of water per day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though this is very popular I have heard about this many times, I have never actually heard about the person who discovered this number and the history related to this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I don't know what these scientists exactly took into account when calculating this. I think this came from measuring how much water your body needs and the calculations said that 2 litres is the correct amount. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So therefore Im asking this over here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the scientist/group of scientists, and when and how did he find this number. &lt;/p&gt;&#xA;" OwnerUserId="257" LastEditorUserId="257" LastEditDate="2014-11-16T02:40:45.063" LastActivityDate="2014-11-16T03:32:02.253" Title="History related to the &quot;2 litres of water per day&quot; theory" Tags="|medicine|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="478" PostTypeId="1" AcceptedAnswerId="493" CreationDate="2014-11-15T16:19:13.733" Score="9" ViewCount="116" Body="&lt;p&gt;In chemistry, especially organic chemistry, molecules and reactions are usually schematically depicted by &lt;a href=&quot;http://en.wikipedia.org/wiki/Lewis_structure&quot;&gt;Lewis structures&lt;/a&gt; which provide a representation that makes it easy to grasp the structure and bonding situation of a molecule. But according to Wikipedia Lewis structures have only been introduced in 1916. What representations have chemists (and other scientists) used prior to 1916 to talk about the structural and electronic changes that molecules undergo during chemical reactions? Was there a generally accepted system or did more or less every scientist have his own style?&lt;/p&gt;&#xA;" OwnerUserId="43" LastActivityDate="2014-11-17T01:49:34.390" Title="Depiction of molecules and reactions prior to the introduction of Lewis structures?" Tags="|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="479" PostTypeId="2" ParentId="474" CreationDate="2014-11-15T18:04:07.740" Score="7" Body="&lt;p&gt;I will try to answer this both in 18th century terms, and in modern dress.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First off, the &lt;em&gt;réduite&lt;/em&gt; equation was introduced on p.208 and named on p.213; it is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y^6 + py^3 - n^3/27 = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The coefficients of this equation are $p$ and $-n^3/27$, thus depending only on the coefficients $n,p$ of the original equation $x^3+nx+p=0$. (Lagrange briefly allows an $mx^2$ term, then transforms it away, resulting in primed coefficients; let's ignore that complication.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now $n,p$ can be expressed in terms of the &lt;em&gt;roots&lt;/em&gt; of the original equation using Vieta's formulas: just expand out&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(x-a)(x-b)(x-c) = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;expressing $n,p$ as so-called elementary symmetric functions of $a,b,c$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since $n,p$ are symmetric functions of $a,b,c$, it follows that the coefficients of the &lt;em&gt;réduite&lt;/em&gt; are too. Thus,&#xA;$-n^3/27$ will be some messy expression in $a,b,c$, but clearly it will be symmetric in those roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange has shown that $y=(a + \alpha b + \beta c)/3$, so everything in sight can be written as a rational function of the roots $a,b,c$. (We regard the cube roots of unity as known constants). Imagine what we get if we substitute these functions for $y,n,p$ in the &lt;em&gt;réduite&lt;/em&gt;: a big messy equation in rational functions of $a,b,c$. If we now &lt;em&gt;permute&lt;/em&gt; $a,b,c$, we will get another such equation. But the &lt;em&gt;coefficients&lt;/em&gt; of the &lt;em&gt;réduite&lt;/em&gt; are symmetric in $a,b,c$, so they don't change. Just $y$ changes, to some new value. Six possible permutations, six values of $y$, all roots of the same equation. Thus the &lt;em&gt;réduite&lt;/em&gt; must have degree 6.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's translate this into modern terms. Lagrange works with two sorts of expressions: rational expressions in the coefficients $n,p$ (or $m,n,p$ if we don't remove the $x^2$ term), and rational expressions in the roots $a,b,c$. Nowadays we like to think set-theoretically, so let's say $E$ is the field of all rational expressions in three letters $a,b,c$. Let $F$ be the subfield generated by the elementary symmetric polynomials &#xA;$s_1=a+b+c$, $s_2=ab+ac+bc$, $s_3=abc$. Then the polynomial equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(X) = (X-a)(X-b)(X-c) = X^3 -s_1 X^2 + s_2 X - s_3 = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;has coefficients in $F$ and roots in $E$. It's not hard, with a modicum of field theory, to show that $E$ is the splitting field of $f(X)$ over $F$.  Furthermore, the automorphism group of $E$ over $F$ is just the permutation group on the letters $a,b,c$. The value $y$ has this property: applying the six automorphism to it yields six different values. A standard trick in Galois theory is to pick an element $y$ of the splitting field and form the equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$g(X) = \prod_{\sigma\in\text{Aut}(E/F)} (X-\sigma y) = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The coefficents of $g(X)$ then lie in the fixed field. That's exactly what Lagrange is doing here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a comment, the OP raises an issue touching on the differences between the modern and 18th century viewpoints. Let $g(Y)$ be the &lt;em&gt;réduite&lt;/em&gt;, where $Y$ is a variable; let $y$ be a rational expression in $a,b,c$ for which $g(y)=0$. Now we permute the roots. The coefficients of $g$ don't change, but $y$ becomes $y'$. How do we know that $g(y')=0$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern answer: permuting the roots induces an automorphism of the field $E$ over $F$, call it $\sigma$. So $\sigma g(y)=g(\sigma y)$ because $\sigma$ is an automorphism leaving all the coefficients of $g$ fixed. So $g(y')=g(\sigma y)=\sigma g(y)=\sigma 0=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange had no explicit notion of the fields $E$ or $F$, still less of an automorphism of $E$ over $F$. But his remarks about the roots entering equally into the coefficients, and being able to exchange the roots at will, show he had an intuitive grasp of the modern concepts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange probably pictured $g(y)=0$ as a big messy rational expression on the left, boiling down &lt;em&gt;formally&lt;/em&gt; to 0. Since the equation is formal, not depending on the particular values of $a,b,c$, permuting the roots will not destroy the validity of the formal identity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I recommend two articles for help with the pre-history of Galois theory (beside Cox's book): &quot;The Development of Galois Theory from Lagrange to Artin&quot;, by B. M. Kiernan (&lt;em&gt;Archive for History of Exact Sciences&lt;/em&gt;, v.8 no. 1/2 p. 40-154; p.45-55 discuss Lagrange's work), and &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CB4QFjAA&amp;amp;url=http%3A%2F%2Fwww.henrikkragh.dk%2Fpdf%2Fpart199911g.pdf&amp;amp;ei=05RnVP61N4ORyQScwID4Dw&amp;amp;usg=AFQjCNERt8E25BPvh9Vl5r150VuitnqbNg&amp;amp;sig2=BDRaFyxap4C5gWZsqAUgHA&amp;amp;bvm=bv.79142246,d.aWw&quot; rel=&quot;nofollow&quot;&gt;&quot;Niels Henrik Abel and the theory of equations&quot;&lt;/a&gt; by H. K. Sørensen. (I've also heard good things about Edwards' book &lt;em&gt;Galois Theory&lt;/em&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-18T14:38:14.900" LastActivityDate="2014-11-18T14:38:14.900" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="480" PostTypeId="1" CreationDate="2014-11-15T18:47:34.460" Score="15" ViewCount="270" Body="&lt;p&gt;The peer-review process is &lt;a href=&quot;https://hsm.stackexchange.com/a/178/29&quot;&gt;old&lt;/a&gt;. I wonder when  anonymous peer review appeared and more importantly what the motivations behind it were (vs. non-anonymous peer review).&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2020-06-01T09:32:04.637" Title="When did anonymous peer review appear and what were the motivations behind it?" Tags="|academia|publishing|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="481" PostTypeId="1" AcceptedAnswerId="484" CreationDate="2014-11-15T19:26:01.917" Score="15" ViewCount="9287" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Caloric_theory&quot; rel=&quot;nofollow&quot;&gt;caloric theory&lt;/a&gt; was replaced by the modern understanding of thermodynamics in the mid-1800's. In this theory, heat is a physical fluid that flows from one object to another; this flow causes objects to warm or cool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia notes that the caloric theory had some successful empirical predictions, and I can even somewhat understand the caloric theory as a loose description of modern thermodynamics. In my reading, I haven't really been able to find a good example of a prediction of the caloric theory that failed. Is there a phenomenon that caloric cannot account for, but the modern theory can? Or did we abandon caloric just because the modern theory is simple to understand and fits better with other mechanical theories?&lt;/p&gt;&#xA;" OwnerUserId="24" LastEditorUserId="6609" LastEditDate="2024-02-05T13:19:28.033" LastActivityDate="2024-02-05T13:19:28.033" Title="What was wrong with the caloric theory?" Tags="|chemistry|thermodynamics|obsolete-theories|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="482" PostTypeId="2" ParentId="481" CreationDate="2014-11-15T19:52:12.507" Score="4" Body="&lt;p&gt;&lt;em&gt;Hint&lt;/em&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See Thomas Kuhn, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0226458040&quot; rel=&quot;nofollow&quot;&gt;The Structure of Scientific Revolutions&lt;/a&gt; (1962 - 3rd ed), page 98 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a theory like energy conservation [...] emerged from a crisis in which an essential ingredient was the incompatibility between Newtonian dynamics and some recently formulated consequences of the caloric theory of heat. Only after the caloric theory had been rejected could energy conservation become part of science.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with reference in a footnote to :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Silvanus P.Thompson, &lt;em&gt;Life of William Thomson Baron Kelvin of Largs&lt;/em&gt;, (London, 1910), I, 266-81.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also by Thomas Kuhn, &lt;a href=&quot;http://www.jstor.org/discover/10.2307/226926?uid=2134&amp;amp;uid=2&amp;amp;uid=70&amp;amp;uid=4&amp;amp;sid=21104550345041&quot; rel=&quot;nofollow&quot;&gt;The Caloric Theory of Adiabatic Compression&lt;/a&gt; (&lt;em&gt;Isis&lt;/em&gt;, 1958).&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-11-15T19:52:12.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="483" PostTypeId="2" ParentId="477" CreationDate="2014-11-15T20:43:12.860" Score="9" Body="&lt;p&gt;This article in &lt;em&gt;Slate&lt;/em&gt;, &lt;a href=&quot;http://www.slate.com/articles/news_and_politics/explainer/2008/04/who_says_you_need_eight_glasses_a_day.html&quot;&gt;Who Says You Need Eight Glasses a Day?&lt;/a&gt;, traces the history of this idea. Quick summary: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[T]he [Slate] Explainer has uncovered evidence of the 8x8 myth going all the way back to 1796, in a German text by Dr. Christoph Wilhelm Hufeland called &lt;a href=&quot;http://books.google.com/books?id=OfAPAAAAYAAJ&amp;amp;pg=PA398&amp;amp;lpg=PA398&amp;amp;dq=dr%20hufeland%20macrobiotic%20theden&amp;amp;source=web&amp;amp;ots=0XSdFDdy0X&amp;amp;sig=BVMW4xO_hwbpagNnVyZQRx0zvfc&amp;amp;hl=en#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;Makrobiotik&lt;/a&gt;. The book includes an anecdote about the surgeon general to the king of Prussia, a vibrant 80-year-old man who had &quot;contracted the habit of drinking daily from seven to eight glasses&quot; of cold water and thus &quot;enjoyed much better health than in his youth.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The article goes on to relate the history of the hydrotherapy craze. It also mentions the most commonly cited source, the 1945 recommendations of the Food and Nutrition Board of the National Research Council (a U.S. organization), which wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A suitable allowance of water for adults is 2.5 liters daily in most instances. An ordinary standard for diverse persons is 1 millimeter for each calorie of food. Most of this quantity is contained in prepared foods.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This last sentence seems to have been ignored in the &quot;drink 8 glasses of water a day&quot; advice.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-16T03:32:02.253" LastActivityDate="2014-11-16T03:32:02.253" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="484" PostTypeId="2" ParentId="481" CreationDate="2014-11-15T21:44:20.787" Score="10" Body="&lt;p&gt;This question is intimately connected with the first law of thermodynamics, aka conservation of energy. Kuhn did a detailed study of the question in his paper &quot;Energy Conservation as an Example of Simultaneous Discovery&quot; (reprinted in &lt;em&gt;The Essential Tension&lt;/em&gt;). Also quite informative: the introduction by Mendoza to the collection &lt;em&gt;Reflections on the Motive Power of Fire, and other Papers on the Second Law of Thermodynamics&lt;/em&gt; (reprint published by Dover books). As Mendoza says in his introduction:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The terms &quot;heat&quot; and &quot;caloric&quot; must be discussed in more detail. ... The most important fact which emerges [from studying papers and textbooks, in the period 1780-1836, in France] is that the caloric theory, which implied that heat was conserved in all thermal processes, and the theory that heat was equivalent to work were &lt;em&gt;both regarded as true&lt;/em&gt; by the French physicists...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He backs this up with quotations from a joint work by Laplace and Lavoisier (1780) and a book by Lamé.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Mendoza explains, one should not unequivocally identify &lt;em&gt;caloric&lt;/em&gt; with &lt;em&gt;heat&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In fact, there are &lt;em&gt;two&lt;/em&gt; heat quantities that it is useful to define [i.e., heat energy and entropy]... Neither quantity is any more fundamental than the other. It is through a historical accident, an arbitrary choice, that we happen to call one of these quantities by the familiar term &quot;heat&quot; and the other by a pseudo-Greek name.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you study Carnot's famous essay carefully, you will find that &lt;em&gt;usually&lt;/em&gt; (not always), his assertions are accurate if you replace the term &lt;em&gt;caloric&lt;/em&gt; with &lt;em&gt;entropy&lt;/em&gt;. One place where this fails is in his discussion of the latent heat of water vapor; however, his conclusions agreed with (erroneous) experimental results by Delaroche and Bérard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 1840s, James Joules' experiments increasingly convinced scientists that energy was conserved, or in other terms, that the mechanical equivalent to heat did not depend on the method of conversion. (This is discussed in great detail in the paper by Kuhn I cited above.) As a result, Thomson (later Lord Kelvin), Clausius, and others took a fresh look at thermodynamics. (The book by Crosbie Smith, &lt;em&gt;The Science of Energy: A Cultural History of Energy Physics in Victorian Britain&lt;/em&gt; gives a thorough account.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper by Clausius &quot;On the Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat&quot; (included in the Dover collection) gives a concise treatment of the issues. He reviews work by Joules and Mayer, and also cites experiments by Regnault that corrected the erroneous results of Delaroche and Bérard. The conclusion seems inevitable that heat cannot be conserved. He then quotes an early paper by Thomson:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If we abandon this principle [conservation of heat], we meet with innumerable other difficulties ... and an entire reconstruction of the theory of heat from its foundation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Clausius responds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I believe that we should not be daunted by these difficulties, but rather should familiarize ourselves as much as possible with the consequences of the idea that heat is a motion, since it is only in this way that we can obtain the means wherewith to confirm or to disprove it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The rest of the paper reworks thermodynamics as it then stood, to be consistent with the first law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summarizing:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Carnot's famous paper, and in the immediate aftermath, heat energy and entropy were not clearly distinguished; &lt;em&gt;caloric&lt;/em&gt; had aspects of each. (By the way, neither of the terms &lt;em&gt;energy&lt;/em&gt; nor &lt;em&gt;entropy&lt;/em&gt; were in use at the time.)&lt;/li&gt;&#xA;&lt;li&gt;As the first law emerged from experimental results and new theoretical considerations, scientists did not, at first, see how to reconcile it with the body of well-established work already developed from the second law (e.g., Clapeyron's equation).&lt;/li&gt;&#xA;&lt;li&gt;Within a relatively short time (a couple of decades at most) classical thermodynamics emerged as we know it today. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Caloric was no longer viable after that: neither heat energy nor entropy is conserved, and the hallmark of caloric was its conservation.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-15T21:44:20.787" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="485" PostTypeId="1" AcceptedAnswerId="487" CreationDate="2014-11-15T23:13:53.123" Score="6" ViewCount="1840" Body="&lt;p&gt;Who invented potential and kinetic energy ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it Newton ?&#xA;Or someone else ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the impression Newton used those ideas but they already existed.&lt;/p&gt;&#xA;" OwnerUserId="313" LastEditorUserId="313" LastEditDate="2014-11-16T19:37:29.010" LastActivityDate="2014-11-16T19:37:29.010" Title="Who invented the concepts of potential and kinetic energy?" Tags="|physics|" AnswerCount="1" CommentCount="7" ClosedDate="2014-11-18T07:19:48.100" ContentLicense="CC BY-SA 3.0" />
  <row Id="486" PostTypeId="2" ParentId="481" CreationDate="2014-11-16T00:24:18.537" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;In this theory, heat is a physical fluid that flows from one object to&#xA;another; this flow causes objects to warm or cool.&lt;/p&gt;&#xA;&lt;p&gt;Is there a phenomenon that caloric cannot account for, but the modern theory can?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Heat flows from the sun to the Earth.  If heat were a physical fluid, there would need to be a medium between Earth and the sun for the fluid to pass through.  If this were the case, there would be turbulence in that medium.  We have seen no evidence of turbulence, or indeed, of any aetheric fluid.&lt;/p&gt;&#xA;&lt;p&gt;Note:  Unsourced at present.  Will update or delete within 48hr.&lt;/p&gt;&#xA;" OwnerUserId="316" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-16T03:50:55.983" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="487" PostTypeId="2" ParentId="485" CreationDate="2014-11-16T00:56:25.420" Score="5" Body="&lt;p&gt;In a way, it goes back to Aristotle and his dual concepts of &lt;a href=&quot;http://plato.stanford.edu/entries/aristotle-metaphysics/#ActPot&quot; rel=&quot;noreferrer&quot;&gt;actuality and potentiality&lt;/a&gt;. My quick description of them is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Potentiality:&lt;/strong&gt; Potentiality, or &lt;em&gt;dunamis&lt;/em&gt;, is the stored ability an object has to do something or be something.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Actuality:&lt;/strong&gt; Actuality, or &lt;em&gt;entelecheia/energeia&lt;/em&gt;, is what an object is doing or is at a point in time.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now, these had broader definitions than the ones we use to day for potential and kinetic energy. For example, according to Aristotle, the potentiality of an object included all the forms it could take (e.g. clay could become a sculpture).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are clearly not the same things as today's notions of kinetic and potential energy, but they were the precursors to our modern ideas.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-16T00:56:25.420" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="488" PostTypeId="2" ParentId="340" CreationDate="2014-11-16T02:20:59.833" Score="3" Body="&lt;p&gt;&lt;strong&gt;Not. At. All.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The plans for the future of the site of the SSC resembled the collider itself in that they were nonexistent. When Congress refused the extend the budget for the project, no alternative ideas for the site went anywhere.&lt;/p&gt;&#xA;&lt;p&gt;From the &lt;a href=&quot;http://news.google.com/newspapers?id=kGAgAAAAIBAJ&amp;amp;sjid=umUFAAAAIBAJ&amp;amp;dq=cost%20overrun%20ssc&amp;amp;pg=3808%2C4981568&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sun-Journal&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Ellis County, Texas, where the collider was being built, owners of condemned houses and others who have invested in businesses and land were wondering what would fill the vacuum left by tunneling the atom smasher.&#xA;&amp;quot;Right now, it's a billion-dollar hole in the ground. And they're arguing about whether to fill it back up,&amp;quot; said Allan Oakley, a Waxahachie police officer and co-owner of the Kountry Café in nearby Maypearl.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From an article &lt;a href=&quot;http://www.bizjournals.com/dallas/stories/2006/08/21/newscolumn6.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; from 2006:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;GVA Cawley has been tapped to market the former Superconducting Super Collider property in Waxahachie. The 134-acre site, which includes 200,000 square feet of space, has been vacant since the super collider project was canceled in 1993.&lt;/p&gt;&#xA;&lt;p&gt;It recently was acquired by Collider Data Center L.L.C., a Dallas-based investment group led by J.B. Hunt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So evidently no plans were made for about 13 years.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.atlasobscura.com/places/superconducting-super-collider&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; suggests that no plans really got anywhere, until a chemical company bought up the site:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Plans for the derelict site have included everything from mushroom farms to data storage in the years since its colossal failure, but despite local protests, the land was purchased by a chemical company who will hopefully find a good use for one of America’s most ambitious scientific missteps.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.wfaa.com/story/local/2014/11/15/13833478/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; article names that company as Magnablend.&lt;/p&gt;&#xA;&lt;p&gt;Finally, &lt;a href=&quot;http://sometimes-interesting.com/2012/01/31/worlds-largest-super-collider-abandoned/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; indicates that the property was given to the county to do what they wished with it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After project cancellation, the site was given to Ellis County Texas. Numerous attempts to sell the property failed until 2006, when a private investment group finally purchased the property. It was rumored there were plans to use the SSC as a tier III or IV data center, but in 2011 the property still sat derelict and abandoned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the answer appears to be that no plans were made. &lt;a href=&quot;http://www2.lbl.gov/Science-Articles/Archive/ssc-and-future.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This report&lt;/a&gt;, though, is illuminating. And a little bit hopeful.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-16T02:20:59.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="489" PostTypeId="2" ParentId="460" CreationDate="2014-11-16T03:23:14.260" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Long before people had knowledge about electric shock from electric ray(2750 B.C.).&lt;/p&gt;&#xA;&lt;p&gt;Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers. Thales of Miletus made a series of observations on static electricity around 600 BC, from which he believed that friction rendered amber magnetic, in contrast to minerals such as magnetite, which needed no rubbing.Thales was incorrect in believing the attraction was due to a magnetic effect, but later science would prove a link between magnetism and electricity.&lt;/p&gt;&#xA;&lt;p&gt;Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber. He coined the New Latin word electricus (&amp;quot;of amber&amp;quot; or &amp;quot;like amber&amp;quot;, from ἤλεκτρον, elektron, the Greek word for &amp;quot;amber&amp;quot;) to refer to the property of attracting small objects after being rubbed. This association gave rise to the English words &amp;quot;electric&amp;quot; and &amp;quot;electricity&amp;quot;, which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;http://en.wikipedia.org/wiki/Electricity#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="247" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-16T03:23:14.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="490" PostTypeId="2" ParentId="146" CreationDate="2014-11-16T04:01:59.963" Score="0" Body="&lt;p&gt;After bit research I got following things:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were many efforts done in order to make compound steam engines by U.S., European countries, Russia, etc in 19th and 20th century. This engines were known as Mallets. Last try was done by Russia in 1954-55. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But By the mid-1950s, with economic recovery from the Second World War, production of diesel locomotives had begun in many countries and the diesel locomotive was on its way to becoming the dominant type of locomotive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also fastest steam engine was not having compound engine which was made in year 1938, thats proves compound engines are not as fast as normal steam engines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So after reading a bit I come to following conclusion:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Due to invention of diesel engines, development of compound steam engines did not take place.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Normal steam engines with aerodynamics model were faster than compound engine.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Due to complexity of compound engine it was difficult and expensive in maintenance point of view.        &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Sources: Wikipedia - &lt;a href=&quot;http://en.wikipedia.org/wiki/Mallet_locomotive&quot; rel=&quot;nofollow&quot;&gt;link1&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/LNER_Class_A4_4468_Mallard&quot; rel=&quot;nofollow&quot;&gt;link2&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Diesel_locomotive&quot; rel=&quot;nofollow&quot;&gt;link3&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="247" LastActivityDate="2014-11-16T04:01:59.963" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="493" PostTypeId="2" ParentId="478" CreationDate="2014-11-17T01:49:34.390" Score="2" Body="&lt;p&gt;I can't leave a comment since I don't have enough rep but I will try to make this an answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that Robert Robinson was the first to use arrows in electron flow pathways in mechanisms and the convention used for radicals and charges. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen some early drawings of phosphorous-oxygen molecules from around the middle 1860's and they looked pretty much like what we use today except there might of been a different symbol used (but the idea of stoichiometry was there). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This wiki page looks like it might answer your question. Particularly the  drawings by Marc Antoine Auguste Gaudin. Before this time you are looking at Dalton's works which seem to be the earliest semblance of modern chemistry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_molecular_theory&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/History_of_molecular_theory&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="330" LastActivityDate="2014-11-17T01:49:34.390" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="494" PostTypeId="2" ParentId="146" CreationDate="2014-11-17T03:03:15.477" Score="3" Body="&lt;p&gt;This question was originally posted to test the limits of what is on and off topic. I was going to eventually post an answer, but I see we have two answers. In my opinion, both answers dance around the subject a little, so I am adding my own answer.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This kind of question is always going to be partially subjective. Locomotive development tended to occur in independent strands (companies, countries) with would often confer with each other. Hence the importance of each reason will vary between different countries and companies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will try to illustrate my reasons with examples, but they are only examples and there will be a bias towards the UK where I have greater knowledge.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Costs&lt;/strong&gt;. Greater efficiency means you can get more power for a set amount of coal or require less coal for a set amount of power. This is important, even in a coal-rich country like the UK. As such, a working compound system should be an easy 'choice'. This would explain the choice in some later (1920s-30s) locomotive developments such as Chapelon in France (&lt;em&gt;Gresley Observer vol. 156&lt;/em&gt;), and US Mallets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However there are multiple costs. In the case of the UK, maintenance costs often  discouraged otherwise 'obvious' money saving issues - eg. feed water heaters (eg. &lt;a href=&quot;http://www.lner.info/locos/B/b12.shtml&quot; rel=&quot;nofollow&quot;&gt;LNER B12&lt;/a&gt;) and boosters (cf. the LNER's experience with the &lt;a href=&quot;http://www.lner.info/locos/S/s1.shtml&quot; rel=&quot;nofollow&quot;&gt;S1&lt;/a&gt;, P1). These developments often had a history of being fitted and then removed a few years later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other answers have hinted at complexity. The reality is that it is possible to squeeze four cylinders into the restricted loading gauge of the UK (US and Continental Europe have more space). However compromises have to be made with steam passages and layouts. This adds inefficiencies and maintenance costs. These can be avoided with the three cylinder layout that was popular in the UK (eg. Stanier Duchess, and Gresley A3 &amp;amp; A4). Gresley's A4 has already been mentioned (one holds the world speed record). The internal streamlining is considered almost as important as the external (as determined by Chapelon and communicated to Gresley), and a complex layout such as four cylinders or compounding would not have allowed for this streamlining.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Effectiveness&lt;/strong&gt;. This basically comes down to good design and train crews who know how to use compounding effectively. Early attempts at compounding in the UK at least, were imperfectly designed. UK companies tended to give limited training to the loco crews, whilst France (especially under Chapelon) implemented a high degree of education - hence they had much more success in the day-to-day use of compounding (&lt;em&gt;Gresley Observer vol. 156&lt;/em&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Culture&lt;/strong&gt;. This is harder to quantify, but it does seem that UK companies were largely poisoned to the idea after the initial failures. Although the North Eastern Railway &#xA;finally managed to get a system developed by &lt;a href=&quot;http://www.lner.info/eng/smith.shtml&quot; rel=&quot;nofollow&quot;&gt;Walter Smith&lt;/a&gt; to work effectively, it never developed them further after his death. This might have been due to patent costs or management decision to move away from compounding. The Midland Railway did make extensive use of this, but only for a mid-power 4-4-0 (of which &lt;a href=&quot;http://www.nrm.org.uk/OurCollection/LocomotivesAndRollingStock/CollectionItem.aspx?objid=1975-7018&quot; rel=&quot;nofollow&quot;&gt;No. 1000 survives in the National Collection&lt;/a&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Requirements&lt;/strong&gt;. The later use of compounding tended to be only when other factors made it more effective. For example, the US usage of compounds in Mallet locomotives. These were usually large freight-hauling locomotives with articulation and higher pressure boilers. Each power truck would have two cylinders. The pressure and space allowed for effective compounding.&#xA;Similarly, Gresley used compounding on his &lt;a href=&quot;http://www.lner.info/locos/W/w1.shtml&quot; rel=&quot;nofollow&quot;&gt;W1 'Hush-Hush'&lt;/a&gt; experimental locomotive. This used an experimental 'water tube boiler' operating at the very high pressure of 400psi. To make effective use of this high pressure, Gresley implemented a compound system with two high pressure and two low pressure cylinders. The W1 never entered production, and it was the only example of a water tube boiler locomotive in the UK.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="25" LastActivityDate="2014-11-17T03:03:15.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="495" PostTypeId="2" ParentId="146" CreationDate="2014-11-17T04:46:02.873" Score="0" Body="&lt;p&gt;Following the (mostly) very interesting links posted in other answers,&#xA;I noticed that according to the Wikipedia page on compound locomotives,&#xA;the use of compound engines on &lt;a href=&quot;http://en.wikipedia.org/wiki/Compound_locomotive#Road_locomotives&quot; rel=&quot;nofollow&quot;&gt;road locomotives&lt;/a&gt; was actually very common.&#xA;This view seems to be borne out by other sources,&#xA;such as &lt;a href=&quot;http://vintagemachinery.org/mfgindex/imagedetail.aspx?id=4014&quot; rel=&quot;nofollow&quot;&gt;this example&lt;/a&gt; and &lt;a href=&quot;http://vintagemachinery.org/mfgindex/imagedetail.aspx?id=4009&quot; rel=&quot;nofollow&quot;&gt;these examples&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was apparently also very common for road locomotives to use one crank per&#xA;cylinder, connected to a crankshaft.&#xA;I believe that was also the case for triple-expansion marine engines.&#xA;On the other hand I believe crankshafts were very rare on steam-powered rail locomotives;&#xA;almost all pictures clearly show cranks attached to the outside of the driving wheels.&#xA;The mechanisms to transfer power and to control the flow of steam would&#xA;have had to be quite different for such engines than for engines with crankshafts.&#xA;(For starters, it seems to have generally involved either multiple cylinders&#xA;driving a single crank, or cylinders placed very far apart, presenting difficulty&#xA;in delivering good quality steam from one cylinder to the other.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some builders succeeded in producing a number of compound rail locomotives&#xA;nevertheless (an interesting account of this is &lt;a href=&quot;http://books.google.com/books?id=dAM-AQAAMAAJ&amp;amp;pg=PA256&amp;amp;lpg=PA256&amp;amp;dq=road+locomotives+compound+engine&amp;amp;source=bl&amp;amp;ots=gc6S04_odY&amp;amp;sig=6pQIV7zPoHqdMDtqrEbTYAOU5Fw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=aXNpVJnfH4q9yQTOmoLwAQ&amp;amp;ved=0CFYQ6AEwCQ#v=onepage&amp;amp;q=road%20locomotives%20compound%20engine&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, continued &lt;a href=&quot;http://books.google.com/books?id=dAM-AQAAMAAJ&amp;amp;pg=PA305&amp;amp;lpg=PA305&amp;amp;dq=road+locomotives+compound+engine&amp;amp;source=bl&amp;amp;ots=gc6S04_odY&amp;amp;sig=6pQIV7zPoHqdMDtqrEbTYAOU5Fw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=aXNpVJnfH4q9yQTOmoLwAQ&amp;amp;ved=0CFYQ6AEwCQ#v=onepage&amp;amp;q=road%20locomotives%20compound%20engine&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;,&#xA;including a panel discussion featuring Samuel M. Vauclain),&#xA;and there were even some compound road locomotives with single cranks,&#xA;but when you consider the investment in new technology that would be required,&#xA;perhaps it is not too surprising that few builders tried to develop it&#xA;and even fewer succeeded before diesel took over.&lt;/p&gt;&#xA;" OwnerUserId="321" LastActivityDate="2014-11-17T04:46:02.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="496" PostTypeId="2" ParentId="419" CreationDate="2014-11-17T10:34:19.043" Score="7" Body="&lt;p&gt;According to the legend, at some time in VI century BC Pythagoras visited a smith shop&#xA;and discovered that musical intervals correspond to simple ratios of small numbers.&#xA;This discovery impressed Pythagoras so much that he concluded that &quot;universe is ruled by numbers&quot;. On my opinion, this can be considered the &quot;birth of exact sciences&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, this legend comes from the authors who lived many centuries after&#xA;Pythagoras and their accounts of what really happened in the smith shop are incorrect.&#xA;(They did not really understand what they were talking about).&#xA;Nevertheless we know that this discovery was made at some time very long ago, and this is reflected in the legend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is certainly a use of numbers &quot;different from counting&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure that measuring&#xA;length belongs to this category. At some later time, in V BC century, followers of&#xA;Pythagoras made a fundamental and disturbing discovery that you cannot measure length with&#xA;numbers. (In modern language: that the diagonal of a square is not commensurable with the side). This discovery led to a crisis not only in mathematics but in the whole worldview&#xA;that &quot;numbers rule the world&quot;. The crisis was resolved at the time of Euclid&#xA;(III BC) in whose book we read &quot;the theory of proportions&quot; which is equivalent to the modern&#xA;theory of real numbers, but exposed in purely geometric language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theory of proportions was used until 19 century.&#xA;Notice for example, that Newton (and many physics books) do not state the gravity law&#xA;or the second law of Newton or Hooke's law as a relation  between numbers. &#xA;They do not say that force equals to the product of such and such things divided&#xA;by such and such things. They only say that something is &quot;proportional&quot; to something. This is a remainder of that epoch which still exists in some physics books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern rigorous theories of real (and complex and other) numbers was created in the second&#xA;half of 19-th century. They are certainly not designed for counting but for measuring.&#xA;Certainly people used numbers for measuring in practice since the times immemorial, without any theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Pythagorean_hammers&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Pythagorean_hammers&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-18T20:45:40.600" LastActivityDate="2017-05-18T20:45:40.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="499" PostTypeId="1" CreationDate="2014-11-17T17:25:29.047" Score="12" ViewCount="1204" Body="&lt;p&gt;Nowadays we can easily prove the following fact using polynomial long division:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $a$ is a root of the polynomial $f$, then there exists a polynomial $g$ such that $f(x) = (x - a)g(x)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I can't imagine how to prove this without polynomial long division. I have two questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who invented polynomial long division, and when?&lt;/li&gt;&#xA;&lt;li&gt;Was the above theorem known before the discovery of polynomial long division? If so, how was it proven?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-04-01T01:47:57.230" Title="Was the &quot;polynomial remainder theorem&quot; known before polynomial long division was discovered?" Tags="|mathematics|elementary-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="500" PostTypeId="1" CreationDate="2014-11-17T19:33:41.540" Score="7" ViewCount="112" Body="&lt;p&gt;Here's an easy lemma:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Any poset $(S, \preceq)$ is order-isomorphic to a subset of the powerset $\mathcal{P}(S)$ ordered by set-inclusion.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I seem to recall having seen this attributed to Dedekind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I right that Dedekind proves this little result? And if so, where does he do it?&lt;/p&gt;&#xA;" OwnerUserId="343" LastEditorUserId="26" LastEditDate="2014-11-17T21:25:05.097" LastActivityDate="2014-11-18T16:46:06.917" Title="Did Dedekind prove this lemma about posets (or an equivalent)?" Tags="|mathematics|set-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="501" PostTypeId="1" AcceptedAnswerId="593" CreationDate="2014-11-17T22:31:14.753" Score="23" ViewCount="1358" Body="&lt;p&gt;For example, do we find something related to the modern energy concept in Ancient China, Ancient India, or the Islamic Golden Age?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among &quot;similarities and differences&quot;, &lt;em&gt;conservation&lt;/em&gt; is obviously important, as is &lt;em&gt;convertibility&lt;/em&gt; and &lt;em&gt;variety of forms&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the word &lt;em&gt;energy&lt;/em&gt; or a close cognate was &lt;em&gt;not&lt;/em&gt; used, what is the justification for regarding it as a related concept?&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-17T22:41:49.750" LastActivityDate="2014-12-06T11:39:13.383" Title="Did ancient/medieval non-European cultures have a concept of energy? If so, what are the similarities and differences to the modern concept?" Tags="|physics|ancient-china|terminology|ancient-india|islamic-science|" AnswerCount="4" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="502" PostTypeId="5" CreationDate="2014-11-17T22:37:53.220" Score="0" Body="&lt;p&gt;Science from the so-called &lt;a href=&quot;http://en.wikipedia.org/wiki/Islamic_Golden_Age&quot; rel=&quot;nofollow&quot;&gt;Islamic Golden Age&lt;/a&gt;, roughly the 8th through the 12th century, and including scholars such as Al-Kwarizmi, Ibn al-Haytham (Alhazan), and Jābir ibn Hayyān.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-12-08T21:16:53.670" LastActivityDate="2014-12-08T21:16:53.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="503" PostTypeId="4" CreationDate="2014-11-17T22:37:53.220" Score="0" Body="Science from the so-called Islamic Golden Age, roughly the 8th through the 12th century, and including scholars such as Al-Kwarizmi, Ibn al-Haytham (Alhazan), and Jābir ibn Hayyān." OwnerUserId="48" LastEditorUserId="2442" LastEditDate="2015-09-01T22:33:20.113" LastActivityDate="2015-09-01T22:33:20.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="505" PostTypeId="1" AcceptedAnswerId="531" CreationDate="2014-11-18T01:29:17.457" Score="18" ViewCount="577" Body="&lt;p&gt;Archimedes constructed a planetarium where as described by Cicero &quot;he had thought out a way to represent accurately by a single device for turning the globe those various and divergent movements with their different rates of speed. And when Gallus moved the globe, it was actually true that the moon was always as many revolutions behind the sun on the bronze contrivance as would agree with the number of days it was behind in the sky. Thus the same eclipse of the sun happened on the globe as would actually happen...&quot; (see &lt;a href=&quot;https://www.math.nyu.edu/~crorres/Archimedes/Sphere/SphereSources.html&quot;&gt;sources&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the Antikythera mechanism was found some &lt;a href=&quot;http://www.giovannipastore.it/ARCHIMEDES1.htm&quot;&gt;suggested&lt;/a&gt; that similar gear system was used by Archimedes. However, the  Antikythera mechanism (dated to 150-100 BC) has an epicyclic gear for the lunar motion, so under the conventional timeline there are two problems with this idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first problem is Apollonius. In Almagest XII,1 Ptolemy credits him with proving theorems about epicycles, and this seems to be the earliest association of epicycles with anybody's name. Some biographers &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Apollonius.html&quot;&gt;say&lt;/a&gt; that &quot;the theory of epicycles certainly predates Apollonius&quot;, but I am not sure where this &quot;certainly&quot; comes from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second problem is Hipparchus. Even Apollonius is only credited with theorems, Hipparchus (again according to Ptolemy) was the first one to apply epicycles to lunar motion, and he did it based on Babylonian data, which Archimedes presumably didn't have a century earlier. By the way, Hipparchus lived around the time the Antikythera was made. However, a gear similar to Antikythera's, but dated to before Hipparchus (200-150 BC), was &lt;a href=&quot;http://bmcr.brynmawr.edu/2012/2012-01-14.html&quot;&gt;found&lt;/a&gt; in a shipwreck that can be linked directly to Syracuse, and hence to Archimedes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if Archimedes did not use epicycles the only model available to him would have been of Eudoxian homocentric spheres. According to &lt;a href=&quot;http://www.jstor.org/stable/41134047&quot;&gt;modern reconstructions&lt;/a&gt; (p.226) reproducing the uneven motion of the Moon even crudely (and predicting eclipses mentioned by Cicero) would require at least three spheres rotating at different speeds under different inclinations, and with outer spheres transferring motion to inner spheres to combine it with their internal rotations. Even more spheres are required to reproduce retrogressions of the planets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy is not exactly a great authority on historical matters (he managed to mention no astronomer between himself and Hipparchus, a span of over 200 years), and his attributions to Apollonius and Hipparchus were specifically &lt;a href=&quot;http://www.pitt.edu/~brg/pdfs/brg_v_3.pdf&quot;&gt;questioned&lt;/a&gt; by some scholars. Life dates of Archimedes (287-212 BC) and Apollonius (262-190 BC) overlap, and we know that both corresponded with Alexandrians, at least two of which, Eratothenes and Aristarchus, performed astronomical measurements (Archimedes wrote to Eratothenes and mentioned Aristarchus in Sand Reckoner). Finally, some arrangements of Eudoxian spheres are &lt;a href=&quot;http://www.jstor.org/stable/41134047&quot;&gt;suggestive&lt;/a&gt; (p. 227) of the simpler epicyclic scheme.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it plausible in the light of recent developments that Archimedes/Alexandrians already developed epicyclic models of the Moon (and possibly planets) a century before Hipparchus? Alternatively, is there a mechanical arrangement of multiple nested spheres with motion transfer combined with internal rotations, that uses gears and can be powered from a single rotational driving source (within Archimedes's capabilities)? Modern reconstructions of the planetarium that I've seen do not seem to be up to the task.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2020-05-13T18:58:24.957" LastActivityDate="2020-05-13T18:58:24.957" Title="Did Archimedes use epicycles in his planetarium?" Tags="|biographical-details|astronomy|ancient-greece|celestial-mechanics|archimedes|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="506" PostTypeId="2" ParentId="31" CreationDate="2014-11-18T02:27:27.887" Score="10" Body="&lt;p&gt;With this kind of &quot;why?&quot; question, one can only speculate. However, it's interesting that Feynman had a similar experience, as recounted in &lt;a href=&quot;http://www.ee.ryerson.ca/~elf/abacus/feynman.html&quot; rel=&quot;noreferrer&quot;&gt;Surely, You're Joking, Mr. Feynman!&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman was sitting in a restaurant, when&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could add numbers faster than any of them could do.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The waiters didn't want to lose face, so they said, &quot;Yeah, yeah. Why don't you go over and challenge the customer over there?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Feynman of course was the customer. He accepted the challenge. Feynman lost with addition and subtraction, even when both contestants were given the numbers written down. But Feynman almost matched him with multiplication, tied with division, and mopped the floor for cube root problems. Feynman remarked, &quot;What he didn't realize was, the harder the problem, the better chance I had.&quot; Feynman's explanation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I realized something: he doesn't &lt;em&gt;know&lt;/em&gt; numbers. With the abacus, you don't have to memorize a lot of arithmetic combinations; all you have to do is to learn to push the little beads up and down. You don't have to memorize 9+7=16; you just know that when you add 9, you push a ten's bead up and pull a one's bead down. So we're slower at basic arithmetic, but we know numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;Wired&lt;/em&gt; article you link to says something similar:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ultimately, the soroban is not as much about mental dexterity as it is about mechanical skill, says soroban user Yannic Piché.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;OK, now I will speculate in force. Compare addition with the abacus to drumming or playing the piano. The activity demands little of memory, or rather, it's mostly muscle memory. As with any physical skill, endless practice turns mental operations into reflex actions. (With drumming and piano playing, this frees the mind to concentrate on expressiveness and musicality.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lots of mental arithmetic tires the brain, far more (I'd wager) than pushing beads up and down.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-11-18T13:33:11.540" LastActivityDate="2014-11-18T13:33:11.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="507" PostTypeId="5" CreationDate="2014-11-18T02:49:05.143" Score="0" Body="&lt;p&gt;Questions pertaining to the &lt;a href=&quot;http://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen&quot; rel=&quot;nofollow noreferrer&quot;&gt;University of Göttingen&lt;/a&gt;, its alumni, or professors, are on topic.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="4251" LastEditDate="2021-03-09T09:53:16.437" LastActivityDate="2021-03-09T09:53:16.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="508" PostTypeId="4" CreationDate="2014-11-18T02:49:05.143" Score="0" Body="Göttingen is a university in Germany. Prior to WW2, Göttingen was a mathematical and scientific powerhouse." OwnerUserId="140" LastEditorUserId="4251" LastEditDate="2021-03-21T18:17:48.067" LastActivityDate="2021-03-21T18:17:48.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="509" PostTypeId="2" ParentId="131" CreationDate="2014-11-18T08:17:31.867" Score="3" Body="&lt;p&gt;A main conclusion presented in the article &lt;a href=&quot;http://aerosociety.com/Assets/Docs/Publications/The%20Journal%20of%20Aeronautical%20History/2011-06Cayley-Ackroyd2.pdf&quot; rel=&quot;nofollow&quot;&gt;Sir George Cayley: The Invention of the Aeroplane near &#xA;Scarborough at the Time of Trafalgar &lt;/a&gt; (Ackroyd, 2011) was that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although certain of his arguments have been shown to be over-simplistic, even &#xA;  on occasions faulty, he nonetheless arrived at many of the basic aerodynamic, structural &#xA;  and stability features of the successful aeroplane, tested them, and showed that they &#xA;  worked&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and interestingly, Ackroyd notes that if people had paid more attention and if there was a wider adoption and expansion of his work, then many of the problems may have been overcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like the power problem, some aspects of stability/control were beyond his time also.  In particular, from the article&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;subject of centre of pressure movement was a further topic to retain its “dark nature” &#xA;  for a considerable period.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to the webpage &lt;a href=&quot;http://aviation-history.com/early/cayley.htm&quot; rel=&quot;nofollow&quot;&gt;Sir George Cayley, The Father of Aviation&lt;/a&gt; (Dwyer, 2009), it seems that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He found that setting the wings at a slight dihedral gave the glider lateral stability and that a tail plane set behind the main wings gave longitudinal stability.3 This was the discovery of Inherent Stability that, although demonstrated but not fully understood by Cayley, would be theorized in greater detail by &lt;a href=&quot;http://aviation-history.com/early/penaud.htm&quot; rel=&quot;nofollow&quot;&gt;Alphonse Pénaud&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A vital aspect that the Wright brothers worked out in 1901.  Another aspect that eluded Cayley was that he&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;failed to grasp the stabilising function of the tail’s &#xA;  fixed surfaces and is rather limited in his understanding of the functions of the tail’s &#xA;  movable surfaces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is something that he realised and applied a bit later in his life, when he successfully tested a flight of the 'Boy Carrier' in 1849 up to the Coachman Carrier around 1853.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue is not so much that Cayley did not make a successful model of a plane (which he did in the form of gliders), he was not able to, to quote Dwyer (2009), achieve&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cayley never achieved his final goal of &lt;strong&gt;sustained&lt;/strong&gt; heavier-than air, manned flight, but his contributions clearly furthered advancement of the modern airplane.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user22" LastEditorUserId="25" LastEditDate="2014-11-18T13:50:50.537" LastActivityDate="2014-11-18T13:50:50.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="510" PostTypeId="1" CreationDate="2014-11-18T09:38:16.567" Score="14" ViewCount="1476" Body="&lt;p&gt;Here are five distinct uses of the word spectrum in physics and mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Spectrum (optics): The range of colors in the rainbow &lt;/li&gt;&#xA;&lt;li&gt;Spectrum (particle physics): The range of electromagnetic frequencies emitted when an electron in an atom moves from a high energy state to a low energy state&lt;/li&gt;&#xA;&lt;li&gt;Spectrum (functional analysis): the set of generalized eigenvalues of a linear map&lt;/li&gt;&#xA;&lt;li&gt;Spectrum (algebra): the set of all prime ideals in a ring equipped with the Zariski topology&lt;/li&gt;&#xA;&lt;li&gt;Spectrum (topology): a sequence $X_n$ of topological spaces equipped with inclusion maps from the suspension of $X_n$ into $X_{n+1}$&lt;/li&gt;&#xA;&lt;li&gt;(Added later): Spectral Sequence (topology): a tool for calculating (co)homology, generalizing exact sequences&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My assumption is that these uses of the word spectrum are all related, and I would like to tell a story about how and why the terminology was adapted to different contexts.  Here is my speculation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 was introduced by Newton in the course of his study of optics.  2 was simply adapted from Newton's use of the word as physicists began to understand the structure of the atom.  3 arose in quantum physics (maybe due to Hilbert or von Neumann?) when it was realized that lines in atomic spectra correspond to eigenvalues of certain linear operators.  4 emerged from the theory of Banach algebras wherein it was realized that if $T$ is a linear operator on Hilbert space which commutes with its adjoint then the maximal ideal space of the commutative C*-algebra generated by $T$ coincides with the spectrum of $T$ in the sense of 3.  I have no idea where 5 or 6 comes from (or if they are related).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone fill in the details of this historical narrative and correct it as necessary?  In particular, does anybody know how the word spectrum came to be used in algebraic topology?&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="35" LastEditDate="2014-11-19T03:58:21.437" LastActivityDate="2023-04-17T12:56:48.297" Title="What is the story behind various uses of the word &quot;spectrum&quot;?" Tags="|mathematics|physics|terminology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="511" PostTypeId="2" ParentId="510" CreationDate="2014-11-18T13:29:40.683" Score="10" Body="&lt;p&gt;The connection between (2) and (3) is one of the surprising curiosities in the history of mathematics and physics.&lt;/p&gt;&#xA;&lt;p&gt;Hilbert worked on integral equations mostly from 1903 to 1910. Here's what Constance Reid has to say about this in her biography:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Courant-Hilbert book on mathematical methods of physics, which had appeared at the end of 1924, before both Heisenberg's and Schrödinger's work, instead of being outdated by the new discoveries, seemed to have been written expressly for the physicists who now had to deal with them....&lt;/p&gt;&#xA;&lt;p&gt;To Hilbert himeself this was yet another example of that pre-established harmony which seemed to him almost the embodiment and realization of mathematical thought.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I developed my theory of infinitely many variables from purely mathematical interests&amp;quot;, he marvelled, &amp;quot;and even called it 'spectral analysis' without any presentiment that it would later find an application to the actual spectrum of physics!&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;http://jeff560.tripod.com/s.html&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt; cited by Jack M in a comment suggests a link between (1) and (3):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There may be a link between Newton and Hilbert for, though the latter cited no previous writer for &amp;quot;Spektrum&amp;quot;, J. Dieudonné History of Functional Analysis (1981, pp. 149-50) suggests he derived the term from W. Wirtinger &amp;quot;Beiträge zu Riemann’s Integrationsmethode für hyperbolische Differentialgleichungen, und deren Anwendungen auf Schwingungsprobleme&amp;quot;, [Contributions to Riemann's integration methods for hyperbolic differential equations, and their application to oscillation problems] Mathematische Annalen, 48, (1897), 365-89. Wirtinger drew upon the similarity with the optical spectra of molecules when he used the term &amp;quot;Bandenspectrum&amp;quot; with reference to Hill’s (differential) equation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-18T13:29:40.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="512" PostTypeId="2" ParentId="500" CreationDate="2014-11-18T13:30:17.823" Score="3" Body="&lt;p&gt;I think that the source is Richard Dedekind's &lt;a href=&quot;https://archive.org/details/wassindundwasso00dedegoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Was sind und was sollen die Zahlen ?&lt;/a&gt; (Vieweg: Braunschweig, 1888).&lt;/p&gt;&#xA;&lt;p&gt;I quote from the English translation : &lt;a href=&quot;http://www.gutenberg.org/files/21016/21016-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;THE NATURE AND MEANING OF NUMBERS&lt;/a&gt;, (The Open Court Publishing Co., 1901) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;98. Definition&lt;/strong&gt; [page 37]. If &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is any number, then will we denote by &lt;span class=&quot;math-container&quot;&gt;$Z_n$&lt;/span&gt; the system [set] of all numbers that are not greater than &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; [...].&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;106. Theorem&lt;/strong&gt; [page 38]. If &lt;span class=&quot;math-container&quot;&gt;$m &amp;lt; n$&lt;/span&gt;, then is  &lt;span class=&quot;math-container&quot;&gt;$Z_m$&lt;/span&gt; proper part of  &lt;span class=&quot;math-container&quot;&gt;$Z_n$&lt;/span&gt; and conversely.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I've found it through :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gregory Moore, &lt;em&gt;Zermelo's Axiom of Choice : Its Origin Development and Influence&lt;/em&gt; (1982), page 26.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;We can easily generalise it to a poset &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; whatever, defining the mapping :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pi : a \to M_a$&lt;/span&gt;, for any &lt;span class=&quot;math-container&quot;&gt;$a \in M.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$M_a = \{ x \in M : x \le a \}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-11-18T16:46:06.917" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="513" PostTypeId="2" ParentId="510" CreationDate="2014-11-18T14:27:13.693" Score="12" Body="&lt;p&gt;One crucial link in 1-5 is missing: spectrum of an atom. (Key words: spectral analysis in chemistry and physics, spectrometer, spectroscope, spectral lines etc.).&lt;/p&gt;&#xA;&lt;p&gt;There is no doubt that all of them are closely related, and I can explain the relations of&#xA;all but 5). The word &amp;quot;spectrum&amp;quot; was first used by Newton when he discovered that white light&#xA;can be dissolved into colors. (I am not 100% sure, but I suppose that it was applied to the rainbow only AFTER that). Spectrum means ghost in Latin.&lt;/p&gt;&#xA;&lt;p&gt;Then more than a century had to pass before Wollaston discovered the spectral lines in the emission spectra in 1802. This triggered a fascinating story of development of spectroscopy.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;S. Sternberg, A history of 19th century spectroscopy, Appendix F in his book Group theory and physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Probably THE most interesting paper I ever read on history of mathematics and physics.&#xA;Strongly recommended to all who is interested in history of physics and mathematics!)&lt;/p&gt;&#xA;&lt;p&gt;Gradually they started using this word in the sense &amp;quot;set of proper frequencies of an oscillator&amp;quot; and apply it to all oscillations. But originally this meant &amp;quot;set of frequencies of light emitted or absorbed by an atom&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Corresponding mathematical theory was developing in parallel starting with Lagrange&#xA;and Fourier. This was known as &amp;quot;linear oscillations&amp;quot; in mathematics.&#xA;Mathematically this is the same as the &amp;quot;set of eigenvalues of a linear operator&amp;quot;. For example, the spectrum of a matrix is the set of its eigenvalues.&#xA;From spectra of atoms we have spectra in quantum mechanics, and in the rest of modern physics which is based on quantum mechanics. This explains the connections 1-3.&lt;/p&gt;&#xA;&lt;p&gt;Next development in mathematics was creation of the abstract theory of Banach Algebras (Gelfand, 1930-th). Original motivation of this theory was a) Fourier analysis and b) theory of linear operators. The revolutionary idea was that elements of any Banach algebra can be considered as functions of the &amp;quot;space of maximal ideals&amp;quot;. This idea quickly penetrated the other area of mathematics, and this is how we obtain 4).&#xA;The definition of the spectrum of a ring 4) is due to Grothendieck, and there is no doubt&#xA;that he was motivated by Gelfand's theory of Banach algebras.&lt;/p&gt;&#xA;&lt;p&gt;I am not an expert in 5) but this is due to J. Leray, and I suppose that it is based on the development of&#xA;same idea in Topology.&lt;/p&gt;&#xA;&lt;p&gt;Added remark. The words &amp;quot;spectrum&amp;quot; and &amp;quot;eigenvalue&amp;quot; were introduced to Mathematics by Hilbert in his series of articles on integral equations in the early 1990s. This was long before quantum mechanics,&#xA;so this was indeed a great prophecy: only in the work of Schrodinger it became clear that spectra of atoms are related to the spectra of certain integral operators&#xA;considered by Hilbert.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-04-17T12:56:48.297" LastActivityDate="2023-04-17T12:56:48.297" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="516" PostTypeId="2" ParentId="458" CreationDate="2014-11-18T21:06:12.353" Score="6" Body="&lt;p&gt;I made a little research for this community: I took the volume of &amp;quot;Cantor's papers on set theory&amp;quot;, and selected from the Index those medieval scholasts whom Cantor mentions&#xA;in his writings on set theory. Here is the list:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Albertus Magnum&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Augustin&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ben Akiba&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Boetius&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ibn Sina (Avizena)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Quintillianus&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nicolaus von Cusanus&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Origenus&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rufinus&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Thomas von Aquinas&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Franciscus von Assisi&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Franciscus von Paula&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(This does not include ancient philosophers, like Aristotle or Archimedes, and&#xA;those of the 17th century and after). As I wrote answering another question, Cantor's first&#xA;publications where set theory (and general topology) appear were on trigonometric series.&#xA;AFTER that he started to write papers on set theory and philosophy. So to determine precisely whether his introduction of infinite sets was MOTIVATED by medieval philosophy or not, one has to penetrate his brain, which I think is impossible.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="14422" LastEditDate="2021-05-12T13:38:37.810" LastActivityDate="2021-05-12T13:38:37.810" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="517" PostTypeId="2" ParentId="343" CreationDate="2014-11-19T00:31:58.843" Score="7" Body="&lt;p&gt;Prior to Isaac Newton's &lt;a href=&quot;http://www.phrases.org.uk/meanings/268025.html&quot; rel=&quot;noreferrer&quot;&gt;tribute&lt;/a&gt; (above) to Rene Descartes and Robert Hooke in his above cited letter to the latter, it was reportedly the 12th century theologian and author John of Salisbury who was recorded as having used an even earlier version of this &lt;a href=&quot;http://en.wikipedia.org/wiki/Standing_on_the_shoulders_of_giants&quot; rel=&quot;noreferrer&quot;&gt;humbling admission&lt;/a&gt;---in a treatise on logic called &lt;i&gt;Metalogicon&lt;/i&gt;, written in Latin in 1159, the gist of which is translatable as:&lt;/p&gt;&#xA;&#xA;&quot;Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size.&lt;/p&gt;&#xA;&#xA;(&lt;i&gt;Dicebat Bernardus Carnotensis nos esse quasi nanos, gigantium humeris insidentes, ut possimus plura eis et remotiora videre, non utique proprii visus acumine, aut eminentia corporis, sed quia in altum subvenimur et extollimur magnitudine gigantea.&lt;/i&gt;)&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond the simplistic interpretation of the remark:&lt;/p&gt;&#xA;&#xA; $\bullet$ 'standing on the shoulders of Giants'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as merely describing:&lt;/p&gt;&#xA;&#xA;$\bullet$ &quot;building on previous discoveries&quot;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it seems to me that what Newton perhaps intended was to echo Bernard of Chartres' dictum that it doesn't necessarily take an exceptional intelligence to see farther; only someone both humble and willing to:&lt;/p&gt;&#xA;&#xA;$\bullet$ first, clamber onto the shoulders of an acknowledged 'giant' and have the self-belief to see things at first-hand as they appear from a higher perspective (achieved more by the nature of height---and the curvature of our immediate space as implicit in such an analogy---than by the nature of genius); and,&lt;/p&gt;&#xA;&#xA;$\bullet$ second, avoid trying to see things first through the eyes of the 'giant' upon whose shoulders one stands (for the 'giant' might indeed be a vision-blinding genius)!&lt;/p&gt;&#xA;" OwnerUserId="362" LastActivityDate="2014-11-19T00:31:58.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="519" PostTypeId="1" AcceptedAnswerId="520" CreationDate="2014-11-19T12:57:33.153" Score="5" ViewCount="929" Body="&lt;p&gt;As a chemist we never really use classical mechanics much instead favouring a quantum description of the world around us. I have been lectured plenty on the origins of quantum mechanics and how the early pioneers forged away to develop ultimately quantum field theory. However I have very little idea about the analogous timeline of classical mechanics! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As such I will list my understanding but would love it if someone could fill in the blanks! (I'm starting at Newton and after more modern scientists :) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Newton &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) d'Alembert&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Lagrange &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Hamilton&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Poisson&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) Liouville&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) Poincare&lt;/p&gt;&#xA;&#xA;&lt;p&gt;8) Noether&lt;/p&gt;&#xA;" OwnerUserId="330" LastEditorUserId="64" LastEditDate="2014-11-20T20:03:40.427" LastActivityDate="2024-02-19T12:43:31.463" Title="Classical physics - A timeline of Mechanics" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="520" PostTypeId="2" ParentId="519" CreationDate="2014-11-19T13:54:00.723" Score="4" Body="&lt;p&gt;Here are some crucial contributors that are missing:&lt;/p&gt;&#xA;&lt;p&gt;Archimedes (statics, including hydrostatics)&lt;/p&gt;&#xA;&lt;p&gt;Stevin (Guido Jorg suggestion)&lt;/p&gt;&#xA;&lt;p&gt;Galileo (relativity principle, pendulum, falling bodies etc.)&lt;/p&gt;&#xA;&lt;p&gt;Huygens (pendulum, oscillations, Huygens principle)&lt;/p&gt;&#xA;&lt;p&gt;Hooke (Hooke's Law, Inverse squares law)&lt;/p&gt;&#xA;&lt;p&gt;Daniel Bernoulli (hydrodynamics)&lt;/p&gt;&#xA;&lt;p&gt;Clairaut (perturbation methods, figures of equilibrium)&lt;/p&gt;&#xA;&lt;p&gt;Euler (contributed to almost everything, motion of rigid bodies, for example, Euler-Lagrange equations, perturbation theory)&lt;/p&gt;&#xA;&lt;p&gt;Laplace (celestial mechanics)&lt;/p&gt;&#xA;&lt;p&gt;Jacobi (integrable systems)&lt;/p&gt;&#xA;&lt;p&gt;Lindstedt (perturbation methods)&lt;/p&gt;&#xA;&lt;p&gt;Hill (perturbation methods)&lt;/p&gt;&#xA;&lt;p&gt;Lyapunov (stability, figures of equilibrium)&lt;/p&gt;&#xA;&lt;p&gt;Birkhoff (closed orbits, ergodic theory)&lt;/p&gt;&#xA;&lt;p&gt;Fatou, Bogolyubov, Krylov (&amp;quot;non-linear mechanics&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;von Neumann (ergodic theory)&lt;/p&gt;&#xA;&lt;p&gt;Siegel (stability)&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov, Arnold, Moser (KAM theory)&lt;/p&gt;&#xA;&lt;p&gt;I omitted: statistical mechanics (which is also mechanics!), most of the fluid mechanics&#xA;after Archimedes and Bernoulli) and relativistic mechanics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-02-19T12:43:31.463" LastActivityDate="2024-02-19T12:43:31.463" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="521" PostTypeId="1" AcceptedAnswerId="526" CreationDate="2014-11-19T14:56:14.427" Score="10" ViewCount="3389" Body="&lt;p&gt;In&lt;/p&gt;&#xA;&#xA;&lt;p&gt;borel, emile, 1922, l'espace et le temps, paris: alcan&lt;/p&gt;&#xA;&#xA;&lt;p&gt;there is a claim that Robert Hooke discovered an inverse square relation of gravitating bodied prior Isaac Newton, who later applied it more successfully in astronomy (Hooke apparently studied bodies on earth).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is said to result in attribution of the principle to Newton, rather than Hooke, since the astronomical example were much more obvious than the small measurements with bodies on earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this true? What primary sources exist? Writing, books or letters, by Hooke, prior 1682, as opposed to later texts &lt;em&gt;about&lt;/em&gt; him discovering the principle, if he really did discover it as Borel claims.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="26" LastEditDate="2014-12-17T12:25:38.323" LastActivityDate="2014-12-17T13:50:15.663" Title="Robert Hooke and the Inverse Square Law" Tags="|biographical-details|discoveries|gravity|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="524" PostTypeId="1" AcceptedAnswerId="536" CreationDate="2014-11-19T18:47:05.857" Score="35" ViewCount="10584" Body="&lt;p&gt;Can you tell me the name of the mathematician, who introduced the Principle of Mathematical Induction for the first time? (with reliable source).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please don't say De Morgan because I have read the name of the mathematician somewhere, but I have forgotten. &lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="55" LastEditDate="2015-05-29T03:42:41.853" LastActivityDate="2022-05-01T19:54:06.947" Title="Who introduced the Principle of Mathematical Induction for the first time?" Tags="|mathematics|discoveries|mathematicians|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="525" PostTypeId="2" ParentId="524" CreationDate="2014-11-19T19:07:21.230" Score="12" Body="&lt;p&gt;Well, you could go with Plato. From &lt;a href=&quot;http://server.math.uoc.gr/%7Etzanakis/Courses/NumberTheory/MathInduction.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; (&lt;a href=&quot;https://web.archive.org/web/20190714035011/http://server.math.uoc.gr/%7Etzanakis/Courses/NumberTheory/MathInduction.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wayback Machine&lt;/a&gt;),&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For a start, although&#xA;the principle itself is not explicitly stated in any ancient Greek text, there are several places that contain precursors of it. Indeed, some historians see the following&#xA;passage from Plato’s (427-347 BC) dialogue Parmenides (§147a7-c3) as the earliest&#xA;use of an inductive argument.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The text quotes Parmenides:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Then they must be two, at least, if there is to be contact. - They must. - And if to the&#xA;two terms a third be added in immediate succession, they will be three, while the&#xA;contacts [will be] two. - Yes. - And thus, one [term] being continually added, one&#xA;contact also is added, and it follows that the contacts are one less than the number of&#xA;terms. For the whole successive number [of terms] exceeds the number of all the&#xA;contacts as much as the first two exceed the contacts, for being greater in number&#xA;than the contacts: for afterwards, when an additional term is added, also one contact&#xA;to the contacts [is added]. - Right. - Then whatever the number of terms, the&#xA;contacts are always one less. -True.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I attempted to find a second translation, and found &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0174%3Atext%3DParm.&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; one, but I have been unable to identify this section, even though the section is supposedly listed. I found another transcript &lt;a href=&quot;http://classics.mit.edu/Plato/parmenides.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, but it, too does not agree with the given text in the pdf. Still, the dialogue closely resembles the technique of mathematic induction, albeit used in a philosophical context.&lt;/p&gt;&#xA;&lt;p&gt;The page goes on to add that Euclid and Pappus, too, played a part:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are, however, several ancient mathematical texts that also contain quasi-inductive arguments. For instance Euclid (~330 - ~ 265 BC) in his &lt;em&gt;Elements&lt;/em&gt; employs one to show that every integer is a product of primes. An argument closer to the modern version of induction is in Pappus' (~290-~350 AD) &lt;em&gt;Collectio&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have been unable to find a similar primary source for Euclid and Pappus. However, &lt;a href=&quot;http://alpha.math.uga.edu/%7Epete/3200induction.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; appears to corroborate Euclid's role (though not as detailed as I would have likes) and also says that Wikipedia's claim about Pascal being important (mentioned below from another source) is accurate. I'm working on more primary sources, but at the moment no others are forthcoming.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/2974308&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; says that Pascal played a role in modern times:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cantor in his Vorlesungen iuber Geschichte der Mathematik' says that Pascal&#xA;was the originator of the method of complete induction&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastEditorUserId="205" LastEditDate="2022-04-29T10:42:57.843" LastActivityDate="2022-04-29T10:42:57.843" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="526" PostTypeId="2" ParentId="521" CreationDate="2014-11-19T21:42:50.723" Score="6" Body="&lt;p&gt;Hooke did propose the inverse square law to Newton, and even the program of ”compounding the celestiall motions of the planetts of a direct motion by the tangent &amp;amp; an attractive motion towards the central body.” These suggestions from Hooke ”occasioned my findings” on planetary motion, Newton admitted. For more details, see &lt;a href=&quot;http://www.amazon.com/review/R3TV1ST7R0BYX5&quot; rel=&quot;nofollow&quot;&gt;my summary of Ofer Gal’s book on this&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added: Hooke stated the law in print in his 1665 Micrographia. See &lt;a href=&quot;http://adsabs.harvard.edu/full/2005HisSc..43..391G&quot; rel=&quot;nofollow&quot;&gt;The Archaeology of the Inverse Square Law&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="370" LastEditorUserId="370" LastEditDate="2014-11-21T09:45:40.277" LastActivityDate="2014-11-21T09:45:40.277" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="527" PostTypeId="2" ParentId="524" CreationDate="2014-11-20T01:24:36.300" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Gersonides#Works_in_mathematics_and_astronomy.2Fastrology&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt; this was Levi ben Gershon (a.k.a. Gershonides, RALBAG). 1288–1344&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The work is notable for its early use of proof by mathematical induction, and pioneering work in combinatorics. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gersonides was also the earliest known mathematician to have used the technique of mathematical induction in a systematic and self-conscious fashion &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Remark. The word &quot;induction&quot; is used in a different sense in philosophy.&#xA;One has to distinguish Mathematical induction from &quot;induction&quot; in philosophy.&#xA;These are very different things. I am not aware of any ancient Greek use of Mathematical induction.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="62" LastEditDate="2014-11-20T03:39:21.590" LastActivityDate="2014-11-20T03:39:21.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="528" PostTypeId="2" ParentId="521" CreationDate="2014-11-20T01:45:07.260" Score="5" Body="&lt;p&gt;The following stories show that the inverse square law was widely discussed at the time of Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First story is about Hooke. He wrote to Newton proposing to determine &quot;how a point will be moving under the inverse square law&quot;. Specifically he discussed the example of an object&#xA;with some initial velocity (NOT directed towards the center) how would it move if the Earth&#xA;did not offer any resitanace. Newton's answer was INCORRECT (he wrote it will move on a spiral winding towards the center of the Earth). In the next letter Hooke wrote the correct answer: it will move on an ellipse. (Hook did not have a mathematical proof of this: he probally EXPERIMENTED!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To this last letter Newton did not reply. Several years later Sir Christopher Wren in a conversation with Hooke and Halley (in a pub:-) proposed to prove that the inverse square law of attraction would imply Kepler orbits, and offered a reward for a proof.&#xA;Halley passed this to Newton, and Newton replied that he has a proof.&#xA;But he could not find it among his papers. Some time later he sent to Halley a manuscript with a &quot;proof&quot;. This is what later become Principia. (It is still discussed whether Newton's proof was a valid one. I share the majority opinion that it was. However, WHEN did Newton obtain it remains a mystery. Certainly he did not know the result, not speaking of the proof at the time when Hooke wrote to him. Later Newton spread rumors that he knew it almost from his childhood:-).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Halley proposed to mention Hooke in Principia. But Newton stubbornly resisted, and his angry answer to Halley is well known and  documented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole story is told in many places, one of them is Arnold's book, Barrow and Huygens,&#xA;Newton and Hooke. Unlike some other stories told by Arnold this story is true, and is well documented: I read the corresponding letters of Newton and Hooke myself.&#xA;And everyone can check this in the published collection of Newton's correspondence. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion. The Inverse square law was discussed at the time of Newton as a plausible CONJECTURE. Newton proved it. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-11-20T01:52:36.630" LastActivityDate="2014-11-20T01:52:36.630" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="529" PostTypeId="2" ParentId="524" CreationDate="2014-11-20T04:22:36.060" Score="5" Body="&lt;p&gt;Victor Katz wrote in a discussion at the Historia mathematica discussion list, &quot;as Barnabas Hughes&#xA;notes, it is difficult to answer the question of the &quot;first instances of&#xA;proof by induction&quot; unless one carefully defines what one means by &quot;proof&#xA;by induction.&quot; (The same general remark applies to lots of questions about&#xA;firsts in the history of mathematics.) Noicomachus certainly has an argument&#xA;which we would find very easy to convert into a modern formal &quot;proof by&#xA;induction.&quot; So do several Islamic authors around the 11th century,&#xA;including al-Karaji and al-Samaw'al. (See my book, pp. 238-242.) Pascal&#xA;may well be the first to state the modern principle of mathematical&#xA;induction explicitly, but even he does not give proofs in the modern style -&#xA;because he has no notation for a general &quot;n&quot;. Thus he generally gives&#xA;proofs by what I call the &quot;method of generalizable example.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In another post Barnabus Hughes suggests yet an earlier &quot;first use&quot; of induction:&#xA;If the essence of math induction lies in a process that begins at some&#xA;small value, which process can be continued to larger values which&#xA;regardless of their size maintain the pattern one wishes to accept, then&#xA;I would hazard that &lt;strong&gt;Nicomachus of Geresa&lt;/strong&gt; used the essence of math&#xA;induction where he discussed figurate numbers (&lt;em&gt;Arithmetica&lt;/em&gt;, Bk. 2, CC.&#xA;7ff.) In C. 7 he states, &quot;Hence, the triangle is elementary among these&lt;/p&gt;&#xA;&#xA;&lt;p&gt;figures; for everything else is resolved into it, but it into nothing&#xA;else.&quot; He then shows how, if the process established for the creation of&#xA;each figurate number is followed, then that number is always seen. C. 12&#xA;shows how the other figures are resolved into triangles, including a&#xA;table which lists the first five polygonal numbers to the tenth degree of&#xA;each. I think that Nick established the pattern by induction. Comment?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Barnabas Hughes&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="115" LastEditDate="2014-11-20T04:49:43.830" LastActivityDate="2014-11-20T04:49:43.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="530" PostTypeId="2" ParentId="521" CreationDate="2014-11-20T04:32:51.000" Score="7" Body="&lt;p&gt;The idea that gravity acted with an inverse square relation was not a &quot;done deal&quot; because Newton or Hooke said so. On Nov 15, 1747 &quot;Clairaut, at a public session of the [French] Academy, announced in rather pompous phrases that the Newtonian Theory of gravity was false!”&lt;br&gt;&#xA;One of the boldest attempts to reconcile the observed and theoretical descriptions of the moon's motion was made not by Euler, but Clairaut, who announced in at a public session in the French Academy of Sciences that Newton's theory of gravity was wrong. Euler and d’Alembert simultaneously came to the same conclusion as both had been working on the motion of the moon as a special case of the three body problem.&lt;br&gt;&#xA;Clairaut suggested that the strength of gravity was proportional not to $1/r^2$, but the more complicated $1/r^2 + c/r^4$ for some constant $c$. Over large distances, the $c/r^4$ term would effectively disappear, accounting for the utility of the inverse square law over large distances. He then began trying to find a value of c which could account for the moon's motion. He would continue to pursue this idea until May 17, 1749, when he made an equally dramatic announcement in which he claimed that Newton was right after all.&lt;/p&gt;&#xA;" OwnerUserId="115" LastEditorUserId="48" LastEditDate="2014-12-17T13:50:15.663" LastActivityDate="2014-12-17T13:50:15.663" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="531" PostTypeId="2" ParentId="505" CreationDate="2014-11-20T12:59:37.140" Score="3" Body="&lt;p&gt;Let me try to summarize this long discussion to an &quot;answer&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that we know about epicycle theory comes from Ptolemy. Ptolemy credits Apollonius&#xA;(262-190 BC) with one mathematical theorem (which says that excentric motion is equivalent to epicycle), and&#xA;Hipparchus (190-120) with using epicycles to describe the motions of Sun, Moon and possibly planets. (Ptolemy credits planets to himself, correct?). Archimedes (287-212) died when Apollonius was 50 years old, so it is very plausible that they knew each other and perhaps even corresponded. (It is known that Archimedes did &#xA;correspond with other Alexandrian scholars).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no historical evidence relating Archimedes to epicycle theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story of the recently found object in Olbia which is presumably a piece of a toothwheel from Archimedes &quot;planetarium&quot; is strange. All information about this object found on the Internet comes from a single person who is an engineer, and who wrote a book about it.&#xA;No photo or description of this object could be found. No reference except on the writings of this single person. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Information about Archimedes planetarium which exists in the surviving literature&#xA;does not permit to judge on what theory it was based.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this is a fair summary. Please correct if necessary.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-20T12:59:37.140" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="533" PostTypeId="1" CreationDate="2014-11-21T01:31:58.523" Score="11" ViewCount="1014" Body="&lt;p&gt;Why was our (statute) mile established as it was?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This happened in 1593, by the order of Elizabeth I which said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A Mile ſhall contain eight Furlongs, every Furlong forty Poles, and every Pole ſhall contain ſixteen Foot and an half.&quot; It was thus 1760 yards (5280 feet, about 1609 meters)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Citation from Wikipedia). The question is why 8 furlongs? Why forty poles, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I conjecture that this was based on some very crude measurement of the Earth circumference,&#xA;and the mile was supposed to be 1' of the meridian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, as late as in 1667 Newton thought that one degree =60 miles, and as a result, that the gravitational force inversely proportional to the square of the distance cannot alone explain the Moon's orbit. (Based on Wiston's recollections of conversations with Newton, cited in&#xA;&lt;a href=&quot;http://books.google.com/books?id=xIkn3eFNy20C&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;http://books.google.com/books?id=xIkn3eFNy20C&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt; this book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nautical  mile is indeed one minute of the meridian.&#xA;And kilometer was originally defined as one &quot;decimal minute&quot; (400 degrees in a circle, 100 minutes in a degree).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I conclude that Elizabeth's statute was based on some very imprecise measurement of the meridian. Which measurement? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Is this not amazing that we are still using this mile established in XVI century on the basis of a wrong measurement? &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="26" LastEditDate="2015-03-01T13:58:46.530" LastActivityDate="2021-11-02T23:41:07.620" Title="Why does the statute mile have the length that it has?" Tags="|terminology|measurement|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="534" PostTypeId="1" AcceptedAnswerId="559" CreationDate="2014-11-21T06:05:58.020" Score="6" ViewCount="1557" Body="&lt;p&gt;According to what we read in footnote #2 of [&lt;strong&gt;I&lt;/strong&gt;], no portrait of Christian Goldbach seems to be known. The author of the article under consideration even adds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... Juskevič &amp;amp; Kopelevič's biography states that none exists [&lt;strong&gt;J&amp;amp;K&lt;/strong&gt;, Christian Goldbach: 1690–1764 (translated from Russian to German by Annerose and Walter Purkert, Vita mathematica, Band 8). Basel: Birkhäuser Verlag, 1994, p. xi], also confirmed by Gerald Alexanderson (personal communication) and Goldbach conjecture expert Yuan Wang (personal communication), who explained that Goldbach was not famous at the time... But Goldbach did in fact eventually attain high rank in the Russian government, first as tutor to the crown prince, then later rising to privy councilor... He was also important for&#xA;  many years at the Petersburg Academy, which did include artists (including both Euler’s brother and the father of Euler’s wife). I therefore hope that a picture might yet be found, though perhaps in a government or art historical collection rather than in scholarly archives. As motivation, I hereby offer a bounty of $100 for the first authentic picture of Goldbach (or complete correct proof of the Goldbach conjecture) that I receive!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it is not rare to find web sites on the internet (e.g., 1) &lt;a href=&quot;http://eljjdx.canalblog.com/archives/2012/08/05/24787409.html&quot; rel=&quot;noreferrer&quot;&gt;http://eljjdx.canalblog.com/archives/2012/08/05/24787409.html&lt;/a&gt;, 2) &lt;a href=&quot;http://images.slideplayer.com.br/3/382572/slides/slide_5.jpg&quot; rel=&quot;noreferrer&quot;&gt;http://images.slideplayer.com.br/3/382572/slides/slide_5.jpg&lt;/a&gt;, 3) &lt;a href=&quot;http://gaussianos.com/la-conjetura-debil-de-goldbach-coloquio-de-harald-helfgott-en-el-icmat/&quot; rel=&quot;noreferrer&quot;&gt;http://gaussianos.com/la-conjetura-debil-de-goldbach-coloquio-de-harald-helfgott-en-el-icmat/&lt;/a&gt;, 4) &lt;a href=&quot;http://paginas.matem.unam.mx/cprieto/index.php/es/matematicos/matematicos-f/126-goldbach-christian&quot; rel=&quot;noreferrer&quot;&gt;http://paginas.matem.unam.mx/cprieto/index.php/es/matematicos/matematicos-f/126-goldbach-christian&lt;/a&gt;, 5) &lt;a href=&quot;http://www.alasayvan.com/bilim-adamlari/400020-christian-goldbach-kristiyan-goldbah-ve-matematige-katkilari.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.alasayvan.com/bilim-adamlari/400020-christian-goldbach-kristiyan-goldbah-ve-matematige-katkilari.html&lt;/a&gt;) that imply that Christian Goldbach was this fellow:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/MuNVP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It follows that either Mr. Haas's sources are somewhat obsolete or the individual in the portrait is actually not Christian Goldbach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you people know about this matter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, this theme has made me recall an amusing 2009 note in the &lt;em&gt;Notices of the AMS&lt;/em&gt; about a similar situation related to a portrait that for many individuals in the past was A. M. Legendre's (cf. [&lt;strong&gt;II&lt;/strong&gt;, pages 1440-1443]).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[&lt;strong&gt;I&lt;/strong&gt;] R. Haas. &lt;em&gt;Goldbach, Hurwitz, and the infinitude of primes: weaving a proof across the centuries.&lt;/em&gt; Math. Intelligencer, Vol. 36, #1, &lt;strong&gt;2014&lt;/strong&gt;, pp. 54-60.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[&lt;strong&gt;II&lt;/strong&gt;] P. Duren. &lt;em&gt;Changing faces: The mistaken portrait of Legendre.&lt;/em&gt; Notices of the AMS, Vol. 56, #11, 2009, pp. 1440-1443.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2014-11-24T20:21:21.307" LastActivityDate="2022-01-26T13:53:55.543" Title="Question related to the legitimacy of a certain portrait of Christian Goldbach" Tags="|debunking|depiction-of-scientists|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="535" PostTypeId="1" AcceptedAnswerId="595" CreationDate="2014-11-21T07:03:39.393" Score="13" ViewCount="628" Body="&lt;p&gt;One of the traditional ideas about personality was Temperament Theory which probably originated from Egypt and was developed by Hippocrates into a medical theory. In the tenth century, it was extended by Avicenna and had been used for many years in medicine. Do you know when this idea was widely replaced by modern methodology in medicine (especially in the West)? Is it still a useful theory in psychology? &lt;/p&gt;&#xA;" OwnerUserId="336" LastEditorUserId="55" LastEditDate="2014-11-29T23:51:08.920" LastActivityDate="2014-12-03T18:53:25.980" Title="Four temperaments and medical methodology" Tags="|medicine|psychology|physiology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="536" PostTypeId="2" ParentId="524" CreationDate="2014-11-21T12:25:11.207" Score="26" Body="&lt;p&gt;&lt;em&gt;The issue is thorny&lt;/em&gt; ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Morris Kline, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B0058C6F0E&quot;&gt;Mathematical Thought from Ancient to Modern Time. Volume I&lt;/a&gt; (1972), page 272 [only entry of the &lt;em&gt;Subject Index&lt;/em&gt; regarding : &lt;em&gt;mathematical Induction&lt;/em&gt;] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The method was recognized explicitly by Maurolycus in his &lt;em&gt;Arithmetica&lt;/em&gt; of 1575 and was used by him to prove, for example, that $1+3+5+ \ldots + (2n+1)=n^2$. Pascal in one of his letters acknowledged Maurolycus's introduction of the method and used it himself in his &lt;em&gt;Traité du triangle arithmétique&lt;/em&gt; (1665), wherein he presents what we now call the Pascal triangle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The modern source is &lt;a href=&quot;http://en.wikipedia.org/wiki/Giovanni_Vacca&quot;&gt;Giovanni Vacca&lt;/a&gt; (1872 –1953) Italian mathematician, assistant to Giuseppe Peano and historian of science in his :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;G.Vacca, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183420503&quot;&gt;Maurolycus, the first discoverer of the principle of mathematical induction&lt;/a&gt; (1909)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;with comments in :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;W.H.Bussey, &lt;a href=&quot;http://www.jstor.org/stable/2974308&quot;&gt;The Origin of Mathematical Induction&lt;/a&gt; (1917).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Acording to Kline :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the method [of mathematical induction] is implicit even in Euclid's proof of the infinitude of the number of primes [IX, 20].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This point is debatable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486600890&quot;&gt;The Thirteen Books of the Elements, Vol. 2: Books III - IX&lt;/a&gt; (T.L.Heath editor), states IX.20 as follows [page 412] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Prime numbers are more than any assigned multitude of prime numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The proof is per impossibile. According to Heath's comment [page 413] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have here the important proposition that the number of prime numbers is infinite.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Neither in the statement of the proposition nor in its proof Euclid uses the work &lt;em&gt;infinite&lt;/em&gt; [&lt;em&gt;apeiros -on&lt;/em&gt; : adj, infinite].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In can be helpful to place it in the context of Grrek debate about the infinite : see &lt;a href=&quot;http://plato.stanford.edu/entries/aristotle-mathematics/&quot;&gt;Aristotle and Mathematics&lt;/a&gt; on the &lt;em&gt;actual infinity&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the Aristotelian philosophy, we cannot legitimately &quot;conceive&quot; &lt;em&gt;actual&lt;/em&gt; infinity; i.e. we have no experience of an infinite &quot;collection&quot; but only of an unlimited iterative process (the &lt;em&gt;potential&lt;/em&gt; infinity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid's statement must be understood in this context : we never have a &quot;complete&quot; infinite set of &lt;em&gt;prime&lt;/em&gt; numbers, but we have a &quot;procedure&quot; that, for a &lt;em&gt;finite&lt;/em&gt; collection of prime numbers whatever, can &quot;produce&quot; a new prime which is not in the collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only occurrence (not regarding straight lines) I've found of the word &lt;em&gt;infinite&lt;/em&gt; (searching into : &lt;a href=&quot;http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf&quot;&gt;EUCLID’S ELEMENTS OF GEOMETRY&lt;/a&gt;, the Greek text of J.L. Heiberg (1883–1885), edited, and provided with a modern English translation, by Richard Fitzpatrick), is VII,31 [Heath ed, page 332] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Any composite number is measured by some prime number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The proof uses the fact that :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For, if it is not found [some prime number which will measure the number before it, which will also measure A], an &lt;em&gt;infinite series of numbers&lt;/em&gt; [emphasis added] will measure the number A, each of which is less than the other: which is impossible in numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems the only reasonable &quot;candidate&quot; for a precursor of the &lt;em&gt;least number principle&lt;/em&gt; [see the comment of Ian Mueller, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486453006&quot;&gt;Philosophy of Mathematics and Deductive Structure in Euclid's Elements&lt;/a&gt; (1981, Dover reprint), page 77]. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;According to :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Reviel Netz, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B000U0SPGQ&quot;&gt;The Shaping of Deduction in Greek Mathematics&lt;/a&gt; (1999), page 215 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the Greeks [...] did not have mathematical induction [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding Pappus, in &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Serafina Cuomo, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521642116&quot;&gt;Pappus of Alexandria and the Mathematics of Late Antiquity&lt;/a&gt; (2000) there are no references to induction.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2014-11-26T07:35:29.200" LastActivityDate="2014-11-26T07:35:29.200" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="537" PostTypeId="1" AcceptedAnswerId="538" CreationDate="2014-11-21T17:11:58.210" Score="6" ViewCount="484" Body="&lt;p&gt;The cross-ratio of four collinear points $A,B,C,D$ in the Euclidean Plane is defined by &#xA;$$(A,B,C,D) = \frac{AC}{AD}\frac{BD}{BC}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And &lt;a href=&quot;http://en.wikipedia.org/wiki/Cross-ratio&quot; rel=&quot;nofollow&quot;&gt;the wikipedia article&lt;/a&gt; states it already appeared in the works of Pappus of Alexandria. Since I can't find any geometrical interpretation for this ratio I am really curious to know how Pappus constructed it, and what his intuition was for it; as I understand, in his epoch constructions of that kind were only made using straightedge-and-compass.&lt;/p&gt;&#xA;" OwnerUserId="346" LastEditorUserId="26" LastEditDate="2014-12-12T11:14:10.653" LastActivityDate="2016-12-20T23:24:34.970" Title="How the Cross-Ratio appears in the Work of Pappus?" Tags="|mathematics|euclidean-geometry|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="538" PostTypeId="2" ParentId="537" CreationDate="2014-11-21T17:34:12.813" Score="4" Body="&lt;p&gt;It appears in the demonstration of Pappus's hexagon theorem as a tool in the demonstration.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(illustrations &lt;a href=&quot;http://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;from the wikipedia article&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note: Pappus' proof is invalid in the case C,c and X are colinear, but the theorem is true nonetheless. )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/3xYzg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="8" LastEditDate="2016-12-20T23:24:34.970" LastActivityDate="2016-12-20T23:24:34.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="539" PostTypeId="1" AcceptedAnswerId="541" CreationDate="2014-11-21T21:54:00.093" Score="17" ViewCount="1518" Body="&lt;p&gt;At the end of 19th century there was a lively discussion about the nature of the second law of thermodynamics, and its relation to Hamiltonian dynamics. Boltzmann developed a &lt;a href=&quot;http://en.wikipedia.org/wiki/Ludwig_Boltzmann#The_Second_Law_as_a_law_of_disorder&quot; rel=&quot;nofollow noreferrer&quot;&gt;position&lt;/a&gt; that the second law is statistical in nature, &quot;ordered&quot; states are less probable, and the entropy is &quot;highly unlikely&quot; to decrease. As a confirmation he offered the H-theorem, a claim that entropy increases in a nearly ideal gas of particles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But first Loschmidt &lt;a href=&quot;http://en.wikipedia.org/wiki/H-theorem#Loschmidt.27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;objected&lt;/a&gt; that Boltzmann's &quot;proof&quot; of the H-theorem had a gap in neglecting correlations due to collisions, and in 1896 Zermelo disputed all of the above. He pointed out that statistical equations of Hamiltonian dynamics (Liouville equations) are just as time reversible as equations for individual particles, and therefore so are the equations for probabilities. Hence entropy is not &quot;likely&quot; to decrease or increase, it has to stay constant (up to deviations due to finite if large number of particles, so-called thermal fluctuations). As an illustration Zermelo brought up the Poincare recurrence theorem, which says that every state of a system nearly recurs infinitely many times in its evolution. He concluded that the second law is mathematically inconsistent with Hamiltonian dynamics, and no amount of probability theory can fix that. Poincare expressed the same opinion earlier (1889).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting that contemporaries thought that Boltzmann won the debate, but Boltzmann felt backed into a corner and &quot;lost faith in himself&quot; (according to Popper's Unended Quest). His response was the &lt;a href=&quot;http://encyclopedia2.thefreedictionary.com/Fluctuation+Hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;fluctuation hypothesis&lt;/a&gt; that our vicinity is a giant fluke in the universe, where the second law happens to hold. Zermelo in his turn suggested that the second law might be a selection rule on initial states rather than a &quot;law of nature&quot;, but commented that such fix contradicts the &quot;spirit of mechanics&quot;, and would not &quot;satisfy anybody for long&quot;. Nonetheless, according to &lt;a href=&quot;http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; &quot;much effort in the field is attempting to understand why the initial conditions early in the universe were those of low entropy, as this is seen as the origin of the second law&quot;. How selection of initial states resolves the issue of the entropy staying constant I am not sure.&#xA;Soon after 1896 the attention of physicists shifted to relativity and quantum mechanics, and the second law dropped out of sight.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was this debate picked up by others later? If so who advocated what position and how? Who was closer to the truth from the modern point of view, Zermelo or Boltzmann?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;EDIT: Here is a modern &lt;a href=&quot;https://web.archive.org/web/20160615152029/http://www.math.ens.fr/~bodineau/GT_2012/kac-ring.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;exposition&lt;/a&gt; of the ideas of Ehrenfests and Mark Kac mentioned in the Alexandre's answer. It turns out that the missing ingredient was coarse-graining (macroscopic averaging) introduced by Gibbs in 1902 in a direct follow-up to the debate. Time reversible microscopic systems have evolutions in which coarse-grained entropy appears to grow for astronomically long periods of time, in the spirit of Boltzmann. Of course, they also have evolutions, where the opposite happens. We only observe the former but not the latter, so there is a selection of initial states suggested by Zermelo. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that they were both half right, but lacked the idea of coarse-graining which reconciles their positions. The nature of selection is apparently still unclear and is traced to low entropy &quot;early in the universe&quot;, which attracts &quot;much effort in the field&quot;, see details &lt;a href=&quot;http://plato.stanford.edu/archives/fall2011/entries/time-thermo/#2&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2020-06-08T01:03:55.317" LastActivityDate="2023-08-23T07:18:23.147" Title="What became of the Boltzmann-Zermelo debate about the second law of thermodynamics?" Tags="|physics|classical-mechanics|thermodynamics|statistical-mechanics|boltzmann|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="540" PostTypeId="2" ParentId="153" CreationDate="2014-11-21T22:27:28.580" Score="3" Body="&lt;p&gt;Stevinus developed the basic equilibrium of forces rules in structures in the late 1500's. All the first editions of his works are available on google books and internet archive, but I highly recommend the technical summary with translation that is in: &lt;a href=&quot;http://www.worldcat.org/title/magic-is-no-magic-the-wonderful-world-of-simon-stevin/oclc/72868813&quot; rel=&quot;nofollow&quot;&gt;MINM&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or you can look at the first editions, most of them are quite short books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One caution: the detailed general mathematical models of loads for material beam configurations are discovered only later by Euler, indeed, and those not so discovered were developed into their reliable and modern form in the 19th century only. There is no single source, but many different papers published in 19th century journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I highly recommend looking into the historical notes and references Clifford Truesdell included in his various technical books (and longer papers, 1950's-1970's) on classical mechanics. They have almost every historical reference (because Truesdell could and did read all the primary sources in the original languages).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's not something that easily fits neatly in a post, requiring a review length treatment, but it exists completely in Truesdell (I'll have to check which of his books has the most concise listing and description of priority references in mechanics---look forward to an edit).&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2014-11-21T22:36:36.000" LastActivityDate="2014-11-21T22:36:36.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="541" PostTypeId="2" ParentId="539" CreationDate="2014-11-22T02:02:58.683" Score="6" Body="&lt;p&gt;Bolzman was closer to the truth. On the modern state of this question, I recommend&#xA;the classics:&lt;/p&gt;&#xA;&lt;p&gt;Ehrenfest, Paul; Ehrenfest, Tatiana The conceptual foundations of the statistical approach in mechanics. Translated from the German by Michael J. Moravcsik. With a foreword by M. Kac and G. E. Uhlenbeck. Reprint of the 1959 English edition. Dover Publications, Inc., New York, 1990.&lt;/p&gt;&#xA;&lt;p&gt;And excellent papers of Mark Kac on the subject, one collection of them is&#xA;Probability and related topics in physical sciences.&#xA;With special lectures by G. E. Uhlenbeck, A. R. Hibbs, and B. van der Pol. Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, Colo., 1957, Vol. I Interscience Publishers, London-New York 1959.&lt;/p&gt;&#xA;&lt;p&gt;Here is a popular (and freely available) article on this discussion:&#xA;&lt;a href=&quot;http://ergodic.ugr.es/statphys_grado/bibliografia/zermelo_boltzmann.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ergodic.ugr.es/statphys_grado/bibliografia/zermelo_boltzmann.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The subject seems somewhat too technical for an explanation here, so I only recommend the literature. It seems to be a general agreement that probability CAN explain the second law of thermodynamics.&lt;/p&gt;&#xA;&lt;p&gt;However, much remains to be done in this area. In modern mathematics this is called &amp;quot;ergodic theory&amp;quot;, and proving that concrete physical systems are indeed ergodic is a  difficult mathematical problem.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="5989" LastEditDate="2023-08-23T07:18:23.147" LastActivityDate="2023-08-23T07:18:23.147" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="542" PostTypeId="2" ParentId="401" CreationDate="2014-11-22T04:22:36.837" Score="4" Body="&lt;p&gt;Another famous controversy is the one about atomism.&#xA;All German philosophers and some scientists (E. Mach, W. Ostwald) denied the existence of atoms as late as in 1904 (!) An information from Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1904 at a physics conference in St. Louis most physicists seemed to reject atoms and he [Boltzmann] was not even invited to the physics section.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Boltzmann died in 1906 (committing suicide), while Einstein and Smoluchowski theory of&#xA;Brownian motion was published in 1905, and as a result the existence of atoms &#xA;was experimentally proved by a series of experiments in 1908-1909, and Avogadro's number&#xA;was measured; finally vindicating Boltzmann. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="26" LastEditDate="2014-11-24T07:03:37.840" LastActivityDate="2014-11-24T07:03:37.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="543" PostTypeId="2" ParentId="401" CreationDate="2014-11-22T04:40:52.420" Score="6" Body="&lt;p&gt;Another famous controversy was about the age of the Earth.&#xA;It was found long ago that the temperature in the very deep mines increases with depth.&#xA;Using the model that the Earth evolved from a hot state when the rocks were melted, and was cooling down ever since, one could estimate the age from the known temperature of&#xA;melting of the rocks and the current gradient of the temperature near the surface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was one of the main motivations of Fourier's seminal book Analytic theory of heat&#xA;(where he invented the Fourier transform). Fourier's idea was further developed by&#xA;W. Thomson (Lord Kelvin) who arrived at the estimate between 20 and 400 million years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geologists and Darwinists could not accept such a short time. By the way, it is interesting to now how exactly the Darwinists estimated the time necessary for the evolution in 19-th century. I will be glad if someone explains this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result Thomson found himself in the camp of anti-evolutionists:-)&#xA;On the other hand he was supported by such people as Helmholtz and astronomer Newcomb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lord Kelvin died in 1907, while radioactivity was discovered in 1896. It took a while&#xA;to understand that  it is radioactivity&#xA;which causes additional heating of the Earth and makes Fourier and Kelvin correct calculations&#xA;non-applicable. Such a sad story. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-22T04:40:52.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="544" PostTypeId="2" ParentId="401" CreationDate="2014-11-22T14:29:20.943" Score="3" Body="&lt;p&gt;I am not sure whether it can be called a conflict or not, but a suitable approach to quantize gravity has caused many discussions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some scientists believe that spacetime itself must be quantized and someones say the bodies behave in a quantized way, not spacetime. Also someones believe that our number sets are incompatible with nature, and so on. As a result, there are several quantum theories of gravity and this problem is one of the greatest problems in physics (or maybe science, for it has a lot of influences on mathematics, astronomy and philosophy at the first level, and on biology at next levels).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These days, string theory (ST) and loop quantum gravity (LQG) with two different viewpoints are the most important candidates to solve the problem, but twistor theory, topos theory, non-commutative gravity and quantum field theory in curved spacetimes, are examples of other theories which have not been as successful as ST or LQG. Almost all of these theories have some inconsistencies or even if they are consistent, they can't be easily tested by experiment.&lt;/p&gt;&#xA;" OwnerUserId="336" LastActivityDate="2014-11-22T14:29:20.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="545" PostTypeId="2" ParentId="537" CreationDate="2014-11-22T14:54:59.397" Score="1" Body="&lt;p&gt;To construct a segment equal to the cross-ratio, you slightly modify the first picture in&#xA;the Wikipedia article. Let us call $P$ the point in the top where the 4 black lines cross. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose you want to construct two segments whose ratio equals&#xA;to the cross-ratio of 4 given points on a line.&#xA;Referring to this picture, suppose that your given points are $A',B',C',D'$, and they are&#xA;on the red line $L'$. Then draw a second red line L, and choose the point $P$ so that&#xA;$PD'$ is parallel to $L$. (Or choose $P$ arbitrarily, and then choose $L$ parallel to&#xA;$PD$). Then use this $P$ as the center of projection and&#xA;project $A',B',C'$ on $L$ from $P$. The ratio $AC/BC$ will be equal to the cross ratio&#xA;of $A',B',C',D'$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To obtain the &quot;numerical&quot; value (the notion foreign to the Greek mathematics; &#xA;after the discovery of non-commensurable segments they did not measure the segments&#xA;with numbers; they were only speaking of proportions, see Euclid), you have to choose the unit of length.&#xA;Suppose that $BC$ has length $1$,&#xA;then $AC$ is the cross ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not checked with Pappus, but I suppose he had some similar construction, if he had any.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-22T14:54:59.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="546" PostTypeId="1" AcceptedAnswerId="547" CreationDate="2014-11-22T18:09:01.520" Score="41" ViewCount="3804" Body="&lt;p&gt;[Mauro Allegranza has answered the question of who introduced the notation &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; (Euler, followed later by Gauss), so I have changed the title. I have also edited the question in other ways to make it clearer what I am asking.]&lt;/p&gt;&#xA;&lt;p&gt;A common bit of mathematical folklore states that &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; was introduced to guard against the fallacy &lt;span class=&quot;math-container&quot;&gt;$1=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=-1$&lt;/span&gt;. (See for example &lt;a href=&quot;https://hsm.stackexchange.com/questions/34/what-mathematical-developments-discoveries-caused-imaginary-numbers-to-gain-acce&quot;&gt;this question&lt;/a&gt;, and this passage from &lt;a href=&quot;http://en.wikipedia.org/wiki/Complex_number#History&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;. Note the [&lt;em&gt;citation needed&lt;/em&gt;] in the Wikipedia entry.)&lt;/p&gt;&#xA;&lt;p&gt;This seems plausible to modern mathematicians because of the work of 19th century mathematicians, especially Riemann. We cannot define a (single-valued) square-root function in the entire complex plane satisfying the identity &lt;span class=&quot;math-container&quot;&gt;$\sqrt{xy}=\sqrt{x}\sqrt{y}$&lt;/span&gt;. (Indeed, the above fallacy, recast, proves that.) So we are naturally suspicious of the expression &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; used to define a value. Which square root is meant, &lt;span class=&quot;math-container&quot;&gt;$\pm i$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;However, modern sensibilities are not a reliable guide to the thoughts of 18th century mathematicians. I will try to cast some doubt on the &amp;quot;fallacy-guarding&amp;quot; explanation below, but first let me elucidate my question. An alternate hypothesis is that Euler simply introduced the symbol &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; for brevity. Perhaps he felt that this fundamental constant deserved a standard name, just as he introduced &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; for the base of natural logarithms. Is there any &lt;em&gt;documentary evidence&lt;/em&gt; to decide between these hypotheses? (For example, if Euler placed a discussion of the above fallacy in close proximity to the introduction &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, that would be evidence for the &amp;quot;fallacy-guarding&amp;quot; hypothesis.)&lt;/p&gt;&#xA;&lt;p&gt;I've poked around on the internet, looking for a definitive answer. Searching math.stackexchange with the tags [math-history][complex-numbers] yielded nothing useful. The &amp;quot;Google full text&amp;quot; function applied to Paul J. Nahin's book &lt;a href=&quot;http://press.princeton.edu/titles/9259.html&quot; rel=&quot;noreferrer&quot;&gt;An Imaginary Tale: The Story of &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;&lt;/a&gt; (with the key &amp;quot;notation&amp;quot;) also failed to answer the question.&lt;/p&gt;&#xA;&lt;p&gt;My reasons for questioning the folklore: First note that introducing &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; does not unequivocally prevent the fallacy, which can be rewritten &lt;span class=&quot;math-container&quot;&gt;$1=\sqrt{(-1)(-1)}=i\cdot i=-1$&lt;/span&gt;. A closely related argument is that the notation &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; is inherently ambiguous. However, the notational definition &amp;quot;We will use &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; to denote one of the two square roots of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;&amp;quot; is no more ambiguous than &amp;quot;We will use &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; to denote one of the two square roots of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;&amp;quot; . The fallacy lies in the unrestricted use of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{xy}=\sqrt{x}\sqrt{y}$&lt;/span&gt;, not the use of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; to stand for an (arbitrarily chosen) square root of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is also a historical reason to question the &amp;quot;fallacy-guarding&amp;quot; hypothesis. The emphasis on single-valued functions was far less prominent in the 18th century. From the Wikipedia article &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_the_function_concept#Euler&quot; rel=&quot;noreferrer&quot;&gt;History of the function concept&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Euler's own definition reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Euler also allowed multi-valued functions whose values are determined by an implicit equation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Later Euler gave another definition, although this was apparently intended as a generalization of his previous definition. Sorting out the issues with single-valued vs. multi-valued functions was largely a task of the 19th century, with Dirichlet and Riemann playing prominent roles.&lt;/p&gt;&#xA;&lt;p&gt;Now, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{xy}=\sqrt{x}\sqrt{z}$&lt;/span&gt; is fine as an equation between multi-valued functions: it simply says that the set of values on the left is equal to the set of values on the right (given the obvious definition for the product of two sets of values). Unbridled use of multi-valued equations has its pitfalls, but this concern seems to belong more to the 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Much of Euler's work was more &amp;quot;formula-centric&amp;quot; than we are used to today. His free-wheeling computations with infinite series are well-known. Although they were (mostly) eventually justified, Euler's standards of rigor were not ours.&lt;/p&gt;&#xA;&lt;p&gt;For these reasons, I find the &amp;quot;fallacy-guarding&amp;quot; argument about as plausible as the &amp;quot;brevity&amp;quot; argument. Is there any contemporaneous historical evidence to help decide the question?&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="4251" LastEditDate="2020-06-22T18:36:54.243" LastActivityDate="2023-10-18T06:19:29.927" Title="What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?" Tags="|mathematics|notation|complex-analysis|euler|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="547" PostTypeId="2" ParentId="546" CreationDate="2014-11-22T18:52:24.297" Score="27" Body="&lt;p&gt;According to Florian Cajori, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486677664&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Mathematical Notations&lt;/a&gt; (1928 - Dover reprint), Vol II, page 128 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;498&lt;/strong&gt;. It was Euler who first used the letter &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;. He gave it in a memoir presented in 1777 to the Academy at St. Petersburg, and entitled &amp;quot;De formulis differentialibus etc.,&amp;quot; but it was not published until 1794 after the death of Euler. [&lt;em&gt;footnote&lt;/em&gt; : The article was published in his &lt;em&gt;Institutiones calculi integralis&lt;/em&gt; (2d ed.), Vol. IV (St. Petersburg, 1794), p. 184. See W.W.Beman in &lt;em&gt;Bull.Amer.Math.Soc.&lt;/em&gt;, Vol. IV (1897-98), p. 274. See also &lt;em&gt;Encyclopédie des scien. math&lt;/em&gt;., Tom. I,&#xA;Vol. I (1904), p. 343, n. 60.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As far as is now known, the symbol &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; did not again appear in print for seven years, until 1801. In that year Gauss [&lt;em&gt;footnote&lt;/em&gt; : K.F.Gauss, &lt;em&gt;Disquisitiones arithmeticae&lt;/em&gt; (Leipzig, 1801), No. 337; translated by A. Ch. M. Poullet-Delisle, &lt;em&gt;Recherches arithmétiques&lt;/em&gt; (Paris, 1807); Gauss, &lt;em&gt;Werke&lt;/em&gt;, Vol. I (Gottingen, 1870), p. 414] began to make systematic use of it; the example of Gauss was followed in 1808 by Kramp. [C.Kramp, &lt;em&gt;Eléments d'arithmétique universelle&lt;/em&gt; (Cologne, 1808), &amp;quot;Notations.&amp;quot;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See page 129 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;499&lt;/strong&gt;. [...] in 1847 Cauchy began to use &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; in a memoir on a new theory of imaginaries [&lt;em&gt;footnote&lt;/em&gt; : A.L.Cauchy, &lt;em&gt;Comptes rendus&lt;/em&gt;, Vol. XXIV (1847), p.1120=(&lt;em&gt;Oeuvres complètes&lt;/em&gt; (1st ser.), Vol. X, p. 313, 317, 318.] where he speaks of &amp;quot; .., le signe symbolique &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;, auquel les geometres allemands substituent la lettre &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;,&amp;quot; and continues: &amp;quot;Mais il est évident que la théorie des &lt;em&gt;imaginaires&lt;/em&gt; [emphasis added] deviendrait beaucoup plus claire encore et beaucoup facile à saisir, qu'elle pourrait etre mise à la portée de toutes les intelligences, si l'on parvenait à réduire les expressions imaginaires, et la lettre &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; elle-meme, a n'etre plus que des quantités réelles.&amp;quot; Using in this memoir the letter &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; as a &lt;em&gt;lettre symbolique&lt;/em&gt;, substituted for the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in the formulas considered, he obtains an &lt;em&gt;équation imaginaire&lt;/em&gt;, in which the symbolic letter &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; should&#xA;be considered as a real quantity, but indeterminate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See page 130 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;501&lt;/strong&gt;. &lt;em&gt;De Morgan's comments on&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;. The great role that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; has played in the evolution of algebraic theory is evident from the statements of Augustus de Morgan. In 1849 he spoke of &amp;quot;the introduction of the &lt;em&gt;unexplained symbol&lt;/em&gt; [emphasis added] &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;,&amp;quot; [&lt;em&gt;footnote&lt;/em&gt; : Augustus de Morgan, &lt;em&gt;Trigonometry and Double Algebra&lt;/em&gt; (London, 1849), p. 41] and goes on to say: &amp;quot;The use, which ought to have been called &lt;em&gt;experimental&lt;/em&gt;, of the &lt;em&gt;symbol&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;, under the name of an &lt;em&gt;impossible&lt;/em&gt; quantity, shewed that; come how it might, the intelligible results (when such things occurred) of the experiment were always true, and otherwise demonstrable. I am now going to try some new experiments.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Comments&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The issue in Cauchy's seems to be : we have here a &amp;quot;name&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;) for an &amp;quot;impossible&amp;quot; quantity, that is for something that cannot exists. This is - I think - the context of Cauchy's concerns: to find a &amp;quot;reference&amp;quot;, that must be some &amp;quot;real&amp;quot; quantity, for the &lt;em&gt;symbolic&lt;/em&gt; expression .&lt;/p&gt;&#xA;&lt;p&gt;We can see in this &lt;a href=&quot;https://math.stackexchange.com/questions/845854/descartes-on-imaginary-unit&quot;&gt;post&lt;/a&gt; some references to Descartes's mathematical works :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;he names some roots 'true', some 'implicit' (that is, less than nothing), and some 'imaginary' (that is, not explicable at all)&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Regarding Euler and the benefits of a &amp;quot;good&amp;quot; symbolism, see L.Euler, &lt;a href=&quot;https://archive.org/details/elementsalgebra00lagrgoog&quot; rel=&quot;noreferrer&quot;&gt;Elements of Algebra&lt;/a&gt; (1765; English translation by John Hewlett : 1822) &lt;strong&gt;§148-149&lt;/strong&gt;, page 43 : &amp;quot;as &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a}$&lt;/span&gt; multiplied by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{b}$&lt;/span&gt; makes &lt;span class=&quot;math-container&quot;&gt;$\sqrt{ab}$&lt;/span&gt;, we shall have &lt;span class=&quot;math-container&quot;&gt;$\sqrt{6}$&lt;/span&gt; for the value of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{−2}$&lt;/span&gt; multiplied by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{−3}$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;This passage supports the conjecture that Euler understood the need of a specific symbol in order to avoid the above &amp;quot;fallacy&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Addendum&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;For &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, see Vol II, page 9 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;396&lt;/strong&gt;. &lt;em&gt;First occurrence of the sign&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. The modern notation for 3.14159 . . . . was introduced in 1706. It was in that year that William Jones [&lt;em&gt;footnote&lt;/em&gt; : William Jones, &lt;em&gt;Synopsis palmariorum matheseos&lt;/em&gt; (London, 1706), p.263.] made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of the circle to its diameter by the letter &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. He took this step&#xA;without ostentation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="251" LastEditDate="2021-02-14T22:06:52.217" LastActivityDate="2021-02-14T22:06:52.217" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="548" PostTypeId="2" ParentId="539" CreationDate="2014-11-22T20:04:52.487" Score="-3" Body="&lt;p&gt;By choosing &quot;a nearly ideal gas&quot; as a model Boltzmann in fact killed thermodynamics - second law violations can only take place in structured systems. Thermodynamics is long dead - nowadays perpetual motion machines of the second kind are being published in prestigious journals but there is no reaction at all from the scientific community: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.researchgate.net/publication/258157913_Electricity_Generated_from_Ambient_Heat_across_a_Silicon_Surface/file/3deec5272604889353.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.researchgate.net/publication/258157913_Electricity_Generated_from_Ambient_Heat_across_a_Silicon_Surface/file/3deec5272604889353.pdf&lt;/a&gt; &#xA;  Electricity generated from ambient heat across a silicon surface, Guoan Tai, Zihan Xu, and Jinsong Liu, Appl. Phys. Lett. 103, 163902 (2013): &quot;We report generation of electricity from the limitless thermal motion of ions across a two-dimensional (2D) silicon (Si) surface at room temperature. (...) ...limitless ambient heat, which is universally present in the form of kinetic energy from molecular, particle, and ion sources, has not yet been reported to generate electricity. (...) This study provides insights into the development of self-charging technologies to harvest energy from ambient heat, and the power output is comparable to several environmental energy harvesting techniques such as ZnO nanogenerator, liquid and gas flow-induced electricity generation across carbon nanotube thin films and graphene, although this remains a challenge to the second law of thermodynamics...&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs10701-014-9781-5&quot; rel=&quot;nofollow&quot;&gt;http://link.springer.com/article/10.1007%2Fs10701-014-9781-5&lt;/a&gt; &#xA;  Experimental Test of a Thermodynamic Paradox, D. P. Sheehan et al, Foundations of Physics, March 2014, Volume 44, Issue 3, pp 235-247: &quot;...there arise between the vane faces permanent pressure and temperature differences, either of which can be harnessed to perform work, in apparent conflict with the second law of thermodynamics. Here we report on the first experimental realization of this paradox, involving the dissociation of low-pressure hydrogen gas on high-temperature refractory metals (tungsten and rhenium) under blackbody cavity conditions. The results, corroborated by other laboratory studies and supported by theory, confirm the paradoxical temperature difference and point to physics beyond the traditional understanding of the second law.&quot;&lt;/p&gt;&#xA;" OwnerUserId="394" LastActivityDate="2014-11-22T20:04:52.487" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="550" PostTypeId="1" AcceptedAnswerId="551" CreationDate="2014-11-23T16:18:35.117" Score="12" ViewCount="488" Body="&lt;p&gt;&lt;strong&gt;BÉZOUT'S THEOREM:&lt;/strong&gt;&#xA;Let $F$ and $G$ be projective plane curves of degree $m$ and $n$ respectively. Assume $F$ and $G$ have no common component. Then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\displaystyle\sum_{P}I(P,F\cap G)=mn$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$I(P,F\cap G)$ is the intersection number of $F$ and $G$ at $P$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to get any reference about the history of this interesting result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="396" LastActivityDate="2015-06-14T19:58:12.153" Title="History of a result from Bézout" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="551" PostTypeId="2" ParentId="550" CreationDate="2014-11-23T17:11:45.980" Score="12" Body="&lt;p&gt;The big surprise I got from my research was that the theorem apparently originated with Maclaurin, whom we remember more for the Maclaurin series than this theorem. From &lt;a href=&quot;http://users.humboldt.edu/pgoetz/Talks/bezout&amp;#39;s.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this pdf&lt;/a&gt; (you have to go to the last page to get to the history portion):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Maclaurin, Euler, Cramer (1700’s) assert the theorem, no valid proof&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But it was Bézout who found a proof, albeit flawed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;2&quot;&gt;&#xA;  &lt;li&gt;Etienne Bezout (1730-1783), flawed proof, didn’t account for multiplicities correctly&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later on, Halphen and van der Waerden came up with accurate proofs that trumped Bézout's.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Bezout.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; biography appears to allude both to Maclaurin's discovery of the theorem and Bézout's &quot;proof&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In this work Bézout also gave the first satisfactory proof of a result of Maclaurin on the intersection of two algebraic curves.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;This work&quot; seems to refer to &lt;em&gt;Théorie générale des équations algébraiques&lt;/em&gt;, published in 1779.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't seem to find a reference to the theorem on the site's &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Maclaurin.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maclaurin page&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As for specific references, I can only give you these (by the way, &lt;a href=&quot;http://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem#References&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; cites &lt;a href=&quot;https://mathoverflow.net/questions/42127/generalization-of-bezouts-theorem&quot;&gt;Math Overflow!&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.jstor.org/discover/10.2307/2374293?uid=3739808&amp;amp;uid=2&amp;amp;uid=4&amp;amp;uid=3739256&amp;amp;sid=21104631220721&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Bezout Theorem&lt;/a&gt;, by Kuo-Tsai Chen&lt;/li&gt;&#xA;&lt;li&gt;Some of the references &lt;a href=&quot;http://www.seminariomatematico.unito.it/rendiconti/cartaceo/43-1/Volume%2043-1%20(1985)%20-%2003%20-%20Old%20and%20new%20problems%20about%20Bezout%20s%20theorem.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Wikipedia does say, though,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bezout's theorem was essentially stated by Isaac Newton in his proof of lemma 28 of volume 1 of his Principia, where he claims that two curves have a number of intersection points given by the product of their degrees. The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques. Bézout, who did not have at his disposal modern algebraic notation for equations in several variables, gave a proof based on manipulations with cumbersome algebraic expressions. From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold. This led to a sentiment, expressed by certain authors, that his proof was neither correct nor the first proof to be given.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The source cited is &lt;em&gt;Complex Algebraic Curves&lt;/em&gt;, by Frances Kirwan.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-06-14T19:58:12.153" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="552" PostTypeId="2" ParentId="533" CreationDate="2014-11-24T13:29:55.440" Score="5" Body="&lt;p&gt;The OED writes this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Originally: the Roman unit of distance of 1000 paces (mīlle passus or passuum), reckoned to have been about 1618 yards (approx. 1479 metres). Hence: a unit of distance derived from this used in the British Isles and in other English-speaking countries, and now equal to 1760 yards (approx. 1609 metres). Frequently with a prefixed numeral forming a phrase used attributively.The length of the mile has varied considerably at different periods and in different localities, chiefly owing to the influence of the agricultural system of measures with which the mile has been brought into relation (see furlong n.). It was fixed by statute at 1760 yards (viz. 8 furlongs of 40 poles, each pole being 16½ feet) in 1592 (Act 35 Eliz. I, c. 6, s. 8), and in Britain is also called a statute mile. This is also the legal mile in the United States. The obsolete Irish mile was 2240 yards (approx. 2048 metres), and the Scottish mile (obsolete by the late 19th century) was 1976 yards (approx. 1807 metres) although values probably varied according to time and place.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This suggests that the mile is originally a military term, a round counting of the paces of marching soldiers, and is not based on a calculation of the size of the earth. Educated Romans were presumably aware of Eratosthenes' rather accurate measurement of the earth's circumference, but it seems unlikely that this should have filtered down to military usage.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2014-11-24T13:29:55.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="553" PostTypeId="1" AcceptedAnswerId="579" CreationDate="2014-11-24T14:34:16.450" Score="18" ViewCount="2489" Body="&lt;p&gt;Anyone who has studied Feynman's path integral will know that it makes quantum mechanics more like classical mechanics. A student who has learned about the Lagrangian will easily understand the concept of quantum mechanics through the path integral (I think). Why do we still keep using Schrödinger's approach? It's history now.&lt;/p&gt;&#xA;&lt;p&gt;Maybe to fully understand the Feynman's approach, you need to, erm, integrate so much? I mean, maybe it is based heavily on math?&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="4251" LastEditDate="2020-07-09T20:43:35.790" LastActivityDate="2020-07-09T20:43:35.790" Title="Why isn't Feynman's path integral taught more widely and earlier in today's academic physics curricula?" Tags="|physics|quantum-mechanics|mathematical-physics|quantum-field-theory|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="554" PostTypeId="1" AcceptedAnswerId="592" CreationDate="2014-11-24T20:05:35.033" Score="4" ViewCount="292" Body="&lt;p&gt;I want to know if, any old civilizations have considered mental disorders as real disease which could be cured? If not, then where it got this status in modern science?&lt;/p&gt;&#xA;" OwnerUserId="245" LastEditorUserId="26" LastEditDate="2014-11-24T21:28:40.513" LastActivityDate="2014-11-29T17:52:36.960" Title="When were mental disorders first recognized as diseases?" Tags="|biology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="555" PostTypeId="2" ParentId="553" CreationDate="2014-11-24T22:02:26.073" Score="3" Body="&lt;p&gt;For quantum mechanics path integral can be done rigorously using &lt;a href=&quot;http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula&quot; rel=&quot;nofollow&quot;&gt;Feynman-Kac formula&lt;/a&gt; and &lt;a href=&quot;http://www.scholarpedia.org/article/Path_integral:_mathematical_aspects#Analytic_continuation&quot; rel=&quot;nofollow&quot;&gt;analytic continuation&lt;/a&gt;, but the approach is by no means elementary. A single look at the Feynman-Kac formula shows why it is not a great place for students to start at. Moreover, understanding where it comes from is itself a chore. While path integral heuristics are intuitively attractive, implementing them in practice - with actual computations - is a different matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While path integral is called &quot;integral&quot; even in the context of quantum mechanics it is something more complicated, a pairing between a distribution and a test function. Analytically continued Feynman-Kac formula is impractical for actual calculations, but what it amounts to is solving an initial-boundary problem for the Schrödinger equation. There are many more methods for doing that, separation of variables and Laplace transform for example, and they are more elementary than the theory of measures and distributions on infinite dimensional spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, teaching Feynman diagrams early was argued convincingly by John Baez and others, not in the context of QFT, but in the context of computing finite-dimensional integrals perturbatively, see &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.206.814&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; for an elementary exposition.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-11-24T23:49:43.900" LastActivityDate="2014-11-24T23:49:43.900" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="556" PostTypeId="2" ParentId="553" CreationDate="2014-11-24T22:12:21.293" Score="6" Body="&lt;p&gt;Let me try to summarize what was said in the comments and add something. (I am a teacher of mathematics, not physics, but this is close). First of all not all students study Lagrangian mechanics at an early stage. It requires much more mathematical sophistication than most undergraduate students have. (I am talking of US here).&lt;/p&gt;&#xA;&lt;p&gt;Second, Feynman's integral itself cannot be called fully mathematically justified. There were many attempts to formulate it rigorously, I am not prepared to analyse these attempts and how successful they are, but one thing is clear: this justification requires by far more advanced mathematics than most physicists know (let alone undergraduates).&#xA;In their daily work most physicists just do not care about rigorous mathematical justification. But teaching is another matter:-)&lt;/p&gt;&#xA;&lt;p&gt;Third. It is not true that Schrodinger's formulation has only historic value. It is simple and convenient, and there is a rich and rigorous mathematical theory behind it. It is widely used by physicists and mathematicians. (Heisenberg's approach (matrix mechanics) is less popular, but even that has much more to it than &amp;quot;historical significance&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Fourth. Education is EXTREMELY conservative. Let me give another example: multivariate calculus (Stokes theorem and its relatives). The modern exposition is based on the formalism of differential forms of E. Cartan, invented in the beginning of 20-th century.&#xA;It is much SIMPLER and more powerful than what we teach to undergraduates. Still undergraduate calculus is taught with 19-th century formalism of &amp;quot;vector analysis&amp;quot; almost everywhere (except France).&#xA;The reason for this is that teachers tend to teach the way they were taught themselves:-)&lt;/p&gt;&#xA;&lt;p&gt;But in the case of Feynman integral, I think that the main reasons are the second and third.&lt;/p&gt;&#xA;&lt;p&gt;By the way, Feynman himself, in his Lectures on Physics explains quantum mechanics without&#xA;his integral. He has one special lecture (a kind of appendix) on the Principle of least action, but warns that this is a much more advanced stuff than the rest of the book:-)&lt;/p&gt;&#xA;&lt;p&gt;P.S. I do not want to register for the Physics site to answer the question whether &amp;quot;Lagrangian&amp;quot; must be capitalized. But my opinion is this: there are rules of the language.&#xA;They may be logical or not, we may like them or not. But there are codified rules, and it is better to obey them. In Russian all these words are written with lower case, in German they write riemannsche Flache, but in English: Riemann surface. So in different languages&#xA;different words in the same expression are capitalized.&lt;/p&gt;&#xA;&lt;p&gt;P.P.S. For many years I was looking in bookstores the book of Feynman and Hibbs, Quantum mechanics and path integrals. The shelves are always full of &amp;quot;Are you joking Mr. Feynman?&amp;quot; and such, but this book is long out of print... Finally I purchased a Russian translation in a used book store in Ukraine.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-07-09T14:41:28.210" LastActivityDate="2020-07-09T14:41:28.210" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="557" PostTypeId="1" CreationDate="2014-11-25T01:46:22.117" Score="7" ViewCount="496" Body="&lt;p&gt;This is going to be tricky because the discovery/synthesis of elements has taken place over centuries. It might be best to restrict this purely to the elements contained on Dmitri Mendeleyev's table, but I'd let the community decide. It may be too broad if all the elements are included, so this may be the best course of action.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, my question is fairly straightforward. When (and by who) were the elements on the periodic table (the modern-day one or simply Mendeleyev's) first standardized? I'd prefer that any answers not use Wikipedia.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="26" LastEditDate="2014-11-25T10:07:44.143" LastActivityDate="2018-03-26T16:00:23.320" Title="When were the abbreviations of chemical elements standardized?" Tags="|chemistry|terminology|naming-conventions|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="558" PostTypeId="2" ParentId="401" CreationDate="2014-11-25T02:12:04.310" Score="5" Body="&lt;p&gt;Let me also add to the list the long conflict between the wave theory and corpuscular theory of light. First serious corpuscular theory was due to Newton (correct me if I am wrong here), it was published in his Opticks. Huygens and Hooke were developing wave theory.&#xA;Soon after the publication of Principia Newton become an absolute authority,&#xA;first in England but some time later, when his gravity theory was brilliantly confirmed, also on the continent. This &quot;cult of Newton&quot; made substantial harm, especially in England.&#xA;The wave theory was forgotten, almost for a century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only in the beginning of 19-th century, wave theory was resurrected and confirmed beyond reasonable doubt by experiment (Fresnel, Th. Young). Newton was &quot;proved wrong&quot; and the wave theory was widely accepted. See about &quot;Poisson's spot&quot; on Wikipedia about one of the most dramatic events in this story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everyone thought that the wave theory is the ultimate one until  1905:-)&#xA;In 1905 Einstein's theory of photoelectric effect was published, which showed again that light&#xA;consists of particles in some sense. The modern theory is a kind of synthesis of waves and particles, so this is perhaps the best example of what was asked:-) &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-25T02:12:04.310" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="559" PostTypeId="2" ParentId="534" CreationDate="2014-11-25T05:19:14.607" Score="4" Body="&lt;p&gt;This image is certainly not of Christian Goldbach. It is attributed to &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Grassmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Grassmann&lt;/a&gt; in all reliable sources I am aware of, and I see no reason to doubt that Grassmann is indeed the one pictured. The text &lt;a href=&quot;http://www.springer.com/birkhauser/history+of+science/book/978-3-0346-0404-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Past to Future: Graßmann's Work in Context&lt;/a&gt; uses it as a cover photo, as do many other texts on Grassmann. It seems to be taken from an 1874 xylograph after a photograph based on the attribution in that text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several reasons we can tell it is not Goldbach. One is that photography was not even invented in Goldbach's time. The earliest attempts at photography were in the early 19th century, and the earliest known photographs are from the 1820s (It didn't become practical until around 1839). Goldbach died in 1764, long before even &lt;a href=&quot;http://www.hrc.utexas.edu/exhibitions/permanent/firstphotograph/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the earliest known photographs&lt;/a&gt;. But the image is quite clearly a photograph, rather than the work of an artist. In addition, the style of dress in the image is very much in the character of 19th century fashion, rather than 18th century. If that isn't enough, doing a &lt;a href=&quot;https://www.google.com/searchbyimage?image_url=https://i.stack.imgur.com/MuNVP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google reverse image search&lt;/a&gt; on the image gives the result of Grassmann; while that isn't an indication that the image is actually of Grassmann, it shows that the internet consensus points to him, casting further doubt onto the (relatively) few sources which identify this image as Goldbach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt any serious historians have bothered to investigate why the two men have been so confused on the internet or where the confusion started. It's likely this mistake has not made its way into the print literature at all, given how easy it is to disprove, but it continues to propagate on poor websites with flimsy fact-checking. It could easily be a case of some writer simply doing an image search for Goldbach, finding an image of Grassmann on some page which discussed both of them, and not bothering to check who was actually in the image (or indeed do any sanity-checking at all). One can find similar confusion today between Goldbach and &lt;a href=&quot;https://i.stack.imgur.com/21jmk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this picture of Riemann&lt;/a&gt; (see  &lt;a href=&quot;http://scitechdaily.com/century-old-goldbach-weak-conjecture-closer-to-being-solved/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) as well as several images of Euler, and once one writer makes a mistake on the internet (or indeed anywhere), it tends to get copied among sources with weaker fact-checking. In any case, it certainly doesn't require serious research to falsify the claim that the image is of Goldbach; common sense does just fine here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the question of whether images of Goldbach exist, I believe there is still no known image of him. However, this is based as much on a lack of evidence to the contrary as actual support. I'd imagine that if an image of Goldbach were found, it would be at least big enough news to get a news article or notice written &lt;em&gt;somewhere&lt;/em&gt;. After reasonably extensive searching, I have found no such article. In fact, the fact that Google is still highly indexing images of other mathematicians on searching for Goldbach is reasonably strong evidence that no image of Goldbach is known to the internet at large.&lt;/p&gt;&#xA;" OwnerUserId="58" LastActivityDate="2014-11-25T05:19:14.607" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="560" PostTypeId="1" AcceptedAnswerId="562" CreationDate="2014-11-25T21:06:54.507" Score="10" ViewCount="820" Body="&lt;p&gt;Thought experiments, imaginary situations designed to ponder conclusions of a theory, have been used throughout history. There is even a Stanford encyclopedia &lt;a href=&quot;http://plato.stanford.edu/entries/thought-experiment/#RecDev&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; about them. But the famous ones that come to mind are either from physics (Archimedes's overturning of the Earth, Newton's bucket, Maxwell's demon, Einstein's elevator, etc.) or philosophy/psychology (Plato's cave, Descartes' evil demon, Leibniz's mill, etc.). The article mentions that more recently &quot;noteworthy contributions have been made exploring the importance of thought experiments in disciplines other than mathematics, philosophy, or physics. They include history, the social sciences, and Christian theology&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conspicuously absent are natural sciences other than physics. Biology is not even mentioned, and chemistry &quot;has none of note at all. Why is this the case? Perhaps it is merely an historical accident that chemists never developed a culture of doing thought experiments&quot;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What kinds of thought experiments appear in biology? Is it true that they were never entertained in chemistry? If so, are there explanations better than &quot;historical accident&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;EDIT: This &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs10698-006-9019-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; discusses possible &quot;fundamental difference&quot; between physics and chemistry as an explanation for scarcity of thought experiments in the latter. Darwin's thought experiments are discussed &lt;a href=&quot;http://www.pitt.edu/~pittcntr/About/center_publications/thought_experiments/13_lennox.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, see also HDE's answer and &lt;a href=&quot;https://biology.meta.stackexchange.com/questions/160/thought-experiments-and-hypothetical-questions&quot;&gt;general discussion&lt;/a&gt; on Biology SE. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-03-16T15:49:46.730" LastActivityDate="2014-11-27T00:21:35.903" Title="Are there historical examples of thought experiments in biology and chemistry?" Tags="|chemistry|biology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="562" PostTypeId="2" ParentId="560" CreationDate="2014-11-26T01:32:44.710" Score="7" Body="&lt;p&gt;In biology, so far I've found &lt;a href=&quot;http://web.archive.org/web/20110523080407/http://www-miller.ch.cam.ac.uk/levinthal/levinthal.html&quot;&gt;Levinthal's Paradox&lt;/a&gt;. The transcript from a lecture given by Prof. Cyrus Levinthal can be found at the previous link. The gist of the paradox is this (not a quote; the indent is used for emphasis):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There exists a multitude of shapes proteins can take, yet they manage to fold into the correct shape extremely quickly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The thought experiment proper is as follows:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Take an unfolded protein that will fold into a certain state in which it can perform its function&lt;/li&gt;&#xA;&lt;li&gt;Observe the various shapes it folds into&lt;/li&gt;&#xA;&lt;li&gt;Note that (this part is the kicker, and supported by experiment) the protein will fold into the correct shape within a fraction of a second.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The idea is explained slightly clearer &lt;a href=&quot;http://biologypop.com/levinthals-paradox/&quot;&gt;here&lt;/a&gt; - at least, clearer to a biology layman like myself.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;There's a long and interesting discourse &lt;a href=&quot;http://www.academia.edu/8767243/Thought_Experiments_in_Biology&quot;&gt;here&lt;/a&gt;. (If it asks you if you want to join, simply click on the 'x' on the left to go straight to the document.) In it the authors (Schlaepfer and Weber) argue that Darwin put forth thought experiments in, among other publications, &lt;em&gt;On the Origin of Species&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After noting that there were quite a few thought experiments in the field of evolutionary biology, the authors then move on to molecular biology, mentioning Levinthal's paradox and a problem regarding protein synthesis - how the body can make so many different types of proteins. I think these are more paradoxes than thought experiments, but that's purely my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The authors go on in a new angle, talking about how some have argued that computational forays in biology count as thought experiments. They then conclude the text.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-11-26T21:50:11.223" LastActivityDate="2014-11-26T21:50:11.223" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="563" PostTypeId="2" ParentId="560" CreationDate="2014-11-26T07:42:41.990" Score="5" Body="&lt;p&gt;&lt;strong&gt;&quot;Daisyworld&quot;&lt;/strong&gt; comes to mind as a thought experiment in ecology/earth-system science (it was also simulated on computer but it is really just a thought experiment).&lt;br&gt;&#xA;For those not familiar with it, it is an experiment designed by James Lovelock (of Gaia Theory fame) and Andrew Watson in 1983.&lt;br&gt;&#xA;The settings, from the paper itself:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Daisyworld is a cloudless planet with a negligible atmospheric greenhouse on which the only plants are two species of daisy of different colours. One species is dark – ground covered by it reflects less light than bare ground – while the other is light and reflects more than the bare ground.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The thought experiment continue in imagining the growth rate of each species (influenced by the planet temperature) and the temperature change on the planet (influenced by the planet albedo, itself influenced by the daisies).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watson, A. J. &amp;amp; Lovelock, J. E. 1983. &lt;a href=&quot;http://www.iai.ga.a.u-tokyo.ac.jp/mizo/iwata/edumaterial/Biological%20homeostasis%20of.pdf&quot;&gt;Biological homeostasis of the global environment: the parable of Daisyworld.&lt;/a&gt; Tellus, 35B, 284-289.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2014-11-26T07:42:41.990" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="564" PostTypeId="1" AcceptedAnswerId="589" CreationDate="2014-11-26T08:03:29.060" Score="8" ViewCount="168" Body="&lt;p&gt;$\mathrm{\delta^{18}O}$ (i. e. the ratio of $\mathrm{^{18}O/^{16}O}$ compared to a standard) is massively used today in paleoclimatology as a global temperature proxy.&lt;br&gt;&#xA;But how did it become a proxy for temperature? When and by whom was the correlation observed for the first time?&lt;br&gt;&#xA;Was the model explaining why it followed global temperature found straight away or was this proxy purely empirical for a long time?&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-12-01T12:31:22.327" LastActivityDate="2014-12-01T12:31:22.327" Title="On the use of $\mathrm{\delta^{18}O}$ as temperature proxy" Tags="|chemistry|earth-sciences|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="565" PostTypeId="2" ParentId="553" CreationDate="2014-11-26T09:55:36.890" Score="7" Body="&lt;p&gt;I would like to add a slightly more technical answer to supplement the other (although it is a bit late). It may be helpful, I think, and it touches a broader issue which interesting. It's a conflict of history of teaching versus history of the mathematical prerequisites for the material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, &lt;em&gt;supposing&lt;/em&gt; the topic material is taught early on, how would that work? I like the following books:---they benefit from the long experience of the authors in teaching this material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cartier P, DeWitt-Morette C&lt;/strong&gt;, 2006, &lt;em&gt;Functional Integration,&lt;/em&gt; Cambridge University Press. (B)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;DeWitt B&lt;/strong&gt;, 2003, &lt;em&gt;Global Approach to Quantum Field Theory,&lt;/em&gt; I, II, Oxford University Press. (C)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prior this, the following textbook would introduce all basics, for example to student starting with basic knowledge of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Choquet-Bruhat Y, DeWitt-Morette C&lt;/strong&gt;, 1982, &lt;em&gt;Analysis, Manifolds, and Physics&lt;/em&gt;, I, Elsevier. (A)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would be a selfcontained series of courses: A, B, C, which would also teach a good chunk of other material, so a pair of introductory mathematical courses, whose content is covered in (A) would be dropped. So, only one more course, in total, would be required. Why isn't this happening?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(X) The answer is indeed historical, due to historically the prerequisite mathematics for the subject, when first developed, gaining the reputation of being &quot;very advanced material&quot; with few introductory texts, and this reputation stuck. Even when no longer justified, it influences how departments construct their curriculum. Mathematics is synergistic. What was once difficult becomes easier when other concepts are discovered and developed. Advances in method over time made the calculations easier and no longer really something difficult in the usual instances, and plenty of introductory books to the material &lt;em&gt;now&lt;/em&gt; exist. However, &lt;em&gt;reputations change more slowly than the environment that generated them.&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;em&gt;A.E.&lt;/em&gt; above brought up the point that analytic continuation is not taught to undergraduates in many places, and complex variables aren't mandatory. That's my experience too. I suggest the same circumstances being the reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For rigorous work with functional integration, distribution methods must be used. (Now I prefer sheaf cohomology methods leading to generalized functions. But the justification and application is the same in this case.) The historical issue is: category theory, sheaves, and generalized functions became widely known and fully developed in the 1970's. (After Feynman's integral was in use for over twenty years!) To know how to do calculations in the arbitrary case, beyond the intuitive setting up of the problems (which was developed 1920's-1950's), these later methods are required. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are plenty of basic introductory books, published 1990's-present, which are not too advanced for undergraduates. E.g., Lawvere's book, MacLane's several books, etc., assume the reader begins with no knowledge whatsoever about even the most elementary mathematical concepts! And the introductory books on generalized functions and distributions begin easily: right hand limits of appropriate series of sums and products applied to 19th century problems, to solve them more easily than is traditional.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue is that they are, in fact, books. Buying a book is not sufficient to learn anything. It must be read. There often (for various institutional reasons) simply isn't enough time in the major curriculum to present all the material without having the undergraduate &lt;em&gt;read something&lt;/em&gt; in &lt;em&gt;some book&lt;/em&gt; that is not presented in class. But in most universities, at this level, use books, if at all, mostly for homework problems... :_( Historically, the education system has been drifting away from required reading the STEM fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the other methods can be presented without required reading. And while the typical undergraduate, coming from a typical high school &lt;em&gt;can&lt;/em&gt; read books, they usually &lt;em&gt;don't&lt;/em&gt;. If they buy them, it's because they contain required homework problems. Many undergraduate courses are not structured to require reading at all. The textbooks are selected only for their changing homework sets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can imagine, and as is indeed the case, whatever cannot be taught this way, or has the reputation of &quot;advanced&quot; isn't mandatory or taught at all until well into graduate coursework. This is merely one case, it seems to me.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2014-11-26T09:55:36.890" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="566" PostTypeId="2" ParentId="419" CreationDate="2014-11-26T15:19:47.997" Score="3" Body="&lt;p&gt;At least for the ancient Greek world, we can see the Pythagorean idea of the &lt;em&gt;tetractys&lt;/em&gt;, the triangular figure made of pebbles, ordered in four rows to form a triangle, which add up to the perfect number ten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Walter Burkert, &lt;a href=&quot;http://books.google.it/books?id=0qqp4Vk1zG0C&amp;amp;printsec=frontcover&amp;amp;hl=it#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Lore and Science in Ancient Pythagoreanism&lt;/a&gt; (1972, German ed : 1962), page 72 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The kernel of pythagorean wisdom is the &quot;tetractys&quot;, or &quot;four-group&quot;, made up of the numbers $1,2,3,4$ which add uo to $10$. They are representen in a pebble figure, in the form of a &quot;perfect triangle&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is clearly a way of conceiving of &lt;em&gt;numbers&lt;/em&gt; as &quot;real&quot; objects with their own properties, and not only as a &quot;counting device&quot;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-11-26T15:19:47.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="576" PostTypeId="1" CreationDate="2014-11-27T01:57:10.497" Score="6" ViewCount="121" Body="&lt;p&gt;Is there any plots, etc. that show performance metrics on various natural language processing tasks as a function of year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example &lt;a href=&quot;http://www.zdnet.com/blog/btl/facial-recognition-technology-shows-improvement/4796&quot; rel=&quot;nofollow noreferrer&quot;&gt;the following figure&lt;/a&gt; shows the evolution of performance of face recognition systems:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Rx0Np.png&quot; alt=&quot;enter image description here&quot;&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for something similar for natural language processing tasks (e.g. evolution of the perplexity of the state-of-the-art language models, BLUE score of translation systems, accuracy of POS tagging, etc.).&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-11-29T01:45:39.550" Title="Evolution of performances in natural language processing tasks" Tags="|artificial-intelligence|linguistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="577" PostTypeId="1" CreationDate="2014-11-27T09:51:23.813" Score="8" ViewCount="1016" Body="&lt;p&gt;while searching for different methods of speed of light measurement, i came across one of the method of fizeau discussed below which i cannot fully understand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short,in Fizeau’s apparatus, a&#xA;beam of light was shone between the teeth of&#xA;a rapidly rotating toothed wheel, so the&#xA;“lantern” was constantly being covered&#xA;and uncovered. Fizeau had a&#xA;mirror, reflecting the beam back, where&#xA;it passed a second time between the&#xA;teeth of the wheel. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do understand the idea i.e. to measure the time during the course of light &quot;from the wheel to mirror and then back to the mirror&quot;.But where are the mirror,lantern and wheel  located? What kind of wheel is that? Does light pass through the holes (or teeth) in the wheel or light gets refracted through the wheel? Finally if a group of light particles goes through one of the hole (or teeth),then how do we know that the same group of light particles came back through which hole (or teeth) after its journey,since there are soo many light particles leaving and entering different holes (or teeth)?&lt;/p&gt;&#xA;" OwnerUserId="427" LastActivityDate="2018-10-02T02:58:11.953" Title="speed of light measurement" Tags="|experimental-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="578" PostTypeId="2" ParentId="577" CreationDate="2014-11-27T14:54:37.330" Score="7" Body="&lt;p&gt;This image:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/pHJaN.jpg&quot; alt=&quot;fizeau speed of light arrangement&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is the most explicit one I've found for the experimental arrangement. Here's another that is more schematic, but illustrates the basic idea:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xFkJc.jpg&quot; alt=&quot;fizeau speed of light arrangement&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now on to your specific questions. The light passes through the gaps between the teeth; it is not refracted. Pulses of light hit the mirror and are reflected back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the experiment, the speed of the wheel is gradually increased. Initially the light making it back through the toothed wheel gets darker and darker. That's because the light pulses arrive in time to hit the teeth instead of the gaps between. As the speed of the wheel is increased, the reflected light starts to get brighter, reaching a maximum when the timing is perfect: a pulse leaves the gap on the way to the mirror, reflects, and makes it back just in time to pass through the next gap. In principle, as the wheel speed is increased, the experimenter should see the reflected light getting darker, then lighter, then darker, etc. Thus, the second brightening should occur when the returning pulse of light goes through two gaps past the original gap.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some more historical details, from the &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Fizeau.html&quot; rel=&quot;noreferrer&quot;&gt;MacTutor biography&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In July 1849 Fizeau set up a mirror at his parents' home at Suresnes and another on Montmartre, the hill on the right bank of Paris. These were 8633 metres apart. Between them he set up a rapidly rotating toothed wheel and determined the speed of rotation necessary for the time taken by the light travelling between the mirrors to equal the time taken for the wheel to rotate by one tooth. He was thus the first to make a successful terrestrial measurement of the velocity of light. We know today that the value he found was almost as accurate as Römer's astronomical measurement having an error of about 5%.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-11-27T14:54:37.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="579" PostTypeId="2" ParentId="553" CreationDate="2014-11-27T16:27:34.787" Score="12" Body="&lt;p&gt;I do not agree with the statement, that the lack of mathematical rigor is a major reason for not teaching the path integral formalism in quantum mechanics. The common physicist is normally not interested in complete mathematical rigor, as long as the concepts make sense from a physical point of view and produce the right results. A good example of this is the Dirac formalism - every physicist knows it, and everyone with an elementary education in functional analysis knows that the mathematical justification cannot rely on pure Hilbert space theory. I don't know any textbook which even does an attempt to formulate Dirac's formalism in a rigorous way (Apart from that usual gibberish about spectral decompositions in finite dimensions, which does of course not apply to most problems in QM), that is because the physicist doesn't care and the mathematical physicist does not use bras and kets. The mathematical formulation needs, apart from spectral theory, also some stuff about locally convex and nuclear spaces, which is also pretty specific.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Feynman integral is a quite intuitive construct from a purely physical point of view, and everyone taking a course in quantum mechanics should have basic education in analytical mechanics, at least it's like this in Germany - usually Lagrange and Hamilton is covered, sometimes also a bit of Hamilton-Jacobi. I don't think this is too much to be asked from an undergrad student who had a course on mechanics. Furthermore, I also don't think that it is too hard for an undergrad to use it, therefore the skills required to introduce and work with the path integral should be given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my point of view, there are three major reasons, for the Schrödinger equation being the &quot;standard approach&quot; to quantum mechanics, and not the path integral formulation. Of course, these are correlated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first reason is of course, that it has always been done like this. For this reason, there exist a lot of good and established textbooks using this approach, whom I can base a lecture on. There is a wide range of good material and problems. Furthermore, depending on in which area you will work in, the Schrödinger approach is usually a good approach to work with, therefore it is by far not outdated - path integrals are primarily used by particle physicists, in other areas, Schrödinger is usually very useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second reason is, that simple problems one can imagine are usually easier to solve using the Schrödinger equation rather than the path integral formulation. For example, try to compute the harmonic oscillator using path integrals. It's complicated, not elegant and you do not get a big physical insight in what happens.&#xA;The hydrogen atom is even worse, and I don't want to start with Delta potentials. Another major disadvantage, which makes it quite complicated to solve certain problems, is that it is difficult to change representation in order to compute something. The Feynman formalism deals well with certain problems which primarily appear in QFT, but no one in his right mind would try to solve the hydrogen atom using path integrals.&#xA;Of course, the choice of what we call &quot;simple and standard problems&quot; is affected by the way we teach quantum mechanics, so this point is correlated to the first one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third reason is, that the Schrödinger approrach (in my opinion) has more educational value. Firstly, because it is an approach we in principle already know from mechanics and electrodynamics: Here you have a differential equation, go solve it. In mechanics, it's Newton/Lagrange/Hamilton, in electrodynamics its Maxwell, and in quantum mechanics its the Schrödinger equation. Furthermore, while solving the SE, you usually solve the eigenvalue problem for the Hamiltonian, which of course has a very direct and experimentally observable interpretation as the energy states of the respective system. The kernel of time evolution is a rather abstract construct, which does not have a direct interpretation like this.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="26" LastEditDate="2014-11-27T19:49:34.790" LastActivityDate="2014-11-27T19:49:34.790" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="580" PostTypeId="2" ParentId="98" CreationDate="2014-11-27T18:24:18.943" Score="6" Body="&lt;p&gt;Before the modern day scientists, there was the evolution concepts in the ancient cultures. For example consider the ancient vedic culture in India. Hinduism is the major religion in India, but it is only an extension of ancient Vedic culture which is widely known as &lt;a href=&quot;http://en.wikipedia.org/w/index.php?title=Sanathana_Dharma&amp;amp;redirect=no&quot; rel=&quot;nofollow&quot;&gt;Sanathana Dharma&lt;/a&gt;. Apart from religious rituals, Hinduism handles many more things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In one of the oldest religion in the world, Hinduism there are some attempts to reconcile evolution theory to the avatars of god Vishu(one of the god among Hindu trinity) in the scriptures. There are lots of internet resources discussing the subject.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Monier Monier-Williams wrote &quot;Indeed, the Hindus were ... Darwinians centuries before the birth of Darwin, and evolutionists centuries before the doctrine of evolution had been accepted by the Huxleys of our time, and before any word like evolution existed in any language of the world.&quot;[25] J. B. S. Haldane suggested that Dashavatara gave a &quot;rough idea&quot; of vertebrate evolution: a fish, a tortoise, a boar, a man-lion, a dwarf and then four men (Kalki is not yet born). Nabinchandra Sen explains the Dashavatara with Darwin's evolution in his Raivatak. C. D. Deshmukh also remarked on the &quot;striking&quot; similarity between Darwin's theory and the Dashavatara.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Some modern interpreters sequence Vishnu's ten main avatars in a definitive order, from simple life-forms to more complex, and see the Dashavataras as a reflection, or a foreshadowing, of the the modern theory of evolution. Such an interpretation was first propounded by Theosophist Helena Blavatsky in her 1877 opus Isis Unveiled, in which she proposed the following ordering of the Dashavataras:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Matsya - fish, the first class of vertebrates; evolved in water&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Kurma - amphibious (living in both water and land; but not to confuse with the vertebrate class amphibians)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Varaha - wild land animal (a form of Boar)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Narasimha - beings that are half-animal and half-human (indicative of emergence of human thoughts and intelligence in powerful wild nature)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Vamana - short, premature human beings&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Parasurama - early humans living in forests and using weapons&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Rama - humans living in community, beginning of civil society&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Krishna - humans practicing animal husbandry, politically advanced societies&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Buddha ( I doubt Buddha is not an accepted avatar of Vishnu among Hindus, directly quoted from wiki) - humans finding enlightenment&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Kalki - advanced humans with great powers of destruction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reference : &#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Hindu_views_on_evolution&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Hindu_views_on_evolution&lt;/a&gt;&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Dashavatara#Evolutionary_interpretation&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Dashavatara#Evolutionary_interpretation&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the Hindu puranas were written thousands of years ago and written by unknown writers, it may not be possible to say that it actually existed as a theory in ancient days. I assume as vedic scriptures are sources of many hidden facts, interested people can have a good research on the same. Anyway Dasavatar(10 avatars) have similarity to the theory of evolution proposed by Darvin and it could be the first among such theories if proper proofs or studies were there on these scriptures and texts. &lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="143" LastEditDate="2014-12-06T12:31:35.757" LastActivityDate="2014-12-06T12:31:35.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="581" PostTypeId="2" ParentId="501" CreationDate="2014-11-27T18:35:55.023" Score="-1" Body="&lt;p&gt;Our word “energy” is borrowed via Latin from the Greek energeia, a term coined by Aristotle from the preposition en (“in”) and ergon (“work”). He developed this concept in his discussion of potentiality versus actuality. For “potentiality” he used the word dynamis “capability”, while for “actuality” he used two apparently interchangeable terms energeia and entelecheia. (The identity of the two terms is expressed by Aristotle is his Metaphysics 1047a30: ἐλήλυθε δ᾽ ἡ ἐνέργεια τοὔνομα, ἡ πρὸς τὴν ἐντελέχειαν συντιθεμένη.) Is this sense dynamis overlaps largely with the modern concept of “potential energy”, while energeia is perhaps rather what we would call “kinetic energy”. Aristotle expresses the difference between the two rather clearly in this passage from Metaphysics 1017a (Tredennick’s translation): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;“For we say that both that which sees potentially and that which sees actually is ‘a seeing thing’. And in the same way we call ‘understanding’ both that which can use the understanding, and that which does ; and we call ‘tranquil’ both that in which tranquillity is already present, and that which is potentially tranquil. Similarly too in the case of substances. For we say that [a statue of] Hermes is in the stone, and the half of the line in the whole; and we call ‘corn’ what is not yet ripe.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you asked about pre-modern non-European cultures. To this we can reply that the mediaeval philosophers in the Islamic world were definitely familiar with the Aristotelian concepts of potentiality versus actuality, but for the former they did not (as far as I can see) use the word energeia, but they did use its synonym entelecheia, which was borrowed directly into Arabic and spelt in Arabic as إنطلاخيا. In this sense you can say that mediaeval Islamic philosophy had a concept of “energy”, but did not use the term “energy” as such.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2014-11-27T18:35:55.023" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="582" PostTypeId="1" AcceptedAnswerId="583" CreationDate="2014-11-27T18:41:24.577" Score="4" ViewCount="406" Body="&lt;p&gt;Recently, I heard that the great &lt;strong&gt;Grothendieck&lt;/strong&gt; has passed away (RIP). But until last year I've been aware that he is lost from the world. So how could they figure out his death? My resources on this brilliant man are too small. Any resource would be appreciated!&lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="4251" LastEditDate="2021-09-17T12:20:58.150" LastActivityDate="2021-09-17T12:20:58.150" Title="Where had Alexander Grothendieck been all this time?" Tags="|mathematicians|grothendieck|" AnswerCount="2" CommentCount="11" ClosedDate="2014-12-05T20:33:26.147" ContentLicense="CC BY-SA 4.0" />
  <row Id="583" PostTypeId="2" ParentId="582" CreationDate="2014-11-27T19:22:11.343" Score="6" Body="&lt;p&gt;He was not &quot;lost&quot; in the first place; it is just that he lived a very private life after his retirement. The general public did not know where he was &lt;em&gt;exactly&lt;/em&gt; but it was not at all that he was a missing person or something along these lines. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He died in a hospital, more precisely at l’hôpital de Saint-Girons (Ariège), in the south of France, near the French-Spanish boarder. This is also the general region that was named all the time for his location. Thus, the answer to the question where he was is in Ariège.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have no information how he spent his last days, and who exactly communicated his death to the general public (and I am rather unsure this is relevant here), but the fact that he died in a hospital in a highly developed country that was moreover his home-country, shows that there is no mystery at all about his death having become known.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Source:&lt;/em&gt; &lt;a href=&quot;http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614&quot; rel=&quot;nofollow&quot;&gt;the obituary of Grothendieck in the major French newspaper Liberation&lt;/a&gt; mentioned the place of his death.&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="98" LastEditDate="2014-12-12T02:00:07.853" LastActivityDate="2014-12-12T02:00:07.853" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="584" PostTypeId="1" AcceptedAnswerId="585" CreationDate="2014-11-27T22:48:22.493" Score="12" ViewCount="706" Body="&lt;p&gt;&lt;strong&gt;I.&lt;/strong&gt; In an &lt;a href=&quot;https://www.youtube.com/watch?v=G8cbIWMv0rI&quot; rel=&quot;noreferrer&quot;&gt;episode&lt;/a&gt; of his &lt;em&gt;Cosmos&lt;/em&gt; series,  Carl Sagan discussed the brilliant argument whereby Eratosthenes purportedly estimated the circumference of our planet. At one point of the episode, Carl Sagan seemed to imply that one of the first observations of Eratosthenes in this matter was that the surface of the Earth &lt;strong&gt;had to be curved&lt;/strong&gt;. However, Carl Sagan didn't elaborate on how it was that Eratosthenes deduced, from what he had concluded about the curvature of the surface of the Earth, that the planet had to have a spherical shape (a point which is essential to the rest of Eratosthenes's argument). So, either the hypothesis on the sphericity of the Earth was rather commonplace in the days of Eratosthenes or the popular accounts on the Eratosthenes's exploit under consideration omit one of the principal ingredients in his determination of the circumference of the planet. What can you people tell me about this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;II.&lt;/strong&gt; Among the modern accounts on the measurement of the Earth's circumference by Eratosthenes, what is in your opinion the most accurate one?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance for your insightful replies!&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="26" LastEditDate="2014-12-12T11:15:02.947" LastActivityDate="2017-10-20T18:11:26.030" Title="Concerning the measurement of the Earth's circumference by Eratosthenes" Tags="|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="585" PostTypeId="2" ParentId="584" CreationDate="2014-11-28T07:40:32.630" Score="10" Body="&lt;p&gt;Certainly that the Earth is spherical was a commonplace (among the educated people) at the time of Eratosthenes. Once you start traveling on sea (or climbing mountains) you immediately notice that the Earth is curved. Observations of the sky from different places confirm that and tell you that it is spherical, at least approximately (before that there were also theories that is is conical or cylindrical). The opinion that Earth is spherical is traditionally credited to Pythagoras (who lived 3 centuries before Eratosthenes), and there is no doubt that it was universally accepted by the Greek and Alexandrian scientists for long time before Eratosthenes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this is described in many books. (Including the details of Eratosthenes measurement). With really great detail in&#xA;Delambre, &lt;strong&gt;Histoire d'Astronomie ancienne&lt;/strong&gt;, vol. I. This is in French and not very modern:-)&#xA;But very comprehensive. A more popular book is Dreyer, History of planetary theories. Later editions are called History of astronomy from Thales to Kepler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We cannot evaluate Eratosthenes accuracy because he gives the dimensions in stadia, and there were many different stadia used at that time, and there is no way to tell how long his stadium exactly was. In any case, the analysis shows that it could not be very accurate, but gave the right order of magnitude. The main source of error was the distance measurement on Earth which could be done only very approximately: a courier walked on a road and counted steps. (It is not even known whether there was a special expedition to measure this distance,&#xA;which is unlikely, or Eratosthenes just relied on travelers accounts). At the time of Eratosthenes (and later in antiquity) there were no precise measurements of the size of the Earth. Much later calculations of Posidonius are even less accurate than those of Eratosthenes. But again, this is a speculation because we do not know how Posidonius measured the distances on the Earth, and what his length unit exactly was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I mentioned in another post, even Newton in his youth thought that one degree is 60  (British) miles. This is probably what he was taught in the university in 17 century! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. The books I mentioned are those which on my opinion are the best. But they can be hardly called &quot;modern&quot;. On the other hand, if you type &quot;Eratosthenes measurement&quot; on Google, you obtain many expositions to choose from.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-20T18:11:26.030" LastActivityDate="2017-10-20T18:11:26.030" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="586" PostTypeId="2" ParentId="582" CreationDate="2014-11-28T07:55:28.250" Score="0" Body="&lt;p&gt;Here is an account of his life, with many details:&#xA;&lt;a href=&quot;http://www.ams.org/notices/200808/tx080800930p.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/notices/200808/tx080800930p.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was not &quot;lost&quot;. He lived in a village in the Pyrenees and did not communicate much with the rest of the world. But he did communicate with some people.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-11-28T07:55:28.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="587" PostTypeId="2" ParentId="501" CreationDate="2014-11-28T20:32:00.730" Score="4" Body="&lt;p&gt;In ancient Indian texts like Vedas and puranas there are so called philosophies or concepts of energy. Actually vedas and puranas are considered as religious texts, but these texts contain many hidden facts and knowledge(I doubt how many of you agree with this statement. Some of the examples of science in &lt;a href=&quot;http://en.wikipedia.org/wiki/Vedic_science&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vedas&lt;/a&gt; are, &lt;a href=&quot;http://en.wikipedia.org/wiki/Indian_mathematics#Vedic_period&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vedic mathematics&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Hindu_astrology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Astrology&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Vastu_shastra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vasthushasthra&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Yoga&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yoga&lt;/a&gt; and many more ). It will be broad to compare the philosophies in Vedas and modern science energy theories in this small answer and also we need experts on both to verify them. Even though there are thousands of websites discussing the science in Vedas which also includes the comparison of energy concepts.&lt;/p&gt;&#xA;&lt;p&gt;Let's check this example,&lt;/p&gt;&#xA;&lt;p&gt;Atom bombs and thermonuclear bombs :  powerful weapons which are practical examples of modern energy concept.&lt;/p&gt;&#xA;&lt;p&gt;So was there an equivalent weapon concept anywhere in the world? Yes, there was and it was in Ancient India. See this &lt;a href=&quot;https://www.google.co.in/search?q=brahmasthra&amp;amp;oq=brahmasthra&amp;amp;aqs=chrome..69i57j69i60.3722j0j7&amp;amp;sourceid=chrome&amp;amp;es_sm=93&amp;amp;ie=UTF-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;google search result&lt;/a&gt; and a quote from wikipedia is below.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Modern day Nuclear weapon and Brahmastra&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Robert Oppenheimer&lt;/strong&gt; (1904-1967) was a scientist, philosopher, bohemian, radical, fanatic of ancient Sanskrit literature,a theoretical physicist and the supervising scientist of the Manhattan Project, and most importantly, &lt;strong&gt;the developer of the atomic bomb&lt;/strong&gt;. Seven years after the first successful atom bomb blast test in New Mexico (Trinity), Dr. Oppenheimer was giving a lecture at Rochester University. To the question &lt;strong&gt;“Was the bomb exploded at Alamogordo during the Manhattan project the first one to be detonated?” he gave a strange reply “Well — yes. In modern times, of course.”&lt;/strong&gt; And as for Oppenheimer’s first words after the &lt;strong&gt;detonation of the bomb&lt;/strong&gt; he quoted from Hindu epic Mahabharata , “If the radiance of a thousand suns were to burst at once into the sky, that would be like the splendor of the mighty one. Now, I am become Death, the destroyer of worlds.” (&lt;strong&gt;Bhagavad Gita&lt;/strong&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The previous quote is for praising Lord Krishna when he displayed his real figure to Arjuna, and this has nothing to do with the energy concept. But we can conclude that Dr.Oppenheimer was familiar with the Bhagavat Gita and the book had it's influence on him which made him say those words from the book. (It is to be noted that, Bhagavat Gita includes messages in the middle of the Mahabharatha war. During this war people could have used the Brahmasthra, Pashupasthra or similar kind of weapons.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Most people agree that no human civilization before us had knowledge of atomic energy and its by-products. The atomic bomb is something completely novel to modern science. But we find in the Vedic literature descriptions of weapons that had a similar amount of energy as the atomic bombs we use today. “The atomic energy fissions the ninety-nine elements, covering its path by the bombardments of neutrons without let or hindrance. Desirous of stalking the head, i. e. the chief part of the swift power, hidden in the mass of molecular adjustments of the elements, this atomic energy approaches it in the very act of fissioning it by the above-noted bombardment. Herein, verily the scientists know the similar hidden striking force of the rays of the sun working in the orbit of the moon.” (Atharva-veda 20.41.1-3).&lt;a href=&quot;http://en.wikipedia.org/wiki/Indian_mathematics#Vedic_period&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See the full &lt;a href=&quot;http://en.wikipedia.org/wiki/Brahmastra&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia link&lt;/a&gt;. From this link it can be understood that there were concepts like these before thousands of years ago in Vedas(which are older than 5000 years ago). Nobody added the Brahmasthra concept just to compare it with modern atom bombs, it was there in the ancient text and Hindu puranas discuss many of such weapons in different stories. Did these weapons used in ancient days? The verification is difficult but one thing is very much sure, the idea was there in these ancient Indian texts much earlier.&lt;/p&gt;&#xA;&lt;p&gt;Many philosophical theories in Vedas and puranas are comparable with modern theories. The May be we all can wonder from where the so called modern energy concept was derived. The actual origin of the modern science energy concept could be from these ancient knowledge. Many scientists like Albert Einstein , Neils Bohr and Oppenheimer were admirers of these ideas and there would have been impact of these ideas in their thought process of deriving modern energy concepts(If there is a verification for the same, we can not actually call it 'modern energy concept', we should call it as just the 'Energy concepts').&lt;/p&gt;&#xA;&lt;p&gt;Another &lt;a href=&quot;http://en.wikiquote.org/wiki/Werner_Heisenberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;wiki link&lt;/a&gt; about Heisenberg who proposed uncertainty principle, see the following quote in the wiki link (not discussing the ideas which influenced him as I am not an expert in both subjects, also I was not present during that discussion:). Just mentioning to prove that such concepts had influenced scientists.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After these conversations with Tagore some of the ideas that had seemed so crazy suddenly made much more sense. That was a great help for me.&#xA;On conversations with Rabindranath Tagore, as quoted in Uncommon Wisdom: Conversations With Remarkable People (1988) by Fritjof Capra, who states that after these &amp;quot;He began to see that the recognition of relativity, interconnectedness, and impermanence as fundamental aspects of physical reality, which had been so difficult for himself and his fellow physicists, was the very basis of the Indian spiritual traditions.&amp;quot;&#xA;Variant: After the conversations about Indian philosophy, some of the ideas of Quantum Physics that had seemed so crazy suddenly made much more sense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here are some basic links which I found in my search.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://cpdarshi.wordpress.com/2011/11/22/modern-physics-found-its-direction-from-vedanta/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cpdarshi.wordpress.com/2011/11/22/modern-physics-found-its-direction-from-vedanta/&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.krishnapath.org/quantum-physics-came-from-the-vedas-schrodinger-einstein-and-tesla-were-all-vedantists/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.krishnapath.org/quantum-physics-came-from-the-vedas-schrodinger-einstein-and-tesla-were-all-vedantists/&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.hinduwisdom.info/Advanced_Concepts.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.hinduwisdom.info/Advanced_Concepts.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="143" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-12-06T11:39:13.383" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="589" PostTypeId="2" ParentId="564" CreationDate="2014-11-28T23:51:36.480" Score="5" Body="&lt;p&gt;You're not going to believe this, but it turns out that the guy behind this was none other than Harold Urey (Yes, &lt;em&gt;the&lt;/em&gt; Harold Urey of the Miller-Urey experiment!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He made the discovery of the correlation sometime in the 1950s, and with his nuclear background, he was well aware of the properties of isotopes. He studied small ancient fossilized creatures called foraminifera, and used the ratio of $^{18}O/^{16}O$ in their shells to figure out ancient temperature patterns. Here's a quote of a passage from &lt;a href=&quot;http://aip.org/history/climate/forams.htm&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; site:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the 1950s, the nuclear chemist Harold Urey devised another way to use the shells to measure ancient temperatures. He found he could take the temperature of an ancient ocean by measuring the oxygen that forams built into their shells. The rare isotope oxygen-18 is a bit heavier than normal oxygen-16, and biologists had shown that the amount of each isotope that a foram takes up varies with the temperature of the water. The isotopes were fossilized with the shells, and the ratio of isotopes (O18 to O16) could be determined with the new and exacting techniques of mass spectrometry. Urey and his team at the University of Chicago refined these tools, developed for nuclear studies, and applied them to calcite in fossils. They found plausible temperatures clear back to the Cretaceous era, more than 100 million years ago.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, the page also says (a bit above this passage) that the person who realized that the foraminifera could give clues as to the temperature of the world at the time of their lives was Wolfgang Schott, working in the 1920s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The page also cites &lt;a href=&quot;http://www.aip.org/history/climate/bib.htm#19&quot; rel=&quot;noreferrer&quot;&gt;two&lt;/a&gt; &lt;a href=&quot;http://www.aip.org/history/climate/bib.htm#23&quot; rel=&quot;noreferrer&quot;&gt;papers&lt;/a&gt; of Urey's, the first from 1947 and the second from 1950, making it seem as if his research was actually in the later 1940s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fun fact: The &lt;a href=&quot;https://en.wikipedia.org/wiki/Harold_Urey#Cosmochemistry_and_the_Miller.E2.80.93Urey_experiment&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; page on Urey hints that this work led him and one of his students to carry out the Miller-Urey experiment. I haven't been able to confirm this, though, so for now it's just a fun fact.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-11-28T23:51:36.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="590" PostTypeId="2" ParentId="576" CreationDate="2014-11-29T01:45:39.550" Score="4" Body="&lt;p&gt;The ACL wiki contains a section on the state-of-the-art systems over time for the main NLP tasks: &lt;a href=&quot;http://aclweb.org/aclwiki/index.php?title=State_of_the_art&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://aclweb.org/aclwiki/index.php?title=State_of_the_art&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of these pages need to be updated but at list there is some attempt of collaborative work to construct a clear overview.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only survey article the point to dates from 1997 (&lt;a href=&quot;http://scholar.google.com/scholar?cluster=1381618411507364126&amp;amp;hl=en&amp;amp;as_sdt=0,22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Survey of the state of the art in human language technology. Vol. 13. Cambridge: Cambridge University Press, 1997&lt;/a&gt;), so I fear there is none more recent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Excerpt from one of the most satisfying page I found on the ACL wiki:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/LCAmq.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2014-11-29T01:45:39.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="591" PostTypeId="1" CreationDate="2014-11-29T04:38:11.587" Score="12" ViewCount="84" Body="&lt;p&gt;Looking at the classic statistical approaches to natural language processing (e.g. tagging, parsing, etc.), I see that they are mostly generative models: n-gram models, Naive Bayes classifiers, hidden Markov models, probabilistic context­‐free grammars, IBM machine translation alignment models and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the opposite, more recent models tend to be discriminative: SVM, conditional log-linear model, conditional random fields, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any reason behind that trend?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2016-05-18T04:02:03.177" Title="Why are the classic statistical approaches to NLP mostly generative models while the most recent ones are mostly discriminative?" Tags="|artificial-intelligence|statistics|linguistics|machine-learning|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="592" PostTypeId="2" ParentId="554" CreationDate="2014-11-29T15:29:24.313" Score="3" Body="&lt;p&gt;For what I know, ancient civilisations did consider mental disorders as a disease. **Hyppocrates (around 400 BC) ** recomended a change in the environment of mentaly-ill persons, giving the diseases a cause - the environment - and a cure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern mental health treatment can be tracked back from &lt;strong&gt;15th century&lt;/strong&gt; with the first european facility for mental health treatment opening in &lt;strong&gt;Valencia, Spain&lt;/strong&gt;. Treatment for the patient and protection for and from the rest of the world remain linked today in many cases, but focus on one or the other of the activities has been shifting ever since. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(main source: &lt;a href=&quot;http://www.dualdiagnosis.org/mental-health-and-addiction/history/&quot; rel=&quot;nofollow&quot;&gt;http://www.dualdiagnosis.org/mental-health-and-addiction/history/&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="8" LastEditDate="2014-11-29T17:52:36.960" LastActivityDate="2014-11-29T17:52:36.960" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="593" PostTypeId="2" ParentId="501" CreationDate="2014-11-29T19:45:24.517" Score="5" Body="&lt;p&gt;&lt;strong&gt;Etymology&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite a few &lt;a href=&quot;http://www.etymonline.com/index.php?term=energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;sources&lt;/a&gt; say that Aristotle was the progenitor of the term, after his use of &lt;em&gt;energeia&lt;/em&gt; (a Latin transliteration of &lt;em&gt;ἐνέργεια&lt;/em&gt;). However, &lt;a href=&quot;http://www.eoht.info/page/Energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;one source&lt;/a&gt; notes that Heraclitus used a similar word, &lt;em&gt;en-ergon&lt;/em&gt;, years prior. Heraclitus wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;En-ergon is the father of everything, king of all things and, out of it, all forms of contrast originate. Since ‘en-ergon’ is common to everything, it is vital for life itself.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This was part of his aforementioned flux theory. However, like Aristotle's ideas, his concepts bear no relation to our modern usage of 'energy', and I doubt the two are related. It is worth noting, though, that Thomas Young's coinage of 'energy' in the early 19th century is thought to have been influenced by Aristotle's usage of &lt;em&gt;energeia&lt;/em&gt;. This would mean that the two, while not the same idea, may be thought of as true cognates.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Comparison&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m going to compare some ancient ideas and explain why they aren’t really precursors to our modern idea of ‘energy’. I suppose this is a &quot;Too long; didn't read&quot; summary; a lot of the other things are extraneous. Note: All information is taken from sources given later in the answer., unless otherwise indicated.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Qi:&lt;/strong&gt; This is occasionally miswritten as 'Qi energy', which is redundant. &lt;em&gt;Qi&lt;/em&gt; is often translated as 'natural energy' or 'energy flow'; adding 'energy' to the end would be incorrect. Qi was the ancient Chinese idea of an all-permeating substance in every living creature. It was responsible for life living. Obviously, this is not 'energy' in the modern sense because it is not conserved (dead things have no Qi).&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;En-ergon and the &lt;a href=&quot;http://plato.stanford.edu/entries/heraclitus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;flux theory&lt;/a&gt;:&lt;/strong&gt; Heraclitus’ brainchild was the flux theory, which states (explained better &lt;a href=&quot;http://www.iep.utm.edu/h/heraclit.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, but sometimes the page goes down and is inaccessible)  that there is some substance that permeates all objects. The reason this isn’t ‘energy’ is that it is uniformly distributed – that is, an object sitting on the floor won’t necessarily have a different energy than an object rolling on a table.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Energeia/ἐνέργεια/potentiality and actuality:&lt;/strong&gt; This was Aristotle's idea. It comes nearer to energy than Heraclitus' ideas, but closer examination reveals major differences. For example, a tree always has potential - to become a wooden bowl! Likewise, a person has potential to be married. These are really just possibilities, not 'energy' found in objects. Aristotle's ideas are not much like modern scientific ideas of energy.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;http://www.britannica.com/EBchecked/topic/630920/vitalism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vitalism&lt;/a&gt;:&lt;/strong&gt; This idea, apparently coming first from Aristotle and surviving for about two millennia, was similar to &lt;em&gt;Qi&lt;/em&gt; - actually it was nearly identical - in that it was the idea of a 'life force'. According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Vitalism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, it gained the support of eminent scientists such as Jöns Jacob Berzelius, one of the foremost figures in 19th century chemistry.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;http://philoscience.unibe.ch/documents/MaterialFS11/PS-Leibniz11/SmithG2006Vis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vis viva&lt;/a&gt;:&lt;/strong&gt; This really isn't part of the list, but it's a true precursor to energy, and worthy of discussion. The term &lt;em&gt;vis viva&lt;/em&gt; was coined by Leibniz in the 17th century, to describe the &quot;living energy&quot; (&lt;em&gt;vis viva&lt;/em&gt; is Latin) of a moving object. This truly was a precursor to kinetic energy, and I'm actually a bit surprised that our 'energy' isn't actually a lingual descendant of &lt;em&gt;vis viva&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;After a week or so of research, I found just about nothing that directly confirms or denies the possibility presented in the question. Zip. Zero. Nada. I've gotten the only null result I've ever gotten in 4 months when researching an answer. It's frustrating, but it's a bit helpful, because it taught me that there are some things that you may never know, no matter how hard you try to find them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the null result. The question asked whether or not any non-European, ancient cultures came up with something related to our modern concept of energy. The answer, as far as I can tell, is that nothing of the sort existed before a few centuries ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you'll pardon this little detour, I'll talk about Europe first. As fdb hit upon, Aristotle is often cited as the creator of a concept similar to energy. I discussed it in &lt;a href=&quot;https://hsm.stackexchange.com/questions/485/who-invented-the-concepts-of-potential-and-kinetic-energy/487#487&quot;&gt;this answer&lt;/a&gt;. The problem is (which I should have expanded upon), &lt;a href=&quot;http://plato.stanford.edu/entries/aristotle-metaphysics/#ActPot&quot; rel=&quot;nofollow noreferrer&quot;&gt;actuality and potentiality&lt;/a&gt; were philosophical ideas. Sure, loose interpretations could make them seem similar to the &quot;energy&quot; of today, but they were quite different from our modern concept. By Aristotle's logic, a tree resting at what we would call a reference point (i.e. where potential energy is 0) would still have potential - the potential to become a wooden bowl!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, we go to Gottfried Leibniz and &lt;a href=&quot;http://www.jstor.org/discover/10.2307/228997?uid=3739808&amp;amp;uid=2&amp;amp;uid=4&amp;amp;uid=3739256&amp;amp;sid=21105322432993&quot; rel=&quot;nofollow noreferrer&quot;&gt;the idea of &lt;em&gt;vis viva&lt;/em&gt;&lt;/a&gt; (behind a paywall and/or registration, but the free first page should give you an idea of what I'm talking about). He argued that a moving object was imbued with some sort of quantity. This could be interpreted as either kinetic energy or momentum, and in fact there was a dispute over this. We might also say that &lt;a href=&quot;http://philoscience.unibe.ch/documents/MaterialFS11/PS-Leibniz11/SmithG2006Vis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes was also involved&lt;/a&gt;. Nevertheless, 350-ish years ago, there was some idea (in Europe) that a quantity in moving object was conserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going to &lt;a href=&quot;https://en.wikipedia.org/wiki/Energy#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; (a short and mostly irrelevant detour, I assure you), we find that the term &quot;energy&quot; (and then &quot;potential energy&quot; and &quot;kinetic energy&quot;) was not coined until the early 19th century, by a few separate scientists working on the idea. At this point, we may consider the concept of energy (in its modern incarnation) to have been born. If you're fine with Wikipedia, you can read more &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Try a few searches (duckduckgo is my engine of choice) on the internet. I recommend the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;History of energy&lt;/li&gt;&#xA;&lt;li&gt;History of energy concept&lt;/li&gt;&#xA;&lt;li&gt;Energy concept&lt;/li&gt;&#xA;&lt;li&gt;Energy ancient [culture]&lt;/li&gt;&#xA;&lt;li&gt;Discovery of energy&lt;/li&gt;&#xA;&lt;li&gt;Energy ancient science concept&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;. . . and many more. All turn up results either about energy generation or (once or twice) crank sites about weird life-force ideas that are decidedly kooky. I was not able, in fact, to get a &lt;em&gt;single mention&lt;/em&gt; of ancient cultures when it came to the concept of energy. This is pretty bad, because a result of &quot;No, there was no idea of energy in ancient cultures&quot; would be definitive. Here, there is nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The one exception I found was from &lt;a href=&quot;http://www.visionlearning.com/en/library/Physics/24/Energy/46&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt;, which gave the completely following (no sources were cited):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ancient cultures associated fire with all forms of energy, and the Greek philosopher Heraclitus (~500 BCE) explained everything in the universe in terms of energy. In the centuries that followed, many scientists proposed theories to explain the concept of energy, but few shed any true light on the subject.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the &quot;ancient cultures&quot; are nowhere to be found. Go ahead a couple hyperlinks to &lt;a href=&quot;http://www.iep.utm.edu/heraclit/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; page. Heraclitus' flux theory is mildly interesting but not too relevant. Further down, we see hints of something along the lines of conservation of mass, but it goes nowhere. The knowledge that &lt;a href=&quot;http://www.iep.utm.edu/aristotl/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle&lt;/a&gt; disagreed with him leads me to believe that there is no hope of a connection between the two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just for fun, I went to &lt;a href=&quot;https://en.wikipedia.org/wiki/Qi&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Wikipedia page&lt;/a&gt; on Qi. The on thing that caught my eye was this sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Notions in the West of energeia, élan vital, or &quot;vitalism&quot; are purported to be similar.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Going to the pages listed reveal nothing of any importance (and none of the ideas are &lt;em&gt;anything&lt;/em&gt; like our modern idea of energy), but the fact that Aristotle is listed nearby further convinces me that his work is not really scientific.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One last stop. Many searches brought me &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S1877042809004704&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, where the author (see the PDF link on the upper left; I've had a hard time copying it here) claims that Joule and Mayer were the originators of the idea of energy. It seems like they certainly made some empirical investigations, but the ideas of Descartes and company seem to predate these men.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll end with &lt;a href=&quot;https://en.wikipedia.org/wiki/Bertrand_Russell&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bertrand Russell&lt;/a&gt;. For those who don't know, &lt;a href=&quot;https://en.wikipedia.org/wiki/Russell&amp;#39;s_teapot&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's Teapot&lt;/a&gt; is the idea that the burden of proof is on the claimant. I could say that there is a teapot floating between the Earth and Mars; it's nearly impossible to disprove unless you search every corner of the space between the solar system and so I can say that it exists. Our situation is similar to that, in that the idea that ancient cultures had a concept similar to energy is hard to disprove. One the one hand, I have found nothing; on the other, there are still chances that I am completely wrong. I'm going to let the community judge this answer and its validity, but at the moment, my answer to the question is &quot;No.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, I'd love some positive and/or negative feedback here. This was an odd experiment for me, and I'm not sure how well it worked.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-12-02T21:22:48.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="594" PostTypeId="2" ParentId="501" CreationDate="2014-11-29T20:07:15.760" Score="2" Body="&lt;p&gt;Weight is an old concept related to energy that was discussed both in &lt;a href=&quot;http://en.wikipedia.org/wiki/Wang_Fuzhi&quot; rel=&quot;nofollow&quot;&gt;not-so-Ancient China&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Jain_philosophy&quot; rel=&quot;nofollow&quot;&gt;Ancient India&lt;/a&gt;, or the &lt;a href=&quot;http://en.wikipedia.org/wiki/Nasir_al-Din_al-Tusi&quot; rel=&quot;nofollow&quot;&gt;Islamic Golden Age&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It meets each and every of your criteria :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Conservation of weight: the belief appeared in all of these cultures that weight was to be conserved on a global scale. (&quot;Pudgala&quot; for the Jain, just &quot;matter&quot; of al-Tusi)&lt;/li&gt;&#xA;&lt;li&gt;Convertibility was widely observed, at least by Golden Age alchimists.&lt;/li&gt;&#xA;&lt;li&gt;As weight and matter where often considered as the same thing, convertibility was often discussed. In a funny way, the same error that was made with energy (&quot;efficiency of the conversion&quot;) is found in discussion of weight.&lt;/li&gt;&#xA;&lt;li&gt;The cognate used had nothing to do with energy (except for chinese's &quot;Qi&quot;, or the &quot;fire/air element&quot;). What makes me believe there is a link are recent (20th century) development of &quot;Conservation of mass-energy&quot;, making mass and energy closely related if not fundamentally identical concepts.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I will document this answer latter as I lack knowledge of Ancient China and Ancient India, but I'm sure I can document most aspects for the three given civilisations.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="8" LastEditDate="2014-11-30T17:10:02.593" LastActivityDate="2014-11-30T17:10:02.593" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="595" PostTypeId="2" ParentId="535" CreationDate="2014-11-29T22:54:08.193" Score="7" Body="&lt;p&gt;Although the original &lt;a href=&quot;http://en.wikipedia.org/wiki/Humorism&quot; rel=&quot;nofollow&quot;&gt;humorism&lt;/a&gt; put forth by Hippocrates and Galen based on Egyptian and Mesopotamian influences was rejected by science as a general theory explaining all diseases and disabilities, a later version restricted to temperaments in psychology, that removed their link to supposed &quot;bodily fluids&quot;, survives until today. The psychological shift can be traced to Avicenna (980–1037), who &lt;a href=&quot;http://en.wikipedia.org/wiki/Four_temperaments#History_and_development&quot; rel=&quot;nofollow&quot;&gt;used&lt;/a&gt; temperaments to describe &quot;emotional aspects, mental capacity, moral attitudes, self-awareness, movements and dreams&quot;, and Culpeper (1616–1654), who discarded the theory that temperaments were influenced by fluids. Modern psychology of temperaments was &lt;a href=&quot;http://muse.jhu.edu/journals/journal_of_the_history_of_philosophy/summary/v039/39.2larrimore.html&quot; rel=&quot;nofollow&quot;&gt;strongly influenced&lt;/a&gt; by Kant (1724-1804). As far as the theory that all ills come from excess or deficit in one of the four humors, its rejection was gradual. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The starting point &lt;a href=&quot;http://en.wikipedia.org/wiki/Medicine#Middle_ages&quot; rel=&quot;nofollow&quot;&gt;seems to be&lt;/a&gt; the Black Plague of 14th and 15th centuries. Anatomical work of Vesalius (1514-1564) undermined many of Galen's traditional assertions, including those about blood and other fluids. However, the transfer to the modern theory of infectious diseases spread by microorganisms took a long time. Even though &quot;little animals&quot; were discovered by Leeuwenhoek in 1677 it seems that only Jenner (1749-1823) definitively &lt;a href=&quot;http://users.stlcc.edu/kkiser/History.page.html&quot; rel=&quot;nofollow&quot;&gt;linked&lt;/a&gt; them to the spread of a disease in 1796. He inoculated patients with a cow-pox virus to prevent them from developing small pox, the term he used for his treatment, &quot;vaccination&quot;, is still in use today. What complicated things was the widely believed &lt;a href=&quot;http://en.wikipedia.org/wiki/Spontaneous_generation#Scientific_method&quot; rel=&quot;nofollow&quot;&gt;spontaneous generation theory&lt;/a&gt; that microorganisms could appear spontaneously under favorable circumstances, which was only fully disproved by Pasteur in 1862 leading to acceptance of sterilization and immunization, anticipated by Jenner. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the psychological side Culpeper observed that temperaments may not be pure, and some people have an admixture of two. This was a precursor of the modern view that temperaments appear in a spectrum rather than as separate poles. After him the theory seemed to fade away for a while, and &lt;a href=&quot;http://muse.jhu.edu/journals/journal_of_the_history_of_philosophy/summary/v039/39.2larrimore.html&quot; rel=&quot;nofollow&quot;&gt;according to&lt;/a&gt; a 19-th century historian Dircksen it was Kant who revived it. In 1760-s he &lt;a href=&quot;http://muse.jhu.edu/journals/journal_of_the_history_of_philosophy/summary/v039/39.2larrimore.html&quot; rel=&quot;nofollow&quot;&gt;suggested&lt;/a&gt; a non-traditional interpretation of &quot;phlegma as strength&quot; and &quot;substitute for wisdom&quot;. Later this led to a theory of &lt;a href=&quot;http://en.wikipedia.org/wiki/Five_Temperaments#From_four_to_five&quot; rel=&quot;nofollow&quot;&gt;five temperaments&lt;/a&gt;, where an additional &quot;neutral&quot; temperament was added to the traditional four, although Kant himself favored the foursome. He also described  melancholy as capable of &quot;genuine virtue&quot;. In the 20-th century on the other hand, some authors characterized melancholy as an affliction to be treated rather than a healthy temperament. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But those were tweaks, after Kant the theory was accepted in one form or another by many 19-th century psychologists. In the 20-th century not only Freudians (Adler, Jung), but even physiologists like Pavlov accepted it as well. Pavlov &lt;a href=&quot;http://webspace.ship.edu/cgboer/eysenck.html&quot; rel=&quot;nofollow&quot;&gt;observed&lt;/a&gt; temperaments in his experiments with  dogs: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;One of the things Pavlov tried with his dogs was conflicting conditioning -- ringing a bell that signaled food at the same time as another bell that signaled the end of the meal. Some dogs took it well, and maintained their cheerfulness. Some got angry and barked like crazy. Some just laid down and fell asleep. And some whimpered and whined and seemed to have a nervous breakdown.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His idea was that temperaments can be recovered from simple binary classification of brain's capabilities for excitation and inhibition. Perhaps the most extensive modern theory of temperaments &lt;a href=&quot;http://webspace.ship.edu/cgboer/eysenck.html&quot; rel=&quot;nofollow&quot;&gt;is due to&lt;/a&gt; Eysenck, a behaviorist who developed Pavlov's idea into a detailed theory, and used statistical factor analysis to support  genetic origins of temperaments.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-12-03T18:53:25.980" LastActivityDate="2014-12-03T18:53:25.980" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="596" PostTypeId="1" AcceptedAnswerId="598" CreationDate="2014-11-30T16:35:07.600" Score="25" ViewCount="4332" Body="&lt;p&gt;Leibniz's concept of &lt;em&gt;vis visa&lt;/em&gt; (literally translated as &lt;em&gt;living force&lt;/em&gt;) was a precursor to our modern concept of kinetic energy. His formula for it was close to the modern non-relativistic one: $mv^2$, but not including factor $\tfrac{1}{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Vis viva&lt;/em&gt; became a bone of contention between the Newtonians and the Cartesians in the 18th century; historians call this the &lt;em&gt;vis viva&lt;/em&gt; debates. The debate mixed technical, philosophical, and even theological issues in a manner that seems very strange to modern sensibilities. However, this kind of argument was characteristic of the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone provide a summary?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: the modern concept of energy (including conservation) only emerged fully in the 19th century, thanks to the work of Helmholtz, Meyer, Joule, William Thomson, Clausius, and others. However, that is a topic for another question (maybe &lt;a href=&quot;https://hsm.stackexchange.com/questions/414/when-were-the-modern-notions-of-work-and-energy-created&quot;&gt;this one&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-12-19T22:26:28.833" Title="What was the vis viva controversy, including its philosophical aspects?" Tags="|physics|classical-mechanics|energy|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="597" PostTypeId="2" ParentId="98" CreationDate="2014-12-01T15:03:39.617" Score="3" Body="&lt;p&gt;It's important to distinguish evolution of organisms, i.e. change of their characters over generational time, and natural selection.  Darwin's greatest achievement was the development of the idea of natural selection in great detail, along with providing a great deal of evidence for its action.  (Alfred Russel Wallace developed a version of the idea of natural selection independently, but only after Darwin had already worked out a more sophisticated, but then unpublished, version.)  In contemporary evolutionary theory, there are multiple mechanisms that play a role in explaining evolutionary change, including random genetic drift.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lucretius (c. 0 CE) claimed only to be reporting the views of Epicurus (c. 300 BCE) in Lucretius's book &lt;em&gt;De Rerum Natura&lt;/em&gt; (&lt;em&gt;On the Nature of Things&lt;/em&gt;).  We have Epicurus's work only in fragmentary form, so it's not clear how much Lucretius has added to Epicurus's philosophy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One idea that people think of as a precursor of &quot;evolution&quot; discussed in Epicurean philosophy appears to come originally from &lt;a href=&quot;http://plato.stanford.edu/entries/empedocles/&quot; rel=&quot;nofollow&quot;&gt;Empedocles&lt;/a&gt; (c. 460 BCE).  It's the idea that at one time, there were animals with body parts put together in random forms.  Only those animals with body parts that functioned well together survived.  Lucretius presents this idea in Book 5, around lines 835-920 (e.g. pp. 182-184 in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/080185055X&quot; rel=&quot;nofollow&quot;&gt;Esolen's translation of &lt;em&gt;De Rerum Natura&lt;/em&gt;&lt;/a&gt;).  (See also &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0872202410&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;The Epicurus Reader&lt;/em&gt;&lt;/a&gt;, by Inwood and Gerson, p. 97.)  Notice that this filtering of animals happens only &lt;em&gt;once&lt;/em&gt;.  The role that natural selection plays in evolution, according to Darwin, requires that traits be heritable, and that natural selection gradually shape them over time.&#xA;(EDIT: My first version said that Lucretius &quot;critiqued&quot; Empedocles' proposal.  Upon more careful reading, it appears that Lucretius &lt;em&gt;endorsed&lt;/em&gt; the proposal, but objected to the view that there were certain specific animals such as centaurs.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At lines 852-874, Lucretius discusses more subtle processes that are reminiscent of natural selection, saying, among other things (p. 183 in Esolen's translation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And many kinds of creatures must have died,&lt;br&gt;&#xA;  Unable to plant out new sprouts of life.&lt;br&gt;&#xA;  For whatever you see that lives and breathes and thrives&lt;br&gt;&#xA;  Has been, from the very beginning, guarded, saved&lt;br&gt;&#xA;  By it trickery or its swiftness or brute strength.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That idea does sound a lot like natural selection, but gradual modification of inherited traits doesn't seem to be a part of it.  Thus, even if the concept that Lucretius (and probably Epicurus) discussed should be considered a concept of &lt;em&gt;natural selection&lt;/em&gt;, there's no reason to think that they had in mind &lt;em&gt;evolution&lt;/em&gt; by natural selection, in the sense of change over a long period of time.  Like Empedocles, Lucretius seems to have been talking about a filtering process that takes place once (see lines 835ff), and then never again, for any given set of species.  Note that it's also not clear that Lucretius had in mind filtering of organisms &lt;em&gt;within species&lt;/em&gt;.  It's plausible that he was talking about some &lt;em&gt;species&lt;/em&gt; failing to survive.  If there is no gradual modification of heritable traits, much of what Darwinian natural selection involves is missing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(BTW, there are many good translations of &lt;em&gt;De Rerum Natura&lt;/em&gt;, which consists of thousands of lines of philosophical poetry in Latin.  Esolen's is my favorite poetic translation.  My favorite prose translation is by &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0872205878&quot; rel=&quot;nofollow&quot;&gt;Martin Ferguson Smith&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="277" LastEditorUserId="277" LastEditDate="2014-12-01T17:22:39.767" LastActivityDate="2014-12-01T17:22:39.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="598" PostTypeId="2" ParentId="596" CreationDate="2014-12-02T03:56:37.417" Score="25" Body="&lt;p&gt;The controversy was ostensibly over what gets to be the &quot;true quantity of motion&quot;, momentum or vis viva (kinetic energy), with Newton and Leibniz on the opposing sides. While there was some philosophical angle at first, a &lt;a href=&quot;http://nature.berkeley.edu/departments/espm/env-hist/articles/2.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;&lt;em&gt;skillful attack by Leibniz&lt;/a&gt; against an inadequate concept, $m|v|$, and its description of the world&lt;/em&gt;&quot;, it quickly deteriorated into a sideshow to the priority dispute over the invention of calculus, which fueled it, with the substantive part getting lost in all the acrimony. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After 1743 Euler and D'Alambert portrayed the controversy as a debate about words, which became the consesus. Here is &lt;a href=&quot;https://archive.org/details/scienceofmechani005860mbp&quot; rel=&quot;noreferrer&quot;&gt;Mach's summary&lt;/a&gt;: &quot;&lt;em&gt;Investigations of Newton really proved that for free material systems not acted on by forces the Cartesian sum $\sum mv$ is a constant, and the investigations of Hyugens showed that also the sum $\sum mv^2$ is a constant... The dispute raised by Leibniz rested, therefore on various misunderstandings. It lasted 57 years till the appearance of D'Alambert's Traite de Dynamique in 1743&lt;/em&gt;&quot;. Some &lt;a href=&quot;http://nature.berkeley.edu/departments/espm/env-hist/articles/1.pdf&quot; rel=&quot;noreferrer&quot;&gt;modern scholars question&lt;/a&gt; this conclusion however, pointing out that the controversy lingered on &quot;through the remainder of the eighteenth century&quot;, that a crucial observation on the issue only appears in the second edition of Traite de Dynamique (1758), and was made earlier by Boscovich (1745). Namely, &quot;&lt;em&gt;vis viva is the measure of a force acting through a distance while momentum is the measure of a force acting through a time&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://books.google.com/books?id=SzSRlYNEm18C&amp;amp;pg=PA158&amp;amp;lpg=PA158&amp;amp;dq=newton+leibniz+%22true+measure+of+motion%22&amp;amp;source=bl&amp;amp;ots=ofiwqdgSJx&amp;amp;sig=5FOMu-W2LV8dnwQhwK3aGOaUAkc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=2SV9VJCFBI2XyQS_gYKQBQ&amp;amp;ved=0CB4Q6AEwAA#v=onepage&amp;amp;q=newton%20leibniz%20%22true%20measure%20of%20motion%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;philosophical side&lt;/a&gt; was only tangentially related to energy and momentum. Leibniz criticized mechanistic Cartesian philosophy for not explaining the &quot;source of the vitality&quot; of matter. Newton concurred, but for this very reason to him force should have remained a fundamental concept of mechanics, irreducible to masses and speeds. So he opposed the elevation of vis viva to a metaphysical status favored by Leibniz. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The starting point of the controversy was &lt;a href=&quot;http://en.wikipedia.org/wiki/Momentum#History_of_the_concept&quot; rel=&quot;noreferrer&quot;&gt;Descartes defining &quot;momentum&quot;&lt;/a&gt; as mass times speed (not velocity) in the tradition of medieval impetus, and claiming that its total value is conserved. Leibniz gave an example with falling bodies demonstrating that Cartesian &quot;momentum&quot; is not conserved. By that time &lt;a href=&quot;http://en.wikipedia.org/wiki/Christiaan_Huygens#Laws_of_motion.2C_impact_and_gravitation&quot; rel=&quot;noreferrer&quot;&gt;Huygens already established&lt;/a&gt; that the sum of masses times speeds squared is conserved in elastic collisions (he also gave a form of &quot;Newton's&quot; second law), so Leibniz declared that the &quot;true quantity of motion&quot; and called it vis viva. In the meantime &lt;a href=&quot;http://nature.berkeley.edu/departments/espm/env-hist/dissertation/ch_2.pdf&quot; rel=&quot;noreferrer&quot;&gt;Wallis gave the correct description&lt;/a&gt; of what happens to velocities in elastic collisions, which is equivalent to &quot;conservation of momentum&quot; (he uses no such language, and there was no notion of vector at the time to define &quot;momentum&quot;). Newton incorporated that into Principia but continued to call Cartesian &quot;momentum&quot;, you guessed it, &quot;quantity of motion&quot;. And so it began.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Partly, the semantic debate was inevitable since the modern definitions of basic mechanical notions weren't established yet. Partly, &lt;a href=&quot;http://books.google.com/books?id=SzSRlYNEm18C&amp;amp;pg=PA158&amp;amp;lpg=PA158&amp;amp;dq=newton+leibniz+%22true+measure+of+motion%22&amp;amp;source=bl&amp;amp;ots=ofiwqdgSJx&amp;amp;sig=5FOMu-W2LV8dnwQhwK3aGOaUAkc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=2SV9VJCFBI2XyQS_gYKQBQ&amp;amp;ved=0CB4Q6AEwAA#v=onepage&amp;amp;q=newton%20leibniz%20%22true%20measure%20of%20motion%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;separate issues got entangled&lt;/a&gt; with the original controversy. In 1724 Paris Academy offered a prize for  the &quot;best&quot; way of calculating collisions between absolutely hard bodies. Johann Bernoulli's submission stated that... there are no absolutely hard bodies, all collisions are elastic, and by the way, vis viva is the true quantity of motion. In return Maclaurin suggested calling &lt;em&gt;mv&lt;/em&gt; the &quot;force of bodies&quot;, and applying Newton's laws to it. Bernoulli's submission was rejected because he rejected the Academy's premise, and Maclaurin won the prize. In 1728-29 there was a scuffle over whose analytic methods are &quot;better&quot; for mechanics, Newton's or Leibniz's. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-22T03:13:43.717" LastActivityDate="2015-05-22T03:13:43.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="599" PostTypeId="1" CreationDate="2014-12-03T14:24:35.687" Score="20" ViewCount="1852" Body="&lt;p&gt;I know that tensor calculus was developed around the same time as general relativity. Tensor calculus was the prime way to deal with geometric objects, based on expliciting all coordinates and doing computations with them (the so called &quot;debauch of indices&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern differential geometry is much more concerned with intrinsic properties, set on stages called &quot;manifolds&quot;. How did this happen? What motivated them to develop these entities? Was the gain immediately noticeable compared to the jump in abstraction involved?&lt;/p&gt;&#xA;" OwnerUserId="44" LastActivityDate="2019-03-30T20:59:18.633" Title="What was the motivation for the development of modern, intrinsic, differential geometry?" Tags="|mathematics|differential-geometry|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="600" PostTypeId="2" ParentId="599" CreationDate="2014-12-03T17:55:17.693" Score="6" Body="&lt;p&gt;I don't have references on hand, but I believe intrinsic geometry began with Gauss' Theorema Egregium, which asserts that the Gaussian curvature of a surface at a point depends only on the intrinsic metric in a neighborhood of that point and not on the embedding of the surface (in other words, it is invariant under local isometry).  This is a bit surprising because the principal curvatures (which are measured along curves in the surface) depend very much on the embedding.  Gauss' observation suggests that it is possible - and desirable - to have a language for describing the geometry of surfaces without reference to a chosen embedding.  I think this helped motivate the development of the metric and curvature tensors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hyperbolic geometry also plays an important role in this story.  The parallel postulate was very much on Gauss' mind when he was thinking about curvature and the geometry of surfaces, and there is evidence that he wrote down the axioms of hyperbolic geometry several years before Lobachevsky but decided not to publish it.  Most likely he kept it secret because he could not construct a model for the hyperbolic plane and thus he was not convinced that he had shown that non-Euclidean geometry &quot;exists&quot; according to the standards of the time.  The problem is that one cannot embed the hyperbolic plane in $\mathbb{R}^3$, though one can in a sense do so locally; thus one needs a geometric language in which a global object is pieced together from well-understood local objects.  The theory of Riemann surfaces was a first step in this direction, but perhaps it was not until the language of Riemannian manifolds was invented that Euclidean geometry and hyperbolic geometry were truly placed on the same footing.&lt;/p&gt;&#xA;" OwnerUserId="35" LastActivityDate="2014-12-03T17:55:17.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="601" PostTypeId="1" CreationDate="2014-12-03T19:50:48.707" Score="8" ViewCount="433" Body="&lt;p&gt;I recall reading years ago in a linear algebra book that Gauss was accused of witchcraft for his (re)discovery of what we now call the row reduction algorithm for solving linear equations.  Has anyone else encountered the same story?  Is there possibly any truth to it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The context, I believe, was the use of the algorithm in &quot;equation solving competitions&quot; that were apparently held in the 19th century; most participants would use substitutions together with some clever tricks, but Gauss' algorithm was much faster for large linear systems.  But I can't verify any of this.&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="26" LastEditDate="2014-12-04T11:33:57.017" LastActivityDate="2014-12-04T15:09:18.013" Title="Gauss accused of witchcraft: apocryphal?" Tags="|biographical-details|debunking|linear-algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="602" PostTypeId="2" ParentId="599" CreationDate="2014-12-03T22:37:13.110" Score="7" Body="&lt;p&gt;It begins with Riemann's lecture in Göttingen in 1854, Über die Hypothesen welche der Geometrie zu Grunde liegen&quot;. Here is an English translation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.emis.de/classics/Riemann/WKCGeom.pdf&quot;&gt;http://www.emis.de/classics/Riemann/WKCGeom.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Paul Siegel said, Gauss theory of surfaces and Lobachevski geometry gave some motivation.&#xA;As another motivation, one can mention Riemann's work on in mathematical physics (heat conduction in solids).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean his work titled &quot;A mathematical work that seeks to answer the question posed by&#xA;the most distinguished academy of Paris&quot;, number XXII in his Collected papers.&#xA;This work is a predecessor of the tensor analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Briefly, the motivation is the following: when you consider a surface, you can always imagine it embedded, or immersed, at least locally. But with 3-D and higher dimensional spaces this does not help much, so one naturally starts to consider intrinsic geometry.&#xA;One early example of this is Lobachevski geometry in dimension 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In turn, motivation for the Gauss theory of surfaces was geodesy. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-04T04:12:40.197" LastActivityDate="2014-12-04T04:12:40.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="603" PostTypeId="1" AcceptedAnswerId="618" CreationDate="2014-12-04T01:12:51.577" Score="7" ViewCount="187" Body="&lt;p&gt;There are some trends to naming celestial objects:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Planets and moons in the solar system may be named after ancient Greek and Roman gods&lt;/li&gt;&#xA;&lt;li&gt;Comets may be named after their discoverer&lt;/li&gt;&#xA;&lt;li&gt;Asteroids may be named after humans.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm curious about the last one, especially since so many famous people have had asteroids named after them (try to find all the Monty Python members &lt;a href=&quot;http://www.minorplanetcenter.net/iau/lists/MPNames.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;!). When did this trend start? If it can be found out, who was the first person to have an asteroid named after them?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The following pages may be relevant: &lt;a href=&quot;http://www.iau.org/public/themes/naming/&quot; rel=&quot;nofollow&quot;&gt;an IAU page&lt;/a&gt; and &lt;a href=&quot;http://www.minorplanetcenter.net/iau/info/OldDesDoc.html&quot; rel=&quot;nofollow&quot;&gt;a Minor Planet Center page&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-12-07T17:22:49.010" LastActivityDate="2014-12-07T19:23:37.847" Title="When did the idea of naming asteroids after humans come about?" Tags="|astronomy|naming-conventions|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="604" PostTypeId="2" ParentId="599" CreationDate="2014-12-04T01:37:28.840" Score="19" Body="&lt;p&gt;Tensor calculus was &lt;a href=&quot;http://en.wikipedia.org/wiki/Tensor#History&quot; rel=&quot;noreferrer&quot;&gt;developed&lt;/a&gt; about 20 years before the general relativity by Ricci and Levi-Civita, starting around 1890, under the name of absolute differential calculus. It was motivated by Riemann's work on manifolds with a metric, and summarized in their comprehensive 1900 book. &lt;a href=&quot;http://repository.cmu.edu/cgi/viewcontent.cgi?article=1338&amp;amp;context=philosophy&quot; rel=&quot;noreferrer&quot;&gt;Einstein learned about it&lt;/a&gt; from a geometer friend Grossman around 1912, and saw it as a good vehicle to express his new ideas for geometrizing mechanics. It took him two years of correspondence with Levi-Civita to master the techniques. He even contributed some simplifications like the use of upper and lower indices, and the &lt;a href=&quot;http://en.wikipedia.org/wiki/Einstein_notation&quot; rel=&quot;noreferrer&quot;&gt;summation convention&lt;/a&gt;. However, coordinate tensor calculus was in principle at odds with Einstein's ideas about relativity and general covariance since it relied on local coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The invariant turn is associated with the &lt;a href=&quot;http://www.math.toronto.edu/mgualt/wiki/samelson_forms_history.pdf&quot; rel=&quot;noreferrer&quot;&gt;work&lt;/a&gt; of Elie Cartan in 1920-s on integrability of so-called Pfaffian systems, where he developed differential forms as an alternative coordinate free notation. In &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Koszul.html&quot; rel=&quot;noreferrer&quot;&gt;1950 Kozsul&lt;/a&gt; used it to eliminate non-tensorial objects like Christoffell symbols from the theory of covariant differentiation, and swift adoption of differential forms by many mathematicians &lt;a href=&quot;http://en.wikipedia.org/wiki/Covariant_derivative#Introduction_and_history&quot; rel=&quot;noreferrer&quot;&gt;followed&lt;/a&gt; in 1950-s. It became a standard due to the Bourbaki project, the expression &quot;debauchery of indices&quot; is theirs, and it tells us how they felt about the coordinate calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of it is that coordinate free notation is more &quot;natural&quot; because there is no global choice of coordinates on manifolds. Part of it is algebraic elegance that allows for cleaner and more conceptual proofs in geometry. Another part is that it fit very well with the framework of algebraic topology, intensively pursued in 1950-s, particularly with deRham cohomology and the theory of characteristic classes. And part of it, especially later, is the strong influence of Bourbaki on standardization of mathematics in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Physicists, who are more often interested in computations than in proofs, were never fully sold on the invariant notation, and continue to use indices alongside it. However, after the publication of Yang-Mills paper in 1954, and the realization that gauge fields are intrinsically connection forms already studied by mathematicians, many physicists invested into learning both languages to understand their counterparts. Also, the ideas of gauge invariance and general covariance are more manifestly expressed by invariant concepts and techniques. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-03-30T20:59:18.633" LastActivityDate="2019-03-30T20:59:18.633" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="605" PostTypeId="2" ParentId="601" CreationDate="2014-12-04T12:24:17.157" Score="8" Body="&lt;p&gt;It seems incredibly unlikely that Gauss was (in a serious way) accused of witchcraft for his contributions to linear algebra. Instead of trying to look for anecdotal evidence, we can reason from general historical context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, note that &lt;a href=&quot;http://en.wikipedia.org/wiki/Witch_trials_in_the_early_modern_period#Sporadic_witch-hunts_after_1750&quot;&gt;it is easily found on Wikipedia&lt;/a&gt; that witch-hunts were mostly a thing of the 17th century and before, and instances of serious accusations were a rarity after 1750. In particular, we find the quote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the later 18th century, witchcraft had ceased to be considered a criminal offense throughout Europe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Secondly, it turns out that &lt;a href=&quot;http://arxiv.org/abs/0907.2397&quot;&gt;'Gaussian elimination' was, &lt;em&gt;in fact&lt;/em&gt;, well known &lt;em&gt;before&lt;/em&gt; Gauss even was born&lt;/a&gt; (more reading material by the same author can be found &lt;a href=&quot;http://www.ams.org/notices/201106/rtx110600782p.pdf&quot;&gt;here&lt;/a&gt;). As the above paper shows, Newton already knew of, and described, the method of solving simultaneous linear equations that we now call Gaussian elimination. The work by Newton predates Gauss' birth by over a hundred years. The method was already widely taught to students before Gauss ever worked on it. It therefore seems quite ridiculous that Gauss would be accused of witchcraft for a mathematical method that had been known for over a hundred years (in the Western world; some sources argue that the technique had already been known for over a millennium by Eastern mathematicians).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the question arises: Why do we even call it &lt;em&gt;Gaussian&lt;/em&gt; elimination? In the same paper by Grcar one finds a section arguing that there was little use for the method introduced by Newton, until the invention of the method of least squares... and &lt;em&gt;that&lt;/em&gt;'s where Gauss comes in:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is&#xA;  not often that a mathematical discovery is immediately useful, as was the case with the method&#xA;  of least squares. And it was the method of least squares which finally created a recurring need&#xA;  to solve simultaneous linear equations. When a use for schoolbook Gaussian elimination finally&#xA;  arose, Carl Friedrich Gauss invented something better for the problem at hand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Gauss' main contribution consisted of inventing a new (bracket) notation for the method, which was more convenient, and subsequently adapted by 'professional computers', i.e. human calculators. Note, however, that Gauss did &lt;em&gt;not&lt;/em&gt; come up with the now universally used matrix notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the second half of the 19th century onward, textbooks started referring to Gauss when treating what was then simply 'elimination'. In the early 20th century, it was realized that Gauss' method could be generalized to solving matrix equations in the way that we all know and love. Grcar also offers an explanation for the fact that the method is now (mistakenly) called Gaussian elimination: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The invention of electronic computers led to new university faculties for instruction in the new&#xA;  way of computing which drew people of diverse training to the field. They extrapolated opinions&#xA;  of its history from whatever heritage they knew. Geodesy at one time accounted for the bulk of&#xA;  the simultaneous linear equations that were solved professionally, so its terminology may have&#xA;  appeared to be authoritative. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The citations that precisely described the contribution of Gauss had&#xA;  already been shortened to an unspecific “Gauss’s procedure.” This usage was misinterpreted as attributing&#xA;  ordinary, common, schoolbook elimination to Gauss, but only after World War II. Von Neumann was apparently the last prominent mathematician to simply write 'elimination'&#xA;  as Lacroix and Gauss had done.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-04T15:09:18.013" LastActivityDate="2014-12-04T15:09:18.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="606" PostTypeId="1" CreationDate="2014-12-04T14:07:05.607" Score="6" ViewCount="667" Body="&lt;p&gt;This question is a sub-question of &lt;a href=&quot;https://math.stackexchange.com/q/751624/103816&quot;&gt;previous question&lt;/a&gt; on MSE. I feel that on this website I have better chances of knowing more things.&lt;/p&gt;&#xA;&lt;p&gt;For quite some time now, I have been searching about the &amp;quot;Principle of permanence&amp;quot;. Mainly it is because of George Peacock and is found in old literature. It is a very powerful argument. But I couldn't found it in any modern literature!, why is that?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why is it not used(perhaps used implicitly but not mentioned) nowadays?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another quite surprising thing that I found is that wikipedea(&lt;a href=&quot;http://en.wikipedia.org/wiki/Principle_of_permanence&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;) mentions an altogether different definition of &amp;quot;Principle of permanence&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The usual definition goes like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;This principle states that we employ rules under circumstances more general than are warranted by the special cases under which the rules were derived and have validity.&amp;quot;--taken from the book &lt;a href=&quot;http://books.google.co.in/books?ei=KlZKU4j1MInBrAfowoDoDw&amp;amp;id=MzJZAAAAYAAJ&amp;amp;dq=Beginning+Algebra+for+College+Students&amp;amp;focus=searchwithinvolume&amp;amp;q=Permanence&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Beginning algebra for college students.&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why it has two different definitions?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I want to study about the &lt;em&gt;Principle Of Permanence&lt;/em&gt; as much as possible so please tell me about all the references/books which mention/explain it. I've already downloaded, Peacock's &amp;quot;&lt;em&gt;A Treatise On Algebra&lt;/em&gt;&amp;quot; vol-1 and vol-2, along with this some other sources, like Felix Klein's &amp;quot;&lt;em&gt;Elementary Mathematics From An Advanced Standpoint&lt;/em&gt;&amp;quot;, etc. The main sources are G.B Fine's  &amp;quot;&lt;em&gt;The Number-System of Algebra&lt;/em&gt;&amp;quot; and George Peacock's book.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I want more references/resources for this term, and want to study all the historical perspective of this term as much as possible.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="141" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2021-05-25T20:30:12.213" Title="Why does the &quot;Principle Of Permanence&quot; have two different definitions?" Tags="|reference-request|elementary-algebra|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="607" PostTypeId="1" AcceptedAnswerId="616" CreationDate="2014-12-04T16:39:48.347" Score="7" ViewCount="958" Body="&lt;p&gt;Were cancer treatments developed using knowledge obtained in Nazi Germany using practices which would have been considered unethical in liberal democracies of the same era?&lt;/p&gt;&#xA;" OwnerUserId="456" LastActivityDate="2018-10-28T06:08:22.280" Title="Cancer treatments development from research in Nazi Germany?" Tags="|medicine|nazi-germany|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="608" PostTypeId="1" AcceptedAnswerId="609" CreationDate="2014-12-04T19:39:48.307" Score="13" ViewCount="3484" Body="&lt;p&gt;When and how did the concept of dimension first emerge — the realisation that there is some “3-ness” about the world we live in, an analogous “2-ness” about the surface of a table, and “1-ness” about a line?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly it must have become clear once Cartesian co-ordinates appeared in their modern form (which as I understand was not actually Descartes’ work, but in reformulations of it by others a few years later).  But had it been realised earlier — in Descartes’ or Fermat’s own first co-ordinate systems, or in earlier approaches to geometry?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It seems a beautiful example of a fact which is both very intuitive and literally all around us, and yet which still would take a major insight to notice for the first time.)&lt;/p&gt;&#xA;" OwnerUserId="458" LastActivityDate="2019-06-16T23:58:35.440" Title="Origin of the concept of dimension?" Tags="|mathematics|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="609" PostTypeId="2" ParentId="608" CreationDate="2014-12-04T21:35:51.407" Score="12" Body="&lt;p&gt;This goes back at least to the Greeks. See the first few pages of the &lt;em&gt;Introduction&lt;/em&gt; to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://archive.org/details/geometryfourdim01manngoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry of Four Dimensions&lt;/strong&gt;&lt;/a&gt; by Henry Parker Manning (1914).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(ADDED 4 DAYS LATER)&lt;/em&gt; Danu requested that I expand my answer, so this morning I looked around my things and found the following papers. I've included relevant excerpts from all of them except for Cajori &lt;strong&gt;[3]&lt;/strong&gt;, which I know I have a copy of somewhere but was not able to locate. However, I did find Cajori &lt;strong&gt;[4]&lt;/strong&gt; (a conference abstract), which is for a talk Cajori gave on &lt;strong&gt;[3]&lt;/strong&gt; (most likely after &lt;strong&gt;[3]&lt;/strong&gt; had been written and submitted). If I happen to come across &lt;strong&gt;[3]&lt;/strong&gt; at some future time, I'll come back here and include some excerpts from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The single best source I managed to find is the &lt;em&gt;Introduction&lt;/em&gt; to Henry Parker Manning's 1914 book. Because that book is freely available on the internet (and quite well known), and the &lt;em&gt;Introduction&lt;/em&gt; is rather lengthy and detailed, I decided to not include excerpts from it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://dx.doi.org/10.13140/RG.2.2.23235.71201&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/strong&gt; Raymond Clare Archibald, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183422749&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Time as a fourth dimension&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Bulletin of the American Mathematical Society&lt;/strong&gt; 20 #8 (May 1914), 409-412.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(first paragraph of the paper, on p. 409)&lt;/strong&gt; [8 of 8 footnotes omitted] WHO first conceived the idea of time as a fourth dimension? In view of the recent development of the theory of relativity the answer to this question is of some interest to both mathematician and physicist. With the definite formulation of our ideas concerning space of $n$ dimensions during the latter part of the last century the conception occurred, doubtless, to many people. But, so far as I am aware, writers who discuss the subject have invariably given the credit to Lagrange (1736-1813), and have referred to a paragraph in either the $\text{first*}$ (1797, Loria† and Enriques‡) or second§ (1813, Dühring‖ and Sommerville¶) editions of his &lt;em&gt;Théorie des Fonctions analytiques.&lt;/em&gt;$\text{*}$ This paragraph, the first of the section &quot;Application de la théorie des fonctions à la mécanique,&quot; is as follows:†† [a passage in French follows]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;(from pp. 410-411)&lt;/strong&gt; [8 of 10 footnotes omitted] No further elaboration of this idea occurs in the published writings of Lagrange. Apparently independent of earlier suggestion, a writer in &lt;em&gt;Nature&lt;/em&gt;$\text{*}$ developed the idea of &quot;time-space.&quot; [footnote: $\text{*}$March 26, 1885, vol. 31, p. 481; &quot;Four-dimensional space,&quot; by &quot;S.&quot;] A decade later the popular English novelist Herbert George Wells wrote a serial, for W. E. Henley's &lt;em&gt;New Review&lt;/em&gt;,† in which an invention for negotiating similar &quot;space&quot; is basic. [footnote: †Jan.-June, 1895, vol. 12; &quot;The time machine,&quot; republished in book form, London, 1895.] Then came the important mathematical developments by R. Mehmke,‡ A. Einstein,§ H. Minkowski,‖ M. Abraham,¶ A. Sommerfeld,$\text{*}$ Wilson and Lewis,†† to mention only a few names.‡‡ But the main purpose of this note is to make a very small contribution to the history of geometry by way of answer to the question with which this paper commences. Long before the publication of the Fonctions analytiques, indeed as far back as 1754, d'Alembert (1717--1783) published the article &quot;Dimension&quot; in the famous Encyclopédie edited by Diderot and himself.§§ Here the idea of fourth dimension is dwelt upon more at length than by Lagrange, and d'Alembert attributes the conception of time as a fourth dimension to &quot;un homme d'esprit de ma connaisance.&quot; It seems questionable that Lagrange, a youth of 18 years [in 1754], was characterized in this way, even though he was in the following year appointed professor of mathematics in the Artillery School at Turin. But at least credit given Lagrange in this connection must, in the future, be rendered with small show of authority. To give appropriate setting to what d'Alembert writes, I quote somewhat fully* from his article in question: [the rest of Archibald's paper consists of a lengthy quote in French]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Henry Parker Manning, &lt;a href=&quot;http://www.jstor.org/stable/2974194&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Geometry of four dimensions&lt;/em&gt;&lt;/a&gt;, Topics for Club Programs column, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 25 #7 (September 1918), 316-320.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from p. 317)&lt;/strong&gt; [the references to &quot;Geometry&quot; are to Manning's 1915 book &lt;strong&gt;Geometry of Four Dimensions&lt;/strong&gt;] 1. The history of the idea of a fourth dimension. It will be convenient to think of three periods: down to 1827, from 1827 to 1870, from 1870 to the present day. In the writings of Aristotle (384-322 B.C.), Pappus (end of the third century), and, according to Simplicius (sixth century A.D.), of Ptolemy (about 150 A.D.) there are speculations about the number of dimensions of space and the possibility or impossibility of there being more than three. The same matter is referred to by the English philosopher Henry Moore (1614-1687), and by Kant (1724-1813).$^2$ [footnote: $^2$Pappus, after speaking of the three-line and four-line locus (see Heath's &lt;em&gt;Apollonius&lt;/em&gt;, Cambridge, 1896, Introduction, chap. v) says that we cannot say of more than six lines, &quot;the ratio of the content of four to that which is contained by the rest, since there is nothing contained by more than three distances. Men a little before our time have allowed themselves to interpret such things, signifying nothing at all comprehensible, speaking of the product of that which is contained by such lines into the square of this or the content of those. These things might, however, be stated and shown generally by means of compound ratios&quot;--Hultsch ed., Vol. II, Berlin, 1877, p. 680. For other references see &lt;em&gt;Geometry&lt;/em&gt;, pp. 1-3.] In the development of algebra, which at first ascribed a geometrical meaning to all of its terms, there came about a reluctant extension of these terms beyond those which describe figures of two and three dimensions.$^3$ [footnote: $^3$ &lt;em&gt;Geometry&lt;/em&gt;, pp. 2-3.] Finally, there is a suggestion by d'Alembert (1717-1783), and after him by Lagrange (1736-1813), that time might be regarded as the fourth dimension.$^4$ [footnote: $^4$ &lt;em&gt;Geometry&lt;/em&gt;, p. 4; R. C. Archibald, &quot;Time as a Fourth Dimension,&quot; &lt;em&gt;Bulletin of the American Mathematical Society&lt;/em&gt;, Vol. 20, 1914, p. 409.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Florian Cajori, &lt;a href=&quot;http://brepols.metapress.com/content/13p0u86502888530/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Early «proofs» of the impossibility of a fourth dimension&lt;/em&gt;&lt;/a&gt;, [Archeion] &lt;strong&gt;Archivio di Storia della Scienza&lt;/strong&gt; 7 #1-2 (March-June 1926), 25-28.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Florian Cajori, &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183486984&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Early &quot;proofs&quot; of the impossibility of a fourth dimension of space&lt;/em&gt;&lt;/a&gt; (conference abstract #1), &lt;strong&gt;Bulletin of the American Mathematical Society&lt;/strong&gt; 32 #3 (May-June 1926), 212.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abstract of a talk given at the 48th regular meeting of the San Francisco Section of the AMS, Stanford University, 3 April 1926. The published abstract follows.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The author cites early proofs of the impossibility of four-space given by Ptolemy, Clavius, Leibniz, Kant, Mellin and Whewell. Of these the Clavius-Mellin proof was the only one elaborated by the authors in detail; it was based upon theorems in Euclid's &lt;em&gt;Elements&lt;/em&gt;, but one of those theorems was given a generality not contemplated by Euclid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Florian Cajori, &lt;a href=&quot;http://www.jstor.org/stable/2298325&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Origins of fourth dimension concepts&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 33 #8 (October 1926), 397-406.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(first paragraph of the paper, on pp. 397-398)&lt;/strong&gt; Inquiries into the possibility of a fourth dimension of space reach as far back as Greek philosophy. Nevertheless, for 2000 years no one dared to proclaim the existence of such a space. Thus Aristotle in his &lt;em&gt;Heaven&lt;/em&gt; says that a solid has magnitude &quot;in three ways and beyond these there is no other magnitude because the three are all.&quot; This is the record of man's observation and every-day experience in our physical universe. In his &lt;em&gt;Metaphysics&lt;/em&gt; [1066b32] he speaks of a body as &quot;that which has dimension every way&quot;; in his &lt;em&gt;Physics&lt;/em&gt; [IV, 1] when considering motion, he regards &quot;dimensions&quot; as six, dividing each of the three into two opposites, &quot;up and down, before and behind, right and left,&quot; these terms being taken relatively. More pretentious was the procedure of Ptolemy who was an astronomer, but dealt also with the philosophy of mathematics. He was the first to offer a &quot;proof&quot; of the unprovable &quot;parallel-postulate&quot; of Euclid. In the same way he &quot;disproved&quot; the possibility of more than three dimensions, because, as Simplicius tells us, &quot;it is possible to take only three lines that are mutually perpendicular, two by which the plane is defined and a third measuring depth.&quot;$^1$ [footnote: $^1$&lt;em&gt;Simplicii in Aristotelis De Coelo Commentaria&lt;/em&gt;, ed. Heiberg, Berlin, 1904, 7a, 33.] The book containing Ptolemy's proof is now lost. Perhaps the first to approach the fourth dimension from the side of physics, was the Frenchman, Nicole Oresme,$^2$ of the fourteenth century. [footnote: $^2$&lt;a href=&quot;https://dx.doi.org/10.13140/RG.2.2.23235.71201&quot; rel=&quot;nofollow noreferrer&quot;&gt;P. Duhem, &lt;em&gt;Études sur Léonard de Vinci&lt;/em&gt;, III$^{\text e}$ série&lt;/a&gt;, Paris, 1913, p. 388; H. Wieleitner, &lt;em&gt;Isis&lt;/em&gt;, vol. 7, 1925, pp. 487, 488.] In a manuscript treatise, he sought a graphic representation of the Aristotelian forms, such as heat, velocity, sweetness, by laying down a line as a basis designated &lt;em&gt;longitudo&lt;/em&gt;, and taking one of the forms to be represented by lines (straight or circular) perpendicular to this either as a &lt;em&gt;latitudo&lt;/em&gt; or an &lt;em&gt;altitudo&lt;/em&gt;. The form was thus represented graphically by a surface. Oresme extended this process by taking a surface as the basis which, together with the &lt;em&gt;latitudo&lt;/em&gt;, formed a solid. Proceeding still further, he took a solid as a basis and upon each point of this solid he entered the increment. He saw that this process demanded a fourth dimension which he rejected; he overcame the difficulty by dividing the solid into numberless planes and treating each plane in the same manner as the plane above, thereby obtaining an infinite number of solids which reached over each other. He uses the phrase &quot;fourth dimension&quot; ($4^{\text {am}}$ &lt;em&gt;dimensionem&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Gerald James Whitrow, &lt;a href=&quot;http://bjps.oxfordjournals.org/content/VI/21/13.short&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Why physical space has three dimensions&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;British Journal for the Philosophy of Science&lt;/strong&gt; 6 #21 (May 1955), 13-31.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(1st paragraph of the paper, on p. 13)&lt;/strong&gt; [this is a quote from an English translation of a work by Galileo] And the first step of the &lt;em&gt;Peripatetick&lt;/em&gt; argument is that, where &lt;em&gt;Aristotle&lt;/em&gt; proveth the integrity and perfection of the World, telling us, that it is not a simple line, nor a bare superficies, but a body adorned with Longitude, Latitude and Profundity; and because there are no more dimensions but these three; the World having them, hath all, and having all, is to be concluded perfect. And again, that by simple length, that magnitude is constituted, which is called a line, to which adding breadth, there is formed a Superficies, and yet further adding the altitude or profundity, there results the Body, and after these three dimensions there is no passing farther, so that in these three the integrity, and to so speak, totality is terminated, which I might but with justice have required &lt;em&gt;Aristotle&lt;/em&gt; to have proved to me by necessary consequences, the rather in regard he was able to do it very plainly and speedily.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;(2nd paragraph of the paper, on p. 13)&lt;/strong&gt; GALILEO's first dialogue on the &lt;em&gt;Two Principal Systems of the World&lt;/em&gt; opens with this query raised by Salvitatus$^1$ concerning the three-dimensional nature of the physical universe. [footnote: $^1$T. Salusbury, &lt;em&gt;Mathematical Collections and Translations&lt;/em&gt;, &lt;strong&gt;I&lt;/strong&gt;, London, 1661, p. 2 sq.] Although the dialogue was ostensibly a debate on the rival merits of the Copernican and Ptolemaic world-systems, Galileo was attacking  the philosophy and cosmology of Aristotle and the Aristotelians. Aristotle's &lt;em&gt;De Caelo&lt;/em&gt; begins with a discussion of the dimensions of spatial objects and of the world, and it was no coincidence that Galileo's polemic is first directed to the same subject.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastEditorUserId="232" LastEditDate="2018-05-30T19:38:08.590" LastActivityDate="2018-05-30T19:38:08.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="610" PostTypeId="2" ParentId="608" CreationDate="2014-12-04T22:34:28.890" Score="7" Body="&lt;p&gt;Euclid's &amp;quot;definitions&amp;quot; of point, line and plane, certainly contain some &amp;quot;concept&amp;quot; (actually an intuitive idea) of dimension. Probably the next step was Cartesian co-ordinate system.&#xA;It was intuitively clear that on a curve you need one coordinate, 2 coordinates on a surface, and 3 in space.&lt;/p&gt;&#xA;&lt;p&gt;However it is very non-trivial to make these intuitive considerations precise.&#xA;Cantor found that the unit square contains &amp;quot;as many&amp;quot; points as the unit segment.&#xA;He tried to prove that there is no continuous one-to-one map between varieties of different dimensions, but failed.&lt;/p&gt;&#xA;&lt;p&gt;The first person who succeeded in proving this (that open sets in &lt;span class=&quot;math-container&quot;&gt;$R^n$&lt;/span&gt; cannot be homeomorphic to the open sets in &lt;span class=&quot;math-container&quot;&gt;$R^m$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$n\neq m$&lt;/span&gt; was L. E. J. Brouwer (1912). Only after that, the notion of dimension became a rigorous mathematical notion.&lt;/p&gt;&#xA;&lt;p&gt;Remark. There are several things called &amp;quot;dimension&amp;quot; in mathematics.&#xA;My previous remarks are related to the topological dimension.&#xA;I do not discuss here the much more elementary notion of dimension of a vector space. But it can also be considered as some rigorous formulation of what we intuitively&#xA;percept as &amp;quot;dimension&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;EDIT. However the idea of a vector space is too restrictive: the vector spaces we &amp;quot;see&amp;quot; are only lines and planes. While when we talk about dimension, even on intuitive level, we mean a much larger class of objects, including curves, surfaces, and finally a 3-space which is not a vector space. Everyone sees that a sphere has dimension 2 but it is not a vector space.&lt;/p&gt;&#xA;&lt;p&gt;On these questions I recommend an excellent popular article of H. Poincare, Pourqoi l'espace a trois dimensions? This was probably written before Brouwer proved his theorem&#xA;(Poincare died in 1912).&lt;/p&gt;&#xA;&lt;p&gt;EDIT2. To address some comments I looked in Euclid:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A point is that which has no parts&lt;/p&gt;&#xA;&lt;p&gt;The ends of a line are points&lt;/p&gt;&#xA;&lt;p&gt;The edges of a surface are lines&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is almost exactly the modern &amp;quot;inductive definition of dimension&amp;quot;! Topological dimension, of course. Euclid makes a distinction between straight lines and general &amp;quot;lines&amp;quot; that is curves, and between planes and surfaces.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, it is an interesting question whether Euclid had any intuitive &amp;quot;concept&amp;quot; of a vector. The answer is a very definite &amp;quot;no&amp;quot;. And for the proof I mention a famous theorem of Apollonius (who certainly knew Euclid by heart) that &amp;quot;an epycycle is equivalent to excentric&amp;quot;. This theorem we know from Ptolemy who explicitly credits it to Apollonius. (And Ptolemy was not very generous with his credits:-)&#xA;If you carefully look what is says, you recognize that it says that addition of vectors in the plane is commutative. For this Apollonius gives a compliated proof in the spirit of Euclid.&lt;/p&gt;&#xA;&lt;p&gt;I conclude that topological dimension is a much more basic, intuitive notion than dimension of vector space.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-12-07T06:06:34.560" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="611" PostTypeId="1" AcceptedAnswerId="612" CreationDate="2014-12-05T01:37:45.487" Score="21" ViewCount="945" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Scientific_priority&quot; rel=&quot;noreferrer&quot;&gt;modern standard&lt;/a&gt; is that &amp;quot;between two or more independent discoverers, the first to make formal publication is the legitimate winner&amp;quot;, colorfully described as publish or perish. This was not always the case. In 16-th century it was common for scientists to hide their discoveries and assert their priority by challenging others to solve problems they could solve, &lt;a href=&quot;https://hsm.stackexchange.com/questions/347/why-is-cardans-formula-wrongly-attributed-to-him/351#351&quot;&gt;Tartaglia's story&lt;/a&gt; being a famous example. Nor was it the case in the 17-th century, when Newton and others coded their discoveries into &lt;a href=&quot;https://hsm.stackexchange.com/questions/679/what-ciphers-did-isaac-newton-use&quot;&gt;anagrams&lt;/a&gt; instead of publicizing them. It is interesting that in some &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_patent_law&quot; rel=&quot;noreferrer&quot;&gt;early versions&lt;/a&gt; of the patent law &amp;quot;actual use of the invention was deemed adequate disclosure to the public&amp;quot; too.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia says that the publish or perish rule came into effect &amp;quot;as soon as modern scientific methods were established&amp;quot; and mentions Newton and Leibniz, which is ironic since in the calculus priority dispute there never was any doubt  that Leibniz published first. The &lt;a href=&quot;http://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy&quot; rel=&quot;noreferrer&quot;&gt;point of contention&lt;/a&gt; rather was whether he saw Newton's private letters sent to Collins and Oldenburg prior to that. Even in 19-th century Gauss is often credited with discovering non-Euclidean geometry based on his private papers, even though he not only never published his findings but made a point of not even mentioning them publicly for fear of &amp;quot;the scream of beotians&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When did publish or perish become the &amp;quot;ironclad&amp;quot; priority rule? Unlike patents and copyrights it was never codified into law, so what was the mechanism of its adoption? Did Academies and Royal Societies play a role? Was it influenced by developments in patent and &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_copyright_law#Early_British_copyright_law&quot; rel=&quot;noreferrer&quot;&gt;copyright&lt;/a&gt; law? When did universities start basing their hiring decisions on publications as they do now?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-10-17T03:48:44.333" Title="How did &quot;publish or perish&quot; become the scientific priority rule?" Tags="|social-context|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="612" PostTypeId="2" ParentId="611" CreationDate="2014-12-05T05:42:17.713" Score="17" Body="&lt;p&gt;The main reason for &quot;publish or perish&quot; is not securing priority.&#xA;(Newton did not publish much, but was very much concerned about priority. Some such people&#xA;exist even now).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason is that &quot;research&quot; is required nowadays from professors at the universities.&#xA;Their salary and even employment depends on the &quot;research&quot; they produce,&#xA;and the simplest way by which the bureaucrats can evaluate our research is the number of publications&#xA;and the rating of the journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another indication of research is the ability to obtain grants. Grants bring to professors and researchers a lot of benefits, and in many cases obtaining them is the condition of employment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The current system developed gradually, mainly in 20-th century and it continues to develop.&#xA;It is related to science becoming a mass enterprise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the middle of 20-th century in many places it was still normal to defend a PhD and to obtain a permanent teaching position. After that your only duty was teaching.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first half of 20-th century there were famous mathematicians who published less than 10 papers in their lifetime. For example, Eduard Helly, whose name is known to every math student.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays one is required to publish several papers every year and in some places they even say that 3 papers per year is not enough.&#xA;Otherwise, even if you have a permanent position, you will not receive grants and will be poorly treated by the administration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grants system was introduced on a massive scale in the second half of 20-th century.&#xA;The expression &quot;publish or perish&quot; appears at the same time.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-17T03:48:44.333" LastActivityDate="2017-10-17T03:48:44.333" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="613" PostTypeId="1" CreationDate="2014-12-05T08:16:59.803" Score="11" ViewCount="3320" Body="&lt;p&gt;I have read (probably) in Kanigel's book &lt;em&gt;The Man Who Knew Infinity&lt;/em&gt; that S. Ramanujan devised his own method of solving the Quartic Equation after he learnt to solve the Cubic Equation. Does anyone know what exactly was Ramanujan's method of solving the Quartic Equation? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have previously posted the question &lt;a href=&quot;https://math.stackexchange.com/questions/1052590/does-anyone-know-about-ramanujans-method-of-solving-the-quartic&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user459" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2014-12-08T16:06:35.143" Title="Does anyone know about Ramanujan's method of solving the quartic?" Tags="|mathematics|reference-request|elementary-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="615" PostTypeId="2" ParentId="613" CreationDate="2014-12-05T14:57:08.980" Score="5" Body="&lt;p&gt;You can find it in &lt;strong&gt;&lt;em&gt;Ramanujan's Notebooks IV&lt;/em&gt;&lt;/strong&gt; by B. Berndt, Chap. 22 Elementary Results, &lt;strong&gt;&lt;em&gt;Entry 20&lt;/em&gt;&lt;/strong&gt;, p.31.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ramanujan starts with this problem. Let $a,b,c,d$ be arbitrary. Solve the system,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2+ay = b\tag{20a}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y^2+cx = d\tag{20b}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eliminating $y$, we find it is equivalent to,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a^2(d-cx) = (b-x^2)^2\tag{20.1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assume without loss of generality that $a=2$. Expanding this out,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^4-2bx^2+4cx+(b^2-4d)=0\tag{20.1a}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By a simple linear substitution, the general quartic equation can be expressed in the &lt;em&gt;depressed form&lt;/em&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^4+px^2+qx+r = 0\tag{20.1b}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Equating coefficients of ${20.1a}$ and ${20.1b}$, you have a system of 3 equations in 3 unknowns {$b,c,d$}. Hence &lt;strong&gt;&lt;em&gt;every&lt;/em&gt;&lt;/strong&gt; quartic can be expressed in the form ${20.1}$. The problem then is to find $x$. Ramanujan defines,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \alpha+\beta+\gamma\tag{20.1c}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y = -(\alpha\beta+\beta\gamma+\gamma\alpha)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$-c/2 = \alpha\beta\gamma$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If you are familiar with cubic equations, you'll already see where Ramanujan is going.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Substitute $x,y,c$ into $(20a)$ and $(20b)$ keeping in mind $a=2$, then,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x^2+ay = \alpha^2+\beta^2+\gamma^2= b$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$y^2+cx = (\alpha\beta)^2+(\beta\gamma)^2+(\gamma\alpha)^2 = d$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(-c/2)^2 = (\alpha\beta\gamma)^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By elementary symmetric polynomials, we then conclude that $\alpha^2,\,\beta^2,\,\gamma^2$ are roots of the cubic equation,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$t^3-bt^2+dt-c^2/4=0\tag{20.1d}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, by solving $(20.1d)$, one can then find $\alpha,\,\beta,\,\gamma$. Using $(20.1c)$, we then further conclude that the four roots of the quartic are,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x = \alpha+\beta+\gamma, \quad\alpha-\beta-\gamma, \quad-\alpha-\beta+\gamma, \quad-\alpha+\beta-\gamma$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; This is similar to Euler's method where he solves a quartic as $x_i = \sqrt{y_1}\pm \sqrt{y_2}\pm \sqrt{y_3}$ and the $y_i$ are roots of a cubic. There's actually a generalization to this for solvable &lt;em&gt;8th deg eqns&lt;/em&gt; as $x_i = \sqrt{y_1}\pm \sqrt{y_2}\pm\dots\pm \sqrt{y_7}$ and the $y_i$ are roots of a &lt;em&gt;7th deg eqn&lt;/em&gt;. See this &lt;a href=&quot;https://mathoverflow.net/questions/145145/on-the-solvable-octic-x8-x729x229-0&quot;&gt;mathoverflow post&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="462" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2014-12-05T16:29:12.973" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="616" PostTypeId="2" ParentId="607" CreationDate="2014-12-06T02:22:04.623" Score="6" Body="&lt;p&gt;&lt;strong&gt;Yes.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I only have one source so far (unfortunately), but it does give quite a bit of information. It's the aforementioned NY Times book review, accessible &lt;a href=&quot;http://www.nytimes.com/books/first/p/proctor-cancer.html&quot;&gt;here&lt;/a&gt;. What is says is amazing: German scientists (&lt;em&gt;not&lt;/em&gt; necessarily Nazis) were the first to do research on many different facets of cancer. Much of this was pre-Third-Reich, in the glory days of late 19th-century Germany.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, despite what the author says, the evidence he gives seems to suggest that Germany was at the forefront of cancer research &lt;em&gt;before the height of the Nazi regime.&lt;/em&gt; He cites numerous studies in the early 1930s and 1920s, which were before Hitler truly reached power and certainly before the infamous Nazi medical experiments. Only two sections seem to indicate that you're correct:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;German scientists in the mid-1930s elaborated on this scientific base. The Reich Anticancer Committee (Reichsausschuss für Krebsbekämpfung) established in 1931 was enlarged, and an ambitious new journal, the Monatsschrift für Krebsbekämpfung (Monthly journal for the struggle against cancer), published by the notoriously antisemitic J. F. Lehmann publishing house, was launched in 1933 to coordinate the anticancer effort.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This supports your position. Continuing on,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;More than a thousand medical doctoral theses explored cancer in one form or another in the twelve years of Nazi rule; only diseases of the blood attracted more attention. Cancer registries were established, including the first German registries to record cancer morbidity (incidence) and not just mortality (deaths).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The 1000+ theses could be a coincidence, and may have not been influenced by Nazi ideology. Finally,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Efforts were made to strengthen prevention-oriented public health measures, including occupational safeguards, laws against the adulteration of food and drugs, bans on smoking, and programs to reduce the use of cancer-causing cosmetics, to name only a few. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Technically, this would have been a Nazi measure, but it may not have been motivated by Nazi ideology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second section is more direct:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A broad range of Nazi officials—from Gauleiter Julius Streicher to Reich Health Führer Leonardo Conti—championed a renewed focus on preventive medicine. The Monatsschrift für Krebsbekämpfung adopted the motto: &quot;The earlier a tumor is treated, the better the likelihood of a cure.&quot; The Zeitschrift für Krebsforschung announced that &quot;prevention is the best therapy.&quot; The ideology of prevention merged with the ideology of &quot;one for all and all for one&quot; (Gemeinnutz geht vor Eigennutz) that was yet another hallmark of Nazi thought: as one anti-tobacco activist put it, nicotine damages not just the individual but the population as a whole.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This all comes after a lengthy section on Erwin Like, a prominent oncologist who vocally supported the Nazis. Whether or not his research was Nazi-motivated may be contested. Overall, though, the Nazis enacted &lt;em&gt;some&lt;/em&gt; cancer treatment and prevention measures, but not a whole lot. Although I only have the one source so far.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-12-06T02:22:04.623" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="618" PostTypeId="2" ParentId="603" CreationDate="2014-12-07T19:23:37.847" Score="3" Body="&lt;p&gt;The asteroid &lt;a href=&quot;http://en.wikipedia.org/wiki/20_Massalia&quot; rel=&quot;nofollow&quot;&gt;20 Massalia&lt;/a&gt; was the first asteroid to not be assigned its own astrological symbol, and was also the first object in the solar system to be given a non-mythological name. It's named after Marseilles, the hometown of the discoverers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that the first person to have an asteroid named after them was Empress Eugenia di Montijo, the wife of Napoleon III. The asteroid, &lt;a href=&quot;http://en.wikipedia.org/wiki/45_Eugenia&quot; rel=&quot;nofollow&quot;&gt;45 Eugenia&lt;/a&gt;, was discovered in 1857 and subsequently named by the astronomer Hermann Goldschmidt.&lt;/p&gt;&#xA;" OwnerUserId="142" LastActivityDate="2014-12-07T19:23:37.847" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="619" PostTypeId="1" CreationDate="2014-12-07T21:19:28.463" Score="1" ViewCount="100" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/613/does-anyone-know-about-ramanujans-method-of-solving-the-quartic&quot;&gt;Elsewhere&lt;/a&gt; it has been suggested that, between 1526 and 1949, that Indian proto-scientists &quot;were more interested in sharing knowledge with the rest of the world,&quot; than in knowledge as private property, and that they &quot;were more interested in the findings than obtaining recognition.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What were the cultural expectations in Indian proto-sciences regarding knowledge as property, recognition, and findings?&lt;/p&gt;&#xA;" OwnerUserId="473" LastEditorUserId="62" LastEditDate="2014-12-08T22:58:09.073" LastActivityDate="2014-12-08T22:58:09.073" Title="What were the cultural attitudes of Indian proto-scientists to findings versus recognition; and knowledge as private property; between 1526 and 1949?" Tags="|scientific-method|india|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="620" PostTypeId="1" AcceptedAnswerId="626" CreationDate="2014-12-08T18:15:43.600" Score="14" ViewCount="216" Body="&lt;p&gt;How was the resemblance between apes and humans explained in pre-Darwinian biology? Who was the first biologist to note the resemblance?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="26" LastEditDate="2014-12-08T18:46:12.150" LastActivityDate="2014-12-10T08:22:02.867" Title="How was the resemblance between apes and humans explained in pre-Darwinian biology?" Tags="|biology|evolution|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="621" PostTypeId="1" AcceptedAnswerId="634" CreationDate="2014-12-09T00:39:21.453" Score="6" ViewCount="96" Body="&lt;p&gt;I remember very well from my (only) class in solid state physics how enthusiastically the professor recounted the discovery of &lt;a href=&quot;http://en.wikipedia.org/wiki/High-temperature_superconductivity&quot;&gt;high-$T_c$ superconductors&lt;/a&gt;. In one particularly vivid anecdote, he recounted how several hundreds of physicists were trying to jam into some giant lecture hall to be able to witness the announcement (bonus points if someone can recall which event this was!). It really sounded like this was an enormously big deal!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a stereotypical 'high energy theorist', I never quite understood his excitement. Sure, high $T_c$-superconductors are a very cool thing to discover, but why would they be &lt;em&gt;that&lt;/em&gt; big a deal? My best guess is that there used to be very strong reasons to believe such things could not exist (which would make for an interesting story of an accepted theory being proven wrong). If this is the case, an answer would ideally give an &lt;em&gt;in-depth&lt;/em&gt; explanation why this result was deemed surprising (i.e. including the physics). In any case, I would love to hear an account of why high-$T_c$ superconductors were such a major breakthrough.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="4251" LastEditDate="2020-07-08T01:28:31.877" LastActivityDate="2020-07-08T01:28:31.877" Title="What was the historical importance of the discovery of high-$T_c$ superconductors?" Tags="|physics|discoveries|experimental-physics|solid-state-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="622" PostTypeId="1" CreationDate="2014-12-09T00:48:10.213" Score="14" ViewCount="3011" Body="&lt;p&gt;I'm a high school student who never studied any relativity before, but I'm just wondering what was &lt;strong&gt;the&lt;/strong&gt; question that Einstein asked himself before going into this field. I knew he has done lots of work such as Brownian motion, photoelectric effect,etc. What was the question that baffled and therefore motivated him to work on relativity?&lt;/p&gt;&#xA;" OwnerDisplayName="user40003" LastEditorUserId="4251" LastEditDate="2020-07-13T01:53:20.347" LastActivityDate="2020-07-13T01:53:20.347" Title="What was Einstein's motivation for relativity theory?" Tags="|physics|theoretical-physics|relativity-theory|einstein|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="624" PostTypeId="2" ParentId="622" CreationDate="2014-12-09T15:57:28.360" Score="2" Body="&lt;p&gt;I suppose you are asking about Special relativity, which Einstein proposed in 1905.&#xA;(Special relativity is about light and kinematics, while General relativity is about gravity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was an apparent contradiction concerning the speed of light. On the one hand there was Maxwell theory which predicted that the speed of light must be the same in all frames of reference. Maxwell's theory was well established and experimentally tested (by Hertz and others. Wireless communication was based on Maxwell theory). On the other hand, this contradicted classical mechanics, which was even better established and tested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the most striking consequences of Maxwell theory (obtained by Lorenz and&#xA;Poincare) were that such things as length&#xA;of a rod and time interval between two events depends on the frame of reference.&#xA;This is why it is called &quot;relativity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's theory of special relativity was essentially a new kinematics which removed all these apparent contradictions. The famous formula &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt; is a consequence&#xA;of the special relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later general relativity was a different theory which was not motivated by any experiment.&#xA;There, the main motivation was the strange fact (well known since Galileo) that the&#xA;&quot;gravitational mass&quot; (or &quot;gravitational charge&quot;, the thing which stands in the gravity law) is identical to the &quot;inertial mass&quot;, the thing which stands in the Newton second law. (For example, electric charge has nothing to do with mass. Why gravitational charge equals mass?)&#xA;This strange coincidence is responsible for the well-known fact that all bodies fall&#xA;with the same acceleration (in vacuum). General relativity was designed to explain this strange fact. As far as I know, nobody before Einstein even tried. The fact was considered as &quot;evident&quot;. Unlike special relativity, the effects of general relativity are small, and difficult to measure. But there were two consequences of the new theory which could be measured: the gravitational lensing, and shift of Mercury perihelion. &#xA;Gravitational lensing was measured and found to conform to general relativity by Eddington&#xA;in 1918, and this immediately made Einstein famous. (Newton's mechanics also predicts gravitational lensing, if light has a mass, but the effect is smaller by a factor of 2).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References. E. Whittaker, A history of the theories of aether and electricity. This is not for a high school student.&#xA;When I was a high school student I read Martin Gardner, Relativity for the million, but there are many other good popular books.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-12T12:38:35.810" LastActivityDate="2020-05-12T12:38:35.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="625" PostTypeId="2" ParentId="622" CreationDate="2014-12-09T15:57:44.260" Score="-2" Body="&lt;p&gt;The main question puzzling &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_synchronisation&quot; rel=&quot;nofollow&quot;&gt;scientists&lt;/a&gt; at the time was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why does the speed of light seem constant in all observational frames?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The main question puzzling &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_synchronisation&quot; rel=&quot;nofollow&quot;&gt;Einstein&lt;/a&gt; at the time was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can I make the train station clocks more accurate?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="261" LastActivityDate="2014-12-09T15:57:44.260" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="626" PostTypeId="2" ParentId="620" CreationDate="2014-12-09T15:58:47.923" Score="13" Body="&lt;p&gt;Perhaps Aristole, as usual ...&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;http://classics.mit.edu/Aristotle/history_anim.2.ii.html&quot; rel=&quot;noreferrer&quot;&gt;Historia Animalium&lt;/a&gt; (translated by &lt;a href=&quot;http://en.wikipedia.org/wiki/D%27Arcy_Wentworth_Thompson&quot; rel=&quot;noreferrer&quot;&gt;D'Arcy Wentworth Thompson&lt;/a&gt;), Book II.8 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some animals share the properties of man and the quadrupeds, as the ape, the monkey, and the baboon. The monkey is a tailed ape. The baboon resembles the ape in form, only that it is bigger and stronger, more like a dog in face, and is more savage in its habits, and its teeth are more dog-like and more powerful.&lt;/p&gt;&#xA;&lt;p&gt;The ape [...] has also arms like man, only covered with hair, and it bends these legs like man, with the convexities of both limbs facing one another. In addition, it has hands and fingers and nails like man, only that all these parts are somewhat more beast-like in appearance. Its feet are exceptional in kind. That is, they are like large hands, and the toes are like fingers, with the middle one the longest of all, and the under part of the foot is like a hand except for its length, and stretches out towards the extremities like the palm of the hand; [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Book II.9 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The monkey, as has been observed, is furnished with a tail. In all such creatures the internal organs are found under dissection to correspond to those of man.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;But I've found no (attempted) explanation proposed by Aristotle.&lt;/p&gt;&#xA;&lt;p&gt;I think that we have to consider the so-called &amp;quot;Ladder of Life&amp;quot; (or &lt;a href=&quot;http://en.wikipedia.org/wiki/Great_chain_of_being&quot; rel=&quot;noreferrer&quot;&gt;Great Chain of Being&lt;/a&gt;) system of classification of organisms as a sort of explanation.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-12-10T08:22:02.867" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="627" PostTypeId="2" ParentId="622" CreationDate="2014-12-09T16:43:05.597" Score="0" Body="&lt;p&gt;I suggest to see these sources:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;O.Darrigol, &lt;a href=&quot;http://www.jstor.org/discover/10.2307/27757763?uid=3738296&amp;amp;uid=2&amp;amp;uid=4&amp;amp;sid=21104806383671&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Electrodynamic Origins of Relativity Theory&lt;/a&gt; (1996)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;as well as :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Olivier Darrigol, &lt;a href=&quot;http://ls.poly.edu/%7Ejbain/histlight/readings/06Darrigol.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Genesis of the Theory of Relativity&lt;/a&gt; (2005).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From this one, page 2 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A first indication of the primary context of the early theory of relativity is found in the very title of Einstein's founding paper: &lt;em&gt;On the electrodynamics of moving bodies&lt;/em&gt;. This title choice may seem bizarre to the modern reader, who defines relativity theory as a theory of space and time. In conformity with the latter view, the first section of Einstein's paper deals with a new kinematics meant to apply to any kind of physical phenomenon.&lt;/p&gt;&#xA;&lt;p&gt;Much of the paper nonetheless deals with the application of this kinematics to the electrodynamics and optics of moving bodies. Clearly, Einstein wanted to solve difficulties he had encountered in this domain of physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2014-12-09T16:43:05.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="628" PostTypeId="2" ParentId="622" CreationDate="2014-12-09T17:26:13.123" Score="14" Body="&lt;p&gt;Let's talk about special relativity (1905) first, then general relativity (1915).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The motivation for special relativity is stated clearly in the first sentence of Einstein's paper &quot;On the Electrodynamics of Moving Bodies&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is known that Maxwell's electrodynamics -- as usually understood at the present time -- when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let me unpack that. Maxwell's equations differ from the equations of Newtonian mechanics in one crucial aspect: Maxwell's equations &lt;em&gt;seem&lt;/em&gt; at first glance to single out a particular reference frame. The clearest example is the speed of electromagnetic waves: this is given by the formula $c=1/\sqrt{\epsilon_0\mu_0}$, where $\epsilon_0$ and $\mu_0$ are universal physical constants. The speed of the source has nothing to do with the speed of the wave, according to this formula. If you turn on a flashlight while standing still, the light beam that comes out should have the same speed as a light beam from a flashlight on a train whizzing along as fast as you please. (The speed of both beams being measured in the same frame.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before special relativity, physicists invoked an invisible medium, the aether, to explain this. Sound waves have a certain speed in air, independent of the source of the wave; likewise water waves on a pond. If EM waves are waves in the so-called luminiferous (&quot;light-carrying&quot;) aether, then the formula $c=1/\sqrt{\epsilon_0\mu_0}$ makes sense. Here, $\epsilon_0$ and $\mu_0$ are constants describing aspects of the aether. And $c$ then describes the speed of light &lt;em&gt;in the rest frame of the aether&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Maxwell's equations also contain hints that there &lt;em&gt;is&lt;/em&gt; no special frame of reference. Right after that first sentence, Einstein gives an example. Move a coil of wire through the field of a magnet. A current will be induced. (This is one of Faraday's most famous discoveries: electromagnetic induction. Also discovered independently by Joseph Henry.) You can calculate the current using Maxwell's equations in two ways: either pick a frame where the wire is at rest, or one in which the magnet is at rest. You get the same current either way!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another famous example: the &lt;a href=&quot;http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment&quot; rel=&quot;noreferrer&quot;&gt;Michelson-Morley experiment&lt;/a&gt;. I won't go into details, but the upshot is that Michelson and Morley failed to detect the speed at which the earth was, supposedly, traveling through the aether. Einstein alludes to this briefly in his 1905 paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Examples of this sort [the wire and magnet], together with the unsuccessful attempts to discover any motion of the earth relatively to the &quot;light medium&quot;, suggest that the phenomena of electrodynamics as well as of mechanics [i.e., Newton's laws] possess no properties corresponding to the idea of absolute rest.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I should say that historians disagree on how critical this experiment was to Einstein's thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And now the punchline:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They suggest rather that ... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the &quot;Principle of Relativity&quot;) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely that light is always propagated in empty space with a definite velocity $c$ which is independent of the state of motion of the emitting body.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The key phrase here is &quot;apparently irreconcilable&quot;. I hope you see the apparent contradiction right away. How can a light beam appear to travel at the same speed $c$ to &lt;em&gt;all observers&lt;/em&gt;, regardless of how they are moving themselves?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That was Einstein's motivation for special relativity (SR).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's turn to general relativity (GR). This started off as an attempt to reconcile Newton's law of gravity with special relativity. Newton's law says that two point masses $m$ and $M$ attract each other with force $F=GmM/r^2$, where $r$ is the distance between them. SR doesn't like this for several reasons. Writing in 1920, Sir Arthur Stanley Eddington described some of the difficulties:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The most serious objection against the Newtonian law as an exact law was that it had become ambiguous. The law refers to the product of the masses of the two bodies; but the mass depends on the velocity -- a fact unknown in Newton's day. Are we to take the variable mass, or the mass reduced to rest? Perhaps a learned judge, interpreting Newton's statement like a last will and testament, could give a decision; but that is scarcely the way to settle an important point in scientific theory.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Further &lt;em&gt;distance&lt;/em&gt;, also referred to in the law, is something relative to an observer. Are we to take the observer travelling with the sun or with the other body concerned, or at rest in the aether or in some gravitational medium? [from &lt;em&gt;Space, Time, and Gravitation&lt;/em&gt;, Eddington's pop-sci treatment]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Eddington was one of first scientists to master GR, and played a key role in its early history, post-1915.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the following point probably loomed as even more problematic:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;According to Newton's law, if you move one mass to a new position, this affects the gravitational force on the other mass &lt;em&gt;instantaneously&lt;/em&gt;, according to Newton's formula. In principle, you could use this to send signals faster than light, indeed, instantaneously.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Not only Einstein, but several other physicists saw these problems, and set about trying to find the correct law of gravity for SR. In 1921, looking back, Einstein described his motivation this way:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When, in 1907, I was working on a comprehensive paper on the special theory of relativity for the &lt;em&gt;Jahrbuch der Radioaktivität und Electronik&lt;/em&gt;, I had also to attempt to modify the Newtonian theory of gravitation in such a way that its laws would fit in the [special relativity] theory. Attempts in this direction did show that this could be done, but did not satisfy me because they were based on physically unfounded hypotheses. [quoted in Pais, &lt;em&gt;Subtle is the Lord&lt;/em&gt;, p.178]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Einstein then described how there occurred to him &quot;the happiest thought of my life&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The gravitational field has only a relative existence in a way similar to the electric field generated by magnetoelectric induction. &lt;em&gt;Because for an observer falling freely from the roof of a house there exists&lt;/em&gt; -- at least in his immediate surroundings -- &lt;em&gt;no gravitational field&lt;/em&gt; [his italics; op cit.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This led to the famous &lt;a href=&quot;http://www.einstein-online.info/spotlights/equivalence_principle&quot; rel=&quot;noreferrer&quot;&gt;Principle of Equivalence&lt;/a&gt;. Roughly speaking, a &lt;em&gt;free-falling&lt;/em&gt; frame of reference &lt;em&gt;in a gravitational field&lt;/em&gt;, is equivalent to a &lt;em&gt;non-accelerating&lt;/em&gt; frame of reference in a &lt;em&gt;gravity-free&lt;/em&gt; field. Also, an accelerating frame of reference in a gravity-free field is equivalent to a non-accelerating frame of reference in a gravitational field (again, roughly speaking).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see why Einstein regarded this as an essential clue. To study what gravity should look like according to SR, we can study accelerating frames of reference without gravity. It turns out that there are useful ways to approach the latter question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the early successes of the Principle of Equivalence was an explanation of why so-called gravitational mass is equal to inertial mass. In Newtonian mechanics, this equality explains why all things fall with the same acceleration (ignoring air resistance). The Principle of Equivalence takes a different tack on this: it replaces the two falling objects in a gravitational field, with freely floating objects viewed from an accelerating frame of reference. So they appear to accelerate at the same rate. You can then argue from that result back to the equality of gravitational and inertial mass. This discovery must have reassured Einstein that he was on the right track.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein wrote his first paper on this new approach in 1907. He did not arrive at the equations of GR until 1915. The &quot;happiest thought of his life&quot; provided his initial motivation (plus the need to reconcile gravity with SR), but the full tale is much longer, with many twists and turns.&lt;/p&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2014-12-09T19:39:12.063" LastActivityDate="2014-12-09T19:39:12.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="629" PostTypeId="1" AcceptedAnswerId="11555" CreationDate="2014-12-09T23:56:00.827" Score="13" ViewCount="1448" Body="&lt;p&gt;In his early years (actually, straight through the early 1910s), Albert Einstein published a lot of &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_scientific_publications_by_Albert_Einstein&quot;&gt;papers&lt;/a&gt; in &lt;a href=&quot;https://en.wikipedia.org/wiki/Annalen_der_Physik&quot;&gt;&lt;em&gt;Annalen der Phyisk&lt;/em&gt;&lt;/a&gt;, a very old and prominent German scientific journal. However, this trend stopped around 1912-ish, and throughout the rest of his career, Einstein spread his wings and skipped around to a whole bunch of journals. Why did he make the switch? The Wikipedia page on &lt;em&gt;Annalen der Physik&lt;/em&gt; notes that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the 1920s, the journal lost ground to the concurrent Zeitschrift für Physik.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, this is certainly later than 1912, and it seems that Einstein was responsible for much of &lt;em&gt;Annalen der Physik&lt;/em&gt;'s popularity in the early 20th century. Why didn't he stay with the journal?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="4251" LastEditDate="2021-09-15T19:30:37.340" LastActivityDate="2021-09-15T19:30:37.340" Title="Why did Einstein stop publishing so many papers in Annalen der Physik?" Tags="|physics|biographical-details|einstein|publishing|academia|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="630" PostTypeId="1" CreationDate="2014-12-10T02:14:09.293" Score="32" ViewCount="3508" Body="&lt;p&gt;Object oriented programming (OOP) is a programming model where code and data are encapsulated into units called objects that behave semi-autonomously. Interaction between objects is arranged through subroutines called methods that are object specific. Modularity of objects makes for an easy transferability between different applications. Objects are arranged into hierarchies of classes with objects at lower levels inheriting methods from the upper level classes but also adding more specialized ones of their own. So classes can be used as object constructors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is hard not to notice the structural analogy between OOP and the mathematical category theory with objects related by specialized morphisms and arranged into categories with transferability corresponding to categorical “up to isomorphism”. The analogy is &lt;a href=&quot;https://mathoverflow.net/questions/17763/how-does-math-relate-to-programming&quot;&gt;well known&lt;/a&gt; and &lt;a href=&quot;http://reperiendi.wordpress.com/2007/11/03/category-theory-for-the-java-programmer/&quot; rel=&quot;noreferrer&quot;&gt;goes deeper&lt;/a&gt;, e.g. there are OOP analogs of some functorial constructions (inheritance, polymorphism, etc.). In fact OOP is somewhat of a constructivist implementation of the category theory. Were such parallel structures really developed independently from completely different motivations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The category theory was &lt;a href=&quot;http://www.cwru.edu/artsci/phil/BJPSMacLane.pdf&quot; rel=&quot;noreferrer&quot;&gt;created&lt;/a&gt; by Eilenberg and Mac Lane in mid 1940-s while formalizing homological algebra, and it seems that most of the motivation came from algebraic topology, and in 1950-s also from algebraic geometry, when Grothendieck and Bourbaki got involved. Not from the theories of algorithms and formal languages, also actively studied at the time. Similarly, OOP was originally &lt;a href=&quot;http://en.wikipedia.org/wiki/Object-oriented_programming#History&quot; rel=&quot;noreferrer&quot;&gt;developed&lt;/a&gt; by MIT AI group (McCarthy, Minsky, Sutherland , etc.) in late 1950-s, and much of the overall structure and terminology is due to Dahl and Nygaard from Norwegian Computing Center in early 1960-s. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were such parallel structures really developed independently from completely different motivations? Mac Lane held a position at Harvard from 1938 to 1947, so perhaps he left a mark in Cambridge, where MIT is also located, but this is very tenuous. McCarthy, who created LISP in 1958 at MIT was explicitly motivated by Church’s lambda calculus, but that belongs more to the general theory of algorithms than categories per se. There is no connection I could find to Dahl or Nygaard. Even later relations between mathematics and programming that are &lt;a href=&quot;https://math.stackexchange.com/questions/600598/oop-and-category-theory-any-connections&quot;&gt;pointed out&lt;/a&gt;, like Curry-Howard correspondence (“every proof is a program”) and Martin-Löf’s work on type theory, are not specifically categorical. Lawvere is mentioned as categorifying the Curry-Howard correspondence later, but I am not sure if even he was motivated by OOP.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Were creators of OOP influenced, directly or by osmosis, by the category theory structures (perhaps lectures on category theory were offered at MIT at the time, and somebody from AI group attended them)? When was the analogy first explicitly made, and how did it influence both areas?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2021-09-08T09:49:58.707" LastActivityDate="2021-09-08T09:49:58.707" Title="Was object oriented programming influenced by the mathematical category theory?" Tags="|mathematics|biographical-details|computer-science|category-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="631" PostTypeId="2" ParentId="630" CreationDate="2014-12-10T02:38:30.333" Score="17" Body="&lt;p&gt;I'm not a programmer, but I'll try to post a coherent answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the major developers of OOP was Alan Kay. Kay himself has mentioned that LISP had an influence on him when he was developing the idea of OOP. from &lt;a href=&quot;http://www.purl.org/stefan_ram/pub/doc_kay_oop_en&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; page:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The second phase of this was to finally understand LISP and then using this understanding to make much nicer and smaller and more powerful and more late bound understructures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;LISP, in turn, was strongly influenced by lambda calculus. &lt;a href=&quot;http://www.cse.scu.edu/~jholliday/COEN171sp03/PZ03J.ppt&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt; powerpoint gives a good idea of how fundamentally related the two are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lambda calculus, in turn, was heavily influenced by category theory. &lt;a href=&quot;http://mathgate.info/cebrown/notes/barendregt.php&quot; rel=&quot;noreferrer&quot;&gt;This page&lt;/a&gt; notes that the two are related; however, the connection is not clearly explained beyond this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Also, there are relationships between typed lambda calculus, proof theory and category theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This answer isn't much, but it's the beginnings of a connection. I will add more tomorrow.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I didn't want to cite Wikipedia, but I should note that some pages led me to the connections. Here they are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Object-oriented_programming#History&quot; rel=&quot;noreferrer&quot;&gt;Object-oriented programming&lt;/a&gt; $\to$ &lt;a href=&quot;https://en.wikipedia.org/wiki/Lisp_(programming_language)&quot; rel=&quot;noreferrer&quot;&gt;LISP&lt;/a&gt; $\to$ &lt;a href=&quot;https://en.wikipedia.org/wiki/Lambda_calculus&quot; rel=&quot;noreferrer&quot;&gt;Lambda calculus&lt;/a&gt; $\to$ &lt;a href=&quot;https://en.wikipedia.org/wiki/Category_theory#Historical_notes&quot; rel=&quot;noreferrer&quot;&gt;category theory&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="402" LastEditDate="2016-05-15T21:49:34.950" LastActivityDate="2016-05-15T21:49:34.950" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="632" PostTypeId="2" ParentId="52" CreationDate="2014-12-10T09:00:50.457" Score="1" Body="&lt;p&gt;Yet another data point, the famous conversation between Newton and Halley a year and half before the Principia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;In 1684 Dr Halley came to visit him at Cambridge. After they had been some time together, the Dr asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it. Why, saith he, I have calculated it. Whereupon Dr Halley asked him for his calculation without any farther delay. Sir Isaac looked among his papers but could not find it, but he promised him to renew it and then to send it him...&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="488" LastActivityDate="2014-12-10T09:00:50.457" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="633" PostTypeId="1" AcceptedAnswerId="768" CreationDate="2014-12-10T13:21:37.700" Score="7" ViewCount="1183" Body="&lt;p&gt;De Sitter space and Anti de Sitter space are two of the most important solutions to the Einstein field equations. One famous method to obtain these spacetimes is to consider a $N$-dimensional hypersurface in $N+1$-dimensional Minkowski space. The surface can, of course, be parametrized in different ways. One option is known as 'Poincaré coordinates'. As is well-known, these coordinates don't really cover the entire spacetime, but only half of it. The sections that are covered by them are known as the &lt;em&gt;Poincaré patches'&lt;/em&gt; of the spacetime. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Why are these things named after Poincaré? Is it an honorary thing, or did he actually work on this? References to original papers would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-30T17:56:57.200" LastActivityDate="2015-01-01T20:00:38.477" Title="How did the 'Poincaré patches' get their name?" Tags="|terminology|theoretical-physics|differential-geometry|relativity-theory|abstract-algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="634" PostTypeId="2" ParentId="621" CreationDate="2014-12-10T14:00:34.110" Score="4" Body="&lt;p&gt;This is what I recall...   (I was a high school kid at the time)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two aspects. First the new ceramic high Tc superconductors could be cooled with liquid nitrogen - this is much cheaper and easier to obtain (and handle) than liquid helium. This should make superconductor applications more practical. It did, but not as much as many people hoped. A flippant upshot is that you are much more likely to have seen a live demo of the Meissner Effect!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second - and this was really pushed by the science TV programs - was the hope that these were just a stepping stone, and by adjusting the formula room temperature superconductors were around the corner. This did not happen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relative lack of practical applications of N2-cooled superconductors? I'm only an outside observer so someone else probably has more information, but my understanding was that it proved difficult to get reliable high current flows - eg. for long distance power lines. Also creating continuous ceramic power lines is &quot;non-trivial&quot;.&lt;/p&gt;&#xA;" OwnerUserId="25" LastActivityDate="2014-12-10T14:00:34.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="635" PostTypeId="1" CreationDate="2014-12-10T14:05:59.330" Score="31" ViewCount="5559" Body="&lt;p&gt;What was Napier's original logic, leading to his invention of logarithms?&lt;br&gt;&#xA;In other words, how did Napier, using the mathematics that was available at that time, derive them?&lt;/p&gt;&#xA;" OwnerUserId="489" LastEditorUserId="26" LastEditDate="2017-04-18T20:08:54.787" LastActivityDate="2017-10-17T03:13:02.440" Title="How did Napier come to invent logarithms?" Tags="|mathematics|computation|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="636" PostTypeId="2" ParentId="635" CreationDate="2014-12-10T15:12:47.453" Score="34" Body="&lt;p&gt;The idea was to simplify multiplicaton of numbers. If you ever tried to multiply $10$-digit numbers by hand you will see what I am talking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is this. We have $a^{m+n}=a^m a^n.$ On the right hand side we have a product&#xA;of $a^m$ and $a^n$, while on the left hand side a sum $m+n$.&#xA;So if you write two progressions, one arithmeric and one geometric in the parallel lines:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$1,2,3,4,...$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a,a^2,a^3,a^4...$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it is very easy to multiply two numbers in the second row: you find the corresponding numbers in the first row, add them and look at the answer which stands below your sum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next idea was to choose the arithmetic progression with very small increment, so that the second row will become &quot;dense&quot; and you can approximate any number&#xA;by some number of the second row. This is the idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then one had to compute the table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may ask why people cared about multiplication of large numbers (with many digits).&#xA;The reason is astronomy. As Kepler said, when he learned of the logarithms: this invention extended the life span of an astronomer many times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before Napier, there was another method called prosthaphaeresis (see &lt;a href=&quot;http://en.wikipedia.org/wiki/Prosthaphaeresis&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;).&#xA;Instead of the simple formula $a^{m+n}=a^ma^n$ it used the more complicated&#xA;formulas&#xA;$$\cos m\cos n=(\cos(m+n)+\cos(m-n))/2$$&#xA;and the analogous formula for the $\sin m\sin n$.&#xA;To find the product of two numbers $a=\cos m$ and $b=\cos n$ one used the tables&#xA;of cosines (backwards) to find $m,n$ then compute $m+n$ and $m-n$, then use the tables of cosines&#xA;again and then perform one more addition and divide by $2$. Division by 2 is easy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This involves more additions/subtractions than the use of the logarithm tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it was used because the tables of trigonometric functions existed long before Napier computed the tables of logarithms. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-17T03:13:02.440" LastActivityDate="2017-10-17T03:13:02.440" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="637" PostTypeId="1" CreationDate="2014-12-10T21:22:35.343" Score="19" ViewCount="2079" Body="&lt;p&gt;In &lt;a href=&quot;https://math.stackexchange.com/questions/1061350/when-contemporary-logical-notation-was-established&quot;&gt;a recent post on math.se&lt;/a&gt;, I suggested that G. Peano's &lt;em&gt;Arithmetices&#xA;principia, nova methodo exposita&lt;/em&gt; (1889) was probably the last important scientific work written in Latin.  This work set forth the so-called &lt;a href=&quot;http://en.wikipedia.org/wiki/Peano_axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peano axioms&lt;/a&gt;, which are still of major importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose there must be something published in Latin by the Vatican astronomer. Let's disqualify Vatican-related holdouts.  Other than these, is there any significant scientific or mathematical work written in Latin after 1889?&lt;/p&gt;&#xA;" OwnerUserId="134" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2022-05-30T19:53:13.017" Title="What was the last important scientific work written in Latin?" Tags="|academia|publishing|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="638" PostTypeId="2" ParentId="637" CreationDate="2014-12-10T23:03:24.737" Score="4" Body="&lt;p&gt;Absence of evidence is poor evidence, particular when the absence is limited to Wikipedia, and shown to be false by information contained in the question itself.  But ignoring those small issues, Wikipedia has two articles:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/New_Latin#Scientific_works&quot; rel=&quot;nofollow&quot;&gt;New Latin&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Contemporary_Latin&quot; rel=&quot;nofollow&quot;&gt;Contemporary Latin&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These seem to try to include all major works in Latin for their period (though Peano 1889 is missing), and the last entry is for 1864.&lt;/p&gt;&#xA;" OwnerUserId="492" LastEditorUserId="62" LastEditDate="2014-12-10T23:06:30.487" LastActivityDate="2014-12-10T23:06:30.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="639" PostTypeId="1" CreationDate="2014-12-10T23:33:39.430" Score="3" ViewCount="1379" Body="&lt;p&gt;Today, research and science is often associated to innovation (particularly by governments and funding agencies). I would like to understand how we got here and when the use of innovation as a buzzword started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we look to &lt;a href=&quot;http://en.wikipedia.org/wiki/Disruptive_innovation&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt; Wikipedia page on innovation, it distinguishes between several types of innovation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;sustaining innovation&lt;/li&gt;&#xA;&lt;li&gt;evolutionnary innovation&lt;/li&gt;&#xA;&lt;li&gt;revolutionary innovation&lt;/li&gt;&#xA;&lt;li&gt;disruptive innovation&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here are my questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the difference between innovation (and these types of innovation) and invention ? (noun)&lt;/li&gt;&#xA;&lt;li&gt;What is the difference between innovate and invent? (verb)&lt;/li&gt;&#xA;&lt;li&gt;What is the difference between an innovator and an inventor? (the one who acts)&lt;/li&gt;&#xA;&lt;li&gt;What is the difference between innovative research and inventive research? (adjective)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Apart from the raw definition, I think that there are differences coming from the ideology behind innovation and how the term was introduced in the recent history of science. Any commentary and thoughts about that are appreciated too.&lt;/p&gt;&#xA;" OwnerUserId="158" LastEditorUserId="26" LastEditDate="2014-12-12T08:56:11.803" LastActivityDate="2020-10-02T12:02:20.273" Title="History of the term innovation and its difference with invention?" Tags="|terminology|philosophy-of-science|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="640" PostTypeId="2" ParentId="637" CreationDate="2014-12-11T07:29:11.607" Score="7" Body="&lt;p&gt;This answer is a bit off because it is not about &lt;em&gt;entire&lt;/em&gt; work written in Latin, but until 2012 (with the &lt;a href=&quot;http://www.iapt-taxon.org/nomen/main.php&quot;&gt;'Melbourne' Code of Botanical Nomenclature&lt;/a&gt;), it was still required for new species of plants to be described in latin (not just the name but the full diagnosis).&lt;br&gt;&#xA;As for the zoological nomenclature, it was still common to describe new species in latin until the end of the 19th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as &lt;em&gt;complete&lt;/em&gt; works in Latin from the field of taxonomy go, the most recent ones I found are various monographies of Malpighiales genera by Franz Josef Niedenzu ranging from 1888 to 1933. The last one that contains more than just new taxa description seems to be a determination key &lt;a href=&quot;http://herbarium.lsa.umich.edu/malpigh/MALPpdf/Nied-MalpPal2-1924.pdf&quot;&gt;&lt;em&gt;Malpighiaceae palaeotropicae II&lt;/em&gt;&lt;/a&gt; (1924). It is however difficult to judge the significance of the work without being a botanist.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2014-12-11T10:06:37.893" LastActivityDate="2014-12-11T10:06:37.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="641" PostTypeId="1" AcceptedAnswerId="642" CreationDate="2014-12-11T10:01:14.563" Score="5" ViewCount="248" Body="&lt;p&gt;I know he wasn't strictly a mathematician, but as I understand, Baudhayana &lt;a href=&quot;http://en.wikipedia.org/wiki/Shulba_Sutras&quot; rel=&quot;nofollow&quot;&gt;recorded information&lt;/a&gt; on (what we know as) the Pythagorean Theorem and other geometrical properties like $\sqrt{2}$, in the Shulba Sutras, c. 800 BCE, about 200 years before Pythagoras. Other than his sutras, do we have any definite historical works on or about him?&lt;/p&gt;&#xA;" OwnerUserId="494" LastEditorUserId="26" LastEditDate="2014-12-12T11:15:38.610" LastActivityDate="2014-12-12T11:15:38.610" Title="Do we have any in depth material on Baudhayana?" Tags="|biographical-details|euclidean-geometry|ancient-india|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="642" PostTypeId="2" ParentId="641" CreationDate="2014-12-11T17:20:03.420" Score="3" Body="&lt;p&gt;You can find a brief discussion in :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Eli Maor, &lt;a href=&quot;https://books.google.it/books?id=Z5VoBGy3AoAC&amp;amp;printsec=frontcover&amp;amp;dq=Eli+Maor+The+Pythagorean+theorem+A+4000-year+history&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ei=f8-JVJu2DJDkatvcgMgG&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Eli%20Maor%20The%20Pythagorean%20theorem%20A%204000-year%20history&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;The Pythagorean theorem : A 4000-year history&lt;/a&gt; (2007), page 66-67.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See also &lt;strong&gt;Ch.8 : Ancient Indian Mathematics&lt;/strong&gt;, page 311-on of :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;George Gheverghese Joseph, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691135266&quot; rel=&quot;nofollow&quot;&gt;The Crest of the Peacock Non-European Roots of Mathematics&lt;/a&gt; (2011).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See also &lt;strong&gt;Ch.IV&lt;/strong&gt; on Geometry, page 155-on, of :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;T.S. Bhanu Murthy, &lt;a href=&quot;https://books.google.it/books?id=R7cJuj1VEJ0C&amp;amp;pg=PR8&amp;amp;lpg=PR8&amp;amp;dq=datta+singh+hindu+mathematics&amp;amp;source=bl&amp;amp;ots=HbKjA27co9&amp;amp;sig=Gy8w7AkrmIoUoDiuQLCQ9SPXiN8&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ei=XdKJVJWdBcHgarzXgbgM&amp;amp;ved=0CC0Q6AEwAjgK#v=onepage&amp;amp;q=datta%20singh%20hindu%20mathematics&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;A Modern Introduction to Ancient Indian Mathematics&lt;/a&gt; (1992).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2014-12-12T07:06:21.120" LastActivityDate="2014-12-12T07:06:21.120" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="643" PostTypeId="1" AcceptedAnswerId="646" CreationDate="2014-12-12T01:30:59.887" Score="10" ViewCount="1106" Body="&lt;p&gt;I was looking at a list of particles, and I noticed that many of them ended in &lt;strong&gt;-on&lt;/strong&gt;.  Prot&lt;em&gt;on&lt;/em&gt;, electr&lt;em&gt;on&lt;/em&gt;, neutr&lt;em&gt;on&lt;/em&gt;, lept&lt;em&gt;on&lt;/em&gt;, etc. Is there a historical (or linguistic) reason behind this naming structure?&lt;/p&gt;&#xA;" OwnerUserId="494" LastActivityDate="2014-12-13T14:01:15.607" Title="Where did the naming structure of particles come from (suffix -on)?" Tags="|terminology|naming-conventions|particle-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="644" PostTypeId="1" AcceptedAnswerId="645" CreationDate="2014-12-12T07:09:09.423" Score="22" ViewCount="2940" Body="&lt;p&gt;We all know the Pythagorean Theorem is one of the most fundamental formulas in mathematics, but it is very interesting to me that this ratio shows up as often as it does.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to have been discovered multiple times (a few times even before the eponymous Pythagoreans!), by &lt;a href=&quot;http://en.wikipedia.org/wiki/Baudhayana#Pythagorean_theorem&quot;&gt;the ancient Indians&lt;/a&gt; as well as &lt;a href=&quot;http://en.wikipedia.org/wiki/Babylonian_mathematics#Geometry&quot;&gt;the Babylonians&lt;/a&gt;, and it has shown up in other works ever since then, sometimes without the writer explicitly meaning to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is it that the Pythagorean theorem is so prominent in basic geometry? Does it show just how fundamental the triangle and the ratios between its sides are to our universe? &lt;/p&gt;&#xA;" OwnerUserId="494" LastEditorUserId="26" LastEditDate="2014-12-12T11:15:18.573" LastActivityDate="2015-05-13T21:30:35.640" Title="Why is the Pythagorean Theorem so ubiquitous?" Tags="|mathematics|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="645" PostTypeId="2" ParentId="644" CreationDate="2014-12-12T08:53:25.277" Score="28" Body="&lt;p&gt;The main question is why the Pythagorean theorem for right triangles:&#xA;$$ a^2+b^2=c^2$$&#xA;is such a central tool of Euclidean geometry. There are many different approaches one can take to this; I'll give it a shot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the key observations is that the triangle is the &lt;em&gt;most basic&lt;/em&gt; 'non-trivial' shape in plane (two-dimensional) geometry. Any three points - one could exclude those that lie on a single straight line, but even that is not strictly necessary as one can even regard a line as a special case of a triangle - &lt;em&gt;uniquely&lt;/em&gt; define a triangle, constructed simply by drawing straight lines between them. As such, it is quite a natural thing to study, and it is therefore perhaps not completely surprising that many scholars independently did so, and discovered things like the Pythagorean theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, why does the Pythagorean theorem find such wide application? I think that the main reason is that many problems can be &lt;em&gt;reduced&lt;/em&gt; to an application of the (simplest, two-dimensional) case of the Pythagorean theorem. Once one has discovered it, it makes sense to try and solve as many problems as possible by using it. This caused the ancients to investigate and discover many connections to more complicated shapes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a first example, we note that any regular polygon in two dimensions can be viewed as a collection of identical triangles arranged in a certain fashion. Another important observation is that the triangle is intimately related to the circle. For instance, if one considers a &lt;a href=&quot;http://en.wikipedia.org/wiki/Pythagorean_triple&quot;&gt;Pythagorean triple&lt;/a&gt; (i.e. three integers which satisfy $a^2+b^2=c^2$) one can divide by $c^2$ to obtain &#xA;$$\frac{a^2}{c^2}+\frac{b^2}{c^2}=1 $$&#xA;which is quickly seen to correspond to a rational point on the unit circle. In fact, this immediately shows that there are infinitely many Pythagorean triples (which are not multiples of any other Pythagorean triple), since the rational points on the circle are dense. This neat little observation (which I got from the first pages of &lt;a href=&quot;http://www.math.cornell.edu/~hatcher/TN/TNbook.pdf&quot;&gt;Hatcher's freely available book 'Topology of numbers'&lt;/a&gt;) shows that the triangle and circle are closely related, a  fact which is often expressed using sine and cosine functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This, combined with the fact that circles are also known to be the set describing all points equidistant to a single point (the center) hints at another very important thing: One can define a notion of &lt;em&gt;distance&lt;/em&gt; by using the Pythagorean theorem. In fact, after making the crucial observation that we can easily generalize to $n$ dimensions by 'decomposing' $\mathbb{R}^n$, $n$-dimensional Euclidean space, into $n$ copies of $\mathbb{R}$ (treating the coordinates independently, so to say), it becomes clear that we can use the Pythagorean theorem to define a notion of distance in any Euclidean space $\mathbb R^n$. This is known as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Euclidean_distance&quot;&gt;'Euclidean distance'&lt;/a&gt; and is one the most important (early) examples of a &lt;em&gt;metric&lt;/em&gt;; the study of metric spaces and related concepts has turned into an entire field of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion, I have tried to illustrate some of the ways that one can naturally relate triangles to more complicated shapes and concepts. I personally think that, in a sense, this process of reducing as much as possible to a simple and well-understood problem is one of the most things in mathematics in general. Therefore, after finding a beautiful yet 'simple' fact like the Pythagorean theorem, it is only natural that the ancient scholars used it again and again in trying to extend their understanding of Euclidean geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One final side-note: In view of my answer, it is natural to ask &lt;em&gt;&quot;But why is Euclidean geometry so important in our world?&quot;&lt;/em&gt; The answer to this can be understood in the context of general relativity. In curved space, Euclidean geometry (and therefore the Pythagorean theorem) no longer holds true. However, since spacetime is (understood to be) a manifold, it can be locally approximated extremely well by flat space. In this context, that means &lt;em&gt;Minkowski space&lt;/em&gt;, not Euclidean space. However, as long as one moves at velocities much lower than the speed of light, space and time can be separated to yield a simple picture of our spatial universe as a 3-dimensional Euclidean space, where we can use Euclidean geometry.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2015-05-13T21:30:35.640" LastActivityDate="2015-05-13T21:30:35.640" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="646" PostTypeId="2" ParentId="643" CreationDate="2014-12-12T18:30:03.040" Score="6" Body="&lt;p&gt;This is a question about etymology. It all started with the genuine Greek words anion “going up” and kation “going down”, both neuter participles of the verb “to go” with different preverbs: an(a)- and kat(a)-. Then we got “ion” on its own as a term encompassing both, and then, by analogy, “proton”, “electron”, “neutron”, and ultimately also “positron” (based on a spurious reanalysis of electr-on as elect-ron, and creation of a pseudo-suffix “ron”.)&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2014-12-13T14:01:15.607" LastActivityDate="2014-12-13T14:01:15.607" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="647" PostTypeId="1" CreationDate="2014-12-12T21:42:00.763" Score="2" ViewCount="135" Body="&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/tpQON.png&quot; alt=&quot;Percentage of Bachelor&amp;#39;s degrees conferred to women in the U.S.A., by major (1970-2012)&quot;&gt;&#xA;&lt;a href=&quot;http://www.randalolson.com/2014/06/14/percentage-of-bachelors-degrees-conferred-to-women-by-major-1970-2012/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the United States, for most majors, the proportion of women either increased or stayed static. By contrast, the proportion of women in computer science declined from a peak of 37% in 1983 to now being in the high teens. What caused this decline?&lt;/p&gt;&#xA;" OwnerUserId="508" LastActivityDate="2014-12-12T21:42:00.763" Title="Why has the proportion of women in computer science degrees declined in the USA?" Tags="|academia|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="648" PostTypeId="1" AcceptedAnswerId="649" CreationDate="2014-12-12T23:53:19.143" Score="11" ViewCount="646" Body="&lt;p&gt;Electron orbitals are referred to as s, p, d and f. I have read that there was a Latinate basis for this, but never found anything more specific. (Obviously, the discovery of &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Electron_configuration&quot;&gt;electron orbitals&lt;/a&gt;&lt;/strong&gt; by Bohr et. al. post-dated the fall of the Roman Empire by nearly 1500 years, so I realize that any such naming convention, if ancient Greek or Latin, would have been inspirational rather than originally sourced.)&lt;/p&gt;&#xA;" OwnerUserId="123" LastActivityDate="2021-07-06T23:14:46.540" Title="What is the historical reasoning for electron orbital names?" Tags="|chemistry|atomic-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="649" PostTypeId="2" ParentId="648" CreationDate="2014-12-13T00:33:27.913" Score="10" Body="&lt;p&gt;The basis is actually German, not Latinate. &lt;a href=&quot;https://books.google.co.uk/books?id=LR8LchaY2L8C&amp;amp;pg=PA138&amp;amp;lpg=PA138&amp;amp;dq=scharf%20prinzipal%20diffus&amp;amp;source=bl&amp;amp;ots=iyZU0PYX6C&amp;amp;sig=Shkl_w6fx6e-wXDFb5aPssuuK9c&amp;amp;h#v=onepage&amp;amp;q=scharf%20prinzipal%20diffus&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;They stand for&lt;/a&gt;:&lt;br /&gt;&#xA;&lt;strong&gt;s&lt;/strong&gt;charf (sharp),&lt;br /&gt;&#xA;&lt;strong&gt;p&lt;/strong&gt;rinzipal (principal),&lt;br /&gt;&#xA;&lt;strong&gt;d&lt;/strong&gt;iffus (diffuse), and&lt;br /&gt;&#xA;&lt;strong&gt;f&lt;/strong&gt;undamental (fundamental).&lt;/p&gt;&#xA;&lt;p&gt;You might be interested in this, &lt;a href=&quot;http://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=5&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0CD4QFjAE&amp;amp;url=http%3A%2F%2Fwww.che.uc.edu%2Fjensen%2FW.%2520B.%2520Jensen%2FReprints%2F137.%2520s%2C%2520p%2C%2520d%2C%2520f.pdf&amp;amp;ei=T4WLVKaNI8m0ggT3o4DACQ&amp;amp;usg=AFQjCNHQI4q4kfiB_7Es9lFrGpDBMiAPNA&amp;amp;sig2=Mvj0NspF0AO9hggwt60LkA&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origin of the s, p, d, f Orbital Labels&lt;/a&gt;, which is a short essay by the historian of chemistry William Jensen.&lt;/p&gt;&#xA;" OwnerUserId="142" LastEditorUserId="123" LastEditDate="2021-07-06T23:14:46.540" LastActivityDate="2021-07-06T23:14:46.540" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="651" PostTypeId="1" AcceptedAnswerId="660" CreationDate="2014-12-14T15:19:00.937" Score="10" ViewCount="468" Body="&lt;p&gt;I've always &quot;known&quot; that the phlogiston theory was naive and unsupported by the facts, which is why it was toppled pretty much instantaneously by Lavoisier's discovery of the role of oxygen. However, &lt;a href=&quot;http://www.scs.illinois.edu/~mainzv/HIST/bulletin_open_access/v30-2/v30-2%20p63-69.pdf&quot;&gt;this article&lt;/a&gt; makes a case that this point of view is naive and that phlogiston anticipated by more than a century Gibbs' discovery of why chemical reactions happen, a question to which Lavoisier's theory had no answer, and thus in a sense was a step backwards. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not at all qualified to judge these claims. Hence my questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How mainstream are the claims in this paper with respect to chemistry and history of chemistry? Are they known facts well-restated? Reasonable, but unorthodox opinions? Extremely controversial and not at all convincing?&lt;/li&gt;&#xA;&lt;li&gt;Assuming the history of the phlogiston theory and Lavoisier's &quot;Chemical Revolution&quot; is more than the naive version one reads in something like a science encyclopedia, can you recommend a history of chemistry that recounts the subtleties and the contemporary spirit of these events? If it all possible, I'd really like to try and understand how chemistry looked like before the atomic theory etc., what did people think, how they worked, what was known and possible.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="377" LastEditorUserId="6609" LastEditDate="2024-02-05T13:19:01.133" LastActivityDate="2024-02-05T13:19:01.133" Title="Was the phlogiston theory ahead of its time?" Tags="|reference-request|chemistry|obsolete-theories|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="652" PostTypeId="1" AcceptedAnswerId="658" CreationDate="2014-12-14T19:12:45.307" Score="6" ViewCount="429" Body="&lt;p&gt;I am currently writing about the &quot;de Morgan's laws&quot; and have seen both &quot;de Morgan&quot; and &quot;De Morgan.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which of these is correct?&lt;/p&gt;&#xA;" OwnerUserId="518" LastEditorUserId="26" LastEditDate="2014-12-14T23:08:07.233" LastActivityDate="2014-12-29T14:25:39.763" Title="Is &quot;de&quot; in &quot;de Morgan&quot; supposed to be capitalized or not?" Tags="|terminology|mathematical-logic|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="653" PostTypeId="2" ParentId="652" CreationDate="2014-12-14T19:46:51.613" Score="1" Body="&lt;p&gt;Why not consult the &quot;History of Mathematics Archive&quot; at &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/index.html&quot; rel=&quot;nofollow&quot;&gt;http://www-groups.dcs.st-and.ac.uk/~history/index.html&lt;/a&gt; ?  It contains mini-biographies of hundreds of mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2014-12-14T19:46:51.613" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="654" PostTypeId="1" AcceptedAnswerId="655" CreationDate="2014-12-14T19:39:37.977" Score="10" ViewCount="294" Body="&lt;p&gt;I'm writing/starting with my bachelor thesis, the subject is about &quot;infinity&quot;: what it is, why do we accept it, etc., but most of all my goal is to give an overview of the history of the interpretation from mathematicians (or maybe philosophic intellectuals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my concrete question is: I am looking for great 'history of mathematics' books, where the books focus on 'infinity' (ancient Greeks, etc.), so that I can show how the interpretation has changed through the past millenia, and why it changed. And what about the concept of infinity now? There are today still math professors who do not want to accept infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody have good resources for the 'historical view of infinity'? Later, if I have time, I will extend it with some philosophical side tracks, but first I need the basis to fully understand, mathematically, what the concept is, how it developed etc.&lt;/p&gt;&#xA;" OwnerUserId="525" OwnerDisplayName="MSKfdaswplwq" LastEditorUserId="26" LastEditDate="2014-12-14T23:07:11.230" LastActivityDate="2018-05-25T11:00:00.973" Title="Request for good resources on 'history of infinity' topics" Tags="|mathematics|reference-request|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="655" PostTypeId="2" ParentId="654" CreationDate="2014-12-15T00:09:34.527" Score="4" Body="&lt;p&gt;The way I would go about this would be to take references from decent online papers and journals on the subject, and then research which of those might seem worth your money. I would pay particular attention to books about &lt;a href=&quot;https://www.google.com/webhp?sourceid=chrome-instant&amp;amp;ion=1&amp;amp;espv=2&amp;amp;ie=UTF-8#q=georg%20cantor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georg Cantor&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an example of such a book chase. A paper titled &lt;a href=&quot;http://www.math.tamu.edu/~dallen/history/infinity.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Infinity&lt;/a&gt; was written at Texas A&amp;amp;M by a professor in the math department (&lt;a href=&quot;http://www.math.tamu.edu/~dallen/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dr. Allen&lt;/a&gt;). In the references to that paper, I see at least two stand out book sources:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dauben, Josheph Warren, Georg Cantor, His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, New Jersey, 1979&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lavine, Shaughan, Understanding the Infinite, Harvard University Press, Cambridge, MA, 1994.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If possible see if you can't survey 20 such references in this manner, secure from your library, possibly through inter library loan, and buy the ones you like the best after making your mind up as to whether the book is something you should own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will know when it is time to stop surveying and start picking acquisitions. Now you are ahead by at least two sources, but you might want to look at Dr. Allen's other references in that paper above as that looks like a pretty decent paper. You can certainly cite that online paper as well, it being from a reputable mathematician with the stamp of a university on it.&lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="2277" LastEditDate="2018-05-25T09:06:23.330" LastActivityDate="2018-05-25T09:06:23.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="656" PostTypeId="1" CreationDate="2014-12-15T05:39:14.540" Score="10" ViewCount="759" Body="&lt;p&gt;Category theory represented a huge change in the way the community thought about mathematics, leaving its the set theoretic nature behind and bringing up the importance of &lt;em&gt;arrows&lt;/em&gt; between the objects rather than the objects themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe this radical change must have originated in philosophy and/or art and was eventually absorbed by mathematics (or even science?). So, my questions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this true? if it is, which philosophical schools are behind the revolutionary ideas of category theory? How did this schools evolve? Did philosophy behind math evolve accordingly?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were Eilenberg and Mac Lane part of any non-exclusively-mathematical community which might have influenced their work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did category theory influence philosophy?&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="4251" LastEditDate="2021-09-08T09:22:29.823" LastActivityDate="2023-06-01T18:57:08.057" Title="Philosophy behind category theory" Tags="|mathematics|philosophy-of-science|category-theory|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="657" PostTypeId="1" AcceptedAnswerId="673" CreationDate="2014-12-15T05:50:02.757" Score="14" ViewCount="699" Body="&lt;p&gt;I'm trying to make a link between important mathematical breakthroughs in history and the important philosophical schools at the time. I realize that this topic is awfully broad and could be the subject of an entire book, so to simplify it a little I'd like to know which philosophy schools were probably behind great mathematical breakthroughs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, which school of philosophy could have motivated Leibniz and Newton, or Riemann to start thinking about manifolds &lt;em&gt;without&lt;/em&gt; relying on an ambient space, or start thinking about spaces with &lt;em&gt;dimensions&lt;/em&gt; greater than $3$?&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="55" LastEditDate="2015-02-16T20:14:32.990" LastActivityDate="2017-07-12T08:31:26.787" Title="Which school of philosophy motivated thinking about spaces of higher dimension?" Tags="|mathematics|philosophy-of-science|examples|" AnswerCount="5" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="658" PostTypeId="2" ParentId="652" CreationDate="2014-12-15T11:04:34.257" Score="5" Body="&lt;p&gt;People have the right to choose how they spell their own names. The person in question spelled his name as &quot;De Morgan&quot;, e.g. here (on page vi):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.co.uk/books?id=zPYDAAAAQAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;http://books.google.co.uk/books?id=zPYDAAAAQAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2014-12-15T11:04:34.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="659" PostTypeId="2" ParentId="657" CreationDate="2014-12-15T11:17:09.863" Score="2" Body="&lt;p&gt;Leibniz was a very famous philosopher. So surely you can say that his mathematical discoveries  were made against the background of his own “school”. Plato reportedly had the words “Let no one ignorant of geometry enter here” inscribed over the threshold of his school in Athens. The great mathematician Eudoxus of Cnidus (who worked out the first ever viable mathematical model for planetary movements) was a pupil of Plato’s.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2014-12-15T11:17:09.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="660" PostTypeId="2" ParentId="651" CreationDate="2014-12-15T12:33:28.153" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://www.hps.cam.ac.uk/people/chang/&quot;&gt;Hasok Chang&lt;/a&gt; is pretty mainstream: former President and Vice President of the British Society for the History of Science. His article &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.2009.00150.x/full&quot;&gt;We Have Never Been Whiggish&#xA;(About Phlogiston)&lt;/a&gt; begins:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From a modern perspective,&#xA;  Lavoisier’s theory is just as wrong as advanced versions of the phlogiston theory. Three of the central&#xA;  pillars of Lavoisier’s system are clearly at odds with modern chemistry: the oxygen theory of acidity,&#xA;  the caloric theory of heat (which also explained the three states of matter), and the theory of combustion&#xA;  (especially with regard to the production of heat and light). On the other hand, if we examine Lavoisier’s&#xA;  lasting contributions, we find that various phlogistonists anticipated or outdid Lavoisier ....&#xA;  The real problem with the much-reviled traditional historiography of the Chemical Revolution (and of&#xA;  much else) is not whiggism, but a crude triumphalism, which unreflectively continues to celebrate what&#xA;  was once victorious in the past.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Section 2 of this paper is a detailed critique of Lavoisier's theories vs. the phlogiston theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another good read is &lt;a href=&quot;http://thonyc.wordpress.com/2013/05/20/the-phlogiston-theory-is-not-equivalent-to-the-aquatic-ape-hypothesis/&quot;&gt;this post&lt;/a&gt; by Thony Christie, who blogs about the history of science. A sample:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is very easy from our standpoint in the twenty-first century to pour scorn on the theory of phlogiston, which viewed whiggishly, that is in comparison to our current knowledge of chemistry, can in some of its aspect appear more than somewhat ridiculous. However viewed within the context of the situation in which it was born the phlogiston theory made a great deal of sense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similar arguments can be made about many discarded theories, prime among them the Ptolemaic theory. Most of these negative evaluations arise from what Michael Bycroft calls the &lt;a href=&quot;http://doublerfraction.blogspot.com/2012/11/different-kinds-of-whig-history-are.html&quot;&gt;fixed-evidence fallacy&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This fallacy is committed by many works of history that assess the reasonableness of past beliefs according to how closely those beliefs resemble our own.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;My problem with this history is not that it assesses past scientists,&#xA;  nor that it assesses past scientists by present-day standards. It is&#xA;  that it is a bad way of assessing past scientists by our standards. It&#xA;  is bad because the evidence for most theories has changed through the&#xA;  past, and the reasonableness of those theories has changed&#xA;  accordingly.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This is a mundane point that does not require any Kuhn-like shifts in&#xA;  norms or standards or research-programs. After reading the weather&#xA;  forecast last night it was reasonable for me to believe that it would&#xA;  not rain today. Now, watching the looming storm clouds, that belief is&#xA;  no longer reasonable. What has changed is not my norms or standards&#xA;  but the state of the evidence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastActivityDate="2014-12-15T12:33:28.153" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="661" PostTypeId="2" ParentId="657" CreationDate="2014-12-15T19:11:51.667" Score="3" Body="&lt;p&gt;One has to decide first what exactly we call philosophy. The meaning of this term substantially changed with time. There was a time when all people doing mathematics and sciences could be called philosophers, and what they were doing could be called philosophy.&#xA;(That the scientific degree in physics and mathematics still called PhD in many countries,&#xA;is a reminder of that time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evidence shows that this practice only ended in the early 19-th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern meaning of &quot;philosophy&quot; can be roughly defined as &quot;speculations on those matters which are not covered by exact science&quot;. So with this definition, the area of philosophy shrinks with time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On my opinion, philosophy ends just where the science begins. For example, physics was a part of philosophy until Descartes (including him, but not including Galileo), but since the time of Newton, it is firmly science, not philosophy.&#xA;After that transition, philosophy&#xA;either stops discussing the subject, or continues, in which case its influence is usually harmful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples: Aristotle's philosophy which Galileo and Kepler had to fight. Attacks of philosophers like E. During on Riemann, non-Euclidean geometry and n&gt;3 dimensional spaces.&#xA;Attacks of philosophers like E. Mach on atomic theory and statistical mechanics, and so on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the positive side I can mention Bolzano, Frege and Russell, who were probably the last mathematicians who were also called philosophers. These were the last examples in mathematics, I guess. (In physics, Einstein was apparently somewhat influenced by philosophy of Hume and perhaps Mach).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics actually separated from philosophy earlier than other sciences: this happened probably at the time of Plato. His contemporaries like Eudoxos and Theaetetus were already mathematicians, not philosophers, when they wrote about mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another matter is that sometimes mathematicians and physicists use the word philosophy for&#xA;the kind of speculations in their sciences which are not rigorous. Examples: Complementarity Principle (Bohr), Bloch's Principle, Continuity Principle (in mathematics), etc., or to take an older example, that &quot;the Nature abhors the vacuum&quot;. But this is different from philosophy done by professional philosophers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-16T00:33:04.207" LastActivityDate="2014-12-16T00:33:04.207" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="662" PostTypeId="1" CreationDate="2014-12-16T00:51:07.900" Score="15" ViewCount="491" Body="&lt;p&gt;I mean in the Byzantine empire, from the transfer of the capital to Constantinople till its conquest by the Turks, spanning about 12 centuries. Unlike the Western Roman Empire, this one was never conquered by &amp;quot;barbarians&amp;quot; (except for a brief period after the Fourth Crusade, and these were already not exactly barbarians :-). It was more Greek than Latin in its language and culture, and it was supposed to inherit the Hellenistic culture.&lt;/p&gt;&#xA;&lt;p&gt;I am interested in any sources about the exact sciences and mathematics in this empire. Did they contribute anything at all during the 1200 years of their existence?&lt;/p&gt;&#xA;&lt;p&gt;Their only well known invention, &amp;quot;Greek fire&amp;quot; is completely lost, and we cannot say what it really was (and it probably belongs to hydraulics or chemistry). And, like the Romans, they also had some magnificent architecture. In nearby Persia, and later in several Islamic countries, some mathematics and astronomy was cultivated. And we know about their achievements.&lt;/p&gt;&#xA;&lt;p&gt;EDIT. In her answer, fdb writes that &amp;quot;the progress in the whole mediaeval period (in Western Europe, Byzantium and Islam) was slight&amp;quot;, which is a well known fact. In my opinion, this is an understatement: there was an actual regress. Is there any convincing explanation why this did happen?&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2020-07-06T12:22:57.033" LastActivityDate="2020-07-06T12:22:57.033" Title="Was there any exact science or mathematics in the Eastern Roman Empire?" Tags="|middle-ages|roman-empire|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="663" PostTypeId="1" AcceptedAnswerId="683" CreationDate="2014-12-16T01:03:38.973" Score="19" ViewCount="1530" Body="&lt;p&gt;this is a counter part to my other question: &lt;strong&gt;&lt;em&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/205/what-led-to-the-fall-of-g%C3%B6ttingen&quot;&gt;What led to the fall of G&amp;#246;ttingen?&lt;/a&gt;.&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Göttingen was a major university in which many famous physicists and mathematicians lived. It was located in Germany, almost in the middle of nowhere to my understanding. So what led to its rise, why did anyone even decide to go there in the first place, the earliest piece of history on it I could find was the &lt;a href=&quot;http://en.wikipedia.org/wiki/G%C3%B6ttingen_Seven&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;Göttingen Seven&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;  which was somewhat irrelevant. So who was the first major professor there, and why did the university become so prevalent.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="4251" LastEditDate="2021-03-08T17:19:27.100" LastActivityDate="2021-03-08T17:19:27.100" Title="What led to the rise of Göttingen?" Tags="|mathematics|physics|gottingen|germany|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="664" PostTypeId="5" CreationDate="2014-12-16T01:08:04.373" Score="0" Body="&lt;p&gt;In the period 1939-1945 a major conflict on a global scale, known as the second World War, occurred. This war had a major impact on every aspect of life in many countries, including academics and hence science and mathematics. Questions about the impact of WW2 on specific scientists and/or mathematicians, universities or academics in general should use this tag.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="26" LastEditDate="2014-12-16T09:03:49.900" LastActivityDate="2014-12-16T09:03:49.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="665" PostTypeId="4" CreationDate="2014-12-16T01:08:04.373" Score="0" Body="The second World War was a major war that took place in the 1940's, which involved many of the world's most powerful states." OwnerUserId="140" LastEditorUserId="26" LastEditDate="2014-12-16T08:58:58.267" LastActivityDate="2014-12-16T08:58:58.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="666" PostTypeId="1" AcceptedAnswerId="672" CreationDate="2014-12-16T08:15:58.640" Score="22" ViewCount="15809" Body="&lt;p&gt;German (and Austrian) scientists of the late 19th - early 20th centuries seem to have been the backbone of most of modern physics - namely quantum theory/mechanics. The following are a few predominant names:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Max Planck&lt;/li&gt;&#xA;&lt;li&gt;Albert Einstein&lt;/li&gt;&#xA;&lt;li&gt;Max Born&lt;/li&gt;&#xA;&lt;li&gt;Werner Heisenberg&lt;/li&gt;&#xA;&lt;li&gt;Erwin Schrödinger (Austrian)&lt;/li&gt;&#xA;&lt;li&gt;Wolfgang Pauli (Austrian/Swiss)&lt;/li&gt;&#xA;&lt;li&gt;and many more&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What was it that caused so many brilliant minds to come out of this area, especially around this time? Was their education system simply that good? Perhaps the culture simply valued knowledge more (we all know of the brilliant German philosophers, as well)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that the Soviet education system (especially towards the mid 1900s) was excellent - Stalin, despite some &lt;a href=&quot;http://en.wikipedia.org/wiki/Suppressed_research_in_the_Soviet_Union#Physics&quot;&gt;scientific quirkiness&lt;/a&gt;, had valued scientific research and advancement, even if only for the advancement of the Soviets' reputation during the Cold War, and adjusted the education system accordingly from secondary school forward to groom scientists. Was the German system using a similar tactic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. - could someone edit the tags to something more fitting? I'm not sure which to properly use.&lt;/p&gt;&#xA;" OwnerUserId="494" LastEditorUserId="4251" LastEditDate="2020-07-13T00:34:06.007" LastActivityDate="2020-07-13T00:34:06.007" Title="Why were 20th Century German scientists so impressive?" Tags="|quantum-mechanics|theoretical-physics|physicists|education|scientists|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="667" PostTypeId="2" ParentId="662" CreationDate="2014-12-16T09:06:42.623" Score="5" Body="&lt;p&gt;The standard work on this subject is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.co.uk/books/about/Die_hochsprachliche_profane_Literatur_de.html?id=YC5wxKY_8wsC&amp;amp;redir_esc=y&quot;&gt;http://books.google.co.uk/books/about/Die_hochsprachliche_profane_Literatur_de.html?id=YC5wxKY_8wsC&amp;amp;redir_esc=y&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be honest, the amount of real scientific progress in the whole mediaeval period (in Western Europe, Byzantium and Islam) was slight, compared with what came before (classical antiquity) and after (Renaissance). This is not a specifically Byzantine issue.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2014-12-16T09:06:42.623" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="668" PostTypeId="5" CreationDate="2014-12-16T11:59:13.090" Score="0" Body="&lt;p&gt;This tag should be used when asking a question about the origin, development or other historical aspects of past or present scientific and mathematical terminology. Note that it should &lt;em&gt;not&lt;/em&gt; be used to indicate that a question contains a lot of terminology.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-16T11:59:13.090" LastActivityDate="2014-12-16T11:59:13.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="669" PostTypeId="4" CreationDate="2014-12-16T11:59:13.090" Score="0" Body="For questions about terms, definitions and related concepts used in science and mathematics." OwnerUserId="26" LastEditorUserId="62" LastEditDate="2015-04-12T16:25:52.907" LastActivityDate="2015-04-12T16:25:52.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="670" PostTypeId="5" CreationDate="2014-12-16T12:03:19.000" Score="0" Body="" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-16T12:03:19.000" LastActivityDate="2014-12-16T12:03:19.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="671" PostTypeId="4" CreationDate="2014-12-16T12:03:19.000" Score="0" Body="For questions about historical aspects of the process of publishing results in academic journals or similar platforms for scientific and/or mathematical knowledge." OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-16T12:03:19.000" LastActivityDate="2014-12-16T12:03:19.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="672" PostTypeId="2" ParentId="666" CreationDate="2014-12-16T18:41:22.403" Score="21" Body="&lt;p&gt;The general question is difficult to answer. Why were the British and Dutch scientists of 17-th century so impressive? Why was French science of the 18-th and 19-th century so impressive, especially in the early part of 19-th century ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only thing which is clearly seen, is the correlation of these periods with vigorous economic development of these countries. One can probably conclude that economic development in the modern era is directly related to the progress of science.&#xA;One cannot deny that since the late 19-th century, that is since the unification of Germany, it was the fastest growing economy in the &quot;developed world&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the second question, the role of education system, it is probably also related to the economic development, and I have to say that Russian/Soviet education system in the late 19 and early 20 centuries was a copy of the German one. They deliberately copied the German education institutions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a saying that the economic and military growth of Germany in the late 19-th and early 20-th century is &quot;due to a German schoolmaster&quot;, I do not remember who said that, but this seems to be correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether the development of science (and education) was a reason or a consequence of the economic development is open to discussion, and I don't think this can be decided.&#xA;All we can say is that all three things, science, education and economic development&#xA;strongly correlate. Or at least correlated in the period from 17-th to the middle of 20-th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. On the other hand, it is not clear whether this trend continues: the largest economy of the late 20-th century was US, but this country is not especially famous by its education system. And it never was. It still imports highly educated people on a large scale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The largest economy now is China, but China is not famous yet by a matching progress in the fundamental science. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-16T18:53:14.183" LastActivityDate="2014-12-16T18:53:14.183" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="673" PostTypeId="2" ParentId="657" CreationDate="2014-12-17T01:39:50.600" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Absolute_time_and_space#Historical_controversy&quot; rel=&quot;noreferrer&quot;&gt;Leibniz and Newton&lt;/a&gt; never thought of manifolds outside of ambient space as far as I know, Newton is credited with forging the concept of absolute space, and Leibniz with reducing it to a relational fiction in the style of Aristotle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most influential philosopher of science in 19-th century was Kant, specifically his &quot;Copernican revolution&quot; of interpreting time and space as derivative from human perception. This led to their relativization by empiricists, neo-Kantians and logical positivists after non-Euclidean geometries were discovered. Kant directly influenced the thinking of Helmholtz, Mach, Poincare and Einstein. His influence on Riemann was through &lt;a href=&quot;http://philpapers.org/archive/BANKHA.pdf&quot; rel=&quot;noreferrer&quot;&gt;Herbart&lt;/a&gt;, German philosopher who assumed Kant's chair at Königsberg after his death, and &quot;was a brilliant clarifier and interpreter who sharpened several lines of thought in Kant and Fichte, making their ideas more acceptable to scientists&quot;. Other than Gauss, Riemann's mentor, Herbart is the only person mentioned by name as an influence in Riemann's famous 1854 lecture &lt;a href=&quot;http://www.emis.de/classics/Riemann/WKCGeom.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;On the Hypotheses which lie at the Bases of Geometry&quot;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another mathematician who played a major role in developing ideas about multiple dimensions, and was influenced by Kant through Herbart, was &lt;a href=&quot;http://en.wikipedia.org/wiki/Hermann_Grassmann#Mathematician&quot; rel=&quot;noreferrer&quot;&gt;Hermann Grassman&lt;/a&gt;, who showed that &quot;once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions&quot;. His main work Ausdehnungslehre is a cross between a mathematical work and a philosophical treatise about the nature of space, which made it very hard to understand, and delayed its appreciation. But Riemann's lecture is written in the same semi-philosophical style, although it is much more readable, and there are some &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=1186816&quot; rel=&quot;noreferrer&quot;&gt;textual similarities&lt;/a&gt; between the two. Grassmann sent a copy of Ausdehnungslehre to Gauss in 1844, and there are some  reasons to believe that Riemann also read Grassmann's 1853 paper, which was part of exchange with Helmholtz on the theory of vision.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kant himself &lt;a href=&quot;http://staffweb.hkbu.edu.hk/ppp/srp/arts/KEGP.html&quot; rel=&quot;noreferrer&quot;&gt;wrote&lt;/a&gt; in 1747: &quot;The ground of the threefold dimension of space is still unknown... property of threefold dimension... is arbitrary, God could have chosen another... law [from which] extension with other properties and dimensions would have arisen. A science of all these possible kinds of space would undoubtedly be the highest enterprise which a finite understanding could undertake in the field of geometry&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although Kant later endorsed Euclidean geometry as a priori true, Helmholtz and Poincare found that it could easily accomodate Riemannian perspective by declaring that spatial intuitions are only precise enough for locally Euclidean assertions. They endorsed empiricist and conventionalist views of geometry respectively, both of which &lt;a href=&quot;http://www.rehseis.cnrs.fr/IMG/pdf/Friedman_RelativizedAPriori.pdf&quot; rel=&quot;noreferrer&quot;&gt;influenced Einstein&lt;/a&gt;. The same goes for Mach's ideas about inertia and relativity of motion. As another example, Bohr's Copenhagen interpretation of quantum mechanics has Kantian flavor, and &lt;a href=&quot;http://unfspb.wordpress.com/2011/07/12/logical-positivism-and-the-copenhagen-interpretation/&quot; rel=&quot;noreferrer&quot;&gt;adopts&lt;/a&gt; conceptual schemes of logical positivism and similar philosophies.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-12-17T23:18:50.663" LastActivityDate="2014-12-17T23:18:50.663" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="674" PostTypeId="1" CreationDate="2014-12-17T06:38:53.837" Score="7" ViewCount="501" Body="&lt;p&gt;This question involves not only those provided by the link, but also those &quot;chess-themed&quot; mathematics and computer science problems which are not included in the link.  Who has published the most peer-reviewed articles on this topic amongst mathematicians and scientists?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know, for example, Euler at least looked at the Knight's tour problem, and I believe Gauss even published a paper in the field!  Granted, those problems weren't that important at the time, I'm certain, but for me, it's fun to look at.  I count 11 from that list from Dr Christine Mynhardt, and she's currently Emeritus from the University of South Africa!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that certainly doesn't count them all, because not all &quot;chess-themed&quot; problems are related to the n-queen's problem.  If that is what this site intended, I know not only mine aren't there, but also one of Dr Doug Chatham's.  It also seems others were missing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.liacs.nl/~kosters/nqueens/nqueens_feb2009.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.liacs.nl/~kosters/nqueens/nqueens_feb2009.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2020-04-15T04:58:24.820" LastActivityDate="2020-04-15T04:58:24.820" Title="What mathematician or scientist has published the most peer-reviewed articles on chess problems?" Tags="|mathematics|computer-science|chess|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="675" PostTypeId="1" AcceptedAnswerId="682" CreationDate="2014-12-17T06:47:58.100" Score="5" ViewCount="173" Body="&lt;p&gt;It's pretty common to call a group, ring or module free when it has a 'basis', but unlike other mathematical definitions whose names can be easily related to the concept they describe (&lt;em&gt;e.g.&lt;/em&gt; the &lt;em&gt;spectrum&lt;/em&gt; of an operator), the name &lt;em&gt;free&lt;/em&gt; hardly tells me anything intuitively about the object itself. So, the questions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What's the origin of this terminology?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any hint on what the original motivation was?&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="26" LastEditDate="2014-12-17T07:13:27.637" LastActivityDate="2014-12-17T19:24:26.577" Title="What's the origin and meaning of the adjective &quot;free&quot; in mathematics?" Tags="|mathematics|terminology|abstract-algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="676" PostTypeId="2" ParentId="657" CreationDate="2014-12-17T07:17:04.950" Score="2" Body="&lt;p&gt;Mathematics is made richer with a diverse set of philosophical opinions, but the paradigm that you think might be &quot;golden&quot; for mathematics changes over time, especially in the partly dialectic process of mathematics, which is constantly synthesizing in unknown ways!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Might there be a &quot;Zeitgeist&quot; or spirit of mathematics then?  Just a little joke.....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering mathematics, and mathematics alone, Platonism might be the most defendable position available for the power it has describing the realism of mathematics.  Then, that just defends realism, not how mathematics is actually done in practice.  &lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-12-17T08:28:41.123" LastActivityDate="2014-12-17T08:28:41.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="677" PostTypeId="5" CreationDate="2014-12-17T09:41:16.950" Score="0" Body="" OwnerUserId="58" LastEditorUserId="58" LastEditDate="2014-12-17T09:41:16.950" LastActivityDate="2014-12-17T09:41:16.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="678" PostTypeId="4" CreationDate="2014-12-17T09:41:16.950" Score="0" Body="Do NOT use this tag. It is too general, and is never necessary because it applies to practically any question that is on-topic on this site. This tag will be deleted shortly." OwnerUserId="58" LastEditorUserId="26" LastEditDate="2014-12-17T21:24:52.553" LastActivityDate="2014-12-17T21:24:52.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="679" PostTypeId="1" AcceptedAnswerId="681" CreationDate="2014-12-17T12:10:38.093" Score="21" ViewCount="3635" Body="&lt;p&gt;A number of sources including &lt;a href=&quot;http://news.google.com/newspapers?nid=1300&amp;amp;dat=18600321&amp;amp;id=TK0QAAAAIBAJ&amp;amp;sjid=8JADAAAAIBAJ&amp;amp;pg=4759,6243737&quot;&gt;this one&lt;/a&gt; assert that Isaac Newton used &lt;a href=&quot;http://en.wikipedia.org/wiki/Cipher&quot;&gt;encrypted messages&lt;/a&gt; to communicate some of his scientific discoveries, and as a way of establishing priority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What cipher(s) did he use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the ciphertext and what was the plaintext?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What discoveries did he obscure in this way?&lt;/p&gt;&#xA;" OwnerUserId="281" LastEditorUserId="4251" LastEditDate="2020-05-12T09:23:51.253" LastActivityDate="2020-05-12T09:23:51.253" Title="What cipher(s) did Isaac Newton use?" Tags="|physics|biographical-details|discoveries|newton|cryptology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="681" PostTypeId="2" ParentId="679" CreationDate="2014-12-17T19:04:34.633" Score="21" Body="&lt;p&gt;Newton used anagrams which are not the usual ciphers. It is not designed for a secret communication, but only for proving at a later time that you knew something. So nobody is supposed to be able to decode the message until you tell what the message was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do this, he used a simple procedure: he wrote a sentence (in Latin) and then just &#xA;counted letters in it. And the anagram consisted of the list of letters and how many times&#xA;each letter occurs in the message.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of such anagram is given on Math Overflow:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram&quot;&gt;https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;together with its decoding, according to Newton's later statements.&#xA;The message was usually sent to some third party (Oldenburg, the secretary of the Royal Society in&#xA;the case of Newton), who then forwarded a copy to the addressee (Leibniz).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discoveries Newton wanted to secure this way was essentially &quot;calculus&quot;,&#xA;more precisely power series.&#xA;There were two coded messages in two different letters to Leibniz via Oldenburg.&#xA;You can see the exact messages in the MO (link above).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar methods were widely used to secure priority in 17-th and 18-th centuries.&#xA;The last time that I know was in 1918&#xA;when Gaston Julia send sealed (not coded) messages to the Academie de Science, and these messages had to be publicly opened and read at some later time on his request, to prove his priority. At that time this practice was already obsolete, and Julia was criticized for using it. However he was able to prove his priority and a Grand Prix of the Academy was awarded to him (for what is known now as &quot;Holomorphic dynamics&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.springer.com/mathematics/history+of+mathematics/book/978-3-642-17853-5&quot; rel=&quot;noreferrer&quot;&gt;http://www.springer.com/mathematics/history+of+mathematics/book/978-3-642-17853-5&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-07T16:57:27.107" LastActivityDate="2020-05-07T16:57:27.107" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="682" PostTypeId="2" ParentId="675" CreationDate="2014-12-17T19:24:26.577" Score="1" Body="&lt;p&gt;&quot;Free&quot; usually means &quot;free of relations&quot;, &quot;independent&quot;.&#xA;For example, a free group is the set of all words in a given alphabet&#xA;$$a, a^{-1}, b, b^{-1}, c, c^{-1}\ldots .$$&#xA;The group operation is concatenation of words, with cancellation of $xx^{-1}$ and $x^{-1}x$.&#xA;the neutral element is the empty word.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every finitely generated group can be presented in a similar way but with certain &quot;relations&quot; that is several words which are equal to the neutral word by definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we have a homeomorphism from the free group to any other group generated by the same&#xA;alphabet. This situation can be formalized in terms of category theory, and in any category&#xA;a &quot;free object&quot; can be defined using the appropriate morphism diagram.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference: S. Lang, Algebra.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-12-17T19:24:26.577" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="683" PostTypeId="2" ParentId="663" CreationDate="2014-12-17T19:38:49.340" Score="10" Body="&lt;p&gt;The university, of course, has &lt;a href=&quot;https://www.uni-goettingen.de/en/history-of-the-university-%E2%80%93-an-overview/90607.html&quot;&gt;a page&lt;/a&gt; on its history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The university was founded in 1734 by George II - yes, the king of great Britain; he was of the House of Hanover, after all - although it was first run by Minister Gerlach Adolph Baron of Münchhausen. While the first lectures took place in 1734, the university did not properly open until 1737. From the start, Münchhausen wanted it to be a place of high standing. Here are the first true intellectuals he brought:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;Albrecht von Haller, physician, natural scientist, and poet (in Göttingen 1736–1756)&lt;/li&gt;&#xA;  &lt;li&gt;Johann David Michaelis, theologist and orientalist (in Göttingen 1746–1791)&lt;/li&gt;&#xA;  &lt;li&gt;Christian Gottlob Heyne, archaeologist and library director (in Göttingen 1763–1812)&lt;/li&gt;&#xA;  &lt;li&gt;Georg Christoph Lichtenberg, physicist, philosopher, and writer (in Göttingen 1770–1799)&lt;/li&gt;&#xA;  &lt;li&gt;August Ludwig von Schlözer, publisher and historian (in Göttingen 1769–1809)&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As you can see, these guys predate the Göttingen Seven by quite a lot!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To add to the university's prestige, it retained a close relationship with The Göttingen Academy of Sciences, founded in 1751. Together, the institutions made Göttingen a renowned center for learning. The university continued to be famous by attracting many great thinkers, among them Benjamin Franklin and Carl Friedrich Gauss.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-12-17T19:38:49.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="684" PostTypeId="2" ParentId="663" CreationDate="2014-12-17T19:49:37.463" Score="11" Body="&lt;p&gt;As it always happens with the great scientific centers, it is a combination of several factors. Money. People (usually governments, sometimes private people) who are willing to spend this money to establish and maintain a scientific center. And their ability to hire the best available people. And to create a good environment for them.&#xA;And a long period of development in the conditions of prevailing peace and prosperity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of Gottingen, it was founded by King George II (of England) who also happened to be the &quot;elector&quot; of the land of Hannover. (This part of Germany belonged to the British crown for long time, I suppose until the unification of Germany). King George and his descendants supported the university with the explicit goal to&#xA;&quot;promote the ideas of academic freedom and enlightenment&quot;. So probably this was the best such place in Germany before the unification. The other German rulers before the unification probably had less money and/or were less willing to spend it on science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They managed to get people like Lichtenberg, Schopenhauer, Heine and Grimm brothers, not speaking of Gauss. &#xA;Gauss spent there most of his long life, so probably he liked the conditions there:-)&#xA;Consult the Wikipedia for a long list of famous people who worked and studied there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you have a person like Gauss for a long time, this makes the place even more attractive, both to the best students and best researchers.&#xA;So in the next generation they had Riemann and Dirichlet....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bismark who was a student there, later unified Germany.&#xA;And I think he was also interested in maintaining the place. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-12-17T19:49:37.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="685" PostTypeId="1" CreationDate="2014-12-18T01:29:09.990" Score="24" ViewCount="1330" Body="&lt;p&gt;Euclid himself already treats it with gloves, it has an unusually precise formulation, and is not used in the first 28 propositions of the Elements. Why? Did he doubt it? It's not like Euclid was a formalist, in the very first proof of &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html&quot;&gt;proposition I.1&lt;/a&gt; (construction of equilateral triangle), he freely concludes from the diagram that two circles with common interior points intersect at two points. This is called &lt;a href=&quot;http://www.jstor.org/stable/10.4169/000298910X480063&quot;&gt;circle-cirle axiom&lt;/a&gt; by Greenberg and is not a postulate. Why is intersection of two circles &quot;self-evident&quot;, but (effectively) intersection of two convergent lines is not? Nor does Euclid complicate the first postulate by stipulating uniqueness of a line through two points, even though it is clear from his proofs that he assumes it. The second postulate is also weaker than what Euclid actually uses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, with parallels Euclid is very pedantic, as if trying to attract attention to them: &#xA;&quot;&lt;em&gt;If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some textbooks blame other instances on Euclid's &quot;flaws&quot; and &quot;gaps&quot;, but this is apparently a &lt;a href=&quot;http://fitelson.org/few/few_07/mumma.pdf&quot;&gt;modernization&lt;/a&gt; based on reading Pasch and Hilbert into Euclid. Euclid's method was not axiomatic, it was synthetic. In other words, when he gives a geometric construction he accepts non-deductive conclusions &quot;from the diagram&quot;, as long as the conclusions would follow from any possible diagram consistent with the construction. Listed postulates were the ones most commonly used and/or most &quot;subtle&quot; perhaps? In principle there was no need to list the parallel postulate at all, like Euclid didn't list the circle-circle or the uniqueness of the line. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if Euclid did want to list something about parallels he could opt for &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI30.html&quot;&gt;I.30&lt;/a&gt; instead, two lines parallel to a third are parallel to each other. Much simpler. The contrapositive to this (two intersecting lines can not both be parallel to a third) is now well known and used as &quot;Playfair's axiom&quot;. Even assuming that Euclid didn't know that it was equivalent, Ptolemy showed it explicitly later, according to Proclus, but that was not deemed satisfactory. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Proof attempts continued for over a thousand years. What proof were geometers looking for? Surely not deducing the fifth postulate &quot;from the other four&quot;, since not much of anything can be deduced from the other four, or all five for that matter, not even I.1. As for synthetic proofs relying on &quot;more self-evident&quot; claims, they were offered &lt;a href=&quot;http://www.math.nmsu.edu/~history/book/geometry.pdf&quot;&gt;starting with Archimedes&lt;/a&gt; apparently, but never accepted. Why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-12-18T01:43:57.500" LastActivityDate="2019-12-27T03:43:07.767" Title="Why were geometers dissatisfied with the parallel postulate?" Tags="|mathematics|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="686" PostTypeId="1" CreationDate="2014-12-18T02:11:09.533" Score="6" ViewCount="522" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Marcel_Grossmann&quot; rel=&quot;nofollow&quot;&gt;Marcel Grossmann&lt;/a&gt; is perhaps best known for helping Einstein learn the Riemannian geometry necessary to formulate general relativity. He was instrumental in its early development. Wikipedia states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The collaboration of Einstein and Grossmann led to a ground-breaking paper: &quot;Outline of a Generalized Theory of Relativity and of a Theory of Gravitation&quot;, which was published in 1913 and was one of the two fundamental papers which established Einstein's theory of gravity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Wikipedia page on Grossmann references that paper and a second paper the duo authored the presented the second part of general relativity, &quot;Covariance Properties of the Field Equations of the Theory of Gravitation Based on the Generalized Theory of Relativity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pitt.edu/~jdnorton/teaching/GR&amp;amp;Grav_2007/pdf/Einstein_Entwurf_1913.pdf&quot; rel=&quot;nofollow&quot;&gt;This&lt;/a&gt; provides an English translation of the first paper. Note that Einstein is credited with the &quot;physical part&quot;, while Grossmann is credited with the &quot;mathematical part&quot;. This suggests that Grossmann was not intimately tied to the theory's physical predictions, merely the math behind it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Grossmann author any other papers on general relativity, with or without Einstein?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2014-12-18T16:31:55.733" LastActivityDate="2016-06-21T07:21:20.977" Title="How many papers on general relativity did Marcel Grossmann author or co-author?" Tags="|physics|publishing|relativity-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="687" PostTypeId="2" ParentId="663" CreationDate="2014-12-18T11:46:53.770" Score="3" Body="&lt;p&gt;&quot;Göttingen, famous for its university and its sausages…&quot; as the poet Heinrich Heine put it. Göttingen was apparently the first university in Germany that abolished the right of the Theological Faculty to “supervise” the other branches of the university. In this sense it anticipated the secularisation of German universities, championed half a century later by Humboldt in Berlin (and later still by Bentham at University College London).&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2014-12-18T11:46:53.770" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="688" PostTypeId="2" ParentId="686" CreationDate="2014-12-18T13:45:09.417" Score="5" Body="&lt;p&gt;Grossmann is spelled with two n in the end. According to the Jahrbuch database he had 31 papers, 3 of them joint with Einstein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jahrbuch is available free:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.emis.de/MATH/JFM/full.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.emis.de/MATH/JFM/full.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a database in Mathematics, covering papers 1868-1942, but most papers in theoretical physics are included.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can look at the complete list there and decide which are in General relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most comprehensive database covering all papers in all sciences&#xA;is Poggendorff, but most of it is not available online, even in my university.&#xA;We have a paper copy, and if this question is considered important (why?) I can go and check.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comment. Einstein had many assistants (he changed them frequently). But one can argue that&#xA;the most famous of them was Cornelius Lanczos. He is not famous as an Einstein assistant&#xA;(he spent only about one year on this position) but for his own contributions.&#xA;Among other things, he invented the Fast Fourier Transform, one of the crucial mathematical inventions which made the modern computer revolution possible.&#xA;He also discovered the Schrodinger equation, but his paper was published few weeks after&#xA;Schrodinger.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-18T14:00:33.860" LastActivityDate="2014-12-18T14:00:33.860" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="689" PostTypeId="2" ParentId="685" CreationDate="2014-12-18T14:17:45.803" Score="13" Body="&lt;p&gt;The reasons are &quot;simple&quot;. &#xA;All other axioms and postulates appeal to our &quot;everyday experience&quot;,&#xA;at least in principle. The straight lines correspond to light rays in everyday experience.&#xA;However it was probably already recognized by Euclid, that the parallel postulate is different from the other axioms. Of course, a degree of mathematical sophistication&#xA;is required to understand this. But probably Euclid already understood this.&#xA;Even when stated accurately, the V postulate looks much more complicated than the other axioms and postulates. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How indeed would you verify it experimentally? The postulate itself says that through every point A not on the line L one can draw only one line parallel to L. How would you propose to check this experimentally? There are clearly many lines through A which intersect L so far away that you cannot see this. So they do not intersect within your field of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or one can try to verify any of its consequences. One of the simplest consequences is that the sum of the angles of a triangle is equal to two right angles. How can you verify that this holds in real life?&#xA;No measurement, no matter how accurate will show you this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss and Lobachevski, who recognized that the postulate does not follow from the rest of the axioms, indeed discuss its possible experimental verification. One has to measure the angles of a very large triangle to do this. And any result that you obtain will have some error in the measurement, and leave the possibility that if you take a larger triangle,&#xA;you will see that the sum is not equal to two right angles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. To obtain a better intuitive understanding what the axioms mean, imagine that you live in a world in which one of the axioms is not satisfied, and explore how different this word looks. For example, suppose that two light rays can intersect at two points.&#xA;This means that under certain conditions you will see the same object as two objects. Our &#xA;everyday&#xA;experience shows that this is not the case in our world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now imagine a world where the parallel postulate does not hold. Will you see something peculiar in your everyday life? The answer is &quot;no&quot;. Everything will look more or less the same. Until you start measuring the angles of large triangles. But we do not really have everyday experience with measuring angles of large triangles. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-12-27T03:43:07.767" LastActivityDate="2019-12-27T03:43:07.767" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="691" PostTypeId="1" CreationDate="2014-12-19T15:43:20.027" Score="14" ViewCount="1158" Body="&lt;p&gt;&lt;a href=&quot;https://academia.stackexchange.com/q/34547/15062&quot;&gt;A question posed on academia.SE&lt;/a&gt; prompts this follow-up question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an example of a famous physical law, constant, equation, theorem etc that was named after its discoverer &lt;em&gt;by the discoverer him/herself&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thinking of things like Newton's Laws, Josephson effect, Kalman filter, Turing machine, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually these things are named by others - but did anyone ever say &quot;I discovered what I propose to call the Jones effect&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For reference, you can find a long list of &quot;laws named after people&quot; at &lt;a href=&quot;http://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/List_of_scientific_laws_named_after_people&lt;/a&gt; (with thanks to J.W. Perry for first posting the link, and to HDE226868 for restoring it when the original comment was deleted)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm sure it's not a complete list...&lt;/p&gt;&#xA;" OwnerUserId="543" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.757" LastActivityDate="2018-03-21T23:30:56.990" Title="What famous laws were named by their discoverer" Tags="|discoveries|naming-conventions|" AnswerCount="3" CommentCount="15" ContentLicense="CC BY-SA 3.0" />
  <row Id="692" PostTypeId="2" ParentId="652" CreationDate="2014-12-19T21:05:24.423" Score="6" Body="&lt;p&gt;I believe I know where the occasional lower case spelling (and thus your confusion) comes from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take for example &lt;a href=&quot;http://press.uchicago.edu/ucp/books/book/distributed/I/bo6166141.html&quot;&gt;this book&lt;/a&gt; which contains &quot;Selected papers from the 7th Augustus de Morgan Workshop, London&quot;. Note the lower case &quot;de&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The editors of the book were Dutch (Amsterdam University Press, Johan van Benthem); and in Dutch, words like &quot;de&quot; or &quot;van&quot; are spelled with a lower case when used in the middle of a person's name, and capitalized when used at the beginning. So a Dutch person, even when writing about an Englishman, would be tempted to write &quot;His biography tells us that De Morgan was born in India&quot; (capital D to indicate the use of &quot;De&quot; as the start of a name) but &quot;Augustus de Morgan entered Trinity College when he was sixteen&quot; (lower case &quot;de&quot; in the middle of a name). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that other sources of the use of &quot;de&quot; can similarly be traced to authors whose culture would naturally use lowercase in the middle of the name; but if you want to get it &quot;right&quot; you should spell the name the way he did - which is &quot;Augustus De Morgan&quot;, as @fdb's answer demonstrates.&lt;/p&gt;&#xA;" OwnerUserId="543" LastActivityDate="2014-12-19T21:05:24.423" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="693" PostTypeId="2" ParentId="662" CreationDate="2014-12-19T23:50:32.273" Score="9" Body="&lt;p&gt;There is a chapter on Byzantine science in a recent Cambridge History of Science &lt;a href=&quot;http://universitypublishingonline.org/cambridge/histories/chapter.jsf?bid=CHO9780511974007&amp;amp;cid=CHO9780511974007A016&quot; rel=&quot;nofollow&quot;&gt;volume&lt;/a&gt;, see also &lt;a href=&quot;http://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=4&amp;amp;ved=0CDkQFjAD&amp;amp;url=http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F247158272_THE_CONTRIBUTION_OF_THE_CHURCH_IN_BYZANTIUM_TO_THE_NATURAL_SCIENCES_BYZANTINE_ASTRONOMERS_AND_SCIENTISTS%2Flinks%2F0deec51dbb60d2c1d8000000&amp;amp;ei=Z6uUVLf5KcWxyASLuYKgAw&amp;amp;usg=AFQjCNFd8QVlmVogjR9fVMm6O6PIwmFywg&amp;amp;sig2=m9bYZWWkKHU1YEMxX4fSHg&amp;amp;bvm=bv.82001339,d.aWw&quot; rel=&quot;nofollow&quot;&gt;this paper&lt;/a&gt;. As to why I'll articulate the non-controversial part that gets drowned in more detailed accounts. Byzantium was driven by two forces, imperial power and orthodox Christian culture, and neither one supported science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Medieval orthodox Christianity was rather millenaristic, it was expected that kingdom come was not long at hand. Salvation of the soul was deemed far more important than trappings of mortal life, including study of nature. In education trivium was emphasized and quadrivium trivialized (pun intended), theorizing was dominated by theological studies. An exception that confirms the rule was &lt;a href=&quot;http://en.wikipedia.org/wiki/John_Philoponus#Life&quot; rel=&quot;nofollow&quot;&gt;John Philoponus&lt;/a&gt;, the founder of impetus theory, who rejected Aristotle's dynamics as erroneous remarking that &quot;our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument&quot;. He was shunned by colleagues, his empirical bent and critique of Aristotle gaining little influence, and forced to abandon philosophy for theology, where his views were declared heresy shortly after his death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why the emperors did not support science is also understandable. Throughout its history Byzantium was a castle under siege. First by barbarian hordes in the West during the &lt;a href=&quot;http://en.wikipedia.org/wiki/Migration_Period&quot; rel=&quot;nofollow&quot;&gt;Great Migration of Peoples&lt;/a&gt;, and no sooner that it subsided the rise of Islam brought Turks to the eastern borders of the empire, the force that haunted it since and eventually brought it down. The &lt;a href=&quot;http://en.wikipedia.org/wiki/East%E2%80%93West_Schism&quot; rel=&quot;nofollow&quot;&gt;East-West schism&lt;/a&gt; of 1054 and subsequent papal crusades undermined its security further, at one point Constantinople was ransacked by the crusaders, even eastern Slavs that adopted orthodox Christianity warred with Byzantium. In such conditions people hunker down, collect and preserve rather than explore, and projects with immediate practical impact are favored over more contemplative and playful pursuits à la Euclid and Archimedes. And indeed architecture, engineering and medicine fared better in Byzantium than science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, the rise of science in Hellenistic and post-Renaissence times was accompanied by the opposite conditions. As described in Lucio Russo's &lt;a href=&quot;http://en.wikipedia.org/wiki/Lucio_Russo#The_Forgotten_Revolution&quot; rel=&quot;nofollow&quot;&gt;famous book&lt;/a&gt;  at its zenith in Hellenistic Egypt the Alexandrian Library was funded and supported by Ptolemies, and Hellenistic cultural outlook was directed towards the world and its workings. Once support was replaced by &lt;a href=&quot;http://en.wikipedia.org/wiki/Ptolemy_VIII_Physcon#Revenge_and_intrigues&quot; rel=&quot;nofollow&quot;&gt;repression under Euergetes II&lt;/a&gt; and then Romans with their &quot;public duty is more important than heavens&quot; attitude Alexandrian science floundered. Max Weber &lt;a href=&quot;http://en.wikipedia.org/wiki/The_Protestant_Ethic_and_the_Spirit_of_Capitalism#Origins_of_the_Protestant_work_ethic&quot; rel=&quot;nofollow&quot;&gt;links&lt;/a&gt; reformed Christian culture in Europe to the enterprising spirit that drove science post-Renaissence, Russo describes traits of Hellenistic Egypt that can be seen as prototypes of capitalism too. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems that science requires stable peace over large territories and an enterprising culture directed outward, the antithesis of Byzantium. Still, Byzantium fulfilled its role as a shield against Muslims, while Europe incubated its new order, and as preserver and transmitter, directly and through Arabs, of Greek and Latin works until the conditions were right for science to rise again.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-03-04T23:54:41.890" LastActivityDate="2015-03-04T23:54:41.890" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="694" PostTypeId="2" ParentId="691" CreationDate="2014-12-20T00:11:24.753" Score="4" Body="&lt;p&gt;In 2013, Harminder Dua announced the discovery of the eponymous &lt;a href=&quot;http://en.wikipedia.org/wiki/Dua&amp;#39;s_layer&quot; rel=&quot;nofollow&quot;&gt;Dua's layer&lt;/a&gt; in the eye.&#xA;Wikipedia writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While some scientists welcomed the announcement, other scientists cautioned that time was needed for other researchers to confirm the discovery and its significance. Others have met the claim &quot;with incredulity&quot;. The choice of the name Dua's Layer has also been criticized.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="549" LastActivityDate="2014-12-20T00:11:24.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="695" PostTypeId="2" ParentId="691" CreationDate="2014-12-20T07:37:13.393" Score="3" Body="&lt;p&gt;Here is an interesting twist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Black-Scholes model for option pricing first appeared in:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Black, Fischer; Scholes, Myron (1973). &quot;The Pricing of Options and Corporate Liabilities&quot;. Journal of Political Economy 81 (3): 637–654.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where they derived and solved the governing partial differential equation, now referred to as the Black-Scholes equation. Robert Merton provided a crucial step in deriving the PDE rigorously using a dynamic hedging argument -- an improvement on an approach considered by Black and Scholes. He was subsequently acknowledged in a footnote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there was no explicit reference to &quot;Black-Scholes model&quot; in the original publication.  Later Merton published a paper where he named the model Black-Scholes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Merton and Scholes were awarded the 1997 Nobel Prize Economics for this work.  The model is now commonly referred to as the Black-Scholes-Merton option pricing model.&lt;/p&gt;&#xA;" OwnerUserId="493" LastActivityDate="2014-12-20T07:37:13.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="696" PostTypeId="1" AcceptedAnswerId="722" CreationDate="2014-12-20T17:07:31.257" Score="10" ViewCount="1319" Body="&lt;p&gt;As far as I know the &lt;a href=&quot;https://en.wikipedia.org/wiki/Vienna_Circle&quot;&gt;Vienna Circle&lt;/a&gt; was very relevant to science in the twentieth century. Why? What was the importance of the members' philosophy in science? Would science be too different from what it is today if it weren't for them? (I know these kinds of question are quite tricky; a comment is enough)&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="62" LastEditDate="2014-12-20T19:44:52.527" LastActivityDate="2017-01-24T00:15:11.937" Title="Why was the Vienna Circle so important?" Tags="|philosophy-of-science|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="698" PostTypeId="2" ParentId="257" CreationDate="2014-12-20T18:55:09.053" Score="-1" Body="&lt;p&gt;Clinical trials is an excellent metaphor explaining this bias issue.  The results of any clinical trial are probability curves with generous standard deviations.  All clinical trials both fail and succeed in curing disease.  Thus bias depends on when the lie is born, which depends on the data analysis performed by an observer.  A bias against publishing lies is oxymoronic, and can't be quantified, as all bias is based on lies. Similarly, the first casualty in war is Truth.  But when have we never not been at war?  &lt;/p&gt;&#xA;" OwnerUserId="555" LastActivityDate="2014-12-20T18:55:09.053" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="701" PostTypeId="1" AcceptedAnswerId="704" CreationDate="2014-12-20T22:27:18.263" Score="7" ViewCount="581" Body="&lt;p&gt;&lt;strong&gt;Context:&lt;/strong&gt; I've been reading a lot about &lt;a href=&quot;http://en.m.wikipedia.org/wiki/Set_theory&quot; rel=&quot;noreferrer&quot;&gt;Set Theory&lt;/a&gt; lately, and how it suddenly sprung onto the mathematical scene in the late 1800's, thanks largely to Cantor. But it seems strange to me that no one had ever done anything similar to it before, so I was wondering...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Are there any historical instances of someone inventing something similar to Set Theory?&lt;/em&gt; Or is it actually the case that no one had considered anything like it until the 1800's?&lt;/p&gt;&#xA;" OwnerUserId="537" LastActivityDate="2024-03-20T13:22:18.897" Title="Historical Instances of Set Theory" Tags="|mathematics|mathematicians|set-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="703" PostTypeId="2" ParentId="303" CreationDate="2014-12-21T02:40:53.057" Score="1" Body="&lt;p&gt;We see objects because objects emit their own light.  How?  The electrons which orbit inside the atoms which comprise the object interact with ambient light radiation.  This causes the electrons to switch to higher orbits.  Then, inexplicably, the electrons return to their original orbit, and in that process emit photons which our eyes detect.  Thus Democritus had the best explanation.  Now science hasn't figured out why the electrons return to their original orbit after they attain the higher orbit when the electrons interact with ambient radiation. However, words like absorption and reflection are misleading, just like the words dark matter have no meaning, as science doesn't know what dark matter is. &lt;/p&gt;&#xA;" OwnerUserId="555" LastEditorUserId="555" LastEditDate="2014-12-21T12:02:51.330" LastActivityDate="2014-12-21T12:02:51.330" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="704" PostTypeId="2" ParentId="701" CreationDate="2014-12-21T09:45:26.343" Score="4" Body="&lt;p&gt;There were no historical instances. This is exactly why this discovery is considered so great. It is a feature of  truly great discoveries that after they are made, people ask: why did nobody think like this before ?&lt;/p&gt;&#xA;&lt;p&gt;Set theory is the best example of this I can think of.&lt;/p&gt;&#xA;&lt;p&gt;Comparable perhaps only to Kepler's First law and General Relativity.&#xA;I do not know any other examples of such radical changes in our thinking, made by one person.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps other members will give other examples.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-20T13:22:18.897" LastActivityDate="2024-03-20T13:22:18.897" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="705" PostTypeId="1" AcceptedAnswerId="707" CreationDate="2014-12-21T12:51:38.877" Score="19" ViewCount="1450" Body="&lt;p&gt;Why do we write $a^n$ instead of $^n\!a$ for exponentiation? What benefit is there to writing the base before the exponent? With addition and multiplication order doesn't matter since $a + b = b + a$, so why was $a^n$ chosen, and who popularised this notation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me the first is preferable since it's a logical choice, as you put the number in front when you add:&#xA;$$a+a+a = 3a$$&#xA;Thus it makes sense to write:&#xA;$$a\times a\times a = \,^3\!a$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would also retain left-associativity as is commonly used with subtraction and division: $$(a\text{^} b)\text{^}c = \,^\left(^ab\right)c$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically the best choice isn't always the one which got popularised, but there must have been some reason why this choice were made and preferred in its period of invention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar question was previously asked on &lt;a href=&quot;https://math.stackexchange.com/questions/1072650/why-do-we-write-an-instead-of-n-a-for-exponentiation&quot;&gt;Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="559" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2014-12-22T09:10:04.217" Title="Who invented the way we write exponentiation?" Tags="|notation|" AnswerCount="2" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="706" PostTypeId="2" ParentId="705" CreationDate="2014-12-21T13:58:57.117" Score="6" Body="&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/operation.html&quot;&gt;Earliest Use of Symbols for Operations&lt;/a&gt; for a discussion.  Summary: Descartes 1637 is the first to use exponents for powers in the modern way:  $a^3$.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2014-12-21T13:58:57.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="707" PostTypeId="2" ParentId="705" CreationDate="2014-12-21T17:50:52.963" Score="16" Body="&lt;p&gt;Browsing the &quot;original&quot; historiographical&quot; source for the &quot;power&quot; symbolisms , i.e. :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Florian Cajori, &lt;a href=&quot;https://books.google.it/books?id=_byqAAAAQBAJ&amp;amp;pg=345&quot;&gt;A History of Mathematical Notations&lt;/a&gt; (1928, Dover reprint), page 335-on&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;can be very istructive, showing how Descartes' choice was the last step of a complex process :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Pietro Antonio Cataldi : $5$ $3(crossed)$ for $5x^3$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Joost Bürgi : with $vi$ on top of $8$ for $8x^6$ (obvious problem : how to handle two variables : $x$ and $y$ ?)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Romanus (Adriaan van Roomen) : $A(4)$ for $A^4$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Pierre Hérigone : $a3$ for $a^3$ and $2b4$ for $2b^4$&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;James Hume (1636) : $A^{iii}$ for $A^3$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thus, Cajori concludes that :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hérigone and Hume almost hit upon the scheme of Descartes. [...] Where Hume would write $5a^{iv}$ and Hérigone would write $5a4$, Descartes wrote $5a^4$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2014-12-22T09:10:04.217" LastActivityDate="2014-12-22T09:10:04.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="708" PostTypeId="2" ParentId="691" CreationDate="2014-12-22T09:01:38.773" Score="6" Body="&lt;p&gt;This isn't really a physical law, nor is it exactly what you're asking for, but the statistical concept usually known as the &lt;a href=&quot;http://en.m.wikipedia.org/wiki/Akaike_information_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Akaike Information Criterion&quot;&lt;/a&gt; (AIC) was indeed called AIC &lt;a href=&quot;http://ieeexplore.ieee.org/document/1100705/?reload=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;by Akaike&lt;/a&gt;, but with the name &quot;an information criterion.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know historically whether that was intentional or not, but when Sumio Watanabe &lt;a href=&quot;http://www.jmlr.org/papers/volume14/watanabe13a/watanabe13a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;called his extension&lt;/a&gt; the &lt;a href=&quot;https://en.wikipedia.org/wiki/Watanabe%E2%80%93Akaike_information_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Widely Applicable Information Criterion&quot;&lt;/a&gt; (WAIC) the standard name of &quot;Watanabe-Akaike Information Criterion&quot; shortly followed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Andrew Gelman &lt;a href=&quot;http://andrewgelman.com/2013/08/09/understanding-predictive-information-criteria-for-bayesian-models/&quot; rel=&quot;nofollow noreferrer&quot;&gt;has joked&lt;/a&gt; that he should develop an information criterion with Aki Vehtari and call it the &quot;Very Good Information Criterion.&quot;&lt;/p&gt;&#xA;" OwnerUserId="568" LastEditorUserId="568" LastEditDate="2018-03-21T23:30:56.990" LastActivityDate="2018-03-21T23:30:56.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="709" PostTypeId="2" ParentId="696" CreationDate="2014-12-22T12:37:03.467" Score="1" Body="&lt;p&gt;The Vienna Circle consisted of two economists - namely the Menger's (among many other heavily credentialed geniuses).  They were influential in the social sciences, namely those that apply game theory.  The son even started his investigation with an exploration as to the success of mathematics in explaining economics from a purely rational point of view.  That sounds like the beginnings of philosophical questioning to me!&#xA;After all, one can easily trade out economics for any other field in which game theory can potentially be applied in such a questioning to obtain a philosophical question - namely how much can game theory apply to the world?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then, it should come as no surprise that when new fields are forged in the sciences there is new history for philosophers to question and contemplate.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Philosophy of science without history of science is empty; history of science without philosophy of science is blind.&quot;&#xA;-Imre Lakatos&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2014-12-22T12:51:32.723" LastActivityDate="2014-12-22T12:51:32.723" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="710" PostTypeId="1" CreationDate="2014-12-22T19:04:20.523" Score="13" ViewCount="4309" Body="&lt;p&gt;Can someone tell me what year was the cogwheel gear invented? I tried searching it up but the answers were too complicated.&lt;/p&gt;&#xA;" OwnerUserId="573" LastEditorUserId="4251" LastEditDate="2020-07-25T12:23:00.077" LastActivityDate="2020-07-25T12:23:00.077" Title="When was the cogwheel gear invented?" Tags="|engineering|technology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="711" PostTypeId="1" CreationDate="2014-12-22T19:24:35.853" Score="9" ViewCount="583" Body="&lt;p&gt;I was looking for some fundamental books on history of science. I picked Thomas Kuhn book &quot;The Structure of Scientific Revolutions&quot; but it's not exactly about history of science - it's more on methodology and philosophy of science. Can you recommend some books? Thank you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;UPD: To be more precise, I wanted to learn more about how humanities in Europe have developed from the &quot;beginning of science&quot; till our days.&lt;/p&gt;&#xA;" OwnerUserId="574" LastEditorUserId="574" LastEditDate="2014-12-23T10:22:44.603" LastActivityDate="2015-01-01T20:35:29.513" Title="Are there any canonical books on history of science?" Tags="|reference-request|philosophy-of-science|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="713" PostTypeId="2" ParentId="711" CreationDate="2014-12-22T20:42:48.043" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Throw away the book of Kuhn. He is not a scientist.&#xA;He is a philosopher. Who has an agenda. And trying to push this agenda.&#xA;This approach has nothing to do with science.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Canonical books&quot; on history of science are (my choice):&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&quot;Dictionary of scientific biography&quot; (multiple volumes by multiple authors)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A history of ancient mathematical astronomy&quot; by O. Neugebauer,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Science awakening&quot; by B. L. van der Waerden, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The Exact sciences in antiquity&quot; by  O. Neugebauer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other list members can complement this list of CANONICAL books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many books on history of specific sciences in specific periods,&#xA;but there is no (and cannot be) a &quot;canonical book&quot; on history of WHOLE SCIENCE&#xA;of ALL periods. Such book is not possible. All books listed above concern either a&#xA;specific area of science or a specific period, except N 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But they are &quot;canonical&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot call any other book in history of science &quot;canonical&quot;,&#xA;but perhaps other members will make more suggestions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is certainly a &quot;big list&quot; question. The kind of question which is not welcome&#xA;(or explicitly prohibited) on this site, for the reasons which I do not understand. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-22T21:08:34.717" LastActivityDate="2014-12-22T21:08:34.717" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="715" PostTypeId="2" ParentId="711" CreationDate="2014-12-23T09:53:12.210" Score="3" Body="&lt;p&gt;A &quot;complete&quot; history of science is hard to find today : the topic is too wide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can try with the multi-volume multi-author :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521594480&quot; rel=&quot;nofollow&quot;&gt;The Cambridge History of Science&lt;/a&gt; (2002-on), 7 vols&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;or the &quot;older&quot; :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;René Taton, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B000V254RG&quot; rel=&quot;nofollow&quot;&gt;Histoire gènèrale des sciences&lt;/a&gt;, (1957-on) 3 vols.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But can be useful to start with recent overviews of some &quot;critical moment&quot;, like the so-called &lt;em&gt;Scientific Revolution&lt;/em&gt;; they are shorter and usually supply a rich bibliography :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;David Lindberg, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0226482057&quot; rel=&quot;nofollow&quot;&gt;The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, Prehistory to A.D. 1450&lt;/a&gt; (2008)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Steven Shapin, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0226750213&quot; rel=&quot;nofollow&quot;&gt;The Scientific Revolution&lt;/a&gt; (1998)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Peter Dear, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691142068&quot; rel=&quot;nofollow&quot;&gt;Revolutionizing the Sciences: European Knowledge and Its Ambitions, 1500-1700&lt;/a&gt; (2nd ed 2009)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John Henry, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0230574386&quot; rel=&quot;nofollow&quot;&gt;The Scientific Revolution and the Origins of Modern Science &lt;/a&gt; (2008).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For connection between European science and other cultures, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Toby Huff, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521529948&quot; rel=&quot;nofollow&quot;&gt;The Rise of Early Modern Science: Islam, China and the West&lt;/a&gt; (2nd ed 2003)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;H.Floris Cohen, &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/9089642390&quot; rel=&quot;nofollow&quot;&gt;How Modern Science Came into the World: Four Civilizations, One 17th-Century Breakthrough&lt;/a&gt; (2011)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some suggestions :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;for Greek science : G.E.R.LLoyd (in general) and Wilbur Knorr, Reviel Netz for mathematics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;for Medieval science : Edward Grant and David Lindberg&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;for Early Modern Science and Galileo/Descartes/Newton : Alexandre Koyré, Stillman Drake, I.B.Cohen.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-12-23T09:53:12.210" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="716" PostTypeId="1" CreationDate="2014-12-23T14:15:01.540" Score="3" ViewCount="299" Body="&lt;p&gt;Who discovered/ defined momentum and impulse?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please, I want suggestions about materials/ books on the development of Classical mechanics? Something like a history of experiments.&lt;/p&gt;&#xA;" OwnerUserId="579" OwnerDisplayName="Eng_Boody" LastEditorUserId="26" LastEditDate="2014-12-23T14:42:36.370" LastActivityDate="2014-12-25T02:15:45.373" Title="Reference needed about history of classical mechanics" Tags="|experimental-physics|reference-request|classical-mechanics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="718" PostTypeId="2" ParentId="257" CreationDate="2014-12-24T00:34:00.920" Score="5" Body="&lt;p&gt;Here some early papers, listed in chronological order:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Sterling TD (1959) Publication Decisions and Their Possible Effects on Inferences Drawn from Tests of Significance--Or Vice Versa. Journal of the American Statistical Association 54: 30–34.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rosenthal R (1979) The file drawer problem and tolerance for null results. Psychological Bulletin 86: 638–641.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simes RJ (1986) Publication bias: the case for an international registry of clinical trials. J Clin Oncol 4: 1529–1541.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dickersin K (1990) The existence of publication bias and risk factors for its occurrence. JAMA 263: 1385–1389.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Chalmers I (1990) Underreporting research is scientific misconduct. JAMA 263: 1405–1408.1.  Sterling TD (1959) Publication Decisions and Their Possible Effects on Inferences Drawn from Tests of Significance--Or Vice Versa. Journal of the American Statistical Association 54: 30–34.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="582" LastActivityDate="2014-12-24T00:34:00.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="719" PostTypeId="1" CreationDate="2014-12-24T02:13:36.790" Score="1" ViewCount="190" Body="&lt;p&gt;Surely the scientists working on (un)manned spaceflight understood the concept of the earth having an upper level to its atmosphere, but how were they able to determine what exactly the physics of space were, before a spacecraft had ever made it out of the atmosphere?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Issues such as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;space is a vacuum&lt;/li&gt;&#xA;&lt;li&gt;traveling at incredibly high speeds while in orbit (due to low friction)&lt;/li&gt;&#xA;&lt;li&gt;atmospheric reentry&lt;/li&gt;&#xA;&lt;li&gt;etc&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;must have been hard to figure out without some sort of actual testing.  Was it simply trial and error?&lt;/p&gt;&#xA;" OwnerUserId="494" LastActivityDate="2014-12-25T10:30:03.703" Title="How did scientists know how to build spacecraft before we got them into space?" Tags="|atmosphere|aerodynamics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="720" PostTypeId="2" ParentId="710" CreationDate="2014-12-25T01:29:06.147" Score="4" Body="&lt;p&gt;Leonardo &lt;a href=&quot;http://answers.google.com/answers/threadview/id/449050.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;thought&lt;/a&gt; it was Archimedes (287-212 BC) who invented the cogwheel gears. This would make a neat story, but history is rarely neat. In a similar matter we know that Archimedes did not exactly invent the  &lt;a href=&quot;http://en.wikipedia.org/wiki/Archimedes%27_screw&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Archimedes’s screw”&lt;/a&gt;, but rather perfected it, which led to its wide use in the Hellenistic period. One can speculate that something similar happened with cogwheel gears as well, especially complex arrangements of them, but this is only an educated guess. Primitive cogwheels might have been known to Egyptian and Mesopotamian engineers prior to that, just like the screw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first written mention of the gear that survives is from Mechanics by Heron of Alexandria (10-70 AD), and Heron &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.1963.tb00551.x/abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;is known&lt;/a&gt; to rely on Archimedes in other parts of his book. Also, the earliest gear known today comes from the famous &lt;a href=&quot;http://en.wikipedia.org/wiki/Antikythera_mechanism#Gearing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Antikythera mechanism&lt;/a&gt;, an ancient astronomical calculator, which contained over 30 of them. It is dated to about 150-100 BC and its maker is not known, although Hipparchus (190-120 BC), the father of astronomy, and Posidonius (131-50 BC), the Hellenistic polymath from Rhodes, are likely possibilities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mechanism is very complex, “technological artifacts approaching its complexity and workmanship did not appear again in Europe until the development of mechanical astronomical clocks in the fourteenth century”. In other words, gears must have been well known and used by the time it was constructed, one can speculate that earlier Archimedes’s planetarium also used gears. There were reports about an older gear dated to 200-150 BC and possibly connected to Archimedes’s Syracuse, but they are disputable, see discussion at &lt;a href=&quot;https://hsm.stackexchange.com/questions/505/did-archimedes-use-epicycles-in-his-planetarium&quot;&gt;Did Archimedes use epicycles in his planetarium?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-12-25T01:29:06.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="721" PostTypeId="2" ParentId="716" CreationDate="2014-12-25T02:15:45.373" Score="2" Body="&lt;p&gt;The formation of the modern notion of momentum is described in the answer to  &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the history of mechanics in addition to Dave Renfro's excellent suggestions there is Dover's &lt;a href=&quot;http://books.google.com/books/about/A_History_of_Mechanics.html?id=jmKxffjQtasC&quot; rel=&quot;nofollow noreferrer&quot;&gt;nice little book&lt;/a&gt; by Rene Dugas. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-12-25T02:15:45.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="722" PostTypeId="2" ParentId="696" CreationDate="2014-12-25T09:36:48.597" Score="8" Body="&lt;p&gt;The Vienna circle, and more broadly logical positivism that it promoted, was the key link connecting philosophy of classical physics in 19-th century, positivism and neo-Kantianism of the Marburg school, to the modern philosophy of science post the discoveries of relativity theory and quantum mechanics. There is a direct line of tradition from Kant, Mach and Cassirer, through Carnap, the most prominent figure of the circle, to Quine, Popper, Kuhn, Feyerabend and Lakatos. Logical positivists were among the first to meet the philosophical challenges presented by the fundamental changes in physics at the turn of the 20-th century, and revise the conceptual framework inherited from their classical predecessors. &lt;a href=&quot;https://books.google.com/books?id=e9TjZc9wNUAC&amp;amp;dq=friedman+reconsidering+logical+positivism&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedman's book&lt;/a&gt; gives a detailed account of the Vienna circle's ideas, their genesis and interaction with the science of the day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps, the most direct influence of the Vienna circle on science proper was in &lt;a href=&quot;http://unfspb.wordpress.com/2011/07/12/logical-positivism-and-the-copenhagen-interpretation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;shaping the Copenhagen interpretation&lt;/a&gt; of quantum mechanics. Both Bohr and Heisenberg were personally acquainted with the circle members in 1920-s, and incorporated &lt;a href=&quot;http://en.wikipedia.org/wiki/Verificationism&quot; rel=&quot;nofollow noreferrer&quot;&gt;verificationism&lt;/a&gt;, one of the central tenets of the logical positivism, into their interpretation. Verificationism asserts that empirically unverifiable statements are meaningless, and questions concerning them are pointless. This was Bohr's main line of defence against Einstein's criticisms of quantum mechanical &quot;incompleteness&quot;. Another direct contribution, to formal semantics, came from the related focus of logical positivists on the logical analysis of language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The development of falsificationism by Popper since 1930s, Quine's criticism of verificationism in &lt;a href=&quot;http://en.wikipedia.org/wiki/Two_Dogmas_of_Empiricism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Two Dogmas of Empiricism&lt;/a&gt; in 1950s, Kuhn's historical analysis of paradigm changes in science in 1960s, etc., undermined the fundamental positions of logical positivists. Although the original doctrines of the Vienna circle in philosophy of science are rarely defended today what came to replace them emerged in no small part from interaction with and revision of their ideas. Carnap himself was instrumental in publishing Kuhn's book about scientific revolutions, and saw the historical aspect of paradigms as an important addition to his own work on analysis of scientific language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But perhaps the circle exerted its greatest influence by osmosis. It was founded in 1922 by Moritz Schlick and officially functioned until 1936, other prominent figures included aforementioned Carnap, Neurath, Hahn (of the Hahn-Banach theorem) and Menger. However many others, Popper, Quine, Reidemeister, Gödel, Tarski, etc. were either associated or actively interacted with the circle at various times. At the time the circle was almost a lone proponent of a rationalist approach to philosophy and science in Europe increasingly dominated by anti-intellectual trends. After the spread of Nazism several key members of the circle emigrated to the United States, where their ideas found far more favorable reception.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-01-24T00:15:11.937" LastActivityDate="2017-01-24T00:15:11.937" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="723" PostTypeId="2" ParentId="719" CreationDate="2014-12-25T10:30:03.703" Score="3" Body="&lt;p&gt;&lt;em&gt;I'll expand a little bit my &quot;oracolar&quot; comment above (with less fun)&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They have studied physics and mathematics, and physics and mathematics, and ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There was &quot;explorations&quot; of &quot;outer layers&quot; of atmosphere with balloons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There ware the &quot;experiments&quot; made with rockets (mainly German) used during WW II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But mainly, there were &lt;em&gt;faith&lt;/em&gt; in the &lt;strong&gt;immense&lt;/strong&gt; power of &lt;em&gt;newtonian mechanics&lt;/em&gt; (and mathematics) to predict the correct behaviour of bodies subject to the &quot;law of motion&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With insight, space travel (think about a 10 years travel of a little &quot;spacecraft&quot; able to &quot;land&quot; on a comet after a journey of millions of km) has been - for me - the &quot;paradigmatic&quot; cased of verification (if any) of a scientific theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean : we can speculate about skepticism, philosophical realism and any other kind of &quot;mental experiment&quot; about the reality of the world &quot;out there&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;But&lt;/strong&gt;, if we are able to compute a ten years long trajcetory with this precision... this means that the theory is &lt;em&gt;definitively&lt;/em&gt; correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See e.g. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ian Hacking, &lt;a href=&quot;https://books.google.it/books?id=4hIQ5fGf-_oC&amp;amp;pg=PA22&quot; rel=&quot;nofollow&quot;&gt;Representing and Intervening: Introductory Topics in the Philosophy of Natural Science&lt;/a&gt; (1983), page 22 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;If you can spray them, then they are real.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2014-12-25T10:30:03.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="724" PostTypeId="1" CreationDate="2014-12-25T19:18:56.167" Score="8" ViewCount="1647" Body="&lt;p&gt;In analysis textbooks and classes I sometimes see the convolution product introduced as a sort of artificial tool - just a clever method for constructing functions that somebody smart came up with at some point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, is this an accurate description of how it first came to light historically? That is, was the convolution product:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Discovered&lt;/em&gt;, in the sense that it arose naturally in a mathematics problem, given to the mathematician by nature, so to speak.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Invented&lt;/em&gt;, somebody went out of their way to develop a construction that would have certain properties.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To illustrate the difference between (1) and (2), consider the complex numbers (discovered accidentally as solutions to equations) and the quaternions (invented after years of painstaking labour specifically in order to suit some purpose).&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-05-09T14:54:14.280" Title="Was the convolution product invented or discovered?" Tags="|mathematics|real-analysis|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="725" PostTypeId="2" ParentId="724" CreationDate="2014-12-25T20:07:08.803" Score="2" Body="&lt;p&gt;This is not a very historically oriented answer, but it grew much too long for a comment and I hope it sheds some light on the issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd find it &lt;em&gt;highly&lt;/em&gt; surprising if the convolution was not what you would call 'discovered', since it arises &lt;em&gt;extremely naturally&lt;/em&gt; in probability theory. In fact, I 'discovered' it in this context a few years ago, even though I was already aware of what a convolution is in other contexts (notably analysis). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a variable $X$ and a variable $Y$, with continuous probability distributions $\rho_X$ and $\rho_Y$ respectively. Then, the probability distribution $\rho_{X+Y}$ for $X+Y$ is given by $\rho_{X+Y}=\rho_X\star \rho_Y$. This is quite obvious from the fact that, if $X+Y=Z$ and $X=A$, then $Y=Z-A$. Thus, &#xA;$$ \rho_{X+Y}(Z)=\int_{-\infty}^\infty \rho_X(A)\rho_Y(Z-A)\ \mathrm{d}A$$&#xA;and we find the convolution to be one of the most basic tools one needs in the context of probability theory.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Historical edit: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through wikipedia, one finds &lt;a href=&quot;http://www.slideshare.net/Alexdfar/origin-adn-history-of-convolution&quot; rel=&quot;nofollow&quot;&gt;this source&lt;/a&gt;, which suggests the convolution actually was used first in the context of analysis, specifically in a work by D'Alembert, while deriving Taylor's theorem. In this case, D'Alembert actually only used the particular case where one of the two functions is just $1$, and he certainly doesn't do anything to develop the general theory that has, by now, been established.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2014-12-25T20:20:24.467" LastActivityDate="2014-12-25T20:20:24.467" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="726" PostTypeId="2" ParentId="724" CreationDate="2014-12-25T20:11:29.917" Score="1" Body="&lt;p&gt;This question does not belong to history of sciences and math in the strict sense. This is a question about philosophy of mathematics and it has nothing to do with convolution. Convolution is just an example of a mathematical object.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can ask about any mathematical object whether it was invented or discovered.&#xA;And the answer depends on the philosophical views of the person who answers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, there cannot be an objective answer to such a question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, like most mathematicians, I am a Platonist, and I believe that mathematics exists independently of us, and we discover it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some others ridicule this point of view. But there is no way to reach a consensus, like on&#xA;any philosophical question. So this question has no answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Convolution is as &quot;natural&quot; as addition or multiplication. So if addition and multiplication (say, of integers) &quot;exists&quot; in the real world, and we discover it, then&#xA;the same can be said about convolution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course there are people who will say that addition and multiplication were invented. (Did not exist before humans started to add and multiply). And history shows that there is no reconciliation between these two sets of people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. The answer of Danu which was simultaneous with my answer gives an example how convolution arises in probability theory. But convolution arises everywhere. If you think for example of multiplication of decimal numbers, this is nothing but a special case of convolution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I said, from my point of view all this &quot;exists&quot; and we discover it. But some people (mostly philosophers) will say that we &quot;invented&quot; probability and multiplication, and they&#xA;&quot;did not exist&quot; before we invented them:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-25T20:23:08.747" LastActivityDate="2014-12-25T20:23:08.747" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="727" PostTypeId="1" AcceptedAnswerId="729" CreationDate="2014-12-25T22:30:12.003" Score="17" ViewCount="1441" Body="&lt;p&gt;Some functions of a real variable are infinitely smooth (have derivatives of all orders) but are not analytic (at some points $a$, the Taylor series at $a$ does not represent the function at any interval around $a$). All sources, such as Wikipedia article &lt;a href=&quot;http://en.wikipedia.org/wiki/Non-analytic_smooth_function&quot;&gt;Non-analytic smooth function&lt;/a&gt;, give about the same example: $f(x)=\exp(-1/x)$ or $f(x)=\exp(-1/x^2)$; the first one is extended by $0$ to the interval $(-\infty,0]$, the other only needs to be defined as $0$ at $x=0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I have not found any references to the origin of these standard examples. Who gave the first example of a smooth non-analytic function of a real variable? Was it a version of the above functions, or something different? &lt;/p&gt;&#xA;" OwnerDisplayName="user145" LastActivityDate="2021-12-23T17:25:36.987" Title="Who discovered smooth non-analytic functions of a real variable?" Tags="|mathematics|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="728" PostTypeId="1" AcceptedAnswerId="1819" CreationDate="2014-12-25T23:26:42.883" Score="21" ViewCount="8261" Body="&lt;p&gt;Note - this question was inspired by &lt;a href=&quot;https://physics.stackexchange.com/q/155005/26969&quot;&gt;this questions on physics.SE&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many (most) physical quantities are denoted with a single letter - latin or greek. For many, the letter chosen makes sense: $t$ for time, $m$ for mass, etc. Often the letter chosen either relates to the word for the quantity (in the language of the person first describing it, or Latin or Greek) or indeed the name of the discoverer. Occasionally, the letter is borrowed from another field with a similar use. For example, the letter $n$ is often used for counting natural numbers, and thus became associated with quantum numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of which leaves me to wonder: why did angular momentum get the letter $L$? I saw a few &quot;answers&quot; online that I found very unsatisfactory - that it related to &quot;Left&quot;, or the right angle made between the velocity and position vectors... I suspect there is a deeper explanation but it has eluded me. I tried looking at some of the early descriptions by Newton, Hooke and Kepler but didn't find an answer.&lt;/p&gt;&#xA;" OwnerUserId="543" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2015-02-13T02:12:59.413" Title="Why did angular momentum get the letter L" Tags="|notation|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="729" PostTypeId="2" ParentId="727" CreationDate="2014-12-25T23:42:15.237" Score="22" Body="&lt;p&gt;An excellent source for the early history of smooth non-analytic functions is the paper&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gerald G. Bilodeau. &lt;em&gt;The origin and early development of non-analytic infinitely differentiable functions&lt;/em&gt;, Arch. Hist. Exact Sci., &lt;strong&gt;27&lt;/strong&gt; (1982), 115-135. &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=677684&quot; rel=&quot;nofollow noreferrer&quot;&gt;MR84g:26017&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, Dave Renfro wrote an excellent historical &lt;em&gt;Essay on nowhere analytic &lt;span class=&quot;math-container&quot;&gt;$C^\infty$&lt;/span&gt; functions&lt;/em&gt; in two parts (with numerous references). See &lt;a href=&quot;https://web.archive.org/web/20161009194815/mathforum.org/kb/message.jspa?messageID=387148&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; (dated May 9, 2002 6:18 PM), and &lt;a href=&quot;https://web.archive.org/web/20161009194829/http://mathforum.org/kb/message.jspa?messageID=387149&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; (dated May 19, 2002 8:29 PM).&lt;/p&gt;&#xA;&lt;p&gt;The first example is Cauchy's (in 1823), and is indeed &lt;span class=&quot;math-container&quot;&gt;$e^{-1/x^2}$&lt;/span&gt;. The point &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is what we now call a &lt;span class=&quot;math-container&quot;&gt;$(C)$&lt;/span&gt;-point (C for Cauchy). Given a smooth function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; (defined on an open interval), a point &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; in its domain is called a &lt;span class=&quot;math-container&quot;&gt;$(C)$&lt;/span&gt;-point iff  the Taylor series of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; about &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; converges in a positive interval, but the (analytic) function it defines does not coincide with &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; in any interval centered at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The other way a smooth function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; may fail to be analytic at a point &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is if the Taylor series at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; has radius of convergence &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. These points we now call &lt;span class=&quot;math-container&quot;&gt;$(P)$&lt;/span&gt;-points (P for Pringsheim). The first example is due to du Bois-Raymond in 1876. His example was the function &lt;span class=&quot;math-container&quot;&gt;$$ \sum_{n=0}^\infty (-1)^{n+1} \frac{x^{2n}}{(2n)! (x^2 + b_n)}, $$&lt;/span&gt; where we start with a sequence &lt;span class=&quot;math-container&quot;&gt;$\{a_n\}$&lt;/span&gt; of non-zero real numbers converging to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, and set &lt;span class=&quot;math-container&quot;&gt;$b_n = (a_n)^2$&lt;/span&gt;. (Again, the &lt;span class=&quot;math-container&quot;&gt;$(P)$&lt;/span&gt;-point is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; in this example.)&lt;/p&gt;&#xA;&lt;p&gt;The classification into &lt;span class=&quot;math-container&quot;&gt;$(C)$&lt;/span&gt;- and &lt;span class=&quot;math-container&quot;&gt;$(P)$&lt;/span&gt;-points is due to Zahorski, who in 1947 published a paper characterizing these points. See &lt;a href=&quot;https://math.stackexchange.com/a/630687/462&quot;&gt;here&lt;/a&gt; for a statement of Zahorski's result, and references.&lt;/p&gt;&#xA;&lt;p&gt;Du Bois-Raymond's example was the basis for a theorem of Peano, from 1884, that nowadays we name after Borel (namely, the result that any sequence is the sequence of coefficients of the Taylor series about &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; of a smooth function). Indeed, Peano's result went essentially unnoticed until very recently, when Besenyei publicized it. Borel's version of the result appeared in his thesis, in 1895. See &lt;a href=&quot;https://math.stackexchange.com/a/387897/462&quot;&gt;here&lt;/a&gt; for some details and references.&lt;/p&gt;&#xA;" OwnerUserId="18" LastEditorUserId="264" LastEditDate="2021-12-23T17:25:36.987" LastActivityDate="2021-12-23T17:25:36.987" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="730" PostTypeId="1" AcceptedAnswerId="732" CreationDate="2014-12-27T05:39:29.690" Score="12" ViewCount="1131" Body="&lt;p&gt;I read that Descartes and some other mathematician figured out a 'double tangent' method (as I think it was called) for calculating a derivative of a conic or some curve &lt;em&gt;without&lt;/em&gt; using the concepts of limits or infinitesimals. There was an article about this in the Mathematical Association of America. It was said this method could find derivatives on many types of curves and the methods of finding derivatives without limits could be applied to basic calculus, such as using the product rule. The problem is that calculus without limits or infinitesimals is not really mentioned in a regular curriculum. I've noticed other so-called &quot;lost theorems&quot; or rarely-mentioned but useful theorems, some of which have been mentioned in math journals. Are there a lot of rarely-mentioned but useful theorems in math and science?&lt;/p&gt;&#xA;" OwnerUserId="597" LastEditorUserId="62" LastEditDate="2014-12-27T21:05:37.890" LastActivityDate="2019-12-28T15:43:17.477" Title="Are there any theorems that become &quot;lost&quot; and discarded over time?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="732" PostTypeId="2" ParentId="730" CreationDate="2014-12-27T19:43:31.323" Score="18" Body="&lt;p&gt;What most mathematicians are doing at a given time is determined by current fashion to a very large extent. I do not know a (fashion-independent) criterion by which a theorem can be &quot;useful&quot;, but theorems can be beautiful. Some deep and beautiful theories can be out of fashion for many years. Sometimes they came into fashion again.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some examples. Holomorphic dynamics created by Fatou and Julia in 1906-1918. It was very well received  at that time. Then it went out of fashion. For about 60 years. Then it suddenly came back to fashion approximately in 1982 and remains one of the hottest areas&#xA;of mathematics to this day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another similar example is Schubert Calculus. Or knot theory. Or integrable systems.&#xA;Or  theory of invariants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I only mentioned some theories which were out of fashion for long time and currently are&#xA;intensively developing. There is no doubt that there are theories which are considered&#xA;&quot;finished&quot; and forgotten, and are still waiting for their revival.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was talking about research above. Teaching is a different matter. We teach Calculus, which is a universal method to solve many problems, and usually do not teach pre-calculus&#xA;methods to solve the same problems. Simply because we are limited in time when we are teaching, so we teach the things which are most important (on our opinion). We do not teach the ingenious methods by which Archimedes and others before Leibniz found the areas.&#xA;And many other similar things which can be done automatically with the powerful methods we have now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Unlike other sciences (except astronomy), mathematics is VERY old. Euclid is 23 centuries old now. By comparison, &quot;physics of 300 years old&quot; and chemistry of 200 years old is of interest only to historians. But mathematics of 2300 years old is of interest to mathematicians. It is the SAME mathematics that we are doing now. If you don't believe me, read Archimedes. Just try. So the amount of accumulated knowledge in mathematics is enormous.&#xA;We cannot teach all these things in our regular courses. We teach only a few selected things which on our opinion are most important.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-28T20:30:29.587" LastActivityDate="2014-12-28T20:30:29.587" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="733" PostTypeId="1" CreationDate="2014-12-28T03:59:29.100" Score="4" ViewCount="173" Body="&lt;p&gt;I think Archimedes had some great non-infinitesimal methods for discovering the area and volume of shapes. Some very visual methods involving his method of exhaustion for the volume of a sphere for example. These ideas were transcribed from some Palimpsest discovered with various drawings on them too but almost hidden. These methods are known now so why aren't they taught in high school (a way to introduce High School student to Calculus)?&lt;/p&gt;&#xA;" OwnerUserId="597" LastEditorUserId="4251" LastEditDate="2020-05-13T18:58:17.980" LastActivityDate="2020-05-13T18:58:17.980" Title="About Archimedes methods in the discovered palimpsest" Tags="|calculus|euclidean-geometry|archimedes|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="734" PostTypeId="2" ParentId="656" CreationDate="2014-12-28T04:43:53.750" Score="3" Body="&lt;p&gt;There are some great references in the comments above, by Nikolajk and Mauro. I recommend them too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as mentioned, mostly inspiration for category theory is philosophy of mathematics, in the sense of logic, at least so far as MacLane talks about it in his autobiography. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I speculate also that the philosophy of Ernst Mach, as evident in his first quasi-popular book, seems to be the only EXTRA-mathematical philosophy (other than the coherence theory of truth, due to Joachim and Blanshard, which however probably had little influence on the mathematicians, unlike Mach) that inspired category theory. (Basically, it's relational realism more or less, itself related to structural realism.) It's true, I agree, that category theory mostly is inspired by philosophy of mathematics, not the extra-mathematical philosophy, which mostly centers on theory of truth, ideas, ethics, and aesthetics (and so has no bearing on the subject, if the theory of truth was not relational fundamentally). Mach was and is very influential on thinking of mathematicians, especially those think about physics, so I think his is the main (extra-mathematical) philosophical influence. Leibnitz seemed to have in turn been an indirect influence on Mach...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To some degree this is by elimination of philosophies directly opposed to the conceptual approach of category theory. The radical empiricism of the instrumentalists and logical atomists (Wittgenstein, Russell, etc.) is generally opposed to the key information being in the arrows, in a static world of related events, so very few popular philosophies really had any influence on category theory, while the classical (Aristotelian realism, abstract Platonism) and neo-classical (Kant, Hegel, Schopenhauer) were not especially interested in putting any emphasis on relations of events instead of the events themselves (the &lt;em&gt;Ens&lt;/em&gt;). Others like Ortega-y-Gasset stressed a return to metaphorical Platonism.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2014-12-28T04:43:53.750" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="737" PostTypeId="1" AcceptedAnswerId="770" CreationDate="2014-12-28T11:02:17.137" Score="11" ViewCount="1303" Body="&lt;p&gt;&amp;quot;All triangles are isosceles&amp;quot; is a famous geometric fallacy (see below). Unlike many other fallacies its flaw is subtle and hard to spot, so it is often used as a cautionary example against the &amp;quot;danger in diagrams&amp;quot;, e.g. in &lt;a href=&quot;http://books.google.com/books/about/Euclidean_and_Non_Euclidean_Geometries.html?id=4uw0dwi7bmQC&quot; rel=&quot;noreferrer&quot;&gt;Greenberg's text&lt;/a&gt;, but always, it seems, without attribution. It feels so much in the spirit of Euclid that one would think it may go back to antiquity, and Euclid did write a book of fallacies called Pseudaria, now lost. But no, only &lt;a href=&quot;http://www.jstor.org/discover/10.2307/41134289?sid=21104946513401&amp;amp;uid=4&amp;amp;uid=70&amp;amp;uid=2129&amp;amp;uid=2&amp;amp;uid=3739256&amp;amp;uid=3739816&quot; rel=&quot;noreferrer&quot;&gt;four geometric fallacies&lt;/a&gt; survive from antiquity, and &amp;quot;all triangles are isosceles&amp;quot; is not one of them.&lt;/p&gt;&#xA;&lt;p&gt;So where did it come from, who came up with it? Or if that can not be traced, what is the earliest known appearence? I am also interested in tracking the origins of other geometric fallacies.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;&amp;quot;Proof&amp;quot;:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hlNO9.gif&quot; alt=&quot;All triangles are isoceles&quot; /&gt;&#xA;If the angle bisector at A and the perpendicular bisector of BC are parallel, then ABC is isosceles. On the other hand, if they are not parallel, they intersect at a point, which we call P, and we can draw the perpendiculars from P to AB at E, and to AC at F. Now, the two triangles labeled &amp;quot;alpha&amp;quot; in this figure have equal angles and share a common side, so they are equal by angle-side-angle. Therefore, PE = PF. Also, since D is the midpoint of BC, the triangles labeled &amp;quot;gamma&amp;quot; are equal right triangles by side-angle-side, and so PB = PC. From this it follows that the triangles labeled &amp;quot;beta&amp;quot; are right triangles with equal leg and hypotenuse, so equal to each other. Thus, we have BE+EA = CF+FA, meaning the triangle ABC is isosceles. (From mathpages)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-01-01T16:43:00.787" Title="Who came up with the &quot;proof&quot; that all triangles are isosceles?" Tags="|mathematics|euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="738" PostTypeId="1" CreationDate="2014-12-28T13:41:19.977" Score="12" ViewCount="531" Body="&lt;p&gt;Japanese scholar Hide Ishiguro published a book in 1990 entitled &amp;quot;Leibniz's philosophy of logic and language&amp;quot; (second edition). Of particular interest, as far as the history of mathematics is concerned, is her Chapter 5.&lt;/p&gt;&#xA;&lt;p&gt;Here she presents an interpretation of Leibniz's infinitesimals in the spirit of a conceptual framework developed by Russell, involving the so-called &amp;quot;logical fictions&amp;quot;. This involves intepreting infinitesimals as non-designating terms, which correspond to propositions quantified over ordinary Archimedean magnitudes. Ishiguro's interpretation usually goes under the name &amp;quot;syncategorematic&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering about the current status of Ishiguro's interpretation of Leibnizian infinitesimals, among Leibniz scholars. Would it be accurate to affirm that this is the dominant interpretation as far as Leibnizian infinitesimals are concerned?&lt;/p&gt;&#xA;&lt;p&gt;Note 1. &lt;em&gt;HOPOS (Journal of the International Society for the History of Philosophy of Science)&lt;/em&gt; just published &lt;a href=&quot;http://dx.doi.org/10.1086/685645&quot; rel=&quot;nofollow noreferrer&quot;&gt;our rebuttal of syncategorematist theories that seek to sweep Leibnizian infinitesimals under a Weierstrassian rug&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note 1.  Further insight into the mentality of some Leibniz scholars can be gleaned from a recent exchange in &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt; (2022-23); see this answer: &lt;a href=&quot;https://math.stackexchange.com/a/4725050/72694&quot;&gt;https://math.stackexchange.com/a/4725050/72694&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-06-25T08:04:03.237" LastActivityDate="2023-06-26T10:55:22.607" Title="Are Leibnizian infinitesimals thought to be logical fictions by Leibniz scholars?" Tags="|mathematics|reference-request|leibniz|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="739" PostTypeId="2" ParentId="733" CreationDate="2014-12-28T21:14:24.320" Score="3" Body="&lt;p&gt;I think I answered this question here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/730/are-there-any-theorems-that-become-lost-and-discarded-over-time&quot;&gt;Are there any theorems that become &amp;quot;lost&amp;quot; and discarded over time?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is very large. There are many beautiful things we could teach. But the time available is limited. Therefore we teach what is considered most important. Geometry (parts of Euclid) and algebra in high school. Calculus and linear algebra in the College. This is a very small part of mathematics. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2014-12-28T21:14:24.320" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="740" PostTypeId="2" ParentId="738" CreationDate="2014-12-28T21:17:42.380" Score="2" Body="&lt;p&gt;The mainstream interpretation of infinitesimals is in the frame of &quot;non-standard analysis&quot;.&#xA;It gives a completely rigorous justification.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2014-12-28T21:17:42.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="741" PostTypeId="1" AcceptedAnswerId="744" CreationDate="2014-12-28T21:20:37.937" Score="4" ViewCount="411" Body="&lt;p&gt;It is well-known that Guiseppe Peano formalized the axioms that, to some extent, motivated mathematical induction. These are known as &lt;a href=&quot;http://en.wikipedia.org/wiki/Peano_axioms&quot; rel=&quot;nofollow&quot;&gt;Peano's axioms&lt;/a&gt;. However, these axioms are often called &lt;em&gt;trivial&lt;/em&gt; as they are quite obvious, whether stated formally or not. Was Peano really the first one to come up with this set of axioms? Is there proof that he was the first one to make such a complete formalization of the natural numbers? &lt;/p&gt;&#xA;" OwnerUserId="554" LastActivityDate="2014-12-29T15:47:38.357" Title="Peano's Axioms: their real origin" Tags="|mathematics|publishing|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="742" PostTypeId="1" AcceptedAnswerId="750" CreationDate="2014-12-28T22:00:10.503" Score="9" ViewCount="343" Body="&lt;p&gt;Today, string theory is considered one of the leading candidates - perhaps &lt;em&gt;the&lt;/em&gt; leading candidate - for a theory of everything. I'm guessing it wasn't always that way, but I haven't figured out just when this hope was kindled in the hearts of string theorists. &lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_everything#String_theory_and_M-theory&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; does say&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Since the 1990s, many physicists believe that 11-dimensional M-theory, which is described in some limits by one of the five perturbative superstring theories, and in another by the maximally-supersymmetric 11-dimensional supergravity, is the theory of everything. However, there is no widespread consensus on this issue.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But that's not too informative. When was string theory first considered a theory of everything? What events led directly to this?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="26" LastEditDate="2015-03-01T14:06:17.920" LastActivityDate="2015-03-01T14:06:17.920" Title="When was string theory first heralded as a theory of everything?" Tags="|theoretical-physics|string-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="743" PostTypeId="2" ParentId="738" CreationDate="2014-12-29T02:06:04.803" Score="3" Body="&lt;p&gt;I haven't read Ishiguro, so I'll just give my impressions of the history from a couple of other secondary sources:&lt;/p&gt;&#xA;&lt;p&gt;Boyer, The History of the Calculus and its Conceptual Development, &lt;a href=&quot;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Blaszczyk, Katz, and Sherry, Ten Misconceptions from the History of Analysis and Their Debunking, &lt;a href=&quot;http://arxiv.org/abs/1202.4153&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://arxiv.org/abs/1202.4153&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The relevant part of Boyer is pp. 210-212. In Boyer's account, Leibniz was a ditherer who wasn't quite sure how to interpret his own dx's. Sometimes he interpreted them in a way that makes them sound like modern differential forms, but various difficulties always lead him back to talking about them as infinitesimals. I'm not sure how much I trust Boyer's interpretation, since he was writing before NSA, and his depiction of the history of calculus is a story in which the bad, evil infinitesimals were driven out by the good, honest limits.&lt;/p&gt;&#xA;&lt;p&gt;Blaszczyk's misconception (or question, actually) #10 is &amp;quot;Is there continuity between Leibniz and Robinson?&amp;quot; This to me seems to swing a little too far in the opposite direction and makes out Leibniz to be overly prescient. However, they do make some very interesting points about close analogies between Leibniz and NSA.&lt;/p&gt;&#xA;&lt;p&gt;My take on the whole thing is that Leibniz's attitude toward and understanding of his infinitesimals was probably analogous to Euclid's relationship with the parallel postulate. It was probably an uneasy relationship, and the definitive clarification wasn't to come until centuries later.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="16318" LastEditDate="2023-06-26T10:55:22.607" LastActivityDate="2023-06-26T10:55:22.607" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="744" PostTypeId="2" ParentId="741" CreationDate="2014-12-29T15:47:38.357" Score="4" Body="&lt;p&gt;A summary from &lt;a href=&quot;http://en.wikipedia.org/wiki/Peano_axioms&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The need for formalism in arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.&lt;a href=&quot;http://en.wikipedia.org/wiki/Peano_axioms&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt; In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.[2] In 1888, Richard Dedekind proposed a collection of axioms about the numbers, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2014-12-29T15:47:38.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="745" PostTypeId="1" AcceptedAnswerId="746" CreationDate="2014-12-29T16:13:50.427" Score="18" ViewCount="2910" Body="&lt;p&gt;We read about nuclear physics and radioactivity in books and we know how to keep safe from their harmful effects, but the physicists who first discovered them didn't have that luxury. Did the pioneers of this type of physics eventually get sick from their experiments? How long did it take for people to realize that the stuff is dangerous?&lt;/p&gt;&#xA;" OwnerUserId="631" OwnerDisplayName="Craig Feinstein" LastEditorUserId="26" LastEditDate="2014-12-29T18:05:30.900" LastActivityDate="2015-01-01T16:58:46.757" Title="Did the pioneers of nuclear physics and radioactivity eventually get sick from their experiments?" Tags="|biographical-details|experimental-physics|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="746" PostTypeId="2" ParentId="745" CreationDate="2014-12-29T16:25:15.043" Score="19" Body="&lt;p&gt;Marie Curie is probably the most famous example of a person who died of the effects of radiation (handling a lot of radium mostly - for the discovery of which she won the Nobel prize in Chemistry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, people were indeed very ignorant of the effects of radiation. A couple of examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Shoe shops used to have an X-ray machine to look at the &quot;fit&quot; of feet inside a shoe - great for children! (image from &lt;a href=&quot;http://placesandplatypie.com/images/022012-5.gif&quot; rel=&quot;noreferrer&quot;&gt;http://placesandplatypie.com/images/022012-5.gif&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/HFbYi.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The earliest X-ray tubes (pre Coolidge tube) relied on the presence of a &lt;em&gt;partial&lt;/em&gt; vacuum to generate a decent cathode current. &quot;Tuning&quot; such a tube required heating a small amount of asbestos wool that was contained inside the vacuum enclosure (the outgassing of the asbestos would provide the partial pressure in the tube), and the quality would be confirmed by holding a piece of anatomy (e.g. one's hand) between the tube and a screen to see if a good image was formed. See for example &lt;a href=&quot;https://books.google.com/books?id=eUAXr11fI-YC&amp;amp;lpg=RA2-PA3&amp;amp;ots=hbJvy-1ZIC&amp;amp;dq=adjust%20xray%20tube%20asbestos&amp;amp;pg=RA2-PA3#v=onepage&amp;amp;q=adjust%20xray%20tube%20asbestos&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;this handbook from 1918&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A very believable summary of the &quot;history of radiation protection&quot; is given at &lt;a href=&quot;http://www.physics.isu.edu/radinf/50yrs.htm&quot; rel=&quot;noreferrer&quot;&gt;http://www.physics.isu.edu/radinf/50yrs.htm&lt;/a&gt; - this suggests that people were aware of some of the dangers quite early on:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However, other reports, describing skin effects similar to those associated with a bad sunburn, began to appear. So frequent and persistent were these reports that in late 1896, less than a year after Roentgen's announcement, Elihu Thomson, an American physicist, deliberately exposed the little finger of his left hand to an x-ray tube for several days, half an hour per day. The resultant effects - pain, swelling, stiffness, erythema and blistering - were convincing for Thomson and others, but not for all. Many prominent physicians still denied that x-rays were in any way harmful, although oft times the denial was tempered by a qualification that the effects noted were attributable to misuse of the x-ray.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That summary also states that the first &quot;film badge&quot; was reported on as early as 1907 - although its inventor died (from radiation exposure?) six months later. Still, it seems to have taken a long time for the message to really sink in. Once the bombs over Hiroshima and Nagasaki had gone off the world finally had a graphical picture of the &quot;dangers of the atom&quot; and started taking radiation safety seriously. Of course there was a lot of secrecy around it at the same time, as described in Feynman's autobiography (where he is describing the transport of nuclear fuel from Brookhaven to Los Alamos, and the people doing the transporting being told to keep the containers a certain distance apart without being told why. Of course there was the risk of enhanced activity by neutron capture if they got too close, and it's hard not to give away secret knowledge about critical mass when explaining this stuff to the guy loading barrels on the train - so they didn't and nearly had a catastrophe on their hands).&lt;/p&gt;&#xA;" OwnerUserId="543" OwnerDisplayName="Floris" LastActivityDate="2014-12-29T16:45:05.353" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="748" PostTypeId="2" ParentId="738" CreationDate="2014-12-29T21:48:24.890" Score="8" Body="&lt;p&gt;My impression is that Leibnitz usually offers several approaches that are alternatives, and are not consistent. (Not surprising, since most of the papers are his notes, written &quot;for the desk&quot;, not intended by him for publication.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wallis in his book on integration, which is prior both Newton and Leibnitz, uses the concept of a right hand limit of a quantity that goes to zero. So it's arbitrarily small but never null.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibnitz in a published short paper, on what infinitesimals are, constructs them in a similar (but not always quite the same) way. Perhaps this may be taken as his &quot;official&quot; opinion, seeing that he published it. (But he treats them inconsistently across his entire body of work.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, he constructs a right triangle in cartesian coordinates, and intersects it with a geodesic in two places. He moves the geodesic continuously toward a vertex of the triangle, so creating another triangle whose sides get shorter and converge to zero (if the geodesic touches the vertex). But he defines a fraction where the numbers are lengths of the sides of the triangle so constructed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case: the denominator cannot be zero, if the system is to be consistent, so the motion of the geodesic is restricted in that it cannot touch the vertex. Whatever can be constructed by such a motion is an infinitesimal, he says. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He defines each infinitesimal as a fraction constructed by such motion. Both denominator and numerator become smaller as the geodesic approaches, but never become null. And because the angle of the geodesic where it intersects each side of the triangle is NOT in general the same, the sides of the triangle constructed are NOT, except in a special case of the construction, equal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously, not all infinitesimals are identical. Infinitesimal to him, therefore, as far as his published work is concerned, refers to any quantity always decreasing, in the same way as Hinchin in his 1950's calculus textbook presented the matter. To be an infinitesimal, as far as Leibnitz is concerned, is to be constructed in a certain way by a certain series of changes in another function, but it says nothing of the quantity itself. (Hinchin nods in approval.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in his discussion of modelling quality by quantity, via continuity, infinitesimals are arbitrarily small quantities, eventually becoming zero, at which point, one quality transforms into another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest treating his opinion of the ontology of infinitesimals as not substantially different from that of Wallis (limit of a variable quantity, going to zero, from the right hand side) which presumably inspired it, and is based on right hand limits of functions that decrease as their input increases. Merely he is not consistent across all his papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking through one of my archive trying to find my translation of the important passages of Wallis's &lt;em&gt;Arithmetica Infinitorum&lt;/em&gt;. From memory, Wallis wrote in one place the equivalent of &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\underset{x\rightarrow 0+}{\lim}x$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as what he meant by infinitesimal, in modern notation. This is the thickness of lines in his diagrams. Summing an infinity of such lines required to fill the figures gave the areas of the figures. But, according to Beeley, in his later communications to Leibniz, he writes they -- the lines -- have literally null thickness, he has decided. Leibniz disagrees. Leibniz's own idea in modern notation is closer to Wallis's first statement, and more precisely the above conception. Where the $x$ is input to a function (the infinitesimal), such that $dx = \frac{f(x)}{g(x)}\rightarrow 0+$ there where $x\rightarrow 0+$.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2014-12-30T06:56:19.220" LastActivityDate="2014-12-30T06:56:19.220" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="750" PostTypeId="2" ParentId="742" CreationDate="2014-12-29T22:27:36.797" Score="8" Body="&lt;p&gt;The history is well-described in &lt;a href=&quot;https://en.wikipedia.org/wiki/String_theory&quot;&gt;Wikipedia's article on string theory&lt;/a&gt;.  There have been many books written about string theory.  Two that I found helpful regarding the history are Davies and Brown's &lt;em&gt;Superstrings: a Theory of Everything?&lt;/em&gt;, published in 1988 (which may provide some indication as to the answer to your question), and Smolin's &lt;em&gt;The Trouble with Physics&lt;/em&gt;.  Here'a a brief summary:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;String theory got its start in 1968 as a theory of hadrons (particles that interact via the strong nuclear force), not as a theory of everything.  The beginning was Gabriele Veneziano's theory explaining certain phenomena in hadron scattering (straight-line Regge trajectories).  His work grew out of the S-matrix program, which eventually lost out to quantum chromodynamics as a theory of the strong interaction.  Veneziano's theory was subsequently understood to be a string theory by Nambu, Nielsen, and Susskind.    After the general acceptance of quantum chromodynamics, around 1973, work on string theory (and the S-matrix program more generally) went into a period of decline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few people continued to investigate string theory during the 1970s.  Originally, string theory was a theory of bosons only, and was only consistent as a relativistic quantum theory in 26 space-time dimensions.  Supersymmetry was developed by Ramond in 1970 in order to introduce fermions, and the resulting theory was consistent in 10 space-time dimensions.  In 1972, Neveu and Scherk discovered that string theory described gauge bosons, such as the photon, and in 1974, Yoneya and, independently, Scherk and Schwartz discovered that it had a spin-2 boson.  It was Scherk and Schwartz who, at that time, proposed that the spin-2 particle could be the graviton, and therefore that string theory was a candidate theory of everything rather than a theory of hadrons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This did not immediately make a strong impression in the physics community.  That only happened 10 years later, in 1984, due the the influence of Ed Witten.  The precipitating event was the discovery of anomaly cancellation by Green and Schwartz in Autumn 1984.  In quantum field theory, an anomaly is a quantum effect that violates a symmetry of a theory.  In order for the theory to be consistent, there must be some effect canceling the anomaly. Here is a quote from an interview with Schwarz in the Davies and Brown book (page 77)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1984, Michael Green and I did a calculation for one of these superstring theories to see whether, in fact, this anomaly occurred or not.  What we discovered was quite surprising to us.  We found that, in general, there was indeed an anomaly that rendered the theory unsatisfactory.  Now there was the freedom to choose the particular symmetry structure that one used in defining a theory in the first place.  In fact, there were an infinite number of possibilities for these symmetry structures.  However, for just one of them the anomaly magically cancelled out of the formulae whereas for all of the others it didn't.  So amid this infinity of possibilities, just one unique one was picked out as being potentially consistent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The name of the symmetry structure is called $SO(32).$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;We also discovered at about the same time that there was a second symmetry structure that seemed to be a consistent possibility which has the name $E_8\times E_8.$  The bizarre thing was that at that time, we didn't have a specific superstring theory that could contain that symmetry. ...  But shortly thereafter a group of four physicists at Princeton University, now known as the Princeton String Quartet, discovered two new superstring theories which they called heterotic strings.  One of these theories incorporated the $E_8\times E_8$ symmetry. The other one was a second example of a superstring theory based on $SO(32).$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The $E_8\times E_8$ theory was the one that aroused the most interest because that's the symmetry structure that looks most promising for accommodating the observed phenomenology of particles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Wikipedia article states that Witten and Alvarez-Gaumé had previously looked at anomalies in superstring theories and had concluded &quot;that type I string theories were inconsistent.&quot;  Smolin's book (page 115) contains an interesting quote from Schwartz about the aftermath of Green and Schwartz's result on anomaly cancelation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[B]efore we even finished writing it up, we got a phone call from Ed Witten saying that he had heard ... that we had a result on canceling anomalies.  And he asked if we could show him our work.  So we had a draft of our manuscript at that point, and we sent it to him by FedEx.  There wasn't e-mail then; it didn't exist, but FedEx did exist.  So we sent it to him and he had it the next day.  And we were told that the following day everyone at Princeton University and at the Institute for Advanced Study, all the theoretical physicists, and there were a large number of them, were working on this.  ... So overnight it became a major industry [laughter], at least in Princeton&amp;mdash;and very soon in the rest of the world.  It was kind of strange, because for so many years we were publishing our results and nobody cared.  Then all of a sudden everyone was extremely interested.  It went from one extreme to the other: the extreme of nobody taking it seriously to the other extreme. ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have a personal recollection about this.  I started as an undergraduate at Princeton in Fall 1983 and in Spring 1984 took freshman electromagnetism from the experimental particle physicist A.J.S. Smith.  Once when he had to travel for one of his experiments (or perhaps it was to lobby congress for funding) he had a young assistant professor, a theorist named Jeff Harvey (later one of the Princeton String Quartet) give a guest lecture on his work on Grand Unification.  The idea was to unify the strong nuclear theory (QCD) with the then-young unified electro-weak theory (the $W$ and $Z$ bosons had been discovered experimentally only a few years prior).  Unfortunately, the most promising candidates predicted proton decay that was rapid enough to be already ruled out by experiment.  There was little, if any, mention that I can recall of quantum gravity, and none of strings, in this talk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning as a sophomore in the Fall, the buzz was all about strings.  A seminar by Witten would lead to an overflow crowd, with people spilling out into the hallway trying to get a glimpse of the proceedings.  By the time I entered my senior year, two years later, there was a palpable sense among my fellow undergraduates that while we had been struggling through coursework, the people around us had been finishing off physics.  Things, of course, were not so simple, and look quite a bit different in retrospect.&lt;/p&gt;&#xA;" OwnerUserId="139" LastActivityDate="2014-12-29T22:27:36.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="751" PostTypeId="1" AcceptedAnswerId="766" CreationDate="2014-12-30T00:27:07.800" Score="12" ViewCount="636" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Tower_of_Hanoi&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia is unequivocal&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The puzzle was invented by the French mathematician Édouard Lucas in 1883.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have no reason to doubt this, but given the many legends surrounding the topic,&#xA;I wonder if there is not an earlier (even &quot;ancient&quot;) source...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/Mf0Hr.jpg&quot; width=&quot;400&quot; /&gt;&#xA;&lt;br /&gt;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;&amp;nbsp;&#xA;(&lt;a href=&quot;http://en.wikipedia.org/wiki/Tower_of_Hanoi&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia image&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="606" LastActivityDate="2014-12-31T23:20:54.553" Title="How certain is it that Lucas invented the Towers of Hanoi puzzle?" Tags="|mathematics|computer-science|mathematical-logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="752" PostTypeId="2" ParentId="745" CreationDate="2014-12-30T01:22:37.530" Score="10" Body="&lt;p&gt;It's hard to quantify the negative effects of radiation. &lt;a href=&quot;https://en.wikipedia.org/wiki/Anatoli_Bugorski&quot; rel=&quot;noreferrer&quot;&gt;In 1978&lt;/a&gt;, a Russian scientist was struck by a particle accelerator beam during routine experimentation. He saw a flash &quot;brighter than a thousand suns&quot;. He survived and even finished his PHD, although the beam left a wake of destruction in its path, giving him seizures and partial paralysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TopQd.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You think consider what type of radiation you're talking about when you contemplate its dangers. &lt;a href=&quot;http://science.howstuffworks.com/radioactivity-info.htm&quot; rel=&quot;noreferrer&quot;&gt;Radioactive&lt;/a&gt; elements naturally emit alpha radiation, beta radiation (which is just electrons), and gamma radiation (which is just photons). Man-made radiation (such as the kind encountered in a particle accelerator) can be just as harmful, or not harmful at all. It depends on how massive the particles are, and how fast they're moving. In other words, it depends on how much energy is in the particles.&lt;/p&gt;&#xA;" OwnerUserId="619" LastActivityDate="2014-12-30T01:22:37.530" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="753" PostTypeId="1" AcceptedAnswerId="757" CreationDate="2014-12-30T16:22:56.950" Score="3" ViewCount="594" Body="&lt;p&gt;It might be famous that $\pi^2$ is a good approximation to the gravitational acceleration in the unit &quot;meter per second squared&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My explanation for this is the seconds pendulum, which was proposed as the definition of &quot;meter&quot; back to the birth of the unit, competing with the meridional definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I don't see why the two competing definitions of meter have to be close to each other.  I don't know any pre-existing unit of length for their reference.  And in the meridional definition, the &quot;ten-millonth&quot; seems to be chosen just for convenience, doesn't it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why the &quot;meter&quot; according to the seconds pendulum and the &quot;meter&quot; according to the meridian turn out to be close? Is it purely a coincidence?&lt;/p&gt;&#xA;" OwnerUserId="629" LastActivityDate="2020-01-02T10:12:53.940" Title="Two competing definitions at the birth of the unit &quot;meter&quot;" Tags="|units|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="754" PostTypeId="2" ParentId="745" CreationDate="2014-12-30T17:00:17.397" Score="7" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Demon_core&quot;&gt;Demon Core&lt;/a&gt;, a subcritical mass of plutonium, went critical during two distinct incidents in 1945 and 1946. Both times, scientists were killed due to exposition to radiation. During the second incident, for example, the two sub-critical masses were separated by a &lt;em&gt;screwdriver&lt;/em&gt; held by the operating scientist. On the fatal day, the screwdriver slipped somewhat, and the two parts of the core assembled, reaching the critical mass. A burst of neutrons were emitted while a blue glow of Tcherenkov radiation surrounded the core.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Radiation was known to be dangerous at the time, but security procedures were not always adequate or enforced. Fermi had warned the operating team several times about the risk to see them &quot;dead within a year&quot;.&lt;/p&gt;&#xA;" OwnerUserId="632" LastEditorUserId="-1" LastEditDate="2014-12-30T19:23:01.607" LastActivityDate="2014-12-30T19:23:01.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="755" PostTypeId="1" CreationDate="2014-12-30T20:15:59.717" Score="10" ViewCount="2357" Body="&lt;p&gt;According to the Gregorian calendar the second month i.e. February have 28 days and in a leap year 29 days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure what calculations(science) it takes to decide the total number of days in each month. But my questions are,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who was the first person to propose that the second month of&#xA;calendar (February) will have 28 days ?&lt;/li&gt;&#xA;&lt;li&gt;Will it make any difference if we swap the total number of days of&#xA;February with any other month in the calendar (say April) ?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="188" LastActivityDate="2017-10-17T04:32:25.620" Title="Why should February have 28 days?" Tags="|astronomy|calendar|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="756" PostTypeId="2" ParentId="755" CreationDate="2014-12-30T20:51:00.273" Score="10" Body="&lt;p&gt;The reason is simple. Baiscaly 365 (approximate number of days in a year)&#xA;is not divisible on 12 (desirable number of months=the number of Zodiac signs).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then it was proposed to have 6 months of 30 days and 6 of 31, alternating. This still&#xA;was not good&#xA;because 6 times 30 + 6 times 31 = 366. So one month has to be 1 day shorter.&#xA;Why it was decided to make February shorter? I suppose because this was the last month of the year,&#xA;which started on March 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This reform was made at the time of Julius Caesar (it is called Julian calendar for this reason) and naturally one month was named after him. But he was followed by an emperor whose  nick name was August, who essentially established some king of monarchy, and it was decided to name a month for him as well (for everyone to know that he was not &quot;smaller&quot; than Julius, but perhaps even greater).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then it did not look good that his month is shorter than July, so one day was added&#xA;to August and subtracted of February. And the lengths of the remaining months&#xA;were changed so they alternate. (Why exactly did they have to name after him the&#xA;month which follows July, I am not sure: on my opinion, if I were charged with this,&#xA;I would simply rename September&#xA;or November in his honor (which are derived from Latin numerals 7 and 9; remember the first month was March!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please do not ask me for a reference: I read the story in my childhood, and do not remember the reference. Some Russian book for children, possibly Children's Encyclopedia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. The above explanation is a very much simplified version of what actually happened.&#xA;You can get a good impression of the complexity of the question from the Wikipedia article &quot;Julian calendar&quot;, especially the chapters &quot;Sacrobosco's theory on month lengths&quot; and &quot;Names of the months&quot;. There are also many references there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another interesting article is &quot;Month&quot; in Wikipedia, the part called &quot;Julian and Gregorean calendars&quot;. In particular, it gives the following interesting interpretation of month lengths: they correspond to the white and black piano keys! (White correspond to 31 days and F corresponds to Jaunary:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-17T04:32:25.620" LastActivityDate="2017-10-17T04:32:25.620" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="757" PostTypeId="2" ParentId="753" CreationDate="2014-12-30T21:06:03.390" Score="5" Body="&lt;p&gt;It is a pure coincidence. And the agreement is not so good. Meter was introduced in connection with decimal system. They wanted all units to be based on decimal system,&#xA;including angular and time units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it was decided to have 100 decimal degrees in the right angle, and 20 hours in a day.&#xA;Each hour was divided into 100 decimal minutes, a decimal (time) minute into 100 decimal&#xA;time-seconds. Similarly with angular measures: one degrees is 100 angular minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The kilometer was supposed to be one (angular) decimal minute of the (Paris) meridian.&#xA;Similarly to the nautical mile which is defined as one angular (sexagesimal) minute of the&#xA;meridian. And meter is 1/10 of one angular decimal second of the meridian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During the rule of the French revolutionary government, they actually made clocks, watches&#xA;and angle measuring instruments with this decimal division. It was abandoned during the&#xA;rule of Napoleon I.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That acceleration of gravity is roughly 10 meters/$sec^2$, where $sec$ is the&#xA;Babilonian (sexagesimal) second, is a pure coincidence.&#xA;A coincidence of the same sort as $\pi^2\approx10$.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2014-12-31T16:00:53.963" LastActivityDate="2014-12-31T16:00:53.963" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="759" PostTypeId="1" AcceptedAnswerId="761" CreationDate="2014-12-31T00:18:22.830" Score="12" ViewCount="6054" Body="&lt;p&gt;Stephen Hawking was &lt;a href=&quot;http://en.wikipedia.org/wiki/Stephen_Hawking#Career&quot; rel=&quot;nofollow noreferrer&quot;&gt;highly prolific&lt;/a&gt;. Compared to a regular person, &lt;strong&gt;how quickly&lt;/strong&gt; could he communicate &lt;strong&gt;language&lt;/strong&gt; to the people assisting him with his research? (e.g. compared to the human average of &lt;strong&gt;40&lt;/strong&gt; or &lt;strong&gt;200&lt;/strong&gt; characters per &lt;strong&gt;minute&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was this process exactly like? Did he rely on any third person to help him visualize and write down equations and develop his theories? &lt;/p&gt;&#xA;" OwnerUserId="638" LastEditorUserId="638" LastEditDate="2018-03-15T02:57:32.037" LastActivityDate="2018-03-15T02:57:32.037" Title="How did Stephen Hawking conduct research?" Tags="|theoretical-physics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="760" PostTypeId="2" ParentId="759" CreationDate="2014-12-31T00:38:14.647" Score="14" Body="&lt;p&gt;The basic answer is yes, he was dependent on a few nurses and a research assistant or two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Stephen_Hawking&quot;&gt;Wikipedia&lt;/a&gt; - the article you brought forth:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nurses were hired for the three shifts required to provide the round-the-clock support he required. One of those employed was Elaine Mason, who was to become Hawking's second wife. For his communication, Hawking initially raised his eyebrows to choose letters on a spelling card. But he then received a computer program called the &quot;Equalizer&quot; from Walt Woltosz. In a method he uses to this day, using a switch he selects phrases, words or letters from a bank of about 2500–3000 that are scanned. The program was originally run on a desktop computer. However, Elaine Mason's husband, David, a computer engineer, adapted a small computer and attached it to his wheelchair.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wall-of-text $\to$ At first, nurses kept him alive and healthy and helped him communicate. Then his condition stabilized, and a computer with a voice synthesizer speaks for him. He controls it with facial movements.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biography.com/people/stephen-hawking-9331710#a-brief-history-of-time&quot;&gt;This&lt;/a&gt; confirms the nurses- and voice-synthesizer- details, and notes specifically that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Through the program, and the help of assistants, Stephen Hawking has continued to write at a prolific rate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://physics.about.com/od/stephenhawking/p/hawking.htm&quot;&gt;here&lt;/a&gt; (emphasis mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As part of his condition, Hawking has lost his ability to speak, so he utilizes device that can translate his &lt;strong&gt;eye movements&lt;/strong&gt; (since he can no longer utilize a keypad) to speak in a digitized voice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Best, of all, from &lt;a href=&quot;http://www.hawking.org.uk/the-computer.html&quot;&gt;the official website&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My main interface to the computer is through a program called EZ Keys, written by Words Plus Inc. This provides a software keyboard on the screen. A cursor automatically scans across this keyboard by row or by column. I can select a character by moving my cheek to stop the cursor. My cheek movement is detected by an infrared switch that is mounted on my spectacles. This switch is my only interface with the computer. EZ Keys includes a word prediction algorithm, so I usually only have to type the first couple of characters before I can select the whole word. When I have built up a sentence, I can send it to my speech synthesizer. I use a separate hardware synthesizer, made by Speech+. It is the best I have heard, although it gives me an  accent that has been described variously as Scandinavian, American or Scottish. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This contradicts the &quot;eye movements&quot; source, although I have read (sorry, I can't find a source for this!) that the mode of operation has changed as Hawking has lost facial mobility.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a nutshell (pun intended): Nurses keep Hawking healthy, a computer lets him &quot;speak&quot;, and his brain does the rest.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2014-12-31T00:38:14.647" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="761" PostTypeId="2" ParentId="759" CreationDate="2014-12-31T04:19:32.747" Score="8" Body="&lt;p&gt;I think this covers the actual question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While working to receive his Ph.D. and make a name for himself at&#xA;  [Oxford], Stephen proved himself able to handle great levels of&#xA;  theoretic equations in his head&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then later&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Most of his [most famous work on black holes and space time] was done while working with Roger Penrose, a&#xA;  mathematician who wrote the equations as Stephen communicated them&#xA;  while they worked together during several years on their theory&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cpp.edu/~nova/scientists/articles/hawk.html&quot;&gt;http://www.cpp.edu/~nova/scientists/articles/hawk.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="639" LastActivityDate="2014-12-31T04:19:32.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="762" PostTypeId="1" AcceptedAnswerId="2714" CreationDate="2014-12-31T08:26:51.403" Score="5" ViewCount="1777" Body="&lt;p&gt;Pretty much every proof of the product or chain rules presented today revolve around the definition of the derivative as a limit (e.g. &lt;a href=&quot;https://math.stackexchange.com/questions/87118/proof-of-product-rule-of-limits&quot;&gt;this post&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when Newton/Leibniz were developing calculus, they would not have had access to the concepts of limits. How, then, were the product and chain rules proved correct? Or was it just generally accepted that, if calculus worked, then the product and chain rules would just have to be in the form they were?&lt;/p&gt;&#xA;" OwnerUserId="589" LastActivityDate="2019-10-05T04:01:21.643" Title="When/How were the product and chain rules first proved?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="763" PostTypeId="2" ParentId="745" CreationDate="2014-12-31T08:53:58.057" Score="2" Body="&lt;p&gt;An illustration of the extent to which people carelessly worked with radioactive material in the early years can be found at &lt;a href=&quot;http://en.wikipedia.org/wiki/Radium_Girls&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Radium_Girls&lt;/a&gt; (glow-in-the-dark watches made with radium!). They were not scientists, but I still think it is a good example for this question, since the workers really had no idea that anything could go wrong until it was sadly too late for most of them. The last worker at that factory passed away just a few days ago (&lt;a href=&quot;http://www.npr.org/2014/12/28/373510029/saved-by-a-bad-taste-the-last-radium-girl-dies-at-107&quot; rel=&quot;nofollow&quot;&gt;http://www.npr.org/2014/12/28/373510029/saved-by-a-bad-taste-the-last-radium-girl-dies-at-107&lt;/a&gt;), and she survived so long only because she quit the job after only a couple of days.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2015-01-01T16:58:46.757" LastActivityDate="2015-01-01T16:58:46.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="765" PostTypeId="2" ParentId="276" CreationDate="2014-12-31T20:58:06.013" Score="3" Body="&lt;p&gt;I went to elementary school in the US.  There we learned that the number zero was invented in Central America ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mediatinker.com/blog/archives/008821.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;HERE&lt;/a&gt; is an example with that location included.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/g7nys.jpg&quot; alt=&quot;map&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2014-12-31T20:58:06.013" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="766" PostTypeId="2" ParentId="751" CreationDate="2014-12-31T23:11:21.650" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://www.springer.com/birkhauser/mathematics/book/978-3-0348-0236-9,&quot; rel=&quot;noreferrer&quot;&gt;The Tower of Hanoi – Myths and Maths&lt;/a&gt; gives a detailed historical account of the game, its predecessors and myths surrounding it, with references to original sources, it also goes in-depth into mathematical issues. The Temple of Brahma story was first reported in 1883 by Professor N. Claus (de Siam), mandarin of college Li-Sou-Stian, and was embellished by Henri de Parville, a popular science writer and chief editor of La Nature at the end of 19-th century. He first published it in Revue des Sciences in 1883, and then a year later, with pictures, in La Nature. If de Parville thought the fable would add to the game's popularity he was certainly right. Later authors made it even more colorful, adding details and modifications. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A mandarin who invents a game about combinations, will incessantly think about combinations&quot;, says de Parville. Permuting the letters in &lt;em&gt;N. Claus (de Siam)&lt;/em&gt; of college &lt;em&gt;Li-Sou-Stian&lt;/em&gt; reveals &lt;em&gt;Lucas d'Amiens&lt;/em&gt; of lycee &lt;em&gt;Saint-Louis&lt;/em&gt;. That would be Francois Lucas born in the city of Amiens, who taught at the lycee. W.W.R. Ball, a mathematician who translated the fable from French into English in &lt;a href=&quot;http://www.jstor.org/stable/3026631?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Mathematical Recreations and Essays&lt;/a&gt; first published in 1892, called it &quot;a sufficiently pretty conceit to deserve repetition&quot;. But if you are not convinced here is a comprehensive &lt;a href=&quot;http://www.cs.wm.edu/~pkstoc/biblio2.pdf&quot; rel=&quot;noreferrer&quot;&gt;bibliography&lt;/a&gt; on the Towers of Hanoi. Not one entry dates from before 1883.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2014-12-31T23:20:54.553" LastActivityDate="2014-12-31T23:20:54.553" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="767" PostTypeId="2" ParentId="711" CreationDate="2015-01-01T00:24:38.290" Score="1" Body="&lt;p&gt;Two works come to mind:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Holton, Gerald, and Stephen G. Brush. &lt;em&gt;Physics, the Human Adventure: From Copernicus to Einstein and Beyond&lt;/em&gt;. Rev. ed. New Brunswick, New Jersey: Rutgers University Press, 2001.&lt;/li&gt;&#xA;&lt;li&gt;Høyrup, Jens. &lt;em&gt;Human Sciences: Reappraising the Humanities through History and Philosophy&lt;/em&gt;. Albany, New York: SUNY Press, 2000.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both works have somewhat misleading titles. &lt;em&gt;Physics, the Human Adventure&lt;/em&gt; is the revised, third edition of &lt;em&gt;Introduction to Concepts and Theories in Physical Science&lt;/em&gt;. &lt;a href=&quot;https://www.physics.harvard.edu/people/facpages/holton&quot; rel=&quot;nofollow&quot;&gt;Holton&lt;/a&gt; and &lt;a href=&quot;http://www.gf.org/fellows/1946-stephen-g-brush&quot; rel=&quot;nofollow&quot;&gt;Brush&lt;/a&gt;, both eminent historians of science in the United States, write in the preface, &quot;Perhaps the most unusual characteristic of our book, compared to other physics textbooks, is the emphasis it puts on the nature of discovery, reasoning, concept-formation and theory-testing in science as a fascinating topic in its own right. This means that the historical and philosophical aspects of the exposition are not merely sugar-coating to enable the reader to swallow the material as easily as possible, but are presented for their own inherent interest&quot; (xiii-xiv). See the book's &lt;a href=&quot;http://www.ipst.umd.edu/Faculty/brush/physicsbibliography.htm&quot; rel=&quot;nofollow&quot;&gt;website&lt;/a&gt; for additional information. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://akira.ruc.dk/~jensh/&quot; rel=&quot;nofollow&quot;&gt;Jens Høyrup&lt;/a&gt;, a historian of mathematics by training, is Professor in the Section for Philosophy and Science Studies at Roskilde Univeristy in Denmark. &lt;em&gt;Human Sciences&lt;/em&gt; &quot;assesses the importance and value of the humanities historically and philosophically, and makes the case for treating them as sciences . . . as systematic, organized bodies of knowledge, rather than as branches of knowledge that should necessarily emulate the quantitative and experimental approach of the natural sciences.&quot;&lt;/p&gt;&#xA;" OwnerUserId="650" LastEditorUserId="650" LastEditDate="2015-01-01T20:35:29.513" LastActivityDate="2015-01-01T20:35:29.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="768" PostTypeId="2" ParentId="633" CreationDate="2015-01-01T05:43:51.730" Score="3" Body="&lt;p&gt;The explanation I learned when I first learned the term Poincare patch on AdS$_{d+1}$ in the context of AdS/CFT is  the following. The metric for such a patch is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation*}&#xA;ds^2 = \frac 1{z^2} \left(dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu \right)&#xA;\end{equation*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $x^\mu$ are coordinates on a $d$-dimensional Minkowski space and $\eta_{\mu \nu}$ is the Minkowski metric, and $z &amp;gt; 0$. An obvious family of isometries of this metric are those which leave $z$ fixed and transform the $x^\mu$ by an isometry of Minkowski space. Of course, these isometries are, as a group, known as the Poincaré group. While we can't necessarily claim that the Poincaré group is a subgroup of the isometry group of AdS, we do certainly get containment at the level of algebras; that is, AdS has a family of Killing fields with the algebra $\mathbf R^{d-1,1} \rtimes \mathfrak{so}(d-1,1)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among physicists, this explanation makes sense from a modern perspective. These isometries exactly give the the Poincaré symmetry on the boundary CFT. Hence, if one wants to break the conformal symmetry while maintaining Poincare symmmetry (as in many applications such as RS models and holographic QCD) the AdS picture is clear. There's no doubt in my mind that the term &quot;Poincaré patch&quot; is so prevalent is at least partially due to this convenience; otherwise it would seem like &quot;AdS-Rindler coordinates&quot; or something similar would fit just as well. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this is &lt;em&gt;not&lt;/em&gt; the origin of the term. Poincaré is involved more directly. In a series of works on hyperbolic space (beginning with [1]), Poincaré found a Riemannian metric (now called the &lt;a href=&quot;http://en.wikipedia.org/wiki/Poincar%C3%A9_metric&quot; rel=&quot;nofollow&quot;&gt;Poincaré metric&lt;/a&gt;) with constant curvature $-1$ on the upper half-space, given by $ds^2 = (dx_1^2 + \cdots + dx_{n-1}^2 + dx_n^2)/x_n^2$. This is most famously known in dimension 2 (where it is related to complex analysis), but easily generalizes to higher dimensions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first application of Poincaré's ideas to de Sitter space which I could find is a rather short note: [2]. The result of this work is showing that the Lorentzian version $ds^2 = (dx_1^2 + \cdots + dx_{n-1}^2 - dx_n^2)/x_n^2$ can be isometrically embedded in de Sitter space, which provides a geodesic completion. While the term &quot;Poincaré patch&quot; doesn't appear explicitly anywhere in this paper, it seems to be a simple leap in terminology from &quot;Lorentz-Poincaré metric on a patch of dS&quot; to &quot;Poincaré patch of dS&quot;, and then analogously to AdS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest paper I could locate using the term &quot;Poincaré patch&quot; is the rather famous paper [3], which gives no citation for it nor an explanation of the meaning, but I'm relatively confident it comes from the Lorentzian version of Poincaré's work on hyperbolic space as described above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] Henri Poincaré (1882) &quot;&lt;a href=&quot;https://archive.org/details/thoriedesgroup00poin&quot; rel=&quot;nofollow&quot;&gt;Théorie des Groupes Fuchsiens&lt;/a&gt;&quot;, &lt;em&gt;Acta Mathematica&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] NOMIZU, Katsumi. The Lorentz-Poincaré metric on the upper half-space and its extension. Hokkaido Math. J. 11 (1982), no. 3, 253--261. doi:10.14492/hokmj/1381757803. &lt;a href=&quot;http://projecteuclid.org/euclid.hokmj/1381757803&quot; rel=&quot;nofollow&quot;&gt;http://projecteuclid.org/euclid.hokmj/1381757803&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[3] Bañados, M., Henneaux, M., Teitelboim, C., Zanelli, J.: Geometry of the 2+1 black hole. &lt;a href=&quot;http://arxiv.org/abs/gr-qc/9302012&quot; rel=&quot;nofollow&quot;&gt;gr-qc/9302012&lt;/a&gt;; Phys. Rev. D48, 1506 (1993)&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="58" LastEditDate="2015-01-01T20:00:38.477" LastActivityDate="2015-01-01T20:00:38.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="769" PostTypeId="1" AcceptedAnswerId="790" CreationDate="2015-01-01T07:02:42.717" Score="9" ViewCount="1950" Body="&lt;p&gt;We know that up to 1726, when the third edition of the &lt;em&gt;Principia&lt;/em&gt; was published, the name for $m\vec v$ was: &lt;em&gt;quantitas motus&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know who substituted that with another Latin word: 'momentum'?&lt;/p&gt;&#xA;" OwnerDisplayName="user646" LastEditorUserId="26" LastEditDate="2015-01-01T14:03:39.947" LastActivityDate="2015-01-06T19:57:49.657" Title="When and by whom was the term 'momentum' introduced?" Tags="|physics|terminology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="770" PostTypeId="2" ParentId="737" CreationDate="2015-01-01T16:43:00.787" Score="9" Body="&lt;p&gt;This is in Rouse Ball's &lt;a href=&quot;https://books.google.com/books?id=IwDvAAAAMAAJ&amp;amp;pg=PA33&quot;&gt;&lt;em&gt;Mathematical recreations and problems&lt;/em&gt; (2nd edition, 1892, p. 33)&lt;/a&gt;, and later editions carry this footnote &lt;a href=&quot;https://archive.org/stream/mathematicalrecr00ball#page/45&quot;&gt;(6th edition, 1914, p. 45)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I believe that this and the fourth of these fallacies were first published in this book. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2015-01-01T16:43:00.787" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="771" PostTypeId="1" CreationDate="2015-01-01T21:45:12.043" Score="38" ViewCount="4118" Body="&lt;p&gt;In his famous estimation of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; by inscribed and circumscribed polygons, Archimedes uses several rational approximations of irrational values; a typical example is that he states, without explanation, &lt;span class=&quot;math-container&quot;&gt;$$\sqrt 3 &amp;gt; \frac{265}{153}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The sudden appearance of &lt;span class=&quot;math-container&quot;&gt;$\frac{265}{153}$&lt;/span&gt; has puzzled many people through the ages.  The page &lt;a href=&quot;http://mathpages.com/home/kmath038/kmath038.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes and the Square Root of 3&lt;/a&gt; lists several such.  This example, attributed to W.W. Rouse Ball, is typical:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It would seem...that [Archimedes] had some (at present unknown) method of extracting the square root of numbers approximately.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To my mind, there is a very easy answer to this seeming puzzle.  If you want rational approximations to &lt;span class=&quot;math-container&quot;&gt;$\sqrt 3$&lt;/span&gt;, the &lt;em&gt;very simplest&lt;/em&gt; thing you can possibly do is as follows:  You want to find integers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; whose ratio is close to &lt;span class=&quot;math-container&quot;&gt;$\sqrt 3$&lt;/span&gt;, or equivalently you want to find &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt; whose ratio is close to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.  So you should tabulate &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3n^2$&lt;/span&gt; and look for integers in the left column that are approximately equal to integers in the right column:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{array}{rr}&#xA;n^2 &amp;amp; 3n^2 \\ \hline&#xA;1 &amp;amp; \color{darkgreen}{3} \\&#xA;\color{darkgreen}{4} &amp;amp; 12 \\&#xA;9 &amp;amp; \color{darkred}{27} \\&#xA;16 &amp;amp; \color{darkblue}{48} \\&#xA;\color{darkred}{25} &amp;amp; 75 \\&#xA;36 &amp;amp; 108 \\&#xA;\color{darkblue}{49} &amp;amp; 147 \\&#xA;64 &amp;amp; 192 \\&#xA;81 &amp;amp; 243 \\&#xA;100 &amp;amp; 300 \\&#xA;121 &amp;amp; \color{purple}{363} \\&#xA;\vdots &amp;amp; \vdots \\&#xA;\color{purple}{361} &amp;amp; 1183 \\&#xA;\vdots &amp;amp; \vdots \\&#xA;\end{array}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The pairs of colored entries give the approximations &lt;span class=&quot;math-container&quot;&gt;$\color{green}{\frac21}, \color{maroon}{\frac53}, \color{darkblue}{\frac74},  $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\color{purple}{\frac{19}{11}}$&lt;/span&gt; respectively.  If you carry the table to 265 entries, you find that &lt;span class=&quot;math-container&quot;&gt;$265^2 = 70225$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3\cdot 153^2 = 70227$&lt;/span&gt; and there is your &lt;span class=&quot;math-container&quot;&gt;$\sqrt 3 \approx \frac{265}{153}$&lt;/span&gt;.  You don't have to be as clever as Archimedes to think of this. No magical technique was required, no sophisticated theory was required.  The calculation is tedious but straightforward and I judge that it could have been carried to 265 entries in at most a few hours, even using whatever awful technology was available at the time.&lt;/p&gt;&#xA;&lt;p&gt;Archimedes  might, of course, have used a better method.  (He also produces the approximation  &lt;span class=&quot;math-container&quot;&gt;$\frac{1351}{780}$&lt;/span&gt;, for which the foregoing is not obviously practical. He might have deployed the so-called “Babylonian method”. Or, having  tabulated the first few approximations from the table above, he might have noticed the simple recurrence that governs them; this would not be hard for a bright high-school student. Or he might have had a some other method.) But it seems to me that since there is a straightforward and simple method by which Archimedes could have produced the required approximations, there is no puzzle to solve.&lt;/p&gt;&#xA;&lt;p&gt;However, Rouse Ball and a number of other eminent authors disagree with me, and consider Archimedes' calculation of &lt;span class=&quot;math-container&quot;&gt;$\sqrt 3\approx \frac{265}{153}$&lt;/span&gt;  to be mysterious. (The page I linked earlier cites: T. Heath, E.T. Bell, C.B. Boyer, M. Kline, P. Beckmann, Sondheimer and Rogerson, and goes on to describe a complicated and elaborate method that Archimedes could have used.)&lt;/p&gt;&#xA;&lt;p&gt;If it was only popular mathematics authors who thought this calculation required  mysterious powers on the part of Archimedes, I would write it off as the authors overlooking the obvious, because the authors of popular books on mathematics can be quite obtuse. But Rouse Ball and Morris Kline are not dummies and I do not quite believe that they could have overlooked the method I described above, which seems not merely obvious but literally the first thing anyone would try.&lt;/p&gt;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there some reason I don't appreciate that Archimedes could not have or certainly did not use the method I described?  Or did all those other guys overlook the obvious?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="134" LastEditorUserId="134" LastEditorDisplayName="user145" LastEditDate="2022-11-29T23:34:28.917" LastActivityDate="2022-11-29T23:34:28.917" Title="What is so mysterious about Archimedes' approximation of $\sqrt 3$?" Tags="|mathematics|computation|archimedes|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="773" PostTypeId="1" AcceptedAnswerId="781" CreationDate="2015-01-02T09:59:38.823" Score="15" ViewCount="1948" Body="&lt;p&gt;The vector notation in physics is a very compact and easy way to write things down, and according to Feynman it also saves print. When exactly did scientists realize that they were summarizing things with vector notation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recap: I know that Oliver Heaviside introduced vector calculus in the 20th century; I am asking about the vector notation in Newton's law, for example, or four-vectors, for that matter.&lt;/p&gt;&#xA;" OwnerUserId="660" LastEditorUserId="62" LastEditDate="2015-01-05T00:09:13.920" LastActivityDate="2015-01-05T00:09:13.920" Title="When was the vector notation in physics and other sciences first introduced?" Tags="|physics|notation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="774" PostTypeId="1" AcceptedAnswerId="775" CreationDate="2015-01-02T10:09:53.993" Score="16" ViewCount="1346" Body="&lt;p&gt;In 1821 Cauchy claimed that the limit of a sequence of continuous functions is continuous. In 1826 Abel gave a complicated trigonometric counterexample. When we teach analysis courses, we usually give the sequence $f_n(x)=x^n$ as a counterexample. Who discovered this counterexample? Why was this counterexample not discovered earlier? I realize that this is a &quot;stupid&quot; question with no simple answer, but every time I teach analysis, this bothers me, so I wanted to see if anybody here could throw some light on this. Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="661" LastEditorUserId="26" LastEditDate="2015-01-02T13:02:04.857" LastActivityDate="2019-11-21T18:46:38.450" Title="Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's &quot;theorem&quot; about limits of continuous functions?" Tags="|mathematics|real-analysis|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="775" PostTypeId="2" ParentId="774" CreationDate="2015-01-02T11:21:54.093" Score="21" Body="&lt;p&gt;&quot;Counterexamples&quot; to Cauchy's theorem were &quot;discovered&quot; as soon as he proved it.&#xA;Of course Cauchy knew all these &quot;counterexamples&quot; but he insisted that his theorem and its proof is correct until his death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, Fourier's book on heat is full of these counterexamples. Fourier tries&#xA;to clarify the matter by defining a continuous function as one &quot;whose graph can be drawn without detaching the pencil from the paper&quot;. So from Fourier's point of view the curve&#xA;consisting of the negative $x$-ray, a vertical segment from $(0,0)$ to $(0,1)$&#xA;and the horizontal ray from $(0,1)$ to the right, is a graph of a continuous function :-).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recall that the definition of &quot;function&quot; your are taught in your Calculus class&#xA;is due to Dirichlet, and this was much later than Cauchy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your function which is zero on $[0,1)$ and $1$ at $1$ simply was not a &quot;function&quot; for Cauchy. About the evolution of the notion of &quot;function&quot;, read Luzin's paper in the Amer. Math. Monthly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that neither the notion of continuous function nor the notion of convergence of functions was sufficiently formalized at that time. If you read Cauchy's proof, it is essentially correct,&#xA;but he really uses uniform convergence. The notion of uniform convergence was not formalized at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these things were clarified only at the time of Weierstrass. Nowadays we are all thinking in the framework created by Cantor, so we cannot understand the difficulties&#xA;experienced by mathematicians before Cantor, Dedekind and Weierstrass. To understand them one has to read the original texts and forget what one was taught in Calculus class.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-02T11:29:48.060" LastActivityDate="2015-01-02T11:29:48.060" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="776" PostTypeId="2" ParentId="755" CreationDate="2015-01-02T16:26:40.780" Score="6" Body="&lt;p&gt;There is an &lt;a href=&quot;http://en.wikipedia.org/wiki/Julian_calendar&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; in Wikipedia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The assertion that August was lengthened by (or in honor of) Augustus comes from Sacrobosco in the 13th century.  (Like Alexandre, I learned this in school.  But in the US.) But apparently this assertion is no longer thought to be true.  For example, there is a painting of a Roman calendar from before the Julian reform showing that February already had 28 days.  The Julian reform just kept that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/rMvHH.jpg&quot; alt=&quot;calendar&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Answer to question:&lt;/strong&gt;&lt;br&gt;&#xA;So, according to this, the length of February was 28 days in the pre-Julian Roman calendar.  Any reply attributing it to Augustus is thus wrong.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-01-02T16:26:40.780" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="777" PostTypeId="2" ParentId="774" CreationDate="2015-01-02T21:13:18.897" Score="13" Body="&lt;p&gt;I'll add to Alexandre's answer that with the way Cauchy thought about continuum and defined convergence, the &quot;counterexamples&quot; did not converge to the limits we ascribe to them today. To Cauchy &quot;points&quot; are &quot;moving points&quot; so he would consider &lt;span class=&quot;math-container&quot;&gt;$x=1-1/n$&lt;/span&gt; with variable &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; going to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; a &quot;point&quot; infinitesimally close to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. But &lt;span class=&quot;math-container&quot;&gt;$(1-1/n)^n$&lt;/span&gt; does not converge to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; does not converge to anything at &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; according to Cauchy. He dismissed Abel's and Fourier's &quot;counterexamples&quot; on the same grounds, see &lt;a href=&quot;https://books.google.com/books?id=VepnMpJXmB8C&amp;amp;pg=PA56&amp;amp;lpg=PA56&amp;amp;dq=cauchy+uniform+convergence+nonstandard+analysis&amp;amp;source=bl&amp;amp;ots=kY29i7yuue&amp;amp;sig=b4xf6H-v8yxvjARvzH00Q3GRwsQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=ewanVMyWNdOBygS7vYGgAw&amp;amp;ved=0CEIQ6AEwBg#v=onepage&amp;amp;q=cauchy%20uniform%20convergence%20nonstandard%20analysis&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lakatos, Cauchy and the Continuum&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With Cauchy's &quot;moving points&quot; his &quot;pointwise&quot; convergence turns out to imply Weierstass's uniform convergence. More on Cauchy's understanding of infinitesimals and continuity in &lt;a href=&quot;https://web.archive.org/web/20150208161531/http://gfisher.org/Cauchy%20and%20the%20Infinitely%20Small.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fisher's Cauchy and the Infinitely Small&lt;/a&gt;, which also gives a modern reconstruction of Cauchy's argument in terms of non-standard analysis, where it is correct.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-21T18:46:38.450" LastActivityDate="2019-11-21T18:46:38.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="778" PostTypeId="2" ParentId="755" CreationDate="2015-01-03T07:27:04.760" Score="1" Body="&lt;p&gt;the 8th century BCE, they used the Calendar of Romulus, a 10-month calendar that kicked the year off in March (with the spring equinox) and ended in December. January and February didn’t even exist:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;`Martius: 31 days&#xA;Aprilius: 30 days&#xA;Maius: 31 days&#xA;Junius: 30 days&#xA;Quintilis: 31 days&#xA;Sextilis: 30 days&#xA;September: 30 days&#xA;October: 31 days&#xA;November: 30 days&#xA;December: 30 days`&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;The Romans believed even numbers were unlucky, so Numa tried to make each month odd. But to reach the quota of 355, one month had to be even. February ended up pulling the short stick, probably because it was simply the last month on the list.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Martius: 31 days&#xA;Aprilius: 29 days&#xA;Maius: 31 days&#xA;Iunius: 29 days&#xA;Quintilis: 31 days&#xA;Sextilis: 29 days&#xA;September: 29 days&#xA;October: 31 days&#xA;November: 29 days&#xA;December: 29 days&#xA;Ianuarius: 29 days&#xA;Februarius: 28 days&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;http://mentalfloss.com/article/55327/why-are-there-only-28-days-february&quot; rel=&quot;nofollow&quot;&gt;Mental Flows&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="247" LastActivityDate="2015-01-03T07:27:04.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="779" PostTypeId="1" AcceptedAnswerId="780" CreationDate="2015-01-03T08:17:41.610" Score="7" ViewCount="387" Body="&lt;p&gt;The definition of what an &lt;em&gt;elementary function&lt;/em&gt; is is quite arbitrary (see what &lt;a href=&quot;https://math.stackexchange.com/questions/118113/what-makes-elementary-functions-elementary&quot;&gt;math.SE&lt;/a&gt; has to say about it) and it makes me wonder why hasn't the mathematical community added other rather natural functions to the list, for example the &lt;a href=&quot;http://en.wikipedia.org/wiki/Error_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;error function&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Logarithmic_integral_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;logarithmic integral function&lt;/a&gt; or &lt;a href=&quot;http://en.wikipedia.org/wiki/Jacobi_elliptic_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Jacobi elliptic functions&lt;/a&gt; among many others. Is there an historical reason for this?.&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-01-04T08:19:03.613" Title="When was the term 'elementary function' first coined and who did it?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="780" PostTypeId="2" ParentId="779" CreationDate="2015-01-03T13:52:57.683" Score="6" Body="&lt;p&gt;I do not know who was the first to use the term, but the rigorous DEFINITION of an elementary function is due to Joseph Liouville (Journal de l'École polytechnique, vol.14 (1833), followed by several other papers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roughly speaking these are functions which are obtained from algebraic functions and&#xA;exponentials and logarithms by using arithmetic operations and superpositions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Liouville's goal was to determine which integrals of elementary functions are also elementary, and which algebraic differential equations have elementary solutions.&#xA;This is called sometimes Liouville's theory (or theories).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These theories had substantial development in 19 and 20-th century (Chebyshev, Hardy, Ritt, Rosenlicht, Picard, Vessiot and others)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other functions you mention (elliptic, error function and logarithmic integral) are&#xA;not elementary according to Liouville's definition. They belong to a class of &quot;special functions&quot;, but unlike elementary functions, this class was never rigorously defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When some function is important enough, and its main properties have been studied in great detail, it is usually called &quot;special&quot;, but I don't think that the class of special functions was ever seriously (mathematically) defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main special functions treated in the books called &quot;Special functions&quot;&#xA;are Gamma-function, Zeta-function, elliptic (and related) functions, and solutions&#xA;of some simple linear differential equations of second order with at most 3 singular points. This set of functions was defined and studied in 19-th century.&#xA;20-th century added to the list solutions of Painleve equations and Heun equation (and its degeneracies) and sometimes various mathematicians suggest adding these functions to the list of special functions. Heun's equation has 4 singular points, and on my opinion it is early to call its solutions or solutions of Painleve equations &quot;special functions&quot;,&#xA;because our knowledge about them is still very incomplete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So one can say that they are on the boundary of the class.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On my opinion, the class of elementary functions is interesting enough (because of the Liouville's theory) so it deserves a name. What the name exactly is, is more or less irrelevant, and everyone seems to be happy with the existing name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, one can also define elementary functions as those which &#xA;are studied in high school and obtain almost the same class:-)&#xA;(&quot;Almost&quot; because the only algebraic functions which occur in high school are those&#xA;expressible in radicals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then this will be similar to &quot;elementary mathematics&quot;, the term which however was never  rigorously&#xA;defined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me also add that elementary functions have been studied in much greater detail than other special functions. For example we know how to compute (in practice!) $e^x$&#xA;or $\log x$ to&#xA;billions of digits. But for most other special functions I do not now how to do this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-04T08:19:03.613" LastActivityDate="2015-01-04T08:19:03.613" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="781" PostTypeId="2" ParentId="773" CreationDate="2015-01-03T14:23:01.467" Score="6" Body="&lt;p&gt;Vector notation was introduced to physics in the end of 19-th century. I am not 100% sure,&#xA;but this is associated in my memory with J. Willard Gibbs (Wikipedia confirms this, and also mentions Heaviside). 3-dimensional vectors were preceded by quaternions, and there was even a discussion what is better: vectors or quaternions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell in his Treatease on Electricity and Magnetism (1873) did not use vectors, and&#xA;the standard mathematical textbook for physicists (Thomson and Tait, first edition 1867) does not mention them. Neither all multiple subsequent editions do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it is still not common knowledge, I will mention that vector notation is out of date.&#xA;It was superseded by the formalism of differential forms of E. Cartan &#xA;(invented in 1920-th). But differential forms still did not penetrate in undergraduate education, which seems very strange to me. And many (not all!) physicists continue to use&#xA;vector notation. Only a few of US universities teach differential forms to undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. To those who do not believe what I wrote in the last paragraph, I recommend to look in the undergraduate textbook which is used in Harvard: Bamberg and Sternberg, A course in mathematics for students in physics, vol. 2, chapter Maxwell equations.&#xA;They explain by the way how notation evolved, from Maxwell's notation to&#xA;&quot;vector calculus&quot; to &quot;tensor calculus&quot; to modern language of differential forms.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-04T17:15:59.390" LastActivityDate="2015-01-04T17:15:59.390" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="783" PostTypeId="1" CreationDate="2015-01-04T19:03:07.113" Score="14" ViewCount="1274" Body="&lt;p&gt;To quote Bertrand Russell, &quot;My Philosophical Development&quot;, Simon and Schuster, N.Y., 1959,&#xA;p. 86:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I used to know of only six people who had read the later parts of the book&#xA;  [Principia Mathematica]. Three of these were Poles .... The other three were Texans ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Jan Woleński in &quot;Polish Logic&quot;, Logical J. of the IGPL 12 (2004), 399-428 &#xA;(reprinted in &quot;Historico-Philosophical Essays&quot;, Copernicus Center Press, Krakow,&#xA;2012), suggests two of the 3 Poles were Chwistek and Leśniewski.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Leon_Chwistek&quot;&gt;Leon Chwistek&lt;/a&gt; wrote about theory of types, got positive reference from Russell while applying for a professorship in Lvov, and is referred to in the introduction to the 2nd edition of &lt;a href=&quot;https://en.wikipedia.org/wiki/Principia_Mathematica&quot;&gt;&quot;Principia Mathematica&quot;&lt;/a&gt;, so he is one of the Poles without doubt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who were the remaining 2 Poles and 3 Texans?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps another strong contender is Tarski.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Stanis%C5%82aw_Ignacy_Witkiewicz&quot;&gt;Stanisław Ignacy Witkiewicz&lt;/a&gt; (a friend of Chwistek) mentions &quot;Principia Mathematica&quot; in his novel &quot;Nienasycenie&quot;, but it is doubtful whether he had really read it, and not being a mathematician or logically inclined philosopher, he, probably, does not qualify.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;Update January 6, 2015&lt;/b&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As was noted by JHD, the full Russell quote continues as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... Three of these were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Indeed, neither of the 3 strong contenders (Chwistek, Leśniewski, Tarski) was &quot;liquidated by Hitler&quot;. I believe Russell is just wrong in that respect, for two reasons: First, at the time of writing this, his interests, for a long time, shifted far away from (mathematical) logic. If he was no longer interested in the subject, it is natural to assume he was also not much uptodate about the fate of its practitioners. Second, since so many Poles, and Polish mathematicians in particular, perished during WWII, it was, probably, natural to assume, not knowing the details, that any particular person &quot;was liquidated by Hitler&quot; than otherwise (btw, as far as I know, Chwistek died in Russia under strange circumstances, so it is quite possible that he was &quot;liquidated by Stalin&quot; instead of Hitler, but this will move us far away from the initial question).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, I am in full ignorance about Texans (and why and how they were &quot;assimilated&quot;).&lt;/p&gt;&#xA;" OwnerUserId="167" LastEditorUserId="167" LastEditDate="2015-01-06T16:06:11.637" LastActivityDate="2020-01-13T14:33:50.837" Title="3 Poles and 3 Texans who had read &quot;Principia Mathematica&quot;" Tags="|mathematics|mathematicians|mathematical-logic|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="784" PostTypeId="2" ParentId="771" CreationDate="2015-01-04T19:51:48.650" Score="37" Body="&lt;p&gt;According to the paper of Davies cited by MJD, Archimedes actually gives a double inequality&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{265}{153}&amp;lt;\sqrt{3}&amp;lt;\frac{1351}{780}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As both of these fractions are not just random approximations,&#xA;but are the 9-th and 12-th convergents&#xA;for the continued fraction expansion of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt;, there is no doubt that Archimedes used&#xA;the continued fraction expansion or some equivalent algorithm, for example Pell's equation. (Otherwise it cannot be explained why Archimedes has chosen these two fractions&#xA;of infinitely many other fractions aproximating &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Continued fraction algorithm was known to Euclid in some form (book 10). Davis's arguments&#xA;against it are totally unconvincing. And his conjecture about Archimedes computation by&#xA;the &quot;averaging&quot; algorithm that he proposes is not plausible, because it does not explain&#xA;the appearance of exactly these fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Davies's argument that the first published theory of Pell's equation appeared in 18-th century is also not convincing. Indeed, there is a well known text which is attributed to&#xA;Archimedes, called &quot;Archimedes Cattle problem&quot; which shows that the author's familiarity&#xA;with very complicated Pell's equations:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is absolutely clear that for the author of the Cattle problem, the Pell equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$m^2-3n^2=\pm1$&lt;/span&gt; was a trivial example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, there are indications that Hellenistic Greeks knew much more of number theory&#xA;than is contained in their work that survived. For example, it was recently discovered that Hipparchus knew the so-called Schroder numbers (rediscovered in 19-th century).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I just found a text whose author shares my opinion and elaborates on the question. So I can add a reference:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathpages.com/home/kmath038/kmath038.htm&quot; rel=&quot;noreferrer&quot;&gt;http://mathpages.com/home/kmath038/kmath038.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I do not know who this author is, but his web pages covering a variety of topics in physics, mathematics and history are so good that I value his opinion.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT2. In the paper by D. H. Fowler, Ratio in early Greek mathematics&#xA;(Bull. AMS, 1(1979) 6, 807-846) the author explains that the continued fraction algorithm was called in Greek anthyphairesis, and that it pre-dates Euclid's Elements.&#xA;He sites Archimedes' approximation of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3}$&lt;/span&gt; as an example of the use of this algorithm.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-30T23:38:41.660" LastActivityDate="2020-05-30T23:38:41.660" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="787" PostTypeId="1" AcceptedAnswerId="789" CreationDate="2015-01-05T22:52:21.083" Score="20" ViewCount="3499" Body="&lt;p&gt;Hesperus (Roman Vesper) is the name ancient Greeks gave to the evening star that appears in the sky for an hour after the Sun sets. Phosphorus (Roman Lucifer, sic!), was the name of the morning star that appears shortly before the dawn. Depending on who you read, at some point Greeks either &lt;a href=&quot;http://www.jackstargazer.com/scripts_feb2009.html&quot;&gt;learned from Babylonians&lt;/a&gt; or &lt;a href=&quot;http://www.jstor.org/stable/2182495?seq=1#page_scan_tab_contents&quot;&gt;figured out on their own&lt;/a&gt; that they are one and the same, and renamed it Aphrodite (Roman Venus). More recently this became a quintessential example in semantics for identity of apparently different objects, &lt;a href=&quot;http://en.wikipedia.org/wiki/Hesperus#.22Hesperus_is_Phosphorus.22&quot;&gt;analyzed&lt;/a&gt; by eminent logicians like Frege and Kripke.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today we have telescopes to discern fine details of distant objects, and satellites that can observe them continuously. But how did Babylonians figure it out watching with a naked eye from the surface? When did it happen? What was it that convinced the Greeks that Hesperus is Phosphorus?&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-03-30T03:49:37.287" Title="How did Babylonians figure out that the evening star is the morning star?" Tags="|astronomy|ancient-greece|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="788" PostTypeId="1" CreationDate="2015-01-06T05:50:46.897" Score="5" ViewCount="306" Body="&lt;p&gt;I was trying to figure out how people came to know about time then I realized that people started keeping track of time to know about sunset and sunrise. But I can't figure out how did time came into the physics world. Other physical quantities can be somewhat directly observed by our 5 senses but what caused time to be considered as physical quantity?&lt;/p&gt;&#xA;" OwnerUserId="672" LastActivityDate="2015-05-20T20:29:20.493" Title="How did the notion of &quot;time&quot; come in the world of physics?" Tags="|physics|time|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="789" PostTypeId="2" ParentId="787" CreationDate="2015-01-06T18:03:46.150" Score="25" Body="&lt;p&gt;I doubt that many Babylonians or Greeks or any others who cared about such things ever thought Hesperus and Phosphorus were different objects any more than we think the Morning Star and Evening Star are different.  Is there any evidence apart from the use of two names?&lt;/p&gt;&#xA;&lt;p&gt;I think we today underrate what it meant to see the stars clear and bright almost every night and at any part of the night you were awake outdoors -- and with many herdsmen and sailors and revelers and soldiers awake outdoors much of the night, besides people deliberately studying the night sky.&lt;/p&gt;&#xA;&lt;p&gt;Every slightly interested party knew the circumpolar stars are up all the time, and just get blotted out by sunlight during the day.  They had every reason to believe the adjacent stars that rise and set in the North are travelling 'behind' the earth from the time they set until they rise again -- and indeed that the whole sphere of fixed stars exists all the time and moves around.&lt;/p&gt;&#xA;&lt;p&gt;They knew that Hesperus, when it is seen, moves away from the sun and then back towards it.  And they knew that shortly after Hesperus disappears towards the sun, Phosphorus appears on the other side of the sun.  Of course stars on one side of the sun are seen just before dawn, moving towards the sun until they disappear and then reappear on the other side just after sunset.&lt;/p&gt;&#xA;&lt;p&gt;All the major wandering planets obviously move past the sun sometimes, appearing just east of it just after sunset at times, with the sun approaching, disappearing as the sun passes it, and and then appearing just west shortly before sunrise.  That is the moon, Mars, Jupiter, and Saturn.&lt;/p&gt;&#xA;&lt;p&gt;I doubt any great number of people who cared about the question ever thought Hesperus was an exception which ceased to exist each time it moved towards the sun and then Phosphorus appeared, or that Hesperus suddenly went &amp;quot;somewhere else&amp;quot; where we could not see it.  These were two names for two roles for one star.&lt;/p&gt;&#xA;&lt;p&gt;Kuhn's book &lt;em&gt;Copernican Revolution&lt;/em&gt; makes these points about the plainly visible motions of the stars -- including the wandering stars.  Well, that is plainly visible before artificial lighting, in parts of the world with generally clear skies.  He has a nice diagram explaining how people could not possibly have missed the constant existence of circumpolar stars, or the very similar motion of the stars near them that do rise and set.&lt;/p&gt;&#xA;&lt;p&gt;B. L. van der Waerden studied Babylonian astronomy and simply reports they knew Venus as a planet -- long before the Greek astronomers that we know.  (You can see this in his &amp;quot;Babylonian Astronomy. II. The Thirty-Six Stars,&amp;quot; Journal of Near Eastern Studies, Vol. 8, No. 1 (Jan., 1949), pp. 6-26) available at  &lt;a href=&quot;http://www.jstor.org/stable/542436&quot; rel=&quot;noreferrer&quot;&gt;www.jstor.org/stable/542436&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;A lot of sources on-line say the Egyptians and Greeks thought the morning star and evening stars were different stars.  But what they actually show is that those people associated different gods to the morning and evening stars.  The Egyptians also associated Khepera to the rising sun and Atum to the setting sun, without thinking the rising and setting suns were two different things.&lt;/p&gt;&#xA;&lt;p&gt;I do not see any evidence of a time when people tried to individuate &amp;quot;stars&amp;quot; as objects but thought the morning star and evening star were two different ones.&lt;/p&gt;&#xA;&lt;p&gt;Added: Plato's &lt;em&gt;Laws&lt;/em&gt; 821-822 has the interlocutors discuss the &amp;quot;impious error&amp;quot; that Greeks make when they believe &amp;quot;Hesperus and Phosphorus and the other planets&amp;quot;  wander rather than following a single course.  One character asserts &amp;quot;The truth is precisely the opposite: each of them always travels in a circle one and the same path.&amp;quot;  The other does not disagree.  The Greek is no more explicit than the English given here.  It does not say anyone thinks &amp;quot;Hesperus and Phoshorus&amp;quot; are two things, and anyway it is discussing an error without saying which Greeks make this mistake.  To believe Hesperus follows one circular orbit you would have to believe half of that orbit goes under the name of  Phosphorus.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2023-03-27T04:43:50.710" LastActivityDate="2023-03-27T04:43:50.710" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="790" PostTypeId="2" ParentId="769" CreationDate="2015-01-06T18:36:50.520" Score="7" Body="&lt;p&gt;I’ve found no secondary sources with specific references to this issue; here are some evidences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding Wiki reference for the term &lt;a href=&quot;http://en.wikipedia.org/wiki/Momentum&quot;&gt;momentum&lt;/a&gt;, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;John Jennings (1721). &lt;a href=&quot;http://books.google.it/books?id=hMsDAAAAQAAJ&amp;amp;pg=PA67&quot;&gt;Miscellanea in Usum Juventutis Academicae&lt;/a&gt;, &lt;strong&gt;n°IX. MECHANICA&lt;/strong&gt;, page  67 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;DEFINITIO I.&lt;/strong&gt; &lt;em&gt;Momentum&lt;/em&gt; seu &lt;em&gt;Quantitas Motus&lt;/em&gt; est Rectangulum ex Quantitate Materiae in Velocitate ducta, i.e. $M=Q \times V$ .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Isaac Newton, &lt;em&gt;Philosophiæ Naturalis Principia Mathematica&lt;/em&gt; (1686), we have only the mathematical term &lt;em&gt;momentum&lt;/em&gt;; see :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Lemma II.&lt;/strong&gt; Momentum Genitæ æquatur momentis Terminorum singulorum generantium in eorundem laterum indices dignitatum &amp;amp; coefficientia continue ductis. […] Has quantitates ut indeterminatas &amp;amp; instabiles, &amp;amp; quasi motu fluxuve perpetuo crescentes vel decrescentes hic considero, &amp;amp; eorum incrementa vel decrementa momentanea sub nomine &lt;em&gt;momentorum&lt;/em&gt; intelligo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://books.google.it/books?id=ySYULc7VEwsC&amp;amp;pg=PA249&quot;&gt;Sir Isaac Newton's Mathematical Principles of Natural Philosophy&lt;/a&gt; (translated into English by Andrew Motte in 1729;  revised by Florian Cajori : 1934), &lt;strong&gt;Lemma II&lt;/strong&gt; [page 249] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These quantities I here consider as variable and indetermined, and increasing or decreasing, as it were, by a continual motion or flux; and I understand their momentary increments or decrements by the name of &lt;em&gt;moments&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've found &lt;strong&gt;no&lt;/strong&gt; usage of &lt;em&gt;momentum&lt;/em&gt; as synonym of &lt;em&gt;quantity of motion&lt;/em&gt; in most of 18th Century texts: here are some refrences :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/William_Whiston&quot;&gt;William Whiston&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=t3hbAAAAQAAJ&amp;amp;pg=PA40&quot;&gt;Sir Isaac Newton's Mathematick Philosophy&lt;/a&gt; (1716), page 40 : &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the &lt;em&gt;Quantity of Motion&lt;/em&gt; is the Measure of the same, arising from the Velocity, and from the Quantity of Matter conjunctly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Leonhard Euler, &lt;a href=&quot;https://books.google.it/books?id=8TAVAAAAQAAJ&amp;amp;pg=PA103&quot;&gt;Mechanica sive Motus scientia analytice exposita&lt;/a&gt; : Tomus I (1736), page 103 [art.n*263] : &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[reffering to Newton’s definition of] vis centripetae quantitatem motricem qua nihil aliud denotat nisi ipsum conatum, quem corpora habent ad centrum virium accedendi; &lt;em&gt;quantitas motus&lt;/em&gt; enim, quam mensurare soliti sunt celeritate ducta in massa,s, quaeque dato tempore generatur, proportionalis est ipsi conatui.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nor we can find the term (linear) &lt;em&gt;momentum&lt;/em&gt; in the first “modern” expressions of Newton’s Law by Euler; neither in his memoir on celestial mechanics of 1747, nor in his &lt;em&gt;Decouverte d'un nouveau principe de Mecanique&lt;/em&gt; (1750), page 196 [see it in &lt;a href=&quot;http://eulerarchive.maa.org/index.html&quot;&gt;Euler archive&lt;/a&gt; and see also Clifford Truesdell, &lt;a href=&quot;https://books.google.it/books?id=uPLqmwEACAAJ&quot;&gt;Essays in the History of Mechanic&lt;/a&gt; (1968), page 167].&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Jean-Baptiste le Rond d'Alembert, &lt;a href=&quot;https://books.google.it/books?id=Kmv6sXij_uUC&amp;amp;pg=PA55&quot;&gt;Traité de dynamique&lt;/a&gt; (nouvelle ed, 1758), page 55 [art.n*46] : &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Le produit de la masse d’un corps par sa vitesse est appellee &lt;em&gt;quantité de Mouvement&lt;/em&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also page 73 : “&lt;strong&gt;DEFINITION.&lt;/strong&gt; […] par &lt;em&gt;quantité de Mouvement&lt;/em&gt;, j’entendrai à l’ordinaire le produit de la masse par la vitesse&quot;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Mathurin_Jacques_Brisson&quot;&gt;Mathurin Jacques Brisson&lt;/a&gt; : &lt;a href=&quot;https://books.google.it/books?id=NwS4s5sjq8MC&amp;amp;pg=PA60&quot;&gt;Traité élémentaire ou Principes de physique&lt;/a&gt;, (Paris 1789–1803), page 60 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;7. Quantité de mouvement.&lt;/strong&gt; La &lt;em&gt;quantité de movement&lt;/em&gt; d’un corps s’estime ou s’évalue en multipliant la masse de ce corps par sa vitesse, car elle y est proportionnelle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also the Italian translation (3rd ed.1804) : &lt;em&gt;Trattato elementare ovvero Principj di fisica : Tomo I&lt;/em&gt;, page 57 : “&lt;strong&gt;VII. Quantità del moto.&lt;/strong&gt; La &lt;em&gt;Quantità del moto&lt;/em&gt; di un corpo si valuta moltiplicando la massa di questo corpo per la velocità.”&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;We can find in Andrew Motte,  &lt;a href=&quot;https://books.google.it/books?id=OnZYAAAAYAAJ&amp;amp;pg=PA8&quot;&gt;A treatise of the mechanical powers&lt;/a&gt;, (1727), page 8 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the Product of these two, the Velocity, and the Weight of a moving Body, which is called the &lt;em&gt;Momentum&lt;/em&gt; of that Body.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also page 53: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two moving Bodies have equal &lt;em&gt;Momenta&lt;/em&gt; or Quantities of Motion, when the Velocity of each Body multiplied into its Weight […].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also in Colin MacLaurin, &lt;a href=&quot;https://books.google.it/books?id=yS1PAAAAcAAJ&amp;amp;pg=PA140&quot;&gt;An Account of Sir Isaac Newton's Philosophical Discoveries&lt;/a&gt; (2nd ed, 1750); see page 140 : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the quantity of motion, or &lt;em&gt;momentum&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and page 111 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;quantity of motion&lt;/em&gt; in a body […] is the compounded ratio of its quantity of matter and of the velocity of the motion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In 19th century, we can find it in “modern” treatises, like :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;William Thomson Baron Kelvin &amp;amp; Peter Guthrie Tait, &lt;em&gt;Treatise on Natural Philosophy&lt;/em&gt; (2nd ed, 1879), art.210, page 221 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;Quantity of Motion&lt;/em&gt;, or the &lt;em&gt;Momentum&lt;/em&gt;, of a rigid body moving without rotation is proportional to its mass and velocity conjointly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;James Clerk Maxwell, &lt;em&gt;Matter and Motion&lt;/em&gt; (1888), art.&lt;strong&gt;LI. - On Momentum&lt;/strong&gt; [page 44] :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The numerical value of the &lt;em&gt;Momentum&lt;/em&gt; of a body is the product of the number of units of mass in the body into the numbers of units of velocity with which it is moving.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An early reference for the English usage in 19th Century is : &lt;a href=&quot;http://en.wikipedia.org/wiki/Thomas_Young_%28scientist%29&quot;&gt;Thomas Young&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=lmM_AAAAcAAJ&amp;amp;pg=PA36&quot;&gt;A Course of Lectures on Natural Philosophy and the Mechanical Arts&lt;/a&gt;, (1807); I’m not able to browse volume I, but the Index point to  : I, page 52 for the definition of quantity of motion and I,53 for the definition of momentum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In volume 2, we can find [page 36] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;n*270 : &lt;strong&gt;DEFINITION.&lt;/strong&gt; The joint ratio of the masses and velocities of any two bodies is the ratio of their &lt;em&gt;momenta&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;n*271 : &lt;strong&gt;THEOREM.&lt;/strong&gt; The &lt;em&gt;momentum&lt;/em&gt; of any body is the true measure of the quantity of its motion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another possible source is Joseph-Louis Lagrange, &lt;em&gt;Mécanique analytique : Tome I&lt;/em&gt; (1811), pag.27, regarding the &lt;em&gt;pricipe des vitesses virtuelles&lt;/em&gt; : let $P,Q,R$ the &lt;em&gt;puissances&lt;/em&gt; and let $dp,dq,dr$ proportional to the &lt;em&gt;vitesses virtuelles&lt;/em&gt; of the respectives &lt;em&gt;puissances&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each term $P dp$ of the general equation of Statics regardings the equilibrium of the system is called :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;le moment de la force $P$, en prenant le mot de &lt;em&gt;moment&lt;/em&gt; dans le sens qui Galilée lui a donné, c’set-à-dire, pour le produit de la force par sa vitesse virtuelle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ref to Galileo Galilei, &lt;a href=&quot;http://it.wikisource.org/wiki/Le_mecaniche/Diffinizioni&quot;&gt;Le mecaniche&lt;/a&gt; (1599) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[…] la velocità del moto sia potente ad accrescere &lt;em&gt;momento&lt;/em&gt; nel mobile, secondo quella medesima proporzione con la quale essa velocità di moto viene augumentata.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Momento è la propensione di andare al basso, cagionata non tanto dalla gravità del mobile, quanto dalla disposizione che abbino tra di loro diversi corpi gravi; mediante il qual momento si vedrà molte volte un corpo men grave contrapesare un altro di maggior gravità: come nella stadera si vede un picciolo contrapeso alzare un altro peso grandissimo, non per eccesso di gravità, ma sì bene per la lontananza dal punto donde viene sostenuta la stadera; la quale, congiunta con la gravità del minor peso, gli accresce momento ed impeto di andare al basso, col quale può eccedere il momento dell'altro maggior grave. È dunque il momento quell'impeto di andare al basso, composto di gravità, posizione e di altro, dal che possa essere tal propensione cagionata.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[…] E da questo discorso possiamo venire in cognizione, come &lt;em&gt;la velocità del moto sia potente ad accrescere momento nel mobile, secondo quella medesima proporzione con la quale essa velocità di moto viene augumentata&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;http://it.wikisource.org/wiki/Discorso_intorno_alle_cose_che_stanno_in_su_l%27acqua_o_che_in_quella_si_muovono&quot;&gt;Discorso intorno alle cose che stanno in su l'acqua&lt;/a&gt; (1612) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Momento&lt;/em&gt;, appresso i meccanici, significa quella virtù, quella forza, quella efficacia,&#xA;  con la quale il motor muove e 'l mobile resiste; la qual virtù depende non solo dalla semplice gravità, ma dalla velocità del moto, dalle diverse inclinazioni degli spazii sopra i quali si fa il moto, perché più fa impeto un grave descendente in uno spazio molto declive che in un meno. Ed in somma, qualunque si sia la cagione di tal virtù, ella tuttavia ritien nome di momento. […]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Come, per esemplo, due pesi d'assoluta gravità eguali, posti in bilancia di braccia eguali, restano in equilibrio, né s'inclina l'uno alzando l'altro; perché l'egualità delle distanze di ambedue dal centro, sopra il quale la bilancia vien sostenuta e circa il quale ella si muove, fa che tali pesi, movendosi essa bilancia, passerebbono nello stesso tempo spazii eguali, cioè si moverieno con eguali velocità, onde non è ragione alcuna, per la quale questo peso più di quello, o quello più di questo, si debba abbassare; e per ciò si fa l'equilibrio, e restano i momenti loro di virtù simili ed eguali.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The source for Galileo’s “dynamical” analysis of the balance is the pseudo-Aristotle, &lt;a href=&quot;http://en.wikipedia.org/wiki/Mechanics_%28Aristotle%29&quot;&gt;Mechanics&lt;/a&gt; [Greek: &lt;em&gt;Μηχανικά&lt;/em&gt;; see also Morris Cohen &amp;amp; IE Drabkin (editor), &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0674823206&quot;&gt;A Source Book in Greek Science&lt;/a&gt; (1948), page 189], where the balance is analyzed in terms of the &lt;em&gt;power&lt;/em&gt; of the agency that sets a body in motion, defined as the product of the weight of the body and the velocity of the motion that the body acquires.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-01-06T19:57:49.657" LastActivityDate="2015-01-06T19:57:49.657" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="791" PostTypeId="1" CreationDate="2015-01-06T21:17:05.020" Score="4" ViewCount="228" Body="&lt;p&gt;Aristotle's natural philosophy was an inquiry into the causal principles of nature. He famously proposed the notions of formal, material, efficient and final causes. At the dawn of modernity, figures such as Galileo and even Newton continued to consider the &quot;causal principles of nature&quot; as the proper subjects of inquiry. Newton said he was interested in the forces of nature (which 'cause' motion), and that he his main goal was&#xA;&quot;to argue from Phaenomena without feigning Hypotheses, and to deduce Causes from Effects.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the dawn of modernity, the concept of 'laws of nature' also appears. Descartes was the first to introduce the notion - Newton also appropriated it. However, the concept did not yet have the priority it has today, and traditional Aristotelean causal explanations seem to linger on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: When in the history of science did 'laws of nature' become the primary/highest form of explanation? When does it become &quot;mainstream&quot; in physics?&lt;/p&gt;&#xA;" OwnerUserId="675" LastEditorUserId="675" LastEditDate="2015-01-07T10:58:26.677" LastActivityDate="2015-03-09T00:29:40.947" Title="When in the history of physics did 'laws of nature' become the primary form of explanation?" Tags="|physics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="792" PostTypeId="2" ParentId="791" CreationDate="2015-01-06T22:36:44.920" Score="1" Body="&lt;p&gt;The early scientists, from Francis Bacon on, were practically all Christians, and they knew the Bible fairly well. I believe they took the concept of &quot;the Laws of Nature and of Nature's God&quot; (to quote the Declaration of Independence) from several famous passages of the Bible. Here are just two examples.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Genesis 8:22 (NASB): &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;While the earth remains,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Seedtime and harvest,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;And cold and heat,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;And summer and winter,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;And day and night&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Shall not cease.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jeremiah 33:25 (NASB): &quot;Thus says the LORD, 'If My covenant for day and&#xA;  night stand not, and the fixed patterns of heaven and earth I have not&#xA;  established, ...&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Therefore, it appears that the concept of the &quot;law of nature&quot; was there from the very beginning of modern science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read several books that make that point, but I do not now own them. One of them is on my wish list, but I haven't gotten it yet. Therefore I cannot give a good citation here.&lt;/p&gt;&#xA;" OwnerUserId="426" LastEditorUserId="426" LastEditDate="2015-01-06T22:52:28.877" LastActivityDate="2015-01-06T22:52:28.877" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1790" PostTypeId="1" CreationDate="2015-01-07T08:44:45.427" Score="2" ViewCount="151" Body="&lt;p&gt;A question arose in my mind about modular inverse. We know that there is no division operator in modular arithmetic, so we figure out the modular inverse of the denominator. For example, if we want to find out the value of $\frac ab  \pmod{M} $, at first we determine the modular inverse of $b$ and then multiply it by$a \pmod{M}$ .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is that who first invented the above the techniques, and what inspired him/her?&lt;/p&gt;&#xA;" OwnerUserId="1676" LastEditorUserId="62" LastEditDate="2015-01-08T01:11:17.193" LastActivityDate="2015-01-08T01:11:17.193" Title="Invention of modular inverse" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1791" PostTypeId="2" ParentId="1790" CreationDate="2015-01-07T16:12:00.143" Score="2" Body="&lt;p&gt;The Euclidean algorithm is in effect the division operator in modular arithmetic, giving the value of  $\frac{a}{b}  \pmod{M}$ when $b$ is relatively prime to $M$, and every other case reduces to that one.  It suffices to find each  $\frac{1}{b}  \pmod{M}$. Then, as Euclid shows, you can find a sum $cb+dM=1$.  And that amounts to  $c=\frac{1}{b}  \pmod{M}$ though of course Euclid did not think of this as modular arithmetic.  He just thought of it as the arithmetic of greatest common divisors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The harder question would be who did first think of this kind of calculation as being modular arithmetic.  I believe Fermat did not think this way, and Euler rather did.  I do not know the details and there is probably no single neat answer to be given.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to intuition, modular arithmetic arose as a neat way of organizing arithmetic facts already known to Euclid and Diophantus. &lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2015-01-07T16:12:00.143" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1792" PostTypeId="1" CreationDate="2015-01-07T19:30:19.270" Score="2" ViewCount="93" Body="&lt;p&gt;I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature of a normal section is determined by the principal curvatures and the angle that the tangent to the curve makes with a principal direction via Euler equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives a rather rigid, albeit infinitesimal, description of the geometry of a surface at a point. I have been wondering how Euler might have thought that something like this should be true. Does it come from linear algebra and results about eigenvalues and eigenvectors of symmetric matrices?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did have a look at Euler's original paper (English translation). It was interesting to see how mathematics research was done and discussed at the time but it doesn't give any insight on Euler's geometric intuition.&lt;/p&gt;&#xA;" OwnerUserId="1678" LastEditorUserId="26" LastEditDate="2015-01-08T11:45:58.377" LastActivityDate="2015-01-08T11:45:58.377" Title="Motivation behind Euler Theorem in differential geometry" Tags="|mathematics|differential-geometry|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-01-10T16:28:32.287" ContentLicense="CC BY-SA 3.0" />
  <row Id="1793" PostTypeId="2" ParentId="791" CreationDate="2015-01-07T20:16:51.327" Score="2" Body="&lt;p&gt;It seems to the that has been a &amp;quot;porcess&amp;quot;, but a quite fast one.&lt;/p&gt;&#xA;&lt;p&gt;In Galileo Galilei we have few occurrences of this term :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Galileo Galilei, &lt;a href=&quot;http://it.wikisource.org/wiki/Le_mecaniche/Della_forza_della_percossa&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le mecaniche&lt;/a&gt; (1599) :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;E questo sarà co ’l ridurci inanzi gli occhi quello, che in ogni altra operazione meccanica s’è veduto accadere: cioè che la forza, la resistenza ed il spazio, per lo quale si fa il moto, si vanno alternamente con tal proporzione seguendo, e con &lt;em&gt;legge&lt;/em&gt; tale rispondendo, che resistenza eguale alla forza sarà da essa forza mossa per egual spazio e con egual velocità di quella che essa si muova.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The term is fundamental in Descartes :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;René Descartes, &lt;a href=&quot;http://books.google.it/books?id=JQs_AAAAcAAJ&amp;amp;pg=PA43&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principia philosophiae&lt;/a&gt; (1644), art.XXXVII and XXXIX :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Prima lex naturae [...] Altera lex natura&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Isaac Newton, Philosophiae Naturalis Principia Mathematica (1686 - latin text) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Axiomata sive Leges Motus&lt;/strong&gt; [...]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;De Mundi Systemate - Liber Tertius&lt;/strong&gt; [...] Hæc sunt &lt;em&gt;motuum &amp;amp; virium leges&lt;/em&gt; &amp;amp; conditiones, quæ ad Philosophiam maxime spectant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After Newton, the diffusion of the term seems impressive :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Leonhard Euler, &lt;a href=&quot;https://books.google.it/books?id=8TAVAAAAQAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mechanica sive Motus scientia analytice exposita&lt;/a&gt; : Tomus I (1736), &lt;strong&gt;Prefatio&lt;/strong&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Expono igitur Capitae promo generales motus proprietates [...] atque demonstro &lt;em&gt;leges naturae universales&lt;/em&gt; [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Jean-Baptiste le Rond d'Alembert, &lt;a href=&quot;https://books.google.it/books?id=Kmv6sXij_uUC&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Traité de dynamique&lt;/a&gt; (nouvelle ed, 1758), &lt;strong&gt;Avertissement&lt;/strong&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] si le &lt;em&gt;loix&lt;/em&gt; de la Statique et de la Méchanique sont de vérité nécessaire ou contingente&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-01-07T20:16:51.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1794" PostTypeId="2" ParentId="791" CreationDate="2015-01-07T23:43:57.123" Score="1" Body="&lt;p&gt;Plato, Timaeus 83e, speaks explicitly of the “laws of nature” (παρὰ τοὺς τῆς φύσεως νόμους). The early modern authors cited by Mauro are presumably alluding intentionally to this famous passage. The basic idea is that nature has its own intrinsic laws, not legislated by god(s).&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-01-07T23:43:57.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1795" PostTypeId="2" ParentId="317" CreationDate="2015-01-09T18:47:15.157" Score="10" Body="&lt;p&gt;A fairly comprehensive treatment of &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Leray&quot; rel=&quot;noreferrer&quot;&gt;Jean Leray&lt;/a&gt;'s work as a prisoner of war can be found in &lt;a href=&quot;http://www.mat.univie.ac.at/~michor/leray.pdf&quot; rel=&quot;noreferrer&quot;&gt;this paper&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among the citations you can see the rather famous &lt;em&gt;Course d’algèbre topologique enseigné&#xA;en captivité&lt;/em&gt; (&quot;A course in algebraic topology taught in prison&quot;). &lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="72" LastEditDate="2015-03-03T10:26:48.670" LastActivityDate="2015-03-03T10:26:48.670" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1796" PostTypeId="1" AcceptedAnswerId="1797" CreationDate="2015-01-09T22:02:33.497" Score="21" ViewCount="14534" Body="&lt;p&gt;Who first measured the distance to the Moon? How was it done? I think it had to happen after Newton, but I am not sure.&lt;/p&gt;&#xA;" OwnerUserId="1688" LastEditorUserId="62" LastEditDate="2015-01-09T22:30:18.340" LastActivityDate="2022-05-18T23:26:35.020" Title="Who first measured the distance to the Moon? How was it done?" Tags="|astronomy|measurement|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1797" PostTypeId="2" ParentId="1796" CreationDate="2015-01-09T23:50:55.283" Score="24" Body="&lt;p&gt;It happened long before Newton. In the second century BC Hipparchus used lunar parallax to calculate a value for the minimum and maximum distance of the earth and moon. His results are very close to the modern calculation of this distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can read about it here: Toomer G.J. (1974), &quot;Hipparchus on the Distances of the Sun and Moon.&quot; Archives for the History of the Exact Sciences 14: 126–142.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-01-09T23:50:55.283" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1798" PostTypeId="2" ParentId="1796" CreationDate="2015-01-10T07:45:16.297" Score="12" Body="&lt;p&gt;This is an ill defined question. It can be interpreted as &amp;quot;Who was the first to TRY to measure the distance to the Moon&amp;quot;, or &amp;quot;Who was the first to give a correct number&amp;quot;, and what is counted as correct number.&lt;/p&gt;&#xA;&lt;p&gt;And even this does not define the question precisely. The crucial question: &amp;quot;in what units&amp;quot;? It is relatively easy to measure the distance in terms of the radius of the Earth.&#xA;But it is another matter, if you want the answer in miles, kilometers or stadia.&lt;/p&gt;&#xA;&lt;p&gt;The earliest written evidence that survived is a purely theoretical work of Aristarchus&#xA;where he explains some mathematics involved. One can conclude from the book that he hardly&#xA;measured anything in practice.&lt;/p&gt;&#xA;&lt;p&gt;Hipparchus who lived later, had already the idea of the order of magnitude of Moon's&#xA;parallax (this is equivalent to measuring the distance in terms of the Earth radius).&#xA;However most ancients did not discuss the distance as we understand it, but only ratios,&#xA;for example the ratio of the distance to the Sun and to the Moon, or the Moon's parallax&#xA;which is sufficiently large to be measured by primitive tools.&#xA;To pass from the Moon parallax to the distance in some conventional length units one needs to know the&#xA;size of the Earth. The size of the Earth was measured by Eratosthenes, but it is still subject of discussion how accurate his measurement was.&lt;/p&gt;&#xA;&lt;p&gt;The problem is that he gives the answer is stadia, and nobody knows how long his stadium was. Few other measurements were made after him, but still when Newton was a student&#xA;he was taught that the degree of the meridian is 60 (British) miles.&#xA;There was no way to measure the distance to the Moon without knowing the radius of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;Speaking of accuracy: Hellenistic Greeks could measure the Moon parallax to minutes (Ptolemy), and this was not improved until good instruments were manufactured at the time of Brahe. Since then accuracy constantly increased. Precise measurements of the size of the Earth are even later, they begin in the end of 17 century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-05-18T23:26:35.020" LastActivityDate="2022-05-18T23:26:35.020" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="1799" PostTypeId="1" AcceptedAnswerId="1807" CreationDate="2015-01-07T10:38:26.460" Score="10" ViewCount="1501" Body="&lt;p&gt;I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature of a normal section is determined by the principal curvatures and the angle that the tangent to the curve makes with a principal direction via Euler equation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives a rather rigid, albeit infinitesimal, description of the geometry of a surface at a point. I have been wondering how Euler might have thought that something like this should be true. Does it come from linear algebra and results about eigenvalues and eigenvectors of symmetric matrices?&lt;/p&gt;&#xA;" OwnerUserId="1678" OwnerDisplayName="Giuseppe" LastEditorUserId="55" LastEditDate="2019-12-13T09:17:55.613" LastActivityDate="2022-12-29T10:43:14.590" Title="Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?" Tags="|mathematics|differential-geometry|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="1800" PostTypeId="2" ParentId="1799" CreationDate="2015-01-07T21:42:53.280" Score="9" Body="&lt;p&gt;You can read about the origin of Euler's 1760 study on curvature in &lt;A HREF=&quot;https://books.google.de/books?id=V_B5SIGTnnoC&amp;amp;pg=PA187&quot;&gt;Mathematical Masterpieces: Further Chronicles by the Explorers&lt;/A&gt; (page 187 and following). The idea of curvature goes back several decades, Alexes Clairaut studied it for one-dimensional curves in his 1731 book &lt;em&gt;Recherces sur les courbes à double courbure.&lt;/em&gt; Euler was the first to apply this concept to higher-dimensional objects. The formula $k=k_1\cos^2\alpha+k_2\sin^2\alpha$ relating the curvature $k$ of a normal section to the principal curvatures $k_1,k_2$ and the inclination $\alpha$ was implicit in Euler's 1760 paper, but first written down in 1813 by Charles Dupin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book I mentioned above gives Euler's step-by-step derivation of his formula in English translation. I show the first paragraph, to appreciate the lucid and systematic way he approached the problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In order to know the curvature of curved lines, the determination of&#xA;  the radius of the osculating circle offers the proper method, which&#xA;  for each point on the curve provides us with a circle whose curvature&#xA;  is the same. But when one asks for the curvature of a surface, the&#xA;  question is rather equivocal, and not at all subject to a definitive&#xA;  answer, as in the previous case. It is only spherical surfaces for&#xA;  which one can measure the curvature, considering that the curvature of&#xA;  a sphere is the same as that of its great circles, and its radius can&#xA;  be considered as the proper measure of curvature. But for other&#xA;  surfaces, one would not even know how to compare the curvature of the&#xA;  surface with that of a sphere, as one can always compare the curvature&#xA;  of a curved line with that of a circle. The reason for this is&#xA;  evident, since through each point of a surface, there are infinitely&#xA;  many different curves. One only need consider the surface of a&#xA;  cylinder, where along directions parallel to the axis, there is no&#xA;  curvature, while cross sections perpendicular to the axis, which are&#xA;  circles, have the same curvature, and all other sections taken&#xA;  obliquely to the axis yield particular values of the curvature.&#xA;  Similarly for all other surfaces, where it can even happen that in one&#xA;  direction the curvature might be convex and in another concave, as for&#xA;  surfaces which resemble a saddle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1697" OwnerDisplayName="Carlo Beenakker" LastActivityDate="2015-01-07T21:57:17.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1801" PostTypeId="1" AcceptedAnswerId="1802" CreationDate="2015-01-10T18:14:58.083" Score="6" ViewCount="660" Body="&lt;p&gt;The first know use of $0$ as it's own number was in India, but what was the equation in which it was used? Also, what was the tablet/scroll/whatever about? &lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2015-04-02T18:29:37.500" Title="First use of zero as a number" Tags="|mathematics|zero|ancient-india|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1802" PostTypeId="2" ParentId="1801" CreationDate="2015-01-10T18:19:48.087" Score="6" Body="&lt;p&gt;I touched upon this in my answer &lt;a href=&quot;https://hsm.stackexchange.com/questions/276/when-was-zero-actually-introduced-in-mathematics/277#277&quot;&gt;here&lt;/a&gt;, near the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This source&lt;/a&gt; indicates that one of the earliest known instances - if not the earliest known instance - is a carving on a tablet discussing gardens in the town of Gwalior. The page says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/samplings/feature-column/fcarc-india-zero&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; shows images of the tablet and gives the inscription as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Om. Adoration to Vishnu! In the year 933, on the second day of the bright half of the month of Magha the whole town gave to the temple which Alla, the son of Vaillabhatta, had caused to be built a piece of land 270 hastas in length and 187 hastas in breadth, for a flower garden the town gave in perpetual endowment for a daily gift of 50 garlands of flowers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While the tablet is from 876 A.D. yet refers to a year 933, this is explainable because the ancient Indians did not use the A.D./B.C. calendar used widely today.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-01-10T19:59:26.343" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1803" PostTypeId="1" CreationDate="2015-01-11T05:01:17.190" Score="7" ViewCount="779" Body="&lt;p&gt;The first (documented) hot-air balloon was made in the late 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The materials necessary (wood to burn, sheets for balloon) were around for thousands of years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why didn't anyone think of this invention earlier?&lt;/p&gt;&#xA;" OwnerUserId="1698" LastActivityDate="2015-02-07T17:55:35.543" Title="Why were hot-air balloons invented so late?" Tags="|discoveries|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1804" PostTypeId="2" ParentId="1801" CreationDate="2015-01-11T18:19:56.657" Score="1" Body="&lt;p&gt;Amir Aczel has a new book [&quot;Finding Zero&quot;]. There he finds zero on an inscription from 7th century Cambodia. Perhaps this is earlier than in India.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(My remark copied from that &lt;a href=&quot;https://hsm.stackexchange.com/q/276/229&quot;&gt;other question&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-01-11T18:19:56.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1805" PostTypeId="1" AcceptedAnswerId="1808" CreationDate="2015-01-11T20:57:59.847" Score="12" ViewCount="3072" Body="&lt;p&gt;This is a pretty straightforward question, when the first observations of the night sky were being made, who was the first person to suggest that a planet, say Mars, was not a star, in the sense that their physical compositions were different? I thought Tycho Brahe, but he lived in the 16th century, whilst a large part of such observations was made in ancient Greece, so could it be someone else?&lt;/p&gt;&#xA;" OwnerUserId="1700" LastEditorUserId="1700" LastEditDate="2015-01-14T10:48:13.607" LastActivityDate="2019-05-05T23:22:30.433" Title="Who first distinguished planets from the stars?" Tags="|astronomy|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1806" PostTypeId="2" ParentId="1805" CreationDate="2015-01-11T21:49:17.320" Score="5" Body="&lt;p&gt;I'm not an expert in the field, but I'll give it a go. Unfortunately, my only source is wikipedia, since I learned most of these things from encyclopedias years ago. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The distinction was certainly made centuries before Tycho Brahe. My humble research hasn't been able to exactly pin down who were the first to observe the difference, but I'd bet Babylonians were already aware of it. In &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_astronomy#Mesopotamia&quot;&gt;this wikipedia article&lt;/a&gt;, we can read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The oldest significant astronomical text that we possess is Tablet 63 of the &lt;em&gt;Enūma Anu Enlil&lt;/em&gt;, the Venus tablet of Ammi-saduqa, which lists the first and last visible &lt;strong&gt;risings of Venus over a period of about 21 years and is the earliest evidence that the phenomena of a planet were recognized as periodic&lt;/strong&gt;. The MUL.APIN, contains catalogues of stars and constellations as well as schemes for predicting heliacal risings and the settings of the planets, lengths of daylight measured by a water clock, gnomon, shadows, and intercalations. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which strongly suggests that Babylonians knew that there two different luminious objects in the night sky: planets and fixed stars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &lt;em&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Planet&quot;&gt;planet&lt;/a&gt;&lt;/em&gt; comes from greek πλανήτης, which means wandering (or errant, don't know which word is better). This is evidence of the greeks knowing about the difference, which is confirmed in the (very complete) &lt;a href=&quot;http://en.wikipedia.org/wiki/Fixed_stars#Origin_of_name&quot;&gt;fixed star Wikipedia article&lt;/a&gt;, where we particularly read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;The phrase [fixed star or &lt;em&gt;stellae fixae&lt;/em&gt;] originated in classical antiquity, when astronomers and natural philosophers divided the lights in the sky into two groups.&lt;/strong&gt; One group contained the fixed stars, which appear to rise and set but keep the same relative arrangement over time. The other group contained the naked eye planets, which they called wandering stars.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt; Even though evidence strongly suggests Babylonians were already aware of the difference around 1000 BC, when the &lt;a href=&quot;http://en.wikipedia.org/wiki/Enuma_anu_enlil&quot;&gt;&lt;em&gt;Enuma anu enlil&lt;/em&gt;&lt;/a&gt; was written, we can be sure the differentiation was made sometime between this date and classical antiquity, which ended around 100 BC.&lt;/p&gt;&#xA;" OwnerUserId="520" LastActivityDate="2015-01-11T21:49:17.320" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1807" PostTypeId="2" ParentId="1799" CreationDate="2015-01-11T23:11:45.577" Score="8" Body="&lt;p&gt;Euler was most certainly not motivated by matrices and eigenvalues, the chain of causation goes the other way. Sylvester only introduced the term &amp;quot;matrix&amp;quot; to denote an array of numbers in 1850, and did not do much with them. Only Cayley discovered in 1857 that each matrix satisfies equation of its own order, which led to &amp;quot;characteristic values&amp;quot;. However, already in 1829 Cauchy &amp;quot;proved that symmetric matrices have real eigenvalues&amp;quot;. How did he accomplish such a feat? What he actually proved was that any quadratic form can be reduced to a weighted sum of squares by a linear change of variables. Cauchy even used Lagrange multipliers to find extremal values of quadratic forms subject to normalization constraint, and obtained equation for the &amp;quot;characteristic values&amp;quot; identical to Cayley's for the corresponding matrix. More on history of matrices and eigenvalues in Victor Katz's article in &lt;a href=&quot;https://books.google.com/books?id=gqGLoh-WYrEC&amp;amp;dq=learn+from+the+masters&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Learn from the Masters&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now back to Euler. Cauchy considered quadratic forms in any number of variables, but the case of two and three was known long before that because they correspond to conics and quadrics. Euler's &lt;a href=&quot;https://math.dartmouth.edu/%7Eeuler/docs/translations/E333.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Curvature of Surfaces paper&lt;/a&gt; appeared in 1760, but 12 years earlier in an appendix to the second volume of the famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Introductio_in_analysin_infinitorum&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introductio in Analysin Infinitorum&lt;/a&gt; Euler showed how to reduce equations of quadrics to the simplest form, and gave explicit but cumbersome formulas for the transformations that accomplish that.&lt;/p&gt;&#xA;&lt;p&gt;So the quadratic expression in sines and cosines in the denominator of the &amp;quot;osculatory radius&amp;quot; (i.e. the curvature) would have been familiar to him from even simpler case of conics in polar coordinates. He knew that ellipses and hyperbolas have perpendicular principal axes, that their equations look simplest with those as coordinate axes, and how to change the coordinates. This also explains the word &amp;quot;principal&amp;quot;, now attached to directions and curvatures. In the paper Euler finds the extrema the modern way, by taking the derivative and setting it equal to zero, and then transforms coordinates by two reflections, rather than a single rotation used today.&lt;/p&gt;&#xA;&lt;p&gt;We should also add that curvature of curves was first introduced in 1659 by Huygens via osculating circles to construct evolutes for his ideal pendulum, and was published in his famous Horologium Oscillatorum of 1673 that Euler was certainly familiar with. Jerry Lodder's &lt;a href=&quot;http://www.math.nmsu.edu/hist_projects/curvature.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Curvature in the Calculus Curriculum&lt;/a&gt; (&lt;a href=&quot;https://web.archive.org/web/20180722030034/http://www.math.nmsu.edu/hist_projects/curvature.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wayback Machine&lt;/a&gt;) gives a step by step breakdown of both Huygens's and Euler's curvature computations along with insightful commentary.&lt;/p&gt;&#xA;&lt;p&gt;What Euler suspected before starting the computation calls for speculation of course, so I will entertain some. If we take a generic point and zoom in on its vicinity it will approximately look like a paraboloid, elliptic or hyperbolic, standing on its vertex. In modern notation we get its equation by keeping terms up to order two in the Taylor expansion of &lt;span class=&quot;math-container&quot;&gt;$z=f(x,y)$&lt;/span&gt;. Cross-sections parallel to the surface will then be approximately ellipses or hyperbolas, and mentally fitting osculating circles perpendicular to the surface at the vertex suggests, with elliptic one at least, that they have the largest and the smallest radii along their principal axes. That Euler had this in mind is supported by the fact that after considering curves obtained by intersecting a surface with an arbitrary plane in general coordinates, he switches to normal planes and chooses &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; in the tangent plane to the surface. Of course, this intuition was made fully explicit only much later in &lt;a href=&quot;https://en.wikipedia.org/wiki/Dupin_indicatrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dupin indicatrix&lt;/a&gt;, but considering Euler's facility with infinitesimals and quadrics it is not too far fetched  that he had some idea of what to expect from geometry.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="205" LastEditDate="2022-12-29T10:43:14.590" LastActivityDate="2022-12-29T10:43:14.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1808" PostTypeId="2" ParentId="1805" CreationDate="2015-01-12T14:53:54.943" Score="8" Body="&lt;p&gt;We can try to find the necessary informations browsing some &lt;em&gt;History of Astronomy&lt;/em&gt; books, like :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Anton Pannekoek, A History of Astronomy (1961, original ed : 1961)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Christopher Linton, From Eudoxus to Einstein A History of Mathematical Astronomy (2004)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and the &quot;classical&quot; studies of Otto Neugebauer on ancient science and astronomy, mainly : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Otto Neugebauer, The Exact Sciences in Antiquity (2nd ed, 1957).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think that it is necessary to separate at least three different &quot;stages&quot; : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(i)&lt;/strong&gt; the &quot;planets&quot; have irregular motions (the retrograde ones) compared to the movement of the &quot;fixed stars&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is due to &lt;a href=&quot;https://books.google.it/books?id=JVhTtVA2zr8C&amp;amp;pg=PA97&quot; rel=&quot;noreferrer&quot;&gt;Babylonian astronomy&lt;/a&gt;; see &lt;a href=&quot;https://books.google.it/books?id=aJuwFLGWKF8C&amp;amp;pg=PA9&quot; rel=&quot;noreferrer&quot;&gt;Lipton&lt;/a&gt;, page  : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ancient astronomers were also aware that five of the star-like objects in&#xA;  the sky changed their position relative to the other stars. These five objects –&#xA;  now named after the Roman gods Mercury, Venus, Mars, Jupiter and Saturn –&#xA;  are the planets, from the Greek for ‘wanderer’. Careful observations of these&#xA;  objects reveal that, like the Sun, as well as participating in the daily rotation of&#xA;  the heavens, they, too, move around the celestial sphere though with differing&#xA;  periods, and also that while they move predominantly in the same direction as the&#xA;  Sun – from west to east – they sometimes switch back and, for a time, move from&#xA;  east to west in so-called &lt;em&gt;retrograde&lt;/em&gt; motions. [...] The planets also remain close to the ecliptic, the maximum deviation for any of them being 8◦, and thus all the wandering heavenly bodies can be found within a strip on the celestial sphere 16◦ across centred on the ecliptic. This strip,therefore, is very important, and is known as the &lt;em&gt;zodiac&lt;/em&gt; and was divided by the Babylonians into twelve equal parts: the signs of the zodiac. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Despite their astronomical and mathematical ability, it seems to me that their is no trace of &quot;physical&quot; Babylonian theorie regerding the nature of &quot;celestial objects&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(ii)&lt;/strong&gt; some planet, like the Moon, does not shine by his own light (compared to the Sun and the stars); see &lt;a href=&quot;https://books.google.it/books?id=O7D9AyU-nLYC&amp;amp;pg=PA100&quot; rel=&quot;noreferrer&quot;&gt;Pannekoek&lt;/a&gt;, page 100 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[the pre-socratic philosopher] Anaxagoas of Clazomenae (c.500-428 BC) [...] was the first to state clearly that the moon shines by the light it receives from the sun and that lunar eclipses occur when the earth (or another dark body) intercepts the sun's light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also Lipton, page 21 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He [Anaxagoras] was the first to think of the seven wandering heavenly bodies in the order Sun, Moon, followed by the five other planets, an arrangement adopted by a number of later astronomers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(iii)&lt;/strong&gt; The last stage is the understanding of the &quot;physical&quot; difference between the fixed stars and others &quot;celestial objects&quot;, like the planets and the comets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This process occurred during the Renaissance, starting with the physical-cosmological debate of the 1580s, stimulated by the apparition of a &lt;a href=&quot;http://en.wikipedia.org/wiki/SN_1572&quot; rel=&quot;noreferrer&quot;&gt;nova in Cassiopeia in 1572&lt;/a&gt;, the comets of 1577, 1580, 1582 and 1585, and followed by the &lt;a href=&quot;http://en.wikipedia.org/wiki/Kepler%27s_Supernova&quot; rel=&quot;noreferrer&quot;&gt;nova of 1604&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Miguel A.Granada, Adam Mosley, Nicholas Jardine &lt;a href=&quot;https://books.google.it/books?id=LUv3AwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Christoph Rothmann's Discourse on the Comet of 1585&lt;/a&gt; (2014)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Patrick Boner, &lt;a href=&quot;https://books.google.it/books?id=xPq0Bv5Kz4EC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Change and Continuity in Early Modern Cosmology&lt;/a&gt; (2011).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This debate culminated with Galileo's telescopic observation of the moon and of the phases of Venus (1610), and of the sunspots (1612)  :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Galileo Galilei &amp;amp; Christoph Scheiner, &lt;a href=&quot;https://books.google.it/books?id=vdDhfPxIcLQC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;On Sunspot&lt;/a&gt; (translated and edited by Eileen Reeves &amp;amp; Albert Van Helden - 2010).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-01-13T15:12:41.880" LastActivityDate="2015-01-13T15:12:41.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1809" PostTypeId="2" ParentId="1803" CreationDate="2015-01-13T03:34:00.880" Score="12" Body="&lt;p&gt;Hot air balloons that can carry people require very large amounts of fabric.  They were first created a few decades after the &quot;flying shuttle&quot; made it possible for a single weaver to weave cloth wider than the weaver's arms -- and then made mechanized looms possible.  On the shuttle and the looms see &lt;a href=&quot;https://en.wikipedia.org/wiki/Loom&quot;&gt;https://en.wikipedia.org/wiki/Loom&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hugely increased availability of fabric has to be one reason the balloons were first made at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am assuming hot air balloons were not used to create the Nazca Lines in Peru 1500 years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related issue: parachutes were clearly conceived centuries before hot air balloons carried people, and probably some were actually used in jumps from high places.  But their well documented use and modern development began when people used them with hot air balloons.     See &lt;a href=&quot;https://en.wikipedia.org/wiki/Parachute&quot;&gt;https://en.wikipedia.org/wiki/Parachute&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2015-02-07T17:55:35.543" LastActivityDate="2015-02-07T17:55:35.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1810" PostTypeId="1" CreationDate="2015-01-07T10:42:30.643" Score="7" ViewCount="148" Body="&lt;p&gt;The fact that a &lt;a href=&quot;https://en.wikipedia.org/wiki/Lorentz_force&quot;&gt;Lorentz force&lt;/a&gt; in a reference frame 1 can become a &lt;a href=&quot;https://en.wikipedia.org/wiki/Coulomb&amp;#39;s_law&quot;&gt;Coulomb force&lt;/a&gt; in another reference frame 2 has always astonished me, especially because the velocities involved are really small. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder: who was the first to discuss this peculiar interpretation (or effect) of Lorentz transformations of the EM-field?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://en.wikipedia.org/wiki/Relativistic_electromagnetism&quot;&gt;this Wikipedia article&lt;/a&gt; a letter from Einstein is mentioned in which he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, this letter dates from 1953. I wonder whether Einstein really discussed this in his earlier articles? Did he not mainly discuss the effects in space-time-coordinates of an invariant and finite speed of light? Further I noted that a publication by E. Purcell from 1965 is often referenced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there also publications known from the beginning of the 20th century? So, not publications that deal with Lorentz-transformations of the EM-field in general, but that specifically discuss changes in charge distiributions due to length contraction etc.&lt;/p&gt;&#xA;" OwnerUserId="1705" OwnerDisplayName="Gerard" LastEditorUserId="62" LastEditDate="2015-01-13T23:19:48.550" LastActivityDate="2015-01-13T23:19:48.550" Title="Who first discussed the Lorentz force with respect to special relativity?" Tags="|physics|relativity-theory|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1811" PostTypeId="1" CreationDate="2015-01-13T15:00:40.313" Score="2" ViewCount="346" Body="&lt;p&gt;The Frobenius Perron Operator $P: L^1 \to L^1$ is defined by the integral equation&#xA;$$ \int_A Pf(x) \mu(dx) = \int_{S^{-1}(A)} f(x) \mu(dx)$$&#xA;for some $\mu$-non singular map $S$. I found it in the book from 1985 called 'Probabilistic Properties of Deterministic Systems'. But I am not sure who originally invented it. Is there any history known about this subject?&lt;/p&gt;&#xA;" OwnerUserId="1706" LastActivityDate="2022-11-30T05:30:54.110" Title="Origin of Perron-Frobenius Operator" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1812" PostTypeId="1" CreationDate="2015-01-13T16:05:40.990" Score="13" ViewCount="331" Body="&lt;p&gt;A while ago, in USSR there were published two very voluminous collections entitled&#xA;&quot;Mathematics in USSR for 30 years. 1917-1947&quot; and &#xA;&quot;Mathematics in USSR for 40 years. 1917-1957&quot;. These collections featured &#xA;surveys for various fields of mathematics (algebra, differential equations, etc.)&#xA; as well as &#xA;biographical and bibliographical data of virtually all Soviet mathematicians&#xA;of that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Throughout the book by A.D. Myshkis, &#xA;&quot;Soviet mathematicians. My reminiscences&quot;, URSS, Moscow, 2007 (in Russian),&#xA;it is mentioned several times that even a more voluminous &#xA;&quot;Mathematics in USSR. 1958-1967&quot; was planned, on which circa&#xA;200 people worked. However, writes Myshkis, the publication was blocked on a &#xA;rather later stage by a group of &#xA;&quot;influential members of USSR Academy of Sciences&quot; who were not&#xA;happy how their research is reflected in those surveys. And indeed, only the &#xA;second volume of &quot;Mathematics in USSR. 1958-1967&quot; was published, containing &#xA;biographical and bibliographical data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who blocked this publication? (At one place, Myshkis mentions the name&#xA;of Pontryagin, though rather equivocally).&lt;/p&gt;&#xA;" OwnerUserId="167" LastEditorUserId="4251" LastEditDate="2020-05-13T12:58:25.420" LastActivityDate="2020-05-13T12:58:25.420" Title="Who blocked publication of &quot;Mathematics in USSR. 1958-1967&quot;?" Tags="|mathematics|mathematicians|publishing|academia|soviet-union|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1813" PostTypeId="1" AcceptedAnswerId="1817" CreationDate="2015-01-13T22:23:44.787" Score="45" ViewCount="9754" Body="&lt;p&gt;&lt;a href=&quot;http://xahlee.info/math/i/Alexander_Grothendieck_cartier.pdf&quot; rel=&quot;noreferrer&quot;&gt;Grothendieck&lt;/a&gt; is arguably the most brilliant mathematician of the 20th century, with his influence felt the most in algebraic geometry, which he transformed. Some time ago the story used to be told was that Grothendieck put it on a rigorous footing after the Italian School starting in the 19-th century (Castelnuovo) gradually neglected rigor (Enriques), and ended up with results which no one could tell were true or not in 1940-50s (Severi). The actual story is much &lt;a href=&quot;http://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry&quot; rel=&quot;noreferrer&quot;&gt;more nuanced&lt;/a&gt;, but more to the point I eventually learned, much to my surprise, that Zariski, who was originally part of the Italian School, left it and published a &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/354058658X&quot; rel=&quot;noreferrer&quot;&gt;classical book&lt;/a&gt;, which was already up to the modern standards of rigor, back in 1935. There is an insightful discussion of the Italian School and its demise in this &lt;a href=&quot;https://mathoverflow.net/questions/19420/what-mistakes-did-the-italian-algebraic-geometers-actually-make&quot;&gt;MO thread &lt;/a&gt;, where one of the commenters quipped: &quot;Grothendieck didn't make algebraic geometry rigorous. Weil and Zariski made algebraic geometry rigorous&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know of Grothendieck's technical innovations (like schemes, motives and topoi), and his achievements (like proving the Weil conjectures), but it is hard to put them into context. What was his role &quot;structurally&quot;? To give an analogy, in set theory we have Cantor exploring a new territory, but running into logical troubles (roughly like the Italian School), and then Zermelo making the set theory rigorous (roughly like Weil and Zariski). The next towering figure is perhaps Gödel, and he transformed set theory by going to a &quot;meta&quot; level, because some deeper problems, like continuum hypothesis or the status of axiom of choice, could not be resolved in Zermelo's framework. Similarly, Weierstrass's analysis was already rigorous but some major issues (like Lebesgue integration) could not be advanced without going to the next level of abstraction. My impression is that Grothendieck's role was similar to that, but the subject is so technical that for a non-expert it's hard to move beyond vague impressions and analogies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the analogy correct? Can somebody explain in accessible terms what was still lacking after &quot;Weil and Zariski made algebraic geometry rigorous&quot;, and why Grothendieck's contribution allowed such rapid progress after him? Why did AG need AG?&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2021-09-13T09:05:23.773" LastActivityDate="2022-01-21T07:09:48.910" Title="Why did algebraic geometry need Alexander Grothendieck?" Tags="|mathematics|algebraic-geometry|grothendieck|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1814" PostTypeId="2" ParentId="1812" CreationDate="2015-01-13T22:34:49.900" Score="15" Body="&lt;p&gt;I cannot name a person. But I can describe what probably happened. (My PhD adviser, A. Goldberg was one of the authors of the second volume (1958-1967), and he told me the story).&#xA;At some point, this publication was criticized by the people who stood much higher in the Soviet hierarchy than the editorial board. Such decisions were usually made secretly by some&#xA;Communist party body. Goldberg cites a report signed by Yu. A. Buslaev (a full member of the Soviet Academy of Science, but not a mathematician). The person who signed the report is not necessarily the person who made the decision. I remember one sentence from this report:&#xA;&quot;The history of Soviet mathematics is reflected in the book like in a curved mirror&quot;.&#xA;It was clear that they did not like the prominent role of mathematicians of Jewish origin &#xA;in the book, though this was never stated explicitly. I do not know any evidence that&#xA;Pontriagin played any role in this. I suppose that the command to stop the publication was issued&#xA;by much higher authorities in the Communist party.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. Actually, 1958-67 was planned in several volumes, and the first of them was already published when the whole project was abandoned. I had it once in my possession. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-04T21:45:41.717" LastActivityDate="2020-05-04T21:45:41.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1815" PostTypeId="2" ParentId="1811" CreationDate="2015-01-13T22:56:21.727" Score="2" Body="&lt;p&gt;The earliest occurrence of this name in Mathscinet database is the paper&#xA;MR0486428 &#xA;Walters, Peter&#xA;A generalized Ruelle Perron-Frobenius theorem and some applications. International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), pp. 183–192. Asterisque, No. 40, Soc. Math. France, Paris, 1976. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The operator itself was introduced by D. Ruelle (Comm. Math. Phys. 9 (1968), 267–278).&#xA;I am not sure whether Ruelle himself&#xA;used this name for it. Some people call it the Ruelle-Perron-Frobenius operator.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-01-13T22:56:21.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1816" PostTypeId="1" AcceptedAnswerId="1820" CreationDate="2015-01-14T14:54:17.723" Score="10" ViewCount="18683" Body="&lt;p&gt;When I was in 9th grade, I learned that Aristotle was responsible for holding back physics for centuries because he said that heavier objects fall faster than lighter objects. Finally, in the 16th century Galileo disproved this theory by dropping two balls of different masses from the Leaning Tower of Pisa showing that they both fell at the same speed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And when I took physics in 12th grade, I learned that Newton's Law of Gravitation explains the results of Galileo's experiment, showing that the acceleration of an object near the earth's surface is always the same $g=GM/R^2=9.80 m/s^2$, where $G$ is the gravitational constant, $R$ is the distance of the object to the center of the earth, and $M$ is the mass of the earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This all seemed to conclusively disprove Aristotle's theory that heavier objects fall faster than lighter objects. However, this line of argument neglects to consider Newton's Third Law, which implies that the falling object forces the earth to move at acceleration proportional to the mass of the falling object. And this will cause the distance between the falling object and the center of the earth to decrease faster for heavier falling objects, implying that the heavier objects do in fact fall faster than lighter objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is why was I taught that Aristotle was completely wrong when his prediction seems to be totally in agreement with Newtonian mechanics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added: If you don't believe me, just check out the differential equations obtained from Newton's Law of Gravitation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$MR''=GmM/|R-r|^2$ and $mr''=-GmM/|r-R|^2$,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $m$ is the mass of the object, $M$ is the mass of the earth, $R$ is the position of the earth, $r$ is the position of the object. Making it simpler, we get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$R''=Gm/|R-r|^2$ and $r''=-GM/|r-R|^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When $m$ is large, $R''$ is large, implying that $|R'-r'|$ becomes large faster than when $m$ is small, implying that the object will eventually move faster towards the earth when $m$ is large than when $m$ is small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See a similar question here: &lt;a href=&quot;https://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity&quot;&gt;https://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="631" LastEditorUserId="4251" LastEditDate="2021-03-03T07:32:20.893" LastActivityDate="2021-03-08T05:26:04.420" Title="Was Aristotle really wrong about gravity?" Tags="|physics|ancient-greece|gravity|aristotle|" AnswerCount="11" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="1817" PostTypeId="2" ParentId="1813" CreationDate="2015-01-14T22:19:35.170" Score="63" Body="&lt;p&gt;Just prior to Grothendieck's entry to the subject, Weil had gotten terrific results in number theory by algebro-geometric arguments, and pointed the way to far more, but some of his methods went beyond existing rigorous foundations. He aimed to supply new foundations adequate to his ideas.  Around the same time Zariski and van der Waerden were also generalizing the foundations of algebraic geometry, and many others were introducing various innovations.&lt;/p&gt;&#xA;&lt;p&gt;In particular the &amp;quot;Weil conjectures&amp;quot; suggested that topological methods of cohomology applied to algebraic geometry might have huge consequences for number theory -- but neither Weil nor anyone else in say 1954 saw how to actually do that.&lt;/p&gt;&#xA;&lt;p&gt;Serre believed more strongly than Weil himself did that such a cohomology could actually be created. He recruited Grothendieck to the project.&lt;/p&gt;&#xA;&lt;p&gt;Grothendieck gave two huge simplifications.&lt;/p&gt;&#xA;&lt;p&gt;First simplification: cohomology in 1954 had numerous alternative forms in topology, geometry, and analysis. Believe me, you do not want the details, but it could be defined by covering spaces, or differential forms, or Čech covers, or simplicial decomposition, or injective resolutions, or flabby resolutions, or bar resolutions, and much more. These had numerous complicated relations to one another.  Grothendieck invented the now-current ideas of Abelian category and derived functor so that essentially all these cohomology theories used one simple kind of resolution: the injective.&lt;/p&gt;&#xA;&lt;p&gt;The other resolutions are all still used today for calculations but Grothendieck's framework is the standard and the others are organized around it. It has led to creating very many more alternative cohomologies.&lt;/p&gt;&#xA;&lt;p&gt;Second simplification: schemes. By mid 20th century leading algebraic geometers all saw that algebraic geometry needed a notion of algebraic space more general than spaces defined by polynomial equations over the complex numbers. Many people gave examples, led by Weil and notably Serre. In these, algebraic spaces could have coordinate functions much more general than polynomials with complex coefficients. As KConrad notes they all stayed fairly close to taking coordinates in fields, generally in algebraically closed fields.&lt;/p&gt;&#xA;&lt;p&gt;This was a serious obstacle to Weil's hopes for algebraic geometry as a tool in number theory, since the integers do not form a field at all. The rational numbers form a field but it is as far from algebraically closed as you can be -- if an integer polynomial with lead coefficient 1 has no integer roots, then it has no rational roots either, but of course it has as many complex roots as its degree.&lt;/p&gt;&#xA;&lt;p&gt;Grothendieck defined &amp;quot;schemes&amp;quot; so that every ring is the coordinate ring of a scheme. Rings that seem to be purely algebraic abstract entities still have geometry meaning in this theory. Some schemes by themselves have little apparent geometric sense, but they have eminently geometric relations to one another. You do not want the details but I give you Deligne's summary of it: &amp;quot;if the decision to let every commutative ring define a scheme gives standing to bizarre &lt;em&gt;schemes&lt;/em&gt;, allowing it gives a &lt;em&gt;category&lt;/em&gt; of schemes with nice properties&amp;quot; (Deligne, &amp;quot;Quelques idees maıtresses de l’œuvre de A. Grothendieck&amp;quot; 1998, p. 13 (his emphasis) see &lt;a href=&quot;http://www.ihes.fr/%7Elandais/smf_sem-cong_3_11-19.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.ihes.fr/~landais/smf_sem-cong_3_11-19.pdf&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;You can define schemes without knowing what polynomials are, let alone knowing what &amp;quot;algebraically closed&amp;quot; means. In lightly over-simplified terms, you only need to know the commutative, associative, and distributive laws so that you can learn what a commutative ring is, and you can define a scheme.&lt;/p&gt;&#xA;&lt;p&gt;That does not make scheme theory entirely easy. In fact it makes scheme theory very general so that there are vast numbers of important special cases to learn. In practice virtually all algebraic geometers who do not want to just use polynomials on the complex numbers, use scheme theory.&lt;/p&gt;&#xA;&lt;p&gt;Schemes are important even for geometry over the complex numbers. Ancient Greek geometers already debated whether or not a circle tangent to a line meets it in a &amp;quot;bigger&amp;quot; point than when two lines cross. In scheme theory the answer is simply &amp;quot;yes.&amp;quot; With schemes a point of tangency includes an infinitesimal neighborhood of an ordinary point. These schemes are not just cute but have a lot of uses in studying singularities of spaces, and degeneracies in families of spaces, and group actions on spaces, and they use the non-zero nilpotents that KConrad mentions.&lt;/p&gt;&#xA;&lt;p&gt;These two ideas, derived functor cohomology, and schemes, let Grothendieck pick up another idea of Serre's (&amp;quot;unramified&amp;quot; covers, on which &amp;quot;isotrivial&amp;quot; fiber spaces become trivial) and turn it into &amp;quot;étale cohomology of schemes,&amp;quot; which with Deligne's help fulfilled the goal of a cohomological proof for the Weil conjectures. Étale cohomology has also turned out to be the right setting for much more geometry and number theory, and it has spun off many variants to do much more geometry. Étale cohomology also led Grothendieck to create toposes, which have spun off into many other quite unrelated uses.&lt;/p&gt;&#xA;&lt;p&gt;Grothendieck originally saw the &amp;quot;étale topos&amp;quot; of a given scheme as an expression of that scheme as a &amp;quot;generalized topological space.&amp;quot; It was a category organizing all the basic data on the space, and Grothendieck came to see it as a mathematical world like the universe of sets, where you can do analogues of all of standard mathematics but it is intrinsically adapted to that one geometric scheme.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Motives&amp;quot; for Grothendieck were a further radical simplification of cohomology that would give the same useful information with even less apparatus -- but that theory would depend on new basic theorems that have not yet been proved today. There has been a lot of work on this but so far as I can see the ideas are very far from taking any stable form.&lt;/p&gt;&#xA;&lt;p&gt;It is entirely impossible now to do justice to Grothendieck's contribution in a few words. I will enter these few words in hopes of helping you see what he did.&lt;/p&gt;&#xA;&lt;p&gt;Much of this answer is now expanded as &lt;a href=&quot;http://www.ams.org/publications/journals/notices/201603/rnoti-p256.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.ams.org/publications/journals/notices/201603/rnoti-p256.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="16082" LastEditDate="2022-01-21T07:09:48.910" LastActivityDate="2022-01-21T07:09:48.910" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="1818" PostTypeId="2" ParentId="1816" CreationDate="2015-01-14T23:58:29.493" Score="18" Body="&lt;p&gt;In short, you were taught that Aristotle was wrong because he was wrong. He didn't make a prediction, he made an observation about rock and feather, and then sloppily generalized it to all objects without a second thought. The subtle effects you are describing weren't even noticable in his time, but that a feather falls slower because it is much more affected by air resistance, would have been obvious to sailors, or anyone who dealt with winds, even then. Already in antiquity John Philoponus &lt;a href=&quot;http://en.wikipedia.org/wiki/John_Philoponus&quot;&gt;pointed out&lt;/a&gt; that if one corrects for that the sole basis for Aristotle's conclusion disappears: &quot;&lt;em&gt;But this is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you were taught that Aristotle was &quot;holding back science&quot; not because he was just wrong about falling bodies. As Philoponus pointed out, it was merely an illustration of  a general attitude, unfortunately adopted by many after him, that facts about nature can be reasoned out of their heads with spotty and misconstrued observations, if any at all. To be fair, Aristotle's contribution wasn't all negative, he gave first systematic descriptions in what now became established natural sciences, and tried to organize and structure what was known about the world in his time. But his method of inquiry was wrong headed, and it took a lot of time and effort to &lt;a href=&quot;http://en.wikipedia.org/wiki/Baconian_method&quot;&gt;overcome&lt;/a&gt; it. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-01-15T00:20:39.627" LastActivityDate="2015-01-15T00:20:39.627" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1819" PostTypeId="2" ParentId="728" CreationDate="2015-01-15T06:01:55.560" Score="28" Body="&lt;p&gt;I just want to comment that the agreement on letters, by which we write $\frac d{dt}\mathbf L=\mathbf M$ for the law of angular momentum, must have come very late -- after 1964. As evidence, note that it is still written&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac d{dt}\mathfrak N=\mathfrak M$ by Sommerfeld in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=4jVWAAAAMAAJ&amp;amp;q=%22Diese+Gleichung+ist+das+genaue%22&quot; rel=&quot;noreferrer&quot;&gt;Mechanik&lt;/a&gt;&lt;/em&gt; (1943, p.63);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac d{dt}\mathbf M=\mathbf L$ by Sommerfeld in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=pbs3BQAAQBAJ&amp;amp;pg=PA72&quot; rel=&quot;noreferrer&quot;&gt;Mechanics&lt;/a&gt;&lt;/em&gt; (1952, p.72);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac d{dt}\mathbf P=\mathbf M$ by Joos in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=wFl2MkpcY6kC&amp;amp;pg=PA110&quot; rel=&quot;noreferrer&quot;&gt;Theoretical Physics&lt;/a&gt;&lt;/em&gt; (3rd edition, 1958, p.110);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac d{dt}\mathbf M=\mathbf K$ by Landau-Lifshitz in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=YxkIAQAAIAAJ&amp;amp;q=dM&quot; rel=&quot;noreferrer&quot;&gt;Mechanics&lt;/a&gt;&lt;/em&gt; (1960, p.108);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac d{dt}\mathbf H=\mathbf L$ by Truesdell (quoting Joos) in &lt;em&gt;&lt;a href=&quot;http://link.springer.com/chapter/10.1007/978-3-642-86647-0_5&quot; rel=&quot;noreferrer&quot;&gt;Whence the Law of Moment of Momentum?&lt;/a&gt;&lt;/em&gt; (1964); &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\frac d{dt}\mathbf M=\mathbf L$ by Truesdell in &lt;em&gt;&lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1002/zamm.19640440402/abstract&quot; rel=&quot;noreferrer&quot;&gt;Die Entwicklung des Drallsatzes&lt;/a&gt;&lt;/em&gt; (1964), which is a translation of 5.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thus, it's likely going to be  impossible to attribute the choice of $\mathbf L$ to anyone in particular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; Much more promising seems to be the hypothesis that the choice was first made in &lt;em&gt;quantum&lt;/em&gt; mechanics, to denote orbital angular momentum. As evidence, we have the following nice unanimity:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;$L_x = \frac1i\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in Weyl, &lt;a href=&quot;https://archive.org/stream/quantumtheoryofa029235mbp#page/n211&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Gruppentheorie und Quantenmechanik&lt;/em&gt;&lt;/a&gt; (1928, p.167);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$L_x = -\frac1i\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in Eckart, &lt;a href=&quot;http://dx.doi.org/10.1103/RevModPhys.2.305&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Application of Group theory...&lt;/em&gt;&lt;/a&gt; (1930, p.350);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\pmb{\mathsf L}_z = \frac1i\bigl(y\frac\partial{\partial x}-x\frac\partial{\partial y}\bigr)$ in Wigner, &lt;a href=&quot;http://dx.doi.org/10.1007/978-3-663-02555-9&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Gruppentheorie und ihre Anwendung...&lt;/em&gt;&lt;/a&gt; (1931, p.219);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$L_x = \frac1i\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in van der Waerden, &lt;a href=&quot;http://dx.doi.org/10.1007/978-3-662-02187-3&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Die gruppentheoretische Methode...&lt;/em&gt;&lt;/a&gt; (1932, p.19);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\pmb l_3 = \frac1i\bigl(x_1\frac\partial{\partial x_2}-x_2\frac\partial{\partial x_1}\bigr)$ in Pauli, &lt;a href=&quot;http://dx.doi.org/10.1007/978-3-642-52619-0_2&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Die allgemeinen Prinzipien der Wellenmechanik&lt;/em&gt;&lt;/a&gt; (1933, p.185);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$L_x = \frac h{2\pi i}\bigl(y\frac\partial{\partial z}-z\frac\partial{\partial y}\bigr)$ in Bauer, &lt;a href=&quot;http://www.numdam.org/item?id=AIHP_1933__4_1_1_0&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Introduction à la théorie des groupes...&lt;/em&gt;&lt;/a&gt; (1933, p.38).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The point was, of course, that the eigenvalue of $L_x^2+L_y^2+L_z^2$ would relate to the &lt;a href=&quot;http://en.wikipedia.org/wiki/Azimuthal_quantum_number&quot; rel=&quot;noreferrer&quot;&gt;azimuthal quantum number&lt;/a&gt; already denoted $l$ — a letter choice which seems attributed by Sommerfeld &lt;a href=&quot;https://books.google.com/books?id=llMNAQAAIAAJ&amp;amp;q=%22Russell+and+Saunders+used+the+converse%22&quot; rel=&quot;noreferrer&quot;&gt;(1926)&lt;/a&gt; to Russell and Saunders, by Russell et al. &lt;a href=&quot;http://dx.doi.org/10.1103/PhysRev.33.900&quot; rel=&quot;noreferrer&quot;&gt;(1929)&lt;/a&gt; to Hund &lt;a href=&quot;http://dx.doi.org/10.1007/978-3-642-49831-2&quot; rel=&quot;noreferrer&quot;&gt;(1927, p.27)&lt;/a&gt;, and by Hund &lt;a href=&quot;http://dx.doi.org/10.1007/BF01328319&quot; rel=&quot;noreferrer&quot;&gt;(1925, p.347)&lt;/a&gt; to Heisenberg &lt;a href=&quot;http://dx.doi.org/10.1007/BF01331720&quot; rel=&quot;noreferrer&quot;&gt;(1925, p.850)&lt;/a&gt;...&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2015-02-13T02:12:59.413" LastActivityDate="2015-02-13T02:12:59.413" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1820" PostTypeId="2" ParentId="1816" CreationDate="2015-01-15T08:01:27.537" Score="24" Body="&lt;p&gt;&lt;em&gt;I'll try with some calculations : please, check it and the formulae used ...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A solid ball with a mass $m$ of $1$ kg falls (with the usual approxiamtions : no drag, etc.) with an acceleration $a$ that is about $10 \ m/sec^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that falling from a tower $80$ meters heigh, it will touch ground after $4$ sec, with a final velocity of about $40 \ m/sec$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are right : in the same time, the Earth will &quot;fall towards&quot; the ball, attracted by the same gravitational force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mass $M$ of the Earth is about $6 \times 10^{24} \ kg$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The acceleration $A$ of the Earth that is proportional to $a$ as the reciprocal of the masses; i.e. :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$A = a \times m/M = 10/(6 \times 10 ^{24}) \approx 2 \times 10 ^{-24} \ m/sec^2$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After $4$ seconds, the Earth reachs a velocity of fall of $8 \times 10 ^{-24} \ m/sec$ and it has traversed a space $s \approx 16 \times 10 ^{-24} \ m$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that, due to the reciprocal attraction, the two bodies will touch each other in slightly less than $4 \ sec$ and that the &quot;real&quot; space traversed by the falling ball with respect to the Earth is about : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$80$ meters minus the space traversed by the &quot;falling&quot; Earth during the short time of the fall.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It can be useful to recall that :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;the size of atoms is measured in &lt;em&gt;picometers&lt;/em&gt; : trillionths ($10 ^{-12}$) of a meter.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If the ball has a mass $m'$ of $1000 \ kg$, the force with which it &quot;pulls&quot; the Earth will be $1000$ times greater, producing an acceleration $A' \approx 2 \times 10 ^{-21} \ m/sec^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This implies that in this second case the space traversed by the Earth duing the fall of the heavier ball will be $s' \approx 16 \times 10 ^{-21} \ m$.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;According to Aristotle (&lt;em&gt;Physiscs&lt;/em&gt;, Book VII) the &quot;law&quot; of dynamics is :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;if a power $\alpha$ moves a body $\beta$ during time $\delta$ for a distance $\gamma$, then an equal power $\alpha$ will move a body half of $\beta$ along a distance twice as $\gamma$ in the same time&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In an anachronistic way, we can say :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$F \propto V$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, if we apply this law to &quot;our model&quot; of free fall, with the weight of the body as the force, we have that - assuming that after an initial short time of acceleration the falling body will reach a constant &quot;terminal velocity&quot; - in the second case the acquired speed $v'$must be $1000$ times the first one : $v$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that after $4$ seconds the heavier ball has traversed a space : $s'= v' \times t$, i.e. $s' = 1000 \times v \times t = 1000 \times s$, where $s$ is the space traversed by the lighter body after a fall of $4$ seconds.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Conclusion&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see the &quot;same factor&quot; : $1000$ acts in a completely different way in the two models.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to the huge mass of the Earth, for bodies of &quot;normal&quot; size (meaning body of our daily life experience) it has no &lt;strong&gt;experimentally verifiable&lt;/strong&gt; effect on the behaviour of different bodies falling due to the gravitational force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Aristotelian model, that factor has an evident &lt;strong&gt;experimentally verifiable&lt;/strong&gt; effect on the behaviour of different bodies falling due to the &quot;tendency towards the centre&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is exactly where is the &quot;conceptual&quot; difference : to perform some sort of experimental test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the answer to :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Was Aristotle really wrong about gravity?&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is a definitive &lt;strong&gt;YES&lt;/strong&gt; if we try to answer the question from the point of view of modern science, a point of view that is &lt;strong&gt;not&lt;/strong&gt; that of aristotelian &lt;em&gt;natural philosophy&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If instead we want to compare two different qualitative &quot;worldviews&quot;, things are different (see at least the philosophical debate involving : &lt;a href=&quot;http://plato.stanford.edu/entries/thomas-kuhn/&quot;&gt;Thomas Kuhn&lt;/a&gt;, &lt;a href=&quot;http://plato.stanford.edu/entries/incommensurability/&quot;&gt;The Incommensurability of Scientific Theories&lt;/a&gt;, &lt;a href=&quot;http://plato.stanford.edu/entries/scientific-revolutions/&quot;&gt;Scientific Revolutions&lt;/a&gt;, &lt;a href=&quot;http://plato.stanford.edu/entries/rationality-historicist/&quot;&gt;Historicist Theories of Scientific Rationality&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Imre_Lakatos&quot;&gt;Imre Lakatos&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/feyerabend/&quot;&gt;Paul Feyerabend&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Notes&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;i)&lt;/em&gt; In our computations we made approximations; &lt;em&gt;approximation&lt;/em&gt; is a modern concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without precise mathematical laws there are no possible approximations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle's &quot;natural laws&quot; are &lt;strong&gt;not&lt;/strong&gt; approximations in the modern sense; they are qualitative description of facts, like the one made by the same Aristotle on botany and zoology (whih were impressively accurate, by the way).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;ii)&lt;/em&gt; Writing the &quot;aristotelian equation&quot; we make an &quot;historical mistake&quot; : he never thinked in terms of mathematical formuale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please, note that the mathematics used by Galileo in his analysis of the free falling body problem was only the theory of proportions of Eudoxus/Euclid, that was already available in Aristotle's time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the &quot;tools&quot; available to ancient natural philosophers were more or less the same compared to the ones available to Galileo (not so with Newton ...).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Postscriptum&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above &quot;experiment&quot; was alredy discussed by Galileo, in advance of the correct formulation of the law of universal gravitation. See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Galileo Galilei, &lt;a href=&quot;https://books.google.it/books?id=n5cbjKmaZBwC&amp;amp;pg=PA64&quot;&gt;Dialogues concerning two new sciences&lt;/a&gt; (1638 - Engl tr Henry Crew &amp;amp; Alfonso de Salvio - Dover ed), pag.64 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;SALV. We infer therefore that large and small bodies move with the same speed provided they are of the same specific gravity.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SIMP. Your discussion is really admirable; yet I do not find it easy to believe that a bird-shot falls as swiftly as a cannon ball.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SALV. Why not say a grain of sand as rapidly as a grindstone? But, Simplicio, I trust you will not follow the example of many others who divert the discussion from its main intent and fasten upon some statement of mine which lacks a hair’s-breadth of the&#xA;  truth and, under this hair, hide the fault of another which is as big as a ship’s cable. Aristotle says that “an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.” I say that they arrive at the same time. You find, on making the experiment, that the larger outstrips the smaller by two finger-breadths, that is, when the larger has reached the ground, the other is short of it by two finger-breadths; now you would not hide behind these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one. Aristotle declares that bodies of different weights, in the same medium, travel (in so far as their motion depends upon gravity) with speeds which are proportional to their weights; this he illustrates by use of bodies in which it is possible to perceive the pure and unadulterated effect of gravity, eliminating other considerations, for example, figure as being of small importance, influences which are greatly dependent upon the medium&#xA;  which modifies the single effect of gravity alone.Thus we observe that gold, the densest of all substances, when beaten out into a very thin leaf, goes floating through the air; the same thing happens with stone when ground into a very fine powder.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;But if you wish to maintain the general proposition you will have to show that the same ratio of speeds is preserved in the case of all heavy bodies, and that a stone of twenty pounds moves ten times as rapidly as one of two; but I claim that this is false and that, if they fall from a height of fifty or a hundred cubits, they will reach the earth at the same moment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-01-26T14:12:49.857" LastActivityDate="2015-01-26T14:12:49.857" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1821" PostTypeId="2" ParentId="1816" CreationDate="2015-01-15T21:18:35.250" Score="8" Body="&lt;p&gt;That Aristotle (and you with your example) are wrong is proved by the following simple argument: imagine two bricks of equal mass. Each of them falls with certain acceleration.&#xA;Now glue them together and let them fall. According to Aristotle two bricks will fall&#xA;faster than each brick separately. It is evident that this is absurd: what difference&#xA;does it make whether the bricks are glued together or not?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose this argument is due to Galileo, but I am not 100% sure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually Aristotle's &quot;physics&quot; says that two bricks will fall twice faster than one brick.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-01-15T21:18:35.250" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1822" PostTypeId="2" ParentId="1816" CreationDate="2015-01-15T22:47:38.363" Score="11" Body="&lt;p&gt;Yes, Aristotle was wrong about gravity. But I think it is unfair to say “that Aristotle was responsible for holding back physics for centuries”. The ones who held back physics for centuries were the late-antique and mediaeval (Christian, Muslim and Jewish) so-called philosophers who transformed Aristotelianism into an ossified dogmatic doctrine. Aristotle himself was always willing to change his mind and to consider alternative explanations.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-01-15T22:47:38.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1824" PostTypeId="1" CreationDate="2015-01-17T09:36:59.853" Score="5" ViewCount="850" Body="&lt;p&gt;I'm interested to know what mathematical and physics topics Einstein learnt during his graduation.Did he know about advanced mathematical topics such as Hamiltonian mechanics ,Lagrangian mechanics, Manifold  and Differential topology .&lt;/p&gt;&#xA;" OwnerUserId="1716" LastActivityDate="2018-05-01T07:48:23.843" Title="Knowledge of Einstein during obtaining his graduation" Tags="|biographical-details|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1825" PostTypeId="2" ParentId="1824" CreationDate="2015-01-17T17:33:55.480" Score="5" Body="&lt;p&gt;It is very difficult to access what he really knew from reading or from doing the things himself, but this paper address the simpler questions, what classes he took and what were his scientific contacts in the student years:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1205/1205.4335.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://arxiv.org/ftp/arxiv/papers/1205/1205.4335.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me also add that &quot;manifolds&quot; and &quot;differential topology&quot; were not formalized at that time, and these words were not used. On the other hand, some physicists and mathematicians has quite good understanding of these things. For example, the preliminary chapter of Maxwell's Treatease on electricity and magnetism shows that he had quite a good understanding of differential forms, homology and cohomology, Morse theory and Hodge theory (all in dimension 3, of course), the things which were formally invented or&#xA;discovered at a much later time. There is little doubt that Einstein knew about this book,&#xA;and it was easily available to him, but how much did he actually read or understand of it, probably cannot be determined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I actually did not read the paper referred above when I wrote this answer. Now I read it carefully, and it is indeed amazing:-) Einstein had Adolf Hurwiz as his math professor, one of the greatest mathematicians of that time. Another professor was Hermann Minkowski. But Einstein's classmates say he skipped their seminars.&#xA;Instead, as a friend recalls:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein felt that &quot;the most fascinating subject at the time that I was a student was&#xA;  Maxwell's theory&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Somehow it happened that mathematicians of 19 and early 20 century did not read Maxwell's book (and of course they do not read it nowadays. Who cares about 19-th century books!) There is a very interesting paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dyson, Freeman J. Missed opportunities. Bull. Amer. Math. Soc. 78 (1972), 635–652. &#xA; &lt;a href=&quot;http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-12971-9/S0002-9904-1972-12971-9.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-12971-9/S0002-9904-1972-12971-9.pdf&lt;/a&gt;  Especially pp. 638-9. My own experience confirms everything Dyson says there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He discusses exactly this subject: what would mathematics gain if mathematicians read Maxwell. Maxwell actually anticipated much of the most important mathematics of the 20-th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is amazing how mature Einstein was in his student years: he knew what to study:-)&#xA;Instead of attending lectures, he studied himself &quot;the works of Bolzmann, Kirchhoff, Helmholtz, Hertz etc.&quot;, according to his own recollection, and perhaps of Maxwell as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I recently found a Russian translation of Einstein's paper &quot;Electrodynamics of moving bodies&quot;, 1905 and read it. It is frequently mentioned that this paper has no reference list, and Einstein gives no credit to anyone. But Maxwell is mentioned frequently, (without exact references). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-30T16:57:36.603" LastActivityDate="2018-04-30T16:57:36.603" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1826" PostTypeId="1" CreationDate="2015-01-17T19:31:52.850" Score="7" ViewCount="533" Body="&lt;p&gt;Some problems in mechanics are simplified by considering the action of &quot;fictitious forces&quot;. These appear in accelerated frames, such as circular movements. When were these first considered so? Did Newton himself deal with these types of problems?&lt;/p&gt;&#xA;" OwnerUserId="44" LastEditorUserId="44" LastEditDate="2015-01-18T10:43:41.800" LastActivityDate="2015-01-20T19:22:24.163" Title="When were fictitious forces introduced" Tags="|physics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1827" PostTypeId="1" CreationDate="2015-01-17T19:49:56.503" Score="10" ViewCount="2162" Body="&lt;p&gt;Some previous questions on Einstein bring the following question: is a complete (or incomplete) list of Einstein assistants available somewhere?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick search on the Internet brings several familiar names, but no list.&#xA;I know many of them,&#xA;but I would like to see a complete list, with dates of employment, if possible. Of course, one can look at the list of his co-authors, but not all assistants co-authored papers with him. Some of them were (or later become) famous mathematicians. As I understand, this was a formal payed position: a German&#xA;professor was eligible to have assistants. This applies to the period when Einstein worked in Berlin. Perhaps he also had an assistant in Princeton, I am not sure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Here are the candidates whom I extracted from the publicaton list until 1938:&#xA;W. Mayer (8), N. Rosen (4), J. Laub (3), L. Hopf (2), W. Ritz, O. Stern, A. Fokker, B. Hoffmann and P. Bergmann. (Other co-authors in this period are Lorenz, Minkowski, de Sitter and Grossmann,&#xA;but they were not his assistants). Later publications add V. Bargmann, B. Kaufman, H. Born, A. Lande, S. Penner, B. Podolsky, and few other people who could not be his assistants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cornelius Lanczos was an assistant in 1928-29 but never co-authored a paper.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-18T14:24:41.760" LastActivityDate="2017-01-22T20:40:46.667" Title="Einstein's assistants" Tags="|biographical-details|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1828" PostTypeId="2" ParentId="1826" CreationDate="2015-01-17T20:56:44.023" Score="4" Body="&lt;p&gt;I've found some pretty old usages:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Coriolis force:&lt;/strong&gt; From &lt;a href=&quot;http://scitation.aip.org/content/aip/magazine/physicstoday/article/64/8/10.1063/PT.3.1195&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Coriolis deflection of moving objects seen from within a rotating frame of reference—important in physics, meteorology, and oceanography—was described by Italian scientists Giovanni Battista Riccioli (1598–1671) and his assistant Francesco Maria Grimaldi (1618–63) nearly two centuries before Gaspard-Gustave Coriolis (1792–1843).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Riccioli and Grimaldi give a detailed description in Riccioli’s 1651 &lt;em&gt;Almagestum Novum&lt;/em&gt; of how Earth’s rotation should cause a rightward deflection in a projectile fired toward the north.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Centrifugal force:&lt;/strong&gt; I can't find a non-Wikipedia citation that doesn't end in a &quot;404&quot; link that is pre-Newton, but I did find &lt;a href=&quot;http://www.archive.org/stream/newtonspmathema00newtrich/newtonspmathema00newtrich_djvu.txt&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt;, a translation of Newton's &lt;em&gt;Principia&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflections conjunctly ; that is (if the species of the polygon be given), as the length described in that given time, and in &#xA;  creased or diminished in the ratio of the same length to the radius of the circle ; that is, as the square of that length applied to the radius ; and therefore the polygon, by having its sides diminished in infinitum, coincides with the circle, as the square of the arc described in a given time applied to the radius. This is the centrifugal force, with which the body impels the circle ; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But above that - aha! It might have been Huygens!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm not positive (in the latter case) precisely who was first, though. But it wasn't Newton.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-01-17T20:56:44.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1829" PostTypeId="2" ParentId="1826" CreationDate="2015-01-18T04:22:00.087" Score="6" Body="&lt;p&gt;It depends on what you mean by &quot;introduced&quot;. The centrifugal and the Coriolis forces are not the only inertial forces, or even the simplest ones. In classical mechanics inertial forces appear in any accelerating frame. The simplest one would be attached to a uniformly accelerated body. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In full generality inertial forces appear in D'Alembert's &lt;a href=&quot;http://www.colorado.edu/engineering/CAS/courses.d/OSE800/Lectures.d/OSE800.Lecture.03.2013_DAlembert.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Traite de Dynamique&lt;/a&gt; of 1743, where he clarified what happens under Newtonian laws in non-inertial frames. Since any accelerating body is at rest in a frame attached to itself the sum total of all forces acting on it must be balanced by an additional force, which he called &quot;the force of inertia&quot;. This is now known as the D'Alembert's principle, and Lagrange, who gave its more common variational formulation, quipped that &quot;D'Alembert had reduced dynamics to statics by means of his principle&quot;. In recognition, inertial forces are occasionally called D'Alembert forces today. Traite de Dynamique was a landmark treatise also because it freed classical mechanics from geometric straightjacket of Newton's Principia, and systematized its basic notions ending &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;terminological disputes&lt;/a&gt; of early 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The centrifugal force was already known in antiquity. There is a passage in Plutarch (1st century AD) describing the motion of a rock in a sling when it is rotated before the discharge, he even compares it to the motion of the Moon around the Earth. This passage is analyzed (perhaps overanalyzed) by Lucio Russo in &lt;a href=&quot;http://books.google.com/books/about/The_Forgotten_Revolution.html?id=MOTpnfz7ZuYC&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Forgotten Revolution&lt;/a&gt;, see also his &lt;a href=&quot;https://www.academia.edu/2119678/The_astronomy_of_Hipparchus_and_his_time_A_study_based_on_pre-ptolemaic_sources&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt;, who suggests that the original source was likely Hipparchus (2nd century BC). He goes on to speculate about the law of inertia and &quot;Hipparchian mechanics&quot;, but not very convincingly. Most likely, this was still understood in the Aristotelian context, where forces were proportional to induced speeds. &lt;a href=&quot;http://muse.jhu.edu/journals/journal_of_the_history_of_ideas/v064/64.4franco.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval impetus&lt;/a&gt; under some interpretations can also be construed as a version of a linear inertial force for a decelerating frame in a resisting medium. For example, Avicenna's self-degrading &quot;impressed force&quot; for a flying arrow almost matches a modern description if we relate it to such a frame.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 17th century the centrifugal force was more or less commonly accepted after  Descartes gave it a prominent place in &lt;a href=&quot;http://plato.stanford.edu/entries/descartes-physics/#6&quot; rel=&quot;nofollow noreferrer&quot;&gt;his physics&lt;/a&gt;, curiously using the same example of rock in a sling. But Huygens was first to derive a formula for it. Today ironically, essentially his very derivation is presented in modern textbooks as the derivation for the centripetal force instead. Newton himself originally used centrifugal force in that sense, but eventually switched (under Hooke's influence) to the centripetal reinterpretation more consistent with the absolute frame of Principia. But when Leibniz wrote the equation for the radial inertial force acting on a planet in the attached frame (with the inverse cube term) he pretended not to understand the context that he himself used before, and declared it wrong quoting the third law (out of place). More on this episode in Aiton's paper from &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0883857030&quot; rel=&quot;nofollow noreferrer&quot;&gt;Learn from the Masters&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose linear inertial forces did not receive more attention because observers rarely found themselves in accelerating frames until recently, but overloads during take offs and nose dives are very real effects of these &quot;fictitious&quot; forces. Beyond linearly accelerating and rotating frames however expressions for the inertial forces are typically too cumbersome to be useful. The linear inertial force received more attention after Einstein proposed his famous &lt;a href=&quot;http://www.pitt.edu/~jdnorton/papers/TE_in_AE.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;elevator thought experiment&lt;/a&gt;. Ironically again, Einstein's conslusion, the equivalence principle, effectively eliminates it from relativistic physics, because there is no physical way to distinguish it from gravity. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-01-20T18:45:58.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1830" PostTypeId="2" ParentId="1827" CreationDate="2015-01-18T08:06:56.357" Score="7" Body="&lt;p&gt;Here is a list (no claim to completeness) covering Einstein's period in Princeton:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Walther Mayer 1930-1934 (see pp. 492-494 of &quot;Subtle is the Lord&quot; by Abraham Pais).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nathan Rosen 1934-1936 (&lt;a href=&quot;http://physics.technion.ac.il/extras/history/memoriam/rosen/Peres-Obituary.pdf&quot;&gt;http://physics.technion.ac.il/extras/history/memoriam/rosen/Peres-Obituary.pdf&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peter Bergmann 1936-1941 (&lt;a href=&quot;http://en.wikipedia.org/wiki/Peter_Bergmann&quot;&gt;http://en.wikipedia.org/wiki/Peter_Bergmann&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Valentine Bargmann 1937-1946 (&lt;a href=&quot;http://relativity.livingreviews.org/Articles/lrr-2014-5/desc_2.html&quot;&gt;http://relativity.livingreviews.org/Articles/lrr-2014-5/desc_2.html&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the book by Pais (pp. 496-497) later assistants were Ernst Straus 1944-1948, John Kemeny 1948-1949, Robert Kraichnan 1949-1950, and Bruria Kaufman 1950-1955.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2015-01-18T08:24:22.750" LastActivityDate="2015-01-18T08:24:22.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1831" PostTypeId="1" CreationDate="2015-01-07T14:29:16.277" Score="6" ViewCount="42" Body="&lt;p&gt;I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature of a normal section is determined by the principal curvatures and the angle that the tangent to the curve makes with a principal direction via Euler equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives a rather rigid, albeit infinitesimal, description of the geometry of a surface at a point. I have been wondering how Euler might have thought that something like this should be true. Does it come from linear algebra and results about eigenvalues and eigenvectors of symmetric matrices?&lt;/p&gt;&#xA;" OwnerUserId="1678" OwnerDisplayName="Giuseppe" LastEditorUserId="98" LastEditDate="2015-01-18T18:13:09.797" LastActivityDate="2015-01-18T18:13:09.797" Title="Motivation behind Euler Theorem in differential geometry" Tags="|mathematics|differential-geometry|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-01-18T18:12:59.103" ContentLicense="CC BY-SA 3.0" />
  <row Id="1832" PostTypeId="2" ParentId="1824" CreationDate="2015-01-18T20:05:28.653" Score="0" Body="&lt;p&gt;Arthur I. Miller's &lt;a href=&quot;http://books.google.com/books?id=-MZCRkCAdeoC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911)&lt;/em&gt;&lt;/a&gt; has on p. 81 a section &quot;On Einstein's Knowledge of Electrodynamics in 1905&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't have access to it at the moment, but if I remember correctly, that section lists the books and monographs by Lorentz, Hertz, Föppl, Boltzmann, Helmholtz, Kirchhoff and others that Einstein had &quot;definitely&quot;, &quot;very probably&quot;, or &quot;maybe&quot; read by that time.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 1:&lt;/strong&gt; Having found the book, here goes (it's mostly electrodynamics, except Kirchhoff's book on mechanics):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. &quot;&lt;em&gt;Definitely read, and evidence&lt;/em&gt;&quot;:&lt;br&gt;&#xA;   1. Lorentz (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?24.1081.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1892&lt;/a&gt;, &lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?25.1632.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1895&lt;/a&gt;).&lt;br&gt;&#xA;   2. Hertz (&lt;em&gt;Collected Works&lt;/em&gt;, &lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?25.1774.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1893&lt;/a&gt;).&lt;br&gt;&#xA;   3. Föppl (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?25.1672.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1894&lt;/a&gt;).&lt;br&gt;&#xA;   4. Boltzmann (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?23.1110.01|25.1671.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1891, 1893&lt;/a&gt;), von Helmholtz (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?28.0739.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1897&lt;/a&gt;) and Kirchhoff (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?08.0542.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1876&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. &quot;&lt;em&gt;Very probably&lt;/em&gt;&quot;:&lt;br&gt;&#xA;   1. Drude (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?31.0786.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900&lt;/a&gt;).&lt;br&gt;&#xA;   2. Abraham-Föppl (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?35.0862.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. &quot;&lt;em&gt;Maybe&lt;/em&gt;&quot;:&lt;br&gt;&#xA;   1. Lorentz (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?35.0861.02|35.0861.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;).&lt;br&gt;&#xA;   2. Lorentz &lt;em&gt;Festschrift&lt;/em&gt; (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?32.0884.01|31.0671.02|31.0751.04|31.0752.01|31.0763.01|31.0766.02|31.0788.02|31.0788.03|31.0792.04|31.0793.03|31.0795.02|31.0826.01|31.0826.02|31.0826.03|31.0827.01|31.0827.02|31.0827.03|31.0829.03|31.0831.02|31.0832.02|31.0833.01|31.0834.01|31.0834.02|31.0835.02|31.0837.02|31.0842.01|31.0846.02|31.0849.03|31.0849.04|31.0851.02|31.0851.03|31.0851.04|31.0852.01|31.0852.02|31.0853.02|31.0855.02|31.0864.01|31.0864.02|31.0884.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arch. Néerl. (2) &lt;strong&gt;5&lt;/strong&gt; (1900)&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 2:&lt;/strong&gt; Einstein’s Collected Papers reproduce his &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol1-trans/47&quot; rel=&quot;nofollow noreferrer&quot;&gt;ETH Record and Grade Transcript&lt;/a&gt; which shows all the classes he took for credit. An &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol1-doc/430&quot; rel=&quot;nofollow noreferrer&quot;&gt;Appendix&lt;/a&gt; has details of the course syllabi. (To the OP’s specific question, see Geiser’s courses on &lt;em&gt;Infinitesimalgeometrie&lt;/em&gt;, and Minkowski’s on &lt;em&gt;Analytische Mechanik + Anwendungen&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit 3:&lt;/strong&gt; There is also an &lt;a href=&quot;//ams.org/mathscinet-getitem?mr=1770300&quot; rel=&quot;nofollow noreferrer&quot;&gt;entire book&lt;/a&gt; on the subject.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-04-30T23:06:40.367" LastActivityDate="2018-04-30T23:06:40.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1833" PostTypeId="1" AcceptedAnswerId="1834" CreationDate="2015-01-18T23:50:24.433" Score="2" ViewCount="771" Body="&lt;p&gt;I am reading these articles and books about science and I find that there are debates on who was the real (and first) inventor of the laser. Who is the real inventor of the laser?&lt;/p&gt;&#xA;&lt;p&gt;Please include reliable links and sources so I won't think your answer is an opinion.&lt;/p&gt;&#xA;" OwnerUserId="1721" LastEditorUserId="4251" LastEditDate="2020-07-20T11:49:10.130" LastActivityDate="2020-07-20T11:49:10.130" Title="Who really was the first inventor of the laser?" Tags="|physics|discoveries|physicists|photonics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1834" PostTypeId="2" ParentId="1833" CreationDate="2015-01-19T02:17:42.493" Score="4" Body="&lt;p&gt;LoganM wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The best we can hope to do here is summarize exactly who did what and explain why the debate exists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's it. That's the best we can do. So I'm going to start gathering evidence for the various claimants. There's no way I can finish it tonight, but I'll start.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Gordon Gould&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gould sketched an idea for a laser in 1957 and then had his idea notarized (at a candy store, apparently). He coined the term &lt;em&gt;laser&lt;/em&gt; in the notes beside it. He didn't build anything until 1958, and so only filed for a patent in 1959, at which point others had already patented the idea. He did announce his idea at a conference that year, but it didn't help him secure the patent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theodore Maiman&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maiman built the first operational laser in 1960 at Hughes Research Laboratories. His design was tiny and used ruby, but it was successful, and HRL claimed that he was the laser's inventor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Charles Townes and Arthur Schawlow&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The duo had already patented their maser design in 1954. In 1958, they came up with theoretical models of a derivative of the idea using visible light, but did no further research. They are able to patent their design in 1960.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's not a non-opinion-based definitive answer here.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://inventors.about.com/od/lstartinventions/a/laser.htm&quot; rel=&quot;nofollow&quot;&gt;about.com&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://worldoflasers.com/laserhistory.htm&quot; rel=&quot;nofollow&quot;&gt;worldoflasers.com&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://national-laser.com/laser-history.htm&quot; rel=&quot;nofollow&quot;&gt;National Laser Company&lt;/a&gt; and &lt;a href=&quot;http://www.photonics.com/Article.aspx?PID=5&amp;amp;VID=61&amp;amp;IID=464&amp;amp;Tag=Features&amp;amp;AID=42279&quot; rel=&quot;nofollow&quot;&gt;photonics.com&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Laser#History&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://laserstars.org/history/&quot; rel=&quot;nofollow&quot;&gt;laserstars.org&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://laserstars.org/history/ruby.html&quot; rel=&quot;nofollow&quot;&gt;laserstars.org&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-01-19T02:17:42.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1835" PostTypeId="1" CreationDate="2015-01-19T09:50:47.343" Score="7" ViewCount="2293" Body="&lt;p&gt;I'm very interested to know how the concept of vectors came in mathematics and physics. How were vector quantities discovered in physics, and how and by whom were they developed?&lt;/p&gt;&#xA;" OwnerUserId="1716" LastEditorUserId="58" LastEditDate="2015-01-23T22:05:49.670" LastActivityDate="2015-01-23T22:05:49.670" Title="How were vector quantities developed?" Tags="|calculus|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1836" PostTypeId="2" ParentId="1835" CreationDate="2015-01-19T10:22:14.787" Score="8" Body="&lt;p&gt;You can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Michael Crowe, &lt;a href=&quot;https://books.google.it/books?id=y5-S5dmVqGIC&amp;amp;printsec=frontcover&quot;&gt;A History of Vector Analysis&lt;/a&gt; (1967).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We have to take into account some &quot;precursors&quot;, like the representation of the parallelogram of the velocities and of the forces, already present into ancient Greek science (see the pseudo-Aristotle, &lt;a href=&quot;http://en.wikipedia.org/wiki/Mechanics_%28Aristotle%29&quot;&gt;Mechanica&lt;/a&gt;) and some attempts by Leibniz with his &lt;a href=&quot;http://fr.wikipedia.org/wiki/Analysis_situs&quot;&gt;Analysis situs&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key achievement was the geometrical representation of complex numbers, discovered independently in 1831 by Wessel, Gauss, Argand, etc.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we have &lt;a href=&quot;http://en.wikipedia.org/wiki/William_Rowan_Hamilton&quot;&gt;William Rowan Hamilton&lt;/a&gt; with the &lt;a href=&quot;http://en.wikipedia.org/wiki/Quaternion&quot;&gt;quaternions&lt;/a&gt; and finally the modern system of vector analysis developed by &lt;a href=&quot;http://en.wikipedia.org/wiki/Josiah_Willard_Gibbs&quot;&gt;Josiah Willard Gibbs&lt;/a&gt; with his lectures of 1881 and 1884 [published as E.B.Wilson, &lt;em&gt;Vector Analysis&lt;/em&gt;, a text-book for the use of students of Mathematics and Physics, founded upon the Lectures of J.Willard Gibbs, (1901)] and &lt;a href=&quot;http://en.wikipedia.org/wiki/Oliver_Heaviside&quot;&gt;Oliver Heaviside&lt;/a&gt;, with his 1882-83 paper on electromagnetism published on &lt;em&gt;The Electrician&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-01-19T10:22:14.787" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1837" PostTypeId="2" ParentId="1835" CreationDate="2015-01-19T14:07:30.917" Score="5" Body="&lt;p&gt;I agree with Mauro's answer, and want to add two interesting details.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The first &quot;use of vectors&quot; in science that I know was the famous theorem of Apollonius which says that epicycles are equivalent to excentrics (in the description of the motion of celestial bodies). This theorem is reproduced in Ptolemy, with a proof, and credited explicitly to Apollonius. Ptolemy was not very generous in giving credits, so I supose this is really due to Apollonius. Translated in modern language the theorem says that vector addition is commutative. (It is amazing that in the ancient times this &quot;evident&quot; thing required a proof).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another curious thing is that quaternions were introduced much earlier than vectors,&#xA;and later introduction of vectors to physics (by Gibbs) generated a substantial controversy.&#xA;This controversy is nicely described in this paper:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01182-2/S0273-0979-07-01182-2.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/journals/bull/2008-45-01/S0273-0979-07-01182-2/S0273-0979-07-01182-2.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;A also want to mention that mathematicians (like Grassmann, Cayley and Silvester) routinely used vectors long before physicists.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Remark. The paper I mentioned above is a part of the Gibbs Lectures collection, which&#xA;on my opinion is a great source on the history of Math and Science. All old lectures are published in the AMS Bulletin which is freely available. Unfortunately they discontinued&#xA;publication of these lectures, so the recent ones do not seem to be available at all. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-19T14:30:22.663" LastActivityDate="2015-01-19T14:30:22.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1838" PostTypeId="1" CreationDate="2015-01-20T09:30:49.733" Score="5" ViewCount="270" Body="&lt;p&gt;I was wondering if there is any book that chronicles the progress of Math over the centuries and also mentions about how/when applications of various theories were discovered/invented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been trying to study university level math and I keep thinking about applications. Applications like principal component analysis, cryptography needed modern computing power. So before computers, how was all this abstract algebra used?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What kind of math was required for the manufacturing industry since the industrial revolution started (manufacturing heavy and precise equipment from steam engines to transistors...)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I saw a lot of such &quot;history of math&quot; books on amazon, but I am looking for something like &quot;this technology is dependent on that math/science theory&quot;, preferably written in somewhat &quot;math history for kids/dummies&quot; style.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had asked this on &lt;a href=&quot;https://math.stackexchange.com/questions/991404/any-book-on-timeline-of-progress-of-math-concepts-and-applications&quot;&gt;math.stackexchange.com&lt;/a&gt; some time back, got no suggestions and now it can't be migrated here. So, posting this as a new question. Please advise.&lt;/p&gt;&#xA;" OwnerUserId="1703" LastEditorUserId="4251" LastEditDate="2021-03-10T21:30:23.377" LastActivityDate="2021-03-10T21:30:23.377" Title="Any book on the timeline of progress of mathematical concepts and applications?" Tags="|mathematics|reference-request|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="1839" PostTypeId="2" ParentId="317" CreationDate="2015-01-20T13:34:42.080" Score="8" Body="&lt;p&gt;I'm going to interpret &quot;jail&quot; and &quot;prison&quot; as including house arrest and concentration camps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alhazen&quot; rel=&quot;noreferrer&quot;&gt;Ibn al-Haytham&lt;/a&gt; wrote his &lt;a href=&quot;https://en.wikipedia.org/wiki/Book_of_Optics&quot; rel=&quot;noreferrer&quot;&gt;Book of Optics&lt;/a&gt; (in seven volumes) while under house arrest.  He&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;was kept under house arrest [by Fatimid Caliph al-Hakim] from 1011 until al-Hakim's death in 1021.  During this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jakow_Trachtenberg&quot; rel=&quot;noreferrer&quot;&gt;Jakow Trachtenberg&lt;/a&gt; created the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trachtenberg_system&quot; rel=&quot;noreferrer&quot;&gt;Trachtenberg System&lt;/a&gt; for quick mental arithmetic calculation while in a concentration camp.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After the Russian Revolutions of 1917, Trachtenberg fled to Germany where he became critical of Nazi policies.  He was imprisoned in a Nazi concentration camp during World War II.  He developed his system of mental arithmetic during his imprisonment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="72" LastEditorUserId="72" LastEditDate="2015-01-21T03:51:09.517" LastActivityDate="2015-01-21T03:51:09.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1840" PostTypeId="2" ParentId="1826" CreationDate="2015-01-20T19:12:29.030" Score="2" Body="&lt;p&gt;Ingoli's letter to Galileo and Galileo's reply give a pretty clear snapshot of one point in the very slow development of this idea over the centuries. See Graney, &lt;a href=&quot;http://arxiv.org/abs/1211.4244&quot; rel=&quot;nofollow&quot;&gt;http://arxiv.org/abs/1211.4244&lt;/a&gt; , esp. pp. 29ff. Mach's critique in &lt;a href=&quot;https://archive.org/stream/scienceofmechani005860mbp#page/n5/mode/2up&quot; rel=&quot;nofollow&quot;&gt;The Science of Mechanics&lt;/a&gt; shows that as late as 1883, nobody had really thoroughly addressed the logical problems with defining the distinction between inertial and noninertial frames. Einstein didn't even get it quite right; modern physicists don't agree with his interpretation of GR as a theory of accelerated frames. In answer to the question, &quot;When were fictitious forces introduced,&quot; I think they were partially understood as early as 1616, and fully understood as late as ca. 1960, when relativists developed what is now considered the correct interpretation of GR. In fact, if you ask a relativist today whether gravity is a fictitious force, you will probably get a nuanced answer. Some might say it's not a force, some might say it's a fictitious force, and still others might say that it's hard to say, because the equivalence principle resists attempts at precise formulation (see Sotiriou et al, &quot;Theory of gravitation theories: a no-progress report,&quot; &lt;a href=&quot;http://arxiv.org/abs/0707.2748&quot; rel=&quot;nofollow&quot;&gt;http://arxiv.org/abs/0707.2748&lt;/a&gt; ).&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2015-01-20T19:22:24.163" LastActivityDate="2015-01-20T19:22:24.163" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1841" PostTypeId="1" AcceptedAnswerId="1844" CreationDate="2015-01-20T20:09:01.290" Score="6" ViewCount="456" Body="&lt;p&gt;Kuhn's theory of paradigm shifts dates back over 50 years. Are there modern critiques of or alternatives to his theory? It seems to have been controversial from the start, with some people apparently claiming that Kuhn was attacking scientific realism. I think a lot of scientists today are familiar with Kuhn's ideas in a vague way and may have read his book, but also simultaneously hold a lot of philosophical or historical beliefs that they may not realize are incompatible with Kuhn, e.g., a belief in the importance of Popper's falsifiability criterion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;related: &lt;a href=&quot;https://hsm.stackexchange.com/questions/187/examples-of-kuhn-loss&quot;&gt;Examples of Kuhn loss?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-01-21T13:22:17.200" Title="Modern critiques of, alternatives to Kuhn's paradigm shifts?" Tags="|philosophy-of-science|kuhn|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1842" PostTypeId="2" ParentId="1838" CreationDate="2015-01-20T21:26:31.367" Score="2" Body="&lt;p&gt;I do not know of such book, and will be glad if someone suggest it.&#xA;But let me attempt a broad sketch of applications of Mathematics before 1900:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first main application was astronomy, there is no doubt in this. It started in the Hellenistic Greece, and probably until the 18-th century this was the main application.&#xA;Astronomy and (after Newton) celestial mechanics. Ability to predict the Moon motion&#xA;and to predict the existence of new planets were probably the greatest &quot;practical&quot; achievements&#xA;of mathematics until the middle 19-th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Serious applications to physics and engineering begin in 17-th century.&#xA;With the invention of calculus, we see an explosive growth of applied mathematics.&#xA;Bernoulli's and Euler tried to model literally everything: blood flow in the blood vessels,&#xA;behavior of a ship at sea, behavior of a projectile inside the gun barrel (not only outside!) and so on. (It is questionable whether these early theories had any influence&#xA;on the industry and technology. The problems they considered were too complicated for mathematics of that time). One early example, of successful applications of mathematics that&#xA;I know besides astronomy is the design of clocks and watches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two entirely new playgrounds were opened in the beginning of 19-th century: electricity/magnetism and heat conduction. Here we already have examples of direct influence of mathematics on industry. I mean design of steam engines, design of rails and wires, and various machines, etc. One especially famous story is Lord Kelvin's&#xA;mathematical contribution to the design of&#xA;the first transatlantic cable. (He became a Lord because of this! This was around 1850-s)&#xA;He also improved the compass for marine navigation using modern mathematical theories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just few examples. But in general, one can say that since the beginning of&#xA;19-th century, mathematics becomes a true productive force. Necessary for engineering and industry (not only for astronomy and physics). Hydrodynamics was successively applied to ship construction in the end of 19-th century, though the first attempts start with&#xA;Bernoulli and Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You ask explicitly about algebra. Algebra had fewer applications before the modern period.&#xA;One of them was to coding. They say this application begins with Vieta, the founder of&#xA;algebraic notation, but I do not have precise references.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did not mention applications of geometry to geodesy, and building construction, and this is&#xA;perhaps as old as applications to astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will indeed be interesting if someone writes a book along these lines.&#xA;But the examples I mentioned are scattered in many books. One notable example is&#xA;T. Korner's books &quot;The pleasures of counting&quot; and &quot;Fourier Analysis&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A remark on computation before electronic computers. Of course, massive computations were always needed. One milestone was invention of logarithms in the early 17 century.&#xA;This was not enough, and people were always working on the invention of computing machines, beginning from 17-th century. There were also &quot;computing centers&quot; where many people (called &quot;computOrs&quot;) computed. Specialized tables for all kinds of computation were developed. There were quite advanced analog computing machines,&#xA;like harmonic analizers (invented by the same Lord Kelvin for tide prediction).&#xA;Of course, all this changed with the wide spread of cheap electronic computers, but this happened only very recently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I think, fdb's comment requires somewhat longer answer than fits in a comment.&#xA;Indeed, Pythagoras lived in Italy, Thales in Turkey:-) and Euclid in Egypt. However I think it will be a ridiculous anachronism to call Thales &quot;a Turk&quot;. All these people are commonly called Greeks, and there are at least some reasons for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, and more important. I use the words &quot;mathematics&quot; and &quot;astronomy&quot; in a narrow sense. (You may use them differently, I do not want to argue about words). All human cultures had some means of counting and measuring the amounts of grain, beer, and the areas of land plots. Many of them also had a tax code. From my point of view this is not &quot;mathematics&quot; yet. Mathematics (in the narrow sense), as far as I know was invented by some Greeks (according to the legend, either by Thales or Pythagoras) and the earliest documentary evidence of it comes to us from Hellenistic times/states. Namely from Alexandria, now in Egypt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Astronomy was first practiced in Babylon (and we have well-documented prime sources of this) but there was no much mathematics yet. Real APPLICATIONS of mathematics to astronomy&#xA;come from Hellenistic Greeks (who lived everywhere on the Mediterranean coast and islands.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eudoxus was a great mathematician, of course (judging by what is attributed to him) but his contribution to astronomy (comparable to the contribution of Aristarchus) is not really an application of mathematics to astronomy. Eudoxus's spheres (and Aristarchus determination of the ratio of distances to Sun and Moon) where nice, beautiful mathematical exercises, but they were not real &quot;successful applications of mathematics to astronomy&quot;. Aristarchus proposal is not realistic, and it is clear that he did not even try to measure the quantities involved. If you do not believe me, try to determine the&#xA;ratio of these distances with Aristarchus method. I tried. Really. That the planets do not actually move as Eudoxus described was probably clear even to is contemporaries, and certainly to anyone who did &#xA;observations. So applications of mathematics to astronomy, to the best of my knowledge, begin with Apollonius and Hypparchus (who lived in Egypt:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, astronomy in the narrow sense (as I understand it), does not include such facts&#xA;that there are stars and planets,  or names of constellations, or the fact that that Sun raises in the East and sets in the West, and so on. The facts that anyone discovers once she looks at the sky regularly for few months.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-21T23:41:09.843" LastActivityDate="2015-01-21T23:41:09.843" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1844" PostTypeId="2" ParentId="1841" CreationDate="2015-01-21T12:33:37.123" Score="3" Body="&lt;p&gt;You can see at least &lt;a href=&quot;http://plato.stanford.edu/entries/thomas-kuhn/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Kuhn&lt;/a&gt;, with extensive bibliography.&lt;/p&gt;&#xA;&lt;p&gt;The debate following the pubblication of &lt;em&gt;SSR&lt;/em&gt; in 1962 was impressive and interesting, and far from having reached a consensus on his main thesis.&lt;/p&gt;&#xA;&lt;p&gt;In the &lt;em&gt;SEP&lt;/em&gt; entry above you can see :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;6.4 Recent Developments&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even if Kuhn's work has not remained at the centre of the philosophy of science, a number of philosophers have continued to find it fruitful and have sought to develop it in a number of directions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to my knowledge, during the 50 years since the first edition of &lt;em&gt;SSR&lt;/em&gt;, there has been no new &amp;quot;approach&amp;quot; to the philosophy of science able to arouse so big a consensus and so much dissent.&lt;/p&gt;&#xA;&lt;p&gt;Nor Kuhn's views regarding science has acquired the status of a new &amp;quot;received view&amp;quot; (i.e. the status achieved during the '30s by the neo-positivist conception of science).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;You can see :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Nickles (editor), &lt;a href=&quot;https://books.google.it/books?id=_mA2PC4WwDkC&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Kuhn&lt;/a&gt; (2003)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;or :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Vasso Kindi &amp;amp; Theodore Arabatzis (editors), &lt;a href=&quot;https://books.google.it/books?id=yAKirh-gGSgC&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kuhn’s The Structure of Scientific Revolutions Revisited&lt;/a&gt; (2013).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-01-21T13:22:17.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1845" PostTypeId="2" ParentId="1838" CreationDate="2015-01-21T14:21:59.873" Score="2" Body="&lt;p&gt;Some suggestions; not a single history, but some &quot;pictures&quot; :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Jacqueline Stedall (editor), &lt;a href=&quot;https://books.google.it/books?id=HIMiWHVKLDkC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;The Oxford Handbook of the History of Mathematics&lt;/a&gt; (2008) [some chapters]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jens Hoyrup, &lt;a href=&quot;https://books.google.it/books?id=koe14BYdsAsC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;In Measure, Number, and Weight : Studies in Mathematics and Culture&lt;/a&gt; (1994)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ad Meskens, &lt;a href=&quot;https://books.google.it/books?id=dZxEAAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Practical mathematics in a commercial metropolis : Mathematical life in late 16th century Antwerp&lt;/a&gt; (2013)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Frank Swetz, &lt;a href=&quot;https://books.google.it/books?id=xu_SVhW9DK0C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Capitalism &amp;amp; Arithmetic : The New Math of the 15th Century&lt;/a&gt; (1987)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I.Bernard Cohen, &lt;a href=&quot;https://books.google.it/books?id=PMN7PwAACAAJ&quot; rel=&quot;nofollow&quot;&gt;The Triumph of Numbers : How Counting Shaped Modern Life&lt;/a&gt; (2006)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Evelyne Barbin &amp;amp; Raffaele Pisano (editors), &lt;a href=&quot;https://books.google.it/books?id=qaREAAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;The Dialectic Relation Between Physics and Mathematics in the XIXth Century&lt;/a&gt; (2013).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-01-21T14:21:59.873" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1846" PostTypeId="1" AcceptedAnswerId="1847" CreationDate="2015-01-21T23:32:24.297" Score="2" ViewCount="404" Body="&lt;p&gt;I am reading this book on Archimedes, famous mathematician of his days and now, and it was found that he caculated the amount of sand in the known Universe at his days, which is from Mercury to Jupiter I believe. Solving how much for a teaspoon and a finger was complicated enough, but the task for the universe seems impossible. Usually he should have some equation or some way to caculate that large number. In the process, he made exponents, I know, but how did Archimedes complete his challange? What mathematical process or equation(s) did he use?&lt;/p&gt;&#xA;" OwnerUserId="1721" LastEditorUserId="4251" LastEditDate="2020-06-04T07:08:50.077" LastActivityDate="2020-06-04T07:08:50.077" Title="How did Archimedes complete this impossible task?" Tags="|mathematics|mathematicians|archimedes|" AnswerCount="1" CommentCount="3" ClosedDate="2015-01-22T22:31:14.993" ContentLicense="CC BY-SA 4.0" />
  <row Id="1847" PostTypeId="2" ParentId="1846" CreationDate="2015-01-21T23:50:22.977" Score="6" Body="&lt;p&gt;I recommend that you read Archimedes himself. This is easily available, translated to most languages, and you will not regret. This is really a very entertaining reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this book of Archimedes was a kind of joke (like the Cattle problem attributed to him). He did not &quot;calculate&quot; the number of pieces of sand. He only estimated it from above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His purpose was only to explain&#xA;how one can represent arbitrarily large numbers. Which is not a big deal. But the details he gives are indeed very interesting and amazing. For example from this book we learn what people thought of the size of the Universe, and various related parameters involved. Especially interesting are some experimental procedures he describes. For example how he found the size of one piece of sand. And visible diameter of the Sun, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Archimedes simply uses the best available at that time estimates of the size of the Earth, distance to the Moon, distance to the Sun, and how much the distance to the &quot;fixed stars&quot; is greater than the distance to the Sun. He obtains the volume of the Universe. Then he measures the size of the sand piece, and does the arithmetic. What is the volume of the ball of given radius he knew:-)&#xA;But the main point is introducing a simple notation which can represent arbitrarily large numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. The Cattle problem (which I believe is also due to Archimedes) shows the same kind of the sense of humor). The point here is that every Pell's equation has a solution. The smallest solution of the Cattle problem is much greater than the number of sand pieces in the Universe:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-22T02:16:12.347" LastActivityDate="2015-01-22T02:16:12.347" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1848" PostTypeId="1" AcceptedAnswerId="1852" CreationDate="2015-01-22T01:29:40.310" Score="13" ViewCount="818" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;To make it more formal, I am looking for striking historical examples of objects or concepts that were well known in a field and perceived as different, but later discovered to be the same. I am interested in how they happen and what impact they have, because they are perhaps the clearest symbols of scientific success, and the simplest cases of what Arthur Koestler called &lt;a href=&quot;https://www.brainpickings.org/2013/05/20/arthur-koestler-creativity-bisociation/&quot; rel=&quot;noreferrer&quot;&gt;bisociation&lt;/a&gt;, &amp;quot;perceiving of a situation or idea in two self-consistent but habitually incompatible frames of reference&amp;quot;, which underlies discovery process in general.&lt;/p&gt;&#xA;&lt;p&gt;This is potentially a &amp;quot;big list&amp;quot; question, but I expect that striking &amp;quot;bidentifications&amp;quot; that had a major impact  are very rare. Below are three examples of what I have in mind, but examples in any natural or social sciences would be interesting.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/787/how-did-babylonians-figure-out-that-the-evening-star-is-the-morning-star&quot;&gt;Hesperus is Phosphorus&lt;/a&gt; (Venus)&lt;/p&gt;&#xA;&lt;p&gt;Perhaps the prototypical such example, the discovery is anecdotally attributed to Pythagoras (6th century BC). Careful observation of the night sky reveals that the bright star appearing in the evening after the sunset is the same as the one seen before the dawn. Had to be done before the motion of Venus could be modeled.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=xg2fBfXUtGgC&amp;amp;pg=PA161&amp;amp;lpg=PA161&amp;amp;dq=greatly+amazed+when+I+say+that+exactly+this+cycloid,+or+tautochrone+of+Huygens,+is+our+required+brachistochrone&amp;amp;source=bl&amp;amp;ots=pPulDL1Vzp&amp;amp;sig=0kKen_faZVsvfVCmvSyDjeCpacQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=ej3AVNHIFcedygTw74KACQ&amp;amp;ved=0CCkQ6AEwAQ#v=onepage&amp;amp;q=greatly%20amazed%20when%20I%20say%20that%20exactly%20this%20cycloid%2C%20or%20tautochrone%20of%20Huygens%2C%20is%20our%20required%20brachistochrone&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Brahistohrone is tautochrone&lt;/a&gt; (cycloid)&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;But the reader will be greatly amazed, when I say that exactly this cycloid, or tautochrone of Huygens, is our required brachistochrone.&amp;quot; With these words Johan Bernoulli in 1697 presented his solution to the problem that attracted the attention of Newton and Leibniz, among others, and launched the calculus of variations. Perhaps, in some part because Bernoulli's example encouraged the motivating idea that solutions might be otherwise known entities. His method of solution, the &lt;a href=&quot;https://books.google.com/books?id=ZWoYYr8wk2IC&amp;amp;pg=PA136&amp;amp;lpg=PA136&amp;amp;dq=optico-mechanical+analogy&amp;amp;source=bl&amp;amp;ots=SC972DBMnP&amp;amp;sig=0XrAwU_QgannbdRQJI5sSxo4ltM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=_kDAVOKqC4qmyASVtIDQCQ&amp;amp;ved=0CFsQ6AEwCQ#v=onepage&amp;amp;q=optico-mechanical%20analogy&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;optico-mechanical analogy&lt;/a&gt;, was itself an example of bisociation, which played a role in discovering the variational principles of mechanics. The brahistochrone is the curve constrained to move along which a heavy particle will reach the end the fastest. And the tautochrone is the curve that the particle starting at any intermediate point of will reach the end of in equal times. Today we should be even more amazed that the fastest curve is the curve of equal times because even a slightest variation in the conditions of the problem &lt;a href=&quot;https://physics.stackexchange.com/questions/16819/is-there-an-intuitive-reason-the-brachistochrone-and-the-tautochrone-are-the-sam&quot;&gt;destroys the correspondence&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/87/123.full.pdf+html&quot; rel=&quot;noreferrer&quot;&gt;Coal is diamond&lt;/a&gt; (carbon)&lt;/p&gt;&#xA;&lt;p&gt;In 1796 after burning diamonds (!) in oxygen Tennant discovered that only carbon dioxide was produced as a result. Which meant that diamond is chemically identical to coal, a conclusion that evaded Lavoisier. A century later in 1893 Moissan stirred a &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/18435987&quot; rel=&quot;noreferrer&quot;&gt;major controversy&lt;/a&gt; claiming that he transmuted coal into diamond in his furnace. Today his claim appears more credible, and of course diamond synthesis is an industry.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2023-06-13T09:27:43.320" Title="Apparently different objects discovered to be the same" Tags="|discoveries|" AnswerCount="7" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1849" PostTypeId="2" ParentId="1848" CreationDate="2015-01-22T02:00:08.707" Score="15" Body="&lt;p&gt;I would say that the most famous discovery of this sort was the explanation, why &quot;gravitational mass&quot; and &quot;inertial mass&quot; are the same. (The fact was so well known since Galileo that nobody really noticed that it is some strange coincidence. This remarkable coincidence is explained in General relativity). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only competing example is that light and electromagnetic waves are the same, but it does not quite fit: electomagnetic waves (other than light) were not detected until after Maxwell predicted them. But anyway, he discovered that optics and electromagnetism are about the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much earlier discovery of the same kind is that Sun is one of the stars, and Earth is one of the planets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. It did not occur to me until I read KCd's answer &#xA;that examples from mathematics also qualify:-) But in mathematics there are so many, that this will be indeed a VERY big list. So I limited myself to physics/astronomy.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-22T22:32:06.450" LastActivityDate="2015-01-22T22:32:06.450" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1850" PostTypeId="2" ParentId="1848" CreationDate="2015-01-22T02:35:08.820" Score="8" Body="&lt;p&gt;My pick is &lt;a href=&quot;http://en.wikipedia.org/wiki/Apatosaurus&quot; rel=&quot;nofollow&quot;&gt;Apatosaurus&lt;/a&gt;/&lt;a href=&quot;http://en.wikipedia.org/wiki/Brontosaurus&quot; rel=&quot;nofollow&quot;&gt;Brontosaurus&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.npr.org/2012/12/09/166665795/forget-extinct-the-brontosaurus-never-even-existed&quot; rel=&quot;nofollow&quot;&gt;A short run-down&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around 1877, Othniel Charles Marsh discovered a dinosaur skeleton that he named &lt;em&gt;Apatosaurus&lt;/em&gt;. In 1879, he found a different skeleton that seemed completely different from the first. Marsh named it &lt;em&gt;Brontosaurus&lt;/em&gt;. Throughout the next few decades or so, skeletons were incorrectly assembled and mixed up due to Marsh's mistake (he had also incorrectly added the Apatosaurus head, as well); eventually, it was realized that a mistake had been made. Exactly 100 years later, the correct head was inserted, thereby definitively dissipating the myth that Brontosaurus was a distinct genus.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.scientificamerican.com/article/the-brontosaurus-is-back1/&quot; rel=&quot;nofollow&quot;&gt;A recent study&lt;/a&gt; has concluded that &lt;a href=&quot;http://www.nytimes.com/2015/04/08/science/earth/the-brontosaurus-a-prehistoric-giant-is-revived-if-only-in-name.html?_r=0&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Brontosaurus&lt;/em&gt; is a different genus than &lt;em&gt;Apatosaurus&lt;/em&gt;&lt;/a&gt;. Peer review is needed to determine the validity of the result, but it seems that the two were different all along.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T23:36:11.727" LastActivityDate="2015-04-12T23:36:11.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1852" PostTypeId="2" ParentId="1848" CreationDate="2015-01-22T15:26:03.370" Score="16" Body="&lt;p&gt;Such &quot;bidentifications&quot; of concepts in math with major impact are what some important mathematical ideas are all about. Here are some examples, and the list could very easily be extended.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Logarithms show multiplication on $\mathbf R_{+}$ is a disguised form of addition on $\mathbf R$, and this is why logarithms were such an important computational aide for centuries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) The introduction of coordinates in the plane by Descartes showed many types of curves are basically the same thing as algebraic equations in two variables, which allowed geometric problems to be solved algebraically, algebraic problems to be solved geometrically, and allowed spaces of dimension greater than three (and the associated geometric language) to be defined by using ordered lists $(x_1,\dots,x_n)$ with $n &amp;gt; 3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Computing areas and tangents are the same problem in reverse, as codified in the Fundamental Theorem of Calculus. The importance of this discovery is clear to anyone who knows calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) Exponential and trigonometric functions are essentially the same thing when allowed to be functions of a complex variable. This streamlines the theory of Fourier series when you allow complex coefficients, since it lets infinite series in sines and cosines be written as series in powers of $e^{ix}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) The discrete groups preserving certain automorphic functions are the same as groups of motions in non-Euclidean geometry. Poincare's discovery of this while stepping onto a bus is famous, and it led to his model of the hyperbolic plane using complex variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) Probability lacked an axiomatic mathematical foundation until the 1930s, when Kolmogorov identified the basic concepts of probability theory such as events, probabilities, random variables, and independence, with notions of measure theory that came out of the work of Lebesgue from thirty years before. Briefly put, probability = measure theory when the whole space has measure 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) Modular forms and representations of Galois groups were studied separately, even by Artin and Hecke in the same department at the same time, but the fact that they are two different ways of getting at the same thing (constructing the same $L$-functions in different ways) was not appreciated until several decades later in the context of the Langlands program.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2015-01-24T01:03:59.843" LastActivityDate="2015-01-24T01:03:59.843" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1853" PostTypeId="1" AcceptedAnswerId="1856" CreationDate="2015-01-23T06:09:28.970" Score="6" ViewCount="1357" Body="&lt;p&gt;All the sources I've read say that Leonhard Euler presented his paper on the Konigsberg bridge problem to the St. Petersburg Academy on August 26, 1735.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, they don't seem to agree on what year the paper was first published in journal form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg#Euler.27s_analysis&quot;&gt;Wikipedia&lt;/a&gt; gives 1741 as the year of the journal publication.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Euler's work was presented to the St. Petersburg Academy on August 26, 1735, and published as &lt;em&gt;Solutio problematis ad geometriam situs pertinentis&lt;/em&gt; (The solution of a problem relating to the geometry of position) in the journal &lt;em&gt;Commentarii academiae scientiarum Petropolitanae&lt;/em&gt; in 1741.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But page 3 of Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198539169&quot;&gt;Graph Theory 1736--1936&lt;/a&gt; (Oxford: Clarendon Press, 1998) states that the paper (L. Euler's &lt;em&gt;Solutio problematis ad geometriam situs pertinentis&lt;/em&gt; in &lt;em&gt;Commentarii Academiae Scientiarum Imperialis Petropolitanae,&lt;/em&gt; &lt;strong&gt;8&lt;/strong&gt;, 128-140) was published in 1736.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.dartmouth.edu/~euler/pages/E053.html&quot;&gt;This wepage&lt;/a&gt; gives the year 1741, &lt;a href=&quot;http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html&quot;&gt;this webpage&lt;/a&gt; gives the year 1736, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was Euler's paper in volume 8, pages 128-140 of the Proceedings of the&#xA;St. Petersburg Academy published in 1736 or in 1741?&lt;/p&gt;&#xA;" OwnerUserId="72" LastActivityDate="2015-02-08T18:03:00.163" Title="What year was Euler's paper on the Konigsberg bridge problem first published?" Tags="|mathematics|publishing|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1854" PostTypeId="2" ParentId="1853" CreationDate="2015-01-23T07:06:55.693" Score="2" Body="&lt;p&gt;You have to check into &lt;a href=&quot;http://eulerarchive.maa.org/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Euler Archive&lt;/a&gt; : search by publications under : &lt;em&gt;Commentarii Academiae Scientiarum Petropolitanae&lt;/em&gt; :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;n*53 : &lt;em&gt;Solutio problematis ad geometriam situs pertinentis&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to the records, it was presented to the St. Petersburg Academy on August 26, 1735.&lt;/p&gt;&#xA;&lt;p&gt;Originally published in &lt;a href=&quot;http://eulerarchive.maa.org/docs/originals/E053.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Commentarii academiae scientiarum Petropolitanae&lt;/a&gt; 8, 1741, pp. 128-140.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-01-23T07:06:55.693" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1855" PostTypeId="2" ParentId="1853" CreationDate="2015-01-23T07:36:37.250" Score="5" Body="&lt;p&gt;Thanks to Mauro Allegranza, I visited the &lt;a href=&quot;http://eulerarchive.maa.org/index.html&quot; rel=&quot;noreferrer&quot;&gt;Euler Archive&lt;/a&gt; and found &lt;a href=&quot;http://eulerarchive.maa.org/docs/other/journals/CASP.html&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The &lt;em&gt;Commentarii academiae scientiarum imperialis Petropolitanae&lt;/em&gt; (CASP) was the first journal series published by the St. Petersburg Academy. Its publication eventaully ran to 14 annual volumes, of which Euler published in numbers 2-14.&lt;/p&gt;&#xA;&lt;p&gt;One feature to note is that the publication lag for each journal was quite large; The years which appear as volume numbers can lag up to 10 years behind the actual year of publication. When possible, both dates are noted below.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And later on:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Volume 8: 1736, published 1741.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the date printed on the journal was 1736, but it was actually printed in 1741.  Perhaps I'll just write 1736(1741) when I cite it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/GGvEW.png&quot; alt=&quot;CASP volume 8 cover&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;(The image above was taken from &lt;a href=&quot;http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01130-X/S0273-0979-06-01130-X.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="72" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-01-23T08:03:18.320" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1856" PostTypeId="2" ParentId="1853" CreationDate="2015-01-23T09:22:33.180" Score="5" Body="&lt;p&gt;With all due respect to the Euler Archive, one should note that they are not a primary source. (Nor are they especially forthcoming in sourcing their data: &amp;quot;according the the records&amp;quot;...) &lt;em&gt;&lt;strong&gt;The&lt;/strong&gt;&lt;/em&gt; original work dating Euler's papers and their oral presentation is the Jacobi-Fuss correspondence &lt;a href=&quot;https://archive.org/stream/derbriefwechsel00fussgoog#page/n133/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Briefwechsel über die Herausgabe der Werke Leonhard Eulers&lt;/em&gt; (Ed. Paul Stäckel and Wilhelm Ahrens; Teubner, Leipzig, 1908)&lt;/a&gt;, which in this case gives&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/d9L1x.png&quot; alt=&quot;Jacobi-Fuss&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Note that the citation is in just the form @JoelReyesNoche suggests.) This information was then subsumed in G. Eneström's &lt;em&gt;Verzeichnis der Schriften Leonhard Eulers&lt;/em&gt; (Jber. Deutsche Math.-Verein. Ergänzungsbände &lt;strong&gt;4&lt;/strong&gt; (1913) 1-388), and of course in the (still ongoing) &lt;em&gt;Herausgabe&lt;/em&gt; of Euler's works.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-01-23T09:58:11.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1857" PostTypeId="1" AcceptedAnswerId="1860" CreationDate="2015-01-23T12:28:59.787" Score="1" ViewCount="245" Body="&lt;p&gt;Some authors, namely Daniel Dennett and Douglas Hofstadter, argue that anything capable of passing the Turing test is necessarily conscious (Hofstadter, D. R., &amp;amp; Dennett, D. C. (2006). The Mind's I: Fantasies and Reflections on Self &amp;amp; Soul).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others use the M Thesis - any physical system can be simulated by Turing Machine simulation - to make statements about the brain and conscience: It can be fully simulated by a machine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Turing work make any impact on Psychology and Neurosciences or it was mostly ignored? If it did have an impact, which author bridged the gap between Math\CS to the other sciences?&lt;/p&gt;&#xA;" OwnerUserId="1739" LastEditorUserId="1739" LastEditDate="2015-01-27T15:01:46.680" LastActivityDate="2015-01-27T15:01:46.680" Title="Did Turing ideas make any impact on Psychology or Brain Science? Why?" Tags="|mathematics|computer-science|artificial-intelligence|neuroscience|psychology|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1858" PostTypeId="2" ParentId="256" CreationDate="2015-01-23T14:17:16.880" Score="7" Body="&lt;p&gt;Andre Weil's approach to algebraic geometry, set out in his book Foundations of Algebraic Geometry, was a breakthrough for its time because it was the first language for algebraic geometry that could handle abstract algebraic varieties that were not a priori subvarieties of affine or projective space (analogous to the distinction between submanifolds of Euclidean space and abstract manifolds). Weil's Foundations provided the terminology and viewpoint of the field for roughly 10 years in the mid-20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grothendieck's approach to algebraic geometry, based on schemes and not directly building on Weil's work, completely supplanted Weil's Foundations to the extent that Weil's Foundations are largely forgotten today and important papers from the 1950s and later written in the language of Weil's Foundations are very hard to read unless they can be translated into the modern language. See &lt;a href=&quot;https://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point&quot;&gt;https://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point&lt;/a&gt; for a discussion of this last point and chapter 8 of Reid's Undergraduate Algebraic Geometry (&lt;a href=&quot;http://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf&lt;/a&gt;) for a comparison of the three main waves of rigor in algebraic geometry in the 20th century. While Grothendieck's approach could be considered a successor to Weil's, it was not a logical descendant and therefore I think this example fits the question.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-01-23T14:25:39.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1859" PostTypeId="2" ParentId="256" CreationDate="2015-01-23T14:37:40.043" Score="1" Body="&lt;p&gt;Let us think of something more controversial: how about Freud's theory of the ego, super-ego and id; does anyone believe in this stuff any more?&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-01-23T14:37:40.043" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1860" PostTypeId="2" ParentId="1857" CreationDate="2015-01-24T11:40:29.340" Score="2" Body="&lt;p&gt;Turings works are highly formal and hardcore mathematical. Actually, not even a simple mathematician would understand it, if he is not on a similar area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Turing-test is a very different thing from the Turing-machines. The Turing-machines aren't about consciousness, they are about computability. Turing reached very… surprising results, from a philosophical viewpoint they are similar in the computer science as Gödels' results were in the mathematics. For example, there are functions which can't be calculated by any computer. Actually, most of them aren't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Turing-test is rather a philosophical construction. Afaik the main problem for the psychologists, that for them is everything which is non-intelligent, per definitionem out-of-scope. &lt;em&gt;(Off-topic, he-he).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;AFAIK the philosophers of the present time like to play a lot with every scientific results, especially on the area of the artificial intelligence. They interact relative heavily with current results / possibilities. Around half of the &lt;a href=&quot;http://aima.cs.berkeley.edu&quot; rel=&quot;nofollow&quot;&gt;leading text-book in most university-grad Artificial Intelligence courses&lt;/a&gt;, is about philosophical questions and not about the actual algorithms.&lt;/p&gt;&#xA;" OwnerUserId="217" LastActivityDate="2015-01-24T11:40:29.340" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1861" PostTypeId="1" AcceptedAnswerId="1878" CreationDate="2015-01-24T12:05:30.697" Score="7" ViewCount="265" Body="&lt;p&gt;When you think of taxonomies, the first thing that comes to mind is Linneaus. But taxonomies were and continue to be used for all kinds of classification, from biology to astronomy and even linguistics and philosophy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's hard, however, to find an overview of the evolution of taxonomies. Didn't Plato already use taxonomies? So maybe you guys, with your diverse backgrounds, can mention some of the early historical examples of taxonomies used in various sciences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks a lot. &lt;/p&gt;&#xA;" OwnerUserId="1744" LastEditorUserId="55" LastEditDate="2015-01-29T00:08:21.470" LastActivityDate="2015-02-01T21:26:35.510" Title="What are the early examples of taxonomies in the history of science?" Tags="|philosophy-of-science|taxonomy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1862" PostTypeId="2" ParentId="1861" CreationDate="2015-01-24T14:01:12.083" Score="4" Body="&lt;p&gt;My favorite one is the ancient Chinese classification of animals from  the&#xA;Encyclopaedia of Benevolent Knowledge:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Celestial_Emporium_of_Benevolent_Knowledge&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Celestial_Emporium_of_Benevolent_Knowledge&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the primary source did not survive&#xA;:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-01-24T14:01:12.083" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1863" PostTypeId="1" AcceptedAnswerId="1872" CreationDate="2015-01-24T15:18:18.220" Score="18" ViewCount="10204" Body="&lt;p&gt;As we know, even Archimedes did soon some experimental calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question were, who calculated first time the exact formulas ($V=\frac{4\pi}{3}r^3$, $A=4\pi r^2$)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I know, these formulas require the higher understanding of differential calculus, thus I think it happened after Newton and Leibnitz. But who did them?&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="55" LastEditDate="2017-05-03T22:30:36.777" LastActivityDate="2024-02-26T21:05:23.733" Title="Who calculated for the first time the volume (and surface area) of the sphere exactly?" Tags="|mathematics|calculus|euclidean-geometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1864" PostTypeId="2" ParentId="1863" CreationDate="2015-01-24T16:15:08.657" Score="14" Body="&lt;p&gt;Archimedes calculated the exact formulas (in the way that the ancient Greeks gave formulas) in his book &lt;a href=&quot;http://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder&quot; rel=&quot;noreferrer&quot;&gt;On the Sphere and Cylinder&lt;/a&gt;. This was not &quot;experimental&quot;: He gave a full geometric proof, rigorous for its time period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He considered this his greatest work. He asked that a &lt;a href=&quot;http://en.wikipedia.org/wiki/Archimedes&quot; rel=&quot;noreferrer&quot;&gt;diagram representing his proof&lt;/a&gt; be inscribed on his tomb. This was apparently done as at least one visitor to later Syracuse reported seeing the diagram.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people think that Archimedes discovered calculus and found the formulas in that way, but hid his discovery. Newton did much the same thing later, using Calculus to discover much about gravity but using geometric proofs when he wrote about gravity. Newton finally revealed his own work in Calculus when forced by Leibniz. Newton realized that Calculus would be controversial. Perhaps Archimedes did as well.&lt;/p&gt;&#xA;" OwnerUserId="426" LastEditorUserId="426" LastEditDate="2015-01-24T18:32:34.950" LastActivityDate="2015-01-24T18:32:34.950" CommentCount="23" ContentLicense="CC BY-SA 3.0" />
  <row Id="1865" PostTypeId="1" AcceptedAnswerId="1868" CreationDate="2015-01-24T17:25:41.143" Score="17" ViewCount="4480" Body="&lt;p&gt;I am very interested in the history of $\pi$. I am first trying to find out who calculated it. Many sources have different answers, from the ancient Egyptians, to Archimedes, to the Babylonians. I still can't find an answer to who first discovered $\pi$, or found a way to calculate it to any degree of accuracy. So who, or which group of people, were the first one(s) to discover/calculate $\pi$?&lt;/p&gt;&#xA;" OwnerUserId="1721" LastEditorUserId="3338" LastEditDate="2018-05-11T21:22:23.057" LastActivityDate="2019-09-10T11:21:06.320" Title="Who was the first to calculate $\pi$?" Tags="|mathematics|discoveries|geometry|euclidean-geometry|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1866" PostTypeId="2" ParentId="1848" CreationDate="2015-01-24T18:30:47.407" Score="6" Body="&lt;p&gt;I hope my memory of this is right...  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;James Clerk Maxwell came up with some differential equations to describe electricity and magnetism.  As usual when differential equations are solved, there was a certain constant that appeared.  Measurements showed the constant had value $3 \times 10^{10}$ cm/sec ... the speed of light.  Nowadays we call light an &quot;electromagnetic radiation&quot;, acknowledging its connection to electricity and magnetism.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2015-01-27T13:16:12.710" LastActivityDate="2015-01-27T13:16:12.710" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1867" PostTypeId="2" ParentId="1865" CreationDate="2015-01-24T21:19:25.987" Score="9" Body="&lt;p&gt;What do you mean &quot;correctly or not&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a brief history. The Bible has a sentence which can be interpreted as implying that $\pi=3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The EXISTENCE of $\pi$ (the ratio of circumference to diameter) was rigorously proved by Archimedes. He also calculated it approximately.&#xA;For many centuries it was called the &quot;Archimedes number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As centuries passed, they calculated it with larger and larger precision. Today it is known&#xA;to more than a billion decimal digits and more. In 1990-th there was an interesting article in New Yorker magazine about the brothers Chudnovskii who evaluated the first billion digits. (With a huge computer they specially assembled for this purpose).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\pi$ is a transcendental number, so you cannot &quot;evaluate&quot; it exactly (there is no finite or periodic decimal (or other) expression). This was proved by Lindemann in 1882.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark.&#xA;Let me also add that there was a crank in 19-th century who &quot;proved&quot; that $\pi=4$.&#xA;I think this is recorded in the Guinness book of records as the LEAST precise value of $\pi$ ever proposed. He offered his work as a gift to the State of Indiana. Indiana parliament had to&#xA;decide whether to accept the gift. They could not decide. The decision was postponed.&#xA;It is still postponed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives some evil (or not informed) people a reason to say that &quot;Indiana parliament&#xA;LEGISLATED that $\pi=4$&quot;. Don't believe these people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: My mistake: certainly &quot;existence of $\pi$&quot; was known to Euclid. Conifold: thanks for the correction. To address the same remark of Conifold: Euclid and Archimedes (and possibly Eudoxus) certainly understood perfectly what is a &quot;real number&quot; though they used somewhat different terminology of &quot;proportions&quot;. Theory of proportions of Euclid is equivalent to the theory of real numbers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-26T21:55:19.503" LastActivityDate="2015-01-26T21:55:19.503" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1868" PostTypeId="2" ParentId="1865" CreationDate="2015-01-24T22:59:50.123" Score="17" Body="&lt;p&gt;It depends on the meaning of &quot;calculate&quot;, since &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is a transcendental number it can not be &quot;calculated&quot; in the usual meaning of the word. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first analytic formula (in the form of an infinite series) that in principle can calculate &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to any required accuracy is probably due to medieval Indian mathematician &lt;a href=&quot;http://en.wikipedia.org/wiki/Madhava_of_Sangamagrama#The_value_of_.CF.80_.28pi.29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Madhava&lt;/a&gt;, who was first to conceive of infinite series explicitly, or one of his successors. In Europe this series was rediscovered by Leibniz, and is usually called Leibniz series in Western literature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A semi-geometric &quot;calculation&quot; procedure capable in principle of producing arbitrary accuracy is much older. It consists of approximating a circle by inscribed and circumscribed polygons, and can be traced to ancient Greek orator &lt;a href=&quot;http://en.wikipedia.org/wiki/Antiphon_%28orator%29#Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Antiphon the Sophist&lt;/a&gt;. This method was mathematically justified by &lt;a href=&quot;http://www.math.tamu.edu/~dallen/masters/Greek/eudoxus.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eudoxus of Cnidus&lt;/a&gt; using what is now called method of exhaustion, and his justification is presented in &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book XII&lt;/a&gt; of Euclid's Elements. Archimedes perfected the method in &lt;a href=&quot;http://www.cs.xu.edu/math/math300/07f/archcircle.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Measurement of the Circle&lt;/a&gt;. He proved rigorously that the ratio of the circle to the square on its radius was the same as the ratio of the circumference to the diameter, so it could be computed both ways. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Approximating the circumference with polygon perimeters is much simpler than approximating the circle area with polygon areas as Antiphon suggested. Using &lt;span class=&quot;math-container&quot;&gt;$96$&lt;/span&gt;-gons, Archimedes obtained what is now presented as the double estimate &lt;span class=&quot;math-container&quot;&gt;$3\frac{10}{71}&amp;lt;\pi&amp;lt;3\frac17$&lt;/span&gt;, although to Archimedes &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; was not a number, and the result was phrased geometrically in terms of &lt;a href=&quot;http://www.lamath.org/journal/vol5no2/Ratio_Proportion.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;ratios of magnitudes&lt;/a&gt;. In 17th century &lt;a href=&quot;https://books.google.com/books?id=nMlKxQ0ZUbMC&amp;amp;pg=PA283&amp;amp;lpg=PA283&amp;amp;dq=Huygens+polygons+circle+pi&amp;amp;source=bl&amp;amp;ots=5-T-rjxsRy&amp;amp;sig=f2WsBLtbhA6UM6pkFlpv8YYLafw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=IyjEVLnENsmiyATX_ICoDg&amp;amp;ved=0CEIQ6AEwBg#v=onepage&amp;amp;q=Huygens%20polygons%20circle%20pi&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huygens&lt;/a&gt; refined the method further, and obtained the most accurate estimates to date, but after that analytic methods became more effective. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, ancient Greeks also had a different concept of calculation, a purely geometric one, that was dominant in their time. A geometric magnitude was considered &quot;calculated&quot; if one could give a geometric construction for it. In the case of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; this meant constructing a square with area equal to the area of a given circle (or, after Archimedes, a segment with the length equal to half the circle's circumference). With straightedge and compass this is impossible (but it was only shown by Lindeman in 19th century), but Greeks also entertained more broad notions of &quot;construction&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this broader sense, the circle was first squared by &lt;a href=&quot;http://www.educ.fc.ul.pt/icm/icm2003/icm14/Dinostratus.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dinostratus&lt;/a&gt; around 350 BC using a curve, now known as the &lt;a href=&quot;http://www.ms.uky.edu/~carl/ma330/hippias/hippias2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadratrix&lt;/a&gt;. The curve itself was invented a century earlier by Hippias, who generated it by combining uniform linear and circular motions, and used it for angle trisection. In modern terms, from a segment of length &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; it produces (&quot;calculates&quot;) a segment of length &lt;span class=&quot;math-container&quot;&gt;$2/\pi$&lt;/span&gt;, and from that it is a simple matter to get a segment of length &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; with straightedge and compass. Later Archimedes &quot;calculated&quot; &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; in the same spirit using another curve generated by uniform linear and circular motions, the &lt;a href=&quot;http://mypages.iit.edu/~maslanka/151notes1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedean spiral&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-10T11:21:06.320" LastActivityDate="2019-09-10T11:21:06.320" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="1870" PostTypeId="1" AcceptedAnswerId="1871" CreationDate="2015-01-26T02:50:38.957" Score="3" ViewCount="270" Body="&lt;p&gt;The term &quot;world-line&quot; is a little odd in English. Google n-grams shows the English term going back to 1915 in the books google has scanned. Is its origin in Minkowski, Raum und Zeit (1909), Jahresberichte der Deutschen Mathematiker-Vereinigung, 75-88 ( &lt;a href=&quot;http://de.wikisource.org/wiki/Raum_und_Zeit_%28Minkowski%29&quot; rel=&quot;nofollow&quot;&gt;http://de.wikisource.org/wiki/Raum_und_Zeit_%28Minkowski%29&lt;/a&gt;, English translation at &lt;a href=&quot;http://en.wikisource.org/wiki/Space_and_Time_%28Prasad%29&quot; rel=&quot;nofollow&quot;&gt;http://en.wikisource.org/wiki/Space_and_Time_%28Prasad%29&lt;/a&gt; )? The Minkowski paper seems to make it somewhat more understandable. Although the title of the paper translates as &quot;space and time,&quot; in the text it seems that Minkowski uses &quot;world&quot; to mean &quot;spacetime:&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Die Mannigfaltigkeit aller denkbaren Wertsysteme x, ,y, ,z, ,t soll die Welt heißen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Prasad's translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The totality of all conceivable systems x, y, z, t may be called the world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it seems like Minkowski didn't coin &quot;Raumzeit,&quot; &quot;spacetime,&quot; and instead used &quot;world&quot; to mean &quot;spacetime,&quot; so that &quot;world-line&quot; means &quot;spacetime-line.&quot; Have I got this right historically?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="26" LastEditDate="2015-01-26T08:17:50.867" LastActivityDate="2015-01-26T09:53:50.527" Title="Origin of &quot;world-line?&quot;" Tags="|terminology|relativity-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1871" PostTypeId="2" ParentId="1870" CreationDate="2015-01-26T09:53:50.527" Score="3" Body="&lt;p&gt;The Oxford English Dictionary has this definition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;world line   n.  [after German Weltlinie (H. Minkowski 1909, in Physikalische Zeitschr. 10 104)] Physics a line in space-time comprising the successive points occupied by a particle, celestial object, etc., throughout its history; also in extended use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and cites as its earliest occurrence in English this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1914   Science 20 Nov. 725/1   Every point in space, even if at rest, describes a world line, which may be referred to and is contained between the two extremities of the time axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words: &quot;world line&quot; (1914) is a calque on &quot;Weltlinie&quot; (1909).&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-01-26T09:53:50.527" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1872" PostTypeId="2" ParentId="1863" CreationDate="2015-01-26T20:01:20.140" Score="35" Body="&lt;p&gt;This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to history, is that only Euler presented volume formulas in this form in his textbooks after 1737.&lt;/p&gt;&#xA;&lt;p&gt;The principal step was no doubt made by Archimedes in &lt;a href=&quot;http://books.google.com/books?id=msyO12v3tqcC&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Sphere and Cylinder&lt;/a&gt;, where he proved rigorously that &lt;span class=&quot;math-container&quot;&gt;$V_S:V_C=2:3$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$V_S$&lt;/span&gt; is the volume of a sphere and &lt;span class=&quot;math-container&quot;&gt;$V_C$&lt;/span&gt; is the volume of the circumscribed cylinder (he also gave proportion for the surface area). &amp;quot;Obviously&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$V_C=\pi r^2\times2r$&lt;/span&gt;, so we get the modern formula, right? Although this is a popular way of projecting modern concepts onto history it sells short the ingenuity of ancient Greeks and the difficulties they managed to overcome, not to mention the work of countless others who brought about our mathematical paradise. Here is one problem: how does one assign a number to a volume, or even to a length for that matter? Today we use real numbers and integration theory, but ancient Greeks had none of that. Their ingenious solution was to make do without both. Geometric magnitudes (volumes, areas, lengths) weren't assigned numbers at all, they were related to other like magnitudes as ratios. These ratios weren't numbers, they could be compared but not added, and only occasionally they could be expressed as ratios of whole numbers, the only numbers proper. This is why Archimedes expressed the &amp;quot;formula&amp;quot; the way he did.&lt;/p&gt;&#xA;&lt;p&gt;But to get to our modern formula even in the ratio form like &lt;span class=&quot;math-container&quot;&gt;$V_S:r^3=4\pi:3$&lt;/span&gt; there is another problem standing in the way. This proportion relates the volume of the sphere not to a cylinder, but to the cube on its radius. Problem is, &lt;span class=&quot;math-container&quot;&gt;$4\pi:3$&lt;/span&gt; is not a ratio of whole numbers, unlike &lt;span class=&quot;math-container&quot;&gt;$2:3$&lt;/span&gt;. Of course, Archimedes didn't know that for sure, but Pythagoreans already got into hot water by assuming it for the side and the diagonal of a square, and later proving otherwise. So Archimedes, like geometers before and after him, did not write it that way, and they did not write &lt;span class=&quot;math-container&quot;&gt;$A=\pi r^2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$A:r^2=\pi$&lt;/span&gt; for the area of a circle either. Not in equations and not in words. Yet again, Greeks rose to the occasion despite the absence of our modern machinery. The theory of proportion, an ingenious invention of Eudoxus of Cnidus presented in Book V of Euclid's Elements, allowed them to make sense of estimates like &lt;span class=&quot;math-container&quot;&gt;$s:t&amp;lt;A:r^2&amp;lt;p:q$&lt;/span&gt; with whole numbers &lt;span class=&quot;math-container&quot;&gt;$p,q,s,t$&lt;/span&gt;. Archimedes and many of his successors did prove many such estimates without invoking mysterious entities, which remained undefined for almost two millenia hence, and without the modern idea that unbeknownst to them those ratios were &amp;quot;approximating&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If this last step seems trivial to us today let me point out that in 17th century Cavalieri and Roberval were still presenting their volumes and areas as ratios to other simpler volumes and areas, and recall the &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of zero&lt;/a&gt;, which was not understood or used as a number for centuries after Babylonians and Alexandrian astronomers were using a symbol for it as a placeholder. With &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; there wasn't even a symbol. Only at the end of middle ages some Arabs and Europeans started thinking of irrationals as some kind of numbers, while giving them telling nicknames, like &amp;quot;deafmute numbers&amp;quot;. And this was for irrationals like &lt;span class=&quot;math-container&quot;&gt;$1+\sqrt{5}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt; given by algebraic formulas.&lt;/p&gt;&#xA;&lt;p&gt;It appears that the first person to contemplate that &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; were also &amp;quot;some kind of numbers&amp;quot; in print was &lt;a href=&quot;http://en.wikipedia.org/wiki/James_Gregory_%28mathematician%29#Vera_Circuli_et_Hyperbolae_Quadratura&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Gregory&lt;/a&gt; in The True Squaring of the Circle and of the Hyperbola published in 1667. He was also the first to suggest a possibility that quadrature of the circle was unsolvable with straightedge and compass, although his argument for it was flawed. Even then it took time for the idea to percolate until &lt;a href=&quot;http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Jones&lt;/a&gt; in 1706 was bold enough to assign a symbol to the new &amp;quot;number&amp;quot;, our modern &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, while still saying &amp;quot;the exact proportion between the diameter and the circumference can never be expressed in numbers&amp;quot;. Integration theory was sufficiently developed by then to be comfortable with volumes and areas as numbers as well, so Jones could dispose with the ratios and write &lt;span class=&quot;math-container&quot;&gt;$A=\pi r^2$&lt;/span&gt;. And when Euler adopted the symbol 30 years later, and made it famous, he could finally write  &lt;span class=&quot;math-container&quot;&gt;$V_S=\frac43\pi r^3$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-02-26T21:05:23.733" LastActivityDate="2024-02-26T21:05:23.733" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="1873" PostTypeId="2" ParentId="1865" CreationDate="2015-01-27T16:55:26.300" Score="8" Body="&lt;p&gt;Since the question specifically says &quot;to any accuracy,&quot; I will assume you mean approximations as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, the first recorded approximation of $\pi$ comes from the Babylonians, who not only had an awareness of it as being a specific constant, had approximated it's value to $3\frac{1}{8}$ or $3.125$. This is recorded in a tablet fond near &lt;a href=&quot;http://en.wikipedia.org/wiki/Susa&quot; rel=&quot;noreferrer&quot;&gt;Susa&lt;/a&gt; dating from 2000 BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both the Egyptians and Archimedes had a value for it but these date from much later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus&quot; rel=&quot;noreferrer&quot;&gt;Rhind Papyrus&lt;/a&gt; dating from around 1650 BC we find an Egyptian's scribe method for obtaining the area of a circle which is equivalent to using the value  $\pi = 3\frac{1}{6}$ or $3.166$ repeating. This is perhaps nearly as good as the Babylonian approximation, but the Egyptians did not apparently have an awareness of it as being a specific constant, this is merely the effective value of it arising from their method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Archimedes there is given an approximation of $\pi$ more accurate than either the Egyptian or Babylonian, but this dates from almost 2000 years after the recorded Babylonian value.&lt;/p&gt;&#xA;" OwnerUserId="1760" LastEditorUserId="1760" LastEditDate="2015-01-27T17:49:06.877" LastActivityDate="2015-01-27T17:49:06.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1874" PostTypeId="1" AcceptedAnswerId="1877" CreationDate="2015-01-28T02:33:29.337" Score="5" ViewCount="4167" Body="&lt;p&gt;I get that Galileo Galilei was a major contributor in proving the helio-centric theory, and same goes to Sir Isaac Newton. The battle for the theory to become law took centuries to prove. People finally agreed that the Earth was not in the center of the Universe. But when?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How long did take for people to finally accept  Copernicus’ theory? The time starts on the year  Copernicus’ theory was introduced to the world.&lt;/p&gt;&#xA;" OwnerUserId="1721" LastEditorUserId="217" LastEditDate="2019-08-16T23:47:49.523" LastActivityDate="2019-08-16T23:47:49.523" Title="How long did it take to prove that the Earth revolves around the Sun?" Tags="|discoveries|astronomy|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="1875" PostTypeId="2" ParentId="1874" CreationDate="2015-01-28T17:19:16.790" Score="3" Body="&lt;p&gt;The biggest factor in the general acceptance of the helio-centric model was probably Kepler's Rudolphine Tables, published in 1627.  These used Kepler's heliocentric system with elliptical orbits* and were a substantial improvement on tables found using the Ptolemaic or Tychonic systems.  Their practical use in predicting planetary positions for astronomers, astrologers, doctors, etc. in the following decades was probably the biggest factor in swaying Europe's educated population in favour of heliocentricism.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*(one of the reasons Copernicus's system wasn't accepted early on was that tables created using his circular orbits weren't any more accurate then those found with the Ptolemaic system).&lt;/p&gt;&#xA;" OwnerUserId="1731" LastActivityDate="2015-01-28T17:19:16.790" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1876" PostTypeId="1" CreationDate="2015-01-28T17:27:58.197" Score="2" ViewCount="907" Body="&lt;p&gt;Was Kepler aware that Venus and Mars have a golden section relationship other than that he attributed to them in his &lt;a href=&quot;https://en.wikipedia.org/wiki/Platonic_solid&quot; rel=&quot;noreferrer&quot;&gt;5 perfect solids&lt;/a&gt; model of the Solar System? The dodecahedron and the icosahedron, which Kepler placed between the Earth and Mars and Venus, both have the golden section in their structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The value for Venus's semi-major axis (in AU) to the power of 21 multiplied by that of Mars to the power of 55 equals $\varphi$ (&lt;a href=&quot;https://en.wikipedia.org/wiki/Golden_ratio&quot; rel=&quot;noreferrer&quot;&gt;the golden section&lt;/a&gt;) to the power of 34! These are &lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci_number&quot; rel=&quot;noreferrer&quot;&gt;Fibonacci numbers&lt;/a&gt; and any such sequence means that using powers above or below in the Fibonacci sequence (or the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lucas_sequence&quot; rel=&quot;noreferrer&quot;&gt;Lucas sequence&lt;/a&gt;) will give an approximate same answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://solarsystem.nasa.gov/planets/index.cfm&quot; rel=&quot;noreferrer&quot;&gt;NASA&lt;/a&gt;: Venus's semi-major axis (0.72332102 AU) to the power of 21 times Mars's semi-major axis (1.52371243 AU) to the power of 55 equals $\varphi$ to the power of 34. The discrepency is 1.000088909, but it would be the 76th root of this, 1.00000117, to correct the equation. Venus's semi-major axis to the power of 13 times Mars's semi-major axis to the power of 34 equals $\varphi$ to the power of 21, which is is 1.002919389. The 47th root is 1.000062026. Venus's semi-major axis to the power of 18 times Mars's semi-major axis to the power of 47 is $\varphi$ to the power of 29, which is 1.008694078. The 65th root is 1.000133186.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What understanding of these relationships and calculations did Kepler demonstrate in his published works?&lt;/p&gt;&#xA;" OwnerUserId="1763" LastEditorUserId="4251" LastEditDate="2021-09-13T09:03:49.357" LastActivityDate="2021-09-13T09:03:49.357" Title="Was Kepler aware of the following golden relationship between the planets?" Tags="|physics|astronomy|17th-century|kepler|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="1877" PostTypeId="2" ParentId="1874" CreationDate="2015-01-28T19:58:28.900" Score="5" Body="&lt;p&gt;Heliocentrism did not start with Copernicus, and Copernicus's theory was not the one accepted, he still had circular orbits and equants. It is Kepler's version that became the consensus after Newton derived elliptic orbits from the inverse square law in Principia (1687). That &quot;proved&quot; it to most scientists. We can count from around 270 BC, when Aristarchus first suggested the idea, or from 1533, when Copernicus reintroduced it in a much more elaborate from, or from 1609 when Kepler replaced circles by ellipses. Wikipedia gives a very detailed &lt;a href=&quot;http://en.wikipedia.org/wiki/Heliocentrism&quot; rel=&quot;noreferrer&quot;&gt;account of history&lt;/a&gt; that would answer the question for any starting date, and under any interpretation of &quot;prove&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep in mind however that under both Newtonian and relativistic physics the question of what revolves around what is subtle. A reference frame can be attached to both the Sun and the Earth, and either one will then revolve around the other in it. We need a way to select a preferred frame, for example according to the simplest description of dynamics. For the two of them that would be the frame attached to the center of their masses (which is inside the Sun because Earth is so much lighter, but not at its center), and they both revolve around it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-01-28T20:07:20.847" LastActivityDate="2015-01-28T20:07:20.847" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1878" PostTypeId="2" ParentId="1861" CreationDate="2015-01-29T18:57:51.863" Score="4" Body="&lt;p&gt;A hierarchic process of producing taxonomies was formalized by 3rd century AD logician Porphyry in his presentation of Aristotle's classification of categories, and became popular during middle ages in a diagrammatic form of &lt;a href=&quot;http://en.wikipedia.org/wiki/Porphyrian_tree&quot; rel=&quot;nofollow&quot;&gt;&quot;Porphyrian tree&quot;&lt;/a&gt;. One starts with the highest category in a field, &quot;summum genus&quot;, and proceeds in successive steps by using &quot;differentia&quot; to introduce lower genera and ultimately species. One easily recognizes the taxonomy scheme implemented in biology by Linneaus, and used until today. But Porphyry only summarized an earlier tradition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the earliest examples, even before Plato, was Greek geometer &lt;a href=&quot;http://en.wikipedia.org/wiki/Oenopides#Geometry&quot; rel=&quot;nofollow&quot;&gt;Oenopides&lt;/a&gt; (c. 450 BC) distinguishing theorems, which demonstrate a property, from problems, which provide a construction. There is a similar distinction between axioms, asserting what is easy to see, and postulates, asserting what is easy to construct. Plato's dialogues are explicitly structured to create such classifications by making dichotomic distinctions in Socratic arguments. One example is the famous quadrivium: mathematical sciences are first divided into those dealing with quantity, and those dealing with &quot;qualified quantity&quot;. The former are subdivided into arithmetic and music, and the latter into geometry and astronomy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geminus (c. 50 BC) gives a much more elaborate classification (quoted from &lt;a href=&quot;https://www.academia.edu/8016013/Two_Approaches_to_Foundations_in_Greek_Mathematics_Apollonius_and_Geminus&quot; rel=&quot;nofollow&quot;&gt;Acerbi's paper&lt;/a&gt;). &quot;The classification envisages eight primary domains: arithmetic, geometry (namely, those dealing with intelligibles), mechanics, astronomy, optics, geodesy, canonics, logistics (namely, those attending to sensibles). Further divisions are set out for geometry (into plane geometry and stereometry), arithmetic (theory of linear, plane, and solid numbers), optics (optics proper, catoptrics, scene-painting), mechanics (construction of war engines, wonder-working, science of equilibrim and centers of gravity, construction of celestial spheres, and in general everything pertaining to kynetics), astronomy (gnomonics, meteoroscopy, and dioptrics)&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is Geminus's classification of &quot;lines&quot; (geometric curves), which starts with dividing them into incomposite/composite.&quot;The second kind includes lines “that are inflected and make an angle”. The first kind is further divided into lines “that are finite and make a figure”/lines “that can be produced indefinitely.” The former class includes the circular line, the contour of the ellipse, the cissoid. The latter class includes parabola, hyperbola, conchoid, straight line etc... Of the latter, some are asymptotic, some are not. Of the former, some are in one single plane, some are not. Of the former, some are equidistant, as parallel straight lines are, some are progressively diminishing the distance between themselves, as an hyperbola does with respect to a straight line and as the conchoid does with respect to a straight line.&quot; Some anticipation of modern topological distinctions there.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-01T21:26:35.510" LastActivityDate="2015-02-01T21:26:35.510" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1881" PostTypeId="1" CreationDate="2015-01-30T09:49:06.640" Score="7" ViewCount="1661" Body="&lt;p&gt;I'm very interested to know how coordinate systems were discovered and why mathematicians discovered them? Actually I want to know what things motivated mathematicians to discover and develop coordinate geometry or analytic geometry. What problems they faced in the past for which they felt the need for discovery and development of coordinate system?&lt;/p&gt;&#xA;" OwnerUserId="1716" LastEditorUserId="26" LastEditDate="2015-03-01T14:05:58.670" LastActivityDate="2015-03-01T14:05:58.670" Title="Discovery and Development of Coordinate Systems" Tags="|discoveries|euclidean-geometry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1882" PostTypeId="2" ParentId="1881" CreationDate="2015-01-30T12:08:04.633" Score="2" Body="&lt;p&gt;The coordinate system, as we know it now, was developed by Descartes. However, a system of spherical coordinates to measure longitude and latitude goes back at least to Eratosthenes and Hipparchus.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-01-30T12:08:04.633" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1883" PostTypeId="2" ParentId="1881" CreationDate="2015-01-30T14:03:09.563" Score="3" Body="&lt;p&gt;The earliest coordinate systems appear in astronomy and geography. I think it is completely clear why they were necessary.  As fdb noticed Eratosthenes and Hipparchus already used them &#xA;but they must have been used much earlier.&#xA;Indeed, Hipparchus discovered precession of equinoxes. How can this be done?&#xA;One has to compare the star coordinates that you measure with the same stars coordinates&#xA;measured many years earlier. So there is no doubt that there exited some star catalog before Hipparchus. (Somewhere I've seen the statement that Eudoxus made one. Neither Hipparchus catalog nor the earlier one survived to our days, but there is no doubt that they existed).&#xA;The person who made the first star catalog must have used some coordinate system. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The accuracy of measurement available at Hipparchus time was not sufficient to discover&#xA;precession from observations spanned over one life time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 1. According to Wikipedia, the earliest known star catalog that lists stars with their coordinates was the one by Timocharsis (320-260 bc). This catalog was used by Hipparchus to discover the precession.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2. In pure mathematics, it is of course credited to Descartes. One of the motivations he apparently had was to simplify the ancient theory of conic sections. Ancient theory of conic sections is known from the books of Apollonius, and this is really a hard reading, even today:-) Descartes was apparently the first to notice that these conic sections can be defined by simple quadratic equations which simplifies the theory so much that it can be even taught to undergraduates nowadays.&#xA;Source: M. Berger, Geometry revealed, Springer 2010.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-01-31T01:08:06.373" LastActivityDate="2015-01-31T01:08:06.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1884" PostTypeId="1" AcceptedAnswerId="1885" CreationDate="2015-01-30T14:34:55.787" Score="20" ViewCount="13034" Body="&lt;p&gt;This is by far one of the most challenging and popular HSM questions on the Net. Proofs are, countless discussions about it in math forums. The answers only led to two theories, which &lt;a href=&quot;http://en.wikipedia.org/wiki/Degree_(angle)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; does a great job at summing up.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.&lt;/p&gt;&#xA;&lt;p&gt;Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The two theories seem very legitimate. With some background search I came up with the fact that the use of &lt;a href=&quot;http://en.wikipedia.org/wiki/Astrolabe&quot; rel=&quot;noreferrer&quot;&gt;astrolabes&lt;/a&gt; by Persian astronomers (which used circles as its fundamental &amp;quot;measurement medium&amp;quot;) could actually lead to invention of such system by them.&lt;/p&gt;&#xA;&lt;p&gt;Nevertheless, Babylonians are known to be the inventors of sexagesimal system and additionally, their more &amp;quot;recent&amp;quot; studies were using such system in the advent of trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;A deeper study also indicates that Persian people could've actually be influenced by Babylonian &amp;quot;culture&amp;quot; to make use of their sexagesimal system. This makes logical decision harder: Maybe it was Babylonians who invented 360 degrees in a circle, or it could be Persians that did so with the sexagesimal system that was introduced to them.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, I'm not able to cite my non-English sources, but it would be great to hear an expert's advice on this.&lt;/p&gt;&#xA;&lt;p&gt;Who were the first to use degrees in a circle and divide it to 360 parts?&lt;/p&gt;&#xA;" OwnerUserId="1769" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-21T18:49:39.170" Title="Origin of 360 degrees?" Tags="|mathematics|archaeoastronomy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1885" PostTypeId="2" ParentId="1884" CreationDate="2015-01-30T15:40:16.540" Score="17" Body="&lt;p&gt;We have Babylonian astronomical texts going back at least to 2000 BC. The history of Babylonian astronomy is thus very well documented. They divided the ecliptic into 360 degrees, and each degree into 60 minutes, etc. Why 360? It is true that the solar year is very approximately 360 days, though it could be noted that the Babylonians used a luni-solar year, and thus had an average of 354 days in a common year, and 383-4 in an intercalated year. The number 360 has lots of factors; in particular 360 degrees can easily be divided into 12 zodiacal signs of 30 degrees each, corresponding to the approximate position of the sun in 12 lunar months.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The results of Babylonian astronomy were adopted by the Greeks in the classical period and the division of the ecliptic into 360 degrees was adopted notably by Ptolemy in his Almagest, the most influential astronomical book in late antiquity and the middle ages. It is likely that some Babylonian mathematical knowledge passed to the Persians in the pre-Islamic period, although this is difficult to document. What is certain is that the translation of Ptolemy’s Almagest into Arabic in the 9th century made Ptolemaic astronomy well-known to learned people in the Islamic world, including Persia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You talk about Persian astrolabes, but you do not make it clear what period you are talking about. There is no evidence for astrolabes in Persia before the Islamic period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Wikipedia is wrong to suggest that the ancient Persians - or indeed anyone - had a year of exactly 360 days. This is an old myth. If you have access to jstor you can read about the ancient Persian calendar here: &lt;a href=&quot;http://www.jstor.org/discover/10.2307/4299943?sid=21105210724371&amp;amp;uid=2&amp;amp;uid=2129&amp;amp;uid=3738032&amp;amp;uid=70&amp;amp;uid=4&quot; rel=&quot;noreferrer&quot;&gt;http://www.jstor.org/discover/10.2307/4299943?sid=21105210724371&amp;amp;uid=2&amp;amp;uid=2129&amp;amp;uid=3738032&amp;amp;uid=70&amp;amp;uid=4&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-01-31T15:12:40.910" LastActivityDate="2015-01-31T15:12:40.910" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1886" PostTypeId="1" CreationDate="2015-01-26T00:21:49.043" Score="5" ViewCount="638" Body="&lt;p&gt;I'd really never studied tensors until I started studying the Einstein Field Equations. Since then, I have realized they are fairly common tool in physics and pretty basic to understanding many areas. But for the majority of my initial introduction to special relativity, I never needed tensors. They never really came up. As I try to think about how Einstein moved from creating special relativity to general relativity, I wonder how he knew that tensors would be useful for general relativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What led him to think tensors would be needed to formulate the EFEs? &lt;/p&gt;&#xA;" OwnerUserId="1751" OwnerDisplayName="Stan Shunpike" LastEditorUserId="62" LastEditDate="2015-01-30T20:39:50.347" LastActivityDate="2015-01-30T20:39:50.347" Title="How did Einstein know tensors would be needed in the EFEs?" Tags="|physics|relativity-theory|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1888" PostTypeId="1" AcceptedAnswerId="1891" CreationDate="2015-01-31T14:47:46.497" Score="6" ViewCount="2117" Body="&lt;p&gt;The Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Bernoulli%27s_inequality&quot; rel=&quot;noreferrer&quot;&gt;“Bernoulli's inequality”&lt;/a&gt; says, that this inequality is called after the mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacob_Bernoulli&quot; rel=&quot;noreferrer&quot;&gt;Jacob Bernoulli&lt;/a&gt;. Was he the first one using the inequality? Why is this inequality named after him? When was the inequality used first?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the Talk page of the Wikipedia article is already &lt;a href=&quot;https://en.wikipedia.org/wiki/Talk:Bernoulli%27s_inequality#Please_add_History&quot; rel=&quot;noreferrer&quot;&gt;a thread&lt;/a&gt; about this. Unfortunately the link provided there does not work anymore.&lt;/p&gt;&#xA;" OwnerUserId="1772" LastEditorUserId="1772" LastEditDate="2015-01-31T17:45:39.317" LastActivityDate="2015-02-04T23:49:50.347" Title="First use of Bernoulli's inequality and its name" Tags="|mathematics|reference-request|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1890" PostTypeId="1" CreationDate="2015-01-31T20:46:31.157" Score="6" ViewCount="187" Body="&lt;p&gt;It s well known that in 1888, Italian Giovanni Schiaparelli announced finding a network of narrow lines on Mars, which he described as &quot;canali,&quot; which a lot of English speakers including American Astronomer Percival Lowell took to mean &quot;canals&quot; and assumed were created by intelligent beings.    Others point out that the word can as well mean &quot;channels,&quot; and even in English &quot;canal&quot; need not mean an artificially built thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Schiaparelli lived until 1910 and there was a huge fuss about these canals and as far as I can find on line there is no record at all of any response by Schiaparelli.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone here know of a response by  Schiaparelli?  Did he ever clarify whether he thought these &quot;canali&quot;  could have been created by intelligent beings?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2015-02-01T00:44:45.583" LastActivityDate="2015-02-02T15:32:53.550" Title="Did Schiaparelli comment on the martian canals debate?" Tags="|astronomy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1891" PostTypeId="2" ParentId="1888" CreationDate="2015-01-31T22:18:40.520" Score="5" Body="&lt;p&gt;&lt;strong&gt;Jacob Bernoulli&lt;/strong&gt; first published the inequality on the second page of his &lt;a href=&quot;https://archive.org/stream/jacobibernoulli00conggoog#page/n494&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Positiones Arithmeticae de Seriebus Infinitis&lt;/em&gt; (Basel, 1689)&lt;/a&gt;, as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\textrm{IV.}$$&#xA;  &lt;em&gt;Si sit Progressio Geometrica quaecunque A, B, C, D, E; &amp;amp; alia Arithmetica totidem terminorum A, B, F, G, H, incipiens ab iisdem terminis A &amp;amp; B, erunt reliquorum singuli in Geometrica singulis ordine sibi respondentibus in Arithmetica majores, tertius tertio, quartus quarto, ultimus ultimo, adeoque omnes omnibus.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bernoulli uses the inequality several times in the rest of his &lt;em&gt;Positiones&lt;/em&gt; (a 100+-pages treatise in 5 parts). Quoth André Weil in &lt;a href=&quot;https://zbmath.org//?q=an:921.01046&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Die Werke von Jakob Bernoulli. Band 4. Reihentheorie&lt;/em&gt; (Basel, 1993)&lt;/a&gt;, p.10: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(Prop. IV) C'est le lemme $(1 + x)^n &amp;gt; 1+nx$, cité encore dans les ouvrages scolaires sous le nom de Bemoulli. Celui-ci en donne une démonstration basée sur le livre V d'Euclide; mais bien entendu ce lemme dit simplement que pour un taux d'intérêt donné il est plus coûteux d'emprunter à intérêts composés qu'à intérêt simple, et Bernoulli, qui à l'exemple de Leibniz avait dû se poser déjà des problèmes d'intérêts composés, avait probablement conscience de ce fait.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt; According to Joseph E. Hofmann, &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=161779&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Über die Exercitatio Geometrica des M. A. Ricci&lt;/em&gt; (1963)&lt;/a&gt;, p.177, the inequality is actually due to &lt;strong&gt;Sluse&lt;/strong&gt; in his &lt;a href=&quot;https://books.google.com/books?id=ugJ1GtyH08kC&amp;amp;pg=PA114&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Mesolabum&lt;/em&gt; (1668 edition)&lt;/a&gt;, Chapter IV &quot;De maximis &amp;amp; minimis&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Es lässt sich (in moderner Umschrift) in drei Zeilen hinschreiben:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ist $x&amp;gt;0$ (...), dann ist&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$\frac{1+px}{1+(p-1)x}&#xA;&amp;lt;\frac{1+(p-1)x}{1+(p-2)x}&#xA;&amp;lt;\dots&#xA;&amp;lt;\frac{1+2x}{1+x}&#xA;&amp;lt;\frac{1+x}1\quad(\dots).\tag{4,11}$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Also ergibt sich durch “Zusammensetzen”, d.h. durch Multiplikation entsprechender Quotienten (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$1+px &amp;lt; (1+x)^p\quad(\dots).\tag{4,12}$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In Form der ersten Gleichung (4,12) steht die sog. Jak. &lt;strong&gt;Bernoulli&lt;/strong&gt;sche Ungleichung erstmals im Druck vor uns. &lt;strong&gt;Bernoulli&lt;/strong&gt; ist durchaus nicht der Erfinder dieser Ungleichung, hat sie jedoch vermutlich nicht direkt aus &lt;strong&gt;Sluse&lt;/strong&gt;, sondern auf dem Umweg über I. &lt;strong&gt;Barrow&lt;/strong&gt; (...) Auch in dessen &lt;a href=&quot;https://archive.org/stream/mathematicalwor00whewgoog#page/n664&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Lectiones geometricae&lt;/em&gt; (London, 1670)&lt;/a&gt; findet sich die Ungleichung, und zwar in der 7. Vorlesung, §13-16. Sie wird dort in einer Form bewiesen, die dem Vorgehen &lt;strong&gt;Sluse&lt;/strong&gt;s sehr ähnlich ist (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2015-02-04T23:49:50.347" LastActivityDate="2015-02-04T23:49:50.347" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1892" PostTypeId="2" ParentId="1890" CreationDate="2015-02-01T01:04:03.800" Score="3" Body="&lt;p&gt;I've found a place where he says that they did &lt;em&gt;not&lt;/em&gt; need to be created by intelligent beings. That's something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The source is Schiaparelli's book &lt;a href=&quot;http://www.gutenberg.org/cache/epub/7781/pg7781.html&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;La Vita Sul Pianeta Marte&lt;/em&gt;&lt;/a&gt; (&lt;em&gt;The Life on the Planet Mars&lt;/em&gt;), published in 1893.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Near the middle (in Part IV), he writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Che del resto le linee dette _canali_siano veramente grandi solchi o depressioni delle superficie del pianeta destinate al passaggio di masse liquide, e costituiscano su di esso un vero sistema idrografico, è dimostrato dai fenomeni che in quelli si osservano durante lo struggersi delle nevi boreali. Già dicemmo che queste, nello sciogliersi appaiono circondate da una zona oscura, formante una specie di mare temporario. In tale epoca i canali delle regioni circostanti si fanno più neri e più larghi, ingrossando al punto da ridurre, in un certo momento, ad isole di poca estensione tutto le aree gialle comprese fra l'orlo della neve e il 60° parallelo nord. Tale stato di cose non cessa, se non quando le nevi, ridotte ormai al loro minimo di estensione, cessano di struggersi. Si attenuano allora le larghezze dei canali, scompare il mare temporario, e le aree gialle riprendono l'estensione primitiva. Le diverse fasi di questa grandiosa operazione si rinnovano ad ogni giro di stagioni ed i loro particolari si son potuti osservare con molta evidenza nelle opposizioni 1882, 1884, 1886, quando il pianeta presentava allo spettatore terrestre il suo polo boreale. L'interpretazione più naturale e più semplice è quella che abbiam riferito, di una grande inondazione prodotta dallo squagliarsi delle nevi; essa è interamente logica, e sostenuta da evidenti analogie con fenomeni terrestri. Concludiamo pertanto, che i canali son tali di fatto, e non solo di nome. La rete da essi formata probabilmente fu determinata in origine dallo stato geologico del pianeta, e si è venuta lentamente elaborando nel corso dei secoli. Non occorre suppor qui l'opera di esseri intelligenti; e malgrado l'apparenza quasi geometrica di tutto il loro sistema, per ora incliniamo a credere che essi siano prodotti dell'evoluzione del pianeta, appunto come sulla Terra il canale della Manica e quello di Mozambico.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't know Italian, but &lt;a href=&quot;https://translate.google.com/&quot; rel=&quot;nofollow&quot;&gt;Google Translate&lt;/a&gt; seems to work reasonably well. It gives me (emphasis mine)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;That the rest of the lines called &quot;canali&quot; [channels] are really great grooves or depressions of the surface of the planet destined to the passage of liquid masses, and constitute about it a real river system, is demonstrated by the phenomena which are observed during those pining snow borealis. We have said that these, in the melt appear surrounded by a dark area, forming a kind of temporary sea. At that time the channels of the surrounding regions are more blacks and wider, swelling to the point of reducing, at a certain moment, to the islands of little extension all the yellow areas ranging from the edge of the snow and the 60th parallel north. This situation does not cease, except when the snows, now reduced to their minimum extension, cease to droop. Then the widths of the channels are reduced, the temporary sea disappears, and yellow areas resume the primitive extension. The different phases of this great operation are renewed with every turn of the seasons and we were able to observe this phenomenon with much evidence during opposition in 1882, 1884, 1886, when the planet showed to the terrestrial viewer its north pole. The most natural and simplest interpretation is the one that we have reported, a great flood produced by melting snow; it is entirely logical, and supported by clear similarities with terrestrial phenomena. &lt;strong&gt;We conclude therefore, that the channels are such in fact, and not only in name. The network formed by them was probably determined in origin by the geological state of the planet, and it slowly developed over the centuries. There is no need to suppose here the work of intelligent beings; and despite the almost geometric appearance of the whole system, up to now we are inclined to believe that they are the product of the evolution of the planet, just as on Earth the English Channel and the Mozambique.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This appears to have been &lt;em&gt;before&lt;/em&gt; Lowell got started, which means that Schaparelli saw this debate coming, and did his best to make it clear that he did &lt;em&gt;not&lt;/em&gt; think that these features were created by intelligent beings.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="131" LastEditDate="2015-02-02T15:32:53.550" LastActivityDate="2015-02-02T15:32:53.550" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1894" PostTypeId="1" CreationDate="2015-02-01T11:18:12.823" Score="5" ViewCount="105" Body="&lt;p&gt;I heard there was a lot of controversy in the royal scientific societies when the first specimens of platypus were shown. How did this affect the taxonomy of animals used at the time, if at all? Has any (other) animal shook the notion of Linneaus' taxonomy, at least temporarily?&lt;/p&gt;&#xA;" OwnerUserId="1744" LastEditorUserId="26" LastEditDate="2015-02-01T15:30:26.330" LastActivityDate="2015-02-01T15:30:26.330" Title="How did the discovery of the platypus affect Linneaus' taxonomy?" Tags="|biology|taxonomy|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1895" PostTypeId="1" AcceptedAnswerId="1898" CreationDate="2015-02-01T19:38:53.877" Score="4" ViewCount="412" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;His exact contributions are difficult to disentangle from those of his mentor Leucippus, as they are often mentioned together in texts. Their speculation on atoms, taken from Leucippus, bears a passing and partial resemblance to the nineteenth-century understanding of atomic structure that has led some to regard Democritus as more of a scientist than other Greek philosophers. &lt;a href=&quot;http://en.wikipedia.org/wiki/Democritus&quot; rel=&quot;nofollow&quot;&gt;Democritus, wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't know if this is only in my home country or is widespread in the world. In here, Democritus is the person that's known to be the &quot;pioneer&quot; of atomic theory. However, Leucippus is well known to be his master and the teachings of Democritus are almost unanimously been linked with that of Leucippus. Then why is Democritus credited with atomic theory invention? Why isn't the student-and-master relationship applicable in here? (If both master and student have brought up an idea into development, shouldn't the master be credited more?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fun trivia: The SE spell checker finds Democritus legit, while Leucippus is a spelling mistake! This should mean that Democritus &lt;em&gt;is&lt;/em&gt; more credited everywhere.&lt;/p&gt;&#xA;" OwnerUserId="1769" LastActivityDate="2015-02-02T00:55:36.680" Title="A reason for why Leucippus isn't credited the original &quot;atomist&quot; in the norm?" Tags="|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1896" PostTypeId="2" ParentId="1890" CreationDate="2015-02-01T22:33:35.227" Score="1" Body="&lt;p&gt;Schiaparelli, unlike Lowell, was cautious not to declare that his canali were artificially made, but he was &lt;a href=&quot;http://www.daviddarling.info/encyclopedia/S/Schiaparelli.html&quot; rel=&quot;nofollow&quot;&gt;sympathetic to the possibility&lt;/a&gt;. In an article of 1893 he wrote &quot;Their singular aspect, and their being drawn with absolute geometrical precision, as if they were the work of rule or compass, has led some to see in them the work of intelligent beings... I am very careful not to combat this supposition, which includes nothing impossible.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Schiaparelli firmly believed that they were bodies of water despite some indications to the contrary. His preferred theory was that they were natural  (and his preferred translation of &quot;canali&quot; into English probably would have been &quot;channels&quot;): &quot;[W]e are inclined to believe them to be produced by an evolution of the planet, just as on the Earth we have the English Channel and the Channel of Mozambique.&quot; Accordingly, he named them after famous rivers: Gehon, Hiddekel, and Phison from Eden, Lethes and Nepenthes from Hades, and even some real ones, Ganges, Euphrates, and Nilus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schiaparelli was later similarly biased in favor of habitability of Mercury. But it is interesting that he was more cautious after originally observing the canali and the &quot;seas&quot; during the Mars opposition of 1877, and disclaimed that &quot;these names may be regarded as a mere artifice... After all, we speak in a similar way of the seas of the Moon, knowing very well that they do not consist of liquid masses.&quot; Only after the  subsequent opposition of 1879 when one of the bright areas darkened, which Schiaparelli interpreted as flooding, did he become convinced and wrote &quot;It is [as] impossible to doubt their existence as that of the Rhine on the surface of the Earth.&quot; In 1888 Schiaparelli published the most detailed map of Mars to date, which included the canali, and remained in use until 1960-s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Schiaparelli's late years were brutal to his signature discovery. In 1892 Pickering saw his &quot;lakes&quot; dissociate into clusters of dark dots, in 1903 experiments involving schoolboys demonstrated that the canali too could arise as an optical illusion, in 1907 Wallace pointed out that spectroscopic analysis detected no traces of water vapor in the Martian atmosphere, and finally in 1909 high quality photos taken at Pic du Midi observatory showed no channels. But by then the idea of &lt;a href=&quot;http://www.daviddarling.info/encyclopedia/M/Marscanals.html&quot; rel=&quot;nofollow&quot;&gt;Martians and their canali&lt;/a&gt; was already firmly entrenched in the popular culture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. Since it's HSM I'll point out off topic that Schiaparelli was also deeply interested in the history of ancient astronomy. In 1877 he published a &lt;a href=&quot;http://www.jstor.org/stable/41134047?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;reconstruction&lt;/a&gt; of the first geometric astronomical model of Eudoxian homocentric spheres based on very scant mentions in only two sources. It remains the consensus to this day.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-01T22:39:56.880" LastActivityDate="2015-02-01T22:39:56.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1897" PostTypeId="1" AcceptedAnswerId="1899" CreationDate="2015-02-02T00:02:25.393" Score="9" ViewCount="747" Body="&lt;p&gt;Galileo's writings describe two experiments involving ships. These are summarized in the Wikipedia article &lt;a href=&quot;http://en.wikipedia.org/wiki/Galileo%27s_ship&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo's ship&lt;/a&gt;. (A lot of the text in the article is mine, and if there are things I'm misunderstanding, then they probably appear in the article as well.) In one of these experiments, a cannonball is dropped from the mast of a moving ship and is observed to land at the base of the mast rather than behind it. In his letter to Ingoli, he insists strenuously both on the fact that he really carried out the experiment and on the outcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This experiment is basically identical to one shown in the classic PSSC film &lt;a href=&quot;https://www.youtube.com/watch?v=pyBNImQkRuk#t=249&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frames of Reference&lt;/a&gt;. For 20 years, I've been showing students this film and accompanying it with a description of Galileo's version, and an anecdote I heard somewhere about a bet. The story is that Galileo got in a bet with two noblemen about the outcome of the experiment, but after the noblemen agreed to the bet, Galileo was unable to collect his money because they didn't agree with him that an experiment was the proper way to settle who was right. As I've learned more about the history in recent years, I've failed to come up with any information to support the anecdote about the bet. Is it totally bogus?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a circumstantial evidence against it, secondary sources such as Drake devote considerable energy to discussing the evidence that the experiment was real rather than a thought experiment, and, e.g., Drake never mentions a bet. It seems that if there were historical sources for the story of the bet, then a specialist like Drake would have considered them important evidence on this point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many other people from that era appear to have discussed the experiment or claimed to have done it. Is the story about the bet about one of them, not Galileo?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-06-10T02:14:53.367" LastActivityDate="2020-06-10T02:14:53.367" Title="Did Galileo bet money on the ship experiment?" Tags="|physics|experimental-physics|debunking|physicists|galileo|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1898" PostTypeId="2" ParentId="1895" CreationDate="2015-02-02T00:28:48.660" Score="2" Body="&lt;p&gt;Wikipedia article on &lt;a href=&quot;http://en.wikipedia.org/wiki/Leucippus&quot; rel=&quot;nofollow&quot;&gt;Leucippus&lt;/a&gt; pretty much answers your question: &quot;a brief notice in Diogenes Laertius’s life of Epicurus says that on the testimony of Epicurus Leucippus never existed. As the philosophical heir of Democritus, Epicurus's word has some weight... Furthermore, in his Corpus Democriteum, Thrasyllus of Alexandria, an astrologer and writer living under the emperor Tiberius (14–37 CE) compiled a list of writings on atomism that he attributed to Democritus to the exclusion of Leucippus.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scholars preferred to give credit to the person that they were confident actually existed. Now the consensus has shifted, and Leucippus's contribution is recognized more independently as in &quot;mechanistic determinism, which seems to go back to Leucippus, and the distinction between primary and secondary qualities, which appears in Democritus&quot; from Russo's &lt;a href=&quot;http://books.google.com/books/about/The_Forgotten_Revolution.html?id=MOTpnfz7ZuYC&quot; rel=&quot;nofollow&quot;&gt;Forgotten Revolution&lt;/a&gt;.  Graham even notes some &lt;a href=&quot;http://plato.stanford.edu/entries/leucippus/#2&quot; rel=&quot;nofollow&quot;&gt;possible differences&lt;/a&gt;: &quot;Rather than logical abstractions, Being and Not-being, Leucippus' atoms would in essence be based on Parmenides' cosomological contraries, night and light. If this line of interpretation is followed, Leucippus' notion of atom and void might have been rather different from Democritus&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over time it will enter more popular accounts, but slowly, many popular texts still repeat myths about Pythagoras despite the contrary consensus among scholars since the Burkert's book of 1972.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-02T00:55:36.680" LastActivityDate="2015-02-02T00:55:36.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1899" PostTypeId="2" ParentId="1897" CreationDate="2015-02-02T09:33:39.663" Score="6" Body="&lt;p&gt;You have correctly pointed at Wiki's quotation from Galileo's &lt;a href=&quot;http://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;nofollow&quot;&gt;Dialogo sopra i due massimi sistemi del mondo&lt;/a&gt; (&lt;em&gt;The Dialogue Concerning the Two Chief World Systems&lt;/em&gt;, 1632) where the &lt;strong&gt;thought experiments&lt;/strong&gt; regarding the &quot;composition of motions&quot; in what we today call &quot;inretial frames&quot; (e.g. the fall of a ball from the mast of a sailing ship) are discussed at lenght (for about 50 pages in the &lt;em&gt;Second Day&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;Dialogue&lt;/em&gt; is &quot;acted&quot; by three fictional characters :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Salviati : the Copernican presenting Galileo's views&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sagredo : the &quot;intelligent layman&quot; who is initially neutral&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simplicio : the follower of Ptolemy and Aristotle, presenting the traditional views regarding the natural world.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It seems to me that the story of &quot;bet with two noblemen about the outcome of the experiment&quot; is only a way to simplify for a child audience the literary dialogic form adopted by Galileo (following his &quot;theacher&quot; Plato).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Addendum&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Galileo Galilei, &lt;a href=&quot;http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?step=thumb&amp;amp;dir=galil_syste_065_en_1661&quot; rel=&quot;nofollow&quot;&gt;Dialogues on two world systems&lt;/a&gt;, (1632, Engl.tr. Thomas Salusbury, 1661), page 125-126 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;SALV&lt;/strong&gt;. Now tell me; if the stone let fall from the Round­ top, when the ship is in a swift course, should fall exactly in the same place of the ship, in which it falleth when the ship is at anchor, what service would these experiments do you, in order to the ascertaining whether the vessel doth stand still or move? [...] Have you ever tryed the experiment of the Ship?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;SIMPL&lt;/strong&gt;. I have not; but yet I believe that those Authors which alledg the same, have accurately observed it; besides that the cause of the disparity is so manifestly known, that it admits of no question.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;SALV&lt;/strong&gt;. That it is possible that those Authors instance in it, without having made tryal of it, you your self are a good testi­mony, that without having examined it, alledg it as certain, and in a credulous way remit it to their authority; as it is now not onely &#xA;  possible, but very probable that they likewise did; I mean, did remit the same to their Predecessors, without ever arriving at one that had made the experiment: for whoever shall examine the same, shall find the event succeed quite contrary to what hath &#xA;  been written of it: that is, he shall see the stone fall at all times in the same place of the Ship, whether it stand still, or move with any whatsoever velocity. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;SIMPL&lt;/strong&gt;. How is this? You have not made an hundred, no nor one proof thereof, and do you so confidently affirm it for true? I for my part will return to my incredulity, and to the confidence I had that the Experiment hath been tried by the principal Au­thors who made use thereof, and that the event succeeded as they affirm.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;SALV&lt;/strong&gt;. I am assured that the effect will ensue as I tell you; for so it is necessary that it should.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-02-05T16:02:50.797" LastActivityDate="2015-02-05T16:02:50.797" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1900" PostTypeId="2" ParentId="629" CreationDate="2015-02-02T17:12:33.720" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www.preposterousuniverse.com/blog/2005/09/16/einstein-vs-physical-review/&quot; rel=&quot;nofollow&quot;&gt;This web page&lt;/a&gt; by Sean Carroll seems to have some relevant information. Einstein was visiting the US in 1933 when Hitler was elected chancellor, and he lived in the US from then on. Because he was in the US, he switched to publishing in American journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The dispute that Daniel refers to in comments appears to have been a 1936 dispute with Physical Review, over an incorrect paper that Einstein wrote with Rosen, claiming that gravitational waves didn't exist. The anonymous reviewer (now known to have been the well-known cosmologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Howard_P._Robertson&quot; rel=&quot;nofollow&quot;&gt;Robertson&lt;/a&gt;) wrote a lengthy critique of the paper, and the paper was not published. Although the paper was wrong, Einstein apparently held a grudge about it and never again published in Physical Review, even though he eventually realized that the paper had been wrong, and published a different version with the opposite conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not a complete answer, because it may be true that Einstein started publishing in different European journals ca. 1917-1933, and if so, then I don't know why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A complete, footnoted source of information on the Phys Rev dispute is an article in Physics Today, September 2005, p. 43, by Daniel Kennefick. The article is available &lt;a href=&quot;http://scitation.aip.org/content/aip/magazine/physicstoday/article/58/9/10.1063/1.2117822&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="26" LastEditDate="2016-06-29T22:03:44.753" LastActivityDate="2016-06-29T22:03:44.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1901" PostTypeId="1" AcceptedAnswerId="1903" CreationDate="2015-02-03T19:33:03.867" Score="10" ViewCount="445" Body="&lt;p&gt;Finite fields, I believe, were introduced by Galois in his paper &quot;Sur la théorie des nombres&quot;, found on page 398 of his Oeuvres. In this paper Galois introduces the idea of taking a polynomial equation $f(x)=0$ having no solution modulo some prime, and introducing an imaginary number $i$ to act as its solution, giving a very concrete construction of a finite field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galois simply says he was led to this idea &quot;by some researches on incommensurable solutions&quot;, and that the goal of the paper is &quot;the classification of these imaginaries, and their reduction to the smallest possible number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the idea of studying how adjoining imaginary roots behaves in the structure $\mathbb Z/p\mathbb Z$ is interesting in its own right, are there any particular applications that Galois might have had in mind? What applications were made of his ideas immediately after their diffusion? Was anyone else studying and applying the concept of finite fields independently of Galois? Or were they introduced purely for their own sake, and had to wait a long time before applications were stumbled upon?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2015-02-03T20:22:01.793" LastActivityDate="2020-04-15T13:51:52.923" Title="What were the initial applications of finite fields?" Tags="|number-theory|abstract-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1902" PostTypeId="1" AcceptedAnswerId="2056" CreationDate="2015-02-03T21:42:02.670" Score="7" ViewCount="3406" Body="&lt;p&gt;I know this might be a silly question, but I think it's worth asking. For his undergraduate degree at MIT, did he do a double major with math, or did he really just get that good at mathematics by himself?&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="4251" LastEditDate="2020-06-05T21:36:02.407" LastActivityDate="2020-06-05T21:36:02.407" Title="Was Richard Feynman a double major?" Tags="|biographical-details|physicists|feynman|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1903" PostTypeId="2" ParentId="1901" CreationDate="2015-02-04T01:45:05.243" Score="7" Body="&lt;p&gt;In Cox's &quot;Galois Theory&quot; (2nd edition) he explains in Chapter 14 why Galois created finite fields: it was to construct solvable primitive permutation groups. Solvable groups were a big deal to Galois, for his theorem on solvability by radicals, and in those days all groups were groups of permutations. See in particular the Historical Notes to sections 14.3 and 14.4 as well as Theorems 14.4.9 and 14.4.10 of Cox's book. The slides &lt;a href=&quot;https://dacox.people.amherst.edu/lectures/bilbao.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://dacox.people.amherst.edu/lectures/bilbao.pdf&lt;/a&gt; from a talk by Cox may be a better place to look first.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2020-04-15T13:51:52.923" LastActivityDate="2020-04-15T13:51:52.923" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1904" PostTypeId="1" AcceptedAnswerId="1906" CreationDate="2015-02-04T05:30:55.240" Score="9" ViewCount="2881" Body="&lt;p&gt;I know the definition and the procedure to calculate the inverse of a matrix, but I want to know the history of starting the idea of an inverse matrix. mathematicians must have faced a real life situation which compelled them to invent the inverse matrix. What was the situation?&lt;/p&gt;&#xA;" OwnerUserId="1789" LastEditorUserId="62" LastEditDate="2015-02-04T16:32:45.203" LastActivityDate="2021-11-19T18:52:01.990" Title="History of the inverse matrix" Tags="|mathematics|linear-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1905" PostTypeId="1" AcceptedAnswerId="2620" CreationDate="2015-02-04T21:59:01.863" Score="7" ViewCount="580" Body="&lt;p&gt;As far as I know, light bulbs existed early in the late 19th century, and the vacuum tube was created based on their design. But I haven't found any information about &lt;em&gt;who&lt;/em&gt; came up with the idea to construct special, light bulb-like devices with the intention of &lt;em&gt;amplification&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this person well-known?&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="4251" LastEditDate="2020-07-06T22:59:23.923" LastActivityDate="2020-07-06T22:59:23.923" Title="Who created the first vacuum tube?" Tags="|electronics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1906" PostTypeId="2" ParentId="1904" CreationDate="2015-02-04T23:49:05.970" Score="9" Body="&lt;p&gt;&lt;strong&gt;Cayley&lt;/strong&gt; defines the inverse $M^{-1}$ of a matrix in the very same note &lt;a href=&quot;https://archive.org/stream/collmathpapers02caylrich#page/n202&quot;&gt;&lt;em&gt;Remarques sur la notation des fonctions algébriques&lt;/em&gt; (1855)&lt;/a&gt; where he first introduces matrix multiplication. He emphasizes the fact that a linear system $Mx = \xi$ is then solved by $x = M^{-1}\xi$, and that seems motivation enough for him. Here, $x,\xi\in \mathbb{R}^n$ and $M\in GL(n,\mathbb{R})$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same thing when he later expounds the theory in &lt;a href=&quot;https://archive.org/stream/collmathpapers02caylrich#page/n492&quot;&gt;&lt;em&gt;A memoir on the theory of matrices&lt;/em&gt; (1858)&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="26" LastEditDate="2015-02-05T08:37:33.523" LastActivityDate="2015-02-05T08:37:33.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1907" PostTypeId="1" AcceptedAnswerId="1913" CreationDate="2015-02-05T11:24:41.840" Score="9" ViewCount="673" Body="&lt;p&gt;In the wiki page about &lt;a href=&quot;http://en.wikipedia.org/wiki/Einstein%27s_unsuccessful_investigations#Unified_field_theory&quot; rel=&quot;nofollow&quot;&gt;unsuccessful investigations of Einstein&lt;/a&gt;, there is only one sentence describe his unsuccessful of unifying general relativity with electromagnetism: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein spent many years pursuing a unified field theory, and published many papers on the subject, without success.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It has more detail in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Albert_Einstein#Unified_field_theory&quot; rel=&quot;nofollow&quot;&gt;Einstein's wiki&lt;/a&gt;, but only focus about his being isolation from mainstream physics. However I can't find any info about what we have learnt from his unsuccessful. By learning I mean the physics knowledge, not life lessons like &quot;Never give up&quot; or YOLO. So far the only thing I can tell about that is the concept of unifying physics that Einstein had inspired other physicists.&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="604" LastEditDate="2023-06-19T13:40:47.297" LastActivityDate="2023-06-19T13:40:47.297" Title="What have we learned from Einstein's unsuccessful dream of unifying electromagnetism and gravity?" Tags="|biographical-details|theoretical-physics|weyl|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1908" PostTypeId="1" AcceptedAnswerId="1909" CreationDate="2015-02-05T12:44:19.323" Score="9" ViewCount="2599" Body="&lt;p&gt;Most of the sources online say that Leibniz invented the sign. There's also &lt;a href=&quot;https://hsm.stackexchange.com/a/95/126&quot;&gt;this&lt;/a&gt; answer on this site which says so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is fine. But recently when I was watching Cosmos, I noticed this:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/0zAq0.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the integration sign on the wall? That's Newton's house. Was it a co-incidence that Leibniz used the same symbol for integration? &lt;/p&gt;&#xA;" OwnerUserId="126" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-02-06T01:18:30.167" Title="Who really invented the integration symbol?" Tags="|mathematicians|calculus|notation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1909" PostTypeId="2" ParentId="1908" CreationDate="2015-02-05T13:11:52.900" Score="9" Body="&lt;p&gt;The sign of the integral was proposed by Leibniz (for indefinite integral). It is a stylized letter $S$, and $S$ stands for the &quot;sum&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The things seen in your picture are probably metal parts of the house construction which can be seen on many old houses built of stones.&#xA;They have nothing to do with the integration symbol:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. The modern notation for definite integral&#xA;$$\int_a^b$$ was invented by J. Fourier. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-05T13:11:52.900" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1910" PostTypeId="1" CreationDate="2015-02-05T18:51:40.207" Score="1" ViewCount="52" Body="&lt;p&gt;Why do we typically use $h$ for&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student asked me this the other day. My guess was that it was originally height, because Newton was originally applying it to gravity and he was looking at change in height, and then it stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone have any insight into this?&lt;/p&gt;&#xA;" OwnerUserId="1792" LastEditorUserId="26" LastEditDate="2015-02-06T07:10:44.500" LastActivityDate="2015-02-06T07:10:44.500" Title="Use of $h$ in the Newton Quotient" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-02-06T07:07:44.520" ContentLicense="CC BY-SA 3.0" />
  <row Id="1911" PostTypeId="2" ParentId="1914" CreationDate="2015-02-05T21:25:22.033" Score="5" Body="&lt;p&gt;Please, note that Newton did not used the &lt;a href=&quot;http://en.wikipedia.org/wiki/Leibniz%27s_notation&quot; rel=&quot;nofollow&quot;&gt;&quot;differential&quot;&lt;/a&gt; notation, due to Leibniz, but the &lt;a href=&quot;http://en.wikipedia.org/wiki/Notation_for_differentiation#Newton.27s_notation&quot; rel=&quot;nofollow&quot;&gt;&quot;fluxional&quot;&lt;/a&gt; one : $\dot x$, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See for example :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Brook_Taylor&quot; rel=&quot;nofollow&quot;&gt;Brook Taylor&lt;/a&gt; [English mathematician known for Taylor's theorem and the Taylor serie], &lt;a href=&quot;http://www.17centurymaths.com/contents/brooktaylor/methodofincrementspartone.pdf&quot; rel=&quot;nofollow&quot;&gt;Methodus Incrementorum Directa et Inversa&lt;/a&gt;, (1715) :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The moment of a fluent such as $x$ is the amount by which it changes in an indefinitely&#xA;  small length of time $o$, given by $\dot xo$ , and all products in an equation that involve orders of $o$ greater than one can be ignored. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Florian Cajori, &lt;a href=&quot;https://books.google.it/books?id=gxrO8ZnMK_YC&amp;amp;pg=RA2-PA263&quot; rel=&quot;nofollow&quot;&gt;A history of mathematical notation&lt;/a&gt; (1919), &lt;strong&gt;FINITE DIFFERENCES&lt;/strong&gt;, para 640-on, Vol II, page 263-on .&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;We can find the use of $h$ in :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;George Boole, &lt;a href=&quot;https://books.google.it/books?id=aYI3AAAAMAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow&quot;&gt;A treatise on the calculus of finite differences&lt;/a&gt; (1860).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My &quot;humble opinion&quot; is that there is no special reason for it ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As long as the symbolism underwent the process of standardization, we can see that the (insufficient number of elements of the) alphabet partecipate to a process of &quot;specialization&quot; :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;$x,y,z$ for the unknown; $a,b,c$ for the known terms, mainly in algebra; $i,j,k$ for numerical indices; $n,m$ for naturals;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in addition, $d$ was used for derivative, $f$ was chosen for the &quot;typical&quot; function; $e$ and $i$ acquired a status of &lt;em&gt;proper nouns&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, there were not many letters available ...&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-02-06T12:58:30.713" LastActivityDate="2015-02-06T12:58:30.713" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1913" PostTypeId="2" ParentId="1907" CreationDate="2015-02-05T23:20:32.383" Score="12" Body="&lt;p&gt;What we learned so far belongs to the realm of mathematics, or more generously formal side of theoretical physics, rather than knowledge of nature in the narrow sense. Namely, what effects different unification constructions can produce and account for, and what in them is problematic. Many tricks originally introduced trying to unify gravity and electromagnetism were later used, and continue to be used, in other contexts, some of which have nothing to do with unification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unification of gravity and electromagnetism wasn't initiated by Einstein, &lt;a href=&quot;https://hsm.stackexchange.com/questions/10/when-and-how-was-the-geometric-understanding-of-gauge-theories-developed/147#147&quot;&gt;Herman Weyl pursued it first&lt;/a&gt; in 1918-1920, and although Einstein noticed right away that the theory was unphysical what came out of it was the idea of gauge invariance, which became central in the Standard Model. In 1921 Kaluza offered a different model elaborated by Klein in 1926 that introduced the idea of using curled dimensions to unify fields. This idea eventually led to string theory and M-theory with their extra dimensions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein got involved in 1928, and his main idea was to explore the influence of keeping torsion in play in addition to curvature (in standard general relativity pseudometric connection is used, which by definition has zero torsion). Cartan already suggested a non-zero torsion generalization in 1922, and Einstein attempted to match torsion to the electromagnetic energy tensor. That was unsuccessful, but the &lt;a href=&quot;http://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory&quot; rel=&quot;noreferrer&quot;&gt;Einstein-Cartan theory&lt;/a&gt; introduces non-linearity into the Dirac equation which allows to avoid the Big Bang singularity. This feature is currently pursued in cosmology under the name of &lt;a href=&quot;http://arxiv.org/abs/1105.6127&quot; rel=&quot;noreferrer&quot;&gt;Big Bounce&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein also went further and replaced the usual metric connections by their direct &quot;opposites&quot;, Weitzenböck connections, that have non-zero torsion but zero curvature. This leads to a theory with global parallelism, or &lt;a href=&quot;http://en.wikipedia.org/wiki/Teleparallelism#cite_note-1&quot; rel=&quot;noreferrer&quot;&gt;teleparallelism&lt;/a&gt;, which is equivalently described by a tetrad of vector fields. Again, although the intended goal was not achieved the machinery proved useful elsewhere. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the theory of gravity itself, for example, Møller showed in 1961 that the tetrad description has some advantages over the metric tensor description, and several theories were proposed that treat gravitational field as a gauge field of the translation group. This approach was later used in &lt;a href=&quot;http://en.wikipedia.org/wiki/Loop_quantum_gravity#Spin_foams_from_BF_theory&quot; rel=&quot;noreferrer&quot;&gt;loop quantum gravity&lt;/a&gt;, which attempts to quantize gravity without unification, ironically. Teleparallelism also came up in two-dimensional non-linear sigma models, where at the infrared limit of renormalization flow curvature disappears but torsion remains. Finally, there is a &lt;a href=&quot;http://en.wikipedia.org/wiki/Teleparallelism#Non-gravitational_contexts&quot; rel=&quot;noreferrer&quot;&gt;mathematical analogy&lt;/a&gt; between curvature and torsion of spacetime on one hand, and disclinations and dislocations of crystals on the other. Einstein's idea allows one to rewrite curvature effects in terms of torsion, which was &lt;a href=&quot;http://www.scielo.br/scielo.php?pid=S0103-97332000000200011&amp;amp;script=sci_arttext&quot; rel=&quot;noreferrer&quot;&gt;fruitful&lt;/a&gt; in both areas.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-02-06T18:52:37.370" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1914" PostTypeId="1" AcceptedAnswerId="1916" CreationDate="2015-02-05T18:45:17.993" Score="6" ViewCount="289" Body="&lt;p&gt;Why do we typically use $h$ for&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A student asked me this the other day. My guess was that it was originally height, because Newton was originally applying it to gravity and he was looking at change in height, and then it stuck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone have any insight into this?&lt;/p&gt;&#xA;" OwnerUserId="1792" OwnerDisplayName="charlotte" LastEditorUserId="62" LastEditDate="2015-02-07T20:08:29.170" LastActivityDate="2015-02-07T20:08:29.170" Title="Use of $h$ in the Newton Quotient" Tags="|mathematics|notation|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1915" PostTypeId="2" ParentId="1914" CreationDate="2015-02-05T18:52:07.190" Score="2" Body="&lt;p&gt;I have see many letters and symbols used, for example, it need not be $h$, but could also be&#xA;\begin{align}&#xA;f'\left(a\right)&amp;amp;=\lim_{\Delta x\rightarrow 0}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x},\tag{1}\\&#xA;f'_{+}\left(a\right)&amp;amp;=\lim_{\Delta x\rightarrow 0+}\frac{f\left(a + \Delta x\right)-f\left(a\right)}{\Delta x},\tag{2}\\&#xA;f'_{-}\left(a\right)&amp;amp;=\lim_{\Delta x\rightarrow 0-}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x},\tag{3}&#xA;\end{align}&#xA;as is shown in Widder's &lt;em&gt;Advanced Calculus&lt;/em&gt; to denote the derivative of $f\left(x\right)$ at  $x=a$, the derivative on the right of $f\left(x\right)$ at $x=a$, and the derivative on the left  of $f\left(x\right)$ at $x=a$.&lt;/p&gt;&#xA;" OwnerUserId="3070" OwnerDisplayName="bd1251252" LastActivityDate="2015-02-05T18:57:52.637" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1916" PostTypeId="2" ParentId="1914" CreationDate="2015-02-05T19:00:34.067" Score="7" Body="&lt;p&gt;Here is an interesting &lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot;&gt;list of first uses&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Small $h$ seems to have been used by Cauchy's Cours d'Analyse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both Cauchy's and Newton's papers are available online, so it should be possible to verify if it showed up there.&lt;/p&gt;&#xA;" OwnerUserId="2420" OwnerDisplayName="mvw" LastActivityDate="2015-02-05T19:00:34.067" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1917" PostTypeId="2" ParentId="1914" CreationDate="2015-02-05T19:12:17.660" Score="2" Body="&lt;p&gt;A quick look at &quot;&lt;a href=&quot;https://archive.org/details/analysisperquan00jonegoog&quot; rel=&quot;nofollow&quot;&gt;Analysis per quantitatum series, fluxiones, ac differentias ...&lt;/a&gt;&quot; didn't find any $h$'s.  In fact Newton didn't use the modern definition of derivative at all.  The rigourous development of calculus in terms of limits didn't happen until the 19'th century.&lt;/p&gt;&#xA;" OwnerDisplayName="Robert Israel" LastActivityDate="2015-02-05T19:12:17.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1918" PostTypeId="1" CreationDate="2015-02-07T04:43:09.027" Score="6" ViewCount="139" Body="&lt;p&gt;I semi-recently went to work for a workplace that employs Agile development instead of Waterfall. I have my own reasons for appreciating it, but who was the original creator of Agile? What problems were they hoping to solve with it?&lt;/p&gt;&#xA;" OwnerUserId="19" LastEditorUserId="62" LastEditDate="2015-09-07T19:17:42.557" LastActivityDate="2015-09-07T19:17:42.557" Title="Who created Agile programming and why?" Tags="|computer-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1919" PostTypeId="1" AcceptedAnswerId="1921" CreationDate="2015-02-07T09:05:38.560" Score="8" ViewCount="870" Body="&lt;p&gt;We all know that the idea of irrational numbers was very much anathema to the Pythagorean school of mathematics, and that its revelation caused quite a stir. But ultimately, after the dust had settled, the Pythagoreans had to accept the proof and move on. How, then, did they incorporate this radical new idea into their mathematics? How did they account for it philosophically?&lt;/p&gt;&#xA;" OwnerUserId="137" LastEditorUserId="643" LastEditDate="2015-02-07T19:46:49.773" LastActivityDate="2015-02-16T01:13:20.097" Title="What was the aftermath of the proof of irrationality of $\sqrt 2$ for the Greeks?" Tags="|mathematics|ancient-greece|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1920" PostTypeId="2" ParentId="1919" CreationDate="2015-02-07T11:16:04.917" Score="3" Body="&lt;p&gt;You can see this post &lt;a href=&quot;https://hsm.stackexchange.com/questions/2/irrationality-of-the-square-root-of-2&quot;&gt;on the discovery of the irrationality of the $\sqrt 2$&lt;/a&gt; and this post on &lt;a href=&quot;https://math.stackexchange.com/questions/571998/did-pythagoras-school-collapse-because-of-their-discovery-of-irrational-numbers&quot;&gt;the collapse of Pythagoras' school because of ther discovery of irrational numbers&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can say, simplifying a little bit, that the discovery of the existence of irrational magnitudes leads Greek mathematicians to the withdrawal of the so-called &quot;commensurability assumption&quot;, i.e. the (implicit) assumption that :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;given two magnitudes, e.g. two segments of lenght $a,b$, it is always possible to find a segment of unit lenght $u$ such that it measures both, i.e. $a=n \times u$ and $b=m \times u$ for suitable integers $n,m$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this originated the need to axiomatize geometry, i.e. the need to explicitly lists all the needed assumptions.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-02-07T11:16:04.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1921" PostTypeId="2" ParentId="1919" CreationDate="2015-02-07T14:44:28.540" Score="7" Body="&lt;p&gt;The answer to this discovery was the theory of proportions (Euclud book 10), where geometric language is used to deal with numbers (and numbers themselves are banned). The theory of proportions was a substitute for the modern theory of real numbers. It is not equivalent because not all numbers can be constructed with a compass and a ruler, but Hellenistic scientists did not know this. Eudoxus and Archimedes developed a method of exhaustion&#xA;which is essentially equivalent to the modern theory of real numbers. (They understood the main property of the real numbers which we call &quot;completeness&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took long time to develop the modern theory starting from Nepier and until Cantor.&#xA;But notice that physicists of 17 and 18 centuries formulated their laws in terms of proportions, not numbers! And this formulation survived till the middle 20-th century school textooks. &quot;The gravitational force is proportional to the mass and inverse proportional to the square of the distance&quot;, etc.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-07T14:44:28.540" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1923" PostTypeId="1" AcceptedAnswerId="1924" CreationDate="2015-02-07T19:56:52.290" Score="7" ViewCount="2780" Body="&lt;p&gt;Galileo's pupil Viviani said that Galileo dropped unequal weights from the Leaning Tower of Pisa and observed them to take equal times to hit the ground. Galileo's own writings do not describe such an experiment, but it is clear that he did quantitative experiments with balls rolling down inclined planes, which would make it natural to extrapolate to the same result for falling objects. (I suppose it's not a matter of pure extrapolation, since the acceleration of a sphere rolling without slipping is strictly less than &lt;span class=&quot;math-container&quot;&gt;$g\sin\theta$&lt;/span&gt;.) &lt;a href=&quot;http://en.wikipedia.org/wiki/Galileo_Galilei#Falling_bodies&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia says&lt;/a&gt; that most historians of science think there was never any more than a thought experiment (e.g., Koyre seems to have thought this), but that Drake disagrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What would be the arguments for and against the actual existence of the experiment? What did Viviani claim Galileo actually did? (Apparently in some versions of the story, Galileo dropped his objects near the heads of professors emerging from their lectures!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the only argument about whether the experiment took place at the Leaning Tower, or is there doubt about whether it happened at all? Galileo does describe such an experiment, in a line spoken by Simplicio:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But I, Simplicio, who have made the test, can assure you that a cannon&#xA;  ball weighing one or two hundred pounds, or even more, will not reach&#xA;  the ground by as much as a span ahead of a musket ball weighing only&#xA;  half a pound, provided both are dropped from a height of 200 cubits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There seems to be room for doubt, since it's fictionalized as a dialog, and the words are spoken by Simplicio rather than by Salviati, who is Galileo's mouthpiece.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side issue, there appears to have been some controversy about what Aristotle had really claimed about this. Apparently &lt;a href=&quot;http://www.nature.com/nature/journal/v158/n4025/abs/158906e0.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;there is a case to be made&lt;/a&gt; that he was only talking about terminal velocities. For translations of the relevant passages from Aristotle see &lt;a href=&quot;https://books.google.com/books?id=LGBLAAAAMAAJ&amp;amp;lpg=PA639&amp;amp;ots=f0epcez6Ht&amp;amp;dq=hardcastle%20nature%201914%20aristotle%20584&amp;amp;pg=PA639#v=onepage&amp;amp;q=hardcastle%20nature%201914%20aristotle%20584&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2018-11-12T20:11:10.560" LastActivityDate="2020-03-10T09:37:49.917" Title="Did Galileo perform an experiment at the Leaning Tower of Pisa?" Tags="|physics|biographical-details|experimental-physics|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="1924" PostTypeId="2" ParentId="1923" CreationDate="2015-02-07T20:37:09.847" Score="9" Body="&lt;p&gt;See the detailed discussion on :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Stillman Drake, &lt;a href=&quot;https://books.google.it/books?id=OwOlRPbrZeQC&amp;amp;pg=PA414&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo at work&lt;/a&gt;, (1978), page 414&amp;ndash;16.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In 1641 Galileo’s successor on the chair of mathematics at Pisa, Vincenzo Renieri, experimented with bodies made of different materials dropped from the leaning tower, finding&#xA;that a lead ball fell faster than a wooden one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Drake discuss at lenght about the fact that at that time Viviani was assistant and amanuensis of the old and blind Galileo; thus, it is quite certain that the lost answer to Ranieri was dictated to Viviani by Galileo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, the lost reply to Renieri may include the account of the &amp;ldquo;real&amp;rdquo; Galileo's early Pisan experiment; if so, Viviani's recollection can be grounded by the original Galileo's words.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Drake's conjecture is plausible, but we still lack a definitive proof.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="5989" LastEditDate="2020-02-12T06:31:36.650" LastActivityDate="2020-02-12T06:31:36.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1925" PostTypeId="1" AcceptedAnswerId="1929" CreationDate="2015-02-08T05:31:57.783" Score="10" ViewCount="871" Body="&lt;p&gt;(Math Stack Exchange suggested that the same question I posted there be migrated here; The one at Math Stack Exchange was thus deleted. The recommendation message of migration can be found &lt;a href=&quot;https://math.stackexchange.com/questions/1136205/how-are-vector-calculus-identities-first-derived-in-history&quot;&gt;here&lt;/a&gt;, though the page is now deleted.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whenever students are asked to derive or prove some vector calculus identities such as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\nabla^2\vec{A}=\nabla(\nabla\cdot \vec{A})-\nabla \times (\nabla \times \vec{A})$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;they are often asked to expand in index notation and rearrange to give the required expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This caused me to wonder how these identities were first derived (since not all of them are consequence of the product rule of derivatives).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A brief search of the &lt;a href=&quot;https://www.math.ucdavis.edu/~temple/MAT21D/SUPPLEMENTARY-ARTICLES/Crowe_History-of-Vectors.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;history&lt;/a&gt; of the topic seemed to suggest the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complex numbers $\to$ quaternions $\to$ Grassmann (exterior algebra) $\to$ Vector analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another brief search on the topic of &lt;a href=&quot;http://en.wikipedia.org/wiki/Tensor#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;tensors&lt;/a&gt; and Albert Einstein showed that the two periods of development of vector analysis and tensor analysis have some overlap thus it might be possible they might be aware of the development of each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, since the vector analysis history account above &lt;strong&gt;does not mentioned about tensors nor index notation&lt;/strong&gt;, how were the vector calculus identities originally derived, possibly &lt;strong&gt;without index notation available&lt;/strong&gt; at that time?&lt;/p&gt;&#xA;" OwnerUserId="1801" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-04-16T19:58:30.060" Title="How were vector calculus nabla ∇ identities first derived?" Tags="|mathematics|calculus|mathematical-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1926" PostTypeId="1" CreationDate="2015-02-08T12:02:47.953" Score="7" ViewCount="629" Body="&lt;p&gt;In Riemanns monumental paper, he expresses a prime counting function as an inverse Mellin transform of the log of the function he analytically continued into the complex plane&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\Pi(x) = \frac{1}{2\pi i} \int \log \zeta(s)\ x^s \frac{\mathrm{d}s}{s}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the zeros of $\zeta$ are consequently of interest (Riemann hypothesis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Associated quantities relate closely to concepts in statistical physics. &lt;a href=&quot;http://en.wikipedia.org/wiki/Planck%27s_law&quot; rel=&quot;noreferrer&quot;&gt;Planck's law&lt;/a&gt; of for the spectral radiance ($B_\nu(\nu, T) = 2 h c^{-2}\frac{\nu^3}{e^{h\nu/k_\mathrm{B}T} - 1}$) Mellin transforms to the Riemann zeta function ($\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x ^ {s-1}}{e ^ x - 1} \mathrm{d}x$) and physicists have their own &lt;a href=&quot;http://en.wikipedia.org/wiki/Polylogarithm&quot; rel=&quot;noreferrer&quot;&gt;polylog&lt;/a&gt; (mind the $z$) in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Complete_Fermi%E2%80%93Dirac_integral&quot; rel=&quot;noreferrer&quot;&gt;Fermi–Dirac integral&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the &lt;a href=&quot;http://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29#Grand_canonical_partition_function&quot; rel=&quot;noreferrer&quot;&gt;grand partition function&lt;/a&gt; $\mathcal{Z}$ relates microscopic statistical physics to thermodynamics via&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$-k_B T \ln \mathcal{Z} = \langle E \rangle - TS - \mu \langle N\rangle$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and is often expressed as a power series in the fugacity $$z=\exp\left(\frac{\mu}{T}\right)$$ as&#xA;$\mathcal{Z}(z) = \sum_{N_i} z^{N_i} Z(N_i).$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The zeros of $\mathcal{Z}$ in $z$ make the log $\ln \mathcal{Z}$ explode and this phenomenon is associated with phase transitions &#xA;and people study and cook up result like the &lt;a href=&quot;http://en.wikipedia.org/wiki/Lee%E2%80%93Yang_theorem&quot; rel=&quot;noreferrer&quot;&gt;Lee-Yang-theorem&lt;/a&gt;. &#xA;The fugacity as a parameter in $\mathbb C$, which controls the zeros, is actually on the cover of one of my favorite books. But I'd think the fugacity was defined without that context in mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I wanted to know is when did &quot;$z$&quot; become a standard name for a //complex// variable and, more importantly, is it coincidence that the names fit?? Who names it and where there other names for the fugaciy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/WoHqv.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="585" LastEditorUserId="585" LastEditDate="2015-02-08T15:29:41.460" LastActivityDate="2015-03-03T14:19:52.577" Title="Who named the fugacity, who coined the variable name and did it already relate to complex analysis?" Tags="|theoretical-physics|number-theory|statistical-mechanics|mathematical-physics|complex-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1928" PostTypeId="2" ParentId="1853" CreationDate="2015-02-08T18:03:00.163" Score="2" Body="&lt;p&gt;Yes, Euler papers have three dates (see the Euler Archive) -- when they were presented to the St Petersburg or Berlin Academy, when they were written, and also when they were published. Euler presented the Konigsberg result to the Academy in August 1735 and it was published in the 1736 issue of the Commentarii which didn't actually appear until 1741. For more on Euler's paper see 'The truth about Konigsberg' by Brian Hopkins and myself, which appeared in the College Mathematics journal in 2004 and won an MAA Polya prize in 2005.&#xA;Robin Wilson&lt;/p&gt;&#xA;" OwnerUserId="1805" LastActivityDate="2015-02-08T18:03:00.163" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1929" PostTypeId="2" ParentId="1925" CreationDate="2015-02-09T02:30:49.803" Score="5" Body="&lt;p&gt;Index notation is just a shorthand for coordinate computations, which becomes necessary in many dimensions, but is merely convenient in dimensions 3 and 4. There is no need to even index variables there, they can just be denoted by different letters. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see how Hamilton dealt with nabla identities in sections 49-50 of his seminal 1844 paper &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Quaternions&lt;/a&gt;. Basically, he expresses them in coordinates and manipulates coordinate expressions according to the rules of quaternion algebra. Maxwell does the same in part VI of his &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/1/19/A_Dynamical_Theory_of_the_Electromagnetic_Field.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dynamical Theory of Electromagnetic Field&lt;/a&gt;, except without quaternions, he derives formulas in coordinates and then rewrites them with nabla. By the way, Maxwell was the one who called nabla &lt;a href=&quot;http://en.wikipedia.org/wiki/Nabla_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;nabla&quot;&lt;/a&gt; in jest, by the name of similarly shaped Egyptian harp. Gauss and Green formulas were originally expressed, and derived, without index notation, or even nabla.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big advantage of Gibbs's symbolic vector calculus, &lt;a href=&quot;http://en.wikipedia.org/wiki/Vector_Analysis#Genesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;which appeared in draft before 1888&lt;/a&gt; and was systematized in his 1901 book with Wilson, was that he listed the basic identities and offered rules by which more complicated ones could be derived from them. His formalism was incomplete however, some identities do not reduce to basic ones and have to be derived from coordinates directly, and it only worked in 3 dimensions. So Einstein couldn't use it in general relativity. Tensor formalism was developed by Ricci and Levi-Civita in 1890-s, and they are the ones who championed the index notation, see more &lt;a href=&quot;https://hsm.stackexchange.com/questions/599/what-was-the-motivation-for-the-development-of-modern-intrinsic-differential-g/604#604&quot;&gt;here&lt;/a&gt;. Einstein appropriated it, and introduced some simplifying conventions, including the summation rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get a formalism completely free of coordinates, that subsumes vector calculus and generalizes to higher dimensions, one needs Cartan's differential forms, the differential, flats, sharps and &lt;a href=&quot;http://en.wikipedia.org/wiki/Hodge_dual#The_codifferential&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hodge star&lt;/a&gt;. It was not developed until 1950-s.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-03-23T18:24:06.090" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1930" PostTypeId="1" AcceptedAnswerId="1932" CreationDate="2015-02-09T03:17:53.740" Score="8" ViewCount="7973" Body="&lt;p&gt;Who invented, or first used, the concept of the mole? I did my own research and the closest I came was Avogadro’s constant, which was made not by Avogadro but rather by Jean Baptiste Perrin. However, I am still not sure of who invented it. I cannot find any information about this, so who, if anyone, invented the mole?&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="26" LastEditDate="2015-03-01T13:38:18.217" LastActivityDate="2015-03-01T13:38:18.217" Title="Who invented the mole?" Tags="|terminology|chemistry|units|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1931" PostTypeId="2" ParentId="1919" CreationDate="2015-02-09T04:32:56.660" Score="10" Body="&lt;p&gt;Irrational numbers were not anathema to Pythagoreans, they never thought of them at all, or of rational numbers for that matter. The only numbers they acknowledged before, during, and after the discovery of incommensurables were positive integers. Euclid in &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookV/defV3.html&quot; rel=&quot;noreferrer&quot;&gt;Elements&lt;/a&gt; writes &quot;ratio is a sort of relation in respect of size between two magnitudes of the same kind&quot;, i.e. ratios are not numbers, nowhere in Elements are ratios added or subtracted, in fact they never appear alone, only as sides of a proportion. He also writes &quot;commensurable magnitudes have to one another the ratio which a number has to a number&quot;, i.e. Pythagoreans discovered the possibility that not all ratios were as integer to integer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is likely but by no means established that they originally thought otherwise, largely because that would simplify the theory of similar figures. However, there is no evidence that Pythagoreans were disturbed by the discovery. Even before incommensurable ratios were studied for their own sake, Hippocrates of Chios and Archytas of Tarentum used them in a solution to the cube duplication problem. The traditional story with Hippasus of Metapontum is a &lt;a href=&quot;http://www.ams.org/notices/201004/rtx100400464p.pdf&quot; rel=&quot;noreferrer&quot;&gt;Neo-Pythagorean myth&lt;/a&gt;. According to &lt;a href=&quot;http://www.jstor.org/stable/230094?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Knorr&lt;/a&gt;, the discovery was a blessing, the study of incommensurables was &quot;a massive project which engaged the best efforts of the most notable fourth century mathematicians: Theodorus, Theaetetus, Archytas and Eudoxus&quot;, all Pythagoreans. As far as similar figures were concerned two resolutions were developed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An earlier approach, formalized by Theaetetus, was based on anthyphairesis, a version of Euclidean algorithm, applied to segments instead of numbers. The algorithm produces a sequence of integers (continued fraction of the ratio in our terms), which for incommensurables does not terminate. However, anthyphaireses of two different ratios could still match. &quot;Such is the definition of the same ratio&quot; writes Aristotle, and goes on to explain how the theory of similar figures can be developed without common measures based on this definition, see &lt;a href=&quot;http://www.ams.org/journals/bull/1979-01-06/S0273-0979-1979-14684-6/&quot; rel=&quot;noreferrer&quot;&gt;Fowler's paper&lt;/a&gt; and &lt;a href=&quot;http://www.maa.org/publications/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt;. To the extent that Pythagorean &quot;everything is a number&quot; required reduction of geometry to integers, this accomplished it. Theaetetus also classified &quot;magnitudes commensurable in square only&quot;, what we call quadratic irrationals, Book X of Elements is based on his work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the anthyphairetic calculations are unwieldy and there is rarely a pattern in the terms, so this approach was abandoned once an alternative emerged. This alternative was due to Eudoxus, who defined two ratios to be equal if they end up on the same side of every integer to integer ratio (Euclid &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookV/defV5.html&quot; rel=&quot;noreferrer&quot;&gt;phrases it&lt;/a&gt; in terms of multiples). Note how again only equality of ratios is defined, not what a ratio &quot;is&quot;. This definition is much handier and allowed Eudoxus, and later Archimedes, to develop a theory of areas and volumes (&quot;method of exhaustion&quot;), that would have been insurmountable with anthyphairesis. Euclid adopted it in Book V of Elements, and redeveloped the theory of similar figures based on it in Book VI, &quot;method of exhaustion&quot; is taken up in &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/bookXII.html&quot; rel=&quot;noreferrer&quot;&gt;Book XII&lt;/a&gt;. Dedekind cuts were inspired by Eudoxus's theory of proportion, however unlike Eudoxus Dedekind  treats ratios as numbers proper, i.e. conceives of them as separate items, rather than placeholders in a proportion, and defines arithmetical operations on them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-16T01:13:20.097" LastActivityDate="2015-02-16T01:13:20.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1932" PostTypeId="2" ParentId="1930" CreationDate="2015-02-09T05:41:09.013" Score="9" Body="&lt;p&gt;August Horstmann first introduced the concept of gram-molecular weight in the sense of today’s mole concept in 1881. In 1865 Loschmidt first estimated the number of molecules in a cubic centimetre of a gas under normal conditions as 1.83 × 10$^{18}$, and in 1889 Than first determined the gram-molecular volume of gases under normal conditions as 22,330 cm$^3$. This leads to the first estimate for the Avogadro number, see &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs10698-011-9128-7&quot; rel=&quot;nofollow&quot;&gt;Sarikaya's paper&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the conventional story focuses on names rather than concepts, see &lt;a href=&quot;http://lrrpublic.cli.det.nsw.edu.au/lrrSecure/Sites/LRRView/13897/applets/12869/12869/timeline12869_text.htm&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; for example. The term &quot;mole&quot; was only introduced in 1900 by Ostwald in his chemistry text. He originally defined it as &quot;the molecular weight of a substance in mass grams&quot;, but later clarified &quot;that amount of any gas that occupies a volume of 22414 mL in normal conditions is called one mole&quot;. Avogadro realized back in 1811 that there is proportionality between the amount of gas and the volume it occupies. Perrin, using moles explicitly, estimated the proportionality constant in 1908 and named it after Avogadro in 1909.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-09T23:56:34.093" LastActivityDate="2015-02-09T23:56:34.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1934" PostTypeId="1" CreationDate="2015-02-10T07:09:14.490" Score="2" ViewCount="180" Body="&lt;p&gt;I found this question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/738/are-leibnizian-infinitesimals-thought-to-be-logical-fictions-by-leibniz-scholars&quot;&gt;Are Leibnizian infinitesimals thought to be logical fictions by Leibniz scholars?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The winning answers had this to say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;This is the thickness of lines in his diagrams. Summing an infinity&#xA;  of such lines required to fill the figures gave the areas of the&#xA;  figures. But, according to Beeley, in his later communications to&#xA;  Leibniz, he writes they -- the lines -- have literally null thickness,&#xA;  he has decided. Leibniz disagrees. Leibniz's own idea in modern&#xA;  notation is closer to Wallis's first statement, and more precisely the&#xA;  above conception.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I haven't read Leibniz directly, but I read &quot;metaphysical principals of the infinitesimal calculus&quot; (1946), and while I am not a historian or mathematician, in my opinion it's a scholastic work reviewing Leibniz's various writings and attempting a broad philosophic understanding. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this book, Guenon says something like this: points, lines, surfaces, and volumes are all of different species... you cannot integrate or differentiate up and down a dimension.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However elsewhere in the book I found that when you differentiate you can turn a volume into a surface, a surface into a line, and a line into a point, but the reverse (integration) is not possible. I may be misinterpreting, but this seemed bizarre.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the different views on this? Is my source disreputable or am I misinterpreting?&lt;/p&gt;&#xA;" OwnerUserId="1810" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-02-22T18:06:06.950" Title="Leibniz: integrating/differentiating up and down a spatial dimension" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1935" PostTypeId="1" CreationDate="2015-02-10T13:51:12.343" Score="10" ViewCount="513" Body="&lt;p&gt;Gauss credits Euler (and I think some others) with having known of the existence of primitive roots. How did these predecessors of Gauss formulate the notion of a primitive root without a concept of congruence? In what works, and in what context? In particular, since the notion of a primitive root seems quite unnatural to me if you don't have a notion of congruence (I mean, the shortest definition I can think of is &quot;A number $a$ such that for every possible remainder $r$ there is an integer $n$ such that $a^n$ is $r$ more than a multiple of $p$&quot;), what led Euler and other pre-Gaussian mathematicians to consider such a concept?&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-02-11T01:40:43.290" Title="How and where was the notion of a primitive root formulated before Gauss?" Tags="|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1936" PostTypeId="1" CreationDate="2015-02-10T14:28:16.903" Score="16" ViewCount="332" Body="&lt;p&gt;The most pictorial geometric thinking I find in Dedekind is the intuitive-geometric idea of a continuum which he criticizes as too vague before he gives his account of the continuum based on cuts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In sharp contrast to Riemann, who Dedekind whole-heartedly praises, Dedekind and Weber prove the Riemann-Roch theorem without so much as using the word &quot;Curve (Kurve);&quot; and using the word &quot;surface (Fläche)&quot; only in describing Riemann's own work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dedekind was a great geometrizer in the sense of formalizing ideas of dimension and genus and differentials.  But so far as I have found he really does not see these pictorially.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not just that he does not trust pictorial thinking in proofs.  Lots of people who do not trust pictures &lt;em&gt;in&lt;/em&gt; proofs still give pictures &lt;em&gt;with&lt;/em&gt; their proofs.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some pictorial thinking in Dedekind that I have missed? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ADDED: &#xA;K.B.'s answer points to a peculiarly non-visual visualization, where Dedekind says he &lt;em&gt;cannot&lt;/em&gt; see the elements of a set.  The best source I can find is a editorial note by Emmy Noether in Dedekind's &lt;em&gt;Gesammelte mathematische Werke&lt;/em&gt; vol. 3 p. 449.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dedekind expressed himself on the concept of a set saying:  one&#xA;  represents a set as a closed sack, containing completely determinate&#xA;  things, but one cannot see them, and one knows only that they are&#xA;  determinate and are there.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The German is a bit more subtle:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dedekind äußerte hinsichtlich des Begriffs der Menge: er stelle sich&#xA;  eine Menge vor wie einen geschlossenen Sack, der ganz bestimmte Dinge&#xA;  enthalte, die man aber nicht sehe, und von denen man nichts wisse,&#xA;  außer daß sie bestimmt und vorhanden seien.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="38" LastEditorUserId="12699" LastEditDate="2017-07-03T12:04:36.910" LastActivityDate="2017-07-03T12:04:36.910" Title="Did Dedekind show any evidence of pictorial geometric sense?" Tags="|mathematics|complex-analysis|algebraic-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1937" PostTypeId="2" ParentId="1934" CreationDate="2015-02-10T15:46:12.493" Score="2" Body="&lt;p&gt;You can see at least the recent overview by John L. Bell in &lt;em&gt;SEP&lt;/em&gt; :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/continuity/#4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Continuity and Infinitesimals&lt;/a&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;according to the intuitive view of the &lt;em&gt;continuum&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;to be continuous is to constitute an unbroken or uninterrupted whole, like the ocean or the sky. A continuous entity — a &lt;em&gt;continuum&lt;/em&gt; — has no “gaps”. [...] While it is the fundamental nature of a continuum to be &lt;em&gt;undivided&lt;/em&gt;, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive &lt;em&gt;division without limit&lt;/em&gt;. This means that the process of dividing it into ever smaller parts will never terminate in an &lt;em&gt;indivisible&lt;/em&gt; or an &lt;em&gt;atom&lt;/em&gt; — that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are &lt;em&gt;divisible without limit&lt;/em&gt; or &lt;em&gt;infinitely divisible&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Closely associated with the concept of a continuum is that of &lt;em&gt;infinitesimal&lt;/em&gt;. An &lt;em&gt;infinitesimal magnitude&lt;/em&gt; has been somewhat hazily conceived as a continuum “viewed in the small,” an “ultimate part” of a continuum. In something like the same sense as a discrete entity is made up of its individual units, its “indivisibles”, so, it was maintained, a continuum is “composed” of infinitesimal magnitudes, its ultimate parts. (It is in this sense, for example, that mathematicians of the 17th century held that continuous curves are “composed” of infinitesimal straight lines.) Now the “coherence” of a continuum entails that each of its (connected) parts is also a continuum, and, accordingly, divisible. Since points are indivisible, it follows that no point can be part of a continuum. Infinitesimal magnitudes, as parts of continua, cannot, of necessity, be points: they are, in a word, &lt;em&gt;nonpunctiform&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The concept of an &lt;em&gt;indivisible&lt;/em&gt; is closely allied to, but to be distinguished from, that of an infinitesimal. An indivisible is, by definition, something that cannot be divided, which is usually understood to mean that it has no proper parts. Now a partless, or indivisible entity does not necessarily have to be infinitesimal: souls, individual consciousnesses, and Leibnizian monads all supposedly lack parts but are surely not infinitesimal. But these have in common the feature of being unextended; extended entities such as lines, surfaces, and volumes prove a much richer source of “indivisibles”. Indeed, if the process of dividing such entities were to terminate [...] it would necessarily issue in indivisibles of a qualitatively different nature. In the case of a straight line, such indivisibles would, plausibly, be points; in the case of a circle, straight lines; and in the case of a cylinder divided by sections parallel to its base, circles. In each case the indivisible in question is infinitesimal in the sense of &lt;em&gt;possessing one fewer dimension than the figure from which it is generated&lt;/em&gt;. In the 16th and 17th centuries indivisibles in this sense were used in the calculation of areas and volumes of curvilinear figures, a surface or volume being thought of as a collection, or sum, of linear, or planar indivisibles respectively.&lt;/p&gt;&#xA;&lt;p&gt;Newton's during the year 1665–66 issued the invention of what he called the “Calculus of Fluxions”, the principles and methods of which were presented in three tracts published many years after they were written : &lt;em&gt;De analysi per aequationes numero terminorum infinitas&lt;/em&gt;; &lt;em&gt;Methodus fluxionum et serierum infinitarum&lt;/em&gt;; and &lt;em&gt;De quadratura curvarum&lt;/em&gt;. Newton's approach to the calculus rests [...] on the conception of continua as being generated by motion. [...] In the &lt;em&gt;Methodus fluxionum&lt;/em&gt; Newton makes explicit his conception of variable quantities as generated by motions, and introduces his characteristic notation.&lt;/p&gt;&#xA;&lt;p&gt;Newton later became discontented with the undeniable presence of infinitesimals in his calculus, and dissatisfied with the dubious procedure of “neglecting” them. In the preface to the &lt;em&gt;De quadratura curvarum&lt;/em&gt; he remarks that there is no necessity to introduce into the method of fluxions any argument about infinitely small quantities [...].&lt;/p&gt;&#xA;&lt;p&gt;Newton developed three approaches for his calculus, all of which he regarded as leading to equivalent results, but which varied in their degree of rigour. The first employed infinitesimal quantities which, while not finite, are at the same time not exactly zero. Finding that these eluded precise formulation, Newton focussed instead on their ratio, which is in general a finite number. If this ratio is known, the infinitesimal quantities forming it may be replaced by any suitable finite magnitudes—such as velocities or fluxions—having the same ratio. This is the method of fluxions. Recognizing that this method itself required a foundation, Newton supplied it with one in the form of the doctrine of prime and ultimate ratios, a kinematic form of the theory of limits.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz's essays [devoted to the development of the infinitesimal calculus] &lt;em&gt;Nova Methodus&lt;/em&gt; of 1684 and &lt;em&gt;De Geometri Recondita&lt;/em&gt; of 1686 may be said to represent the official births of the differential and integral calculi, respectively. His approach to the calculus, in which the use of infinitesimals, plays a central role, has combinatorial roots, traceable to his early work on derived sequences of numbers. Given a curve determined by correlated variables &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt;, he wrote &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; for infinitesimal differences, or differentials, between the values &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;: and &lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt; for the ratio of the two, which he then took to represent the slope of the curve at the corresponding point. This suggestive, if highly formal procedure led Leibniz to evolve rules for calculating with differentials, which was achieved by appropriate modification of the rules of calculation for ordinary numbers.&lt;/p&gt;&#xA;&lt;p&gt;Although the use of infinitesimals was instrumental in Leibniz's approach to the calculus, in 1684 he introduced the concept of differential without mentioning infinitely small quantities, almost certainly in order to avoid foundational difficulties.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz's attitude toward infinitesimals and differentials seems to have been that they furnished the elements from which to fashion a formal grammar, an algebra, of the continuous. Since he regarded continua as purely ideal entities, it was then perfectly consistent for him to maintain, as he did, that infinitesimal quantities themselves are no less ideal—simply useful fictions, introduced to shorten arguments and aid insight.&lt;/p&gt;&#xA;&lt;p&gt;Although Leibniz himself did not credit the infinitesimal or the (mathematical) infinite with objective existence, a number of his followers did not hesitate to do so. Among the most prominent of these was Johann Bernoulli [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-02-10T15:46:12.493" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1938" PostTypeId="2" ParentId="1935" CreationDate="2015-02-10T19:01:06.067" Score="5" Body="&lt;p&gt;According to &lt;a href=&quot;https://books.google.com/books?id=DQXvAAAAMAAJ&amp;amp;pg=PA181&amp;amp;lpg=PA181&amp;amp;dq=Lagrange+primitive+root+1769&amp;amp;source=bl&amp;amp;ots=E0XVJ25r3F&amp;amp;sig=xRrWNxy0qOKoyq5LOjP9wgfP8OM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=q1DaVOTxEJH4yQSruoLQAQ&amp;amp;ved=0CEgQ6AEwBg#v=onepage&amp;amp;q=Lagrange%20primitive%20root%201769&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Dickson's history book&lt;/a&gt; Lambert in 1769 was the first to grasp the concept by claiming that for any prime $p$ there was a number $g$ such that $g^{p-1}-1$ was divisible by $p$, but  $g^e-1$ was not for any $0&amp;lt;e&amp;lt;p-1$. Euler coined the term &quot;primitive root&quot; in 1773 when he attempted to prove Lambert's claim in 1773, but his proof had a flaw. He also listed primitive roots for primes up to 41, but noted that he had no general way of finding them. Lagrange gave a result about polynomials in 1777 that fills the gap, and Lagrange in 1785 connected primitive roots to roots of unity, where the concept arises very naturally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss himself was partly motivated by roots of unity since he wanted to prove that certain polygons were constructible when a primitive root of unity could be expressed using only square roots. Gauss only defined congruences in their modern form in Disquisitiones Arithmeticae (completed in 1798 but published only in 1801), where he gave two proofs of their existence for primes. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-11T01:40:43.290" LastActivityDate="2015-02-11T01:40:43.290" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1939" PostTypeId="1" CreationDate="2015-02-10T21:22:07.547" Score="10" ViewCount="3139" Body="&lt;p&gt;UPDATE: I've updated the title of the question from &lt;code&gt;Why did India lag behind in discovering Maths and Science?&lt;/code&gt; to &lt;code&gt;Why did India miss the Great Revolution in Maths and Science?&lt;/code&gt; after some of the answers and comments provided.&lt;/p&gt;&#xA;&lt;p&gt;NOTE: This question lies more in the &amp;quot;discussion&amp;quot; category rather than an &amp;quot;objective answer&amp;quot; category. But bear with me. I am asking this here because I want an objective answer. Or at least some reference to some research paper which answers this question.&lt;/p&gt;&#xA;&lt;p&gt;The Question is simply this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;India had a lot of time in her hands. They, alongwith Greece, developed a mature Philosophy in the times of BC. Everything went well for a good thousand after that (i.e. till 1000 AD). India lived a peaceful Golden age during this time where no philosophy was violently suppressed.&lt;/p&gt;&#xA;&lt;p&gt;Enter West. They went through Dark age for those good 1000 years. But as soon as they recovered from the Dark age, they started discovering a good deal of Maths and Science in just a couple of centuries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, some would argue that various crucial Maths/Science was also discovered before those times, like the contributions of Pythagoras (Greece) or Brahmagupta(India). Sure, they were good contributions but what I am talking here about is that advanced Maths which revolutionized the whole scenario. &lt;strong&gt;Calculus&lt;/strong&gt; would be one such revolutionary candidate, apart from others.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;PS:&lt;/strong&gt; This question is not meant to incite any debate or hurt anyone. This is a sincere academic question.&lt;/p&gt;&#xA;&lt;h3&gt;A Few Reasons Upfront&lt;/h3&gt;&#xA;&lt;p&gt;We came up with a few answers which I would like to mention upfront:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;India was always a very rich and content country, while Europe had to suffer through a lot. So, probably while India laid back, Europe HAD to discover and invent things.&lt;/li&gt;&#xA;&lt;li&gt;India always hanged on to its religion and culture. While Europe was almost completely devastated by &amp;quot;The Plague&amp;quot; which lowered the faith of people on Church, nothing as devastating for the Indian religions/culture happened in India. After the Plague, the Science of Reason and Logic (which was already present in Europe but suppressed) rose above the Church.&lt;/li&gt;&#xA;&lt;li&gt;Maybe there was some small but very crucial discovery which changed everything. Had it not happened, things would have been different. After that discovery, everything escalated pretty quickly in the West while Indian laid back.&lt;/li&gt;&#xA;&lt;li&gt;The Westerners are a superior race. &lt;em&gt;(I am mentioning this reason purely for the sake of mentioning all the possible options. No intention to incite any debate or hurt anyone. FYI, I am an Indian.)&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;In his book &amp;quot;The Shape of Ancient Thought&amp;quot;, Mcevilley says that both Greek and Indian philosophies developed together and enriched each other. And both had the same amount of rationalities and irrationalities. However, there was one crucial difference: While Indian philosophies had rationality in them, they never quite cracked the code of Logic and Deduction. So, maybe this caused major difference in thinking processes of the Indians and the Westerners.&lt;/li&gt;&#xA;&lt;li&gt;The famous Nalanda university was decimated completely and with it, perhaps, all the knowledge was gone. Several other instances also happened in India where various Mulsim invaders devastated various knowledge centers.&lt;/li&gt;&#xA;&lt;li&gt;The West had an expansionist outlook, approach or way of life. They wanted to conquer, expand and enforce themselves. This demanded desperately innovations in technology. While India never had any such approach.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1816" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-10-28T13:14:09.680" Title="Why did India miss the Great Revolution in Maths and Science?" Tags="|mathematics|ancient-india|india|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1940" PostTypeId="2" ParentId="1939" CreationDate="2015-02-10T23:21:05.347" Score="12" Body="&lt;p&gt;India did not &quot;lag behind&quot;. &#xA;It was the ancient Greece, its successor Hellenistic states, and Renaissance Western Europe where Math jumped ahead.&#xA;(More precisely, one can say, one Hellenistic state:-))&#xA;India was not much different in comparison with most of the rest of the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The correct question would be &quot;Why in ancient Greece and renaissance Europe mathematics&#xA;suddenly developed so much&quot;. By comparison there was almost no development in Europe in&#xA;the period from 2-nd AD to 17 century, like in the rest of the world. &quot;Almost none&quot; in comparison with those short periods of&#xA;intensive development.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One has to conclude that intensive development of mathematics is not a rule but a rare exception which happened only in two cultures, and took relatively short time.&#xA;The first period took essentially 300 years, and the second (modern) 400 years.&#xA;These are short periods for history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the same applies to science, though I understand that many people will object.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-10-28T13:14:09.680" LastActivityDate="2019-10-28T13:14:09.680" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="1941" PostTypeId="5" CreationDate="2015-02-11T02:39:01.890" Score="0" Body="&lt;p&gt;Questions with this tag should be about the history of (the use of) units, which are standard measures of some quantity, e.g. length, weight or current. &lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="26" LastEditDate="2015-02-11T07:57:50.170" LastActivityDate="2015-02-11T07:57:50.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1942" PostTypeId="4" CreationDate="2015-02-11T02:39:01.890" Score="0" Body="A unit is a quantity chosen as a standard in terms of which other quantities may be represented. Examples include Joules, Moles, Seconds, etc." OwnerUserId="140" LastEditorUserId="140" LastEditDate="2015-02-11T02:45:39.717" LastActivityDate="2015-02-11T02:45:39.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1943" PostTypeId="5" CreationDate="2015-02-11T02:44:21.937" Score="0" Body="&lt;p&gt;Constants are especially used in math and science. Examples include pi, speed of light, and Avogadro's number.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2015-02-11T02:45:44.147" LastActivityDate="2015-02-11T02:45:44.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1944" PostTypeId="4" CreationDate="2015-02-11T02:44:21.937" Score="0" Body="A constant is a quantity that never changes." OwnerUserId="140" LastEditorUserId="62" LastEditDate="2015-02-11T02:45:33.393" LastActivityDate="2015-02-11T02:45:33.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1945" PostTypeId="1" AcceptedAnswerId="1947" CreationDate="2015-02-11T09:04:58.170" Score="4" ViewCount="245" Body="&lt;p&gt;As we know, the revolutionizing step of the Special Relativity was, to solve the contradiction between electromagnetism and the Newtonian mechanics, not the Electromagnetism should be altered (which, as a new thing, had been the &quot;natural&quot; option), but the Newtonian mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the problems leading to that existed long before (f.e., but not exclusively, the Michelson-Morley experiment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does any concurrent theory existed at the time, which was also based on this surprising idea?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did any of them had signs, with time it could have developed into a Special Relativity, if Einstein didn't exist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.s. Aether and aether-dragging theories are not okay - they tried to solve the contradictions not by changing the classical mechanics. They suspected a new form of matter, which has uncommon properties, &lt;em&gt;but&lt;/em&gt; behaved on the classical mechanics.&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="26" LastEditDate="2015-03-01T14:00:12.437" LastActivityDate="2015-03-01T18:38:08.157" Title="Tries to alter classical mechanics before Einstein?" Tags="|theoretical-physics|relativity-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1946" PostTypeId="2" ParentId="186" CreationDate="2015-02-11T12:00:48.967" Score="2" Body="&lt;p&gt;From &lt;a href=&quot;http://www.oxforddictionaries.com/words/was-the-first-computer-bug-a-real-insect&quot; rel=&quot;nofollow&quot;&gt;Oxford Dictionaries&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term in fact originates not with computer pioneers, but with engineers of a much earlier generation. The first example cited in the 20-volume historical Oxford English Dictionary is from the Pall Mall Gazette of 11 March 1889:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Mr. Edison, I was informed, had been up the two previous nights discovering 'a bug' in his phonograph - an expression for solving a difficulty, and implying that some imaginary insect has secreted itself inside and is causing all the trouble.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the &lt;a href=&quot;http://www.computerhistory.org/tdih/September/9/&quot; rel=&quot;nofollow&quot;&gt;computing context&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In September 1947, operators of the Mark II Aiken Relay Computer found something curious trapped between points at Relay #70, Panel F: A moth. The term became popular after Grace Hopper logged the “first actual case of bug being found” in her diary, and stuck the culprit to the page. &lt;/p&gt;&#xA;" OwnerUserId="1821" LastActivityDate="2015-02-11T12:00:48.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1947" PostTypeId="2" ParentId="1945" CreationDate="2015-02-11T13:39:59.040" Score="6" Body="&lt;p&gt;To answer the question: &quot;if Einstein did not exist&quot;: the special relativity was gradually developing independently of Einstein (by Lorentz and Poincare). If Einstein did not exist it would be developed at approximately the same time by the others.&#xA;By the way, in the book of Whittaker, History of theories of electricity and magnetism,&#xA;the chapter on special relativity is called &quot;Relativity theory of Lorentz and Poincare&quot;,&#xA;and Einstein does not play a very prominent role.&#xA;But of course this is not a &quot;concurrent theory&quot;, but the same theory.&#xA;Special relativity is a logical conclusion from the theory of electromagnetism.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(With General relativity, it is different: there were no important experiments or observations that required it, and the idea is completely due to Einstein).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-01T18:38:08.157" LastActivityDate="2015-03-01T18:38:08.157" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1948" PostTypeId="1" CreationDate="2015-02-11T05:27:09.090" Score="21" ViewCount="2202" Body="&lt;p&gt;It is amazing that Kepler determined his three laws by looking at data, without a calculator and using only pen and paper. It is conceivable how he proved his laws described the data after he had already conjectured them, but what I do not understand is how he guessed them in the first place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will focus in particular on Kepler's third law, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of the orbit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume that Kepler was working with data about the planets only, plus our own moon, and the sun. I make this assumption because I don't think Kepler had data about other moons, comets, or asteroids, which had not been observed by telescope yet. If this is true, knowing that Neptune, Uranus, and Pluto were not yet discovered when Kepler was alive, this means Kepler had less than 9 data points to work with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My friend claims that it is totally concievably how Kepler guessed this relationship (although he provides no method of how Kepler might have done it), and also that Kepler's observations are &quot;not that hard&quot;.  As a challenge, I gave my friend a data table with one column labeled $x$, the other $y$, and 9 coordinates $(x,y)$ which fit the relationship $x^4=y^3$. I said &quot;please find the relationship between $x$ and $y$&quot;, and as you might expect he failed to do so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please explain to me how in the world did Kepler guess this relationship working with so few data points. And if my assumption that the number of data points Kepler had at his disposal is small, is wrong, then I still think its quite difficult to guess this relationship without a calculator. &lt;/p&gt;&#xA;" OwnerDisplayName="Joshua Benabou" LastEditorUserId="4251" LastEditDate="2021-03-02T16:21:04.217" LastActivityDate="2022-01-20T05:03:59.860" Title="How did Kepler &quot;guess&quot; his third law from data?" Tags="|physics|astronomy|17th-century|kepler|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1950" PostTypeId="2" ParentId="1948" CreationDate="2015-02-11T19:56:37.053" Score="13" Body="&lt;p&gt;I agree that it is amazing and a credit to Kepler's insight into numerical patterns, which is reminiscent of Euler's. It took Kepler extra 12 years to discover the third law after discovering the first two, perhaps exactly because of the comparative scarcity of data points.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathpages.com/rr/s8-01/8-01.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Kepler&lt;/a&gt;, after years of searching for additional patterns on March 8 of 1618 a &amp;quot;marvelous&amp;quot; idea suddenly &amp;quot;appeared&amp;quot; in his head, that &amp;quot;the proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances&amp;quot;. In other words, he conceived of what we would call a linear log log fit between mean distances and periods. It didn't fit though... at first, because of a computational error. But on March 15 the idea &amp;quot;came&amp;quot; to him again, and he got the computations right.&lt;/p&gt;&#xA;&lt;p&gt;A plausible speculation is that Kepler got his miraculous inspiration from reading Napier's 1614 work on logarithms at the end of 1616. It is unlikely that he used graphs or plots but if he converted mean distances and periods into their logarithms using Napier's tables the constant proportionality ratio of &lt;span class=&quot;math-container&quot;&gt;$1\frac12$&lt;/span&gt; between the two would have been noticeable. His account suggests however that he first conceived of a pattern in this form, and only then did the computations. He later wrote his own book about logarithms (published in 1621), and used them in computations for &lt;a href=&quot;http://en.wikipedia.org/wiki/Rudolphine_Tables&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rudolphine Tables&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A word of caution on Kepler's self reporting though. He told a breathtakingly fascinating story of discovering the first two laws in Astronomia Nova. But... &amp;quot;recent research, especially that of William H. Donahue, has shown that the account Kepler offers his readers is not a true history of the course of his research--something Kepler never claimed--but is rather a didactic or rhetorical pseudohistory&amp;quot;, see &lt;a href=&quot;https://www.google.com/books/edition/The_Composition_of_Kepler_s_Astronomia_n/SLQEEAAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voelkel, The Composition of Kepler's Astronomia Nova&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-01-20T05:03:59.860" LastActivityDate="2022-01-20T05:03:59.860" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="1951" PostTypeId="1" CreationDate="2015-02-11T23:00:11.653" Score="6" ViewCount="416" Body="&lt;p&gt;I imagine it to be Isaac Newton, who brought the discussion of &quot;force&quot; rather than &quot;center&quot; into the modeling of orbits. I imagine that prior to Newton's forces, one had to be either a geocentrist or a heliocentrist. &lt;/p&gt;&#xA;" OwnerUserId="1826" LastActivityDate="2015-02-16T17:17:29.117" Title="Who was the first person to use physics to suggest that both the Sun and the Earth move?" Tags="|astronomy|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1952" PostTypeId="2" ParentId="1951" CreationDate="2015-02-12T00:33:48.577" Score="1" Body="&lt;p&gt;Galileo and Kepler used physics (whatever physics was available at that time) to prove that Earth moves around the Sun. This is the difference between Copernicus on one side and Galileo and Kepler on another side. For Copernicus, heliocentrism was a kinematic theory (probably kinematics does not belong to &quot;physics&quot;, or does it?)&#xA;But Galileo and Kepler used dynamical theories, attraction etc. in the case of Kepler, and other arguments like inertia in the case of Galileo. This is certainly physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another question is that most of that physics was wrong. The correct physics was indeed discovered by Newton. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-12T00:33:48.577" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1953" PostTypeId="2" ParentId="1948" CreationDate="2015-02-12T00:53:28.253" Score="16" Body="&lt;p&gt;You can read Kepler's Harmonia Mundi (there is an abbreviated English translation, but it includes that part.)&#xA;Kepler was looking for all kinds of numerical relations for many years (most of his life).&#xA;Most of the relations he found in Harmonia Mundi are accidental and are of no value for modern science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with your friend, that given 6 pairs of numbers (5 planets were known +Earth), after some trials one could discover&#xA;the 3-d law. (There were more striking discoveries of this sort in history of science,&#xA;for example Balmer's series, which is even more impressive than the Third law).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler's greatest achievement was the First Law. This was really one of the greatest breakthogh's in the whole history of science. And he described himself in great detail all steps which he made to discover this. Including all mistakes that he made. This description is in his Astronomia Nova. And there is also an&#xA;English translation, complete! &#xA;So you can follow each step that he made. If you have time and patience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Logarithms were just invented when Kepler was writing Astronomia Nova.&#xA;The third Law was discovered 10 years later. If you plot logarithms of periods against&#xA;logarithms of distances, you get a straight line. So it would be surprising if someone&#xA;knowing the data and logarithms, and trying to discover some relation missed this:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-12T00:53:28.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1954" PostTypeId="1" AcceptedAnswerId="1961" CreationDate="2015-02-12T02:46:10.303" Score="6" ViewCount="348" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What are some of the earliest graphics (drawings/carvings/etc.) clearly attempting to depict some concept of the infinite (infinite divisibility or embeddings, infinite extension, infinite number)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are some of the earliest depictions of a line or curve extending to infinity in one direction? opposing directions?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Update:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest depictions - don't ignore the plural form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, the earliest examples are related naturally to temporal extension and religion, not surprisingly, given the diurnal, lunar, and seasonal cycles and the cycle in nature of birth and death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theoretical arguments concerning infinite geometric divisibility and extension are associated with the Greeks some 2,500 years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest mirrors were made at least 6,000 years ago, I expect some of the prosperous with exceptional mirrors at some time played with multiple reflections, which continue to amuse movie audiences today, made the mental leap to an infinite number, and felt compelled to record their observations and idea for others in schematics/drawings that may have survived somewhere, particularly since religious significance was often associated with people's reflections in mirrors. (I'd be interested in a verbal record of this as well.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Look at the highly suggestive recursion of triangles in two columns of these 5,000-year-old [temple mosaics][1] of Uruk in ancient Sumeria.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1]: &lt;a href=&quot;http://math.wikia.com/wiki/Iraqi_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://math.wikia.com/wiki/Iraqi_mathematics&lt;/a&gt;&lt;img src=&quot;https://i.stack.imgur.com/RlGpp.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2018-09-02T22:27:58.050" LastActivityDate="2018-09-02T22:27:58.050" Title="Earliest depictions of the infinite" Tags="|mathematics|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="1955" PostTypeId="1" CreationDate="2015-02-12T07:14:16.963" Score="9" ViewCount="205" Body="&lt;p&gt;Popular works of fiction claim that after breaking the Enigma in Bletchley, some sophisticated mathematics or statistical techniques were used to hide this fact of breaking (not necessarily by the same people). This may have included at looking at probabilities of certain war time events, and probabilities of their chance intervention, to estimate whether some acquired information could be acted upon and by whom. Popular history writing is filled with anecdotes of sending spy-submarines regularly to &quot;spot&quot; the already known locations or sending letters to fake spies for blame.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are the stories of sophisticated or even simple data analysis or model building for this purpose at some level of british intelligence for these reasons a myth? Or are some historical details known on this (these kind of  historical materials are assumedly very difficult to preserve due to their sensitive nature)? It has been given a surprisingly specific form in popular fiction writings.&lt;/p&gt;&#xA;" OwnerUserId="1827" LastActivityDate="2015-02-12T13:44:15.497" Title="Mathematical counterintelligence at Bletchley during World War 2" Tags="|mathematics|statistics|world-war-2|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1956" PostTypeId="1" CreationDate="2015-02-12T11:51:03.713" Score="3" ViewCount="72" Body="&lt;p&gt;It is well-known that in each $U(1)$-gauge theory one can define, in analogy with electromagnetism, a 1-form connection and an associated 2-form of curvature on an appropriate (principal) bundle, dependent upon the concrete problem. Nowadays this fact has become common lore of theoretical physics, however I'd like to know, for reference purpose, who was the first that pointed out such an analogy. Does anyone know and, if possible, provide a reference to the original article? &lt;/p&gt;&#xA;" OwnerUserId="1833" OwnerDisplayName="Federico" LastEditorUserId="1833" LastEditDate="2015-02-12T12:42:22.753" LastActivityDate="2015-02-12T12:42:22.753" Title="Who was the first pointing out the $U(1)$-gauge theories common structure?" Tags="|reference-request|mathematical-physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1958" PostTypeId="2" ParentId="1954" CreationDate="2015-02-12T13:45:29.517" Score="4" Body="&lt;p&gt;Certainly not the earliest, but Escher liked these hyperbolic tessellations, which show the infinite in a finite space...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/lqs93.jpg&quot; alt=&quot;circleLimit3&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-02-12T13:45:29.517" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1959" PostTypeId="2" ParentId="1951" CreationDate="2015-02-12T14:37:59.573" Score="1" Body="&lt;p&gt;from wikipedia heliocentrism page: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1687, Isaac Newton published Philosophiæ Naturalis Principia Mathematica, which provided an explanation for Kepler's laws in terms of universal gravitation and what came to be known as Newton's laws of motion. This placed heliocentrism on a firm theoretical foundation, although Newton's heliocentrism was of a somewhat modern kind. Already in the mid-1680s he recognized the &quot;deviation of the Sun&quot; from the centre of gravity of the solar system. For Newton it was not precisely the centre of the Sun or any other body that could be considered at rest, but &quot;the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World&quot;, and this centre of gravity &quot;either is at rest or moves uniformly forward in a right line&quot;. Newton adopted the &quot;at rest&quot; alternative in view of common consent that the centre, wherever it was, was at rest.&lt;/p&gt;&#xA;" OwnerUserId="1826" LastActivityDate="2015-02-12T14:37:59.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1960" PostTypeId="2" ParentId="1951" CreationDate="2015-02-12T20:30:58.447" Score="2" Body="&lt;p&gt;The answer depends on what is meant by &quot;physics&quot;. If it is mathematical physics in the modern sense then the answer is perhaps Newton. But Newton's universe was static with stars performing only minor rotations around centers of mass of their planetary systems. There is a deeper sense in which Newton was first however. Before him mathematical modeling of celestial motions was always kinematic, although it was sometimes supported by a dynamical theory, as in the case of Aristotle or &lt;a href=&quot;http://www.new-science-theory.com/johannes-kepler.php&quot; rel=&quot;nofollow&quot;&gt;Kepler&lt;/a&gt;, but always a qualitative one, only kinematics was mathematized. Newton was first to mathematize the dynamics as well, and to derive kinematics from it. But if it is &quot;physics&quot; in the original sense of Aristotle, what is now called &quot;natural philosophy&quot;, then it is most certainly not Newton. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case one did not have to be a geocentrist or a heliocentrist before Newton. The third most popular alternative is now called plurality of worlds, or &lt;a href=&quot;http://en.wikipedia.org/wiki/Cosmic_pluralism&quot; rel=&quot;nofollow&quot;&gt;cosmic pluralism&lt;/a&gt;, and it predates heliocentrism by at least two centuries. &lt;a href=&quot;http://en.wikipedia.org/wiki/Anaximander#Multiple_worlds&quot; rel=&quot;nofollow&quot;&gt;Anaximander&lt;/a&gt; speculated about the possibility of multiple worlds. Although he himself preferred geocentrism, the idea was whole heartedly embraced by early atomists, Leucippus, Democritus and Epicurus, the latter writing about an infinite universe filled with systems of worlds, &quot;some like ours, some unlike it&quot;. This was accompanied by a developed system of &quot;physics&quot;, but of qualitative nature. Since atoms were constantly on the move so presumably was everything made of them. A peculiarity of Epicurus is that he considered the Sun to be approximately of its visible size, and rejected contemporary geometric models in astronomy (homocentric spheres of Eudoxus). He even claimed that &quot;all of geometry is wrong&quot;, because its principles were inconsistent with atomism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During the Renaissence Nicolas of Kusa revived the infinite universe idea saying that the universe had &quot;center nowhere and circumference everywhere&quot;. Later Giordano Bruno offered a more elaborate version of cosmic pluralism and was burned for it in 1600. Perhaps the most elaborate plural system prior to Newton was &lt;a href=&quot;http://plato.stanford.edu/entries/descartes-physics/#7&quot; rel=&quot;nofollow&quot;&gt;Descartes's vortex theory&lt;/a&gt;, where everything moves (although not according to Descartes' concept of &quot;moving&quot;). Universe is partioned into rotating vortices with stars in the middle, and planets circling around them. But like other natural philosophers before Newton Descartes did not provide a mathematical formulation of his dynamics, despite some important insights about the centrifugal force and momentum, which doomed it eventually. For a while in 18th century Cartesian physics and cosmology competed with Newton's, but as Newton's theory provided ever more precise numerical predictions Cartesian vortices remained only an attractive intuitive picture.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-16T17:17:29.117" LastActivityDate="2015-02-16T17:17:29.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1961" PostTypeId="2" ParentId="1954" CreationDate="2015-02-12T21:36:17.887" Score="4" Body="&lt;p&gt;I will start with the second question first. Obviously, diagrams can only depict a finite pice of a line or curve, so we wouldn't &quot;clearly&quot; know if infinite extension is meant unless there is an accompanying text saying so. It is unlikely that practical geometrical concerns of ancient Egyptians and others encouraged recording such thoughts even if they did occur, which is likely. Ancient Greeks were first to start recording abstractions of this sort, and as far as I know the earliest surviving record is postulate 2 in Euclid's Elements: &quot;let it be given that a finite straight line can be produced continuously in a straight line&quot;. Postulate 2 most likely dates to late Pythagoreans like Hippocrates and Oenopides of Chios (5th century BC), and the tradition attributes an even earlier proposition (about parallels cutting an angle), which can be generously read as implying indefinite extension, to Thales of Miletus (6th century BC). But was Thales, if it was him, the first to conceive of an infinitely extending line? Probably not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This illustrates the difficulties with finding &quot;the earliest symbol&quot; in general. Here are two contenders for the first question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.crossroad.to/Books/symbols1.html#an&quot; rel=&quot;nofollow&quot;&gt;Ankh&lt;/a&gt;, ancient Egyptian symbol for &quot;mythical eternal life&quot; and &quot;rebirth&quot;, similar in meaning to also ancient Egyptian scarab, and later &lt;a href=&quot;http://en.wikipedia.org/wiki/Ouroboros&quot; rel=&quot;nofollow&quot;&gt;Ouroboros&lt;/a&gt;, unbroken circle, &quot;snake biting its own tail&quot;. Medieval alchemists occasionally twisted the circle into a horizontal figure 8, which may or may not have inspired John Wallis to introduce $\infty$ into mathematics in 1667. Alternative theory is that he simply closed Greek $\omega$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Endless_knot&quot; rel=&quot;nofollow&quot;&gt;Endless knot&lt;/a&gt;, Shrivatsa in sanskrit, &quot;an ancient symbol representing the interweaving of the Spiritual path, the flowing of Time and Movement within That Which is Eternal&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-02-12T21:36:17.887" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1962" PostTypeId="1" CreationDate="2015-02-13T18:20:59.680" Score="9" ViewCount="2151" Body="&lt;p&gt;I'm very interested to learn about historical phenomenology of proportion and ratios, i.e. I want to know how and why people in the past introduced these concepts. As far as I understand symmetric objects or patterns in geometric figures found in nature suggest the idea of proportion and ratios to human mind. But can someone explain how exactly they came to be studied with examples and details?                             &lt;/p&gt;&#xA;" OwnerUserId="1841" LastEditorUserId="55" LastEditDate="2016-10-05T20:46:20.683" LastActivityDate="2019-11-06T05:41:01.163" Title="Did geometric patterns in nature suggest the early notions of proportion and ratios?" Tags="|mathematics|ancient-greece|ancient-egypt|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1963" PostTypeId="1" AcceptedAnswerId="1968" CreationDate="2015-02-14T06:59:29.960" Score="12" ViewCount="1798" Body="&lt;p&gt;Are there any 'lost' theorems of calculus that could be used to 'simplify' it? For example, are there ways to calculate derivatives without using limits, maybe by some forgotten methods in calculus?&lt;/p&gt;&#xA;" OwnerUserId="597" LastEditorUserId="55" LastEditDate="2015-02-15T03:35:33.823" LastActivityDate="2021-03-29T12:05:32.347" Title="Is there a 'lost calculus'?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="1964" PostTypeId="1" CreationDate="2015-02-14T08:38:12.757" Score="3" ViewCount="472" Body="&lt;p&gt;We know that $(A \times B) \mod C = (A \mod C \times B \mod C) \mod C$. I know its proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my question is: Who first noticed this modular property. Which problems led him to introduce the rule? How did he discover this rule?  &lt;/p&gt;&#xA;" OwnerUserId="1841" LastEditorUserId="62" LastEditDate="2015-02-14T17:28:10.467" LastActivityDate="2015-02-14T17:28:10.467" Title="How was the modular arithmetic law invented?" Tags="|mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1965" PostTypeId="1" AcceptedAnswerId="1966" CreationDate="2015-02-14T15:57:49.660" Score="3" ViewCount="445" Body="&lt;p&gt;As we know, there is a difference between the (infinite) size (or &lt;em&gt;cardinality&lt;/em&gt;) of the integer numbers and the size of the reals ($\aleph_0$ and $\mathfrak c=2^{\aleph_0}$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who discovered it first?&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="4251" LastEditDate="2021-02-19T09:03:52.550" LastActivityDate="2021-02-19T09:03:52.550" Title="Who discovered the difference between the infinities?" Tags="|mathematics|mathematicians|real-analysis|set-theory|cantor|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1966" PostTypeId="2" ParentId="1965" CreationDate="2015-02-14T17:54:47.030" Score="11" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Georg_Cantor&quot; rel=&quot;noreferrer&quot;&gt;Georg Cantor&lt;/a&gt; discovered it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see at least : &lt;a href=&quot;http://plato.stanford.edu/entries/settheory-early/&quot; rel=&quot;noreferrer&quot;&gt;The Early Development of Set Theory&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with &lt;a href=&quot;http://en.wikipedia.org/wiki/Richard_Dedekind&quot; rel=&quot;noreferrer&quot;&gt;Dedekind&lt;/a&gt;, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-one correspondence. In reply, Dedekind offered a surprising proof that the set $A$ of all algebraic numbers is denumerable (i.e., there is a one-to-one correspondence with $\mathbb N$). A few days later, Cantor was able to prove that the assumption that $\mathbb R$ is denumerable leads to a contradiction. To this end, he employed the Bolzano-Weierstrass principle of completeness. Thus he had shown that there are more elements in $\mathbb R$ than in $\mathbb N$ or $\mathbb Q$ or $A$, in the precise sense that the &lt;a href=&quot;http://en.wikipedia.org/wiki/Cardinality&quot; rel=&quot;noreferrer&quot;&gt;cardinality&lt;/a&gt; of $\mathbb R$ is strictly greater than that of $\mathbb N$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Georg Cantor, &lt;a href=&quot;https://archive.org/details/contributionstot003626mbp&quot; rel=&quot;noreferrer&quot;&gt;Contributions to the Founding of the Theory of Transfinite Numbers&lt;/a&gt; (Engl.transl.by Philip Jourdain of &quot;&lt;em&gt;Beiträge zur Begründung der transfiniten Mengenlehre&lt;/em&gt; (I, 1895) and (II, 1897), 1915) , page 86,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;for the definition of &lt;em&gt;power&lt;/em&gt; or &lt;em&gt;cardinal number&lt;/em&gt; of a set.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-02-15T12:31:18.600" LastActivityDate="2015-02-15T12:31:18.600" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1967" PostTypeId="2" ParentId="1939" CreationDate="2015-02-14T20:20:08.197" Score="7" Body="&lt;p&gt;At one time, India led rather than lagged. The Indo-Arabic numerals that are now used worldwide originated in India. We count &lt;strong&gt;0&lt;/strong&gt;, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1&lt;strong&gt;0&lt;/strong&gt;, 11, 12, ... Note that I made &lt;strong&gt;0&lt;/strong&gt; (zero) bold. The concept of representing nothing with something is uniquely Indian. The world owes a lot, a whole lot, to India and to Brahmagupta in particular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The real question here is why India (and also China, the middle East, pre-European Americas, and even Greece and Rome) were intellectually and  economically overtaken by the undereducated, unsophisticated, and brutal barbarians from northwestern Europe. This is perhaps a better question for the sister SE site, &lt;a href=&quot;http://history.stackexchange.com&quot;&gt;http://history.stackexchange.com&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-02-14T20:20:08.197" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="1968" PostTypeId="2" ParentId="1963" CreationDate="2015-02-15T01:01:55.107" Score="19" Body="&lt;p&gt;As a matter of fact, there was something now called &quot;lost calculus&quot; or &quot;algebraic calculus&quot; in the 17th century, that avoided concepts like limits or infinitesimals, which where problematic at the time. It was developed by Descartes, Hudde and others, and is described in Suzuki's award winning paper &lt;a href=&quot;http://www.maa.org/programs/maa-awards/writing-awards/the-lost-calculus-1637-1670-tangency-and-optimization-without-limits&quot;&gt;The Lost Calculus (1637-1670): Tangency and Optimization without Limits&lt;/a&gt;. However, it only applied to algebraic functions, and it &quot;simplifies&quot; calculus only in the sense of avoiding more abstract concepts, rather than necessarily making computations simpler. So it was abandoned in favor of more general calculus of Newton and Leibniz based on infinitesimals, and later formalized using limits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Descartes first introduced the idea in La Geometrie in 1637, and later simplified it to &quot;the method of tangents&quot; in 1638. Suppose we want to find the slope of the tangent to $y=x^2$ at $x=1$. The general equation of a line passing through $(1,1)$ is $y-1=m(x-1)$, and $m$ is the slope we are looking for. Since nearby lines intersect the graph at two points and the tangent only touches it at one algebraically the system&lt;br&gt;&#xA;$$y=x^2,\ \ y-1=m(x-1)$$ must have a double root at $x=1$. Eliminating $y$ and factoring we get $$x^2-1-m(x-1)=(x-1)(x+1-m)=0,$$ so for $x=1$ to be a double root we must have $1-m=-1$ or $m=2$, which is the sought slope of the tangent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This approach works for any polynomial $y=p(x)$, and more generally for rational and even algebraic functions given implicitly. For example, to find slope of the tangent at $x=a$ we write $y-p(a)=m(x-a)$ and look for the value of $m$ that makes $a$ a double root of $p(x)-p(a)=m(x-a)$. This only requires long division of $p(x)-p(a)$ by $x-a$,  no limits or infinitesimals, and if the quotient is $q(x)$ then $m=q(a)$. To make the method computationally viable an algorithm more efficient than long division is needed for detecting double roots. Such method was provided by Jan Hudde, a talented Dutch mathematician who had to abandon mathematics for politics to save Netherlands from Spanish invasion, in two letters included into 1659 edition of Descartes' La Geometrie. It involves a clever modular reduction of polynomials that anticipates methods of modern algebraic geometry.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-16T03:20:08.147" LastActivityDate="2015-02-16T03:20:08.147" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1969" PostTypeId="2" ParentId="1962" CreationDate="2015-02-15T03:23:15.897" Score="9" Body="&lt;p&gt;It seems that originally ratio and proportion emerged not so much &quot;from nature&quot;, as from human activities, first practical and later more theoretical. We often find things in nature only after we already know what to look for. I will give several early examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In construction of the pyramids ancient Egyptian builders needed to maintain constant slope. Special measure called &lt;a href=&quot;http://en.wikipedia.org/wiki/Seked&quot; rel=&quot;nofollow noreferrer&quot;&gt;seked&lt;/a&gt; was introduced for this purpose, which measured it as rise over run. The use of seked implies practical understanding of the fact that sides of similar triangles are in the same ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the tradition, the same fact was used more consciously by Thales of Miletus to measure the height of a pyramid, and determine the distance to ships at sea, &lt;a href=&quot;http://www.math.tamu.edu/~dallen/masters/Greek/thales.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;described here&lt;/a&gt;. Although the attribution may be anecdotal, it shows that Greeks were aware of such methods already in 5th century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Pythagorean_hammers&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Xenocrates&lt;/a&gt; (4th century BC), Pythagoras noticed that different sounds appear in consonance to us when the lengths of strings producing them are &lt;span class=&quot;math-container&quot;&gt;$2:1$&lt;/span&gt;, and in dissonance when they are &lt;span class=&quot;math-container&quot;&gt;$9:8$&lt;/span&gt;, etc. Attribution to Pythagoras in this case is very likely fabricated, but Pythagoreans did develop a system of &lt;a href=&quot;http://en.wikipedia.org/wiki/Pythagorean_tuning&quot; rel=&quot;nofollow noreferrer&quot;&gt;musical tuning&lt;/a&gt; based on such observations. This one is the closest to coming from nature, but still involves instruments. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The famous &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Golden_ratio.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;golden ratio&lt;/a&gt;, that some find everywhere nowadays, does not seem to have emerged from direct observations of nature, but rather from Pythagorean obsession with &quot;mystical figures&quot;. Namely, the diagonals in the regular pentagon (which form the pentagram) cut each other in the golden ratio. The idea that the ratio represents &quot;ideal proportions&quot; of the human body was introduced into sculpture by Phidias only after the fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around 350 BC Eudoxus of Cnidus wrote a book, On Speeds, where he was trying to reproduce motions of the planets using combined rotations of several spheres. He noticed that the path of the planet depends not on the rotation speeds of the spheres, but only on their ratios. &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF00776065&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riddell speculates&lt;/a&gt; that this work led Eudoxus to his famous theory of proportion, presented in Book V of Euclid's Elements, which became a cornerstone of Greek mathematics.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-06T05:41:01.163" LastActivityDate="2019-11-06T05:41:01.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="1970" PostTypeId="1" AcceptedAnswerId="1971" CreationDate="2015-02-15T17:56:25.650" Score="0" ViewCount="182" Body="&lt;p&gt;Afaik, even in the middle ages was it impossible to melt iron. Thus, alloys had to be also impossible. Next to that, the ancient Indians surely didn't knew nickel, which is alloyed into the today's rust-free steels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, how was it reached?&lt;/p&gt;&#xA;" OwnerUserId="217" LastActivityDate="2015-02-15T19:34:57.350" Title="How the ancient iron Pillar of Delhi was created?" Tags="|ancient-india|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1971" PostTypeId="2" ParentId="1970" CreationDate="2015-02-15T18:19:46.110" Score="2" Body="&lt;p&gt;The Iron Age in India began around &lt;a href=&quot;http://en.wikipedia.org/wiki/Painted_Grey_Ware_culture&quot; rel=&quot;nofollow&quot;&gt;1200 BCE&lt;/a&gt;. The Pillar of Delhi was created around &lt;a href=&quot;http://en.wikipedia.org/wiki/Chandragupta_II&quot; rel=&quot;nofollow&quot;&gt;400 CE&lt;/a&gt;, 1600 years after Indians had first learned to melt iron. Alloys weren't impossible. In fact, Indians had learned to make steel (not just wrought iron) by 200 CE, and possibly as early as &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_metallurgy_in_the_Indian_subcontinent#Wootz_and_steel&quot; rel=&quot;nofollow&quot;&gt;300 BCE&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="1843" LastEditDate="2015-02-15T19:34:57.350" LastActivityDate="2015-02-15T19:34:57.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1972" PostTypeId="2" ParentId="1918" CreationDate="2015-02-15T19:28:34.030" Score="8" Body="&lt;p&gt;A tiny bit of research is all it takes to find who and when. Agile was conceived between February 11-13, 2001 by Kent Beck, Mike Beedle, Arie van Bennekum, Alistair Cockburn, Ward Cunningham, Martin Fowler, James Grenning, Jim Highsmith, Andrew Hunt, Ron Jeffries, Jon Kern, Brian Marick, Robert C. Martin, Steve Mellor, Ken Schwaber, Jeff Sutherland, and Dave Thomas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to why: The waterfall model doesn't work. It never has; it's pure fiction. Per the waterfall model, all an organization has to do is specify the requirements. After that everything is easy. The design necessarily follows from the requirements, the implementation necessarily follows from the design, and testing demonstrates that everything went according to plan. The waterfall model assumes that what the ultimate software product is supposed to do is fully known from the onset, and that nothing will never, ever go bump in the night (&lt;em&gt;From ghoulies and ghosties / And long-leggedy beasties / And things that go bump in the night, / Good Lord, deliver us!&lt;/em&gt;) during the design, development, and test processes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good luck with that!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In reality, people are close to clueless with regard to what the product is supposed to do early on, and things always go bump in the night. Every once in a while, projects developed per the waterfall model are completed on-time and on-budget, but the typical result is that software projects are late, over budget, and buggy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, I'm not a big fan of Agile, either. Some of the Agile concepts drive me crazy. These include big huge dehumanizing cube farms, pair programming, meetings galore, and an utter lack of requirements and documentation.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-02-15T19:28:34.030" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1973" PostTypeId="1" AcceptedAnswerId="1975" CreationDate="2015-02-15T21:50:22.933" Score="13" ViewCount="1233" Body="&lt;p&gt;The surface area of a sphere is $4\pi r^2$ and when you increase the distance to a point charge the force diminishes like the $r^{-2}$. Who was the first person to realize this?&lt;/p&gt;&#xA;" OwnerUserId="1849" LastEditorUserId="55" LastEditDate="2015-07-20T19:01:24.427" LastActivityDate="2022-08-23T17:35:11.520" Title="Who was first to explain intuitively the inverse square law of gravity?" Tags="|theoretical-physics|gravity|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1974" PostTypeId="1" CreationDate="2015-02-15T21:27:31.940" Score="0" ViewCount="28" Body="&lt;p&gt;the surface area of a sphere is 4pir^2 and when you increase it the energy disperses by the square. Who was the first person to realize this? &lt;/p&gt;&#xA;" OwnerUserId="1849" OwnerDisplayName="Ray Kay" LastActivityDate="2015-02-15T21:52:12.477" Title="Who was the first person to intuitively explain why gravity weakens by the square of the distance?" Tags="|gravity|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-02-15T22:29:16.453" ContentLicense="CC BY-SA 3.0" />
  <row Id="1975" PostTypeId="2" ParentId="1973" CreationDate="2015-02-16T00:57:34.470" Score="12" Body="&lt;p&gt;Curiously, Kepler thought that gravity falls as &lt;span class=&quot;math-container&quot;&gt;$1/r$&lt;/span&gt;, and he had a peculiar &lt;a href=&quot;http://www.new-science-theory.com/johannes-kepler.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;ether vortex theory&lt;/a&gt; borrowed from Gilbert's work on magnetism, to support it. But he argued that the intensity of light falls as &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt; along the same lines that others later applied to gravity: &amp;quot;&lt;em&gt;there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there&lt;/em&gt;&amp;quot;. French astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Inverse-square_law#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bullialdus seems to be the first&lt;/a&gt; to do so in 1645:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;... it turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But... he did not believe that such a force existed:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I say that no kind of motion presses upon the remaining planets … indeed [I say] that the individual planets are driven round by individual forms with which they were provided&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is unclear if Hooke knew of Bullialdus when he gave a &lt;a href=&quot;http://en.wikipedia.org/wiki/Inverse-square_law#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;similar argument&lt;/a&gt;, which he did believe, in his Micrographia of 1666. However, Bullialdus was elected to the Royal Society in 1667, and Newton credits him in &lt;em&gt;Principia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;After Huygens's &lt;a href=&quot;https://www.princeton.edu/%7Ehos/mike/texts/huygens/centriforce/huyforce.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De vi centrifuga&lt;/em&gt;&lt;/a&gt; of 1659 the inverse square law &lt;em&gt;could be&lt;/em&gt; inferred from combining Huygens's formula &lt;span class=&quot;math-container&quot;&gt;$a=v^2/r$&lt;/span&gt; for the centrifugal acceleration with the Kepler's third law. However, &lt;em&gt;De vi centrifuga&lt;/em&gt; was not published until 1703, Newton independently rederived the relation in 1669, also privately, and it only became public with Huygens's &lt;em&gt;Horologium Oscillatorum&lt;/em&gt; (1673), see &lt;a href=&quot;https://hsm.stackexchange.com/a/7289/55&quot;&gt;Who first derived &lt;span class=&quot;math-container&quot;&gt;$a=v^2/r$&lt;/span&gt;?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Assuming that the planets moved around the Sun in circles, to get the Kepler's relation between the radii and periods the balancing attraction force had to fall as &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt;. Indeed, if &lt;span class=&quot;math-container&quot;&gt;$a=v^2/r\propto1/r^n$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$v^2\propto1/r^{n-1}$&lt;/span&gt;, and since &lt;span class=&quot;math-container&quot;&gt;$T=2\pi r/v$&lt;/span&gt; we get &lt;span class=&quot;math-container&quot;&gt;$T\propto r^{(n+1)/2}$&lt;/span&gt;. For squares of periods to be in the same ratio as cubes of radii we must have &lt;span class=&quot;math-container&quot;&gt;$n+1=3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$n=2$&lt;/span&gt;. The general concern was if it still holds for elliptic orbits, and that was only resolved by Newton in Principia.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20150402114141/http://www.itpa.lt/%7Eacus/Knygos/Clifford_algebra_books/%5BI._M._Yaglom%5D_Felix_Klein_and_Sophus_Lie_Evoluti(BookFi.org).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yaglom in Felix Klein and_Sophus Lie (p.184)&lt;/a&gt; quotes Kant's 1747 paper Thoughts on the True Estimation of Living Forces which takes Bullialdus' dimensional reasoning further:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;substances in our universe interact with each other so that the acting force is inversely proportional to the square of the distance... If the number of dimensions were different, the forces of attraction would have different properties and dimensions.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-23T17:35:11.520" LastActivityDate="2022-08-23T17:35:11.520" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="1976" PostTypeId="2" ParentId="1973" CreationDate="2015-02-16T00:58:02.693" Score="2" Body="&lt;p&gt;One possibility is Johannes Kepler, however one could argue that &lt;strong&gt;John Dumbleton&lt;/strong&gt; was homing in on the law as well perhaps about 250 years before Kepler. It's amazing it was being realized that long ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Inverse-square_law&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Inverse-square_law&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested in how the inverse square law can be thought of as a point source that is emanating a known quantity of &quot;whatever&quot; which propagates outward and causes effects on a per unit area basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$F = \frac{GM_1M_2}{r^2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could be rewritten as;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$F = \frac{KM_1M_2}{4{\pi}r^2}$$&lt;/p&gt;&#xA;" OwnerUserId="1851" LastEditorUserId="1851" LastEditDate="2015-02-16T01:14:28.010" LastActivityDate="2015-02-16T01:14:28.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1977" PostTypeId="1" CreationDate="2015-02-16T07:08:39.323" Score="4" ViewCount="119" Body="&lt;p&gt;The first correct proof of the insolvability of the quintic is due to Abel. But my question   is who gave the proof of insolvability of the quintic using Galois theory? Does Abel know about Galois theory? &lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="26" LastEditDate="2015-02-16T09:07:31.900" LastActivityDate="2015-02-16T09:07:31.900" Title="Who is the first to give the proof of insolvability of quintic functions using Galois theory?" Tags="|mathematicians|galois-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1978" PostTypeId="1" CreationDate="2015-02-16T09:18:08.270" Score="3" ViewCount="661" Body="&lt;p&gt;I know the principle of inclusion-exclusion and solved some problems related to this principle but I don't know how this principle was discovered?&#xA;Whenever I use this principle I was amazed thinking of how the concept of the principle of inclusion-exclusion came into the mathematicians head?&#xA;Can anyone tell me how mathematician discovered the principle of inclusion-exclusion in combinatorics? What problems led them to introduce the the principle of inclusion-exclusion?&lt;/p&gt;&#xA;" OwnerUserId="1841" LastActivityDate="2015-02-16T14:18:00.453" Title="Invention of Principle of Inclusion-Exclusion" Tags="|discoveries|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1979" PostTypeId="1" AcceptedAnswerId="1982" CreationDate="2015-02-16T09:18:54.303" Score="12" ViewCount="7143" Body="&lt;p&gt;This might be more a question for historians, but it's a question I've given some thought to.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By using what was essentially Euclidean geometry, Aristarchus was able to calculate, with some measure of accuracy, the distance and the size of both the Moon and the Sun. His calculations were off, mostly due to the difficulty in perceiving the variation of shadow on the Moon to Earth, but his method was both sound and reproducible. I've also read that Archimedes knew about and agreed with Aristarchus' theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's my question, in 2 parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, since the theory was sound and the results reproducible, why didn't it catch on?  I know that Aristotle was perhaps more respected and he believed in the geocentric model.   Aristarchus followed Aristotle, and science usually moves forward. I would think that good scientists who followed would have known about and recognized the validity of Aristarchus' work, especially since Archimedes agreed with him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Logically, there's a few possibilities.   One, a number of the scientists of the time, 1st and 2nd century BC might have agreed with Aristarchus and it wasn't until later that the Geocentric model became the &quot;accepted truth&quot;, or two is that Aristotle was held above reproach, or three, word simply didn't get around very well. I hope anyone can shed some light on why this sound and simple theory didn't catch on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, I read that Ptolemy, a couple century's later, found the idea of Earth spinning at &lt;span class=&quot;math-container&quot;&gt;$1,000\ \frac{\text{miles}}{\text{hour}}$&lt;/span&gt; too improbable, so he became the champion of geocentrism, even had the model named after him, but in the 300 or so years between Aristarchus and Ptolemy, why didn't Aristarchus' theory stick?&lt;/p&gt;&#xA;" OwnerUserId="1852" LastEditorUserId="1852" LastEditDate="2019-07-19T02:34:28.883" LastActivityDate="2019-07-19T02:34:28.883" Title="Why didn't Aristarchus' theory of Heliocentrism stick?" Tags="|discoveries|astronomy|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="1980" PostTypeId="2" ParentId="1979" CreationDate="2015-02-16T13:48:41.570" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Firstly, since the theory was sound and the results reproducible, why didn't it catch on?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Aristarchus' heliocentric model was not sound and the results were not reproducible.  Not only was there an utter lack of compelling evidence for Aristarchus' heliocentric point of view, there was an immense amount of compelling evidence against that point of view given the theoretical and observational science of the time. Evidence for? None. Evidence against? No parallax (it wasn't until 1838 that astronomers (Bessel) finally did see parallax) and nonsensical physics given the physics everyone at the time &quot;knew&quot; was right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern view naively presents Aristotle and Ptolemy as incredibly dumb. They were not. They rank amongst the most brilliant people of all time. Their science withstood the test of time for two millennia. Newton? His science withstood the test of time for a mere 300 years. Einstein? His special relativity withstood the test of time for just a decade, when Einstein himself introduced general relativity. Everyone knows that general relativity too is somehow wrong, they just don't know how.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-02-16T13:48:41.570" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1981" PostTypeId="2" ParentId="1978" CreationDate="2015-02-16T14:18:00.453" Score="2" Body="&lt;p&gt;You can see &lt;a href=&quot;http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle&quot; rel=&quot;nofollow&quot;&gt;Inclusion–exclusion principle&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. This concept is attributed to Abraham de Moivre (1718); but it first appears in a paper of &lt;a href=&quot;http://en.wikipedia.org/wiki/Daniel_da_Silva_(mathematician)&quot; rel=&quot;nofollow&quot;&gt;Daniel da Silva&lt;/a&gt; (&quot;&lt;em&gt;Proprietades geraes et resolucao das Congruencias biniomis&lt;/em&gt;&quot;, Lisbon 1854), and later in a paper by J.J. Sylvester (1883). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The source seems reliable; see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Leonard Eugene Dickson, &lt;a href=&quot;https://books.google.it/books?id=FYWn8XWrAI0C&amp;amp;pg=PA119&quot; rel=&quot;nofollow&quot;&gt;History of the Theory of Numbers, Volume I: Divisibility and Primality&lt;/a&gt; (1919), page 119.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-02-16T14:18:00.453" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1982" PostTypeId="2" ParentId="1979" CreationDate="2015-02-16T20:04:44.920" Score="13" Body="&lt;p&gt;I agree with the answer of David, but I would like to add few points to it:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is a common misconception that Aristarchus found (or attempted to find) the sizes of Sun and Moon or distances to them. What his surviving book contains is a method of finding the RATIO of the distance to the Sun to the distance to to the Moon. He estimated this ratio&#xA;(with a gross error) and it is likely that he made no actual observations.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The only way to find the distance to the Moon available to the ancients was measuring its parallax and finding the radius of the Earth. The early attempts to estimate the distance led only to the statements that &quot;Moon is probably bigger than Peloponnese&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Archimedes (in Psammites) briefly mentions various opinions on the sizes of Sun and Moon,&#xA;and says that &quot;according to majority of astronomers Sun is bigger than the Earth&quot;. Ptolemy, who wrote much later computed that Sun is 5 1/2 bigger than Earth (diameter).&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;I am sure that astronomers of antiquity as well as later ones understood that  kinematically the heliocentric system is EQUIVALENT to the geocentric one, and the choice between them is a question of mathematical convenience.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;To decide what &quot;really rotates about what&quot; one needs physics (dynamics). And all reasons which Ptolemy gives against rotation of the Earth (I mean rotation about its axis) are based on physics. And his physics was wrong. Galileo's physics which he used to justify&#xA;heliocentric system was also wrong by the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To refute that Earth has any other motion, Ptolemy refers to the absence of parallax.&#xA;And he is right. More than 1000 years had to pass before parallax could be observed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The correct physics was discovered by Newton, and the acceleration of the Earth was confirmed experimentally only in 18-th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The argument that Sun is bigger than the Earth does not prove anything: who knows what&#xA;the Sun is made of? Perhaps it is bigger but the Earth is heavier? Comet tails are much bigger in size than the Sun, btw. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude, there was no real reason in antiquity to prefer the heliocentric system&#xA;in comparison with geocentric one. Physics was not sufficiently developed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And without physics the question is meaningless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Let me use this opportunity to explain one general misunderstanding which is very common (see the comments of David Hammen). &#xA;All scientific theories are APPROXIMATIONS to reality. Every scientific theory explains certain class of phenomena. We do not have and probably will never have a &quot;theory of everything&quot; which will explain everything, with absolute precision. So it is WRONG to say that Newton mechanics is&#xA;&quot;incorrect&quot;. It is correct. It is one of the MOST SUCCESSFULL scientific theories ever found. It explains the motion of planets perfectly (except ONE very small correction related to Mercury). It explains a lot of other things, and a lot of technology is based on it. But it was never supposed to explain such things as light or electricity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is why it is CORRECT, and this is why it is taught in all physics courses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Electromagnetism and the workings inside the atom are simply out of the scope of classical mechanics. And we have different theories which explain these things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question was about motion of planets. Motion of planets is COMPLETELY explained by Newtonian mechanics (except one little correction that I already mentioned).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, Ptolemy's theory is CORRECT. It explains and permits to predict the motion of the planets. Look at the modern Astronomical Almanac. It describes the motion of the planets&#xA;in Ptolemy's terms! And to compute this almanac, trigonometric series are used (which is the same as epicycles). It is another matter that we have different theories about planets which are better for other purposes. But Ptolemy's theory does what it is supposed to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Aristotle's physics is INCORRECT. It cannot be used to predict anything quantitatively. It contradicts the experiments.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-02-17T21:51:34.487" LastActivityDate="2015-02-17T21:51:34.487" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="1983" PostTypeId="2" ParentId="1979" CreationDate="2015-02-17T10:44:40.577" Score="5" Body="&lt;p&gt;Alexandre has given a very good answer to this question, but I have some remarks about his conclusion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“To conclude, there was no real reason in antiquity to prefer the heliocentric system in comparison with geocentric one. Physics was not sufficiently developed.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is true not only “in antiquity”, but also for Copernicus, and even for Kepler, don’t you think? For Copernicus the attraction of the heliocentric model was that it is more economical with regard to perceived motions, not that it was better physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle’s development of the Eudoxian model makes sense within the framework of Aristotle’s own physics (the element aether has an inherent circular motion around the centre of the cosmos), but it was soon rejected in favour of Ptolemy’s cycle/epicycle theory, which is a purely mathematical model that makes no pretence of offering a physical explanation of the movements. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason the heliocentric model was rejected is that it is counterintuitive. It says that the earth is spinning around its own axis, but our sensory perception is that we are standing still. Why don’t we feel the earth rotating?&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-02-17T12:56:05.850" LastActivityDate="2015-02-17T12:56:05.850" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1984" PostTypeId="1" CreationDate="2015-02-17T18:28:58.983" Score="9" ViewCount="1063" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Heaviside_step_function&quot; rel=&quot;noreferrer&quot;&gt;Heaviside step function&lt;/a&gt; is usually defined as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\theta(x)=\left\{\begin{array}{ll}0&amp;amp;\text{if }x&amp;lt;0\\\tfrac12&amp;amp;\text{if }x=0\\1&amp;amp;\text{if }x&amp;gt;0.\\\end{array}\right.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is remarkably simple and doesn't take a lot of work to define, though if used properly it does play an important role in a number of fields, including notably the study of &lt;a href=&quot;http://en.wikipedia.org/wiki/Distribution_%28mathematics%29&quot; rel=&quot;noreferrer&quot;&gt;distributions&lt;/a&gt;. It is nevertheless named after &lt;a href=&quot;http://en.wikipedia.org/wiki/Oliver_Heaviside&quot; rel=&quot;noreferrer&quot;&gt;Oliver Heaviside&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was his involvement with this function, and why does it merit this honour? In particular, I'm looking for well-documented sources which show his use of the concept and the context in which he used it. Ideally, I would also like references to the initial works which associate his name with the function, and their reasons for doing so.&lt;/p&gt;&#xA;" OwnerUserId="272" LastEditorUserId="4251" LastEditDate="2020-06-07T13:41:15.857" LastActivityDate="2020-06-07T13:41:15.857" Title="Why is the Heaviside step function named after Heaviside?" Tags="|mathematics|terminology|naming-conventions|heaviside|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="1986" PostTypeId="2" ParentId="1984" CreationDate="2015-02-17T21:29:00.897" Score="6" Body="&lt;p&gt;Oliver Heaviside invented what is called &quot;symbolic calculus&quot; which was a mathematically non-rigorous (at that time) but very effective way of solving differential equations that occur in physics and engineering. Later it was justified using Laplace transform and distributions theory. Heaviside function plays an important role in his formalism, analogous to the delta-function in Dirac's formalism. (Derivative of the Heaviside function in the sense of distributions is the delta-function. It has no derivative in the usual, &quot;high-school&quot; sense). Of course his contribution cannot be described&#xA;as &quot;inventing the Heaviside function&quot; :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference: any book which has &quot;Laplace transform&quot; or &quot;Operational calculus&quot; in the title.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-17T21:29:00.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="1987" PostTypeId="1" AcceptedAnswerId="1999" CreationDate="2015-02-17T22:17:13.767" Score="41" ViewCount="2532" Body="&lt;p&gt;While physics and astronomy sported mathematical models for centuries mathematical chemistry and biology appeared relatively recently. Most of the interaction seems to go one way, established mathematical theories (differential equations, combinatorics, graph theory, etc.) are applied to formalize and solve chemical and biological problems. I am interested in the reverse effect: development of new mathematical theories inspired by chemistry and biology, like trigonometry was inspired by astronomy, or calculus by physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example I know of is &lt;a href=&quot;http://www.ams.org/journals/bull/1997-34-02/S0273-0979-97-00712-X/&quot;&gt;genetic algebras of Etherington&lt;/a&gt; that describe the structure of genetic inheritance. They are structurally different from algebras that emerged from physical applications or inner workings of mathematics, for example non-associative in ways distinct from Lie, Jordan or alternative algebras, and rely on different intuitions. Are there other such examples?&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-02-03T05:10:58.770" Title="What new mathematics was inspired by biology and chemistry?" Tags="|mathematics|discoveries|chemistry|biology|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1988" PostTypeId="2" ParentId="1987" CreationDate="2015-02-18T00:35:02.173" Score="12" Body="&lt;p&gt;This is a correct observation. Chemistry and biology indeed contributed very little to mathematics itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the examples of chemistry contribution is the &quot;Belousov-Zhabotinsky reaction&quot;.&#xA;This was an experimental discovery whose explanation stimulated to some extent the development of the theory of dynamical systems (known as &quot;chaos theory&quot; in the popular literature). A lot of sophisticated mathematics was invented to explain the Periodic Table. But this was mostly applications of mathematics TO chemistry; I cannot say that chemistry brought new ideas to mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All examples from biology which come to my mind are about population genetics, the thing mentioned in the question. This is also related to dynamical systems and related algebra&#xA;(Bernstein algebras, for example). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Volterra-Lottka systems (also from population biology) stimulated the qualitative theory of differential equations in the first half of 20-s century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But most of it again works one-way: applications of mathematics TO biology etc.&#xA;In the best case, chemistry and biology provide QUESTIONS to which mathematicians sometimes find answers. But no really new ideas in mathematics that originate in&#xA;chemistry/biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that there are no new mathematical theories (of importance to mathematics itself,&#xA;with no regard to applications) which came from chemistry or biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is indeed very different from physics which constantly feeds mathematics with new ideas. So I just confirm your observation.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-18T00:35:02.173" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1989" PostTypeId="1" CreationDate="2015-02-18T01:19:01.387" Score="3" ViewCount="75" Body="&lt;p&gt;As the world developed new technology, astronomers discovered more celestial bodies, such as asteroids and black holes and such. Galaxies were probably discovered long ago before these modern telescopes came along. Even with the lack of technology, someone has discovered about the galaxies. Who and how did he/she find out about the galaxies' existance?&lt;/p&gt;&#xA;" OwnerUserId="1721" LastActivityDate="2015-02-18T01:19:01.387" Title="Who found out about galaxies?" Tags="|astronomy|" AnswerCount="0" CommentCount="4" ClosedDate="2015-02-18T22:52:43.997" ContentLicense="CC BY-SA 3.0" />
  <row Id="1990" PostTypeId="2" ParentId="1987" CreationDate="2015-02-18T21:40:35.313" Score="7" Body="&lt;p&gt;How about: (1) the logistic equation of &lt;a href=&quot;http://en.wikipedia.org/wiki/Pierre_Fran%C3%A7ois_Verhulst&quot;&gt;Pierre François Verhulst&lt;/a&gt; (describing the change in a population over time, published 1838) leads to (2) Feigenbaum's work on chaos.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2015-02-18T22:14:36.980" LastActivityDate="2015-02-18T22:14:36.980" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="1991" PostTypeId="1" AcceptedAnswerId="1992" CreationDate="2015-02-19T13:53:28.063" Score="5" ViewCount="2789" Body="&lt;p&gt;Some sources on the Internet state that the steam engine was known in ancient Egypt, although it wasn't used very widely.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it true?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If the answer is yes, it was known, or at least suspected, why wasn't it used more widely? For example, it could have helped to build more and bigger pyramids.&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="281" LastEditDate="2015-03-02T23:24:18.953" LastActivityDate="2015-03-02T23:24:18.953" Title="Was the steam engine really known in ancient Egypt?" Tags="|discoveries|ancient-egypt|steam-engines|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="1992" PostTypeId="2" ParentId="1991" CreationDate="2015-02-19T19:01:03.853" Score="8" Body="&lt;p&gt;Like Mauro I suspect that your sources make a conflation. The earliest surviving description of a steam engine is in &lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1101/1101.3470.pdf&quot;&gt;Pneumatica by Heron (or Hero) of Alexandria&lt;/a&gt;, who calls it aeolipile after the Greek god of air. One of the described applications is to automated opening of temple doors by lighting a fire on the altar. Alexandria is in Egypt, which is famous for its priests, so with some creativity we get an ancient Egyptian wonder. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Russo makes a case in &lt;a href=&quot;http://books.google.com/books/about/The_Forgotten_Revolution.html?id=MOTpnfz7ZuYC&quot;&gt;Forgotten Revolution&lt;/a&gt; based on analysis of Heron's text and other sources that the invention actually goes back to Ctesibius of Alexandria (3rd century BC), some other devices described by Heron are known to be his. But that is still Hellenistic Egypt, not ancient. The only earlier mention of a steam powered device that I know of is a flying pigeon of &lt;a href=&quot;http://en.wikipedia.org/wiki/Archytas#Life_and_work&quot;&gt;Archytas of Tarentum&lt;/a&gt; (4th century BC), but its credibility is low since it is only mentioned in a collection of science related folk tales by &lt;a href=&quot;http://www.studylight.org/encyclopedias/bri/view.cgi?w=Aulus+Gellius&quot;&gt;Aulus Gellius&lt;/a&gt;, who lived over five centuries after Archytas. Still, Archytas is known as a brilliant mechanic from other sources, so it's possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to why steam engines weren't used more widely in antiquity the reason is that they were too weak. Toys and contraptions described by Heron are just about all they could power. Only in 1663 &lt;a href=&quot;http://en.wikipedia.org/wiki/Timeline_of_steam_power&quot;&gt;John Somerset&lt;/a&gt; invented upgrades (based on condensing steam) that allow for &quot;industrial grade&quot; engines. Why wasn't it done in antiquity? One reason is that Hellenistic science had a very narrow window to develop, about two centuries between Alexandrian and Roman conquests. Around 150 BC scientists were chased away from Alexandria, and when better times returned much of the tradition was lost. On top of that during the Imperial period that followed the culture appreciated &quot;wonders&quot; rather than the science that produced them, and toys and contraptions were just right for that. Again, Forgotten Revolution is a good reference for this period.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-19T22:48:20.223" LastActivityDate="2015-02-19T22:48:20.223" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="1993" PostTypeId="2" ParentId="1939" CreationDate="2015-02-20T06:46:36.333" Score="6" Body="&lt;p&gt;In the discussion of how different cultures developed mathematics, it is important to remember that patterns of cultural exchange influenced this development. Mathematics did not just independently pop up in Greece, then India, then Europe; rather, the legacy of mathematics was inherited, refined, and passed on to descendants and neighbors in mostly continuous fashion, and it is in this sort of context that one should view the history of Indian mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For much of ancient history India did not at all &quot;lag behind&quot; its western contemporaries. Besides the invention of zero, Indian mathematicians independently discovered the Pythagorean theorem, the series expansions of the trigonometric functions, the quadratic formula, several theorems in geometry, and more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If any culture could lay claim to the title of &quot;foremost in mathematics&quot; before the advent of calculus per Newton and Leibniz, it would be the Muslim world. The golden age of Muslim civilization saw numerous advances in mathematics and the true rise of a symbolic calculus. All of this was aided by the fact that Muslim scholars had the works of the Greeks and Egyptians to draw from, and also were in contact with the work of the Indians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Europe around this time was somewhat mathematically bankrupt. The legacy of the Greeks had long passed by the end of the first millenium CE, and the Romans were only ever interested in practical mathematics. Quoth C. Edwards, on Cicero's restoration of the tomb of Archimedes in Syracuse: &quot;The Romans had so little interest in pure mathematics that this action by Cicero was probably the greatest single contribution of any Roman to the history of mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as we approach the Rennaisance years there was an increase in communication and trade between Europe and the Muslim world, and with this Europeans catch up quickly in their mathematical knowledge. Moreover, this coincides with many other important advances in European science and technology, the printing press being one of the more notable ones. It is important to consider that the Europeans only got wide access to Euclid around this time - until this time it had remained with the Muslim scholars in its original Greek (with Ptolemy's commentary) and its Arabic translation. One should also consider the enlargement of the European economy and the Age of Discovery in this time period.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of European mathematics around this time was practically motivated - Napier's invention of the logarithm, Fermat's work on optics, Pascal's work on probability, etc. The highlight, of course, is the invention of calculus by Newton and Leibniz (though preceding mathematicians, like Archimedes and Fermat, had already hinted at several notions like the derivative or integral), following which European mathematics and science exploded in sophistication. As to why &quot;pure&quot; mathematics exploded likewise...who knows? I haven't the foggiest why Fermat was so motivated to do number theory, though surely he found it vastly entertaining.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to why so little of this explosion made its way back to the Muslim world, who knows? But it's not hard to imagine why India never got to see the European advances - with the Ottoman Empire standing in the way, India was in extremely infrequent contact with the Europeans, and what contact they did have was largely mercantile, not scholarly. And unfortunately, much of the later contact between the Indians and the Europeans was of an exploitative or colonial nature - hardly a breeding ground for intellectual exchange.&lt;/p&gt;&#xA;" OwnerUserId="1873" LastEditorUserId="1873" LastEditDate="2015-02-20T06:52:21.103" LastActivityDate="2015-02-20T06:52:21.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1994" PostTypeId="1" CreationDate="2015-02-20T17:41:51.887" Score="5" ViewCount="369" Body="&lt;p&gt;In modern physics, a quantity is said to behave like a wave if it satisfies a &lt;a href=&quot;http://en.wikipedia.org/wiki/Wave_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;wave equation&lt;/a&gt;. For a contemporary instance of this type of reasoning, see e.g. &lt;a href=&quot;https://physics.stackexchange.com/a/166122/24653&quot;&gt;this&lt;/a&gt;. With some quick googling, it is quite simple to find out in what context the wave equation was first derived and applied (e.g. &lt;a href=&quot;http://en.citizendium.org/wiki/Wave_equation_(classical_physics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, it is known that throughout the 18th and 19th century, the wave equation kept on popping up everywhere and that, by the time Maxwell put the final touch on what are now known as Maxwell's equations, there was some kind of consensus that his demonstration that the electromagnetic fields satisfy a wave equation is enough to proclaim the discovery that certain kinds of radiation (most notably visible light; pun intended) should be interpreted as electromagnetic &lt;em&gt;waves&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that, somewhere in this period, there must have been a certain shift in the understanding of waves. I would imagine that initially, one would observe wavelike behavior and, &lt;em&gt;a posteriori&lt;/em&gt;, derive a wave equation that turned out to describe the phenomenon. However, by the time Maxwell did his most famous work, the approach had apparently reversed its order: Finding a wave equation now implied wavelike behavior! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is simple: Is it known when people started &lt;em&gt;defining&lt;/em&gt; a wave via the wave equation?&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2015-02-20T22:01:05.703" Title="When did physicists start defining waves as 'solutions of a wave equation'?" Tags="|mathematics|theoretical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1995" PostTypeId="2" ParentId="1994" CreationDate="2015-02-20T21:37:35.327" Score="7" Body="&lt;p&gt;Let's sort out a few things first. It is true that originally people called things waves if they behaved like waves in the colloquial sense. This is still true today even in physics, waves do not have to be solutions to the wave equation, solutions to the Schrödinger or Korteweg–de Vries equations for example are also called waves, as is any &quot;disturbance or oscillation (of a physical quantity), that travels through matter or space, accompanied by a transfer of energy&quot;, regardless of what equation it satisfies if any.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now concerning light. The idea that light is a wave came up before the wave equation was written down. &lt;a href=&quot;http://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle#History&quot;&gt;Huygens offered&lt;/a&gt; wave theory of light in 1678. However, his version of it was essentially equivalent to geometric optics, which could just as well be explained by the corpuscular theory of Newton. On Newton's authority, and due to difficulties with explaining polarization in wave theory, the corpuscular optics dominated in the 18-th century. Experience with surface waves however brought about the realization that there were effects that distinguished the wave theory, such as diffraction and interference. The wave nature of light was accepted after 1816 experiments of Fresnel showed that light exhibited diffraction and interference patterns, and in quantitative agreement with wave theory's predictions. The &lt;a href=&quot;http://en.wikipedia.org/wiki/Fresnel_diffraction#Early_treatments_of_this_phenomenon&quot;&gt;dark spot at the center&lt;/a&gt; of diffraction pattern, straight across the hole through which the light is shone, also looked very convincing. This started the theory of the &quot;luminiferous ether&quot;, a hypothetical medium in which optical waves were supposed to propagate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Maxwell's contribution was not establishing the wave nature of light, it was establishing that light was a particular type of electromagnetic field. The deciding factor in proving it was not that light and electromagnetic waves both satisfied the wave equation,  so did acoustic and elastic waves among others. It was that &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_electromagnetic_theory#Maxwell&quot;&gt;Maxwell showed in 1862&lt;/a&gt; that the propagation speed of electromagnetic waves predicted by his equations matched the known speed of light. The formula was $c=\frac1{\sqrt{\varepsilon_0\mu_0}}$, where $c$ is the speed of light, and $\varepsilon_0$ and $\mu_0$ (electric permittivity and magnetic permeability of vacuum) are constants obtained from purely electric and magnetic measurements. It was quite a tell.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-02-20T22:01:05.703" LastActivityDate="2015-02-20T22:01:05.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="1996" PostTypeId="1" CreationDate="2015-02-21T01:59:49.120" Score="8" ViewCount="1847" Body="&lt;p&gt;My sketchy understanding of the (no doubt long) history of integration theory is that the first integration theory was created by Riemann as part of his work on trigonometric series (&quot;Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe&quot;, according to Wikipedia).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other integration theories followed. The standard one today is the Lebesgue integral, which is used, among other places, in the theory of probability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Riemann integral is not as useful as the Lebesgue integral, but is still a &quot;modern&quot; integration theory, and is comparable to the Lebesgue integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is whether this was really the first &quot;modern&quot; integration theory. Did Riemann come up with it out of the blue? Were there precursors? And if it was his original creation, how much is known about how he developed the notion?&lt;/p&gt;&#xA;" OwnerUserId="1878" LastEditorUserId="4251" LastEditDate="2021-09-10T10:05:15.047" LastActivityDate="2021-09-10T10:05:15.047" Title="Was the Riemann Integral the first integration theory?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="1997" PostTypeId="2" ParentId="1996" CreationDate="2015-02-21T14:05:56.483" Score="13" Body="&lt;p&gt;The first rigorous integration theory in due to Eudoxus and Archimedes. It is called the method of exhaustion, and it allowed Archimedes to find the volumes of the balls, pyramids, cones, areas of segments of parabolas etc.&#xA;(By the way, it was proved in 20th century that one cannot find the volume&#xA;of a pyramid by pure geometric methods, some form of integral is indeed necessary here).&lt;/p&gt;&#xA;&lt;p&gt;In the modern era, the first rigorous definition of the definite integral is due to Cauchy.&#xA;Cauchy's definition of the integral is close to that of Riemann, the main difference is that&#xA;he takes equidistant points in the partition to define the integral sums, and assigns&#xA;specific points where the values of the function are taken. This gives the same result for&#xA;continuous functions as Riemann's integral.&#xA;Cauchy's approach is used sometimes in elementary calculus courses, because it is conceptually simpler than the Riemann approach.&lt;/p&gt;&#xA;&lt;p&gt;It is not true that Riemann integral is &amp;quot;not as useful&amp;quot; as Lebesgue's integral. Because in many cases the functions we integrate are continuous. For example in all books on Complex Analysis, Riemann integral is used. It is also taught in almost all courses of real analysis, with very few exceptions, before the Lebesgue's integral, even in the Bourbaki course.&lt;/p&gt;&#xA;&lt;p&gt;Sources. Medvedev, Development of the concept of integral, Moscow 1974.&#xA;I. N. Pesin, Development of the concept of integral, Moscow 1966 (There is an English translation).&#xA;N. Bourbaki,  Elements of the history of mathematics. Translated from the 1984 French original by John Meldrum. Springer-Verlag, Berlin, 1994.&#xA;N. Bourbaki, Functions of a real variable, Elementary theory. Translated from the French, Springer 2004.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-07T18:59:52.527" LastActivityDate="2021-09-07T18:59:52.527" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="1998" PostTypeId="2" ParentId="1996" CreationDate="2015-02-21T22:54:12.037" Score="4" Body="&lt;p&gt;I seem to remember a description of the origin of the Riemann integral that goes like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Integrals were widely used ever since Newton and Leibniz.  But everyone just assumed all functions (or at least bounded functions on a finite interval) had integrals.  In due time, the abstract notion of &quot;function&quot; came in.  Then it was not so clear that all functions have integrals.  At one point Cauchy published an argument that continuous functions have integrals.  (Looking back today, we can see the notion of uniform continuity in the proof.)  When Riemann looked at the proof he said: Here is what's going on: if we take such-and-such as the &lt;em&gt;definition&lt;/em&gt; for &quot;integral&quot;, then Cauchy showed that continuous functions satisfy it.  But look, some discontinuous functions also satisfy it.  So Riemann didn't just invent the definition from scratch.  He distilled it from a proof of Cauchy.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-02-21T22:54:12.037" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="1999" PostTypeId="2" ParentId="1987" CreationDate="2015-02-22T16:53:52.630" Score="27" Body="&lt;p&gt;In 1959, Eugene Wigner presented a talk on &lt;em&gt;The Unreasonable Effectiveness of Mathematics in the Natural Sciences.&lt;/em&gt; Many papers on the unreasonable effectiveness of mathematics in this field, that field, and some other field quickly followed suit. As an opposing point of view, the unreasonably effective (800+ papers, 30+ books) mathematician I.M.Gelfand noted that (emphasis mine)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. &lt;strong&gt;He meant physics, of course.&lt;/strong&gt; There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is &lt;strong&gt;the unreasonable ineffectiveness of mathematics in biology&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I.M. Gelfand said this after developing an interest in biology due to the premature death of his son. He organized a weekly seminar that attracted the best minds in Russian biology and mathematics. Gelfand's interest in biology resulted in pioneering work in the field of biomathematics. Not only is mathematics applicable to some aspects of biology, biology has motivated many new developments in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Just a few of the areas where biology has inspired new mathematics:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Mathematics inspired by population modeling&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Modeling population dynamics has been a fruitful application of mathematics starting with Euler, who studied age distributions in stable populations (Euler 1760). What about unstable populations? One of the seminal papers (if not the seminal paper) in the development of chaos theory was written by Robert May in 1976. May was educated as a physicist (where mathematics is unreasonably effective), but then switched to biology (where mathematics is supposedly unreasonably ineffective). His paper on population dynamics (May 1976) discusses the logistic map, &lt;span class=&quot;math-container&quot;&gt;$x_{n+1} = \lambda x_n(1-x_n)$&lt;/span&gt;, which he used to model population dynamics. This paper marked the beginning of chaos theory.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical techniques inspired by evolution and animal behavior&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The contributions of biology to optimization theory are immense. Evolution has motivated a number of techniques used in mathematics and artificial intelligence, including evolutionary programming (Fogel 1966), evolutionary strategy (Rechenberg 1973), genetic algorithms (Holland 1975).&lt;/p&gt;&#xA;&lt;p&gt;Emulating animal behavior has provided a number of other optimization techniques. These include the critter of the month optimization techniques, starting with ant colony optimization (Dorigo 1996). Ant colony optimization has been used to attack a large number of problems from very diverse fields. Now there are bees algorithms, bacterial colony optimization algorithms, foraging algorithms, all based on the behaviors of simple creatures. I won't give references for all of these; there are now entire journals dedicated to this subject (e.g., the &lt;a href=&quot;http://cis.ieee.org/ieee-transactions-on-evolutionary-computation.html&quot; rel=&quot;noreferrer&quot;&gt;IEEE Transactions on Evolutionary Computation&lt;/a&gt;). Collectively, these fall into the category of &lt;a href=&quot;http://en.wikipedia.org/wiki/Swarm_intelligence&quot; rel=&quot;noreferrer&quot;&gt;swarm intelligence&lt;/a&gt;. One last technique that I will give a reference on is particle swarm optimization (Kennedy 1995, Eberhart 1996).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Mathematics inspired by DNA sequencing&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The above areas are new areas of applied mathematics. The problem of how to &lt;a href=&quot;http://en.wikipedia.org/wiki/DNA_sequencing_theory&quot; rel=&quot;noreferrer&quot;&gt;sequence DNA&lt;/a&gt; produced not only new applied mathematics (e.g., the image below from Letunic 2007) but new theoretical mathematics as well.&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/1/11/Tree_of_life_SVG.svg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;All of these developments led Joel Cohen (Cohen 2004) to conjecture that &lt;strong&gt;mathematics is biology's next microscope, only better; biology is mathematics' next physics, only better&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Joel E. Cohen (2004), &amp;quot;Mathematics is biology's next microscope, only better; biology is mathematics' next physics, only better.&amp;quot; &lt;em&gt;PLoS Biology&lt;/em&gt; 2.12:e439.&lt;/p&gt;&#xA;&lt;p&gt;Marco Dorigo, et al. (1996), &amp;quot;Ant system: optimization by a colony of cooperating agents,&amp;quot; &lt;em&gt;IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics&lt;/em&gt;  26.1:29-41.&lt;/p&gt;&#xA;&lt;p&gt;Leonard Euler (1760), &amp;quot;Recherches générales sur la mortalité et la multiplication,&amp;quot; &lt;em&gt;Mémoires de l’Académie Royal des Sciences et Belles Lettres&lt;/em&gt; 16:144–164.&lt;/p&gt;&#xA;&lt;p&gt;Russell Eberhart and James Kennedy, &amp;quot;A new optimizer using particle swarm theory,&amp;quot; &lt;em&gt;Proceedings of the Sixth International Symposium on Micro Machine and Human Science&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Lawrence J. Fogel, et al. (1966), &amp;quot;Artificial intelligence through simulated evolution,&amp;quot; Wiley.&lt;/p&gt;&#xA;&lt;p&gt;John H. Holland, (1975), &amp;quot;Adaptation in Natural and Artificial Systems,&amp;quot; University of Michigan Press (second edition, MIT Press, 1992).&lt;/p&gt;&#xA;&lt;p&gt;James Kennedy and Russel Eberhart (1995), &amp;quot;Particle swarm optimization,&amp;quot; &lt;em&gt;Proc. IEEE International Conf. on Neural Networks&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ivica Letunic and Peer Bork (2007), &amp;quot;Interactive Tree Of Life (iTOL): an online tool for phylogenetic tree display and annotation,&amp;quot; &lt;em&gt;Bioinformatics&lt;/em&gt; 23.1:127-128.&lt;/p&gt;&#xA;&lt;p&gt;Robert M. May (1976), &amp;quot;Simple mathematical models with very complicated dynamics,&amp;quot; &lt;em&gt;Nature&lt;/em&gt; 261.5560:459-467.&lt;/p&gt;&#xA;&lt;p&gt;Ingo Rechenberg (1973), &amp;quot;Evolutionsstrategie Optimierung technischer Systeme nach Prinzipien der biologischen Evolution,&amp;quot; (PhD thesis), Friedrich Frommann Verlag, Struttgart-Bad Cannstatt.&lt;/p&gt;&#xA;&lt;p&gt;Eugene P. Wigner (1960), &amp;quot;The unreasonable effectiveness of mathematics in the natural sciences,&amp;quot; &lt;em&gt;Communications on Pure and Applied Mathematics&lt;/em&gt; 13.1:1-14.&#xA;Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="4251" LastEditDate="2020-07-13T13:09:25.780" LastActivityDate="2020-07-13T13:09:25.780" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2000" PostTypeId="1" CreationDate="2015-02-22T20:00:30.607" Score="9" ViewCount="154" Body="&lt;p&gt;Starting somewhere in the 19th century, mathematics turned from the study of concrete objects to the study of objects satisfying enough properties to lead to interesting theorems. For example:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;From permutation groups and groups of geometrical transfomations we got the group axioms.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From ad hoc methods of approximating functions in analysis we got metric spaces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;From metric spaces we got &lt;em&gt;topological&lt;/em&gt; spaces.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Are there documented examples of mathematicians from this period explicitly noticing and commenting on this trend (either in mathematics in general, or in a particular field)? In particular, are there examples of resistance to the idea?&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-02-22T20:00:30.607" Title="Contemporary reactions to the rise of axiomatization in the 19th/20th centuries" Tags="|mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2002" PostTypeId="1" CreationDate="2015-02-23T02:57:55.510" Score="19" ViewCount="3085" Body="&lt;p&gt;Single authors referring to themselves as &quot;we&quot; is still commonplace today, and already Newton was we-ing in Principia. There is even a Latin term for we-ing, nosism, from &quot;nos&quot;, which is Latin for &quot;we&quot;. Googling &quot;royal we in scientific writing&quot; returns surprisingly many hits, including a &lt;a href=&quot;http://en.wikipedia.org/wiki/Royal_we&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;, &lt;a href=&quot;https://academia.stackexchange.com/questions/15344/what-to-use-instead-of-academic-we-when-describing-an-experiment&quot;&gt;two questions&lt;/a&gt; on Academia.SE, and a &lt;a href=&quot;http://scienceblogs.com/evolgen/2007/08/15/the-notsoroyal-we/&quot; rel=&quot;noreferrer&quot;&gt;hilarious thread&lt;/a&gt; on scienceblogs with passages like &quot;&lt;em&gt;I’m we-ing all over myself&lt;/em&gt;&quot;, &quot;&lt;em&gt;it is apparently meant to include the author and the reader, and presupposes that the reader always agrees with anything the author might say&lt;/em&gt;&quot;, and &quot;&lt;em&gt;As kids we were schooled to avoid ‘I’, lest we be look like we had big egos. Instead we should be humble, and therefore all first person was the plural. This was taught to us by teachers who thought their own opinions were pure gold and more important than anything else on earth&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess this tells us why the practice persists, but not how it originated. The best I could find is &lt;a href=&quot;http://www.grammarphobia.com/blog/2011/07/nosism.html&quot; rel=&quot;noreferrer&quot;&gt;Grammarphobia blog&lt;/a&gt; quoting Oxford English Dictionary, whose earliest example of nosism is &quot;Old English translation of a 5th-century Christian history written in Latin by Paulus Orosius, a student of St. Augustine&quot;. We doubt however that scientists of all people would have been swayed by an obscure student of St. Augustine. More importantly, at least in the English translation, &lt;a href=&quot;http://cerebro.xu.edu/math/math147/02f/ptolemy/ptolemytext.html&quot; rel=&quot;noreferrer&quot;&gt;Ptolemy is already we-ing&lt;/a&gt; in the Almagest three centuries earlier. Of course, we don't know if we can trust that modern translators did not insert modern usage, but they did translate &lt;a href=&quot;http://www.wilbourhall.org/pdfs/archimedes/archimedesHeiberg.pdf&quot; rel=&quot;noreferrer&quot;&gt;Archimedes&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=xXk6AAAAIAAJ&amp;amp;pg=PR75&amp;amp;lpg=PR75&amp;amp;dq=apollonius+conics+appendix+preface+to+book+5&amp;amp;source=bl&amp;amp;ots=OpKWN68JzW&amp;amp;sig=JeO1brINZkfFJuoxgTCyOzf0VRY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=55LqVMvQF46NyASBg4LIAg&amp;amp;ved=0CC4Q6AEwAw#v=onepage&amp;amp;q=apollonius%20conics%20appendix%20preface%20to%20book%205&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Apollonius&lt;/a&gt; referring to themselves as &quot;I&quot;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So where did nosism come from and how did it take hold? Is Ptolemy we-ing in the original, and was he the one who started the practice in science (that would make more sense than Orosius)? Or was it somebody else?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.757" LastActivityDate="2015-02-23T17:20:44.193" Title="How did &quot;royal we&quot; become a standard of scientific writing?" Tags="|naming-conventions|social-context|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2003" PostTypeId="2" ParentId="2002" CreationDate="2015-02-23T13:25:51.237" Score="12" Body="&lt;p&gt;The &quot;we&quot; in the scientific papers does not mean the &quot;royal we&quot;. This &quot;we&quot; means &quot;you and I&quot;, that is &quot;the author and the reader&quot;. For example, in proving something they say &quot;let us consider&quot;... and then &quot;we conclude&quot;. A teacher addressing the class says: &quot;we write&quot;,&#xA;&quot;we draw&quot;, inviting her students to do the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some people write instead &quot;I consider...&quot; &quot;I obtain&quot;,&#xA;but this sounds more arrogant and rarely used.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-23T13:25:51.237" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2005" PostTypeId="1" AcceptedAnswerId="2006" CreationDate="2015-02-23T14:44:04.890" Score="7" ViewCount="497" Body="&lt;p&gt;I know this question sounds like it should be in the sci-fi site (and maybe it should), but I'm more interested in the state of metallurgy etc at the time of the Roman Empire (let's say around 0 AD) and if a modern electrical engineer, if somehow magically transported back to that time, would be able to create and store electricity in a way that would be useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, did the Romans have the technology for drawing copper wire thin enough to create a rotor/stator setup, and did they have the ability to create a mechanical crank and gears to turn this type of primitive electromagnetic motor, as well as all concomitant parts (housing, shafts, etc.)? Would they have had the grease? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="1891" LastEditorUserId="26" LastEditDate="2015-03-01T13:29:53.627" LastActivityDate="2015-03-01T13:29:53.627" Title="Was it technologically possible to create electricity, batteries, and electrical devices at the time of the Roman Empire?" Tags="|electricity|roman-empire|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2006" PostTypeId="2" ParentId="2005" CreationDate="2015-02-23T19:18:35.487" Score="3" Body="&lt;p&gt;The simplest electric generator contains copper wire, and permanent magnet (loadstone).&#xA;You need no grease. There is no doubt that copper wire could be made, but probably was not readily available. With some effort, you can make it yourself, using technology available to the Romans. (They did make very fine wire of gold, for example). It will be a bit difficult&#xA;to insulate the wire properly (otherwise you cannot make a coil). Other parts could be certainly made. So you can make simplest electric motor/generator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Batteries or accumulators will be more difficult: chemistry was in very primitive state,&#xA;and making the necessary chemicals yourself is much harder, but probably not completely impossible. Volta's first batteries used plates of copper and zink, and salt water brine&#xA;as electrolite. Everything available in Rome. The general conclusion is that given enough time and money, this is probably possible. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-02-23T19:18:35.487" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2007" PostTypeId="2" ParentId="184" CreationDate="2015-02-24T01:30:45.947" Score="9" Body="&lt;p&gt;&lt;a href=&quot;http://nautil.us/issue/21/information/the-man-who-tried-to-redeem-the-world-with-logic&quot; rel=&quot;noreferrer&quot;&gt;Amanda Gefter's recent article in &lt;em&gt;Nautilus&lt;/em&gt;&lt;/a&gt; takes a very unambiguous position on the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There was just one person who wasn’t happy about the reunion: Wiener’s wife.&#xA;  Margaret Wiener was, by all accounts, a controlling, conservative prude—&#xA;  and she despised McCulloch’s influence on her husband. McCulloch&#xA;  hosted wild get-togethers at his family farm in Old Lyme, Connecticut,&#xA;  where ideas roamed free and everyone went skinny-dipping. It had been&#xA;  one thing when McCulloch was in Chicago, but now he was coming to Cambridge&#xA;  and Margaret wouldn’t have it. And so she invented a story. She sat&#xA;  Wiener down and informed him that when their daughter, Barbara, had&#xA;  stayed at McCulloch’s house in Chicago, several of “his boys” had&#xA;  seduced her. Wiener immediately sent an angry telegram to Wiesner:&#xA;  “Please inform [Pitts and Lettvin] that all connection between me and&#xA;  your projects is permanently abolished. They are your problem. Wiener.”&#xA;  He never spoke to Pitts again.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Gefter references Conway &amp;amp; Siegelman's 2006 &lt;a href=&quot;http://darkherooftheinformationage.stillpointpress.net/&quot; rel=&quot;noreferrer&quot;&gt;Dark Hero of the Information Age&lt;/a&gt;, a biography of &lt;a href=&quot;https://en.wikipedia.org/wiki/Norbert_Wiener&quot; rel=&quot;noreferrer&quot;&gt;Norbert Wiener&lt;/a&gt;. It's a great pity that Pitts' life isn't better documented, because he was, from any point of view, a fascinating genius, as was recognized at the time.&lt;/p&gt;&#xA;" OwnerUserId="1892" LastEditorUserId="4419" LastEditDate="2016-07-15T07:36:11.647" LastActivityDate="2016-07-15T07:36:11.647" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2008" PostTypeId="1" AcceptedAnswerId="2009" CreationDate="2015-02-24T14:09:25.647" Score="5" ViewCount="918" Body="&lt;p&gt;Gauss introduced the Gaussian integers in an 1832 Latin work named &lt;em&gt;Theoria residuorum biquadraticorum&lt;/em&gt;. I believe there is a German translation available. Is there an English, or possibly French translation? If not, is there any exposition of the work available in English in some more modern article or textbook?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="10106" LastEditDate="2020-06-07T13:41:08.827" LastActivityDate="2020-06-07T13:41:08.827" Title="Is there a translation of Gauss' work on Gaussian integers?" Tags="|mathematics|reference-request|number-theory|translations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2009" PostTypeId="2" ParentId="2008" CreationDate="2015-02-24T14:52:48.133" Score="4" Body="&lt;p&gt;&lt;em&gt;Theoria residuorum biquadraticorum&lt;/em&gt; was in fact published in two parts. Gauss wrote separate German reviews of many of his published papers and his review for the second part, where he introduces the complex integers, is translated into English by Ewald in &lt;em&gt;From Kant to Hilbert&lt;/em&gt;, vol. 1, p. 307 (originally in &lt;em&gt;Werke&lt;/em&gt;, 2, p. 169).&lt;/p&gt;&#xA;" OwnerUserId="1817" LastEditorUserId="1817" LastEditDate="2015-03-21T22:47:58.847" LastActivityDate="2015-03-21T22:47:58.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2010" PostTypeId="1" AcceptedAnswerId="2011" CreationDate="2015-02-24T17:17:40.433" Score="14" ViewCount="773" Body="&lt;p&gt;I seem to fail at googling a simple question: In which language did Pólya originally write “How To Solve It” (or “Schule des Denkens”, as is the German title)? Did he translate the book himself to other languages?&lt;/p&gt;&#xA;" OwnerUserId="412" OwnerDisplayName="k.stm" LastEditorUserId="98" LastEditDate="2015-02-25T00:35:16.373" LastActivityDate="2015-02-25T01:01:30.793" Title="Was Pólya’s “How To Solve It” originally written in German or English (or some other language)?" Tags="|mathematics|biographical-details|mathematicians|publishing|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2011" PostTypeId="2" ParentId="2010" CreationDate="2015-02-25T01:01:30.793" Score="23" Body="&lt;p&gt;&lt;em&gt;How To Solve It&lt;/em&gt; was originally published in English in 1945 by Princeton University Press in English, after being rejected by three other U.S. publishers.&lt;/p&gt;&#xA;&lt;p&gt;However, the original text at least started out in German as a draft. Pólya began writing the draft prior to 1940, while he was living in Zürich, and presumably initially intended for the text to be published in Germany. However, the political situation in Europe at the time forced him to relocate to the U.S. Naturally, there weren't many opportunities in the U.S. to publish a book in German, so he published it in English instead. The English version was substantially different from the German draft, not solely a translation.&lt;/p&gt;&#xA;&lt;p&gt;Here is a quote from an interview with Pólya saying the same thing, reproduced from [1]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;MP&lt;/strong&gt;: What I would like to ask about that, then, is: When did you develop an interest in teaching at the earlier levels and was this in any way related to your work with secondary school teachers in the teacher institute programs?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Pólya&lt;/strong&gt;:  No, I was interested in it fairly early. I have also a teaching certificate, as I told you, for the lower classes, in Latin and Hungarian. In fact I have a teaching certificate for all classes of the gymnasium in mathematics, physics, and even philosophy, and in order to get the certificate you must be a practice teacher. So I taught in the high-school level. They call it gymnasium, for kids between ten and eighteen, but it is different in several respects from a high school. I was a practice teacher for a year. So I had experience. I must tell you, &lt;em&gt;How to Solve It&lt;/em&gt; was written really twice. I wrote something, a draft, in German while I was still in Zürich. Then I came to America and in this respect, my coming to America was, I think, useful, because here, in this country, there is more interest in the “How to” books. And, by the way, Hardy predicted it to me. When I told him about the “How to” book, he said, “Oh, you must go to America.” And then I rewrote it, it appeared in final form in English. It is considerably different from my original German version, and I think to its advantage.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Reference:&lt;/p&gt;&#xA;&lt;p&gt;[1]: D J Albers and G L Alexanderson (eds.), 2008. &lt;em&gt;Mathematical People: Profiles and Interviews&lt;/em&gt; (2nd ed.) (pp. 257-258). p. Wellesley, MA:  A K Peters.&lt;/p&gt;&#xA;" OwnerUserId="58" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-02-25T01:01:30.793" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2012" PostTypeId="1" AcceptedAnswerId="2013" CreationDate="2015-02-25T15:18:03.960" Score="10" ViewCount="2554" Body="&lt;p&gt;When did primates, protohumans, or humans make the connection between sexual activity and pregnancy?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I notice that many ancient statues depict &lt;a href=&quot;http://en.wikipedia.org/wiki/Venus_figurines&quot;&gt;pregnancy&lt;/a&gt; or &lt;a href=&quot;http://www.ibtimes.co.uk/100-prehistoric-cult-sites-penis-stone-structures-vulva-shapes-discovered-israel-1487432&quot;&gt;penises&lt;/a&gt;, this may (tenuously) suggest that the connection was known at the time that statue-making became known. Do we have any other evidence or knowledge of when the connection was made, or at least have bounds for a time interval in which drawing the connection would have been likely?&lt;/p&gt;&#xA;" OwnerUserId="261" LastEditorUserId="55" LastEditDate="2015-02-26T22:31:56.287" LastActivityDate="2015-02-26T22:31:56.287" Title="When did humans realize that sex leads to pregnancy?" Tags="|discoveries|medicine|physiology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2013" PostTypeId="2" ParentId="2012" CreationDate="2015-02-25T16:16:51.290" Score="14" Body="&lt;p&gt;The question asked in the title is not at all the same as the fist question in the body of the text. Because any meaningful answer to the latter seemingly presupposes (wrongly in my opinion) the existence of a mechanism to ascertain if a non-human animal has made an abstract connection in the scientific consensus, I will simply note that paternal care is &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF02192783&quot;&gt;rare&lt;/a&gt; among non-human primates and apparently &lt;a href=&quot;http://pin.primate.wisc.edu/factsheets/entry/chimpanzee/behav&quot;&gt;non-existent&lt;/a&gt; among our closest kin and so that the connection between sexual activity and fecundation is in all likelihood a learned fact, not part of our core evolved cognitive abilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, parietal art, though it contains exaggerated depiction of both sexual organs and of gravid women (&lt;a href=&quot;http://en.wikipedia.org/wiki/Venus_figurines&quot;&gt;Venus figures&lt;/a&gt;), gives no indication of a link between the two (though it is somewhat hard to decide what would count as uncontroversial evidence).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Egyptian papyri abounds in detailed gynecological knowledge, complete with accurate early prognosis for pregnancy and contraception advice. So the absolute upper historical bound for this connection is 1800 BCE, the time of writing of the &lt;a href=&quot;http://www.reshafim.org.il/ad/egypt/timelines/topics/kahunpapyrus.htm&quot;&gt;Kahun papyrus&lt;/a&gt;. I will also note that the &lt;a href=&quot;http://en.wikipedia.org/wiki/Code_of_Ur-Nammu&quot;&gt;Code of Ur-Nammu&lt;/a&gt;, written around 2100BCE and one of the oldest known written text, deals largely with deflowering, offsprings produced by a couple of slaves, by a couple formed of a slave and a free person etc. This strongly suggests that the connection between sex and procreation was perfectly well understood by the writers of this code. Likewise, paternal as well as maternal lineage is very clearly understood in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Pyramid_Texts&quot;&gt;Pyramid texts&lt;/a&gt; dating from around 2400 BCE. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, there are no evidence that the connection was understood in pre-historical time, just as there is no or scarce evidence for any abstract knowledge in pre-historical time almost by definition. However, evidence that the connection was very well understood is almost contemporary with the oldest known written texts.      &lt;/p&gt;&#xA;" OwnerUserId="189" LastActivityDate="2015-02-25T16:16:51.290" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2014" PostTypeId="2" ParentId="557" CreationDate="2015-02-26T00:07:20.477" Score="3" Body="&lt;p&gt;Elements were standardized before we were even talking about chemistry: The alchemical elements were all well-known to those dabbling in that &quot;science&quot;.  This should put the answer to your question around 1670 according to the &lt;a href=&quot;http://webapp1.dlib.indiana.edu/newton/fonts/Alchemy%20Unicode%20Proposal---March%2031%202009.pdf&quot; rel=&quot;nofollow&quot;&gt;Proposal for Alchemical Symbols in Unicode&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for modern chemical symbols, the first notation approximation of modern elements I could find are &lt;a href=&quot;http://www.uh.edu/engines/jdsymbols.jpg&quot; rel=&quot;nofollow&quot;&gt;John Dalton's elements&lt;/a&gt; in 1808 where we can still clearly see alchemical influences on his representations of elements, though evolving more towards modern notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the &lt;a href=&quot;http://www.mhs.ox.ac.uk/visit/virtual-tour/#&quot; rel=&quot;nofollow&quot;&gt;virtual tour&lt;/a&gt; of the basement of the Museum of the History of Science in Oxford, you will find Charles Daubeny's &quot;&lt;a href=&quot;http://www.meta-synthesis.com/webbook/35_pt/Daubeny1.png&quot; rel=&quot;nofollow&quot;&gt;Symbols of Simple bodies&lt;/a&gt;&quot; dated to 1831.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So not an exact date, but depending on your definition of &quot;(al)chemical elements&quot; middle of the 17th century or early 19th century.&lt;/p&gt;&#xA;" OwnerUserId="1893" LastEditorUserId="1893" LastEditDate="2015-02-27T08:56:28.063" LastActivityDate="2015-02-27T08:56:28.063" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2015" PostTypeId="1" CreationDate="2015-02-26T03:33:31.343" Score="9" ViewCount="94" Body="&lt;p&gt;I am interested in the historical priority in population biology, essays or monographs, discussing the concept of territoriality prior to 1950. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is it? In the early 18th century discussions of mortality due to conflict or starvation resulting from the scarcity of food, there was an implicit belief that the end result in the surviving population is widespread malnutrition, and some mortality due to starvation. Later, biologists and demographers realized that often no malnutrition in the surviving part of the population occurs, but a much greater mortality rate from starvation is observed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there are two stronger individuals, two weaker individuals and one pie, the outcome often is not that each individual takes a quarter of the pie due to common competition. That would result in general malnutrition if the food supply is sufficiently scarce, in turn causing us to observe starvation equally distributed in the population. Instead, in species with territoriality, the stronger (typically more aggressive) individuals each eat their fill (how much they would consume in the case food was not scarce) let's say half the pie, and the weaker individuals starve. This occurs because the stronger individuals expel the weaker individuals from larger foraging areas in proportion to the scarcity of food. Eventually, weaker individuals are expelled to areas without food, or are left with insufficient foraging areas to feed themselves, rather than all individuals competing for food in a single area.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="26" LastEditDate="2015-03-01T13:29:03.257" LastActivityDate="2015-03-01T13:29:03.257" Title="On the history of population dynamics of territorial species" Tags="|biology|statistics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2016" PostTypeId="2" ParentId="1987" CreationDate="2015-02-26T17:39:34.303" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The famous Polya's 1937 paper &quot;Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen&quot; is considered, as far as I understand, as one of pillars of the modern theory of combinatorial enumeration. The title of the paper suggests that, at least to some degree, it was motivated by a question from chemistry.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Genetic algorithms.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some algebro-geometric questions from algebraic statistics seem to be &#xA;motivated by biological considerations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;My personal impression (based mainly on hearing people from a few IHES &#xA;conferences devoted to interaction of mathematics and biology, and personal&#xA;experience from working in a biotech industry) confirms &#xA;Alexandre Eremenko's comment that &quot;biology is too young&quot;: the structure of&#xA;biological knowledge is mainly descriptive, as opposed to the physical one,&#xA;organized in &quot;theories&quot;.&lt;/p&gt;&#xA;" OwnerUserId="167" LastEditorUserId="167" LastEditDate="2015-02-27T08:12:56.170" LastActivityDate="2015-02-27T08:12:56.170" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2017" PostTypeId="1" CreationDate="2015-02-26T19:44:23.990" Score="3" ViewCount="380" Body="&lt;p&gt;It is just a notation, but it is so economical and so systematic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So who invented them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A handy notation should be helpful for the development of the whole field. &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="62" LastEditDate="2015-03-05T15:28:18.970" LastActivityDate="2015-03-05T15:28:18.970" Title="How was the use of upper and lower indices in tensor notation developed?" Tags="|mathematics|differential-geometry|notation|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2018" PostTypeId="2" ParentId="1801" CreationDate="2015-02-28T18:52:00.757" Score="-3" Body="&lt;p&gt;I am surprised that no one has considered the number of times that the zero is used in the 5 books of Moses in the Hebrew language, which were written down by Moses in about  1273 BCE, as being even older that India or anywhere else.    And considering that the first 11 chapters far predate the Hebrew people and Abraham, then you can push that date back maybe another 2000 years at least.  In Genesis chapter 5, the first clear use of zero is used for how old people were when their first sons were born and how long they lived, we see that zero is used 14 times!  And by the way, archeology continues to validate the historicity and dates given in Genesis.&lt;/p&gt;&#xA;" OwnerUserId="1914" LastActivityDate="2015-02-28T18:52:00.757" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2019" PostTypeId="2" ParentId="1801" CreationDate="2015-02-28T22:38:05.650" Score="0" Body="&lt;p&gt;The use of zero in India must be a lot older than the inscription from AD 876 mentioned by HDE. The use of the &quot;Indian numbers&quot; and specifically zero was introduced into the Muslim world by al-Khwarizmi, who died about AD 850.And he had them from some Indian text.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-02-28T22:38:05.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2020" PostTypeId="2" ParentId="1991" CreationDate="2015-03-01T15:23:47.793" Score="2" Body="&lt;p&gt;To answer the &quot;why&quot; part of your question: In Greek and Roman antiquity there was no economic incentive to produce labour-saving devices due to the ready availability of slave labour. The technological advances were mainly in the field of military technology (think of Archimedes and his burning mirrors).&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-03-01T15:23:47.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2021" PostTypeId="2" ParentId="287" CreationDate="2015-03-01T18:19:58.977" Score="7" Body="&lt;p&gt;The theorem that any compass and straightedge construction can be performed by a compass alone is called the &lt;a href=&quot;http://en.wikipedia.org/wiki/Mohr%E2%80%93Mascheroni_theorem&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Mohr-Mascheroni Theorem&lt;/strong&gt;&lt;/a&gt;. It was first described by the Danish mathematician &lt;strong&gt;Georg Mohr&lt;/strong&gt; (1640-1697) in &lt;strong&gt;1672&lt;/strong&gt;. However, Mohr's proof was lost until 1928 and the theorem was independently proved by the Italian mathematician &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Mascheroni.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Lorenzo Mascheroni&lt;/strong&gt;&lt;/a&gt; in &lt;strong&gt;1797&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mascheroni's interest in the problem was in fact inspired by ad hoc compass-only constructions used for astronomical quadrants at the time. Quoting from &lt;em&gt;Dictionary of Scientific Biography&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;In the preface Mascheroni recounts the genesis of his work. He was moved initially by a desire to make a contribution to elementary geometry. It occurred to him that ruler and compass could perhaps be separated, as water can into two gases; but he was assailed by doubts and fears often attendant upon research. &lt;strong&gt;He then chanced to reread an article on the way Graham and &lt;a href=&quot;http://cudl.lib.cam.ac.uk/view/ES-LON-00015/1&quot; rel=&quot;noreferrer&quot;&gt;Bird&lt;/a&gt; [who supplied instruments to Maupertuis] had divided their great astronomical quadrant, and he realised that the division had been made by compass alone, although, to be sure, by trial and error&lt;/strong&gt;. This encouraged him and he continued his work with two purposes in mind: to give a theoretical solution to the problem of constructions with compasses alone and to offer practical constructions that might be of help in making precision instruments. The second concern is shown in the brief solutions of many specific problems and in a chapter on approximate solutions.&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sophiararebooks.com/pages/books/3508/lorenzo-mascheroni/la-geometria-del-compasso&quot; rel=&quot;noreferrer&quot;&gt;This summary&lt;/a&gt; notes that the problem of carrying out geometric constructions under additional restrictions was previously also considered by &quot;da Vinci, Dürer, Cardano and Tartaglia, among others&quot;. However, I don't know if these too explicitly linked to astronomical research.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastActivityDate="2015-03-01T18:19:58.977" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2022" PostTypeId="1" AcceptedAnswerId="2025" CreationDate="2015-03-01T22:01:59.840" Score="6" ViewCount="375" Body="&lt;p&gt;Johannes Hevelius of Danzig (1611-1687) was the most famous astronomer of his time, and also&#xA;the last famous astronomer who did not use telescope. Most books on the history of astronomy&#xA;(and even the Wikipedia article on Hevelius) mention a dispute between Hooke and Hevelius in which Hooke tried to convince him to use telescope. Now let me cite Rees's Cyclopedia&#xA;or the universal dictionary (1819):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is somewhat extraordinary that the dispute alluded to terminated with the public opinion in favor of plain sights, in preference to telescopic ones, as to the accuracy of observations made respectively with each...&lt;/p&gt;&#xA;&lt;p&gt;In 1679, Dr. Halley traveled to Dantzic to settle the dispute between Dr. Hooke and Hevelius...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This dispute is always mentioned but without any detail.&#xA;My question: what were the arguments of Hevelius against the telescopic sites?&lt;/p&gt;&#xA;&lt;p&gt;Of course I understand that without a telescope you cannot see Jupiter satellites, phase of Venus, and other things discovered since Galileo. But here we are talking about&#xA;&amp;quot;classical&amp;quot; astronomical observation, that is of measuring coordinates of stars&#xA;and other celestial objects.&lt;/p&gt;&#xA;&lt;p&gt;Reed refers to the book of Hevelius, Cometographia, I don't know whether it is available, and in any case, my Latin is not good enough to read it. Perhaps there exists some exposition of this discussion in a modern language?&lt;/p&gt;&#xA;&lt;p&gt;Remark. I did a lot of this kind of observations myself, with marine sextants. And I can confirm that even a poor quality telescope helps. A sextant has a scale which can be read&#xA;with 6&amp;quot; to 10&amp;quot; accuracy. With a naked eye I could measure the distance between two stars&#xA;to 30&amp;quot; in the average (same accuracy as Hevelius). With a simple telescope 30x6 (without any micrometer) the accuracy is about 10&amp;quot;-15&amp;quot;, almost the same as Hooke claimed.&#xA;The difference is that Hevelius instruments were enormous, and the modern one is portable,&#xA;and does not require a firm foundation.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-03-03T21:21:35.460" Title="Hevelius-Hooke dispute" Tags="|astronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2023" PostTypeId="2" ParentId="138" CreationDate="2015-03-02T06:09:26.793" Score="1" Body="&lt;p&gt;Ignorance of the dangers of ionizing radiation is something you would have seen ca. 1905, when Marie Curie was working. (She was killed by the radiation she worked with.) The idea of a nuclear chain reaction was not proposed until much, much later -- by Szilard in 1933. By this time physicists had fairly good knowledge of the nucleus, neutrons and protons, the line of stability, the interaction of ionizing radiation with matter, and the health effects of ionizing radiation. In particular, we know almost nothing today that was not known in 1945 about the effects of low population doses of radiation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What underwent further evolution after 1945 was not physicists' understanding of these issues but the conception of these issues in the minds of politicians and laypeople. In the popular consciousness, the 1950s, for example, were the era of monster movies in which radioactivity acquired various scary voodoo powers.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2015-03-02T06:09:26.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2024" PostTypeId="1" AcceptedAnswerId="2026" CreationDate="2015-03-02T19:39:39.680" Score="4" ViewCount="295" Body="&lt;p&gt;Some friend from Georgia told me that in every village of the Soviet Union there was an expert in nuclear physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, the Soviet Union did not invest into biology at all. As a consequence, people in Russia now do not know how to do biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it really so crazy?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Soviet Union is fascinating in some sense, I would say. &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="4251" LastEditDate="2020-05-13T11:19:59.250" LastActivityDate="2020-05-13T11:19:59.250" Title="Did the Soviet Union emphasize nuclear physics over biology?" Tags="|physics|biology|social-context|soviet-union|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2025" PostTypeId="2" ParentId="2022" CreationDate="2015-03-02T21:56:25.790" Score="6" Body="&lt;p&gt;The debate was not about using telescopes, Hevelius was using telescopes since 1647, but about &quot;telescopic sights&quot;, micrometers with attached lenses and crosshairs introduced by Hooke, Wren and in France Auzout in 1660-s. It started innocently enough as an exchange of letters in 1665-68 through Oldenburg, a secretary of the Royal Society, and then escalated into a public spat after Hooke's 1674 pamphlet in response to Hevelius's book Machina Coelestis Pars Prior published in 1673 (Cometographia was written before the debate took shape). The context, the substance and the consequences of the debate are described and analyzed in great detail in chapter 4 of &lt;a href=&quot;http://scholar.lib.vt.edu/theses/available/etd-11162001-113447/&quot; rel=&quot;nofollow&quot;&gt;Saridakis's 2001 thesis&lt;/a&gt; available online. The last episode of the controversy, which reignited in 1685, is covered in the &lt;a href=&quot;https://royalsociety.org/events/2012/triangulating-positions/&quot; rel=&quot;nofollow&quot;&gt;Moxham's video lecture&lt;/a&gt;, which focuses on social context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As early as 1665 Oldenburg communicated Hooke's opinion that &quot;few persons can distinguish an angle less than 1$'$&quot;, and that telescopic sights allow to make &quot;instruments of much less bulk, to do ten times more&quot;. Hevelius replied in 1667 and asked for  full description of telescopic sights fitted to sextants, and for inexpensive lenses of long focal length. But he also made his first arguments against the telescopic sights, and asked for a proof that telescopic sights are more accurate by comparing eight observations of angular distances between stars. Hooke never sent the lenses, despite being prompted to do so by the Royal Society, or took up Hevelius's challenge. The dispute escalated in 1673 after Hevelius gave an expanded version of his arguments in Machina Coelestis without mentioning Hooke. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hevelius was an old school astronomer in the tradition of Tycho Brahe, he believed in using large instruments, including large telescopes, and &quot;combining measurements from many instruments over a long period&quot; to improve accuracy. In his 1674 pamphlet Animadversions Hooke didn't confine himself to rebutting Hevelius's arguments, but resorted to slights, implying that Hevelius's instruments were no better than Tycho's, and that he wasted his life when he could do so much better using telescopic sights. In this Hooke overestimated telescopic sights of his time, and underestimated Hevelius's instruments and skill. He achieved the accuracy of about 30$''$, whereas Flamsteed using telescopic sights only improved it to 15$''$-20$''$, hardly &quot;ten times&quot;. When Halley was dispatched in 1679 to convert Hevelius to the use of telescopic sights his observations with a 2-foot quadrant did not prove superior to Hevelius's, who used 6 or even 9 feet quadrants with open sights. Only years after Hevelius’ death, when Flamsteed completed his new star catalog, did the advantage of telescopic sights become apparent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The acerbic tone Hooke took, wrapping himself into a mantle of the defender of progress against Hevelius's &quot;attacks&quot;, also rubbed many the wrong way including Molyneux and Wallis, who rebuked him publicly for not delivering on the promises of greater precision. Others, like Halley and Flamsteed, were put into an uncomfortable position of agreeing with Hooke on substance but having to distance themselves from his hostile attitude. The Royal Society got involved because such public opposition to telescopic sights from one of the Europe's leading astronomers threatened their general acceptance and use, but favored resolving the issue in a non-confrontational manner, hence the Halley's 1679 mission. According to Saridakis the controversy &quot;entailed such issues as witnessing, etiquette of communication, and the role of the Royal Society as a forum in which individuals could express dissent.&quot; Moxham characterizes it as &quot;not just a clash between two notable and notoriously egotistical natural philosophers but a complex set of negotiations between three British learned societies which developed into an open competition for ascendancy.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hevelius argued that &quot;lenses cannot be maintained in the same position. Consequently, an observer could not acquire the same exact measurements each time he used telescopic sights. On the other hand, with open sights he can repeat the same observation with equal precision each time because he did not have to worry about shifting telescopes caused by their lack of being “firmly fixed.” Besides, lenses could easily break, especially in cold weather. Furthermore, winter weather could fog up or dull lenses when the breath of an observer came into contact with them. As a result, the lenses needed to be frequently removed and cleaned... Cross-hairs (their intersection) tended to conceal small stars so that an observer had difficulty distinguishing them... contributed to inaccurate measurements because of their close proximity to the observer’s eyes (a few inches). Therefore, telescopic sights were less effective than open sights in which the sights themselves were six, or even eight feet from the eyes... proximity between the eye and cross-hairs in telescopic lenses could not ensure an accurate line of collimation.&quot;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-03-03T21:21:35.460" LastActivityDate="2015-03-03T21:21:35.460" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2026" PostTypeId="2" ParentId="2024" CreationDate="2015-03-03T01:16:22.633" Score="8" Body="&lt;p&gt;Your friend told you an anecdote, possibly in jest, because the truth seems comically twisted beyond recognition. Historical events dating back to 1930-1960s did influence the development of Soviet science in ways more favorable to physics than to biology. Much has changed since then, including cultural attitudes and government funding, but those events did leave a lasting mark on the tradition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For ideological reasons Stalin opposed genetics and cybernetics as &quot;foreign influences&quot;, and promoted agricultural pseudoscience of the likes of &lt;a href=&quot;https://books.google.com/books?id=e2x9FwBE3u8C&amp;amp;pg=PA275&amp;amp;lpg=PA275&amp;amp;dq=physicists+and+lyricists+soviet+union&amp;amp;source=bl&amp;amp;ots=sZgirq8nAD&amp;amp;sig=X7BsqUdBP6W8p4sGQq8ZmOaesXY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=iP70VIXyNtCToQSdnoDgCw&amp;amp;ved=0CB4Q6AEwAA#v=onepage&amp;amp;q=physicists%20and%20lyricists%20soviet%20union&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Trofim Lysenko&lt;/a&gt;, who promised results in the spirit of &quot;transforming the world&quot;, the official Soviet ideology at the time.  According to Sakharov, &quot;he is responsible for the shameful backwardness of Soviet biology and of genetics in particular, for the dissemination of pseudo-scientific views, for adventurism, for the degradation of learning, and for the defamation, firing, arrest, even death, of many genuine scientists.&quot; The poisoned atmosphere in Soviet biology during the late Stalin years is portrayed in a celebrated 1987 novel &lt;a href=&quot;https://books.google.com/books?id=cjTMsnasswEC&amp;amp;pg=PA245&amp;amp;lpg=PA245&amp;amp;dq=white+garments+dudintsev&amp;amp;source=bl&amp;amp;ots=wOqk7hl07U&amp;amp;sig=lODf-vk2ABG76tpCLtJiZJjbLLY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=AAH2VNMHw6qiBLiDgLAN&amp;amp;ved=0CD4Q6AEwCTgK#v=onepage&amp;amp;q=white%20garments%20dudintsev&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;White Garments&lt;/a&gt; by Vladimir Dudintsev, which is translated into English. Unfortunately, Lysenko managed to engratiate himself to Khrushchev as well, so he remained a smothering influence in biology until Khrushchev's removal in 1964.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Physics and mathematics, on the other hand, enjoyed a somewhat greater degree of research freedom because Stalin was interested in potential military applications, the nuclear bomb in particular, more than in the ideological purity of physicists and mathematicians. This attracted many bright and independent people to these fields. In 1960s nuclear physics enjoyed broad popularity, physics degree programs at universities even became some of the most competitive ones for a while. Unlike high minded intellectual debates about Snow's &quot;two cultures&quot; the Soviet version of the controversy, nicknamed &lt;a href=&quot;https://books.google.com/books?id=e2x9FwBE3u8C&amp;amp;pg=PA275&amp;amp;lpg=PA275&amp;amp;dq=physicists+and+lyricists+soviet+union&amp;amp;source=bl&amp;amp;ots=sZgirq8nAD&amp;amp;sig=X7BsqUdBP6W8p4sGQq8ZmOaesXY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=iP70VIXyNtCToQSdnoDgCw&amp;amp;ved=0CB4Q6AEwAA#v=onepage&amp;amp;q=physicists%20and%20lyricists%20soviet%20union&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&quot;physicists and lyricists&quot;&lt;/a&gt;, became a truly popular culture phenomenon. Interestingly enough, it was public denunciation in 1962 of Lysenko's work as pseudoscience, and of his use of political repression to silence and eliminate opponents, by three prominent Soviet physicists, Zel'dovich, Ginzburg and Kapitsa, that hastened his demise. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-03-03T18:52:59.797" LastActivityDate="2015-03-03T18:52:59.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2027" PostTypeId="2" ParentId="2024" CreationDate="2015-03-03T03:42:16.653" Score="7" Body="&lt;p&gt;I can add to the answer of Conifold some of my own experience (I was a student and then a teacher in Ukraine in 1970-80-s). Soviet Union had excellent education system, on all levels from elementary school to University. Special emphasis was on science education&#xA;(mathematics, physics, chemistry and biology). All these subjects were mandatory for all high school students, a rigorous curriculum was approved by a government body, and textbooks were written by outstanding scientists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Comparing this with the US education system where I work now was a real shock).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides this there were &quot;olimpiads&quot; in all these subjects, encouraging students to learn more, and selecting the most talented of them for future careers in science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Physics and mathematics were especially popular in 1960-s and 1970-s and a career in these areas was&#xA;considered desirable and had high prestige. Biology was also highly regarded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intrusion of the Communist ideology to natural sciences was not felt much after 1970-s. &#xA;(Unlike to the social and humanitarian sciences like History and Economy, where it was very strong).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most well-known example of this intrusion indeed happened in biology, but by the 1970-s it was over, and genetics returned to the curriculum. On the other hand, the scientific school in genetics was destroyed in the previous period, and such things leave a long lasting effect. On research, not on &quot;K-12&quot; education.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-03-03T03:42:16.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2028" PostTypeId="1" CreationDate="2015-03-03T11:40:03.903" Score="14" ViewCount="7026" Body="&lt;p&gt;What is the easiest evidence that there is no air on the Moon?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Frankly speaking, I myself have no idea how to prove it. &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="62" LastEditDate="2015-03-03T19:35:38.140" LastActivityDate="2018-05-29T21:39:35.990" Title="When did humans realize that there is no air on the Moon?" Tags="|astronomy|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2029" PostTypeId="2" ParentId="2028" CreationDate="2015-03-03T13:27:30.177" Score="14" Body="&lt;p&gt;There is no sharp difference between &quot;no air&quot; and &quot;very little air&quot;.&#xA;That it is &quot;very little&quot; is seen when we observe the Moon with a telescope. An atmosphere&#xA;creates a visible haze, especially on the edge of the object (because of refraction). Especially well visible it would be during Solar eclipses. So when people started to observe&#xA;the Moon with telescopes it was clear that there is very little air. The more powerful the telescopes that were used, the smaller this &quot;very little&quot; became.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For practical purposes it was clear that there is no air by the end of the 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For comparison, comets usually have atmospheres (visible with a naked eye) but they are so rarified that &quot;there is no air&quot; on the comets, for all practical purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Of course when I say that &quot;it was known in 18-th century&quot; this means &quot;it was known to the experts, scientists&quot;. It is easy to show that this was NOT known to the &quot;general public&quot;. Many writers (and some movies in the beginning of 20-th centiry) described travels to the Moon, and they did not know or did not care that a human cannot breath there. Jules Verne's travelers, for example has no provision for breathing on the Moon.&#xA;Jules Verne wrote in the second half of 19-s century. (That air is necessary for survival&#xA;was shown in 17-s century).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-03T22:25:25.113" LastActivityDate="2015-03-03T22:25:25.113" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2030" PostTypeId="2" ParentId="1926" CreationDate="2015-03-03T14:19:52.577" Score="4" Body="&lt;p&gt;Yes, it is a coincidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of &lt;strong&gt;fugacity&lt;/strong&gt; (from the Latin for &quot;fleetness, tendency to flee&quot;) was originally introduced by &lt;a href=&quot;http://en.wikipedia.org/wiki/Gilbert_N._Lewis&quot; rel=&quot;nofollow&quot;&gt;Gilbert Lewis&lt;/a&gt; in his 1901 paper &quot;&lt;a href=&quot;https://archive.org/details/jstor-20021635&quot; rel=&quot;nofollow&quot;&gt;The Law of Physico-Chemical Change&lt;/a&gt;&quot; for the pressure of an ideal gas which has the same chemical potential as a real gas. Lewis notated this with &lt;strong&gt;ψ&lt;/strong&gt;, though these days the letter &lt;strong&gt;f&lt;/strong&gt; is used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of &lt;strong&gt;complex fugacity&lt;/strong&gt; was introduced by Lee and Lang in their 1952 paper &quot;&lt;a href=&quot;http://ergodic.ugr.es/statphys/bibliografia/yang1.pdf&quot; rel=&quot;nofollow&quot;&gt;Statistical Theory of Equations of State and Phase Transitions&lt;/a&gt;&quot;. The complex fugacity plane was originally notated &lt;strong&gt;y&lt;/strong&gt;, though as you note these days the letter &lt;strong&gt;z&lt;/strong&gt; is used. I'm not sure precisely how old the use of z for a generic complex variable is but it dates to at least the 1890s (e.g. see &lt;a href=&quot;https://books.google.co.uk/books?ei=LL_1VMiLHoa0sATxhYKwDA&amp;amp;id=xVmqAAAAIAAJ&amp;amp;dq=%22complex%20variable%20z%22&amp;amp;focus=searchwithinvolume&amp;amp;q=%22complex%20variable%20z%22&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of &lt;strong&gt;partition function&lt;/strong&gt; was introduced by Planck in 1921. Its use of the letter &lt;strong&gt;Z&lt;/strong&gt; comes from its original German name,  Zustandssumme (&quot;sum over states&quot;). The English name is due to Darwin and Fowler (1922).&lt;/p&gt;&#xA;" OwnerUserId="1912" LastActivityDate="2015-03-03T14:19:52.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2031" PostTypeId="2" ParentId="2028" CreationDate="2015-03-03T16:10:43.213" Score="3" Body="&lt;p&gt;You could also ask when people decided that there was air on Earth. The concept of “air”, as opposed to “wind”, is a relatively advanced (and relatively abstract) notion, which probably came up at about the time of Empedocles (or whoever it was that first posited four elements: earth, fire, water, air). Aristotle taught that air fills the whole space between the surface of the Earth and the sphere of the moon; above that sphere everything is made of the fifth element, aether. This implies that there is no air on the moon. Somehow I don’t think anyone ever thought there was air on the moon.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-03-03T16:10:43.213" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2032" PostTypeId="2" ParentId="2028" CreationDate="2015-03-03T19:17:05.277" Score="7" Body="&lt;p&gt;Here are three evidences that do not require a telescope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) No clouds are ever seen on the moon's surface. This is the weakest evidence, since the clouds could be too small to see with the naked eye, or there could be too little water in the air to form clouds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) When the moon moves over a star, occulting it, the star winks out quickly. If the moon had air, the air would take a little time to cause the star to loose its brightness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) The terminator, the dividing line between light and darkness on the face of the moon, is jagged due to craters and mountains, but it is very sharp. An atmosphere would make it fuzzy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, there was good reason to conclude the lack of air on the moon long before the telescope was invented.&lt;/p&gt;&#xA;" OwnerUserId="426" LastActivityDate="2015-03-03T19:17:05.277" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2033" PostTypeId="1" CreationDate="2015-03-03T23:26:04.643" Score="11" ViewCount="424" Body="&lt;p&gt;I am looking not for documentaries or biographies, those are easier to find, but for well written fictionalized but realistic portrayals of scientists and their work. These are hard to search for because of the large number of hits about science fiction and scientific literature, and because scientists often appear in &lt;a href=&quot;http://bigthink.com/age-of-engagement/reconsidering-the-image-of-scientists-in-film-television&quot; rel=&quot;nofollow&quot;&gt;film&lt;/a&gt; and &lt;a href=&quot;http://www.unpopularscience.co.uk/scientists-in-fiction-the-good-the-bad-and-the-poorly-represented/&quot; rel=&quot;nofollow&quot;&gt;fiction&lt;/a&gt; riddled with cliches and stereotypes like &quot;mad scientist&quot; (Dr. Strangelove), &quot;hero scientist&quot; (Day After Tomorrow), &quot;boy genius&quot; (Numbers) or &quot;comic relief nerd&quot; (Big Bang Theory).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 20th century witnessed great advances in nuclear physics, molecular biology, organic chemistry, geology, archaeology, aerospace and software engineering, etc. I am particularly interested in works about the second half of 20th century, when science became more of a collective enterprise, that convey the &quot;air of the times&quot;. What it &quot;felt like&quot; to be a regular scientist (not Einstein, Nash or Crick and Watson) living and working on projects at a university, a research lab, or a field expedition during those times. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a 2008 Scientific American article &lt;a href=&quot;http://www.scientificamerican.com/article/the-mad-scientist-myth/&quot; rel=&quot;nofollow&quot;&gt;We Need More Novels about Real Scientists&lt;/a&gt; Mark Alpert mentions Roger’s Version by John Updike, and Intuition by Allegra Goodman. Any other suggestions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2020-07-13T07:58:09.303" LastActivityDate="2020-07-13T07:58:09.303" Title="What films and fiction give realistic historical portrayal of science and scientists?" Tags="|discoveries|social-context|scientists|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2034" PostTypeId="2" ParentId="2033" CreationDate="2015-03-04T04:44:15.803" Score="3" Body="&lt;p&gt;Good serious fiction about scientists is extremely rare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example is the novel by Sinclair Lewis, Arrowsmith (1925). It is about a medical scientist, a regular one, not a Nobel prize winner, and&#xA;I know professional medical scientists who consider it realistic and praise it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know of a single good movie or a novel about a physicist, astronomer or mathematician. Which is not so surprising: the life of most of these people is very ordinary, and does not contain events that can be effectively shown in the movies or novels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, there is a movie &quot;Einstein and Eddington&quot;, which is tolerable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several good biographies, of real, famous people, but this is not what you are&#xA;asking about. There are also several excellent and very interesting autobiographies of 20th century mathematicians (Walter Rudin, Laurent Schwartz, Andre Weil), and Richard Feynmann, of course. They really read  like novels.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-04T04:51:46.200" LastActivityDate="2015-03-04T04:51:46.200" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2035" PostTypeId="1" AcceptedAnswerId="2038" CreationDate="2015-03-05T11:39:35.987" Score="18" ViewCount="1550" Body="&lt;p&gt;I've found Lagrange's &lt;em&gt;Sur la résolution des équations algébriques&lt;/em&gt; to be a very confusing and difficult read, and I think I'm starting to see why: it seems that Lagrange thinks of algebra in a much more formal/symbolic way than I'm used to. Whereas I think of a symbol $x$ as referring to a &lt;em&gt;specific number&lt;/em&gt; (which may be unknown), and I think of $2x + 3x = 5x$ as being justified because it would be true no matter which number $x$ is, Lagrange seems to just view $x$ as a sort of symbol on which certain rules of operation are defined, and $2x + 3x = 5x$ as being justified because it's one of those rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When explaining Cardan's method for solving the cubic, Lagrange blithely divides by a variable $y$ with no consideration for whether it is or isn't zero. At first I thought this was sloppy reasoning but now I wonder if he basically thinks of this as a calculation in the field of rational functions $\mathbb C(y)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Where $a, b, c$ are the roots of a particular cubic, Lagrange states in passing that $\frac {a+\alpha b + \beta c} 3$ will have $3!$ different values when we permute $a, b, c$ in every possible way. This is correct if we think of &quot;different values&quot; as meaning different &lt;em&gt;forms&lt;/em&gt;, but possibly incorrect if $a, b, c$ are standing in for specific numbers (if they're all zero, for instance).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With $\alpha$ a primitive cube root of unity, Lagrange concludes from $\alpha Aa+\alpha Bb + \alpha Cc=Aa+Bc+Cb$ that $\alpha A=A$. This may be incorrect if $A, B, C$ and $a, b, c$ are specific numbers, but correct if we think of them as symbols and just want the expression on the left to be the same as the expression on the right.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Am I correct that 18th century algebra, or at least Lagrange, was done in this more &quot;symbolic&quot; way? If so, how can this be made rigorous in modern terminology? Where can I learn more about the way Lagrange and his contemporaries thought about their subject?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="4251" LastEditDate="2020-07-06T07:11:40.220" LastActivityDate="2020-12-23T07:02:59.670" Title="Was 18th century algebra more symbolic/formal than the modern conception?" Tags="|mathematics|elementary-algebra|18th-century|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2036" PostTypeId="2" ParentId="2035" CreationDate="2015-03-05T12:59:37.740" Score="8" Body="&lt;p&gt;The answer depends on what you consider the &quot;modern concepts&quot;. In modern algebra, they usually do not consider polynomials or rational functions as functions of the variable $x$,&#xA;but exactly as you say in $1$, as elements of the field $C(y)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Lagrange's writing is not more symbolic/abstract than modern algebra, but more abstract&#xA;than modern &quot;high school algebra&quot;. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-06T04:14:59.977" LastActivityDate="2015-03-06T04:14:59.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2037" PostTypeId="1" CreationDate="2015-03-05T18:00:04.133" Score="2" ViewCount="78" Body="&lt;p&gt;Apologies if this isn't quite the right place to ask this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The popular science sites like &lt;a href=&quot;http://www.wired.com/&quot; rel=&quot;nofollow&quot;&gt;Wired&lt;/a&gt; and &lt;a href=&quot;http://io9.com/&quot; rel=&quot;nofollow&quot;&gt;io9&lt;/a&gt; are too biased and reactionary for my tastes. I'm looking for sites or blogs that cover historical and contemporary science without injecting the writer's bias or agendas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any reading suggestions?&lt;/p&gt;&#xA;" OwnerUserId="1928" LastEditorUserId="26" LastEditDate="2015-03-05T23:37:46.533" LastActivityDate="2015-03-06T03:31:47.223" Title="What blogs or websites cover historical or contemporary science?" Tags="|discoveries|resource-recommendation|" AnswerCount="1" CommentCount="2" ClosedDate="2015-03-16T22:15:51.497" ContentLicense="CC BY-SA 3.0" />
  <row Id="2038" PostTypeId="2" ParentId="2035" CreationDate="2015-03-05T23:00:15.877" Score="20" Body="&lt;p&gt;There is indeed a difference between 18th century and modern concepts of algebra. For example, Lagrange and his contemporaries did not define algebraic structures into existence by specifying axioms for operations on abstract sets. Most of algebra was about integers and polynomials, which were assumed to pre-exist, and they were not thought as manifestations of &amp;quot;algebraic structures&amp;quot;, although similarities between them were known. However, I do not believe that this is what is causing the conceptual block.&lt;/p&gt;&#xA;&lt;p&gt;Lagrange's formal manipulation of polynomials is actually in the spirit of modern algebra, but a modern algebraist would distinguish between treating polynomials formally and as functions. Lagrange did not. He treated not only polynomals and rational functions formally, but transcendental functions and their Taylor series as well. This is because not only the concept of algebra was different in the 18th century, so was the concept of analysis.&lt;/p&gt;&#xA;&lt;p&gt;Some historians refer to 18th century analysis as &amp;quot;algebraic analysis&amp;quot;, see e.g. &lt;a href=&quot;https://www.jstor.org/stable/41133857&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fraser, The Calculus as Algebraic Analysis&lt;/a&gt;. Lagrange gave it its completed form in his 1797 &lt;em&gt;Théorie des fonctions analytique&lt;/em&gt;. The modern notion of a function, as assignment of values to arguments, only appeared later in &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Functions.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier's work&lt;/a&gt;, and was made explicit by Dirichlet (1829). To Euler, Lagrange and others &amp;quot;function&amp;quot; was analytic expression formed by arithmetical rules and compositions, from a set of basic functions. Roughly, it corresponded to our &amp;quot;elementary function&amp;quot;, although functions given implicitly were also admitted, only occasionally more general expressions (e.g. integrals of elementary functions) were considered.&lt;/p&gt;&#xA;&lt;p&gt;Additionally, unlike modern algebraists, Lagrange did not distinguish between polynomials, formal or not, over real and complex fields. It was assumed that algebraic expressions can be operated upon according to the rules of algebra independently of such distinctions. If a formula is derived algebraically it is valid in general, for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; real or complex, infinite or infinitesimal. The idea can be traced back to &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_continuity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz's law of continuity&lt;/a&gt; and was turned into a broader &lt;a href=&quot;https://en.wikipedia.org/wiki/Generality_of_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;generality of algebra&amp;quot;&lt;/a&gt; principle that Cauchy ascribed to Euler and Lagrange. In particular, the sum of a geometric series was considered equal to &lt;span class=&quot;math-container&quot;&gt;$\frac1{1-x}$&lt;/span&gt; &amp;quot;in general&amp;quot;, essentially by definition. Convergence was studied, but more as a practical matter of obtaining approximations and estimates.&lt;/p&gt;&#xA;&lt;p&gt;Can it be made rigorous? Yes and no. As long as we do not read modern idea of a function into 18th century works much of what needs to be &amp;quot;made rigorous&amp;quot; goes away. However, sometimes rules established formally were used to compute values with implicit identification of the two function concepts. To some extent, &amp;quot;generality of algebra&amp;quot; is vindicated by analytic continuation theory, summation methods for divergent series, and the transfer principle in nonstandard analysis, but not to the extent assumed by Lagrange and Euler, and sometimes it led to outright self-contradictory results, as with the &lt;a href=&quot;http://eulerarchive.maa.org/docs/translations/E807en.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;value of &lt;span class=&quot;math-container&quot;&gt;$\ln(i)$&lt;/span&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-12-23T07:02:59.670" LastActivityDate="2020-12-23T07:02:59.670" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="2039" PostTypeId="2" ParentId="2037" CreationDate="2015-03-06T02:38:59.670" Score="2" Body="&lt;p&gt;See links at &lt;a href=&quot;http://www.aip.org/history/links.html&quot; rel=&quot;nofollow&quot;&gt;Online resources for history of physics&lt;/a&gt;, &lt;a href=&quot;https://home.comcast.net/~rmillstein1/philbio.html&quot; rel=&quot;nofollow&quot;&gt;History and Philosophy of Biology Resources&lt;/a&gt;, &lt;a href=&quot;http://archives.math.utk.edu/topics/history.html&quot; rel=&quot;nofollow&quot;&gt;Chemical Heritage Foundation&lt;/a&gt;, &lt;a href=&quot;http://archives.math.utk.edu/topics/history.html&quot; rel=&quot;nofollow&quot;&gt;Math Archives: History&lt;/a&gt; and &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/Links/&quot; rel=&quot;nofollow&quot;&gt;Websites relevant to the History of Mathematics&lt;/a&gt;, also &lt;a href=&quot;http://www.math.nmsu.edu/~history/&quot; rel=&quot;nofollow&quot;&gt;Teaching with Original Historical Sources in Mathematics&lt;/a&gt; puts an interesting spin on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like &lt;a href=&quot;http://www.preposterousuniverse.com/blog/&quot; rel=&quot;nofollow&quot;&gt;Sean Carroll's blog&lt;/a&gt; on physics,  &lt;a href=&quot;http://www.slate.com/blogs/bad_astronomy.html&quot; rel=&quot;nofollow&quot;&gt;Phil Plait's blog&lt;/a&gt; on astronomy, and &lt;a href=&quot;https://terrytao.wordpress.com/&quot; rel=&quot;nofollow&quot;&gt;Terence Tao's blog&lt;/a&gt; on mathematics, they occasionally write about historical issues. Some reputable science news sites are &lt;a href=&quot;http://discovermagazine.com/&quot; rel=&quot;nofollow&quot;&gt;Discover Magazine&lt;/a&gt;, &lt;a href=&quot;http://arstechnica.com/science/&quot; rel=&quot;nofollow&quot;&gt;Ars Technica&lt;/a&gt;, &lt;a href=&quot;https://www.sciencenews.org/&quot; rel=&quot;nofollow&quot;&gt;Science News&lt;/a&gt; and &lt;a href=&quot;https://www.sciencenews.org/&quot; rel=&quot;nofollow&quot;&gt;Science Daily&lt;/a&gt;, Discover Magazine also hosts &lt;a href=&quot;http://discovermagazine.com/rss&quot; rel=&quot;nofollow&quot;&gt;many blogs&lt;/a&gt;, so does &lt;a href=&quot;http://blogs.scientificamerican.com/&quot; rel=&quot;nofollow&quot;&gt;Scientific American&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-03-06T03:31:47.223" LastActivityDate="2015-03-06T03:31:47.223" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2040" PostTypeId="1" CreationDate="2015-03-06T13:34:10.590" Score="5" ViewCount="130" Body="&lt;p&gt;The &quot;fundamental theorem of symmetric polynomials&quot; states that any symmetric polynomial can be expressed as a polynomial in the &lt;em&gt;elementary&lt;/em&gt; symmetric polynomials. This, or at least variants on it or corollaries of it, was definitely known to Lagrange in the 18th century (probably many others) and possibly even to Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't see anything &quot;elementary&quot; about the elementary symmetric polynomials - in the sense that I don't see anything about them that immedaitely suggests that &lt;em&gt;all&lt;/em&gt; symmetric polynomials should be expressible by them. What do we know about how and when such an obscure yet powerful fact was discovered?&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-03-06T13:34:10.590" Title="How were the phenomena relating to symmetric polynomials discovered?" Tags="|mathematics|elementary-algebra|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2041" PostTypeId="1" AcceptedAnswerId="2050" CreationDate="2015-03-06T13:59:25.173" Score="6" ViewCount="1174" Body="&lt;p&gt;I am looking for a good book on Noether's life. Not only a biography, but a book that also explains her life's work to a general, somewhat mathematically mature audience. If such a book is not available, then a book aimed at graduate students would be the next best thing.&lt;/p&gt;&#xA;" OwnerUserId="422" LastActivityDate="2015-05-26T11:06:21.643" Title="A book on Emmy Noether's life" Tags="|reference-request|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2042" PostTypeId="1" CreationDate="2015-03-06T15:43:12.757" Score="13" ViewCount="427" Body="&lt;p&gt;In Grassmann's famous article &lt;em&gt;&lt;a href=&quot;http://ncatlab.org/nlab/show/Ausdehnungslehre&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ausdehnungslehre&lt;/a&gt;&lt;/em&gt; from 1844 (the one where he introduces what has come to be famous as &lt;em&gt;&lt;a href=&quot;http://ncatlab.org/nlab/show/exterior+algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grassmann algebra&lt;/a&gt;&lt;/em&gt;) he uses the termionology &quot;form&quot; in place of, as he explains in his introduction, &quot;quantity&quot; or really &quot;magnitude&quot; (namely German: &lt;em&gt;Größe&lt;/em&gt;). He says that he does so because he finds &quot;quantity&quot;/&quot;magnitude&quot; to be too restrictive to capture what he has in mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, from a modern perspective what he has in mind and what he discusses at length in the bulk of the article is what today we call &lt;em&gt;differential forms&lt;/em&gt; on Euclidean space. Maybe what he considers are only the constant differential forms on Euclidean spaces, but anyway. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly one sees from this why he found &quot;magnitude&quot; to be too restrictive. But besides this it seems striking that where he says &quot;form&quot; we &lt;em&gt;still&lt;/em&gt; say &quot;form&quot;. Or is it &lt;em&gt;again&lt;/em&gt; instead of &lt;em&gt;still&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's my question: is there a continuity of usage from Grassmann's terminology &quot;form&quot; to modern terminology? Or is it a coincidence that when we say &quot;differential form&quot; today, deriving from &quot;linear form&quot;, that this happens to use the same word for the same concept as Grassmann did, who seems to have been motivated from more general ideas about &quot;forms of thought&quot;.&lt;/p&gt;&#xA;" OwnerUserId="1934" LastEditorUserId="1934" LastEditDate="2017-08-18T16:11:06.613" LastActivityDate="2017-10-27T01:46:31.807" Title="Grassmann's &quot;forms&quot;" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2043" PostTypeId="1" CreationDate="2015-03-06T20:26:33.820" Score="28" ViewCount="4173" Body="&lt;p&gt;It is a common opinion that Romans did not contribute anything to exact sciences, but did contribute much to engineering. (How can it be otherwise, anyone who has been on the territory of the former empire could see all these marvelous buildings, temples, walls, roads, bridges, aqueducts and bathhouses everywhere).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Can anyone name any Roman engineer, except Vitruvius?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To be more precise, there was a lot of excellent mathematics, physics and astronomy in the Roman empire. But this was done by people who wrote in Greek and had Greek names (though it is possible that some of them, like perhaps Ptolemy, were Roman citizens).&lt;/p&gt;&#xA;&lt;p&gt;I do not count popular writers like Pliny (the elder) or Varro, who wrote huge low-quality encyclopedias: this is not science but popularization. (Same applies to Vitruvius btw).&lt;/p&gt;&#xA;&lt;p&gt;I recently read the correspondence of Pliny the Younger (who was a governor of a province) with his princeps (Trajan). One of the main recurring  topics is Pliny asking to send him an architect or a surveyor for his architectural or hydraulic works. To all these repeated requests Trajan replies that engineers are in great shortage in Rome, and why does not Pliny find one himself nearby, he is in Greece, after all!&lt;/p&gt;&#xA;&lt;p&gt;Another piece of evidence is Wikipedia article on Greek and Roman artillery. 5 books on&#xA;artillery survived. All Greek. Except Vitruvius, again.&lt;/p&gt;&#xA;&lt;p&gt;Could it be that all these engineering marvels were really created by Greek engineers?&#xA;Could it be that even in the area of artillery and siege engines the Roman empire relied exclusively on the&#xA;Greek engineers?&lt;/p&gt;&#xA;&lt;p&gt;EDIT. The author of the Wikipedia article on Vitruvius kindly made a &amp;quot;list of references&amp;quot; in Vitruvius book. Of the &amp;quot;writers who WROTE on architecture&amp;quot; there are 3 Latin names.&#xA;Two of them are known only from this reference, the third one is Varro, who was certainly not an architect himself.&#xA;Of the &amp;quot;architects&amp;quot; all 5 are Greeks.  Of the &amp;quot;temple builders&amp;quot; one of the 9 names sounds Latin, other 8 Greek. On all other engineering - only Greek names.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-21T03:09:53.880" LastActivityDate="2021-09-21T03:09:53.880" Title="Roman engineers" Tags="|engineering|roman-empire|antiquity|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="2044" PostTypeId="1" CreationDate="2015-03-08T05:40:39.760" Score="2" ViewCount="325" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Lev_Landau&quot; rel=&quot;nofollow noreferrer&quot;&gt;Landau&lt;/a&gt; was famous for his abusive personality. Are there any examples of his students leaving his tutelage because of that and still achieving great success in physics? Any similar examples with other teachers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, are there historical examples correlating students' success with teachers' personality at the &lt;strong&gt;high&lt;/strong&gt; end of the spectrum? (I.e., &lt;em&gt;very&lt;/em&gt; talented graduate students, &lt;em&gt;very&lt;/em&gt; famous professors, success is measured in Nobel Prizes or something similar). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is motivated by the movie Whiplash which justifies the abusive behavior of a teacher by the future success of the student. The idea is that if a student fails to become fanatically driven to the degree that (s)he ignores/survives/enjoys teacher's abuse, then (s)he is no potential genius and his/her departure from the field is no loss to the field and Humanity. Is this idea supported by history?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. I tried to expand on the question &lt;a href=&quot;https://academia.stackexchange.com/q/41273/5829&quot;&gt;here&lt;/a&gt; (as recommended by a commenter).&lt;/p&gt;&#xA;" OwnerUserId="1940" LastEditorUserId="-1" LastEditDate="2017-04-13T12:49:57.757" LastActivityDate="2017-09-25T06:57:25.837" Title="Can abuse break a genius?" Tags="|teaching|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2045" PostTypeId="2" ParentId="264" CreationDate="2015-03-08T12:30:33.473" Score="22" Body="&lt;p&gt;The 16th (1995) edition of &lt;a href=&quot;http://www.kayelaby.npl.co.uk/&quot; rel=&quot;noreferrer&quot;&gt;Kaye and Laby&lt;/a&gt; includes the &lt;a href=&quot;https://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4250137/B6P1%20ch-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;following progression&lt;/a&gt; of the accepted values for the charge of an electron. The first value &quot;is essentially Milikan's oil drop value&quot; and the second is &quot;the 'X-ray grating' value&quot;. I couldn't find any more details about the actual experiments. The standard errors are often over-optimistic, indicating unexpected systematic errors; in Milikan's case this was due to an inaccurate value for the viscosity of air.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1.5911 ± 0.0024 [Birge, 1929]&lt;/li&gt;&#xA;&lt;li&gt;1.60203 ± 0.00050 [Birge, 1942]&lt;/li&gt;&#xA;&lt;li&gt;1.60210 ± 0.00002 [Dummond and Cohen, 1963]&lt;/li&gt;&#xA;&lt;li&gt;1.602192 ± 0.000007 [Taylor et al, 1969]&lt;/li&gt;&#xA;&lt;li&gt;1.6021892 ± 0.0000046 [Cohen and Taylor, 1973]&lt;/li&gt;&#xA;&lt;li&gt;1.6021773 ± 0.00000049 [Cohen and Taylor, 1987]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/xx1Fm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, this doesn't really seem to bear out Feynmann's assertion. However, it would be useful to see the progression of values between 1929 and 1942: e.g. from K&amp;amp;L 8th edition (1936).&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2015-03-08T18:26:41.660" LastActivityDate="2015-03-08T18:26:41.660" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2046" PostTypeId="1" CreationDate="2015-03-08T13:28:35.417" Score="3" ViewCount="283" Body="&lt;p&gt;Why is the &lt;a href=&quot;http://en.wikipedia.org/wiki/Wreath_product&quot; rel=&quot;nofollow&quot;&gt;wreath product&lt;/a&gt; so named?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If possible, please provide a citation.&lt;/p&gt;&#xA;" OwnerUserId="1941" LastEditorUserId="26" LastEditDate="2015-03-08T13:32:12.907" LastActivityDate="2015-03-08T13:32:12.907" Title="History of the Wreath product" Tags="|mathematics|terminology|abstract-algebra|group-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2047" PostTypeId="2" ParentId="2044" CreationDate="2015-03-08T13:47:11.540" Score="3" Body="&lt;p&gt;If a genius is &quot;broken&quot; at an early stage, we will never know that s/he was a genius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand there are abundant positive examples.&#xA;For example, Hungarian mathematician Fejer made an important contribution to mathematics himself, but also as a teacher had such students as von Neumann, Erdos, Polya and Turan.&#xA;(There was one school in Budapest in the early 20-th century whose graduates are called collectively &quot;Hungarian miracle&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Russia, in the early 20-th century, there was Luzin who taught in Moscow University.&#xA;Perhaps more than 1/2 of the most famous Russian mathematicians were his students or students&#xA;of his students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most incredible fact of this sort is that Gauss (in  Braunschweig) and Lobachevski (in Kazan) had the same teacher,  Johann Christian Martin Bartels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So there is no doubt that there is some strong correlation between a teacher personality and his students success.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-03-08T13:47:11.540" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2048" PostTypeId="1" CreationDate="2015-03-08T16:47:20.643" Score="3" ViewCount="4582" Body="&lt;p&gt;Who discovered that Helium affects your voice?&lt;/p&gt;&#xA;" OwnerDisplayName="Diego Allan Reyna Rodriguez" LastEditorUserId="62" LastEditDate="2015-03-08T17:39:03.470" LastActivityDate="2018-10-19T13:16:35.347" Title="Who discovered that helium affects your voice?" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2050" PostTypeId="2" ParentId="2041" CreationDate="2015-03-09T05:39:33.287" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0801896940&quot; rel=&quot;nofollow&quot;&gt;Emmy Noether's Wonderful Theorem&lt;/a&gt; by Dwight E. Neuenschwander.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The title says it all: the book explains Noether's theorem in detail. Everything is derived and explained from scratch: the action principle, symmetries, Lagrangians, Hamiltonians,... The style very reader-friendly, it is not a textbook. However, the book is only meant for those who have enough mathematical background (in particular maths and physics undergrads). The introduction contains a short (4-page) biography.&lt;/p&gt;&#xA;" OwnerUserId="220" LastActivityDate="2015-03-09T05:39:33.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2051" PostTypeId="1" CreationDate="2015-03-09T12:51:44.343" Score="0" ViewCount="680" Body="&lt;p&gt;Math is critical in most science related subjects, such as physics and astronomy. People would agree that math has helped and even influenced science. Yet no one mentioned the other way around: science influencing and helping math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is that has science influenced and/or helped math in any way? All help is appreciated!&lt;/p&gt;&#xA;" OwnerUserId="1721" LastEditorUserId="26" LastEditDate="2015-10-11T10:54:22.143" LastActivityDate="2015-10-11T10:54:22.143" Title="How Has Science Influenced or Helped Mathematics" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ClosedDate="2015-03-09T18:22:32.570" ContentLicense="CC BY-SA 3.0" />
  <row Id="2052" PostTypeId="2" ParentId="2051" CreationDate="2015-03-09T13:37:57.640" Score="3" Body="&lt;p&gt;On sciences other than physics and astronomy, there was already such question:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/1987/what-new-mathematics-was-inspired-by-biology-and-chemistry/1988#1988&quot;&gt;What new mathematics was inspired by biology and chemistry?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning physics and astronomy, examples are so abundant that no list of reasonable length&#xA;can be made.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are just some random examples. In fact more than half of the whole mathematics comes from physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trigonometry was invented for the needs of astronomy. (First spherical, then flat).&#xA;Calculus was developed with one of the main motivation coming from physics.&#xA;Almost all theory of differential equations, ordinary and partial was invented for applications to physics (in 18 century the main motivation came from celestial mechanics,&#xA;later also from other parts of physics). Fourier transform was invented by Fourier when he studied heat conduction in solids. Ergodic theory and &quot;chaos theory&quot; originate in celestial mechanics.&#xA;Tensor calculus originates from Riemann's work on continuous mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quantum mechanics still makes an increasing influence on pure mathematics. First it stimulated the development of functional analysis, operator theory, then quantum algebra.&#xA;Statistical mechanics gave rise to many new mathematical theories.&#xA;Of the modern physical theories, let me mention string theory and quantum gravity which make very large influence on the modern mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it is not an exaggeration to say that more than 1/2 of the whole body of mathematics&#xA;comes from physics and would not be invented otherwise. Perhaps much more than 1/2. When I write &quot;comes from&quot; I mean it: not only problems are suggested by physics but also methods of their solution, and in many cases physics suggests solution of the problems which DO NOT come from physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting case study is contained in the paper of Ruelle, Is our mathematics natural?&#xA;in Bull. Amer. math. Soc. (available free online). He discusses the following question: if an extraterrestrial civilization will be ever discovered, will its mathematics be similar to ours? The answer is probably &quot;yes&quot;, exactly for the reason that mathematics comes from physics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-03-09T13:50:55.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2053" PostTypeId="2" ParentId="2048" CreationDate="2015-03-09T23:28:41.087" Score="12" Body="&lt;p&gt;With helium the knowledge of voice change goes back perhaps to as early as 1919, when  US Navy started experimenting with alternative gas mixes for deep diving. Before this helium was very rare and expensive, so it is unlikely that sizable quantities of it were inhaled. The idea was due to Elihu Thompson, an electronics engineer and inventor, who speculated in 1919 that nitrogen narcosis could be avoided if oxygen was mixed with other gases. Around the same time  C.J. Cooke applied for a patent on the use of helium in a breathing gas mix. A series of experimental dives was done on the U.S.S. Falcon, which included at least one dive to 150 feet on a heliox mix. However, the first record of voice change is from more &lt;a href=&quot;http://www-personal.umich.edu/~lpt/mixhistory.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;systematic experiments done in 1925&lt;/a&gt;:&quot;Divers commented on the ease of breathing helium, but noticed that they always felt chilled while breathing heliox mixes. The change in voice characteristics often made communication at depth difficult.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that the effect became common knowledge later due to inhaling &lt;a href=&quot;https://biology.stackexchange.com/questions/8907/can-breathing-in-helium-from-balloons-for-example-cause-damage-to-the-throat-i&quot;&gt;helium from toy balloons&lt;/a&gt; at parties, a bad idea that can lead to asphyxiation, but popular nonetheless. However, helium is not the only gas that raises voice timbre. Inhaling any gas lighter than air would do the same because sound waves in it travel faster, while vocal cords oscillate at the same frequency. So hydrogen would work even &quot;better&quot;. Rubber balloons filled with hydrogen were originally &lt;a href=&quot;https://www.balloonhq.com/faq/history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;used by Michael Faraday in 1824&lt;/a&gt; in physics experiments. Toy rubber balloons were introduced the next year by Hancock, and vulcanized ones by Ingram in 1847. I am not aware of any record that Faraday took gulps from his balloons and talked squeaky, but who knows. By the way, inhaling hydrogen is an even worse idea than inhaling helium because its mix with oxygen is highly explosive, from a smallest spark.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:48:15.817" LastActivityDate="2015-03-12T21:36:56.007" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2054" PostTypeId="1" AcceptedAnswerId="2055" CreationDate="2015-03-10T03:11:18.443" Score="14" ViewCount="971" Body="&lt;p&gt;I am looking for examples of great minds (on any personally defined level) in the history of science and mathematics that had a considerably lesser pre/mid-career reputation/lifestyle/upbringing/etc. Here are a few examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/317/what-are-some-scientific-breakthroughs-that-have-been-done-during-jail-time&quot;&gt;The answers to this question about people that produced scientific/mathematical breakthroughs during prison.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/George_Green_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Green&lt;/a&gt; was a landmark mathematician in human history that lived in a small village and self-taught himself almost everything he knew (and produced). He had an unnoticeable reputation for quite a while.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm particularly looking for people with a not-so-great childhood (i.e. not being a prodigy as a child, growing up in a very poor place, being a delinquent, etc.)&lt;/p&gt;&#xA;" OwnerUserId="238" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-03-15T21:04:01.157" Title="Who are some great scientists/mathematicians with not-so-great pre/mid-career credentials?" Tags="|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2055" PostTypeId="2" ParentId="2054" CreationDate="2015-03-10T10:08:26.007" Score="16" Body="&lt;p&gt;One of the most recent famous examples is &lt;a href=&quot;http://en.wikipedia.org/wiki/Yitang_Zhang&quot;&gt;Yitang Zhang&lt;/a&gt;, &lt;a href=&quot;http://annals.math.princeton.edu/2014/179-3/p07&quot;&gt;who proved in 2013&lt;/a&gt; that, if $p_i$ denote primes, then&#xA;$$ \liminf_{n\to\infty} (p_{n+1}-p_n)&amp;lt;7\times 10^7$$&#xA;i.e. there are infinitely many primes that are less than $7\times 10^7$ apart. In particular, there are infinitely primes that separated by less than some &lt;em&gt;finite&lt;/em&gt; number. This was an amazing breakthrough in number theory, resolving a long-standing conjecture, and using Zhang's methods his result was &lt;a href=&quot;http://arxiv.org/abs/1409.8361&quot;&gt;quickly refined by a large community project that ensued&lt;/a&gt;, sharpening the bound to $246$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yitang Zhang was 58 when he published his proof, which is regarded as extremely old in the community. He had, up until that moment, had quite an unremarkable academic career, publishing only &lt;em&gt;a single paper&lt;/em&gt; in 2001. &lt;a href=&quot;http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty&quot;&gt;An interesting profile&lt;/a&gt; in the New Yorker details some of his personal life. It explains that Zhang never had much success finding jobs at universities:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;His adviser, T. T. Moh, with whom he parted unhappily, recently wrote a description on his Web site of Zhang as a graduate student: “When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the North Pole.” Zhang left Purdue without Moh’s support, and, having published no papers, was unable to find an academic job. He lived, sometimes with friends, in Lexington, Kentucky, where he had occasional work, and in New York City, where he also had friends and occasional work. [...] A member of the group [...] opened a Subway franchise as a means of raising money. [...] Zhang kept the books. “Sometimes, if it was busy at the store, I helped with the cash register,” Zhang told me recently. “Even I knew how to make the sandwiches, but I didn’t do it so much.” When Zhang wasn’t working, he would go to the library at the University of Kentucky and read journals in algebraic geometry and number theory. “For years, I didn’t really keep up my dream in mathematics,” he said.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is clear that Zhang was not regarded as what is known as a 'first-rate mind', much less a prodigy. After his breakthrough paper &lt;a href=&quot;http://annals.math.princeton.edu/2014/179-3/p07&quot;&gt;'Bounded gaps between primes'&lt;/a&gt;, Zhang immediately became famous in mathematical circles, receiving a host of prizes, and got to spend time at prestigious academic institution such as the Institute for Advanced Study at Princeton, NJ. &lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2015-03-15T21:04:01.157" LastActivityDate="2015-03-15T21:04:01.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2056" PostTypeId="2" ParentId="1902" CreationDate="2015-03-11T14:13:08.250" Score="5" Body="&lt;p&gt;Read the Feynman interview at aip.org/history/ohilist/5020_2.html. Pretty early in the interview he says that he started as a math major, but it was getting abstract so he asked one of the professors in the MIT math department what abstract math was good for besides teaching it. He did not get a satisfactory answer, so he switched to electrical engineering and later moved into physics. He only got a physics degree in college.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2015-03-11T14:13:08.250" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2057" PostTypeId="1" AcceptedAnswerId="2059" CreationDate="2015-03-11T16:26:28.747" Score="5" ViewCount="514" Body="&lt;p&gt;I know that James Gregory came up with the idea of using a parabolic mirror to eliminate spherical aberration in 1663. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theory behind it had actually been around since classical antiquity I think. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, no one could make a parabolic mirror at the time, so eventually, Newton made his own design and utilized a spherical one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to figure out how long it took before a Gregorian telescope was actually made. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most sources say that Hooke made one in 1673. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am reading &lt;a href=&quot;http://amazing-space.stsci.edu/resources/explorations/groundup/lesson/scopes/hadley/&quot; rel=&quot;noreferrer&quot;&gt;this souce&lt;/a&gt; which says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first person to actually create a parabolic mirror was another&#xA;  English mathematician, John Hadley. In 1721, he built a Gregorian&#xA;  reflector whose mirror had very little spherical aberration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1951" LastActivityDate="2015-03-13T02:16:00.737" Title="When was the first true Gregorian telescope built?" Tags="|astronomy|telescope|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2058" PostTypeId="2" ParentId="1902" CreationDate="2015-03-11T19:26:35.450" Score="5" Body="&lt;p&gt;To save you the bother, here's the relevant excerpt from the link posted in the other answer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Weiner:&#xA;  Were you majoring in mathematics?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Feynman:&#xA;  Oh, yes. That’s interesting. At first I was in the mathematics course. It doesn’t make any difference what course you are in, really, very much. For the first year or so you take more or less the same thing. You take physics and chemistry and electrical engineering, mathematics, and so on. English. Somewhere around the first year I began to get upset. This wasn’t right. The mathematics, I looked at it, the mathematical things, were too abstract. They weren’t connected to anything, mathematics. And I went to the head of the mathematics department. This was in 1936, now, so you know it’s still in the Depression. I said to him, “Sir, what is the use of higher mathematics besides teaching more higher mathematics.” So he said, “Well, you could become an actuary,” calculating the insurance rates for an insurance company. This didn’t sit well with me, see. He also said that a man who asked that kind of a question is perhaps not right for mathematics. And I thought the thing I ought to do — I mean, I liked to get my hands dirty. I’d had a laboratory. The physical world was real, and the mathematics, I had become enthralled with, but not for itself, really — you know what I mean? It was fascinating, but my real heart was somewhere else. So I decided, I have to get my hands dirty, I can’t stand these abstract things. So I changed to electrical engineering, because there was something that was real. But then some few months later, I realized I’d gone too far, and that somewhere in between — that physics was the right place. So I moved around a little bit at the beginning, and ended up with the physics course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1817" LastActivityDate="2015-03-11T19:26:35.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2059" PostTypeId="2" ParentId="2057" CreationDate="2015-03-12T21:07:48.353" Score="3" Body="&lt;p&gt;There is a confusion in your question. ALL reflecting telescopes ideally should have parabolic mirrors. The problem is that a parabolic mirror is difficult to make. So they were&#xA;made with spherical or approximately parabolic mirrors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this has nothing to do with the difference between Newtonian, Gregorian and Cassegrain telescopes. The distinction between these three is in the optical scheme.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newtonian uses one concave and one flat mirror (or just one concave).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gregorian uses two concave mirrors, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cassegrain uses one concave and one convex mirror.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these mirrors should be parabolic in an ideal situation. First Gregorian was made by Hooke in 1673.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was 5 years before John Hadley was born. Hadley made a parabolic mirror in 1721,&#xA;for a Newtonian telescope btw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The principle on which the parabolic mirror works was indeed known in antiquity.&#xA;However they did not know in antiquity how to make good mirrors, neither flat, not parabolic nor spherical (at least no evidence survived that they did know. The legend of Archimedes mirrors burning ships at a distance appears at a much later period. This is not possible even with modern mirrors).&#xA;This was a problem even in Newton's time, what metal to use. Good mirrors with silver coating on glass were produced much later. They also did not know in antiquity that one can use this principle to make telescopes. (Or, if they knew, this knowledge did not survive).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. History of (optical) telescopes is explained in great detail in the book &#xA;Telescopes, by Gerard P. Kuiper and Barbara M. Middlehurst, U. Chicago Press, 1961.&#xA;Very interesting reading. Includes the history of glass making and mirror polishing technology.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-13T02:16:00.737" LastActivityDate="2015-03-13T02:16:00.737" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2060" PostTypeId="1" AcceptedAnswerId="2062" CreationDate="2015-03-13T12:13:35.683" Score="15" ViewCount="1023" Body="&lt;p&gt;At least two 18th century writers wrote that Mars has two satellites: Swift in Gulliver's travels (1726) and Voltaire in Micromégas (1752).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did they guess this? Was Voltaire repeating Swift's prediction?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was known at that time that Earth has 1, Jupiter 4 and Saturn has 5 satellites, so simple interpolation does not give a definite result. None of them suggested that Venus&#xA;has any satellites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The additional 3 satellites of Saturn were discovered only in 1789, and Mars's satellites only in 1877.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2021-03-01T07:19:35.677" LastActivityDate="2021-03-01T07:19:35.677" Title="Why did 18th century writers think that Mars had 2 satellites?" Tags="|physics|astronomy|18th-century|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2061" PostTypeId="1" AcceptedAnswerId="2092" CreationDate="2015-03-13T14:40:11.213" Score="7" ViewCount="212" Body="&lt;p&gt;The earliest text I've been able to find that explain the GR energy conditions is &quot;The large scale structure of space-time&quot; (1973) by Hawking and Ellis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However in Barcelo and Visser's paper &quot;&lt;a href=&quot;http://arxiv.org/pdf/gr-qc/0205066.pdf&quot; rel=&quot;nofollow&quot;&gt;Twilight for the energy conditions?&lt;/a&gt;&quot; they speak of the trace energy condition which was popular in the 60s and has since now been abandoned. This is obviously before 1973 so who actually created the energy conditions? &lt;/p&gt;&#xA;" OwnerUserId="1957" OwnerDisplayName="Rammus" LastEditorUserId="55" LastEditDate="2015-03-24T00:17:05.647" LastActivityDate="2015-03-24T17:23:24.183" Title="Who created the energy conditions?" Tags="|physics|astronomy|relativity-theory|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2062" PostTypeId="2" ParentId="2060" CreationDate="2015-03-14T06:00:33.873" Score="18" Body="&lt;p&gt;It is a funny story, a comedy of errors, that started with &lt;a href=&quot;http://www.mathpages.com/home/kmath151/kmath151.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kepler and Galileo&lt;/a&gt;, and was likely picked up by Swift as a target of satire. It is assumed that &lt;a href=&quot;http://www.uapress.arizona.edu/onlinebks/MARS/CHAP14.HTM&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voltaire followed Swift&lt;/a&gt;, but it paid off for both, two Martian craters are now named after them. How ironic if the 'prediction' was indeed meant as a mockery. Another irony is that perhaps Swift should not have been so swift to mock, it is similar fitting of few data points to data that &lt;a href=&quot;https://hsm.stackexchange.com/questions/1948/how-did-kepler-guess-his-third-law-from-data/1950#1950&quot;&gt;led Kepler to his famous third law&lt;/a&gt;, which Swift himself appears to have used. False positives are part of the process. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In January 1610 Galileo discovered the four moons of Jupiter with a newly built telescope, and informed his correspondents, including Kepler, about it. Combined with the Earth's one moon this formed a pattern, to Kepler anyway: 1,2,4..., and it meant two moons for Mars. &lt;a href=&quot;http://www.astropa.unipa.it/versione_inglese/Hystory/BODE%27S_LAW.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kepler tried geometric sequences before&lt;/a&gt;, for the planetary distances to the Sun, but he did not succeed even after placing extra planets between Mercury and Venus, and Mars and Jupiter. This is surprising because for the five then known planets a close fit does exist, it is now known as the Titius-Bode rule. After that Kepler turned to inscribed and circumscribed Platonic solids of Mysterium Cosmographicum (1596), another false pattern. But the story doesn't end there. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On July 25, 1610 Galileo made another discovery, and this times &lt;a href=&quot;http://www.daviddarling.info/encyclopedia/S/Swift.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;coded it in an anagram&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;s m a i s m r m i l m e p o e t a l e u m i b u n e n u g t t a u i r a s&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This looked like a challenge made just for Kepler, and he applied himself. After a while he cracked it, or so he thought:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Salue umbistineum geminatum Martia proles&#xA;(Hail, twin companionship, children of Mars)&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, Kepler was off by one letter, but he possibly assumed that it was a transcription error, especially since the &quot;twin companionship&quot; matched his earlier hunch. It wasn't. The correct unscrambling was&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Altissimum planetam tergeminum observavi&#xA;(I observed the most distant planet to have a triple form)&lt;/em&gt;, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and referred to Saturn surrounded by profiles of its rings (but Galileo did not know that at the time). When Galileo later sent another anagram Kepler pleaded: &quot;&lt;em&gt;I adjure you not to leave us long in doubt of the meaning. For you see you are dealing with real Germans. Think in what distress you place me by your silence.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Swift's Laputa was a &lt;a href=&quot;http://www.theguardian.com/science/the-h-word/2013/mar/14/science-history-satire-politics-swift-gulliver&quot; rel=&quot;nofollow noreferrer&quot;&gt;satire of science and scientists&lt;/a&gt; of his time, and he couldn't do better than Kepler, with his patterns that are not always there, as a real life prototype. That he chose moons of Mars deliberately is confirmed by the fact that the orbital distances and periods he gives for them have approximately the same ratio $R^3/T^2$, as the Kepler's third law requires. Swift apparently wanted whole numbers, and noticed that $(5^3)(20^2) = 50000$ is approximately $(3^3)(43^2) = 49923$, so for his moons he chose distances of $3$ and $5$ Martian diameters with periods of $10$ and $21.5$ hours respectively (actual distances to Phobos and Deimos are 1.4 and 3.5 diameters, with periods 7.6 and 30.3 hours). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is curious about this reconstruction is the degree of familiarity assumed of Swift with astronomy and its history (Gulliver was published in 1726), not to mention his prowess with solving Diophantine equations, I guess he did his research. But it is better than another theory, advanced by a spacecraft designer Perminov among others, that Swift found and deciphered records that Martians left on Earth.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-03-14T10:21:30.107" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2063" PostTypeId="1" CreationDate="2015-03-14T06:00:34.567" Score="4" ViewCount="517" Body="&lt;p&gt;I'm very interested to know how the concept of vector field was created. What problems or phenomena led mathematicians to introduce the concept of vector field?&lt;/p&gt;&#xA;" OwnerUserId="1958" LastEditorUserId="123" LastEditDate="2015-03-15T13:07:07.897" LastActivityDate="2015-03-15T13:07:07.897" Title="Invention of the concept of vector field" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2064" PostTypeId="1" AcceptedAnswerId="8402" CreationDate="2015-03-14T13:14:06.147" Score="4" ViewCount="437" Body="&lt;p&gt;Can anyone suggest me some book/article explaining how and why the quantity named 'Energy' made its way into physics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have gone through &quot;Lectures on Physics&quot; (Vol. 1) by R.P. Feynman and have been convinced that at first scientists were in search of a quantity which remains constant w.r.t. any other internal change,in a closed system.The quantity later turned out to be 'Force x displacement'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we know that momentum is also conserved in closed systems. So why don't we invent a scalar 'mass x speed' (in order to fix the problem that momentum is a vector) and use this instead of 'Energy'? While we could take the advantage that 'mass x speed' is much simpler than '1/2 (mass x square of velocity)'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my teachers, on being asked this question, said that energy is more fundamental than momentum. And giving the example of a field force, he wrote:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\vec{F}=\frac{\partial \varphi}{\partial x}\hat{\imath}+\frac{\partial \varphi}{\partial y}\hat{\jmath}+\frac{\partial \varphi}{\partial z}\hat{k}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And showed that the quantity $\varphi$ turns out to be the potential energy of a particle (e.g. a point mass in case of a gravitational field) in the field at the point $\vec r = (x,y,z)$. But what is the thought behind the approach to find out such a quantity whose change w.r.t. position will describe the force? And at which point 'Force x displacement' becomes more fundamental than momentum?&lt;/p&gt;&#xA;" OwnerUserId="1960" LastEditorUserId="2442" LastEditDate="2016-01-27T02:20:09.640" LastActivityDate="2019-03-21T18:17:20.883" Title="History of Energy" Tags="|physics|energy|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2065" PostTypeId="2" ParentId="2063" CreationDate="2015-03-14T14:04:46.947" Score="3" Body="&lt;p&gt;I would point to Euler's &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1755&amp;amp;seite:int=282&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Principes généraux du mouvement des fluides&lt;/em&gt;&lt;/a&gt;, Mém. Acad. Sci. Berlin &lt;strong&gt;11 (1755)&lt;/strong&gt; (1757) 274-315 for a pretty clear notion of a fluid's velocity field (whose integral curves are the lines of flow, and satisfying what we now call Euler's equations and the continuity equation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;O. Darrigol in &lt;a href=&quot;http://books.google.com/books?id=H-E0_-X_h2wC&amp;amp;pg=PA25&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl&lt;/em&gt;&lt;/a&gt; has some of this anticipated in earlier memoirs of Euler and D'Alembert, but I find the modern concept quite a bit harder to identify there.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2015-03-14T14:04:46.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2066" PostTypeId="1" AcceptedAnswerId="2072" CreationDate="2015-03-14T23:40:15.347" Score="15" ViewCount="1359" Body="&lt;p&gt;Near the end of Einstein's 1905 paper on $E=mc^2$ he states (in the translation by Perrett and Jeffery):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This suggests that there were no pre-1905 experiments that would have pointed to mass-energy equivalence. Are there there one or more experiments from the era immediately following 1905 to which we could point as the earliest confirmations of Einstein's prediction?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[EDIT] A couple of possibilities are the Cockroft-Walton experiment (1932) and electron-positron annihilation (positrons were discovered in 1932, but not through their annihilation radiation). WP's article on mass-energy equivalence claims that Cockroft-Walton verified $E=mc^2$ to within 0.5% and gives a link to scans of the paper, but I can't find any such result in the paper.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="55" LastEditDate="2016-07-22T23:27:14.323" LastActivityDate="2016-07-22T23:27:14.323" Title="What were the early empirical tests of Einstein's mass-energy equivalence $E=mc^2$?" Tags="|experimental-physics|relativity-theory|energy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2067" PostTypeId="1" CreationDate="2015-03-15T16:57:05.870" Score="7" ViewCount="841" Body="&lt;p&gt;As everyone knows, there are dozens of proofs of Binet's Fibonacci Number Formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, what was the first approach used by Binet? What let people to say this result (the Binet formula) was known to Euler and Bernoulli? &lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="62" LastEditDate="2015-05-03T18:31:43.167" LastActivityDate="2017-07-18T13:59:40.870" Title="With what kind of proof was the Binet formula derived for the first time?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2069" PostTypeId="1" CreationDate="2015-03-17T04:45:55.210" Score="5" ViewCount="143" Body="&lt;p&gt;Who was the first to publish a Hankel contour integral?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See notes in my answer to the MO-Q  &lt;a href=&quot;https://mathoverflow.net/questions/58004/how-does-one-motivate-the-analytic-continuation-of-the-riemann-zeta-function/97401#97401&quot;&gt;How does one motivate the analytic continuation of the Riemann zeta function?&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-05-15T10:42:56.563" Title="Origin of the Hankel contour?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2072" PostTypeId="2" ParentId="2066" CreationDate="2015-03-19T00:41:52.043" Score="11" Body="&lt;p&gt;Let me clarify first that there are deep &lt;a href=&quot;http://plato.stanford.edu/entries/equivME/#2&quot; rel=&quot;nofollow&quot;&gt;conceptual issues with what $E=mc^2$ means&lt;/a&gt;, and what it means to verify it. That energy contributes to inertial mass was known before Einstein. In 1900 &lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1204/1204.6575.pdf&quot; rel=&quot;nofollow&quot;&gt;Poincare showed&lt;/a&gt; that electromagnetic field has momentum, hence effectively mass, and implicitly gave it as $E/c^2$. Austrian physicist &lt;a href=&quot;http://physicsworld.com/cws/article/news/2011/aug/23/did-einstein-discover-e-equals-mc-squared&quot; rel=&quot;nofollow&quot;&gt;Hasenöhrl even wrote explicitly&lt;/a&gt; $E=3/8\,mc^2$ in a 1904 paper, while German physicist Abraham was interpreted as giving $E=3/4\,mc^2$ (wrong proportionality constants were due to subtle mistakes in calculations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1901-1915 Kaufman and Bucherer conducted a series of measurements of electrons' mass-to-charge ratio, and electrons traveling between 0.25–0.75c indicated an increase in mass in agreement with $E=mc^2$. If we also accept the equivalence of inertial and gravitational mass, &lt;a href=&quot;http://en.wikipedia.org/wiki/Gravitational_lens#History&quot; rel=&quot;nofollow&quot;&gt;bending of the light by the Sun observed by Eddington and Dyson&lt;/a&gt; during a solar eclipse of 1919 can be seen as another verification of energy to mass contribution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, &lt;a href=&quot;http://arxiv.org/pdf/1407.8507v1.pdf&quot; rel=&quot;nofollow&quot;&gt;Einstein's 1905 paper&lt;/a&gt; not only extended this effect to all matter, but also boldly claimed the converse, that any mass can be transmuted into (kinetic, and hence any) energy. This is what Cockcroft and Walton are traditionally credited with first experimentally verifying in 1932. The reason it is hard to locate it in their paper is that their intentions were different. They bombarded lithium nucleus by protons to verify that two $\alpha$ particles are produced as a result, not that mass is converted into energy. The rest mass of proton+nucleus was $8.0176$ amu, but of the two $\alpha$ particles only $8.0022$ amu. They noted that the mass defect of $0.0154$ amu “is equivalent to an energy liberation of (14.3 ± 2.7) ×106 Volts” based on $E=mc^2$, which is consistent with about $17$ MeV, the observed value of the kinetic energy of two $\alpha$ particles. However, Cockcroft and Walton also give independent evidence that the products are the $\alpha$ particles, so in hindsight their result can be interpreted as confirming the transmutation of mass into energy. &lt;a href=&quot;http://plato.stanford.edu/entries/equivME/#4&quot; rel=&quot;nofollow&quot;&gt;Stanford article&lt;/a&gt; has more details and analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another hindsight verification came from the famous &lt;a href=&quot;http://www.conservapedia.com/E%3Dmc%C2%B2#A_Famous_Example_--_Nuclear_Fission_of_Uranium&quot; rel=&quot;nofollow&quot;&gt;Hahn-Strassmann nuclear fission experiment&lt;/a&gt;, where uranium 235 decayed first into xenon 140, and then into barium 140, with some byproducts. The appearence of barium was unexpected because it is so much lighter than uranium, but Lise Meitner in December of 1938 surmised what the reaction was. She also noted the much more sizable mass defect ($0.177427$ amu equivalent to $165$ MeV based on $E=mc^2$, the modern value is $200$ MeV), and predicted massive release of energy, which was observed shortly thereafter. This &quot;splitting of the atom&quot; was a key step towards the nuclear bomb. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above two experiments are singled out because of precise mass measurements, but in principle conversion of mass into energy was apparent in other early observations. Perhaps the most &quot;philosophically&quot; satisfactory were &lt;a href=&quot;http://encyclopedia2.thefreedictionary.com/Annihilation+and+Creation+of+Particle-Antiparticle+Pairs&quot; rel=&quot;nofollow&quot;&gt;electron-positron creation and annihilation&lt;/a&gt; observed in 1933 by Irène and Frédéric Joliot-Curie using a Wilson chamber in a magnetic field. Since photons have zero rest mass these events signify clean and symmetric conversion of mass into energy and back. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is $\beta$ decay, which led Pauli to conjecture the existence of neutrino in 1930, so named by Fermi in 1933. However, an independent confirmation of neutrino's existence, required to confirm mass to energy conversion, did not come until &lt;a href=&quot;http://en.wikipedia.org/wiki/Neutrino#Direct_detection&quot; rel=&quot;nofollow&quot;&gt;Cowan-Reines experiment&lt;/a&gt; in 1956. Of course, various reactions observed in 1950-60s, when a host of new particles were discovered, also qualify as confirmations. One of the most accurate recent measurements is due to &lt;a href=&quot;http://jilawww.colorado.edu/thompson/sites/default/files/pdf/4381096a.pdf&quot; rel=&quot;nofollow&quot;&gt;Rainville et al. in 2005&lt;/a&gt;, which was explicitly designed to test the mass to energy conversion, it is based on &#xA;$\gamma$ ray emission when a nucleus captures a neutron. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-05-02T20:03:44.080" LastActivityDate="2016-05-02T20:03:44.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2074" PostTypeId="1" AcceptedAnswerId="2075" CreationDate="2015-03-20T05:13:04.200" Score="4" ViewCount="211" Body="&lt;p&gt;Many of the 'famous' papers in the scientific community have resulted in Nobel prizes, but certainly that represents only a small fraction of notably important papers throughout scientific history. What are some of the most famous/important non-Nobel prize winning papers/scientific-works?&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2015-03-20T22:07:40.190" Title="What are some of the most important scientific works done that have not received a Nobel prize?" Tags="|physics|chemistry|publishing|biology|nobel-prize|" AnswerCount="1" CommentCount="4" ClosedDate="2015-03-21T06:04:21.637" ContentLicense="CC BY-SA 3.0" />
  <row Id="2075" PostTypeId="2" ParentId="2074" CreationDate="2015-03-20T08:11:34.137" Score="8" Body="&lt;p&gt;Here are some prominent examples of unrecognized discoveries; the actual publications should be easily findable online. Note that I've only included omitted discoveries, not omitted individuals (such as Lise Meitner's omission for nuclear fission, or Chien-Shiung Wu's for parity conservation). I've also only included work whose inventor was still alive when the Nobels were around (so no Newton!).&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Chemistry&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The periodic table of elements (1869) by Dmitri Mendeleev (1834-1907).&lt;/li&gt;&#xA;&lt;li&gt;Covalent bonds and electron pairs (1916) by Gilbert Lewis (1875-1946).&lt;/li&gt;&#xA;&lt;li&gt;Transition state theory (1935) by Henry Eyring (1901-1981).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h2&gt;Physics&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Special relativity (1905), Brownian motion (1905) and general relativity (1915) by Albert Einstein (1879-1955)&lt;/li&gt;&#xA;&lt;li&gt;Bose-Einstein statistics (1924) by Satyendra Nath Bose (1894-1974) and Einstein.&lt;/li&gt;&#xA;&lt;li&gt;Big Bang theory and Hubble's law (1927) by Georges Lemaître (1896-1966)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h2&gt;Physiology or Medicine&lt;/h2&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;DNA as the building block of genes (1944) by Oswald Avery (1877-1955)&lt;/li&gt;&#xA;&lt;li&gt;Molecular phylogeny (1977) by Carl Woese (1928-2012)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="64" LastEditDate="2015-03-20T22:07:40.190" LastActivityDate="2015-03-20T22:07:40.190" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2078" PostTypeId="1" CreationDate="2015-03-21T06:58:23.113" Score="8" ViewCount="1815" Body="&lt;p&gt;I'd like to know about examples of prominent scientists having exceptionally good memory, with anecdotes if possible, and how it affected their work. It would also be interesting to see examples of the complete opposite, prominent scientists having exceptionally bad memory, and how they made up for it.&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="4251" LastEditDate="2020-07-11T17:15:56.917" LastActivityDate="2023-12-05T20:44:08.517" Title="Scientists with exceptionally good or bad memory" Tags="|biographical-details|examples|scientists|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2079" PostTypeId="2" ParentId="2078" CreationDate="2015-03-21T12:12:47.207" Score="8" Body="&lt;p&gt;The standard example of &quot;complete opposite&quot; was N. Wiener. Anecdotes:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Wiener walking on campus is stopped by someone and they start a mathematical conversation. After the end of the conversation Wiener asks: &quot;In which direction I was walking before I met you?&quot; &#xA;-&quot;This direction.&quot;&#xA;-&quot;OK, this means I already had my lunch!&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wiener bought a new house. On the way back home from his office, he forgot where it was. He sees a little girl and asks: &quot;Do you know where the Wieners live? They recently moved to this neighborhood&quot;.&#xA;-&quot;Yes, Daddy! My mother sent me to meet you!&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A student comes to the post office. He sees Wiener walking back and forth, in deep thought. For some time the student hesitates to distract him, but finally&#xA;addresses him: &quot;Professor Wiener, ...&quot;&#xA;-&quot;Oh, yes! Wiener! (Addressing the post office clerk) Wiener! Do you have something for Wiener?&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As an example of good memory, Hardy surprised his friends by remembering all cricket scores for many years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking seriously, from what I read and from my own contacts with outstanding physicists and mathematicians, I can conclude that ability to do original work in science is not correlated with good or bad memory. Like it does not correlate with body weight or height. Unlike weight and height, memory cannot be objectively measured because it is highly selective. (A person may remember cricket scores for many years but poorly memorize verses etc.) Experience shows that this kind of ability simply does not correlate with the ability to do exact sciences. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-21T16:49:10.270" LastActivityDate="2015-03-21T16:49:10.270" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2080" PostTypeId="2" ParentId="596" CreationDate="2015-03-21T16:47:16.467" Score="5" Body="&lt;p&gt;I must put up a clarifying answer. They groped towards different concepts and talked past each other, in the modern perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of what we call Newtonian mechanics is due to Euler, who called it such in his early papers on the topic, but Euler wrote after the controversy. (See Truesdell)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is, as is clearly seen from the primary texts of Newton and Leibnitz, that Newton was working with the concept we call force (indeed, in the later edition of his &lt;em&gt;optics&lt;/em&gt;, he discovered repulsive forces, in addition to attractive forces he postulated). This would lead to the progress made by Euler and Boscovich. Cartesius worked with momentum prior both.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibnitz was however concerned primarily with mechanics only so far as it concerned his &quot;the most diverse one the one possible world that exists&quot; philosophy. He argued that since all laws of nature can be written (truly modelled analogously) as equalities, the &quot;whole effect&quot; is always conserved. One side is cause and the other effect. This was intended to eliminate the concept and problem of &quot;first cause&quot; from an practical significance in his philosophical system of the world. Which side in an equation is the whole cause and which is the whole effect?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is loosely similar to the modern position, because we no longer care or even assume existence of first causes, and it is not impossible that a time reverse situation occurs—for the cause and the effect in this case are in an equivalence class—but rather it is highly improbable, as determined by the thermodynamics of the situation. From here we have Boltzmann and Mach picking up the stick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The confusion early on was caused by the fact that &lt;em&gt;modern&lt;/em&gt; theory in thermodynamics did not exist prior to Waterston, Mayer, and Boltzmann. Indeed, Leibnitz could not explain where the &quot;effect&quot; went when a body dropped from a height fell onto the floor, which ended its fall. That was the Mayer experiment 150 years later. And 15 years after than, even scientists such as Kelvin still had to be shown errors in their papers where they claimed they'd found violations of energy conservation—such as when Waterson pointed out to Kelvin that phase transformations do in fact require energy and cannot be ignored in well constructed experiments. The energy conservation concept was broadly spread at that point by the popular Spencer with his &lt;em&gt;synthetic philosophy&lt;/em&gt;.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibnitz simply extrapolated from a pendulum experiment that $mv^2$ is conserved in all cases. Such extrapolation is not logically kosher, true, but that was his hypothesis in his &lt;em&gt;dynamics&lt;/em&gt;. He was essentially groping toward the modern energy concept, which was finally clarified by Mayer and Clausius and Mach and Einstein, in that order. It was Mayer who gave the first clear discussion of kinetic as distinguished from potential as distinguished from thermal energy. Maupertuis and D'Alembert and Euler worked on clarifying the concepts of kinetic and potential energy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without thermodynamics the controversy could not have been resolved given the state of knowledge when it occurred—and so indeed it wasn't. There are no miracles.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="6780" LastEditDate="2017-12-19T22:26:28.833" LastActivityDate="2017-12-19T22:26:28.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2081" PostTypeId="2" ParentId="2078" CreationDate="2015-03-21T23:18:51.933" Score="3" Body="&lt;p&gt;There are similar stories about Einstein. There are two sides to this. On the one hand: all people, even geniuses, have problems with their memory when they get old. On the other hand: Both Wiener and Einstein were outspoken critics of the cold war hysteria. These stories were part of an agenda to depict them as harmless eccentrics whose opinions on political and ethical matters did not need to be taken seriously.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-03-21T23:18:51.933" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2082" PostTypeId="2" ParentId="187" CreationDate="2015-03-22T08:42:10.907" Score="4" Body="&lt;p&gt;One example sometimes given is the lack of &lt;strong&gt;an explanation for hybrid sterility&lt;/strong&gt; in Darwin's original theory of evolution by natural selection, something that &lt;em&gt;was&lt;/em&gt; explained by theistic evolution. Another is the lack of &lt;strong&gt;an explanation for dispersion and selective absorption&lt;/strong&gt; in the wave theory of light, something that &lt;em&gt;was&lt;/em&gt; explained by the emission theory. However, like many of Kuhn's examples, neither of these was much of a 'loss' in practice, as neither of the displaced theories had genuine predictive power.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps a better example is the classical description of fluid dynamics given by the &lt;strong&gt;Poiseuille equation&lt;/strong&gt;, which cannot (yet) be derived in the modern quantum physics paradigm. However, its usefulness means that it continues to be taught alongside the quantum model, despite the fact that it can no longer be 'justified' in the accepted paradigm.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2015-06-27T15:32:12.737" LastActivityDate="2015-06-27T15:32:12.737" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2083" PostTypeId="1" CreationDate="2015-03-22T09:14:22.877" Score="2" ViewCount="175" Body="&lt;p&gt;Let $ e $ be the base of natural logarithms. Then $ e^3 = 20.08\dots\approx 20$. Was this convenient fact for back of the envelope calculations noted in the literature or in the historical record, and if so how early could it have appeared? &lt;/p&gt;&#xA;" OwnerUserId="144" LastActivityDate="2015-03-23T20:02:40.670" Title="History of $ e^3\approx 20$" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2085" PostTypeId="1" AcceptedAnswerId="2090" CreationDate="2015-03-22T17:24:43.083" Score="7" ViewCount="3523" Body="&lt;p&gt;I have my respect for Mr Tesla, but it seems weird that &amp;quot;he&amp;quot; was chosen to be the units of magnetic flux density. I mean, he didn't contribute much to magnetic fields theory, nor did he work with them a lot. So why was it named after him? Why not, for example, after Hendrik Lorentz, who formulated Lorentz Force?&lt;/p&gt;&#xA;" OwnerUserId="1991" LastEditorUserId="4251" LastEditDate="2021-12-26T09:04:21.443" LastActivityDate="2021-12-26T09:04:21.443" Title="Why is magnetic flux density named after Nikola Tesla?" Tags="|physics|terminology|physicists|magnetism|tesla|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2086" PostTypeId="2" ParentId="2085" CreationDate="2015-03-22T20:10:01.050" Score="5" Body="&lt;p&gt;According to &lt;a href=&quot;http://www.journal.ftn.kg.ac.rs/Vol_3-2/08-Spaic-Spiridonovic.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;, the name 'tesla' was first suggested to the Technical Committee of the International Electrotechnical Commission in 1950 by two professors from the &lt;strong&gt;Belgrade Faculty of Electrical Engineering&lt;/strong&gt;. The suggestion was accepted by the committee in 1954, and by the International Committee for Weights and Measures in 1956, which also happened to be the &lt;strong&gt;centenary of Tesla's birth&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The choice of 'tesla' was publicly justified due to Tesla's &quot;&lt;strong&gt;fundamental scientific work&lt;/strong&gt;&quot; in the &quot;electrical world&quot;, rather than anything more specifically connected to magnetic flux density.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="2277" LastEditDate="2019-04-22T21:53:30.187" LastActivityDate="2019-04-22T21:53:30.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2087" PostTypeId="1" CreationDate="2015-03-23T02:37:14.150" Score="7" ViewCount="2841" Body="&lt;p&gt;A friend of mine claims that vector calculus was invented to do electrodynamics.  I'm dubious.  I know that Maxwell first wrote down the so-called Maxwell's equations in scalar form and only later converted them into their vector forms.  Because all of classical electrodynamics is derivable from Maxwell's equations, I wouldn't see the need to develop a branch of mathematics to simply condense the equations of a theory that was pretty much done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Vector_calculus&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; oddly does not have a &quot;History&quot; section for vector calculus.  So can anyone either confirm or deny this?  What is the history of vector calculus?  Why was it invented (/discovered)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;: I specifically want to know about how E&amp;amp;M may (or may not) have affected the development of vector calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of something that might be addressed: Apparently Heaviside's &lt;em&gt;The Electrian&lt;/em&gt; was written a couple of decades before Gibbs' &lt;em&gt;Vector Analysis&lt;/em&gt; -- which apparently is the book that codified the modern notations.  How influential was &lt;em&gt;The Electrian&lt;/em&gt; on Gibbs's work?&lt;/p&gt;&#xA;" OwnerUserId="1992" LastEditorUserId="8" LastEditDate="2015-04-30T18:24:18.087" LastActivityDate="2022-03-06T21:15:24.933" Title="Electromagnetics and vector calculus" Tags="|mathematics|physics|calculus|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2088" PostTypeId="1" AcceptedAnswerId="2089" CreationDate="2015-03-23T06:45:16.263" Score="4" ViewCount="6590" Body="&lt;p&gt;The Michelson and Morley experiment always seems to be the experiment that caused the dismissal of the aether and that lead to special relativity. I cannot see why this is, granted it contributed to these two points but in no more away then say the Fizeau experiments or experiments into stellar aberration. In fact, I think it was the contradiction between the results of these three experiments that lead to the dismal of the aether (as they thought of it at the time). So here is my question: Why is the Michelson and Morley experiment seen as the most important of these (and other experiments like them)? And how did this lead to the special theory of relativity?&lt;/p&gt;&#xA;" OwnerUserId="1994" LastActivityDate="2022-05-19T09:23:42.483" Title="Significance of the Michelson and Morley experiment?" Tags="|physics|experimental-physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2089" PostTypeId="2" ParentId="2088" CreationDate="2015-03-23T12:37:27.913" Score="4" Body="&lt;p&gt;The crucial thing to dismissal of the aether was not this experiment but the special relativity. There were alternative theories explaining the outcome of this experiment. The main base of the special relativity was not this experiment but  Maxwell's theory of electromagnetism, from which it is a logical consequence.&#xA;(And Maxwell's theory of magnetism was very well established, not only by experiments but by many practical applications.)&#xA;But the experiment nicely confirms it, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Michelson-Morley experiment, unlike the earlier experiments measures the speed of light with very high accuracy, due to the novel technology based on interference.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-03-23T12:37:27.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2090" PostTypeId="2" ParentId="2085" CreationDate="2015-03-23T19:25:02.943" Score="8" Body="&lt;p&gt;Actually, Tesla's first major invention was based on his discovery of rotating magnetic field around 1882. To quote &lt;a href=&quot;http://www.teslauniverse.com/nikola-tesla-article-teslas-egg-of-columbus&quot; rel=&quot;noreferrer&quot;&gt;colorful description from 1919 Electrical Experimenter&lt;/a&gt;:&quot;There is a vast difference between an ordinary electro-magnet and that invented by Tesla. In the former the lines are stationary, in the latter they are made to whirl around at a furious rate. The first attracts a piece of iron and holds it fast; the second causes it to spin in any direction and with any speed desired.&quot; Look at &lt;a href=&quot;https://www.youtube.com/watch?v=aWTXgQ_IdtI&quot; rel=&quot;noreferrer&quot;&gt;Tesla's Egg of Columbus&lt;/a&gt; video to see what it can do. This was the source of all modern AC motors, Tesla patented the first one in 1887, that before him were &lt;a href=&quot;http://science.howstuffworks.com/innovation/famous-inventors/famous-nikola-tesla-inventions.htm&quot; rel=&quot;noreferrer&quot;&gt;believed to be impossible&lt;/a&gt; because &quot;the alternating current was but imperfectly understood and had no standing with engineers or electricians&quot;. Nuclear magnetic resonance underlying the modern Magnetic Resonance Imaging (MRI) technology is based on properties of magnetic nuclear spins, a quantum analog of rotating magnetic field discovered by Rabi in 1937. Modern &lt;a href=&quot;http://www.teslasociety.com/mri.htm&quot; rel=&quot;noreferrer&quot;&gt;MRI machines are calibrated in tesla units&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A word on terminology, technically tesla is not a unit of magnetic flux, that would be &lt;a href=&quot;http://en.wikipedia.org/wiki/Weber_%28unit%29&quot; rel=&quot;noreferrer&quot;&gt;weber&lt;/a&gt;, but a unit of magnetic flux density per area. The genesis of tesla was somewhat controversial, disagreements over whether magnetic field strength and magnetic flux density were quantities of different nature lasted until 1930 when the International Electrotechnical Commission declared that they were, and tasked the Technical Committee 1 to come up with new units. In 1935 it passed the task to the Technical Committee 24, which dragged its foot until 1956 (in fairness World War II happened, and they had to settle on units for several other quantities as well). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a &lt;a href=&quot;http://encyclopedia2.thefreedictionary.com/Lorentz+unit&quot; rel=&quot;noreferrer&quot;&gt;unit named after Lorentz&lt;/a&gt;, but a very obscure one, &quot;a unit of reciprocal length used to measure the difference, in wave numbers, between a (zero field) spectrum line and its Zeeman components&quot;. It is fitting since he is the founder of the (classical) electron theory. But Lorentz is honored much more meaningfully by having the electromagnetic force acting on charges named after him.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-03-29T20:04:35.453" LastActivityDate="2015-03-29T20:04:35.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2091" PostTypeId="2" ParentId="2083" CreationDate="2015-03-23T20:02:40.670" Score="3" Body="&lt;p&gt;I don't recall seeing this pointed out in 1800s literature, and I've looked at a HUGE amount of it in the past 10 years or so (over 1000 books, all volumes of over 15 journals). At least, if it was pointed out, it was only as a curiosity. Also, if you think about it, there is little use for this approximation. Unless you want rough approximations to constants times $e^{3},$ constants times $e^{6},$ constants times $e^{-3},$ etc. and their sums/differences, it won't help much. Unlike the case with $2^{10} \approx 10^{3},$ which is useful because &lt;em&gt;positive integer powers&lt;/em&gt; of $2$ are what show up in practice (e.g., $23$ doublings is equivalent to multiplication by $2^{23} = 2^3 \times 2^{20} \approx 8 \times 10^{6}),$ the exponents that show up for base $e$ tend to be non-integers (because of the prevalence of functions involving exponentiating with base $e).$ Thus, I don't see how knowing $e^3 \approx 20$ is of much help in getting quick estimates (with around 5% accuracy) for things like $e^{2/3},$ $\;e^{1/2 + 1/3 + 1/4},$ $\;e^{-2.34},$ etc.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2015-03-23T20:02:40.670" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2092" PostTypeId="2" ParentId="2061" CreationDate="2015-03-24T00:13:19.703" Score="7" Body="&lt;p&gt;I emailed Matt Visser to ask for help, his reply is reproduced below with his permission:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;From:&lt;/strong&gt; Matt Visser&lt;br /&gt;&#xA;&lt;strong&gt;Sent:&lt;/strong&gt; Saturday, March 21, 2015 11:49 PM&lt;/p&gt;&#xA;&lt;p&gt;I think the name &amp;quot;trace energy condition&amp;quot; was actually invented by Carlos Barcelo and myself; but the condition (without the explicit name) was used extensively --- for instance, in the neutron star community.... &lt;a href=&quot;https://books.google.co.nz/books?id=fgj_TZ06niYC&amp;amp;pg=PA211&amp;amp;lpg=PA211&amp;amp;dq=zeldovich+trace+stress+energy&amp;amp;source=bl&amp;amp;ots=QX1YXvCxxC&amp;amp;sig=62e5C2-uGq0r0FVBISZydiwBulA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=nkYOVa6qMcXf8AWuw4GwDQ&amp;amp;redir_esc=y#v=onepage&amp;amp;q=zeldovich%20trace%20stress%20energy&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;see for instance&lt;/a&gt;, &lt;a href=&quot;https://books.google.co.nz/books?id=Vq787qC5PWQC&amp;amp;pg=PA226&amp;amp;lpg=PA226&amp;amp;dq=zeldovich+trace+stress+energy&amp;amp;source=bl&amp;amp;ots=Y-6-_c64g_&amp;amp;sig=fFi65LeeLtICUUjc0UeYXJ5Pk5k&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=nkYOVa6qMcXf8AWuw4GwDQ&amp;amp;redir_esc=y#v=onepage&amp;amp;q=zeldovich%20trace%20stress%20energy&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;see also&lt;/a&gt;. A google search on &amp;quot;Zeldovich trace stress energy&amp;quot; will pick up these references and link back to even older stuff...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Following up on Matt's links one finds that the &amp;quot;trace energy condition&amp;quot; goes back further than 1960-s. In a 1939 paper On Massive Neutron Cores Oppenheimer and Volkoff obtained the limiting mass of &lt;span class=&quot;math-container&quot;&gt;$0.71$&lt;/span&gt; solar masses for a neutron star, only half of the corresponding Chandrasekhar's limit for white dwarves. They used the stress energy tensor of a free Fermi gas of neutrons in their calculations, but remarked that their conclusions &amp;quot;would not be appreciably affected by any modification of these equations at supranuclear densities that was physically reasonable in the sense that the trace of the stress-energy tensor did not become negative&amp;quot;. This relied on an unpublished result of von Neumann mentioned by Chandrasekhar in a 1935 paper.&lt;/p&gt;&#xA;&lt;p&gt;However, the free model discarded the effects of nuclear interaction, and while von Neumann's result gave a perfectly fine asymptotic, treating it as a bound proved to be an overreach. In 1959 Skyrme constructed a model of nuclear interaction, which now bears his name, and Cameron applied it to model a gas of interacting neutrons. The Skyrme model was not relativistic however. Finally, in 1961 Zeldovich constructed an interacting gas model compatible with special relativity, but with a stress-energy tensor whose trace wasn't always non-negative. Correspondingly, &lt;a href=&quot;http://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_limit#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;the mass limit went up&lt;/a&gt; to &lt;span class=&quot;math-container&quot;&gt;$2.0$&lt;/span&gt;, and eventually to &lt;span class=&quot;math-container&quot;&gt;$3.0$&lt;/span&gt; solar masses, which is much more consistent with modern observations.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-03-24T17:23:24.183" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2093" PostTypeId="1" AcceptedAnswerId="2108" CreationDate="2015-03-24T12:12:31.983" Score="14" ViewCount="15272" Body="&lt;p&gt;I am curious who came up with algorithms that we use today to manually solve mathematical division problems, such as short or long division; how were they  established or standardized that way and why? Who first used or invented them, and how did they come up with it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also curious about the origin of the tableau sign, roughly like so )¯¯¯¯ for example, where did it come from?&lt;/p&gt;&#xA;" OwnerUserId="1998" LastEditorUserId="9523" LastEditDate="2021-02-09T07:47:15.757" LastActivityDate="2021-02-09T07:47:15.757" Title="Who invented short and long division?" Tags="|mathematics|mathematicians|discoveries|notation|algorithm|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2094" PostTypeId="1" AcceptedAnswerId="2095" CreationDate="2015-03-24T12:57:39.447" Score="15" ViewCount="619" Body="&lt;p&gt;As far as I understand, the Gamma function was developped as a way of calculating &quot;the&quot; factorial of a non-integer number. Why did this problem interest 18th century mathematicians? Was it just a puzzle, or did they have some specific application in mind?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="4251" LastEditDate="2020-10-02T06:18:09.683" LastActivityDate="2024-03-27T07:43:27.470" Title="Why were 18th century mathematicians interested in extending the factorial to non-integers?" Tags="|mathematics|18th-century|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2095" PostTypeId="2" ParentId="2094" CreationDate="2015-03-24T15:07:55.253" Score="15" Body="&lt;p&gt;The problem can be seen as part of more general attempts to extend the domain of operations defined initially for naturals (integers, or rationals), only. A prominent natural example would be taking an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th power (another would be the binomial coefficients, and related Newtonian series).&#xA;The problem is, thus, not an isolated one, but rather part of a general and natural project of that time.&lt;/p&gt;&#xA;&lt;p&gt;An application that directly motivated the invention of the Gamma function was Goldbach trying to find something like a closed form for &lt;span class=&quot;math-container&quot;&gt;$\sum_{k=1}^n k!$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This answer is based on the information given in Section 1 of:&lt;/p&gt;&#xA;&lt;p&gt;Gronau, D. &lt;a href=&quot;https://imsc.uni-graz.at/gronau/TMCS_1_2003.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why is the gamma function so as it is.&lt;/a&gt; Teaching Mathematics and Computer Science, 1, 43-53 (2003).&lt;/p&gt;&#xA;&lt;p&gt;The reference mentioned there to support this point of view specifically is :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;J. Christoph Scriba, &amp;quot;Von Pascals Dreieck zu Eulers Gamma-Funktion, Zur Entwicklung der Methodik der Interpolation&amp;quot;, in:&#xA;Mathematical Perspectives, Essays on Mathematics and Its Historical Development, (Joseph W. D auben, ed.), Academic Press, New York, 1981&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I could not read this reference, but even only the title (my translation) &amp;quot;From Pascal's triangle to Euler's Gamma function, On the developpment of the method of interpolation.&amp;quot; supports it.&lt;/p&gt;&#xA;&lt;p&gt;The MO-question &lt;a href=&quot;https://mathoverflow.net/questions/9746/who-invented-the-gamma-function&quot;&gt;Who invented the gamma function?&lt;/a&gt; contains related information.&lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="55" LastEditDate="2024-03-27T07:43:27.470" LastActivityDate="2024-03-27T07:43:27.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2096" PostTypeId="2" ParentId="2094" CreationDate="2015-03-24T17:25:18.513" Score="6" Body="&lt;p&gt;The reasons are the same as extending the binomial formula to non-positive exponents (Newton binomial). It involves generalized binomial coefficients which cannot be expressed in factorials. Binomial formula with non-integer exponent arises, for example in the computation of the arc length of an ellipse, and in other natural problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first part of the survey by J. Lagarias, Euler's constant, in Bull AMS, 50 (2013) 527-628 (freely available online) contains a detailed analysis of Euler's work, and in particular addresses the need to interpolate various functions of integers with analytic functions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-03-25T02:27:09.150" LastActivityDate="2015-03-25T02:27:09.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2098" PostTypeId="2" ParentId="2094" CreationDate="2015-03-26T01:28:29.713" Score="7" Body="&lt;p&gt;&lt;strong&gt;Euler's motivation&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;R. Hilfer on pg. 18 of &quot;&lt;a href=&quot;http://www.icp.uni-stuttgart.de/~hilfer/publikationen/pdfo/ZZ-2008-ATFaA-17.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Threefold Introduction to Fractional Derivatives&lt;/a&gt;&quot; states, &quot;Derivatives of non-integer (fractional) order motivated Euler to introduce the Gamma function ....&quot; Euler introduced in the same reference given by Hilfer essentially &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related &lt;a href=&quot;https://math.stackexchange.com/questions/1537/why-is-eulers-gamma-function-the-best-extension-of-the-factorial-function-to/133194#133194&quot;&gt;MSE-Q&amp;amp;A&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-03-26T22:35:29.847" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2100" PostTypeId="1" AcceptedAnswerId="2104" CreationDate="2015-03-27T15:11:21.853" Score="7" ViewCount="624" Body="&lt;p&gt;What are the earliest recorded acknowledgements of the concept that motivates time zones - that the sun and other celestial objects appear in different parts of the sky to people at different longitudes, observing at the same moment?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would imagine that this would come out as a theoretical consequence of understanding that the world is round before it would be practically observed, since the latter requires either long-distance near-instantaneous communications or some sort of sun-independent clock that you can bring from place to place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question was motivated by consideration of a &lt;a href=&quot;https://judaism.stackexchange.com/questions/56811/is-there-any-evidence-of-time-zones-in-talmud-tanaim-and-amoraim-only&quot;&gt;question&lt;/a&gt; on Mi Yodeya asking for evidence of such as realization in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Talmud&quot; rel=&quot;nofollow noreferrer&quot;&gt;Talmud&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2019" LastEditorUserId="55" LastEditDate="2020-04-03T01:33:00.347" LastActivityDate="2020-04-03T01:33:00.347" Title="When was the issue of time zones at different longitudes first described?" Tags="|discoveries|astronomy|time|geography|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="2101" PostTypeId="1" CreationDate="2015-03-27T19:17:15.560" Score="8" ViewCount="3301" Body="&lt;p&gt;Why did Laplace invent the Laplace transform? What was the context? Was it before Fourier or after? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="62" LastEditDate="2015-03-27T20:46:06.827" LastActivityDate="2015-03-28T19:35:32.290" Title="Why did Laplace invent the Laplace transform?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2102" PostTypeId="2" ParentId="2101" CreationDate="2015-03-28T19:35:32.290" Score="9" Body="&lt;p&gt;Laplace &quot;invented&quot; Laplace transform for applications to probability, namely to prove the special case of what is known now as the central Limit theorem (1785).&#xA;According to Wikipedia, he used a special case that is called the $z$-transform nowadays (another, more common name is generating function). He also used other forms of the same idea, for example what is called Mellin's transform today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In no way was Laplace the first. Generating functions  can be found in the work on de Moivre on combinatorics/probability (Doctrine of chance, 1718). This is the earliest use of the general idea of harmonic analysis that I know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laplace transform (in the modern sense) was probably used for the first time by Euler in 1744 (again, the information from Wikipedia).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this was long time before Fourier whose Analytic theory of heat was published in 1822. (There was a very long delay before the publication, but certainly Fourier made all these inventions in 19th century, until 1801 he had more important things to do:-).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-03-28T19:35:32.290" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2103" PostTypeId="2" ParentId="2100" CreationDate="2015-03-29T14:16:04.173" Score="5" Body="&lt;p&gt;This is an interesting question because the answer depends on how you actually conceptualise time. In the ancient world, the usual system of time keeping involved “seasonal hours”. This means that the time from sunrise to sunset is divided into 12 equal “hours of the day” and the time from sunset to sunrise is divided into 12 equal “hours of the night”. The time of sunset will be (by definition) “the beginning of the first hour of the night” at all locations on Earth. The &quot;time&quot; of sunset is thus the same everywhere and at every time of the year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, for astronomical calculations the ancients also had the concept of “equinoctial hours”, where the time from midnight to midnight (where midnight means the halfway point between sunset and sunrise) is divided into 24 equal hours (as in modern clock time). In this system sunset will be at exactly the same time in all locations on the same latitude, but will be different at different latitudes. This fact was used by Ptolemy when he defined the latitude of different locations not in degrees north or south of the equator, but by indicating the maximum local length of daylight in equinoctial hours (and subdivisions of the same). Similarly, he was fully aware that astronomical phenomena like equinoxes or new moons occur at different times at different longitudes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that Ptolemy had the mathematical and astronomical knowledge that could have enabled him to slice the globe into 24 equal time zones, counting full hours east and west of Alexandria. But there was no practical advantage in doing this, especially since for astronomers it is much more useful to know the exact longitude (in degrees, minutes, seconds…) of a locality. A system of time zones, and of standard time within each zone, becomes of interest only with the invention of the railway and other means of rapid transportation.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-03-29T14:24:17.873" LastActivityDate="2015-03-29T14:24:17.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2104" PostTypeId="2" ParentId="2100" CreationDate="2015-03-29T19:57:34.117" Score="5" Body="&lt;p&gt;The first written account of the phenomenon might be due to Eratosthenes (276-195 BC), the head of Alexandrian Museum and contemporary of Archimedes, with whom he corresponded. Eratosthenes is credited with proposing to use longitude and latitude to specify locations on Earth, an idea &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Longitude1.html&quot; rel=&quot;nofollow&quot;&gt;developed further by Hipparchus&lt;/a&gt; (190-120 BC), &quot;the father of astronomy&quot;, and later used by Ptolemy in his maps. Hipparchus appears to be the first to suggest explicitly that local time determines longitude. Interestingly enough, Ctesibius (285–222 BC) invented the first fairly precise and constantly working clock, the &lt;a href=&quot;http://www.mlahanas.de/Greeks/Clocks.htm&quot; rel=&quot;nofollow&quot;&gt;clepsydra&lt;/a&gt;, around the same time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is likely that differences in local occurrences of astronomical events were noticed much earlier by seafarers. &lt;a href=&quot;http://www.pbs.org/wgbh/nova/ancient/secrets-of-ancient-navigators.html&quot; rel=&quot;nofollow&quot;&gt;NOVA mentioned&lt;/a&gt; in their special that Phoenicians knew, possibly as far back as late second millenium BC, that &quot;at any one time in the year at any one point on the globe, the sun and stars are found above the horizon at certain fixed &quot;heights&quot;—a distance that mariners can measure with as simple an instrument as one's fingers&quot;, and used it for navigation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, there was little systematic development of navigational uses of local time until the 16-th century. In 1502 &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_longitude#Problem_of_longitude&quot; rel=&quot;nofollow&quot;&gt;Amerigo Vespucci wrote&lt;/a&gt; &quot;I maintain that I learned [my longitude]... by the eclipses and conjunctions of the Moon with the planets&quot;, and Johannes Werner developed the method systematically in his 1514 commentary to a translation of Ptolemy's Geography. Subsequent proposals for longitude determination based on astronomical observations were given by Galileo and Halley among others. However, the calculations involved were too complicated for regular use, and the issue came to be known as &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_longitude#Problem_of_longitude&quot; rel=&quot;nofollow&quot;&gt;the problem of longitude&lt;/a&gt;. Spanish kings even offered a prize for solving it in 1567, and increased it in 1598. The idea of using chronometers, which ultimately resolved the longitude problem, is usually attributed to &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Longitude1.html&quot; rel=&quot;nofollow&quot;&gt;Gemma Frisius&lt;/a&gt; (1508-1555), but it gained little currency until Huygens built precise pendulum clocks in late 17th century.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-01T00:37:02.390" LastActivityDate="2015-04-01T00:37:02.390" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2105" PostTypeId="1" AcceptedAnswerId="2106" CreationDate="2015-03-31T00:07:49.300" Score="67" ViewCount="8476" Body="&lt;p&gt;Years ago, I read a story about a mathematician who found a numerical counterexample to some conjecture long believed to be true. He gave a talk during which he didn't utter a single word but simply wrote out the arithmetic computation on the board and, as I recall the narrative, he &quot;sat down to thunderous applause&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone happen to know the story I'm talking about? &lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="26" LastEditDate="2015-03-31T14:13:21.373" LastActivityDate="2018-05-10T05:30:44.960" Title="What's the famous story about a mathematician who gave a talk without saying a word?" Tags="|biographical-details|discoveries|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2106" PostTypeId="2" ParentId="2105" CreationDate="2015-03-31T01:07:36.590" Score="84" Body="&lt;p&gt;You are most likely referring to the 1903 presentation by American mathematician &lt;a href=&quot;http://en.wikipedia.org/wiki/Frank_Nelson_Cole&quot; rel=&quot;noreferrer&quot;&gt;Frank Cole&lt;/a&gt;. The original false conjecture was that the 67-th Mersenne number $M_{67}:=2^{67}-1$ is prime, and it goes back to the preface to &lt;a href=&quot;https://primes.utm.edu/mersenne/&quot; rel=&quot;noreferrer&quot;&gt;Mersenne's own&lt;/a&gt; Cogitata Physica-Mathematica (1644). However, Cole was already confirming rather than disproving, that $M_{67}$ is composite was shown indirectly by Lucas in 1876, using what is now known as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Lucas_primality_test&quot; rel=&quot;noreferrer&quot;&gt;Lucas primality test&lt;/a&gt;. The dramatic account of Cole's presentation is due to Eric Temple Bell, who attended it. But although Bell's account is often repeated, and embellished, there are reasons to doubt its accuracy, see below.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bell writes in his 1951 book &lt;a href=&quot;https://archive.org/details/MathematicsQueenAndServantOfScience&quot; rel=&quot;noreferrer&quot;&gt;Mathematics; Queen and Servant of Sciences&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;At the October, 1903, meeting in New York of the American Mathematical Society, Cole had a paper on the program with the modest title On the factorization of large numbers. When the chairman called on him for his paper, Cole—who was always a man of few words—walked to the board and, saying nothing, proceeded to chalk up the arithmetic for raising $2$ to the sixty-seventh power. Then he carefully subtracted $1$. Without a word, he moved over to a clear space on the board and multiplied out, by longhand,&#xA;      $193,707,721\times 761,838,257,287.$&#xA;  The two calculations agreed. … For the first and only time in record, an audience of the American Mathematical Society vigorously applauded the author of a paper delivered before it. Cole took his seat without having uttered a word. Nobody asked him a question.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again according to Bell, when he wrote to Cole in 1911 to ask how long it had taken him to crack $M_{67}:=2^{67}-1$, Cole reportedly answered: ‘three years of Sundays.’ However, according to &lt;a href=&quot;http://rutherfordjournal.org/article030105.html&quot; rel=&quot;noreferrer&quot;&gt;Corry's article&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;Historians of mathematics tend to distrust the historical reliability of most of Bell’s accounts, and in this case there are good reasons to stick to this habitude. For one thing, the Bulletin of the American Mathematical Society records the talks presented at its meeting of December 31, 1903, in New York, including Cole’s, precisely with the name mentioned by Bell. The text is much more elaborate than simply two arithmetic operations whose results are equated, and it contains some interesting ideas about the importance of the result and about how Cole went about finding the factors involved in his calculation (more on this below). One may certainly agree that Cole deserved the standing ovation, and indeed the ovation may have actually taken place. None of this, however, is mentioned in the Bulletin. As for the amount of time spent on the calculation, there seems to be no other source of information about this than Bell. His account, at any rate, became an accepted mathematical urban legend that has been repeated over and over again, often extending the three years of Bell to ‘twenty years of Sunday afternoons’...&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Briefly stated, Cole relied on techniques such as introduced by Legendre and used existing tables of quadratic remainders... He discussed thoroughly the possible candidates of factors obtained with the help of this technique, together with some specific considerations for the case in point, and gradually focused on a reduced number of candidates which he tried one by one until he found the result. Cole was aware of Lucas’ announcement that  $2^{67}-1$ and $2^{89}-1$ are composite... From all we know, the calculations he did, whether they took three years of Sunday afternoons or not, were done manually and without the aid of any mechanical device.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-10T05:30:44.960" LastActivityDate="2018-05-10T05:30:44.960" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2107" PostTypeId="2" ParentId="98" CreationDate="2015-03-31T10:21:41.777" Score="7" Body="&lt;p&gt;I remember reading something about it in Bertrand's Russell's &quot;A History of Western Philosophy&quot;. Although it's a disputed work in its historical rigour, it does point out some predecessors to Darwin's theory of evolution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding Anaximander:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There was evolution also in the animal kingdom. Living creatures arose&#xA;  from the moist element as it was evaporated by the sun. Man,  like&#xA;  every other animal, was descended from fishes. He must be  derived&#xA;  from animals of a different sort, because, owing to his  long infancy,&#xA;  he could not have survived, originally, as he is now.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another quaint one regarding Empedocles:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He knew that there is sex in plants, and he had a theory (some-what&#xA;  fantastic, it must be admitted) of evolution and the survival  of the&#xA;  fittest. Originally, &quot;countless tribes of mortal creatures were &#xA;  scattered abroad endowed with all manner of forms, a wonder to &#xA;  behold.&quot; There were heads without necks, arms without shoulders,  eyes&#xA;  without foreheads, solitary limbs seeking for union. These  things&#xA;  joined together as each might chance ; there were shambling  creatures&#xA;  with countless hands, creatures with faces and breasts  looking in&#xA;  different directions, creatures with the bodies of oxen  and the faces&#xA;  of men, and others with the faces of oxen and the  bodies of men.&#xA;  There were hermaphrodites combining the natures  of men and women, but&#xA;  sterile. In the end, only certain forms  survived.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's the full source of the text:&#xA;&lt;a href=&quot;https://archive.org/details/westernphilosoph035502mbp&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/westernphilosoph035502mbp&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2037" LastEditorUserId="26" LastEditDate="2015-03-31T14:16:31.087" LastActivityDate="2015-03-31T14:16:31.087" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2108" PostTypeId="2" ParentId="2093" CreationDate="2015-03-31T21:05:22.583" Score="15" Body="&lt;p&gt;The history of the idea underlying the short/long/synthetic division turned out to be far more complicated than I expected, somewhat reminiscent of the history of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, with no single inventor. According to the &lt;a href=&quot;http://www.angelfire.com/ma4/khmath/timeline.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Angelfire timeline&lt;/a&gt; the modern long division symbol of English-speaking countries is first used in the 1888 teacher's edition of The Elements of Algebra by G. A. Wentworth. However, &lt;a href=&quot;http://en.wikipedia.org/wiki/Long_division#Notation_in_non-English-speaking_countries&quot; rel=&quot;nofollow noreferrer&quot;&gt;as Wikipedia attests&lt;/a&gt;, its use is far from universal even today. The arrangement of calculations grew out of two much earlier preceding methods, the Italian and the galley. Originally, the divisor was written on one side of the dividend, and the quotient on the other. Frank Swetz suggests in &lt;a href=&quot;http://books.google.com/books/about/Capitalism_and_Arithmetic.html?id=xu_SVhW9DK0C&quot; rel=&quot;nofollow noreferrer&quot;&gt;Capitalism and Arithmetic&lt;/a&gt; that the quotient remained on the right by custom after the galley method gave way to the Italian method in the 17th century. And only with the advent of decimal division, and the greater need to align the decimal places, the quotient was moved to above the dividend.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A detailed account of early paper-and-pencil division algorithms is given on &lt;a href=&quot;https://pballew.blogspot.com/2019/11/some-notes-on-division-and-its-history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pat Ballew's blog&lt;/a&gt; and in &lt;a href=&quot;https://www.semanticscholar.org/paper/An-Historical-Analysis-of-the-Division-Concept-and-Windsor-Booker/1fbf13e7fe8d7549a870cf1fd430af5b4d8691d4&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Historical Analysis of the Division Concept and Algorithm by Windsor and Booker&lt;/a&gt;. Not all of them are &quot;equivalent&quot; to long division even speaking broadly, and many of them survived into 20-th century. For example, one method described by the famous Fibonacci in his Liber Abaci of 1202, required prime factoring the dividend first. The closest predecessor of the modern long division is the Italian method, which simply omits writing the partial products, so it is closer to the short division. It was first described by Calandri in a 1491 book, and nicknamed &quot;danda&quot; (&quot;giving&quot;) by Cataneo in 1546. Cataneo noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder. Henry Briggs, the first professor of geometry at Gresham college is credited with transforming danda into long division in early 1597. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Danda is more economical in terms of paper space, an important consideration at the times when paper was expensive, but the galley or scratch method was even better on this score, as parts of the calculation could be successively erased and written over. According to &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History of Mathematical Notations&lt;/a&gt;:&quot;&lt;em&gt;It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure&lt;/em&gt;&quot;. Apparently, Al-Khwarizmi, the founder of algebra, also used this method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://new.math.uiuc.edu/im2008/rogers/algebra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some authors&lt;/a&gt; credit medieval Arabic mathematician al-Samawal (1130-1180) for inventing long division, see also &lt;a href=&quot;http://books.google.com/books/about/The_History_of_Mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Victor Katz's book (7.2.3)&lt;/a&gt;. Al-Samawal was first to use tables of coefficients to write and perform calculations with polynomials, he even allowed negative powers. He described a division method underlying the polynomial long division, although his record keeping is more akin to what is now called synthetic division. He also noted the analogy between his way of writing polynomials and the decimal positional notation, and transferred his algorithms to decimal numbers by simply replacing the variable with &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, so he may well be the first to provide a mathematical justification for a positional division algorithm.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are even older twists to the story. Many advantages of positional notation are delivered by the abacus, a manual counting device representing numbers by successive rows originally filled with beads or pebbles, which has the added advantage of needing no paper at all. &lt;a href=&quot;https://skeptics.stackexchange.com/questions/15130/did-only-a-handful-of-people-in-europe-know-how-to-do-division-before-the-13th-c&quot;&gt;Gerbert improved the abacus&lt;/a&gt; by placing symbols at the top of each column, and developed a division algorithm for it, first in Europe, around 972-982 AD, while working at a cathedral school at Rheims. Gerbert is better known for being the only mathematician to be eventually elected the pope, Sylvester II. Division on the abacus became widely known in Europe long before the Arab numerals were introduced. See how an &lt;a href=&quot;http://scienceblogs.com/goodmath/2006/09/25/division-on-the-abacus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;abacus algorithm compares to long division&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The abacus itself goes much further back. Sumerian abacus appeared as far back as 2700–2300 BC, its successive rows reflecting the successive orders of magnitude of the sexagesimal number system. Babylonians likely derived their addition and subtraction algorithms from abacus, and developed one for multiplication. However, according to &lt;a href=&quot;http://books.google.com/books/about/The_History_of_Mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Victor Katz's book (1.2.2)&lt;/a&gt;, they had no division algorithm, and used tables of reciprocals and multiplication instead. So long division doesn't go that far back. It is unclear when a division algorithm for the abacus was first invented, a likely place is China after 190 AD. &lt;a href=&quot;http://en.wikipedia.org/wiki/Abacus#Chinese&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chinese abacus is called suanpan&lt;/a&gt;, and originally used the hexadecimal system. Roman abacus is sufficiently similar to suanpan to suggest transmission, albeit unconfirmed, it is the Roman abacus that Gerbert improved upon.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-14T09:31:59.527" LastActivityDate="2020-04-14T09:31:59.527" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2109" PostTypeId="2" ParentId="499" CreationDate="2015-04-01T01:47:57.230" Score="4" Body="&lt;p&gt;Up to notational conventions polynomial division algorithm was first described by an Arabic mathematician al-Samawal (1130-1180), who can also be credited with defining what we call polynomials today, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2093/who-invented-short-and-long-division/2108#2108&quot;&gt;Who invented short and long division?&lt;/a&gt; In light of that the factorization theorem does not predate polynomial division.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-04-01T01:47:57.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2110" PostTypeId="1" AcceptedAnswerId="2112" CreationDate="2015-04-01T23:42:45.063" Score="14" ViewCount="1497" Body="&lt;p&gt;$\sin^{-1} x$ means the inverse of $\sin x$ (it is also often called $\arcsin x$), but it can fairly easily be confused with $\sin(x)^{-1}$. Why is it used, when $\arcsin x$ is easier to type &lt;em&gt;and&lt;/em&gt; is not easily confused with other things?&lt;/p&gt;&#xA;" OwnerUserId="2053" LastEditorUserId="26" LastEditDate="2015-04-02T08:13:51.013" LastActivityDate="2017-06-22T20:55:32.563" Title="Why is the notation $\sin^{-1} x$ so common?" Tags="|mathematics|notation|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2111" PostTypeId="2" ParentId="2110" CreationDate="2015-04-02T00:26:37.410" Score="4" Body="&lt;p&gt;Florian Cajori has it in &lt;em&gt;A History of Mathematical Notations Vol. II&lt;/em&gt; Paragraph 533 that &quot;John Herschel's notation for inverse functions $\sin^{-1}x$,... etc., was published by him in the &lt;em&gt;Philosophical Transactions of London&lt;/em&gt;, for the year 1813.&quot; Cajori goes on to indicate that Hershel gave specific instructions to not confuse the notation with the multiplicative inverse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His justification for the notation, revolves around a &quot;symmetry&quot; with other notations like, for example, writing $\Delta\Delta\Delta x$ as $\Delta^3x$, and he lists several others. I refer you to the literature for the full description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now in the same, Cajori notes that &quot;Some years later, Hershel explained that in 1813 he used ...[these notations]... as he then supposed for the first time. The work of a German analyst, Burmann, has however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is much more detail in this reference, in particular explaining the logic of the notation.   &lt;/p&gt;&#xA;" OwnerUserId="110" LastActivityDate="2015-04-02T00:26:37.410" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2112" PostTypeId="2" ParentId="2110" CreationDate="2015-04-02T00:40:34.107" Score="7" Body="&lt;p&gt;If you want someone to blame blame John Herschel, son of the famous astronomer William Herschel. The father in 1781 gave us the planet Uranus, the &lt;a href=&quot;http://www.pballew.net/arithm14.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;son in 1813 gave us the dubious symbol $\sin^{-1}(x)$&lt;/a&gt; for the inverse sine. As precedent, he cited the 'operator powers' notation, like ${d^n}f$ from calculus, so according to Herschel we should write $d^{-1}f$ for $\int f$. &quot;&lt;em&gt;The symmetry of this notation, and above all the new and most extensive views it opens seem to authorize its universal adoption&lt;/em&gt;&quot;, he wrote. Herschel was aware of the potential confusion with arithmetical reciprocals, but apparently thought that the benefits outweighed the downsides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First notation for the inverse sine was proposed by Daniel Bernoulli in 1729, it was $AS$, Euler shortened it to $A$ alone. Lagrange was first to introduce $\operatorname{arc}.\!\sin$, and with the dot eventually removed it became the standard in the continental Europe. And US and Britain would have followed suit too... if it wasn't for Herschel Jr.! When he published Traite de la Luminere in French in 1829 he himself used Lagrange's $arc.sin$. Perhaps it was a leftover of the Newton-Leibniz priority dispute and/or cultural rivalry between Britain and the continent, but British often insisted on being notationally different, feet and pounds are another example, as are the &lt;a href=&quot;https://hsm.stackexchange.com/questions/2093/who-invented-short-and-long-division/2108#2108&quot;&gt;long division symbol&lt;/a&gt;, $\tan$ vs $\text{tg}$ for tangent, and $\sinh$ vs $\text{sh}$ for hyperbolic sine.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-04-07T08:30:13.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2113" PostTypeId="2" ParentId="2110" CreationDate="2015-04-02T02:12:59.140" Score="1" Body="&lt;p&gt;Notations of this kind are determined by high school textbooks. I noticed that&#xA;$\sin^{-1}$ is common in the English-speaking countries, while in many European countries they prefer $\arcsin$. This is consistent with the remark in the end of Conifold's answer about nationalistic reasons for this difference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was educated in Ukraine (then a part of the former Soviet Union) and I have never seen $\sin^{-1}$ before I came to the US. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-04-02T02:12:59.140" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2114" PostTypeId="2" ParentId="2110" CreationDate="2015-04-02T14:26:16.047" Score="11" Body="&lt;p&gt;Here's some detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Herschel's '&lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/103/8.full.pdf&quot; rel=&quot;noreferrer&quot;&gt;On a Remarkable Application of Cotes's Theorem&lt;/a&gt;', &lt;em&gt;Philosophical Transactions&lt;/em&gt;, 1813:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/yMYlr.png&quot; alt=&quot;enter image description here&quot;&gt;'&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss wrote &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/download/PPN236005081/LOG_0029.pdf&quot; rel=&quot;noreferrer&quot;&gt;a review of this article&lt;/a&gt; in the &lt;em&gt;Göttingsche Gelehrte Anzeigen&lt;/em&gt; (reprinted in Werke, vol. 4, p. 361):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/5c3sN.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will translate the last sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What the author says about the notation $\cos^2 A$, which some newer mathematical writers use for the square of $\cos A$, totally against all analogy, as accordingly this should mean the cosine of an angle $= \cos A$, has our total approbation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1817" LastEditorUserId="1817" LastEditDate="2015-09-29T20:16:18.380" LastActivityDate="2015-09-29T20:16:18.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2115" PostTypeId="1" CreationDate="2015-04-02T13:11:44.150" Score="23" ViewCount="2188" Body="&lt;p&gt;I do not remember the name/source of this paradox,but I remember I have discussed this with mathematicians and non mathematicians at least 5 times.&lt;br&gt;&#xA;It goes like this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Every point of a line has length $0$ and every line segment consists&#xA;  of points.     So the length of the line segment is the sum&#xA;  $0+0+\ldots=0$&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which seems like a paradox.&lt;br&gt;&#xA;Of course the number of points on a line segment is uncountable so the &quot;summation&quot; actually does not contain every point,so there is an obvious mistake there and we are done.&lt;br&gt;&#xA;My question is:&lt;br&gt;&#xA;Was this paradox discussed (or answered) before Cantor?&lt;/p&gt;&#xA;" OwnerUserId="2056" OwnerDisplayName="Konstantinos Gaitanas" LastEditorUserId="58" LastEditDate="2015-04-02T16:49:04.200" LastActivityDate="2015-04-03T23:09:05.320" Title="What was the answer to this paradox before Cantor?" Tags="|mathematics|real-analysis|set-theory|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2116" PostTypeId="2" ParentId="2115" CreationDate="2015-04-02T14:20:05.743" Score="20" Body="&lt;p&gt;&lt;A HREF=&quot;https://en.wikipedia.org/wiki/Zeno_of_Elea&quot;&gt;Zeno&lt;/A&gt; (around 500 BC) raised this paradox to argue against the notion of &quot;plurality&quot;, arguing that a belief in the existence of many things rather than only one leads to absurd conclusions: &lt;em&gt;If there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See section 2.2 of &lt;A HREF=&quot;http://plato.stanford.edu/entries/paradox-zeno/#ArgDen&quot;&gt;Zeno's paradoxes&lt;/A&gt; in the Stanford Encyclopedia of Philosophy. The following quote is from a discussion of Zeno's book on paradoxes by &lt;A HREF=&quot;https://en.wikipedia.org/wiki/Simplicius_of_Cilicia&quot;&gt;Simplicius&lt;/A&gt; (who writing one thousand year after Zeno apparently still had his book, which now is lost):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a thing has no magnitude or bulk or mass, it would not exist. For&#xA;  if it should be added to something else that exists, it would not make&#xA;  it any bigger. For if it were of no size and was added, it cannot&#xA;  increase in size. And so it follows immediately that what is added is&#xA;  nothing. But if when it is subtracted, the other thing is no smaller,&#xA;  nor is it increased when it is added, clearly the thing being added or&#xA;  subtracted is nothing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This seems to be the earliest source for the paradox, but not its resolution, which had to wait several millennia for the development of the notion of a distance function, which allows you to compute the length of an uncountable infinity of points.&lt;/p&gt;&#xA;" OwnerUserId="1697" OwnerDisplayName="Carlo Beenakker" LastActivityDate="2015-04-02T14:44:13.240" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2117" PostTypeId="2" ParentId="2115" CreationDate="2015-04-02T15:55:54.887" Score="4" Body="&lt;p&gt;You might look at Galileo's discussion of infinity in  &quot;Discourses and Mathematical Demonstrations Relating to Two New Sciences&quot;: e.g. look up &lt;a href=&quot;http://en.wikipedia.org/wiki/Galileo%27s_paradox&quot; rel=&quot;nofollow&quot;&gt;Galileo's paradox&lt;/a&gt;&quot;. &lt;/p&gt;&#xA;" OwnerDisplayName="Robert Israel" LastActivityDate="2015-04-02T15:55:54.887" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2118" PostTypeId="2" ParentId="2115" CreationDate="2015-04-02T16:17:05.203" Score="2" Body="&lt;p&gt;See the book discussed in &lt;a href=&quot;https://math.stackexchange.com/q/661999/442&quot;&gt;this post&lt;/a&gt; .&lt;br&gt;&#xA; Amir Alexander, &lt;em&gt;Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World&lt;/em&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 17th century there was a dispute about whether a plane region consists of an infinite family of parallel line segments.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Argument against:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do these line segments have width zero?  Then the whole region would have area zero.  Do they have positive width?  Then there can be only finitely many of them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The whole book is interesting.&lt;/p&gt;&#xA;" OwnerUserId="229" OwnerDisplayName="Gerald Edgar" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-04-02T16:17:05.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2122" PostTypeId="2" ParentId="2115" CreationDate="2015-04-02T20:34:51.887" Score="15" Body="&lt;p&gt;Aristotle gave the first systematic rebuttal of Zeno, in particular &lt;a href=&quot;http://www.iep.utm.edu/zeno-par/#H4&quot; rel=&quot;noreferrer&quot;&gt;he wrote in Physics&lt;/a&gt;: &quot;&lt;em&gt;…a line cannot be composed of points, the line being continuous and the point indivisible&lt;/em&gt;&quot;. According to Aristotle, a line can be composed only of smaller, indefinitely divisible lines, and not of points without magnitude. This was the mainstream view until the &quot;dissociation&quot; of continuum by Cantor and Dedekind at the end of 19-th century. Under the new paradigm the paradox was resolved by the Lebesgue measure theory, which postulated that length is only countably additive, so the continuum wide summation of point lengths is invalid. This vindicated Aristotle's view: in terms of magnitude continuum can not be assembled from points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that continuum is a &quot;set of points&quot; leads to a number of conceptual problems, this being one of them. Others include its well-orderability, which strains credulity and implies existence of Lebesgue non-measurable sets, and the undecidability of the continuum hypothesis. A lot of effort was expended, by Zermelo, Lebesgue and others, to reconcile the intuition of continuum with &quot;arithmetical&quot; set theory. However, despite the objections of intuitionists like Hermann Weyl, in the end the benefits to analysis far outweighed the costs for most practitioners. &lt;a href=&quot;http://publish.uwo.ca/~jbell/Hermann%20Weyl.pdf&quot; rel=&quot;noreferrer&quot;&gt;Weyl's views echoed Aristotle's&lt;/a&gt;:&lt;em&gt;&quot;The notion that a set is a “gathering” brought together by infinitely many individual arbitrary acts of selection, assembled and then surveyed as a whole by consciousness, is nonsensical; “inexhaustibility” is essential to the infinite... Exact time- or space-points are not the ultimate, underlying atomic elements of the duration &#xA;or extension given to us in experience.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-03T23:09:05.320" LastActivityDate="2015-04-03T23:09:05.320" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2126" PostTypeId="1" CreationDate="2015-04-04T14:30:38.743" Score="9" ViewCount="230" Body="&lt;p&gt;I was doing some reading on stellar structure, and I noticed the phrase &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Standard_solar_model&quot;&gt;the standard solar model&lt;/a&gt;&quot; used to describe the structure of the Sun, and stars in general. It reminded me of &lt;a href=&quot;https://en.wikipedia.org/wiki/Standard_Model&quot;&gt;the Standard Model&lt;/a&gt; of particle physics. Also, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lambda-CDM_model&quot;&gt;$\Lambda$-CDM model&lt;/a&gt; is sometimes referred to as &quot;the standard model of cosmology.&quot; There are &lt;a href=&quot;https://en.wikipedia.org/wiki/Standard_model_(disambiguation)&quot;&gt;other uses&lt;/a&gt; of the phrase.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who first used the phrase &quot;the standard model&quot; to describe a theory? Was it first applied to the Standard Model of particle physics, or was it used somewhere else?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2020-11-26T03:16:21.783" Title="Who was the first to use the phrase &quot;the standard model&quot; of something?" Tags="|terminology|naming-conventions|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2127" PostTypeId="1" AcceptedAnswerId="2128" CreationDate="2015-04-03T21:01:58.133" Score="13" ViewCount="372" Body="&lt;p&gt;As the LHC begins its second run this year, the physics community seems to be significantly split between those who are confident that SUSY exists and those who believe it doesn't, given its failure to show up at the LEP, Tevatron and LHC during run 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was there also a period of time when there was a significant split in the belief of the Higgs mechanism before its confirmation in 2012 at the LHC?&lt;/p&gt;&#xA;" OwnerUserId="2461" OwnerDisplayName="Larry Harson" LastEditorUserId="26" LastEditDate="2015-04-05T07:43:17.540" LastActivityDate="2022-02-14T21:09:43.107" Title="Was there a period of doubt in the history of the Higgs mechanism?" Tags="|theoretical-physics|particle-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2128" PostTypeId="2" ParentId="2127" CreationDate="2015-04-05T22:32:34.463" Score="10" Body="&lt;p&gt;Not only was there such a period, but it lasted up until the LHC detection of the Higgs boson in 2012, and in a way it continues until today. Higgs boson is the quantum carrier of a scalar field, the Higgs field, and other elementary particles acquire mass through symmetry breaking by interacting with it, this is called the Higgs mechanism. The indirect experimental evidence for Higgs was quite compelling since W and Z bosons were detected at CERN in 1983 with masses as predicted by the theory, and especially after the &lt;a href=&quot;http://arxiv.org/abs/hep-ph/0404165&quot; rel=&quot;nofollow noreferrer&quot;&gt;electroweak precision tests&lt;/a&gt; that followed since 1989. But there were always theoretical issues with this particle (see below), and direct detection proved elusive for about 40 years.&lt;/p&gt;&#xA;&lt;p&gt;Since 1979 this led to the development of a number of alternatives. Although known under the name of &lt;a href=&quot;http://en.wikipedia.org/wiki/Alternatives_to_the_Standard_Model_Higgs#List_of_alternative_models&quot; rel=&quot;nofollow noreferrer&quot;&gt;Higgsless theories&lt;/a&gt; many of them include an effective version of the Higgs boson, and therefore are not ruled out by the LHC detection. Moreover, some physicists almost rooted for Higgs not to be found. They saw finding &lt;a href=&quot;http://www.sciencemag.org/content/315/5819/1657.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Higgs and nothing else&amp;quot;&lt;/a&gt; at LHC as a nightmare scenario, &amp;quot;because that would mean there wouldn't need to be any new physics all the way up to the Planck scale&amp;quot;, according to CERN's Ellis.&lt;/p&gt;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;http://www.extremetech.com/extreme/95097-what-does-a-higgsless-universe-mean-for-science&quot; rel=&quot;nofollow noreferrer&quot;&gt;typical sceptical account&lt;/a&gt; from 2011 pointing out that the predicted particle has not been found after surveying 95% of its suspected energy range. Of course, after the LHC detection the next year few doubt Higgs boson's existence anymore, but the Standard Model makes further claims about it, that are still in dispute. Namely, that Higgs boson is a fundamental particle, like photon or gluon, rather than an effective manifestation of some deeper physics. This is emotionally charged because in the latter case much of its hype, including &amp;quot;the God particle&amp;quot; nickname, would be misplaced.&lt;/p&gt;&#xA;&lt;p&gt;The Higgs mechanism was not immediately accepted after being proposed either. The &lt;a href=&quot;http://en.wikipedia.org/wiki/1964_PRL_symmetry_breaking_papers#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;seminal 1964 papers&lt;/a&gt; by  Brout-Englert, Higgs and Guralnik-Hagen-Kibble, anticipated by Anderson in 1962 in a non-relativistic setting, &amp;quot;were at first largely ignored, because it was widely believed that the (non-Abelian gauge) theories in question were a dead-end, and in particular that they could not be renormalised&amp;quot;. &amp;quot;Essentially no-one paid any attention&amp;quot;  also in 1967, when Weinberg and Salam independently showed how the Higgs mechanism can break the symmetry in Glashow's electroweak theory. &lt;a href=&quot;http://www.nobelprize.org/nobel_prizes/physics/laureates/2004/politzer-lecture.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Politzer&lt;/a&gt;, the mainstream acceptance came only after 1971, when 't Hooft and Veltman showed renormalisability of Yang-Mills gauge fields, and Lee developed and popularized their ideas.&lt;/p&gt;&#xA;&lt;p&gt;In Higgsless theories symmetry breaking is effected by new interactions that manifest above the electroweak scale, rather than by an extra scalar field. The most popular ones appear to be &lt;a href=&quot;http://en.wikipedia.org/wiki/Technicolor_%28physics%29#Introduction&quot; rel=&quot;nofollow noreferrer&quot;&gt;technicolor&lt;/a&gt;, first proposed by Weinberg and Susskind in 1979, and &lt;a href=&quot;https://www.ias.edu/news/press-releases/2012/arkani-hamed&quot; rel=&quot;nofollow noreferrer&quot;&gt;extra-dimensional theories&lt;/a&gt;. Early technicolor was ruled out by the electroweak precision tests, but not the later &amp;quot;walking technicolor&amp;quot; versions, like 2004 one by Sannino and Tuominen, and extra dimensions, first advanced by &lt;a href=&quot;http://arxiv.org/abs/hep-ph/9803315&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arkani-Hamed, Dimopoulos and Dvali&lt;/a&gt; in 1998, remain as popular as ever.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://arxiv.org/abs/1305.5110&quot; rel=&quot;nofollow noreferrer&quot;&gt;This recent paper&lt;/a&gt; gives a detailed analysis of theoretical objections to the Higgs mechanism of the Standard Model, that stimulated doubts and motivated development of alternative theories, from the modern perspective. The principal ones are.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Free parameters problem&lt;/strong&gt;: Neither Higgs field parameters, nor its couplings to other fields can be derived from the Standard Model, they have to be fitted to the experimental data. That includes the mass of the Higgs boson, which is one reason why it took so long to detect it. Because &amp;quot;with four parameters I can fit an elephant&amp;quot;, according to von Neumann's quip, many physicists find the theory distasteful. Similar objections are raised against the string theory.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Scalar field problem&lt;/strong&gt;: This objection goes back at least to 1979, when it was raised by Slavnov. The Higgs field is the only scalar field in the Standard Model, the rest are gauge fields, and in the absence of Higgs they predict a number of zero mass particles, which are not observed. All other known scalar fields are only effective, not fundamental. So to some the Higgs field looks like a &amp;quot;convenient hypothesis to save the appearances&amp;quot; rather than a fundamental object. Moreover, scalar fields in general have serious conceptual problems.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/43303/relationship-between-hierarchy-problem-and-higgs-fine-tuning&quot;&gt;&lt;strong&gt;Fine tuning/ hierarchy/ naturalness problem&lt;/strong&gt;&lt;/a&gt; is perhaps the most severe one, and is a consequence of Higgs field being a scalar field. For scalar particles quantum corrections to the bare mass are particularly large, and 34(!) orders of magnitude of them will have to be canceled out by the bare mass to produce the measured physical mass of &lt;span class=&quot;math-container&quot;&gt;$125$&lt;/span&gt; GeV. This miraculous fine tuning is similar to that in Lorentz's explanation of the Michelson-Morley experiment, where the lengths contract and the clocks retard by just the right amount to make the ether wind undetectable. Both technicolor and extra-dimensional theories were primarily aimed at avoiding this fine-tuning. By the way, supersymmetry was also proposed as a way to protect the small Higgs bare mass from huge quantum corrections.&lt;/p&gt;&#xA;&lt;p&gt;All three issues underscore the idea that the Higgs mechanism has an air of ad hocness to it, and none of them is addressed by the LHC detection. It seems that the Standard Model is following Quantum Mechanics into becoming a victim of its own success. It describes everything we observe so well that there is no hope of experimental guidance to resolving its conceptual problems in the foreseeable future.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="11054" LastEditDate="2022-02-14T21:09:43.107" LastActivityDate="2022-02-14T21:09:43.107" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2129" PostTypeId="1" AcceptedAnswerId="2130" CreationDate="2015-04-06T04:57:46.853" Score="9" ViewCount="8761" Body="&lt;p&gt;There is an anecdote that appears in various popular books on physics, about a physics professor who advised a student of his not to pursue a career in physics since almost all that was to be discovered had been discovered, or something to that effect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, the story unfolds just before 1900 and the discoveries of radioactivity and quantum physics that overturned much of the classical thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the professor and who was the student? I seem to remember the professor and student both being notable physicists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, is this story true or just one of those things that get made up and spread around?&lt;/p&gt;&#xA;" OwnerUserId="2075" LastEditorUserId="26" LastEditDate="2015-04-06T16:16:24.270" LastActivityDate="2019-02-27T16:35:11.520" Title="Who said that essentially everything in theoretical physics had already been discovered?" Tags="|biographical-details|theoretical-physics|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2130" PostTypeId="2" ParentId="2129" CreationDate="2015-04-06T11:47:12.410" Score="10" Body="&lt;p&gt;The anecdote you have in mind concerns &lt;a href=&quot;http://en.wikipedia.org/wiki/Max_Planck&quot;&gt;Max Planck&lt;/a&gt; and his advisor &lt;a href=&quot;http://en.wikipedia.org/wiki/Philipp_von_Jolly&quot;&gt;Philipp von Jolly&lt;/a&gt; who, in 1874, told the young Planck that it was probably not a great idea to study theoretical physics, since there was not much left to do. With the power of hindsight this is of course extremely ironic since Von Jolly was addressing the man who would personally (though reluctantly) play a large role in the quantum revolution of physics in the early 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears that the source of this anecdote is Max Planck himself. In a 1924 lecture, he told the story as follows: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When I began my physical studies [in Munich in 1874] and sought advice from my venerable teacher Philipp von Jolly...he portrayed to me physics as a highly developed, almost fully matured science...Possibly in one or another nook there would perhaps be a dust particle or a small bubble to be examined and classified, but the system as a whole stood there fairly secured, and theoretical physics approached visibly that degree of perfection which, for example, geometry has had already for centuries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I found a little collection of different versions of the story compiled &lt;a href=&quot;http://www.ccl.net/cgi-bin/ccl/message-new?2002+08+30+008&quot;&gt;here&lt;/a&gt;. One of them gives the anecdote as I reproduced it above, with a reference where one can (presumably) find the original lecture by Planck: A 1996 issue of Scientific American. Regrettably, I do not have access to this journal. &lt;a href=&quot;http://www.todayinsci.com/P/Planck_Max/PlanckMax-Quotations.htm&quot;&gt;Another source&lt;/a&gt; which has an identical version refers the reader to a book called &quot;Year Million: Science at the Far Edge of Knowledge&quot;, which consists of essays by scientists and apparently also contains this lecture by Planck. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though I was unable to gain access to this either, I think we can be content that it does exist and that at least Planck himself believed the story to be true. &lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2015-04-06T11:57:07.330" LastActivityDate="2015-04-06T11:57:07.330" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2131" PostTypeId="2" ParentId="2129" CreationDate="2015-04-06T12:00:51.423" Score="2" Body="&lt;p&gt;Max Planck told this story about his teacher von Jolly in a lecture. I found the following reference in the German Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Planck: Wege zur Physikalischen Erkenntnis. Reden und Vorträge, Band 1. Leipzig 1943&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://de.wikipedia.org/wiki/Philipp_von_Jolly#cite_note-5&quot; rel=&quot;nofollow&quot;&gt;http://de.wikipedia.org/wiki/Philipp_von_Jolly#cite_note-5&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(typically for Wikipedia without the page number).&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-04-06T12:16:39.730" LastActivityDate="2015-04-06T12:16:39.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2132" PostTypeId="2" ParentId="2129" CreationDate="2015-04-06T16:28:50.240" Score="7" Body="&lt;p&gt;The title question is different from the one in the text so I'll add a couple of things. Professor Jolly was not alone in his sentiment. Albert Michelson of the Michelson-Morley experiment fame said in 1894 that &quot;it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice&quot;. And added, &lt;a href=&quot;http://www.philsoc.org/1999Fall/2107minutes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to some sources&lt;/a&gt;, &quot;An eminent physicist remarked that the future truths of physical science are to be looked for in the sixth place of decimals&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;eminent physicist&quot; is believed to be Lord Kelvin, who on April 27, 1900 gave a now infamous speech titled &lt;a href=&quot;http://physics.about.com/od/physicshistory/p/KelvinsDarkClouds.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Nineteenth-Century Clouds over the Dynamical Theory of Heat and Light&quot;&lt;/a&gt;, into which the above quote is often retroactively inserted. The two &quot;clouds&quot; were the absence of the ether wind, and the ultraviolet catastrophe in the black body radiation, which led to the special relativity, and the quantum mechanics, respectively. Some add derisively that if Kelvin also mentioned the anomalous precession of Mercury he could go three for three with general relativity.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These stories are used in many popular books to create an impression of staleness and complacency in the late 19th century physics, before Einstein and Planck &quot;set everybody straight&quot;. However, [modern historians are skeptical][1] of such extrapolations. Most theoretical phycisists did not share Michelson's view in 1894, and Michelson himself sounded a different tune in 1902, see &lt;a href=&quot;https://books.google.com/books?id=ELrFDIldlawC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kragh's Quantum Generations, p. 4&lt;/a&gt;:&quot;the day seems not far distant when the converging lines from many apparently remote regions of thought will meet on ... common ground.&quot; And it wasn't because of Planck's 1900 discovery, Michelson was inspired instead by &quot;one of the grandest generalizations of modern science ... that all the phenomena of the physical universe are only different manifestations of the various modes of motion of one all-pervading substance — the ether.&quot; &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-02-27T09:05:14.323" LastActivityDate="2019-02-27T09:05:14.323" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="2133" PostTypeId="1" CreationDate="2015-04-06T21:49:30.180" Score="10" ViewCount="619" Body="&lt;p&gt;The proofs of quadratic reciprocity based on quadratic &lt;a href=&quot;http://en.wikipedia.org/wiki/Gauss_sum&quot; rel=&quot;noreferrer&quot;&gt;Gaussian sums&lt;/a&gt; involve sums of the form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\sum_{k=1}^{p-1}\left(\frac k p \right)\zeta^{ak}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where $\left(\frac k p \right)$ is a Legendre symbol and $\zeta$ is a $p$-th root of unity. Where were these sums considered (if in the DA, which section) ? Can someone summarize what motivated these seemingly arbitrary expressions?&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2015-05-07T15:09:20.407" Title="What motivated Gauss' quadratic sums?" Tags="|mathematics|number-theory|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2134" PostTypeId="2" ParentId="2133" CreationDate="2015-04-07T13:47:56.207" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Can someone summarize what motivated these &lt;em&gt;seemingly&lt;/em&gt; arbitrary expressions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They are far from arbitrary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because quadratic reciprocity is about whether a prime $p$ is a square modulo another prie $q$, it is natural to look for an actual square-root when it is (an idea that was already familiar to Euler). The use of quadratic Gauss sums is to construct the desired square-root (but possibly in a finite extension). With that explicit goal in mind, it is in fact not so hard to see patterns emerging when doing computations, especially if you have been thinking long and hard about Legendre's symbols and even if you don't know the answer beforehand. I have sometimes asked students to guess them for instance and they do succeed (occasionally).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But though the students do not know the result, they have a tremendous advantage: they are conducting the guessing in a course on finite extension of finite fields or algebraic number theory. That idea, &lt;em&gt;i.e&lt;/em&gt; the idea of looking for the square-root in extensions (of the integers or of finite fields) and of studying the behavior of primes in such extensions, was absolutely groundbreaking.&lt;/p&gt;&#xA;" OwnerUserId="189" LastActivityDate="2015-04-07T13:47:56.207" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2135" PostTypeId="1" AcceptedAnswerId="2136" CreationDate="2015-04-07T16:17:55.197" Score="18" ViewCount="2253" Body="&lt;p&gt;My question is why has Jean Buridan faded into obscurity while Newton is venerated as a God by scientists? Here is a description of Buridan's impetus theory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The concept of inertia was alien to the physics of Aristotle.&#xA;Aristotle, and his peripatetic followers, held that a body was only&#xA;maintained in motion by the action of a continuous external force.&#xA;Thus, in the Aristotelian view, a projectile moving through the air&#xA;would owe its continuing motion to eddies or vibrations in the&#xA;surrounding medium, a phenomenon known as antiperistasis. In the&#xA;absence of a proximate force, the body would come to rest almost&#xA;immediately.&lt;/p&gt;&#xA;&lt;p&gt;Jean Buridan, following in the footsteps of John Philoponus and&#xA;Avicenna, proposed that motion was maintained by some property of the&#xA;body, imparted when it was set in motion. Buridan named the&#xA;motion-maintaining property impetus. Moreover, he rejected the view&#xA;that the impetus dissipated spontaneously (this is the big difference&#xA;between Buridan's theory of impetus and his predecessors), asserting&#xA;that a body would be arrested by the forces of air resistance and&#xA;gravity which might be opposing its impetus. Buridan further held that&#xA;the impetus of a body increased with the speed with which it was set&#xA;in motion, and with its quantity of matter. Clearly, Buridan's impetus&#xA;is closely related to the modern concept of momentum. Buridan saw&#xA;impetus as causing the motion of the object. Buridan anticipated Isaac&#xA;Newton when he wrote:&lt;/p&gt;&#xA;&lt;p&gt;...&lt;em&gt;after leaving the arm of the thrower, the projectile would be moved by n impetus given to it by the thrower and would continue to be moved as long as the impetus remained stronger than the resistance, and would be of infinite duration were it not diminished and corrupted by a contrary force resisting it or by something inclining it to a contrary motion&lt;/em&gt; (Questions on Aristotle's Metaphysics XII.9).&lt;/p&gt;&#xA;&lt;p&gt;Buridan used the theory of impetus to give an accurate qualitative&#xA;account of the motion of projectiles but he ultimately saw his theory&#xA;as a correction to Aristotle, maintaining core peripatetic beliefs&#xA;including a fundamental qualitative difference between motion and&#xA;rest.&lt;/p&gt;&#xA;&lt;p&gt;The theory of impetus was also adapted to explain celestial phenomena&#xA;in terms of circular impetus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This sounds a lot to me like some of the theories that made Newton a household name.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jean_Buridan&quot; rel=&quot;noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Jean_Buridan&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2094" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2024-02-19T11:35:45.290" Title="Why don't we learn Buridan's laws of motion?" Tags="|physics|discoveries|classical-mechanics|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2136" PostTypeId="2" ParentId="2135" CreationDate="2015-04-07T20:22:31.407" Score="20" Body="&lt;p&gt;The next to last sentence has all the reasons in a nutshell:&quot;Buridan used the theory of impetus to give an accurate qualitative account of the motion of projectiles but he ultimately saw his theory as a correction to Aristotle&quot;. Buridan's account, as Aristotle's or Avicenna's before him, was qualitative, he never put it into equations, which would allow for predictions and experimental verification. It only applied to projectiles, there was no notion of universally valid dynamical laws that governed motion in general, so no &quot;laws of motion&quot;. There was a separate &quot;circular impetus&quot; to explain celestial motions, and &quot;natural places&quot; to explain gravity on Earth. And finally, being a correction to Aristotle, Buridan's theory retained its central problem, the murky and/or erroneous relation between speed and force, with little understanding of the role of acceleration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This being said, I do agree that the impetus theory played a positive role in the genesis of classical mechanics, and deserves to be better known. It directly &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;influenced Descartes&lt;/a&gt;, for example, and played a role in the discovery of the conservation of momentum law. Its roots can be traced back to Hipparchus, the father of astronomy, and medieval authors&#xA;learned it from a commentary of &lt;a href=&quot;http://self.gutenberg.org/articles/John_Philoponus#cite_note-3&quot; rel=&quot;noreferrer&quot;&gt;John Philoponus&lt;/a&gt;, who by the way also anticipated the modern idea of empirical inquiry. Avicenna introduced the idea that projectile impetus can be self-degrading, which makes his description comparable to the modern one in the decelerating frame attached to the projectile. &lt;a href=&quot;http://muse.jhu.edu/journals/journal_of_the_history_of_ideas/v064/64.4franco.pdf&quot; rel=&quot;noreferrer&quot;&gt;Franco's paper&lt;/a&gt; gives a good overview of the history of the impetus theory.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-04-08T01:08:22.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2137" PostTypeId="2" ParentId="2135" CreationDate="2015-04-07T20:30:53.103" Score="13" Body="&lt;p&gt;I have not read Buridan, but I am sure he was a philosopher, like Aristotle. The laws of nature are usually not named after philosophers. Philosophers can state all kinds of opinions, but this does not really add to the body of positive knowledge about nature. The difference between Newton and Buridan, is that Newton not just stated an opinion, but developed a useful theory which can predict the motion of planets, and projectiles and almost everything else. Without this,&#xA;philosophical opinions about nature are worthless. (Even if the later development of science later proves that philosophers accidentally guessed something correctly).&lt;/p&gt;&#xA;&lt;p&gt;You may compare this with atoms. Democritus and Leucippus speculated about some atoms. There was no proofs whatsoever and no concrete predictions which could be tested. Other philosophers speculated that atoms do not exist. Such speculations can go on forever without any useful consequences. When atoms were finally incorporated into a scientific theory (in the end of 19th century) nobody teaches this as a &amp;quot;Leucippus-Democritus theory&amp;quot;. Same with Buridan.&lt;/p&gt;&#xA;&lt;p&gt;There are many cases when philosophers had wrong opinions, the most famous examples are Aristotle and Hegel (but also Marx and Engels:-) Sometimes they accidentally guess. The problem with philosophy is that it has no means to test, to verify a statement.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="10608" LastEditDate="2024-02-19T11:35:45.290" LastActivityDate="2024-02-19T11:35:45.290" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2138" PostTypeId="1" CreationDate="2015-04-08T03:39:21.843" Score="6" ViewCount="6999" Body="&lt;p&gt;Recently read the book &quot;Gravity&quot; by George Gamow, in which he says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For centuries Aristotelian philosophy and scholasticism dominated human thought. Scientific questions were answered by dialectic arguments (i.e. by just talking), and no attempt was made to check, by direct experiments, the correctness of the statements made.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I was wondering if there are any contemporary accounts of the realization of the horrendous mistake that was made, lamenting the wasted centuries of unquestioning acceptance of the Aristotlean view of science. I mean, weren't people of Galileo's time shocked at the thought that something so simple yet so wrong lay right beneath their noses for centuries?!&lt;/p&gt;&#xA;" OwnerUserId="2097" LastEditorUserId="26" LastEditDate="2015-04-08T09:41:03.127" LastActivityDate="2015-04-12T21:08:19.350" Title="How did people react to the realization that Aristotle's ideas had gone without question for way too long?" Tags="|physics|experimental-physics|philosophy-of-science|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2139" PostTypeId="1" CreationDate="2015-04-08T12:31:32.053" Score="4" ViewCount="629" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Life of &lt;strong&gt;Galileo Galilei&lt;/strong&gt; was not simple, predictable, or rational. Being as great as he was, Galileo, for example, did not accept Kepler's laws of planetary motion. At the same time, he believed with all his heart in the Copernicus`s theory of the solar system. Because of his personal belief in the heliocentric doctrine, Galileo was heavily criticized by the church, and in 1616 was sentenced by the Inquisition to life imprisonment in his home in Florence.&lt;/p&gt;&#xA;&lt;p&gt;It was outrageous that a person such as Galileo was forced to spend the remainder of his days in house exile. Nevertheless, he continued to work and write books while in exile. One of the books was the famous &amp;quot;Dialogue on the Two Chief World Systems.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Galileo died in 1642 after having suffered blindness in the last four years of his life and leaving his inventions, discoveries, books, and his devotion to science to future generations, that is, to us.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://cse.ssl.berkeley.edu/lessons/indiv/nellie/galileo.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;[source]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Are there any other examples in the history where a physicist was criticised socially or by his peers for his idea/theory/ work, but was later accepted wholeheartedly by the society?&lt;/p&gt;&#xA;&lt;p&gt;One such instance that I know is the case of &lt;strong&gt;Galileo Galilei&lt;/strong&gt;. I don't know any other example of such a case and I had no idea where to ask. (I am also afraid for the duplication of this question.)&lt;/p&gt;&#xA;" OwnerUserId="2100" OwnerDisplayName="a.s." LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2021-02-25T16:51:12.943" Title="Physicists who were criticised for their works but later celebrated by the people" Tags="|biographical-details|physicists|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2140" PostTypeId="2" ParentId="2138" CreationDate="2015-04-08T18:17:56.090" Score="4" Body="&lt;p&gt;The main reasons Aristotle dominated European thought for centuries were the canonization of his physics and cosmology by the Christian church, and the general decline of science and education in Europe. People were not shocked in part because acceptance of Aristotle's views was not based on their empirical merits. Scholasts and theologians, the learned of the time, were more interested in the internal coherence of the church's teachings, and especially their conformity with the scripture. Once Aristotle's natural philosophy was assimilated, rejecting it could be seen as a challenge to the church's authority, a risky move. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition to that, what is &quot;simple&quot; and &quot;beneath your nose&quot; is often in the eye of the beholder, and depends on a cultural context. Aristotle based his view of falling bodies on comparing a rock and a feather. Like many others in antiquity he rejected existence of a void, so Galileo's idea that air resistance should be &quot;disregarded&quot; was alien to them. For comparison, Babylonians, and even mathematically savvy astronomers like Hipparchus and Ptolemy, were &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html&quot; rel=&quot;nofollow&quot;&gt;using a zero placeholder symbol&lt;/a&gt; for centuries, but it never occurred to them to treat it as a number. It did to medieval Indians. Kepler mentioned that he did not initially consider ellipses when fitting the orbit of Mars because ellipses were known since antiquity, and he assumed that somebody tried them before him. Nobody did, it was all deferents and epicycles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Aristotle's views did not go unquestioned for all those centuries. His theories of projectile motion, and of falling bodies were disputed already in antiquity, by Hipparchus (2nd century BC) and John Philoponus (6th century AD) among others. They originated an alternative theory of impetus, which became &lt;a href=&quot;http://muse.jhu.edu/journals/journal_of_the_history_of_ideas/v064/64.4franco.pdf&quot; rel=&quot;nofollow&quot;&gt;popular with Arabic scholars&lt;/a&gt; during middle ages, and in 14th century back in Europe. Philoponus rejected Aristotle's view and gave a modern account of falling bodies long before Galileo:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, Philoponus happened to live in a wrong place (Byzantine empire) at a wrong time. For his trouble he was shunned by colleagues, his empirical bent and critique of Aristotle considered too radical, and forced to abandon philosophy for theology, where his views were declared heresy shortly after his death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The final irony is that most historians believe now that Galileo's famous &lt;a href=&quot;http://en.wikipedia.org/wiki/Galileo%27s_Leaning_Tower_of_Pisa_experiment&quot; rel=&quot;nofollow&quot;&gt;leaning tower of Pisa experiments&lt;/a&gt; never took place, they were thought experiments (&lt;a href=&quot;http://sites.uci.edu/trabing/files/2013/11/Buridan-copy.pdf&quot; rel=&quot;nofollow&quot;&gt;possibly extracted from Philoponus&lt;/a&gt; or a later commentator). He reasoned the correct law of falling bodies out of a &quot;sort of verbal argument&quot; that Philoponus criticized Aristotle for. Perhaps, Galileo even thought that such an argument would be more convincing to people accustomed to Aristotle and scholasticism. But since such arguments can not possibly deduce empirical laws &lt;a href=&quot;http://philsci-archive.pitt.edu/2524/1/Galileo_vs_Aristotle_on_Free_Falling_Bodies.pdf&quot; rel=&quot;nofollow&quot;&gt;Galileo's had logical mistakes in it&lt;/a&gt; too. However, it was based on a much better empirical intuition and understanding of accelerated motion than Aristotle's, the insights not available to Hipparchus or Philoponus. Some of them developed earlier in Europe by &lt;a href=&quot;http://www.jstor.org/stable/41133336?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Oresme and the Merton school&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="62" LastEditDate="2015-04-12T21:08:19.350" LastActivityDate="2015-04-12T21:08:19.350" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2141" PostTypeId="2" ParentId="2139" CreationDate="2015-04-08T20:41:32.483" Score="7" Body="&lt;ol&gt;&#xA;&lt;li&gt;Boltzmann. He is one of the main founders of atomic theory. Atomic theory was strongly criticized on philosophical grounds, by some prominent philosophers, including Mach. Boltzmann committed suicide in a state of depression in 1906. I do not claim that criticism was the only reason of this depression, but I suspect it contributed to it.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The paper of Einstein was published in 1905 and Smoluchowski in 1906.&#xA;Experiments of Perrin in 1908 confirmed the theory of Brownian motion, so the&#xA;final proof of existence of atoms was obtained.&lt;/p&gt;&#xA;&lt;p&gt;By the way, Mach was not only a philosopher. He made some contribution to physics&#xA;as well (Mach's number, for example).&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Influential philosopher Eugene Duhring (and other philosophers) strongly criticized  Riemann (they worked in the same university), and I suppose created some troubles for him. The thing which philosophers did not like were non-Euclidean geometries and multi-dimensional spaces. Few people remembers Duhring today. (Thanks to Conifold who taught me how to spell his name correctly).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;By the way, Galileo was prosecuted and criticized mainly not by his peers but&#xA;by the Catholic Church. Of course not all scientists accepted his theories, but discussions within science are normal. It is mainly philosophers, church and communist authorities that prosecute scientists.&lt;/p&gt;&#xA;&lt;p&gt;EDIT. To answer the objection of Tom Crowel. I read the article he refers to.&#xA;Does not look very convincing to me. I cite&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Boltzmann was involved in various disputes. But this is not to say that he was the innocent victim of hostilities. In many cases he took the initiative by launching a polemic attack on his colleagues.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Exactly the same can be said about Galileo. Some of his enemies were also respected scientists (Scheiner, for example). History shows that Galileo was right in the main dispute. Same can be said about Boltzmann. Therefore I think that comparison is legitimate.&lt;/p&gt;&#xA;&lt;p&gt;Further, I find in the article the following argument:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He was elected to membership or honorary membership in many academies (cf. Höflechner 1994, 192), received honorary doctorates, and was also awarded various medals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is about Boltzmann. But this also applies to Galileo:-)&lt;/p&gt;&#xA;&lt;p&gt;But thanks for the reference anyway.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-02-25T16:51:12.943" LastActivityDate="2021-02-25T16:51:12.943" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="2142" PostTypeId="2" ParentId="2138" CreationDate="2015-04-08T21:56:38.607" Score="8" Body="&lt;p&gt;I would not call this &quot;horrendous mistake&quot;, and I disagree with Conifold's statement that the reason was canonization by the Catholic Church.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is easier to disprove Conifold's statement, so let me begin with this.&#xA;Canonization of Aristotle begins at the times of Thomas Aquinas (13 century),&#xA;AFTER 1000 years of the Dark Age, and about 200 years before the renaissance of science started. Actually incorporation of Aristotle into the official church teaching is considered a great progress by some historians (when compared with the Dark Age). Aristotle at least did not doubt that the Earth is spherical, as some Christian &quot;scientists&quot; did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, the Muslim scientific thought was dominated by Aristotle, as much as&#xA;the later Christian thought,&#xA;since much earlier time, since its origin, and this has nothing to do with Christian church, of course. Let me add that Aristotle was dominating the thought already in the late antiquity, long before the introduction of Christianity in the Empire.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer the question itself is somewhat more difficult because OUR thought is dominated by something else, let me call it &quot;Baconian tradition&quot;, I do not insist on the name, this is just a label.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The statements that scientific theories have to be verified by experiments and observations, and that this is the main criterion of truth, is FAR from being self-evident. It was actually a great discovery, made approximately in 16 century, and the precise formulation is associated with the name of Francis Bacon&#xA;in the English-speaking world, at least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I emphasize that this was a GREAT DISCOVERY, the thing which is hard to understand to modern people, because of the way we were educated. We tend to think that this is something self-evident.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But science, as we understand it now, really never existed before in this form.&#xA;Of course there was some great science in antiquity but it was different. People THOUGHT differently from what we do. The true experimental science in the modern sense did not exist. (If you disagree, tell me of ONE true physical experiment described in ancient Greek or Roman books. I know only one, but a closer examination proves that this really was a &quot;thought experiment&quot; :-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern period in science lasts less 400 years. Most of the time in history, in most of the places the idea that &quot;experiment is the criterion of truth&quot;&#xA;would sound weird and incomprehensible. What we call &quot;Dark Age&quot; is more normal and typical than what we see in the last 400 years, from the point of view of the whole history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even at the times of Galileo, one could hear objections to what he said of the type: This cannot be true BECAUSE this contradicts Aristotle (or worse, contradicts the Scriptures). And some people refused to look in the telescope, saying that all these are &quot;tricks&quot; etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even at this time there are people who would say this, that something contradicts the Scriptures, or Quran, or Engels, or whatever other authority, THEREFORE it cannot be true.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-04-09T06:39:06.633" LastActivityDate="2015-04-09T06:39:06.633" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2143" PostTypeId="1" CreationDate="2015-03-10T18:37:13.540" Score="23" ViewCount="10465" Body="&lt;p&gt;This is more of a philosophical question, but I want a mathematical explanation. During ancient times, it was well accepted that the surface of Earth was spherical. People first observed this when they saw that starting from a point going to any direction they ultimately reach the same point, but by this logic how did they conclude that or even guess that this might be a sphere? Why did nobody guess that, say, a torus might also  be possible? As far as I know, at those time there were lots of mathematicians. If they were convinced by this logic then they must have had some logical explanation, and I do not think that for them just staying on the surface would at all be possible to guess the structure of a surface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my logical thoughts for this is the following: If I take a telescope and stand on a field and look up, my vision will never  get limited by the surface of Earth. Mathematically, if I draw a line perpendicular to the surface at a point, then that line will never intersect the surface. But doing the same experiment at each point on Earth is not very convincing. Can anybody please provide me a better logical explanation of how people of that time concluded that the shape of Earth is a sphere?    &lt;/p&gt;&#xA;" OwnerUserId="2103" OwnerDisplayName="AnubhaV" LastEditorUserId="2103" LastEditDate="2015-11-12T21:58:33.223" LastActivityDate="2019-07-25T23:29:47.950" Title="In ancient times, how did people conclude that the shape of Earth is a sphere?" Tags="|mathematics|physics|philosophy-of-science|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2144" PostTypeId="2" ParentId="2143" CreationDate="2015-03-10T18:42:44.623" Score="12" Body="&lt;p&gt;The ancients understood that a lunar eclipse is caused when the earth gets between the sun and the moon.  They saw that the shadow the earth casts on the moon is round.  From that it wasn't too far of a leap for them to conclude the earth is a sphere.&lt;/p&gt;&#xA;" OwnerDisplayName="Gregory Grant" LastActivityDate="2015-03-10T18:42:44.623" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2145" PostTypeId="2" ParentId="2143" CreationDate="2015-03-10T18:47:03.220" Score="2" Body="&lt;p&gt;I believe the answer to your question lies mostly in &lt;a href=&quot;http://en.wikipedia.org/wiki/Timaeus_(dialogue)&quot; rel=&quot;nofollow&quot;&gt;Greek's cosmic philosophy&lt;/a&gt; of the way the universe should be. They saw spheres as one of the most symmetrical simplistic shapes, and if our earth, which for them was at the center of the cosmos, would be the shape of anything it would be a sphere. Of course, the calculations they used and observations they made lined up with a sphere, so it seemed that it was the most reasonable thing for earth to be. Now, many philosophers of science might argue that the only reason one would choose some model over another, which calculate the same and take account of all phenomena, is mostly due to simplicity. Not because one necessarily represents reality more so than the other. Hence, I'm sure a creative philosopher could argue that your torus world view can't be shown to be wrong if you find clever ways to account for all the various phenomena.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I'm not even sure that the the ancients had the mathematical concept of a torus. I'm probably entirely wrong about this. The greeks were fascinating.&lt;/p&gt;&#xA;" OwnerUserId="2208" OwnerDisplayName="Valentino" LastEditorUserId="8" LastEditDate="2015-04-30T18:24:35.427" LastActivityDate="2015-04-30T18:24:35.427" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2146" PostTypeId="2" ParentId="2139" CreationDate="2015-04-09T17:40:24.043" Score="4" Body="&lt;p&gt;Another well documented example is Subrahmanyan Chandrasekhar ... &quot;Chandra&quot; who gave the first mathematical description of black holes.  His conflict with Sir Arthur Eddington is portrayed by Arthur I. Miller in &lt;em&gt;Empire of the Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes&lt;/em&gt;.  Also see K.C. Wali, &quot;Chandrasekhar vs. Eddington: An Unanticipated Confrontation&quot;, &lt;em&gt;Physics Today&lt;/em&gt;, vol. 35, no. 10, pp. 33–40 (October, 1982).&lt;/p&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2015-04-09T17:40:24.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2147" PostTypeId="2" ParentId="343" CreationDate="2015-04-10T04:16:43.977" Score="14" Body="&lt;p&gt;Newton implicitly refers to the medieval physicists who had such a profound affect on the intellectual atmosphere of Galileo et al. that they took their discoveries as common knowledge. Some of the most famous of these physicists were:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/people/philosophy-and-religion/philosophy-biographies/saint-thomas-aquinas&quot; rel=&quot;nofollow noreferrer&quot;&gt;St. Thomas Aquinas&lt;/a&gt;, &lt;a href=&quot;https://sententiaedeo.blogspot.com/2010/05/sciences-light-ages.html#4&quot; rel=&quot;nofollow noreferrer&quot;&gt;the first to distinguish weight, mass, and the resisting medium&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The French bishop, physicist, and &lt;a href=&quot;https://mises.org/library/nicholas-oresme-and-first-monetary-treatise&quot; rel=&quot;nofollow noreferrer&quot;&gt;economist&lt;/a&gt; &lt;a href=&quot;http://www.encyclopedia.com/topic/Nicole_Oresme.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicole Oresme&lt;/a&gt;, who determined the mean speed theorem of uniformly accelerated bodies: &lt;span class=&quot;math-container&quot;&gt;$v_\mathrm{avg} = v_f / 2$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Nicole_Oresme.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bishop Oresme&lt;/a&gt; posed the famous &lt;em&gt;Gedankenexperiment&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I posit that the Earth is pierced clear through and that we can see&#xA;  through a great hole farther and farther right up to the other end&#xA;  where the antipodes [poles] would be if the whole of this Earth were&#xA;  inhabited; I say, first of all, that if we dropped a stone through&#xA;  this hole, it would fall and pass beyond the center of the earth,&#xA;  going straight on toward the other side for a certain limited&#xA;  distance, and that then it would turn back going beyond the center on&#xA;  this side of the Earth; afterward, it would fall back again, going&#xA;  beyond the center but not so far as before; it would go and come this&#xA;  way several times with a reduction of its reflex motions until finally&#xA;  it would come to rest as the center of the Earth....&lt;br&gt;&lt;br&gt;{Quoted by K. V. Magruder from &lt;em&gt;&lt;a href=&quot;http://www.worldcat.org/oclc/577410807&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le Livre du Ciel et due Monde&lt;/a&gt;&lt;/em&gt; (Madison: University of Wisconsin Press, 1968), translated by D. Menut, pg. 573.}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Nicole_Oresme.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bishop Oresme&lt;/a&gt; wrote (before Galilean relativity): “If air were enclosed in a moving ship, it would seem to the person situated in this air that it was not moved.” &lt;em&gt;Book of the Heavens&lt;/em&gt;, Book II chapter 25, from Grant, &lt;em&gt;&lt;a href=&quot;http://garrigou.us.to/browse/book/3471&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Source Book of Medieval Science&lt;/a&gt;&lt;/em&gt;, pg. 505, Harvard, 1974&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Jean_Buridan.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Buridan&lt;/a&gt; (d. ca. 1359) invented/discovered the concept of momentum and the equation &lt;span class=&quot;math-container&quot;&gt;$p=mv$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Thomas_Bradwardine.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas of Bradwardine&lt;/a&gt; (c. 1295-1349) distinguished mean and instantaneous velocity.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Thomas_Bradwardine.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bradwardine&lt;/a&gt; determined in 1300 that for uniformly accelerated objects, &lt;span class=&quot;math-container&quot;&gt;$d = \frac{1}{2} a t^2$&lt;/span&gt;, which &lt;a href=&quot;http://www.encyclopedia.com/doc/1G2-2830904089.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Soto, O.P.&lt;/a&gt;, (b. ca. 1494) applied to free-falling objects; Bradwardine thus wrote the first physics equation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jordanus de Nemore and Torricelli influenced Galileo's treatment of inclined planes.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Grosseteste.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Grosseteste&lt;/a&gt; (c. 1168-1253) did experiments (not yet of course with modern rigor) and was keen on using mathematics; he is known for his work on understanding the rainbow. &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Bradwardine.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas of Bradwardine&lt;/a&gt; (c. 1295-1349) at Merton College Oxford introduced the distinction between mean velocity (x/t) and instantaneous velocity (dx/dt) [and he was the first to write a physics equation]. Bradwardine had an enthusiasm for empiriometric physics that started a whole school called the Merton school (his successors include: William Heytesbury, Richard Swineshead, and John Dumbleton) that was extremely influential throughout Europe. Among other things, they were known for the Merton mean speed theorem, by which they proved the correct formula for free fall distance was given by &lt;span class=&quot;math-container&quot;&gt;$s=\frac{1}{2}at^2$&lt;/span&gt;. Interestingly, both Bradwardine and Grosseteste at some point in their lives were Archbishops of Canterbury. &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Oresme.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicole Oresme&lt;/a&gt; (&amp;lt;1348-1382) and &lt;a href=&quot;http://en.wikipedia.org/wiki/Giovanni_di_Casali&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giovanni di Casali&lt;/a&gt; (c. 1350) independently developed use of 2-D graphs [long before &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Descartes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes&lt;/a&gt; (1596-1650)]. Oresme described all change using these graphs in particular local motion, including calculating area (integrating) under velocity curves to get distance. Oresme's arguments for the sun-centered and moving earth were widely known: he said, for example, that &quot;...not only is the earth so moved diurnally, but with it the water and the air, as was said, in such a way that the water and the lower air are moved differently than they are by winds and other causes. It is like this situation If air were enclosed in a moving ship, it would seem to the person situated in this air that it was not moved.&quot; (p. 133, Dales.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;—A. Rizzi's &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=7249&quot; rel=&quot;nofollow noreferrer&quot;&gt;Science Before Science&lt;/a&gt;&lt;/em&gt; pgs. 199-200&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, I'm very surprised some here think Newton was referring to Descartes. Newton's physics was radically different than Descartes' (cf. &lt;a href=&quot;http://www.thomist.org/jourl/1968%2C%20vol.%2032/Jan/1968%20Jan%20A%20Wallace.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even further back, &lt;a href=&quot;http://sententiaedeo.blogspot.com/2013/05/john-philoponus-facts-information.html?m=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philoponus&lt;/a&gt; (late 5th, 2nd ½ of 6th century A.D.) is impressive:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He argued that the sun is fire and of terrestrial-like, corruptible matter. He devised a precursor to the notion of impetus which &lt;a href=&quot;http://www.encyclopedia.com/topic/Jean_Buridan.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Buridan&lt;/a&gt; later developed, that which keeps moving bodies in motion even after the mover ceases being in contact with them; air does not keep projectiles in motion. He discovered that light rays travel the same both backwards and forwards. He invented functions of variables and their &quot;courses&quot; (what we'd call &quot;first derivatives&quot; in modern calculus). He discovered the law of inertia, that bodies in motion remain in motion unless something impedes their movement, literally a thousand years before Galileo, Newton, et al.!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He's certainly one of the &quot;&lt;em&gt;grands génies de l'Antiquité&lt;/em&gt;&quot; (&quot;great geniuses of Antiquity&quot;) and &quot;&lt;em&gt;principaux précurseurs de la Science moderne&lt;/em&gt;&quot; (&quot;principle precursers to modern Science&quot;), &lt;a href=&quot;https://archive.org/stream/lesystmedumond01duhe#page/398/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;as Pierre Duhem wrote&lt;/a&gt; in his magisterial, 10 volume work in the history of medieval physics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/#panel=book_list&amp;amp;search=title:Le%20Syst%C3%A8me%20du%20Monde&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le système du monde: Histoire des doctrines cosmologiques de Platon à Copernic&lt;/a&gt;&lt;/em&gt;. 10 vols. Paris: A. Hermann, 1913-1959.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Partially translated in:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4757&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds&lt;/a&gt;&lt;/em&gt;. Edited and translated by Roger Ariew. Chicago: University of Chicago Press, 1985.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;cf. also:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hannam, James. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4182&quot; rel=&quot;nofollow noreferrer&quot;&gt;God’s Philosophers: How the Medieval World Laid the Foundations of Modern Science&lt;/a&gt;&lt;/em&gt;. London: Icon Books, 2009.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-06-19T23:45:57.053" LastActivityDate="2019-06-19T23:45:57.053" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2148" PostTypeId="1" CreationDate="2015-04-10T05:22:57.407" Score="9" ViewCount="13019" Body="&lt;p&gt;Discovery of Sine and Cosine of an angle, the intuition behind it is always intriguing. Apart from &quot;that is the way they were defined&quot;, could someone explain how the discovery happened? I have read &quot;What is Mathematics?&quot; by R. Courant and related material on web. To this particular question, haven't found an answer for long.&lt;/p&gt;&#xA;" OwnerUserId="2110" LastActivityDate="2015-04-10T14:07:05.200" Title="Discovery of Sine and Cosine" Tags="|calculus|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2149" PostTypeId="1" AcceptedAnswerId="2155" CreationDate="2015-04-10T10:45:37.970" Score="11" ViewCount="689" Body="&lt;p&gt;In studying vector spaces we learn about linear transformations from one vector space to another and in particular the kernel of such a transformation. When learning about group theory we also learn of kernels of homomorphisms. I know that group theory generalises the idea of kernel to work with other groups, but I was wondering which came first... was the kernel originally defined from the study of vectors spaces or from the more general group theory?&lt;/p&gt;&#xA;" OwnerDisplayName="user2114" LastActivityDate="2017-08-26T09:03:57.947" Title="What came first? The kernel from vector spaces or from group theory?" Tags="|group-theory|linear-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2150" PostTypeId="1" AcceptedAnswerId="2226" CreationDate="2015-04-10T11:55:01.703" Score="5" ViewCount="265" Body="&lt;p&gt;There is a construct very useful to measure the efficiency taking into account both quantity and quality, which states something like&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;N is the highest number that fulfils the statement &quot;in this set, there are at least N elements having at least size/quality/goodness N&quot;.&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This construct is the basis of the &lt;a href=&quot;https://en.wikipedia.org/wiki/H-index&quot; rel=&quot;nofollow&quot;&gt;h-index&lt;/a&gt; (2005),&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A scientist has index h if h of his/her Np papers have at least h citations each, and the other (Np − h) papers have no more than h citations each.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But had been previously used by Arthur Eddington to value a cyclist's performance with his &lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Eddington#Eddington_number_for_cycling&quot; rel=&quot;nofollow&quot;&gt;Eddington number&lt;/a&gt; (before 1944):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;the maximum number E such that the cyclist has cycled E miles on E days&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This resembles also the Pareto principle (1896), when he first stated that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;approximately 80% of the land in Italy was owned by 20% of the population&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is a sort of &quot;inverted&quot; version of the &quot;h-index/Eddington number&quot; construct (however, Pareto's principle, using that formulation, does not use the original construct).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Is there an earlier use of the h-index/Eddington number construct? Is this construct known with a more general name, or a more general definition?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2115" LastEditorUserId="2115" LastEditDate="2015-04-10T21:54:03.233" LastActivityDate="2015-04-23T12:43:01.343" Title="construct for h-index and Eddington number" Tags="|mathematics|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2151" PostTypeId="2" ParentId="2139" CreationDate="2015-04-10T12:20:07.893" Score="3" Body="&lt;p&gt;Hugh Everett was heavily criticised by his peers, for the &quot;Theory of Universal Wave Function&quot; also known as &quot;Many World Interpretation&quot; of quantum physics. &#xA;To a point where he decided to end his academic career in physics after his PhD.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His work, published in 1957, was only recognized during 1970's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the documentary &lt;a href=&quot;https://vimeo.com/58603054&quot; rel=&quot;nofollow&quot;&gt;Parallel Worlds, Parallel Lives&lt;/a&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="1831" LastEditorUserId="55" LastEditDate="2015-04-10T22:42:49.977" LastActivityDate="2015-04-10T22:42:49.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2152" PostTypeId="2" ParentId="2148" CreationDate="2015-04-10T13:00:51.787" Score="5" Body="&lt;p&gt;Trigonometry was developed for the needs of astronomical computation.&#xA;(The whole idea was foreign to the classical Greek mathematics (Euclides, Apollonius). They did not have even a fully developed number system, and did very little numerical computation, preferring pure geometric methods). Astronomical computations became possible after the adoption of the Babylonian sexagesimal notation for numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Trigonometry is indispensable in most astronomical computations,&#xA;the simplest one is passing from equatorial to ecliptic coordinates. This is needed for example when you compute such things as rising times, length of daylight, etc.&#xA;Spherical trigonometry was developed simultaneously with the plane trigonometry,&#xA;and the main emphasis was on the spherical one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This happened in the the very end of the 1st century and is due to Menelaus.&#xA;(His book Spherics can be dated by an observation made in the year 98).&#xA;His trigonometric tables, if he made them, did not survive, and the earliest&#xA;existing work where trigonometry is fully developed is Ptolemy's Almagest.&#xA;Almagest contains the first known trigonometric tables, and Neugebauer thinks that&#xA;Ptolemy was the first to make them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At that time our usual trigonometric functions were not used. The only trigonometric function was the chord $\mathrm{chd}\ x=2\sin(x/2)$.&#xA;This function is tabulated in Almagest. The role of cosine played $\mathrm{chd}(\pi-x)$, the cord of the complementary angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sine was introduced by Abu'l Wafa in 8th century, as a more convenient function, and gradually spread first in the Muslim world,&#xA;and then to the West. (But apparently it was used in India centuries before him), as a more convenient function.&#xA;However this new notation was adopted very slowly, it took centuries.&#xA;Copernicus (16th century) in fact used sine but calls it &quot;half of the chord of double angle&quot;. But at the time before Copernicus there were already more advanced&#xA;treatments of trigonometry, by J. Muller (Regiomontanus). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best reference for all this information is the books&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neugebauer, History of ancient mathematical astronomy, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sverdlov and Neugebauer, Mathematical astronomy in Copernicus's De Revolutionibus. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-04-10T13:09:39.963" LastActivityDate="2015-04-10T13:09:39.963" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2153" PostTypeId="2" ParentId="2148" CreationDate="2015-04-10T14:07:05.200" Score="8" Body="&lt;p&gt;If I may expand on Alexandre’s answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is correct that Ptolemy operated not with sines, but with chords. The use of sines (half-chords) was introduced (not “apparently” but definitely) by the Indian mathematicians, who used the Sanskrit word jīva- “bow string” (which is actually a translation of Greek chordē, but which Indians used not for the whole chord, but the half-chord). When Indian astronomical works were translated into Arabic the Indian term was adopted as jīb. This word is written identically, and pronounced almost identically, with the native Arabic word jayb “breast pocket of a garment”, and this is why it is (mis)translated by Latin sinus “breast”.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-04-10T14:07:05.200" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2154" PostTypeId="2" ParentId="2143" CreationDate="2015-04-10T15:17:39.617" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://www.academia.edu/27928173/How_the_Ancient_Egyptians_had_Calculated_the_Earths_Circumference_between_3750-1500_BC_a_revision_of_the_method_used_by_Eratosthenes&quot; rel=&quot;nofollow noreferrer&quot;&gt;How the Ancient Egyptians had Calculated the Earth's Circumference between 3750-1500 BC:a revision of the method used by Eratosthenes&lt;/a&gt;  Christian Irigaray&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eratosthenes (276-194 BC) is credited as &quot;the first&quot; person to have measured the circumference of the Earth in our modern history books, but a serious study of the circumstances shows that Eratosthenes did not actually measure anything, and that he simply copied an earlier method of the Egyptians. Eratosthenes was a Greco-Egyptian director of the Library of Alexandria at the time, and he had access to the ancient records of science collected during the Alexandrian era which contained not only Egyptian but Mesopotamian and Persian knowledge of a variety of kinds. Among these documents was one from which Eratosthenes copied a method of deriving the Earth’s circumference, but the method itself (as we will see) shows that the actual experiment and record of this in Egypt occurred as&#xA;  early as 3750 BC, making “the first” to actually measure a portion of the Earth’s&#xA;  circumference and calculating were most probably Predynastic Egyptians living up to 3500 years before Eratosthenes’ time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Cartography.html#s25&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Cartography&lt;/a&gt; &quot;Around 350 BC Aristotle put forward six arguments to prove that the Earth was spherical and from that time on scholars generally accepted that indeed it was a sphere.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=vzImgJCgUkg&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Aristotle Proved the Earth is Spherical&lt;/a&gt; (explains the lunar eclipse observation too)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://perseus.mpiwg-berlin.mpg.de/GreekScience/Students/Tom/AristotleAstro.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle's Astronomy by Thomas Fowler&lt;/a&gt;:  &quot;To prove that the earth is a sphere&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To prove that the earth is a sphere, he produced the argument that all&#xA;  earthly substances move towards the center, and thus would eventually&#xA;  have to form a sphere. He also used evidence based on observation. If&#xA;  the earth were not spherical, lunar eclipses would not show segments&#xA;  with a curved outline. Furthermore, when one travels northward or&#xA;  southward, one does not see the same stars at night, nor do they&#xA;  occupy the same positions in the sky. (De Caelo, Book II, chapter 14)&#xA;  That the celestial bodies must also be spherical in shape, can be&#xA;  determined by observation. In the case of the stars, Aristotle argued&#xA;  that they would have to be spherical, as this shape, which is the most&#xA;  perfect, allows them to retain their positions. (De Caelo, Book II,&#xA;  chapter 11)&#xA;  ...&#xA;  In &lt;a href=&quot;http://books.google.com/books/about/Science_before_Socrates.html?id=OOYgLSiEpJoC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Science before Socrates&lt;/em&gt;&lt;/a&gt;, Daniel Graham argues against the&#xA;  prevalent belief that the Presocratic philosophers did not produce any&#xA;  empirical science and that the first major Greek science, astronomy,&#xA;  did not develop until at least the time of Plato. Instead, Graham&#xA;  proposes that the advances made by Presocratic philosophers in the&#xA;  study of astronomy deserve to be considered as scientific&#xA;  contributions.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Virtually all philosophers came to accept Anaxagoras' theory of lunar&#xA;  light and eclipses. Aristotle endorsed Anaxagoras' theory of eclipses&#xA;  as a paradigm of scientific explanation. Anaxagoras' theories launched&#xA;  a geometrical approach to astronomy and were accepted as foundational&#xA;  principles by all mathematical astronomers from Aristarchus to Ptolemy&#xA;  to Copernicus and Galileo-and to the present day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&#xA; Also D.R. Dicks in &lt;em&gt;Early Greek Astronomy to Aristotle&lt;/em&gt;, says speculation about the spherical earth can be dated to at least the 6th century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Supporting historical references found on the History Wiki: &lt;em&gt;&lt;a href=&quot;http://history.wikia.com/wiki/Spherical_Earth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spherical Earth&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The earliest evidence for a spherical Earth came from an ancient&#xA;  Phoenician expedition for ancient Egypt. The Egyptian pharaoh Necho&#xA;  II, during his reign from 610 BCE to 595 BCE, employed Phoenician&#xA;  sailors to circumnavigate around the entire African continent, then&#xA;  known as &quot;Libya&quot;. In The Histories (written c. 431 BCE - 425 BCE),&#xA;  Herodotus described how the Phoenicians reported the sun being&#xA;  observed shining from the north. ... With this expedition, the&#xA;  Phoenicians and Egyptians were thus the first to discover evidence of&#xA;  the Earth being curved and therefore spherical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://plus.google.com/+SacredGeometryDecoded/posts/PeEdSXPjBhH&quot; rel=&quot;nofollow noreferrer&quot;&gt;The ancient Egyptians ahead of Eratosthenes&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eratosthenes gets the credit for the first earth measure of any&#xA;  accuracy. The Egyptians had the same knowledge some 2000 years&#xA;  earlier. The height of the Great Pyramid x 43200 = Earth Polar Radius&#xA;  The base perimeter of the Great Pyramid x 43200 = Earth&#xA;  Circumference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.academia.edu/4807717/The_Egyptian_Heritage_in_the_Ancient_Measurements_of_the_Earth&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Egyptian Heritage in the Ancient Measurements of the Earth&lt;/a&gt;, by Gyula Priskin, Published in  &lt;em&gt;Göttinger Miszellen&lt;/em&gt;, 208 (2006), pp. 75-88. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abstract:  The comparison of an Egyptian text recording the&#xA;  north-south extent of the country with the descriptions of&#xA;  Eratosthenes’ and Posidonius’ experiments to measure the earth’s&#xA;  circumference reveals that early Hellenic mathematical geography&#xA;  borrowed much information from Egyptian science. A further analysis of&#xA;  all the figures circulated for the length of the meridian in antiquity&#xA;  reinforces the case that Hellenic geographers were greatly influenced&#xA;  by Egyptian ideas when they formed their opinions on the size of the&#xA;  earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.metrum.org/key/pyramids/estimates.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Egyptian Estimates of the Size and Shape of the Earth&lt;/a&gt;&#xA;From Stecchini's &lt;em&gt;The Pyramids of Egypt&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=Historical%20Metrology,%20a%20New%20Analysis%20of%20the%20Archaeological%20and%20the%20Historical%20Evidence%20Relating%20to%20Weights%20and%20Measures,%20by%20A.%20E.%20Berriman&amp;amp;num=20&amp;amp;newwindow=1&amp;amp;source=lnms&amp;amp;tbm=bks&amp;amp;sa=X&amp;amp;ei=Q9ZMVe-WGsehNuDWgWA&amp;amp;ved=0CA4Q_AUoAQ&amp;amp;biw=1264&amp;amp;bih=803&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historical Metrology, a New Analysis of the Archaeological and the Historical Evidence Relating to Weights and Measures&lt;/a&gt;, by &lt;a href=&quot;http://scholar.google.com/scholar?cluster=3537861579905702604&amp;amp;hl=en&amp;amp;as_sdt=5,37&amp;amp;sciodt=0,37&quot; rel=&quot;nofollow noreferrer&quot;&gt;A. E. Berriman&lt;/a&gt;,...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.academia.edu/6478703/The_Acropolis_Width_and_Ancient_Geodesy&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Acropolis Width and Ancient Geodesy Nicholas Kollerstrom&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Was the Earth measured in remote antiquity?” This was the stirring&#xA;  question with which Berriman opened his book, &lt;em&gt;Ancient Metrology.&lt;/em&gt; To&#xA;  be sure, the question had earlier been tackled in Nicholson’s &lt;em&gt;Men and&#xA;  Measures&lt;/em&gt;, as to whether a knowledge of Earth’s dimension had afforded&#xA;  the original basis for units of measure. Here we inquire,&#xA;  specifically, as to whether the ancient Greek units of measure were&#xA;  related to the circumference of the Earth. This hypothesis tends to be&#xA;  related to the notion that a global, maritime civilization had once&#xA;  existed in prehistory. Our inquiry is therefore in some degree related&#xA;  to the thesis propounded by Francis Bacon, in his &lt;em&gt;New Atlantis&lt;/em&gt;: “You&#xA;  shall understand (that which you will scarce think credible) that&#xA;  about three thousand years ago, or somewhat more, the navigation of&#xA;  the world (especially for remote voyages) was greater than at this&#xA;  day.” Could there have been a civilization of prehistory which&#xA;  vanished, but left behind its geodetically-defined units? ....&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The position here advocated could be described as Newtonian, insofar&#xA;  as Isaac Newton believed, that one function of ancient temples was&#xA;  that of expressing the proportions of the world ....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From his study of the Hermetic texts of the Egyptians, Newton stated in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1512181153&quot; rel=&quot;nofollow noreferrer&quot;&gt;The System of the World&lt;/a&gt; what some controversial authors are trying to say today: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was the ancient opinion of not a few, in the earliest ages of&#xA;  philosophy, that the fixed stars stood immoveable in the highest parts&#xA;  of the world; that, under the fixed stars the planets were carried&#xA;  about the sun; that the earth, as one of the planets, described an&#xA;  annual course about the sun, while by a diurnal motion it was in the&#xA;  mean time revolved about its own axis;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As Steven Weinberg says in his latest book, &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0062346652&quot; rel=&quot;nofollow noreferrer&quot;&gt;To Explain the World: The Discovery of Modern Science&lt;/a&gt;&lt;/em&gt;, (the Dendera Zodiac is shown on the book cover) &quot;At several points in this book I suggest that, as great as is the progress that has been made in the methods of science, we may today be repeating some of the errors of the past.&quot; Indeed, it seems as though some forgotten history is repeating itself in the controversy over early knowledge of precession, the spherical earth and what this means for modern science. For instance, in &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691145768&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Zodiac of Paris&lt;/a&gt;&lt;/em&gt; by Jed Z. Buchwald &amp;amp;  Diane Greco Josefowicz &quot;The Dendera zodiac--an ancient bas-relief temple ceiling adorned with mysterious symbols of the stars and planets--was first discovered by the French during Napoleon's campaign in Egypt, and quickly provoked a controversy between scientists and theologians.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This is a fascinating study of how politics, science, and religion&#xA;  intersected in the heated debates over the meanings of the&#xA;  hieroglyphics on a pair of stones brought from Egypt to Paris in 1821.&#xA;  At the heart of the tale is the question of how we know the past. It&#xA;  has the excitement of a real-life archeology mystery combined with a&#xA;  clash between science and theology that has great resonance for&#xA;  today. --Walter Isaacson, author of &lt;em&gt;Einstein: His Life and Universe&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See, Buchwald, J.Z. &quot;Egyptian Stars under Paris Skies.&quot; &lt;em&gt;Engineering and Science&lt;/em&gt; 66.4 (2003): 20-31. Notice of long standing errors on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Talk:Dendera_zodiac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dendera Zodiac English Wiki talk page&lt;/a&gt;: &quot;It might be that the Dendera Zodiac is academic hotstuff that not many academics dare touching, ... said Rursus.&quot; ...and to be compared with the &lt;a href=&quot;http://fr.wikipedia.org/wiki/Zodiaque_de_Dend%C3%A9rah&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dendera Zodiac Google French translation&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jean-Baptiste Biot, astronomer and Egyptologist, meanwhile, looks in&#xA;  detail at the Egyptian circular zodiac.... and shows that the&#xA;  celestial arrangement goes well beyond the Roman era. ... According to&#xA;  Sylvie Cauville, Egyptologist, author of numerous books including &lt;em&gt;The&#xA;  Eye of Re&lt;/em&gt;, the founding charter of the Dendera temple is part of the&#xA;  ancient writings of the library of Cheops.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ancient-wisdom.co.uk/egyptdenderra.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Temple of Hathor at Dendera:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The temple complex, as it stands today, was built on the site of an&#xA;  older temple, and is a replica of the original. The present building&#xA;  was first initiated by Ptolemy III, with numerous additions by&#xA;  subsequent Ptolemaic and Roman rulers. The inscription on the present&#xA;  temple states that the original building was erected in the far&#xA;  pre-dynastic times, by the followers of Heru (Horus).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Axial_precession&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Rice&lt;/a&gt;, in &lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0415268761&quot; rel=&quot;nofollow noreferrer&quot;&gt;Egypt's Legacy&lt;/a&gt;&lt;/em&gt;, &quot;the precession is fundamental to an understanding of what powered the development of Egypt&quot; (p.10). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update comment: One item in reference to the torus is the discovery of earth's magnetospheric lines having a dodecahedral vortex configuration. So from the etheric the earth is indeed something of a torus. Author John Michell's ancient metrology discusses the blueprint of megalithic monuments as being related to what he called the New Jerusalem or Cosmological Circle, which has the form of a dodecahedron in three dimensions! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another update to connect some of the references: The Cosmological Circle is also known today as Plato's Wheel, drawn from his &lt;em&gt;Timaeus&lt;/em&gt;. Ernest Pecci shows how the scalene triangles within Plato's Wheel are fundamental to the geometry of the Great Pyramid and that the builders knew the precise circumference of the planet. See &lt;em&gt;The Sacred Geometry of the Great Pyramid: From the Drawing Board of Its Architects&lt;/em&gt; by Ernest F. Pecci.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More conservative references show the &quot;Pythagoreans were the first ... to say the earth is round.&quot; - George Sarton (noted founder of the academic discipline of the History of Science) &lt;em&gt;Ancient Science Through the Golden Age of Greece&lt;/em&gt;.  Pythagoras was an Egyptian initiate and some of his knowledge was directly attributed to the Egyptians. Both Kepler and Newton were also privy to some of the secret oral tradition. Kepler acknowledged his debt to the Egyptians. Newton went to great lengths in a paper trying to figure out the Sacred Cubit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another &quot;source&quot;: &lt;a href=&quot;http://www.ancient-wisdom.co.uk/egyptastronomy.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient Wisdom&lt;/a&gt;  &lt;strong&gt;Discovery that the Earth is spherical&lt;/strong&gt; - &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first (official) measurement of the radius of the earth was made&#xA;  by Erasthenes (b. 275 B.C.), who was the head of the great library of&#xA;  Alexandria. He was born in Cyrene, now Libya. It seems likely that the&#xA;  ancient Egyptians, much before Egypt's conquest by Alexander the&#xA;  great, had already grasped the idea of a spherical Earth, and it was&#xA;  from them that this doctrine was adopted by Pythagoras, who, as we&#xA;  know, spent many years of study in Egypt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.amatterofmind.us/constructive-geometry/advanced-constructive-geometry/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Egyptian Pyramid and the Archytas Doubling of the Cube&lt;/a&gt; by Pierre Beaudry&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... the Great Pyramid included in its very construction frame the idea&#xA;  of doubling the cube.  ... Now, as you all know, the geometric problem&#xA;  that Archytas had to resolve was formulated as follows: {find two mean&#xA;  proportionals between two extremes in the ratio of two to one}. In&#xA;  order to find those two mean proportionals, the Archytas construction&#xA;  for the doubling of the cube required a {&lt;strong&gt;cone, a torus, and a&#xA;  cylinder&lt;/strong&gt;}. ... However, this Greek discovery was based on the more&#xA;  ancient Egyptian discovery as its {necessary predecessor}.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Thus, the Egyptian doubling of the cube is simply a derivative of two&#xA;  astrophysical observations that had to be made at the site of the&#xA;  Great Pyramid in order to establish its architectural design. Those&#xA;  two conic projections, from the North Pole and from the Ecliptic,&#xA;  generate the frame-shadow of the Great Pyramid whose triangular&#xA;  meridian angle, PAM, shows that the two proportional segments, AM and&#xA;  AP, respectively represent the sides of two cubes whose volumes are in&#xA;  the ratio of 2/1. So, it becomes clear that this {is} where the&#xA;  Archytas construction took its origins. ... Ironically, this Great&#xA;  Pyramid triangular frame-shadow of 90°, 52°, and 38° degrees, with its&#xA;  harmonically conjugated segments, AB, AM, and AP, not only reflects&#xA;  the power of successively doubling the cube, but also reflects the&#xA;  golden section, the Great Pyramid paradox of squaring the circle, and&#xA;  the 256 series behind the well tempered musical system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://amatterofmind.org/Pierres_PDFs/ANCIENT_ASTRONOMY_I/BOOK_I/3._PYTHAGOREAN_SPHAERICS_PART_I.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Constructive Geometry of Pythagorean Sphaerics: Part I&lt;/a&gt;  by Pierre Beaudry&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pythagoras had established his Astronomy on the original&#xA;  accomplishments of the Egyptians, who, themselves had received their&#xA;  legacy from Atlas, the original Trans-Atlanticist founder of ancient&#xA;  Astronomy, and the first inventor of the celestial sphere, which,&#xA;  according to Jean Sylvain Bailly, can be dated at about 4,000 BC. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The discovery of Precession Astronomy can be traced back, in an&#xA;  architectural documented form, to the construction of the Step Pyramid&#xA;  of Zoser at Saqqara (circa 3400 B.C.)....    Between the years 495 and&#xA;  491 B.C.,... Khnum-Ab-R'a, who was chief minister of works in Egypt,&#xA;  had left an inscription on a public monument of the valley of Wadi&#xA;  Hammamat, which put on record his 24 architect predecessors, leading&#xA;  back to Imhotep .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more on the &lt;strong&gt;Torus and Sphere of the Earth&lt;/strong&gt;, with animations and illustrations:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://wlym.com/antidummies/part42.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archytas from the Standpoint of Cusa, Gauss, and Riemann&lt;/a&gt; &amp;amp; &lt;a href=&quot;http://wlym.com/antidummies/part33.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyperbolic Functions -- A Fugue Across 25 Centuries&lt;/a&gt; &amp;amp; &lt;a href=&quot;http://schillerinstitute.org/fidelio_archive/2003/fidv12n02-2003Su/fidv12n02-2003Su_067-on_principles_and_powers.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Principles and Powers&lt;/a&gt;   by Bruce Director&#xA;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.scribd.com/doc/242632395/THALES-THEOREM-AND-THE-ARCHYTAS-MODEL-pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thales Theorem and the Archytas Model&lt;/a&gt; by Pierre Beaudry &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ancient-wisdom.co.uk/egyptgeodesy.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prehistoric Egyptian Geodesy:&lt;/a&gt; &quot;Giza (Heliopolis) to the Equator = 1/12th the circumference of the earth. (360°/12 = 30° 00’).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.geoman.com/jim/pyramid.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Precession and the Pyramid Astronomical Knowledge in Ancient Egypt&lt;/a&gt; by&#xA;&lt;a href=&quot;http://www.geoman.com/jim/Fournier.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jim Fournier&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is one point about ancient Egypt which stands out above all&#xA;  others, an insight critical not only to our understanding of Egypt,&#xA;  but also to our overall understanding of history. The ancient&#xA;  Egyptians observed, and to an important degree understood, the&#xA;  precession of the equinoxes. This point is really a subsidiary&#xA;  correlate to the realization that at least circa 2500 BC the Egyptians&#xA;  knew the size of the earth very precisely. Precise geodetic knowledge&#xA;  is contingent upon precise astronomical observations, and both taken&#xA;  together imply an advanced understanding of geometry, as well as&#xA;  precession. It follows that the ancient Greeks should be taken at&#xA;  their word when they claim that their knowledge is of great antiquity&#xA;  and was derived from Egyptian sources. Indeed it is nothing if not&#xA;  bizarre that modern scholars of the Greek world should go to great&#xA;  lengths to dismiss such claims on the part of the authors of the&#xA;  primary texts themselves, to instead rely on the advice of modern&#xA;  Egyptologists that the ancient Egyptians had no such knowledge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;....&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After reviewing the opinions and work of the best of the Egyptian&#xA;  astronomical tradition: Sir Issac Newton, Sir John Herschel, and Sir&#xA;  Norman Lockyer, Neugebauer &amp;amp; Parker, Livio Catullo Stecchini, Robert&#xA;  Bauval, and even Schwaller de Lubicz, and finally, visiting the key&#xA;  sites myself, I believe the situation we are faced with is one in&#xA;  which it can be demonstrated that c2500 BC someone designed and&#xA;  oversaw the construction of an object, the Great Pyramid, which&#xA;  encoded exceedingly accurate geodetic information along with profound&#xA;  geometric insight and subtly. While it is disorienting to recognize&#xA;  that so early in the chronology of human civilization there stands&#xA;  such a discontinuous alpha point, it exists, and attempting to dismiss&#xA;  its implications is no substitute for grappling honestly with them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;_&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=_C644p_kvVIC&amp;amp;pg=PA48&amp;amp;dq=height%20of%20the%20Great%20Pyramid%20x%2043200%20%3D%20Earth%20Polar%20Radius&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=WoRHVfizBIK1ggS27YHwBQ&amp;amp;ved=0CCQQ6AEwAQ#v=onepage&amp;amp;q=scale%20model&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Culture of Astronomy&lt;/a&gt;&lt;/em&gt; by Thomas Karl Dietrich: &quot;The Great Pyramid at Giza demonstrates that it is a scale model of the Earth in the proportion of 43,200 Great Pyramids equal to one Earth.&quot;&#xA;Also, from &lt;a href=&quot;http://www.cosmomyth.com/Accomplishments.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cosmo Myth&lt;/a&gt;, Thomas Karl Dietrich discovered  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Substantiating evidence that Eratosthenes did not discover the size of&#xA;  the Earth, but only read about it in the Library of Alexandria&#xA;  (&lt;em&gt;Culture of Astronomy&lt;/em&gt;). John Michell already discusses this fallacy in&#xA;  some detail in his famous &lt;em&gt;The New View over Atlantis&lt;/em&gt; (p.135-6). Thomas&#xA;  G. Brophy in &lt;em&gt;The Origin Map&lt;/em&gt; (p.110-11) actually states that&#xA;  Eratosthenes used data from the latitude and longitude of the ancient&#xA;  Nabta Playa which was also used in the Piri Re'is Map from Istanbul&#xA;  noticed by Charles Hapgood in &lt;em&gt;Maps of the Ancient Sea Kings&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And ... though a mathematical explanation was asked for, Richard Grossinger says in &lt;em&gt;The Night Sky: The Science and Anthropology of the Stars and Planets&lt;/em&gt;- &quot;History changes everything, and we must struggle to regain primary experience, whether visionary or scientific.&quot; (p.100) The shape of the earth was commonly known in the ancient practice of shamanism (including the Egyptians). Also, E. C. Krupp, &lt;em&gt;Skywatchers, Shamans &amp;amp; Kings: Astronomy and the Archaeology of Power&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;OK ... a little math :-)  ... The approximate diameter of the earth in miles: &lt;a href=&quot;http://www.wolframalpha.com/input/?i=8%20x%209%20x%2010%20x%2011&quot; rel=&quot;nofollow noreferrer&quot;&gt;7920 = 8 x 9 x 10 x 11&lt;/a&gt;. These numbers are encoded in the geometry of the Cosmological Circle. Michael Schneider, &lt;em&gt;Constructing the Cosmological Circle&lt;/em&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Cosmological Circle is a geometric diagram that has appeared in&#xA;  the arts, crafts, architecture, religion and literature of cultures&#xA;  around the world, and is associated with their golden ages. Because&#xA;  it's the visual representation of the harmony naturally inherent in&#xA;  the structure of the numbers 1 through 12, it encodes the ideal&#xA;  patterns and proportions toward which nature's forms strive. ... Its&#xA;  dimensions are described as the Heavenly City described in The Book of&#xA;  Revelation, seen in the plan of the Buddhist Brobodur temple,&#xA;  Stonehenge, Glastonbury, and described as Plato's ideal city Magnesia.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;http://www.wolframalpha.com/input/?i=55%20x%20144&quot; rel=&quot;nofollow noreferrer&quot;&gt;7920 = 55 x 144&lt;/a&gt;. Notice the &lt;a href=&quot;http://www.wolframalpha.com/input/?i=Fibonacci%20numbers&amp;amp;a=*C.Fibonacci%20numbers-_*IntegerSequence-&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fibonacci numbers&lt;/a&gt; and that &lt;a href=&quot;http://www.wolframalpha.com/input/?i=144%20%2F%2055&quot; rel=&quot;nofollow noreferrer&quot;&gt;144/55&lt;/a&gt; is approximately equal to the &lt;a href=&quot;http://www.wolframalpha.com/input/?i=Golden%20Number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Golden Number&lt;/a&gt; squared. Half the value of the the Golden Number squared is equal to the midradius of the &lt;a href=&quot;http://www.wolframalpha.com/input/?i=dodecahedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;dodecahedron&lt;/a&gt;. Schneider says the geometry is just a mask for the &lt;em&gt;Number Canon&lt;/em&gt;. Now, the question is how did they discover the &lt;em&gt;Number Canon&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;Number Canon&lt;/em&gt; removes the geometric mask and looks into the numbers&#xA;  at the heart of the Cosmological Circle. The way the twelve numbers&#xA;  combine and organize into an all-encompassing whole was the model for&#xA;  most of the ancient world as a microcosmic representation of the&#xA;  harmonious universe. It was used as a standard for defining&#xA;  relationships among weights, measures, music and the proportions of&#xA;  sacred art and architecture.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://www.wolframalpha.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WolframAlpha&lt;/a&gt;, ... this answer requested by &lt;em&gt;AnubhaV&lt;/em&gt; ... is mathematically correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Supporting references -- &lt;em&gt;Thoth: Architect of the Universe&lt;/em&gt;, by Ralph Ellis, in a review by &lt;a href=&quot;http://www.amazon.com/review/R3EHGTCUJQYRRG/ref=cm_cr_dp_title?ie=UTF8&amp;amp;ASIN=0953191303&amp;amp;nodeID=283155&amp;amp;store=books&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elliot Malach&lt;/a&gt; he says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The author ties the measurements and mathematics of the pyramids,&#xA;  Stonehenge and Avebury with the myth of Thoth, who educated mankind in&#xA;  math and the mysteries of the heavens, leaving repositories of&#xA;  knowledge throughout the Earth. Those repositories may not be &quot;inside&quot;&#xA;  these megalithic structures, but instead the fundamental mathematics&#xA;  encoded in the architecture of these structures themselves.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Thoth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thoth&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Egyptians credited him as the author of all works of science,&#xA;  religion, philosophy, and magic. The Greeks further declared him the&#xA;  inventor of astronomy, astrology, the science of numbers, mathematics,&#xA;  geometry, land surveying, medicine, botany, theology, civilized&#xA;  government, the alphabet, reading, writing, and oratory. They further&#xA;  claimed he was the true author of every work of every branch of&#xA;  knowledge, human and divine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1950" LastEditorUserId="1950" LastEditDate="2017-05-29T02:23:18.050" LastActivityDate="2017-05-29T02:23:18.050" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2155" PostTypeId="2" ParentId="2149" CreationDate="2015-04-10T17:59:27.303" Score="5" Body="&lt;p&gt;According to an entry attributed to &lt;a href=&quot;http://www.economics.soton.ac.uk/staff/aldrich/aldrich.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Aldrich&lt;/a&gt; in &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;, the word &quot;kernel&quot; in the algebraic sense was first used in the &lt;a href=&quot;http://books.google.com/books/about/Topological_Groups_Translated_from_the_R.html?id=0ClpNAEACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;1946 translation of Pontryagin's book Topological Groups&lt;/a&gt; by Emma Lehmer. So kernels of group homomorphisms came first. An unrelated use as in &quot;integral kernel&quot; occurs earlier, in Hilbert's 1904 Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen, in German, and in Bochner's 1909 Introduction to the Study of Integral Equations, in English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;https://math.stackexchange.com/questions/500920/word-origin-meaning-of-kernel-in-linear-algebra&quot;&gt;Mathematics Stack Exchange thread&lt;/a&gt;, that has speculations about Pontryagin's reasons for the name.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2017-04-04T21:54:59.707" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2156" PostTypeId="2" ParentId="1816" CreationDate="2015-04-10T20:09:19.853" Score="2" Body="&lt;p&gt;You are right that a bigger object will reach the ground in a miscroscopically smaller time if dropped independently. Note however that if dropped at the same time, this will still not be the case, as the earth will be attracted to the force of gravity created by both of those objects.&lt;/p&gt;&#xA;" OwnerUserId="2118" LastActivityDate="2015-04-10T20:09:19.853" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2157" PostTypeId="1" CreationDate="2015-04-11T01:06:33.887" Score="7" ViewCount="2980" Body="&lt;p&gt;The Michel electron is what we call the electron produced from muon decay, and it's named after Louis Michel. I mention this in a paper I'm writing, and I was told that I need to cite it. I can't find anything that explains the origin of the name, much less a paper that's citeable. Does anyone have any insights into this, or even more appreciated, knowledge of a paper that I could cite for this?&lt;/p&gt;&#xA;" OwnerUserId="2119" LastEditorUserId="26" LastEditDate="2015-04-13T05:17:59.137" LastActivityDate="2015-04-13T05:17:59.137" Title="How did the term &quot;Michel electron&quot; come about?" Tags="|terminology|theoretical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2158" PostTypeId="1" CreationDate="2015-04-09T00:09:36.880" Score="5" ViewCount="599" Body="&lt;p&gt;I recently tried to ask about when the last book was published that contained logarithmic tables. From what I understand, the question was unclear because it is unclear what published means.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, let me try with what I hope is a well-defined questions. What edition of CRC mathematical Tables was the last to contain logarithmic tables? The current edition (&lt;a href=&quot;http://www.crcpress.com/product/isbn/9781584882916&quot; rel=&quot;nofollow&quot;&gt;31st edition&lt;/a&gt;) does not contain the logarithmic tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: For those who don't know, logarithmic tables are tables with values of logarithms. As an example see &lt;a href=&quot;http://www.sliderules.info/a-to-z/log-tables.htm&quot; rel=&quot;nofollow&quot;&gt;this webpage&lt;/a&gt; or this &lt;a href=&quot;http://en.wikipedia.org/wiki/Mathematical_table#Tables_of_logarithms&quot; rel=&quot;nofollow&quot;&gt;Wikipedia article&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="John Doe" LastEditorUserId="26" LastEditDate="2015-04-11T17:56:25.153" LastActivityDate="2015-04-11T17:56:25.153" Title="What edition of CRC mathematical Tables was the last to contain logarithmic tables?" Tags="|mathematics|reference-request|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2159" PostTypeId="2" ParentId="2158" CreationDate="2015-04-09T02:13:30.780" Score="1" Body="&lt;p&gt;According to the table of contents,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mathtable.com/smtf/smtf30_toc.html&quot; rel=&quot;nofollow&quot;&gt;http://www.mathtable.com/smtf/smtf30_toc.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;30-th edition contains logarithms.&lt;/p&gt;&#xA;" OwnerUserId="64" OwnerDisplayName="Alexandre Eremenko" LastActivityDate="2015-04-09T02:13:30.780" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2160" PostTypeId="2" ParentId="2158" CreationDate="2015-04-09T03:24:36.987" Score="1" Body="&lt;p&gt;Bradis mathematical tables, 13th edition, published in 2010, contains logarithms:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.alleng.ru/d/math/math863.htm&quot; rel=&quot;nofollow&quot;&gt;http://www.alleng.ru/d/math/math863.htm&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2203" OwnerDisplayName="Anixx" LastActivityDate="2015-04-09T03:24:36.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2161" PostTypeId="1" AcceptedAnswerId="2172" CreationDate="2015-04-11T08:31:56.620" Score="8" ViewCount="930" Body="&lt;p&gt;As it is well known, every platonic solid has a dual (obtained by interchanging vertices and faces), which also happens to be a platonic solid. I would like to know who was the first person to discover this, or in which period it probably happened. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As we can possibly argue from this &lt;a href=&quot;http://en.wikipedia.org/wiki/Carved_Stone_Balls&quot;&gt;carved stone figures&lt;/a&gt;, platonic solids were already known long before ancient Greece (~600BC). So it's plausible the discovery wasn't recorded. On the other hand, in the &lt;a href=&quot;http://en.wikipedia.org/wiki/Platonic_solid#History&quot;&gt;Wikipedia article&lt;/a&gt; we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And proceeds to note that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which makes me believe that the discovery was either made by Euclid or before him.&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="62" LastEditDate="2015-04-11T12:56:23.023" LastActivityDate="2016-01-20T12:02:52.747" Title="Who discovered the duality between platonic solids?" Tags="|mathematics|discoveries|euclidean-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2163" PostTypeId="2" ParentId="2158" CreationDate="2015-04-11T15:38:05.610" Score="1" Body="&lt;p&gt;I have the 28th Edition of CRC Standard Mathematical Tables (1988) and the log tables begin on page 161.&lt;/p&gt;&#xA;" OwnerUserId="2121" LastActivityDate="2015-04-11T15:38:05.610" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2164" PostTypeId="1" CreationDate="2015-04-11T21:24:14.833" Score="5" ViewCount="516" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Tau_(particle)&quot; rel=&quot;noreferrer&quot;&gt;tau particle&lt;/a&gt; (so named because it was the third charged lepton, behind the electron and muon) was discovered in the 1970s by Martin Perl and colleagues. In &lt;a href=&quot;http://slac.stanford.edu/pubs/slacpubs/1750/slac-pub-1923.pdf&quot; rel=&quot;noreferrer&quot;&gt;one of the SLAC papers&lt;/a&gt;, Perl refers to the particle as the &quot;$\tau$ lepton&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've heard the particle referred to as the &quot;tau&quot;, the &quot;tau lepton&quot;, the &quot;tau particle&quot;, and other variants substituting in &quot;$\tau$&quot; for &quot;tau&quot;. I've also heard it referred to as the &quot;tauon&quot;, in keeping with the mu&lt;em&gt;on&lt;/em&gt; and electr&lt;em&gt;on&lt;/em&gt;. I've only heard it used from a few sources, though. I have no idea where the term originally came from, though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an interesting side note, &lt;a href=&quot;http://en.wikipedia.org/wiki/Talk:Tau_(particle)#name&quot; rel=&quot;noreferrer&quot;&gt;the Wikipedia talk page&lt;/a&gt; is in a furor over this. I have no idea if anyone on that page is actually who they say they are, or if any sources they give are true, but it seems that regardless of whether or not &quot;tauon&quot; is a valid term, nobody seems to know where it comes from.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where did the term &quot;tauon&quot; come from? Was it from Perl, or was it from someone completely different?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt; In theory, &lt;a href=&quot;http://xkcd.com/1167/&quot; rel=&quot;noreferrer&quot;&gt;there have been worse disputes&lt;/a&gt;, but I find the &quot;tau&quot;/&quot;tauon&quot; one more amusing.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-11T21:37:29.813" LastActivityDate="2015-04-13T02:04:02.923" Title="Where did the term &quot;tauon&quot; come from?" Tags="|naming-conventions|particle-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2165" PostTypeId="5" CreationDate="2015-04-12T16:25:11.827" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:25:11.827" LastActivityDate="2015-04-12T16:25:11.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2166" PostTypeId="4" CreationDate="2015-04-12T16:25:11.827" Score="0" Body="For questions about the history and development of how symbols and related objects are written." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:25:11.827" LastActivityDate="2015-04-12T16:25:11.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2167" PostTypeId="5" CreationDate="2015-04-12T16:34:43.677" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:34:43.677" LastActivityDate="2015-04-12T16:34:43.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2168" PostTypeId="4" CreationDate="2015-04-12T16:34:43.677" Score="0" Body="For questions regarding either the special or general theories of relativity." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:34:43.677" LastActivityDate="2015-04-12T16:34:43.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2169" PostTypeId="5" CreationDate="2015-04-12T16:35:11.740" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:35:11.740" LastActivityDate="2015-04-12T16:35:11.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2170" PostTypeId="4" CreationDate="2015-04-12T16:35:11.740" Score="0" Body="For questions about the history and/or rationale behind the names of objects or groups of objects." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-12T16:35:11.740" LastActivityDate="2015-04-12T16:35:11.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2171" PostTypeId="2" ParentId="2157" CreationDate="2015-04-12T19:53:54.173" Score="4" Body="&lt;p&gt;In 1935 Yukawa proposed that nucleon interaction is mediated by a new particle, which he termed &quot;meson&quot; because its mass was predicted to be intermediate between electron and nucleon. In 1937 Neddermeyer and Anderson discovered a candidate for such a particle in cosmic rays, which was called &quot;$\mu$ meson&quot;. However, after the war new experiments showed in 1947 that &quot;$\mu$ meson&quot; neither interacted strongly with matter, nor decayed as predicted. So it couldn't have been Yukawa's particle. Rabi even quipped &quot;who ordered that?&quot; about it, and the name &quot;$\mu$ meson&quot; was shortened to the modern &quot;muon&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Experiments soon revealed the manner of muon's decay, into an electron and a pair of neutrinos, and the shape of the electron spectrum was first calculated by Louis Michel in 1950. This is where &quot;Michel electron&quot; comes from. Michel introduced a single parameter $\rho$ to describe the shape. However, after the non-conservation of parity in weak interactions proposal of Lee and Yang three more parameters ($\eta$, $\xi$ and $\delta$) had to be added, and now all four are referred to collectively as &quot;Michel parameters&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.physicsegypt.org/nuppac07/npc7005.pdf&quot; rel=&quot;nofollow&quot;&gt;Depommier's paper&lt;/a&gt; gives more historical details, and a reference to the Michel's 1950 paper &lt;a href=&quot;http://iopscience.iop.org/0370-1298/63/5/311&quot; rel=&quot;nofollow&quot;&gt;Interaction between Four Half-Spin Particles and the Decay of the $\mu$-Meson&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-12T23:56:01.673" LastActivityDate="2015-04-12T23:56:01.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2172" PostTypeId="2" ParentId="2161" CreationDate="2015-04-12T20:25:59.233" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;http://www.newworldencyclopedia.org/entry/polyhedron#Prehistory&quot; rel=&quot;nofollow&quot;&gt;New World Encyclopedia&lt;/a&gt; people may have showed some awareness of the duality already in prehistoric times:&#xA;&quot;&lt;em&gt;Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland and may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyhedra, but also the relations of duality amongst some of them (that is, that the centers of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University.&lt;/em&gt;&quot; However, as Marius Kempe pointed out, the stones in question were &lt;a href=&quot;http://www.math.ucr.edu/home/baez/icosahedron/&quot; rel=&quot;nofollow&quot;&gt;altered to look more like Platonic solids&lt;/a&gt;. And even interpreting &quot;polyhedron&quot; loosely &lt;a href=&quot;http://archive.bridgesmathart.org/2014/bridges2014-441.pdf&quot; rel=&quot;nofollow&quot;&gt;icosahedron is still missing among them&lt;/a&gt;:&quot;&lt;em&gt;while the&#xA;symmetry of some of the Platonic solids is present, these are not polyhedra...  no balls with twenty knobs have icosahedral symmetry. Many of the carved stone balls have six knobs, and thus have approximate octahedral symmetry in their knob placement. Some have four knobs and tetrahedral symmetry.&lt;/em&gt;&quot; While duality is not restricted to Platonic solids, and some idea of it is possible without icosahedron, it is unclear what exactly &quot;relations of duality amongst some of them&quot; refers to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are differing accounts on whether Greeks knew of all five regular polyhedra before Theaetetus. But even for Theaetetus the duality, if he  noticed it, was likely little more than an afterthought or a curious fact rather than a construction/principle. When Archimedes later discovered semi-regular polyhedra, a.k.a. Archimedean solids, as far as we know neither he, nor anybody else for over two thousand years thought to swap their faces and vertices and see what happens. There was so little interest in polyhedra that the work of Archimedes did not survive the middle ages, and we only know of it from a mention in Pappus's Collection. Only in 1430 Paolo Uccello depicted two new &lt;a href=&quot;http://mathworld.wolfram.com/Kepler-PoinsotSolid.html&quot; rel=&quot;nofollow&quot;&gt;(non-convex) regular polyhedra&lt;/a&gt; in a mosaic on the floor of San Marco cathedral in Venice, and Kepler described them mathematically only in 1619 &lt;a href=&quot;http://en.wikipedia.org/wiki/Harmonices_Mundi#Content&quot; rel=&quot;nofollow&quot;&gt;Harmonices Mundi&lt;/a&gt;, two more were found by Poinsot in 1809. But again neither Kepler nor Poinsot made any use of duality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first use of duality as a principle appears to be in Catalan's Memoire sur la Theorie des Polyedres of 1865, where he constructed the duals to Archimedean solids, and noted that they are not semi-regular (faces are not regular polygons). These are now called Catalan solids. And only Brückner in his classic 1900 textbook Vielecke und Vielflache: Theorie und Geschichte, that summarized everything known about polyhedra at the time, defines what we refer to as &quot;duality&quot; explicitly. George Hart assembled &lt;a href=&quot;http://www.georgehart.com/virtual-polyhedra/references.html&quot; rel=&quot;nofollow&quot;&gt;annotated bibliography&lt;/a&gt; on polyhedra that may be useful for further digging. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-04T05:52:44.993" LastActivityDate="2015-05-04T05:52:44.993" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2173" PostTypeId="2" ParentId="2164" CreationDate="2015-04-12T21:07:40.583" Score="2" Body="&lt;p&gt;This Wikipedia thread is hilarious, but it is not hard to figure out where &quot;tauon&quot; came from after muon, pion, kaon, etc. When the same type of abbreviation is used enough times people start treating it as a generating rule, it's the same with other types of linguistic formation. It's a wonder we don't have rhoons, omegaons as well, maybe they just don't sound as good. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tauon is a natural abbreviation of &quot;tau-lepton&quot;, so it wouldn't surprise me if the original editor of the Wikipedia page and/or the astronomers mentioned in the thread came up with it on their own. It wouldn't surprise me either if eventually it becomes accepted usage. The reason for pushback may be that initially such abbreviations were applied to mesons, starting with &lt;a href=&quot;https://hsm.stackexchange.com/questions/2157/how-did-the-term-michel-electron-come-about/2171#2171&quot;&gt;&quot;mu-meson&quot;, originally mistaken for a meson,&lt;/a&gt; and shortened to &quot;muon&quot;, when it became clear. In fact, muon is a mu-lepton, although it is not usually called that, but it is only natural to want uniform naming for both mu and tau leptons. &#xA;People who know that muon was originally mu-meson are unlikely to infer from the name that &quot;tauon&quot; is a meson, if that's the concern.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This thread reminded me of &lt;a href=&quot;http://www.jstor.org/stable/10.4169/college.math.j.45.5.400&quot; rel=&quot;nofollow noreferrer&quot;&gt;Origins of Mathematical Words&lt;/a&gt; by Anthony Lo Bello, where he rages against word formation in mathematics. For example, &quot;repunit&quot; is &quot;a ludicrous word, a comical abbreviation of repeated unit... One should not do this sort of thing unless one's purpose is to be ridiculous&quot;. &quot;Nonhomogeneous&quot; is terrible because &quot;non&quot; is Latin, and &quot;homogeneous&quot; is Greek, and &quot;isometry&quot; is because although both parts are Greek, they are not spliced together as in Greek. I am not sure if Lo Bello realizes how comical his own objections are. But fortunately in the end common usage always wins over lexicographers.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-04-13T02:04:02.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2174" PostTypeId="1" AcceptedAnswerId="2175" CreationDate="2015-04-12T23:53:32.697" Score="5" ViewCount="1119" Body="&lt;p&gt;In his experiments to disprove the theory of &lt;a href=&quot;http://en.wikipedia.org/wiki/Spontaneous_generation&quot; rel=&quot;nofollow&quot;&gt;spontaneous generation&lt;/a&gt;, &lt;a href=&quot;http://en.wikipedia.org/wiki/Louis_Pasteur&quot; rel=&quot;nofollow&quot;&gt;Louis Pasteur&lt;/a&gt; used a &lt;a href=&quot;http://en.wikipedia.org/wiki/Swan_neck_duct&quot; rel=&quot;nofollow&quot;&gt;swan-necked flask&lt;/a&gt;. One &lt;a href=&quot;http://www.daviddarling.info/encyclopedia/P/Pasteur.html&quot; rel=&quot;nofollow&quot;&gt;source&lt;/a&gt; implies that Pasteur invented it himself, but I haven't been able to find anything definite.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To address the criticism leveled at Spallanzani's early experiments, namely that boiling might destroy some &quot;vital principle&quot; in air, Pasteur devised a long swan-necked flask. Air could reach the flask through the opening but dust particles and microorganisms could not, because the curved neck served as a trap.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="62" LastEditorUserId="26" LastEditDate="2015-04-13T05:14:40.957" LastActivityDate="2015-04-13T23:09:47.493" Title="Who invented the swan-necked flask?" Tags="|biology|engineering|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2175" PostTypeId="2" ParentId="2174" CreationDate="2015-04-13T02:02:47.407" Score="4" Body="&lt;p&gt;No, Pasteur did not invent swan necked bottles, he only invented a novel use for them as described in the quote. Venetian craftsmen produced swan necked glass bottles for decorative purposes back in 16th century. They were imported to Iran under the reign of Shah Abbas I (1587–1629) and became a traditional glassware there, the Persian name is &lt;a href=&quot;http://en.wikipedia.org/wiki/Swan-neck_bottle&quot; rel=&quot;nofollow&quot;&gt;ashkdān&lt;/a&gt;. Their necks are not as bent as Pasteur's however.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 17th century some experimenters used inverted bottles to collect gas products of a chemical reaction, e.g. Boyle did in 1671 to collect hydrogen, which he called &quot;certain kind of air&quot;. A major improvement was &lt;a href=&quot;https://books.google.com/books?id=1VckAQAAMAAJ&amp;amp;pg=PA345&amp;amp;lpg=PA345&amp;amp;dq=history+swan+neck+Bottle++-pasteur&amp;amp;source=bl&amp;amp;ots=WPHJENdLg5&amp;amp;sig=PrvJdhF7Z89G_z67umlUb4pCgBU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=exkrVeH8EYfwoASF94CICw&amp;amp;ved=0CFkQ6AEwEA#v=onepage&amp;amp;q=history%20swan%20neck%20Bottle%20%20-pasteur&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;pneumatic trough&lt;/a&gt;, invented by &lt;a href=&quot;https://medicalphysiologyonline.wordpress.com/2011/10/&quot; rel=&quot;nofollow&quot;&gt;Stephen Hales&lt;/a&gt; around 1727, which allowed collecting gases over water in a different vessel from the one where the reaction takes place. A swan necked bottle was part of the apparatus, and had a flexible neck, which was shaped more like Pasteur's. &lt;a href=&quot;http://www.jstor.org/stable/229488?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Lavoisier, Cavendish and Priestley&lt;/a&gt; used it among others.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-13T23:09:47.493" LastActivityDate="2015-04-13T23:09:47.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2176" PostTypeId="1" AcceptedAnswerId="2209" CreationDate="2015-04-13T15:29:53.657" Score="7" ViewCount="607" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/George_de_Hevesy&quot;&gt;George de Hevesy&lt;/a&gt; famously dissolved the Nobel Prizes of Max von Laue and James Franck during the German invasion of Denmark. The Nazis had prohibited any German scientist from receiving or keeping a Nobel Prize, and de Hevesy hid the solution in a laboratory, keeping them safe until after the war.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't been able to find any information about cases where Nobel Prizes were actually seized by the Nazis, but I assume that there were other cases. Which German scientists had their Nobel Prizes seized during World War II?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="4251" LastEditDate="2020-07-12T14:42:49.653" LastActivityDate="2020-07-12T14:42:49.653" Title="Which German scientists had their Nobel Prizes seized during World War II?" Tags="|nobel-prize|world-war-2|nazi-germany|scientists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2177" PostTypeId="1" CreationDate="2015-04-13T15:51:37.963" Score="-1" ViewCount="210" Body="&lt;p&gt;Has anyone ever considered the connection between the five perfect solids and the three most important music intervals of 2, 1.5, 1.25 and their two counterparts, 1.33333 and 1.6? The tetrahedron could be assigned to the octave 2, but what would be the best fit for the others, given there is a sort of duality between them, e.g. 1.5 x 1.33333 = 2?&lt;/p&gt;&#xA;" OwnerUserId="1763" LastEditorUserId="62" LastEditDate="2015-04-15T01:46:00.503" LastActivityDate="2015-04-15T01:46:00.503" Title="Five perfect solids" Tags="|euclidean-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2178" PostTypeId="1" AcceptedAnswerId="2199" CreationDate="2015-04-14T01:34:23.993" Score="5" ViewCount="985" Body="&lt;p&gt;As far as I know, not many (if any) &lt;a href=&quot;http://en.wikipedia.org/wiki/Alchemical_symbol&quot; rel=&quot;nofollow&quot;&gt;alchemical symbols&lt;/a&gt; have survived in modern nomenclature of science, either in chemistry or any other. I think $\LaTeX$ doesn't even support most of them! I know some symbols are quite cumbersome to draw, but not all of them, so I'd like to know:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Examples of surviving symbols which can be traced back to alchemy (not the obvious planetary ones in astronomy!).&lt;/li&gt;&#xA;&lt;li&gt;An explanation of why most of them didn't survive. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="520" LastEditorUserId="55" LastEditDate="2015-04-16T18:37:06.533" LastActivityDate="2016-02-05T21:38:39.873" Title="Modern usage of alchemical symbols" Tags="|chemistry|notation|terminology|alchemy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2179" PostTypeId="2" ParentId="2139" CreationDate="2015-04-14T03:21:34.223" Score="0" Body="&lt;p&gt;Julius Mayer&#xA;-conservation of energy and kinetic origin of heat&#xA;-gravity is not a force (prior to Clifford and Einstein, who showed it's curvature)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bernard Riemann (AE mentioned him below; his work inspired Clifford on gravity and was directly important for differential geometry, the core of relativity theory)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;John Waterston (see Haldane, J. (ed.) (1928) The Collected Scientific Papers of John James Waterston. This is all his works in one place, but an incredibly rare edition. You'll only get it from a university library. Would be nice if somebody scanned it to make more widely available---although too late for me as I made notes.)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;kinetic theory of heat in mathematical detail (the only predecessor was a page in Bernoulli's hydrodynamics, a diagram in fact)&lt;/li&gt;&#xA;&lt;li&gt;neural networks / connectionism prior Darwin and Ziehen&lt;/li&gt;&#xA;&lt;li&gt;first statement of the ergodic problem&lt;/li&gt;&#xA;&lt;li&gt;phase transitions must be accounted for energetically (Kelvin ignored them and thought for a couple months he found disproof of energy conservation ... )&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="163" LastActivityDate="2015-04-14T03:21:34.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2180" PostTypeId="2" ParentId="2177" CreationDate="2015-04-14T09:59:54.133" Score="1" Body="&lt;p&gt;The notion that the proportions of geometrical shapes, musical harmony, and other natural relationships are all interconnected and reflect some all-embracing harmony of the spheres goes back to classical antiquity. It is expressed most clearly (in particular with regard to musical intervals) in the pseudo-Platonic Epinomis.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-04-14T09:59:54.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2181" PostTypeId="1" CreationDate="2015-04-14T14:56:00.560" Score="6" ViewCount="1997" Body="&lt;p&gt;Ten is special to humans, as there are 10 fingers on two hands, and fingers are still the basic counting medium for people.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;So, was there any &lt;strong&gt;digit representing the number &quot;10&quot;&lt;/strong&gt; in a positional system historically used by any civilizations?&lt;/h2&gt;&#xA;&#xA;&lt;h2&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;We could have taken &lt;strong&gt;A&lt;/strong&gt; as a digit to represent number 10, and thus the numbers would have been:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;0   1   2   3   4   5   6   7   8   9   A&#xA;10  11  12  13  14  15  16  17  18  19  1A&#xA;...&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Obviously, there would have been new methods of calculation.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Why was 10 not given its own digit, and instead was represented by using other digits?&lt;/h2&gt;&#xA;" OwnerUserId="2127" LastEditorUserId="55" LastEditDate="2015-04-17T18:10:25.607" LastActivityDate="2022-06-13T03:18:45.807" Title="Has a digit ever been used to represent the number &quot;10&quot;?" Tags="|mathematics|notation|computation|" AnswerCount="6" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="2182" PostTypeId="1" AcceptedAnswerId="2184" CreationDate="2015-04-14T15:02:06.840" Score="7" ViewCount="305" Body="&lt;p&gt;During the introductory lecture to a cosmology course I'm currently taking, there was a brief discussion of some of the &quot;highlights&quot; of the Big Bang model. One of these is, of course, &lt;a href=&quot;http://en.wikipedia.org/wiki/Recombination_%28cosmology%29&quot; rel=&quot;noreferrer&quot;&gt;recombination&lt;/a&gt;. This is the first moment in the history of the Universe that electrons bonded to nuclei, forming atoms: The Universe then became transparent, and this is when the cosmic microwave background radiation &quot;last scattered&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everybody in the room noted the strangeness of this name: The prefix &lt;em&gt;re-&lt;/em&gt; is wholly unjustified! The wikipedia page offers no explanation and although I found &lt;a href=&quot;https://books.google.de/books?id=t8O-yylU0j0C&amp;amp;pg=PA284&amp;amp;lpg=PA284&amp;amp;dq=recombination+cosmology+misnomer&amp;amp;source=bl&amp;amp;ots=zC3ZyJJpdK&amp;amp;sig=UFyisFg6pgrN6i-ORHC0-1QUHRE&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=KCotVeb_Ac-V7AbI44CICw&amp;amp;sqi=2&amp;amp;ved=0CD4Q6AEwBg#v=onepage&amp;amp;q=recombination%20cosmology%20misnomer&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;some comments that seemed relevant in a book&lt;/a&gt;, it is still quite unclear to me how the term came into mainstream usage. Would anyone have the original reference, or a plausible explanation (possibly based on the above link)?&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2015-04-14T20:53:00.213" LastActivityDate="2015-04-16T23:09:05.700" Title="What is the origin of the term recombination?" Tags="|terminology|theoretical-physics|cosmology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2183" PostTypeId="2" ParentId="2181" CreationDate="2015-04-14T15:02:17.543" Score="6" Body="&lt;p&gt;Decimal &quot;10&quot; = Hexadecimal &quot;A&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you don't consider that a &quot;digit&quot;, but we'll necessarily have to broaden the definition of &quot;digit&quot; to include anything that's not 0-9.&lt;/p&gt;&#xA;" OwnerUserId="2128" LastActivityDate="2015-04-14T15:02:17.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2184" PostTypeId="2" ParentId="2182" CreationDate="2015-04-14T18:00:02.487" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Dissociation_%28chemistry%29&quot; rel=&quot;nofollow&quot;&gt;Recombination&lt;/a&gt; is a general term in chemistry for assembling molecules from atoms, ions and radicals, whether or not they were dissociated into them first. So the analogy holds. And the term is directly in line with &lt;a href=&quot;http://en.wikipedia.org/wiki/Plasma_recombination&quot; rel=&quot;nofollow&quot;&gt;usage in plasma physics&lt;/a&gt;, where it means &quot;a process by which positive ions of a plasma capture a free (energetic) electron and combine with electrons&quot;, again regardless of whether ionization occurred prior to that. This is pointed out on the &lt;a href=&quot;http://en.wikipedia.org/wiki/Talk%3ARecombination_%28cosmology%29&quot; rel=&quot;nofollow&quot;&gt;Wikipedia talk page&lt;/a&gt; that discusses the &quot;misnomer&quot;. &lt;a href=&quot;http://www.jetp.ac.ru/cgi-bin/dn/e_028_01_0146.pdf&quot; rel=&quot;nofollow&quot;&gt;Zeldovich, Kurt and Syunyaev&lt;/a&gt;, for example, write in their 1968 paper:&quot;&lt;em&gt;In the hot model of the universe it is assumed that at an early stage of the expansion the fully ionized plasma is in equilibrium with radiation. Cooling as a result of expansion leads to recombination&lt;/em&gt;&quot; with reference to &lt;a href=&quot;http://journals.aps.org/pr/abstract/10.1103/PhysRev.70.572.2&quot; rel=&quot;nofollow&quot;&gt;Gamow's famous &quot;hot Big Bang&quot; paper&lt;/a&gt; from 1946. &lt;a href=&quot;https://books.google.com/books?id=eq7TfxZOzSEC&amp;amp;pg=PA109&amp;amp;lpg=PA109&amp;amp;dq=Gamow+physics+review+1946&amp;amp;source=bl&amp;amp;ots=Sg0NCNmWY-&amp;amp;sig=BKNJ5wxrJqAH3aAGP8XzF-xdam8&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=di0wVeGyGYmGsAXXoYH4Cg&amp;amp;ved=0CFsQ6AEwCA#v=onepage&amp;amp;q=Gamow%20physics%20review%201946&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Cosmology and Controversy&lt;/a&gt; gives a detailed historical account of Gamow's discovery. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_the_Big_Bang_theory#1950_to_1980s&quot; rel=&quot;nofollow&quot;&gt;according to Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before the late 1960s, many cosmologists thought the infinitely dense and physically paradoxical singularity at the starting time of Friedmann's cosmological model could be avoided by allowing for a universe which was contracting before entering the hot dense state, and starting to expand again. This was formalized as Richard Tolman's oscillating universe. In the sixties, Stephen Hawking and others demonstrated that this idea was unworkable, and the singularity is an essential feature of the physics described by Einstein's gravity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So if they believed that the Big Bang followed the Big Crunch then electrons and nuclei literally recombined. Gamow for instance favored this picture, he stressed that &quot;from the physical point of view we must forget entirely about the precollapse period&quot;. More recently, the oscillating universe has been revived under the name &lt;a href=&quot;http://en.wikipedia.org/wiki/Big_Bounce#Recent_developments_in_the_theory&quot; rel=&quot;nofollow&quot;&gt;Big Bounce&lt;/a&gt; in some non-Einstein gravity theories.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-16T23:09:05.700" LastActivityDate="2015-04-16T23:09:05.700" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2185" PostTypeId="2" ParentId="2181" CreationDate="2015-04-14T18:15:52.963" Score="5" Body="&lt;p&gt;Mauro has already mentioned the Roman numerals. The Greeks of the classical period used the letters of the alphabet to represent numbers. The first nine letters (A to Θ)  stand for the units from 1 to 9, the next nine letters (I to Ϙ) stand for the multiples of ten (10, 20, 30…), the next nine (from P to ϡ) are the hundreds (100, 200…). You can write (for example) 111 as PIA.  If these are not digits, then I do not know what a digit is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Hellenistic period the Jews and Arabs and others adopted this system, using the letters of the Hebrew and Arabic alphabets to represent numbers in exactly the same way. This continues throughout the Middle Ages.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-04-14T18:15:52.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2186" PostTypeId="2" ParentId="2181" CreationDate="2015-04-14T21:30:53.757" Score="14" Body="&lt;p&gt;Yes, it has been: &lt;img src=&quot;https://i.stack.imgur.com/eUqLJ.gif&quot; alt=&quot;cuneiform 10&quot;&gt;, or more stylized &lt;img src=&quot;https://i.stack.imgur.com/CGpvd.png&quot; alt=&quot;Babylonian 10&quot;&gt;, the depression made by the tip of a Babylonian wedge shaped stylus on a clay tablet. When a circular stylus was used (rarely) the symbol was just $\bigcirc$. The earliest positional system was &lt;a href=&quot;http://en.wikipedia.org/wiki/Sexagesimal#Babylonian_mathematics&quot; rel=&quot;noreferrer&quot;&gt;sexagesimal&lt;/a&gt;, with base 60, so it had cuneiform symbols for all digits from 1 to 59. Babylonians used it since before 2000 BC for commercial bookkeeping, etc. The absence of zero caused ambiguities, e.g. 1 and 60 had the same symbol. In the medial positions, however, zero was represented by a blank and later by a placeholder symbol &lt;img src=&quot;https://i.stack.imgur.com/2oLtA.png&quot; alt=&quot;enter image description here&quot;&gt;. Greek astronomers replaced it with $o$ in the second century BC, that could be used at the end of a number also, removing the ambiguities. They also used their letter $\iota$ for 10 instead of Babylonian cuneiform, even though it was the 9th letter of their alphabet (archaic letter $\digamma$ was used for 6).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the use of zero as a placeholder much predates its use as a number, and Indians learned about it from Ptolemy's Almagest, if not earlier. Therefore, they would have needed no separate symbol for 10 when they switched to decimal notation. But in any case, zero is already used in 3rd century AD even as a number, while the &lt;a href=&quot;http://en.wikipedia.org/wiki/0_%28number%29#India&quot; rel=&quot;noreferrer&quot;&gt;decimal notation first appears&lt;/a&gt; much later, around 458 AD, so the problem never arose.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But 10 was not special among sexagesimal digits, so perhaps more in the spirit of the question is the symbol $\big|$ used in &lt;a href=&quot;http://books.google.com/books/about/The_History_of_Mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;noreferrer&quot;&gt;Chinese proto-decimal system&lt;/a&gt; before 4th century BC. That system had hieroglyphs for digits from 1 to 9, and for the powers of 10. Although digits were written in order we write them today to form a number, the symbol for a power of ten was placed above or below each, so its value wasn't indicated by position alone. This allowed for non-ambiguous representation without even placeholder 0, and the number could still be recovered if the digits got scrambled. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, &lt;a href=&quot;http://en.wikipedia.org/wiki/Suanpan&quot; rel=&quot;noreferrer&quot;&gt;hexadecimal system was also used in China&lt;/a&gt;, mostly for calculations with weights. This was done on abacus (counting board) starting around 190 AD, and since abacus is not paper what represented 10 was not a symbol, but an arrangement of beads.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-08T21:19:47.143" LastActivityDate="2016-09-08T21:19:47.143" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2187" PostTypeId="5" CreationDate="2015-04-14T23:42:28.050" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:42:28.050" LastActivityDate="2015-04-14T23:42:28.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2188" PostTypeId="4" CreationDate="2015-04-14T23:42:28.050" Score="0" Body="For questions about the attractive force between two objects which possess gravitational mass or energy." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:42:28.050" LastActivityDate="2015-04-14T23:42:28.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2189" PostTypeId="5" CreationDate="2015-04-14T23:43:23.150" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:43:23.150" LastActivityDate="2015-04-14T23:43:23.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2190" PostTypeId="4" CreationDate="2015-04-14T23:43:23.150" Score="0" Body="For questions about the mathematical field studying functions, focusing on infinitesimals and rates of change." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:43:23.150" LastActivityDate="2015-04-14T23:43:23.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2191" PostTypeId="5" CreationDate="2015-04-14T23:45:13.323" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:45:13.323" LastActivityDate="2015-04-14T23:45:13.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2192" PostTypeId="4" CreationDate="2015-04-14T23:45:13.323" Score="0" Body="For questions about events occurring in Greece and the immediate areas around it between 800 B.C. and 600 A.D." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:45:13.323" LastActivityDate="2015-04-14T23:45:13.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2193" PostTypeId="5" CreationDate="2015-04-14T23:46:32.343" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:46:32.343" LastActivityDate="2015-04-14T23:46:32.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2194" PostTypeId="4" CreationDate="2015-04-14T23:46:32.343" Score="0" Body="For questions about the mathematical study of shapes and space based on the works of Euclid." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:46:32.343" LastActivityDate="2015-04-14T23:46:32.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2195" PostTypeId="5" CreationDate="2015-04-14T23:47:59.827" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:47:59.827" LastActivityDate="2015-04-14T23:47:59.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2196" PostTypeId="4" CreationDate="2015-04-14T23:47:59.827" Score="0" Body="For questions about the branch of philosophy studying science, scientists, the scientific method and related topics." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:47:59.827" LastActivityDate="2015-04-14T23:47:59.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2197" PostTypeId="5" CreationDate="2015-04-14T23:49:34.020" Score="0" Body="" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:49:34.020" LastActivityDate="2015-04-14T23:49:34.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2198" PostTypeId="4" CreationDate="2015-04-14T23:49:34.020" Score="0" Body="For questions about geology, volcanology, seismology and related fields of science." OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-04-14T23:49:34.020" LastActivityDate="2015-04-14T23:49:34.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2199" PostTypeId="2" ParentId="2178" CreationDate="2015-04-15T01:34:26.397" Score="3" Body="&lt;p&gt;By the end of 18th century the theory of four elements or the mystic duality between planets and metals, which among other things informed the formation of alchemic symbols, did not have much currency with chemists. Neither did traditional alchemic objectives like the elixir of life everlasting, panacea (universal cure), or transmutation of metals. The old notation suggested connections that weren't there, and missed the ones that were. There was a &lt;a href=&quot;http://www.reddit.com/r/chemistry/comments/2sd5xa/alchemy_symbols_use_for_modern_chemistry/&quot; rel=&quot;nofollow&quot;&gt;similar question on Reddit&lt;/a&gt;, where a commenter put it bluntly:&quot;&lt;em&gt;Until after 1800, they had no clue of chemical structures nor a notation for writing them. Although there were lots of empirical observations, alchemy actually contributed very little to the modern science of chemistry as a theoretical structure.&lt;/em&gt;&quot; He sources &lt;a href=&quot;http://pubs.acs.org/cen/books/8251/8251books1.html&quot; rel=&quot;nofollow&quot;&gt;Buckingham's Chasing the Molecule&lt;/a&gt; for the sentiment.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end of of 18th century chemists started figuring out real relations between different compounds, and anticipating &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_molecular_theory#18th_century&quot; rel=&quot;nofollow&quot;&gt;molecular structure and valency bonds&lt;/a&gt;. In 1803 John Dalton explicitly introduced the idea of atoms &quot;hooking up&quot; together to form molecules, and in 1808 replaced most of alchemic notation &lt;a href=&quot;http://www.vanderkrogt.net/elements/chemical_symbols.php&quot; rel=&quot;nofollow&quot;&gt;with his own&lt;/a&gt;, incorporating the idea of elements forming compounds. Berzelius introduced more or less modern notation in 1813-14, flushing out what little was left of alchemic symbology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the symbols survive for incidental reasons. &lt;a href=&quot;http://www.ancient-symbols.com/alchemy_symbols.html&quot; rel=&quot;nofollow&quot;&gt;Diamond symbol&lt;/a&gt; is used by jewelers because it literally resembles a cut diamond. Caduceus (the rod of Hermes), appropriated by alchemists to represent amalgamation of sulphur and mercury, is used as a symbol of commerce, and in the US of medicine due to a &lt;a href=&quot;http://en.wikipedia.org/wiki/Caduceus_as_a_symbol_of_medicine&quot; rel=&quot;nofollow&quot;&gt;historical comedy of errors&lt;/a&gt;. It is unknown if John Wallis had Ouroboros in mind when he picked $\infty$ for infinity. Few mathematicians know (or care) that $\triangle$ is fire, $\nabla$ is water, $\square$ is urine, $\ominus$ is salt, $\oplus$ is sal armoniac (a mineral form of ammonium chloride), and $\odot$ is gold. Dalton, interestingly enough, used $\oplus$ for sulphur, and $\odot$ for hydrogen, they are that flexible. It is likely that mathematicians made up those symbols independently rather than picked them up from alchemic tables. But here is one seen in some logos, which is definitely borrowed, &lt;a href=&quot;http://chemistry.about.com/od/periodictableelements/ig/Alchemy-Symbols/Philosopher-s-Stone.htm&quot; rel=&quot;nofollow&quot;&gt;the Squared Circle&lt;/a&gt;, the 17th century glyph for the Phiolosopher's Stone. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-05T21:38:39.873" LastActivityDate="2016-02-05T21:38:39.873" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2200" PostTypeId="1" AcceptedAnswerId="2202" CreationDate="2015-04-15T09:44:16.517" Score="6" ViewCount="4354" Body="&lt;p&gt;Most of you know what I mean, but I will define it broadly: the binary search algorithm consists in searching iteratively for an element within an ordered set, by asking yes/no questions that will exclude about half of the remaining elements each loop, until there remains only a possible element. For example &quot;think of a number between 1 and 100; is it bigger than 50? &gt; no; is it bigger than 25? yes; is it bigger than 37?&quot; and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This algorithm is probably as old as child games, but there must be a first historical reference to it. Which one?&lt;/p&gt;&#xA;" OwnerUserId="2115" LastActivityDate="2018-04-11T16:07:15.767" Title="What is the first historical reference to the binary search algorithm?" Tags="|mathematics|computer-science|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2202" PostTypeId="2" ParentId="2200" CreationDate="2015-04-16T13:11:37.897" Score="9" Body="&lt;p&gt;In &quot;The Art of Computer Programming, Vol 3,  second edition, p422&quot;, it is said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Binary search was first first mentioned by John Mauchly, in what was perhaps the first published discussion of nonnumerical programming methods [&lt;em&gt;Theory and Techniques for the dsign of Electronic Digital computers&lt;/em&gt;, edited by G.W. Patterson, &lt;strong&gt;1&lt;/strong&gt; (1946), 9.7-9.8; &lt;strong&gt;3&lt;/strong&gt; (1946),22.8-22.9]. The method became well known to programmers, but nobody seems to have worked out the details of what should be done when N does not have the special form $2^n-1$. [See A.D.Booth, &lt;em&gt;Nature&lt;/em&gt; &lt;strong&gt;176&lt;/strong&gt; (1955), 565; A. I. Dumey, &lt;em&gt;Computers and Automation&lt;/em&gt; &lt;strong&gt;5&lt;/strong&gt; (December 1956), 7, where binary search is called &quot;Twenty Questions&quot;; Daniel D. McCracken, &lt;em&gt;Digital Computer Programming&lt;/em&gt; (Wiley, 1957), 201-203; and M. Halpern, &lt;em&gt;CACM&lt;/em&gt; &lt;strong&gt;1&lt;/strong&gt;, 1 (February 1958), 1-3.]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;D.H.Lehmer [&lt;em&gt;Proc. Symp. Appl. Math.&lt;/em&gt; &lt;strong&gt;10&lt;/strong&gt; (1960), 180-181] was apparently the first to publish a binary search algorithm that works for all N. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Binary trees similar to Fibonacci trees appeared in the pioneering work of the Norwegian mathematician Axel Thue as early as 1910.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On Google Scholar, the oldest article that I could find that has a reference to the binary search algorithm seems to be an article of 1962 by Thomas N. Hibbard called &quot;Some Combinatorial Properties of Certain Trees With Applications to Searching and Sorting&quot;. I do not have access to the full text, but according to google it is said inside the article &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The well-known binary search algorithm [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1897" LastActivityDate="2015-04-16T13:11:37.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2203" PostTypeId="2" ParentId="2200" CreationDate="2015-04-16T17:57:25.667" Score="2" Body="&lt;p&gt;Like skysurf I could not find any ancient record for binary search, so this is more of an extended comment. The problem may be that before computer science algorithms were either practical, or games for mathematically inclined, and the binary search wasn't attractive either practically or recreationally. Of course, if we cast the net wider something ancient can be found. Already Plato and Aristotle constructed classifications based on binary dichotomies, which were later depicted as binary &lt;a href=&quot;https://hsm.stackexchange.com/questions/1861/what-are-the-early-examples-of-taxonomies-in-the-history-of-science/1878#1878&quot;&gt;Porphyrian trees&lt;/a&gt;, and reincarnated in computer science as &lt;a href=&quot;http://en.wikipedia.org/wiki/Binary_search_tree&quot; rel=&quot;nofollow noreferrer&quot;&gt;sorted binary trees&lt;/a&gt;, that are custom made for binary searches. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A closer relative concerns continuous cousin of binary search, the method of bisection. It appears that &lt;a href=&quot;https://books.google.com/books?id=ilrlBwAAQBAJ&amp;amp;pg=PA308&amp;amp;lpg=PA308&amp;amp;dq=method+of+bisection+history+edwards+bolzano&amp;amp;source=bl&amp;amp;ots=GXUkK5RSrk&amp;amp;sig=x6NZn-Cj0o7y0l1t5KOVmH5ilHQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=oK4uVa6tMcr1oASdtICgCg&amp;amp;ved=0CC4Q6AEwAw#v=onepage&amp;amp;q=method%20of%20bisection%20history%20edwards%20bolzano&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;it was used by Bolzano in 1817&lt;/a&gt; to prove the intermediate value property (but see Mikhail's comment below on Stevin's prior use), and the numerical method based on it &lt;a href=&quot;https://sites.google.com/site/knowyourrootsmaxima/introduction/bisectionmethod&quot; rel=&quot;nofollow noreferrer&quot;&gt;was developed&lt;/a&gt; shortly thereafter. To find a zero of a continuous function on an interval where it takes values of opposite signs at the ends one bisects the interval, takes the subinterval with opposite signs, bisects again, etc. Unlike binary search bisection may never halt, but it converges to a zero, and produces approximations with obvious error bounds at each step. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I could not find an early reference for a yes/no number guessing game, although &quot;think of a number&quot; tricks with extra information occur as early as the Rhind papyrus (c. 1800 BC). When extra information is provided on how badly a guess fails, even in a simple form like cold/warmer/hot, it is also inefficient since it doesn't take it into account. The same goes for bisection, if we take into account end values rather than just signs &lt;a href=&quot;http://www.maa.org/sites/default/files/Boman-CMJ0941923.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;double false position&lt;/a&gt; converges much faster for practically relevant functions. And it is indeed ancient, occurring already in Chinese Nine Chapters from before 220 AD. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-03T06:42:51.440" LastActivityDate="2018-02-03T06:42:51.440" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2205" PostTypeId="1" AcceptedAnswerId="2206" CreationDate="2015-04-16T21:32:01.713" Score="11" ViewCount="934" Body="&lt;p&gt;I know that Part I was published while John McCarthy was working at MIT in 1960, in which he describes the LISP language, which he had invented 2 years prior, and is today considered one of the most seminal papers withing Computer Science. However, I can't seem to find any information about part II, other than an unreferenced sentence on Wikipedia stating that it &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lisp_%28programming_language%29#History&quot;&gt;was never published&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If somebody has any information concerning the paper, I'd be very interested to hear it. For example, was it even started? What were its contents supposed to be (about)? Are there any drafts in existence? Why was it never published?&lt;/p&gt;&#xA;" OwnerUserId="2142" LastActivityDate="2015-04-17T11:32:20.760" Title="Is anything known about part II of John McCarthy's Recursive Functions of Symbolic Expressions and Their Computation by Machine Part II?" Tags="|computer-science|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2206" PostTypeId="2" ParentId="2205" CreationDate="2015-04-17T11:32:20.760" Score="15" Body="&lt;p&gt;In &quot;History of Programming Languages&quot; by Richard L. Wexelblat page 178:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Part II was never written but was intended to contain applications to computing with algebraic expressions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that this part of the book was written by John MacCarthy himself.&lt;/p&gt;&#xA;" OwnerUserId="1897" LastActivityDate="2015-04-17T11:32:20.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2207" PostTypeId="1" AcceptedAnswerId="2208" CreationDate="2015-04-17T14:39:57.670" Score="9" ViewCount="330" Body="&lt;p&gt;We are familiar with the instances when experimental results disproved physics theories, such as the &lt;a href=&quot;http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment&quot;&gt;Michelson-Morley experiment&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What about computational physics results? To date, is there any computational physics simulation that cast a strong doubt or straightaway disprove a physics theory?&lt;/p&gt;&#xA;" OwnerUserId="2143" LastActivityDate="2015-04-20T23:38:02.163" Title="Have the results from Computational Physics disprove any physics theory?" Tags="|physics|computation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2208" PostTypeId="2" ParentId="2207" CreationDate="2015-04-17T19:00:53.483" Score="13" Body="&lt;p&gt;Perhaps one of the most famous computational model discoveries of the 20th century is Lorenz's observation in 1961 of chaotic behavior in a weather model. It overturned the existing consensus in meteorology, and led him to the discovery of a strange attractor in a simplified model of atmospheric convection. The implications are described in the now famous 1963 paper &lt;a href=&quot;http://eaps4.mit.edu/research/Lorenz/Deterministic_63.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Deterministic Nonperiodic Flow&lt;/a&gt;, that started modern chaos theory, where he stated &quot;&lt;em&gt;In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.&lt;/em&gt;&quot; In a &lt;a href=&quot;https://books.google.com/books?id=0yaM63MuqckC&amp;amp;pg=PA15&amp;amp;lpg=PA15&amp;amp;dq=vladimir+arnold+weather+prediction+week&amp;amp;source=bl&amp;amp;ots=0I5Gly9Mbf&amp;amp;sig=w_cWge1dru7ZGqEqRFcbYCQTfG8&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=RlQxVZm5HJLVoATjiYCgBQ&amp;amp;ved=0CDwQ6AEwBA#v=onepage&amp;amp;q=vladimir%20arnold%20weather%20prediction%20week&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1966 paper Arnold&lt;/a&gt; gave a general theorem about a class of models that confirmed Lorenz's insight, in his words he &quot;&lt;em&gt;used it to show that weather prediction is impossible for periods longer than two weeks&lt;/em&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Chaos_theory#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia describes the circumstances of Lorenz's discovery:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lorenz was using a simple digital computer, a Royal McBee LGP-30, to run his weather simulation... To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 was printed as 0.506. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However, Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But I have to quibble a little with the idea that an experiment can &quot;disprove&quot; a theory. Modern theories are too sophisticated to be disproved by a single experiment, or even by a group of them. Experimental data is based not only on a theory but also on a number of extra assumptions (coming from multiple other theories and &quot;common sense&quot;) that can be changed or discarded leaving the theory intact. The Michelson-Morley experiment for example was quickly reconciled with the ether theory by Lorentz, who proposed that (contrary to &quot;common sense&quot;) rulers contract and clocks slow down when moving through ether. In fact, &lt;a href=&quot;https://hsm.stackexchange.com/questions/2129/who-said-that-essentially-everything-in-theoretical-physics-had-already-been-dis/2132#2132&quot;&gt;Michelson himself still believed in 1902 that the ether theory was triumphing&lt;/a&gt;, which was the consensus at the time. Developing a scientific theory is a complicated process, that involves experiments, theoretical analysis, numerical simulations (or just computations and estimates by hand), etc., and any of these stages can provide what is later seen as &quot;the critical step&quot; in disproving a theory.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-04-20T23:38:02.163" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2209" PostTypeId="2" ParentId="2176" CreationDate="2015-04-18T13:33:51.937" Score="7" Body="&lt;p&gt;This is not a definitive answer, bu I did not find any proof that the Nazis had prohibited any German scientists from &lt;em&gt;keeping&lt;/em&gt; a Nobel Prize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story of how their medals were kept safe breaks into two parts: the first part is why they sent their medals to Denmark, and the second is why they were dissolved. There seems to be few years gap between the two events. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;About the sources:&lt;/strong&gt;&#xA;The best source I could find seems to be following : &lt;a href=&quot;http://www.gamma.nbi.dk/Galleri/gamma150/stephanschwarz.pdf&quot;&gt;The case of the bottled Nobel medals.&#xA;&lt;/a&gt; This note was written fairly recently (the most recent source was published in 2007), and the author seemed to have had access to some unpublished correspondance between some of the protagonists of the story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Sending the medals to Denmark.&lt;/strong&gt; It is probable that Franck left his medal at the NBI (Niels Bohr Institute) when he was a visitor there, that is &lt;a href=&quot;http://www.nbi.ku.dk/english/www/nobel/associated_with_nbi/&quot;&gt;between 1933 and 1935&lt;/a&gt;. The date at which von Laue left his is however unknown. In any case it is probable that Frank and von Laue left their medals at the NBI during the affair surrounding Ossietzky’s peace prize (1935-1936). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, note that Hitler only forbid German scientists to accept a nobel prize on &lt;a href=&quot;http://www.nature.com/nature/journal/v139/n3510/abs/139228c0.html&quot;&gt;January 30, 1937&lt;/a&gt;. That means that Franck and von Laue sent their medals because they &lt;em&gt;feared that their medals may get confiscated&lt;/em&gt; (as opposed to &lt;em&gt;because they would have been confiscated&lt;/em&gt;), either because of a general law, or because they could both be targeted as political opponents (Franck because he was Jewish, von Laue because of his vocal opposition to Nazism). I did not find any clue that a confiscation law was ever passed concerning Nobel Prizes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Dissolving the medals.&lt;/strong&gt; This part is better documented so here is what de Hevesy wrote: (emphasis mine)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;”My work was interrupted only one day during the enemy occupation of Denmark. When, in the morning of Denmark’s occupation [9 April 1940], I arrived in the laboratory, I found Bohr worrying about Max von Laue’s Nobel medal, which Laue had sent to Copenhagen for safe-keeping. &lt;strong&gt;In Hitler’s empire it was almost a capital offense to send gold out of the country and, Laue’s name being engraved into the medal, the discovery of this by the invading forces would have had very serious consequences for him.&lt;/strong&gt; (Three years later the invading army occupied Bohr’s Institute.) I suggested that we should bury the medal, but Bohr did not like this idea as the metal might be unearthed. I decided to dissolve it. While the invading forces marched in the streets of Copenhagen, I was busy dissolving Laue’s and also James Franck’s medals. After the war, the gold was recovered and the Nobel Foundation generously presented Laue and Franck with new Nobel Medals.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular Bohr and Hevesy weren't afraid of the medals being confiscated, bit of the fact that exporting gold was forbidden. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion.&lt;/strong&gt;&#xA;I think it is improbable that a law confiscated nobel prizes was ever passed in Nazi Germany (a definitive answer may be in &lt;a href=&quot;http://www.nature.com/nature/journal/v139/n3510/abs/139228c0.html&quot;&gt;this article&lt;/a&gt; but I do not have access to the full text) . This does not mean however that no medal was confiscated, only that it was done on an individual basis. In particular there is the case of Ossietzky's medal. However there are two things that may me think that it wasn't confiscated:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The first one is that he was already in jail by the time he was awarded the Nobel Peace prize, so it is unlikely that he ever received the medal.&lt;/li&gt;&#xA;&lt;li&gt;The second is that Shirin Ebadi is said to be the first Nobel Peace Prize winner whose medal has been confiscated by national authorities (&lt;a href=&quot;http://en.wikipedia.org/wiki/Shirin_Ebadi&quot;&gt;wikipedia&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1897" LastActivityDate="2015-04-18T13:33:51.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2211" PostTypeId="2" ParentId="2181" CreationDate="2015-04-18T23:56:50.727" Score="7" Body="&lt;p&gt;&lt;strong&gt;Posting History&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was already mentioned that the Mesopotamian (hexagezimal system) used a special symbol for our &quot;10&quot;. Also, In the modern hexadecimal system we have A for &quot;10&quot;. The Roman and the Greek specialties were also mentioned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;New stuff&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To add something to the already existing posts I copy here a paragraph from my own lecture notes about numbers systems:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://zgyorfi.weebly.com/uploads/4/5/7/3/45737565/mayan_numerals.png&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if these systems (Mezopotamian, the two Mayan systems and the hexadecimal system) use(d) special symbol for ten, those symbols, however, were not special like our nine in our modern decimal version of counting. Using a special symbol for ten did not make these systems of base eleven.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Base &quot;eleven&quot; could have been the most natural number system...  To support this idea I copy here another paragraph from the same lecture notes:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://zgyorfi.weebly.com/uploads/4/5/7/3/45737565/finger_numbers.png&quot; alt=&quot;&quot;&gt; symbols. That is, the number system of base eleven would have been the most natural one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking into account our toes the number system of base 21 seems to be even more natural, uncomfortable though...&lt;/p&gt;&#xA;" OwnerUserId="2151" LastEditorUserId="2151" LastEditDate="2018-02-17T16:29:48.587" LastActivityDate="2018-02-17T16:29:48.587" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2213" PostTypeId="2" ParentId="2181" CreationDate="2015-04-19T07:16:49.657" Score="2" Body="&lt;p&gt;Actually you can use any symbol to represent &quot;Zero&quot;. Suppose we take it &quot;X&quot;.&#xA;But in whatever index you are working with (let us say our index is &quot;n&quot;). the $n^{th}$ number in that index has to be (#symbol for &quot;1&quot;)(symbol for &quot;0&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example if your arithmetic is based on index of &quot;7&quot; and we use &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A&quot; for &quot;1&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;B&quot; for &quot;2&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;C&quot; for &quot;3&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;D&quot; for &quot;4&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;E&quot; for &quot;5&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;F&quot; for &quot;6&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And say &quot;X&quot; for &quot;0&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;then seventh number (equivalent to &quot;10&quot; in deimal arithmetic) will be &quot;AX&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answers why &quot;ten&quot; is represented just by two digits &quot;1&quot; and &quot;0&quot;. in a given order?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chek that any other representation say &quot;8&quot;&quot;9&quot; will collapse whole arithmetic!&lt;/p&gt;&#xA;" OwnerUserId="2154" LastActivityDate="2015-04-19T07:16:49.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2214" PostTypeId="1" AcceptedAnswerId="2217" CreationDate="2015-04-19T16:53:30.503" Score="6" ViewCount="219" Body="&lt;p&gt;We know that the electron was discovered in 1897 by J.J. Thomson. So was the knowledge about charge, current, electricity and all those terms related to it which we study today discovered after it, or did we have that knowledge from early times?&lt;/p&gt;&#xA;" OwnerUserId="2168" OwnerDisplayName="Shubham Prateek" LastActivityDate="2015-04-20T23:06:49.727" Title="Since when do humans know about electric charge?" Tags="|electricity|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2215" PostTypeId="1" CreationDate="2015-04-20T16:29:34.070" Score="1" ViewCount="45166" Body="&lt;p&gt;Every quantity has some SI units like distance, time, speed etc. Why do we prefer SI units for these quantities?&lt;/p&gt;&#xA;" OwnerUserId="2168" LastActivityDate="2019-07-02T01:48:53.633" Title="What is the importance of SI units in Physics?" Tags="|units|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2216" PostTypeId="2" ParentId="2215" CreationDate="2015-04-20T20:03:39.773" Score="5" Body="&lt;p&gt;The &lt;em&gt;Système International d'Unités&lt;/em&gt; (SI) is not just a collection of units but rather a &lt;em&gt;system&lt;/em&gt; of units*. The units all work together, so if you use some quantities in SI units, other quantities that you calculate from them will also be in SI units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You give the examples &quot;distance, time, speed etc.&quot; These units are related: if you give distance in meters (m) and time in seconds (s), the calculated speed found by dividing these units will be given in the SI unite meters per second (m/s). If you had an object and measured its mass in kilograms (kg) and calculated work or energy, you would get the SI unit Joule (J) which is equivalent to kg m^2/s^2. You know in advance just which unit you will get, with no possibility of getting calories, Calories, BTU's, or ergs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are other systems, such as cgs (centimeters, grams, seconds), and these are useful in some situations. But at the human-sized level, SI measures well most things that we work with. (The main exception is volume, where m^3 is too much for common use.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This deals with which particular metric system is used. Other, non-metric systems such as Imperial (or Avoirdupois or American customary units), are best dealt with separately. Suffice it to say that almost every country in the world has settled on metric systems.&lt;/p&gt;&#xA;" OwnerUserId="426" LastActivityDate="2015-04-20T20:03:39.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2217" PostTypeId="2" ParentId="2214" CreationDate="2015-04-20T20:37:21.890" Score="6" Body="&lt;p&gt;One manifestation of static electricity, the lightning, is known since the prehistoric times, a theoretical idea of electric charge first explicitly appears in &lt;a href=&quot;http://en.wikipedia.org/wiki/William_Gilbert_%28astronomer%29#Life_and_work&quot; rel=&quot;nofollow noreferrer&quot;&gt;works of William Gilbert around 1600&lt;/a&gt;. We already have a thread &lt;a href=&quot;https://hsm.stackexchange.com/questions/460/when-and-where-was-electricity-used-for-the-first-time?rq=1&quot;&gt;When and where was Electricity used for the first time?&lt;/a&gt;, that discusses history of electricity in more detail, so I will only add what is not mentioned there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first record of a practical use of electricity appears to be in medicine by Roman physician &lt;a href=&quot;http://www.painbytes.com/history/electroanalgesia-and-electromedicine/90-ancient-medicine-electricity-uses&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scribonius Largus around 46-47 AD&lt;/a&gt;, who prescribed electric shocks for treating foot-gout (arthritis), and severe headaches (migraine). The source of electric charge was the black torpedo fish, a kind of electric ray. Largus writes:&quot;&lt;em&gt;For any sort of podagria (foot-gout): when the pain comes on, it is good for one to put a living black torpedo-fish under his feet while standing on a beach (not dry but one on which the sea washes), until he feels that his whole foot and shank are numb just up to the knees. This will both relieve the current pain and alleviate future recurrences&lt;/em&gt;&quot;. Numbing the nerves provided short term pain relief for headaches also, and minus the torpedo fish the &lt;a href=&quot;http://www.nytimes.com/2006/11/07/health/07migr.html?pagewanted=all&amp;amp;_r=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;treatment is in use today&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is likely that Largus was reporting an already known treatment, but it is hard to tell how far back it goes. In Plato's dialogue named after him &lt;a href=&quot;http://en.wikipedia.org/wiki/Electric_ray#Relationship_to_humans&quot; rel=&quot;nofollow noreferrer&quot;&gt;Meno accuses Socrates&lt;/a&gt; of &quot;stunning&quot; people with his puzzling questions like a torpedo fish. Egyptian texts from around 2750 BC refer to electric catfish as &lt;a href=&quot;http://electricity.how-old-is-this.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;the thunderer of the Nile&quot;&lt;/a&gt;, that  &quot;protects&quot; all other fish, and there is a depiction of an electric catfish on a &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/00222938500770031#.VTWFY5OE6ew&quot; rel=&quot;nofollow noreferrer&quot;&gt;Narmer's palette from about 3100 BC&lt;/a&gt; &quot;if the pictogram has been correctly interpreted&quot;. So was &quot;the thunderer of the Nile&quot; used medicinally as early as that? Put that into a maybe category, despite claims on many internet sites there is no direct evidence. But it still appears that electric fish were the first practical sources of electricity. A connection between them and the lightning was made  some time between 12th and 15th centuries AD by Arabs, who started &lt;a href=&quot;http://electricity.how-old-is-this.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;referring to electric rays as &quot;raad&quot;&lt;/a&gt;, their word for lightning. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-04-20T23:06:49.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2218" PostTypeId="1" AcceptedAnswerId="2219" CreationDate="2015-04-21T00:23:32.547" Score="4" ViewCount="4796" Body="&lt;p&gt;I often see the &lt;a href=&quot;http://en.wikipedia.org/wiki/Dirac_delta_function&quot; rel=&quot;nofollow&quot;&gt;Dirac delta function&lt;/a&gt; defined as&#xA;$$\delta(x)=\lim_{a \to 0}\frac{1}{a \sqrt{\pi}}e^{-x^2/a^2}$$&#xA;Yet this is clearly equivalent to&#xA;$$\delta(x)=\lim_{b \to \infty}\frac{b}{\sqrt{\pi}}e^{-b^2x^2}$$&#xA;and you might say that it's a bit neater, getting rid of the division in the exponent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is the first definition so commonly used?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="26" LastEditDate="2015-04-27T08:25:30.570" LastActivityDate="2015-04-27T11:51:05.260" Title="Why is the Dirac delta &quot;function&quot; often presented as the limit of a Gaussian with a fraction in the exponent?" Tags="|mathematics|theoretical-physics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2219" PostTypeId="2" ParentId="2218" CreationDate="2015-04-21T09:30:47.367" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Paul_Dirac&quot; rel=&quot;nofollow&quot;&gt;Paul Dirac&lt;/a&gt; was a physicist and I think it's in that regard that a division in the exponent seems much more interesting and common than a product, because it is easier to assign a physical interpretation: $a$ can have the same dimension as $x$ which allows you to give $a$ a significance - for example the width of a potential well which area is constant.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="26" LastEditDate="2015-04-27T08:26:26.437" LastActivityDate="2015-04-27T08:26:26.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2220" PostTypeId="2" ParentId="2215" CreationDate="2015-04-21T13:33:48.620" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the importance of SI units in Physics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If by &quot;physics&quot; you mean units used by professional physicists and related disciplines, it's not as much as one would think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While motivated by science, SI units exist primarily to benefit commerce, not science. The ordinary, everyday phenomena described well in terms of SI units aren't the phenomena that physicists currently study. (They used to study those phenomena, but that was long ago. Physicists have moved on to studying much, much smaller and much, much bigger things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A widely used system is theoretical physics is &lt;a href=&quot;http://en.wikipedia.org/wiki/Planck_units&quot; rel=&quot;nofollow&quot;&gt;Planck units&lt;/a&gt;, where the gravitational constant $G$, the speed of light $c$, the reduced Planck constant $\hbar=h/(2\pi)$, the Coulomb constant $k_e = 1/(4\pi\varepsilon_0)$, and the Boltzmann constant $k_B$ all have numerical values of one. These choices lead to rather different values for the unit of length, unit of time, unit of mass, and unit of charge than in SI. The relationship isn't all that precise because the SI value of $G$ is only good to four decimal places of accuracy. That problem disappears in Planck units, where $G$ is exactly one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Astronomers and astrophysics who study very remote phenomena use redshift as their measure of time. They might translate this to years, but that requires a specific model of the expansion of the universe. Redshift is observable. The time at which some phenomenon that occurred long ago is not. What about distance? Distance is even more problematic. In their professional writing (i.e. peer-reviewed journal papers), you'll rarely see distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Astronomers and astrophysicists who study not quite so remote phenomena do use distance, but their unit of choice is the parsec. The issue of expansion of space is less problematic for something observed a few galaxies away. The unit of time to these astronomers and physicists is the Julian year, or 365.25 $\cdot$ 86400 SI seconds. The unit of mass is the solar mass. Astronomers and astrophysicists who study solar system bodies also use the solar mass as the unit mass, but the unit of distance used by these scientists is the astronomical unit, and the unit of time is typically a day (but sometimes a Julian year). The large uncertainty in $G$ once again gets in the way of translating to/from SI units.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a much, much smaller scale, particle physicists working at the atomic scale oftentimes use the electron charge and electron mass as the units of charge and mass; the other units are chosen to make $\hbar$, $k_e$, and $k_B$ all have numerical values of one. Nuclear physicists tend to make $c$, $\hbar$, and $k_B$ have numerical values of one. Some nuclear physicists use the proton mass as the unit of mass, others use the electron-volt as the unit of mass-energy (since $c=\hbar=k_B=1$, energy, mass, and temperature all have the same dimensions).&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="4567" LastEditDate="2016-08-30T01:10:09.547" LastActivityDate="2016-08-30T01:10:09.547" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2221" PostTypeId="2" ParentId="2218" CreationDate="2015-04-21T22:07:22.637" Score="3" Body="&lt;p&gt;Perhaps people are following the Wikipedia, but this form is by no means exclusive, &lt;a href=&quot;https://math.okstate.edu/people/binegar/4233/4233-l15.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; the one you prefer is used instead. If there is a physically canonical one it is $\delta(x)=\lim_{\sigma \to 0}\frac{1}{\sigma\sqrt{2\pi}}e^{-x^2/2\sigma^2}$, which contains the Gaussian distribution density with the standard deviation $\sigma$, see &lt;a href=&quot;http://hitoshi.berkeley.edu/221a/delta.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, or a version with the Planck constant in it. Other characteristics of the distribution, like entropy for example, are traditionally expressed in terms of $\sigma$ as well, making this form most convenient for look ups. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In quantum mechanics the Gaussian distribution represents a localized &quot;wave packet&quot; with &quot;characteristic width&quot; $\sigma$, so the limit is a transition to the complete localization of the packet in the &quot;position space&quot;. Gaussian wave packets are often used as &quot;typical&quot; in heuristic quantum mechanical calculations, and the ground state of the quantum oscillator happens to be of this form. The Heisenberg uncertainty principle is also usually expressed in terms of standard deviations, $\sigma_x\sigma_p\geq1/2$ (setting the Planck constant to $1$), where $\sigma_x$, $\sigma_p$ are the standard deviations of a wave function and its Fourier transform, which represents the wave function in the &quot;momentum space&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the $\sigma$ version is most natural from Dirac's point of view, my guess is that rescaling to $a=\sigma\sqrt{2}$ is done by industrious mathematicians, who are less interested in physical meanings. But reversing to $1/a$ is perhaps a step too far even for many of them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-21T22:22:12.910" LastActivityDate="2015-04-21T22:22:12.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2223" PostTypeId="1" AcceptedAnswerId="8273" CreationDate="2015-04-22T02:22:57.783" Score="29" ViewCount="3495" Body="&lt;p&gt;I used to think that the answer is Kolmogorov. So the &lt;a href=&quot;http://arxiv.org/pdf/math/0606533.pdf&quot;&gt;Shafer-Vovk's review&lt;/a&gt; of Kolmogorov's famous 1933 axiomatization of probability surprised me a bit:&quot;&lt;em&gt;Today, what Frechet and his contemporaries knew is no longer known. We know  Kolmogorov  and  what  came  after; we have  mostly  forgotten  what  came before... Our review of it will confirm what Frechet said in 1937 and what Kolmogorov says in the preface: it was a synthesis and a manual, not a report on new research. Like any textbook, its mathematics was novel for most of its readers, but its real originality was rhetorical and philosophical&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, Kolmogorov's book was a lot like Euclid's Elements, it systematized everything already known, and wiped out the predecessors. A quick look at Wikipedia confirms the latter, its &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_probability&quot;&gt;history of probability&lt;/a&gt; has a gaping void between Laplace (1812) and Kolmogorov, and manages to skip even those Kolmogorov himself names: Borel, Bernstein and von Mises. Another exhibit is the name of Gerovitch's popular article: &lt;a href=&quot;http://nautil.us/issue/4/the-unlikely/the-man-who-invented-modern-probability&quot;&gt;The Man Who Invented Modern Probability&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The review has a lot of interesting information. For example, Borel defined countably additive measures in 1898, and in 1905-1909 explored connections between them and geometric probability. Frechet generalized Borel/Lebesgue measures to abstract spaces, Hausdorff equated probability with them by 1914, and Bernstein gave a version of probability axioms in 1917. But it does not say much about random variables. The classical authors (Bernoulli, de Moivre, Poisson, Laplace) talk about random events, trials and distributions, but not something so abstract. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who and when introduced the concept of random variable, was it a &quot;basic notion&quot; before measure theory? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://arxiv.org/abs/1309.6488&quot;&gt;Seneta's paper&lt;/a&gt; is also interesting, but it rewrites everything, starting with Bernoulli's 1713 Ars Conjectandi, in modern notation with random variables, so it is hard to tell where either they, or the notation, originated.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who introduced the set builder notation involving random variables, like Pr$(|\xi|&amp;gt;\varepsilon)$, and when did it become common to express results in it? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-04-30T22:24:34.430" LastActivityDate="2020-07-31T23:13:44.580" Title="Who introduced random variables into probability?" Tags="|mathematics|discoveries|statistics|probability|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2224" PostTypeId="2" ParentId="317" CreationDate="2015-04-22T14:51:28.903" Score="8" Body="&lt;p&gt;Jean-Victor Poncelet was a prisoner of war in Russia in the years 1812–1814, and wrote Traité des propriétés projectives des figures there. This was an important work in projective geometry. More info on &lt;a href=&quot;http://en.wikipedia.org/wiki/Jean-Victor_Poncelet&quot; rel=&quot;noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Jean-Victor_Poncelet&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="661" LastActivityDate="2015-04-22T14:51:28.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2225" PostTypeId="2" ParentId="2223" CreationDate="2015-04-22T23:16:29.453" Score="8" Body="&lt;p&gt;This question has no definitive answer, because people were operating with random variables long before any rigorous definition was given.&lt;/p&gt;&#xA;&lt;p&gt;Probability theory begins in 16-th century, if not earlier. Cardano wrote a book on it, for example. In 1773 de Moivre wrote an important book where he essentially introduced the principal method of modern probability (harmonic analysis). This was much developed by Laplace. Important development happened in 19-th century (Chebyshev, for example). All these people were talking about random variables without giving&#xA;a precise definition. I mean a &amp;quot;precise definition&amp;quot; from the point of view of modern mathematician who only understands definitions given in terms of set theory. Of course there was no set theory yet.&lt;/p&gt;&#xA;&lt;p&gt;By the beginning of 20-th century probability was a highly developed theory, with substantial applications to physics (Brownian motion, for example). All essential applications of probability already existed by the beginning of 20-th century, including statistical mechanics, financial mathematics and, of course statistics.&lt;/p&gt;&#xA;&lt;p&gt;Attempts to extablish a rigorous justification in the spirit of newly developed set theory started in the second decade of 20-th century. One early system of axioms was due to S. Bernstein, another to J. Schauder. Still another system of foundations was due to von Mises.&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov's system was published in 1933. Gradually (not immediately!) it took over.&lt;/p&gt;&#xA;&lt;p&gt;To answer your other question, no this cannot be compared to Euclid. Euclid gave a systematic and comprehensive exposition of all mathematics that existed at that time. Kolmogorov's book is a tiny booklet, where he only proposes a new system of axioms, proves a few very general theorems. It is a little research monograph, not an exposition of the known things.&lt;/p&gt;&#xA;&lt;p&gt;Remark. Modern education system makes many young mathematicians think that there was no probability theory before Kolmogorov and no algebraic geometry before Grothendieck. Developing this principle one step further, one can say that there was no mathematics before Cantor:-) Because all modern mathematics is based on set theory.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-07-28T02:54:43.333" LastActivityDate="2020-07-28T02:54:43.333" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2226" PostTypeId="2" ParentId="2150" CreationDate="2015-04-23T12:43:01.343" Score="2" Body="&lt;p&gt;A &lt;a href=&quot;http://en.wikipedia.org/wiki/Durfee_square&quot; rel=&quot;nofollow&quot;&gt;Durfee square&lt;/a&gt; can be defined for every integer partition. It was introduced before 1883 (which makes it anterior to Pareto's work). It is the construct at the root of both Eddigton number and h-index.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example of a Durfee square :&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;x x x * * * *&#xA;x x x * *&#xA;x x x *&#xA;* * *&#xA;*&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Here the Durfee square of the set {1,3,4,5,7} is the 3x3 square highlighted as there are not 4 elements greater than 4 in the set.&lt;/p&gt;&#xA;" OwnerUserId="8" LastActivityDate="2015-04-23T12:43:01.343" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2227" PostTypeId="1" AcceptedAnswerId="2228" CreationDate="2015-04-17T07:45:39.323" Score="15" ViewCount="667" Body="&lt;p&gt;Statistical mechanics is a subject with a particularly rich history. I think of the early debates of Boltzmann and Loschmidt, the rather confusing differences between the approaches of Gibbs and Boltzmann, the philosophical efforts of E.T. Jaynes to phrase entropy in terms of information theory, all the way through to more modern work like Evan and Searles' fluctuation theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A beginner is swamped not only by the intrinsically difficult ideas, but also by the feeling that even the experts can't agree on the most basic definitions. I'm sure that's not true, but when you read Gibbs and Boltzmann it certainly feels that way!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what I'm looking for a book (ideally technical, like a textbook) which develops the ideas historically. Perhaps similar to Lanczos' book on mechanics. It would discuss historical controversies, and to what extent the modern theory has resolved them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does such a book exist? Thanks. &lt;/p&gt;&#xA;" OwnerUserId="3406" OwnerDisplayName="tom" LastEditorUserId="4251" LastEditDate="2020-06-06T12:58:48.117" LastActivityDate="2020-06-06T12:58:48.117" Title="Historical Survey of Statistical Mechanics" Tags="|physics|thermodynamics|statistical-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2228" PostTypeId="2" ParentId="2227" CreationDate="2015-04-17T23:34:22.803" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://www.researchgate.net/profile/Stefano_Bordoni/info&quot;&gt;Stefano Bordoni&lt;/a&gt;'s 2012 &lt;a href=&quot;http://garrigou.us.to/browse/book/4326&quot;&gt;&lt;strong&gt;&lt;em&gt;Taming Complexity&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt; (&lt;a href=&quot;https://www.researchgate.net/profile/Stefano_Bordoni/publication/234081619_Taming_Complexity._Duhem%27s_third_pathway_to_Thermodynamics/file/79e41511c9e8bd432e.pdf&quot;&gt;e-book from ResearchGate&lt;/a&gt;; &lt;a href=&quot;http://garrigou.us.to/misc/Physics%20papers%20and%20books/Duhem/Modern%20Duhem%20research/refs%20etc.%20from%20Bordoni%27s%20%C2%ABTaming%20Complexity%C2%BB/Needham%27s%20review%20of%20Bordoni.pdf&quot;&gt;review&lt;/a&gt;) is a good place to start.&lt;br&gt;&lt;sup&gt;(Bordoni has a master's degree in physics and &lt;em&gt;three&lt;/em&gt; PhDs, in the history of science, anthropology and epistemology of complexity, and philosophy.)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bordoni refers to&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Brush's 1986 &lt;strong&gt;&lt;em&gt;&lt;a href=&quot;http://garrigou.us.to/browse/book/4437&quot;&gt;The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century&lt;/a&gt;&lt;/em&gt;&lt;/strong&gt; and&lt;/li&gt;&#xA;&lt;li&gt;Brush's &amp;amp; Hall's 2003 &lt;strong&gt;&lt;em&gt;&lt;a href=&quot;http://garrigou.us.to/get/djvu/Kinetic%20Theory%20of%20Gases_%20An%20Anthology%20of%20Classic%20Papers%20with%20Historical%20Commentary%20-%20Brush%2C%20Stephen%20G.%20%26amp%3B%20Hall%2C%20Nancy%20S__4437.djvu&quot;&gt;Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary&lt;/a&gt;&lt;/em&gt;&lt;/strong&gt;,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;which are both excellent &quot;surveys of statistical mechanics,&quot; especially regarding the problem of whether reversible physical theory (e.g., mechanics) can account for irreversible phenomena (cf. Poincaré's famous &quot;&lt;em&gt;Le mécanisme et l’expérience&lt;/em&gt;&quot; in the aforementioned anthology).&lt;/p&gt;&#xA;" OwnerUserId="232" OwnerDisplayName="Geremia" LastActivityDate="2015-04-18T09:51:07.973" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2229" PostTypeId="1" AcceptedAnswerId="2231" CreationDate="2015-04-26T08:44:05.190" Score="4" ViewCount="200" Body="&lt;p&gt;It seems that both Bohr and Rutherford assumed circular orbits of electrons around the atom. But why did they do so, instead of assuming e.g. elliptical orbits?&lt;/p&gt;&#xA;" OwnerUserId="1994" LastEditorUserId="26" LastEditDate="2015-04-27T08:20:00.407" LastActivityDate="2015-04-27T08:20:00.407" Title="Why did the &quot;old quantum theory&quot; start off considering circular electron orbits?" Tags="|physics|quantum-mechanics|atomic-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2231" PostTypeId="2" ParentId="2229" CreationDate="2015-04-26T10:35:57.957" Score="3" Body="&lt;p&gt;It's a common assumption for planetary - ie central Newtonian field - models to consider the orbits to be circular. I don't think Bohr or Rutherford ever believed in circular orbits for electrons, but rather decided to use a circle to describe an orbit they believed was elliptic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rutherford-Bohr Model use the planetary system as an analogy, where the movement is in fact elliptic but can be approached by circular paths. Using circles allows you to handle discrete set of radii rather than discrete areas for the movement, hence making the description of orbital jumps easier, but I don't believe Bohr or Rutherford had any fundamental reason to believe the orbits where circular. Check the maths (it's an exercise we did in uni) and everything works fine for fixed areas and elliptic orbits. Actually modern view of the ground state of an electron tend to represent it as orbiting in a path of nil-area, so you would need elliptic orbits if you want it to have a speed. I believe Sommerfeld published a more refined classic model for electrons using elliptic orbits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Short answer: they wanted to keep it simple &lt;/p&gt;&#xA;" OwnerUserId="8" LastActivityDate="2015-04-26T10:35:57.957" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2232" PostTypeId="2" ParentId="195" CreationDate="2015-04-26T10:52:53.207" Score="0" Body="&lt;p&gt;&lt;strong&gt;Yes&lt;/strong&gt;, atomic bomb is a good name. And so is nuclear bomb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no reason to make a distinction between the two, and this distinction doesn't exist in some languages, which tends to show it is not based on scientific facts but on English usage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In French for exemple, we make a distinction between fission bomb (A-type) and fusion bomb (H-type), but both are called atomic/nuclear bombs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion A-bomb are called atomic bombs because they were the first to be developed, and when the name was taken and people developed H-bombs they had to use a less generic name to make the distinction. H-bombs use A-bombs to start the reaction, so there is no explanation why a fusion bomb should be less atomic than a fission bomb.&lt;/p&gt;&#xA;" OwnerUserId="8" LastActivityDate="2015-04-26T10:52:53.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2234" PostTypeId="1" AcceptedAnswerId="2238" CreationDate="2015-04-26T11:15:32.987" Score="3" ViewCount="1458" Body="&lt;p&gt;I'm researching historical use of geometry to find solutions to polynomial equations. I'd like to ask for those familiar with this topic, could you describe the use of geometry by early mathematicians to solve polynomial equations? Could you also offer some examples to illustrate the ideas?&lt;/p&gt;&#xA;" OwnerUserId="2188" LastEditorUserId="55" LastEditDate="2016-08-31T22:42:46.507" LastActivityDate="2018-05-05T22:50:57.983" Title="How was geometry historically used to solve polynomial equations?" Tags="|mathematics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2235" PostTypeId="1" AcceptedAnswerId="2237" CreationDate="2015-04-27T07:44:21.273" Score="15" ViewCount="642" Body="&lt;p&gt;I've searched, and found some interesting articles on the subject such as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Iron_catastrophe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;, which is a nice summary. But most of the references in these articles are to other popular culture articles/documentaries on the subject. I can't actually find any details on the original research and geologists who proposed the idea and when/where the first evidence that this event took place was unearthed (obviously the magnetic field is a glaring piece of that evidence) and how it was eventually interpreted to come to this conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would have thought that the first significant propositions for this would have perhaps come around the beginning of the 20th century... maybe....&lt;/p&gt;&#xA;" OwnerUserId="182" LastEditorUserId="182" LastEditDate="2018-04-27T08:19:12.353" LastActivityDate="2018-04-27T08:19:12.353" Title="When did we first begin to suspect that the Earth had an Iron Catastrophe?" Tags="|earth-sciences|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2236" PostTypeId="2" ParentId="2218" CreationDate="2015-04-27T11:51:05.260" Score="1" Body="&lt;p&gt;Writing the density as&#xA;$$&#xA;x\mapsto \frac 1 {a\sqrt{2\pi}} e^{-x^2/(2a^2)}&#xA;$$&#xA;is a natural way to do it since that makes $a$ the &lt;b&gt;standard deviation&lt;/b&gt; of the distribution.  If it is written as&#xA;$$&#xA;x\mapsto \frac 1 {a\sqrt{\pi}} e^{-x^2/a^2}&#xA;$$&#xA;then $a$ is not the standard deviation but $a$ is still a scale parameter.&lt;/p&gt;&#xA;" OwnerUserId="2193" LastActivityDate="2015-04-27T11:51:05.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2237" PostTypeId="2" ParentId="2235" CreationDate="2015-04-27T13:06:24.293" Score="16" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I would have thought that the first significant propositions for this would have perhaps come around the beginning of the 20th century, maybe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would think it was quite a bit later. Victor Goldschmidt published his classification of elements as siderophile, chalcophile, lithophile, or atmophile in the late 1920s. This marked the beginning of modern geochemistry, and I would put the date well after this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geologists of the 19th and early 20th century were very skeptical of &quot;contributions&quot; to geology by the physical sciences. Physicists and astronomers in the late 19th century thought the Earth was no more than a few tens of millions years old. Geologists knew it had to be much older than that. Geology, physics, and astronomy wouldn't agree on an age of the Earth until the mid 1950s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One problem with the concept of an iron catastrophe is that it was a catastrophic event. Geologists had spent a good chunk of the 19th century overturning biblical-based geology in favor of uniformitarianism (or gradualism). By the late 19th century / early 20th century, a biblical, catastrophic view of geology was deemed non-scientific. There were no catastrophes. (As an aside, this was one of the key drivers in geology's rejection of Alfred Wegener's theory of continental drift.) It wasn't until the 1960s that the theory of plate tectonics took hold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It wasn't until the 1960s were when geologists seriously started considering catastrophic events such as the formation of the Earth's core. There were some predecessors, and these can be found as references to 1960 era articles. Francis Birch investigated the energy released by the formation of the Earth's core in his 1965 publication &quot;Energetics of core formation,&quot; &lt;em&gt;Journal of Geophysical Research&lt;/em&gt; 70.24:6217-6221. He references earlier material, the earliest of which is Kuhn and Rittmann, &quot;Über den Zustand des Erdinnern und seine Entstehung aus einem homogenen Urzustand&quot;, Geol. Rundschau 32, 215-256, 1941 (translated title: &quot;About the state of the Earth's interior and its origination from a homogeneous original state&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know whether Kuhn and Rittmann investigated the energetic consequences of the formation of the Earth's core in 1941. Birch most definitely did do so in 1965.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="12699" LastEditDate="2017-06-22T19:25:05.157" LastActivityDate="2017-06-22T19:25:05.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2238" PostTypeId="2" ParentId="2234" CreationDate="2015-04-27T18:54:35.013" Score="14" Body="&lt;p&gt;Modern style &lt;a href=&quot;https://hsm.stackexchange.com/questions/499/was-the-polynomial-remainder-theorem-known-before-polynomial-long-division-was/2109#2109&quot;&gt;polynomials were introduced in 12th century&lt;/a&gt; by Arabic mathematician Al-Samawal, so before that we can only talk about problems that reduce to polynomial equations, some obviously, some not so much. Many such problems came from geometry, so naturally they were solved using geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quadratic &quot;equations&quot;, i.e. geometric problems today convertible into them, were solved by Sumerians and Egyptians as early as 2000 BC. Some computational prescriptions for doing it, like false position, survived, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2837/55&quot;&gt;The origin of quadratic equation in actual practice&lt;/a&gt;.&#xA;Pythagoreans did it by literally completing squares with straightedge and compass, Euclid describes it in book II of Elements. This used to be mischaracterized as &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_algebra#Greek_geometric_algebra&quot; rel=&quot;noreferrer&quot;&gt;&quot;geometric algebra&quot;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The famous Delphic problem of duplicating the volume of a cube shaped altar reduces to solving $x^3=2$. One of the first solutions was given by Menaechmus, a friend of Plato's, by intersecting two conic sections. In modern notation they are two parabolas $x^2=y$ and $y^2=2x$, or parabola and hyperbola $x^2=y$ and $xy=2$. In On Sphere and Cylinder Archimedes posed a problem of cutting a sphere in a given ratio, which reduces to $x^2-x^3=s$, where $s$ is a parameter, and solved it also by intersecting conic sections. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method was perfected by famous Persian poet and mathematician Omar Khayyám. Unlike the predecessors who started from a geometric problem and proceeded geometrically, he began with an algebraic classification of cubics, and only then turned to geometry and constructed conic section solutions for all 14 non-trivial cases. For example, Archimedes's equation is of the type $x^3+a^3=bx^2$ in Khayyám's classification, and (modernizing) he solves it by intersecting the hyperbola $xy=a^2$ with the circle $x^2+y^2=bx$. &lt;a href=&quot;https://books.google.com/books?id=ijghAwAAQBAJ&amp;amp;dq=netz+transformation+of+mathematics&quot; rel=&quot;noreferrer&quot;&gt;Netz's Transformation of Mathematics in the Early Mediterranean World&lt;/a&gt; is a good source on these topics, its subtitle is From Problems to Equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem of inscribing regular polygons into a circle also reduces to solving a polynomial equation, although that was discovered much later, namely finding complex roots of $z^n=1$. For $n=3,4,5,6,8,9,10$ it was solved with straightedge and compass already by Pythagoreans, which is described in book IV of Euclid's Elements. For $n=7$ Archimedes gave a solution with compass and &lt;em&gt;marked&lt;/em&gt; straightedge using a construction known as neusis or verging. &lt;a href=&quot;http://books.google.com/books/about/Episodes_from_the_Early_History_of_Mathe.html?id=5wGzF0wPFYgC&quot; rel=&quot;noreferrer&quot;&gt;Aaboe's Episodes from the Early History of Mathematics&lt;/a&gt; gives a nice exposition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Inscribing regular polygons is a particular case of $n$-secting an angle into equal parts, which corresponds to solving $z^n=a$, where $a$ is a complex number of absolute value $1$. Trisection with $n=3$ is another famous ancient problem, which is known to be unsolvable with straightedge and compass, but a neusis solution was also given by Archimedes. &lt;a href=&quot;https://faculty.unlv.edu/baragar/papers/monthly151-164.pdf&quot; rel=&quot;noreferrer&quot;&gt;Baragar's Constructions Using a Compass and Twice-Notched Straightedge&lt;/a&gt; gives a modern perspective on neusis, and its relation to algebraic equations. Other ancient solutions used intersections of conic sections (Apollonius), and mechanically generated curves (conchoid of Nicomedes and cissoid of Diocles). Other curves, quadratrix of Hippias and Archimedean spiral, can be used to $n$-sect any angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. Ironically, Khayyám was dissatisfied with his geometric solutions despite perfecting them. He wrote in Al-jabr w'al-Muqabala: &quot;&lt;em&gt;When however the object of the problem is an absolute number neither we nor any of those who are concerned with algebra have been able to solve this equation - perhaps others who follow us will be able to fill the gap...&lt;/em&gt;&quot; Others tried in vain, and  in 1494 Summa Arithmeticae Luca Pacioli went so far as to say that algebraically &quot;&lt;em&gt;solving the cubic is as impossible as the quadrature&lt;/em&gt;&quot;. Only 19 years later &lt;a href=&quot;https://hsm.stackexchange.com/questions/347/why-is-cardans-formula-wrongly-attributed-to-him/351#351&quot;&gt;Scipione del Ferro algebraically solved the first cubic&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-05T22:50:57.983" LastActivityDate="2018-05-05T22:50:57.983" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2240" PostTypeId="2" ParentId="2143" CreationDate="2015-04-28T11:39:34.113" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;In ancient times, how did people conclude that the shape of Earth is a&#xA;  sphere?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I believe also that many theorized that if the world was flat then a ship appearing on the horizon would not do it bit by bit but would appear all at once.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sailors at least knew the earth was not flat.&lt;/p&gt;&#xA;" OwnerUserId="2094" LastActivityDate="2015-04-28T11:39:34.113" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2242" PostTypeId="2" ParentId="2087" CreationDate="2015-04-28T17:16:42.903" Score="1" Body="&lt;p&gt;William Rowan Hamilton invented the Del or Nabla operator when working with quaternions.  ee &lt;a href=&quot;https://books.google.com/books?id=iuoZSkSOBQsC&amp;amp;pg=PA142&amp;amp;dq=%22William+Rowan+Hamilton%22+and+Nabla&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=u78_VaGAGsWEsAWevoGoCg&amp;amp;ved=0CDkQ6AEwBA&quot; rel=&quot;nofollow&quot;&gt;https://books.google.com/books?id=iuoZSkSOBQsC&amp;amp;pg=PA142&amp;amp;dq=%22William+Rowan+Hamilton%22+and+Nabla&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=u78_VaGAGsWEsAWevoGoCg&amp;amp;ved=0CDkQ6AEwBA&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2201" LastActivityDate="2015-04-28T17:16:42.903" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2243" PostTypeId="1" AcceptedAnswerId="2247" CreationDate="2015-04-28T19:44:36.540" Score="8" ViewCount="425" Body="&lt;p&gt;Briot and/or Cauchy are often said to have written the first papers on holomorphic functions, explicitly discussing them as such and their special properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which papers are these? When and where were these published?&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="26" LastEditDate="2015-05-01T19:00:57.303" LastActivityDate="2015-05-01T19:00:57.303" Title="First papers on holomorphic functions" Tags="|mathematics|biographical-details|discoveries|complex-analysis|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2244" PostTypeId="1" AcceptedAnswerId="2246" CreationDate="2015-04-28T04:24:09.053" Score="4" ViewCount="6436" Body="&lt;p&gt;Speed is defined to be distance divided by time; when and who by was this definition first put forward? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The obvious guess would be Galileo in early modern physics; is this right, and can it be pushed further back?&lt;/p&gt;&#xA;" OwnerUserId="2324" OwnerDisplayName="Mozibur Ullah" LastEditorUserId="55" LastEditDate="2015-04-28T21:30:25.110" LastActivityDate="2020-07-09T11:35:17.297" Title="When and by whom was the earliest definition of speed given?" Tags="|physics|discoveries|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2245" PostTypeId="2" ParentId="2244" CreationDate="2015-04-28T04:58:08.763" Score="1" Body="&lt;p&gt;Yes, your guess looks correct:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Italian physicist Galileo Galilei is credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.m.wikipedia.org/wiki/Speed#Definition&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2216" OwnerDisplayName="Freddy" LastActivityDate="2015-04-28T04:58:08.763" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2246" PostTypeId="2" ParentId="2244" CreationDate="2015-04-28T05:34:15.317" Score="8" Body="&lt;p&gt;Aristotle (Physics, Book VII, Chapter 5) made use of the concept of speed in his laws of motion written around 330 BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He said that a force which moves a mass a distance D in time T will move half the mass twice the distance in the same time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the result of his equation is not correct where the force is gravity (he was concerned with preserving proportionality), it and his other rules of motion indicate that he employed the concept of average speed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See this summary:  &lt;a href=&quot;http://faculty.poly.edu/~jbain/mms/lectures/15.GalileoMotion.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://faculty.poly.edu/~jbain/mms/lectures/15.GalileoMotion.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="Ernie" LastEditorUserId="-1" LastEditDate="2019-09-13T05:21:16.347" LastActivityDate="2019-09-13T05:21:16.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2247" PostTypeId="2" ParentId="2243" CreationDate="2015-04-28T20:27:14.777" Score="5" Body="&lt;p&gt;According to F.Klein's &quot;Lectures on the development of mathematics in the 19th century&quot; Cauchy created this theory (by studying the convergence of series in the complex plane) in his &lt;em&gt;Cours d'Analyse&lt;/em&gt; from 1821 (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k90195m.r=.langFR&quot;&gt; Series 2, Volume 3 of his collected works&lt;/a&gt;). A direct predecessor (but with a lack of rigour seems to be Lagrange) and also Klein declares that these things were known at this time - but seemingly as &quot;folklore&quot;. &lt;/p&gt;&#xA;" OwnerUserId="1764" LastActivityDate="2015-04-28T20:27:14.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2248" PostTypeId="2" ParentId="2243" CreationDate="2015-04-28T23:25:22.840" Score="5" Body="&lt;p&gt;The theory of analytic (holomorphic) functions was indeed created by Cauchy. Briot-Bouquet book was also very influential, in particular they introduced the modern terminology (&quot;meromorpic functions&quot; for sure, perhaps also the term &quot;holomorphic&quot;, and such things as poles, removable and essential singularities etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their book is Theorie des fonctions doublement periodiques et en particulier, des fonctions elliptiques, 1859 (first edition). Available free on Internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. In the old books they used to distinguish analytic, holomorphic, regular and monogenous functions. Nowadays this is all the same. Some people like analytic, others holomorphic, while regular and monogenous are out of date.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cauchy's papers on holomorphic functions start in 1814 and continue to the 1840-th. Wikipedia says that he &quot;created this theory single handed&quot; and most mathematicians would agree. You can see some references on Cauchy's early work in Wikipedia, and I suppose his collected papers are available online (He was only slightly less prolific than Euler:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-04-28T23:48:48.547" LastActivityDate="2015-04-28T23:48:48.547" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2249" PostTypeId="2" ParentId="2244" CreationDate="2015-04-28T23:40:30.610" Score="3" Body="&lt;p&gt;The exact notion of speed, and its precise definition are due to Galileo. But long time before him people had a good intuitive concept of speed, and could use it. (The notions of momentary speed and average speed were confused, of course). One example where the notion of speed was used before Galileo is &amp;quot;dead reckoning&amp;quot; in navigation, which is the computation of the place of the ship using time traveled, speed and direction.&lt;/p&gt;&#xA;&lt;p&gt;Columbus (who lived roughly a century before Galileo) was a great master of this art, one of his extraordinary achievements was steering his fleet in the second transatlantic voyage exactly to the place where he left a party in his first voyage. So he had a very clear concept of speed, though the means of measuring it were very primitive. He also tried to use astronomy in his navigation but without great success.&lt;/p&gt;&#xA;&lt;p&gt;Ref. Samuel Morrison, Admiral of the Ocean sea: a life of Christopher Columbus, 1991.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-07-09T11:35:17.297" LastActivityDate="2020-07-09T11:35:17.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2250" PostTypeId="1" AcceptedAnswerId="2251" CreationDate="2015-04-28T23:46:10.913" Score="4" ViewCount="439" Body="&lt;p&gt;Kronecker is generally credited with the formalization of &quot;adjoining a root to $f(x)=0$&quot;. Nowadays it is interpreted as the quotient $K[x]/(f)$, where $K$ is some appropriate algebraic structure in a wide range of structures. But if $K=\mathbb Q$ or $\mathbb Z$, some of the simplest examples possible, then the construction appears unnecessary, since our imaginary root is simply an element of $\mathbb C$, and there is no need to construct anything (assuming one is already satisfied that $\mathbb C$ exists).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, for which structure(s) $K$ did Kronecker (or whoever actually first thought of this construction) develop his original construction, and why did he feel it was necessary?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="4251" LastEditDate="2022-04-27T05:33:11.033" LastActivityDate="2024-03-03T04:51:22.260" Title="Why did Kronecker develop the &quot;adjoining a root&quot; construction?" Tags="|mathematics|discoveries|abstract-algebra|kronecker|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2251" PostTypeId="2" ParentId="2250" CreationDate="2015-04-29T01:56:31.237" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.jstor.org/stable/41133600&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edwards, The Genesis of Ideal Theory&lt;/a&gt;, in his 1887 &lt;a href=&quot;https://books.google.com/books?id=8MK_AwAAQBAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fundamental Theorem of General Arithmetic&lt;/a&gt; Kronecker was working with polynomials with integer coefficients. He proclaimed, however, that &amp;quot;&lt;em&gt;the so-called fundamental theorem of algebra is replaced&lt;/em&gt;&amp;quot; with the title theorem, and that he &amp;quot;&lt;em&gt;put foundations of algebra in an entirely new form&lt;/em&gt;&amp;quot;. Indeed the proof applies much more generally, providing that any polynomial can be split into linear factors in some extension of its field of coefficients. This is now known as the Kronecker's theorem, or the splitting field theorem, and that the proof does not depend on any analytic properties of &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;. This was of great appeal in Kronecker's eyes. He names as an inspiration his &amp;quot;&lt;em&gt;sharp critique of the method of defining quantities that began with Heine&lt;/em&gt;&amp;quot;, a stand-in for transcendental arguments of Cantor, Dedekind and Weierstrass.&lt;/p&gt;&#xA;&lt;p&gt;However, Kronecker did not come up with the modern quotient ring construction, being an early constructivist he had philosophical objections to Dedekind's ideals themselves, and even offered divisors as an alternative to them earlier. Instead, he formalized a more &amp;quot;explicit&amp;quot; construction of Galois, who called it &amp;quot;adjoining a root&amp;quot;, arguably completing the long process of proving the FTA &amp;quot;algebraically&amp;quot;. Gauss already pointed out the circularity of reasoning that roots are elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt; in Euler's and Lagrange's attempted proofs of FTA. They needed to perform computations with roots, which required treating them as elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, in the course of &lt;em&gt;proving&lt;/em&gt; that they are elements of &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;. One either has to presuppose that the roots exist, which is circular, or to use continuity arguments to prove existence, which is &amp;quot;non-constructive&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Galois still tacitly assumed that computations with roots were valid, but he usually expressed the adjoined roots as polynomials with rational coefficients in a new indeterminate &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, specifying &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; to be a root of a known irreducible polynomial. Kronecker called this construction &amp;quot;&lt;em&gt;the precious Galois principle&lt;/em&gt;&amp;quot;, and proved that such presentation is always possible for polynomials with integer coefficients. Namely, given a polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; it is possible to find a polynomial &lt;span class=&quot;math-container&quot;&gt;$G(t)$&lt;/span&gt; such that adjoining its root allows to express all roots of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; constructively as polynomials in &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. Thus, adjoining a single element splits &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; into linear factors, and computations with roots become justified.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-03-03T04:51:22.260" LastActivityDate="2024-03-03T04:51:22.260" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2252" PostTypeId="2" ParentId="2087" CreationDate="2015-04-29T12:54:39.467" Score="8" Body="&lt;h2&gt;Before the vector&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Before vector calculus was introduced, a few landmarks have to be considered to understand vector history. These are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;complex numbers and their geometrical interpretation&lt;/li&gt;&#xA;&lt;li&gt;Leibniz's work on the geometry of position&lt;/li&gt;&#xA;&lt;li&gt;the parallelogram representation of force and velocity&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The first one can be traced back to Cardan's &lt;code&gt;Ars Magna&lt;/code&gt; published in 1545 as the guy was the first to introduce roots of negative numbers for themselves. One has to wait for two centuries before witnessing approval of this strange writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz, in a letter to Huygens, express the will to give position a mathematical expression, just as amplitude has one. This is exactly what vectorial analysis is about, right? Unfortunately, I don't know what Leibniz' answer to his own question was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's quote a genius&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A body, acted on by two forces simultaneously, will describe the&#xA;  diagonal of a parallelogram in the same time as it would describe the&#xA;  sides by those forces separately.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Newton in 1678, did not have the idea of a vector, but this does look like the sum of two vectors, right? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Between 1799 and 1828, three pairs of two authors simultaneously and independently worked on the geometry of complex numbers. Wessel and Gauss in 1799 wrote independently on how to represent direction analytically, Buéé and Argand follow up in 1806 with geometrical interpretations for complex numbers, and Warren and Mourey both published in 1828 extensive books describing such representations.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Going 3-D: the quaternion&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Hamilton (we're getting closer to electromagnetism) published a philosophical-ish paper in 1837 in which he expresses his hope to come up with a &quot;theory of triplets&quot; to describe 3-D geometry. In 1843 he finally comes up with a multiplication operation on what is now known as quaternions. He was very happy with that and proceeded to spend the rest of his life writing on quaternions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamilton (him again) introduced in a subsequent paper (1846) the terms scalar and vector, to describe the real and imaginary parts of his quaternions. The vector part of the quaternion product of two purely vectorial quaternions is equal to the opposite of what is know the vector/dot product, and the scalar part is what is now known as the cross product.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamilton died but left a successor to his cause, Tait. Tait published numerous papers on quaternions, including extensive description of the use of operator Nabla which earned him Maxwell's praise as the &quot;Chief Musician upon Nabla&quot;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Maxwell, or answering the original question&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;In 1873, Maxwell published his &lt;code&gt;Treatise on Electricity and Magnetism&lt;/code&gt;, a paper that had a huge impact on 19th century science. In this paper, Maxwell presents many of his results not only in the then-usual cartesian form, but also in their quaternionic forms. Maxwell defended and advertise the use of quaternions, not only as a practical tool (he seemed to be more at ease with cartesian geometry), but as a more effective way to think space-related quantities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what I know, Maxwell did not contribute anything to vector calculus, but his endorsement of the then praised, but not used, quaternions in a breakthrough paper allowed vector calculus to become a widespread object in physics. Modern (post-Hamilton) Vector Analysis is mostly based on Gibbs and Heaviside work at the turn of the 20th century, but Maxwell sure contributed a lot to vector calculus by using them in what is, perhaps, the most read paper of the 19th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer is based both for structure and content on a &lt;a href=&quot;https://www.math.ucdavis.edu/~temple/MAT21D/SUPPLEMENTARY-ARTICLES/Crowe_History-of-Vectors.pdf&quot; rel=&quot;nofollow&quot;&gt;talk&lt;/a&gt; that I invite you to read as it's a very entertaining/informative piece of writing.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="2905" LastEditDate="2015-11-15T22:51:59.487" LastActivityDate="2015-11-15T22:51:59.487" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2253" PostTypeId="2" ParentId="2143" CreationDate="2015-04-29T17:37:05.990" Score="12" Body="&lt;p&gt;Eratosthenes of Cyrene did an &lt;a href=&quot;http://scienceblogs.com/startswithabang/2011/09/21/who-discovered-the-earth-is-ro/&quot;&gt;experiment that confirmed that the Earth was roughly spherical&lt;/a&gt;, and estimated its circumference around 200 BC.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the lunar eclipse observation mentioned by Gregory Grant suggests that the Earth is round in 2-dimensions (an Earth shaped like a flat disc is consistent with this observation), but does not provide evidence regarding the Earth's 3-dimensional shape.  &lt;/p&gt;&#xA;" OwnerUserId="2207" LastActivityDate="2015-04-29T17:37:05.990" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2254" PostTypeId="1" AcceptedAnswerId="2258" CreationDate="2015-04-30T01:11:02.967" Score="10" ViewCount="9915" Body="&lt;p&gt;In many fields, it's common for $\Delta$ (the Greek letter &lt;em&gt;delta&lt;/em&gt;) to represent a change or difference.  Math uses it, physics uses it, engineering uses it, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why was $\Delta$ chosen for this?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I realize that &quot;difference&quot; starts with &quot;d&quot;, but I'm wary to assume that just because I can think of a word starting with &quot;d&quot;, that that's why the notation $\Delta$ was introduced.&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="26" LastEditDate="2015-05-01T08:46:45.313" LastActivityDate="2021-10-07T22:41:06.970" Title="Why was delta ($\Delta$) chosen to represent change of a quantity?" Tags="|terminology|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2255" PostTypeId="2" ParentId="2254" CreationDate="2015-04-30T05:26:03.587" Score="3" Body="&lt;p&gt;I actually made this question a couple of months ago and deleted it after a comment by Alexandre Eremenko, if I recall correctly, which, in my opinion, made it perfectly clear. I'll try to expand it a little:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's used because of the word &lt;strong&gt;&lt;em&gt;d&lt;/strong&gt;ifference&lt;/em&gt;, whose latin etymology happens to start with a &quot;d&quot; also. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Differ comes from the latin word &lt;em&gt;differre&lt;/em&gt;, derived from the word &lt;em&gt;ferre&lt;/em&gt; which can often be translated by &quot;carry&quot;. Pretty much every word which contains the word &quot;differ&quot; is derived from this.${}^*$ In &lt;a href=&quot;http://www.etymonline.com/index.php?allowed_in_frame=0&amp;amp;search=difference&amp;amp;searchmode=none&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; you can find a similar explanation of the etymology of the word.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this it's clear that $\Delta$, $\delta$, $d$, and even $D$, are all associated with the change of a given quantity (except when talking about a &lt;a href=&quot;http://en.wikipedia.org/wiki/River_delta&quot; rel=&quot;nofollow&quot;&gt;river delta&lt;/a&gt;, in which case it has to do with how the greek uppercase symbol looks). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though this is most likely the reason, asking who was the first one to use it, or to set a bound would be very interesting. In that sense this isn't a complete answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$(*)$ I'm translating this from an etymological dictionary in spanish and assuming the roots can't be that &lt;em&gt;different&lt;/em&gt;, given the &lt;a href=&quot;http://en.wikipedia.org/wiki/List_of_English_words_of_French_origin&quot; rel=&quot;nofollow&quot;&gt;overwhelming influence&lt;/a&gt; french had in it.&lt;/p&gt;&#xA;" OwnerUserId="520" LastEditorUserId="520" LastEditDate="2015-05-02T02:06:06.810" LastActivityDate="2015-05-02T02:06:06.810" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2256" PostTypeId="1" AcceptedAnswerId="2260" CreationDate="2015-04-30T12:21:12.247" Score="8" ViewCount="1971" Body="&lt;p&gt;I found this definition of &quot;Markov operator&quot; in the book &lt;a href=&quot;http://www.springer.com/gp/book/9780387940496&quot; rel=&quot;noreferrer&quot;&gt;Chaos, Fractals, and Noise by Lasota and Mackey&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Denote by $L^1(\mu)$ the space of Lebesgue integrable functions according to the measure $\mu$. Then, an operator $$ P: L^1(\mu) \to L^1(\mu)$$ is called Markov Operator when &#xA;$$ f\geq 0 \Rightarrow Pf \geq 0$$&#xA;and&#xA;$$ \int (Pf)(x) \mu(dx) = \int f(x) \mu(dx) $$&#xA;holds for any $f \in L^1(\mu)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But where does this originate from? Who invented it?&lt;/p&gt;&#xA;" OwnerUserId="1706" LastEditorUserId="26" LastEditDate="2015-05-01T19:00:29.273" LastActivityDate="2015-05-06T21:00:15.563" Title="Where does Markov operator come from?" Tags="|mathematics|discoveries|statistics|probability|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2257" PostTypeId="1" CreationDate="2015-04-30T15:30:09.553" Score="19" ViewCount="7556" Body="&lt;p&gt;In describing black body radiation Planck assumed that the energy that can be absorbed or emitted by charges is quantized, i.e., they can only absorb or emit certain quantities of energy. But it was Einstein, who postulated the photon. I have two questions relating to this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Did Planck think these energies travelled in wave packets (if not, then how)? And why is his quantization of the energy of light different from Einstein's quantization of light?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;If Planck thought that energies of EM radiation, which weren't multiples of Planck's constant with frequency could exist, what if one of these interacted with a charge that could according to him only absorb certain energies of light? What did he think would happen to this radiation?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1994" LastEditorUserId="10106" LastEditDate="2020-06-21T16:42:21.660" LastActivityDate="2023-12-05T23:06:43.620" Title="What was different about Planck's quantization of light compared to Einstein's?" Tags="|physics|biographical-details|quantum-mechanics|planck|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2258" PostTypeId="2" ParentId="2254" CreationDate="2015-04-30T17:54:58.750" Score="17" Body="&lt;p&gt;There was a &lt;a href=&quot;https://math.stackexchange.com/questions/775603/who-introduced-the-finite-difference-notation-using-delta&quot;&gt;related question on Math.SE&lt;/a&gt;, which Mauro Allegranza answered with reference to Cajori's classic &lt;a href=&quot;http://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf&quot; rel=&quot;noreferrer&quot;&gt;History of Mathematical Notations&lt;/a&gt; (v.II, p.205). It is a great source and is freely available online. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surprisingly, it was not Leibniz, the notational lion of calculus, who introduced it. &quot;&lt;em&gt;A provisional, temporary notation $\Delta$ for differential coefficient or différences des fonctions was used in 1706 by Johann Bernoulli. Previously he had used the corresponding Latin letter D&lt;/em&gt;&quot;. The Math.SE thread also claims about Leibniz that &quot;in developing his calculus for infinitesimal differences he was inspired by his previous work on finite differences&quot;. But I was unable to confirm that, in the &lt;a href=&quot;http://curvebank.calstatela.edu/derrules/derrules.htm&quot; rel=&quot;noreferrer&quot;&gt;famous 1684 paper&lt;/a&gt; he manipulates infinitesimals from the get go. The foundational Treatise on Finite Differences by George Boole, the inventor of Boolean algebra, first appeared in print in 1860.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The choice of $\Delta$ was natural since Latin $d$ was already taken for differentials, and the Greek word for &quot;difference&quot;, which both Leibniz and Bernoulli were likely familiar with, is διαφορά. Its transliterated negation, adiaphora (&quot;indifference&quot;), is in modern use, mostly by philosophers and theologians, referring to morally neutral actions, neither good nor bad. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-05-03T18:32:00.227" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2259" PostTypeId="2" ParentId="2257" CreationDate="2015-04-30T19:06:56.440" Score="22" Body="&lt;p&gt;Unlike Einstein, Planck did not quantize electromagnetic waves themselves, only the exchanged energies, and even them only statistically. So the other two questions have no satisfactory answer because he was not dealing with specifics of emission/absorption at the level of individual quanta. The quanta were meant as mathematical fictions for the purposes of a statistical count, like Boltzmann's boxes. It was Einstein, who acknowledged them as real physical entities in 1905, postulating that electromagnetic waves split into wave packets carrying quanta of energy. But this was not a coherent picture either since Maxwell's electrodynamics does not allow such stable packets, and &lt;a href=&quot;http://www.daviddarling.info/encyclopedia/E/Einstein_and_photoelectric_effect.html&quot; rel=&quot;noreferrer&quot;&gt;Einstein declared in 1911&lt;/a&gt; &quot;&lt;em&gt;I insist on the provisional character of this concept, which does not seem reconcilable with the experimentally verified consequences of the wave theory&lt;/em&gt;.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Planck's derivation of the radiation formula was a convoluted affair, that started in 1894 with him taking interest in emission experiments conducted in Berlin's Physikalisch-Technische Reichenstalt by a number of young researchers, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/5452/did-light-bulb-companies-commission-planck-to-study-black-body-radiation/5454#5454&quot;&gt;Did light bulb companies commission Planck to study black body radiation?&lt;/a&gt; After several unsuccessful attempts Planck finally derived the correct formula in October 1900, without any energy quantization. But in his eyes it lacked foundational insight. He considered thermodynamics to be more fundamental than Boltzmann's statistical mechanics, a common sentiment at the time, but could not extract the correct formula from it despite trying for 6 years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contrary to a popular belief, &lt;a href=&quot;http://www.math.lsa.umich.edu/~krasny/math156_article_planck.pdf&quot; rel=&quot;noreferrer&quot;&gt;Planck was not concerned by the &quot;ultraviolet catastrophe&quot;&lt;/a&gt; (Ehrenfest's nickname from 1911) in black body radiation, when the energy gets converted into higher and higher frequency waves in ether, which is a commonly given back engineered motivation. This &quot;catastrophe&quot; is related to the equipartition theorem of statistical mechanics, and does not occur in the thermodynamical approach of Wien, which Planck originally followed.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Turning to discrete statistics with &quot;quanta&quot; of energy in November 1900 was &quot;an act of despair&quot;, and he hoped to dispense with them in the end by taking a continuous limit. The last part did not work out. This &lt;a href=&quot;https://physics.stackexchange.com/questions/72578/plancks-law-where-does-it-come-from&quot;&gt;thread on Physics.SE&lt;/a&gt; discusses Planck's struggles in more detail. A key qoute comes from his 1931 letter to Wood:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...one finds that the continuous loss of energy into radiation can be prevented by assuming that energy is forced at the outset to remain together in certain quanta. &lt;strong&gt;This was purely a formal assumption and I really did not give it much thought&lt;/strong&gt; except that no matter what the cost, I must bring about a positive result.&lt;/em&gt;&quot; (boldface mine)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If &quot;energy is forced at the outset to remain together in certain quanta&quot; then presumably non-multiple energies do not occur. But this is a later self-interpretation, long after Einstein's and Bohr's inputs. In &lt;a href=&quot;http://books.google.com/books/about/Black_Body_Theory_and_the_Quantum_Discon.html?id=zYUhVHzlr2IC&quot; rel=&quot;noreferrer&quot;&gt;Black-Body Theory and the Quantum Discontinuity, 1894-1912&lt;/a&gt; Kuhn paints a messier picture based on the inspection of contemporary evidence, which became the consensus among historians.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-11T01:56:52.197" LastActivityDate="2020-04-11T01:56:52.197" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="2260" PostTypeId="2" ParentId="2256" CreationDate="2015-04-30T21:56:38.820" Score="8" Body="&lt;p&gt;This is a continuous analog of (transposed) stochastic matrix, the transition matrix in a Markov chain with discrete set of outcomes. These were introduced in 1906 by Markov apparently to disprove Nekrasov's claim that central limit theorems only applied to independent events, but later found many practical applications. Entries of a stochastic matrix are probabilities covering a complete set of outcomes in each row. So they have to be positive and add up to $1$. A transposed matrix will have positive entries and column sums $1$, which means that it transforms positive vectors into positive vectors, and preserves their entry sums.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The continuous version of Markov chains replaces vectors with functions, matrices with operators, and sums with integrals, the stochasticity conditions are transformed accordingly. These were &lt;a href=&quot;http://en.wikipedia.org/wiki/Kolmogorov_equations#History&quot; rel=&quot;noreferrer&quot;&gt;first introduced by Kolmogorov&lt;/a&gt; in 1931 based on the analysis of Markov's discrete case.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-06T21:00:15.563" LastActivityDate="2015-05-06T21:00:15.563" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2261" PostTypeId="2" ParentId="2087" CreationDate="2015-05-02T00:30:45.440" Score="3" Body="&lt;p&gt;According to Thomas Hankin's biography of William Rowan Hamilton, Michael Faraday met with Hamilton in Dublin in 1834.  Faraday is famous for his electromagnetic  experiments and read about the field ideas of Boscovich.  Faraday's experiments suggested to him that they could best be explained by the idea of fields.  Faraday realized that field treatment must involve geometry and numerical analysis.  Hamilton was known as a mathematical genius and Faraday convinced him of the need for a geometric Algebra.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took Hamilton 9 years, in 1843 he came up with Quaternions and in 1844 biquaternions (complex quaternions)[proceedings of Royal Irish Academy]. Maxwell used quaternions learning from his boyhood friend Peter Guthrie Tait. Maxwell wrote to Tait that he wanted to let quaternions &quot;leaven electromagnetism&quot;,[Michael Crowe History of Vector Analysis]&#xA;Unfortunately, Maxwell died young of cancer and Electromagnetism was kidnapped by telegraph operator Oliver Heaviside who hated quaternions, titling one of his book's sections &quot;On the abstrusity of quaternions and the advantages gained in ignoring them&quot;.  Heaviside with the Help J. Willard Gibbs, truncated the quaternion to make vector analysis. This lead Physics down a wrong path that still exists today in spite of a small number of people championing quaternions since Hamilton's time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ludwik Silberstein was first to publish the (bi)quaternion version of Maxwell's Equation where all 4 Maxwell's are written as one biquaternion equation in the same paper he gave the quaternion Lorentz transformation [L.Silberstein &quot;The Quaternion form of Relativity&quot; , Philosophical Magazine,1913] The quaternion form of Dirac's equation was discovered by Cornelius Lanzcos 1n 1929.&lt;/p&gt;&#xA;" OwnerUserId="2220" LastEditorUserId="26" LastEditDate="2015-05-02T14:58:07.340" LastActivityDate="2015-05-02T14:58:07.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2263" PostTypeId="1" AcceptedAnswerId="2266" CreationDate="2015-05-02T18:05:10.297" Score="9" ViewCount="1068" Body="&lt;p&gt;I recently heard a story of a clever method of approximating an integral which,  instead of using numerical techniques, relied on physically drawing out the graph of a function, cutting it out, and weighing it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not been able to find very much information about it (partially because I haven't been able to find if the technique has a specific name), but I have found an anecdote in which Galileo uses the method to find the area of a cycloid (See &lt;a href=&quot;http://mathworld.wolfram.com/Cycloid.html&quot;&gt;Cycloid from Wolfram Mathworld&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do we know who came up with the cut and weigh technique, or if not, is Galileo's use the first to have been recorded?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2212" LastEditorUserId="55" LastEditDate="2018-12-31T11:59:02.260" LastActivityDate="2018-12-31T11:59:02.260" Title="What is the origin of the cut and weigh method of integration, is it Galileo's?" Tags="|mathematics|calculus|debunking|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2264" PostTypeId="1" AcceptedAnswerId="2265" CreationDate="2015-05-02T12:15:52.510" Score="20" ViewCount="4200" Body="&lt;p&gt;This question arose in&#xA;a conversation with a teacher who was introducing square roots to her students.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know from the website&#xA;&lt;a href=&quot;http://jeff560.tripod.com/operation.html&quot;&gt;Earliest Uses of Symbols of Operation&lt;/a&gt;&#xA;that the symbol $\sqrt{}$ has its origin in medieval times.&#xA;But I have not found why the word &quot;radical&quot; is used (in English) for this symbol.&#xA;The English connotation of &quot;far-reaching&quot; or &quot;extreme&quot; strikes students as odd (and might add to the mystery of roots).&#xA;Perhaps in other languages there is not an analogous association?&lt;/p&gt;&#xA;" OwnerUserId="606" OwnerDisplayName="Joseph O'Rourke" LastEditorUserId="98" LastEditDate="2015-05-02T19:58:01.830" LastActivityDate="2015-05-04T23:33:04.740" Title="Why is the radical symbol $\sqrt{}$ called &quot;radical&quot;?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2265" PostTypeId="2" ParentId="2264" CreationDate="2015-05-04T06:44:26.267" Score="19" Body="&lt;p&gt;&quot;Radical&quot; comes from Latin &quot;radicalis&quot;, having roots, an equivalent &quot;radix&quot; was also commonly used earlier. While Rudolff did use the radix in 1525 his did not have the overbar on top, now called vinculum, Latin for bond. That innovation was added by Descartes in La Geometrie (1637). Before that all sorts of tricks had to be resorted to for grouping terms &quot;under the root&quot;. Even before Rudolff Luca Pacioli in Summa Arithmeticae (1494) would write $\mathcal{RU}$ for &quot;radix universalis&quot;, indicating that it applied to everything that follows. Today radix is still occasionally used to refer to the base (&quot;root&quot;) of a positional system, the separation point is called the radix point, and fractions of the form $\frac1{b}+\frac3{b^2}-\frac5{b^3}$ are called radix fractions with radix $b$, see Eves's &lt;a href=&quot;http://books.google.com/books/about/An_Introduction_to_the_History_of_Mathem.html?id=PXvwAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to the History of Mathematics&lt;/a&gt; and Math SE &lt;a href=&quot;https://math.stackexchange.com/questions/93315/name-for-decimals-in-other-bases&quot;&gt;Name for “decimals” in other bases?&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This explains why $x$ in $x^n=a$ is called root or radical, it is the hidden &quot;base&quot; of $a$, and the name likely spread to general polynomial equations by association. To counter the negative connotation it might help to point out to students that &quot;radical&quot; means not generically &quot;far reaching&quot; or &quot;extreme&quot; , but  foundational, reaching to the root causes. One may also point out that in medicine, for example, a radical treatment like surgery, aimed at the root causes of an ailment rather than its symptoms, is often the best course of action, if available. And of course in mathematics the radical symbol is a long happy marriage of radix and vinculum, root and bond.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another Math SE thread &lt;a href=&quot;https://math.stackexchange.com/questions/15787/how-did-the-square-root-get-its-shape&quot;&gt;How did the square root get its shape?&lt;/a&gt; is relevant. The commonly repeated idea that the shape of the radical comes from Latin &quot;r&quot; is Euler's speculation from Institutiones Calculi Differentialis (1775). Cajori, the author of the classic History of Mathematical Notations, is sceptical of it. The resemblance is only there when the vinculum is attached, without it Rudolff's radix resembles &quot;v&quot; more than &quot;r&quot;, and the vinculum was attached only over 100 years later. One of the posters offered an alternative speculation, &quot;Arabic letter jeem ج which is the first letter in the Arabic Jathr جذر meaning root&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-05-04T23:33:04.740" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2266" PostTypeId="2" ParentId="2263" CreationDate="2015-05-04T08:00:43.763" Score="7" Body="&lt;p&gt;To work the weighing method also needs a standard of area, cut from the same sheet. Galileo was allegedly balancing the cycloid cutout with that of its generating circle, and found that they were as &lt;span class=&quot;math-container&quot;&gt;$3:1$&lt;/span&gt;, but did not believe it because he expected that they are incommensurable. The first association that comes to mind is &lt;a href=&quot;http://www.maa.org/publications/periodicals/convergence/archimedes-method-for-computing-areas-and-volumes-cylinders-cones-and-spheres&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes's Method&lt;/a&gt;, where he cuts cone, sphere and cylinder into thin slices, and then balances them on a lever. Of course, this was a purely mental exercise, but in the famous Eureka incident (if Vitruvius's tell tale is believed) Archimedes was presumably balancing the suspicious crown and a slab of gold on physical scales. Of course, that was to compare densities rather than areas or volumes, but the re-purposing is straightforward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Galileo story is not exactly an anecdote. It is reported by his loyal assistant (some say &quot;desciple&quot;) Torricelli in Opera Geometrica (1644), only two years after Galileo's death, citing a 1639 letter of Galileo's to him. Torricelli and his student Viviani spent the last years of Galileo's life (1639-1642) helping him, and were two of the three people present at his deathbed, the third one was his son. However, after Galileo's death Viviani wrote his first biography, where he told another story, you heard of it, about the balls dropped from the leaning tower of Pisa. &lt;a href=&quot;http://www.numericana.com/answer/record.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;He clearly embellished things a little...&quot;&lt;/a&gt;. It doesn't mean that the student took after the teacher, and Torricelli's cycloid story is also made up, but it does raise some doubts. Roberval, one of Torricelli's correspondents, found the area of cycloid geometrically back in 1634, so he likely knew what the correct ratio was, even without the Galileo's letter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a risk of giving material to conspiracy theorists I'll point out another quirk of history. Another &quot;desciple&quot; of Galileo's, Cavalieri, reinvented Archimedes's method of &quot;indivisibles&quot; in 1627, and in a streamlined form that did not require levers and balancing. In 1644, the same year that Torricelli published his cycloid story, Cavalieri used indivisibles to show that cycloid is to the generating circle as &lt;span class=&quot;math-container&quot;&gt;$3:1$&lt;/span&gt;. Torricelli lavished Cavalieri with praise for indivisibles, but wrote: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;I should not dare affirm that this geometry of indivisibles is actually a new discovery. I should rather believe that the ancient geometricians availed themselves of this method in order to discover the more difficult theorems, although in their demonstration they may have preferred another way, either to conceal the secret of their art or to afford no occasion for criticism by invidious detractors&quot;&lt;/em&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this he was a prophet. In 1906 Heiberg found a palimpsest, later lost again and rediscovered with much hype in 1998, containing the only extant copy of Archimedes's Method. It is a letter to Eratosthenes, where he &quot;availed himself of this method&quot; of balancing those indivisible slices of cone, sphere and cylinder on a lever to discover their volume ratios. The method he &quot;concealed&quot; in On the Sphere and Cylinder by giving an unilluminating but impeccable exhaustion proof, &quot;to afford no occasion for criticism&quot;. So far as we know neither Cavalieri, nor Torricelli, nor anybody else in the West knew of the Method before 1906. Sometimes history winks at us. ;-)&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-12-31T11:57:50.927" LastActivityDate="2018-12-31T11:57:50.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2267" PostTypeId="1" AcceptedAnswerId="2272" CreationDate="2015-05-05T14:38:37.090" Score="10" ViewCount="525" Body="&lt;p&gt;So this photo of the Cavendish Laboratory is from 1939, and I was wondering if anyone knew any more about who was in the photo. (Click to enlarge)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zhZnU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iu0gK.jpg&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From notes I've been given, I believe the front row is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;DE Lea?; Unknown; ME Oliphant; Norman Feather; &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Frederick_Charles_Searle&quot; rel=&quot;nofollow noreferrer&quot;&gt;GFC Searle&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Lawrence_Bragg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bragg Jr&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Cockcroft&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Cockroft&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Philip_Dee&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philip Dee&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Chadwick&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Chadwick&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Thomson_Rees_Wilson&quot; rel=&quot;nofollow noreferrer&quot;&gt;CTR Wilson&lt;/a&gt;? Miss Davies; Unknown&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second row definitely has &lt;a href=&quot;https://en.wikipedia.org/wiki/Samuel_Devons&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sam Devons&lt;/a&gt; (5th from left; dark shirt), and there's some suggestion that the man to his left (i.e. 4th from left on 2nd row) is &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Bondi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Bondi&lt;/a&gt;; Eric Wild is in the 3rd row (4th from left)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Context of the photo. I don't know anything more about why it was taken other than it being the yearly (1939) photo of the department's Faculty Staff and Graduates; a family member has inherited it as his father is in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help, contributions or suggestions appreciated!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="2236" OwnerDisplayName="Savara" LastEditorUserId="2236" LastEditDate="2015-05-06T08:55:33.937" LastActivityDate="2015-05-06T11:56:17.633" Title="Cavendish Laboratories Photo, 1939" Tags="|physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2268" PostTypeId="1" AcceptedAnswerId="2270" CreationDate="2015-05-05T23:12:09.397" Score="12" ViewCount="12219" Body="&lt;p&gt;Did the ancient Greeks understand that the moon shines by reflection of sunlight, and that this was the explanation of its phases? Did this allow them to conclude that the moon was a sphere? From what little I know of ancient Greek science, I thought they didn't actually understand the notion of seeing by reflection of light.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="55" LastEditDate="2015-05-06T02:37:18.710" LastActivityDate="2015-05-07T01:10:37.660" Title="How did ancient Greeks explain moon phases without reflection of sunlight?" Tags="|astronomy|ancient-greece|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2269" PostTypeId="2" ParentId="2268" CreationDate="2015-05-06T00:10:31.367" Score="12" Body="&lt;p&gt;&lt;strong&gt;Yes.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://www.moonzoo.org/Moon_Culture_and_History&quot;&gt;this&lt;/a&gt;, we don't know &lt;em&gt;who&lt;/em&gt; it was to explain the phases using a spherical model, though it was before 600 B.C:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first person to correctly explain the phases of the Moon is lost in history. By the time Pythagoras wrote in 600 B.C., the ancient Greeks knew that the Moon is spherical and that it revolves around the Earth. The Greeks understood how that motion causes the monthly changes in the Moon’s appearance. In fact, they even had pretty good measurements of the Moon’s relative size and distance from the Earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I dread going to &lt;a href=&quot;https://en.wikipedia.org/wiki/Moon#Early_studies&quot;&gt;Wikipedia&lt;/a&gt;, but it &lt;em&gt;is&lt;/em&gt; helpful:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ancient Greek philosopher Anaxagoras (d. 428 BC) reasoned that the Sun and Moon were both giant spherical rocks, and that the latter reflected the light of the former.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One of the sources cited there is from &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Anaxagoras.html&quot;&gt;here&lt;/a&gt;, which has biographies of scientists; it has proved useful to HSM in the past.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-05-06T00:10:31.367" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2270" PostTypeId="2" ParentId="2268" CreationDate="2015-05-06T00:27:18.510" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;http://www.space.com/6650-moon-phases-work.html&quot; rel=&quot;nofollow&quot;&gt;explanation of Moon phases&lt;/a&gt; (and eclipses) is basically the same under all theories of vision that reduce to geometric optics, because under all such theories illumination of the Moon is due to sunlight. The round shadow on the Moon during eclipses was used as one of the arguments for spherical Earth (e.g. by Aristotle in De Caelo), but of course it is hardly conclusive. Spherical Earth was accepted mostly because it fit into the &quot;perfection of cosmos&quot; doctrine that originated with Pythagoreans and Plato, but the round shadow didn't hurt. Anaximander was the only one who went with a cylinder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Details of vision theories vary, but perceiving incoming light, including reflected light, was one of them circulating in antiquity. It is now called the intromission theory and was favored by Aristotle and the atomists. It was not the most popular one however. Most pre-Socratics, Plato, and geometers and astronomers from Euclid to Ptolemy seemed to prefer the extramission theory, where the visual rays coming out of the eyes &quot;feel&quot; distant objects, or some mixture of the two. Plato for example writes in Timaeus:&quot;&lt;em&gt;...whenever there is daylight round about, the visual current issues forth, like to like, and coalesces with it and is formed into a single homogeneous body in a direct line with the eye&lt;/em&gt;&quot;. Surprised? Don't be, according to 2002 paper &lt;a href=&quot;http://people.auc.ca/brodbeck/4007/article7.pdf&quot; rel=&quot;nofollow&quot;&gt;Fundamentally Misunderstanding Visual Perception&lt;/a&gt; &quot;&lt;em&gt;...when  adults  were  repeatedly  asked  specifically  to  draw whether something comes into or goes out of the eyes when  person  sees  a  balloon,  69%  placed  outward-pointing arrows...&lt;/em&gt;&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I digress. Even under the strict extramission theory what visual rays &quot;feel&quot; depends on what is there, and that depends on other sources of light, like the Sun or a lantern. If they 'light up' other objects that is &quot;felt&quot;, and if the Sun is badly positioned, or the Earth is in the way, that is &quot;felt&quot; too. Empedocles, one of the first extramissionists, explicitly likens eyes to lanterns, Ptolemy is less poetic:&quot;&lt;em&gt;Illumination by itself, however, is a sort of excess-condition in luminous objects, so it hurts and offends the [visual] sense. Illumination is also created along with coloring in objects that are struck by light from outside.&lt;/em&gt;&quot; See &lt;a href=&quot;http://www.jstor.org/stable/3185879?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Ptolemy and the Foundations of Ancient Mathematical Optics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-07T01:10:37.660" LastActivityDate="2015-05-07T01:10:37.660" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2271" PostTypeId="1" AcceptedAnswerId="2281" CreationDate="2015-05-06T07:09:49.260" Score="9" ViewCount="1670" Body="&lt;p&gt;From what date can we trace the knowledge (or the hypothesis) that sound has a finite speed of transmission through air? Thunder/lightning is the most striking clue, but echoes would be the readiest source of controlled observation. However, I would appreciate some historical data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speed of travel is implicit in the idea that sound is transmitted via &lt;i&gt;oscillations&lt;/i&gt; - which are discussed in some useful answers to &lt;a href=&quot;https://hsm.stackexchange.com/questions/454/when-and-how-it-was-discovered-that-sound-is-related-to-some-kind-of-oscillation&quot;&gt; this question&lt;/a&gt;, so perhaps there is not much more to be said, but I am particularly interested in the simpler idea that sound travels with a finite speed.&lt;/p&gt;&#xA;" OwnerUserId="2229" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2019-03-26T13:58:00.643" Title="When was it first realized that sound travels with finite speed?" Tags="|discoveries|experimental-physics|acoustics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2272" PostTypeId="2" ParentId="2267" CreationDate="2015-05-06T11:56:17.633" Score="8" Body="&lt;p&gt;So in answer to my own question, I've since emailed Cambridge and had a fantastically helpful reply: they've sent me a photo of the the fully named / annotated original photo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BDEdF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/peUzi.jpg&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this might be of interest to some.&lt;/p&gt;&#xA;" OwnerUserId="2236" LastActivityDate="2015-05-06T11:56:17.633" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2273" PostTypeId="2" ParentId="480" CreationDate="2015-05-06T13:34:05.873" Score="5" Body="&lt;p&gt;I believe that in the 1930s, peer-review was mostly done directly by editors-as-peers. This belief is based on this letter by Albert Einstein to the &lt;strong&gt;Physical Review&lt;/strong&gt;, 1936:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dear Sir,&#xA;  We (Mr. Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed. I see no reason to address the in any case erroneous comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Respectfully,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;P.S. Mr. Rosen, who has left for the Soviet Union, has authorized me to represent him in this matter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This seems to indicate that Einstein, who publied about 300 papers in the first half of the 20th century, was unfamiliar with anonymus peer-reviewing (for the sake of the story, it turned out the reviewer's opinion was right on this matter).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Nature&lt;/strong&gt; &lt;a href=&quot;http://www.nature.com/nature/history/timeline_1960s.html&quot; rel=&quot;noreferrer&quot;&gt;documents&lt;/a&gt; the introduction of a formal, anonymous, systematic peer-review system in 1967, mainly to deal with the accumulation of unpublished-yet-not-rejected papers. Before this, reviewing was performed by editors, eventually seeking for advice outside.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what I have read on the subject, it seems to me that non-anonymous peer review never really existed: today's peer review is derived from reviewing comities in publications, that were allowed to evolve into more flexible, distributed reviewing groups in the 60s/70s because both of the possibility of doing it (Xerox copier comes to mind) and the necessity of doing it (to face accumulation of paper, as stated in Nature example above). Nobody ever thought of giving the name of the reviewers, are they were not considered important. There are &lt;a href=&quot;http://michaelnielsen.org/blog/three-myths-about-scientific-peer-review/&quot; rel=&quot;noreferrer&quot;&gt;trends&lt;/a&gt; today to make the reviewing process more anonymous (double-blind, the reviewer doesn't know who is the author) and less anonymous (disclosing the name of the reviewers), but I don't believe the system will change in a close future, as there is no consensus in the scientific community about this.&lt;/p&gt;&#xA;" OwnerUserId="8" LastActivityDate="2015-05-06T13:34:05.873" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2274" PostTypeId="1" AcceptedAnswerId="2283" CreationDate="2015-05-06T13:37:40.057" Score="12" ViewCount="811" Body="&lt;p&gt;I've heard that Euler actually never managed to prove $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ &#xA;If this is true, then where does the common misconception come from?&lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="55" LastEditDate="2022-08-15T19:28:18.160" LastActivityDate="2022-08-16T03:53:55.263" Title="Is it true that Euler did not prove the sum of the Basel series $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$?" Tags="|mathematics|discoveries|euler|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2275" PostTypeId="2" ParentId="2271" CreationDate="2015-05-06T13:53:09.340" Score="3" Body="&lt;p&gt;I'm not sure if there was ever a doubt that sound has a finite speed. Your reference to thunder/lightning may have allowed people throughout history to realize that sound has a finite speed, slower than light.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was Ole Roemer who first discovered that light traveled at a finite speed, people before him believed that light traveled at an infinite speed.&#xA;&lt;a href=&quot;https://www.amnh.org/learn-teach/curriculum-collections/cosmic-horizons/profile-ole-roemer-and-the-speed-of-light&quot; rel=&quot;nofollow noreferrer&quot;&gt;Article about Ole Roemer&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back to the sound question, while it may have been intuitive for people to realize that sound travels slower than light, Pierre Gassendi was the first known person to try to measure the speed of sound. He got the speed wrong, but did figure out that the speed is not dependent on frequency. His experiments used cannon fire, observing the difference between the flash and the sound at a known distance. He did assume that light was traveling at infinite speed when performing his calculations. Here's an article about him in general, with only a small bit about the cannon experiments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/gassendi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Large article about Pierre Gassendi&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, Aristotle posited that the speed of sound was dependent on frequency. Gassendi had religious/philosophical issues with Aristotle's views, and one reason for the sound experiments was to prove Aristotle incorrect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More on that at www.encyclopedia.com/topic/Pierre_Gassendi.aspx&lt;/p&gt;&#xA;" OwnerUserId="2245" LastEditorUserId="9325" LastEditDate="2019-03-26T13:58:00.643" LastActivityDate="2019-03-26T13:58:00.643" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2276" PostTypeId="2" ParentId="2274" CreationDate="2015-05-06T16:46:40.250" Score="5" Body="&lt;p&gt;No it's not.&lt;/p&gt;&#xA;&lt;p&gt;I think the misconception comes from the fact that Euler announced the result five years before providing a correct proof (Euler, Opera Omnia Series 1, Volume 14, p. 177-186 (E63)). &lt;a href=&quot;https://plus.maths.org/content/infinite-series-surprises&quot; rel=&quot;nofollow noreferrer&quot;&gt;See this webpage&lt;/a&gt; for the original proof Euler used (&lt;strong&gt;the 1735 version, not the 1741 amended version&lt;/strong&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Euler is well-known for unjustified mathematics manipulations (think of the sum of the alternate series being worth &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt;), and his first &amp;quot;proof&amp;quot; relied on extending properties of the product expression of polynomials to infinite polynomial series. It took him 10 years, and development in the understanding of series, to show a second version, that was less litigious, hence the misconception you're talking about: between publishing his discovery in 1735 and proving it rigorously in 1741.&lt;/p&gt;&#xA;&lt;p&gt;However, &amp;quot;proof&amp;quot; clearly has a different meaning today than in Euler's mind, as Conifold’s answer addresses.&lt;/p&gt;&#xA;&lt;p&gt;Note that Cauchy provided a simpler, more elegant proof in 1820.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="12330" LastEditDate="2022-08-16T03:53:55.263" LastActivityDate="2022-08-16T03:53:55.263" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2277" PostTypeId="2" ParentId="2274" CreationDate="2015-05-06T16:53:27.603" Score="4" Body="&lt;p&gt;It really depends on what you think constitutes a 'proof'.  Euler gave a proof for $\zeta(2)=\pi^2/6$ with an argument that $\sin(\pi x)/\pi x$ should have a factorization like a polynomial in terms of its (infinitely many) roots.  This would be convincing to most calculus students.  He gave the values of $\zeta(2n)$ by manipulating series expansions of $\cot(z)$.  This would be convincing to undergraduate math majors.  In both cases there are certainly subtleties that need to be considered.  For example, in his general proof there are double limits that are interchanged formally, and this needs justification.&lt;/p&gt;&#xA;" OwnerUserId="2061" LastActivityDate="2015-05-06T16:53:27.603" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2278" PostTypeId="1" CreationDate="2015-05-06T19:24:32.887" Score="11" ViewCount="550" Body="&lt;p&gt;Today we see the photoelectric effect as one of the simplest pieces of empirical evidence that leads to quantum physics. The historical development of the subject, however, seems to have involved much more complicated and indirect evidence such as the blackbody radiation curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In textbooks and laboratory manuals, a fact about the photoelectric effect that is often presented prominently is that the stopping voltage is independent of the intensity of the light. This fact is easy to explain with the photon model. &#xA;One also often sees in textbooks statements that according to &quot;the classical theory,&quot; the stopping voltage should have increased with the intensity of the light. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is not at all obvious to me that any such classical theory can be constructed. Specifically, I doubt that one can construct a classical model that makes the following predictions: (P1) the photoelectric effect occurs with solid metals, (P2) there is a finite stopping voltage, and (P3) this stopping voltage increases with the intensity of the light. Note that if the model doesn't predict P1 and P2, then we can't even meaningfully ask whether it predicts P3.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What was the actual historical situation? I can imagine at least five scenarios.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(a) The photoelectric effect in a solid metal cathode was a total Kuhnian anomaly. Physicists trained within the classical paradigm would not have seen any promising lines of attack, and would have been unable to construct even a partially successful model. Therefore they ignored it and worked on other problems.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(b) There was some actual classical model that was published or publicly discussed, which predicted P1-P3.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(c) People built and discussed models that gave predictions similar to P1-P3 in the case of a free electron, or perhaps a gaseous cathode. The models just couldn't be extended to a solid metal cathode.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(d) Textbook authors who refer to &quot;the classical model&quot; are using sloppy language or have not studied the actual history; there is actually some model that predicts P1-P3, but it's not a purely classical model. It's some kind of hybrid such as the Bohr-Kramers-Slater theory.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(f) Looking through the &lt;a href=&quot;http://en.wikipedia.org/wiki/Photoelectric_effect#20th_century&quot; rel=&quot;noreferrer&quot;&gt;history section&lt;/a&gt; of the WP article on the photoelectric effect, it seems that textbook presentations may be giving a misleading impression of the historical sequence. It seems that the evidence prior to 1905 was mostly very crude and qualitative. Because of the paucity of quantitative data, there may have been little motivation to embark on serious model-building.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Can anyone clarify what actually happened?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Einstein's 1905 paper has the following (translation by D. Ter Haar?) on this topic:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;As far as I can see, our ideas are not in contradiction to the properties of the photoelectric action observed by Mr. Lenard. [Under the quantum hypothesis] the velocity distribution of the electrons ... will be independent of the intensity of the incident light ...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He does not follow up by claiming that such an observation would contradict the classical theory. Einstein references Lenard's 1902 paper, which showed that the stopping potential increased with the frequency of the light but was independent of intensity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Briefly, here is the reason why it's not obvious to me that any classical model can produce P1-P3. I'm not claiming that I'm right -- I'm just trying to explain why I have doubts about the existence of any such model. In a classical model of a solid metal cathode, the conduction electrons are a classical gas with a Maxwellian velocity distribution. Visible light has a wavelength much longer than the mean separation between electrons, and therefore all it really does is heat the cathode. Regardless of temperature, the Maxwellian distribution has a high-velocity tail whose energy is greater than the work function, so electrons are emitted from the surface at some rate. There is no sharp cutoff to this tail, and therefore there is no well-defined stopping voltage.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="55" LastEditDate="2015-05-10T22:07:06.673" LastActivityDate="2019-11-15T11:34:25.167" Title="Were there serious attempts to model the photoelectric effect classically?" Tags="|physics|discoveries|experimental-physics|quantum-mechanics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2279" PostTypeId="1" AcceptedAnswerId="2280" CreationDate="2015-05-06T20:10:32.950" Score="44" ViewCount="7677" Body="&lt;p&gt;As I was reading some papers written by Schrödinger and Heisenberg back in 1920s, I noticed that the symbols they use such as the integral or summation sign or calligraphic letters are as if printed out of a LaTeX document. I don't think that they had characters such as these in a typewriter. I was wondering how they could have achieved writing so flawlessly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: What I am talking about is something like this:&lt;img src=&quot;https://i.stack.imgur.com/DsuqM.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that this is from „Quantisierung als Eigenwertproblem“ by Erwin Schrödinger, which was written in 1926.&lt;/p&gt;&#xA;" OwnerUserId="660" LastEditorUserId="26" LastEditDate="2015-05-06T21:43:11.543" LastActivityDate="2023-12-05T23:49:49.160" Title="Writing Mathematical Symbols in 20th century" Tags="|mathematics|physics|notation|publishing|" AnswerCount="5" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="2280" PostTypeId="2" ParentId="2279" CreationDate="2015-05-06T21:47:28.757" Score="44" Body="&lt;p&gt;Documents like the one you linked to were not typed on a typewriter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When writing on typewriters, it was common to leave some space in the document in which the formulas could be inserted by hand. For professional publications, this was then given to a printer (the person, not the machine on your desk ;-)) who hot metal typeset the document.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Printers had special boxes for all kinds of types:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/734DL.jpg&quot; alt=&quot;Box of types for mathematical formulas&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://www.hoechst-schoen.de/bleisatz-raritaeten/&quot; rel=&quot;noreferrer&quot;&gt;Source of the picture&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in the contents sheet of this box, you can see the integral sign and many more types for mathematical formulas.&lt;/p&gt;&#xA;" OwnerUserId="2224" LastActivityDate="2015-05-06T21:47:28.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2281" PostTypeId="2" ParentId="2271" CreationDate="2015-05-06T22:02:43.970" Score="10" Body="&lt;p&gt;The question is delicate because of the phrasing that assumes empirical approach, which did not emerge until 17th century. And the notion of &quot;speed&quot; as applied to sound presupposes the concept of wave in a medium, which dates to late Renaissance. The use of &quot;speed&quot; in antiquity was mostly confined to uniform motions of localized objects, not light or sound.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the idea that sound takes time to travel is ancient if not prehistoric. The earliest surviving written record is perhaps &lt;a href=&quot;http://www.islandnet.com/~see/weather/elements/thunder1.htm&quot; rel=&quot;noreferrer&quot;&gt;Aristotle's Meteorology (c.334 BC)&lt;/a&gt;, where he gives his theory of thunder and lightning: &quot;..&lt;em&gt;.the dry exhalation that gets trapped when the air is in the process of colliding is forcibly ejected as the clouds condense and in its course strikes the surrounding clouds, and the noise caused by the impact is what we call thunder... the ejected wind burns with a fine and gentle fire and it is then what we call lightning, which occurs when the falling wind appears to us to be coloured. &lt;strong&gt;Lightning is produced after the impact and so later than thunder, but it appears to us to precede it because we see the flash before we hear the noise.&lt;/em&gt;&lt;/strong&gt;&quot; (boldface mine)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;General &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/09608788.2013.792239&quot; rel=&quot;noreferrer&quot;&gt;theory of sound is described in Aristotle's De Anima&lt;/a&gt;, but he mostly elaborates on what was a consensus among Greek natural philosophers since early pre-Socratics: sound is some kind of movement of air (with each part &quot;&lt;em&gt;striking the air which is next to it&lt;/em&gt;&quot;), and no movement can be instantaneous because the same thing can not be two places at once. This was suggested by experience with noisy falls and hits, vibrating strings and air filled pipes in musical instruments, etc. The collision theory of thunder relies on such intuitions, and goes back to Anaximander (c. 611-547 BC) and Anaximenes (c. 585-528 BC). According to Aristotle, &lt;a href=&quot;http://www.ldolphin.org/chistory.html&quot; rel=&quot;noreferrer&quot;&gt;Empedocles (c.450 BC) even thought of light&lt;/a&gt; as &quot;traveling or being at any given moment between the earth and its envelope, its movement being unobservable to us&quot;. On this account  Aristotle disagrees because he considers light a state rather than a movement. But on sound there was complete agreement, I am not aware of anyone who thought that sound spreads instantaneously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, nothing resembling the expression &quot;finite speed&quot; was used anywhere in antiquity to describe either light or sound. Leonardo is credited with suggesting c.1500 that sound travels in waves in the modern sense (&quot;&lt;em&gt;we see the waves running across the field while the grain remains in place&lt;/em&gt;&quot;). This allows one to make sense of speed and frequency. However, &lt;a href=&quot;http://www3.nd.edu/~powers/ame.20231/finn1964.pdf&quot; rel=&quot;noreferrer&quot;&gt;first measurements of the speed of sound&lt;/a&gt; were only attempted in 1630s by Mersenne and Gassendi, the first &quot;authoritative&quot; one was the one performed by the Florentine Academy in 1660. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-07T19:05:31.680" LastActivityDate="2015-05-07T19:05:31.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2282" PostTypeId="2" ParentId="2279" CreationDate="2015-05-06T22:30:49.113" Score="16" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I don't think that they had characters such as these in a typewriter. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First off, that wasn't typewritten. Look at the &quot;i&quot;s in $\text{Variationsproblem}$ or the &quot;fl&quot; ligature in $\it\text{Kugelflächenfunction}$ (on the next page of the article). Typewriters use monospaced fonts, and they don't do ligatures. That article was typeset, printed in the venerable journal Annalen der Physik. Publishers have had the ability to create perfectly typeset mathematics for a long time. Newton's &lt;em&gt;Principia&lt;/em&gt; had typeset mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, typewriters did have characters such as those. The IBM Selectric had changeable print balls. Every organization that submitted technical articles to conferences, journals, and books had secretaries who knew how to use those beasts. Before that, numerous typewriter manufacturers sold typewriters with switchable heads. The demand was apparently strong enough that Sears sold &quot;Change-A-Type&quot; kits for mathematics, engineering, and Greek symbols.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-05-06T22:30:49.113" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2283" PostTypeId="2" ParentId="2274" CreationDate="2015-05-07T00:23:21.620" Score="15" Body="&lt;p&gt;The question can be answered &amp;quot;yes&amp;quot; or &amp;quot;no&amp;quot; depending on how the word &amp;quot;prove&amp;quot; is interpreted.  Euler was working in the context of &lt;a href=&quot;https://hsm.stackexchange.com/questions/2035/was-18th-century-algebra-more-symbolic-formal-than-the-modern-conception/2038#2038&quot;&gt;&amp;quot;algebraic analysis&amp;quot; of 18th century&lt;/a&gt;, so he wasn't primarily concerned with issues that underlie modern definition of convergence, despite performing many numerical computations.  &lt;a href=&quot;http://www.jstor.org/stable/41134263?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historian Ferraro writes&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The Eulerian definition of the sum of a series should be understood in such a context. According to this definition, the value of the sum of a series was the value of the function from the development of which the series was generated.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For example, according to Euler, &lt;span class=&quot;math-container&quot;&gt;$$1+2+2^2+\dots=\frac{1}{1-2}=-1,$$&lt;/span&gt; by definition! Deciding whether his proofs are &amp;quot;rigorous&amp;quot; by modern standards is like fitting square peg into round hole, he operated in a different environment and under different rules.&lt;/p&gt;&#xA;&lt;p&gt;Euler gave several derivations for the sum of the Basel series. The first one, where he manipulated infinite series as if they were polynomials to factorize &lt;span class=&quot;math-container&quot;&gt;$1-\sin s$&lt;/span&gt;, naturally &amp;quot;is not considered valid today&amp;quot;. However, some of its intuitions can be justified using non-standard analysis, see &lt;a href=&quot;http://www1.american.edu/cas/mathstat/People/kalman/pdffiles/Sixways.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kalman's Six Ways to Sum a Series&lt;/a&gt;. At the time it did not satisfy even Euler himself as it raised some serious questions. How can we be sure that &lt;span class=&quot;math-container&quot;&gt;$1−\sin s$&lt;/span&gt; does not have other roots besides the obvious ones? How come &lt;span class=&quot;math-container&quot;&gt;$\sin s$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e^s\sin s$&lt;/span&gt; have the same roots, but presumably different factorizations?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It took him about ten years, but he finally succeeded in obtaining the famous product formula for &lt;span class=&quot;math-container&quot;&gt;$\sin s$&lt;/span&gt;... The proof... was beautiful and direct.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But... it still involved formal manipulation of complex numbers, poorly understood at the time, and was missing justifications for uniform convergence, absolute convergence, or even a notion of convergence for that matter. So it wasn't &amp;quot;rigorous&amp;quot; by modern standards either.&lt;/p&gt;&#xA;&lt;p&gt;Cauchy gave a more elementary proof in &lt;em&gt;Cours d'Analyse&lt;/em&gt; (1821), which is also not beyond modern reproach. Euler's &amp;quot;gaps&amp;quot; were eventually filled by Weierstrass, but they were what is called &amp;quot;technical&amp;quot;, see &lt;a href=&quot;http://www.ams.org/journals/bull/2007-44-04/S0273-0979-07-01175-5/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Varadarajan's Euler and his Work on Infinite Series&lt;/a&gt; for details.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-15T19:26:59.133" LastActivityDate="2022-08-15T19:26:59.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2284" PostTypeId="2" ParentId="2279" CreationDate="2015-05-07T01:48:32.193" Score="39" Body="&lt;p&gt;It is really funny to read that in the beginning of 21-st century, some young people may think that journals and books printing had something to do with typewriters:-)&lt;/p&gt;&#xA;&lt;p&gt;If you look attentively at the page you scanned you will easily see that this is not a TeX font and not a typewriter. It is more beautiful.&lt;/p&gt;&#xA;&lt;p&gt;Before the middle 1990s we lived in the &amp;quot;Gutenberg Universe&amp;quot;. In the universe where everything was PRINTED using lead fonts. This was the case from XVI century to th 1990s. The authors (Heisenberg, for example) would submit their paper to the journal, HANDWRITTEN, or text typed on a typewriter, and formulas inserted by hands. Until the beginning of 1990-th mathematicians inserted formulas by handwriting in a typed manuscript.&#xA;Typewriters which could type Greek and Latin letters, and some formulas were common only in the US, and even there I am not sure how common.&lt;/p&gt;&#xA;&lt;p&gt;This handwritten manuscript would go to a typography. There professional typesetters would make &amp;quot;pages&amp;quot; composed of lead letters (they had thousands of fonts in good typographies). Typesetters in the beginning of the 20-th century took these letters (made of lead) from boxes, and connected them together&#xA;into lines and into pages (called &amp;quot;matrices&amp;quot;). These lead blocks called &amp;quot;matrices&amp;quot; contained mirror images of pages, one of each page. They would be used for printing, then the output was&#xA;sent to the author for correction, then corrected and sent for correction again,&#xA;several times.&lt;/p&gt;&#xA;&lt;p&gt;In 1960s I visited such a print shop. Printing the local newspaper. The labor condition were terrible: enormous noise, lead dust everywhere... This was a very hard work. And take into account that the&#xA;typesetter had to compose lines right-to-left. Some mechanization was introduced later, like the machine called linotype, (see &amp;quot;Linotype machine&amp;quot; in Wikipedia) which was something like a typewriter but enormous. And only professionals worked on them, not the authors, not Heisenberg:-)&lt;/p&gt;&#xA;&lt;p&gt;As Gerald Edger commented, typesetting mathematics was a very expensive task which required most experienced typesetters. Printing anything was very expensive, especially math.&lt;/p&gt;&#xA;&lt;p&gt;So it is really strange that the prices of mathematical books and journals&#xA;skyrocketed AFTER the journals switched to TeX. Nowadays the publisher has practically nothing to do: the major expense of typesetting has been eliminated.&lt;/p&gt;&#xA;&lt;p&gt;Added remark: whatever the publishers do nowadays, before 1990s they did the same AND typesetting in addition to this. And now compare the prices, just out of curiosity. At least in 1980s many good universities were able to afford subscription to all or almost all math journals.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-03-06T20:40:40.177" LastActivityDate="2021-03-06T20:40:40.177" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="2285" PostTypeId="1" CreationDate="2015-05-07T07:26:44.440" Score="7" ViewCount="203" Body="&lt;p&gt;The &lt;i&gt;point charge&lt;/i&gt; concept is clearly a very useful &lt;i&gt;mathematical fiction&lt;/i&gt;, but it is also problematic from the point of view of &quot;physical intuition&quot;. Even a layperson would feel that an explanation is required as to why a point charge is not split asunder by self-repulsion. A friend of mine who knows far more physics than me said that Maxwell's equations would require a point electric charge to rapidly attenuate (on a time-scale of something like $10^{-18}$ sec!). She refused to be drawn into discussing the following question, however:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How did classical physicists soothe their anxieties on this matter, and is the conundrum really dissipated, or merely sidelined, in quantum theories of elementary particles?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2229" LastEditorUserId="55" LastEditDate="2015-05-07T20:19:08.390" LastActivityDate="2015-05-10T18:54:05.780" Title="Did classical physicists feel ill-at-ease about point charges?" Tags="|physics|quantum-mechanics|particle-physics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2286" PostTypeId="1" CreationDate="2015-05-07T12:30:27.927" Score="5" ViewCount="1840" Body="&lt;p&gt;Does logical-AND have precedence over logical-OR because of a reason or was it an arbitrary choice made sometime in the distant past?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Perhaps it could have been the other way around: OR-terms evaluated first, then AND'ed together?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've Googled and can't seem to find an answer; most mathematical and computer language references state the precedence rankings of various operations but not the origins.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NOTE: I don't believe this question has anything to do with notation, but rather the order that operations are performed.&lt;/p&gt;&#xA;" OwnerUserId="2254" LastEditorUserId="2254" LastEditDate="2015-05-07T17:24:05.577" LastActivityDate="2022-09-25T09:14:24.663" Title="Why does logical-AND take operator precedence (evaluated first) over logical-OR?" Tags="|mathematical-logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2287" PostTypeId="2" ParentId="2286" CreationDate="2015-05-07T14:16:01.673" Score="7" Body="&lt;p&gt;&amp;quot;And&amp;quot; is often seen as the multiplication in Boole's algebra, sometime written &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; and &amp;quot;Or&amp;quot; is seen as the addition (and sometime written &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;, though &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; is more often used for xor than for or), which is why they commonly inherit the precedance of their namesakes.&lt;/p&gt;&#xA;&lt;p&gt;Note that both are distributive over the other, and it's the neutral element (0 for or, 1 for and) that gives the correspondance.&lt;/p&gt;&#xA;&lt;p&gt;So the next question that has to be addressed is why multiplication usually has precedence over addition (note that some programming languages, notably, do not follow this convention).&lt;/p&gt;&#xA;&lt;p&gt;It &lt;a href=&quot;https://web.archive.org/web/20180111045655/http://mathforum.org/library/drmath/view/52582.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;seems&lt;/a&gt; to have arisen from a global consensus, mainly over concerns of simplicity of writing polynomials, but wasn't formalised until the 20th century. This is because mathematicians don't care so much about precedence, but wider teaching of mathematics in textbooks, as well as the rise of IT-science, have made it compulsory to formalise these rules.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="2277" LastEditDate="2022-09-25T09:14:24.663" LastActivityDate="2022-09-25T09:14:24.663" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="2288" PostTypeId="2" ParentId="2094" CreationDate="2015-05-07T15:01:33.933" Score="3" Body="&lt;p&gt;Interpolation became a big thing after Wallis's Algebra, where he first interpolated various sequences just for the fun of it. Since Wallis's &quot;proofs&quot;&#xA;were hardly more than a sequence of educated guesses, Goldbach and Euler used&#xA;Wallis's work as a collection of open problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, Euler would use the same methods for extending functions such as the zeta&#xA;function to negative numbers and could even deduce (conjecturally) the functional equation of zeta(s) as well as for the Dirichlet L-function for the Dirichlet&#xA;character $(-1/p)$ with conductor 4. &lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2015-05-07T15:01:33.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2289" PostTypeId="2" ParentId="2133" CreationDate="2015-05-07T15:09:20.407" Score="7" Body="&lt;p&gt;One answer is that Gauss sums are special cases of Lagrange resolvents; these in turn are expressions that Lagrange used for analysing the solution of equations by radicals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another source are Euler's and Lagrange's factorizations of the cyclotomic equations $4(x^p-1)/(x-1)$ in the form $Y^2 - p^*Z^2$, where &#xA;$p^* = (-1)^{\frac{p-1}2}p$, for $p = 3, 5, 7$ in connection with problems on &#xA;special cases of the quadratic reciprocity law. Both of them were unable to &#xA;find the factorization for $p = 11$, but Gauss realized that the roots of &#xA;the linear factors of the quadratic polynomials are roots of unity $\zeta^r$&#xA;and $\zeta^s$, respectively, where $r$ and $n$ denote the quadratic residues &#xA;and nonresidues modulo $p$. Once you start writing down the polynomials&#xA;$(X - \zeta)(X-\zeta^4)(X-\zeta^9)\cdots$ and look at the coefficient of the&#xA;second largest power of $X$, you will discover quadratic Gauss sums.&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2015-05-07T15:09:20.407" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2290" PostTypeId="1" AcceptedAnswerId="2296" CreationDate="2015-05-07T16:02:11.147" Score="1" ViewCount="1036" Body="&lt;p&gt;When was the reduced planck constant $\hbar= h/2\pi$ first introduced and what was the reason behind introducing such a constant? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that $E=\hbar \omega$ and $p=\hbar k$ and writing again and again the $2\pi$ in $E=(h/2\pi)\omega$ $p=(h/2\pi) k$ etc. must really annoy the physicist of the time. However there are alternatives to these quantities namely $E=h \nu$ and $p=h/\lambda$. What was the reason preferring $\omega$ and $k$ over $\nu$ and $\lambda$ at the cost of introducing a new constant &lt;/p&gt;&#xA;" OwnerUserId="660" LastEditorUserId="55" LastEditDate="2019-09-14T06:24:02.400" LastActivityDate="2019-09-14T06:24:02.400" Title="Why was the reduced Planck constant introduced and when?" Tags="|physics|notation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2291" PostTypeId="2" ParentId="2285" CreationDate="2015-05-07T20:09:10.807" Score="5" Body="&lt;p&gt;Yes, they did, and the problem persists in quantum field theory to the extent that electrons can be called &quot;point charges&quot; (they are neither waves nor particles despite common terminology, and technically each one is smeared all over the universe). The classical electron theory of Lorentz-Abraham had multiple issues, some related to electron's point like nature, like infinite self-energy and self-stress, some not, like pre-acceleration (charge 'feels' application of force from the future) and internal instability. Lorentz tried to 'dissolve' electron in ether reducing it to an artifact of electromagnetic field, Abraham  attempted to build a theory where electron was a small sphere of finite radius in &lt;a href=&quot;http://en.wikisource.org/wiki/Translation:Principles_of_the_Dynamics_of_the_Electron_%281902%29&quot; rel=&quot;nofollow&quot;&gt;Principles of the Dynamics of the Electron (1902)&lt;/a&gt;, and Poincaré suggested some attractive non-electromagnetic force to hold it together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is from &lt;a href=&quot;http://www.philsoc.org/1962Spring/1526transcript.html&quot; rel=&quot;nofollow&quot;&gt;Rohrlich's lecture&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;... when the radius shrinks to zero, the electron mass becomes infinitely large... This is the famous problem of the electron self-energy. It exists in the classical theory of the electron as well as in the quantum theory. A satisfactory solution of it is not known. An at best temporary solution is provided by the renormalization procedure... It is therefore not surprising that many attempts were made to keep the electron of finite extension and carry along throughout the development of the theory an explicit electron structure.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And that's not all:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Obviously, an accumulation of negative charge is not a stable configuration: Any finite charge distribution of only one sign would explode... A solution to it was proposed by Poincaré who showed that an attractive and consequently non-electromagnetic force can always be added to the theory so as to just balance the stresses and establish stability. It is a very ad hoc solution and not at all fitting a &quot;fundamental&quot; theory. On the other hand, in the point electron theory the electromagnetic self-stress is infinite and so must be the Poincaré's cohesive force.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Electrons in quantum field theory are still &quot;point like&quot; in the sense that they technically collapse into points when positions are measured. In perturbative expansions their paths are represented by lines in Feynman diagrams, which leads to divergences. &lt;a href=&quot;http://math.ucr.edu/home/baez/renormalization.html&quot; rel=&quot;nofollow&quot;&gt;Renormalization&lt;/a&gt; is roughly a consistent way of &quot;canceling&quot; infinities coming from all relevant diagrams. This depends on a &quot;renormalization scale&quot;, and to fix it measured quantities, like electron mass, then have to be substituted into theoretical formulas. With gravity the trick doesn't work, it is not renormalizable, which is a big reason why we still do not have a theory of quantum gravity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;String theory is an attempt to deal with this issue in a principled way. Points are replaced with segments or circles, and line diagrams with smooth two dimensional &quot;fat graphs&quot;, whose contributions are expected to be finite. If string theory works out perhaps electrons will no longer be point like but have an internal stringy structure.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-10T18:54:05.780" LastActivityDate="2015-05-10T18:54:05.780" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2294" PostTypeId="1" AcceptedAnswerId="2299" CreationDate="2015-05-08T00:10:56.240" Score="3" ViewCount="301" Body="&lt;p&gt;Who was/were first one(s) to recognize that what we see in the night sky are not just points of lights but full blown lands(in case of moon and some planets) and so on? I am guessing it would not have been possible before telescopes.&lt;/p&gt;&#xA;" OwnerUserId="2262" LastActivityDate="2015-05-08T14:52:03.910" Title="Planets/stars as more than points of light" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2296" PostTypeId="2" ParentId="2290" CreationDate="2015-05-08T12:05:07.763" Score="1" Body="&lt;p&gt;&lt;em&gt;As none of the answer on physics give the historical significance of the introduction, I'll give it a try.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bohr&lt;/strong&gt; first introduced $\hbar$ as the quantum of angular momentum (when $h$ is the quantum of action). Bohr believed it to be the angular momentum of each electron in an atom. Despite this being false, both Heisengberg and Schrödinger parallel work on quantization rules for electron (circa 1925) kept the reduced Plank constant as the fundamental quantum of angular momentum. For this reason, Heisenberg was familiar with $\hbar$, and he stated his 1927 uncertainty principle in terms of $\hbar$ rather than $h$, ensuring a bright future to the constant, that is sometime named &lt;strong&gt;Dirac constant&lt;/strong&gt; because Dirac was the first one to use the notation (Bohr referred to it as $M_0$).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that some physicist believe it to be the angular derivative of the (not reduced) Planck constant.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="8" LastEditDate="2015-05-08T21:19:24.327" LastActivityDate="2015-05-08T21:19:24.327" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2299" PostTypeId="2" ParentId="2294" CreationDate="2015-05-08T14:52:03.910" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Cosmic_pluralism&quot; rel=&quot;nofollow&quot;&gt;plurality of worlds&lt;/a&gt; had been discussed since ancient times, but there was little evidence, and the other worlds were not necessarily imagined to be the planets. The ancients had some idea of the relative sizes of the different planets' orbits (around the earth), but they had no way of tying these to any absolute distance scale, so they didn't know how big the planets were.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo was the first to resolve the disk of Jupiter, detect the moons of Jupiter, and observe the phases of Venus. The phases of Venus implied that it was a sphere and that it circled the sun rather than traveling on a geocentric epicycle. The motion of the Jovian moons was seen as a challenge to Aristotelianism, since according to Aristotelianism they should have followed their natural motion rather than being affected by Jupiter. From an Aristotelian point of view it wouldn't make sense to imagine, for example, an explorer standing on another planet. The explorer would have fallen back to earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the moon has features that are clearly visible to the naked eye, apparently it was believed before Galileo that these were markings on a perfectly smooth surface. Galileo observed the moon telescopically and published a description of mountains and craters. Through a telescope you can see that shadows are being cast.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once Kepler's laws had been found, and the apparent angular sizes of the planets measured, it became possible to find the sizes of the planets in comparison with the earth.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2015-05-08T14:52:03.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2302" PostTypeId="2" ParentId="724" CreationDate="2015-05-09T14:54:14.280" Score="3" Body="&lt;p&gt;Neither was discovered nor invented. A particular form of the convolution operation arised &quot;naturally&quot; in the solution of diferential equations by certain definite integrals, as described on Section 1049 of Euler's book &quot;Institutionum Calculi Integralis (Vol. 2).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more complete history of convolution product may be read on the paper &quot;A History of the Convolution Operation&quot; (&lt;a href=&quot;http://pulse.embs.org/january-2015/history-convolution-operation/&quot; rel=&quot;nofollow&quot;&gt;http://pulse.embs.org/january-2015/history-convolution-operation/&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="2270" LastActivityDate="2015-05-09T14:54:14.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2303" PostTypeId="1" AcceptedAnswerId="2315" CreationDate="2015-05-10T15:51:41.720" Score="5" ViewCount="300" Body="&lt;p&gt;I am looking for books which include a mathematical interpretation of Aristotle's hypotheses about mechanics. I heard that there are a few books which interpret his mechanical ideas mathematically, but I couldn't find any. &lt;/p&gt;&#xA;" OwnerUserId="2274" LastEditorUserId="55" LastEditDate="2015-05-11T20:59:20.140" LastActivityDate="2015-05-11T20:59:20.140" Title="Mathematical interpretation of Aristotelian mechanics" Tags="|physics|reference-request|ancient-greece|classical-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2304" PostTypeId="5" CreationDate="2015-05-10T17:19:06.113" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-05-10T17:19:06.113" LastActivityDate="2015-05-10T17:19:06.113" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2305" PostTypeId="4" CreationDate="2015-05-10T17:19:06.113" Score="0" Body="For questions about the science that deals with classification, analysis and interpretation of numerical facts and data." OwnerUserId="8" LastEditorUserId="8" LastEditDate="2015-05-10T17:30:40.910" LastActivityDate="2015-05-10T17:30:40.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2306" PostTypeId="2" ParentId="2278" CreationDate="2015-05-10T20:08:50.073" Score="11" Body="&lt;p&gt;&amp;quot;Serious&amp;quot; in the OP sense is probably too high a bar. In 1900-s the situation was very much in flux as to what classical physics could and could not explain. Even Planck's and Einstein's ideas, that we now associate with quantum mechanics, were incorporated into seemingly classical approaches at the time. But what  dominated the scene were qualitative hypotheses, rather than carefully constructed theories with spelled out predictions.&lt;/p&gt;&#xA;&lt;p&gt;Between 1902, when Lenard established the independence of photoeffect on the intensity of light, and 1911, the prevailing explanation was Lenard's own &amp;quot;triggering hypothesis&amp;quot;. According to it, &amp;quot;&lt;em&gt;the light only has an initiating action, rather like that of the fuse in firing a loaded gun&lt;/em&gt;&amp;quot;, as Lenard himself put it in his &lt;a href=&quot;https://www.nobelprize.org/uploads/2018/06/lenard-lecture.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906 Nobel lecture&lt;/a&gt;. It went with, his also, model of the atom that anticipated Bohr's in some aspects, but both were qualitative, not quantitative.&lt;/p&gt;&#xA;&lt;p&gt;Although Lenard's experiments inspired Einstein's 1905 &amp;quot;photoeffect&amp;quot; paper, his light quanta did not get much traction until 1911, when Lenard himself disproved the triggering hypothesis by demonstrating that ionization in gases by ultraviolet light did not produce strong ionization unaccompanied by absorption of the light. More precise experiments with gas were only performed in 1908 by Langevin and Bloch, and with metals in 1914 by Millikan. A detailed historical account is &lt;a href=&quot;https://www.jstor.org/stable/27757381&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wheaton's Philipp Lenard and the Photoelectric Effect, 1889-1911&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Lenard's largely empty atom, which emits characteristic spectral lines when free electrons return to equilibrium in the dynamic atomic structure, is suggestive of the atom proposed five years later by Niels Bohr. But there are fundamental differences. Lenard's was never a quantitative atom theory. It was primarily because of his remarkable experimental skill that his ideas about atoms drew support in Germany. His triggering hypothesis for the photoelectric effect required that each electron within the atom has a velocity which is related in a consistent way to its mechanical frequency. The specific relation of frequency to velocity for each cathode material could only be determined by experiment. The photoelectric effect did not predict what the relationship would be, it only made it accessible to observation. The triggering hypothesis formed the bridge which connected the photoelectric effect to the issues of atomic structure.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1909 Lenard's triggering hypothesis was included by a knowledgeable reviewer among &amp;quot;the generally accepted truths of physics.&amp;quot; Albert Einstein, who characterized Lenard's experimental work on the photoeffect as &amp;quot;pathfinding,&amp;quot; proposed an alternative interpretation in 1905. But Einstein's light quantum, which explained Lenard's results at the cost of rejecting classical wave radiation, had little support early in the century... Einstein had predicted that a continuous linear relation between frequency and stopping potential would be found. But the empirical data was in doubt until 1915. With an appropriate atom model, like those proposed by Thomson in 1910 and 1913, even the Einstein law could be reconciled with the triggering mechanism.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even Einstein's light quantum ideas seemed to be reconcilable with classical physics at the time. Einstein and Lorentz corresponded about light quanta in 1909. Lorentz just worked out the classical theory of blackbody radiation in painstaking detail, and sharply criticized &amp;quot;&lt;em&gt;point-like quantities of energy or at least energy quantities concentrated in very small spaces&lt;/em&gt;&amp;quot;, that Einstein seemed to postulate. Classical explanations for discretization of energy in the photoeffect seemed achievable. Even Einstein distanced himself from  &amp;quot;point-like light quanta&amp;quot; and offered a description of them as &amp;quot;&lt;em&gt;singularities that are surrounded by a vector field whose strength decreases with increasing distance. The field energy is then related somehow to the number of these singularities.&lt;/em&gt;&amp;quot; See &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs00407-012-0107-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kox's Hendrik Antoon Lorentz’s struggle with quantum theory&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Similar dynamic happened with &lt;a href=&quot;https://hsm.stackexchange.com/questions/2257/what-was-different-about-plancks-quantization-of-light-compared-to-einsteins/2259#2259&quot;&gt;Planck's blackbody radiation formula&lt;/a&gt;. Statistical mechanics in general, and the equipartition theorem in particular, were under suspicion. Planck, and even Jeans and Lorentz, had doubts about the  universality of the equipartition theorem. It did not become clear that classical theory leads to the &amp;quot;ultraviolet catastrophe&amp;quot; until Lorentz's 1908 Rome lecture. According to Kuhn's source study, Planck too was under the impression that he was doing classical physics until 1906. In 1908, when Lorentz admitted that &amp;quot;&lt;em&gt;only  with  the  help  of  the  hypothesis  of  energy elements one can arrive at the correct radiation-law&lt;/em&gt;&amp;quot;, the focus shifted from purely classical approach to identifying the source of &amp;quot;discontinuity&amp;quot;: was it in the matter, in its interaction with electromagnetic field, or in the field itself? And later to reconciling Maxwell's waves with Planck's quanta, or even getting rid of waves altogether, as Duane suggested in connection with the X-ray diffraction.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-11-15T11:34:25.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2307" PostTypeId="1" CreationDate="2015-05-10T20:13:35.793" Score="5" ViewCount="503" Body="&lt;p&gt;In studying the solutions proposed for Russell's Paradox on naive set theory (mainly &lt;a href=&quot;http://plato.stanford.edu/entries/russell-paradox/&quot; rel=&quot;noreferrer&quot;&gt;the corresponding entry in the Stanford Encyclopedia of Philosophy&lt;/a&gt;) I came across the following considerations:&#xA;von Neumann theory of class solves Russell's Paradox as does Zermelo's solution, but with the advantage of treating directly proper classes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question then is why Zermelo's solution (or precisely Zermelo-Frankel + axiom of choice) still more widely used (to my knowledge) than von Neumann-Bernay-Gödel's?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was told it is an historical accident, but I am not sure if it is only the period of discovery that made ZFC stick as grounding of set theory or if there are other reasons.&lt;/p&gt;&#xA;" OwnerUserId="2275" LastEditorUserId="520" LastEditDate="2015-05-10T22:43:59.177" LastActivityDate="2015-05-10T22:43:59.177" Title="Why is ZFC used more widely than NBG?" Tags="|mathematical-logic|set-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2308" PostTypeId="1" AcceptedAnswerId="2313" CreationDate="2015-05-10T20:56:07.223" Score="27" ViewCount="2304" Body="&lt;p&gt;The story is usually told starting with Einstein's 1915 paper &lt;a href=&quot;http://www.gsjournal.net/old/eeuro/vankov.pdf&quot; rel=&quot;noreferrer&quot;&gt;Explanation of the Perihelion Motion of Mercury from General Relativity Theory&lt;/a&gt;, or at least its drafts from 1913-14. It was the first triumph of general relativity. But how did Einstein zero in on Mercury in the first place? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's early work seems to be rather distant from astronomy. Special relativity grew out of classical electrodynamics, as did the photoelectric effect, the work on heat and Brownian motion is far from astronomy too. Even general relativity was driven by philosophical concerns about general covariance, etc., rather than empirical issues in astronomy. I am not sure how well the Mercury's anomaly was known at the time, but Kelvin and others did not mention it as one of the &quot;clouds&quot;. While developing general relativity Einstein learned of tensor calculus from a geometer friend, perhaps he had an astronomer friend as well?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who or what attracted Einstein's attention to Mercury, and when? What alerted him to the idea that Mercury's case was different from all those other cases, when a mundane explanation was involved? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Another twist is that special relativity combined with the inverse square law already causes elliptic orbits to precess. &lt;a href=&quot;http://philsci-archive.pitt.edu/5547/&quot; rel=&quot;noreferrer&quot;&gt;Sommerfeld knew that&lt;/a&gt; at least by 1916, when he refined Bohr's model of the hydrogen atom by using precessing elliptic orbits under special relativity kinematics. A &lt;a href=&quot;https://physics.stackexchange.com/questions/64221/mercurys-orbital-precession-in-special-relativity?rq=1&quot;&gt;calculation on Physics SE&lt;/a&gt; shows that this effect can account for about 7&quot; out of 43&quot; per century of Mercury's anomalous precession.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was Einstein or his contemporaries aware of this effect for Mercury before 1915, and how did it reflect on special relativity if they were?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; VicAche's answer below led me to &lt;a href=&quot;http://mathpages.com/rr/s8-10/8-10.htm&quot; rel=&quot;noreferrer&quot;&gt;Conquering the Perihelion chapter in Kevin Brown's book Reflections on Relativity&lt;/a&gt;, which gives the whole story summarized below. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein first mentions Mercury in a letter to Habicht in 1907:&quot;&lt;em&gt;At the moment I am working on a relativistic analysis of the law of gravitation by means of which I hope to explain the still unexplained secular changes in the perihelion of Mercury.&lt;/em&gt;&quot; He most likely got the idea from Mach's Science of Mechanics, which mentions &quot;&lt;em&gt;Paul Gerber alone, from the perihelial motion of Mercury, forty-one seconds in a century, finds the velocity of propagation of gravitation to be the same as that of light.&lt;/em&gt;&quot; This directly suggested that Mercury's anomaly, unlike many others, was relativistic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in 1906 Seeliger gave an alternative solar corona explanation, which convinced many of his contemporaries, but not Einstein. And not Poincare, who mentioned in 1908 book Science and Method that special relativity already predicts an advance of 7&quot; for Mercury’s perihelion. He further wrote (answering my second question):&quot;&lt;em&gt;This cannot be regarded as an argument in favor of the new dynamics, since we still have to seek another explanation of the greater part of the anomaly connected with Mercury, but still less can it be regarded as an argument against it.&lt;/em&gt;&quot; This likely confirmed the relativistic nature of the anomaly in Einstein's eyes. Einstein's original version of a new gravity theory (&quot;Entwurf&quot;) first predicted negative precession, and then 18&quot; instead of 45&quot;, which was one of three reasons he cited for abandoning it in favor of what we now call general relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; Although Mercury's is the only one featured in the story orbital anomalies are plenty, as are proposals to change the laws of gravity for them. Newton could not account for all of lunar precession, and Clairaut suggested a modification to inverse square law before discovering that the issue was not retaining enough terms in the Taylor series for the solution. The anomaly of Uranus famously led to Leverier's prediction of Neptune, and Leverier, who also discovered the anomalous precession of Mercury, suggested a new planet Vulcan to explain it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vulcan did not materialize by 1915, but neither did Pluto until 1930, and it was blamed for orbital anomalies of Uranus and Neptune known in Einstein's time. After the discovery it turned out that little Pluto wasn't responsible after all, a misestimation of Neptune's mass was. Also in Einstein's time there was still a known lunar anomaly, which was not resolved until 1940s, the non-uniformity of Earth's rotation was the culprit. Zodiacal cloud and another modification of inverse square law were suggested instead of Vulcan for Mercury by Newcomb. A more recent &lt;a href=&quot;http://spectrum.ieee.org/aerospace/astrophysics/finding-the-source-of-the-pioneer-anomaly&quot; rel=&quot;noreferrer&quot;&gt;Pioneer anomaly&lt;/a&gt; also caused a lot of speculation about new gravity physics (MOND), but was finally traced to subtle thermal pressure. Some suggested it was responsible early on.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2022-11-11T07:39:37.990" Title="What attracted Einstein to the anomalous precession of Mercury?" Tags="|physics|discoveries|biographical-details|astronomy|relativity-theory|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2309" PostTypeId="1" AcceptedAnswerId="2310" CreationDate="2015-05-10T22:45:28.213" Score="33" ViewCount="3911" Body="&lt;p&gt;How much is known, or can be conjectured, about why the Greeks originally became interested in the somewhat arbitrary construction of intersecting a plane with a cone? The folklore that I've heard is that conics were forgotten until they turned out to be useful in describing the orbits of planets, making it seem even more unusual that the Greeks would have considered it worth writing entire books about them.&lt;/p&gt;&#xA;" OwnerUserId="50" LastActivityDate="2019-11-30T01:53:54.110" Title="Why did the ancient Greeks originally become interested in conic sections?" Tags="|mathematics|ancient-greece|euclidean-geometry|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2310" PostTypeId="2" ParentId="2309" CreationDate="2015-05-10T23:14:33.530" Score="32" Body="&lt;p&gt;The truth is that we do not know. &lt;a href=&quot;http://www.robertnowlan.com/pdfs/Menaechmus.pdf&quot; rel=&quot;noreferrer&quot;&gt;We do know of the person who is credited with the discovery, Menaechmus (c. 350 BC),&lt;/a&gt; a student of Eudoxus of Cnidus and a friend of Plato's, one of the most prominent mathematicians of his time. The names ellipse, parabola and hyperbola were given to them by Apollonius of Perga over a century later however. Menaechmus called them literally acute section, right section and obtuse section because they can be obtained by sectioning cones with the corresponding apex angles perpendicularly to the generator. Others also called them Menaechmian triads. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The traditional answer is that the reason was the Delian problem, the most famous of the &quot;three construction problems of antiquity&quot;. The oracle of Delphi allegedly told the citizens of Delos to double the size of a cube shaped altar to stop a plague, and when they doubled the sides and nothing happened, specified that it was the volume that was to be doubled. The part about the plague in Greece at about the right time can be confirmed, as can be the Greek habit of consulting the Delphic oracle. The rest is neither here nor there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking the side of the original cube as a unit doubling of the cube reduces in modern algebraic notation to solving &lt;span class=&quot;math-container&quot;&gt;$x^3=2$&lt;/span&gt;. Greeks had no algebraic notation, and their favored method of solving geometric problems was using straightedge and compass. As we know today such an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; can not be constructed with those tools. Hippocrates of Chios noticed however that one could find such &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by solving the double proportion &lt;span class=&quot;math-container&quot;&gt;$1:x=x:y=y:2$&lt;/span&gt;, or as Greeks put it, &quot;inserting two mean proportionals&quot; between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a trivial manipulation today to see that Hippocrates's proportion is equivalent to a pair of equations &lt;span class=&quot;math-container&quot;&gt;$x^2=y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y^2=2x$&lt;/span&gt;, describing two parabolas, or &lt;span class=&quot;math-container&quot;&gt;$x^2=y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$xy=2$&lt;/span&gt;, describing parabola and hyperbola. If they can be constructed then the intersection point will give the solution to the Delian problem. Menaechmus's task was considerably harder. He could not manipulate coordinate formulas, Greeks only had a crude prototype of them called &quot;symptoms&quot;, nor did he presumably know of curves with such symptoms ahead of time. So he had to reverse engineer symptoms from Hippocrates's proportion, and then have an insight that curves with exactly these symptoms can be obtained by sectioning cones. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this seems hard to believe, it is. It is possible that Menaechmus got some clues from previous, mechanical, solutions to the Delian problem by his teacher Eudoxus, and by his teacher's teacher Archytas of Tarentum. It is also possible that he experimented with sections of the cone for other reasons, ellipses implicitly appear in Eudoxus's homocentric astronomical models, for example, and noticed that they have the requisite properties to solve the Delian problem. See &lt;a href=&quot;https://link.springer.com/article/10.1007/s004070100039&quot; rel=&quot;noreferrer&quot;&gt;Yavetz's A New Role for the Hippopede of Eudoxus&lt;/a&gt; and &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF00776065&quot; rel=&quot;noreferrer&quot;&gt;Riddel's Eudoxan Mathematics and the Eudoxan Spheres&lt;/a&gt; for geometric details of such speculations.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-27T07:16:35.817" LastActivityDate="2019-11-27T07:16:35.817" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2311" PostTypeId="2" ParentId="2309" CreationDate="2015-05-11T02:07:07.797" Score="14" Body="&lt;p&gt;This question has been discussed several times on math overflow:&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/191909/discovery-and-study-of-conic-sections-in-ancient-greece&quot;&gt;https://mathoverflow.net/questions/191909/discovery-and-study-of-conic-sections-in-ancient-greece&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also has references.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One theory is that they appeared when the Greeks started to think how to make&#xA;accurate sundial. This theory is developed in several books and articles on the subject, and the top scoring answer in the MO question I cite above is about this theory. However on my opinion, shared by some historians of mathematics,&#xA;this theory is not sufficiently justified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more plausible theory is that they were discovered when &#xA;attempting to double the cube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That conic sections were forgotten with the decline &#xA;of Hellenistic mathematics is correct. But this applies also to the other great works of that epoch (Archimedes). This followed by a complete collapse of science, and for about a millennium there were simply no people who were able to understand Apollonius or Archimedes.  Then Kepler discovered that planets move on&#xA;conic sections, and a bit later Pascal proved first new theorems about them.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-11-30T01:53:54.110" LastActivityDate="2019-11-30T01:53:54.110" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="2312" PostTypeId="2" ParentId="2303" CreationDate="2015-05-11T15:24:32.827" Score="0" Body="&lt;p&gt;To be honest, Galilei's &lt;a href=&quot;http://galileoandeinstein.physics.virginia.edu/tns61.htm&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Two New Sciences&lt;/em&gt;&lt;/a&gt; is what you're looking for. This is as close to quantitative analysis as you'll find, because Aristotle did not quantify his observations even in form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The entire &lt;a href=&quot;https://books.google.com/books?id=tG1bAAAAQAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;book&lt;/a&gt; contains more discussion, but in general, you can reconstruct Aristotle's arguments from his own &lt;em&gt;Physics&lt;/em&gt; and &lt;em&gt;Metaphysics&lt;/em&gt;, with a couple of algebraic equations, which is what I recommend. It's the most straightforward method. Writings of Huygens are recommended, as well. There isn't much detailed discussion afterwards, mostly examples selected as illustrations of theories of history of science.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2015-05-11T15:24:32.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2313" PostTypeId="2" ParentId="2308" CreationDate="2015-05-11T15:32:29.663" Score="19" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Who or what attracted Einstein's attention to Mercury, and when? What&#xA;alerted him to the idea that Mercury's case was different from all&#xA;those other cases, when a mundane explanation was involved?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I know for sure that &lt;a href=&quot;http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henri Poincaré&lt;/a&gt; was aware of the problem and of its singularity - had he been in Kelvin's place, he would have added it to the list, and Einstein was most certainly aware of the work of Poincaré (Étienne Klein reported &lt;em&gt;La science et l'hypothèse&lt;/em&gt; was one of the subject of discussion in Olympia Academy).&lt;/p&gt;&#xA;&lt;p&gt;Poincaré studied the stability of the solar system &lt;em&gt;in extenso&lt;/em&gt; during the 1880s and 1890s. I know no work mentioning Mercury before his address of the problem in 1910 (see page 424 of &lt;a href=&quot;https://books.google.fr/books?id=w2Tya9gOKqEC&amp;amp;pg=PA425&amp;amp;lpg=PA425&amp;amp;dq=Poincar%C3%A9%20precession%20of%20mercury&amp;amp;source=bl&amp;amp;ots=Us5PslJKjK&amp;amp;sig=vmmYIHXis0LFbZMtxAEDxAhVraQ&amp;amp;hl=fr&amp;amp;sa=X&amp;amp;ei=ysdQVYHdBoaSsgGsvIHgDQ&amp;amp;ved=0CFQQ6AEwBw#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poincaré's biography by Jeremy Gray&lt;/a&gt;), but despite my deception in failing to find explicit address prior 1905, this is still a valid answer.&lt;/p&gt;&#xA;&lt;p&gt;Note that the study of Mercury and the solar system by Poincaré led to another whole new theory, the Chaos Theory, which makes this ridiculous advance in the perihelion of Mercury on of truly productive ground for XX&lt;sup&gt;th&lt;/sup&gt; century science.&#xA;Why? Because Einstein didn't use this example to advert his theory until &lt;a href=&quot;http://mathpages.com/rr/s8-10/8-10.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;1915&lt;/strong&gt;&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Was Einstein or his contemporaries aware of this effect for Mercury&#xA;before 1915, and how did it reflect on special relativity if they&#xA;were?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Langevin says that Poincaré proposed several propagation equations for gravitational waves at the speed of light, that all had in common  &amp;quot;that they reduced the gap between the laws of physics and the facts, in the move of the perihelia of Mercury for example&amp;quot;. This proves that, when working on similar laws, the French academy of Science was not unused to test them on Mercury (Poincaré's failure to find special relativity can be explained by his obsession to keep his laws Lorentz-compliant. Poincaré's work led to the &lt;a href=&quot;http://en.wikipedia.org/wiki/Relativity_priority_dispute&quot; rel=&quot;nofollow noreferrer&quot;&gt;Relativity priority dispute&lt;/a&gt; . Poincaré attributes special relativity to Lorentz, Lorentz to Poincaré and Einstein, and Einstein to himself realizing that Lorentz's local time was in fact just time. It is sure the three of them looked after each other, and Poincaré was the astrophysicist of the lot.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="16318" LastEditDate="2022-11-11T07:39:37.990" LastActivityDate="2022-11-11T07:39:37.990" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="2314" PostTypeId="2" ParentId="2308" CreationDate="2015-05-11T15:33:56.913" Score="9" Body="&lt;p&gt;I think the answer is rather anticlimactic. The tl;dr is: The anomalous precession of Mercury was in fact a well-known problem amongst astronomers at that time. Correctly predicting the anomalous precession was a big and well-known test for any theory of gravitation which was not Newton's &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt; law.&lt;/p&gt;&#xA;&lt;p&gt;In the 1840's, Le Verrier, based on his calculations on the orbit of the back-then known planet Uranus, proposed the existence of another planet we today know as Neptune which was indeed discovered just shortly after. Around the same time, he also calculated the orbit of Mercury, noticing, that the theoretically predicted result was largely differing from the real one. In a letter to Hervé Faye, he considered this to be&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] a difficult problem, worthy of the attention of all astronomers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Being one of the leading astronomers of his time, I am pretty sure that this encouraged many people to search for solutions. Le Verrier himself doubted the existence of a undiscovered intra-Mercury planet (Vulcan). The only solution within the known effects was a at least 10%-increase of the Venus mass, which seemed pretty absurd.&lt;/p&gt;&#xA;&lt;p&gt;As you already mentioned, there were two major approaches to solve this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Taking into account unknown celestial bodies or altering the properties of the known one. Known examples are the existence of the planet Vulcan, or the suggestion to flatter the sun. These examples sticked to the Newtonian law of gravitation. However, none of these phenomenons were ever observed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As you already mentioned, also many theorists tried to explain this by modifying Newton's holy &lt;span class=&quot;math-container&quot;&gt;$1/r^2$&lt;/span&gt;-law. Two major approaches were popular here:&#xA;Avoiding a completely static law, or correcting the Newtonian law by a velocity-dependent term (e.g. the Lorentzian theory of gravitation). All of these approaches were either unsuccessful or contained arbitrary parameters.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Einstein's approach is obviously one of the theoretical kind, but, apart from the fact that it actually worked, it had major advantages over all the other approaches — in particular, unlike many other approaches, it was &lt;em&gt;not&lt;/em&gt; specifically designed to solve this problem and Einstein did not take into account any additional hypotheses apart from his laws of gravitation. Therefore, it was a very successful test for Einstein's theory of gravitation.&lt;/p&gt;&#xA;&lt;p&gt;See also IV.14c in Pais' Einstein Biography.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="11059" LastEditDate="2022-11-01T07:30:28.240" LastActivityDate="2022-11-01T07:30:28.240" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2315" PostTypeId="2" ParentId="2303" CreationDate="2015-05-11T20:27:31.407" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1930972830&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Groot's book Aristotle's Empiricism: Experience and Mechanics in the 4th Century B.C.&lt;/a&gt; puts Aristotelian mechanics in the context of mathematical developments of his time. &lt;a href=&quot;http://arxiv.org/pdf/1312.4057.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rovelli's article Aristotle’s Physics: a Physicist’s Look&lt;/a&gt; gives a modernized and mathematized view of Aristotelian mechanics, interpreting him generously, and availing him of an overall dynamical framework that did not emerge until Newton. Still, one can agree that in some sense and partially the classical mechanics does reduce to Aristotle's in the limit of highly resistive medium. If by &quot;Aristotle's hypothesis&quot; you simply mean &quot;velocity is proportional to the force&quot; there is some discussion of it on &lt;a href=&quot;https://physics.stackexchange.com/questions/69790/interpreting-aristotles-law-of-motion-vec-f-r-vec-v&quot;&gt;Physics SE&lt;/a&gt;, and a bit more in Rovelli's article, but still not enough to fill a book I am afraid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the spirit of modernization, some physicists are having even more fun with Aristotelian mechanics, &lt;a href=&quot;http://arxiv.org/pdf/1412.0010v1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;relating it to special relativity&lt;/a&gt;, or even &lt;a href=&quot;https://physics.stackexchange.com/questions/80562/can-we-quantize-aristotelian-physics&quot;&gt;quantizing it!&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2015-05-11T20:53:02.957" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2316" PostTypeId="1" AcceptedAnswerId="2320" CreationDate="2015-05-12T11:48:16.610" Score="5" ViewCount="879" Body="&lt;p&gt;Electrons orbits are referred to as s, p, d, and f, then alphabetically. These &lt;a href=&quot;https://hsm.stackexchange.com/questions/648/what-is-the-historical-reasoning-for-electron-orbital-names&quot;&gt;stand&lt;/a&gt; for sharp, principal, diffuse and fundamental respectively (or maybe the German translations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was any other naming convention ever used to describe them in the beginning of the 20th century? The question dates the use back to the late 1920s, which leaves 10/15 years for alternative conventions to exist.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-05-13T16:27:12.997" Title="Were there alternative names for electron orbitals?" Tags="|chemistry|notation|atomic-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2317" PostTypeId="1" AcceptedAnswerId="2318" CreationDate="2015-05-12T12:28:01.257" Score="6" ViewCount="961" Body="&lt;p&gt;Was Galileo a plagiarizer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we where to apply to the works of Galileo the general standards of plagiarism that we conform to today at our local institutions, would he be considered a plagiarizer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take, for instance, the mean speed theorem. From &lt;a href=&quot;http://en.wikipedia.org/wiki/Mean_speed_theorem&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The now published sources prove to us, beyond contention, that the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics texts, were discovered and proved by scholars of Merton college.... In principle, the qualities of Greek physics were replaced, at least for motions, by the numerical quantities that have ruled Western science ever since. The work was quickly diffused into France, Italy, and other parts of Europe. Almost immediately, Giovanni di Casale and Nicole Oresme found how to represent the results by geometrical graphs, introducing the connection between geometry and the physical world that became a second characteristic habit of Western thought ..&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yet Galileo makes no mention of these predecessors on whose work his is based. He even comes very nearly to claiming that these were his own inventions.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So my question is: would such a vagrant disregard of proper crediting be considered plagiarism if this piece of academia was to happen today?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2094" LastEditorUserId="4251" LastEditDate="2020-05-15T00:24:14.817" LastActivityDate="2020-05-15T00:24:14.817" Title="Was Galileo a plagiarizer?" Tags="|discoveries|classical-mechanics|social-context|galileo|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2318" PostTypeId="2" ParentId="2317" CreationDate="2015-05-12T20:23:32.837" Score="13" Body="&lt;p&gt;See &lt;a href=&quot;http://plato.stanford.edu/entries/heytesbury/#3.5.4&quot; rel=&quot;noreferrer&quot;&gt;Heytesbury and the Physical Sciences&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/nicole-oresme/&quot; rel=&quot;noreferrer&quot;&gt;Nicole Oresme&lt;/a&gt; for detailed information about the so-called &lt;a href=&quot;http://en.wikipedia.org/wiki/Oxford_Calculators&quot; rel=&quot;noreferrer&quot;&gt;Oxford Calculators&lt;/a&gt; and their contribution (mainly) to mathematics.&lt;/p&gt;&#xA;&lt;p&gt;The issue is not so clearly assessed by modern historiography:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There has been some discussion of the meaning of the work of Heytesbury and the other Calculators for the development of the physical sciences. Here it must first be noted that Heytesbury did not think of himself as doing natural science. [...] There is no direct evidence that Galileo knew of or used their results, but it is apparent from the presence of this work in the logic curriculum in Italy that these results were in the air.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You must at least take into account that medieval philosphical treatises were not available in print during XVI and XVII Centuries ...&lt;/p&gt;&#xA;&lt;p&gt;For a detailed discussion of Galileo (and Descartes) works on kinematics and possible connections with the so-called &lt;em&gt;Merton Rule&lt;/em&gt; (the medieval forerunner of mean-speed theorem) see :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Peter Damerow &amp;amp; Gideon Freudenthal &amp;amp; Peter McLaughlin &amp;amp; Jurgen Renn, &lt;a href=&quot;https://books.google.it/books?id=TYTdBwAAQBAJ&amp;amp;pg=PA242&quot; rel=&quot;noreferrer&quot;&gt;Exploring the Limits of Preclassical Mechanics&lt;/a&gt; (2nd ed - 2004): see Index.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;See also:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Wilbur Applebaum (editor), &lt;a href=&quot;https://books.google.it/books?id=k43Q9RHuGXgC&quot; rel=&quot;noreferrer&quot;&gt;Encyclopedia of the Scientific Revolution : From Copernicus to Newton&lt;/a&gt; (2003): &lt;strong&gt;Medieval Science and the Scientific Revolution&lt;/strong&gt; (by W.R.Laird), page 659-on:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At Oxford, the so-called Calculators, many of them associated with Merton College, devised a number of quantitative methods for the logical examination and description of qualitative change and motion. [...] Another was the mean-speed theorem, which equated the overall speed of a uniformly accelerated motion to its mean, or middle, degree of speed. Usually attributed to William Heytesbury (fl.1335), the mean-speed theorem was widely used and is found in the works of other fourteenth-century Mertonians.&lt;/p&gt;&#xA;&lt;p&gt;These fourteenth-century ideas continued to be taught in various forms through the sixteenth century, when they finally fell under the criticism of the humanists, who ridiculed them for their trivial logic-chopping and their authors for their barbarous Latin and bizarre names. Their works lay unread until the French physicist and historian of science Pierre Duhem (1861–1916) rediscovered them at the end of the nineteeenth century.&lt;/p&gt;&#xA;&lt;p&gt;Subsequent scholars often tacitly assumed some degree of continuity between late-medieval and early-modern science without necessarily adhering to the whole Duhem thesis. [...] Ernest A. Moody, however, suggested that Galileo’s early ideas concerning motion were not derived from the fourteenth century Mertonians and Parisians but from anti-Aristotelian alternatives reported in the sixteenth-century sources that Galileo is known to have read. Since these sources contained little about such fourteenth-century developments in physics as the mean-speed theorem, Moody suggested that Galileo later rediscovered them on his own.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-04-23T09:07:23.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2319" PostTypeId="2" ParentId="2317" CreationDate="2015-05-12T20:40:19.850" Score="13" Body="&lt;p&gt;By this standard why single out Galileo? Euclid &quot;plagiarized&quot; Elements, there isn't a single theorem in it that can be reliably attributed to him, and there are entire books that can be attributed to early Pythagoreans, Eudoxus or Theaetetus. In the whole 13 books he does not credit a single person by name. Descartes &quot;plagiarized&quot; analytic geometry, after all Oresme and Casale were drawing coordinate graphs before him. And Newton sure &quot;plagiarized&quot; most of Principia, the first law of motion is due to Galileo, the second one was formulated by Huygens (in the integrated form), and the inverse square law was discussed by many astronomers in the 17th century, not all of whom he credited. The same with calculus, Archimedes was finding areas and volumes, and Apollonius finding tangents in antiquity, not to mention Fermat and others in the 17th century, and even the &quot;fundamental theorem of calculus&quot; was already proved geometrically by Newton's teacher, Barrow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all cases the difference is between isolated ideas and a systematic theory. Galileo not only derived the time squared law without the mean speed theorem, but put it into a broader context of new mechanics, and applied it to physical reality, none of which Merton school, or even Oresme, did. In a 1639 letter he writes:&quot;&lt;em&gt;But in this, I may say, I have been lucky; for the motion of heavy bodies, and the properties thereof, correspond point by point to the properties demonstrated by me&lt;/em&gt;&quot;. Here is from &lt;a href=&quot;http://sites.uci.edu/trabing/files/2013/11/Buridan-copy.pdf&quot; rel=&quot;noreferrer&quot;&gt;Screnes's essay&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Galileo’s calculations relied on an ancient Greek theory of proportion that was not preserved in standard medieval translations of Euclid, and didn’t become available until a better translation was made in 1543. Recent research confirms his claim that he derived the law of acceleration arithmetically, independently of experiment. &lt;strong&gt;His derivation was also independent of the mean speed theorem of the Merton school&lt;/strong&gt;... In the case of the 14th  century mathematicians at Oxford, a mathematical treatment of the concept of uniform acceleration was developed from which the times-squared law can be derived directly. &lt;strong&gt;No one in the Merton school actually derived the law, however, and there was little concern for applying the mathematics to physical systems&lt;/strong&gt;.&lt;/em&gt;&quot; &#xA;  (boldface mine)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Science is a multi-generational enterprise. Some of this sentiment comes from the popular but unfortunate manner of presenting its history as a sequence of discoveries and inventions that came out of nothing in a flash of insight. Every idea has precursors, and every genius has predecessors. Not all of them can be cited, or even clearly identified, so in a sense everybody is a &quot;plagiarizer&quot;. When Newton said that he was &quot;standing on the shoulders of giants&quot; he unintentionally admitted it (the intention was a slight at Hooke's small stature).&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-27T22:36:52.333" LastActivityDate="2015-05-27T22:36:52.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2320" PostTypeId="2" ParentId="2316" CreationDate="2015-05-13T08:46:30.597" Score="2" Body="&lt;p&gt;After a bit research I can not find anything about alternative names for quantum model of atom.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Bohr's model 1, 2, 3... was used but it was for orbit not for orbitals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According by According &lt;a href=&quot;http://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/137.%20s,%20p,%20d,%20f.pdf&quot; rel=&quot;nofollow&quot;&gt;William B. Jensen&lt;/a&gt; Hund had followed Bohr’s practice of labelling the various shells and subshells in terms of their corresponding numerical&#xA;quantum numbers as $3_1$, $3_2$, $3_3$ etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But that was not followed(broadly speaking accepted) by other, So he replaced it by  series notations (s, p, d, and f).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reasons behind no alternative name according to me are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The discovery is new compared to many discoveries which have different names during different time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Modern Periodic Table was made during same period so they had to find something which can be used for years without changing it. So if they keep  had change the name then most probably there should be change in modern periodic table(can't find any such reference).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="247" LastEditorUserId="8" LastEditDate="2015-05-13T16:27:12.997" LastActivityDate="2015-05-13T16:27:12.997" CommentCount="2" CommunityOwnedDate="2015-05-13T08:46:30.597" ContentLicense="CC BY-SA 3.0" />
  <row Id="2321" PostTypeId="1" AcceptedAnswerId="2322" CreationDate="2015-05-13T11:15:10.287" Score="5" ViewCount="256" Body="&lt;p&gt;I've read Skolem's paper on his non-standard models of the arithmetic (&quot;Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I noticed there that he is not mentioning Gödel's incompleteness theorems, though it is known that one of their consequences is the existence of a non-standard model of the arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know who was the first to derive that connection (between Gödel's theorem and the non-standard model)? I believe that Skolem was aware eventually was Gödel's theorems, no?&lt;/p&gt;&#xA;" OwnerUserId="2294" LastEditorUserId="247" LastEditDate="2015-05-13T15:49:07.383" LastActivityDate="2015-05-15T20:34:40.953" Title="Non-standard model of arithmetic and Gödel's theorem" Tags="|mathematics|reference-request|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2322" PostTypeId="2" ParentId="2321" CreationDate="2015-05-13T12:39:50.573" Score="3" Body="&lt;p&gt;Some preliminary references :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Thoralf Skolem, (1929), Uber einige Grundlagenfragen der Mathematik. &lt;em&gt;Skrifter utgit av Det Norske Videnskaps-Akademi i Oslo I. Matematisk-naturvidenskabelig klasse&lt;/em&gt;, 4, pp. 1-49.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Kurt_G%C3%B6del&quot; rel=&quot;nofollow&quot;&gt;Kurt Gödel&lt;/a&gt;, (1931), Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. &lt;em&gt;Monatshefte für Mathematik und Physik&lt;/em&gt; 38: 173–98.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Thoralf_Skolem&quot; rel=&quot;nofollow&quot;&gt;Thoralf Skolem&lt;/a&gt; (1934) Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. &lt;em&gt;Fundam. Math.&lt;/em&gt; 23, 150-161.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://www.hf.uio.no/ifikk/forskning/publikasjoner/tidsskrifter/njpl/vol1no2/skobio.pdf&quot; rel=&quot;nofollow&quot;&gt;Jens Erik Fenstad&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Towards the end of the 1929 paper Skolem expressed some doubts about the complete axiomatizability of mathematical concepts. His scepticism was based on the set-theoretic relativism which follows from the Löwenheim-Skolem theorem. In 1929 he could give only some partial results, but in a paper from 1934 (and a previous one from 1933) &quot;Über die Nichtcharacterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen&quot; he could prove that there is no finite or countably infinite&#xA;  set of sentences in the language of Peano arithmetic which characterizes the natural numbers. Today, this follows as a simple consequence of Gödel's completeness theorem. The technique used by Skolem was a more direct model-theoretic construction. And this technique, suitably refined to the so-called &quot;ultraproduct&quot; construction, has been an important tool in recent work on model theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;You can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Stephen Cole Kleene, &lt;a href=&quot;https://books.google.it/books?id=4GzCAgAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Mathematical logic&lt;/a&gt; (1967 - Dover ed  2002), page 326-27 for a discussion of Skolem's theorem (1934) on nonstandard models of arithmetic :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is surprising that the existence of nonstandard models of the usual &#xA;  axioms of elementary number theory was not widely recognized very early &#xA;  by juxtaposing Gödel's completeness theorem 1930 and his incompleteness &#xA;  theorem 1931 [footnote].&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;[footnote] Not however with quite the version in Gödel 1931, where the incompleteness is given for &quot;&lt;em&gt;Principia Mathematica&lt;/em&gt; [Whitehead and Russell, 1910—13] and related systems&quot;. PM is not of the form {(first-order) predicate calculus} + {number-theoretic axiom system}. Number-theoretic systems of that form came in with Hilbert 1928, so they are sometimes called Hilbert arithmetic. In 1931-2 (reporting a colloquium held January 22, 1931), Gödel states his incompleteness theorem for such a formal system, with all the usual axioms of elementary number theory.&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;When the author pointed out the connection between Skolem's and Gödel's theorems in &lt;strong&gt;Introduction to metamathematics&lt;/strong&gt; (1952) p. 430, he knew of no earlier mention of it in the literature. Recently, he has come across 2 lines in Gödel's review of Skolem 1933 [Uber die Unmbglichkeit einer vollstdndigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems] (K.Gödel, 1934, Review of Skolem 1933, &lt;em&gt;Zentralblatt fiir Mathematik und ihre Grenzgebiete&lt;/em&gt;, vol. 7, pp. 193-194, p.194 lines 10-11) and 3 in Henkin 1950 (L.Henkin, &lt;a href=&quot;http://www.jstor.org/discover/10.2307/2266967?uid=3738296&amp;amp;uid=2&amp;amp;uid=4&amp;amp;sid=21106414969211&quot; rel=&quot;nofollow&quot;&gt;Completeness in the theory of types&lt;/a&gt;, &lt;em&gt;JSL&lt;/em&gt;, vol.15, pp. 81-91, p.91 lines 8-10) alluding to such a connection. &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-05-15T20:34:40.953" LastActivityDate="2015-05-15T20:34:40.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2323" PostTypeId="1" AcceptedAnswerId="2324" CreationDate="2015-05-13T20:20:26.177" Score="7" ViewCount="326" Body="&lt;p&gt;After Gauss helped relocate Ceres, he studied the orbit of the asteroid Pallas and discovered (1812) that Jupiter and Pallas have an orbital resonance that is nearly equal to 18:7. For instance, using the modern estimates of their orbital periods as 4332.59 days and 1684.87 days respectively, their ratio has continued fraction expansion $[2,1,1,2,1,511,2,\ldots]$, which is quite clearly approximated well by $[2,1,1,2,1] = 18/7$. Of course Gauss did not have the modern values for the orbital periods. If we use the cruder estimates of 4333 and 1685 we get a ratio with continued fraction expansion $[2, 1, 1, 2, 1, 240, 113, \ldots]$, which again is begging to be approximated by  $[2,1,1,2,1]$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What was the ratio computed by Gauss that he then determined is nearly 18/7? Maybe it was not the orbital period directly but some other astronomical measure whose ratio would come out to the same thing.&lt;/li&gt;&#xA;&lt;li&gt;Did he use continued fractions, or was his great familiarity with decimal expansions of &quot;small&quot; fractions enough? I think this example is a really super illustration of continued fractions, but I would like to know for sure if it's how Gauss attacked the problem.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I can find a number of sources discussing this resonance, but they mention the value 18/7 without giving the Gaussian calculation behind it (i.e., what ratio did he estimate as 18/7), and they don't indicate his method either.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="4251" LastEditDate="2020-05-14T20:13:43.057" LastActivityDate="2020-05-14T20:13:43.057" Title="Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas" Tags="|mathematics|astronomy|gauss|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2324" PostTypeId="2" ParentId="2323" CreationDate="2015-05-13T23:12:31.063" Score="5" Body="&lt;p&gt;Gauss's ratio was that of the &lt;em&gt;mean motions&lt;/em&gt; $\mathit{[M]}$ (⚴) and $\mathit{[M']}$(♃) of Pallas and Jupiter. (Instead of these letters he used planet symbols, which are shown in their unicode version in brackets.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As can be seen in his &lt;em&gt;Nachlass&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/werkehrsgvonderg07gausuoft#page/553&quot; rel=&quot;nofollow&quot;&gt;Werke, vol. 7, e.g. p. 553&lt;/a&gt;), he was expressing the perturbations of Pallas elements as sums of dozens of trigonometric terms $A\sin(k\mathit{[M']-\ell [M]}+\delta)$. On &lt;a href=&quot;https://archive.org/stream/werkehrsgvonderg07gausuoft#page/604&quot; rel=&quot;nofollow&quot;&gt;p. 604&lt;/a&gt; the editors comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On one of the sheets which contain the integration of the perturbations of the epoch $\varepsilon$, one finds the following small computation, in which the first number given is Pallas's mean motion $\mathit{[M]}$:&#xA;  $$&#xA;\begin{gather}&#xA;\mathit{769'',202079}\\&#xA;\textit{das 7-fache} = \mathit{5384'',414553}\\&#xA;\underline{\mathit{18\ m. m. [M'] = 5384'',392272}}\\&#xA;\mathit{18 [M']- 7[M] = -0,022281}&#xA;\end{gather}&#xA;$$&#xA;  This seems to be the only hint at the first step in Gauss's discovery of the commensurability of both periods; from this one can't however conclude anything more than that he had just found the quantity $\mathit{18 [M']- 7[M]}$ extremely small.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I read this as saying that this (&quot;small divisor&quot;?) stood out enough as it is, &lt;em&gt;numerically&lt;/em&gt; among the $\mathit{k[M']- \ell[M]}$'s he was looking at, with  no need for an appeal to, e.g., continued fractions. &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="8" LastEditDate="2015-05-15T18:35:02.483" LastActivityDate="2015-05-15T18:35:02.483" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2325" PostTypeId="1" AcceptedAnswerId="2326" CreationDate="2015-05-14T00:00:50.607" Score="5" ViewCount="436" Body="&lt;p&gt;Having borrowed from the library an English translation of Newton's Principia (Motte's), I read the begining sections, Part 1 and the Systems of the world, and noticed that Newton did physics completely differently than is currently taught in a classical mechanics course. He uses synthetic geometry with certain theorems on limits, or a geometric calculus, which is very different from the symbolic calculus we used. Furthermore, he only uses scalar quantities instead of vector ones. Finally, he seems to define three types of forces, accelerative, absolute and motive forces. His derivation of Kepler's second law is completely different than the ones currently taught. When did physics texts then begin to discuss or teach newtonian mechanics in the modern analytical way using symbolic calculus, coordinate systems and vectors with free body diagrams?  &lt;/p&gt;&#xA;" OwnerUserId="2171" LastActivityDate="2018-10-07T05:05:00.657" Title="On the development of Newtonian Mechanics" Tags="|physics|classical-mechanics|mathematical-physics|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2326" PostTypeId="2" ParentId="2325" CreationDate="2015-05-14T01:49:33.873" Score="3" Body="&lt;p&gt;One cannot answer &quot;when&quot; because this was a &quot;slow&quot;, gradual development.&#xA;And it continues. Some milestones are&#xA;the work of d'Alembert, Euler and Clairaut, then &#xA;Lagrange's Mechanique analytique. It does not use a single figure, only formal calculus, and Lagrange was very proud of this. Then comes Laplace, Poisson, Hamilton and Hamiltonian mechanics. Vectors is a much later invention (end of 19 century, in physics). The modern formulation uses the language of manifolds, cotangent spaces and differential forms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern high school and elementary university courses reflect approximately the level reached in the beginning of 20-s century, except vectors. Vectors penetrated elementary education only in the second half of 20-s century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. If you read what Newton wrote on Calculus, you will not recognize absolutely what is taught in modern courses. Modern calculus, as taught nowadays&#xA;owes more to Leibniz, Bernoulli and Euler than to Newton. But here too there was a gradual development. Two parallel developments: on the advanced level, and another one, in elementary education which lags the advanced level by 50-100 years.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-05-14T01:49:33.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2327" PostTypeId="2" ParentId="2325" CreationDate="2015-05-14T02:19:47.697" Score="6" Body="&lt;p&gt;According to Truesdell [&lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=70568&quot; rel=&quot;nofollow noreferrer&quot;&gt;1954&lt;/a&gt;]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(p. xliii:) As far as I can ascertain, it is &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1750&amp;amp;seite:int=200&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler [1750, p. 196]&lt;/a&gt; which contains the first general statement of “Newton’s equations”. (p. xlii:) The axioms which Euler asserts “include all principles of mechanics” are&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;2M\frac{d^2x}{dt^2}=P,\qquad&#xA;2M\frac{d^2y}{dt^2}=Q,\qquad&#xA;2M\frac{d^2z}{dt^2}=R.&#xA;$$&lt;/span&gt;&#xA;  (...) Anyone who has looked at Newton's &lt;em&gt;Principia&lt;/em&gt; knows that no such equations occur in it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think he's right, except that Euler actually did it earlier in &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1747&amp;amp;seite:int=132&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1747, p. 103]&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cela posé, prenant l'element du tems &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt; pour constant, le changement instantané du mouvement du Corps sera exprimé par ces trois équations:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\textrm{I.}\quad \frac{2ddx}{dt^2}=\frac XM;\qquad&#xA;\textrm{II.}\quad \frac{2ddy}{dt^2}=\frac YM;\qquad&#xA;\textrm{III.}\quad \frac{2ddz}{dt^2}=\frac ZM&#xA;$$&lt;/span&gt;&#xA;  d'où l'on pourra tirer pour chaque tems ecoulé &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; les valeurs &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;, &amp;amp; par conséquent l'endroit où le Corps se trouvera. C. Q. F. T.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Maltese [&lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=2052763&quot; rel=&quot;nofollow noreferrer&quot;&gt;2003&lt;/a&gt;, &lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=2193097&quot; rel=&quot;nofollow noreferrer&quot;&gt;2006&lt;/a&gt;] has more on e.g. Euler and vectors.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;[1747] Leonhard Euler. Recherches sur le mouvement des corps celestes en general. &lt;em&gt;Hist. Acad. Roy. Berlin&lt;/em&gt; &lt;strong&gt;3&lt;/strong&gt; (1749), 93–143. (“Presented on June 8, 1747”. Reprint: &lt;em&gt;Opera Omnia&lt;/em&gt; (2) &lt;strong&gt;25&lt;/strong&gt; (1960), 1–44.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[1750] Leonhard Euler. Decouverte d’un nouveau principe de Mecanique. &lt;em&gt;Hist. Acad. Roy. Berlin&lt;/em&gt; &lt;strong&gt;6&lt;/strong&gt; (1752), 185–217. (“Presented on September 3, 1750”. Reprint: &lt;em&gt;Opera Omnia&lt;/em&gt; (2) &lt;strong&gt;5&lt;/strong&gt; (1957), 81–108.) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[1954] Clifford A. Truesdell. Rational fluid mechanics, 1687–1765. Leonhardi Euleri &lt;em&gt;Opera Omnia&lt;/em&gt; (2) &lt;strong&gt;12&lt;/strong&gt; (1954), ix–cxxv.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[2003] Giulio Maltese. The ancients’ inferno: the slow and tortuous development of ‘Newtonian’ principles of motion in the eighteenth century. In A. Becchi et al. (eds) &lt;em&gt;Essays on the history of mechanics&lt;/em&gt;, Birkhäuser, Basel, pp. 199–221.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[2006] Giulio Maltese. On the changing fortune of the Newtonian tradition in mechanics. In K. Williams (ed.) &lt;em&gt;Two cultures&lt;/em&gt;, Birkhäuser, Basel, pp. 97–113.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-07T05:05:00.657" LastActivityDate="2018-10-07T05:05:00.657" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2329" PostTypeId="1" AcceptedAnswerId="2331" CreationDate="2015-05-15T22:37:13.160" Score="5" ViewCount="511" Body="&lt;p&gt;In cosmology, &quot;dust&quot; refers to a pressureless perfect fluid, which essentially means a continuum of nonrelativistic material particles, such as galaxies. This is a picturesque and unusual piece of terminology, and in western culture it has the inevitable connotation from Genesis: &quot;In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who invented this term? I notice that, for example, the title of van Stockum's 1937 paper describing what we now call &quot;van Stockum dust,&quot; was &quot;The gravitational field of a distribution of particles rotating around an axis of symmetry.&quot; So presumably the term didn't exist in 1937, or he would have used it in the title.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2015-05-18T01:17:27.950" Title="Origin of &quot;dust&quot; in cosmology?" Tags="|terminology|relativity-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2330" PostTypeId="1" CreationDate="2015-05-15T22:53:55.650" Score="17" ViewCount="806" Body="&lt;p&gt;The story was circulating in early 2000s, so presumably it happened in 1990s. Kontsevich, it goes, opened a lecture course on mirror symmetry with:&quot;&lt;em&gt;This course is about two categories. One I will not define, the other nobody knows how to define, and we will be proving that they are equivalent&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like the story because it highlights the informal side of mathematics, and counters the impression that what mathematicians do is logically deduce theorems from ready made definitions and axioms. But did it actually happen? There is no doubt that Kontsevich was giving lectures on mirror symmetry with categories in them, categorical mirror symmetry is his field after all. But did he say anything resembling the quote, or was it embellished like Eric Temple Bell embellished &lt;a href=&quot;https://hsm.stackexchange.com/questions/2105/whats-the-famous-story-about-a-mathematician-who-gave-a-talk-without-saying-a-w/2106#2106&quot;&gt;Cole's &quot;talk without saying a word&quot; about Mersenne numbers.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-05-15T22:53:55.650" Title="Did Kontsevich start a lecture with &quot;one I will not define, the other nobody knows how to define, and we will be proving that they are equivalent&quot;?" Tags="|mathematics|biographical-details|mathematicians|debunking|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2331" PostTypeId="2" ParentId="2329" CreationDate="2015-05-16T00:51:56.913" Score="5" Body="&lt;p&gt;The prime suspect for injecting Biblical references into cosmology would be Georges Lemaître, a cosmologist and... a Catholic priest. His &quot;Cosmic Egg&quot; and &quot;Primeval Atom&quot; lost to Hoyle's sarcastic &quot;Big Bang&quot; however, and it seems he passed up on introducing &quot;dust&quot;. His 1927 Ph.D. thesis The Gravitational Field in a Fluid became the basis for the now famous 1927 paper &lt;a href=&quot;http://mnras.oxfordjournals.org/content/91/5/483&quot; rel=&quot;nofollow&quot;&gt;Homogeneous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae&lt;/a&gt;, written in French, and self-translated into English in 1931 with minor edits. Lemaître talks about nebulae and introduces the usual assumption of negligible pressure, but does not call his fluid &quot;dust&quot; even descriptively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems the 'priority' goes to Tolman's &lt;a href=&quot;http://www.webcitation.org/5w35fIoge&quot; rel=&quot;nofollow&quot;&gt;Effect of Inhomogeneity on Cosmological Models (1934)&lt;/a&gt;, who writes &quot;&lt;em&gt;the fluid in the models was taken as dust exerting negligible pressure&lt;/em&gt;&quot;, and repeats &quot;dust&quot; 16 times on 8 pages. In the assumptions on the particular form of stress-energy tensor he follows Lemaître:&quot;&lt;em&gt;we shall confine our attention to very simple models composed of dust particles (nebulae) which exert negligible pressure... This will permit  us to employ expressions for the line element and its consequences which are equivalent to those recently developed by Lemaître for investigating the formation of nebulae&lt;/em&gt;&quot;. Some authors define &quot;dust&quot; more broadly, Sachs and Wu in General Relativity for Mathematicians (3.13) simply give a kind of divergence free condition on stress energy tensor, and note that pressureless perfect fluid satisfies it. The prototypical example seems to be nebulae, clouds of particles interacting only gravitationally, literally dust.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the association with stress energy tensors is clearly specific to relativity more loosely &quot;nebulae&quot; and &quot;dust&quot; were already used by Newtonian astronomers, and there seems to be continuity with Tolman's usage. The reference to nebulae reminds the &lt;a href=&quot;http://en.wikipedia.org/wiki/Nebular_hypothesis#History&quot; rel=&quot;nofollow&quot;&gt;nebular hypothesis&lt;/a&gt;, according to which a rotating dust cloud formed the solar system. It goes back to Kant (1755), and in a more detailed form to Laplace (1796), and was revived by Chamberlin and Moulton in 1901. Both van Stockum's and Gödel's early &quot;dust solutions&quot; to the Einstein equations involve rotating dust. The nebular hypothesis is (metaphorically) in line with the Biblical origin story, but nebular dust is also literally analogous to the ordinary dust, so we can only speculate what contributed to the spread of usage. As an alternative reference, &lt;a href=&quot;https://books.google.com/books?id=kvpVkTyBynwC&amp;amp;pg=PA193&amp;amp;lpg=PA193&amp;amp;dq=lucretius+dust&amp;amp;source=bl&amp;amp;ots=y-8bQdJNHa&amp;amp;sig=JthRX_nW37bjz07RL2Xm7S7Jp3Q&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=bZZWVY-tGoaRsAWXi4DABg&amp;amp;ved=0CCAQ6AEwATgK#v=onepage&amp;amp;q=lucretius%20dust&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Democritus envisioned a universe filled with &quot;&lt;em&gt;fine dust&lt;/a&gt; which is divided in the breeze and becomes visible in the sun's beam&lt;/em&gt;&quot;. Lucretius expressed it poetically in De Rerum Natura, well familiar to people with classical education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are other examples of classical usage. The paradox that in an infinite universe uniformly filled with stars the night sky would appear to be bright, stated most eloquently by Olbers in 1826 and named after him, was already pointed out by Digges back in 1576. Kepler took it as an argument for finite universe, but Halley in 1705, Cheseaux in 1744 and Olbers himself suggested &lt;a href=&quot;http://www.mathpages.com/home/kmath141/kmath141.htm&quot; rel=&quot;nofollow&quot;&gt;interstellar dust obscuring distant stars&lt;/a&gt; as a resolution. In 1831 Herschel pointed out however that the dust would simply be heated up until it glowed as brightly as the stars (further reflection shows that the sky should in fact be infinitely bright). Finally, zodiacal light is explained by an interplanetary dust cloud, which attracted much attention at the turn of the centuries, because both Newcomb (1895) and Seeliger (1906) tried to use it to explain the anomalous precession of Mercury, before Einstein gave the general relativity explanation in 1915.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-18T01:17:27.950" LastActivityDate="2015-05-18T01:17:27.950" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2332" PostTypeId="1" AcceptedAnswerId="2338" CreationDate="2015-05-17T00:11:12.440" Score="21" ViewCount="1351" Body="&lt;p&gt;When was the last analog computer developed? What about one that is still in use (as in computation or experiment)? Were there major improvement done from the day digital computers became so popular?&lt;/p&gt;&#xA;" OwnerUserId="2312" LastEditorUserId="4251" LastEditDate="2020-05-13T11:22:59.370" LastActivityDate="2021-10-01T06:16:25.380" Title="What happened to analog computers?" Tags="|computer-science|computers|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2334" PostTypeId="2" ParentId="2332" CreationDate="2015-05-17T10:36:54.783" Score="11" Body="&lt;p&gt;The &lt;a href=&quot;http://turnstiletours.com/uss-iowa-brooklyn-built-battleship-lives-on-in-la/&quot; rel=&quot;nofollow noreferrer&quot;&gt;USS Iowa&lt;/a&gt; has mechanical analog computers for firing that survived the 1980-refit of the ship.&lt;/p&gt;&#xA;&lt;p&gt;An analog computer of everyday use in the XXIst century are the &lt;a href=&quot;http://en.wikipedia.org/wiki/Flight_computer&quot; rel=&quot;nofollow noreferrer&quot;&gt;flight computers&lt;/a&gt; used on a wide range of aircraft daily.&lt;/p&gt;&#xA;&lt;p&gt;Electronical analog computers were used until the 70s because they're very good at analytics, but I think even electronic analog computers have now lost the race.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="4251" LastEditDate="2021-10-01T06:16:25.380" LastActivityDate="2021-10-01T06:16:25.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2336" PostTypeId="1" CreationDate="2015-05-17T14:43:50.817" Score="7" ViewCount="312" Body="&lt;p&gt;The US and the UK uses a baseline dot as the decimal mark. Most other European countries use the comma. &lt;a href=&quot;http://www.councilscienceeditors.org/wp-content/uploads/v31n2p042-043.pdf&quot;&gt;Here&lt;/a&gt; is a short piece on the history of this matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: When did the use of the comma as the decimal mark become (largely) extinct in the US? I ask because I happened to come across &lt;a href=&quot;https://books.google.com/books?id=c0VIAQAAMAAJ&amp;amp;pg=PA121&amp;amp;dq=%22Judgment%20for%20the%20plaintiffs%20for%20$95,18%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=9qhYVYHIPMi5sAWAuoCYBg&amp;amp;ved=0CCcQ6AEwAA#v=onepage&amp;amp;q=%22Judgment%20for%20the%20plaintiffs%20for%20%2495%2C18%22&amp;amp;f=false&quot;&gt;this 1868 document (Maine Supreme Court ruling) where the comma was used&lt;/a&gt;, suggesting that it was still in some regular use in the US as late as 1868.&lt;/p&gt;&#xA;" OwnerDisplayName="user67" LastActivityDate="2015-06-28T18:48:33.073" Title="When did the use of the comma as a decimal mark become extinct in the US?" Tags="|notation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2338" PostTypeId="2" ParentId="2332" CreationDate="2015-05-17T22:27:46.423" Score="20" Body="&lt;p&gt;Rutherford journal gives a &lt;a href=&quot;http://www.rutherfordjournal.org/article020106.html&quot; rel=&quot;noreferrer&quot;&gt;nice chronology of analog computing up to 1970&lt;/a&gt;, when they went out of mass use. &quot;The demise of the analogue computer was a gradual process. Apart from reductions in size and improvements in speed, the machines on sale up until 1970 were all essentially electronic differential analysers.&quot; Aside from living relics there has been something of a resurgence in the use of analog computing in the last 10 years however. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Artificial neural networks remained a subject of attention even after 1970s. They would be analog computers if implemented directly, rather than digitally simulated, &lt;a href=&quot;http://en.wikipedia.org/wiki/Artificial_neural_network#History&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia gives some history&lt;/a&gt;. In 1990s Siegelmann showed mathematically that theoretically &lt;a href=&quot;http://www.sciencemag.org/content/268/5210/545&quot; rel=&quot;noreferrer&quot;&gt;analog neural networks are strictly stronger than the universal Turing machine&lt;/a&gt; (although implementation requires non-physical assumptions).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in 1971 Chua suggested based on symmetry arguments that in addition to three known passive elements of electric circuits, resistors, inductors and capacitors, there should be a fourth, which he called memristor (memory resistor). Unfortunately, no physical model was known at the time. Only in 2008 Strukov, Snider, Stewart and Williams found a &lt;a href=&quot;http://inst.eecs.berkeley.edu/~ee129/sp09/handouts/TheMissingMemristorFound.pdf&quot; rel=&quot;noreferrer&quot;&gt;physical realization of memristor&lt;/a&gt;, and pointed out that it explains &quot;hysteretic current–voltage behavior observed in many nanoscale electronic devices&quot;. In 2009-2015 researchers &lt;a href=&quot;http://www.nature.com/nature/journal/v521/n7550/full/521037a.html&quot; rel=&quot;noreferrer&quot;&gt;used memristors to build artificial neural networks for machine learning and pattern recognition&lt;/a&gt;. Nanotechnology is generally expected to make wide use of analog computing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another motivation for resurgence of analog is the power efficiency wall for digital computers. &quot;Moore’s law — the maxim that processing power will double every 18 months or so — continues, but battery lives just haven’t kept up&quot;. This opened a new  niche, &quot;low-power, high-performance computing, especially in areas where the answer doesn’t have to be completely perfect — image rendering for example&quot;. &lt;a href=&quot;http://www.wired.com/2012/08/upside/&quot; rel=&quot;noreferrer&quot;&gt;US Department of Defense funded UPSIDE program&lt;/a&gt; led by Hammerstrom, Intel's chipmaker back in 1980s, &quot;to build analog processors that can do probabilistic math without forcing transistors into an absolute one-or-zero state, a technique that burns energy&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2012 Rice University designed a &quot;low-power chip that uses probabilistic computing techniques to do energy-efficient, if occasionally inexact, calculations&quot;. At Stanford researchers built biological transistors inside cells, and at MIT created &lt;a href=&quot;http://www.extremetech.com/computing/155994-mit-constructs-synthetic-analog-computers-inside-living-cells&quot; rel=&quot;noreferrer&quot;&gt;&quot;synthetic analog computers that run genetic machinery&quot;&lt;/a&gt;, they turn on or suppress a gene that manufactures green fluorescent protein, an optically detectable molecule.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-30T22:00:48.563" LastActivityDate="2015-10-30T22:00:48.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2339" PostTypeId="1" CreationDate="2015-05-17T22:30:19.480" Score="10" ViewCount="4572" Body="&lt;p&gt;I want to know what the first command-line interface was, and when it was. That is, the first time one could type in a command on a keyboard or similar, and receive feedback via a screen or teleprinter, as opposed to entering commands via paper tape, etc.&lt;/p&gt;&#xA;" OwnerUserId="423" LastActivityDate="2015-05-20T23:45:53.667" Title="When was the first command-line interface developed?" Tags="|computer-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2340" PostTypeId="2" ParentId="2339" CreationDate="2015-05-18T02:04:55.343" Score="9" Body="&lt;p&gt;According to a wikibook &lt;a href=&quot;http://en.wikibooks.org/wiki/Operating_System_Design/Command_line_interfaces&quot; rel=&quot;noreferrer&quot;&gt;Operating System Design&lt;/a&gt;: &quot;A command line interface or CLI is a method of interacting with a computer by giving it lines of textual commands (that is, a sequence of characters) either from keyboard input or from a script. They were originally developed for interfacing with computers over teletype machines in the 1950s&quot;. &lt;a href=&quot;https://i.stack.imgur.com/Cc9DR.jpg&quot; rel=&quot;noreferrer&quot;&gt;Landley in The Art of Unix Usability&lt;/a&gt; explains further that CLI &quot;were closely associated with the rise of timesharing computers. The concept of timesharing dates back to the 1950s; the most influential early experiment was the MULTICS operating system after 1965; and by far the most influential of present-day command-line interfaces is that of Unix itself, which dates from 1969 and has exerted a shaping influence on most of what came after it... The widespread adoption of video-display terminals (VDTs) in the mid-1970s ushered in the second phase of command-line systems.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The teletype itself predates electronic computers, mechanical engineer Charles Krum filed patents for &quot;typebar page printer&quot;in 1903, and for &quot;type wheel printing telegraph machine&quot; in 1904, both based on the 1902 idea of electrical engineer Pearne. In 1906 Krum's son Howard developed the start-stop synchronizing method for code telegraph systems, which made possible the practical teleprinter. In 1908 a company eventually called Teletype Corporation was founded to mass produce the devices. Here is what ASR-33 Teletype (1963), used to interact with &lt;a href=&quot;http://www.computersciencelab.com/ComputerHistory/HistoryPt4.htm&quot; rel=&quot;noreferrer&quot;&gt;time-sharing IBM mainframes&lt;/a&gt;, looks like: &#xA;&lt;img src=&quot;https://i.stack.imgur.com/Cc9DR.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-05-20T23:45:53.667" LastActivityDate="2015-05-20T23:45:53.667" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2341" PostTypeId="1" AcceptedAnswerId="2342" CreationDate="2015-05-18T02:52:01.747" Score="3" ViewCount="133" Body="&lt;p&gt;How did Aristotle define life using his elemental system?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because it seems that Earth, Fire, Air, and Water, only defined non-living matter, and Quintessence was solely used as the element of the heavens...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or am I wrong and Earth, Fire, Air, and Water were also used to define life?&lt;/p&gt;&#xA;" OwnerUserId="2313" LastEditorUserId="2313" LastEditDate="2015-05-18T10:19:44.407" LastActivityDate="2018-07-15T14:37:19.550" Title="How did Aristotle define life using his elemental system?" Tags="|physics|ancient-greece|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2342" PostTypeId="2" ParentId="2341" CreationDate="2015-05-18T14:13:24.350" Score="1" Body="&lt;p&gt;In &lt;a href=&quot;http://en.wikipedia.org/wiki/On_the_Soul&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Anima&lt;/em&gt;&lt;/a&gt;, Aristotle gives a thorough description of his vision of life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First think to note to answer your question is that Aristotle does not make a distinction between the body and the soul we know today:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the affections of soul are inseparable from the material substratum of animal life&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The question you're asking did not really occur to Aristotle, but he does address it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;and things are formed out of the principles or elements, so that soul must be so too&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But he then gives a state-of-the-art of soul research and come to the following (terribly funny, if you want my opinion) conclusion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Each of the elements has thus found its partisan, except earth-earth has found no supporter unless we count as such those who have declared soul to be, or to be compounded of, all the elements. All, then, it may be said, characterize the soul by three marks, Movement, Sensation, Incorporeality, and each of these is traced back to the first principles. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, to answer your question: souls are, to Aristotle, made up of about 3 of the 4 elements. It doesn't seem much of an issue to him, but he does mention it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to read more about this, there are very readable translations of &lt;em&gt;De Anima&lt;/em&gt; &lt;a href=&quot;http://classics.mit.edu/Aristotle/soul.1.i.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;available out there&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="2313" LastEditDate="2018-07-15T14:37:19.550" LastActivityDate="2018-07-15T14:37:19.550" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2343" PostTypeId="1" AcceptedAnswerId="2344" CreationDate="2015-05-20T00:26:47.683" Score="31" ViewCount="4181" Body="&lt;p&gt;I'm studying High School Science teaching in Australia. In our Science curriculum there are &quot;cross-curriculum&quot; priorities &quot;Aboriginal and Torres Strait Islander histories and cultures&quot; and &quot;Asia and Australia’s engagement with Asia&quot;. (see &lt;a href=&quot;http://www.australiancurriculum.edu.au/science/cross-curriculum-priorities&quot;&gt;http://www.australiancurriculum.edu.au/science/cross-curriculum-priorities&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our lecturer told us that teaching science means teaching Western science based around the Scientific Method and that the above priorities should be taught in parallel with Western science. Specifically, we shouldn't try to integrate them too much into our teaching. For example, teaching Western medicine and Aboriginal Medicine comparatively rather than to trying to unify them. Similarly, for the Asian priority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, China and Aboriginal Australia developed many technologies  and observed many phenomena but, perhaps, not in a scientific way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &quot;Did other cultures use/develop principles analogous to the western scientific method?&quot;&lt;/p&gt;&#xA;" OwnerUserId="2321" LastEditorUserId="2321" LastEditDate="2015-05-20T01:59:14.963" LastActivityDate="2023-06-22T04:56:31.070" Title="Is the Scientific Method uniquely Western?" Tags="|philosophy-of-science|scientific-method|ancient-china|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2344" PostTypeId="2" ParentId="2343" CreationDate="2015-05-20T01:49:29.237" Score="23" Body="&lt;p&gt;In a word, yes, but only interpreting &amp;quot;western&amp;quot; broadly. The tradition of precise and systematic astronomical observations comes from ancient Babylonians, i.e. from Mesopotamia, and even Alexandria, where many scientists worked during Hellenistic times, is in the middle east. John Philoponus (c. 490 – 570) working there wrote things like &amp;quot;&lt;em&gt;but this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument&lt;/em&gt;&amp;quot;. One of the first practitioners of &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_scientific_method#Ibn_al-Haytham&quot; rel=&quot;nofollow noreferrer&quot;&gt;controlled experiments and field studies&lt;/a&gt; were an Arab polymath Alhazen (c. 965 –1040), and Persians Al-Biruni (973 – 1048) and Avicenna (c. 980 –1037). Ancient Babylon, Hellenistic Alexandria, and medieval middle east deeply influenced the scientific method tradition. They are often counted as &amp;quot;western&amp;quot; because they culminated in a completed form of scientific method, explicitly outlined by Francis Bacon (1620), in western Europe during and after the Renaissance,  and then exported all over the world during the colonial period.&lt;/p&gt;&#xA;&lt;p&gt;There was a &lt;a href=&quot;http://en.wikipedia.org/wiki/Science_and_technology_of_the_Song_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;remarkable period of scientific and technological innovation in China during Song dynasty (960–1279)&lt;/a&gt;, that in some ways compares to Hellenistic period and Renaissance. The invention of gunpowder and nautical compass, as well as widespread use of printing press and paper money in China date to this period. Chinese Leonardo, Shen Kuo (1031–1095), the inventor of compass, also measured the distance between the North star and the true north, improved designs of the armillary sphere, gnomon, sighting tube, and water clock, devised a geological hypothesis for land formation, and proposed a hypothesis of gradual climate change. And yet Song China was a near miss, it did not lead to development of systematic theoretical and empirical science as we understand it today. But it shows that much can be accomplished in other traditions. A very sympathetic, anti-eurocentric, account is given by &lt;a href=&quot;https://books.google.com/books/about/Why_the_West_Rules_for_Now.html?id=4gFfSyXG2_wC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Morris in Why the West Rules - For Now&lt;/a&gt;, for balance here is a &lt;a href=&quot;http://www.history.ac.uk/reviews/review/1091&quot; rel=&quot;nofollow noreferrer&quot;&gt;critical review&lt;/a&gt; of his book. But even Morris acknowledges that the East did not formulate a scientific method of its own.&lt;/p&gt;&#xA;&lt;p&gt;Ancient Egypt and China, medieval India, American Indian civilizations, etc. also made many technological advances, but they were practically oriented traditional societies that developed their tools and procedures through accumulation of incremental glacial changes, generational trial and error, rather than through intentional innovation guided by scientific inquiry. Traditional medicine, aboriginal one in particular, is a typical product of such development, but so is ancient Babylonian or medieval Chinese mathematics.&#xA;Here is &lt;a href=&quot;http://strangebeautiful.com/other-texts/stein-eudoxos-dedekind.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stein's characterization of Babylonian mathematics, that applies to science in traditional societies&lt;/a&gt; more broadly:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...for the Babylonians, mathematics was not an enterprise, but a lore and a skill possessed by the priestly or scribal class, for use in essentially administrative functions, and passed on from generation to generation much as were the techniques of the craft&#xA;guilds in the middle ages. Of course, the techniques had to have been&#xA;discovered; and innovations might occur from time to time; but&#xA;invention or discovery was no more the business of those trained in the&#xA;lore, than it was the business of the medieval master of a craft.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&#xA;" OwnerUserId="55" LastEditorUserId="424" LastEditDate="2023-06-22T04:56:31.070" LastActivityDate="2023-06-22T04:56:31.070" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2345" PostTypeId="1" AcceptedAnswerId="2361" CreationDate="2015-05-20T16:17:35.640" Score="6" ViewCount="2012" Body="&lt;p&gt;In the contemporary axiomatic description the conservation laws in mechanics are deduced from Noether's Theorem using the homogeneity and isotropy of space and time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did Newton, if he did, establish the conservation of linear momentum and energy in the &lt;em&gt;Principia&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not looking for the most general concept of these as conservation laws; but possibly them in a limited form; say elastic collision.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've had a brief look and the first surprise is that he eschews the use of calculus and uses instead the theory of conic sections, i.e. he is relying on Euclid's &lt;em&gt;Elements&lt;/em&gt;; I take it because this was the most favored method as calculus was in a way too new for natural philosophers then. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="55" LastEditDate="2015-05-22T03:29:44.550" LastActivityDate="2015-06-17T22:59:01.410" Title="How did Newton establish the conservation laws in the Principia?" Tags="|theoretical-physics|classical-mechanics|energy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2346" PostTypeId="2" ParentId="788" CreationDate="2015-05-20T20:29:20.493" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I was trying to figure out how people came to know about time then I realized that people started keeping track of time to know about sunset and sunrise.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;People first kept track of time to know about the seasons, not sunrise and sunset. Knowing when to plant and when to harvest was a critical step in humanity's neolithic advancement. This knowledge was quite lucrative, and it most likely precedes writing. Understanding of how the constellations change over the course of a year and of the cyclical nature of the phases the Moon was much more important in this regard than was understanding when sunrise and sunset occur.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Knowledge of time of day only became lucrative after one of the worst inventions ever developed by mankind, those awful things we  now call &quot;meetings.&quot; Even then, the notion of accurate time of day is a fairly recent invention, circa the 16th or 17th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two distinct notions of time of day, one based on the position of the Sun and the other based on the position of the stars. Sundials &quot;ticked&quot; with the Earth's rotation with respect to the Sun. Modern clocks tick with the Earth's rotation with respect to the stars, but with the tick rate adjusted so that, on average, there are 86,400 seconds from noon to noon. (A mean sidereal day is 236 seconds shorter than a mean solar day.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As with many other things in science, our understanding of time has changed over the course of centuries. Galileo, Kepler, and Newton were critical in our classical (non-relativistic) understanding of time. While the &lt;a href=&quot;http://en.wikipedia.org/wiki/Equation_of_time&quot; rel=&quot;nofollow&quot;&gt;equation of time&lt;/a&gt; was known to the ancients, Kepler was one of the first to argue that time as measured by the stars (or by a mechanical clock) was better than time as measured by the Sun (or by a sundial). Galileo performed experiments in which time was the independent variable of motion. Newton cemented these ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Relativity turned these very solid notions upside down. Contrary to what Newton thought, time is not the absolute, universal independent variable of motion.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-05-20T20:29:20.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2348" PostTypeId="2" ParentId="2345" CreationDate="2015-05-21T01:07:28.467" Score="3" Body="&lt;p&gt;In Principia Newton presents a picture based on forces rather than energy and momentum, and he did not have the concepts of  vector and of mechanical energy at his disposal. Moreover, &lt;a href=&quot;https://books.google.com/books?id=SzSRlYNEm18C&amp;amp;pg=PA158&amp;amp;lpg=PA158&amp;amp;dq=newton+leibniz+%22true+measure+of+motion%22&amp;amp;source=bl&amp;amp;ots=ofiwqdgSJx&amp;amp;sig=5FOMu-W2LV8dnwQhwK3aGOaUAkc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=2SV9VJCFBI2XyQS_gYKQBQ#v=onepage&amp;amp;q=newton%20leibniz%20%22true%20measure%20of%20motion%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton opposed Leibniz's idea of putting kinetic energy (&quot;vis viva&quot;) at the center of dynamics&lt;/a&gt; on philosophical grounds, because he considered forces to be more fundamental. There was no notion of potential energy at the time, and some of the forces treated in Principia, such as air resistance, are not potential, so when they operate even total mechanical energy is not conserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conservation of kinetic energy and momentum was mostly considered at the time in the context of elastic collisions, and &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;there was &quot;vis viva controversy&quot;&lt;/a&gt; as to which of them was the &quot;true&quot; quantity of motion. Neither could be considered a universal law at the time, because the nature of heat was not understood, and frame dependent components of a vector were not an item. In fact, &lt;a href=&quot;http://www.mathpages.com/home/kmath106/kmath106.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton worked with two different notions of momenta&lt;/a&gt;, mass times speed, going back to Descartes, and the modern one, but only in terms of components. He explicitly notes that Cartesian &quot;motion may be got or lost&quot;, and &lt;a href=&quot;https://archive.org/stream/100878576#page/86/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;derives conservation of components&lt;/a&gt; from the third law of motion:&quot;if the bodies meet with contrary motions there will be an equal deduction from the motions of both&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an anticipation of modern view &lt;a href=&quot;http://nature.berkeley.edu/departments/espm/env-hist/articles/1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boscovich in 1745 characterized vis viva&lt;/a&gt; as the measure of force acting through a distance, and momentum as the measure of a force acting through a time, and D'Alembert included this characterization into the second edition of his Traite de Dynamique (1758). But neither the concept of a vector, nor the general notion of energy, required to give conservation laws centrality in physics, emerged until the 19th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for use of calculus, there is evidence that &lt;a href=&quot;http://adsabs.harvard.edu/full/1970JHA.....1..116W&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton originally worked out Principia in terms of Euclidean geometry&lt;/a&gt;, and connected it to calculus only after the first publication. Conversion of Principia into the calculus notation was done by Euler in Mechanica (1736). &lt;a href=&quot;https://archive.org/details/mcaniqueanalyt02lagr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange in Mecanique Analytique (1788) introduced potential energy&lt;/a&gt; (without the name, proposed by Rankine in 1853), and explicitly derived conservation of mechanical energy from Euler's reformulation in essentially the modern way. Modernizing the notation, Newton's second law $m_i\frac{d^2x_i}{dt^2}=-\frac{\partial U}{\partial x_i}$, where $U$ is the potential of forces, is multiplied on both sides by $\frac{dx_i}{dt}$ and summed over the (generalized) coordinates. Using the product rule on the left, the mutivariable chain rule on the right, and moving everything to the left gives: &#xA;$$\frac{d}{dt}\left(\sum_i\frac12m_i\left(\frac{dx_i}{dt}\right)^2+U\right)=0,$$&#xA;i.e. the sum of kinetic and potential energies is conserved.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-06-17T22:59:01.410" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2349" PostTypeId="1" CreationDate="2015-05-21T12:26:30.637" Score="7" ViewCount="129" Body="&lt;p&gt;Galileo's Dialogue Concerning the Two Chief World Systems opens with a seemingly sarcastic &amp;quot;&lt;a href=&quot;http://www.tc.umn.edu/%7Eallch001/galileo/library/Dialog/front.htm&quot; rel=&quot;noreferrer&quot;&gt;Note to the discerning reader&lt;/a&gt;&amp;quot;, beginning:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Several years ago there was published in Rome a salutary edict which, in order to obviate the dangerous tendencies of our present age, imposed a seasonable silence upon the Pythagorean opinion that the earth moves There were those who impudently asserted that this decree had its origin not injudicious inquire, but in passion none too well informed Complaints were to be heard that advisers who were totally unskilled at astronomical observations ought not to clip the wings of reflective intellects by means of rash prohibitions.&lt;/p&gt;&#xA;&lt;p&gt;Upon hearing such carping insolence, my zeal could not be contained. Being thoroughly informed about that prudent determination, I decided to appear openly in the theater of the world as a witness of the sober truth. I was at that time in Rome; I was not only received by the most eminent prelates of that Court, but had their applause; indeed this decree was not published without some previous notice of it having been given to me. Therefore I propose in the present work to show to foreign nations that as much is understood of this matter in Italy, and particularly in Rome, as transalpine diligence can ever have imagined Collecting all the reflections thai properly concern the Copernican system, I shall make it known that everything was brought before the attention of the Roman censorship, and that there proceed from this clime not only dogmas for the welfare of the soul, but ingenious discoveries for the delight of the mind as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I can't quite tell how sarcastic Galileo is being here - in the first paragraph he seems to be facetiously praising the Vatican, but in the second he seems to drops the irony and state his opinions directly. More importantly though, I don't actually have a clear picture what events from &amp;quot;several years ago&amp;quot; he's referring to. Can someone clarify precisely what story this all fits into, and maybe assess the level of irony in Galileo's tone?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-05-21T12:26:30.637" Title="What events does the note at the beginning of Dialogue Concerning Two Systems refer to?" Tags="|biographical-details|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2351" PostTypeId="1" CreationDate="2015-05-21T21:14:14.627" Score="8" ViewCount="250" Body="&lt;p&gt;In 1968 NASA, through the assistance of the Israel Program for Scientific Translation, translated from Russian the 1964 book &quot;&lt;a href=&quot;http://archive.org/stream/nasa_techdoc_19690003053/19690003053_djvu.txt&quot; rel=&quot;nofollow&quot;&gt;The Earth in the Universe&lt;/a&gt;&quot;, with chapters written by top Soviet scientists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a chapter written by &lt;a href=&quot;http://en.wikipedia.org/wiki/Alexander_Chizhevsky&quot; rel=&quot;nofollow&quot;&gt;Prof. A. L. Chizhevskii&lt;/a&gt;, who was honorary president of the first International Congress of Biophysics and Biocosmics held in New York, wrote (quoting pages 295-296):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is known that a minimum amount of electro- &#xA;  magnetic energy may markedly lower the stability of the dispersed phase &#xA;  of certain colloidal fluids. Experiments have shown that radio waves from &#xA;  telegraphic transmitters caused relatively rapid precipitation of the solid &#xA;  phase of certain colloids, yet the field intensity is only of the order &#xA;  of 0.001 to 0.00001 of the intensity of the so-called electrical &#xA;  interference. When placed in sheet-iron chambers the same colloids were &#xA;  not affected by radio transmitters (Wilke and Müller). &lt;strong&gt;It has been known &#xA;  for a long time that milk curdles considerably faster on days with &#xA;  marked atmospheric- electrical disturbances (thunderstorms) than on other &#xA;  days. It was experimentally proved that the curdling is quite independent &#xA;  of bacterial processes. Evidently, under the effect of the factors mentioned above, syneresis and disruption of the protein- colloid system occurs in the milk. Experimental coagulation of milk by treatment with short waves was reported, thus excluding from the process any thermal phenomena (Kerber, Goetinck).&lt;/strong&gt; Similar observations were made on various gels and emulsions in which the suspended phases precipitated during thunder storms (Wedekind and others).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question is, what are the sources that this passage is citing?  Who are Kerber and Goetinck? What publication(s) could this being referring to?&lt;/p&gt;&#xA;" OwnerUserId="2331" LastEditorUserId="2331" LastEditDate="2015-05-28T10:29:15.147" LastActivityDate="2015-06-02T03:26:47.000" Title="History of Coagulation of Milk by Short Waves" Tags="|chemistry|reference-request|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="2352" PostTypeId="5" CreationDate="2015-05-21T23:09:29.187" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-05-21T23:09:29.187" LastActivityDate="2015-05-21T23:09:29.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2353" PostTypeId="4" CreationDate="2015-05-21T23:09:29.187" Score="0" Body="For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems." OwnerUserId="8" LastEditorUserId="8" LastEditDate="2015-05-22T01:11:44.407" LastActivityDate="2015-05-22T01:11:44.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2354" PostTypeId="5" CreationDate="2015-05-21T23:10:28.313" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-05-21T23:10:28.313" LastActivityDate="2015-05-21T23:10:28.313" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2355" PostTypeId="4" CreationDate="2015-05-21T23:10:28.313" Score="0" Body="For question regarding the applications of formal logic to mathematics." OwnerUserId="8" LastEditorUserId="8" LastEditDate="2015-05-22T01:11:26.653" LastActivityDate="2015-05-22T01:11:26.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2356" PostTypeId="5" CreationDate="2015-05-21T23:11:33.810" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-05-21T23:11:33.810" LastActivityDate="2015-05-21T23:11:33.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2357" PostTypeId="4" CreationDate="2015-05-21T23:11:33.810" Score="0" Body="For questions about the mathematical branch that is based on the study of sets, i.e. collections of objects." OwnerUserId="8" LastEditorUserId="62" LastEditDate="2015-05-22T01:11:40.910" LastActivityDate="2015-05-22T01:11:40.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2358" PostTypeId="5" CreationDate="2015-05-22T09:27:47.873" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-05-22T09:27:47.873" LastActivityDate="2015-05-22T09:27:47.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2359" PostTypeId="4" CreationDate="2015-05-22T09:27:47.873" Score="0" Body="For questions about specific occurrences of a distinct character (to be specified in the question) in the history of science or mathematics." OwnerUserId="8" LastEditorUserId="26" LastEditDate="2015-05-22T12:13:50.000" LastActivityDate="2015-05-22T12:13:50.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2360" PostTypeId="1" AcceptedAnswerId="2364" CreationDate="2015-05-22T11:44:39.160" Score="0" ViewCount="517" Body="&lt;p&gt;Numbers were introduced in our lives far before we had the capability to understand their beauties. But as mathematics aspirant I really want to know how they all evolved and in what way number theory took its own path. May be the answer will be too descriptive,  So any link or book will work. Please if anybody know as such link and book refer it  (except Wikipedia). &lt;/p&gt;&#xA;" OwnerUserId="2334" LastActivityDate="2015-05-22T21:00:51.243" Title="What is the history of number theory?" Tags="|mathematics|" AnswerCount="1" CommentCount="7" ClosedDate="2015-06-03T00:10:57.410" ContentLicense="CC BY-SA 3.0" />
  <row Id="2361" PostTypeId="2" ParentId="2345" CreationDate="2015-05-22T20:06:24.643" Score="3" Body="&lt;p&gt;Newton wrote the Principia in 1684-1686. At that time, he would not have expected any reader in Europe (with the possible exception of a few friends) to have known the method of fluxions (calculus), which he had never published. In order to have his work understood, he needed to write it without using either the techniques or the notation of fluxions. (Leibniz's first publication of the techniques of calculus was not until 1684, so Newton also would not have anticipated that many of his readers could be familiar with the ideas through Leibniz, and in any case readers who knew Leibniz's notation would not have understood Newton's.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton clearly understood what we would today call vectors, as you can clearly tell from the &lt;a href=&quot;https://archive.org/stream/100878576#page/84/mode/2up&quot; rel=&quot;nofollow&quot;&gt;Principia&lt;/a&gt;. On pp. 84-86, he presents what we would today call the parallelogram method of vector addition (Cor. I), and states that the momentum vector is conserved (Cor. III). The fact that he doesn't use the term &quot;vector&quot; (which was invented much later) is irrelevant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He doesn't discuss conservation of energy because conservation of energy wasn't discovered until the 19th century. In the 17th century, there was no notion of the mechanical equivalent of heat, and thermal energy was not conceived of as the kinetic energy of the random motion of molecules. In this period, the phlogiston theory was brand-new.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2015-05-22T20:47:01.910" LastActivityDate="2015-05-22T20:47:01.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2362" PostTypeId="1" AcceptedAnswerId="2366" CreationDate="2015-05-22T20:33:57.530" Score="8" ViewCount="3756" Body="&lt;p&gt;I'm not suggesting that Newton did not discover calculus - the question is written this way to express my surprise that the &lt;em&gt;Principia&lt;/em&gt; does not use the methods of calculus (or 'fluxions'). He instead uses plane and conic geometry; and of course the methods of differential calculus are implicit in it in the way that Euclid's demonstration of the area of a circle contains a limiting argument that expresses the notion of integration.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, even if he chose not to use calculus so as to get the widest possible audience it seems strange not to introduce it in an addendum or appendix to show that the same results can be deduced with greater conceptual clarity, and shorter proofs.&lt;/p&gt;&#xA;&lt;p&gt;Are there lesser known published or unpublished works in which the novel techniques were advertised?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I mean for example  the notation that is introduced in classical mechanics of &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; for the derivative of function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, or with a dot above it &lt;span class=&quot;math-container&quot;&gt;$\dot{f}$&lt;/span&gt;; which I recall reading was ascribed to Newton; whereas the notation &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dt}$&lt;/span&gt; is usually ascribed to Leibniz.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2023-12-27T20:44:22.373" Title="Why is calculus missing from Newton's Principia?" Tags="|notation|calculus|classical-mechanics|euclidean-geometry|newton|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2363" PostTypeId="2" ParentId="2362" CreationDate="2015-05-22T20:54:14.707" Score="2" Body="&lt;p&gt;In your question you implicitly assume that Newton wrote only Principia.&#xA;Which is strange. Look at the Mathematical Works of Newton:&#xA;&lt;a href=&quot;http://www.newtonproject.sussex.ac.uk/prism.php?id=147&quot; rel=&quot;nofollow&quot;&gt;http://www.newtonproject.sussex.ac.uk/prism.php?id=147&lt;/a&gt;&#xA;which contain abundant evidence of his mathematical discoveries.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-05-22T21:36:41.537" LastActivityDate="2015-05-22T21:36:41.537" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2364" PostTypeId="2" ParentId="2360" CreationDate="2015-05-22T21:00:51.243" Score="1" Body="&lt;p&gt;There is a very comprehensive (several volumes) History of number theory by Dickson, &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick&quot; rel=&quot;nofollow&quot;&gt;https://archive.org/details/historyoftheoryo01dick&lt;/a&gt;&#xA;Here is a much shorter and more modern one:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weil, André&#xA;Number theory. An approach through history from Hammurapi to Legendre. Reprint of the 1984 edition.&#xA;Basel: Birkhäuser (ISBN 978-0-8176-4565-6/pbk; 978-0-8176-4571-7/ebook). xxii, 376 p. EUR 34.90/net; SFR 59.90 (2007).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-05-22T21:00:51.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2365" PostTypeId="2" ParentId="2343" CreationDate="2015-05-22T21:15:17.303" Score="3" Body="&lt;p&gt;Yes, the scientific method is a Western invention. It was formulated in the end of the 16th century. In the English-speaking world, its invention is associated with &lt;a href=&quot;https://en.wikipedia.org/wiki/Francis_Bacon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francis Bacon&lt;/a&gt;, but as it always happens, many other people contributed, all of whom lived in Europe in the 16th and 17th centuries.&lt;/p&gt;&#xA;&lt;p&gt;The ancients (Babylonians, Hellenistic Greeks) did not have a fully developed scientific method in the modern sense of the word. Their achievements in mathematics and astronomy can be considered as a beginning of science (if you include mathematics and astronomy into science, which not everyone does), but none of them formulated or used the scientific method as we understand it now, though Archimedes was very close.&lt;/p&gt;&#xA;&lt;p&gt;And yes, most of the science until the end of the 19th century was done in Europe. More precisely, in Italy, England, France and Germany.&lt;/p&gt;&#xA;&lt;p&gt;Some other countries also had important achievements, this cannot be denied, but the body of exact sciences which we study now in schools and universities was created in Western Europe. This applies to the universities in China, India, Russia and elsewhere. Their ancient science is of interest only to historians, not to the students of science.&lt;/p&gt;&#xA;&lt;p&gt;Let me give just one example to clarify what I am talking about. There are two very well documented first-hand accounts of Tamil astronomy (one written by a French in the 18th century, another by  an English traveler in the 19th century). They describe in great detail the methods used by Tamil astronomers to predict eclipses. The accuracy of their predictions was comparable to the European methods of the 18th century. These astronomers were illiterate (by European standards) and they used some development of the ancient Babylonian system (it passed to India after Alexander's conquest). Nowadays this indigenous Tamil astronomy does not exist (is not practiced) but we know it very well because of the records of European travelers. And historians of astronomy study it. But no one teaches or uses Tamil astronomy &lt;strong&gt;as astronomy.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;And let me add that what I wrote concerns only science, not inventions. We do use some ancient Chinese (and other nations') inventions in our daily life, e.g., paper, the compass, gunpowder, fireworks, kites and several others.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-11T00:42:59.177" LastActivityDate="2021-09-11T00:42:59.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2366" PostTypeId="2" ParentId="2362" CreationDate="2015-05-23T00:15:41.010" Score="10" Body="&lt;p&gt;There are two separate issues here. The method of fluxions and fluents, Newton's version of calculus, is amply represented in Newton's extant papers, starting with &lt;a href=&quot;http://www.britannica.com/EBchecked/topic/1084435/De-Analysi-per-Aequationes-Numero-Terminorum-Infinitas&quot; rel=&quot;nofollow noreferrer&quot;&gt;1669 On Analysis by Equations with an Infinite Number of Terms&lt;/a&gt; sent as a letter to John Collins, and disseminated by him to multiple correspondents, including Leibniz. The dotted shorthand was only invented in 1691, after the publication of Principia (1687), and Newton did publish his account of calculus in 1693. Before that his methods were mostly known in Europe from letters sent through Collins and Oldenburg.&lt;/p&gt;&#xA;&lt;p&gt;As to why he did not use calculus in Principia the answer is controversial. To begin with, it is not clear that calculus in its original cumbersome form would have introduced greater clarity or shorter proofs. It could instead compound the difficulty of understanding new mechanics by the difficulty of understanding new mathematics. There is also evidence that Newton worked out Principia in the very form that they were written, Euclidean, rather than back translated it from a calculus version as he later claimed.  &lt;a href=&quot;http://adsabs.harvard.edu/full/1970JHA.....1..116W&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whiteside&lt;/a&gt; gives an illuminating discussion.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="643" LastEditDate="2023-12-27T20:44:22.373" LastActivityDate="2023-12-27T20:44:22.373" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2367" PostTypeId="1" CreationDate="2015-05-23T02:00:34.100" Score="4" ViewCount="2463" Body="&lt;p&gt;Electricity history starting from William Gilbert in the 17th century.&lt;/p&gt;&#xA;" OwnerUserId="2335" LastActivityDate="2015-05-23T06:39:03.777" Title="Is there any good book about the history of electricity?" Tags="|electricity|resource-recommendation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2368" PostTypeId="2" ParentId="2367" CreationDate="2015-05-23T06:39:03.777" Score="4" Body="&lt;p&gt;Yes, there is:&#xA;&lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich&quot; rel=&quot;nofollow&quot;&gt;https://archive.org/details/historyoftheorie00whitrich&lt;/a&gt;&#xA;A history of the theories of aether and electricity : from the age of Descartes to the close of the nineteenth century&#xA;by Whittaker, E. T.&#xA;It also contains history of optics, quantum mechanics and relativity.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-05-23T06:39:03.777" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2369" PostTypeId="1" CreationDate="2015-05-23T09:18:02.480" Score="14" ViewCount="1186" Body="&lt;p&gt;How did Poincare discover the &lt;a href=&quot;http://en.wikipedia.org/wiki/Fundamental_group&quot;&gt;fundamental group&lt;/a&gt;? What was the first instance that led Poincaré to discover this amazing theory?&lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="26" LastEditDate="2015-05-23T10:13:04.467" LastActivityDate="2020-03-09T18:02:35.873" Title="How did Poincaré discover the fundamental group?" Tags="|mathematics|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2370" PostTypeId="2" ParentId="2369" CreationDate="2015-05-23T10:27:06.310" Score="19" Body="&lt;p&gt;The concept of fundamental group was introduced in Poincaré's seminal paper &lt;em&gt;Analysis situs&lt;/em&gt;, so that is a natural place to look for some motivation for this concept. In &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf&quot; rel=&quot;noreferrer&quot;&gt;Stillwell's nice translation of Poincaré's Analysis situs and related papers&lt;/a&gt; he explains the motivation in one of the introductory paragraphs, written by himself: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Poincaré set the agenda for his 1895 Analysis situs paper with a short announcement [...] In it he raises the question whether the Betti numbers suffice to determine the topological type of a manifold, and introduces the fundamental group to further illuminate this question. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Fortunately, Stillwell also included a translation of this anouncement of &lt;em&gt;Analysis situs&lt;/em&gt;. In it, after Poincaré introduced the concept of Betti numbers, we find the following paragraphs (with footnotes from Stillwell): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One may ask whether the Betti numbers suffice to determine a closed surface from the viewpoint of Analysis situs. That is, given two surfaces with the same Betti numbers, we ask whether it is possible to pass from one to the other by a continuous deformation. This is true in the space of three dimensions, and we may be inclined to believe that it is again true in any space. The contrary is true.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In order to explain, I want to approach the question from a new viewpoint.&#xA;  Let $$x_1, x_2,\dots,x_{n+1}$$ be the coordinates of a point on the surface. These $n+1$ quantities are connected by the equation of the surface. Now let&#xA;  $$F_1, F_2,\dots,F_p$$ be any $p$ functions of the $n + 1$ coordinates $x$ (which I always suppose to be connected by the equation of the surface, and which I suppose to take only real values).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I do not assume that the functions $F$ are uniform, but I suppose that if the&#xA;  point $(x_1 , x_2 , \dots , x_{n+1})$ describes an inﬁnitely small contour on the surface then each of the functions $F$ returns to its initial value. This being so, we suppose that our point now describes a ﬁnite closed contour on the surface. It may then happen that the $p$ functions do not return to their initial values, but instead become $$F_1',F_2',\dots, F_p'$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In other words, they undergo the substitution&#xA;  $$(F_1,F_2,\dots,F_p;F_1',F_2',\dots, F_;').$$ &#xA;  All the substitutions corresponding to the different closed contours that we&#xA;  can trace on the surface form a group which is discontinuous (at least as far as its form is concerned). &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This group evidently depends on the choice of functions $F$. We suppose ﬁrst&#xA;  that these functions are the most one can imagine, other than being subject to the condition imposed above, and let $G$ be the corresponding group. If $G'$ is the group corresponding to another choice of functions, then $G'$ will be isomorphic to $G$—holoedrically in general but meriedrically in special cases$^1$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The group $G$ can then serve to define the form of the surface and it is called the group of the surface$^2$. It is clear that if two surfaces can each be transformed to the other by a continuous transformation, then their groups are isomorphic. The converse, though less evident, is again true for closed surfaces, so that what defines a closed surface, from the viewpoint of Analysis situs, &lt;em&gt;is its group&lt;/em&gt;$^3$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$^1$ The terminology &quot;holoedric isomorphism&quot; and &quot;meriedric isomorphism&quot; correspond to the modern concepts of isomorphism and homomorphism, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$^2$ This would later be called the &lt;em&gt;fundamental group&lt;/em&gt; in §12 of Analysis situs itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$^3$ This is true for surfaces in the traditional, two-dimensional, sense, but not for manifolds of three dimensions. Near the end of §14 of Analysis situs, Poincaré raised the question whether two manifolds with the same group are necessarily homeomorphic.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In conclusion, it appears that the concept appeared quite naturally in Poincaré's attempts to classify surfaces/manifolds up to homeomorphism.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2015-12-13T16:18:22.917" LastActivityDate="2015-12-13T16:18:22.917" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2371" PostTypeId="1" CreationDate="2015-05-23T18:32:38.327" Score="5" ViewCount="84" Body="&lt;p&gt;If you look at the original pair of 1948 papers[Bondi 1948,Hoyle 1948] that proposed the steady-state model, it's clear that they knew they were throwing out a lot of cherished principles of physics. The Bondi paper in particular reads like a catalog of ideas to be given up: Lorentz invariance, the equivalence principle, possibly conservation of charge and mass-energy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although they clearly recognized that they were breaking Lorentz symmetry, there is no discussion of discrete symmetries. It seems that they were also violating time-reversal (T) and charge conjugation (C), and that they were violating CPT. The steady-state model persisted well into the 1960s, by which time I think the study of these discrete symmetries and their violation were more clearly understood by physicists to be at the core of their discipline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did anyone in either camp point out at some point that these breakings of discrete symmetries were present and possibly an important feature of the model?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose that the CPT theorem is irrelevant, since the CPT theorem limits what you can do if you want to maintain Lorentz invariance, but they'd already broken Lorentz invariance. Hoyle and Narlikar didn't actually attempt to add dynamics to the theory by making it a quantum field theory (which, after all, would require Lorentz invariance). They instead tried to do it as a direct field theory.[Hoyle 1964]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bondi and Gold, &quot;The Steady-State Theory of the Expanding Universe,&quot; MNRAS 108 (1948) 252 &lt;a href=&quot;http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1948MNRAS.108..252B&quot; rel=&quot;noreferrer&quot;&gt;http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1948MNRAS.108..252B&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hoyle, &quot;A New Model for the Expanding Universe,&quot; MNRAS 108 (1948) 372. &lt;a href=&quot;http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1948MNRAS.108..372H&quot; rel=&quot;noreferrer&quot;&gt;http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1948MNRAS.108..372H&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hoyle and Narlikar, &quot;The C-field as a direct particle field,&quot; Proc Roy Soc London 282 (1964) 178&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2015-05-23T19:10:38.617" LastActivityDate="2017-01-04T00:50:55.410" Title="Did breaking of discrete symmetries play a role in the development of steady-state cosmology?" Tags="|particle-physics|cosmology|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2372" PostTypeId="1" AcceptedAnswerId="2374" CreationDate="2015-05-25T01:06:40.680" Score="12" ViewCount="867" Body="&lt;p&gt;I read in a not always reliable source (David Foster Wallace's &lt;em&gt;Everything and More&lt;/em&gt;, p.104), that Leibniz introduced the terms &lt;em&gt;constant&lt;/em&gt;, &lt;em&gt;variable&lt;/em&gt;, and &lt;em&gt;function&lt;/em&gt;, the latter as an alternative to Newton's &lt;em&gt;fluent&lt;/em&gt;. (Perhaps Leibniz used the German words Konstante, Veränderliche, Funktion?)&#xA;Can anyone verify or undermine this attribution of fundamental terms of our&#xA;current mathematical language?&lt;/p&gt;&#xA;" OwnerUserId="606" LastEditorUserId="658" LastEditDate="2015-05-26T16:26:37.843" LastActivityDate="2015-05-26T16:26:37.843" Title="Is it true that Leibniz introduced &quot;constant,&quot; &quot;variable,&quot; and &quot;function&quot;?" Tags="|mathematics|terminology|calculus|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2373" PostTypeId="2" ParentId="2372" CreationDate="2015-05-25T10:05:32.830" Score="6" Body="&lt;p&gt;You can see into :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gottfried Wilhelm Leibniz, &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;printsec=frontcover&quot;&gt;The Early Mathematical Manuscripts&lt;/a&gt; (Dover ed), page 22-on the English translation of the :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://ia802704.us.archive.org/17/items/historiaetorigo00gerhgoog/historiaetorigo00gerhgoog.pdf&quot;&gt;Historia et Origo Calculi Differentialis&lt;/a&gt; (1714-1716).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original text is in Latin and you can find there occurrences of :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;considerando differentias $dx$ [...] funciones quasdam ipsarum&lt;/em&gt; $x$&quot; [&quot;considering the differences $dx$ [...] as functions of the $x$'s&quot;] (page 2)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;ita jam quantitates [consideratae] ut variantes, habere novas functiones nempe differentias.&quot;&lt;/em&gt; [&quot;also such quantities [considered] as were variable had new functions, namely, differences.&quot;] (page 17).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Newton instead, into his &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Newton/Quadratura/IntQuad.pdf&quot;&gt;Introductio ad Quadraturam Curvarum&lt;/a&gt; (published 1704), writes : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Quantitates Mathematicas non ut ex partibus quam minimis constantes, sed ut motu continuo descriptas hic considero.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He consider &lt;em&gt;curves&lt;/em&gt; as described by continuous motion, and define his :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Method of determining the Quantities from the Velocities of their Motions or Increments, by which they are generated; and by calling the Velocities of the Motions, or of the Augments, by the Name of &lt;em&gt;Fluxions&lt;/em&gt;, and the generated&#xA;  Quantities &lt;em&gt;Fluents&lt;/em&gt; [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;But I think that the main difference between the two approach was not regarding &quot;terminology&quot;, but &quot;symbolism&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;algebraic&quot; symbols invented by Leibniz was &lt;strong&gt;one of&lt;/strong&gt; the strong point supporting the success of his new calculus.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-05-25T10:05:32.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2374" PostTypeId="2" ParentId="2372" CreationDate="2015-05-25T10:56:31.200" Score="5" Body="&lt;p&gt;Wallace may have read Kline's &lt;em&gt;Mathematical thought from ancient to modern times&lt;/em&gt;, which on p. 340 makes this attribution to Leibniz in &lt;a href=&quot;https://books.google.com/books?id=bK1RAAAAcAAJ&amp;amp;pg=PA168&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu&lt;/em&gt;&lt;/a&gt;, Acta Eruditorum (1692) 168-171. This short text, of which a French translation is available &lt;a href=&quot;https://books.google.com/books?id=lfEy-OzaWkQC&amp;amp;pg=PA210&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, has indeed &lt;em&gt;functiones&lt;/em&gt;, &lt;em&gt;constans&lt;/em&gt;, &lt;em&gt;variabilis&lt;/em&gt; (and more) italicized by Leibniz himself.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2015-05-26T16:13:41.627" LastActivityDate="2015-05-26T16:13:41.627" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2375" PostTypeId="2" ParentId="56" CreationDate="2015-05-25T11:34:07.223" Score="6" Body="&lt;p&gt;An early example is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=I-5hAAAAcAAJ&amp;amp;pg=PR12&amp;amp;dq=easy+square+roots+children&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=XgFjVcX8PMLKsAWo1ICQDA&amp;amp;ved=0CDgQ6AEwBTgK#v=onepage&amp;amp;q=%20roots&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Arithmetick Made fo Easy, That it may be Learned without a Master: After a new and concife method; the Like not yet Extant&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked second edition was published in 1740.  The first edition was 1727. (1725 for the original French: L'arithmétique rendue facile de façon à la pouvoir apprendre sans Maître)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The translator's preface explains it is for teaching children.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculating square roots digit by digit is pages 347-350.&lt;/p&gt;&#xA;" OwnerUserId="2331" LastEditorUserId="2331" LastEditDate="2015-05-25T14:35:34.863" LastActivityDate="2015-05-25T14:35:34.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2377" PostTypeId="2" ParentId="2067" CreationDate="2015-05-25T23:11:48.557" Score="8" Body="&lt;p&gt;The original proof by Binet was published in 1843 in Paris.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Binet, J. P.: &lt;em&gt;Mémoire sur l'intégration des équations linéaires aux&#xA;  différences finies, d'un ordre quelconque à coefficients variables&lt;/em&gt;,&#xA;  Comptes Rendus des Séances de l'Académie des Sciences 17 (1843),&#xA;  559-567.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You could find the online version of the paper &lt;a href=&quot;http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&amp;amp;O=NUMM-2976&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;https://books.google.com/books?id=DGpEAQAAIAAJ&amp;amp;lpg=PA1375&amp;amp;ots=MKh3d4ADZK&amp;amp;dq=M%C3%A9moire%20sur%20l&amp;#39;int%C3%A9gration%20des%20%C3%A9quations%20lin%C3%A9aires%20aux%20diff%C3%A9rences%20finies%2C%20d&amp;#39;un%20ordre%20quelconque%20%C3%A0%20coefficients%20variables&amp;amp;pg=PA559#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, and a list of publications by Binet &lt;a href=&quot;http://www-gap.dcs.st-and.ac.uk/~history/Extras/Binet_Papers.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cs.cas.cz/portal/AlgoMath/NumberTheory/Sequences/RecurrentSequences/LinearRecurrentSequences/BinaryRecurrentSequences/LucasSequences/FibonacciNumbers/BinetFormula.htm&quot; rel=&quot;noreferrer&quot;&gt;cs.cas.cz&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Like many results in mathematics it is often not the original&#xA;  discoverer who gets the glory of having their name attached to the&#xA;  result. J. P. M. Binet (1786-1856) published [Bin] this result now known&#xA;  as the Binet’s formula in 1843 although the result was known earlier.&#xA;  It seems that Daniel Bernoulli (1700-1782) discovered and proved this&#xA;  formula in 1726 ([Ber], §7). Two years later also Euler mentioned the&#xA;  formula in a letter to Bernoulli, but he published [Eul] it only in&#xA;  1765. A. de Moivre (1667-1754) published it in the general form in the first systematic account of linear recurrences ([Mo1], pp. 26-42) and it&#xA;  seems that Binet’s formula was known to him already in 1718 [Mo2]. From&#xA;  the later authors let us mention for instance E.Lucas who may&#xA;  rediscovered it but did not report it before 1876.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Bin] Binet, J. P. (1843). Mémoire sur l’intégration des équations&#xA;  linéaires aux différences finies, d’un ordre quelconque, à&#xA;  coefficients variables. Comptes Rendus hebdomadaires des séances de&#xA;  l’Académie des Sciences (Paris), 17, 559-567.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Ber] Bernoulli, D. (1728). &lt;a href=&quot;https://archive.org/stream/commentariiacade03impe#page/90&quot; rel=&quot;noreferrer&quot;&gt;Comment. Acad. Sci. Petrop., 3, 85-100.&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Eul] Euler, L. (1765). Observationes analyticae. Novi commentarii&#xA;  ascaemiae scientiarum imperialis Petropolotanae, 11, 124-143.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Mo1] de Moivre, A. (1730). &lt;a href=&quot;https://archive.org/stream/bub_gb_TFX1165yEc4C#page/n104&quot; rel=&quot;noreferrer&quot;&gt;Miscellanea analytica de seriebus et&#xA;  quadraturis . London.&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Mo2] de Moivre, A. (1722). &lt;a href=&quot;http://dx.doi.org/10.1098/rstl.1722.0029&quot; rel=&quot;noreferrer&quot;&gt;Philos. Trans., 32, 162-178.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html#finder&quot; rel=&quot;noreferrer&quot;&gt;maths.surrey.ac.uk&lt;/a&gt; says:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Many authors say that this formula was discovered by J. P. M. Binet&#xA;  (1786-1856) in 1843 and so call it Binet's Formula. Graham, Knuth and&#xA;  Patashnik in Concrete Mathematics (2nd edition, 1994) mention that&#xA;  Euler had already published this formula in 1765. But Don Knuth in The&#xA;  Art of Computer Programming, Volume 1 Fundamental Algorithms, section&#xA;  1.2.8, traces it back even further, to A de Moivre (1667-1754). He had written about &quot;Binet's&quot; formula in 1730 and had indeed found a method&#xA;  for finding formulae for any general series of numbers formed in a&#xA;  similar way to the Fibonacci series. Like many results in Mathematics,&#xA;  it is often not the original discoverer who gets the glory of having&#xA;  their name attached to the result, but someone later!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Concrete Mathematics (2nd edition, 1994) by Graham, Knuth and&#xA;  Patashnik, Addison-Wesley.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The Art of Computer Programming D E Knuth,&#xA;  Volume 1: Fundamental Algorithms (now in its Third Edition, 1997).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html&quot; rel=&quot;noreferrer&quot;&gt;mathworld.wolfram.com&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was derived by Binet in 1843, although the result was known to&#xA;  Euler, Daniel Bernoulli, and de Moivre more than a century earlier.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Livio, M. The Golden Ratio: The Story of Phi, the World's Most&#xA;  Astonishing Number. New York: Broadway Books, p. 108, 2002.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p.&#xA;  21, 2000.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Wells, D. The Penguin Dictionary of Curious and Interesting Numbers.&#xA;  Middlesex, England: Penguin Books, p. 62, 1986.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="100" LastEditorUserId="658" LastEditDate="2017-07-18T13:59:40.870" LastActivityDate="2017-07-18T13:59:40.870" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2378" PostTypeId="1" AcceptedAnswerId="2382" CreationDate="2015-05-26T02:51:30.127" Score="9" ViewCount="1076" Body="&lt;p&gt;I want to see an exact image of the first use of the Cartesian plane. I guess it came the first time with Descartes.&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="55" LastEditDate="2019-04-20T02:50:44.647" LastActivityDate="2023-01-25T13:51:06.783" Title="When do we see for the first time the use of the Cartesian coordinates?" Tags="|mathematics|euclidean-geometry|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="2379" PostTypeId="2" ParentId="2041" CreationDate="2015-05-26T11:06:21.643" Score="1" Body="&lt;p&gt;Some books about Emmy Noether:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1468405373&quot; rel=&quot;nofollow&quot;&gt;A. Dick: &lt;em&gt;Emmy Noether 1882-1935&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1568814305&quot; rel=&quot;nofollow&quot;&gt;M. B. W. Tent: &lt;em&gt;Emmy Noether, The Mother of Modern Algebra&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0801896940&quot; rel=&quot;nofollow&quot;&gt;D. E. Neuenschwander: &lt;em&gt;Emmy Noether's Wonderful Theorem&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/038787867X&quot; rel=&quot;nofollow&quot;&gt;Y. Kosmann-Schwarzbach: &lt;em&gt;The Noether Theorems, Invariance and Conservation Laws in the Twentieth Century&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The first two is more biographical, the last two contain formal details too.&lt;/p&gt;&#xA;" OwnerUserId="100" LastActivityDate="2015-05-26T11:06:21.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2380" PostTypeId="1" AcceptedAnswerId="2381" CreationDate="2015-05-26T16:09:53.447" Score="6" ViewCount="1656" Body="&lt;p&gt;I've just finished reading &lt;em&gt;Quantum: Einstein, Bohr, and the Great Debate About the Nature of Reality&lt;/em&gt; by &lt;em&gt;Manjit Kumar&lt;/em&gt;, which deals with the &lt;strong&gt;history&lt;/strong&gt; of QM from Plank's quantisation for black body problem to Bell's inequality focusing especially on the debate between Bohr and Einstein as indicated in the title. Studying QM with mathematical background the book provides a perfect way to follow the advancements of QM step by step, though a layman could also understand the book since he explains very briefly the physical concepts aside. I'm looking for such books, with which I can study the history of physics and not the physics of some subject, which I already study in a more rigorous way.  &lt;/p&gt;&#xA;" OwnerUserId="660" LastEditorUserId="26" LastEditDate="2015-10-11T10:54:04.390" LastActivityDate="2015-10-13T23:06:16.780" Title="Books on the History of Physics" Tags="|physics|resource-recommendation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2381" PostTypeId="2" ParentId="2380" CreationDate="2015-05-26T18:17:48.720" Score="3" Body="&lt;p&gt;If you're interested in reading more about the history of quantum mechanics, one of the most comprehensive works is &lt;a href=&quot;http://www.springer.com/book/9780387952628&quot; rel=&quot;nofollow&quot;&gt;Mehra's six-volume epic&lt;/a&gt;. Other than that, I can personally recommend reading &lt;em&gt;scientific&lt;/em&gt; (rather than &quot;informal&quot;) biographies of some of the individually involved scientists. I thoroughly enjoyed Pais' accounts of both &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198520484&quot; rel=&quot;nofollow&quot;&gt;Bohr's&lt;/a&gt; and &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0192806726&quot; rel=&quot;nofollow&quot;&gt;Einstein's&lt;/a&gt; (which naturally also discusses relativity at length) life, as well as Moore's book on &lt;a href=&quot;https://books.google.de/books/about/A_Life_of_Erwin_Schr%C3%B6dinger.html?id=n6XZlrJxgFkC&amp;amp;redir_esc=y&quot; rel=&quot;nofollow&quot;&gt;Schrödinger&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the slightly more modern topic of quantum field theory and QED more specifically, I've found Mehra's biography of &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198539487&quot; rel=&quot;nofollow&quot;&gt;Feynman&lt;/a&gt; quite good, though stylistically inferior to the above-mentioned books. I'm also currently reading his book on &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198506589&quot; rel=&quot;nofollow&quot;&gt;Schwinger&lt;/a&gt;, and found it to be quite good so far.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The great Dirac falls somewhere in between these two (as he mastered both!), and the recent book by Farmelo &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0465022103&quot; rel=&quot;nofollow&quot;&gt;on his life&lt;/a&gt; was decent, though a bit short and less in-depth than the other books I mentioned. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these go (as most scientific biographies) into considerable detail, both historically and scientifically, and I think it's exactly this type of books that you need to look out for.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2015-05-26T18:17:48.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2382" PostTypeId="2" ParentId="2378" CreationDate="2015-05-26T22:30:27.780" Score="17" Body="&lt;p&gt;There are several ways to answer this depending on what &amp;quot;Cartesian plane&amp;quot; means. Most literally, Cartesius is the Latinization of Descartes name, so one can look at &lt;a href=&quot;http://2.bp.blogspot.com/-xEExWlptL1k/TwvqJ80sl-I/AAAAAAAAAio/sQ52gE44L08/s1600/La-Geometrie-Extract.png&quot; rel=&quot;noreferrer&quot;&gt;pictures in Descartes's La Geometrie&lt;/a&gt;, which was the first systematic use of coordinates to solve geometric problems. However, coordinate graphs were introduced before Descartes by Oresme, to plot speed against time for example, &lt;a href=&quot;https://web.archive.org/web/20210914052837/http://www.jpowered.com/graphs-and-charts/images/nicole-oresme-bar-charts-420x615.png&quot; rel=&quot;noreferrer&quot;&gt;Oresme's were bar graphs&lt;/a&gt; however. &lt;a href=&quot;http://www.dam.brown.edu/people/mumford/beyond/coursenotes/2006PartIb.pdf&quot; rel=&quot;noreferrer&quot;&gt;Oresme calls&lt;/a&gt; the axes &lt;em&gt;longitude&lt;/em&gt; and &lt;em&gt;latitude&lt;/em&gt;, the independent variable &lt;em&gt;extension&lt;/em&gt;, and the dependent one &lt;em&gt;intension&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Descartes got the idea of coordinates from Apollonius of Perga, &lt;em&gt;abscissas&lt;/em&gt; and &lt;em&gt;ordinates&lt;/em&gt; are Latin translations of Apollonius's terms. In Conica Apollonius uses tangent to a conic section and its diameter as what we now call coordinate axes, in general they are oblique, draws segments parallel to them, abscissas and ordinates, and relates their lengths by what we now call coordinate equations, he calls them symptoms. He even &amp;quot;changes coordinates&amp;quot; by passing to a new diameter, see &lt;a href=&quot;http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1226&amp;amp;context=jhm&quot; rel=&quot;noreferrer&quot;&gt;Pierce's Abscissas and Ordinates&lt;/a&gt;. A suggestive figure is on p.176 of &lt;a href=&quot;https://web.archive.org/web/20151017073910/https://www.stmarys-ca.edu/sites/default/files/attachments/files/Conic_Sections.pdf&quot; rel=&quot;noreferrer&quot;&gt;Heath's translation&lt;/a&gt;, here is another, &lt;a href=&quot;http://www.mlahanas.de/Greeks/images/ApolloniusBook.jpg&quot; rel=&quot;noreferrer&quot;&gt;from an Arabic copy of book V&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, this was specific to conic sections, Apollonius has no symptoms of this kind for general curves, and his &amp;quot;coordinates&amp;quot; are curve specific, he does not consider even different conics in the same coordinates. So I am not sure if this qualifies as Cartesian. On the other hand, geometrically &lt;em&gt;Cartesian plane&lt;/em&gt; is the same as &lt;em&gt;Euclidean plane&lt;/em&gt;, only with numerical labels attached, so perhaps Euclid's diagrams already qualify.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-01-25T13:51:06.783" LastActivityDate="2023-01-25T13:51:06.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2383" PostTypeId="1" AcceptedAnswerId="2384" CreationDate="2015-05-27T00:17:27.293" Score="4" ViewCount="238" Body="&lt;p&gt;I was wondering, what book did people use in the 17 th century to learn trigonometry as it was before ? Just general curiosity...&lt;/p&gt;&#xA;" OwnerUserId="2310" LastActivityDate="2015-05-27T06:38:58.150" Title="What are typical trigonometry textbooks in the 17 th century?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2384" PostTypeId="2" ParentId="2383" CreationDate="2015-05-27T01:54:14.780" Score="4" Body="&lt;p&gt;As it turns out, &lt;a href=&quot;http://en.wikipedia.org/wiki/Bartholomaeus_Pitiscus&quot; rel=&quot;nofollow&quot;&gt;Bartholomaeus Pitiscus&lt;/a&gt; is known as the person who introduced the word trigonometry into the English language in the sense that we understand the word today. In 1595 he produced &lt;em&gt;Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus&lt;/em&gt;, and this seminal work was, throughout the 17th century, translated into English several times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A copy of the Latin original can be found at &lt;a href=&quot;http://www.e-rara.ch/doi/10.3931/e-rara-4035&quot; rel=&quot;nofollow&quot;&gt;e-rara&lt;/a&gt;, and here is another copy at &lt;a href=&quot;http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10164580_00005.html&quot; rel=&quot;nofollow&quot;&gt;bsb&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this book was translated to English a few times throughout the 17th century, but I can't manage to find a good download link at the moment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pitiscus' book was certainly not the only trigonometry book used in the 17th century, but it stands out as being a very important historical example, and one which I think deserves mention.&lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2015-05-27T06:38:58.150" LastActivityDate="2015-05-27T06:38:58.150" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2385" PostTypeId="1" CreationDate="2015-05-27T17:28:21.400" Score="3" ViewCount="442" Body="&lt;p&gt;Why is it that JPL (Jet Propulsion Lab) at NASA gives astronomical unit values for the planets for two different time periods, namely 3000 B.C. to 3000 A.D., and 1850 to 2050?&lt;/p&gt;&#xA;" OwnerUserId="1763" LastEditorUserId="1815" LastEditDate="2017-10-23T14:20:56.087" LastActivityDate="2017-10-23T14:20:56.087" Title="Why does JPL at NASA give two different values in astronomical units for planetary distances?" Tags="|astronomy|units|celestial-mechanics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2386" PostTypeId="2" ParentId="2351" CreationDate="2015-05-28T00:53:46.790" Score="2" Body="&lt;p&gt;Here is a direct link to &lt;a href=&quot;http://gluedideas.com/The-Earth-in-the-Universe-1968/One-Aspect-of-the_P6.html&quot; rel=&quot;nofollow&quot;&gt;One Aspect of the Specific Bioactive or Z-Radiation of the Sun&lt;/a&gt;, Chizhevsky's article in the book where the quote comes from. Some partial results are below. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Electrocoagulation#Coagulation_process&quot; rel=&quot;nofollow&quot;&gt;Electrocoagulation of colloids&lt;/a&gt; is a known phenomenon, ionization destabilizes colloids because they are stabilized by electrostatic repulsion. One of the methods, albeit not very common, is short wave electrolysis. In 1979 a patent &lt;a href=&quot;http://www.google.com/patents/CA1060839A1?cl=en&quot; rel=&quot;nofollow&quot;&gt;Electrolysis Using Electromagnetic Radiation&lt;/a&gt; was granted to Peter Horvath. From description: &lt;em&gt;The transformer generates a magnetic field in the electrolyte which provides preferred paths for the high speed electrons of the short wave electromagnetic radiation and also for the ions in the electrolyte thereby increasing the possibility for collision between the electrons and ions with subsequent improved radiolysis yield&lt;/em&gt;&quot;. The usual applications are to wastewater and medical treatments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Non-thermal effects of irradiating milk are mentioned in an &lt;a href=&quot;http://www.avaate.org/IMG/doc/Microwaving_Our_Planet_firstenberg.doc&quot; rel=&quot;nofollow&quot;&gt;AVAATE Report with refernces to the original papers&lt;/a&gt;:&quot;&lt;em&gt;Muth in 1927 was the first to observe the formation of chains of emulsified fat particles in 20 kHz to 2 MHz electromagnetic fields. This is now a well-known phenomenon and is called the pearl chain effect. It occurs at all microwave frequencies and was captured on film by Liebesny and Pace in 1937, using milk, blood and yeast suspensions... Back in the 1930s Krasny-Ergen demonstrated that the minimum field strength for pearl chain formation... is equivalent to a power density of only about 26 µW / cm2, eminently &quot;non-thermal&quot;...&lt;/em&gt;&quot;. Alas, no mention of &quot;Kerber&quot; or &quot;Goetinck&quot; (the names are possibly misspelled, see below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=Yb03BQAAQBAJ&amp;amp;pg=PA233&amp;amp;lpg=PA233&amp;amp;dq=radio+frequency+diathermy+colloids&amp;amp;source=bl&amp;amp;ots=faVUYAqzSa&amp;amp;sig=N3qr2h7Wsax_qR_hIJXiJrdyGsM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=fUtmVdnjH4TFsAXPqIOIBQ&amp;amp;ved=0CEEQ6AEwBA#v=onepage&amp;amp;q=radio%20frequency%20diathermy%20colloids&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;This book&lt;/a&gt; describes molecular denaturation caused in the body by strong electromagnetic fields, and mentions that &quot;&lt;em&gt;considerable work done in the USSR has been concerned with much lower levels of radiation intensity&lt;/em&gt;&quot;,  referencing papers from 1957-1965. It also cites a 1963 paper of Wilkins, who may be misspelled &quot;Wilke&quot;, and Heller, who may be misspelled &quot;Miiller&quot;, in Journal of Chemical Physics called &lt;a href=&quot;http://adsabs.harvard.edu/abs/1963JChPh..39.3401W&quot; rel=&quot;nofollow&quot;&gt;Effect of Radio-Frequency Fields on the Electrophoretic Mobility of Some Colloids&lt;/a&gt;. I was unable to locate any joint paper by Wilkins (or Wilke) and Muller, as &quot;Miiller&quot; is spelled in the book. However, another 1963 paper, in Nature, by Heller, Wilkins and Freeborn, &lt;a href=&quot;http://www.nature.com/nature/journal/v197/n4871/abs/197997a0.html&quot; rel=&quot;nofollow&quot;&gt;Effect of Radio-frequency Fields on the Zeta-potential of a Colloidal Suspension&lt;/a&gt; reads:&quot;&lt;em&gt;Reports from this laboratory, afterwards confirmed elsewhere, have shown non-thermal effects of radio-frequency on living organisms and non-living material&lt;/em&gt;&quot;, which is similar to Chizhevsky's claim about Wilke and Muller. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://discovery.ucl.ac.uk/2223/1/2223.pdf&quot; rel=&quot;nofollow&quot;&gt;Biological Effects of Electromagnetic Radiation&lt;/a&gt; gives a nice survey of effects of short wave radiation used in medical treatments, the experimentation goes back to d'Arsonval, who placed animals, and later humans into induction coils. Another survey &lt;a href=&quot;http://discovery.ucl.ac.uk/2223/1/2223.pdf&quot; rel=&quot;nofollow&quot;&gt;Early Research on the Biological Effects of Microwave Radiation&lt;/a&gt; explains that by 1930s there emerged a consensus that all short wave medical effects were thermal, with some prominent experimenters reversing their earlier position on the issue. Apparently, pearl chain effect, etc. did not make much of a dent in it. Extensive bibliographies on research up to 1960s are also available &lt;a href=&quot;http://www.totalitaer.de/Waffen/referencesmarha.htm&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=9KrTBwAAQBAJ&amp;amp;pg=PA8&amp;amp;lpg=PA8&amp;amp;dq=liebesny+athermic+short+wave+therapy&amp;amp;source=bl&amp;amp;ots=dQSzecLye1&amp;amp;sig=pKiHYLKmQvLIBx0jKX1PaafAjzg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=NCFtVdbjHpf-yQSRsYHQBw&amp;amp;ved=0CB4Q6AEwAA#v=onepage&amp;amp;q=liebesny%20athermic%20short%20wave%20therapy&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A word on Chizhevsky. He is one of so-called Russian &quot;scientific cosmists&quot;, Vernadsky and Tsiolkovsky, with whom he worked on space biology in 1926, are two other prominent representatives. Unlike philosophical cosmists, like Fedorov, who sought mystic connections between the Cosmos and life on earth, scientific cosmists were interested in finding connections mediated by physical phenomena. This explains Chizhevsky's interest in &quot;bioactive radiation&quot;, in fact he was one of the early proponents of &quot;heliobiology&quot;, study of sun’s effects on living organisms. In 1942 he was told to retract his writings about effects of solar cycles on mass psychology because they contradicted &quot;historical materialism&quot; explanations of the Russian revolutions. He refused, in Stalin's Soviet Union (!), was arrested and spent eight years in a forced labor camp. He was incredibly lucky to get released in 1950, still under Stalin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://www.academia.edu/6794926/George_M._Young._The_Russian_Cosmists_The_Esoteric_Futurism_of_Nikolai_Fedorov_and_His_Followers._Oxford_Oxford_University_Press_2012&quot; rel=&quot;nofollow&quot;&gt;review of Young's book Russain Cosmists&lt;/a&gt; characterizes them as &quot;&lt;em&gt;visionary Cosmists who, being mostly scientists, conceive of cosmos as an ordered or an orderable universe to be explored and explained&lt;/em&gt;&quot;. And generally, &quot;&lt;em&gt;Cosmism, both philosophically and scientifically, gravitates toward a metaphysical vision of reality. Scientific knowledge and metaphysical knowledge (including the oddest forms of the occult) are united in one pursuit of comprehensive theory of everything&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-02T03:26:47.000" LastActivityDate="2015-06-02T03:26:47.000" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2387" PostTypeId="2" ParentId="2385" CreationDate="2015-05-28T03:08:54.960" Score="0" Body="&lt;p&gt;Because &quot;&lt;em&gt;the expected positions and distances of objects at an established time are calculated (in au) from these laws, and assembled into a collection of data called an ephemeris&lt;/em&gt;&quot;. According to &lt;a href=&quot;http://www.cv.nrao.edu/~rfisher/Ephemerides/ephem_descr.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Description of JPL Solar System Ephemeris&lt;/a&gt; &quot;&lt;em&gt;the computer calculations have been extended as far as 3000 BC to 3000 AD, but positions for the 1850-2050 range are the most accurate, of course... one needs to be very careful about comparing observations reduced on the basis of different ephemerides and coordinate systems.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a separate issue with the astronomical unit itself. Until 2012 IAU (International Astronomical Union) &lt;a href=&quot;http://en.wikipedia.org/wiki/Astronomical_unit#Development_of_unit_definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;defined astronomical unit as the mean distance from the earth to the sun&lt;/a&gt;, and their best 2009 estimate &quot;&lt;em&gt;was still derived from observation and measurements subject to error, and based on techniques that did not yet standardize all relativistic effects, and thus were not constant for all observers.&lt;/em&gt;&quot; This should not be relevant anymore because in 2012 IAU finally saw the light, and &quot;&lt;em&gt;simply used the 2009 estimate to redefine the astronomical unit as a conventional unit of length directly tied to the meter (exactly 149597870700 m) and assigned it the official symbol au&lt;/em&gt;&quot;. The meter is currently tied to the speed of light in the vacuum, so that is as ironclad as it gets. But... &lt;a href=&quot;http://www2.jpl.nasa.gov/basics/units.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;stay tuned for JPL to get up to speed&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Physics SE David Hammen describes a previous &lt;a href=&quot;https://physics.stackexchange.com/questions/186186/resolving-general-relativity-and-newtonian-mechanics-to-a-computer/186196#186196&quot;&gt;standardization &quot;war&quot; between JPL and Europeans&lt;/a&gt;, over relativisitic time measurements, which affected the astronomical unit under the old definition, by the way. &quot;&lt;em&gt;Relativistic religious wars were fought over this issue. Europeans (Russia and France, the only other countries with a stake in the game) wanted to use a (theoretical) clock far removed from the Sun. One second on that clock would always tick faster than an Earth-based clock. JPL's ephemeris time is essentially that far-removed clock, scaled to tick at Earth rate (on average). JPL won that religious war, but perhaps only because they got slightly better results than did the Russian Academy or the Paris Observatory.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2015-05-28T04:07:42.900" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2388" PostTypeId="1" CreationDate="2015-05-28T06:43:00.753" Score="8" ViewCount="780" Body="&lt;p&gt;Regarding General Theory of Relativity, I'm interested to find out whether there are some contributors to this theory in 19th century or not. In fact I want to know whether there are some physicists before Einstein who had approached even experimentally or theoretically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any kind of books on the development of General theory of Relativity?&lt;/p&gt;&#xA;" OwnerUserId="2364" LastEditorUserId="55" LastEditDate="2015-05-29T02:16:27.147" LastActivityDate="2018-07-05T18:48:55.840" Title="What 19th century developments contributed to the General theory of Relativity?" Tags="|physics|relativity-theory|gravity|celestial-mechanics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2389" PostTypeId="1" CreationDate="2015-05-28T11:18:04.777" Score="17" ViewCount="6753" Body="&lt;p&gt;The fact that space is a vacuum and that the Earth's atmosphere only extends a short way above the surface is accepted as obvious. However, there is nothing &lt;em&gt;a priori&lt;/em&gt; obvious about it: you first have to understand the nature of a vacuum and its differences from air, show that it can exist and convince yourself that it is a thing, and &lt;em&gt;then&lt;/em&gt; go on to speculate about whether the vast stretches of outer space have air in them or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the first person to make this particular leap? What made them think that?&lt;/p&gt;&#xA;" OwnerUserId="272" LastActivityDate="2021-05-05T15:14:39.860" Title="Who was the first to postulate that space was a vacuum?" Tags="|physics|astronomy|atmosphere|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2391" PostTypeId="2" ParentId="2389" CreationDate="2015-05-28T11:40:06.470" Score="4" Body="&lt;p&gt;Henning Ganz in his book &lt;em&gt;Nothingness: The science of empty space&lt;/em&gt; contends that the pre-Socratic philosopher Thales of Miletus was the first to consider the question of something vs nothing. But Thales did not ask about empty space. The physics question about the existence and properties of empty space can, in principle, be answered with experiments. In Ganz' view, Evangelista Torricelli, a 17th c student of Galileo, was the first to experimentally demonstrate that nature opposes the formation of a space devoid of air (empty space) with a finite force that can be overcome.&lt;/p&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2015-05-28T11:40:06.470" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2392" PostTypeId="2" ParentId="2343" CreationDate="2015-05-28T12:05:29.170" Score="4" Body="&lt;p&gt;The whole question of the provenance of a cultural artifact such as the &quot;scientific method&quot; is fraught with hazard. For instance, alchemy has ancient origins. The late historian of science, Derek J. de Sola Price, viewed alchemy as a pre-scientific methodology since many of its practitioners documented their use of quasi-experimental designs in an exhaustive combinatorics of the elements hypothesized to transmute lead into gold. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others, notably 50s and 60s French Structuralists, have argued that mythology was another form of pre-scientific endeavor, particularly insofar as myth relates to the cosmos and origins of existence. If you believe Giorgio de Santillana and Hertha von Dechen in their book, &lt;em&gt;Hamlet’s Mill&lt;/em&gt; (1965), the word “hamlet” originated in Persia thousands of years ago in the linguistic form of “amlodhi,” a cognate of the Persian word for necessity. Then it diffused through time and culture including Old Icelandic myths and other vestigial forms to appear in the version closest to Shakespeare’s in a 13th c work by the Danish author Saxo Grammaticus where “amlodhi” had morphed into “amleth.” The authors of &lt;em&gt;Hamlet’s Mill&lt;/em&gt; viewed these ancient creation myths as the technical language of a pre-scientific elite, finding their analogues in biblical “floods,” West African (Dogon) cosmologies and MesoAmerican oral histories where lurking beneath these mythic artifacts are cosmological phenomena such as the 26,000 year precession of the equinoxes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bottom line is that it's only a kind of ethnocentric and arrogant hubris that demands fealty to modern, Western definitions of &quot;science.&quot;&lt;/p&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2015-05-28T12:05:29.170" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2393" PostTypeId="2" ParentId="2385" CreationDate="2015-05-28T17:44:45.273" Score="3" Body="&lt;p&gt;I believe the question is asking about the following tables, one for the period 1800 AD to 2050 AD, the other two for 3000 BC to 3000 AD. Note that in these tables, &quot;EM Bary&quot; refers to the Earth-Moon barycenter, the center of mass of the Earth-Moon system. &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Table 1.&#xA;&#xA;Keplerian elements and their rates, with respect to the mean ecliptic&#xA;and equinox of J2000, valid for the time-interval 1800 AD - 2050 AD.&#xA;&#xA;               a              e               I                L            long.peri.      long.node.&#xA;           AU, AU/Cy     rad, rad/Cy     deg, deg/Cy      deg, deg/Cy      deg, deg/Cy     deg, deg/Cy&#xA;-----------------------------------------------------------------------------------------------------------&#xA;Mercury   0.38709927      0.20563593      7.00497902      252.25032350     77.45779628     48.33076593&#xA;          0.00000037      0.00001906     -0.00594749   149472.67411175      0.16047689     -0.12534081&#xA;Venus     0.72333566      0.00677672      3.39467605      181.97909950    131.60246718     76.67984255&#xA;          0.00000390     -0.00004107     -0.00078890    58517.81538729      0.00268329     -0.27769418&#xA;EM Bary   1.00000261      0.01671123     -0.00001531      100.46457166    102.93768193      0.0&#xA;          0.00000562     -0.00004392     -0.01294668    35999.37244981      0.32327364      0.0&#xA;Mars      1.52371034      0.09339410      1.84969142       -4.55343205    -23.94362959     49.55953891&#xA;          0.00001847      0.00007882     -0.00813131    19140.30268499      0.44441088     -0.29257343&#xA;Jupiter   5.20288700      0.04838624      1.30439695       34.39644051     14.72847983    100.47390909&#xA;         -0.00011607     -0.00013253     -0.00183714     3034.74612775      0.21252668      0.20469106&#xA;Saturn    9.53667594      0.05386179      2.48599187       49.95424423     92.59887831    113.66242448&#xA;         -0.00125060     -0.00050991      0.00193609     1222.49362201     -0.41897216     -0.28867794&#xA;Uranus   19.18916464      0.04725744      0.77263783      313.23810451    170.95427630     74.01692503&#xA;         -0.00196176     -0.00004397     -0.00242939      428.48202785      0.40805281      0.04240589&#xA;Neptune  30.06992276      0.00859048      1.77004347      -55.12002969     44.96476227    131.78422574&#xA;          0.00026291      0.00005105      0.00035372      218.45945325     -0.32241464     -0.00508664&#xA;Pluto    39.48211675      0.24882730     17.14001206      238.92903833    224.06891629    110.30393684&#xA;         -0.00031596      0.00005170      0.00004818      145.20780515     -0.04062942     -0.01183482&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;Source: &lt;a href=&quot;http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Table 2a.&#xA;&#xA;Keplerian elements and their rates, with respect to the mean ecliptic and equinox of J2000,&#xA;valid for the time-interval 3000 BC -- 3000 AD.  NOTE: the computation of M for Jupiter through&#xA;Pluto *must* be augmented by the additional terms given in Table 2b (below).&#xA;&#xA;               a              e               I                L            long.peri.      long.node.&#xA;           AU, AU/Cy     rad, rad/Cy     deg, deg/Cy      deg, deg/Cy      deg, deg/Cy     deg, deg/Cy&#xA;------------------------------------------------------------------------------------------------------&#xA;Mercury   0.38709843      0.20563661      7.00559432      252.25166724     77.45771895     48.33961819&#xA;          0.00000000      0.00002123     -0.00590158   149472.67486623      0.15940013     -0.12214182&#xA;Venus     0.72332102      0.00676399      3.39777545      181.97970850    131.76755713     76.67261496&#xA;         -0.00000026     -0.00005107      0.00043494    58517.81560260      0.05679648     -0.27274174&#xA;EM Bary   1.00000018      0.01673163     -0.00054346      100.46691572    102.93005885     -5.11260389&#xA;         -0.00000003     -0.00003661     -0.01337178    35999.37306329      0.31795260     -0.24123856&#xA;Mars      1.52371243      0.09336511      1.85181869       -4.56813164    -23.91744784     49.71320984&#xA;          0.00000097      0.00009149     -0.00724757    19140.29934243      0.45223625     -0.26852431&#xA;Jupiter   5.20248019      0.04853590      1.29861416       34.33479152     14.27495244    100.29282654&#xA;         -0.00002864      0.00018026     -0.00322699     3034.90371757      0.18199196      0.13024619&#xA;Saturn    9.54149883      0.05550825      2.49424102       50.07571329     92.86136063    113.63998702&#xA;         -0.00003065     -0.00032044      0.00451969     1222.11494724      0.54179478     -0.25015002&#xA;Uranus   19.18797948      0.04685740      0.77298127      314.20276625    172.43404441     73.96250215&#xA;         -0.00020455     -0.00001550     -0.00180155      428.49512595      0.09266985      0.05739699&#xA;Neptune  30.06952752      0.00895439      1.77005520      304.22289287     46.68158724    131.78635853&#xA;          0.00006447      0.00000818      0.00022400      218.46515314      0.01009938     -0.00606302&#xA;Pluto    39.48686035      0.24885238     17.14104260      238.96535011    224.09702598    110.30167986&#xA;          0.00449751      0.00006016      0.00000501      145.18042903     -0.00968827     -0.00809981&#xA;------------------------------------------------------------------------------------------------------&#xA;&#xA;&#xA;&#xA;Table 2b.&#xA;&#xA;Additional terms which must be added to the computation of M&#xA;for Jupiter through Pluto, 3000 BC to 3000 AD, as described&#xA;in the related document.&#xA;&#xA;                b             c             s            f&#xA;---------------------------------------------------------------&#xA;Jupiter   -0.00012452    0.06064060   -0.35635438   38.35125000&#xA;Saturn     0.00025899   -0.13434469    0.87320147   38.35125000&#xA;Uranus     0.00058331   -0.97731848    0.17689245    7.67025000&#xA;Neptune   -0.00041348    0.68346318   -0.10162547    7.67025000&#xA;Pluto     -0.01262724&#xA;---------------------------------------------------------------&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;Source: &lt;a href=&quot;http://ssd.jpl.nasa.gov/txt/p_elem_t2.txt&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ssd.jpl.nasa.gov/txt/p_elem_t2.txt&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt;&#xA;It's important to note that the above tables are approximations. More accurate techniques exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For details on how to use these tables, and on the errors that result, see &lt;a href=&quot;http://ssd.jpl.nasa.gov/txt/aprx_pos_planets.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ssd.jpl.nasa.gov/txt/aprx_pos_planets.pdf&lt;/a&gt; and &lt;a href=&quot;http://ssd.jpl.nasa.gov/?planet_pos&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ssd.jpl.nasa.gov/?planet_pos&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The latter shows the errors in the two:&#xA;$$\begin{matrix}&#xA;&amp;amp;&amp;amp;1800&amp;amp;\text{to}&amp;amp;2050 &amp;amp;&amp;amp; 3000\ \text{BC}&amp;amp;\text{to} &amp;amp; 3000\ \text{AD} \\ \\&#xA;\text{Planet} &amp;amp;\ &amp;amp; \text{RA} &amp;amp; \text{Dec} &amp;amp; \text{r} &amp;amp;\quad&amp;amp; \text{RA} &amp;amp; \text{Dec} &amp;amp; \text{r} \\&#xA;&amp;amp;&amp;amp; \text{arcsec} &amp;amp; \text{arcsec} &amp;amp; \text{km} &amp;amp;\quad&amp;amp; \text{arcsec} &amp;amp; \text{arcsec} &amp;amp; \text{km} \\&#xA;-----&amp;amp;&amp;amp;---&amp;amp;---&amp;amp;---&amp;amp;&amp;amp;---&amp;amp;---&amp;amp;---\\&#xA;\text{Mercury} &amp;amp;&amp;amp; 15 &amp;amp; 1 &amp;amp; 1 &amp;amp;&amp;amp; 20 &amp;amp; 15 &amp;amp; 1\\&#xA;\text{Venus} &amp;amp;&amp;amp; 20 &amp;amp; 1 &amp;amp; 4 &amp;amp;&amp;amp; 40 &amp;amp; 30 &amp;amp; 8\\ &#xA;\text{Earth-Moon} &amp;amp;&amp;amp; 20 &amp;amp; 8 &amp;amp; 6 &amp;amp;&amp;amp; 40 &amp;amp; 15 &amp;amp; 15\\&#xA;\text{Mars} &amp;amp;&amp;amp; 40 &amp;amp; 2 &amp;amp; 25 &amp;amp;&amp;amp; 100 &amp;amp; 40 &amp;amp; 30\\&#xA;\text{Jupiter} &amp;amp;&amp;amp; 400 &amp;amp; 10 &amp;amp; 600 &amp;amp;&amp;amp; 600 &amp;amp; 100 &amp;amp; 1000\\&#xA;\text{Saturn} &amp;amp;&amp;amp; 600 &amp;amp; 25 &amp;amp; 1500 &amp;amp;&amp;amp; 1000 &amp;amp; 100 &amp;amp; 4000\\&#xA;\text{Uranus} &amp;amp;&amp;amp; 50 &amp;amp; 2 &amp;amp; 1000 &amp;amp;&amp;amp; 2000 &amp;amp; 30 &amp;amp; 8000\\ &#xA;\text{Neptune} &amp;amp;&amp;amp; 10 &amp;amp; 1 &amp;amp; 200 &amp;amp;&amp;amp; 400 &amp;amp; 15 &amp;amp; 4000\\&#xA;\text{Pluto} &amp;amp;&amp;amp; 5 &amp;amp; 2 &amp;amp; 300 &amp;amp;&amp;amp; 400 &amp;amp; 100 &amp;amp; 2500&#xA;\end{matrix}&#xA;$$&#xA;&lt;sup&gt;Source: &lt;a href=&quot;http://ssd.jpl.nasa.gov/?planet_pos&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ssd.jpl.nasa.gov/?planet_pos&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the errors are considerably greater for tables 2a and 2b than they are for table 1, even with the extra complexity involved with using tables 2a and 2b. This says you should use table 1 if the time point of interest is between 1800 AD and 2050 AD, tables 2a and 2b if the time point of interest is outside that interval but between 3000 BC and 3000 AD.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for two tables is that a narrow span of time around the present is of interest to most people, but a broad span of time is of interest to some.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These tables were constructed using some kind of optimal fit (e.g., least squares) that best matched a set of observations. It is very bad form to extrapolate an optimal fit outside of the region over which the estimation was performed. The errors that result from using table 1 (1800 AD to 2050 AD) on the span 3000 BC to 3000 AD would be immense. Never extrapolate curve-fit data!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The data used to generate table 1 was based on (somewhat) modern instrumentation. The measurements are much more accurate and much more plentiful than the older data needed to generate the long ephemeris. The narrow time span combined with plentiful, accurate measurements is what makes the 1800 AD to 2050 AD table considerably more accurate than the long ephemeris.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of that older data goes back to 2800 BC. Chinese, Babylonian, and Egyptian astrologers granted immense powers in the stars and planets to influence kings and calamities such as war, famine, drought, and disease. That irrational behavior led to great record keeping. Those ancient astrological observations (particularly solar eclipses) provide a fantastic anchor for long ephemerides.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="1843" LastEditDate="2017-10-22T14:41:46.223" LastActivityDate="2017-10-22T14:41:46.223" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2394" PostTypeId="2" ParentId="2389" CreationDate="2015-05-28T21:45:34.757" Score="16" Body="&lt;p&gt;The idea of vacuum, or void as it was called in antiquity, did not originally come from considerations about air and Earth's atmosphere, but rather from natural philosophic speculations about the nature of Cosmos, which predate  discovering the relation between void and air. Presocratic philosophers starting with Thales considered nothing or void speculatively, but rejected it as a physical possibility, until the atomists (c. 450 BC), most notably Leucippus, Democritus, and later Epicurus, embraced it, and &lt;a href=&quot;http://plato.stanford.edu/entries/atomism-ancient/#2&quot; rel=&quot;noreferrer&quot;&gt;developed a systematic view of nature&lt;/a&gt; based on it. Their entire universe consisted of atoms and void in between, air was a particular arrangement of atoms.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;These ancient atomists theorized that the two fundamental and oppositely characterized constituents of the natural world are indivisible bodies—atoms—and void. The latter is described simply as nothing, or the negation of body. Atoms are by their nature intrinsically unchangeable; they can only move about in the void and combine into different clusters. Since the atoms are separated by void, they cannot fuse, but must rather bounce off one another when they collide.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They also postulated that &quot;&lt;em&gt;there is an infinite number of worlds, some like this world, others unlike it&lt;/em&gt;&quot; (&lt;a href=&quot;http://www.epicurus.net/en/herodotus.html&quot; rel=&quot;noreferrer&quot;&gt;Epicurus&lt;/a&gt;), separated by the void, and &lt;a href=&quot;http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511552540&amp;amp;cid=CBO9780511552540A042&quot; rel=&quot;noreferrer&quot;&gt;confined earth, sea, air, etc. to our world&lt;/a&gt;. However, they also saw the fixed stars as a part of our world, in fact the outer edge of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That air was a substance might have been demonstrated by a presocratic &lt;a href=&quot;https://books.google.com/books?id=oabrCbbMx6oC&amp;amp;pg=PT16&amp;amp;lpg=PT16&amp;amp;dq=empedocles+hydra+pressure+vacuum&amp;amp;source=bl&amp;amp;ots=bYeVIBI7YV&amp;amp;sig=pYqqU6hzt8YymVXDmhKw6v2cFWo&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=uo1nVYa2DYnvtQW5g4HYDw&amp;amp;ved=0CC8Q6AEwBQ#v=onepage&amp;amp;q=empedocles%20hydra%20pressure%20vacuum&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Empedocles (c. 490–430 BC) with a glass tube called hydra&lt;/a&gt;, but this account is controversial, and Aristotle's supposed demonstration by weighing an inflated bladder is known to be an artifact. It was only much later that Galileo measured air weight, and suggested that existence of a vacuum can be tested in experiments. One year after Galileo's death, in 1643, his assistant Torricelli famously gave the experimental proof he suggested, and invented the barometer to measure atmospheric pressure. Thus, it became possible to measure atmospheric pressure at different heights (qualitatively, 'thinning' of the air with altitude was observed much earlier). Pascal performed a public demonstration at a fair of the effects of altitude on a column of mercury, and &lt;a href=&quot;http://www4.ncsu.edu/~kimler/hi322/air.html&quot; rel=&quot;noreferrer&quot;&gt;surmised existence of a vacuum above the atmosphere in 1648&lt;/a&gt;. And soon &lt;a href=&quot;http://adsabs.harvard.edu/full/2005HisSc..43..391G&quot; rel=&quot;noreferrer&quot;&gt;Hooke presented an estimate of how far the atmosphere extends in Micrographia (1665)&lt;/a&gt;, after conjecturing the inverse square law for gravity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The presocratic tradition culminated in Aristotle (384-322 BC) postulating that &quot;nature abhors vacuum&quot;, which is how he explained why water rises in a suction pump, the phenomenon that Torricelli explained using the concepts of atmospheric pressure and air vacuum. But it does not mean that he believed in air stretching out into the whole Cosmos. &lt;a href=&quot;http://arxiv.org/pdf/1312.4057v2.pdf&quot; rel=&quot;noreferrer&quot;&gt;According to Aristotle&lt;/a&gt; the natural places of &quot;four elements&quot;, earth, water, air and fire, are confined to the sublunar sphere. Everything above the moon is permeated by a &quot;nobler&quot; substance, ether. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-21T04:04:05.613" LastActivityDate="2018-02-21T04:04:05.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2395" PostTypeId="2" ParentId="2388" CreationDate="2015-05-29T01:53:47.543" Score="10" Body="&lt;p&gt;There are several major ideas that went into general relativity: finite speed of gravity propagation, not necessarily Euclidean geometry of space, identification of inertia and gravity, mechanics as geometry, and uniformity of physical laws in all coordinate frames, even accelerated ones (general covariance). The last two ideas are specifically Einstein's, and only emerged after he formulated special relativity in 1905. The equivalence of inertia and gravity, &quot;the happiest thought of my life&quot;, occured to him in 1907, but he did have a 19th century inspiration in Ernst Mach, who vaguely speculated that inertial mass arises from the body's interaction with the rest of the universe. But the first two ideas have the deepest 19th century roots. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dissatisfaction with action at a distance goes back to Newton, but the first physicist to systematically consider a gravity theory with finite propagation speed was &lt;a href=&quot;http://en.wikipedia.org/wiki/Speed_of_gravity#Laplace&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laplace in his Celestial Mechanics (1799)&lt;/a&gt;. He introduced velocity dependence, so called non-zero latency, when forces are directed to retarded positions of bodies rather than the current ones... and discovered that planets will fly off of their orbits in a hurry, unless gravity was millions of times faster than light. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next attempt was made in 1830 by Mossotti, a French physics teacher at the University of Buenos Aires, who had an ingenious idea that electric attraction and repulsion do not balance each other exactly, &lt;a href=&quot;http://en.wikisource.org/wiki/Scientific_Memoirs/1/On_the_Forces_which_regulate_the_Internal_Constitution_of_Bodies&quot; rel=&quot;nofollow noreferrer&quot;&gt;and the difference is gravity&lt;/a&gt;. This idea did not gain much currency &lt;a href=&quot;http://en.wikipedia.org/wiki/Lorentz_ether_theory#Gravitation&quot; rel=&quot;nofollow noreferrer&quot;&gt;until 1900, when Lorentz&lt;/a&gt;, Einstein's predecessor in special relativity also, used Maxwell's electrodynamics to show that it did not suffer from Laplace's latency problem, because the correction to force's direction was of the order $v^2/c^2$ rather than $v/c$ as in Laplace's case. That was the consequence of Lorentz invariance compensating the effects of retardation in the first order, as &lt;a href=&quot;http://en.wikipedia.org/wiki/Speed_of_gravity#Lorentz_covariant_models&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poincare pointed out in 1905&lt;/a&gt;, and an indication that a relativistic theory with gravity propagating at the same speed as light was viable. However, Lorentz's theory did not predict the correct value for the anomalous precession of Mercury. Lorentz's theory was also mentioned by Poincare in 1908, soon after Einstein got interested in extending relativity to gravity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author most commonly mentioned as Einstein's &quot;predecessor&quot; is Gerber, who in 1898 inserted into Newton's potential a velocity dependent term that does produce the correct value of Mercury's precession. Gerber's theory was ad hoc however, and he could not explain how his particular correction can be physically implemented. Einstein was aware of Gerber's gravity, it was mentioned in books of Mach and Poincare, but there was not much there to influence him, except perhaps for suggesting the idea that the precession was relativistic in origin. His work was however used by Einstein's detractors to accuse him of plagiarism, and to argue against awarding him a Nobel prize for general relativity. See &lt;a href=&quot;http://www.mathpages.com/home/kmath527/kmath527.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerber's Gravity&lt;/a&gt; for details. I am leaving out  Heaviside's 1893 theory, which mimicked Maxwell's electrodynamics for gravity and is essentially equivalent to a &lt;a href=&quot;http://en.wikipedia.org/wiki/Gravitoelectromagnetism&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern approximation of general relativity, gravitoelectromagnetism&lt;/a&gt;, because there is no evidence that Einstein was aware of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;General metrics were of course introduced by Riemann in his famous lecture, and by 1900 Riemannian geometry was systematized using &lt;a href=&quot;https://hsm.stackexchange.com/questions/599/what-was-the-motivation-for-the-development-of-modern-intrinsic-differential-g/604#604&quot;&gt;tensor calculus developed by Ricci and Levi-Civita&lt;/a&gt;. Einstein learned of them from Grossman, who was a geometer, and providing the most important mathematical tool used in general relativity. The original version of relativistic gravity theory, called Entwurf, was originally worked out by Einstein and Grossman in 1913. But alas, just like Lorentz's it did not predict Mercury's precession correctly. In 1912 Mie, and in 1913 Nordstrom suggested &lt;a href=&quot;https://books.google.com/books?id=vpW_sBxwr88C&amp;amp;pg=PA142&amp;amp;lpg=PA142&amp;amp;dq=nordstrom+mie+gravity&amp;amp;source=bl&amp;amp;ots=QusdxkQKwc&amp;amp;sig=633z7wrNSPwEBlHqf2IGLGLYZU4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=kH1nVe7yKMnEsAWy5IHADg&amp;amp;ved=0CEQQ6AEwBg#v=onepage&amp;amp;q=nordstrom%20mie%20gravity&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;relativistic gravity theories alternative to general relativity&lt;/a&gt; finalized by Einstein in 1916. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=2&amp;amp;ved=0CCUQFjAB&amp;amp;url=http%3A%2F%2Fwww.springer.com%2Fcda%2Fcontent%2Fdocument%2Fcda_downloaddocument%2F9780817649395-c1.pdf%3FSGWID%3D0-0-45-1265138-p174265357&amp;amp;ei=S7pnVZ_THIqtyASLnYDIDA&amp;amp;usg=AFQjCNGuzFaAuIlsrP3mze1WDZp5fC11ww&amp;amp;sig2=h3DKe6VGv1TWV3QJXXzy1A&quot; rel=&quot;nofollow noreferrer&quot;&gt;Renn and Schemmel's Theories of Gravitation in the Twilight of Classical Physics&lt;/a&gt; gives a broad historical overview, see &lt;a href=&quot;http://www.daviddarling.info/encyclopedia/G/genrel.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Darling's Encyclopedia&lt;/a&gt; for a nice popular account. A cranky but well referenced account of some lesser known 19th century developments is Hecht's &lt;a href=&quot;http://21sci-tech.com/articles/spring01/Electrodynamics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Suppressed Electrodynamics of Ampère-Gauss-Weber&lt;/a&gt;. &lt;a href=&quot;http://repository.cmu.edu/cgi/viewcontent.cgi?article=1338&amp;amp;context=philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earman and Glymour's Lost in the Tensors&lt;/a&gt; focuses on the mathematical side of history of general relativity. &#xA;You can read more on the final stages of its creation in &lt;a href=&quot;http://mathpages.com/rr/s8-10/8-10.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kevin Brown's Conquering the Perihelion&lt;/a&gt; and &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/PrintHT/General_relativity.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor History&lt;/a&gt;. See also related &lt;a href=&quot;https://hsm.stackexchange.com/questions/2308/what-attracted-einstein-to-the-anomalous-precession-of-mercury&quot;&gt;What attracted Einstein to the anomalous precession of Mercury?&lt;/a&gt; thread. A good book is &lt;a href=&quot;https://books.google.com/books?id=KYpU4AWwJ3UC&amp;amp;dq=mossotti+lorentz+gravity+1900&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Renn's The Genesis of General Relativity&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-07-05T18:48:55.840" LastActivityDate="2018-07-05T18:48:55.840" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2396" PostTypeId="1" CreationDate="2015-05-29T01:54:02.303" Score="5" ViewCount="76" Body="&lt;p&gt;The technique of &quot;mathematical induction&quot; is a method of proof where you show some theorem is true for some starting integer and prove also that it holding at any arbitrary integer implies it must hold at the following integer. From this you conclude it must hold for all integers from the starting point on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think most people would consider this &quot;obvious&quot;. So obvious in fact that I always considered it as dating back to the earliest history of mathematics, such as to the ancient Greeks. But I was teaching a course in Discrete Math and was surprised to read in the text that the first explicit explanation of the principle was from a mathematician in the 1500's named Francesco Maurolico:&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/Francesco_Maurolico&quot; rel=&quot;nofollow&quot;&gt;http://en.wikipedia.org/wiki/Francesco_Maurolico&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How could it have taken so long for mathematicians to discover it? It makes you wonder if there are also other obvious principles out there that have not been yet explicitly enunciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another point: European math and science in general was in deep decline during the Dark Ages. But Chinese and Indian and Arabic math continued to make great advances during this period. I wouldn't be surprised if we had greater records of the mathematicians in these other cultures during that period that they had stated the principle earlier.&lt;/p&gt;&#xA;" OwnerUserId="2368" OwnerDisplayName="Robert Clark" LastEditorUserId="58" LastEditDate="2015-05-29T03:06:31.813" LastActivityDate="2015-05-29T03:06:31.813" Title="Mathematical induction only dates from the Middle Ages?" Tags="|mathematics|mathematical-logic|" AnswerCount="0" CommentCount="4" ClosedDate="2015-05-29T20:33:32.383" ContentLicense="CC BY-SA 3.0" />
  <row Id="2397" PostTypeId="1" CreationDate="2015-05-29T03:25:28.167" Score="12" ViewCount="836" Body="&lt;p&gt;I am about halfway through reading E.T. Bell's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0671628186&quot;&gt;Men of Mathematics&lt;/a&gt;, and I absolutely love it. I'm a mathematician, and I enjoy learning about the lives behind the names that I know and use so often. (I also rather like the distinct opinionatedness and bias).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I consider &lt;em&gt;Men of Mathematics&lt;/em&gt; to be a sort of canonical introduction to the history of mathematicians. I do not know this to be true, but it was recommended to me so strongly and so widely that I feel it must be somewhat canonical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But &lt;em&gt;Men of Mathematics&lt;/em&gt; simply does not cover contemporary mathematicians at all, or even relatively recent mathematicians. Are there &quot;canonical&quot; references for lives of more recent mathematicians, or even exposés or profiles of recent mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="2369" LastEditorUserId="55" LastEditDate="2016-11-02T00:14:35.473" LastActivityDate="2020-06-05T18:35:03.317" Title="What resources are available for lives of recent mathematicians besides E.T. Bell's Men of Mathematics?" Tags="|biographical-details|mathematicians|reference-request|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2398" PostTypeId="2" ParentId="2397" CreationDate="2015-05-29T07:25:05.470" Score="7" Body="&lt;p&gt;It is hard to imagine a book like Bell's one written about modern mathematicians, but a nice substitute is some mathematicians autobiographies. Some outstanding examples are those of Andre Weil, Laurent Schwartz, Walter Rudin.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-05-29T07:25:05.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2400" PostTypeId="2" ParentId="2397" CreationDate="2015-05-29T22:50:57.747" Score="14" Body="&lt;p&gt;I feel somewhat conflicted writing this because Bell's book inspired many people to become mathematicians, including some prominent ones. However, it is not a canonical introduction to the history of mathematics and mathematicians, &lt;a href=&quot;http://rutherfordjournal.org/article030105.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;historians of mathematics tend to distrust the historical reliability of most of Bell’s accounts&quot;&lt;/a&gt;(Leo Corry). We recently had a question &lt;a href=&quot;https://hsm.stackexchange.com/questions/2105/whats-the-famous-story-about-a-mathematician-who-gave-a-talk-without-saying-a-w/2106#2106&quot;&gt;What&amp;#39;s the famous story about a mathematician who gave a talk without saying a word?&lt;/a&gt;, which illustrates why. He embellishes facts, retells anecdotes, and makes up some of his own. Even normally bland Wikipedia has some &lt;a href=&quot;http://en.wikipedia.org/wiki/Men_of_Mathematics#cite_note-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;choice words&lt;/a&gt; for Men of Mathematics.&#xA;&lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/missed.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Siegmund-Schultze summarizes it&lt;/a&gt; well:&quot;&lt;em&gt;E.T. Bell’s book on Men of Mathematics is known to be notoriously unreliable and sketchy, but it is a good read and can instill enthusiasm for mathematics in the beginner, not least because it aims to show that “mathematicians can be as human as anybody else” and because Bell wants to “lead up to some of the dominating ideas governing vast tracts of mathematics as it exists today”&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Bell's books are in the genre of historical fiction (he also wrote science fiction), which in itself is valuable, but it should not be taken for history. If you like the genre some call &lt;a href=&quot;https://books.google.com/books?id=uV3rJkmnQhsC&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Muir's Of Men and Numbers&lt;/a&gt;&#xA;&quot;Bell's clone&quot;, although it still ends with Cantor. A much better book on 19th century is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0817633162&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yaglom's Felix Klein and Sophus Lie&lt;/a&gt;, which despite the title gives a panoramic view of 19th century mathematicians, and is more accurate. Like Bell Yaglom is a good writer, but he is captivating in ways different from Bell's and Muir's. More scholarly works on 19th century are Parshall's &lt;a href=&quot;https://books.google.com/books?id=OZNNee_H1hEC&amp;amp;dq=inauthor:%22Karen+Hunger+Parshall%22&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sylvester&lt;/a&gt; and &lt;a href=&quot;http://www.jstor.org/stable/41133752?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wedderburn&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Researching mathematician's life is a lot of work, especially given the number of sources for recent times, and individual biographies written by original researchers are the way to go if you want real history.  Something along the lines of &lt;a href=&quot;http://www.ams.org/notices/201110/rtx111001458p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Constance Reid's Hilbert, Courant and Julia&lt;/a&gt;. I am afraid there are no &quot;canonical&quot; compilations. By the way, instead of Hilbert Reid was supposed to write an &quot;update&quot; of Bell's Men of Mathematics, but changed her mind after traveling to Göttingen. In that she was encouraged by her sister, Julia Robinson, the heroine of the last book. Later she wrote a book about Bell himself, &quot;&lt;em&gt;more of a detective story than a true biography&lt;/em&gt;&quot;, uncovering some details that not even his family knew about. You may also go for first hand accounts, like &lt;a href=&quot;https://zelalemkibret.files.wordpress.com/2012/01/the-autobiography-of-bertrand-russell.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's Autobiography&lt;/a&gt;, touching on his interactions with early 20th century mathematicians, or deeply personal and candid &lt;a href=&quot;http://attach3.bdwm.net/attach/0Announce/groups/GROUP_3/Mathematics/D4335A993//M.1104367920.A/RS.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grothendieck's Récoltes et Semailles&lt;/a&gt; featuring French mathematicians of 1950-60s, but keep in mind that first hand accounts are by definition opinionated and biased.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=hmath-23&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800-1945&lt;/a&gt; is a collection of 20 scholarly essays on the period.&#xA;James's &lt;a href=&quot;https://books.google.com/books/about/Remarkable_Mathematicians.html?id=fptj8AbgUDgC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Remarkable Mathematicians: From Euler to Von Neumann&lt;/a&gt; goes into 20th century, and is less anecdotal, but he is not as good a writer, and see linked Siegmund-Schultze's essay about his history. &lt;a href=&quot;http://math.sjtu.edu.cn/course/skymath/skymathinfo/referencebook/The%20Poincar%C3%A9%20Conjectur,%20Donal%20O%27Shea,%202007.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;O’Shea's Poincare Conjecture&lt;/a&gt; (touches on many 20th century mathematicians) gets a nod despite &quot;&lt;em&gt;many minor factual mistakes&lt;/em&gt;&quot;, it also has further reading pointers. &quot;&lt;em&gt;O’Shea, unlike James, is a mathematician who uses the work of historians with profit, gives credit where credit is due, immerses himself in the spirit of the times, and, crucially, imparts his mathematical knowledge in a form which the public can understand.&lt;/em&gt;&quot; &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/BiogIndex.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor History has chronologically indexed biographies&lt;/a&gt; &quot;&lt;em&gt;useful as a first orientation for further research. Unlike in James’s biographies, there is also some mathematical literature for further reading added, although the manner of quoting within the biographical entries and the reliability of information provided are often similarly dubious&lt;/em&gt;&quot;. &#xA;&lt;a href=&quot;https://www.maths.cam.ac.uk/undergrad/admissions/files/reading-list.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cambridge's Mathematical Reading List&lt;/a&gt; has a history section with brief characterizations of some mathematicians' biographies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also an &lt;a href=&quot;https://math.stackexchange.com/questions/28885/anecdotes-about-famous-mathematicians-or-physicists&quot;&gt;assorted collection of modern anecdotes on Math SE&lt;/a&gt;, and this unsigned &lt;a href=&quot;http://sinbad.bplaced.net/dwnldir/greatest_mathematicians.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Greatest Mathematicians of 20th Century&lt;/a&gt;, read at your own risk.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="8068" LastEditDate="2020-06-05T18:35:03.317" LastActivityDate="2020-06-05T18:35:03.317" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="2402" PostTypeId="2" ParentId="404" CreationDate="2015-05-30T22:16:20.637" Score="1" Body="&lt;p&gt;Alfred Nobel was also one of the world´s biggest weapon manufacturers after having bought the Bofors Industries and started production of cannons. He also invenyed the smokeless gunpowder thus making it possible to see the soldiers at the battlefields and kill them more efficiently.&lt;/p&gt;&#xA;" OwnerUserId="2376" LastActivityDate="2015-05-30T22:16:20.637" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2403" PostTypeId="1" CreationDate="2015-05-31T18:15:29.373" Score="5" ViewCount="346" Body="&lt;p&gt;There are two typical visualizations for one of the results of the Millikan oil-drop experiment to illustrate that only integer multiples of some elementary charge occur in nature.&lt;/p&gt;&#xA;&lt;p&gt;The first one is as a histogram like this one (&lt;a href=&quot;https://iopscience.iop.org/article/10.1088/0143-0807/33/5/1227/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/YJyVn.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;The second just a simple scatter plot like this (&lt;a href=&quot;https://docplayer.org/76243825-Physik-iv-einfuehrung-in-die-atomistik-und-die-struktur-der-materie.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qtrqn.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;I just skimmed over some of Millikan's original papers but didn't find any visualization like the above ones. So did Millikan actually use such kind of plots? In which paper did those typical textbook diagrams of the Millikan experiment appear for the first time?&lt;/p&gt;&#xA;" OwnerUserId="2380" LastEditorUserId="2380" LastEditDate="2022-06-08T19:28:29.757" LastActivityDate="2022-06-08T19:28:29.757" Title="In which paper did the typical textbook diagrams of the Millikan experiment appear the first time?" Tags="|physics|experimental-physics|particle-physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2404" PostTypeId="1" CreationDate="2015-05-31T19:45:28.123" Score="0" ViewCount="24465" Body="&lt;p&gt;What are the most famous polymaths of the previous century? I care mostly about scientists and scholars.&lt;/p&gt;&#xA;" OwnerUserId="2364" LastEditorUserId="62" LastEditDate="2015-05-31T20:31:40.847" LastActivityDate="2015-06-01T17:21:16.780" Title="What are the most famous polymaths of the 20th century?" Tags="|polymath|" AnswerCount="1" CommentCount="4" ClosedDate="2015-06-02T03:07:51.293" ContentLicense="CC BY-SA 3.0" />
  <row Id="2406" PostTypeId="2" ParentId="2404" CreationDate="2015-06-01T17:21:16.780" Score="2" Body="&lt;p&gt;Polymaths from the 20th century.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Ott%C3%B3_Herman&quot; rel=&quot;nofollow&quot;&gt;Ottó Herman&lt;/a&gt; (1835–1914) was a Hungarian zoologist, ethnographer, archaeologist, and politician; a polymath recognized as a pioneer of Hungarian natural history research. He made numerous studies on Hungarian spiders, birds, and fishes, founded the journal Natural History Notebooks, which became one of the most popular scientific publications of Hungary, and the ornithological journal Aquila. He is called &quot;the Father of the birds&quot; in Hungary. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Charles_Sanders_Peirce&quot; rel=&quot;nofollow&quot;&gt;Charles Sanders Peirce&lt;/a&gt; (1839–1914) was an American philosopher, logician, mathematician, statistician, semiotician and chemist who founded the philosophical tradition of pragmatism. He made major contributions to logic, but logic for him encompassed much of that which is now called epistemology and philosophy of science. He saw logic as the formal branch of semiotics, of which he is a founder. As early as 1886 he saw that logical operations could be carried out by electrical switching circuits, the same idea as was used decades later to produce digital computers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Rabindranath_Tagore&quot; rel=&quot;nofollow&quot;&gt;Rabindranath Tagore&lt;/a&gt; (1861–1941) was an Indian Bengali poet, novelist, musician, playwright and painter who reshaped Bengali literature and music in the late 19th and early 20th centuries. As author of Gitanjali and its &quot;profoundly sensitive, fresh and beautiful verse&quot;, and as the first Asian to win the Nobel Prize in Literature, Tagore was perhaps the most widely regarded Indian literary figure of all time. He was a mesmerizing representative of the Indian culture whose influence and popularity internationally perhaps could only be compared to that of Gandhi, whom Tagore named &lt;em&gt;Mahatma&lt;/em&gt; out of his deep admiration for him.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Edward_Heron-Allen&quot; rel=&quot;nofollow&quot;&gt;Edward Heron-Allen&lt;/a&gt; (1861–1943) was not only a lawyer by trade, he also wrote, lectured on and created violins, was an expert on the art of chiromancy or palmistry, having read palms and analysed the handwriting of luminaries of the period. He wrote on musical, literary and scientific subjects ranging from foraminifera, marine zoology, meteorology, as a Persian scholar translated Classics such as the Rubaiyat of Omar Khayyam and The Lament of Baba Tahir, also wrote on local history, archaeology, Buddhist philosophy, the cultivation, gourmet appreciation of and culture of the asparagus, as well as a number of novels and short stories of science fiction and horror written under his pseudonym of &quot;Christopher Blayre.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/C._B._Fry&quot; rel=&quot;nofollow&quot;&gt;C. B. Fry&lt;/a&gt; (1872–1956) was an International and record-breaking athlete at sports including football, cricket, athletics and rugby union, also a politician, publisher, teacher and writer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Bertrand_Russell&quot; rel=&quot;nofollow&quot;&gt;Bertrand Russell&lt;/a&gt; (1872–1970) was a British philosopher, logician, mathematician, historian, and social critic. He is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege and his friend Ludwig Wittgenstein, and is widely held to be one of the 20th century's premier logicians. He co-authored, with A. N. Whitehead, Principia Mathematica, an attempt to ground mathematics on logic. His philosophical essay &quot;On Denoting&quot; has been considered a &quot;paradigm of philosophy.&quot; His work has had a considerable influence on logic, mathematics, set theory, linguistics, computer science (see type theory and type system), and philosophy, especially philosophy of language, epistemology, and metaphysics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Albert_Schweitzer&quot; rel=&quot;nofollow&quot;&gt;Albert Schweitzer&lt;/a&gt; (1875–1965) was an Alsatian theologian, musician, philosopher, physician, humanitarian, and peace activist. Influential in all of these fields, he wrote theological and philosophical texts ranging from the historical Jesus to J.S. Bach and German/French organ-building which influenced the Organ reform movement. He founded the Albert Schweitzer Hospital in Lambaréné, and was honored with the 1952 Nobel Prize for his philosophy of &quot;Reverence for Life.&quot; He spent the last years of his life campaigning against the use of nuclear weapons.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger&quot; rel=&quot;nofollow&quot;&gt;Erwin Schrödinger&lt;/a&gt; (1887–1961) was a physicist and theoretical biologist. Schrödinger was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933. In 1935, after extensive correspondence with personal friend Albert Einstein, he proposed the Schrödinger's cat thought experiment.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jean_Cocteau&quot; rel=&quot;nofollow&quot;&gt;Jean Cocteau&lt;/a&gt; (1889–1963) was a French poet, novelist, artist, filmmaker, dramatist, designer, boxing manager, diarist, ballet scenarist, illustrator and playwright. The &quot;polymath par excellence&quot; was an internationally renowned figure who was an influence on Modernism and the 20th century avant-garde.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/J._R._R._Tolkien&quot; rel=&quot;nofollow&quot;&gt;J. R. R. Tolkien&lt;/a&gt; (1892–1973) was an English writer, poet, philologist, polyglot, and university professor who is best known as the author of the classic high-fantasy works The Hobbit, The Lord of the Rings, and The Silmarillion. Tolkien learned Latin, French, and German from his mother, and while at school he learned Middle English, Old English, Finnish, Gothic, Greek, Italian, Old Norse, Spanish, Welsh, and Medieval Welsh. He was also familiar with Danish, Dutch, Lombardic, Norwegian, Russian, Serbian, Swedish and older forms of modern Germanic and Slavonic languages, revealing his deep linguistic knowledge, above all of the Germanic languages.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/William_James_Sidis&quot; rel=&quot;nofollow&quot;&gt;William James Sidis&lt;/a&gt; (1898–1944) was a child prodigy who wrote on such varied subjects as mathematics, cosmology, psychology, Native American history, and public transportation. &quot;His sister, Helena, said of him that, as an adult, he could learn a new language in one day, and as an adult, he was a true polymath, a 'Renaissance man'.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Paul_Robeson&quot; rel=&quot;nofollow&quot;&gt;Paul Robeson&lt;/a&gt; (1898–1976) was an internationally renowned American bass-baritone concert singer, actor of film and stage, All-American and professional athlete, writer, multi-lingual orator, scholar and lawyer who was also noted for his wide-ranging social justice activism.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Vladimir_Nabokov&quot; rel=&quot;nofollow&quot;&gt;Vladimir Nabokov&lt;/a&gt; (1899–1977) was a Russian-born novelist, lepidopterist, professor, and chess problem composer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Friedrich_Hayek&quot; rel=&quot;nofollow&quot;&gt;Friedrich Hayek&lt;/a&gt; (1899–1992) was a Austrian economist, philosopher, psychologist and political scientist important contributor to the Austrian School of economics and winner of the Nobel Memorial Prize in Economic Sciences in 1974.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/John_von_Neumann&quot; rel=&quot;nofollow&quot;&gt;John von Neumann&lt;/a&gt; (1903–1957) was a Hungarian and later American pure and applied mathematician, physicist, inventor, polymath, and polyglot. He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and fluid dynamics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics. He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor, and the digital computer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Howard_Hughes&quot; rel=&quot;nofollow&quot;&gt;Howard Hughes&lt;/a&gt; (1905–1976) was an American business tycoon, investor, aviator, aerospace engineer, inventor, filmmaker and philanthropist. His life story provided the inspiration for the film &lt;em&gt;The Aviator&lt;/em&gt; starring Leonardo Dicaprio as Hughes, which was released in 2004. His legacy is maintained through the Howard Hughes Medical Institute.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jacob_Bronowski&quot; rel=&quot;nofollow&quot;&gt;Jacob Bronowski&lt;/a&gt; (1908–1974) was a mathematician, poet, inventor, humanist and historian of science. Author of &lt;em&gt;The Ascent of Man&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Herbert_A._Simon&quot; rel=&quot;nofollow&quot;&gt;Herbert A. Simon&lt;/a&gt; (1916–2001) is famed for work in psychology and computer science, philosophy of science, a leader in artificial intelligence, and a Nobel Prize winner in Economics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Isaac_Asimov&quot; rel=&quot;nofollow&quot;&gt;Isaac Asimov&lt;/a&gt; (1920–1992) having written or edited more than 500 books and an estimated 9,000 letters and postcards, his works have been published in nine of the ten major categories of the Dewey Decimal System. &quot;While most prolific writers tend to concentrate almost exclusively on a single genre, Asimov was a polymath who wrote widely on a variety of subjects.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Boris_Vian&quot; rel=&quot;nofollow&quot;&gt;Boris Vian&lt;/a&gt; (1920–1959) was a French polymath, writer, poet, musician, singer, translator, critic, actor, inventor and engineer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Noam_Chomsky&quot; rel=&quot;nofollow&quot;&gt;Noam Chomsky&lt;/a&gt; (born 1928) is an American linguist, philosopher, cognitive scientist, historian, and activist. He is an Institute Professor and Professor (Emeritus) in the Department of Linguistics &amp;amp; Philosophy at MIT, where he has worked for over 50 years. Chomsky has been described as the &quot;father of modern linguistics&quot; and a major figure of analytic philosophy. His work has influenced fields such as computer science, mathematics, and psychology.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Desmond_Morris&quot; rel=&quot;nofollow&quot;&gt;Desmond Morris&lt;/a&gt; (born 1928) is a British zoologist and ethologist, who is also a surrealist painter, popular author, and television presenter. &quot;Desmond Morris already has a reputation for being something of a polymath. Zoologist, Man Watcher, gesture expert and even television celebrity...&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Umberto_Eco&quot; rel=&quot;nofollow&quot;&gt;Umberto Eco&lt;/a&gt; (born 1932) is a professor of semiotics, novelist, and an expert on literature, medieval philosophy and pop culture.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jonathan_Miller&quot; rel=&quot;nofollow&quot;&gt;Jonathan Miller&lt;/a&gt; (born 1934) is a British theatre and opera director, author, television presenter, humorist, and sculptor, who trained as a doctor. &quot;The polite polymath.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jared_Diamond&quot; rel=&quot;nofollow&quot;&gt;Jared Diamond&lt;/a&gt; (born 1937) is an anthropologist, geographer, physiologist, author, ornithologist.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Douglas_Hofstadter&quot; rel=&quot;nofollow&quot;&gt;Douglas Hofstadter&lt;/a&gt; (born 1945) is a professor of cognitive science, author, philosopher, aesthetist, academic musician and artist, mathematician, physicist.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Hamlet_Isakhanli&quot; rel=&quot;nofollow&quot;&gt;Hamlet Isakhanli&lt;/a&gt; (born 1948) is a professor of mathematical-physical sciences, mathematician, poet, social scientist, scholar of history of education, literature and culture, founder of university, founder and co-founder of various organizations, entrepreneur, writer, translator.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Steve_Jobs&quot; rel=&quot;nofollow&quot;&gt;Steve Jobs&lt;/a&gt; (1955–2011) was an entrepreneur, innovator, inventor, designer. CEO of Pixar &amp;amp; Apple Inc.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="100" LastActivityDate="2015-06-01T17:21:16.780" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2407" PostTypeId="1" CreationDate="2015-06-02T22:55:18.813" Score="8" ViewCount="467" Body="&lt;p&gt;Viète's equations are used in some proofs of the Basel problem, which was allegedly solved by Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Viète's equations include the following: given a polynomial,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a_0 + a_1x+a_2x^2 + ... + a_nx^n$$ with roots $\alpha_1, \alpha_2,...\alpha_n,$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a_{n-1}= -a_n\,(\alpha_1+\alpha_2\,+...+\,\alpha_n).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be applied to a polynomial of the form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$b_0 - b_1\, x^2 +b_2\,x^4-\cdots+(-1)^n\,b_n\,x^{2n},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in which case we find that, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$b_0 - b_1\, x^2 +b_2\,x^4-\cdots+(-1)^n\,b_n\,x^{2n}=b_0\left(1-\frac{x^2}{\beta_1^2}\right)\,\left(1-\frac{x^2}{\beta_2^2}\right)\, \cdots\,\left(1-\frac{x^2}{\beta_n^2}\right).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$b_1 = b_0\,\left(\frac{1}{\beta_1^2}\right)+\,\left(\frac{1}{\beta_2^2}\right)\,+\cdots+\,\left(\frac{1}{\beta_n^2}\right).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This form allows expressing the second coefficient in:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{sin (x)}{x}= 1 - \frac{x^2}{2\cdot 3}+\frac{x^4}{2\cdot 3\cdot 4\cdot 5}-\frac{x^6}{2\cdot3\cdots7}+\cdots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{1}{2\cdot3}= \frac{1}{\pi^2}+\frac{1}{4\pi^2}+\frac{1}{9\pi^2}+\cdots.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is whether Euler truly resorted to Viète's equations, and how big of a mathematical figure Viète name truly is in his own right.&lt;/p&gt;&#xA;" OwnerUserId="2352" LastActivityDate="2015-06-03T22:29:20.730" Title="Viète's Relevance and his Connection to Euler" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2408" PostTypeId="2" ParentId="2407" CreationDate="2015-06-03T01:03:34.767" Score="9" Body="&lt;p&gt;According to &lt;a href=&quot;http://www.ams.org/journals/bull/2007-44-04/S0273-0979-07-01175-5/S0273-0979-07-01175-5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Varadarajan's Euler and his Work on Infinite Series&lt;/a&gt; (p.518) &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Euler’s idea was based on an audacious generalization of Newton’s formula for&#xA;  the sums of powers of the roots of a polynomial to the case when the polynomial&#xA;  was replaced by a power series&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;formula&quot; is the one relating power sums to elementary symmetric polynomials, and &lt;a href=&quot;http://en.wikipedia.org/wiki/Vieta%27s_formulas#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Girard is usually credited&lt;/a&gt; with deriving it in full generality in 1629 (Newton independently rederived it in 1666). Simpler formulas relating elementary symmetric polynomials of the roots to coefficients of the polynomial were given by Viète (Vieta in Latin) for positive real roots, and in general again by Girard. While Newton may have been unaware of Girard &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...to some extent Newton cut his mathematical teeth on Viète, under the keen eye of Isaac Barrow, and Vol. 1 of D.T. Whiteside's monumental athematical Papers of Isaac Newton contains some relevant material from Newton's notes: if nothing else, these demonstrate the calibre of the young Newton, who had transcribed some of the theorems of this work of Viète and others to aid his own understanding.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;This work&quot; is &lt;a href=&quot;http://www.17centurymaths.com/contents/Angular%20Sections.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ad Angulares Sectiones&lt;/a&gt;, which contained polynomial expansions for sine and cosine of multiple angles, among other things. It was also the basis of Viète's trigonometric solution to the irreducible case of the cubic, with three real roots, &lt;a href=&quot;https://hsm.stackexchange.com/questions/347/why-is-cardans-formula-wrongly-attributed-to-him/351#351&quot;&gt;none of which is given by the Cardano formula&lt;/a&gt; (it involves extracting cube root of a complex number, which as Bombelli showed earlier, requires solving another cubic). Viète showed that the problem reduces to angle trisection, so in modern terms all three roots can be found by inverting cosines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Viète's significance, his &lt;a href=&quot;http://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te#A_new_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isagoge (1591) did for algebra what Descartes's La Géométrie (1637) did for geometry&lt;/a&gt;, in other words, he was the father of symbolic algebra. Modern style symbolic notation, especially systematic use of letter notation for both variables and parameters, enabled wide use of algebraic manipulation rules (Viète still uses words for powers, these were symbolized by Descartes, but they are attached to variables). Familiar approach of converting problems to equations, and then solving them symbolically, was also coined by Viète. Hartshorne gives a nice description of &lt;a href=&quot;https://math.berkeley.edu/~robin/Viete/construction.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Viète applying it to classical geometric problems&lt;/a&gt; in Supplementum Geometriae (1593). &lt;a href=&quot;https://books.google.com/books?id=ApbfBwAAQBAJ&amp;amp;pg=PA146&amp;amp;lpg=PA146&amp;amp;dq=viete%27s+isagoge&amp;amp;source=bl&amp;amp;ots=SR_3I5cU3d&amp;amp;sig=mMcuObvHVwT0fTlZr-zn4ThUiG4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=hU5uVcf6NND-yQTfi4H4BQ&amp;amp;ved=0CFMQ6AEwCQ#v=onepage&amp;amp;q=viete%27s%20isagoge&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bos in Redefining Geometrical Exactness&lt;/a&gt; gives a detailed historical analysis of Viète's contributions: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was Viète who first introduced and promoted the idea that algebra was proper method for analysis of problems both in geometry, and in the theory of numbers... Viète did not see algebra as a technique concerning numbers... but as a method of symbolic calculation concerning abstract magnitudes.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For more on his influence see &lt;a href=&quot;http://logica.ugent.be/philosophica/fulltexts/87-5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Esteve's paper&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-06-03T22:29:20.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2409" PostTypeId="1" CreationDate="2015-06-03T03:00:48.033" Score="3" ViewCount="83" Body="&lt;p&gt;In my answer to &lt;a href=&quot;https://mathoverflow.net/questions/145555/why-is-there-a-connection-between-enumerative-geometry-and-nonlinear-waves/181534#181534//&quot;&gt;MO-Q: Enumerative geometry and nonlinear waves&lt;/a&gt;, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for the compositional inverse of a formal power series, noted by Loday in his paper referenced in the answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the first to note this relation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ancillary question: Who was the first to note the relation between the dissections of convex polygons (or, closely related, Cayley trees depicting the repeated action of $g(x)D_x$) and compositional inversion?&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-06-03T03:06:28.277" Title="First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?" Tags="|mathematics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2411" PostTypeId="1" AcceptedAnswerId="2420" CreationDate="2015-06-06T19:39:29.423" Score="4" ViewCount="319" Body="&lt;p&gt;In Einstein's biography written by Walter Isaacson, by the way wonderful book that I recommend to everyone, there are few pages referring vaguely to his first child with Mileva Marić, a daughter named Lieserl being born a year before they married. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are only few speculations mentioned to the effect that the child was born out of wedlock, and a possibility of some kind of disability, which made the couple send the child to Mileva's parents in Serbia, whereafter the track is lost.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there more information related to this? What impact did social norms of the time, and being forced to abandon his child, have on Einstein 's scientific work?    &lt;/p&gt;&#xA;" OwnerUserId="2351" LastEditorUserId="55" LastEditDate="2015-06-07T23:26:25.943" LastActivityDate="2015-06-09T02:27:16.320" Title="How did the loss of Einstein's first child affect his work?" Tags="|biographical-details|social-context|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2412" PostTypeId="1" AcceptedAnswerId="2417" CreationDate="2015-06-06T21:26:13.573" Score="3" ViewCount="163" Body="&lt;p&gt;It's a quite common belief that emotions are a factor that may cause cancer. I don't &lt;em&gt;believe&lt;/em&gt; this can be scientifically verified, which makes the idea just a myth. I'd like to know where does it come from.&lt;/p&gt;&#xA;" OwnerUserId="520" LastActivityDate="2015-06-07T21:58:26.390" Title="Origin of the idea that 'bad' emotions cause cancer" Tags="|medicine|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2413" PostTypeId="2" ParentId="2412" CreationDate="2015-06-06T23:43:44.107" Score="3" Body="&lt;h1&gt;Possibly China.&lt;/h1&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.itmonline.org/arts/cancemo.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; seems to take the idea that emotions can contribute to illness quite seriously, but it does talk a bit about the history of the idea.&lt;/p&gt;&#xA;&lt;p&gt;According to the author, traditional Chinese medicine, going back a couple thousands years (that time given &lt;a href=&quot;http://www.itmonline.org/arts/cancer.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), states that emotions can influence diseases of all source. There are some books written today that support the theory, and they all focus on these guiding principles. Though the exact supposed date of the idea is impossible to determine, there is evidence that a 16th/17th century doctor may have had it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In their section on treatment of breast cancer, the authors refer to a discussion in a Ming Dynasty text by the surgeon Chen Shigong (1555-1636 A.D.) indicating that breast cancer &amp;quot;results from anxiety, emotional depression, and overthinking which impairs the liver, spleen, and heart and causes the obstruction of the channels.&amp;quot; This text is also mentioned in The Treatment of Cancer by Integrated Chinese-Western Medicine (4), translated this way:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Breast cancer is due to worry and melancholy. Lots of ideas hanging around make one feel dissatisfied. Perverse flow of liver qi to the spleen leads to the obstruction of the channels and collaterals and congealations due to excessive accumulation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The page is nicely cited, though some of the sources might not be the best, given the idea they subscribe to. Just because their medical ideas might be wrong doesn't preclude their historical research from being wrong, though.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-06-06T23:43:44.107" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2414" PostTypeId="1" AcceptedAnswerId="2416" CreationDate="2015-06-07T15:37:07.720" Score="7" ViewCount="827" Body="&lt;p&gt;Everyone will encounter the notion of &lt;a href=&quot;http://en.wikipedia.org/wiki/Connection_(mathematics)#Historical_survey_of_connections&quot; rel=&quot;noreferrer&quot;&gt;connection&lt;/a&gt; in differential geometry. But who gave this name of connection( or affine connection)? Why is this derivative operator called connection? What object does it connect? I'm just confused about the origin of the name. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="2411" LastActivityDate="2015-06-28T09:11:45.720" Title="The origin of the name &quot;connection&quot; in differential geometry" Tags="|differential-geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2416" PostTypeId="2" ParentId="2414" CreationDate="2015-06-07T20:51:00.830" Score="7" Body="&lt;p&gt;According to &lt;a href=&quot;http://soar.wichita.edu/bitstream/handle/10057/3953/t11015_Freeman.pdf&quot; rel=&quot;nofollow&quot;&gt;Freeman's Historical Overview of Connections in Geometry&lt;/a&gt; Hermann Weyl introduced the term &quot;connection&quot; in his 1918 text Reine Infinitesimal Geometrie. His definition explains the connection part, affine connection is &quot;&lt;em&gt;that which determines into which vector at $P'$ a vector at an infinitesimally close point $P$ will transform under parallel displacement from $P$ to $P'$&lt;/em&gt;&quot;. The affine part comes from Weyl requiring that the &quot;parallel displacement&quot; be an affine transformation, i.e. preserve collinearity and ratios of distances but not necessarily distances themselves, from &quot;totality of vectors&quot; at $P$ to that at $P'$. The year before Levi-Civita (and independently Hessenberg and Schouten) introduced the notion of parallel directions and gave a geometric interpretation of the Riemann curvature in terms of parallel transport, writing &quot;&lt;em&gt;once the law is known according to which one passes from a point to a point infinitely close to it, one is able immediately to accomplish the displacement of parallel directions along any arbitrary curve $C$&lt;/em&gt;&quot;. This is the first expression of the idea, and connections associated to Riemannian metrics are now called Levi-Civita connections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formally Riemannian connections appeared much earlier, but relation to parallelism remained unknown. Back in 1869 Christoffel introduced the symbols now named after him, that in modern terminology represent coefficients of a connection. However, he was interested only in classifying invariants and &quot;&lt;em&gt;conditions that differential expressions of degree 2 must verify to be able to transform one into another by a change of variables&lt;/em&gt;&quot;, not in any geometric interpretations. In 1890s Ricci-Curbastro noticed that because of their transformation properties Christoffel symbols of a  Riemannian metric could be used to create an &quot;absolute differential calculus&quot;, now known as tensor calculus, but he used the terms &quot;covariant and contravariant derivatives&quot;, not &quot;connections&quot;. In 1901 Ricci and his student Levi-Civita wrote a long expose of the new calculus, but according to Levi-Civita, &quot;&lt;em&gt;for many more years it was used almost exclusively by its inventor and a few of his students&lt;/em&gt;&quot;, until in general relativity &quot;&lt;em&gt;Ricci's calculus revealed itself to be not only useful but truly indispensable&lt;/em&gt;&quot;. By the way, Levi-Civita corresponded with Einstein extensively after 1916 helping him with the technical side of tensor calculus as it applies to general relativity.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-15T22:14:41.273" LastActivityDate="2015-06-15T22:14:41.273" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2417" PostTypeId="2" ParentId="2412" CreationDate="2015-06-07T21:58:26.390" Score="4" Body="&lt;p&gt;I'll add to HDE's answer that modern spread of the idea that emotional stress &quot;causes&quot; cancer is largely due to &quot;eclectic medical doctor&quot; Eli Jones, who named &quot;worriment of mind&quot; number one cause of cancer in his &lt;a href=&quot;https://books.google.com/books?id=qaxuIglnhZoC&amp;amp;pg=PA15&amp;amp;source=gbs_toc_r&amp;amp;cad=3#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908 book&lt;/a&gt;. It is clear from the context that he meant it as a number one cause of cancer &lt;em&gt;spread&lt;/em&gt;, but the distinction is lost on &lt;a href=&quot;http://www.healthbanquet.com/eli-jones.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern adherents&lt;/a&gt;, who also claim the good doctor had 95% success rate in curing cancer if &quot;taken on by him before any other form of treatment&quot;, &quot;at one time, had fifteen physicians under his treatment&quot;, and was known as a &quot;doctor's doctor&quot;. The guy supposedly was a miracle worker, and has a cult following today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The myth also fits well with &quot;holistic&quot; view of health common in &quot;eclectic&quot; medicine. Incidentally, it draws on &lt;a href=&quot;http://www.cancer.org/cancer/cancerbasics/thehistoryofcancer/the-history-of-cancer-cancer-causes-theories-throughout-history&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hippocrates's original theory&lt;/a&gt; (c. 400 BC) that cancer is caused by excess of black bile, one of the four humors. Once Avicenna (980–1037) &lt;a href=&quot;https://hsm.stackexchange.com/questions/535/four-temperaments-and-medical-methodology&quot;&gt;linked humors&lt;/a&gt; to &quot;emotional aspects, mental capacity, moral attitudes, self-awareness, movements and dreams&quot; emotional disturbance as a cause of imbalance became plausible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This being said, a link between emotional state and cancer spread is not a myth. According to &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/21142861&quot; rel=&quot;nofollow noreferrer&quot;&gt;Impact of Stress on Cancer Metastasis paper&lt;/a&gt;:&quot;&lt;em&gt;Whereas evidence for the role of psychosocial factors in cancer initiation is limited and somewhat contradictory, support is stronger for links between psychological factors such as stress, depression and social isolation and disease progression&lt;/em&gt;&quot;. There is evidence that stress hormones may accelerate tumor growth or facilitate metastasis. For example, a &lt;a href=&quot;http://www.medicalnewstoday.com/articles/265254.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;recent study&lt;/a&gt; showed that &quot;&lt;em&gt;cancer cells are able to switch on ATF3 [gene] in immune cells that have been summoned to tumor sites. The result is ATF3 then causes the immune cells to malfunction and allow cancer cells to escape from the tumor and spread to other parts of the body&lt;/em&gt;&quot;.  &lt;a href=&quot;http://www.cancer.gov/about-cancer/coping/feelings/stress-fact-sheet&quot; rel=&quot;nofollow noreferrer&quot;&gt;National Cancer Institute&lt;/a&gt; discusses other mechanisms. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-06-07T21:58:26.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2418" PostTypeId="1" AcceptedAnswerId="2419" CreationDate="2015-06-07T23:02:07.453" Score="13" ViewCount="643" Body="&lt;p&gt;&lt;em&gt;Kontinuum&lt;/em&gt; is German for continuum, but Cantor used $\mathfrak{c}$.&#xA;&lt;hr /&gt;&#xA;&lt;strong&gt;&lt;em&gt;Revision&lt;/em&gt;&lt;/strong&gt;. J.W.Perry questions whether or not Cantor ever in fact&#xA;used the symbol $\mathfrak{c}$. I must admit I just assumed that he did&#xA;from statements that say that Cantor proved the cardinality of $\mathfrak{c}$&#xA;is larger than that of $\aleph_0$ (and he certainly introduced the $\aleph$ notation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the proper question is now: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who introduced the symbol $\mathfrak{c}$ for the continuum,&#xA;  and when?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="606" LastEditorUserId="4251" LastEditDate="2020-11-18T01:47:13.790" LastActivityDate="2020-11-18T01:47:13.790" Title="Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?" Tags="|mathematics|notation|mathematical-logic|set-theory|cantor|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2419" PostTypeId="2" ParentId="2418" CreationDate="2015-06-07T23:37:36.233" Score="7" Body="&lt;p&gt;Cantor did not invent the term, it goes back to antiquity. &amp;quot;&lt;em&gt;&lt;a href=&quot;http://www.latinlanguage.org/latin/history.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Latin was a lingua franca&lt;/a&gt;, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th&lt;/em&gt;&amp;quot;. In Latin, French and English continuum starts with &amp;quot;c&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Cantor saw himself as confronting the dogma of medieval scholasts &amp;quot;infinitum actu non datur&amp;quot; (actual infinity is not given), which originates in Aristotle, and the arguments supporting it, such as &amp;quot;annihilation of numbers&amp;quot; by infinity. From &lt;a href=&quot;https://math.dartmouth.edu/%7Ematc/Readers/HowManyAngels/Cantor/Cantor.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dauben's book&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Cantor condemned this kind of argument, however, on the grounds that it was fallacious to assume that infinite numbers must exhibit the same arithmetic characteristics as did finite numbers... Having dealt with Aristotle and the scholastics, Cantor undertook an investigation of other works by some of the most impressive thinkers of the seventeenth century, a century that witnessed serious and often profound analysis of the nature of infinity. He suggested that anyone interested in such things would do well to consult Locke, Descartes, Spinoza, and Leibniz, while Hobbes and Berkeley were highly recommended as additional reading.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not many of Cantor's contemporaries were interested in the subtleties of actually infinite (one exception is Dedekind), so most of Cantor's intellectual companions wrote, or were translated into, Latin.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; After J.W. Perry's comment I looked through Medvedev's book Early History of the Axiom of Choice, where he quotes Cantor's set theoretic papers and letters to Dedekind from 1872 to 1899, and also did not find any instance of him using &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{c}$&lt;/span&gt;. The earliest usage Medvedev quotes is from Bernstein's paper Über die Reihe der transfiniten Ordnungszahlen in Mathematische Annalen, v.60 (1905), 187-193, where he writes (my translation, &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0060&amp;amp;DMDID=DMDLOG_0023&amp;amp;LOGID=LOG_0023&amp;amp;PHYSID=PHYS_0204&quot; rel=&quot;nofollow noreferrer&quot;&gt;direct link to the paper, see p.192&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Although it remains very &lt;strong&gt;likely&lt;/strong&gt; that &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}=\mathfrak{c}=\aleph_1$&lt;/span&gt;, so far nobody managed to prove that &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}&amp;gt;2^{\aleph_1}$&lt;/span&gt; [sic!]. Therefore, it is not ruled out that &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}=2^{\aleph}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\aleph$&lt;/span&gt; is any aleph. In that case &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}$&lt;/span&gt; would contain all alephs as subsets...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In his 1901 dissertation Untersuchungen aus der Mengenlehre, later published in Mathematische Annalen, v.61 (1905), 117-155, but circulated among experts earlier, Bernstein uses Latin transcription for &amp;quot;c&amp;quot; instead, writing &amp;quot;&lt;em&gt;Bezeichnet &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; de Mächtigkeit des Kontinuums...&lt;/em&gt;&amp;quot; (Let &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; refer to the cardinality of the continuum), despite &amp;quot;Kontinuum&amp;quot; with a &amp;quot;K&amp;quot;. &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0061&amp;amp;DMDID=DMDLOG_0015&amp;amp;LOGID=LOG_0015&amp;amp;PHYSID=PHYS_0141&quot; rel=&quot;nofollow noreferrer&quot;&gt;Direct link, see p. 133&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-10T05:27:41.910" LastActivityDate="2020-08-10T05:27:41.910" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="2420" PostTypeId="2" ParentId="2411" CreationDate="2015-06-08T01:16:30.707" Score="4" Body="&lt;p&gt;The only piece we know for sure is a cryptic passage in 1903 letter from Einstein to Mileva:&quot;&lt;em&gt;I am very sorry about what has happened to Lieserl. Scarlet fever often leaves some lasting trace behind&lt;/em&gt;&quot;. An amateur scholar Michele Zackheim got fascinated by the incident, and did a copious amount of research over the course of five years, visiting Switzerland, Germany, England, Hungary, and Serbia, including Mileva's ancestral villages, where she talked to anyone close to her family and went through countless baptismal records and archives. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Her resulting book &lt;a href=&quot;http://content.time.com/time/magazine/article/0,9171,31490,00.html&quot; rel=&quot;nofollow&quot;&gt;Einstein's Daughter: The Search for Lieserl&lt;/a&gt; &quot;&lt;em&gt;is a colorful glimpse of rural Serbian culture, with its patrimonial society, strong family loyalties, female subservience, slow, leisurely discourse. Zackheim does manage to eliminate a number of women as possible Lieserls, including a melodramatic Berlin actress who claimed in the 1930s to be Einstein's daughter&lt;/em&gt;&quot;. The rest is explicitly speculative:&quot;&lt;em&gt;a Greenwich Village painter turned writer, argues that the toddler was severely retarded and probably had Down syndrome. A simpleton child, in the language of the time, she would have been considered uneducable... Mileva, unable to place the little girl for adoption or send her to an orphanage, left her with her parents at their home in Serbia's rural Vojvodina region on the fertile Danube plain. And there Lieserl's life was poignantly short... the little girl died at 21 months after a bout of scarlet fever&lt;/em&gt;&quot;. Many consider this speculation plausible, but all agree that it is only a speculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the influence on Einstein's work, so little is known about the matter that it is hard to even speculate. We do know that Einstein soon entered the most productive period of his scientific life, with special relativity and photoelectric effect papers coming out in 1905. Einstein's stormy private life (&quot;&lt;em&gt;his flirtations, his stormy divorce from Mileva, his possible dalliance with the daughter of the woman who would become his second wife, his estrangement from his two sons, one of whom was schizophrenic&lt;/em&gt;&quot;) only came out after publication of his collected works started in 1987. One could say perhaps that he was &quot;&lt;em&gt;as roguishly independent in his personal life as he would be in his science&lt;/em&gt;&quot; in early 1900s, refusing to break up with Mileva despite his mother's warning:&quot;&lt;em&gt;If she gets a child, you'll be in a pretty mess&lt;/em&gt;&quot;. Something more substantive emerges from the &lt;a href=&quot;http://content.time.com/time/magazine/article/0,9171,1211594-3,00.html&quot; rel=&quot;nofollow&quot;&gt;letters from the time he worked on general relativity&lt;/a&gt;:&quot;&lt;em&gt;His letters... show how his personal and scientific struggles intertwined in 1915, culminating in his great triumph that fall&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-09T02:27:16.320" LastActivityDate="2015-06-09T02:27:16.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2421" PostTypeId="2" ParentId="2418" CreationDate="2015-06-08T09:01:30.747" Score="9" Body="&lt;p&gt;Latin loan words in German usually retained their &quot;c&quot; until about the end of the 19th century. The modern spellings like Kontinuum (for Continuum) and Zentrum (for Centrum) result from a fairly recent spelling reform.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-06-08T09:01:30.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2422" PostTypeId="1" CreationDate="2015-06-08T11:53:11.903" Score="4" ViewCount="653" Body="&lt;p&gt;Medicine is often traced back to &lt;a href=&quot;http://en.wikipedia.org/wiki/Hippocrates&quot; rel=&quot;nofollow&quot;&gt;Hippocrates&lt;/a&gt;, history to &lt;a href=&quot;http://en.wikipedia.org/wiki/Herodotus&quot; rel=&quot;nofollow&quot;&gt;Herodotus&lt;/a&gt;. These claims of course tell only part of the story, but I've always been fascinated by figures which inspired successors to draw a boundary between themselves and their precursors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any claims regarding the first attempt at discussing the history of science?&lt;/p&gt;&#xA;" OwnerUserId="8" LastEditorUserId="26" LastEditDate="2015-10-11T10:53:54.593" LastActivityDate="2015-10-11T10:53:54.593" Title="How far can history of science be traced back?" Tags="|ancient-greece|medicine|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2423" PostTypeId="2" ParentId="2418" CreationDate="2015-06-08T15:39:55.193" Score="3" Body="&lt;p&gt;&lt;em&gt;Long comment&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I agree with Perry's comment.&lt;/p&gt;&#xA;&lt;p&gt;In Cantor's &lt;a href=&quot;http://web.archive.org/web/20140423224341/http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&amp;amp;&amp;amp;action=pdf&amp;amp;metsFile=PPN235181684_0046&amp;amp;divID=LOG_0044&amp;amp;pagesize=original&amp;amp;pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN235181684_0046%7C&amp;amp;targetFileName=PPN235181684_0046_LOG_0044.pdf&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beiträge zur Begründung der transfiniten Mengenlehre (1)&lt;/a&gt; &lt;em&gt;Mathematische Annalen&lt;/em&gt;, 46:481–512 (1895), page 488  (and see the English (1915) translation, &lt;a href=&quot;https://books.google.it/books?id=flb_nnYJJYkC&amp;amp;pg=PA96&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 96&lt;/a&gt; ) the symbol used for &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_0}$&lt;/span&gt; is clearly &lt;strong&gt;not&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathfrak c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In previous pages &lt;span class=&quot;math-container&quot;&gt;$\mathfrak a, \mathfrak b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathfrak c$&lt;/span&gt; are used for &lt;em&gt;cardinal&lt;/em&gt; numbers.&lt;/p&gt;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gregory Moore, &lt;a href=&quot;https://books.google.it/books?id=3RLGKcEjVIoC&amp;amp;pg=PA42&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zermelo's Axiom of Choice : Its Origin Development and Influence&lt;/a&gt; (1982 - also Dover reprint), page 42.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Some early sources :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Felix Hausdorff, &lt;a href=&quot;https://books.google.it/books?id=yvVIdH16k0YC&amp;amp;pg=PA44&quot; rel=&quot;nofollow noreferrer&quot;&gt;Set theory&lt;/a&gt; (English transl of the 3rd German ed : 1937), page 44, uses &lt;span class=&quot;math-container&quot;&gt;$\aleph$&lt;/span&gt; [without subscript] for &amp;quot;the cardinality of the continuum&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Wacław Sierpiński, &lt;a href=&quot;https://books.google.it/books?id=gwDvAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cardinal and ordinal numbers&lt;/a&gt; (English transl of the 3rd Polish ed : 1928), page 372&#xA;: &amp;quot;The power of the set &lt;span class=&quot;math-container&quot;&gt;$Z_2$&lt;/span&gt; of all numbers of the 2-nd class is denoted by &lt;span class=&quot;math-container&quot;&gt;$\aleph_1$&lt;/span&gt; (read &lt;em&gt;aleph-one&lt;/em&gt;).&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;According to :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Michael Hallett, &lt;a href=&quot;https://books.google.it/books?id=TM3AKPYdQVgC&amp;amp;pg=PA68&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantorian set theory and limitation of size&lt;/a&gt; (1986), page 68 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the aleph notation was first introduced to the mathematical public in Cantor's &lt;em&gt;Beiträge I&lt;/em&gt; [1895]. &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt; is characterized as 'the power of the natural numbers', i.e. of the [&lt;em&gt;first&lt;/em&gt; number] class (I). [...] Thus, in &lt;em&gt;Beiträge II&lt;/em&gt; [1897], &lt;span class=&quot;math-container&quot;&gt;$\aleph_1$&lt;/span&gt; is introduced as the power of [the &lt;em&gt;second&lt;/em&gt; number class] (II). The aleph&#xA;notation appears to have been referred to first in a letter to Vivanti of 13 December 1893, though here Cantor has &lt;span class=&quot;math-container&quot;&gt;$\aleph_1, \aleph_2$&lt;/span&gt;, etc. in place of the later &lt;span class=&quot;math-container&quot;&gt;$\aleph_0, \aleph_1$&lt;/span&gt; etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Possible sources :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Henry_Young&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Henry Young&lt;/a&gt; &amp;amp; Grace Chisholm Young, &lt;a href=&quot;https://books.google.it/books?id=TIzfPwAACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Theory of Sets of Points&lt;/a&gt; (1906), page 46 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We shall now see that, in order to measure a perfect set we only have to take the linear continuum itself as unit; in this way we obtain a new potency, which is more than countable, viz. the potency &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; of the linear continuum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Vermilye_Huntington&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward V.Huntington&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=pPjDD5QnRd4C&amp;amp;pg=PA80&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Continuum, and Other Types of Serial Order: With an Introduction to Cantor's Transfinite Numbers&lt;/a&gt; (1917 - also Dover reprint), page 80 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Perhaps the most famous result in this algebra [tha algebra of the cardinal numbers] is the formula [ref in footnote to &lt;em&gt;Beiträge I&lt;/em&gt;, page 488]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$c=2^{\aleph_0}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; stands for the cardinal number of the continuum [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Finally, see : Patrick Suppes, &lt;a href=&quot;https://books.google.it/books?id=skTCAgAAQBAJ&amp;amp;pg=PA193&quot; rel=&quot;nofollow noreferrer&quot;&gt;Axiomatic set theory&lt;/a&gt; (1960 - also Dover reprint), page 193 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$\mathfrak c$&lt;/span&gt; is the standard one [&lt;em&gt;sic !&lt;/em&gt;] for the cardinality of the continuum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-09-12T13:33:35.003" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2424" PostTypeId="1" AcceptedAnswerId="2425" CreationDate="2015-06-08T18:39:22.293" Score="8" ViewCount="249" Body="&lt;p&gt;So, I was wondering, how did Mersenne come up with the formulae $2^p-1$? Do we have any ideas of how it came to be?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="26" LastEditDate="2015-06-09T10:03:41.520" LastActivityDate="2022-01-20T15:00:45.927" Title="How did Mersenne discover Mersenne primes?" Tags="|number-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2425" PostTypeId="2" ParentId="2424" CreationDate="2015-06-09T02:48:13.877" Score="9" Body="&lt;p&gt;In &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX36.html&quot;&gt;Elements IX.36&lt;/a&gt; Euclid proves:&quot;&lt;em&gt;If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect&lt;/em&gt;&quot;. Translation: if $1+2+\dots+2^{n-1}=2^n-1$ is prime then $2^{n-1}(2^n-1)$ is perfect, meaning that it is equal to the sum of its proper divisors like $6=1+2+3$. It is straightforward today, and known to Fermat and Mersenne, that if $2^n-1$ is prime so is $n$, because otherwise $2^n-1$ factorizes as a difference of powers. In other words, find a Mersenne prime and you find a perfect number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For religious reasons perfect numbers fascinated Pythagoreans, the theorem itself may go back to them, and it came to fascinate mathematicians and numerologists alike for centuries thereafter. A Neo-Pythagorean Nichomachus of Gerasa (c. 100 AD) lists four &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Perfect_numbers.html&quot;&gt;perfect numbers known in antiquity&lt;/a&gt;:&quot;...&lt;em&gt;Only one is found among the units, only one other among the tens, and a third in the rank of the hundreds, and a fourth within the limits of the thousands... And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even&lt;/em&gt;&quot;. They are $6, 28, 496, 8128$, corresponding to Mersenne primes $2,3,5,7$, but two generalizations Nichomachus implies are incorrect (even perfect numbers do end in 6 or 8, but not alternatively). In 17th century the interest in number theory was rekindled after the translation of Diophantus's Arithmetica, and Mersenne (1644) was the first to extend Nichomachus's list. Eventually Euler proved that all even perfect numbers are given by Euclid's formula. We still do not know if there are infinitely many of them, or if there are any odd ones.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-09T17:46:55.813" LastActivityDate="2015-06-09T17:46:55.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2426" PostTypeId="2" ParentId="2422" CreationDate="2015-06-09T03:21:49.877" Score="6" Body="&lt;p&gt;I agree with Mauro's comment that Aristotle was first to take systematic account of the work of his predecessors. Many of his scientific works, Physics, Meteorology, etc., start with detailed description of his predecessors' opinions on the subject, along with Aristotle's own analysis and criticism. Early sources are mostly known in fragments, but nothing comparable appears even in Plato's dialogues or Euclid's Elements (Euclid mentions no names whatsoever despite organizing earlier material, and writing after Aristotle).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the first exclusive historian of science (that we know of) was &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Eudemus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eudemus of Rhodes (c. 370-300 BC)&lt;/a&gt;, a student of Aristotle's, who upon his encouragement wrote History of Arithmetic, History of Geometry and History of Astronomy (exclusive in the sense of writing first works specifically dedicated to history of science, not in the sense of writing them only). Not one of them survives, and we know next to nothing about the first, but the last two are our main source of information on Greek geometry and astronomy before 4th century BC. Through quoted fragments and retellings by later commentators such as Proclus, Pappus, Eutocius, Theon of Smyrna, Simplicius, Diogenes Laërtius, Clement of Alexandria and others, Eudemus had that kind of reach. In particular, largely due to him we know some details about geometric and astronomical studies of Anaximander, Anaximenes, Pythagoreans, Oenopides and Hippocrates of Chios, Archytas and Eudoxus. And largely due to him we know more about astronomy before Hipparchus than about it between Hipparchus and Ptolemy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a term for the study of history of science, historiography of science. Wikipedia even has an &lt;a href=&quot;http://en.wikipedia.org/wiki/Historiography_of_science&quot; rel=&quot;nofollow noreferrer&quot;&gt;article about it&lt;/a&gt;, although the earliest work they mention is Whewell's History of the Inductive Sciences (1837). However, there is an excellent book &lt;a href=&quot;http://www.degruyter.com/view/product/36576&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origin of the History of Science in Classical Antiquity by Zhmud&lt;/a&gt; that goes much further back. Here is from the abstract:&#xA;&lt;em&gt;The first part of the book presents an analysis of those trends in Presocratic, Sophistic and Platonic thought that contributed to the development of the history of science. The second part provides a detailed study of Eudemus' writings in their relationship with the scientific literature of his time, Aristotelian philosophy and the other historiographic genres practiced at the Lyceum: biography, medical and natural-philosophical doxography. Although Peripatetic historiography of science failed in establishing itself as a continuous genre, it greatly contributed both to the birth of the Arabic medieval historiography of science and to the development of this genre in Europe in the 16th-18th centuries&lt;/em&gt;&quot;. So perhaps Eudemus is the father of historiography of science. If we had a logo he should be on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/7e96k.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, it did not take in antiquity. Zhmud writes:&quot;&lt;em&gt;Taken as a whole, the surviving texts of Hellenistic mathematicians and astronomers demonstrate that they could not care less about presenting their predecessors’ contribution in a historical perspective. What interested them as a rule was their precursors’ errors and failures, rather than the problems the latter had succeeded in solving&lt;/em&gt;&quot;. Among Alexandrian scientists Eratosthenes seems to be the only one to follow Aristotle's example:&quot;&lt;em&gt;Eratosthenes approached&#xA;geography as a scientist and started it, therefore, not with Homer, as&#xA;Strabo would later insist, but with Anaximander and Hecataeus of Miletus... Joining praise with criticism, Eratosthenes did his best to point out his forerunners’ individual contributions to geography...&lt;/em&gt;&quot; However, &quot;&lt;em&gt;unlike mathematicians and astronomers, the Greek engineers of the third century, Ctesibius, Philo of Byzantium, and Biton, the authors of treatises on siege and shooting engines, show a much greater interest in the history of their discipline&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-10T01:05:55.290" LastActivityDate="2015-06-10T01:05:55.290" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2427" PostTypeId="1" CreationDate="2015-06-09T21:59:54.903" Score="9" ViewCount="243" Body="&lt;p&gt;Was science really the driving force behind the Apollo program?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or was the &quot;race to the moon&quot; the primary reason and the science objectives were put together as a secondary step?&lt;/p&gt;&#xA;" OwnerUserId="2424" LastActivityDate="2015-06-28T08:49:44.147" Title="Was science the driving force behind the Apollo program?" Tags="|astronomy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2428" PostTypeId="1" CreationDate="2015-06-10T03:25:22.287" Score="5" ViewCount="674" Body="&lt;p&gt;I'd like to ask a similar question from &lt;a href=&quot;https://math.stackexchange.com/q/699002&quot;&gt;Math.SE&lt;/a&gt; for the natural sciences (physics, chemistry, biology and allied disciplines).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are examples of scientific results that were discovered surprisingly late in history? For instance, a paper where all the technology has been available for years, and the theoretical underpinnings necessary are not recent, so in principle the same research could have been conceived and published years ago. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am particularly interested in &lt;strong&gt;actual publications&lt;/strong&gt; in peer-reviewed journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By way of example, I think &lt;a href=&quot;https://en.wikipedia.org/wiki/Gibson_assembly&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gibson assembly&lt;/a&gt; would be a good one: It is a simple modification of the standard PCR technique invented in the 70s. I imagine by the end of the 80s polymerases and buffers had been developed to the point that they could handle a simple Gibson assembly, and sequencing to confirm integrity of the product (or alternatively screening a library of products until a correct one is found) would also have been feasible. Whether the scientific community at large would have been more or less receptive to this idea at the time is another matter -- and something I specifically want to disregard for this question (ie. being &quot;late&quot; means late in terms of only technology and theory, not politics, funding and similar non-scientific prerequisites of scientific research).&lt;/p&gt;&#xA;" OwnerUserId="544" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2023-04-10T11:38:46.387" Title="Scientific discoveries that were made &quot;late&quot;" Tags="|discoveries|reference-request|examples|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2429" PostTypeId="2" ParentId="2427" CreationDate="2015-06-10T15:20:27.907" Score="8" Body="&lt;p&gt;The driving force behind the Apollo program was leadership in space over the Soviet Union. Kennedy asked Vice President Johnson to investigate whether &quot;we have a chance of beating the Soviets by placing a laboratory in space, or by a trip around the moon, or by a rocket to land on the moon, or by a rocket to go to the moon and back with a man. Is there any other space program which promises dramatic results in which we could win?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;Source: &lt;a href=&quot;http://www.hq.nasa.gov/office/pao/History/Apollomon/apollo1.pdf&quot;&gt;John F. Kennedy, Memorandum for Vice President, 20 April 1961&lt;/a&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt;&#xA;The decision to go to the Moon was political, beating the Soviets. NASA recommended that Kennedy pursue the manned mission to the Moon because of a perceived Soviet Union primacy over the US in terms of large rocket technology. A target after the mid 1960s was needed to give the US a chance at catching up with and surpassing the Soviet Union.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Science was an add-on, only done after the decision was made to go to the Moon.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-06-10T15:20:27.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2430" PostTypeId="2" ParentId="2428" CreationDate="2015-06-11T10:42:09.080" Score="2" Body="&lt;p&gt;One &amp;quot;late&amp;quot; contribution might be the discovery in the late 20th c that Rome's &lt;em&gt;Pantheon&lt;/em&gt;, built in the 2nd c AD, was not &amp;quot;replicable&amp;quot; using modern technology and construction principles without the aid of a computer. In other words, the intricacies of design and materials that went into the Pantheon's construction were so rigorously conceived that our civilization's advanced machinery could not emulate the Romans' skill without the invention and addition of computer-aided design.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;On the Structure of the Roman Pantheon&lt;/em&gt;&#xA;Robert Mark and Paul Hutchinson&#xA;&lt;em&gt;The Art Bulletin&lt;/em&gt;&#xA;Vol. 68, No. 1 (Mar., 1986), pp. 24-34&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Pantheon,_Rome&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://en.wikipedia.org/wiki/Pantheon,_Rome&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This video does much to explain just how remarkable the construction and design of the Pantheon was...and still is, &amp;quot;even for modern times.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Pantheon: Ancient High Technology on the Next Level&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=wYgne_mxjUs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=wYgne_mxjUs&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2365" LastEditorUserId="2365" LastEditDate="2023-04-10T11:38:46.387" LastActivityDate="2023-04-10T11:38:46.387" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="2431" PostTypeId="2" ParentId="19" CreationDate="2015-06-11T10:54:42.930" Score="6" Body="&lt;p&gt;FamousScientists.org has a thread titled, &quot;7 Great Examples of Scientific Discoveries Made in Dreams.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.famousscientists.org/7-great-examples-of-scientific-discoveries-made-in-dreams/&quot; rel=&quot;noreferrer&quot;&gt;http://www.famousscientists.org/7-great-examples-of-scientific-discoveries-made-in-dreams/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The examples include those already discussed as well as:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mendeleev and the Periodic Table&lt;/li&gt;&#xA;&lt;li&gt;AR Wallace and natural selection&lt;/li&gt;&#xA;&lt;li&gt;Descartes' scientific method&lt;/li&gt;&#xA;&lt;li&gt;Loewi and synaptic nerve transmissions&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Another article adds Watson's vision of the double helix structure of DNA&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/List_of_dreams&quot; rel=&quot;noreferrer&quot;&gt;http://en.wikipedia.org/wiki/List_of_dreams&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2015-06-11T10:54:42.930" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2432" PostTypeId="1" CreationDate="2015-06-11T20:22:48.720" Score="3" ViewCount="76" Body="&lt;p&gt;In modern arithmetic textbooks, students are taught about alternate numeric bases. The notation for indicating the base of a number is to attach the base as a subscript. The subscript is itself a numeral in base ten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, when was that notation first invented? Who invented it? Where did it first appear in print?&lt;/p&gt;&#xA;" OwnerUserId="2431" LastActivityDate="2015-06-11T20:22:48.720" Title="Origin of alternate base annotation" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2433" PostTypeId="1" AcceptedAnswerId="2441" CreationDate="2015-06-12T19:11:44.660" Score="3" ViewCount="310" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Magma_(algebra)&quot; rel=&quot;nofollow&quot;&gt;According to Wikipedia&lt;/a&gt;, the choice of vocabulary was made partially to avoid overloading the term &quot;groupoid&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that still does not explain etymologically speaking, &quot;magma&quot; was chosen instead of any other word that isn't &quot;groupoid&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you explain?&lt;/p&gt;&#xA;" OwnerUserId="2134" LastActivityDate="2015-06-15T13:23:33.540" Title="From the perspective of etymology, why was the word &quot;magma&quot; chosen to describe a set with a single binary operation defined on it?" Tags="|mathematics|terminology|abstract-algebra|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2434" PostTypeId="2" ParentId="9" CreationDate="2015-06-13T15:14:54.700" Score="18" Body="&lt;p&gt;I'd lile to mention two interesting papers written by Olivier Courcelle, that have been very recently published on the excellent &lt;strong&gt;Image des mathématiques&lt;/strong&gt; website. The papers, written in French, contain numerous and previously unpublished details about the identity of Galois' duel opponent. They may not directly answer the original question, but I think they are worth mentioning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first paper, &lt;a href=&quot;http://images.math.cnrs.fr/L-adversaire-de-Galois-I.html&quot;&gt;&lt;em&gt;L'Adversaire de Galois (I)&lt;/em&gt;&lt;/a&gt;, he mostly addresses the question of who the opponent was. He mentions that the newspaper &lt;em&gt;Le Précurseur&lt;/em&gt; of June 4th 1832 reports about Galois' death and mentions the initals of his opponent, L. D. (centre column of the referenced page). In his &lt;em&gt;Mémoires&lt;/em&gt;, Alexandre Dumas names the opponent as being Pescheux d’Herbinville.&#xA;According to Courcelle, these elements do not conflict, as Pescheux was sometimes called Lepescheux, just as Galois has been referenced as Legallois in some situations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning the reasons for a duel, the author points to various sources, including a letter by Galois himself, none of them supporting the love story theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the second paper, &lt;a href=&quot;http://images.math.cnrs.fr/L-adversaire-de-Galois-II.html&quot;&gt;&lt;em&gt;L'Adversaire de Galois (II)&lt;/em&gt;&lt;/a&gt;, published today, Courcelle reveals previously unpublished biographical elements about Pescheux d’Herbinville, that he was able to find by scouring numerous archives and registers. He concludes his paper with the full identity of the man: Étienne-François Pecheux d’Herbenville (Paris, April 5th 1809 – Paris, march 23rd 1871).&lt;/p&gt;&#xA;" OwnerDisplayName="user2441" LastEditorUserId="1878" LastEditDate="2015-06-14T14:26:39.727" LastActivityDate="2015-06-14T14:26:39.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2435" PostTypeId="1" AcceptedAnswerId="2436" CreationDate="2015-06-13T17:46:43.113" Score="18" ViewCount="1061" Body="&lt;p&gt;Did they screw this up? It says that Fermat's Last Theorem was the longest open problem - with only 365 years. See &lt;a href=&quot;http://www.guinnessworldrecords.com/world-records/longest-standing-maths-problem-(ever)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guinness Book of Records&lt;/a&gt;. However, there are Greek problems that were longer open:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Squaring the circle, proposed before 428 BCE (Anaxagoras worked on it, who died in 428 BCE), solved 1882, open for&#xA;at least 2.310 years.&lt;/li&gt;&#xA;&lt;li&gt;Doubling the cube, proposed before 430 BCE, solved 1837, open for at least 2.267 years.&lt;/li&gt;&#xA;&lt;li&gt;Archimedes Cattle Problem, proposed before 212 BCE, solved 1880, open for at least 2.092 years.&lt;/li&gt;&#xA;&lt;li&gt;Angle trisection, solved 1837.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there a reason why Guinness Book of Records listed Fermat's Last Theorem?&lt;/li&gt;&#xA;&lt;li&gt;Are there problems that were even longer open than the above (Egyptian, Chinese, Indian, Babylonian?), or can you find a reference that the problems above were proposed earlier?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Crossposted&lt;/strong&gt; from &lt;a href=&quot;https://math.stackexchange.com/questions/1302183/did-guinness-book-of-records-screw-this-up&quot;&gt;Mathematics Stack Exchange&lt;/a&gt;. Progress so far there: Someone e-mailed Guinness Book of records, no reaction.&lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-07-31T20:28:34.910" Title="Did Guinness Book of Records screw up on the &quot;longest-standing maths problem (ever)&quot;?" Tags="|mathematics|ancient-greece|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2436" PostTypeId="2" ParentId="2435" CreationDate="2015-06-14T00:27:53.090" Score="9" Body="&lt;p&gt;There is an issue with &quot;proposed&quot; and &quot;solved&quot;. Of all the examples the Archimedes Cattle Problem is the only one that was truly proposed and solved. The oracle of Delphi allegedly &quot;proposed&quot; duplication to the Athenians, but it could not have been very precise since they originally took it to mean doubling the sides. Who proposed quadrature and trisection is hidden in the mists of time (in a practical context &lt;a href=&quot;http://www-math.ucdenver.edu/~wcherowi/courses/history2/quad.pdf&quot;&gt;finding the area of a circle already appears as problem 50 in the Rhind papyrus c. 1600 BC&lt;/a&gt;), but they apparently did not specify that it had to be done with straightedge and compass either, because all three were solved within a hundred years of being &quot;proposed&quot; by other means (e.g. &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Dinostratus.html&quot;&gt;quadrature was solved by Dinostratus c. 350 BC, using quadratrix&lt;/a&gt;). And one has to be really precise, because already Archimedes knew how to solve duplication and trisection with marked straightedge and compass. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That with such precise restrictions there were no solutions was suspected in late antiquity, as follows from &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Pappus.html&quot;&gt;Pappus's classification of problems into plane (solved with straightedge and compass), solid (solved with conic sections), and other&lt;/a&gt;, but the idea of proving something like that does not emerge as a problem until 17th century, when &lt;a href=&quot;https://hal-insu.archives-ouvertes.fr/tel-01098493/document&quot;&gt;Gregory attempted proving the impossibility of quadrature in 1667&lt;/a&gt;. So what was &quot;solved&quot; in 1837 and 1882 is not what was &quot;proposed&quot; in 5th century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, Euclid did not &quot;propose&quot; to derive the parallel postulate from other postulates, although many took his verbose phrasing and avoiding the use of it for half of book I as an invitation to do so. Fermat did not propose the Last Theorem as a problem either, he stated it, and mentioned  on the margins of his copy of Diophantus's Arithmetica that he &quot;discovered a truly marvelous proof of this, which this margin is too narrow to contain&quot;. Only later Euler and others realized that something was amiss when they tried to reproduce this &quot;marvelous proof&quot;. It is more accurate to say that all these problems, as we know them today, emerged over time from multiple people playing with a common theme.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is Guinness Book of Records, so we shouldn't be too picky. In which case &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Perfect_numbers.html&quot;&gt;Pythagoreans already cared about perfect numbers&lt;/a&gt; in 6th century BC, Euclid gives a familiar characterization of them in Elements IX.36 (c.300 BC). They probably wondered if there is the largest among them, Euclid answers such question about primes in Elements IX.20, Nichomachus implied the same answer for perfects c. 100 AD. In any case this is what emerged after centuries of playing with them, so the problem &quot;originates&quot; before all of the above, and it is still open.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-14T03:07:01.280" LastActivityDate="2015-06-14T03:07:01.280" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2437" PostTypeId="1" AcceptedAnswerId="2438" CreationDate="2015-06-14T16:08:16.360" Score="8" ViewCount="1384" Body="&lt;p&gt;I have read, in many places, statements like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Heaviside was able to greatly simplify Maxwell's 20 equations in 20&#xA;  variables, replacing them by four equations in two variables. Today we&#xA;  call these 'Maxwell's equations' forgetting that they are in fact&#xA;  'Heaviside's equations'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... but nowhere can I find exactly what these equations look like in the form Heaviside produced. There are many versions of the equations, most with more than just four.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(From &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Heaviside.html&quot; rel=&quot;nofollow&quot;&gt;this biographical article&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please can anyone tell me exactly what are the original &quot;four equations in two variables&quot;?&lt;/p&gt;&#xA;" OwnerUserId="2444" LastEditorUserId="4251" LastEditDate="2020-06-07T18:36:43.177" LastActivityDate="2023-10-22T18:44:05.397" Title="What is Heaviside's version of Maxwell's equations?" Tags="|physics|theoretical-physics|electromagnetism|mathematical-physics|heaviside|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2438" PostTypeId="2" ParentId="2437" CreationDate="2015-06-14T16:49:04.833" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://www.testandmeasurementtips.com/new-articles/oliver-heaviside-and-maxwells-equations/&quot; rel=&quot;nofollow&quot;&gt;This&lt;/a&gt; gives the four equations in the form Heaviside came up with:&#xA;$$\varepsilon E = \rho$$&#xA;$$\nabla \times E = - \mu \frac{\partial H}{\partial t}$$&#xA;$$\nabla \cdot \mu H = 0$$&#xA;$$\nabla \times H = k E + \varepsilon \frac{\partial E}{\partial t}$$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;where $E$ represents the electric field, $H$ represents the magnetic field, $\varepsilon$ is the permittivity, $\mu$ is the permeability, $\rho$ is the charge density, and $k$ is the conductivity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, $\times$ and $\cdot$ denote the cross product and dot product. Also, &lt;a href=&quot;http://en.wikipedia.org/?title=Magnetic_field#The_H-field&quot; rel=&quot;nofollow&quot;&gt;$H$ is not the same as the perhaps more commonly used $B$&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: The first equation should properly be $\varepsilon \nabla \cdot E = \rho$, though the source leaves it out, probably due to a typo.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-06-14T19:49:30.590" LastActivityDate="2015-06-14T19:49:30.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2439" PostTypeId="1" AcceptedAnswerId="2442" CreationDate="2015-06-14T18:59:46.857" Score="5" ViewCount="3781" Body="&lt;p&gt;In a book, I saw a short anecdote about something called the &quot;Horus-eye&quot; hieroglyph, supposedly used in parts to represent fractions (the book dates to 1997). Wikipedia has &lt;a href=&quot;http://en.wikipedia.org/wiki/Eye_of_Horus&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article&lt;/a&gt; on the symbol and states&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Studies from the 1970s to this day in Egyptian mathematics have clearly shown this theory was fallacious and Jim Ritter definitely showed it to be false in 2003. The evolution of the symbols used in mathematics, although similar to the different parts of the Eye of Horus, is now known to be distinct.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I detect a slight lilt towards a bias against the symbol there, given that I've read other (more recent than the book) sources that either omit the &quot;studies from the 1970s&quot; and Ritter's work or barely mention it altogether - not that I'm doubting it; I'm only noting that perhaps there isn't as much certainty as Wikipedia claims.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we use the hypothesis that the symbol did &lt;em&gt;not&lt;/em&gt; represent fractions then someone in the modern era must have come up with the theory. Who was it? Wikipedia alludes to &lt;a href=&quot;http://en.wikipedia.org/wiki/Gardiner%27s_sign_list&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gardiner's sign list&lt;/a&gt;, but I have no idea if the hieroglyph is represented there merely in an encyclopedic sense, e.g. Gardiner catalogued it from someone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, by the way, is a diagrammed representation of the symbol:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Oudjat.SVG/640px-Oudjat.SVG.png&quot; alt=&quot;&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;a href=&quot;http://commons.wikimedia.org/wiki/File:Oudjat.SVG&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikimedia Commons&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-03-10T09:42:49.827" LastActivityDate="2023-08-27T09:59:03.200" Title="How did the theory of the Horus-eye fractions originate?" Tags="|mathematics|notation|ancient-egypt|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2441" PostTypeId="2" ParentId="2433" CreationDate="2015-06-15T11:52:21.410" Score="3" Body="&lt;p&gt;Regarding the mathematical usage of &lt;em&gt;magma&lt;/em&gt;, it seems that the origin is the &quot;Bourbaki circle&quot;, but it is not so easy to trace exactly the date ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.cmat.edu.uy/~marclan/TM/Algebra%20i%20-%20Bourbaki.pdf&quot; rel=&quot;nofollow&quot;&gt;first English translation&lt;/a&gt; of Nicolas Bourbaki, &lt;a href=&quot;https://books.google.it/books?id=4RIDPH1ULXcC&quot; rel=&quot;nofollow&quot;&gt;Algèbre: Chapitres 1 à 3&lt;/a&gt; (1970) dates the first pubblication to 1943. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, the definition of &lt;em&gt;magma&lt;/em&gt; is the first one (page I-1), under the heading : &lt;strong&gt;1. Lois de composition&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The corresponding &lt;a href=&quot;http://sites.mathdoc.fr/archives-bourbaki/PDF/033bis_iecnr_041.pdf&quot; rel=&quot;nofollow&quot;&gt;draft&lt;/a&gt; (rédaction n°033 bis : what date ?) does not have the term &lt;em&gt;magma&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not able to find a copy od the original French edition; thus, I'm not able to check if the term was already there. It seems not ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Hubert Kiechle, &lt;a href=&quot;https://books.google.it/books?id=J3id_WLGgKMC&amp;amp;pg=PA23&quot; rel=&quot;nofollow&quot;&gt;Theory of K-Loops&lt;/a&gt;, page 23 : the term magma was introduced in the 1970-edition of Bourbaki's &lt;em&gt;Algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have also a &lt;a href=&quot;https://www.google.com/search?tbm=bks&amp;amp;tbo=p&amp;amp;hl=it&amp;amp;q=magma&amp;amp;safe=active#hl=it&amp;amp;tbm=bks&amp;amp;q=magma+Bourbaki++Alg%C3%A8bre+I+(Paris%2C+1942)++&quot; rel=&quot;nofollow&quot;&gt;review&lt;/a&gt; of the 1970 edition that seemingly alludes to &lt;em&gt;magma&lt;/em&gt; as a &quot;new entry&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For sure, the term is used by Serre in 1964 : Jean-Pierre Serre, &lt;a href=&quot;https://books.google.it/books?id=ChJqCQAAQBAJ&amp;amp;pg=PA18&quot; rel=&quot;nofollow&quot;&gt;Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University&lt;/a&gt;, page 18 : &lt;strong&gt;1. Free magmas&lt;/strong&gt;. &lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-06-15T13:23:33.540" LastActivityDate="2015-06-15T13:23:33.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2442" PostTypeId="2" ParentId="2439" CreationDate="2015-06-15T22:02:20.033" Score="3" Body="&lt;p&gt;Ritter himself gives a detailed historical account in &lt;a href=&quot;https://www.academia.edu/11310442/_Closing_the_Eye_of_Horus_The_Rise_and_Fall_of_Horus-Eye_Fractions_in_J._Steele_and_A._Imhausen_eds._Under_One_Sky_AOAT_297_._M%C3%BCnster_Ugarit-Verlag_2003_298_323&quot; rel=&quot;nofollow&quot;&gt;Closing the Eye of Horus: The Rise and Fall of &quot;Horus-eye Fractions&quot;&lt;/a&gt;. Egyptians had two forms of writing, hieratic (cursive) and hieroglyphic (pictographic). Ritter traces the origin of the myth to &quot;&lt;em&gt;“central dogma” of our ideas about Egyptan writing: all hieratic signs go back ultimately to pictographic signs. The idea of organizing hieratic signlists by hieroglyph is one expression of this and, as we have seen, it was precisely in just such a context that the search for the “hieroglyphic originals” of the capacity metrological signs led to the Horus-eye hypothesis&lt;/em&gt;&quot;. But analysis of documents shows that in Egypt, as in Mesopotamia, numerical and non-numerical signs evolved quite differently. Ritter's view remains the consensus, see e.g. &lt;a href=&quot;https://books.google.com/books/about/The_Mathematics_of_Egypt_Mesopotamia_Chi.html?id=3ullzl036UEC&quot; rel=&quot;nofollow&quot;&gt;The Mathematics of Egypt, Mesopotamia, China, India, and Islam edited by Katz&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story begins with Möller's 1911 Hieratische Paläographie (&quot;CSM signs&quot; below are what Ritter calls the &quot;fraction&quot; symbols):&#xA;&quot;&lt;em&gt;Möller presents essentially two arguments for the identification of the CSM signs with parts of the completed right eye of Horus: 1° the proximity of some hieratic and some hieroglyphic forms of the signs on two unpublished votive cubits and 2° the visual similarities of the remaining signs, which “can be traced back to parts of the eye”&lt;/em&gt;&quot;. Peet already called this idea into question in 1923, because &quot;&lt;em&gt;1° the hieratic forms “show but slight resemblance to the parts of the eye” and 2° the hieroglyphic forms are much later, they do not predate Dynasty 18 or, for some forms, Dynasty 20. The frequent use of the hieratic forms in the medical texts leads Peet to speculate that the Horus-eye identification was “an ingenious discovery of some [medical] scribe” since the sacred eye was “so closely connected in Egyptian eyes with healing power.”&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nonetheless, Gardiner accepted Möller's &quot;strong thesis&quot;, as Ritter calls it, in Egyptian Grammar (1927), even reflecting the eye from right to left to make the signs fit. After the publication of Moscow papyrus in 1930 Neugebauer objected to that, and pointed out that there was little resemblance in 1/16 and 1/32, arriving at Peet's &quot;weak thesis&quot; that identification was of &quot;late date&quot;.&#xA;Still, &quot;&lt;em&gt;...by the beginning of the nineteen-thirties, the Horus-eye interpretation had entered the two canonical grammars and, especially in the form given to it by Gardiner, was to become the standard interpretation for Egyptologists and historians of science in the succeeding decades. This was true very early; indeed neither Struve nor Neugebauer nor Vogel cites Möller’s article but only Gardiner&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Things changed after publication of new documents in 1960-1970s, including those unpublished by Möller, and a new look in their light at the old documents. Ritter gives a detailed discussion, summarized as follows:&#xA;&quot;&lt;em&gt;The trend of the evidence of third-millennium sign forms that have accumulated since the end of the debate in the nineteen-thirties is clear. The further back we go the further the hieratic signs diverge from their supposed Horus-eye counterparts. The inconsistency between the hieratic signs for “1/16” and “1/32” and their purported hieroglyphic originals, already signaled by Neugebauer..., is even greater for the original third-millennium forms and the others reveal similar difficulties. The hieroglyphic “1/2” signs vary widely among themselves and all differ from their New Kingdom descendants&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So much for the strong thesis. And the weak thesis? &#xA;&quot;&lt;em&gt;If it, unlike the strong thesis, cannot be ruled out completely, the positive evidence for it seems to be meagre in the extreme... The beginning of the replacement of the special CSM signs by their general fractional equivalents is to be located just at this period when, in the weak thesis, the idea that the classical CSM signs form an important conceptual and religious unity begins its career. The votive cubits that are supposed to enshrine this idea have proven to be much more ambiguous and unclear than foreseen... But even if the cubits did represent an attempt to reinterpret the CSM signs in terms of Horus-eye parts, a speculation in the milieu which produced votive cubits does not automatically mean that “the Egyptians” thought like that; for example, those Egyptians whose task it was to engrave hieroglyphic inscriptions on temple walls. Theological or any other constructs of one community do not necessarily propagate to every other...&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-06-15T22:02:20.033" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2443" PostTypeId="1" CreationDate="2015-06-15T23:57:01.867" Score="3" ViewCount="596" Body="&lt;p&gt;I was going through the documentation of the manual page in the Linux Operating System, when, I encountered the term &quot;typesetting&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I surfed the web for answers and found it in &lt;a href=&quot;http://en.wikipedia.org/wiki/Typesetting&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I am actually confused over how the computer came into existence. Is it through the invention of the linotype printing machine (The first keyboard that came into exsistence. Please correct me if I am wrong.) and the invention of typewriter, or through the idea of inventing a calculating machine.I like to know the history and origin of how modern computer as we know today came into exsistence (with the keyboard, mouse and CPU). Actually, how did the computer come into its true form?&lt;/p&gt;&#xA;" OwnerUserId="2454" LastEditorUserId="62" LastEditDate="2015-06-16T00:14:13.413" LastActivityDate="2015-06-16T16:04:24.627" Title="Was the computer invented through the influence of the printing press or through the influence of the calculator?" Tags="|computer-science|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2444" PostTypeId="2" ParentId="2443" CreationDate="2015-06-16T01:54:47.023" Score="7" Body="&lt;p&gt;I am immediately reminded of a great 2010 documentary called &lt;a href=&quot;http://www.imdb.com/title/tt1587359/&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Top Secret Rosies: The Female 'Computers' of WWII&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &quot;computer&quot; has had a definition since at least the 17th century. In the traditional sense, this from the &lt;em&gt;Oxford Dictionary of English Etymology (1992)&lt;/em&gt;, &quot;compute&quot; came from French, meaning &quot;determine by mathematical reckoning.&quot; The word computer would refer to a profession wherein one makes their living doing mathematical and arithmetic calculations. The Wiki on &lt;a href=&quot;https://en.wikipedia.org/wiki/Human_computer&quot; rel=&quot;nofollow&quot;&gt;human computer&lt;/a&gt; should give you loads of insight into this old profession. An nice additional piece of historical perspective is encapsulated in the late 19th century job advertisement, &lt;a href=&quot;http://query.nytimes.com/mem/archive-free/pdf?res=9F07E0D81438E233A25751C0A9639C94639ED7CF&quot; rel=&quot;nofollow&quot;&gt;A Computer Wanted&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, when some large computations for some large complicated process needed to be done, and the person wanting them done had the cash to hire the people, human computers would be hired, organized and instructed to take on various parts of long computations in an organized assembly line style. Computer was still a human profession right in to world war II, and the first electronic computers like &lt;a href=&quot;https://en.wikipedia.org/wiki/ENIAC&quot; rel=&quot;nofollow&quot;&gt;ENIAC&lt;/a&gt; were actually programmed by these human computers. The people who were in this profession typically had math degrees. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ENIAC was not the only electronic computer of the 1940s. An earlier computer, &lt;a href=&quot;https://en.wikipedia.org/wiki/Colossus_computer&quot; rel=&quot;nofollow&quot;&gt;Colossus&lt;/a&gt;, was used by British cryptologists during the war, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bombe&quot; rel=&quot;nofollow&quot;&gt;Bombe&lt;/a&gt; computer was used to help decipher the German Enigma machine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recall Richard Feynman telling stories about his days working on the Manhattan Project. In &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Feynman#Manhattan_Project&quot; rel=&quot;nofollow&quot;&gt;Feynman - Manhattan Project&lt;/a&gt; the author states, &quot;The greater part of his work was administering the computation group of human computers in the theoretical division.&quot; Much more first person anecdotal stories of his time here working with his human computers, as I recall, can be found in his book &lt;em&gt;Surely You're Joking, Mr. Feynman!&lt;/em&gt; That is a much recommended read, I do not possess a copy; I read it in a library. I vaguely recall him describing his process as one where these students in an orderly fashion, an adder, a multiplier, a divider, and so forth, would, with the use of a punch card system, build up big computations over many hours and days, maybe even weeks. One human misstep would obviously have disastrous results. So I guess he was sort of using a combination of human computers, and machine style punchcard devices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially, the answer to your question is (mostly I think) the latter; electronic computers arose straight from a desperate demand for more computational power. Being able to automate a huge algorithm formerly done by a team of directed humans in less time with less error became possible, and it was important at the time. They were really pushed along out of necessity in a big way by the demands of the second world war, and the cold war funding immediately thereafter kept this advancement moving forward. I am not saying that WWII gave us the very first examples of computers, but this would be the age when they were dramatically ushered in, and the word computer really began to transfer its meaning from &quot;one (a person) who computes,&quot; to these machines that, in a decade, made the job of human computer obsolete, and eventually turned into that thing currently sitting on your desktop.&lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2015-06-16T16:04:24.627" LastActivityDate="2015-06-16T16:04:24.627" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2445" PostTypeId="1" AcceptedAnswerId="2446" CreationDate="2015-06-15T00:04:54.737" Score="5" ViewCount="206" Body="&lt;p&gt;Could you recommend a historically rigorous biography of L. Euler (if possible with discussions and examples of the mathematics he was doing)? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I'd rather prefer a book (not necessarily about Euler alone) or a thorough article.&lt;/p&gt;&#xA;" OwnerUserId="2426" OwnerDisplayName="gerd" LastActivityDate="2015-06-16T04:31:17.123" Title="Biography of L. Euler" Tags="|reference-request|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2446" PostTypeId="2" ParentId="2445" CreationDate="2015-06-15T11:16:58.890" Score="5" Body="&lt;p&gt;The standard reference seems to be&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.maa.org/publications/books/euler-the-master-of-us-all&quot;&gt;Euler: The Master of Us All&lt;/a&gt; by William Dunham.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.springer.com/br/book/9783764375386&quot;&gt;Leonhard Euler&lt;/a&gt;&#xA;  by Emil A. Fellmann.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is a short article here: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/2968971&quot;&gt;Biography: Leonhard Euler&lt;/a&gt;, &#xA;  by B. F. Finkel.&lt;br&gt;&#xA;  &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;,&#xA;  Vol. 4, No. 12 (Dec., 1897), pp. 297-302&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;http://eulerarchive.maa.org&quot;&gt;The Euler Archive&lt;/a&gt;, in particular its &lt;a href=&quot;http://eulerarchive.maa.org/resources.html&quot;&gt;Historical and Biographical Resources&lt;/a&gt; page.&lt;/p&gt;&#xA;" OwnerUserId="186" OwnerDisplayName="lhf" LastActivityDate="2015-06-15T11:24:02.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2447" PostTypeId="2" ParentId="2443" CreationDate="2015-06-16T10:31:48.527" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Was the computer invented through the influence of the printing press or through the influence of the calculator?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Neither.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Electronic typesetting began in the mid 1960s with &lt;a href=&quot;https://en.wikipedia.org/wiki/TYPSET_and_RUNOFF&quot; rel=&quot;nofollow&quot;&gt;TYPSET and RUNOFF&lt;/a&gt;, which was a couple of decades after the invention of digital electronic computers. Desktop publishing followed a couple of decades later. Until the late 1960s, all that calculators could do was add, subtract, multiply, and divide. By the late 1960s, some calculators had a &quot;memory&quot;, but that memory was limited to a single number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Up until the 1940s, computers were people. While those human computers did use mechanical calculators to aid in the computations (the electronic calculator was yet to be invented), they did something that mere calculators cannot do. Those human computers could follow a complicated algorithm that involve decision-making. Calculators that can only add, subtract, multiple, and divide can't do that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was automating those decision processes that enabled the transition from human computers to electronic computers in the 1940s and beyond.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-06-16T10:31:48.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2449" PostTypeId="2" ParentId="2380" CreationDate="2015-06-16T19:18:45.300" Score="0" Body="&lt;p&gt;You will find history of development of solid state physics in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/019505329X&quot; rel=&quot;nofollow&quot;&gt;Out of crystal maze&lt;/a&gt;. It is a very well researched book.&lt;/p&gt;&#xA;" OwnerUserId="2459" LastActivityDate="2015-06-16T19:18:45.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2450" PostTypeId="1" CreationDate="2015-06-16T22:44:44.607" Score="4" ViewCount="366" Body="&lt;p&gt;On page 71 of &lt;a href=&quot;http://www.amazon.co.uk/Absolute-Differential-Calculus-Tensors-Mathematics/dp/0486634019&quot; rel=&quot;nofollow&quot;&gt;The Absolute Differential Calculus by Levi-Civita&lt;/a&gt;, a very clear definition of a tensor is given in terms of how the coefficients of a multi-linear form transform, such that the product with its dual multi-linear form remains &lt;strong&gt;invariant&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An m-fold covariant is an m-fold system which is transformed in the same way as the coefficients of a multilinear form in point variables; an m-fold contravariant is one which is transformed in the same way as the coefficients of a multilinear form in dual variables; more generally, a mixed system or &lt;strong&gt;tensor&lt;/strong&gt; is one which is transformed in the same way as the coefficients of a multilinear form in both point and dual variables (including also as particular cases both purely covariant and purely contravariant systems).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Were there other related definitions defined around this time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How has today's definition of a tensor changed compared to its beginnings?&lt;/p&gt;&#xA;" OwnerUserId="2461" LastEditorUserId="2324" LastEditDate="2020-03-16T14:58:03.533" LastActivityDate="2020-07-08T22:06:40.600" Title="How has the definition of a tensor today changed compared to its original definition?" Tags="|mathematics|mathematical-physics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2451" PostTypeId="1" CreationDate="2015-06-17T13:41:53.503" Score="2" ViewCount="225" Body="&lt;p&gt;I intend to review the different concepts related to laws of motion floating around the publication of &lt;em&gt;Principia&lt;/em&gt; and hopefully would like to comment on these (It's an essay project). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've gathered couple of books for this purpose, mostly the usual suspects(Galileo's &lt;em&gt;Two New Sciences&lt;/em&gt;, Newton's &lt;em&gt;Principia&lt;/em&gt;, Mach's &lt;em&gt;Science of Mechanics&lt;/em&gt;, Dugas's &lt;em&gt;History of Mechanics&lt;/em&gt;, Chandrashekhar's &lt;em&gt;Newton's Principia for Common Readers&lt;/em&gt;, Maxwell's &lt;em&gt;Matter and Motion&lt;/em&gt;, and a host of common classical mechanics books like Landau, Marion, Summerfield, Taylor, etc. which usually comment on the logical structure of these three laws plus gravitation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it would have been really nice to find some articles which directly discuss Newton's laws from philosophical and historical point of views. Searching in Google Scholar got me a paper which discusses Newton's laws with Kant's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I really need some references for my project.&lt;/p&gt;&#xA;" OwnerUserId="443" LastEditorUserId="62" LastEditDate="2015-06-21T21:48:16.587" LastActivityDate="2015-06-21T21:48:16.587" Title="Reading material suggestion for history of Newton's laws of motion" Tags="|physics|reference-request|classical-mechanics|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2454" PostTypeId="1" AcceptedAnswerId="2455" CreationDate="2015-06-18T21:47:33.330" Score="10" ViewCount="6885" Body="&lt;p&gt;In &lt;em&gt;The Norton History of the Mathematical sciences&lt;/em&gt;, Ivor Grattan-Guinness writes of Newton's &lt;em&gt;Principia&lt;/em&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Indeed, in working drafts for the book Newton considered up to six laws, and this trio are wanting in their (non-) handling of angular momentum and frames of reference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By &quot;this trio&quot;, he means the three laws we normally consider to be Newton's Three Laws of Motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What were the other three?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="2671" LastEditDate="2015-08-01T21:32:37.970" LastActivityDate="2023-09-13T01:40:39.223" Title="What were Newton's six laws of motion?" Tags="|physics|classical-mechanics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2455" PostTypeId="2" ParentId="2454" CreationDate="2015-06-20T15:26:43.377" Score="7" Body="&lt;p&gt;The &quot;classical&quot; references for the sources and evolution of Newton's mechanics are : &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexandre_Koyr%C3%A9&quot; rel=&quot;nofollow&quot;&gt;Alexandre Koyré&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_S._Westfall&quot; rel=&quot;nofollow&quot;&gt;Richard S.Westfall&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/?title=I._Bernard_Cohen&quot; rel=&quot;nofollow&quot;&gt;I.Bernard Cohen&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ivor Grattan-Guinness' comment is [in the Norton edition of the book] at page 259 refers to Richard S. Westfall, &lt;a href=&quot;https://books.google.it/books?id=nqYNAQAAIAAJ&quot; rel=&quot;nofollow&quot;&gt;Force in Newton's physics: the science of dynamics in the seventeenth century&lt;/a&gt; (1971), &lt;strong&gt;Ch.8&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've no acces to it, but we can see a detailed discussion of Newton's drafts of &lt;em&gt;Principia&lt;/em&gt; into :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;R.S.Westfall, &lt;a href=&quot;https://books.google.it/books?id=3ngEugMMa9YC&amp;amp;pg=PA411&quot; rel=&quot;nofollow&quot;&gt;Never at Rest : A Biography of Isaac Newton&lt;/a&gt; (1983), page 409-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'll try to sumamrize his argument :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum&quot; rel=&quot;nofollow&quot;&gt;De motu&lt;/a&gt; started with two definitions and two hypotheses. Definition 1 drew upon the lesson about circular motion that Hooke had taught him in 1679 to introduce a new word into the vocabulary of mechanics. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;I call that by which a body is impelled or attracted toward some point which is regarded as a center centripetal force. &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;[...] The legacy of &lt;a href=&quot;http://williambarclayallen.com/translations/De_Gravitatione_et_Aequipondio_Fluidorum_translation.pdf&quot; rel=&quot;nofollow&quot;&gt;De gravitatione&lt;/a&gt; appeared in Definition 2, about rectilinear motion. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;And [I call] that by which it endeavors to persevere in its motion in a right line the force of a body or the force inherent in a body. &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Hypothesis 2 extended the definition into a general conception of motion. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;By its inherent force alone, every body proceeds uniformly in a right line to infinity unless something extrinsic hinders it.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;[...] The tract attempted to derive orbital motion from the interaction of two forces: inherent force, which maintains rectilinear motion, and centripetal force, which continually diverts it. To compound the two, Newton employed the parallelogram of forces, which he inserted later as Hypothesis 3, although he made use of it before he added its formal statement. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;When it is acted upon by [two] forces simultaneously, a body is carried in a given time to that place to which it would be carried by the forces acting separately in succession during equal times. [411]&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Three versions of &lt;em&gt;De motu&lt;/em&gt; survive. The &lt;a href=&quot;https://books.google.it/books?id=uvMGAAAAcAAJ&amp;amp;pg=PA111&quot; rel=&quot;nofollow&quot;&gt;second version&lt;/a&gt;, late 1684, which is written in Halley's hand and is identical to the copy in the register of the Royal Society, was merely a fair copy of the first with small additions (such as Hypothesis 4) which Newton had indicated his intention to insert. The third version, on the other hand, contained the beginning of Newton's reconstruction of his dynamics. [412]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the &lt;a href=&quot;http://www.newtonproject.sussex.ac.uk/view/texts/normalized/NATP00091&quot; rel=&quot;nofollow&quot;&gt;third version&lt;/a&gt;, early 1685, Newton moved Hypotheses 3 and 4, the parallelogram of force and the measure of distance described under a centripetal force, to the status of lemmas. To the remaining two hypotheses modestly altered, he added three more and then rejected the name &quot;Hypothesis&quot; in favor of a new one, &#xA;  &quot;Lex.&quot; &lt;strong&gt;Hence Newton's laws of motion initially numbered five&lt;/strong&gt; [&lt;em&gt;emphasis added&lt;/em&gt;]. From its new position at the head of the list, Law 1 continued to &#xA;  assert that a body moves uniformly by its inherent force alone. In Law 2, Newton attempted to define the action of force. &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;The change of motion is proportional to the impressed force and is made in the direction of the right line in which that force is impressed. [414]&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Law 3 asserted that the motions among themselves of bodies included in a given space are the same whether the space is at rest or moves uniformly in a straight line without circular motion. Law 4 asserted that the mutual actions of bodies do not change the state of motion or of rest of their common center of gravity. The first draws on the principle of inertia; the second extends it to systems of bodies considered as unities. Newton later demoted the two to the rank of corollaries to the laws of motion, but both made their way into the Principia. Law 5, a restatement of Hypothesis 1 of the first version, contained a supposedly empirical statement about the resistance of media. [414] &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Added August 2015&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grattan-Guinness's comments refers to Westfall's book on &lt;em&gt;Force in Newton's physics&lt;/em&gt;, where the discussion of Newton's 1684 manuscripts on &lt;em&gt;De Motu&lt;/em&gt; takes place in page 432-on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Page 432 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As far as his [Newton's] dynamics is concerned, [...] this development appears to have been completed during roughly the six months following the redaction of the original version of &lt;em&gt;De motu&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the original version of &lt;em&gt;De motu&lt;/em&gt;, Newton derived the law of areas, the connection of the inverse square relation with elliptical orbits, and basic propositions concerning motion through a uniformly resisting medium from a dynamics less satisfactory than that he had employed in 1679, a dynamics inconsistent with itself and utterly inadequate to the load of demonstration it was asked to carry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Page 433 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the MS. in which it has survived, the first version of &lt;em&gt;De motu&lt;/em&gt; contains two stages, an original form and a set of emendations and additions. The original form started with the two definitions cited above and two hypotheses. Hypothesis one said that bodies are not impeded by a medium or by other external causes which would prevent them from exactly obeying the inherent and centripetal forces. [...] he inserted a third definition - of resistance as the force of a medium uniformly impeding motion - and altered hypothesis one to say that resistance is assumed to be zero in the first nine propositions and proportional to the velocity of the body and the density of the medium in the remaining ones.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Sometime after the original redaction, and probably at the same time as the additions mentioned above, he entered a third hypothesis in the margin and the heading ‘Hyp 4,’ although he did not write in the fourth hypothesis itself. The third one stated explicitly the dynamics already employed in the following propositions.&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Hyp. 3. When it is acted upon by [two] forces simultaneously, a body is carried in a given time to that place to which it would be carried by the forces acting separately in succession during equal times.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Page 439 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As far as Newton’s ultimate dynamics was concerned, the central problem that the first version of &lt;em&gt;De motu&lt;/em&gt; left for solution was the resolution of the contradiction posed by his acceptance of two radically different concepts of force, inherent force which maintains uniform motion, and centripetal force which alters it. To the solution of the problem, version two, which was little more than a fair copy of the amended form of version one, added nothing. &lt;em&gt;Version three, in contrast, marked a further stage in Newton’s thought&lt;/em&gt; [emphasis added]. Hypotheses 3 and 4 of the earlier versions were moved to the status of Lemmas, enhanced by demonstrations, but not otherwise altered. To the remaining two hypotheses Newton added three more, decided that the word ‘Hypothesis’ did not express his meaning, struck it out in all five cases, and replaced it with a new word, ‘Lex.’&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In their original incarnation, then, the laws of motion numbered five&lt;/em&gt; [emphasis added]. The first remained substantially unchanged from version one, still asserting that by its inherent force alone a body proceeds with a uniform motion in a right line. Law two, however, underwent a significant change. The word ‘impressed’ replaced the word ‘centripetal,’ not only increasing the generality of the law, but also expressing the dichotomy that Newton was seeking to exploit. The inherent or internal force of a body maintains it in uniform motion. The force external to a body, impressed on it from without, alters its uniform motion.&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;The change of motion is proportional to the impressed force and is made in the direction of the right line in which that force is impressed.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;With the exception of one word (the adjective ‘motive’ modifying impressed force), the law is identical to that which appeared in the &lt;em&gt;Principia&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Of the remaining three hypotheses - or laws - added in this version, the last of them formalised what he had discovered in version one about the resistance of a medium. It asserted that the resistance of a meditun is proportional to its density, to the surface of the body moved, and to the velocity, all three taken conjointly. This peculiar ‘law’ experienced a peculiar fate. In the next revision, it suffered the indignity of an additional note in which Newton confessed that he did not affirm the law to be exact; it was sufficient that it be approximate. Whereupon he recognised the absurdity of what he was saying, and with a stroke of the pen removed it forever, as it deserved, from the laws of motion.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The other two laws of version three were less anomalous but in the end more troublesome. One asserted that the ‘relative motions of bodies contained in a given space are the same whether the space in question rests or moves perpetually and uniformly in a straight line without circular motion.’ The other complemented it by affirming that the common centre of gravity of a system of bodies does not change its state of motion or rest because of the mutual actions of the bodies on each other.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In a word, these two laws returned to the insights of the Waste Book and its realisation that in an inertial system two bodies isolated from outside influences can be considered as one concentrated at their common centre of gravity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For more details, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;John Herivel, &lt;a href=&quot;https://books.google.it/books?id=vn2BMgEACAAJ&quot; rel=&quot;nofollow&quot;&gt;The Background to Newton's Principia : A Study of Newton's Dynamical Researches in the Years 1664-84&lt;/a&gt; (1965), &lt;strong&gt;Ch.IX ; THE TRACT&lt;/strong&gt; &lt;em&gt;DE MOTU&lt;/em&gt;, page 257-on, with discussion, texts, translations and commentary of the &quot;five known versions of the tract &lt;em&gt;de Motu&lt;/em&gt; [the three MSS.IXa, IXb and IXc of the Cambridge U.L., named : &lt;em&gt;De Motu Corporutn in Gyrum&lt;/em&gt;].&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;From page 304-on there is the text with translation of MSS.Xa of the Cambridge U.L., named : &lt;em&gt;De Motu Corportim in medijs regulariter cedentibus&lt;/em&gt; and of MS.Xb, named &lt;em&gt;De Motu Corporum&lt;/em&gt;. The last one includes Definitions only, while the first one has six &lt;em&gt;Leges Motus&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is also probable that they succeeded Version III of the tract &lt;em&gt;de Motu&lt;/em&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here are the six &lt;em&gt;Leges Motus&lt;/em&gt; of MS.Xa [Herivel transl.] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Law 1. By reason of its innate force every body preserves in its state of rest or of moving uniformly in a straight line unless in so far as it is obliged to change its state by forces impressed on it. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Law 2. The change of motion is proportional to the force impressed and takes place along the straight line in which the force is impressed. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Law 3. As much as any body acts on another so much does it experience in reaction. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Law 4. The relative motion of bodies enclosed in a given space is the same whether that space rests absolutely or moves perpetually and uniformly in a straight line without circular motion. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Law 5. The common centre of gravity of [a number of] bodies does not change its state of rest or motion by reason of the mutual actions of the bodies. This law and the two above mutually confirm each other.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Law 6. The resistance of a medium is jointly proportional to the density of that medium, the area of the moved spherical body and the velocity. I do not assert this law to be exact. It suffices that it should be approximately true. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-08-04T08:25:50.710" LastActivityDate="2015-08-04T08:25:50.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2456" PostTypeId="1" AcceptedAnswerId="2457" CreationDate="2015-06-20T04:56:00.097" Score="2" ViewCount="775" Body="&lt;p&gt;Scientist predict that light moves 3x10&lt;sup&gt;8&lt;/sup&gt; m/s in a vacuum. Then, when light hits a medium like air, it goes into a medium with a different index and travels slower (I believe it indexes at 1.007 in air). So my question is, how have scientists come to the conclusion that light travels as fast as it does in a vacuum, because can't we only observe it through a medium?&lt;/p&gt;&#xA;" OwnerUserId="2474" OwnerDisplayName="Dashboarrd" LastEditorUserId="62" LastEditDate="2015-06-21T19:43:31.180" LastActivityDate="2015-06-21T19:43:31.180" Title="How did we come to the conclusion that light moves as fast as it does?" Tags="|physics|light|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2457" PostTypeId="2" ParentId="2456" CreationDate="2015-06-21T06:26:01.587" Score="0" Body="&lt;p&gt;The very fact that light has finite speed was first discovered in astronomy. See for example the Wikipedia article on Aberration of light. Light which is responsible for this phenomenon travels in the empty space. By the way the difference of speed in the air and in vacuum is small, and the first precise measurement of this speed were made in the air. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude from these measurements what is the speed in vacuum is easy by combining with an astronomical observation. Indeed, the difference of light speed in vacuum and in the air is responsible for the phenomenon which is called (astronomical) refraction. And refraction of the light coming from a star can be very exactly measured. If you know the speed of light in the atmosphere, you can compute the speed in vacuum. This is the general principle.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-06-21T06:26:01.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2458" PostTypeId="2" ParentId="2454" CreationDate="2015-06-21T23:03:56.683" Score="4" Body="&lt;p&gt;&amp;quot;De Motu&amp;quot; that Mauro describes is De Motu Corporum in Gyrum, which first appeared in November 1684. In the early 1685 Newton also produced another draft called &lt;a href=&quot;http://www.newtonproject.sussex.ac.uk/view/texts/normalized/NATP00091&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Motu Corporum in Mediis Regulariter Cedentibus&lt;/a&gt;, which is available on the Newton Project website. The main difference is that Law 3 is now the familiar law of action and reaction, and Laws 3-5 of the previous version become Laws 4-6. The last Law 6 does not appear in the transcribed version because it is crossed out in the &lt;a href=&quot;https://www.repository.cam.ac.uk/bitstream/handle/1810/194266/Add.3965.Sec5a.24%2cr.jpg?sequence=3&quot; rel=&quot;nofollow noreferrer&quot;&gt;handwritten copy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Law 4 of 1685 is the Galileo's principle of relativity, and some modern authors call it &lt;a href=&quot;http://www.mathsman.co.uk/Special%20Relativity%20and%20Electromagnetism.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Newton's fourth law&amp;quot;&lt;/a&gt;. Newton demoted it to a corollary in Principia (despite keeping Law 1, which is also a corollary) because he no longer believed it to be foundational. Concluding that relativistic notions of Galileo and Descartes did not offer solid enough foundation for dynamics, Newton decided to introduce the absolute space. The rest and the  motion in De Motu and in Principia are always understood in absolute terms. Commenting on the revision of De Motu, Newton remarked that &amp;quot;&lt;em&gt;all phenomena depend on absolute quantities... Ordinary people who fail to abstract thought from sensible appearances always speak of relative quantities so much so that it would be absurd for wise men or even Prophets to speak to them otherwise... In philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them&lt;/em&gt;&amp;quot;. See &lt;a href=&quot;https://koleksibuku191177.files.wordpress.com/2015/03/rob_iliffe_a_very_short_introductionbookza-org.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Iliffe's Newton: A Very Short Introduction&lt;/a&gt; (p. 94).&lt;/p&gt;&#xA;&lt;p&gt;Here is the 1685 list:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lex 1: By virtue of the law inherent in it every body persists in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by impressed forces.&lt;/p&gt;&#xA;&lt;p&gt;Lex 2: Alteration of motion is always proportional to the force impressed; and is made in the straight line along which it is impressed.&lt;/p&gt;&#xA;&lt;p&gt;Lex 3: As much as any body acts on another, so much does it experience in reaction. Whatever draws or presses another is as much drawn or pressed by it.&lt;/p&gt;&#xA;&lt;p&gt;Lex 4: The movements of bodies included in a given space are the same among themselves, whether that space is at rest or moves in absolute terms uniformly in a straight line without any circular motion.&lt;/p&gt;&#xA;&lt;p&gt;Lex 5: The movement or rest of the common center of gravity does not change its state by mutual actions of the bodies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-06-25T04:21:35.590" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2459" PostTypeId="1" CreationDate="2015-06-23T16:07:50.450" Score="3" ViewCount="82" Body="&lt;p&gt;If you search for the definition of analytic and holomorphic functions in books and online, you get crosses between two definitions (one involving taylor series and one differential) I can find no clear consensus on which definition holds for analytic and which for holomorphic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know they are mathematically equivalent but I want to know how they where originally defined? (please could you give a reliable source with your answer)   &lt;/p&gt;&#xA;" OwnerUserId="1994" LastActivityDate="2015-06-23T16:07:50.450" Title="Analytic and holomorphic functions, definitions and foundations?" Tags="|mathematics|complex-analysis|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2460" PostTypeId="1" CreationDate="2015-06-23T17:51:34.050" Score="7" ViewCount="6096" Body="&lt;p&gt;I was studying Aristotle's laws of motion and comparing them to Newton's. He states that heavier bodies fall faster than lighter ones. I really can't understand how he could have committed such a mistake. It seems obvious that two stones with different weights would reach the ground at the same time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, he believed that after applying a force to an object that object will move with a constant speed. But in the falling experiment we see that the speed will increase for normal objects like stones (not something like leather whose speed is influenced by air resistance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did Aristotle make these mistakes?&lt;/p&gt;&#xA;" OwnerUserId="2274" LastEditorUserId="4251" LastEditDate="2021-09-14T17:37:30.930" LastActivityDate="2021-09-14T17:37:30.930" Title="Why did Aristotle make mistakes in his laws of motion?" Tags="|physics|philosophy-of-science|ancient-greece|aristotle|antiquity|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2462" PostTypeId="2" ParentId="2460" CreationDate="2015-06-23T19:25:12.583" Score="15" Body="&lt;p&gt;Air or more generally medium resistance was not yet treated as a separate effect in Aristotle's time. Nor was there a clear idea of motion in a vacuum, in fact most ancient Greek philosophers, including Aristotle, did not believe that vacuum exists. So he had to explain phenomena as they are observed, resistance and all, and without the benefit of experiments where different factors can be controlled for. His prototypical example was a rock and a feather, which do not fall at the same speed. That different weights would fall simultaneously if effects of resistance can be neglected is not obvious at all unless you perform an experiment, and that did not become common practice in science until 17th century, two millennia after Aristotle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://arxiv.org/pdf/1312.4057.pdf&quot;&gt;Aristotle’s Physics: a Physicist’s Look&lt;/a&gt; Rovelli explains that in a resisting medium the steady speed attained by a body after a transient phase is indeed proportional to the force causing the motion, so his &quot;law of motion&quot; does hold in the limit of high friction/resistance. As for the free fall, it is not that easy to notice the increase in speed over the short periods of time that everyday objects take to fall from small heights (people did not deliberately drop stones from tall towers in his time). Aside from that, Aristotle distinguished between violent motions, caused by a force, and natural motions, that do not require a force, free fall is one of the natural ones. Another one is the circular motion of stars in the heaven. Aristotle's &quot;law&quot; therefore applies to neither.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This being said, Aristotle's theories of free fall and projectile motion were criticized already in late antiquity, e.g. by &lt;a href=&quot;http://en.wikipedia.org/wiki/John_Philoponus&quot;&gt;John Philoponus&lt;/a&gt; (c. 490 –  570 AD), who anticipated modern empirical approach:&quot;&lt;em&gt;But this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small&lt;/em&gt;&quot;.   &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-23T19:38:32.727" LastActivityDate="2015-06-23T19:38:32.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2463" PostTypeId="1" CreationDate="2015-06-23T20:29:42.433" Score="20" ViewCount="614" Body="&lt;p&gt;Who discovered the non-obvious identity $\int \frac 1x dx=\ln(x)+c$ ? Were power series involved? The series look similar on the opposite sides of $1$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.wolframalpha.com/input/?i=1/x&quot;&gt;$$ \frac 1x =\sum_{n=0}^\infty (-1+x)^n \text{ for } |x-1|&amp;lt;1 $$&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.wolframalpha.com/input/?i=ln%28x%29&quot;&gt;$$ \ln(x) = \ln(-1+x)-\sum_{k=1}^\infty \frac{(-1)^k}{k(-1+x)^k} \text{ for } |x-1|&amp;gt;1 $$&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \ln(x) = -\sum_{k=1}^{\infty}\frac{(-1)^k(-1+x)^k}k \text{ for } |x-1|&amp;lt;1 $$&lt;/p&gt;&#xA;" OwnerUserId="2483" LastEditorUserId="55" LastEditDate="2015-06-23T21:30:42.067" LastActivityDate="2015-06-23T21:30:42.067" Title="Who first realized that $\int \frac 1x dx =\ln(x)+c$?" Tags="|mathematics|discoveries|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2464" PostTypeId="2" ParentId="2463" CreationDate="2015-06-23T20:33:20.423" Score="24" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;By about 1640, the solution to the &amp;quot;area problem&amp;quot; for curves with equation &lt;span class=&quot;math-container&quot;&gt;$Y^n = aX^m$&lt;/span&gt;  was known by Fermat  for all  integer cases except when &lt;span class=&quot;math-container&quot;&gt;$n = 1, m = -1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I.e., the only unsolved area problem was for &lt;span class=&quot;math-container&quot;&gt;$Y = \frac 1X$&lt;/span&gt;  - the standard equation for the graph of a hyperbola.&lt;/p&gt;&#xA;&lt;p&gt;In 1647, &lt;a href=&quot;http://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincent&quot; rel=&quot;noreferrer&quot;&gt;Gregoire de St. Vincent&lt;/a&gt; showed the following special property for hyperbolas:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/DFzn4.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;If: &lt;span class=&quot;math-container&quot;&gt;$\frac ab = \frac cd$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\int_a^b\frac 1x dx=\int_c^d\frac 1x dx$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1649 &lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Alphonse_Antonio_de_Sarasa&quot; rel=&quot;noreferrer&quot;&gt;Alfonso Antonio de Sarasa&lt;/a&gt;&lt;/strong&gt; recognized this feature in Gregoire's work  and connected it to the properties of logarithms. In particular he recognized the following additive property of these measurements &lt;strong&gt;which had previously be a key feature in the study of common logarithms (base 10)&lt;/strong&gt;: the area determined by a product of two numbers , bd,  is equal to the sum of the areas determined by b and d separately.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\frac ab = \frac cd$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c=1$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\frac 1b \not= \frac 1d$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$\frac 1b + \frac 1d = \frac {1}{bd}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is the same as saying what RGB did:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{1}^{bd}\frac{\text{d}t}{t}=\int_{1}^{b}\frac{\text{d}t}{t}+\int_{1}^{d}\frac{\text{d}t}{t}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(bd)=f(b)+f(d)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ A^b A^d=A^{b+d}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus a logarithmic relationship was implied and A was found to be e.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_1^b \frac 1x dx = \ln(b)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int_1^d \frac 1x dx = \ln(d)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \int_1^{bd} \frac 1x dx = \ln(b)+\ln(d)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://users.humboldt.edu/flashman/Presentations/Estimations.html&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Hyperbolic_angle&quot; rel=&quot;noreferrer&quot;&gt;Hyperbolic Angle&lt;/a&gt; &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;&amp;quot; is defined in relation to the function &lt;span class=&quot;math-container&quot;&gt;$\frac 1x$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZeBqg.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;The magnitude of the hyperbolic angle &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;&amp;quot; is the area of the corresponding &lt;a href=&quot;http://en.wikipedia.org/wiki/Hyperbolic_sector&quot; rel=&quot;noreferrer&quot;&gt;hyperbolic sector&lt;/a&gt; (red above, yellow below) which is ln x since it is defined as the integral of the projection onto the axis from a=1 or y=x to b: &lt;span class=&quot;math-container&quot;&gt;$\int_1^b \frac 1x dx=\ln(b)$&lt;/span&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/oyrzm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;While power series were not involved in the discovery of the natural logarithm, they can be used to show the &lt;a href=&quot;http://www.freestudy.co.uk/analytical/outcome1t2.pdf&quot; rel=&quot;noreferrer&quot;&gt;relationship between &lt;span class=&quot;math-container&quot;&gt;$\frac 1x$&lt;/span&gt; and e&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;If a function is its own derivative then: &lt;span class=&quot;math-container&quot;&gt;$ y=f(x), \frac{dy}{dx}=y$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Such a function is a series: &lt;span class=&quot;math-container&quot;&gt;$ y=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy}{dx}=0+1+\frac{2x}{2!}+\frac{3x^2}{3!}+\frac{4x^3}{4!}\dots=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+ \dots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we differentiate &lt;span class=&quot;math-container&quot;&gt;$y^p$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$ \frac{d(y^p)}{dx}=py^{p-1}\frac{dy}{dx}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$ \frac{dy}{dx}=y$&lt;/span&gt; it follows: &lt;span class=&quot;math-container&quot;&gt;$ \frac{d(y^p)}{dx}=py^{p-1}y=py^p$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Given: &lt;span class=&quot;math-container&quot;&gt;$y_1=1+z+\frac{(z)^2}{2!}+ \frac{(z)^3}{3!}+\dots$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy_1}{dz}= 1+z+\frac{(z)^2}{2!}+ \frac{(z)^3}{3!}+\dots=y_1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$z=ax$&lt;/span&gt;,&lt;span class=&quot;math-container&quot;&gt;$\frac{dy_1}{d(ax)}=y_1$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{dy_1}{d(x)}=ay_1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$y_1=y^p= 1+px+\frac{(px)^2}{2!}+ \frac{(px)^3}{3!}+\dots=y_1$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we let &lt;span class=&quot;math-container&quot;&gt;$p=\frac 1x$&lt;/span&gt; then, &lt;span class=&quot;math-container&quot;&gt;$y^{\frac 1x}=1+1+\frac{(1)^2}{2!}+ \frac{(1)^3}{3!}+\dots=e$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And Hence replacing &lt;span class=&quot;math-container&quot;&gt;$e^1$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt; yeilds:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$e^x=1+x+\frac{(x)^2}{2!}+ \frac{(x)^3}{3!}+\dots$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This all lead to the development of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Slide_rule&quot; rel=&quot;noreferrer&quot;&gt;Slide Rule&lt;/a&gt; as well as &lt;a href=&quot;https://math.stackexchange.com/questions/455480/how-were-hyperbolic-functions-derived-discovered&quot;&gt;hyperbolic trig functions&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2483" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-06-23T20:33:20.423" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2465" PostTypeId="1" AcceptedAnswerId="2467" CreationDate="2015-06-24T21:08:27.303" Score="25" ViewCount="2861" Body="&lt;p&gt;While browsing on Physics Stack Exchange, I discovered numerous posts (&lt;a href=&quot;https://physics.stackexchange.com/questions/190959/why-is-the-momentum-of-a-particle-gamma-mv&quot;&gt;these&lt;/a&gt; &lt;a href=&quot;https://physics.stackexchange.com/questions/178553/what-is-a-bodys-momentum-really-equal-to&quot;&gt;two&lt;/a&gt; are examples) which assert that relativistic mass is an outdated concept used in older textbooks. Looking through my own physics textbooks, I noticed the same dichotomy, of relativistic mass being treated in older texts. I then searched up this concept in &lt;a href=&quot;http://en.wikipedia.org/wiki/Mass_in_special_relativity&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; wherein it outlined both approaches. It includes a quote from Arnold B Arons:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a troublesome and dubious concept&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which seems to corroborate the idea that relativistic mass is outdated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on these points, I have the following questions: What are the origins of this movement which question the concept of relativistic mass? When did textbooks begin to abandon the old approach and adopt the new approach? What are the reasons for the abandoning of relativistic mass, beyond those outlined by Arnold B. Arons and Taylor and Wheeler (which are provided in wikipedia, whose explanation I still don't completely understand)?&lt;/p&gt;&#xA;" OwnerUserId="2171" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2023-03-28T17:35:06.160" Title="When and why did the concept of relativistic mass become outdated?" Tags="|theoretical-physics|relativity-theory|education|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2467" PostTypeId="2" ParentId="2465" CreationDate="2015-06-25T00:38:55.763" Score="26" Body="&lt;p&gt;The &amp;quot;movement&amp;quot; against relativistic mass was started by Adler in 1987 with &lt;a href=&quot;https://web.archive.org/web/20210507001927/http://sites.fas.harvard.edu/%7Ephys191r/References/b5/Adler1987.pdf&quot; rel=&quot;noreferrer&quot;&gt;Does Mass Really Depend on Velocity, Dad?&lt;/a&gt; (his answer, &amp;quot;&lt;em&gt;actually no, but don't tell your teacher&lt;/em&gt;&amp;quot;). It got a boost from Okun's two 1989 papers. In 1990 American Journal of Physics solicited Okun's contribution on the relativistic mass to be published alongside Sandin's, who defended it. Okun declined because he was helping Taylor and Wheeler write a dialog on the subject in the second addition of their book, and published a &lt;a href=&quot;https://books.google.com/books?id=mVrGSMPBSn8C&amp;amp;dq=okun+relations+of+particles&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;book of his own&lt;/a&gt; that addresses it.&lt;/p&gt;&#xA;&lt;p&gt;Two decades later in &lt;a href=&quot;http://www.stat.physik.uni-potsdam.de/%7Epikovsky/teaching/stud_seminar/einstein_okun.pdf&quot; rel=&quot;noreferrer&quot;&gt;The mass versus relativistic and rest masses&lt;/a&gt; Okun did not sound as optimistic as Physics SE:&amp;quot;&lt;em&gt;The article “In defense of relativistic mass” by Sandin was published in 1991 and propagated the concept of relativistic mass for almost 20 years. In August 2008 I read a recent Wikipedia article “Mass in special relativity” which confronts the arguments of Adler and myself with arguments of Sandin. The unknown author in Wikipedia seems to share the views that relativistic mass and rest mass are ‘pedagogically useful’. The idea that according to the theory of relativity the mass of a body increases with its velocity is spread almost as widely as twenty years ago. It is promoted by the very popular books by Hawking and Jammer&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Here is the gist of Adler-Okun's argument as expressed by Okun in 2008:&amp;quot;&lt;em&gt;The concept of relativistic mass, which increases with velocity, appears only in the framework of a language which is not compatible with the standard language of relativity theory and therefore impedes the understanding and learning the theory by beginners... The four equations... defining relativistic and rest masses completely ignore the space-time relativistic symmetry. Relativistic mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; in these equations is a kind of a stub of a four-vector.These equations&#xA;are based on a naive but unfounded requirement that masses in relativity must have the property of additivity. They are based on the purely non-relativistic concept of mass as the measure of inertia. It is well known that in relativity this role is played not by mass but by energy&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;He then traces the root of the problem to Einstein's famous equation:&amp;quot;&lt;em&gt;Sandin keeps stressing that relativistic mass is the best concept for introducing relativity to ‘introductory students’. He thinks so because all of them know ‘the famous equation &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt;. This equation... is indeed deeply ingrained in the mass culture and widely believed to represent relativity theory. But unfortunately it misrepresents the standard theory of relativity as known to a majority of relativity experts. One consequence of the equation is the mass increasing with velocity. Though in his popular writings Einstein sometimes resorted to this equation, he carefully explained that ‘it is not good’ to introduce velocity dependent mass. He kept insisting that mass increases with energy but not with velocity and that this is expressed by the formula &lt;span class=&quot;math-container&quot;&gt;$E_0=mc^2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$E_0$&lt;/span&gt; is the rest energy of a body&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-05-06T17:40:58.203" LastActivityDate="2022-05-06T17:40:58.203" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="2468" PostTypeId="1" CreationDate="2015-06-25T10:49:54.717" Score="8" ViewCount="9578" Body="&lt;p&gt;With the restriction of physicists after the year 1900, were there any self-taught physicists that achieved fame without having gone to university?&lt;/p&gt;&#xA;" OwnerUserId="2521" OwnerDisplayName="buzhidao" LastEditorUserId="2515" LastEditDate="2015-06-29T18:32:19.973" LastActivityDate="2015-07-22T03:07:05.587" Title="Are there any famous physicists that never went to university?" Tags="|physics|biographical-details|education|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2469" PostTypeId="1" CreationDate="2015-06-25T13:17:35.077" Score="3" ViewCount="272" Body="&lt;p&gt;The most common proof of the &lt;a href=&quot;http://en.wikipedia.org/wiki/Inverse_function_theorem#Statement_of_the_theorem&quot; rel=&quot;nofollow&quot;&gt;Inverse Function Theorem&lt;/a&gt; in textbooks relies on the &lt;a href=&quot;http://en.wikipedia.org/wiki/Banach_fixed-point_theorem&quot; rel=&quot;nofollow&quot;&gt;Banach fixed-point theorem&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;QUESTION.&lt;/strong&gt; &#xA;  It is possible to say, using historical references, which mathematician came up with the idea of using Banach's Contraction Principle to prove the inverse function theorem? If not, what is the earliest known source where such proof appeared?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I did a search on the Mathematics history books specialized in the subject, but I did not find any clues. For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Krantz, Steven G.; Parks, Harold R. &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/146145980X&quot; rel=&quot;nofollow&quot;&gt;The implicit function theorem. History, theory, and applications. Birkhäuser Boston, Inc., Boston, MA, 2002.&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dontchev, Asen L.; Rockafellar, R. Tyrrell &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387878203&quot; rel=&quot;nofollow&quot;&gt;Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht, 2009&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1846" LastEditorUserId="55" LastEditDate="2015-06-25T20:44:58.390" LastActivityDate="2015-06-25T20:44:58.390" Title="Who first used Banach's Contraction Principle to prove the Inverse Function Theorem?" Tags="|mathematics|discoveries|real-analysis|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2470" PostTypeId="1" AcceptedAnswerId="2471" CreationDate="2015-06-25T19:04:24.590" Score="3" ViewCount="248" Body="&lt;p&gt;Ideally, I would prefer not a book on a specific topic (like QM) but a general survey, analogous to &lt;em&gt;A History of Mathematics&lt;/em&gt; by Carl Boyer. If such a work does not exist, can someone suggest works (multiple volumes if need be) which can satisfy my curiosity about the history of physics? Thanks. &lt;/p&gt;&#xA;" OwnerUserId="2410" LastActivityDate="2015-06-26T00:21:05.413" Title="Reference Request: Books on the (general) History of Physics" Tags="|reference-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2471" PostTypeId="2" ParentId="2470" CreationDate="2015-06-25T20:20:09.550" Score="3" Body="&lt;p&gt;The authoritative source is &lt;a href=&quot;https://global.oup.com/academic/product/the-oxford-handbook-of-the-history-of-physics-9780199696253?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow&quot;&gt;The Oxford Handbook of the History of Physics&lt;/a&gt;. From description:&quot;&lt;em&gt;Comprehensive coverage, treating all aspects of physics over a period of more than four hundred years... brings together cutting-edge writing by more than twenty leading authorities on the history of physics from the seventeenth century to the present day&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A popular &quot;general&quot; book getting good reviews is &lt;a href=&quot;https://books.google.com/books/about/A_Cultural_History_of_Physics.html?id=Bmcpsgp-Ml4C&quot; rel=&quot;nofollow&quot;&gt;Cultural History of Physics by Simonyi&lt;/a&gt;. I have not read it personally, but it has &lt;a href=&quot;https://edge.org/conversation/a-cultural-history-of-physics&quot; rel=&quot;nofollow&quot;&gt;Dyson's&lt;/a&gt; and &lt;a href=&quot;http://www.cut-the-knot.org/books/Reviews/CulturalHistory.shtml&quot; rel=&quot;nofollow&quot;&gt;Bogomolny's&lt;/a&gt; approval. Here is Bogomolny:&quot;&lt;em&gt;the coverage of the scientific developments in the West is undoubtedly comprehensive. After discussing philosophy of science and historiography, the author notes one of the first documented abstractions, to wit, an image of bison on cave walls, and ends with the Standard Model of quantum mechanics; mentions of quantum chromodynamics, Grand Unified Theory, and TOE - theory of everything... One characteristic of the book is abundance of diagrams that represent various ideas in compact, visual format&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shorter general histories by &lt;a href=&quot;http://archive.org/stream/historyofphysics030356mbp/historyofphysics030356mbp_djvu.txt&quot; rel=&quot;nofollow&quot;&gt;von Laue&lt;/a&gt; and &lt;a href=&quot;https://www.trentu.ca/physics/history_895.html&quot; rel=&quot;nofollow&quot;&gt;Slavin&lt;/a&gt; are available online.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-06-26T00:21:05.413" LastActivityDate="2015-06-26T00:21:05.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2472" PostTypeId="1" CreationDate="2015-06-27T11:06:17.983" Score="3" ViewCount="1268" Body="&lt;p&gt;Who came up with the laws of conservation of momentum? I'm more specifically interested in  the conservation of angular momentum.&lt;/p&gt;&#xA;" OwnerUserId="1700" LastEditorUserId="62" LastEditDate="2015-06-27T17:40:40.587" LastActivityDate="2016-04-29T10:11:56.057" Title="Who came up with the laws of conservation of momentum?" Tags="|mathematics|physics|classical-mechanics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2473" PostTypeId="2" ParentId="2427" CreationDate="2015-06-28T08:49:44.147" Score="4" Body="&lt;p&gt;It isn't entirely clear that the cynical view that science was just tacked onto a political project is correct. The whole issue of science in the Apollo program is discussed very well indeed in &quot;&lt;a href=&quot;http://store.doverpublications.com/0486478882.html&quot; rel=&quot;nofollow&quot;&gt;Where no man has gone before&lt;/a&gt;&quot; by William David Compton, a serious historian who looked into this question in detail. To really answer the question, you have to say &lt;em&gt;who's&lt;/em&gt; driving force. If you're talking about JFK, he had one motivation. If you're talking about the members of congress, they had other motivations. If you're talking about the engineers at NASA, they had other motivations. And then there were the astronauts and the scientists who participated in the design of experiments, and then the various top scientific bodies in the USA. And of course there were also the taxpayers who paid for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was in fact a constant to-and-fro between NASA and the scientists. NASA leadership was very insistent that the absolute maximum of science must come out of the project. There were very intensive studies to determine the best sites on the Moon and the best equipment to take with them in order to answer the maximum number of scientific questions. There was also a clear and explicit statement from NASA that engineering came first until the first landing was achieved. After that, the objectives were to be scientific. If the goal had been only political, they would have stopped after the first landing! But multiple landings had always been planned from the very beginning.&lt;/p&gt;&#xA;" OwnerUserId="2502" LastActivityDate="2015-06-28T08:49:44.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2474" PostTypeId="2" ParentId="2414" CreationDate="2015-06-28T09:11:45.720" Score="2" Body="&lt;p&gt;If you look at Weyl's 1918 &lt;a href=&quot;http://www.springer.com/gp/book/9783540182900&quot; rel=&quot;nofollow&quot;&gt;Raum, Zeit, Materie&lt;/a&gt;, you will see in the index three kinds of connection listed under the word &quot;Zusammenhang&quot;, which we now translate as &quot;connection&quot;. His three kinds of connections were continuous, affine and metric. Weyl clearly thought of the affine connection as an intermediate stage of connectivity between topological connectivity and the full Riemannian metric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ricci and Levi-Civita had introduced the covariant derivative much earlier, about 1900/1901, but Levi-Civita did not talk about connections even in his 1925/1926 book on the absolute differential calculus. Levi-Civita's first mention of parallel transport was in a 1917 paper &quot;Nozione di parallelismo ...&quot;, but he did not use any word like &quot;connection&quot;. Since we nowadays identify parallel transport with an affine (or other kind of) connection, it could be said that the notion of a connection originated in 1917 with Levi-Civita. Weyl directly acknowledge Levi-Civita for this on page 94 of &quot;Raum, Zeit, Materie&quot;. But it was Weyl who coined the term &quot;Connection&quot; (&quot;Zusammenhang&quot;) by analogy with topological connectivity to mean the idea of local parallel transport of vectors. Thus the motivation for the name &quot;connection&quot; was the analogy with topological connectivity.&lt;/p&gt;&#xA;" OwnerUserId="2502" LastActivityDate="2015-06-28T09:11:45.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2475" PostTypeId="1" AcceptedAnswerId="2477" CreationDate="2015-06-28T11:35:49.330" Score="15" ViewCount="1915" Body="&lt;p&gt;Over the years, what language was spoken at the International Mathematical Congress? I assume it is English right now, but it was founded by two German mathematicians, so I assume the language was German once, especially since Göttingen (Germany) was the mathematical center of that time. Is this true? If so, when and why did it change form German to English? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I've written an own answer to this question lately, using the links and information others provided in answers. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-07-27T12:59:12.740" LastActivityDate="2015-07-27T12:59:12.740" Title="What language was spoken at the International Mathematical Congress?" Tags="|mathematics|language|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2476" PostTypeId="2" ParentId="2336" CreationDate="2015-06-28T12:12:18.793" Score="1" Body="&lt;p&gt;Lets look to some old $10,000 bills. &#xA;I think there is a good chance that it is between 1900 and 1918, looking to these bills. However it is just one case and I assume that is just changed gradually. Of course, when one starts using the comma is thousands separator, one uses the dot as decimal mark, otherwise it would become highly confusing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one from 1878 with a dot:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/FMO5e.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one from 1900 with a dot:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/SWfEA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is one from 1918 with a comma:&#xA;&lt;img src=&quot;https://i.stack.imgur.com/vqReZ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-06-28T18:48:33.073" LastActivityDate="2015-06-28T18:48:33.073" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2477" PostTypeId="2" ParentId="2475" CreationDate="2015-06-28T13:00:20.890" Score="18" Body="&lt;p&gt;The proceedings of the first congress were published as “Verhandlungen des ersten Internationalen Mathematiker- Kongresses in Zürich vom 9. bis 11. August 1897”. Most of the papers were in German, a good number were in French, and one only was in Italian. There do not seem to have been any contributions in English. At that time, professional mathematicians could be expected to understand German and French at the least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proceedings of this and the following congresses can be found here: &lt;a href=&quot;http://www.mathunion.org/ICM/ICM1897/&quot;&gt;http://www.mathunion.org/ICM/ICM1897/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-06-28T13:00:20.890" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2478" PostTypeId="2" ParentId="2475" CreationDate="2015-06-28T21:34:58.537" Score="17" Body="&lt;p&gt;Purely on the basis of the link given, the first congress in a non English speaking location which used an English title was Amsterdam 1954. The last one which used a local-language title was Nice 1970. From Vancouver 1974 onwards, all the titles are in English. The last one to specifically use a German title was Zürich 1932.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not that this proves anything about the lingua franca at each congress. The book &quot;Mathematicians of the World, Unite!&quot; by Guillermo Curbera mentions six languages including Latin that were used at the Bologna conference in 1928 (p88).&#xA;&lt;a href=&quot;https://books.google.co.uk/books?id=9uDqBgAAQBAJ&quot;&gt;https://books.google.co.uk/books?id=9uDqBgAAQBAJ&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user2507" LastActivityDate="2015-06-28T21:34:58.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2479" PostTypeId="2" ParentId="2475" CreationDate="2015-06-29T06:31:16.913" Score="8" Body="&lt;p&gt;Official languages of the early congresses used to be English, French, German and Italian. Also Russian for the congress in Moscow. Since 1970s most talks are given in English but other languages are not prohibited. For example, in Beijin 2002,&#xA;plenary speaker Lafforgue was speaking in French and showing transparencies in Chinese.&#xA;(I was present at the talk).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-06-29T06:31:16.913" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2480" PostTypeId="2" ParentId="2468" CreationDate="2015-06-29T10:38:49.243" Score="1" Body="&lt;p&gt;I'm not sure if you count applied physics (aerodynamics) as well, but if you do, I have an answer. &#xA;The Wright Brothers never finished high school and never went to university, and they invented their machine in 1905, so that is after 1900.&lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2015-06-29T10:38:49.243" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2481" PostTypeId="2" ParentId="2468" CreationDate="2015-06-29T17:35:57.870" Score="7" Body="&lt;p&gt;A similar question was asked to the theoretical physicist Gerald 't Hooft on his page &lt;a href=&quot;http://www.staff.science.uu.nl/~hooft101/theorist.html&quot;&gt;'Theoretical Physics as a Challenge'.&lt;/a&gt; It is mentioned at the bottom of the linked page:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mr. Hisham Kotry came with an important question:&lt;/strong&gt;&#xA;&quot;... Two years ago I decided to self-study theoretical physics by following the syllabus of a renown university and the advice from your page and now I'm half-way through the journey but I was wondering about what happens next? ...Do you know of anyone who got tenure at a physics department or any research institute based on studies he did in his own time without holding a university degree?&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;T'Hooft quoted the following message from a contributor to answer a part of the question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3/04/06: Message received from John Glasscock, Bloomington, IN:&lt;/strong&gt; &#xA;The only one I know of currently is John Moffatt at U Toronto, who was a student of Abdus Salam at Imperial College, London. He started life as a painter in Paris, had no undergraduate degree, taught himself, corresponded with Einstein, and was admitted, based on his demonstrated original work, at IC. (Source: Jo�o Magueijo, &lt;em&gt;Faster than the Speed of Light&lt;/em&gt;. Perseus Publishing, Cambridge, MA. 2003.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I'm not sure if the example I quoted satisfies the &quot;famous&quot; criteria you stated in your question, but I hope this helped! &lt;/p&gt;&#xA;" OwnerUserId="2515" LastEditorUserId="26" LastEditDate="2015-06-29T18:31:24.817" LastActivityDate="2015-06-29T18:31:24.817" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2482" PostTypeId="1" CreationDate="2015-06-29T01:16:38.153" Score="8" ViewCount="321" Body="&lt;p&gt;The main-stream view seems to be that &lt;a href=&quot;http://www.tgeorgiev.net/Gravitation_and_Electricity.pdf&quot;&gt;Weyl's 1918 paper &lt;em&gt;Gravitation and Electricity&lt;/em&gt;&lt;/a&gt; was initially considered a failure for reasons first pointed out by Einstein. But these initial ideas were reapplied to quantum mechanics giving rise to the idea of gauge-invariance as a symmetry principle in physics in his 1929 paper, &lt;em&gt;Electron and Gravitation&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With hindsight today, did any of the ideas contained in his 1918 paper apply to classical physics?&lt;/p&gt;&#xA;" OwnerUserId="2461" OwnerDisplayName="Larry Harson" LastEditorUserId="520" LastEditDate="2015-06-30T15:09:14.147" LastActivityDate="2015-06-30T15:09:14.147" Title="How did Weyl's 1918 paper; Gravitation and Electricity, influence classical physics?" Tags="|electromagnetism|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2485" PostTypeId="2" ParentId="2475" CreationDate="2015-06-30T00:38:56.107" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://www.bbc.com/news/magazine-29543708&quot; rel=&quot;nofollow&quot;&gt;Here&lt;/a&gt; is a nice article entitled &quot;Nobel Prize: How English beat German as language of science&quot; explaining why the language of the scientific world wasshifted from German to English. The most compact explanantion could be put in a single word, namely &quot;war&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in this article historian of science Michael Gordin fromPrinceton University gives fine details. I give you just a taste:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;German is criminalised in 23 [American] states. You're not allowed to speak it in public, you're not allowed to use it in the radio, you're not allowed to teach it to a child under the age of 10,&quot; Gordin explains.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2520" LastEditorUserId="62" LastEditDate="2015-07-01T23:46:19.953" LastActivityDate="2015-07-01T23:46:19.953" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2486" PostTypeId="1" AcceptedAnswerId="2494" CreationDate="2015-06-30T05:40:04.693" Score="21" ViewCount="3250" Body="&lt;p&gt;Reportedly this was uttered at a banquet in which Hilbert was seated next to the new Minister of Education, Bernhard Rust, in response to Rust inquiring as to the state of mathematics in Göttingen now that it had been &quot;freed of the Jewish influence&quot;. I have seen this quote in several places but it seems to be related as more of an anecdote with no particular reference. Is there any such reference that can serve as evidence for the quote, and if possible with any commentary on the reaction of those present at the banquet to this statement (which was rather inflammatory, given the context)?&lt;/p&gt;&#xA;" OwnerUserId="137" LastEditorUserId="4251" LastEditDate="2021-09-13T14:23:31.240" LastActivityDate="2021-09-13T14:23:31.240" Title="Source for Hilbert's famous quote &quot;Mathematics in Göttingen? There really is none anymore&quot;" Tags="|mathematics|reference-request|quote|hilbert|gottingen|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2487" PostTypeId="2" ParentId="2486" CreationDate="2015-06-30T11:12:26.867" Score="4" Body="&lt;p&gt;The German Wikipedia cites &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;D. Nachmansohn, R. Schmidt: Die große Ära der Wissenschaft in Deutschland 1900–1933, 1988, S. 55 &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;as the source of the original quote. However I am unable to find a free online version of this book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here S.55 means Seite 55, so page 55. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2015-06-30T11:12:26.867" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2488" PostTypeId="1" CreationDate="2015-06-30T16:36:00.963" Score="8" ViewCount="702" Body="&lt;p&gt;There is a notation used in differential geometry and general relativity in which the partial derivative operators &lt;span class=&quot;math-container&quot;&gt;$\partial_\mu$&lt;/span&gt; are used as the basis for the space of contravariant vectors, and their duals, notated &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;, are the basis for covariant vectors. I've found a couple of sources that attribute this notation to &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Cartan.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Élie Cartan&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;for [covectors] a cute notational game due to Élie Cartan, is played&amp;quot; -- Mathematics for Physics II, Michael Stone, cns.gatech.edu/~predrag/courses/PHYS-6124-12/StGoChap11.pdf&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Élie Cartan proposed to use differential coordinates &lt;span class=&quot;math-container&quot;&gt;$dx^i$&lt;/span&gt; as a convenient basis of 1-forms.&amp;quot; -- Kosyakov, Classical Theory of Particles and Fields, quoted at &lt;a href=&quot;https://physics.stackexchange.com/questions/144089/is-partial-derivative-a-vector-or-dual-vector&quot;&gt;https://physics.stackexchange.com/questions/144089/is-partial-derivative-a-vector-or-dual-vector&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cartan published his first paper in 1899 and lived until 1951. Can anyone give date, or a reference to the paper in which he first introduced this notation?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-12-12T07:15:19.787" Title="Where did Cartan introduce his notation for basis vectors and covectors?" Tags="|notation|differential-geometry|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2490" PostTypeId="1" CreationDate="2015-07-01T18:08:58.393" Score="8" ViewCount="1028" Body="&lt;p&gt;According to P. M. Cohn's &lt;em&gt;Classic Algebra&lt;/em&gt;, for historical reasons we call a linear mapping &amp;quot;linear mapping&amp;quot;. What are the historical reasons that led to the adoption of the term &amp;quot;linear mapping&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="2530" OwnerDisplayName="Chilote" LastEditorUserId="4251" LastEditDate="2021-11-24T12:23:19.447" LastActivityDate="2021-11-24T12:23:19.447" Title="Why do we call a linear mapping &quot;linear mapping&quot;?" Tags="|mathematics|terminology|reference-request|linear-algebra|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2491" PostTypeId="2" ParentId="2490" CreationDate="2015-07-01T18:40:48.873" Score="0" Body="&lt;p&gt;This is a technical rather than a historical reason.&lt;br&gt;&#xA;In a power series like $$A + Bx + Cx^2 + \dotsb $$ you've got a constant term $A$ and a linear term $Bx$ and a quadratic term $Cx^2$. Generalizing appropriately to higher dimensions gives you the notion of constant, linear and quadratic mappings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to be the source of the term &lt;em&gt;linear&lt;/em&gt; in &lt;em&gt;linear map&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15627" OwnerDisplayName="NaN" LastActivityDate="2015-07-01T19:09:59.733" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2492" PostTypeId="2" ParentId="2490" CreationDate="2015-07-02T05:26:50.613" Score="12" Body="&lt;p&gt;The theory of Linear Algebra, along with the associated concept of linear mapping, was named as &quot;linear&quot; by its creator, Hermann Graßmann, which he developed in his 1844 linear algebra manifesto, &lt;em&gt;Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik&lt;/em&gt;  [The Theory of Linear Extension, a New Branch of Mathematics], and also later in &lt;em&gt;Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet&lt;/em&gt; [The Theory of Extension, Thoroughly and Rigorously Treated], which was a thorough revision of the former. With his theory of Linear Extension, Graßmann was looking to provide an algebraic framework for geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In linear algebra, we deal with objects called &lt;em&gt;vectors&lt;/em&gt;, which geometrically often represent points in space as measured with reference to some fixed point which is labelled the &lt;em&gt;origin&lt;/em&gt;. The collection of all these vectors is called a &lt;em&gt;vector space&lt;/em&gt;. We can perform &lt;em&gt;scalar multiplication&lt;/em&gt; on these vectors, by stretching, shrinking, and/or reflecting them along the line passing through the origin and the point. Hence, the name of &lt;em&gt;linear extension&lt;/em&gt; seems rather fitting. A &lt;em&gt;linear map&lt;/em&gt; is thus a map which respects the &lt;em&gt;linear structure&lt;/em&gt; of its domain and range, as vector spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is the historical reason we call linear maps &lt;em&gt;linear&lt;/em&gt;. But that may not be the kind of linear map of which you may be thinking, as is evidenced by NaN's answer. Sometimes, the term &lt;em&gt;linear&lt;/em&gt; is applied to a somewhat  related, but different kind of map, namely a function of the form $ax+b$. This is linear in the linear algebra sense if and only if $b=0$. In that case all I can say is that they're probably just called linear because the graph of such a function in the plane is a straight line.&lt;/p&gt;&#xA;" OwnerUserId="137" LastActivityDate="2015-07-02T05:26:50.613" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2493" PostTypeId="2" ParentId="2490" CreationDate="2015-07-02T07:25:18.107" Score="6" Body="&lt;p&gt;The expression &quot;linear mapping&quot; can have different meaning in different contexts. In many high school textbooks $x\mapsto ax+b$ is called a &quot;linear function&quot;. I suppose, because the graph is a straight line. According to the modern terminology, which comes from linear algebra only $x\mapsto ax$ should be called linear, while $x\mapsto ax+b$ must be called affine.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-07-02T07:25:18.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2494" PostTypeId="2" ParentId="2486" CreationDate="2015-07-02T14:38:32.783" Score="18" Body="&lt;p&gt;The following is how far we get from the direct English and German Wikipedia references. I had a look at the German Wikipedia reference wythagoras pointed out. In D. Nachmansohn, R. Schmidt: Die große Ära der Wissenschaft in Deutschland 1900–1933, 1988, Stuttgart : Wiss. Verl.-Ges., I could only find one reference regarding Hilbert, namely, the Hilbert biography by Constance Reid, which is the reference for the entry on the English Wikipedia as well. Reid's report is a little less detailed than the English Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sitting next to the Nazis' newly appointed minister of education at a banquet, he was asked, &quot;And how is mathematics in Göttingen now that it has been freed of the Jewish influence?&quot;&#xA;  &quot;Mathematics in Göttingen?&quot; Hilbert repiled. &quot;There is really none any more.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(Page 205 of C. Reid, Hilbert, 2nd printing, 1972 Springer.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, Nachmansohn and Schmid seem to have had another reference (or they were exaggerating), because they give a different dialogue which shows a much more offensive Hilbert. Page 55 says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bei einer Zusammenkunft in Göttingen im Jahre 1934 fragte ihn der Reichsminister für Wissenschaft Bernhard Rust, wie es seiner Fakultät gehe. Hilbert entgegnete, welche Fakultät er meine. Als Rust sagte, er meine natürlich die Mathematisch-physikalische Fakultät, antwortete Hilbert, daß sie nicht mehr existiere, denn er, Rust, habe sie zerstört, indem er die besten Wissenschaftler fortgejagt habe [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The [...] just says for more details see the Hilbert biography by C. Reid 1970.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It translates as follows.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1934, Hilbert and the Minister of Science, Education and National Culture Bernard Rust met in Göttingen, where Rust asked ihm, how his department was. Hilbert asked what department he meant. When Rust said he certainly meant the Mathematics and Physics Departement, Hilbert replied it wouldn't exist any more, because Rust himself destroyed it by expelling the best scientists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The major difference is the following. According to Reid, it was Rust who asked not just how the mathematics department was, but how it was &quot;now that it has been freed of the Jewish influence&quot;. So here it was Rust who pointed out that there were no Jewish scientists any more, whilst according to Nachmansohn-Schmid, Rust only asked how the mathematics- and physics department was, and Hilbert replied, that there was none any more, and that Rust himself had destroyed it, by expelling the best scientists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have two references now. Unfortunately, neither of them gives informations about how the surrounding people (or Rust) reacted, they tell slightly different stories, and both don't give useful further references.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of the difference between the dialogues, of which I don't know is more trustworthy, I begin to doubt at least that Hilbert's words were (a German analogue of) &quot;Mathematics in Göttingen? There really is none anymore&quot;. I'd like to note that indeed Rust became the Minister of Science, Education and National Culture in 1934, but as a matter of fact, he kept his position; it's just the name of the Ministry that changed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, certainly, nothing about this story appears in the short biography written by Otto Blumenthal which appeared in the third volume of Hilbert's collected works in 1935. Nevertheless I'm advertising it for those interested in Hilbert's life and work!&lt;/p&gt;&#xA;" OwnerUserId="79" LastActivityDate="2015-07-02T14:38:32.783" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2495" PostTypeId="1" CreationDate="2015-07-03T04:56:37.633" Score="20" ViewCount="9697" Body="&lt;p&gt;I knew about linear approximations, quadratic approximations and the use of Taylor polynomials to approximate a function. Furthermore, I was aware of other applications of Taylor polynomials and the intuition behind them from this &lt;a href=&quot;http://davidlowryduda.com/?p=1520&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, the concept of Taylor series was discovered by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my main curiosity is about the problems and situations that resulted in a need to  approximate a function using Taylor series. Would someone please enlighten me?&lt;/p&gt;&#xA;" OwnerUserId="2534" LastEditorUserId="55" LastEditDate="2016-07-19T02:55:23.483" LastActivityDate="2023-08-13T06:07:47.560" Title="What was the historical context of the development of Taylor series?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2496" PostTypeId="2" ParentId="2488" CreationDate="2015-07-04T01:59:36.707" Score="2" Body="&lt;p&gt;Some sluething revealed to me that the first source was really Cartan's &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/MSM/MSM_1925__9_/MSM_1925__9__1_0/MSM_1925__9__1_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;La Géométrie des espaces de Riemann&lt;/em&gt;&lt;/a&gt;, published in 1925 three years before the &lt;a href=&quot;https://books.google.com/books?id=zBuxzExmSAkC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Lecons sur la geometrie des espaces de Riemann&lt;/em&gt;&lt;/a&gt;. Looking at the list of Cartan's works, this is the first work on the Riemannian differential geometry. The link for the first source is in the original French; About 60-80% of the text is understandable with only knowledge of English (from my experience, for I do not know French) because the French words for mathematical terms are very similar. For the basis of covariant vectors being $dx$, here are the relevant sections (three sections, close together but with paragraphs in between, images have been posted in French):&#xA;&lt;img src=&quot;https://i.stack.imgur.com/GCsFu.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/iQFdV.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/TFj8P.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This is the source for the statement &quot;&quot;Élie Cartan proposed to use differential coordinates dxi as a convenient basis of 1-forms,&quot; though his terminology is outdated in this context (no mention of basis or of differential forms, even though both are being discussed). I believe that in the second source linked (the one given by Mauro ALLEGRANZA) the notation is more modern (being more 'official' than this one) but I haven't checked. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is where the cute notational game is played. Although the partial derivate operators aren't explicitly shown, because the inner product of the contravariant and covariant bases equals the metric tensor (second image), by definition the contravariant base would be equivalent to the partial derivative operator. I believe that the more modern notation is in the second source linked, but as I have before stated, I haven't checked.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As an addition to this answer, wythagoras translated the French text to English. Note that he had to reform some of the sentences or split them in two sentences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Citation 1.&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An entire system of curvlinear coordinates $u^1$, $u^2$, ... $u^n$ in $n$-dimensional Euclidean space correspond to a determined form of linear elements of the space $$ds^2=\sum_{i,j} g_{ij} du^i du^j$$ where the coefficients $g_{ij}$ are certain functions in $u^1$, $u^2$, ... $u^n$. &lt;br&gt;&#xA;  The two fundamental problems that arise are the following:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The linear element of the Euclidean space is&lt;/em&gt; a priori &lt;em&gt;supposed to give rise to a point such that the curvlinear coordinates can be compared to the Cartesian coordinates.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Find the conditions under which the functions $g_{ij}$ are able to represent the $ds^2$ from the Euclidean space.&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Citation 2.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Imagine for each point $M$ the curvlinear coordinates $u^1$, $u^2$, ... $u^n$ a system of cartesian coordinates (R) with the origin on that point. Now consider the point on infinitesimal distance of $M$, namely $M'$, and compare it to  $du^1$, $du^2$, ... $du^n$. The coordinate axes are therefore tangents to the &lt;em&gt;line coordinates&lt;/em&gt; and the scalar products of the vectors of the coordinates $e_1, e_2, e_n$ is the quantity $g_{ij}$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Citation 3.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The conditions for integrability of equation (4) provide the other equations. First notice that the first member of equation (4), which we call $Dx^i$, represent the components of motion of a mobile point, of which the coordinates, comparing to (R), would be $x^i$. Likewise the components of the geometric variation of a vector $X^i$ would be: $$DX^i = dX^i + \sum_{k,r} \Gamma^i_{k,r}X^k du^r$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2171" LastEditorUserId="2442" LastEditDate="2015-07-05T08:53:28.400" LastActivityDate="2015-07-05T08:53:28.400" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2498" PostTypeId="1" AcceptedAnswerId="2499" CreationDate="2015-07-05T07:57:13.443" Score="7" ViewCount="4079" Body="&lt;p&gt;Why is Einstein's mass-energy relation usually written as $E=mc^2$, and not $\Delta E=\Delta m c^2$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you calculate the energy $\Delta E$ released during nuclear fission, you take the difference $\Delta m$ between the mass of the particles before and after nuclear fission to find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the second formula follows from the first, but in most contexts the second one is used. Which brings me to the question: why did the form $E=mc^2$ become so much more popular than $\Delta E=\Delta m c^2$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a historical reason?&lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="55" LastEditDate="2015-07-05T20:29:51.610" LastActivityDate="2019-09-14T05:50:33.933" Title="Why is Einstein's mass-energy relation usually written as $E=mc^2$, and not $\Delta E=\Delta m c^2$?" Tags="|physics|relativity-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2499" PostTypeId="2" ParentId="2498" CreationDate="2015-07-05T20:23:27.707" Score="11" Body="&lt;p&gt;Einstein is the one who removed the increments from the formula back in 1905, and one could say that the &quot;historical reason&quot; is his influence on theoretical physics. But his position was amply justified by subsequent experimental evidence. To explain, let us assume for now that energy and mass are considered in a frame where the object is stationary relative to the observer, and write &lt;span class=&quot;math-container&quot;&gt;$E_0=m_0c^2$&lt;/span&gt; to reflect that (there is a separate issue with &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt; that complicates the answer, I will address it below). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's original calculation in &lt;a href=&quot;http://arxiv.org/pdf/1407.8507v1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Does the inertia of a body depend on its energy content (1905)&lt;/a&gt; only considered mass changes effected by emission and absorption of electromagnetic radiation, which means that, indeed, he only derived &lt;span class=&quot;math-container&quot;&gt;$\Delta E_0=\Delta m_0c^2$&lt;/span&gt;, and only for electromagnetic processes. As pointed out by a number of authors, &lt;a href=&quot;http://arxiv.org/pdf/1407.8507v1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematically we only get &lt;span class=&quot;math-container&quot;&gt;$E_0-K=(m_0-q)c^2$&lt;/span&gt;&lt;/a&gt;, where &lt;span class=&quot;math-container&quot;&gt;$K, q$&lt;/span&gt; are some constants. &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; can be set to zero by calibration, but not &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Setting &lt;span class=&quot;math-container&quot;&gt;$q = 0$&lt;/span&gt; involves a hypothesis concerning the nature of matter, because it rules out the possibility that there exists matter that has mass, but which is such that some of its mass can never be “converted” into energy... Einstein's argument allows for the possibility that once a body's energy store has been entirely used up (and subtracted from the mass using the mass-energy equivalence relation) the remainder is not zero&lt;/em&gt;&quot;: &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, Einstein made two conjectures in his derivation of &lt;span class=&quot;math-container&quot;&gt;$E_0=m_0c^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1) Effect of energy transfer on mass is the same for all processes, as it is for electromagnetic ones.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2) There is no matter with residual mass, which never participates in energy transfers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first conjecture is, of course, a logical consequence of special relativity (and its predecessor, the hypothesis of molecular forces, introduced by Lorentz). It was introduced the same year in another paper, where Einstein interpreted relations derived from Maxwell's electrodynamics as kinematic relations that apply to all objects uniformly, and is vindicated by overwhelming evidence accumulated since then. But if that was not enough, Einstein gave another derivation in 1935, which is manifestly based on special relativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second conjecture is also supported by evidence, but it is independent of special relativity and the case for it is not as strong, although so far we have not encountered any matter with residual mass. The most direct evidence is annihilation experiments in particle physics, where massive particles are entirely reduced to radiation, but there are types of matter (exotic, dark) that we are scarcely familiar with, so residual mass is not entirely ruled out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, &lt;a href=&quot;https://physics.stackexchange.com/questions/91501/what-keeps-mass-from-turning-into-energy&quot;&gt;to get &lt;span class=&quot;math-container&quot;&gt;$E = mc^2$&lt;/span&gt;, one has to introduce the so-called &quot;relativistic mass&quot;&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$m:=\frac{m_0}{\sqrt{1-v^2/c^2}}$&lt;/span&gt;, which is ironically named because it is not Lorentz invariant (depends on velocity, and hence on reference frame), and therefore can not represent a physical property in relativity. Then &lt;span class=&quot;math-container&quot;&gt;$m_0$&lt;/span&gt;, which is in fact invariant, is called &quot;rest mass&quot;. Experts in relativity, starting with Einstein himself, consider velocity dependent mass an artificial construction, that should be dispensed with, and call the &quot;rest mass&quot; just mass denoted &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. In recent decades, this began to be reflected in textbooks. However, then &lt;span class=&quot;math-container&quot;&gt;$E = mc^2$&lt;/span&gt; becomes false and should also be dispensed with, but it is hard to overcome its entrenched status in popular culture, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2465/when-and-why-did-the-concept-of-relativistic-mass-become-outdated/2467#2467&quot;&gt;When and why did the concept of relativistic mass become outdated?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-14T05:50:33.933" LastActivityDate="2019-09-14T05:50:33.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2500" PostTypeId="2" ParentId="2488" CreationDate="2015-07-05T22:13:54.540" Score="1" Body="&lt;p&gt;Differentials and their linear combinations already appear in full glory in Cartan's first paper on Pfaffian systems &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1899_3_16_/ASENS_1899_3_16__239_0/ASENS_1899_3_16__239_0.pdf&quot; rel=&quot;nofollow&quot;&gt;Sur certaines expressions différentielles et le problème de Pfaff&lt;/a&gt; (1899), where he works out some of their algebra. But he was not the first to use them, the paper mentions  Pfaff, Natani, Clebsch, Lie, Frobenius and Darboux, and in &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1901_3_18_/ASENS_1901_3_18__241_0/ASENS_1901_3_18__241_0.pdf&quot; rel=&quot;nofollow&quot;&gt;L’intégration des systèmes d’équations aux différentielles totales&lt;/a&gt; (1901) he references Biermann's 1885 paper Ueber n simultané Differenlialgleichungen der Form $\sum X_\mu dx_\mu$. &quot;&lt;em&gt;It was Pfaff who pioneered the study of exterior differential systems by his formulation of the Pfaff problem in [Pfaff 1814-15]. The exterior derivative of a pfaffian form, called the bilinear covariant, was introduced by Frobenius in 1877 and efficiently used by Darboux in [Darboux 1882]&lt;/em&gt;&quot;, see &lt;a href=&quot;http://library.msri.org/books/Book18/MSRI-v18-Bryant-Chern-et-al.pdf&quot; rel=&quot;nofollow&quot;&gt;Exterior Differential Systems&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did they think of $dx_\mu$ as &quot;basis of $1$-forms&quot;? Certainly not Pfaff, linear algebra as we know it did not emerge until Sylvester and Cayley later into the century, but it does not mean that he could not play the &quot;notational game&quot; without the terminology. Cartan's innovation, which came in his 1922 book &lt;a href=&quot;https://archive.org/details/leonssurlesinv00cartuoft&quot; rel=&quot;nofollow&quot;&gt;Lecons sur les Invariants Integraux&lt;/a&gt;, was extending the exterior differential, and the game, to higher order forms, but he inherited manipulation of $d$'s from much earlier. In fact, at a formal level some of this game goes back to Leibniz, who would write differential equations in differentials to separate variables, etc. As for $\partial$'s the game, if not the notation, goes back to at least moving trihedrons in &lt;a href=&quot;https://archive.org/details/leconsurlagendes01darbrich&quot; rel=&quot;nofollow&quot;&gt;Darboux's Leçons sur la Théorie Générale des Surfaces&lt;/a&gt;, although it was Cartan who greatly advanced the method and generalized it to higher dimensions.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-07-05T22:13:54.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2502" PostTypeId="2" ParentId="2495" CreationDate="2015-07-06T20:00:09.657" Score="20" Body="&lt;p&gt;First Taylor series, specifically for sine, cosine and arctangent, were developed by Indian astronomers of Kerala school to facilitate astronomical calculations based on certain astronomical models which developed over time. The school existed from 13th to 16th century The first written reference is a book by Jyesthadeva from early 1500s. Another book by Nilakantha from before 1545 attributes the discovery to Madhava (1349-1425). The derivations used iteration of trigonometric sum and difference formulas, a method that goes back to Aryabhata (c.490) and  Brahmagupta (c.665). Unfortunately, the discovery remained confined to a small circle of practitioners in India, and did not become known in Europe until long after these series were independently rediscovered in 17th century. See &lt;a href=&quot;https://www.researchgate.net/publication/254616376_Was_Calculus_Invented_in_India&quot; rel=&quot;nofollow noreferrer&quot;&gt;Was Calculus Invented in India? by Bressoud&lt;/a&gt;, &lt;a href=&quot;http://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall14/128/CalcIslamIndia.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ideas of Calculus in Islam and India by Katz&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Madhava series&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In 17th century Europe the principal motivation for using power series came from the need to do effective computations with transcendental functions appearing in geometry, astronomy, cartography, navigation, and later mechanics. Briggs discovered the series for &lt;span class=&quot;math-container&quot;&gt;$(1-x)^{1/2}$&lt;/span&gt; in 1620s, while working with newly introduced logarithms, and in 1668 Mercator published Logarithmotechnia that contained the series for &lt;span class=&quot;math-container&quot;&gt;$\ln(1+x)$&lt;/span&gt;. Around 1664-1665 in his tour de force discovery of calculus Newton found what he called a &amp;quot;universal method&amp;quot; for expanding any function into a series. His method was quite convoluted however, it involved inversion, binomial expansion, long division and termwise integration. Newton used the series to approximate quadratures (areas), interpolate values of functions, and more generally solve what we now call differential equations. In 1669 he produced a manuscript De Analysi per Aequationes Numero Terminorum Infinitas (&lt;a href=&quot;http://www.britannica.com/topic/Newton-and-Infinite-Series-1368282&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Analysis by Equations with an Infinite Number of Terms&lt;/a&gt;), which however was rejected by the Royal Society. So only examples for trigonometric functions and &amp;quot;binomial expansion of fractional exponents&amp;quot;, i.e. &lt;span class=&quot;math-container&quot;&gt;$(1+x)^\alpha$&lt;/span&gt;, communicated through letters to Collins in 1670, came to be known.&lt;/p&gt;&#xA;&lt;p&gt;Like Newton, Gregory was interested in expanding general functions into a series, and in 1671 he thought he figured out Newton's method. But he did better, he actually discovered the familiar relation between values of derivatives and Taylor coefficients, about forty years before Taylor. Unfortunately, Gregory was isolated in his late years, and his work did not gain much recognition. Curiously, although the arctangent series was known to Gregory since 1670 he did not think to plug &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; into it and get an infinite sum for &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt;. Leibniz thought of it several years later, after rederiving the expansion, it was his first major mathematical work. See &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roy's Discovery of the Series formula for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; by Leibniz, Gregory and Nilakantha&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-08-13T06:07:47.560" LastActivityDate="2023-08-13T06:07:47.560" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="2503" PostTypeId="1" CreationDate="2015-07-06T21:57:27.283" Score="13" ViewCount="4202" Body="&lt;p&gt;How did theoretical physicist Jun John Sakurai die? The only result from extensive googling is that he died &quot;suddenly&quot; while working at CERN. Does anyone know anything more specific?&lt;/p&gt;&#xA;" OwnerUserId="2544" LastEditorUserId="4251" LastEditDate="2021-12-02T08:06:31.430" LastActivityDate="2021-12-02T08:06:31.430" Title="How did Jun John Sakurai die?" Tags="|biographical-details|physicists|death|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2504" PostTypeId="1" AcceptedAnswerId="2509" CreationDate="2015-07-07T05:16:10.590" Score="16" ViewCount="13468" Body="&lt;p&gt;This may be quite a broad question, but lately I've been wondering about the history behind polynomials. Nowadays these are pretty much the simplest kind of functions to work with, but I'd like to know how this came to be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd also like to know who was/were (probably) the first to translate practical problems into solving polynomial equations, not using geometric methods like the greeks, but something closer to the modern approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://hsm.stackexchange.com/a/2502/520&quot;&gt;this answer&lt;/a&gt;, Conifold mentions Indian scientists knew some Taylor series already around the fifth century, so the importance of polynomial equations was already known by then. On the other hand, I &lt;em&gt;believe&lt;/em&gt; greeks didn't know about it (I know this assertion is too broad).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summarising:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What led to the realisation that polynomials were so important and when did it happen?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What philosophical implications did it have?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="520" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2018-05-05T22:38:13.040" Title="What is the origin of polynomials and notation for them?" Tags="|mathematics|discoveries|notation|philosophy-of-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2505" PostTypeId="2" ParentId="2504" CreationDate="2015-07-07T06:55:30.207" Score="6" Body="&lt;p&gt;Polynomial equations are among the primary objects of mathematics since the beginning. Babylonians knew how to solve quadratic equations though they did not use modern notation of course. The theory of conic sections is the study of quadratic equations in 2 variables, and it was performed by Apollonius geometrically. (At that time algebraic notation did not exist, and the Greeks did not think of the conic sections as polynomial equations). Polynomial equations of higher degree and in many variables were&#xA;first systematically studied by Diophantus, over the field of rational numbers, so in a sense Apollonius and Diophantus can be considered as creators of algebraic geometry (over the reals and over the rationals, respectively).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Muslim scientists continued the study of polynomials during the &quot;Dark Age&quot; in Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was natural to search of a general formula for solution of the polynomial equations in one variable of degree higher than 2. This was accomplished in XVI century (Tartaglia, Cardano, Ferrari). Modern notation for polynomials was introduced by Vieta. Finally Descartes invented his coordinate&#xA;method and connected the geometric approach of the Greeks with the algebraic approach. So Descartes was probably the first who studied systems of polynomial equations as we know them now. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So by the time of Newton, study of polynomial equations was very well established.&#xA;Newton himself contributed to this by classifying the real algebraic curves&#xA;of degree 3 (for degree 2 this was done by Apollonius, and in algebraic notation&#xA;by Descartes). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me add that polynomials are not just the &quot;simplest functions&quot; but remain one&#xA;of the principal objects of study in mathematics nowadays. The area called algebraic geometry is a central part of mathematics. It studies polynomial equatons. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-07-07T06:55:30.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2506" PostTypeId="1" CreationDate="2015-07-07T12:44:46.980" Score="10" ViewCount="2037" Body="&lt;p&gt;It is currently accepted that a year is equal to the time it takes for the earth to revolve around the sun. However around Roman times, Ptolemy's geocentric model was the widely accepted view of nature. Under the geocentric model the sun and all the stars moved around the earth once per day while the earth stayed fixed. But around the same time they used the Julian calendar which considered a year as 365.25 days long, a highly accurate number for the length of a year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is how exactly did they arrive at this figure? What exactly did a year mean to them if it wasn't the sun revolving around the earth? If the earth was allowed to rotate while the sun revolved around the earth, then I can see how they could calculate that. But it seems like the earth rotating was not an acceptable proposition at that time.&lt;/p&gt;&#xA;" OwnerUserId="2547" LastActivityDate="2021-03-18T03:53:47.960" Title="What is the historical basis for the length of a year?" Tags="|physics|astronomy|time|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2507" PostTypeId="2" ParentId="2506" CreationDate="2015-07-07T14:43:56.130" Score="7" Body="&lt;p&gt;The ancients distinguished two separate motions of the sun. On the one hand, the sun and all the stars seem to rotate around the Earth from East to West once every day. This was explained by the daily rotation of the sphere of the fixed stars. At the same time, the sun, moon and planets move in the opposite direction, along the ecliptic, around the Earth, each at its own pace. In the case of the sun, it takes one year to return to the same position with regards to the ecliptic. By taking an average of the apparent time required for this rotation Hipparchus (2nd century BC) calculated the tropical year as 365.24667 days (though he expressed this figure not with decimal fraction, but with a sexagesimal fraction). This is more accurate than the figure later adopted by the Julian calendar.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-07-07T14:49:31.770" LastActivityDate="2015-07-07T14:49:31.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2508" PostTypeId="2" ParentId="2506" CreationDate="2015-07-07T15:06:15.977" Score="17" Body="&lt;p&gt;First of all, whether the earth is rotating or everything else is rotating around it, is completely irrelevant for the question. This is just the matter of point of view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As seen from the earth, the fixed stars rotate with period day+night. On this background, the Sun describes a circle slowly with period &quot;one year&quot;. When one looks closely, one sees that there are TWO approximately equal (but not exactly equal) periods of the sun:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) The time in which the Sun returns to the same position WITH RESPECT TO THE STARS. This is called &quot;sideral year&quot;, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) The time in which the Sun travels from one spring equinox to another. This is called &quot;tropical year&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difference is roughly 50 angular seconds per year.&#xA;The difference was discovered by Hypparchus in the 2-nd century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For practical purposes (agriculture) tropical year is relevant (it is the period of change of seasons). To compute its length in days, one has to observe equinoxes (or solstices). The difference in days from say a spring equinox to the next one is the tropical year. Ancients could not observe and time &#xA;them with high precision, but there is a way around this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose that you observe a solstice and another one, not the next one but $m$-th one (so that $m$ years passed from one to another). And you count days and you find that $n$ days passed from the first solstice to the other one.&#xA;Then the length of the year will be approximately $n/m$ days. If $m$ is very large,&#xA;you obtain a very good approximation. Larger the $m$, more precise the approximation is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Systematic observations of sky were made by Babylonians since 8 century BC. At least this is the time from which continuous records survive. So at the time of&#xA;Hipparchus the data for about 6 centuries were available. (In principle. It is disputed how much did he use or know Babylonian observations. He apparently had in&#xA;his possessions the Greek records of solstices for a few centuries. Practically nothing survived of Hipparchus writings. We know about them from Ptolemy who wrote  3 centuries later).&#xA;This is how the very good approximation of the length of the tropical year was found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then making a calendar is a technical (but non-trivial) problem. Egyptians had a calendar with exactly 365 days in the year. So the beginning of the year was floating among the seasons. A reform was made in the 1-st century AD (Julian calendar) which decreed the year of 365 1/4 days. Though a more precise value was already known. More than a millenium later the need for a better calendar was recognized, and eventually the modern calendar (called Gregorian) was introduced.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-07-07T16:16:00.973" LastActivityDate="2015-07-07T16:16:00.973" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2509" PostTypeId="2" ParentId="2504" CreationDate="2015-07-08T03:26:30.030" Score="16" Body="&lt;p&gt;Although many problems that we now reduce to polynomial equations were solved since time immemorial early occurences are coached in verbal and/or geometric terms, and polynomials are not treated as separate items. For early occurences of geometric problems that lead (today) to quadratic equations see &lt;a href=&quot;https://hsm.stackexchange.com/a/2837/55&quot;&gt;The origin of quadratic equation in actual practice&lt;/a&gt;. The equations proper, let alone polynomials themselves, do appear only relatively late.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first breakthrough was made by Diophantus in Arithmetica (c.250 AD), who introduced notation for an indeterminate $\varsigma$, the first $x$, and its powers up to sixth, $\Delta^{\upsilon}$ is $\varsigma$ squared, $K^{\upsilon}$ cubed, etc. So for the first time it became possible to write down polynomials, albeit only up to degree six. He even had symbols for constant terms and reciprocals of powers. Diophantus was also first to convert text problems into polynomial equations, and use some rudimentary (verbal) algebra to solve them. However, Diophantus's &quot;polynomials&quot; are not yet separate items either, they only appear as sides in equations. This is analogous to Babylonian and Hellenistic use of zero symbol as a placeholder. Here is $3x^2+1-(10x^3+2x)=4$ in Diophantus's notation: $\Delta^{\upsilon}\gamma\mathring{M}\alpha\pitchfork K^{\upsilon}\iota\varsigma\beta\iota^{\sigma}\mathring{M}\delta$. See &lt;a href=&quot;http://scholarship.claremont.edu/jhm/vol5/iss1/8/&quot; rel=&quot;noreferrer&quot;&gt;Hettle's The Symbolic and Mathematical Influence of Diophantus's Arithmetica&lt;/a&gt; for details and notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next great step was made by medieval Islamic mathematicians. The familiar rules of algebra were formalized by Al-Khowarizmi (ca. 800-847 AD), who also derived first algebraic solutions to linear and quadratic equations (before him they were either given as prescriptions, or geometrically concealed as in Euclid). Perhaps the first person to conceive of general polynomials is Al-Karaji (953-1029 AD), who can be called their &quot;discoverer&quot;. He realized that the series of powers (and their reciprocals) extends indefinitely. Correspondingly, he extended Diophantus's notation to write polynomials of arbitrary degree, and gave rules for their addition, subtraction and multiplication. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But polynomials truly come to shine in the work of Al-Samawal (1130-1180) The Shining Book on Calculation. He dispensed with verbal and semi-verbal notation, and used tables of coefficients to write and perform calculations with polynomials, including those with negative powers (Laurent polynomials). This greatly simplified all algebraic calculations with them because &quot;laws of exponents&quot; are applied automatically. And he gives the first polynomial division algorithm, the grandfather of modern long and synthetic division. He also noted the analogy between his way of writing polynomials and the decimal positional notation, and transferred his algorithms to decimal numbers by replacing the variable with $10$. This was the first mathematical justification of a positional division algorithm. See &lt;a href=&quot;http://new.math.uiuc.edu/im2008/rogers/algebra.html&quot; rel=&quot;noreferrer&quot;&gt;Islamic Mathematics&lt;/a&gt; and &lt;a href=&quot;http://books.google.com/books/about/The_History_of_Mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;noreferrer&quot;&gt;Katz's History of Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, Renaissence Europe did not absorb Al-Samawal's innovations, and instead proceeded through incremental improvement of notation. In particular, the notation used by del Ferro, Tartaglia and even Cardano to solve the cubic in 1500s was largely verbal, and inferior to Al-Samawal. Even so, in Ars Magna (1545) Cardano introduced the technique of substitutions that not only solved the cubic and quartic, but became indispensable in polynomial algebra later. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/347/why-is-cardanos-formula-wrongly-attributed-to-him/351#351&quot;&gt;Why is &amp;quot;Cardano&amp;#39;s Formula&amp;quot; (wrongly) attributed to him?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Viète's Isagoge (1591) introduced modern style symbolic notation and algebraic manipulation rules Viète still uses words for powers, these were symbolized by Descartes, but they are attached to variables. In particular the use of letters for parameters allowed general consideration of polynomials rather than example by example. And philosophically, in Viète's works we for the first time encounter a systematic use of the method where problems are converted to equations, and then solved algebraically. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/2407/vi%C3%A8tes-relevance-and-his-connection-to-euler/2408#2408&quot;&gt;Vi&amp;#232;te&amp;#39;s Relevance and his Connection to Euler&lt;/a&gt;, and references therein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method was further sharpened by Descartes's analytic geometry in La Géométrie (1637). He connected it to the classical method of analysis and synthesis described by Pappus, but with conversion to algebraic equations in the middle. This not only streamlined solutions of many classical problems, but also covered many new ones, expressible by higher order equations. Moreover, Descartes considers polynomials in two variables, which represented algebraic curves in analytic geometry, and this is where algebraic geometry takes its root. Descartes's formalization of construction methods, and classification of problems based on their algebraic representation, led to techniques required for impossibility proofs, starting with Gregory's (unsuccessful) attempt to prove algebraic unsolvability of quadrature (1667). See &lt;a href=&quot;https://hal-insu.archives-ouvertes.fr/tel-01098493/document&quot; rel=&quot;noreferrer&quot;&gt;Crippa's Impossibility Results: from Geometry to Analysis&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-05T22:38:13.040" LastActivityDate="2018-05-05T22:38:13.040" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2510" PostTypeId="1" AcceptedAnswerId="2514" CreationDate="2015-07-07T09:18:28.843" Score="13" ViewCount="1161" Body="&lt;p&gt;The realisation that the atom was divisible, and thus that elements were not the &quot;fundamental building blocks&quot; arguably began around 1897 with J. J. Thomson's &lt;a href=&quot;https://en.wikipedia.org/wiki/J._J._Thomson#Discovery_of_the_electron&quot; rel=&quot;noreferrer&quot;&gt;discovery of the electron&lt;/a&gt; and proceeded through the early 20C to Rutherford's experiments with hydrogen nuclei, described as &lt;a href=&quot;https://en.wikipedia.org/wiki/Proton#History&quot; rel=&quot;noreferrer&quot;&gt;the discovery of the proton&lt;/a&gt; (although William Prout [&lt;a href=&quot;https://en.wikipedia.org/wiki/Proton#History&quot; rel=&quot;noreferrer&quot;&gt;ibid&lt;/a&gt;] had suggested that all atoms were composed of hydrogen atoms as early as 1815).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Chemical_element#Chemical_definitions&quot; rel=&quot;noreferrer&quot;&gt;From this Wiki page&lt;/a&gt;, Robert Boyle proposed the idea that matter was made of irreducible atoms in 1661, rejecting earlier ideas of &quot;the elements&quot;, and in 1789, Antoine Lavoisier published a list of 33 elements (although some were questionable).  By 1818, atomic weights were known for 45 of the 49 accepted elements; in 1869, Mendeleev had 66 elements in his periodic table; and by 1900, all but three of the naturally occurring stable elements had been discovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is: as the number of known elements grew (&lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_chemical_element_discoveries#Graphics&quot; rel=&quot;noreferrer&quot;&gt;see graph here&lt;/a&gt;), did anyone (ideally with at least some degree of credibility) use what might be termed a more &quot;philosophic&quot; argument to question the fundamental nature of elements and atoms, based on the idea that there were &quot;too many&quot; elements for them to be &quot;fundamental&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;thought process&quot; (not &lt;em&gt;necessarily&lt;/em&gt; &quot;logical objection&quot;) that I was wondering whether anyone had gone through was along the lines of:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;So, we've now got 50 [or however many] &lt;em&gt;fundamental particles&lt;/em&gt;, with more being found every year... maybe they are not &lt;em&gt;quite so fundamental&lt;/em&gt; as we thought... maybe there is &lt;em&gt;something else&lt;/em&gt; [protons / neutrons / electrons in the first descent] going-on underneath?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is, I suppose, no intrinsic reason why there &lt;em&gt;couldn't&lt;/em&gt; be a large / infinite number of &quot;fundamental particles&quot;, and possibly such a situation only &quot;feels wrong&quot; to me because of the knowledge that atoms &lt;em&gt;aren't&lt;/em&gt; fundamental.  In essence, the question is about whether anyone else &quot;felt this was wrong&quot; &lt;em&gt;before&lt;/em&gt; it was known that atoms were not fundamental.  I'm suspecting the answer is &quot;no&quot;.&lt;/p&gt;&#xA;" OwnerUserId="2552" OwnerDisplayName="TripeHound" LastEditorUserId="2552" LastEditDate="2015-07-08T15:49:28.300" LastActivityDate="2015-08-11T19:48:39.317" Title="Did anybody question the indivisibility of the atom because there were &quot;too many&quot; elements?" Tags="|chemistry|atomic-theory|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2511" PostTypeId="2" ParentId="2510" CreationDate="2015-07-08T13:49:44.703" Score="0" Body="&lt;p&gt;The idea that matter consists of indivisible atoms was first proposed by Democritus and Leucippus in the 5th century BC. In fact, the word “atomos” means “un-cut”. And Democritus at least seems to have believed that there is an infinite number of types of atoms. I do not see any logical objection against indivisibility on the basis of there being “too many” elements. In any case, it does not seem that anyone has ever raised that particular objection.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-07-08T13:49:44.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2512" PostTypeId="1" AcceptedAnswerId="2516" CreationDate="2015-07-08T20:30:38.837" Score="17" ViewCount="1890" Body="&lt;p&gt;Recently, I learned that &lt;span class=&quot;math-container&quot;&gt;$\emptyset$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\varnothing$&lt;/span&gt; is a relatively new notation for the empty set and was &lt;a href=&quot;http://en.wikipedia.org/wiki/Empty_set#Notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;created in 1939&lt;/a&gt;. I know &lt;span class=&quot;math-container&quot;&gt;$\{\}$&lt;/span&gt; is also used along with &lt;span class=&quot;math-container&quot;&gt;$\{\cdot\}$&lt;/span&gt; to denote empty sets. What was their origin? In general, what was the historical development of notation for the empty set?&lt;/p&gt;&#xA;&lt;p&gt;Since set theory was given a rigorous treatment starting in the late 19th century, did mathematicians have different, non-standardized notations prior?&lt;/p&gt;&#xA;" OwnerUserId="2259" LastEditorUserId="4251" LastEditDate="2021-09-26T13:09:44.023" LastActivityDate="2021-09-27T17:23:07.767" Title="How did mathematicians notate the empty set before $\varnothing$?" Tags="|mathematics|notation|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2513" PostTypeId="1" CreationDate="2015-07-08T20:45:28.207" Score="2" ViewCount="266" Body="&lt;p&gt;I read in many sites that Gauss worked in theoretical physics in the areas of mechanics, crystallography, acoustics, capillarity, electromagnetism and optics, but I couldn't find any place in his heritage in which he made contributions to acoustics. I'm very interested to learn of his contributions in this field.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="26" LastEditDate="2015-07-09T10:41:33.013" LastActivityDate="2015-07-09T10:41:33.013" Title="What are Carl Friedrich Gauss's contributions to acoustics?" Tags="|physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2514" PostTypeId="2" ParentId="2510" CreationDate="2015-07-08T22:12:03.787" Score="5" Body="&lt;p&gt;There were both historical and experimental reasons for maintaining indivisibility of atoms in chemistry despite large number of elements. What confirmed compositeness of molecules were transformations of compounds into each other in chemical reactions. Analogous idea for atoms suggested transmutation of elements, which was not observed. In particular, it resembled the alchemic idea of transmuting metals into gold, many metals were elements and alchemists failed to transmute them for centuries. Only at the &lt;a href=&quot;http://www.vanderkrogt.net/elements/chemical_symbols.php&quot; rel=&quot;nofollow&quot;&gt;start of 19th century chemistry&lt;/a&gt; fully freed itself from alchemic mentality and notation and became an established science, scientific chemists had no appetite for going back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So ideas that led to the discovery of atomic structure came from different quarters. In 1845 Fechner suggested that electric currents were composed of small electric charges, named electrons by Stoney in 1891, that moved in opposite directions. Weber in 1846 developed the first theory of them, as did Stoney and Clausius later. The most successful electron theory was proposed in 1892 by Lorentz, who combined it with Maxwell's electrodynamics and explained many observed electric properties of matter based on it. Lorentz imagined a gas of electrons filling material bodies, but the question remained as to where electrons came from, and how they fit together with atoms and molecules. This is what prompted Thomson's experiments in 1897, who then proposed the first model of composite atom with electrons as negatively charged raisins stuck in a positively charged pudding (his analogy), but occasionally break out and form Lorentz's gas. See &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich&quot; rel=&quot;nofollow&quot;&gt;Whittaker's History of the Theories of Aether and Electricity&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say that there weren't philosophical objections to atoms, but they did not concern the large number of elements. They had more to do with the rise of philosophical empiricism, and a preference for phenomenological theories over mechanistic ones like atomism. The most prominent opponents, Mach and Ostwald, denied reality of atoms altogether. They considered them theoretical fictions only useful in organizing calculations of observable quantities like energy, entropy, etc. Mach stopped mentioning atoms in his works in 1863, and openly attacked them in his 1872 book. Ostwald, a very influential chemist in his time, published a chemistry textbook with no mention of atoms in 1892. The chief defender of atomic theory was not a chemist but Boltzmann, the founder of the kinetic theory of gases, who engaged both of them in public debates in 1890s. Interestingly, he considered atoms changeable and perishable, but without suggesting a specific mechanism. See &lt;a href=&quot;http://www.physics.umd.edu/~einstein/HONR228K/MachBoltzmann.pdf&quot; rel=&quot;nofollow&quot;&gt;Blackmore's note&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-08-11T19:48:39.317" LastActivityDate="2015-08-11T19:48:39.317" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2515" PostTypeId="1" AcceptedAnswerId="2517" CreationDate="2015-07-08T22:20:38.657" Score="3" ViewCount="230" Body="&lt;p&gt;Gottfried Leibniz developed a system of propositional logic in the late 1600's, which wasn't published until 1903, when it was &lt;a href=&quot;http://www.iep.utm.edu/leib-log/&quot; rel=&quot;nofollow&quot;&gt;discovered&lt;/a&gt; in the Royal Library of Hanover by Louis Couturat. How did this unpublished work end up in that particular library, where is it now, and did any of his work in this area influence other contemporary mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="2487" LastActivityDate="2015-07-09T10:04:42.903" Title="Timeline of Leibniz' Propositional Logic" Tags="|mathematicians|publishing|mathematical-logic|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2516" PostTypeId="2" ParentId="2512" CreationDate="2015-07-08T22:53:03.613" Score="21" Body="&lt;p&gt;George Boole introduced the concept of empty set, or &amp;quot;nothing&amp;quot; as he called it,  as the complement to the &amp;quot;universe&amp;quot; in his Mathematical Analysis of Logic (1847). His notations for them were somewhat boring, &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; respectively. Cantor wrote in 1880 &amp;quot;for the absence of points we choose the letter &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;&amp;quot;. Frege, the founder of mathematical logic, interpreted &amp;quot;null class&amp;quot; as extension of the concept &amp;quot;not identical with itself&amp;quot; rather than a collection of objects, and used &lt;span class=&quot;math-container&quot;&gt;$\{\}$&lt;/span&gt; for it. Peano used &lt;span class=&quot;math-container&quot;&gt;$\Lambda$&lt;/span&gt; for both null class and &amp;quot;false&amp;quot; in his axiomatization of arithmetic in 1889, but Zermelo axiomatizing set theory in 1908 went back to Boole's &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, and so did Hausdorff in his influential book Grundzuge der Mengenlehre (1914). Hausdorff was the first to use &amp;quot;empty set&amp;quot; in its modern purely extensional meaning, before him there was always an air of intensional &amp;quot;null class&amp;quot; to it, and even Zermelo stipulates extensionality as a separate assumption. See &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.9839&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kanamori's The empty set, the singleton, and the ordered pair&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The modern symbol &lt;span class=&quot;math-container&quot;&gt;$\emptyset$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\varnothing$&lt;/span&gt; was indeed &lt;a href=&quot;https://web.archive.org/web/20150310160945/http://mathforum.org/library/drmath/view/62382.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;introduced in 1939 by Andre Weil&lt;/a&gt;, and gained wide acceptance after its use in the Bourbaki volumes.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="14365" LastEditDate="2021-09-27T17:23:07.767" LastActivityDate="2021-09-27T17:23:07.767" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2517" PostTypeId="2" ParentId="2515" CreationDate="2015-07-09T09:57:14.903" Score="3" Body="&lt;p&gt;Regarding Leibniz's unpublished papers in Hanover Library, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gottfried Wilhelm Leibniz&lt;/a&gt; (July 1, 1646 – November 14, 1716) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1673, the Duke of Brunswick offered him the post of Counsellor which Leibniz very reluctantly accepted two years later, only after it became clear that no employment in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court was forthcoming.&lt;/p&gt;&#xA;&lt;p&gt;In 1677, he was promoted, at his request, to Privy Counselor of Justice, a post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for the period.&lt;/p&gt;&#xA;&lt;p&gt;The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting the calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz is the most important logician between Aristotle and 1847, when George Boole and Augustus De Morgan each published books that began modern formal logic. [...] Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts. In his book &lt;em&gt;History of Western Philosophy&lt;/em&gt;, Bertrand Russell went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later.&lt;/p&gt;&#xA;&lt;p&gt;While serving as librarian of the ducal libraries in Hanover and Wolfenbuettel, Leibniz effectively became one of the founders of library science. The latter library was enormous for its day, as it contained more than 100,000 volumes, and Leibniz helped design a new building for it, believed to be the first building explicitly designed to be a library. He also designed a book indexing system in ignorance of the only other such system then extant, that of the Bodleian Library at Oxford University.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz mainly wrote in three languages: scholastic Latin, French and German. During his lifetime, he published many pamphlets and scholarly articles, but only two &amp;quot;philosophical&amp;quot; books, the &lt;em&gt;Combinatorial Art&lt;/em&gt; and the &lt;em&gt;Théodicée&lt;/em&gt;. (He published numerous pamphlets, often anonymous, on behalf of the House of Brunswick-Lüneburg, most notably the &amp;quot;De jure suprematum&amp;quot; a major consideration of the nature of sovereignty). One substantial book appeared posthumously, his &lt;em&gt;Nouveaux essais sur l'entendement humain&lt;/em&gt;, which Leibniz had withheld from publication after the death of John Locke. Only in 1895, when Bodemann completed his catalogue of Leibniz's manuscripts and correspondence, did the enormous extent of Leibniz's &lt;em&gt;Nachlass&lt;/em&gt; become clear: about 15,000 letters to more than 1000 recipients plus more than 40,000 other items. Moreover, quite a few of these letters are of essay length. Much of his vast correspondence, especially the letters dated after 1700, remains unpublished, and much of what is published has been so only in recent decades.&lt;/p&gt;&#xA;&lt;p&gt;The systematic cataloguing of all of Leibniz's &lt;em&gt;Nachlass&lt;/em&gt; began in 1901. It was hampered by two world wars and decades of German division in two states with the cold war's &amp;quot;iron curtain&amp;quot; in between, separating scholars, and also scattering portions of his literary estates. The ambitious project has had to deal with seven languages contained in some 200,000 pages of written and printed paper. In 1985 it was reorganized and included in a joint program of German federal and state (Länder) academies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-07-09T10:04:42.903" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2518" PostTypeId="1" CreationDate="2015-07-09T23:18:15.667" Score="5" ViewCount="350" Body="&lt;p&gt;I am curious to learn in more detail why Rene Descartes was not knighted like Isaac Newton and many other great mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="2561" LastEditorUserId="62" LastEditDate="2015-07-11T12:29:32.600" LastActivityDate="2017-10-25T05:50:31.457" Title="Why was Rene Descartes not knighted?" Tags="|biographical-details|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2520" PostTypeId="1" AcceptedAnswerId="3689" CreationDate="2015-07-10T10:32:40.813" Score="6" ViewCount="645" Body="&lt;p&gt;I know that a special case of the Bonnet theorem, called the Theorema Elegantissimum, was proved by Gauss in his 1827 treatise on differential geometry. This was a theorem that dealt with the connection between total curvature and angular deficit, but only in the case of geodesic lines. I know also that he described the notion of geodesic curvature in an unpublished paper. But is there a place (letters, private notes) where he directly referred to the full version of Bonnet theorem, e.g the connection between geometry and topology (in the sense of Euler's characteristic)?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-09-10T22:10:27.413" LastActivityDate="2023-09-16T15:18:52.973" Title="Did Gauss formulate, or at least know of, the full essence of the Gauss-Bonnet Theorem?" Tags="|mathematics|biographical-details|differential-geometry|topology|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2521" PostTypeId="1" CreationDate="2015-07-10T13:00:23.860" Score="2" ViewCount="127" Body="&lt;p&gt;I read in Gauss's biography - Gauss:Titan of Science, in the section &quot;milestones&quot; that &quot;in particular, he found in a note dated December, 1844, the numerous forms which closed curves with four knots can exhibit&quot;. What's the meaning of this sentence? does it mean that he found the number of links which four knots of a certain kind can create? Where is it in his nachlass?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2015-07-10T15:07:57.187" LastActivityDate="2015-07-10T15:07:57.187" Title="What is the complete list of Gauss's contributions to topology and knot theory?" Tags="|biographical-details|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2522" PostTypeId="1" CreationDate="2015-07-10T13:08:01.107" Score="3" ViewCount="105" Body="&lt;p&gt;I wonder if Gauss gave mathematical expressions for abberations in some of his optical papers (papers others than his Dioptrische Untersuchungen)? I know that he worked out the theory of Gaussian optics (paraxial approximation), but I ask if he did something else about optics in addition.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2905" LastEditDate="2015-12-27T14:47:25.710" LastActivityDate="2015-12-27T14:47:25.710" Title="Did Gauss work on the theory of optical abberations?" Tags="|biographical-details|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2523" PostTypeId="1" AcceptedAnswerId="2524" CreationDate="2015-07-10T17:07:34.607" Score="3" ViewCount="250" Body="&lt;p&gt;When saying graph in mathematics, it can be either a graph of a function, or a graph in graph theory. However mathematically they have nothing in common. How did they get the same name?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know graph theory has been invented by Euler (the seven bridges of Königsbergen, 1736), but I haven't been able to find the history of graphing a function. My guess is that Leibniz had done that as well while discovering functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, they both have something related to graphical things, but you might as well say that about geometry, topology and often also combinatorics, so that isn't really a statistifactory argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did 'graphing' a function and 'graph' theory get the same name? Was it a confusion, or did they have a common ancestor? &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-07-10T18:49:57.453" LastActivityDate="2015-07-10T21:58:02.217" Title="Do the words 'graphing' a function and 'graph' theory have a common ancestor?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2524" PostTypeId="2" ParentId="2523" CreationDate="2015-07-10T21:58:02.217" Score="7" Body="&lt;p&gt;All these words come from the same Greek root, &quot;graphein&quot; means to write in Greek.&#xA;Later the meaning extended to include drawing.&#xA;Also graphite, graphology, graphics (drawing), graphoman etc. Also photography, biography, histogram, cardiogram, grammar and telegraph have the same root.&#xA;Graphs of functions and graphs in combinatorics are something that we draw, pictures. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-07-10T21:58:02.217" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2525" PostTypeId="1" AcceptedAnswerId="2533" CreationDate="2015-07-11T03:00:44.197" Score="8" ViewCount="1858" Body="&lt;p&gt;I have read a variety of articles and books about string theory that relate the various initial discoveries that ultimately lead to a theory we know now as &quot;string theory&quot; (and its descendants such as M-theory, branes, etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, is any single individual noted as being the father (or, mother) of string theory. Could it be Susskind? Who was the first person to recognize that actual vibrating strings were at the core of the original work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this is rather recent history but that makes it easier -- someone here lived through this time.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastEditorUserId="4251" LastEditDate="2020-07-06T12:24:05.647" LastActivityDate="2020-07-06T12:24:05.647" Title="Who is the father (if there is one) of string theory?" Tags="|physics|theoretical-physics|physicists|string-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2526" PostTypeId="1" CreationDate="2015-07-11T07:23:00.613" Score="7" ViewCount="686" Body="&lt;p&gt;Like many mathematicians, I have long been fascinated by &lt;a href=&quot;https://en.wikipedia.org/wiki/Srinivasa_Ramanujan&quot;&gt;Ramanujan&lt;/a&gt;'s work and also by what little I know of his life. I would like to learn more, but have found the standard book on this subject (Kanigel's &lt;em&gt;The Man Who Knew Infinity&lt;/em&gt;), while well-written, to be lacking in mathematical depth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an account of Ramanujan's life that places it in the context of his mathematical work, and actually discusses his results (and where they came from) rather than describing them poetically as &quot;formulas of a genius never seen before&quot; or similar phrases that serve to mystify, rather than illuminate, what is actually going on?&lt;/p&gt;&#xA;" OwnerUserId="2568" LastActivityDate="2017-03-27T12:54:07.193" Title="Is there a biography of Ramanujan for mathematicians?" Tags="|mathematics|biographical-details|reference-request|mathematicians|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2527" PostTypeId="2" ParentId="2526" CreationDate="2015-07-11T09:31:12.303" Score="8" Body="&lt;p&gt;You could find the original manuscript by Ramanujan in the following work.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Srinivasa Ramanujan: &lt;em&gt;Notebooks of Srinivasa Ramanujan&lt;/em&gt;. Tata Institute of Fundamental Research, Bombay, Springer-Verlag Berlin Heidelberg, 1984.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This work contains 390 pages of scanned version of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ramanujan%27s_lost_notebook&quot;&gt;Ramanujan's lost notebook&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two famous book series which contains proofs and added details of Ramanujan's work are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bruce_C._Berndt&quot;&gt;Bruce C. Berndt: &lt;em&gt;Ramanujan's Notebooks.&lt;/em&gt; Springer, 1985–2005. (five parts)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/George_Andrews_%28mathematician%29#Publications&quot;&gt;George E. Andrews, Bruce C. Berndt: &lt;em&gt;Ramanujan's Lost Notebook.&lt;/em&gt; Springer, 2005–2013. (four parts)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;At last:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.springer.com/us/book/9788132207696&quot;&gt;M. R. Murty, V. K. Murty: &lt;em&gt;The Mathematical Legacy of Srinivasa Ramanujan&lt;/em&gt;. Springer India, 2013.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;An online collection of Ramanujan's problems and papers is &lt;a href=&quot;http://www.imsc.res.in/~rao/ramanujan/collectedindex.html&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="100" LastEditorUserId="100" LastEditDate="2015-07-11T09:38:37.127" LastActivityDate="2015-07-11T09:38:37.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2528" PostTypeId="2" ParentId="2518" CreationDate="2015-07-11T10:03:43.330" Score="6" Body="&lt;p&gt;&lt;em&gt;This answer was based on an earlier version of the question which asked why Descartes was not made a Sir&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The primary reason is that &quot;Sir&quot; denotes a knighthood in the British honour system, and is normally open only to British subjects(*).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;René Descartes was a French subject and therefore not eligible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As noted by Bernard Massé in his comment, the equivalent in 16th-century France would have been to make him a count, marquis or similar title. There are at least two factors which may have contributed to this not happening: firstly he lived a good part of his life outside France (in the Netherlands and Sweden), and secondly his philosophy was in direct opposition to that of the Scholastics who still ruled the Sorbonne.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(*) According to the &lt;a href=&quot;http://www.royal.gov.uk/MonarchUK/Honours/Knighthoods.aspx&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;official website of the British Royal Family&quot;&gt;official website of the UK monarchy&lt;/a&gt;, honorary knighthoods are occasionally bestowed on foreign citizens, but they are not entitled to call themselves &quot;Sir&quot;.&lt;/p&gt;&#xA;" OwnerUserId="536" LastEditorUserId="536" LastEditDate="2017-10-25T05:50:31.457" LastActivityDate="2017-10-25T05:50:31.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2529" PostTypeId="1" CreationDate="2015-07-11T10:07:13.797" Score="5" ViewCount="537" Body="&lt;p&gt;When I was learning classical mechanics, I was quite baffled by the Hamilton's principle, since it involves a quantity named after Lagrange. So, it seems that the principle was not discovered by Lagrange himself. A natural question is then, what did Lagrange do with his quantity?  &lt;/p&gt;&#xA;" OwnerUserId="2570" LastEditorUserId="26" LastEditDate="2015-07-13T20:18:16.060" LastActivityDate="2015-07-13T21:19:53.920" Title="What did Lagrange do with his quantity (the Lagrangian in classical mechanics)?" Tags="|mathematics|mathematicians|classical-mechanics|mathematical-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2530" PostTypeId="2" ParentId="2518" CreationDate="2015-07-11T17:08:25.110" Score="16" Body="&lt;p&gt;The French word for &quot;knight&quot; is &quot;chevalier&quot;. Descartes did not need to be knighted; he inherited the title &quot;chevalier&quot; and was frequently called &quot;Chevalier du Peron&quot; or &quot;Chevalier Seigneur du Peron&quot;, as in this portrait:&#xA;&lt;a href=&quot;http://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=3610101&amp;amp;partId=1&amp;amp;people=88521&amp;amp;peoA=88521-1-6&amp;amp;page=1&quot;&gt;http://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=3610101&amp;amp;partId=1&amp;amp;people=88521&amp;amp;peoA=88521-1-6&amp;amp;page=1&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-07-11T17:08:25.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2531" PostTypeId="2" ParentId="2518" CreationDate="2015-07-12T09:37:56.833" Score="3" Body="&lt;p&gt;Another point is that Isaac Newton, as well as other important mathematicians in England and France, was also a civil servant (Master of the Mint and other offices). He was president of the Royal Society and Member of Parliament.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, Viete was privy councillor to two Kings of France and a high-level civil servant.  Brouncker was already noble by birth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Descartes went to the Netherlands for freedom of thought and study as well as to learn from Dutch scientists and was not member of the french establishment, although most members of his family were distinguished civil servants (member of provincial parlements), with land properties, mostly in Britany and Poitou. &lt;/p&gt;&#xA;" OwnerUserId="312" LastEditorUserId="312" LastEditDate="2015-07-14T12:49:14.677" LastActivityDate="2015-07-14T12:49:14.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2532" PostTypeId="2" ParentId="2526" CreationDate="2015-07-13T00:15:52.557" Score="4" Body="&lt;p&gt;The English mathematician Hardy, who was close to Ramanujan, authored &quot;&lt;a href=&quot;http://www.ams.org/bookstore-getitem/item=CHEL/136.H&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="2524" LastEditorUserId="26" LastEditDate="2017-03-27T12:54:07.193" LastActivityDate="2017-03-27T12:54:07.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2533" PostTypeId="2" ParentId="2525" CreationDate="2015-07-13T18:38:21.660" Score="6" Body="&lt;p&gt;I will reduce the &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_string_theory&quot; rel=&quot;nofollow&quot;&gt;detailed history&lt;/a&gt; to principal highlights. Any of the physicists mentioned below can be considered &quot;fathers&quot; of string theory depending on which aspect of it is considered &quot;the core&quot;. The problem with a straightforward answer is that the mathematical formalism was originally developed for purposes that had little to do with modern string theory, and there were several additional insights by different people that made it viable, any of which can be seen as its &quot;true birth&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest candidate for a &quot;father&quot; is perhaps Gabriele Veneziano, who used Euler's beta function as a scattering amplitude in his 1968 paper. At the time unification of quantum fields and gravity was not on the agenda, the issue was working out a theory of strong interaction, and that is what Veneziano introduced his formula for. By 1970, Nambu, Nielsen and Susskind concluded that Veneziano's formula would be based on more than a mere analogy if particles were treated as vibrating strings. This led to the so-called dual resonance models of strong interaction, but they were abandoned in 1973 in favor of quantum chromodynamics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original string theory was purely bosonic, fermions were added in 1971 by Ramond and led to the discovery of supersymmetry and superstrings. The number of required dimensions was reduced from 26 in bosonic theory to 10. One of the bosonic vibration modes was a massless particle of spin two, which as Schwarz,  Scherk, and independently Yoneya, pointed out in 1974, were exactly the properties of hypothesized graviton. Moreover, in late 1970s supersymmetry was used to unify field theories with gravity independently of strings, in the so-called supergravity. So while original string theory of strong interactions was discarded, the mathematical formalism lived on as a candidate for quantum gravity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was not taken seriously however until 1984, when Green and Schwarz discovered that superstring theories produce counterterms that eliminate so-called quantum anomalies. These are mathematical complications that prevent quantized field theories from manifesting classical symmetries, and make them inconsistent. This came to be called the &quot;first superstring revolution&quot;, which attracted many new researchers to the field, including Witten, who initiated the &quot;second superstring revolution&quot; in 1995.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-07-20T19:51:40.363" LastActivityDate="2015-07-20T19:51:40.363" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2534" PostTypeId="2" ParentId="2529" CreationDate="2015-07-13T21:13:34.157" Score="5" Body="&lt;p&gt;Lagrange and Hamilton principles are different ways to formulate the same Newtonian mechanics. One uses positions and momenta as variables, the other positions and velocities. But they both use kinetic $T$ and potential $V$ energies to write down Hamiltonian $\mathcal{H}=T+V$ and Lagragian $L=T-V$ functions respectively, which lead to Hamilton and Euler-Lagrange equations of motion respectively. Lagrange's formulation is variational, he integrates $L$ and takes the variation to obtain the equations, in this he was &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Louis_Maupertuis#Least_Action_Principle&quot; rel=&quot;noreferrer&quot;&gt;anticipated by Maupertuis&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamilton's formulation is not directly variational, although in hindsight it can be interpreted in terms of optimal control theory developed in 1950s. Being equivalent one can always be transformed into the other of course, but Hamilton did discover the formulation attributed to him in 1834. See &lt;a href=&quot;http://homes.chass.utoronto.ca/~cfraser/HJ.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Early History of Hamilton- Jacobi Dynamics 1834-1837&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-07-13T21:19:53.920" LastActivityDate="2015-07-13T21:19:53.920" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2535" PostTypeId="2" ParentId="2525" CreationDate="2015-07-14T18:05:42.337" Score="4" Body="&lt;p&gt;This seems relevant: ${}{}{}{}{}{}{}{}{}{}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/fX1wT.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Image obtained from Chapter 9 of &lt;em&gt;A Brief History of String Theory: From Dual Models to M-Theory&lt;/em&gt; by Dean Rickles. Its legend in the book reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Extract from Weinberg's letter of support for John Schwarz's nomination for California Scientist of the Year. &lt;em&gt;Image source&lt;/em&gt; Caltech archives [letter dated Feb. 13, 1986; Gell-Mann papers: Box 5, Folder 10]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="520" LastEditorUserId="520" LastEditDate="2015-07-14T18:15:04.397" LastActivityDate="2015-07-14T18:15:04.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2537" PostTypeId="2" ParentId="2468" CreationDate="2015-07-15T08:51:48.567" Score="3" Body="&lt;p&gt;The best example I can think of is &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Green.html&quot; rel=&quot;nofollow&quot;&gt;George Green&lt;/a&gt;. George Green quit his studies at the age of 9 to work with his father and eventually was owner of a mill. He spent the majority of his life as a miller but found time to work on advanced math and developed concepts far ahead of his own time. He published an &lt;a href=&quot;https://en.wikipedia.org/wiki/An_Essay_on_the_Application_of_Mathematical_Analysis_to_the_Theories_of_Electricity_and_Magnetism&quot; rel=&quot;nofollow&quot;&gt;essay&lt;/a&gt; on electromagnetism in a little known journal that would eventually have a wide reaching impact in nearly every branch of physics. He is the namesake for &lt;a href=&quot;https://en.wikipedia.org/wiki/Green%27s_theorem&quot; rel=&quot;nofollow&quot;&gt;Green's theorem&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Green%27s_function&quot; rel=&quot;nofollow&quot;&gt;Green's function&lt;/a&gt; as well as some other important mathematical objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventually he did go to Cambridge at the age of 40 and earned his degree but this was well after his most important work. So it can be argued that he achieved this without going to university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting case to mention is &lt;a href=&quot;https://en.wikipedia.org/wiki/Freeman_Dyson&quot; rel=&quot;nofollow&quot;&gt;Freeman Dyson&lt;/a&gt;. While he did earn a masters degree, he never earned a doctorate but was still able to work as a mathematical physicist. Many people, including Steven Weinberg, believe he should have been awarded a Nobel Prize for the work he's done.&lt;/p&gt;&#xA;" OwnerUserId="2547" LastActivityDate="2015-07-15T08:51:48.567" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2539" PostTypeId="1" CreationDate="2015-07-16T03:39:31.567" Score="5" ViewCount="234" Body="&lt;p&gt;While reading the book of Burnside, the history gave interest to me to see further the old books on group theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It will be a great pleasure if one can suggest some books on group theory published between 1885 to 1900.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will not worry for the languages of the books (German, English, etc.)&lt;/p&gt;&#xA;" OwnerUserId="2594" LastEditorUserId="2594" LastEditDate="2015-07-16T15:41:12.733" LastActivityDate="2015-07-16T23:34:35.943" Title="Books on Group Theory between 1885-1900" Tags="|reference-request|group-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2541" PostTypeId="2" ParentId="2539" CreationDate="2015-07-16T07:25:51.970" Score="0" Body="&lt;p&gt;Several sources include this:&#xA;&lt;a href=&quot;https://www.math.lsu.edu/~adkins/m7200/GroupHistory.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.lsu.edu/~adkins/m7200/GroupHistory.pdf&lt;/a&gt; (Which specifies certain lectures and resources)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And this:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_important_publications_in_mathematics#Abstract_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_important_publications_in_mathematics#Abstract_algebra&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pardon my French, but it appears most resources in that time were written in it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also see &lt;a href=&quot;https://math.stackexchange.com/questions/16765/historical-textbook-on-group-theory-algebra&quot;&gt;this&lt;/a&gt; question.&lt;/p&gt;&#xA;" OwnerUserId="2553" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-07-16T07:25:51.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2542" PostTypeId="1" CreationDate="2015-07-16T20:31:36.593" Score="6" ViewCount="2844" Body="&lt;p&gt;Water is a pretty basic molecule that people used since early science. What were the steps in figuring out what water is made of and what is its structure?&lt;/p&gt;&#xA;" OwnerUserId="1876" LastActivityDate="2015-08-18T22:32:03.587" Title="How was the composition and structure of water determined?" Tags="|discoveries|chemistry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2543" PostTypeId="2" ParentId="2542" CreationDate="2015-07-16T20:58:07.700" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Composition (i.e. water is not an element)&lt;/strong&gt;:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;1783:&lt;/strong&gt; The Scottish inventor James Watt published a paper on the composition of water in 1783; Cavendish had performed the experiments first but published second. Controversy about priority ensued. (source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Cavendish&quot; rel=&quot;nofollow&quot;&gt;Wikipedia - Cavendish&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;1783:&lt;/strong&gt; Lavoisier began a series of experiments on the composition of water which were to prove an important capstone to his combustion theory and win many converts to it. Many investigators had been experimenting with the combination of Henry Cavendish's inflammable air, which Lavoisier termed hydrogen (Greek for &quot;water-former&quot;), with dephlogisticated air (oxygen) by electrically sparking mixtures of the gases. All of the researchers noted the production of water, but all interpreted the reaction in varying ways within the framework of the phlogiston theory. In cooperation with mathematician Pierre Simon de Laplace, Lavoisier synthesized water by burning jets of hydrogen and oxygen in a bell jar over mercury. (source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Antoine_Lavoisier&quot; rel=&quot;nofollow&quot;&gt;Wikipedia - Lavoisier&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Ratio of oxygen to hydrogen&lt;/strong&gt;:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;1805:&lt;/strong&gt; Gay-Lussac and Humboldt discover that water is formed by two parts of hydrogen and one part of oxygen (by volume). (source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Louis_Gay-Lussac&quot; rel=&quot;nofollow&quot;&gt;Wikipedia - Gay Lussac&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;1842:&lt;/strong&gt; Dumas fixed the ratio H:O as 2:16. (source: &lt;a href=&quot;http://hydrogen.atomistry.com/history_weight_of_hydrogen.html&quot; rel=&quot;nofollow&quot;&gt;Atomistry - Atomic weight of hydrogen history&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1876" LastActivityDate="2015-07-16T20:58:07.700" CommentCount="1" CommunityOwnedDate="2015-07-16T20:58:07.700" ContentLicense="CC BY-SA 3.0" />
  <row Id="2544" PostTypeId="2" ParentId="2539" CreationDate="2015-07-16T23:22:56.463" Score="2" Body="&lt;p&gt;The he earliest one seems to be&#xA;C. Jordan, Traite des substitutions algebriques:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. Jordan,  Traite des substitutions et des equations algebriques. (French)&#xA;Paris. Gauthier-Villars. 1870. Published: 1870&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most comprehensive one covering this period is&#xA;Dickson, Linear groups with an exposition of the Galois field theory. (German)&#xA;Leipzig: B. G. Teubner. X + 312 S. (Teubners Sammlung No. VI.) Published: 1901&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are also interested in papers, read C. Jordan, for example,&#xA;Memoire sur les groupes des mouvements.&#xA;Brioschi Ann. (2). II. 168-215. 320-345. Published: 1868&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and Oeuvres Complete of Lie and Killing, Also Cartan:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cartan, E.&#xA;Sur la structure des groupes de transformations finis et continus. (French)&#xA;These presentee a la Faculte des Sciences. 156 S. Paris. Nony et Co. Published: 1894,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sur la reduction a sa forme canonique de la structure d'un groupe de transformations fini et continu. (French)&#xA;American J. 18, 1-61. Published: 1896&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beside this, papers of&#xA;Arthur Cayley, Sylov, Holder, Mattheu, Klein, Burnside and Frobenius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To find their papers use Jahrbuch database,&#xA;&lt;a href=&quot;http://www.emis.de/cgi-bin/jfmen/MATH/JFM/full.html&quot; rel=&quot;nofollow&quot;&gt;http://www.emis.de/cgi-bin/jfmen/MATH/JFM/full.html&lt;/a&gt;&#xA;which is available free on Internet, unlike Mathscinet and Zentralblatt, and covers the late 19s century very well.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-07-16T23:34:35.943" LastActivityDate="2015-07-16T23:34:35.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2546" PostTypeId="2" ParentId="2475" CreationDate="2015-07-17T06:39:25.020" Score="6" Body="&lt;p&gt;I want to share some data and interesting facts:&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/aj9vQ.png&quot; alt=&quot;Table of paper languages each year. Red: Most used, green: Second most used. Source: Own work&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Erratum: 1897 host is Switzerland, not Germany. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The table shows the percentage of papers in a given language in a given year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Main observations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;At least in the early days, the language of the host country was the most used language, unless it was a rarely spoken language. (e.g. in Oslo 1936 Swedish wasn't used)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Main reason: Traveling was more difficult.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The 1% German in 1920 can be explained, because German and Austria were suspended from the ICM.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;1958 and further years, English was the main language together with a good number of papers in German and French, and in Russian if the Soviet government allowed them to attend the congress. (You must realize that it was the middle of the cold war)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-07-17T16:58:37.717" LastActivityDate="2015-07-17T16:58:37.717" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2547" PostTypeId="1" CreationDate="2015-07-17T10:22:22.660" Score="3" ViewCount="128" Body="&lt;p&gt;This is quite an obscure question but one that I have been curious over for some time. My great-grandfather was a shipwright at the turn of the 20th century (1890s-1925) and family lore says that because of his good work he was presented with a book about Euclid's theorems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My dad and I have never been able to find a copy and so we wondered if anyone knew of such a book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All my great-grandfather apparently said about it was that &quot;Yeah it is very good but it was all wrong&quot;. I am not too sure what he meant by this, maybe referring to the modern mathematics of the day on manifolds?&lt;/p&gt;&#xA;" OwnerUserId="198" LastEditorUserId="62" LastEditDate="2015-07-18T20:49:31.420" LastActivityDate="2015-07-18T20:49:31.420" Title="Book on Euclid's theorem from ~1910-1920" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2548" PostTypeId="1" CreationDate="2015-07-17T06:33:10.260" Score="5" ViewCount="105" Body="&lt;p&gt;What were the physics curricula for 19th century French universities?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for something akin to &lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=hvd.32044079809943;view=1up;seq=400&quot; rel=&quot;nofollow&quot;&gt;this distribution of courses at the &lt;em&gt;École Polytechnique&lt;/em&gt;&lt;/a&gt;, but for other French universities, like the &lt;em&gt;École Normale&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="232" OwnerDisplayName="Geremia" LastEditorUserId="4251" LastEditDate="2020-07-05T14:50:35.243" LastActivityDate="2020-07-05T14:50:35.243" Title="Physics Curricula of 19th Century French Universities?" Tags="|physics|academia|education|19th-century|france|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2549" PostTypeId="2" ParentId="2547" CreationDate="2015-07-17T12:08:35.030" Score="2" Body="&lt;p&gt;Could the &quot;book about Euclid's Theorems&quot; be &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_Elements&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Euclid's Elements&lt;/em&gt;&lt;/a&gt;? That was the standard school textbook on geometry until the 1900's.&lt;/p&gt;&#xA;" OwnerUserId="426" LastActivityDate="2015-07-17T12:08:35.030" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2550" PostTypeId="2" ParentId="2397" CreationDate="2015-07-17T13:43:08.663" Score="2" Body="&lt;p&gt;For biographies of Mathematicians I believe one of the best online resources is maintained by the University of St. Andrews.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/&quot; rel=&quot;nofollow&quot;&gt;http://www-history.mcs.st-and.ac.uk/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2547" LastActivityDate="2015-07-17T13:43:08.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2551" PostTypeId="2" ParentId="2547" CreationDate="2015-07-17T17:04:29.483" Score="2" Body="&lt;p&gt;Maybe he meant &lt;em&gt;Foundations of Geometry&lt;/em&gt; by David Hilbert? This is about Euclidean Geometry, but it gives arguments against the Euclidean axiomation of it, which is what he could have meant by 'it was all wrong'. It was published in 1899. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-07-18T18:25:16.987" LastActivityDate="2015-07-18T18:25:16.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2552" PostTypeId="1" AcceptedAnswerId="2566" CreationDate="2015-07-18T08:55:34.763" Score="9" ViewCount="565" Body="&lt;p&gt;Is Coulomb's law the earliest formal equation of electricity?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before Coulomb, many scientists and engineers conducted experiments of electricity.&lt;/p&gt;&#xA;" OwnerUserId="2605" OwnerDisplayName="user2986288" LastEditorUserId="55" LastEditDate="2016-11-25T21:18:45.320" LastActivityDate="2016-11-25T21:18:45.320" Title="Is Coulomb's law the earliest mathematical formula describing electricity?" Tags="|physics|electricity|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2553" PostTypeId="1" AcceptedAnswerId="2554" CreationDate="2015-07-18T18:54:18.730" Score="5" ViewCount="1072" Body="&lt;p&gt;Where did the basic physics law $P=VI$ come from? Here, $P$ is power, $V$ is voltage and $I$ is current.  It doesn't have a name like Ohm's law, as far as I could find. So where did it originally come from, who invented it and when? Or what are early references for it? &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="62" LastEditDate="2015-07-18T21:00:53.933" LastActivityDate="2022-07-18T08:42:02.677" Title="Where did $P=VI$ come from?" Tags="|physics|electricity|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2554" PostTypeId="2" ParentId="2553" CreationDate="2015-07-18T20:44:16.640" Score="7" Body="&lt;p&gt;This is a form of &lt;a href=&quot;http://en.wikipedia.org/wiki/Joule_heating&quot;&gt;a law&lt;/a&gt; known as &lt;strong&gt;Joule heating&lt;/strong&gt;, &lt;strong&gt;Joule's first law&lt;/strong&gt;, or the &lt;strong&gt;Joule-Lenz law&lt;/strong&gt;. It is the simplest form of the law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.britannica.com/science/Joules-law&quot;&gt;Joule discovered, in the 1840s,&lt;/a&gt; that the power in a circuit was proportional to the current squared times the resistance. &lt;a href=&quot;http://www.electricalfacts.com/Neca/People/history/joule.shtml&quot;&gt;He published a short paper&lt;/a&gt; entitled &lt;a href=&quot;http://archive.org/stream/jstor-110721/110721_djvu.txt&quot;&gt;&quot;On Production of Heat by Voltaic Electricity&quot;&lt;/a&gt;. Joule used the form&#xA;$$P=I^2R$$&#xA;One version of &lt;a href=&quot;http://en.wikipedia.org/wiki/Ohm%27s_law&quot;&gt;Ohm's law&lt;/a&gt;, of course, is&#xA;$$V=IR$$&#xA;Doing some substitution yields the form you give.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that it is impossible to determine who first wrote Joule's law by substituting in Ohm's law. It would be akin to asking, &quot;Who was the first person to write Ohm's law as $R=\frac{V}{I}$?&quot; &lt;a href=&quot;http://archive.org/stream/scientificpapers01joul#page/82/mode/2up&quot;&gt;This&lt;/a&gt; might be worth looking through, although I do not know if it is comprehensive.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-07-18T20:44:16.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2557" PostTypeId="1" AcceptedAnswerId="2561" CreationDate="2015-07-19T00:59:24.213" Score="4" ViewCount="1069" Body="&lt;p&gt;Why did the ancients believe celestial matter is of a totally different type than terrestrial matter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a footnote in Christopher A. Decaen's &lt;em&gt;&lt;a href=&quot;http://www.thomist.org/jourl/explore.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Thomist&lt;/a&gt;&lt;/em&gt; &lt;strong&gt;68&lt;/strong&gt; (2004): 375-429 article &quot;&lt;a href=&quot;http://www.thomist.org/jourl/2004/July/2004%20July%20A%20Dec.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle's Aether and Contemporary Science&lt;/a&gt;:&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;7&quot;&gt;&#xA;  &lt;li&gt;Emphasizing the dialectical character of these arguments for aether, St. Thomas notes that although all the evidence suggests that the heavenly bodies and their motions are incorruptible, it remains a possibility that we have not observed them long enough (&lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#1-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;I De Caelo&lt;/em&gt;, lect. 7&lt;/a&gt;, nn. 5-6). Aristotle himself does not try to show that the Earth is immobile at the center of the universe until &lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#2-13&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Caelo&lt;/em&gt;, 2.13&lt;/a&gt;-&lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#2-14&quot; rel=&quot;nofollow noreferrer&quot;&gt;14&lt;/a&gt;, and St. Thomas points out that until this issue is settled it is possible that the Earth, not the heavens, is moving (&lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#2-11&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;II De Caelo&lt;/em&gt;, lect. 11&lt;/a&gt;, n. 2). While Aristotle could not be accused of circular reasoning here, since the later arguments against the Earth's mobility do not appear to rest on the assumption that the heaven is a different kind of matter, nevertheless these arguments are not demonstrative.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, one argument &quot;that the heavenly bodies and their motions are incorruptible&quot; is the uniformity and constancy of their apparent motions over long periods of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did ancient astronomers advance any other arguments &quot;that the heavenly bodies and their motions are incorruptible&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A plausible argument, which I don't know if any ancient astronomers have made, is that astronomy's objects of study are so far removed from the senses; thus, why would they be akin to touchable matter?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-01-07T19:34:47.473" LastActivityDate="2022-08-22T21:32:43.630" Title="Why did the ancients believe celestial matter was of a different type than terrestrial matter?" Tags="|astronomy|ancient-greece|archaeoastronomy|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2559" PostTypeId="2" ParentId="2526" CreationDate="2015-07-19T02:20:59.470" Score="-1" Body="&lt;p&gt;The BBC has a podcast called &lt;a href=&quot;http://www.bbc.co.uk/programmes/b00srz5b/episodes/downloads&quot; rel=&quot;nofollow&quot;&gt;A Brief History of Mathematics&lt;/a&gt; which gives 10 short (approximately 10 minute) biographies of famous mathematicians, including Ramanujan and Hardy.  It discusses the famous letter from Ramanujan to Hardy (which Hardy initially thought was a fraud), how Hardy brought Ramanujan to Britain, and discusses the collaboration, and, later personal problems of both men.  Be forewarned the ending is rather sad.&lt;/p&gt;&#xA;" OwnerUserId="2458" LastActivityDate="2015-07-19T02:20:59.470" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2561" PostTypeId="2" ParentId="2557" CreationDate="2015-07-19T06:06:23.597" Score="7" Body="&lt;p&gt;One argument is that terrestrial objects are made up of the four elements earth, water, air, fire, and out of these the first two are &quot;heavy&quot; elements whose natural motion is rectilinear motion toward the center of the universe, while the latter two are &quot;light&quot; elements whose natural motion is rectilinear motion upwards. The natural motion of the heavenly bodies, on the other hand, is evidently circular. Therefore they are unlike any terrestrial matter. If they were made of stone they would fall down, and if they were made of fire they would fly away. There is a discussion more or less to this effect in Aristotle, &lt;em&gt;De Caelo&lt;/em&gt; I.2.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2015-07-19T06:06:23.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2562" PostTypeId="1" AcceptedAnswerId="2882" CreationDate="2015-07-19T07:10:16.977" Score="6" ViewCount="587" Body="&lt;p&gt;On page 10 of &lt;a href=&quot;https://rmu.sbm.org.br/wp-content/uploads/sites/27/2018/03/n46_Noticias.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article (in Portuguese), I read that &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89variste_Galois&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galois&lt;/a&gt; used the following phrase when his memoirs weren't accepted by the French Academy of Sciences:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Oh! Innocent cherubs!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Could someone give references or details on this quote?&lt;/p&gt;&#xA;" OwnerUserId="2611" LastEditorUserId="2611" LastEditDate="2021-09-16T22:48:41.743" LastActivityDate="2021-09-20T07:09:35.950" Title="Galois quote about his refused work" Tags="|mathematics|mathematicians|biographical-details|quote|galois|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2563" PostTypeId="1" CreationDate="2015-07-19T17:06:10.053" Score="4" ViewCount="1288" Body="&lt;p&gt;Why is Lagrange buried in the Panthéon in Paris, but not any other French mathematicians or theoretical physicists? I would understand if they were not represented at all, but it seems absurd to choose just one from a selection of many great names.&lt;/p&gt;&#xA;" OwnerUserId="2150" LastActivityDate="2015-07-20T06:28:47.237" Title="Lagrange buried in the Panthéon" Tags="|mathematicians|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2564" PostTypeId="2" ParentId="2563" CreationDate="2015-07-19T19:17:27.633" Score="15" Body="&lt;p&gt;A definitive list might be useful. I went through the entire &lt;a href=&quot;https://en.wikipedia.org/wiki/Panth%C3%A9on#Full_list_of_people_interred_or_commemorated&quot;&gt;list&lt;/a&gt;, and here are my results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematicians and physicists buried in the Panthéon:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Marie_Curie&quot;&gt;Marie Curie&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Curie&quot;&gt;Pierre Curie&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin&quot;&gt;Jean Baptiste Perrin&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Langevin&quot;&gt;Paul Langevin&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Painlev%C3%A9&quot;&gt;Paul Painlevé&lt;/a&gt;. (Also a politician)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Lazare_Carnot&quot;&gt;Lazare Carnot&lt;/a&gt;. (Also a politician)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange&quot;&gt;Joseph-Louis Lagrange&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Marquis_de_Condorcet&quot;&gt;Marquis de Condorcet&lt;/a&gt;. (Also a philosopher). His remains have been lost.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaspard_Monge&quot;&gt;Gaspard Monge&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Other scientists:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The chemist &lt;a href=&quot;https://en.wikipedia.org/wiki/Marcellin_Berthelot&quot;&gt;Marcellin Berthelot&lt;/a&gt;. (Also a politician)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The physiologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Jean_George_Cabanis&quot;&gt;Pierre Jean George Cabanis&lt;/a&gt;. (Also a philosopher)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Physician and scientist (and other things) &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean-Paul_Marat&quot;&gt;Jean-Paul Marat&lt;/a&gt;. (Later disinterred from the Panthéon)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;At this point I have gone through the entire list, every biography at least three times. Several interred have no direct biography, but I chased and read their French biographies in every case. I now believe that this is complete. My inclusions in the &quot;Other Scientists&quot; category were a judgement call. I omitted a couple of physicians and cartographers (and that sort in general) in the spirit of the original post. I would be pleasantly surprised to hear I missed someone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visiting the Panthéon biographically was a historically enriching way to spend a Sunday morning for me. I will be sure to take pictures of all of these memorials the next time I find myself in Paris.&lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2015-07-20T06:28:47.237" LastActivityDate="2015-07-20T06:28:47.237" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2565" PostTypeId="1" CreationDate="2015-07-20T15:32:54.610" Score="3" ViewCount="88" Body="&lt;p&gt;According to &lt;a href=&quot;http://en.wikipedia.org/wiki/Comoving_distance&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Comoving distance factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space (though this may change due to other, local factors such as the motion of a galaxy within a cluster)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Who was the person who came up with the idea of comoving distance?&lt;/p&gt;&#xA;" OwnerUserId="2574" LastEditorUserId="62" LastEditDate="2015-07-20T21:36:38.637" LastActivityDate="2015-10-19T02:19:00.530" Title="Who introduced the notion of comoving distance in cosmology?" Tags="|cosmology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2566" PostTypeId="2" ParentId="2552" CreationDate="2015-07-20T19:19:52.330" Score="4" Body="&lt;p&gt;Sort of, if we interpret &quot;electricity&quot; and &quot;formal equation&quot; narrowly. Then the only other contender is Cavendish, who in 1775 did experiments with various capacitors, established that charge on them was proportional to voltage, and introduced the notion of capacitance as the proportionality constant, so $Q=CV$. Of course, he did not call voltage &quot;voltage&quot;, but &quot;degree of electrification&quot;, and he is the one who introduced that too. Unfortunately, Cavendish did not publish his results, they were only published by Lord Kelvin in 1879, so the &quot;priority&quot; goes to Coulomb in 1785. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Coulomb's achievement was more experimental than theoretical. That electric force falls as inverse square was already conjectured by Priestley in 1766 based on Franklin's observation that a charge is not attracted to the walls of electrified can when inside it. Priestley recalled that Newton proved in Principia that a mass inside a spherical shell is not attracted to its walls either, and that the force has to fall as inverse square of the distance for this to happen. Cavendish came to the same conclusion in 1771, but &quot;indifferent to fame he neglected to communicate this and other work of importance&quot;, as we already know. His anticipations of the notions of current, resistance and the Ohm's law (albeit only qualitatively, he &quot;measured&quot; current by electrocuting himself) shared the same fate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laws of magnetic force were also established earlier. Newton wrote in Principia that the &quot;&lt;em&gt;power of magnetism... in receding from the magnet, decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations&lt;/em&gt;&quot;, the result confirmed by the publishers of the 1742 edition Le Seur and Jacquier. And in 1750 Michell clarified that in a magnet &quot;&lt;em&gt;each pole attracts or repels exactly equally&lt;/em&gt;&quot; and &quot;&lt;em&gt;attraction and repulsion of magnets increases, as the squares of the distances from the respective poles increase&lt;/em&gt;&quot;. See &lt;a href=&quot;http://www.electricityforum.com/a-timeline-of-history-of-electricity.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whittaker's History of the Theories of Aether and Electricity&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=CGJDAAAAIAAJ&amp;amp;pg=PA52&amp;amp;lpg=PA52&amp;amp;dq=electricity+history+cavendish&amp;amp;source=bl&amp;amp;ots=S-dh0pHFTg&amp;amp;sig=hqA8utwgmB_7yM5ggXJCVomQf-M&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CCwQ6AEwBTgKahUKEwik-cveq-rGAhWQl4gKHchWDNA#v=onepage&amp;amp;q=electricity%20history%20cavendish&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Timeline Of History Of Electricity&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although that light is an electromagnetic wave was not established until 19th century by Maxwell, the law of reflection was already known to Euclid c. 300 BC if not earlier, one of the first quantitative laws of physics to be discovered. Ptolemy knew the approximate law of refraction (for small incidence angles) c. 150 AD, and measured some refraction coefficients, and the exact law (with sines) was established by Snell and Descartes in 17th century. And in 1619 Kepler gave the famous geometric argument for the inverse square law applied to the intensity of light from point sources:&quot;&lt;em&gt;there is as much light in the narrower spherical surface, as in the wider... therefore the form will be made weaker in unequal spheres in the ratio of the square of its distance&lt;/em&gt;&quot;. Bullialdus extended it to gravity in 1645, and eventually it reached Newton (possibly through Hooke), who confirmed it for gravity based on Kepler's planetary laws, and influenced Michell and Priestley. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/1973/who-was-first-to-explain-intuitively-the-inverse-square-law-of-gravity/1975#1975&quot;&gt;Who was first to explain intuitively the inverse square law of gravity?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-07-20T19:19:52.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2567" PostTypeId="1" AcceptedAnswerId="2572" CreationDate="2015-07-20T19:51:39.927" Score="20" ViewCount="36612" Body="&lt;p&gt;Why is $\mathrm{kg}$ the standard unit for mass and not $\mathrm{g}$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that there is the &lt;em&gt;kilogramme des Archives&lt;/em&gt; which is a kilogram and not a gram. But originally on April 7, 1795 the gram was defined as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The absolute weight of a volume of pure water equal to the cube of the hundredth part of the metre, and at the temperature of melting ice. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is the reason that they switched to the $\mathrm{kg}$ when using the &lt;em&gt;kilogramme des Archives&lt;/em&gt;? Perhaps it was easier to make and less sensitive to mistakes? Are there other reasons?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify why I think this is weird: The other six standards, namely metre, second, ampere, Kelvin, mole and candela don't have a SI prefix when used as standard unit. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2710" LastEditDate="2015-08-10T05:58:36.320" LastActivityDate="2020-06-16T23:00:13.310" Title="Why is kg the standard unit for mass and not g in SI?" Tags="|physics|units|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2568" PostTypeId="2" ParentId="2565" CreationDate="2015-07-21T00:24:44.127" Score="1" Body="&lt;p&gt;From &lt;a href=&quot;https://books.google.com/books?id=z0vlYHQZHJcC&amp;amp;pg=PA115&amp;amp;lpg=PA115&amp;amp;dq=1934+Milne+and+McCrea+comoving+distance&amp;amp;source=bl&amp;amp;ots=UjpFKH3yQc&amp;amp;sig=6THued47Tsa-WxwD_GpOmybnZyI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CC0Q6AEwAmoVChMImMe-sfPqxgIVVDWICh3O-Q6i#v=onepage&amp;amp;q=1934%20Milne%20and%20McCrea%20comoving%20distance&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Longair's Cosmic Century&lt;/a&gt; it appears that Milne and McCrea introduced comoving radial distance in 1934 in their analysis of Friedmann–Lemaître–Robertson–Walker expanding universe solutions to general relativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;They showed that despite the fact that Newtonian mechanics can not provide a fully consistent cosmological model, simple ideas from Newtonian physics can provide insight into solutions&lt;/em&gt;&quot;. Using Newtonian gravity they &quot;&lt;em&gt;derived formulae of exactly the same form as in full theory&lt;/em&gt;&quot; with comoving distance to a galaxy as a coordinate. &quot;&lt;em&gt;Not unexpectedly the Newtonian argument can not cope with the curvature of space, and a relativistic expression for the inertial mass density is needed, which includes the pressure of the gas... The analysis of Milne and McCrea was important because it showed that, despite the problems with boundary conditions at infinity the Newtonian model can be used successfully on large scales in the Universe...&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-07-21T00:24:44.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2570" PostTypeId="1" CreationDate="2015-07-21T22:50:36.510" Score="9" ViewCount="1638" Body="&lt;p&gt;This &lt;a href=&quot;https://youtu.be/hUnhrFHe2AA?t=58&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurzgesagt YouTube video&lt;/a&gt; claims that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Without [the Hubble Space Telescope], we would never have known that our galaxy is one of billions in an enormous universe. But, the actual human, Mr. Hubble, discovered another galaxy in 1929 (Andromeda).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I found this claim surprising (note: I do not have much more than high school physics). According to Wikipedia, the Hubble Space Telescope was launched only in 1990. I would have thought that we already knew that there were very many other galaxies, long before 1990. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, when did scientists know (with a great deal of confidence) that our galaxy was merely one amongst billions in the universe?&lt;/p&gt;&#xA;" OwnerDisplayName="user67" LastEditorUserId="-1" LastEditDate="2017-07-10T11:19:42.943" LastActivityDate="2023-12-01T22:59:01.557" Title="When did we learn that our galaxy was merely one amongst many billions?" Tags="|astronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2571" PostTypeId="2" ParentId="2570" CreationDate="2015-07-21T23:33:01.540" Score="6" Body="&lt;p&gt;That video comment is not precisely true in the sense that, the idea that we were one of billions of galaxies was not new to 1990 (quite the contrary), but it has an element of truth to it. Galaxies were known as &quot;extragalactic-nebulae&quot; in the 1920's. By 1929, when Hubble wrote his seminal work &lt;a href=&quot;http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?letter=.&amp;amp;classic=YES&amp;amp;bibcode=1926ApJ....64..321H&amp;amp;page=&amp;amp;type=SCREEN_VIEW&amp;amp;data_type=PDF_HIGH&amp;amp;send=GET&amp;amp;filetype=.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Extra-Galactic Nebulae&lt;/em&gt;&lt;/a&gt; we had already photographed several of them. Hubble's paper has many pictures. Here is a &lt;a href=&quot;https://cosmology.carnegiescience.edu/timeline/1929&quot; rel=&quot;noreferrer&quot;&gt;supplementary article&lt;/a&gt; (light reading), sketching the events from 1919 to Hubble's 1929 paper highlighting the issue, to wit, the universe is expanding. At the time, astronomers were identifying galaxies many millions of light years away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the thing about the Hubble telescope is, it quite simply has allowed us to discover more galaxies. There is a nifty little article on hubblesite titled &lt;a href=&quot;http://hubblesite.org/hubble_discoveries/breakthroughs/cosmology&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Taking the Universe's Baby Pictures&lt;/em&gt;&lt;/a&gt; that has some of these results. The &lt;a href=&quot;http://imgsrc.hubblesite.org/hu/hubble_discoveries/breakthroughs/assets/cosmo_large/HUDF.png&quot; rel=&quot;noreferrer&quot;&gt;first picture&lt;/a&gt; on that page claims to have about 10,000 galaxies in one image. That's the sort of thing made possible by the telescope. Here is another article, &lt;a href=&quot;http://news.nationalgeographic.com/news/2012/121214-hubble-oldest-galaxy-discovered-space-science/&quot; rel=&quot;noreferrer&quot;&gt;Hubble Discovers Oldest Known Galaxy&#xA;&lt;/a&gt;. Again, a breakthrough made possible by the telescope. We can now see galaxies billions of light years away.   &lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2015-07-22T02:18:48.990" LastActivityDate="2015-07-22T02:18:48.990" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2572" PostTypeId="2" ParentId="2567" CreationDate="2015-07-21T23:34:34.217" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Why is kg the standard unit for mass and not g?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Tongue in cheek answer: Because a foolish consistency is the hobgoblin of little minds?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More seriously, none of the immediate predecessors of the SI bothered to have all of their base units be consistent with the prefix-free units. Gauss proposed a millimeter-gram-second system in the 1830s. Maxwell and Thomson modified this to a centimeter-gram-second system in the 1860s. There was a lot of infighting over the electromagnetic units in those CGS systems. Giorgi proposed yet another system in 1901, the meter-kilogram-second-ampere system. This system is the immediate predecessor to the current International System.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the reason that they switched to the kg when using the kilogramme des Archives? Perhaps it was easier to make and less sensitive to mistakes? Are there other reasons?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You have it backwards. The original concept of mass by the French revolutionaries working on the metric system was the mass of a liter of water. This unit of mass was to be called the grave. French scientists worked on making this realizable (the mass of a volume water turned out not to form a good basis). The Republican government that followed the French Revolution thought this grave was too big for practical uses, so they invented the gramme as the mass of a milliliter of water. The work on the grave prototype continued, only now this would be called the kilogram prototype.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-07-21T23:34:34.217" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2576" PostTypeId="1" AcceptedAnswerId="2577" CreationDate="2015-07-22T12:16:48.950" Score="4" ViewCount="360" Body="&lt;p&gt;I know that Mathematics is nearly as old if not AS old as humanity itself, but which empire/nation can be credited as the cornerstone of what we know today as Mathematics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My initial guess is Mesopotamia as it contains some of the earliest historical evidence of what we called Mathematics today. If anyone has some hard primary source information, I'd love to hear it.&lt;/p&gt;&#xA;" OwnerUserId="2624" LastActivityDate="2017-06-20T13:53:56.233" Title="Where did Mathematics establish its roots?" Tags="|mathematics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2577" PostTypeId="2" ParentId="2576" CreationDate="2015-07-22T14:12:19.373" Score="4" Body="&lt;p&gt;The word &quot;mathematics&quot; can have several different meanings.&#xA;If you include things like counting (all sort of things, say sheep or beer or days of the year, stars in the sky or solstices and other astronomical events) or making the corners of a building right and the walls straight, this kind of mathematics is as old as mankind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you are asking of mathematics &quot;as it is understood now&quot;, in the narrow sense, that is a system of knowledge based on rigorous proofs, this was invented in one country, probably in one place approximately in 6-th  century BC.&#xA;We do not now exactly, but according to the tradition this invention was made&#xA;by Thales (624-546 BC, dates from Wikipedia but they are only guesses). He lived in Miletus, on the territory of modern Turkey,&#xA;but was Greek by language. He is credited with the crucial discovery: that some true things can be PROVED. (Not just verified by experience). Some simple things he proved are still taught in schools, essentially with the same proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a semi-legendary person, his writings do not survive, and we know about them only from much later sources. So one cannot be sure about the details. But the fact is that mathematics in the modern sense was invented by the ancient Greeks&#xA;approximately 6 centuries BC. There is no record in history of an independent&#xA;invention anywhere else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. We cannot be certain, of course that Thales really proved any theorems, this statement is based on ancient histories written many centuries after him.&#xA;But we can be reasonably certain that mathematics was invented by some Greek, and&#xA;probably this happened in 6-th century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest surviving complete mathematical works, with proofs, come, as I understand, from the 3-d century BC. (Correct me if I missed some earlier work). But this is already fully developed, essentially modern mathematics. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-07-22T14:42:00.427" LastActivityDate="2015-07-22T14:42:00.427" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2578" PostTypeId="1" CreationDate="2015-07-23T01:42:28.910" Score="8" ViewCount="1571" Body="&lt;p&gt;In his famous address at the Royaumont Seminar in 1959, Jean Dieudonné famously called for the elimination of Euclidean geometry from the secondary school curriculum.  In the published (English-language) version of his remarks (&quot;New Thinking in School Mathematics&quot;, 1959), Dieudonné expressed this call with the slogan &quot;Euclid must go!&quot;, and most subsequent references to this speech quote him with that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But a different (and stronger) version of this slogan is quoted in Siobhan Roberts's book &lt;em&gt;King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry&lt;/em&gt;.  In that book (p. 157) Dieudonné is quoted as saying (in French) &quot;A bas Euclide! Mort aux triangles!&quot; (&quot;Down with Euclid!  Death to triangles!&quot;).  Roberts cites three sources for this quote -- but two of them are to books that include the other version (&quot;Euclid must go!&quot;), and the third seems to be to an unpublished interview the author conducted with Imre Toth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We know what Dieudonné wrote in the published version of his Royaumont address, but (apparently) Imre Toth remembers him saying something different.  Does anybody know if there is any other published source, other than Roberts, that can shed light on what Dieudonné actually said at Royaumont?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="294" LastEditorUserId="26" LastEditDate="2015-07-23T20:39:17.317" LastActivityDate="2021-07-12T12:52:23.087" Title="Did Dieudonné say &quot;Euclid must go!&quot; or &quot;Down with Euclid! Death to triangles!&quot;?" Tags="|biographical-details|euclidean-geometry|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2579" PostTypeId="2" ParentId="2576" CreationDate="2015-07-23T11:02:08.140" Score="4" Body="&lt;p&gt;Alexandre Eremenko's answer is all good, but here are some extra things : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics did sort of develop independently several times in history, I believe basically three times during the Iron Age : Greece (as mentioned above), India and China. Before then, people didn't care all that much about proofs and mostly had ad-hoc formulas and big tables of values (Sumerian math is famous for giant tables).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;China had such works as &quot;The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven&quot; during the Zhou dynasty (1122–256 BCE), which contains a proof of the Pythagorean theorem, and the Mo Jing in 330 BCE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit : Some more things, I guess : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot find much in the ways of sources indicating foreign origins of chinese mathematics, except perhaps &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0083665688903169&quot; rel=&quot;nofollow&quot;&gt;babylonian&lt;/a&gt; (Chinese mathematics was pretty tied to astronomy). For dates, a comment from Mathematics across cultures : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;On the whole, if Greek and Chinese mathematical astronomy are compared from a merely technical and non-historical point of view, as an abstract catalogue of isolated results, it clearly appears that Chinese developments, such as those connected with the discovery of the precession, or the solar and lunar inequality, occur much later than the former, usually five or more century&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Indian mathematics, the oldest geometry texts are the Sulbasūtras, parts of religious texts used for altar construction. The seven big ones are the Bodhāyana, Āpasthamba, Kātyāyana, Mānava, Maitrāyana, Varāha and Vidhūla, written between 800 and 500 BCE. They include the proofs of such theorems as : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The diagonal of a rectangle divides it into equal parts&lt;/li&gt;&#xA;&lt;li&gt;The diagonals of a rectangle bisecting each other and opposite areas are equal&lt;/li&gt;&#xA;&lt;li&gt;The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into two equal halves&lt;/li&gt;&#xA;&lt;li&gt;A rectangle and a paralleloram on the same base and between the parallels are equal in area&lt;/li&gt;&#xA;&lt;li&gt;The diagonals of a rhombus bisect each otbher at right angles&lt;/li&gt;&#xA;&lt;li&gt;The pythagorean theorem&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="2628" LastEditDate="2015-07-24T08:25:54.503" LastActivityDate="2015-07-24T08:25:54.503" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2580" PostTypeId="1" AcceptedAnswerId="2581" CreationDate="2015-07-24T01:09:36.253" Score="6" ViewCount="171" Body="&lt;p&gt;Is there any current English translation of the mathematical works of &lt;a href=&quot;https://en.wikipedia.org/wiki/Omar_Khayy%C3%A1m&quot; rel=&quot;nofollow&quot;&gt;Omar Khayyám&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="337" LastEditorUserId="10106" LastEditDate="2020-06-07T13:40:58.453" LastActivityDate="2020-06-07T13:40:58.453" Title="English translation of Omar Khayyám's mathematical work" Tags="|mathematics|mathematicians|reference-request|translations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2581" PostTypeId="2" ParentId="2580" CreationDate="2015-07-25T13:53:32.990" Score="2" Body="&lt;p&gt;I am not sure that there is an adequate English translation of al-Khayyāmī's famous treatise on Algebra. If you read French there is a good scholarly translation by Roshdi Rashed et Ahmad Djebbar, &quot;L’Oeuvre algébrique d’al-Khayyām&quot;, Aleppo 1981&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-07-25T13:53:32.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2582" PostTypeId="1" CreationDate="2015-07-25T17:16:03.783" Score="12" ViewCount="299" Body="&lt;p&gt;In &lt;em&gt;The Equation that Couldn't be Solved&lt;/em&gt;, Mario Livio writes of academia in 16th century Bologna. Apparently, mathematicians would take part in public debates, sometimes involving solving problems. They were extremely well-known, and could result in great discoveries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the format of a mathematical debate? Do any accounts exist of one?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="4251" LastEditDate="2020-07-08T17:06:10.350" LastActivityDate="2020-07-08T17:06:10.350" Title="What was the typical format of a 16th century mathematical debate?" Tags="|mathematicians|academia|16th-century|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2583" PostTypeId="2" ParentId="2557" CreationDate="2015-07-26T06:31:01.000" Score="5" Body="&lt;p&gt;It's difficult for us who have learned that the laws of physics are constant for all time and space to put ourselves in the minds of &quot;the ancients&quot;. But all the reading I've done in this area, of the classical Greeks and the early, middle and late Middle Ages, indicates that they had absolutely no reason to believe that the sky had the same laws as the Earth. The idea of unified physics was unknown. Even two hundred years ago, anyone who claimed that electricity, magnetism and light were the &quot;same stuff&quot; would not have been believed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More importantly, the ancient Greeks strictly split physics and astronomy into mutually exclusive subjects. The Heath book (in Dover Publications) about &quot;Greek Astronomy&quot; contains passages which scold astronomers for going outside their job of making numerical calculations to speculate on the actual &lt;em&gt;nature&lt;/em&gt; of what they were describing numerically. So all speculations about the nature of things were non-numerical. So they never got to the point that Newton did, where he showed that the &lt;em&gt;numbers&lt;/em&gt; for describing terrestrial and extra-terrestrial physics were a near-perfect match. It was this numerical match in the gravity equations which finally unified sky physics with Earth physics. And even then, most people found this difficult to accept. It is only because we are taught the modern viewpoint in school that we find it difficult to see how odd it used to be to think of earth and sky as a combined system obeying the same laws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the answer to your question is that it is the wrong question. The &quot;correct&quot; question is when and how did people get the idea that a single set of physical laws governs both Earth and sky. And the answer to that is that the calculations of astronomers finally led to Newton demonstrating that apples and moons obey the same laws. That clinched it!&lt;/p&gt;&#xA;" OwnerUserId="2502" LastActivityDate="2015-07-26T06:31:01.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2585" PostTypeId="1" CreationDate="2015-07-25T19:25:08.280" Score="11" ViewCount="425" Body="&lt;p&gt;What is the first appearance in print of the modern definition of an abstract group? To qualify, it should be a formal definition, contain the word &quot;elements&quot; (so Burnside's 1897 restriction to &quot;operations&quot;, where associativity is automatic, does not qualify), and contain the 4 standard properties: closure, associativity, identity, inverses (so Weber's 1882 two-sided cancellation does not qualify).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find all these requirements in Harold Hilton, An Introduction to the Theory of Groups of Finite Order, published in 1908. Is there an earlier appearance?&lt;/p&gt;&#xA;" OwnerDisplayName="David Callan" LastEditorUserId="62" LastEditDate="2015-07-26T19:03:59.243" LastActivityDate="2015-07-28T01:21:29.280" Title="First appearance of modern definition of a group" Tags="|mathematics|group-theory|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2586" PostTypeId="2" ParentId="2585" CreationDate="2015-07-25T19:29:07.127" Score="0" Body="&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Group_(mathematics)#History&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;, Walther von Dyck gave the first modern definition of a group.&lt;/p&gt;&#xA;" OwnerDisplayName="mathers101" LastActivityDate="2015-07-25T19:38:25.850" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2587" PostTypeId="2" ParentId="2585" CreationDate="2015-07-25T20:37:13.930" Score="0" Body="&lt;p&gt;&quot;... and the notion of abstract groups was introduced by A.Cayley in three papers starting in 1849...&quot; (from &lt;a href=&quot;https://books.google.se/books?id=q7sPg5cuwtAC&amp;amp;lpg=PP1&amp;amp;hl=sv&amp;amp;pg=PA50#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, &lt;a href=&quot;http://static-content.springer.com/lookinside/art%3A10.1007%2FBF01443322/000.png&quot; rel=&quot;nofollow&quot;&gt;the original von Dyck's paper 1882&lt;/a&gt; does carry the epigraph of Cayley's quote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;UPDATE: Another historical paper from 2007 &quot;&lt;a href=&quot;http://www.rbhm.org.br/issues/RBHM%20-%20Festschrift/3%20-%20Wussing%20-%20final.pdf&quot; rel=&quot;nofollow&quot;&gt;Arthur Cayley, the multiplication table and the notion of abstract group&lt;/a&gt;&quot; (in German). Cannot read German myself unfortunately...&lt;/p&gt;&#xA;" OwnerUserId="2640" OwnerDisplayName="A.G." LastActivityDate="2015-07-25T20:49:09.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2588" PostTypeId="1" AcceptedAnswerId="2597" CreationDate="2015-07-24T23:18:22.523" Score="8" ViewCount="274" Body="&lt;p&gt;I was looking up some ancient ways of writing numbers, to use as part of a lesson on the history of discovery of new numbers, for some young students.&#xA;In particular I looked at the notation used by the ancient Babylonians, Mayans, Chinese, and Egyptians. According to what I read, all of them had a concept of place value except the Egyptians. Those that did have a place value used different bases, respectively of 60, 20, and 10.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What surprised me is the fact that in all three of these examples, the place values are structured so that powers of the base increase from right to left, just as they do in the current standard notation. I see nothing canonical about this decision. It could just as easily have been left to right, top to bottom, bottom to top, along some diagonal, or even indicated by size of the symbol. In fact we find most of these variations and others when we look at different inventions of writing words. So is there an explanation for this common choice?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick search yields no examples with an alternative place value system. It seems the ancient Indians also developed place value the same way, though I didn't come across a picture of the exact notation. Maybe this question seems lazy but I'm in the middle of teaching an intense program where this would be useful, and would like a quick answer if possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an example of an alternative development of place-value in the ancient history of inventing symbols for numbers? If not, is there some archaeological justification for this cross-cultural phenomenon?&lt;/p&gt;&#xA;" OwnerUserId="2652" OwnerDisplayName="j0equ1nn" LastEditorUserId="62" LastEditDate="2015-07-29T22:23:34.813" LastActivityDate="2019-07-21T21:16:46.713" Title="How can we account for the independent inventions of place value all using the same direction?" Tags="|mathematics|number-theory|teaching|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2589" PostTypeId="2" ParentId="2588" CreationDate="2015-07-27T09:23:35.430" Score="5" Body="&lt;p&gt;In Arabic the so-called Indian numbers are written with the highest place value on the left, exactly as they are in India and in the West. However, in the sexagesimal notation used by mediaeval Arabic astronomical authors the highest place value is on the right. For example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;لٮ  مط  لر ح&lt;/p&gt;&#xA;&#xA;&lt;p&gt;30+2 40+9 30+7 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;= 32 x 216000 + 49 x 3600 + 37 x 60 + 8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;= 7 090 628&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-07-27T09:23:35.430" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2590" PostTypeId="2" ParentId="537" CreationDate="2015-07-27T09:36:22.217" Score="2" Body="&lt;p&gt;Proposition 129 of Pappus' Book VII says that the projection from one line to another keeps the &quot;cross ratio&quot;. Geometric interpretation is the hexagon theorem, geometric construction can be found possibly in von Staudt. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A nice lecture about Pappus and the cross ratio is on &lt;a href=&quot;https://www.youtube.com/watch?v=JJbh0iJ1Agc&quot; rel=&quot;nofollow&quot;&gt;youtube&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2647" LastEditorUserId="26" LastEditDate="2015-07-28T09:03:37.127" LastActivityDate="2015-07-28T09:03:37.127" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2593" PostTypeId="2" ParentId="2585" CreationDate="2015-07-28T01:21:29.280" Score="1" Body="&lt;p&gt;Maybe Dedekind could have done what you are looking for. The sources for the early development apparently include Cauchy, Galois, Abel, Cayley, etc. However, according to Bell &quot;Men of Mathematics&quot; (enjoyable to read, but not a really scholarly work), page 282:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;This abstract point of view is that now current. It was not Cauchy's, but was introduced by Cayley in 1854. Nor were completely satisfactory sets of&#xA;  postulates for groups stated till the first decade of the twentieth century.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bell says the following on page 518:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Dedekind was one of the first to appreciate the fundamental importance of the concept of a group in algebra and arithmetic. In this early work Dedekind&#xA;  already exhibited two of the leading characteristics of his later thought,&#xA;  abstractness and generality. Instead of regarding a finite group from the&#xA;  standpoint offered by its representation in terms of substitutions [....],&#xA;  Dedekind defined groups by means of their postulates [....] and sought to&#xA;  derive their properties from this distillation of their essence. This is in the modern manner: abstractness and therefore generality.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm not sure if Dedekind is the &quot;first decade of the twentieth century&quot; mathematician Bell was talking about. I think a more interesting secondary question here is when did group theory explicitly include infinite and continuous groups. I'm guessing that in the early days, finite groups were the focus. The combination of the purely algebraic finite groups with analysis to study infinite groups was much later, and I guess Sophus Lie gets most of the credit for that.&lt;/p&gt;&#xA;" OwnerUserId="2502" LastActivityDate="2015-07-28T01:21:29.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2594" PostTypeId="1" CreationDate="2015-07-28T15:00:07.263" Score="9" ViewCount="4606" Body="&lt;p&gt;Dmitri Mendeleev arranged his first periodic table according to the number of protons and valency electrons. How did he, and others of his age, know of these subatomic particles and electron shell capacities? Nowadays, we have technology such as electron microscopes that can help us see (referring to another Physics Stack Exchange post regarding whether we can actually see atoms) these subatomic particles, but how did chemists and scientists in general know?&lt;/p&gt;&#xA;" OwnerUserId="2656" LastEditorUserId="4251" LastEditDate="2021-09-07T21:06:54.487" LastActivityDate="2021-09-07T21:06:54.487" Title="Dmitri Mendeleev's Periodic Table" Tags="|chemistry|atomic-theory|mendeleev|chemists|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2596" PostTypeId="2" ParentId="2557" CreationDate="2015-07-28T17:19:39.330" Score="3" Body="&lt;p&gt;From my studies on the Western, Chinese and Meso-American  cultures(still working on ancient India) from the classical eras up to early Renaissance, the ancient deduced that heavenly bodies were made of different materials than that of the earth because the observers qualities and behaviors of heavenly objects differed radically from the observed qualities and behavior of mundane objects and materials. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prior to the invention of modern materials like stainless steel and plastics, all known materials save gold began to degraded constantly. All metals, even bronze oxidized, woods dried, split or decayed under microbial attack, stones cracked and eroded. Producing pure substances was difficult (save for gold and sulfur) and the vast majority of pure substance either degraded or became contaminated. Pure white substances such a fresh snow, ice or purified salt, seemed to have special properties. Salt was considered a magical and material purification agent because of its pure white color and its know ability to prevent decay. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By contrast, all observable celestial objects except the moon, appeared as eternally unblemished and never decaying objects. For many traditions and schools, the observable features on the face of moon indicated it belonged to the mundane realm. Medieval western (Judaeo-christian-Islamic) astronomy divided the heavens into the sub-lunar, the corruptible earth, atmosphere  and moon and the trans-lunden, the incorruptible regions beyond. The sun appeared to be made of molten cold and the fixed and wandering stars (planets) appeared a dots of pure usually white light. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the available observed data, the only logical deduction was that that the materials of the heavens differed from the materials of the mundane. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Likewise, the behavior of the mundane world seemed utterly chaotic and unpredictable while the behavior of the heavens appeared a paragon of order and predictability. Cartography as we know it today did not exist so all ancient cultures had better maps of the entire heavens than they did of the earth in just a couple of hundred miles away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Astronomy became the first science precisely because the heavens could be universally and repeatable observed. Different cultures with no communication developed the same predictive models for the motions of the sun, moon, planets and stars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, based on the available observed data, the only logical deduction was that the material objects of the heavens followed different rules than material objects in the mundane realm. (It helped that measurements in astronomy are made in angles, which can be easily and repeatedly measures with tools made of many different materials but which still produce the same measure angles. Linear measurement tools are far more susceptible to the error from the composition of the tool and its deformation from changing environmental conditions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observations about the different quality of materials of the heavens and their different behaviors &lt;strong&gt;&lt;em&gt;preceded&lt;/em&gt;&lt;/strong&gt; the philosophical and theological explanations for those differences. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a famous anecdote, usually attributed to the great British scientist Lord Kelvin. After giving a speech on ptolemaic geocentrism, a member of audience came up after and said, &quot;It sure was silly of the Ptolemy to say that the sun went around the earth. Why to do you think he believed something so silly.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Supposedly, Lord Kelvin blinked and replied, &quot;Because good Sir/Madam that's what it looked like.&quot; He meant that prior to the invention of the telescope, no scientific evidence existed that the sun didn't orbit the earth. All heliocentric arguments up to that point e.g. Bruno, were based on philosophy or theology, not observation and science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because Ptolemy got caught up in the scientific wars of the Reformation, he has gone down in history a foolish foil to Copernicus and Kepler but in reality he was one of humanities greatest scientist. His complex model of the heavens remained the most predictive and useful model of the heavens up for centuries up to Kepler. His model even made better predications than Copernicus. That stands as humanities longest standing scientific model. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's important to remember that science isn't about producing absolute truth but rather producing the most predictive model possible. All scientific knowledge is only true or useful within a strict range of precision. Newtonian physics is only valid to a certain precision of IIRC, 6 decimal points in the motions of of all the planets save mercury and fails utterly to predict the orbit of Mercury because the Newton did not know to account for gravities distortion of space itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the usually answer answer as to why the ancient held some particular model of natural phenomena usually boils down to Lord Kelvin's answer, &quot;because that's what it looked like to them.&quot; That divergent cultures often arrived at very similar models supports this assertions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There ancient were hampered not by superstition, a lack of imagination of intellectual courage but by a lack of sufficiently precise and accurate scientific instruments with which to extend human senses and make repeatable measurements. They usually made the correct logical deductions and inferences based on what they could actually observe with the tools they had. The same way science works today.&lt;/p&gt;&#xA;" OwnerUserId="2658" LastActivityDate="2015-07-28T17:19:39.330" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2597" PostTypeId="2" ParentId="2588" CreationDate="2015-07-28T18:20:12.670" Score="5" Body="&lt;p&gt;I don't think the position of place value qualifies as a &quot;discovery&quot; as it's not a natural phenomena. I would qualify it as a convention and conventions usually evolve as matter of connivence attached to some previously existing technology or biases built into the human body structure, neurology or preceptive field.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this case, its likely the predominance of right handedness in humans which in turn biases the visual attention to attend first to the the right visual field and for to the upper right of objects. So for computer screens, abaci, counting bowls, pits tables etc or even patches of sand or other flat writing service, our built in bias is to start in the upper right hand corner. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it also important that placeholder numbers were actually seldom written in the vast majority of their day-to-day uses but more often entered into some kind of physically tallying/calculating tool like an abacus or counting bowls/pits/tables. In all such traditional tools I can think of, the user puts the first markers in the upper right hand corner and works to the left and/or down. Only when all the counting and calculation had been performed was a final number transferred to writing and in most cases, they simply remembered the number.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, class bias likely played a role making a divide between those whose used placeholders for numerical data and calculations but seldom wrote versus those who seldom used numerical data of any kind but who wrote a lot more often.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well up to the time of Newton in most of the world's major cultures, numerical data was for commoners, usually &quot;grubby&quot; merchants, while the elites concerned themselves with the &quot;higher purer&quot; world of largely dimensionless geometry which involved little numerical data.  Counting things, making tallies and sums was undignified grunt work and fit only for plebes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result written numerical data, tallies and especially calculations are extremely rare in the written historical records up until the 1600s. (Newton's first proofs of his laws of motion were purely geometrical.) So even in cultures that wrote left-to-right or vertically, the actual work of counting and calculating used tallying and calculating tools instead of writing down numbers and performing calculations on paper. All those tools started their operations in the upper right hand corner owing to built in anatomical bias. Later, when such tools or their outputs were converted to writing on horizontal surfaces, paper, tablets, walls etc. the right to left placeholder convention followed from the tool even in languages that wrote left-to-right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Anecdotally, I have observed that unless told otherwise, children who write in English still have a tendency to start place holder calculation on the right or towards the right. You also see people start columnar placeholder calculation on the right side of chalkboards but start linear, non-placeholder centered algebraic calculation on the left as if writing text.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I doubt the convention is truly universal as other technological factors or previously existing convention will override the bias in some case. For example, many early astronomers trained themselves to write with their left hands without looking so their right hand could manipulate their instruments while their eyes remained fixed on their celestial object under observations. The text of such writing usually went right-to-left even in languages that normally wrote left-to-right. It was simply easier to blindly position the left hand at the right edge of the paper or tablet and move to the left. Try it yourself and you'll see. &lt;/p&gt;&#xA;" OwnerUserId="2658" LastActivityDate="2015-07-28T18:20:12.670" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2598" PostTypeId="2" ParentId="1874" CreationDate="2015-07-28T18:33:58.830" Score="-2" Body="&lt;p&gt;Depends on when people started thinking about it the issue. Modern humans evolved some 50,000-100,000 years ago, so somebody probably started mulling the issue over at that point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A better way to phrase the question would be, when did humans acquire the observational tools and the math to determine that despite appearances, the earth orbited the sun and not the other way around. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because the issue got caught up in the propaganda wars of the Reformation, it's easy to think that the movement from geocentrism to heliocentrism occurred because of some shift in philosophy or theology. It didn't. It happened when technology provided better observations. From the invention of the telescope, the matter was settled in less than 50 years. Pretty quick for the times and the ongoing conflicts.&lt;/p&gt;&#xA;" OwnerUserId="2658" LastActivityDate="2015-07-28T18:33:58.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2599" PostTypeId="2" ParentId="2542" CreationDate="2015-07-28T18:45:16.223" Score="0" Body="&lt;p&gt;I am pretty sure that Lavoisier was the the first to experimentally prove water a compound by running it down super heated gun barrels. The oxygen bound to the iron, and hydrogen came out as a gas. By careful weighing he determined that the barrel had gained mass, indicating something from the water had bound to the metal in the barrel, and hence that the water had two components. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cavendish and Watt came at the problem from another direction, producing hydrogen first from acids and metals and only later connecting hydrogen with water, thus inferring it was a compound. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's actually kind of hard to tell as no journals existed at the time and most scientific discoveries were conveyed by chain letters. Nationalism played a great role as well, particularly with the English and French each claiming priority for this or that.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lavoisier also spent most of the 1770s and 1780s perfecting French methods for producing stable and reliable gunpowder (which played a major role in the American revolution) so much of his work during that time was considered state secrets. Many of his papers were destroyed when he was murdered in the French revolution. As a result, we don't have a clear picture of exactly when he did what. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All in all though, it was a surprisingly small group of individuals, really a few dozen at most, that jump started chemistry in the late 1700s. They all communicated and built on each other's works making a collective snowball of progress, so determining priority down the wire is really more of a game than useful history. &lt;/p&gt;&#xA;" OwnerUserId="2658" LastEditorUserId="1878" LastEditDate="2015-08-18T22:32:03.587" LastActivityDate="2015-08-18T22:32:03.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2600" PostTypeId="2" ParentId="2428" CreationDate="2015-07-28T19:27:04.837" Score="-1" Body="&lt;p&gt;Don't have recent peer reviewed examples but just for interest there are some odd detours in the history of science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In biology a famous example is Mendelian genetics which was made in the mid-1800s but utterly unrecognized at the time only to be independently rediscovered by two different researchers circa 1904. It's possible a lot political ugliness of the 20th century might have been blunted if population genetics and thus synthetic Darwinism had got an earlier start. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read that relativity could have been deduced from Maxwells equations on electromagnetism but I don't have the source off the top of my head. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big bang is innate in Newtonian physics. When ask why the universe didn't collapse as his models predicted, Newton said god had made the universe balanced precisely to be stable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The big bang was evident in Hubble's work, but speculation on it was ignored until Bell labs stumbled into the cosmic background radiation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edison, Tesla and other electrical inventors of their day were repeatedly told by scientist that many of the inventions would not work because the primary scientific model for electricity, especially in plasma like Edison and Cathode ray tubes, was seriously flawed. Not sure if that counts as late discovery, as the inventors made things thought impossible which sent scientist scrambling. &lt;/p&gt;&#xA;" OwnerUserId="2658" LastActivityDate="2015-07-28T19:27:04.837" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2601" PostTypeId="2" ParentId="2594" CreationDate="2015-07-28T19:32:02.940" Score="16" Body="&lt;p&gt;Of course Mendeleev had no idea about protons or electrons. &#xA;It is he who discovered the integer which we call atomic number, and which was later found to be the number of protons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story of this discovery is approximately this. He was preparing to his chemistry course where he was supposed to give a survey of elements.&#xA;He had to choose some logical order. For this he made little paper cards on which he wrote the element names and their main properties (like chemical valency). Then he tried to arrange the cards in some logical order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally he found that if the cards are ordered according to the atomic weight,&#xA;then their chemical properties change roughly periodically. This was called the&#xA;periodic law. It suggested that the elements are arranged in a table...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were several exceptions to ordering according to the weight. With the corresponding changes the&#xA;table looked so nice, that Mendeleev was convinced that he discovered some law. &#xA;The table also had some gaps. Mendeleev had courage to predict that these&#xA;gaps correspond to the elements which are not discovered yet. He PREDICTED the existence&#xA;of new elements and predicted some of their properties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When these new elements were discovered, this essentially proved that the law was correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the time of discovery this was a purely empirical law. Explanation of this law had to wait for the discovery of quantum mechanics and spin and Pauli Principle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. These sources make it more precise: he did not do this in one evening while preparing to a lecture, but worked on a textbook (as a part of his duties as a professor of chemistry). The sources give his personal recollections, unfortunately all only in Russian:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.hemi.nsu.ru/ucheb145.htm&quot;&gt;http://www.hemi.nsu.ru/ucheb145.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.chem.msu.su/rus/elibrary/trifonov/ivan-mendeleev.html&quot;&gt;http://www.chem.msu.su/rus/elibrary/trifonov/ivan-mendeleev.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-08-03T21:38:04.960" LastActivityDate="2015-08-03T21:38:04.960" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2603" PostTypeId="2" ParentId="533" CreationDate="2015-07-28T21:40:51.943" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Remark. Is this not amazing that we are still using this mile&#xA;  established in XVI century on the basis of a wrong measurement?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not really, all measurements are arbitrary at their base so there are no &quot;wrong measurements.&quot; All measurement simply come down to defining an arbitrary object as a reference against which all other objects are compared to. The point of measurement is to create a common reference for communication and memory. What the reference object actually is has no bearing on its utility. Our entire system could be based on length of a halibut caught 1272 and it would work just the same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As long as everyone can and will use the same reference object, the system works. If we visited an alien world, we'd just ask for their reference object and we'd be good to go. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evolved traditional systems have a better track record than designed &quot;rational&quot; systems. The meter was supposed to be all rational and derived from natural measurements, (ignoring the fact that people pushing the system at the time where lopping off heads right and left) but owing to simple error one of science history's epic bitter personality feuds, its was originally wildly off.  Now it's just an arbitrary length. Even if it were based on an actual consistent physical phenomena, it wouldn't really matter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took more than a century for the metric system to really catch on despite being official in most of Europe since the Napoleonic era. The metric system relies on users having precise linear measurements as well as a need for intensive computations. We take such precision for grandted today but mass produced precision is a mid-to-industrial era phenomena at best.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A metal ruler made of the metals of any time up to the mid industrial era would expand and contract in length by several millimeters as the ambient temperature changed and in any case, manufacturing could not mass produce meter rulers to to less than than a 5mm tolerance at best. So every meter ruler in use could be off as much as 10mm from any other ruler in use. The same ruler would give different measurements at different times. That's a whole lot of accumulated error. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's why the metric system was slow to catch on and mostly ended up being imposed by force officially and was used largely only by elites and military while artisans and manufactures continued to use traditional measures for a good century or more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Revolutionary and Napoleonic France the government issued specs and had shipwrights present plans in metric, and then the shipwrights went right back and used their traditional reference based tools. When the ships all came out slightly different sizes and clearly not based on metric measurements, the shipwrights just gave a gallic shrug and something like, &quot;the ship she still floats and sails, yes?&quot;  (Funny story, the French official specification were based on mathematically flawed concept of a ship's meta center. Had the shipwrights actually follow official instructions, the ships would have been disastrously unstable. The snoots were so confident in their math they didn't bother to test it on real ships.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Metric worked fine for elite computational intensive work like early 18th century astronomy or even surveying in most cases but good luck trying to reliably and repeatedly measure a 41.5cm length of wood for table day after day around the year with early linear measure made with 18th century metallurgy. (Even today it sucks for traditional woodworking. I was amazed to find out how many European woodworkers don't use metric for anything more complicated than cranking out the plywood boxes of the modern style.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lacking precise linear measurements, people in the past made precise measurements with angles instead. A compass/divider can be made of many different materials and be used under many different conditions of heat and humidity and still give the same consistent measurement of angle. Angular measurement dominated all measurement in all human cultures for thousands of years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But converting angles to linear lengths is computationally expensive, especially when you don't have computers or even a handy sheet of paper, so instead, people used compass/dividers to take measures of lengths off local reference objects and then just made everything various multiple or divisions of those lengths.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original &quot;ruler&quot; was block of stone whose dimensions where used as a references for all the measurements of a particular project e.g. a cathedral or castle. It was said to &quot;rule&quot; all the other dimensions. Later, &quot;rulers&quot; became standardized lengths of plain unmarked metal of various sizes and lengths made of alloys that did expand or contract to much in temperature changes. Only in the late 1800s were gradation marks made on them and engineers still use old reference rulers in some cases. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phrase &quot;rule of thumb&quot; means to use one's thumb as a reference object. Since the length of people's thumbs vary, it means that measurement with a lot of potential for variation but which will serve for an immediate purpose, if you use the same thumb. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other traditional measurements evolved to meet local needs and which could be made with local tools and reference objects. An acre was originally defined as the amount of land that could plowed by a team of oxen in a set number of hours in particular types of soil. Sounds silly to modern ears but for taxation and figuring work back at the time, it was a very useful measure. The original concept of acre included the amount of food a unit of land could produce per man hours per year. It allowed for the convertibility between units of poor and good land. A simple modern numerical measurement of area like hectare would have told them nothing useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In theory, the best unit of measurement on the earth's surface is the nautical mile, because it can be in theory be derived from solar observation. Of course, since the earth is not actually round and its axis wobbles, the nautical mile is always off as well. But since it is based on an angular measurement, its very easy to convert from sextant sightings and to make accurate measurements and calculation off charts without dragging out Dr. Babbage's difference engine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have an odd tendency to see lack of power in technology, in this case a lack of mass produced precision, with stupidity in our ancestors. In fact, they had fair less margin for error than we do and their solutions had to work well and reliably where and when they were needed. Smirk all you like but if they hadn't got things right consistently year after year, decade after decade, century after century, we wouldn't be sitting here all fat, happy, snug and smug. &lt;/p&gt;&#xA;" OwnerUserId="2658" LastActivityDate="2015-07-28T21:40:51.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2604" PostTypeId="1" AcceptedAnswerId="2606" CreationDate="2015-07-29T05:52:12.987" Score="6" ViewCount="800" Body="&lt;p&gt;While looking up the old &lt;a href=&quot;http://en.wikipedia.org/wiki/Copernican_Revolution&quot; rel=&quot;noreferrer&quot;&gt;Copernican model&lt;/a&gt; for orbits, I encountered the following image (courtesy Wikipedia):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uL8vy.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uL8vy.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems... weird. Not only would it be an odd thing to come up with or express in mathematics, it also doesn't make much sense intuitively. Even if they weren't using rigorous mathematics to develop the model... why would they ever consider this a reasonable orbit?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that not much about orbiting bodies was known at the time, but it still leads me to wonder why they would consider this viable. In particular:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In such a model, wouldn't Mars change size in the sky?&lt;/li&gt;&#xA;&lt;li&gt;Wouldn't Mars tend to delay at a particular point in the sky for a measurable length of time? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Why would these things not be immediately obviously wrong to anyone who could observe these bodies? Of note, I'm assuming they had some method of observing the location of Mars, since the model includes its existence.&lt;/p&gt;&#xA;" OwnerDisplayName="user2285" LastEditorUserId="62" LastEditDate="2015-07-29T13:47:40.573" LastActivityDate="2015-07-29T15:36:42.630" Title="Why did scientists think the orbit of Mars had the shape of a limaçon in the geocentric model?" Tags="|astronomy|celestial-mechanics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2605" PostTypeId="1" CreationDate="2015-07-29T07:19:52.980" Score="5" ViewCount="1389" Body="&lt;p&gt;The adjoint of a matrix is also called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Adjugate_matrix&quot; rel=&quot;noreferrer&quot;&gt;adjugate&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the word &quot;adjoint&quot; is also used in Linear Algebra for linear transformations, and the &quot;adjoint of a matrix&quot; and &quot;adjoint of a linear map&quot; are very different notions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; For matrices, which word was used first - &quot;adjoint&quot; or &quot;adjugate&quot; (can one give a historical reference)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; Can one give the historical reference for the identity &#xA;$$A\, \mathrm{adj}(A)=\mathrm{det}(A)I$$&lt;/p&gt;&#xA;" OwnerUserId="2594" LastEditorUserId="2594" LastEditDate="2015-07-29T08:15:22.053" LastActivityDate="2017-05-01T16:05:50.640" Title="The terminologies &quot;Adjoint&quot; and &quot;Adjugate&quot;" Tags="|linear-algebra|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2606" PostTypeId="2" ParentId="2604" CreationDate="2015-07-29T11:34:53.840" Score="7" Body="&lt;p&gt;The right hand side picture shows qualitatively what you obtain from Ptolemy's model.&#xA;The actual curve is a cycloid, in the first approximation.&#xA;It uses rigorous mathematics. Moreover it describes quite well the actual motion of planets, including Mars. Just look at the actual motion of Mars in the sky (or in the Internet:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your specific questions: yes Mars actually changes its visible size when you&#xA;look at it. (And this is explained in both models)&#xA;Yes, Mars tends to delay and stop at certain moments (called stationary points)&#xA;and then moves in the opposite direction (retrogradation). All these phenomena are very well familiar to anyone who looks at the sky, and Ptolemy's model was designed exactly for the purpose to explain them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT.&#xA;Moreover, if you look at the left hand side picture, and imagine how the same motion of Mars looks FROM THE EARTH, you obtain the right hand side picture.&#xA;If it is difficult to imagine, write a computer program and make a plot.&#xA;Or make a plot by hand. In fact both descriptions (the one of Copernicus and the one of Ptolemy) are equivalent, if you ignore smaller effects which are not visible in these pictures anyway. As it should be, because both pictures describe the reality correctly. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-07-29T15:36:42.630" LastActivityDate="2015-07-29T15:36:42.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2607" PostTypeId="2" ParentId="2215" CreationDate="2015-07-29T14:50:19.587" Score="4" Body="&lt;p&gt;SI units are important because:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;They are common to the people of the entire world, so that people from different countries can communicate with each other conveniently regarding business&#xA;and science.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It makes systematic use of prefixes, making it easy to express very large or very small numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It makes calculations very fast.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2662" LastEditorUserId="26" LastEditDate="2015-07-31T23:00:04.717" LastActivityDate="2015-07-31T23:00:04.717" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2608" PostTypeId="1" AcceptedAnswerId="2609" CreationDate="2015-07-30T13:44:15.053" Score="12" ViewCount="2223" Body="&lt;p&gt;When the Greeks invented science and mathematics in around 600 BC, it was considered as a part of philosophy. Thales of Miletus was a mathematician and philosopher. Aristotle was a philosopher, biologist and physicist. Eratosthenes was a earth scientist, mathematician and philosopher. There are of course many more examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays, physics, chemistry, mathematics, biology, earth science are all separated from philosophy. However, ethics, philosophical anthropology, epistemology and metaphysics are part of philosophy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why and when did physics, chemistry, mathematics and earth science&#xA;merit an own field? &lt;/li&gt;&#xA;&lt;li&gt;Why don't ethics and metaphysics merit an own field?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I think that it certainly has to do something with size, since the former fields grew a lot in the 16th century. But the latter fields already were very large, since they were studied a lot more in ancient Greek times. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="4287" LastEditDate="2016-06-14T19:09:40.427" LastActivityDate="2016-06-15T12:39:58.650" Title="Why and when did some areas separate themselves from philosophy and some not?" Tags="|mathematics|physics|ancient-greece|philosophy-of-science|earth-sciences|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2609" PostTypeId="2" ParentId="2608" CreationDate="2015-07-30T14:24:19.890" Score="11" Body="&lt;p&gt;The area of knowledge separates itself from philosophy as soon  as a reliable method of obtaining exact knowledge in this area is invented. Thus mathematics separated from philosophy at its very beginning. In astronomy, there was an area covered by exact knowledge (based on observations) and another, speculative part.&#xA;As exact knowledge expanded, the speculative part shrinked. Ancient Greek work in statics is exact knowledge. Dynamics remained a part of philosophy until Galileo and Newton. Newton's title &quot;Mathematical principles of natural philosophy&quot; signifies the transition point. Since this book, most of physics is not philosophy anymore. (Though the name &quot;natural philosophy&quot; existed for another 200 years,&#xA;and even now a degree in mathematics is called PhD in some countries).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last part of astronomy which separated from philosophy was cosmology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In areas like ethics, aesthetics, metaphysics, theology, etc., a method of obtaining exact, objective knowledge still does not exist. So they remain parts of philosophy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true, Thales was both a mathematician and a philosopher. Pascal, Descartes, Leibniz and even Cantor, Russell and Thom did philosophy as well. But their mathematical work is very clearly separated from philosophy. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-07-30T14:50:04.103" LastActivityDate="2015-07-30T14:50:04.103" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2610" PostTypeId="2" ParentId="2567" CreationDate="2015-07-30T19:49:11.980" Score="12" Body="&lt;p&gt;The kilogram is the base unit of mass because electrical engineers in the late 19th century chose a particular set of practical electrical units. Their practical units were a success, and we are still using them today: ohm, volt, and ampere. In 1881 the International Electrotechnical Commission (IEC) created two sets of units: a set of theoretical units, and a set of practical units. The theoretical electrical units, abampere, abvolt, abohm were coherent with the mechanical  units cm, g, s. Coherence in this case primarily means that electrical energy and mechanical energy have identical units: $V\cdot I\cdot t = F \cdot L$. Unfortunately, the abvolt and abohm were inconveniently small. On the other hand, the &lt;em&gt;practical&lt;/em&gt; electrical units, ampere, volt, and ohm, were &lt;em&gt;not&lt;/em&gt; coherent with cm, g, s, nor with m, g, s. However, by coincidence they were coherent with m, kg, s. That is why the kilogram was chosen as the base unit of mass in the SI system, in 1960.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2442" LastEditDate="2015-08-03T20:32:59.720" LastActivityDate="2015-08-03T20:32:59.720" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2611" PostTypeId="1" AcceptedAnswerId="2614" CreationDate="2015-07-30T21:51:07.010" Score="9" ViewCount="3076" Body="&lt;p&gt;Is there a book anybody here can recommend on the history of measure theory? Or, maybe a book that contains the history of measure theory?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most books that are supposed to cover this material, i.e. books on the integration theory that lead to the measure theory, explain the integration quite well, but barely mention the measure theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I want to learn the history of measure theory is to better understand the theory itself. So, I want a book that goes beyond just the history of measure theory, and also delves into the mathematics of the theory. The things I am most interested in are the definitions, like $\sigma$-algebra, and measurability, and the rationale and the evolution of these concepts.&lt;/p&gt;&#xA;" OwnerUserId="2666" LastEditorUserId="2666" LastEditDate="2015-07-31T23:43:37.850" LastActivityDate="2019-09-23T07:59:51.630" Title="History of measure theory" Tags="|mathematics|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2612" PostTypeId="2" ParentId="2608" CreationDate="2015-07-31T12:07:35.120" Score="5" Body="&lt;p&gt;I often say the following, especially to philosophers.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Philosophy is the study of problems which cannot be solved.&lt;br&gt;&#xA;  As soon as a problem &lt;em&gt;can&lt;/em&gt; be solved, it moves to the science faculty.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But one philosopher told me that at least in ethics, philosophers (i.e. philosophers in the academic/university system) &lt;em&gt;do&lt;/em&gt; solve problems. Well, I can't agree with that. Ethics is the study of what &lt;em&gt;should&lt;/em&gt; be. Science is the study of what &lt;em&gt;is&lt;/em&gt;. So ethics can never be a science. One can, however, systematize ethics in terms of an axiomatic system or various rule-based systems. But just because something looks very technical doesn't mean it's a science. (Take astrology for example!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, many subjects were actually a part of religion, not philosophy. As more has become known about geology, biology, medicine, cosmology and psychology, these subjects have moved to the science faculty. People used to give gifts to the god Asclepius to cure diseases, and they gave gifts to the gods of harvests to get a successful crop. Now people give their gifts to doctors and seed and fertilizer companies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to answer the question, some subjects never acquired objective means of obtaining knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think also that many areas of philosophy are actually conducted within each field, like philosophy of mathematics, philosophy of physics, and so forth. It's a bit like mathematics because mathematics is a tool which is applied to dozens of subjects. Philosophy can also be regarded as a &lt;em&gt;method&lt;/em&gt;, which can be applied to any subject. So every subject has its own philosophy now. The philosophy department of a university focuses mainly on applications of philosophical thinking to problems that are either not related to the real world, or are just not amenable to objective methods yet, like: &quot;What is the nature of consciousness and awareness?&quot;&lt;/p&gt;&#xA;" OwnerUserId="2502" LastActivityDate="2015-07-31T12:07:35.120" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2614" PostTypeId="2" ParentId="2611" CreationDate="2015-08-01T03:04:09.557" Score="7" Body="&lt;p&gt;Yes, there is such a book. &#xA;MR0264015 Pesin, Ivan N. Classical and modern integration theories. Translated from the Russian and edited by Samuel Kotz. Probability and Mathematical Statistics, No. 8 Academic Press, New York-London 1970 xviii+195 pp.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original Russian title, literally translated is: &quot;Develpment of the notion of Integral&quot;. I read the book, and can recommend it.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-08-01T12:45:10.307" LastActivityDate="2015-08-01T12:45:10.307" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2615" PostTypeId="2" ParentId="2608" CreationDate="2015-08-01T14:00:38.537" Score="3" Body="&lt;p&gt;As a second answer to the main question, I just remembered that &lt;strong&gt;logic&lt;/strong&gt; is an &quot;outlier&quot; as a subject which used to lie inside the philosophy department because of Aristotle's &quot;Logic&quot; book. (Philosophy used to cover all of the topics written about by Aristotle and Plato.) But then logic moved into mathematics, more or less, from about 1890 up to about 1965, because it was introduced into mathematics by Peano, a mathematician, in the late 19th century with his 1899 &quot;Arithmetices principia: Nova methodo exposita&quot;, which included an axiomatization of logic (and set/class theory) written entirely in Latin. This was then followed by the Whitehead/Russell 1910, 1912, 1913 &quot;Principia mathematica&quot;, which also axiomatized logic. There were many other logic axiomatizations, which sort of moved between the philosophy and mathematics departments. However, after Paul Cohen's solution of the problem of the independence of the axiom of choice (and the continuum hypothesis) relative to Zermelo-Fraenkel set theory, logic seems to have moved back into the philosophy departments again.&lt;/p&gt;&#xA;" OwnerUserId="2502" LastEditorUserId="3791" LastEditDate="2016-06-15T12:39:58.650" LastActivityDate="2016-06-15T12:39:58.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2616" PostTypeId="1" CreationDate="2015-08-01T17:40:09.020" Score="7" ViewCount="232" Body="&lt;p&gt;This question was sparked by the observation that the rebranding of the field called &quot;neural networks&quot; into &quot;deep learning&quot; is quite impressive. I wonder whether there have been similar renamings in the past.&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="26" LastEditDate="2015-08-02T07:38:29.137" LastActivityDate="2015-08-08T19:45:13.023" Title="What are the most famous examples of theory rebranding?" Tags="|terminology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2617" PostTypeId="1" CreationDate="2015-08-02T02:48:11.463" Score="17" ViewCount="974" Body="&lt;p&gt;Admittedly, this may not be a research level question, but I am deeply curious about this.&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x) \in \mathbb{R}[x]$&lt;/span&gt;, and write &lt;span class=&quot;math-container&quot;&gt;$d = \deg f$&lt;/span&gt;. It is well known that if &lt;span class=&quot;math-container&quot;&gt;$\deg f &amp;gt; 2$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is reducible over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;. This fact can easily be proved with the fundamental theorem of algebra. Indeed, by the fundamental theorem of algebra, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; splits over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;, and since &lt;span class=&quot;math-container&quot;&gt;$f(x) = \overline{f(x)}$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is real, it follows that the linear factors of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; must be real or come in conjugate pairs. Therefore, irreducible polynomials over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; are only of degree 1 or 2, and it is easy to find examples of degree two polynomials with real coefficients which are irreducible.&lt;/p&gt;&#xA;&lt;p&gt;This fact has a profound impact on the theory of partial fractions, a staple in first year calculus. Indeed, Calculus II (at least at my university) tends to spend an inordinate amount of time on 'integration' via the method of anti-derivatives, which I believe most mathematicians know is ineffective in solving the majority of problems, considering most functions do not have elementary anti-derivatives. However, in the context of &lt;em&gt;rational&lt;/em&gt; functions, anti-differentiation and partial fractions &lt;em&gt;completely&lt;/em&gt; solves the problem, as any rational function can be written as the sum of simpler rational functions, each with an elementary anti-derivative.&lt;/p&gt;&#xA;&lt;p&gt;However, this fact is (I believe) far from obvious if you do not know that the only irreducible polynomials over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; are linear or quadratic. That said, it seems to me that the method of partial fractions is much older than the fundamental theorem of algebra.&lt;/p&gt;&#xA;&lt;p&gt;So when did people know (perhaps before the first proof of the fundamental theorem of algebra) that the only irreducible polynomials over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; are linear or quadratic? When was the 'complete' solution of anti-derivatives of rational functions obtained? Is this history accounted for anywhere?&lt;/p&gt;&#xA;&lt;p&gt;I apologize in advance if this question is trivial.&lt;/p&gt;&#xA;" OwnerUserId="2681" OwnerDisplayName="Stanley Yao Xiao" LastEditorUserId="16318" LastEditDate="2022-12-18T18:38:09.823" LastActivityDate="2022-12-18T18:38:09.823" Title="When did people know that all real polynomials of degree greater than 2 were reducible?" Tags="|real-analysis|abstract-algebra|complex-analysis|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="2618" PostTypeId="1" CreationDate="2015-08-02T20:15:48.920" Score="3" ViewCount="1576" Body="&lt;p&gt;As stated in the article &lt;a href=&quot;https://www.britannica.com/topic/binomial-distribution&quot; rel=&quot;nofollow&quot;&gt;binomial distribution&lt;/a&gt; by britannica.com the binomial distribution was used by Jakob Bernoulli when he said that &quot;the probability of $k$ ... outcomes in $n$ repetitions is equal to the $k$th term ... in the expansion of the binomial expression $(p + q)^n$, where $q = 1 − p$&quot; (quote from &lt;a href=&quot;https://www.britannica.com/topic/binomial-distribution&quot; rel=&quot;nofollow&quot;&gt;this article&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question:&lt;/strong&gt; Is it right to say that Jakob Bernoulli is the inventor of the binomial distribution or was this distribution already used before? I guess he's the first one who uses the skewed binomial distribution but I am not sure whether the symmetric binomial distribution is used before or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I have found out:&lt;/strong&gt; For example &lt;a href=&quot;https://en.wikipedia.org/wiki/Blaise_Pascal&quot; rel=&quot;nofollow&quot;&gt;Blaise Pascal&lt;/a&gt; investigated the &lt;a href=&quot;https://en.wikipedia.org/wiki/Problem_of_points&quot; rel=&quot;nofollow&quot;&gt;problem of points&lt;/a&gt; (together with Fermat). In this problem one has to decide how to split the stakes of a game when a game is interrupted before it ends. Pascal proved that if person A needs $r$ points and $B$ needs $s$ rounds to win, the stakes should be split as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\sum_{k=0}^{s-1} \binom{r+s-1}{k} \text{ to } \sum_{k=s}^{r+s-1} \binom{r+s-1}{k}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently I read &quot;A history of Probability and Statistics and their Applications before 1750&quot; by Anders Hald but it is not clear to me whether Pascals knows the symmetric binomial distribution or not.&lt;/p&gt;&#xA;" OwnerUserId="1772" LastActivityDate="2016-05-31T04:28:29.630" Title="First usage of binomial distribution" Tags="|mathematics|reference-request|statistics|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2620" PostTypeId="2" ParentId="1905" CreationDate="2015-08-03T06:01:52.757" Score="5" Body="&lt;p&gt;Looking at a variety of sources, it seems that Thomas Edison was the first to create vacuum tubes when trying to find better light bulbs in 1883, and was the first to discover the Edison effect where if you heat a conductor and pass charge through it, the charge can jump gaps in a vacuum. J.J Thomson in 1897 was the first to discover and explain how the Edison effect works while working on his experiments on electrons.  To prove J.J Thomson right though, current had to flow one way in the tube (and the only way most electronics would have been able to work using the tubes); which is where John Ambrose Fleming invented the Fleming valve in 1904 which forced current to go one way in the tube. The new invention of radios started to be made using vacuum tubes, but they were still unreliable because they were unamplified and passed weak signals. In 1906 Lee De Forest created the Audion tube  which had a metal grid on one side that a second current was passed to which acted to control the strength of the current going through the tube, thus being the first triode tube (meaning the outputs were all on, all off, and anywhere between). Later in 1912 Lee came up with the idea of using multiple triode arrays to amplify the signal over all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;sources:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pbs.org/transistor/science/events/vacuumt.html&quot; rel=&quot;nofollow&quot;&gt;pbs_vacuum_tubes&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ethw.org/Electron_(or_Vacuum)_Tubes&quot; rel=&quot;nofollow&quot;&gt;engineering history&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://ethw.org/Diode&quot; rel=&quot;nofollow&quot;&gt;engineering history:diodes&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ee.umd.edu/~taylor/Electrons3.htm&quot; rel=&quot;nofollow&quot;&gt;University of Maryland&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the book &lt;em&gt;Father Of The Radio&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2684" LastEditorUserId="2684" LastEditDate="2015-08-03T14:44:20.643" LastActivityDate="2015-08-03T14:44:20.643" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2621" PostTypeId="1" CreationDate="2015-08-03T11:58:21.560" Score="5" ViewCount="218" Body="&lt;p&gt;It is well known that the theory of reflections  was considerably developed during the 19th century with the development of group theory (e.g. Klein) and the theory of transformations. However, I'm interested to know who were the mathematicians that dealt with reflections during the 18th century and the beginning of the 19th century. A colleague of mine mentioned that several mathematicians looked on reflections in terms of foldings, in order to see if curves intersect themselves while being reflected; However, I didn't manage to find any sources for that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, any reference would be appreciated!&lt;/p&gt;&#xA;" OwnerUserId="2687" LastEditorUserId="62" LastEditDate="2015-08-03T17:31:14.543" LastActivityDate="2015-08-03T17:31:14.543" Title="Reflections in the 18th century" Tags="|reference-request|mathematicians|group-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2622" PostTypeId="1" AcceptedAnswerId="2623" CreationDate="2015-08-03T15:18:31.813" Score="9" ViewCount="528" Body="&lt;p&gt;My understanding is that the term &quot;field&quot; in science was first used in physics, while the mathematical term, at least the algebraic one, was more recent. &#xA;Does anybody know when the first occurrence of the term appeared, and if there was any connection with the physical meaning?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="-1" LastEditDate="2015-08-03T20:33:06.210" LastActivityDate="2015-08-03T20:33:06.210" Title="When was the term &quot;field&quot; first used in maths?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2623" PostTypeId="2" ParentId="2622" CreationDate="2015-08-03T16:37:57.227" Score="10" Body="&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/f.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;, sub voce : &lt;em&gt;Field&lt;/em&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term &lt;em&gt;Zahlenkörper&lt;/em&gt; (body of numbers) is due to Richard Dedekind (1831-1916) (Kline, page 1146). Dedekind used the term in his lectures of 1858 but the term did not come into general use until the early 1890s.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/E._H._Moore&quot; rel=&quot;noreferrer&quot;&gt;Eliakim Hastings Moore&lt;/a&gt; (1862-1932) was apparently the first person to use the English word &lt;em&gt;field&lt;/em&gt; in its modern sense and the first to allow for a finite field. He coined the expressions &amp;quot;field of order &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;&amp;quot; and &amp;quot;Galois-field of order &lt;span class=&quot;math-container&quot;&gt;$s = q^n$&lt;/span&gt;.&amp;quot; These expressions appeared in print in December 1893 in the &lt;em&gt;Bulletin of the New York Mathematical Society&lt;/em&gt; III.75. The paper was presented to the Congress of Mathematics at Chicago on Aug.25, 1893.&lt;/p&gt;&#xA;&lt;p&gt;At any event, a decade later &lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Vermilye_Huntington&quot; rel=&quot;noreferrer&quot;&gt;Edward Vermilye Huntington&lt;/a&gt; [in 1904] wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Closely connected with the theory of groups is the theory of fields, suggested by GALOIS, and due, in concrete form, to DEDEKIND in 1871. The word &lt;em&gt;field&lt;/em&gt; is the English equivalent for DEDEKIND’s term &lt;em&gt;Körper&lt;/em&gt;; KRONECKER’s term &lt;em&gt;Rationalitätsbereich&lt;/em&gt;, which is often used as a synonym, had originally a somewhat different meaning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It does not seem to me that there is indications of a relation with Maxwell's concept of the &lt;em&gt;electromagnetic field&lt;/em&gt; ...&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-08-03T16:37:57.227" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2624" PostTypeId="1" AcceptedAnswerId="2625" CreationDate="2015-08-03T23:50:57.920" Score="8" ViewCount="470" Body="&lt;p&gt;In &lt;em&gt;Stalking the Riemann Hypothesis&lt;/em&gt;, Rockmore discusses how Bernhard Riemann, as per custom, submitted three potential areas of research for his &lt;a href=&quot;http://en.wikipedia.org/wiki/Habilitation&quot;&gt;habilitation&lt;/a&gt;. Gauss was the chairman of the committee responsible for picking which of the three Riemann would work on, and, perhaps because of his interest in non-Euclidean geometry, he selected the third one, regarding the &quot;foundations of geometry&quot;. Riemann's work there would, or course, lead to the development of Riemannian geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What were Riemann's other two submissions? Did he go on later in life to research those topics?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-08-04T14:39:37.123" Title="What were Riemann's other two submissions for his habilitation?" Tags="|mathematics|riemannian-geometry|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2625" PostTypeId="2" ParentId="2624" CreationDate="2015-08-04T10:32:07.823" Score="7" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Detlef Laugwitz, &lt;a href=&quot;https://books.google.it/books?id=7U6njBdp_yoC&amp;amp;pg=PA219&quot;&gt;Bernhard Riemann 1826-1866 : Turning Points in the Conception of Mathematics&lt;/a&gt; (1996 - German ed.1996), page 182; we have :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Riemann's habilitation paper &quot;&lt;a href=&quot;http://www.emis.de/classics/Riemann/Trig.pdf&quot;&gt;Uber die Darstellbarkeit einer Funktion durch &#xA;  eine trigonometrische Reihe&lt;/a&gt;&quot; (W.227-265) (&quot;On the representability of a function by means of a trigonometric series&quot;) was completed in 1853, but was published &#xA;  by Dedekind only in 1868. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and page 219 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Riemann [...] habilitation lecture &quot;&lt;a href=&quot;http://www.emis.de/classics/Riemann/Geom.pdf&quot;&gt;Uber die Hypothesen die der Geometrie zu Grunde liegen&lt;/a&gt;&quot; (&quot;On the hypotheses which lie at the foundation of geometry&quot;) (W.272-287), presented in the summer of 1854 [also published by Dedekind in 1868]. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;As is done even today, Riemann was supposed to submit titles of three lectures, from which the faculty of philosophy was to choose one. The first title was &quot;Geschichte der Frage uber die Darstellbarkeit einer Function durch eine trigonometrische Reihe&quot; (&quot;The history of the question of representability of a function as a trigonometric series&quot;), and the second, &quot;Uber die Auflosung &#xA;  zweier Gleichungen zweiten Grades mit zwei unbekannten Grossen&quot;  (&quot;Solution of two quadratic equations in two unknowns&quot;). Of course, the third title was &quot;Uber die Hypothesen...&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-08-04T14:39:37.123" LastActivityDate="2015-08-04T14:39:37.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2626" PostTypeId="2" ParentId="333" CreationDate="2015-08-05T17:36:16.300" Score="2" Body="&lt;p&gt;Although this story is frequently repeated, if you think about it you will realise that it is highly implausible. Let us start off with him sleeping eight hours a night. We will see him sinking from a normal sleep pattern to oblivion in just 4 x 16 = 64 days, or just over two months. Is such a thing possible?&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-08-05T17:36:16.300" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2627" PostTypeId="2" ParentId="2616" CreationDate="2015-08-08T16:33:15.467" Score="4" Body="&lt;p&gt;I suggest &lt;em&gt;continental drift&lt;/em&gt; to &lt;em&gt;plate tectonics&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2706" LastEditorUserId="2442" LastEditDate="2015-08-08T19:45:13.023" LastActivityDate="2015-08-08T19:45:13.023" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2629" PostTypeId="1" AcceptedAnswerId="2634" CreationDate="2015-08-09T06:26:16.090" Score="11" ViewCount="2477" Body="&lt;p&gt;Thomas Young is famous for his double slit experiment, but I can't seem to find his experimental setup (such as how is prepared the light before it  went through the apparatus. Does anyone know his experimental arrangement? If so please can you describe it and provide a source (if possible). Also did Young publish any papers on his experiment? &lt;/p&gt;&#xA;" OwnerUserId="1994" LastEditorUserId="55" LastEditDate="2015-08-11T23:40:34.480" LastActivityDate="2016-04-14T00:24:19.637" Title="How did Young perform his double slit experiment?" Tags="|physics|reference-request|experimental-physics|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2630" PostTypeId="1" CreationDate="2015-08-09T07:56:27.740" Score="3" ViewCount="105" Body="&lt;p&gt;I mean the Brillouin in the WKB method. I want to read his original paper. But it is in French. Is it translated into English? It should be.&lt;/p&gt;&#xA;" OwnerUserId="2641" LastEditorUserId="10106" LastEditDate="2020-06-07T18:39:19.323" LastActivityDate="2020-06-07T18:39:19.323" Title="Are Brillouin's papers translated into English?" Tags="|physics|reference-request|translations|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2631" PostTypeId="1" AcceptedAnswerId="2632" CreationDate="2015-08-10T02:36:25.047" Score="22" ViewCount="2731" Body="&lt;p&gt;Which were the first mathematical developments to state that the product of two negative numbers is a positive number, and what was their justification for this choice? I am not interested in a modern explanation for why this choice is sensible, but want to know how mathematicians from the past arrived at this convention.&lt;/p&gt;&#xA;" OwnerUserId="2712" LastActivityDate="2015-08-12T19:22:18.100" Title="Historically, how did people define multiplication for negative numbers?" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2632" PostTypeId="2" ParentId="2631" CreationDate="2015-08-10T06:06:22.967" Score="17" Body="&lt;p&gt;&lt;strong&gt;Brahmagupta&lt;/strong&gt; (598-670) was one of the first mathematicians to introduce negative numbers. He called them debts. The rule was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The product or quotient of two debts is one fortune.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This was logical, because you have a debt of 3 by 3 people, then you have a debt by 9 people. In modern language, $3 \times -3 = -9$, however Brahmagupta wouldn't write that. So fortune times debt is debt. Fortune times fortune is a fortune. So changing fortune to debt once als changes the outcome. Therefore changing it again would result to a fortune. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-08-12T19:22:18.100" LastActivityDate="2015-08-12T19:22:18.100" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2633" PostTypeId="1" AcceptedAnswerId="2636" CreationDate="2015-08-10T18:29:11.403" Score="5" ViewCount="398" Body="&lt;p&gt;Today it is well known even by the non-scientific community that there is a part of physics that deals with the very small called quantum physics. My question is how this became &quot;common knowledge&quot;. I don't imagine there was a headline in a newspaper that read something like &quot;Matter has a wave nature, read all about it&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know therefore &lt;strong&gt;how&lt;/strong&gt; and &lt;strong&gt;when&lt;/strong&gt; (roughly speaking) this shifted both in the wider scientific and non-scientific community?&lt;/p&gt;&#xA;" OwnerUserId="198" LastActivityDate="2018-08-06T12:21:21.790" Title="How did quantum mechanics become well known by (non)-scientific community." Tags="|quantum-mechanics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2634" PostTypeId="2" ParentId="2629" CreationDate="2015-08-10T20:46:15.610" Score="11" Body="&lt;p&gt;Young's original setup demonstrating interference of light was not double slit but sunbeam splitting with a single thin card. He presented a paper On the theory of Light and Color to the Royal Society in May 1801 published Proceedings of the Royal Society of London A 92 (1802) (see &lt;a href=&quot;http://www.aps.org/publications/apsnews/200805/physicshistory.cfm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://skullsinthestars.com/2009/03/28/optics-basics-youngs-double-slit-experiment/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), and in November 1803 gave a public talk Experimental Demonstration of the General Law of the Interference of Light published in Philosophical Transactions of the Royal Society of London, v.94 (1804) and reprinted in Morris Shamos, ed., Great Experiments in Physics, pp. 96-101, Holt Reinhart and Winston, New York, 1959.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cavendishscience.org/phys/tyoung/tyoung.htm&quot; rel=&quot;noreferrer&quot;&gt;Scheider describes the setup&lt;/a&gt;: &quot;&lt;em&gt;A narrow beam of sunlight was split with what Young described as &quot;a slip of card, about one thirtieth of an inch in breadth (thickness).&quot; The slip of card was held edgewise into the sunbeam, which was made to enter the room horizontally by means of a &quot;looking glass&quot; (mirror) and a tiny hole in a window shutter&quot;. The sunbeam had a diameter slightly greater than the thickness of the card. When the card was placed properly it split the beam into two slivers, one passing on each side of the slip of card&lt;/em&gt;&quot;. Young wrote:&quot;&lt;em&gt;I made a small hole in a window-shutter, and covered it with a piece of thick paper, which I perforated with a fine needle... It will not be denied by the most prejudiced, that the fringes are produced by the interference of two portions of light&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even after the 1803 demonstration Brougham, a supporter of Newton's theory,  called light interference &quot;&lt;em&gt;absurdity and one of the most incomprehensible suppositions that we remember to have met with in the history of human hypothesis&lt;/em&gt;&quot;, which might have inspired Young to modify the experiment. Back in 1802 he built water &quot;ripple tank&quot; with a more recognizable double slit setup to demonstrate interference of water waves, apparently inspired by Newton's double prism - double perforation setup for color splitting. The ripples were produced by a horizontally moved board, then passed through a single slit in the first shutter to create coherence, and then through two slits in the second shutter to produce the interference pattern, see &lt;a href=&quot;http://www.eoht.info/page/Double-slit+experiment&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around 1804 Young decided to convince the detractors by reproducing this exact setup with light. Results were published as Experiments and Calculations Relative to Physical Optics in the Proceedings A 94 (1804), and are reproduced in his 1807 Course of Lectures on Natural Philosophy and the Mechanical Arts, you can see the exact setup in its &lt;a href=&quot;https://books.google.com/books?id=lyQEAAAAQAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;1845 edition here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-04-14T00:24:19.637" LastActivityDate="2016-04-14T00:24:19.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2635" PostTypeId="2" ParentId="2631" CreationDate="2015-08-10T21:28:32.043" Score="13" Body="&lt;p&gt;Alas, we will never know. The earliest mention of negative numbers occurs in Liu Hui's c. 260 AD commentary to Chinese classic Nine Chapters on the Mathematical Art, a compendium of prescriptions for solving practical mathematical problems. The context is solution of linear systems by what we call Gaussian elimination, and the numbers are represented by red and black counting rods (contrary to the modern accounting convention red is positive):&quot;&lt;em&gt;Rods of the same name multiplied by each other make positive. Rods of different names multiplied by each other make negative&lt;/em&gt;&quot;. In the style of the book there is no explanation or justification for the rule. If anything, the context suggests that it was adopted because the algebra worked, not for any &quot;intuitive&quot; reason:&quot;&lt;em&gt;Interchanging the red and black rods in any column is immaterial. So one can make the first entries of opposite sign&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brahmagupta's Correctly Established Doctrine of Brahma (c. 630 AD) gives the rule in the same prescription style, along with many others:&quot;&lt;em&gt;The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. A positive divided by a positive and negative divided by a negative is positive; a zero divided by a zero is zero...&lt;/em&gt;&quot;. Literally, he says &quot;fortunes&quot; and &quot;debts&quot; rather than positive and negative, but while such merchant interpretation provides intuition for addition and subtraction of debts their product makes little sense, as does $0/0=0$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mumford comments&lt;/a&gt; that even the merchant interpretation is pure speculation:&quot;&lt;em&gt;It would be wonderful to know what considerations led Indian mathematicians in the late centuries BCE or the early centuries  CE  to  these  conclusions  –  especially  for  the  multiplication  of  negative  numbers. The  predominately  oral  transmission  of  knowledge  in  the  Vedic tradition – and perhaps the difficulty of preserving perishable writing materials through yearly monsoons – has not left us with any record of these discoveries. They just appear full blown in Brahmagupta’s summary. R.Mattessich has developed at length the idea that it was the highly developed tradition of accounting which&#xA;led to the full understanding of negative numbers, but unfortunately no evidence&#xA;for this plausible conjecture exists&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The father of algebra, Al-Khwarizmi (c. 790-840 AD), explicitly cites Brahmagupta on negatives (and zero) in the famous Al-jabr w’al Muqabala, but he has little use for them in the geometric interpretations he favors. In the whole book he mentions them only once, when explaining the identity (in modern notation) $(a-b)(c-d)=ac-ad-bc+bd$. This might be a clue as to why he accepts the Brahmagupta’s rule, it has to hold for the last term to come up with the right sign. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mumford's essay&lt;/a&gt; gives a good survey of historical struggles with the nature of negatives (which continued into the middle of 19th century!) with references and attempted reconstructions of their meaning.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-08-11T23:48:57.757" LastActivityDate="2015-08-11T23:48:57.757" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2636" PostTypeId="2" ParentId="2633" CreationDate="2015-08-10T23:21:41.993" Score="6" Body="&lt;p&gt;I can tell &quot;how&quot; but cannot tell exactly &quot;when&quot;. It was indeed different from the general relativity which really had the newspapers headlines, after Sir Arthur Eddington made&#xA;an observation confirming it (see the nice movie &quot;Einstein and Eddington&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the case of quantum mechanics, everything was much more calm. It gradually spread through the popular expositions, and then penetrated high school textbooks.&#xA;When exactly it penetrated high school textbooks, is a good question, and I do not now the exact answer on this. I conjecture that this happened only after WW2, &#xA;when there was a sudden public interest to physics in general and to atomic bombs in particular. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. My experience with undergraduate students shows that it is still not much &#xA;familiar to the non-scientific community. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT2. Perhaps one reason is that QM developed more gradually, and through the effort of many people, and it is much harder to explain.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-08-06T12:21:21.790" LastActivityDate="2018-08-06T12:21:21.790" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2637" PostTypeId="1" CreationDate="2015-08-11T12:00:21.680" Score="2" ViewCount="25" Body="&lt;p&gt;About an year ago, I had seen an article somewhere on the internet which discussed Newton's Principia Mathematica and the rigor (or lack thereof) of the arguments presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have forgotten who the author was and am unable to find the article back.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a published book in which similar work has been done?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am trying to write a term paper in my History of Math course for which I think this could be a suitable topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="2721" LastActivityDate="2015-08-11T12:00:21.680" Title="Looking for a Book Which Discusses the Standards of Rigor in Newton's Principia" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="1" ClosedDate="2015-08-14T00:02:16.280" ContentLicense="CC BY-SA 3.0" />
  <row Id="2638" PostTypeId="1" AcceptedAnswerId="2666" CreationDate="2015-08-11T12:07:52.150" Score="4" ViewCount="178" Body="&lt;p&gt;About an year ago, I had seen an article somewhere on the internet which discussed Newton's Principia Mathematica and the rigor (or lack thereof) of the arguments presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have forgotten who the author was and am unable to find the article back.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is there a published book in which similar work has been done?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am trying to write a term paper in my History of Math course for which I think this could be a suitable topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I accidently created the &lt;a href=&quot;https://hsm.stackexchange.com/questions/2637/looking-for-a-book-which-discusses-the-standards-of-rigor-in-newtons-principia?rq=1&quot;&gt;same post&lt;/a&gt; with a different account and was unable to delete it.&lt;/p&gt;&#xA;" OwnerUserId="2722" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-08-17T11:22:34.527" Title="Looking for a Book Which Discusses the Rigor in Newton's Principia Mathematica" Tags="|physics|reference-request|classical-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2639" PostTypeId="1" AcceptedAnswerId="2641" CreationDate="2015-08-11T16:27:21.720" Score="10" ViewCount="1637" Body="&lt;p&gt;Was this model widely accepted in the time as a  model for the atom? or was it just a model between many alternatives?&lt;/p&gt;&#xA;" OwnerUserId="2504" LastEditorUserId="2442" LastEditDate="2015-08-14T20:39:44.133" LastActivityDate="2017-10-06T11:10:06.500" Title="Was Thomson's model of Plum Pudding widely accepted as a model of atom?" Tags="|physics|chemistry|atomic-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2640" PostTypeId="1" AcceptedAnswerId="2642" CreationDate="2015-08-11T16:55:25.397" Score="9" ViewCount="1242" Body="&lt;p&gt;In Wikipedia I found this claim by E.T. Bell in his &lt;em&gt;Men of Mathematics&lt;/em&gt;. However in the next paragraph it says that &quot;it is doubtful that Gauss put Eisenstein in the same league as Newton&quot;, which makes me wonder why is he not considered to be in the same league as the aforementioned giants?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone with better knowledge clarify this for me please? &lt;/p&gt;&#xA;" OwnerUserId="2866" LastEditorUserId="4251" LastEditDate="2020-06-04T07:08:27.303" LastActivityDate="2020-06-04T07:08:27.303" Title="Did Gauss say &quot;there have been but three epoch-making mathematicians, Archimedes, Newton and Eisenstein&quot;?" Tags="|mathematicians|reference-request|debunking|gauss|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2641" PostTypeId="2" ParentId="2639" CreationDate="2015-08-11T20:48:47.743" Score="9" Body="&lt;p&gt;No, it was not the only one, but 19th century models were purely speculative, and Lewis did not publicize his 1902 &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubical_atom&quot; rel=&quot;nofollow&quot;&gt;cubical atom&lt;/a&gt; until 1916. So in 1904, when Thomson proposed his plum pudding, the only real competition was Nagaoka's 1903 &lt;a href=&quot;http://link.springer.com/chapter/10.1007%2F978-3-540-70626-7_10&quot; rel=&quot;nofollow&quot;&gt;Saturnian atom&lt;/a&gt;, a large, massive, positively charged sphere, encircled by hundreds of light-weight electrons arranged into flat rings, and bound by electrostatic forces (analogous to rings of Saturn). It was an inspiration for Rutherford's 1911 model, and shared the problem of all &quot;planetary&quot; models: rotating electrons radiate energy according to Maxwell's theory, so the atom would promptly collapse. Nagaoka's rings were also unstable to disturbances orthogonal to their plane, as Thomson pointed out. Nagaoka abandoned his model in 1908, and Rutherford's was saved by Bohr in 1913 using the new idea of quanta.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in 1909 Rutherford's scattering experiments showed that atoms have large empty space within them rather than Thomson's pudding filler, so we only have a span of five years for the plum pudding to stay accepted. It was more or less, as the best available offer and provisionally, and as such it motivated more experiments including Rutherford's. The model itself was provisional, Thomson was silent on what the pudding is made of, or what held it together. But there were similar issues with the electron at the time. In Maxwell's theory point-like accelerating charge has infinite self-interaction energy, so little ball electrons were considered by Abraham for example, and Poincare even conjectured a new &lt;a href=&quot;https://www.academia.edu/1589499/Mechanics_and_Electromagnetism_in_the_Late_Nineteenth_Century_The_Dynamics_of_Maxwells_Ether._MARTINS_Roberto_de_Andrade&quot; rel=&quot;nofollow&quot;&gt;non-electromagnetic attractive force&lt;/a&gt; to stabilize them. After Planck's and Einstein's discoveries the general sentiment was that physics is in flux, and little is settled.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Atoms were considered indivisible for most of 19th century, but Fechner speculated about &quot;small suns, floating at comparatively large distances from one another, such that each or several of them together are surrounded by orbiting planetary atoms” as early as 1828 (he also suggested that electric current is a flow of corpuscles in 1845), and Weber proposed an atom with small positively charged mass rotating around a large negatively charged one in 1871. Weber based his model on his own version of electrodynamics, different from Maxwell's, so it became obsolete by 1890s. Stoney (who &lt;a href=&quot;http://www.askaboutireland.ie/reading-room/life-society/science-technology/irish-scientists/george-johnstone-stoney-f/&quot; rel=&quot;nofollow&quot;&gt;gave electron its name in 1891&lt;/a&gt;) also suggested a &quot;planetary&quot; model. Perhaps, the most extravagant was Lord Kelvin's 1867 model of &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_knot_theory#Early_Modern&quot; rel=&quot;nofollow&quot;&gt;atoms as knotted vortices in ether&lt;/a&gt; with chemical elements corresponding to different knot and link types (e.g. sodium was the Hopf link). It did not work out as physics, but it did give a starting impulse to the mathematical knot theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://books.google.com/books?id=W5kyppVPyesC&amp;amp;pg=PA169&amp;amp;dq=Wilhelm+Weber+atomic+model&amp;amp;hl=en&amp;amp;ei=6Bk2TKDiFtexnAeH65mMBA&amp;amp;sa=X&amp;amp;oi=book_result&amp;amp;ct=result&amp;amp;resnum=2&amp;amp;ved=0CDYQ6AEwAQ#v=onepage&amp;amp;q=Wilhelm%20Weber%20atomic%20model&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Jagdish and Rechenberg, Historical Development of Quantum Theory&lt;/a&gt;, and Baily's &lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1208/1208.5262.pdf&quot; rel=&quot;nofollow&quot;&gt;Early Atomic Models – From Mechanical to Quantum&lt;/a&gt; for more details.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-03-29T01:55:48.683" LastActivityDate="2016-03-29T01:55:48.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2642" PostTypeId="2" ParentId="2640" CreationDate="2015-08-11T21:07:55.983" Score="10" Body="&lt;p&gt;Wikipedia also says:&quot;&lt;em&gt;this is not a quote by Gauss, but is (a translation of) the end of a sentence from the biography of Eisenstein by Moritz Cantor (1877), one of Gauss's last students and a historian of mathematics, who was summarizing his recollection of a remark made by Gauss about Eisenstein in a conversation many years earlier&lt;/em&gt;&quot;. Human memory is tricky with context and nuance after many years, especially when one admires a person enough to write their biography, and is focused predominantly on this person. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for why, Archimedes founded mechanics as a science, and anticipated ideas of integral calculus by two thousand years, Newton revolutionized physics and invented calculus, Eisenstein's work had a much more narrow impact, mostly in algebra. But it was of high quality, and we know from Gauss's own writings that he thought very highly of Eisenstein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And generally, &lt;a href=&quot;https://hsm.stackexchange.com/questions/2397/what-resources-are-available-for-lives-of-recent-mathematicians/2400#2400&quot;&gt;take everything E.T. Bell writes with a grain of salt&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-08-11T21:16:45.740" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2643" PostTypeId="1" CreationDate="2015-08-12T03:28:40.927" Score="19" ViewCount="1365" Body="&lt;p&gt;In the old days. stars of physicists like Einstein$^{[1]}$, Poincare, Heisenberg, Pauli, $^{[2]}$ Bohr and so on are quite philosophical mind, and like philosophy. $^{[3]}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But now, it seems to me a totally different matter - physicists like Steven Weinberg, Lawrence Krauss ${^{[4]}}$ and many others have a strong disrespect for philosophy $^{[5]}$, I wonder:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did philosophy become unpopular in the circle of physics$^{[6]}$? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize this question can be opinion based, so please provide a.) real major history events like my &lt;a href=&quot;http://rationallyspeaking.blogspot.tw/2012/04/lawrence-krauss-another-physicist-with.html&quot;&gt;[4]&lt;/a&gt; below that probably increased tension between physics and philosophy, b.) any events that giving us a more complete picture (such as: I could be under a wrong impression after all!). In case of it gradually happened... well, I don't know how you can prove that, but please provide explanation and strong argument.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;{1}:&quot;I fully agree with you about the significance and educational value of methodology as well as history and philosophy of science. So many people today—and even professional scientists—seem to me like somebody who has seen thousands of trees but has never seen a forest. A knowledge of the historic and philosophical background gives that kind of independence from prejudices of his generation from which most scientists are suffering. This independence created by philosophical insight is—in my opinion—the mark of distinction between a mere artisan or specialist and a real seeker after truth.&quot; (Einstein to Thornton, 7 December 1944, EA 61-574)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;{2}: They have written different articles and books on philosophy of science, see &quot;Physics and Philosophy&quot; by Pauli, &quot;Science and Hypothesis&quot; by Poincare, &quot;Physics and Philosophy&quot; by Heisenberg. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;{3}: The fact that they like philosophy doesn't mean their work are inspired by philosophy though, despite a few of their work does seem to be so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;{4}: see &lt;a href=&quot;http://rationallyspeaking.blogspot.tw/2012/04/lawrence-krauss-another-physicist-with.html&quot;&gt;this&lt;/a&gt;. Also, don't get me wrong, I like Krauss and philosophy quite a lot .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;{5}: The only exception I found is: Sean Carroll and Anthony Zee&lt;/p&gt;&#xA;&#xA;&lt;p&gt;{6}: To be more specific, I mean the majority of physics circle. Having said all these, this question is just based on personal observation, I could be wrong though! Given my observation samples are not board enough. &lt;/p&gt;&#xA;" OwnerUserId="2713" LastEditorUserId="26" LastEditDate="2017-04-27T16:27:19.243" LastActivityDate="2017-05-29T16:17:49.783" Title="Conflict between physics and philosophy" Tags="|physics|philosophy-of-science|social-context|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2644" PostTypeId="2" ParentId="2643" CreationDate="2015-08-12T05:19:52.357" Score="10" Body="&lt;p&gt;The modern conflict is not so much between physics and philosophy, as between physics and &quot;half&quot; of philosophy, the continental philosophy. Scientists, and physicists in particular, see culturally relativistic attitudes of continental philosophy as leading to ignorance and outright hostility to science. &lt;a href=&quot;https://en.wikipedia.org/wiki/Fashionable_Nonsense&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sokal and Bricmont's Fashionable Nonsense&lt;/a&gt; gives an idea why, in works of some prominent continental philosophers they find &quot;&lt;em&gt;mystification, deliberately obscure language, confused thinking and the misuse of scientific concepts&lt;/em&gt;&quot;. The other &quot;half&quot;, analytic philosophy, is closely associated with science and is viewed much more favorably, Karl Popper for example is quite popular among physicists. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prominent physicists like Bohm, Penrose and Smolin wrote books philosophizing about the nature of mind and reality, Bell and Wigner wrote a number of papers. Philosophy always had the most currency with scientists working at the &quot;frontlines&quot;, and even today &lt;a href=&quot;http://philsci-archive.pitt.edu/5387/1/Rickles_QG.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to Rickles&lt;/a&gt; &quot;&lt;em&gt;quantum gravity is... one of the few areas of contemporary physics where (some of) the physicists who are central figures in the field actively engage with philosophers, collaborating with them, participating in philosophy conferences, and contributing chapters to philosophical books and journals...&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The philosophical split was prompted by an exchange between Carnap and Heidegger, and solidified in 1930s according to Friedman:&quot;&lt;em&gt;But the thoroughgoing intellectual estrangement of these two traditions, their almost total lack of mutual comprehension, is a product of the National Socialist seizure of power in 1933 and the resulting intellectual migration&lt;/em&gt;&quot;. Many analytic philosophers (and physicists too) had to emigrate to the US, severing  personal ties that held the philosophical community together. See more in the &lt;a href=&quot;https://philosophy.stackexchange.com/questions/17765/can-philosophy-overcome-the-two-cultures-divide&quot;&gt;Philosophy SE thread&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as Snow pointed out in his famous 1959 lecture &lt;a href=&quot;https://en.wikipedia.org/wiki/Fashionable_Nonsense&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Two Cultures&lt;/a&gt; there is a broader cultural split:&#xA;&quot;&lt;em&gt;A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is the scientific equivalent of: Have you read a work of Shakespeare’s?... So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their neolithic ancestors would have had&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Snow's lecture blamed the British educational system, which overemphasized humanities at the expense of sciences since the Victorian era, and left British elites unprepared for the modern scientific world. He contrasted it with the situation in the US and Germany. As a specifically modern factor one can also point out increasingly narrow specialization in all areas, which makes it harder for scientists and non-scientists to understand each other's principles and methodologies, which is reflected in philosophical attitudes.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:42:23.043" LastActivityDate="2015-08-16T00:51:11.687" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2645" PostTypeId="1" CreationDate="2015-08-12T09:37:39.190" Score="6" ViewCount="397" Body="&lt;p&gt;I know that before the beginning of the 20th century it was thought that the Milky Way galaxy was the whole universe. Then there was the discovery of galaxy redshift, and the size of the Universe was considered to be much larger. But who was the person or persons who arrived at the figure of 46 billion ly (proper distance)? What year was it?&lt;/p&gt;&#xA;" OwnerUserId="2574" LastEditorUserId="2905" LastEditDate="2015-09-24T22:41:31.670" LastActivityDate="2015-09-24T22:41:31.670" Title="Who discovered that the radius of the observable Universe is 46 billion ly?" Tags="|discoveries|cosmology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2646" PostTypeId="2" ParentId="2645" CreationDate="2015-08-12T12:32:38.237" Score="5" Body="&lt;p&gt;There was no single person who discovered the current estimate of the radius of the observable universe, nor was it discovered at one particular moment in history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A primitive version of the big bang model was first proposed in 1927 by Georges Lemaître. Early observational work that helped to build up the model was done by Vesto Slipher, Henrietta Swan Leavitt, and Edwin Hubble. The first mathematical models of big bang cosmology were the Friedmann–Lemaître–Robertson–Walker models, created in the 1920s and 30s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These models describe the big bang as the beginning of time itself, and that immediately implies that since light has only had a finite time to travel, the observable universe has a finite radius. Although this may seem conceptually straightforward, keep in mind that there were competing models such as the steady state model, which were taken seriously as late as the 1960s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roughly speaking, the radius of the observable universe is on the same order of magnitude as the inverse of the Hubble constant. The Hubble constant was first estimated by Hubble and others around the 1920s and 30s, but the estimates were extremely rough. Its accuracy was gradually improved over the decades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get the modern figure of about 46 billion light years for the radius of the observable universe, you need an accurate cosmological model of the past expansion of the universe. That requires knowledge of the acceleration of cosmological expansion and dark energy, which came from a number of different workers, using several independent methods, starting around 1998. That work is still going on.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2015-08-12T13:13:51.113" LastActivityDate="2015-08-12T13:13:51.113" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2650" PostTypeId="2" ParentId="2643" CreationDate="2015-08-12T23:30:46.463" Score="7" Body="&lt;p&gt;On my opinion, this is related to the shift of the centers of research in mathematics and physics from Germany to the English-speaking world. The British and American physicists and mathematicians in most cases looked at philosophical speculation with contempt. (Bertrand Rusell is an exception). This is an old tradition, going back at least to 17 century, and it influenced education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you look at the most famous physicists of the first half of the 20s century, many of them were writing on philosophical questions. And mathematicians too (Hermann Weyl, for example). But most of them were educated in Germany (Switzerland, Austria, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the recent times there was one famous mathematician (Rene Thom, French) who switched from mathematics to philosophy, but this case is an exception.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-08-15T05:20:17.720" LastActivityDate="2015-08-15T05:20:17.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2651" PostTypeId="1" CreationDate="2015-08-14T11:00:40.640" Score="5" ViewCount="129" Body="&lt;p&gt;In the Elements of Euclid a plane number (i.e. a composite number), was represented by a line AB.&#xA;But, being a plane number a multiplication of two numbers (i.e. two lines, in the mind of a pythagorean) it should have been represented by an area.&#xA;In fact it is believed that the pythagoreans represented plane numbers by means of rectangles.&#xA;When did begin to be conceived that a surface could be also represented by a line, to divide it in fractions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Babylonian tablet Plimpton 322 (1800 B.C.) there is a list of very big pythagorean triples ($a^2 + b^2 = c^2$). They must have been generated by means of some formula. But if the formula is the classic $a=p^2-q^2$, $b=2pq$, $c=p^2+q^2$, then they would have added/subctracted two squares to obtain a side of the triangle. Did the babylonians have this capacity for abstraction?&lt;/p&gt;&#xA;" OwnerUserId="118" LastActivityDate="2015-11-14T01:44:54.163" Title="Plane/composite numbers as lines?" Tags="|ancient-greece|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2652" PostTypeId="1" CreationDate="2015-08-14T03:01:02.353" Score="5" ViewCount="159" Body="&lt;p&gt;A Google search for the first academic computer science program a couple of years ago cited a half-dozen programs in France and England that were considered proto-computer science. Current searches yield no such answers. Can anyone suggest an avenue for recovering the names of those programs? The earliest was in the 1800s.&lt;/p&gt;&#xA;" OwnerDisplayName="david west" LastEditorUserId="62" LastEditDate="2015-08-15T21:14:05.890" LastActivityDate="2015-08-15T21:14:05.890" Title="History of computing/computation in higher education" Tags="|computer-science|education|computation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2653" PostTypeId="2" ParentId="2652" CreationDate="2015-08-14T05:46:54.397" Score="1" Body="&lt;p&gt;There was study of computation without (analog or digital) computers. Initially, computers were &lt;a href=&quot;https://en.wikipedia.org/wiki/Human_computer&quot; rel=&quot;nofollow&quot;&gt;people&lt;/a&gt; who computed. A more theoretical question is that of &lt;a href=&quot;https://en.wikipedia.org/wiki/Computability&quot; rel=&quot;nofollow&quot;&gt;computatability&lt;/a&gt;. If Gödel's answer to Hilbert's question counts, how about the &lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen#18th_.E2.80.93_19th_centuries&quot; rel=&quot;nofollow&quot;&gt;Göttingen School&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="o.m." LastActivityDate="2015-08-14T05:46:54.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2654" PostTypeId="1" CreationDate="2015-08-14T22:25:08.353" Score="5" ViewCount="728" Body="&lt;p&gt;Why was the V-2 rocket made with ethanol and liquid oxygen, which had a specific impulse of &lt;a href=&quot;http://www.astronautix.com/lvs/v2.htm&quot; rel=&quot;nofollow&quot;&gt;215 seconds at sea level&lt;/a&gt;, when a solid rocket using APCP gets &lt;a href=&quot;http://www.astronautix.com/stages/shulesrb.htm&quot; rel=&quot;nofollow&quot;&gt;237 seconds&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, a solid rocket engine would have almost no moving parts, so it would be a lot cheaper and quicker to manufacture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm guessing that what I'm effectively asking is: Why was APCP (or a similar solid fuel) not available or not developed at that time? What is the developmental/scientific history of solid rockets?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="26" LastEditDate="2015-08-15T08:08:34.103" LastActivityDate="2015-08-15T08:08:34.103" Title="Why was the V-2 not a solid rocket, and/or what is the scientific history of solid rockets?" Tags="|nazi-germany|world-war-2|engineering|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2655" PostTypeId="2" ParentId="2654" CreationDate="2015-08-15T01:41:42.100" Score="9" Body="&lt;p&gt;Unlike the short-range rockets which are all solid fueled, making a solid-fuel long range rocket is very challenging. The main technical problem is that the fuel block which is very large can develop a crack under mechanical and thermal stress. The flame will immediately spread into the crack, and the whole thing will explode.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second difficult problem is how to cool down the nozzle. In the liquid-propelled rockets it is cooled by the fuel. In short range rockets the engine works very short time and no cooling is needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the history of German rockets, and in general for the early rocket history (till the late 40-s) there is an excellent classical book:&#xA;Willey Ley, Rockets, Missiles and Space travel. (Strongly recommended).&#xA;The Russian side of the story is described in the books of Chertok (4 volumes in Russian).&#xA;He mentions a breakthrough in solid fuel development made in Caltech in the early 50-s, the thing they just could not match in Soviet Union.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All early Soviet long range rockets, even those launched from submarines were liquid fueled. Because the Soviets could not develop solid fuel technology needed for large enough engines. (Just imagine a liquid fuel rocket launched from a submarine!!) Earlier models could not be kept fueled inside in a submarine, so they had to undergo a long and extremely dangerous fueling procedure immediately before launch. If the launch was aborted after the fueling began, the rocket had to be just damped overboard:-) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This shows that the problem is highly non-trivial.&#xA;Roughly speaking the fuel block must be mechanically strong and elastic.&#xA;And the nozzle has to withstand enormous temperature for relatively long time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am afraid that a complete scientific story of (long range, intercontinental) solid fuel rockets is not written yet: the Russians used liquid fuel rockets until recently, and maybe still using some of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. 1. I should mention that liquid fuel rockets have also some advantages: you can regulate the engine power during the flight. It is much more difficult with solid fuel.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Actually all three currently operational Russian submarine based ICBM use&#xA;liquid fuel, one of them has liquid and solid on various stages. See &#xA;Wikipedia article &quot;Submarine-launched ballistic missile&quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-08-15T03:51:11.760" LastActivityDate="2015-08-15T03:51:11.760" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2656" PostTypeId="1" CreationDate="2015-08-15T11:04:27.177" Score="10" ViewCount="2390" Body="&lt;p&gt;Let me start by saying that I've just started studying algebraic geometry. I have a habit of trying to find out something about the motivation behind new concepts that I'm studying, in this case that means that I'm trying to find out something about how the concept of schemes arose in algebraic geometry, and how to view schemes from a historical perspective.&lt;/p&gt;&#xA;" OwnerUserId="2741" LastEditorUserId="26" LastEditDate="2015-08-15T21:12:37.480" LastActivityDate="2023-01-22T08:02:16.007" Title="What is the history behind the concept of &quot;schemes&quot; in algebraic geometry?" Tags="|mathematics|algebraic-geometry|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2657" PostTypeId="2" ParentId="2656" CreationDate="2015-08-15T13:41:45.043" Score="5" Body="&lt;p&gt;The history is simple:-) Schemes were invented by Grothendieck. The purpose was unification and simplification of the foundations of algebraic geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The general idea is to shift from considering points of a manifold to certain ideals of a ring of functions on this manifold. This general idea evolved gradually. An early predecessor was Gelfand's theory of commutative Banach algebras, where he starts with an algebra and recovers a space so that this algebra is the algebra of functions on this space. The points of the space are defined as maximal ideals. Grothendieck did similar thing in algebraic geometry.&#xA;He proposed to consider prime ideals of a ring as points of certain space. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the reference on Grothendieck's original text see&#xA;&quot;Elements de geometrie algebrique&quot; on Wikipedia. I believe it was never completely published but various versions were widely circulated. So there is no well-defined publication year. Grothendieck worked and reworked it for almost 10 years ending around 1970.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-08-18T13:32:02.560" LastActivityDate="2015-08-18T13:32:02.560" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2658" PostTypeId="2" ParentId="2651" CreationDate="2015-08-16T00:26:18.783" Score="2" Body="&lt;p&gt;The geometric turn recorded in Euclid comes with late Pythagoreans, it is likely that it was prompted by the discovery of incommensurability. Early Pythagoreans, like ordinary Greeks, did not represent natural numbers by line segments even if they arranged them into geometric shapes (anecdotally, Pythagoras used pebbles). In fact, they tried to reduce everything, including geometry, to natural numbers and their ratios. Even the original &lt;a href=&quot;http://www.jstor.org/stable/2690123?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;even/odd proof of incommensurability&lt;/a&gt; might have been arithmetical in nature, see &lt;a href=&quot;http://projecteuclid.org/download/pdf_1/euclid.bams/1183544897&quot; rel=&quot;nofollow&quot;&gt;Fowler&lt;/a&gt;. Books VII-IX of Euclid's Elelments are generally attributed to early Pythagoreans, who knew four out of five Platonic solids even before Theaetetus, so plane and solid numbers were likely considered already by them. Late Pythagoreans, like Archytas of Tarentum, were already quite adept at dealing with surfaces and curves produced by their intersections. Unfortunately, the sources for pre-Euclidian period are too scarce to say much more than that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Babylonians their system of numerals was positional with base $60$ and excellent for doing calculations, which they were very good at, even their astronomical models used numerical sequences rather than geometry. So they are much more likely to have found small Pythagorean triples by numerical experimentation and then figured out the pattern without squares or triangles. A &lt;a href=&quot;http://www.ams.org/samplings/happ5-history.pdf&quot; rel=&quot;nofollow&quot;&gt;suggestion of Robson&lt;/a&gt; based on refined translations is that for pedagogical purposes they were interested in reciprocal pairs of numbers $x=p/q$ and $y=q/p$ with only $2,3,5$ (divisors of $60$) as prime divisors of $p,q$. Pythagorean triples then come as a side effect: if $xy=1$ then $1+\big(\frac{x-y}{2}\big)^2=\big(\frac{x+y}{2}\big)^2$, and in terms of $p,q$ we get the formula for the triples.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-08-16T00:26:18.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2659" PostTypeId="1" AcceptedAnswerId="2660" CreationDate="2015-08-16T02:17:58.147" Score="1" ViewCount="54" Body="&lt;p&gt;What is the decade of experimentation that start to verify the formulas from classical mechanics of vibration?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People unable to &quot;see&quot; or &quot;hear&quot; the vibration of the objects in the late 19 century...&lt;/p&gt;&#xA;" OwnerUserId="299" LastEditorUserId="299" LastEditDate="2015-08-17T04:33:18.133" LastActivityDate="2015-08-17T04:33:18.133" Title="What is the decade of experimentation that start to verify the formulas from mechanics of vibration?" Tags="|classical-mechanics|" AnswerCount="1" CommentCount="2" ClosedDate="2015-08-21T15:30:17.703" ContentLicense="CC BY-SA 3.0" />
  <row Id="2660" PostTypeId="2" ParentId="2659" CreationDate="2015-08-16T03:12:50.590" Score="1" Body="&lt;p&gt;In the early development of the theory of oscillations (vibrations), music played an important role. In fact, according to the legend, mathematical theory of vibrations begins with Pythagoras, and it was closely related to music theory. In 17s century Marine Mersenne discovered the precise law of vibration of a sting. He was a music theorist. Music (that you can hear) was the experimental verification of this mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the later development Galileo and Huygens are important, they were interested in improving clocks. Good clocks were needed for determination of longitude at sea. Experimental verification of these theories were better cocks based on them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the invention of calculus, the theory of oscillations became a major area of mathematics with the range of applications to the real life so wide that I cannot even list the areas: almost all physical theories use it in some form. Electricity and magnetism, acoustics, all sorts of waves, quantum theory, etc.&#xA;because of this extremely large number of applications, the theory is verified literally by everyone every day. Almost all devices around us use it in some form.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-08-16T03:12:50.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2662" PostTypeId="1" CreationDate="2015-08-16T21:42:24.180" Score="7" ViewCount="266" Body="&lt;p&gt;There are some real odd names for functions in mathematics, but &lt;a href=&quot;http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function&quot;&gt;Minkowski's question mark function&lt;/a&gt;, denoted by $?(x)$, may be the oddest one I have ever seen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://ada00.math.uni-bielefeld.de/ICM/ICM1904/Main/icm1904.0164.0173.ocr.pdf&quot;&gt;&lt;em&gt;Zur Geometrie der Zahlen&lt;/em&gt;&lt;/a&gt;, Minkowski doesn't discuss his choice of notation (at least, not in the relevant sections that I could find and use a translator on).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is the function denoted by $?(x)$? Most related graphs of it (I've played around with some) fail to show anything like a question mark.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-08-16T21:42:24.180" Title="Why is Minkowski's question mark function denoted by a question mark?" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2663" PostTypeId="1" CreationDate="2015-08-16T16:27:09.993" Score="5" ViewCount="6453" Body="&lt;p&gt;I'm interested in understanding the history of the geometric series, especially how it was discovered and whether there exists continued fraction representations for the geometric series, just as there are continued fractions for many elementary functions in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also appreciate some historical perspective on how geometric series played a role in the development of calculus.&lt;/p&gt;&#xA;" OwnerDisplayName="math-enthusiast" LastEditorUserId="62" LastEditDate="2015-09-24T23:37:11.087" LastActivityDate="2022-03-12T06:23:13.757" Title="History of the geometric series" Tags="|mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2664" PostTypeId="1" CreationDate="2015-08-17T04:34:16.690" Score="3" ViewCount="30" Body="&lt;p&gt;When does the first vibration meter/tester exist in the earth?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was too hard to find the answer in Google in few hours&lt;/p&gt;&#xA;" OwnerUserId="299" LastActivityDate="2015-08-17T04:34:16.690" Title="When does the first vibration meter/tester exist in the earth?" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2666" PostTypeId="2" ParentId="2638" CreationDate="2015-08-17T11:22:34.527" Score="2" Body="&lt;p&gt;The following might be of interest to you:&#xA;&lt;a href=&quot;https://mitpress.mit.edu/books/isaac-newton-mathematical-certainty-and-method&quot; rel=&quot;nofollow&quot;&gt;https://mitpress.mit.edu/books/isaac-newton-mathematical-certainty-and-method&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2748" LastActivityDate="2015-08-17T11:22:34.527" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2667" PostTypeId="1" CreationDate="2015-08-17T18:45:14.400" Score="13" ViewCount="530" Body="&lt;p&gt;&lt;p&gt;I'm puzzled why the noted scientist Jan Ingenhousz's FRS reported study in the title has been almost universally interpreted by modern writers (in books, papers since ca. 1968, and now online resources) as Brownian motion being unwittingly observed in an inert substance. Thus giving him priority over Robert Brown by over 40 years because Ingenhousz's use of an inert material rules out any confusion with reporting a study of live microorganisms in motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The primary source is a short article of five pages, (quoted in part below). He is describing the benefits of what we now call&#xA;a coverslip, a very thin glass plate to cover a subject on a microscope slide, for which he is duly credited as inventing.&#xA;He illustrates the problems with uncovered liquids by a&amp;nbsp;simple demonstration of fine charcoal suspended in an uncovered alcohol which evaporates so rapidly that convection currents cause &amp;quot;everything which is contained in it&amp;quot; to move i.e. implying of any size. This confuses any microscopy studies, in particular if assessing whether a moving small subject is alive or not.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;p&gt;I'm not a science historian, just an optical microscopy hobbyist, but do have a particular interest in and experience of Brownian motion demonstrations in both the liquid and gas phase. &lt;p&gt;Brownian motion in the liquid phase is only observed in suspended particles below ca. 4 - 5 microns and requires an optical microscope typically with at least 200 - 400X total optical mag to observe it e.g. a classic demonstration is fat globules in dilute&#xA;milk. It is a distinctive jiggling motion of fat globules ca. 0.5 - 3 microns in diameter. &lt;p&gt;I've attempted comparable experiments to Ingenhousz with fine charcoal and uncovered ethanol at various temps. The convection currents are so large that particles up to 100 microns in size are moving and exactly as Ingenhousz describes, i.e. in a &amp;quot;confused, continuous and violent&amp;quot; way. Particles are in macroscopic motion almost two orders of magnitude greater in size than that in which Brownian&#xA;motion is observed and which would overwhelm the chance of observing the distinctive microscopic Brownian motion. Only the low&#xA;magnification of a 10X hand lens or low power on a microscope is needed to see this gross motion. We don't know his room&#xA;temp but Ingenhousz does note that his &amp;quot;droplets&amp;quot; of alcohol have evaporated &amp;quot;in a few minutes&amp;quot; so it was rapid. &lt;p&gt;If Ingenhousz suspected that there was an underlying microscopic motion present in the smallest particles beyond the macroscopic motion&#xA;of all sized particles caused by evaporation, wouldn't such an astute researcher (noted for his studies on photosynthesis) have tried the device he was advocating, the coverslip, to remove evaporative effects and also use water not alcohol to decrease evaporation rates?&#xA;&lt;p&gt;After my own studies, I recently found one source who questions the near universal opinion that Ingenhousz was unwittingly observing and describing Brownian motion. This is Beale and Beale in their biography of Ingenhousz 'Echoes of Ingen Housz' 2011, pp.344-6 who remark when discussing one of the earliest interpretations of the work as Brownian motion: &#xA;&lt;p&gt;&amp;quot;The justification [of Ingen Housz observing Brownian motion] is, however far less convincing than it is for Ingen Housz having devised the cover slip and is almost certainly an over-enthusiastic conjecture.&amp;quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These authors also discuss a later part of Ingenhousz's article not quoted above on his described benefits of the use of a coverslip which they note offers &quot;clinching evidence that Ingenhousz was not observing Brownian motion&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But these authors are the exception and don't understand why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;David&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;p&gt;Source material:&#xA;&lt;p&gt;Extract from English translation source (my use of bold): P W van der Pas, 'The Discovery of the Brownian Motion', Scientiarum Historia, 1971, vol. 13, pp.27-35.&#xA;&lt;p&gt;Primary French work and source for above translation. 'Nouvelles Expériences et Observations Sur Divers Objets De Physique' volume 2 by Jan Ingenhousz, 1789. pp.1-5.&#xA;&lt;a href=&quot;https://books.google.co.uk/books?id=kcQLi7BLX8IC&amp;amp;printsec=frontcover&amp;amp;dq=Nouvelles+Exp%C3%A9riences+et+Observations+Sur+Divers+Objets+De+Physique&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=ojqiVf_jL66P7AaK0Z-ACQ&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Nouvelles%20Exp%C3%A9riences%20et%20Observations%20Sur%20Divers%20Objets%20De%20Physique&amp;amp;f=false&quot;&gt;Copy online&lt;/a&gt;.&#xA; &lt;br&gt;A German edition was published in 1784. &#xA;&lt;p&gt;[Pages 1-2] &amp;quot;REMARKS ON THE USE OF THE MICROSCOPE. &#xA;&lt;p&gt;N.B. I have thought it appropriate to precede the following paper with these considerations which, although of no importance for those have frequently used the microscope, will at least be a guide to others. &#xA;&lt;p&gt;I have often troubled my head about the problem to find a method to avoid the too rapid evaporation of a drop of water, or any other liquid, in which I wanted to observe the insects, and I know that other observers are plagued with the same problem. Even if one wishes to observe the shape and size of some of these corpuscules for even the short time during which such a droplet lasts in the focal point of a microscope, one must agree that, as long as the droplet lasts, &lt;B&gt;the entire&#xA;liquid and consequently everything which is contained in it, is kept in continuous motion by the evaporation&lt;/B&gt;, and that this motion can give the impression that some of these corpuscules are living, even if they have not the slightest life in them. &#xA;&lt;p&gt;To see clearly how one can deceive one's mind on this point, if one is not careful; one has only to place a drop of alcohol in the focal point of a microscope and introduce a little finely ground charcoal therein, and one will see these corpuscules in a &lt;B&gt;confused, continuous and violent motion&lt;/B&gt; as if they were animalcules which move rapidly around. &#xA;&lt;p&gt;If the droplet is rather large, it has a convex surface which refracts the light more or less; if it is very small, it lasts hardly long enough to enable observing its contents at one's leisure.&amp;quot; &#xA;&lt;p&gt;... [page 4] &amp;quot;Under these films, the vaporisation is so slow that a &lt;B&gt;droplet which would evaporate in a few minutes&lt;/B&gt;, hardly vaporises in the course of hours.&amp;quot; ... &#xA;&lt;p&gt;[continues to page 5.] &#xA;&lt;p&gt;&amp;nbsp;&lt;/p&gt;&#xA;" OwnerUserId="2744" LastEditorUserId="2744" LastEditDate="2015-08-17T18:58:21.097" LastActivityDate="2015-08-19T03:23:05.100" Title="Why is Ingenhousz's 1784 study of fine charcoal in uncovered alcohol interpreted as Brownian motion, thus giving him priority over Brown?" Tags="|discoveries|experimental-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2668" PostTypeId="2" ParentId="2633" CreationDate="2015-08-18T08:47:43.640" Score="4" Body="&lt;p&gt;The answer to this question cannot but be very vague and much country-dependent. Even when on limits itself to the scientific environment it has to be considered that quantum mechanichs, shared with relativity a curious fate: it was termed as &quot;non arian physics&quot; or &quot;hebrew physics&quot; and so much opposed by sort of &quot;conservative&quot; physicists in Germany and Italy, but also in some other European countries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we can put some landmarks, here and there. In 1932 Heisenberg was awarded the Nobel prize and in 1933 Schrodinger and Dirac as well. So we can fairly say that apart from this conservative resistance I mentioned above, by that moment quantum mechanichs was mainstream in modern physics. On the other hand in 1929 the poet D.H. Lawrence wrote the following poem  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;RELATIVITY by D.H. Lawrence&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I like relativity and quantum theories&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;because I don't understand them&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;and they make me feel as if space shifted about like a swan that&#xA;can't settle,&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;refusing to sit still and be measured;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;and as if the atom were an impulsive thing&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;always changing its mind.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(from Pansies: Poems, 1929)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which shows that the theory was already circulating in cultural circles. It should be no surprise since at the time it was normal for famous scientists to be invited to give &quot;open conferences&quot; to a general audience (which for example Bohr did the first time he visited USA).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So if we consider that Heisenberg's uncertainity principle is from 1922 (EDIT 1927), De Broglie PhD thesis from 1924, the famous Vth Solvay Conference in 1927 we can say that during the 20ies the theory both rapidly spread in scientific circles and in cultural ones. The thirties, together with the opposition I mentioned, brought this somewhat to a stop, The theory was still perceived to be controversial, and this might explain why it took much more to have it so widely accepted to arrive to high school textbooks.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="1689" LastEditDate="2018-08-04T09:16:53.250" LastActivityDate="2018-08-04T09:16:53.250" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2669" PostTypeId="1" AcceptedAnswerId="2673" CreationDate="2015-08-18T17:24:34.963" Score="9" ViewCount="4911" Body="&lt;p&gt;I want to understand more about &lt;a href=&quot;http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors&quot;&gt;the history of eigenvectors&lt;/a&gt;. Was the discovery of eigenvectors inspired from an application to achieve a result in a historical context, was there a phenomenon which operated in some related way that was observed, was it discovered from the formulation and explored and mentioned till applications were found, or from an indirect usage in some other area of mathematics for solving equations?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Did they solve a problem which was not solvable previously?&lt;/li&gt;&#xA;&lt;li&gt;Where can someone read the original track of their derivation from basic principles?&lt;/li&gt;&#xA;&lt;li&gt;Who were the main mathematicians to contribute to the development and use of eigenvectors and in which texts are they described?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1870" LastEditorUserId="4251" LastEditDate="2023-12-07T14:51:30.290" LastActivityDate="2023-12-07T14:51:30.290" Title="The history and motivation of eigenvectors" Tags="|mathematics|mathematicians|reference-request|linear-algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2670" PostTypeId="1" AcceptedAnswerId="2671" CreationDate="2015-08-18T18:48:27.703" Score="6" ViewCount="190" Body="&lt;p&gt;In my 10th class school material, it is given that &lt;strong&gt;Aryabhatta&lt;/strong&gt;  discovered the following formulas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sum n=\dfrac{n(n+1)}{2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sum n^2=\dfrac{n(n+1)(2n+1)}{6}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\sum n^3=\dfrac{n^2(n+1)^2}{4}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is he the first person to discover these formula? If this is true, then who discovered the formula for the generalization $\sum n^k$ ?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastEditorUserId="62" LastEditDate="2015-08-22T19:27:57.257" LastActivityDate="2015-08-22T19:27:57.257" Title="History of summing integers with natural powers" Tags="|discoveries|mathematicians|ancient-india|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2671" PostTypeId="2" ParentId="2670" CreationDate="2015-08-18T20:07:27.267" Score="6" Body="&lt;p&gt;&lt;strong&gt;No.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.math.nmsu.edu/~davidp/bridge.pdf&quot; rel=&quot;nofollow&quot;&gt;The bridge between the continuous and the discrete via original sources in &lt;em&gt;Study the Masters: The Abel-Fauvel Conference&lt;/em&gt;&lt;/a&gt;, Pengelley writes that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In the 6th century B.C., the Pythagoreans knew about the formula for $\sum n$.&lt;/li&gt;&#xA;&lt;li&gt;In the 3rd century B.C., Archimedes figured out the formula for $\sum n^2$.&lt;/li&gt;&#xA;&lt;li&gt;In the 1st century A.D., Nichomachus figured out the formula for $\sum n^3$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-08-20T19:27:40.533" LastActivityDate="2015-08-20T19:27:40.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2672" PostTypeId="2" ParentId="2667" CreationDate="2015-08-18T21:44:18.207" Score="7" Body="&lt;p&gt;Since the question calls for speculation here is one. I think the reason is the confusion between Brownian motion in the strict sense, as understood by Einstein and Smoluchowski, chaotic motion of small particles in fluids &lt;em&gt;caused exclusively by their collisions with molecules&lt;/em&gt;, and loosely used &quot;Brownian motion&quot;, where the italicized part is neglected. The &quot;dancing&quot; of dust particles in a sunbeam described in the oft cited passage from Lucretius's De Rerum Natura (c. 75 BC) is often referred to as &quot;Brownian motion&quot; (the observation of &quot;&lt;em&gt;motes in a sunbeam&lt;/em&gt;&quot; goes back to &lt;a href=&quot;https://books.google.com/books?id=_cpZxx1bQokC&amp;amp;pg=PA33&amp;amp;lpg=PA33&amp;amp;dq=Democritus+sunbeam&amp;amp;source=bl&amp;amp;ots=YR43y53Vjm&amp;amp;sig=6ODsIfS5XEQiDmmV6Wf3BD1KuWY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CDAQ6AEwBGoVChMIiZz7nLyzxwIVBUKSCh1aqQBy#v=onepage&amp;amp;q=Democritus%20sunbeam&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Democritus c. 400 BC, and possibly Leucippus c. 475 BC&lt;/a&gt;). Of course, the dust particles are visible with a naked eye and so too large for exclusivity, their motion is a combination of &quot;mingling motion&quot; caused by small air currents, and &quot;glittering, tumbling motion&quot; caused by molecular collisions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is Brownian motion in the strict sense that played a historical role at the beginning of 20th century, because it provided empirical confirmation of the kinetic theory, and by implication of Boltzmann's statistical account of the second law of thermodynamics. Both were controversial at the time with some prominent figures (Mach and Ostwald) even denying the existence of atoms.&#xA;But the looser &quot;Brownian motion&quot; also played a historical role, indeed it led Leucippus and Democritus to suggest the existence atoms for the first time. There is a nice parallelism there, so what I think happened is that the kinetic theory story and the ancient atomism story got bundled together at the expense of using &quot;Brownian motion&quot; loosely. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=CkeGYQ8xT_UC&amp;amp;pg=PA182&amp;amp;lpg=PA182&amp;amp;dq=ingenhousz+brownian+motion+photosynthesis&amp;amp;source=bl&amp;amp;ots=7okerLe6CA&amp;amp;sig=mTWpU65LPt-2Ept7qC-7uggcArM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CFwQ6AEwDGoVChMI67zy_8OzxwIVEhaSCh22zA0x#v=onepage&amp;amp;q=ingenhousz%20brownian%20motion%20photosynthesis&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Einstein knew nothing of Ingenhousz&lt;/a&gt;, and for a while the story followed his account that starts with Brown in 1827. But once the loose usage became common someone had to recall Ingenhousz and his fine charcoal particles, apparently Peter van der Pas was first in 1968, according to the &lt;a href=&quot;http://www.microscopy-uk.org.uk/mag/indexmag.html?http://www.microscopy-uk.org.uk/mag/artaug15/dw-ingenhousz.html&quot; rel=&quot;noreferrer&quot;&gt;OP's online article&lt;/a&gt; on this issue. Still, why credit Brownian motion to him rather than to Leucippus and Democritus? The difference is that they were philosophers speculating about nature based on sporadic observations, whereas Ingenhousz was a reputable experimenter conducting systematic observations &lt;em&gt;under a microscope&lt;/em&gt;. There is an element of novelty to him, unlike to Brown or Democritus, going against the &quot;conventional wisdom&quot;, and sympathy for an unjustly forgotten discoverer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is an appealing story, so appealing that the mention of the microscope makes it plausible enough, and the numerical details of particle sizes are easily overlooked. Indeed, &lt;a href=&quot;https://en.wikipedia.org/wiki/Brownian_motion#Intuitive_metaphor&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia warns us&lt;/a&gt; that pollen particles few micrometres in size observed by Brown are &quot;&lt;em&gt;not to be confused with the actual pollen particle which is about 100 micrometres&lt;/em&gt;&quot;, just a few scrolls down from quoting Lucretius and calling it Brownian motion.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-08-19T03:23:05.100" LastActivityDate="2015-08-19T03:23:05.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2673" PostTypeId="2" ParentId="2669" CreationDate="2015-08-18T23:41:40.050" Score="15" Body="&lt;p&gt;Eigenvectors (but not the word for them!) gradually appeared in 18s century in solving differential equations&#xA;which we write now as $y'=Ay$ describing all sorts of oscillatory phenomena in the nature (mechanical vibrations, light, sound, etc.) Of course this was long before the words &quot;matrix&quot; and &quot;vector&quot; appeared. The simplest of these equations is $y''+k^2y=0$, it was completely understood by Huygens and its general solution is $y(t)=c_1\sin kt+c_2\cos kt$. Now we state this as &quot;the operator of second derivative has eigenvalues $-k^2$ and eigenfunctions&#xA;$\cos kt,\sin kt$. It may sound strange but eigenvectors in infinite dimensional spaces (eigenfunctions) appeared under various names long before linear algebra, and before the word &quot;vector&quot; came into common use.&#xA;They played central role in the theory of small oscillations, and in Fourier's work on partial differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The words &quot;eigenvector&quot; and &quot;eigenfunction&quot; were not standard until well into 20s century (all kinds of other words were used like proper vectors characteristic vectors, etc.). The term &quot;eigenvector&quot; comes from German, I think under the influence of the book of Hilbert-Courant, first edition in 1920s.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-08-23T20:17:59.533" LastActivityDate="2015-08-23T20:17:59.533" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2675" PostTypeId="1" CreationDate="2015-08-19T15:18:56.157" Score="0" ViewCount="24" Body="&lt;p&gt;I'm interested in finding something out about the history of the problem of the delta potential barrier in QM. Who was the first to propose this problem, and perhaps any particular motivation for it. References that I could read and quote would be excellent. I am aware of some of the usual motivations for using it, but any examples that come to mind are appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="2762" LastActivityDate="2015-08-19T15:18:56.157" Title="History of the delta potential barrier in Quantum Mechanics" Tags="|quantum-mechanics|mathematical-physics|probability|" AnswerCount="0" CommentCount="1" FavoriteCount="0" ClosedDate="2015-08-19T21:31:49.847" ContentLicense="CC BY-SA 3.0" />
  <row Id="2676" PostTypeId="1" CreationDate="2015-08-19T14:58:59.737" Score="4" ViewCount="245" Body="&lt;p&gt;I'm interested in finding something out about the history of the problem of the delta potential barrier in quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which was the first study to propose this problem, and perhaps any particular motivation for it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of some of the usual motivations for using it, but any examples that come to mind are appreciated.&lt;/p&gt;&#xA;" OwnerUserId="2762" OwnerDisplayName="Ed Wolf" LastEditorUserId="62" LastEditDate="2015-08-20T00:11:55.687" LastActivityDate="2015-08-21T08:50:15.117" Title="History of delta barrier in quantum mechanics" Tags="|quantum-mechanics|mathematical-physics|probability|" AnswerCount="2" CommentCount="6" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2677" PostTypeId="2" ParentId="2676" CreationDate="2015-08-19T15:21:10.903" Score="1" Body="&lt;p&gt;Intriguingly, I have never used references prior to the eighties. The oldest I used were:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;R.G. Newton Inverse scattering by a local impurity in a periodic potential in one dimension, JMP 24 (1983)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P. Seba, The generalized point interaction in one dimension, Cz J Phys B 36:667 (1986)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;M. Carreau; Four-parameter point-interaction in 1D-quantum systems, J Phys A: Math Gen, v 26  (1993)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a book on the topic in 1988, by Albeverio et al, that instead closing the topic, caused some subsequent work due to issues on the denomination of the different solutions. While the plain &quot;delta barrier&quot; was clear to everyone, some different barriers were expected to appear when using the $\delta '$ distribution, and it was controversial if the book had choosen the right solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematically the question is about self-adjoint extensions of hermitian operators, and I'd expect it to appear in textbooks addressing this issue.&lt;/p&gt;&#xA;" OwnerUserId="2763" OwnerDisplayName="arivero" LastEditorUserId="2763" LastEditDate="2015-08-20T00:07:17.437" LastActivityDate="2015-08-20T00:07:17.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2678" PostTypeId="1" AcceptedAnswerId="2680" CreationDate="2015-08-19T18:48:02.640" Score="5" ViewCount="935" Body="&lt;p&gt;The classic &lt;a href=&quot;http://www.pitt.edu/~jdnorton/Goodies/Dome/&quot; rel=&quot;nofollow&quot;&gt;Norton's dome problem&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=UchIDBrAQEMC&amp;amp;pg=PA202&amp;amp;lpg=PA202&amp;amp;dq=space%20invaders%20classical%20mechanics&amp;amp;source=bl&amp;amp;ots=2b2dkwJ3Wp&amp;amp;sig=nY289Hqp-2APICbr9JGqUfEwo5o&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CCcQ6AEwAmoVChMIjq7Wiua1xwIVBJmICh1gqASz#v=onepage&amp;amp;q=space%20invaders%20classical%20mechanics&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;space invaders&lt;/a&gt; and other examples, show that Classical Mechanics, held as the paragon of determinism for ages having inspired Laplace's statements on determinism, is not necessarily deterministic, as explained &lt;a href=&quot;http://plato.stanford.edu/entries/determinism-causal/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; and to a smaller extent &lt;a href=&quot;http://jamesowenweatherall.com/SCPPRG/FletcherSam2010Man_Dome.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, which also discusses what counts as a Newtonian system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it appears that these examples of indeterminism in classical mechanics were all found in the 20th/21st century. Are there any indications that before the 20th century there were any investigations on indeterministic classical systems? When did people first realize that classical mechanics can be indeterministic? A short timeline on major milestones in the investigation of indeterminism in classical mechanics would be nice. Also, an overview of the controversy would be nice.&lt;/p&gt;&#xA;" OwnerUserId="2171" LastEditorUserId="2171" LastEditDate="2015-08-21T23:32:09.797" LastActivityDate="2017-05-12T19:49:07.900" Title="History of the study of indeterminism in classical mechanics" Tags="|physics|classical-mechanics|philosophy-of-science|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2679" PostTypeId="1" CreationDate="2015-08-20T21:09:47.943" Score="3" ViewCount="671" Body="&lt;p&gt;In his 1975 paper, &lt;a href=&quot;http://adsabs.harvard.edu/abs/1975MSRSL...8..369R&quot; rel=&quot;nofollow&quot;&gt;Circumstellar absorption lines and mass loss from red giants&lt;/a&gt;, Reimers gave a relation for mass loss from red giants via stellar winds as&#xA;$$\dot{M}\propto\frac{L}{gR}$$&#xA;where $\dot{M}$ is mass loss, $L$ is luminosity, $g$ is the acceleration due to gravity at the surface of the star and $R$ is the radius of the star.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reimers wrote that this was &quot;the simplest quantity&quot; based on the three parameters that had the units of g s&lt;sup&gt;-1&lt;/sup&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using observational data, Reimers was able to determine a coefficient, $\eta$ that created the law the bears his name. However, assuming I'm understanding the paper correctly, he seems to have chosen it as an approximation, just because it was simple and it worked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Reimers have any theoretical evidence for believing that the law took the form it did, or was this assumption simply based on a conjecture?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-07-18T14:30:44.020" Title="What theoretical basis did Reimers base his mass loss law on?" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2680" PostTypeId="2" ParentId="2678" CreationDate="2015-08-21T00:39:43.203" Score="9" Body="&lt;p&gt;Perhaps, the most insightful analysis (possibly to this day) of indeterminism in classical mechanics and its implications was given by Joseph Boussinesq, best known for his work on solitons, in a book long essay Reconciliation of Mechanical Determinism with Moral Freedom (1878). His ideas were based on the general theory of solutions to differential equations that was undergoing systematization at the time. In particular, solutions were classified into regular, which depend on initial data continuously, and singular, e.g. unstable equilibria (like the circular pendulum in the upright position), where infinitesimal deviations produce large changes in the outcome. Boussinesq further distinguished between asymptotic and singular solutions proper, like equilibrium at the top of Norton dome, which can be reached in finite time, and produces non-uniqueness of solutions under time reversal. In fact he used an example almost identical to the Norton dome according to &lt;a href=&quot;https://en.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_20.djvu/461&quot; rel=&quot;nofollow noreferrer&quot;&gt;Popular Science Monthly (1882)&lt;/a&gt;:&quot;&lt;em&gt;It then reaches the apex with a velocity of zero, and remains there till it pleases some guiding principle residing there to give it an impulse in a required direction, which, although it is equal to nothing, shall yet be competent to let it glide down the paraboloid again&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boussinesq saw these singular &quot;bifurcations&quot; (his word) as creating gaps in causal chains, and suggesting an additional &quot;guiding principle&quot; operating in living organisms. This answered the objection, raised earlier by Helmholtz, Du Bois-Reymond and others against such a principle, that conservation of energy precludes its operation. &lt;a href=&quot;http://www.informationphilosopher.com/freedom/mechanisms.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxwell praised the idea&lt;/a&gt; in a letter to Galton (1879):&quot;&lt;em&gt;it may at any instant, at its own sweet will, without exerting any force or spending any energy, go off along that one of the particular paths which happens to coincide with the actual condition of the system at that instant. In most of the former methods... there was a certain small but finite amount of... trigger-work for the Will to do. Boussinesq has managed to reduce this to mathematical zero... I think Boussinesq’s method is a very powerful one against metaphysical arguments about cause and effect and much better than the insinuation that there is something loose about the laws of nature...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To put this into perspective note that non-uniqueness of solutions to equations of motion is neither necessary nor sufficient for indeterminism. When two identical rigid balls collide moving along a line, conservation of energy and momentum are not enough to determine their motion after the collision, unless we also assume that they remain on the same line. That extra assumptions are needed to resolve collisions was known even before Newton to Huygens, Wren and Wallis, but it was seen as reflection of the model being too idealized and not fully describing the physics involved, rather than indeterminism. Poisson took it the same way in 1806, when he considered medium resistance laws of the form $m\dot{v}=-av^n$, and discovered that for $0&amp;lt;n&amp;lt;1$ there are non-trivial solutions when the initial velocity is zero (in modern terms, because $v^n$ in non-Lipschitz). This is the viewpoint reflected in the 19th century textbooks of Duhamel and Cournot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term determinism did not even come into common usage until 1860s (Kant mentions it as unwelcome neologism in 1793, and so does Mill in 1865), and Laplace's famous quote was likely little more than a flowery metaphor. Only in 1865 Bernard in his influential 1865 Introduction to the Study of Experimental Medicine publicized the word &quot;determinism&quot; as reducing &quot;the features of living beings to physical-chemical features”, but implied that the reduction may not be complete. And only in 1872 Du Bois-Reymond fully articulated modern (mechanical) determinism of “reducing all transformations taking place in the material world to atomic motions” ruled by “mechanical necessity”, turning Laplace's metaphor into a doctrine. According to Bordoni, &quot;&lt;em&gt;it seems reasonable to think that the mythology of Laplacian determinism was a late reconstruction, and the physiologist Emile Du Bois-Reymond played an important role in the emergence of that mythology&lt;/em&gt;&quot;. The subsequent philosophical discussions involved Renouvier, Peirce, Boutroux and James. James is usually credited with the first two-stage model of free will, outlined in his famous Dilemma of Determinism (1884), but Boussinesq's 1878 model is also two-stage, although his determined/free stages are reversed compared to James's.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was against this backdrop that Boussinesq advanced his idea. Earlier free will issues were in the domain of philosophers, who largely assumed, along with the common public, that mechanical laws did not apply to the mental. Bertrand, in his 1878 critique of Boussinesq gave a mathematician's version of this position (which incidentally rejects Laplace's extrapolation): &quot;&lt;em&gt;the results of equations could not attain absolute precision&lt;/em&gt;”, and “&lt;em&gt;the certainty of equations cannot be greater than the certainty of principles from which they stem&lt;/em&gt;”, so it is naive to expect that solutions will be &quot;loyally followed&quot; even along regular segments. Bertrand's criticism was off the mark, Boussinesq did not suggest that mechanical description was precise, he explicitly described bifurcations as only structural toy models for what might be happening in the living organisms, but toy models with the benefit of mathematical precision. In 1880 Du Bois-Reymond acknowledged Boussinesq, and his precursors Saint-Venant and Cournot, but deemed their position unsatisfactory. Soon attention turned to new physics, and Boussinesq's programme of exploring the nature of &quot;guiding principle&quot; scientifically fell into oblivion until recent times. Although some of his ideas about the role of causal gaps and the guiding principle in the mind-body problem were re-explored since 1950s by Heisenberg and Wigner (later Penrose and others) with quantum mechanics in place of classical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most detailed historical account is given by &lt;a href=&quot;http://pubman.mpiwg-berlin.mpg.de/pubman/item/escidoc:698512:2/component/escidoc:698510/P460.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bordoni in Unexpected Convergence between Science and Philosophy&lt;/a&gt;, and philosophical side by &lt;a href=&quot;http://www.jstor.org/stable/2709176?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hacking in Nineteenth Century Cracks in the Concept of Determinism&lt;/a&gt;. See also van Strien's shorter &lt;a href=&quot;http://www.tandfonline.com/doi/pdf/10.1080/00033790.2014.935954&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vital Instability: Life and Free Will in Physics and Physiology&lt;/a&gt; and &lt;a href=&quot;http://conferences.phs.uoa.gr/andhps/Files/Abstracts/van%20Strien.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Norton Dome and the Nineteenth Century Foundations of Determinism&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-05-12T19:49:07.900" LastActivityDate="2017-05-12T19:49:07.900" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2682" PostTypeId="2" ParentId="2676" CreationDate="2015-08-21T08:50:15.117" Score="3" Body="&lt;p&gt;After looking for a while this is what I have found, although it does not mean that perhaps earlier references do not exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the review article by M. Belloni and R. W. Robinett, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics&quot;, PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS  Volume: 540   Issue: 2   Pages: 25-122   Published: JUL 10 2014,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;there is a section in which the delta potential is discussed at some length. They seem to find the first references to the use of the delta barrier in the classic article by R. de L. Kronig and W. G. Penney&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Quantum Mechanics of Electrons in Crystal Lattices&quot; Proc. R. Soc. Lond. Ser. A  Volume: 130   Pages: 499-513   Published: 1931.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There, the delta potential is introduced as the limit of zero width but constant area of square potential barriers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Morse and Feshbach's treatise &quot;Methods of Theoretical Physics&quot; in v. 2 (published in 1953) discusses the single delta potential problem (as a well). Belloni and Robinett above make a note of this. &lt;/p&gt;&#xA;" OwnerUserId="2762" LastActivityDate="2015-08-21T08:50:15.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2683" PostTypeId="1" CreationDate="2015-08-19T18:51:33.617" Score="13" ViewCount="397" Body="&lt;p&gt;The following passage has been extracted from the Newton's (John Stewart's English translated version) &quot;&lt;a href=&quot;http://archive.org/details/sirisaacnewtons00stewgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms&lt;/a&gt;&quot;:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I consider mathematical quantities in this place not as consisting of parts; but as described by a continued motion. Lines are described, and there by generated not by the apposition of parts, but by the continued motion of points; superficies's by the motion of lines; Solids by the motion of superfices's; Angles by the rotation of the sides; Portion of time by a continual flux: and so in other quantities. These geneses really take place in nature of things, and are daily seen in the motion of bodies. And after this manner the ancients, by drawing moveable right lines along immoveable right lines taught the genesis of reflection...    &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here Newton doesn't provide any reason on why he wants to describe lines to be generated by the &quot;continued&quot; motion rather than by the appositon of parts (= points??). &lt;strong&gt;Is there any reason for his preference for motion view?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I noticed that Newton doesn't define point. I don't understand whether he is following Euclid's method of having some of the terms to be undefined, or some other philosophy. &lt;strong&gt;I want to know Newton's view on mysterious points&lt;/strong&gt;. I will be really happy if sources on this regard (Newton's view on points) is provided.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Meaning of Apposition from &quot;The New Oxford American Dictionary&quot;: The positioning of things or the condition of being side by side or close together. So, I interpret apposition of parts to be positioning of points/parts side by side or close together to form a line.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;References to the full latin text:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Newton drafts, (&lt;a href=&quot;http://cudl.lib.cam.ac.uk/view/MS-ADD-03962/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;MS Add.3962&lt;/a&gt;) &lt;/li&gt;&#xA;&lt;li&gt;In Whiteside collection, &lt;a href=&quot;http://books.google.es/books?id=EqlWllD_H8MC&amp;amp;pg=PA106&amp;amp;lpg=PA106&amp;amp;dq=partibus%20quam%20minimis&amp;amp;source=bl&amp;amp;ots=IJ7ICF9UEd&amp;amp;sig=qMyuybvdeqn52QlRmfneyMUlpJw&amp;amp;hl=es&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=partibus%20quam%20minimis&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;vol VIII&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I have asked same question in &lt;a href=&quot;https://physics.stackexchange.com/questions/201530/why-newton-wanted-lines-to-be-generated-by-continued-motion-of-points-rather-tha&quot;&gt;PSE&lt;/a&gt; too.&lt;/p&gt;&#xA;" OwnerUserId="2172" OwnerDisplayName="Feynman" LastEditorUserId="2172" LastEditDate="2019-01-08T12:54:12.190" LastActivityDate="2019-01-22T12:12:30.227" Title="Why did Newton want lines to be generated by continued motion of points rather than by apposition of parts?" Tags="|mathematicians|calculus|philosophy-of-science|geometry|" AnswerCount="4" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="2684" PostTypeId="2" ParentId="2683" CreationDate="2015-08-19T19:20:18.637" Score="5" Body="&lt;p&gt;Seeing as Newton developed a lot of what we now define as Calculus I, it makes sense that he would consider lines as continuous/differentiable, as opposed to his contemporaries who might have only considered them a finite collection of smaller line segments or points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus is, after all, the analysis of change of &quot;mathematical quantities&quot; at infinitesimally small distances. My interpretation: Newton is claiming that he works with lines under the assumption that there is a &quot;continuous motion&quot; or differentiability at every point.&lt;/p&gt;&#xA;" OwnerDisplayName="NoseKnowsAll" LastActivityDate="2015-08-19T19:20:18.637" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2685" PostTypeId="2" ParentId="2683" CreationDate="2015-08-21T09:45:45.610" Score="10" Body="&lt;h2&gt;Problem: classical geometry is not happy with infinitesimals&lt;/h2&gt;&#xA;&lt;p&gt;Newton is systematically trying to avoid basing calculus on infinitesimal geometric quantities.  We can see this from how he emphasizes that his method is consistent with the &amp;quot;ancient&amp;quot; standard of rigor:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To institute an Analysis after this manner in finite Quantities... is consonant to the Geometry of the Ancients... in the Method of Fluxions, there is no necessity of introducing Figures infinitely small into Geometry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Page 4 of &lt;a href=&quot;https://archive.org/stream/sirisaacnewtons00stewgoog&quot; rel=&quot;noreferrer&quot;&gt;the English translation cited in the question&lt;/a&gt;, page 6 of &lt;a href=&quot;http://reader.digitale-sammlungen.de/en/fs1/object/context/bsb10053711_00002.html&quot; rel=&quot;noreferrer&quot;&gt;this scanned Latin edition&lt;/a&gt; - note that the page numbers refer to the numbering as printed on the actual scanned dead-tree pages, not the electronic numbering.)&lt;/p&gt;&#xA;&lt;p&gt;Recall that before the modern era, the standard for rigor in mathematics was Euclid, and Newton followed the tradition by using geometric figures and arguments whenever possible.  Now sure, classical geometry had what we now call the method of exhaustion.  But it did not have the notion of limits or infinitesimals the way Newton did.  Especially since the foundations of calculus were &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Analyst&quot; rel=&quot;noreferrer&quot;&gt;controversial&lt;/a&gt; from the start, Newton needed his ideas to be agreeable with the standards of classical geometry.&lt;/p&gt;&#xA;&lt;h2&gt;Solution: frame the foundation in terms of motion, not geometry&lt;/h2&gt;&#xA;&lt;p&gt;Calculus has both a geometric interpretation and a dynamic interpretation - think tangent lines vs. rates of change.  You learn this early on.  But arguments directly in terms of infinitesimal geometric figures don't fit into the classical geometry.  If you appeal to motion and change, though, you can use perfectly reasonable and familiar concepts.  For example, a secant line that moves and becomes a tangent line sounds a lot more believable than supposing you can just take two &amp;quot;infinitely close&amp;quot; points and take the secant line through them.  There's some hand-waving going on either way, but the latter flies directly in the face of classical geometry, which, remember, was the standard of rigor.  &lt;strong&gt;Framing calculus in terms of motion shifts the controversial foundation further from geometry&lt;/strong&gt;, where it would face stricter scrutiny.&lt;/p&gt;&#xA;&lt;p&gt;With this in mind, we can interpret the quote as establishing the emphasis on the dynamic viewpoint.  Essentially, Newton is saying: Don't worry, we're not gonna do infinitesimal geometry.  We're not gonna break a line or anything down into absurd infinitesimal parts which may not really be possible.  Just view a line as the path traced by a moving point.  With this viewpoint, we can appeal to concepts like velocity to justify our arguments.  We can usefully apply calculus to geometry, without having overtly infinitesimal geometry.&lt;/p&gt;&#xA;&lt;p&gt;And of course the perspective helps teach the important interplay between the geometric and dynamic interpretations of calculus.&lt;/p&gt;&#xA;&lt;h2&gt;P.S. Notes on the quote&lt;/h2&gt;&#xA;&lt;p&gt;For completeness, here is the quote in Latin:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quantitates Mathematicas non ut ex partibus quam minimis constantes, sed ut motu continuo descriptas hic considero.  Lineae describantur ac describendo generantur non per appositionem partium, sed per motum continuum punctorum; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(page 3)&lt;/p&gt;&#xA;&lt;p&gt;You had some hypotheses about some particular words and phrases in the quote, so here are my comments on them specifically.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;You guessed that &amp;quot;mathematical quantities&amp;quot; (&amp;quot;Quantitates Mathematicas&amp;quot;) means curves.  It looks it just means quantities.  For example, compare with page 5 of the Latin, page 4 of the English: &amp;quot;&lt;em&gt;Fluat quantitas &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; uniformiter &amp;amp; invenienda sit fluxio quantitatis &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;.&lt;/em&gt;&amp;quot;  (&amp;quot;&lt;em&gt;Let the Quantity &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; flow uniformly, and let it be proposed to find the fluxion of &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;.&lt;/em&gt;&amp;quot;)  Clearly &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt; are numerical quantities.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;Parts&amp;quot; and &amp;quot;points&amp;quot; are not the same word (&amp;quot;partium&amp;quot; vs. &amp;quot;punctorum&amp;quot;); in fact, Newton is contrasting the two.  Parts are pieces you divide something into.  Points are points, although since you're thinking of them as moving, you might say point particles.  I'm sure there's more to say about what a point is according to Newton.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2772" OwnerDisplayName="echinodermata" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-08-21T09:45:45.610" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2686" PostTypeId="2" ParentId="2639" CreationDate="2015-08-21T16:43:37.913" Score="2" Body="&lt;p&gt;The book &quot;Physics&quot; by Cutnell and Johnson, says:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In the early part of the twentieth century a widely accepted model, due to the English physicist Joseph J. Thomson (1856-1940), pictured the atom very differently. In Thomson's view there was no nucleus at the center of an atom. Instead, the positive charge was assumed to be spread throughout the atom, forming a kind of &quot;paste&quot; or &quot;pudding,&quot;...&quot;&lt;/p&gt;&#xA;" OwnerUserId="2574" LastActivityDate="2015-08-21T16:43:37.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2687" PostTypeId="1" AcceptedAnswerId="2688" CreationDate="2015-08-22T01:30:42.280" Score="6" ViewCount="1292" Body="&lt;p&gt;Was Hooke's law first coming from experiment or from math derivation of which Newtonian mechanical laws are the only prerequisite? Also can the law itself be reinvented in this way, or is it impossible to derive it mathematically from Newton's fundamental laws?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to know if Hooke's law is an axiom or a theorem, even though it was called a law...&lt;/p&gt;&#xA;" OwnerUserId="299" LastEditorUserId="55" LastEditDate="2015-08-22T17:04:16.783" LastActivityDate="2022-03-11T13:53:54.787" Title="Did Hooke's law come from experiments, or was it mathematically derived from Newtonian mechanics?" Tags="|discoveries|experimental-physics|classical-mechanics|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2688" PostTypeId="2" ParentId="2687" CreationDate="2015-08-22T16:51:32.973" Score="11" Body="&lt;p&gt;Newtonian mechanics was not yet in place when Hooke published his De Potentia Restitutiva (On Restoring Force) in 1678, Newton's Principia only came out in 1687. Hooke inferred the law from experiments not only with springs but also with wood and a &quot;body of air&quot;, concluding: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The power of any spring is in the same proportion with the tension thereof: that is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward. Now as the theory is very short, so the way of trying it is very easie… it is very evident that… in every springing body… the force or power thereof to restore itself to its natural position is always proportionate to the distance or space it is removed therefrom&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This being said, Hooke did attempt a theoretical justification by imagining a mechanical model of springs as composed of tiny particles vibrating around some fixed positions, colliding with their neighbors and surrounded by &quot;subtle matter&quot;. When the body is compressed the rate of collisions increases, accounting for the restoring force. When the body is stretched the force is provided by the pressure from &quot;subtle matter&quot;. Although Hooke claims that the model leads to his law, modern analysis of his reasoning shows that it produces inverse square rather than proportional relationship. Today Hooke's law can be derived from  more fundamental principles, but that requires assumptions about the molecular composition of matter and the nature of intermolecular forces, in addition to Newton's fundamental laws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.jstor.org/stable/230074?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moyer's Robert Hooke's Ambiguous Presentation of &quot;Hooke's Law&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-06T05:55:00.830" LastActivityDate="2019-06-06T05:55:00.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2689" PostTypeId="2" ParentId="2683" CreationDate="2015-08-22T18:58:32.997" Score="5" Body="&lt;p&gt;Although Newton was certainly trying to avoid infinitesimals there is some distance between &quot;no infinitesimals&quot; and &quot;no points&quot;, after all we now have calculus with points but without infinitesimals. Moreover, Newton's goal was not to invent calculus, but to analyze motion, so his reasons were somewhat deeper. &lt;a href=&quot;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Arrow_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zeno's paradox Arrow&lt;/a&gt; argues that if time consisted of instants then flying arrow would never move, because within an instant there is no difference between arrow in flight and arrow at rest. Zeno, and Plato after him, concluded that motion is but an illusion and should be banished from geometry, position loyally followed in Euclid's Elements, but obviously unacceptable to Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle offered an alternative resolution that &quot;saved&quot; motion: time does not consist of instants, a magnitude can only be broken into smaller magnitudes, but never into points. This was the price of making the arrow fly, see &lt;a href=&quot;https://philosophy.stackexchange.com/questions/26441/aristotles-objection-to-zenos-paradox/26443#26443&quot;&gt;Aristotle's objection to Zeno's paradox&lt;/a&gt;. After Euclid some prominent geometers, notably Archimedes and Apollonius, considered curves generated by combining linear and circular motions. Newton was very familiar with such curves, as well as with Aristotle and Euclid. Fermat and others tried to get around the issue with infinitesimals, but motion itself provided Newton with a natural alternative, what we now call limits, and what he called &quot;first and last ratios&quot;. As he wrote in Principia:&quot;&lt;em&gt;Quantities and ratios of quantities which in finite time converge continually to equality and before the end of that time approach nearer to each other than by any given difference, become ultimately equal&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's definition sounds similar to the modern explanation of limits, but to make it a definition today we use three nested quantifiers $\forall\varepsilon&amp;gt;0\,\exists\delta\,\forall x: |x-x_0|&amp;lt;\delta\Rightarrow|f(x)-a|&amp;lt;\varepsilon$. Aristotle's syllogistic, the only logic known to Newton, had no quantifiers, and while $\forall x$ can be simulated with variables, $\exists x$ with constants, and even $\forall x\exists y\,P(x,y)$ with implicitly defined function $y=f(x)$, the combination $\forall\exists\forall$ needed to express convergence to limits is beyond it. But it is not beyond kinematic interpretation, which simulates the missing quantifier dependence, but requires that lines and other magnitudes be generated by motions in the first place. So the quote in the OP is essentially Newton's definition of &quot;quantity&quot;. Newton's &quot;first and last ratios&quot; manage to express the modern limit concept without quantifiers, but at Aristotelian expense of sacrificing decomposition of magnitudes into points. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is &lt;a href=&quot;http://www.jstor.org/stable/2185244?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;explained by Friedman&lt;/a&gt; (pp. 477-481) Kästner in his influential mathematics textbooks, remarkably for an 18th century German author, &quot;&lt;em&gt;argues that Newton's fluxions are in some respects clearer and more perspicuous than Leibniz's differentials. Further, he explicitly applauds Collin Maclaurin's attempt, in his monumental Treatise of Fluxions (1742), to develop the calculus on the basis of a kinematic conception of the limit operation&lt;/em&gt;&quot;. And &quot;&lt;em&gt;when the kinematic interpretation was explicitly criticized by mathematicians like D'Alembert and l'Huilier in the late eighteenth century, it was not on grounds of coherence and consistency but because it was thought to import a &quot;foreign&quot; or &quot;physical&quot; element into pure mathematics&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:42:23.043" LastActivityDate="2015-08-22T18:58:32.997" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2691" PostTypeId="2" ParentId="2687" CreationDate="2015-08-23T00:18:06.537" Score="5" Body="&lt;p&gt;There is no derivation of Hooke's Law from Newtonian mechanics. By Hooke or anyone else. At the time of Hooke, this was an experimental law. If Hooke had some theoretical explanation, it was incorrect. The derivation of this law from other more fundamental laws is a matter of Solid State physics, and I do not think this is achieved even today.&lt;/p&gt;&#xA;&lt;p&gt;Remark: If &amp;quot;Newtonian mechanics&amp;quot; means the contents of his book Principia, then Hooke's law was published earlier.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-03-11T13:53:54.787" LastActivityDate="2022-03-11T13:53:54.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2692" PostTypeId="2" ParentId="2669" CreationDate="2015-08-23T03:01:08.137" Score="3" Body="&lt;p&gt;The history of eigenvectors is more or less the same as the history of eigenvalues. I recommend looking at &lt;a href=&quot;http://www.steen-frost.org/Steen/Papers/73spectral.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Steen's paper &quot;Highlights in the history of spectral theory&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2019-12-15T03:49:29.243" LastActivityDate="2019-12-15T03:49:29.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2693" PostTypeId="1" AcceptedAnswerId="2700" CreationDate="2015-08-23T11:38:05.623" Score="12" ViewCount="1716" Body="&lt;p&gt;Pi to 7 decimal digits is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;3.1415926&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many people are familiar with &lt;a href=&quot;https://en.wikipedia.org/wiki/Pi_Day&quot;&gt;Pi day&lt;/a&gt;. Celebrated on March 14 as per American date format, the holiday brings attention to the fact that the date resembles the decimal value of the natural constant Pi. This year was notable in particular, as Pi was approximated to four decimal places when written in two-digit year date notation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;3-14-15&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reviewing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pi#Polygon_approximation_era&quot;&gt;history of Pi&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gregorian_calendar&quot;&gt;history of the Gregorian calendar&lt;/a&gt;, it seems that Pi may have been known to sufficient digits to mark &lt;code&gt;3-14-1592&lt;/code&gt; just as the relevant calendar was becoming available. &lt;strong&gt;Might there have been anybody involved with both projects, who may have noticed that the date matched Pi to six decimal places at the time?&lt;/strong&gt; Since the seventh Pi digit is &lt;code&gt;6&lt;/code&gt;, it would round up the previous digit. Therefore, for purposes of this question &lt;code&gt;3-14-1593&lt;/code&gt; could also be a valid date.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, hte premise of this question hinges on the use of the MM-DD-YYYY date format being in use. I cannot find any reference to this, so references supporting or denying usage of this date format would be appreciate.&lt;/p&gt;&#xA;" OwnerUserId="261" LastActivityDate="2017-06-19T18:39:14.343" Title="Did anybody know Pi well enough in 1592 to celebrate Pi day?" Tags="|mathematics|terminology|computation|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2694" PostTypeId="1" AcceptedAnswerId="2695" CreationDate="2015-08-23T15:02:04.230" Score="5" ViewCount="518" Body="&lt;p&gt;I found out that the &lt;strong&gt;symbol&lt;/strong&gt; for union, &amp;cup;, was created in 1895 by Giuseppe Peano in his &lt;em&gt;Formulario Mathematico&lt;/em&gt; but of course the use of &lt;strong&gt;the word &quot;union&quot;&lt;/strong&gt; in mathematics was older. Do you have a source for the earliest occurrence? &lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="81" LastEditDate="2015-08-23T17:58:29.660" LastActivityDate="2023-05-17T14:07:10.503" Title="When was the term &quot;union&quot; first used?" Tags="|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2695" PostTypeId="2" ParentId="2694" CreationDate="2015-08-23T17:34:17.087" Score="5" Body="&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow&quot;&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Intersection and union. The symbols $\cap$ and $\cup$ were used by Giuseppe Peano (1858-1932) for intersection and union in 1888 in &lt;a href=&quot;https://books.google.it/books?id=5LJi3dxLzuwC&quot; rel=&quot;nofollow&quot;&gt;Calcolo geometrico secondo l'&lt;em&gt;Ausdehnungslehre&lt;/em&gt; di H.Grassmann&lt;/a&gt; (Cajori vol. 2, page 298).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See page 2 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;Colla scrittura $A \cup B \cup C \cup \ldots$ intenderemo la minima classe che contiene le classi $A, B, C,\ldots$, ossia la classe formata dagli enti che &#xA;  sono o $A$ o $B$ o $C$, ecc. Il segno $\cup$ si leggerà &lt;em&gt;o&lt;/em&gt;; l'operazione indicata col segno $\cup$ chiamasi in logica &lt;em&gt;disgiunzione&lt;/em&gt;; noi la diremo &#xA;  anche &lt;em&gt;addizione logica&lt;/em&gt;; le classi $A, B,\ldots$ si diranno i &lt;em&gt;termini&lt;/em&gt; della &#xA;  &lt;em&gt;somma&lt;/em&gt; $A \cup B \cup \ldots$ [With the symbol $A \cup B \cup C \cup \ldots$ we mean the least class containing the classes $A, B, C,\ldots$, i.e. the class formed by the entities that are either $A$ or $B$ or $C$, etc. The symbol $\cup$ will be read &quot;&lt;em&gt;or&lt;/em&gt;&quot;; the operation denoted by the symbol $\cup$ is named in logic &lt;em&gt;disjunction&lt;/em&gt;; we will call it also &lt;em&gt;logical sum&lt;/em&gt;; the classes $A, B,\ldots$ will be called &lt;em&gt;terms&lt;/em&gt; of the &lt;em&gt;sum&lt;/em&gt; $A \cup B \cup \ldots$.]&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For the &lt;em&gt;term&lt;/em&gt;, we have to search : W&amp;amp;R's &lt;a href=&quot;https://books.google.it/books?id=ke9yGmFy24sC&amp;amp;pg=PA27&quot; rel=&quot;nofollow&quot;&gt;Principia&lt;/a&gt; is still under Peano's influence; see page 27 : &quot;Similarly the &lt;em&gt;logical sum&lt;/em&gt; of two classes $\alpha$ and $\beta$ ...&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some early occurrences are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Felix Hausdorff, &lt;a href=&quot;https://books.google.it/books?id=yvVIdH16k0YC&amp;amp;pg=PA18&quot; rel=&quot;nofollow&quot;&gt;Set theory&lt;/a&gt; (Engl.transl (1957) of the 3rd German ed. 1937), page 18 : &quot;If $A, B$, are two sets, then by their &lt;em&gt;sum&lt;/em&gt;, or union ...&quot; [but it is necessary to check on the earlier German editions.]&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Waclaw Sierpinski, &lt;a href=&quot;https://books.google.it/books?id=fZFsAAAAMAAJ&quot; rel=&quot;nofollow&quot;&gt;Algèbre des ensembles&lt;/a&gt; (1951), page 62 : &quot;&lt;em&gt;somme&lt;/em&gt; (ou &lt;em&gt;réunion&lt;/em&gt;) des ensembles&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-08-23T18:39:43.703" LastActivityDate="2015-08-23T18:39:43.703" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2696" PostTypeId="2" ParentId="2693" CreationDate="2015-08-23T18:53:40.883" Score="5" Body="&lt;p&gt;François Viète &lt;a href=&quot;https://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80&quot; rel=&quot;noreferrer&quot;&gt;computed&lt;/a&gt; $\pi$ to nine decimal places in 1573. He was &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te#Gregorian_calendar&quot; rel=&quot;noreferrer&quot;&gt;also&lt;/a&gt; intimately aware of the Gregorian calendar, though as a denouncer rather than a supporter of the calculations used to obtain it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(That said, you could just as well celebrate Ultimate Pi Day in the Julian calendar. What's more important is that you were using Anno Domini years, which &lt;a href=&quot;https://en.wikipedia.org/wiki/Anno_Domini#Popularization&quot; rel=&quot;noreferrer&quot;&gt;were widespread&lt;/a&gt; in Europe from the time of the Carolingian Renaissance — six centuries earlier.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know for sure whether the MM-DD-YYYY convention was widespread or even existent in 1592, but it would surprise me very much if it was.&lt;/p&gt;&#xA;" OwnerUserId="2780" LastActivityDate="2015-08-23T18:53:40.883" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2697" PostTypeId="1" CreationDate="2015-08-24T06:52:18.907" Score="-1" ViewCount="745" Body="&lt;p&gt;Was watching &lt;a href=&quot;http://www.sciencechannel.com/tv-shows/dark-matters-twisted-but-true/&quot; rel=&quot;nofollow&quot;&gt;Dark Matters: Twisted but true&lt;/a&gt; the other night. It was about Einstein and how he turned the knowledge of physics up side down and pretty much said that &quot;Newton was wrong&quot;, simplified. Because of this, he was the target of much slander and anti semitic propaganda and he eventually moved to the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Throughout history scientific theories have been presented and either proven or debunked. But what about theories that HAVE been proven and then later new knowledge has proven them wrong? Like Einstein that proved that time is not absolute. And what of our proven theories today? Will there in some near or distant future be other theories that will disprove what we &quot;know&quot; today? Will Einstens E=mc² someday prove to be wrong and open a whole new World for us?&lt;/p&gt;&#xA;" OwnerUserId="2784" LastActivityDate="2015-08-25T17:31:30.800" Title="How many proven theories have been debunked?" Tags="|experimental-physics|philosophy-of-science|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2698" PostTypeId="1" CreationDate="2015-08-24T11:17:29.083" Score="3" ViewCount="106" Body="&lt;p&gt;In  Fisher's paper, he did not include the error term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://psychclassics.yorku.ca/Fisher/Methods/chap5.htm&quot; rel=&quot;nofollow&quot;&gt;http://psychclassics.yorku.ca/Fisher/Methods/chap5.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Durbin  &amp;amp; Watson suggested the error term, and also made the matrix form of the modern version of regression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://scholar.google.com.hk/scholar_url?url=http://ies.fsv.cuni.cz/default/file/download/id/26916&amp;amp;hl=zh-TW&amp;amp;sa=X&amp;amp;scisig=AAGBfm3WtUN0lM-Oyh-uzYyDSLBt0_G54g&amp;amp;nossl=1&amp;amp;oi=scholarr&amp;amp;ved=0CCAQgAMoADAAahUKEwid8bqOio3HAhVDI5QKHfx5CNw&quot; rel=&quot;nofollow&quot;&gt;http://scholar.google.com.hk/scholar_url?url=http://ies.fsv.cuni.cz/default/file/download/id/26916&amp;amp;hl=zh-TW&amp;amp;sa=X&amp;amp;scisig=AAGBfm3WtUN0lM-Oyh-uzYyDSLBt0_G54g&amp;amp;nossl=1&amp;amp;oi=scholarr&amp;amp;ved=0CCAQgAMoADAAahUKEwid8bqOio3HAhVDI5QKHfx5CNw&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I found this sentence in Fisher's paper: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If errors occur in the heights, this will not influence the regression of height on age, provided that at all ages positive and negative errors are equally frequent, so that they balance in the averages. &quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means Fisher included the error term implicitly and take the average of the equation. So, the error term on average is zero in his paper. Durbin &amp;amp; Watson suggested the error term explicitly.&lt;/p&gt;&#xA;" OwnerUserId="2605" LastEditorUserId="2605" LastEditDate="2015-10-13T01:22:28.617" LastActivityDate="2015-10-13T01:22:28.617" Title="Why do we see the modern version of regression as &quot;Fisher's regression&quot;?" Tags="|statistics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2699" PostTypeId="1" AcceptedAnswerId="2703" CreationDate="2015-08-24T16:56:44.677" Score="10" ViewCount="375" Body="&lt;p&gt;The thing with mathematics is that on one if you define something, you are completely free in choosing any name you want, and on the other hand you should find a meaningful name that evokes some intuitive ideas that coincide with what you actually wanted to do. So long story short, one thing that always bothered me is the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What is the origin of the term chain? When was it first used and what explanation (if any) was given?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be a bit more precise, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Chain_%28algebraic_topology%29&quot;&gt;chain&lt;/a&gt; I mean a finite linear combination of (oriented) simplices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked into Dieudonné's &quot;A History of Algebraic and Differential Topology&quot; but there he only points out in a footnote that the term was coined by J.W. Alexander without giving any source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe related to this is the notion of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Chain_complex&quot;&gt;chain complex&lt;/a&gt;, which I think at least once someone explained to me as being a chain of arrows, eventhough these arrows are actually the things mapping chains to chains. This is of course not helped by the fact that some useful abstractions of a chain are called complexes (as in CW or simplicial). I guess that the term chain complex was coined some time afterwards, but in some cases concepts are found and given a name much later, so maybe there is a chance they are related.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally I may need to add that my background is in mathematical analysis, my primary interest in chains is integrating something along them. I am not very well versed in abstract algebra, so there may be some simple source that I missed. However I am more interested in the basic geometrical objects not in the abstract generalisations. (only there is no algebraic geometry tag...)&lt;/p&gt;&#xA;" OwnerUserId="2785" LastActivityDate="2015-08-25T18:12:57.663" Title="On the notion of a chain (as for example in chain complex)" Tags="|mathematics|abstract-algebra|geometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2700" PostTypeId="2" ParentId="2693" CreationDate="2015-08-24T18:47:47.790" Score="19" Body="&lt;p&gt;There could not have been a $\pi$ day in 1592 regardless of calendar conventions for the simple reason that there was no such thing as $\pi$ back then. The &lt;a href=&quot;http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi&quot; rel=&quot;nofollow noreferrer&quot;&gt;symbol was introduced by William Jones in 1706&lt;/a&gt; and did not come into common usage until after 1737, when Euler popularized it in his texts. This was &lt;a href=&quot;http://www.math.harvard.edu/~engelwar/MathE300/A%20history%20of%20Zero.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;similar to zero&lt;/a&gt;, which got a placeholder symbol long before it was recognized as a number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, they could have celebrated the number without the name, right? Still no, because &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Gregory_%28mathematician%29#Vera_Circuli_et_Hyperbolae_Quadratura&quot; rel=&quot;nofollow noreferrer&quot;&gt;$\pi$ and $e$ were first acknowledged as &quot;some kind of numbers&quot;&lt;/a&gt; in print by James Gregory in The True Squaring of the Circle and of the Hyperbola published in 1667, and even Jones wrote in 1706 that &quot;&lt;em&gt;the exact proportion between the diameter and the circumference can never be expressed in numbers&lt;/em&gt;&quot;. Even irrationals like $1+\sqrt{5}$ or $\sqrt[3]{2}$ were treated with gloves at the time and called &quot;deafmute&quot;, and for them one could at least write formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what was $\pi$ in 1592, what was Viète approximating? As Jones said, and as Euclid defined in the Elements, it was a proportion or a ratio of circumference to diameter, and &quot;approximating&quot; is a modernization of what Viète, and Archimedes before him, were doing. According to Euclid, a ratio is a &quot;relation&quot; of magnitudes or numbers, and ratios can be compared to other ratios using an ingenious procedure invented by Eudoxus and described in the Elements. The inequalities with bounds like $3/1 &amp;lt; \pi &amp;lt; 22/7$ were understood as such comparisons rather than &quot;approximations&quot; of a non-existent &quot;number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, as can be seen from Viète's approach, he was relying on geometry and expressed the bounds in terms of radicals (Archimedes did in terms of integer ratios), so even if something so abstract as a relation between two geometric magnitudes got to be celebrated, decimal digits in the bounds would not have stood out for pointing to the occasion. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-06-19T18:39:14.343" LastActivityDate="2017-06-19T18:39:14.343" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2701" PostTypeId="2" ParentId="2697" CreationDate="2015-08-24T19:17:27.443" Score="4" Body="&lt;p&gt;Einstein did not say that Newtonian mechanics was wrong, special and general relativity reaffirm it within its domain of application, he showed that this domain is restricted, namely Newtonian mechanics does not apply to objects moving at speeds close to the speed of light, or in strong gravitational fields. Absolute space and time were part of Newton's philosophical interpretation of his theory rather than part of the theory itself. As for anti-semitism it was common in Europe for centuries, and especially prominent at the time due to the rise of Nazism, so Einstein's science was less important than his race.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can some elements of modern scientific theories be eventually abandoned? Certainly, especially interpretational parts, even big ones. For example, consider a common belief of late 19th century as expressed by &lt;a href=&quot;http://press.princeton.edu/chapters/s6683.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michelson in 1902&lt;/a&gt;:&quot;&lt;em&gt;The day seems not far distant when the converging lines from many apparently remote regions of thought will meet... Then the nature of the atoms, and the forces called into play in their chemical union... the explanation of cohesion, elasticity, and gravitation — all these will be marshaled into a single compact and consistent body of scientific knowledge... one of the grandest generalizations of modern science ... that all the phenomena of the physical universe are only different manifestations of the various modes of motion of one all-pervading substance — the ether.&lt;/em&gt;&quot; Ironically, it was Michelson's own experiment that eventually led Einstein to reject ether, and the absolute space it embodied. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Maxwell's electrodynamics, which was originally phrased in terms of ether, was not only retained but became a foundation for special relativity. Because empirical evidence which it proved to describe and organize did not change, only its interpretation did. The same goes for proven modern theories. We may discover that they do not apply universally (in fact, we already know that general relativity and quantum mechanics do not), or choose to revise some of their concepts, but they will still describe the phenomena that they proved to describe. This goes for E=mc² just as for Newtonian mechanics and Maxwell's electrodynamics. How it manifests itself in gravitational bending of light or splitting of the atom will remain true in whatever theory that replaces Einstein's, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2066/early-empirical-tests-of-e-mc2/2072#2072&quot;&gt;What were the early empirical tests of Einstein&amp;#39;s mass-energy equivalence $E=mc^2$?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-08-25T17:31:30.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2702" PostTypeId="2" ParentId="2697" CreationDate="2015-08-25T00:32:00.943" Score="2" Body="&lt;p&gt;To this simply stated question there is a simply stated answer: None.&#xA;If the theory is proven, then it is correct. So it cannot be &quot;debunked&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of concrete example of Newtonian mechanics, it has never been &quot;debunked&quot;.&#xA;It is a proven and CORRECT theory, which nowadays is taught in schools and universities, which would not be the case if it was not correct.&#xA;Engineers base their designs on Newton's mechanics, and the machines designed by them work. This is the ultimate proof. It was not, and will never be &quot;debunked&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell theory has nothing to do with aether. Maxwell's theory is correct, (and proven), aether was always a hypothesis its existence was never proven, (and now&#xA;it is &quot;debunked&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You should apply some skepticism and common sense when you listen to journalists&#xA;and movie makers (and philosophers) who speak on the matters thay do not understand themselves.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-08-25T00:32:00.943" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2703" PostTypeId="2" ParentId="2699" CreationDate="2015-08-25T02:32:25.843" Score="6" Body="&lt;p&gt;One learns quite a bit by searching I. M. James' &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1674906&quot;&gt;&lt;em&gt;History of Topology&lt;/em&gt; (1999)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The name:&lt;/strong&gt; &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1674924&quot;&gt;Lefschetz (p. 546)&lt;/a&gt; confirms that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The felicitous term &quot;chain&quot; is due to Alexander&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;while &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1674911&quot;&gt;Katz (p. 119)&lt;/a&gt; specifies that it was coined in the paper &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1501346&quot;&gt;&lt;em&gt;Combinatorial analysis situs&lt;/em&gt; (1926)&lt;/a&gt;. &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1721123&quot;&gt;Weibel (p. 801)&lt;/a&gt; adds that &quot;chain complex&quot; was introduced by Mayer in &lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?55.0963.02&quot;&gt;&lt;em&gt;Über abstrakte Topologie&lt;/em&gt; (1929)&lt;/a&gt;. Finally &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1674909&quot;&gt;Vanden Eynde (p. 97)&lt;/a&gt; points to the earlier use of chains (&quot;Ketten&quot;) with a different meaning by Brouwer (following Cantor) in &lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?43.0569.03&quot;&gt;&lt;em&gt;Beweis der Invarianz der geschlossenen Kurve&lt;/em&gt; (1912)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The thing:&lt;/strong&gt; &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1674912&quot;&gt;Sarkaria (p. 141)&lt;/a&gt; traces it to Poincaré's &lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?26.0541.07&quot;&gt;&lt;em&gt;Analysis Situs&lt;/em&gt; (1895)&lt;/a&gt; where it is among the four(!) different things called &quot;variety&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In § 4 &quot;varieties&quot; are oriented, and then in §§ 5 and 6 Poincaré considers integral or rational linear combinations of the oriented varieties of an $M$ to define its homologies and Betti numbers: in this context it is best to think of his &quot;varieties&quot; as &lt;strong&gt;smooth chains&lt;/strong&gt; of $M$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(As @CatherineRay notes, Poincaré's paper ironically also used the term &quot;(connected) chain&quot; to mean something altogether different.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2015-08-25T02:40:31.953" LastActivityDate="2015-08-25T02:40:31.953" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2704" PostTypeId="1" AcceptedAnswerId="2705" CreationDate="2015-08-25T09:48:54.923" Score="7" ViewCount="390" Body="&lt;p&gt;Various sources, such as Cicero's Republic state that Archimedes had made a machine consisting of glass spheres that represented the Eudoxian system of the world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering that Callipus died over a decade before Archimedes was born, would Archimedes have been aware of Callipus' work and therefore would his mechanism use the full 34 spheres needed to account for planetary movement as Callipus suggested instead of the 27 used by Eudoxus?&lt;/p&gt;&#xA;" OwnerUserId="2702" LastEditorUserId="4251" LastEditDate="2020-05-14T03:16:44.333" LastActivityDate="2020-05-14T03:16:44.333" Title="Did Archimedes know about Callipus?" Tags="|astronomy|ancient-greece|archaeoastronomy|archimedes|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2705" PostTypeId="2" ParentId="2704" CreationDate="2015-08-25T12:23:35.177" Score="5" Body="&lt;p&gt;There is no definite answer on this question. About the Archimedes machine we only know a very imprecise description from the authors who did not understand much about this machine, or about science in general. What Archimedes knew and what he did not we cannot tell for sure. Probably he knew about Callipus. Probably he visited Alexandria. Alexandria at that time had a library, and one may suppose it contained all important writings on mathematics and astronomy. There are no reasons to assume that Callipus writings were missing since he lived recently....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is no firm ground for all these speculations. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-08-25T12:23:35.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2706" PostTypeId="2" ParentId="2699" CreationDate="2015-08-25T17:47:39.403" Score="4" Body="&lt;p&gt;I believe the term chain was introduced by Poincare as part of his second definition of the term manifold; the definition I am referring to is on pg. 24-25 of &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stillwell's wonderful translation of Analysis Situs&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;We then say that the two manifolds V and V ′ are analytic&#xA;  continuations of each other. In this way we can form a chain of&#xA;  manifolds V1, V2, ..., Vn such that each is an analytic continuation&#xA;  of its predecessor, and there is a common part between any two&#xA;  consecutive manifolds of the chain. I shall call this a connected&#xA;  chain.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Poincare goes on and mentions something akin to a combination of oriented simplices: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;...There are, in fact, manifolds (and we shall see examples later)&#xA;  which can be decomposed into a certain number of partial manifolds&#xA;  forming a connected chain or network and such that each of them can be&#xA;  defined by equations of the form (8)...&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[(8) references that a collection of equations of the form&#xA;$x_1= \theta_1(y_1,...y_m) ; x_2=\theta_2(y_1,...y_m) ;...; x_n=\theta_n(y_1,...y_m) $ represents a manifold of m dimensions (if the $y$ are considered as independent variables).]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note, Alexander's definition of the term chain (page 301 of Combinatorial Analysis Situs) cites Poincare's formula for describing the cell structure of a manifold in terms of an incidence matrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iX62j.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iX62j.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;In his first long paper on topology Poincare (1895) continued to&#xA;  explore the Betti numbers. He set up machinery for computing them by&#xA;  assuming that each manifold has a decomposition into cells&#xA;  homeomorphic to simplices, reading off linear equations he called&#xA;  homologies, and computing the Betti numbers by linear algebra.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-- &lt;a href=&quot;http://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01385-X/S0273-0979-2012-01385-X.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical perspectives&lt;/a&gt;, Stillwell&lt;/p&gt;&#xA;" OwnerUserId="2787" LastEditorUserId="2787" LastEditDate="2015-08-25T18:12:57.663" LastActivityDate="2015-08-25T18:12:57.663" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2707" PostTypeId="2" ParentId="2704" CreationDate="2015-08-26T01:07:07.450" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.academia.edu/3051043/Epicurus_and_the_mathematicians_of_Cyzicus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sedley's Epicurus and the mathematicians of Cyzicus&lt;/a&gt;, &quot;&lt;em&gt;Archimedes's celebrated planetarium was said to reproduce all the planetary motions simultaneously, but whether it followed a system of concentric spheres or that of eccentrics or epicycles is unknown. Our text... suggests that comparable, though undoubtedly much simpler, mechanisms existed a century earlier&lt;/em&gt;&quot;. The text comes from writings of Epicureans recovered at Herculanum about a school of astronomy founded by Eudoxus at Cyzicus. Calippus &quot;&lt;em&gt;while at Cyzicus encountered the mechanical difficulties  involved in constructing a unified working model of all the concentric spheres, and was familiar with whatever solutions had been attempted. The insertion of counteracting spheres was probably one of these solutions... Cyzicenes may have limited each model to depicting just one or two orbits.&lt;/em&gt;&quot; Aristotle already describes the model completed with counterspheres in Metaphysics as if it was reality, and even Epicureans, their philosophical opponents, were aware of what Cyzicenes were up to. Eudoxian sphere building, as it was called, and Calippus's innovations were quite well known, and Eudoxians in general had a strong influence on Alexandrian geometry, as can be seen from Euclid's Elements. It is hard to imagine that Archimedes, who corresponded with Alexandrians and wrote his own book on sphere building missed it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Archimedes apparently made a big step forward, his planetarium is described as combining motions of all seven planets and powered by a single source, so if he used Calippean spheres he would have to insert counterspheres that cancel planet specific motions when the motion is transmitted from outer to inner spheres. That would increase the total to at least 47 spheres, if not all 54 Aristotle had, and make the device rather unwieldy. But Cyzicene planetaria were also quite schematic, Epicurus criticized them for failing to reduce &quot;&lt;em&gt;irregularities of sun's appearence&lt;/em&gt;&quot; to regularity. If Archimedes's device, like its predecessors, was meant to demonstrate the motion of the planets only qualitatively there would be no need for extra Calippean spheres. Eudoxus's original model could already do that, including even the retrograde loops, and Calippean additions did not make it quantitatively feasible anyway. For additional discussion see &lt;a href=&quot;https://hsm.stackexchange.com/questions/505/did-archimedes-use-epicycles-in-his-planetarium&quot;&gt;Did Archimedes use epicycles in his planetarium?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-08-26T02:50:55.353" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2708" PostTypeId="1" CreationDate="2015-08-26T23:28:16.790" Score="5" ViewCount="191" Body="&lt;p&gt;In discussing Lie derivatives in his book Advanced Combinatorics (Reidel, 1974, &lt;a href=&quot;https://books.google.com/books?id=PJzuCAAAQBAJ&amp;amp;pg=PA220&amp;amp;lpg=PA220&amp;amp;dq=comtet%20faa%20di%20bruno%20pourchet&amp;amp;source=bl&amp;amp;ots=FaagQD6xv-&amp;amp;sig=RJ2zyaXsh0TNaED2kEGgPrqWxo8&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CDIQ6AEwAmoVChMIgJney7bHxwIV0jWICh0fMQbg#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;pg. 220&lt;/a&gt;), Comtet refers to a formula of Pourchet related to the celebrated Faa di Bruno formula. Other references to Pourchet's formula that I've found always refer back to Comtet. I suspect that the actual author was Eugene &lt;a href=&quot;http://sites.mathdoc.fr/cgi-bin/rbsm?au=PROUHET&quot; rel=&quot;nofollow&quot;&gt;Prouhet&lt;/a&gt;, but I have no access to his publications in the 1800s nor Lukacs &lt;a href=&quot;http://www.jstor.org/stable/2307040?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;paper&lt;/a&gt; on the FdB formula that might reference Prouhet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Edit 8/29/15 to clarify my interest: To whom was Comtet actually referring, i.e., to whom was Comtet attempting to attribute the formula in his publications? Ideally, I would like confirmation of the formula in a publication--primary source--by Prouhet or Pourchet.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know to whom Comtet was actually referring; i.e., is anyone aware of the primary source for Comtet's attribution? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Edit after Conifold's initial answer: Note that I am not interested in the Faa di Bruno formula per se, just the primary source of Comtet's attribution.)&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2015-08-29T21:52:08.127" LastActivityDate="2015-08-31T17:40:05.373" Title="Primary reference for a formula related to the Faa di Bruno formula" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2709" PostTypeId="2" ParentId="2708" CreationDate="2015-08-27T01:58:41.080" Score="1" Body="&lt;p&gt;Some preliminary remarks. Lukacs's paper does not refer to or mention Pourchet,  Prouhet, or any similarly spelled name. Johnson gives a detailed &lt;a href=&quot;http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Johnson217-234.pdf&quot; rel=&quot;nofollow&quot;&gt;history of Faa di Bruno's formula&lt;/a&gt;. He does not mention such a name either, but comments &quot;&lt;em&gt;one is struck by the obscurity of many of the names in our story... It would be nice to have more biographical information about some of them...&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a feeling that the only way to confirm would be to find Comtet's formula in Prouhet's papers. &lt;a href=&quot;http://www.plouffe.fr/simon/math/AdvancedComb.pdf&quot; rel=&quot;nofollow&quot;&gt;Comtet's book (p.331)&lt;/a&gt; references one Prouhet's paper in particular, in Nouvelles Annales M[athématiques], 5 (1866) 384, and nothing by &quot;Pourchet&quot;. Unfortunately, &lt;a href=&quot;http://www.numdam.org/numdam-bin/browse?id=NAM_1866_2_5_&quot; rel=&quot;nofollow&quot;&gt;Numdam shows no Prouhet's paper&lt;/a&gt; in 1866 Nouvelles Annales, on p.384 or elsewhere, and the &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/NAM/NAM_1867_2_6_/NAM_1867_2_6__337_0/NAM_1867_2_6__337_0.pdf&quot; rel=&quot;nofollow&quot;&gt;one from 1867&lt;/a&gt; has nothing of the sort in it. Then again Numdam does not show his 1864 and 1865 papers, which appear on &lt;a href=&quot;http://sites.mathdoc.fr/cgi-bin/rbsm?au=Prouhet&quot; rel=&quot;nofollow&quot;&gt;Gallica-Math list&lt;/a&gt;. Both agree on nothing from Prouhet in 1866 though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if it is there he may not have been the first to derive the formula. Faa di Bruno published 'his' formula in December of 1855, but &quot;&lt;em&gt;that several other mathematicians found different expressions for the m-th derivative of $g(f(t))$ in the 19th century has been forgotten; these are all independent of Faa di Bruno's work and a few of them predate it. Most of all, Faa di Bruno was neither the first to state the formula that bears his name, nor the first to prove it... Other mathematical accomplishments... have caused him to receive more credit for the higher chain rule than he deserves...&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first proof appeared in volume 3 of Lacroix's Traite du Calcul, &quot;&lt;em&gt;a snapshot (or perhaps a mural) of 18th century analysis&lt;/em&gt;&quot;, 2nd edition only, which came out in 1819. His source appears to be 1800 Du Calcul des De'rivations of Arbogast, who &quot;&lt;em&gt;Arbogast gives a prose rule for writing down the general case, and on p. 313 he illustrates his rule by writing down the case n = 6 again, twice, followed by n = 7. Although he never gives a general formula, nor a proof of his rule... there is a formula for the coefficient of $x^m$ in $(\zeta + \gamma x + \delta x^2 + \varepsilon x^3 + \text{etc.})^n$. Thus Arbogast had most of the ingredients of Lacroix's argument at hand, but he seems never to have written down Faa di Bruno's formula as such.&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-08-31T17:40:05.373" LastActivityDate="2015-08-31T17:40:05.373" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2711" PostTypeId="1" AcceptedAnswerId="2712" CreationDate="2015-08-27T20:41:51.123" Score="10" ViewCount="595" Body="&lt;p&gt;J. Alexander's 1926 paper, Combinatorial Analysis Situs, doesn't refer to the field as combinatorial topology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He mentions that combinatorial analysis situs is concerned with topological invariants and topological constants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, he refers to the subject of the paper itself as a subfield of analysis situs, considered as a branch of geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When/why did we transition from calling &quot;the study of topological invariants&quot; by the name &quot;analysis situs&quot; to calling it by the name &quot;topology&quot;?&lt;/p&gt;&#xA;" OwnerUserId="2787" LastEditorUserId="4251" LastEditDate="2020-07-06T15:39:22.033" LastActivityDate="2020-07-06T15:39:22.033" Title="What caused the name change from &quot;analysis situs&quot; to &quot;topology&quot;?" Tags="|mathematics|terminology|topology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2712" PostTypeId="2" ParentId="2711" CreationDate="2015-08-27T22:10:43.420" Score="7" Body="&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Listing.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to MacTutor&lt;/a&gt;, &quot;the subject was known as analysis situs for many years and only in the late 1920s was the English word topology used by Lefschetz&quot;. Lefschetz's 1924 work is titled Analysis Situs and Algebraic Geometry, but his influential 1930 monograph is already Topology, which he defines as &quot;&lt;em&gt;doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity..&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Why&quot; is speculative. From an &lt;a href=&quot;https://math.stackexchange.com/questions/70634/why-algebraic-topology-is-also-called-combinatorial-topology&quot;&gt;answer on MSE&lt;/a&gt;:&quot;&lt;em&gt;Lefshetz&lt;/em&gt; [Topology, Amer. Math. Soc. Colloq. Publ 12 (1930), page 361] &lt;em&gt;wrote that Poincaré had tried to develop the subject along `analytic' lines, but had turned instead to combinatorial methods because the analytic approach failed for example in the Poincaré duality theorem&lt;/em&gt;&quot;. Since the focus shifted from analytic to combinatorial methods &quot;analysis&quot; was not a fitting name. Listing published a book Vorstudien zur Topologie in 1847 and used the word since 1836. According to his 1883 obituary it was meant to  distinguish &quot;...&lt;em&gt;qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated&lt;/em&gt;&quot;, which clearly resonates with Lefschetz's description. So it was a natural choice to make a point. Later, a similar change in perspective was reflected in the switch from &quot;combinatorial topology&quot; to &quot;algebraic topology&quot;,  &lt;a href=&quot;https://en.wikipedia.org/wiki/Combinatorial_topology&quot; rel=&quot;nofollow noreferrer&quot;&gt;attributed to Hopf, under Noether's influence, Mayer and Vietoris&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-08-27T22:10:43.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2713" PostTypeId="1" CreationDate="2015-08-28T02:21:48.253" Score="5" ViewCount="2363" Body="&lt;p&gt;The surname of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Ising&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ernst Ising&lt;/a&gt;, of Ising model fame, is often pronounced as EYE-zing by non-German speakers, but as EE-zing by German speakers. It is frequently &lt;a href=&quot;http://backreaction.blogspot.kr/2008/01/germenglish.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;claimed&lt;/a&gt; that EE-zing is the correct pronunciation, as Ising was German. However, he emigrated to the US and lived there for over half his (long) life. It therefore seems possible that in later life he pronounced his name in an Anglicised style as EYE-zing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reasonable point of view is that the word &quot;Ising&quot; in &quot;Ising model&quot; should be pronounced in the same way as Ising himself would have done. So my question is: &lt;b&gt;how did Ising pronounce his own name?&lt;/b&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although this is a trivial point, given the number of people working on the Ising model worldwide, it would be nice to have some evidence for standardising the pronunciation of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Originally posted on &lt;a href=&quot;https://physics.stackexchange.com/questions/202435/how-did-ising-pronounce-his-name&quot;&gt;physics.SE&lt;/a&gt;, where it was suggested the question should be moved here.)&lt;/p&gt;&#xA;" OwnerUserId="2796" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2015-08-28T08:30:51.987" Title="How did Ising pronounce his name?" Tags="|physics|biographical-details|" AnswerCount="0" CommentCount="6" ClosedDate="2015-08-28T08:31:08.990" ContentLicense="CC BY-SA 3.0" />
  <row Id="2714" PostTypeId="2" ParentId="762" CreationDate="2015-08-28T16:07:22.673" Score="5" Body="&lt;p&gt;This is not a complete answer, but the chain rule apparently was not even stated explicitly until 1797, by Lagrange. So says &lt;a href=&quot;http://www.math.umt.edu/tmme/vol7no2and3/9_RodriguezFernandez_TMMEvol7nos2and3_pp.321_332.pdf&quot; rel=&quot;noreferrer&quot;&gt;this reference&lt;/a&gt; by Rodríguez &amp;amp; Fernández. Footnote 5 in the paper reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As far as we can tell, the first “modern” version of the chain rule appears in Lagrange’s 1797 &lt;em&gt;Théorie des fonctions&#xA;  analytiques&lt;/em&gt;, (Lagrange, J. L., 1797, §31, pp. 29); it also appears in Cauchy’s 1823 &lt;em&gt;Résumé des Leçons données a&#xA;  L’École Royale Polytechnique sur Le Calcul Infinitesimal&lt;/em&gt;, (Cauchy, A. L., 1899, Troisième Leçon, pp. 25).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This footnote appears in section 2 of the paper, titled &quot;History of the Chain Rule&quot;. According to this section, the chain rule is nowhere explicitly stated in Euler's books on analysis, nor even the notion of a composite function. (Wikipedia agrees with this, but their source seems to be the paper just mentioned.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chain rule appears &lt;em&gt;implicitly&lt;/em&gt; in a memoir by Leibniz in 1676 (according to these authors, who cite &lt;em&gt;The Early Mathematical Manuscripts of Leibniz&lt;/em&gt;, translated by J.M. Child). The idea seems to be the free use of differentials, presumably something like this computation:&#xA;$$&#xA;  d\sqrt{a+bz+cz^2}=\frac{b+2cz}{2\sqrt{a+bz+cz^2}}dz&#xA;$$&#xA;Differentials are treated by Leibniz as infinitesimal differences. In L’Hospital's 1696 textbook &lt;em&gt;Analyse des infiniment petits&lt;/em&gt;, the rule $dx^r=rx^{r-1}dx$ is given (our authors even use the word &quot;proved&quot;, though they don't say how). L'Hospital then uses it pretty much the way a modern textbook would use the chain rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short, the free use of Leibnizian differentials can serve the same purpose as the chain rule.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2015-08-28T16:07:22.673" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2715" PostTypeId="1" CreationDate="2015-08-30T09:24:26.617" Score="6" ViewCount="2782" Body="&lt;p&gt;It was to my understanding that the constant $e$ came about as a result of simplifying the differentiation of an exponential. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the derivative of $2^x$ is $2^x \cdot \ln 2$; for $3^x$ it's $3^x \cdot \ln 3$ etc. and essentially a number was required such that the natural logarithm at the end would equal to 1. Such a number would be called $e$. At least, that's how I thought it originated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I didn't realize that the expression $\ln$ required $e$ as a base. Furthermore, to differentiate exponentials like $2^x$ and $3^x$ you need to have some knowledge that $\frac {d(e^{cx})} {dx} = ce^{cx}$ (as you bring $2^x$ in terms of $e$), and bringing it in terms of any other base like $10$, would essentially be putting you back to square one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So essentially, it seems like $e$  goes back much further than I thought. &#xA;I was just wondering on where the idea of natural logarithms came to be and why, and whether the knowledge of the derivative of $e$ dates back further than the derivative of other exponents.&lt;/p&gt;&#xA;" OwnerUserId="2806" LastEditorUserId="55" LastEditDate="2020-05-01T06:15:56.177" LastActivityDate="2020-05-01T06:15:56.177" Title="What are the uses and the origin of the constant $e$?" Tags="|mathematics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2716" PostTypeId="1" CreationDate="2015-08-30T14:55:22.503" Score="2" ViewCount="58" Body="&lt;p&gt;In &lt;a href=&quot;http://geo.mff.cuni.cz/vyuka/Martinec-ContinuumMechanics.pdf&quot; rel=&quot;nofollow&quot;&gt;these lecture notes&lt;/a&gt;, I read about two different ways of looking at the position of a particle $\mathcal{X}$ in some body $\mathcal{B}$, &lt;a href=&quot;http://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field&quot; rel=&quot;nofollow&quot;&gt;Lagrangian and Eulerian coordinates&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Lagrangian coordinates, the particle is identified by an ordered triple based on the reference configuration $\kappa_0$. In Eulerian coordinates, the particle is identified by an ordered triple based on the present configuration $\kappa_t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While both systems can be used concurrently, and are connected by some useful identities, I'm curious as to which one was used first.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-08-30T15:00:35.293" LastActivityDate="2015-08-30T15:00:35.293" Title="Were Lagrangian coordinates or Eulerian coordinates used first in continuum mechanics?" Tags="|physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2717" PostTypeId="2" ParentId="2715" CreationDate="2015-08-30T20:48:49.677" Score="4" Body="&lt;p&gt;Constant $e$ (but not the name) appeared long before any differentiation or exponential function were invented. It probably appears for the first time in the work on Napier on the logarithms. His logarithms are (equivalent to, more or less)&#xA; the natural logarithms, and the old name of the constant was&#xA;the &quot;base of the natural logarithms&quot;. For practical use, natural logarithms were soon displaced with logarithms based on $10$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formal definition, and modern notation of the constant and of the function $e^x$&#xA;are due to Euler, and it was called sometimes &quot;Euler's constant&quot;, but now this usage is out of date because there is another famous constant which is called&#xA;the &quot;Euler (Masceroni) constant&quot; and denoted by $\gamma$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The constant $e$ was discovered because it is indeed the base of &quot;natural&quot; logarithms.&#xA;You may look here &lt;a href=&quot;https://hsm.stackexchange.com/questions/42/which-came-first-the-natural-logarithm-or-the-base-of-the-natural-logarithm/47#47&quot;&gt;Which came first, the natural logarithm or the base of the natural logarithm?&lt;/a&gt; for more detail on&#xA;Napier and natural logarithms.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-08-30T20:55:10.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2718" PostTypeId="1" CreationDate="2015-08-30T21:54:10.423" Score="2" ViewCount="258" Body="&lt;p&gt;I'm looking for problems about quadratic function across the ages. For example, in the Babylonian civilization, there are problems which are related with quadratic equation. Besides that, the concept of function was developed through relation between numbers. On the other hand, in Greek Culture the problems were focusing in geometrical interpretation for solving quadratic equation. Al-Khwarizmi, in the Islamic Culture, was the most important scholar because he posed a formula to solve quadratic equation. I've been reading some Al-Khwarizmi's problems and his solution , which could be solved through algebraic ideas.&lt;/p&gt;&#xA;&lt;p&gt;My principal references are&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Swetz, F. J. (2012). Mathematical expeditions: Exploring word problems across the ages. JHU Press.&lt;/p&gt;&#xA;&lt;p&gt;Kline, M. (1990). Mathematical thought from ancient to modern times (Vol.1, 2 y 3). Oxford University Press.&lt;/p&gt;&#xA;&lt;p&gt;Boyer, C. B., &amp;amp; Merzbach, U. C. (2011). A history of mathematics. John Wiley &amp;amp; Sons.&lt;/p&gt;&#xA;&lt;p&gt;Irving, R. (2013). Beyond the quadratic formula. Washington, D.C.: Mathematical Association of America.&lt;/p&gt;&#xA;&lt;p&gt;Bashmakova, I. G., &amp;amp; Smirnova, G. S. (2000). The beginnings and evolution of algebra (No. 23). Cambridge University Press.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My focus is to study problems posed about quadratic function in some scenes of history of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Please, could you help me in this endeavour?&lt;/p&gt;&#xA;" OwnerUserId="2412" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-09-05T15:02:25.457" Title="Exploring problems about quadratic function (in one variable) across the ages" Tags="|mathematics|calculus|elementary-algebra|middle-ages|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2719" PostTypeId="2" ParentId="2718" CreationDate="2015-08-30T23:39:23.023" Score="0" Body="&lt;p&gt;Toomer’s excellent article ‘al-Khwārizmī’ in the Complete Dictionary of Scientific Biography offers a rather more cautious evaluation of al-Khwārizmī’s contribution to the history of algebra. He shows in particular that “both Greek and Hindu algebra had advanced well beyond the elementary stage of al-Khwārizmī’s work”.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-08-30T23:39:23.023" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2720" PostTypeId="2" ParentId="2718" CreationDate="2015-08-31T01:47:30.433" Score="1" Body="&lt;p&gt;If you read the papers under with keywords &quot;Holomorphic dynamics&quot;, &quot;Mandelbrot set&quot;, and MLC conjecture, you discover that quadratic functions are still a hot research topic.&#xA;A. Douady and J. Hubbard recalled in the late 90s, that when in the early 80-s&#xA;they were asked &quot;what problems are you working on now ?&quot;, they replied &quot;we study quadratic polynomials (of one variable)&quot;. A typical reaction was: &quot;Do you hope to find something new about them???&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since then, two Fields medals were awarded for results on quadratic polynomials:-) So you do not have to restrict yourself to Babylonian mathematics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-08-31T01:47:30.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2721" PostTypeId="2" ParentId="2715" CreationDate="2015-09-01T03:28:11.443" Score="3" Body="&lt;p&gt;Number $e$ had several &quot;near discoveries&quot; by Napier (1614), St. Vincent (1647) and Huygens (1661) in connection with natural logarithms before it was finally explicitly introduced... without any connection to them. Napier's was led to (natural) logarithms by considering motion of a point, whose velocity is equal to its distance to the origin (in modern notation $dx/dt=x$). St. Vincent considered areas under a rectangular hyperbola ($y=1/x$), which as Huygens observed later have the same multiplicative property as logarithms. Huygens even introduced general exponential function $ka^x$, which he called &quot;logarithmic&quot;, and calculated what we call &quot;logarithm to base 10 of $e$&quot; to 17 decimal places. But he did not &quot;see&quot; $e$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/e.html&quot; rel=&quot;nofollow&quot;&gt;MacTutor describes&lt;/a&gt; it, &quot;&lt;em&gt;in 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of $(1 + 1/n)^n$ as n tends to infinity. He used the binomial theorem to show that the limit had to lie between 2 and 3...  if we accept this as a definition of e, it is the first time that a number was defined by a limiting process.&lt;/em&gt;&quot; It was also Jacob Bernoulli who started treating logarithms as functions, rather than numerical aids, and realized that they were the inverses of exponential functions (at the level of numbers Gregory might have known it before 1683). The first notation for $e$ appears in Leibniz's 1690 letter to Huygens, and it was $b$. Thus, $e$ got a proper name 16 years earlier than $\pi$ when Jones christened it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus for exponential functions first appears in Johann Bernoulli's (Jacob's brother) 1697 Principia Calculi Exponentialium seu Percurrentium, only then derivatives come into play. But only Euler pulled logarithmic, area under hyperbola, compound interest, exponential function, approximation by series, etc. threads together into a systematic treatment in his He also gave $e$ its modern name in a letter to Goldbach in 1731, not because of his own name, and possibly not even because of &quot;exponential&quot;, but because the preceding vowel $a$ was already taken. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more on Napier and St. Vincent see Fauvel's and Katz's articles in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0883857030&quot; rel=&quot;nofollow&quot;&gt;Learn from the Masters&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-01T03:48:49.520" LastActivityDate="2015-09-01T03:48:49.520" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2722" PostTypeId="1" CreationDate="2015-09-01T12:42:32.237" Score="5" ViewCount="1842" Body="&lt;p&gt;The 18th century had sought a huge, immense progress in mathematics. As late as 17th century people still wrote algebraic equations with words. But by the end of 18th century we had&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mathematical analysis and theory of differential equations&lt;/li&gt;&#xA;&lt;li&gt;Complex numbers fully formalized and the connection between trigonometric and hyperbolic functions discovered&lt;/li&gt;&#xA;&lt;li&gt;Gamma and Zeta functions discovered and described&lt;/li&gt;&#xA;&lt;li&gt;Calculus of variations&lt;/li&gt;&#xA;&lt;li&gt;Theory of graphs&lt;/li&gt;&#xA;&lt;li&gt;Probability theory and combinatorics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;etc etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that the most of mathematics that could be used in today's engineering applications was developed in 18th century. Neither 19th nor 20th century had saught such a major breakthrough despite the rise of industry, technology and appearance of computers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder what was the drive behind these successes in mathematics at a century when it would not be expected to find a lot of applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it indeed driven by the needs of industry, military and technological progress?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or was it developed as a form of amusement because some nobility had a lot of spare time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or was it due to personality traits of some of government leaders who sponsored mathematics and made it a very prestigious profession of the time?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="4251" LastEditDate="2020-07-05T23:50:30.470" LastActivityDate="2020-07-05T23:50:30.470" Title="What was the reason for the immense progress of mathematics in 18th century?" Tags="|mathematics|18th-century|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2723" PostTypeId="1" CreationDate="2015-09-01T14:42:59.630" Score="2" ViewCount="107" Body="&lt;p&gt;I've tried searching everywhere, but I can't seem to find anything related to how the quantities got named!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Base quantities:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Mass&lt;/li&gt;&#xA;&lt;li&gt;Distance&lt;/li&gt;&#xA;&lt;li&gt;Time&lt;/li&gt;&#xA;&lt;li&gt;Temperature&lt;/li&gt;&#xA;&lt;li&gt;Electric current&lt;/li&gt;&#xA;&lt;li&gt;Quantity of a substance&lt;/li&gt;&#xA;&lt;li&gt;Light intensity&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2816" LastEditorUserId="62" LastEditDate="2015-09-01T17:45:45.623" LastActivityDate="2015-09-01T22:03:40.213" Title="How did each base quantity get its name?" Tags="|physics|units|" AnswerCount="1" CommentCount="3" ClosedDate="2015-09-20T15:27:48.500" ContentLicense="CC BY-SA 3.0" />
  <row Id="2724" PostTypeId="5" CreationDate="2015-09-01T16:35:51.157" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-01T16:35:51.157" LastActivityDate="2015-09-01T16:35:51.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2725" PostTypeId="4" CreationDate="2015-09-01T16:35:51.157" Score="0" Body="For questions about linear algebra, a mathematical field studying vector spaces and matrices. " OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-09-01T22:32:58.220" LastActivityDate="2015-09-01T22:32:58.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2726" PostTypeId="5" CreationDate="2015-09-01T16:38:19.030" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-01T16:38:19.030" LastActivityDate="2015-09-01T16:38:19.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2727" PostTypeId="4" CreationDate="2015-09-01T16:38:19.030" Score="0" Body="Elementary algebra, also called algebra precalculus, is the mathematical field studying basic properties of linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, graphs, and the solving of equations and systems of equations. " OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-09-01T22:33:10.243" LastActivityDate="2015-09-01T22:33:10.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2728" PostTypeId="5" CreationDate="2015-09-01T16:40:16.483" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-01T16:40:16.483" LastActivityDate="2015-09-01T16:40:16.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2729" PostTypeId="4" CreationDate="2015-09-01T16:40:16.483" Score="0" Body="For questions about the mathematical field abstract algebra that studies algebraic structures, most notably groups, rings and fields. " OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-09-01T22:33:37.360" LastActivityDate="2015-09-01T22:33:37.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2730" PostTypeId="5" CreationDate="2015-09-01T16:42:48.857" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-01T16:42:48.857" LastActivityDate="2015-09-01T16:42:48.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2731" PostTypeId="4" CreationDate="2015-09-01T16:42:48.857" Score="0" Body="For questions about events occurring in Egypt and the immediate areas around it between 3500 B.C. and 0 A.D." OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-09-01T22:32:53.947" LastActivityDate="2015-09-01T22:32:53.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2732" PostTypeId="5" CreationDate="2015-09-01T16:46:23.290" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-01T16:46:23.290" LastActivityDate="2015-09-01T16:46:23.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2733" PostTypeId="4" CreationDate="2015-09-01T16:46:23.290" Score="0" Body="For questions about events occurring in India and the immediate areas around it between 2500 B.C. and 1500 A.D." OwnerUserId="2442" LastEditorUserId="2442" LastEditDate="2015-09-01T22:33:29.883" LastActivityDate="2015-09-01T22:33:29.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2734" PostTypeId="2" ParentId="2722" CreationDate="2015-09-01T16:55:43.437" Score="4" Body="&lt;p&gt;One can easily name one main intrinsic reason: invention of Calculus in the very end of the previous century, and &quot;invention of mathematical physics&quot; by Newton.&#xA;It happened in the very beginning of 18-s century that sufficiently many people suddenly realized that mathematics can effectively explain the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course, there were also outside reasons, like development of industry and capitalism. That is people were interested in the explanation of the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;invention of physics&quot; I mean rigorous formulation of principal laws of mechanics and explanation of Kepler laws, explanation of tides, explanation of the shape of the Earth. All this was shortly confirmed by measurements, and these discoveries made an enormous impression. For the first time it was evident to many people that mathematics can really explain the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the external reasons were also important: people ( also kings and governments) were really INTERESTED in these questions, they payed for expeditions to check the true shape of the Earth and built observatories to observe stars and planets. They were interested in supporting mathematicians and several well payed jobs appeared.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laws of motions of stars and planets had one important application at that time: to navigation. And reliable navigation was important for overseas trade&#xA;and functioning of the navies. Another very important application&#xA;was military science (gunnery and fortification).&#xA;True industrial applications of mathematics&#xA;came a bit later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But people were so very much impressed by physics and calculus that they immediately started applying mathematics literally everywhere, not always with success, initially. For example, D. Bernoulli developed his famous laws of hydrodynamics trying to model the blood flow in the blood vessels. This could not give interesting results in 18s century, but the laws of hydrodynamics were created. Similarly, Euler wrote a lot about interior ballistics and shipbuilding. It was too early for any success in such complicated things. But this shows how great was enthusiasm about applications of mathematics they had at that time.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-09-01T17:14:00.427" LastActivityDate="2015-09-01T17:14:00.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2735" PostTypeId="2" ParentId="2722" CreationDate="2015-09-01T20:15:25.673" Score="1" Body="&lt;p&gt;The progress in 18th century followed two important breakthroughs in 17th. Algebraic notation which was essentially modern appeared in Descartes's Geometry published already in 1637, and the calculus was invented by Newton and Leibniz by 1680s. Some methods of the calculus of variations were also developed in 17th century, in connection with the famous brachistochrone problem. If 17th century can be compared to breaking the wall the 18th was more of gathering of the stones, systematization and advancement of previously made discoveries. This progress accelerated in the 19th century and led to another major breakthrough at the end of it, comparable to that in the 17th century, creation of set theory by Cantor and establishment of logical foundations of modern mathematics by Weierstrass, Dedekind, Frege, Hilbert and others. Creation of new domains of mathematics that followed in the 20th century, mathematical logic, abstract algebra, algebraic geometry, algebraic topology, functional analysis, category theory, theory of computation, etc., is no less impressive if not more impressive than the progress in the 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for reasons, it is hard to miss that both revolutions in 17th and 19th centuries were contemporary with revolutions in natural sciences, especially in physics, breakthroughs in technology, and major social and cultural changes (transitions from medieval to capitalist, and from national to global societies) that fueled them directly and indirectly. Some of it was intrinsic, many ancient mathematical works became available in Europe during Renaissance and stimulated great improvements in algebraic notation and methods, by Cardano and Viete especially, as well as ideas about tangents and volumes that led to calculus; some was technological, invention of the printing press made dissemination of scientific works much easier and broader, so more people could be pulled into doing mathematics and science; some was political, 17th and 18th centuries saw creation of government funded scientific societies like the Royal Society in England and Academy of Sciences in France; and some was economical, emerging capitalist trade and industry stimulated quantitative methods in mechanical engineering, cartography, chemistry, etc. that created steady supply of new problems and demand for new methods to solve them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-09-01T20:15:25.673" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2736" PostTypeId="2" ParentId="2723" CreationDate="2015-09-01T22:03:40.213" Score="2" Body="&lt;p&gt;Most of these derive their names from colloquial usage, and their origins can be found in etymology dictionaries, which also usually give pointers on the first scientific usage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;http://www.etymonline.com/index.php?term=temperature&quot; rel=&quot;nofollow noreferrer&quot;&gt;temperature&lt;/a&gt; derives from Latin &quot;temperatus&quot;, due proportion, and was in usage since 1530s, it was first used to denote the degree of heat by Boyle in 1670s. &lt;a href=&quot;http://www.etymonline.com/index.php?term=mass&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mass&lt;/a&gt; comes from Latin &quot;massa&quot;, kneaded dough, lump, old French &quot;masse&quot; was used to denote bulk size/quantity since 14th century. &lt;a href=&quot;https://hsm.stackexchange.com/questions/2665/when-were-the-inertial-and-gravitational-masses-distinguished-and-why/2674#2674&quot;&gt;Newton writes in Principia (1687)&lt;/a&gt;:&quot;&lt;em&gt;it is this quantity that I mean everywhere under the name of body or mass. And the same is known by the weight of each body for it is proportional to the weight&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Electric current is two words. Gilbert used Latin word “electricus” (amber) in 1600 to describe attraction that certain materials exhibit when rubbed against each other. A few years later Browne wrote several books based on Gilbert’s work, and he used the word “electricity” to describe the phenomenon.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Watson_%28scientist%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Around 1746 Watson&lt;/a&gt; and Franklin suggested that positive and negative charges were surpluses and deficiencies of a single fluid, Watson called it electrical ether, and the flow of it electric current. In 1747 he sent a current through a 6,732 foot long wire at Shooter's Hill in London.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And so on.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-09-01T22:03:40.213" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2737" PostTypeId="1" CreationDate="2015-09-02T04:34:38.437" Score="4" ViewCount="217" Body="&lt;p&gt;Coming from outside the field but with healthy curiosity I have been struck by what may be regarded as a discrepancy between the impact on science at large of the mathematics developed in the field of statistics, and the relative lack of popular recognition of its most outstanding figures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here come to mind the ubiquitous use of &lt;em&gt;t-tests&lt;/em&gt; and &lt;em&gt;ANOVA&lt;/em&gt; in medicine, biology, and psychology for example. Certainly, a lot of life-altering decisions on treatment of individuals, as well as health policies, have been informed by results considered scientific and hence trustworthy due to their foundation on these mathematical tools passed along by figures such as RA Fisher (or William Gosset). Another prominent name who has changed our understanding of science, and practically issued an epistemology of empiricism in its own right, can be found in reverend Thomas Bayes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anecdotally, a Google search of &lt;code&gt;top mathematicians in history&lt;/code&gt; yields all sort of returns on the first page, one of them broad enough to be promising: &lt;a href=&quot;http://www.storyofmathematics.com/mathematicians.html&quot; rel=&quot;nofollow&quot;&gt;&quot;List of Important Mathematicians&quot;&lt;/a&gt;, without mention of RA Fisher, or Karl Pearson, John Tukey... Under &lt;a href=&quot;http://famous-mathematicians.org&quot; rel=&quot;nofollow&quot;&gt;&quot;Famous Mathematicians&quot;&lt;/a&gt; even CF Gauss is left out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In trying to sort out a reason for this differential appreciation of historical figures in mathematics and science by the public at large, a suggestion was advanced in one of the comments that perhaps this was extensive to all applied mathematics, not just statistics, in contradistinction to pure mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a plausible explanation, and invites the reformulation of the initial question as: What has been the relation and mutual regard of pure and applied branches of mathematics throughout history?&lt;/p&gt;&#xA;" OwnerUserId="2352" LastEditorUserId="2352" LastEditDate="2015-09-03T00:59:50.407" LastActivityDate="2015-09-06T23:41:38.890" Title="Secular Status Difference Between Applied and Pure Mathematics" Tags="|mathematics|statistics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2739" PostTypeId="2" ParentId="2737" CreationDate="2015-09-02T13:00:50.700" Score="2" Body="&lt;p&gt;The answer is certainly &quot;no&quot;, they are not under-appreciated. A simple proof of this is the salary survey in mathematical sciences published yearly by the Notices of the American Mathematical society. From this survey you can see that the average salary (on all levels) is higher in statistics than in pure mathematics,&#xA;and the highest one is in &quot;bio statistics&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The articles like those you refer to are probably written by the people who read more on history of pure mathematics, and do not consider statistics a part of it.&#xA;And indeed it is somewhat separate subject: in many universities, department of&#xA;statistics is separate from the department of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your observation actually applies to several other mathematical areas, which are frequently not included: mathematical physics, applied mathematics, mechanics, computer science.&#xA;They have their separate histories, and rarely included in the popular papers on history of mathematics of the kind you refer to.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I doubt that George Dantzig (the inventor of the simplex method), or&#xA;Cornelius Lanczos (inventor of the Fast Fourier Transform) are included in the lists of &quot;great mathematicians&quot;. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-02T13:00:50.700" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2740" PostTypeId="1" AcceptedAnswerId="2743" CreationDate="2015-09-03T03:01:59.987" Score="23" ViewCount="1713" Body="&lt;p&gt;I know that the notion of irrational number (in one form or another) goes back to the Pythagoreans, and therefore far predates the decimal system, and certainly the representation of non-integer quantities with decimal fractions.  I also know that al-Samaw'al ibn Yahya ibn Yahuda al-Maghribi (1125-1174), in &lt;em&gt;Treatise on Arithmetic&lt;/em&gt; (1172), explained that quantities such as $\sqrt{2}$ can be approximated to as much precision as desired using decimal fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: My source the claim about al-Samaw'al is Katz, &lt;em&gt;A History of Mathematics&lt;/em&gt; (3rd edition).  (Sorry I don't have access to any primary sources.)  On p. 270 we find the following explanation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... Similarly, he [al-Samaw'al] calculated the square root of 10 verbally to be 3 plus 1 part of 10 plus 6 parts of 100 plus 2 parts of 1000 plus 2 parts of 10,000 plus 7 parts of 100,000 plus 7 parts of 1,000,000.  Unlike his predecessor [al-Uqlidisi], he still used words to describe the various places. Nevertheless, he understood the value of using decimal fractions for approximating rational numbers or irrational numbers.  In fact, when al-Samaw-al calculated higher roots by a method similar to that used in China, he explicitly noted the purpose of the successive steps of the algorithm:  &quot;And thus we operate to determine the side of a cube, of a square-square, a square-cube and other [powers]. This method enables us... to obtain an infinite number of answers, each one being more precise and closer to the truth than the preceding one.&quot;  Al-Samaw'al evidently realized that, in theory at least, one can calculate an infinite decimal expansion of a number, and that the finite decimals of this expansion &quot;converge&quot; to the exact value, a value not expressible in any finite form.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But who was the first to explicitly state (and prove?) that rational numbers have decimal expansions that terminate or repeat, and irrational numbers therefore have non-terminating and non-repeating decimal representations?&lt;/p&gt;&#xA;" OwnerUserId="294" LastEditorUserId="294" LastEditDate="2015-09-03T13:29:12.690" LastActivityDate="2019-11-04T00:11:07.993" Title="When did it become understood that irrational numbers have non-repeating decimal representations?" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2741" PostTypeId="1" AcceptedAnswerId="2752" CreationDate="2015-09-03T15:57:51.953" Score="10" ViewCount="1090" Body="&lt;p&gt;Recently I came across several papers of Monge and Lagrange, around the end of the 18th century, considering developable surface as 'folded' planes, using specifically the word &quot;plié&quot; (i.e. folded). Now, as these surface are now considered as part of differential geometry, is there any &lt;em&gt;connection&lt;/em&gt; between the term Manifold, coined by Riemann as &quot;Mannigfaltigkeit&quot; in 1854, and the terms fold &quot;Falte&quot; and folding &quot;falten&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &lt;em&gt;connection&lt;/em&gt; I mean not just a conceptual one but in fact something that is said explicitly (by Gauss, Riemann etc); i.e. that these mathematicians meant that the term &quot;Mannigfaltigkeit&quot; and the new domain of research (differential geometry) are inherently and deliberately connected to a folded space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many thanks!&lt;/p&gt;&#xA;" OwnerUserId="2687" LastEditorUserId="26" LastEditDate="2015-09-03T17:56:52.407" LastActivityDate="2017-06-15T21:16:17.357" Title="Was the term &quot;manifold&quot; (or its German equivalent) chosen with the verb &quot;to fold&quot; in mind?" Tags="|mathematics|terminology|differential-geometry|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2743" PostTypeId="2" ParentId="2740" CreationDate="2015-09-03T22:34:00.143" Score="20" Body="&lt;p&gt;Before approximating roots Al-Samawal performs long division of 210 by 13 to five decimal places, not enough to notice that digits cycle after the sixth. And this is the problem with discovering it experimentally in general, rationals may have arbitrarily long periods (repetends), not to mention preperiods. &lt;a href=&quot;https://books.google.com/books?id=DQXvAAAAMAAJ&amp;amp;pg=PA176&amp;amp;lpg=PA176&amp;amp;dq=%22proved%22+-continued+periodic+fractions&amp;amp;source=bl&amp;amp;ots=E0ZTHW9m6H&amp;amp;sig=LRodNxj1gNiuVz9fa8gDjxP7iv0&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CCcQ6AEwAWoVChMI1YWB8uzbxwIVwwiSCh2Y7gX6#v=onepage&amp;amp;q=%22proved%22%20-continued%20periodic%20fractions&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Dickson&lt;/a&gt;, Al-Maridini in 15th century was first to note periodicity, in a particular example with sexagesimals. Leibniz observed in 1677 that expansion of $1/n$ is periodic for any base (he later added - relatively prime to $n$). Using this it is straightforward to show that all decimal expansions of common fractions terminate or are repeating decimals (a.k.a. periodic fractions), although Leibniz did not spell it out. From this point on patterns in decimal and other base expansions were studied in the context of number theory, Wallis, Robertson, Lambert and Euler were the early contributors, Jean Bernoulli summarized and built on their work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest mention of the converse, that all repeating decimals are rationals, comes late, surprisingly late, by Lambert in 1758, he justifies it by using the now familiar trick with the geometric series. In &lt;a href=&quot;https://books.google.com/books?id=X8yv0sj4_1YC&amp;amp;dq=euler+elements+of+algebra&amp;amp;printsec=frontcover&amp;amp;source=bn&amp;amp;hl=en&amp;amp;ei=KiwwS8bdEcutlAe2waCoBw&amp;amp;sa=X&amp;amp;oi=book_result&amp;amp;ct=result&amp;amp;resnum=5&amp;amp;ved=0CCQQ6AEwBA#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's Elements of Algebra (1765)&lt;/a&gt;, &quot;&lt;em&gt;one of the earliest books to set out algebra in the modern form... and one of Euler's few writings that are accessible to the general public&lt;/em&gt;&quot; we find the same trick $9.999...=9+9/10+9/10^2+...=\frac{9}{1-1/10}=10$. However, according to the formal attitude of the 18th century, &lt;a href=&quot;https://hsm.stackexchange.com/questions/2274/is-it-true-that-euler-didnt-prove-sum-n-1-infty-frac1n2-frac-pi2/2283#2283&quot;&gt;e.g. Euler's&lt;/a&gt;, we also have $1+2+2^2+\dots=\frac{1}{1-2}=-1$, by definition. It took some time before Cauchy tied summing series to convergence. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems surprising today but that rational numbers have decimal expansions that terminate or repeat, and that irrational numbers have non-terminating and non-repeating expansions are two distinct problems. This is because the connection between them relies on two things: that there are not just examples like $\sqrt{2}$ or $\pi$, but a species of things called &quot;irrational numbers&quot;, and that completed, infinitely extending strings of decimal digits represent anything at all, and specifically their specimen. These realizations only crystallized in late 19th century. A symbolic year is 1872, when Cantor, Dedekind and Heine independently published their constructions of real numbers as a species. After 1872 acceptance was quick, a &lt;a href=&quot;https://books.google.com/books?id=BiQAAAAAYAAJ&amp;amp;pg=PA241&amp;amp;lpg=PA241&amp;amp;dq=%22proved%22+repeating+decimal&amp;amp;source=bl&amp;amp;ots=AW0Ky2FiT5&amp;amp;sig=5kivDmW_ra23U-1hjU6tN30MbPU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CDkQ6AEwBGoVChMI2tv80-rbxwIVzAWSCh0DPgTp#v=onepage&amp;amp;q=%22proved%22%20repeating%20decimal&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;reply to a question in an 1889 issue of The Bizzarre&lt;/a&gt; shows that by then they were already commonly known as numbers having non-terminating and non-repeating decimal expansions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://books.google.com/books/about/The_history_of_mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;another book of Katz's&lt;/a&gt; the quoted passage is reproduced almost verbatim, except for one word:&quot;&lt;em&gt;one can &lt;strong&gt;potentially&lt;/strong&gt; calculate an infinite decimal expansion of a number&lt;/em&gt;&quot;. This goes back to the view of Aristotle and Euclid that all infinity is only potential, not actual, not completed, which continued to exert heavy influence on mathematicians until late 19th century. Infinite strings of digits were mostly contemplated as processes, not finished objects. A notable exception was &lt;a href=&quot;https://en.wikipedia.org/wiki/De_Thiende&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stevin's Arithmetic (1585)&lt;/a&gt;, largely responsible for acceptance of the decimal system in Europe (it helped that Stevin was more engineer than mathematician). But rejection of actual infinities infected even the identification of repeating decimals with rationals, let alone of &quot;lawless&quot; strings with irrationals. This attitude is deep seated, even today many students have Aristotelian intuitions of infinity in repeating decimals. &lt;a href=&quot;http://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Tall&lt;/a&gt; &quot;&lt;em&gt;interviews revealed that students continued to conceive of  $0.999...$ as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are'&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting to compare this to a similar issue for continued fractions. &lt;a href=&quot;http://www.ams.org/journals/bull/1979-01-06/S0273-0979-1979-14684-6/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fowler argues&lt;/a&gt; that already Theaetetus (c. 417–369 BC) had a heuristic understanding that simple terminating continued fractions correspond to commensurable ratios and periodic ones to ratios &quot;commensurable in square&quot; &#xA;(rationals and quadratic irrationals in modern terms). But it was &lt;a href=&quot;http://www.csun.edu/~hbund408/math%20history/math.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;stated and proved explicitly only around the same time as for repeating decimals&lt;/a&gt;, by Euler in De Fractionlous Continious (1737) and Lagrange in Sur la Resolution des Equations Numeriques (1767), respectively.  &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-01-11T22:58:20.783" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2744" PostTypeId="2" ParentId="2741" CreationDate="2015-09-04T09:42:45.673" Score="2" Body="&lt;p&gt;I am not sure whether this question is about mathematics or etymology. In the latter case the answer is: Yes, the suffix -fold, -faltig, -fältig derives from the noun fold, Falte. You can look them up in any etymological dictionary, for example here: &lt;a href=&quot;http://www.dwds.de/?qu=Falte&quot; rel=&quot;nofollow&quot;&gt;http://www.dwds.de/?qu=Falte&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-09-04T09:55:17.683" LastActivityDate="2015-09-04T09:55:17.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2745" PostTypeId="5" CreationDate="2015-09-04T16:53:52.863" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-04T16:53:52.863" LastActivityDate="2015-09-04T16:53:52.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2746" PostTypeId="4" CreationDate="2015-09-04T16:53:52.863" Score="0" Body="For questions on the development of science in Nazi Germany and the influence of Nazi Germany on science. " OwnerUserId="2442" LastEditorUserId="54" LastEditDate="2016-04-24T16:59:45.023" LastActivityDate="2016-04-24T16:59:45.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2747" PostTypeId="5" CreationDate="2015-09-04T16:58:35.583" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-04T16:58:35.583" LastActivityDate="2015-09-04T16:58:35.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2748" PostTypeId="4" CreationDate="2015-09-04T16:58:35.583" Score="0" Body="For questions about the history and origin of atomic models and atomic theory. Topics include the classical Greek atomism and the development towards the atom models of Dalton and Rutherford. " OwnerUserId="2442" LastEditorUserId="54" LastEditDate="2016-04-24T16:59:40.303" LastActivityDate="2016-04-24T16:59:40.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2749" PostTypeId="5" CreationDate="2015-09-04T17:00:57.563" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2015-09-04T17:00:57.563" LastActivityDate="2015-09-04T17:00:57.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2750" PostTypeId="4" CreationDate="2015-09-04T17:00:57.563" Score="0" Body="For questions about the development of evolution, including evolution before Darwin, Darwinian evolution through natural selection and Neo-Darwinism. " OwnerUserId="2442" LastEditorUserId="54" LastEditDate="2016-04-24T16:59:36.677" LastActivityDate="2016-04-24T16:59:36.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2751" PostTypeId="2" ParentId="2741" CreationDate="2015-09-04T19:55:39.050" Score="5" Body="&lt;p&gt;The word &quot;&lt;a href=&quot;https://de.wiktionary.org/wiki/Mannigfaltigkeit&quot; rel=&quot;nofollow&quot;&gt;Mannigfaltigkeit&lt;/a&gt;&quot; itself was not invented by Riemann. It is a word meaning (great) variety or richness of types, in an every day sense. It might be used to describe that there are a variety of different things of a related type, like many different trees, or something along these lines. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I suppose there is an etymological connection, in the common usage of the word this idea seems rather lost. Thus, as long as there is no actual evidence for it, I would be skeptical at that connection. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A further piece of evidence for the lack of a relation to &quot;fold&quot; is the example of a &quot;real world&quot; occurrence of &quot;continuous manifold&quot; that Riemann gives in &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Ueber die Hypothesen, welche der Geometrie zu Grunde liegen,&quot;&lt;/a&gt;, the article linked to in a comment, it is: &lt;strong&gt;colors.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a general explication of Riemann was up to, see Conifold's nice answer.        &lt;/p&gt;&#xA;" OwnerUserId="98" LastEditorUserId="98" LastEditDate="2015-09-05T13:08:50.923" LastActivityDate="2015-09-05T13:08:50.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2752" PostTypeId="2" ParentId="2741" CreationDate="2015-09-04T22:03:44.520" Score="6" Body="&lt;p&gt;Notion of manifold is often motivated today by examples of simple surfaces, including developable ones, so this is a natural guess, but it is unlikely to be the case historically. &lt;a href=&quot;http://blog.ierna.name/2012/04/24/husserls-notion-of-manifold/&quot; rel=&quot;nofollow noreferrer&quot;&gt;German Mannigfaltigkeit&lt;/a&gt; is associated not so much with geometric folding as with &quot;many folded into one&quot;, alternatively translated as &quot;multiplicity&quot;. Cantor even used Mannigfaltigkeit for general sets until 1890s, when he switched to Menge (literally &quot;quantity&quot;), although according to Husserl &quot;&lt;em&gt;this conception does not coincide with that of Riemann and as used elsewhere in the theory of geometry, according to which a manifold is a collection not of merely united, but also ordered elements, and on the other hand not merely united, but continuously connected elements&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann himself in his lecture&lt;/a&gt; uses Mannigfaltigkeit as almost synonymous with &quot;magnitude&quot;, saying that he set himself &quot;&lt;em&gt;the task of constructing the notion of a multiply extended magnitude&lt;/em&gt;&quot;, and invokes different intuitions when first using the word, Mannigfaltigkeit can equally be discrete:&quot;&lt;em&gt;according as there exists... a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements&lt;/em&gt;&quot;. He is also quite clear on his influences:&quot;&lt;em&gt;besides some very short hints on the matter given by Privy Councillor Gauss in his second memoir on Biquadratic Residues, in the Gottingen Gelehrte Anzeige, and in his Jubileebook, and some philosophical researches of Herbart, I could make use of no previous labours&lt;/em&gt;&quot;. Herbart is a German philosopher who clarified and popularized Kant's philosophy of space, he also &lt;a href=&quot;https://hsm.stackexchange.com/questions/657/which-school-of-philosophy-motivated-thinking-about-spaces-of-higher-dimension&quot;&gt;influenced Grassman and his &quot;science of magnitude&quot;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The motivation he cites is related to complex analysis:&quot;&lt;em&gt;Such researches have become a necessity for many parts of mathematics, e.g., for the treatment of many-valued analytical functions; and the want of them is no doubt a chief cause why the celebrated theorem of Abel and the achievements of Lagrange, Pfaff, Jacobi for the general theory of differential equations, have so long remained unfruitful&lt;/em&gt;&quot;. Finally, the main focus of the lecture are local metric properties of manifolds, especially curvature, but developable surfaces are folded from flat ones, i.e. they are locally flat and have zero curvature. So they wouldn't be interesting for what Riemann has in mind.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-09-04T22:03:44.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2753" PostTypeId="1" CreationDate="2015-09-05T00:29:09.120" Score="2" ViewCount="90" Body="&lt;p&gt;Ideas such as Geocentrism were widely accepted in historic times. Models and theories were set out by some of the most prolific scientists and/or philosophers of the time. Those with opposing ideas were ridiculed, often even killed. Now we have much evidence against these theories, showing clearly they were incorrect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was there a scientific flaw with the &lt;strong&gt;way&lt;/strong&gt; the ideas were presented? What concepts do we have in place today where such flaws are outed and not regarded as science?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the theory of evolution is widely accepted as a fact, as much as the theory of gravity is. Are these theories susceptible to similar discredit in the future?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially I am looking for a counter-argument against the following statement:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Many ideas such as geocentrism were widely accepted as fact through science, and now have been completely debunked. Evolution is another one of these cases.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: This is not a question about the contents of the theory of evolution, but about progression of scientific thinking that allow us to have more credible scientific theories.&lt;/p&gt;&#xA;" OwnerUserId="2830" LastActivityDate="2015-09-05T03:31:57.797" Title="Was there a flaw with scientific thinking that theories such as geocentrism were widely accepted in the past?" Tags="|evolution|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2754" PostTypeId="1" AcceptedAnswerId="2767" CreationDate="2015-09-05T00:35:23.290" Score="5" ViewCount="1156" Body="&lt;p&gt;We know, of course, that zoonoses – diseases transmitted from animals to humans – have existed about as long as humans and animals have co-existed, but what was the earliest documented case of a zoonosis? In other words, who first attributed a human disease to an animal source?&lt;/p&gt;&#xA;" OwnerUserId="2826" LastActivityDate="2016-02-28T01:11:06.800" Title="What was the earliest documented zoonosis?" Tags="|discoveries|medicine|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2755" PostTypeId="2" ParentId="2753" CreationDate="2015-09-05T02:03:18.830" Score="1" Body="&lt;p&gt;There was no &quot;flow of scientific thinking&quot; in the work of those who developed geocentric system. The flow is in the education of modern philosophers and writers &#xA;who propagate the idea that &quot;geocentrism&quot; was somehow &quot;incorrect&quot;. Geocentric theory was a valid scientific theory which predicted some phenomena correctly, within the accuracy of the measurement at that time. No proof of motion of the Earth existed&#xA;at that time, neither it existed 1000 years later. The geocentric theory quite well described the phenomena with respect to the frame of reference connected with Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many modern computations are made in the same frame of reference. If you want to compute how to aim your telescope, you have to use geocentric system (if your telescope stands on Earth).&#xA;If you want to launch a space craft to Mars, you use some other theory. But Ptolemy was not concerned with launching a craft to Mars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theory is accepted if it is well enough tested. As science develops new facts are discovered, and new theories are developed to explain these facts. Sometimes these new theories contain old theories as a special case. Sometimes they give a more precise predictions, but old (tested) theories remain valid approximations, within the range of phenomena for which they were designed and tested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, quantum mechanics generalizes Newtonian mechanics. But to say that Newtonian mechanics is incorrect is nonsense. It is tested, is correct, and will always be correct, no matter what new theories will be invented. Moreover, all&#xA;these new theories must be consistent with it, and any generalization should contain it as a special case. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-09-05T03:31:57.797" LastActivityDate="2015-09-05T03:31:57.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2756" PostTypeId="1" AcceptedAnswerId="2759" CreationDate="2015-09-05T08:47:12.847" Score="25" ViewCount="5207" Body="&lt;p&gt;I read in my mechanics textbook written by Goodstein that Robert Millikan cherry-picked his data in his famous oil drop experiment, and now I'm left wondering about the scientific value of his results. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems &quot;okay&quot; to me if one gets rid of one or two ridiculous data points: One doesn't get infinite money and time to perform perfect experiments. On the other hand, scientists have to be able to trust the work of others (or reproduce the results themselves, but that's not always feasible). Should Millikan's &quot;creative&quot; way of handling his data be regarded as fraud? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be particularly interested in an answer by someone who has professional experience in experimental science.&lt;/p&gt;&#xA;" OwnerUserId="2713" LastEditorUserId="55" LastEditDate="2017-03-24T23:16:38.047" LastActivityDate="2017-08-22T01:45:30.037" Title="Is Millikan's famous oil drop experiment a fraud?" Tags="|physics|experimental-physics|debunking|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2757" PostTypeId="1" AcceptedAnswerId="2758" CreationDate="2015-09-05T14:09:32.457" Score="9" ViewCount="972" Body="&lt;p&gt;To my knowledge, the symbol for &quot;is defined as&quot; historically has been the notation, $\equiv$ or $\triangleq$. More recently, however, the notation $:=$ seems to have overtaken the latter two notations (at least, this seems to be the case in modern texts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has been suggested that this notation was borrowed from &lt;a href=&quot;http://en.wikipedia.org/wiki/Assignment_%28computer_science%29&quot;&gt;the assignment operator in computer programming languages&lt;/a&gt; &amp;mdash; in particular, from Pascal. Does there exist evidence of the use of this notation in mathematics for &quot;is defined as&quot; prior to its widespread use as the assignment operator in programming?&lt;/p&gt;&#xA;" OwnerUserId="2835" LastEditorUserId="62" LastEditDate="2015-09-05T14:14:57.943" LastActivityDate="2015-09-05T16:32:53.580" Title="Is the modern &quot;is defined as&quot; notation from computer programming?" Tags="|mathematics|notation|computers|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2758" PostTypeId="2" ParentId="2757" CreationDate="2015-09-05T16:32:53.580" Score="3" Body="&lt;p&gt;I did a bit of snooping, and followed the Wikipedia link you named to &lt;a href=&quot;http://en.wikipedia.org/wiki/Relational_operator#Confusion_with_assignment_operators&quot; rel=&quot;nofollow&quot;&gt;another subsection&lt;/a&gt;, which states&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;International Algebraic Language and ALGOL (1958 and 1960) therefore introduced &quot;&lt;code&gt;:=&lt;/code&gt;&quot; for assignment, leaving the standard &quot;&lt;code&gt;=&lt;/code&gt;&quot; available for equality, a convention followed by CPL, Algol W, BCPL, Simula, Algol 68, SETL, Pascal, Smalltalk, Modula2, Ada, Standard ML, OCaml, Eiffel, Delphi, Oberon, Dylan, VHDL, and several other languages.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problem, of course, was that there are case where expressions such as&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;code&gt;Variable = Variable + NewVariable&lt;/code&gt;&lt;br&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;are valid. In mathematics, if the same statement was used, it would obviously be false if $\text{NewVariable}\neq1$. Therefore, it was decided to use &lt;code&gt;:=&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found &lt;a href=&quot;http://www.softwarepreservation.org/projects/ALGOL/report/Algol58_preliminary_report_CACM.pdf/&quot; rel=&quot;nofollow&quot;&gt;the text of ALGOL 1958&lt;/a&gt; (although not the subsequent 1960 text), and it admittedly does not explicitly support this interpretation anywhere. However, it strongly emphasizes the user of &lt;code&gt;:=&lt;/code&gt; in statements, while it does not list &lt;code&gt;=&lt;/code&gt; as a building block for the language, so I think it's safe to say that this marked the transition, as claimed. We therefore have a date of the early use of &lt;code&gt;:=&lt;/code&gt;: 1958.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've found various texts and sites that use roughly the same wording when discussing the history of &quot;is defined as&quot; in mathematics. &lt;a href=&quot;http://www.math.ucdavis.edu/~anne/WQ2007/mat67-Common_Math_Symbols.pdf&quot; rel=&quot;nofollow&quot;&gt;This&lt;/a&gt;, for example, states that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$:=$(the equal by definition sign) means “is equal by definition to”. This is a common alternate form of the symbol “$=_{\text{Def}}$”, which appears in the 1894 book Logica Matematica by the logician Cesare Burali-Forti (1861–1931). Other common alternate forms of the symbol “$=_{\text{Def}}$” include “[a]” and “$\equiv$”, the latter being especially common in applied mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, [a] is a stand-in for a symbol which does not properly render. On page 7 of &lt;a href=&quot;http://archive.org/stream/logicamatematic00buragoog#page/n18/mode/2up&quot; rel=&quot;nofollow&quot;&gt;a version of the book&lt;/a&gt;, I see a statement that appears to end with&#xA;$$=1:=:D(xy,z)=1$$&#xA;which includes $:=:$, a possible variant of the symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On page 26, however, he writes an expression that includes&#xA;$$a=b:=_{\text{Def}}$$&#xA;However, I think that it is more likely that it is simply a use of the $:$ with $=_{\text{Def}}$, which he introduced further up the page.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that it is likely that Burali-Forti uses the symbol somewhere inside, but I'm, not certain. At any rate, it seems likely that &quot;$=$&quot;, or some variant thereof, was used prior to 1958.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-09-05T16:32:53.580" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2759" PostTypeId="2" ParentId="2756" CreationDate="2015-09-06T21:41:26.387" Score="18" Body="&lt;p&gt;In Millikan's publications, he stated categorically that every single&#xA;oil drop observed had had a charge that was a multiple of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, with no&#xA;exceptions or omissions. But his notebooks are full of notations&#xA;such as &amp;quot;beautiful data, keep,&amp;quot; and &amp;quot;bad run, throw out.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Richard Feynman wrote an essay called &amp;quot;Cargo Cult Science,&amp;quot; in which he pointed out:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Millikan measured the charge on an electron by an experiment with falling oil drops, and got an answer which we now know not to be quite right. It's a little bit off because he had the incorrect value for the viscosity of air. It's interesting to look at the history of measurements of the charge of an electron, after Millikan. If you plot them as a function of time, you find that one is a little bit bigger than Millikan's, and the next one's a little bit bigger than that, and the next one's a little bit bigger than that, until finally they settle down to a number which is higher.&lt;/p&gt;&#xA;&lt;p&gt;Why didn't they discover the new number was higher right away? It's a thing that scientists are ashamed of--this history--because it's apparent that people did things like this: When they got a number that was too high above Millikan's, they thought something must be wrong--and they would look for and find a reason why something might be wrong. When they got a number close to Millikan's value they didn't look so hard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So basically, the answer is yes. The experiment was fraudulent.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-09-06T21:41:26.387" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2760" PostTypeId="2" ParentId="2756" CreationDate="2015-09-07T12:40:58.827" Score="7" Body="&lt;p&gt;Working as a physicist, I have to say that it depends quite a bit on exactly what happened in those runs, and there isn't enough information to judge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps he failed to focus on a droplet or follow it correctly, and knew that the data would be 'bad'. In my judgement, this is okay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if he was calculating the charge for each run, and deciding on this basis to reject runs, that would be fraudulent.&lt;/p&gt;&#xA;" OwnerUserId="2840" LastActivityDate="2015-09-07T12:40:58.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2761" PostTypeId="1" AcceptedAnswerId="2768" CreationDate="2015-09-07T15:28:18.960" Score="5" ViewCount="329" Body="&lt;p&gt;I always thought of Skorokhod associated with the topology for convergence in the space of càdlàg (right continuous with left limits) functions in probability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it seems that he made several other contributions. He studied the reflected Brownian motion, proved the first existence theorem for stochastic differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What else did he do in the field?&lt;/p&gt;&#xA;" OwnerUserId="2630" LastEditorUserId="55" LastEditDate="2015-09-08T00:44:12.390" LastActivityDate="2015-09-08T00:44:12.390" Title="Skorokhod's contribution to probability theory" Tags="|mathematics|mathematicians|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2762" PostTypeId="1" CreationDate="2015-09-07T15:35:08.557" Score="10" ViewCount="3420" Body="&lt;p&gt;The natural logarithm is often represented by several different notations:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$\log_e x$&lt;/li&gt;&#xA;&lt;li&gt;$\log x$ (although this is also used for logarithms with a base of 10)&lt;/li&gt;&#xA;&lt;li&gt;$\ln x$&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It is the third notation that has me wondering. Why is $\ln$ used, and not, say, $\text{nl}$? My two theories about this are&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It is an abbreviation for &quot;natural logarithm&quot; in a non-English language&lt;/li&gt;&#xA;&lt;li&gt;It is meant to correspond with the &quot;$l$&quot; in typical logarithmic notation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Why is the notation $\ln$?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-09-07T18:25:40.370" LastActivityDate="2020-02-11T00:42:09.467" Title="Why is the natural logarithm represented by $\ln$?" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2763" PostTypeId="2" ParentId="2762" CreationDate="2015-09-07T18:47:56.393" Score="6" Body="&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/functions.html&quot; rel=&quot;nofollow&quot;&gt;Earliest Uses of Function Symbols&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\ln$ (for natural logarithm) was used in 1893 by Irving Stringham (1847-1909) in &lt;a href=&quot;https://books.google.it/books?id=BTY7AQAAIAAJ&quot; rel=&quot;nofollow&quot;&gt;Uniplanar Algebra&lt;/a&gt; (&lt;a href=&quot;https://books.google.it/books?id=bT5suOONXlgC&amp;amp;pg=PA107&quot; rel=&quot;nofollow&quot;&gt;Cajori&lt;/a&gt; vol. 2, page 107).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thnaks to KCd's reference in his comment, we have an earlier occurrence :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Anton Steinhauser, &lt;a href=&quot;https://books.google.it/books?hl=it&amp;amp;id=ZzU7AQAAIAAJ&amp;amp;dq=Anton+Steinhauser+Lehrbuch+der+Mathematik&amp;amp;focus=searchwithinvolume&amp;amp;q=ln&quot; rel=&quot;nofollow&quot;&gt;Lehrbuch der Mathematik für höhere Gewerbeschulen&lt;/a&gt; (1875), page 277.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-09-10T15:38:41.690" LastActivityDate="2015-09-10T15:38:41.690" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2764" PostTypeId="1" AcceptedAnswerId="2769" CreationDate="2015-09-07T11:50:02.397" Score="22" ViewCount="2136" Body="&lt;p&gt;Bourbaki, for example, was the name of a set of mathematicians, rather than a single person, under which several books were published.&lt;/p&gt;&#xA;&lt;p&gt;Out of curiosity, I wonder if there is any historical evidence that makes one maintain that the author of the &lt;em&gt;Elements&lt;/em&gt;, Euclid, was a single person instead of a set of ancient scholars?&lt;/p&gt;&#xA;" OwnerUserId="2839" OwnerDisplayName="Gudson Chou" LastEditorUserId="10106" LastEditDate="2023-12-23T15:35:54.107" LastActivityDate="2023-12-23T15:35:54.107" Title="Is there any historical &quot;evidence&quot; maintaining that Euclid was a single person?" Tags="|mathematics|mathematicians|ancient-greece|euclid|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="2765" PostTypeId="2" ParentId="2764" CreationDate="2015-09-07T12:26:47.983" Score="0" Body="&lt;p&gt;A &quot;corporation&quot; as Bourbaki  is, from one perspective, a sign of great scientific progress: his great purpose was to unify somehow  mathematics understood as a science that had acquired a vastness covering &quot;many&quot; quite developed branches each of which was autonomous.&#xA;In contrast, unlike, in times of Euclid's purpose it was found in a unit just a branch of mathematics in different subjects but the same &quot;spirit&quot; which is ancient geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could be that Euclid was &quot;A set of ancient scholars&quot; as you imagine, I do not know but personally, I would say no, he was a genius of his time living in a single human body. &lt;/p&gt;&#xA;" OwnerUserId="2120" OwnerDisplayName="Ataulfo" LastActivityDate="2015-09-07T12:26:47.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2766" PostTypeId="2" ParentId="2764" CreationDate="2015-09-07T12:39:38.327" Score="0" Body="&lt;p&gt;Actually as &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid#Life&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt; mentions:    &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;This hypothesis is not well accepted by scholars and there is little&#xA;  evidence in its favor&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;All evidence shows that Euclid was a real person.&lt;br&gt;&#xA;Actually I am not sure if such an idea (of naming a group of scientists with an actual name) would be accepted at 300 B.C. and in addition, I do not believe there was such a big amount of mathematicians at this period (and so good).&lt;br&gt;&#xA;Also,to my best knowledge there is no known similar group of scientists in the ancient times so why should &quot;Euclid&quot; be a group of mathematicians?&lt;br&gt;&#xA;I believe he was a single person. &lt;/p&gt;&#xA;" OwnerUserId="2056" OwnerDisplayName="Konstantinos Gaitanas" LastActivityDate="2015-09-07T12:39:38.327" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2767" PostTypeId="2" ParentId="2754" CreationDate="2015-09-07T19:51:20.073" Score="3" Body="&lt;p&gt;The answer depends on definition of &quot;documented&quot;. There are documented cases of diseases that we &lt;em&gt;now&lt;/em&gt; know transmit from other vertebrates to humans going as far back as 18th century BC, Babylonian codex Eshuna mentions “mad dogs” most likely infected by rabies. The Bible mentions an epidemic of (what appears to be) plague among the Philistines around 1320 BC, along with overpopulation of “mice” at the same time. See &lt;a href=&quot;http://download.springer.com/static/pdf/666/chp%253A10.1007%252F978-90-481-9657-9_3.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-90-481-9657-9_3&amp;amp;token2=exp=1441654683~acl=%2Fstatic%2Fpdf%2F666%2Fchp%25253A10.1007%25252F978-90-481-9657-9_3.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fchapter%252F10.1007%252F978-90-481-9657-9_3*~hmac=267dd9bed1d73cf4385fdd6cd988acbd7c48945f2380a976b09b27471b429a1b&quot; rel=&quot;nofollow&quot;&gt;History of Zoonoses and Sapronoses&lt;/a&gt; for multiple other events.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Documented&quot; in the modern sense could not have happened before 1546, when Fracastro first proposed that infectious diseases are caused by germs (&quot;seminaria morbi&quot;) in De Contagione et Contagiosis Morbis et Eorum Curatione, and then it took a while. The priority seems to go to &lt;a href=&quot;http://pdml.net/pipermail/pdml_pdml.net/2011-October/292097.html&quot; rel=&quot;nofollow&quot;&gt;Russian military doctor Stepan Andrievsky&lt;/a&gt;:&quot;&lt;em&gt;To prove that it was the same disease in animals and humans, &lt;strong&gt;he injected himself with the spores of anthrax taken from a sick animal in 1788 (or 1787)&lt;/strong&gt;. His diary helped him in the accurate disease description. He also originated the Russian name for anthrax: &quot;Sibirskaya iazva&quot; that is translated as &quot;Siberian ulcer&quot;... In all English sources&#xA;the discovery of anthrax etiology is attributed to Robert Koch, and sometimes to Casimir Davaine &amp;amp; Aloys Pollender&lt;/em&gt;&quot;. Some English sources now do mention Andrievsky, although &lt;a href=&quot;https://en.wikipedia.org/wiki/Bacillus_anthracis#Historical_background&quot; rel=&quot;nofollow&quot;&gt;not yet Wikipedia&lt;/a&gt;. Koch's 1876 study is considered  more conclusive than Andrievsky's, Davaine's or Pollender's, by then however other zoonoses (e.g. dermatomycosis) were documented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should be noted that although vaccination in the modern sense was only introduced by Jenner in 1796 (smallpox) more primitive inoculations originated in China and India as early as 16th century. &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0025556402001220&quot; rel=&quot;nofollow&quot;&gt;According to Dietz and Heesterbeek&lt;/a&gt;:&quot;&lt;em&gt;infectious material from smallpox cases was transferred into the skin of susceptibles with the intention to induce lifelong immunity by a mild infection with a low case fatality. In 1721 this method was introduced from Turkey into England by the wife of the English Ambassador to Constantinople, Lady Mary Wortley Montague. It is not widely known that inoculation was even attempted against measles, plague and several diseases of animals (rinderpest, sheep pox, contagious bovine pleuropneumonia)&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-28T01:11:06.800" LastActivityDate="2016-02-28T01:11:06.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2768" PostTypeId="2" ParentId="2761" CreationDate="2015-09-07T20:34:56.647" Score="2" Body="&lt;p&gt;Perhaps the best known contributions are the method of a single probability space, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Skorokhod_integral&quot; rel=&quot;nofollow&quot;&gt;Skorokhod integral&lt;/a&gt;, which extends the Itō integral to non-adapted processes and is adjoint to the Malliavin derivative. He also did some foundational work on stochastic processes on manifolds with boundary. The Institute of Mathematics in Kiev maintains a &lt;a href=&quot;http://www.imath.kiev.ua/~random/A_short_sketch_of_life_and_research___of_A_V__Skorokhod.html&quot; rel=&quot;nofollow&quot;&gt;Skorokhod webpage&lt;/a&gt; with a &quot;short sketch of life and research&quot;, which is much longer than the &lt;a href=&quot;https://en.wikipedia.org/wiki/Anatoliy_Skorokhod&quot; rel=&quot;nofollow&quot;&gt;Wikipedia entry&lt;/a&gt;. &quot;&lt;em&gt;His absence at international scientific conferences gave birth to the opinion among foreign scientists that &quot;Skorokhod&quot;  was the collective name of a group of Soviet scientists, just as the group  of French mathematicians united under the name &quot;Bourbaki&quot;&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some details on his research from the webpage:&quot;&lt;em&gt;The first series of Skorokhod's works that gained him wide recognition was devoted to the limit theorems for random processes  constructed on the basis of sums of independent random variables. These works accomplished the series of attempts of numerous mathematicians aimed at the generalization of the famous Donsker invariance principle to the case where the limit process is an arbitrary, not necessarily continuous, process with independent  increments...  Skorokhod proposed the method of a single probability space (mentioned earlier) and introduced several topologies in the space of functions that do not have discontinuities of the second kind, one of which is now widely known  as the  Skorokhod topology... One  should mention his proof of the theorem on existence of solutions of stochastic differential equations by the method of a single  probability space under the assumption that the coefficients of these equations are continuous functions (i.e., they may not satisfy  the Lipschitz condition)...&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;In 1966, he proved that a sufficiently broad class of continuous Markov processes can be reduced to quasidiffusion processes by a random change of the time variable. In the monograph &quot;Stochastic Equations for Complex Systems&quot;, he constructed stochastic differential equations for quasidiffusion processes taking values in spaces of complex structure (e.g., manifolds with boundary, manifolds with variable dimensionality,  etc.)... In the 1970s, Skorokhod introduced several notions, which are now  widely used not only by mathematicians, but also by physicists.  Among them, one should mention the notions of  extended  stochastic integral  (the  Skorokhod integral),  strong (weak)  random linear operator, and  stochastic semigroup.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-09-07T20:34:56.647" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2769" PostTypeId="2" ParentId="2764" CreationDate="2015-09-07T23:31:41.587" Score="7" Body="&lt;p&gt;I too do not see any reason to doubt that the works of Euclid were written by one person, apart, naturally, from some interpolations. However, I must disagree with Konstantinos when he writes that “there is no known similar group of scientists in the ancient times”. The medical writings attributed to Hippocrates are well-known to have been written over a period of centuries and have very clear differences in content and style. But maybe medicine is not a science in the way that mathematics is.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-09-08T08:56:20.587" LastActivityDate="2015-09-08T08:56:20.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2770" PostTypeId="2" ParentId="2764" CreationDate="2015-09-08T05:36:30.610" Score="13" Body="&lt;p&gt;There's little question that some parts of &quot;The Elements&quot; were added by later authors.  Heath's English translation gives a lengthy commentary on what is likely in the original work and what is known or suspected to be from later editors.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were also two entire books tacked onto the end that were often attributed to Euclid, but are now generally recognized as being from later authors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in that sense the author of the Elements wasn't just one person.  But there almost certainly was an actual Euclid who wrote the bulk of what's come down to us as The Elements.  Pappus and Proclus both discuss him as an actual person, and while they lived several centuries after The Elements were written, we know they were drawing from near-contemporary sources.  &lt;/p&gt;&#xA;" OwnerUserId="1731" LastActivityDate="2015-09-08T05:36:30.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2771" PostTypeId="1" AcceptedAnswerId="2773" CreationDate="2015-09-08T20:48:26.470" Score="-2" ViewCount="208" Body="&lt;p&gt;Do we have any information about definition of topology . Definition is not intuitive for me .Please share information about definition of topology .&lt;/p&gt;&#xA;" OwnerUserId="2222" LastEditorUserId="2222" LastEditDate="2015-09-08T21:15:16.087" LastActivityDate="2015-09-08T21:56:21.377" Title="questions about definition of topology" Tags="|mathematics|topology|" AnswerCount="1" CommentCount="3" ClosedDate="2015-10-08T01:13:05.470" ContentLicense="CC BY-SA 3.0" />
  <row Id="2772" PostTypeId="1" CreationDate="2015-09-08T21:06:03.393" Score="5" ViewCount="212" Body="&lt;p&gt;Some years ago, I read the Newton biography &quot;Never at Rest&quot; by Richard Westfall. However, I don't currently have access to a copy, so the following is from memory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the biography, Newton has no formal mathematical background to speak of, nor was there really much by way of mathematical background to be had at the time his career began. However, after becoming a student at Cambridge, it took him approximately 18 months, from a standing start, surrounded by a whole lot of nothing (ie. mid-seventeenth century Cambridge) to start making original mathematical discoveries, including the Binomial Theorem, and the early stages of the calculus. Much of this work emerged from things like formal polynomial manipulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His background was essentially his own reading of people like Descartes, and Fermat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recollection is that the main historical record of this is his notebook (called by him, his &quot;waste&quot; book), which was given to him by his stepfather, Barnabas Smith. But it is unclear to me how historians know what period the notes therein were made.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The upshot is, I'm wondering if my description above is still the standard &#xA;understanding of the Newton mathematics chronology, or at the least the beginning of the chronology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The description given above just seems a little improbable. It seems more likely to me that such substantial mathematical advances came at a later period of his life, and with more effort than would be suggested by an 18 month run up from a standing start to original mathematical discoveries.&lt;/p&gt;&#xA;" OwnerUserId="1878" LastActivityDate="2015-09-17T00:55:16.453" Title="What is the modern understanding of the chronology of Newton's mathematical work?" Tags="|mathematics|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2773" PostTypeId="2" ParentId="2771" CreationDate="2015-09-08T21:56:21.377" Score="2" Body="&lt;p&gt;I suppose you are asking for the definition of topology as a mathematical term, not about the name of the part of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern definition of topology is a product of long development. Originally open and closed sets were defined through accumulation points (Cantor).&#xA;This is very intuitive. The basic notion was the notion of limit.&#xA;Later, the operation of closure was introduced, described by axioms and a set was&#xA;called closed if it coincides with its closure. This approach is used in the comprehensive book by Kuratowski, &quot;Topology&quot; written in the late 1930-s.&#xA;Later it was found more convenient to define topology by simple axioms of open sets. Because of the Bourbaki influence this approach became standard. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern definition indeed looks very abstract to a beginner, but it has certain advantages. A good book which can be recommended as an introduction to topology&#xA;is Chinn and Steenrod, First concepts of topology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I learned these first notions from this book as a freshman undergraduate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that you can read some undergraduate textbook on topology, like Kelley,&#xA;General topology, or some more modern undergraduate book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. For most questions studied in the undergraduate curriculum, the general definition of topology is not needed. So it is reasonable to familiarize yourself first with the special case of metric spaces (instead of the general case of topological spaces). metric spaces are explained in any good (rigorous) calculus textbook, in the beginning chapters. In metric spaces everything is defined in terms of the distance, a very intuitive notion.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-08T21:56:21.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2774" PostTypeId="2" ParentId="2772" CreationDate="2015-09-09T00:39:41.420" Score="4" Body="&lt;p&gt;This paper&#xA;&lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020729.02p0067y.pdf&quot; rel=&quot;nofollow&quot;&gt;https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020729.02p0067y.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;addresses the question. Comprehensive commentaries of Whiteside to Mathematical Papers of Newton give more details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most important authors Newton read in his young age were&#xA;Euclid, Descartes, Oughtred, Schooten, Viete and Wallis. He was in personal contact with another outstanding mathematician, Barrow. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-09T00:39:41.420" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2776" PostTypeId="1" CreationDate="2015-09-09T14:01:03.540" Score="22" ViewCount="2280" Body="&lt;p&gt;Let's take an obvious example. I'm sure that amateur mathematicians (and some professionals) will continue to search for Fermat's 'marvellous proof' of his Last Theorem. This is despite the fact that Andrew Wiles has proved the assertion using much more sophisticated techniques than Fermat could possibly have known.&#xA;(Facts drawn from &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem&quot;&gt;Fermat's Last Theorem&lt;/a&gt; From Wikipedia, the free encyclopedia, and other sources.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any proof in the history of mathematics that has substantially been simplified from one that only specialised mathematicians would understand to a shorter simpler proof that someone with a good understanding of maths (let's say up to beginning undergraduate level) would be able to comprehend? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more specific terms, is there any evidence in history that supports the idea that finding a simple proof of Fermat's Last Theorem is a realistic hope?&lt;/p&gt;&#xA;" OwnerUserId="2849" LastActivityDate="2017-06-16T00:51:36.950" Title="Has any 'difficult' proof ever been superseded by a 'simple' one?" Tags="|mathematics|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2777" PostTypeId="2" ParentId="2776" CreationDate="2015-09-09T17:35:49.323" Score="13" Body="&lt;p&gt;The original proof of the Lasker-Noether Theorem by Emanuel Lasker took 98 pages, but now there are proofs known of less than a page. You can see it on &lt;a href=&quot;https://en.wikipedia.org/wiki/Primary_decomposition#Proof&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We also have the Resolution of singularities, which was simplified from 216 pages (Hironaka, 1964) to a few dozen pages. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also noteworthy is Ruffini's attempt at proving the Abel-Ruffini theorem, spanning over 500 pages, but it was not complete and thus cannot be considered as a proof. Abels proof was only 6 pages. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are not really undergratuate concepts, but they are interesting (I think) for the massive decrease in number of pages. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2015-09-09T17:35:49.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2778" PostTypeId="2" ParentId="2776" CreationDate="2015-09-09T19:05:30.960" Score="24" Body="&lt;p&gt;This happens all the time, mathematicians are fond of publishing papers titled Elementary Proof of Such and Such [hard] Theorem. Original proofs are often tour de force (feats of strength), they introduce new concepts along the way, make convoluted constructions, involve heavy calculations, just to get to the final result somehow. Once a new framework is developed around them ready made concepts become available, some detours are eliminated, constructions are optimized, and calculations streamlined and minimized. Sometimes a different approach allows to short circuit many original complications completely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prototypical example is the Pythagorean theorem, &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookI/propI47.html&quot;&gt;Euclid's &quot;bride's chair&quot; proof&lt;/a&gt; is a pain to follow, now there are proofs that can be understood even without words. The same goes for his volume of a pyramid proof by &quot;method of exhaustion&quot;, even many authors writing about history prefer to replace it by a version with limits. Many Apollonius's proofs in Conic Sections are  characterized (by Heath for example) as almost impenetrable even to professional geometers, but they can be streamlined using coordinate geometry.&#xA;A simple proof of focal properties based on inscribing a sphere into the cone, that &lt;em&gt;could&lt;/em&gt; have been given by Apollonius but wasn't, was &lt;a href=&quot;http://nonagon.org/ExLibris/dandelin-spheres-conic-sections&quot;&gt;only found by Dandelin in 1822&lt;/a&gt;. To a lesser extent the same applies to Newton's Euclidean proofs in Principia, for example his &lt;a href=&quot;https://books.google.com/books/about/Second_Year_Calculus.html?id=aeg48f5xk8EC&quot;&gt;derivation of the inverse square law from Kepler's laws&lt;/a&gt;, a much more accessible calculus based proof is given by Bressoud in Second Year Calculus (along with an exposition of Newton's for comparison). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Abel's original proof of unsolvability of higher order equations in radicals was quite involved, Alexeev wrote a book &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/abel.pdf&quot;&gt;Abel’s Theorem&#xA;in Problems and Solutions&lt;/a&gt; based on Arnold's lectures to bright high schoolers that uses complex analysis. The same happened with the Fundamental Theorem of Algebra, elementary complex analysis proofs are now accessible to undergraduates, Gauss's original proofs are not so much. Gödel's 1930 proofs of incompleteness theorems are a technical maze (not the least because he was using notation and terminology of Russell's Principia), he simplified them himself somewhat in 1934, and now many popular books present (often somewhat sloppy) elementary proofs.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-09T19:20:52.340" LastActivityDate="2015-09-09T19:20:52.340" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2780" PostTypeId="2" ParentId="2762" CreationDate="2015-09-09T22:58:21.167" Score="8" Body="&lt;p&gt;As far as I am aware, the mathematical operator commonly spelled &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; is an abbreviation of the Latin term,  &lt;a href=&quot;https://en.wiktionary.org/wiki/natural_logarithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;b&gt;l&lt;/b&gt;ogarithmus &lt;b&gt;n&lt;/b&gt;ātūrālis&lt;/em&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure who first used this abbreviation, but I suppose it very well may have been Napier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember having seen &lt;em&gt;ln&lt;/em&gt; written as an abbreviation of two words in the form &lt;span class=&quot;math-container&quot;&gt;$l.n.$&lt;/span&gt; &lt;/p&gt;&#xA;" OwnerUserId="2852" LastEditorUserId="4961" LastEditDate="2020-02-11T00:42:09.467" LastActivityDate="2020-02-11T00:42:09.467" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2781" PostTypeId="2" ParentId="2776" CreationDate="2015-09-10T01:52:51.470" Score="8" Body="&lt;p&gt;A proof of Fermat's Last Theorem for polynomials in one variable over the complex numbers was first found in the 19th century, using algebraic geometry. Nowadays it can be proved more simply as a consequence of the Mason-Stothers theorem, which has a very simple proof accessible to someone who knows only how to (formally) differentiate a polynomial. &lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2015-09-10T01:52:51.470" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2782" PostTypeId="1" AcceptedAnswerId="2784" CreationDate="2015-09-10T17:31:02.343" Score="5" ViewCount="279" Body="&lt;p&gt;I was wondering why Greeks chose to have Olympic games every four years. Now, since we usually every fourth year is a leap one, it makes sense; but the reform of the calendar which stated this is due to Julius Caesar, and besides (see &lt;a href=&quot;https://hsm.stackexchange.com/q/2506/81&quot;&gt;here&lt;/a&gt;) Greeks had a lunisolar calendar based on Metonic cycle, and a very good appoximation of the tropical year came only with Hipparcus, who lived in the 2nd century BC. I naively thought that a 5-year period should have been more logical, but maybe they had some scientific reason to choose a 4-year period. Does any evidence exist?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-09-12T23:51:11.020" Title="Why did Greek Olympic games take place every fourth year?" Tags="|physics|astronomy|time|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2783" PostTypeId="1" CreationDate="2015-09-10T21:13:40.510" Score="4" ViewCount="326" Body="&lt;p&gt;Because of recent hype surrounding the &lt;a href=&quot;https://physics.stackexchange.com/q/25930/2540&quot;&gt;&quot;end&quot; of the Mayan calendar&lt;/a&gt; it is nearly impossible to find an objective quantitative assessment of the accuracy and sophistication of Mayan astronomy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, how accurate were there &quot;ephemerides&quot; in their time, and extended into the present? Most popular descriptions suggest that we understand enough of their models to apply them to present times, but could we really use them to, say, predict the time of a sunrise or the right ascension of a planet. If so, with what accuracy?&lt;/p&gt;&#xA;" OwnerUserId="2027" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2019-10-02T15:00:52.737" Title="How accurate are Mayan astronomical &quot;ephemerides&quot;?" Tags="|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2784" PostTypeId="2" ParentId="2782" CreationDate="2015-09-11T01:21:09.667" Score="2" Body="&lt;p&gt;It is unknown why the 4 year period, a.k.a. the olympiad, was chosen. Most of the writings concerning the origins of the games come from 2nd century AD historians, nine centuries after the fact, and are largely mythological. &lt;a href=&quot;https://en.wikipedia.org/wiki/Ancient_Olympic_Games#Origins&quot; rel=&quot;nofollow&quot;&gt;According to one of them, Pausanias&lt;/a&gt;, &quot;dactyl&quot; Herakles and his four brothers, Paeonaeus, Epimedes, Iasius and Idas, raced at Olympia to entertain the newborn Zeus, who then crowned the victor with an olive branch, which became a peace symbol. Each year of the olympiad (inclusively) is to celebrate each brother. This is most likely a late fable invented to &quot;explain&quot; traditions the true source of which was forgotten. What little can be plausibly extracted from the myths is that the games were tied to worship of the gods, and were intended to promote and maintain peace and cooperation across Greece. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only known astronomical connection is that the games began on the full moon closest to the summer solstice, which marked the start of a new year. The date was somewhat altered after the introduction of the Metonic cycle c. 432 BC. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Metonic_cycle&quot; rel=&quot;nofollow&quot;&gt;Metonic cycle&lt;/a&gt; is close to a common multiple of the solar year and the synodic lunar month, and it is 6,940 days, about 19 years. Dividing gives 365 + 1⁄4 + 1⁄76 days per year, which comes close to rounding up the fraction every four years. Whether this played a role in the choice made earlier one can only speculate. If there was something tied to gods that all Greeks could agree on it would be what is seen in the heavens. We do know that after three centuries Greek historians started using the olympiad as a universal interval for reckoning time independently of individual conventions in city-states. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-12T23:51:11.020" LastActivityDate="2015-09-12T23:51:11.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2785" PostTypeId="2" ParentId="2783" CreationDate="2015-09-11T01:56:30.290" Score="3" Body="&lt;p&gt;An comprehensive source on this issue is &lt;a href=&quot;https://books.google.com/books?id=-kWPZwEACAAJ&amp;amp;dq=inauthor:%22Harvey+M.+Bricker%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CB8Q6wEwAGoVChMI8vD9kPLtxwIVFI6SCh0AkQKS&quot; rel=&quot;nofollow&quot;&gt;Brickers' Astronomy in the Maya Codices&lt;/a&gt;. They meticulously correlated Mayan calendar with the Western one, and systematically analyzed their Venus, Mars and eclipse tables. It turned out that a table dated to 11-12th century predicted a solar eclipse in 1991 within a day of actual occurence. Another measure of accuracy is given by &lt;a href=&quot;http://ircamera.as.arizona.edu/NatSci102/NatSci102/text/extmayaastronomy.htm&quot; rel=&quot;nofollow&quot;&gt;comparing Mayan values to Ptolemy's and the modern ones&lt;/a&gt;, and one can see that they did essentially as well as Ptolemy:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parameter (days) | Maya | Ptolemy | Modern&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lunar (synodic) month | 29.53086 | 29.53337 | 29.53059&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Synodic period of Venus | 583.92027 | 583.94267 | 583.93    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Synodic period of Mars | 780 | 779.94 | 779.94 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Solar (tropical) year | 365.242 | 365.24667 | 365.24198&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mayan observations were performed with a naked eye assisted by a pair of crossed sticks, and they did not have any kind of angular coordinates to record and predict positional data such as found in ephemerides. All the tabulated data in the surviving codices is on cycles, time periods between recurring astronomical events, solstices and equinoxes, zenial passages, eclipses, risings and settings, stations and so on, and it is about this data that the accuracy claims are made. For instance, the prediction error on eclipses and Venus's revolutions is under one day in 6000 years. Apparently, they even noticed a regularity in the motion of Mars that went unnoticed until recently found in the codices. Some numerical &lt;a href=&quot;http://www.authenticmaya.com/maya_astronomy.htm&quot; rel=&quot;nofollow&quot;&gt;data from the codices is reviewed here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-12T21:50:40.310" LastActivityDate="2015-09-12T21:50:40.310" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2786" PostTypeId="1" AcceptedAnswerId="2796" CreationDate="2015-09-11T03:43:53.240" Score="9" ViewCount="1207" Body="&lt;p&gt;Several authors (z.B.: Arthur Engel in his &lt;em&gt;Problem-Solving Strategies&lt;/em&gt;, Alexander Bogomolny in &lt;a href=&quot;http://www.cut-the-knot.org/blue/SophieGermainIdentity.shtml&quot; rel=&quot;nofollow&quot;&gt;this&lt;/a&gt; entry of the &lt;em&gt;Cut the Knot&lt;/em&gt; website) refer to the following (straightforward) consequence of &lt;em&gt;the difference of squares factorization formula&lt;/em&gt; as the Sophie Germain Identity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A^{4}+4B^{4} = (A^{2}+2B^{2}+2AB)(A^{2}+2B^{2}-2AB)$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anybody here explain why it is that this identity was christened thus? Do we actually have any evidence that it can be found somewhere in the Nachlaß of M. Le Blanc or the said identity is yet another instance of the famed &lt;a href=&quot;https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy&quot; rel=&quot;nofollow&quot;&gt;Stigler law of eponymy&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="3869" LastEditDate="2022-04-23T03:06:02.133" LastActivityDate="2022-04-23T03:06:02.133" Title="Why is the Sophie Germain Identity called thus?" Tags="|terminology|number-theory|debunking|sophie-germain|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2787" PostTypeId="2" ParentId="2782" CreationDate="2015-09-11T12:02:56.363" Score="2" Body="&lt;p&gt;The estimate of the length of the tropical year and the synodic month by Meton, Hipparchus and others has no bearing on the system of four-year Olympiads. The use of a 365-day year with a leap day every four years was introduced by Julius Caesar in the first century BC. A similar calendar had previously been decreed in Egypt in 238 BC, but never actually implemented. The dating of events according to Olympiads can be traced back at the latest to Thucydides (5th century BC).&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-09-11T16:04:32.367" LastActivityDate="2015-09-11T16:04:32.367" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2788" PostTypeId="1" AcceptedAnswerId="2795" CreationDate="2015-09-11T18:08:15.973" Score="8" ViewCount="455" Body="&lt;p&gt;According to &lt;a href=&quot;http://www.cut-the-knot.org/proofs/AfterEuler.shtml&quot; rel=&quot;nofollow&quot;&gt;this website&lt;/a&gt;, we see that Euler formally proved (observed?) his result connecting the harmonic series and a product over primes. Riemann wrote his historical paper in 1859 which considers the same observation, but now with $$\zeta(s)=\sum_{n=1}^\infty n^{-s}=\prod_p\left(1-p^{-s}\right)^{-1}$$where $s\in\Bbb C$. According to the same website above, it is said that Kronecker proved in 1876 the identity $\sum_{n=1}^\infty n^{-s}=\prod_p\left(1-p^{-s}\right)^{-1}$ for $s&amp;gt;1$. If this is the case, how well was Riemann's paper received by modern mathematicians at the time? It would seem to be based on something unproven. Or is it more likely something that all of the professional mathematicians would find obvious; hence, not worth publishing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've attempted to find some references regarding Kronecker, but MathSciNet didn't pull anything up (the publications jump from 1869 to 1881).&lt;/p&gt;&#xA;" OwnerUserId="2860" LastEditorUserId="55" LastEditDate="2015-09-15T02:46:07.297" LastActivityDate="2015-09-15T02:46:07.297" Title="Was Riemann's zeta function paper based on Euler's unproven observation?" Tags="|mathematics|mathematicians|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2789" PostTypeId="1" CreationDate="2015-09-11T19:06:33.943" Score="7" ViewCount="568" Body="&lt;p&gt;There are quite a few Nobel Prize winners (e.g. Physics) who worked on the Manhattan project in the development of the atomic bomb.  But, did any Nobel Prize result from direct research performed on the Manhattan Project?  For example, although Feynman was a key participant in the Manhattan project, his Nobel did not derive from work specific to the development of the bomb.  But, are their other Nobel Prize winners in Physics (or, Chemistry) whose work can be considered a descendent of work performed in support of the work on the atomic bomb?&lt;/p&gt;&#xA;" OwnerUserId="2075" LastEditorUserId="26" LastEditDate="2015-09-11T19:36:30.177" LastActivityDate="2015-09-12T18:02:59.223" Title="Nobel Prize work derived from Manhattan Project research?" Tags="|physics|nobel-prize|world-war-2|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2790" PostTypeId="2" ParentId="2788" CreationDate="2015-09-11T20:45:03.017" Score="3" Body="&lt;p&gt;The formula you wrote is indeed due to Euler. Whether he did have a rigorous proof or not is not important. At the time of Riemann giving a rigorous proof of this was a relatively easy exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riemann's famous paper contains MUCH MORE than this formula, which is only a starting point. You should look to his paper (or to the enormous literature generated by it) before asking such question. Open any book titled Analytic number theory. It will contain an exposition of this paper, many of these books in English.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-11T20:45:03.017" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2791" PostTypeId="1" CreationDate="2015-09-12T12:07:45.687" Score="6" ViewCount="242" Body="&lt;p&gt;I'm looking for a &quot;pop maths&quot; book on the subject. Something much more accessible than &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0412993716&quot; rel=&quot;noreferrer&quot;&gt;Dieudonné&lt;/a&gt;: &quot;light reading&quot; with emphasis on the history, personalities and general ideas and minimal technicalities. It needn't be comprehensive, or even get as far as the 20th century developments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does such a thing exist?&lt;/p&gt;&#xA;" OwnerUserId="2864" LastEditorUserId="26" LastEditDate="2015-09-12T14:29:04.187" LastActivityDate="2016-08-30T14:23:39.280" Title="History of Algebraic Geometry for a general readership?" Tags="|algebraic-geometry|resource-recommendation|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2792" PostTypeId="1" AcceptedAnswerId="2794" CreationDate="2015-09-12T17:42:34.947" Score="3" ViewCount="1073" Body="&lt;p&gt;In the &lt;a href=&quot;https://www.youtube.com/watch?v=9rIy0xY99a0&quot; rel=&quot;nofollow&quot;&gt;What is Random?&lt;/a&gt; vlog of the Vsauce channel, Michael says (start from 3:25):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the 1300s, random meant running or at great speed. Later, it would be used to describe things that have no definite purpose. It wasn't until the 1800s that random took on a particular mathematical definition. Then in the 1970s, MIT's student paper popularised the use of the word random to simply mean strange.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, it must after 1970s that the word &quot;random&quot; started to have the meaning as we know today. So before that, what word was used instead, both for mathematicians/scientists and normal people?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And why did the word &quot;random&quot; spread out, when there was already another exactly term for that?&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="658" LastEditDate="2015-09-14T22:26:38.783" LastActivityDate="2015-09-16T21:47:34.277" Title="What word meaning &quot;random&quot; was used before the word &quot;random&quot; got popularized?" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2793" PostTypeId="2" ParentId="2789" CreationDate="2015-09-12T18:02:59.223" Score="7" Body="&lt;p&gt;Seaborg received a Chemistry Nobel in 1951 for exploring the chemistry of transuranic elements.  That presumably would include his work with Plutonium during the Manhattan project.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm pretty sure that's the only one (and it admittedly only sort of counts).  Most of the scientific discoveries necessary for the bomb were done in the late 30's.  The Manhattan project itself was mainly concerned with the resulting engineering problems.  &lt;/p&gt;&#xA;" OwnerUserId="1731" LastActivityDate="2015-09-12T18:02:59.223" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2794" PostTypeId="2" ParentId="2792" CreationDate="2015-09-12T18:36:01.150" Score="5" Body="&lt;p&gt;I would say &lt;strong&gt;hazard&lt;/strong&gt; or &lt;strong&gt;hasard&lt;/strong&gt; (or for an adjective, &lt;strong&gt;fortuitous&lt;/strong&gt; or &lt;strong&gt;fortuit&lt;/strong&gt;), as in&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;C. Huygens, &lt;a href=&quot;http://dbnl.nl/tekst/huyg003oeuv14_01/huyg003oeuv14_01_0004.php&quot; rel=&quot;nofollow&quot;&gt;Du calcul dans les jeux de &lt;strong&gt;hasard&lt;/strong&gt;&lt;/a&gt; (Manuscript, 1656-1657).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A. Furetière, &lt;a href=&quot;https://books.google.com/books?id=3blKAAAAcAAJ&amp;amp;pg=PT140&quot; rel=&quot;nofollow&quot;&gt;Dictionnaire Universel&lt;/a&gt; (Leers, 1690): &quot;&lt;strong&gt;Hasard&lt;/strong&gt;. f.m. Cas &lt;strong&gt;fortuit&lt;/strong&gt;, ce qui arrive sans cause apparente ou nécessaire. (...) Ce mot vient de l'Espagnol &lt;a href=&quot;https://books.google.com/books?id=cnlFAAAAcAAJ&amp;amp;pg=PT82&amp;amp;dq=Azar&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;azar&lt;/em&gt;&lt;/a&gt;, qui signifie un &lt;em&gt;as&lt;/em&gt;, et qui se prend aussi pour le &lt;em&gt;hasard&lt;/em&gt; du dé.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;J. Arbuthnot, &lt;a href=&quot;http://quod.lib.umich.edu/cgi/t/text/text-idx?c=eebo;idno=A25748&quot; rel=&quot;nofollow&quot;&gt;Of the Laws of Chance, or, A Method of Calculation of the &lt;strong&gt;Hazards&lt;/strong&gt; of Game&lt;/a&gt; (London, 1692) (a translation of Huygens' 1657 &lt;em&gt;De Ratiociniis in Ludo Aleae&lt;/em&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;P. R. de Montmort, &lt;a href=&quot;http://dx.doi.org/10.3931/e-rara-5352&quot; rel=&quot;nofollow&quot;&gt;Essay d'Analyse sur les Jeux de &lt;strong&gt;Hazard&lt;/strong&gt;&lt;/a&gt; (Paris, 1708).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;J. Bernoulli, &lt;a href=&quot;https://books.google.com/books?id=LG1UAAAAYAAJ&amp;amp;pg=PA384&quot; rel=&quot;nofollow&quot;&gt;Mémoire sur un problème de la doctrine du &lt;strong&gt;hazard&lt;/strong&gt;&lt;/a&gt;, Hist. Acad. Roy. Berlin &lt;strong&gt;1768&lt;/strong&gt; 384-408.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;P. S. de Laplace, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k77596b/f10&quot; rel=&quot;nofollow&quot;&gt;Mémoire sur les suites récurro-récurrentes et sur leurs usages dans la theorie des &lt;strong&gt;hasards&lt;/strong&gt;&lt;/a&gt;, Mém. Acad. Roy. Sci. Paris &lt;strong&gt;6&lt;/strong&gt; (1774) 353-371.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;J. L. Lagrange, &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1775&amp;amp;seite:int=265&quot; rel=&quot;nofollow&quot;&gt;Recherches sur les suites récurrentes dont les termes varient de plusieurs manières différentes, ou sur l'intégration des équations linéaires aux différences finies et partielles; et sur l'usage de ces équations dans la théorie des &lt;strong&gt;hasards&lt;/strong&gt;&lt;/a&gt;, Nouv. Mém. Acad. Roy. Sci. Berlin &lt;strong&gt;1775&lt;/strong&gt; 183-272.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;P. Prévost, &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1780&amp;amp;seite:int=501&quot; rel=&quot;nofollow&quot;&gt;Sur les principes de la théorie des gains &lt;strong&gt;fortuits&lt;/strong&gt;&lt;/a&gt;, Nouv. Mém. Acad. Roy. Sci. Berlin &lt;strong&gt;1780&lt;/strong&gt; 430-472. (&lt;a href=&quot;https://books.google.com/books?id=cQIwAAAAYAAJ&amp;amp;pg=PA19&amp;amp;dq=fortuitous&quot; rel=&quot;nofollow&quot;&gt;Review&lt;/a&gt;: &quot;The art of calculating &lt;strong&gt;fortuitous&lt;/strong&gt; events is not above a century and a half old.&quot;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;At random&lt;/em&gt; (which seems to have been originally an &lt;a href=&quot;https://books.google.com/books?id=Ru-o5SnehxkC&amp;amp;pg=PT533&amp;amp;dq=random&quot; rel=&quot;nofollow&quot;&gt;artillery&lt;/a&gt; &lt;a href=&quot;https://books.google.com/books?id=CFBGAAAAYAAJ&amp;amp;pg=PT700&amp;amp;dq=random&quot; rel=&quot;nofollow&quot;&gt;expression&lt;/a&gt;) translates into &lt;em&gt;au hasard&lt;/em&gt; in &lt;a href=&quot;https://books.google.com/books?id=1IZaAAAAQAAJ&amp;amp;pg=PT455&amp;amp;dq=at%20random&quot; rel=&quot;nofollow&quot;&gt;Boyer&lt;/a&gt; (1711) and again in Laplace's &lt;a href=&quot;https://books.google.com/books?id=f7Kv2iFUNJoC&amp;amp;pg=PA359&amp;amp;dq=random&quot; rel=&quot;nofollow&quot;&gt;System of the World&lt;/a&gt; (1809; &lt;a href=&quot;https://archive.org/stream/ed6expositiondusy00lapl#page/444&quot; rel=&quot;nofollow&quot;&gt;original&lt;/a&gt;); the phrase &lt;em&gt;to hazard at random&lt;/em&gt; is found in &lt;a href=&quot;https://books.google.com/books?id=uzlfAAAAcAAJ&amp;amp;pg=PA132&amp;amp;dq=random&quot; rel=&quot;nofollow&quot;&gt;Voltaire&lt;/a&gt; (1760) and &lt;a href=&quot;https://books.google.com/books?id=SXQSAAAAIAAJ&amp;amp;pg=PA309&amp;amp;dq=random&quot; rel=&quot;nofollow&quot;&gt;Adams&lt;/a&gt; (1794).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt; I now believe that the &lt;em&gt;adjective&lt;/em&gt; you are really after is the French &lt;a href=&quot;http://www.cnrtl.fr/definition/al%C3%A9atoire&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;aléatoire&lt;/strong&gt;&lt;/a&gt; or its German equivalent &lt;a href=&quot;http://www.dict.cc/deutsch-englisch/zuf%C3%A4llig.html&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;zufällig&lt;/strong&gt;&lt;/a&gt; (whichever came first), as in&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;événement aléatoire (Lacroix &lt;a href=&quot;https://books.google.com/books?id=yg8OAAAAQAAJ&amp;amp;pg=PA260&amp;amp;dq=al%C3%A9atoires#v=onepage&amp;amp;q=al%C3%A9atoires&quot; rel=&quot;nofollow&quot;&gt;1816&lt;/a&gt;, Cournot &lt;a href=&quot;https://books.google.com/books?id=_fk3AAAAMAAJ&amp;amp;q=%22%C3%A9v%C3%A9nement%20al%C3%A9atoire%22&quot; rel=&quot;nofollow&quot;&gt;1843&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;zufällige Werthe (Hauber &lt;a href=&quot;https://books.google.com/books?id=LiIFAAAAQAAJ&amp;amp;pg=PA26&amp;amp;dq=zuf%C3%A4lligen&quot; rel=&quot;nofollow&quot;&gt;1830&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;zufällige Ereignisse (Cournot &lt;a href=&quot;https://books.google.com/books?id=AP8GAAAAcAAJ&amp;amp;pg=PA24&amp;amp;dq=zuf%C3%A4lligen&quot; rel=&quot;nofollow&quot;&gt;1849&lt;/a&gt;, Herr &lt;a href=&quot;https://books.google.com/books?id=hENLAAAAMAAJ&amp;amp;pg=PA2&amp;amp;dq=%22zuf%C3%A4lligen%20Ereignisses%22&quot; rel=&quot;nofollow&quot;&gt;1887&lt;/a&gt;, Bernoulli &lt;a href=&quot;https://books.google.com/books?id=i1ReAAAAcAAJ&amp;amp;pg=PA73&amp;amp;dq=zuf%C3%A4lligen&quot; rel=&quot;nofollow&quot;&gt;1899&lt;/a&gt;, Czuber &lt;a href=&quot;https://books.google.com/books?id=g5cjAQAAMAAJ&amp;amp;pg=PA764&amp;amp;dq=zuf%C3%A4lligen&quot; rel=&quot;nofollow&quot;&gt;1901&lt;/a&gt;, v. Mangoldt &lt;a href=&quot;https://books.google.com/books?id=e7NXAAAAYAAJ&amp;amp;pg=PA36&amp;amp;dq=zuf%C3%A4lligen&quot; rel=&quot;nofollow&quot;&gt;1911&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;zufällige Grösse (Hagen &lt;a href=&quot;https://books.google.com/books?id=g-cGAAAAYAAJ&amp;amp;pg=PA53&amp;amp;dq=%22zuf%C3%A4lligen%22#v=onepage&amp;amp;q=%22zuf%C3%A4lligen%22&quot; rel=&quot;nofollow&quot;&gt;1867&lt;/a&gt;, Kolmogorov &lt;a href=&quot;http://dx.doi.org/10.1007/978-3-642-49888-6&quot; rel=&quot;nofollow&quot;&gt;1933&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;nombre aléatoire (Fréchet &lt;a href=&quot;https://books.google.com/books?id=eUBtAAAAMAAJ&amp;amp;q=al%C3%A9atoire&quot; rel=&quot;nofollow&quot;&gt;1924&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;variable aléatoire (Borel &lt;a href=&quot;https://books.google.com/books?id=njDvAAAAMAAJ&amp;amp;q=%22variable+al%C3%A9atoire%22&quot; rel=&quot;nofollow&quot;&gt;1925&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It would be interesting if you could pinpoint when &quot;in the 1800s&quot; the adjective &lt;em&gt;random&lt;/em&gt; took on a mathematical meaning comparable to the above.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2015-09-16T21:47:34.277" LastActivityDate="2015-09-16T21:47:34.277" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="2795" PostTypeId="2" ParentId="2788" CreationDate="2015-09-12T20:15:50.943" Score="7" Body="&lt;p&gt;Here is &lt;a href=&quot;http://www.claymath.org/sites/default/files/ezeta.pdf&quot; rel=&quot;noreferrer&quot;&gt;Wilkins's translation of Riemann's paper&lt;/a&gt;. Remarks at the start of &lt;a href=&quot;http://www.claymath.org/sites/default/files/ezeta.pdf&quot; rel=&quot;noreferrer&quot;&gt;the paper&lt;/a&gt; show that Riemann was well aware of the convergence issues:&quot;&lt;em&gt;For this investigation my point of departure is provided by the observation of Euler that the product&lt;/em&gt; $\prod\frac{1}{1-\frac{1}{p^{s}}}=\sum\frac{1}{n^{s}}$ &lt;em&gt;if one substitutes for $p$ all prime numbers, and for $n$ all whole numbers.&#xA;The function of the complex variable $s$ which is represented by these two expressions, wherever they converge, I denote by $\zeta(s)$. Both expressions converge only when the real part of $s$ is greater than $1$; at the same time an expression for the function can easily be found which always remains valid&lt;/em&gt;&quot;. The reference here is not to Euler's 1737 &quot;factorization&quot; of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity for $s&amp;gt;1$ appears. Although Euler did not work with convergence in the modern sense he already knew the difference between $s=1$ and $s&amp;gt;1$ cases since he summed the Basel series in 1735 ($s=2$, later all even $s$), and used unlimited growth of &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2006-03.pdf&quot; rel=&quot;noreferrer&quot;&gt;harmonic series to prove infinitude of primes&lt;/a&gt; in that 1737 paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cauchy addressed convergence of the $p$-series for real $s$ in his texts, and the infinite product reduces to a sum by logarithming, a trick used by Riemann later in the paper, and by others before him. In fact, Dirichlet used analogous identity for $L$ functions, with a character $\chi$ on top of the fractions, in his famous 1837 paper on primes in arithmetic progressions. And in 1848 Chebychev presented to St. Petersburg Academy his results, published in early 1850s, on the asymptotics of the count of primes (a version of the &quot;prime number theorem&quot;), where he logarithms the Euler product for real $s&amp;gt;1$, and derives an identity for $\zeta(s)$ (not under this name, of course). In 1859, when the paper was published, Riemann was familiar with the work of Dirichlet and likely Chebychev (who is mentioned in his unpublished notes), so he saw replacing real $s$ with complex one, and rephrasing the convergence in terms of real parts, as a minor issue. It appears that Kronecker simply wrote down what was a well known folklore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main insight of the paper is a close relation discovered by Riemann between specific traits of the distribution of primes and locations of the zeros of the $\zeta$ function, which lead to strengthened formulations of the prime number theorem. The problem was of high interest to contemporary mathematicians, with Gauss and Bertrand in addition to Dirichlet and Chebychev making important contributions prior to Riemann's work. The paper redirected the further work on the prime number theorem into analytic vein leading to Hadamard's and Vallee Poussin's 1896 results on $\zeta$ zeros. See &lt;a href=&quot;http://www.claymath.org/sites/default/files/official_problem_description.pdf&quot; rel=&quot;noreferrer&quot;&gt;Bombieri's exposition&lt;/a&gt; in the millenium problem description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This being said, the standards of rigor weren't modern ones even after Gauss. In another paper Riemann used the Dirichlet principle to prove existence of a holomorphic function with specified poles on a Riemann surface, the kind of variational existence argument later famously criticized by Weierstrass. According to &lt;a href=&quot;http://www.ams.org/journals/bull/1928-34-04/S0002-9904-1928-04589-5/S0002-9904-1928-04589-5.pdf&quot; rel=&quot;noreferrer&quot;&gt;Klein's Development of Mathematics in the 19th Century&lt;/a&gt;, this did not prompt anyone, Weierstrass included, to doubt his results. To paraphrase, rigor is a friend but insight is a better friend. This hasn't changed if Witten's Fields medal and the vast literature on mirror symmetry are any indications. Perelman left out many technical steps and conspicuously declined to publish his proof of geometrization in refereed journals, and so on.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-13T02:18:05.313" LastActivityDate="2015-09-13T02:18:05.313" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2796" PostTypeId="2" ParentId="2786" CreationDate="2015-09-12T21:22:28.707" Score="6" Body="&lt;p&gt;This is one of those cases where the significance of a result is not in its content but in its use. The identity itself is trivial to verify, but behind it is a trick that is unreasonably useful in number theoretic reasoning. It is that that Sophie Germain pointed out in her famous analysis of the last Fermat theorem, and it is that that apparently prompted Dickson to assign the authorship to her. The identity has since found many other applications, including to &lt;a href=&quot;http://www.cut-the-knot.org/blue/SophieGermainIdentity.shtml&quot; rel=&quot;nofollow&quot;&gt;olympiad problems&lt;/a&gt; like showing that $2015^{4}+4^{2015}$ isn't prime or solving $3^r+4^s=5^t$ in non-negative integers, &lt;a href=&quot;https://topologicalmusings.wordpress.com/2008/02/20/sophie-germain-identity&quot; rel=&quot;nofollow&quot;&gt;more here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sophie Germain does not quite state it in Dickson's form however. She writes &quot;&lt;em&gt;No number of the form $p^4+4$ except $5$ is a prime number, because $p^4+4=(p^2-2)^2+4p^2$ and consequently these numbers are representable in more than one way as a sum of two squares. In general $p^4+q^4=(p^2-q^2)^2+2p^2q^2=(p^2+q^2)^2-2p^2q^2$. Thus when $p^4+q^4$ is a prime one is sure that the number is only thus expressible as the sum of a square $+$ or $-$ twice a square. The numbers $2^{2^i}+1$ are just a special case of the form $p^4+q^4$&lt;/em&gt;&quot;. The identity used here follows from $p^4+4q^4=(p^2+2q^2)^2-4(pq)^2$ by factorising the difference of two squares.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dickson also found a prior occurence of the same identity in a letter from Euler to Goldbach in 1742, but only in a special case $1+4x^4=(2x^2+2x+1)(2x^2-2x+1)$. See the original manuscript and references at &lt;a href=&quot;http://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf&quot; rel=&quot;nofollow&quot;&gt;Theorem of the Day&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-06T19:24:26.167" LastActivityDate="2015-10-06T19:24:26.167" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2797" PostTypeId="1" AcceptedAnswerId="2801" CreationDate="2015-09-13T17:56:39.733" Score="6" ViewCount="553" Body="&lt;p&gt;David Hilbert wrote a couple of anecdotal paragraphs regarding &quot;the limits of science.&quot; He recalled that in the early 19th Century the position of a philosopher - or philosophy in general - was that empirical science would never be able to know what stars are made of, because they are just too far away. Hilbert pointed out that just a couple of years before that remark had been made, the Fraunhofer lines had been discovered in the spectrum of the Sun (1814). Just a few decades later, Kirchhoff and Bunsen identified the same features in the spectrum of hydrogen, showing that the Sun is made out of hydrogen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember reading this in an online version of a book by Hilbert, but I don't remember which, or even it it was in English or German. I haven't been able to find a source, to quote properly. I would be grateful for any clues!&lt;/p&gt;&#xA;" OwnerUserId="2870" LastEditorUserId="4251" LastEditDate="2020-07-12T09:42:55.403" LastActivityDate="2020-07-12T09:42:55.403" Title="David Hilbert and the limits of science" Tags="|mathematicians|philosophy-of-science|hilbert|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2798" PostTypeId="2" ParentId="2792" CreationDate="2015-09-14T09:49:46.373" Score="4" Body="&lt;p&gt;I do not quite understand whether this is a question about history of science or about English usage. The Oxford English Dictionary points out that the phrase &quot;at random&quot; (earlier: &quot;randon&quot;) occurs in English since the 16th century with this meaning:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;haphazardly, without aim, purpose, or fixed principle; heedlessly, carelessly; (also) erratically, indiscriminately, unsystematically&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example in this quotation from 1543:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A certaine licenciousnes, or leude libertie..to ronne at randon hither, and thyther, through the wyde worlde perteygninge to no body.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-09-15T08:39:43.560" LastActivityDate="2015-09-15T08:39:43.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2799" PostTypeId="1" AcceptedAnswerId="2802" CreationDate="2015-09-14T16:43:31.190" Score="9" ViewCount="1940" Body="&lt;p&gt;I am reading Karatzas and Shreeve (Brownian Motion and Stochastic Calculus) and on page 128, at the beginning of chapter 3 on Stochastic Integrals, one reads:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BQCl8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BQCl8.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't think of any social problem that was treated by Newton and Leibniz.  Speaking of differential equations generally financial applications would count as a social problem, but that's only at the turn of the XX century. Are there any earlier applications? Do you know any? Which?&lt;/p&gt;&#xA;" OwnerUserId="2630" LastEditorUserId="4251" LastEditDate="2020-06-21T17:39:35.517" LastActivityDate="2020-06-21T17:39:35.517" Title="What are the early applications of differential equations to social sciences?" Tags="|discoveries|calculus|differential-equations|social-sciences|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2800" PostTypeId="2" ParentId="2797" CreationDate="2015-09-14T17:51:40.907" Score="4" Body="&lt;p&gt;This famous quote occurs at the end of &lt;a href=&quot;http://dx.doi.org/10.1007/BF01492194&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Naturerkennen und Logik&lt;/em&gt;, Die Naturwissenschaften &lt;strong&gt;18&lt;/strong&gt; (1930) 959-963&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Einst sagte der Philosoph COMTE — in der Absicht, ein gewiß unlösbares Problem zu nennen —, daß es der Wissenschaft nie gelingen würde, das Geheimnis der chemischen Zusammensetzung der Himmelskörper zu ergründen. Wenige Jahre später wurde durch die Spektralanalyse von KIRCHHOFF und BUNSEN dieses Problem gelöst, and heute können wir sagen, daß wir die entferntesten Sterne als wichtigste physikalische und chemische Laboratorien in Anspruch nehmen, wie wir solche auf der Erde gar nicht finden. Der wahre Grund, warum es COMTE nicht gelang, ein unlösbares Problem zu finden, besteht meiner Meinung nach darin, daß es ein unlösbares Problem überhaupt nicht gibt. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung:&lt;/p&gt;&#xA;&lt;p&gt;Wir müssen wissen,&lt;/p&gt;&#xA;&lt;p&gt;Wir werden wissen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-09-14T17:51:40.907" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2801" PostTypeId="2" ParentId="2797" CreationDate="2015-09-14T18:29:43.343" Score="2" Body="&lt;p&gt;This bit comes from Hilbert’s address Naturerkennen und Logik given in Königsberg on 8 September 1930 to the annual meeting of the Society of German Natural Scientists and Physicians. A &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/david-hilberts-radio-address&quot; rel=&quot;nofollow&quot;&gt;four minute excerpt from the speech was played on the radio&lt;/a&gt;, the recording, German transcript, and English translation are available online, courtesy of James Smith and MAA. Unfortunately the excerpt does not include the relevant passage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is &lt;a href=&quot;http://www.oswego.edu/~delancey/309_DIR/LLT_LECTURES/Hilbert.html&quot; rel=&quot;nofollow&quot;&gt;quoted in Reid's book&lt;/a&gt;:&quot;&lt;em&gt;In an effort to give an example of an unsolvable problem, the philosopher Comte once said that science would never succeed in ascertaining the secret of the chemical composition of the bodies of the universe. A few years later this problem was solved... The true reason, according to my thinking, why Comte could not find an unsolvable problem lies in the fact that there is no such thing as an unsolvable problem&lt;/em&gt;&quot;. Auguste Comte is the father of sociology and the founder of philosophical positivism who wrote in his &#xA;Cours de Philosophie Positive (1835):&quot;&lt;em&gt;We understand the possibility of determining their shapes, their distances, their sizes and their movements; whereas we would never know how to study by any means their chemical composition, or their  mineralogical  structure,  and,  even  more  so,  the  nature  of  any organized   beings   that   might   live   on   their  surface. I  persist  in  the  opinion  that  every  notion  of  the  true mean  temperatures  of he  stars  will  necessarily  always be concealed from us&lt;/em&gt;&quot;. As it happens,  Joseph Fraunhofer studied  the  line  spectra  of  the  Sun  and  some bright stars in 1814 and 1823.  In 1849 Foucault observed the coincidence of D lines in solar spectrum with the lines emitted by sodium in  the  laboratory, which led Kirchhoff to the expected conclusion a decade later. In 1860 Bunsen  published  a  study  of  the  emission line spectra of many chemical element, opening the way to systematic analysis of solar composition. See &lt;a href=&quot;http://adsabs.harvard.edu/abs/2010JAHH...13...90H&quot; rel=&quot;nofollow&quot;&gt;Hearnshaw's &#xA;Auguste Comte's blunder&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Comte was not the primary target of Hilbert's speech, it was a philosophizing physiologist Emile du Bois-Reymond (not to be confused with his brother Paul, a mathematician), and his motto &quot;ignoramus et ignoramibus&quot;, we are ignorant and we shall remain ignorant. Hilbert ends his address in a forceful rejection of the sentiment, now famous:&quot;&lt;em&gt;We must not believe those, who today with philosophical bearing and a tone of superiority prophesy the downfall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion even none whatever in natural science. In place of the foolish ignorabimus let stand our slogan: We must know, We will know&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-14T18:35:57.277" LastActivityDate="2015-09-14T18:35:57.277" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2802" PostTypeId="2" ParentId="2799" CreationDate="2015-09-15T00:30:16.933" Score="7" Body="&lt;p&gt;One of the earliest works containing a &quot;mathematical model&quot; in sociology was &lt;a href=&quot;http://www.esp.org/books/malthus/population/malthus.pdf&quot; rel=&quot;nofollow&quot;&gt;Malthus's An Essay on the Principle of Population (1798)&lt;/a&gt;. It did not however contain differential equations, or even formulas for that matter, Malthus expressed that population multiplies geometrically and food arithmetically in words. Malthus's model did influence early work in political economy and sociology, and even Darwin's work. Only in 1838 Comte, the founder of philosophical positivism, declared that &quot;social physics&quot; was ready to join the ranks of empirical sciences, and dubbed it &quot;sociology&quot; (Sieyès used the word in 1780, but not in print). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But like Malthus the early sociologists, Durkheim and Weber, did not use differential equations. One does appear in &lt;a href=&quot;https://en.wikipedia.org/wiki/Logistic_function#In_ecology:_modeling_population_growth&quot; rel=&quot;nofollow&quot;&gt;Verhulst's 1838 work on population growth&lt;/a&gt;, which was motivated by Malthus's model, the differential equation for the now ubiquitous logistic growth is derived and solved there. And in &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323640_0018&amp;amp;DMDID=DMDLOG_0008&amp;amp;LOGID=LOG_0008&amp;amp;PHYSID=PHYS_0018&quot; rel=&quot;nofollow&quot;&gt;Mathematical Researches into the Law of Population Growth Increase (1845)&lt;/a&gt;  Verhulst gave the logistic its name. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_economics#Marginalists_and_the_roots_of_neoclassical_economics&quot; rel=&quot;nofollow&quot;&gt;Marginalists applied calculus methods to economics&lt;/a&gt; (marginals are economists' name for derivatives). In Cournot's Researches into the Mathematical Principles of Wealth (1838) differentiation is used to find equilibrium quantities by maximizing profit, he also anticipates the idea of non-cooperative games and the Nash equilibruium in his models of oligopoly and duopoly. In Walras's Elements of Pure Economics (1874) Cournot's approach is generalized to multiple market agents and the global equilibrium. Edgeworth modeled market behavior based on utility function in Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (1881), and later even criticized others for lack of mathematical rigor. But neither Walras, nor Edgeworth, nor Marshall in Principles of Economics (1890) use differential equations proper in their models. That had to wait until later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly the earliest entry was &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Bachelier&quot; rel=&quot;nofollow&quot;&gt;Bachelier's PhD thesis Theory of Speculation (1900)&lt;/a&gt; that somewhat informally introduced mathematical model that we now call stochastic process of Brownian motion to evaluate stock options even before Einstein and Smoluchowski first studied it in kinetic theory in 1905-06. The model leads to diffusion equations. Poincare, Bachelier's instructor, called the work &quot;&lt;em&gt;very original, and all the more interesting in that Fourier's reasoning can be extended with a few changes to the theory of errors&lt;/em&gt;&quot;. Bachelier's model of option pricing closely anticipated the Black-Scholes model widely used today, see &lt;a href=&quot;http://arxiv.org/pdf/0711.1272.pdf&quot; rel=&quot;nofollow&quot;&gt;Schachermayer-Teichmann's How Close Are the Option Pricing Formulas of Bachelier and Black-Merton-Scholes?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; While epidemiology may be borderline between biology and social sciences I found out that Bernoulli's work, pointed out in skol's answer, led to early applications of differential equations to actuarial science, which clearly belongs to the latter. According to &lt;a href=&quot;http://www.jstor.org/stable/41140783&quot; rel=&quot;nofollow&quot;&gt;Daw's Smallpox and the Double Decrement Table: a Piece of Actuarial Pre-history&lt;/a&gt; Lambert (1772) &quot;&lt;em&gt;gives the earliest application known to me of what would now be called actuarial formulae for dealing with mortality data&lt;/em&gt;&quot;. Although Lambert's formulae were based on finite differences he was well aware of Bernoulli's work and their relation to the continuous case. In his 1806 book one of the first French actuaries Duvillard combines Bernoulli type differential equations and Lambert's methods to discriminate between effects of different causes of death, his tables came to be used to estimate risk exposure in life insurance calculations. Interestingly, when later the data collection became more systematic calculations based on Bernoulli's equations became ineffective, and new ones were developed starting with Wittstein (1862), see &lt;a href=&quot;http://www.jstor.org/stable/2345318?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Multiple Decrements or Competing Risks by Seal&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-28T02:17:13.983" LastActivityDate="2016-02-28T02:17:13.983" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2804" PostTypeId="1" CreationDate="2015-09-16T16:33:44.707" Score="9" ViewCount="246" Body="&lt;p&gt;When the original beta decay experiments seemed to suggest that energy was not a conserved quantity, Landau (among others) proposed that energy conservation was a statistical law rather then a absolute one. However, I came across this in an article about Landau:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Unfortunately, the co-founder of Marxism, Friedrich Engels, had declared in&#xA;  the 19th century that the law of conservation of energy was to be forever fundamental to science, and Landau was severely castigated in the local papers&#xA;  for his (temporary) blasphemy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;http://academic.evergreen.edu/z/zita/articles/history/landau.pdf&quot;&gt;link&lt;/a&gt; to the article, the quote is on page 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone provide a link to an article that contains some more information about this story?&lt;/p&gt;&#xA;" OwnerUserId="2890" LastEditorUserId="4251" LastEditDate="2021-09-06T17:35:18.157" LastActivityDate="2021-09-06T21:24:54.773" Title="Landau doubting conservation of energy---and what followed" Tags="|physics|biographical-details|physicists|energy|landau|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2805" PostTypeId="1" AcceptedAnswerId="3669" CreationDate="2015-09-16T16:49:32.783" Score="5" ViewCount="153" Body="&lt;p&gt;I don't recall the reference, but I read once in a work of Hardy's that &quot;number-theoretic&quot; should not be used when talking about number theory, however tempting it may be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what do we say? I read &lt;a href=&quot;https://en.wikipedia.org/wiki/Number_theory&quot; rel=&quot;nofollow&quot;&gt;on Wikipedia&lt;/a&gt; that the word &quot;arithmetical&quot; is preferred.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is &quot;number-theoretical&quot; wrong? Is &quot;arithmetical&quot; the correct word to use? Recently I've been looking into the Number Theoretical Transform (NNT). Ought this be called the &quot;Arithmetical Transform&quot; instead?&lt;/p&gt;&#xA;" OwnerDisplayName="user2114" LastEditorDisplayName="user2114" LastEditDate="2015-10-19T06:33:53.967" LastActivityDate="2016-04-17T15:19:22.573" Title="&quot;Number Theoretic&quot; is wrong ? &quot;Arithmetical&quot; is right?" Tags="|mathematics|language|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2806" PostTypeId="1" AcceptedAnswerId="2808" CreationDate="2015-09-16T18:34:32.867" Score="6" ViewCount="230" Body="&lt;p&gt;&quot;Any function addition law is due to an elliptic curve lurking in the background.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was reading about the origin of the concept of a genus, I came across a quote along these lines, I believe the quote was attributed to Weierstrass. I have been unable to find it again! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the source/precise wording of this quote?&lt;/p&gt;&#xA;" OwnerUserId="2787" LastEditorUserId="26" LastEditDate="2015-09-21T06:54:46.190" LastActivityDate="2015-09-21T06:54:46.190" Title="Source for Weierstrass's quote &quot;Any function addition law is due to an elliptic curve lurking in the background.&quot;" Tags="|examples|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2808" PostTypeId="2" ParentId="2806" CreationDate="2015-09-17T01:19:05.710" Score="4" Body="&lt;p&gt;This is an informal statement of the Weierstrass theorem: If $F$ is a polynomial in three variables, and $f$ solves $F(f(x+y),f(x),f(y))=0$, then $f$ is an elliptic function, possibly degenerate. Weierstrass did not publish it: apparently it comes from his lectures. Unfortunately, many of his lectures were not recorded, or no record survives. The first published proof that I know is due to Phragmen. &#xA;But it is widely known as Weierstrass theorem.&#xA;There was a lot of generalizations.&#xA;A distant descendent is the Chevalley Structure theorem on algebraic groups &lt;a href=&quot;https://en.wikipedia.org/wiki/Chevalley%27s_structure_theorem&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Chevalley%27s_structure_theorem&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-17T01:19:05.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2809" PostTypeId="1" AcceptedAnswerId="2816" CreationDate="2015-09-17T02:43:28.887" Score="3" ViewCount="491" Body="&lt;p&gt;I am trying to find a classical source for the following theorem about parallel lines and transversals:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If three parallel lines are cut by two transversals, then the segments&#xA;  between the parallels along one of the transversals are in the same&#xA;  ratio as the corresponding segments along the other transversal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(For an illustration of this theorem, see &lt;a href=&quot;http://hotmath.com/hotmath_help/topics/parallel-lines-and-transversals.html&quot; rel=&quot;nofollow&quot;&gt;http://hotmath.com/hotmath_help/topics/parallel-lines-and-transversals.html&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This theorem is a standard part of the conventional secondary Geometry course in the United States (and probably elsewhere?), not least because it lends itself naturally to computational problems that can easily be tested on standardized assessments. An added benefit from a pedagogical perspective is that it has a naturally-stated converse that is superficially plausible but false.  (It also has a more clumsily-stated converse that is true.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assumed that I would find this in Euclid's &lt;em&gt;Elements&lt;/em&gt;, but the closest I can find there is Book VI, Proposition 2, which establishes an analogous result for a line parallel to one side of a triangle and intersecting the other two sides. The theorem I am interested in can be proved fairly easily as a corollary of VI.2, but as far as I can see Euclid does not state it explicitly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is a classical source for this theorem?  (I am also interested in any historical background on its converse.)&lt;/p&gt;&#xA;" OwnerUserId="294" LastActivityDate="2015-09-17T22:04:29.243" Title="Classical source for theorem on three parallel lines cut by two transversals" Tags="|mathematics|reference-request|euclidean-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2810" PostTypeId="1" AcceptedAnswerId="2812" CreationDate="2015-09-17T08:27:53.203" Score="10" ViewCount="967" Body="&lt;p&gt;According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Russell%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's paradox&lt;/a&gt;, which ultimately lead to the development of Zermelo-Fraenkel set theory, thus putting mathematics back on solid formal foundations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, how much of a revolution did this process cause in the rest of mathematics, outside of the formal foundations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine two extremes, with the truth presumably lying somewhere in between. On the one hand one could imagine that a crack in the foundations would break the whole of mathematics, with most theorems even in quite applied topics needing to be re-derived along quite different lines within the new system, in a process that would take many years. On the other hand I can imagine it not really making much difference at all, with most of the higher-level results being somehow independent of the low-level stuff below them, so that the old foundations could be swapped out and new ones put in without disturbing the structures that were built on top of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I had to guess, I would say it was closer to the latter, since when reasoning about higher or applied mathematics we rarely have to go right down to the axioms of set theory. But I would appreciate knowing historically how it played out.&lt;/p&gt;&#xA;" OwnerUserId="595" LastEditorUserId="595" LastEditDate="2020-07-06T02:21:27.267" LastActivityDate="2020-07-06T02:21:27.267" Title="How much of mathematics did Russell's paradox really break?" Tags="|mathematics|mathematical-logic|set-theory|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2811" PostTypeId="1" CreationDate="2015-09-17T10:41:26.473" Score="8" ViewCount="1030" Body="&lt;p&gt;In Academia SE, there is a question about the credibility of Prof. Richard Duffin, who wrote the notorious letter of recommendation for John Nash, who later received the Nobel Memorial Prize in Economic Sciences. Its answers focus on the credibility of Duffin to the academic/mathematic world. However, since this letter was sent directly to Prof. Solomon Lefschetz, not the adcom, so I have another theory that Duffin had directly credibility to Lefchetz, that is they were friends, or at least know each other personally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know the relationship between these two professors? I have check their Wikipedia page, but I can't find any evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;• &lt;a href=&quot;https://academia.stackexchange.com/q/29642/14341&quot;&gt;Will a recommendation letter such as that written for John F. Nash be convincing?&lt;/a&gt; on Academia&lt;br&gt;&#xA;• &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Forbes_Nash,_Jr.&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Forbes Nash, Jr&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Duffin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard Duffin&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Solomon_Lefschetz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solomon Lefchetz&lt;/a&gt; on Wikipedia&lt;br&gt;&#xA;Image taken from the &lt;a href=&quot;http://findingaids.princeton.edu/collections/AC105/c3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graduate Alumni Records&lt;/a&gt; of Princeton University:&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p7gtk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p7gtk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="4251" LastEditDate="2022-04-26T16:15:36.800" LastActivityDate="2022-04-26T16:15:36.800" Title="About the LOR of John Nash, was there any relationship between Richard Duffin and Solomon Lefschetz?" Tags="|mathematicians|mathematics-social-history|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="2812" PostTypeId="2" ParentId="2810" CreationDate="2015-09-17T12:24:45.253" Score="16" Body="&lt;p&gt;Your guess is right. Russell's paradox broke only what people considered at that time as foundations of set theory. More specifically, the logical foundation system built by Frege. Of course this was very disturbing, because many people understood that&#xA;logic and set theory is really the foundation of the whole mathematics. However it did not &quot;invalidate&quot; any theorems outside set theory and some closely connected areas, like the new theory of functions of the real variable. I am sure that most mathematicians, doing for example differential equations or functions of complex variable or group theory or geometry, did not care much about Russell's paradox. Soon, many theories of foundations were developed to avoid Russell's paradox and similar paradoxes. One is due to Russell himself (it is called the theory of types), another is Intuitionism. Eventually most mathematicians settled with ZF system.&#xA;Intuitionism (which later evolved to Constructive Mathematics) founded by Brouwer was the most radical attempt to save the foundations. It indeed rejected much of the classical mathematics. Discussions about Intuitionism continued well into the second half of the 20s century.&#xA;But most mathematicians working in other areas than foundations were not really very interested in these discussions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-17T12:24:45.253" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2814" PostTypeId="2" ParentId="2799" CreationDate="2015-09-17T20:14:11.870" Score="1" Body="&lt;p&gt;What about Adolphe Quetelet? Didn't his work with social sciences data inspire Maxwell and Boltzmann's efforts modeling the distribution of velocities in a population of gas particles? See: &lt;a href=&quot;https://en.wikipedia.org/wiki/Adolphe_Quetelet&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Adolphe_Quetelet&lt;/a&gt;   &lt;/p&gt;&#xA;" OwnerUserId="2141" LastActivityDate="2015-09-17T20:14:11.870" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2816" PostTypeId="2" ParentId="2809" CreationDate="2015-09-17T22:04:29.243" Score="2" Body="&lt;p&gt;This is essentially the &quot;intercept theorem&quot; in the broad formulation of older textbooks (e.g. Pogorelov's): parallels cut the opposite sides of an angle in the same ratios. That transversals form an angle is unimportant. The minimal number of parallels required to make this non-vacuous is three, making two segments, unless we use the vertex (as Wikipedia does) to make do with just two. The &quot;tradition&quot; attributes this theorem to Thales, but it is not listed by Eudemus, who is most reliable on early geometers, and of three accounts on which the attribution is based descriptions in two do not require it. &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Thales.html&quot; rel=&quot;nofollow&quot;&gt;The third is Plutarch&lt;/a&gt;, who, even if we take him at his word, only writes:&quot;&lt;em&gt;without trouble or the assistance of any instrument merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the impact of the sun's rays... showed that the pyramid has to the stick the same ratio which the shadow has to the shadow&lt;/em&gt;&quot;. This is the formulation in terms of similar triangles of which Elements VI.2 is a precise statement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is little doubt that the property in terms of cutting parallels would have been easily recognized by Greek geometers. &lt;a href=&quot;https://projecteuclid.org/euclid.bams/1183544897&quot; rel=&quot;nofollow&quot;&gt;Aristotle discusses such cutting&lt;/a&gt; for parallel transversals, but in relation to areas, and it occurs in vertical section when cutting pyramids and cones by planes a la Democritus. Archimedes does it in the Method, but he does not need anything like the &quot;intercept theorem&quot; specifically. I suspect that the earliest source for the intercept formulation will not be classical because the word &quot;intercept&quot; (Latin &quot;taken in between&quot;) only came into usage c. 1400.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the converse, the simplest case would be the &quot;midsegment theorem&quot;: line through the midpoints of the sides of a triangle is parallel to its base. Euclid does not state anything like this either, neither does Archimedes, but a use of it is buried in his &lt;a href=&quot;http://jwilson.coe.uga.edu/emt668/EMAT6680.F99/Erbas/emat6690/essay1/essay1.html&quot; rel=&quot;nofollow&quot;&gt;Quadrature of the Parabola&lt;/a&gt; (when proving that consecutive triangles are as 1:4). Combing through proofs in Apollonius's Conic Sections one can probably find more of both since he is fond of cutting conics by multiple parallels (in anticipation of oblique coordinates), but I doubt that he so much as nods at this specifically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Singling out these statements and giving them names most likely occured during &quot;didactization&quot; of Euclid by textbook authors in 18-19th centuries, if not later. What is focused on and how it is parsed was different in the age of standardized education, and this is not the only example. Playfair's axiom (two intersecting lines are not both parallel to a third) is just the contrapositive of Elements I.30, and Proclus gives a lemma that if a line cuts one of the parallels it cuts the other also. But if we want exactly the modern formulation we have to go all the way to Playfair in 1795. Similarly, &#xA;Acerbi writes that &quot;&lt;em&gt;the historiographical debate about the&lt;/em&gt; &lt;a href=&quot;http://www.jstor.org/stable/41287704?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;first appearance of the focus–directrix property&lt;/em&gt;&lt;/a&gt; &lt;em&gt;is biased by the relevance that this characterization has assumed in the modern characterization of conic sections. Of this property, there is no trace in Apollonius’ Conica&lt;/em&gt;&quot; and it is &quot;&lt;em&gt;reduced to the status of mere functionality to a pointwise determination of&#xA;a parabola&lt;/em&gt;&quot; in Diocles’ On Burning Mirrors.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-09-17T22:04:29.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2817" PostTypeId="1" AcceptedAnswerId="2822" CreationDate="2015-09-18T01:47:08.947" Score="3" ViewCount="109" Body="&lt;p&gt;In the English Language section, &lt;a href=&quot;https://english.stackexchange.com/q/260495/129806&quot;&gt;this question&lt;/a&gt; was asked. I posted the beginnings of an answer, and invited others to comment so that I could edit my answer and improve it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to know about the history of optics, and the field of optics.  My questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If someone today says they are in the field of optics, I would think that they work with lenses, light, lasers, etc. I would not think that they work with the human eye and the visual system. Is this true?&lt;/li&gt;&#xA;&lt;li&gt;At one time, it seems that the field of optics did include interest in how humans see. At what point did this interest in the eye start shifting to medicine and away from physics?&lt;/li&gt;&#xA;&lt;li&gt;Many sources say the ancients were interested in optical illusions.  I cannot find what these were. What were they?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2900" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:16.393" LastActivityDate="2015-09-26T13:58:24.890" Title="History of optics" Tags="|optics|" AnswerCount="2" CommentCount="5" ClosedDate="2015-09-26T00:36:37.557" ContentLicense="CC BY-SA 3.0" />
  <row Id="2818" PostTypeId="1" AcceptedAnswerId="2819" CreationDate="2015-09-18T05:19:26.593" Score="11" ViewCount="1622" Body="&lt;p&gt;Though the planets of Venus, Earth, Mars, Jupiter, and Saturn seemed to obey Kepler's laws, &lt;a href=&quot;http://archive.ncsa.illinois.edu/Cyberia/NumRel/EinsteinTest.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mercury did not&lt;/a&gt;. I just found &lt;a href=&quot;http://io9.com/the-200-year-old-mystery-of-mercurys-orbit-solved-1458642219&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt; which states that the discrepancy was noted very soon after newton published his laws of motion. Since a law of nature is obviously not true if there exists a counter example, &lt;strong&gt;why were Newton's laws accepted despite the obvious counter example?&lt;/strong&gt; Was science less rigorous during the time of Newton?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note that this question was translated into English by my father, who is the one typing it. I'm eight years old and &lt;a href=&quot;https://aviation.meta.stackexchange.com/questions/1693/move-a-question-to-a-new-user&quot;&gt;I don't use SE unsupervised&lt;/a&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2902" LastEditorUserId="-1" LastEditDate="2017-03-16T15:52:28.347" LastActivityDate="2015-09-20T19:45:51.740" Title="Why were Newton's laws accepted when there exists a counter-example?" Tags="|scientific-method|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2819" PostTypeId="2" ParentId="2818" CreationDate="2015-09-18T06:28:59.603" Score="7" Body="&lt;p&gt;&lt;strong&gt;Every&lt;/strong&gt; scientific theory has some counter-examples or &quot;discrepancies&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, in general, a &quot;good&quot; theory will not be rejected until a &quot;better&quot; new theory is available.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Little&quot; discrepancies, like that involving Mercury's orbit were far less relevant that &quot;big&quot; successes like the correct prediction of the existence of a previously unseen planet; see &lt;a href=&quot;https://en.wikipedia.org/wiki/Discovery_of_Neptune&quot; rel=&quot;nofollow&quot;&gt;Discovery of Neptune&lt;/a&gt; (1846).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to take into account that, also after the availability of a new &quot;better&quot; theory, like Relativity, that is able to explain Mercury's orbital anomalies, Newtonian' mechanics is so &quot;reliable&quot; (being an approximation of Relativity that fit very well when the velocities involved are very little when compared to the speed of light) that human being have been able to safely &quot;land&quot; on the Moon (mission: &lt;a href=&quot;https://en.wikipedia.org/wiki/Apollo_11&quot; rel=&quot;nofollow&quot;&gt;Apollo 11&lt;/a&gt; : July 20, 1969) and come back based on calculation that are not so different in principle from what Newton himself had done in his masterpiece of 1687: &lt;a href=&quot;https://it.wikipedia.org/wiki/Philosophiae_Naturalis_Principia_Mathematica&quot; rel=&quot;nofollow&quot;&gt;Philosophiae Naturalis Principia Mathematica&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scientifc knowledge is not &quot;perfect&quot;: we have to continue to revise and improve it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, good luck for your student career: we need future scientists that can contribute to the process of continuously improvement of human knowledge.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt; : suggestion for your future studies in depth :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/scientific-progress/&quot; rel=&quot;nofollow&quot;&gt;Scientific Progress&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/popper/&quot; rel=&quot;nofollow&quot;&gt;Karl Popper&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="2905" LastEditDate="2015-09-20T19:45:51.740" LastActivityDate="2015-09-20T19:45:51.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2820" PostTypeId="2" ParentId="2817" CreationDate="2015-09-18T06:36:58.027" Score="0" Body="&lt;p&gt;On &lt;em&gt;optics&lt;/em&gt;, &lt;em&gt;telescope&lt;/em&gt; and &lt;em&gt;illusions&lt;/em&gt; at the origin of modern science, see &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_telescope&quot; rel=&quot;nofollow&quot;&gt;History of the telescope&lt;/a&gt; and e.g. :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Eileen Adair Reeves, &lt;a href=&quot;https://books.google.it/books?id=WWu8D4yoh_UC&amp;amp;pg=PA97&amp;amp;lpg=PA1&quot; rel=&quot;nofollow&quot;&gt;Galileo's Glassworks: The Telescope and the Mirror&lt;/a&gt; (2009).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-09-26T13:58:24.890" LastActivityDate="2015-09-26T13:58:24.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2821" PostTypeId="2" ParentId="2818" CreationDate="2015-09-18T12:09:47.593" Score="4" Body="&lt;p&gt;Other bodies also do not obey Kepler's laws strictly. Only approximately.&#xA;The most well-known exception is the Moon, not Mercury.&#xA;The reasons of this were clear to Newton and to his readers. Kepler's laws describe the motion of TWO bodies. When a third (or more) bodies are taken into account, it is difficult to compute from the law of gravitation and the laws of mechanics how will they move. Newton himself could not fully explain the observed motion of the Moon, its irregularities known at that time. And until this motion was explained (in the second half of 18s&#xA;century) there were indeed doubts about exactness of Newton's gravitation theory.&#xA;This was the most serious test.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another crucial test was his correct prediction of the shape of the Earth.&#xA;The most famous test came in the 19 century, when a new planet (Neptune) was discovered on the basis of calculations using Newton's theory, and after that Neptune was found in the sky. The small disagreement for Mercury had to wait for the beginning of 20s century and invention of general relativity.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-18T12:09:47.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2822" PostTypeId="2" ParentId="2817" CreationDate="2015-09-18T12:25:39.803" Score="0" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This is right. Optics is a part of physics, not biology not medicine. But the person who prescribes eyeglasses is called an optometrist which has the same root.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It still includes. This part is called optics of the eye. It studies the eye optical system. Like there is an instrumental optics, and many other parts.&#xA;There are also optical studies of other (non-human) eyes in biology.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why most of our scientific terminology uses Greek and Latin? Because science was invented by the Greeks. All our civilization comes from Greek and Roman civilization. During a very long period, until 17s century, Latin was the&#xA;universal language of science and education in Europe. They did not study English of French or Polish in schools and universities. They studied Latin. But Latin itself used Greek science terms. Because science was invented in Greek.&#xA;Nowadays scientists and engineers do not study Latin or Greek, so they invent terms mixing Latin and Greek roots.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-09-18T12:25:39.803" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2823" PostTypeId="1" AcceptedAnswerId="2829" CreationDate="2015-09-18T12:53:48.883" Score="7" ViewCount="304" Body="&lt;p&gt;Consider the following figure taken from Wikipedia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/k9ioi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/k9ioi.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was Einstein the first person to think of the equivalence of a ball drop in an elevator with acceleration of $g$ and a ball drop on Earth?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be honest, I was also thinking of it before I learned relativity in high school.&lt;/p&gt;&#xA;" OwnerUserId="2903" LastEditorUserId="62" LastEditDate="2015-09-20T00:21:23.657" LastActivityDate="2015-09-20T23:21:41.300" Title="Was Einstein the first person to think of the equivalence of a ball drop in an accelerated elevator and a ball drop on Earth?" Tags="|physics|relativity-theory|gravity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2825" PostTypeId="2" ParentId="2799" CreationDate="2015-09-19T07:28:18.757" Score="3" Body="&lt;p&gt;One of the first appearances of DE in this area must be in D. Bernoulli's study on the impact of variolation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l'inoculation pour la prévenir, Mem. Math. Phys. Acad. R. Sci., 1766&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is &lt;a href=&quot;http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/competingRisks/DanielBernoulli.pdf&quot; rel=&quot;nofollow&quot;&gt;Daniel Bernoulli’s Epidemiological Model Revisited by Dietz and Heesterbeek&lt;/a&gt;, which states that &quot;In the second edition of Bailey’s book full tribute is given to Bernoulli’s work and since then he is frequently quoted as originator of the first epidemiological model for an infectious disease&quot;.&lt;/p&gt;&#xA;" OwnerUserId="2906" LastEditorUserId="55" LastEditDate="2016-01-28T22:07:11.863" LastActivityDate="2016-01-28T22:07:11.863" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2826" PostTypeId="1" CreationDate="2015-09-19T13:06:58.100" Score="5" ViewCount="1194" Body="&lt;p&gt;I read that the famous screw pumps were used before Archimedes (in the hanging gardens of Babylon for example), and that the Archimedean screw is named after him because he &quot;developed a rigorous theory of the kinematics of the screw&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone provide the details? Where did he develop his theory? I thought that kinematics began with Galileo's research, so I am curious to know more about the theory Archimedes developed.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2021-12-14T20:29:44.127" LastActivityDate="2021-12-14T20:29:44.127" Title="What are Archimedes's contributions to the principle of the screw pump?" Tags="|physics|discoveries|ancient-greece|archimedes|antiquity|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2828" PostTypeId="2" ParentId="2826" CreationDate="2015-09-20T21:25:36.147" Score="2" Body="&lt;p&gt;The full quote appears to be &quot;&lt;em&gt;developed a rigorous theory of levers and the kinematics of the screw&lt;/em&gt;&quot;, and is repeated by many authors after History of Technology by Dimarogonas. The rigorous theory of levers is developed in Archimedes's only surviving mechanical work On the Equilibrium of Plane Figures, along with the law of buoyancy, but it is hard to say what Dimarogonas means by &quot;kinematics of the screw&quot;. We know (from Pappus's Collection) of a classical work that analyzes screw motion as a composition of uniform linear and circular motions, About the Screw, but it is by Apollonius rather than Archimedes, although it was likely motivated in part by Archimedes's earlier work On Spirals. Its content is discussed in detail in &lt;a href=&quot;https://www.academia.edu/8016021/Homeomeric_Lines_in_Greek_Mathematics&quot; rel=&quot;nofollow&quot;&gt;Acerbi's Homeomeric Lines in Greek Mathematics&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Archimedes's main contribution was creating a first mechanical theory, the theory of simple machines, which can be applied to the screw just as to the lever. It is best characterized not as kinematics, since it dealt with forces and weights, but as statics, or better quasi-statics, since motion was assumed to be of a kind where at each instant the moving body is at equilibrium to a good approximation. Here is assessment of available sources on this theory in &lt;a href=&quot;https://books.google.com/books?id=MOTpnfz7ZuYC&quot; rel=&quot;nofollow&quot;&gt;Russo's Forgotten Revolution&lt;/a&gt;:&quot;&lt;em&gt;we can reconstruct some features of third century B.C. mechanics by combining information from three sources: the one surviving Archimedean book; documents, particularly works of military technology, that mention machines actually built; and treatises written centuries later, above all Pappus' Collection and Heron's works. Among these the most useful is the Mechanics, which describes the five simple machines (the winch, the lever, the pulley, the wedge/ramp and the screw) as well as a number of composite machines designed for various uses. The pseudo-Aristotelian Mechanics, which share many features with Heron's Mechanics, also has interesting information&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here is his reconstruction of the theory itself:&quot;&lt;em&gt;The problem, given a maximum available force F and the need to lift a weight W, is to design a machine having the appropriate mechanical advantage&lt;/em&gt; [the ratio W/F] &lt;em&gt;and configuration, so the weight can be lifted by applying the available force at a convenient point and in a convenient direction. All devices of this type can ultimately be traced back to the simplest such device, the lever, which Archimedes uses as the starting point of his scientific theory of mechanics. Of course problems of this type had always been around and had often been solved practically, as far back as paleolithic times... The qualitative leap made possible by science lay in that now one could compute the mechanical advantage theoretically, and so for the first time design a machine from first principles. This leap surely took place as early as the third century B.C. Pappus and Plutarch tell us that Archimedes had solved the problem of lifting a given weight with a given force; in other words, he knew how to design a machine with a specified mechanical advantage&lt;/em&gt;&quot;. As applied to the screw this would mean that given its length and height for example Archimedes could calculate the cross section and the handle size to make it possible for a person to operate it, and solve other such problems.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-22T20:49:21.973" LastActivityDate="2015-09-22T20:49:21.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2829" PostTypeId="2" ParentId="2823" CreationDate="2015-09-20T23:21:41.300" Score="4" Body="&lt;p&gt;Einstein also considered weightlessness in free fall as a manifestation of equivalence between inertial forces and gravity. Although simple drum-rope elevators were constructed as early as 236 BC (by Archimedes) they started to spread after 1854, when Otis gave a dramatic demonstration of his safety catch invention by cutting holding ropes while people were inside. By 1900s a good number of people probably did drop balls in some of them, but as far as we know none of them analyzed the physics involved, or expressed their thoughts in print. This being said, there were authors who imagined what it would be like in a vertically accelerating vehicle, but they wither did not focus on the physics, or did not get it quite right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly, the first was Islamic polymath &lt;a href=&quot;https://en.wikipedia.org/wiki/Floating_man&quot; rel=&quot;nofollow&quot;&gt;Avicenna (c. 980-1037) with his Floating Man argument&lt;/a&gt;, the literal translation is &quot;falling man&quot;. Avicenna was not concerned with physics, he argued that a person can be self-aware even if from birth deprived of all sensory input, including weight. It is unclear however what Avicenna thought cancels the effect of weight, free fall or air buoyancy. First definite account of weightlessness is attributed to &lt;a href=&quot;http://www.technovelgy.com/ct/content.asp?Bnum=683&quot; rel=&quot;nofollow&quot;&gt;Francis Godwin's novel Man in the Moone (1638)&lt;/a&gt; written before Newtonian mechanics was even established. The character is carried upward by birds, who &quot;&lt;em&gt;strooke bolt upright, and never did linne towring upward, and still upward, for the space&lt;/em&gt;&quot; until &quot;&lt;em&gt;the Lines slacked; neither I, nor the Engine moved at all, but abode still as having no manner of weight&lt;/em&gt;&quot;. But Godwin's weightlessness is due to weakening of gravity rather than to inertia countering it:&quot;&lt;em&gt;those things which wee call heavie, do not sinke toward the Center of the Earth, as their naturall place, but as drawen by a secret property of the Globe of the Earth, or rather some thing within the same, in like sort as the Loadstone draweth Iron, being within the compass of the beames attractive&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is unlikely that Einstein was aware of Godwin, but he might have encountered &lt;a href=&quot;http://www.technovelgy.com/ct/content.asp?Bnum=596&quot; rel=&quot;nofollow&quot;&gt;Jules Verne's novels From Earth to the Moon (1865), and Around the Moon (1870)&lt;/a&gt;. They were quite popular at the time, and even inspired a 1902 silent film. In them travelers are launched to the Moon from a moongun in a projectile called Columbiad. Verne was aware of the overload forces, and protected his passengers from them with hydraulic and frangible shock absorbers, but it did not occur to him that they would be working like gravity. Moreover, he expected the passengers to feel the pull of gravity throughout the journey despite being mostly in free fall. Only near the &quot;neutral point&quot;, where the gravity forces of the Earth and the Moon balance out, did he expect weightlessness:&quot;&lt;em&gt;But that day, about eleven o'clock in the morning, Nicholl having accidentally let a glass slip from his hand, the glass, instead of falling, remained suspended in the air. &quot;Ah!&quot; exclaimed Michel Ardan, &quot;that is rather an amusing piece of natural philosophy&quot;&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://books.google.com/books?id=zndYLKa26wAC&amp;amp;pg=PA3&amp;amp;lpg=PA3&amp;amp;dq=Tsiolkovsky+Exploration+of+Cosmic+Space+by+Means+of+Reaction+Devices&amp;amp;source=bl&amp;amp;ots=4LQZaBkzsC&amp;amp;sig=aOpD6S2-Vfy4BKLyW9TM1Ww-T_g&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CEMQ6AEwB2oVChMI4q7Vgd-GyAIVQQ-SCh3KjQK7#v=onepage&amp;amp;q=Tsiolkovsky%20Exploration%20of%20Cosmic%20Space%20by%20Means%20of%20Reaction%20Devices&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Exploration of Outer Space by Means of Rocket Devices (1903) Tsiolkovsky&lt;/a&gt; presented detailed engineering calculations related to launching rockets into space. In particular, he concluded that the Verne's moongun would have to be impossibly long for the passengers to survive the acceleration overloads at take off, and offered rockets as the alternative. But he did not comment either on the gravity like behavior of the inertial forces, or on weightlessness.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-09-20T23:21:41.300" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2830" PostTypeId="1" AcceptedAnswerId="2832" CreationDate="2015-09-21T01:13:54.640" Score="1" ViewCount="77" Body="&lt;p&gt;I'm interested in tracing the history of the word &quot;Optical Illusion.&quot;  See these two questions &lt;a href=&quot;https://hsm.stackexchange.com/q/2817/2900&quot;&gt;History of optics&lt;/a&gt;&#xA; and &lt;a href=&quot;https://english.stackexchange.com/q/260495/129806&quot;&gt;https://english.stackexchange.com/q/260495/129806&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems Greek and Latin words were mixed together to make new words / ideas.  Optical Illusion is one: television is another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why were words mixed together?  Why not keep the words in the same language?  It appears Greek has a separate word for &quot;optical illusion;&quot; ophtalmarati.  Why was this renamed?&lt;/p&gt;&#xA;" OwnerUserId="2900" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-09-21T09:05:14.717" Title="What was the system for naming scientific ideas / inventions?" Tags="|terminology|optics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2831" PostTypeId="1" CreationDate="2015-09-21T04:04:06.277" Score="2" ViewCount="137" Body="&lt;p&gt;What scientific experimentations, methods of thinking, ideas and so forth necessitate each individual definition of their acids / bases?&lt;/p&gt;&#xA;" OwnerUserId="2915" LastEditorDisplayName="user5903" LastEditDate="2017-06-13T11:14:48.563" LastActivityDate="2017-06-13T11:14:48.563" Title="How did the theories of Lavoisier, Davy, Arrhenius then Bronsted &amp; Lowry come about?" Tags="|discoveries|chemistry|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2832" PostTypeId="2" ParentId="2830" CreationDate="2015-09-21T09:05:14.717" Score="2" Body="&lt;p&gt;&quot;Optical illusion&quot; is a phrase combining two separate words. Each was already a word in English, adapted (directly or indirectly) from respective Latin and Greek sources. Combining two English words to coin an English phrase is not at all controversial. &lt;/p&gt;&#xA;" OwnerUserId="536" LastActivityDate="2015-09-21T09:05:14.717" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2833" PostTypeId="1" CreationDate="2015-09-21T12:02:58.137" Score="2" ViewCount="126" Body="&lt;p&gt;In order to understand my motivation to answer this question, I'll remark that Vitruvius said that the screw was inclined by the small angle of a 3:4:5 triangle. So it means that perhaps there was knowledge of how to optimize the design of a screw, and I guess this knowledge is of Archimedean origin. So my question is, is there any evidence that Archimedes improved the screw? It's important, since modern research has found that the proportions described by Vitruvius are near-optimal, so if these proportions were discovered by Archimedes, one can argue that Archimedes developed a complicated geometric theory of the flow of water through the screw.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2021-12-14T20:29:25.360" LastActivityDate="2021-12-14T20:29:25.360" Title="What were Archimedes's improvements to the screw pump?" Tags="|biographical-details|ancient-greece|engineering|archimedes|antiquity|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2836" PostTypeId="1" CreationDate="2015-09-22T15:20:39.393" Score="12" ViewCount="3173" Body="&lt;p&gt;I read that in ancient times the quadratic equation of this kind $$x^2+10x=39$$ had been solved long ago. I read that this kind of equation originated in the geometric question of &quot;Given an area of 39, find the side x s.t. the square of x plus the rectangle of x and 10 fit the area.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this condition &quot;Given...area&quot; seems artificial to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: How was this condition related to the actual needs of the real world in ancient times?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any evidence that this kind of condition REALLY grows out of some geometric construction problem for the base of some temple or something like that?&lt;/p&gt;&#xA;" OwnerUserId="2921" LastEditorUserId="26" LastEditDate="2015-09-23T08:31:37.217" LastActivityDate="2018-05-05T22:49:15.793" Title="The origin of quadratic equation in actual practice" Tags="|mathematics|ancient-greece|euclidean-geometry|elementary-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2837" PostTypeId="2" ParentId="2836" CreationDate="2015-09-22T20:31:55.327" Score="6" Body="&lt;p&gt;For a broader perspective see &lt;a href=&quot;https://hsm.stackexchange.com/a/2238/55&quot;&gt;How was geometry historically used to solve polynomial equations?&lt;/a&gt; For early practical problems that would lead (today) to quadratic equations see e.g. &lt;a href=&quot;http://cdli.ucla.edu/pubs/cdlj/2009/cdlj2009_003.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friberg's discussion of cuneiform tablet YBC 3879 (c. 2000 BC)&lt;/a&gt;, a judicial document from third Sumerian Ur period, that describes field division problems leading to quadratic equations. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The field plan shows a nearly square field, apparently a “temen”... A temen is a regularly shaped field, typically made up of one or several rectangles or trapezoids... In connection with a field plan, the word appears to refer to the initial form of a field, before various modifications. In the field plan on the reverse of YBC 3879, six fields of various shapes (three triangles and three (quasi-) trapezoids) have been removed from the temen, leaving as residue an irregularly shaped field&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The geometry in YBC 3879 is rather messy, &quot;&lt;em&gt;the area B that was computed above is also equal to the area of the rectangle with the sides a and u1' minus the area of the right triangle with the sides a and b = f · a&lt;/em&gt;&quot;, and the OP example seems like a simplified problem of this sort. Babylonians, Egyptians, etc., also had to deal with dividing fields into simple shapes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another early occurence is in the &lt;a href=&quot;https://themathematicaltourist.wordpress.com/2014/05/24/a-piece-of-the-berlin-papyrus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Berlin papyrus 6619 (c.1800 BC)&lt;/a&gt; with the problem stated as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares?&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The equation is solved by false position, a common maneuver in the absence of notation for unknowns. It could also be motivated by field division, but not necessarily. It is known that papyri were written by Egyptian scribes and contained practice problems for training other scribes in their craft. Some problems were explicitly related to construction, such as &lt;a href=&quot;https://www.math.washington.edu/~greenber/Rhind.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;reckoning a pyramid in Rhind papyrus&lt;/a&gt;, some not, and were specifically meant to teach them how to perform certain computations. Many practical problems involved finding Aha (unknowns) by false position, so mastering this technique was essential.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-05T22:49:15.793" LastActivityDate="2018-05-05T22:49:15.793" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2838" PostTypeId="1" CreationDate="2015-09-23T02:44:46.160" Score="14" ViewCount="1124" Body="&lt;p&gt;Now I am starting to learn Quantum Mechanics. In the class I am taught about operators, postulates and all other basic stuff.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand operators to be +, -, /, etc; but quantum mechanical operators are entirely different; to understand them, I think, I need to know the historical development of the physics operators. So, I want to know how these operators were discovered/invented; some of the historical figures on this subject would also help along with some first text books from the original authors (modern texts are also okay).&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="55" LastEditDate="2017-12-04T20:51:59.647" LastActivityDate="2018-06-10T19:12:26.883" Title="How did quantum mechanics operators come into being?" Tags="|mathematics|physics|quantum-mechanics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2839" PostTypeId="2" ParentId="2838" CreationDate="2015-09-23T04:16:34.187" Score="10" Body="&lt;p&gt;It is a good question, how such a weird theory could be invented:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In its present form the theory was developed by Dirac (physicist) &#xA;and von Neumann (mathematician). von Neumann essentially developed the mathematical operator theory which is needed. They both wrote books on quantum mechanics which explain the motivation for operators. (You may prefer one book or another depending on your background: whether you are a physicist or a mathematician.)&#xA;But they gave the theory its final form, and the history is really complicated. To understand some history, I recommend van der Waerden, &lt;strong&gt;&lt;em&gt;Sources in the history of quantum mechanics&lt;/em&gt;&lt;/strong&gt;, which is a collection of original articles, translated&#xA;into English, with his commentaries. To go deeper in the history, you have to learn German, first of all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Some people who knew von Neumann say that he was an alien, extraterrestrial: a human mind could not invent such things.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. As I said, discovery of quantum mechanics is a very long story, and this is a result of collective effort. It starts at least from the discovery by Balmer of the empirical formula for the spectral lines of hydrogen, which triggered the whole process. (But really the story can be traced back to Newton.) Then Rydberg and Ritz should be mentioned, and Planck and Einstein.&#xA;A milestone was Bohr's theory which is sometimes called &quot;old quantum mechanics&quot;.&#xA;It describes many phenomena correctly, but there are no operators yet.&#xA;Then Heisenberg, Schrödinger, Born, Jordan and Dirac made a &quot;quantum leap&quot;, and gave it the more or less modern form. Operators were introduced by Dirac but without a rigorous mathematical justification. Then von Neumann developed the rigorous mathematical theory that was required. On the early stage (which can be called pre-history, that is before Bohr), there is an excellent exposition in the book by Shlomo Sternberg,&#xA;Group theory and physics, Appendix F &lt;strong&gt;&lt;em&gt;“A history of 19th century spectroscopy”&lt;/em&gt;&lt;/strong&gt;. It covers the development from Newton to Bohr. I am not familiar with any exposition of similar quality and clarity for&#xA;the period between Bohr and Heisenberg. Sommerfeld's book &lt;strong&gt;&lt;em&gt;Atombau und Spektrallinien&lt;/em&gt;&lt;/strong&gt; is close to it, but he was more interested in explaining the current state of the theory, rather than history.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="658" LastEditDate="2018-06-10T19:12:26.883" LastActivityDate="2018-06-10T19:12:26.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2840" PostTypeId="2" ParentId="2756" CreationDate="2015-09-23T08:38:39.853" Score="23" Body="&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;:  I present here some information defending Millikan, but please note that I do not necessarily agree with the article it came from.&lt;/p&gt;&#xA;&lt;p&gt;From the feature article &amp;quot;&lt;a href=&quot;http://www.americanscientist.org/issues/feature/2001/1/in-defense-of-robert-andrews-millikan&quot; rel=&quot;noreferrer&quot;&gt;In Defense of Robert Andrews Millikan&lt;/a&gt;&amp;quot; by David Goodstein (&lt;em&gt;American Scientist&lt;/em&gt;, January-February 2001):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Awkwardly, an examination of Millikan's private laboratory notebooks indicates that he did not in fact include every droplet for which he recorded data. He published the results of measurements on just 58 drops, whereas the notebooks reveal that he studied some 175 drops in the period between November 11th, 1911 and April 16th, 1912. In a classic case of cooking, the accusation goes, he reported results that supported his own hypothesis of the smallest unit of charge and discarded those contrary results that would have supported Ehrenhaft's position. And, to make matters very much worse, he lied about it. Millikan's 1913 paper contains this explicit assertion: &amp;quot;&lt;em&gt;It is to be remarked, too, that this is not a selected group of drops, but represents all the drops experimented upon during 60 consecutive days&lt;/em&gt;, during which time the apparatus was taken down several times and set up anew.&amp;quot; (Emphasis in the original). Thus, Millikan is accused of cheating and then compounding his cheating by lying about it in one of the most important scientific papers of the 20th century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The author defends some of Millikan's actions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] More than one of the entries in his notebooks show the result of a computation and then the comment &amp;quot;very low something wrong,&amp;quot; perhaps with an indication of what Millikan thought might have disturbed the measurement. Needless to say, such entries were not included in the 58 drops Millikan published.&lt;/p&gt;&#xA;&lt;p&gt;At first glance, this procedure certainly appears questionable. But one needs to dig deeper. The notebooks also contain a calculation with the comment &amp;quot;This is almost exactly right, the best one I ever had!!!&amp;quot; And yet Millikan did not include this drop either in his crucial 1913 paper. These discarded measurements, the good and the bad, were all part of a warm-up period during which Millikan gradually refined his apparatus and technique, in order to make the best determination possible of the unit of electric charge. The first observation that passed muster and made it into print was taken on February 13th, 1912, and all of the published data were taken between then and April 16th. This period of roughly two months is what Millikan refers to when he talks about &amp;quot;60 consecutive days,&amp;quot; although the interval was actually a bit longer (63 days), in part because 1912 was a leap year.&lt;/p&gt;&#xA;&lt;p&gt;During these nine weeks Millikan recorded in his notebooks measurements on roughly 100 separate drops. Of these, about 25 series are obviously aborted during the run, and so cannot be counted as complete data sets. Of the remaining 75 or so, he chose 58 for publication. Millikan's standards for acceptability were exacting. If a drop was too small, it was excessively affected by Brownian motion, or at least by inaccuracy in Stokes's law for the viscous force of air. If it was too large, it would fall too rapidly for accurate measurement. He also preferred to have a drop capture an ion a number of times in the course of observation, so that he could investigate changes as well as total charge, which had to be an integer multiple of the fundamental unit, &lt;em&gt;e&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;[...] He had no special bias in choosing which drops to discard: Allan Franklin of the University of Colorado reanalyzed Millikan's raw data in 1981 and discovered that his final value for &lt;em&gt;e&lt;/em&gt; and for its margin of error would barely have changed had he made use of all the data he had, rather than just the 58 drops he selected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is actually much more to this than what I quoted.  The article contains the complete story.&lt;/p&gt;&#xA;" OwnerUserId="72" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-09-23T08:38:39.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2842" PostTypeId="2" ParentId="2838" CreationDate="2015-09-23T18:54:34.693" Score="7" Body="&lt;p&gt;The turn to operators occurred in Heisenberg's 1925 paper, and it was a compromise between positions taken by Bohr and Born. Bohr wanted minimal modifications to classical mechanics, like his quantization rules for the hydrogen atom, while Born wanted a new discrete mechanics governed by difference equations. Heisenberg's idea was to keep equations of motion more or less classical, but to reinterpret symbols in them. Those symbols represent observables (position, momentum, energy, etc.), and in Hamiltonian dynamics they are functions of positions $x$ and momenta $p$, which form the phase space. Applying this idea Heisenberg discovered that his symbols would not commute, for example $xp-px=i\hbar$ with the Planck constant $\hbar$, so they could not be numbers. A mathematician friend of his pointed out that matrices do not commute, and could satisfy such relations, and this gave Heisenberg's proposal its name, matrix mechanics. Except that since electrons in an atom have infinitely many energy levels these &quot;matrices&quot; had to be of infinite size.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At about the same time Schrodinger was working from a different perspective, he wanted to reduce quantum effects to wave dynamics. So he represented states of quantum systems by wave functions $\psi$ on the phase space and looked for equations of motion. His idea was that quantum particles are wave packets, and he originally thought of $|\psi|^2$ as charge density, only later Born identified it with probability density. After matrix mechanics came out Schrodinger realized that he needed to introduce observables into his picture, but they could not be mere symbols. Since states are wave functions on the phase space rather than its points, observables can not be functions of them, they have to act on wave functions like matrices act on vectors. After some experimenting in 1926 he came up with representing position by a multiplication operator $\psi\mapsto x\psi$, and momentum by differentiation $\psi\mapsto-i\hbar\frac{\partial \psi}{\partial x}$. The big clue was that if we think of $x,p$ as these operators then $(xp-px)\psi=i\hbar\,\psi$. Then we can form other observables from these ones. Classical kinetic energy is $p^2/2m$, replacing $p$ by its operator we get $-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$ from the Schrodinger equation. The total energy of a classical oscillator is $p^2/2m+kx^2/2$, replacing gives the Hamiltonian of the quantum oscillator $-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+\frac{kx^2}2$. And so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1930 Dirac conceptualized this picture in his textbook &lt;a href=&quot;http://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/57355817/57355817.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principles of Quantum Mechanics (1930)&lt;/a&gt;, where he introduced the bra-ket notation among other things. Then von Neumann gave an axiomatic formulation based on the notion of abstract Hilbert space that he introduced, and established that matrix and wave mechanics were talking about the same thing in two different ways. In Schrodinger's picture the states became elements of a Hilbert space, namely the space $L^2$ of square integrable (wave) functions on the position space. The observables became self-adjoint operators on it. Upon choice of an orthonormal basis in this space however the states become (infinite) vectors, and the operators become (infinite) matrices. This is the Heisenberg's picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a relevant thread on Physics SE &lt;a href=&quot;https://physics.stackexchange.com/questions/153672/how-did-the-operators-come-about&quot;&gt;How did the operators come about?&lt;/a&gt; Landsman gives a nice short survey of historical developments in &lt;a href=&quot;http://arxiv.org/pdf/quant-ph/0506082v2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Between Classical and Quantum&lt;/a&gt;. More comprehensive references are &lt;a href=&quot;http://press.princeton.edu/chapters/s6683.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kragh's Quantum Generations&lt;/a&gt;, and &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B0006BO3FG&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jammer's classic Conceptual Development of Quantum Mechanics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-12-04T20:58:08.357" LastActivityDate="2017-12-04T20:58:08.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2845" PostTypeId="1" AcceptedAnswerId="2849" CreationDate="2015-09-24T17:20:47.377" Score="5" ViewCount="238" Body="&lt;p&gt;This &lt;a href=&quot;https://www3.nd.edu/~powers/ame.20231/carnot1897.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF file&lt;/a&gt; contains an English translation of Sadi Carnot's &lt;strong&gt;Refelections on the motive power of fire&lt;/strong&gt;. It also contains commentary by Lord Kelvin. And some chapters about the work and life of Carnot, made by the editor R. H. Thruston.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Under the section, &lt;strong&gt;The Work of Sadi Carnot&lt;/strong&gt;, the editor talks about how Carnot discovered what's called the principle of equivalence between energy and heat. He also talks about the units Carnot employed to measure energy and heat. He writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In making his measures of heat-energy, he assumes as a unit a measure&#xA;  not now common, but &#xA;   one which may be easily and conveniently reduced to the now general system of measurement. He takes the amount of&#xA;  power required to exert an energy equal to that needed to raise one cubic meter of water through a height of one meter, as his unit; this is 1000 kilogrammeters, taken as his unit of motive power; while he says that this is the equivalent of 2.7 of his units of heat; which latter quantity would be destroyed in its production of this amount of power, or rather work. His unit of heat is thus seen to be 1000 /2.7, or 370 kilogram meters. This &#xA;  is almost identical with the figure obtained by &#xA;  Mayer, more than ten years later, and from &#xA;  presumably the same approximate physical data, &#xA;  the best then available, in the absence of a Regnault to determine the exact values. Mayer obtained 365, a number which the later work of &#xA;  Regnault enabled us to prove to be 15 per cent. &#xA;  too low, a conclusion verified experimentally by &#xA;  the labors of Joule and his successors. Carnot was thus a discoverer of the equivalence of the units of heat and work...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Carnot defined a unit of energy as that energy that raises a meter cube of water one meters high. This is equivalent to $1\text {kg} m^2s^{-2}=\text {joule}$. But the editor says this is &quot;&lt;em&gt;1000 kilogrammeters&lt;/em&gt;&quot;. I'm puzzled by this assertion, since kilogrammeters is not a dimensionally correct unit of energy. Also where does this $1000$ come from? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Why Carnot uses a unit of heat that's not equal to a unit of energy (different by a factor of $2.7$), although he knew that energy and heat are the same physical quantities. Why not choose the same units for energy and heat? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: I just learned from user Mauro that kilogram-meter=joule. In page 100, Carnot defined a unit of heat as the amount of heat needed to raise a body of one kilogram by 1 degree. The specific heat of water is 4.186 joule/gm.celsius or 4.186 kilogram-meter/gm.celsius. $Q=cm\Delta t$. So a unit of heat according to his definition is= 4.186 kilogram-meter/gm.celsius *1000 gm*1 celsius= 4186 kilogram-meter or 4186 joule (1kilo-calorie) .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So his definition of energy is energy needed to lift one kilogram one meter high which is equal to is 1 joule or 1 kilogram-meter(as stated in the quotation). So the ratio between his unit of heat and energy should be 4186/1=4186; a factor of 4186 not 2.7; so where does 2.7 come from? And where does the factor of 1000 stated by the editor come from? &lt;/p&gt;&#xA;" OwnerUserId="2423" LastEditorUserId="12699" LastEditDate="2017-07-03T12:16:34.820" LastActivityDate="2017-07-03T12:16:34.820" Title="On the work of Sadi Carnot" Tags="|thermodynamics|units|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2846" PostTypeId="1" AcceptedAnswerId="2850" CreationDate="2015-09-24T17:49:02.840" Score="5" ViewCount="1119" Body="&lt;p&gt;When was $\sum$ introduced as the notation for a sum and who was the first person to solve a infinite sum other than 0+0+0+...?&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="140" LastEditDate="2015-09-24T18:12:04.740" LastActivityDate="2015-09-24T23:54:15.340" Title="History of infinite series" Tags="|mathematics|mathematicians|notation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2847" PostTypeId="2" ParentId="2845" CreationDate="2015-09-24T18:54:51.157" Score="1" Body="&lt;p&gt;The &quot;heat-energy&quot; [page 8] is a &lt;em&gt;motive power&lt;/em&gt;, i.e. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;work = force $\times$ distance&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and the &lt;a href=&quot;https://en.wiktionary.org/wiki/kilogram-meter&quot; rel=&quot;nofollow&quot;&gt;kilogram-meter&lt;/a&gt; is a unit of work.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-09-26T14:00:55.630" LastActivityDate="2015-09-26T14:00:55.630" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2849" PostTypeId="2" ParentId="2845" CreationDate="2015-09-24T22:47:07.230" Score="5" Body="&lt;p&gt;Alright after doing some research I figured out the answers to all my questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1)Carnot's definition of energy is as follows: it's the energy needed to lift a cubic meter of water one meters high. By definition a cubic meter of water is 1000 kg, and following from the definition of work according to Carnot is: $W=Fd=mgh$ , where $g=9.8 ms^{-2}$ and $h=1 m$ therefore a unit of work = $1000 \text{kg}*9.8ms^{-2}*1 m=9800$  $\text{joule}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) $\text{kilogrammeter}$ (also more correctly known as &#xA;$\text{kilogram-force meter [kgf*m]}$) is a unit of work. $1$ $\text{joule}=0.101971621$ $\text{kilogrammeter}$; Therefore  $9800$ $\text{joule}=999.3 \approx1000$ $\text{kilogrammeter}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3)Carnot's definition of unit of heat is stated in page 100 of the linked PDF as follows: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If, then, we take for the unit of heat the quantity necessary to raise&#xA;  1 kilogram of water 1 degree&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which is = $1$ $\text{Kcal}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4)In a posthumous notes transmitted to the french academy of science, Carnot writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to some ideas that I have formed on the theory of heat, the&#xA;  production of a unit of motive power necessitates the destruction of&#xA;  2.70 units of heats.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also thanks for the help presented by user Mauro.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;source: &lt;strong&gt;&lt;a href=&quot;http://geosci.uchicago.edu/~moyer/GEOS24705/Readings/Carnot_article_1998.pdf&quot; rel=&quot;noreferrer&quot;&gt;Sadi Carnot, ‘Founder of the Second&#xA;Law of Thermodynamics’&lt;/a&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2423" LastActivityDate="2015-09-24T22:47:07.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2850" PostTypeId="2" ParentId="2846" CreationDate="2015-09-24T23:36:50.940" Score="5" Body="&lt;p&gt;According to Florian Cajori's &lt;em&gt;History of Mathematical Notations Vol II&lt;/em&gt; paragraph 438, the symbol $\displaystyle \sum$ for summation is attributed to Leonard Euler. In Euler's own words, &quot;summam indicabimus signo $\sum$&quot;. Cajori cites Euler's &lt;em&gt;Institutiones calculi differentialis (1755)&lt;/em&gt; in his sourcing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the first known infinite sum, I do see some online sources attributing the first known infinite sum to Archimedes. In particular, on the site &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html&quot; rel=&quot;nofollow&quot;&gt;A history of the calculus&lt;/a&gt;, the authors &lt;a href=&quot;https://risweb.st-andrews.ac.uk/portal/da/persons/john-joseph-oconnor(d8ca4d9f-4b1b-4627-93f9-c035fa1e1b6c).html&quot; rel=&quot;nofollow&quot;&gt;John J. O'Connor&lt;/a&gt; and &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/~edmund/&quot; rel=&quot;nofollow&quot;&gt;Edmund Robertson&lt;/a&gt; cite Archimedes' showing around 225 BC that &quot;the area of a segment of a parabola is $\frac{4}{3}$ the area of a triangle with the same base and vertex and $\frac{2}{3}$ of the area of the circumscribed parallelogram,&quot; as the first known example of an infinite sum. Archimedes lived from 287 to 212 BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personal opinion has it that while this may be the first known infinite summation, there may well have been one before that not known to us today. &lt;/p&gt;&#xA;" OwnerUserId="110" LastEditorUserId="110" LastEditDate="2015-09-24T23:54:15.340" LastActivityDate="2015-09-24T23:54:15.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2852" PostTypeId="1" CreationDate="2015-09-25T14:13:59.500" Score="2" ViewCount="115" Body="&lt;p&gt;Entry 93 in Gauss's famous mathematical diary states - &quot;solution of a problem in ballistics&quot;. What is the problem in ballistics that Gauss solved? can someone give a reliable source to what Gauss actually solved? &lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2015-09-25T14:13:59.500" Title="Are there any gaussian contributions to ballistics?" Tags="|biographical-details|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2853" PostTypeId="1" CreationDate="2015-09-25T21:32:11.907" Score="5" ViewCount="292" Body="&lt;p&gt;I'm reaching the point in my mathematical career that the names aren't so well known. Everybody knows that Euler was great and Gauss was even better, and it's not hard to learn that if Riemann died too young, then Abel died when he was only 26.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, there's a lot of things named after a lot of people, and my education just feels incomplete without knowing who these people are and what they were up to. Is there such a thing as an anthology or maybe a good MOOC I can look at in my spare time?&lt;/p&gt;&#xA;" OwnerDisplayName="user2942" LastEditorUserId="2442" LastEditDate="2015-09-28T22:29:52.763" LastActivityDate="2015-10-01T02:01:59.710" Title="Where can I learn more about lesser known mathematicians?" Tags="|mathematicians|reference-request|education|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2854" PostTypeId="2" ParentId="2853" CreationDate="2015-09-25T23:17:22.030" Score="6" Body="&lt;p&gt;There is a &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/BiogIndex.html&quot; rel=&quot;nofollow&quot;&gt;large list of links to biographies here&lt;/a&gt;. Most mathematicians on the list are not as well known as Gauss and Euler, and they link to MacTutor, which is more reliable than Wikipedia.&lt;/p&gt;&#xA;" OwnerUserId="2944" LastEditorUserId="55" LastEditDate="2015-10-01T02:01:59.710" LastActivityDate="2015-10-01T02:01:59.710" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2856" PostTypeId="1" AcceptedAnswerId="2863" CreationDate="2015-09-26T18:25:19.827" Score="6" ViewCount="368" Body="&lt;p&gt;I've seen from a number of sources that both the Colossus and the cryptological bombes operating for England were dismantled after the war ended. The &lt;a href=&quot;http://en.wikipedia.org/wiki/Colossus_computer#Influence_and_fate&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; even says that all of Colossus' documentation was burnt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would guess that at that time it would be clear to anyone that the need for cryptanalysis would not end with the end of the war. What is the rationale behind this?&lt;/p&gt;&#xA;" OwnerUserId="2810" LastEditorUserId="8068" LastEditDate="2020-05-11T20:56:32.127" LastActivityDate="2020-05-13T13:22:11.780" Title="Why were British WWII computing machines and their projects destroyed after the war ended?" Tags="|computation|computers|world-war-2|cryptology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="2857" PostTypeId="1" AcceptedAnswerId="3200" CreationDate="2015-09-26T18:45:29.790" Score="31" ViewCount="2050" Body="&lt;p&gt;As a research mathematician, working in number theory, who is interested in the history of his own field, I have done some reading in the History of Mathematics, particularly that of Ancient Greek and Hellenistic mathematics. When I compare the way history of mathematics is done now with the way it was done, say 50 years ago, I am struck by certain distinct changes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be specific, I have noticed a certain trend, among historians today, to attack &quot;naive&quot; or &quot;Whiggist&quot; approaches to their subject, which they perceive as having dominated the field for too long. Put roughly, at least as far as I understand it, a naive way of doing historiography would be to force a &quot;modern&quot; reading upon historical mathematical texts. In contrast, a more refined, say &quot;postmodern&quot;, approach would be to deliberately &lt;em&gt;refrain&lt;/em&gt; from forcing ancient terminology in (what would be perceived to be) a modern straitjacket. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An infamous example that is central to this debate is the so-called &quot;Geometrical Algebra,&quot; which is an interpretation of Book II of Euclid's Elements that originated in the work of Zeuthen in the 19th century. Indeed, the critique of &quot;Geometrical Algebra&quot; seems to be the flagship of the postmodern movement (as I will call it for ease of reference). This discussion is already quite old (1970s I think), but it is still quite often referred to in order to make a point. Here is the debate in a nutshell: in Book II of the Elements, Euclid discusses propositions such as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;II.1. If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments.&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;II.4. If a straight line is cut at random, the square on the whole equals the squares on the segments plus twice the rectangle contained by the segments.&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now the proponents of &quot;Geometrical Algebra&quot; would say that II.1 is just Euclid's way of stating that $a(b_1+\ldots+b_n)=ab_1+\ldots+ab_n$, in other words the distributive law, and II.4 is just Euclid's way of stating that $(a+b)^2=a^2+b^2+2ab$. Personally, I find these readings quite convincing. On the other hand, postmodern historians of mathematics would label this line of reasoning &quot;anachronistic&quot;, and that is of course to some extent simply true. At the same time, there seems to be a certain vacuity about just labeling a certain reading anachronistic, without substituting a new reading in its place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without getting into this whole discussion any further, I was wondering the following things: &lt;blockquote&gt;&#xA;1. What is the current status of this debate? &#xA;&lt;/blockquote&gt;&#xA;I presume that it has died down somewhat, since the 70s are a while back. So I would also like to know:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;2. How are opinions divided nowadays? (For example, on the Geometrical Algebra issue.) Has one camp &quot;won&quot; the debate, or are there several &quot;camps&quot; still carrying on independently from each other?&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I look forward to reading your comments.&lt;/p&gt;&#xA;" OwnerUserId="2353" LastEditorUserId="2353" LastEditDate="2015-09-26T19:49:26.090" LastActivityDate="2019-06-20T04:32:18.803" Title="Current ways of thinking in the History of Mathematics" Tags="|mathematics|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2858" PostTypeId="2" ParentId="2857" CreationDate="2015-09-27T07:56:33.340" Score="7" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Historians of mathematics currently are trying, IMHO, to be overly exact; I've seen some of them criticize the use of the word &quot;triangle&quot; in connection with Babylonian mathematics since Babylonians did not have the concept of angle. To me, this is ridiculous, not only because in German triangle is Dreieck (3-gon). I do regret the current trade of making history of mathematics a playground for four of five specialists. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As for the geometric algebra debate: the original idea was that there was some sort of algebra before the Greeks, i.e. Euclid, translated it into geometry. This position, as far as I can see, is dead. I still like to think of Euclid II as geometric algebra because these results are used by Euclid not in geometric contexts, but in algebraic ones (e.g. in the construction of Pythagorean triples). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Let me add something I just have seen in an article by Jens Hoyrup (Les Lais: or, or, What Ever Became of Mesopotamian Mathematics?):&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&quot;In the main, Neugebauer was thus fully right; but what went into&#xA;Greek &quot;geometrical algebra&quot; was not the sophisticated scribal algebra of&#xA;the Old Babylonian era but the anonymous riddle tradition of Mesopotamian&#xA;origin.&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastEditorDisplayName="user2255" LastEditDate="2016-01-21T18:03:12.970" LastActivityDate="2016-01-21T18:03:12.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2859" PostTypeId="1" AcceptedAnswerId="2862" CreationDate="2015-09-27T10:42:32.987" Score="2" ViewCount="656" Body="&lt;p&gt;Concerning the Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Root_test&quot; rel=&quot;nofollow&quot;&gt;&quot;root test&quot;&lt;/a&gt;, this convergence criterion for series was first proved by &lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy&quot; rel=&quot;nofollow&quot;&gt;Augustin-Louis Cauchy&lt;/a&gt;. What is the name of the treatise where he proved the root test? Which page?&lt;/p&gt;&#xA;" OwnerUserId="1772" LastActivityDate="2015-09-27T21:37:09.120" Title="Where is the root test first proved" Tags="|mathematics|reference-request|calculus|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2860" PostTypeId="1" AcceptedAnswerId="2861" CreationDate="2015-09-27T11:58:39.990" Score="2" ViewCount="1103" Body="&lt;p&gt;Obviously &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy&amp;#39;s_convergence_test&quot; rel=&quot;nofollow&quot;&gt;Cauchy's convergence test&lt;/a&gt; is named after &lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy&quot; rel=&quot;nofollow&quot;&gt;Augustin-Louis Cauchy&lt;/a&gt;. Is he the person who first proved this criterion or is it another &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_misnamed_theorems&quot; rel=&quot;nofollow&quot;&gt;misnamed theorem&lt;/a&gt;? If so: In which treatise?&lt;/p&gt;&#xA;" OwnerUserId="1772" LastActivityDate="2015-09-27T12:18:55.730" Title="Origin of Cauchy convergence test" Tags="|mathematics|reference-request|calculus|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2861" PostTypeId="2" ParentId="2860" CreationDate="2015-09-27T12:18:55.730" Score="2" Body="&lt;p&gt;You can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Bradley &amp;amp; CEdward Sandifer (editors), &lt;a href=&quot;https://books.google.it/books?id=M0or-HGe7D0C&amp;amp;pg=PA84&quot; rel=&quot;nofollow&quot;&gt;Cauchy's Cours d’analyse : An Annotated Translation&lt;/a&gt; (2009), page 85-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Cauchy's textbook, was first published in 1821; the above is a translation of the second edition (appeared in 1897 into Cauchy’s &lt;em&gt;Oeuvres complètes&lt;/em&gt;) of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cours_d%27Analyse&quot; rel=&quot;nofollow&quot;&gt;Cours d’analyse&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-09-27T12:18:55.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2862" PostTypeId="2" ParentId="2859" CreationDate="2015-09-27T21:37:09.120" Score="3" Body="&lt;p&gt;It's Theorem 1 from Chapter 6 Section 2 of Cauchy's &lt;a href=&quot;https://en.wikipedia.org/wiki/Cours_d%27Analyse&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Cours d'analyse&lt;/em&gt;&lt;/a&gt;, published in 1821. You can read the original &lt;a href=&quot;https://archive.org/details/coursdanalysede00caucgoog&quot; rel=&quot;nofollow&quot;&gt;at archive.org&lt;/a&gt; or have a look at the 2009 English translation by Rob Bradley and Ed Sandifer (published by Springer).&lt;/p&gt;&#xA;" OwnerUserId="2950" LastActivityDate="2015-09-27T21:37:09.120" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2863" PostTypeId="2" ParentId="2856" CreationDate="2015-09-27T22:39:50.380" Score="6" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To keep secret the level of expertise in cryptanalysis so future opponents wouldn't put effort into improving their own codes. Probably pointless because it was inevitable that some details of Bletchley would leak to the USSR, in the same way as secrets of the Manhattan project. And probably equally pointless in that, although every side in WWII broke almost all of the opponents codes - they all believed their own codes were unbreakable so put little effort in counter-crypt-analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The UK government gave captured Enigma machines to their allies - claiming that they were unbreakable. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Allied leaders wanted the glory of people believing that they had won through their own bravery and military genius. Not because they had been reading all the opposition's orders before they did.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Standard government procedure to keep everything secret until compelled otherwise.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally and possibly the most important: To not give the losers a simple excuse for why they lost. If the message becomes: Our invincible army only lost because of poor crypto then we would win if we fixed this problem and and so should try again.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2838" LastEditorUserId="4251" LastEditDate="2020-05-13T13:22:11.780" LastActivityDate="2020-05-13T13:22:11.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2864" PostTypeId="2" ParentId="2428" CreationDate="2015-09-28T12:40:11.940" Score="5" Body="&lt;p&gt;More of an engineering feat, but I'd go for the Steam Engine (or at least a Segner Wheel), the basics had been there for about 1500 years (The use of compressed air in pumps, the idea that boiling water produces steam) and had been been used for sort of toys/novelties but the jump to powering machinery wasn't made.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aeolipile&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Aeolipile&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Segner_wheel&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Segner_wheel&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2952" LastActivityDate="2015-09-28T12:40:11.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2865" PostTypeId="1" AcceptedAnswerId="2866" CreationDate="2015-09-29T02:45:27.920" Score="4" ViewCount="257" Body="&lt;p&gt;When and how was water (in ice or any other state) first observed outside of Earth? What is the high-level chronology of this discovery, and how did the scientific community first reach reasonable consensus about it?&lt;/p&gt;&#xA;" OwnerUserId="640" LastEditorUserId="640" LastEditDate="2015-09-29T02:54:41.647" LastActivityDate="2015-09-29T23:03:14.767" Title="When and how was water first detected outside of Earth?" Tags="|physics|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2866" PostTypeId="2" ParentId="2865" CreationDate="2015-09-29T13:02:11.317" Score="7" Body="&lt;p&gt;Mars polar caps were discovered in 17s century  from the beginning it was speculated that they consist of ice (later observations confirmed that they are 70% water ice). The evidence of presence of water on Mars was obtained in 1872 by C. Vogel in 1872 by spectral analysis. This was disputed until the conclusive results of Adams in 1925 (also by spectral analysis). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comet cores also contain water ice, and this was plausible since 19s century, by spectral analysis, but confirmed only in 1950s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jupiter's satellite Gannimede has oceans of frozen water, they probably contain more water that all oceans on the Earth, but this is a much more recent discovery which was made using spacecraft.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I understand, the discovery of the liquid, flowing water was announced yesterday for the first time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. The difficulty with finding water by spectral analysis is that some spectral lines come from the water vapor in the Earth atmosphere, when you observe from the Earth. So many of the early results until the middle of 20s century were questioned. In particular, Internet search does not give a clear reference for the first definite proof that comets contain water (ice). In any case this was widely believed in the first half of 20s century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Sometimes pieces of ice fall from the sky:-) The intriguing question whether they really come from meteorites (not from the Earth atmosphere) is discussed here:&#xA;   &lt;a href=&quot;http://uregina.ca/~astro/Ice_Mets.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://uregina.ca/~astro/Ice_Mets.pdf&lt;/a&gt;&#xA;and here &lt;a href=&quot;https://en.wikipedia.org/wiki/Megacryometeor&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Megacryometeor&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-09-29T23:03:14.767" LastActivityDate="2015-09-29T23:03:14.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2867" PostTypeId="1" AcceptedAnswerId="2868" CreationDate="2015-09-29T16:44:42.467" Score="6" ViewCount="1196" Body="&lt;p&gt;I read about Napoleon's Theorem in geometry. Was this the same Napoleon as the great warrior? Was he a mathematician too?&lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="55" LastEditDate="2015-09-29T17:47:28.143" LastActivityDate="2015-09-30T01:22:31.793" Title="Who discovered Napoleon's theorem?" Tags="|mathematics|biographical-details|discoveries|debunking|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2868" PostTypeId="2" ParentId="2867" CreationDate="2015-09-29T17:46:04.987" Score="7" Body="&lt;p&gt;The name of the theorem (that centers of equilateral triangles erected externally on the sides of a triangle form another equilateral triangle) does refer to the famous French general and later emperor, which likely contributed to its popularity. However, its earliest attribution to him comes from an Italian textbook of elementary geometry by Faifofer published in 1911, almost a century after Napoleon's death. There is just a single phrase in parentheses without any named source:“&lt;em&gt;theorem proposed for demonstration by Napoleon to Lagrange&lt;/em&gt;”. The earliest known occurence is as a challenge problem in Rutherford's Ladies Diary of 1825, four years after Napoleon's death. But the theorem was rediscovered multiple times by amateur geometers, so Rutherford may not have been the first either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coxeter and Greitzer claimed that &quot;&lt;em&gt;the possibility of [Napoleon] knowing enough geometry for this feat is as questionable as the possibility of his knowing enough English to compose the famous palindrome, ABLE WAS I ERE I SAW ELBA&lt;/em&gt;&quot;, but this is unwarranted. Napoleon was reported by biographers to have mathematical talent since boyhood, and closely interacted with prominent mathematicians and scientists of his day, including Fourier, Monge, Laplace and Lagrange. Monge and Fourier accompanied him on 1798 expedition to Egypt, along with 150 other scientists and engineers, with whom Napoleon held lengthy discussions on scientific matters. Laplace even briefly served as his Minister of Interior (for 6 weeks), but did not leave a good impression. Harris comments that “&lt;em&gt;after the revolution, [Lagrange] fell into the favor of Napoleon Bonaparte, who enjoyed sharing geometrical puzzles with Lagrange and Laplace&lt;/em&gt;”, but without any specific reference to the Napoleon's theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.mathpages.com/home/kmath270/kmath270.htm&quot; rel=&quot;nofollow&quot;&gt;Mathpages&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-09-30T01:22:31.793" LastActivityDate="2015-09-30T01:22:31.793" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2869" PostTypeId="2" ParentId="2857" CreationDate="2015-09-29T22:03:20.250" Score="19" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Historiography_of_science#The_Hessen_thesis_and_the_birth_of_externalism&quot; rel=&quot;noreferrer&quot;&gt;Histories of science and mathematics are very young&lt;/a&gt;, only emerging as disciplines in the early 20th century. This means that most of the received view (tradition) was created by non-historians, who often had expository, didactic and human interest purposes as their primary motivation. This encouraged a view of history as a sequence of bursts of genius, and rational reconstruction as the method of analyzing it, reading past results as imperfect prototypes of modern concepts, perhaps in a different guise, towards which they develop. Grattan-Guinness coined the term &quot;heritage&quot;, as opposed to &quot;history&quot;, for this kind of approach in &lt;a href=&quot;https://www.jstor.org/stable/4145010&quot; rel=&quot;noreferrer&quot;&gt;History or Heritage? An Important Distinction in Mathematics and for Mathematics Education&lt;/a&gt;. It often produces conceptually cleaned up and notationally modernized sketches, and for many purposes this may well be desired. Both tendencies were taken to the extreme however in E.T. Bell's controversial Men of Mathematics, and they largely dominate historical parts of many mathematical textbooks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main changes occurred in two stages, neither exclusive to the history of mathematics. In 1930s the traditional historical internalism was challenged by demands for attention to social and cultural context, the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Historiography_of_science#The_Hessen_thesis_and_the_birth_of_externalism&quot; rel=&quot;noreferrer&quot;&gt;Hessen thesis&lt;/a&gt;. The second stage was a turn from rational reconstruction to detailed analysis of contemporary sources, and more sceptical view of mathematical speculations. For the history of mathematics specifically the watershed was &lt;a href=&quot;http://www.lrb.co.uk/v29/n04/mf-burnyeat/other-lives&quot; rel=&quot;noreferrer&quot;&gt;Burkert's Lore and Science: Studies of Pythagoras, Philolaus and Plato&lt;/a&gt;, that exposed almost all of traditional story about Pythagoras as a myth fabricated by Plato's successors at the Academy. &quot;&lt;em&gt;Mystic, yes... but a mathematician, no&lt;/em&gt;&quot;. Burkert's is an impeccable work of classical scholarship with exhaustive analysis of multiple original sources, many of them obscure, that left little room for doubt and set a standard for subsequent work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometric algebra was a natural hypothesis under the rational reconstruction methodology. It is very appealing if modern algebra is taken as the final cause, and gained wide popularity after van der Waerden's eloquent popularization. But its speculative nature and conflation of Babylonian arithmetic with algebra led Szabo to question it in late 60s, and Unguru  to reject it forcefully in 1975, &lt;a href=&quot;http://users.auth.gr/~nioka/Files/GEOMALGE.pdf&quot; rel=&quot;noreferrer&quot;&gt;more on the debate here&lt;/a&gt;. A modern view can be found in &lt;a href=&quot;http://www.maa.org/press/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction&quot; rel=&quot;noreferrer&quot;&gt;Fowler's Mathematics of Plato's Academy&lt;/a&gt;, which is directly indebted to Burkert for its methodology and many insights:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The idea that a length is a number is so deeply ingrained in our thought that it takes a conscious effort to conceive of an approach to geometry that does not make such an assumption. It is such an arithmetized interpretation that led historians to describe Book II of the elements as &quot;geometric algebra&quot;. Fowler argues that Greek geometry was completely non-arithmetized. The strongest evidence comes from his analysis of the very difficult Book X, where he shows, I think successfully, that the way Euclid (or Theaetetus?) structures the argument precludes an arithmetical approach. Most historians would agree with this description of Greek geometry, I think. Many, however, see this characteristic as something to be explained... Fowler turns this on its head, and argues that in fact the non-arithmetized nature of Greek geometry should be read as evidence that Greek geometry does not derive from Babylonian sources. Essentially, Fowler argues that this approach to geometry is just as &quot;natural&quot; as the arithmetized one that we now use, and that it therefore is not necessary to explain why the Greeks used it&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While the geometric algebra debate may have been the loudest the points made apply more broadly. This is what Unguru argued, for example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;To read ancient mathematical texts with modern mathematics in mind is the safest method for misunderstanding the character of ancient mathematics, in which philosophical presuppositions and metaphysical commitments played a much more fundamental and decisive role than they play in modern mathematics&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For example, one finds Euclid interpreted by Hilbert and Klein as a founder of axiomatic method in the popular textbooks of Greenberg and Hartshorne. But Euclid's Elements read in the context of the times (known from multiple mathematical and non-mathematical sources), turn out to have more in common with modern constructivism and intuitionism than with Hilbert's axiomatic formalization. See the historical part of &lt;a href=&quot;http://www.jstor.org/stable/2185244?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Friedman's Kant's Theory of Geometry&lt;/a&gt; and section 3 of &lt;a href=&quot;https://www.academia.edu/8016051/Euclid_s_Pseudaria&quot; rel=&quot;noreferrer&quot;&gt;Acerbi's Euclid's Pseudaria&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-20T04:32:18.803" LastActivityDate="2019-06-20T04:32:18.803" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="2870" PostTypeId="1" CreationDate="2015-09-30T15:49:49.027" Score="6" ViewCount="306" Body="&lt;p&gt;For years, I have been perplexed that the expression $\Gamma^i_{jk}$ is often referred to in the plural as &quot;the Christoffel symbols&quot;, although sometimes it is referred to in the singular as &quot;the Christoffel symbol&quot;. I have found this even in the 1926 book &quot;The absolute differential calculus&quot;, which is a translation of Levi-Civita's 1925 Italian book &quot;Lezioni di calcolo differenziale assoluto&quot;. Since Levi-Civita approved the translation, I imagine he approved the use of &quot;Christoffel's symbol of the first kind&quot; and &quot;Christoffel's symbol of the second kind&quot; which appear in there, pages 109–110, although he changed to the plural on page 169.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another early writer, Hermann Weyl, in his 1918 &quot;Raum, Zeit, Materie&quot;, page 91, referred to &quot;Christoffelsche Dreiindizes-Symbole&quot;, in the plural.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the original 1869 paper by Christoffel, &quot;Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades&quot;, the word &quot;symbol&quot; was not really used, and the words which &lt;em&gt;are&lt;/em&gt; used don't give a clear indication either way of the singular or the plural.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern authors seem to have a slight preference for the plural, although it doesn't sound quite right, and there is quite a lot of inconsistency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now here are my theories.&lt;br&gt;&#xA;Theory 1: Because initially there was the symbol of the first kind and symbol of the second kind, the plural was adopted to refer to the two &lt;em&gt;kinds&lt;/em&gt; of symbols. I.e. there were two symbols. Nowadays we mostly only use the second kind. But the use of the plural has stuck, erroneously, due to everyone just following everybody else like sheep. In other words, it's a misunderstanding, which everyone copies because the original reason for the plural is lost in the mists of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theory 2: Since the array of coefficients is not a tensor, it could not be called &quot;the Christoffel tensor&quot;. It could have been called &quot;the Christoffel array&quot;, but almost no one does that. So since there is no &quot;container&quot; term for the full array, people use the plural &quot;symbols&quot; to mean the &lt;em&gt;elements&lt;/em&gt; of the array. In other words, each &lt;em&gt;element&lt;/em&gt; of the array is one symbol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem with both of these theories is that so many authors alternate between singular and plural, with no obvious reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So here is my question:&lt;br&gt;&#xA;Is there a &lt;em&gt;correct&lt;/em&gt; terminology, with a definitive ruling on which usage is correct?&lt;/p&gt;&#xA;" OwnerUserId="2502" LastEditorUserId="2502" LastEditDate="2015-10-01T04:17:50.240" LastActivityDate="2015-10-01T04:17:50.240" Title="Is $\Gamma^i_{jk}$ the Christoffel symbol or the Christoffel symbols?" Tags="|terminology|differential-geometry|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2875" PostTypeId="1" AcceptedAnswerId="2880" CreationDate="2015-10-01T16:33:57.623" Score="20" ViewCount="21831" Body="&lt;p&gt;Did Ramanujan have any formal training in mathematics? I read that he did not. Then how did he learn to do mathematics? Mathematics does not come from instinct. You have to learn the basic facts. How did Ramanujan manage to solve hard mathematical problems with so little knowledge of theorems?&lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="4251" LastEditDate="2020-07-06T06:59:13.213" LastActivityDate="2020-07-06T06:59:13.213" Title="How did Ramanujan learn to do mathematics?" Tags="|mathematics|mathematicians|education|ramanujan|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="2876" PostTypeId="1" AcceptedAnswerId="2881" CreationDate="2015-10-01T18:41:23.197" Score="8" ViewCount="5957" Body="&lt;p&gt;There are several uses of 12 in some old systems of measurement. Some of them make sense given current context (There are 12 lunar cycles per year), however some of them seem to be arbitrarily chosen. Just to name a few...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There are 12 inches in a foot&lt;/li&gt;&#xA;&lt;li&gt;Day and night are each divided into 12 hours&lt;/li&gt;&#xA;&lt;li&gt;There are 12 signs in both Western and Chinese Zodiacs&lt;/li&gt;&#xA;&lt;li&gt;We have a word specifically for twelve (dozen)&lt;/li&gt;&#xA;&lt;li&gt;We have a word for 12 12's (gross)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2962" LastActivityDate="2018-12-31T10:25:28.057" Title="Was there a very early culture that's number system was 12-based, like ours is 10-based?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2878" PostTypeId="5" CreationDate="2015-10-02T01:44:35.733" Score="0" Body="&lt;p&gt;This tag can be used for two purposes, either for resources for learning, or for to learn more about a scientist's/mathematician's educational background. The first can be quickly asked in an answer's comments if it is not already discussed in the answer. This tag applies to all levels of the person's education, so if possible use a more specific tag.&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="62" LastEditDate="2015-10-02T02:08:17.093" LastActivityDate="2015-10-02T02:08:17.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2879" PostTypeId="4" CreationDate="2015-10-02T01:44:35.733" Score="0" Body="This tag can be used for two purposes, either for resources for learning, or for to learn more about a scientist's/mathematician's educational background." OwnerUserId="140" LastEditorUserId="62" LastEditDate="2015-10-02T02:08:27.530" LastActivityDate="2015-10-02T02:08:27.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2880" PostTypeId="2" ParentId="2875" CreationDate="2015-10-02T02:06:31.277" Score="31" Body="&lt;p&gt;First hand testimony and insightful thoughts on Ramanujan's background and way of doing mathematics can be found in &lt;a href=&quot;https://www.jstor.org/stable/2301659&quot; rel=&quot;noreferrer&quot;&gt;Hardy's lecture Indian Mathematician Ramanujan&lt;/a&gt;. Hardy is the British mathematician who first appreciated the full extent of Ramanujan's talent, knew him well personally, and had a fruitful mathematical collaboration with him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is Hardy's review of Ramanujan's mathematical education:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He was sent at 7 to the High School of Kumbakonam, and remained there nine years. His exceptional abilities had begun to show themselves before he was 10, and by the time that he was 12 or 13 he was recognised as a quite abnormal boy. His biographers tell some curious stories of his early years. They say for example that, soon after he had begun the study of trigonometry, he discovered for himself &quot;Euler's theorems for the sine and cosine&quot; (by which I understand the relations between the circular and exponential functions), and was very disappointed when he found later, apparently from the second volume of Loney's Trigonometry, that they were known already. Until he was 16 he had never seen a mathematical book of any higher class... He had had no real teaching at all; there was no one in India from whom he had anything to learn. He can have seen at the outside three or four books of good quality, all of them English. There had been periods in his life when he had access to the library in Madras, but it was not a very good one&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, what Ramanujan lacked in education he made up for with his natural knack for mathematics, especially his ability to recognize and relate highly non-obvious abstract and symbolic patterns. These abilities, while not instinctive, likely have an underlying biological component. But it was not just natural ability that distinguished Ramanujan, but also hard work and intensity of concentration:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He was so absorbed in the study of Mathematics that in all lecture hours-whether devoted to English, History, or Physiology - he used to engage himself in some mathematical investigation, unmindful of what was happening in the class&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And there was a book that made a difference, &lt;a href=&quot;https://archive.org/details/asynopsiselemen00carrgoog/page/n7/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Carr's Synopsis of Elementary Results in Pure and Applied Mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was a book of a very different kind, Carr's Synopsis, which first aroused Ramanujan's full powers... The book is not in any sense a great one, but Ramanujan has made it famous, arid there is no doubt that it influenced him profoundly and that his acquaintance with it marked the real starting point of his career... Carr has sections on the obvious subjects, algebra, trigonometry, calculus and analytical geometry, but some sections are developed disproportionally, and particularly the formal side of the integral calculus. This seems to have been Carr's pet subject, and the treatment of it is very full and in its way definitely good. There is no theory of functions... What is more surprising, in view of Carr's own tastes and Ramanujan's later work, is that there is no elliptic functions. However Ramanujan may have acquired his very peculiar knowledge of this theory, it was not from Carr. On the whole, considered as an inspiration for a boy of such abnormal gifts, Carr was not too bad, and Ramanujan responded amazingly.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nonetheless, the lack of education had obvious adverse effects on Ramanujan's mathematical accomplishments:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He worked, for most of his life, in practically complete ignorance of modern European mathematics, and died when he was a little over 30 and when his mathematical education had in some ways hardly begun... Whittaker's Modern Analysis had not yet spread so far, and Bromwich's Infinite Series did not exist. There can be no doubt that either of these books would have made a tremendous difference to him if they could have come his way... He had been carrying an impossible handicap, a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe... It was impossible to teach him systematically, but he gradually absorbed new points of view. In particular he learnt what was meant by proof, and his later papers, while in some ways as odd and individual as ever, read like the works of a well-informed mathematician. His methods and his weapons, however, remained essentially the same.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But we can also never know how originality of his thought would have been affected had he been educated more traditionally.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-29T10:37:25.213" LastActivityDate="2020-01-29T10:37:25.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2881" PostTypeId="2" ParentId="2876" CreationDate="2015-10-02T02:35:21.213" Score="6" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Duodecimal#Origin&quot; rel=&quot;nofollow noreferrer&quot;&gt;duodecimal system&lt;/a&gt; as a developed positional system of numerals does not appear historically. However, ancient Egyptians, Babylonians and Chinese used duodecimal units in astronomical and time keeping matters. &lt;a href=&quot;http://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Scientific American&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;As early as 1500 B.C., the Egyptians had developed a more advanced sundial. A T-shaped bar placed in the ground, this instrument was calibrated to divide the interval between sunrise and sunset into 12 parts. This division reflected Egypt's use of the duodecimal system--the importance of the number 12 is typically attributed either to the fact that it equals the number of lunar cycles in a year or the number of finger joints on each hand (three in each of the four fingers, excluding the thumb), making it possible to count to 12 with the thumb...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;During the era when sundials were first used, however, Egyptian astronomers also first observed a set of 36 stars that divided the circle of the heavens into equal parts. The passage of night could be marked by the appearance of 18 of these stars, three of which were assigned to each of the two twilight periods when the stars were difficult to view. The period of total darkness was marked by the remaining 12 stars, again resulting in 12 divisions of night (another nod to the duodecimal system). During the New Kingdom (1550 to 1070 B.C.), this measuring system was simplified to use a set of 24 stars, 12 of which marked the passage of the night... Once both the light and dark hours were divided into 12 parts, the concept of a 24-hour day was in place&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Modern fixed-length hours, however, were only first suggested by Hipparchus c.140 BC, who used their length on the days of equinox. Seasonally varying hours remained in use for many centuries hence.&lt;/p&gt;&#xA;&lt;p&gt;Babylonians also originally had twelve divisions in a day, eventually increased to modern 24. Traditional Chinese calendars, clocks, and compasses are based on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Earthly_Branches&quot; rel=&quot;nofollow noreferrer&quot;&gt;twelve Earthly Branches&lt;/a&gt;, a calendar system derived from the motion of Suìxīng, the Year Star, our Jupiter. The celestial circle was divided into 12 sections, and the orbital period was rounded from 11.86 to 12 years. The duodecimal weight units go back to Romans, Latin  uncia gave birth to modern English ounce and inch.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-12-31T10:25:28.057" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2882" PostTypeId="2" ParentId="2562" CreationDate="2015-10-02T15:13:07.413" Score="12" Body="&lt;p&gt;&lt;a href=&quot;http://bibliotheque-institutdefrance.fr/sites/default/files/memoire-equations-radicaux-galois-evariste.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; is a scanned PDF of Evariste's work, that includes the memoir you're mentioning (&lt;a href=&quot;http://www.calames.abes.fr/pub/#details?id=IF2B14057&quot; rel=&quot;nofollow noreferrer&quot;&gt;parent page&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;See the left part of page #3:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/svpTf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/svpTf.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The only explanation I could find is &lt;a href=&quot;http://fr.wikisource.org/wiki/Page:Galois_-_Manuscrits,_%C3%A9dition_Tannery,_1908.djvu/13&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (French).&lt;/p&gt;&#xA;&lt;p&gt;Galois wrote this &amp;quot;Oh ! Chérubins&amp;quot; on the back of the cover sheet after the memoir had been (again) rejected. Presumably this is addressed to Mr. Lacroix and Mr. Poisson, whom Galois considers to be responsible for the rejection.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Oh ! Chérubins&amp;quot; may be considered as a both condescending and sarcastic cry.&lt;br /&gt;&#xA;However, even for a French speaker, this is hard to interpret, since this expression is not common at all. In French, the word &amp;quot;Chérubin&amp;quot; is a kind of angel, or, in the colloquial language, an innocent child. Anyway, this word is not very common.&lt;/p&gt;&#xA;&lt;p&gt;[EDIT]&#xA;I've found another version of the paper in an old magazine I had bought 20 years ago. Oddly, the stains do not look the same, but the text itself matches perfectly. I've scanned this from &amp;quot;Pour la Science - Les génies de la science - n°14 - Evariste Galois&amp;quot; (February 2003):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vWfGC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vWfGC.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2963" LastEditorUserId="2963" LastEditDate="2021-09-20T07:09:35.950" LastActivityDate="2021-09-20T07:09:35.950" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2883" PostTypeId="1" AcceptedAnswerId="2885" CreationDate="2015-10-02T15:14:53.047" Score="12" ViewCount="1329" Body="&lt;p&gt;Did Evariste Galois create the entire &lt;strong&gt;group structure&lt;/strong&gt;  concept?  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If yes, were &quot;super-sets&quot; of groups (e.g. rings or vector&#xA;spaces) created on top of Galois's work? When and by who?&lt;/li&gt;&#xA;&lt;li&gt;If no, did Galois rely on a previously existing concept of groups to create &quot;permutation groups&quot;? Was that concept something fully common/known at that time?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2963" LastEditorUserId="4251" LastEditDate="2021-06-22T08:21:03.540" LastActivityDate="2021-06-22T08:21:03.540" Title="Did Evariste Galois create the entire group structure concept?" Tags="|mathematics|group-theory|19th-century|galois-theory|galois|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2884" PostTypeId="2" ParentId="2883" CreationDate="2015-10-02T18:09:47.773" Score="13" Body="&lt;p&gt;Galois's introduced the term group in 1831 in a paper on solutions of polynomials by radicals, but he only considered permutation groups. Without an explicit concept group properties of permutations were used by Lagrange, Ruffini and Cauchy before him. In particular in Lagrange's Reflexions sur la Resolution Algebrique des Equation (1770-71) a theorem on permutations fixing a function is proved, which is now identified with the coset theorem. Cauchy's 1815 paper Memoir on the Number of Values that a Function can Acquire (When One Permutes in all Possible Ways the Quantities it Contains) is considered to be the first paper on permutation groups as such. Galois does not mention Lagrange in 1831, but in his famous last letter he included an example of coset decomposition. He was also likely familiar with Cauchy's work. However, group related notions remained little known until Galois' works were republished in Liouville's Journal in 1846. An abstract definition of (finite) groups appears  in Cayley's On the theory of Groups, as Depending on the Symbolic Equation $θ^n = 1$ (1854). See &lt;a href=&quot;http://www.jstor.org/stable/2690624?seq=1#page_scan_tab_contents&quot;&gt;History of Lagrange's Theorem on Groups&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The kind of abstraction that we associate with algebraic structures (as in Bourbaki) did not exist in Galois' time and for some time since. Grassmann introduced (what we now call) vector spaces and exterior algebra in his famous 1844 book, but his approach is in the style of &quot;concrete algebra&quot; and infused with geometry.  Dedekind was perhaps first to treat abstract algebraic concepts in their modern spirit, in particular he studied rings of integers and ideals in them. The term number ring was coined by Hilbert in 1892 (published in 1897), and Peano gave the modern definition of vector spaces and linear maps in 1888. Emmy Noether and van der Waerden were largely responsible for promoting the abstract approach before Bourbaki canonized it. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-10-02T18:09:47.773" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2885" PostTypeId="2" ParentId="2883" CreationDate="2015-10-02T18:15:30.563" Score="13" Body="&lt;p&gt;The group concept (and all other important concepts) evolved gradually. Galois did not define the group as we know it now. For him, a group was a group of substitutions on a finite set. Substitutions on finite sets were considered by Cauchy and Lagrange before Galois. Including premutations of the roots of algebraic equations.  Galois contribution is that he created a powerful theory (called Galois theory) which uses these permutation groups to analyse algebraic equations. He certainly also had the notion of a field, &#xA;(domain of rationality) though not formalized in the modern way.&#xA;The general group concept was formalized during approximately a century (19) with research of Galois and (much later) S. Lie as primary examples of groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same happened with other notions you mention. The notion of vector evolved for almost 2 millenia, beginning with Apollonius, whose famous theorem means exactly that addition of vectors in the plane is commutative. The modern definition of vector space is quite recent (20s century, I suppose) though the ideas which belong to this theory were used by many, including Galois. The development of vector spaces (linear algebra) was not much dependent on the notion of group,&#xA;because the most interesting examples of groups are non-commutative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is important to understand that mathematics did not develop in the way that we learn it in schools and universities. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-10-02T18:15:30.563" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2886" PostTypeId="1" CreationDate="2015-10-05T01:03:51.493" Score="4" ViewCount="186" Body="&lt;p&gt;Whenever a (major) new result is proved, it is typically only understood only by a handful of experts, then by people who work in close vicinity, gradually understood by everyone in the general subject area, and then by a broad mathematical audience, and if the result is particularly important, eventually trickle down to mainstream science. This process is a more realistic version of the adage &quot;there are only two kinds of mathematical problems: trivial and impossible. It is impossible until you solve it and then it becomes trivial.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking what is the expected amount of time (in a modern sense) before a major breakthrough in mathematics is 'digested'. For instance, the invention of calculus was no doubt at the time a tour de force, but by now it is understood by almost all professional mathematicians and is taught to a broad audience at a relatively low level. Doron Zeilberger once remarked, on his opinion page, that eventually the proof of Fermat's Last Theorem can be understood by undergraduates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I am asking for the average time, as there are great many examples of variances. I believe that the proof that there exist infinitely many bounded gaps between primes will be digested far faster than, say, the proof of Weil's conjectures, due to the relatively simple principles and machinery that goes into the former. Thus a discussion on the variance and the reliance on particular subject matters might be enlightening as well.&lt;/p&gt;&#xA;" OwnerUserId="2681" LastActivityDate="2015-10-05T17:42:11.663" Title="Expected time before a theorem or theory becomes 'easy'" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2887" PostTypeId="2" ParentId="2810" CreationDate="2015-10-05T08:06:36.420" Score="4" Body="&lt;p&gt;You say&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] due to Russell's paradox [&lt;strong&gt;1901&lt;/strong&gt;], which ultimately led to the development of Zermelo-Fraenkel set theory, thus &lt;strong&gt;putting mathematics back on solid formal foundations&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question is, how much of a revolution did this process cause in the rest of mathematics, outside of the formal foundations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but be aware that Frege's &lt;a href=&quot;https://en.wikipedia.org/wiki/Begriffsschrift&quot; rel=&quot;nofollow noreferrer&quot;&gt;Begriffsschrift&lt;/a&gt; is only form 1879! The idea of using &lt;span class=&quot;math-container&quot;&gt;$\forall, \exists$&lt;/span&gt; is younger than the photoelectric effect and so is the line of thinking that you should actually use formal logic for mathematics in this way.&lt;/p&gt;&#xA;&lt;p&gt;Russel's issue arose a few years after the &amp;quot;logic people&amp;quot; took a look into set theory for the first time. What would be the &amp;quot;solid foundations you put mathematics back on&amp;quot; that mathematicians in different fields would already know (much less trust and care) about?&lt;/p&gt;&#xA;" OwnerUserId="585" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-06-17T20:03:51.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2888" PostTypeId="2" ParentId="2886" CreationDate="2015-10-05T13:01:03.203" Score="5" Body="&lt;p&gt;This time widely varies, so the &quot;average time&quot; is not well-defined. Once I asked myself: what is the most recent theorem that we teach to  undergraduates (in my university). This is the duality theorem in linear programming by J. Nash which is about 50 years old. (Next goes the fast Fourier Transform). Most of the undergraduate math curriculum was essentially discovered in 18s century, &#xA;with some&#xA;improvements made in the 19s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there are theorems whose proof does not simplify over very long periods. One example is the Uniformization Theorem which is central in the subject of Conformal Mapping. It was discovered by Klein in the late 19s century, and rigorously proved by Poincare and Koebe in the very beginning of the 20s century. The proof was somewhat simplified since then but to give a complete proof is an advanced course remains a challenge. On the other hand, everyone knows this theorem and many people use it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Classification of simple Lie algebras is another example. It was obtained in 1890s by Cartan and Killing, many simplifications and improvements were made since then, but the result remains difficult and takes long time to explain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want an older example, take a fundamental result of ancient Greek geometry: classification of regular polytopes (including their construction). Of course now, after more than 2000 years passed we have much better understanding of this matter, and we can explain it to undergraduates in 2 or 3 hours, but still I would not say that this is too easy.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-10-05T17:42:11.663" LastActivityDate="2015-10-05T17:42:11.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2889" PostTypeId="1" AcceptedAnswerId="2890" CreationDate="2015-10-06T07:10:38.307" Score="4" ViewCount="1142" Body="&lt;p&gt;Euler mentions in his preface of the book &quot;Foundations of Differential Calculus&quot; (Translated version of Blanton):   &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;....Even now there is more that remains obscure than what we see clearly. As differential calculus is extended to all kinds of functions, no matter how they are produced, it is not immediately known that method is to be used to the vanishing increments of absolutely all kinds of functions. Gradually this discovery has progressed to more and more complicated functions, the ultimate ratio that the vanishing increments attain could be assigned long before the time of Newton and Leibniz, so that the differential calculus applied to only these rational functions must be held to have been invented long before that time. However, there is no doubt that Newton must be given credit for that part of differential calculus concerned with irrational functions...  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational fuctions long before (Newton and Leibniz), without knowing application of the method to the vanishing icrements; if it was already invented, how does that differ from that of the study-of-ultimate ratio (differentail calculus) for irrational functions, it must be the same method (??).&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="26" LastEditDate="2016-02-23T15:08:46.193" LastActivityDate="2016-02-23T15:08:46.193" Title="Who invented differential calculus for rational functions?" Tags="|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2890" PostTypeId="2" ParentId="2889" CreationDate="2015-10-06T09:23:35.360" Score="9" Body="&lt;p&gt;The differential calculus for rational functions can be reduced to finding roots of polynomials. Such methods were used by Fermat and Descartes. See problems 12.3-12.5 of &lt;a href=&quot;http://intellectualmathematics.com/history-of-mathematics/&quot;&gt;my history of mathematics notes&lt;/a&gt; for a summary of their methods. These methods are not based on infinitesimals but rather on the idea that a tangent line is a line for which two points of intersection with a curve have degenerated into one. This translates into a double-root algebraic property for a certain polynomial, from which the tangent can be found. Such methods do not work for &quot;irrational functions,&quot; meaning functions that cannot be reduced to a polynomial equation.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2015-10-06T09:23:35.360" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2892" PostTypeId="1" CreationDate="2015-10-06T15:24:45.360" Score="8" ViewCount="1538" Body="&lt;p&gt;I read that several principles of Al-Jazari's monumental water clocks were based upon earlier designs of water clocks by Archimedes, for example the use of valves, feedback system and flow control regulator. Apparently, Archimedes even wrote a treatise On the Construction of Water Clocks. So what innovations did Archimedes introduce in his hydraulic clock?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-05-13T12:21:48.173" LastActivityDate="2021-04-07T15:14:34.843" Title="How did Archimedes's water clock work?" Tags="|biographical-details|ancient-greece|engineering|archimedes|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2894" PostTypeId="2" ParentId="2892" CreationDate="2015-10-06T19:41:58.257" Score="7" Body="&lt;p&gt;Jazari is referring to &quot;&lt;em&gt;an Arabic treatise of unknown date and authorship&lt;/em&gt;&quot; that describes a monumental water-clock. It is not listed among Archimedes's works in any ancient sources, and &lt;a href=&quot;http://www.history-science-technology.com/articles/articles%206.html&quot; rel=&quot;noreferrer&quot;&gt;according to  Hill&lt;/a&gt;, most of it was written by several Arabic authors. The &quot;author&quot; is now referred to as Pseudo-Archimedes. Ridwan al-Saati built &lt;a href=&quot;https://en.wikipedia.org/wiki/Jayrun_Water_Clock&quot; rel=&quot;noreferrer&quot;&gt;Jayrun water clock&lt;/a&gt; based on the Pseudo-Archimedes's design. He also mentions some Hormuz, who invented a water clock used by his father in the construction of the Damascus clock:“&lt;em&gt;the design continued in the land of Fars for a long time, and was transmitted from there to the land of the Greeks, and its construction spread out in the land until it was transmitted to Damascus, where it was constructed up to the days of the Byzantines and after that in the days of Banu Umayya, according to what is mentioned in the histories&lt;/em&gt;&quot;. For details see &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B0006E69V4&quot; rel=&quot;noreferrer&quot;&gt;Hill's Arabic Water Clocks&lt;/a&gt;, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is known that a public water clock was erected in Gaza in fifth century AD, so Hormuz might have lived before the rise of Islam, and possibly re-invented the water clock independently. But simple water clocks were already used by ancient Babylonians and Egyptians. A perfected version, clepsydra (&quot;water thief&quot;) is due to Archimedes's contemporary Ctesibius (c.285-222 BC), a renowned engineer, the founder of pneumatics, and possibly the first head of the Alexandrian Museum. Unfortunately, none of his works survived to our times, but his inventions are described by Vitruvius, Athenaeus, Philo of Byzantium, Hero and Proclus. A detailed analysis of Ctesibius's work is given in &lt;a href=&quot;https://books.google.com/books?id=MOTpnfz7ZuYC&quot; rel=&quot;noreferrer&quot;&gt;Russo's Forgotten Revolution&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some modern engineers credit clepsydra as the first automatic feedback device. &#xA;The problem Ctesibius faced was that the speed of water escaping a container through a hole depends on the water level in it. To make it work for keeping time he needed a container where the water level remains fixed. This is accomplished by a float with a valve that blocks the inflow when the level rises, and lets it through when it falls, a feedback controller in modern terms. The clock involves three containers, the first one serves as a water reservoir, and empties into the second, which has the float with a valve ensuring that its water level remains constant. So the speed at which it empties into the third container is also constant, and a float in the third container will rise uniformly. A pointer attached to it indicates time. See a nice &lt;a href=&quot;http://www.mlahanas.de/Greeks/Clocks.htm&quot; rel=&quot;noreferrer&quot;&gt;illustration at Lahanas's site&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-07T21:16:45.913" LastActivityDate="2015-10-07T21:16:45.913" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2896" PostTypeId="1" AcceptedAnswerId="2898" CreationDate="2015-10-07T18:02:40.667" Score="2" ViewCount="29337" Body="&lt;p&gt;Why was it decided to differentiate kcals from &lt;a href=&quot;https://en.wikipedia.org/wiki/Calorie&quot; rel=&quot;nofollow&quot;&gt;calories&lt;/a&gt; with a capital 'C'? It seems kind of odd to me.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;  1 Cal = 1 kcal = 1000 cal  &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;What were the reasons behind that decision and who decided to that?&lt;/p&gt;&#xA;" OwnerUserId="2980" OwnerDisplayName="HSchmale" LastEditorUserId="26" LastEditDate="2015-10-08T07:29:02.720" LastActivityDate="2023-02-13T18:45:02.347" Title="&quot;Calories&quot; vs &quot;calories&quot;" Tags="|terminology|notation|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2897" PostTypeId="2" ParentId="2896" CreationDate="2015-10-07T18:17:40.953" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Calorie&quot; rel=&quot;nofollow noreferrer&quot;&gt;The small calori&lt;/a&gt;e or gram calorie (symbol: cal) is the approximate amount of energy needed to raise the temperature of one gram of water by one degree Celsius at a pressure of one atmosphere.&lt;/p&gt;&#xA;&lt;p&gt;The large calorie, kilogram calorie, dietary calorie, nutritionist's calorie, nutritional calorie, Calorie (capital C)[2] or food calorie (symbol: Cal) is approximately the amount of energy needed to raise the temperature of one kilogram of water by one degree Celsius. The large calorie is thus equal to 1000 small calories or one kilocalorie (symbol: kcal)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Simplicity for diets, not to carry so many significant figures which make no difference in dietary programs.&lt;/p&gt;&#xA;" OwnerUserId="3848" OwnerDisplayName="anna v" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-10-07T18:17:40.953" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2898" PostTypeId="2" ParentId="2896" CreationDate="2015-10-07T18:56:03.677" Score="4" Body="&lt;p&gt;The rocky history of the calorie is explained fairly well in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does the history of food energy units suggest a solution to &quot;Calorie confusion&quot;? J. L. Hargrove, &lt;a href=&quot;http://dx.doi.org/10.1186/1475-2891-6-44&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Nutr. J.&lt;/em&gt; &lt;strong&gt;6&lt;/strong&gt;, 44 (2007)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In short, it appears that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;both the &quot;small&quot; and the &quot;large&quot; calories (the energy used to raise the temperature of 1 g and 1 kg of water by 1°C, respectively) were in use, and&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;there was no standard spelling for either,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;for the long stretch of time between their initial uses in the first half of the 19th century until the systematization of unit systems in the mid-twentieth century. In particular, large stretches of nutritional science and the public were using the name Calorie (I believe for the large calorie) for several decades during the twentieth century, in parallel with a limited use of the small calorie by scientists and engineers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time nutritionists sat down to sort things out in the 1960s, it was pretty much too late. They resolved to&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;keep both units,&lt;/li&gt;&#xA;&lt;li&gt;indicate the large calorie with a capital C, as Calorie,&lt;/li&gt;&#xA;&lt;li&gt;emphasize that this is equivalent to 1000 small calories, i.e. 1 kcal, and&lt;/li&gt;&#xA;&lt;li&gt;phase out the term Cal in favour of kcal.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first three are easy, as they make the system confusing but at least it is unambiguous. The fourth point is a lot harder, because it requires the public to make widespread use of this weird kcal thing. (Moreover, it was already behind the times, and they should have switched to kJ while they were at it.) Because of these difficulties, the fourth point was not really completed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, the use of the large calorie as a heat unit is pretty strongly discouraged, but this does depend on where you are. In the UK, for example, nutrition facts labels must indicate energy content using kcal and kJ (&lt;a href=&quot;http://www.nutrition.org.uk/healthyliving/healthyeating/labels.html&quot; rel=&quot;nofollow&quot;&gt;source&lt;/a&gt;). The &lt;a href=&quot;http://www.ecfr.gov/cgi-bin/retrieveECFR?gp=1&amp;amp;SID=4bf49f997b04dcacdfbd637db9aa5839&amp;amp;ty=HTML&amp;amp;h=L&amp;amp;mc=true&amp;amp;n=pt21.2.101&amp;amp;r=PART#se21.2.101_160&quot; rel=&quot;nofollow&quot;&gt;USA regulations&lt;/a&gt;, on the other hand, are pretty ambiguous, and there probably are a lot of brands out there that use the large calorie in their labels to avoid big, intimidating numbers.&lt;/p&gt;&#xA;" OwnerUserId="272" OwnerDisplayName="Emilio Pisanty" LastActivityDate="2015-10-07T18:56:03.677" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2899" PostTypeId="1" AcceptedAnswerId="2900" CreationDate="2015-10-08T10:52:54.583" Score="8" ViewCount="2359" Body="&lt;p&gt;Who replaced the Halmos solid QED symbol $\blacksquare$ with the open square $\square$? In his 1955 &quot;general topology&quot; book, page vi, John Kelley attributed the original solid square to Paul Halmos.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The end of each proof is signalized by $\blacksquare$. This notation is also due to Halmos.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To be precise, Kelley used a solid rectangle, about 7 points height above the line, 2 points depth below the line, and a width of 3.2 points. But I can't find such a solid rectangle in this TeX symbol manual for use in Mathjax: &lt;a href=&quot;http://www.math.harvard.edu/texman/node21.html&quot;&gt;http://www.math.harvard.edu/texman/node21.html&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1958, Howard Eves in &quot;Foundations and fundamental concepts of mathematics&quot;, page 149, wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The modern symbol $\square$, suggested by Paul R. Halmos, or some variant of it, is frequently used to signal the end of a proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even though Eves used the open square, he didn't attribute the change from solid to open to any particular author. The other books where I have seen any historical comments at all on this subject all simply refer to Halmos. No one gives any credit for who used the open square first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q1. Does anyone have any references which use the open square earlier than 1958?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q2. Does anyone know of a definitive attribution for the open square?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q3. Is it possible that the open square came first, before the solid square?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you would expect, there are several mentions in wikipedia, but none of them definitive. The closest to an explanation is this apparently self-contradictory page: &lt;a href=&quot;https://en.wikipedia.org/wiki/Tombstone_(typography)&quot;&gt;https://en.wikipedia.org/wiki/Tombstone_(typography)&lt;/a&gt;. I don't have full confidence in the accuracy of the Halmos quote on that page. There's also &lt;a href=&quot;https://en.wikipedia.org/wiki/Q.E.D.&quot;&gt;Q.E.D.&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_proof&quot;&gt;Mathematical proof&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_mathematical_symbols&quot;&gt;List of mathematical symbols&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2502" LastEditorUserId="2502" LastEditDate="2015-10-08T10:58:29.910" LastActivityDate="2015-10-09T00:22:32.490" Title="Who replaced the solid Halmos QED symbol with the open one?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2900" PostTypeId="2" ParentId="2899" CreationDate="2015-10-09T00:22:32.490" Score="3" Body="&lt;p&gt;Interesting, I thought that the origin was a Latin pun: &quot;quod&quot; in &quot;quod erat demonstrandum&quot; is similar to &quot;quad&quot;, the root in &quot;quadratus&quot;, square. It is possible that Halmos did not specify whether the square needed to be filled, and Kelley, or his typographers, simply chose the filled version. &lt;a href=&quot;http://www.magforum.com/blog/?p=1751&quot; rel=&quot;nofollow&quot;&gt;End marks in articles&lt;/a&gt; that Halmos refers to as inspiration were not necessarily filled, or squares. Reporters often use &lt;a href=&quot;http://ajrarchive.org/Article.asp?id=4408&quot; rel=&quot;nofollow&quot;&gt;-30- to sign off&lt;/a&gt;, and &quot;&lt;em&gt;in the late 1930s, the popular weekly Woman used a right-facing double chevron to show when its fiction continued over a page (», like a guillemet or French punctuation mark) and simply said ‘THE END’ at, well, the end. Algamated’s news magazine Pictorial Weekly also said ‘THE END.’ on its long pieces (1933)&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Halmos' quote is accurate, and appears on p. 403 of his &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387960783&quot; rel=&quot;nofollow&quot;&gt;I Want to Be a Mathematician&lt;/a&gt;. The symbol in the text is a hollow square.  Here is the full paragraph: &quot;&lt;em&gt;My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented &quot;iff&quot;, for &quot;if and only if&quot; - but I could never believe that I was really its first inventor. I am quite prepared to believe that it existed before me, but I don't know that it did, and my invention (re-invention ?) of it is what spread it through the mathematical world. The symbol is definitely not my invention - it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like&lt;/em&gt; ▯ &lt;em&gt;and is used to indicate an end, usually the end of a proof. It is most frequently called the &quot;tombstone&quot;, but at least one generous author referred to it as the &quot;halmos&quot;&lt;/em&gt;&quot;. The use of &quot;iff&quot; was also pioneered in Kelley's General Topology (1955).&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-10-09T00:22:32.490" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2901" PostTypeId="1" CreationDate="2015-10-10T00:08:36.263" Score="15" ViewCount="11437" Body="&lt;p&gt;&lt;a href=&quot;https://english.stackexchange.com/questions/278272/how-to-say-its-not-rocket-science-before-rockets-existed#comment616722_278275&quot;&gt;This comment&lt;/a&gt; cites &lt;a href=&quot;http://en.wikipedia.org/wiki/History_of_calculus#Newton_and_Leibniz&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; in stating that, before the development of the modern-day subject of calculus (i.e. analysis), the term &quot;calculus&quot; referred to general mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who first used the term &quot;calculus&quot; in mathematics, and what specifically as it applied to? I do know that &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=maqhmatiko/s&quot; rel=&quot;noreferrer&quot;&gt;the Greeks used the word &quot;mathematics&quot;&lt;/a&gt; in ancient times to refer to what we would today call &quot;mathematics&quot;, and the word took on different meanings at different points in time.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:16.393" LastActivityDate="2021-03-29T05:59:46.577" Title="Who first used the word &quot;calculus&quot;, and what did it describe?" Tags="|mathematics|terminology|calculus|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2902" PostTypeId="2" ParentId="2901" CreationDate="2015-10-10T04:15:35.093" Score="14" Body="&lt;p&gt;According to Carl B. Boyer, &quot;The history of the calculus and its conceptual development&quot;, Dover Publications 1959, page 98,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The improved notation led also to methods which were so much more facile in application than the cumbrous geometrical procedures of Archimedes, of which they were modifications, that these methods were eventually recognized as forming a new analysis—the calculus. The period during which this transformation took place may be considered as the century preceding the work of Newton and Leibniz.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The question is complicated by the fact that most mathematicians were writing in Latin.) If you are asking when the word &quot;calculus&quot; was used to refer to the differential and/or integral calculus, that must surely post-date the development of those subjects. For example, Richard Suiseth was known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Swineshead&quot;&gt;Calculator&lt;/a&gt; in the 14th century, and experts on what we now call arithmetic were called &quot;reckoners&quot; in the middle ages. The word &quot;reckon&quot; is really the German word &quot;Rechen&quot;, which means &quot;calculate&quot;. So &quot;Calculator&quot; is just Latin for &quot;Rechner&quot;. So the word &quot;calculus&quot; in one form or another was around for a while in regard to arithmetic. The question is somewhat confused by the fact that what we call &quot;mathematics&quot; was actually called &quot;geometry&quot; until only a few centuries ago, and mathematicians were called &quot;geometers&quot;, even into the 19th century. The term &quot;mathematics&quot; in Greek refers to knowledge in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the better question to ask is when were the terms &quot;differential calculus&quot; and &quot;integral calculus&quot; first used. The guy who taught Newton his basic calculus (before Newton developed it further) was Isaac Barrow, who wrote a book &quot;Geometrical Lectures&quot;, showing that this subject was still considered to be part of geometry at that time. According to Boyer, page 190, Newton used the word &quot;analysis&quot; for his first publication on calculus.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first notice of his calculus was given, however, in 1669, in &lt;em&gt;De analysi per aequationes numero terminorum infinitas&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And integration was originally called &quot;quadrature&quot;, as in Newton's &lt;em&gt;De quadratura&lt;/em&gt;. (Boyer, page 205.) It seems to me that it was Leibniz who first promoted the term &quot;calculus&quot; in the modern sense. He wrote a book &lt;em&gt;Historia et origo calculi differentialis&lt;/em&gt; a year or two before his death. (Boyer, page 215.) In 1712, the Royal Society committee reported that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;differential method&lt;/em&gt; is one and the same with the &lt;em&gt;method of fluxions&lt;/em&gt;, excepting the name and mode of notation [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Boyer, page 221.) This shows that Newton was not even using the term &quot;differential calculus&quot;. (The term &quot;differential method&quot; referred to Leibniz's method.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Boyer, page 67, the word &quot;integral&quot; originated with Leibniz also, but on the suggestion of the Bernoulli brothers. So it seems to me that the term &quot;calculus&quot;, with that name, developed only during the late 17th century, more in Europe than in Britain, where the term was resisted for some time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. If you just want to know when the word &quot;calculus&quot; was first used for calculating (as opposed to the meaning of small stones or pebbles), my Latin dictionary (John T. White, 1902, 1923) gives quotes from Cicero and Livius where the word was used in the sense of &quot;reckoning, computing, calculating&quot;. That puts it back to 2000 years ago, plus or minus about 30 years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PPS. Perhaps I could also mention that what was called &quot;calculus&quot; in ancient Roman times and up until the Renaissance in Europe was not considered to be part of the same subject as geometry, which was taught at the universities and the precursors of universities. There was a sharp divide between the advanced mathematics of Euclid, Apollonious and Archimedes and the market-place arithmetic which is well-documented since the beginning of writing 5500 years ago. Everyone in these 5.5 millennia would have learned the basic marketplace arithmetic as part of growing up. It was not a study as such. It did not need a university education. Nor was it generally learned from books. So to call that marketplace &quot;calculus&quot; or &quot;reckoning&quot; part of mathematics is not really correct. If you ask a mathematician right now to add up a supermarket invoice with 30 items, they might have some difficulties, but we would not think that this indicates a lack of mathematical ability. That's just the &quot;reckoning&quot; of the marketplace, and carpentry, book-keeping, and dozens of other trades. So in my opinion, &quot;calculus&quot; was not used for a really &lt;em&gt;mathematical&lt;/em&gt; subject until the 16th or 17th century. Until then, it was only applied to marketplace addition, subtraction, multiplication and division. Calling that mathematics would be like calling a road-sign literature.&lt;/p&gt;&#xA;" OwnerUserId="2502" LastEditorUserId="2502" LastEditDate="2015-10-10T12:35:00.887" LastActivityDate="2015-10-10T12:35:00.887" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2903" PostTypeId="2" ParentId="2901" CreationDate="2015-10-10T09:47:26.287" Score="6" Body="&lt;p&gt;Your question was: Who was the first person to use the term “calculus”?&#xA;I cannot say who the first individual was, but in any case he or she was an ancient Roman: the Latin word “calculus” means “small stone, pebble”, then “a stone used in reckoning on the counting-board” and then “reckoning, computing, calculating”. This usage is known from the time of Cicero (1st century BC) onwards. The word “calculus” itself comes from “calx” “limestone” (as in “calcium”), which is probably a borrowing from Greek “chalix”, with the same meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You might want to look here: &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dcalculus&quot; rel=&quot;noreferrer&quot;&gt;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dcalculus&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2015-10-10T09:47:26.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2904" PostTypeId="1" AcceptedAnswerId="2909" CreationDate="2015-10-10T21:12:36.557" Score="18" ViewCount="1898" Body="&lt;p&gt;Britannica is terse:&quot;&lt;em&gt;Möbius discovered this surface in 1858. The German mathematician Johann Benedict Listing had discovered it a few months earlier, but he did not publish his discovery until 1861&lt;/em&gt;&quot;. Möbius never published the discovery, it was found in his 1865 memoir and published posthumously after 1868. Derbyshire comments &quot;&lt;em&gt;It had previously been described by another mathematician, Johann Listing, also in 1858. Listing published, and Möbius didn’t, so according to the academic rules it should really be called “the Listing strip”. There is no justice in this world&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure how Britannica's &quot;few months&quot; are known, considering that publications came years after, but it is not unheard of for two people to make the same discovery almost simultaneously, when an idea is &quot;in the air&quot;. Newton-Leibniz and Leverrier-Adams are famous examples. What seems to be missing is what exactly was in the air, how did the Möbius strip come up?&#xA;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Mobius.html&quot;&gt;MacTutor&lt;/a&gt; says only &quot;&lt;em&gt;this discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Académie&lt;/em&gt;&quot;. This is more or less repeated everywhere else. But the context is missing, and some strange fables are spun. The most outlandish one is that Möbius and Listing discovered the strip &quot;together&quot;, and &quot;decided&quot; to give priority to Möbius. Another, from an old popular book that I can not recall, is that Möbius was inspired by the shape of a twisted stock garter dropped by his maid. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Möbius strip is not exactly a polyhedron, what is the connection? Möbius worked in Leipzig, and Listing in Göttingen. Listing was not exactly unknown, he was recommended for the Göttingen position by Gauss himself. Did they both work on the Académie problem? were they aware of each other's work? did they correspond or meet? why did the name &quot;Möbius strip&quot; stuck? Can somebody fill in the context?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-12-24T19:16:05.227" Title="How was the Möbius strip discovered?" Tags="|mathematics|biographical-details|discoveries|debunking|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2905" PostTypeId="1" CreationDate="2015-10-09T15:13:32.533" Score="9" ViewCount="15360" Body="&lt;p&gt;If &quot;millimeter waves&quot; have a wavelength of about 1 mm, one might linguistically expect microwaves to have a wavelength of about three orders of magnitude less, not the same or greater. &lt;strong&gt;How did microwaves get their name?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I did consider posting this on &lt;a href=&quot;https://english.stackexchange.com/&quot;&gt;EL&amp;amp;U&lt;/a&gt; but think here it might get the benefit of more specialized knowledge or people more familiar with the concept of microwaves and wavelengths and the concepts being communicated by these terms.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2672" OwnerDisplayName="WBT" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:16.393" LastActivityDate="2016-08-23T06:05:51.000" Title="Why are microwaves called &quot;microwaves&quot;, when they are much longer than a micrometer?" Tags="|physics|terminology|electromagnetism|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2906" PostTypeId="2" ParentId="2905" CreationDate="2015-10-09T18:12:53.480" Score="6" Body="&lt;p&gt;The micro- in micro-waves, as far as I know, comes from the way that we produced electromagnetic waves at the time: following Hertz, people generally used currents cycling through antennas. Generally for broadcast the antennas would be huge affairs and would radiate in all directions. The radiated wavelength goes proportional to the antenna length because you want the antenna to resonate; so you got very long wavelengths. So, for example, according to Wikipedia, a 50m wavelength was used by Bowen's first antenna (the first radar antenna) in 1935.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, microwaves existed before this point, with the first microwave generator usually credited to the Germans in 1920 (the &lt;a href=&quot;https://en.wikipedia.org/wiki/Barkhausen%E2%80%93Kurz_tube&quot; rel=&quot;noreferrer&quot;&gt;Barkhausen-Kurz tube&lt;/a&gt;), and the &lt;a href=&quot;http://www.etymonline.com/index.php?term=microwave&quot; rel=&quot;noreferrer&quot;&gt;online etymology dictionary&lt;/a&gt; listing a first usage of the term &quot;micro-wave&quot; to describe them in English from 1931. The OED expands on this with a quote that you &lt;a href=&quot;http://www.grayswandir.net/modules/olpc/wikislice-chemistry-en/files/articles/Microwave.htm&quot; rel=&quot;noreferrer&quot;&gt;can also read here&lt;/a&gt;, a 1931 article in the British Postmaster-General's &quot;Telegraph and Telephone Journal&quot;, where they generated 18cm waves and reported immediate &quot;undisguised surprise... that the problem of the micro-wave had been solved so soon.&quot; But the point was that at that time, it was very common to generate much longer wavelengths, maybe hundreds of meters long; it was not nearly so common to create smaller ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This terminology continued into World War II: more precisely, in 1940, &lt;a href=&quot;http://www.makingthemodernworld.org.uk/icons_of_invention/technology/1939-1968/IC.051/&quot; rel=&quot;noreferrer&quot;&gt;a new device was created by John Randall and Harry Boot&lt;/a&gt; in the UK, called the &quot;cavity magnetron.&quot; It was immediately applied to radar with positive results: it shrank the size of a radar antenna to the point where it could be mounted on an aircraft and used to detect a sub. At this point &quot;microwave&quot; seems to have been well-established usage for a wave which was much smaller than the large radio waves but larger than the infra-red opacity barrier of air.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those cavity magnetrons became &quot;microwave ovens&quot; in the conventional sense, though the idea of using some sort of electromagnetic radiation to cook a steak was apparently demonstrated &lt;a href=&quot;http://www.scottedelman.com/2011/12/18/the-first-microwave-was-unveiled-in-1931/&quot; rel=&quot;noreferrer&quot;&gt;much earlier in 1931 in Berlin&lt;/a&gt;, with the chief innovation claimed there being a spiraling waveguide which transmitted the radiation through a ceramic pot into whatever wet objects were placed therein. These could well have been microwaves, given that in 1933 the above microwave tubes were old enough that they were &lt;a href=&quot;https://en.wikipedia.org/wiki/Rudolf_K%C3%BChnhold#Radar_research&quot; rel=&quot;noreferrer&quot;&gt;being used by a German researcher to try to spot ships at sea&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3012" OwnerDisplayName="Chris Drost" LastEditorUserId="62" LastEditDate="2015-10-11T00:28:33.080" LastActivityDate="2015-10-11T00:28:33.080" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2907" PostTypeId="2" ParentId="2901" CreationDate="2015-10-11T12:23:04.233" Score="5" Body="&lt;p&gt;As per fdb's answer, the latin word &lt;a href=&quot;http://webalt.net/Calculus-2004/Various/MeaningOfCalculus.htm&quot; rel=&quot;nofollow&quot;&gt;calculus&lt;/a&gt; was used with the arithmetical meaning early on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See Anthony Lo Bello, &lt;a href=&quot;https://books.google.it/books?id=LfEEAQAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Origins of mathematical words : A Comprehensive Dictionary of Latin, Greek, and Arabic Roots&lt;/a&gt; (2013), page 50 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;calculate&lt;/strong&gt; The Latin word for stone is &lt;em&gt;calx, calcis&lt;/em&gt;. The addition of the&#xA;  ending &lt;em&gt;-ulus&lt;/em&gt; to the stem produces the diminutive &lt;em&gt;calculus&lt;/em&gt;, which means a &lt;em&gt;small stone&lt;/em&gt; or &lt;em&gt;pebble&lt;/em&gt;. As a medical term, it is used of bladder, gall, and kidney stones, and even the gritty accumulation on the teeth. Since such pebbles were used as counters in counting, the verb &lt;em&gt;calculo, calculare, calculavi, calculatus&lt;/em&gt; came into existence with the meaning to count. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;calculus&lt;/strong&gt; This is the Latin word for &lt;em&gt;a small stone&lt;/em&gt;. It came to be used by the late seventeenth-century mathematicians as a technical term for any theory that laid the foundations of a general method to calculate the solutions of certain types of problems and then, to those theories that solved the problems of tangent lines (differential calculus) and quadrature (integral calculus). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But the use of &lt;em&gt;calculus&lt;/em&gt; in mathematical books is not easy to trace.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have take into account :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing&quot; rel=&quot;nofollow&quot;&gt;The Compendious Book on Calculation by Completion and Balancing&lt;/a&gt; (Arabic: &lt;em&gt;Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala&lt;/em&gt;), written by &lt;a href=&quot;https://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB&quot; rel=&quot;nofollow&quot;&gt;Muḥammad ibn Mūsā al-Khwārizmī&lt;/a&gt; around 820 and translated into Latin by &lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_of_Chester&quot; rel=&quot;nofollow&quot;&gt;Robert of Chester&lt;/a&gt; in the mid-12th century as &lt;em&gt;Liber algebrae et almucabola&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;nofollow&quot;&gt;Leonardo Fibonacci&lt;/a&gt; or Leonardo Pisano, &lt;a href=&quot;https://en.wikipedia.org/wiki/Liber_Abaci&quot; rel=&quot;nofollow&quot;&gt;Liber Abaci&lt;/a&gt;, composed in 1202; we can translate it as &quot;The Book of Calculation&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Swineshead&quot; rel=&quot;nofollow&quot;&gt;Richard Swineshead&lt;/a&gt;, &lt;em&gt;Liber calculationum&lt;/em&gt; (&quot;Book of Calculations&quot;), written around 1350.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;During the Renaissance, we have a tension between re-discovery of ancient Greek mathematics and new discoveries. Thus, we have a flourishing of new names for rediscovered or new-founded disciplines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first one is &lt;em&gt;algebra&lt;/em&gt;; we can list :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gerolamo_Cardano&quot; rel=&quot;nofollow&quot;&gt;Girolamo Cardano&lt;/a&gt;, with his &lt;a href=&quot;https://en.wikipedia.org/wiki/Ars_Magna_(Gerolamo_Cardano)&quot; rel=&quot;nofollow&quot;&gt;Artis magnae, sive de regulis algebraicis&lt;/a&gt; of 1545 (&quot;The Great Art&quot;) dedicated to the new &lt;em&gt;algebra&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Rafael_Bombelli&quot; rel=&quot;nofollow&quot;&gt;Rafael Bombelli&lt;/a&gt;, with his  1572 book, &lt;a href=&quot;http://mathematica.sns.it/media/volumi/9/L&amp;#39;algebra_bw.pdf&quot; rel=&quot;nofollow&quot;&gt;L'Algebra&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow&quot;&gt;François Viète&lt;/a&gt;, with his &lt;a href=&quot;https://en.wikipedia.org/wiki/New_algebra&quot; rel=&quot;nofollow&quot;&gt;In Artem Analyticien Isagoge (Introduction to the art of analysis)&lt;/a&gt; (1591).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In Viète, &lt;em&gt;Analysis&lt;/em&gt; is not yet the modern discipline; for its &quot;birth&quot;, we have to see Newton's 1669 manuscript : &lt;a href=&quot;https://en.wikipedia.org/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas&quot; rel=&quot;nofollow&quot;&gt;De analysi per aequationes numero terminorum infinitas&lt;/a&gt; (published in 1704).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The first usage of &lt;em&gt;calculus&lt;/em&gt; with the new meaning is due to Leibniz; see G.W.Leibniz (editor : J.M.Child), &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;The Early Mathematical Manuscripts&lt;/a&gt; (1920 - also Dover reprint) :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Analysis Tetragonistica Ex Centrobarycis&lt;/em&gt; (Analytical quadrature by means of centers of gravity), dated 1675 (page 65)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Methodi tangentium directae compendium calculi ...&lt;/em&gt; (Compendium of the calculus of the direct method of tangents ...), dated 1675, page 111 &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Calculus Tangentium differentialis&lt;/em&gt;(Differential calculus of tangents), dated  1676, page 124&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Elementa Calculi Novi pro differentiis et summis, tangentibus et quadraturis, maximis et minimis, dimensionibus linearum, superficierum, solidorum, allisque communem calculum transcendentibus&lt;/em&gt; (The Elements of a New Calculus for Differences and Sums, Tangents and Quadratures, maxima and minima, the measurement of lines, surfaces and solids, and other things which transcend the usual sort of calculus); the manuscript is undated, but appears to have been compiled sometime later than 1677 and prior to 1680, page 136.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See the &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Newton/CommerciumAccount/CommAcc.pdf&quot; rel=&quot;nofollow&quot;&gt;Commercium Epistolicum&lt;/a&gt; for a variety of terms : &lt;em&gt;Analysis&lt;/em&gt;, Newton's &lt;em&gt;Method of Series&lt;/em&gt;, &lt;em&gt;Method of Fluxions&lt;/em&gt;, Dr.Barrow [and] Mr.Leibnitz : their &lt;em&gt;Methods of Tangents&lt;/em&gt;, Leibniz's &lt;em&gt;calculo differentiali&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Presumably, the success of &lt;em&gt;Analysis&lt;/em&gt; is due to :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital&quot; rel=&quot;nofollow&quot;&gt;Marquis de L'Hopital&lt;/a&gt;'s tretise : &lt;a href=&quot;https://en.wikipedia.org/wiki/Analyse_des_Infiniment_Petits_pour_l%27Intelligence_des_Lignes_Courbes&quot; rel=&quot;nofollow&quot;&gt;Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes&lt;/a&gt; (1696, several editions).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-05-10T14:33:10.947" LastActivityDate="2016-05-10T14:33:10.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2908" PostTypeId="1" CreationDate="2015-10-11T15:07:12.583" Score="3" ViewCount="265" Body="&lt;p&gt;I am working on a project in which I want to show how the thoughts of some non-scientists, poets and artists in particular, matched with science. To instill interest in science among my peers I want to show that science (and math) is a way of thinking and not only a subject of fat books. I have worked on it and found some examples:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The Salvador Dali painting &quot;Persistence of Memory&quot; shows melting clocks, relating to the 4th dimension.&lt;/li&gt;&#xA;&lt;li&gt;The Rayleigh-Taylor instability.. is used in observations from a painting by artist David Alfaro Siqueiros that relates to major discoveries that now relate to astrophysics.&lt;/li&gt;&#xA;&lt;li&gt;Van Gogh has a painting that depicting turbulence.&lt;/li&gt;&#xA;&lt;li&gt;Sanskrit poets used math to rhyme sequences in their poems.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Please help me with a few more examples and help me to spread the interest about science and mathematics to all. Hope you will help me to succeed me in this quest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To say it shortly, I want historical examples of thinking related to science and mathematics.  &lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="55" LastEditDate="2015-10-11T21:43:42.980" LastActivityDate="2015-10-12T01:07:32.333" Title="Historical examples of non-scientists who thought scientifically" Tags="|mathematics|philosophy-of-science|social-context|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2909" PostTypeId="2" ParentId="2904" CreationDate="2015-10-11T20:25:27.457" Score="8" Body="&lt;p&gt;My answer is based on guesses, not on knowledge of original work. Those years were the years were polyhedra were studied in connection with the problem of understanding to what extent Euler's formula $V-L+F=2$ was valid. In particular it was clear that there were some exceptions to the rule that needed, in the prevalent opinion, not be considered as polyhedra (see Lakatos &quot;Proofs and refutations&quot; for a modern discussion, with historical references). Since Listing numbers, one of the things he is remembered for, are topological invariants of homological nature, it will not surprise me if he discovered the Möbius band while trying to understand characteristic of a triangulation that was violating Euler's formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, it is not a polyhedra, but it is easily constructed from a triangulated polyhedra. But the original polyhedra with no identification satisfies Euler's formula and the triangulation on the strip does not. So this is my guess about the link between polyhedra and Möbius strip. But one should read original papers at this point...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ADDED: I finally received and read Tripodi's papers but didn't gat much out of them (at least no explicit reference to the Möbius strip). But I found a &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN599484047_0002&amp;amp;DMDID=DMDLOG_0027&amp;amp;LOGID=LOG_0030&amp;amp;PHYSID=PHYS_0088&quot; rel=&quot;nofollow&quot;&gt;very nice paper&lt;/a&gt; by Cayley which corroborates my hypothesis. The paper refers, from the title, to Listing's work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;it concerns a generalization of Euler's formula for &quot;a figure of any sort whatever&quot;; it counts lines on the figure which may be curved, for example. So it clearly extend out of polyhedra. In fact, in reading, one can easily see homology theory in disguise (and I have to say, personally, it is thrilling to read some remarks from which you really can see topology in its childhood...). All examples from 13 starts from a rectangular piece of paper and says and construct a space after identifications of opposite side. Even though there is no explicit mention of the strip, if the work of Listing was exemplified by such constructions it would be quite reasonable that he found the twisted strip in connection to this problem. I cannot go any further from here, since I am not able to read German.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="2442" LastEditDate="2015-12-24T19:16:05.227" LastActivityDate="2015-12-24T19:16:05.227" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2910" PostTypeId="2" ParentId="2908" CreationDate="2015-10-11T21:35:33.430" Score="3" Body="&lt;p&gt;The ancient &lt;a href=&quot;http://www.ancientgreece.com/s/People/Phidias/&quot; rel=&quot;nofollow&quot;&gt;Greek sculptor Phidias&lt;/a&gt; (c. 480-430 BC), who designed the statue of Zeus at Olympia and of Athena in Parthenon, apparently used golden proportions in his sculptures and made them a part of Greek artistic canon. The notation $\varphi$ for golden ratio is said to come from the first letter of his name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Islamic architects used aperiodic tilings of the plane in their decorative designs since c. 1200 using a set of &lt;a href=&quot;https://en.wikipedia.org/wiki/Girih_tiles#Mathematics_of_girih_tilings&quot; rel=&quot;nofollow&quot;&gt;Girih tiles&lt;/a&gt;. In modern times they were rediscovered and mathematically studied by Penrose and others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Italian architect Brunelleschi (1377-1446), one of the founding fathers of Renaissence, developed the techniques of &lt;a href=&quot;https://en.wikipedia.org/wiki/Perspective_%28graphical%29#History&quot; rel=&quot;nofollow&quot;&gt;linear perspective&lt;/a&gt;, a practical application of geometric optics, in modern times (it was already used in some ancient Roman paintings) and employed it in designing the dome of the Florence Cathedral. Many other prominent Renaissence figures contributed to the development of the technique. Alberti's De Pictura (&quot;On Painting&quot;, 1436) was first to treat perspective mathematically, but only for objects on the ground plane, della Francesca in 1470s extended it to solids anywhere in the picture plane, Luca Pacioli's 1509 De Divina Proportione (On Divine Proportion), illustrated by Leonardo da Vinci, summarized the use of perspective in painting, as well as promoted the use of golden ratio. Development of linear perspective eventually led to a new mathematical discipline, projective geometry, in the hands of French engineer and architect Desargues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the early 20th century new developments in physics inspired a number of poets and artists see &lt;a href=&quot;http://www.surrealismcentre.ac.uk/papersofsurrealism/journal8/acrobat%20files/Book%20Reviews/Ambrosio%20final%2017.05.10.pdf&quot; rel=&quot;nofollow&quot;&gt;Parkinson's Surrealism, Art and Modern Science&lt;/a&gt;. Valery visited physical laboratories and befriended Bohr, Perrin, Langevin and de Broglie. Breton and Soupault’s surrealist painting Magnetic Fields (1919) is meant to capture the spirit of relativity, Paalen’s 1940 Figure Pandynamique tries to picture the wave/particle duality artistically, it depicts a set of repeating rings culminating in spheres.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the most famous mathematics inspired modern artist is &lt;a href=&quot;https://en.wikipedia.org/wiki/M._C._Escher#Works&quot; rel=&quot;nofollow&quot;&gt;Escher&lt;/a&gt; (1898–1972), who wrote &quot;mathematicians have opened the gate leading to an extensive domain&quot;. His initial inspiration was Polya's paper on plane symmetry groups, and his discussions with Coxeter, a prominent Canadian geometer, led to Escher's famous tessellations of the hyperbolic plane. Aside from tessellations Escher is known for using impossible objects, conformal maps, and other mathematical constructions in his paintings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is plenty of &lt;a href=&quot;http://www.math.uconn.edu/~glaz/My_Articles/PoetryInspiredByMathematics.Bridges10.pdf&quot; rel=&quot;nofollow&quot;&gt;poetry inspired by mathematics&lt;/a&gt;. English poet Coleridge (1772–1834) wrote a poem inspired by Euclid's construction of equilateral triangle. Lyrics to Coulton's song Mandelbrot Set go like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pathological monsters! cried the terrified mathematician   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every one of them is a splinter in my eye &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hate the Peano Space and the Koch Curve   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I fear the Cantor Ternary Set                      &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the Sierpinski Gasket makes me want to cry &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And a million miles away a butterfly flapped its wings &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a cold November day a man named Benoît Mandelbrot was born &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-11T21:44:48.450" LastActivityDate="2015-10-11T21:44:48.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2911" PostTypeId="2" ParentId="2908" CreationDate="2015-10-12T01:07:32.333" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://www.mcescher.com&quot; rel=&quot;noreferrer&quot;&gt;Escher&lt;/a&gt; was very intrigued by symmetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole area of perspective was explored and put on a solid mathematical foundation by renaissance artists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leonardo da Vinci (and others) studied anatomy (something that was frowned upon, or even directly harshly punished at the time). In his time artists where just an aspect of what we'd call &quot;engineers&quot;, in any case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are beautifully illustrated tomes on plants and animals, done by artists striving for maximal fidelity with nature. That might be near the other extreme, of scientists doing art.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2015-10-12T01:07:32.333" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2912" PostTypeId="1" CreationDate="2015-10-12T14:16:37.167" Score="7" ViewCount="693" Body="&lt;p&gt;I'm trying to learn tensor calculus on my own, and I thought it would help if I could get some historical perspective, like who developed the component-free approach to tensors, and around what time it was developed.&lt;/p&gt;&#xA;" OwnerUserId="2574" LastEditorUserId="62" LastEditDate="2015-10-12T14:37:05.297" LastActivityDate="2015-10-15T11:41:22.117" Title="History of the component-free approach to tensors" Tags="|mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2913" PostTypeId="1" CreationDate="2015-10-12T23:28:27.140" Score="5" ViewCount="410" Body="&lt;p&gt;The Bernoulli numbers appear in the Harer-Zagier formula enumerating gluings of polygons, the Kervaire-Milnor formula for the order of homotopy groups for n-spheres, and (with the connection to the Riemann zeta function at positive integer values) Witten's formulas for volumes of curvature forms in moduli spaces for quantum guage theories in two dimensions. Hirzebruch used his apparently intimate knowledge of them and the Norlund genearlized (convolved) Bernoulli numbers in setting up his theory of genera, and Milnor and Stasheff devote an appendix to them in their book Characteristic Classes. Lie theory is tied up with the structure of manifolds, and the Bernoulli numbers enter Lie theory through the BCH theorem. So, there are many cameo appearances of the numbers in topology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who first introduced the Bernoulli numbers into topology and how?&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2015-10-13T02:58:09.763" LastActivityDate="2015-10-13T02:58:09.763" Title="Injection of Bernoulli numbers into topology" Tags="|mathematics|number-theory|group-theory|topology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2914" PostTypeId="1" AcceptedAnswerId="2916" CreationDate="2015-10-13T01:30:12.603" Score="4" ViewCount="196" Body="&lt;p&gt;What is the earliest formal equation of magnetism?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The history of discovery of magnetism can trace back to Thales of Miletus. Although many people had discovered, they did not formalize an equation.&lt;/p&gt;&#xA;" OwnerUserId="2605" LastEditorUserId="55" LastEditDate="2016-11-25T21:19:52.357" LastActivityDate="2016-11-25T21:19:52.357" Title="What is the earliest mathematical formula describing magnetism?" Tags="|physics|magnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2915" PostTypeId="2" ParentId="2913" CreationDate="2015-10-13T01:40:01.143" Score="3" Body="&lt;p&gt;The earliest occurence Milnor mentions in his survey &lt;a href=&quot;http://www.math.stonybrook.edu/~jack/hed-again.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Differential Topology Forty-six Years Later&lt;/a&gt; is &lt;a href=&quot;http://www.jstor.org/stable/1968956?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whitehead's paper On the Homotopy Groups of Spheres and Rotation Groups (1942)&lt;/a&gt;. If $J_n$ is the image of the stable Whitehead homomorphism from the $n$th stable homotopy group of rotation groups to the $n$th stable homotopy group of spheres, then $J_n$ is cyclic of order $B_k/4k$ for $n=4k-1$. Whitehead's paper is the source for the Milnor-Stasheff's appendix, and the starting point of Milnor's, Brumfiel's, Kervaire's, Browder's, etc. 1950-1960s results on homotopy groups of spheres, that was the boom age of the subject. Their work eventually led to computing the number of exotic spheres, homeomorphic but not diffeomorphic to the standard Euclidean sphere. The discovery of exotic spheres in 1956 is what earned Milnor his Fields medal in 1962. By the way, here is a video of &lt;a href=&quot;https://www.math.stonybrook.edu/Videos/IMS/Differential_Topology/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Milnor delivering Hedrik lecture on exotic spheres back in 1965&lt;/a&gt;!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On Bernoulli numbers in general see Math Overflow thread &lt;a href=&quot;https://mathoverflow.net/questions/61252/why-do-bernoulli-numbers-arise-everywhere&quot;&gt;Why do Bernoulli numbers arise everywhere?&lt;/a&gt; Sums of powers of integers can be expressed in their terms, as Faulhaber discovered for first 17 powers in 1631, before Bernoulli, and their generating function is $t/(e^t-1)=\sum_{n=0}^{\infty}\frac{B_n\;t^n}{n!}$. According to Henry Cohn &quot;&lt;em&gt;the reason why $t/(e^t-1)$ is a natural generating function to consider is that we sometimes want to invert $e^t-1$ (the factor of t is just to make it holomorphic), and the most important reason I know of to invert it is that we want to invert $\Delta=e^D-1$&lt;/em&gt;&quot;, where $D$ is the derivative and $\Delta$ is the finite difference. But this does not quite explain their appearance in topology.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-10-13T01:40:01.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2916" PostTypeId="2" ParentId="2914" CreationDate="2015-10-13T02:04:58.343" Score="4" Body="&lt;p&gt;As with the &lt;a href=&quot;https://hsm.stackexchange.com/questions/2552/is-coulombs-law-the-earliest-formal-equation-of-electricity&quot;&gt;previous question on electricity&lt;/a&gt; it is ambiguous as to what constitutes &quot;formal equation&quot; or &quot;magnetism&quot;. Light is an electromagnetic wave, and geometric optics became mathematical in antiquity, Descartes even introduced wave optics in luminiferous ether in 1638. Peregrinus discovered that lodestones align needles along lines of longitude between poles on the stone back in 1269, which one can turn into an &quot;equation&quot; if one so wishes. Gilbert's theory of magnetism appeared in 1600 and was qualitative, but many of his observations can also be written as &quot;equations&quot;, as can be Descartes's later on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the first explicit mathematical law is mentioned in Newton's Principia:&quot;&lt;em&gt;power of magnetism... in receding from the magnet, decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations&lt;/em&gt;&quot;. The law was confirmed by the publishers of Principia's 1742 edition Le Seur and Jacquier. In 1750 Michell clarified that in a magnet &quot;&lt;em&gt;each pole attracts or repels exactly equally&lt;/em&gt;&quot; and &quot;&lt;em&gt;attraction and repulsion of magnets increases, as the squares of the distances from the respective poles increase&lt;/em&gt;&quot;. See &lt;a href=&quot;https://books.google.com/books?id=CGJDAAAAIAAJ&amp;amp;pg=PA52&amp;amp;lpg=PA52&amp;amp;dq=electricity+history+cavendish&amp;amp;source=bl&amp;amp;ots=S-dh0pHFTg&amp;amp;sig=hqA8utwgmB_7yM5ggXJCVomQf-M&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CCwQ6AEwBTgKahUKEwik-cveq-rGAhWQl4gKHchWDNA#v=onepage&amp;amp;q=electricity%20history%20cavendish&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whittaker's History of the Theories of Aether and Electricity&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-10-13T02:04:58.343" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2918" PostTypeId="1" AcceptedAnswerId="3103" CreationDate="2015-10-13T19:59:17.317" Score="10" ViewCount="1374" Body="&lt;p&gt;I am interested in this question: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the first to show that for every circle the fraction $$\frac{\text{circumference}}{\text{diameter}}$$ is always constant?    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not interested in $\pi$ approximations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;proof&quot; I mean any argument that could be &quot;fixed&quot; in some way to work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(because I suppose that an ancient proof would not fit the modern standards)&lt;/p&gt;&#xA;" OwnerUserId="2056" LastEditorUserId="62" LastEditDate="2015-10-13T22:17:18.930" LastActivityDate="2018-10-24T00:47:21.910" Title="First proof that circumference/diameter $=\pi$" Tags="|mathematics|geometry|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2919" PostTypeId="2" ParentId="2918" CreationDate="2015-10-13T20:22:03.393" Score="6" Body="&lt;p&gt;It must be old at least as much as the first written Egyptian and Babylonian approximations : if you do not think that it is a &quot;specific&quot; number, why try to compute it ?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Written sources :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/defIII1.html&quot;&gt;Euclid's Elements&lt;/a&gt; , &lt;strong&gt;Book III&lt;/strong&gt;, &lt;em&gt;Def.1&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Equal circles are those whose diameters are equal, or whose radii are equal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Book III&lt;/strong&gt;, &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII26.html&quot;&gt;Prop.26&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Book XII&lt;/strong&gt;, &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html&quot;&gt;Prop.2&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Circles are to one another as the squares on their diameters.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Archimedes' &lt;a href=&quot;https://en.wikipedia.org/wiki/Measurement_of_a_Circle&quot;&gt;Measurement of a Circle&lt;/a&gt; : &lt;em&gt;Prop.1&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, if we assume that two circles of equal radii have equal area (by Euclid, XII.2), then the two circumferences must be equal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Prop.3&lt;/em&gt; is the well-known Archimedes' approximations of $\pi$ :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ratio of the circumference of any circle to its diameter is greater than $3\tfrac{10}{71}$ but less than $3\tfrac{1}{7}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-10-13T20:58:16.997" LastActivityDate="2015-10-13T20:58:16.997" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2920" PostTypeId="2" ParentId="2918" CreationDate="2015-10-13T20:57:01.380" Score="5" Body="&lt;p&gt;Before ancient Greeks no notion of &quot;proof&quot; existed, mathematical works such as Babylonian tablets and Egyptian papyri, gave practical recipes. &quot;&lt;em&gt;Circles are to one another as the squares on their diameters&lt;/em&gt;&quot; is &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html&quot; rel=&quot;nofollow&quot;&gt;proposition 2 of Book XII of Euclid's Elements&lt;/a&gt;, but is broadly believed to be due to Eudoxus of Cnidus (c.400 - 350 BC), the inventor of the method of exhaustion, which is used to prove it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid (and likely Eudoxus) did not consider the ratio of circumference to diameter. The invariance of that ratio, along with bounds for it, is proved in &lt;a href=&quot;http://www.ams.org/samplings/feature-column/fc-2012-02&quot; rel=&quot;nofollow&quot;&gt;Archimedes's Measurement of a Circle&lt;/a&gt;, also by method of exhaustion. He had to use additional postulates however, in particular the famous &quot;&lt;em&gt;line segment between two points is shorter than any other path between them&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-15T21:02:21.170" LastActivityDate="2015-10-15T21:02:21.170" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2921" PostTypeId="2" ParentId="2380" CreationDate="2015-10-13T23:06:16.780" Score="0" Body="&lt;p&gt;I'm currently reading &quot;The History of the theories of aether and electricity&quot; by Whittaker. A truly illuminating book. Found it in recommended readings section in Jackson, where he described it as one of the most authoritative sources of the history of electromagnetism and science as a whole. It actually includes some history of nuclear physics, relativity and QM, with many amusing examples of historical accidents that lead to great discoveries. Like Becquerel believing radiation came only after the exposure of his crystals to the Sun, since he couldn't believe something could radiate without energy, and energy should have come from somewhere, right? So it went until one cloudy day, when it turned out he was wrong. &lt;/p&gt;&#xA;" OwnerUserId="3014" LastActivityDate="2015-10-13T23:06:16.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2922" PostTypeId="1" AcceptedAnswerId="2924" CreationDate="2015-10-14T15:57:47.897" Score="19" ViewCount="1010" Body="&lt;p&gt;This is my first question for HSM. If it is consider too specialized for HSM, perhaps it can be migrated to MathOverflow. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In algebraic number theory, one frequently denotes the ring of algebraic integers in a number field $K$ by $O_K$. Similarly in algebraic geometry, where one studies algebraic varieties and later schemes, it is common to denote the ring of regular functions over an open set $U$ by $O(U)$ or $\mathcal{O}(U)$, and similarly one often denotes the stalk at a point $p$ by something like $O_p$ or $\mathcal{O}_p$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where does this $O$ come from, or where was it first used? My best guess is that it originally had something to do with &lt;a href=&quot;https://en.wikipedia.org/wiki/Order_%28ring_theory%29&quot; rel=&quot;noreferrer&quot;&gt;Order in commutative ring theory&lt;/a&gt;, starting primarily in algebraic number theory and later spilling over into algebraic geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'm asking as a follow-up to a comment I made at MO &lt;a href=&quot;https://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18641#comment544737_18641&quot;&gt;here&lt;/a&gt;, where I explain a private pun I've developed to give a sense to O-notation as it is used in asymptotic analysis; I realized after making that comment that I had no idea where the notation $O$ in the sense of this question comes from.) &lt;/p&gt;&#xA;" OwnerUserId="1987" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-10-15T01:11:08.210" Title="Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory" Tags="|mathematics|notation|number-theory|algebraic-geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2923" PostTypeId="2" ParentId="2912" CreationDate="2015-10-14T22:38:15.573" Score="4" Body="&lt;p&gt;Roger Penrose has something to say about tensor calculus in his popular book &quot;The Road to Reality&quot;. He discerns a &quot;mathematician's notation&quot; and a &quot;physicists's notation&quot; (which is &quot;abstract index calculus&quot; and his invention) plus gives a graphical notation (also his invention).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best quick introduction to abstract index calculus is R.Wald's General Relativity. Roughly speaking it is just a different reading of component-notation tensor calculus, or, Ricci calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The historical landmark book on Ricci calculus is Tullio Levi-Civita's The Absolute Differential Calculus. First english edition 1926. Levi-Civita was a student of Gregorio Ricci-Curbastro, who developed this calculus 1887-1896. A nice quick introduction into what E.Cartan called &quot;the debauch of indices&quot; is E.Schrödinger's Space-Time Structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The component-free (but not mess-free!) &quot;mathematician's notation&quot; $\nabla_X$ is due to J.L.Koszul, first presented by himself in his famous &quot;Lectures on Fibre Bundles and Differential Geometry&quot; at the Tata Institute Bombay 1960. (I have the 1965 issue, one of my most beloved math books.) But Koszul's calculus was first publicly used in a paper by Nomizu: &quot;Invariant affine connections on homogenous spaces&quot;, Amer.J.Math. 76 (1954). -- This is the history according to M.Berger: &quot;A Panoramic View of Riemannian Geometry&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More history is in Kobayashi&amp;amp;Nomizu Vol.I p.287 - but they don't mention Koszul!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Last but not least, here is the best mathematical book on the history of the subject:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;M.Spivak, A Comprehensive Introduction to Differential Geometry Vol.II&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;with translations of historical texts and maths from Gauss to Riemann and a comparative presentation of all diverse tensor calculi from Ricci, Cartan, Koszul, ending with Ehresmann connections in principal fibre bundles. Except abstract index calculus. He also attributes the $\nabla_X$ notation to Koszul, but doesn't quote his landmark Tata lectures.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I find abstract index calculus superior to Koszul's for a few reasons, e.g. a) Koszul's is a partial directional derivative b) stuff tends to get presented upside-down (compare the definitions of symmetric connection) and c) left-right messed up (higher covariant derivatives) and d) due to a) there are lots of superfluous Lie brackets (e.g. definition of curvature and torsion)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Crazily, I'm currently writing up my own tensor calculus, which is abstract index calculus with them abstract indices replaced by abstract multilinear algebra, and bridling the horse from the right end.&lt;/p&gt;&#xA;" OwnerUserId="3024" LastEditorUserId="3024" LastEditDate="2015-10-15T11:41:22.117" LastActivityDate="2015-10-15T11:41:22.117" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2924" PostTypeId="2" ParentId="2922" CreationDate="2015-10-15T00:58:15.507" Score="16" Body="&lt;p&gt;Your guess is right: the notation $\mathfrak o$ goes back to Dedekind. If you get a copy of Dirichlet-Dedekind's Vorlesungen über Zahlentheorie and look in Dedekind's famous XI-th Supplement, which was the first systematic development of algebraic number theory, you'll see $\mathfrak o$ starting in section 170 when Dedekind defines Ordnung (= Order).&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2015-10-15T01:11:08.210" LastActivityDate="2015-10-15T01:11:08.210" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2925" PostTypeId="1" CreationDate="2015-10-15T19:16:45.720" Score="3" ViewCount="145" Body="&lt;p&gt;I believe i have been looking in the Internet once for the origin of the term &quot;spectrum&quot; in functional analysis and saw that the term was proposed by someone (by Hilbert?) with no relation to physics, but later by pure coincidence it turned out to be related to radiation spectrum in physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this really so? I would like to have some references about the origin of the term and about the possible relation of operator spectrum to radiation spectrum.&lt;/p&gt;&#xA;" OwnerUserId="3030" OwnerDisplayName="Alexey" LastActivityDate="2015-10-17T00:32:27.063" Title="Origin of the term &quot;operator spectrum&quot; and its relation to spectrum in physics" Tags="|reference-request|mathematical-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2926" PostTypeId="2" ParentId="2925" CreationDate="2015-10-16T11:02:52.890" Score="3" Body="&lt;p&gt;The terminology in math goes back to Hilbert, and the link to spectrum of atoms was indeed a coincidence. See section 7 of &lt;a href=&quot;http://www.stolaf.edu/people/steen/Papers/73spectral.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.stolaf.edu/people/steen/Papers/73spectral.pdf&lt;/a&gt;, esp. the top of p. 369.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2015-10-16T11:02:52.890" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2927" PostTypeId="1" AcceptedAnswerId="2930" CreationDate="2015-10-16T13:26:12.687" Score="11" ViewCount="2351" Body="&lt;p&gt;In the Golden Age of Islam, Muslim scholars and scientists wrote a lot of books and manuscripts in many sciences such as medicine.&lt;/p&gt;&#xA;&lt;p&gt;My question is, are there any books which have had some longer (or future) influence on the development modern theories?&lt;/p&gt;&#xA;&lt;p&gt;What books have been translated into other languages?&lt;/p&gt;&#xA;&lt;p&gt;As an example, I know that Avicenna's (Ibn Sina) book about Medicine &amp;quot;&lt;a href=&quot;http://en.wikipedia.org/wiki/The_Canon_of_Medicine&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Canon of Medicine&lt;/a&gt; القانون في الطب‎ al-Qānūn fī aṭ-Ṭibb) was used as a medical textbook through the 18th century in Europe!&lt;/p&gt;&#xA;&lt;p&gt;Well, if I need to focus my question, my focus here could be on Medicine!&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If i don't need to focus this would be great!&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2821" LastEditorUserId="4251" LastEditDate="2021-12-26T09:06:57.427" LastActivityDate="2021-12-26T09:06:57.427" Title="I'm looking for books of Muslim scholars/scientists who have had some influence in the development of modern science" Tags="|book-recommendation|middle-ages|islamic-science|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2928" PostTypeId="2" ParentId="2927" CreationDate="2015-10-16T14:00:05.173" Score="9" Body="&lt;p&gt;In mathematics, perhaps the best known is&#xA;&lt;a href=&quot;http://en.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing&quot;&gt;&quot;The Compendious Book on Calculation by Completion and Balancing&quot;&lt;/a&gt; by the Persian scholar &lt;a href=&quot;http://en.wikipedia.org/wiki/Mu%E1%B8%A5ammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB&quot;&gt;Muḥammad ibn Mūsā al-Khwārizmī&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The English words &quot;algebra&quot; and &quot;algorithm&quot; derive from this.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The date is around 820.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="62" LastEditDate="2015-10-16T23:41:24.697" LastActivityDate="2015-10-16T23:41:24.697" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2930" PostTypeId="2" ParentId="2927" CreationDate="2015-10-16T21:32:02.570" Score="12" Body="&lt;p&gt;Islamic science and mathematics experienced a boom during middle ages, contributions in mathematics, mechanics, astronomy and medicine were especially prominent, and had a deep impact on Renaissance Europe. There are many historical articles on &lt;a href=&quot;http://www.muslimheritage.com/science&quot;&gt;Muslim Heritage: Science&lt;/a&gt;, see also links on &lt;a href=&quot;http://www.geni.com/projects/Muslim-Philosophers-Mathematicians-Scientists/13304&quot;&gt;Muslim Philosophers, Mathematicians &amp;amp; Scientists&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics aside from Khwarizmi's famous Hisab Al-Jabr wal Muqabala we have Omar Khayyam's Treatise on the Demonstration of Problems of Algebra, that has complete classification and geometric solution of all cubics. His also Explanations of the Difficulties in the Postulates of Euclid is considered the first work on the parallel postulate that avoids circular attempts to prove it, and anticipates a path to the hyperbolic geometry using a quadrilateral, later rediscovered by Saccheri and named after him in the West. Al-Karaji's s Al-Fakhri Fil-Jabr wal-Muqabala (Glorious on Algebra) introduced notations for all positive and negative powers of a variable and gave laws of exponents among them. Al-Samawal's Shining Book on Calculation gave the first version of long/synthetic division of polynomials. Al-Tusi's Treatise On The Quadrilateral was the most authoritative medieval work on trigonometry, and the first one to treat it independently of astronomy. See &lt;a href=&quot;http://www.fountainmagazine.com/Issue/detail/Muslim-Contributions-to-Mathematics&quot;&gt;Kadyrov's Muslim Contributions to Mathematics&lt;/a&gt; for more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Maragheh_observatory#Maragha_Revolution&quot;&gt;Meragha school&lt;/a&gt; reformed epicyclic astronomy of Ptolemy making it consistent with (Aristotelian) dynamics, improving its precision and computational techniques. Al-Tusi’s Tadhkira i ilm al-Haya (Memoir on Astronomy) was a thorough critique of Ptolemy, especially of eccentrics and equants that appeared as ad hoc tricks. Al-Tusi found a way to model oscillatory linear motion with epicycles (the so-called &lt;a href=&quot;http://www.geni.com/projects/Muslim-Philosophers-Mathematicians-Scientists/13304&quot;&gt;Tusi couple&lt;/a&gt;), and used it in place of equants. In the Final Quest Concerning the Rectification of Principles al-Shatir eliminated eccenters and equants  completely, without sacrificing precision, by mounting epicycles on epicycles. Although there is no direct confirmation, numerical coincidences strongly suggest that Copernicus used al-Shatir's models in his heliocentric reformulation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his influential commentary on Aristotle's physics &lt;a href=&quot;http://www.jstor.org/stable/3654219?seq=1#page_scan_tab_contentshttp://www.jstor.org/stable/3654219?seq=1#page_scan_tab_contents&quot;&gt;Avempace revived the impetus theory&lt;/a&gt; of Philoponus (Avicenna discussed it as well), an alternative to Aristotle's theory of projectile motion, and an inspiration for momentum in modern dynamics. His commentary made its way to Europe, where the impetus theory was taken up by Buridan, Oresme and the merton school, who eventually influenced Galileo and Descartes. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-20T01:05:31.420" LastActivityDate="2015-10-20T01:05:31.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2932" PostTypeId="1" CreationDate="2015-10-17T04:16:34.380" Score="11" ViewCount="4476" Body="&lt;p&gt;The term &quot;gas giant&quot; was invented in the 1950s, but I think the concept has been known longer than that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did we first learn that Jupiter was a gas giant, not a terrestrial-type planet?&lt;/p&gt;&#xA;" OwnerUserId="60" LastEditorUserId="60" LastEditDate="2015-10-18T16:15:04.517" LastActivityDate="2019-05-05T15:26:57.957" Title="When did we first learn that Jupiter was a gas giant?" Tags="|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2933" PostTypeId="1" CreationDate="2015-10-17T08:20:23.897" Score="10" ViewCount="1015" Body="&lt;p&gt;The halting problem is a very famous example from computability theory of a problem that is undecidable. It is often said that the proof of its undecidability was given by Alan Turing, indeed &lt;a href=&quot;https://en.wikipedia.org/wiki/Halting_problem&quot;&gt;wikipedia&lt;/a&gt; repeatedly says so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Turing's paper &quot;On Computable Numbers, with an Application to the Entscheidungsproblem&quot; does not deal with the halting problem at all. As Charles Petzold writes in &quot;The Annotated Turing&quot;, page 234:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The halting problem has subsequently become closely identified with Turing Machines, but the concept is foreign to Turing's original paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problems that Turing's paper &lt;em&gt;does&lt;/em&gt; show are undecidable are the problem of deciding whether a Turing machine is circle-free (which is not the same as halting, since both circle-free and circular machines can keep printing forever), that of deciding whether a machine ever prints some specific symbol, and that of the Entscheidungsproblem. Of course, the problem of deciding whether a Turing machine is circle-free is very close to the halting problem, but it is nonetheless not the halting problem. I am aware that the term &quot;halting problem&quot; was coined by Martin Davis in 1958.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, if Alan Turing never discussed the halting problem, why do so many sources attribute it to him?&lt;/p&gt;&#xA;" OwnerUserId="467" LastEditorUserId="4251" LastEditDate="2020-07-09T20:43:53.980" LastActivityDate="2020-07-09T20:43:53.980" Title="Why is the Halting problem attributed to Alan Turing?" Tags="|mathematics|computer-science|turing|computer-scientists|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2934" PostTypeId="1" AcceptedAnswerId="2945" CreationDate="2015-10-17T20:25:12.813" Score="15" ViewCount="2159" Body="&lt;p&gt;As a student of physics one will, on several occasions, indubitably hear professors or other physicists (&lt;a href=&quot;https://physics.stackexchange.com/questions/95065/why-must-the-dirac-equation-multiplied-by-its-complex-conjugate-give-the-kg-equa#comment194447_95065&quot;&gt;here&lt;/a&gt; is an example, from Physics.SE's highest-rep user John Rennie) tell the famous story that Paul Dirac came upon his relativistic wave equation for the electron, now called the Dirac equation, by &quot;trying to take the square root of the Klein-Gordon operator&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story comes in many different versions, some dressing it up in a funny anecdote about Dirac, when asked what he was working on, casually replying with &lt;em&gt;&quot;Oh, just trying to take the square root of something&quot;&lt;/em&gt;. Others simply claim that this was the train of thought he followed in his derivation. &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirac_equation#Dirac.27s_coup&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; has another version, which features Dirac pondering by the fireplace...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A quick look at the first few sections of &lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/117/778/610&quot; rel=&quot;noreferrer&quot;&gt;the paper where Dirac introduced his wave equation&lt;/a&gt;, however, shows no trace of &quot;taking square roots&quot; of any kind. Of course, it is clear that Dirac &lt;em&gt;was&lt;/em&gt; looking for an operator that was linear---as opposed to the quadratic Klein-Gordon equation---but that doesn't really justify saying that he was looking to take a square root. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nonetheless, this doesn't really debunk the entire story, since it seems quite reasonable to assume that Dirac may have streamlined his derivation considerably before publishing. Thus, the intuitive idea (maybe he thought it would sound a little silly to publish something so vague-sounding) may have been carefully hidden behind a more rigorous derivation and justification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;em&gt;What is this story about &quot;taking the square root of the Klein-Gordon operator&quot; based on? Is there any evidence that Dirac was indeed thinking along these lines? If not, can the story be traced back to any other source?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="4251" LastEditDate="2020-08-15T07:39:24.990" LastActivityDate="2020-08-15T07:39:24.990" Title="Was Dirac really trying to take the square root of the Klein-Gordon operator?" Tags="|physics|biographical-details|quantum-mechanics|theoretical-physics|dirac|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2935" PostTypeId="1" CreationDate="2015-10-17T22:25:32.117" Score="7" ViewCount="4244" Body="&lt;p&gt;Maxwell's equations in their modern differential form are: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\nabla \cdot \mathbf{E} = \dfrac {\rho} {\varepsilon_0}$ (Gauss's law for electricity)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\nabla \cdot \mathbf{B} = 0$ (Gauss's law for magnetism) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}} {\partial t}$ (Maxwell–Faraday equation) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}} {\partial t} \right)$ (Ampère's circuital law)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware that Maxwell generalized Faraday's law and Ampère's law by adding displacement current, but why are these four equations named after him although they were discovered by three other people: Gauss, Faraday and Ampère? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that his addition to the laws (two of them) are not significant enough to make it bear his name.&lt;/p&gt;&#xA;" OwnerUserId="2423" LastEditorUserId="4251" LastEditDate="2020-06-04T17:42:00.430" LastActivityDate="2020-06-04T17:42:00.430" Title="Why do Maxwell's equations bear his name?" Tags="|physics|electromagnetism|naming-conventions|maxwell|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2936" PostTypeId="2" ParentId="2935" CreationDate="2015-10-17T22:59:44.183" Score="8" Body="&lt;p&gt;As you noticed, separate equations have other names as well. Maxwell's adding the displacement term made the system complete, with all important consequences, in particular, existing of electromagnetic waves. So the name of the whole system after Maxwell is completely justified.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-10-17T22:59:44.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2939" PostTypeId="1" AcceptedAnswerId="2942" CreationDate="2015-10-18T12:26:04.190" Score="1" ViewCount="96" Body="&lt;p&gt;Who discovered the Rayleigh-Taylor instability? Was it artist David Alfaro Siquerios? Or was he the first to observe it?&lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="62" LastEditDate="2015-10-18T15:42:14.150" LastActivityDate="2015-10-18T16:17:00.360" Title="Who discovered the Rayleigh-Taylor instability?" Tags="|physics|discoveries|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2940" PostTypeId="1" AcceptedAnswerId="2941" CreationDate="2015-10-18T13:28:28.120" Score="92" ViewCount="17460" Body="&lt;p&gt;We know that there were/are many famous female mathematicians who influenced the mathematics as we know it today, but their numbers are few compared to male mathematicians. While we have numerous famous results by many male mathematicians like Gauss, Euler and many others, what are famous results bearing the name of a female mathematician which also have a very deep impact on our understanding of mathematics?&lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="2891" LastEditDate="2015-11-30T13:36:17.220" LastActivityDate="2019-06-13T18:56:29.040" Title="What famous theorems or results were proven by female mathematicians?" Tags="|mathematicians|" AnswerCount="14" CommentCount="19" ContentLicense="CC BY-SA 3.0" />
  <row Id="2941" PostTypeId="2" ParentId="2940" CreationDate="2015-10-18T14:34:56.643" Score="92" Body="&lt;p&gt;See at least &lt;a href=&quot;https://en.wikipedia.org/wiki/Emmy_Noether&quot;&gt;Emmy Noether&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;was a German mathematician known for her contributions to abstract algebra and theoretical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras. In physics, &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether%27s_theorem&quot;&gt;Noether's theorem&lt;/a&gt; explains the connection between symmetry and conservation laws.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course, this is &lt;em&gt;only&lt;/em&gt; an example; there are many others: see answers below.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-10-25T12:35:30.537" LastActivityDate="2015-10-25T12:35:30.537" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="2942" PostTypeId="2" ParentId="2939" CreationDate="2015-10-18T16:09:09.250" Score="2" Body="&lt;p&gt;Rayleigh published his findings in 1883. Siqueiros (thus, not &quot;Siquerios&quot;) was born in 1896. Does that answer your question?&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-10-18T16:17:00.360" LastActivityDate="2015-10-18T16:17:00.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2943" PostTypeId="2" ParentId="2932" CreationDate="2015-10-18T16:13:44.753" Score="8" Body="&lt;p&gt;We need to clear one thing up here, and that is the definition of a gas giant. As an example, take &lt;a href=&quot;http://en.wikipedia.org/wiki/Gas_giant&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia's definition&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A gas giant is a giant planet composed mainly of hydrogen and helium.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;All we have to do to classify Jupiter as a gas giant is to prove that it's made of . . . gas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer leads to &lt;a href=&quot;http://www.space.com/18902-giovanni-cassini.html&quot; rel=&quot;noreferrer&quot;&gt;Giovanni Cassini&lt;/a&gt;, who observed Jupiter's &lt;a href=&quot;http://www.space.com/18902-giovanni-cassini.html&quot; rel=&quot;noreferrer&quot;&gt;differential rotation&lt;/a&gt; in the 1690s, thus showing that it could not be solid.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="62" LastEditDate="2015-10-18T16:18:09.990" LastActivityDate="2015-10-18T16:18:09.990" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2944" PostTypeId="1" AcceptedAnswerId="2947" CreationDate="2015-10-18T17:05:17.677" Score="6" ViewCount="1187" Body="&lt;p&gt;Every kid knows $1+2\times3$ is equal to $1+(2\times3)$, not $(1+2)\times3$. But the more I think about it, the more counterintuitive it seems. You have to tell kids to memorize the rule, instead of just following the literal order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think the operator precedence might be &quot;invented&quot; somewhere, by someone or some cultures. Is it a part of Arabic numerals? Did ancient Greek and Egyptians have it? And is/was there a culture where $1+2\times3$ equals $9$?&lt;/p&gt;&#xA;" OwnerUserId="3040" LastEditorUserId="62" LastEditDate="2015-10-18T17:28:29.650" LastActivityDate="2015-10-18T19:01:05.020" Title="When was &quot;operator precedence&quot; invented? Did any culture use a different rule?" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2945" PostTypeId="2" ParentId="2934" CreationDate="2015-10-18T17:25:22.113" Score="11" Body="&lt;p&gt;One routinely speaks of the Dirac operator as a &lt;a href=&quot;https://www.google.com/search?q=Dirac+%22square+root+of+the+Laplacian%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;square root of the Laplacian&lt;/a&gt; (or Dalembertian as the case may be), and Dirac himself rather supports this heuristic in his &lt;em&gt;Recollections of an exciting era&lt;/em&gt;, &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=441654&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of twentieth century physics&lt;/a&gt;, Academic Press 1977, pp. 109-146:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I was playing around with the three components $\sigma_1, \sigma_2,\sigma_3,$ which I had used to describe the spin of the electron, and I noticed that if you formed the expression $\sigma_1p_1+\sigma_2p_2+\sigma_3p_3$ and squared it, $p_1$, $p_2$ and $p_3$ being the three components of momentum, you got just $\smash{p_1^2+p_2^2+p_3^2}$, the square of the momentum (&lt;sup&gt;1&lt;/sup&gt;). This was a pretty mathematical result. I was quite excited over it. It seemed that it must be of some importance. (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I suddenly realized that there was no need to stick to the quantities $\sigma$, which can be represented by matrices with just two rows and columns. Why not go over to four rows and columns? Mathematically there was no objection to this at all. Replacing the $\sigma$-matrices by four-row and column matrices, one could easily &lt;strong&gt;take the square root of the sum of four squares&lt;/strong&gt;, or even five squares if one wanted to.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Accordingly, as I am sure is well known, in &lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt; (4th ed., §67; it would be interesting to compare the first of 1930, which I can’t find online) he starts with “the relativistic Hamiltonian [that] leads to the wave equation&#xA;$$&#xA;\{p_0 - (m^2c^2+p_1^2+p_2^2+p_3^2)^{\frac12}\}\psi=0,&#xA;\tag5&#xA;$$&#xA;where the $p$'s are interpreted as operators in accordance with [$p_\mu=i\hbar\partial/\partial x^\mu$]”, then multiplies by the conjugate to obtain&#xA;$$&#xA;\{p_0^2 - m^2c^2-p_1^2-p_2^2-p_3^2\}\psi=0,&#xA;\tag6&#xA;$$&#xA;and then seeks “a wave equation that is linear in $p_0$ and that is roughly equivalent to (6)” in the form&#xA;$$&#xA;\{p_0 - \alpha_1p_1-\alpha_2p_2-\alpha_3p_3 -\beta\}\psi=0,&#xA;\tag7&#xA;$$&#xA;which precisely leads him to the “square root of the sum of four squares” problem.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;(&lt;sup&gt;1&lt;/sup&gt;) An identity found explicitly, &lt;em&gt;with&lt;/em&gt; $p_r=-i\partial/\partial x_r$, in the 1928 paper you quote (bottom of p. 618, case $\mathbf A=0$). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Dirac’s first edition (1930) is now online, and its &lt;a href=&quot;https://archive.org/stream/in.ernet.dli.2015.177580#page/n250&quot; rel=&quot;nofollow noreferrer&quot;&gt;§74&lt;/a&gt; already has the exact same development as above (but for minimal differences in notation).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-03-26T06:41:33.017" LastActivityDate="2018-03-26T06:41:33.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2946" PostTypeId="2" ParentId="2944" CreationDate="2015-10-18T17:44:51.263" Score="6" Body="&lt;p&gt;&lt;em&gt;Supplement to HDE's answer.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A problem sent from &lt;a href=&quot;https://en.wikipedia.org/wiki/Lodovico_Ferrari&quot; rel=&quot;noreferrer&quot;&gt;Ludovico Ferrari&lt;/a&gt; to &lt;a href=&quot;https://en.wikipedia.org/wiki/Niccol%C3%B2_Fontana_Tartaglia&quot; rel=&quot;noreferrer&quot;&gt;Tartaglia&lt;/a&gt; in 1547 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Find me two numbers such that when they are added together, they make as much &#xA;  as the cube of the lesser added to the product of its triple with the square of the greater; and the cube of the greater added to its triple times the square of the lesser makes $64$ more than the sum of these two numbers. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[i.e. Find $a, b$ such that &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$a + b = b^3 + 3ba^2$ and &lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;$a^3 + 3ab^2 = 64 + a + b$ (from : John Fauvel &amp;amp; Jeremy Gray (editors), &lt;a href=&quot;https://books.google.it/books?id=kFwPAQAAMAAJ&quot; rel=&quot;noreferrer&quot;&gt;The History of Mathematics : A Reader&lt;/a&gt; (1987), page 257).] &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These were among the &quot;leading&quot; &lt;a href=&quot;http://www.scit.wlv.ac.uk/~cm1993/maths/mm2217/ar.htm&quot; rel=&quot;noreferrer&quot;&gt;algebraist of the Reanaissance&lt;/a&gt;; it is clear that, until the modern algebraic symbolsim were developed (some millenia after ancient Egyptians and Babylonians) the issue of the convention about &quot;operators precedence&quot; was quite meaningless.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-10-18T19:01:05.020" LastActivityDate="2015-10-18T19:01:05.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2947" PostTypeId="2" ParentId="2944" CreationDate="2015-10-18T17:45:07.600" Score="7" Body="&lt;p&gt;The idea of operator precedence - also known as &lt;a href=&quot;http://en.wikipedia.org/wiki/Order_of_operations&quot;&gt;order of operations&lt;/a&gt; - is a relatively recent concept, in part because the common mathematical operators for the four basic arithmetical operations - $+,-,\times,\div$ - weren't all in use until a few hundred years ago. For example, as Florian Cajori writes in &lt;a href=&quot;http://books.google.com/books?id=bBoPAAAAIAAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;&lt;em&gt;A History of Mathematics&lt;/em&gt;&lt;/a&gt;, one of the symbols for multiplication ($\times$) was created by William Oughtred in the 17th century. This means that older cultures didn't have the same system we use today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, according to &lt;a href=&quot;http://jeff560.tripod.com/operation.html&quot;&gt;this source&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The convention that multiplication precedes addition and subtraction was in use in the earliest books employing symbolic algebra in the 16th century. The convention that exponentiation precedes multiplication was used in the earliest books in which exponents appeared.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The debate about the order of multiplication/division came later, although given that the two operations are inverses of each other, it is not as important.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-10-18T17:45:07.600" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="2948" PostTypeId="2" ParentId="2940" CreationDate="2015-10-18T19:56:51.310" Score="45" Body="&lt;p&gt;Perhaps because of its youth, the mathematical end of Computer Science has several notable women in its history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sheila_Greibach&quot;&gt;Sheila Greibach&lt;/a&gt; was a pioneer in the field of formal language theory, particularly in the area of context-free languages. At the time, that would have been considered more a branch of mathematics, as Computer Science wasn't really a thing of its own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, she developed &lt;a href=&quot;https://en.wikipedia.org/wiki/Greibach_normal_form&quot;&gt;Greibach Normal Form&lt;/a&gt;, which is fairly instrumental in the theory of parsing, which is extremely critical to modern programming languages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Continuing down programming language theory, &lt;a href=&quot;https://en.wikipedia.org/wiki/Barbara_Liskov&quot;&gt;Barbara Liskov&lt;/a&gt; developed the &lt;a href=&quot;https://en.wikipedia.org/wiki/Liskov_substitution_principle&quot;&gt;Liskov Substitution Principle&lt;/a&gt;, which was critical in developing a formalized model for object-oriented languages. She won the Turing Award (CS equivalent of the Fields medal) for her contributions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did these have a &quot;deep impact&quot; on our understanding of mathematics? Not in the classical sense, but they've led to some amazing developments, arguably as many as the 400 years of calculus/analysis theory.&lt;/p&gt;&#xA;" OwnerUserId="3044" LastActivityDate="2015-10-18T19:56:51.310" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="2949" PostTypeId="1" CreationDate="2015-10-19T02:15:04.830" Score="2" ViewCount="187" Body="&lt;p&gt;I was looking for some material on the history of fractional calculus and googled Pincherle Amaldi to find a source. Up popped a map of the environs of the University of Rome and there was the intersection of Via Amaldi and Via Pincherle. One day, should I be in the area, I'll have to visit a coffee shop near the intersection and pay homage to the two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where can similar intersections be found? &lt;/p&gt;&#xA;" OwnerUserId="1830" LastActivityDate="2016-02-16T21:48:25.287" Title="The intersection of history, mathematics, and geography" Tags="|mathematicians|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2950" PostTypeId="2" ParentId="2940" CreationDate="2015-10-19T13:21:30.707" Score="32" Body="&lt;p&gt;There is &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sophie_Germain&amp;#39;s_theorem&quot;&gt;Sophie Germain's theorem&lt;/a&gt;&lt;/strong&gt;, a theorem in number theory, related to Fermat's last theorem and proved by the French mathematician &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sophie_Germain&quot;&gt;Sophie Germain&lt;/a&gt;&lt;/strong&gt; (1776-1831).&lt;/p&gt;&#xA;" OwnerUserId="3058" LastActivityDate="2015-10-19T13:21:30.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2951" PostTypeId="2" ParentId="2940" CreationDate="2015-10-19T13:31:27.053" Score="8" Body="&lt;p&gt;There are not female mathematicians who have had &lt;em&gt;quite&lt;/em&gt; so much impact as Gauss or Euclid, for example, but this is to be expected because of historical reasons which everyone is familiar with. A quick google will have told the questioner that there are many important female mathematicians in history, but I think the question is asking for a &lt;em&gt;really&lt;/em&gt; important mathematician, or at least a &lt;em&gt;well known&lt;/em&gt; mathematician, like Galois who has Galois Theory named after him, or Hilbert, who has Hilbert spaces named after him. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first person who I thought of when I read this question was Emmy Noether, who, in terms of fame, isn't quite Newton (obviously) but is &lt;em&gt;at least&lt;/em&gt; Galois.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You probably aren't going to see the names of any women in the titles of any undergrad maths courses, but if you are, it will probably be Noether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is possible that many Ancient mathematicians were actually woman. This is certainly the case for Egypt but probably less so for Greece (although who knows?) Even in more modern times, a few woman have worked under male pseudonyms and there may be many more that we do not know about (although it is unlikely that any of the 'big name' mathematicians like Newton and Euler were actually women because their lives have been well documented). It is also possible (in fact very likely) that work by women has been plagiarised by men, so some theorems named after males may have actually been developed by females, we probably will never know how many. &lt;/p&gt;&#xA;" OwnerUserId="3059" LastEditorUserId="26" LastEditDate="2015-10-19T18:20:41.410" LastActivityDate="2015-10-19T18:20:41.410" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2952" PostTypeId="2" ParentId="2940" CreationDate="2015-10-19T14:22:32.227" Score="25" Body="&lt;p&gt;I found an &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kowalevski_theorem&quot;&gt;existence theorem&lt;/a&gt; for the Cauchy Problem in partial differential equations which has been proven by &lt;a href=&quot;https://en.wikipedia.org/wiki/Sofia_Kovalevskaya&quot;&gt;Sofia Vasilyevna Kovalevskaya&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="3338" LastEditDate="2015-12-05T23:13:55.843" LastActivityDate="2015-12-05T23:13:55.843" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2953" PostTypeId="2" ParentId="2940" CreationDate="2015-10-19T14:59:32.033" Score="31" Body="&lt;p&gt;There is the work by &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Lovelace.html&quot;&gt;Ada Lovelace&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the annotations, which were called &quot;Notes&quot;, Ada Lovelace described how the analytical engine could be programmed and gave what many consider to be the first ever computer program. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular, she found and corrected a bug in Babbage's algorithm for computing Bernoulli numbers:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We discussed together the various illustrations that might be introduced: I suggested several, but the selection was entirely her own. So also was the algebraic working out of the different problems, except, indeed, that relating to the numbers of Bernoulli, which I had offered to do to save Lady Lovelace the trouble. This she sent back to me for an amendment, having detected a grave mistake which I had made in the process.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(from C Babbage, Passages from the Life of a Philosopher (London, 1864).)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the &lt;a href=&quot;http://www.adaic.org/&quot;&gt;programming language Ada&lt;/a&gt; is named after &lt;a href=&quot;https://en.wikipedia.org/wiki/Ada_%28programming_language%29&quot;&gt;her&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="93" LastEditorUserId="-1" LastEditDate="2015-10-21T11:59:53.537" LastActivityDate="2015-10-21T11:59:53.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2954" PostTypeId="1" AcceptedAnswerId="2961" CreationDate="2015-10-19T15:52:40.627" Score="7" ViewCount="242" Body="&lt;p&gt;In 2014 BICEP2 made a very controversial &lt;a href=&quot;https://www.cfa.harvard.edu/news/2014-05&quot;&gt;press release&lt;/a&gt;. It was rushed, given before any papers where published, because they feared they would be scooped by another similar experiment. In the end it turned out that their analysis of their data was flawed. It was particularly dramatic because if they had been correct it would have been a significant development.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any similar cases in other sciences? Not necessary an identical situation, but a high profile result that was released in a hurry that later turned out to be incorrect.&lt;/p&gt;&#xA;" OwnerUserId="3066" LastEditorUserId="55" LastEditDate="2016-11-12T23:02:59.090" LastActivityDate="2020-07-19T03:18:20.963" Title="What are some major examples of rushed scientific publications based on flawed data?" Tags="|publishing|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2956" PostTypeId="2" ParentId="2940" CreationDate="2015-10-19T19:37:48.110" Score="16" Body="&lt;p&gt;Danica McKellar is the McKellar in the Chayes-McKellar-Winn theorem,&lt;/p&gt;&#xA;" OwnerUserId="3071" LastActivityDate="2015-10-19T19:37:48.110" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2957" PostTypeId="2" ParentId="2940" CreationDate="2015-10-20T11:03:40.360" Score="17" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya&quot;&gt;Olga Ladyzhenskaya&lt;/a&gt; proved a result related to the Navier-Stokes equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result by itself is not very famous, but the Navier-Stokes equations are. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2015-10-20T11:03:40.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2958" PostTypeId="1" CreationDate="2015-10-20T13:15:16.960" Score="3" ViewCount="200" Body="&lt;p&gt;I am reading the (seminal?) paper of R. Courant on Finite elemtent methods (FEM)  and in it I found the following phrase:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;But only the spectacular success of Walther Ritz and its tragic circumstances caught the general interest.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;that is on the first page of the article Variational methods for the solution of problems of equilibrium and vibrations of 1943 by R. Courant&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does it mean? what are the tragic circumstances to which Courant refers to?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did those circumstances caught the general interest on the subject discussed?(FEM)&lt;/p&gt;&#xA;" OwnerUserId="2630" LastEditorUserId="26" LastEditDate="2015-10-22T09:15:07.110" LastActivityDate="2016-06-11T04:25:14.447" Title="Courant and Finite element methods" Tags="|mathematics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2959" PostTypeId="2" ParentId="2940" CreationDate="2015-10-20T15:16:05.173" Score="9" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Denjoy%E2%80%93Young%E2%80%93Saks_theorem&quot; rel=&quot;nofollow&quot;&gt;Denjoy–Young–Saks theorem&lt;/a&gt; gives some possibilities for the Dini derivatives of a function that hold almost everywhere. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Grace_Chisholm_Young&quot; rel=&quot;nofollow&quot;&gt;Grace Chisholm Young&lt;/a&gt; extended Denjoy's result on continuous functions to measurable functions. Her husband was &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Henry_Young&quot; rel=&quot;nofollow&quot;&gt;William Henry Young&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="100" LastEditorUserId="2442" LastEditDate="2016-08-12T15:44:46.193" LastActivityDate="2016-08-12T15:44:46.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2960" PostTypeId="2" ParentId="59" CreationDate="2015-10-20T15:44:44.320" Score="4" Body="&lt;p&gt;This is not really an answer, but is too long to fit in a comment. The question presupposes that according to 19th-century &lt;em&gt;definitions&lt;/em&gt;, it is false that the intermediate value property implies continuity. It's far from clear to me that this was the case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many possible ways of formulating the definition of the function. Three examples would be to define the notion as a formula, to use point-set notions, or to proceed as in modern smooth infinitesimal analysis (SIA). As far as I can tell from the WP article &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_function_concept&quot; rel=&quot;nofollow&quot;&gt;History of the function concept&lt;/a&gt;,&quot; the point-set version didn't become fully developed and universally accepted until well into the 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we have a counterexample to the claim, then for every real $y$, we have a set $S_x$ of $x$ values equinumerous to the rationals, with all the $S_x$ disjoint and lying in a finite interval. This is equivalent to a proof that $\mathbb{R}$ is equinumerous to $\mathbb{R}\times\mathbb{Q}$. This requires at least the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) We accept the existence of functions that are discontinuous everywhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) We accept the Cantorian analysis of infinities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are both significant philosophical &lt;em&gt;choices&lt;/em&gt;, not inevitable truths. #1 is false in SIA, for example. #2 was highly controversial at the &lt;em&gt;end&lt;/em&gt; of the 19th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think a better way to ask the question might be more like this: at what point in the 19th or 20th century did there develop a sufficient consensus about definitions and philosophy to make it false, according to the standard choices, that the intermediate value property implies continuity?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2015-10-20T15:44:44.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2961" PostTypeId="2" ParentId="2954" CreationDate="2015-10-20T17:57:37.390" Score="12" Body="&lt;p&gt;A textbook case of this is the famous &lt;a href=&quot;http://undsci.berkeley.edu/article/cold_fusion_01&quot; rel=&quot;nofollow noreferrer&quot;&gt;cold fusion debacle&lt;/a&gt;. It explains the temptation, if cold fusion worked out, producing a cheap and abundant energy source, there would be a Nobel Prize and world fame. Would all the cut corners still have mattered then? Maybe not that much.&lt;/p&gt;&#xA;&lt;p&gt;Back in the 1960s, Fleischmann noticed that palladium can absorb copious amounts of hydrogen, which meant that the packing of hydrogen atoms in it had to be very dense. This gave him an idea that if deuterium, whose atoms are larger, is used in place of hydrogen the packing will be dense enough to fuse two atoms together and produce helium-4, an unstable element that quickly decays into helium-3 releasing large amount of heat. If it worked, cold fusion would have solved the energy crisis. In 1983, in collaboration with Pons, Fleischmann performed an experiment with palladium and platinum rods submerged in heavy water and deuterium was produced by electrolysis. They did detect a large amount of excess heat, not accounted for by chemistry, and applied for an NSF grant. One of the reviewers, Jones, was working on a similar setup, but tried to detect neutrons produced by the helium-4 decay rather than excess heat. The amounts he detected, however, weren't nearly enough to account for the heat Pons and Fleischmann reported. Still, it looked promising and Jones proposed a collaboration.&lt;/p&gt;&#xA;&lt;p&gt;Pons and Fleischmann did not take to Jones kindly, thought he stole their idea, and the race was on. They decided to measure the neutrons themselves, which was not in their area of expertise. The results did not match either Jones's or their own excess heat measurements. Nevermind. In 1989, Jones proposed submitting publications simultaneously to the same journal to share the credit, the Journal of Electroanalytical Chemistry, in 18 days. Pons and Fleischmann agreed despite the fact that by their own estimation they needed 18 more months to gather data. In only 5 days they submitted their paper. Given the historical importance of the paper and about-to-be-solved energy crisis, the editor put it through an expedited review process, giving reviewers only one week instead of several. Even that was not expedited enough apparently. With a nudge from the University of Utah, Pons and Fleischmann held a press conference where they announced their results without providing much detail on the experiments.&lt;/p&gt;&#xA;&lt;p&gt;Then the troubles began. The unpublished paper was circulated and several groups tried to reproduce the cold fusion heat generation. Most did not succeed and the few that did later retracted due to an equipment error, or because they could not reproduce their own results. The paper also did not have enough details to reproduce the original experiment exactly, and Pons refused to provide them, the University of Utah did not want disclosure until a patent could be filed. Even when palladium rods from his own laboratory were inspected the results were negative. Pons &amp;quot;explained&amp;quot; it by heat output being smaller than he originally reported. In the meantime, errors in Pons and Fleischmann's experimental design came to light, and the fact that fusion in it was at variance with nuclear physics. Finally, Salamon was allowed to perform independent measurements on the original apparatus, and detected no required neutrons even when Pons claimed the fusion was on. Pons threatened him with legal action if he did not retract. After a year the headlines read: &lt;a href=&quot;http://content.time.com/time/magazine/article/0,9171,957604,00.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fusion or Illusion&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="4251" LastEditDate="2020-07-19T03:18:20.963" LastActivityDate="2020-07-19T03:18:20.963" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="2962" PostTypeId="2" ParentId="783" CreationDate="2015-10-21T08:27:12.750" Score="4" Body="&lt;p&gt;One of the Poles was Emil Leon Post, who discussed the incompleteness of Principia.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Emil_Leon_Post&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While at Princeton, he came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931.&lt;/p&gt;&#xA;&lt;p&gt;In his doctoral thesis, Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post also devised truth tables independently of Wittgenstein and C.S. Peirce and put them to good mathematical use. Jean Van Heijenoort's well-known source book on mathematical logic (1966) reprinted Post's classic article setting out these results.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3095" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-10-21T23:46:14.197" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2963" PostTypeId="1" AcceptedAnswerId="2964" CreationDate="2015-10-21T08:53:41.790" Score="4" ViewCount="4180" Body="&lt;p&gt;The question is already in the title.&lt;br&gt;&#xA;I am interested in who was the first to introduce the symbol $\sqrt {.}$&lt;br&gt;&#xA;I was asked this question from one of my students -since as he said the symbol is bizarre-&lt;br&gt;&#xA;If it is possible I would also like to see other notations that have been used for square roots through time. &#xA;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="2056" LastActivityDate="2015-10-21T10:26:14.180" Title="Who introduced the symbol $\sqrt {x}$?" Tags="|mathematics|mathematicians|notation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2964" PostTypeId="2" ParentId="2963" CreationDate="2015-10-21T09:08:07.873" Score="7" Body="&lt;p&gt;The easiest way is to check on this &lt;a href=&quot;http://jeff560.tripod.com/operation.html&quot; rel=&quot;noreferrer&quot;&gt;site&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Square root&lt;/strong&gt;. The first use of a capital R with a diagonal line was by &lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;noreferrer&quot;&gt;Leonardo of Pisa (or Fibonacci)&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;p&gt;The first appearance of the abbreviation R or R(with line) for &lt;em&gt;radix&lt;/em&gt; is in his &lt;em&gt;Pradica geometriae&lt;/em&gt; (1220), where one finds the R meaning &amp;quot;square root&amp;quot; in an expression &amp;quot;et minus R. 78125 dragme, et diminuta radice 28125 dragme.&amp;quot;(&lt;a href=&quot;https://books.google.it/books?id=7juWmvQSTvwC&amp;amp;pg=PA90&quot; rel=&quot;noreferrer&quot;&gt;Cajori&lt;/a&gt; vol. 1, page 90).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The radical symbol first appeared in 1525 in &lt;em&gt;Die Coss&lt;/em&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Christoph_Rudolff&quot; rel=&quot;noreferrer&quot;&gt;Christoff Rudolff&lt;/a&gt; (1499-1545). He used the&#xA;symbol (without the vinculum) for square roots. He did not use indices to indicate higher roots, but instead modified the appearance of the radical symbol for higher roots.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The fundamental work on these topics is still :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Florian Cajori, &lt;a href=&quot;https://books.google.it/books?id=7juWmvQSTvwC&amp;amp;pg=PA369&quot; rel=&quot;noreferrer&quot;&gt;A history of mathematical notations&lt;/a&gt;, Volumes 1-2 (1919 - Dover reprint).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-10-21T10:26:14.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2965" PostTypeId="2" ParentId="2940" CreationDate="2015-10-21T12:10:29.530" Score="17" Body="&lt;p&gt;The following would be my top picks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;Sophie Germain identity&lt;/strong&gt; says that $a^4+4b^4=(a^2+2b^2+2ab)(a^2+2b^2-2ab)$ for $a, b \in \mathbb{Z}$. This is a very simple identity but is quite useful in many problems of elementary number theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;strong&gt;Noether normalization lemma&lt;/strong&gt; is a result in commutative algebra that is taught probably in the very first week of a graduate level course in algebraic geometry. One version of the result says that, for any field $\mathbb{K}$ and any f.g. commutative $\mathbb{K}$-algebra $A$, there exists a non-negative integer $k$ and algebraicly independent elements $y_1, y_2, \ldots, y_k \in A$ such that $A$ is a f.g. module over the ring $\mathbb{K}[y_1, y_2, \ldots, y_k]$.&lt;/p&gt;&#xA;" OwnerUserId="102" LastActivityDate="2015-10-21T12:10:29.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2966" PostTypeId="2" ParentId="2526" CreationDate="2015-10-21T12:23:14.323" Score="1" Body="&lt;p&gt;For some bits of mathematics did and/or inspired by Ramanujan, I would recommend the following books in addition to the already mentioned ones:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0821841785&quot; rel=&quot;nofollow&quot;&gt;B. C. Berndt, &lt;em&gt;Number Theory in the Spirit of Ramanujan&lt;/em&gt;, published by American Mathematical Society.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ramanujanmathsociety.org/publications/rms-lecture-notes-series/number14-ramanujanrediscovered&quot; rel=&quot;nofollow&quot;&gt;N. D. Baruah, B. C. Berndt, T. Huber, M. Schlosser (eds.), &lt;em&gt;Ramanujan Rediscovered&lt;/em&gt;, published by Ramanujan Mathematical Society.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.springer.com/gp/book/9788132207665&quot; rel=&quot;nofollow&quot;&gt;K. Alladi, &lt;em&gt;Ramanujan's Place in the world of Mathematics&lt;/em&gt;, published by Springer.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="102" LastActivityDate="2015-10-21T12:23:14.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2967" PostTypeId="2" ParentId="2927" CreationDate="2015-10-21T13:34:36.140" Score="2" Body="&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Alhazen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham (Arabic: أبو علي، الحسن بن الحسن بن الهيثم‎; c. 965 – c. 1040 CE), better known by the Latinization Alhazen or Alhacen or as Ibn al-Haytham (ابن الهيثم), was an Arab Muslim polymath and philosopher who is widely considered as one of the most influential scientists of all time. Referred to as the father of experimental physics and modern optics and scientific methodology, he made significant contributions to the principles of optics, astronomy, mathematics, meteorology, visual perception and the scientific method.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His most famous work is his &lt;a href=&quot;https://en.wikipedia.org/wiki/Book_of_Optics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book of Optics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Book of Optics (Arabic: Kitāb al-Manāẓir‎ (كتاب المناظر); Latin: De Aspectibus or Perspectiva; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen (965–c. 1040 AD).&lt;/p&gt;&#xA;&lt;p&gt;The Book of Optics presented experimentally founded arguments against the widely held extramission theory of vision (as held by Euclid in his Optica) and in favor of intromission theory, as supported by thinkers such as Aristotle, the now accepted model that vision takes place by light entering the eye. Alhazen's work extensively affected the development of optics in Europe between 1260 and 1650.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="72" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-10-21T13:34:36.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2968" PostTypeId="1" AcceptedAnswerId="2971" CreationDate="2015-10-21T20:45:46.750" Score="23" ViewCount="4537" Body="&lt;p&gt;Momentum and energy play very similar roles in mechanics, each being changed by the application of force over a interval. For energy the interval is in space and for momentum it is in time. Both have associated conservation laws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, energy units are named in many systems and momentum units generally are not (and in particular are not named in SI).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a clear historical reason for the difference?&lt;/p&gt;&#xA;" OwnerUserId="178" LastEditorUserId="55" LastEditDate="2016-01-24T03:56:36.673" LastActivityDate="2017-02-16T22:43:01.507" Title="Why is there no named unit for momentum but there is one for energy?" Tags="|physics|units|terminology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2970" PostTypeId="1" CreationDate="2015-10-22T17:29:44.707" Score="5" ViewCount="288" Body="&lt;p&gt;I am interested in how multiplication symbols came about, and where different ones were used. When were $x\cdot y$ and $xy$ first used and by whom?&lt;/p&gt;&#xA;" OwnerUserId="3112" LastEditorUserId="55" LastEditDate="2015-10-24T19:43:33.013" LastActivityDate="2015-11-23T20:07:08.833" Title="When and by whom were the different symbols for multiplication used?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2971" PostTypeId="2" ParentId="2968" CreationDate="2015-10-22T17:59:08.067" Score="21" Body="&lt;p&gt;There is a historical reason. But it was not a fluke of history, the underlying reason is that energy comes up in non-mechanical (thermal, electric) contexts whereas momentum does not. Derived alternative, newton-meter in SI, did not arise naturally in such contexts, and alternative units, like calories, were used prior to the discovery of the general energy conservation law. It made sense to replace them with named standardized units. &lt;a href=&quot;https://books.google.com/books?id=CrmuSiCFyikC&amp;amp;dq=history+of+measurement&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Klein's Science of Measurement&lt;/a&gt; is a nice historical survey.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Standardization and naming of units happened in two waves. First started during the French revolution at the behest of Taleyrand (known as bishop turned politician who later managed to serve as foreign minister under both Napoleon and Bourbons after Waterloo). French Academy appointed a committee on weights and measures, including Condorcet, Lagrange, Laplace, Monge, and unofficially Lavoisier before he was guillotined, came up with the foundation of SI. At that point only the base units of length, time, weight and temperature received single names. The derived units of area, volume, speed, acceleration, work/energy, torque, momentum, etc., were combinations of the base ones even if they had commonly named units like liters or knots. Future kilogram, called kilograve, was the unit of weight, i.e. force, not mass. Mechanical energy and momentum received equal treatment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second wave, started in 1860s and formalized by 1880s in both SI and its competitor CGS, was meant to catch up with developments in thermodynamics and electromagnetism, and gave us ohms, volts, farads, watts, etc. Kilograve renamed into kilogram became the unit of mass, the unit of force was named dyne in CGS (from Greek dynamis - force), and newton in SI. What of energy? Its mechanical manifestation received no new attention, but in 1864 Clausius suggested erg (from Greek ergon - work) to replace calorie as a unit of thermal energy, a remnant of by then discredited caloric/phlogiston theory. It was adopted in CGS. The unit for power, watt, was suggested even before joule, by Siemens in 1882, to replace Watt's own horsepower used to measure output of steam engines. Siemens was an electric engineer. Joule himself was honored by a unit name for determining the mechanical equivalent of heat. Momentum was out of luck.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-01-24T03:51:48.743" LastActivityDate="2016-01-24T03:51:48.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2972" PostTypeId="2" ParentId="2970" CreationDate="2015-10-22T22:10:12.097" Score="2" Body="&lt;p&gt;Apparently, early Babylonian tablets used an ideogram for multiplication (depicted &lt;a href=&quot;https://motivate.maths.org/content/sites/motivate.maths.org/files/Multiplication.pdf&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;). But &quot;&lt;em&gt;Babylonian scribes did not use the ‘times’ symbol every time they did a calculation. Instead, they just drew a vertical line on their clay tablet, putting numbers to be multiplied on the left of the line and the answers on the right&lt;/em&gt;&quot;. Diophantus (c.250 AD) follows suit without even a line, he simply writes the result. What of juxtaposition? Generally he uses it for addition rather than multiplication, but he does write coefficients next to power symbols, and means that the two are multiplied. This convention remains in some 15th century European manuscripts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest known Indian mathematical work, the so-called Bakhshali manuscript (date estimates range from 300 BC to 400 AD), uses juxtaposition for multiplication. Bhaskara I (c.650 AD) usually also juxtaposes factors, but occasionally inserts a dot between them. There is no explanation of the usage in the text however, and he has an alternative notation that does not function like an operation. In Europe Stifel (1545) used $M$ between factors, Vieta $in$, and Oughtred (1631) the familiar St. Andrews cross $\times$. Modern juxtaposition convention became standard after Descartes's Geometry (1637).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp&quot; rel=&quot;nofollow&quot;&gt;Cajori's History of Mathematical Notations&lt;/a&gt; details many such issues. It is in the public domain and is available online. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-22T23:58:15.320" LastActivityDate="2015-10-22T23:58:15.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2973" PostTypeId="1" CreationDate="2015-10-23T07:28:35.547" Score="2" ViewCount="292" Body="&lt;p&gt;Recently I have been familiarized with the concept of 'Möbius function'. As far I&lt;br&gt;&#xA;know the Möbius function is defined by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\mu(n) =&#xA;\begin{cases}&#xA;1  &amp;amp; \small \text{if  $n$  is a square-free positive integer with an even number of prime factors.} \\&#xA;-1 &amp;amp; \small \text{if $n$ is a square-free positive integer with an odd number of prime factors.}\\&#xA;0  &amp;amp; \small \text{if n has a squared prime factor.}&#xA;\end{cases} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'm very interested to know how the concept of Möbius function developed historically. What situations or problems led the mathematicians to initiate the concept of Möbius function?&#xA;Can anyone explain?&lt;/p&gt;&#xA;" OwnerUserId="3116" LastEditorUserId="2442" LastEditDate="2015-10-23T12:18:12.127" LastActivityDate="2015-10-23T13:11:15.230" Title="Historical development of Möbius function" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2974" PostTypeId="1" CreationDate="2015-10-23T10:08:34.687" Score="5" ViewCount="1053" Body="&lt;p&gt;The d'Ocagne's identity is normally stated as $(-1)^n F_{m-n} = F_m F_{n+1}-F_n F_{m+1}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every book about the Fibonacci numbers has this formula in it, but I can't find any context about it. When it was first published, why it deserves a name or even if originally $m,n$ were considered to be any integers.&lt;/p&gt;&#xA;" OwnerUserId="3117" LastEditorUserId="2442" LastEditDate="2015-10-23T12:18:07.877" LastActivityDate="2015-10-25T03:02:51.277" Title="History of the d'Ocagne's identity for Fibonacci numbers" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2975" PostTypeId="2" ParentId="2973" CreationDate="2015-10-23T13:11:15.230" Score="1" Body="&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/M%C3%B6bius_function&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Möbius_function&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The German mathematician August Ferdinand Möbius introduced it in 1832.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$ $&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hardy &amp;amp; Wright, Notes on ch. XVI: &quot;... μ(n) occurs implicitly in the works of Euler as early as 1748, but Möbius, in 1832, was the first to investigate its properties systematically.&quot;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$ $&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the Disquisitiones Arithmeticae (1801) Carl Friedrich Gauss showed that the sum of the primitive roots (mod p) is μ(p − 1), (see #Properties and applications) but he didn't make further use of the function. In particular, he didn't use Möbius inversion in the Disquisitiones.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-10-23T13:11:15.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2976" PostTypeId="1" AcceptedAnswerId="2977" CreationDate="2015-10-23T14:05:17.273" Score="3" ViewCount="224" Body="&lt;p&gt;Recently I got some knowledge of the so-called hyper-determinant discovered by Cayley.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It arose a lot of interest of determinant in my mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is there any good account of the history of determinant? &lt;/p&gt;&#xA;" OwnerUserId="2570" LastActivityDate="2020-01-09T06:49:11.253" Title="Any account of the history of determinant of a matrix?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2977" PostTypeId="2" ParentId="2976" CreationDate="2015-10-23T18:07:45.207" Score="8" Body="&lt;p&gt;At &lt;a href=&quot;http://www-igm.univ-mlv.fr/~al/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alain Lascoux's web page&lt;/a&gt; you can find .pdf files of Thomas Muir's encyclopedic history of determinants. Muir devoted quite a bit of his long life to this work, which is only one of a very few essentially complete literature surveys ever done of a mathematical field of study. FYI, Dickson's multi-volume treatise &lt;strong&gt;History of the Theory of Numbers&lt;/strong&gt; is another such literature survey that is essentially complete up to the time it was written.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-01-09T06:49:11.253" LastActivityDate="2020-01-09T06:49:11.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="2979" PostTypeId="1" AcceptedAnswerId="2983" CreationDate="2015-10-13T22:37:53.573" Score="6" ViewCount="1005" Body="&lt;p&gt;In response to quantum mechanics, so the story goes, Einstein proposed a machine, that, based on the uncertainty principle, was a perpetual motion. This showed that quantum mechanics was at odds with evidence that energy is conserved. Bohr later showed that the analysis was flawed; Einstein had failed to take gravity into account (ironically).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this story true? If so, what was the machine Einstein proposed, how was it supposed to work, and what did Bohr reveal about it?&lt;/p&gt;&#xA;" OwnerUserId="4349" OwnerDisplayName="PyRulez" LastActivityDate="2021-04-09T19:22:17.350" Title="Did Einstein propose a perpetual motion machine to try to disprove quantum mechanics?" Tags="|quantum-mechanics|gravity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2980" PostTypeId="2" ParentId="2979" CreationDate="2015-10-13T22:43:54.090" Score="2" Body="&lt;p&gt;Can't find the source for the Einstein quote but it is in the introduction to a proof that he was wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://arxiv.org/abs/1205.6894v1&quot; rel=&quot;nofollow&quot;&gt;A violation of the uncertainty principle implies a violation of the second law of thermodynamics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2838" OwnerDisplayName="Martin Beckett" LastActivityDate="2015-10-13T22:43:54.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2981" PostTypeId="1" AcceptedAnswerId="3000" CreationDate="2015-10-24T18:04:59.727" Score="4" ViewCount="811" Body="&lt;p&gt;How did Saint Vincent prove that if &lt;span class=&quot;math-container&quot;&gt;$\frac{a}{b} = \frac{c}{d}$&lt;/span&gt;, then the area of a hyperbola &lt;span class=&quot;math-container&quot;&gt;$y = \frac{1}{x}$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; equals the area from &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;? What references (pdfs, links, books) are there on this specific subject? Are there any English translations of the relevant work of Saint Vincent? &lt;/p&gt;&#xA;" OwnerUserId="2171" LastEditorUserId="10106" LastEditDate="2020-06-07T18:40:15.017" LastActivityDate="2020-06-07T18:40:15.017" Title="How did Saint Vincent prove the logarithmic property of areas under hyperbolas?" Tags="|mathematics|reference-request|euclidean-geometry|translations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2982" PostTypeId="2" ParentId="2974" CreationDate="2015-10-25T01:30:45.603" Score="3" Body="&lt;p&gt;A special case for $m=n-1$ was discovered by Cassini in 1680, it was rediscovered by Simson in 1753. &lt;a href=&quot;http://www.cut-the-knot.org/arithmetic/algebra/CassinisIdentity.shtml&quot; rel=&quot;nofollow&quot;&gt;Cassini's identity&lt;/a&gt; $(-1)^{n+1} = F_{n-1} F_{n+1}-F_n^2$ was generalized in two ways: one was found by Catalan in 1879, and reads $(-1)^{n+k} = F_{n-k} F_{n+k}-F_n^2$, the other by d'Ocagne (1862-1938). It is unclear when exactly, but &lt;a href=&quot;https://books.google.com/books?id=rxzsCAAAQBAJ&amp;amp;pg=PA176&amp;amp;lpg=PA176&amp;amp;dq=Philbert+Maurice+d%27Ocagne+fibonacci&amp;amp;source=bl&amp;amp;ots=fu6T9g40ef&amp;amp;sig=VTcQ7MRhsuJ4a-dGGo4Y-WzjLso&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0CDIQ6AEwBGoVChMI8qW9_LncyAIVyOAmCh3S4AMC#v=onepage&amp;amp;q=Philbert%20Maurice%20d%27Ocagne%20fibonacci&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;in 1889 he studied recurrence relations&lt;/a&gt; of Fibonacci and Lucas type, in connection with periodic continued fractions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for context, both identities are examples of determinantal identities for Fibonacci numbers, and show close relation between recurrence relations and matrix algebra. Koshy's survey book Fibonacci and Lucas Numbers with Applications has a chapter on determinantal identities. Cassini's identity can be used to generate &lt;a href=&quot;http://www.cut-the-knot.org/Curriculum/Fallacies/FibonacciCheat.shtml&quot; rel=&quot;nofollow&quot;&gt;Curry's paradox&lt;/a&gt; invented by Curry in 1953, a dissection of $8×8$ square &quot;rearranged&quot; into a $5×13$ rectangle despite $64\neq65$. Other cases of d'Ocagne's identity can be used to &lt;a href=&quot;http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml&quot; rel=&quot;nofollow&quot;&gt;generate similar paradoxes&lt;/a&gt;. Recently, Spivey used determinants to derive an identity that &lt;a href=&quot;http://www.maa.org/sites/default/files/Spivey-CMJ-2006.pdf&quot; rel=&quot;nofollow&quot;&gt;generalizes both d'Ocagne's and Catalan's cases&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, &lt;a href=&quot;https://en.wikipedia.org/wiki/Philbert_Maurice_d&amp;#39;Ocagne&quot; rel=&quot;nofollow&quot;&gt;d'Ocagne's Wikipedia entry&lt;/a&gt; does not even mention this identity. He served as the president of the Mathematical Society of France in 1901 and the chair of geometry at École Polytechnique since 1912. But he is most remembered for inventing nomography, a method of graphic calculation popular with engineers before the onset of pocket calculators.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-25T03:02:51.277" LastActivityDate="2015-10-25T03:02:51.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2983" PostTypeId="2" ParentId="2979" CreationDate="2015-10-25T01:49:53.333" Score="8" Body="&lt;p&gt;Actually what Einstein proposed was not a perpetual motion machine but was an experiment design to take down the Uncertainty principle. So Einstein design an experiment known as &quot;Einstein's box&quot;. It was a thought experiment. He said that consider a box (ideal one) lined up with mirrors so that it contains light indefinitely. Also there is a shutter (ideal one) in the box so that one can open and let go a photon. So he said that, first weighed the box before a clockwork mechanism and chose a proper instant and open the shutter to let go one photon and measure the time. (It was design to work like that, in a clockwork mechanism). Then he proposed to weighed the box again, the change in the mass will tell us the energy emitted and that's how one can determine with precision the time and amount of energy released simultaneously, contradicting the uncertainty principle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Bohr pointed out that Einstein reasoning was flawed due to his own theory of gravity, as gravity effects the time so there is an uncertainty about time due to the gravity of earth and similar other reasoning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Uncertainty_principle#Einstein.27s_box&quot; rel=&quot;noreferrer&quot;&gt;See here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2015-10-25T01:49:53.333" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2984" PostTypeId="1" CreationDate="2015-10-25T07:03:17.787" Score="3" ViewCount="674" Body="&lt;p&gt;I'm interested in the impact of the discovery of non-Euclidean geometry on math, philosophy, and the attitudes of the general public.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know anything about how things changed right after the discoveries in the 1840's. My knowledge starts at Cantor and continues forward. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By way of examples: What did mathematicians think about the loss of certainty in their field? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did philosophers process this development?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As regards the public; my understanding is that this when postmodernism and the culture wars began. If there are mutually inconsistent geometries, then perhaps we're each entitled to our own subjective reality. How did the world get from there to here? Is that a generally accepted view of the social impact? Where can I read more about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've heard of Kline's book on the Loss of Certainty. Is that something I should read?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks much.&lt;/p&gt;&#xA;" OwnerUserId="340" LastActivityDate="2015-10-27T20:14:03.173" Title="History of impact of non-Euclidean geometry on math, philosophy, and the public" Tags="|philosophy-of-science|euclidean-geometry|geometry|riemannian-geometry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2985" PostTypeId="2" ParentId="2984" CreationDate="2015-10-25T10:41:16.370" Score="3" Body="&lt;p&gt;Regarding :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Morris Kline, &lt;a href=&quot;https://books.google.it/books?id=RNwnUL33epsC&amp;amp;pg=PA69&quot; rel=&quot;nofollow&quot;&gt;Mathematics: The Loss of Certainty&lt;/a&gt; (1982)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;the answer is : yes; &lt;strong&gt;Ch.IV The First Debacle : The Withering of Truth&lt;/strong&gt; is dedicated to non-Eucliidean geometry, and it is worth to be read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More details into :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;BA Rosenfeld, &lt;a href=&quot;https://books.google.it/books?id=3wzSBwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;A history of non-euclidean geometry&lt;/a&gt; (1988)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jeremy Gray, &lt;a href=&quot;https://books.google.it/books?id=3UeSCvazV0QC&amp;amp;pg=PR3&quot; rel=&quot;nofollow&quot;&gt;Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century&lt;/a&gt; (2007)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jeremy Gray, &lt;a href=&quot;https://books.google.it/books?id=ldzseiuZbsIC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Plato's Ghost: The Modernist Transformation of Mathematics&lt;/a&gt; (2008).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For a more &quot;philosophical&quot; point of view, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Roberto Torretti, &lt;a href=&quot;https://books.google.it/books?id=EcLrCAAAQBAJ&amp;amp;pg=PR4&quot; rel=&quot;nofollow&quot;&gt;Philosophy of Geometry from Riemann to Poincaré&lt;/a&gt; (1978).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For an overview, see also &lt;a href=&quot;http://plato.stanford.edu/entries/geometry-19th/&quot; rel=&quot;nofollow&quot;&gt;Nineteenth Century Geometry&lt;/a&gt;, &lt;a href=&quot;http://plato.stanford.edu/entries/kant-spacetime/&quot; rel=&quot;nofollow&quot;&gt;Kant's Views on Space and Time&lt;/a&gt;, &lt;a href=&quot;http://plato.stanford.edu/entries/hermann-helmholtz/&quot; rel=&quot;nofollow&quot;&gt;Hermann von Helmholtz&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/poincare/&quot; rel=&quot;nofollow&quot;&gt;Henri Poincaré&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-10-25T12:39:58.493" LastActivityDate="2015-10-25T12:39:58.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2986" PostTypeId="1" AcceptedAnswerId="2987" CreationDate="2015-10-25T10:21:03.117" Score="7" ViewCount="3709" Body="&lt;p&gt;Are there any examples of mathematicians have produced good research, having done poorly in mathematics exams?&lt;/p&gt;&#xA;" OwnerUserId="2839" OwnerDisplayName="Gudson Chou" LastEditorUserId="27" LastEditDate="2015-10-29T15:43:12.897" LastActivityDate="2019-05-11T19:00:45.917" Title="Examples of good research mathematicians that did poorly in exams" Tags="|education|mathematicians|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2987" PostTypeId="2" ParentId="2986" CreationDate="2015-10-26T02:21:41.483" Score="14" Body="&lt;p&gt;Well Ramanujan was such a mathematician. He was not so poor in math exams but he scored somewhat unbelievable marks in mathematics. In $1907$, he appeared in FA Examination at Pachaiyappa College, after studying privately. He got $85$ out of $150$ in mathematics and failed in English, Sanskrit, Physiology and History. Now one would expect that Ramanujan would get $100\%$ by analyzing his ability to tackle hard problems, but it was not so. But as we all know, he was very good at research. The reason for his scoring low marks in examinations are discussed &lt;a href=&quot;http://www.math.uiuc.edu/~berndt/articles/exams.pdf&quot;&gt;here&lt;/a&gt; by Berndt.  &lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2015-10-26T02:21:41.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2989" PostTypeId="2" ParentId="2986" CreationDate="2015-10-26T12:33:37.020" Score="9" Body="&lt;p&gt;The most famous case of this sort is Galois failing the entrance exam to Ecole Polytechnique.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-10-26T12:33:37.020" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="2990" PostTypeId="2" ParentId="2986" CreationDate="2015-10-26T14:31:54.957" Score="6" Body="&lt;p&gt;Newton somewhat infamously &lt;a href=&quot;https://books.google.co.uk/books?id=h3KgCgAAQBAJ&amp;amp;lpg=PT28&amp;amp;ots=kYOg9UUeng&amp;amp;dq=isaac%20newton%20euclid%20examination&amp;amp;pg=PT28#v=onepage&amp;amp;q=examination&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;failed his examinations at Cambridge&lt;/a&gt;: he was questioned orally about the proofs in Euclid, and since he had looked at Euclid once and thought it a total waste of time, and had just rederived all the results himself, he had no idea how to answer things like &quot;How does Euclid prove X.1?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Somehow he was awarded the scholarship anyway - presumably someone noticed that they were dealing with &lt;em&gt;Isaac Newton&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1939" LastActivityDate="2015-10-26T14:31:54.957" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="2991" PostTypeId="1" AcceptedAnswerId="2999" CreationDate="2015-10-26T16:24:06.870" Score="5" ViewCount="229" Body="&lt;p&gt;Today, I've heard someone speak of a &lt;em&gt;basis&lt;/em&gt; (of an ideal), meaning a generating set. All the time, I was fine with the term &lt;em&gt;Gröbner-basis&lt;/em&gt;, but when it comes without the prefix, it's a bit funny, since &lt;em&gt;basis&lt;/em&gt;, morally, is earmarked for something generating freely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I wondered which term was used first, and how, as well as by whom of course. The first thing that came to my mind was &lt;em&gt;Hilbert's basis theorem&lt;/em&gt;, but Hilbert did not speak about bases. The next names to consider were Gröbner and Buchberger. And in fact, Gröbner used &lt;em&gt;Basis&lt;/em&gt; for generating systems of ideals whereas speaking about a module, a basis was supposed to generate freely. (See, e.g., &lt;em&gt;Moderne Algebraische Geometrie&lt;/em&gt;, 1949, Springer Wien &amp;amp; Innsbruck.)&#xA;Not surprisingly, being a student of Gröbner, Buchberger also called generating sets of ideals bases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realise that a full description of the evolution of &lt;em&gt;basis&lt;/em&gt; could be too much to ask for, so I would already be happy to read an answer which builds a bridge between Hilbert's basis theorem (&lt;em&gt;Über die Theorie der algebraischen Formen&lt;/em&gt;, 1890) and Gröbner's terminology, possibly by pointing at the first one to refer to it as &lt;em&gt;basis theorem&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="79" LastEditorUserId="55" LastEditDate="2015-12-16T19:48:36.653" LastActivityDate="2015-12-16T19:48:36.653" Title="What was the evolution of &quot;basis&quot; and &quot;generating set&quot; in algebra?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2992" PostTypeId="2" ParentId="2986" CreationDate="2015-10-26T17:34:46.563" Score="5" Body="&lt;p&gt;G.H. Hardy.  From  &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Hardy.html&quot; rel=&quot;noreferrer&quot;&gt;http://www-history.mcs.st-and.ac.uk/Biographies/Hardy.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;While at Winchester Hardy won an open scholarship to Trinity College, Cambridge, which he entered in 1896. At Cambridge Hardy was assigned to the most famous coach R R Webb. He quickly realised that the point of the training was simply to achieve the best possible marks in the examinations by learning all the tricks of the trade. He was shocked to discover that Webb was not interested in the subject of mathematics, only in the tricks of examinations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hardy was placed as fourth wrangler in the Mathematical Tripos of 1898, a result which continued to annoy him for, despite feeling that the system was very silly, he still felt that he should have come out on top.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia adds&#xA;&quot;Years later, he sought to abolish the Tripos system, as he felt that it was becoming more an end in itself than a means to an end.&quot;&lt;/p&gt;&#xA;" OwnerUserId="2061" LastActivityDate="2015-10-26T17:34:46.563" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="2993" PostTypeId="1" CreationDate="2015-10-26T20:49:54.773" Score="2" ViewCount="131" Body="&lt;p&gt;When was (and in what paper) Paley-Wiener integral invented?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, what was first? Paley-Wiener integral or Multiple Wiener Integral?&lt;/p&gt;&#xA;" OwnerUserId="3145" LastActivityDate="2017-06-13T17:10:49.307" Title="When was (and in what paper) Paley-Wiener integral invented?" Tags="|mathematics|probability|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="2994" PostTypeId="2" ParentId="2984" CreationDate="2015-10-27T19:52:42.000" Score="2" Body="&lt;p&gt;Have you taken a look at Richard Trudeau's &lt;strong&gt;&lt;em&gt;The non-Euclidean revolution&lt;/em&gt;&lt;/strong&gt; (with an introduction by H. S. M. Coxeter. Birkhäuser Boston, Inc., Boston, MA, 1987. xiv+269 pp. ISBN: 0-8176-3311-1)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first two paragraphs of the review of this book, which K. Strubecker contributed to MathSciNet, read thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Starting from a very detailed, critical overview of plane geometry as&#xA;  axiomatically based by Euclid in his Elements, the author, in this&#xA;  remarkable book, describes in an incomparable way the fascinating path&#xA;  taken by the geometry of the plane in its historical evolution from&#xA;  antiquity up to the discovery of non-Euclidean geometry. This&#xA;  discovery, characterized by the names of Gauss, Bolyai and&#xA;  Lobachevskiĭ, signified a revolution not just for geometry; the&#xA;  philosophical views of space—as shaped predominantly by Kant and&#xA;  generally accepted—and also epistemology underwent an unforeseen&#xA;  upheaval. This &quot;non-Euclidean revolution'', in all its aspects, is&#xA;  described very strikingly here.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The book begins with a detailed critical discussion of Euclidean&#xA;  explanations of concepts, definitions and axioms. The fifth axiom, in&#xA;  Euclid's version not very transparent, then, in the intuitive&#xA;  formulation named after John Playfair, appeared very much clearer and&#xA;  convincing. The many futile attempts at proving Euclid's &quot;parallel&#xA;  axiom'' and the final solution of the parallel problem through the&#xA;  discovery of non-Euclidean geometry are described in detail. Finally,&#xA;  the relative freedom from contradiction in non-Euclidean geometry is&#xA;  proved. Much attention is paid to thus newly acquired horizons of&#xA;  knowledge and their influence on problems in natural philosophy and&#xA;  particularly in physics...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Seems to me that this book might be right up your alley!&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2015-10-27T20:14:03.173" LastActivityDate="2015-10-27T20:14:03.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2995" PostTypeId="1" AcceptedAnswerId="2997" CreationDate="2015-10-28T00:52:18.633" Score="3" ViewCount="68" Body="&lt;p&gt;Clausius somewhere wrote (in German): ``The energy of the world is constant.'' This is a well known quotation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the source, publication, the rest of the paper?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Obviously he was summarizing the conservation of energy principle of Mayer by making a trivial inference. Mayer in turn had actually done the water dropping experiment Carnot suggested as a definition and used the mechanical heat equivalent this implied. As far as that energy is conserved but having no idea how energy connects to heat, or proof, that was Leibniz (in his Dynamics). Mayer used the Leibniz terminology. Clausius introduced the word energy later.]&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2015-10-28T02:53:12.383" Title="Clausius regarding energy" Tags="|classical-mechanics|thermodynamics|energy|resource-recommendation|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2996" PostTypeId="1" CreationDate="2015-10-28T02:23:25.870" Score="8" ViewCount="552" Body="&lt;p&gt;Sophie Germain's identity is only about factorising &lt;span class=&quot;math-container&quot;&gt;$a^4+4b^4$&lt;/span&gt; as product of two squares.It's not quite difficult. So,why is it so popular?&lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="3869" LastEditDate="2022-04-23T03:06:22.377" LastActivityDate="2022-04-23T03:06:22.377" Title="Why is Sophie-Germain's Identity popular?" Tags="|mathematics|mathematicians|sophie-germain|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="2997" PostTypeId="2" ParentId="2995" CreationDate="2015-10-28T02:53:12.383" Score="3" Body="&lt;p&gt;There is a second sentence to the quote:&quot;&lt;em&gt;The energy of the universe is constant. The entropy of the universe is increasing&lt;/em&gt;&quot;. The conservation of energy law. The second law of thermodynamics. The quote became famous because from the conjuction of the two conclusion about the heat death of the universe appears to follow. The idea was expressed earlier by Thomson in On the Age of the Sun’s Heat (1862), but Clausius's quip had a nice ring to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote appears in  Ueber Verschiedene für die Anwendung Bequeme Formen der Hauptgleichungen der Mechanischen Wärmetheorie (On Several Applications of the Main Equations of the Mechanical Theory of Heat), Annalen der Physik und Chemie., vol. 125 No. 7 (1865): pp. 353–400. Here is a &lt;a href=&quot;https://books.google.com/books?id=qAg4AAAAMAAJ&amp;amp;pg=PA353#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;direct link&lt;/a&gt; to the paper.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-10-28T02:53:12.383" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="2998" PostTypeId="2" ParentId="2996" CreationDate="2015-10-28T04:38:14.970" Score="6" Body="&lt;p&gt;Because of it's simplicity I guess. The identity is $a^4+4b^4=(a^2+2b^2-2ab)(a^2+2b^2+2ab)$. The identity is easy to verify but difficult to arrive at in the first place. We are familiar with $a^2-b^2=(a+b)(a-b)$, but not with the kind $a^2+b^2$ splitting as two factors. Also knowing the sum can be factored by such a identity, it is helpful in number theory, specially factoring numbers, determining a large number is prime or not. You can check &lt;a href=&quot;http://www.cut-the-knot.org/blue/SophieGermainIdentity.shtml&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for an example.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="658" LastEditDate="2015-10-28T15:06:17.293" LastActivityDate="2015-10-28T15:06:17.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="2999" PostTypeId="2" ParentId="2991" CreationDate="2015-10-28T19:03:11.687" Score="4" Body="&lt;p&gt;I do not have conclusive evidence, but I will name likely suspects and make a circumstantial case. Hopefully, it will get the ball rolling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Basis&quot; as generating set of ideals may have been introduced before Hilbert by Dedekind, even if Hilbert did not use the word. Dedekind introduced &quot;ideals&quot; in (what we call) number rings only in the supplements to 1879 edition of &lt;a href=&quot;https://archive.org/details/vorlesungenberz02dedegoog&quot; rel=&quot;nofollow&quot;&gt;Vorlesungen über Zahlentheorie&lt;/a&gt;, but used &quot;modul&quot; back in 1871. A generating set of the ideal is then naturally a basis of the &quot;modul&quot; by analogy to vector space basis. In 1882 Dedekind and Weber &quot;related geometric ideas with rings of polynomials and extended the use of modules&quot; according to &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Ring_theory.html&quot; rel=&quot;nofollow&quot;&gt;MacTutor's Ring Theory&lt;/a&gt;. By the way, while &quot;ideal&quot;, &quot;module&quot; and &quot;field&quot; are Dedekind's neologisms, &quot;ring&quot; is Hilbert's from 1893 (in print 1897).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we know for sure is that van der Waerden in his celebrated Modern Algebra uses &quot;basis&quot; as a matter of course, and explicitly relates that to the module interpretation. He also calls Hilbert basis theorem by this name, and is most likely Gröbner's and Buchberger's source. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two volumes of Modern Algebra were published in 1930 and 1931, but written in 1926-1929 under the heavy influence, by van der Waerden's own admission, of Emmy Noether. Van der Waerden became the modern algebraist when working under Noether in Göttingen in 1924-1925. &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Noether_Emmy.html&quot; rel=&quot;nofollow&quot;&gt;Noether herself&lt;/a&gt; was converted to the Dedekind-Hilbert methods by Fischer back in Erlangen since 1911, and quickly became such an authority on them that Hilbert and Klein invited her to Göttingen in 1915. Her seminal paper &lt;a href=&quot;https://eudml.org/doc/158855&quot; rel=&quot;nofollow&quot;&gt;Idealtheorie in Ringbereichen (1921)&lt;/a&gt; is the seed of Modern Algebra, and gives the decomposition of ideals into intersections of primary ideals in commutative rings with ascending chain condition. Now these are called &quot;Noetherian rings&quot;, and the Hilbert &quot;basis&quot; theorem is proved for polynomial rings over them. My guess is that van der Waerden got his usage of &quot;basis&quot; from Noether, and that she is the one who christened the Hilbert's theorem by its modern name. She might also have been the first to use &quot;basis&quot; this way, but it could be Dedekind or somebody in between.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-28T19:18:19.640" LastActivityDate="2015-10-28T19:18:19.640" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3000" PostTypeId="2" ParentId="2981" CreationDate="2015-10-29T00:03:13.787" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/3482652?seq=6#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Dhombres's Is One Proof Enough?&lt;/a&gt; gives English translation of the Latin text with original figures, and explains why St. Vincent gave two proofs:&quot;&lt;em&gt;the Jesuit mathematician prefers to describe precisely the road, the method of discovery, rather than to show the shortest way to the summit&lt;/em&gt;&quot;. There is a nice parallel to Archimedes, who gave a heuristic mechanical argument for the quadrature of the parabola in the &lt;a href=&quot;https://en.wikiquote.org/wiki/Archimedes#The_Method_of_Mechanical_Theorems&quot; rel=&quot;nofollow&quot;&gt;Method&lt;/a&gt;, and then reproved it by the method of exhaustion. Of course St. Vincent could not possibly know of the Method, but his motivation echo's Archimedes's &quot;&lt;em&gt;certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, St Vincent's first proof is also heuristic, and the other parallels Archimedean method of exhaustion quadrature! &lt;a href=&quot;http://www.jstor.org/stable/3620779?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Burn in Gregory of St Vincent and the Rectangular Hyperbola&lt;/a&gt; gives a modernization of it, replacing Apollonian lemmas with coordinate geometry:&quot;&lt;em&gt;Gregory provided two proofs, A and B; proof A used inscribed parallelograms of equal area and of increasing multiplicity, by repeated insertion of geometric means. For the rectangular hyperbola these become inscribed rectangles. Proof A (in its structure) is a recognisable precursor of the later theory of integration, though the existence of the limit is not quite proved. Proof B, which is the basis of the proof described below, bears a remarkable similarity to Archimedes' quadrature of the parabola and is as complete a proof as that of Archimedes, but to my knowledge it has not been reproduced in any English book on the history of mathematics. Gregory's work was based on expert knowledge of the conic sections, and did not use coordinate geometry&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-10-29T00:03:13.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3001" PostTypeId="1" CreationDate="2015-10-29T01:46:02.563" Score="15" ViewCount="1076" Body="&lt;p&gt;Are there any examples of mathematical ideas being communicated in an &lt;em&gt;anonymous&lt;/em&gt; manner that had a substantial impact on mathematics at the time. It seems to me that it is very rare for an author to not take credit for their work, rarer still for it to have a great impact.&lt;/p&gt;&#xA;" OwnerUserId="27" LastActivityDate="2022-04-23T00:56:35.130" Title="Are there any anonymous contributions to mathematics that had a great impact?" Tags="|mathematics|mathematicians|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3002" PostTypeId="2" ParentId="3001" CreationDate="2015-10-29T03:12:12.967" Score="16" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Student%27s_t-test&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland... The t-test work was submitted to and accepted in the journal &lt;em&gt;Biometrika&lt;/em&gt; and published in 1908. Company policy at Guinness forbade its chemists from publishing their findings, so Gosset published his statistical work under the pseudonym &quot;Student&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course eventually the secret came out of the bag.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-10-29T03:12:12.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3003" PostTypeId="1" AcceptedAnswerId="3007" CreationDate="2015-10-29T05:55:13.497" Score="11" ViewCount="1896" Body="&lt;p&gt;I am interested in the early history of curvature. Who defined it first and when, who came up with the name, how was it calculated &lt;em&gt;before&lt;/em&gt; mathematicians used calculus to define $k=|α''(s)|$? Are there books or articles containing such early calculations?&lt;/p&gt;&#xA;" OwnerUserId="3161" LastEditorUserId="4251" LastEditDate="2021-09-16T05:47:02.283" LastActivityDate="2022-07-20T21:37:29.270" Title="How was curvature originally defined and calculated?" Tags="|mathematics|reference-request|calculus|differential-geometry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3004" PostTypeId="2" ParentId="3001" CreationDate="2015-10-29T18:01:45.993" Score="10" Body="&lt;p&gt;Anonymity can be motivated by a variety of reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest example is early Pythagoreans, who had a rule of attributing all discovering to &quot;the Man&quot;, as they called Pythagoras. As a result, we do not know the names of the discoverers of the Pythagorean theorem, incommensurability of side and diagonal of a square, etc. although it is fairly certain that it wasn't Pythagoras. And those results were the foundation of Greek geometry and formed a large portion of Euclid's Elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sophie Germain corresponded first with Lagrange and then Gauss under the pseudonym Monsieur Antoine-August Le Blanc, the name of a former student she tutored. The 1804-1807 letters to Gauss presented early stages of her celebrated work on Fermat's Last Theorem, which influenced subsequent developments in the field, notably Kummer. She explained the reason when disclosing her identity to Gauss:&quot;&lt;em&gt;fearing the ridicule attached to a female scientist&lt;/em&gt;&quot;. Gauss's reply indicates his appraisal:&quot;&lt;em&gt;when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are lighter examples that are not too great in terms of impact, but interesting nonetheless.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When Bernoulli challenged European mathematicians to find the curve of fastest descent, the brachistochrone, one of the solutions arrived anonymously. But the solution itself was identification enough. &quot;&lt;em&gt;Ex unge leonem&lt;/em&gt;&quot;, Bernoulli famously wrote outing Newton, &quot;&lt;em&gt;the lion is recognized by his claws&lt;/em&gt;&quot;. Biographers later found out Newton's reason:&quot;&lt;em&gt;I do not love to be dunned and teased by foreigners about mathematical things...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we call the towers of Hanoi first saw the light of day in 1883, when Professor N. Claus (de Siam), mandarin of college Li-Sou-Stian described a version of the puzzle complete with a temple of Brahma story asserting its ancient origins. Permuting the letters of &lt;em&gt;N. Claus (de Siam) Li-Sou-Stian&lt;/em&gt; reveals the real author's name Lucas d'Amiens of lycee Saint-Louis, of the Lucas numbers fame. Sometimes it is a hoax for publicity effect. Some online sources are still under the impression that the temple of Brahma was real.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-10-29T21:54:20.517" LastActivityDate="2015-10-29T21:54:20.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3005" PostTypeId="2" ParentId="2996" CreationDate="2015-10-29T21:05:12.650" Score="7" Body="&lt;p&gt;For any field $K$, $c \in K^\times$ and integer $n &amp;gt; 2$, when is $X^n - c$ irreducible over $K$?  A necessary condition is that for each prime number $p$ dividing $n$, $c$ is not a $p$th power in $K$; otherwise $X^p - c$ would be reducible over $K$ (a linear factor), so $X^n - c$ would be reducible over $K$ too. It turns out that this necessary condition is sufficient &lt;em&gt;except&lt;/em&gt; if $4\mid n$, in which case we also need to check $c$ is not of the form $-4d^4$ for some $d$ in $K$ precisely because of the &quot;unexpected&quot; factorization of $X^4+4d^4$ over $K$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus the only subtlety in determining irreducibility of $X^n-c$ over $K$ is due to the type of factorization going back to Sophie Germain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lang's &lt;em&gt;Algebra&lt;/em&gt; devotes a section to the study of $X^n - a$, and its irreducibility is the first main result he describes.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2015-10-29T21:05:12.650" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3006" PostTypeId="1" CreationDate="2015-10-29T23:18:44.373" Score="8" ViewCount="1649" Body="&lt;p&gt;It is common for mathematicians to use counterclockwise (ccw) as positive, and&#xA;clockwise (cw) as negative.&#xA;For example, trigonometric functions increase from $0^\circ$ along the positive $x$-axis (East)&#xA;to $90^\circ$ along the positive $y$-axis (North), sweeping out the first (NE) quadrant of&#xA;the Cartesian coordinate system.&#xA;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Right-hand_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;right-hand-rule&lt;/a&gt; rules such coordinate systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, non-mathematicians are exposed to cw advancement as increasing time in all(?) analog clocks,&#xA;and so for many of them, cw is the natural sense of positive angular advance.&#xA;I myself, as a mathematician, view the annual calendar as advancing &#xA;ccw&amp;mdash;summer @$(0,-1)$ South, fall @$(1,0)$ East, Christmas @$(0,1)$ North&amp;mdash;but&#xA;I learned when making a presentation to random people (faculty) that many (most?) view the annual&#xA;calendar as advancing cw.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q1&lt;/em&gt;&lt;/strong&gt;. Where did the mathematician's ccw convention originate, and where did the clockwise convention originate, and why are they opposite one another?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And secondarily,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q2&lt;/em&gt;&lt;/strong&gt;. Are there data on how we mentally view the annual calendar advancing&amp;mdash;linearly: L/R, R/L, circularly: cw, ccw?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Likely the answer to Q2 depends on nationality/culture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;(Although I've wondered this a long time, this posting was triggered&#xA;by the MESE posting, &quot;&lt;a href=&quot;https://matheducators.stackexchange.com/q/9874/511&quot;&gt;Why do we conventionally treat trig functions as going anti-clockwise from the right?&lt;/a&gt;.&quot;)&lt;/p&gt;&#xA;" OwnerUserId="606" LastEditorUserId="-1" LastEditDate="2017-04-13T12:50:24.327" LastActivityDate="2022-03-20T14:09:53.777" Title="Counterclockwise vs. clockwise" Tags="|mathematics|naming-conventions|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3007" PostTypeId="2" ParentId="3003" CreationDate="2015-10-30T00:30:46.943" Score="14" Body="&lt;p&gt;Apollonius (c. 262–190 BC) &amp;quot;calculated&amp;quot; curvature of conic sections implicitly when solving the problem of drawing normals to them in book V of Conica, but he did not think of it as a property of a curve, and his &amp;quot;calculations&amp;quot; are constructions of segments. The first person to &amp;quot;see&amp;quot; curvature was Oresme (c. 1320-1382), Descartes's precursor in introducing coordinates. He described it as a local measure of curve's bending, and christened it with the Latin &amp;quot;curvitas&amp;quot;. Later he proposed that for circles it can be quantified by the reciprocal of the radius, our modern convention. Kepler vaguely suggested how to define curvature for general curves by considering the &amp;quot;closest&amp;quot; circle at a point, named osculating (&amp;quot;kissing&amp;quot;) circle by Leibniz in 1680s. But it was Huygens, who first found a way to calculate curvature for general curves, and Newton who gave the concept its modern form.&lt;/p&gt;&#xA;&lt;p&gt;The calculus was yet to be invented so Huygens relied on Descartes's ideas about multiplicity of intersections and Fermat's ideas of infinitely small. Descartes in Geometry (1637) gives a method of drawing tangents and normals to a curve by finding circles with centers on the axis, whose two intersection points with the curve &amp;quot;coalesce&amp;quot;, and so intersection gives a double root. This was a foundation of &lt;a href=&quot;https://hsm.stackexchange.com/questions/1963/is-there-a-lost-calculus/1968#1968&quot;&gt;Descartes's algebraic calculus&lt;/a&gt;, that preceded Newton's and Leibniz's. Huygens realized in 1653-4 that for any point on a curve one can consider two &amp;quot;coalescing&amp;quot; normals, and gave a semi-geometric way to calculate their point of intersection. That of course was the center of curvature, and its distance to the curve was the radius of curvature, although Huygens did not use these terms. Van Schooten included the method in appendices to the second edition of Descartes' Geometry (1659), which is where Newton learned it from. Later in a remarkable tour de force of Horologium Oscillatorum (1673) Huygens named the locus of the centers of curvature to a curve its evolute, and showed how to construct a perfect pendulum, whose period does not depend on its amplitude. The construction was based on the fact that evolute to a cycloid is congruent to it.&lt;/p&gt;&#xA;&lt;p&gt;It was Newton who developed Huygens's ideas into a general method for computing curvature (which he initially called &amp;quot;crookedness&amp;quot;) in 1664. He used that two nearby normals &amp;quot;coalescing&amp;quot; is equivalent to three nearby intersection points with the circle &amp;quot;coalescing&amp;quot; on the curve, and used Descartes's method, with Hudde's 1659 improvements to find its center by looking for a triple root. Newton also realized that at inflexion points, where the radius of curvature &amp;quot;blows up&amp;quot;, one should assign to curvature value zero. Later in the &lt;a href=&quot;http://gymportalen.dk/sites/lru.dk/files/lru/2_1_a_treatise.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Method of Fluxions and Infinite Series; with its Applications to the Geometry of Curve-Lines (1671)&lt;/a&gt; he transformed the algebraic approach to curvature into a more recognizable one of calculus. But it is interesting that Newton was well aware, and facile with, the &amp;quot;lost&amp;quot; algebraic calculus when developing his own. Some modern textbooks, e.g. Struik's Lectures on Classical Differential Geometry, still use the language of coalescing points and normals when defining curvature.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.nmsu.edu/%7Edavidp/hist_projects/curvature.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lodder's Curvature in Calculus Curriculum&lt;/a&gt; gives a step by step guide through Huygens's calculation of curvature and evolute of the cycloid, he also describes Euler's 1760 calculation of the principal curvatures of a surface. An overview of early history (with some inaccuracies) is given in &lt;a href=&quot;https://blogs.goucher.edu/verge/files/2016/01/Curvature_of.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Curvature of Surfaces in 3-Space&lt;/a&gt;, the source is &lt;a href=&quot;https://www34.homepage.villanova.edu/robert.jantzen/maple/misc/history_of_curvature/k.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Margalit's History of Curvature&lt;/a&gt;. &lt;a href=&quot;http://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0020.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nauenberg's Huygens and Newton on Curvature and its Applications to Dynamics&lt;/a&gt; gives a detailed account based on original sources, it briefly describes Newton's alternative method for calculating curvature a la Descartes.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-09-16T04:11:10.227" LastActivityDate="2021-09-16T04:11:10.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3008" PostTypeId="2" ParentId="381" CreationDate="2015-10-30T05:19:14.347" Score="4" Body="&lt;p&gt;The use of a base 60 number system in Babylon is mentioned at &lt;a href=&quot;http://www.storyofmathematics.com/sumerian.html&quot; rel=&quot;nofollow&quot;&gt;http://www.storyofmathematics.com/sumerian.html&lt;/a&gt; where it says, &quot;Sumerian and Babylonian mathematics was based on a sexegesimal, or base 60, numeric system, which could be counted physically using the twelve knuckles on one hand, the five fingers on the other hand.&quot; (That presumably was supposed to mean the four fingers plus the thumb, rather than five fingers plus thumb but I have read ramblings about people (and alien creatures) with six rather than five digits or toes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read several years ago (although I cannot now remember where) that the number base 60 was (is?) still being used in modern-day Iraq by street market traders.  The twelve knuckles (or more simply the twelve finger bones) do not include the thumb; fingers only.  I don't know how this system might be used to perform calculations like multiplication or division.  Perhaps it is used simply to count items, literally, by the handful.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Counting to twelve can be easily achieved on just one hand by using the thumb and the twelve finger bones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It does make one wonder about ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) the origin of the word 'dozen'.  Is that an ancient Babylonian word or at least Arabic?  It is commonly claimed to derive ultimately from the Latin duodecim corrupted into the old French word, douzaine.  However, and this is pure conjecture on my part, there are Arabic words in English which contain letter 'z' (eg azimuth, azure, gauze, gazelle ...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) the origin and extent of 12 used in numbers in Britain, before decimalisation was introduced, and elsewhere.  There were 12 pennies in a shilling, there are 12 inches in a foot (still in use of course), 12 ounces in a troy pound (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Troy_weight&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Troy_weight&lt;/a&gt; for a suggested connection to the arabic unit Dirhem - an interesting story in itself it seems), 12 dozen in a gross (144), and half of the length of a day is 12 hours from midnight to midday or midday to midnight.&lt;/p&gt;&#xA;" OwnerUserId="2219" LastActivityDate="2015-10-30T05:19:14.347" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3009" PostTypeId="2" ParentId="3006" CreationDate="2015-10-30T13:38:52.040" Score="9" Body="&lt;p&gt;I suppose that all these notions and terminology come from the early astronomy.&#xA;It so happened that in the ancient times astronomy was mostly practiced in the Northern hemisphere. If you observe from Northern hemisphere, the sky rotates about you clockwise (East to West). Sun in particular. This determines the clockwise direction, as the time was always measured by the Sun.&lt;/p&gt;&#xA;&lt;p&gt;However interesting astronomy begins when you start to appreciate and describe the slower motions than the daily rotation: the progress of the planets, Sun and Moon with respect to the &amp;quot;fixed&amp;quot; stars. As all they move generally in the same direction (opposite to the direction of the daily rotation), this direction was traditionally called &amp;quot;positive&amp;quot; or &amp;quot;forward&amp;quot; in astronomy. (Planets sometimes &amp;quot;retrograde&amp;quot; but they retrograde with respect to this positive, natural direction).&lt;/p&gt;&#xA;&lt;p&gt;So positive direction is counter-clockwise.&lt;/p&gt;&#xA;&lt;p&gt;As mathematics was always closely connected to astronomy (well, until 20s century), it was natural for mathematicians to take the same direction of rotation as positive. In particular trigonometry was invented and developed for the needs of astronomy. Choosing the other direction in trigonometry as positive would be extremely inconvenient for the main purpose of it.&lt;/p&gt;&#xA;&lt;p&gt;EDIT All pictures illustrating Ptolemy are made with Sun, Moon and planets moving&#xA;counterclockwise (as they actually move with respect to the fixed stars, as seen from the Northern hemisphere). Of course I understand that these are not original pictures, but it is hard to imagine that the original pictures looked differently. So for the inventors of trigonometry there was probably no choice but to count argument counterclockwise.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-03-20T14:09:53.777" LastActivityDate="2022-03-20T14:09:53.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3010" PostTypeId="1" CreationDate="2015-10-31T07:58:53.887" Score="1" ViewCount="184" Body="&lt;p&gt;My history of maths lecturer claims Pythagoras never existed. Is this true? Is there any evidence he existed? He also claims the Pythagoreans did not discover or prove the Pythagorean theorem.&lt;/p&gt;&#xA;" OwnerUserId="3170" OwnerDisplayName="LTS" LastEditorUserId="62" LastEditDate="2015-10-31T16:56:07.760" LastActivityDate="2015-10-31T16:56:07.760" Title="Did Pythagoras exist?" Tags="|biographical-details|ancient-greece|" AnswerCount="0" CommentCount="1" ClosedDate="2015-10-31T08:26:34.340" ContentLicense="CC BY-SA 3.0" />
  <row Id="3011" PostTypeId="1" CreationDate="2015-10-31T08:07:49.677" Score="1" ViewCount="55" Body="&lt;p&gt;My history of maths lecturer claims the Greeks did not discover the irrational numbers. Is this true? &lt;/p&gt;&#xA;" OwnerUserId="3170" OwnerDisplayName="LTS" LastEditorUserId="26" LastEditDate="2015-10-31T10:38:10.353" LastActivityDate="2015-10-31T10:38:10.353" Title="Did the Greeks discover the irrational numbers?" Tags="|mathematics|ancient-greece|number-theory|" AnswerCount="0" CommentCount="0" ClosedDate="2015-10-31T10:37:52.347" ContentLicense="CC BY-SA 3.0" />
  <row Id="3012" PostTypeId="1" AcceptedAnswerId="5250" CreationDate="2015-10-31T13:25:07.613" Score="2" ViewCount="52" Body="&lt;p&gt;For my philosophy of science class, I'm looking for references that will provide a historical / science philosophical insight on the development and advances of lithium-ion batteries. For now, I imagine two types of such sources:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;encyclopedias on technology. I will read &lt;a href=&quot;http://store.elsevier.com/Encyclopedia-of-Electrochemical-Power-Sources/isbn-9780444520937/&quot; rel=&quot;nofollow&quot;&gt;Encyclopedia of Electrochemical Power Sources&lt;/a&gt;, will do my best to find something in the Encyclopedia of History of Technology or Encyclopedia of Electrochemistry. Any more books that I could look into?&lt;/li&gt;&#xA;&lt;li&gt;review articles. For now, I have &lt;a href=&quot;http://pubs.acs.org/doi/abs/10.1021/cm901452z&quot; rel=&quot;nofollow&quot;&gt;Goodenough, 2009&lt;/a&gt;, but that doesn't seem to be sufficient to me.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Am I missing any other types of references? I feel that I will possibly need something more on the philosophy of science side of things, not purely technical.&lt;/p&gt;&#xA;" OwnerUserId="3166" LastActivityDate="2016-10-04T15:31:03.897" Title="Where to look for information on technological advances in lithium-ion batteries?" Tags="|reference-request|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3013" PostTypeId="1" AcceptedAnswerId="3021" CreationDate="2015-11-01T14:12:48.643" Score="8" ViewCount="840" Body="&lt;p&gt;I've read about a Lucasian Professor called Joshua King. From the Wiki page, he seems like a mysterious man to me&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Joshua King came to Cambridge from Hawkshead Grammar School. It was soon evident that the school had produced someone of importance. He became Senior Wrangler, and his reputation in Cambridge was immense. It was believed that nothing less than a second Newton had appeared. They expected his work as a mathematician to make an epoch in the science. At an early age he became president of Queens’; later, he was Lucasian Professor. He published nothing; in fact, he did no mathematical work. But as long as he kept his health, he was an active and prominent figure in Cambridge, and he maintained his enormous reputation. When he died, it was felt that the memory of such an extraordinary man should not be permitted to die out, and his papers should be published. So his papers were examined, and nothing whatever worth publishing was found.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/edYor.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/edYor.gif&quot; alt=&quot;Him&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have any of you guys heard anything more about him? What great things had he done to merit the title of the &quot;second Newton&quot; (except being a Senior Wrangler, which is of course a great title to achieve by itself)? Why didn't the man of his caliber published anything? Is there a place where I can read more about his fascinating story? Thank you in advance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Joshua_King&quot; rel=&quot;noreferrer&quot;&gt;Joshua King&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="3179" LastEditorUserId="26" LastEditDate="2015-11-01T18:27:55.650" LastActivityDate="2017-05-14T05:14:32.770" Title="Who is Joshua King?" Tags="|mathematics|biographical-details|mathematicians|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3014" PostTypeId="1" CreationDate="2015-11-01T19:34:47.173" Score="21" ViewCount="3612" Body="&lt;p&gt;Alexander Grothendieck is known to have revolutionized several areas of mathematics. His insights were very deep, original and revolutionary. Time after time he showed that he could see in ways that no one else could. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that brings up the question. How could he do that? &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it because some people are more intellectually gifted? I have not seen anyone producing tens of thousands pages of abstract deep mathematics in 1-2 decades. If you divide by time, it comes down to something like 5 pages of new highly abstract mathematics every day (a conference paper!).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Or is it because he took a special approach that most of us don't take?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Or was it because he had an anarchist upbringing? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I would like to better understand what makes some people great scientists. &#xA;I look forward to your comments. &lt;/p&gt;&#xA;" OwnerUserId="3182" LastEditorUserId="4251" LastEditDate="2021-09-21T18:58:53.443" LastActivityDate="2021-09-21T18:58:53.443" Title="Grothendieck's approach to solving problems" Tags="|mathematicians|biographical-details|algebraic-geometry|grothendieck|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3015" PostTypeId="2" ParentId="3014" CreationDate="2015-11-02T01:05:23.470" Score="18" Body="&lt;p&gt;A person's life and behavior are always shaped by a number of factors, not (in general) just one. I think it highly unlikely that any one of the three points you bring up is responsible for Grothendieck's success. At the same time, all of them may have contributed to his life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point that I find singular about Alexander Grothendieck is his overarching approach to mathematical research. As Allyn Jackson writes in &lt;a href=&quot;http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf&quot;&gt;&lt;em&gt;Comme Appelé du Néant&lt;/em&gt;—As If Summoned from the Void: The Life of Alexandre Grothendieck&lt;/a&gt; (all quotes from this work unless otherwise noted),&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Grothendieck changed the landscape of mathematics with a viewpoint that is “cosmically general”, in the words of Hyman Bass of the University of&#xA;  Michigan.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;. . .&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;He had an extremely powerful, almost other-worldly ability of abstraction that allowed him to see problems in a highly general context, and he used this ability with exquisite precision.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Grothendieck had an incredible capacity for abstraction, even more so than other leading mathematicians who studied in the same fields as he.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Allyn contrasts Grothendieck with John Forbes Nash. Nash went after specific problems that he found interesting, while Grothedieck searched for a more general framework:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If Nash is an ideal example of a problem-solver, then Grothendieck is an ideal&#xA;  example of a theory-builder.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, one clearly can't use this facet to argue that Grothendieck's approach was better than others. Nash's work in game theory revolutionized many disciplines. Grothendieck's strategy and use of abstraction simply worked for him.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-11-02T01:05:23.470" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3016" PostTypeId="1" AcceptedAnswerId="3017" CreationDate="2015-11-02T16:17:24.057" Score="4" ViewCount="589" Body="&lt;p&gt;Euclid is widely touted as the &quot;inventor&quot; of the axiomatic method, as his Elements are build around it. But there are earlier reported Elements (seemingly completely lost), and parts of the Elements are attributed to earlier mathematicians.  So the question is if he was the inventor of this approach, or just the great systematizer/organizer of earlier work, which his Elements so thoroughly incorporated that it all but faded from view?&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2015-11-03T14:41:04.147" Title="What is the origin of the axiomatic method?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3017" PostTypeId="2" ParentId="3016" CreationDate="2015-11-02T16:58:52.207" Score="5" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Heath (editor), &lt;a href=&quot;https://books.google.it/books?id=lKulMgEACAAJ&quot;&gt;The Thirteen Books of Euclid's Elements&lt;/a&gt; (Dover reprint, Volume 1) : &lt;strong&gt;Introduction, Ch.IX, §2 : ELEMENTS ANTERIOR TO EUCLID'S&lt;/strong&gt;, page 116 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;for references to earlier &lt;em&gt;Elements&lt;/em&gt; now lost :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hippocrates_of_Chios&quot;&gt;Hippocrates of Chios&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Leon_(mathematician)&quot;&gt;Leon&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus&quot;&gt;Eudoxus of Cnidus&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Theudius&quot;&gt;Theudius of&#xA;  Magnesia&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Heath, &lt;a href=&quot;https://books.google.it/books?id=J-kaAgAAQBAJ&amp;amp;pg=PA319&quot;&gt;A History of Greek Mathematics. Volume I&lt;/a&gt; (1921), page 319, lists :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[the] names mentioned in the summary of &lt;a href=&quot;https://en.wikipedia.org/wiki/Proclus&quot;&gt;Proclus&lt;/a&gt; [in his influential commentary on the first book of Euclid's &lt;em&gt;Elements&lt;/em&gt;, §66] :&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&quot;Younger than &lt;a href=&quot;https://en.wikipedia.org/wiki/Leodamas_of_Thasos&quot;&gt;Leodamas&lt;/a&gt; were Neoclides and his pupil &lt;a href=&quot;https://en.wikipedia.org/wiki/Leon_(mathematician)&quot;&gt;Leon&lt;/a&gt;, who added many things to what was known before their time, so that Leon was actually able to make a collection of the elements more carefully designed in respect both of the number of proposition proved and of their utility, besides which he invented &lt;em&gt;diorismi&lt;/em&gt; (the object of which is to determine) when the problem under investigation is possible of solution and when impossible.&quot;&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;[...] &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus&quot;&gt;Eudoxus of Cnidos&lt;/a&gt; [...]was the first to increase the number of the so-called general theorems [...]. Amyclas [more correctly Amyntas] of Heraclea [...] &lt;a href=&quot;https://en.wikipedia.org/wiki/Menaechmus&quot;&gt;Menaechmus&lt;/a&gt;, a pupil of Eudoxus [...] and &lt;a href=&quot;https://en.wikipedia.org/wiki/Dinostratus&quot;&gt;Dinostratus&lt;/a&gt;, his brother, made the whole of geometry still more perfect. &lt;a href=&quot;https://en.wikipedia.org/wiki/Theudius&quot;&gt;Theudius of Magnesia&lt;/a&gt; [...] put together the elements admirably and made many partials theorems more general.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See in the complete &lt;a href=&quot;https://books.google.it/books?id=JZEHj2fEmqAC&amp;amp;pg=PA53&quot;&gt;translation of Proclus&lt;/a&gt;, &lt;strong&gt;Prologue&lt;/strong&gt;, §65-on, Proclus' historical account, from &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudemus_of_Rhodes&quot;&gt;Eudemus of Rhodes&lt;/a&gt;' history of Greek mathematics, concluding with  :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of&#xA;  Theaetetus, and putting in irrefutably demonstrable form propositions&#xA;  that had been rather loosely established by his predecessors.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In conclusion, you are right :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he was &lt;strong&gt;not&lt;/strong&gt; the inventor of this approach, but just the great systematizer/ organizer of earlier work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-11-03T14:41:04.147" LastActivityDate="2015-11-03T14:41:04.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3018" PostTypeId="1" CreationDate="2015-11-02T21:16:42.563" Score="4" ViewCount="148" Body="&lt;p&gt;In the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schiehallion_experiment#Surveying&quot; rel=&quot;nofollow&quot;&gt;Schiehallion experiment&lt;/a&gt; (&lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/68/689&quot; rel=&quot;nofollow&quot;&gt;original paper&lt;/a&gt;), Charles Hutton computed the density of the Earth, and from there estimated the densities of the major bodies of the Solar System.  His numbers for most bodies were within 30% of currently-accepted values, with the exception of Mercury, which he over-estimated by 70%.  Why was he so far off?&lt;/p&gt;&#xA;" OwnerUserId="3186" LastActivityDate="2015-11-02T21:56:11.943" Title="Why was Charles Hutton's estimate of the density of Mercury so far off?" Tags="|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3019" PostTypeId="2" ParentId="3018" CreationDate="2015-11-02T21:56:11.943" Score="1" Body="&lt;p&gt;I think you may be misinterpreting the WP article. Although I could be wrong, this is too long for a comment, so I'll write it as an answer. It looks to me like Lalande found the relative masses of various bodies in the solar system, but this was before Cavendish, so Lalande had no way to express these masses in terrestrial units such as kilograms. I think Hutton simply found the conversion factor between Lalande's astronomical units and terrestrial units of mass. This was a single number, which he then applied to every body in the solar system. So the poor precision in the case of Mercury is to be blamed on Lalande, not Hutton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is then why Lalande got Mercury's mass wrong by so much. This seems not very surprising. Mercury doesn't have any moons, so its mass can only be estimated by looking for its perturbing effect on the other planets. But because Mercury's mass is small, that perturbation would be quite small and hard to measure.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2015-11-02T21:56:11.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3020" PostTypeId="1" CreationDate="2015-11-03T06:42:42.197" Score="5" ViewCount="214" Body="&lt;p&gt;&quot;Why is the sky blue&quot; is a question that everybody seems to know, and in modern times is associated with children's innate curiosity. If I casually flip through a few of the children's science books I have, there's always some mention of it. Of course, one could argue that simply because the sky exists, and is blue, the question must have existed forever. However, the question has become so popular, that personally I believe I must have heard the question from other sources before I had even asked it myself. So my question is, who was the person who popularized the question (through science broadcasting perhaps), or who published an attempt to explain why the sky is blue? &lt;/p&gt;&#xA;" OwnerUserId="3188" LastEditorUserId="26" LastEditDate="2015-11-03T14:12:55.440" LastActivityDate="2015-12-30T08:04:51.740" Title="Who popularized the question &quot;why is the sky blue?&quot;" Tags="|physics|social-context|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3021" PostTypeId="2" ParentId="3013" CreationDate="2015-11-03T13:33:22.473" Score="3" Body="&lt;p&gt;For a profile of Joshua King as Cambridge professor and &quot;administrator&quot;, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kevin C.Knox &amp;amp; Richard Noakes (editors), &lt;a href=&quot;https://books.google.it/books?id=QGX_rAeia4kC&amp;amp;pg=PA302&quot; rel=&quot;nofollow&quot;&gt;From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics&lt;/a&gt; (2003), &lt;strong&gt;Ch.7 Arbiters of Victorian science : G.G.Stokes and Joshua King&lt;/strong&gt; by David B.Wilson, page 295-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-11-03T13:33:22.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3023" PostTypeId="2" ParentId="2927" CreationDate="2015-11-03T16:50:50.340" Score="3" Body="&lt;p&gt;Another great contribution to astronomy came from al-Sūfī (903-986, better known as Azophi in the West) with his book &lt;em&gt;Book on the Constellations&lt;/em&gt; (&lt;a href=&quot;http://www.wdl.org/en/item/2484/&quot; rel=&quot;nofollow&quot;&gt;Here is a copy of his book&lt;/a&gt;). In his book, he gave details of more than a thousand stars based on Ptolemy's &lt;em&gt;Almagest&lt;/em&gt; and his own observations and provided beautiful illustrations of constellations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Book on the Constellations&lt;/em&gt; was very influential in especially the naming of the stars. It was translated into Latin and used by many Muslim scholars as well as Europeans, such as Albrecht Dürer (1515), Peter Apian, J.A. Colom (1635), Thomas Hyde,  Giuseppe Piazzi, Giovanni Riccioli (1651), V. Coronelli (1665). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://link.springer.com/referenceworkentry/10.1007/978-0-387-30400-7_1346&quot; rel=&quot;nofollow&quot;&gt;The Biographical Encyclopedia of Astronomers, p 1110&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2428" LastEditorUserId="2428" LastEditDate="2015-11-03T17:31:35.163" LastActivityDate="2015-11-03T17:31:35.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3024" PostTypeId="1" AcceptedAnswerId="3025" CreationDate="2015-11-03T23:19:42.133" Score="11" ViewCount="4945" Body="&lt;p&gt;Who discovered the general rule for differentiating polynomials, in particular that the derivative of $x^n$ is $n x^{n-1}$, and when? I appreciate the answer may not be a clear-cut individual and year, perhaps more a succession of deeper geometric and notational insights. The key point I'm looking for, though, is a dawning awareness that $nx^{n-1}$ represents a &lt;em&gt;general&lt;/em&gt; rule for the gradient of tangents to polynomials of arbitrary degree, not just &lt;em&gt;specific&lt;/em&gt; results for the parabola and cubic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not so concerned &lt;em&gt;linearity&lt;/em&gt;, which allows us to differentiate polynomials with several terms, nor at what point the discovery was &lt;em&gt;proven&lt;/em&gt; (to whatever degree of rigour), but if someone wants to include such detail in the answer for completeness that would, of course, be welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is generally known as the &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Power_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;power rule&lt;/a&gt;&quot;, though that might refer to both the integral or derivative of a polynomial. Wikipedia suggests that the power rule for the derivative was discovered by Newton and Leibniz, though that would surprise me. Fermat had clearly explored similar territory &amp;mdash; &lt;a href=&quot;http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this student project&lt;/a&gt; gives some details, but sadly lacks citations, particularly for its key claim that Fermat extended his method to show that the tangent to $y=x^n$ had gradient $n x^{n-1}$. &lt;a href=&quot;https://en.wikipedia.org/wiki/Isaac_Barrow&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isaac Barrow&lt;/a&gt; also investigated tangents extensively, using infinitesimal triangles that presaged subsequent developments more closely than Fermat's method, and  &lt;a href=&quot;https://en.wikipedia.org/wiki/Johannes_Hudde&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johannes Hudde&lt;/a&gt; worked in this area too, though I know less about his work. His work on polynomials, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hudde%27s_rules&quot; rel=&quot;nofollow noreferrer&quot;&gt;particularly that a double-root of a polynomial is also a root of what we would call its derivative, and that a maximum or minimum value occurs at a root of the derivative&lt;/a&gt;, comes startlingly close to what I am looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have sometimes seen the power rule for differentiation called &quot;Wallis'[s] rule&quot; or &quot;Wallis'[s] law&quot;, which also suggests an earlier date. However, I have also occasionally seen that phrase attached to &lt;a href=&quot;https://en.wikipedia.org/wiki/Cavalieri%27s_quadrature_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cavalieri's quadrature formula&lt;/a&gt; instead &amp;mdash; which would be fair enough, since according to Wikipedia, Wallis extended Cavalieri's discovery (in modern notation, that $\int_0^a x^n \mathrm{d}x = \frac{1}{n+1}a^{n+1}$, for $n \in \mathbb{N}$) to rational and negative indices $n$ in his work &lt;em&gt;Arithmetica Infinitorum&lt;/em&gt;. Effectively Wallis had filled in many of the details of the power rule for integration, &lt;a href=&quot;https://hsm.stackexchange.com/q/2463/3191&quot;&gt;though the exceptional case $n=-1$ was dealt with by others&lt;/a&gt;. Of course, once we are armed with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fundamental Theorem of Calculus&lt;/a&gt; then the power rules for integration and differentiation are singing the same song, but before this development, the fact somebody could (in modern parlance) integrate $x^n$ does not mean they could differentiate it (or as they would likely see it, find its tangent). Did Wallis know &lt;em&gt;both&lt;/em&gt; forms of the power rule?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for &quot;Cavalieri's&quot; formula for integration, I wouldn't be surprised if the answer to my question about differentiation is somewhat different for the separate cases of $n$ being a positive integer, negative integer or rational number.&lt;/p&gt;&#xA;" OwnerUserId="3191" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2019-09-13T00:46:21.307" Title="Who discovered the power rule for derivatives?" Tags="|mathematics|discoveries|calculus|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3025" PostTypeId="2" ParentId="3024" CreationDate="2015-11-04T04:28:40.767" Score="12" Body="&lt;p&gt;Cavalieri was presumably the first to state the &quot;power rule&quot; for areas under parabolas with positive integer exponents, but he only derived it up to &lt;span class=&quot;math-container&quot;&gt;$n=4$&lt;/span&gt;, beyond that his methods became intractable. &quot;Presumably&quot;, because the timeline is spotty due to absence of publications, and Fermat might have done it earlier. In any case Fermat was likely the first to generalize it to rational exponents somewhere between 1635 and 1643. Toricelli, Roberval and Wallis arrived at the same result independently by mid 1640-s. There were no integrals at the time of course, so the rule was expressed and derived in geometric terms, although some derivations, especially Fermat's, anticipated the ideas of Riemann integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for derivatives the name &quot;power rule&quot; is somewhat more tenuous before Newton, who derived it in the familiar form in De Analysi (1669) using fluxions. Since antiquity the tradition was to characterize tangents in terms of so-called subtangents rather than slopes, which in modern terms are segments connecting the abscissa to the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-intercept of the tangent. Toricelli considered parabolas with positive integer exponents, and &quot;hyperbolas&quot; &lt;span class=&quot;math-container&quot;&gt;$x^my^n=k$&lt;/span&gt; derived their subtangents in terms of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, which amounts to the power rule for slopes if we help ourselves to the modern notation. But the most interesting part is &lt;em&gt;how&lt;/em&gt; he did it. It was not through Fermat's infinitesimals or &lt;a href=&quot;https://hsm.stackexchange.com/questions/1963/is-there-a-lost-calculus/1968#1968&quot;&gt;Descartes-Hudde algebraic calculus&lt;/a&gt;, but through kinematic description of the curves as compositions of motions using the parallelogram of velocities that goes back to Aristotle and Archimedes. Archimedes possibly found the tangent to his spiral using such kinematic description, and Toricelli was of course familiar with such description of the ordinary parabola due to his teacher Galileo. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through Barrow this approach influenced Newton, and became the foundation of his calculus of fluxions. In later works Newton defined quantities as traced by motions, which allowed him to define &quot;first and last ratios&quot; kinematically, without infinitesimals. So Toricelli was Newton's precursor in the same way that Fermat was Leibniz's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The emergence of the general power rule is discussed in detail in &lt;a href=&quot;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment/page/n1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyer's History of Calculus&lt;/a&gt;, pp.120-172, with references to the original sources.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10194" LastEditDate="2019-09-13T00:46:21.307" LastActivityDate="2019-09-13T00:46:21.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3027" PostTypeId="1" AcceptedAnswerId="3033" CreationDate="2015-11-04T13:10:52.943" Score="3" ViewCount="1623" Body="&lt;p&gt;There are many &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_scientific_units_named_after_people&quot; rel=&quot;nofollow&quot;&gt;scientific units named after people&lt;/a&gt;, of which most were given their names a relatively long time ago (Watt, Newton, Celsius, etc).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if the scientists who &quot;discovered&quot; these units/laws usually get to know that their name is being used in such a way, i.e. if it is common to name the discovery while the person is still alive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course I expect there to be many individual cases, but maybe there is some kind of standard or guideline when it comes to naming units. These are the cases I can think of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The scientist names the unit after himself. This rarely happens: &lt;a href=&quot;https://hsm.stackexchange.com/questions/691/what-famous-laws-were-named-by-their-discoverer&quot;&gt;What famous laws were named by their discoverer?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The unit is given its name by someone else to honor the discoverer who is  still alive.&lt;/li&gt;&#xA;&lt;li&gt;The unit is given its name by someone else (a long time) after the discoverer has died. Maybe to honor the person, or just because nobody could come up with a better name.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;How is this usually handled, if there is any usual way at all?&lt;/p&gt;&#xA;" OwnerUserId="3196" LastActivityDate="2017-02-24T14:34:45.630" Title="When are units that are named after persons given their names?" Tags="|discoveries|naming-conventions|units|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3028" PostTypeId="2" ParentId="2143" CreationDate="2015-11-04T14:19:32.110" Score="1" Body="&lt;p&gt;In this following link I've found some mathematical argument, and some of those are really interesting to solve our this doubt. Interested people are welcome.&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/854380/how-to-distinguish-walking-on-a-sphere-or-on-a-torus&quot;&gt;https://math.stackexchange.com/questions/854380/how-to-distinguish-walking-on-a-sphere-or-on-a-torus&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="2103" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2015-11-04T14:27:25.133" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3030" PostTypeId="2" ParentId="3027" CreationDate="2015-11-05T16:21:18.440" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/SI_derived_unit&quot; rel=&quot;nofollow&quot;&gt;HERE&lt;/a&gt; is a listing of the SI derived units.  As far as I know, none of these were someone naming it for himself.  If your name is &lt;em&gt;Henry Gray&lt;/em&gt; then you are a &lt;em&gt;double unit&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-11-05T16:21:18.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3031" PostTypeId="1" CreationDate="2015-11-06T04:20:50.877" Score="4" ViewCount="142" Body="&lt;p&gt;Sometime in 1980 George Polya gave a lecture at the University of Minnesota about solutions of algebraic equations that have symmetry in the appearance of the variables in the equation (any permutation of the variables resulted in the same system).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My recall is far from complete, but I suspect he began with a 2-dimensional system like x + y = 1 and $x^2 + y^2 = 4$.  Simple algebra without computational error shows this system has solutions (x,y) = $([1+\sqrt{7}]/2) , [1-\sqrt{7}]/2)$ and (x,y) = $([1-\sqrt{7}]/2) , [1+\sqrt{7}]/2)$.  The solution sets for x and for y are equal.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He generalized to a class of 2-dimensional systems whose solution sets for each variable were the same set.  I recall thinking his conclusions were obvious (my arrogance and/or my ignorance).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He repeated in three dimensions, and then in four, while I wondered why I had come to the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Polya, in his eighties at the time, was still a master.  Eventually he displayed a system (in six or seven dimensions) with the required symmetry, in which the solution sets for some of the variables were not equal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume he gave this lecture elsewhere.  Does anyone know more details?&lt;/p&gt;&#xA;" OwnerUserId="2960" LastEditorUserId="2960" LastEditDate="2015-11-07T17:57:32.087" LastActivityDate="2015-11-07T17:57:32.087" Title="A Lecture by Polya on Symmetric Algebraic Equations with an Unexpected Conclusion" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3032" PostTypeId="1" CreationDate="2015-11-06T15:21:08.080" Score="4" ViewCount="86" Body="&lt;p&gt;There are various different situations in mathematics in which, in order to find a fixed point of $f$, we iterate $f$ on an arbitrary starting value, and show that the process must converge to a fixed point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the earliest known instance of this line of reasoning? I'm looking for examples in which the iteration and the convergence to a fixed point was &quot;explicit&quot;, in other words, where the mathematician was consciously aware of what they were doing.&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="50" LastEditDate="2015-11-06T18:00:54.047" LastActivityDate="2015-11-06T22:04:45.197" Title="What is the earliest result saying that the limit of $x_{n+1}=f(x_n)$ must be a fixed point of $f$?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3033" PostTypeId="2" ParentId="3027" CreationDate="2015-11-06T19:37:54.313" Score="3" Body="&lt;p&gt;Much of the personal naming occured in 1860-70s for CGS and 1870-1880s for SI, see &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_metric_system#Development_of_a_coherent_metric_system&quot; rel=&quot;nofollow&quot;&gt;History of the Metric System&lt;/a&gt;. Most of the units named referred to deceased people: newtons, volts, ohms, amperes, pascals, coulombs, watts, etc. Some passed away a while ago, like Pascal and Newton, some relatively recently, like Ampere and Ohm. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Farads, joules and webers might have entered discussions while their name bearers were still around, but by the time the names were officially recommended for adoption (informal adoption began earlier) in 1873 for CGS and 1893 for SI they passed away as well (Faraday in 1867, Joule in 1889 and Weber in 1891). None were self-named.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not aware of a formal rule but the tradition seems to be that you do have to die to get a unit to your name. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-11-06T19:45:44.673" LastActivityDate="2015-11-06T19:45:44.673" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3034" PostTypeId="2" ParentId="3032" CreationDate="2015-11-06T22:04:45.197" Score="2" Body="&lt;p&gt;It is always difficult, in most cases impossible, to answer the question of the type &quot;who was the first&quot;, especially who was the first to discover some trivial thing, which everyone can see once s/he looks at it. By &quot;trivial&quot; in this case I mean that once you start considering iterations&#xA;of a continuous function, you immediately discover that they can converge only to a fixed point. Thus a more reasonable question would be &quot;who was the first to consider infinite iteration of continuous functions&quot;. The earliest instance that I know is Euler's paper&lt;/p&gt;&#xA;&#xA;&lt;p&gt;De formulis exponentialibus replicatis. Acta Acad. Petropolitanae 1 (1777), 38–60.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where he considers the existence of the limit&#xA;$$x^{x^{x^{...}}}$$&#xA;This can be easily reformulated as iteration of exponential function.&#xA;Euler himself says in the paper that he follows the remarkable work of Marquis de Condorcet. But the paper of Condorcet does not address the question of convergence, he just iterates some formal expressions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But of course I cannot prove that &quot;Euler was the first&quot;, and whatever other answer to this question is given, it can never be proved. That's why I think this is a meaningless question, as all questions of this type are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 1. Of course, Babylonians had an algorithm of extracting the square root,&#xA;which is equivalent to the Newton method for the equation $x^2-a=0$.&#xA;But to say that Babylonians were considering &quot;convergence to a fixed point of iterations of a continuous function&quot; would be a ridiculous anachronism.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler at least understood the words &quot;function&quot;, &quot;limit&quot; and &quot;fixed point&quot; more or less as&#xA;we understand them now, though not exactly in the same way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 2.&#xA;Perhaps it is interesting to mention that Euler found when the limit exists for real $x$. In the subsequent centuries there was a great interest to these problems in complex domain, and complete understanding of possible limits was achieved only in 2009 (J. Hubbard, D. Schleicher and M. Shishikura &#xA;Journal: J. Amer. Math. Soc. 22 (2009), 77-117).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-06T22:04:45.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3035" PostTypeId="1" AcceptedAnswerId="5044" CreationDate="2015-11-07T02:45:14.020" Score="4" ViewCount="279" Body="&lt;p&gt;Saunders Mac Lane published his thesis (&lt;em&gt;Abgekuerzte Beweise in Logikkalkul&lt;/em&gt;) which anticipated to some degree constructing proofs of theorems by constructing programs, giving a sufficient input, and proving the program would prove the theorem if executed, but not actually following out or executing the program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Did anybody cite his thesis, prior the late 1970's? What is the first edition, if it was printed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Mac Lane published a note in english (prior to coming back to the USA) that communicated his ideas to those who did not read his dissertation? What is the reference? And did anybody cite &lt;em&gt;it&lt;/em&gt; prior the late 1970's?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He writes in his autobiography, and a chapter after the reprint, that it seemed to have no influence but anticipated much of the subject. Perhaps some mathematicians did read it but he was unaware they did?&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2016-07-25T06:26:56.307" LastActivityDate="2016-07-25T06:26:56.307" Title="History of Mac Lane's dissertation on abbreviated proving" Tags="|mathematics|mathematicians|publishing|mathematical-logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3036" PostTypeId="2" ParentId="3014" CreationDate="2015-11-07T08:30:33.220" Score="6" Body="&lt;p&gt;This citation, form Grothendieck himself, to me shows a little bit why, a part from him being exceptionally gifted, his approach to problems was radically different. He describes the process of solving a math problem as that of opening a nut:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*The ... analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more ﬂexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!&#xA;A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet it finally surrounds the resistant substance.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this paper you may find some description of the intellectual environment and the ideas that allowed such approach to problems to work incredibly well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cwru.edu/artsci/phil/RisingSea.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.cwru.edu/artsci/phil/RisingSea.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2015-11-07T08:30:33.220" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3037" PostTypeId="1" CreationDate="2015-11-08T14:34:54.137" Score="10" ViewCount="1328" Body="&lt;p&gt;Who first proposed the distinctive shapes and the rectangular shape labels?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the consideration behind the shapes and the labels?&lt;/p&gt;&#xA;" OwnerUserId="629" LastEditorUserId="62" LastEditDate="2015-11-08T18:07:01.923" LastActivityDate="2020-03-23T19:37:44.230" Title="The origin of logic gate symbols" Tags="|computer-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3038" PostTypeId="1" CreationDate="2015-11-08T01:56:58.297" Score="1" ViewCount="148" Body="&lt;p&gt;Epistemology and philosophy of science reference intellectual, sagacious, scholarly vanguards such as &lt;a href=&quot;http://en.wikipedia.org/wiki/Michael_Servetus&quot; rel=&quot;nofollow&quot;&gt;Michael Servetus&lt;/a&gt;  (who correctly explicated pulmonary circulation) and &lt;a href=&quot;http://www.reddit.com/r/todayilearned/comments/31egn3/til_of_ignaz_semmelweis_who_made_doctors_wash&quot; rel=&quot;nofollow&quot;&gt;Dr Ignaz Semmelweis&lt;/a&gt; (who correctly advocated for handwashing) wrongfully condemned and contemned for averring their beliefs 'ahead of their time'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wish to learn more about such vanguards; so are there any books or lists or resources? &lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastEditorUserId="4251" LastEditDate="2020-07-13T11:29:20.167" LastActivityDate="2020-07-15T05:59:32.673" Title="Where can I learn more about intellectuals who were wrongfully harmed for their correct theories?" Tags="|scientists|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3039" PostTypeId="2" ParentId="3038" CreationDate="2015-11-09T04:47:29.867" Score="2" Body="&lt;p&gt;One of the famous incident is of &lt;strong&gt;Giordano Bruno&lt;/strong&gt; who was burned at the stake in $1600$ in Rome for his theories about the Universe. He said that stars are just distant SUNs and may have their own planetary systems which probably holds intelligent life also. Also he said that there is no preferable center of the Universe. This ideas were true but against the many religious beliefs of that time. &lt;strong&gt;Galileo&lt;/strong&gt; was also a victim of the wrong beliefs that the society had. He was kept as house arrest for opposing several ideas including geocentric astronomical system.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2015-11-09T04:47:29.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3040" PostTypeId="1" AcceptedAnswerId="3046" CreationDate="2015-11-09T09:11:03.677" Score="2" ViewCount="1522" Body="&lt;p&gt;I am not here to ask why &quot;minus times minus is plus&quot;, this is a basic arithmetic fact. The related question most people ask is: &lt;em&gt;why&lt;/em&gt; does &lt;span class=&quot;math-container&quot;&gt;$-\times-=+$&lt;/span&gt;. Of, course there may be several explanations for this fact. But I want to know who first wrote down that particular rule? Is there any historical evidence in some papers or books where it appeared for the first time? &lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="9499" LastEditDate="2019-08-16T23:39:14.647" LastActivityDate="2019-08-16T23:39:14.647" Title="Who wrote down minus times minus is equal to plus?" Tags="|mathematics|terminology|notation|mathematical-logic|" AnswerCount="2" CommentCount="2" ClosedDate="2015-11-12T02:14:58.010" ContentLicense="CC BY-SA 4.0" />
  <row Id="3041" PostTypeId="2" ParentId="3038" CreationDate="2015-11-09T14:28:30.130" Score="4" Body="&lt;p&gt;I would just like to point out a debatable element of selection in the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you say &quot;correct theory,&quot; you are implicitly seeking &quot;scientists&quot; martyred by religious authorities, which is our modern view of the matter. Hypatia, Giordano Bruno, and Galileo are the &quot;poster-children&quot; for this view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One would never make this sort of claim for Jesus, St. Paul, Socrates, Robespierre, the Paris Communards, or the many socialists killed in counter-revolutionary actions. They did not have what we would call &quot;correct theories.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other cases are borderline. What about Newton's attacks on Leibniz, whose calculus notation in fact proved superior? Were Cantor and Boltzmann driven to their deaths for scientific nonconformism? What about the various scientists suppressed by Stalin and Lysenko or by the McCarthy hearings?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, one of the first &quot;scientific martyrs&quot; reveals the antiquity of the rift and the early divorce between the sacred and the secular. This was the purported slaying of Hippasus by the Pythagoreans for revealing proofs of the irrationals. The Pythagoreans were, in turn, slain for their cultish hoarding of knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But let's return to Galileo, the most famous case. To be gratuitously provocative, one can make the case the Galileo was, in fact, persecuted by the Church for being insufficiently &quot;scientific.&quot; He claimed his solar-centric views were &quot;the truth,&quot; whereas the Church had allowed Copernican views, as long as they were explicitly qualified as &quot;hypothesis only.&quot; In this respect, one could say the Church was closer to strict scientific methodology.         &lt;/p&gt;&#xA;" OwnerUserId="3212" LastActivityDate="2015-11-09T14:28:30.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3042" PostTypeId="2" ParentId="3040" CreationDate="2015-11-09T18:35:55.503" Score="2" Body="&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Negative_number&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-11-09T18:35:55.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3044" PostTypeId="1" CreationDate="2015-11-10T03:21:26.167" Score="0" ViewCount="49" Body="&lt;p&gt;I'm trying to find the original papers describing Young's double slit experiments including his experimental set-ups and findings. I've tried looking and all I can find is &lt;a href=&quot;http://www.biodiversitylibrary.org/item/63005#page/498/mode/1up&quot; rel=&quot;nofollow&quot;&gt;http://www.biodiversitylibrary.org/item/63005#page/498/mode/1up&lt;/a&gt; and &lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/92/12&quot; rel=&quot;nofollow&quot;&gt;http://rstl.royalsocietypublishing.org/content/92/12&lt;/a&gt; But neither are papers describing the experiments.&lt;/p&gt;&#xA;" OwnerUserId="3219" LastEditorUserId="2891" LastEditDate="2015-11-10T03:44:30.240" LastActivityDate="2015-11-10T03:44:30.240" Title="Original paper for Young's double slit experiments" Tags="|physics|reference-request|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-11-10T12:27:18.967" ContentLicense="CC BY-SA 3.0" />
  <row Id="3045" PostTypeId="1" AcceptedAnswerId="3737" CreationDate="2015-11-10T07:46:00.220" Score="1" ViewCount="899" Body="&lt;p&gt;Despite this is being a very trivial question, I would like to know who was the first (with reference evidence) to show that the following quintic equation has five radical roots: &lt;span class=&quot;math-container&quot;&gt;$$x^5 + x + 1 = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;What brings up this question is that &amp;quot;the radical real root for this specific example is impossible to be contained in a much wider range formula given as an independent following question for any polynomial degree as a trinomial polynomial (with all coefficients are zeros or (+/-) one (for only three terms),&lt;/p&gt;&#xA;&lt;p&gt;however, that question which is an answer itself may not have been received well enough by some mathematicians mentioning that I'm not a professional mathematician which makes hard enough to convey the simplest idea to the professionals in this field&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Is this a NEW CORRECT One real approximated root in a rational form of the following &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th degree polynomial equation &lt;span class=&quot;math-container&quot;&gt;$$ x^n + x^m = 1,$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$(n &amp;gt; m)$&lt;/span&gt; are two distinct positive integers, is given by the following series formula as an approximation where no real root ever exists exactly as we may imagine with our imaginative roots that are strictly associated or derived from that real root that never exists&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;x&#xA;&amp;amp;=  1-\frac{1}{n}+\sum_{k=2}^N \frac{(-1)^k \prod_{i=1}^{k-1}(km-in+1)}{k!n^k} \\&#xA;&amp;amp;= 1-\frac{1}{n}+\frac{2m-n+1}{2!n^2}-&#xA;\frac{(3m-2n+1)(3m-n+1)}{3!n^3}&#xA;+\frac{(4m-3n+1)(4m-2n+1)(4m-n+1)}{4!n^4}&#xA;-\frac{(5m-n+1)(5m-2n+1)(5m-3n+1)(5m-4n+1)}{5!n^5}&#xA;+\dotsb&#xA;\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is a relatively small suitable chosen integer up to our practical needs and  capability of making our real roots more sensible (only in rational form) for any other purpose since it is absolutely and perpetually impossible to make the sum of our terms go with no number like infinity since then our real root in our minds would simply become like a ratio of two no existing numbers (each with an uncountable number of digits) which isn't any number, especially that root also is impossible to be constructed exactly by any means&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note that in very rare cases, the above trinomial polynomial has real existing roots in constructible number form and strictly in their surd form (and never in their approximations in decimal or rational forms),  when &lt;span class=&quot;math-container&quot;&gt;$(n = 2m)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is a power of two&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;REFERENCE&lt;/strong&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quora.com/What-are-the-ways-to-understand-the-proof-that-there-is-no-formula-for-expressing-the-roots-of-the-general-quintic-equation-via-radicals/answer/Bassam-Karzeddin-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/What-are-the-ways-to-understand-the-proof-that-there-is-no-formula-for-expressing-the-roots-of-the-general-quintic-equation-via-radicals/answer/Bassam-Karzeddin-1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For a reference to the more general solution of this Trinomial equation &lt;span class=&quot;math-container&quot;&gt;$$ax^n +bx^m +c = 0$$&lt;/span&gt; click, OK, then click National then type in the search button &amp;quot;solution of equations by power series&amp;quot; at this link  &lt;a href=&quot;http://opac.nl.gov.jo/uhtbin/cgisirsi.exe/zJ7a8B3i0n/MAIN/263930002/123&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://opac.nl.gov.jo/uhtbin/cgisirsi.exe/zJ7a8B3i0n/MAIN/263930002/123&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="26" LastEditDate="2021-02-20T15:22:09.377" LastActivityDate="2021-02-21T01:46:58.480" Title="Who was the first to show that this quintic equation has five radical roots?" Tags="|mathematics|number-theory|elementary-algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3046" PostTypeId="2" ParentId="3040" CreationDate="2015-11-10T08:37:44.647" Score="2" Body="&lt;p&gt;For &lt;a href=&quot;https://en.wikipedia.org/wiki/Diophantus&quot; rel=&quot;nofollow&quot;&gt;Diophantus&lt;/a&gt; (born probably sometime between AD 201 and 215; died aged 84, probably sometime between AD 285 and 299) &lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmetica&quot; rel=&quot;nofollow&quot;&gt;Arithmetica&lt;/a&gt;, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Heath, &lt;a href=&quot;https://books.google.it/books?id=jbY8AAAAIAAJ&amp;amp;pg=PA130&quot; rel=&quot;nofollow&quot;&gt;Diophantus of Alexandria : A Study in the History of Greek Algebra&lt;/a&gt; (1910), page 140 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A minus multiplied by a minus makes a plus; a minus multipliedby a plus makes a minus ; and the sign of a minus is a truncated $\Psi$ turned upside down, thus $\Lambda$ [with a third central leg]. [&lt;em&gt;Footnote&lt;/em&gt;. The literal rendering would be &quot;A wanting multiplied by a wanting makes a forthcoming.&quot;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Heath, &lt;a href=&quot;https://books.google.it/books?id=41s9fPYjxJQC&amp;amp;pg=PA459&quot; rel=&quot;nofollow&quot;&gt;A History of Greek Mathematics. Volume II&lt;/a&gt; (1921), page 459 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For subtraction alone is a sign used. The full term for &lt;em&gt;wanting&lt;/em&gt; is λείψις  as opposed to ύπαρξις, a &lt;em&gt;forthcoming&lt;/em&gt;, which denotes a &lt;em&gt;positive&lt;/em&gt; term.  The symbol used to indicate &lt;em&gt;wanting&lt;/em&gt;, corresponding to our sign for &lt;em&gt;minus&lt;/em&gt;, is $\Lambda$ [with a third central leg]，which is described in the text as a ‘$\Psi$ turned downwards and truncated’.The description is evidently &#xA;  interpolated, and it is now certain that the sign has nothing to do with $\Psi$.Nor is it confined to Diophantus, for it appears in practically the same form in Heron's &lt;em&gt;Metrica&lt;/em&gt; [...]. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Clearly, also if the rule is &quot;correctly described&quot;, there is no hint to its modern symbolic formulation for lack of the modern symbols, that dates to the late 16th- early 17th Centuries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the modern version, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Wallis&quot; rel=&quot;nofollow&quot;&gt;John Wallis&lt;/a&gt; (23 November 1616 – 28 October 1703), &lt;a href=&quot;https://books.google.it/books?id=EuzpN1t5SOcC&amp;amp;pg=PA78&quot; rel=&quot;nofollow&quot;&gt;A Treatise of Algebra&lt;/a&gt; (1685), page 78-79 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Unde patet ratio tum hujus regulae, $+$ in $+$ facit $+$; tum hujus $-$ in $+$ facit $-$. [...] Indeque patet ratio tum hujus regulae, $+$ in $-$ facit $-$; tum hujus, $-$ in $-$ facit $+$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;justification&quot; is that to &lt;em&gt;multiply&lt;/em&gt; a given quantity by a &lt;em&gt;positive&lt;/em&gt; factor is &lt;em&gt;ponendi&lt;/em&gt; (&quot;ubi $+2$ significat &lt;em&gt;bis ponere&lt;/em&gt;&quot;) while to &lt;em&gt;multiply&lt;/em&gt; it by a &lt;em&gt;negeative&lt;/em&gt; factor is &lt;em&gt;tollendi&lt;/em&gt; (&quot;ibidem $-2$ est &lt;em&gt;bis tollere&lt;/em&gt;, seu &lt;em&gt;bis ponere contrarium&lt;/em&gt;&quot;).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Contra vero, $-A$ per $-2$ multiplicare, est bis tollere $-A$, seu defectum $-A$ bis supplere, quod est $+A$ bis ponere, facitque $+2A$, (adeoque $-$ in $-$ facit $+$.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-11-10T14:08:15.733" LastActivityDate="2015-11-10T14:08:15.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3048" PostTypeId="1" CreationDate="2015-11-10T09:36:47.873" Score="9" ViewCount="631" Body="&lt;p&gt;As a follow up question with respect to : &lt;a href=&quot;https://hsm.stackexchange.com/questions/3040/who-wrote-down-minus-times-minus-is-equal-to-plus&quot;&gt;Who wrote down minus times minus is equal to plus?&lt;/a&gt; and to : &lt;a href=&quot;https://hsm.stackexchange.com/questions/2631/historically-how-did-people-define-multiplication-for-negative-numbers&quot;&gt;Historically, how did people define multiplication for negative numbers?&lt;/a&gt;, it can be interesting to trace the first &amp;quot;modern&amp;quot; &lt;em&gt;justification&lt;/em&gt; for the rule :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;minus&lt;/em&gt; multiplied by a &lt;em&gt;minus&lt;/em&gt; makes a &lt;em&gt;plus&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If it is quite intuitive that &amp;quot;if you have a debt of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; people, then you have a debt of &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;&amp;quot; (we can justify it reducing &lt;em&gt;multiplication&lt;/em&gt; to repeated &lt;em&gt;sum&lt;/em&gt;) it is not so easy to imagine a debt of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; by &amp;quot;minus&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; people.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For a (negative) reference, see :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Isaac Newton, &lt;a href=&quot;https://books.google.it/books?id=3_s2AAAAMAAJ&amp;amp;pg=PA16&quot; rel=&quot;nofollow noreferrer&quot;&gt;Universal Arithmetick : or, A Treatise of Arithmetical Composition and Resolution. To which is Added, Dr.Halley's Method of Finding the Roots of Equations Arithmetically&lt;/a&gt; (latin manuscript : &lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmetica_Universalis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arithmetica Universalis&lt;/a&gt; (edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge, in 1707, based on Newton's lecture notes).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;See page 3 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quantities are either Affirmative, or greater than nothing; or Negative, or less than nothing. Thus in humane Affairs, Possessions or Stock may be calld &lt;em&gt;affirmative&lt;/em&gt; Goods, and Debts &lt;em&gt;negative ones&lt;/em&gt;. [...] A negative Quantity is denoted by the Sign &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; ; the Sign &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; is prefix'd to an affirmative one [...].&lt;/p&gt;&#xA;&lt;p&gt;In an Aggregate of Quantities the Note &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; signifies, that the Quantity it is prefix'd to, is to be added, and the Note &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;, that it is to be subtracted. And we usually express these Notes by the Words &lt;em&gt;Plus&lt;/em&gt; (or &lt;em&gt;more&lt;/em&gt;) and &lt;em&gt;Minus&lt;/em&gt; (or &lt;em&gt;less&lt;/em&gt;). Thus &lt;span class=&quot;math-container&quot;&gt;$2+3$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; more &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, denotes the Sum of the Numbers &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, that is &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;. And &lt;span class=&quot;math-container&quot;&gt;$5-3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; less &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, denotes the Difference which arises by subducting &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;, that is &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. [&lt;em&gt;Note the clear distinction of the two usages of the signs &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;&lt;/em&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then see page 16 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Simple Algebraick Terms are multiply'd by multiplying the Numbers into the Numbers, and the Species into the Species, and by making the Product Affirmative, if both the Factors are Affirmative, or both Negative : and Negative if otherwise. Thus &lt;span class=&quot;math-container&quot;&gt;$2a$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$3b$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$- 2a$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$- 3b$&lt;/span&gt; make &lt;span class=&quot;math-container&quot;&gt;$6ab$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$6ba$&lt;/span&gt;; for it is no Matter in what Order they are plac'd. Thus also &lt;span class=&quot;math-container&quot;&gt;$2a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$- 3b$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$- 2a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$3b$&lt;/span&gt; make &lt;span class=&quot;math-container&quot;&gt;$- 6ab$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Useful &amp;quot;post-Newtonian&amp;quot; references :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Willem_%27s_Gravesande&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Jacob s'Gravesande&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=lmtbAAAAQAAJ&amp;amp;pg=PA9&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Elements of Universal Mathematics or Algebra&lt;/a&gt; (1728 - Latin original : 1727), page 9 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if the Signs of Multiplicand and Multiplicator are familiar (or the same) the Product will be Affirmative, but Negative if they are different.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The explanation of the &lt;em&gt;minus times minus&lt;/em&gt; case is in terms of &amp;quot;symmetry&amp;quot;, followed by an intuitive example :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is taken away in this Case a negative Quantity, by which the Negation vanishes. &lt;em&gt;So to take away a Debt is to pay it&lt;/em&gt; [emphasis added].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;It is useful to compare with the fully &amp;quot;algebraic&amp;quot; explanation by one of the most brilliant Newton's followers:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Colin_Maclaurin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Colin MacLaurin&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=QoAHAAAAcAAJ&amp;amp;pg=PA13&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Treatise of Algebra&lt;/a&gt; (posthumous : 1748 - 3rd ed 1771), page 13 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By the definitions, &lt;span class=&quot;math-container&quot;&gt;$+a-a=0$&lt;/span&gt;; therefore, if we multiply &lt;span class=&quot;math-container&quot;&gt;$+a-a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, the product must vanish or be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; because the factor &lt;span class=&quot;math-container&quot;&gt;$a-a$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. [...] Therefore &lt;span class=&quot;math-container&quot;&gt;$-a$&lt;/span&gt; multiplied by &lt;span class=&quot;math-container&quot;&gt;$+n$&lt;/span&gt; gives &lt;span class=&quot;math-container&quot;&gt;$-na$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In like manner, if we multiply &lt;span class=&quot;math-container&quot;&gt;$+a-a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$-n$&lt;/span&gt;, the first term of the product being &lt;span class=&quot;math-container&quot;&gt;$-na$&lt;/span&gt;, the latter term of the product must be &lt;span class=&quot;math-container&quot;&gt;$+na$&lt;/span&gt;, because the two together must destroy each other, or they amount be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, since one of the factors (&lt;em&gt;viz&lt;/em&gt;.&lt;span class=&quot;math-container&quot;&gt;$a-a$&lt;/span&gt;) is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Therefore &lt;span class=&quot;math-container&quot;&gt;$-a$&lt;/span&gt; multiplied by &lt;span class=&quot;math-container&quot;&gt;$-n$&lt;/span&gt; must give &lt;span class=&quot;math-container&quot;&gt;$+na$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-12-01T16:03:05.340" Title="Historical roots of the justification for the rule for multiplication of negative numbers" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3049" PostTypeId="2" ParentId="2933" CreationDate="2015-11-10T10:18:38.297" Score="4" Body="&lt;p&gt;In my opinion, it's because Turing's proof of the &quot;circularity question&quot; in &lt;a href=&quot;https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf&quot; rel=&quot;nofollow&quot;&gt;On Computable Numbers, with an Application to the Entscheidungsproblem&lt;/a&gt; is close enough to the Halting Problem to warrant giving him credit. As you note, the phrasing is in terms of circle-free nonhalting machines rather than machines that &quot;halt&quot; but the idea is exactly the same. Rephrasing the problem in terms of halting rather than circularity (which Kleene and Davis, who like Turing were both students of Church, did as early as 1952) doesn't change the essence of the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is Turing's actual statement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The fallacy in this argument lies in the assumption that § is computable. It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the &lt;strong&gt;problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps.&lt;/strong&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2015-11-10T11:16:13.953" LastActivityDate="2015-11-10T11:16:13.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3050" PostTypeId="1" CreationDate="2015-11-10T03:06:34.670" Score="4" ViewCount="3239" Body="&lt;p&gt;I'm trying to find the original papers describing Young's double slit experiments including his experimental set-ups and findings. I've tried looking and all I can find is &lt;a href=&quot;http://www.biodiversitylibrary.org/item/63005#page/498/mode/1up&quot; rel=&quot;nofollow&quot;&gt;http://www.biodiversitylibrary.org/item/63005#page/498/mode/1up&lt;/a&gt; and &#xA;&lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/92/12&quot; rel=&quot;nofollow&quot;&gt;http://rstl.royalsocietypublishing.org/content/92/12&lt;/a&gt; But neither are papers describing the experiments. &lt;/p&gt;&#xA;" OwnerUserId="3219" OwnerDisplayName="TheStrangeQuark" LastEditorUserId="62" LastEditDate="2015-11-12T02:11:27.547" LastActivityDate="2016-01-11T03:07:53.230" Title="Original paper for Young's double slit experiments" Tags="|physics|experimental-physics|quantum-mechanics|" AnswerCount="1" CommentCount="9" ClosedDate="2016-01-11T23:20:26.903" ContentLicense="CC BY-SA 3.0" />
  <row Id="3051" PostTypeId="2" ParentId="3050" CreationDate="2015-11-10T06:29:03.760" Score="0" Body="&lt;p&gt;In this &lt;a href=&quot;https://en.wikipedia.org/wiki/User:HolgerFiedler/Ecxerpt_Young&quot; rel=&quot;nofollow noreferrer&quot;&gt;sandbox&lt;/a&gt; I made an excerpt for myself. All citations are originally in English, some comments are in German. At the end there is a link to &quot;A Course of Lectures on Natural Philosophy and the Mechanical Arts&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW, diffraction of light was known long before Young. &lt;a href=&quot;https://en.wikipedia.org/wiki/Francesco_Maria_Grimaldi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francesco Maria Grimaldi&lt;/a&gt; wrote in his book &quot;Physicomathesis de lumine, coloribus et iride aliisque annexis, libri II&quot; in 1665 about diffraction from a rod and a pinhole.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An amazing fact is, that &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Young_Course_V.2_Plate_XX_liquid_waves.PNG&quot; rel=&quot;nofollow noreferrer&quot;&gt;Young's sketches&lt;/a&gt; about water waves and their interferences sometimes is interpreted as a sketch for light interference (Fig. 267).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WXLa0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WXLa0.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3223" OwnerDisplayName="HolgerFiedler" LastActivityDate="2015-11-10T06:29:03.760" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3053" PostTypeId="2" ParentId="3048" CreationDate="2015-11-10T13:24:26.600" Score="3" Body="&lt;p&gt;Possible source :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow&quot;&gt;Francois Viète&lt;/a&gt;  (1540 – 1603), , &lt;a href=&quot;https://books.google.it/books?id=omHqrYBzMpAC&amp;amp;pg=PA19&quot; rel=&quot;nofollow&quot;&gt;The Analytic Art&lt;/a&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/New_algebra&quot; rel=&quot;nofollow&quot;&gt;In artem analyticam isagoge&lt;/a&gt; 1591), &lt;strong&gt;RULE III : To multiply one magnitude by another&lt;/strong&gt;, page 19 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If a positive term of one quantity is multiplied by a positive term of another quantity, the product will be positive and if by a negative the result &#xA;  will be negative. The consequence of this rule is that multiplying a negative &#xA;  by a negative produces a positive, as when $A - B$ is multiplied by $D - G$. &#xA;  The product of $+ A$ and $- G$ is negative, but this takes away or subtracts too &#xA;  much since $A$ is not the exact magnitude to be multiplied. Similarly the &#xA;  product of $- B$ and $+ D$ is negative, which takes away too much since $D$ is &#xA;  not the exact magnitude to be multiplied. The positive product when $- B$ is &#xA;  multiplied by $- G$ makes up for this. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The geometrical interpretation can be traced back to ancient Greek &lt;a href=&quot;https://books.google.it/books?id=4sA8AAAAIAAJ&amp;amp;pg=PA372&quot; rel=&quot;nofollow&quot;&gt;geometrical algebra&lt;/a&gt;; see Euclid's &lt;em&gt;Elements&lt;/em&gt; : &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII7.html&quot; rel=&quot;nofollow&quot;&gt;II.7&lt;/a&gt; and &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII4.html&quot; rel=&quot;nofollow&quot;&gt;II.4&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For a &quot;pure&quot; algebraic exposition, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Wallis&quot; rel=&quot;nofollow&quot;&gt;John Wallis&lt;/a&gt; (23 November 1616 – 28 October 1703), &lt;a href=&quot;https://books.google.it/books?id=EuzpN1t5SOcC&amp;amp;pg=PA78&quot; rel=&quot;nofollow&quot;&gt;A Treatise of Algebra&lt;/a&gt; (1685), page 78-79 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Unde patet ratio tum hujus regulae, $+$ in $+$ facit $+$; tum hujus $-$ in $+$ facit $-$. [...] Indeque patet ratio tum hujus regulae, $+$ in $-$ facit $-$; tum hujus, $-$ in $-$ facit $+$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;justification&quot; is that to &lt;em&gt;multiply&lt;/em&gt; a given quantity by a &lt;em&gt;positive&lt;/em&gt; factor is &lt;em&gt;ponendi&lt;/em&gt; (&quot;ubi $+2$ significat &lt;em&gt;bis ponere&lt;/em&gt;&quot;) while to &lt;em&gt;multiply&lt;/em&gt; it by a &lt;em&gt;negeative&lt;/em&gt; factor is &lt;em&gt;tollendi&lt;/em&gt; (&quot;ibidem $-2$ est &lt;em&gt;bis tollere&lt;/em&gt;, seu &lt;em&gt;bis ponere contrarium&lt;/em&gt;&quot;).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Contra vero, $-A$ per $-2$ multiplicare, est bis tollere $-A$, seu defectum $-A$ bis supplere, quod est $+A$ bis ponere, facitque $+2A$, (adeoque $-$ in $-$ facit $+$.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-11-10T14:12:16.707" LastActivityDate="2015-11-10T14:12:16.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3054" PostTypeId="2" ParentId="3037" CreationDate="2015-11-10T14:06:28.820" Score="7" Body="&lt;p&gt;The &quot;&lt;em&gt;distinctive shapes&lt;/em&gt;&quot; specified in &lt;strong&gt;IEEE Std 91/91a-1991&lt;/strong&gt; derive from &lt;strong&gt;MIL-STD-806 Graphical Symbols for Logic Diagrams&lt;/strong&gt;, which was originally a US Air Force standard. This in turn was based on a 1960 paper sponsored by the American Institute of Electrical Engineers and American Society of Mechanical Engineers which begins:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Because the development of acceptable logic symbols has not kept pace with the use of electronic data-processing equipment by both the military and industry, a serious state of emergency now exists. This was the basis for calling together on September 8, 1960, individuals representing the many groups working in this area to consider the desirability of forming a task group under ASA Sectional Committee Y32 to propose a solution.&quot; (&lt;a href=&quot;https://books.google.co.uk/books?id=Miw9AAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Magazine of Standards, Volumes 30-31&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The foreword to &lt;strong&gt;IEEE Std 91-1984&lt;/strong&gt; gives some more background, but still doesn't describe the earliest evolution of the symbols:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;It is the latest step in a program that began in 1956 within the IEEE to develop a comprehensive single standard, consistent with ongoing developments in technology and logic symbology, from several ad hoc, industry, military, and international standards. In 1960, an ad hoc group on logic diagram graphic symbols was formed within the American National Standards Institute in order to develop a draft American Standard. In 1961, this committee became a permanent subcommittee, Y32.14, of the Graphic Symbols Committee, Y32, under the cosecretariat of ASME and IEEE.&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;&lt;em&gt;rectangular shapes&lt;/em&gt;&quot;, meanwhile, are based on a gradual evolution of standards developed in the US (and to a lesser extent the UK) during the 60's, 70's and 80's, starting with &lt;strong&gt;IEE Std 91-1962&lt;/strong&gt; and &lt;strong&gt;IEC 117-15 of 1972&lt;/strong&gt;, which codified some of the basic ideas, and continuing with &lt;strong&gt;ANSI Y32.14-1973&lt;/strong&gt;, &lt;strong&gt;BS 3939-1977&lt;/strong&gt;, &lt;strong&gt;IEC 617-12 of 1983&lt;/strong&gt; and &lt;strong&gt;IEEE Std 91-1984&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's worth noting that there is also a third style of symbols still occasionally used in Europe and probably German in origin. This is codified in &lt;strong&gt;DIN 40700&lt;/strong&gt;: see for example &lt;a href=&quot;https://de.wikipedia.org/wiki/Logikgatter#Typen_von_Logikgattern_und_Symbolik&quot; rel=&quot;nofollow noreferrer&quot;&gt;this table on German Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="72" LastEditDate="2020-03-23T19:37:44.230" LastActivityDate="2020-03-23T19:37:44.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3057" PostTypeId="1" AcceptedAnswerId="3063" CreationDate="2015-11-12T07:56:27.793" Score="8" ViewCount="2108" Body="&lt;p&gt;I am trying to understand the uses of the word &quot;Geometric&quot; throughout mathematics.  I suspect that there may be some historical reasons which would tie things together and help my understanding.  Supposedly, &quot;Geometric&quot; means &quot;of or relating to Geometry&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Geometry&quot; is the study of points, lines, surfaces, shapes, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &quot;Geometric mean&quot; of n numbers is the nth root of the product of the numbers -- how is this related to &quot;Geometry&quot; (shapes, lines, points, etc.)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &quot;Geometric series&quot; consists of a sequence of numbers, with each successive number the product of the previous number and some constant -- how does this relate to Geometry?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that there have been some attempts by mathematicians to combine Geometry and Algebra into a single framework.  Could terms like &quot;Geometric mean&quot; and &quot;Geometric series&quot; have come from these attempts?&lt;/p&gt;&#xA;" OwnerUserId="2458" LastActivityDate="2015-11-20T12:25:25.883" Title="What is the history of the meanings behind the word &quot;Geometric&quot;?" Tags="|geometry|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3058" PostTypeId="1" AcceptedAnswerId="3083" CreationDate="2015-11-12T09:18:02.930" Score="7" ViewCount="782" Body="&lt;p&gt;As we know, it was Weierstrass who gave the first (published) example, in 1872, of  a function which is continuous but everywhere non-differentiable. However, in his paper &quot;Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen&quot; there is no figure  of this function or an attempt to draw it.&#xA;Hence, is it known who was first to draw it? Was its depiction possible only to computer graphics or were there attempts before?&lt;/p&gt;&#xA;" OwnerUserId="3232" LastEditorUserId="26" LastEditDate="2015-11-12T10:39:44.213" LastActivityDate="2016-03-22T18:17:58.980" Title="Who first drew the Weierstrass function?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3059" PostTypeId="2" ParentId="3057" CreationDate="2015-11-12T09:56:28.507" Score="1" Body="&lt;p&gt;&quot;Geometry&quot; means literally &quot;Earth measuring&quot;, i.e., what we today would call &quot;topography&quot;. The term &quot;geometric mean&quot; is ancient Greek, while the use of algebra to solve geometric problems dates from the times of Descartes (1596-1650). No relation.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2015-11-12T09:56:28.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3061" PostTypeId="2" ParentId="3057" CreationDate="2015-11-12T14:49:47.103" Score="5" Body="&lt;p&gt;&lt;em&gt;Geometric mean&lt;/em&gt; can be understood in terms of geometry. The &lt;em&gt;geometric mean&lt;/em&gt; (or &lt;em&gt;mean proportional&lt;/em&gt;) of two numbers, $a$ and $b$, is the length of the side of a square whose area is equal to the area of a rectangle with sides of lengths $a$ and $b$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other terms, is a number $c$ such that :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$a \times b = c \times c$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;that comes from :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\frac a c = \frac c b.$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See Euclid's &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI13.html&quot; rel=&quot;nofollow&quot;&gt;Elements VI.13&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-11-13T14:07:48.803" LastActivityDate="2015-11-13T14:07:48.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3062" PostTypeId="2" ParentId="3057" CreationDate="2015-11-12T15:41:47.120" Score="1" Body="&lt;p&gt;For first uses of mathematical terms, you can try &lt;a href=&quot;http://jeff560.tripod.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;HERE&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;some items found there:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;GEOMETRIC MEAN. The geometric mean is one of the three Pythagorean means. ...&lt;/p&gt;&#xA;&lt;p&gt;The term geometrical mean is found in E. Halley &amp;quot;A Most Compendious and Facile Method for Constructing the Logarithms, Exemplified and Demonstrated from the Nature of Numbers, without any Regard to the Hyperbola, with a Speedy Method for Finding the Number from the Logarithm Given,&amp;quot; Philosophical Transactions of the Royal Society of London, Vol. 19. (1695 - 1697), pp. 58-67.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term GEOMETRIC PROGRESSION was used by Michael Stifel in 1543: &amp;quot;Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term GEOMETRY was in use in the time of Plato and Aristotle, and “doubtless goes back at least to Thales,” according to Smith (vol. 2, page 273). The Greek word &lt;span class=&quot;math-container&quot;&gt;$\gamma\epsilon\omega\mu\epsilon\tau\rho\hat\iota\alpha$&lt;/span&gt; was formed from &lt;span class=&quot;math-container&quot;&gt;$\gamma\hat\eta$&lt;/span&gt; earth and &lt;span class=&quot;math-container&quot;&gt;$\mu\epsilon\tau\rho\hat\iota\alpha$&lt;/span&gt; measuring; transliteration produced the Latin gemetria which became géométrie in French and thence geometry in English.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-11-12T15:41:47.120" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3063" PostTypeId="2" ParentId="3057" CreationDate="2015-11-12T18:29:45.790" Score="10" Body="&lt;p&gt;&quot;Geometry&quot; literally translates from Greek as &quot;land measurement&quot;, and reflected a popular Greek belief that they learned it from Egyptian &quot;rope stretchers&quot;, land meausurers, who used ropes to perform what came to be called straightedge and compass constructions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometric mean for $n=2$, or mean proportional as Greeks called it, was likely so named because it answers a geometric question: what is the side of a square having the same area as a rectangle with given sides. Euclid gives a straightedge and compass construction for it in Elements VI.13, but it was already known to early Pythagoreans in 6th century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometric series was likely so named because each of its terms is the geometric mean of its immediate neighbors $a_n=\sqrt{a_{n-1}a_{n+1}}$ (assuming all terms are positive). The same is true of arithmetic and harmonic series, their terms are also the corresponding means of their neighbors. The notion if not the name is also attributed to early Pythagoreans, Euclid expresses in words what we call partial sum of the geometric series in Elements IX.35.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-11-12T18:29:45.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3064" PostTypeId="2" ParentId="3048" CreationDate="2015-11-13T17:02:35.440" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Historical roots of the justification for the rule for multiplication of negative numbers&quot;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This was an on going evolution process, it's impossible to pinpoint someone and give him all the credit. For hundreds of years (maybe thousands) merchants and accountants relied a lot on such rules for their bookkeeping, even without a formal description from the mathematicians about what negative numbers was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mentioned this, probably the first one to state such rules was the Indian mathematician Brahmagupta in the 7th century:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At that time there was no concept of number line (John Wallis at 17th century) or vectors (20th century). His proof should have been based only on simpler concepts. The most simple proof is that all the properties of the elementary arithmetic operations must be the same whether the number is positive or negative. In other words, the distributive property must be valid also for negative numbers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(-1) . 0 = 0&lt;br&gt;&#xA;(-1) . (1 + (-1)) = 0&lt;br&gt;&#xA;(-1).1 + (-1).(-1) = 0&lt;br&gt;&#xA;(-1) + (-1).(-1) = 0&lt;br&gt;&#xA;(-1).(-1) = 1  &lt;/p&gt;&#xA;" OwnerUserId="3238" LastEditorUserId="3238" LastEditDate="2015-11-20T13:19:08.420" LastActivityDate="2015-11-20T13:19:08.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3065" PostTypeId="1" AcceptedAnswerId="3067" CreationDate="2015-11-14T01:40:06.673" Score="3" ViewCount="475" Body="&lt;p&gt;I am writing a research paper on the interplay between science and religion, and need some examples of times when religion has benefited or inspired science and technology. &lt;/p&gt;&#xA;" OwnerUserId="3242" LastEditorUserId="62" LastEditDate="2015-11-14T02:42:35.560" LastActivityDate="2015-11-20T14:39:40.923" Title="In which ways has religion helped the progress of science?" Tags="|philosophy-of-science|" AnswerCount="4" CommentCount="2" ClosedDate="2015-11-22T23:57:08.227" ContentLicense="CC BY-SA 3.0" />
  <row Id="3067" PostTypeId="2" ParentId="3065" CreationDate="2015-11-14T05:15:51.680" Score="4" Body="&lt;p&gt;Geology -- the science of Geology was basically invented by the search for proof of the world-wide Noah's Flood as depicted in the book of Genesis chapter 6+.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This search by both religious (Christian) churchmen and laymen who had interests in science used their emergent knowledge of rocks, strata, and other features common to Geologic structures and artifacts to find evidence that there was a world-wide flood.  They believed that the earth would show evidence of flood geology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During this process which took decades and was passed down from one generation of &quot;flood scientists&quot; to the next started to discover not only evidence of new geologic science but indications that the Earth was much older than they first thought and also that evidence proving a world-wide flood could not be found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result of their work and the work of others in parallel discovery, the science of geology was born and matured to a point that it was totally separate from its early Flood Geology roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This entire story is told in a very excellent way by a book, &quot;The Rocks Don't Lie -- A Geologist Investigates Noah's Flood&quot; by David R. Montgomery, Professor of Earth and Space Sciences at University of Washington (Seattle).  I highly recommend the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to point out that this is not an Anti-Christianity book nor does it use this history to defame Christianity.  It is a fair and balanced story of the history of the birth of Geology in this hunt for Noah's Flood.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastEditorUserId="2905" LastEditDate="2015-11-15T02:04:22.883" LastActivityDate="2015-11-15T02:04:22.883" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3068" PostTypeId="1" CreationDate="2015-11-14T16:37:51.377" Score="3" ViewCount="124" Body="&lt;p&gt;&lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/royprsa/117/778/610.full.pdf&quot; rel=&quot;nofollow&quot;&gt;Dirac (1928)&lt;/a&gt;,&#xA;(&lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/royprsa/126/801/360.full.pdf&quot; rel=&quot;nofollow&quot;&gt;http://rspa.royalsocietypublishing.org/content/royprsa/126/801/360.full.pdf&lt;/a&gt;)&#xA;explained the negative-energy solutions of his equations as holes in a sea of electrons. &#xA;The concept seems very similar to that of hole conduction in semi-conductor physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Commonly, the transistor (also the semiconductor diode) is explained in terms of electrons and holes. According to Wikipedia the transistor was patented 1925 by Julius Edgar Lilienfeld, but not realized experimentally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did he build on the concept of a Dirac sea to theorize the transistor? Less specifically: Did the concept of the Dirac Sea influence (lead to) thedevelopment of semiconductors like the transistor?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or, in view of the order of the dates: Was Dirac inspired by semiconductor physics?&lt;/p&gt;&#xA;" OwnerUserId="3246" LastEditorUserId="4251" LastEditDate="2020-07-07T06:27:40.787" LastActivityDate="2020-07-07T06:27:40.787" Title="Did the concept of the Dirac Sea influence the development of semiconductors like the transistor?" Tags="|physics|quantum-mechanics|electronics|solid-state-physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3069" PostTypeId="2" ParentId="3065" CreationDate="2015-11-14T17:21:17.260" Score="2" Body="&lt;p&gt;There is a line of thought that says the reforms of John Calvin and other Protestant leaders lead to a change in the idea of &amp;quot;work&amp;quot;, which in turn made the Industrial Revolution possible.&lt;/p&gt;&#xA;&lt;p&gt;There are many works expounding this hypothesis.  Here is one I found by googling...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Prior to Calvin, work was not considered a good thing. It was something the lower classes did. Anyone who was anyone didn’t do physical work. They lived lives of leisure and contemplation. They devoted themselves to music and poetry and literature and art. Such was their &amp;quot;higher calling.&amp;quot; The lower classes did not aspire to much beyond their laborious lower class callings. That is something of a simplification, but it is, in a nutshell, the way it was.&lt;/p&gt;&#xA;&lt;p&gt;Calvin and the other Protestant Reformers of 16th century Europe challenged all that. They postulated that work, physical work, was good, that God created man to work, that all men, regardless of their position in society, were called to work, that there was dignity in work, and that work was a form of worship. This teaching is best summed up in the term “Protestant work ethic.” It started with the Reformation Protesters. It started with John Calvin. &lt;a href=&quot;http://thedeliberateagrarian.blogspot.com/2006/11/john-calvin-father-of-industrial.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-11-14T17:21:17.260" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3070" PostTypeId="2" ParentId="205" CreationDate="2015-11-14T22:02:59.387" Score="14" Body="&lt;p&gt;It's worth noting that we can't just pin it down to &amp;quot;The Nazi regime&amp;quot; and we may have to just say &amp;quot;The Nazis.&amp;quot; Take for instance the case of &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Biographies/Landau.html&quot; rel=&quot;noreferrer&quot;&gt;Landau&lt;/a&gt;. He could not be purged as such but when he tried to teach in Fall 1933 he faced a student boycott led by &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Biographies/Teichmuller.html&quot; rel=&quot;noreferrer&quot;&gt;Oswald Teichmuller&lt;/a&gt;. A letter from Landau sets the scene.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On 2 November, about 11.15, as I wished to leave my office and go to the large lecture theatre to begin my lecture, the entrance hall was filled with about 80 to 100 students who let me pass through unhindered. In the lecture hall was one person. Clearly therefore, there was a boycott with sentries at the door who had prevented (without force) those students who wanted to work from setting foot in the lecture room.&lt;/p&gt;&#xA;&lt;p&gt;What happened - and it happened with the collaboration of many who were my pupils - leads me to believe that the only consequence must be my application to become emeritus or pensioned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The article notes that it's reasonable to assume the boycott was actually organized by the SA but given his later history, it seems likely that Teichmuller was perfectly happy to be the face of the boycott. He was a staunch Nazi who frequently published in Deutsche Mathematik and indeed met his end fighting on the Eastern front after the German defeat at Stalingrad. Much more on this and similar topics can be found in &amp;quot;Mathematicians under the Nazis&amp;quot; by Segal.&lt;/p&gt;&#xA;" OwnerUserId="3248" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-11-14T22:02:59.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3071" PostTypeId="2" ParentId="3065" CreationDate="2015-11-15T01:34:29.307" Score="2" Body="&lt;p&gt;During the &quot;dark ages&quot; science was kept alive essentially in monasteries. Benedictine and Cistecians studied and catalogued properties of soils and resulting wines.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2015-11-15T01:34:29.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3073" PostTypeId="2" ParentId="3057" CreationDate="2015-11-15T08:45:56.730" Score="1" Body="&lt;p&gt;The word 'geometry' originates from the Sanskrit word gyaamiti - &quot;measuring the earth&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;gya means &quot;earth&quot; &#xA;miti means &quot;to measure&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the link:&#xA;&lt;a href=&quot;http://veda.wikidot.com/sanskrit-origin-names&quot; rel=&quot;nofollow&quot;&gt;http://veda.wikidot.com/sanskrit-origin-names&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3249" LastEditorUserId="3249" LastEditDate="2015-11-20T12:25:25.883" LastActivityDate="2015-11-20T12:25:25.883" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3074" PostTypeId="1" AcceptedAnswerId="3076" CreationDate="2015-11-15T14:23:57.213" Score="1" ViewCount="84" Body="&lt;p&gt;The German Wikipedia article &lt;a href=&quot;https://de.wikipedia.org/wiki/Bernoulli-Verteilung&quot; rel=&quot;nofollow&quot;&gt;&quot;Bernoulli-Verteilung&quot;&lt;/a&gt; (&quot;Bernoulli distribution&quot;) claims that the term &quot;Bernoullian trials&quot; occurs first in the textbook &quot;Introduction to Mathematical Probability&quot; by J.V. Uspensky (published 1937, page 45). I have found the cited statement there:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;In this chapter we shall discuss the simplest case of series of independent trials with constant probability. They are often called &quot;bernoullian series of trials&quot; in honor of Jacob Bernoulli who, in his classical book, &quot;Ars conjectandi&quot; (1713) made a profound study of such series and was led to the discovery of one of the most important theorems in the theory of probability.&quot; -- J.V. Uspensky&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For me this paragraph does not support the claim that Uspensky was the first who used the term &quot;Bernoullian trial&quot;. This leads me to my more general question: How can a historian be sure that a certain scientist or mathematician used a phrase for the first time? And more precisely: Is it right, that Uspensky was the first mathematician who used the term &quot;Bernoullian trial&quot;?&lt;/p&gt;&#xA;" OwnerUserId="1772" LastEditorUserId="2905" LastEditDate="2015-11-15T19:24:15.043" LastActivityDate="2015-11-15T20:08:54.193" Title="How can I be sure that a certain term ocurred first in a certain textbook?" Tags="|mathematics|terminology|publishing|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3075" PostTypeId="2" ParentId="160" CreationDate="2015-11-15T17:11:07.780" Score="3" Body="&lt;p&gt;Linnean taxonomy was very much the progression and refinement of earlier taxonomic thoughts. This answer draws heavily from &quot;Linnaeus as an Intermediary between Ancient and Modern Zoölogy; His Views on the Class Mammalia&quot; by W. K. Gregory (1908).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gregory notes several major figures in the history of taxonomic nomenclature before Linnaeus:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Aristotle&lt;/strong&gt; (384–322 BC): In &lt;a href=&quot;http://classics.mit.edu/Aristotle/parts_animals.html&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;On the Parts of Animals&lt;/em&gt;&lt;/a&gt;, Aristotle began to group and classify animals by certain characteristics, such as teeth, hooves, etc. Aristotle considered the state of nomenclature in &lt;a href=&quot;http://classics.mit.edu//Aristotle/history_anim.html&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;The History of Animals&lt;/em&gt;&lt;/a&gt; (Book I, part 6), where he wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the genus that combines all viviparous quadrupeds are many species,&#xA;  but under no common appellation. They are only named as it were one by&#xA;  one, as we say man, lion, stag, horse, dog, and so on; though, by the&#xA;  way, there is a sort of genus that embraces all creatures that have&#xA;  bushy manes and bushy tails, such as the horse, the ass, the mule, the&#xA;  jennet, and the animals that are called Hemioni in Syria,-from their&#xA;  externally resembling mules, though they are not strictly of the same&#xA;  species. And that they are not so is proved by the fact that they mate&#xA;  with and breed from one another. For all these reasons, we must take&#xA;  animals species by species, and discuss their peculiarities severally'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Andrea Cesalpino&lt;/strong&gt; (1519-1603): A botanist, who in &lt;em&gt;De Plantis&lt;/em&gt; (1583), divided plants into ten large classes based on morphology such as fruits and seeds, and then further subdivided them. He gave them monomial names.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;John Ray&lt;/strong&gt; (1627-1705): An English naturalist, he made important steps in taxomony. For example, he classified many plants in his publication of &lt;em&gt;Historia Plantarum&lt;/em&gt;. Gregory describes that:&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ray also used the term &quot;species&quot; in quite a Linnaean manner, as in the&#xA;  names &lt;em&gt;Ovis laticauda&lt;/em&gt;, &lt;em&gt;Ovis strepsiceros&lt;/em&gt; and &lt;em&gt;Ovis domestica&lt;/em&gt;. In form,&#xA;  at least, this foreshadows the binomial system of nomenclature and the&#xA;  recognition of the species in general as supposedly objective reality&#xA;  and the unit of classification.The form of Ray's specific definitions&#xA;  seems, however, to imply that the term &quot;species&quot; in Ray's mind was&#xA;  often more a &quot;differentia,&quot; or specific adjective modifying the&#xA;  generic concept than a fully developed substantive name, and Ray did&#xA;  not apparently realize the convenience of applying the binomial method&#xA;  of nomenclature universally.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;While there were many other figures involved in the development of modern taxonomy and taxonomial nomenclature, these individuals illustrate well that Linnaeus's binomial nomenclature, while an important step, was not an enormous leap. The idea of classifying species pre-dates Aristotle, and while monomial names for species were mostly used up until the seventeenth century, Linneaus was not the first to use binomial names for species, although he was the first to do so systematically across many classes of living things.  &lt;/p&gt;&#xA;" OwnerUserId="2826" LastActivityDate="2015-11-15T17:11:07.780" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3076" PostTypeId="2" ParentId="3074" CreationDate="2015-11-15T20:08:54.193" Score="4" Body="&lt;p&gt;Of course you cannot be sure it is the first.  Merely that it is the oldest one we have found.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://jeff560.tripod.com/mathword.html&quot; rel=&quot;noreferrer&quot;&gt;Mathword&lt;/a&gt; has your same 1937 citation for &quot;Bernoulli trials&quot;.  It has a 1934 citation for &quot;Bernoulli distribution&quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-11-15T20:08:54.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3077" PostTypeId="1" AcceptedAnswerId="3079" CreationDate="2015-11-16T06:23:21.940" Score="10" ViewCount="968" Body="&lt;p&gt;...and what was the general view on it before?&lt;/p&gt;&#xA;&lt;p&gt;Plagiarism in modern day academia is totally prohibited, and people doing such a misconduct would be banned from academia, or would at least get into serious troubles. And definitely lose a lot of credit and reputation.&lt;/p&gt;&#xA;&lt;p&gt;However, the Wikipedia article on academic plagiarism seems to indicate that it is a recent ethical concern: see &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_plagiarism_incidents#Notable_cases_in_academia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plagiarism in academia&lt;/a&gt;. Except a very early case, all examples are from the 20th and 21th centuries. This does not show it didn't happen before, but may shows that they were not considered as a problem in earlier times.&lt;/p&gt;&#xA;&lt;p&gt;My question: did plagiarism become a concern only in recent history? If so, what were the views towards plagiarism in earlier history?&lt;/p&gt;&#xA;&lt;p&gt;Or is it just that Wikipedia lacks of old examples?&lt;/p&gt;&#xA;&lt;p&gt;PS: the second part of the question may be too broad. Please ignore it in that case, and focus on the turning point that changed the views on plagiarism.&lt;/p&gt;&#xA;" OwnerUserId="372" LastEditorUserId="372" LastEditDate="2022-04-22T00:26:18.343" LastActivityDate="2022-04-22T00:26:18.343" Title="When did plagiarism become a major misconduct in academia?" Tags="|academia|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="3078" PostTypeId="1" AcceptedAnswerId="3085" CreationDate="2015-11-16T13:51:08.347" Score="8" ViewCount="405" Body="&lt;p&gt;Vladimir Arnold &lt;a href=&quot;http://www.mccme.ru/edu/viarn/obscur.htm&quot; rel=&quot;noreferrer&quot;&gt;wrote&lt;/a&gt; that Andrei Kolmogorov told him the names of the two mathematicians in conversation with whom he &quot;feels the presence of higher intelligence&quot;. One of them is Israel Gelfand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who is the second one?&lt;/p&gt;&#xA;" OwnerUserId="3253" LastEditorUserId="4251" LastEditDate="2022-08-16T12:27:14.240" LastActivityDate="2022-08-16T12:27:14.240" Title="Who is the second person Kolmogorov supposedly thought possessed some kind of &quot;higher intelligence&quot;?" Tags="|mathematicians|biographical-details|kolmogorov|vladimir-arnold|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3079" PostTypeId="2" ParentId="3077" CreationDate="2015-11-16T22:01:11.140" Score="13" Body="&lt;p&gt;Plagiarism was a concern already in antiquity. In the preface to &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Archimedes.html&quot; rel=&quot;noreferrer&quot;&gt;On Spirals Archimedes&lt;/a&gt; mentions that he was sending Alexandrians statements of his latest theorems, but some had claimed them as their own. Then on occasion he included two false ones &quot;&lt;em&gt;so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most famous case of ancient plagiarism however showcases the differences between then and now. The earliest documented concern about the data in Ptolemy's Almagest came from Kepler and Severin in 17th century, over a millennium after Ptolemy's death. Later Delambre in History of Astronomy (1817) pointed out that observations of solstices and equinoxes that Ptolemy reports as his own appear to be back-engineered from Hipparchus's data. In the 20th century thorough inspection of Almagest showed that there are the same problems with elongations of Venus and various other data that Ptolemy reports. The biggest problem is Ptolemy's star catalogue, analysis of which suggests that &quot;his&quot; observations were made 5° to the north of Alexandria, likely at Rhodes where Hipparchus worked. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In modern times this sustained a debate between &quot;scientists&quot; van der Waerden, Thurston, Rawlins and Richard Newton, and &quot;historians&quot; Neugebauer, Swerdlow and Gingerich on Ptolemy's conduct. Newton especially did not mince words in the tellingly named book The Crime of Claudius Ptolemy, concluding that he &quot;&lt;em&gt;is not the greatest astronomer of antiquity, but he is something still more unusual: he is the most successful fraud in the history of science&quot;&lt;/em&gt;. And Rawlins writes that &quot;&lt;em&gt;defenders invariably excuse Ptolemy's star thefts by noting that modern cataloguers commonly use other astronomers' stellar data. But such apologia just as invariably fail to mention that none of these borrowers claim such gleanings as their own outdoor labor&lt;/em&gt;&quot;. However, the defenders do have a point that some of this indignation comes from projecting modern standards onto antiquity, and that Ptolemy had to make something of sporadic and conflicting data without modern statistical methods. For more see &lt;a href=&quot;http://www.jstor.org/stable/10.1086/380660&quot; rel=&quot;noreferrer&quot;&gt;debate in ISIS&lt;/a&gt; sparked by &lt;a href=&quot;http://documents.mx/documents/greek-mathematical-astronomy-reconsidered-hugh-thurston.html&quot; rel=&quot;noreferrer&quot;&gt;Thurston's Greek Mathematical Astronomy reconsidered&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Accusations of plagiarism continued into middle ages, Cardano for instance published &lt;a href=&quot;https://hsm.stackexchange.com/questions/347/why-is-cardanos-formula-wrongly-attributed-to-him/351#351&quot;&gt;solution to cubics given to him by Tartaglia&lt;/a&gt; after promising not to do so (he did give him credit though), and especially into 17th century, when Hooke alone made multiple plagiarism accusations, e.g. against Newton over the inverse square law, and even lobbied the Royal Society to stop sending its publications to non-members to prevent theft. The atrocious Newton-Leibniz priority dispute also had its share of plagiarism accusations from both sides. So already the first professional organizations, like the Academy of Sciences and the Royal Society, were very mindful of the plagiarism issue, and tried to address it from the start through institutional mechanisms. The main problem was distinguishing between true plagiarism, on which there was trivially a consensus, from &quot;fair use&quot; and uncertain timeline cases. For instance, the priority by publication date convention emerged after the Newton-Leibniz dispute to remedy the latter.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-11-17T02:29:20.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3081" PostTypeId="1" CreationDate="2015-11-17T10:24:46.950" Score="8" ViewCount="320" Body="&lt;p&gt;This has bugged me for some time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tycho Brahe's data on planetary observations, presumably, consisted of the direction in which a planet was observed at a given date and time, but &lt;strong&gt;not&lt;/strong&gt; the distance to the planet. What techniques did Kepler use to add a depth dimension to these observations, to create the three-dimensional data that one can start studying to arrive at his three laws?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/161333/how-did-kepler-infer-three-dimensional-positions-from-tycho-brahes-data&quot;&gt;Cross-posted from Physics Stack Exchange.&lt;/a&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="272" LastEditorUserId="4251" LastEditDate="2021-02-28T08:10:33.100" LastActivityDate="2021-02-28T08:10:33.100" Title="How did Kepler infer three-dimensional positions from Tycho Brahe's data?" Tags="|physics|astronomy|17th-century|kepler|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3082" PostTypeId="1" CreationDate="2015-11-17T13:04:04.403" Score="5" ViewCount="759" Body="&lt;p&gt;You saw the question right. Why isn't it so? &lt;a href=&quot;https://en.wikipedia.org/wiki/Aryabhata#Mathematics&quot; rel=&quot;noreferrer&quot;&gt;Aryabhata&lt;/a&gt; had done more things than him. Is it because of the 400 or 500 years of difference between their existence? &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagoras#Mathematics&quot; rel=&quot;noreferrer&quot;&gt;Pythagoras&lt;/a&gt; is famous most for his just one theorem. Is there something about Pythagoras that I don't know?&lt;/p&gt;&#xA;" OwnerUserId="3222" LastEditorUserId="2905" LastEditDate="2015-11-17T19:14:18.867" LastActivityDate="2015-11-17T19:14:18.867" Title="Why isn't Aryabhata more famous than Pythagoras?" Tags="|mathematicians|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3083" PostTypeId="2" ParentId="3058" CreationDate="2015-11-17T16:02:55.183" Score="8" Body="&lt;p&gt;I looked through some of my stuff this morning and found the following publications that seem to be relevant to your question.&lt;/p&gt;&#xA;&lt;p&gt;Moore &lt;strong&gt;[1]&lt;/strong&gt; (1900) and Koch &lt;strong&gt;[2]&lt;/strong&gt; (1904), &lt;strong&gt;[3]&lt;/strong&gt; (1906) are the earliest publications I could find that have diagrams of somewhat accurate approximating curves to nowhere differentiable continuous functions. Until fairly recently (1960s and later), almost all publications on this topic have no figures, and the few that do have figures tend to have only very superficial figures such as absolute value graphs or step functions that are used in various ways to generate such functions. Lang &lt;strong&gt;[4]&lt;/strong&gt; (1961) is the earliest publication I could find that attempts to show the &amp;quot;final state&amp;quot; of the construction of a nowhere differentiable continuous function. Hailpern &lt;strong&gt;[5]&lt;/strong&gt; (1976) is the earliest publication I could find on this topic that gives computer generated graphs, the graphs being for high level approximating curves for such a function. Tall &lt;strong&gt;[6]&lt;/strong&gt; (1982), Dubuc &lt;strong&gt;[7]&lt;/strong&gt; (1989), Hata &lt;strong&gt;[8]&lt;/strong&gt; (1991), Duistermaat &lt;strong&gt;[9]&lt;/strong&gt; (1992) also give various computer generated graphs, but none yet show the Weierstrass function. Baouche/Dubuc &lt;strong&gt;[10]&lt;/strong&gt; (1992) is the earliest paper I could find with the Weierstrass function shown.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Eliakim Hastings Moore, &lt;a href=&quot;http://www.ams.org/journals/tran/1900-001-01/S0002-9947-1900-1500526-4/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On certain crinkly curves&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Transactions of the American Mathematical Society&lt;/strong&gt; 1 (January 1900), 72-90.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;See the various approximations to the coordinate functions of Peano/Hilbert's square-filling curves on pp. 81, 83.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Helge von Koch, &lt;a href=&quot;https://books.google.com/books?id=KZ81AQAAIAAJ&amp;amp;pg=PA681&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Arkiv för Matematik, Astronomi och Fysik&lt;/strong&gt; 1 (1904), 681-702.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;See Figure 4 on p. 698. Incidentally, Section III (pp. 697-702) of the paper involves a modification of the now-called Koch curve to obtain the graph of a (nowhere differentiable continuous) function. Incidentally, in the title of Section III translates to: &amp;quot;Transformation of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to a curve &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt; where the ordinate (i.e. &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate) is a uniform function (i.e. single-valued function) of the abscissa (i.e. &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate)&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Helge von Koch, &lt;a href=&quot;https://books.google.com/books?id=nLxLAAAAYAAJ&amp;amp;pg=PA145&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Acta Mathematica&lt;/strong&gt; 30 (1906), 145-174.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I believe this is a republication of Koch's 1904 paper. Figure 4 and Section III mentioned above are on p. 167 and pp. 166-174.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Lester Henry Lange, &lt;a href=&quot;http://www.jstor.org/stable/2688066&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Successive differentiability&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematics Magazine&lt;/strong&gt; 34 #5 (May-June 1961), 275-279.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Figure II on p. 279 is &amp;quot;An attempt to picture the function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; so defined&amp;quot;, where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a nowhere differentiable continuous function given in Olmsted's 1956 book &lt;strong&gt;Intermediate Analysis&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Brent Hailpern, &lt;a href=&quot;http://brent.hailpern.com/PMEJ_6_5_1976.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Continuous non-differentiable functions&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Pi Mu Epsilon Journal&lt;/strong&gt; 6 #5 (1976), 249-260.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This paper has some computer generated graphs of approximations to Perkins' function (Amer. Math. Monthly 34, 1927, pp. 476-478) and Van der Waerden's function.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; David Tall, &lt;a href=&quot;http://www.jstor.org/stable/3617301&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The blancmange function. Continuous everywhere but differentiable nowhere&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 66 #435 (March 1982), 11-22.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;See Figure 1 on p. 11, Figure 5 on p. 14, Figure 6 on p. 15, Figure 7 on p. 16.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Benoit Dubuc, &lt;a href=&quot;http://cms.math.ca/10.4153/CMB-1989-055-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On Takagi fractal surfaces&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Canadian Mathematical Bulletin&lt;/strong&gt; 32 #3 (September 1989), 377-384.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Figure 2.1 on p. 379 gives graphs of two Takagi-type functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Masayoshi Hata, &lt;a href=&quot;http://link.springer.com/chapter/10.1007%2F978-94-015-7931-5_6&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Topological aspects of self-similar sets and singular functions&lt;/em&gt;&lt;/a&gt;, pp. 255-276 in Jacques Bélair and Serge Dubuc (editors), &lt;strong&gt;Fractal Geometry and Analysis&lt;/strong&gt;, Kluwer Academic Publishers [later published by Springer], 1991.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proceedings of a 3-21 1989 conference in Montreal (Canada). Figure 3 on p. 271 shows the Besicovitch function (has at no point a one-sided finite or a one-sided infinite derivative). [Various piece-wise continuous approximations of this function can be found much earlier, but this appears to be a computer generated graph.] Figure 4 on p. 271 shows the Takagi function.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; J. J. Duistermaat, &lt;em&gt;Selfsimilarity of 'Riemann's nondifferentiable function'&lt;/em&gt;, &lt;strong&gt;Nieuw Archief voor Wiskunde&lt;/strong&gt; (4) 9 (1991), 303-337.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Figure 1.1 on p. 304 gives a graph of Riemann's function. Figure 4.2 on p. 321 shows the behavior at the right side of a point where the derivative exists and is negative -- it looks like a choppier version of what &lt;span class=&quot;math-container&quot;&gt;$-x + x^2 \sin (x^{-2})$&lt;/span&gt; looks like at the right side of &lt;span class=&quot;math-container&quot;&gt;$x=0.$&lt;/span&gt; See also Figures 4.3, 4.4-4.5, 6.1, 6.2 on pp. 322, 323, 335, 336.&lt;/p&gt;&#xA;&lt;p&gt;The following is from pp. 303-304: &amp;quot;Already for many years, a picture of the graph of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; [the Riemann function], made by A.J. de Meijer, adorns the cover of the notes of the first semester analysis course in Utrecht. In it, the aforementioned differentiability properties at the rational points with not too large denominators can be distinguished quite clearly.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The following is from p. 308: &amp;quot;As for the role of the computer, of course in the old days computer pictures were not available to put one on the track. On the other hand, the mathematical analysis definitely is the more essential part of the story. Without it, one may look at the pictures with equal fascination, but with less understanding.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Amar Baouche and Serge Dubuc, &lt;a href=&quot;http://retro.seals.ch/digbib/view?pid=ensmat-001:1992:38::53&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;La non-dérivabilité de la fonction de Weierstrass&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;L'Enseignement Mathématique&lt;/strong&gt; (2) 38 #1-2 (January-June 1992), 89-94.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Figure 1 on p. 90 gives a graph of the Weierstrass function.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;(ADDED ABOUT 4 MONTHS LATER)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recently came across another paper that is old enough and relevant enough to belong to the above list (between &lt;strong&gt;[5]&lt;/strong&gt; and &lt;strong&gt;[6]&lt;/strong&gt;):&lt;/p&gt;&#xA;&lt;p&gt;Michael Victor Berry and Zinaida V. Lewis, &lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/370/1743/459&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On the Weierstrass-Mandelbrot fractal function&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Proceedings of the Royal Society&lt;/strong&gt; A 370 #1743 (April 1980), 459-484.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This paper gives many computer generated graphs of a variation on the Weierstrass function that the authors call the Weierstrass-Mandelbrot function.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-03-22T18:17:58.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3084" PostTypeId="2" ParentId="3081" CreationDate="2015-11-17T17:54:27.463" Score="3" Body="&lt;p&gt;At the time of Kepler distances were not measured or observed directly. Or if measured, the results were completely wrong. All his laws are stated in terms of RATIOS of distances, and these ratios can be in principle measured from the geometry&#xA;of the situation. When you describe everything as seen from the Earth (as the ancients did), distances are completely irrelevant. But when one uses heliocentric system, all ratios of distances to the distance from Earth to Sun&#xA;can be obtained from angular observation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this is only the general principle. The details are MUCH more complicated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. But the idea is the following: suppose for simplicity that everything happens in the same plane, and that the planets move on circles.&#xA;(This is actually a good approximation because the inclinations of the orbits are small and excentricities are also small. As seen from the Earth, the Sun rotates on a circle of radius $r$ uniformly, with center at the Earth. The planet rotates around the Sun on a circle of radius $R$, also uniformly. Suppose that at three different times you observe the direction on the planet, this essentially means that you measure two angles on your picture. And two other angles you know because you know the time of observations and Sun's rotation speed. From these angles by pure geometry you can find the ratio $r/R$. Just make a picture, and consider this a high school geometry exercise (of moderate difficulty). This was already known to Ptolemy, by the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These ratios were known to Kepler, and by playing with them he discovered his 3-d&lt;br&gt;&#xA;law. But his greatest achievement is the FIRST law: he was able to derive from&#xA;the observations (of the same kind that I described) that the planets really&#xA;do not move on circles but on ellipses. How exactly he did this I cannot explain within the space allowed here:-) But his own explanation is available in English btw. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-11-17T18:49:12.340" LastActivityDate="2015-11-17T18:49:12.340" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3085" PostTypeId="2" ParentId="3078" CreationDate="2015-11-17T21:56:19.430" Score="15" Body="&lt;p&gt;It is indeed the author. In an &lt;a href=&quot;http://www.pseudology.org/science/ArnoldInterview2008.pdf&quot;&gt;interview&lt;/a&gt; (see page 41) Arnold is more explicit:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;он [Gelfand] был, как и я, учеником Колмогорова – Андрей Николаевич говаривал даже, что «только в беседах с этими двумя учениками испытывал ощущение присутствия высшего разума»&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... Gelfand, like me, was Kolmogorov's student - Andrey Nikolayevich even used to say that &quot;only talking to those two students he experienced presence of higher intelligence.&lt;/p&gt;&#xA;" OwnerUserId="1779" LastActivityDate="2015-11-17T21:56:19.430" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3087" PostTypeId="1" AcceptedAnswerId="3097" CreationDate="2015-11-18T05:08:19.890" Score="7" ViewCount="587" Body="&lt;p&gt;I am looking for the translated manuscript of Abel where he proved the unsolvability of the quintic. Can anyone give me a pointer? I tried Google, but nothing came up.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="10106" LastEditDate="2020-06-07T18:45:05.363" LastActivityDate="2023-10-09T21:59:43.450" Title="Where can I find the translated manuscript of Abel?" Tags="|mathematicians|reference-request|abstract-algebra|translations|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3088" PostTypeId="1" AcceptedAnswerId="3090" CreationDate="2015-11-18T08:28:54.730" Score="3" ViewCount="130" Body="&lt;p&gt;Google Ngram shows that the expression &amp;quot;open problem&amp;quot; started to be in use around the end of the 19th century.&lt;/p&gt;&#xA;&lt;p&gt;My question is then 2-fold:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who coined the expression? Wikipedia doesn't seem to know.&lt;/li&gt;&#xA;&lt;li&gt;What was the intended meaning of the adjective &amp;quot;open&amp;quot; (in comparison to the straightforward &amp;quot;unsolved problem&amp;quot;? It seems to be opposite to &amp;quot;closed problem&amp;quot;, similar to a &amp;quot;closed file&amp;quot; in workplace. Or is it supposed to mean &amp;quot;available to everybody (to solve it)&amp;quot;, like in &amp;quot;open source&amp;quot; or &amp;quot;open space&amp;quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="372" LastEditorUserId="4251" LastEditDate="2022-09-08T10:08:03.507" LastActivityDate="2022-09-08T10:08:03.507" Title="Origin of the expression &quot;Open problem&quot;" Tags="|terminology|open-problems|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3089" PostTypeId="1" CreationDate="2015-11-18T15:57:31.720" Score="5" ViewCount="1874" Body="&lt;p&gt;I'm looking for examples of ideas/discoveries/concepts in Maths or Science that had no practical application at first and were maybe considered as nothing but a theoretical concept but they turned out to be useful years after?&lt;/p&gt;&#xA;&lt;p&gt;I think that imaginary numbers may be such a thing - it's written on Wikipedia that &amp;quot;The name &amp;quot;imaginary number&amp;quot; was coined in the 17th century as a derogatory term, as such numbers were regarded by some as fictitious or useless.&amp;quot; - and now they are very common in Physics and Maths, right? What about some other examples?&lt;/p&gt;&#xA;" OwnerUserId="3268" LastEditorUserId="6609" LastEditDate="2023-03-10T09:10:34.703" LastActivityDate="2023-03-10T09:10:34.703" Title="What are some concepts/discoveries in mathematics and science that found practical application years after they had been formulated/discovered?" Tags="|mathematics|physics|theoretical-physics|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3090" PostTypeId="2" ParentId="3088" CreationDate="2015-11-18T16:39:11.403" Score="4" Body="&lt;p&gt;In the &lt;em&gt;Oxford English Dictionary&lt;/em&gt;, we find&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;open&lt;/strong&gt; adj, definition 27a:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Of a matter, discussion, etc.: not finally settled or determined; that may be decided according to circumstances or at will; (hence) still admitting of debate, uncertain. Freq. in &lt;em&gt;open question&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with many examples.  One in Old English:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gif he ðonne byrigan forwærne, xii scillingas agylde þam cyninge, &amp;amp; sio se[o] sacy swa open swa hio ær wes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;T. More 1532:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;maketh the matter open &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;W. Blackstone 1769&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An open verdict &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;W. Cruse 1804&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The other question...was still left open.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They have many more &quot;open question&quot; examples, but they do not have an &quot;open problem&quot; example.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-11-18T16:39:11.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3091" PostTypeId="1" CreationDate="2015-11-19T11:27:04.193" Score="3" ViewCount="1868" Body="&lt;p&gt;Greek letters are used everywhere in science and mathematics. When did this tradition start? If it is because the Greeks were pioneers of maths? why did the Indian literals, for example, not become popular?&lt;/p&gt;&#xA;" OwnerUserId="3270" LastEditorUserId="62" LastEditDate="2015-11-19T21:24:23.373" LastActivityDate="2015-12-14T02:04:49.980" Title="Whose idea was it to use Greek symbols?" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3092" PostTypeId="2" ParentId="3089" CreationDate="2015-11-19T11:31:09.293" Score="3" Body="&lt;p&gt;Most of the stuff Einstein worked on can come under this category. For example, people did have any use of the relativity theory in daily life before Global Positioning System came around. Similarly, we do not yet know how to exploit the wormholes (which are allowed by modern physics) but one day might.&lt;/p&gt;&#xA;" OwnerUserId="3271" LastActivityDate="2015-11-19T11:31:09.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3093" PostTypeId="1" AcceptedAnswerId="3106" CreationDate="2015-11-19T16:21:52.403" Score="6" ViewCount="1042" Body="&lt;p&gt;As best as I can tell Dedekind's paper was published in 1887 already several years after Cantor's flurry of papers on Set Theory between 1879-1883. With this in mind my central questions are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) What did Dedekind contribute that was not already present in Cantor's work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Why has Dedekind chosen a different terminology from that of Cantor though he certainly knew of Cantor's work? (Maybe the terminology difference is due to differing translations but from the few examples I've been able to pick out they are different in the German as well) &lt;/p&gt;&#xA;" OwnerUserId="3273" LastActivityDate="2015-11-21T11:12:38.833" Title="What did Dedekind's The Nature and Meaning of Numbers contribute to the founding of Set Theory?" Tags="|terminology|number-theory|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3094" PostTypeId="2" ParentId="3091" CreationDate="2015-11-19T17:58:29.767" Score="1" Body="&lt;p&gt;It is not the idea of any individual I suppose.Greeks were one of the most advanced of all in science and all other fields in the ancient times.They had their own list of alphabets which they used.Slowly,it spread to some neighboring groups who followed their alphabets and methods.This then spread all around the globe and there has been perhaps no change or modifications in the language since then and till now widespreadly used. &lt;/p&gt;&#xA;" OwnerUserId="2955" LastActivityDate="2015-11-19T17:58:29.767" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3095" PostTypeId="2" ParentId="3087" CreationDate="2015-11-19T20:35:44.730" Score="6" Body="&lt;p&gt;I have doubts that it was translated into English. French and German works often go untranslated because it is assumed that most English speaking mathematicians can make out enough from the original. The original publication of Abel's collected works was in French. MAA hails old &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/mathematical-treasure-works-of-abel-and-galois&quot; rel=&quot;noreferrer&quot;&gt;German translation&lt;/a&gt; without mentioning anything about English one, and promotes English translation of &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/abel-on-elliptic-integrals-a-translation&quot; rel=&quot;noreferrer&quot;&gt;Abel's elliptic functions&lt;/a&gt; paper, but not the quintic one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I took pdf of the &lt;a href=&quot;http://www.abelprize.no/c54178/binfil/download.php?tid=53608&quot; rel=&quot;noreferrer&quot;&gt;French manuscript&lt;/a&gt;, put it through Adobe Acrobat's OCR and saved it as a Word document, then translated that using Google Tranlate &lt;a href=&quot;https://translate.google.com/toolkit&quot; rel=&quot;noreferrer&quot;&gt;translator toolkit&lt;/a&gt;. The &lt;a href=&quot;https://translate.google.com/toolkit/workbench?did=002oujh00rvcclugwvls&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;result&lt;/a&gt; is messy but looking at it next to the French text and formulas I can understand most of it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-11-19T21:52:54.750" LastActivityDate="2015-11-19T21:52:54.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3096" PostTypeId="2" ParentId="3089" CreationDate="2015-11-19T21:12:42.030" Score="5" Body="&lt;p&gt;Radon's transform is a good recent example. It was discovered in 1917 by Johann Radon. Computer tomography is based on it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But actually there are plenty of examples. The most famous is the theory of conic sections which had no practical purpose until Kepler found that they describe orbits of celestial bodies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Linear programming can be traced back to Cauchy and Fourier, but real practical applications begin in the middle 20s century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-19T21:12:42.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3097" PostTypeId="2" ParentId="3087" CreationDate="2015-11-19T21:54:36.287" Score="8" Body="&lt;p&gt;Peter Pesic's popular math book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0262661829&quot;&gt;Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability&lt;/a&gt; contains the author's own translation of Abel's 1824 paper as an appendix. He also has some annotated notes which make understanding the exposition easier. &lt;/p&gt;&#xA;" OwnerUserId="3273" LastActivityDate="2015-11-19T21:54:36.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3098" PostTypeId="2" ParentId="3065" CreationDate="2015-11-20T14:39:40.923" Score="2" Body="&lt;p&gt;The Royal Society, founded in 1662, perhaps the first scientific society  ... the majority of the founding members were Puritans.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Not bound by tradition, Puritan schools fostered science. Theodore Haak, a professor at the largely Puritan Gresham College, initiated the meetings of the &quot;invisibles.&quot; Other Protestant schools revolutionized medicine about the same time, and it was a Protestant school which later trained John Dalton, author of modern atomic theory.&#xA;  &lt;a href=&quot;http://www.christianity.com/church/church-history/timeline/1601-1700/scientists-of-faith-found-royal-society-11630137.html&quot; rel=&quot;nofollow&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-11-20T14:39:40.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3099" PostTypeId="1" AcceptedAnswerId="3100" CreationDate="2015-11-20T15:41:04.227" Score="11" ViewCount="2194" Body="&lt;p&gt;Now-a-days a research paper can have many pages (the paper claiming the proof of abc conjecture is of $500$ pages long!). But is there any paper in the history of mathematics which we can say as the shortest paper ever published in a well-known journal which has a remarkable impact? &lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2022-07-16T18:32:41.460" Title="What is the shortest paper in the history of mathematics?" Tags="|mathematics|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3100" PostTypeId="2" ParentId="3099" CreationDate="2015-11-20T20:12:29.760" Score="19" Body="&lt;p&gt;There are many research announcements, sometimes containing few lines, so the question is not very meaningful. French Comptes rendus and old issues of AMS Bulletin have many papers of&#xA;one page length or less. So they can be only compared by counting letters, and&#xA;I do not think this counting is a reasnoable occupation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if we are talking of serious papers, with complete proofs, and containing great discoveries, then this one is hard to beat:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. Milnor, Eigenvalues of the Laplace operator on certain manifolds,&#xA;Proc. Nat. Acad. Sci. 51 N4, 1964, p. 542.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It occupies substantially less than one page, including title, author and reference list. And it is really a great discovery, and frequently cited.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-20T20:12:29.760" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3101" PostTypeId="1" AcceptedAnswerId="3105" CreationDate="2015-11-20T21:33:59.360" Score="-2" ViewCount="583" Body="&lt;p&gt;I somewhere read that ancient Indian science was so advanced even in 2000 B.C. (or may be even before this) that Indian scientist and mathematicians calculated the value of distance between sun and Earth. They even found the value of $\pi$. They made a piller made of iron which resists rusting.They did even more. It is said that they wrote all their discoveries in scriptures which can be found even today. They called it Vedic science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: Is it possible that they could have found and done all these things? If yes, how? If no, how?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not claiming that they did those things. But it has confused me for so long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some sites in the favour:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://forum.davidicke.com/showthread.php?t=202595&quot; rel=&quot;nofollow&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=uKyn3FDaTYc&quot; rel=&quot;nofollow&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.dandavats.com/?p=10946&quot; rel=&quot;nofollow&quot;&gt;3&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here are some against it : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.tifr.res.in/~dani/vmtimeart.pdf&quot; rel=&quot;nofollow&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://nirmukta.net/Thread-Links-on-Pseudoscience-and-Superstition&quot; rel=&quot;nofollow&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://asfak-ur-rahman.blogspot.in/2011/11/list-of-scientific-errors-in-hindu.html&quot; rel=&quot;nofollow&quot;&gt;3&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3222" LastEditorUserId="3222" LastEditDate="2015-11-21T05:27:16.017" LastActivityDate="2020-01-02T23:00:56.530" Title="Is Vedic Science true?" Tags="|mathematicians|mathematical-physics|" AnswerCount="1" CommentCount="2" ClosedDate="2015-11-25T00:24:27.080" ContentLicense="CC BY-SA 3.0" />
  <row Id="3102" PostTypeId="2" ParentId="3099" CreationDate="2015-11-21T01:29:51.330" Score="14" Body="&lt;p&gt;After few searching I found &lt;a href=&quot;http://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf&quot;&gt;this paper&lt;/a&gt; by L. J. Lander and T. R. Parkin on counterexample to Euler's conjecture on the sum of like powers. May be this is the shortest paper?&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2015-11-21T01:29:51.330" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3103" PostTypeId="2" ParentId="2918" CreationDate="2015-11-21T08:33:41.353" Score="-1" Body="&lt;p&gt;It seems that the answer is &lt;a href=&quot;http://www.jstor.org/stable/10.4169/college.math.j.46.3.162?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.The author has the same question with me.&lt;/p&gt;&#xA;" OwnerUserId="2056" LastActivityDate="2015-11-21T08:33:41.353" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3104" PostTypeId="2" ParentId="3099" CreationDate="2015-11-21T08:37:34.140" Score="3" Body="&lt;p&gt;If you see at the bottom of the contents of American Mathematical Monthly it has a section called &quot;Mathbits&quot;.&lt;br&gt;&#xA;These are papers less than a page in length.&lt;br&gt;&#xA;Maybe there you can find many short papers.&lt;/p&gt;&#xA;" OwnerUserId="2056" LastActivityDate="2015-11-21T08:37:34.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3105" PostTypeId="2" ParentId="3101" CreationDate="2015-11-21T10:41:38.940" Score="3" Body="&lt;p&gt;There is &lt;strong&gt;no&lt;/strong&gt; &quot;Vedic science&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an ancient &lt;a href=&quot;https://en.wikipedia.org/wiki/Indian_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indian mathematics&lt;/a&gt; as well as an ancient &lt;a href=&quot;https://en.wikipedia.org/wiki/Indian_astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indian astronomy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are comparable (from an historical point of view; not in term of &quot;better or worse&quot;) to ancient &lt;a href=&quot;https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Egyptian mathematics&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Babylonian_astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Babylonian astronomy&lt;/a&gt;,  &lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chinese astronomy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Regarding the question :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is Vedic Science true?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;if you consider e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Hindu_astrology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jyotisha&lt;/a&gt;, i.e. the traditional Hindu system of astrology, we have the &lt;em&gt;Siddhānta&lt;/em&gt;:  the branch of Jyotisha devoted to calculating the position of the planets and other heavenly bodies :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to this theory, the earth is motionless and is not considered a planet. Seven planets (Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn) revolve around the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, this theory is false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding &lt;em&gt;Samhitā&lt;/em&gt; : Mundane astrology, and &lt;em&gt;Horā&lt;/em&gt; : Predictive astrology, they are clearly not scientific at all in any &quot;modern&quot; sense.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-01-02T23:00:56.530" LastActivityDate="2020-01-02T23:00:56.530" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3106" PostTypeId="2" ParentId="3093" CreationDate="2015-11-21T11:12:38.833" Score="3" Body="&lt;p&gt;See &lt;a href=&quot;http://plato.stanford.edu/entries/dedekind-foundations/&quot; rel=&quot;nofollow&quot;&gt;Dedekind's Contributions to the Foundations of Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Letting aside the issue of &quot;priority&quot; (that are not interesting), we can say in a nutshell that Dedekind's contribution was fundamental to support the &lt;a href=&quot;http://plato.stanford.edu/entries/settheory-early/&quot; rel=&quot;nofollow&quot;&gt;early development of set theory&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the context of his work on algebraic number theory [1871], Richard Dedekind introduced an essentially set-theoretic viewpoint, defining fields and ideals of algebraic numbers. These ideas were presented in a very mature form, making use of set operations and of structure-preserving mappings [Cantor employed Dedekind's terminology for the operations in his own work on set theory around 1880].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[In 1872] Dedekind published a paper in which he provided an axiomatic analysis of the structure of the set $\mathbb R$ of real numbers. He defined it as an &lt;a href=&quot;https://en.wikipedia.org/wiki/Ordered_field&quot; rel=&quot;nofollow&quot;&gt;ordered field&lt;/a&gt; that is also &lt;a href=&quot;https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers&quot; rel=&quot;nofollow&quot;&gt;complete&lt;/a&gt;; completeness in that sense has the &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedean_property&quot; rel=&quot;nofollow&quot;&gt;Archimedean axiom&lt;/a&gt; as a consequence. Cantor too provided a definition of $\mathbb R$ in 1872, employing &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy_sequence&quot; rel=&quot;nofollow&quot;&gt;Cauchy sequences&lt;/a&gt; of rational numbers, which was an elegant simplification of the definition offered by &lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Weierstrass&quot; rel=&quot;nofollow&quot;&gt;Karl Weierstrass&lt;/a&gt; in his lectures. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The Cantor and Dedekind definitions of the real numbers relied implicitly on set theory, and can be seen in retrospect to involve the assumption of a Power Set principle. Both took as given the set of rational numbers, and for the definition of $\mathbb R$ they relied on a certain totality of infinite sets of rational numbers (either sequences, or &lt;a href=&quot;https://en.wikipedia.org/wiki/Dedekind_cut&quot; rel=&quot;nofollow&quot;&gt;Dedekind cuts&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But while Cantor's works was (mainly) involved with &lt;a href=&quot;https://en.wikipedia.org/wiki/Transfinite_number&quot; rel=&quot;nofollow&quot;&gt;transfinite arithmetic&lt;/a&gt; and the problem of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cardinality_of_the_continuum&quot; rel=&quot;nofollow&quot;&gt;Cardinality of the continuum&lt;/a&gt;, Dedekind's work of 1888 dedicated to the theory of the natural numbers (&lt;a href=&quot;https://archive.org/details/wassindundwasso00dedegoog&quot; rel=&quot;nofollow&quot;&gt;Was sind und was sollen die Zahlen?&lt;/a&gt;) applies the set theoretic concepts to &quot;elucidate&quot; and develop a very fundamental piece of our &quot;basic&quot; mathematical knowledge : the natural numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Richard Dedekind, &lt;a href=&quot;https://books.google.it/books?id=zfTlPWe5QFAC&amp;amp;pg=frontcover&quot; rel=&quot;nofollow&quot;&gt;Essays on the Theory of Numbers&lt;/a&gt; (Engl.transl.1901 - Dover reprint). Ref to Cantor (and Weierstrass): &lt;em&gt;Preface to the 1st Ed&lt;/em&gt;, page 36 and &lt;em&gt;Preface to the 2Nd Ed&lt;/em&gt;, page 41;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;I. SYSTEMS OF ELEMENTS&lt;/strong&gt; [page 44] :&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;ol&gt;&#xA;    &lt;li&gt;&lt;p&gt;In what follows I understand by &lt;em&gt;thing&lt;/em&gt; every object of our thought. In order to be able easily to speak of things, we designate them by symbols, e.g.,&#xA;    by letters, and we venture to speak briefly of the thing $a$ or of $a$ simply, when we mean the thing denoted by $a$ and not at all the letter $a$ itself. A thing is completely determined by all that can be affirmed or thought concerning it. A thing $a$ is the same as $b$ (identical with $b$), and $b$ the same as $a$, when all that can be thought concerning $a$ can also be thought concerning&#xA;    $b$, and when all that is true of $b$ can also be thought of $a$. That $a$ and $b$ are only symbols or names for one and the same thing is indicated by the notation $a = b$, and also by $b = a$. If further $b = c$, that is, if $c$ as well as $a$ is a symbol for the thing denoted by $b$, then is also $a = c$. If the above coincidence of the thing denoted by $a$ with the thing denoted by $b$&#xA;    does not exist, then are the things $a, b$ said to be different, $a$ is another thing than $b$, $b$ another thing than $a$; there is some property belonging to the one that does not belong to the other.&lt;/p&gt;&lt;/li&gt;&#xA;    &lt;li&gt;&lt;p&gt;It very frequently happens that different things, $a, b, c, \ldots$ for some reason can be considered from a common point of view, can be associated in the&#xA;    mind, and we say that they form a &lt;em&gt;system&lt;/em&gt; $S$; we call the things $a, b, c, \ldots$ &lt;em&gt;elements&lt;/em&gt; of the system $S$, they are &lt;em&gt;contained&lt;/em&gt; in $S$; conversely, $S$ &lt;em&gt;consists&lt;/em&gt; of these elements. Such a system $S$ (an aggregate, a manifold,&#xA;    a totality) as an object of our thought is likewise a thing; it is completely determined when with tespect to every thing it is determined whether it is an element of $S$ or not. The system $S$ is hence the same as the system $T$, in symbols $S = T$, when every element of $S$ is also element of $T$, and every&#xA;    element of $T$ is also element of $S$. &lt;/p&gt;&lt;/li&gt;&#xA;    &lt;/ol&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We can find quite &lt;em&gt;verbatim&lt;/em&gt; the same introduction in every modern algebra or calculus textbook.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-11-21T11:12:38.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3107" PostTypeId="1" CreationDate="2015-11-21T14:20:27.040" Score="2" ViewCount="128" Body="&lt;p&gt;Are there any functions which were proposed by various mathematicians to be included in the set of elementary functions because of their properties but not considered elementary as of now?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2015-11-21T14:20:27.040" Title="Are there any functions which were proposed as elementary by mathematicians but not considered elementary now?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3108" PostTypeId="1" AcceptedAnswerId="5903" CreationDate="2015-11-22T02:15:14.383" Score="3" ViewCount="318" Body="&lt;p&gt;Stillwell mentions in his &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387942904&quot; rel=&quot;nofollow&quot;&gt;Elements of Algebra&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first to use the term &quot;isomorphism&quot; was Jordan, in his Traite des Substitutions [1870], the first textbook on group theory...Jordan used the word &quot;isomorphism&quot; for both isomorphisms and homomorphisms, but distinguished between the two by calling them &quot;isomorphismes holoedriques&quot; and &quot;isomorphismes meriedriques&quot; respectively.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What do the words holoedriques and meriedriques mean?                       &lt;/p&gt;&#xA;" OwnerUserId="2134" LastEditorUserId="10106" LastEditDate="2020-06-07T18:36:34.007" LastActivityDate="2020-06-07T18:36:34.007" Title="Jordan called isomorphisms (iso.) and homomorphisms &quot;iso. holoedriques&quot; and &quot;iso. meriedriques&quot; respectively; translation of holoe/meried-driques?" Tags="|mathematics|terminology|abstract-algebra|translations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3109" PostTypeId="1" AcceptedAnswerId="3110" CreationDate="2015-11-22T02:48:43.000" Score="2" ViewCount="323" Body="&lt;p&gt;The Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Ratio_test&quot; rel=&quot;nofollow&quot;&gt;ratio test&lt;/a&gt; states that it was first published by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert&quot; rel=&quot;nofollow&quot;&gt;Jean le Rond d'Alembert&lt;/a&gt;. In which of his works did he state the ratio test?&lt;/p&gt;&#xA;" OwnerUserId="1772" LastActivityDate="2015-11-22T06:17:01.777" Title="Where did d'Alembert published the ratio test?" Tags="|mathematics|reference-request|calculus|publishing|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3110" PostTypeId="2" ParentId="3109" CreationDate="2015-11-22T06:17:01.777" Score="2" Body="&lt;p&gt;Following the link given in the Wikipedia, I reached &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k62424s/f192.image&quot; rel=&quot;nofollow&quot;&gt;Opuscules&lt;/a&gt;, I think here he published the ratio test.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2015-11-22T06:17:01.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3112" PostTypeId="2" ParentId="3089" CreationDate="2015-11-23T01:42:44.297" Score="0" Body="&lt;p&gt;Perhaps somewhat the other way around: Arthur C. Clarke (yes, the science fiction writer) invented communication satellites and published about them in 1945. He didn't bother to patent the idea, he thought it would take longer than the patent's duration. First communication satellite was launched in 1958.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2015-11-23T01:42:44.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3113" PostTypeId="2" ParentId="3089" CreationDate="2015-11-23T11:24:57.473" Score="6" Body="&lt;p&gt;I would probably say number theory, which had quite a lot of work on theorems regarding prime numbers that (as far as I know) did not get much use until the development of cryptography, such as the RSA algorithm from the 70's using Fermat's little theorem from the 1640's.&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2015-11-23T11:24:57.473" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3115" PostTypeId="1" CreationDate="2015-11-25T12:15:27.780" Score="1" ViewCount="123" Body="&lt;p&gt;Did we humans actually invent mathematics? Or did we just stumble upon this huge gift and discovered it? I read a book on this question called, &quot;Is God A Mathematician&quot; by Mario Livio but still couldnt ascertain the answer as the book presented the best of both ideas.&lt;/p&gt;&#xA;" OwnerUserId="3284" LastActivityDate="2015-11-25T12:47:36.203" Title="Was mathematics invented or discovered?" Tags="|mathematics|" AnswerCount="0" CommentCount="7" ClosedDate="2015-11-25T22:05:55.837" ContentLicense="CC BY-SA 3.0" />
  <row Id="3117" PostTypeId="2" ParentId="3099" CreationDate="2015-11-25T13:17:17.860" Score="6" Body="&lt;p&gt;&amp;quot;The shortest paper ever published in a serious math journal&amp;quot;, according to &lt;a href=&quot;https://fermatslibrary.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermat's Library&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;J. H. Conway and A. Soifer, &lt;a href=&quot;https://fermatslibrary.com/s/shortest-paper-ever-published-in-a-serious-math-journal-john-conway-alexander-soifer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Can n² + 1 unit equilateral triangles cover an equilateral triangle of side &amp;gt; n, say n + ε?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2924" LastEditorUserId="2924" LastEditDate="2022-02-24T05:39:49.370" LastActivityDate="2022-02-24T05:39:49.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3118" PostTypeId="1" CreationDate="2015-11-25T14:21:03.880" Score="3" ViewCount="269" Body="&lt;p&gt;I have been looking into the history of topology. One thing I am very curious about is the history of the open cover definition of compactness. According to Raman-Sundström, this goes back to a lemma in Borels 1894 thesis. However, I have not been able to find what problem Borel was trying to solve or what kind of insight mathematicians like Heine, Borel and Lebesgue were trying to gain in studying open sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What motivated the open cover definition of compactness?&lt;/p&gt;&#xA;" OwnerUserId="2666" LastActivityDate="2015-11-26T14:42:55.297" Title="History of open cover compactness definition" Tags="|mathematics|topology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3120" PostTypeId="1" CreationDate="2015-11-26T10:25:54.473" Score="7" ViewCount="4186" Body="&lt;p&gt;This is a copy of a &lt;a href=&quot;https://physics.stackexchange.com/q/220558/99592&quot;&gt;question I just asked&lt;/a&gt; at Physics Stack Exchange. From reading the text on the related questions, it seems that Maxwell equated light with the carrier of electromagnetic force just by the fact the two had the same propagation speed. That doesn't seem very rigorous; the two phenomena could have had the same speed by coincidence. Now, later scientists and engineers did find better proof connecting light and electromagnetism, but was Maxwell's assertion just a lucky guess? (AFAIK, light was the only known thing that had the propagation speed, but we have found other particles since then that share that speed.)&lt;/p&gt;&#xA;" OwnerUserId="3289" LastEditorUserId="4251" LastEditorDisplayName="user1709" LastEditDate="2020-06-06T12:59:27.313" LastActivityDate="2020-06-06T12:59:27.313" Title="How did Maxwell conclude that light is an electromagnetic wave?" Tags="|physics|electromagnetism|light|maxwell|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3121" PostTypeId="2" ParentId="3120" CreationDate="2015-11-26T10:41:53.593" Score="2" Body="&lt;p&gt;I think it was speculated even before Maxwell that light might be an electromagnetic wave, or that there is a connection between electromagnetism  and light. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that Maxwell did not prove it, but - as you say - his equations for the first time rigorously predicted electromagnetic waves (and also all other electromagnetic phenomena). Perhaps the stronger indication is that it also predicted the speed of light correctly with only two constants (epsilon and mu) measured in static benchtop experiments. I would say this is - although not a proof - a strong indication and perhaps the people of that time also felt this way. And back than I think the speed of light was the only speed known to be this high.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To my knowledge electromagnetic waves were rigorously proven by direct experiment by Heinrich Hertz, which was after Maxwell write down his equations.&lt;/p&gt;&#xA;" OwnerUserId="2087" LastActivityDate="2015-11-26T10:41:53.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3123" PostTypeId="2" ParentId="3120" CreationDate="2015-11-26T14:37:51.310" Score="8" Body="&lt;p&gt;Maxwell had at least three arguments in favor of the conjecture on electromagnetic nature of light. The first one was philosophical (Chap. XX, section 781). He could not imagine waves propagating in empty space, so for the electromagnetic wave he had to assume the existence of some medium that fills the space. Then he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To fill the space with a new medium whenever any new phenomenon is t be explained is by no means philosophical, but if the study of two different branches of science has independently suggested the idea of a medium and if the properties which must be attributed to the medium in order to account for electromagnetic phenomena are of the same kind as those which we attribute to the luminiferous medium in order to account for the phenomena of light, the evidence for the physical existence of the medium will be considerably strengthened&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then he derives the speed of propagation of electromagnetic waves, not only in vacuum but in different substances, so in fact he computes the index of refraction, in terms of electromagnetic constants, and gets a fair numeric coincidence (at least 2 digits in all cases).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally he discusses Faraday's experiments demonstrating the action of magnetism on light. Faraday was able to alter the plane of polarization of light by electro-magnet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these arguments discussed in Chapter 20 of his treatise make a very convincing case for electromagnetic nature of light. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-26T14:37:51.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3124" PostTypeId="2" ParentId="3118" CreationDate="2015-11-26T14:42:55.297" Score="3" Body="&lt;p&gt;There is a famous paper by Alexandroff &amp;amp; Urysohn, I believe, where they analyze many compactness-type definitions for the recently-defined notion of &quot;topological space&quot;.  Since &quot;compact&quot; was already used to mean: every sequence has a convergent subsequence, they came up with other names.  They used the term &quot;bicompact&quot; for: every open cover has a finite subcover.  In subsequent years, mathematicians came to see that this was the more fundamental notion, and started to use &quot;compact&quot; for it.  But you may still find the term &quot;bicompact&quot; used in some Russian papers.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-11-26T14:42:55.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3125" PostTypeId="1" CreationDate="2015-11-26T18:59:38.427" Score="6" ViewCount="1681" Body="&lt;p&gt;If we have an annual interest rate of $r$, meaning that each year we multiply our capital by $1+r$, but we want to compound it $n$ times throughout the year, then the usual formula for the amount of money at the end of the year is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\left(1+\frac r n\right)^n$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which corresponds to compounding an interest rate of $r/n$ every $n$-th of a year. To me a more logical procedure would be to multiply the capital each $n$-th of a year by $\sqrt[n]{1+r}$, that way you're just interpolating the original annual interest rate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me the usual formula seems like exactly the kind of mistake someone who's not very good at mathematics might make - say a mediocre high school student. &quot;Oh, we're applying an interest rate of $r$ a total of $n$ times throughout the year, so we should divide $r$ by $n$ to even things out&quot;. Err, no, that would make sense if we were &lt;em&gt;adding&lt;/em&gt;, but we're &lt;em&gt;multiplying&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My hypothesis is that this might really be how the formula for compound interest came about - invented by a medieval banker who didn't entirely understand mathematics beyond what he needed to count money. If this is true, then you might say banks are damn lucky that $\lim_{n\to\infty}(1+\frac 1 n)^n$ turned out to be finite! Someone suggested to me that it might have just been that $n$-th roots were too hard to calculate, but surely it wouldn't have been too bad with a log table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, my idle speculation aside (and arrogant criticism of a field I know little about), is it known how old this formula is (even if used implicitly)? Was it invented independantly several times? Do we have any indication what the logic behind it was?&lt;/p&gt;&#xA;" OwnerUserId="50" LastEditorUserId="2442" LastEditDate="2016-08-12T15:46:16.577" LastActivityDate="2016-08-12T15:46:16.577" Title="Where does the formula $(1+\frac r n)^n$ for compound interest come from?" Tags="|mathematics|economics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3126" PostTypeId="1" AcceptedAnswerId="3166" CreationDate="2015-11-26T19:18:08.540" Score="2" ViewCount="290" Body="&lt;p&gt;When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111… Later, Koenig found a way to save the diagonal argument in binary digits (a reference would be appreciated). Meanwhile, the most popular form is to use decimal digits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My main question: Who was the first person to mention that in the diagonal argument digits should not be replaced by 9's because of 0.1999⋯ = 0.2000… ?&lt;/p&gt;&#xA;" OwnerDisplayName="user3292" LastActivityDate="2017-06-11T06:16:35.143" Title="Who was the first to mention the nine problem?" Tags="|set-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3129" PostTypeId="2" ParentId="3126" CreationDate="2015-11-27T14:02:28.543" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is &lt;strong&gt;not&lt;/strong&gt; relevant for the proof.&lt;/p&gt;&#xA;&lt;p&gt;See :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Joseph Dauben, &lt;a href=&quot;https://books.google.it/books?id=-cpFeTPJXDIC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georg Cantor : His Mathematics and Philosophy of the Infinite&lt;/a&gt; (1990)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;José Ferreiros, &lt;a href=&quot;https://books.google.it/books?id=-WL0BwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Labyrinth of thought : A history of set theory and its role in modern mathematics&lt;/a&gt; (2ed 2007)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For detailed discussion of Cantor's proofs:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1873 proof: &lt;a href=&quot;https://books.google.it/books?id=-cpFeTPJXDIC&amp;amp;pg=PA50&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dauben&lt;/a&gt; page 50 and &lt;a href=&quot;https://books.google.it/books?id=-WL0BwAAQBAJ&amp;amp;pg=PA179&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferreiros&lt;/a&gt; page 179.&lt;/p&gt;&#xA;&lt;p&gt;1877 proof: &lt;a href=&quot;https://books.google.it/books?id=-WL0BwAAQBAJ&amp;amp;pg=PA190&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferreiros&lt;/a&gt; page 190-2. See page 190, for the presumibely source of the misconception.&lt;/p&gt;&#xA;&lt;p&gt;1891 proof (the &lt;em&gt;diagonalization&lt;/em&gt; one): &lt;a href=&quot;https://books.google.it/books?id=-cpFeTPJXDIC&amp;amp;pg=PA165&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dauben&lt;/a&gt; page 165 and &lt;a href=&quot;https://books.google.it/books?id=-WL0BwAAQBAJ&amp;amp;pg=PA287&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferreiros&lt;/a&gt; page 287.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For an early textbook version of the diagonal proof, see:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Henry_Young&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Henry Young&lt;/a&gt; &amp;amp; Grace Chisholm Young, &lt;a href=&quot;https://books.google.it/books?id=TIzfPwAACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Theory of Sets of Points&lt;/a&gt;, (1906) page 43 : decimal notation used, and the rule for &amp;quot;diagonalization&amp;quot; is :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;we now define a number [...] &lt;span class=&quot;math-container&quot;&gt;$b=0.b_1b_2b_3 \ldots$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$b_r$&lt;/span&gt; is never the same as &lt;span class=&quot;math-container&quot;&gt;$a_{rr}$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Regarding the purported &amp;quot;mistake&amp;quot; in the diagonal proof, see e.g. Robert Gray &lt;a href=&quot;http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Gray819-832.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georg Cantor and Transcendental Numbers&lt;/a&gt;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, vol. 101, 1994, page 823.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;general&amp;quot; theorem consider an enumeration of an infinite sequnce of infinite strings of symbols, with only two symbols : &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The proof does &lt;strong&gt;not&lt;/strong&gt; rely on any property of real numbers (or their representation).&lt;/p&gt;&#xA;&lt;p&gt;It is easily &amp;quot;applied&amp;quot; to the real numbers through their binary representation.&lt;/p&gt;&#xA;&lt;p&gt;In the case of number with a &amp;quot;dual&amp;quot; representation, both representation must be included into the enumeration.&lt;/p&gt;&#xA;&lt;p&gt;We can consider a simplified case :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a_1=0.000 \ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a_2=0.011 \ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a_3=0.100 \ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and apply Young's procedure; we will get :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$b=0.101 \ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;because: &lt;span class=&quot;math-container&quot;&gt;$b_1 \ne a_{11}=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b_2 \ne a_{22}=1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b_3 \ne a_{33}=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The presence of &lt;span class=&quot;math-container&quot;&gt;$a_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_3$&lt;/span&gt;, that are two &amp;quot;dual&amp;quot; representation of the same number, does not invalidate the diagonalization.&lt;/p&gt;&#xA;&lt;p&gt;With this simple example we see that also if two members of the list, say &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_m$&lt;/span&gt;, are two distinct &lt;em&gt;representations&lt;/em&gt; of the same &lt;em&gt;number&lt;/em&gt; we have that :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b \ne a_n$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$b_n \ne a_{nn}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$b \ne a_m$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$b_m \ne a_{mm}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and this &amp;quot;works&amp;quot; irrespectively of the fact that &lt;span class=&quot;math-container&quot;&gt;$a_n=a_m$&lt;/span&gt; or not.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-06-11T06:16:35.143" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3130" PostTypeId="2" ParentId="3125" CreationDate="2015-11-27T16:46:41.417" Score="0" Body="&lt;p&gt;I am not sure if I am answering the OP's question but let me say this anyway.  I wouldn't say that the formula for compound interest was invented or derived, it is actually a kind of common sense.  Or, at least that is what it seems to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us suppose I have a principle amount $P$ earning interest at a given rate I will call $r$ over a given period of time.  This period of time is not necessarily a day, a month, a week, a year or anything.  Just a period of time.  Let's say the period is every 14 days just for the heck of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the close of the first 14 day period, the principle has grown by the amount of the interest rate $r$.  The formula is:&#xA;$$&#xA;P(1+r)&#xA;$$&#xA;At the end of the second period the new principle amount which is $P(1+r)$ has grown by the interest rate again:&#xA;$$&#xA;P(1+r)(1+r)&#xA;$$&#xA;At the end of $n$ period of 14 days each period the total principle has grown to the amount:&#xA;$$&#xA;P(1+r)(1+r)\cdots(1+r) \quad\textrm{$n$ times}&#xA;$$&#xA;Or,&#xA;$$&#xA;P(1+r)^n&#xA;$$&#xA;Now, the reason for the $\frac{r}{n}$ in the OP's post is of course because interest rates are quoted at annual rate figures.  And, the OP's formula is assuming $n$ such periods in a year.  But, the actual formula that I posted here is merely applying interest per interval which like I said seems to be common &quot;interest&quot; sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Added Edit for Clarification]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the rate $r$ is a annual percentage interest rate, say something like $.03$ for 3 percent annual interest then the monthly interest is simply $\frac{r}{12}$ to account for 12 months per year.  Thus, the formula is now:&#xA;$$&#xA;P\left(1+\frac{r}{12}\right)^n&#xA;$$&#xA;Note that I did not divide $r$ by $n$.  The reason is that $n$ is the number of intervals or in this particular case the number of months.  It is not representative of a year unless you cap $n$ at 12 in the exponent.  The formula the OP posted is technically, small detail though, not general because it assumes that $n=12$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Follow Up Edit]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your multiplicative factor each $n$th of a year, $(1+r)^{\frac{1}{n}}$ would be simple interest, not compound.  That interest paid each &quot;month&quot; over 12 months is just $(1+r)$ times the principle.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastEditorUserId="2075" LastEditDate="2015-11-27T19:51:17.747" LastActivityDate="2015-11-27T19:51:17.747" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3131" PostTypeId="1" CreationDate="2015-11-27T17:08:03.823" Score="47" ViewCount="15808" Body="&lt;p&gt;When I first thought of this question, I wanted to ask: If you could give one mathematician his remaining life until an average age for the time he lived in, who would you choose? However, this question is a bit too subjective for SE. So I decided to modify it to: Which mathematicians died very young or in a tragic way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two obvious candidates are Galois and Abel. I included more detail about them in an answer. But I'd like to know whether there are more examples. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="9325" LastEditDate="2019-03-03T07:46:43.637" LastActivityDate="2024-02-23T16:26:38.090" Title="Which mathematicians died very young or in a tragic way?" Tags="|mathematicians|big-list|" AnswerCount="45" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="3132" PostTypeId="2" ParentId="3131" CreationDate="2015-11-27T17:08:03.823" Score="50" Body="&lt;p&gt;Évariste Galois (1811 - 1832), aged 20, was killed in a duel. He is known for Galois theory and he wrote his most notable results down in the night before the duel. You can also find more information about him and why he was killed in &lt;a href=&quot;https://hsm.stackexchange.com/questions/9/why-was-%C3%89variste-galois-killed&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Niels Henrik Abel (1802 - 1829), aged 26, died of tuberculosis. He is mainly known for proving the Abel-Ruffini theorem, that had been open for hunderds of years, but also for several other results. However, he was barely recognized during his lifetime and because of that he was very poor. He died two days before he got a letter that he was appointed as a professor at the Berlin university. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-11-27T17:08:03.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3133" PostTypeId="2" ParentId="3131" CreationDate="2015-11-27T17:36:37.433" Score="42" Body="&lt;p&gt;Srinivasa Ramanujan (1887 - 1920) died at the age of $32$, according to Wikipedia the cause was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D.A.B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver widespread in Madras, where Ramanujan had spent time. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis, a difficult disease to diagnose, but once diagnosed readily cured.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="2442" LastEditDate="2015-11-28T01:44:27.250" LastActivityDate="2015-11-28T01:44:27.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3134" PostTypeId="2" ParentId="3125" CreationDate="2015-11-27T23:33:43.233" Score="5" Body="&lt;p&gt;Good accountants are aware of the difference between &lt;a href=&quot;https://en.wikipedia.org/wiki/Effective_interest_rate#Calculation&quot; rel=&quot;nofollow&quot;&gt;nominal and effective annual rates&lt;/a&gt;, but it is certainly likely that in practice they were often confused. The formula $\left(1+\frac r n\right)^{nt}$ is incorrect if $r$ is understood as the effective rate, the correct formula is $\left(1+\frac{i^{(n)}} n\right)^{nt}$, where $i^{(n)}$ is the (annualized) nominal rate for $n$ compounding cycles per year. In other words, if the compounding cycle is not a year the rate $i=i^{(n)}/n$ is set directly for it rather than derived from an annual rate, division by $n$ is for convenience. This is simpler, and more transparent, than calculating nominal rates for different $n$ from some prescribed effective rate. The correct relation between rates follows from  $\left(1+\frac{i^{(n)}} n\right)^{n}=1+r$, and simple calculus shows that even in the limit $n\to\infty$ the nominal rates $i^{(n)}$ converge to $\ln(1+r)$, the so-called &quot;force of interest&quot;, rather than $r$. If one wishes to use the effective rate directly the correct formula is $\left(1+r \right)^{t}$ with fractional $t$, which is equivalent to the OP suggestion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for history, &quot;interest on interest&quot; is much older than middle ages, much much older. It is mentioned, along with sample calculations, on Babylonian clay tablets dated to c. 2000–1600 BC. Although there are no compounding cycles of less than a year on the tablets there is one where the interest is compounded once in five years, and scribes already appear to make the &quot;mistake&quot; of adding up annual $20\%$ to five year $100\%$. More likely the agreement was directly that the amount doubles in five years since fractional time explicitly appears in calculations (there are no formulas of course). See &lt;a href=&quot;http://arxiv.org/abs/1510.00330&quot; rel=&quot;nofollow&quot;&gt;Muroi's Oldest Example of Compound Interest in Sumer&lt;/a&gt; (p.7), who also argues that the use of compound interest can be traced to even earlier, c. 2600–2350 BC. Interestingly enough, compound interest was condemned by Aristotle as &quot;usury&quot;, and mostly banned in the Christian world during middle ages. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nominal rates in formulas like $\left(1+\frac{i^{(n)}} n\right)^{nt}$ appear in 17th century, see &lt;a href=&quot;http://www.actuaries.org.uk/documents/compound-interest-seventeenth-century&quot; rel=&quot;nofollow&quot;&gt;Lewin's Compound Interest in Seventeenth Century&lt;/a&gt;. Harriot uses just such a formula to compute interest c. 1615 (p. 428), and even &quot;approximates the limit&quot; $n\to\infty$ using binomial expansion. His description of the answer is &quot;&lt;em&gt;the sum of interest on interest continually for every instant in seven years with the principal of £100 after the rate of 10 in the 100 for a year&lt;/em&gt;&quot;. Harriot keeps $i^{(n)}$ at $10\%$ throughout, which would amount to the same &quot;mistake&quot; if we insist that he keep the effective annual rate fixed rather than the nominal one. Witt's landmark Arithmetical Questions (1613) was the first book entirely devoted to compound interest, and is often called the first accounting textbook. It set many conventions that endure to our day, and is praised for accuracy, attention to detail and clarity of exposition.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-11-27T23:59:46.063" LastActivityDate="2015-11-27T23:59:46.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3135" PostTypeId="2" ParentId="3131" CreationDate="2015-11-28T02:34:32.370" Score="25" Body="&lt;p&gt;Mikhail Yakovlevich Suslin (1894-1919).&#xA;Known for &quot;Suslin sets&quot; and the &quot;Suslin hypothesis&quot;.&#xA;He died of typhus following the Russian Revolution.  He was 25.&#xA;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Suslin.html&quot; rel=&quot;noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently read a very interesting book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0674032934&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Naming Infinity&lt;/em&gt;&lt;/a&gt; about the Moscow school of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2018-06-19T12:34:16.310" LastActivityDate="2018-06-19T12:34:16.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3136" PostTypeId="1" AcceptedAnswerId="3138" CreationDate="2015-11-28T13:57:07.183" Score="5" ViewCount="654" Body="&lt;p&gt;I want to know what lead to the notion of covering spaces, and the evolution of the concept. I understand that topology was not developed to solve problems, but to gain insight into the foundation of mathematics. So, what insight lead to the development of covering spaces?&lt;/p&gt;&#xA;" OwnerUserId="2666" LastEditorUserId="2666" LastEditDate="2017-04-17T06:51:15.823" LastActivityDate="2017-04-17T06:51:15.823" Title="History of covering spaces" Tags="|topology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3138" PostTypeId="2" ParentId="3136" CreationDate="2015-11-28T16:38:12.880" Score="4" Body="&lt;p&gt;You wrote &quot;topology was not developed to [solve?] problems but to gain insight to foundations of mathematics&quot;. This is completely wrong, and the notion of covering is a good example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Klein describes in his memoirs how he discovered what is now called the Uniformization theorem. (This is a very concrete statement from the theory of analytic functions). Then he wrote to Schwarz and Poincaré. Then Schwarz proposed the notion of covering, in particular the universal covering. So it was invented in the attempts to solve a concrete problem.&#xA;The story is described in detail in this paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;William Abikoff, The uniformization theorem. Amer. Math. Monthly 88 (1981), no. 8, 574–592. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, the research of the uniformization theory led to the rigorous dimension theory, one of the earliest deep results of topology (Brouwer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, one of the first results of topology, classification of compact surfaces, comes from Riemann's theory of algebraic functions and Abelian integrals. (Riemann visited Italy where he met Betti, and made him interested in topology.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the &quot;founding father&quot; of modern topology (Poincaré) was clearly stimulated by concrete problems of differential equations and by uniformization theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of the founders of topology (Riemann, Betti, Möbius, Schwarz, Brouwer) was thinking of it as a &quot;foundation of mathematics&quot;. All were solving concrete, specific problems of analysis.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="2442" LastEditDate="2016-02-15T16:25:53.003" LastActivityDate="2016-02-15T16:25:53.003" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3139" PostTypeId="1" CreationDate="2015-11-28T17:27:47.713" Score="4" ViewCount="921" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/%28%CE%B5,_%CE%B4%29-definition_of_limit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; Bolzano and later Weierstrass were the first who gave an &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; definition of continuity and convergence.&#xA;But did they already use the letters &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in their formulation and, if yes, did they have any particular reason for their choice?&lt;/p&gt;&#xA;" OwnerUserId="3297" LastEditorUserId="15817" LastEditDate="2021-12-13T01:17:17.133" LastActivityDate="2021-12-13T01:17:17.133" Title="$\epsilon$-$\delta$ definition of continuity" Tags="|mathematics|notation|real-analysis|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3140" PostTypeId="2" ParentId="3139" CreationDate="2015-11-28T18:53:35.097" Score="7" Body="&lt;p&gt;Regarding the &lt;em&gt;use&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; in the context of the definition of continuity of &lt;em&gt;functions&lt;/em&gt;, we can say that they &amp;quot;were in the air&amp;quot; since Cauchy.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Augustin-Louis Cauchy&lt;/a&gt;'s definition of &lt;em&gt;limit&lt;/em&gt; in terms of &lt;em&gt;infinitesimals&lt;/em&gt; in his &lt;a href=&quot;https://en.wikipedia.org/wiki/Cours_d%27Analyse&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cours d'Analyse&lt;/a&gt; (1821):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Bradley &amp;amp; C.Edward Sandifer (editors), &lt;a href=&quot;https://books.google.it/books?id=M0or-HGe7D0C&amp;amp;pg=PA21&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy's Cours d’analyse : An Annotated Translation&lt;/a&gt; (2009).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As we can see, the definition is purely &amp;quot;verbal&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://books.google.it/books?id=M0or-HGe7D0C&amp;amp;pg=PA26&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 26&lt;/a&gt; for &lt;em&gt;continuity&lt;/em&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; be a function of the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and suppose that for each value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; between two given limits [i.e. in an &lt;em&gt;interval&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$[x_0,x_1]$&lt;/span&gt;], the function always takes a unique finite value. If, beginning with a value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; contained between these limits, we add to the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; an infinitely small increment &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, the function itself is incremented by the difference&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x+\alpha)− f(x),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which depends both on the new variable a and on the value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Given this, the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is a continuous function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; between the assigned limits if, for each value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; between these limits, the numerical value of the difference&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x+\alpha)− f(x)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;decreases indefinitely with the numerical value of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;. In other words, &lt;em&gt;the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is continuous with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We can see in &lt;a href=&quot;https://books.google.it/books?id=M0or-HGe7D0C&amp;amp;pg=PA22&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 22&lt;/a&gt;, in the context of the discussion about &lt;em&gt;infinitely small and infinitely large quantities&lt;/em&gt;, usually denoted with &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; denotes a finite quantity different from zero and &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; denotes a variable number that decreases indefinitely with the numerical value of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, the general form of infinitely small quantities of the first order is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$k \alpha \ \text {or at least} \ k \alpha (1 ± \epsilon).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Regarding &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; occurrences, see  A.-L.Cauchy, &lt;em&gt;Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal&lt;/em&gt;, Paris, 1823; in Oeuvres, series 2, vol.4, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f47&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 44&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a &lt;em&gt;continuous&lt;/em&gt; function between the limits &lt;span class=&quot;math-container&quot;&gt;$x=x_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=X$&lt;/span&gt; [...].&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\delta, \epsilon$&lt;/span&gt; be two very small numbers, the first is chosen so that for all numerical values of &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; less than &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; and for any value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; between the limits &lt;span class=&quot;math-container&quot;&gt;$x_0, X$&lt;/span&gt;, the ratio&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac {f(x+i)− f(x)}{i}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;will always be greater than &lt;span class=&quot;math-container&quot;&gt;$f'(x) - \epsilon$&lt;/span&gt; and lesser than &lt;span class=&quot;math-container&quot;&gt;$f'(x) + \epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Judith V.Grabiner, &lt;a href=&quot;http://www.mr-ideahamster.com/classes/assets/a_evepsilon.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus&lt;/a&gt;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, March 1983, Volume 90, Number 3, pp.185–194.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For &lt;a href=&quot;https://en.wikipedia.org/wiki/Bernard_Bolzano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernard Bolzano&lt;/a&gt;, see the English translation (by S.B.Russ) of Bolzano's paper, &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086080900361&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege&lt;/a&gt; (Prague 1817), page 162 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to a &lt;em&gt;correct definition&lt;/em&gt;, the expression &lt;em&gt;that a function &lt;span class=&quot;math-container&quot;&gt;$fx$&lt;/span&gt; varies according to the law of continuity for all values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; inside or outside certain limits&lt;/em&gt; means just that: if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is some such value, the difference &lt;span class=&quot;math-container&quot;&gt;$f(x + \omega) - fx$&lt;/span&gt; can be made smaller than any given quantity provided &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; can be taken as small as we please.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gert Schubring, &lt;a href=&quot;https://books.google.it/books?id=rMWe3okqPOcC&amp;amp;pg=PA535&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conflicts Between Generalization Rigor and Intuition in the Development of Analysis in 17th-19th Century France and Germany&lt;/a&gt; (2006), page 535, for another source : &lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Georg_Tralles&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johann Georg Tralles&lt;/a&gt; (1763 – 1822).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In the manuscript “&lt;em&gt;On the founding of higher analysis&lt;/em&gt;” to which is attached the note, “&lt;em&gt;Draft of Lecture&lt;/em&gt; (dated 17.3.1808)”&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Tralles described discontinuity as a “jump” in the values of the function (Fol.&#xA;10v), but he also gave an algebraic description, in an early use of the letters &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; , by which he stated that “for however small a number &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; of the radical &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;” (he considers the function to be dependent on an increase &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;) “and for the value &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; of the function corresponding to &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; the inequality &lt;span class=&quot;math-container&quot;&gt;$\alpha^{\epsilon} &amp;gt; (1 - \omega)^{\epsilon}$&lt;/span&gt; must hold” (Fol. 19v).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-04-20T14:15:50.493" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3141" PostTypeId="1" AcceptedAnswerId="3143" CreationDate="2015-11-28T23:19:41.533" Score="5" ViewCount="369" Body="&lt;p&gt;In &lt;a href=&quot;https://books.google.com/books?id=oSioAM4wORMC&amp;amp;pg=PA39&amp;amp;hl=en#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;one of Srinivasa Ramanujan's writings&lt;/a&gt;, he discusses the perimeter of an ellipse, $p$. He finds two approximations (page 39):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;16.&lt;/strong&gt; The following approximations for $p$ were obtained empirically:&#xA;   $$p=\pi\left[3(a+b)-\sqrt{(a+3b)(3a+b)}+\epsilon\right]\tag{49}$$&#xA;   where $\epsilon$ is about $ak^{12}/1048576$;&#xA;   $$p=\pi\left\{(a+b)+\frac{3(a-b)^2}{10(a+b)+\sqrt{a^2+14ab^+b^2}}+\epsilon\right\}\tag{50}$$&#xA;   where $\epsilon$ is about $3ak^{20}/68719476736$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, $a$ and $b$ are the semi-major and semi-minor axes, and $k$ is the eccentricity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did Ramanujan &quot;empirically&quot; find these error values? Did he use geometric measurements, numerical analysis, or something completely different?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2015-11-29T02:40:37.580" Title="How did Ramanujan empirically obtain these errors?" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3143" PostTypeId="2" ParentId="3141" CreationDate="2015-11-29T02:23:30.710" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;http://arxiv.org/abs/math/0506384&quot; rel=&quot;nofollow&quot;&gt;Villarino's paper Ramanujan's Perimeter of an Ellipse&lt;/a&gt; &quot;&lt;em&gt;Ramanujan never explained his “empirical” method of obtaining this approximation, nor ever subsequently returned to this approximation, neither in his published work, nor in his Notebooks. Indeed, although the Notebooks does contain the above approximation (see Entry 3 of Chapter XVIII) the statement there does not even mention his asymptotic&#xA;error estimate...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Almkvist and Berndt in &lt;a href=&quot;http://www.math.illinois.edu/~berndt/gausslandenmonthly.pdf&quot; rel=&quot;nofollow&quot;&gt;Gauss, Landen, Ramanujan, the Arithmetic-eometric Mean, Ellipses, $\pi$, and the Ladies Diary&lt;/a&gt; (1988) speculated that Ramanujan developed a continued fraction expansion for the perimeter based on the infinite series found by Ivory in 1796. That produces his error estimates (p.601). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for proofs, already back &lt;a href=&quot;http://www.jstor.org/stable/3607944&quot; rel=&quot;nofollow&quot;&gt;in 1933 Watson claimed&lt;/a&gt; &quot;&lt;em&gt;I have proved that in these formulae the errors (which are zero in the limiting case of the circle) are in defect and that they increase when the eccentricity is increased&lt;/em&gt;&quot;. But alas like Fermat, and Ramanujan, he withheld the method. Villarino gave a proof of a strengthened version of Watson's claim in 1978.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-11-29T02:40:37.580" LastActivityDate="2015-11-29T02:40:37.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3144" PostTypeId="2" ParentId="3139" CreationDate="2015-11-29T03:01:02.177" Score="5" Body="&lt;p&gt;See the answer at &lt;a href=&quot;https://mathoverflow.net/questions/82302/why-do-we-use-epsilon-and-delta&quot;&gt;https://mathoverflow.net/questions/82302/why-do-we-use-epsilon-and-delta&lt;/a&gt;. In particular, $\varepsilon$ is from the first letter of the French word &quot;erreur&quot; (error). Probably $\delta$ is from the first letter of the French word &quot;différence.&quot;&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2015-11-29T03:01:02.177" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3145" PostTypeId="1" AcceptedAnswerId="3157" CreationDate="2015-11-29T17:31:18.993" Score="12" ViewCount="1051" Body="&lt;p&gt;I am really curious about who was the first one to define the &lt;strong&gt;gyromagnetic ratio&lt;/strong&gt; $\gamma$ of a body, i.e. the ratio of its magnetic dipole momento to its angular momentum. It is a very important concept in quantum mechanics, but I was wondering whether it first appeared in a classical context or in a quantum one, who defined it, and why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can see it appears in the formula for the Larmor precession:&#xA;$$\omega=\gamma B\tag{1}$$ which tells us the frequency of precession of a body in the presence of a magnetic field $B$. Was Larmor the first person to highlight the importance of this ratio? was he the first one that used it in a formula? Was $(1)$ the first formula where $\gamma$ appeared? If he was indeed the first one, could you tell me what was the name of the paper in which he published the results?&lt;/p&gt;&#xA;" OwnerUserId="3301" LastEditorUserId="55" LastEditDate="2015-12-01T04:10:44.437" LastActivityDate="2016-12-16T19:36:45.487" Title="How did gyromagnetic ratio come up before quantum mechanics, and who introduced it?" Tags="|physics|classical-mechanics|quantum-mechanics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3146" PostTypeId="2" ParentId="3131" CreationDate="2015-11-29T18:31:47.300" Score="18" Body="&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Joseph_Liberman&quot; rel=&quot;noreferrer&quot;&gt;Joseph Liberman&lt;/a&gt;&lt;/strong&gt; was killed in WW2 at the age of 23. He defended his thesis during a short vacation in Leningrad. The idea in his thesis is one of the most fruitful in comparison geometry, which was developed further by Alexandrov's school. &lt;/p&gt;&#xA;" OwnerUserId="3303" LastEditorUserId="4251" LastEditDate="2018-05-16T18:09:49.213" LastActivityDate="2018-05-16T18:09:49.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3147" PostTypeId="1" AcceptedAnswerId="13637" CreationDate="2015-11-29T18:41:25.400" Score="4" ViewCount="1828" Body="&lt;p&gt;According to &quot;Squaring the Circle&quot; by Ernest Hobson,&#xA;this formula&#xA;$$\theta =\frac{\sin\theta}{\cos\frac\theta2\cdot \cos\frac\theta4\cdot \cos\frac\theta8\dots}$$&#xA;for $\theta&amp;lt;\pi$ is due to Euler. He gives no reference.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Does anyone knows the reference to the original Euler's work?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Comments.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Likely the formula was obtained by applying&#xA;$$\sin(2{\cdot}x)=2\cdot\sin x\cdot \cos x$$&#xA;recursively. (It seems to be the easiest way.)&#xA;If you plug in $\theta=\tfrac\pi2$ then you get the Vieta's identity &#xA;$$\frac\pi2=&#xA;\frac2{\sqrt{2}}\cdot&#xA;\frac2{\sqrt{2+\sqrt2}}\cdot&#xA;\frac2{\sqrt{2+\sqrt{2+\sqrt2}}} \ldots$$&#xA;This formula seems to be the first known &quot;explicit&quot; expression for $\pi$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;So far it seems that Hobson (and others after Hobson, including Cajori) made a mistake by attributing this formula to Euler.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3303" LastEditorUserId="3303" LastEditDate="2015-12-23T14:30:31.740" LastActivityDate="2021-09-04T16:51:17.150" Title="Euler's formula for product of cosines" Tags="|mathematics|reference-request|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3149" PostTypeId="2" ParentId="3131" CreationDate="2015-11-30T01:38:46.190" Score="24" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gotthold_Eisenstein&quot; rel=&quot;noreferrer&quot; title=&quot;Gotthold Eisenstein&quot;&gt;Gotthold Eisenstein&lt;/a&gt; is a classic example. This prodigious young mathematician was held in the highest esteem by none other than Gauss himself, and he was also given encouragement and support by Jacobi and Kummer. He worked in number theory and analysis, publishing, for example, proofs on the laws of quadratic, cubic and quartic reciprocity. He always suffered from ill health, however, and died at the age of 29 of tuberculosis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pavel_Urysohn&quot; rel=&quot;noreferrer&quot; title=&quot;Pavel Urysohn&quot;&gt;Pavel Urysohn&lt;/a&gt;, known for his eponymous Metrization Theorem and also eponymous Lemma. He died at the age of 26 in a drowning accident off the coast of Brittany, France. He did fundamental work in the then nascent field of point-set topology.&lt;/p&gt;&#xA;" OwnerUserId="137" LastEditorUserId="137" LastEditDate="2015-11-30T01:55:47.037" LastActivityDate="2015-11-30T01:55:47.037" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3150" PostTypeId="1" CreationDate="2015-11-30T03:18:42.907" Score="3" ViewCount="2283" Body="&lt;p&gt;Why is Jan. 1, the start of a new year, several days &lt;em&gt;after&lt;/em&gt; the Winter Solstice, instead of coinciding with a solstice or equinox or other natural annual event? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note: The question does &lt;em&gt;not&lt;/em&gt; ask why New Year's is on January 1 instead of March 1, June 1, etc. &lt;/p&gt;&#xA;" OwnerUserId="2672" LastEditorUserId="2672" LastEditDate="2015-11-30T14:25:02.870" LastActivityDate="2015-12-03T21:48:20.743" Title="Why does the start of the calendar year not correspond to a natural event?" Tags="|terminology|astronomy|naming-conventions|units|time|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3151" PostTypeId="2" ParentId="3147" CreationDate="2015-11-30T05:13:27.660" Score="5" Body="&lt;p&gt;This formula is an immediate consequence of the product expansion of the sine:&#xA;$$\frac{\sin x}{x}=\prod_{n=1}^\infty\left(1-\frac{x^2}{\pi^2n^2}\right),$$&#xA;and of the well-known fact that every integer is a product of a power of $2$&#xA;and an odd integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The infinite product formula for the sine was proved by Euler in&#xA;De summis serierum reciprocarum, Comment. Acad. Sci. Petrop. 7 (1740), 123–&#xA;134. (Read Dec. 5, 1735). (Opera Omnia, Series 1, Vol. 14, pp. 73–86.)&#xA;The formula can be seen in paragraph 5 of Euler's work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good source for Euler's work on this subject is the survey of Lagarias&#xA; Euler's constant: Euler's work and modern development, in the Bulletin AMS.&#xA;50, 4 2013, 527-628.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-11-30T05:13:27.660" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3153" PostTypeId="2" ParentId="3150" CreationDate="2015-11-30T15:13:40.943" Score="3" Body="&lt;p&gt;Because the various &amp;quot;calendar modifications&amp;quot; in Western Europe during the ages were driven by political and religious &amp;quot;interests&amp;quot; and &lt;strong&gt;not&lt;/strong&gt; by scientific ones.&lt;/p&gt;&#xA;&lt;p&gt;European calendar is basically due to Ancient Rome; see &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_calendar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roman calendar&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Roman calendar changed its form several times between the founding of Rome and the fall of the Roman Empire.&lt;/p&gt;&#xA;&lt;p&gt;The original Roman calendar is believed to have been a lunar calendar, which may have been based on one of the Greek lunar calendars. [...] Roman writers attributed the original Roman calendar to Romulus, the mythical founder of Rome around 753 BC. Romulus' calendar had ten months with the spring equinox in the first month : &lt;a href=&quot;https://en.wikipedia.org/wiki/Martius_(month)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Martius&lt;/a&gt; (31 days). [...] The regular calendar year consisted of 304 days, with the winter days after the end of December and before the beginning of the following March not being assigned to any month.&lt;/p&gt;&#xA;&lt;p&gt;Numa Pompilius, the second of the seven traditional kings of Rome, reformed the calendar of Romulus around 713 BC. The Romans considered even numbers to be unlucky, so Numa took one day from each of the six months with 30 days, reducing the number of days in the 10 previously defined months by a total of six days. There were 51 previously unallocated winter days, to which were added the six days from the reductions in the days in the months, making a total of 57 days. These he &lt;em&gt;made into two months, January and February, which he prefixed to the previous 10 months&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;].&lt;/p&gt;&#xA;&lt;p&gt;Julius Caesar, as Pontifex Maximus, reformed the calendar in 46 BC. The new calendar became known as the Julian calendar. Quintilis was renamed as Iulius (July) in honour of Julius Caesar in 44 BC by Mark Antony. The calendar reforms were completed during the reign of his successor Augustus, when the Senate renamed Sextilis as Augustus (August) in 8 BC.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The calendar year originally began on 1 March&lt;/strong&gt;, as is shown by the names of the six months following June (Quintilis = fifth month, Sextilis = sixth month, September = seventh month, etc.). &lt;strong&gt;It is not known when the start of the calendar year was changed to 1 January&lt;/strong&gt;. Ancient authors attributed it to Numa Pompilius. [...] A surviving calendar from the late Republic proves the calendar year started in January before the Julian reform.&lt;/p&gt;&#xA;&lt;p&gt;How years were identified during the Roman monarchy is not known. During the Roman Republic, years were named after the consuls, who were elected annually. Thus, the name of the year identified a consular term of office, not a calendar year. For example, 205 BC was &amp;quot;The year of the consulship of Publius Cornelius Scipio Africanus and Publius Licinius Crassus&amp;quot;, who took office on 15 March of that year, and their consular year ran until 14 March 204 BC. [...] The first day of the consular term changed several times during Roman history. The Senate changed it to 1 January in 153 BC in order to allow consul Quintus Fulvius Nobilior to attack the city of Segeda (in Aragon, Spain) during the Celtiberian Wars. Before then, it was 15 March. Earlier changes are a little less certain. There is good reason to believe it was 1 May for most of the third century BC, until 222 BC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For more details, see :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Hannah, &lt;a href=&quot;https://books.google.it/books?id=qTWPAQAAQBAJ&amp;amp;pg=PA43&quot; rel=&quot;nofollow noreferrer&quot;&gt;Greek and Roman Calendars&lt;/a&gt; (2005);&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;page 43 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In fifth-century Athens [...] The 12 lunar months in Athens were: 1. Hekatombaion, 2. Metageitnion, [...]. The year started on Hekatombaion 1, which occurred on the evening of the first sighting of the new moon's crescent following the summer solstice. Another way of expressing this practice is that the last month of the year included the summer solstice. [...] To maintain alignment with the seasons, a lunar calendar eventually requires the intercalation of a 13th month. In Athens it is usually thought that this was achieved by repeating the sixth month, Poseideon, but the evidence is not so emphatic on this point [...].&lt;/p&gt;&#xA;&lt;p&gt;A fundamental component of the Athenian democracy was a Council of 500&#xA;(the Boule), comprising 50 citizens drawn by lot annually from each of the&#xA;city-state's ten tribes. The representatives of each tribe acted as a Standing&#xA;Committee (prytany) of the Council for a tenth of the year. The year&#xA;was thus divided into ten 'months' (prytanies). In the political calendar of&#xA;Athens, dates were counted according to these ten prytanies of each year's&#xA;Council. [...] How long the two calendars were independent of each other we do not know.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;page 98 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Roman calendar still forms the core of our own western calendar, in terms of the month-names, the lengths of those months and - give or take a tweak or two since antiquity - the length of the year. But the route to the present-day version of the Roman calendar is, as ever, not straightforward.&lt;/p&gt;&#xA;&lt;p&gt;The sequence of the Roman months was the same as it still is in the western calendar: January, February, March, April, May, June, July, August, September, October, November, December. All but July and August derive directly from the calendar of the Roman Republic - those from September to December are in fact the Latin forms of the names, while the rest are anglicised versions of the originals. July and August, on the other hand, stem from two changes in the late Republic/early Empire, when the original names of Quintilis and Sextilis were altered to reflect the names of the two leading political figures of the time, Julius Caesar and Augustus (the former Octavian). In 44 BC, before Julius Caesar's assassination, the Roman Senate decreed that the month Quintilis should be called Iulius after him, because he was born in that month. The month Sextilis was named Augustus, in the lifetime of the emperor of that name, because it was the month in which he gained his first consulship and most important victories.&lt;/p&gt;&#xA;&lt;p&gt;[...] The Romans ascribed the creation of their own calendar to the shadowy figures of Romulus and Numa, respectively the founder and the second king of Rome (ostensibly in the eighth-seventh centuries BC). According to Macrobius, Numa made the calendar lunar, by increasing the Roman year first to 354 days and then to 355, and divided the year into 12 months. The length of the year was increased to 354 days to accord with the time 'in which 12 circuits of the moon are completed', but then Numa afterwards added an extra day 'in honour of the odd number'. To the pre-existing ten months were added January, which was made the first of the year, and February to follow it ahead of March.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;page 102 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The sole surviving Republican calendar predating the reforms of Julius Caesar is that from Antium, south of Rome, the &lt;em&gt;Fasti Antiates Maiores&lt;/em&gt;. This dates to 84-55 BC, and its preserved fragments cover just over half the year. [...] The whole calendar starts on 1 January [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;page 112 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 46 BC Julius Caesar ordered a wholesale revision of the calendar to the point of dismissing the quasi-lunar calendar of the Republic and adopting a purely solar one. The new calendar had an average year of &lt;span class=&quot;math-container&quot;&gt;$365 \frac 1 4$&lt;/span&gt; days,&#xA;with the quarter-day being absorbed into an extra single whole day added&#xA;every fourth year. [...] Significantly, in his own calendar reform Caesar had the services of Egyptian astronomers, whose expertise he called on while he was in Alexandria helping to put Cleopatra back on the throne.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-12-01T16:55:39.970" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3154" PostTypeId="2" ParentId="3131" CreationDate="2015-11-30T16:05:09.487" Score="27" Body="&lt;p&gt;Eugenio Elia Levi (1883-1917), the one of Levi decomposition in Lie algebras, was killed in action during WWI to which he participated as a volunteer. It is often said that his early death played a role in the Italian school of geometry leavng largely unexplored the subject of Lie algebras and Lie groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Andreas Floer (1956-1991) who introduced what is nowadays called Floer homology, ICM plenary speaker in 1990, commited suicide at age of 34. He opened up a completely new subject (symplectic topology) and, in fact, the list of posthomous paper speaks in itself.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2015-11-30T16:05:09.487" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3155" PostTypeId="1" CreationDate="2015-11-30T18:10:10.513" Score="2" ViewCount="67" Body="&lt;p&gt;I have been recently trying to find context and motivation for my Ph.D. Thesis in probability theory. I have not much background on the history of my theme. I just studied it aiming at solving a particular problem involved. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my understanding of the matter is very poor. I lack context. I've been looking at the &lt;a href=&quot;http://www.math.ucla.edu/~biskup/Prague-school/figs/Koenig/PAMsurvey.pdf&quot; rel=&quot;nofollow&quot;&gt;PAM survey from König and Wolff&lt;/a&gt;. But despite their effort to approach the theme in a pedagogical way, I feel that context is not very well established in those notes and the whole subject is much more devoted to abstract notions and mathematical motivations that are alien to my knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I come to you hopping that you might be able to help me in this quest for historical meaning. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've looked in Wikipedia, and as far as I see, the work of Phillip W. Anderson is related to the study of particle physics. This particle physics area, however demands quite some knowledge on quantum and relativistic physics. I am willing to learn those subjects (There is the classes from Leonard Susskind on YouTube) but I am afraid I might miss the connection it has with this equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, the stochastic differential equation I am studying comes from scaling limits of reaction diffusion models in particle systems and have the form:&#xA;$$&#xA;dX_t(p) = (\Delta X_t(p) - X_t^k(p)) d t + X_t^{l/2} dB_t\,.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When $k = 0$ and $l = 2$ this looks a lot with the PAM with the difference that the random field is induced by the state of the system. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first phrase of the intro of the survey is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Random motions in random media are an important subject in probability theory since there have a lot of applications to real-world problems in the sciences, like astrophysics, magnetohydrodynamics, chemical reactions. For these reasons and also because of their mathematical interest, they have been studied a lot for decades, with a particular intensity in the last twenty years. There is a number of different models of random motions in random media, like random walk in random environment, the random conductance model, random walk in random scenery, random walk in a random potential. Because of the variety of models, there is also a variety of questions and of mathematical methods to solve the questions, like homogenisation, subadditive ergodic theorems, Lyapunov exponents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Can you help me find a way to understand those examples and motivations?&lt;/p&gt;&#xA;" OwnerUserId="2630" LastEditorUserId="55" LastEditDate="2015-11-30T21:26:45.937" LastActivityDate="2015-11-30T21:26:45.937" Title="Trying to learn some history of mathematics with parabolic Anderson model (PAM) as theme" Tags="|reference-request|probability|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3156" PostTypeId="2" ParentId="3150" CreationDate="2015-11-30T21:48:28.850" Score="0" Body="&lt;p&gt;There are calendars where the New Year does correspond with a “natural” (or more specifically: astronomical) event. In luni-solar calendars (Babylonian, Ancient Greek, Chinese, Jewish, Indian etc.) the year begins in principle with the new moon following the vernal (or autumnal) equinox. Such calendars are however difficult to work with, as you need to intercalate a thirteenth month at irregular intervals and also because the length of each month varies depending on the time of the new moon. By contrast, the Roman calendar operates with a fixed year of 365 days, adding an extra day every four years. This means that it is relatively easy to manipulate, but it also means that the beginning of the year (and of the months) does not and indeed cannot correspond to any astronomical event.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS. In theory, it should be possible to begin the year at the winter solstice. But to determine the date of the solstice you need to be able to measure the length of the day and the night, which was not easy before the invention of the clock. It is possible that the Roman New Year on 1 January is based on a faulty calculation of the solstice.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2015-12-01T11:25:30.397" LastActivityDate="2015-12-01T11:25:30.397" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3157" PostTypeId="2" ParentId="3145" CreationDate="2015-12-01T04:06:24.007" Score="10" Body="&lt;p&gt;This is a very good question, I wish it got more attention. My answer will only be partial for I had difficulty finding early details on gyromagnetic effect and ratio. The concept comes up every time we have a rotating system of charged particles, because it has angular momentum and creates a magnetic dipole field, and it played an important historical role already in classical physics, but ended up paving the way for the quantum one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Larmor.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Larmor's precession formula and theorem&lt;/a&gt; first appeared in 1894-97 papers summarized in his comprehensive book Aether and Matter (1900), in a way the final summa of classical electrodynamics. Larmor was first to conjecture that electrons orbit a center, based on his precession formula he could then explain splitting and polarisation of spectral lines in a magnetic field. Ironically, according to Warwick, &quot;&lt;em&gt;Larmor is widely remembered by scientists for just two formulae and one theorem which, although correctly attributed to him, have been seen by historians of science as tangential to his main research interests&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least implicitly, the ratio came up earlier in &lt;a href=&quot;http://dafix.uark.edu/~danielk/History/Group/Galison.doc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxwell's work on gyromagnetic effect&lt;/a&gt;. He noted that &quot;electricity circulates with something like momentum or inertia&quot; (even earlier Ampere conjectured that magnetism is caused by &quot;molecular currents&quot; in matter), and concluded that magnetization should cause slight rotation of the magnetized material. Maxwell's attempts to measure it were unsuccessful, but the mantle was picked up by none other than Einstein himself, who wanted to confirm Ampere's hypothesis. Together with de Haas he conducted a series of experiments in 1915-16 with theoretical predictions featuring the gyromagnetic ratio prominently. They denote it $\lambda$ and show that the theory predicts $\lambda=2e/m$, where $m,e$ are the mass and the charge of the electron. Of course, the formula would have been pointless earlier because the mass/charge ratio wasn't known. Their experiments confirmed the prediction, but... only due to a measurement error. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The truth is that the gyromagnetic effect in their experiments was due not to molecular currents, but to electron spin, and because it is $1/2$ the correct value is about half of their classical prediction. Curiously, Barnett who conducted similar experiments at the same time originally got values that differed from Einstein-de Haas ones by a factor of $2-2.3$, but after their publication &quot;corrected&quot; himself down to $1.1-1.4$, see &lt;a href=&quot;http://arxiv.org/pdf/physics/0508199.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeng's Selected History of Expectation Bias in Physics&lt;/a&gt;. However, the discrepancy soon came to light, and went under the name &quot;gyromagnetic anomaly&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On how the anomalous gyromagnetic ratio led to the discovery of electron spin see &lt;a href=&quot;http://www.physics.umd.edu/grt/taj/411c/EinsteindeHaas.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frenkel's paper On the History of Einstein-de Haas Effect&lt;/a&gt;, and &lt;a href=&quot;https://books.google.com/books?id=DN-9m2jSo8YC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galison's book How Experiments End&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-12-16T19:36:45.487" LastActivityDate="2016-12-16T19:36:45.487" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3158" PostTypeId="1" AcceptedAnswerId="3172" CreationDate="2015-12-02T05:02:59.673" Score="4" ViewCount="681" Body="&lt;p&gt;Frequently, 19th century physicists—e.g., Helmholtz or Maxwell—did not use modern-day vector notation, which Gibbs contributed in large part to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Helmholtz in his famous paper on the conservation of energy writes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$X=m\frac{du}{dt},\hspace{1em}Y=m\frac{dv}{dt},\hspace{1em}Z=m\frac{dw}{dt},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $X,Y,Z$ are components of a force and $u=dx/dt$, $v=dy/dt$, $w=dz/dt$ are the components of the tangential velocity $q$. Instead, he could've written this much more concisely as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\vec{F}=m\vec{a},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $\vec{F}=(X,Y,Z)$ and $\vec{a}=(u,v,w)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, he could've written&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$d(q^2)=\frac{d(q^2)}{dx}dx+\frac{d(q^2)}{dy}dy+\frac{d(q^2)}{dz}dz$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;much more concisely as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$d(q^2)=\nabla q^2\cdot d\vec{r}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Why did those physicists use such a cumbersome, redundant way of&#xA;writing what today we'd write with vector notation?&lt;/li&gt;&#xA;&lt;li&gt;Are there some&#xA;expressions that cannot be expressed in vector notation but must be&#xA;expressed in this seemingly cumbersome notation that predated vector&#xA;notation?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In summary:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;What were the criticisms against the introduction of &quot;vector analysis&quot;?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2015-12-04T04:57:13.910" LastActivityDate="2015-12-04T04:57:13.910" Title="What were the criticisms against the introduction of &quot;vector analysis&quot;?" Tags="|physics|notation|mathematical-physics|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3159" PostTypeId="2" ParentId="3158" CreationDate="2015-12-02T12:39:16.140" Score="4" Body="&lt;p&gt;You can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Michael Crowe, &lt;a href=&quot;https://books.google.it/books?id=y5-S5dmVqGIC&amp;amp;pg=PA155&quot; rel=&quot;nofollow&quot;&gt;A History of Vector Analysis&lt;/a&gt; (1967) &lt;strong&gt;Ch.5.IV. Gibbs' Elements of Vector Analysis&lt;/strong&gt;, page 155-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs&quot; rel=&quot;nofollow&quot;&gt;Josiah Willard Gibbs&lt;/a&gt;' &lt;em&gt;Elements&lt;/em&gt; was privately printed in 1881 and 1884. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His pupil &lt;a href=&quot;https://en.wikipedia.org/wiki/Edwin_Bidwell_Wilson&quot; rel=&quot;nofollow&quot;&gt;Edwin Bidwell Wilson&lt;/a&gt; compiled the textbook &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_Analysis&quot; rel=&quot;nofollow&quot;&gt;Vector Analysis&lt;/a&gt;, based on Gibbs' lectures, as Gibbs was at the time busy preparing his book on thermodynamics :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://archive.org/stream/vectoranalysiste00gibbiala#page/n7/mode/2up&quot; rel=&quot;nofollow&quot;&gt;Vector Analysis&lt;/a&gt;, first published in 1901 and based on the lectures that Josiah Willard Gibbs had delivered on the subject at Yale University, did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus, as used by physicists and mathematicians. It went through seven editions (1913, 1916, 1922, 1925, 1929, 1931, and 1943). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oliver_Heaviside&quot; rel=&quot;nofollow&quot;&gt;Oliver Heaviside&lt;/a&gt; was the other co-founder of vector analysis :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Heaviside did much to develop and advocate vector methods and the vector calculus. Maxwell's formulation of electromagnetism consisted of 20 equations in 20 variables. Heaviside employed the curl and divergence operators of the vector calculus to reformulate 12 of these 20 equations into four equations in four variables, the form by which they have been known ever since.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See again &lt;a href=&quot;https://books.google.it/books?id=y5-S5dmVqGIC&amp;amp;pg=PA163&quot; rel=&quot;nofollow&quot;&gt;Crowe's book&lt;/a&gt; : &lt;strong&gt;Ch.5.VII. Heaviside's Electrical Papers&lt;/strong&gt;, page 163-on :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first paper in which Heaviside introduced vector methods was his 1882-1883 paper &quot;&lt;a href=&quot;https://books.google.it/books?id=4d820gph6z0C&amp;amp;pg=PA195&quot; rel=&quot;nofollow&quot;&gt;The Relations between Magnetic Force and Electric Current&lt;/a&gt;,&quot; published in the Electrician. The way in which vectors were introduced by Heaviside is somewhat surprising. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;J.C.Maxwell's &lt;a href=&quot;https://en.wikipedia.org/wiki/A_Treatise_on_Electricity_and_Magnetism&quot; rel=&quot;nofollow&quot;&gt;A Treatise on Electricity and Magnetism&lt;/a&gt; was published in 1873, prior to Gibbs's pamphlet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same for &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_von_Helmholtz&quot; rel=&quot;nofollow&quot;&gt;Hermann von Helmholtz&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Helmholtz studied the phenomena of electrical oscillations from 1869 to 1871, and in a lecture delivered to the Naturhistorisch-medizinischen Vereins zu Heidelberg (Natural History and Medical Association of Heidelberg) on April 30, 1869 titled &lt;em&gt;On Electrical Oscillations&lt;/em&gt; [...]. In 1871, Helmholtz moved from Heidelberg to Berlin to become a professor in physics. He became interested in electromagnetism and the Helmholtz equation is named for him. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding Maxwell's attitude, see Crowe : &lt;strong&gt;Ch.5.V. James Clerk Maxwell: Critic of Quaternions&lt;/strong&gt;, page 127-on :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Of the four books written by Maxwell after 1873 only one mentioned vectors; this is his elementary work on mechanics published in 1876 and entitled &lt;a href=&quot;https://books.google.it/books?id=KfC1BshMZ7QC&amp;amp;pg=PA4&quot; rel=&quot;nofollow&quot;&gt;Matter and Motion&lt;/a&gt;. Herein Maxwell included a short section on the idea of and the addition and subtraction of vectors. Quaternions were not mentioned [page 138].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-02T15:23:26.997" LastActivityDate="2015-12-02T15:23:26.997" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3162" PostTypeId="1" AcceptedAnswerId="3164" CreationDate="2015-12-02T15:15:08.110" Score="3" ViewCount="413" Body="&lt;p&gt;American Mathematical Society lists 25 according to one of the comments in the thread below: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/207477/john-nashs-mathematical-legacy&quot;&gt;https://mathoverflow.net/questions/207477/john-nashs-mathematical-legacy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as indicated in the thread, a user indicated that almost half of these are reprints.  Wikipedia.org gives the number as 21. Also in the above thread one of the authors list 13 as the answer to the question.  Who's right? &lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="4251" LastEditDate="2022-04-26T16:16:01.203" LastActivityDate="2022-04-26T16:16:01.203" Title="How many mathematical publications containing non-reprinted, original mathematics did John Nash publish?" Tags="|mathematics|mathematicians|biographical-details|john-nash|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3164" PostTypeId="2" ParentId="3162" CreationDate="2015-12-02T19:09:24.800" Score="2" Body="&lt;p&gt;Mathscinet database gives all his publications and publications related to him personally. Total is 54. If one removes reprints, translations and publications about him, 14 or 15 original papers remain.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-12-02T19:36:12.270" LastActivityDate="2015-12-02T19:36:12.270" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3165" PostTypeId="1" CreationDate="2015-12-03T01:47:37.257" Score="10" ViewCount="531" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lise_Meitner&quot;&gt;https://en.wikipedia.org/wiki/Lise_Meitner&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Encouraged by her father and backed by his financial support, she went to the Friedrich-Wilhelms-Universität in Berlin where famous physicist Max Planck allowed her to attend his lectures, an unusual gesture by Planck, who until then had rejected any woman wanting to attend his lectures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Did Max Planck really forbid women from attending his lectures (Wikipedia does not give any reference), and if so what was the reason?&lt;/p&gt;&#xA;" OwnerUserId="29" LastEditorUserId="4251" LastEditDate="2020-06-07T20:16:51.753" LastActivityDate="2020-06-07T20:16:51.753" Title="Why did Max Planck forbid women from attending his lectures?" Tags="|biographical-details|physicists|teaching|planck|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3166" PostTypeId="2" ParentId="3126" CreationDate="2015-12-03T09:48:28.520" Score="4" Body="&lt;p&gt;The first mention of the &quot;nine-problem&quot; was by Felix Klein, Vorträge über ausgewählte Fragen der Elementargeometrie, Teubner 1895, p. 42.&#xA;&lt;a href=&quot;http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACV2370.0001.001&quot; rel=&quot;noreferrer&quot;&gt;http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ACV2370.0001.001&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2015-12-03T09:48:28.520" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3167" PostTypeId="2" ParentId="3165" CreationDate="2015-12-03T10:37:28.737" Score="7" Body="&lt;p&gt;For some detail, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;John L. Heilbron, &lt;a href=&quot;https://books.google.it/books?id=hNb427xI-bcC&amp;amp;pg=PA37&quot; rel=&quot;nofollow&quot;&gt;The Dilemmas of an Upright Man : Max Planck As Spokesman for German Science&lt;/a&gt; (1987), page 37, regarding Planck's attitude towards women and academic life.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Regarding Boltzmann's student, Lise Meitner, see page 38-39 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[After Boltzmann's suicide in 1906] Meitner went to Berlin, where with Planck's help she obtained a place in the chemical institute of his fellow professor and academician Emil Fischer. In 1912 Planck appointed her his own assistant, and two years later, together with Fischer, he defeated an effort to draw her away to the University of Prague.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Relevant dates (from Wiki) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Max_Planck&quot; rel=&quot;nofollow&quot;&gt;Max Planck&lt;/a&gt; in &lt;strong&gt;June 1880&lt;/strong&gt; presented his habilitation thesis [...]. With the completion of his habilitation thesis, Planck became an unpaid private lecturer in Munich, waiting until he was offered an academic position.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In &lt;strong&gt;April 1885&lt;/strong&gt; the University of Kiel appointed Planck as associate professor of theoretical physics. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;1891&lt;/strong&gt; - &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_women%27s_rights_(other_than_voting)&quot; rel=&quot;nofollow&quot;&gt;Germany&lt;/a&gt; : Women are allowed to attend university lectures, which makes it possible for individual professors to accept female students if they wish.&#xA;  [But regarding the too broad denomination of &quot;Germany&quot; in 19th Century, see Alexandre's post.]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Women were not allowed to attend public institutions of higher education in Vienna around 1900, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Lise_Meitner&quot; rel=&quot;nofollow&quot;&gt;Lise Meitner&lt;/a&gt; was able to achieve a private education in physics in part because of her supportive parents, and she completed in &lt;strong&gt;1901&lt;/strong&gt; with an &quot;externe Matura&quot; examination at the Akademisches Gymnasium.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Meitner studied physics and went on to became the second woman to obtain a doctoral degree in physics at the University of Vienna in &lt;strong&gt;1905&lt;/strong&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-05T17:11:02.523" LastActivityDate="2015-12-05T17:11:02.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3168" PostTypeId="2" ParentId="3165" CreationDate="2015-12-03T13:55:40.220" Score="11" Body="&lt;p&gt;It is difficult to say what Max Planck really thought on the topic, but &#xA;I suspect that there is nothing special here about Max Planck:&#xA;exclusion of women was common at that time. In many German and other European universities women were explicitly excluded by university regulations. Apparently a professor could make an exception for some particular student, but he definitely had to make some effort for this. So I interpret the phrase &quot;unusual gesture by Planck&quot; not in the sense that &quot;he forbid other women&quot; but as &quot;he did not care to obtain a permission&quot; for most women.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My opinion is based on a similar case (which happened earlier)  when Weierstrass had to fight his university administration to allow Sophia Kowalewski to attend his lectures. Some details of this discussion have been preserved. Weierstrass failed in his fight with the university administration, and had to teach Kowalewski privately.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember one argument made by the opponents of Weierstrass: that presence of a woman in class would distract young male students from concentrating on the lecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I think it would be interesting to make a list of the dates when women were&#xA;admitted for the first time as students in top universities of various countries.&#xA;I mean admitted in a normal way, not as an exception. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to this page:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.hu-berlin.de/en/about/history/huben_html&quot;&gt;https://www.hu-berlin.de/en/about/history/huben_html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;women were admitted to Prussian universities since 1908. This is somewhat different date from what is given tor &quot;Germany&quot; in the answer of Mauro Allegranza who cites Wikipedia, and the difference is relevant for the case we are discussing. The statement in Wikipedia about &quot;Germany&quot; is suspicious, because these questions were decided separately in each German land.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-12-03T19:13:23.020" LastActivityDate="2015-12-03T19:13:23.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3169" PostTypeId="1" CreationDate="2015-12-03T15:44:21.983" Score="3" ViewCount="615" Body="&lt;p&gt;The telegraph model of Weber-Gauss, relying on &lt;a href=&quot;https://en.wikipedia.org/wiki/Weber_electrodynamics&quot; rel=&quot;nofollow&quot;&gt;Weber's electrodynamics&lt;/a&gt;, modeled instantaneous action at a distance of the electric scalar potential (Coulomb potential) &lt;a href=&quot;http://ppp.unipv.it/Collana/Pages/Libri/Saggi/Volta%20and%20the%20History%20of%20Electricity/V%26H%20Sect3/V%26H%20267-286.pdf&quot; rel=&quot;nofollow&quot;&gt;manifesting as propagation of signals in a resistanceless wire&lt;/a&gt;, through the collective forces of charge carriers in the wire. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite seeing many allusions to the notion that there was work on modeling wireless transmission of signals with Weber's electrodynamics I have been unable to locate such a model even for a simple Hertzian dipole acting on a remote test charge. Does such a wireless signaling model exist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some relevant background: Maxwell cited Weber's model in a positive light and objected to it because of a faulty critique by Helmholtz to the effect that it necessarily violated conservation of energy. Weber debunked this critique and later Maxwell retracted his critique but, presumably due to the absence of a way to describe electromagnetic energy propagation through space, he left it at that. This is most ironic not only because the speed of light was first described in Weber's work, but also because the vector potential originated with Weber's colleague, Kirchhoff. And that, presumably, contributed to Maxwell's development of his equations. &lt;/p&gt;&#xA;" OwnerUserId="3328" LastEditorUserId="55" LastEditDate="2015-12-04T01:16:22.737" LastActivityDate="2017-11-18T21:40:38.387" Title="Wireless transmission and speed of light in Weber Electrodynamics" Tags="|theoretical-physics|classical-mechanics|electromagnetism|radiation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3170" PostTypeId="2" ParentId="3150" CreationDate="2015-12-03T21:48:20.743" Score="0" Body="&lt;p&gt;The Roman year started when consuls entered office or vice versa. When the Julian reform was done this was Jan. 1st.&#xA;The position of the sun on the celestial sphere changes by 1 degree per day; around the 1st c BCE it was known that the vernal equinox is at 7-8 degrees from the beginning of the zodiacal constellation Aries, that is, there was a displacement of one week between the 12 sideral/zodiacal divisions and the beginnings of months.&#xA;The Gregorian reform ensures that equinoxes and solstices fall within a two day fixed interval, e.g. Dec 21-22 for winter solstice; achieving absolute precision would be impractical.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2015-12-03T21:48:20.743" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3171" PostTypeId="2" ParentId="3169" CreationDate="2015-12-04T00:36:38.463" Score="1" Body="&lt;p&gt;Maxwell did indeed &lt;a href=&quot;http://www.mathpages.com/home/kmath613/kmath613.htm&quot; rel=&quot;nofollow&quot;&gt;criticize Weber’s law&lt;/a&gt; for electric force in 1864 based on a confusion: “&lt;em&gt;mechanical difficulties, however, which are involved in the assumption of particles acting at a distance with forces which depend on their velocities are such as to prevent me from considering this theory as an ultimate one&lt;/em&gt;”. The “mechanical difficulties” refer to the fact that velocity &lt;em&gt;only&lt;/em&gt; dependent forces must violate the conservation of energy. Weber’s force however also depends on acceleration, and so is consistent with conservation. After realizing his mistake Maxwell shifted his reasoning to what we now call the principle of locality, which he inherited from Faraday and favored already in 1864. The  issue was that in Weber’s electrodynamics the electromagnetic energy and momentum emitted by one body and absorbed by another had no place to be in between, there was no medium of transmission:”&lt;em&gt;I have therefore preferred to seek an explanation of the facts… without assuming the existence of forces capable of acting directly at sensible distances&lt;/em&gt;”. It is worth mentioning that the speed of light plays a different role in Weber’s electrodynamics since there are no waves in it, and the forces act instantaneously at &quot;sensible&quot; distances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF00665930&quot; rel=&quot;nofollow&quot;&gt;Wesley’s Weber Electrodynamics&lt;/a&gt; (which is very sympathetic to Weber) it was exactly its inability to explain Hertz’s electromagnetic signaling that led to Maxwell’s field theory superseding it. So presumably there weren’t satisfactory models at the time. Wesley himself develops what he calls “Weber field theory” by introducing potentials, and develops Weber style model of Pappas-Vaughan experiment with Z-antenna.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-12-04T00:49:43.467" LastActivityDate="2015-12-04T00:49:43.467" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3172" PostTypeId="2" ParentId="3158" CreationDate="2015-12-04T04:55:18.007" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx#1&quot; rel=&quot;noreferrer&quot;&gt;Pierre Duhem&lt;/a&gt; (1861-1916) gave in his &lt;a href=&quot;https://books.google.com/books?id=5mVPK7QBdTkC&amp;amp;pg=PA77&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Aim &amp;amp; Structure of Physical Theory&lt;/em&gt; (1906) p. 77&lt;/a&gt;—commenting on what he considered a characteristic of, generally, English thinkers, i.e., their &quot;ampleness of mind&quot; (Pascal's &lt;em&gt;&lt;a href=&quot;http://sententiaedeo.blogspot.com/2014/05/esprit-de-geometrie-esprit-de-finesse.html?m=1&quot; rel=&quot;noreferrer&quot;&gt;esprit de géométrie&lt;/a&gt;&lt;/em&gt;, vs. the &lt;em&gt;&lt;a href=&quot;http://sententiaedeo.blogspot.com/2014/05/esprit-de-geometrie-esprit-de-finesse.html?m=1&quot; rel=&quot;noreferrer&quot;&gt;esprit de finesse&lt;/a&gt;&lt;/em&gt; characteristic of a &quot;French [or German] mind&quot;)—a criticism of &quot;vector analysis&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But only among the English is ampleness of mind found so frequently as an endemic, traditional habit; thus it is only among English men of science that symbolic algebras, the calculus of quaternions, and &quot;vector analysis&quot; are customary, most of the English treatises making use of these complex and shorthand languages. French and German mathematicians do not learn these languages readily; they never succeed in speaking them fluently or, above all, in thinking directly in the forms which constitute these languages. In order to follow a calculation based on the method of quaternions or of &quot;vector analysis&quot; they have to translate it into a version of classical algebra. One of the French mathematicians, Paul Morin, who had studied most profoundly the different kinds of symbolic calculi, once told me: &quot;I am never sure of a result obtained by the method of quaternions until I have checked it by our old Cartesian algebra.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2015-12-04T04:55:18.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3174" PostTypeId="1" AcceptedAnswerId="3175" CreationDate="2015-12-06T02:15:20.847" Score="17" ViewCount="3636" Body="&lt;p&gt;When proving them the &quot;modern&quot; way (from first principles) it seems impossible to get around proving the identities $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ and the related $\cos$ limit. This itself requires some geometric reasoning and uses the definition of radians that we have today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But surely people knew that $(\sin \theta)'=\cos \theta$ and so forth long before limits came about. Is this just obvious? How would they have justified it?&lt;/p&gt;&#xA;" OwnerDisplayName="user3339" LastActivityDate="2020-07-10T12:42:22.107" Title="How were derivatives of trigonometric functions first discovered?" Tags="|mathematics|calculus|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3175" PostTypeId="2" ParentId="3174" CreationDate="2015-12-06T05:09:44.707" Score="12" Body="&lt;p&gt;Did you try looking in any books on the history of calculus? The following is taken from &quot;The Historical Development of the Calculus&quot; by C. H. Edwards (p. 205 ff).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The inverse sine function (for radian angles in the first quadrant) can be related to an area under an arc of the unit circle, which is $y = \sqrt{1-x^2}$ in the first quadrant. Newton knew from his discovery of the binomial expansion for rational exponents how to write $\sqrt{1-x^2}$ as a power series, which he could then integrate termwise. In this way he found the power series for $\arcsin x$. Then he inverted that series until he recognized the pattern to &quot;establish&quot; the power series of $\sin x$, from which he could find the power series for $\cos x$ as the series for $\sqrt{1-\sin^2x}$ with constant term $1$. From the power series expansions of $\sin x$ and $\cos x$ it is clear that $\sin'x = \cos x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The power series for $\sin x$ and $\cos x$ were known in India long before they were found by Newton. See &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Madhava_series&quot; rel=&quot;noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Madhava_series&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the power series formulas reveal the derivatives of the sine and cosine to us, it is not the case that Newton or Leibniz in the 1600s took this step. The trigonometric functions were not systematically thought of as functions to be graphed and have a derivative. Euler, in the 1700s, was the one who first incorporated the trigonometric functions into calculus. See &quot;The calculus of the trigonometric functions&quot; by Victor Katz, which is available online at &lt;a href=&quot;http://ac.els-cdn.com/0315086087900644/1-s2.0-0315086087900644-main.pdf?_tid=a15b6d9e-9c37-11e5-b17b-00000aacb35e&amp;amp;acdnat=1449420025_6f4ed309a5aed67668cf37b14c276280&quot; rel=&quot;noreferrer&quot;&gt;http://ac.els-cdn.com/0315086087900644/1-s2.0-0315086087900644-main.pdf?_tid=a15b6d9e-9c37-11e5-b17b-00000aacb35e&amp;amp;acdnat=1449420025_6f4ed309a5aed67668cf37b14c276280&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2015-12-14T17:16:36.567" LastActivityDate="2015-12-14T17:16:36.567" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3176" PostTypeId="1" AcceptedAnswerId="3186" CreationDate="2015-12-06T07:27:36.563" Score="23" ViewCount="5733" Body="&lt;p&gt;I try to make my question short. I am familiar with Popper’s philosophy as well as with statistical hypothesis testing after Fisher and Neyman-Pearson. I am not so familiar with the Bayesian approach except that this methodology is defined by including prior probabilities which will influence posterior probabilities. &#xA;Very often, I find that null hypothesis testing is being linked to Popper although it is the result of the works of Fisher and Neyman-Pearson. Also, if I understood Popper’s version of hypothesis testing correctly, he says that one should sincerely try to &lt;strong&gt;disproof&lt;/strong&gt; hypotheses – and I am quite certain that he didn’t mean the &lt;em&gt;null hypothesis&lt;/em&gt; that Fisher formulated but rather the hypothesis that is of critical importance to us. Besides that, I think the timing when Fisher introduced the &lt;em&gt;null hypothesis&lt;/em&gt; was a few years before Popper formulated his account on hypothesis testing.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1&lt;/strong&gt;: If my understanding is correct, isn’t this &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pubmed/23249368&quot;&gt;abstract&lt;/a&gt; partly incorrect? &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;J Sports Sci. 2013;31(9):919-20. doi: 10.1080/02640414.2012.753636. Epub 2012 Dec 19.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Testing the null hypothesis: the forgotten legacy of Karl Popper?&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Wilkinson M.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Abstract&#xA;  Testing of the null hypothesis is a fundamental aspect of the scientific method and has its basis in the falsification theory of Karl Popper. Null hypothesis testing makes use of deductive reasoning to ensure that the truth of conclusions is irrefutable. In contrast, attempting to demonstrate the new facts on the basis of testing the experimental or research hypothesis makes use of inductive reasoning and is prone to the problem of the Uniformity of Nature assumption described by David Hume in the eighteenth century. Despite this issue and the well documented solution provided by Popper's falsification theory, the majority of publications are still written such that they suggest the research hypothesis is being tested. This is contrary to accepted scientific convention and possibly highlights a poor understanding of the application of conventional significance-based data analysis approaches. Our work should remain driven by conjecture and attempted falsification such that it is always the null hypothesis that is tested. The write up of our studies should make it clear that we are indeed testing the null hypothesis and conforming to the established and accepted philosophical conventions of the scientific method.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2&lt;/strong&gt;: Is there any record on what Popper's position was regarding null hypothesis testing (after Fisher and Neyman-Pearson) and the Bayesian approach as tools to gain knowledge in the sciences?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 3&lt;/strong&gt;: Are there any records of communication between Popper and Fisher regarding hypothesis testing?&lt;/p&gt;&#xA;" OwnerUserId="3206" LastEditorUserId="3206" LastEditDate="2015-12-07T05:10:54.773" LastActivityDate="2021-01-05T12:49:26.967" Title="Hypothesis testing: Fisher vs. Popper vs. Bayes" Tags="|philosophy-of-science|statistics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3177" PostTypeId="1" CreationDate="2015-12-06T10:32:48.550" Score="1" ViewCount="97" Body="&lt;p&gt;Where I should look for a biographical note on Yitang Zhang's life. I read the wikipedia article but it does not provide any in depth perception on his life. There is nothing on &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. If anybody knows some sources then please give me a pointer. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2015-12-06T10:32:48.550" Title="Is there any online resource on Yitang Zhang's biography?" Tags="|mathematicians|reference-request|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3178" PostTypeId="2" ParentId="3176" CreationDate="2015-12-06T11:51:46.233" Score="5" Body="&lt;p&gt;Into &lt;a href=&quot;http://plato.stanford.edu/entries/popper/#ProKnoVer&quot; rel=&quot;noreferrer&quot;&gt;Popper&lt;/a&gt;'s &lt;a href=&quot;https://books.google.it/books?id=Yq6xeupNStMC&amp;amp;pg=PA133&quot; rel=&quot;noreferrer&quot;&gt;The Logic of Scientific Discovery&lt;/a&gt; (1934 - 1st Engl.ed.1959) you can find &lt;strong&gt;CH.VIII : Probability&lt;/strong&gt; and several &lt;em&gt;Appendces&lt;/em&gt; devoted to probability theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bayes [Bayes,Th. 144, 168n, 288–9] and Fisher [Fisher,R.A. 18&amp;amp;n, 326, 334, 384,&#xA;394n, 403] are presents into the &lt;em&gt;Index of names&lt;/em&gt;, also if Keynes and von Mises are prominent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The basic &quot;axiom&quot; of Popper's initial view is that probability cannot solve the &lt;a href=&quot;http://plato.stanford.edu/entries/induction-problem/&quot; rel=&quot;noreferrer&quot;&gt;Problem of Induction&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I do not think that it is possible to produce a satisfactory theory of what&#xA;  is traditionally called ‘induction’. On the contrary, I believe that any such theory — whether it uses classical logic or a probability logic — must for purely logical reasons either lead to an infinite regress, or operate with an aprioristic principle of induction, a synthetic principle which cannot be empirically tested.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Scientific theories can never be ‘justified’, or verified. But in spite of&#xA;  this, a hypothesis $A$ can under certain circumstances achieve more than&#xA;  a hypothesis $B$ — perhaps because $B$ is contradicted by certain results of&#xA;  observations, and therefore ‘falsified’ by them, whereas $A$ is not falsified;&#xA;  or perhaps because a greater number of predictions can be derived with the help of $A$ than with the help of $B$. The best we can say of a hypothesis is that up to now it has been able to show its worth, and that it has been more successful than other hypotheses although, in principle, it can never be justified, verified, or even shown to be probable. This appraisal of the hypothesis relies solely upon deductive consequences (predictions) which may be drawn from the hypothesis. &lt;em&gt;There is no need even to mention induction&lt;/em&gt; [ &lt;a href=&quot;https://books.google.it/books?id=Yq6xeupNStMC&amp;amp;pg=PA316&quot; rel=&quot;noreferrer&quot;&gt;page 316-17&lt;/a&gt; ].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-12-06T11:51:46.233" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3179" PostTypeId="1" CreationDate="2015-12-05T16:21:50.487" Score="0" ViewCount="46" Body="&lt;p&gt;When proving them the &quot;modern&quot; way (from first principles) it seems impossible to get around proving the identities $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ and the related $\cos$ limit. This itself requires some geometric reasoning and uses the definition of radians that we have today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But surely people knew that $(\sin \theta)'=\cos \theta$ and so forth long before limits came about. Is this just obvious? How would they have justified it?&lt;/p&gt;&#xA;" OwnerDisplayName="user3339" LastEditorUserId="62" LastEditDate="2015-12-06T14:30:22.437" LastActivityDate="2015-12-06T14:30:22.437" Title="How were derivatives of trigonometric functions first discovered? ($(\sin \theta)'=\cos \theta$ and such)" Tags="|mathematics|calculus|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-12-06T14:31:02.920" ContentLicense="CC BY-SA 3.0" />
  <row Id="3180" PostTypeId="1" CreationDate="2015-12-07T11:11:09.153" Score="2" ViewCount="451" Body="&lt;p&gt;Lots of books and web sites claim that the lituus was first defined by Roger Cotes (correct) but named by Maclaurin in his &quot;Harmonia mensurarum&quot; in 1722. I believe the latter is incorrect on two accounts: 1) Cotes was the first to use the word lituus, not Maclaurin; 2) Harmonia mensurarum (1722) was the book by Cotes where the lituus was defined, and Maclaurin never wrote a book with that title. The Wikipedia artcle on the lituus does not make this error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is why Maclaurin has been pulled into this story? Did Maclaurin ever write about the lituus at all? Or was he somehow involved with the publishing of Cotes' Harmonia mensurarum?&lt;/p&gt;&#xA;" OwnerUserId="3351" LastActivityDate="2016-02-10T18:45:21.993" Title="Maclaurin and the Lituus spiral" Tags="|mathematicians|debunking|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3181" PostTypeId="2" ParentId="3180" CreationDate="2015-12-07T12:32:00.820" Score="1" Body="&lt;p&gt;I think that a possible source of error is the following entry (dated : Jan.1997) of the authoritative web -site of &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/&quot; rel=&quot;nofollow&quot;&gt;MacTutor History of Mathematics archive&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Curves/Lituus.html&quot; rel=&quot;nofollow&quot;&gt;Lituus&lt;/a&gt; : The &lt;em&gt;lituus&lt;/em&gt; curve originated with Cotes in 1722. Lituus means a &lt;em&gt;crook&lt;/em&gt;, for example a bishop's crosier. Maclaurin used the term in his book &lt;em&gt;Harmonia Mensurarum&lt;/em&gt; in 1722. The lituus is the locus of the point $P$ moving in such a manner that the area of a circular sector remains constant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But MacTutor list of Maclaurin's &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Extras/Maclaurin_publications.html&quot; rel=&quot;nofollow&quot;&gt;publications&lt;/a&gt; does not list &lt;a href=&quot;https://books.google.it/books?id=J6BGAAAAcAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Harmonia mensurarum&lt;/a&gt;, while it is named as &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Cotes.html&quot; rel=&quot;nofollow&quot;&gt;Cotes&lt;/a&gt;' posthumous work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://books.google.it/books?id=J6BGAAAAcAAJ&amp;amp;pg=PA85&quot; rel=&quot;nofollow&quot;&gt;Scholium&lt;/a&gt;, page 85 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;quam &lt;em&gt;Litui&lt;/em&gt; Figuram appello propter formae similitudinem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This source can explain why the mistake is repeated into many recent books and web-pages (see e.g. &lt;a href=&quot;http://mathworld.wolfram.com/Lituus.html&quot; rel=&quot;nofollow&quot;&gt;Wolfram MathWorld's entry&lt;/a&gt;, with explicit reference to MacTutor Archive).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can comapre it with some &quot;old&quot; source, like :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Jephson, &lt;a href=&quot;https://books.google.it/books?id=rGo9AAAAYAAJ&amp;amp;pg=PA381&amp;amp;lpg=PA381&quot; rel=&quot;nofollow&quot;&gt;The Fluxional Calculus: An Elementary Treatise&lt;/a&gt; (1826), page 381.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-12T17:20:49.937" LastActivityDate="2015-12-12T17:20:49.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3182" PostTypeId="2" ParentId="3147" CreationDate="2015-12-07T13:49:13.640" Score="1" Body="&lt;p&gt;The formula Hobson is reporting on is unrelated to Euler's famous infinite product formula for sine. The constants in Hobson's formula vary geometrically whereas in Euler's famous product formula they vary quadratically. The latter result is considerably deeper than the former (though perhaps the former is also due to Euler). A discussion of the famous formula can be found in our Notices article here:  &lt;a href=&quot;http://www.ams.org/notices/201307/rnoti-p886.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/notices/201307/rnoti-p886.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2015-12-07T13:49:13.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3183" PostTypeId="1" CreationDate="2015-12-07T16:05:01.620" Score="3" ViewCount="116" Body="&lt;p&gt;The meter and second, at least, have been redefined several times since they became common in science. Sometimes when referencing historical documents, it's useful to know how the units of the time relate to units in the modern day, but I'm finding it hard to locate that information by web searches. Does anyone know a good source (or can provide the information directly)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when referencing this chart (shared by Livermore National Lab),&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.flickr.com/photos/llnl/9403051123/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cW0dW.jpg&quot; alt=&quot;test&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd want to know what the length of a 1944 meter is in current day meters, and similarly for the second, to check whether they account for the difference in the reported values of the physical constants $c$ and $h$.&lt;/p&gt;&#xA;" OwnerUserId="485" LastActivityDate="2015-12-08T16:26:33.907" Title="Former definitions of MKS units" Tags="|units|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3184" PostTypeId="1" AcceptedAnswerId="3185" CreationDate="2015-12-07T16:50:36.877" Score="5" ViewCount="371" Body="&lt;p&gt;Some books like Griffith's begin quantum mechanics with the Schrödinger equation as a postulate while some other text books derive it and state $[x,p]=i \hbar$ as an axiom. I'm not sure which one came first as the operator $\hat{p}$ can be derived from both. Also I'd like to know how did the commutation relation came about because the textbooks I have don't mention about it.&lt;/p&gt;&#xA;" OwnerUserId="3353" LastActivityDate="2015-12-07T22:43:02.007" Title="What came first? The Schrödinger equation or commutation relation between $p$ and $x$" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3185" PostTypeId="2" ParentId="3184" CreationDate="2015-12-07T22:35:36.290" Score="5" Body="&lt;p&gt;If you are asking &quot;which came first&quot; historically, the answer is clear:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Commutation relation was written for the first time in the paper of Born and Jordan, On Quantum Mechanics, submitted on September 27, 1925. It is formula (38) in this paper. English translation is available in Van der Waerden's book, Sources on quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schrodinger equation was published in Annalen der Physik by Scrodinger in his paper Quantisierung als Eigenwertproblem, submitted in January 1926. (I don't know whether there is an English translation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are asking &quot;what comes first&quot; logically, it depends on the system of exposition. In every theory we are free what to take as axioms and how to develop&#xA;it from the axioms. In most modern expositions, Schrodinger equation is taken as an axiom. The system where the commutation relation &quot;comes first&quot; is called &quot;matrix&quot; mechanics, and you can read about it in Dirac'a book.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-12-07T22:43:02.007" LastActivityDate="2015-12-07T22:43:02.007" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3186" PostTypeId="2" ParentId="3176" CreationDate="2015-12-08T02:31:12.210" Score="23" Body="&lt;p&gt;According to Mayo, Popper did not designate statistical tests implementing his logic of falsification, or as Hilborn and Mangel put it &amp;quot;Popper supplied the philosophy, and Fisher, Neyman and colleagues supplied the statistics&amp;quot;, see references in &lt;a href=&quot;https://books.google.com/books?id=VtU3-y7LaLYC&amp;amp;dq=popper+fisher+hypothesis+testing&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quinn and Keough's Experimental Design and Data Analysis for Biologists&lt;/a&gt; (Ch. 3). Popper viewed probability somewhat dismissively because &amp;quot;&lt;em&gt;as scientists, we do not seek highly probable theories, but explanations; that is to say, powerful and improbable theories&lt;/em&gt;&amp;quot;. Nonetheless, the corroboration measure he proposed in 1954 is surprisingly Bayesian (surprisingly because Bayesianism is usually associated with induction Popper rejected).&lt;/p&gt;&#xA;&lt;p&gt;Fisher emphasized that we can never prove a theory, but only refute it, already back in 1925, before Popper's &lt;em&gt;Logik der  Forschung&lt;/em&gt; (1934), but he implemented this principle in a rather peculiar way. Instead of testing the actual hypothesis directly Fisher suggested testing its &amp;quot;negation&amp;quot;, the null hypothesis, if the null hypothesis comes out as unlikely after the test then our actual hypothesis &amp;quot;survived the test&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Why such a detour? Because Fisher's approach can not directly provide what a scientist actually wants, &lt;span class=&quot;math-container&quot;&gt;$p(H|D)$&lt;/span&gt;, the probability of the hypothesis given the test data. It only provides the converse &lt;span class=&quot;math-container&quot;&gt;$p(D|H)$&lt;/span&gt;, the probability of data given the hypothesis is true, this is the famous &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value, a.k.a. statistical significance. &lt;a href=&quot;http://bayesfactor.blogspot.co.uk/2015/03/the-frequentist-argument-against.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;In his 1952 criticism Neyman&lt;/a&gt; framed Fisherian testing as follows, if &lt;span class=&quot;math-container&quot;&gt;$p(D|H)$&lt;/span&gt; is low &amp;quot;&lt;em&gt;what we actually observed would be a miracle. We don't believe in miracles nowadays and therefore we do not believe in H being true&lt;/em&gt;&amp;quot;. Unfortunately, a simple computation shows that &lt;span class=&quot;math-container&quot;&gt;$p(H|D)=\frac{p(D|H)}{p(D)}p(H)$&lt;/span&gt;, and the relation of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value to the probability of interest depends on the prior probability &lt;span class=&quot;math-container&quot;&gt;$p(H)$&lt;/span&gt;, a.k.a. the &amp;quot;base rate&amp;quot;, and reliability of data &lt;span class=&quot;math-container&quot;&gt;$p(D)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Fisher was aware that his &amp;quot;no miracles&amp;quot; argument for the null hypothesis significance testing (NHST) may &lt;a href=&quot;http://polmeth.wustl.edu/media/Paper/gill99.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;break down in some circumstances&lt;/a&gt; (see also the &lt;a href=&quot;http://polmeth.wustl.edu/media/Paper/gill99.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;base rate fallacy&lt;/a&gt;), especially when low probability hypotheses are involved, and emphasized the importance of choosing the right kind of hypotheses.&lt;/p&gt;&#xA;&lt;p&gt;However, a different type of significance testing involving the null and the &amp;quot;alternative&amp;quot; hypothesis, proposed by Neyman and Pearson, is arguably closer to &amp;quot;sophisticated falsificationism&amp;quot; of Lakatos than to Popper's own. Lakatos differs from Popper on two key issues: he allows auxiliary hypotheses to &amp;quot;save&amp;quot; a theory in view of a &amp;quot;falsifying&amp;quot; test, and he encourages keeping a &amp;quot;falsified&amp;quot; theory until a clear alternative emerges. Accordingly, the Neyman-Pearson procedure presupposes adjudication between the null and the &amp;quot;alternative&amp;quot; hypothesis, the one of interest, unlike Fisherian testing that delivers a (potentially flawed) up or down verdict. As with Popper, Neyman and Pearson developed their procedure well before Lakatos formulated his methodology. Despite their differences, Fisher and Neyman more or less came to an understanding in 1970s, and both types of significance testing became &amp;quot;the standard procedure&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;More recently, there has been something of a revolt against significance testing of both types, especially in soft sciences. &lt;a href=&quot;https://books.google.com/books?id=LekYBwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Prevost&lt;/a&gt; (p.25), in 1999&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;the APA Task Force on Statistical Inference seriously thought about banishing tests of significance from all APA journals... Even though they decided not to do it, they still recommended a reduction of the use of tests of significance et suggested to use of more useful methods (e.g. effect sizes, confidence intervals, Bayesian estimations,...)&lt;/em&gt;&amp;quot; [APA is American Psychological Association].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A frequent criticism is that &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-values have little actual &amp;quot;significance&amp;quot; to decision making about research, and in particular, that significance testing does not really implement the &amp;quot;logic of falsification&amp;quot;. See &lt;a href=&quot;https://web.archive.org/web/20170812054959/http://psych.colorado.edu/%7Ewillcutt/pdfs/Nickerson_2000.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nickerson&lt;/a&gt; and &lt;a href=&quot;https://web.archive.org/web/20170808204908/http://people.hss.caltech.edu/%7Ejiji/confirmation-induction/readings/Giere.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giere&lt;/a&gt; for opposing points of view. The root of the problem seems to be that &lt;span class=&quot;math-container&quot;&gt;$p(H|D)$&lt;/span&gt; can not be directly computed, so direct implementation of Popper's philosophy in probabilistic contexts may not be feasible. Bayesian inference is commonly suggested as an alternative.&lt;/p&gt;&#xA;&lt;p&gt;Wilkinson overreaches in his identification of NHST with Popper's falsification, and his critique of induction methods depends on this overreach. The traditional position is more diffused. It suggests that NHST encourages formulation of falsifiable hypotheses and evaluates them based on test data, not that it represents the statistical implementation of Popper's criterion. Fisher's deductive reasoning gets one from the null hypothesis to the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value. But interpreting the latter to judge the actual hypothesis, and formulating the null hypothesis itself, depend on pragmatic considerations that are, in essence, inductive. A Bayesian would object that in her approach these pragmatic considerations are, at least,  quantified, through assignment of prior probabilities, etc., whereas in NHST they are hidden from view. While there are good arguments for preferring NHST to Bayesian inference, Wilkinson's is not one of them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-05T12:49:26.967" LastActivityDate="2021-01-05T12:49:26.967" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3187" PostTypeId="1" AcceptedAnswerId="3191" CreationDate="2015-12-08T06:29:38.187" Score="3" ViewCount="108" Body="&lt;p&gt;I am going to give a lecture to school children on the lives of Newton, Galileo and Einstein. I have enough literature on Einstein but not enough on the lives of Galileo and Newton. So I am looking for some articles written by prolific physicist or historian about the lives of Newton and Galileo. So can anyone suggest me some articles? Thanks folks in advance.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="4251" LastEditDate="2020-06-07T18:36:11.910" LastActivityDate="2020-06-07T18:36:11.910" Title="Biographical articles on Newton and Galileo by physicist or historian" Tags="|biographical-details|reference-request|newton|galileo|physicists|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3188" PostTypeId="1" AcceptedAnswerId="3189" CreationDate="2015-12-08T08:05:44.707" Score="9" ViewCount="252" Body="&lt;p&gt;I recently wondered: Even when diving to just 2-3 meters without any modern equipment one can feel the change in pressure. Do we have any evidence how this was perceived and explained back in ancient times throughout middle ages, besides ascribing deeper water any religious or mythical property? (Archimedes came to my mind, but I wouldn't know how to research this question in detail). &lt;/p&gt;&#xA;" OwnerUserId="3299" LastActivityDate="2018-03-14T15:47:05.047" Title="How was difference in water pressure perceived in ancient cultures or the middle ages?" Tags="|physics|philosophy-of-science|middle-ages|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3189" PostTypeId="2" ParentId="3188" CreationDate="2015-12-08T14:30:01.367" Score="10" Body="&lt;p&gt;One has to distinguish static pressure from the dynamic pressure in a flow.&#xA;Static pressure was understood to some extent by Archimedes and later Hellenistic writers like Hero and Ctesibius. At least their texts that we know do not contain mistakes: all that they wrote was essentially correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dynamic pressure was very poorly understood before the modern times (that is before Bernoulli). Of course one cannot be sure what Archimedes really knew, besides what he wrote in the texts that survived, but a survived Roman engineering work on hydraulics shows very poor understanding of the matter. There is a &lt;a href=&quot;http://penelope.uchicago.edu/Thayer/E/Journals/LacusCurtius/Note_on_Water_Measurements_by_Frontinus*.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern commentary analysing this work&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A translation of the original book is on the same site. The author (Frontinus) was the official responsible for the water supply of Rome, so he probably used the&#xA;best available knowledge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the ancients actually did many complicated hydraulic works and they must have known a lot from experience, but this knowledge is not reflected in the surviving books.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. In connection with this question I started to investigate when was the diving bell used for the first time. Many sources refer to Aristotle who described how Alexander the Great used the diving bell in the siege of Tyre.&#xA;Same story is contained in some 12 century book on Alexander.&#xA;I did not find the reference on diving bell in Aristotle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle also wrote this (in his book on animals):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Just as divers are sometimes provided with instruments for respiration, through which they can draw air from above the water, and thus may remain a long time under the sea, so also have elephants been furnished by nature with their lengthened nostril; and, whenever they have to traverse water, they lift this up above the surface and breathe through it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On my opinion, this passage shows that Aristotle completely misunderstood the problem related to water pressure: what he describes is just non-realistic, neither for a human nor for an elephant:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-03-14T15:47:05.047" LastActivityDate="2018-03-14T15:47:05.047" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3190" PostTypeId="2" ParentId="3183" CreationDate="2015-12-08T16:26:33.907" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Sometimes when referencing historical documents, it's useful to know how the units of the time relate to units in the modern day, but I'm finding it hard to locate that information by web searches.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You can't find that information because it doesn't exist; the values haven't changed. The various redefinitions of the meter did not change what a meter represents. They instead made the definition more precise. The current definition of the meter is consistent with the prototype metre bar constructed in 1889. Maintaining consistency across definitions is a key precept in metrology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Violating this concept is something that should be done with great care and great fanfare. This happened with the liter, which was redefined in 1964 to be exactly a cubic decimeter (or 1/1000 of a cubic meter). This was inconsistent with the pre-1964 liter, which was based on the volume occupied by a kilogram of pure water at 4 °C. Per this prior definition, a liter was a bit larger than a cubic decimeter, by about 28 parts per million. The inconsistent redefinition in 1964 was only made after significant debate and consensus, and it received a lot of fanfare. &lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2015-12-08T16:26:33.907" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3191" PostTypeId="2" ParentId="3187" CreationDate="2015-12-08T23:31:17.563" Score="2" Body="&lt;p&gt;I am not sure how much detail you are looking for. MacTutor has decent short biographies of both &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Galileo.html&quot; rel=&quot;nofollow&quot;&gt;Galileo&lt;/a&gt; and &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Newton.html&quot; rel=&quot;nofollow&quot;&gt;Newton&lt;/a&gt; by O'Connor and Robertson. Einstein wrote a &lt;a href=&quot;http://www.pbs.org/wgbh/nova/physics/einstein-on-newton.html&quot; rel=&quot;nofollow&quot;&gt;eulogy of Newton in 1927&lt;/a&gt;, which also mentions Galileo, but focuses on science rather than personal life. If you want something more detailed and personal there is recent &lt;a href=&quot;https://global.oup.com/academic/product/newton-a-very-short-introduction-9780199298037?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow&quot;&gt;Newton by Iliffe&lt;/a&gt;, who is the historian in charge of the NEWTON project. There is analogous book about &lt;a href=&quot;https://books.google.com/books/about/Galileo.html?id=XcgfAQAAIAAJ&quot; rel=&quot;nofollow&quot;&gt;Galileo by Drake&lt;/a&gt;. It is not by physicist or historian but I'll mention it anyway, Bertold Brecht has a famous play &lt;a href=&quot;http://www.stagework.org.uk/resources/assets/s/synopsis_10076.pdf&quot; rel=&quot;nofollow&quot;&gt;Life of Galileo&lt;/a&gt;, a literary adaptation highlighting his personal struggles.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-12-08T23:31:17.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3192" PostTypeId="2" ParentId="3187" CreationDate="2015-12-09T17:15:43.933" Score="2" Body="&lt;p&gt;Historian Pietro Redondi wrote a fascinating account of Galileo's tumultuous career in a book entitled &lt;em&gt;Galileo Heretic&lt;/em&gt;.  There is certainly more detail here than you will need for your lecture but on the other hand it does not contain any technical material in physics except for some very elementary arguments.  You can use some of the anecdotes to spice up your lecture. See for example &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/069102426X&quot; rel=&quot;nofollow&quot;&gt;http://www.amazon.com/Galileo-Heretic-Pietro-Redondi/dp/069102426X&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2015-12-09T17:15:43.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3193" PostTypeId="1" AcceptedAnswerId="3198" CreationDate="2015-12-10T04:48:37.597" Score="5" ViewCount="750" Body="&lt;p&gt;This question comes from my attempts to give a good answer to the previous question,  &lt;a href=&quot;https://hsm.stackexchange.com/questions/3188/how-was-difference-in-water-pressure-perceived-in-ancient-cultures-or-the-middle&quot;&gt;How was difference in water pressure perceived in ancient cultures or the middle ages?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know from my childhood that &amp;quot;Alexander the Great used a diving bell (some say submarine:-) during the siege of Tyre, even submerged himself&amp;quot;. Trying to find the source of this story I searched Internet for &amp;quot;diving history&amp;quot;, &amp;quot;diving bell history&amp;quot; and so on. Everyone repeats this story, some with reference on Aristotle's &amp;quot;Problemata&amp;quot;, others on&#xA;the Medieval Arabic &amp;quot;Alexandria&amp;quot; (around 1200). Some also mention Roger Bacon (who also wrote on Alexander), without a precise reference.&lt;/p&gt;&#xA;&lt;p&gt;Search in &amp;quot;Problemata&amp;quot; found no mentioning of Alexander or Tyre or siege, not speaking of the diving bell.&#xA;Other works of Aristotle (on animals) show that he had no clue about the principle on which the diving bell works.&lt;/p&gt;&#xA;&lt;p&gt;The citation from Alexandria says that Alexander used a &amp;quot;barrel made of glass&amp;quot;, and there are even pictures of this:-)&lt;/p&gt;&#xA;&lt;p&gt;So I have two specific questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Where did Aristotle write ANYTHING about diving (in a bell or otherwise) during the siege of Tyre by Alexander? (Title and page number).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any credible mentioning of a diving bell before, say 1500, or ANY evidence that it existed or that someone understood how it works?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;EFDIT. Let me try to state shortly what I understood from the long answer of Conifold. One sentence of Aristotle (highlighted in the answer) can be interpreted as that he has heard of something like a diving bell, without himself understanding much about water and air pressure (as his other statements show). The story of Alexander in Tyre is made up by medieval authors.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2021-09-19T14:02:43.323" LastActivityDate="2021-09-19T14:02:43.323" Title="When was the diving bell invented?" Tags="|reference-request|technology|middle-ages|antiquity|aristotle|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3194" PostTypeId="1" CreationDate="2015-12-10T14:46:55.187" Score="3" ViewCount="147" Body="&lt;p&gt;Complement of the set B in A is indicated as A\B (see$_1$). Is there any significance for this slanted version of the symbol? Who first introduced it?     &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Source: $_1$ Fundamental Concepts of Abstract Algebra by Gertrude Ehrlich.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2015-12-10T19:52:19.320" Title="Development of the symbol for the complement of the set B in A" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3195" PostTypeId="2" ParentId="3194" CreationDate="2015-12-10T15:40:13.503" Score="3" Body="&lt;p&gt;Originally a set difference was written $A - B$.  But of course when $A,B$ are sets of numbers, that may also mean $\{a-b : a \in A , b \in B\}$.  So it is confusing.  To reduce confusion, one can see $A \sim B$ for this subtraction, or (most common nowadays) $A \setminus B$ as you note.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I have no information on who first used it.)&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-12-10T15:40:13.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3196" PostTypeId="2" ParentId="3194" CreationDate="2015-12-10T19:15:01.590" Score="5" Body="&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/ISO_31-11#Sets&quot; rel=&quot;noreferrer&quot;&gt;ISO 31-11&lt;/a&gt; part of  international standard ISO 31 (1992) that defines mathematical signs and symbols for use in physical sciences and technology :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$A \setminus B$ : &lt;em&gt;difference&lt;/em&gt; between $A$ and $B$; $A$ &lt;em&gt;minus&lt;/em&gt; $B$. The set of elements which belong to $A$ but not to $B$.&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$A \setminus B = \{ x ∣ x ∈ A ∧ x ∉ B \}$&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;$A − B$ should not be used.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Originally was $A-B$; from &lt;a href=&quot;https://books.google.it/books?id=yvVIdH16k0YC&amp;amp;pg=PA14&quot; rel=&quot;noreferrer&quot;&gt;Hausdorff&lt;/a&gt; (1927, 1937), &lt;a href=&quot;https://books.google.it/books?id=WZLOBQAAQBAJ&amp;amp;pg=PA2&quot; rel=&quot;noreferrer&quot;&gt;Kuratowski&lt;/a&gt; (1933) and &lt;a href=&quot;https://books.google.it/books?id=7eclBQAAQBAJ&amp;amp;pg=PA71&quot; rel=&quot;noreferrer&quot;&gt;Bourbaki&lt;/a&gt; (1939-1957), to &lt;a href=&quot;https://books.google.it/books?id=jV_aBwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Halmos&lt;/a&gt; (1960) and &lt;a href=&quot;https://books.google.it/books?id=hCv-vFu4jskC&amp;amp;pg=PA4&quot; rel=&quot;noreferrer&quot;&gt;Devlin&lt;/a&gt; (1979, 1993).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-10T19:52:19.320" LastActivityDate="2015-12-10T19:52:19.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3197" PostTypeId="2" ParentId="3193" CreationDate="2015-12-10T19:16:22.880" Score="4" Body="&lt;p&gt;For Aristotle :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/workstranslatedi07arisuoft#page/n353/mode/2up&quot; rel=&quot;nofollow&quot;&gt;Problemata&lt;/a&gt; Book XXXII, 960b,30-31 : “divers to respire equally well by letting down a cauldron”.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/partsofanimals00aris#page/192/mode/2up&quot; rel=&quot;nofollow&quot;&gt;Parts of Animals&lt;/a&gt;, Book II, 659a,5 : “Some divers, when they go down into the sea, provide themselves with a breathing-machine, by means of which they can inhale the air from above the surface while they remain for a long time in the water.”&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For Roger Bacon, the &lt;em&gt;locus&lt;/em&gt; rferring to Alexander is &lt;em&gt;De mirabili&lt;/em&gt;; see &lt;em&gt;Friar BACON his Discovery of the Miracles of Art, Nature, and Magick&lt;/em&gt;, transcribed by Dr.Alan R.Young (1659), &lt;a href=&quot;https://archive.org/stream/FriarBaconHisDiscoveryOfTheMiraclesOfArtNatureAndMagick/Miracle#page/n29/mode/2up&quot; rel=&quot;nofollow&quot;&gt;page 18&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For the &lt;em&gt;colimpha&lt;/em&gt; attributed to Alexander the Great, it seems is due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Jerome&quot; rel=&quot;nofollow&quot;&gt;Jerome&lt;/a&gt;, see &lt;a href=&quot;http://www.uni-mannheim.de/mateo/camenaref/ducange/bd1/jpg/s1157.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;. Presumibely, the attribution &quot;migrated&quot; subsequently from Alexander to his teacher Aristotle.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-11T10:19:07.820" LastActivityDate="2015-12-11T10:19:07.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3198" PostTypeId="2" ParentId="3193" CreationDate="2015-12-10T21:07:07.150" Score="5" Body="&lt;p&gt;Historians believe that the &lt;a href=&quot;https://archive.org/stream/worksofaristotle07arisuoft/worksofaristotle07arisuoft_djvu.txt&quot; rel=&quot;nofollow&quot;&gt;extant version of Problemata&lt;/a&gt; was not penned by Aristotle personally, but “while the Problemata is not the genuine Aristotelian work, it nevertheless contains an element derived from such a work”.  Problemata XXXII.5 discusses breathing under water, including the oft quoted passage interpreted as referring to a diving bell:”&lt;em&gt;Why do sponge-divers slit their ears and nostrils? Is it in order that the breath may pass more freely? For it is by this way that the breath seems to pass out; for it is said that they suffer more from difficulty of breathing by being unable to expel the breath, and they are relieved when they can as it were vomit the breath forth. It is strange, then, that they cannot achieve respiration for the sake of its cooling effect; this appears to be a greater necessity. Is it not quite natural that the strain should be greater when the breath is held, since then they are swollen and distended? But there appears to be a spontaneous passage of the breath outwards; and we must next consider whether breathing inwards is so also. Apparently it is; &lt;strong&gt;for they enable the divers to respire equally well by letting down a cauldron;  for this does not fill with water, but retains the air, for it is forced down straight into the water; since, if it inclines at all from an upright position, the water flows in&lt;/em&gt;&lt;/strong&gt;”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Alexander story does not appear in Problemata, and has all the hallmarks of a late anecdote. The siege of Tyre did happen in 332 BC, and divers were used to destroy underwater defenses, the same happened even earlier at the 413 BC siege of Syracuse according to Thucydides. But the &quot;&lt;em&gt;very fine barrel made entirely of white glass&lt;/em&gt;&quot; called calimpha, the two companions, the bottom of the sea, the &quot;&lt;em&gt;large and powerful fish devour the small fry&lt;/em&gt;&quot; are likely Alexandriad’s own fable. It got better as time went by, Alexander went 600 feet underwater in a glass case when he was 11, see &lt;a href=&quot;http://www.allempires.com/forum/printer_friendly_posts.asp?TID=22009&quot; rel=&quot;nofollow&quot;&gt;Bachrach’s History of the Diving Bell&lt;/a&gt;. The conflation likely happened by associating Aristotle as Alexander’s mentor and the cauldron of Problemata with the calimpha fable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presumably sponge divers had some working understanding of the bell’s principle already in Aristotle’s time, and presumably it was reinvented multiple times in different places. A documented account of bell’s use does not appear until 16th century however, Bacon’s alleged contraption c. 1240 notwithstanding. &lt;a href=&quot;https://books.google.com/books?id=fg0AAAAAMAAJ&amp;amp;dq=bacon+diving+bell+1240&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow&quot;&gt;Tasnier reported&lt;/a&gt; (p. 94) that in Toledo in 1538 he saw two Greeks lowering themselves under water in a large inverted kettle in the presence of Charles V, the inventor might have been de Lorena who is credited with &quot;inventing&quot; the bell in 1535 by some sources. Norwood took out a patent on diving devices in 1632, but by then Drebbel already built a working submersible (1620), and by the end of 1620s improved it enough to transport king James I. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anything like modern understanding of water pressure etc. does not appear until Torricelli, even &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768090/&quot; rel=&quot;nofollow&quot;&gt;Galileo had trouble explaining siphons&lt;/a&gt; before that. In 1683 Sinclair published a book called &lt;a href=&quot;http://www.abebooks.com/servlet/BookDetailsPL?bi=17001723919&amp;amp;searchurl=tn%3Dnew%2520diving%2520bell&quot; rel=&quot;nofollow&quot;&gt;Natural Philosophy Improven by New Experiments Touching the Mercurial Weather-Glass, the Hygroscope, Eclipsis, Conjunctions of Saturn and Jupiter, by New Experiments, Touching the Pressure of Fluids, the Diving-Bell, and All the Curiosities Thereof&lt;/a&gt; (sic!), which might have been the first modern work on physics of diving bells. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-12-10T22:47:44.947" LastActivityDate="2015-12-10T22:47:44.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3199" PostTypeId="1" AcceptedAnswerId="3248" CreationDate="2015-12-09T17:20:04.870" Score="5" ViewCount="50" Body="&lt;p&gt;When was the word mesosynteny first used to describe the unique features of fungal genomes? How old is the discovery?&lt;/p&gt;&#xA;" OwnerUserId="3474" OwnerDisplayName="rosysnake" LastEditorUserId="2905" LastEditDate="2015-12-20T08:39:47.027" LastActivityDate="2015-12-21T22:24:37.510" Title="How old is the idea of effector islands in fungi?" Tags="|biology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3200" PostTypeId="2" ParentId="2857" CreationDate="2015-12-11T09:38:42.433" Score="15" Body="&lt;p&gt;I just wrote &lt;a href=&quot;http://link.springer.com/article/10.1007/s00407-015-0169-5&quot;&gt;a paper going through the case against the geometrical algebra and arguing that the modern consensus against it is ill-founded&lt;/a&gt; and &lt;a href=&quot;http://intellectualmathematics.com/blog/geometrical-algebra-a-parable/&quot;&gt;an associated blog post&lt;/a&gt;. But my opinion is a minority one. The geometrical algebra hypothesis has indeed been declared dead with considerable confidence by the modern generation of historians. In my introduction I quote these statements from establishment historians, which are a very accurate description of the state of the matter in recent years: “Unguru’s position [against geometrical algebra] could now be regarded as the accepted orthodoxy.” “It is clear that the old historiography has been overcome. … There are very few who still believe in such historiographical artefacts as … geometric algebra.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in the comments, see also &lt;a href=&quot;http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1199&amp;amp;context=jhm&quot;&gt;my critique the modern consensus in the historiography of mathematics in general&lt;/a&gt;, as well as its &lt;a href=&quot;http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1235&amp;amp;context=jhm&quot;&gt;reply from a representative of the modern establishment opinion&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="370" LastEditorUserId="370" LastEditDate="2015-12-11T09:45:28.577" LastActivityDate="2015-12-11T09:45:28.577" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3201" PostTypeId="2" ParentId="3091" CreationDate="2015-12-11T10:59:26.107" Score="3" Body="&lt;p&gt;There was no one person who decided one day to use Greek letters for all of their mathematics. It was a very slow process. The person to popularize the use of $ \pi $ to signify the mathematical value was Euler. He also started the use of $ \Sigma $ for summations.&#xA;Alternatively, Francois Viete was the first to use letters to represent unknowns. This in all likelihood led to the introduction of Greek letters when Latin characters ran out.&lt;/p&gt;&#xA;" OwnerUserId="3365" LastEditorUserId="3365" LastEditDate="2015-12-11T11:10:56.533" LastActivityDate="2015-12-11T11:10:56.533" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3202" PostTypeId="2" ParentId="3089" CreationDate="2015-12-11T11:04:46.390" Score="3" Body="&lt;p&gt;One mathematical idea that went unused in physics until decades later was the idea on non-Euclidean space. Gauss and Riemann among others had formulated the idea of space that did not follow Euclidean geometry and defined rules for how the two other geometries (hyperbolic and elliptic) would work. It was not until Einstein that it was utilized by physics to model the universe because it was Einstein that theorized/made popular that the universe is a non-Euclidean space, and that Euclidean geometry is simply a good enough model at the scales we had been looking at.&lt;/p&gt;&#xA;" OwnerUserId="3365" LastActivityDate="2015-12-11T11:04:46.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3203" PostTypeId="1" CreationDate="2015-12-12T11:10:14.643" Score="4" ViewCount="2411" Body="&lt;p&gt;What was behind accepting the existence of irrational numbers historically? Especially numbers that are not constructible on the real number line, say for example $\sqrt[3]{2}$. Was it a (somewhat) rigorous proof, or was it only an agreement or convention among mathematicians in those days?&lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="55" LastEditDate="2019-10-26T07:41:28.007" LastActivityDate="2023-01-23T03:51:11.323" Title="How were irrational numbers accepted by mathematicians?" Tags="|mathematics|mathematicians|philosophy-of-science|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3204" PostTypeId="2" ParentId="3089" CreationDate="2015-12-12T15:24:49.527" Score="2" Body="&lt;p&gt;I am not an expert on the history of the transistor, but I know enough for the purposes of this question (If anyone knows more, I would love to hear about).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://en.wikipedia.org/wiki/Transistor&quot; rel=&quot;nofollow&quot;&gt;transistor&lt;/a&gt; (the piece of electronics found in effectively everything you use) came from an obscure branch of quantum mechanics which was called &lt;a href=&quot;http://en.wikipedia.org/wiki/Surface_science&quot; rel=&quot;nofollow&quot;&gt;surface physics&lt;/a&gt;.  The theoretical work began in the early 1900's during the beginning of quantum and had no immediate (or even foreseeable) application/use.  By the late 1940's to early 1950's, the first prototypes of the transistor were developed (I believe the first was built at &lt;a href=&quot;https://en.wikipedia.org/wiki/Bell_Labs&quot; rel=&quot;nofollow&quot;&gt;Bell Labs&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the year 2006, the transistor alone would be responsible for over 30 trillion dollars (that is \$30,000,000,000,000 USD) in revenue (this comes from a 2007 MIT Masters thesis on a market evaluation of semiconductor technologies).  A more recent report by &lt;em&gt;McKinsey&amp;amp;Company&lt;/em&gt; found that search engines alone generate some $780 billion USD GDP annually across global markets.  Thus, a seemingly intangible theory can result in very profitable applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a recent economic study [van Bochove, 2012] that showed the financial benefits of pure research.  In that study, they state:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Directing basic research towards economic opportunities is detrimental to growth and may reduce the growth rate by as much as one half. The steady state is shown to be globally stable; in the steady state, the growth rate is independent of the research intensity, but the level of income depends on it. Given current OECD levels of R&amp;amp;D spending and saving, a one dollar increase of applied R&amp;amp;D spending will increase national income with 6-25 dollars and one dollar extra basic research by 20-100 dollars. These rates of return are ten and thirty times higher, respectively, than those on physical capital investment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;br&gt;&#xA;van Bochove, C.A. &quot;Basic Research and Prosperity: Sampling and Selection of Technological Possibilities and of Scientific Hypotheses as an Alternative Engine of Endogenous Growth,&quot; &lt;em&gt;CWTS Working Paper Series&lt;/em&gt;, Leiden University, 2012.&lt;/p&gt;&#xA;" OwnerUserId="3373" LastActivityDate="2015-12-12T15:24:49.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3205" PostTypeId="2" ParentId="3203" CreationDate="2015-12-12T17:52:52.803" Score="4" Body="&lt;p&gt;For details, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Doubling_the_cube&quot; rel=&quot;nofollow&quot;&gt;Doubling the cube&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A significant development in finding a solution to the problem was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths $a$ and $2a$, the duplication of the cube is equivalent to finding segments of lengths $r$ and $s$ so that&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$a : r = r : s = s : 2a.$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In turn, this means that : $r=a \sqrt[3]{2}$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the cube involve the cissoid of Diocles, the conchoid of Nicomedes, or the &lt;a href=&quot;https://en.wikipedia.org/wiki/Philo_line&quot; rel=&quot;nofollow&quot;&gt;Philo line&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also : Thomas Heath, &lt;a href=&quot;https://books.google.it/books?id=J-kaAgAAQBAJ&amp;amp;pg=PA262&quot; rel=&quot;nofollow&quot;&gt;A History of Greek Mathematics. Volume I&lt;/a&gt; (1921), page 262-on for details and discussion.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;If I read correctly the solution, the construction (that obviously is &lt;strong&gt;not&lt;/strong&gt;   constructible by compass and straightedge only) ends up with two segments $RS$ and $RW$ that are in proportion $1 : \sqrt[3]{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, subject to Conifold's remarks above about the fact that, for ancient Greek mathematics, $\sqrt[3]{2}$ is not a number, it seems evident to me that this is a &quot;sufficient&quot; basis for accepting the existence of a segment of that length.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-15T13:37:49.040" LastActivityDate="2015-12-15T13:37:49.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3206" PostTypeId="1" AcceptedAnswerId="3208" CreationDate="2015-12-12T18:24:56.280" Score="7" ViewCount="171" Body="&lt;p&gt;I came across a WWI service record for a relative which includes his medical history. It notes that he was vaccinated in infancy:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/M9hks.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/M9hks.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The person in question was born in 1897. I note on this simple &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_vaccines&quot; rel=&quot;noreferrer&quot;&gt;timeline of vaccines&lt;/a&gt; that there were several vaccines that had been developed by then:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1796: First vaccine for smallpox&lt;/li&gt;&#xA;&lt;li&gt;1879: First vaccine for cholera&lt;/li&gt;&#xA;&lt;li&gt;1885: First vaccine for rabies&lt;/li&gt;&#xA;&lt;li&gt;1890: First vaccine for tetanus&lt;/li&gt;&#xA;&lt;li&gt;1896: First vaccine for typhoid fever&lt;/li&gt;&#xA;&lt;li&gt;1897: First vaccine for bubonic plague&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Which vaccines did infants in England routinely receive in the late nineteenth century? How commonly were these vaccines used, and were any mandatory?&lt;/p&gt;&#xA;" OwnerUserId="2826" LastActivityDate="2015-12-13T08:52:45.863" Title="What vaccines did infants receive in late nineteenth century England?" Tags="|medicine|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3208" PostTypeId="2" ParentId="3206" CreationDate="2015-12-13T08:52:45.863" Score="6" Body="&lt;p&gt;Vaccination was made compulsory in England and Wales in 1853, with 1853 &lt;a href=&quot;https://en.wikipedia.org/wiki/Vaccination_Act&quot; rel=&quot;noreferrer&quot;&gt;Vaccination Act&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.gla.ac.uk/schools/socialpolitical/research/economicsocialhistory/historymedicine/scottishwayofbirthanddeath/vaccination/&quot; rel=&quot;noreferrer&quot;&gt;Vaccination against smallpox&lt;/a&gt; was one of the first major achievements of modern medicine, stemming from Edward Jenner's experiments in 1796. [...]  The reform of the English Poor Law in 1834, followed by the Registration Act of 1836, gave England a basis for a public vaccination service, and in 1840 the first Vaccination Act enabled the new Poor Law guardians to set up public vaccination services, usually through their own Poor Law medical officers.  A year later, this service became free.&lt;/p&gt;&#xA;&lt;p&gt;Vaccination was not compulsory in England until 1853, when the law began to involve the local registrars of births, marriages and deaths.  [...] Further legislation in 1867 tried to tighten up the law: to force guardians to provide adequate services, to compel doctors to fill in the certificates efficiently, and, in particular, to make guardians prosecute parents who did not have their infants vaccinated.  In spite of all this, there were few prosecutions, many children were not vaccinated, and during the  smallpox epidemic in 1871, over 23,000 people died.&lt;/p&gt;&#xA;&lt;p&gt;As with other aspects of registration, Scotland was slow to follow England, but then tried to improve on the English model.  The English Vaccination Act of 1840 could not have been applied to Scotland, which at that time had neither substantial Poor Law authorities, nor registrars. [...] Scotland did finally get a Vaccination Act in 1863.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-12-13T08:52:45.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3209" PostTypeId="1" CreationDate="2015-12-13T10:13:26.277" Score="6" ViewCount="435" Body="&lt;p&gt;The direction in which leading research is heading in these subjects (Math, Physics) is very much different and don't seem to be in tandem. Is this something that developed in more recent times? This is strictly my belief. This query is not meant to target any particular field, but to clarify this stereotype for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically we have seen many mathematicians contribute to physics and vice versa. For example Newton, Gauss, Einstein, Hilbert, Poincaré, etc. But now this communication between mathematicians and physicists seems to have reduced. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just want to know if this is true or a misconception. Provide some resources if possible with your opinion. Also, if true, how does history suggest that we work towards their reunification?&lt;/p&gt;&#xA;" OwnerUserId="3377" LastEditorUserId="3377" LastEditDate="2019-02-24T04:29:07.793" LastActivityDate="2019-02-24T04:29:07.793" Title="Nowadays I see a distinct &quot;line&quot; dividing people working in Mathematics and the Physical Sciences. Why?" Tags="|mathematicians|astronomy|philosophy-of-science|mathematical-physics|cosmology|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3210" PostTypeId="1" CreationDate="2015-12-13T19:39:03.263" Score="2" ViewCount="96" Body="&lt;p&gt;This is a follow up question to &lt;a href=&quot;https://hsm.stackexchange.com/q/3120/2380&quot;&gt;How did Maxwell conclude that light is an electromagnetic wave?&lt;/a&gt; where the arguments of Maxwell were explained why he considered light as an electromagnetic wave. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the furhter milestones towards more and more direct and clearer experimental tests of the electric and magnetic properties of light waves?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure if the following experiment would consider as such a milestone because it demonstrated the magnetic property of light rather directly: &lt;a href=&quot;http://physics.aps.org/story/v26/st13&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://physics.aps.org/story/v26/st13&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2380" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-12-13T19:39:03.263" Title="Historical milestones in experimental &quot;verification&quot; that light is a electromagnetic phenomenum" Tags="|physics|electromagnetism|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3211" PostTypeId="2" ParentId="3209" CreationDate="2015-12-14T01:50:32.523" Score="9" Body="&lt;p&gt;You do not say what field of mathematics you are working in, and perhaps there are signs of separation there. Overall however, lively interaction between mathematics and physics is alive and well. John Baez has a blog &lt;a href=&quot;http://math.ucr.edu/home/baez/twfshort.html&quot;&gt;This Week's Finds in Mathematical Physics&lt;/a&gt;, that is full of contemporary examples of it, so does &lt;a href=&quot;https://terrytao.wordpress.com/&quot;&gt;Terence Tao&lt;/a&gt;. Nature, a leading journal in empirical sciences, just published a mathematical paper &lt;a href=&quot;http://www.nature.com/nature/journal/v528/n7581/full/nature16059.html&quot;&gt;Undecidebility of the Spectral Gap&lt;/a&gt; related to one of the millenium problems, which in turn came from quantum field theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example is the Fields medal awarded to Witten for his insights into the mathematical structure of gauge theories and low-dimensional topology that were obtained by combining advanced mathematics with heuristic physical reasoning. Converse example is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Homological_mirror_symmetry&quot;&gt;mirror symmetry conjecture&lt;/a&gt; in algebraic geometry that came from string theory physics and led to foundational work by some of our leading mathematical lights like Givental, Kontsevich, Fukaya, etc., which is ongoing. The Gromov-Witten theory, a major mathematical advance in algebraic and symplectic geometry, grew out of this circle of ideas. &lt;a href=&quot;http://arxiv.org/abs/math/0609847&quot;&gt;Okounkov received Fields medal&lt;/a&gt; in 2006 for his mathematical work in it partially done in collaboration with leading physicists like Vafa. More recently, large N duality and &lt;a href=&quot;https://www.newton.ac.uk/event/hol&quot;&gt;holography conjectures&lt;/a&gt; received the same treatment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not restricted to frontier quantum physics. General relativity continues to present deep and intricate problems in (pseudo-)Riemannian geometry, and cosmologists like Hawking are adept at proving mathematical theorems about pseudo-Riemannian manifolds, see e.g. work on the &lt;a href=&quot;http://journals.aps.org/prd/abstract/10.1103/PhysRevD.46.603&quot;&gt;chronology protection conjecture&lt;/a&gt;. Physical insights by Mandelbrot and Feigenbaum in 1970s led to the modern theory of fractals and chaotic dynamical systems. Perelman's work on the Ricci flow and the geometrization conjecture was as driven by physical inspiration as was Riemann's, one of the central notions in his proof is a &lt;a href=&quot;https://terrytao.wordpress.com/tag/perelman-entropy/&quot;&gt;new kind of entropy&lt;/a&gt; in a statistical canonical ensemble.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the best evidence for the link is the 1990s controversy, where some mathematicians became concerned that the link between mathematics and physics became so strong that it was undermining the integrity of mathematics by blurring the lines between mathematical and physical &quot;levels of rigor&quot;, and threatening to make mathematical results unreliable. The controversy was sparked by Jaffe and Quinn, who sugested explicit separation of &quot;theoretical mathematics&quot; with lowered standars of rigor, from the body of mathematics proper. Take a look at &lt;a href=&quot;http://arxiv.org/abs/math/9404229&quot;&gt;responses from who's who in modern mathematical physics&lt;/a&gt;: Atiyah, Glimm, Maclane, Mandelbrot, Thom, Uhlenbeck, Witten, etc.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2015-12-14T01:50:32.523" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3213" PostTypeId="2" ParentId="3203" CreationDate="2015-12-14T02:57:13.053" Score="25" Body="&lt;p&gt;Let me clarify a couple of things. No student of Pythagoras discovered irrational numbers, although this is a common misconception, Pythagoreans and even Euclid did not associate numbers with geometric points or segments, the only numbers available were positive integers. Instead they had magnitudes of different dimensions (segments, areas, volumes), and ratios of numbers and magnitudes. The latter produce what we call rational numbers, but also things like the ratio of circumference to diameter, which to them was &lt;em&gt;not&lt;/em&gt; a number. But ratios could be compared, and what Pythagoreans discovered is that the ratio of the diagonal to the side of a square was not equal to any ratio of integers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the problem was in fact the converse, geometry and segments were the starting point, and integer ratios did not provide enough equipment to account for all geometric segments. Eudoxus then worked out a theory of ratios, presented in Euclid's Elements, that allowed to deal with ratios of magnitudes without relating them to any number ratios at all. This being said, if we modernize &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt; in the same vein that what Pythagoreans did is modernized in terms of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; a construction of it was known since Archytas of Tarentum (c.428–347 BC), it just wasn't a straightedge and compass construction. Archytas constructed two segments that (we would say) have this ratio by intersecting a torus, a cone, and a cylinder. Greeks knew it as solving duplication of the cube problem. Later Menaechmus (c.380-320 BC) gave a simpler construction by intersecting two parabolas, or a parabola and a hyperbola, you can see details here &lt;a href=&quot;https://hsm.stackexchange.com/questions/2309/why-did-the-ancient-greeks-originally-become-interested-in-conic-sections/2310#2310&quot;&gt;Why did the ancient Greeks originally become interested in conic sections?&lt;/a&gt; Greeks and later mathematicians had no reason to doubt that conics intersect any more than that lines and circles do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the problem later mathematicians faced was not finding something like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt; on a geometric line, but of accepting what they already knew produced such ratios as new kinds of &quot;numbers&quot;. The process was indeed slow, medieval Islamic and European mathematicians called things like &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2}$&lt;/span&gt; &quot;absurd&quot; (deaf-mute, unheard of), &quot;irrational&quot;, &quot;irregular&quot;, &quot;inexplicable&quot;, etc., and even Cardano in 16th century uses them sparingly. &lt;a href=&quot;https://books.google.fr/books/about/L_arithmetique.html?hl=fr&amp;amp;id=1dU5AAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;Stevin's  L'Arithmétique (1585)&lt;/a&gt; is considered the first work to advocate what we now call real numbers in their totality, by identifying them with infinite decimal fractions, and using iterated tenfold subdivision to locate the corresponding point on the line. According to van der Waerden this &quot;&lt;em&gt;general notion of a real number was accepted, tacitly or explicitly, by all later scientists&lt;/em&gt;&quot;. For details see &lt;a href=&quot;http://arxiv.org/pdf/1107.3688.pdf&quot; rel=&quot;noreferrer&quot;&gt;Stevin Numbers and Reality by Katz and Katz&lt;/a&gt;, who add that &quot;&lt;em&gt;concerns  about the reality of numbers generally preoccupy cognitive scientists and philosophers more than mathematicians&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-05T08:38:02.743" LastActivityDate="2019-11-05T08:38:02.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3214" PostTypeId="1" AcceptedAnswerId="3221" CreationDate="2015-12-14T08:49:22.133" Score="9" ViewCount="1628" Body="&lt;p&gt;The notion of a matrix appeared far ahead of that of a vector space. So when did people start considering a matrix as a linear transformation between two vector spaces? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="55" LastEditDate="2015-12-16T00:41:42.843" LastActivityDate="2015-12-21T21:38:01.910" Title="When did people start viewing a matrix as a linear transformation between two vector spaces?" Tags="|mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3215" PostTypeId="1" AcceptedAnswerId="3216" CreationDate="2015-12-14T08:22:01.970" Score="4" ViewCount="268" Body="&lt;p&gt;Modern atomic theory of matter is often attributed to Dalton in 1803, with roots going all the way back to Democritus, but I was wondering about early modern antecedents of Dalton's theory, perhaps in the 18th or even early 19th century. Were the general outlines of the atomic theory of matter accepted already at the dawn of the scientific revolution, who were the physicists involved, what were the elements of Dalton's theory that were already generally accepted before he came on the scene, and what were the genuinely new aspects in Dalton's formulation?&lt;/p&gt;&#xA;" OwnerUserId="604" OwnerDisplayName="katz" LastEditorUserId="604" LastEditDate="2015-12-14T16:00:47.623" LastActivityDate="2015-12-15T17:39:46.773" Title="Sources of early modern atomic theory of matter?" Tags="|physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3216" PostTypeId="2" ParentId="3215" CreationDate="2015-12-14T19:06:25.883" Score="3" Body="&lt;p&gt;Regarding &lt;em&gt;Early Modern Atomism&lt;/em&gt; and chemistry, some possible sources of information are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;for &lt;a href=&quot;http://plato.stanford.edu/entries/atomism-modern/&quot; rel=&quot;nofollow&quot;&gt;Atomism&lt;/a&gt;  :Robert Hugh Kargon, &lt;a href=&quot;https://books.google.it/books/about/Atomism_in_England_from_Hariot_to_Newton.html?id=MBfQAAAAMAAJ&quot; rel=&quot;nofollow&quot;&gt;Atomism in England from Hariot to Newton&lt;/a&gt; (1970) and Antonio Clericuzio, &lt;a href=&quot;https://books.google.it/books?id=qO4sBAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Elements, Principles and Corpuscles: A Study of Atomism and Chemistry in the Seventeenth Century&lt;/a&gt; (2000)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;for &lt;a href=&quot;http://plato.stanford.edu/entries/gassendi/#5&quot; rel=&quot;nofollow&quot;&gt;Pierre Gassendi&lt;/a&gt; : Lynn Sumida Joy, &lt;a href=&quot;https://books.google.it/books?id=ZjU4nq2ARzAC&amp;amp;pg=PR7&quot; rel=&quot;nofollow&quot;&gt;Gassendi the Atomist: Advocate of History in an Age of Science&lt;/a&gt; (1987) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and, of course,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;for &lt;a href=&quot;http://plato.stanford.edu/entries/boyle/&quot; rel=&quot;nofollow&quot;&gt;Robert Boyle&lt;/a&gt; :  Steven Shapin &amp;amp;Simon Schaffer, &lt;a href=&quot;https://books.google.it/books?id=U0WYRhODPLAC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life&lt;/a&gt; (2nd ed, 2011) and William R. Newman, &lt;a href=&quot;https://books.google.it/books?id=hy5sxK7pHGIC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Atoms and Alchemy: Chymistry &amp;amp; the Experimental Origins of the Scientific Revolution&lt;/a&gt; (2006).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also very useful :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Katharine Park &amp;amp; Lorraine Daston (editors), &lt;a href=&quot;https://books.google.it/books?id=2kvjAp92zcYC&amp;amp;pg=PA497&quot; rel=&quot;nofollow&quot;&gt;The Cambridge History of Science. Volume 3 : Early Modern Science&lt;/a&gt; (2006); see : &lt;strong&gt;Ch. 21 From Alchemy to “Chymistry”&lt;/strong&gt;, by William R.Newman, page 497-on&lt;/li&gt;&#xA;&lt;li&gt;Roy Porter (editor), &lt;a href=&quot;https://books.google.it/books?id=KDSqLsOHc9UC&amp;amp;pg=PA375&quot; rel=&quot;nofollow&quot;&gt;The Cambridge History of Science. Volume 4 : Eighteenth-Century Science&lt;/a&gt; (2003);  see : &lt;strong&gt;Ch.16 : Chemistry&lt;/strong&gt;, by Jan Golinski, page 375-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2015-12-14T19:06:25.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3217" PostTypeId="2" ParentId="3214" CreationDate="2015-12-14T19:19:22.300" Score="4" Body="&lt;p&gt;This happened gradually, one cannot pinpoint a precise moment. Probably the most important step in this respect was publication of the first volume of Hilbert Courant,&#xA;Methods of Mathematical Physics in 1924. Until the invention of quantum mechanics, few people have heard of matrices. But after that, all started reading Hilbert Courant, and gradually this point of view prevailed.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-12-14T19:19:22.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3218" PostTypeId="2" ParentId="3215" CreationDate="2015-12-14T21:55:27.557" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://scholastic.us.to/Dictionary%20of%20Scientific%20Biography/13.%20Bo%C5%A1kovi%C4%87%20b.%201711%20(Markovi%C4%87).pdf&quot; rel=&quot;nofollow&quot;&gt;Rudjer J. Bošković&lt;/a&gt; (1711-1787), who predates Dalton, could be considered the father of modern atomic physics. See his works &lt;a href=&quot;https://www.worldcat.org/identities/lccn-n80119415/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, especially his &lt;em&gt;&lt;a href=&quot;https://archive.org/details/theoryofnaturalp00boscrich&quot; rel=&quot;nofollow&quot;&gt;A Theory of Natural Philosophy&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mention Bošković (also spelled Boscovich) because he was after Newton (and his corpuscles) and before Faraday, who introduced the field concept and said in 1844 that&lt;blockquote&gt;the safest course appears to be to assume as little as possible, and that is why the atoms of Boscovich appear to me to have a great advantage over the more usual notion,&lt;/blockquote&gt;and Maxwell, who said in 1877 that&lt;blockquote&gt;the best thing we can do is to get rid of the rigid nucleus and substitute an atom of Boscovich.&lt;/blockquote&gt;Kelvin wrote in 1899:&lt;blockquote&gt;Hooke’s exhibition of the forms of crystals by piles of globes, Navier’s and Poisson’s theory of the elasticity of solids, Maxwell’s and Clausius’ work in the kinetic theory of gases…all developments of Boscovich’s theory pure and simple.&lt;/blockquote&gt;Kelvin wrote in 1905:&lt;blockquote&gt;My present assumption is Boscovichianism pure and simple.&lt;/blockquote&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(See: &lt;em&gt;&lt;a href=&quot;https://www.worldcat.org/title/roger-joseph-boscovich-sj-frs-1711-1787-studies-of-his-life-and-work-on-the-250th-anniversary-of-his-birth/oclc/609777521&quot; rel=&quot;nofollow&quot;&gt;Roger Joseph Boscovich&lt;/a&gt;&lt;/em&gt; by Whyte.)&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2015-12-15T17:39:46.773" LastActivityDate="2015-12-15T17:39:46.773" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3219" PostTypeId="2" ParentId="98" CreationDate="2015-12-15T10:03:51.013" Score="4" Body="&lt;p&gt;This is adding to previous answers, but the philosopher Immanuel Kant is another interesting example. In the 1780-1790s, he appears to have had rather well-developed ideas about evolutionary change, a common origin of life, that also included humans:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The agreement of so many genera of animals in a certain common schema, which appears to be fundamental not only in the structure of their bones but also in the disposition of their remaining parts,—so that with an admirable simplicity of original outline, a great variety of species has been produced by the shortening of one member and the lengthening of another, the involution of this part and the evolution of that,—allows a ray of hope, however faint, to penetrate into our minds, that here something may be accomplished by the aid of the principle of the mechanism of nature (without which there can be no natural science in general). This analogy of forms, which with all their differences seem to have been produced according to a common original type, strengthens our suspicions of an actual relationship between them in their production from a common parent, through the gradual approximation of one animal-genus to another—from those in which the principle of purposes seems to be best authenticated, i.e. from man, down to the polype, and again from this down to mosses and lichens, and finally to the lowest stage of nature noticeable by us, viz. to crude matter. And so the whole Technic of nature, which is so incomprehensible to us in organised beings that we believe ourselves compelled to think a different principle for it, seems to be derived from matter and its powers according to mechanical laws (like those by which it works in the formation of crystals).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(from: Kant. 1790. &lt;em&gt;Critique of Judgment&lt;/em&gt;, MacMillian 1914 (translated by Bernard), &lt;a href=&quot;https://www.gutenberg.org/ebooks/48433&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gutenberg.org/ebooks/48433&lt;/a&gt;) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hat-tip to &lt;a href=&quot;https://philosophy.stackexchange.com/a/9116/3762&quot;&gt;this answer at Philosophy-SE&lt;/a&gt; for finding the quote.&lt;/p&gt;&#xA;" OwnerUserId="376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:42:23.043" LastActivityDate="2015-12-15T12:19:00.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3220" PostTypeId="1" CreationDate="2015-12-15T13:00:10.450" Score="1" ViewCount="69" Body="&lt;p&gt;I have two questions.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What is the importance of Sophie Germain's work with Fermat's last theorem, if her only contribution was Sophie Germain prime numbers? Euler was able to calculate on the problem before, right?&lt;/li&gt;&#xA;&lt;li&gt;What was the general outlook on science during Sophie's lifetime (including the French Revolution)?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3390" LastEditorUserId="62" LastEditDate="2015-12-16T22:43:21.713" LastActivityDate="2015-12-16T22:43:21.713" Title="Sophie Germain and the outlook on science" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3221" PostTypeId="2" ParentId="3214" CreationDate="2015-12-16T00:40:26.333" Score="8" Body="&lt;p&gt;The notion of a matrix did not appear ahead of the notion of a linear transformation by that far. Although matrices feature implicitly in Cramer’s work on determinants (1750), and  Euler’s (1760) and Cauchy’s (1829) work on quadratic forms, Sylvester only introduced the term &quot;matrix&quot; to denote an array of numbers in 1850. He did not do much with them, but Cayley developed a theory of them starting in 1857, when he discovered that each matrix satisfies equation of its own order and defined &quot;characteristic values&quot;. See more in &lt;a href=&quot;https://hsm.stackexchange.com/questions/1799/motivation-behind-euler-theorem-in-differential-geometry/1807#1807&quot;&gt;Motivation behind Euler Theorem in differential geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of linear transformations appears implicitly at around the same time in Grassman’s Ausdehnungslehre (1844), and especially in Hamilton’s work on quaternions (1845-), which made extensive use of quaternions’ ability to represent rotations in space. In 1875 Darboux gave the first axiomatization of vector spaces, quite different looking from the modern one, he was motivated by representing forces in statics. And in 1888 Peano gave an axiomatization that was essentially modern, but like Sylvester he did little with it, and few paid attention to it. Nonetheless, &quot;&lt;em&gt;Peano did see Carvallo's linear systems (i.e., matrices) as instances of his own linear systems, without explicitly stating that square matrices of fixed order were an example of a vector space&lt;/em&gt;&quot; according to &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086085710257&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore’s  Axiomatization of Linear Algebra: 1875-1940&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the few was Pincherle, who in 1901 published a book about operators on what he called linear sets or linear spaces, focusing on infinite-dimensional ones like spaces of analytic functions. Another was Burali-Forti, who applied vector spaces in projective geometry in 1896, and then in vector calculus in 1910. In 1912 he published a book Transformations Lineaires, co-authored with Marcolongo, which opens with:“&lt;em&gt;We briefly set forth the foundations of the general theory of linear systems and linear operators. Generally, these matters are familiar in large part&lt;/em&gt;”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is tempting to say that people started viewing matrices as linear transformations from that point on, but history is rarely so straightforward. As Moore writes, “&lt;em&gt;although this general notion [of vector space] was first formulated by Peano in a geometric context in 1888, it remained almost unknown at the time. Three decades later it was rediscovered independently in two quite distinct branches of mathematics: functional analysis and ring theory. In ring theory, its roots were in Dedekind's work on algebraic number theory… The period in question can be considered to end with the publication in 1941 of A Survey of Modern Algebra by the young algebraists Garrett Birkhoff and Saunders Mac Lane, who were then at Harvard&lt;/em&gt;”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first branch was deeply influenced by &lt;a href=&quot;https://books.google.com/books/about/Mathematical_Foundations_of_Quantum_Mech.html?id=JLyCo3RO4qUC&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Neumann's Mathematical Foundations of Quantum Mechanics (1932)&lt;/a&gt; that formalized the subject in terms of Hilbert spaces and self-adjoint operators, and the Polish school of functional analysis. &lt;a href=&quot;http://kielich.amu.edu.pl/Stefan_Banach/e-operations.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Banach's Théorie des Opérations Linéaires&lt;/a&gt; also came out in 1932, and his point was that it is much more efficient to prove a result once about all vector spaces than to prove it repeatedly for 10 different function spaces. The second branch owes its rediscovery to the work of Emmy Noether and van der Waerden in 1920s, which culminated in der Waerden's highly influential two volume Modern Algebra (1930,31). A new generation of algebraists was raised on it, and it featured vector spaces and linear transformations among many other innovations, see more in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2991/distinction-between-basis-and-generating-set/2999#2999&quot;&gt;What was the evolution of &amp;quot;basis&amp;quot; and &amp;quot;generating set&amp;quot; in algebra?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-12-21T21:38:01.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3222" PostTypeId="1" AcceptedAnswerId="3224" CreationDate="2015-12-16T10:50:41.660" Score="6" ViewCount="544" Body="&lt;p&gt;The celebrated &lt;a href=&quot;https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem&quot; rel=&quot;noreferrer&quot;&gt;Nielsen-Schreier&lt;/a&gt; theorem in group theory says subgroup of a free group is free. This was proved for finitely generated subgroups of free groups by Jakob Nielsen in 1921, which involved the machinery of Nielsen transformations. It was proven in full generality by Otto Schreier in 1926.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the standard proof of the theorem is done using topology in modern days. If $F$ is a free group, one can realize $F$ as fundamental group of wedge of circles $X$ with circles labelled by the generators of $F$. If $H$ is a subgroup of $F$, then using covering space theory pick a covering space $p : E \to X$ with $p_*\pi_1(E) = H$. Since covering space of a graph is a graph, and fundamental group of a graph is free (quotient the maximal tree out), $H$ is free and we're done. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whom was this proof originally done by? I always thought it was done by Serre in his famous &lt;em&gt;Trees&lt;/em&gt;. But apparently there are controversies about this, as someone recently told me that this might have been done by Dehn earlier than Serre. I'd appreciate a definitive answer to this one. &lt;/p&gt;&#xA;" OwnerUserId="3392" LastActivityDate="2015-12-17T00:54:42.547" Title="Who discovered the topological proof of Nielsen-Schreier theorem?" Tags="|mathematics|abstract-algebra|topology|group-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3223" PostTypeId="1" AcceptedAnswerId="3225" CreationDate="2015-12-16T13:55:19.763" Score="1" ViewCount="218" Body="&lt;p&gt;What is historically first meaningful theorem about general convex sets?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(My guess would be Hyperplane separation theorem, but I can not see any reliable information on who and when it is proved.)&lt;/p&gt;&#xA;" OwnerUserId="3303" LastActivityDate="2015-12-16T17:50:35.420" Title="The first meaningful theorem about convex sets" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3224" PostTypeId="2" ParentId="3222" CreationDate="2015-12-16T14:06:51.287" Score="6" Body="&lt;p&gt;See the end of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem#History&quot;&gt;Wikipedia link&lt;/a&gt; in your first sentence. The source is &lt;a href=&quot;http://shi.matmor.unam.mx/algebra/Rotman%20J.J.%20Introduction%20to%20the%20theory%20of%20groups.pdf&quot;&gt;Rotman's Introduction to the Theory of Groups (1995)&lt;/a&gt;, which reads on p.383:&quot;&lt;em&gt;There are today several different proofs of this theorem, some &quot;algebraic&quot; and some &quot;geometric&quot;. The first geometric proof was given by Baer and Levi in 1936, and this is the proof we present. There is another elegant geometric proof, due to J.-P.Serre, using the notion of groups acting on trees&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="55" LastEditDate="2015-12-17T00:54:42.547" LastActivityDate="2015-12-17T00:54:42.547" CommentCount="2" CommunityOwnedDate="2015-12-16T23:50:33.793" ContentLicense="CC BY-SA 3.0" />
  <row Id="3225" PostTypeId="2" ParentId="3223" CreationDate="2015-12-16T15:33:51.563" Score="2" Body="&lt;p&gt;It is not completely clear what &quot;general&quot; means in your question. In arbitrary dimension? Descartes theorem in dimension 3 gives a relation between solid and flat angles&#xA;of a convex polytope which is equivalent to the Gauss--Bonnet for this case, and implies Euler's theorem.&#xA;The first results on convex polytopes in arbitrary dimension are probably due to Cauchy and Fourier in connection with linear programming (Cauchy actually had something like simplex method). Also Steiner was working on convex polytopes in the first part of 19s century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if &quot;general&quot; really means arbitrary convex sets in any dimension, then it is possible indeed that it is the separation theorem. According to Wikipedia it was proved by Edward Helly in 1912, in an infinite dimensional space but not in full generality. Hahn and Banach proved the general version in 1920. What is usually called &quot;Helly's theorem&quot; in the theory of convex sets in finite dimension  was proved in 1913.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Geometrie der Zahlen of Minkowski is 1896 (vol. I) ans 1910 (vo. II).&lt;br&gt;&#xA;Search in the Jahrbuch for &quot;konvex&quot; in the title gives only one 19 century paper, and it is about convex curves, not general convex sets.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-12-16T17:50:35.420" LastActivityDate="2015-12-16T17:50:35.420" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3226" PostTypeId="2" ParentId="3223" CreationDate="2015-12-16T17:16:18.177" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow&quot;&gt;Earliest Known Uses of&#xA;Some of the Words of Mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Convex set&lt;/strong&gt;: The German term appears in E. Steinitz,&#xA;  &quot;Bedingt konvergente Reihen und konvexe Systeme, I,&quot; &lt;em&gt;J. Reine Angew. Math.&lt;/em&gt;, &lt;strong&gt;143&lt;/strong&gt;, (1913) 128-175&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;so perhaps that is a place to look for a &quot;meaningful theorem&quot;.  Then there would remain the question of such meaningful theorems about convex sets from before the terminology &quot;convex set&quot; was coined.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-12-16T17:16:18.177" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3227" PostTypeId="1" CreationDate="2015-12-17T03:07:13.520" Score="9" ViewCount="6902" Body="&lt;h2&gt;Part I&lt;/h2&gt;&#xA;&lt;p&gt;In E. Maor's book [2, p. 117] we read that, somewhere in his &lt;em&gt;Autobiographical Notes&lt;/em&gt;, Einstein wrote this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An uncle told me about the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in &amp;quot;proving&amp;quot; this theorem on the basis of the similarity of triangles; in doing so it seemed to me &amp;quot;evident&amp;quot; that the relations [ratios] of the sides of the right-angled triangles would have to be completely determined by one of the acute angles...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;E. Maor adds that Einstein's proof of the Pythagorean theorem was reconstructed by Einstein's biographer and collaborator Banesh Hoffmann (for more information in this regard, E. Maor points his readers to &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt; 1 &lt;/a&gt;). Then, E. Maor mentions that what B. Hoffmann put forward as Einstein's proof of the Pythagorean theorem turns out to be basically &amp;quot;the first of the 'algebraic proofs' in Elisha Scott Loomis's book (attributed there to [a certain David] Legendre but actually being Euclid's second proof; see [4, p. 24] or look for &amp;quot;proof using similar triangles&amp;quot; in &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;this webpage&lt;/a&gt;)&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Having said all this, I would like to ask you the following questions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I.&lt;/strong&gt; a) How did B. Hoffmann manage to &amp;quot;reconstruct&amp;quot; Einstein's proof of the Pythagorean theorem? b) Do we know which his references were? c) Did the &amp;quot;reconstruction&amp;quot; in question was actually recognized as &lt;em&gt;the one&lt;/em&gt; by Einstein in his lifetime?&lt;/p&gt;&#xA;&lt;h2&gt;Part II&lt;/h2&gt;&#xA;&lt;p&gt;S. Strogatz in &lt;a href=&quot;http://www.newyorker.com/tech/elements/einsteins-first-proof-pythagorean-theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;, published a month ago in &lt;em&gt;The New Yorker&lt;/em&gt;, defies, based on [5, pp. 3-4], the consensus among several biographers of Einstein (Hoffmann included) as to how it was that Einstein's proof of the Pythagorean theorem actually went. According to Schroeder (and Strogatz), Einstein considered, just as in the figure below, the altitude to the hypotenuse &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; of the right-angled triangle &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gc3m6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gc3m6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then, on the one hand, Einstein has that &lt;span class=&quot;math-container&quot;&gt;$\triangle ABC \sim \triangle CBD \sim \triangle ACD$&lt;/span&gt; and that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathrm{area}(\triangle CBD) + \mathrm{area}(\triangle ACD) = \mathrm{area}(\triangle ABC). \qquad \mbox{(*)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathcal{A}:= \mathrm{area}(\triangle CBD),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathrm{area}(\triangle ACD) = \left(\frac{b}{a}\right)^{2}\mathcal{A}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathrm{area}(\triangle ABC) = \left(\frac{c}{a}\right)^{2} \mathcal{A}$$&lt;/span&gt; (it has to be recalled that, according to Eucl. VI-19, &lt;em&gt;the ratio of the areas of two similar triangles is equal to the square of the ratio of any two corresponding sides&lt;/em&gt;). From this and &lt;span class=&quot;math-container&quot;&gt;$(*)$&lt;/span&gt;, it follows that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathcal{A} + \left(\frac{b}{a}\right)^{2}\mathcal{A} = \left(\frac{c}{a}\right)^{2} \mathcal{A};$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;obviously, the Pythagorean theorem is an immediate consequence of the above equality.&lt;/p&gt;&#xA;&lt;p&gt;In the opinion of Strogatz, this proof is neater than the one typically attributed to Einstein; naturally, I agree with him in this respect. Besides, it has to be noted that it is basically through this approach that B. Mazur proves in &lt;a href=&quot;https://i.stack.imgur.com/gc3m6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt; 3 &lt;/a&gt; a much more general version of the Pythagorean theorem (which Mazur refers to as &lt;em&gt;the blob Pythagorean theorem&lt;/em&gt;). Nevertheless, Strogatz's article originated in my psyche the following questions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;II.&lt;/strong&gt; a) How did Einstein's proof of the Pythagorean theorem actually go? b) Will we ever know it? c) The proof of the Pythagorean theorem that Schroeder (and Strogatz) ascribe to Einstein can actually be found in [4, pp. 230-231]; in point of fact, E. S. Loomis mentions in page 230 of that book that the proof of the Pythagorean theorem --along those lines-- was communicated to him on June 4, 1934 by Stanley Jashemski (from Youngstown, Ohio, USA), &amp;quot;a young man of superior intellect&amp;quot;. Since there is no mention whatsoever to Mr. Jashemski in Strogatz's article, how seriously should we take this article of his on the &amp;quot;genuine&amp;quot; einsteinian proof of the Pythagorean theorem?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Albert Einstein: Historical and Cultural Perspectives.&lt;/em&gt; Eds. Gerald Holton and Yehuda Elkana, Princeton University Press, 1982, pp. 92-93.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Eli Maor, &lt;em&gt;The Pythagorean Theorem: a 4000-year story.&lt;/em&gt; Princeton University Press, USA, 2007.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Barry Mazur, &lt;a href=&quot;https://youtu.be/ItiFO5y36kw&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A mathematical fable&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Elisha Scott Loomis, &lt;em&gt;The Pythagorean Proposition.&lt;/em&gt; National Council of Teachers of Mathematics, 2nd. Edition, Ann Arbor, Michigan, USA, 1940.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Manfred Schroeder, &lt;em&gt;Fractals, chaos, power laws: minutes from an infinite paradise.&lt;/em&gt; Dover Publications, Inc. Mineola, New York, USA, 2009.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://drive.google.com/file/d/1Z8ACedrsq10Wd0K1Ba57tXqCzNe6krny/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sobre el artículo &lt;em&gt;On Einsteins' Proof&lt;/em&gt; de Stephen Strogatz.&lt;/a&gt; (My first impressions on Strogatz's article)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2021-04-08T19:22:58.627" LastActivityDate="2023-09-04T14:49:36.043" Title="On Einstein's proof of the so-called Pythagorean theorem" Tags="|mathematics|euclidean-geometry|debunking|pythagorean-proposition|" AnswerCount="2" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="3228" PostTypeId="2" ParentId="2227" CreationDate="2015-12-17T11:04:38.907" Score="4" Body="&lt;p&gt;The following two books are also very good resources about this topic ; the first one has a more historical approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Carlo Cercignani, &lt;em&gt;Ludwig Boltzmann. The man who trusted atoms.&lt;/em&gt; Oxford University Press, Oxford, 1998. xviii+329 pp. ISBN: 0-19-850154-4&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Giovanni Gallavotti, &lt;em&gt;Statistical mechanics. A short treatise.&lt;/em&gt; Texts and Monographs in Physics. Springer-Verlag, Berlin, 1999. xiv+339 pp. ISBN: 3-540-64883-6&lt;/p&gt;&#xA;" OwnerUserId="3401" LastActivityDate="2015-12-17T11:04:38.907" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3229" PostTypeId="1" CreationDate="2015-12-18T03:24:06.140" Score="3" ViewCount="143" Body="&lt;p&gt;Who has proved Boltzmann's H-theorem (which tells that entropy of a closed system grows as the system approaches thermal equilibrium) for the kinetic equation with Landau collisions operator? One proof is provided by F. Hinton in 1st volume of &quot;Basic plasma physics&quot; (edited by Sudan and Galeev) but I wonder if Hinton was the first. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE&lt;/strong&gt;: I asked same question on &lt;a href=&quot;https://www.researchgate.net/post/Who_was_first_to_prove_the_Boltzman_H-theorem_for_the_Landau_kinetic_equation&quot; rel=&quot;nofollow&quot;&gt;https://www.researchgate.net/post/Who_was_first_to_prove_the_Boltzman_H-theorem_for_the_Landau_kinetic_equation&lt;/a&gt; and was answered that a derivation was provided by N.Bogolubov but I don't believe in that. &lt;/p&gt;&#xA;" OwnerUserId="3407" LastEditorUserId="3407" LastEditDate="2015-12-23T04:06:08.550" LastActivityDate="2016-02-13T14:30:57.777" Title="Who was the first to prove the Boltzman H-theorem for the Landau kinetic equation?" Tags="|physics|statistical-mechanics|" AnswerCount="1" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3230" PostTypeId="1" CreationDate="2015-12-18T21:23:48.977" Score="17" ViewCount="1691" Body="&lt;p&gt;If I understand correctly, the concept of a Minkowski space/metric was already known before Einstein's paper on special relativity. Was there any physical motivation for studying this type of metric spaces and how did Einstein become aware of these studies?&lt;/p&gt;&#xA;" OwnerUserId="3414" LastEditorUserId="2324" LastEditDate="2020-12-05T21:38:01.767" LastActivityDate="2024-03-24T12:48:31.703" Title="What was the motivation for Minkowski spacetime before special relativity?" Tags="|discoveries|theoretical-physics|relativity-theory|mathematical-physics|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3231" PostTypeId="2" ParentId="3230" CreationDate="2015-12-18T22:27:36.450" Score="34" Body="&lt;p&gt;Not quite. Minkowski had the idea of representing special relativity as geometry in 1907 under the direct influence of Einstein's 1905 paper, and he developed it in Raum und Zeit (1907) and Zwei Abhand lungen über die Grundgleichungen der Elektrodynamik (1909). See &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Minkowski.html&quot; rel=&quot;noreferrer&quot;&gt;Minkowski on MacTutor&lt;/a&gt;. Before that only classical &amp;quot;spacetime&amp;quot; appeared, and only superficially. D'Alembert wrote in the Dimension entry of the Encyclopedia (1756):&amp;quot;&lt;em&gt;One could consider time as a fourth dimension, so that the product of time by volume would, in a certain sense, be the product of four dimensions; this idea is perhaps debatable, but I feel that it has certain merits...&lt;/em&gt;&amp;quot; At the end of 19th century the idea became quite fashionable in the popular culture. In 1885 Nature published an unsigned letter &lt;a href=&quot;http://www.nature.com/nature/journal/v31/n804/abs/031481a0.html&quot; rel=&quot;noreferrer&quot;&gt;Four-Dimensional Space&lt;/a&gt;, which read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Since this fourth dimension cannot be introduced into space, as commonly understood, we require a new kind of space for its existence, which we may call time-space… We must, therefore, conceive that there is a new three-dimensional space for each successive instant of time; and, by picturing to ourselves the aggregate formed by the successive positions in time-space of a given solid during a given time, we shall get the idea of a four-dimensional solid, which may be called a sur-solid... Let any man picture to himself the aggregate of his own bodily forms from birth to the present time…&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And in 1895 Wells, inspired by a fellow student at the Imperial college, immortalized it in his Time Machine, see &lt;a href=&quot;https://medium.com/starts-with-a-bang/when-einstein-met-h-g-wells-425372d21821&quot; rel=&quot;noreferrer&quot;&gt;When Einstein met H.G. Wells by Halpern&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Before Minkowski, what we call Minkowski geometry in two and three dimensions appeared on a long list of geometries in Klein's classification under his &lt;a href=&quot;https://en.wikipedia.org/wiki/Erlangen_program&quot; rel=&quot;noreferrer&quot;&gt;Erlangen program (1872)&lt;/a&gt;. But it only came up as a collection of objects invariant under all transformations that preserve a quadratic form with indefinite signature. Erlangen Program in general was about interpreting geometries as invariants of various transformation groups acting on projective space, it greatly expanded the number of known homogeneous geometries from the originally discovered hyperbolic and elliptic ones. Unlike them however, it had no spatial interpretation. Before 1910 in his popularizations of the Erlangen program Klein only treated spatial geometries in any detail. Earlier he did not even count the total number of geometries his construction produced, and mentioned higher dimensional generalizations required for spacetime only in single sentences.&lt;/p&gt;&#xA;&lt;p&gt;There is a connection to &lt;a href=&quot;https://en.wikipedia.org/wiki/Split-complex_number#History&quot; rel=&quot;noreferrer&quot;&gt;split complex numbers&lt;/a&gt; introduced by Cockle (1848) and studied by Clifford in 1873 under the name of double numbers. But that was mostly algebraic playing around, and the connection to Minkowski geometry was pointed out only after Minkowski. This is despite the fact that Clifford was deeply involved with non-Euclidean geometries, and his short note &lt;a href=&quot;https://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter&quot; rel=&quot;noreferrer&quot;&gt;On the Space-Theory of Matter (1876)&lt;/a&gt; anticipated Einstein's idea of reducing matter to curvature (but of space, not spacetime). Even &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_Lorentz_transformations#Poincar.C3.A9_.281900.2C_1905.29&quot; rel=&quot;noreferrer&quot;&gt;Poincare, who in 1905&lt;/a&gt; expressed Lorentz's theory of ether (mathematically equivalent to special relativity) in terms of what we call Poincare group of transformations, and was well familiar with the Erlangen program, did not make the geometric connection either. In Science and Hypothesis (1902) he even mentioned &amp;quot;&lt;em&gt;forth geometry&lt;/em&gt;&amp;quot; of space, in addition to elliptic, hyperbolic and Euclidean ones, and pointed out that in it lines can be perpendicular to themselves. But according to &lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1204/1204.6576.pdf&quot; rel=&quot;noreferrer&quot;&gt;Weinstein's Poincaré's Dynamics of the Electron&lt;/a&gt;, in his famous 1904-06 papers &amp;quot;&lt;em&gt;Poincaré did not associate this quadratic form with propagation of light in order to define a null interval like Einstein or a metric like Minkowski&lt;/em&gt;&amp;quot;. It took Minkowski to take that extra step of treating time as space.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0817633162&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yaglom's Felix Klein and Sophus Lie&lt;/a&gt; gives many historical details on late 19th century geometry and its relation to relativity with references to the original sources.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="5989" LastEditDate="2024-03-22T07:38:41.850" LastActivityDate="2024-03-22T07:38:41.850" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3232" PostTypeId="2" ParentId="3230" CreationDate="2015-12-19T00:01:44.150" Score="9" Body="&lt;p&gt;Minkowski space time was considered by mathematicians before Einstein and before Minkowski. Of course the name &amp;quot;space-time&amp;quot; was not used. The reasons were purely mathematical, not physical. The most important application was the Klein model of the hyperbolic geometry (non-Eucludean geometry of Bolyai and Lobachevski).  More presicely,&#xA;Klein considered the metric&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ds^2=-dx_0^2+dx_1^2+\ldots+dx_n^2$$&lt;/span&gt;&#xA;in the &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt; space. This metric, restricted to the upper portion of the hyperboloid&#xA;&lt;span class=&quot;math-container&quot;&gt;$$-x_0^2+x_1^2+\ldots+x_n^2=-1$$&lt;/span&gt;&#xA;is the Klein model of the hyperbolic geometry. In physics when &lt;span class=&quot;math-container&quot;&gt;$n=3$&lt;/span&gt; the space&#xA;with metric &lt;span class=&quot;math-container&quot;&gt;$-ds^2$&lt;/span&gt; is the Minkowski space-time. So Klein's model of a hyperblic space is a hypersurface in the Minkowski space time, and this is how it was&#xA;introduced.&lt;/p&gt;&#xA;&lt;p&gt;When I wrote that the reasons were pure mathematical, I did not take into account&#xA;that Lobachevski and Gauss admitted the possibility that hyperbolic geometry&#xA;holds in the real physical space. Both of them realized that no experiment really&#xA;tells us that the geometry of our physical space is Euclidean. An experiment can tell us the opposite: you can never be sure that the sum of the angles of a triange exactly equals 180 degrees but an experiment can show that it is less,&#xA;if it is less indeed). There are indication that Gauss really tried this sort of experiment (measuring angles of a really big triangle between mountain tops). Lobachevski discussed in one of his papers the data on stellar parallaxes trying to determine whether the geometry of the real space is Euclidean. But apparently connections with physics were not among Klein's motivations.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-24T12:48:31.703" LastActivityDate="2024-03-24T12:48:31.703" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3233" PostTypeId="1" CreationDate="2015-12-16T16:07:11.537" Score="1" ViewCount="9" Body="&lt;p&gt;Who has proved &lt;a href=&quot;http://en.wikipedia.org/wiki/H-theorem&quot; rel=&quot;nofollow&quot;&gt;Boltzmann's H-theorem&lt;/a&gt; (which tells that entropy of a closed system grows as the system approaches thermal equilibrium) for the kinetic equation with Landau collisions operator? One proof is provided by F. Hinton in 1st volume of &quot;Basis plasma physic&quot; (edited by Sudan and Galeev) but I wonder if Hinton was the first. &lt;/p&gt;&#xA;" OwnerUserId="3407" OwnerDisplayName="Igor Kotelnikov" LastActivityDate="2015-12-19T04:51:47.677" Title="Who was first to prove the Boltzman H-theorem for the Landau kinetic equation?" Tags="|statistical-mechanics|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2015-12-19T14:54:00.727" ContentLicense="CC BY-SA 3.0" />
  <row Id="3234" PostTypeId="1" AcceptedAnswerId="13379" CreationDate="2015-12-19T12:12:45.167" Score="8" ViewCount="386" Body="&lt;p&gt;Cohomology is a stronger invariant than homology because it can be equipped with a ring structure. To be precise, if one starts with the singular cohomology groups &lt;span class=&quot;math-container&quot;&gt;$H^\bullet(-; R)$&lt;/span&gt; with coefficients in a ring &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, then there is a map &lt;span class=&quot;math-container&quot;&gt;$$H^n(X; R) \times H^m(X; R) \stackrel{\smile}{\to} H^{m+n}(X; R)$$&lt;/span&gt; which is graded commutative - that is, &lt;span class=&quot;math-container&quot;&gt;$\alpha \smile \beta = (-1)^{mn} \beta \smile \alpha$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\alpha \in H^n(X; R)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta \in H^m(X; R)$&lt;/span&gt;. This makes &lt;span class=&quot;math-container&quot;&gt;$\bigoplus_i H^i(X; R)$&lt;/span&gt; into a graded commutative ring, which we denote as &lt;span class=&quot;math-container&quot;&gt;$H^*(X; R)$&lt;/span&gt;, known as the graded cohomology ring of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; with coefficients in &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Now, the cup product map above can be defined in several ways.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;One standard way to define it is as &lt;span class=&quot;math-container&quot;&gt;$$H^n(X; R) \times H^m(X; R) \to H^{n+m}(X \times X; R) \to H^{n+m}(X; R)$$&lt;/span&gt; where the first map in the composition is induced from the &lt;a href=&quot;https://ncatlab.org/nlab/show/Eilenberg-Zilber+map&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eilenberg-Zilber map&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$C^n(X) \times C^m(Y; R) \to C^{n+m}(X \times Y; R)$&lt;/span&gt; and the second is induced from the diagonal inclusion &lt;span class=&quot;math-container&quot;&gt;$\Delta : X \hookrightarrow X \times X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Originally, Poincaré defined this for smooth manifolds using &lt;a href=&quot;https://en.wikipedia.org/wiki/Transversality_%28mathematics%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;transversality&lt;/a&gt;. That is, if &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is a smooth &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-manifold, &lt;span class=&quot;math-container&quot;&gt;$N, L$&lt;/span&gt; are submanifolds of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; of codimension &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; respectively, then &lt;span class=&quot;math-container&quot;&gt;$[N], [L]$&lt;/span&gt; represent homology classes in &lt;span class=&quot;math-container&quot;&gt;$H_{n-k}(M; \Bbb R)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$H_{n-l}(M; \Bbb R)$&lt;/span&gt; respectively. Assume further than &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; are transverse. Under the Poincaré duality, &lt;span class=&quot;math-container&quot;&gt;$[N]$&lt;/span&gt; corresponds to &lt;span class=&quot;math-container&quot;&gt;$[N]^*$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$H^k( M; \Bbb R)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$[L]$&lt;/span&gt; corresponds to &lt;span class=&quot;math-container&quot;&gt;$[L]^*$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$H^l(M; \Bbb R)$&lt;/span&gt;. Poincaré then defined cup product of these two classes as &lt;span class=&quot;math-container&quot;&gt;$[N]^* \smile [L]^* := [N \pitchfork L]^*$&lt;/span&gt;, the dual of the class represented by &lt;span class=&quot;math-container&quot;&gt;$N \pitchfork L$&lt;/span&gt;.&lt;span class=&quot;math-container&quot;&gt;$(\star)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;However, there is a much easier definition one can see in standard algebraic topology books (e.g., Hatcher) which goes as follows. Take a &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-cochain &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$C^k(X; R)$&lt;/span&gt;, an &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;-cochain &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$C^l(X; R)$&lt;/span&gt;, and then define the &lt;span class=&quot;math-container&quot;&gt;$k+l$&lt;/span&gt;-cochain &lt;span class=&quot;math-container&quot;&gt;$\psi \smile \varphi$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$$(\psi \smile \varphi)(\sigma) := \psi(\sigma|_{[v_0, \cdots, v_k]}) \varphi(\sigma|_{[v_k, \cdots, v_{k+l}]})$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\sigma : \Delta^{k+l} \to X$&lt;/span&gt; is a &lt;span class=&quot;math-container&quot;&gt;$(k+l)$&lt;/span&gt;-singular simplex. One can easily see that this cochain-level map  &lt;span class=&quot;math-container&quot;&gt;$C^k(X; R) \times C^l(X; R) \to C^{k+l}(X; R)$&lt;/span&gt; descends to homology. This is precisely the cup product.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;All the definitions above are equivalent. The first definition was discovered by Eilenberg &amp;amp; Zilberg in &lt;a href=&quot;https://www.jstor.org/stable/2372629?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; paper, the second by Poincaré in &lt;em&gt;Analysis Situs&lt;/em&gt;. But who discovered the third?&lt;/p&gt;&#xA;&lt;p&gt;Remark&lt;span class=&quot;math-container&quot;&gt;$(\star)$&lt;/span&gt;: It is not true however that every sinuglar cycle of a smooth manifold can be represented by a submanifold, see Thom's &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF02566923&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt;. Although, this is a high dimensional phenomenon and the first counterexample appears at dimension &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3392" LastEditorUserId="10608" LastEditDate="2024-03-12T09:22:00.707" LastActivityDate="2024-03-12T09:22:00.707" Title="Who discovered the singular cup product?" Tags="|mathematics|topology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3238" PostTypeId="1" CreationDate="2015-12-20T03:33:21.757" Score="2" ViewCount="2579" Body="&lt;p&gt;I am trying to understand the conceptual development of the &quot;torque&quot;. I saw that Archimedes was the one from whom this idea took birth. I wanted to know from whom the torque got its mathematical form.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will also be happy to know further reading materials and sources. &lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2015-12-21T03:39:42.527" Title="Who gave mathematical expression for torque?" Tags="|mathematics|physics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3239" PostTypeId="2" ParentId="3238" CreationDate="2015-12-20T05:55:15.663" Score="1" Body="&lt;p&gt;The concept of torque, also called moment of a force, as you've pointed out correctly, originated with the studies of Archimedes on levers. The term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884, as suggested &lt;a href=&quot;https://books.google.co.in/books?id=YGVDAAAAIAAJ&amp;amp;source=gbs_similarbooks%20&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Mathematical form of Torque as we know it now is $\vec{\tau} = \vec{r} \times \vec{F}$. Or $\tau =  r \cdot F \cdot \sin{\theta}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While studying/using levers the &lt;b&gt;Law Of The Lever&lt;/b&gt; was used which just gave us the mechanical advantage of the lever. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take this scenario: &#xA;If &lt;strong&gt;a&lt;/strong&gt; and &lt;strong&gt;b&lt;/strong&gt; are &lt;em&gt;distances&lt;/em&gt; from the fulcrum to points A and B and the force F&lt;sub&gt;A&lt;/sub&gt; applied to A is the input and the force F&lt;sub&gt;B&lt;/sub&gt; applied at B is the output, the ratio of the velocities of points A and B is given by $\frac{a}{b}$, so we have the ratio of the output force to the input force, or mechanical advantage, is given by $\frac{F_B}{F_A} = \frac{a}{b}.$ &#xA;Archimedes proved it using Geometric Reasoning, as stated &#xA;&lt;a href=&quot;https://books.google.co.in/books?id=Zt4Aw9wKjm8C&amp;amp;pg=PA94&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After which Newton's Law's helped grasp rotational dynamics better.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To Learn more about &lt;a href=&quot;http://farside.ph.utexas.edu/teaching/301/lectures/node104.html&quot; rel=&quot;nofollow&quot;&gt;Torque&lt;/a&gt;. Also check Wikipedia.&lt;/p&gt;&#xA;" OwnerUserId="3377" LastActivityDate="2015-12-20T05:55:15.663" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3240" PostTypeId="1" CreationDate="2015-12-20T06:54:50.320" Score="2" ViewCount="111" Body="&lt;p&gt;I believe it's a great application of Pascal's Law. Are there ways to overcome it? I mean, not having the fluid drop through the tube. One way I Know is to put a denser fluid in first and on top of it a comparatively less dense one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Answer if you want to&lt;/em&gt;&lt;/strong&gt;: Also are there other toys which can aid a 5-7 year old learn concepts of science and also the equipment required should be very minimal. &lt;/p&gt;&#xA;" OwnerUserId="3377" LastEditorUserId="2442" LastEditDate="2016-08-12T15:44:55.490" LastActivityDate="2016-08-12T15:44:55.490" Title="Are there items similar to the Pythagorean cup?" Tags="|teaching|" AnswerCount="0" CommentCount="6" ClosedDate="2015-12-20T18:13:58.730" ContentLicense="CC BY-SA 3.0" />
  <row Id="3241" PostTypeId="2" ParentId="3238" CreationDate="2015-12-20T12:26:06.630" Score="2" Body="&lt;p&gt;Regarding the &lt;em&gt;word&lt;/em&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Torque&quot; rel=&quot;nofollow noreferrer&quot;&gt;torque&lt;/a&gt;, Wiki refers to : &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Thomson_(engineer)&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Thomson&lt;/a&gt;, (&lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Larmor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Larmor&lt;/a&gt; editor, 1912), &lt;em&gt;Collected Papers in Physics and Engineering&lt;/em&gt;, &lt;a href=&quot;https://archive.org/stream/collectedpapersi00thomrich#page/n51/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;page xlvii&lt;/a&gt; (dated 1888).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Vector analysis&lt;/em&gt; was quite slow to be used in physics at the end of 19th Century [see this &lt;a href=&quot;https://hsm.stackexchange.com/questions/3158/what-were-the-criticisms-against-the-introduction-of-vector-analysis&quot;&gt;post&lt;/a&gt; for references].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For some 19th Century textbooks, and the &quot;verbal&quot; definition, see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;James Clerk Maxwell, &lt;a href=&quot;https://archive.org/stream/mattermotion00maxwiala#page/50/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matter and motion&lt;/a&gt; (1st ed, 1877) : &lt;strong&gt;§70. Moment of a force about a point&lt;/strong&gt;:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the product of a force into the perpendicular from the origin on its line of action is called the Moment of the force about the origin. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Thomson, Baron Kelvin&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Peter_Tait_(physicist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peter Guthrie Tait&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/elementsnatural04taitgoog#page/n24/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements of Natural Philosophy&lt;/a&gt; (1872), page 13: &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;§46.&lt;/strong&gt; The &lt;em&gt;Moment&lt;/em&gt; of a velocity or of a force about any point is the product of its magnitude into the perpendicular from the point upon its direction. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-12-20T14:27:39.310" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3243" PostTypeId="2" ParentId="3238" CreationDate="2015-12-21T03:39:42.527" Score="1" Body="&lt;p&gt;Archimedes did provide a mathematical expression for torque in his Law of Levers - but it could be argued that the Peripatetic School beat him to it in the book &lt;em&gt;Mechanica&lt;/em&gt; (&lt;a href=&quot;https://www.math.nyu.edu/%7Ecrorres/Archimedes/Lever/LeverLaw.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Archimedes wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Magnitudes are in equilibrium at distances reciprocally proportional to their weights.&lt;/p&gt;&#xA;&lt;p&gt;ΤΑ ΜΕΓΕΘΕΑ ΙΣΟΡΡΟΠΕΟΝΤΙ ΑΠΟ ΜΑΚΕΩΝ ΑΝΤΙΠΕΠΟΝΘΟΤΩΣ ΤΟΝ ΑΥΤΟΝ ΛΟΓΟΝ ΕΧΟΝΤΩΝ ΤΟΙΣ ΒΑΡΕΣΙΝ.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;em&gt;Mechanica&lt;/em&gt; we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The ratio of the weight moved to the weight moving it is the inverse ratio of the distances from the centre.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Both seem to me to be mathematical statements of torque - although that word had not yet been invented&lt;/p&gt;&#xA;" OwnerUserId="543" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2015-12-21T03:39:42.527" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3244" PostTypeId="1" AcceptedAnswerId="3247" CreationDate="2015-12-21T07:44:11.423" Score="4" ViewCount="193" Body="&lt;p&gt;Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his &lt;em&gt;Cum Prodiisset&lt;/em&gt; that was analyzed in detail by H. Bos in a seminal text dating from the 1970s.  The distinction between assignable and inassignable number seems to appear out of nowhere in Leibniz (as does the term &lt;em&gt;infinitesimal&lt;/em&gt; itself).  Historically is there an earlier source for this?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2015-12-22T07:51:52.737" LastActivityDate="2015-12-23T07:28:31.137" Title="What is the origin of the distinction between assignable and inassignable number?" Tags="|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3245" PostTypeId="1" AcceptedAnswerId="3249" CreationDate="2015-12-21T12:09:53.170" Score="6" ViewCount="1202" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any ancient civilizations which had concept of &lt;em&gt;zero&lt;/em&gt; but didn't not use &lt;em&gt;positional numerals&lt;/em&gt; for any somewhat non-negligible (from historical point of view) amount of time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If there are such civilizations, why didn't they adopt/come up with &lt;em&gt;positional numeral system&lt;/em&gt; shortly after introducing &lt;em&gt;zero&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;h2&gt;edit:&lt;/h2&gt;&#xA;&lt;p&gt;As pointed out in multiple answers, my original question had incorrect statement that the zero is necessary for the positional number systems. This statement is not true.&lt;/p&gt;&#xA;" OwnerUserId="3029" LastEditorUserId="3029" LastEditDate="2023-05-14T23:10:48.737" LastActivityDate="2023-05-14T23:10:48.737" Title="Was the concept of zero ever developed without relation to positional number systems?" Tags="|mathematics|notation|zero|numbers|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3247" PostTypeId="2" ParentId="3244" CreationDate="2015-12-21T20:00:10.227" Score="1" Body="&lt;p&gt;For some references, see : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Paolo Mancosu, &lt;a href=&quot;https://books.google.it/books?id=60qaEePdqcoC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century&lt;/a&gt; (1999).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See &lt;strong&gt;5.1 Indivisibles and Infinitely Small Quantities&lt;/strong&gt;, page 119-on, and see &lt;a href=&quot;https://books.google.it/books?id=60qaEePdqcoC&amp;amp;pg=PA126&quot; rel=&quot;nofollow&quot;&gt;page 126&lt;/a&gt; for Leibniz &lt;em&gt;De Quadratura Arithmetica&lt;/em&gt; (ca.1676), regarding Theorem 6 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;that a curvilinear figure can be approximated with an arbitrary degree of accuracy by step-figures, in such a way that the difference between the area under the curve and the area of the step-figures can be made less than any assigned quantity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The source can be found in Cavalieri geometry of indivisibles and the revival of the &lt;em&gt;exhaustion&lt;/em&gt; method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;strong&gt;Ch.6 Leibniz's Differential Calculus and Its Opponents&lt;/strong&gt;, &lt;a href=&quot;https://books.google.it/books?id=60qaEePdqcoC&amp;amp;pg=PA150&quot; rel=&quot;nofollow&quot;&gt;page 150-on&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See : Gottfried Wilhelm Leibniz, &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;pg=PA19&quot; rel=&quot;nofollow&quot;&gt;The Early Mathematical Manuscripts of Leibniz&lt;/a&gt; (J.M.Child ed, 1920 - Dover reprint), &lt;em&gt;Letter to Bernoulli&lt;/em&gt; (1703), page 19 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I filled some hundreds of sheets with them [the results that I obtained] in that year [the year 1672]. These I divided into two classes of &lt;em&gt;assignables&lt;/em&gt; and &lt;em&gt;inassignables&lt;/em&gt;. Among assignables I placed everything I obtained by the methods &#xA;  previously used by Cavalieri, Guldinus, ToricelIi, Gregory St.Vincent and Pascal, such as sums, sums of sums, transpositions, &quot;ductus,&quot; cylinders truncated by a plane, and lastly by the method of the center of gravity; and among inassignables I placed all that I obtained by the use of the triangle which I at that time called &quot;the characteristic triangle&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And see &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;pg=PA146&quot; rel=&quot;nofollow&quot;&gt;page 146 : draft manuscript of an answer to Bernhard Nieuwentijt&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Since I introduced not only the first differences, but also the second, &#xA;  third and other higher differences, inassignable or incomparable with these first differences [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See H.J.M.Bos, &lt;a href=&quot;http://www.tau.ac.il/~corry/teaching/toldot/download/Bos1974.pdf&quot; rel=&quot;nofollow&quot;&gt;Differentials and Higher-Order Differentials in the Leibnizian Calculus&lt;/a&gt; (1974), pag.13-on : &lt;em&gt;inassignable&lt;/em&gt; means &lt;em&gt;incomparable&lt;/em&gt; [&quot;less than any assigned quantity&quot;].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that is a new term introduced by Leibniz :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I agree with Euclid &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookV/bookV.html#defs&quot; rel=&quot;nofollow&quot;&gt;Book V, Def.5&lt;/a&gt; that only those homogeneous quantities are comparable, of which the one can become larger than the other if multiplied by a number, that is, a finite number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the case of an increment in a numerical (i.e. discrete) series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz &quot;extrapolate&quot; the method of taking the sum of a series to a continuous quantity (i.e. a function) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I feel that this method and others in use up till now can all be deduced from a&#xA;  general principle which I use measuring curvilinear figures, &lt;em&gt;that a curvilinear figure must be considered to be the same as a polygon with infinitely many sides&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;if $x$ or $y$ were not discrete terms, but continual terms, that is, not numbers whose differences are assignable intervals, but straight line abscissas increasing continually or by elements, that is, by inassignable intervals, so that the sequence of terms constitutes the figure, [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, &quot;infinitesimal&quot; increments are &lt;em&gt;inassignable&lt;/em&gt; quantity because they are incomparable to the quantity &quot;increasing continually&quot;, because they violate Euclide's definition.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Finally, see Gottfried Wilhelm Freiherr von Leibniz, &lt;a href=&quot;https://books.google.it/books?id=dwVsQgAACAAJ&quot; rel=&quot;nofollow&quot;&gt;The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686&lt;/a&gt; (Richard Arthur editor, 2001), page 89, from the ms &lt;em&gt;Infinite Numbers&lt;/em&gt; (ca.1676) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The circle - as a polygon greater than any assignable [&lt;em&gt;quolibet assignabili maius&lt;/em&gt;] [...]. So when something is said about the circle we understand it to be true of any polygon such that there is some polygon in which the error is less than any assigned amount $a$ [&lt;em&gt;error minor sit quovis assignato&lt;/em&gt; $a$] [...]. And so if certain polygons are able to increase according to some law, and something is true of them the more they increase, our mind imagines some ultimate polygon; and whatever it sees becoming more and more so in the individual polygons, it declares to be perfectly so in this ultimate one. And even though this ultimate polygon does not exist in the nature of things, one can still give an expression for it, for the sake of abbreviation of expressions. [See &lt;a href=&quot;http://plato.stanford.edu/entries/leibniz/#PriCont&quot; rel=&quot;nofollow&quot;&gt;Leibniz's Principle of Continuity&lt;/a&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2015-12-23T07:28:31.137" LastActivityDate="2015-12-23T07:28:31.137" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3248" PostTypeId="2" ParentId="3199" CreationDate="2015-12-21T22:24:37.510" Score="3" Body="&lt;p&gt;Use of &lt;em&gt;mesosynteny&lt;/em&gt; was proposed in 2011 in this paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://www.genomebiology.com/2011/12/5/R45#B14&quot; rel=&quot;nofollow&quot;&gt;Hane JK, Rouxel T, Howlett BJ, Kema GH, Goodwin SB, Oliver RP. A novel mode of chromosomal evolution peculiar to filamentous Ascomycete fungi. Genome Biol. 2011 May 24;12(5):R45.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This pattern is characterized by the conservation of gene content in chromosomes, without conservation of gene order or orientation; that is, synteny without co-linearity. We propose to call this sequence conservation 'mesosynteny' to distinguish it from micro- and macrosynteny. Mesosynteny appears to be peculiar to the filamentous Ascomycetes (syn. Pezizomycotina), particularly in the class Dothideomycetes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2015-12-21T22:24:37.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3249" PostTypeId="2" ParentId="3245" CreationDate="2015-12-21T23:12:08.947" Score="2" Body="&lt;p&gt;The concept of zero (as a number) is not necessary for positional notation, one can do with a placeholder symbol, used like a punctuation mark, or even with just a blank space. This is exactly what Sumerians and Babylonians did in their sexagesimal system of numerals since 3rd millenium BC, the earliest known positional system. This system was later transmitted to Hellenistic Greece in late 3rd or second century BC, and was adopted by Greek astronomers including Hipparchus for their calculations. Ptolemy uses the placeholder symbol in his Almagest, it looks like letter o (one speculation is that it stood for &quot;obol&quot;, a coin of smallest value). But as &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor in its History of Zero&lt;/a&gt; writes &quot;&lt;em&gt;one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself&lt;/em&gt;&quot;. You can see placeholders in the sexagesimal and some other historical positional systems in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2181/has-a-digit-ever-been-used-to-represent-the-number-10/2186#2186&quot;&gt;Has a digit ever been used to represent the number &amp;quot;10&amp;quot;?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indians most likely encountered the sexagesimal system in their contacts with the Hellenistic world, although there is no direct evidence of transmission. Around 500 AD Aryabhata used a decimal positional system without even a placeholder zero, and some other Indian manuscripts used a dot as a placeholder. Aryabhata's word &quot;kha&quot; for the blank will later be used as the name for zero. But by the time Brahmagupta started talking about zero as a number over a century later positional notation was already established for millenia. I am not aware of any place where the development went the other way, from number zero to positional notation. Zero number appears to be a far more challenging concept than a punctuation mark.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2015-12-21T23:20:52.450" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3250" PostTypeId="1" CreationDate="2015-12-22T06:20:49.183" Score="3" ViewCount="163" Body="&lt;p&gt;The modern definition of group representation is a homomorphism between a group $G$ and the group $GL(V, K)$ of some vector space over the field $K$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as far as I know, when Frobenius developed his character theory, he did not have this picture in his mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, who invented (or discovered) this definition?&lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="2905" LastEditDate="2015-12-22T20:19:29.300" LastActivityDate="2015-12-24T03:21:27.523" Title="Who defined group representation in the modern way?" Tags="|mathematics|group-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3251" PostTypeId="2" ParentId="3227" CreationDate="2015-12-22T17:13:20.120" Score="5" Body="&lt;p&gt;Regarding II (c): Strogatz/Schroeder does offer a string of connections to Einstein through Shneior Lifson and Ernst Straus, one of Einstein's assistants at IAS.  But it would be hard to connect Einstein and the young Jashemski, since Einstein came to the US in 1933.  There are many instances of people coming up with the same proof independently, especially on something as well studied as the Pythagorean theorem (I understand your rationale for &amp;quot;so-called,&amp;quot; but I think that's a losing battle--see also &lt;em&gt;Was Pythagoras Chinese?&lt;/em&gt; by Frank Swetz, 1977).&lt;/p&gt;&#xA;&lt;p&gt;I wonder if Loomis's &amp;quot;young man of superior intellect&amp;quot; is the same Stanley Jashemski who graduated from U. Chicago in 1943, worked as a physicist for the Navy, and is best known as the husband of Wilhelmina Feemster Jashemski, a U. Maryland historian who specialized in the gardens of Pompeii (Stanley took the pictures for her research).&lt;/p&gt;&#xA;" OwnerUserId="3429" LastEditorUserId="136" LastEditDate="2023-09-04T14:49:36.043" LastActivityDate="2023-09-04T14:49:36.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3253" PostTypeId="2" ParentId="3250" CreationDate="2015-12-22T21:54:21.953" Score="3" Body="&lt;p&gt;It is true that Frobenius did not have the notion of a group representation in his celebrated &quot;trilogy&quot; of 1896 papers on factorization of the group determinant problem, posed to him by Dedekind in their correspondence the same year. Instead, he defined characters by working with a commutative complex algebra (which he later identified as the center of the group algebra), and taking traces of some projection maps. These are the papers where he proved the orthogonality relations and related characters to the structure constants, among other things. Some inspiration came from the work of Weierstrass, Dedekind, and Study on commutative hypercomplex systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Frobenius did however give a more or less modern definition of a group representation shortly thereafter, also on Dedekind's suggestion:&quot;&lt;em&gt;Let $H$ be an abstract group, $A, B, C\dots$ be its elements. One associates to the element $A$ the matrix $(A)$, to the element $B$ the matrix $(B)$, etc., in such away that the group $\mathfrak{H}'$ is isomorphic to the group $\mathfrak{H}$, that is, $(A)(B)=(AB)$. Then I say that the substitutions or the matrices $(A),(B),(C)\dots$ represent the group $\mathfrak{H}$&lt;/em&gt;&quot; (&quot;isomorphic&quot; did not yet mean one-to-one, so it corresponds to modern &quot;homomorphic&quot;). He then pointed out that the characters are given by the traces of matrices in &quot;primitive&quot; representations, defined as those with irreducible determinants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Within two years of his initial trilogy Frobenius introduced the “composition” (now tensor product) of characters, established relations between the characters of a group and its subgroups, introduced induced representations, and proved the reciprocity law for them. He did acknowledge as inspiration two works of Molien (1893, 1897), who analyzed group algebras as hypercomplex systems. Molien anticipated methods of Maschke, Wedderburn, and Noether in modern group representation theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Frobenius's work is described in detail in &lt;a href=&quot;http://www.ams.org/notices/199803/lam.pdf&quot; rel=&quot;nofollow&quot;&gt;Lam's Representations of Finite Groups: a Hundred Years&lt;/a&gt; in AMS Notices, which references the original papers.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2015-12-24T03:21:27.523" LastActivityDate="2015-12-24T03:21:27.523" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3254" PostTypeId="1" AcceptedAnswerId="3255" CreationDate="2015-12-23T19:18:08.690" Score="13" ViewCount="2180" Body="&lt;p&gt;Having done some searches on the internet, seems like the term &quot;power&quot; is a mistranslation. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponentiation#History_of_the_notation&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; links to an article in the &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Miscellaneous/Mathematical_notation.html&quot; rel=&quot;noreferrer&quot;&gt;MacTutor History of Mathematics archive&lt;/a&gt; where it is written&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The notation and terminology for powers and exponents is interesting. Power is first used for the square. Euclid uses the phrase in power , for example he says that magnitudes are commensurable in power when their squares are commensurable. Of course Euclid thought geometrically and the square to him was the geometrical square not a new number formed by multiplying the number by itself. Henry Billingsley, the first English translator of Euclid in 1570, makes the definition precise in his translation of Euclid's Second book:-&#xA;  The power of a line is the square of the same line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Are these the intended meanings? For the first instance (concerning commensurability) -&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;εὐθεῖαι δυνάμει σύμμετροί εἰσιν, ὅταν τὰ ἀπ᾽ αὐτῶν τετράγωνα τῷ αὐτῷ χωρίῳ μετρῆται&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I could not find the corresponding place in the Book 2 for Billingsley's  translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would translate &quot;δυνάμει&quot; in the above context as &quot;potentially&quot;, which must be closely related to &quot;δύνατον&quot; (&quot;possible&quot;) frequently used across Euclid's books. Cf. German &quot;Potenz&quot; for &quot;power&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus although &quot;potence&quot; is sort of a synonym for &quot;power&quot;, that another meaning of potentiality is lost when using &quot;power&quot; I think.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is my interpretation correct? Are there any authoritative opinions on this?&lt;/p&gt;&#xA;" OwnerUserId="3439" LastEditorUserId="18903" LastEditDate="2023-10-28T07:11:38.333" LastActivityDate="2023-10-28T07:11:38.333" Title="Etymology of &quot;power&quot; (math.)" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3255" PostTypeId="2" ParentId="3254" CreationDate="2015-12-23T19:51:23.577" Score="14" Body="&lt;p&gt;&lt;em&gt;Comments&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The correct source is Euclid : &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookX/bookX.html#defsI&quot; rel=&quot;noreferrer&quot;&gt;Book X, Def.2&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &lt;em&gt;δυνάμει&lt;/em&gt;, it is a word used in Aristotle : see &lt;a href=&quot;https://en.wikipedia.org/wiki/Potentiality_and_actuality&quot; rel=&quot;noreferrer&quot;&gt;Potentiality and actuality&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/aristotle-metaphysics/#ActPot&quot; rel=&quot;noreferrer&quot;&gt;Aristotle's Metaphysics : Actuality and Potentiality&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the Aristotelian distinction between potentiality (&lt;em&gt;dunamis&lt;/em&gt;) and actuality (&lt;em&gt;entelecheia&lt;/em&gt; or &lt;em&gt;energeia&lt;/em&gt;). This distinction is the main topic of &lt;strong&gt;Book $\Theta$&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, from the Greek &lt;em&gt;dunamis&lt;/em&gt; to the Latin &lt;a href=&quot;https://en.wiktionary.org/wiki/potentia&quot; rel=&quot;noreferrer&quot;&gt;potentia&lt;/a&gt; and finally to &lt;em&gt;power&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can transalte &lt;em&gt;dunamei summetros&lt;/em&gt; as &quot;potential commensurability&quot; : two segments of lenght $1$ and $\sqrt 2$ are &lt;em&gt;incommensurable&lt;/em&gt; but &quot;commensurable in square&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See this post on &lt;a href=&quot;https://philosophy.stackexchange.com/questions/28356/aristotelian-terminology-dynamis-and-entelechy&quot;&gt;Aristotelian terminology, dynamis and entelechy&lt;/a&gt; for some comments on the multiplicity of meaning of &lt;em&gt;dunamis&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, if &lt;em&gt;dunamis&lt;/em&gt; stands for &lt;em&gt;potentiality&lt;/em&gt; and &lt;em&gt;capacity&lt;/em&gt;, it has some &quot;nuances&quot; that also &lt;a href=&quot;http://www.etymonline.com/index.php?term=power&quot; rel=&quot;noreferrer&quot;&gt;power&lt;/a&gt; has : &quot;ability to act or do&quot;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-06-19T07:17:48.550" LastActivityDate="2018-06-19T07:17:48.550" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="3256" PostTypeId="1" CreationDate="2015-12-23T23:36:00.017" Score="5" ViewCount="369" Body="&lt;p&gt;For some reason, everybody refers to the function &#xA;$$\lambda(x,y,z) = x^2 + y^2 + z^2 - 2xy -2xz-2yz$$&#xA;as &quot;Källén's triangle function.&quot;  (see for example &lt;a href=&quot;https://en.wikipedia.org/wiki/Gunnar_K%C3%A4ll%C3%A9n&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Gunnar_Källén&lt;/a&gt;).  I understand why its a &lt;em&gt;triangle&lt;/em&gt; function (because it's proportional the area of a triangle of sides $x$ $y$ and $z$).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But how did Källén get his name attached to it?  Is there a paper of his where he used that combination a lot?&lt;/p&gt;&#xA;" OwnerUserId="3441" LastEditorUserId="3441" LastEditDate="2016-08-04T12:22:18.937" LastActivityDate="2016-08-04T22:45:42.203" Title="How did Källén get his name attached to $x^2 + y^2 + z^2 - 2 x y -2 x z- 2 y z$?" Tags="|physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3257" PostTypeId="1" CreationDate="2015-12-24T15:57:52.110" Score="6" ViewCount="2267" Body="&lt;p&gt;When and who was the first person to discover a correct formula for the real number $r$ in terms of any &lt;strong&gt;&lt;em&gt;given&lt;/em&gt;&lt;/strong&gt; three positive distinct integers $x&amp;lt;y&amp;lt;z$ such that $$x^r + y^r = z^r\,.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formula: let $$t=\log(y/x) / \log(z/x)$$ and define $f(t)$ as function of real variable $t$ and positive integer $n$ when $n$ tends to infinity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(t) = \frac{3t}{2} + (2t-1)\left(\frac{4t}{3} - 1\right) + \left(\frac{5t}{2} - 1\right)\left(\frac{5t}{3} - 1\right)\left(\frac{5t}{4} - 1\right) + \cdots + \left(\frac{nt}{2} - 1\right)\left(\frac{nt}{3} -1 \right)...\left(\frac{nt}{n-1} -1 \right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where $t$ is positive real number less than one, &#xA;then: $$(x/z)^r = (\log(z/y) / \log(z/x))\cdot f(t)\,,$$or simply $r = \log \left((1-t)f(t)\right) / \log(x/z)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So $r$ is therefore obtainable as a function of the three positive &lt;em&gt;distinct given&lt;/em&gt; integers $x&amp;lt;y&amp;lt;z$.&lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="2442" LastEditDate="2016-08-12T17:07:37.743" LastActivityDate="2019-11-19T11:38:43.387" Title="Sum of like powers in real numbers" Tags="|mathematics|discoveries|abstract-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3258" PostTypeId="2" ParentId="783" CreationDate="2015-12-25T02:51:11.990" Score="3" Body="&lt;p&gt;Even as a Polish-trained mathematician with active interests in philosophy and history of mathematics and science (who took two courses from Prof. Woleński, among others), I am hard pressed to think of three mathematicains/logicians/philosophers who would have good motivation for reading Principia and were “liquidated by Hitler”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be sure, Leon Chwistek is the strongest contender. He worked on theory of types and corresponded with Russell, who (successfully) recommended him for the chair of logic at the Lwów University in 1928. Mark Kac, a student in Lwów in 1930s, wrote the following words about Chwistek in his autobiography “Enigmas of chance”: “Rumor had it that he was the only one who, except for the authors, had read all three volumes of Principia Mathematica (pp. 42-43).” However, Chwistek died of a kidney failure in Moscow in 1944 (there were rumors about his poisoning).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As noted by other participants in this discussion, several strong candidates cannot be counted because of when and how they died. But it is still worth recalling some of the evidence of their reading Principia, along with a little of biographical information, since not all are household names.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jan Sleszyński (1854-1931), a Pole who was first a professor in Odessa (under the name of Ivan Sleshinskii), then in Kraków, started out working in number theory and analysis, later turning to logic. His lecture notes in proof theory, published in Kraków by the Mathematics and Physics Student Circle in 1925-29, discuss extensively the history of logic up to (and including) the contributions of Russell and Whitehead. (see the book by Roman Murawski, Philosophy of mathematics and logic in the inter-war Poland). According to A. Hoborski, O Śleszyńskim, wspomnienie pośmiertne [On Śleszyński, an after-death reminiscence], Wiadomości Matematyczne, 36 (1934), he sought to eliminate any hidden rules from deductive reasoning, which prompted him to study Principia Mathematica (private information from Lidia Obojska).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Zygmunt Zawirski (1882-1948) was a philosopher with some mathematical background, a student of Kazimierz Twardowski. In 1937 he became a professor in Kraków. He was interested in application of logic to natural sciences. He had hoped that Russell’s theory of types would help eliminate logical antinomies arising in quantum physics (this information after Irena Szumilewicz-Lachman: Zygmunt Zawirski. His life and work. With selected writings on time, logic and the methodology of science. Kluwer Academic Publishers, 1994).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the names of Alfred Tarski and Stanisław Leśniewski are much better known among mathematicians. Regarding whether Tarski read Principia, one can note that he considered the concept of membership and in a lecture in 1966 he quoted Principia Mathematica to support his view that it is a logical notion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Using this method [of Principia Mathematica], it is clear that the membership relation is certainly a logical notion. It occurs in several types, for individuals are elements of classes of individuals, classes of individuals are elements of classes of classes of individuals, and so on. And by the very definition of an induced transformation it is invariant under every transformation of the world onto itself” (quotation after Stanford Encyclopedia of Philosophy, &lt;a href=&quot;http://plato.stanford.edu/entries/tarski/&quot; rel=&quot;nofollow&quot;&gt;http://plato.stanford.edu/entries/tarski/&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tarski’s PhD advisor Leśniewski tried to provide foundations for mathematics in a very unique manner, bypassing the set theory and the type theory. He referred to Principia Mathematica at least by criticizing its terminology, which he found confusing (“Foundations of a General Theory of Sets I”, 1916; after &lt;a href=&quot;http://plato.stanford.edu/entries/lesniewski/&quot; rel=&quot;nofollow&quot;&gt;http://plato.stanford.edu/entries/lesniewski/&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My personal bets (if speculation is allowed), besides Chwistek, are Władysław&#xA;Hetper and Witold Wilkosz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hetper was a student and coauthor of Chwistek (and a close friend and roommate of Mark Kac in their student days), who received his PhD in 1937 and his habilitation in 1939. He was imprisoned by the Soviets and executed by them, probably in Kharkov in 1941.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wilkosz was a professor of mathematics in Kraków, Banach’s high school classmate, a PhD student of Giuseppe Peano in Turin, and a polymath with multiple interests, including logic. Not quite ”liquidated”, he died in 1941 (during the German occupation of Poland) of natural causes, but the time he spent in German prison in 1939 and subsequent wartime deprivation made his condition worse. Wilkosz explicitly referred to Russell and his arguments with Peano when discussing equivalence relations in the paper “O definicji przez abstrakcjȩ” [On defining through abstraction], Kwartalnik Filozoficzny 14 (1938), 1-13 (see Murawski’s book cited above). Hetper in his work credited some unpublished results of Wilkosz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All three (Chwistek, Hetper, Wilkosz) died during WWII. Russell either did not know the details of their death (in the cases of Chwistek and Hetper, these are still not fully confirmed and were not public knowledge in communist Poland) or else he was reluctant to blame the Soviets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was also Stanisław Bilski, a graduate in mathematics and a doctor in philosophy (from the Jagilellonian University in Kraków) in 1926, whose thesis title was “A priori knowledge in Bertrand Russell’s epistemology”. he was a communist activist in Poland, subsequently liquidated in the USSR in the Great Purge of 1934 (hence again by Stalin, not by Hitler). But his name was rather unknown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, a popular account in Polish of Russell’s antinomy was written in 1927 (Przegla̧d Filozoficzny, vol. 30, issue 4, pp. 291-292) by the Lwów mathematician Lucjan Emil Boettcher, who worked mainly in iteration theory. He died of natural causes in 1937, before WWII and German occupation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only sure conclusion of all this is that Russell’s works were studied by&#xA;Poles (and in Poland, when it was reborn in 1918) before 1939.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="-1" LastEditDate="2015-12-27T14:47:18.503" LastActivityDate="2015-12-27T14:47:18.503" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3259" PostTypeId="1" CreationDate="2015-12-25T12:51:02.637" Score="4" ViewCount="178" Body="&lt;p&gt;As we know $$c^2\text{curl}\;\bf B= \frac{\bf J}{\epsilon_0}$$ is incomplete &amp;amp; in many cases like capacitor give contradictory result against Law of conservation of charge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Feynman writes;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Until Maxwell’s work &lt;em&gt;[...]&lt;/em&gt; the equation for the magnetic field of steady currents was known only as&#xA;  $$\boldsymbol ∇×\boldsymbol B=\frac{\boldsymbol j}{ϵ_0c^2}.\tag{18.1}$$&#xA;  Maxwell began by considering these known laws and expressing them as differential equations, as we have done here. (Although the $\bf ∇$ notation was not yet invented, it is mainly due to Maxwell that the importance of the combinations of derivatives, which we today call the curl and the divergence, first became apparent.) He then noticed that there was something strange about Eq. (18.1). If one takes the divergence of this equation, the left-hand side will be zero, because the divergence of a curl is always zero. So this equation requires that the divergence of $\boldsymbol j$ also be zero. But if the divergence of $\boldsymbol j$ is zero, then the total flux of current out of any closed surface is also zero.&#xA;  The flux of current from a closed surface is the decrease of the charge inside the surface. This certainly cannot in general be zero because we know that the charges can be moved from one place to another. The equation&#xA;  $$\boldsymbol ∇\cdot \boldsymbol j=−\frac{∂ρ}{∂t}\tag{18.2}$$&#xA;  has, in fact, been almost our definition of $\boldsymbol j.$ This equation expresses the very fundamental law that electric charge is conserved—any flow of charge must come from some supply. &lt;strong&gt;&lt;em&gt;Maxwell appreciated this difficulty and proposed that it could be avoided by adding the term $∂\boldsymbol E/∂t$ to the right-hand side of Eq. (18.1)&lt;/em&gt;&lt;/strong&gt;; he then got the fourth equation in Table 18–1:&#xA;  $$\text{IV}.c^2\boldsymbol ∇×\boldsymbol B=\frac{\boldsymbol j}{ϵ_0}+\frac{∂\boldsymbol E}{∂t}.$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now what did Maxwell do to conclude that indeed $\frac{\partial \bf E}{\partial t}$ was required to solve the problem? Did he conduct experiment to conclude or just some of his obscure calculations? &lt;/p&gt;&#xA;" OwnerDisplayName="user1709" LastEditorUserId="4251" LastEditDate="2020-06-06T17:20:25.023" LastActivityDate="2020-06-06T17:20:25.023" Title="How did Maxwell conclude that $\frac{\partial\bf E}{\partial t}$ was necessary to complete $c^2\nabla \times \bf B= \frac{\bf J}{\epsilon_0}\;?$" Tags="|physics|theoretical-physics|electromagnetism|maxwell|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3265" PostTypeId="1" AcceptedAnswerId="3267" CreationDate="2015-12-28T15:50:23.950" Score="4" ViewCount="95" Body="&lt;p&gt;I was wondering when eigenmodes (modes of vibration) were first mentioned in mechanics. Any reference would be appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I expect them to date back to the second half of the 19th century because of the breakthrough in linear algebras, but I'm not sure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Additional notes&lt;/em&gt; This question was initially asked on &lt;a href=&quot;http://physics.stackexchange.com&quot;&gt;physics.SE&lt;/a&gt;. I was suggested to post it here, but I'm keeping track of the comments below:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;[DanielSank] &lt;em&gt;Fourier's big publication was 1822, and although the big example there was heat diffusion it definitely gets at the idea of normal modes. Interestingly, Wikipedia points out that decomposition into basis functions was kind of sort of known in antiquity, albeit without any true understanding of what was going on.&lt;/em&gt; But I don't think there was anything about vector space structure at that time some we can't speak &quot;modal analysis&quot;, which is more general: any motion of a linear n-dimensional system (even non-periodic) can be decomposed as a linear combination of n modes. &lt;/li&gt;&#xA;&lt;li&gt;[CuriousOne] &lt;em&gt;You may want to look at the references in Gram-Schmidt wiki. I think one should link an understanding of eigenmodes to an interest in orthogonalization of bases in linear spaces, which would put it formally closer to the end of the 19th, early 20th century. As it has been pointed out, these fairly trivial ideas have been invented multiple times, so it's not clear who was &quot;first&quot;, since you have to pick based on the context.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3458" LastActivityDate="2015-12-28T19:56:08.833" Title="First mention of modes of vibration in mechanics" Tags="|classical-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3266" PostTypeId="2" ParentId="774" CreationDate="2015-12-28T16:10:08.783" Score="7" Body="&lt;p&gt;Cauchy's formulation of the theorem in 1821 is ambiguous and at any rate in a 1853 article Cauchy seems to state that it was incorrect as stated. On the other hand, in the 1853 article Cauchy modifies the hypotheses and claims that this results in a true theorem, providing a proof thereof.  The existence of a correct result (modulo modified hypotheses) could explain why it may not have been obvious that the theorem is incorrect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much ink has been spilled on this issue. User Conifold seems to employ Lakatos' terminology who sought to explain Cauchy in terms of so-called &quot;moving points&quot;. I personally don't understand what these moving points are, and prefer the explanation given by D. Laugwitz in his 1987 paper in &lt;em&gt;Historia Mathematica&lt;/em&gt; here: &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086087900450&quot; rel=&quot;noreferrer&quot;&gt;http://www.sciencedirect.com/science/article/pii/0315086087900450&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his paper Laugwitz emphasizes that he gives an interpretation using Cauchy's own concepts, and that no nonstandard numbers are needed to explain Cauchy, contrary to Fisher cited by User Conifold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our recent paper deals with some aspects of this debate: &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-015-9473-4&quot; rel=&quot;noreferrer&quot;&gt;http://dx.doi.org/10.1007/s10699-015-9473-4&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2015-12-28T16:10:08.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3267" PostTypeId="2" ParentId="3265" CreationDate="2015-12-28T19:56:08.833" Score="3" Body="&lt;p&gt;As every question which involves &quot;who was the first&quot; or &quot;when was the first&quot;, this one is unlikely to have a definite answer. (Read the first chapter of Th. Mann, &quot;Joseph and his brothers&quot; where this is explained in great detail).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is clear that the modes where well known long before the 19th century.&#xA;(I mean the phenomenon itself, I am not discussing the term &quot;mode&quot;).&#xA;An exact solution of the 1-dimensional wave equation in the form of superposition of eigenfunctions was obtained by Daniel Bernoulli in 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the fact itself (as an experimental fact) was certainly known to Marine Mersenne, (1637) and to Kepler (1619), and in some form to the Pythagoreans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the ancient times this fact was considered as belonging to music theory, but music theory was a part of mathematics, probably since Pythagoras himself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the fact is indeed known since antiquity, but for a rigorous mathematical formulation probably Bernoulli can be credited.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2015-12-28T19:56:08.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3268" PostTypeId="1" CreationDate="2015-12-29T15:05:48.317" Score="6" ViewCount="13309" Body="&lt;p&gt;I am a programmer that has been working for fixed income financial companies for the past couple of years. As such, I have learned about a weird notation on fixed income products.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the US, bonds are priced in 1/32 increments (&quot;a tick&quot;) or 1/64 increments (&quot;a plus&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Tick_size&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia explains the usage&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...a price is quoted as 99-30+, meaning 99&#xA;  and 61/64 percent (or 30.5/32 percent) of the face value. As an&#xA;  example, &quot;par the buck plus&quot; means 100% plus 1/64 of 1% or 100.015625%&#xA;  of face value.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But in Europe, they are priced in 1/100 increments, represented in decimal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't find much on the history of this system - why it's priced this way, notated this why, and why it only seems to be the US.&lt;/p&gt;&#xA;" OwnerUserId="3463" LastActivityDate="2022-12-02T08:13:22.457" Title="Why are US Bonds measured in '32nd' ticks?" Tags="|notation|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3269" PostTypeId="1" CreationDate="2015-12-30T00:48:08.130" Score="4" ViewCount="715" Body="&lt;p&gt;Unlike other organs, such as the heart or intestines, it seems much harder to discern what the brain is doing in the human body. How did scientists figure out the brain was responsible for information processing, thought, and control of the rest of the body?&lt;/p&gt;&#xA;" OwnerUserId="3467" LastActivityDate="2015-12-30T05:03:06.987" Title="How and when did we discover the role of the brain?" Tags="|discoveries|neuroscience|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3270" PostTypeId="1" AcceptedAnswerId="3271" CreationDate="2015-12-30T03:27:19.387" Score="12" ViewCount="303" Body="&lt;p&gt;If you look at old books like Euclid, and Archimedes it seems that there weren't any problem set exercises. They just presented the all information. There were no &quot;end chapter&quot; problems. In short, it seems that most ancient books were written as treatises. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, for contemporary textbooks, they often leave exercises at the end of the chapter, often with no solutions. I wonder when did exercises were appended into textbooks. Specifically, exercises which do not contain answers/derivations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I conjecture maybe from the the industrial revolution, but that's just my guess. I want to know the palpable answer to this. In other words, I wonder when did books begin to include exercises with omitted solutions.&lt;/p&gt;&#xA;" OwnerUserId="3469" LastEditorUserId="3469" LastEditDate="2015-12-30T15:35:40.917" LastActivityDate="2018-03-14T15:45:40.787" Title="When were problem sets included in Science/math textbooks?" Tags="|education|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3271" PostTypeId="2" ParentId="3270" CreationDate="2015-12-30T04:26:03.880" Score="10" Body="&lt;p&gt;I can only talk about math books. Problems in the end of the chapters is a British-American custom. German, French and Russian books do not have them.&#xA;Instead they publish separate problem books, usually by a different author. &#xA;Problem books contain only problems, and in the introduction they mention&#xA;which textbook(s) are recommended.&#xA;British/American textbooks with&#xA;problems at the ends of the chapters appear since the beginning of 20th century,&#xA;perhaps there are some late 19th century but I have not seen them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, in the end of 20th century this British/American custom spreads over the world, like many other British/American customs, and you can see textbooks in other languages that follow it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples from my speciality (where I know most of the existing textbooks).&#xA;Whittaker-Watson (first edition 1902) has problems.&#xA;K. Knopp (German, many textbooks) no problems. Separate problem books by the same author.&#xA;Hurwitz-Courant (German, first ed. 1922, and many later editions) no problems.&#xA;H. Cartan (French, first ed. 1963) no problems.&#xA;Markushevich, Privalov, Lavrentiev-Shabat (Russian, multiple editions) no problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most famous problem books in the same subject: Polya-Szego (German), Volkovyski (Russian). On the other hand, there is no such genre as a &quot;problem book&quot; by a British/American author, with very rare exceptions like Halmos.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-03-14T15:45:40.787" LastActivityDate="2018-03-14T15:45:40.787" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3272" PostTypeId="2" ParentId="3269" CreationDate="2015-12-30T04:42:54.483" Score="2" Body="&lt;p&gt;Some functions of the brain are actually easier to discover than functions of the heart or kidneys or liver, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do this you hit a human at the head with a mallet (wooden hammer) and see what happens. When s/he recovers her or his consciousness, ask him or her to describe &#xA;his/her experience. (Disclaimer: I do NOT recommend to anyone to perform a real experiment, one needs a lot of knowledge and experience to calculate the appropriate force of the hit, if one does not want to make a permanent damage. Consider this as a thought experiment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This procedure was used for millenia in cases when we use modern anaesthesia at the time&#xA;of surgery. It was still in use in the middle 19 century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore I suppose that at least some functions of the brain were well-known well before the beginning of history. One cannot tell exactly when this method was discovered, but we know that surgery was performed very long before the beginning of writing, and certainly many people had similar experience without any surgery. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2015-12-30T05:03:06.987" LastActivityDate="2015-12-30T05:03:06.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3273" PostTypeId="2" ParentId="3020" CreationDate="2015-12-30T08:04:51.740" Score="4" Body="&lt;p&gt;It may have been, indirectly, Charles Dickens who popularized the question &quot;Why is the sky blue?&quot;  In the 1850s Dickens edited a weekly titled &lt;a href=&quot;https://en.wikipedia.org/wiki/Household_Words&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Household Words&lt;/em&gt;&lt;/a&gt; which was inexpensive and had a wide circulation.  Dickens had recruited &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Morley&quot; rel=&quot;nofollow&quot;&gt;Henry Morley&lt;/a&gt; as a contributor to the magazine.  Morley wrote a piece called &quot;Light and Air&quot; which appeared in the 20 September 1851 issue of &lt;em&gt;Household Words&lt;/em&gt;, and &lt;a href=&quot;http://www.djo.org.uk/household-words/volume-iii/page-614.html&quot; rel=&quot;nofollow&quot;&gt;in it he says&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Now, why is the sky blue? Not only does the air absorb light; it&#xA;  reflects it also. The particles of air reflect, however, most&#xA;  especially the blue ray, while they let the red and his companions&#xA;  slip by. This constant reflection of the blue ray causes the whole air&#xA;  to appear blue;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This same article later appeared in &lt;a href=&quot;https://books.google.com/books?id=APAvAAAAMAAJ&amp;amp;pg=PA813&amp;amp;dq=%22why%20is%20the%20sky%20blue%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiIr5rs-4LKAhXJKGMKHe1kAhUQ6AEITjAJ#v=onepage&amp;amp;q=%22why%20is%20the%20sky%20blue%22&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Harper's Magazine&lt;/em&gt;&lt;/a&gt;, and the question later being appearing in publications like &lt;a href=&quot;https://books.google.com/books?id=dTMIAAAAIAAJ&amp;amp;printsec=frontcover&amp;amp;dq=%22why%20is%20the%20sky%20blue%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwj70pOZ-ILKAhUBRmMKHZDoB6wQ6AEIVjAJ#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Peterson's Familiar Science&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=iYQDAAAAQAAJ&amp;amp;pg=PA98&amp;amp;dq=%22why%20is%20the%20sky%20blue%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiIr5rs-4LKAhXJKGMKHe1kAhUQ6AEIHTAA#v=onepage&amp;amp;q=%22why%20is%20the%20sky%20blue%22&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;The Reason Why?&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A reason to think the publication in Dickens' &lt;em&gt;Household Words&lt;/em&gt; was instrumental in popularizing the question is that the &lt;a href=&quot;https://books.google.com/ngrams/graph?content=why%20is%20the%20sky%20blue%2Cwhat%20makes%20the%20sky%20blue&amp;amp;case_insensitive=on&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=0&amp;amp;share=&amp;amp;direct_url=t4%3B%2Cwhy%20is%20the%20sky%20blue%3B%2Cc0%3B%2Cs0%3B%3BWhy%20is%20the%20sky%20blue%3B%2Cc0%3B%3Bwhy%20is%20the%20sky%20blue%3B%2Cc0%3B%3BWhy%20Is%20the%20Sky%20Blue%3B%2Cc0%3B%3BWhy%20is%20the%20Sky%20Blue%3B%2Cc0%3B%3BWHY%20IS%20THE%20SKY%20BLUE%3B%2Cc0%3B.t4%3B%2Cwhat%20makes%20the%20sky%20blue%3B%2Cc0%3B%2Cs0%3B%3BWhat%20makes%20the%20sky%20blue%3B%2Cc0%3B%3Bwhat%20makes%20the%20sky%20blue%3B%2Cc0&quot; rel=&quot;nofollow&quot;&gt;Google Books N-Gram viewer shows&lt;/a&gt; the phrase first appearing in its corpus about 1852 and then growing in frequency.&lt;/p&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2015-12-30T08:04:51.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3274" PostTypeId="1" AcceptedAnswerId="3292" CreationDate="2015-12-30T08:43:31.153" Score="4" ViewCount="209" Body="&lt;p&gt;Gauss proved that the so-called Gaussian curvature is an intrinsic invariant of the surface, even though it is defined extrinsically as the product of the principal curvatures of a Euclidean embedding.  But apart from the issue of its intrinsic nature, did anyone consider the invariant given by the product of the principal curvatures before Gauss?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="26" LastEditDate="2015-12-31T16:40:18.503" LastActivityDate="2016-01-04T21:06:58.693" Title="Did anybody consider the product of the principal curvatures before Gauss?" Tags="|mathematics|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3276" PostTypeId="1" CreationDate="2015-12-30T21:10:40.890" Score="3" ViewCount="357" Body="&lt;p&gt;While doing the tedious work of documenting my software I tried to find the original source of the divide and conquer method for the conversion of numbers of one base to a number in another base (mostly binary to decimal and vice versa these days). The obligatory &quot;If in doubt, it was Knuth.&quot; was not very helpful: In his TAoCP[^1] in the answer to exercise 4.4.14 he just gave the hint&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Similarly Schönhage has observed that we can convert a ($2^n\log 10$)-bit number $U$ from binary to decimal, in $O(nM(2^n))$ steps.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The single place where Arnold Schönhage said something in that direction was in a book[^2] he co-authored&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is a standard divide-and-conquer method (here mentioned as a nice exercise) by which the full speed of the of the fast multiplication routines can be utilized so that a time bound of order $\Theta\cdot n \cdot(\log n)^2\log \log n$ for the conversion of $n$ word operands [&amp;hellip;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are other findings, for example Richard Brent in an article[^3] but he pointed to D. Knuth at[^1].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method is most probably not very old because it needs asymptotically fast multiplication (There is the Nikhilam method from India[^4], Karatsuba[^5] in the West was much later) to function properly, so where did it originate, who can I cite? Or is it lost, deep in the dusty abyss of some obscure mathematical almanac, long forgotten?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[^1] Knuth, Donald-E. &quot;The Art of Computer Programming. Vol. 2.&quot; (1998).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[^2] Schönhage, Arnold, Andreas FW Grotefeld, and Ekkehart Vetter. &quot;Fast algorithms: a multitape turing machine implementation&quot;. BI Wissenschaftsverlag, 1994.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[^3] Brent, Richard P. &quot;The complexity of multiple-precision arithmetic.&quot; The Complexity of Computational Problem Solving (1976): 126-165.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[^4] Tirtha, Swami Bharati Krishna, Vasudeva Sharana Agrawala, and V. S. Agrawala. Vedic mathematics. Vol. 10. Motilal Banarsidass Publ., 1992.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[^5] Karatsuba, Anatolii, and Yu Ofman. &quot;Multiplication of multidigit numbers on automata.&quot; Soviet physics doklady. Vol. 7. 1963.&lt;/p&gt;&#xA;" OwnerUserId="3472" LastActivityDate="2016-01-06T02:47:18.200" Title="Where did the divide and conquer method for radix conversion come from?" Tags="|reference-request|computer-science|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3277" PostTypeId="2" ParentId="3268" CreationDate="2015-12-30T22:59:54.057" Score="-1" Body="&lt;p&gt;I can only speculate that it is the legacy of peso de ocho popularity. The &quot;piece of eight&quot; coin was often &lt;a href=&quot;http://www.history.org/history/teaching/enewsletter/volume3/march05/iotm.cfm&quot; rel=&quot;nofollow&quot;&gt;physically cut&lt;/a&gt; into 8 &lt;em&gt;bits&lt;/em&gt; (mostly to obtain change - a solid peso contained too much value for everyday circulation, almost an ounce of fine silver). The bit was still large enough to accurately halve and even quarter it, giving 1/16 and 1/32 of original value.&lt;/p&gt;&#xA;" OwnerUserId="1779" LastActivityDate="2015-12-30T22:59:54.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3278" PostTypeId="2" ParentId="3203" CreationDate="2015-12-31T13:51:25.593" Score="4" Body="&lt;p&gt;I do not think there was any problem in &quot;accepting the existence of irrational numbers&quot; before the clarification of the axiomatic method (1910-1940).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book X of Euclid's Elements looks more like an attempt to classify numbers (=size, magnitude) according to the complexity they were built. It was easier to add/subtract with the ruler, than to multiply/divide with the compass. And the famous problems like doubling the cube, squaring the circle, trisecting the angle were more related to finding the construction than to doubt about the existence of a solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think mathematicians who were computing numbers on a board or abacus, like the Chinese and Indian, accepted rather quickly what we now call algebraic numbers, and invented algorithms to compute root of n-th degree equation long ago. Again, the complexity of the computation was much more of a concern than the existence of the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematicians who were still computing with geometric instruments, like medieval arabo-islamic mathematician, made some progress in solving fractions, quadric, cubic, bi-quadratic with the claim of inventing new methods, not new numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With Al-Khwarizmi (Algorithm) and his book on computation with al-jabr (algebra) to compute geometric constructions with numbers, starts a language which will need 10 centuries for P. Wantzel to express (and to prove) that the duplication of the cube and trisection of the angle were impossible to solve with a straightedge and compass. Yet, Wantzel never doubted that $\sqrt[3]{2}$ and $\cos 10$ were otherwise perfectly standard numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the modern position is that, with the language of mathematical logic, you can rigorously classify numbers in set of increasing &quot;complexity&quot;, and look back in history to see how these class of complexity did appear in time. You can also (mathematically) define what is an &quot;undecidable&quot; question, and then have yet a new class of complexity classes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, to come back to your question, I think that was about behind irrational numbers is not accepting their existence, but rather the complexity of computing with them, and first of all to define them.&lt;/p&gt;&#xA;" OwnerUserId="3475" LastActivityDate="2015-12-31T13:51:25.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3279" PostTypeId="2" ParentId="3270" CreationDate="2015-12-31T14:03:13.897" Score="2" Body="&lt;p&gt;Mezopotamian and ancient Egyptian math &quot;textbook&quot; contains only problems and and solution. Something like &quot;Adding a length to a square give 1. What was the number?&quot; followed by actual computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seams that their conception of mathematics was to accumulate solved problems, like astronomers accumulate observations, without never thinking of making a general theory of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first to write a text-book in the modern form was Euclid.&lt;/p&gt;&#xA;" OwnerUserId="3475" LastActivityDate="2015-12-31T14:03:13.897" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3280" PostTypeId="2" ParentId="3245" CreationDate="2015-12-31T14:16:46.067" Score="2" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/0_(number)&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia article on the Number 0&lt;/a&gt;, by 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Egyptians did not used a positional numeration system before its introduction by the Persian mathematician Al Khwarizmi in 813 AD. &lt;/p&gt;&#xA;" OwnerUserId="3475" LastEditorUserId="1723" LastEditDate="2019-07-12T13:57:59.967" LastActivityDate="2019-07-12T13:57:59.967" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3281" PostTypeId="2" ParentId="3270" CreationDate="2015-12-31T15:36:03.833" Score="7" Body="&lt;p&gt;Here is one data point.  (It does not contradict Alexandre's assertion that this is an American innovation, but does provide an earlier starting date.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Day's Algebra&lt;/em&gt;  (I picked this volume up in a used book store once.)&#xA;&lt;a href=&quot;https://i.stack.imgur.com/c3kIi.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/c3kIi.jpg&quot; alt=&quot;Days&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This textbook does have lists of exercises at the end of each section.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2015-12-31T15:36:03.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3282" PostTypeId="1" AcceptedAnswerId="3283" CreationDate="2016-01-01T15:32:36.797" Score="2" ViewCount="699" Body="&lt;p&gt;Ferromagnetism is everyday life. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Paramagnetism is not rare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But diamagnetism is rare and weak. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, who first discovered diamagnetism? in which material?&lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="26" LastEditDate="2016-01-01T15:42:51.140" LastActivityDate="2016-01-01T16:08:28.390" Title="Who discovered diamagnetism first?" Tags="|physics|experimental-physics|electromagnetism|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3283" PostTypeId="2" ParentId="3282" CreationDate="2016-01-01T16:08:28.390" Score="2" Body="&lt;p&gt;Credit for discovering diamagnetism is given to different people. &lt;a href=&quot;https://en.wikipedia.org/wiki/Diamagnetism&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt; states that Sebald Justinus Brugmans was the first to discover it, in 1778, when he observed bismuth and antimony being repelled by magnetic fields. Interestingly enough, some sources (e.g. &lt;a href=&quot;http://revue.elth.pub.ro/upload/292688art01Gerald.pdf&quot; rel=&quot;nofollow&quot;&gt;Kustler (2007)&lt;/a&gt;) instead credit his father, Anton Brugmans, with the discovery. The elder Brugmans was also a scientist, and investigated electricity and magnetism, which may have led to the confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless of which Brugmans deserves credit, a more rigorous treatment of diamagnetism was given in 1845 by Michael Faraday (again, see Kustler for more information), who noted that a magnetic field could cause a piece of glass to rotate. Faraday's work was soon built upon by John Tyndall (see &lt;a href=&quot;http://www.tandfonline.com/doi/pdf/10.1080/00033790.2014.929743&quot; rel=&quot;nofollow&quot;&gt;Jackson (2014)&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-01-01T16:08:28.390" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3285" PostTypeId="1" CreationDate="2016-01-02T01:51:36.313" Score="1" ViewCount="611" Body="&lt;p&gt;Ferromagnetism is experienced by everyone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But paramagnetism is not easy to perceive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, who discovered paramagnetism first? The question is actually, who distinguished paramagnetism with ferromagnetism? &lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2018-09-11T12:26:56.717" Title="Who discovered paramagnetism first?" Tags="|physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3288" PostTypeId="1" CreationDate="2016-01-02T17:09:51.760" Score="4" ViewCount="1247" Body="&lt;p&gt;I would like to know the history and the original meaning of the word &quot;vector&quot;. Thank you for any hints.&lt;/p&gt;&#xA;" OwnerUserId="3485" LastEditorUserId="62" LastEditDate="2016-01-02T17:19:38.050" LastActivityDate="2016-01-02T21:46:58.383" Title="Origin of the word &quot;vector&quot;" Tags="|mathematics|physics|terminology|" AnswerCount="1" CommentCount="2" ClosedDate="2016-01-02T23:12:54.903" ContentLicense="CC BY-SA 3.0" />
  <row Id="3289" PostTypeId="2" ParentId="3288" CreationDate="2016-01-02T21:46:58.383" Score="2" Body="&lt;p&gt;The &lt;em&gt;Oxford English Dictionary&lt;/em&gt; has this:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. Harris &lt;em&gt;Lexicon Technicum I&lt;/em&gt;, 1704  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A Line supposed to be drawn from any Planet moving round a Center, or the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the New Astronomy, called the Vector; because 'tis that Line by which the Planet seems to be carried round its Center.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-01-02T21:46:58.383" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3290" PostTypeId="2" ParentId="3203" CreationDate="2016-01-03T01:16:26.850" Score="3" Body="&lt;p&gt;Greek mathematicians (e.g. Euclid and followers) accepted &lt;em&gt;points on a line&lt;/em&gt; that aren't constructible (or at least they didn't know to be constructible) as a matter of course. Lines were considered to be continuous, no &quot;holes&quot; in them. They didn't identify line lengths with numbers at all, as has been noted. That is a much later development, where initially the people starting to define the calculus assumed without second thought that &quot;numbers&quot; (real numbers we would say today) corresponded one to one with points on a line (e.g. for analytic geometry). That this has it's logical snares became known rather slowly, until we got today's definition of real numbers by Dedekind and colleagues to resolve the matter.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2016-01-03T01:16:26.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3291" PostTypeId="1" CreationDate="2016-01-03T15:46:38.167" Score="7" ViewCount="884" Body="&lt;p&gt;When &amp;amp; who was the first mathematician to discover the following simple triangle with a unique property that it has one angle is equal to one third of another angle in the same triangle?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The triangle sides are the following: $$a^3, a(b^2 – a^2), b(b^2 – 2a^2)$$ where, $(b/a) &amp;gt; \sqrt{2}$, &amp;amp;  $(a, b)$ are positive real constructible numbers&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know this seemingly a very simple question that may not be in Wikipedia!, also this is not a claim of solution of angle trisection&lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="55" LastEditDate="2016-01-10T22:13:33.517" LastActivityDate="2017-01-20T09:48:28.973" Title="Who discovered integer triangles with one angle trisecting another?" Tags="|mathematics|mathematicians|ancient-greece|euclidean-geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3292" PostTypeId="2" ParentId="3274" CreationDate="2016-01-04T00:34:56.573" Score="2" Body="&lt;p&gt;The answer is not really. Before &lt;a href=&quot;http://eulerarchive.maa.org/pages/E333.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's Recherches sur la Courbure des Surfaces (1760, published 1767&lt;/a&gt;, the publication and partial translation are available on Euler Archive) only curvature of curves was considered, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3003/how-was-curvature-originally-defined-and-calculated/3007#3007&quot;&gt;How was curvature originally defined and calculated?&lt;/a&gt; Euler introduced principal directions and curvatures by cutting a surface with perpendicular sections, and explicitly stated that both numbers are required to characterize curvature of a surface at a point in contrast to curves. The product of principal curvatures does appear in some formulas, but Euler places no special significance on it. Euler returned to surfaces in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E419.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Solidis Quorum Superficiem in Planum Explicare Licet (1771)&lt;/a&gt;, where he introduced parametric representation for the first time, and studied developable surfaces. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to one source, which I do not trust too much, Gauss named the product of principal curvatures &quot;total curvature&quot; after Euler, &lt;a href=&quot;https://books.google.com/books?id=TIXg4hSXmFAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Struik's Lectures on Classical Differential Geometry&lt;/a&gt; only confirms that the term was used in this way. It was later renamed into modern &quot;Gaussian curvature&quot;, and total curvature was appropriated for the latter's integral over the entire surface. If so this is not in Euler's 1760 paper, one would have to check the 1771 paper which is only available in Latin, and it seems that it lacked applications. Monge studied developable surfaces starting in 1770s, his work is summarized in &lt;a href=&quot;https://books.google.com/books/about/Application_de_l_analyse_%C3%A0_la_g%C3%A9om%C3%A9tr.html?id=Nf5zhlffjd0C&quot; rel=&quot;nofollow noreferrer&quot;&gt;Application de L'analyse à la Géométrie&lt;/a&gt; written in 1795-1801. He wrote a differential equation for developables, which is equivalent to Gaussian curvature being 0, but it would be a stretch to say that he &quot;considered&quot; it. Monge represented surfaces by implicit equations, so principal curvatures are not apparent in his differential equation, and one does not need to multiply them to express that one of them is 0.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another combination of principal curvatures however, their average now called mean curvature, had better luck. In 1760, the same year as Euler, Lagrange also published a paper about surfaces that minimize area among those with the same boundary. Like Monge he used implicit equations and his differential equation for minimal surfaces was messy, he could not produce any non-trivial solutions (trivial solutions are flat surfaces). But according to Struik some time in early 1770s Monge showed Euler's papers to his student Meusnier (later French revolutionary general who died at the siege of Mayence). Meusnier wrote the now well known paper in 1776 (published in Mém. prés. par div. Etrangers. Acad. Sci. Paris, 10 (1785) pp. 477–510), where he proved the theorem about the curvature of non-perpendicular sections, now named after him, and simplified Lagrange's differential equation to show that it amounted to the mean curvature being 0 (it is unclear if he used the name). This allowed him to produce the first non-trivial examples of minimal surfaces, catenoid and helicoid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mean curvature then came up in the work of Sophie Germain that won her a kilogram of gold prize in 1816. It was a generalization of Euler-Bernoulli theory of elastic beams to thin plates, the first work to correctly derive elastic force in them, see &lt;a href=&quot;http://www.jstor.org/stable/3647744?origin=crossref&amp;amp;seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lodder's Curvature in the Calculus Curriculum&lt;/a&gt;. In 1821 she reformulated it by introducing the &quot;referent sphere&quot; to represent the curvature of surfaces in the same way that the osculating circle represents it for curves, but the curvature it represents is the mean curvature. &lt;a href=&quot;https://books.google.com/books?id=hEtTBQAAQBAJ&amp;amp;pg=PA113&amp;amp;lpg=PA113&amp;amp;dq=mean+curvature+germain+sphere&amp;amp;source=bl&amp;amp;ots=kGHjOENL34&amp;amp;sig=vqTeEjAlydA4sHXwAonsV00tCXU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjM8YHk747KAhWDKWMKHafuAroQ6AEILjAD#v=onepage&amp;amp;q=mean%20curvature%20germain%20sphere&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Germain's correspondence with Gauss&lt;/a&gt; in 1820s shows that his treatment of surface geometry was novel even to her. Using parametric representation of surfaces to discover their intrinsic geometry was Gauss's major breakthrough in 1825-27 papers. Gaussian curvature is an important invariant of intrinsic geometry, but there does not seem to have been much use for it before that.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-01-04T21:06:58.693" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3293" PostTypeId="1" CreationDate="2016-01-04T16:26:55.687" Score="1" ViewCount="41" Body="&lt;p&gt;Ethnography &amp;amp; ethnomethodology, as both being around for quite a while there must be theories/sub-fields which have developed out of these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coming from an interactionist point of view, I'm aware of social network theory (Milroy &amp;amp; Milroy, 1978) &amp;amp; Conversation Analysis, the sub field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I'm looking for more theories/sub-fields to really drive my point about it being a useful method historically, and still is useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is primarily from social sciences, such as anthropology, sociology, linguistics etc. etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any suggestions/guidance are greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="3494" LastActivityDate="2016-01-04T16:26:55.687" Title="What theories/sub-fields have developed out of ethnography?" Tags="|social-sciences|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3294" PostTypeId="1" AcceptedAnswerId="3295" CreationDate="2016-01-04T22:36:06.763" Score="7" ViewCount="1383" Body="&lt;p&gt;The Cartesian coordinate system is what I have been told provided the first link between algebra and geometry. However, I also have learned that, for instance, Omar Khayyam solved cubic equations as the intersection between parabolas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how was this visualization done before Descartes? Also, what is so fundamentally different between the Cartesian coordinate system and the older methods?&lt;/p&gt;&#xA;" OwnerUserId="2666" LastActivityDate="2016-01-04T23:59:19.407" Title="Visualizing algebra before Descartes" Tags="|mathematics|algebraic-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3295" PostTypeId="2" ParentId="3294" CreationDate="2016-01-04T23:48:35.963" Score="9" Body="&lt;p&gt;Cartesian coordinates provided the first systematic way of converting geometric problems into algebraic ones and vice versa, but one can do that in elementary geometry without any coordinates simply by denoting sides and angles of polygons by letters and using trigonometry. This is more or less what ancient astronomers did when computing with their geometric models after the father of astronomy Hipparchus introduced chords, the predecessors of sine and cosine, in 2nd century BC. This approach was generalized and perfected by the founder of modern symbolic algebra Viete, who already used modern trigonometry, even before Descartes. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/2407/vi%C3%A8tes-relevance-and-his-connection-to-euler/2408#2408&quot;&gt;Vi&amp;#232;te&amp;#39;s Relevance and his Connection to Euler&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The use of geometry to solve (what we would call) polynomial equations is also ancient. Omar Khayyam was following in the footsteps of Menaechmus, Archimedes and Diocles, the first of whom lived before Euclid. Of course, ancient geometers did not start with equations, usually the initial problem was itself geometric, e.g. Menaechmus used intersection of curves to duplicate the cube, which in algebraic terms translates into solving $x^3=2$. Once Islamic mathematicians developed algebraic notation however, starting with al-Khwarismi in the 9th century AD, they started converting geometric problems into equations and back, and using ancient geometric methods to solve them. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/2234/historical-use-of-geometry-to-solve-polynomial-equations/2238#2238&quot;&gt;How was geometry historically used to solve polynomial equations?&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 3rd century BC Apollonius of Perga already used predecessors of Cartesian coordinates to study conic sections, but again he used them to solve geometric problems since there was no algebra then. Only in hidsight can we interpret him as visualizing algebra. Oresme used primitive coordinate graphs to visualize motion during middle ages, the algebra was in diapers back then. Both of them influenced Descartes, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2378/when-do-we-see-for-the-first-time-the-use-of-the-cartesian-plane/2382#2382&quot;&gt;When do we see for the first time the use of the Cartesian plane?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-01-04T23:59:19.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3296" PostTypeId="2" ParentId="188" CreationDate="2016-01-05T18:21:04.767" Score="9" Body="&lt;p&gt;Many German scientists were either captured by the Soviet army or were surrended to the Soviets after their capture by Western armies. All of these scientists were enslaved by the Soviets and forced to work on various scientific projects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The experience of one of Germany's greatest scientists, &lt;a href=&quot;https://en.wikipedia.org/wiki/Manfred_von_Ardenne&quot; rel=&quot;noreferrer&quot;&gt;Manfred von Ardenne&lt;/a&gt;, is representative. After voluntarily foregoing capture by Americans he was seized by the Red Army and moved to the Soviet Union. Shortly thereafter, at secret police compound of &lt;a href=&quot;https://en.wikipedia.org/wiki/Lubyanka_Building&quot; rel=&quot;noreferrer&quot;&gt;Lubyanka&lt;/a&gt; he was brought before a commission of Soviet scientists headed up by &lt;a href=&quot;https://en.wikipedia.org/wiki/Lavrentiy_Beria&quot; rel=&quot;noreferrer&quot;&gt;Lavrenty Beria&lt;/a&gt;. Beria made a veiled threat to kill him if he did not work for them. He agreed to work on a project to devise electromagnetic methods for the purification of Uranium 235, a subject in which he was expert. In 1954, after working on the Soviet nuclear program for nine years he was allowed to leave and settle in East Germany. See Ardenne's biography, &quot;Mein Leben für Forschung und Fortschritt&quot; for details.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastEditorUserId="3497" LastEditDate="2016-01-06T11:45:19.217" LastActivityDate="2016-01-06T11:45:19.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3297" PostTypeId="2" ParentId="2343" CreationDate="2016-01-05T18:50:55.683" Score="4" Body="&lt;p&gt;Generally speaking every culture has a unique way of gathering knowledge, what is called &quot;epistemology&quot;. Even in Anglo-American culture there is a lot of variance in the scientific method. For example, in the 17th century there was a huge dispute between Hobbes and Boyle about the proper way to validate a theory (see the book, &lt;em&gt;Leviathan and the Air Pump&lt;/em&gt;). Today in Japan, for example, scientific research has aspects and methods that differ somewhat from Western methods. The West is not even consistent in its own methods. For example, it is very common for Western scientists to keep their data secret and only publish their results, which is contrary to Boyle's methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ancient Greece had attitudes towards science which had many similarities with modern methods and in fact the word &lt;em&gt;epistemology&lt;/em&gt; is a Greek word. So did the ancient Egyptians and Babylonians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aboriginal cultures are too diverse to categorize in any fixed way, but in general tended to be (and still are) much more superstitious, attributing natural phenomena to &quot;spirits&quot;. Much of Asiatic culture is the same way. For example, most of the documents recovered from the manuscript cave in China by Aurel Stein and date from 1000 AD are either religious texts or involve fortune telling or augury, which mirrors their society. Even in Japan, which is the most rationalist of the Asiatic nations, superstition is universal and most people carry around lucky charms in their wallets or purses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, lack of rationalism does not mean a culture cannot excel in scientific discovery. As I wrote before, each culture's method is unique. For example, in Japan there is the concept of &lt;em&gt;honte&lt;/em&gt;, what is proper. We have this now in America, too, where we call it &quot;best practices,&quot; but the Japanese concept is far more developed and has resulted in them making many discoveries that Americans have failed to make.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2016-01-05T18:50:55.683" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3298" PostTypeId="2" ParentId="4" CreationDate="2016-01-05T19:14:09.710" Score="0" Body="&lt;p&gt;The value of a digital system of recording numbers is that it allows for a more concise and understandable method of doing arithmetical computations. Before Hindu numerals were used, the use of composite systems (like Roman numerals) were usual. So, for example, 83 is LXXXIII. The modern digital system was originated by Fibonacci in the &lt;em&gt;Liber Abaci&lt;/em&gt;, but it took literally hundreds of years before it became used systematically. For a long time there was a fight between the &quot;abacists and algorithmists&quot; over which system was better. There were even contests and tournaments where the two different schools would compete head to head in speed adding matches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventually it became clear that adding and multiplying digital numbers was easier than working with fractional or composite values. This occurred with the publication of &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Napier&quot; rel=&quot;nofollow&quot;&gt;Napier&lt;/a&gt;'s Mirifici Logarithmorum, as edited and improved by Edward Wright in 1616. At first Napier's book was just noticed by calculators, but after John Wallis adopted its methods around 1660 the use of digital notation and decimals became universal and decisive.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2016-01-05T19:14:09.710" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3299" PostTypeId="1" AcceptedAnswerId="3300" CreationDate="2016-01-06T21:51:36.903" Score="2" ViewCount="282" Body="&lt;p&gt;I was browsing through Paul Halmos' &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387900888&quot; rel=&quot;nofollow noreferrer&quot;&gt;classic book&lt;/a&gt; on measure theory from 1950, when I came by the following definition of separability on page 3 in the chapter on prerequisites:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fNHip.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fNHip.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today a separable space is one, which contains a countable, dense subset, while a second countable space is defined as a space having a countable base (and a second countable space is thus separable).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the other direction, from &lt;a href=&quot;https://en.wikipedia.org/wiki/Separable_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;this wiki page&lt;/a&gt; on separability, a separable space is second countable, iff it is Lindelöf.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the reason that Halmos used this definition of separability? Is it because the term has evolved over time? Or perhaps all relevant spaces in measure theory are Lindelöf, so it is not important?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He has only discussed general topological spaces up to this point, and has made no mention of metric spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also posted this question on &lt;a href=&quot;https://math.stackexchange.com/questions/1592742/separability-and-second-countability-is-the-same-thing-to-halmos&quot;&gt;math.stackexchange&lt;/a&gt; a week ago, but so far there are no answers.&lt;/p&gt;&#xA;" OwnerUserId="3504" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-03-16T20:23:26.417" Title="Separability and second countability is the same thing to Halmos" Tags="|mathematics|terminology|topology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3300" PostTypeId="2" ParentId="3299" CreationDate="2016-01-07T03:05:19.130" Score="2" Body="&lt;p&gt;J. L. Kelley, &lt;em&gt;General Topology&lt;/em&gt; (1955) defines the &lt;strong&gt;second axiom of countability&lt;/strong&gt; then notes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(The terms &lt;strong&gt;separable&lt;/strong&gt; and &lt;strong&gt;perfectly separable&lt;/strong&gt; are also used in this connection but we shall use neither.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-01-07T03:05:19.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3301" PostTypeId="1" CreationDate="2016-01-10T04:24:12.903" Score="7" ViewCount="780" Body="&lt;p&gt;I've read that the original approach to demonstrating the efficacy of calculus left something to be desired among mathematicians. What exactly was the problem? Does the later accepted definition illuminate anything about the original problem or avoid it?&lt;/p&gt;&#xA;" OwnerUserId="3515" LastEditorUserId="62" LastEditDate="2016-01-10T18:10:53.743" LastActivityDate="2016-01-11T16:09:56.657" Title="Did Newton and Leibniz lack rigour?" Tags="|mathematics|calculus|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3302" PostTypeId="2" ParentId="3301" CreationDate="2016-01-10T07:04:15.610" Score="8" Body="&lt;p&gt;Problems were abundant. There was no rigorous definitions of limits, convergence, and even functions and real numbers. And without definitions there could be no real, rigorous proofs. All this was not achieved until 19 century. Newton's and Leibniz results were correct, but they were not proved to the standards of rigor which already existed in mathematics since the times of Euclid and Archimedes, not speaking of the later standards. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And mathematicians of 17 and 18 century understood this. In many arguments they had to rely on intuition, as modern physicists frequently do, when they use mathematical tools which are not fully justified. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As calculus developed further, this lack of rigor led to paradoxes and controversies. Only in 19s century, starting with Gauss, Abel, Cauchy, and later Weierstrass, Dedekind and Cantor, a satisfactory and rigorous foundation of&#xA;calculus was established.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Here are two examples of 18 and early 19 century calculus:&#xA;$$x=2\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sin nx.$$&#xA;This can be verified numerically by plugging some values, and even experimentally. But this makes no sense because the RHS is periodic while the LHS is not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second example is from Euler. By multiplication term by term we obtain&#xA;$$(1-x)(1+x+x^2+x^3+\ldots)=1.$$&#xA;Putting $x=2$ we obtain&#xA;$$1+2+4+8+16+\ldots=-1.$$&#xA;This can be justified from the modern point of view, but at the time of Euler this led to controversies of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, the situation is very similar. Physicists discover new mathematical results using methods which make no sense to pure mathematicians. Mathematicians consider these results as conjectures and sometimes prove them with their rigorous methods. As it was in 17s and 18s centuries the process is very beneficial to both physics and mathematics. Actually Archimedes also used physical reasoning to discover new mathematical results, and he always made it very clear what is really proved and what is just ``discovered''.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-01-10T16:34:55.370" LastActivityDate="2016-01-10T16:34:55.370" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3303" PostTypeId="2" ParentId="606" CreationDate="2016-01-10T14:20:56.983" Score="5" Body="&lt;p&gt;As you correctly noted, the first &quot;principle or law of the &lt;em&gt;permanence of equivalent forms&lt;/em&gt;&quot; was formulated by the English algebraist &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Peacock&quot; rel=&quot;noreferrer&quot;&gt;George Peacock&lt;/a&gt; in his book &lt;a href=&quot;https://books.google.it/books?id=mcVc5v8_kyoC&amp;amp;pg=PA104&quot; rel=&quot;noreferrer&quot;&gt;A treatise of Algebra&lt;/a&gt; (1st ed.1830), page 104 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Peacock was an eminent member of the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Analytical_Society&quot; rel=&quot;noreferrer&quot;&gt;Analytical Society&lt;/a&gt; with &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Babbage&quot; rel=&quot;noreferrer&quot;&gt;Charles Babbage&lt;/a&gt; and was contemporary to other eminent English algebraists of that time, like &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Rowan_Hamilton&quot; rel=&quot;noreferrer&quot;&gt;William Rowan Hamilton&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/boole/&quot; rel=&quot;noreferrer&quot;&gt;George Boole&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The said principle was a powerful heuristic tool in the discovery of new &quot;algebraic structures&quot; like &lt;a href=&quot;http://plato.stanford.edu/entries/boolalg-math/&quot; rel=&quot;noreferrer&quot;&gt;Boolean Algebra&lt;/a&gt; but it is clearly not &quot;absolutely sound&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can think at &lt;a href=&quot;https://en.wikipedia.org/wiki/Quaternion&quot; rel=&quot;noreferrer&quot;&gt;Quaternions&lt;/a&gt;; they are: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is &lt;em&gt;noncommutative&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this case, contrary to what we may expect on the basis of the principle, we have a &lt;em&gt;number system&lt;/em&gt; featuring all the &quot;symbolic patterns&quot; common to previous number systems &lt;strong&gt;except for&lt;/strong&gt; commutativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The principle has been forsaken in modern mathematics due to the systematic adoption of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Axiomatic_system#Axiomatic_method&quot; rel=&quot;noreferrer&quot;&gt;axiomatic method&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_structure&quot; rel=&quot;noreferrer&quot;&gt;mathematical structure&lt;/a&gt; is now defined by its axioms: there are structures endowed with a &lt;a href=&quot;https://en.wikipedia.org/wiki/Binary_operation&quot; rel=&quot;noreferrer&quot;&gt;binary operation&lt;/a&gt; that is commutative and structures where the operation is not so; they are simply characterized by differnt axioms.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-01-10T14:32:09.540" LastActivityDate="2016-01-10T14:32:09.540" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3304" PostTypeId="2" ParentId="3291" CreationDate="2016-01-10T22:12:56.290" Score="5" Body="&lt;p&gt;This is a neat observation. I was unable to find historical references for it in particular, but there is a rich history of solving this type of problems. The problem is clearly reminiscent of finding the Pythagorean triples, right triangles with integer sides, for which &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookX/propX29.html&quot; rel=&quot;nofollow&quot;&gt;Elements X.29&lt;/a&gt; gives a solution without a hint of how it was obtained (of course once the rule is known it is easy to check that it works). A method first appears in Diophantus's Arithmetica (c. 250 AD) when solving the famous problem II.8, &quot;&lt;em&gt;partition a given square into two squares&lt;/em&gt;&quot;, the one whose margins were too narrow in Fermat's copy of the book. &lt;a href=&quot;http://grmath4.phpnet.us/mathimatikoi/diofantos/dioph_text.pdf?ckattempt=1&quot; rel=&quot;nofollow&quot;&gt;Schappacher's article&lt;/a&gt; (pp.12-13) gives a nice commentary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In modern retelling if $p,q,r$ are the sides then for $x:=p/r$, $y:=q/r$ we have $x^2+y^2=1$ by the Pythagorean theorem. Write $y^2=1-x^2$ and make substitution $y=t(1-x)$ then equation for $x$ is linear and gives $x=(t^2-1)/(t^2+1)$, $y=2t/(t^2+1)$. If $t$ is rational so are $x$ and $y$, and setting $t=n/m$ gives the Pythagorean triples $n^2-m^2$, $2nm$, $n^2-m^2$. The trick can be interpreted geometrically as intersecting the circle with lines of slope $-t$ passing through a point $(1,0)$, and it works because each line has exactly one other intersection point with the circle. Diophantus describes nothing of the sort of course. Schappacher's remarks (p.27) suggest that Poincare might have come up with this interpretation in 1901, it is now called &lt;a href=&quot;http://people.reed.edu/~jerry/131/conics.pdf&quot; rel=&quot;nofollow&quot;&gt;rational parametrization in algebraic geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To extend this to self-trisecting triangles we first need a &quot;Pythagorean theorem&quot;. Just like having a right angle makes triangle's angles into $\alpha,\pi/2-\alpha,\pi/2$ the trisection condition makes them into $\alpha,3\alpha,\pi-4\alpha$. In the first case writing equations of the theorem of sines and eliminating $\alpha$ from them leads to the Pythagorean equation $p^2+q^2=r^2$, in the second case after some fun with more esoteric trig identities one gets $q^2r=(p+r)(p-r)^2$. This is the &quot;Pythagorean theorem&quot; for self-trisecting triangles. With the same notation as above the corresponding curve is a cubic $$y^2=(x+1)(x-1)^2.$$ Normally cubics do not have rational parametrizations, but this one is special because of the squared factor, it means that it has a node (self-intersection point) at $(1,0)$. Drawing lines through the node, i.e. making the substitution $y=t(x-1)$, immediately gives $x=t^2-1$ and $y=t(t^2-2)$. Setting $t=n/m$ gives us $$p=m(n^2-m^2),\ q=n(n^2-2m^2),\ r=m^3,$$ as in the OP up to notation. So not only does the formula work, it actually lists all self-trisecting triples. The nodal cubic looks something like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Folium_of_Descartes&quot; rel=&quot;nofollow&quot;&gt;folium of Descartes&lt;/a&gt; (rotated by $135^\circ$ clockwise, scaled, and shifted by $1$ to the right so that the $x$-intercepts move to $(-1,0)$ and $(1,0)$), but it does not have an asymptote.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-07-20T02:23:41.857" LastActivityDate="2016-07-20T02:23:41.857" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3305" PostTypeId="1" CreationDate="2016-01-11T13:16:34.893" Score="-5" ViewCount="572" Body="&lt;p&gt;The problem is the infinite or endless repeated digits of $9's$ after zero digit and the decimal notation,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite its apparent simplicity &amp;amp; the huge talk about it every where in mathematics or scientific community, one wonders if it is of all that importance,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What its background, Is it settled as being equal to one or meaningless, how can this problem affects mathematics, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How many proofs or disproofs for this puzzle?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is a bet different in the sense of its absolute legend truthiness, where one may easily spot the illusion in its absolute truthiness by means of approximation, limits, ambiguous use of infinity, convergence, famous cuts, ...etc,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where all these tools are good for calculating approximately (also in our own sense only), the area of a circle for example, but the exactness sense in mathematics doesn't require any kind of approximation, it doesn't consider terms being large as $10^n$ or being little as $10^-n$ when $n$ tends to infinity, it consider them existing, regardless of our own needs or sense &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate further, if we define the accuracy of approximating $pi$ by how many digits can be obtained,(instead of area of a circle of radius one), then all the formulas of $ pi $ are useless    &lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="2855" LastEditDate="2016-01-12T09:16:22.613" LastActivityDate="2016-01-12T19:08:40.570" Title="When or why &amp; who originated this puzzle, $0.999... = 1$" Tags="|mathematics|physics|mathematicians|philosophy-of-science|mathematical-logic|" AnswerCount="2" CommentCount="8" ClosedDate="2016-01-15T00:22:54.400" ContentLicense="CC BY-SA 3.0" />
  <row Id="3306" PostTypeId="2" ParentId="3305" CreationDate="2016-01-11T13:23:30.300" Score="1" Body="&lt;p&gt;The infinite series $9 \cdot \sum_{n \ge 1} 10^{-n} = 9 \cdot 10^{-1} \cdot \frac{1}{1 - 10^{-1}} = 1$. Just a run of the mill geometric series. No mystery, no puzzle.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastEditorUserId="26" LastEditDate="2016-01-12T17:14:07.730" LastActivityDate="2016-01-12T17:14:07.730" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3307" PostTypeId="1" CreationDate="2016-01-11T15:17:33.333" Score="1" ViewCount="170" Body="&lt;p&gt;Frequently when electromagnetic waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an ongoing discussion on Physics Stack Exchange of how (far) this is true (see: &lt;a href=&quot;https://physics.stackexchange.com/questions/181277/do-the-electric-and-magnetic-components-of-an-electromagnetic-wave-really-genera&quot;&gt;Do the electric and magnetic components of an electromagnetic wave really generate each other?&lt;/a&gt; or &lt;a href=&quot;https://physics.stackexchange.com/questions/228796/how-can-one-meaningfull-say-that-one-field-generates-the-other-in-an-em-wave&quot;&gt;How can one meaningfully say that one field generates the other in an EM-wave?&lt;/a&gt;). However, in this question, I am interested in the historic point of view about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there passages in the papers or books of the founding fathers of the electromagnetic theory (Maxwell, Hertz, ...) which express this statement (in words, not formulas) and, even further, use it as a main idea in developing the theory?&lt;/p&gt;&#xA;" OwnerUserId="2380" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2016-01-11T23:22:45.963" Title="Mutual induction in electromagnetic wave theory" Tags="|physics|electromagnetism|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3308" PostTypeId="2" ParentId="3305" CreationDate="2016-01-11T16:00:12.063" Score="3" Body="&lt;p&gt;One of the earliest places where an infinite string of 9s is rounded off to a finite string is Euler's text on algebra around 1777.  Here he gets the answer 9.999... and makes a comment that this is virtually indistinguishable from 10.  That was &lt;em&gt;one&lt;/em&gt; of the earliest places as I mentioned.  Conifold found an earlier source (see comment above).&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-01-12T19:08:40.570" LastActivityDate="2016-01-12T19:08:40.570" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3309" PostTypeId="2" ParentId="3301" CreationDate="2016-01-11T16:09:56.657" Score="3" Body="&lt;p&gt;When speaking about rigor in a historical context, one must be careful not to apply modern habits of thought to historical developments where they are inappropriate.  For example, one distinction that must be made is between the set-theoretic foundations we take for granted today that were not available before the second half of the 19th century.  More specifically, modern punctiform continua (i.e., continua made of points) were certainly not the &quot;foundational&quot; background before, say, 1870.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This does not mean, however, that one can't meaningfully discuss the work of earlier authors without calling it unrigorous.  For example, the work of Gauss is generally considered rigorous.  The key distinction helpful in disengaging ourselves from the too-easy dismissal of historical mathematicians as unrigorous is the distinction between procedure and ontology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was dealt with by authors varying from Benacerraf to Quine to Wartofsky but it will suffice to say that the term &quot;procedure&quot; refers to the actual inferential moves as they appear in the work of those mathematicians, whereas &quot;ontology&quot; refers to the justification of the entities like number, point, function, etc. used by those authors.  The set-theoretic ontology commonly taken for granted today was not there in the work of Gauss and others, but this shouldn't prevent a scholar from analyzing their &lt;em&gt;procedures&lt;/em&gt;, which often turn out to be rigorous to a satisfactory degree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, Cauchy used infinitesimals in much the way they would be used today, and his procedural definition of the continuity of a function is essentially indistinguishable from a modern one (an infinitesimal increment assigned to the variable always produces an infinitesimal change in the function), even though the ontological foundation that would be expected today was not there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer your question about Leibniz more specifically, his infinitesimal procedures were more soundly founded than the generally touted critique thereof by George Berkeley.  This overlooks the procedure/ontology distinction.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-01-11T16:09:56.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3310" PostTypeId="2" ParentId="606" CreationDate="2016-01-11T16:17:04.543" Score="3" Body="&lt;p&gt;The principle of permanence was a heuristic principle serving an important role for discovery of new results, historically speaking.  In modern mathematics, it has been formulated as a rigorous logical principle called the transfer principle.  This exists in a number of contexts but is most famously present in Robinson's framework which involves a hyperreal extension $\mathbb{R}\subseteq{}^{\ast}\mathbb{R}$.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-01-11T16:17:04.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3312" PostTypeId="1" AcceptedAnswerId="3316" CreationDate="2016-01-13T06:55:55.183" Score="4" ViewCount="861" Body="&lt;p&gt;In the english translation of Newton's work &quot;Enumeratio linearum tertii ordinis&quot; by C.R.M. Talbot, we can see &lt;a href=&quot;https://books.google.gr/books?id=6I97byFB3v0C&amp;amp;hl=el&amp;amp;pg=PA86#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;in a figure&lt;/a&gt; the depiction of a Cartesian coordinate system pretty much as we know it today: two perpendicular axes labeled X, Y intersected at the origin O. Is it possible that this figure corresponds to some of the original Newton's drawings? I cannot find it in the &lt;a href=&quot;https://archive.org/stream/isaacinewtonien00newtgoog#page/n211/mode/2up&quot; rel=&quot;nofollow&quot;&gt;latin edition&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2906" LastEditorUserId="131" LastEditDate="2016-01-14T17:41:52.630" LastActivityDate="2016-01-14T22:29:23.503" Title="Cartesian coordinate system in Newton's work" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3313" PostTypeId="1" CreationDate="2016-01-13T09:12:17.060" Score="6" ViewCount="183" Body="&lt;p&gt;Ian Hacking's 2014 book *Why is there philosophy of mathematics at all?&amp;quot;, see &lt;a href=&quot;http://www.cambridge.org/il/academic/subjects/philosophy/philosophy-science/why-there-philosophy-mathematics-all&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, contains many interesting ideas.  One of the ideas is the dichotomy of two distinct models for the development of a science. One is the butterfy model. A biological organism follows a course of development that is predetermined to a certain extent by its genetic make-up. The example of butterfly was chosen I think because of the dramatic changes a butterfly undergoes (cocoon, etc), unlike, say, an elephant; yet even those dramatic changes are written into the script so to speak.&lt;/p&gt;&#xA;&lt;p&gt;The other model is the Latin model of development of a natural language like Latin. Such development obviously is affected by historical factors and contingencies of human behavior.&lt;/p&gt;&#xA;&lt;p&gt;Hacking seems to think that scholars often view mathematics as following a butterfly model, whereas he himself seems to argue in favor of the Latin model.&lt;/p&gt;&#xA;&lt;p&gt;So to summarize, my question is: does Hacking's Latin model really apply to mathematics?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-12-05T13:40:48.970" LastActivityDate="2023-12-05T13:40:48.970" Title="Different models for the development of mathematics: Latin versus butterfly" Tags="|mathematics|philosophy-of-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3314" PostTypeId="2" ParentId="3313" CreationDate="2016-01-13T12:55:14.073" Score="3" Body="&lt;p&gt;Mathematics is a language (and a culture), as such it isn't constrained by any &quot;predetermined path&quot; mandated by some genome makeup. I'd strongly say mathematics (and all science, as part of human culture) follows the &quot;Latin&quot; scheme (which itself is also part of the same cultural development).&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2016-01-13T12:55:14.073" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3315" PostTypeId="1" CreationDate="2016-01-13T19:09:27.397" Score="-1" ViewCount="805" Body="&lt;p&gt;The purpose of the question is to understand why the number $\sqrt[3]{2}$, that was proven rigorously by ancient Greek is an impossible number (even at infinity), by their three famous impossibility problems,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;this one was simply the impossibility of doubling the cube,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, much later and oddly after many centuries, particularly (in the middle ages), mathematicians fond $\sqrt[3]{2}$ also at infinity, and they widely accepted it as a real number, which would imply the invalidity of the most famous Ancient Greek rigorous proof!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course it so clear to even a school student that $$a^3 = 2b^3$$&#xA;in positive constructible numbers is &lt;strong&gt;&lt;em&gt;impossible&lt;/em&gt;&lt;/strong&gt; equation, thus $\sqrt[3]{2}$ is also &lt;strong&gt;&lt;em&gt;impossible&lt;/em&gt;&lt;/strong&gt; number as a result, this was the basis of Greek famous rigorous proof&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had chosen this number $\sqrt[3]{2}$ in particular since it was the oldest famous number that encountered the Greek with a tremendous challenge that is still valid!&lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="2855" LastEditDate="2016-01-26T16:37:23.130" LastActivityDate="2016-01-26T16:37:23.130" Title="When and who was the first mathematicians to prove rigorously that $\sqrt[3]{2}$ was impossible number?" Tags="|mathematics|physics|philosophy-of-science|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="3" ClosedDate="2016-01-22T00:19:01.100" ContentLicense="CC BY-SA 3.0" />
  <row Id="3316" PostTypeId="2" ParentId="3312" CreationDate="2016-01-13T20:12:30.143" Score="6" Body="&lt;p&gt;The standard way we represent Cartesian coordinates with orthogonal axis is not present into Descartes' &lt;a href=&quot;https://fr.wikipedia.org/wiki/La_G%C3%A9om%C3%A9trie_(Descartes)&quot; rel=&quot;nofollow noreferrer&quot;&gt;La Géométrie&lt;/a&gt; nor into Newton's works.&lt;/p&gt;&#xA;&lt;p&gt;An early occurrence of the &amp;quot;basic&amp;quot; definitions are in :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Colin_Maclaurin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Colin McLaurin&lt;/a&gt;'s &lt;a href=&quot;https://books.google.it/books?id=QoAHAAAAcAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Treatise on Algebra&lt;/a&gt; (1748 - two years after his death); see page 297-on:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§2. We are now to consider quantities as represented by &lt;em&gt;lines&lt;/em&gt;. [ &lt;a href=&quot;https://books.google.it/books?id=QoAHAAAAcAAJ&amp;amp;pg=PA301&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 301&lt;/a&gt; ] So that, if &lt;span class=&quot;math-container&quot;&gt;$AP$&lt;/span&gt; represent &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and the prependicular &lt;span class=&quot;math-container&quot;&gt;$PM$&lt;/span&gt; represent the corresponding values of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, then there will be as many points (&lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;,) the extremities of these perpendiculars or &lt;em&gt;ordinates&lt;/em&gt; [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The modern representation (as usual), is due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt; into his &lt;a href=&quot;https://books.google.it/books?id=q9br6CFATEcC&amp;amp;pg=PA403&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introductio in analysin infinitorum&lt;/a&gt;, (1748), Tomus Secundus.&lt;/p&gt;&#xA;&lt;p&gt;See page 3 for an &amp;quot;elucidation&amp;quot; of &lt;em&gt;function&lt;/em&gt; and page 4:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sit igitur &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; quantitas variabilís, quae per rectam indefinitam &lt;span class=&quot;math-container&quot;&gt;$RS$&lt;/span&gt;  repraesentetur, atque manifestum est omnes valores determinatos ipsius &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, qui quidem sint reales, per portiones in recta &lt;span class=&quot;math-container&quot;&gt;$RS$&lt;/span&gt; abscindendas repraesentari posse. [...] Vocantur аutem haec intervalla &lt;span class=&quot;math-container&quot;&gt;$АР$&lt;/span&gt;, ABSCISSAE. Atque ideo Abscissae exhibent variabilis &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; valores determinatos.&lt;/p&gt;&#xA;&lt;p&gt;[page 7] Portiones autem Axis &lt;span class=&quot;math-container&quot;&gt;$АР$&lt;/span&gt;, quibus determinati ipsius &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; valores indicantur, vocari solent &lt;span class=&quot;math-container&quot;&gt;$ABSCISSAE$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Et perpendiculares &lt;span class=&quot;math-container&quot;&gt;$PM$&lt;/span&gt;, ex terminis &lt;em&gt;Abscissarum&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; ad lineam curvam pertingentes, nomen &lt;span class=&quot;math-container&quot;&gt;$APPLICATARUM$&lt;/span&gt; obtinuerunt.&lt;/p&gt;&#xA;&lt;p&gt;Vocantur autem hoc casu Applicatae &lt;em&gt;normales&lt;/em&gt; seu &lt;em&gt;orthogonales&lt;/em&gt;, quin cum Axe angulum rectum constituunt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In general, we can see :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Carl Boyer, &lt;a href=&quot;https://books.google.it/books?id=2T4i5fXZbOYC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of analytic geometry&lt;/a&gt; (1956 - also Dover reprint).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-01-13T20:27:03.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3317" PostTypeId="2" ParentId="3315" CreationDate="2016-01-13T20:24:07.580" Score="6" Body="&lt;p&gt;Gauss considered the algebraic background behind straightedge and compass constructions, and from his work it is clear that e.g. $\sqrt[3]{2}$ is impossible to construct, as it can't be expressed via a finite number of sums, products, and square roots starting with $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That it can't be &lt;em&gt;constructed&lt;/em&gt; with some random selection of tools doesn't mean it doesn't &lt;em&gt;exist&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2016-01-13T20:24:07.580" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3319" PostTypeId="2" ParentId="3229" CreationDate="2016-01-14T13:33:54.643" Score="1" Body="&lt;p&gt;Much to my dismay, the first proof of a global solution to the Boltzmann equation was not published until 2010.  The &lt;a href=&quot;http://www.pnas.org/content/107/13/5744.abstract&quot; rel=&quot;nofollow&quot;&gt;paper&lt;/a&gt; was:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gressman, P.T., and R.M. Strain &quot;Global classical solutions of the Boltzmann equation with long-range interactions,&quot; &lt;em&gt;Proc. Natl. Academy Sci.&lt;/em&gt; &lt;strong&gt;107&lt;/strong&gt;, pp. 5744-5749, 2010.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, all previous proofs had been for local interactions.  I should note that &lt;a href=&quot;https://en.wikipedia.org/wiki/C%C3%A9dric_Villani&quot; rel=&quot;nofollow&quot;&gt;Cédric Villani&lt;/a&gt; was awarded the 2010 Math Fields Medal for his work on this topic and the generation of entropy, specifically in a collisionless medium like a &lt;a href=&quot;https://en.wikipedia.org/wiki/Plasma_(physics)&quot; rel=&quot;nofollow&quot;&gt;plasma&lt;/a&gt;.  There was work by &lt;a href=&quot;https://en.wikipedia.org/wiki/Lev_Landau&quot; rel=&quot;nofollow&quot;&gt;Lev Landau&lt;/a&gt;, of course, but others worked on this subject long before Villani like &lt;a href=&quot;https://en.wikipedia.org/wiki/Roald_Sagdeev&quot; rel=&quot;nofollow&quot;&gt;Roald Sagdeev&lt;/a&gt; and &lt;a href=&quot;https://cass.ucsd.edu/index.php/faculty:Mmalkov&quot; rel=&quot;nofollow&quot;&gt;Mikhail Malkov&lt;/a&gt;.  Yet I am not sure if they provided proofs since Villani was given the Fields Medal for his work on the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless, Villani has some very detailed and good material on his website &lt;a href=&quot;http://cedricvillani.org&quot; rel=&quot;nofollow&quot;&gt;http://cedricvillani.org&lt;/a&gt; including a book that has some very detailed notes and, I believe, some history on the subject in which you are interested.&lt;/p&gt;&#xA;" OwnerUserId="3373" LastActivityDate="2016-01-14T13:33:54.643" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3320" PostTypeId="1" AcceptedAnswerId="3321" CreationDate="2016-01-14T19:29:15.263" Score="0" ViewCount="441" Body="&lt;p&gt;Finally I found the source of the dictum &amp;quot;debauches of indices&amp;quot;. It is most often used in singular &amp;quot;&amp;quot;debauch&amp;quot;, as in Spivaks's Vol.II p.211. The original is from the first preface in E. Cartan's &amp;quot;Lecons sur la Geometrie des Espaces de Riemann&amp;quot;. Alas I don't understand enough French. Can somebody please translate it? (Google translator is worthless.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Les services éminents qu'a rendus et que rendra encore le Calcul différentiel absolu du Ricci et Levi-Civita ne doivent pas nous empécher d'éviter les calculs trop exclusivement formels, ou les débauches d'indices masquent une realité géométrique souvent tres simple. C'est cette realité que j'ai cherché a mettre partout en evidence.&lt;/p&gt;&#xA;" OwnerUserId="3024" LastEditorUserId="55" LastEditDate="2021-09-16T04:22:17.907" LastActivityDate="2021-09-16T04:22:17.907" Title="The debauch of indices: translation request" Tags="|mathematics|differential-geometry|translations|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="3321" PostTypeId="2" ParentId="3320" CreationDate="2016-01-14T21:48:02.690" Score="1" Body="&lt;p&gt;superabundance of indexes &#xA;There is a triple negation &quot;ne doivent pas nous empêcher d'éviter&quot; which makes the text hard to grasp, that is ../they/ should not prevent us from avoiding, but the sense is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Notable oppurtunities that we have received and will continue to receive further from le Calcul différentiel absolu de Ricci et Levi-Civita should not prevent us from avoiding purely formal calculations where a superabundance of indexes hides an often simple geometrical reality.&quot;&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-01-14T21:48:02.690" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3322" PostTypeId="2" ParentId="3312" CreationDate="2016-01-14T22:29:23.503" Score="1" Body="&lt;p&gt;Philippe de la Hire &lt;em&gt;Nouveaux elements de sections coniques&lt;/em&gt; published in 1679 seems a good candidate for inventing cartesian coordinates: he was strongly influenced by Descartes and draw conics by referring them to an origin &lt;em&gt;O&lt;/em&gt;, a vertical axis called a stem ('tige')  with  knots 'noeuds' &lt;em&gt;N&lt;/em&gt; and horizontal branches 'rameaux' &lt;em&gt;NL&lt;/em&gt;. This is probably the source for the popular view that Descartes invented the left handed coordinate system.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-01-14T22:29:23.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3323" PostTypeId="1" AcceptedAnswerId="3338" CreationDate="2016-01-15T15:28:42.147" Score="20" ViewCount="1097" Body="&lt;p&gt;This &lt;a href=&quot;https://math.stackexchange.com/questions/1612437/notation-of-the-second-derivation-where-does-the-d-go&quot;&gt;MSE question&lt;/a&gt; made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the &lt;em&gt;second&lt;/em&gt; derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. Did Leibnitz use it himself? Or was it introduced later?&lt;/p&gt;&#xA;" OwnerUserId="3529" LastActivityDate="2018-10-26T18:55:38.490" Title="Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?" Tags="|mathematics|notation|calculus|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3324" PostTypeId="1" CreationDate="2016-01-15T16:30:12.840" Score="9" ViewCount="5373" Body="&lt;p&gt;I'm very interested to know about historical development of power series, i.e.&#xA;$$\sum_{n=0}^{\infty}a_n(x-c)^n=a_0+a_1(x-c)+a_2(x-c)^2+\dots$$&#xA;What was the situation and historical context that necessitated the discovery of power series? Why did mathematicians originally need power series? &lt;/p&gt;&#xA;" OwnerUserId="3531" LastEditorUserId="62" LastEditDate="2017-11-22T16:19:05.410" LastActivityDate="2017-12-19T22:26:19.937" Title="Historical development of power series" Tags="|mathematics|calculus|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3325" PostTypeId="2" ParentId="3324" CreationDate="2016-01-15T21:29:54.457" Score="10" Body="&lt;p&gt;Some power series, like the geometric progression were indeed encountered since the ancient times, but the first person who used them systematically was I. Newton. Actually Newton considered this his main mathematical discovery: that any equation (algebraic, differential, etc.) can be solved by substituting a power series with undetermined coefficients, and then the coefficients can be found successively. This is the essential message of his two coded sentences which he sent to Leibniz via Oldenburg. You may read here a discussion about these coded messages:&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram&quot;&gt;https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram&lt;/a&gt;&#xA;It is clear from Newton's mathematical papers, that for him &quot;calculus&quot; was essentially the use of power series. Mathematicians of 18s century started immediately using them for all sorts of problems. However they mostly did formal manipulations. The rigorous theory of convergence goes back to Cauchy and Abel.&#xA;After that the theory essentially merged to the theory of analytic functions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-23T14:42:43.953" LastActivityDate="2017-11-23T14:42:43.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3327" PostTypeId="2" ParentId="3323" CreationDate="2016-01-16T01:04:08.757" Score="1" Body="&lt;p&gt;I think  $\frac{d^2y}{dx^2}$ comes from multiplying $\frac{dy}{dx}$ by $\frac{d}{dx}$. &#xA;In the Notation(&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Abuse_of_notation#Derivitive&quot; rel=&quot;nofollow&quot;&gt;https://en.m.wikipedia.org/wiki/Abuse_of_notation#Derivitive&lt;/a&gt;) multiplication signifies iteration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Disclaimer; This is a very rough response. There have not been any other answers yet, I will look for the notation in a textbook.)&lt;/p&gt;&#xA;" OwnerUserId="196" LastActivityDate="2016-01-16T01:04:08.757" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3328" PostTypeId="1" CreationDate="2016-01-16T01:19:17.070" Score="8" ViewCount="6890" Body="&lt;p&gt;Although Darwin died a year before the term &quot;eugenics&quot; was coined by his cousin, Francis Galton, Galton had already been working on the idea of using selective breeding to improve humans genetically for many years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presumably, Darwin was aware of Galton's work, as it was inspired in part by his work on evolution. Did Darwin ever express his views on Galton's ideas before his death? If so, what were they?&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="4251" LastEditDate="2020-07-12T14:42:37.140" LastActivityDate="2020-07-12T14:42:37.140" Title="Did Darwin ever express his views on eugenics?" Tags="|biology|evolution|darwin|biologists|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3329" PostTypeId="2" ParentId="3328" CreationDate="2016-01-16T02:08:25.827" Score="1" Body="&lt;p&gt;Unfortunately, Darwin was a Victorian, and while he did not view the races as arising from different genetic origins (monogenic versus polygenic), he did view the white races as evolutionarily more advanced than the black races. He had similar Victorian values on gender norms as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This short article, &lt;a href=&quot;http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2672903/&quot; rel=&quot;nofollow&quot;&gt;Darwin, race and gender&lt;/a&gt;, by Steven Rose provides a detailed overview on Darwin's stance, and some of his writing on the issue. It states&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He enthusiastically endorsed his cousin Francis Galton's view of hereditary genius transmitted down the male line, and nodded cautiously towards eugenics. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, Darwin's views on race are no longer the official views of the scientific community. In the American Association of Physical Anthropology's position statement on the &lt;a href=&quot;http://physanth.org/about/position-statements/biological-aspects-race/&quot; rel=&quot;nofollow&quot;&gt;Biological Aspects of Race&lt;/a&gt;, it outlines what science understands about the human species today and states that there is no scientific foundation for the ideas that were perpetuated in the 19th and early 20th century with regards to races. In their analysis, then do not feel that one can clearly define what a race is scientifically and differs from how we would be able to categorize breeds in domesticated animals.&lt;/p&gt;&#xA;" OwnerUserId="3148" LastActivityDate="2016-01-16T02:08:25.827" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3330" PostTypeId="2" ParentId="3328" CreationDate="2016-01-16T02:37:52.803" Score="9" Body="&lt;p&gt;Even Weikart, whose book From Darwin to Hitler: Evolutionary Ethics, Eugenics and Racism in Germany greatly overplays the connection, admits that Darwin generally tried to stay away from discussions of social, political and economic issues. In an 1869 letter to Hugo Thiel Darwin wrote &quot;&lt;em&gt;you  apply to moral and social questions analogous views to those which I have used in regard to the modification of species. It did not occur to me formerly that my views could be extended to such widely different, and most important subjects&lt;/em&gt;&quot;. In an 1873 letter to Marx, thanking him for a copy of Das Kapital, he mentions that he was ignorant of economics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This being said, Darwin did express some views that can be interpreted as sympathetic to social Darwinism and eugenics. There is a passage from Descent of Man (1871) oft quoted by opponents of evolution:&quot;&lt;em&gt;There is reason to believe that vaccination has preserved thousands, who from a weak constitution would formerly have succumbed to small-pox. Thus the weak members of civilised societies propagate their kind. No one who has attended to the breeding of domestic animals will doubt that this must be highly injurious to the race of man. It is surprising how soon a want of care, or care wrongly directed, leads to the degeneration of a domestic race; but excepting in the case of man himself, hardly any one is so ignorant as to allow his worst animals to breed...&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interpretation of this passage is controversial, and Darwin goes on to write:&quot;&lt;em&gt;...but if we were intentionally to neglect the weak and helpless, it could only be for a contingent benefit, with an overwhelming present evil. Hence we must bear without complaining the undoubtedly bad effects of the weak surviving and propagating their kind; but there appears to be at least one check in steady action, namely the weaker and inferior members of society not marrying so freely as the sound; and this check might be indefinitely increased, though this is more to be hoped for than expected&lt;/em&gt;&quot;. See Wikiquote for the &lt;a href=&quot;https://en.wikiquote.org/wiki/Notable_Charles_Darwin_misquotes&quot;&gt;full context and discussion&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weikart also makes much of 1872 letter that reads &quot;&lt;em&gt;I fear that Cooperative Societies, which many look at as the main hope for the future, likewise exclude competition. This seems to me a great evil for the future progress of mankind. Nevertheless under any system, temperate and rugal workmen will have an advantage and leave more offspring than the drunken and reckless.&lt;/em&gt;&quot; See &lt;a href=&quot;https://www.csustan.edu/sites/default/files/History/Faculty/Weikart/Recently-Discovered-Darwin-Letter.pdf&quot;&gt;Recently Discovered Darwin Letter on Social Darwinism&lt;/a&gt; for the full text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bottom line is that Darwin was not particularly interested in these issues, but he did make passing comments that look offensive today.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-01-16T05:14:08.103" LastActivityDate="2016-01-16T05:14:08.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3331" PostTypeId="2" ParentId="2776" CreationDate="2016-01-16T02:42:31.720" Score="5" Body="&lt;p&gt;After reading other answers, I just like to add one more example. The famous prime number theorem. The proof of the theorem was given (independently) by  &lt;strong&gt;Jacques Hadamard&lt;/strong&gt; and &lt;strong&gt;Charles Jean de la Vallée-Poussin&lt;/strong&gt; in $1896$ using ideas introduced by &lt;strong&gt;Bernhard Riemann&lt;/strong&gt;, particularly, the &lt;em&gt;Riemann zeta function&lt;/em&gt;. Now the proof was hard for someone with a undergraduate level of understanding (I am not saying that no one can!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After almost $50$ years, in $1948$, &lt;strong&gt;Atle Selberg&lt;/strong&gt; and &lt;strong&gt;Paul Erdős&lt;/strong&gt; gave a rather &quot;elementary&quot; proof of the PNT.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.lsu.edu/~mahlburg/.../Selberg-ElemPNT1949.pd&quot; rel=&quot;noreferrer&quot;&gt;See here for Selberg's paper&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Prime_number_theorem#Elementary_proofs&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia_link&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2016-01-16T02:42:31.720" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3332" PostTypeId="1" AcceptedAnswerId="3339" CreationDate="2016-01-16T07:29:06.407" Score="8" ViewCount="200" Body="&lt;p&gt;A simple question(s):&#xA;When and by who was Fermat's famous conjecture (proved by Wiles et al.) called his &quot;last theorem&quot;? Was the original motivation the same as is given nowadays, that this would be the last-&lt;em&gt;proved&lt;/em&gt; of his theorems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting that so many accounts of its history were written which did not really mention the issue of its relatively mysterious naming.&lt;/p&gt;&#xA;" OwnerUserId="3481" LastActivityDate="2016-01-17T15:11:49.470" Title="The first of Fermat's last theorem" Tags="|number-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3333" PostTypeId="2" ParentId="3323" CreationDate="2016-01-16T13:05:53.670" Score="8" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Differential_(infinitesimal)&quot; rel=&quot;noreferrer&quot;&gt;differential&lt;/a&gt; symbold $dx$ is due to Leibniz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He introduced also &quot;iterated&quot; differentials; see  :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;H.J.M. Bos, &lt;a href=&quot;http://www.tau.ac.il/~corry/teaching/toldot/download/Bos1974.pdf&quot; rel=&quot;noreferrer&quot;&gt;Differentials and Higher-Order Differentials in the Leibnizian Calculus&lt;/a&gt; (1974), page 17:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Moreover, to introduce higher-order differentials, first-order differentials have to be conceived as variables ranging over an ordered sequence; if only a single $dx$ is considered, $ddx$ does not make sense. The following quotation&#xA;  from Leibniz [&quot;Monitum de characteribus algebraicis&quot;, 1710] illustrates this:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Further, $ddx$ is the element of the element or the &lt;em&gt;difference of the differences&lt;/em&gt;, for the quantity $dx$ itself is not always constant, but usually increases or decreases continually.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;The Early Mathematical Manuscripts of Leibniz&lt;/a&gt; (J.M. Child ed., 1920 - also Dover reprint): manuscript of an answer to Bernhard Nieuwentijt, &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;pg=PA158&quot; rel=&quot;noreferrer&quot;&gt;page 144-on&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$dx, ddx, dv, ddv, dy, ddy \ldots$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We have to note that Leibniz has $xx$ for $x^2$; see &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;pg=PA151&quot; rel=&quot;noreferrer&quot;&gt;page 151&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Then, since $y-xx : a \ldots$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-01-16T13:05:53.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3334" PostTypeId="1" AcceptedAnswerId="3335" CreationDate="2016-01-16T17:04:32.660" Score="6" ViewCount="495" Body="&lt;p&gt;I know about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Plimpton_322&quot; rel=&quot;noreferrer&quot;&gt;Plimpton 322 tablet and Pythagorean triples&lt;/a&gt;. But can we be sure that this is the first instance of mathematical reasoning? I am talking about notions, propositions, or  questions about mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask you if you can say what the milestone of mathematical reasoning or mathematical knowledge is, as accepted by the scientific community.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have two ideas as to what ancient notions of mathematics could be like. Let's say that humans were a band of Paleolithic hunters or gatherers. If the group gets $N$ items of food, then, leaving out social synergies, the appropriate distribution is the &lt;em&gt;Euclidean division&lt;/em&gt; between the participants. If there are paintings of two animals in a cave, does this mean that Paleolithic men/women had the notion of the integer $2$? Are there scientific thoughts from the viewpoint of history of mathematics? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the first occurrence of mathematical reasoning or mathematical knowledge?&lt;/p&gt;&#xA;" OwnerDisplayName="user3537" LastEditorUserId="62" LastEditDate="2016-01-16T17:49:26.783" LastActivityDate="2017-01-24T02:38:20.877" Title="What is the most ancient milestone of mathematical reasoning or mathematical knowledge?" Tags="|mathematics|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3335" PostTypeId="2" ParentId="3334" CreationDate="2016-01-16T19:15:56.827" Score="5" Body="&lt;p&gt;To address this question, one has to define more precisely what &quot;mathematical reasoning&quot; is. When you start to count things, this already can be called &quot;mathematical reasoning&quot;. And there is no doubt that this kind of mathematical reasoning is much older than the invention of writing. So it does not belong to history in the proper sense of this word, and there are no written records by definition. Therefore, it is impossible to say when &quot;mathematical reasoning&quot; really started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are old Egyptian texts, like Rhind's Papyrus, and even older &quot;Moscow Papyrus&quot;,&#xA;dated roughly as 1850 BC. They certainly contain some sort of mathematical reasoning. Manipulation with integers and fractions, measuring of areas and volumes, as well as the fact that some numerical relations between side lengths of triangles imply a right angle, all this belongs to this kind of mathematical reasoning. The observation that some numbers are even and others are odd, and that the sum of even numbers is even, as well as the sum of two odd numbers,&#xA;is also a mathematical reasoning. If one sheep exchanges for two widgets, then&#xA;five sheep exchange for 10 widgets, this is also a mathematical reasoning, and it is certainly very ancient. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So mathematical reasoning is done practically by all humans and is probably as old as humankind itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This has to be distinguished from Mathematics in the narrow sense. Mathematics as mathematicians understand it is a special kind of reasoning which involves proofs.&#xA;As far as we know this was invented by ancient Greeks. There is no slightest evidence that this was invented anywhere else independently. No primary written record survives, but according to tradition which goes back to Hellenistic times,&#xA;Thales and Pythagoras (or his disciples) in early 6 century BC are credited with this invention.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-01-16T23:43:30.470" LastActivityDate="2016-01-16T23:43:30.470" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3336" PostTypeId="2" ParentId="3332" CreationDate="2016-01-17T01:38:31.347" Score="2" Body="&lt;p&gt;Fermat stated lots of conjectures, and announced to have proofs of many startling theorems without sharing said proofs in his correspondence. Famously, he scribbled on the margins of a copy of Diophantus &quot;Arithmetic&quot;, scribblings which his son published after Fermat's death. Euler set to work on the missing proofs, and when he finished almost all his voluminous statements had been proven or disproven. One of the few that weren't resolved, and which caught the imagination, was the &quot;last theorem&quot; (&quot;last&quot; because it still had no proof, not because of any temporary order).&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2016-01-17T01:38:31.347" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3337" PostTypeId="2" ParentId="3334" CreationDate="2016-01-17T11:54:29.857" Score="3" Body="&lt;p&gt;I agree that there are types of reasoning that have been in use for many millenia before writing and which could be described as &quot;mathematical&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are notions, such as transitivity, that seem fundamental to human reasoning.&#xA;For example, suppose I owe Juan a share of the next mammoth hunt. If I give &#xA;Alexandre a mammoth steak and ask him to give it to Juan, then it is reasonable for me to expect my debt to Juan to have been fulfilled.&#xA;Certain human transactions are commutative, others are not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll hazard a guess that certain kinship structures could be viewed as a type of mathematical reasoning. These relationships (e.g. incest taboos, marriage eligibility, inheritance, reciprocity etc.) have been in use for a very long time, and can involve some very intricate, sophisticated reasoning and&#xA;analysis.&lt;/p&gt;&#xA;" OwnerUserId="3540" LastActivityDate="2016-01-17T11:54:29.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3338" PostTypeId="2" ParentId="3323" CreationDate="2016-01-17T13:39:38.647" Score="23" Body="&lt;p&gt;Leibniz did use this notation for instance in his paper Supplementum geometriae practicae, &lt;em&gt;Acta Eruditorum&lt;/em&gt;, April 1693, p. 179 (&lt;a href=&quot;https://books.google.nl/books?id=jz7vFXu8ODUC&amp;amp;dq=%22ddx%20dy2%22&amp;amp;pg=PA459#v=onepage&amp;amp;q=%22ddx%20dy2%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Google Books link&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YpGx1.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YpGx1.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="370" LastEditorUserId="370" LastEditDate="2017-12-15T14:15:57.677" LastActivityDate="2017-12-15T14:15:57.677" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3339" PostTypeId="2" ParentId="3332" CreationDate="2016-01-17T15:11:49.470" Score="8" Body="&lt;p&gt;My guess is that it was G. Lam&amp;eacute;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Memoire sur le dernier theoreme de Fermat&#xA;C.R. Acad. Sci. Paris, 9, 1839, 45-46.&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-01-17T15:11:49.470" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3340" PostTypeId="1" AcceptedAnswerId="3341" CreationDate="2016-01-17T21:47:32.070" Score="4" ViewCount="927" Body="&lt;p&gt;I have read Disquisitiones Arithmeticae and I have read somewhere that he introduced the concept of congruence with this book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wasn't this concept already used by other Mathematicians in the past?&lt;/p&gt;&#xA;" OwnerUserId="3544" LastActivityDate="2024-02-24T21:17:35.827" Title="Is Gauss the first who introduced congruences?" Tags="|mathematics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3341" PostTypeId="2" ParentId="3340" CreationDate="2016-01-18T03:56:38.570" Score="2" Body="&lt;p&gt;Yes, the idea of congruences is due to Gauss. Case closed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earlier mathematicians had to write out divisibility relations or full equations, without the convenience of the congruence notation that make divisibility relations look like (and can be manipulated as) equations from algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do you think it might have been introduced by someone earlier?&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2016-01-18T03:56:38.570" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3342" PostTypeId="1" AcceptedAnswerId="3345" CreationDate="2016-01-18T09:36:37.373" Score="4" ViewCount="396" Body="&lt;p&gt;I am curious about how the majority of non-mathematicians felt about mathematics throughout history. When did the general opinion on math start to become what it is today--namely, that math is boring and for geniuses and other things of this nature?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course that answer depends on the culture and time period. But I have a guess on when it started in the United States of America.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose that my assumption that the general opinion towards mathematics is hostile was not granted. Most people know on some level of the importance of mathematics even if it isn't exactly their cup of tea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After more thought, I realize that most people don't hate mathematics, but are merely apathetic towards it. Most know that it is important in some vague way, but it is not clear how what they learned in class actually translates to real life--besides basic arithmetic. However, I have witnessed that those who are currently schooling usually have one of two attitudes towards math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are those who are rather fond of it because they have found something in math that makes them happy and they keep finding new things. Then there are those who were once fond of math but then lose interest either because it became too repetitive or too challenging or unmotivated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nonetheless, the stereotypes I originally asked about are still prevalent in popular culture, especially in the U.S. (I don't know about the rest of the world.) Then a more accurate question would be this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Were there similar stereotypical attitudes towards mathematics throughout history? Was there any strong general attitudes towards mathematics?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once again the answer depends on the time and culture, so an answerer could just pick a time period they like or one could attempt to answer for many time periods. Whatever you wish.&lt;/p&gt;&#xA;" OwnerUserId="3546" LastActivityDate="2019-04-29T16:11:37.857" Title="Attitude towards mathematics throughout History" Tags="|mathematics|social-context|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3343" PostTypeId="2" ParentId="3342" CreationDate="2016-01-18T15:30:23.800" Score="4" Body="&lt;p&gt;Consider the &lt;strong&gt;liberal arts&lt;/strong&gt; (that is, arts suitable for practice by a free man).  Of the seven liberal arts, two are &lt;em&gt;geometry&lt;/em&gt; and &lt;em&gt;arithmetic&lt;/em&gt;.  Can we conclude that, up to medieval times, these (along with the other five) were highly respected? but only men who did not have to work for a living would be likely practice them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/I8R8v.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I8R8v.jpg&quot; alt=&quot;quadrivium&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-01-18T15:30:23.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3344" PostTypeId="2" ParentId="3081" CreationDate="2016-01-18T15:35:21.183" Score="1" Body="&lt;p&gt;This is not really a complete answer, but is too long to be a comment. Alexandre Eremenko has written a nice answer, which this is meant to supplement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One point to understand is that what Kepler did was an exercise in curve-fitting. There are 6 parameters needed in order to describe a Keplerian orbit. (You can tell there are 6 by counting degrees of freedom. An initial position vector and an initial momentum vector are enough to define such an orbit.) When you observe an object's position on the sky, in principle you need 6 numbers in order to define its orbital elements. For example, you could do this by finding its declination and right ascension on three different nights.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of this only works if you &lt;em&gt;assume&lt;/em&gt; a Keplerian orbit. For example, suppose that the sun violated the laws of physics, as currently understood, by moving randomly along the line connecting the sun to the earth. This would have no effect at all on the sun's right ascension and declination, so we could never detect this motion from measurements of those coordinates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another thing that may help with intuition is to realize that this is directly analogous to the parallax method for measuring the distances to stars. The only difference is that we take a star to be distant and at rest, which is a degenerate case of Keplerian motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What techniques did Kepler use to add a depth dimension to these observations, to create the three-dimensional data that one can start studying to arrive at his three laws?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I think this part of the question makes an incorrect assumption, which is that Kepler first found the three-dimensional motion, and then inferred Kepler's laws from it. The three-dimensional motion was already known and modeled using epicycles, with the earth arbitrarily taken to be at rest. Kepler was refining this previously developed 3-d model and also (trivially, from a mathematical point of view) changing the origin of coordinates.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2016-01-18T15:35:21.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3345" PostTypeId="2" ParentId="3342" CreationDate="2016-01-18T18:54:36.140" Score="6" Body="&lt;p&gt;Since mathematics was invented, it was highly respected, as a part of the education curriculum. I mean those who establish curricula in education think&#xA;it is very important. European curriculum in the Middle ages contained trivium and quadrium; they included arithmetic, geometry, astronomy and music (the last two items were considered as parts of mathematics too). Mathematics and music were also included in the 6 Noble Arts of Confucius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the present time, in elementary, middle and high school, more hours is spent on mathematics then on any other single subject. I suppose this is the case in all countries, even in the US (where math curriculum in general education is very weak is comparison with other countries that I know).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics is a mandatory subject in almost all college curricula, no matter what your major is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, because of the nature of mathematics, it is considered a difficult subject by most students, and quite intimidating too. That's why they say that they hate it. I suppose this is what most student said since mathematics became one of the main parts of the mandatory curriculum. There is an evidence of this since the Hellenistic times (the anecdote about Euclid teaching some king).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason why it is difficult was very well explained by Euclid himself in this anecdote: in most other subjects you can just learn, memorize something, and think that you know the subject. In mathematics you have to understand. Another reason is that mathematics develops continuously already 2600 years, and the accumulated knowledge never becomes obsolete, unlike in other sciences.&#xA;When teaching chemistry or biology we may safely ignore everything which was done before 19 century. But in mathematics we have to begin with the very beginning:&#xA;Thales, Pythagoras and Euclid. This makes it difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Certainly, the &quot;general public&quot; is as ignorant in mathematics as in any other science. Perhaps mathematics is hated more, because it is taught much more than any other science. So students take this as a kind of torture.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-01-19T05:31:27.823" LastActivityDate="2016-01-19T05:31:27.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3346" PostTypeId="2" ParentId="3340" CreationDate="2016-01-18T23:55:34.337" Score="6" Body="&lt;p&gt;Gauss was anticipated by Euler by about fifty years, however his notation and terminology were somewhat more rudimentary, and he did not publish his work. The ideas are contained in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E792.html&quot; rel=&quot;noreferrer&quot;&gt;Euler's Tractatus de Numerorum Doctrina (Treatise on the Doctrine of Numbers)&lt;/a&gt; that he started writing around 1750 but never finished, it was only published posthumously in 1849, long after Disquisitiones Arithmeticae. In it Euler defined &quot;residue&quot; with respect to an integer $d$ as the remainder from the division by it, noted that integers are thereby divided into $d$ classes, with all numbers in a class regarded as “equivalent” (unlike Gauss Euler does not have a symbol for this equivalence). He then defined arithmetical operations on classes and showed that they are preserved by assigning the residue to its number. Other algebraic ideas also appear in the manuscript, in particular Euler shows that residues relatively prime to $d$ have multiplicative inverses, and uses what is now called coset argument in group theory to prove a generalization of the little Fermat theorem to non-prime $d$ (sometimes called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler&amp;#39;s_theorem&quot; rel=&quot;noreferrer&quot;&gt;totient theorem&lt;/a&gt;). Euler's residue classes are of course what Gauss called congruence classes, but some modern textbooks (e.g. Rosen's Elementary Number Theory) still use the terms residue and residue class. See detailed commentary in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0321387007&quot; rel=&quot;noreferrer&quot;&gt;Katz's History of Mathematics&lt;/a&gt; (19.1.3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two numbers being congruent can be expressed in terms of divisibility or remainders, so a lot of prior work could be converted into the language of congruences once Gauss introduced it. Pythagorean even-odd argument, Chinese remainder theorem, Little Fermat theorem, etc. use arguments now associated with congruences. However, to say that Pythagoreans, &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Mathematical_Classic_of_Sunzi&quot; rel=&quot;noreferrer&quot;&gt;Sunzi&lt;/a&gt; or Fermat already had a &quot;concept&quot; of congruence would be misleading, it would be like saying that ancient Greeks already used polynomials, limits and integrals by other names. The kind of abstraction of classes, equivalence and algebraic manipulations that went into the concept of congruence as Gauss presented it in Disquisitiones Arithmeticae (and Euler somewhat anticipated) was a major new step, not contemplated and probably unnatural in earlier times.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-01-18T23:55:34.337" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3347" PostTypeId="2" ParentId="3334" CreationDate="2016-01-19T00:52:34.483" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://scribol.com/anthropology-and-history/what-the-lascaux-cave-paintings-tell-us-about-how-our-ancestors-understood-the-stars&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cave paintings in Lascaux&lt;/a&gt;, a late Upper Paleolithic cave complex in southwestern France, dated back to c.15–18,000 B.C. and associated with the Magdalenian culture are often seen as some of the earliest artifacts displaying the beginnings of scientific and mathematical thought. &quot;&lt;em&gt;Mathematical aspects in late Upper Paleolithic mythopoetics derived from observational astronomy. Some cave paintings recorded an understanding of the path the moon takes around the sun – the ecliptic. An understanding of the ecliptic leads to the discovery of the zodiac, the annual path of the sun through the celestial sphere... An important cave painting at Lascaux, France depicts six large dots above a magnificent portrait of an Auroch that is known as Bull #18... At Lascaux in 15,300 B.C. the Pleiades were very near the point of the autumn equinox... The six stars in the Salle des Taureaux therefore represent a striking and excellent heavenly marker for the beginning of autumn and of spring&lt;/em&gt;&quot;. There is some speculation involved here, but the helical rise and fall of the Pleiades has been used for calendaric purposes in many later cultures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://afrolegends.com/2013/08/29/the-ishango-bone-craddle-of-mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ishango bone&lt;/a&gt; dating back to about 9000-6500 BC (Upper Paleolith) is another candidate. It has an intricate arrangement of notches that suggests mathematical facility that goes beyond simple counting. &lt;a href=&quot;http://www.math.buffalo.edu/mad/AMU/amu_chma_09.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerdes writes in On the History of Mathematics in Africa South of the Sahara&lt;/a&gt;:&quot;&lt;em&gt;The bone's discoverer, De Heinzelin, interpreted the patterns of notches as an &quot;arithmetical game of some sort, devised by a people who had a number system based on 10 as well as a knowledge of duplication and of prime numbers&quot;. Marshack, on the contrary, explains the bone as early lunar phase count&lt;/em&gt;&quot;. &lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1204/1204.1019.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pletser recently argued&lt;/a&gt; that it is a primitive slide rule. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much later, but still before Egyptian papyruses, Sumerian and Babylonian sources show a relatively advanced practical mastery of mathematics. &lt;a href=&quot;http://www.math.buffalo.edu/mad/AMU/amu_chma_09.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Muroi's Oldest Example of Compound Interest in Sumer&lt;/a&gt; traces the origins of compound interest back to c. 2400 BC, when an inscription on the foundation cone of a temple reads that the city of Lagash lent barley at interest to its neighbor Umma, which Umma did not repay. The interest calculation involves computing the seventh power of four-thirds. A cuneiform tablet YBC 3879 from the third Sumerian Ur period c. 2000 BC was studied by Freiberg, and shows proficiency with elementary geometry, arithmetic, measurement relation between the two, and even ability to solve quadratic equations, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2836/the-origin-of-quadratic-equation-in-actual-practice/2837#2837&quot;&gt;The origin of quadratic equation in actual practice&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-01-24T02:38:20.877" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3348" PostTypeId="2" ParentId="3342" CreationDate="2016-01-19T01:55:59.030" Score="2" Body="&lt;p&gt;The attitude towards mathematics expressed as ``it is not clear how what they learned in class actually translates to real life&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;is probably as old as teaching of mathematics. On a more substantial note, the relevance of mathematics and its notions was challenged by the Greek philosopher and physician Sextus Empiricus (about 160-210 CE), a representative of scepticism.&lt;br&gt;&#xA;Among other things, he wrote a treatise known under the common title ``Against the Mathematicians (Πρὸς μαθηματικούς, Pros mathematikous, Adversus Mathematicos)&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the &lt;code&gt;mathematicians&quot; &#xA;in the title mean the&#xA;&lt;/code&gt;learned ones&quot; (&lt;code&gt;mathesis&quot;=&lt;/code&gt;learning, science, knowledge&quot;), but among the six books of the treatise one finds &lt;code&gt;Against the Geometers&quot; and&lt;/code&gt;Against the Arithmeticians&quot;. To follow the descriptions after Stanford Encyclopedia of Philosophy, &lt;a href=&quot;http://plato.stanford.edu/entries/sextus-empiricus/#AgaMatAdvMat&quot; rel=&quot;nofollow&quot;&gt;http://plato.stanford.edu/entries/sextus-empiricus/#AgaMatAdvMat&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;``Against the Geometers stands at 116 sections. It takes geometers to task for their employment of hypotheses in establishing theorems, and then attacks the hypotheses themselves, such as the definitions of body, point, line (particularly the definition of the line as ‘length without breadth’), and angle.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;``Against the Arithmeticians&quot; is a specific polemic with the Pythagorean school regarding the existence of numbers and their properties. Sextus presents the Pythagorean viewpoint and then counterargues.The conclusion is ‘But if number is conceived as subsisting through addition, as I said, and subtraction, and we have shown that neither of these exists, one must declare that number is nothing’ . So even the basic arithmetics is put into question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this was a part of the philosophical program of scepticism, which proposed  total suspension of judgment on things  when  no reason could be found to favor one side of argument over the other. Still, one can say that respect towards mathematics and its results was not universal and that ambivalent attitudes had their serious adherents. &lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2016-01-19T01:55:59.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3349" PostTypeId="1" AcceptedAnswerId="3350" CreationDate="2016-01-19T07:36:18.527" Score="4" ViewCount="368" Body="&lt;p&gt;There have been some recent developments in reevaluation of the geometric algebra interpretation of ancient greek mathematics, where Reviel Netz is an expert. I don't have enough background in the field but would like to know what Reviel Netz's position is.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-01-20T01:20:40.600" Title="Reviel Netz's views on geometric algebra" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3350" PostTypeId="2" ParentId="3349" CreationDate="2016-01-19T22:33:50.760" Score="5" Body="&lt;p&gt;The issue certainly still inspires passions, for context see our previous discussion in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2857/current-ways-of-thinking-in-the-history-of-mathematics/2869#2869&quot;&gt;Current ways of thinking in the History of Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Netz's position is expressed in his panoramic book &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0521041740&quot; rel=&quot;nofollow noreferrer&quot;&gt;Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations&lt;/a&gt;. It is not so much a return to the geometric algebra debate as such, as to the conflict of philosophies behind it. Netz favors a compromise between the liberties of rational reconstruction and the strictures of context of the times, somewhat reminiscent of Knorr's views, for the sake of evolutionary view of history. In particular, he criticizes Unguru for neglecting &quot;&lt;em&gt;the dynamics of the transformation from the ancient to the modern. Unguru’s premise was that of a great divide separating ancient from modern thinking. The assumption of a great divide, in itself, is not conducive to the study of the dynamics leading from one side of the divide to the other&lt;/em&gt;&quot;. Netz tries to identify the trends behind historical dynamics rather than simply present snapshots of separate epochs in their contexts:&quot;&lt;em&gt;...Late Antiquity and the Middle Ages were characterized by a culture of books-referring-to-other-books (what I call a deuteronomic culture). This emphasized ordering and arranging previously given science: that is, it emphasized the systematic features of science. Early Greek mathematics,on the other hand, was more interested in the unique properties of isolated problems. The emphasis on the systematic led to an emphasis on the relations between concepts, giving rise to the features we associate with 'algebra'&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As expected, Netz's historiography is controversial. &lt;a href=&quot;http://math.harvard.edu/archive/archimedes09/pdf/Acerbi_Phantom_Paths.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi's very long review of the book&lt;/a&gt; disputes Netz's translations and even parsing of the sources, and criticizes him for &quot;&lt;em&gt;unsatisfactory command and highly tendentious use of primary sources and of secondary literature, unwarranted resort to overgeneralization and to modern mathematical concepts... and treating the received texts and diagrams as if they were what their alleged author wrote in the first place&lt;/em&gt;&quot;. Despite praising the book for &quot;&lt;em&gt;fascinating and deep theses, brilliant argumentation, an unmistakably flamboyant style, lucid and rhetorically very effective expositions of the difficult proofs presented, wide-ranging interpretative perspectives, and refined tools of analysis&lt;/em&gt;&quot; he calls it &quot;utterly disappointing&quot; because in his opinion Netz adjusts facts to fit his picture of historical development. Acerbi's conclusion is:&quot;&lt;em&gt;The main thesis itself is a true masterpiece of interpretative insight; yet, I believe, it is simply not supported by the textual evidence adduced. The only conclusion one draws after reading Netz’ book is that the Greek tradition is a dead end, and that Arabic mathematicians reconsidered the whole issue on entirely new grounds. Despite the author’s efforts, there is no continuous trajectory from problems to equations...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-01-20T01:20:40.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3351" PostTypeId="2" ParentId="2161" CreationDate="2016-01-20T12:02:52.747" Score="1" Body="&lt;p&gt;Kepler knew about the duality: he described the five solids as two pairs male/female and called the self-dual tetrahedron 'androgyne' (See Stephenson's book &lt;em&gt;The Music of the Heavens&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a good ground to suspect that indeed Theaetetus already knew about the duality. If, as the story is told, the series tetrahedron, cube, dodecahedron was already known, he might have made his discovery by noting the duals; and the self-duality seems to hint that there are no more to be found - which prompted him to invent the proof. Apparently he told Plato who, as his &lt;em&gt;Timaeus&lt;/em&gt; suggests, seems to be aware of both the finite number of solids and their dualities (see e.g. &lt;a href=&quot;http://www.academia.edu/18133136/Misreading_the_Timaeus_55c-d&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-01-20T12:02:52.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3352" PostTypeId="2" ParentId="2857" CreationDate="2016-01-20T17:21:18.460" Score="3" Body="&lt;p&gt;You wrote &quot;To be specific, I have noticed a certain trend, among historians today, to attack 'naive' or 'Whiggist' approaches to their subject, which they perceive as having dominated the field for too long. Put roughly, at least as far as I understand it, a naive way of doing historiography would be to force a 'modern' reading upon historical mathematical texts.&quot; You mentioned geometric algebra as an example, but did not limit your question to this context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The received views in current historiography tend to (1) accept dogmas that have been developed when the state of mathematics was very different from what it is today, and strongly affected by what mathematicians believed then; and (2) be suspicious of relying on procedures in modern mathematics to shed light on procedures of historical mathematics. In this way, received historiography has developed its own set of what you refer to as &quot;Whiggist&quot; (or to be more polite, presentist) positions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aside from the issue of geometrical algebra, we have challenged such received views on the history of analysis in a number of recent publications, including &lt;a href=&quot;http://www.ams.org/notices/201307/rnoti-p886.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Is mathematical history written by the victors?&lt;/em&gt;&lt;/a&gt; and some follow-up articles. &lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-01-21T08:06:03.800" LastActivityDate="2016-01-21T08:06:03.800" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3353" PostTypeId="1" CreationDate="2016-01-22T15:11:23.093" Score="8" ViewCount="1690" Body="&lt;p&gt;Could someone please explain how the concept of energy was originally conceived and how it evolved over time to our current understanding of it? Also how did people come about various ways of measuring, mathematically expressing and precisely calculating various forms of energy, like kinetic energy, potential energy, thermal energy, etc.?&lt;/p&gt;&#xA;" OwnerUserId="3564" LastEditorUserId="55" LastEditDate="2016-01-24T04:13:08.593" LastActivityDate="2019-03-22T16:03:16.487" Title="What is the history of the energy concept and its measurement?" Tags="|physics|classical-mechanics|thermodynamics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3354" PostTypeId="1" CreationDate="2016-01-22T20:02:30.180" Score="7" ViewCount="927" Body="&lt;p&gt;Checking modern quotes from Aristotle almost everywhere the word &lt;code&gt;astronomy&lt;/code&gt; is found to be replacing  the original Greek or Latin astrology'  (astrologia). In Plato's original texts only &lt;code&gt;astronomy&lt;/code&gt; (astronomia) is found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One hypothesis is that Aristotle was greatly impressed by Eudoxus' model and decided that this mechanical and causal approach to sky science should be called differently as to emphasize the distinction from the previous geometrical and descriptive stance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other hypothesis is that his texts, after being withheld from circulation for nearly 200 years, have been heavily edited and updated.&#xA;It was during this time that the Chaldean divinatory practice which &lt;em&gt;we&lt;/em&gt; call 'astrology' became widespread.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most scholars apparently refrain from comments on this word usage,  perhaps fearing that they would only bring some unnecessary confusion, replacements being made silently and according to their own wits.  I know  just &lt;a href=&quot;http://articles.adsabs.harvard.edu/full/2012JAHH...15...42L&quot; rel=&quot;nofollow&quot;&gt;one recent paper&lt;/a&gt; on this question about which I would like to hear more.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3617" LastEditDate="2016-02-03T20:53:03.020" LastActivityDate="2016-02-03T20:53:03.020" Title="Why does Aristotle write 'astrology' when Plato writes 'astronomy'?" Tags="|reference-request|ancient-greece|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3355" PostTypeId="1" CreationDate="2016-01-23T10:06:20.913" Score="2" ViewCount="75" Body="&lt;p&gt;Einstein's 1905 paper discussed electromagnetism and the laws of mechanics.  But did not discuss gravity, and of course did not discuss the strong or weak nuclear forces.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Principle of Relativity - (from translation of Einstein's 1905 paper) &quot;the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;1) At what point did physicists interpret this to mean all physical laws should have these symmetries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems like physicists were trying to get a relativistic version of gravity fairly soon after this, so maybe immediately?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'm not sure when the symmetry point of view took hold.  For example, would physicists back then interpret conservation of energy and momentum as a consequence of special relativity (translation invariance), or would they still consider that a consequence of the individual physical laws?  As a particular example, consider the initial confusion around decays involving neutrinos. When it appeared to be a possible violation of energy conservation, did they also view this as a potential violation of relativity? Or did they not think of it in that context?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Since the &quot;equations of mechanics hold good&quot; under parity transformations, did physicists historically believe special relativity predicted parity symmetry?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if parity violation was discovered in 1906, would this have been considered a blow to special relativity?&lt;/p&gt;&#xA;" OwnerUserId="3565" LastActivityDate="2016-02-22T15:00:42.890" Title="Historical interpretation of symmetries required by Principle of Relativity" Tags="|theoretical-physics|relativity-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3357" PostTypeId="1" CreationDate="2016-01-23T16:09:56.807" Score="3" ViewCount="70" Body="&lt;p&gt;I've read this &lt;a href=&quot;https://en.wikipedia.org/wiki/Antonie_van_Leeuwenhoek&quot; rel=&quot;nofollow&quot;&gt;article&lt;/a&gt;, and I know that he made lenses but it's not clear what tools he used to build them. Specifically, if he used mathematical tools of if he did that only by trial and error. I was unable to find a copy of the references in the end of the article.&lt;/p&gt;&#xA;" OwnerUserId="332" LastEditorUserId="26" LastEditDate="2016-01-23T18:20:53.270" LastActivityDate="2016-01-23T19:33:57.987" Title="How much mathematics did Van Leeuwenhoek know, and did he use mathematics in this fabrication of lenses?" Tags="|mathematics|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3358" PostTypeId="2" ParentId="3357" CreationDate="2016-01-23T19:33:57.987" Score="2" Body="&lt;p&gt;The same Wikipedia article only mentions that he &quot;attended school&quot; for a short time. After that he became a bookkeeper's apprentice. So he probably learned some&#xA;mathematics at school and as a bookkeeper's apprentice. Then we learn that he was appointed a land surveyor, which certainly requires more mathematics. This implies that he learned somehow the necessary mathematics for this job. Either at school or (more likely) by self-education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is not so much mathematics a lens maker has to know, besides what a land surveyor has to know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most 17s and 18s and 19s century craftsmen who made important inventions had no formal mathematical training (I mean at the university level). They were self-educated, and they knew enough mathematics for their trade. I mean people like&#xA;Jost Burgi (one inventor of logarithms), James Watt, or   John Harrison (chronometer), and many others. They obtained the necessary mathematical knowledge in the process&#xA;of training for the trade and self-education.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-01-23T19:33:57.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3359" PostTypeId="2" ParentId="3353" CreationDate="2016-01-24T04:02:14.137" Score="9" Body="&lt;p&gt;The word originated with Aristotle, whose &quot;energeia&quot; and &quot;entelecheia&quot; can roughly be translated as enaction, that which makes matter move, and embodiment, that which makes matter take form, respectively. Over the middle ages the attention was focused on &quot;impetus&quot;, roughly associated with mass times speed and usually taken to be the precursor of modern momentum, but natural philosophers did not mean mathematically phrases that look that way to us, so there was enough vagueness to say that the idea incubated both modern concepts. Kinetic energy under the name of vis viva (living force) first properly entered the stage in 17th century, when Huygens showed that it is conserved in elastic collisions, while impetus, finally mathematized by Descartes and called &quot;quantity of motion&quot;, was not. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This last part was pointed out by Leibniz, the main proponent of vis viva far beyond mechanics only, who begged to differ on what was more deserving of the name &quot;quantity of motion&quot;. Newton did not take kindly to the fact that a quantity hardly fundamental to his Principia got unwarranted attention, and a controversy that lasted for over half a century began. We have a detailed account of it here &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt; It wasn't a total loss, many basic mechanical notions got clarified in the process, including the notion of work, and in 1740s Euler (1736) and D'alambert (1743) produced systematic monographs on mechanics that finally deflated the controversy. Potential energy joined the club in Lagrange's Mecanique Analytique (1788), see more in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2345/how-did-newton-establish-the-conservation-laws-in-the-principia&quot;&gt;How did Newton establish the conservation laws in the Principia?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around the start of 18th century systematic analysis of thermal energy began under the auspices of Stahl's caloric theory that treated heat as a fluid (&quot;phlogiston&quot;), with no inkling of course about a relation to mechanical energy. This did not stop Newcomen from inventing a new steam engine in 1712, and Watt from improving it in 1763-1775 (he also introduced a unit of power, &quot;horsepower&quot;) After Lavoisier's work on oxygenation (1783) phlogiston theory was modified into caloric one, which remained in place until mid 19th century,  Carnot even developed the basics of the theory of heat engines based on it (1824). But measurements of the mechanical equivalent of heat first performed by Rumford in 1798, and more conclusively by Joule in 1840s, originally very coldly received, the formulation of the general energy conservation law by Mayer in 1841, and the reformulation of Carnot's work by Clausius in 1850 led to the acceptance of heat as another form of energy. In 1851 Thomson already could write that “&lt;em&gt;heat is not a substance, but a dynamical form of mechanical effect, we perceive that there must be an equivalence between mechanical work and heat, as between cause and effect&lt;/em&gt;”. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/3470/what-are-the-major-flaws-of-the-caloric-theory-of-heat/3474#3474&quot;&gt;What are the major flaws of the “caloric” theory of heat?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time SI and CGS were undergoing major overhaul of units in 1860-1880s mechanical, thermal and electric energy were treated uniformly, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2968/why-is-there-no-named-unit-for-momentum/2971#2971&quot;&gt;Why is there no named unit for momentum but there is one for energy?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-02T18:54:53.843" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3360" PostTypeId="1" CreationDate="2016-01-24T04:39:45.953" Score="6" ViewCount="672" Body="&lt;p&gt;Or who first solved the harmonic oscillator in the algebraic method?&lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2021-09-19T22:40:05.757" Title="Who introduced the creation and annihilation operators for the harmonic oscillator?" Tags="|physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3361" PostTypeId="2" ParentId="3353" CreationDate="2016-01-24T10:35:27.460" Score="7" Body="&lt;p&gt;There is a chapter &quot;the history of the energy concept&quot; of some 80 pages in Philip Mirowski's book &lt;em&gt;More heat than light&lt;/em&gt; which is rather  informative. The main point of the story appears to be that energy is something that is conserved, that is the concept really makes sense as an &quot;invariant&quot;. The end of the chapter mentions various energetist and energetisms, largerly forgotten today.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Energy became the novel organizing principle of physical research, linking the previously disjunct and disparate studies of motion, light, heat, electricity, and magnetism. After this &quot;discovery,&quot; the very science&#xA;  of physics was redefined to be the reduction of all phenomena to their energetic foundations, and hence implicitly the reduction of all phenomena to mechanics (Harman 1982a, p. 158). Only after 1850 did physics become the king of the sciences, usurping the throne from physical astronomy (Cannon 1978, p. 2). Energy was the reason.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thomas Kuhn also researched the history and Mirowski reports that he came with some 16 names claiming credit for the discovery, the list of most promising being&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;J. R. Mayer, James Joule, Hermann von Helmholtz, and Ludwig Colding.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After a dozen descriptive pages it is pointed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The assertion of the &quot;discovery&quot; of the conservation of energy actually conflates four distinct ideas: (1) the formation of a concept of energy; (2) an ontological claim that there was an energy &quot;out there&quot; and &quot;in here&quot; waiting to be found; (3) the mathematical statement that this energy is neither created nor destroyed; and (4) some procedure&#xA;  of justification for ideas (2) and (3). The incongruity of claiming&#xA;  a simultaneous discovery of energy conservation in the 1840s is that each individual so far identified only achieved a subset of the above conditions, and no individual succeeded in implementing them all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-01-24T10:35:27.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3362" PostTypeId="1" AcceptedAnswerId="3363" CreationDate="2016-01-24T11:21:31.370" Score="4" ViewCount="857" Body="&lt;p&gt;One of the comments on &lt;a href=&quot;https://math.stackexchange.com/questions/551074/are-all-mathematicians-human-calculators&quot;&gt;this  Math.SE question&lt;/a&gt; states that Hilbert had dyscalculia. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I have not been able to find a reliable source that confirms this, and I do not really believe it. Of course, there were no tests that he could have done to confirm or deny it, but is there more evidence than just the classical, unsourced story that Hilbert could not add seven and five in a lecture? Or is there a good source for that story?&lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="4251" LastEditDate="2020-07-12T02:06:33.570" LastActivityDate="2020-07-12T02:06:33.570" Title="Did Hilbert really have dyscalculia?" Tags="|mathematicians|biographical-details|hilbert|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3363" PostTypeId="2" ParentId="3362" CreationDate="2016-01-24T14:26:06.040" Score="7" Body="&lt;p&gt;No sources, no history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Constance Reid, &lt;a href=&quot;https://books.google.it/books?id=pPNirfJL86MC&amp;amp;pg=PA8&quot; rel=&quot;noreferrer&quot;&gt;Hilbert&lt;/a&gt; (1970, reprinted as Hilbert-Courant, 1986), page 7-8 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the Wilhelm Gymnasium, David was much happier than he had been at Friedrichskolleg. The teachers seemed to recognize and encourage his originality, and in later years he was often to recall them with affection. &#xA;  Grades improved — &quot;good's&quot; in almost everything (German, Latin, Greek, &#xA;  theology and physics) and in mathematics &quot;vorzüglich, [&lt;em&gt;excellent&lt;/em&gt;]&quot; the highest &#xA;  possible mark given at the time. He did so well on his written examinations that &#xA;  he was excused from taking the final oral examination for the leaving certificate. The evaluation which appeared on the back of the certificate rated his deportment as &quot;exemplary&quot; commented on his industry and &quot;serious scientific interest,&quot; and then concluded: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&quot;For mathematics he always showed a very lively interest and a &#xA;    penetrating understanding: he mastered all the material taught in the school in a very pleasing manner and was able to apply it with sureness and ingenuity.&quot; &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;This is the earliest recorded glimpse of the mathematician Hilbert. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So, the conclusion is: it is not confirmed. After all, does it really matter?&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="2905" LastEditDate="2016-01-24T20:29:16.163" LastActivityDate="2016-01-24T20:29:16.163" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3364" PostTypeId="1" AcceptedAnswerId="3369" CreationDate="2016-01-24T18:30:49.600" Score="8" ViewCount="1078" Body="&lt;p&gt;Reading the biographies of &lt;a href=&quot;http://www-gap.dcs.st-and.ac.uk/history/Biographies/Lagrange.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt; and &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Euler.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler&lt;/a&gt; by all accounts they were on good terms and correspondence. However it is well known that Lagrange would not accept a position at the Berlin Academy of sciences because &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;it is not suitable while Euler is there&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;It is correspondence with d'Alembert and King Frederick coupled with the knowledge that Euler was leaving for St. Petersburg that convinced him to move to Prussia. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why then did Lagrange not accept the position when Euler worked there? Were they not on friendly terms or was it a respectful gesture? There is a clear animosity between Euler and d'Alembert. Does the correspondence between Lagrange and d'Alembert indicate a shift in allegiance?&lt;/p&gt;&#xA;" OwnerUserId="198" LastEditorUserId="5989" LastEditDate="2018-10-13T09:52:56.187" LastActivityDate="2018-10-13T19:07:47.567" Title="The relationship between Euler and Lagrange" Tags="|mathematics|biographical-details|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3365" PostTypeId="1" CreationDate="2016-01-24T19:00:00.437" Score="4" ViewCount="216" Body="&lt;p&gt;&lt;a href=&quot;http://en.wikipedia.org/wiki/Jacopo_Riccati&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; has a jpeg of the front cover of the book Opere (1761) by Jacopo Riccati, author of the celebrated Riccati equation. What is the symbolism incorporated in this illustration?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/D3xWS.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See discussion of the possible symbolism and a higher resolution image at the cross-posted question at History SE &lt;a href=&quot;https://history.stackexchange.com/questions/34977/symbolism-in-illustration-for-a-book-by-riccati&quot;&gt;https://history.stackexchange.com/questions/34977/symbolism-in-illustration-for-a-book-by-riccati&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="-1" LastEditDate="2017-04-13T12:47:53.943" LastActivityDate="2017-01-16T19:18:20.073" Title="Symbolism in illustration for a book by Riccati" Tags="|biographical-details|mathematicians|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3366" PostTypeId="2" ParentId="3285" CreationDate="2016-01-24T22:33:09.990" Score="1" Body="&lt;p&gt;On page 423 of Helmut Rechenberg's &quot;The Historical Development of Quantum Theory&quot;, it is stated that Paul Langevin first described paramagnetism as being due to the presence of unpaired electrons in atomic or molecular orbitals in a 1905 publication.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2016-01-24T22:33:09.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3367" PostTypeId="2" ParentId="3285" CreationDate="2016-01-25T04:45:47.380" Score="4" Body="&lt;p&gt;It appears that Faraday discovered it and that W. Whewell coined the term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;a href=&quot;http://www.oed.com/view/Entry/137506?redirectedFrom=paramagnetism&quot; rel=&quot;nofollow&quot;&gt;paramagnetism, n.&lt;/a&gt;&quot;. &lt;em&gt;OED Online&lt;/em&gt;. December 2015. Oxford University Press. (accessed January 24, 2016).:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;W. Whewell &lt;em&gt;Let. c&lt;/em&gt; 10 Dec. in M. Faraday &lt;em&gt;Corr.&lt;/em&gt; (1996) III. 442,   I think you may keep &lt;em&gt;diamagnetic&lt;/em&gt; for this class, and give to the opposite class a name implying that they &lt;em&gt;rank along with&lt;/em&gt; magnetic bodies. I propose &lt;em&gt;paramagnetic&lt;/em&gt;... Then this new branch of science, for so, of course, it will soon become, will be &lt;em&gt;Paramagnetism&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-01-25T04:45:47.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3368" PostTypeId="1" AcceptedAnswerId="3370" CreationDate="2016-01-25T05:09:00.420" Score="8" ViewCount="321" Body="&lt;p&gt;I mean, was there anyone who tried (before Gödel) to encode meta-mathematics? Or, was the idea of constructing &lt;strong&gt;&lt;em&gt;formal&lt;/em&gt;&lt;/strong&gt; sentences which informally refer to itself completely new when Gödel introduced his work? Or were there some work before which makes such a step quite natural and not surprising?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I mean, how did he come up with this very idea of arithmetizing logic and constructing self-referential sentences? Was there any effort done before him which makes such a step not very surprising? Or is it better to say that &quot;he was just able to see it where no other mathematician was able to&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that, by &quot;in the air&quot; I mean that there was previous work that indicates that such thing could be fruitful to do or that the idea of encoding things was already thought of by other people (or closely related ideas which could naturally lead to encoding).&lt;/p&gt;&#xA;" OwnerUserId="2504" LastEditorUserId="55" LastEditDate="2016-01-25T19:49:15.373" LastActivityDate="2016-01-25T20:54:29.027" Title="Was the idea of encoding meta-mathematics into arithmetic in the air when Gödel proved incompleteness theorems?" Tags="|mathematicians|mathematical-logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3369" PostTypeId="2" ParentId="3364" CreationDate="2016-01-25T08:01:41.467" Score="7" Body="&lt;p&gt;&lt;em&gt;Some suggestions&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the Euler-Lagrange correspondence, &lt;a href=&quot;http://eulerarchive.maa.org/correspondence/letters/OO1373.pdf&quot;&gt;Euler to Lagrange, Berlin, le 3 Mai 1766&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler is leaving Berlin for Pétersbourg and it seems that he is trying to &quot;enroll&quot; Lagrange for the project [&quot;le dessain de l'Impératrice&quot;] of &quot;rétablir l'Académie de Russie&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If so, we can conjecture that, in addition to Lagrange's shyness, the motivation behing Lagrange's refusal was the same of his previous refusal to join Euler in Berlin: a tacit refusal to be the &quot;second great Mathematicians&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only when Eluer leaved Berlin, Lagrange agreed on d'Alembert proposal to leave Turin and move to Berlin. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I'm not able to access d'Alembert-Lagrange corresponsence; but we can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pierre Crépel &amp;amp; Guillaume Jouve, &lt;a href=&quot;http://download.springer.com/static/pdf/487/art%253A10.1007%252Fs40329-014-0039-z.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs40329-014-0039-z&amp;amp;token2=exp=1453709890~acl=%2Fstatic%2Fpdf%2F487%2Fart%25253A10.1007%25252Fs40329-014-0039-z.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252Fs40329-014-0039-z*~hmac=f5b376dd68615617a4bee7a9a67a98a9e68ee80cd7f032214328d502707e4851&quot;&gt;1765: A year in the correspondence between d’Alembert and Lagrange&lt;/a&gt; (2014):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before 1766, Frederick II of Prussia had more than once invited both d’Alembert and Lagrange to move to Berlin. The encyclopaedist had declined the offer and suggested the name of his Turinese friend. But Lagrange, even though he was on good terms with Euler, did not relish a &quot;cohabitation&quot; with him in the Berlin Academy. The situation changed at the beginning of 1766, and in a letter of 6&#xA;  March 1766 d’Alembert spoke of Lagrange about it in these terms:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;M.Euler s’en va, dit-on, à Pétersbourg pour quelque mécontentement qu’il a eu à Berlin. Je lui ai écrit pour l’en dissuader. S’il s’en va, et que vous vouliez&#xA;    le remplacer, vous n’avez qu’à m’écrire un mot et je ferai de mon mieux pour vous servir. (Mr Euler is leaving, he says, for St.Petersburg because of some unhappiness he has had in Berlin. I wrote to him to dissuade him. If he leaves, and you want to replace him, you have only to write me and I will do my best to serve you.)&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;D’Alembert also wrote to Frederick II in praise of Lagrange, and in his letter of 26 April 1766, he announced to the erudite Turinese scholar:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Mon cher et illustre ami, le roi de Prusse me charge de vous écrire que, si vous voulez venir à Berlin pour y occuper une place dans l’Académie, il vous donne 1500 écus de pension qui sont 6000 livres argent de France; … M.Euler mécontent pour des raisons dont je ne sais pas bien le détail, mais dans lesquelles je vois que tout le monde lui donne le tort, demande son&#xA;    congé et veut s’en aller à Pétersbourg. Le roi, qui n’a pas trop d’envie de le lui accorder, le lui donnera certainement si vous acceptez la proposition qu’on&#xA;    vous fait;…. (My dear and illustrious friend, the king of Prussia has charged me to write you that, if you would like to come to Berlin to occupy a place in the Academy, he would give you a pension of 1,500 crowns, which are 6,000 French pounds … Mr Euler, unhappy for reasons of which I do not know the details, but in which I see that everyone thinks him wrong, requests permission to leave and wants to go to St. Petersburg. The king, who was not too anxious to grant it, would definitely give it to him if you accept the proposition that he has made; ….)&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Reading the epistolary exchanges between the two scientists shows that they shared a common correspondence of 1764–1765, one can feel the rise in Lagrange’s bitterness that will lead him into exile. As Marco Segala [ &lt;a href=&quot;http://booksandjournals.brillonline.com/content/journals/10.1163/182539110x00037&quot;&gt;Lagrange. Why I left Turin and never went back again&lt;/a&gt;, Nuncius, Volume 25-1, pages 23–40, 2010] shows, a low, stagnant salary forced him to teach in a rather hostile environment, and uninterested&#xA;  in theory, the contempt of Piedmont authorities for science and remoteness of European intellectual centres led Lagrange to leave his country and accept the offer of d’Alembert and Frederick II in Berlin. He regrets that in Turin, &quot;&lt;em&gt;on regarde la science dont [il s’]occupe comme très inutile et meme ridicule&lt;/em&gt; (one regards the science with which I am occupied as quite unnecessary and ridiculous,&quot; letter of 10 May 1766). Fearing that his letters to d’Alembert&#xA;  might be intercepted, he also used roundabout methods to deliver correspondence to Paris. But the King of Sardinia did not allow him to leave very easily. He had made up his mind to accept Frederick II’s offer, but he was&#xA;  not granted permission to leave Turin until early July 1766. Lagrange left Turin definitively on 21 August 1766 and arrived to Berlin at the end of October, after a short time spent in Paris and London. Thus began for Lagrange a new&#xA;  period of intense scientific production.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-01-25T08:23:08.853" LastActivityDate="2016-01-25T08:23:08.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3370" PostTypeId="2" ParentId="3368" CreationDate="2016-01-25T19:44:49.717" Score="9" Body="&lt;p&gt;Let me open with a quote from &lt;a href=&quot;https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hofstadter's classic Godel-Escher-Bach&lt;/a&gt;:&quot;&lt;em&gt;In my opinion, if one is interested in understanding Gödel's proof in a deep way, then one must recognize that the proof, in its essence, consists of a fusion of these two main ideas. Each of them alone is a master stroke; to put them together took an act of genius. If I were to choose, however, which of the two key ideas is deeper, I would unhesitatingly pick the first one-the idea of Godel-numbering, for that idea is related to the whole notion of what meaning and reference are, in symbol-manipulating systems. This is an idea which goes far beyond the confines of mathematical logic, whereas the Cantor trick, rich though it is in mathematical consequences, has little if any relation to issues in real life&lt;/em&gt;&quot;. The Gödel numbering gets us a predicate Prov(x,y), which obtains when x is the Gödel number of a proof whose conclusion has Gödel number y. The Cantor's diagonal argument, non-trivially converted by Gödel into a fixed point theorem, gets us a sentence G with Gödel number n, provably equivalent to ∀x¬Prov(x,n). This is the Gödel sentence, and it is provably equivalent to the one &quot;saying&quot; that it is unprovable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gödel's insights are his own, his closest precursors weren't too close. What was &quot;in the air&quot; by the end of 1920s is that formal mathematics probably can not prove its own consistency, like Hilbert wanted, but not a particular path to it. And it was the air breathed by select few (Skolem, von Neumann), even Russell, Wittgenstein and Zermelo did not belong to the club. When Gödel's proof came out they were surprised by it and apparently mistook it for a paradox, like Russell's in Frege's original logic, see &lt;a href=&quot;https://philosophy.stackexchange.com/questions/29288/what-are-some-resources-about-russells-response-to-g%C3%B6dels-incompleteness-theor/29304#29304&quot;&gt;What sources discuss Russell's response to Gödel's incompleteness theorems?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that symbols are just like numbers, and manipulating them is just like doing arithmetic, was Hilbert's, it was the basis of his formalist programme of mathematical foundations. At the time the prevailing wisdom was that while Kant was mistaken about the a priori nature of geometry he was right about arithmetic (although Hilbert and Frege disagreed whether it was a priori synthetic like geometry or a priori analytic like logic). Hilbert's innovation was to analogize arithmetic to formal deductions in axiomatic theories, and conceive of that as the ultimate basis for all of mathematics. Mathematics as a priori reasoning about symbols according to formal rules, see &lt;a href=&quot;https://philosophy.stackexchange.com/questions/28154/is-there-a-kantian-influence-on-hilberts-formalist-programme/28157#28157&quot;&gt;Is there a Kantian influence on Hilbert's formalist programme?&lt;/a&gt; It is a supreme irony that the very idea that drove Hilbert's programme carried the seed of its demise. But Hilbert thought of it as extension, it did not occur to him to literally code meta-mathematics into arithmetic. Nor did it occur to Russell in Principia Mathematica. Skolem perhaps came closer in his 1923 paper on primitive recursive arithmetic, where he pioneered the language/meta-language distinction employed by Gödel. Skolem first introduced a formal system with objects defined by primitive recursion, and then introduced another one on top of it that proved properties of objects in the first system. But even Skolem did not code one into the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When later asked when he became interested in the completeness problem and what where his influences Gödel replied &quot;1928&quot;, and &quot;Hilbert Ackermann: Introduction to math Logic, Carnap: Lectures of math Logic&quot;. Carnap, the rising star of logical positivism who Gödel met at the Vienna Circle in 1928, introduced him to logic, in particular to the issue of completeness. The book presented Hilbert's programme, and stated completeness of the first order logic as an open problem, Gödel proved it the next year (it was missed by Skolem, from whose results it quickly follows). And some time in early 1930 he had the incompleteness breakthrough, in August he mentioned his negative result to Carnap. The immediate run up to the proof of the incompleteness theorem is described in &lt;a href=&quot;http://projecteuclid.org/euclid.bsl/1120231629&quot; rel=&quot;nofollow noreferrer&quot;&gt;Goldfarb's On Gödel's Way In&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:42:23.043" LastActivityDate="2016-01-25T20:54:29.027" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3371" PostTypeId="2" ParentId="3354" CreationDate="2016-01-25T21:12:32.937" Score="3" Body="&lt;p&gt;The issue is difficult, but can be useful to clarify that there were no &quot;stable&quot; terminology in ancient Greece science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus&quot; rel=&quot;nofollow&quot;&gt;Eudoxus of Cnidus&lt;/a&gt;'s (scholar of Plato : 408–355 BC) lost astronomical works were titled : &lt;em&gt;Phaenomena&lt;/em&gt; (Φαινόμενα) and &lt;em&gt;Entropon&lt;/em&gt; (Ἔντροπον).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The lost book of Aristotle's pupils, &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudemus_of_Rhodes&quot; rel=&quot;nofollow&quot;&gt;Eudemus of Rhodes&lt;/a&gt; (c.370 BC - c.300 BC): &lt;em&gt;History of Astronomy&lt;/em&gt; (Άστρολογικὴ ἱστορία), is clearly influenced by Aristotle terminology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can list also the didactic poem Φαινόμενα (&quot;Appearances&quot;), &lt;a href=&quot;https://en.wikipedia.org/wiki/Aratus&quot; rel=&quot;nofollow&quot;&gt;Aratus&lt;/a&gt; (c.315 BC – 240 BC) and the only preserved work of the great astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Hipparchus&quot; rel=&quot;nofollow&quot;&gt;Hipparchus&lt;/a&gt; (c.190 – c.120 BC): Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις (&quot;Commentary on the Phaenomena of Eudoxus and Aratus&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Theodosius_of_Bithynia&quot; rel=&quot;nofollow&quot;&gt;Theodosius of Bithynia&lt;/a&gt; (c.160 BC – c.100 BC) wrote the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sphaerics&quot; rel=&quot;nofollow&quot;&gt;Sphaerics&lt;/a&gt; (Σφαιρικά).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Geminus&quot; rel=&quot;nofollow&quot;&gt;Geminus of Rhodes&lt;/a&gt; (1st century BC) only surving work is &lt;em&gt;Introduction to the Phenomena&lt;/em&gt; (Εἰσαγωγὴ εἰς τὰ Φαινόμενα).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ptolemy&quot; rel=&quot;nofollow&quot;&gt;Claudius Ptolemy&lt;/a&gt; (c.AD 100 – c.170) main works are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the astronomical treatise now known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Almagest&quot; rel=&quot;nofollow&quot;&gt;Almagest&lt;/a&gt;, although it was originally entitled the &quot;Mathematical Treatise&quot; (Μαθηματικὴ Σύνταξις) and then known as the &quot;Great Treatise&quot; (Ἡ Μεγάλη Σύνταξις)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the astrological treatise sometimes known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tetrabiblos&quot; rel=&quot;nofollow&quot;&gt;Apotelesmatika&lt;/a&gt; (Ἀποτελεσματικά) but more commonly known as the &lt;em&gt;Tetrabiblos&lt;/em&gt; (Τετράβιβλος) meaning &quot;Four Books&quot; or by the Latin &lt;em&gt;Quadripartitum&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In conclusion, during six centuries of Greek astronomy, we have very few tratises with &lt;em&gt;Astronomy&lt;/em&gt; (or &lt;em&gt;Astrology&lt;/em&gt;) in the title.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, my personal suggestion is do not read too much about a definite disciplinary separation in ancient science.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;More specificaly, about Plato and Aristotle, see :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Plato, &lt;em&gt;Rep.&lt;/em&gt;, VII, 528E-on :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Astronomy (&lt;em&gt;astronomìa&lt;/em&gt;) will accordingly be fourth in order, and Stereometry third. [...] as studied at present [Socrates replies], Astronomy turn the soul's eye down, though the bodily eyes looks upward. True astronomy is not observation of the visible heavens [...]; it is a mathematical science, which studies the true movements of intelligible stars and uses the visible firmament as its orrery. We shall therefore pursue Astronomy by making use of problems and leave the heavens alone.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Aristotle, &lt;em&gt;Met&lt;/em&gt;, B, 997a34-on :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And astronomy (&lt;em&gt;astrologìa&lt;/em&gt;) also cannot be dealing with perceptible magnitudes nor with this heaven above us. [...] nor are the movements and complex orbits in the heavens like those of which astronomy talks. [The context is a critique of Plato's theory of &lt;em&gt;Forms&lt;/em&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We can speculate about what are the different points of view discussed here, but it seems quite clear that it is not the difference between :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;astronomy : the description of apparent motions of the stars and planetary paths&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;astrology :  the study of the effects of astronomical cycles on earthly matters.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt; see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Westman, &lt;a href=&quot;https://books.google.it/books?id=iEueQqLQyiIC&amp;amp;pg=PA36&quot; rel=&quot;nofollow&quot;&gt;The Copernican Question: Prognostication, Skepticism, and Celestial Order&lt;/a&gt; (2011), for an overview and, in particular, page 36 for the schema of the &quot;science of the stars&quot; according to Ptolemy.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-01-26T07:43:21.583" LastActivityDate="2016-01-26T07:43:21.583" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3372" PostTypeId="1" AcceptedAnswerId="3411" CreationDate="2016-01-25T21:24:26.117" Score="5" ViewCount="802" Body="&lt;p&gt;The constant G in Newton's law $F = G m_1m_2/r^2$ is, as far as I know, absent from Newton's work - who introduced this constant?&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-02-13T09:22:56.020" Title="Introduction of the Gravitational constant" Tags="|physics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3373" PostTypeId="1" AcceptedAnswerId="3375" CreationDate="2016-01-26T18:15:33.970" Score="8" ViewCount="974" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Master_equation&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; about the Master equations describes pretty well how many there are and what kind of equations are called &quot;Master equations&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know where the term originates, why there are so many, and what is so masterful about them?&lt;/p&gt;&#xA;" OwnerUserId="3584" LastEditorUserId="55" LastEditDate="2016-01-26T23:16:11.233" LastActivityDate="2016-03-13T04:01:04.017" Title="Where did Master equations come from, and why are there so many of them?" Tags="|physics|terminology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3375" PostTypeId="2" ParentId="3373" CreationDate="2016-01-26T21:55:19.923" Score="6" Body="&lt;p&gt;There are so many of them because they are phenomenological equations derived from exact theory by using any number of simplifying heuristic assumptions about statistical behavior, and there is a great variety of assumptions one can make in any given context, as well as a variety of different contexts. Even to &quot;derive&quot; the Boltzmann equation in kinetic theory through master equations, which is apparently a prototype for using them, one can come up with plenty, see e.g. &lt;a href=&quot;http://www.math.csi.cuny.edu/~carlo/ttsp04&quot; rel=&quot;nofollow&quot;&gt;On the Master-Equation Approach to Kinetic Theory by Kiessling and Lancellotti&lt;/a&gt;, who mention Kac's 1956 work on it as &quot;pioneering&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest mention in the MathSciNet is in Kac's entry to the 1954-55  Berkeley Symposium on Mathematical Statistics and Probability. The reviewer writes:&quot;&lt;em&gt;This paper is an attempt to justify and derive the Boltzmann equation by the use of the &quot;master equation&quot; [A. Nordsieck, W. E. Lamb, Jr. and G. E. Uhlenbeck, Physica 7 (1940), 344–360; MR0007519 (4,152b); A. J. F. Siegert, Phys. Rev. (2) 76 (1949), 1708–1714] which governs the time rate of change of the N-particle distribution function. The master equation is linear and further, incorporates Boltzmann's Stosszahlansatz [molecular chaos]&lt;/em&gt;&quot;. The use of scare quotes indicates that the term was not yet established then, and the cited Nordsieck-Lamb-Uhlenbeck article uses &quot;master function&quot; also with scare quotes:&quot;&lt;em&gt;If now a single electron of energy E0 is incident normally upon a plate of homogeneous matter, the nature of the emerging shower of electrons is specified by the &quot;master function&quot; giving the probability that, after crossing a thickness x of matter, any given number of electrons emerge in any given energy interval&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are no mentions of &quot;master&quot; in the relevant sense before 1940, but the use of &quot;master&quot; in the review of Lehmer's 1939 article indicates that the word was used for something like &quot;main&quot; or &quot;all-encompassing&quot; rather than &quot;masterful&quot;:&quot;&lt;em&gt;The automatic nature of modern card punching equipment is responsible for the remarkable fact only one error was detected in the master set for the new edition in the final comparison with the original edition&lt;/em&gt;&quot;. Brahana's 1940 article uses it in the same sense:&quot;&lt;em&gt;This group is called the master group for k, since every group of the class considered with the same k is a homomorphic picture of the master group&lt;/em&gt;&quot;. Translation of &quot;master equation&quot; to some languages literally means &quot;main equation&quot;. This is plainly because it usually is the underlying mathematical device one works with, from which more specific properties and characteristics are derived, so it &quot;rules&quot; over them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-03-13T04:01:04.017" LastActivityDate="2016-03-13T04:01:04.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3376" PostTypeId="2" ParentId="1816" CreationDate="2016-01-27T10:59:02.663" Score="12" Body="&lt;p&gt;No. Aristotle was not necessarily wrong. This is in substance &lt;a href=&quot;https://en.wikipedia.org/wiki/Carlo_Rovelli&quot; rel=&quot;noreferrer&quot;&gt; Carlo Rovelli's&lt;/a&gt; view in &lt;a href=&quot;http://arxiv.org/abs/1312.4057&quot; rel=&quot;noreferrer&quot;&gt; &lt;em&gt;Aristotle’s Physics: a Physicist’s Look&lt;/em&gt;&lt;/a&gt;. As the abstracts announces it&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Aristotelian physics is a correct and non-intuitive approximation of Newtonian physics in the suitable domain (motion in fluids), in the same technical sense in which Newton theory is an approximation of Einstein’s theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If one agrees that falling occurs in a fluid, then it is not different from 'sinking'. Heavier bodies sink faster. (Buoyancy which can play a crucial role is also due to gravity).&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-01-27T10:59:02.663" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3377" PostTypeId="2" ParentId="3372" CreationDate="2016-01-27T21:19:23.773" Score="-1" Body="&lt;p&gt;According to Wikipedia: &quot;one of the first references to $G$ is in 1873, 75 years after Cavendish's work.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton assumed an inverse square law, as had already been proposed.  Inverse square laws are usually due to spherical propagation: something is emitted in all directions from a point, so that at later times it is distributed on the surface of a sphere.  This implies velocity, which Newton ignored (or assumed was too fast to mention).  He could have conjectured that gravity was spherically propagated, and used $4\pi\cdot r \cdot r,$ instead of just $r\cdot r.$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G is not merely a constant of proportionality.  It has units which deserve further attention.  It is roughly ($c\cdot c \cdot R / M)/(4\pi),$ where $c=3\mathrm{E}8 ~\textrm{m/s}$ = speed of light (or gravity); $R=4.6\mathrm{E}26~\textrm{meters}$ = radius of visible universe; $M=3\mathrm{E}52~\textrm{kg};$ and $4\pi$ is the neglected spherical factor. &lt;/p&gt;&#xA;" OwnerUserId="3589" LastEditorDisplayName="user1709" LastEditDate="2016-02-13T09:22:56.020" LastActivityDate="2016-02-13T09:22:56.020" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3378" PostTypeId="1" CreationDate="2016-01-29T01:18:29.713" Score="4" ViewCount="404" Body="&lt;p&gt;I'm interested in the history of ellipses. When were they discovered, what uses (if any) did they have before the true shape of orbits were found (by Kepler I think)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are some interesting ways to construct an ellipse. The common way is to anchor a string at the two focii and trace a circle with a pencil, but there is another way called a &lt;a href=&quot;https://en.wikipedia.org/wiki/Trammel_of_Archimedes&quot; rel=&quot;nofollow&quot;&gt;trammel of Archimedes&lt;/a&gt; that looks like it might be useful for something.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did the ancients ever construct ellipses or ellipsoids for anything useful? Did they want to but were just too difficult to build?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2016-01-30T02:11:27.497" Title="What were the applications of ellipses before elliptical orbits were discovered?" Tags="|euclidean-geometry|engineering|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3379" PostTypeId="2" ParentId="3378" CreationDate="2016-01-29T02:25:40.833" Score="2" Body="&lt;p&gt;I am not going to trace back to the antiquity the whole history of ellipses and the instruments that drew them; I trust that more knowledgeable users would comment on Meneachmus, Proclus and others. Instead, let me offer some clear purpose for which ellipses were (and still are) used, namely for depicting circles in perspective in paintings and drawings. One of the pioneers of this use was Albrecht Dürer (1471-1528; hence he comes before Kepler, who was born in 1571). He even found a method of constructing an ellipse as a section of a right circular cone (which however was based on a mistaken assumption). More information about this, along with pictures from Dürer's &quot;Treatise on mensuration&quot; can be found e.g. in the article by Frank J. Swetz and Victor J. Katz,&quot;`Mathematical treasures. Albrecht Dürer's treatise on mensuration&quot;, at &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/mathematical-treasures-albrecht-durers-treatise-on-mensuration&quot; rel=&quot;nofollow&quot;&gt;this link&lt;/a&gt; to the (free) online magazine &quot;Convergence&quot; published by MAA:&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="3434" LastEditDate="2016-01-30T02:11:27.497" LastActivityDate="2016-01-30T02:11:27.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3380" PostTypeId="1" CreationDate="2016-01-28T16:30:02.203" Score="6" ViewCount="329" Body="&lt;p&gt;I was working on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Morse_code&quot; rel=&quot;nofollow&quot;&gt;Morse code Wikipedia article &lt;/a&gt;, when I noticed something strange.&#xA;&lt;br&gt;&#xA;There might be a reason, but I can't find it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, the code symbol &lt;code&gt;...-.&lt;/code&gt; is not used, however the code symbol &lt;code&gt;-----&lt;/code&gt; is. &lt;strong&gt;Why?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first symbol is shorter (&lt;code&gt;11&lt;/code&gt;) than the second one (&lt;code&gt;19&lt;/code&gt;).&#xA;&lt;br&gt;&#xA;This is not the only code symbol unused.&lt;/p&gt;&#xA;" OwnerDisplayName="TomatoPink" LastEditorUserId="2277" LastEditDate="2019-10-11T13:19:50.133" LastActivityDate="2021-10-02T14:02:18.680" Title="Why doesn't Morse code try to achieve more compression?" Tags="|information-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3381" PostTypeId="2" ParentId="3380" CreationDate="2016-01-28T17:07:26.133" Score="10" Body="&lt;p&gt;Three possible reasons:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;code&gt;...-.&lt;/code&gt; is used by a &lt;a href=&quot;http://en.wikipedia.org/wiki/Prosigns_for_Morse_code#Table_of_Morse_Prosigns&quot; rel=&quot;noreferrer&quot;&gt;prosign&lt;/a&gt; meaning &lt;em&gt;understood&lt;/em&gt;;&lt;/li&gt;&#xA;&lt;li&gt;using &lt;code&gt;-----&lt;/code&gt; for the digit 0 makes the encoding of digits regular, thus easier to memorize;&lt;/li&gt;&#xA;&lt;li&gt;&lt;code&gt;-----&lt;/code&gt; for the digit 0 could be replaced by the shorter &lt;code&gt;---&lt;/code&gt; for the letter O, in contexts where it is clear that a digit is thought.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I conjecture that 2 is the historical reason.&lt;/p&gt;&#xA;" OwnerUserId="3594" OwnerDisplayName="fgrieu" LastEditorUserId="3594" LastEditDate="2016-01-30T11:15:46.947" LastActivityDate="2016-01-30T11:15:46.947" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3382" PostTypeId="2" ParentId="3380" CreationDate="2016-01-29T11:55:57.017" Score="4" Body="&lt;p&gt;Morse code was defined by Morse himself, and later (somewhat) extended. Using letter frequency for dense coding was a consideration (&lt;code&gt;.&lt;/code&gt; for the commonest letter &lt;code&gt;e&lt;/code&gt;, and on down), trying to make it easy to learn clearly was a goal, and (some) resistance to garbling; but all this happened long before Shannon's study of communication or any systematic study of coding, error detection and such.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastEditorUserId="4251" LastEditDate="2021-10-02T14:02:18.680" LastActivityDate="2021-10-02T14:02:18.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3383" PostTypeId="2" ParentId="3378" CreationDate="2016-01-29T13:34:10.227" Score="3" Body="&lt;p&gt;This question has been discussed on Math overflow:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653&quot;&gt;https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is about conic sections in general. In my answer I gave them as example&#xA;of &quot;useless mathematics which became useful centuries later&quot;. Some tried to challenge this statement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the conjectures mentioned that conic sections were discovered when the Greeks tried to develop a theory of sundials.&#xA;(Normally you get a hyperbola rather than ellipse, an ellipse will arise only in polar regions on the days when Sun never sets). This conjecture was much discussed in the history literature but on my opinion this is an unlikely explanation how conic sections were discovered. Another conjecture says that Menachem used conic sections for doubling the cube. This conjecture is nore plausible. See the references in the link above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. There is of course a well-known legend about Archimedes burning Roman ships with parabolic mirrors, and it is clear that this is only a legend: no serious scientist believes that this can be actually performed. (But you can see this in the movies).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-01-29T20:02:45.377" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3386" PostTypeId="1" CreationDate="2016-01-31T14:26:48.550" Score="2" ViewCount="126" Body="&lt;p&gt;Detlef Laugwitz is mainly known for his by-now-standard biography of Riemann:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Detlef Laugwitz, &lt;a href=&quot;https://books.google.it/books?id=7U6njBdp_yoC&amp;amp;pg=PA219&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernhard Riemann 1826-1866 : Turning Points in the Conception of Mathematics&lt;/a&gt; (1996 - German ed.1996)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;as noted in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2624/what-were-riemanns-other-two-submissions-for-his-habilitation/2625#2625&quot;&gt;this answer&lt;/a&gt;. He has written on Cauchy, as for example this text:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laugwitz, Detlef. Infinitely small quantities in Cauchy's textbooks. Historia Math. 14 (1987), no. 3, 258–274.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How is Laugwitz's work on Cauchy viewed by historians today and why? I am interested in the &lt;em&gt;historians&lt;/em&gt;' own position on the matter (rather than the opinions of SE editors).&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-02-18T17:20:43.630" Title="How is Laugwitz's work on Cauchy viewed by historians?" Tags="|mathematics|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3387" PostTypeId="2" ParentId="3364" CreationDate="2016-01-31T14:46:42.130" Score="0" Body="&lt;p&gt;Apparently the visibly good relations between Euler and Lagrange were all to Euler's credit.  Euler always spoke well of Lagrange, whereas the latter permitted himself dismissive comments concerning the former; see &lt;a href=&quot;http://arxiv.org/abs/1409.4736&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-01-31T14:46:42.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3388" PostTypeId="1" CreationDate="2016-02-01T11:00:11.763" Score="2" ViewCount="408" Body="&lt;p&gt;Egyptians and Babylonians must have had some ideas about primes but what are the earliest mentions or comments? Where is the first list of primes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After Burkert's work on Pythagoreanism an answer about 'unidentified pythagoreans' would not count much. &#xA;Euclide's proof (about the infinity of primes IX.20) is a landmark in mathematics well known enough to make later writings historically unintersting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically did early Greek philosophers mention primes and are there possibly different words and/or confusions?&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-06-08T15:28:06.900" Title="What are the earliest mentions of Prime Numbers?" Tags="|reference-request|ancient-greece|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3389" PostTypeId="2" ParentId="3388" CreationDate="2016-02-01T12:39:09.793" Score="4" Body="&lt;p&gt;For some references, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Heath, &lt;a href=&quot;https://books.google.it/books?id=J-kaAgAAQBAJ&amp;amp;pg=PA72&quot; rel=&quot;nofollow&quot;&gt;A History of Greek Mathematics. Vol I&lt;/a&gt; (1921), page 72 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Prime&lt;/em&gt; or &lt;em&gt;incomposite&lt;/em&gt; numbers and &lt;em&gt;secondary&lt;/em&gt; or &lt;em&gt;composite&lt;/em&gt; numbers are distinguished in a fragment by &lt;a href=&quot;https://en.wikipedia.org/wiki/Speusippus&quot; rel=&quot;nofollow&quot;&gt;Speusippus&lt;/a&gt; based upon works of &lt;a href=&quot;https://en.wikipedia.org/wiki/Philolaus&quot; rel=&quot;nofollow&quot;&gt;Philolaus&lt;/a&gt; . We are told [ &lt;a href=&quot;https://en.wikipedia.org/wiki/Iamblichus&quot; rel=&quot;nofollow&quot;&gt;Iamblichus&lt;/a&gt;, &lt;em&gt;in Nicom.&lt;/em&gt;] that &lt;a href=&quot;https://en.wikipedia.org/wiki/Thymaridas&quot; rel=&quot;nofollow&quot;&gt;Thymaridas&lt;/a&gt; called a prime number &lt;em&gt;rectilinear&lt;/em&gt;, the ground being that it can only be set out in one dimension (since the only measure of it, excluding the number itself, is $1$). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-02-01T12:39:09.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3390" PostTypeId="1" CreationDate="2016-02-01T16:19:07.453" Score="4" ViewCount="133" Body="&lt;p&gt;In 1202 Leonardo of Pisa, later known as Fibonacci, wrote, in his book &quot;liber abbaci&quot;, his explanation of a means of expressing numbers, and manipulating them, that gave us arithmetic as we know it today. My question is: how was that book distributed, since the printing press did not arrive in Europe for another two hundred years? How many copies were produced?&lt;/p&gt;&#xA;" OwnerUserId="2444" LastEditorUserId="26" LastEditDate="2016-02-01T23:35:48.723" LastActivityDate="2016-02-02T12:28:20.227" Title="How was Leonardo of Pisa's book liber abbaci distributed?" Tags="|biographical-details|number-theory|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3391" PostTypeId="2" ParentId="3390" CreationDate="2016-02-02T10:40:49.193" Score="1" Body="&lt;p&gt;Some English Sources for &lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;nofollow&quot;&gt;Leonardo Piasno&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Laurence Sigler (editor), &lt;a href=&quot;https://books.google.it/books?id=PilhoGJeKBUC&amp;amp;pg=PR4&quot; rel=&quot;nofollow&quot;&gt;Fibonacci’s Liber Abaci&lt;/a&gt; (2002): unfortunately, the &lt;em&gt;Introduction&lt;/em&gt; does not give details on the manuscripts. Only :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Liber abbaci&lt;/em&gt;, or the Book of Calculation, appeared first in 1202, and then&#xA;  again in a second version in 1228. [...] This English translation is prepared from Baldassarre Boncompagni's &lt;a href=&quot;http://lhldigital.lindahall.org/cdm/compoundobject/collection/math/id/8734/rec/1&quot; rel=&quot;nofollow&quot;&gt;Latin edition of 1857&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Barnabas Hughes (editor), &lt;a href=&quot;https://books.google.it/books?id=aEOZThAbCKIC&amp;amp;pg=PR11&quot; rel=&quot;nofollow&quot;&gt;Fibonacci's De Practica Geometrie&lt;/a&gt; (2008): good introduction.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jens Høyrup, &lt;a href=&quot;https://books.google.it/books?id=XZhPC1Y-SOUC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Jacopo da Firenze's Tractatus Algorismi and Early Italian Abbacus Culture&lt;/a&gt; (2007): &lt;strong&gt;Fibonacci and the Abbaco&lt;/strong&gt;, page 41-44.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-02-02T12:28:20.227" LastActivityDate="2016-02-02T12:28:20.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3392" PostTypeId="2" ParentId="3324" CreationDate="2016-02-03T17:13:23.020" Score="8" Body="&lt;p&gt;As noted by Victor J. Katz, the source of Newton's introduction of infinite series is unending decimals.  Victor J. Katz provides a quotation from Newton where the latter explicitly states that unending decimals were the source of his inspiration, and express wonder why nobody else tried to study these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unending decimals were pioneered by Simon Stevin.  While Stevin is not mentioned explicitly by Newton in this quotation, decimal notation and unending decimals were by then in increasingly common use among European mathematicians, even though it took a while before they become commonplace.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore the sources of infinite series can be sought in the work of Stevin, as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stevin also pioneered the viewpoint of considering all numbers (rational or irrational) as being given by their unending decimal representation, in his book &lt;em&gt;L'Arithmetique&lt;/em&gt;.  He was thus a pioneer of the real number system, as well, nearly three centuries before Cantor and Dedekind.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-02-03T17:13:23.020" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3393" PostTypeId="1" AcceptedAnswerId="3397" CreationDate="2016-02-04T03:31:18.037" Score="3" ViewCount="115" Body="&lt;p&gt;With enough experimentation to test observational results, a science could be thought of as 'maturing' into prediction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some sciences, however, remain absent from meaningful experiment due to practical reasons (Geologists cannot 'experiment' with plate tectonics, at least before the days of global models).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of these 'observational' sciences, which has progressed farthest?&lt;/p&gt;&#xA;" OwnerUserId="3627" LastActivityDate="2016-03-06T02:43:40.953" Title="What is the most mature observational science?" Tags="|earth-sciences|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3395" PostTypeId="1" AcceptedAnswerId="3396" CreationDate="2016-02-04T17:34:42.613" Score="3" ViewCount="610" Body="&lt;p&gt;Any research about real measurements with objects moving faster than me - like GPS, for example - leads to a mix of different sideway phenomenons like different gravitational potentials on Earth and in orbit or accelerations during liftoff and landing.  I'm curious; was Einstein had able to answer this thought experiment?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_special_relativity&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Langevin solved the paradox by alluding to the fact that one twin accelerates and changes direction&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;, so perhaps it is more precise to ask about when the twin paradox was simplified to &quot;a travelling twin and a twin in rest&quot;.&#xA;Perhaps the question more specific has to be: When the simplification of the twin thought experiment to the twin paradox took place? When did this happens?&lt;/p&gt;&#xA;" OwnerUserId="3223" OwnerDisplayName="HolgerFiedler" LastEditorUserId="3223" LastEditDate="2016-02-05T16:10:04.383" LastActivityDate="2016-02-05T16:10:04.383" Title="When was the twin paradox first formulated?" Tags="|physics|relativity-theory|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3396" PostTypeId="2" ParentId="3395" CreationDate="2016-02-04T22:34:38.663" Score="4" Body="&lt;p&gt;The alluded to Wikipedia has a rather well referenced &lt;a href=&quot;https://en.wikipedia.org/wiki/Twin_paradox#History&quot; rel=&quot;nofollow&quot;&gt;article on Twin paradox&lt;/a&gt; specifically. Einstein himself formulated the &quot;paradox&quot; in terms of clocks  already back in 1905. He considered it just &quot;peculiar&quot; however rather than a paradox, because it led to no contradictions that needed to be resolved. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1911 Einstein replaced clocks with living organisms, perhaps to make the peculiarity more striking:&quot;&lt;em&gt;If we placed a living organism in a box... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations&lt;/em&gt;&quot;. The same year Langevin replaced organisms with human travelers, and von Laue turned them into twins.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-02-04T22:34:38.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3397" PostTypeId="2" ParentId="3393" CreationDate="2016-02-05T00:12:27.910" Score="5" Body="&lt;p&gt;There can be no doubt that the the most &quot;mature&quot; observational science is astronomy. Whatever is the exact meaning of &quot;mature&quot;.&#xA;First, it is the oldest science among all exact sciences. There was no physics to talk about, (not speaking of chemistry biology, geology) when astronomy was already quite mature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, astronomy makes predictions. Very precise predictions, and for very long time. The most exact measurements available also belong to astronomy. Astronomy can predict events that will happen thousand years from now with accuracy to a fraction of a second. Time measurements in astronomy are the most precise measurements (in terms of the number of significant digits) in any science.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-02-05T00:12:27.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3398" PostTypeId="1" CreationDate="2016-02-05T07:41:39.960" Score="5" ViewCount="1782" Body="&lt;p&gt;Henry Cavendish &lt;a href=&quot;https://en.wikipedia.org/wiki/Hydrogen#History&quot; rel=&quot;nofollow&quot;&gt;isolated hydrogen&lt;/a&gt; in 1766, but I want to know if he knew the gas was actually $H_2$ instead of just a plain $H$ gas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would make a major difference in the atomic weights, which would have implications for the periodic table once people like Mendeleev noticed that atomic weights were usually integer multiples of hydrogen's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If he knew it was $H_2$ gas, how did he know? If not, then when and how did we figure out that hydrogen gas is actually two $H$ atoms in one molecule?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on the answer I may ask the same thing for Oxygen and Nitrogen.&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="55" LastEditDate="2016-09-14T00:04:38.760" LastActivityDate="2016-09-14T00:04:38.760" Title="How did chemists figure out atomic weights, when hydrogen, etc., were first discovered?" Tags="|discoveries|chemistry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3399" PostTypeId="2" ParentId="3398" CreationDate="2016-02-05T17:21:20.150" Score="6" Body="&lt;p&gt;The question would not have made sense in 1766, and for a while after. At the time chemistry was still transitioning from the dominance of alchemic ideas, notation and terminology. Lavoisier's 1789 list of elements for example still bears the hallmarks of Aristotle's four elements with multiple &quot;earths&quot; and &quot;airs&quot;, and includes caloric fluid (heat) and &quot;dephlogisticated air&quot;. Hydrogen is called &quot;base of inflammable air&quot;. Some modern ideas about chemical composition start to appear in the work of Cullen, who drew some rudimentary reaction diagrams in 1757, and his student Black even drew schematic circles (later &quot;molecules&quot;) split into constitutive parts (later &quot;atoms&quot;). Swedish chemist Bergman systematized these ideas in his influential 1775 chemistry textbook, which included 64 diagrams of chemical reactions. However, the theory was not yet sophisticated enough to assign (or even make sense of) numbers to constitutive parts, and distinguish say $H_2O$ from $HO$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The atomic theory in a form resembling classical chemistry was only proposed by Dalton in 1803. In particular, he made the distinction between &quot;simple atoms&quot; (elements), &quot;compound atoms&quot; (molecules), and &quot;complex atoms&quot; (complex molecules), which would make the question meaningful. Dalton took the atomic weight of hydrogen as unity and determined the ratios of elements in some compounds based on it, e.g. the ratio for nitrous anhydride was 2 to 3 corresponding to modern $N_2O_3$. But his symbol for water $\bigodot\!\!\bigcirc$, with the circles representing hydrogen and oxygen respectively, indicates that he did not always have such information. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What enabled determination of correct chemical formulas was Avogadro's law formulated in 1811: equal volumes of gases contain equal numbers of molecules under standard conditions. This meant that ratios of molecular masses could be calculated by measuring gas samples. Moreover, Avogadro first clearly stated that even element molecules can be composed of several atoms, allowing for the possibility of $H_2$ not contemplated by Dalton. Expanding on the work of  Avogadro Gaudin published many familiar chemical formulas in 1833, including the ones for $H_2$, $N_2$, $O_2$ and $H_2O$. The notation was also essentially modern, Berzelius introduced it to replce Dalton's in 1813-1814. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.vanderkrogt.net/elements/chemical_symbols.php&quot;&gt;Development of the chemical symbols and the Periodic Table&lt;/a&gt;, &lt;a href=&quot;http://www.eoht.info/page/History+of+chemical+equations&quot;&gt;History of chemical equations&lt;/a&gt;, &lt;a href=&quot;http://www.eoht.info/page/History+of+chemical+bonding+theory&quot;&gt;History of chemical bonding theory&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_molecular_theory&quot;&gt;History of molecular theory&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-05T21:48:33.150" LastActivityDate="2016-02-05T21:48:33.150" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3400" PostTypeId="1" AcceptedAnswerId="3402" CreationDate="2016-02-06T10:55:21.003" Score="6" ViewCount="598" Body="&lt;p&gt;I found &lt;a href=&quot;https://books.google.ch/books?id=eKYSBwAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=Hitler%E2%80%99s+Uranium+Club:+The+Secret+Recordings+at+Farm+Hall&amp;amp;hl=fr&amp;amp;sa=X&amp;amp;ved=0ahUKEwiP6vbO--LKAhUKPxoKHa-HD90Q6AEIIDAA#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;there&lt;/a&gt; (on page 98) the following quote :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Heisenberg: During the war I had five calls for help in cases where people were murdered by our people. One was Soloman, Hoffman's son-in law. I could do nothing in his case as he had already been killed when I got the letter. The second one was Cousyns the Belgian cosmic ray man; he disappeared in a Gestapo camp and I couldn't even find out through Himmler's staff whether he was alive or dead. I presume he is dead too. Then there was the mathematician &lt;strong&gt;Cammaille&lt;/strong&gt;; I tried to do something about him through Sethel but it was no good he was shot. Then from among the Polish professors there was a logistician with a Jewish name — and then with the other Poles the following happened: his name was &lt;strong&gt;Schouder&lt;/strong&gt;, a mathematician. He had written to me and I had put out feelers in order to see what could be done. I wrote to Scholz who had something to do with Poland. [...] I heard nothing more about Schouder and I have now been told that he was murdered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does anyone know who Cammaille and Schouder are ? I tried to search some information about them but I found nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you !&lt;/p&gt;&#xA;" OwnerUserId="3454" LastEditorUserId="26" LastEditDate="2016-02-06T22:05:31.807" LastActivityDate="2021-02-13T14:25:02.833" Title="Who are the mathematicians Cammaille and Schouder, mentioned by Heisenberg when recalling mathematicians who died during the Second World War?" Tags="|biographical-details|mathematicians|nazi-germany|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3401" PostTypeId="2" ParentId="3400" CreationDate="2016-02-06T11:14:18.670" Score="7" Body="&lt;p&gt;For the second one, see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Maria Georgiadou, &lt;a href=&quot;https://books.google.it/books?id=lpprCQAAQBAJ&amp;amp;pg=PA584&amp;amp;lpg=PA584&quot;&gt;Constantin Carathéodory: Mathematics and Politics in Turbulent Times&lt;/a&gt; (2013), page 584:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the mathematician Schauder [...]. The name is wrongly spelled as &quot;Schouder&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Juliusz_Schauder&quot;&gt;Juliusz Paweł Schauder&lt;/a&gt; (September 21, 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland).&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-02-06T11:14:18.670" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3402" PostTypeId="2" ParentId="3400" CreationDate="2016-02-06T13:48:46.233" Score="8" Body="&lt;p&gt;Julius Pawel Schauder was a very famous mathematician. He worked in Lwow (Lviv, Lvov, Lemberg). He is remembered for the &amp;quot;Leray-Schauder degree&amp;quot;, and&#xA;&amp;quot;Leray-Schauder Theorem&amp;quot;, this is a cornerstone of&#xA;the important area called Non-linear Functional Analysis,&#xA;&amp;quot;Schauder basis&amp;quot; and many other things. He worked in functional analysis, and in fact was one of the main creators of this discipline.&lt;/p&gt;&#xA;&lt;p&gt;When the German troops occupied Lwow, he was confined in the ghetto. He wrote letters to German mathematicians, asking to help him. The story about these letters is told in detail in the book on Caratheodory cited in Mauro Allegranza answer. Nobody did anything to help him, and he perished. It is not known exactly when and where he died, but apparently he was deported with the others to&#xA;one of the extermination camps.&lt;/p&gt;&#xA;&lt;p&gt;There is a Wikipedia article about him, and also a biography in MacTutor.&lt;/p&gt;&#xA;&lt;p&gt;As his name is misspelled in the citation you give, I suppose that the other name is also misspelled. But I cannot determine who is the other person Heisenberg mentioned. There is a (very incomplete) list of&#xA;&amp;quot;Mathematicians who died in the Holocaust&amp;quot; in Wikipedia. A much more complete list is included in the book&#xA;by A. W. Goodman, Univalent functions, chapter 18. (Mariner Publishing Co., Inc., Tampa, FL, 1983.) I also could not determine who was &amp;quot;a logistician with a Jewish name&amp;quot;. Shortly after occupation of Lwow, the Germans killed 25 Polish professors, see&#xA;&amp;quot;Massacre of Lviv Professors&amp;quot; on Wikipedia, but there is no logician among them.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-02-13T14:25:02.833" LastActivityDate="2021-02-13T14:25:02.833" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="3403" PostTypeId="1" CreationDate="2016-02-07T00:19:07.120" Score="1" ViewCount="114" Body="&lt;p&gt;I read in some places that Daniel Bernoulli anticipated van der Waals equation by a century in section 10 of his Hydrodynamica. Most sources state that he only discovered one idea in the model; that of taking into account the size of the particles. But did he also consider the attraction between the particles? I'm unable to read his Latin work so I need someone to answer.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2905" LastEditDate="2016-02-07T09:37:28.453" LastActivityDate="2016-02-07T13:50:22.690" Title="Did Daniel Bernoulli state van der Waals equation?" Tags="|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3404" PostTypeId="1" AcceptedAnswerId="3408" CreationDate="2016-02-07T08:31:12.490" Score="13" ViewCount="4153" Body="&lt;p&gt;It is so funny that science never develops as in the textbooks. Bose only introduced his statistics in 1924, so Planck could not possibly have used it to derive the radiation formula in 1900. So how did he do it? Also, are there any translations of his original paper? &lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="4251" LastEditDate="2020-06-08T06:00:30.043" LastActivityDate="2021-04-09T09:49:57.870" Title="How did Planck derive the black body radiation formula without using the Bose statistics?" Tags="|physics|discoveries|quantum-mechanics|translations|planck|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3405" PostTypeId="2" ParentId="3403" CreationDate="2016-02-07T13:43:33.370" Score="1" Body="&lt;p&gt;&lt;em&gt;References&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Section 10, §1-8, of Daniel Bernoulli, &lt;a href=&quot;https://books.google.it/books?id=3yRVAAAAcAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Hydrodynamica sive de viribus et motibus fluidorum commentarii&lt;/a&gt;, Agentorati, 1738, are discussed into:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Clifford Truesdell, &lt;a href=&quot;https://books.google.it/books?id=_crsCAAAQBAJ&amp;amp;pg=PA277&quot; rel=&quot;nofollow&quot;&gt;Essays in the History of Mechanics&lt;/a&gt; (1968), &lt;strong&gt;Ch.VI. Early Kinetic Theories of Gases&lt;/strong&gt;; see page 277-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See also: Clifford Truesdell, &lt;a href=&quot;https://books.google.it/books?id=sT1cHAAACAAJ&quot; rel=&quot;nofollow&quot;&gt;Rational fluid mechanics. 1687-1765&lt;/a&gt; (1954), &lt;strong&gt;Part IV. The hydrodynamics of Daniel and John BERIOULLI&lt;/strong&gt;, page XXIII, as well as: Julian Simon Calero, &lt;a href=&quot;https://books.google.it/books?id=pyZIotXTX2sC&amp;amp;pg=PA320&quot; rel=&quot;nofollow&quot;&gt;The genesis of fluid mechanics. 1640-1780&lt;/a&gt; (2008), page 320, where it is discussed the chapter of &lt;em&gt;Hydrodynamica&lt;/em&gt; subtitled ‘Concerning Properties and Motions of Elastic Fluids, but especially of Air’.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-02-07T13:50:22.690" LastActivityDate="2016-02-07T13:50:22.690" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3406" PostTypeId="1" AcceptedAnswerId="3407" CreationDate="2016-02-07T13:59:28.320" Score="1" ViewCount="1935" Body="&lt;p&gt;A typical quote used to explain the theory of evolution by natural selection is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;I&gt;&quot;Variation is a feature of natural populations and every population produces more progeny than its environment can manage. The consequences of this overproduction is that those individuals with the best genetic fitness for the environment will produce offspring that can more successfully compete in that environment. Thus the subsequent generation will have a higher representation of these offspring and the population will have evolved.&quot;&lt;/I&gt; - Charles Darwin&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does anyone know the exact source for this quote (i.e. where Darwin wrote or said it)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. I have tried 'On the Origin of Species' and some of his other works but I can't seem to find it.&lt;/p&gt;&#xA;" OwnerUserId="1994" LastEditorUserId="4251" LastEditDate="2020-07-13T01:54:19.730" LastActivityDate="2020-07-13T01:54:19.730" Title="Darwin's theory of evolution quote?" Tags="|biology|quote|evolution|darwin|biologists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3407" PostTypeId="2" ParentId="3406" CreationDate="2016-02-07T16:03:00.023" Score="2" Body="&lt;p&gt;&lt;em&gt;(What I'm trying to say is too long for a comment, but I'm not sure it's an answer...)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are you sure this is an exact quote? &#xA;A fast research in Google books of the text or of part of it has provided no results. The concept expressed is very similar to the following, from &lt;a href=&quot;http://darwin-online.org.uk/content/frameset?pageseq=77&amp;amp;itemID=F373&amp;amp;viewtype=image&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On the Origin of Species&lt;/em&gt;, Chapter III: &lt;em&gt;Struggle for existence&lt;/em&gt;, page 61:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Owing to this struggle for life, any variation, however slight and from whatever cause proceeding, if it be in any degree profitable to an individual of any species, in its infinitely complex relations to other organic beings and to external nature, will tend to the preservation of that individual, and will generally be inherited by its offspring. The offspring, also, will thus have a better chance of surviving, for, of the many individuals of any species which are periodically born, but a small number can survive. I have called this principle, by which each slight variation, if useful, is preserved, by the term of Natural Selection, in order to mark its relation to man's power of selection.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;On a less serious note...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is &lt;a href=&quot;http://www.smbc-comics.com/index.php?id=71&quot; rel=&quot;nofollow noreferrer&quot;&gt;something completely original&lt;/a&gt;: &lt;img src=&quot;https://i.stack.imgur.com/Ygf0m.jpg&quot; alt=&quot;http://www.smbc-comics.com/index.php?id=71&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;alt-text:&lt;/em&gt; &lt;img src=&quot;https://i.stack.imgur.com/jpoSs.jpg&quot; alt=&quot;alt-text&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="3006" LastEditorUserId="3006" LastEditDate="2016-02-23T07:33:21.943" LastActivityDate="2016-02-23T07:33:21.943" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3408" PostTypeId="2" ParentId="3404" CreationDate="2016-02-08T20:17:39.380" Score="19" Body="&lt;p&gt;Bose derived the black body radiation formula in early 1924 by considering the ideal gas of light quanta. Nothing could be further from Planck's mind in 1900. The idea of light quanta did not appear until Einstein's 1905 photoeffect paper, in 1908 he thought of them as vortices in EM field rather than localized energy packets, and in 1911 remarked that the idea &amp;quot;&lt;em&gt;does not seem reconcilable with the experimentally verified consequences of the wave theory&lt;/em&gt;&amp;quot;. Planck, Nernst, Rubens and Warburg recommending Einstein's admission to the Prussian Academy of Sciences wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;That he might sometimes have overshot his target in his speculations, as for example in his light quantum hypothesis, should not be counted against him too much&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even after Bohr's 1913 model of the atom Einstein was one of very few who took light quanta seriously. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/2257/what-was-different-about-plancks-quantization-of-light-compared-to-einsteins/2259#2259&quot;&gt;What was different about Planck&amp;#39;s quantization of light compared to Einstein&amp;#39;s?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Planck treated radiation classically, where statistics came in was in what emitted it. Like many physicists he was partly influenced by Mach's and Ostwald's skepticism about atoms, so he assumed that the radiation is produced by some ideal oscillators of unknown nature. In 1899 following the ideas of kinetic theory he found an expression for the entropy of his oscillators that reproduced Wien's phenomenological radiation law, but experiments quickly showed it to be incorrect at low frequencies.&lt;/p&gt;&#xA;&lt;p&gt;Planck originally obtained the correct law in October 1900 by simply modifying his entropy expression, without any statistics. Two months later he was forced to turn to statistics to justify his entropy formula, and formulated the &amp;quot;Boltzman equation&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$S=k\ln W$&lt;/span&gt;. To find the disorder &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; he had to count the number of ways that a given energy can be distributed among the oscillators. It is for that that he needed &amp;quot;energy droplets&amp;quot;, which were supposed to be fictitious and go away when the continuous limit was taken. But instead of taking the limit Planck was forced to introduce a &amp;quot;new constant of nature&amp;quot;, to fix the amount of energy in the droplets, &amp;quot;&lt;em&gt;a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result&lt;/em&gt;&amp;quot;, as he wrote in 1931. See &lt;a href=&quot;http://www.math.lsa.umich.edu/%7Ekrasny/math156_article_planck.pdf&quot; rel=&quot;noreferrer&quot;&gt;Kragh's Max Planck: the Reluctant Revolutionary&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Planck continued to not give it much thought for five years, and he certainly did not think that his counting was in conflict with classical physics. The source of discreteness was unclear, Lorentz, who was first to give it a thought in 1903, subsequently pondered various classical mechanisms (in interactions of the oscillators with ether, atoms, electrons, etc.). Rayleigh in June 1900 argued that equipartition of energy in classical statistical mechanics leads to a discrepancy with radiation experiments, and in a 1905 paper with Jeans derived a different radiation formula. As Kragh writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The result is an energy density that keeps on increasing as the frequency gets higher and higher, becoming &amp;quot;catastrophic&amp;quot; in the ultraviolet region. In spite of its prominent role in physics textbooks, the formula played no part at all in the earliest phase of quantum theory. Planck did not accept the equipartition theorem as fundamental, and therefore ignored it&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;ultraviolet catastrophe&amp;quot; only became an issue when Lorentz showed in 1908 that the Rayleigh-Jeans law obtains even under the most general (classical) assumptions about a system of atoms, electrons and ether. The nickname was introduced by Ehrenfest at Solvay in 1911. So much for textbooks. See &lt;a href=&quot;https://web.archive.org/web/20210208082442/http://www.einstein.caltech.edu/news/Kox-HistExactSci.2013.pdf&quot; rel=&quot;noreferrer&quot;&gt;Kox's Hendrik Antoon Lorentz’s Struggle with Quantum Theory&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One last bit of historical peculiarity. Bose was counting microstates in a funny way, which implied heavy statistical dependence between particles, apparently without realizing it. Einstein, to whom Bose sent his paper, did not notice the funny math either until Ehrenfest pointed it out to him later in 1924. Only in the January 1925 paper he finally acknowledged it explicitly with an &amp;quot;explanation&amp;quot; that it is &amp;quot;completely mysterious&amp;quot;, but hey, it gives the right answer. Only after the wave and matrix mechanics were introduced a year later could the funny math be justified by indistinguishability of particles. See &lt;a href=&quot;http://users.physik.fu-berlin.de/%7Epelster/Vorlesungen/WS1213/delbrueck.pdf&quot; rel=&quot;noreferrer&quot;&gt;Delbruck's Was Bose-Einstein Arrived at by Serendipity?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-04-09T09:49:57.870" LastActivityDate="2021-04-09T09:49:57.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3409" PostTypeId="1" CreationDate="2016-02-09T03:19:24.683" Score="3" ViewCount="176" Body="&lt;p&gt;I've recently come across the idea of the &quot;conservation of information&quot; in the so called &lt;a href=&quot;https://en.wikipedia.org/wiki/Black_hole_information_paradox&quot; rel=&quot;nofollow&quot;&gt;Black Hole Information Paradox&lt;/a&gt;. Yet &lt;a href=&quot;http://www.evolutionnews.org/2012/08/conservation_of063671.html&quot; rel=&quot;nofollow&quot;&gt;this article&lt;/a&gt; claims that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Conservation of information is a term with a short history. Biologist Peter Medawar used it in the 1980s to refer to mathematical and computational systems that are limited to producing logical consequences from a given set of axioms or starting points, and thus can create no novel information (everything in the consequences is already implicit in the starting points). His use of the term is the first that I know, though the idea he captured with it is much older. Note that he called it the &quot;Law of Conservation of Information&quot; (see his The Limits of Science, 1984).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2461" LastEditorUserId="26" LastEditDate="2016-02-09T11:47:26.430" LastActivityDate="2020-03-16T18:53:31.773" Title="Who first developed the idea of conservation of information in physics?" Tags="|theoretical-physics|biology|evolution|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3410" PostTypeId="2" ParentId="3409" CreationDate="2016-02-09T09:34:35.810" Score="4" Body="&lt;p&gt;The two notions that you refer to in your post are not necessarily the same: The &quot;conservation of information&quot; that is used as a principle in theoretical physics, particular in gedankenexperiments involving black holes, is more commonly known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Unitarity_%28physics%29&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;unitarity&lt;/em&gt;&lt;/a&gt;. Unitarity became important with the advent of quantum mechanics around 1925 and was investigated rigorously in the late 1920's to early 1930's, resulting in e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Stone&amp;#39;s_theorem_on_one-parameter_unitary_groups&quot; rel=&quot;nofollow&quot;&gt;Stone's theorem on one-parameter unitary groups&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that this has very little to do with the &quot;conservation of information&quot; that your second linked article refers to, which is indeed a relatively &lt;em&gt;new&lt;/em&gt; concept in (evolutionary) biology, it seems.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2016-02-09T09:34:35.810" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3411" PostTypeId="2" ParentId="3372" CreationDate="2016-02-10T14:47:17.960" Score="4" Body="&lt;p&gt;As a constant using ordinary units (e.g., metric), the gravitational constant G doesn't occur until late in the 19th century. Scientists in Newton's time and into the 18th century were quite happy to work in terms of proportionalities and ratios; the constant of proportionality never needed to be written down. For example, one does not see a gravitational constant, in any form, in Cavendish's description of his experiments to &quot;determine the density of the Earth&quot; (1798). This practice of not using a gravitational constant with regard to earthly matters continued throughout much of the 19th century, e.g., Pratt (1855).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One does see something much akin to the Newtonian gravitational constant in the works of Laplace (1799) and Gauss (1809). Using modern nomenclature, Newton's law of gravitation using the gravitational constant of Laplace and Gauss is $$F = k^2 \frac{m_1 m_2}{r^2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key differences between that and $F=G \frac{m_1 m_2}{r^2}$ are that the Laplace's and Gauss's $k$ is the square root of the Newtonian constant, and that the system of units is more apropos to modeling the solar system. Gauss explicitly specified his system of units: One mean solar day as the unit of time, one solar mass as the unit of mass, and one astronomical unit (the mean distance between the Earth and the Sun) as the unit of length. The Gaussian gravitational constant has a numerical value of 0.01720209895, as reported by Gauss. This value persisted as a defined constant until very recently (2012, and perhaps later).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aside: Note that making $k$ a defined constant effectively made the astronomical unit a derived quantity, divorcing it from the size of the Earth's orbit. Keeping $k$ at the value established by Gauss was standard practice throughout the 19th and 20th centuries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The push to measure $G$ using earthly units didn't occur until after physicists saw the value of the metric system and its predecessors, was largely driven by electromagnetism. A flurry of publications occurred late in the 19th century regarding the Newtonian gravitational constant, apparently starting with Cornu and Baille (1873). (The reason I wrote &quot;apparently&quot; is because I can't access that paper, and because Poynting describes that paper as &quot;brief&quot;.) Judging a book by its cover (or a scientific paper by its title), the title &quot;A new determination of the constant of attraction and of the mean density of the Earth&quot; certainly does appear that Cornu and Baille made the connection between assessing the mean density of the Earth and the gravitational constant. Boys (1889) makes the connection very explicit; Boys announced he was going to use a Cavendish experiment to measure the gravitational constant as the primary goal. That this measurement also would yield an estimate of the density of the Earth was secondary.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rspl.royalsocietypublishing.org/content/46/280-285/253&quot; rel=&quot;nofollow&quot;&gt;C.V.Boys (1889), &quot;On the Cavendish experiment&quot;, &lt;em&gt;Proc. R. Soc. London&lt;/em&gt;, 46, 253-268&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=ZrloHemOmUEC&amp;amp;pg=PA353#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;C.V. Boys (1894), &quot;The Newtonian constant of gravitation,&quot; &lt;em&gt;Notices R. Inst.&lt;/em&gt; 14 353-377.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.phys.ufl.edu/courses/phy3221/spring10/1798Cavendish.pdf&quot; rel=&quot;nofollow&quot;&gt;H. Cavendish (1798), &quot;Experiments to determine the Density of the Earth,&quot; &lt;em&gt;Phil. Trans. R. Soc. London,&lt;/em&gt; 88 469-526.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Cornu et B. Baille (1873), &quot;Détermination nouvelle de la constante de l'attraction et de la densité moyenne de la Terre,&quot; &lt;em&gt;Comptes Rendus&lt;/em&gt; lxxvi, 954-8.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://ia801409.us.archive.org/3/items/bub_gb_ORUOAAAAQAAJ/bub_gb_ORUOAAAAQAAJ.pdf&quot; rel=&quot;nofollow&quot;&gt;Gauss (1809), &quot;Theoria motus corporum coelestium in sectionibus conicis solem ambientium&quot;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/treatiseofcelest12lapl&quot; rel=&quot;nofollow&quot;&gt;Laplace (1799), &quot;Traité de mécanique céleste&quot;.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/meandensityofear00poynuoft&quot; rel=&quot;nofollow&quot;&gt;Poynting (1894), &quot;The mean density of the Earth&quot;.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2016-02-10T14:47:17.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3412" PostTypeId="1" CreationDate="2016-02-10T15:33:38.253" Score="4" ViewCount="2418" Body="&lt;p&gt;It is commonplace that it is about 6000 Kelvin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But who came to this value first?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And with what method? Based on the black body radiation theory?&lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="55" LastEditDate="2016-02-11T21:55:19.887" LastActivityDate="2023-07-27T10:04:53.570" Title="Who determined the temperature of the Sun first?" Tags="|physics|astronomy|experimental-physics|thermodynamics|radiation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3413" PostTypeId="2" ParentId="3412" CreationDate="2016-02-11T01:08:43.123" Score="7" Body="&lt;p&gt;As is often the case, nobody was first, but both early and modern estimates of the surface temperature are based on comparisons to the black body radiation, the modern value is about 5800 K. The &lt;a href=&quot;http://solar-center.stanford.edu/FAQ/Qcorona.html&quot;&gt;solar corona is however much hotter&lt;/a&gt;, several million K. That was discovered by Grotrian (1939) and independently Edlen, who noticed that the observed spectral lines are emitted by iron, calcium, and nickel in very high stages of ionization. Why it is so hot is still an open problem, one promising hypothesis is the so-called &lt;a href=&quot;http://astrogeo.oxfordjournals.org/content/43/4/4.16.full&quot;&gt;&quot;magnetic carpet&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the surface temperature the older method used measurements of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Solar_constant&quot;&gt;solar constant&lt;/a&gt;, the mean energy flux per unit area at Earth's distance, and the inverse square law, to estimate the mean energy flux per unit area at the Sun's surface. First solar constant measurements were made by Pouillet in France and Herschel at the Cape of Good Hope in 1837. Subsequent measurements by Violle on Mont Blanc (1875) and Langley on Mount Whitney in California (1884) turned out to be further from modern values. See &lt;a href=&quot;https://books.google.com/books?id=PS7aBAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot;&gt;Concise History of Solar and Stellar Physics by Tassoul and Tassoul&lt;/a&gt; (p. 41).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But to get the temperature one needed to relate it to Sun's luminosity (total emitted energy). The modern law, which says that the luminosity is proportional to the fourth power of surface temperature, was only formulated by Stefan in 1879, and Boltzmann derived it from thermodynamics in 1884. Before that the two were related semi-empirically using laboratory experiments, with estimates based on Newton's and Dulong-Petit's cooling laws giving, respectively, overestimates and underestimates. In 1884, when the Stefan's law was still not accepted, &lt;a href=&quot;https://books.google.com/books?id=qrRz-sXyXJYC&amp;amp;source=gbs_navlinks_s&quot;&gt;Langley wrote in New Astronomy&lt;/a&gt; that &quot;&lt;em&gt;it is probable from all experiments made to date, that the solar effective temperature is not less than 3000 nor more than 30000 degrees of the centigrade thermometer&lt;/em&gt;&quot;. Later estimates were based on matching empirical peak intensity to &lt;a href=&quot;http://www.astronomynotes.com/starsun/s2.htm&quot;&gt;Wien's (1896) or Planck's (1900)&lt;/a&gt; black body radiation curves, and analyzing the line spectrum of solar radiation. A consensus close to the modern value emerged by the end of 19th century, but only later advances in radiation theory firmed it up. See &lt;a href=&quot;https://books.google.com/books?id=qrRz-sXyXJYC&amp;amp;source=gbs_navlinks_s&quot;&gt;Astrophysics and Twentieth-Century Astronomy to 1950&lt;/a&gt; (pp. 66-67).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More details on modern measurements can be found at &lt;a href=&quot;http://eaae-astronomy.org/WG3-SS/WorkShops/SunLuminosity.html&quot;&gt;Luminosity and Temperature of the Sun workshop&lt;/a&gt; website.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-11T03:53:32.513" LastActivityDate="2016-02-11T03:53:32.513" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3414" PostTypeId="1" CreationDate="2016-02-12T08:09:30.013" Score="4" ViewCount="4021" Body="&lt;p&gt;I'm doing some research about the beginning of trigonometry. I want to know why and who draw the first time the sine function. Do you have one site or something that can help me ?&lt;/p&gt;&#xA;" OwnerUserId="3661" LastActivityDate="2016-02-26T18:27:17.123" Title="History of sine function" Tags="|mathematics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3415" PostTypeId="2" ParentId="3414" CreationDate="2016-02-12T12:47:57.463" Score="4" Body="&lt;p&gt;See at least :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer&quot; rel=&quot;nofollow&quot;&gt;Albrecht Dürer&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Albrecht_D%C3%BCrer#Four_Books_on_Measurement&quot; rel=&quot;nofollow&quot;&gt;Underweysung der Messung mit dem Zirckel und Richtscheyt&lt;/a&gt; (or &lt;em&gt;Instructions for Measuring with Compass and Ruler&lt;/em&gt;), Nürnberg 1525, &lt;a href=&quot;https://de.wikisource.org/wiki/Seite%3ADuerer_Underweysung_der_Messung_021.jpg&quot; rel=&quot;nofollow&quot;&gt;page 17&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;An overview of the early history of trigonometry is into:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Enrique González Velasco, &lt;a href=&quot;https://books.google.it/books?id=0sTd4qJgOmsC&amp;amp;pg=PA1&quot; rel=&quot;nofollow&quot;&gt;Journey through Mathematics&lt;/a&gt; (2011), &lt;strong&gt;Ch.1 Trigonometry&lt;/strong&gt;, page 1-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The first modern textbook: &lt;a href=&quot;https://en.wikipedia.org/wiki/Bartholomaeus_Pitiscus&quot; rel=&quot;nofollow&quot;&gt;Bartholomaeus Pitiscus&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=LRk8AAAAcAAJ&amp;amp;pg=PA123&quot; rel=&quot;nofollow&quot;&gt;Trigonometriae sive de dimensione triangulorum libri quinque&lt;/a&gt; (1600, first edition 1595) does not seem to have graphs, but only tables.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;We may suppose that to draw the graph of the trigonometric functions, we must have a &quot;modern&quot; idea of &lt;em&gt;function&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first (quite) clear understanding of it is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow&quot;&gt;Leonhard Euler&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=_zpRAAAAYAAJ&amp;amp;pg=PA4&quot; rel=&quot;nofollow&quot;&gt;Introductio in analysin infinitorum&lt;/a&gt; (1748), &lt;strong&gt;CAPUT PRIMUM&lt;/strong&gt; &lt;em&gt;De functionibus in genere&lt;/em&gt;, page 4:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Functio quantitatis variabilis, est expressio analytica quomodocunque composita ex illa quantitate variabili, et numeris seu quantitatibus constantibus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also &lt;strong&gt;CAPUT VIII&lt;/strong&gt; &lt;em&gt;De quantitatibus trascendentibus ex Circulo оrtis&lt;/em&gt;, &lt;a href=&quot;https://books.google.it/books?id=_zpRAAAAYAAJ&amp;amp;pg=PA90&quot; rel=&quot;nofollow&quot;&gt;page 90-on&lt;/a&gt;, for trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe in the Plates... see &lt;a href=&quot;https://books.google.it/books?id=_zpRAAAAYAAJ&amp;amp;pg=PA353&quot; rel=&quot;nofollow&quot;&gt;Planche 12, Fig.104&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-02-13T17:51:18.057" LastActivityDate="2016-02-13T17:51:18.057" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3416" PostTypeId="1" AcceptedAnswerId="3418" CreationDate="2016-02-12T13:09:28.983" Score="7" ViewCount="1008" Body="&lt;p&gt;Artillery was established in Europe around the year 1400. But physics, the mathematical and systematical description of how objects fall, the foundation of engine construction, didn't dawn until about 1600 with Johannes Kepler's explanation of planetary movements. And physics didn't become (potentially) practically useful for artillery until the better part of that century later with Newton's math describing forces and vectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a battery of sub-questions for inspiration. Please aim, fire and duck as you please among these bullet points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How did early artillery work in the times before Galileo when&#xA;everybody still believed Aristotle's false idea that heavy objects&#xA;fall faster than light objects? How did this serious misconception affect the design and aiming of&#xA;early artillery? Was it just trial and error?&lt;/li&gt;&#xA;&lt;li&gt;How did artillery change when physics discovered how projectiles&#xA;actually move? Did for example the fortification genius de Vauban use&#xA;Newtonian calculations in 1700?&lt;/li&gt;&#xA;&lt;li&gt;Did the advent of artillery in Europe maybe even contribute to the&#xA;breakthrough of physical science? I've never heard of it, just a very&#xA;speculative question. One might've expected that physics came before&#xA;artillery, but it was the other way around. And remember Newton's&#xA;drawing of a cannon ball being fired into Earth's orbit, as&#xA;illustration of a consequence of his physics. Artillery came quickly&#xA;to mind while I suspect that the falling apple is a biblically inspired popular myth.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tzVCp.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tzVCp.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorUserId="4251" LastEditDate="2021-03-02T16:18:46.227" LastActivityDate="2021-03-02T16:18:46.227" Title="How did artillery and physics co-evolve during 1400-1700?" Tags="|physics|technology|17th-century|16th-century|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3417" PostTypeId="2" ParentId="3414" CreationDate="2016-02-12T14:07:00.197" Score="5" Body="&lt;p&gt;Ancient Greeks did not have the sine function; their only trigonometric function was the chord which is related to sine by the formula $\mathrm{chd(x)}=2\sin(x/2)$.&#xA;The first known tables of chords are due to Ptolemy. Greek trigonometry penetrated to India during the Hellenistic period. Sine was introduced Aryabhata in India (476-550).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table&quot;&gt;https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From India it penetrated to the Muslim world, where several astronomers developed trigonometry further, one notable figure was Abu al Wafa (940-998), who composed sine tables. He also introduced tangent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Europe people were not interested in mathematics at that time. Later the knowledge penetrated Europe, and the first European sine tables (other than copies of the Arab tables) were made by&#xA;Johannes Muller (Regiomontanus, 1436-1476).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See Wikipedia articles on Abu al Wafa and Johann Muller.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-02-12T14:07:00.197" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3418" PostTypeId="2" ParentId="3416" CreationDate="2016-02-12T14:36:02.083" Score="11" Body="&lt;p&gt;The interaction of physics and artillery begins with Galileo (there was no physics to talk about before that time except the statics). There are pictures in the books before Galileo which show completely unrealistic trajectories of projectiles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo developed a computational device which he called the Geometric and Military Compass, which he produced and sold, and trained people to use it for profit. From this time artillery is connected with physics. Earlier artillerists used their experience to aim their guns but there was no science to help them, except trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Application of physics to artillery is called ballistics. There are two kinds of ballistics: interior and exterior. Exterior ballistics studies the motion of a projectile in the air, and it is based on mechanics. Interior ballistics studies the motion of the projectile inside the gun barrel, and it uses much more advanced physics. As I said, scientific exterior ballistics begins at the time of&#xA;Galileo when the laws of motion were discovered. Interior ballistics begins with the attempts of Euler to describe what happens inside the barrel. These attempts were unsuccessful, because physics at the time of Euler was not sufficiently developed. One had to wait for thermodynamics which is a 19s century creation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On interaction of physics/mathematics and artillery. Probably there was a lot of interaction because many leading physicists and mathematicians were interested in&#xA;perfection of artillery (like Galileo, Descartes, Euler, and many later mathematicians). Of course, once a physics law is discovered, the rest of the work, application of the law is the business of mathematicians.&#xA;Some famous 20s century mathematicians worked on ballistics, for example, J. E. Littlewood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the modern times, the development of mathematics and physics was of course far ahead of&#xA;the development of ballistics, so this interaction was one-way: application of known physics and mathematics to ballistics. For example, the correction for Coriolis force was introduced to artillery only in 20s century, when they started to use big naval guns in the Southern hemisphere.  But there is no doubt that in&#xA;17s century there was some influence of ballistics on development of physics and mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the most important contribution of artillery to physics and mathematics&#xA;was indirect: governments are generally reluctant to spend money on science and education, but military needs force them. Take for example, the&#xA;establishment of the famous educational institutions in&#xA;France, like Ecole Polytechnique. The original purpose was to train the army officers. But one side result was strong development of physics mathematics in&#xA;19s century. Of course this pattern continues nowadays.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-03-14T15:43:16.520" LastActivityDate="2018-03-14T15:43:16.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3419" PostTypeId="1" AcceptedAnswerId="3447" CreationDate="2016-02-13T09:33:01.610" Score="4" ViewCount="447" Body="&lt;p&gt;As the title says I was wondering if the first time humans and other apes were thought of as a group was in Linnaeus' work as this is the first explicit reference I am aware of. I know of course that humans were already grouped as animals in many older works but that isn't the specific detail that I'm interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So can somebody who is familiar with the history of Biology tell me where the first known reference of humans and other apes being a tight family occurred?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of Aristotle's passages on apes but they seem to still be making a marked difference between what is a human and what is an ape. Aristotle does not seem to me to be suggesting a deeper relationship between non-human primates and humans. I'm looking for something on the scale of the words of Linnaeus, who stated that he knows of no criteria to separate apes and humans. I was wondering if this kind of deep similarity was spoken of before, not the (to me) superficial (yet still relevant and impressive) description of Aristotle.&lt;/p&gt;&#xA;" OwnerUserId="3652" LastEditorUserId="3652" LastEditDate="2016-02-13T14:13:04.290" LastActivityDate="2016-02-26T09:55:10.120" Title="Was Carl Linnaeus the first one to group humans and other apes together?" Tags="|biology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3420" PostTypeId="1" AcceptedAnswerId="3427" CreationDate="2016-02-14T03:14:48.450" Score="8" ViewCount="2663" Body="&lt;p&gt;Moment generating functions provide an alternative specification for a probability distribution function (pdf), often making it very convenient to calculate expectations, variances, etc. of said pdf.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A moment generating function is defined as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$M_X(t) = \mathbb{E}(e^{tX}), t\in\mathbb{R}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A caveat to moment generating functions is that their integrals might not always exist. The more general characteristic function gets around this, and they are defined as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$C_X(t) = \mathbb{E}(e^{itX}), t\in\mathbb{R}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given their tremendous capacity for simplifying the calculation of the moments of a probability distrubtion, moment-generating and characteristic functions seem almost magical -- a wonderful rabbit pulled, from thin air, out of a magician's hat. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Few things are truly that way in the history of mathematics, so I am curious to know how this particular rabbit was born: how did human beings zero in on them?&lt;/p&gt;&#xA;" OwnerUserId="2134" LastActivityDate="2023-09-30T16:02:04.757" Title="What is the history of moment generating functions, and the more general characteristic functions?" Tags="|mathematics|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3421" PostTypeId="1" CreationDate="2016-02-14T04:59:37.603" Score="10" ViewCount="375" Body="&lt;p&gt;The recent announcement of the detection of gravitational waves by LIGO comes almost exactly 100 years after Einstein predicted gravitational waves in 1916. Are there other examples (in the era of modern science) of such a long delay between a theoretical prediction and an experimental confirmation?&lt;/p&gt;&#xA;" OwnerUserId="3670" LastEditorUserId="55" LastEditDate="2016-07-22T23:11:04.837" LastActivityDate="2016-07-22T23:11:04.837" Title="What was the longest delay between prediction and confirmation of a theory?" Tags="|experimental-physics|relativity-theory|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3422" PostTypeId="1" CreationDate="2016-02-14T11:52:10.777" Score="5" ViewCount="772" Body="&lt;p&gt;In &lt;a href=&quot;https://youtu.be/s86-Z-CbaHA?t=19m33s&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt; about the Banach-Tarski paradox the host stipulates that history is full of examples of abstract mathematical theories that were later found to be applicable to the physical world. Is he right?&lt;/p&gt;&#xA;&lt;p&gt;(I realise this question may not specific enough for this forum but I am dying to learn more about this subject)&lt;/p&gt;&#xA;" OwnerUserId="3672" LastEditorUserId="4251" LastEditDate="2021-06-13T02:40:46.443" LastActivityDate="2021-06-13T02:40:46.443" Title="Example of abstract math theory that was later found to be applicable to physical world?" Tags="|mathematics|mathematical-physics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3423" PostTypeId="1" CreationDate="2016-02-14T13:25:27.697" Score="2" ViewCount="49" Body="&lt;p&gt;I am new to astrophysics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a question about the celebrated Harvard classification. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can people at 1900 determine the temperature of a remote star using the black body radiation formula? I mean, by fitting the continuum spectrum of a star to a black body? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question hinges on whether people at that time could measure the continuum spectrum.&lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2016-02-14T13:25:27.697" Title="Determining the temperature of a remote star by using the black body radiation formula" Tags="|physics|astronomy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3424" PostTypeId="2" ParentId="3422" CreationDate="2016-02-14T13:32:02.053" Score="9" Body="&lt;p&gt;&lt;strong&gt;Group theory.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A story I heard (perhaps enhanced over the years):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1910, Princeton was about to begin a major in physics.  The physicists sat down to decide what would be required for those majors to study.  As part of the exercise, they went through a list of mathematics courses to determine which would be used in physics.  They came to &quot;Group Theory&quot;, they all laughed.  &quot;There is a subject that will never be used in physics!&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But of course nowadays group theory is important in physics.*&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another story (you can probably find if you look).  An interview with physicist Richard Feynman.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(discussion of group theory used in physics)&lt;br&gt;&#xA;  &lt;em&gt;Interviewer:&lt;/em&gt; What would have happened if group theory had not already been developed by mathematicians?&lt;br&gt;&#xA;  &lt;em&gt;Feynman:&lt;/em&gt; Some physicist would have taken a week off to develop group theory, then progress in physics would have continued.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;*J. Cornwell, &lt;em&gt;Group Theory in Physics&lt;/em&gt;, volumes I and II&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-02-14T13:32:02.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3425" PostTypeId="2" ParentId="3422" CreationDate="2016-02-14T14:00:22.287" Score="7" Body="&lt;p&gt;Essentially the same question was asked on Math Overflow:&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653&quot;&gt;https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653&lt;/a&gt; , and many examples were given. Such examples are really abundant. It is more difficult to name a major mathematical theory which did NOT eventually find an application in the real world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Philosophers who think that mathematicians do mathematics just for their own entertainment, were always puzzled by this. But &quot;physical world&quot; somewhat restricts the areas of applications. I would say instead &quot;real world&quot;. For example, number theory (prime integers etc.) for two and half millenia was the purest part of the pure mathematics, with no apparent relation to the real world. In the end of 20s century important applications to coding were discovered. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-02-14T14:05:25.547" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3426" PostTypeId="2" ParentId="3421" CreationDate="2016-02-14T14:25:23.147" Score="4" Body="&lt;p&gt;It depends on what you call &quot;modern&quot; science. In one case, prediction was made in the antiquity, and confirmation had to wait for more than 2000 years: I mean &quot;atomic theory&quot; of Leukippe and Demokritus. The final confirmation was obtained only in 1905. (ON my opinion, 1905 is the &quot;modern science). One can argue, of course whether this was a real scientific theory. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-02-14T14:25:23.147" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3427" PostTypeId="2" ParentId="3420" CreationDate="2016-02-14T14:47:18.663" Score="11" Body="&lt;p&gt;The general idea of generating function has much wider scope than its applications&#xA;to probability. The proper setting is ``harmonic analysis'' which is one of the central and most developed parts of mathematics. The birth of the idea can be traced back to&#xA;Abraham de Moivre (1667-1754), and his book Doctrine of Chances.&#xA;Later the same idea was developed and applied in number theory (Euler), and most importantly in mathematical physics (Fourier). (Characteristic function is a special case of Fourier transform). Laplace transform (the moment generating function) belongs to the same circle of ideas, and its original use was also in probability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this powerful set of ideas (Generating function, Fourier, Laplace and other similar transforms) is called (linear) Harmonic analysis and it is one of the most powerful methods of mathematics, with applications practically everywhere&#xA;(in almost all areas of mathematics and sciences). If you add a generalization of it,&#xA;called non-linear harmonic analysis, you obtain most of the existing mathematics:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an excellent historical survey:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/S0273-0979-1980-14783-7.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mackey, George W. Harmonic analysis as the exploitation of symmetry—a historical survey. Bull. Amer. Math. Soc. 3 (1980), no. 1, part 1, 543–698&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="2134" LastEditDate="2016-02-16T15:23:13.323" LastActivityDate="2016-02-16T15:23:13.323" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3428" PostTypeId="2" ParentId="3421" CreationDate="2016-02-14T16:49:32.647" Score="7" Body="&lt;p&gt;I think Guido makes a good point in a comment that the analogue in math would be proving an old conjecture, and for this there are many examples that were settled after over 100 years. Besides Fermat's Last Theorem  there is: &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;the Poincaré conjecture, posed around 1900 and settled over 100 years later by Perelman,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the prime number theorem, conjectured independently by Legendre and Gauss in the 1790s, and proved independently by Hadamard and de la Valee Poussin in 1896,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the Gauss class number one problem, posed by Gauss around 1800 and settled in the second half of the 20th century independently by Heegner, Baker, and Stark, with further aspects settled later by work of Gross and Zagier,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the transcendence of e and pi, which was conjectured by Lambert in 1761 but proved in the 1870s and 1880s by Hermite and Lindemann. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Another old problem was the classification of constructible regular polygons. This was initiated by the ancient Greeks and settled 2000+ years later by Gauss, although the ancient Greeks did not predict which polygons were constructible (or for that matter if there were any constraints at all).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want to allow disproofs of a conjecture after a long time, Euler's conjecture from 1761 that for n &gt; 1 a sum of n-th powers is not an n-th power in positive integers when we allow only fewer than n summands (and of course require at least two summands) was disproved for n = 4 and 5 by computer searches in the 20th century: Elkies for n = 4 and Lander and Parkin for n = 5.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="-1" LastEditDate="2016-02-15T15:49:05.133" LastActivityDate="2016-02-15T15:49:05.133" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3429" PostTypeId="2" ParentId="3421" CreationDate="2016-02-15T00:50:23.897" Score="9" Body="&lt;p&gt;There is inherent vagueness in dating &quot;predictions&quot; and &quot;confirmations&quot; in many cases. For instance, who predicted heliocentrism? Copernicus, Kepler, Newton, perhaps Aristarchus? When was it confirmed? Another difficulty, also present in this example, is with predictions across major shifts in science (which happened about every 100 years since 17th century). There is more or less clear sense of confirming a prediction formulated within a theory that stays in place for a period of time, as was the case with the detection of gravitational waves, but over longer periods of time it is not so clear. Clean cases of confirmation show a strong selection bias towards short periods of time, and this one is indeed unusual.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dalton conjectured atomic bonds in forming chemical compounds in 1803 by analyzing known chemical reactions, development of chemistry throughout 19th century supported it indirectly, but many physicists (perhaps most) rejected their existence until early 20th century. When atoms were finally confirmed experimentally though they weren't the kinds of indivisible balls with hooks that Dalton imagined. Does it count? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another curious example is &lt;a href=&quot;http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...12...90M.pdf&quot; rel=&quot;nofollow&quot;&gt;Michell's prediction of black holes in 1784&lt;/a&gt;, he even anticipated how they might be detected:&quot;&lt;em&gt;If there should really exist in nature any bodies, whose density  is  not  less than  that  of  the  sun,  and  whose diameters  are  more  than  500  times  the  diameter  of  the sun,  since  their  light  could  not  arrive  at  us...  we could have   no   information   from   light;   yet,   if   any   other luminous  bodies  should  happen  to  revolve  about  them we   might   still perhaps from the motions of these revolving bodies infer the existence  of the central ones with some degree of probability...&lt;/em&gt;&quot;. Of course, Michell was working with Newtonian gravity and corpuscular optics, and when black holes were finally found the governing theories were general relativity and quantum electrodynamics (with wave optics rising and falling in between). Should we count the prediction from Schwarzschild's 1916 paper instead? Dating confirmation is also tricky: &lt;a href=&quot;https://en.wikipedia.org/wiki/Cygnus_X-1#Discovery_and_observation&quot; rel=&quot;nofollow&quot;&gt;Cygnus X-1&lt;/a&gt; was discovered in 1964 and suspected to be a black hole since. But the consensus about that only started forming in 1970s, and even in 1980s its status was still somewhat controversial.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-21T02:31:15.293" LastActivityDate="2016-02-21T02:31:15.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3430" PostTypeId="1" CreationDate="2016-02-15T09:41:03.147" Score="1" ViewCount="553" Body="&lt;p&gt;When I'm thinking of the most crucial discoverys of science than I think of the lens invented in the 16e century. By using a lens man could make microscopes and telescopes. When I think about that than I can't imagine that people had gone to the moon, had a mobile telephone, can listen to the radio watch tv, use GPS. But is through that the invention of the lens was crucial for futher technological inventions or could man invent those things via an other way and was it not really necessary?&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="3677" LastEditDate="2016-02-15T11:48:53.613" LastActivityDate="2016-02-15T13:57:49.960" Title="If the lens wasn't discovered/invented what current technologies wouldn't be possible" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="2" ClosedDate="2016-02-16T15:23:27.093" ContentLicense="CC BY-SA 3.0" />
  <row Id="3431" PostTypeId="2" ParentId="3430" CreationDate="2016-02-15T13:36:48.070" Score="2" Body="&lt;p&gt;First of all, the lens was discovered much earlier than 16s century. The properties of the convex lens were described by Alhazen in his Book of Optics in 1021. The eyeglasses were invented in Italy in 13s century. This is by far the most important use of lens.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that most humans after the age of 40 need eyeglasses just to be able read and write (and do many other jobs). And many people need them before 40. So this invention substantially prolonged the active life of people of many professions, including scientists and inventors. So certainly many discoveries in all areas would not be made without this invention. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another invention based on the lens, which affected the life of most humans is&#xA;photography, with all its subsequent derivatives including cinema, video and TV. &#xA;All this would be impossible without lens.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-02-15T13:57:49.960" LastActivityDate="2016-02-15T13:57:49.960" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3432" PostTypeId="2" ParentId="3421" CreationDate="2016-02-16T09:36:28.893" Score="3" Body="&lt;p&gt;Democritus did not formulate his ideas on atomism as conjecture but rather as fact or theory, as do many researchers :-) In line with other editors' comments that &lt;em&gt;the analogue in math would be proving an old conjecture&lt;/em&gt;, I would point out that Leibnizian theory of infinitesimals, assignable vs inassignable quantities, a more general relation of equality &quot;up to&quot;, etc. was not &quot;confirmed&quot; until the 20th century with the work of Hewitt, Los, and Robinson. Thus the delay was almost as long as with Fermat's last theorem, and perhaps even longer if one takes into account that infinitesimal approaches go back to Kepler and Galileo (and of course even earlier to Archimedes).&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-02-16T09:36:28.893" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3433" PostTypeId="1" AcceptedAnswerId="6131" CreationDate="2016-02-16T16:02:54.730" Score="3" ViewCount="1397" Body="&lt;p&gt;Leandro Panizzon &lt;a href=&quot;http://serendip.brynmawr.edu/bb/neuro/neuro05/web1/a1jordan.html&quot; rel=&quot;nofollow&quot;&gt;1&lt;/a&gt; &lt;a href=&quot;http://www.swissinfo.ch/eng/ritalin-use-is-on-the-rise/29155620&quot; rel=&quot;nofollow&quot;&gt;2&lt;/a&gt; &lt;a href=&quot;http://inhn.org/drugs/methylphenidate.html&quot; rel=&quot;nofollow&quot;&gt;3&lt;/a&gt; synthesised methylphenidate or something like that back in 1944. Ritalin is named after his wife Rita.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He doesn't seem to have a Wikipedia page or anything. What happened to him? I'm guessing he's dead now. When did he die? Any heirs or relatives?&lt;/p&gt;&#xA;" OwnerUserId="3338" LastEditorUserId="3338" LastEditDate="2016-02-23T22:59:30.930" LastActivityDate="2017-07-08T14:42:50.753" Title="What ever happened to the methylphenidate synthesiser and his family?" Tags="|chemistry|biology|death|psychology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3434" PostTypeId="2" ParentId="469" CreationDate="2016-02-16T17:24:09.480" Score="4" Body="&lt;p&gt;Let me add another piece of history going slightly back in time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that polyhedra satisfies the formula $V-L+F=2$, where $V$ is the number of vertices, $L$ is the number of sides and $F$ is the number of faces, goes back to Descartes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formula is nowadays called Euler's formula (yet again, Euler!) because he was the first one to attempt a proof. I say &quot;to attempt&quot; because his proof was &quot;almost&quot; correct. But exactly in this almost layed the interesting part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the nineteenth century various mathematicians proposed example of &quot;polyhedra&quot; that were not satisfying the formula: a cube with inside a cubic cavity, a smaller cube sitting on the side of a larger cube (which satisfies or not the formula dependind on how you count faces) and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wrote &quot;polyhedra&quot; because the usual objection to such examples was to refine the notion of polyhedra as to exclude them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally it is exactly in studying this kind of anomalies that Listing and Möbius came across the Möbius band.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it turned out the the definition of polyhedra became so complicated to counter intuition, it solwly became clear that such anomalies could in fact bring in new informations... it is also in this way that the Euler-Poincaré (genus) invariant was born. A discover which sits at the core of further developments of homotopy thoery and later on homology theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Alexander wrote general topology was born more in connection with clarifying the foundations of real analysis. I just want to stress that also in this case a notable role was played by &quot;monsters&quot;, i.e. by all sorts of strange, intuition defying math objects that were developed throughout he second half of the nineteenth century and the first half of the twentieth (Cantor set, Sierpinski gasket ecc...)&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="2442" LastEditDate="2016-05-09T18:18:47.963" LastActivityDate="2016-05-09T18:18:47.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3435" PostTypeId="1" AcceptedAnswerId="3436" CreationDate="2016-02-16T20:55:28.083" Score="5" ViewCount="1044" Body="&lt;p&gt;I am asking about the reason for the use of this letter, if known, as well as the initial occasion of its use.  Ideas that have been suggested concerning the former include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;That it stands for geometry or Geometrie&lt;/li&gt;&#xA;&lt;li&gt;That it stands for some other German word&lt;/li&gt;&#xA;&lt;li&gt;That it is in homage to Gauss&lt;/li&gt;&#xA;&lt;li&gt;That it refers to Gravitation&lt;/li&gt;&#xA;&lt;li&gt;That it refers to the Gram matrix&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As for the latter, I'm guessing it originates somewhere between Riemann and Einstein.&lt;/p&gt;&#xA;" OwnerUserId="3669" LastActivityDate="2016-02-18T03:09:00.357" Title="What is the origin of the use of &quot;g&quot; for a Riemannian metric?" Tags="|terminology|relativity-theory|differential-geometry|riemannian-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3436" PostTypeId="2" ParentId="3435" CreationDate="2016-02-17T01:24:05.320" Score="8" Body="&lt;p&gt;You are guessing correctly. Riemann did not use $g$ for the metric tensor, he writes things like $ds^2$ or $\sum dx^2$ instead, see his 1854 lecture &lt;a href=&quot;http://www.emis.de/classics/Riemann/WKCGeom.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;On the Hypotheses which lie at the Bases of Geometry&quot; (1854)&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Originally, &quot;g&quot; was for gravity. Einstein and Marcel Grossman, his mathematician friend who introduced him to tensor calculus and collaborated with him on prototypes of general relativity, adopted it after realizing some time between 1909 and 1913 that the metric tensor depended on the distribution of gravitating matter in a region of spacetime. After the success of Einstein's final version of the theory the notation was widely adopted by physicists and spread into tensor calculus and differential geometry as well. See &lt;a href=&quot;https://books.google.com/books?id=wtoKZODmoVsC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Foster and Nightingale's Short Course in General Relativity&lt;/a&gt; (p.112).&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-02-18T03:09:00.357" LastActivityDate="2016-02-18T03:09:00.357" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3437" PostTypeId="1" CreationDate="2016-02-19T15:30:17.133" Score="4" ViewCount="12611" Body="&lt;p&gt;Before Newton many phycisists try to understand nature and the rotations of planets. But Newton founded his laws of gravity. But was he the first who used the word gravity or when was it first used? And comes the word gravity from the word 'grievous'? Or was Aristotle or Galilei before him using that word? Of course they spoke another language but perhaps the already used βαρύτητα or gravità?&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="131" LastEditDate="2016-02-19T20:42:35.547" LastActivityDate="2022-07-21T12:39:49.107" Title="Was the word 'gravity' an invention of Newton?" Tags="|terminology|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3438" PostTypeId="2" ParentId="3437" CreationDate="2016-02-19T16:01:35.700" Score="4" Body="&lt;p&gt;Gravity meant weight or heaviness before Newton. Thus Newton writes: “That force by which the moon is held back in its orbit is that very force which we usually call ‘gravity’.” (Principia, Book III, Prop. IV.)&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2016-02-19T16:01:35.700" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3439" PostTypeId="2" ParentId="3437" CreationDate="2016-02-19T20:20:13.607" Score="12" Body="&lt;p&gt;&lt;a href=&quot;https://en.wiktionary.org/wiki/gravitas#Latin&quot; rel=&quot;noreferrer&quot;&gt;Gravitas&lt;/a&gt; is the latin word used by Medieval natural philosophers to translate the Aristotelian &lt;em&gt;heaviness&lt;/em&gt;, i.e. the quality of a body &quot;having weight&quot; (&lt;a href=&quot;https://en.wiktionary.org/wiki/pondus#Latin&quot; rel=&quot;noreferrer&quot;&gt;pondus&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let us then apply the term ‘heavy’ (βάρος) to that which naturally moves towards the centre, and ‘light’ to that which moves naturally away from the centre. (&lt;a href=&quot;https://archive.org/stream/decaeloleofric00arisuoft#page/n17/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;De Caelo&lt;/a&gt;, 269b)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Isaac Newton uses &lt;em&gt;gravity&lt;/em&gt; in his &lt;a href=&quot;https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica&quot; rel=&quot;noreferrer&quot;&gt;Philosophiae Naturalis Principia Mathematica&lt;/a&gt; (1687, 3rd Latin ed, 1726):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://books.google.it/books?id=WeZ09rjv-1kC&amp;amp;pg=PA3&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;DEFINITION V&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Of this sort is gravity [&lt;em&gt;gravitas&lt;/em&gt;], by which bodies tend to the centre of the earth. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Thus, Newton did not invent the word, but was the author of a crucial &quot;shift of meaning&quot;: previously &quot;gravity&quot; denoted a &lt;em&gt;quality&lt;/em&gt;, then a &lt;em&gt;force&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For sure, Newton &quot;invented&quot; the &lt;em&gt;force of gravity&lt;/em&gt; [&lt;em&gt;vi gravitatis&lt;/em&gt;].&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note.&lt;/em&gt; Galileo Galilei in his &lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;noreferrer&quot;&gt;Dialogue&lt;/a&gt; (in Italian) uses: &lt;em&gt;gravità&lt;/em&gt; and &lt;em&gt;leggerezza&lt;/em&gt; for heaviness and lightness, respectively.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-02-21T16:52:08.807" LastActivityDate="2016-02-21T16:52:08.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3440" PostTypeId="2" ParentId="3174" CreationDate="2016-02-22T01:12:18.090" Score="4" Body="&lt;p&gt;Knowledge of the specific fact that $(\sin x)' = \cos x$ actually predates the general knowledge of calculus and derivatives. It was known in the following form: that for very small $\Delta x$, when you increase $x$ to $x + \Delta x$, the increase in value of the sine, from $\sin x$ to $\sin (x + \Delta x)$, is proportional to $\Delta x$ times $\cos x$. In other words, that&#xA;$$\frac{\sin (x + \Delta x) - \sin x}{\Delta x} \approx \cos x$$&#xA;The approximation being exact in the limit as $\Delta x \to 0$ is of course the modern definition of the derivative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This happened historically in Indian mathematics, where Muñjala (around 932), Āryabhaṭa II (around 950), Prashastidhara (around 958) all give the above rule for calculating $\sin(x + \Delta x)$, and an explicit geometric reasoning / justification is given by Bhāskara II (around 1150) in his Siddhanta Shiromani. I have not found a perfectly good reference to these, but you can start with the following article:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Use of Calculus in Hindu Mathematics&lt;/em&gt;, by Bibhutibhusan Datta and Awadhesh Narayan Singh, revised by Kripa Shankar Shukla, &lt;em&gt;Indian Journal of the History of Science,&lt;/em&gt; &lt;strong&gt;19&lt;/strong&gt; (2): 95–104 (1984). (&lt;a href=&quot;http://www.dli.gov.in/rawdataupload/upload/insa/INSA_1/20005abd_95.pdf&quot; rel=&quot;nofollow&quot;&gt;PDF&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It was first pointed out by Bapu Deva Shastri in &lt;em&gt;Bhaskara's knowledge of the Differential Calculus,&lt;/em&gt; Journal of the Asiatic Society of Bengal, Volume 27, 1858, &lt;a href=&quot;https://books.google.com/books?id=kOrRAAAAMAAJ&amp;amp;pg=PA213#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;pp. 213–6&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="432" LastActivityDate="2016-02-22T01:12:18.090" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3442" PostTypeId="1" CreationDate="2016-02-23T16:15:02.217" Score="5" ViewCount="729" Body="&lt;p&gt;In the early 1800s Thomas Young introduced (a thought-?) experiment also known as the two slit experiment. He discovered the strange way photons created a interference pattern on a screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also an extra experiment done by detecting through which slit those photons had gone. But was it also Young who did that or was it someone else later?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And how did they think of detecting a photon. Because it looks crazy work to detect them because there had to be a kind of absorbtion of the photon. And we know that influenced the results important. &lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="4251" LastEditDate="2021-10-10T09:15:35.553" LastActivityDate="2021-10-10T09:15:35.553" Title="Who was first to observe or detect photons in the double slit experiment, and how did they do that?" Tags="|physics|quantum-mechanics|experimental-physics|physicists|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3443" PostTypeId="1" CreationDate="2016-02-23T19:04:03.267" Score="1" ViewCount="358" Body="&lt;p&gt;I need information about Hotelling and Bodewig, who they were and why they developed this method. anything will help, references to articles, links, or any other information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link to the method: &lt;a href=&quot;http://ads.harvard.edu/books/1990fnmd.book/chapt2.pdf&quot; rel=&quot;nofollow&quot;&gt;http://ads.harvard.edu/books/1990fnmd.book/chapt2.pdf&lt;/a&gt; page 41 (2.3.10)&lt;/p&gt;&#xA;" OwnerUserId="3726" LastEditorUserId="2905" LastEditDate="2016-02-25T21:29:31.260" LastActivityDate="2016-02-25T21:29:31.260" Title="Need information about the history of the Hotelling and Bodewig method" Tags="|linear-algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3444" PostTypeId="2" ParentId="3442" CreationDate="2016-02-23T23:14:44.860" Score="7" Body="&lt;p&gt;To the contrary, Young's point was to disprove the then dominant Newton's corpuscular theory of light by demonstrating light's wave properties, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2629/how-did-young-perform-his-double-slit-experiment/2634#2634&quot;&gt;How did Young perform his double slit experiment?&lt;/a&gt; The first &quot;quantum&quot; version of the experiment, with emission-absorption events creating a dotted pattern on the back screen, was performed by Taylor in 1909, based on Thomson's earlier idea, see &lt;a href=&quot;https://www.physics.wisc.edu/undergrads/courses/spring09/205/handouts/GITaylor_Interference.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Taylor's Interference Fringes with Feeble Light&lt;/a&gt;. Keep in mind however that at the time even old quantum mechanics of Bohr did not yet exist, and nobody took the idea of light quanta as traveling packets of energy seriously, not even Einstein, who thought they were vortices in a continuous field, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3404/how-did-planck-derive-the-black-body-radiation-formula-without-using-the-bose-st/3408#3408&quot;&gt;How did Planck derive the black body radiation formula without using the Bose statistics?&lt;/a&gt; So the familiar paradoxical descriptions, as e.g. in the popular &lt;a href=&quot;https://www.youtube.com/watch?v=rWTo2Gk5iU0&amp;amp;ebc=ANyPxKpk_olIyo-zvd2eJLsU9-kYEafBaVK5lFsAZv80zEtMDLdnxGOjpVeygBIZdfECwqOGlPLLZjToVoaq4jXqyaExbL0BQg&quot; rel=&quot;nofollow noreferrer&quot;&gt;What The Bleep Do We Know&lt;/a&gt; video, did not appear until much later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first experiment with electrons was performed in 1961, by Jönsson of the University of Tübingen, before that only experiments with photons were performed, here is his &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF01342460&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elektroneninterferenzen an Mehreren Künstlich Hergestellten Feinspalten&lt;/a&gt;. The most popular, and paradoxical, one electron at a time version of the experiment, where emission-absorption are time separated so single electrons apparently &quot;interfere with themselves&quot;, was performed by Merli, Missiroli, and Pozzi in 1974. This electron version is the one most often featured in popular expositions, see &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs00016-011-0079-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rosa's survey Merli–Missiroli–Pozzi Two-Slit Electron-Interference Experiment&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is Taylor's description of the technique in the 1909 experiment:&quot;&lt;em&gt;So far all the evidence brought forward in support of the theory has been of an indirect nature; for all ordinary optical phenomena are average effects, and are therefore incapable of differentiating between the usual electromagnetic theory and the modification of it that we are considering. Sir J.J. Thomson however suggested that if the intensity of light in a diffraction pattern were so greatly reduced that only a few of these indivisible units of energy should occur on a Huygens zone at once the ordinary phenomena of diffraction would be modified. Photographs were taken of the shadow of a needle, the source of light being a narrow slit placed in front of a gas flame. The intensity of the light was reduced by means of smoked glass screens&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-02-25T18:37:18.563" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3445" PostTypeId="1" AcceptedAnswerId="3446" CreationDate="2016-02-24T01:55:56.503" Score="3" ViewCount="1313" Body="&lt;p&gt;I was wondering who was the first person to&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Use curly braces to represent a finite set. Exempli gratia, $\{1,2,3\}$.&lt;/li&gt;&#xA;&lt;li&gt;Use set builder notation. Such as $\{2n:n \in \mathbb{Z}\}$ to represent the even numbers. I am not sure if the colon or vertical pipe is the original form for set builder notation, I have seen both used, but the colon comes out better in $\TeX$.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3727" LastActivityDate="2016-02-24T15:51:08.703" Title="First use of curly braces to denote a set?" Tags="|notation|set-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3446" PostTypeId="2" ParentId="3445" CreationDate="2016-02-24T07:59:42.017" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Georg_Cantor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georg Cantor&lt;/a&gt; was the first; see at least:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.mat.univie.ac.at/%7Emichael/EmA/EmA_Briefe/EmA_Kapitel_4/ema_21_Cantor.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Beiträge zur Begründung der transfiniten Mengenlehre (1)&amp;quot;&lt;/a&gt; (1895), &lt;em&gt;Mathematische Annalen&lt;/em&gt;, 46, Engl.transl.&lt;a href=&quot;https://en.wikipedia.org/wiki/Philip_Jourdain&quot; rel=&quot;nofollow noreferrer&quot;&gt;P.E.B.Jourdain&lt;/a&gt; (&lt;a href=&quot;https://archive.org/details/contributionstof00cant&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915&lt;/a&gt;, also Dover reprint):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By an &amp;quot;aggregate&amp;quot; (&lt;em&gt;Menge&lt;/em&gt;) we are to understand any collection into a whole &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; of definite and separate objects &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; of our intuition or our thought. These objects are called the &amp;quot;elements &amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In signs we express this thus :&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$M = \{ m \}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The formal &lt;a href=&quot;https://en.wikipedia.org/wiki/Set-builder_notation#Formal_set_builder_notation_sets&quot; rel=&quot;nofollow noreferrer&quot;&gt;set builder&lt;/a&gt; notation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\{ x \mid \varphi(x) \}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is an evolution of W&amp;amp;R's &lt;a href=&quot;http://plato.stanford.edu/entries/pm-notation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principia Mathematica&lt;/a&gt; notation for &amp;quot;abstraction&amp;quot;: &lt;span class=&quot;math-container&quot;&gt;$x \in \hat z(\psi z) \equiv \psi x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;An early example is in:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Bernays&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Bernays&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=YVKVAwAAQBAJ&amp;amp;pg=PA46&quot; rel=&quot;nofollow noreferrer&quot;&gt;Axiomatic set theory&lt;/a&gt; (1958, Dover reprint), page 46:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The class operator &lt;span class=&quot;math-container&quot;&gt;$\{ x \mid \mathfrak A(x) \}$&lt;/span&gt; &amp;quot;the class of the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\mathfrak A(x)$&lt;/span&gt;&amp;quot; [&lt;em&gt;footnote&lt;/em&gt; : This symbol (used in some newer papers) is here taken, for the sake of easier print, instead of Russell's class symbol &lt;span class=&quot;math-container&quot;&gt;$\hat x \mathfrak A(x)$&lt;/span&gt;, whose adoption was first intended.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The 2nd French edition of:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nicolas Bourbaki, &lt;a href=&quot;https://books.google.it/books?id=VDGifaOQogcC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Théorie des ensembles&lt;/a&gt; (2éme ed., 1970), page E.II.4 adopts it:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On introduira [...] un symbole fonctionnel; dans ce qui suit, nous utiliserons le symbole &lt;span class=&quot;math-container&quot;&gt;$\{ x \mid R \}$&lt;/span&gt;; le terme correspondant ne contient pas &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. C'est de ce terme qu'il s'agira quand on parlera de «l'ensemble des &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; tels que &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;». [...] par suite la relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; est équivalente à &lt;span class=&quot;math-container&quot;&gt;$x \in \{ x \mid R \}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The notation was not present in the &lt;a href=&quot;https://books.google.it/books?id=7eclBQAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation of 1968&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-02-24T15:51:08.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3447" PostTypeId="2" ParentId="3419" CreationDate="2016-02-26T09:55:10.120" Score="3" Body="&lt;p&gt;Though the similarity between humans and apes had been observed before, it appears that &lt;strong&gt;Linnaeus&lt;/strong&gt; was indeed the first to classify them together. &lt;strong&gt;John Ray&lt;/strong&gt;, one of the founders of taxonomy, introduced a group called &lt;em&gt;Anthropomorpha&lt;/em&gt; in 1693 (a name which Linnaeus later borrowed) but this didn't include humans. Humans were likewise excluded from the other early taxonomic systems, including &lt;strong&gt;Edward Tyson&lt;/strong&gt;'s, despite Tyson clearly documenting the anatomic parallels between humans and apes. Indeed his &lt;em&gt;Orang-Outang, sive Homo Sylvestris&lt;/em&gt; (1699) contains an exhaustive account of the similarities, but does so to prove the thesis that &lt;em&gt;inter hominem et non-hominem tertium non datur&lt;/em&gt; (there is nothing between man and non-man).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Visual similarities between humans and apes had been observed for much of the previous century, often by travelers to Asia and Africa. Furthermore, in 1616 the Italian free-thinker &lt;strong&gt;Lucilio Vanini&lt;/strong&gt; apparently suggested that humans were descended from apes. I have seen some claims that this was based on skeletal similarities, and others that it was based on visual similarities between Ethiopians and apes; I don't know how true these claims are.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastActivityDate="2016-02-26T09:55:10.120" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3448" PostTypeId="1" AcceptedAnswerId="3450" CreationDate="2016-02-26T11:26:15.957" Score="8" ViewCount="308" Body="&lt;p&gt;There are different conventions around whether $0 \in \mathbb N$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that $\mathbb N = \{0,1,2,3, \cdots\}$ is called the Bourbakian definition of $\mathbb N$. But did this convention really come form Bourbaki?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the axioms of Peano, which were earlier, 0 was also taken as a natural number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So where did each convention ($0 \in \mathbb N$ and $0 \not\in \mathbb N$) come form?&lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2017-06-08T14:10:46.843" Title="History of $0 \in \mathbb N$." Tags="|mathematics|number-theory|set-theory|zero|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3449" PostTypeId="1" CreationDate="2016-02-26T12:07:39.000" Score="2" ViewCount="229" Body="&lt;p&gt;Einstein explained the dependence of stopping voltage on photon wavelength by quantizing energy of photon and in his model increasing the intensity will increase the number of photons keeping the frequency same and thereby not transferring enough energy to electrons to overcome threshold. Did he consider the possibility that electrons can keep on gaining energy from lower frequency photon and can eventually escape surface when they accumulate enough energy to overcome energy barrier. Fact that this is not observed experimentally may be due to the fact that electrons are not able to absorb arbitrary amount of energy and therefore electron energy levels are quantized and not the photon energy level.  &lt;/p&gt;&#xA;" OwnerUserId="2459" LastActivityDate="2016-02-26T17:48:42.600" Title="In explaining photoelectric effect, why is photon quantized and not energy levels of electrons?" Tags="|physics|quantum-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3450" PostTypeId="2" ParentId="3448" CreationDate="2016-02-26T12:31:30.223" Score="11" Body="&lt;p&gt;For modern times:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Dedekind&quot; rel=&quot;noreferrer&quot;&gt;Richard Dedekind&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/wassindundwasso00dedegoog#page/n42/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Was sind und was sollen die Zahlen?&lt;/a&gt; (1888), page 20:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§71. &lt;strong&gt;Definition&lt;/strong&gt;. A system &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; [&lt;em&gt;Ein system&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;] is said to be &lt;em&gt;simply infinite&lt;/em&gt; when there exists a similar transformation &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; in itself such that &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; appears as chain of an element not contained in &lt;span class=&quot;math-container&quot;&gt;$\phi(N)$&lt;/span&gt;. We call this element, which we shall denote in what follows by the symbol &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, the &lt;em&gt;base-element&lt;/em&gt; of N and say the simply infinite system &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is set in order by this transformation &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§73. &lt;strong&gt;Definition&lt;/strong&gt;. If in the consideration of a simply infinite system &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; set in order by a transformation &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; [...] these elements [are] called &lt;em&gt;natural numbers&lt;/em&gt; or &lt;em&gt;ordinal numbers&lt;/em&gt; or simply &lt;em&gt;numbers&lt;/em&gt;, and the base-element &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is called the &lt;em&gt;base-number&lt;/em&gt; of the &lt;em&gt;number-series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Giuseppe_Peano&quot; rel=&quot;noreferrer&quot;&gt;Giuseppe Peano&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/arithmeticespri00peangoog&quot; rel=&quot;noreferrer&quot;&gt;Arithmetices Principia Novo Methodo Exposita&lt;/a&gt; (1889), page 1:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Axiomata&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$1 \in N$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/frege/&quot; rel=&quot;noreferrer&quot;&gt;Gottlob Frege&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/diegrundlagende00freggoog#page/n123/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl&lt;/a&gt; (1884):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§74. Null [&lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;] ist die Anzahl, welche dem Begriffe &amp;quot;sich selbst ungleich&amp;quot; zukommt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/russell/&quot; rel=&quot;noreferrer&quot;&gt;Bertrand Russell&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/principlesofmath005807mbp#page/n165/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;The Principles of Mathematics&lt;/a&gt; (1903):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§120. Peano’s three indefinables are &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, &lt;em&gt;finite integer&lt;/em&gt; and &lt;em&gt;successor of&lt;/em&gt;. [...] Peano’s primitive propositions are then the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is a number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mario_Pieri&quot; rel=&quot;noreferrer&quot;&gt;Mario Pieri&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=zZoYihwdxDwC&amp;amp;pg=PA309&quot; rel=&quot;noreferrer&quot;&gt;Sopra gli assiomi aritmetici&lt;/a&gt; (1907):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\gamma.$&lt;/span&gt; There is at least one number that is not the successor of any number. [...]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$f.$&lt;/span&gt; The number &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; that is not the successor of any number [...] is given the name &lt;em&gt;zero&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For Bourbaki, see:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nicolas Bourbaki, &lt;a href=&quot;https://books.google.it/books?id=VDGifaOQogcC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Théorie des ensembles&lt;/a&gt; (2nd ed 1970), &lt;strong&gt;Ch.III, §3&lt;/strong&gt;:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;1. Le cardinal d'un ensemble&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On note &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; le cardinal &lt;span class=&quot;math-container&quot;&gt;$\text {Card} (\emptyset)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-02-28T08:32:21.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3451" PostTypeId="1" AcceptedAnswerId="3463" CreationDate="2016-02-26T14:21:01.503" Score="8" ViewCount="466" Body="&lt;p&gt;The question of the distribution of the leading digits of the sequence $2^n$ is called Gelfand's problem or Gelfand's question. Is there any source that indicates Gelfand's own work on this, or the exact context where it arose with him? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References I have looked at say the first place connecting Gelfand's name with this question is on p. 37 of Avez's book &quot;Ergodic Theory of Dynamical Systems, Vol. 1,&quot; but this is not available to me. I tried searching the internet with some Russian terms that seemed relevant (вопрос гельфанда, распределение цифр) and nothing special came up.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="55" LastEditDate="2016-03-17T02:10:21.963" LastActivityDate="2016-03-17T02:11:07.517" Title="What is the original source for Gelfand's problem on leading digits of the powers of 2?" Tags="|mathematics|mathematicians|debunking|probability|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3452" PostTypeId="2" ParentId="3449" CreationDate="2016-02-26T17:48:42.600" Score="3" Body="&lt;p&gt;No, I don't think Einstein or his contemporaries ever considered the idea you proposed, because it would have been obvious to them that it wouldn't have worked. Einstein was proposing an interpretation of both the photoelectric effect and Planck's ad hoc calculation of the blackbody spectrum. Your idea would not work for explaining the blackbody spectrum. (Your idea also would not work for explaining the photoelectric effect, since energy is effectively continuous, not quantized, both for conduction electrons and for electrons in the vacuum. However, he wouldn't realistically have been able to figure this out in 1905.)&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2016-02-26T17:48:42.600" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3453" PostTypeId="2" ParentId="3414" CreationDate="2016-02-26T18:27:17.123" Score="0" Body="&lt;p&gt;In the second century CE, Ptolemy created a &lt;a href=&quot;https://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords&quot; rel=&quot;nofollow&quot;&gt;table of chords&lt;/a&gt; that is in some sense equivalent to the sine function.&lt;/p&gt;&#xA;" OwnerUserId="2193" LastActivityDate="2016-02-26T18:27:17.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3454" PostTypeId="2" ParentId="2776" CreationDate="2016-02-26T18:55:54.437" Score="7" Body="&lt;p&gt;From Gian-Carlo Rota's book &lt;em&gt;Indiscrete Thoughts&lt;/em&gt; (not to be confused with his other book &lt;em&gt;Discrete Thoughts&lt;/em&gt;), pages 114–116:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One might think that once the prime number theorem was proved other attempts at proving it by altogether different techniques would be abandoned as fruitless.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[Actually, no one who knows what typically happens would think that.]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But this is not what happened after Hadamard and de la Vallée Poussin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;for about fifty years thereafter, paper after paper began to appear in the best mathematics journals that provided nuances, simplification, alternative routes, slight generalizations, and eventually alternative proofs of the prime number theorem.  For example, in the thirties, the American mathematician Norbert Wiener developed an extensive theory of Tauberian theorems that unified a great number of disparate results in classical mathematical analysis.  The outstanding application of Wiener's theory, widely acclaimed throughout the mathematical world, was precisely a new proof of the prime number theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wiener's proof had a galvanizing effect.  From that time on, it was believed that the proof of the prime number theorem could be made elementary.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It took another ten years and a few hundred research papers to  remove a farrago of irrelevancies from Wiener's proof.  The first elementary proof of the prime number theorem, one that &quot;in principle&quot; used only elementary estimates of the relative magnitudes of primes was finally obtained by the mathematicians Erdős and Selberg.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Erdős and Selberg's proof added up to a good fifty pages of elementary but thick reasoning and was longer and harder to follow than any of the preceding ones.  It did, however, have the merit of relying only upon notions that were &quot;intrinsic&quot; to the definition of prime number, as well as on a few other elementary facts going back to Euclid and Eratosthenes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It took another few hundred research papers, whittling down Erdős and Selberg's argument to the barest core, until, in the middle sixties, the American mathematician Norman Levinson (who was Norbert Wiener's research student) published a short note bearing the title &quot;An elementary proof of the prime number theorem.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;that can be followed by careful reading by anyone with no more knowledge of mathematics than that of undergraduates at an average American college.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2193" LastActivityDate="2016-02-26T18:55:54.437" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3456" PostTypeId="1" CreationDate="2016-02-27T22:07:36.143" Score="3" ViewCount="68" Body="&lt;p&gt;Over the years, and long before the discovery of the causative agent, tuberculosis was known by various names, including &quot;phthisis&quot; and &quot;consumption&quot;. Robert Koch discovered Mycobacteria in 1882.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We now know that there are several species of Mycobacteria that cause tuberculosis, the key ones historically in England being &lt;em&gt;Mycobacterium tuberculosis&lt;/em&gt; and &lt;em&gt;Mycobacterium bovis&lt;/em&gt; (the latter of which causes bovine tuberculosis and can also cause disease in humans).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pre-nineteenth century historical records will not, of course, have possibly distinguished between TB in people caused by &lt;em&gt;M. tuberculosis&lt;/em&gt; versus &lt;em&gt;M. bovis&lt;/em&gt;. However, have there been any studies to retrospectively assess what proportion of TB in people was caused by each of these organisms? In other words, has it been possible to estimate the human mortality due to each of these organisms in pre-nineteenth century England?&lt;/p&gt;&#xA;" OwnerUserId="2826" LastActivityDate="2016-02-27T22:07:36.143" Title="Assessing TB mortality due to M. bovis vs M. tuberculosis infections in pre-19th century England?" Tags="|medicine|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3457" PostTypeId="2" ParentId="391" CreationDate="2016-02-28T14:07:40.713" Score="10" Body="&lt;p&gt;I'd like to present my point of view on the history of Greek mathematics from  Thales to Euclid, which differs from Colin's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we know about Greek mathematics we have learnt from sources&#xA;that were written after 300 BC, and of course we do not have any&#xA;originals but copies of copies of copies. In particular, everything&#xA;we seem to know about Thales or Pythagoras is second hand knowledge&#xA;coming from histories composed at least 200 years after their death.&#xA;Just because legends about Pythagoras have been embellished a few&#xA;centuries later does not mean, however, that there is no substance &#xA;to these legends at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that there are many possible interpretations, all of which are more or less compatible with the &quot;facts&quot;. The two extreme positions are held by&#xA;Burkert, who views Pythagoras as some kind of medicine man, and e.g.&#xA;by Zhmud (Pythagoras as a mathematician). Just because they are &#xA;extreme does not mean that they must be wrong, or that the truth lies&#xA;somewhere in between. But there is certainly not an unanimous opinion &#xA;concerning the &quot;historical truth&quot;. For a balanced view (whatever that &#xA;means) one may consult Kahn's book &quot;Pythagoras and the Pythagoreans&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, honesty requires using the phrase &quot;we do not know for certain&quot; a lot more often than did historians in the last century. There is an &quot;ignoramus&quot; in history of mathematics, and unless someone comes up with a time machine, there also is an &quot;ignorabimus&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-02-28T14:07:40.713" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3458" PostTypeId="1" CreationDate="2016-02-28T14:26:07.500" Score="1" ViewCount="131" Body="&lt;p&gt;Seeing &lt;a href=&quot;https://hsm.stackexchange.com/questions/2776/has-any-difficult-proof-ever-been-superseded-by-a-simple-one&quot;&gt;this question&lt;/a&gt;, a related one is to ask when a simple proof or derivation was ever replaced by a complex one, and what the reasons were. This probably makes the most sense in the context of expository material, like textbooks.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-02T18:21:17.290" Title="Simple proofs/derivations replaced by complex ones?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3459" PostTypeId="2" ParentId="3458" CreationDate="2016-02-28T15:29:06.390" Score="2" Body="&lt;p&gt;Such cases exist indeed. Sometimes a simple proof is considered unsatisfactory on&#xA;philosophical grounds. And people search for another proof, which can be more complicated than the original one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example is Picard's theorem. The original proof was essentially two lines. However many people thought that this proof does not really reveal the reasons why it is true. Very sophisticated theories were constructed to &quot;explain&quot; (and generalize) Picard's theorem. See for example &quot;Nevanlinna theory&quot;. This research had the general title: search of an&#xA;&quot;elementary&quot; proof of Picard's theorem. It is somewhat difficult to explain what the word &quot;elementary&quot; exactly means in this context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A different example is known under the title &quot;Elementary methods in number theory&quot;. Since Euler, Dirichlet and Riemann, the most powerful methods in the study of&#xA;distribution of primes were based on the theory of functions of a complex variable. However it was considered desirable to give &quot;elementary&quot; proofs of&#xA;theorems like the asymptotic law of prime numbers. Such proof was obtained&#xA;by Erdos and Selberg in 1940-s, and this brought to Selberg a Fields medal.&#xA;&quot;Elementary&quot; here means &quot;not using complex function theory&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In both cases, the reasons for search of new proof were philosophical. But at least in the first case, the search of a new proof really led to much progress and interesting and important new results. I am not so sure about the second example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference: Gelʹfond, A. O.; Linnik, Yu. V. Elementary methods in the analytic theory of numbers, Pergamon Press, Oxford-New York-Toronto, Ont. 1966.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Another classical example is the Quadratic Reciprocity Law of Legendre and Gauss in number theory. Gauss himself published 6 proofs, and according to Wikipedia 200 various proofs have been published since. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-02-28T16:55:03.787" LastActivityDate="2016-02-28T16:55:03.787" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3460" PostTypeId="2" ParentId="3451" CreationDate="2016-02-28T16:47:38.780" Score="3" Body="&lt;p&gt;The book of Avez you refer to is indeed rare: it is not listed in the common databases Mathscinet and Zentralblatt (which is very strange). So I cannot say anything about the relation of Gelfand to this problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A better known book Arnold and Avez, Ergodic Problems of Classical Mechanics,&#xA;has this problem as a problem 3.2 in Ch.1.3, with solution in Appendix 12.5&#xA;(pp. 134-135 of the Russian edition), but it never mentions Gelfand in connection with this problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what concerns the problem itself, its origin is clear and the problem has been completely solved. For more general problems of this sort, the&#xA;keywords are &quot;Benford law&quot;, and &quot;distribution of first digits&quot;, and a solution is given in this paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P. Diaconis, The Distribution of Leading Digits and Uniform Distribution Mod 1,&#xA;Ann. Prob., Volume 5, Number 1 (1977), 72-81.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-02-28T19:04:34.303" LastActivityDate="2016-02-28T19:04:34.303" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3461" PostTypeId="1" CreationDate="2016-02-29T17:42:31.313" Score="3" ViewCount="103" Body="&lt;p&gt;I'm sure this question has been asked somewhere, but I have been unable to find it. Why is it that when we have some set $X$ with an equivalence relation $\equiv$, and $X$ has some structure (e.g. a group operation, a topology, a vector space), we refer to the structure induced by $\equiv$ on $X / \equiv$ a &quot;quotient&quot; (group, topology, vector space, etc.). My guess would be that it comes from a prior use of &quot;quotient&quot; to talk about division on $\mathbb{R}$, and so the field of fractions on $\mathbb{Z}$ (with an equivalence relation in the from of $x / y \equiv ax / ay$) was the source of using &quot;quotient&quot;. Is this correct? If not, what is the origin behind referring to the induced structure as a &quot;quotient&quot;?&lt;/p&gt;&#xA;" OwnerUserId="2448" LastActivityDate="2016-02-29T21:22:26.770" Title="Origin of use of &quot;quotient&quot; to describe structures induced by equivalence classes" Tags="|mathematics|terminology|language|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3462" PostTypeId="2" ParentId="3461" CreationDate="2016-02-29T21:22:26.770" Score="4" Body="&lt;p&gt;Probably the origin is with the theory of groups: see &lt;a href=&quot;https://en.wikipedia.org/wiki/Quotient_group#Motivation_for_the_name_.22quotient.22&quot; rel=&quot;nofollow noreferrer&quot;&gt;quotient group&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Burnside&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Burnside&lt;/a&gt; in:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://archive.org/details/theorygroupsfin00burngoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theory of groups of finite order&lt;/a&gt; (1897), page 38:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Herr &lt;a href=&quot;https://en.wikipedia.org/wiki/Otto_H%C3%B6lder&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hölder&lt;/a&gt; [&lt;em&gt;footnote&lt;/em&gt;:  &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0034&amp;amp;DMDID=DMDLOG_0008&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen&amp;quot;&lt;/a&gt;, &lt;em&gt;Math.Ann.&lt;/em&gt;,‎ 1889] has introduced the symbol&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac G \Gamma$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;to represent this group; he calls it the &lt;em&gt;quotient&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt; and a &lt;em&gt;factor-group&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-02-29T21:22:26.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3463" PostTypeId="2" ParentId="3451" CreationDate="2016-03-01T00:09:33.163" Score="4" Body="&lt;p&gt;I assume that one of the sources is  &lt;a href=&quot;https://mathworld.wolfram.com/GelfandsQuestion.html&quot; rel=&quot;nofollow&quot;&gt;MathWorld&lt;/a&gt;. But the question they claim Avez attributes to Gelfand is not the distribution of the leading digits generally, but specifically &quot;&lt;em&gt;will the digit 9 ever occur&lt;/em&gt;&quot; as the leading digit in $2^n$ (the answer is yes, but the smallest $n$ is 54). They also link to &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B0007G1HCM&quot; rel=&quot;nofollow&quot;&gt;Avez's 1966 book&lt;/a&gt;, which is their source. I was unable to access the book so far, but &lt;a href=&quot;http://www.math.rug.nl/~top/Monthly2015.pdf&quot; rel=&quot;nofollow&quot;&gt;Eising, Radcliffe and Top in Simple Answer to Gelfand’s Question&lt;/a&gt; confirm that the problem is attributed in it to Gelfand, and add that Arnold-Avez's book published two years later replaced 9 by 7.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it appears that Avez is everybody's source for the attribution to Gelfand. What about Avez? Here is from &lt;a href=&quot;https://dmv.mathematik.de/index.php/publikationen/dmv-jahresbericht/dmv-jahresb-onl/jahrgang-2011/heft-4-2/240-leonid-polterovich-inna-scherbak/file&quot; rel=&quot;nofollow&quot;&gt;Arnold's obituary&lt;/a&gt;:&quot;&lt;em&gt;Arnold spent  1965  in  Paris  as  a  postdoctorate  at the  Sorbonne.  At  the  request of  his  supervisor,  J.  Leray,  Arnold  delivered  a  one-semester  course  on  dynamical systems. The audience included many renowned mathematicians (Cartan, Douady,&#xA;Fréchet, Godement, Leray, Schwarz, Serre, Thom). One of the participants, Andre&#xA;Avez, recorded the lectures and then published them as a book&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am going to speculate that Gelfand never published the problem, that Avez got it from Arnold's 1965 lectures, and Arnold, who was a student at the Moscow State University and took classes with Gelfand, did not need a publication to get it from Gelfand directly. In particular, Arnold gave multiple &lt;a href=&quot;http://www.mccme.ru/gelfand/notes/all-links.html&quot; rel=&quot;nofollow&quot;&gt;talks at Gelfand's seminar in 1964-65&lt;/a&gt;, handwritten notes (in Russian) are available online. On the other hand, Arnold-Avez do not attribute the problem (with 7) to anybody, so either Arnold forgot, or Avez misinterpreted his French in 1965, and the problem was Arnold's own. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-03-17T02:11:07.517" LastActivityDate="2016-03-17T02:11:07.517" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3464" PostTypeId="1" CreationDate="2016-03-01T01:20:11.967" Score="1" ViewCount="148" Body="&lt;p&gt;There are about &lt;a href=&quot;http://www.wolframalpha.com/input/?i=world%20population&quot; rel=&quot;nofollow&quot;&gt;7.13 billion humans alive today&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When installing Java, it says that 3 billion devices run Java.  The question is, out of how many?  How many computing devices that have enough hardware to run Java will be on at some point on a non-holiday Tuesday?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Servers count, smartphones count, and if you see a clear way to refine the definition in the question please feel free to propose an edit or state the slightly different definition in your answer.  &lt;/p&gt;&#xA;" OwnerUserId="2672" LastActivityDate="2016-03-01T01:20:11.967" Title="How many computers are there?" Tags="|computer-science|computers|technology|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="3465" PostTypeId="2" ParentId="38" CreationDate="2016-03-01T03:26:31.297" Score="0" Body="&lt;p&gt;Had they not been so optimistic, they would have pursued other endeavors instead, and not became the founders of AI.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's a selection bias that things may go the other way around from what seems to be behind the question.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, they were solving the easier problems first, so progress seemed fast, and they just projected that speed of progress out even beyond where it was accurate to do so, because the harder parts take longer to solve.  Thanks to much better tools and the work that's been done already, progress is being made faster than before, but the challenges being tackled are also quite hard.&lt;/p&gt;&#xA;" OwnerUserId="2672" LastEditorUserId="2672" LastEditDate="2016-03-01T03:33:11.817" LastActivityDate="2016-03-01T03:33:11.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3466" PostTypeId="1" CreationDate="2016-03-01T04:40:26.787" Score="8" ViewCount="20680" Body="&lt;p&gt;The story is that Einstein was shown a German newspaper that claimed &quot;One hundred German physicists claim Einsteins theory of relativity is wrong.&quot; Einsteins reply was supposedly, &quot;If I were wrong, it would only take one.&quot; I have looked and looked for this story and I can't find it anywhere. Did I just dream of it?&lt;/p&gt;&#xA;" OwnerUserId="3763" LastEditorUserId="4251" LastEditDate="2022-01-07T15:08:43.440" LastActivityDate="2024-03-27T00:40:00.447" Title="Did Einstein really say: &quot;If I were wrong, it would only take one.&quot;" Tags="|physics|einstein|physicists|quote|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3467" PostTypeId="1" AcceptedAnswerId="3469" CreationDate="2016-03-01T05:50:20.450" Score="5" ViewCount="334" Body="&lt;p&gt;I know that sine and cosine functions are used ubiquitously today, but I got to wondering about what sort of unsolved or difficult to solve problems required inventing the trigonometric functions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, the complex number $i$ was invented for the purpose of solving quadratics unsolvable in the reals - i.e. solving the equation $x^2=-1$. The number $0$ was invented to represent absence, making possible much more efficient, unambiguous, and reliable number keeping. The Laplace Transform was invented to transform entire difference equations and make solving DEs much easier and faster in many cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What problem did someone have to invent the sine and cosine functions to solve, or to make much easier?&lt;/p&gt;&#xA;" OwnerUserId="3764" LastEditorUserId="604" LastEditDate="2016-04-13T21:07:12.443" LastActivityDate="2016-04-13T21:07:12.443" Title="What were the problems that necessitated the introduction of the sine and cosine functions, and how long ago were they introduced?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3468" PostTypeId="1" AcceptedAnswerId="5313" CreationDate="2016-03-01T09:33:52.800" Score="6" ViewCount="1842" Body="&lt;p&gt;The prefix &quot;ortho-&quot; means straight or right; &quot;meta-&quot; means beyond or after; &quot;para-&quot; means beside or along. How, then, did ortho-, meta- and para- come to refer to the carbon positions one, two, and three positions away from a reference point on a benzene?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we have two groups on a benzene ring immediately beside one another, we do not denote the positions by para-, which would seem logical given its meaning, but instead ortho-. For instance, a benzene ring with two hydroxyl (-OH) groups on immediate positions of a benzene ring can be called para-hydroxybenzene (with the common name &quot;quinone&quot;). What is the etymological history that led to the rather counterintuitive use of these prefixes in organic chemistry?&lt;/p&gt;&#xA;" OwnerUserId="3766" LastActivityDate="2016-10-24T22:20:06.493" Title="What is the origin of &quot;ortho-,&quot; &quot;meta-,&quot; &quot;para-,&quot; in chemistry?" Tags="|terminology|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3469" PostTypeId="2" ParentId="3467" CreationDate="2016-03-01T13:06:39.533" Score="4" Body="&lt;p&gt;Trigonometric functions were invented by astronomers to solve problems of spherical trigonometry. The main problem was passing from the ecliptic to equatorial coordinate system and back. Plane trigonometry was a secondary aplication which appeared as a byproduct. The original trigonometric function was the chord,&#xA;$\mathrm{chd}\, t=2\sin(t/2)$. The first table of chords was made by the Greek astronomer Hipparchus. The first table of chords which survived is due to another astronomer, Ptolemy. In the medieval times trigonometry was developed in India, they introduced&#xA;the modern functions, sine and cosine. It is not known who exactly and when introduced them. (See Wikipedia, &quot;Surya Siddhanta&quot;). Later the Arabs, or Indians added tangent. All this development happened for the needs of astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The statement that $i=\sqrt{1}$ was invented to solve quadratic equations which have no real solutons is not quite correct. People were happy with some quadratic equations having solutions, and others not. $i$ was invented to solve cubic equations. Cardan's formula involves square roots and cubic roots. When you try to solve a simple cubic equation with three real roots with Cardan's formula, for example,&#xA;$x^3-x=x(x-1)(x+1)=0$, the square root in Cardan's formula has to be extracted from&#xA;a negative quantity. So to obtain the correct result from Cardans's formula in such case, one needs to compute with square roots of negative numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Until the late 1970s astronomy and geodesy remained the most important areas of application of trigonometric functions. Every ship (and a trans-ocean airplane) had a navigator whose main job was to determine the position of the ship by astronomical observations. He used trigonometry, to solve essentially the same problem for which trigonometry was invented. (He used tables of logarithms of sines or tables specially designed for this specific problem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the late 1970-s two revolutions happened, almost simultaneosly, which eliminated this profession. 1. Electronic calculators (and later computers), and almost at the same time 2. Satellite technology. These made the profession of navigator obsolete. Nowadays there is no mass profession where the working knowledge of trigonometric functions is required.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-03-02T05:55:14.017" LastActivityDate="2016-03-02T05:55:14.017" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3470" PostTypeId="1" AcceptedAnswerId="3474" CreationDate="2016-03-01T13:11:11.703" Score="6" ViewCount="4457" Body="&lt;p&gt;I was reading about the history of thermodynamics and came across Lavoisier's idea of heat. He proposed that heat was a fluid. I am curious to know what are the major drawbacks of this theory. I know that if heat is fluid then it must have mass and this means that if a body is heated then its mass must increase. However, this is not the case. Are there any other major contentions that can be added to this list?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank You.&lt;/p&gt;&#xA;" OwnerUserId="3767" LastActivityDate="2020-04-24T11:50:26.070" Title="What are the major flaws of the “caloric” theory of heat?" Tags="|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3471" PostTypeId="2" ParentId="38" CreationDate="2016-03-01T15:44:06.873" Score="1" Body="&lt;p&gt;There is a possible answer to this which is pretty generic. I came across this idea in an essay by Robert Heinlein -- can't find the title or other info right now, unfortunately. The idea is that technology tends to follow an exponential curve, but when people imagine the future, they tend to assume linear progress.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bcSYi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bcSYi.png&quot; alt=&quot;exponential and linear curves&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result is something like the graph drawn above. In the short term, the line predicts progress that's too rapid, but in the long term it's the other way around.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, if you look at science fiction books from the 20th century, they project that human space travel will be much more common and economically viable than it really turned out to be. For example, in the 1960s they projected lunar colonies in the 1970s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;AI could just be another example of the same phenomenon.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2016-03-01T15:44:06.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3472" PostTypeId="2" ParentId="38" CreationDate="2016-03-01T16:58:17.453" Score="0" Body="&lt;p&gt;One effect that is visible e.g. in programming, is that one of the holy grails was &quot;automatic programming&quot;, i.e., having the computer write a program from some high(er) level description. Nobody calls that &quot;artificial intelligence&quot; anymore, it is programming language design, compiler building (In the sixties there was a project to build a compiler giving &lt;em&gt;better&lt;/em&gt; code than experienced assembly language programmers... and they succeeded. Doing this is bread-and-butter for modern optimizing compilers.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The curse of AI is that when they find out how to solve a problem, it suddenly isn't AI anymore. Heuristics, search in humongous spaces getting cut down to manageable sizes, all that just melts away.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastEditorUserId="2905" LastEditDate="2016-03-01T19:20:21.163" LastActivityDate="2016-03-01T19:20:21.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3473" PostTypeId="1" CreationDate="2016-03-01T21:22:28.107" Score="2" ViewCount="266" Body="&lt;p&gt;From Bruno Belhoste's &lt;a href=&quot;http://dx.doi.org/10.1007/978-1-4612-2996-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Augustin-Louis Cauchy — A Biography&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Even if we should doubt the remark Valson attributed to Lagrange, it is likely that he did advise Louis-François about to his son's education in mathematics: Do not allow him even to open a mathematics book nor write a single number before he has completed his studies in literature.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why did Lagrange say that Cauchy should concentrate his efforts on literature before studying mathematics?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="4251" LastEditDate="2021-06-23T07:03:16.613" LastActivityDate="2021-06-23T07:03:16.613" Title="Why did Lagrange say that Cauchy should learn literature before mathematics?" Tags="|mathematicians|biographical-details|quote|lagrange|cauchy|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3474" PostTypeId="2" ParentId="3470" CreationDate="2016-03-01T23:09:35.857" Score="5" Body="&lt;p&gt;Modern theories reproduce caloric in the effective sense (similar to effective &quot;particles&quot; like phonons) for most phenomena it was used to explain. Mass was not really an issue because the fluid was assumed to be very &quot;subtle&quot; to easily permeate ordinary matter, and therefore to have negligible mass (from modern perspective, heating does add some minuscule mass due to mass-energy equivalence). One weakness emerged in 1820-s with the adoption of wave optics, it became hard to keep interpreting radiant heat as caloric particles streaming through space. But the key weakness was the caloric explanation of heating from friction, namely that friction forces out some fluid from between the atoms. If that were the case the released heat would obey a material conservation law, not grow proportionally to mechanical work as predicted by the mechanical theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This motivated Rumford's experiments in 1798. He was relatively successful, but caloric theory had advantages in other areas, and defenders pointed out flaws in Rumford's experiments that he could not compensate for. So in 1820-s Carnot was still inspired by the caloric water wheel analogy in his work on steam engines, and Joules' improved experiments on the mechanical equivalent of heat were still coldly received in 1842. The articulation of the energy conservation law by Mayer in 1841, Thomson paying close attention to Joule's experiments in 1847, and Clausius's mechanical reinterpretation of Carnot's theory in 1850 had to come together to displace the caloric theory in 1850-s. See &lt;a href=&quot;http://galileoandeinstein.physics.virginia.edu/more_stuff/TeachingHeat.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fowler's Teaching Heat: the Rise and Fall of the Caloric Theory&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; &#xA;There were two different theories of heat fluid, the earlier one, phlogiston, was introduced by Stahl in 1703, and the later one, caloric, put forth by Lavoisier in 1780-s. The theory was successful at explaining several types of physical and chemical phenomena, namely heat exchange, combustion, latent heat in phase transitions, and decalcination (formation of metals). In decalcination, for example, adding phlogiston to &quot;earths&quot; (ores) explained their acquisition of metalic properties. Black's work on latent and specific heats in 1757-1764, and Wilcke's in 1772, was seen as decisive in preferring phlogiston over the mechanical theory. It was hard to explain mechanically where the latent heat went, why capacity for heat was not proportional to density, or how phase transitions took place. Black, on the other hand, naturally explained it all, latent heat was due to phlogiston's ability to combine chemically with matter, and gaseous state was a solution of liquid in phlogiston. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the addition of mass in combustion (burning) eventually became noticable, and was meticulously measured by Lavoisier. This led to his criticism of the phlogiston theory and its explanation of combustion in 1780s: according to Stahl phlogiston escaped a substance, whereas according to Lavoisier oxygen was added to it. While this part is mentioned in modern expositions, the other part is often omitted, Lavoisier's oxygenation theory still required a heat fluid. In the notes on Kirwan's Essay on Phlogiston (1784) Lavoisier writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...When a metal is heated to a certain temperature, and when its particles are separated from each other to a certain distance by heat, and their attraction to each other is sufficiently diminished, it becomes capable of decomposing vital air, from which it seizes the base, namely oxygen, and sets the other principle, namely the caloric, at liberty... If therefore we attach to the word Inflammability the idea of disengagement of caloric and light, such as takes place in the phenomena of combustion and calcination, we must conclude, that vital air or oxygenous gas is the inflammable body most eminently, since it is principally and almost entirely from this substance that the caloric and light are disengaged&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Moreover, the caloric was needed to retain Black's part of the theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia's history article is not very good, unfortunately, see &lt;a href=&quot;https://books.google.com/books/about/Kant_and_the_Exact_Sciences.html?id=cBtOFUg4tHAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chemical Revolution chapter in Friedman's book&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-24T11:50:26.070" LastActivityDate="2020-04-24T11:50:26.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3475" PostTypeId="1" CreationDate="2016-03-01T23:35:20.293" Score="3" ViewCount="345" Body="&lt;p&gt;The space of sections of a bundle $\pi: E \to B$ is commonly denoted $\Gamma(E)$. (Note that the graph of a function $f$ is $\Gamma(f):=\left\{(x,f(x))\right\}$, and a particular section $\sigma: B \to E$ &quot;is&quot; a graph $\left\{(b,\sigma(b)) : b \in B\right\}$; but it is the &lt;em&gt;space of all such objects&lt;/em&gt; that is called $\Gamma(E)$.  I find this a bit confusing, etymologically.) If knowable, what is the reason for this letter (and where and by whom was this notation introduced)?&lt;/p&gt;&#xA;" OwnerUserId="3669" LastEditorUserId="26" LastEditDate="2016-03-02T09:47:04.183" LastActivityDate="2016-03-02T09:47:04.183" Title="Why is the space of sections of $E$ called $\Gamma(E)$?" Tags="|notation|differential-geometry|topology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3476" PostTypeId="1" CreationDate="2016-03-01T23:48:49.480" Score="0" ViewCount="307" Body="&lt;p&gt;Who did create the notions of acid, base, and salt first?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many chemists had mentioned these 3 terms before Svante Arrhenius defined them in 1884. For example, Antoine Lavoisier mentioned the notions of acid and base in 17XX; Paracelsus mentioned salt in 15XX; litmus was used for the first time about 1300 AD by Spanish alchemist Arnaldus de Villa Nova; related chemicals were used in ancient time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, who are the first to define the notions of acid, base, and salt respectively?&lt;/p&gt;&#xA;" OwnerUserId="2605" LastEditorUserId="2905" LastEditDate="2016-03-02T14:51:26.330" LastActivityDate="2016-03-02T15:23:33.147" Title="Who did create the notions of acid, base, and salt first?" Tags="|chemistry|alchemy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3477" PostTypeId="2" ParentId="3476" CreationDate="2016-03-02T08:43:37.520" Score="1" Body="&lt;p&gt;We have to go back at least to &lt;a href=&quot;https://en.wikipedia.org/wiki/Alchemy#Renaissance_and_early_modern_Europe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Renessaince alchemy&lt;/a&gt;; see &lt;a href=&quot;https://en.wikipedia.org/wiki/Paracelsus#Chemistry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paracelsus' chemistry&lt;/a&gt; with the &lt;em&gt;tria prima&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the idea of triparite alternatives to explain the nature of medicine, taking the place of a combustible element (&lt;em&gt;sulphur&lt;/em&gt;) a fluid and changeable element (&lt;em&gt;mercury&lt;/em&gt;) and a solid, permanent element (&lt;em&gt;salt&lt;/em&gt;.) The first mention of the mercury, sulphur, salt model was in the &lt;em&gt;Opus paramirum&lt;/em&gt; dating to about 1530 Paracelsus believed that the principles sulphur, mercury, and salt contained the poisons contributing to all diseases.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This doctrine evolved slowly into the modern chemical theory of &lt;a href=&quot;https://en.wikipedia.org/wiki/Principle_(chemistry)&quot; rel=&quot;nofollow noreferrer&quot;&gt;principles&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;See William Jensen, &lt;a href=&quot;http://www.che.uc.edu/Jensen/W.%20B.%20Jensen/Reprints/129.%20Base.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origin of the term &amp;quot;base”&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term “base” appears to have been first used in 1717 [“Mémoires sur le nitre,” Mém.Acad.R.Sci. (Paris), 1717] by the French chemist, &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_L%C3%A9mery&quot; rel=&quot;nofollow noreferrer&quot;&gt;Louis Lémery&lt;/a&gt; (1672-1743), as a synonym for the older Paracelsian term “matrix”. In keeping with 16th-century animism, Paracelsus had postulated that naturally occurring salts grew within the earth as a result of a universal acid or seminal principle having impregnated an earthy matrix or womb. By the early 1730s the newer term had largely replaced the older Paracelsian terminology and was being used by such French chemists as &lt;a href=&quot;https://en.wikipedia.org/wiki/Henri-Louis_Duhamel_du_Monceau&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henri Louis Duhamel du Monceau&lt;/a&gt; (1700-1782) [“Sur le sal ammoniac,” Mém.Acad.R.Sci. (Paris), 1735].&lt;/p&gt;&#xA;&lt;p&gt;Its modern meaning and general introduction into the chemical vocabulary, however, is usually attributed to the French chemist, &lt;a href=&quot;https://en.wikipedia.org/wiki/Guillaume-Fran%C3%A7ois_Rouelle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guillaume-François Rouelle&lt;/a&gt;, who used the term in a memoir on salts written in 1754 [“Memoire sur les sels neutres,” Mém.Acad.R.Sci. (Paris), 1754].&lt;/p&gt;&#xA;&lt;p&gt;In this paper, which was an extension of an earlier memoir on the same subject written in 1744 [“Memoire sur les sels neutres,” Mém.Acad.R.Sci. (Paris), 1744], Rouelle pointed out that the number of known salts had increased significantly during the 17th- and early 18th-centuries, due not only to the preparation of new salts, but also to an increasing ability to distinguish between sodium and potassium compounds, and to a generalization of the concept so as to include many substances, such as the alums and vitriols (i.e., sulfates), that had been previously excluded.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also the affinity table: “Table of the different relationships observed between different substances” presented by &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89tienne_Fran%C3%A7ois_Geoffroy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Etienne François Geoffroy&lt;/a&gt; to the Paris Academy in 1718.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mary Jo Nye, &lt;a href=&quot;https://books.google.it/books?id=zOXB_AVT08kC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Chemical Philosophy to Theoretical Chemistry&lt;/a&gt; (1994), page 79-on on &amp;quot;affinity&amp;quot; and Lavoisier's &amp;quot;Tableau des substances simples&amp;quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-03-02T15:23:33.147" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3478" PostTypeId="2" ParentId="3422" CreationDate="2016-03-02T12:24:39.823" Score="1" Body="&lt;p&gt;Perhaps the answer to that is that a physicist will grab the best (mathematical) tool for the job at hand. Sometimes this means the tool is reshaped (and often extended out any recognition) to fit the use better. Other times it is found it doesn't cut it, and is discarded for one custom-built.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastActivityDate="2016-03-02T12:24:39.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3480" PostTypeId="1" CreationDate="2016-03-02T16:42:25.797" Score="16" ViewCount="2436" Body="&lt;p&gt;Morris Kline, in Mathematical Thought from Ancient to Modern Time, writes in chapter 40 (The Installation of Rigor in Analysis), &quot;Though Bolzano and Cauchy had rigorized (somewhat) the notions of continuity and the derivative, Cauchy and nearly all mathematicians of his era believed and many texts &quot;proved&quot; for the next fifty years that a continuous function must be differentiable.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it true that Cauchy believed that a continuous function must be differentiable as Kline claims?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1.  As user Renfro correctly pointed out &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=272289&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; the mathematician who attempted to &quot;prove&quot; that continuity implies differentiability was Ampere (rather than Cauchy).  The first one to confuse the two names seems to have been Boyer. I can't vouch for Kline's philosophy but at any rate his history can only be as good as the historians he relies upon, and in this case he seems to have copied the claim about Cauchy from Boyer's confused passage.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 2. Cauchy in his 1823 book mentions the example of the function $y=x^{\frac{1}{3}}$ and points out that the derivative is infinite at the origin.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="4251" LastEditDate="2021-06-23T07:03:11.193" LastActivityDate="2023-08-20T03:23:03.653" Title="Is Kline right that Cauchy believed that continuous functions must be differentiable?" Tags="|mathematics|mathematicians|real-analysis|19th-century|cauchy|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3481" PostTypeId="2" ParentId="3480" CreationDate="2016-03-03T00:09:10.453" Score="10" Body="&lt;p&gt;Many mathematicians did believe that continuous functions are mostly differentiable (except for some isolated points perhaps) until Weierstrass made a splash by publishing an example of a nowhere differentiable continuous function in 1875. But it was not so much for the rigor reason implied by Kline, and repeated in older history books, as for the difference in the understanding of what a function is. Kline's phrasing is very misleading because &amp;quot;Cauchy and nearly all mathematicians&amp;quot; meant the concept differently, and for their version the issue was of little importance, see &lt;a href=&quot;http://www.springer.com/us/book/9783764325725&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medvedev's Scenes from the History of Real Functions&lt;/a&gt; (Ch. 5).&lt;/p&gt;&#xA;&lt;p&gt;The 18-th century function concept was more or less of an analytic expression in closed form, and those of course were both continuous and differentiable (except for isolated points) in the modern sense, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2035/was-18th-century-algebra-more-symbolic-formal-than-the-modern-conception/2038#2038&quot;&gt;Was 18th century algebra more symbolic/formal than the modern conception?&lt;/a&gt; So for the good part of 19-th century those were still the functions that most mathematicians had in mind. Obvious counterexamples certainly known to Cauchy and others, like the square root function at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, showed that continuity did not imply differentiability at some points, but all that coming apart of functions, continuous functions, and differentiable functions was taking place at few isolated points only. And since it was accepted that functions may become &amp;quot;infinite&amp;quot; at isolated points the supposed &amp;quot;belief&amp;quot; was correct (even for our analytic functions), although that would be an odd way to put it, and it is more appropriate to say that the question was moot.&lt;/p&gt;&#xA;&lt;p&gt;The view of functions started to change after Gauss's work on hypergeometric functions, and especially after Fourier's work on trigonometric series. Only in 1822 in &lt;em&gt;Théorie Analytique de la Chaleur&lt;/em&gt; (Analytic Theory of Heat) Fourier for the first time gave something close to the modern definition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In general, the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; represents a succession of values or ordinates each of which is arbitrary... We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as it were a single quantity&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But, as &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/HistTopics/Functions.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor's function concept history&lt;/a&gt; article points out, &amp;quot;when he begins to prove theorems about expressing an arbitrary function as a Fourier series, he uses the fact that his arbitrary function is continuous in the modern sense!&amp;quot; It is Dirichlet who started using Fourier's definition in a consistent manner, and popularized it. But even after the new function concept took hold, and Weierstrass's and other pathological counterexamples were accepted, Poincare was not happy about it as late as 1899 (quoted from MacTutor):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose... Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show that our ancestor's reasoning was at fault, and we shall never get anything more than that out of them&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-08-20T03:23:03.653" LastActivityDate="2023-08-20T03:23:03.653" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="3482" PostTypeId="2" ParentId="3480" CreationDate="2016-03-03T13:14:07.467" Score="5" Body="&lt;p&gt;&lt;em&gt;New edition&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Frank Smithies, &lt;a href=&quot;https://books.google.it/books?id=RWf3-PwokpMC&amp;amp;pg=PA188&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy and the creation of complex function theory&lt;/a&gt; (1997), page 188:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Cauchy] first contemplates the possibility that a continuous function may fail to have a continuous derivative in [1839], remarking that one cannot be certain that a function and its derivative necessarily have discontinuities or infinities for the same values of the independent variable. Here he may perhaps have been influenced by some comments made by Frisiani and Piola in the appendix to their Italian translation (1834) of the first Turin memoir [&lt;em&gt;Extrait du Mémoire presenté à l'Academie de Turin, le 11 Octobre 1831&lt;/em&gt;]. It is clear that he was concerned only by the possibility that the derivative might have discontinuities at isolated points. In any case, he did insert the additional hypothesis that the derivative should be continuous when [the memori] was republished in his &lt;em&gt;Exercices d'Analyse&lt;/em&gt; in 1841, although it appears that he did not do so consistently in later publications.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See: A.L.Cauchy, &lt;em&gt;Note sur l'intégration des equations différentielles des mouvements planétaires&lt;/em&gt; (1839), &lt;a href=&quot;https://books.google.it/books?id=4VpDAAAAcAAJ&amp;amp;pg=PA27&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exercices d'Analyse et de Physique Mathématique, Vol. I&lt;/a&gt;, page 27–32.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For subsequent developments, see:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ivor Grattan-Guinness (editor), &lt;a href=&quot;https://books.google.it/books?id=OLNeNIbD3jUC&amp;amp;pg=PA138&quot; rel=&quot;nofollow noreferrer&quot;&gt;From the Calculus to Set Theory: 1630-1910&lt;/a&gt; (1980), &lt;strong&gt;Ch.&amp;amp;nbsp:3 The emergence of mathematical analysis and its foundational progress, 1780–1880&lt;/strong&gt;, page 94–on:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[page 138] The Weierstrassians took up the construction of analytic expressions&#xA;which defined functions with infinities of oscillations and discontinuous within a finite interval, and they examined the consequences for mathematical analysis. [...] Further such examples were studied in a long paper [of] 1875 ['&lt;a href=&quot;http://culturemath.ens.fr/nodeimages/images/Darboux1875.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur les fonctions discontinues'&lt;/a&gt;, Ann.sci.Ecole Norm.Sup., (2)4(1875)] on various aspects of Weierstrassian analysis and Riemannian problems by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Gaston_Darboux&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaston Darboux&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;We thus obtain a continuous function of which the derivative is discontinuous for all commensurable [rational] values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but exists for all values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; he concluded (p. 109). After considering this and other examples of functions constructed in his paper, he remarked understandably that &amp;quot;it seems difficult to indicate a general character which allows the recognition of whether a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; has a primitive function, that is to say, if it is the derived [function] of another function' (p. 111). [...] With functions such as these we see the question of continuity vis-a-vis differentiability arise in a stark form. Could there be a continuous function which is differentiable nowhere ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Weierstrass answered this question in the affirmative in a manuscript [of] 1872, with the example [...]. He published it in his 1880 ['Zur Functionenlehre', Monatsb.Preuss.Akad.Jifliss. Berlin, (1880)], art. 6, but it appeared first in  &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_du_Bois-Reymond&quot; rel=&quot;nofollow noreferrer&quot;&gt;du Bois Reymond&lt;/a&gt; 1875 ['Versuch einer Classification der willktirlichen Functionen...&amp;quot; J.rei.ang.Math., 79(1875), 21-37]. An extensive discussion of such functions was carried on in print as well as in correspondence, and a large number of continuous non-differentiable functions were invented over the following decades. [See Thomas Hawkins, &lt;a href=&quot;https://books.google.it/books?id=oV1aLqag6WwC&amp;amp;pg=PA44&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lebesgue's Theory of Integration: Its Origins and Development&lt;/a&gt;, (1970), page 44-50].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And see Johan Thim's theses: &lt;a href=&quot;https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Continuous Nowhere Differentiable Functions&lt;/a&gt;, page 6, for a useful list and an &lt;a href=&quot;https://fr.wikipedia.org/wiki/Fonction_continue_nulle_part_d%C3%A9rivable&quot; rel=&quot;nofollow noreferrer&quot;&gt;abstract&lt;/a&gt; in fr.wiki.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For some recent sources, see:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Umberto Bottazzini &amp;amp; Jeremy Gray, &lt;a href=&quot;https://books.google.it/books?id=X08dgn3YcLwC&amp;amp;pg=PR8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hidden Harmonies: The Rise of Complex Function Theory&lt;/a&gt; (2013)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jeremy Gray, &lt;a href=&quot;https://books.google.it/books?id=zpDDCgAAQBAJ&amp;amp;pg=PR9&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Real and the Complex: A History of Analysis in the 19th Century&lt;/a&gt; (2015)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="5989" LastEditDate="2022-12-22T08:53:33.940" LastActivityDate="2022-12-22T08:53:33.940" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="3483" PostTypeId="1" CreationDate="2016-03-03T15:54:27.213" Score="2" ViewCount="193" Body="&lt;p&gt;I would like to know if there was an approach of ancient astronomers to describe what happens during sunset in the following way: that the sun is going toward west during the day and in the end moving &lt;strong&gt;upward&lt;/strong&gt; of the horizon and disappear(and then making all the way back to east &lt;strong&gt;above&lt;/strong&gt; the sky and not under earth).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Maybe they thought so because the upper western sky is red after sunset so they've tried to explain it by saying that the sun entered to the sky [didn't move under earth])&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i'm aware that there was among many astronomers a conception of flat earth, but i'm particulary asking for that specific way of percieving sunset. who ever hold such an explanation of sunset??&lt;/p&gt;&#xA;&#xA;&lt;p&gt;who ever said such a thing in the history of astronomy? i mean greek astronomers BCE and also maybe astronomers in medival europe CE.&#xA; thanks!&lt;/p&gt;&#xA;" OwnerUserId="3780" LastEditorUserId="2905" LastEditDate="2016-03-06T11:02:41.153" LastActivityDate="2016-03-25T20:51:42.637" Title="Old perception of the sunset: the sun is going ABOVE the horizen. does anyone heard such a thing?" Tags="|earth-sciences|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3484" PostTypeId="1" CreationDate="2016-03-03T22:21:24.127" Score="2" ViewCount="33" Body="&lt;p&gt;Today, the commonly recognized authority on the naming of astronomical bodies is the &lt;a href=&quot;http://www.iau.org/&quot; rel=&quot;nofollow&quot;&gt;International Astronomical Union (IAU)&lt;/a&gt;. It was created in 1919, and quickly rose to a position of prominence within the astronomical community.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before the creation of the IAU, what body (if any) was considered the leading international organization for astronomical naming?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-03-03T22:21:24.127" Title="Before the IAU, what was the international governing body for naming astronomical bodies?" Tags="|astronomy|naming-conventions|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3485" PostTypeId="2" ParentId="3466" CreationDate="2016-03-04T05:23:55.263" Score="7" Body="&lt;p&gt;From &lt;a href=&quot;https://books.google.com/books?id=G_iziBAPXtEC&amp;amp;pg=170&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Ultimate Quotable Einstein&lt;/em&gt; p. 170&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If I were wrong, then one [author] would have been enough!&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;Einstein’s retort with regard to his theory when he heard that a book titled &lt;a href=&quot;https://archive.org/details/HundertAutorenGegenEinstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;100 Authors against Einstein&lt;/em&gt;&lt;/a&gt; was published in Germany. Quoted in Stephen Hawking, &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=oZhagX6UWOMC&amp;amp;pg=193&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Brief History of Time&lt;/a&gt;&lt;/em&gt; (London: Bantam, 1988), &lt;a href=&quot;https://books.google.com/books?id=oZhagX6UWOMC&amp;amp;pg=193&quot; rel=&quot;nofollow noreferrer&quot;&gt;178&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-25T21:54:29.717" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3486" PostTypeId="1" CreationDate="2016-03-04T12:17:16.123" Score="11" ViewCount="776" Body="&lt;p&gt;I heard in wave optics and electromagnetism that Hamilton could have discovered the Schrödinger equation, or that he was the first man who used the expression&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \Psi(x)= \exp(i S(x)/\hbar)\,. $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also heard that Hamilton got the idea from the eikonal equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ (\nabla S)^{2}=n(x)\,, $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but that he couldn't complete it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was trying to show that light particles could also be described as a wave, but had no way to prove it. Is this what happened?&lt;/p&gt;&#xA;" OwnerUserId="3786" LastEditorUserId="55" LastEditDate="2016-10-20T17:23:00.190" LastActivityDate="2019-01-30T07:17:22.947" Title="Did wave optics anticipate quantum mechanics?" Tags="|quantum-mechanics|classical-mechanics|optics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3487" PostTypeId="1" AcceptedAnswerId="3488" CreationDate="2016-03-04T15:45:50.960" Score="-1" ViewCount="373" Body="&lt;p&gt;Newton changed the way we look at physics by giving a simple rules such like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;F=ma&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In these equation we don't count with specific numbers, we are only stating what are the dependences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the truly first scientist who counted only with dependences of one factor on another?&lt;/p&gt;&#xA;" OwnerUserId="3630" LastEditorUserId="3630" LastEditDate="2016-03-05T20:21:54.673" LastActivityDate="2016-03-05T22:26:28.557" Title="Who was the first scientist who calculated with physical quantities?" Tags="|physics|" AnswerCount="1" CommentCount="1" ClosedDate="2016-03-06T11:03:30.030" ContentLicense="CC BY-SA 3.0" />
  <row Id="3488" PostTypeId="2" ParentId="3487" CreationDate="2016-03-04T16:18:31.067" Score="2" Body="&lt;p&gt;Newton's formulation of his &lt;strong&gt;II Law of Motion&lt;/strong&gt; into the &lt;a href=&quot;https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principia&lt;/a&gt; is not &amp;quot;symbolic&amp;quot;; see:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sir Isaac Newton's &lt;a href=&quot;https://books.google.it/books?id=ySYULc7VEwsC&amp;amp;pg=PA13&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Principles of Natural Philosophy and His System of the World&lt;/a&gt;, Andrew Motte's translation (1729) revised by Florian Cajori (1934), page 13.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The first &amp;quot;algebraic&amp;quot; formulation of Newton's law of motion is due to:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Leonhard Euler, &lt;a href=&quot;https://math.dartmouth.edu/%7Eeuler/docs/originals/E177.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Decouverte d'un nouveau principe de Mecanique&lt;/a&gt; (1752), page 196:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;XXII. [...] after having decomposed all the forces acting on the body into the three perpendicular components &lt;span class=&quot;math-container&quot;&gt;$P, Q, R$&lt;/span&gt; [...] the movement of the body will be described by the three following formulae:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$I. \ \ 2M ddx=P dt^2 \ \ \ II. \ \ 2M ddy=Q dt^2 \ \ \ III. \ \ 2M ddz=R dt^2.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;An early example of dependency between physical magnitudes is Aristotle (wrong) law of motion:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If, then, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is the mover, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; the moved, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; the distance moved, and &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; the time, then in the same time the same force &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; will move &lt;span class=&quot;math-container&quot;&gt;$\dfrac 1 2 B$&lt;/span&gt; twice the distance &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, and in &lt;span class=&quot;math-container&quot;&gt;$\dfrac 1 2 D$&lt;/span&gt; it will move &lt;span class=&quot;math-container&quot;&gt;$\dfrac 1 2 B$&lt;/span&gt; the&#xA;whole distance &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;; for thus the rules of proportion will be observed [&lt;em&gt;Physics&lt;/em&gt;, &lt;strong&gt;Book VII&lt;/strong&gt;, 249b27-250a9].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-03-05T22:26:28.557" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3489" PostTypeId="2" ParentId="3480" CreationDate="2016-03-05T15:21:52.030" Score="0" Body="&lt;p&gt;This is to supplement the two other answers. It is worth reading in the context of the question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/journals/bull/2007-44-02/S0273-0979-07-01140-8/home.html&quot; rel=&quot;nofollow&quot;&gt;http://www.ams.org/journals/bull/2007-44-02/S0273-0979-07-01140-8/home.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Absolutely&lt;/em&gt; continuous functions are the ones that all allow for calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This type of thing was also used long ago by Daniell for his integration, etc.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2016-03-05T15:21:52.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3490" PostTypeId="2" ParentId="3486" CreationDate="2016-03-05T19:16:07.110" Score="2" Body="&lt;p&gt;What Hamilton discovered is a mathematical &quot;Hamiltonian formalism&quot;. It was applied to those parts of physics which were known at the time of Hamilton: classical mechanics and optics. There was no slightest reasons at the time of Hamilton to suspect that matter on small scale does not obey the laws of classical mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That this is so, is a late 19s century discovery. However it turned out that Hamiltonian formalism is so general that it applies to quantum mechanics as well.&#xA;Thus many equations written by Hamilton (and not only by Hamilton, but by other late 18s and early 19s century researchers in mechanics, like Lagrange) actually apply, if they are correctly interpreted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, Calculus, a mathematical tool invented in 17s century serves not only&#xA;mechanics, but all physics discovered later. But nobody claims on this ground that Newton and Leibniz discovered all physics which was discovered later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Of course this is a miracle that the same mathematical tool applies to a very wide class of phenomena in the universe, including those which were not known when the tool was invented/discovered, but this is the way our world is created.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-03-05T19:21:19.733" LastActivityDate="2016-03-05T19:21:19.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3491" PostTypeId="1" AcceptedAnswerId="3526" CreationDate="2016-03-06T02:19:33.690" Score="5" ViewCount="89" Body="&lt;p&gt;In 1942 the famous British biochemist Joseph Needham was sent to China to help their scientists because the Japanese had been targetting Chinese universities, academies and technical institutions since about 1938.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that England was in diabolical trouble itself in 1942, I wonder who made the case that convinced Churchill and his government to fund the mission.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it a particular scientist in the Royal Society? Or a politician? Or someone &#xA;else?&#xA;(I couldn't find the answer in Winchester's biography of Needham, &quot;The Man Who&#xA;Loved China&quot;.)&lt;/p&gt;&#xA;" OwnerUserId="3540" LastEditorUserId="3540" LastEditDate="2016-03-12T14:46:29.160" LastActivityDate="2016-04-11T15:32:49.743" Title="Who convinced Churchill to fund Needham's trip to China in 1942?" Tags="|social-context|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3492" PostTypeId="2" ParentId="3393" CreationDate="2016-03-06T02:43:40.953" Score="0" Body="&lt;p&gt;Many different kinship structures have arisen in small human societies from prehistoric times in order to avoid a variety of genetic disorders.&#xA;Maybe I'm drawing a long bow here, but I'd say observations of the result of close (and distant) genetic relationships led to deliberate attempts to improve plants and animals through selective breeding, and via Mendel and others, to the modern science of genetics.&lt;/p&gt;&#xA;" OwnerUserId="3540" LastActivityDate="2016-03-06T02:43:40.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3493" PostTypeId="2" ParentId="3486" CreationDate="2016-03-06T02:46:36.927" Score="10" Body="&lt;p&gt;It is more accurate to say that Hamilton anticipated some of the ideas of &lt;em&gt;mathematics and heuristics&lt;/em&gt; of quantum mechanics, that would later inspire Schrödinger to produce his formulation of wave mechanics. The reason he was able to anticipate those ideas is that the quantum wave-particle duality had a classical predecessor, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hamilton&amp;#39;s_optico-mechanical_analogy&quot; rel=&quot;nofollow noreferrer&quot;&gt;optico-mechanical analogy&lt;/a&gt;. Indeed, it was optics, not mechanics, that originally inspired the Hamiltonian formalism. But no, Hamilton was not about to discover the Schrödinger equation. There is a qualitative difference between the first order Hamilton-Jacobi equation, and the second order Schrödinger equation, the connection is only recovered in the  quasi-classical limit. And it is only for the latter that the stationary phase approximation for integrals of &lt;span class=&quot;math-container&quot;&gt;$\exp(i S(x)/\hbar)$&lt;/span&gt; is used. The approximation itself was only introduced by Kelvin in 1887, Hamilton's approach was more geometric. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a simple form the analogy was &lt;a href=&quot;https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;discovered by Huygens around 1670&lt;/a&gt; (published in Traitė de la Lumiere, 1678), who noticed that propagation of waves could be dually described in terms of wavefronts and rays (&quot;characteristics&quot;) perpendicular to them. The latter can be considered as trajectories of particles, and massive amounts of particles spreading along characteristics can create the appearence of a continuous wave. But it also works vise versa, and Huygens suggested that light may well be a wave, with geometric optics of rays being only the first approximation. In particular, Huygens showed how Fermat's least time principle follows from the analogy, Johann Bernoulli used it to solve the famous brachistochrone problem in 1696, and in 1818 Fresnel showed how not only geometric optics but also diffraction and interference can be explained by wave optics, which led to its wide acceptance in the 19th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it was Hamilton who explored the analogy in its full generality. As &lt;a href=&quot;http://www.freeinfosociety.com/media/pdf/2287.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guillemin writes in Geometric Asymptotics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1828 Hamilton published his fundamental paper on geometrical optics, introducing his &quot;characteristics&quot;, as a key tool in the study of optical instruments. It wasn't until substantially later that Hamilton realized that his method applied equally well to&#xA;  the study of mechanics. Hamilton's method was developed by Jacobi and has been a cornerstone of theoretical mechanics ever since... It is interesting to note that although Hamilton was aware of the work of Fresnel, he chose to ignore it completely in his fundamental papers on geometrical optics.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The eikonal equation is only the simplest case of the Hamilton-Jacobi equation, when the Hamiltonian of the mechanical system only has the standard kinetic energy term. In general, the familiar equations of Hamilton dynamics are solved by the characteristics of the corresponding Hamilton-Jacobi equation. Although the method of stationary phase is often used to derive geometric asymptotics today, it was not available to Hamilton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the conflict between the wave optics and the newly introduced light quanta Schrödinger, a big fan of Hamiltonian dynamics, was reminded of the optico-mechanical analogy, and started thinking about classical mechanics being the limit of a new mechanics along the lines of geometric optics being a limit of wave optics. This led him to his celebrated equation. In the quasi-classical limit the surfaces of equal phase are the &quot;wave fronts&quot;, and the trajectories of the particles are the characteristics. A systematic development of this approach is known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/WKB_approximation&quot; rel=&quot;nofollow noreferrer&quot;&gt;WKB (Wentzel–Kramers–Brillouin) method.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-01-30T07:17:22.947" LastActivityDate="2019-01-30T07:17:22.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3494" PostTypeId="1" AcceptedAnswerId="3502" CreationDate="2016-03-06T07:56:33.260" Score="5" ViewCount="2271" Body="&lt;p&gt;There is no information on wiki. Is it known to Newton? &#xA;It seems that this has not been mentioned in most textbooks.&lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2016-03-18T03:31:38.927" Title="Who discovered total internal reflection?" Tags="|physics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3496" PostTypeId="2" ParentId="3494" CreationDate="2016-03-06T14:54:34.743" Score="1" Body="&lt;p&gt;A quick search shows the references on Kepler (1611, the book Dioptrice)&#xA;&lt;a href=&quot;http://kepler.nasa.gov/Mission/JohannesKepler/&quot; rel=&quot;nofollow&quot;&gt;http://kepler.nasa.gov/Mission/JohannesKepler/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-03-06T18:33:00.130" LastActivityDate="2016-03-06T18:33:00.130" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3497" PostTypeId="2" ParentId="3494" CreationDate="2016-03-06T16:00:08.440" Score="4" Body="&lt;p&gt;For Early Modern Science we can find some sources into:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A.I. Sabra, &lt;a href=&quot;https://books.google.it/books?id=nB84AAAAIAAJ&amp;amp;pg=PA108&quot; rel=&quot;nofollow&quot;&gt;Theories of Light: From Descartes to Newton&lt;/a&gt;, page 108, referring to Descartes' &lt;a href=&quot;https://fr.wikipedia.org/wiki/La_Dioptrique_(Descartes)&quot; rel=&quot;nofollow&quot;&gt;Dioptrique&lt;/a&gt; (1637), A.T.VI, &lt;a href=&quot;https://fr.wikisource.org/wiki/Page%3A%C5%92uvres_de_Descartes%2C_%C3%A9d._Cousin%2C_tome_V.djvu/28&quot; rel=&quot;nofollow&quot;&gt;page 99&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://books.google.it/books?id=nB84AAAAIAAJ&amp;amp;pg=PA111&quot; rel=&quot;nofollow&quot;&gt;page 111&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although Descartes is careful to mark the place for the phenomenon of total reflection, his investigation lacks the systematic approach which characterized Kepler's experimental research.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://books.google.it/books?id=nB84AAAAIAAJ&amp;amp;pg=PA319&quot; rel=&quot;nofollow&quot;&gt;page 319&lt;/a&gt; for reference to Newton's comment that &quot;it was known that at a sufficiently great angle of incidence, the whole of the incident light was reflected back into the glass&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For Kepler &lt;a href=&quot;https://archive.org/stream/DioptriceByJohannesKeplerAkaIoannisKepleri#page/n41/mode/2up&quot; rel=&quot;nofollow&quot;&gt;Dioptrice&lt;/a&gt; (1611), see page 4, &lt;strong&gt;Ax.IX&lt;/strong&gt; and &lt;strong&gt;Prop XIII&lt;/strong&gt;. &lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-03-09T08:40:22.127" LastActivityDate="2016-03-09T08:40:22.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3498" PostTypeId="1" CreationDate="2016-03-03T23:05:03.670" Score="28" ViewCount="1825" Body="&lt;p&gt;From time to time one sees insistence that the inequality name &quot;Cauchy-Schwarz&quot; should include Buniakovsky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is based on a paragraph in a &lt;a href=&quot;http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/bunyakovsky.pdf&quot;&gt;note to the St Petersburg Academy&lt;/a&gt; from 1859, where Buniakovsky takes Cauchy's inequality to be well known (&quot;la relation bien connue&quot;, Cauchy is not named or cited), and fills it with values of continuous functions to get, in the limit, the analogous inequality $\int f(x)^2 dx \int g(x)^2 dx \geq (\int f(x)g(x) dx)^2$ for functions on an interval.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By today's standards the entire paper of Buniakovsky contains nothing more than the idea that a limit of inequalities on finite sums gives an inequality on integrals.  He derives the integral formula for the geometric mean of a function on an interval this way, and introduces $G$ and $M$ as notation for functionals that compute the geometric and arithmetic means of a function on an interval. As definitions and notation those may have been novel in some way at the time, but as a contribution to the theory of the Cauchy-Schwarz inequality, Buniakovsky's paper did not introduce any new method or result that would not have been obvious to Cauchy (or any mathematician of the times knowing Cauchy's inequality).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;is the tendency to attach Buniakovsky's name to the inequality particular to the Russian sphere of influence (mathematical, linguistic, political) or is also attested in independent Western sources&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;was Buniakovsky's paper the first to treat nonlinear integral inequalities as a stand-alone subject?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3791" OwnerDisplayName="zyx" LastEditorUserId="58" LastEditDate="2016-03-06T23:24:21.080" LastActivityDate="2017-02-15T08:12:26.317" Title="Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?" Tags="|mathematics|terminology|real-analysis|" AnswerCount="3" CommentCount="18" ContentLicense="CC BY-SA 3.0" />
  <row Id="3499" PostTypeId="2" ParentId="38" CreationDate="2016-03-06T21:28:36.717" Score="1" Body="&lt;p&gt;In short, they under-estimated the prime importance of context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;AI can be superior to human effort in some small, well-defined, self-contained worlds, e.g. small vehicles and machines running around a warehouse.&#xA;However, in the real world, they are lost, and can be completely out-performed by small children. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't imagine any AI in the near future being able to understand teenagers' language, slang and argot. Not only does it change quickly, but it includes irony, sarcasm, implied negatives, and much, much more, things that a computer program, no matter how sophisticated, won't be able to keep up with, for the main reason that teenagers don't necessarily publicize the manner in which their slang is to be understood. You have to be part of the &quot;in-crowd&quot;. They do, in fact, go out of their way to ensure that others don't understand the subtleties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's much more, of course, but human language is one aspect where computer programs are completely inferior, and will remain so, possibly forever.&lt;/p&gt;&#xA;" OwnerUserId="3540" LastActivityDate="2016-03-06T21:28:36.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3500" PostTypeId="2" ParentId="3498" CreationDate="2016-03-07T16:54:38.760" Score="6" Body="&lt;p&gt;This is the &lt;em&gt;elephant&lt;/em&gt; phenomenon that Russian mathematicians like to joke about.  It is widely known that Russia is the homeland of the elephants (&quot;Rossia--rodina slonov&quot;).  Similarly any important mathematical result ultimately must have a Russian source and since Buniakovski translated some of Cauchy's work into Russian, he was a natural choice for promotion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally I had occasion to study Buniakovski's translations of Cauchy and they are uniformly atrocious.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-03-07T16:54:38.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3501" PostTypeId="2" ParentId="3483" CreationDate="2016-03-07T18:23:03.263" Score="2" Body="&lt;p&gt;Such beliefs existed in various solar mythologies, see e.g. Sun Lore of All Ages, by William Tyler Olcott, [1914], (&lt;a href=&quot;http://sacred-texts.com/astro/slaa/index.htm&quot; rel=&quot;nofollow&quot;&gt;available at sacred-texts.com&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Romans actually believed that the sun was the wheel of Apollo's chariot. Each morning the god rose from the eastern sea, and drove his four spirited steeds across the sky, and in the evening he descended into the western sea. At night, he reposed in a golden boat which was borne along the northern edge of the earth to the rising point in the east.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Whether anyone was trying to use astronomy to support these beliefs, i do not know.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="3889" LastEditDate="2016-03-25T20:51:42.637" LastActivityDate="2016-03-25T20:51:42.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3502" PostTypeId="2" ParentId="3494" CreationDate="2016-03-08T01:43:37.237" Score="6" Body="&lt;p&gt;Although Kepler was first to study the phenomenon systematically it was known before him, and even used to explain the appearance of rainbow. It was done c. 1300 AD by &lt;a href=&quot;https://en.wikipedia.org/wiki/Kam%C4%81l_al-D%C4%ABn_al-F%C4%81ris%C4%AB#Optics&quot; rel=&quot;nofollow&quot;&gt;Al-Farisi&lt;/a&gt;, and independently c.1310 AD by &lt;a href=&quot;https://en.wikipedia.org/wiki/Theodoric_of_Freiberg#Physics&quot; rel=&quot;nofollow&quot;&gt;Theodoric of Freiberg&lt;/a&gt; in De Iride et Radialibus Impressionibus (On the Rainbow and the Impressions Created by Irradiance), and was one of the early examples of combining experimental study with mathematics. Both relied on a relatively early experimental tradition in optics manifested in &lt;a href=&quot;http://www.jstor.org/stable/3185879?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Ptolemy's Optics (c.150 AD)&lt;/a&gt;, which reports a series of meticulous experiments to measure refraction from air to water, air to glass, and water to glass, with refracted angles tabulated against incident angles at ten degree intervals from 0° to 80°. Ptolemy's work was expanded upon by &lt;a href=&quot;https://en.wikipedia.org/wiki/Book_of_Optics&quot; rel=&quot;nofollow&quot;&gt;Alhazen in Book of Optics (c.1000 AD)&lt;/a&gt;, which was the main inspiration for Theodoric's and Al-Farisi's experimental approach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is from &lt;a href=&quot;https://books.google.com/books?id=ZMNkAwAAQBAJ&amp;amp;pg=PT683&amp;amp;lpg=PT683&amp;amp;dq=theodoric+of+freiberg+total+internal+reflection&amp;amp;source=bl&amp;amp;ots=JYCc40MbaY&amp;amp;sig=4Qsi0RfCPE1yQHJvL9EbvpG_6mA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi0uZ3M8a_LAhVpu4MKHUWvCp4Q6AEIHDAA#v=onepage&amp;amp;q=theodoric%20of%20freiberg%20total%20internal%20reflection&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;The Cambridge History of Science: Volume 2, Medieval Science&lt;/a&gt;:&quot;&lt;em&gt;In one of the most remarkable experimental investigations of the Middle Ages Theodoric projected rays of light through crystalline spheres and water filled flasks meant to simulate the passage of sunlight through a drop of moisture in a rain cloud. He concluded that the primary bow resulted from refraction as a solar ray entered the raindrop, an internal reflection at the back of the raindrop, and a second refraction as it emerged from the raindrop&lt;/em&gt;&quot;. This is essentially the modern explanation often attributed to Descartes. It is unclear if Descartes was aware of Theodoric's work, it was &quot;rediscovered&quot; by Venturi in 1814.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-03-18T03:31:38.927" LastActivityDate="2016-03-18T03:31:38.927" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3503" PostTypeId="1" AcceptedAnswerId="3510" CreationDate="2016-03-08T21:29:08.217" Score="14" ViewCount="2281" Body="&lt;p&gt;The modern axiomatic approach to probability was established by Kolmogorov nearly 70 years ago. According to &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_probability&quot;&gt;Wikipedia&lt;/a&gt;, the first ideas connected to a mathematical theory of probability arose with Cardano, Fermat, Pascal during the 16th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find quite surprising that probability as a mathematical theory has emerged so late in time. In particular, as far as I know, ancient Greeks, that were skilled geometers (and mathematicians in general), did not even try to address this problem. One could argue that probability was not necessary at that time from a practical viewpoint while, for instance, geometry was more useful in everyday life. I think that this is not completely true. Indeed, it is well-known that ancient Greeks were used to bet or to play &lt;a href=&quot;https://en.wikipedia.org/wiki/Knucklebones&quot;&gt;games&lt;/a&gt; that involved dices and, therefore, probability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, my question: is it true that the first attempts to connect probability and mathematics date back to the 16th century? If the answer is affirmative, what are the most probable reasons of this late development? And in particular, why didn't the ancient Greeks give any contribution to this branch of mathematics despite its practical usefulness?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="3715" LastEditorUserId="55" LastEditDate="2016-03-10T04:42:41.173" LastActivityDate="2017-08-24T02:19:59.263" Title="Why wasn't probability developed in ancient Greece?" Tags="|ancient-greece|probability|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3504" PostTypeId="1" AcceptedAnswerId="3505" CreationDate="2016-03-07T23:47:36.170" Score="7" ViewCount="8848" Body="&lt;p&gt;According to the &lt;a href=&quot;http://science.jrank.org/pages/634/Atomic-Weight-History.html&quot;&gt;Science Encyclopedia&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Because atoms were much too small to be seen or measured by any common methods, absolute weights of atoms could not be determined. Rather, these first measurements were made by comparing weights of various atoms to hydrogen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How, though, in the early 1800's, were atomic weights compared to hydrogen?  For example, given a bottle of hydrogen gas, how can we use this to determine the atomic weight of gold?&lt;/p&gt;&#xA;" OwnerUserId="3811" OwnerDisplayName="Doubt" LastActivityDate="2024-01-04T18:51:31.300" Title="How were atomic weights originally measured?" Tags="|experimental-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3505" PostTypeId="2" ParentId="3504" CreationDate="2016-03-08T01:28:05.900" Score="5" Body="&lt;p&gt;Do you know about Lomonosov's and Gay Lussac's contribution in understanding of chemical reactions and Avogadro's contribution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These three chemists contributed to a great extent to the understanding of absolute formulas of substances. Lomonosov earlier was the first to come up to explain four types of chemical reactions in mid of 1700s and was the one to explain law of mass conservation during chemical reactions. Gay-Lussac later in early 1800s mentioned that gases react with each other in certain proportions. &quot;For example, at the same temperature and pressure, two volumes of hydrogen react with one volume of oxygen and form two volumes of water.&quot;[1].&#xA;&quot;Later, Amedeo Avogadro proposed that equal volumes of gases have the same number of particles if measured at the same temperature and pressure. The difficulty of explaining how one volume of oxygen could form two volumes of water without violating the current theory that atoms were indivisible was not resolved until the 1850s when Avogadro's explanation that molecules of gases, such as hydrogen and oxygen, existed as diatomic molecules (molecules with two atoms joined together) was finally accepted. If each oxygen molecule was composed of two oxygen atoms, then it was the molecule and not the atom that split apart to form two volumes of water.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;J. J. Berzelius did a great deal of experimental work in establishing atomic weights and he published his list of the weights of 54 elements along Unlike Dalton's atomic weights, the weights published by Berzelius match quite well the atomic weights used today.&quot; [1]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mendeleev later came up with periodic classification based on his systematic understanding and analysis of the then known elements and classified them in the periodic table. This later led to coming up with the long form of periodic table which is currently in use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;source:&#xA;1. &lt;a href=&quot;http://science.jrank.org/pages/634/Atomic-Weight-History.html&quot; rel=&quot;noreferrer&quot;&gt;http://science.jrank.org/pages/634/Atomic-Weight-History.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3813" OwnerDisplayName="user118008" LastActivityDate="2016-03-08T04:20:26.163" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3506" PostTypeId="2" ParentId="3498" CreationDate="2016-03-09T03:57:33.437" Score="11" Body="&lt;p&gt;Some reflections of &lt;a href=&quot;https://en.wikipedia.org/wiki/J._Michael_Steele&quot; rel=&quot;noreferrer&quot;&gt;J. Michael Steele&lt;/a&gt; (cf. The Cauchy-Schwarz Master Class. Cambridge University Press, 2004, pp. 10-12) on this matter: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;THE PACE OF SCIENCE -- THE DEVELOPMENT OF EXTENSIONS&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Augustin-Louis Cauchy (1789-1857) published his famous inequality in 1821 in the second of two notes on the theory of inequalities that formed the final part of his book &lt;em&gt;Cours d'Analyse Algébrique&lt;/em&gt;, a volume which was perhaps the world's first rigorous calculus text. Oddly enough, Cauchy did not use his inequality in his text, except in some illustrative exercises. The first time Cauchy's inequality was applied in earnest  by anyone was in 1829, when Cauchy used his inequality in an investigation of Newton's method for the calculation of the roots of algebraic and transcendental equations. This eight-year gap provides an interesting gauge of the pace of science; now, each month, there are hundreds--perhaps thousands--of new scientific publications where Cauchy's inequality is applied in one way or another.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A great many of those applications depend on a natural analog of Cauchy's inequality where sums are replaced by integrals,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$ \int_{a}^{b} f(x)g(x) \, dx \leq \left(\int_{a}^{b} f^{2}(x)\, dx\right)^{\frac12}\left(\int_{a}^{b} g^{2}(x)\, dx\right)^{\frac12} \quad ... (*)$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This bound first appeared in print in a &lt;em&gt;Mémoire&lt;/em&gt; by Victor Yacovlevich Bunyakovsky which was published by the Imperial Academy of Sciences of St. Petersburg in 1859. Bunyakovsky (1804-1889) had studied in Paris with Cauchy, and he was quite familiar with Cauchy's work on inequalities; so much so that by the time he came to write his &lt;em&gt;Mémoire&lt;/em&gt;, Bunyakovsky was content to refer to the classical form of Cauchy's inequality for finite sums simply as &lt;em&gt;well-known&lt;/em&gt;. Moreover, Bunyakovsky did not dawdle over the limiting process; he took only a single line to pass from Cauchy's inequality for finite sums to his continuous analog in $(*)$. By ironic coincidence, one finds that this analog is labelled as inequality $(\mathbf{C})$ in Bunyakovsky's &lt;em&gt;Mémoire&lt;/em&gt;, almost as though Bunyakovsky had Cauchy in mind.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Bunyakovsky's &lt;em&gt;Mémoire&lt;/em&gt; was written in French, but it does not seem to have circulated widely in Western Europe. In particular, it does not seem to have been known in Göttingen in 1885 when Hermann Amandus Schwarz (1843-1921) was engaged in his fundamental work on the theory of minimal surfaces.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the course of this work, Schwarz had the need for a a two-dimensional integral analog of Cauchy's inequality. In particular, he needed to show that if $S \subseteq \mathbb{R}^{2}$ and $f \colon S \to \mathbb{R}$ and $g \colon S \to \mathbb{R}$, then the double integrals&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$ A = \iint_{S} f^{2} \, dxdy, \quad B = \iint_{S} fg \, dxdy \quad C = \iint_{S} g^{2} \, dxdy$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;must satisfy the inequality &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$ |B| \leq \sqrt{A} \cdot \sqrt{C}, $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;and Schwarz also needed to know that the inequality is strict unless the functions $f$ and $g$ are proportional.&#xA;  An approach to this result via Cauchy's inequality would have been problematical for several reasons, including the fact that the strictness of a discrete inequality can be lost in the limiting passage to integrals. Thus, Schwarz had to look for an alternative path, and, faced with necessity, he discovered a proof whose charm has stood the test of time.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Schwarz based his proof on one striking observation. Specifically, he noted that the real polynomial&#xA;  $$ p(t) = \iint_{S} \left(tf(x,y)+g(x,y)\right)^{2} \, dxdy = At^{2}+2Bt+C$$&#xA;  is always nonnegative, and, moreover, $p(t)$ is strictly positive unless $f$ and $g$ are proportional. The binomial formula then tells us that the coefficients must satisfy $B^{2}\leq AC$, and unless $f$ and $g$ are proportional, one actually has the strict inequality $B^{2} &amp;lt; AC$. Thus, from a single algebraic insight, Schwarz found everything he needed to know.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Schwarz's proof requires the wisdom to consider the polynomial $p(t)$, but, granted that step, the proof is lightning quick. Moreover, ... Schwarz's argument can be used almost without change to prove the inner product form of Cauchy's inequality, and even there Schwarz's argument provides one with a quick understanding of the case of equality. Thus, there is a little to reason to wonder why Schwarz's argument has become a textbook favorite, even though it does require one to pull a rabbit--or at least a polynomial--out of a hat.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;THE NAMING OF THINGS -- ESPECIALLY INEQUALITIES&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In light of the clear historical precedence of Bunyakovsky's work over that Schwarz, the common practice of referring to the bound $(*)$ as Schwarz's inequality may seem unjust. Nevertheless, by modern standards, both Bunyakovsky and Schwarz might count themselves lucky to have their names so closely associated with such a fundamental tool of mathematical analysis. Except in unusual circumstances, one garners little credit nowadays for crafting a continuous analog to a discrete inequality, or vice versa...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ultimately, one sees that inequalities get their names in a great variety of ways. Sometimes the name is purely descriptive, such as one finds with the triangle inequality... Perhaps, more often, an inequality is associated with the name of a mathematician, but even then there is no hard-and-fast rule to govern that association. Sometimes the inequality is named after the first finder, but other principles may apply--such as the framer of the final form, or the provider of the best known application.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If one were to insist on the consistent use of the rule of the first finder, then Hölder's inequality would become Roger's inequality, Jensen's inequality would become Hölder's inequality, and only riotous confusion would result. The most practical rule--and the one used here--is simply to use the traditional names. Nevertheless, from time to time, it may be scientifically informative to examine the roots of those traditions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="136" LastEditorUserId="658" LastEditDate="2017-02-15T01:33:32.970" LastActivityDate="2017-02-15T01:33:32.970" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3508" PostTypeId="1" AcceptedAnswerId="3509" CreationDate="2016-03-09T18:54:55.010" Score="7" ViewCount="4087" Body="&lt;p&gt;Having started to learn about quantum behavior, this formula came up:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$E = hf $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where $E$ is energy, $h$ is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck_constant&quot;&gt;Planck constant&lt;/a&gt; and $f$ is the frequency.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My physics teacher suggested an experiment involving LEDs as a way of calculating $h$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However given that the Planck constant was calculated long before the invention of LEDs, how was it calculated? I know from doing some research Planck used something to do with black body radiation, but I cannot seem to find out the specifics of his method.&lt;/p&gt;&#xA;" OwnerUserId="3820" OwnerDisplayName="Darth Vader" LastEditorUserId="4251" LastEditDate="2020-06-08T03:31:27.387" LastActivityDate="2020-06-08T03:31:27.387" Title="How did Planck calculate the Planck constant?" Tags="|physics|quantum-mechanics|physicists|planck|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3509" PostTypeId="2" ParentId="3508" CreationDate="2016-03-09T23:23:15.953" Score="8" Body="&lt;p&gt;See &lt;a href=&quot;https://hsm.stackexchange.com/questions/3404/how-did-planck-derive-the-black-body-radiation-formula-without-using-the-bose-st/3408#3408&quot;&gt;How did Planck derive the black body radiation formula without using the Bose statistics?&lt;/a&gt; for general circumstances of Planck's discovery. Specifically, Planck was trying to reproduce the expression for entropy of an ideal oscillator involved in the original derivation of the black body radiation law. To do so, he had to assume that energy is split into &quot;droplets&quot; and count the number of ways it could be distributed. The idea was to take continuous limit at the end. But that did not reproduce the expression Planck needed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an &quot;act of despair&quot; (&quot;&lt;em&gt;no matter what the cost, I must bring about a positive result&lt;/em&gt;&quot;) he fixed the energy of his droplets in proportion to the frequency of light, and introduced the constant of proportionality as a &quot;new constant of nature&quot;. To get the value he fit it to the expression he had for the oscillator entropy, which in turn was fit to the experimental radiation measurements of 1899, the ones that showed that Wien's law, which Planck derived earlier, was not followed at low frequencies.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-09T23:23:15.953" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3510" PostTypeId="2" ParentId="3503" CreationDate="2016-03-10T04:29:40.360" Score="13" Body="&lt;p&gt;This is a good point, I mused about it too. First, Pythagoreans and Plato had a very high minded idea of mathematics, gambling would have been seen as a lowly pursuit. This in itself does not explain it however. &lt;a href=&quot;https://www.pdcnet.org//pdc/bvdb.nsf/purchase?openform&amp;amp;fp=ancientphil&amp;amp;id=ancientphil_1983_0003_0001_0012_0029&amp;amp;onlyautologin=true&quot;&gt;Plato's successors at the Academy&lt;/a&gt; considered mechanics lowly too, for it &quot;&lt;em&gt;uses bodies needing much vulgar manual labour, mechanics was thus distinguished as falling outside of geometry, and, since it was for a long time disregarded by philosophy, it has become one of the military arts&lt;/em&gt;&quot;, as Plutarch relates. That did not stop Ctesibius (the inventor of the water organ, first feedback controlled water clock, and an early head of the Alexandrian Museum), Archimedes and Apollonius. But gambling is not even part of the military art, and we should not underestimate the scarcity of ancient sources, none of Ctesibius's or Apollonius's works on mechanics survive, and of Archimedes's only one does, On Floating Bodies, and even that one was preserved for centuries in a single copy, &lt;a href=&quot;https://www.academia.edu/2119678/The_astronomy_of_Hipparchus_and_his_time_A_study_based_on_pre-ptolemaic_sources&quot;&gt;according to Russo&lt;/a&gt;. One can only imagine how medieval monks might have felt about copying manuscripts on gambling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was also a strong streak of fatalism, both in Greek mythology and philosophy, recall Homer's Iliad, the three Fates, the myth of Oedipus, the oracle of Delphi, and the influential Stoic philosophy, which validated divination by asserting unbreakable causal chains through which the present predetermines the future (what we call hard determinism). Chrysippus, the leader of Stoics, was not a superstitious hack, he invented propositional logic two millenia before Leibniz and Frege, and was held on a par with Aristotle. His works did not survive either, by the way. Nor did &lt;a href=&quot;http://mathpages.com/home/kmath397/kmath397.htm&quot;&gt;Hipparchus's rather sophisticated combinatiorial computation&lt;/a&gt; related to Chrysippus's logic (the number of compound propositions that can be formed from ten simple ones). If people thought that &quot;random&quot; outcomes could be divined or influenced, the oracle of Delphi and sibyls were a better option than binomial coefficients. Epicureans, who opposed fatalism and recognized true chance (&quot;atoms swerve&quot;), were in turn hostile to mathematics, because... Plato was for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gambling was not the only place, where ancients could have used probability and statistics, &lt;a href=&quot;http://www.jstor.org/stable/10.1086/343243?seq=1#page_scan_tab_contents&quot;&gt;according to Gingerich&lt;/a&gt; &quot;&lt;em&gt;the real world of observed data is rarely as clean as the pure mathematics of Euclid. As Ptolemy wrestled with errors of measurement without any error theory, he was repeatedly forced into compromises to reconcile discordant observations...  Clearly Ptolemy’s notion of an observation involved a certain level of processing and not raw data. Interpolation between other observations to produce an “observation” on a critical day required for symmetry is by no means excluded&lt;/em&gt;&quot;. Traces of this &quot;processing&quot; are multiple in the Almagest, but Ptolemy's &quot;methods&quot; appear to be sporadic, and he is silent on the subject. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &quot;blind spot&quot; is not confined to Greeks, we find no probabilistic ideas in Babylonian sources, whose commercial prowess seems to invite them, nor in ancient China where the &lt;a href=&quot;https://en.wikipedia.org/wiki/I_Ching&quot;&gt;Book of Change, I Ching,&lt;/a&gt; with its elaborate permutations of sticks, does too. Ironically, I Ching was used for divination. In the end, I am not sure that we can be more definitive than &quot;the stars just weren't aligned for it&quot; before Cardano.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-03-10T04:29:40.360" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3511" PostTypeId="1" CreationDate="2016-02-29T02:44:22.700" Score="10" ViewCount="1040" Body="&lt;p&gt;How/why was the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaussian_function&quot; rel=&quot;noreferrer&quot;&gt;Gaussian function&lt;/a&gt; invented?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The function reminds me of nothing and I don't understand why Gauss would want to derive such thing, but we can see that it has numerous applications, so is there some connection between applicability and the way it's derived?&lt;/p&gt;&#xA;" OwnerUserId="5603" OwnerDisplayName="mavavilj" LastEditorUserId="62" LastEditDate="2016-03-10T23:21:57.333" LastActivityDate="2016-03-11T11:09:26.217" Title="How/why was the Gaussian function invented?" Tags="|mathematics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3512" PostTypeId="1" AcceptedAnswerId="3513" CreationDate="2016-03-10T09:44:46.090" Score="3" ViewCount="5025" Body="&lt;p&gt;As I understand it, pretty much everybody uses the &quot;Babylonian/sexagesimal&quot; time format: 12/24 hours in a day, 60 minutes in an hour and 60 seconds in a minute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have there been alternative systems to &lt;em&gt;accurately&lt;/em&gt; tell time?&#xA;Or maybe there are still some in existence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;accurately&quot; I mean that one can tell an exact time, i.e. not the &quot;some time during the morning&quot;-type.&#xA;Also, let's exclude inventions that were never properly used over a longer period, e.g. decimal time during the French Revolution.&lt;/p&gt;&#xA;" OwnerUserId="3826" OwnerDisplayName="Gerhard" LastEditorUserId="62" LastEditDate="2016-03-12T17:16:15.190" LastActivityDate="2021-12-05T12:07:34.657" Title="Have there been accurate alternative clocks/ways to tell time?" Tags="|time|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3513" PostTypeId="2" ParentId="3512" CreationDate="2016-03-10T18:09:37.783" Score="4" Body="&lt;p&gt;&quot;Swahili time&quot; is essentially the same as contemporary standard time, synchronized at sunrise / sunset (6:00), so it has the same 12/24 system. There is a Chinese time unit the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ke_%28unit%29&quot;&gt;ke&lt;/a&gt;, 14.4 minutes, but it apparently co-existed with a duodecimal system so one can suspect that it was added to a duodecimal system, used exclusively (as far as I can determine) with months.&lt;/p&gt;&#xA;" OwnerDisplayName="user6726" LastActivityDate="2016-03-10T18:09:37.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3514" PostTypeId="2" ParentId="3511" CreationDate="2016-03-11T11:09:26.217" Score="2" Body="&lt;p&gt;Look at:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Saul Stahl, The Evolution of the Normal Distribution,&#xA;  Mathematics Magazine, Vol. 79 (2006), pp. 96-113&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The paper won the Carl B. Allendoerfer Award in 2007 and can be found &lt;a href=&quot;http://www.maa.org/programs/maa-awards/writing-awards/the-evolution-of-the-normal-distribution&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3827" LastActivityDate="2016-03-11T11:09:26.217" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3515" PostTypeId="1" CreationDate="2016-03-11T12:05:05.823" Score="-1" ViewCount="1083" Body="&lt;p&gt;For instance, Cardano published the formula for the solution to the cubic after promising Tartaglia he would not, and yet in textbooks the formula is often named after Cardano. What is the reason for this?&lt;/p&gt;&#xA;" OwnerUserId="3129" LastActivityDate="2016-03-12T17:17:18.253" Title="Why are stolen mathematical results attributed to the thief?" Tags="|mathematics|mathematicians|publishing|" AnswerCount="1" CommentCount="1" ClosedDate="2016-03-18T21:59:22.357" ContentLicense="CC BY-SA 3.0" />
  <row Id="3516" PostTypeId="2" ParentId="3512" CreationDate="2016-03-11T13:16:04.280" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Decimal_time#France&quot; rel=&quot;noreferrer&quot;&gt;French Revolutionary time&lt;/a&gt; had ten hours to the day, 100 minutes to the hour, and 100 secnds to the minute. That same article describes a couple of other instances of decimal time.&lt;/p&gt;&#xA;" OwnerUserId="3828" LastActivityDate="2016-03-11T13:16:04.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3517" PostTypeId="2" ParentId="3515" CreationDate="2016-03-11T13:22:20.523" Score="7" Body="&lt;p&gt;The case of Cardano is clear: unlike Tartaglia, he published the result.&#xA;By modern standards a result has to be published for any claim of priority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People read books and papers and refer on them. Normally mathematicians (and others) are not very much concerned with the question &quot;who really did this first&quot;. One reason is that this question rarely has a simple answer, because most discoveries evolve gradually. (A had the idea but his proof had gaps, B had a correct proof, C generalized this, D shows how important it is, sometimes D read&#xA;A,B,C, sometimes discovers this independently, and so on. A may be an obscure person, nobody reads his writings, and C may be famous). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, the result does not have to be &quot;stolen&quot;. In many cases A and B discover something independently, say A was earlier, but the publication of B attracted larger attention for some reasons, and the result is named after B.&#xA;In other cases, B really gets the idea from A, develops it further, or exposes better, and does not mention A. Before the 20th century mathematical papers had no reference lists, and many people did not consider it necessary to give accurate credits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For most mathematicians, the names of the theorems are just the labels which permit to recognize them easily. They are not concerned about fair credit.&#xA;They leave this to historians, and continue to use the recognized names&#xA;of the well-known results. There are thousands of such cases besides Cardano,&#xA;and I do not want to make a list. I mean something called &quot;Theorem of X&quot;, while&#xA;everyone knows that X was not the first, or even has nothing to do with this theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arnold once stated &quot;Arnold's Principle&quot;: If something is named after X, this indicates that X had nothing to do with this. (Someone added: &quot;Arnold's Principle applies to itself as well&quot; :-) ) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. In most cases it is impossible to answer the question &quot;who was the first to...&quot;, so this question so frequently asked on this site has no meaning.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="2442" LastEditDate="2016-03-12T17:17:18.253" LastActivityDate="2016-03-12T17:17:18.253" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3521" PostTypeId="1" CreationDate="2016-03-11T16:17:34.223" Score="4" ViewCount="485" Body="&lt;p&gt;On the Wikipedia page about &lt;a href=&quot;https://en.wikipedia.org/wiki/Laurent_Schwartz&quot; rel=&quot;nofollow&quot;&gt;Laurent Schwartz&lt;/a&gt;, one can find the following quote: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;What are mathematics helpful for ? Mathematics are helpful for physics. Physics helps us make fridges. Fridges are made to contain spiny lobsters, and spiny lobsters help mathematicians who eat them and have hence better abilities to do mathematics, which are helpful for physics, which helps us make fridges which...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The source of the quote refers to a French webpage that is now dead (I tried to use the Internet Archive with no success). Do you know a reference (the original reference might be in French) for this quote? A book, a conference, a paper, another webpage...&lt;/p&gt;&#xA;" OwnerUserId="3833" LastEditorUserId="3833" LastEditDate="2016-03-11T16:39:22.283" LastActivityDate="2016-03-11T20:20:47.413" Title="Source of Laurent Schwartz's quote about lobster and math" Tags="|mathematicians|quote|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3522" PostTypeId="2" ParentId="3521" CreationDate="2016-03-11T16:33:26.413" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://theconversation.com/les-langoustes-des-mathematiciens-et-le-frigidaire-des-philologues-52945&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; it is in the original language&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;« pourquoi faire des mathématiques ? ». La première médaille Fields française affirmait alors : « Parce que les mathématiques, ça sert à faire de la physique. La physique, ça sert à faire des frigidaires. Les frigidaires, ça sert à y mettre des langoustes, et les langoustes, ça sert aux mathématiciens, qui les mangent et sont alors dans de bonnes dispositions pour faire des mathématiques, qui servent à la physique, qui sert à faire des frigidaires, qui… ».&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That doesn't seem to provide a reference either.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-03-11T16:33:26.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3523" PostTypeId="2" ParentId="3521" CreationDate="2016-03-11T19:15:54.263" Score="5" Body="&lt;p&gt;This is a paraphrase of a common joke (in Germany, England and France).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Eat more fish. It is useful for your brain. Use your brain to make more money. Which will enable you to buy more fish&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose that it is of ancient Roman origin, as many common jokes, but cannot prove this. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-03-11T19:15:54.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3524" PostTypeId="2" ParentId="3512" CreationDate="2016-03-11T19:32:50.080" Score="0" Body="&lt;p&gt;It is not clear what you are asking about: clocks (devices to measure time) or systems to record the time. Ancient Babylonians had no clocks. Time recording was needed for astronomy. As true (quantitative) observational astronomy began in Babylon, their time recording system spread universally. Clocks were a later invention (water clocks), and this had little to do with the system of units; the system of units only tells you how to divide the dial. The French, when they introduced a universal decimal system of units for measuring everything during the revolution, tried to introduce it for time and angles as well. This did not work because the Babylonian units of time and angles were already universally accepted.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="10106" LastEditDate="2021-12-05T12:07:34.657" LastActivityDate="2021-12-05T12:07:34.657" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3525" PostTypeId="2" ParentId="3521" CreationDate="2016-03-11T20:20:47.413" Score="6" Body="&lt;p&gt;Citations in Wikipedia and in &lt;a href=&quot;http://medlibrary.org/medwiki/Laurent_Schwartz#cite_note-2&quot;&gt;Schwartz's obituary&lt;/a&gt; (from which the quote is copied to many other places) refer to now defunct French site of Michel-Amadry, &lt;a href=&quot;http://archive.is/bChQ#selection-255.202-255.212&quot;&gt;archived here&lt;/a&gt;. There the French quote is accompanied by &quot;citation est extraite du livre &quot;Le facteur temps ne sonne...&quot; (quote is taken from the book &quot;Time never rings...), which appears to be &lt;a href=&quot;http://www.amazon.fr/facteur-temps-sonne-jamais-deux/dp/2081220229&quot;&gt;Le Facteur Temps ne Sonne Jamais Deux Fois&lt;/a&gt; (Time Never Rings Twice) by Étienne Klein. The English translation seems to be named &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1560257083&quot;&gt;Chronos: How Time Shapes Our Universe&lt;/a&gt;, so you may want to look there.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-03-11T20:20:47.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3526" PostTypeId="2" ParentId="3491" CreationDate="2016-03-12T11:43:25.637" Score="2" Body="&lt;p&gt;According to &lt;a href=&quot;https://books.google.it/books?id=lNXZGQVdz_gC&amp;amp;pg=PA11&quot; rel=&quot;nofollow&quot;&gt;footnote a, page 11&lt;/a&gt; of Vol.I of Needham's &lt;a href=&quot;https://en.wikipedia.org/wiki/Science_and_Civilisation_in_China&quot; rel=&quot;nofollow&quot;&gt;Science and Civilization in China&lt;/a&gt;, was the diplomat and historian &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Bailey_Sansom&quot; rel=&quot;nofollow&quot;&gt;Sir George Sansom&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-03-12T12:27:03.963" LastActivityDate="2016-03-12T12:27:03.963" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3527" PostTypeId="2" ParentId="3503" CreationDate="2016-03-12T16:49:07.860" Score="3" Body="&lt;p&gt;I hope another answer is okay. Absolutely nothing of what follows is mine. The source[1] is given at the end. That main concern of that article, however, is de Finetti's theory of subjective probability and Hume's problem of induction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has been suggested that an important step for the development of ideas of probability is the notion of what Jerzy Neyman termed Fundamental Probability Sets (FPS): a set of outcomes that are all equally likely. Such sets naturally allow us to assign numbers to every outcome. In fact, this is how the subject is still introduced in school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we suppress what we understand about probability, and how we currently often use terms such as equally possible, equally probable and equally likely interchangeably; and consider the following heuristic, it still seems very suggestive:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If all alternative outcomes are equally likely then &lt;em&gt;none&lt;/em&gt; of them should obtain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The idea being that if any one of the alternatives &lt;em&gt;were&lt;/em&gt; to obtain it would break the symmetry that led them to be included in the FPS in the first place. (I think this exposes the circularity at the heart of Bernoulli's defining equally probable on the basis of equipossibility, that Reichenbach talks about)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to G.E.L. Owen (&lt;a href=&quot;https://en.wikipedia.org/wiki/Gwilym_Ellis_Lane_Owen&quot; rel=&quot;nofollow&quot;&gt;wiki&lt;/a&gt;) the above heuristic is a 'very Greek pattern of argument'. It has a name: &lt;strong&gt;Ou Mallon&lt;/strong&gt;. It is employed, for example, by Anaximander to argue that the Earth is fixed at the centre of the universe, by Parmenides to argue that the universe is uncreated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, even if the Greeks were mathematically proficient, their existing pattern of reasoning blocked them from identifying FPSs and using it to develop notions of probability, since for them 'equally likely' means none of the outcomes should be possible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A counter view of sorts is voiced in the first chapter of Hacking's brilliant The Emergence of Probability, called an Absent Family of Ideas (this whole chapter is worth reading in connection to your question). While not addressing this particular point, Hacking dismisses the general argument that people in ancient times failed to notice FPSes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] Zabell, S. L. Symmetry and Its Discontents; &#xA;Skyrms, B., &amp;amp; Harper, W. L. (eds). Causation, Chance, and Credence, Vol. I, 155-190. (1988). Kluwer Academic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt;: An interesting flip in perspective seems to happen by the 1700s. The symmetry conditions, that for the Greeks would block an outcome from obtaining, becomes the very grounds for chance to operate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider this passage from Hume, A Treatise of Human Nature, 1739:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Since therefore an entire indifference is essential to chance, no one chance can possibly be superior to another, otherwise than as it is compos’d of a superior number of equal chances. For if we affirm that one chance can, after any other manner, be superior to another, we must at the same time affirm, that there is something, which gives it the superiority, and determines the event rather to that side than the other: That is, in other words, we must allow of a cause, and destroy the supposition of chance; which we had before establish'd. A perfect and total indifference is essential to chance, and one total indifference can never in itself be either superior or inferior to another. This truth is not peculiar to my system, but is acknowledg'd by every one, that forms calculations concerning chances. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3827" LastActivityDate="2016-03-12T16:49:07.860" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3529" PostTypeId="2" ParentId="2958" CreationDate="2016-03-12T18:03:23.827" Score="2" Body="&lt;p&gt;Reiterating on @Mauro ALLEGRANZA's linked response above, note that this historical research was later expanded upon and published in the 2012 article &lt;i&gt;&lt;a href=&quot;http://epubs.siam.org/doi/abs/10.1137/100804036&quot; rel=&quot;nofollow&quot;&gt;From Euler, Ritz, and Galerkin to Modern Computing&lt;/a&gt;&lt;/i&gt; (SIAM Review, Vol. 54, No. 4).  Courant's reference to &quot;&lt;i&gt;tragic circumstances&lt;/i&gt;&quot; may simply have been referring to Ritz's early death from tuberculosis, but it could also have much to do with the lack of appreciation that Ritz received during his lifetime for his work as well as Lord Rayleigh's unjustified accusations of plagiarism against Ritz (on both points, see pages 654-655 and 658-659 of SIAM Review, Vol. 54, No. 4).&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2016-03-13T02:33:26.480" LastActivityDate="2016-03-13T02:33:26.480" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3530" PostTypeId="2" ParentId="3483" CreationDate="2016-03-13T01:31:18.077" Score="2" Body="&lt;p&gt;When I was taught celestial navigation and the art of using a sextant at sea, we learned to take two sightings at sunset: first when the lower limb of the sun met the horizon, and the second for the upper limb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes one or the other is obscured by clouds; taking both gives two chances at a measurement. But there is an additional advantage to sighting the lower limb of the sun: it is unambiguous when you see it, and it gives a very precise time. The upper limb is less clear, and is hard to anticipate at sunrise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus there a practical reasons for choosing the lower limb of the sun for observational work. &lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-03-13T01:31:18.077" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3531" PostTypeId="1" AcceptedAnswerId="3538" CreationDate="2016-03-13T18:47:32.863" Score="8" ViewCount="2187" Body="&lt;p&gt;I found some rumors on the internet regarding &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Stueckelberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stückelbergs&lt;/a&gt; manuscript and it's role for the development of QED.&#xA;In a comment on &lt;a href=&quot;https://www.math.columbia.edu/~woit/wordpress/?p=311&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; somebody writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Crease and Mann say in their book The Second Creation (revised edition, Rutgers 1996) (page 143) that Stueckelberg “… in the army, was almost totally isolated from physics. Nonetheless, he apparently wrote up a lengthy paper – in English, for once – that outlined a complete and correct description of the renormalization procedure for quantum electrodynamics. Sometime in 1942 or 1943, he apparently mailed it to the Physical Review. It was rejected. “They said it was not a paper, it was a program, an outline, a proposal,” Stueckelberg remembered. … Stueckelberg was not a bitter man. … We asked if he had the manuscript, which would help him establish priority. “I never cared much about that question,” he replied. “I don’t know what happened to the original copy. I lost it, it completely disappeared. …”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On this page while &lt;a href=&quot;http://phys-merger.physik.unibas.ch/~aste/stueckelberg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; I find the quote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Via anonymous email somebody informed me about the following anecdote: After Feynman won the Nobel Prize, Schwinger called him up and asked:  &quot;Now are you going to give Stueckelberg his notes back?&quot; &#xA;  (The unknown author: This is anecdotal from a friend who is a theoretical physicist living in Boston whom I have known for 35 years. He told me this several years ago and I believe it to be true.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And finally &lt;a href=&quot;https://pseudopodo.wordpress.com/2006/10/03/la-increible-y-triste-historia-de-ecg-stueckelberg/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; Richard Bentley admits&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I was the person who sent you an email some time ago regarding a physicist who called Feynman after he won the Nobel prize to ask him if he would now give Stueckelberg’s notes back. I was incorrect in the identification. My understanding now is that it was Sidney Coleman.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So my question is this: As implied by these quotes, could it really be that Stueckelbergs manuscript in fact disappeared, because &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Feynman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feynman&lt;/a&gt; got hold of it, never returned it and used it to develop QED? And could it be that this was in fact an open secret in the physics community, so that Sidney Coleman could have known about this and Murray Gell-Mann &lt;a href=&quot;https://www.math.columbia.edu/~woit/wordpress/?p=311&quot; rel=&quot;nofollow noreferrer&quot;&gt;apparently even referred to Feynman diagrams as “Stueckelberg diagrams”&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="3841" OwnerDisplayName="asmaier" LastEditorUserId="4251" LastEditDate="2020-06-06T13:01:54.043" LastActivityDate="2020-06-06T13:01:54.043" Title="Did Feynman develop QED based on Stueckelberg's manuscript?" Tags="|physics|biographical-details|theoretical-physics|physicists|feynman|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3532" PostTypeId="2" ParentId="3503" CreationDate="2016-03-13T21:25:19.770" Score="4" Body="&lt;p&gt;Imho the Greeks would have considered probabilities as a sophism - the attempt to produce knowledge out of ignorance. And even today they are still not far from truth: except for the frequentist interpretation which is objective, the rest is subjective (mostly the Bayesian kind). Kolmogoroff axiomatic evades the interpretation problem but so it is also known as  theory of measure and thus would not be recognized as a separate endeavor from geometry. And one should not forget Bertrand's paradoxes that continue to poison probabilistic arguings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we agree that Plato has had some influence on people  with scientific-philosophic interests from the next few generations, it is rather obvious why they could not conceive the idea of probability except as ignorance. For Plato it was the maths and ideas (forms) that were really real while the changing physical world consisted of mere shadows. Time being excluded from this reality it is difficult to conceive probabilities. And of course there is nothing negative, - unknown or uncertain, in the domain of true knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle had no interest in mathematics so we cans skip him and his early followers. The popular Stoics held a doctrine of eternal recurrence which is obviously incompatible with probabilities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So except for the atomists it is difficult to see who would have entertained seriously the idea of probability theory but apparently they had an other agenda.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Re:refs. Aristotle's Rhetoric (Book III ch5) gives advice for good style: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;strong&gt;avoid ambiguities&lt;/strong&gt;, unless, indeed, you definitely desire to be ambiguous, as those do who have nothing to say but are pretending to mean something. &#xA;  Diviners use these vague generalities about the matter in hand because their predictions are thus, as a rule, less likely to be falsified. &lt;em&gt;We are more likely to be right, in the game of &quot;odd and even,&quot; if we simply guess &quot;even&quot; or &quot;odd&quot; than if we guess at the actual number&lt;/em&gt;; and the oracle-monger is more likely to be right if he simply says that a thing will happen than if he &#xA;  says when it will happen, and therefore he refuses to add a definite date. All these ambiguities have the same sort of effect, and are to be avoided unless we have some such object as that mentioned&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ian Hacking has asserted that the emergence of (modern) probabilities is an 'epsitemic shift', that is a conceptual revolution and his book &lt;em&gt;The Emergence of probabilities&lt;/em&gt; (1975) remains a (much discussed) classic. A more recent book is &lt;em&gt;The Science of Conjecture: Evidence and Probability Before Pascal&lt;/em&gt; by James Franklin (2001).&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2016-03-15T20:01:03.420" LastActivityDate="2016-03-15T20:01:03.420" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3533" PostTypeId="1" AcceptedAnswerId="3537" CreationDate="2016-03-14T01:21:27.570" Score="5" ViewCount="510" Body="&lt;p&gt;When, and where, was the earliest record of non-Euclidean geometry? It was always my impression that Riemann created them, but did it really take that much time for someone to try to create an alternate system of mathematics?&lt;/p&gt;&#xA;" OwnerUserId="140" LastEditorUserId="2193" LastEditDate="2016-03-25T20:51:31.183" LastActivityDate="2017-06-07T15:51:29.210" Title="Who do I blame for non-Euclidean geometry?" Tags="|differential-geometry|riemannian-geometry|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3534" PostTypeId="2" ParentId="3531" CreationDate="2016-03-14T04:49:57.247" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;As implied by these quotes, could it really be that Stückelbergs manuscript in fact disappeared, because Feynman got hold of it, never returned it and used it to develop QED? And could it be that this was in fact an open secret in the physics community, so that Sidney Coleman could have known about this and Murray Gell-Mann apparently even referred to Feynman diagrams as “Stueckelberg diagrams”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Feynman had eidetic memory and had no need to keep any manuscript.  I heard him say that he could read off formulas on a page of a book he had read without opening it again. So the scenario is extreme.&lt;/p&gt;&#xA;&lt;p&gt;If Wheeler was the referee of the paper when it was submitted to physical review and Feynman was asked his opinion, that would be enough for his brain's data banks.&lt;/p&gt;&#xA;&lt;p&gt;We have a proverb in Greece: if my aunt had wheels she would be called a carriage.&lt;/p&gt;&#xA;&lt;p&gt;It is also a well known fact that when ideas are ripe, they appear in more than one place independently when similar minds work on the same problems, for example calculus by &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton and Leibniz&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In this &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/Stueckelberg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A big advance in theoretical physics was the renormalization programme in quantum field theory. At the 1948 Solvay congress, Oppenheimer insisted on preserving covariance in all steps of the calculation if one wants to eliminate the infinities which otherwise occur. He then quoted Stueckelberg's 1934 paper as giving an example of such a covariant theory. This was not Stueckelberg's only contribution to the renormalization programme, however, for in the early 1940s he wrote a long paper outlining a complete and correct description of the renormalization procedure for quantum electrodynamics. He sent it to the Physical Review, but it was rejected. As Stueckelberg later recalled:-&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;They said it was not a paper, it was a programme, an outline, a proposal ...&lt;/p&gt;&#xA;&lt;p&gt;He then set about filling in all the details but Schwinger and Feynman published their version first and Stueckelberg received no recognition for his remarkable contributions. In 1965 Sin-Itiro Tomonaga, Julian Schwinger and Richard P Feynman were jointly awarded the Nobel Prize for Physics:-&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The peer review system developed in the west ensures the dates of results. Without a published source or not even the manuscript and with the author not interested in retaining rights, the question is moot. The credit anyway went to three people.&lt;/p&gt;&#xA;&lt;p&gt;It is a bit like a formal athletic event. The rules have to be followed for the prizes to be given. Even if the jump was longer, the wrong stepping discredits it. And in physics there are no next year races.&lt;/p&gt;&#xA;" OwnerUserId="3846" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-03-14T04:49:57.247" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3535" PostTypeId="2" ParentId="3533" CreationDate="2016-03-14T08:47:32.190" Score="5" Body="&lt;p&gt;Usually this is attributed to, in alphabetical order, Bolyai, Gauss, and Lobachevski all working at about the same time in the first third of the 19th century.  This certainly predates Riemann.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A colorful Harvard mathematician and guitarist &lt;a href=&quot;https://en.wikipedia.org/wiki/Tom_Lehrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tom Lehrer&lt;/a&gt; seems to suggest in one of his songs that Lobachevski was guilty of plagiarism.  However, &#xA;In the liner notes to his album &quot;The Tom Lehrer Collection&quot; about the song &quot;Lobachevsky,&quot; Lehrer wrote this: &quot;Incidentally, Nicolai Ivanovich Lobachevsky (1793-1856) was a genuine mathematician, best known for his development of non-Euclidean geometry. His name was chosen here solely for prosodic reasons and was not intended as a slur on his character.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further discussion can be found &lt;a href=&quot;https://math.stackexchange.com/questions/119123&quot;&gt;at this post&lt;/a&gt; where it transpires (in a deleted &lt;em&gt;answer&lt;/em&gt;) that Gauss wrote a letter on non-Euclidean geometry in 1824, i.e., six years before Lobachevski published. Similarly Bolyai was apparently already working on the problem in the early 1820s. The fact that Buniakovski was sent to study with Cauchy in the early 1820s indicates that Russia was aggressively pursuing contacts with the West at the time. Furthermore both the other page linked above and other internet sources indicate that there do exist historians who hold that Lobachevski may have plagiarized his results, but the views of these historians are rejected out of hand without being discussed. I haven't seen this first hand but if someone can come up with a reference one could check the reliability of publisher/author/etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should be mentioned that earlier work by Saccheri in the 17th century and Lambert in the 18th came very close but since it was thought inconceivable that the postulate might be independent such a conclusion was never pursued.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2017-06-07T15:51:29.210" LastActivityDate="2017-06-07T15:51:29.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3536" PostTypeId="1" CreationDate="2016-03-14T11:35:37.823" Score="2" ViewCount="80" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;“Whereas the chemico-chemists always find in industry a beautiful field of gold-laden soil, the physico-chemists stand somewhat farther off, especially those who seek only the greatest dilution, for in general there is little to make with watery solutions.”&lt;br&gt;&#xA;  -Jacobius Henricus Van’t Hoff.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is the difference between the two according to Van't Hoff?&lt;/p&gt;&#xA;" OwnerUserId="3850" LastEditorUserId="140" LastEditDate="2016-03-14T21:26:32.277" LastActivityDate="2017-06-26T03:07:20.253" Title="What is the difference between chemico-chemists and physico-chemists according to Van't Hoff?" Tags="|physics|chemistry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3537" PostTypeId="2" ParentId="3533" CreationDate="2016-03-14T13:25:42.583" Score="11" Body="&lt;p&gt;Technically, the first was Lobachevski (published in 1829-30).&#xA;Bolyai was independent (published in 1832).&#xA;Gauss discovered it independently of both (not published).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more complicated research is needed to find out when it was actually discovered by each person, and we can never be 100% sure of the result. Perhaps Gauss was the first. But the usual way to establish &#xA;priority is the date of publication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All three discovered ONE non-Euclidean geometry (hyperbolic geometry). Riemann's contribution was a more general approach, introducing infinitely many possible geometries (Riemannian geometry).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-03-16T01:17:01.493" LastActivityDate="2016-03-16T01:17:01.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3538" PostTypeId="2" ParentId="3531" CreationDate="2016-03-14T22:11:11.063" Score="11" Body="&lt;p&gt;There seems to be enough evidence to believe that people were joking around about Stueckelberg's notes in 1965, when the Nobel prize was awarded to Feynman, Schwinger and Tomonaga, but I do not see how that implies conclusions in the last paragraph. If Schwinger or Coleman took it seriously they wouldn't be privately joking about it with Feynman, they'd be accusing him of plagiarism publicly. So would others. Crease and Mann's phrasing is odd, something can be an &quot;outline&quot; or a &quot;complete and correct description&quot;, it can not be both. It also seems that the source of their assessment is Stueckelberg's recollection 50 years after the fact, and Stueckelberg himself makes no accusations against Feynman.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking deeper makes it even less credible. General accounts of renormalization were published by six people independently in 1946-48: Tomonaga, Kramers, Bethe, Lewis, Schwinger and Feynman. Tomonaga's paper appeared two years before Feynman's three 1948 papers, Bethe's one year before, and in that same Physical Review. Two out of three Feynman's papers were also in Physical Review. You'd think someone there would remember Stueckelberg's manuscript. Stueckelberg himself did not perish in World War II either. In fact, he published (with Rivier) a paper on renormalization in the same volume of Physical Review as the last two Feynman papers. But rather than being &quot;complete and correct description&quot;, or even a general outline, it was a specific application to the magnetic moment of the neutron. Three years later, in 1951, he published another paper in Physical Review (with Peterman), on non-renormalizability of magnetic moment interactions with the electromagnetic field. It would be very strange for Stueckelberg to keep publishing in a journal he thought let Feynman plagiarize his manuscript, while staying silent about it. We'd have to believe in a vast cover-up conspiracy of physicists, in which Stueckelberg himself took part. An authoritative source on the history of renormalization is &lt;a href=&quot;https://books.google.com/books/about/Conceptual_Developments_of_20th_Century.html?id=l4PtgYXpb_oC&quot; rel=&quot;noreferrer&quot;&gt;Cao's Conceptual Developments of 20th Century Field Theories&lt;/a&gt;, see also his &lt;a href=&quot;http://www.jstor.org/stable/20117830&quot; rel=&quot;noreferrer&quot;&gt;long survey paper with Schweber.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman's account was somewhat different from the other five, they directly identified divergent terms involving masses and charges, and removed them by redefining masses and charges. Feynman's algorithm was better structured and more efficient, he gave explicit rules of regularization known as &quot;relativistic cutoff&quot;, under which redefinition of parameters does not involve divergent terms directly, and then let the cutoff parameter go to infinity to obtain the renormalized theory. This idea also appears in Stueckelberg-Rivier 1948 paper, which perhaps gave fuel to the anecdote. Pauli and Villars also used this approach in their 1949 paper, likely submitted before they read Feynman's. Stueckelberg's 1942 manuscript was not the first to anticipate renormalization. Subtraction of divergent terms was proposed by Dirac and Heisenberg in 1934, Weisskopf in 1936, and Kramers in 1938. But precision measurements of the spectrum of hydrogen and deuterium by Lamb-Retherford and Rabi, used to corroborate renormalization, were only obtained in late 1940s with instruments developed during the war, before that it was just an abstract idea. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These things happen, Fermat and Torricelli anticipated Leibniz's and Newton's calculus, respectively, Bolzano anticipated Weierstrass's analysis, Bullialdus, Borelli, Halley, Hooke and Wren &lt;a href=&quot;https://hsm.stackexchange.com/questions/1973/who-was-first-to-explain-intuitively-the-inverse-square-law-of-gravity/1975#1975&quot;&gt;conjectured the inverse square law for gravity before Newton&lt;/a&gt;, as he acknoweledged in Principia (except for Hooke), &lt;a href=&quot;http://philsci-archive.pitt.edu/4950/1/MatterFromSpace.pdf&quot; rel=&quot;noreferrer&quot;&gt;Clifford speculated in 1870&lt;/a&gt; that gravity and matter are manifestations of the curvature of space. Some ideas are in the air and every discovery has precursors, it doesn't make it plagiarism.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-07-22T01:15:24.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3540" PostTypeId="1" AcceptedAnswerId="3545" CreationDate="2016-03-15T16:02:21.510" Score="5" ViewCount="483" Body="&lt;p&gt;Archimedes's work &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Sand_Reckoner&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Sand Reckoner&lt;/em&gt;&lt;/a&gt; (also known by its original Greek title &lt;em&gt;Ψαμμίτης&lt;/em&gt; or by the Latin translation thereof, &lt;em&gt;Arenarius&lt;/em&gt;) sets out to prove that the number of grains of sand in the Universe is finite by providing an upper bound.  From my point of view, this text is remarkable because not only it introduces a kind of scientific notation for numbers but also for the idea of putting an upper bound on the order of magnitude of something we cannot directly count (Archimedes arrives at an upper bound of $10^{63}$), and for the introduction of truly large numbers: Archimedes considers numbers like $10^{8\cdot 10^8}$ and $10^{8\cdot 10^{16}}$ (according to Wikipedia — I didn't check) which I &lt;em&gt;presume&lt;/em&gt; were far beyond any finite quantity explicitly described by anyone thitherto.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, now, truly huge numbers have been constructed and used for various reasons in mathematics: some have even proposed the word &lt;a href=&quot;http://googology.wikia.com/wiki/Googology&quot; rel=&quot;noreferrer&quot;&gt;&quot;Googology&quot;&lt;/a&gt; for this art(?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question would be about the history of &quot;Googology&quot; and what the main steps after Archimedes were in the description of large numbers.  Specifically: when was the first time in history that a number greater than $10^{8\cdot 10^{16}}$ was explicitly mentioned (that is sufficiently precisely described so that it can be compared)?  But generally speaking, any comments on the history of large numbers interest me.&lt;/p&gt;&#xA;" OwnerUserId="3616" LastActivityDate="2016-10-29T20:31:48.610" Title="After Archimedes's *Sand Reckoner*, who considered very large numbers?" Tags="|mathematics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3541" PostTypeId="2" ParentId="3540" CreationDate="2016-03-15T23:40:56.183" Score="2" Body="&lt;p&gt;The Central Asian mathematician al-Biruni, writing in the year AD 1000, calculated the solution to the “chess problem”: Take a chessboard with 64 squares, put one grain on the first square, double this on the second square, double it again on the third, and so on. How many grains are there on the whole chessboard? His answer is (in decimal notation) 18,446,744,073,709,551,615, or (in sexagesimal notation) : 30. 30. 27. 9. 5. 3. 50. 40. 31. 0. 15. This is of course much less than the figures discussed by Archimedes, but still quite substantial.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is perhaps interesting is that neither Archimedes nor al-Biruni was actually interested in grains of sand. Both of them are concerned only with proving that it is possible to express very large numbers in some sort of notation.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-03-16T15:20:26.063" LastActivityDate="2016-03-16T15:20:26.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3542" PostTypeId="1" CreationDate="2016-03-16T13:23:21.167" Score="2" ViewCount="171" Body="&lt;p&gt;Well when it comes to Muslim scholars who had influence in medicine the major books and scholars which are referred to is &lt;a href=&quot;https://en.wikipedia.org/wiki/Avicenna&quot; rel=&quot;nofollow&quot;&gt;ibn Sina&lt;/a&gt; and his Canon القانون في الطب.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But many other Muslim scholars have contributed in the field of Medicine.&#xA;I want to ask especially about &lt;a href=&quot;https://en.wikipedia.org/wiki/Averroes&quot; rel=&quot;nofollow&quot;&gt;ibn Rushd&lt;/a&gt; and his book the &lt;a href=&quot;https://en.wikipedia.org/wiki/Colliget&quot; rel=&quot;nofollow&quot;&gt;Colliget&lt;/a&gt; الكليات في الطب al Kulliyat about general Medicine or medication in which he also refers to the work of his teacher, colleague and friend &lt;a href=&quot;https://en.wikipedia.org/wiki/Ibn_Zuhr&quot; rel=&quot;nofollow&quot;&gt;ibn Zuhr&lt;/a&gt; Kitab al-taysir &lt;a href=&quot;https://ar.wikisource.org/wiki/%D9%85%D9%84%D9%81:%D8%A7%D9%84%D8%AA%D9%8A%D8%B3%D9%8A%D8%B1_%D9%81%D9%8A_%D8%A7%D9%84%D9%85%D8%AF%D8%A7%D9%88%D8%A7%D8%A9_%D9%88%D8%A7%D9%84%D8%AA%D8%AF%D8%A8%D9%8A%D8%B1_%D8%AE.pdf&quot; rel=&quot;nofollow&quot;&gt;التيسير في المداواة والتدريب&lt;/a&gt; = the book of simplification of (medical) treatment and diet sheet (May own translation -as i didn't find any- please take it with care) which has been translated into latin &lt;em&gt;Rectificatio medicationis et regiminis&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was ibn Sina the only Muslim physician who had some influence on modern medicine or to what extent have ibn Zuhr and ibn Rushd contributed to a modern medicine?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &lt;strong&gt;modern medicine&lt;/strong&gt; i mean a medicine or medication method that at least outlasted the middle-age a few centuries! As i read that the Canon was still a reference in the 18th Century. For example medication methods like diets and music therapy ...&lt;/p&gt;&#xA;" OwnerUserId="2821" LastEditorUserId="2821" LastEditDate="2016-03-17T06:23:45.563" LastActivityDate="2016-03-17T12:02:35.063" Title="Was ibn Sina the only Muslim physician who influenced modern medicine?" Tags="|medicine|middle-ages|islamic-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3544" PostTypeId="2" ParentId="3480" CreationDate="2016-03-16T17:53:28.967" Score="20" Body="&lt;p&gt;I do not believe Cauchy wrote anything explicit concerning the extent to which we can deduce differentiability from continuity. However, in many places it seems that Cauchy assumes continuity and then introduces derivatives. From what I can determine, this has led historians to &amp;quot;read Cauchy&amp;quot; in different ways according to the embedded contextual world-view the historian felt was appropriate. In what follows I have assembled a number of relevant publications and excerpts that give a sense of what people have written about this and related issues for the period of time between Ampère (1806) and Weierstrass (1872).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Umberto Bottazzini, &lt;strong&gt;The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass&lt;/strong&gt;, translated by Warren Van Egmond, Springer-Verlag, 1986, vi + 332 pages.&lt;/p&gt;&#xA;&lt;p&gt;Section 3.6. &lt;em&gt;Cauchy's Definitions of Derivative and Differential&lt;/em&gt; (pp. 117-122) contains a discussion of the issue at hand.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from pp. 118-119)&lt;/strong&gt; In the same volume of the &lt;em&gt;Journal de l'Ecole polytechnique&lt;/em&gt; that contained Lagrange's supplement, A. M. Ampere (1775-1836) published a long note whose object was &amp;quot;a new demonstration of the Taylor series.&amp;quot; In criticizing Lagrange's assumption, Ampère sought to demonstrate that it is possible for every function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to be expanded in a series of increasing powers&lt;span class=&quot;math-container&quot;&gt;$.^{23}$&lt;/span&gt; [Bottazzini cites Grabiner's 1978 paper and 1981 book.] He began from a definition of the derivative that was based on a property of derived functions found by Lagrange. This property was stated by Ampère as, [begin indented text] Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; be the values of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; corresponding to &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x=k.$&lt;/span&gt; One can always suppose that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; are chosen in such a way that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; never becomes infinite in the interval and that neither &lt;span class=&quot;math-container&quot;&gt;$A=k$&lt;/span&gt; nor &lt;span class=&quot;math-container&quot;&gt;$a=k,$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$\frac{K-A}{k-a}$&lt;/span&gt; is neither zero nor infinite, and that one can take this quantity for that above or below which one can always take &lt;span class=&quot;math-container&quot;&gt;$[f(x+i)-f(x)]/i,$&lt;/span&gt; giving to &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; a value as small as necessary (1806a, p. 151). [end indented text] From this property Ampère drew the definition of derivative that to him &amp;quot;seemed the most general and the most rigorous possible&amp;quot; (&lt;em&gt;ibid&lt;/em&gt;., p. 156). &amp;quot;The derived function of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$[f(x+i)-f(x)]/i$&lt;/span&gt; is always included between two of the values that this derived function takes between &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+i,$&lt;/span&gt; whatever &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; may be (&lt;em&gt;ibid&lt;/em&gt;.). Ampère's work and his lectures at the École Polytechnique did not go without influence on Cauchy, who cites Ampère among those to whom he is indebted in both the &lt;em&gt;Cours d'analyse&lt;/em&gt; and in the &lt;em&gt;Résumé&lt;/em&gt;. But Cauchy's approach reverses the terms of the problem: by defining the derivative of a function (if it exists) as a suitable limit, what Ampère had given as a definition became a property of the derivative, expressible as a theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Carl Benjamin Boyer, &lt;strong&gt;The Concepts of the Calculus. A Critical and Historical Discussion of the Derivative and the Integral&lt;/strong&gt;, Columbia University Press, 1939, vii + 346 pages. [Reprinted by Dover Publications in 1959, with the title changed to &lt;strong&gt;The History of the Calculus and Its Conceptual Development&lt;/strong&gt;.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from middle of p. 282)&lt;/strong&gt; The geometrical intuitions which intruded themselves into Cauchy's view of irrational number likewise led him erroneously to believe that the continuity of a function was sufficient for its geometrical representation and for the existence of a derivative&lt;span class=&quot;math-container&quot;&gt;$.^{45}$&lt;/span&gt; [Jourdain's 1905 paper is cited.] A. M. Ampère also had been led by geometric preconceptions similar to those of Cauchy to try to demonstrate the false proposition that every continuous function has a derivative, except for certain isolated values in the interval&lt;span class=&quot;math-container&quot;&gt;$.^{46}$&lt;/span&gt; [Pringsheim/Molk is cited.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Louis Philippe Gilbert, &lt;a href=&quot;https://books.google.com/books?id=r21fAAAAcAAJ&amp;amp;pg=PR1&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mémoire sur l'existence de la dérivée dans les fonctions continues&lt;/em&gt;&lt;/a&gt; [Memoir on the existence of the derivative in continuous functions], &lt;strong&gt;Mémoires Couronnés et Autres Mémoires, publiés par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique&lt;/strong&gt; 23 #3 (1873), vi + 31 pages.&lt;/p&gt;&#xA;&lt;p&gt;See also Eugène Charles Catalan's 1872 &lt;a href=&quot;https://books.google.com/books?id=qE0-AQAAMAAJ&amp;amp;pg=PA560&quot; rel=&quot;noreferrer&quot;&gt;advance report on Gilbert's memoir&lt;/a&gt; (9 pages, in French) and Gilbert's 1872 &lt;a href=&quot;https://books.google.com/books?id=ImHwAAAAMAAJ&amp;amp;pg=PA498&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur une objection proposée par M. Catalan&lt;/em&gt;&lt;/a&gt; [On an objection proposed by Mr. Catalan] (5 pages, with a 1-paragraph response by Catalan on p. 502) and Gilbert's 1872 &lt;a href=&quot;https://books.google.com/books?id=ZWLwAAAAMAAJ&amp;amp;pg=PA709&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Rectification au sujet d'un mémoire précédent&lt;/em&gt;&lt;/a&gt; [Correction to the subject of the preceding memoir] (9 pages).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Gilbert originally argued (in his earlier written memoir, which has the &amp;quot;official publication&amp;quot; year of 1873) that there were errors in Hankel's 1870 work and that Hankel did not actually show there exist continuous functions that are non-differentiable on dense sets (or &amp;quot;everywhere&amp;quot; non-differentiable; I am not sure to what extent Gilbert originally made a distinction between non-differentiable on every interval and being non-differentiable at every point). Gilbert's approach was based on Lamarle's methods and was intended to conclusively prove that a continuous function is differentiable except for an isolated set of points (see Gilbert's final theorem at the bottom of p. 31). However, after Gilbert learned of the results in Hermann Amandus Schwarz's 1873 paper &lt;a href=&quot;https://books.google.com/books?id=UFsWAQAAIAAJ&amp;amp;pg=PA252&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Neues beispiel einer stetigen nicht differentiirbaren function&lt;/em&gt;&lt;/a&gt; [New example of a continuous non-differentiable function] &lt;a href=&quot;https://books.google.com/books?id=HoVCAQAAMAAJ&amp;amp;pg=RA2-PA33&quot; rel=&quot;noreferrer&quot;&gt;French version&lt;/a&gt;, which gives a particularly simple example of a strictly increasing continuous function that is not differentiable at each nonzero dyadic rational number (in fact, at each such number the right derivative is &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt; and the left derivative is positive and finite), Gilbert came to realize that Hankel's examples were possible. (Schwarz presented his example on 19 August 1873, but since Gilbert's &lt;em&gt;Rectification&lt;/em&gt; was presented in a session dated 7 June 1873, Gilbert obviously knew about it before Schwarz's presentation. Incidentally, although Weierstrass' presentation of an everywhere non-differentiable continuous function was on 18 July 1872, I can find no mention of Weierstrass in Gilbert's &lt;em&gt;Rectification&lt;/em&gt;.) In fact, some of Hankel's reasoning was &lt;em&gt;not correct&lt;/em&gt;. However, I do not know to what extent Gilbert's criticism of Hankel were based on actual errors that Hankel made and to what extent Gilbert's criticism of Hankel were a result of errors in Gilbert's own reasoning. I believe Gilbert, in his &lt;em&gt;Rectification&lt;/em&gt;, still felt that it was not possible for a continuous function to be non-differentiable at each point. With this in mind, it is perhaps worth noting that Gilbert repented his early 1870s views by giving a lengthy survey of this topic -- beginning with Ampère and including a detailed discussion of Hankel's and Schwarz's examples, and even the Weierstrass nowhere differentiable continuous function -- in &lt;em&gt;Note II&lt;/em&gt; (pp. 467-475) in the 1878 2nd edition of his textbook &lt;a href=&quot;https://books.google.com/books?id=CskKAAAAYAAJ&amp;amp;pg=PA467&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Cours d'Analyse Infinitésimale. Partie Élémentaire&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Judith Victor Grabiner, &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086078902082&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The origins of Cauchy's theory of the derivative&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Historia Mathematica&lt;/strong&gt; 5 #4 (November 1978), 379-409.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(first full paragraph on p. 381)&lt;/strong&gt; Cauchy first presented his definition of the derivative to the mathematical world in his &lt;em&gt;Leçons sur le calcul infinitésimal&lt;/em&gt; of 1823. He defined the derivative &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; of a continuous function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; as the limit, when it exists, of the ratio &lt;span class=&quot;math-container&quot;&gt;$f(x+i)-f(x)/i$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; went to zero. But it is not the mere definition of the derivative as the limit of the quotient of differences which constitutes Cauchy's achievement. Newton, after all, had described some of his results in terms of limits [Newton 1934, Scholium to Lemma XI]. Jean-le-Rond D'Alembert, under Newton's influence, had explicitly defined the differential quotient as the limit of the quotient of differences [1789, article &amp;quot;Différentiel.&amp;quot; The definition used by D'Alembert was fairly common by the end of the eighteenth century. See Boyer 1949, Chapter VI.] The difference between Cauchy's work and that of men like D'Alembert lies in the understanding and the use of the definition. D'Alembert did not have what we now call a delta-epsilon translation of the limit-concept; as we shall see, Cauchy did. The only real use D'Alembert made of his definition was to illustrate the finding of a tangent to a parabola as the limit of secants [&lt;em&gt;op. cit.&lt;/em&gt;]. In contrast, Cauchy's definition was the beginning of his task, not the end; his achievement was to produce an extended body of proved results about derivatives.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(middle of p. 383)&lt;/strong&gt; [This appears after a translated quote from pp. 22-23 of Cauchy's 1823 book] Cauchy's phrase &amp;quot;this limit, when it exists&amp;quot; exemplifies his attitude toward rigor. Perhaps his use of the phrase was motivated only by the behavior of known functions at isolated points, but the language was sufficiently general to open the whole question of the existence or non-existence of derivatives. And, though his definition of derivative, like that of limit, is verbal, we shall see immediately that he translated the definition into the algebra of inequalities for use in proofs.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from p. 396, lines 6-13)&lt;/strong&gt; [Ampère's 1806 paper is discussed on pp. 395-400] Cauchy knew Ampère personally, and had once been his student. Cauchy said in the &amp;quot;Introduction&amp;quot; to his &lt;em&gt;Cours d'analyse&lt;/em&gt; that he had &amp;quot;profited several times from the observations of M. Ampère, as well as from the methods that he [= Ampère] has developed in his lectures on analysis&amp;quot; [1821, vii-viii]. He more than once acknowledged Ampère's assistance in a general way [1821, vii; 1826, 10]. And Cauchy explicitly referred to Ampère's 1806 paper on the occasion of his own proof of (1) [1823, 44n; 1829, 268]. [&lt;strong&gt;Note:&lt;/strong&gt; (1) is the statement &lt;span class=&quot;math-container&quot;&gt;$\min \{f'(x): \; a \leq x \leq b \} \; \leq \; \frac{f(b) - f(a)}{b - a} \; \leq \; \max \{f'(x): \; a \leq x \leq b \}.$&lt;/span&gt;]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from lower middle of p. 396)&lt;/strong&gt; Thus it is natural for us, in our search for Cauchy's sources, to examine Ampère's paper. The paper is, unfortunately, confusing to read and not very well organized. It has on occasion been misinterpreted as an attempt to prove that every continuous function is differentiable. The major source of this interpretation seems to be [Pringsheim 1909, 44]. [&lt;strong&gt;Note:&lt;/strong&gt; I disagree. There are &lt;em&gt;many references&lt;/em&gt; in the 19th century literature to Ampère's paper being an attempt to prove that continuous functions are differentiable, except at certain isolated points.] Ampère himself is partly responsible for this error; in his paper he used the term &amp;quot;exist&amp;quot; to describe a derivative when he meant that it was finite and non-zero [1806, 149]. Given the prevailing view on the part of many historians about the lack of rigor and sophistication in analysis before Cauchy, together with Ampère's unusual use of the term &amp;quot;exist,&amp;quot; the misinterpretation of Ampère's words is not surprising. Nevertheless, this error has led to an unfortunate neglect of Ampère's paper by historians of the calculus. Since Ampère's innovations helped Cauchy learn to prove theorems about derivatives, they deserve to be looked at again with care.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I believe I have encountered the use of &amp;quot;number exists&amp;quot; to mean &amp;quot;not zero&amp;quot; (and not negative) several times in very old papers and books (first half of 1800s), but I was not able to locate any specific such uses of &amp;quot;exists&amp;quot; at this time. My feeling is that the use of &amp;quot;exists&amp;quot; in this way is very similar to the use of &amp;quot;exists&amp;quot; for a set to mean that the set is not the empty set (this usage was very common from roughly the 1880s to the 1920s).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Ivor Grattan-Guinness, &lt;em&gt;The emergence of mathematical analysis and its foundational progress, 1780-1880&lt;/em&gt;, pp. 94-148 in Ivor Grattan-Guinness (editor), &lt;strong&gt;From the Calculus to Set Theory, 1630--1910. An Introductory History&lt;/strong&gt;, Gerald Duckworth and Company, 1980, vi + 306 pages. [Reprinted by Princeton University Press in 2000.]&lt;/p&gt;&#xA;&lt;p&gt;Some relevant comments about Cauchy and Ampère on the derivative appear on pp. 112-114, but everything of relevance to the issue at hand is contained in the excerpts I have given elsewhere. Nonetheless, the following comment may be of interest:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from bottom of p. 112)&lt;/strong&gt; An immediate problem is the question of whether or not a continuous function is differentiable. To us the answer is undoubtedly in the negative, for a function with a corner is continuous but not differentiable there. But in the unrigorous systems of the 18th and early 19th centuries the properties of continuity and differentiability were not so clearly defined anyway, and infinitesimals might allow a corner to be interpreted as an infinitely tight (but differentiable) loop.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Ivor Grattan-Guinness, &lt;strong&gt;Convolutions in French Mathematics, 1800-1840&lt;/strong&gt;, 3 volumes, Science Networks / Historical Studies #2, Birkhäuser Verlag, 1990, 1602 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from Volume 2, p. 749, 2nd paragraph)&lt;/strong&gt; Cauchy's treatment of the calculus began in lecture 3. [This pertains to Cauchy's 1823 book &lt;strong&gt;Résumé ...&lt;/strong&gt;] The difference quotient of a continuous function &lt;span class=&quot;math-container&quot;&gt;$y = f(x),\;$&lt;/span&gt; (1152.1) &lt;span class=&quot;math-container&quot;&gt;$\;\Delta y / \Delta x \; = \; (f(x + \Delta x) - f(x))/\Delta x,$&lt;/span&gt; 'may converge towards another limit, either positive, or negative' when &lt;span class=&quot;math-container&quot;&gt;$\Delta x \rightarrow 0.$&lt;/span&gt; [&lt;strong&gt;Note:&lt;/strong&gt; Of course, Cauchy did not use the arrow notation for limits, since this only slowly came into use from about 1908 to the 1920s. Grattan-Guinness explicitly points out on p. 717 that although Cauchy did not use the arrow notation, Grattan-Guinness will use it in his discussion of Cauchy.] 'This limit, when it exists, has a determinate value' (including &lt;span class=&quot;math-container&quot;&gt;$0,$&lt;/span&gt; surely) a 'new function' of &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt; He used Lagrange's notation '&lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;' and name 'derived function' for this new function, but of course it was essentially the derivative that &lt;em&gt;we&lt;/em&gt; recognise, not Lagrange's coefficient (322.8) in the Taylor expansion. The phrase 'when it exists' shows that Cauchy was aware that a continuous function need not have a derivative, although the examples which he then gave &lt;em&gt;were&lt;/em&gt; all differentiable &lt;span class=&quot;math-container&quot;&gt;$(\sin x,\;A^{x},$&lt;/span&gt; and so on); he did not give examples with corners (such as Fourier had shown with his series in Figure 926.1) or define left- and right-hand derivatives.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Thomas William Hawkins, &lt;strong&gt;Lebesgue's Theory of Integration. Its Origins and Development&lt;/strong&gt;, 2nd edition, Chelsea Publishing Company, 1975, xvi + 227 pages. [For the part I'm quoting, the 1970 1st edition, 1979 corrected 2nd edition, and the 2001 AMS reprint of the 1979 corrected 2nd edition are the same.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from pp. 43-44)&lt;/strong&gt; A. M. Ampère [1806] sought to establish this corollary independently of (4) [(4) is the statement that &lt;span class=&quot;math-container&quot;&gt;$f(x+i)=f(x)+pi+qi^2+\cdots,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$p,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$q,$&lt;/span&gt; etc. are functions of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; only] and then use it to establish (4) without recourse to Lagrange's argument. It is his proof that became the paradigm for subsequent ones. What Ampère actually showed is that if &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; exists for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$(f'(x)$&lt;/span&gt; can be infinite for some values of &lt;span class=&quot;math-container&quot;&gt;$x)$&lt;/span&gt; and if the difference quotient converges uniformly to &lt;span class=&quot;math-container&quot;&gt;$f'(x),$&lt;/span&gt; then any interval &lt;span class=&quot;math-container&quot;&gt;$(a,k)$&lt;/span&gt; must contain a point &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; at which &lt;span class=&quot;math-container&quot;&gt;$f'(x_0)$&lt;/span&gt; is finite and nonzero. This would imply that &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; exists as a finite, nonzero quantity on a dense set. But Ampère interpreted his result to mean that &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; is neither zero nor infinite except for &amp;quot;certain particular and isolated values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&amp;quot; [1806: 149]. Between the appearance of Ampère's paper and 1870, the proposition that any (continuous) function is differentiable in general was stated and proved in most of the leading texts on the calculus&lt;span class=&quot;math-container&quot;&gt;$.^{11}$&lt;/span&gt; [10 references are given, none are Cauchy] Frequently Ampère's original argument was simplified by making use of the &amp;quot;intuitively evident&amp;quot; fact that a function which varies continuously must be piecewise monotonic. Thus, differentiability and monotonicity were linked together, albeit tenuously. This is particularly interesting because Cauchy in his &lt;em&gt;Leçons sur le calcul différentiel&lt;/em&gt; [1829] had considered the functions &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^{3}\sin(1/x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g(x)=x\sin(1/x),$&lt;/span&gt; both of which have removable discontinuities at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and yet are not monotonic in any neighborhood of &lt;span class=&quot;math-container&quot;&gt;$0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Philip Edward Bertrand Jourdain, &lt;a href=&quot;https://books.google.com/books?id=IgYMAAAAYAAJ&amp;amp;pg=PA190&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The theory of functions with Cauchy and Gauß&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Bibliotheca Mathematica&lt;/strong&gt; (3) 6 (1905), 190-207.&lt;/p&gt;&#xA;&lt;p&gt;I am including this because Boyer makes reference to Jourdain's discussion in Section 6 on p. 206. Jourdain's discussion mainly concerns the differentiability of functions of a complex variable. However, the nature of Jourdain's discussion, and especially the last sentence on p. 206 (which mentions the Weierstrass nowhere differentiable continuous function), suggests that Jourdain's discussion also concerns the differentiability of functions of a real variable.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from p. 206, lines 5-14)&lt;/strong&gt; In the memoir of 1825, CAUCHY required only one-valuedness and continuity of &lt;span class=&quot;math-container&quot;&gt;$f(z);$&lt;/span&gt; whether he regarded continuity as sufficient for the existence of a derivative cannot be told from his writings, for, as for [&amp;quot;far&amp;quot; intended?] as I am aware, he nowhere made a pronouncement on this point, -- perhaps he did not regard such a pronouncement as necessary. But it is difficult to doubt, from other indications in CAUCHY's works, that CAUCHY imagined that his conception of 'continuity' was in any case sufficient for the 'geometrical representability' of a function, and so for the existence of a derivate [= derivative; Dini derivates are not considered anywhere in Jourdain's paper]. In fact, CAUCHY's conceptions in analysis rested on &lt;em&gt;geometrical&lt;/em&gt; grounds (except at isolated points).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; Philip Edward Bertrand Jourdain, &lt;em&gt;The origin of Cauchy's conceptions of a definite integral and of the continuity of a function&lt;/em&gt;, &lt;strong&gt;Isis&lt;/strong&gt; 1 #4 (1913), 661-703.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from Section XVIII, on pp. 701-702)&lt;/strong&gt; But on the other hand, LACROIX (5) gave a «proof» of the existence of differential quotients. [footnote (5) gives a citation to &lt;a href=&quot;https://books.google.com/books?id=hSBKAAAAMAAJ&amp;amp;pg=PA241&quot; rel=&quot;noreferrer&quot;&gt;pp. 241-242 of Volume I&lt;/a&gt;, and to &lt;a href=&quot;https://books.google.com/books?id=9g_WVK4TVkcC&amp;amp;pg=PA712&quot; rel=&quot;noreferrer&quot;&gt;p. 712 of Volume III&lt;/a&gt;] The first attempt of this kind was due to AMPÈRE. AMPÈRE (1) began a memoir on TAYLOR'S series by a «proof» that the function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; [namely] &lt;span class=&quot;math-container&quot;&gt;$[f(x+i)-f(x)]/i$&lt;/span&gt; can become, when we put &lt;span class=&quot;math-container&quot;&gt;$i=0,$&lt;/span&gt; neither zero nor infinite for all values of &lt;span class=&quot;math-container&quot;&gt;$x.$&lt;/span&gt; [footnote (1) gives a citation to Ampère's 1806 paper] [several sentences omitted] A point which seems to be of very great interest is the implied supposition that &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is continuous. Many authors, like DINI (3), REIFF (4) and PRINGSHEIM (5) just say that this proof is to apply to «continuous» functions, and do not add that the word «continuous» had a different meaning in 1806, and was not used by AMPÈRE. [(3) is a citation to pp. 67-68 &amp;amp; 219 of Dini's 1878 book &lt;a href=&quot;https://books.google.com/books?id=jgQAAAAAMAAJ&amp;amp;pg=PA67&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Fondamenti per la Teorica delle Funzioni di Variabili Reali&lt;/strong&gt;&lt;/a&gt;, (4) is a citation to p. 156 of Reiff's 1889 book &lt;a href=&quot;https://books.google.com/books?id=rWE9AAAAYAAJ&amp;amp;pg=PA156&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Geschichte der Unendlichen Reihen&lt;/strong&gt;&lt;/a&gt;, (5) is p. 449 of Pringsheim's 1890 paper &lt;a href=&quot;https://books.google.com/books?id=lc_uAAAAMAAJ&amp;amp;pg=PA449&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Zur Geschichte des Taylorschen Lehrsatzes&lt;/em&gt;&lt;/a&gt;] The «proof» of the existence of the limit of the ratio &lt;span class=&quot;math-container&quot;&gt;$\; \frac{f(x')-f(x)}{x'-x} \;$&lt;/span&gt; [typo corrected] given by LACROIX, was due to «M. BINET aîné, professeur de mathématiques transcendantes au Lycée de Rennes» (6). [footnote (6) extends to p. 703 and is the following: Presumably this would be the father of JACQUES PHILIPPE MARIE BINET who was born at Rennes in 1786 and died at Paris in 1856 (&lt;em&gt;Poggendorff's biog.-lit. Handwörterbuch&lt;/em&gt;, vol. I, p. 194. For other early &amp;quot;proofs,, of the existence of derivatives, see GALOIS' (1830-1831) &lt;em&gt;Œuvres&lt;/em&gt;, Paris, 1897, &lt;a href=&quot;https://books.google.com/books?id=ZOkGAAAAYAAJ&amp;amp;pg=PA9&quot; rel=&quot;noreferrer&quot;&gt;p. 9&lt;/a&gt;; DE MORGAN, &lt;em&gt;Calculus&lt;/em&gt;, &lt;a href=&quot;https://books.google.com/books?id=mbw4AAAAMAAJ&amp;amp;pg=PA47&quot; rel=&quot;noreferrer&quot;&gt;p. 47-48&lt;/a&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Augustus De Morgan's attempted proof that Jourdain mentions is not included among the 10 references given in a footnote on p. 44 of Hawkins' &lt;strong&gt;Lebesgue's Theory of Integration&lt;/strong&gt;. Since De Morgan's book is fairly well known, I do not think Hawkins overlooked it. Instead, I think Hawkins did not include De Morgan's attempted proof because De Morgan expressed doubt about the proof in a paragraph (upper middle of p. 48) after his attempted proof.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Morris Kline, &lt;strong&gt;Mathematical Thought from Ancient to Modern Times&lt;/strong&gt;, Oxford University Press, 1972, xvii + 1238 pages. [Reprinted as a 3-volume paperback edition in 1990.]&lt;/p&gt;&#xA;&lt;p&gt;Section 40.3. &lt;em&gt;The Derivative&lt;/em&gt; (pp. 954-956 of the 1990 edition) includes the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(top of p. 955)&lt;/strong&gt; Though Bolzano and Cauchy had rigorized (somewhat) the notions of continuity and the derivative, Cauchy and nearly all mathematicians of his era believed and many texts &amp;quot;proved&amp;quot; for the next fifty years that a continuous function must be differentiable (except of course at isolated points such as &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$y=1/x).$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; This is perhaps not a very good example, since &lt;span class=&quot;math-container&quot;&gt;$y=1/x$&lt;/span&gt; is not defined at &lt;span class=&quot;math-container&quot;&gt;$x=0.$&lt;/span&gt; This manner of conflating (i.e. combining together) what we now consider to be two different ideas was common until the late 1800s. However, I do not know whether Kline is using this example to be in keeping with historical context or whether Kline simply overlooked the fact that his example could easily cause confusion with present-day readers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[11]&lt;/strong&gt; Morris Kline, &lt;strong&gt;Mathematics. The Loss of Certainty&lt;/strong&gt;, Oxford University Press, 1980, viii + 366 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(p. 176, lines 2-5)&lt;/strong&gt; Like his contemporaries, he [= Cauchy] believed that continuity implied differentiability (Chapter VII) and so he stated many theorems in which the hypotheses called only for continuity, but where he used differentiability, and even after his attention was called to his mistake, he persisted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I cannot find anything in Kline's book to support this claim about &amp;quot;his attention was called to his mistake&amp;quot;. There are no footnotes at the back of Kline's book and the next few pages do not re-visit this claim about Cauchy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[12]&lt;/strong&gt; Anatole Henri Ernest Lamarle, &lt;a href=&quot;https://books.google.com/books?id=iERkfcMCps8C&amp;amp;pg=PA1&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Étude approfondie sur les deux équations fondamentales&lt;/em&gt; Lim &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x+h) - f(x)}{h} = f'(x)$&lt;/span&gt; &lt;em&gt;et&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$dy = f'(x).\;\Delta x$&lt;/span&gt;&lt;/a&gt; [Extensive study of the two fundamental equations Lim &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x+h) - f(x)}{h} = f'(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy = f'(x).\;\Delta x$&lt;/span&gt;], &lt;strong&gt;Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique&lt;/strong&gt; 29 (1855), 118 pages.&lt;/p&gt;&#xA;&lt;p&gt;See also Lamarle's 1854 summary &lt;a href=&quot;https://books.google.com/books?id=HmPwAAAAMAAJ&amp;amp;pg=PA140&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Note sur les deux équations fondamentales&lt;/em&gt; lim. &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x+h) - f(x)}{h} = f'(x)$&lt;/span&gt; &lt;em&gt;et&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$dy = f'(x).\;\Delta x$&lt;/span&gt;&lt;/a&gt; [Note on the two fundamental equations lim. &lt;span class=&quot;math-container&quot;&gt;$\frac{f(x+h) - f(x)}{h} = f'(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy = f'(x).\;\Delta x$&lt;/span&gt;] (22 pages) and Mathieu Schaar's &lt;a href=&quot;https://books.google.com/books?id=9K9JAAAAcAAJ&amp;amp;pg=PA817&quot; rel=&quot;noreferrer&quot;&gt;1854 6 page summary&lt;/a&gt;. For a 51 page biography of Lamarle (in French), with a bibliography of 40 items published by Lamarle, see &lt;a href=&quot;https://books.google.com/books?id=98g9AQAAMAAJ&amp;amp;pg=PA207&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Notice sur la vie et les travaux de A.-H.-E. Lamarle&lt;/em&gt;&lt;/a&gt; [Notice on the life and works of A.-H.-E. Lamarle].&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[13]&lt;/strong&gt; Jesper Lützen, &lt;em&gt;The foundation of analysis in the 19th century&lt;/em&gt;, pp. 155-195 in Hans Niels Jahnke (editor), &lt;strong&gt;A History of Analysis&lt;/strong&gt;, History of Mathematics #24, American Mathematical Society, 2003, x + 422 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from bottom of p. 169)&lt;/strong&gt; In the definition of the derivative Cauchy [in his 1823 book] started out by assuming that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous, but having formed the difference quotient and its limit, he was cautious to state &amp;quot;if it exists&amp;quot;. This seems to set the stage for the introduction of the concept of differentiability. However, Cauchy did not introduce this concept and in the subsequent chapters of his book he simply assumed that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; was continuous (or he assumed nothing), even though he differentiated it any number of times. It is as though Cauchy still clung to the 18th-century idea of a &amp;quot;safe domain&amp;quot; in which analysis was more or less universally valid. With Euler and d'Alembert this domain consisted of all functions; with Cauchy it consisted of the continuous functions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from p. 176, lines 17-19)&lt;/strong&gt; Though Cauchy did not try to prove the erroneous theorem that any continuous function could be differentiated, he gave, as we have seen above, his readers the impression that it was true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[14]&lt;/strong&gt; Fyodor Andreyevich Medvedev, &lt;strong&gt;Scenes from the History of Real Functions&lt;/strong&gt;, translation by Roger Lee Cooke of the 1975 Russian edition, Science Networks / Historical Studies #7, Birkhäuser Verlag, 1991, 265 pages.&lt;/p&gt;&#xA;&lt;p&gt;Section 5.2 (Ampère's theorem, pp. 214-219) presents the viewpoint, which I agree with, that it is &lt;em&gt;ahistorical&lt;/em&gt; (Medvedev, at least in translation, calls this &amp;quot;modernization&amp;quot;) to interpret Ampère's paper as an attempt to prove that continuous functions are differentiable (except perhaps at certain isolated points).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from middle of p. 216)&lt;/strong&gt; It is difficult for the modern reader to judge the content of Ampère's main proposition and the methods of reasoning he used to prove it. The majority of authors who have written about this are for some reason inclined to believe that in this paper he was attempting to prove that a derivative exists everywhere [in a bounded interval] except at a finite number of points for any continuous function or at least for every continuous function satisfying certain supplementary restrictions&lt;span class=&quot;math-container&quot;&gt;$.^4$&lt;/span&gt; [The footnote mentions F. Riesz writing in 1932 that Ampère may have had in mind piece-wise monotone functions.] This opinion can hardly be considered correct. After all, the concept of a continuous function as it is known today, cannot be dated earlier than Lobachevskii and Dirichlet (cf. Sec. 8 of Essay II), or at most to Bolzano (1817) and Cauchy (1821). Thus it is clear a priori that Ampère could not have stated any such general considerations about such functions in 1806, and in fact he makes no such statements. He himself nowhere says explicitly that the functions he is considering belong to any general class of functions, and any judgment as to what he had in mind must be based on indirect reasoning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[15]&lt;/strong&gt; Zbigniew Piotrowski, &lt;a href=&quot;http://zpiotrowski.people.ysu.edu/papers/genesisseperatevsjoint.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The genesis of separate versus joint continuity&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Tatra Mountains Mathematical Publications&lt;/strong&gt; 8 (1996), 113-126.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from middle of p. 115)&lt;/strong&gt; Like his contemporaries, he [= Cauchy] also believed that continuity implied differentiability (?!) (see Chapter VII of [Ca]), and even after his attention was called to his mistake, he ... persisted (compare [K2] p. 176).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; [Ca] is Cauchy's 1821 &lt;strong&gt;Cours d'Analyse&lt;/strong&gt; and [K2] is Kline's book &lt;strong&gt;Mathematics. The Loss of Certainty&lt;/strong&gt;. The ellipsis &amp;quot;...&amp;quot; is in the original and it does not indicate that I have omitted something while quoting from Piotrowski's paper.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[16]&lt;/strong&gt; Avadhesh [Avaresh] Narayan Singh, &lt;a href=&quot;https://andrescaicedo.files.wordpress.com/2014/10/a-n-singh-the-theory-and-construction-of-non-differentiable-functions.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Theory and Construction of Non-Differentiable Functions&lt;/strong&gt;&lt;/a&gt;, Lucknow University Studies #1, Newul Kishore Press [Lucknow University Press], 1935, vii + 110 pages. [Reprinted in &lt;strong&gt;Squaring the Circle and Other Monographs&lt;/strong&gt;, Chelsea Publishing Company, 1953.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from First Lecture, Article 4: 2nd paragraph on pp. 4-5)&lt;/strong&gt; [(1) and (30) are bibliographic citations in Singh's 104 item bibliography, and they refer to Ampère's 1806 paper and Gilbert's 1873 paper.] An attempt to prove that continuity was a sufficient condition for differentiability was made by Ampère (1) in 1806. Although Ampère's proof was defective yet his result was believed in by most mathematicians for a long time. Mention may be made of Duhamel, Bertrand and Gilbert (30) among those who definitely expressed their belief in Ampère's result. Even in the writings of such eminent mathematicians as Gauss, Cauchy, and Dirichlet there is nothing to show that they held a different opinion. Although they did not endorse Ampère's statement none of them seems to have had the conviction that a function which was everywhere continuous but nowhere differentiable could exist. Darboux in his memoir on &amp;quot;Discontinuous functions,&amp;quot; published in 1875, mentions only one auditor, M. Bienaimé, who said that he was unconvinced by Ampére's proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[17]&lt;/strong&gt; Klaus Thomas Volkert, &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF00738359&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Zur differenzierbarkeit stetiger funktionen -- Ampère's beweis und seine folgen&lt;/em&gt;&lt;/a&gt; [On differentiability of continuous functions -- Ampère's proof and its consequences], &lt;strong&gt;Archive for History of Exact Sciences&lt;/strong&gt; 40 #1 (March 1989), 37-112.&lt;/p&gt;&#xA;&lt;p&gt;This appears to be a very complete study of Ampère's 1806 paper and its influence on later efforts by Lamarle, Hankel, Gilbert, du Bois-Reymond, Dini, and others. Unfortunately, the paper is in German and I cannot read German, and there seems to be nothing on the internet (at least, nothing freely available) that discusses (in English) Volkert's paper.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-03-16T18:25:26.437" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3545" PostTypeId="2" ParentId="3540" CreationDate="2016-03-16T18:31:11.013" Score="7" Body="&lt;p&gt;&lt;em&gt;(comments assembled to form an answer, as OP suggested)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been collecting appearances of large numbers in the literature for many years, and one thing that I've found quite interesting is that I have not found found ANYONE who came close to Archimedes until after 1900. Unfortunately, my extensive literature collection on this topic is not easily available to me right now (I moved a couple of months ago and my stuff on this particular topic has still not been unpacked). However, I do recall that the &quot;number curiosity&quot; $9^{9^9},$ which is approximately $10^{3.7 \times 10^{8}},$ began appearing in popular math articles and puzzle books by the 1890s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 1910 (first) edition of Hardy's &lt;a href=&quot;https://archive.org/details/ordersofinfinity00harduoft&quot;&gt;&lt;strong&gt;Orders of Infinity&lt;/strong&gt;&lt;/a&gt; is an early appearance of some large numbers, which can be found at the end of the book. The largest seems to be $10^{10^{10^{10.3}}}$ which can be found on p. 62. Incidentally, as a possibly interesting historical footnote that I don't recall anyone ever mentioning before, note that $10^{10^{100}}$ (now called a googleplex, but not then) appears on p. 61. &lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2016-03-16T18:31:11.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3546" PostTypeId="2" ParentId="3299" CreationDate="2016-03-16T20:23:26.417" Score="4" Body="&lt;p&gt;I don't know exactly who introduced the terminology, but &quot;separable&quot; comes from functional analysis, probably Hilbert spaces, where &quot;having a countable dense subset&quot; and &quot;having a countable basis of open sets&quot; are equivalent (as for metric spaces in general).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree that now most, if not all, authors seem to use &quot;separable&quot; for topological spaces with the weaker property of the two, one should keep in mind that terminology in general topology fluctuates a lot (e.g., as to whether &quot;compact&quot; implies Hausdorff).  So any reasonable paper will redefine every term for which there is even the slightest possibility of confusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &quot;second countable&quot; is horrendous terminology (as anything that entails remembering what is &quot;first&quot; and what is &quot;second&quot;, e.g., &quot;first category&quot; instead of &quot;meager&quot;); Steen and Seebach's 1970 book &lt;em&gt;Counterexamples in Topology&lt;/em&gt; define the terms &quot;perfectly separable&quot; and &quot;completely separable&quot; to be synonymous with &quot;second-countable&quot; (i.e., &quot;having a countable basis of open sets&quot;), which is probably better terminology if the term has to be redefined in any case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, I remember finding at least one author who used &quot;locally separable&quot; for &quot;first-countable&quot; (i.e., &quot;having a countable basis of neighborhoods at each point&quot;), which would be better if it were not also confusing (because it could mean something like every neighborhood having a separable subneighborhood).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An even better terminology is probably &quot;of countable density&quot; (for &quot;separable&quot;) and &quot;of countable weight&quot; (for &quot;perfectly separable&quot;, i.e. &quot;second-countable&quot;), because density and weight are very standard names for the smallest infinite cardinal of a dense subset, resp. a basis for the topology.  Similarly, a space &quot;of countable character&quot; would be a first-countable one.&lt;/p&gt;&#xA;" OwnerUserId="3616" LastActivityDate="2016-03-16T20:23:26.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3547" PostTypeId="1" AcceptedAnswerId="3550" CreationDate="2016-03-16T20:41:25.763" Score="10" ViewCount="2008" Body="&lt;p&gt;I know that ultimately GR was validated by experiment. I'd like to know however, if there was any experimental discrepancy at the time that pointed to a conclusive flaw in Newtonian gravity, and which led Einstein to say &quot;Hmm...you know what? Newtonian gravity can't be correct. Let me see what I can come up with&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read that SR contradicted Newtonian gravity since the latter postulated instantaneous action at a distance. But surely we don't need to throw the whole thing over right? I mean, I can simply modify Newtonian theory to say that the action of gravity travels at the speed of light - introduce a time component, create your equations to include that and everyone's happy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now if you say that the time delay approach to Newtonian gravity is contradicted by experiments, would you happen to know which experiments exactly? And were those experiments conducted before Einstein concocted GR?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that the Advance of Mercury's perihelion was a problem, but I've not heard that physicists were ready to throw away Newtonian mechanics based on that alone. Certainly I've never read that Einstein was inspired by Mercury to develop GR...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because I'm guessing that the instinctive and first approach to resolving a contradiction in a long standing and established theory, would be to try and tweak what works. Not just happily throw over the entire table and start from scratch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or did Einstein not have a solid reason to develop GR at all? Was he just doing his own thing for kicks, and not looking to actually solve any existing experimental contradiction between Newtonian gravity and experimentation?&lt;/p&gt;&#xA;" OwnerUserId="3861" LastEditorUserId="4251" LastEditDate="2020-06-05T19:07:36.180" LastActivityDate="2020-06-05T19:07:36.180" Title="Why did Einstein develop General Relativity?" Tags="|physics|theoretical-physics|relativity-theory|einstein|gravity|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3548" PostTypeId="2" ParentId="3547" CreationDate="2016-03-16T21:15:07.357" Score="3" Body="&lt;p&gt;Not really an answer, but a few relevant remarks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newtonian gravity is inconsistent with special relativity in various ways (e.g., it describes an instantaneous action at a distance and &quot;instantaneous&quot; can only make sense with respect to a given reference frame).  There are, also, physical reasons to believe that gravity should produce a redshift on light when it loses energy in leaving a gravitational field.  So a number of people tried to formulate relativistic theories of gravitation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nordstrøm was &lt;a href=&quot;https://en.wikipedia.org/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation&quot; rel=&quot;nofollow&quot;&gt;the first to come up&lt;/a&gt; with theories of gravitation that are compatible with relativity.  Nordström's second theory can be reformulated in a way that is very similar to GR (and also satisfies the equivalence principle, background independence, etc.), but it does not have any effect on light.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riemann had already tried, in his time, to describe gravitation &quot;geometrically&quot; by curving space.  So it probably made sense, given Minkowski's reformulation of special relativity by using an indefinite quadratic form, to try to curve this structure.&lt;/p&gt;&#xA;" OwnerUserId="3616" LastActivityDate="2016-03-16T21:15:07.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3549" PostTypeId="2" ParentId="3542" CreationDate="2016-03-16T21:48:08.477" Score="2" Body="&lt;p&gt;As you yourself indicated, many works of mediaeval medicine from the Islamic world were translated into Latin,  and they were indeed among the first printed books in Europe. Apart from Avicenna and the others mentioned by you there is (for example) ar-Rāzī (Rhazes) and his Liber continens. So obviously there was a strong influence of Muslim writers about medicine on mediaeval and early modern European medicine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you are really asking about their influence on MODERN medicine, then we would have to admit that MODERN medicine (from around the end of the 18th century onwards) developed largely out of the critique and rejection of the theories of mediaeval Christian and Islamic medicine, and its main sources (the books attributed to Hippocrates and Galen) and in particular their theoretical basis, the doctrine of the four humours. In this sense, I would contend that “Islamic” medicine had no positive input into modern “scientific” medicine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, neither mediaeval nor modern medicine has solved the fundamental problem of human existence: we still all die in the end. In this sense, the whole history of medicine is ultimately the history of folly and deception.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-03-17T12:02:35.063" LastActivityDate="2016-03-17T12:02:35.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3550" PostTypeId="2" ParentId="3547" CreationDate="2016-03-16T22:14:12.577" Score="12" Body="&lt;p&gt;&quot;Simply modifying&quot; Newtonian gravity to have it spread at finite speed does not work if the finite speed is the speed of light. It was attempted by &lt;a href=&quot;https://en.wikipedia.org/wiki/Speed_of_gravity#Laplace&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laplace in his Celestial Mechanics (1799)&lt;/a&gt;, who found that  the planets will promptly fly off their orbits and the Solar system will disintegrate in seconds, unless the propagation speed is $7×10^6$ times greater than the the speed of light. This is because of the aberration of the direction of attractive force due to delay in transmission, see &lt;a href=&quot;https://physics.stackexchange.com/questions/186186/resolving-general-relativity-and-newtonian-mechanics-to-a-computer/186228&quot;&gt;Resolving General relativity and Newtonian mechanics on a computer&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more sophisticated modification follows from &lt;a href=&quot;https://en.wikisource.org/wiki/Scientific_Memoirs/1/On_the_Forces_which_regulate_the_Internal_Constitution_of_Bodies&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mossotti's electromagnetic gravity hypothesis&lt;/a&gt;: electric attraction and repulsion do not balance each other exactly, and the difference is gravity. &lt;a href=&quot;https://www.21stcenturysciencetech.com/articles/spring01/Electrodynamics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;In 1864-72 Seegers, Scheibner and Tisserand&lt;/a&gt; experimented with applying the velocity and acceleration dependent correction to Newton's law imported from Weber's electrodynamics to the precession of the perihelion of Mercury. Around 1900 Lorentz, Einstein's precursor on special relativity, showed that under the Maxwell electrodynamics the Laplacian aberration problem is eliminated because the correction is of the order $v^2/c^2$ rather than $v/c$ that Laplace assumed, so the attraction between masses moving with constant relative velocity is always toward the instantaneous position of the other mass. It is the Lorentz invariance of the Maxwell electrodynamics that cancels the effects of transmission delay to the first order, as Poincare pointed out in 1905. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/2388/what-19th-century-developments-contributed-to-the-general-theory-of-relativity/2395#2395&quot;&gt;What 19th century developments contributed to the General theory of Relativity?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Lorentz's theory did not work either, and this time exactly because of the perihelion of Mercury. Poincare mentioned in Science and Method (1908) that it gave the advance of 7&quot; for Mercury’s perihelion (the value accepted at the time was 38&quot;, the modern value is 43&quot;), but wrote:&quot;&lt;em&gt;This cannot be regarded as an argument in favor of the new dynamics, since we still have to seek another explanation of the greater part of the anomaly connected with Mercury, but still less can it be regarded as an argument against it.&lt;/em&gt;&quot; Seeliger offered an ad hoc solar corona explanation for the rest in 1906, which some astronomers accepted, but neither Einstein, nor Lorentz and Poincare, took it seriously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein first mentions Mercury in a letter to Habicht in 1907:&quot;&lt;em&gt;At the moment I am working on a relativistic analysis of the law of gravitation by means of which I hope to explain the still unexplained secular changes in the perihelion of Mercury.&lt;/em&gt;&quot; Some of Mach's remarks in Science of Mechanics possibly led him to believe that the effect was relativistic. The original versions of his new gravity theory, the so-called &quot;Entwurf&quot; theory Einstein and Grossman developed in 1913, first predicted precession in the wrong direction, and then 18&quot; instead of 45&quot;. This was one of the reasons Einstein cited for abandoning it in favor of what we now call general relativity. Alternative relativistic theories of gravity proposed by Mie and Nordström in 1912 did not fare much better with the perihelion, nor did they meet Einstein's more philosophical invariance requirements. See more in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2308/what-attracted-einstein-to-the-anomalous-precession-of-mercury&quot;&gt;What attracted Einstein to the anomalous precession of Mercury?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The correction to the Newton's gravity force is of the order $v^4/c^4$ in general relativity, which in hindsight explains why Newton's gravity worked so well. And in the end it was the perihelion of Mercury, along with philosophical considerations related to the Mach's principle, which Einstein morphed into the general covariance requirement, that took it down. See more in &lt;a href=&quot;http://mathpages.com/rr/s8-10/8-10.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Concquering the Perihelion chapter in Kevin Brown's book Reflections on Relativity.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-11-13T02:53:14.787" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3551" PostTypeId="1" CreationDate="2016-03-17T10:49:22.867" Score="4" ViewCount="362" Body="&lt;p&gt;Having recently read a review that expounded on the rigour and depth of books by 'earlier' British mathematicians (mainly analysts) I would like to know more. My own research yielded the following tentative list, which I believe to be relatively incomplete and shallow:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;G. H. Hardy, &lt;em&gt;A Course of Pure Mathematics&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;G. H. Hardy, E. M. Wright, &lt;em&gt;An Introduction to the Theory of Numbers&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;E. C. Titchmarsh, &lt;em&gt;The Theory of Functions&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;E. W. Hobson, &lt;em&gt;The Theory of Functions of a Real Variable and the Theory of Fourier's Series (2 vols.)&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;E. T. Copson, &lt;em&gt;An Introduction to the Theory of Functions of a Complex Variable&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Alfred North Whitehead, Bertrand Russell, &lt;em&gt;Principia Mathematica&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;G. S. Carr, &lt;em&gt;Synopsis of Pure Mathematics (2 vols.)&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'm certain that there are many other classics, concentrated most probably in the early or mid-20th century. Any suggestions would be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="395" OwnerDisplayName="seeker" LastEditorUserId="4251" LastEditDate="2021-09-08T16:03:05.400" LastActivityDate="2021-09-08T16:03:05.400" Title="Classic mathematics books by British mathematicians" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="3552" PostTypeId="2" ParentId="3547" CreationDate="2016-03-17T13:56:55.413" Score="0" Body="&lt;p&gt;Einstein developed General Relativity because his initial analysis of motion was only able to produce Special Relativity.  That is, in his 'first cut' analysis of relativity of motion, he had to limit himself to translating motion between inertial reference frames.  Since not all reference frames are inertial, Relativity was 'incomplete' (hence the term 'Special') and needed to be 'General'-ized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From that perspective, yes, he was doing this 'just for kicks', except that he was more driven than the word 'kicks' suggests.  He was always compelled to find a theory that removed what he saw as inconsistencies of existing theory; that is what led him to develop Special Relativity in the first place.&lt;/p&gt;&#xA;" OwnerUserId="3863" LastActivityDate="2016-03-17T13:56:55.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3553" PostTypeId="1" CreationDate="2016-03-17T18:34:44.010" Score="3" ViewCount="2597" Body="&lt;p&gt;which one has been discovered earlier? Coulomb's law or Gauss's law?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read that Gauss's law is more fundamental and of course we can derive Gauss's law using Coulomb's law and vice versa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my question is that which one is discovered earlier?&lt;/p&gt;&#xA;" OwnerUserId="3864" LastEditorUserId="55" LastEditDate="2016-03-17T23:14:53.270" LastActivityDate="2021-02-03T15:29:31.220" Title="Which came first, Coulomb's law or Gauss's law?" Tags="|physics|discoveries|electromagnetism|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3554" PostTypeId="2" ParentId="3553" CreationDate="2016-03-17T23:13:36.673" Score="4" Body="&lt;p&gt;Cavendish verified the Coulomb's law in the early 1770s, but did not publish,  Coulomb published it in 1785 in his first three reports on electricity and magnetism. For magnetic poles the law was established even earlier by Michell in 1750, explaining Newton's inverse cube law for the magnetic dipole in Principia (1687), see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2552/is-coulombs-law-the-earliest-formal-equation-of-electricity/2566#2566&quot;&gt;Is Coulomb&amp;#39;s law the earliest mathematical formula describing electricity?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when Gauss was born in 1777 Coulomb's law was already known. He only formulated &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss&amp;#39;s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss's law&lt;/a&gt; in 1835, and it was not published until 1867, when Maxwell already stated it as one of his four equations in 1861-62. It is interesting however that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Divergence_theorem#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;divergence theorem&lt;/a&gt; which relates one law to the other in modern expositions was discovered by Lagrange in 1762, and then rediscovered multiple times, by Gauss in 1813, Poisson in 1824, Ostrogradsky in 1826, and Green and Sarrus in 1828. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-17T23:48:11.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3556" PostTypeId="2" ParentId="3553" CreationDate="2016-03-18T01:44:51.703" Score="2" Body="&lt;p&gt;It seems to me that there is some confusion here. Coulomb Law is a law of physics.&#xA;It is about electricity. It was discovered experimentally and can be proved ONLY by experiment. It is a statement about REAL WORLD.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not completely sure what &quot;Gauss Law&quot; in the question is. But I suppose this is Gauss' THEOREM about flux. This is a mathematical theorem. Which is established by mathematical proof and logically speaking has nothing to do with electricity or anything in the real world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I checked the Wikipedia article and it indeed shows the same confusion. These are two different things, from different realities, and one cannot follow from another. Gauss theorem applies to vector fields (which is a mathematical notion).&#xA;Gravity field, electrostatic field, are described as vector fields (this is an experimental fact) and THEREFORE must satisfy Gauss' theorem (and all other theorems about vector fields). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of course the chronological question has trivial answer: Gauss was too young when Coulomb law was discovered.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-03-18T01:44:51.703" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3557" PostTypeId="2" ParentId="3551" CreationDate="2016-03-18T01:52:26.987" Score="1" Body="&lt;p&gt;You really want a list of British early 20th century books in mathematics?&#xA;What for? It will be too long, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two classical books by Titchmarsh are missing in your list:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riemann zeta function,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and Eigenfunction expansions associated with second-order differential equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another notable omission in your list are the books of Watson,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bessel functions, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Course of Modern Analysis, joint with Whittaker.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The last book deserves an entry in the Guinnes book of records as a textbook&#xA;continuously in print for more than 100 years. Perhaps unique in all sciences. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-03-18T01:52:26.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3558" PostTypeId="1" AcceptedAnswerId="3564" CreationDate="2016-03-18T02:54:05.650" Score="10" ViewCount="1756" Body="&lt;p&gt;I've read in several books, including Milnor's &lt;em&gt;Morse Theory&lt;/em&gt; and Petersen's &lt;em&gt;Riemannian Geometry&lt;/em&gt;, that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry)&quot;&gt;exponential map in Riemannian geometry&lt;/a&gt; is named so because it agrees with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)&quot;&gt;exponential map in Lie theory&lt;/a&gt;, at least for a certain choice of metric on the Lie group. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this the real reason why Riemannian geometers originally called the exponential map by that name? Or does the exponential map of Riemannian geometry fulfill some relation of the form $f=f'$ that characterizes &quot;exponential&quot; functions? The thing with Lie groups seems to be a highly nontrivial theorem and is not really covered in standard introductions to the subject like do Carmo and Lee. &lt;/p&gt;&#xA;" OwnerUserId="3558" LastEditorUserId="3558" LastEditDate="2016-03-18T03:40:15.450" LastActivityDate="2016-03-20T23:38:05.720" Title="How did the exponential map of Riemannian geometry get its name?" Tags="|mathematics|differential-geometry|riemannian-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3559" PostTypeId="1" CreationDate="2016-03-19T12:03:15.233" Score="1" ViewCount="163" Body="&lt;p&gt;I have always had a hard time explaining to my students the term one-to-one. After making sure my students understand &amp;quot;in&amp;quot;, &amp;quot;sur&amp;quot; and &amp;quot;bi&amp;quot;, the Bourbaki terms, injective, surjective and bijective make sense to them. But to me, one-to-one sounds like it should mean bijective, not injective. I sometimes tell my students that injective should be called two-to-two, meaning that two different elements are mapped to two different elements.&lt;/p&gt;&#xA;&lt;p&gt;On &lt;a href=&quot;http://jeff560.tripod.com/mathword.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;  it says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;ONE-TO-ONE CORRESPONDENCE is found in H. G. Zeuthen, &amp;quot;Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un,&amp;quot; C. R. LXX. 742. (1870).&lt;/p&gt;&#xA;&lt;p&gt;One-to-one correspondence is found in English in 1873 in Proc. Lond. Math. Soc. IV. 252: &amp;quot;The equations .. being supposed to establish a 'one-to-one' correspondence between the two integral spaces.&amp;quot; In his Principles of Mathematics Bertrand Russell (1903, p. 113) states, “Two classes have the same number...when their terms can be correlated one to one, so that any one term of either corresponds to one and one only term of the other.” (OED).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It sounds to me like Russel is talking about a bijection. Who and when started using &amp;quot;one-to-one&amp;quot; to mean injection?&lt;/p&gt;&#xA;" OwnerUserId="661" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-03-20T21:43:30.837" Title="How did &quot;one-to-one&quot; come to be used to refer to injective functions?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3560" PostTypeId="1" AcceptedAnswerId="3561" CreationDate="2016-03-19T23:40:09.237" Score="2" ViewCount="105" Body="&lt;p&gt;In Jammer's book &lt;em&gt;Concepts of Simultaneity&lt;/em&gt; on page 108, there is a reference to two of Poincaré's papers:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;...his 1904 lecture on St. Louis...&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;...his contribution to the &lt;em&gt;Lorentz Festschrift&lt;/em&gt;.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Unfortunately there is no bibliography in the book, and an internet search does not reveal anything. How can I find a pdf of these papers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: I am not sure if such bibliography type of questions are welcome here. But given the importance of the persons involved and the historical importance of the topic, I guess it can be tolerated.&lt;/p&gt;&#xA;" OwnerDisplayName="user2831" LastEditorUserId="26" LastEditDate="2016-03-21T22:43:29.077" LastActivityDate="2016-03-21T22:43:29.077" Title="Two articles of Poincaré" Tags="|reference-request|relativity-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3561" PostTypeId="2" ParentId="3560" CreationDate="2016-03-19T23:53:31.817" Score="5" Body="&lt;p&gt;Poincare's &lt;a href=&quot;http://projecteuclid.org/euclid.bams/1183418771&quot;&gt;lecture given during the 1904 World's Fair was published in 1906.&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://arxiv.org/ftp/arxiv/papers/1204/1204.6575.pdf&quot;&gt;Lorentz Festschrifft talk is discussed in this paper&lt;/a&gt;, which has a footnote to the original published version, and a translation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Poincaré, Henri, &quot;La théorie de Lorentz et le principe de réaction&quot;, Archives&#xA;néerlandaises des sciences exactes et naturelles. Recueil de travaux offerts par les&#xA;auteurs à H. A. Lorentz (The Hague: Nijhoff) Ser II, 1900, 5, pp. 252-278. See my&#xA;discussion of Poincaré and his work on the violation of the principle of reaction in&#xA;Lorentz's electron theory: Granek, Galina, &quot;Poincaré’s Contributions to Relativistic&#xA;Dynamics&quot;, Studies in History and Philosophy of Modern Physics 31, 2000, pp. 15-&#xA;48.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When looking for items like this include more terms: e.g., &quot;Poincare Lorentz Festschrifft&quot;.  The paper that is linked above show up first in the list.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-03-19T23:53:31.817" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3562" PostTypeId="2" ParentId="402" CreationDate="2016-03-20T02:02:38.053" Score="0" Body="&lt;p&gt;Although as Felix says that the idea of zero is known from as early as Sumeria, and although this is true, the actual widespread first use was by the ancient Babylonians. The Greeks wrote the Babylonian zero as omicron (&quot;o&quot;) with a slash over it. The Indians later adopted this from Hellenistic cultures in what is now Syria and Iraq.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Felix's contention that the Indian's used a dot for a zero is incorrect. A dot is the Arabic symbol for zero. As I said above, our symbol for zero comes from the Greeks.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2016-03-20T02:02:38.053" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3563" PostTypeId="2" ParentId="9" CreationDate="2016-03-20T11:54:29.880" Score="8" Body="&lt;p&gt;The same Galois gives &lt;a href=&quot;https://plus.maths.org/content/genius-stupidity-and-genius-again&quot; rel=&quot;noreferrer&quot;&gt;two different reasons for the duel&lt;/a&gt; in its two last letters:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the &quot;Letter to all Republicans&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;I beg patriots and my friends not to reproach me for dying otherwise than for my country. I die the victim of an infamous coquette. It is in a miserable brawl that my life is extinguished. Oh! Why die for so trivial a thing, die for something so despicable! Pardon for those who have killed me, they are of good faith.&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the letter to two un-named friends:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;I have been challenged by two patriots - it was impossible for me to refuse. I beg your pardon for having advised neither of you. But my opponents had put me on my honour not to warn any patriot. Your task is very simple: prove that I fought in spite of myself.... Preserve my memory since fate has not given me life for my country to know my name. I die your friend.&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="488" LastActivityDate="2016-03-20T11:54:29.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3564" PostTypeId="2" ParentId="3558" CreationDate="2016-03-20T21:29:03.970" Score="8" Body="&lt;p&gt;History does not often develop in the order of textbook expositions. Today the exponential map is introduced early in both Riemannian geometry and Lie group theory, but many results it is used to derive were originally derived without it. There is no &quot;exponential map&quot; in &lt;a href=&quot;http://onlinebooks.library.upenn.edu/webbin/gutbook/lookup?num=36856&quot; rel=&quot;noreferrer&quot;&gt;Gauss's General Investigations of Curved Surfaces (1825,27)&lt;/a&gt; or &lt;a href=&quot;http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf&quot; rel=&quot;noreferrer&quot;&gt;Riemann's On the Hypotheses which lie at the Bases of Geometry (1854)&lt;/a&gt;, even though they introduce special coordinates (Gauss for surfaces, Riemann generally) which can be interpreted as combining the inverse of the exponential map with a choice of orthonormal frame in the tangent space. Those are now called geodesic polar coordinates, although Bonnet only used &quot;geodesic curvature&quot; in 1848, and &quot;geodesic&quot; as a curve does not appear until Stäckel in 1893, see &lt;a href=&quot;https://books.google.com/books?id=TIXg4hSXmFAC&quot; rel=&quot;noreferrer&quot;&gt;Struik's Lectures on Classical Differential Geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nor does one find &quot;exponential map&quot; or &quot;matrix exponential&quot; (or matrix groups for that matter) in Lie's various papers and monographs of 1880-90s on continuous groups. He was talking about groups of transformations acting on domains of &#xA;$\mathbb{R}^n$, and his &quot;infinitesimal transformations&quot; generating them were vector fields. He does write expressions like $f+X(f)+\frac1{2!}X^2(f)+\cdots$, but without symbology or much fuss about convergence. Schur added the last bit around 1890, when he gave a more rigorous proof of Lie's theorem on reconstructing Lie group from its Lie algebra, along with implicit version of the Campbell-Hausdorff formula for $Z$ in $e^Xe^Y=e^Z$. But only Campbell in two short notes of 1897 on that formula invokes exponentials directly:&quot;&lt;em&gt;If x and y are operators which obey the ordinary laws of algebra, we know that &#xA;$e^ye^x = e^{y+x}$. I propose to investigate the corresponding theorem when the operators obey the distributive and associative laws, but not the commutative&lt;/em&gt;&quot;. Poincare arrived at the same ideas independently, and developed the exponential formalism more fully in On Continuous Groups (1899), where he also introduced what is now called &quot;universal enveloping algebra&quot; (rigorous proof of the formula by modern standards was only given by Hausdorff in 1906). See &lt;a href=&quot;https://projecteuclid.org/euclid.bams/1183548683&quot; rel=&quot;noreferrer&quot;&gt;Schmid's lecture Poincaré and Lie groups&lt;/a&gt;, which cites the original sources.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the Riemannian exponential map it took considerably longer to appear, and it was motivated by the Lie group examples rather than by some formal property (there isn't any straightforward one). General Riemannian geometry remained an obscure subject in 19th century with only a small group of Italian geometers like Bianchi, Ricci and Levi-Civita working on absolute (now tensor) calculus. The fad of the day were Kleinian geometries (quotients of projective groups), and especially space-forms, the ones of constant curvature. Killing realized that to classify the latter one needs to classify algebras of Lie groups first, and in a remarkable tour de force of 1885-1890 he managed to do just that, albeit with wobbly rigor. Cartan's grand arrival on the scene was his thesis of 1894, where he reworked Killing's classification rigorously. This meant that Cartan was well aware of the Killing form, the bi-invariant metric on a Lie group. See &lt;a href=&quot;http://www.ams.org/notices/200306/rev-rowe.pdf&quot; rel=&quot;noreferrer&quot;&gt;Rowe's review of Hawkins's Emergence of the Theory of Lie Groups&lt;/a&gt; and &lt;a href=&quot;http://www.ams.org/journals/bull/1999-36-03/S0273-0979-99-00787-9/&quot; rel=&quot;noreferrer&quot;&gt;Petersen's Aspects of Global Riemannian Geometry&lt;/a&gt; for further historical details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians rushed into Riemannian geometry only after Einstein introduced general relativity in 1915. He was lucky that his friend Grossman was even aware of absolute calculus, and even luckier that Levi-Civita took it upon himself to tutor him on it via correspondence later. Cartan was one of the first to get involved too, in 1922 he proposed a non-zero torsion version of general relativity, which Einstein himself later pursued, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/1907/what-have-we-learnt-from-einsteins-unsucessful-dream/1913#1913&quot;&gt;What have we learned from Einstein&amp;#39;s unsuccessful dream?&lt;/a&gt; And in 1925 he published the first major work on global Riemannian geometry, &lt;a href=&quot;https://books.google.com/books/about/Geometry_of_Riemannian_Spaces.html?id=-YvvVfQ7xz4C&quot; rel=&quot;noreferrer&quot;&gt;Geometry of Riemannian Spaces&lt;/a&gt;. By then the notion of parallel transport was already worked out by Levi-Civita and others, and in 1923 Cartan himself introduced affine connections, generalizing Weyl's notion (see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2414/the-origin-of-the-name-connection-in-differential-geometry/2416#2416&quot;&gt;The origin of the name &amp;quot;connection&amp;quot; in differential geometry&lt;/a&gt;). So it would have been obvious (to Cartan anyway) that a vector spread around by group action along a geodesic would be parallel transported. The coincidence of the two exponential maps then follows directly from the definitions. So we should see both exponential maps so named in Cartan's monograph, right? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But we do not. The Riemannian exponential map along with much other modern lingo first appears in the work of his student, Ehresmann, in 1935-1960. Hermann in his notes to the English translation of the second edition of Cartan's book makes interesting remarks on the origin of Ehresmann's innovations:&quot;&lt;em&gt;Motivated partly by Cartan, partly by Lie and Vessiot, and in part by his own mathematical intuition (which was more algebraic and topological than Cartan's), he introduced many geometric superstructures built into this basic &quot;calculus on manifolds&quot;... in his classic work [Les connexions infinitésimales dans un espace fibré différentiable], formalized this Cartan's connection material in a certain form... However, there is a certain difference in methodology between Cartan and these later workers. They have tried to be determinedly index-free...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-20T23:38:05.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3565" PostTypeId="1" CreationDate="2016-03-21T16:58:39.693" Score="2" ViewCount="123" Body="&lt;p&gt;In &lt;em&gt;&lt;a href=&quot;https://worldcat.org/oclc/565662199&quot; rel=&quot;nofollow&quot;&gt;A Discipline of Programming&lt;/a&gt;&lt;/em&gt;, &lt;a href=&quot;http://www.encyclopedia.com/doc/1G2-2830905624.html&quot; rel=&quot;nofollow&quot;&gt;Dijkstra&lt;/a&gt; wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Two circumstances have changed the scene since then.* The one is the insight that, even in the case of fully deterministic machines, program testing is hardly helpful. As I have now said many times and written in many places: program testing can be quite effective for showing the presence of bugs, but is hopelessly inadequate for showing their absence. The other one is the discovery that in the meantime it has emerged that &lt;strong&gt;any design discipline must do justice to the fact that the design of a mechanism that is to have a purpose must be a goal-directed activity&lt;/strong&gt;. In our special case it means that we can expect our post-condition to be the starting point of our design considerations. In a sense we shall be &quot;working backwards&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*[i.e., since the time of irreproducibility/indeterminism (real bugs!) in computation]&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Monte Carlo algorithm is only partially &quot;a goal-directed activity&quot;: We know very little about how the algorithmic form of the pseudorandom number generator relates to the ultimate goal of the Monte Carlo algorithm.&lt;br&gt;&lt;sup&gt;(which, for example, might be computing the area of a circle)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, it would seem Dijkstra would be opposed to Monte Carlo algorithms, either for problem-solving or for testing, because it is unclear what the algorithmic form of a random number generator has to do with what Monte Carlo tasks are designed to accomplish.&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-03-21T16:58:39.693" Title="What did Dijkstra think about Monte Carlo algorithms?" Tags="|computer-science|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3566" PostTypeId="1" CreationDate="2016-03-22T09:28:53.500" Score="3" ViewCount="102" Body="&lt;p&gt;A counterintuitive result in probability theory that may warrant the description of a veridical paradox is the fact that repeating an experiment with a nonzero chance of success infinitely often does not always guarantee eventual success. This is the case if the probability of success shrinks with each iteration quickly enough to compensate for the repetitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a name for this paradox and has it been explicitly discussed anywhere? The paradox arises in branching processes used to model extinction probabilities, including Galton and Watson's 1874 work on the extinction of family names. It also occurs in 3D random walks, where the probability of return to the origin is less than 1, proved by Pólya in 1921.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do these or any other sources discuss the apparent paradox?&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2016-03-23T04:51:15.760" LastActivityDate="2016-03-24T23:59:28.187" Title="Name and history of probabilistic non-inevitability paradox?" Tags="|mathematics|statistics|probability|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3567" PostTypeId="1" CreationDate="2016-03-22T11:42:04.197" Score="2" ViewCount="86" Body="&lt;p&gt;I was wondering where in the literature the standard textbook argument (appearing in Young and Freedman and Purcell and Morin, for example) for displacement current first appears. It seems especially odd to me that Maxwell didn't include it in either his Treatise or research papers seeing as he certainly knew of Stoke's Theorem, Ampere's Law and capacitors&lt;/p&gt;&#xA;" OwnerUserId="3880" LastActivityDate="2016-08-13T00:04:33.697" Title="Origin of Textbook Argument for Displacement Current as Generalization of Ampere's Law" Tags="|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3569" PostTypeId="1" AcceptedAnswerId="3570" CreationDate="2016-03-22T19:30:08.013" Score="6" ViewCount="489" Body="&lt;p&gt;As per the title: where did the notation $a\wedge b$ for the exterior product of $a$ and $b$ originate, and/or who popularised it? I'm especially interested in motivation for the choice of this symbol (which also means &quot;and&quot; in logic and &quot;meet&quot; in order theory).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've moved this question here from MSE, where it got very little attention despite offering a bounty; the original version is pasted below the line because it includes a bit about my reason for asking.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It seems odd that something that looks so much like a join [see below] would get given &quot;the wrong symbol&quot;. It's even worse when you dualise it and get something called (quite reasonably) the &quot;meet product&quot;, but write &lt;em&gt;that&lt;/em&gt; as $A\vee B$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware the answer &lt;em&gt;might&lt;/em&gt; be &quot;it just turned out that way&quot;, but was this choice of symbol motivated by a natural way to think of $A\wedge B$ as a meet in some lattice or other? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not, which historical figure should I blame for this situation? I understand Grassmann wrote $A\wedge B$ as $[A, B]$, which would let him off the hook.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Intuitively, if I have two vectors $a$ and $b$, $a\wedge b$ is the smallest object that contains both $a$ and $b$. If you order elements of the exterior algebra by inclusion, which seems very natural if you're doing geometry, you get a lattice where the join operation is the exterior product.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This mostly seems to come to the fore in Clifford-algebra-related approaches to geometry. Examples of people mentioning this relationship and complaining about / altering the standard notation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.co.uk/books?id=UHeCBwAAQBAJ&amp;amp;pg=PT317&amp;amp;lpg=PT317#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.co.uk/books?id=UHeCBwAAQBAJ&amp;amp;pg=PT317&amp;amp;lpg=PT317#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.co.uk/books?id=y_lvbI70L_YC&amp;amp;pg=PA367&amp;amp;lpg=PA367#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.co.uk/books?id=y_lvbI70L_YC&amp;amp;pg=PA367&amp;amp;lpg=PA367#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.co.uk/books?id=PfWpCAAAQBAJ&amp;amp;pg=PA98&amp;amp;lpg=PA98#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.co.uk/books?id=PfWpCAAAQBAJ&amp;amp;pg=PA98&amp;amp;lpg=PA98#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;While digging up these references I also turned up a claim that Peano wrote the exterior product using $\vee$, whereas Cartan used $\wedge$, but it's not clear to me whether Cartan originated (or popularised) this notation or whether he had a particular reason for choosing it [or indeed whether this claim is correct -- it was an offhand remark with no citation].&lt;/p&gt;&#xA;" OwnerUserId="2864" LastEditorUserId="2864" LastEditDate="2016-03-22T20:31:42.120" LastActivityDate="2020-12-11T22:24:24.270" Title="How did the exterior product get its symbol?" Tags="|notation|differential-geometry|linear-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3570" PostTypeId="2" ParentId="3569" CreationDate="2016-03-22T21:04:10.377" Score="5" Body="&lt;p&gt;Cajori gives the early uses of logical symbols in volume 2 of &lt;a href=&quot;http://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Mathematical Notations&lt;/a&gt;. Neither Boole nor Schröder used $\wedge$ and $\vee$ in Boolean algebra, but rather $\cdot$ (or blank) and $+$, the idea was to make it as &quot;algebra looking&quot; as possible. &lt;a href=&quot;http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093882673&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peirce in his 1865 lectures&lt;/a&gt; was apparently first to give Boolean algebra its modern form, by introducing the non-strict disjunction denoted $\vee$, and the material conditional. Peano uses $\smile$ for &quot;&lt;em&gt;the smallest class containing all the classes&lt;/em&gt;&quot; and $\frown$ for &quot;&lt;em&gt;the greatest class contained in all the classes&lt;/em&gt;&quot; (there was no sharp distinction between logical operations and operations on classes at the time) in Formulaire de Logique (1891). In fact, they already appear in his book &lt;a href=&quot;http://mathematica.sns.it/opere/138/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calcolo Geometrico Secondo l'Ausdehnungslhere di H. Grassmann (1888)&lt;/a&gt;, along with the first modern axiomatization of vector spaces and exterior calculus. But they are for classes only, not as wedges or vees, for multivectors Peano writes concatenations occasionally preceded by $gr$. Russell and Whitehead went back to Peirce's use in Principia (1910-13), with $\vee$ for disjunction and $\cdot$ for conjunction. The meet/join notation for lattices came even later, so it was not established when Cartan was developing exterior calculus in 1899-1925.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it is common to assume that Cartan introduced wedges one finds none of them in his papers and monograph on the subject, linked in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2488/where-did-cartan-introduce-his-notation-for-basis-vectors-and-covectors/2500#2500&quot;&gt;Where did Cartan introduce his notation for basis vectors and covectors?&lt;/a&gt; He mostly just concatenates forms without any symbol, and very occasionaly writes brackets. My guess is that the responsible party is his student Ehresmann, who  came up with most of modern symbology and lingo in the calculus on manifolds in his 1935-1960 transformation of Cartan's work, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3558/how-did-the-exponential-map-of-riemannian-geometry-get-its-name/3564#3564&quot;&gt;How did the exponential map of Riemannian geometry get its name?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the reason, while the use of $\wedge$ and $\vee$ in Boolean algebra is symmetric it is Boole's and Schröder's product that became Peano's $\frown$, reinforced as product by Russell and Whitehead. Since the exterior product is definitely a product and not a sum perhaps $\wedge$ seemed more suitable. Alexander and Kolmogorov used the &quot;correct&quot; $\cup$ for the corresponding cup product in cohomology in 1935. But your kindred spirit is Rota, who boldly reverses the direction even in exterior calculus, see &lt;a href=&quot;http://kalx.net/dsS2011/BarBriRot1985.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Exterior Calculus of Invariant Theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-24T00:39:54.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3571" PostTypeId="2" ParentId="3533" CreationDate="2016-03-23T00:56:29.447" Score="3" Body="&lt;p&gt;Regarding first developments in non- euclidean geometry, Eugenio Beltrami considered Lobachevsky-Bolyai geometry as nothing else but euclidean geometry on a space with (constant) negative curvature. In 2 memoirs published in 1868 he proposed some realizations of such geometries: one by using geodesics on a pseudosphere and others, including what is now known as the Beltrami-Klein model. He also gave a proof of equiconsistency of the Euclidean and hyperbolic geometry.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2016-03-23T00:56:29.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3572" PostTypeId="2" ParentId="3566" CreationDate="2016-03-23T21:08:43.230" Score="1" Body="&lt;p&gt;If we have a trial with non-zero probability of success $p&amp;gt;0$, and we repeat the trial $n$ times, the probability of at least one success is $p(n)=1-(1-p)^n$, which converges to $1$ when $n\to\infty$. An example would be tossing a fair coin &#xA;(with $p=1/2$), and expecting to get heads eventually. Even if the coin is loaded for tails as long as $p&amp;gt;0$ we can still expect getting heads eventually. Of course, if the probability $p$ changes from trial to trial then it is possible that $p(n)$ does not converge to $1$, but that would not be repeating &quot;the same&quot; trial, nor would it be paradoxical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probability of eventual return in 3D random walks is &lt;a href=&quot;https://mathoverflow.net/questions/45098/when-do-3d-random-walks-return-to-their-origin&quot;&gt;discussed on Math Overflow&lt;/a&gt; for example, or in this &lt;a href=&quot;http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter12.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;book chapter&lt;/a&gt;, neither of which describes its values as paradoxical. The paradoxes mentioned are unrelated: the &lt;a href=&quot;http://arxiv.org/pdf/0712.0220.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;birthday paradox&lt;/a&gt; and the &lt;a href=&quot;http://arxiv.org/pdf/cond-mat/9905106.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Levinthal paradox of protein folding&lt;/a&gt;. Similarly, the only paper I found that mentions a &lt;a href=&quot;http://people.bath.ac.uk/maspm/condi.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;paradox&quot; in connection with the Galton-Watson process&lt;/a&gt; refers to something else. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are not however cases of &quot;repeating the same trial&quot; for eventual success. In 3D walks for example the outcomes of random &quot;trials&quot; are individual steps $X_i$ chosen at random, but &quot;success&quot; is defined in terms of the final position $S_n=X_1+\cdots+X_n$, not individual $X_i$, which is clearly different from success in coin tosses. Moreover, each step is performed in different locations at different distances from the origin, so they are not &quot;the same&quot; trials by any stretch. The probability of eventual return is $1$ in 1D and 2D, but only about $0.65$ in 3D, which in principle is not very surprising since there is a lot more room to wander around in 3D (frankly, it might be more surprising that it is still $1$ in 2D). MO thread gives a more detailed intuitive explanation for the difference based on convergence/divergence of $1/n^{d/2}$ series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more general phenomenon at play here is the difference between recurrent and transient states in Markov chains. A state is called recurrent (persistent) if the probability of eventual return to it is $1$, and transient otherwise. Independent trials form a so-called Bernoulli scheme, which is a very special case of a Markov chain, where the next state does not depend even on the previous one (which is what makes them &quot;the same&quot; trial). By the first paragraph above in a Bernoulli scheme all states are recurrent, but general Markov chains are free to have transient ones.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-03-24T23:59:28.187" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3573" PostTypeId="2" ParentId="635" CreationDate="2016-03-23T23:26:41.773" Score="8" Body="&lt;p&gt;Multiplication is a lot of work; in numerical computing it is considered evil and many tricks are used to avoid it.  So Napier created the table of logarithms. (Briggs worked with Napier to make the table more useful.)  IIRC, Napier's logarithms were to base 0.9999999.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, keep in mind that not all multiplication is evil.  Particularly multiplication by 1+10^-n and 1-10^-n, which is merely a shift and add-- much less work than a full blown multiplication.  Multiplication by 0.9999999 would be a shift and subtract.  This is one of the tricks that Napier used.  Similar trickery was used by Robert Flower in 1771, by Euler (year unknown), and in the CORDIC algorithm of Jack Volder (1959, eventually incorporated into most handheld calculators).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Robert Flower claimed that he could compute a logarithm to 20 decimal places in 7 or 8 minutes, with nothing more than a pencil and paper. &lt;strong&gt;* Back in the 1960's, before we had calculators, I would use an abacus.  I wondered if there could be a way to compute logarithms on an abacus.  I later found out that it is not possible.  You need two of them.  What about a recursive method that converges quadratically?  You could get ten digit logs from your calculator, do one iteration, and have them to twenty digits.  Yep, there is such an algorithm.  *&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Math teachers in elementary school dare not offer this knowledge to their students!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*** Left as an exercise for the student, of course!&lt;/p&gt;&#xA;" OwnerUserId="3890" LastActivityDate="2016-03-23T23:26:41.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3574" PostTypeId="1" AcceptedAnswerId="3577" CreationDate="2016-03-25T13:19:02.600" Score="3" ViewCount="651" Body="&lt;p&gt;I have read that Galileo was able to start observing the four large satellites of Jupiter in 1610. Did he ever attempt to estimate the relative sizes of the four orbits, and their periods? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made a plot of the period of the four satellites versus their orbit size, as a ratio to the diameter of Jupiter at the equator. That would have been easier to sketch/estimate than absolute size.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any record of a plot or graph somewhat like this - to visualize even qualitatively what kind of relationship there was? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today one would make a log-log plot and immediately &quot;discover&quot; the 3/2 slope, but logarithms weren't available yet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;About eight years later, Kepler &quot;articulated&quot; what we now call Kepler's third law, the period squared being proportional to the diameter (semi-major axis) cubed. Did that realization come from these moons, or from looking at planetary motion instead?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One reason I ask about the Galilean moons - the eccentricity of orbits of these four moons is very low - their motion would be easily interpreted using just circles, while several planets have significant eccentricity and their positions would require more sophisticated math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the relationship of the planets and that of the Jovian moons would have a different constant - was it immediately interpreted as a measure of the ratio of masses of Jupiter and the Sun?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is my first question here - the wording may seem colloquial, but my question is serious.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qYzGC.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qYzGC.gif&quot; alt=&quot;Galilean Moons of Jupiter Kepler&amp;#39;s Third Law&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="4251" LastEditDate="2021-02-27T19:05:42.090" LastActivityDate="2021-02-27T19:05:42.090" Title="Was &quot;Kepler's third law&quot; deduced from the Galilean moons, or from planetary motion?" Tags="|astronomy|classical-mechanics|celestial-mechanics|17th-century|kepler|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3575" PostTypeId="1" AcceptedAnswerId="3588" CreationDate="2016-03-25T13:41:50.180" Score="8" ViewCount="1059" Body="&lt;p&gt;Today if I want to calculate an elliptical orbit of some object due to the gravity of another, I use &lt;a href=&quot;https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#The_Runge.E2.80.93Kutta_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;Runge-Kutta&lt;/a&gt; integration, and I can see the individual steps I use shown in that article.&lt;/p&gt;&#xA;&lt;p&gt;Back &amp;quot;in the day...&amp;quot; Kepler, Newton, and others might have used what we now call &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler%27s_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kepler's Equation&lt;/a&gt;: &lt;span class=&quot;math-container&quot;&gt;$M=E-e \sin(E)$&lt;/span&gt; where given M one had to solve for E.&lt;/p&gt;&#xA;&lt;p&gt;I would like to &lt;em&gt;see&lt;/em&gt; how Kepler, Newton, and similar contemporaries &lt;em&gt;actually performed the calculation&lt;/em&gt; of the position in the orbit for a specified time. Are there reproductions, or images of their notes, showing the &lt;em&gt;actual steps&lt;/em&gt; each of them used to solve this equation?&lt;/p&gt;&#xA;&lt;p&gt;Did Newton or Leibniz also use some form of &lt;em&gt;integration&lt;/em&gt; as an independent method to obtain the motion of an elliptical orbit as well? Are there images or reproductions of those calculations available anywhere?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="4251" LastEditDate="2021-02-28T12:58:21.263" LastActivityDate="2021-02-28T12:58:21.263" Title="How did Newton and Kepler (actually) do it?" Tags="|calculus|classical-mechanics|newton|celestial-mechanics|kepler|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="3576" PostTypeId="2" ParentId="3575" CreationDate="2016-03-25T17:20:51.040" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://plus.maths.org/content/origins-proof-ii-keplers-proofs&quot; rel=&quot;nofollow&quot;&gt;Kepler's Proofs&lt;/a&gt; will get you started on your quest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This article mentions 987 folio pages of arithmetic; you should also look at the tables and methods of Copernicus.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastEditorUserId="3821" LastEditDate="2016-03-25T22:14:47.373" LastActivityDate="2016-03-25T22:14:47.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3577" PostTypeId="2" ParentId="3574" CreationDate="2016-03-25T17:55:19.610" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Johannes_Kepler#Harmonices_Mundi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article on Harmonices Mundi&lt;/a&gt; states that Kepler gave only the conclusion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since he had taken all of Brahe's observations, the presumption is that he used this data, for he was very familiar with it, and it was more than adequate for the task.  His published result describes the relationship in terms of the sun and planets, but not planets and moons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://physics.stackexchange.com/questions/35809/was-jupiters-mass-guessed-at-by-kepler-or-galileo&quot;&gt;Was Jupiter's mass “guessed at” by Kepler or Galileo?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3821" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2016-03-25T18:09:50.443" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3578" PostTypeId="2" ParentId="3574" CreationDate="2016-03-25T19:30:21.817" Score="4" Body="&lt;p&gt;Kepler's third law was discovered on the basis of comparison of periods and distances of the planets. This was in 1619. Only in 1621 Kepler noticed that Galileo moons of Jupiter also satisfy this law. This fact was later used by Galileo as an argument in favor of Copernican system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Kepler was one of the first astronomers who used logarithms.&#xA;(Napier's discovery was communicated to Ticho Brahe in 1590, and Kepler was Ticho's&#xA;assistant in 1600-1601.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-03-25T19:30:21.817" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3579" PostTypeId="2" ParentId="3575" CreationDate="2016-03-25T19:35:53.323" Score="3" Body="&lt;p&gt;At the time of Kepler, not only Runge-Kutta, but the very notion of the differential equation was not avalailable:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you really would like to see how Kepler calculated the orbits, why don't you look at his own&#xA;work, Astronomia Nova, which is available in English translation?&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-03-25T19:35:53.323" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3580" PostTypeId="1" AcceptedAnswerId="3584" CreationDate="2016-03-25T23:50:27.517" Score="12" ViewCount="293" Body="&lt;p&gt;After reading &lt;a href=&quot;https://hsm.stackexchange.com/a/1950/3869&quot;&gt;this&lt;/a&gt; answer and writing &lt;a href=&quot;https://hsm.stackexchange.com/questions/3574/was-keplers-third-law-deduced-from-the-galilean-moons-or-from-planetary-moti#comment7014_3578&quot;&gt;this&lt;/a&gt; comment, I decided to ask this question: &lt;em&gt;When and where was the earliest known use of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Log%E2%80%93log_plot&quot; rel=&quot;noreferrer&quot;&gt;log-log plot&lt;/a&gt; to demonstrate &lt;a href=&quot;https://en.wikipedia.org/wiki/Power_law&quot; rel=&quot;noreferrer&quot;&gt;power-law&lt;/a&gt; behavior?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2022-11-07T00:18:24.887" Title="When was the earliest use of log-log plots to demonstrate power-law behavior?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3581" PostTypeId="1" AcceptedAnswerId="3593" CreationDate="2016-03-26T03:22:28.167" Score="4" ViewCount="100" Body="&lt;p&gt;Truesdell, C. &lt;em&gt;Rational Thermodynamics&lt;/em&gt;. New York, NY: Springer New York, 1984. &lt;a href=&quot;http://dx.doi.org/10.1007/978-1-4612-5206-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://dx.doi.org/10.1007/978-1-4612-5206-1&lt;/a&gt;, p. 41 says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;… in the numerous papers and books whose titles and topics include &quot;irreversible&quot;—the learned clerk HRÆðRAFORD, writing in modern chemical Latin, has recently called their field &quot;thermodynamice inreparabilis&quot;—…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Who is this &quot;learned clerk&quot; Hræðraford? Below, the relevant paragraph is shown: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/B5cGM.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/B5cGM.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="26" LastEditDate="2016-03-26T18:13:19.473" LastActivityDate="2016-03-28T23:55:44.527" Title="Who is Hræðraford, the &quot;learned clerk&quot; &quot;writing in modern chemical Latin&quot;?" Tags="|chemistry|thermodynamics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3582" PostTypeId="1" CreationDate="2016-03-26T15:33:00.537" Score="2" ViewCount="96" Body="&lt;p&gt;Lore says that Edward Lorenz found out that atmosphere fluidodynamics was very sensitive to initial conditions when he stopped a numerical simulation, dutifully copied the partial results, and after inputting them again he got a result vastly different from what it was supposed. (He then went to talk about the butterfly effect, but that's another story) &#xA;Do you know if there is a first-hand recollection of this story?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="26" LastEditDate="2016-03-26T19:10:18.250" LastActivityDate="2016-03-26T19:10:18.250" Title="Edward Lorenz and sensitivity to initial conditions" Tags="|biographical-details|discoveries|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3584" PostTypeId="2" ParentId="3580" CreationDate="2016-03-26T22:13:38.760" Score="11" Body="&lt;p&gt;Napier introduced the logarithms in 1614, and Gunter invented the slide rule somewhere in 1620s. One might think that log log plots came soon after, but no. According to &lt;a href=&quot;https://deadreckonings.files.wordpress.com/2009/12/calendar2010_no_holidays.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Age of Graphical Computing:&lt;/a&gt; &amp;quot;&lt;em&gt;In 1844 &lt;a href=&quot;https://en.wikipedia.org/wiki/L%C3%A9on_Lalanne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leon Lalanne&lt;/a&gt; succeeded in linearizing the curves &lt;span class=&quot;math-container&quot;&gt;$y=x^p$&lt;/span&gt; by plotting the first log-log plot in history, thereby creating his Universal Calculator, chock-full of lines for common engineering calculations and capable of graphically computing formulas in powers or roots of x ( or of trigonometric functions in x) with ease... Lalanne envisioned copies of his Universal Calculator posted in public squares and business meeting places for popular use&lt;/em&gt;&amp;quot;. Lallane's successor, &lt;a href=&quot;https://fr.wikipedia.org/wiki/Maurice_d%27Ocagne&quot; rel=&quot;nofollow noreferrer&quot;&gt;d'Ocagne&lt;/a&gt;, also credits him with the invention of logarithmic graph paper. The first link has a nice image of Lalanne's Universal Calculator, which looks like a fanciful version of it.&lt;/p&gt;&#xA;&lt;p&gt;Lalanne's approach was not specific to logarithms, he came up with a general idea of transforming curved graphs into straight line ones by modifying the scales on the axes, which he called &amp;quot;&lt;em&gt;geometrical anamorphosis&lt;/em&gt;&amp;quot;. d'Ocagne developed Lalanne's ideas after 1880 under the name of &lt;a href=&quot;https://en.wikipedia.org/wiki/Nomogram&quot; rel=&quot;nofollow noreferrer&quot;&gt;nomography&lt;/a&gt;, and nomograms remained a popular computing tool, especially among engineers, until the onset of pocket calculators. Smith's Source Book in Mathematics has large excerpts from &lt;a href=&quot;https://books.google.com/books?id=bNDDAgAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;d'Ocagne's Treatise on Nomography (1899)&lt;/a&gt;, which describes his and Lalanne's methods of graphical computing. The slide rule, logarithmic paper and nomograms, those nostalgic pieces of the old lore of scientific and technical computing wiped out by the advent of electronic calculators and computers...&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-11-07T00:18:24.887" LastActivityDate="2022-11-07T00:18:24.887" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="3585" PostTypeId="2" ParentId="252" CreationDate="2016-03-27T03:10:54.907" Score="1" Body="&lt;p&gt;Just to add. After giving a standard, modern definition of the function, Stephen Abbott (in p.7 of &lt;em&gt;Understanding Analysis&lt;/em&gt;) notes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This definition of function is more or less the one proposed by Peter Lejeune Dirichlet (1805–1859) in the 1830s. Dirichlet was a German mathematician who was one of the leaders in the development of the rigorous approach to functions that we are about to undertake. His main motivation was to unravel the issues surrounding the convergence of Fourier series. Dirichlet’s contributions figure prominently in Section 8.3, where an introduction to Fourier series is presented, but we will also encounter his name in several earlier chapters along the way. What is important at the moment is that we see how Dirichlet’s definition of function liberates the term from its interpretation as a type of “formula.” In the years leading up to Dirichlet’s time, the term “function” was generally understood to refer to algebraic entities such as $f(x) = x^2+1$ or $g(x) = \sqrt{x^4 + 4}$. [The above definition] allows for a much broader range of possibilities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user67" LastActivityDate="2016-03-27T03:10:54.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3586" PostTypeId="1" CreationDate="2016-03-27T06:35:35.457" Score="0" ViewCount="173" Body="&lt;p&gt;Its power is amazing. For a Hamiltonian, you define the Green function as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$G = \frac{1}{\lambda E-H} .$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who first come up with this definition? What was the motivation?&lt;/p&gt;&#xA;" OwnerUserId="3903" LastEditorUserId="55" LastEditDate="2016-03-28T22:19:02.940" LastActivityDate="2016-03-28T22:19:02.940" Title="Who introduced the Green function method into quantum mechanics?" Tags="|mathematics|physics|mathematical-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3587" PostTypeId="1" AcceptedAnswerId="3590" CreationDate="2016-03-27T13:31:47.527" Score="13" ViewCount="3046" Body="&lt;p&gt;When was the &lt;span class=&quot;math-container&quot;&gt;$3x+1$&lt;/span&gt; problem first proposed?&lt;/p&gt;&#xA;&lt;p&gt;At &lt;a href=&quot;https://en.wikipedia.org/wiki/Collatz_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, I found the following statement (without citation):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is the following article&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Lothar Collatz, &lt;em&gt;On the Motivation and Origin of the (&lt;span class=&quot;math-container&quot;&gt;$3n+1$&lt;/span&gt;)-Problem&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;which is very short and was translated into Chinese and  published there. By the way, why would a German mathematician translate an article to Chinese and publish it there and not in Germany? A translation to English can be found in the book &lt;em&gt;The Ultimate Challenge: The &lt;span class=&quot;math-container&quot;&gt;$3x+1$&lt;/span&gt; Problem&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the article, Collatz claims that he discovered the problem, and mentions three years:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Through the lectures of [...] from 1928 through 1933, I became in interested in problems in the areas of number theoretic functions and graph theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have experimented with many numbers. Before 1950 computers were not as well-developed as they are today. Therefore I could not try any large numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, it sounds to me that the problem must have arisen between 1928-1950. But how does Wikipedia know its from the year 1937?&lt;/p&gt;&#xA;" OwnerUserId="1706" LastEditorUserId="4251" LastEditDate="2020-07-14T04:19:48.353" LastActivityDate="2020-07-14T04:19:48.353" Title="What is the origin of the Collatz conjecture?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3588" PostTypeId="2" ParentId="3575" CreationDate="2016-03-27T16:59:21.593" Score="7" Body="&lt;p&gt;Goldstine, &lt;a href=&quot;https://books.google.com/books?id=r7ffBwAAQBAJ&quot;&gt;&lt;em&gt;A History of Numerical Analysis from the 16th through the 19th Century&lt;/em&gt;&lt;/a&gt; (1977), describes Kepler's approach (p. 47), which may be found in Kepler's &lt;a href=&quot;https://books.google.com/books?id=KUE1AAAAcAAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;&lt;em&gt;Epitome Astronomiae Copernicanae&lt;/em&gt;&lt;/a&gt; (1618), Ch. 4, Bk. V., pp. 665f.  It is an iterative numerical algorithm Kepler called &lt;em&gt;regula positionum&lt;/em&gt;.  Goldstine describes the steps of an example, which begins on p. 686 of the &lt;em&gt;Epitome&lt;/em&gt;, but does not develop the theory (probably because it was not seminal?).&#xA;Goldstine's numbers do not always match Kepler's; he may have corrected them or taken them from an excerpt containing this section printed (and translated) in &lt;em&gt;Great Books of the World&lt;/em&gt;, vol. 16 (1952), pp. 998-999 (Goldstine's reference).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goldstine (p. 64) also refers to Newton's solutions, which may be found in &lt;a href=&quot;https://books.google.com/books?id=Tm0FAAAAQAAJ&amp;amp;pg=PA148#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;&lt;em&gt;Principia&lt;/em&gt;&lt;/a&gt;, Book I, Prop. XXXI (Prob. XXIII) and Scholium.  (Note Goldstine's reference &quot;Problem XXII&quot; is a typo.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton first gives a solution involving a rolling wheel, to which the elliptical orbit is fixed. The wheel is rolled a distance that depends on the time elapsed, while a point on the ellipse fixed to the wheel traces a cycloid.  He then constructs the point on the orbit and shows it has swept out the correct area.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then admitting that foregoing method is hard to solve, Newton gives a solution by approximation in the scholium.  Newton's method is, what else, Newton's Method (often called the Newton-Raphson Method today), although it is presented in the geometric terms of the problem.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2016-03-27T19:29:05.617" LastActivityDate="2016-03-27T19:29:05.617" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3589" PostTypeId="1" CreationDate="2016-03-28T04:46:46.820" Score="1" ViewCount="72" Body="&lt;p&gt;Im currently reading the book &lt;em&gt;A Short History of Nearly Everything&lt;/em&gt; by Bill Bryson, and as I'm reading about the history of many fields of science (e.g. paleontology, geology, astronomy, etc.) it seems that many groundbreaking discoveries were shunned by their respective communities despite very real evidence on the new claim. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't mean to say that wild claims regardless of evidence should be considered, but rather that no matter how groundbreaking the claim is, if there is reasonable evidence to justify it, it should be considered and further looked into. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were a few accounts in the book of how the opinions of a scientific community seem to take precedence over logical and rational thinking when evaluating new claims. I feel like this is a lack of understanding of the scientific method, or merely a poor way of applying the scientific method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question is:&lt;/strong&gt; Have scientists collectively gotten better at being rational when being presented new groundbreaking claims in their respective fields? In other words, has the scientific community gotten better over the years at not allowing opinions and human emotion cloud their judgement when evaluating new evidence? Have scientists gotten better at &quot;doing science&quot;? &lt;/p&gt;&#xA;" OwnerUserId="3908" LastActivityDate="2016-03-28T04:46:46.820" Title="Have humans gotten better at the methods of science over time?" Tags="|discoveries|philosophy-of-science|" AnswerCount="0" CommentCount="2" ClosedDate="2016-03-29T15:09:58.540" ContentLicense="CC BY-SA 3.0" />
  <row Id="3590" PostTypeId="2" ParentId="3587" CreationDate="2016-03-28T21:08:36.783" Score="12" Body="&lt;p&gt;Wikipedia does not actually know that it is from 1937, my guess is that they picked a number between 1929 and 1950. According to Lagarias, who compiled an &lt;a href=&quot;http://arxiv.org/pdf/math/0309224.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;exhaustive bibliography&lt;/a&gt; on the problem, the first publication only appears in 1963, and is by Klamkin, he also gives other names it acquired: Syracuse Problem, Hasse’s Algorithm, Kakutani’s Problem and Ulam’s Problem. Early history is addressed more specifically in his &lt;a href=&quot;http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;American Mathematical Monthly article&lt;/a&gt;, which cites first hand accounts (and links to some of them) that are not publications. Thwaites, who calls it Ulam's problem, recalled that he stated the $3x+1$ problem already back in 1952. Hasse promoted the problem during his visit to the Syracuse university in 1950s, Kakutani got involved around 1960:&lt;em&gt;&quot;For about a month everybody at Yale worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Collatz himself circulated similar iteration problems at the International Congress of Mathematicians in 1950, although it is unclear if $3x+1$ specifically was among them. One such iterative function already appears in his notebook entry from July 1, 1932. Collatz wrote several letters in 1976-1980, where he reminisced about those early days, but as Lagarias points out &quot;&lt;em&gt;in none of these letters does L. Collatz actually state that he proposed the $3x+1$ problem&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He implies it though in a letter to Mays written in 1980 and &lt;a href=&quot;http://www.cecm.sfu.ca/organics/papers/lagarias/paper/goodies/ubersetzung/html/ubersetzung.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;translated here&lt;/a&gt;, and explains his motivation for studying such functions:&quot;&lt;em&gt;In 1929 I attended classes of E. Land and lectures in number theory given by Lettenmeyer, both in Göttingen, and in 1930 I attended classes given by O. Perron in Munich and Isai Schur in Berlin. I found it interesting to sketch the graphs of number theoretic functions $f(n)$ by drawing an arrow from $n$ to $f(n)$, or more simply, by writing $f(n)$ under $n$. (In this way) one can find various concepts which are well known in the theory of digraphs such as trees, cycles, bifurcation, etc. I don't know who the first person was to make these connections to graph theory; I, however, have seen neither in lectures nor in published form this type of representation of number theoretic functions. I enjoy observing the various patterns, and I computed the graphs (at the time of these investigations) of many interesting number theoretic functions for values of $n$ up to about $100$. I examined the example that I mention above&lt;/em&gt; [the conventional $3n+1$] &lt;em&gt;in this way too...&lt;/em&gt;&quot; This is as first hand and definitive as it gets without a contemporary document, and there is no 1937 in it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-02-21T21:44:11.597" LastActivityDate="2017-02-21T21:44:11.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3591" PostTypeId="1" AcceptedAnswerId="3595" CreationDate="2016-03-28T14:53:06.850" Score="1" ViewCount="327" Body="&lt;p&gt;Who made first the connection between light rays and wave fronts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact is stated in many places, e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Ray_%28optics%29&quot; rel=&quot;nofollow&quot;&gt;wikipedia&lt;/a&gt;, but I couldn't find a history.&lt;/p&gt;&#xA;" OwnerUserId="3915" OwnerDisplayName="Arnold Neumaier" LastEditorUserId="1815" LastEditDate="2016-03-29T15:09:20.600" LastActivityDate="2016-03-29T15:09:20.600" Title="light rays and wave fronts" Tags="|physics|discoveries|" AnswerCount="1" CommentCount="4" ClosedDate="2016-03-29T16:50:37.847" ContentLicense="CC BY-SA 3.0" />
  <row Id="3592" PostTypeId="2" ParentId="3586" CreationDate="2016-03-28T22:18:25.533" Score="2" Body="&lt;p&gt;This is not quite the definition of the Green function, but rather of the resolvent, more traditional notation is $R_{\lambda}:=(\lambda E-H)^{-1}$ with $H$ the Hamiltonian of the system and $E$ the identity operator. When $H$ is a differential operator, as the Schrodinger operator with some boundary conditions for example, this resolvent can be represented by an integral operator, whose kernel is called the Green function, i.e. $(R_{\lambda}f)(x)=\int G_{\lambda}(x,y)f(y)\,dy$. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Resolvent_formalism#History&quot; rel=&quot;nofollow&quot;&gt;resolvent formalism&lt;/a&gt; for integral operators was originally developed by Fredholm in 1903, the name is due to Hilbert. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the Green function, &lt;a href=&quot;https://books.google.com/books?id=CVRZEXFVsZkC&amp;amp;pg=PA204&amp;amp;lpg=PA204&amp;amp;dq=green+function+history&amp;amp;source=bl&amp;amp;ots=idAE4yzxYh&amp;amp;sig=L6AKWL5CrBVrGDRop3A6ZMeLFNk&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwix7_uLseTLAhVMsoMKHVlRBVwQ6AEISDAH#v=onepage&amp;amp;q=green%20function%20history&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Green introduced it back in 1828&lt;/a&gt; for solving boundary value problems, and there was no need to &quot;introduce&quot; it into quantum mechanics specifically. The kinds of boundary value problems that come up there are essentially the problems that physicists and mathematicians were solving in the context of classical mechanics throughout 19th century (e.g. the Sturm-Liouville problems), and Schrödinger, Dirac, Weyl, etc., were well familiar with the Green function method of solving them in 1920s, when quantum mechanics was created. The same goes for the Fourier transform methods and the operational calculus.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-03-28T22:18:25.533" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3593" PostTypeId="2" ParentId="3581" CreationDate="2016-03-28T23:55:44.527" Score="2" Body="&lt;p&gt;This is not exactly an answer, but too long for a comment. The quote, even in the attached context, is somewhat incomplete. It follows description of the few (including Duhem and Eckart), who in Truesdell's opinion positively contributed to thermodynamics after Gibbs. In other parts of the book he has pages long critique of what he calls &quot;Onsagerism&quot;, i.e.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Onsager_reciprocal_relations&quot; rel=&quot;nofollow&quot;&gt;Onsager's reciprocal relations&lt;/a&gt; in non-equilibrium (a.k.a. irreversible) thermodynamics. Actually, critique is too mild a word, one gets the idea from the second half of the displayed passage. &quot;HRÆðRAFORD, writing in modern chemical Latin&quot; refers not to Hraedraford, whoever he is, so writing (as I originally misread), but to Truesdell writing his name in &quot;modern chemical Latin&quot;. The whole passage is dripping with sarcasm about Onsager's formalism and notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was unable to find any references to this Hraedraford, but this is perhaps not so surprising because the only reason he is cited is for a Latin pun on &quot;irreversible&quot;: &quot;thermodynamice inreparabilis&quot;, the &lt;em&gt;irreparable thermodynamics&lt;/em&gt;, which is what Onsager's formalism is in Truesdell's opinion. The &quot;numerous papers and books&quot; are not Hraedraford's either, so he is likely just some witty learned citizen who came up with a pun Truesdell liked. Interestingly, the introduction and Hraedraford in it only appear in the &lt;a href=&quot;https://books.google.com/books?id=lMrcBwAAQBAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;second 1984 edition of Rational Thermodaynamics&lt;/a&gt;, the first 1969 edition does not mention him despite already including a chapter long rage against the &quot;Onsagerism&quot; with an approving mention of Eckart.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-03-28T23:55:44.527" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3594" PostTypeId="1" CreationDate="2016-03-29T11:55:57.163" Score="5" ViewCount="464" Body="&lt;p&gt;How did he come up with this idea before there was any experimental data to prove this? &lt;/p&gt;&#xA;" OwnerUserId="3917" LastActivityDate="2016-04-17T00:23:09.033" Title="What made Einstein believe (or know) that time was affected by speed and gravity?" Tags="|physics|theoretical-physics|relativity-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3595" PostTypeId="2" ParentId="3591" CreationDate="2016-03-29T14:42:42.227" Score="1" Body="&lt;p&gt;Paragraph 2 of &lt;a href=&quot;https://hsm.stackexchange.com/questions/3486/did-hamilton-anticipate-quantum-mechanics/3493#3493&quot;&gt;Did Hamilton anticipate quantum mechanics?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In a simple form the analogy was discovered by Huygens around 1670 (published in Traitė de la Lumiere, 1678), who noticed that propagation of waves could be dually described in terms of wavefronts and rays (&quot;characteristics&quot;) perpendicular to them. The latter can be considered as trajectories of particles, and massive amounts of particles spreading along characteristics can create the appearence of a continuous wave. But it also works vise versa, and Huygens suggested that light may well be a wave, with geometric optics of rays being only the first approximation. In particular, Hyugens showed how Fermat's least time principle follows from the analogy, Johann Bernoulli used it to solve the famous brachistochrone problem in 1696, and in 1818 Fresnel showed how not only geometric optics and also diffraction and interference can be explained by wave optics, which led to its wide acceptance.&quot;&lt;/p&gt;&#xA;" OwnerUserId="3821" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-03-29T14:42:42.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3596" PostTypeId="2" ParentId="3594" CreationDate="2016-03-29T15:00:04.630" Score="5" Body="&lt;p&gt;See &lt;a href=&quot;https://www.fourmilab.ch/etexts/einstein/specrel/www/&quot; rel=&quot;noreferrer&quot;&gt;ON THE ELECTRODYNAMICS&#xA;OF MOVING BODIES&lt;/a&gt;, by A. Einstein, where he explains the prior experimental evidence, and then shows the logic for Special Relativity.  He was not the first to show the relative timekeeping of moving clocks, this was already known from the analysis of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment&quot; rel=&quot;noreferrer&quot;&gt;Michelson-Morley experiments.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein recognized early on that gravitation was inconsistent with Special Relativity, and sought new principles, including the &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalence_principle&quot; rel=&quot;noreferrer&quot;&gt;Equivalence Principle&lt;/a&gt;. By applying this to his &quot;Elevator&quot; thought experiment, he was able to predict the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gravitational_redshift#History&quot; rel=&quot;noreferrer&quot;&gt;gravitational red shift&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, he used prior experimental results plus Maxwell's equations, and sought for logical, self-consistent theory with predictive power.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-03-29T15:00:04.630" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3597" PostTypeId="1" CreationDate="2016-03-29T12:38:58.483" Score="14" ViewCount="1758" Body="&lt;p&gt;In my field of research (theoretical physics), titles of publications occasionally begin with &quot;On the ...&quot; (a German language equivalent which was also in use is &quot;Über ... &quot; meaning &quot;About ...&quot;.) &#xA;This seems a bit artificial because in these cases usually also slightly modified phrases can make a title.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Titles beginning with &quot;On the ...&quot; suggest a certain character of the work as being of great relevance or presenting a conclusive result (my notion).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important historic work with this title is &quot;Περὶ οὐρανοῦ&quot; by Aristotle. It is usually translated as &quot;On the Heavens&quot;. There are many other important works in physics that start out like this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering if a connection can be made between the modern use of such titles and such historic works. I.e. did scientists start to use such titles inspired by the titles of early works?&lt;/p&gt;&#xA;" OwnerUserId="3246" OwnerDisplayName="highsciguy" LastEditorUserId="55" LastEditDate="2016-05-06T00:03:09.917" LastActivityDate="2016-05-06T00:03:09.917" Title="Do the titles starting with &quot;On the ...&quot; have a historical tradition?" Tags="|publishing|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="3598" PostTypeId="1" CreationDate="2016-03-29T18:14:48.463" Score="3" ViewCount="1448" Body="&lt;p&gt;I got confused by a question which is : &lt;strong&gt;&lt;code&gt;&quot;How did Stern-Gerlach experiment change the concept of space quantization and electron spin?&quot;&lt;/code&gt;&lt;/strong&gt; Its the question asked in our previous year examination. And I cannot make sense of it. In almost all of books its written that Stern-Gerlach experiment proved the concept of space quantization and electron spin. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I have read about the experiment, and also about its history, and at that time &lt;strong&gt;Stern and Gerlach didn't know about the concept of spin&lt;/strong&gt;. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The magnetic moment they measured of silver atoms was not due to&#xA;  orbital angular momentum but due to spin angular momentum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So how can it change the &lt;strong&gt;concept of spin&lt;/strong&gt;? Does the question want to ask how the stern-gerlach experiment would have changed the concept of spin if both experimenters knew something about the spin. Since Uhlenbeck and Goudsmit proposed somewhat classical concept of spin so if the experiment would have been performed after Uhlenbeck's hypothesis then the classical concept would have been changed to quantum one.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;And also it proved the &lt;strong&gt;concept of space quantization&lt;/strong&gt; which was proposed by Sommerfeld :&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The plane in which the orbit could be located in a magnetic field&#xA;  could have only certain definite orientations in space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It proved the concept so why its asked that it changed the concept of space quantization?, What the question actually means?&lt;/p&gt;&#xA;" OwnerUserId="3921" LastEditorUserId="3921" LastEditDate="2016-04-04T08:20:57.200" LastActivityDate="2016-04-04T08:20:57.200" Title="How did Stern-Gerlach experiment change the concept of space quantization and electron spin?" Tags="|quantum-mechanics|particle-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3599" PostTypeId="2" ParentId="3597" CreationDate="2016-03-29T22:31:40.350" Score="6" Body="&lt;p&gt;Yes, it does. Ignoring the differences between Greek Περὶ, Latin De, etc., the tradition of starting titles with a preposition predates Aristotle, e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus#Eudoxan_planetary_models&quot; rel=&quot;nofollow&quot;&gt;Eudoxus wrote On Speeds&lt;/a&gt;, where he described his astronomical system based on homocentric spheres (the one that Aristotle was elaborating in On Heavens). Modern influence, as for many other things, goes through the Renaissance and 17th century authors, e.g. Amico resurrected Eudoxian-Aristotelian system in his 1536 work titled &lt;a href=&quot;http://adsabs.harvard.edu/full/1972JHA.....3...36S&quot; rel=&quot;nofollow&quot;&gt;On the Motions of Heavenly Bodies according to Peripatetic Principles without Eccentrics or Epicycles&lt;/a&gt;. That Aristarchus titled his only surviving work On the Sizes and Distances of the Sun and Moon might also have helped Copernicus to name his opus magnum De Revolutionibus Orbium Coelestium (1543).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it was not just Aristotle and Eudoxus who were imitated, but also (and perhaps more) Archimedes and Galen. Archimedes's On Spirals, and On the Sphere and Cylinder were instrumental in the development of differential and integral calculus ideas by Cavalieri and Torricelli, respectively, and On the Equilibrium of Planes and On Floating Bodies were equally instrumental in the development of mechanics by Descartes, Pascal and Newton among others. Two of Newton's early drafts of Principia are titled starting with De (&lt;a href=&quot;https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum&quot; rel=&quot;nofollow&quot;&gt;De Motu Corporum in Gyrum&lt;/a&gt; and De Motu Corporum in Mediis Regulariter Cedentibus). There is a number of De titles in &lt;a href=&quot;https://en.wikipedia.org/wiki/Galenic_corpus#I:_Physiology_and_Anatomy&quot; rel=&quot;nofollow&quot;&gt;Galenic corpus&lt;/a&gt; as well (including Περί κράσεων translated into Latin as De Temperamentis), which perhaps caught an eye of Renaissance physicians. &lt;a href=&quot;https://en.wikipedia.org/wiki/Girolamo_Fracastoro&quot; rel=&quot;nofollow&quot;&gt;Fracastoro&lt;/a&gt; titled his most famous work De Contagione et Contagiosis Morbis (1546), that's the one where he introduced the germ theory of diseases. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-04-01T00:18:43.083" LastActivityDate="2016-04-01T00:18:43.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3600" PostTypeId="2" ParentId="3594" CreationDate="2016-03-30T22:53:57.180" Score="4" Body="&lt;p&gt;That time is affected by speed was &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Larmor&quot; rel=&quot;nofollow noreferrer&quot;&gt;suggested before Einstein by Larmor&lt;/a&gt;, and was motivated by experimental data, namely the Michelson-Morley (later also Trouton-Noble) experiments to detect the ether wind. As for gravity its influence on matter came out of general relativity as a side effect, its development was driven by different issues: equivalence of inertial and gravitational masses and the Mach's principle, which led to the general covariance requirements (equivalence of all frames, inertial or not) philosophically, and the perihelion of Mercury experimentally, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3547/why-did-einstein-develop-general-relativity/3550#3550&quot;&gt;Why did Einstein develop General Relativity?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To explain the negative result of the experiments Fitzgerald in 1889 and Lorentz in 1892 introduced the &lt;a href=&quot;https://en.wikipedia.org/wiki/Length_contraction#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;length contraction&lt;/a&gt; for moving objects. But as Larmor pointed out in 1897 the length contraction by itself was not enough to explain the effect, the time had to be dilated as well. Larmor then showed that if the intermolecular forces were of electromagnetic origin obeying the Maxwell electrodynamics then  length contraction and time dilation (as measured by devices so composed) will follow. As we would now say this is because the Maxwell electrodynamics is invariant under the Lorentz transformations, which Larmor also introduced in 1897, two years ahead of Lorentz. He presented his theory systematically in his book Aether and Matter (1900). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, to Fitzgerald, Lorentz and Larmor contraction and dilation were just apparent effects caused by dynamics of clocks and rulers that in no way affected the &quot;true&quot; absolute time and absolute space. In turn, Einstein's insight was that it is the &quot;apparent&quot; effects that were the physical ones, and the ether with the absolute space and time attached to it were the unphysical. Einstein therefore &quot;transformed&quot; the time dilation effect from illusion to reality, and from dynamic to kinematic and universal (not just affecting electromagnetic phenomena), rather than came up with it. Perhaps unsurprisingly, Lorentz and Larmor (and even Poincare) rejected Einstein's re-interpretation, they saw no need to abandon their old school theories on the basis of experiments they already explained.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-17T00:23:09.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3601" PostTypeId="2" ParentId="3598" CreationDate="2016-03-31T00:20:44.257" Score="2" Body="&lt;p&gt;Stern and Gerlach indeed did not know anything about the electron spoin when they conducted their experiments in 1921-22. Their experiments rather were originally misinterpreted to confirm the old quantum theory of Bohr-Sommerfeld, without spin, and eventually led to the emergence of the concept of spin proposed by Goudsmit and Uhlenbeck in 1925-26. Interestingly Goudsmit and Uhlenbeck did not even mention Stern-Gerlach results, their motivation was the anomalous Zeeman effect, two many spectral lines observed in a magnetic field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Larmor’s classical theory predicted that the beam of silver atoms would show  maximum intensity in the center, even in a magnetic field, whereas Sommerfeld's theory predicted spatial separation into two peaks (&lt;em&gt;assuming&lt;/em&gt; silver atoms were in a state with angular momentum 1). Originally they did not detect any separation, but after 1922 refinements it was detected. This meant that angular momentum is quantized, hence the excitement, as Pauli put it:&quot;&lt;em&gt;Hopefully now even the incredulous Stern will be convinced about directional quantization&lt;/em&gt;&quot;. This is what &quot;space quantization&quot; refers to, quantization with respect to direction in space, but the name is highly misleading today when literal quantization of space at Planck length is being discussed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As it turned out, while they were right that something directional was quantized in the experiment they were mistaken as to what it was. Contra Stern and Gerlach the silver atoms were actually in a state with angular momentum 0, not 1, and Sommerfeld's spinless theory predicted no separation, like Larmor's. So in fact their experiment disconfirmed both. See &lt;a href=&quot;http://plato.stanford.edu/entries/physics-experiment/app5.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Right Experiment, Wrong Theory: The Stern-Gerlach Experiment by Franklin and Perovic.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was not the first case of spin implicitly appearing in early experiments, which were misinterpreted to confirm a theory they did not in fact confirm. Einstein-de Haas and Barnett conducted a series of experiments in 1915-16 designed to measure the gyromagnetic effect due to Ampere's molecular currents. The effect they did measure was in fact due to spin, and only a measurement error prevented them from detecting the discrepancy, which got the name &quot;gyromagnetic anomaly&quot;, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3145/how-did-gyromagnetic-ratio-come-up-before-quantum-mechanics-and-who-introduced/3157#3157&quot;&gt;How did gyromagnetic ratio come up before quantum mechanics, and who introduced it?&lt;/a&gt; However, the discrepancy came to light by 1919, and went under the name &quot;gyromagnetic anomaly&quot;, Beck and Barnett even discussed it with Einstein before the introduction of spin. The irony is that the spin origin of the anomaly meant that the gyromagnetic effect received no contribution from the orbital motion of electrons, the very &quot;molecular currents&quot; it was taken to detect.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-01T00:03:14.190" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3602" PostTypeId="1" AcceptedAnswerId="3603" CreationDate="2016-03-31T13:06:52.647" Score="5" ViewCount="5730" Body="&lt;p&gt;I've been trying to find the famous article, &quot;One hundred authors against Einstein&quot; (100 Autoren gegen Einstein), of various objections to special relativity, which is quite often referenced, but finding a copy proved to be quite hard (there is one available &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/3226005375&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;), and I can't seem to find it in English at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was this text ever translated, and where might I be able to find a copy of it?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="55" LastEditDate="2016-05-06T00:00:53.577" LastActivityDate="2023-12-06T12:53:59.867" Title="An English copy of One Hundred Authors Against Einstein?" Tags="|physics|biographical-details|reference-request|relativity-theory|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3603" PostTypeId="2" ParentId="3602" CreationDate="2016-03-31T20:46:11.920" Score="11" Body="&lt;p&gt;Here is a link to &lt;a href=&quot;http://bourabai.kz/articles/100/page01.htm&quot; rel=&quot;noreferrer&quot;&gt;online version in German&lt;/a&gt;. As for the English translation, according to Milena Wazeck's &lt;a href=&quot;https://books.google.com/books?id=w-RRAgAAQBAJ&amp;amp;pg=PA291&amp;amp;lpg=PA291&amp;amp;dq=100+authors+against+Einstein+english+translation&amp;amp;source=bl&amp;amp;ots=w8c-zaZ5f1&amp;amp;sig=dW4GljRXr-8ps21tLWUd8ySS85M&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiAgZWj4OvLAhUK5GMKHYA3C7gQ6AEIKTAC#v=onepage&amp;amp;q=100%20authors%20against%20Einstein%20english%20translation&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Einstein's Opponents: The Public Controversy about the Theory of Relativity&lt;/a&gt; &quot;&lt;em&gt;nothing ever came of an English language version of 100 Autoren - no such book exists&lt;/em&gt;&quot;. This is despite the fact that Reuterdahl (one of the contributors) and Ruckhaber (one of the German edition editors) wanted to publish one in the US already back in 1931. But they had great difficulty finding a publisher because of the Great Depression and lack of scientific backers. According to Reuterdahl it was because scientists were afraid &quot;&lt;em&gt;of loosing scientific caste, and perhaps their position&lt;/em&gt;&quot;. Einstein offered a different explanation in his response to 100 Autoren (&lt;a href=&quot;https://hsm.stackexchange.com/a/3485/232&quot;&gt;source&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I were wrong, then one [author] would have been enough!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="232" LastEditDate="2016-04-01T17:26:37.553" LastActivityDate="2016-04-01T17:26:37.553" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3604" PostTypeId="1" CreationDate="2016-03-31T21:28:23.890" Score="2" ViewCount="129" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/DSM-5&quot; rel=&quot;nofollow&quot;&gt;Wikipedia page on DSM-5&lt;/a&gt; says that the &lt;a href=&quot;https://en.wikipedia.org/wiki/American_Psychiatric_Association&quot; rel=&quot;nofollow&quot;&gt;American Psychiatric Association&lt;/a&gt; (APA) mandated that DSM-5 task force members sign a nondisclosure agreement (NDA). What was APA's public rationale behind this NDA?&lt;/p&gt;&#xA;" OwnerUserId="29" LastActivityDate="2018-04-22T19:16:03.060" Title="Why did the APA require a NDA from DSM-5 task force members?" Tags="|medicine|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3605" PostTypeId="1" CreationDate="2016-04-01T15:15:57.550" Score="8" ViewCount="356" Body="&lt;p&gt;I have started reading Rene Dugas' History of Mechanics book. I want to understand physics from the spectacle of seeing its intellectual evolution instead of directly jumping into its general theories/paradigms, for I don't want leaps in the knowledge, which I think is going to leave in the unstable situation in the scientific field of genuine analysis.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For this project, I have taken Dugas book as the aid to see the persons contributing mechanics, then reading their books. Dugas book starts from Arsitotle, so at the first, I want to know the scientific environment in the antiquity, at least from Aristotle. Any book suggestions or the suggestion on the matter considered here will be really helpful.  &lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2018-09-13T17:58:25.573" Title="Which book gives a thorough understanding on the scientific environment of antiquity?" Tags="|physics|classical-mechanics|book-recommendation|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3606" PostTypeId="2" ParentId="3605" CreationDate="2016-04-01T16:27:54.220" Score="6" Body="&lt;p&gt;Not much is really known about general &quot;scientific environment&quot;. What we really have is only books and fragments of scientists and references on them in other works. It is commonly acknowledged that most writings did not survive. So we cannot have an adequate picture of the general scientific environment. The most developed sciences were mathematics and astronomy, and there are many good books on this, for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exact sciences in antiquity by O. Neugebauer, and Science awakening by B. L. van der Waerden. (They cover Babylonian science, not only Greek).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One general comprehensive book on Hellenistic science is&#xA;L. Russo, The Forgotten Revolution, but it covers only the Hellenistic period. There are many more specialized books, especially on mathematics and astronomy. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-04-01T16:27:54.220" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3607" PostTypeId="2" ParentId="3597" CreationDate="2016-04-01T17:19:27.257" Score="0" Body="&lt;p&gt;It seems that originally it was a descriptive way to refer to  an author's text: &lt;em&gt;Somebody in writings about something&lt;/em&gt;, that is e.g. Aristotle's book(s) about the heavens is &lt;em&gt;Peri Ouranos&lt;/em&gt; ; in Latin it was understood to be an ellipsis:  &lt;em&gt;De Entia et essentia&lt;/em&gt; is in fact &lt;em&gt;Liber de Entia et essentia&lt;/em&gt;. In French  the Latin form came naturally, e.g. &lt;em&gt;De la Grammatologie&lt;/em&gt; but one also wrote &lt;em&gt;(un livre) sur un sujet&lt;/em&gt; that is a book or a text 'on' some topic, or in German &lt;em&gt;uber&lt;/em&gt; etwas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the origin is in  a practice not in an original. Probably it was during the Renaissance that authors started giving titles to their writings and, further, they came to be respected by people who referred to the text.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-04-01T17:19:27.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3608" PostTypeId="2" ParentId="3605" CreationDate="2016-04-01T18:12:56.090" Score="3" Body="&lt;p&gt;I find &lt;a href=&quot;http://www.maa.org/press/maa-reviews/the-forgotten-revolution-how-science-was-born-in-300-bc-and-why-it-had-to-be-reborn&quot; rel=&quot;nofollow&quot;&gt;Russo's Forgotten Revolution&lt;/a&gt; very instructive for the breadth of sources he draws upon if nothing else, although some of his reconstructions are controversial. &lt;a href=&quot;https://www.academia.edu/3480555/The_Origin_of_the_History_of_Science_in_Classical_Antiquity&quot; rel=&quot;nofollow&quot;&gt;Zhmud's Origin of the History of Science in Classical Antiquity&lt;/a&gt; is also panoramic, and covers the pre-Hellenistic period. &lt;a href=&quot;http://courses.unt.edu/rdecarvalho/h5040/StudentPapers/Sambursky01,Samuel.htm&quot; rel=&quot;nofollow&quot;&gt;Sambursky's Physical World of the Greeks&lt;/a&gt; and other works are somewhat more narrowly focused.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-04-01T18:12:56.090" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3609" PostTypeId="2" ParentId="3605" CreationDate="2016-04-01T19:17:33.417" Score="4" Body="&lt;p&gt;For different perspectives:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Richard Olson, &lt;a href=&quot;https://books.google.it/books?id=0vP50ra6SYAC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Technology and Science in Ancient Civilizations&lt;/a&gt; (2010)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/G._E._R._Lloyd&quot; rel=&quot;nofollow&quot;&gt;G.E.RLloyd&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=wN5qQgAACAAJ&quot; rel=&quot;nofollow&quot;&gt;The Revolutions of Wisdom: Studies in the Claims and Practice of Ancient Greek Science&lt;/a&gt; (1989)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;G.E.R. Lloyd &amp;amp; Nathan Sivin, &lt;a href=&quot;https://books.google.it/books?id=PwB_IH0blNYC&quot; rel=&quot;nofollow&quot;&gt;The Way and the Word: Science and Medicine in Early China and Greece&lt;/a&gt; (2002).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;More specifically on &lt;em&gt;mechanics&lt;/em&gt;, see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sylvia Berryman, &lt;a href=&quot;https://books.google.it/books?id=5UohAwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;The Mechanical Hypothesis in Ancient Greek Natural Philosophy&lt;/a&gt; (2009).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-04-02T08:40:47.927" LastActivityDate="2016-04-02T08:40:47.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3610" PostTypeId="1" CreationDate="2016-04-02T22:28:22.217" Score="14" ViewCount="2129" Body="&lt;p&gt;According to the editor of the &lt;a href=&quot;https://books.google.com/books?id=8WmeBgAAQBAJ&amp;amp;pg=PA182&amp;amp;lpg=PA182&amp;amp;dq=Er+macht+es+wie+der+Fuchs,+der+seine+Spuren+im+Sande+mit+dem+Schwanz+ausl%C3%B6scht&amp;amp;source=bl&amp;amp;ots=MggDRxp0or&amp;amp;sig=Tdc8VDhqcgYnfwKLUW0BJMmcDgU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwis3OCCu_HLAhVimoMKHeHwAe4Q6AEIHDAA#v=onepage&amp;amp;q=Er%20macht%20es%20wie%20der%20Fuchs%2C%20der%20seine%20Spuren%20im%20Sande%20mit%20dem%20Schwanz%20ausl%C3%B6scht&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;German version of Ian Stewart's &quot;The Problems of Mathematics&quot;&lt;/a&gt;, on page 226 of the biography of Gauß authored by Erich Worbs (C. F. Gauß: Ein Lebensbild. Koehler &amp;amp; Amelang, Leipzig, 1955) we can find the following comment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... schreibt Abel in seinen Reisebriefen über Gauß: „&lt;em&gt;Er macht es wie der Fuchs, der seine Spuren im Sande mit dem Schwanz auslöscht.&lt;/em&gt;“ (Abel wrote in his letters about Gauss:&quot;&lt;em&gt;He is like the fox, who effaces his tracks in the sand  with his tail&lt;/em&gt;&quot;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wonder if it is actually known whom was the aforementioned letter addressed to or, failing that, if you know of a website wherein one can find a retrodigitized copy of Abel's Briefwechsel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please, let me thank you in advance for your knowledgeable replies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE:&lt;/strong&gt; I have spotted 5 letters from Abel to Prof. Christopher Hansteen in &lt;a href=&quot;http://www.abelprize.no/nedlastning/verker/abel_festskrift/abel_festskrift_06_kap03_breve_fra_og_til_abel_opt.pdf&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; document (Michael E2: thanks a bunch for the tip-off!) If I am correct, in &lt;strong&gt;none&lt;/strong&gt; of those letters one can find the famed „Er macht es wie der Fuchs, der seine Spuren im Sande mit dem Schwanz auslöscht“ quote. Nevertheless, almost at the end of the preantepenultimate paragraph of the fourth letter in the aforecited document (written on December 5, 1825), we do read this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hos Crelle var før ogsaa een Gang om Ugen en Samling af Mathematike men han var nødt til at ophøre dermed da der var een ved Navn Ohm, som ingen kunde komme ud af det med formedelst hans skrækkelige Arrogance. Det er tilforladelig tungt at en enkelt Mand saaledes lægger Hindringer i Veien for Videnskabelighed. Det er over al Maade hvor de unge Mathematikere her i Berlin og som jeg hører over alt i Tyksland ligesom forguder Gauss. Han er for dem Indbegrebet af al mathematisk Fortræffelighed, men lad være at han vist nok er et stort Genie saa er det ligesaa vist at han er [!] et slet Foredrag. &lt;strong&gt;Crelle siger at alt hvad Gauss skriver er Gräuel, da det er saa dunkelt at det næsten ikke er mueligt at forstaae det.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I guess this means that C. A. Bjerknes's account isn't altogether inaccurate. Besides, according to the author of &lt;a href=&quot;https://books.google.com.mx/books?id=ycvuCAAAQBAJ&amp;amp;pg=PA443&amp;amp;dq=hansteen%20letter%20from%20abel&amp;amp;hl=es-419&amp;amp;sa=X&amp;amp;ved=0ahUKEwjZ8u3Cw_PLAhVix4MKHQaCAZQQ6AEIGzAA#v=onepage&amp;amp;q=hansteen%20edit&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Niels Henrik Abel and his times: called too soon by flames afar&lt;/em&gt;&lt;/a&gt;, Prof. Hansteen published some letters by Abel in the &lt;a href=&quot;https://no.wikipedia.org/wiki/Illustreret_Nyhedsblad&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Illustreret Nyhedsblad&lt;/em&gt;&lt;/a&gt; in 1862: surfing the official website of the Abel Prize, we learn that the exact reference to these letters in the said journal(?) is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ch. Hansteen. &lt;em&gt;Niels Henrik Abel&lt;/em&gt;, Illustreret Nyhedsblad 11 (1862), 9-10, 37-38, 41-42 (2nd and 9th of March).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that, at one point of time, the archives of the &lt;em&gt;Illustreret Nyhedsblad&lt;/em&gt; were available online here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://goo.gl/GboE9v&quot; rel=&quot;noreferrer&quot;&gt;http://goo.gl/GboE9v&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have they been taken down or do I need to add something to my web browser in order for the corresponding files to be displayed correctly?&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2016-04-05T03:10:35.500" LastActivityDate="2019-10-16T17:12:48.940" Title="What is the original source for Abel's quote about Gauss:&quot;He is like the fox, who effaces his tracks in the sand with his tail&quot;?" Tags="|biographical-details|reference-request|debunking|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3611" PostTypeId="2" ParentId="3605" CreationDate="2016-04-03T03:55:03.600" Score="2" Body="&lt;p&gt;René Dugas was influenced by &lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre Duhem&lt;/a&gt; (1861-1916), the founder of the field of the history of medieval physics. Duhem's fundamental thesis in the history of physics is known as the &quot;continuity thesis&quot; (cf. Hannam's &lt;a href=&quot;http://jameshannam.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Genesis of Science&lt;/em&gt;&lt;/a&gt; for a semi-popular introduction) which is in contrast to what could be called Kuhn's &quot;discontinuity or rupture thesis&quot; of paradigm shifts. Duhem likened the history of physics to the evolution of the construction of a cathedral: a continuous, centuries-long development and refinement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Duhem's &lt;em&gt;magnum opus&lt;/em&gt; in the history of physics is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pierre Maurice Marie Duhem, &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/search?query=syst%C3%A8me%20du%20monde&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le système du monde: Histoire des&#xA;doctrines cosmologiques de Platon à Copernic&lt;/a&gt;&lt;/em&gt;, 10 vols. (Paris: A.&#xA;Hermann, 1913-1959).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;He begins with Plato and Aristotle and works up until right before the Renaissance. Despite the title, he was never able to complete the work up until Copernicus's time, but he did write about Copernicus and held a very unique view of the &quot;Galileo affair&quot; in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=UofBybolmREC&amp;amp;pg=PA131&quot; rel=&quot;nofollow noreferrer&quot;&gt;To Save the Phenomena&lt;/a&gt;&lt;/em&gt;.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Poignant excerpts are translated in:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pierre Maurice Marie Duhem, &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4757&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Cosmology: Theories of&#xA;Infinity, Place, Time, Void, and the Plurality of Worlds&lt;/a&gt;&lt;/em&gt;, ed.&#xA;and trans. Roger Ariew (Chicago: University of Chicago Press, 1985).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;He not only reports the historical facts but also keenly interprets them from the standpoint of a modern physicist and shows how the scientists of the High Middle Ages, especially, laid the foundations of Newtonian physics. Duhem's works are much deeper philosophically and historiographically than Dugas's; Dugas focuses more on the theoretical and mathematical content of the theories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This work influenced Dugas profoundly:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pierre Duhem, &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4320&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origins of Statics: The Sources of Physical&#xA;Theory&lt;/a&gt;&lt;/em&gt; (Dordrecht: Springer Netherlands, 1991). trans. of &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/search?query=Les%20origines%20de%20la%20statique%20vol.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Les&#xA;origines de la statique&lt;/a&gt;&lt;/em&gt;, 2 vol. (1905).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There's also this on the Parisian precursors of Galileo, which &lt;a href=&quot;https://dx.doi.org/10.13140/RG.2.2.23235.71201/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;has been translated into English&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pierre Maurice Marie Duhem, &lt;em&gt;&lt;a href=&quot;https://archive.org/details/tudessurlona03duhe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Études sur Léonard de Vinci, ceux qu’il&#xA;a lus et ceux qui l’ont lu&lt;/a&gt;&lt;/em&gt;, vol. 3 (&lt;em&gt;Les Précurseurs Parisiens&#xA;de Galilée&lt;/em&gt;) of 3. (Paris: A. Hermann, 1906).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-09-09T03:40:15.070" LastActivityDate="2018-09-09T03:40:15.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3613" PostTypeId="2" ParentId="3610" CreationDate="2016-04-03T14:32:05.033" Score="14" Body="&lt;p&gt;Bjerknes cites a letter from Abel to Christopher Hansteen, a fellow professor of Bjerknes at Christiania/Oslo, who had put up and mentored Abel in the beginning of his career.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Den omgangskreds af aeldre, hvortil han hörte, var vel selvfölgelig heller ikke sa enthusiastisk Gauss-stemt som Berlinerungsdommen. Sjelden er det derhos, at ikke den stigende ros og berommelse, idet den går over sine bredder, fremkalder modsigelser.  Indenfor den Crelleske kreds blev derfor så naturligt en og anden gang også den anden side fremholdt; og det hedte da om Gauss: hans foredrag var slet; or at han &amp;quot;Er macht es wie der Fuchs, der wischt mit dem Schwanse seine Spurten im Sands aus&amp;quot;. (1)&lt;/p&gt;&#xA;&lt;p&gt;(1) Citat af Hansteen i anledning at den naevnte bemaerkning i brev fra Abel&lt;/p&gt;&#xA;&lt;p&gt;-- C.A. Bjerknes, &lt;a href=&quot;https://books.google.com/books?id=itMyAQAAMAAJ&amp;amp;pg=PA61#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Niels Henrik Abel. En skildring af hans liv og videnskabelige virksomhed&lt;/em&gt; (Stockholm. 1880)&lt;/a&gt;, p. 61&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My Norwegian is completely dependent on Google Translate.  Here is a 1885 revision by Bjerknes in French:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;La société des personnes âgées, qu'il fréquentait, ne partageait pas, cela va sans dire, le gaussianisme enthousiaste de la jeunesse berlinoise.  It est rare, en effet, que la renommée croissante d'un nom vanté, lorsqu'elle dépasse les bornes, n'attire pas la contradiction. Dans la société de Crelle, it était tout naturel qu'une fois ou l'autre on soutint la seconde opinion; et l'on disait que «sa manière d'exposer était mauvaise», ou encore qu'«it faisait comme le renard, qui efface avec sa queue les traces de ses pas sur le sable.»&lt;/p&gt;&#xA;&lt;p&gt;-- C.A. Bjerknes, &lt;a href=&quot;https://books.google.com/books?id=-EFBAAAAYAAJ&amp;amp;pg=PA91#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Niels-Henrik Abel: tableau de sa vie et son action scientifique&lt;/em&gt; (Paris, 1885)&lt;/a&gt;, pp. 91-92&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Roughly, it says: &lt;em&gt;The elder mathematicians did not share in the enthusiasm for the &amp;quot;Gaussianism&amp;quot; of the younger generation in Berlin.  Seldom does a name become vaunted without the opposite opinion also arising.  So that in Crelle's society, it was quite natural that now and then someone would say about Gauss that his lectures were bad or &amp;quot;he makes [his mathematics] like a fox, wiping out the traces in the sand with his tail.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;From this account it appears that the saying was something Abel reported, but perhaps did not make up himself.&lt;/p&gt;&#xA;&lt;p&gt;Bjerknes goes on to say that Abel reports that Crelle says that everything Gauss writes is &amp;quot;abomination&amp;quot; (&amp;quot;Gräuel&amp;quot;), since &amp;quot;it is so obscure that it is almost impossible to understand it.&amp;quot; (That is a friend-of-a-friend-of-a-friend-of-a-friend report, from Crelle to Abel to Hansteen to Bjerknes).&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure where the letters from Abel to Hansteen are.  Some have been published, but I don't know if the one referred to by Bjerknes has been.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-04-03T17:36:30.307" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3614" PostTypeId="1" AcceptedAnswerId="3619" CreationDate="2016-04-03T18:13:39.643" Score="3" ViewCount="233" Body="&lt;p&gt;I am writing a Wikipedia article (in a certain language) in which I intend to include a summary of Carl Friedrich Gauss's contributions to capillarity. I know that Gauss derived&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Young%E2%80%93Laplace_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Young-Laplace&lt;/a&gt; equation using a new variational approach&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the relation for the wetting angle of a liquid and solid (liquid drops on solid surface)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Did Gauss derive completely new results?&lt;/p&gt;&#xA;&lt;p&gt;I'll be glad if someone will give an outline of his contributions.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2021-03-09T07:35:14.893" LastActivityDate="2021-03-09T07:35:14.893" Title="What were Gauss's contributions to capillarity?" Tags="|physics|biographical-details|gauss|fluid-dynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3615" PostTypeId="1" AcceptedAnswerId="3618" CreationDate="2016-04-04T01:11:18.893" Score="10" ViewCount="12508" Body="&lt;p&gt;The best thing I could find on the internet was this apparently forgotten article from 2004:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;N. David Mermin, &lt;a href=&quot;https://doi.org/10.1063/1.1768652&quot; rel=&quot;nofollow noreferrer&quot;&gt;Could Feynman have said this?&lt;/a&gt;, Physics Today 57 (5), 2004.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="83" LastEditorUserId="4251" LastEditDate="2023-12-09T19:53:59.620" LastActivityDate="2023-12-10T08:33:42.960" Title="Who was the first to say &quot;Shut up and calculate!&quot;?" Tags="|physics|reference-request|quantum-mechanics|philosophy-of-science|theoretical-physics|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3616" PostTypeId="2" ParentId="3615" CreationDate="2016-04-04T06:02:37.567" Score="9" Body="&lt;p&gt;N. David Mermin (born March 30, 1935, in New Haven, Connecticut, USA) is Horace White Professor of Physics Emeritus at Cornell University&lt;/p&gt;&#xA;&#xA;&lt;p&gt;his quote was like this&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be 'Shut up and calculate!'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3503" LastEditorUserId="26" LastEditDate="2016-04-05T17:45:44.737" LastActivityDate="2016-04-05T17:45:44.737" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3617" PostTypeId="2" ParentId="3615" CreationDate="2016-04-04T09:24:01.487" Score="15" Body="&lt;p&gt;Some centuries before Mermin, Leibniz in the 17th century was seeking a solution to some of the denominational quarrels that were plaguing his generation by envisioning a &lt;em&gt;calculus ratiocinator&lt;/em&gt; that would make it possible for the quarreling parties to &quot;sit down and calculate&quot;. A hypothetical science he envisioned was called &lt;em&gt;Mathesis Universalis&lt;/em&gt; and included his infinitesimal calculus as a first stage.  See also &lt;a href=&quot;https://en.wikipedia.org/wiki/Characteristica_universalis&quot; rel=&quot;noreferrer&quot;&gt;Characteristica Universalis&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-04-04T09:58:44.813" LastActivityDate="2016-04-04T09:58:44.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3618" PostTypeId="2" ParentId="3615" CreationDate="2016-04-04T20:26:01.327" Score="13" Body="&lt;p&gt;Mermin has a thorough analysis&lt;sup&gt;&lt;a href=&quot;https://doi.org/10.1063/1.1768652&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/sup&gt; and traces the phrase to himself in a 1989 Physics Today column&lt;sup&gt;&lt;a href=&quot;https://doi.org/10.1063/1.2810963&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/sup&gt; &amp;amp; makes a case that the numerous attributions to Feynman may or may not be mistaken.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Mermin, &lt;a href=&quot;https://doi.org/10.1063/1.1768652&quot; rel=&quot;nofollow noreferrer&quot;&gt;Could Feynman have said this?&lt;/a&gt;, Physics Today 57 (5), 2004.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Mermin, &lt;a href=&quot;https://doi.org/10.1063/1.2810963&quot; rel=&quot;nofollow noreferrer&quot;&gt;What's wrong with this pillow?&lt;/a&gt;, Physics Today 42 (4), 1989.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1829" LastEditorUserId="4251" LastEditDate="2023-12-10T08:33:42.960" LastActivityDate="2023-12-10T08:33:42.960" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3619" PostTypeId="2" ParentId="3614" CreationDate="2016-04-05T01:23:43.753" Score="3" Body="&lt;p&gt;Gauss derived a new condition for the angle of contact of the capillary surface with the surface of a solid, that was a consequence of his new variational approach to the theory. Before him Laplace and Poisson derived the capillarity theory from hypotheses about molecular forces, which were little known at the time, and so open to criticisms, by Young among others. Even worse, Poisson's expressions for capillary pressure in terms of molecular forces were different from those of Laplace, while there was no difference in observable predictions. Arago called it a &quot;mathematical scandal&quot;, it was an early example of empirical underdetermination of theories. See &lt;a href=&quot;https://books.google.com/books?id=f5FqsDPVQ2MC&amp;amp;pg=PA1205&amp;amp;lpg=PA1205&amp;amp;dq=gauss+capillarity+history&amp;amp;source=bl&amp;amp;ots=k8WEGj3aQH&amp;amp;sig=A5by_iONVjWxWMzRgXYJhSrI_qU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwizqtuWofbLAhVK1GMKHUUnCiQQ6AEINDAD#v=onepage&amp;amp;q=gauss%20capillarity%20history&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Grattan-Guinness's Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences&lt;/a&gt;, vol 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Principia Generalia Theoriae Figurae Fluidorum in Statu Aequilibrii (1830) Gauss came up with an &quot;energetist&quot; approach that was far less sensitive to hypotheses about specifics of molecular forces. &lt;a href=&quot;https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Capillary_Action&quot; rel=&quot;nofollow&quot;&gt;According to Britannica&lt;/a&gt;:&quot;&lt;em&gt;The principle which he adopted is that of virtual velocities, a principle which under his hands was gradually transforming itself into what is now known as the principle of the conservation of energy. Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles. This expression has been called the force-function. With its sign reversed it is now called the potential energy of the system... The condition of equilibrium is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum. This condition when worked out gives not only the equation of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid. Gauss thus supplied the principal defect in the great work of Laplace&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The energetist method was developed by Dupre (1869) and Gibbs (1878) completely purged it of any molecular notions leaving only thermodynamic ones. While it was useful in reducing dependence of phenomenological theories on molecular hypotheses at the end of 19th century Mach and Ostwald took it too far under the name of &quot;energetics&quot;. They turned energetics into a positivist alternative to atomism in general, and Boltzmann's kinetic theory in particular, which they saw as metaphysical speculations. It was largely abandoned (except by Mach) after Einstein and Smoluchowski developed a theory of Brownian motion based on kinetic theory around 1905, which was experimentally confirmed. See &lt;a href=&quot;http://link.springer.com/article/10.1023%2FA%3A1005287003138&quot; rel=&quot;nofollow&quot;&gt;Deltete's Helm and Boltzmann: Energetics at the Lübeck Naturforscherversammlung&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-04-05T20:47:53.267" LastActivityDate="2016-04-05T20:47:53.267" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3620" PostTypeId="1" CreationDate="2016-04-05T05:17:29.323" Score="6" ViewCount="1985" Body="&lt;p&gt;Tried to find it online, but nothing. Everyone refers to it and that it's named after the famous James Clerk Maxwell (of the Maxwell electromagnetic laws and some other things), but there is no direct reference, and it's like the experiment has always been done (although you need some concepts of rigid bodies to do it).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you don't remember, the &lt;a href=&quot;http://www.4physics.com/catalog/product_info.php/products_id/294&quot; rel=&quot;noreferrer&quot;&gt;Maxwell's wheel&lt;/a&gt; is a classic mechanics experiment that is usually done in college laboratory practices. The experiment consists of a wheel with a bar through its radius that is hung on two different threads, one at each side. If you roll upwards the threads turning the wheel and then release it all the potential energy of the wheel converts to Kinetic energy of movement + Kinetic energy of rotation. If you measure falling times for different positions you can get the momentum of inertia of the wheel easily using the principle of conservation of energy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So who came up with it first and when?&lt;/p&gt;&#xA;" OwnerUserId="3945" LastEditorUserId="4251" LastEditDate="2020-06-07T06:57:41.683" LastActivityDate="2020-06-07T06:57:41.683" Title="Did Maxwell invent the Maxwell's wheel?" Tags="|physics|discoveries|classical-mechanics|maxwell|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3623" PostTypeId="1" AcceptedAnswerId="3624" CreationDate="2016-04-05T23:05:21.943" Score="9" ViewCount="818" Body="&lt;p&gt;If you were a student of physics of let's say between 14-20 years old, what textbook would you be using to learn physics from ? &lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="2172" LastEditDate="2016-04-08T13:03:34.693" LastActivityDate="2017-03-22T16:45:13.417" Title="What would have been the popular textbook to learn mechanics in the 19 th century?" Tags="|physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3624" PostTypeId="2" ParentId="3623" CreationDate="2016-04-06T00:04:31.517" Score="8" Body="&lt;p&gt;19s century was long. In the second half of it, if I were an English speaking person, I would study Treatise on Natural Philosophy by Willian Thomson and Peter Tait published in 1867. Before that, I would probably study mechanics from some primary source, rather than a textbook. An excellent source is Laplace,&#xA;Mécanique céleste, 4 vols (translated into English by Bowditch in 1829-39).&#xA;Or from the works of Lagrange or Hamilton, depending on my preferred language.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;George Green (of the Green formula and the Green Function) grew up in Nottingham and had no formal education at the time he made his principal discoveries.  Apparently he had access to only one book on mathematics/physics&#xA;(this was determined by historians who studied the collection of the only library to which he had access). This was Laplace's book, mentioned above. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-06T00:37:23.450" LastActivityDate="2016-04-06T00:37:23.450" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3625" PostTypeId="2" ParentId="3623" CreationDate="2016-04-06T01:06:39.373" Score="8" Body="&lt;p&gt;Here is an interesting &lt;a href=&quot;https://www.siam.org/pdf/news/226.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;morsel about 19th century Cambridge&lt;/a&gt;:&quot;&lt;em&gt;Mechanics was an important part of the curriculum. Newton’s version, despite its relaunching by William Whewell against French competitors in the late 1810s, was not necessarily adopted at Cambridge. In the 1840s Whewell also advocated inclusion of engineering mechanics in the curriculum, which is largely ignored in this book&lt;/em&gt;&quot;. Whewell's own text came out in 1819, and was apparently quite popular since it went through 5 editions by 1836, here is a link to the &lt;a href=&quot;https://archive.org/stream/elementarytreati00whew#page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;5th edition&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;French competitors&quot; no doubt refers to (popularizations of) &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/mathematical-treasures-lagranges-analytical-mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange's Analytical Mechanics (1788)&lt;/a&gt;, &quot;&lt;em&gt;admired for its clarity and structure, this book has been described as a great mathematical classic and a “scientific poem”&lt;/em&gt;&quot;, according to MAA. A somewhat later entry is &lt;a href=&quot;https://archive.org/details/coursdemcanique00duhagoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhamel's Cours de Mécanique&lt;/a&gt;, which also went through multiple editions (1845, 1853, 1862). Some of the late 19th and early 20th century texts, including Rankine's and Wright's, are &lt;a href=&quot;http://onlinebooks.library.upenn.edu/webbin/book/browse?type=lcsubc&amp;amp;key=Mechanics%20--%20Textbooks%20--%201885-1887&amp;amp;c=x&quot; rel=&quot;nofollow noreferrer&quot;&gt;listed here&lt;/a&gt;. And here are some &lt;a href=&quot;http://www4.ncsu.edu/~kimler/hi322/textbooks.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;from the 18th century&lt;/a&gt;, just in case.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="131" LastEditDate="2017-03-22T16:45:13.417" LastActivityDate="2017-03-22T16:45:13.417" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3626" PostTypeId="2" ParentId="55" CreationDate="2016-04-06T11:40:12.667" Score="9" Body="&lt;p&gt;Beyond the issue of notation, Newton experimented with a number of foundational approaches. One of the earliest ones involved infinitesimals, whereas later he shied away from them because of philosophical resistance of his contemporaries, often stemming from sensitive religious considerations closely related to inter-denominational quarrels. Leibniz also was aware of the quarrels, but he used infinitesimals and differentials systematically in developing the calculus, and for this reason was more successful in attracting followers and stimulating research--or what he called the &lt;em&gt;Ars Inveniendi&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-05-01T15:15:17.213" LastActivityDate="2016-05-01T15:15:17.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3627" PostTypeId="2" ParentId="55" CreationDate="2016-04-06T22:55:57.347" Score="10" Body="&lt;p&gt;You should definitely take a look at the second chapter of Arnold's &lt;em&gt;Huygens &amp;amp; Barrow, Newton &amp;amp; Hooke&lt;/em&gt;. The late Prof. Arnold summarized therein the difference between Newton's approach to mathematical analysis and Leibniz's as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton's analysis was the application of power series to the study of&#xA;motion... For Leibniz, ... analysis was a more formal algebraic study of&#xA;differential rings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Arnold's overview of Leibniz's contributions to the theme is spiced up with a non-negligible number of thought-provoking remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the work of other geometers--e.g., Huygens and Barrow--many objects connected with a given curve also appeared [for example: abscissa, ordinate, tangent, the slope of the tangent, the area of a curvilinear figure, the subtangent, the normal, the subnormal, and so on]... Leibniz, with his individual tendency to universality [he considered necessary to discover the so-called characteristic, something universal, that unites everything in science and contains all answers to all questions], decided that all these quantities should be considered in the same way. For this he introduced a single term for any of the quantities connected with a given curve and fulfilling some function in relation to the given curve--the term &lt;em&gt;function&lt;/em&gt;...&lt;/p&gt;&#xA;&lt;p&gt;Thus, according to Leibniz many functions were associated with a curve. Newton had another term--fluent--which denoted a flowing quantity, a variable quantity, and hence associated with motion. On the basis of Pascal's studies and his own arguments Leibniz quite rapidly developed formal analysis in the form in which we now know it. That is, in a form specially suitable to teach analysis by people who do not understand it to people who will never understand it... Leibniz quite rapidly established the formal rules for operating with infinitesimals, whose meaning is obscure.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz's method was as follows. He assumed that the whole of mathematics, like the whole of science, is found inside us, and by means of philosophy alone we can hit upon everything if we attentively  take heed of processes that occur inside our mind. By this method he discovered various laws and sometimes very successfully. For example, he discovered that &lt;span class=&quot;math-container&quot;&gt;$d(x+y) = dx+dy$&lt;/span&gt;, and this remarkable discovery immediately forced him to think about what the differential of a product is. In accordance with the universality of his thoughts he rapidly came to the conclusion that differentiation [had to be] a ring homomorphism, that is, that the formula &lt;span class=&quot;math-container&quot;&gt;$d(xy) = dx dy$&lt;/span&gt; must hold. But after some time he verified that this leads to some unpleasant consequences, and found the correct formula &lt;span class=&quot;math-container&quot;&gt;$d(xy) = xdy + y dx$&lt;/span&gt;, which is now called Leibniz's rule. None of the inductively thinking mathematicians--neither Barrow nor Newton, who as a consequence was called an empirical ass in the Marxist literature--could [have ever gotten] Leibniz's original hypothesis into his head, since to such a person it was quite obvious what the differential of a product is, from a simple drawing...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="136" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-04-06T23:08:33.637" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3628" PostTypeId="1" CreationDate="2016-04-06T23:28:49.230" Score="8" ViewCount="873" Body="&lt;p&gt;Is there any written work by Euclid about consecutive primes differing by two? What work was done on the problem from the time of Euclid about 2,300 years ago to the time of Polignac in 1849?&lt;/p&gt;&#xA;" OwnerUserId="3951" LastActivityDate="2018-05-09T19:56:34.907" Title="What Is the Origin of the Twin Primes Conjecture?" Tags="|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3629" PostTypeId="2" ParentId="3628" CreationDate="2016-04-07T04:44:30.507" Score="5" Body="&lt;p&gt;I am not sure how the association of Euclid with twin primes got started, but  &lt;a href=&quot;http://www.britannica.com/topic/twin-prime-conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;even Brittanica writes&lt;/a&gt; (although &lt;a href=&quot;https://en.wikipedia.org/wiki/Talk%3ATwin_prime_conjecture#Claim_to_be_from_Euclid&quot; rel=&quot;nofollow noreferrer&quot;&gt;not, to its credit, Wikipedia&lt;/a&gt;):&quot;&lt;em&gt;Greek mathematician Euclid (flourished c. 300 bce) gave the oldest known proof that there exist an infinite number of primes, and he conjectured that there are an infinite number of twin primes&lt;/em&gt;&quot;. The first part does occur in &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookIX/bookIX.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements IX.20&lt;/a&gt;, and IX.36 further describes how to produce perfect numbers from &quot;Mersenne primes&quot;, but the ending is a fable. Or as Jason Dyer charmingly put it on MO:&quot;&lt;em&gt;It is possible to guess that he was making a conjecture on the basis of his text but it requires wishful thinking&lt;/em&gt;&quot;. The whole &lt;a href=&quot;https://mathoverflow.net/questions/7639/twin-prime-conjecture-reference&quot;&gt;Twin Prime Conjecture Reference&lt;/a&gt; thread there is instructive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dickson's History of the Theory of Numbers&lt;/a&gt; mentions &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/NAM/NAM_1849_1_8_/NAM_1849_1_8__423_1/NAM_1849_1_8__423_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;de Polignac's Six Arithmetical Propositions Deduced from the Sieve of Eratosthenes (1849)&lt;/a&gt; as the earliest source conjecturing the infinitude of twin primes. But even then according to that same Britannica:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Very little progress was made on this conjecture until 1919, when Norwegian mathematician Viggo Brun showed that the sum of the reciprocals of the twin primes converges to a sum, now known as Brun’s constant... The next big breakthrough occurred in 2003, when American mathematician Daniel Goldston and Turkish mathematician Cem Yildirim published a paper, “Small Gaps Between Primes”, that established the existence of an infinite number of prime pairs within a small difference (16, with certain other assumptions)&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-09T19:56:34.907" LastActivityDate="2018-05-09T19:56:34.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3630" PostTypeId="1" CreationDate="2016-04-07T12:33:52.530" Score="6" ViewCount="321" Body="&lt;p&gt;Arthur's recent article&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arthur, Richard T. W.&#xA;Leibniz's actual infinite in relation to his analysis of matter.  G. W. Leibniz, interrelations between mathematics and philosophy,  137–156, &#xA;Archimedes, 41, Springer, Dordrecht, 2015&lt;/p&gt;&#xA;&#xA;&lt;p&gt;was recently reviewed by Knobloch, who wrote in part:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Despite his rejection of an infinite number and infinite collections, he admitted an actual, but syncategorematic, infinite. There is an infinity of things; that is, there are always more of them than can be specified. Secondly, Arthur refutes Rescher's two-tiered ontology and Russell's pseudo-dilemma. One has to distinguish between division and resolution, between composable, resolvable, and divisible. Arthur essentially bases his deductions on Leibniz's correspondence with Arnauld and de Volder. A body is actually divided into parts by its different motions, but resolved into unities. There are actually infinitely many monads. This infinite is understood syncategorematically. &quot;There are actually infinite aggregates but—in contrast to Cantor—there are no infinite numbers&quot; (p. 155).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems surprising to me since Leibniz on many occasions spoke precisely of &lt;em&gt;infinite numbers&lt;/em&gt; (while rejecting infinite collections). Have Arthur and Knobloch gotten it right?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They don't fully address the distiction between infinite multitudes/wholes/collections, on the one hand, and infinite numbers, on the other. I haven't seen anything in Leibniz that would suggest that he thought that infinite numbers led to contradiction, and on the contrary Leibniz often exploits them as &quot;useful fictions&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a recent article in &lt;em&gt;HOPOS: History of Philosophy of Science&lt;/em&gt; we defend the thesis that Leibniz used infinite numbers galore. We also argue against the so-called syncategorematic interpretation first developed by Ishiguro allegedly following Russell (according to which Leibniz did nothing of the sort). See &lt;a href=&quot;http://dx.doi.org/10.1086/685645&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="4251" LastEditDate="2020-06-06T00:40:47.817" LastActivityDate="2020-06-06T00:40:47.817" Title="Did Leibniz use infinite numbers?" Tags="|mathematics|mathematicians|debunking|leibniz|infinity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3631" PostTypeId="1" AcceptedAnswerId="3655" CreationDate="2016-04-07T12:43:56.083" Score="9" ViewCount="1869" Body="&lt;p&gt;I have been looking into the measurement problem that arises when considering different interpretations of quantum mechanics. Nowadays it seems to be considered a solved problem (in fact some people on physics.SE seem to be quite annoyed at people asking about if it is solved). Please correct me if that is not true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to have a look through the papers relevant to the topic. Some milestones seem to be papers by Legget &amp;amp; Caldeira, Joos &amp;amp; Zeh, Zurek and others. &lt;strong&gt;My question is at which time exactly the measurement problem was solved? Which paper or similar publication was the final brick?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&#xA;People were asking about a precise statement of the problem. I consider the measurement problem to be the fact that there used to be &lt;em&gt;two evolution laws of the wavefunction&lt;/em&gt;: a continuous one (Schrödinger equation) and a discontinuous one (measurement/collapse). The problem lies in how to reconcile these two.&lt;/p&gt;&#xA;" OwnerUserId="3584" LastEditorUserId="4251" LastEditDate="2021-09-16T05:46:05.450" LastActivityDate="2021-09-16T05:46:05.450" Title="When was the measurement problem solved?" Tags="|physics|quantum-mechanics|measurement|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3632" PostTypeId="2" ParentId="3631" CreationDate="2016-04-07T21:02:29.653" Score="1" Body="&lt;p&gt;This problem is a philosophical problem. And philosophical problems are almost never solved. This is probably the reason why some physicists are annoyed with it.&#xA;Strictly speaking this is not a physics problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. There are different opinions on what constitutes physics and what does not.&#xA;And this changes with time too.&#xA;To begin with, would you care to give a precise STATEMENT of the problem?&#xA;Of course one cannot discuss when and how was it solved without an exact statement.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-08T04:29:35.243" LastActivityDate="2016-04-08T04:29:35.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3633" PostTypeId="2" ParentId="3630" CreationDate="2016-04-07T22:26:46.480" Score="2" Body="&lt;p&gt;It appears that they are right in the &quot;metaphysical&quot; sense, here is a passage from Leibniz's letter to Bernoulli (1699):&quot;&lt;em&gt;I concede an infinite multitude, but this multitude forms neither a number nor one whole. It only means that there are more terms than can be designated by a number; just as there is for instance a multitude or complex of all numbers; but this multitude is neither a number nor one whole&lt;/em&gt;&quot;. The main problem for Leibniz was the violation of the part-whole axiom, an infinite whole would end up being equal to a proper part of itself, as in &lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo&amp;#39;s_paradox#Galileo_on_infinite_sets&quot; rel=&quot;nofollow&quot;&gt;Galileo's example with natural numbers and their squares&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in &lt;a href=&quot;https://logic-library.berkeley.edu/catalog/detail/1846&quot; rel=&quot;nofollow&quot;&gt;On Some Relations between Leibniz’ Monadology and Transfinite Set Theory Friedman&lt;/a&gt; argued that Leibniz nonetheless accepted infinite wholes and numbers as useful fictions for calculation (he also saw imaginary roots and infinitesimals as useful fictions). In a letter to Des Bosses (1706) Leibniz wrote:&quot;&lt;em&gt;...properly speaking, an infinity consisting of parts is neither one nor a whole, and can only be conceived of as a quantity by a mental fiction&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See more in &lt;a href=&quot;http://www.springer.com/cda/content/document/cda_downloaddocument/9783319100302-c2.pdf?SGWID=0-0-45-1491952-p176879826.&quot; rel=&quot;nofollow&quot;&gt;van Atten's Note on Leibniz’s Argument Against Infinite&#xA;Wholes&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-04-07T22:26:46.480" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3634" PostTypeId="1" AcceptedAnswerId="3635" CreationDate="2016-04-08T03:41:57.530" Score="26" ViewCount="1949" Body="&lt;p&gt;I am wondering how did Maxwell in the 19th century draw such figures as the one shown? What tools or procedures did he need?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dJHfO.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dJHfO.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it all compass and ruler drawing?&lt;/p&gt;&#xA;" OwnerUserId="3955" LastEditorUserId="4251" LastEditDate="2020-07-05T14:50:09.200" LastActivityDate="2021-02-09T02:47:15.277" Title="How did scientists plot complicated graphs in the 19th century?" Tags="|physics|electromagnetism|19th-century|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3635" PostTypeId="2" ParentId="3634" CreationDate="2016-04-08T04:22:50.800" Score="23" Body="&lt;p&gt;I do not know how exactly was this picture made, but there are at least two methods. The first one is to compute this potential (which is not too difficult), plot sufficiently many points and connect them by smooth curves. This was still quite common in pre-computer era, in 1970s when a special drawing tool was used, in the shape of a curved ruler of variable curvature. (I just found that it is called a &amp;quot;French curve&amp;quot; in English:-) You attach an appropriate part of this &amp;quot;ruler&amp;quot; to several points to draw a smooth curve.&lt;/p&gt;&#xA;&lt;p&gt;Here is how it looks:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/French_curve&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/French_curve&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The second method which could be used was experimental. The picture of equipotential lines can be obtained with a metal dust on a properly charged plate. Then the plate can be photographed and carefully copied from the photo. Or even drawn without photographing on a transparent paper layed over the metal dust. I've seen such method used on more complicated potentials (which would be difficult to compute numerically). An example of such photo is here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Field_line&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Field_line&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Both methods were still used in the 1970s.&lt;/p&gt;&#xA;&lt;p&gt;Remark. It is sometimes difficult to tell by eye whether a picture in an old book really represents the exact data, or is just made by an artist, that is whether the shapes and distances are true. I recomputed some pictures from old books using a computer, and sometimes they look very different, which suggests that they are just hand drawings, showing essential features but not real representations of an object. The picture you show seems to be real, because it represents quite a simple potential which is not difficult to compute.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-02-09T02:47:15.277" LastActivityDate="2021-02-09T02:47:15.277" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3636" PostTypeId="1" AcceptedAnswerId="3637" CreationDate="2016-04-08T12:14:29.837" Score="4" ViewCount="276" Body="&lt;p&gt;As I know, Tarski had proved his undefinability theorem by $1936$, $5$ years after Gödel's incompleteness theorems had been discovered. I wonder whether his original proof was built upon the work of Gödel or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could the incompleteness theorems have been proved by Tarski by $1936$ if Gödel had not already discovered them? &lt;/p&gt;&#xA;" OwnerUserId="2504" LastEditorUserId="2442" LastEditDate="2016-07-26T12:07:14.413" LastActivityDate="2016-07-26T12:07:14.413" Title="Could the incompleteness theorems have been proved by Tarski if Gödel had not proved them first?" Tags="|mathematics|mathematical-logic|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3637" PostTypeId="2" ParentId="3636" CreationDate="2016-04-08T13:05:38.160" Score="7" Body="&lt;p&gt;In principle, yes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Roman Murawski, &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/01445349808837306?journalCode=thpl20#.Vwerb_mLTIU&quot;&gt;Undefinability of Truth. The Problem of Priority: Tarski vs Gödel&lt;/a&gt; (1998):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tarski made clear his indebtedness to Gödel's methods. [...] On the other hand Tarski strongly emphasized the fact that his results were obtained independently.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Tarski did not claim any priority for Gödel's own results. In footnote 2 to the paper &lt;em&gt;Some observations on the concept of omega-consistency and omega-completeness&lt;/em&gt; (1933) he wrote:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;Already, in the year 1927 (...) I also communicated the example of a consistent and yet not omega-consistent system which I give in the present article in a slightly altered form. Naturally it is not hereby claimed that I already knew then the results later obtained by Gödel or had even foreseen them. On the contrary, I had personally felt that the publication of the work of Gödel was a most exciting scientific event.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Though Tarski saw the similarity of the system $P$ used by Gödel in the paper of 1931 and his own system used in 1933 he adds [...] the remark that (...) the abstract character of the methods used by Gödel renders the validity of his&#xA;  results independent to a high degree of the specific peculiarities of the science investigated.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It is known that Tarski's theorem on undefinability of truth implies the existence of undecidable sentences, hence Gödel's first incompleteness theorem. So one can ask whether Tarski was close to this theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-04-08T15:27:24.670" LastActivityDate="2016-04-08T15:27:24.670" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3638" PostTypeId="2" ParentId="3634" CreationDate="2016-04-08T15:16:48.970" Score="18" Body="&lt;p&gt;Adding to Alexandre's answer.  My father, educated in the 1930s, used a set of &quot;French curves&quot; like this  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KG1Vi.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KG1Vi.jpg&quot; alt=&quot;French curves&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Secondly ... perhaps the picture in the post was not drawn by Maxwell himself, but by a professional technical artist.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-04-08T15:16:48.970" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3639" PostTypeId="1" CreationDate="2016-04-09T10:13:43.643" Score="3" ViewCount="1587" Body="&lt;p&gt;I noticed that Dirac bra-kets and their algebra are very much like the linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A ket is like a vector, a bra is like the conjugate transpose of a vector, a bra-ket is like a complex inner product, a ket-bra is like an outer product, and operator is like a matrix, and operator acting on a state is like matrix product.&#xA;Even we speak about eigenstates which are very much like eigenvectors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how is it happened, that we invented a new notation for linear algebra instead of sticking to the usual notation of vectors and matrixes as usual? &lt;/p&gt;&#xA;" OwnerUserId="2178" LastActivityDate="2016-04-09T17:31:25.767" Title="How did the bra-ket notation become mainstream in quantum mechanics?" Tags="|notation|quantum-mechanics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3640" PostTypeId="2" ParentId="3639" CreationDate="2016-04-09T17:31:25.767" Score="4" Body="&lt;p&gt;At the time when Dirac introduced his notation, linear algebra was not widely taught. The clear evidence of this is that Heisenberg had to invent matrix multiplication when he created quantum mechanics. It was not a standard part of the curriculum neither for physicists not for mathematicians. Actually there is&#xA;probably a causal relation between the invention of quantum mechanics and the spread of linear algebra as a part of the standard curriculum. See the discussion of this here: &lt;a href=&quot;https://hsm.stackexchange.com/questions/198/when-exactly-and-why-did-matrices-become-a-part-of-the-undergraduate-curriculu&quot;&gt;When exactly (and why) did matrices become a part of the undergraduate curriculum?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In modern undergraduate courses column vectors $x$ are used. They stand on the right of the matrix: $Ax$. Column vectors correspond to ket vectors in Dirac's&#xA;notation: $|x&amp;gt;$. Then you can write $A|x&amp;gt;$. Row vectors will be $x^T$ or in&#xA;Dirac's notation bra-vectors $&amp;lt;x|$. Then when we want to write the standard dot product&#xA;it will be $x^Ty$ or $&amp;lt;x|y&amp;gt;$, respectively. More generally when we want to write&#xA;a quadratic form, it is either $x^TAy$ or $&amp;lt;x|A|y&amp;gt;$. &#xA;Here I assumed for simplicity that the vectors are real. The correct thing to use instead of $x^T$ is of course the Hermitean transpose $x^*$. Dirac's notation $&amp;lt;x|A|y&amp;gt;$ is unambiguous only if $A$ is Hermitean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many variants of both notations. The point was to make the clear distinction between vectors and co-vectors (linear functionals).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-09T17:31:25.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3641" PostTypeId="2" ParentId="3630" CreationDate="2016-04-10T10:56:05.743" Score="0" Body="&lt;p&gt;Richard Arthur's article can be complemented with a Leibnizian fragment edited into:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gottfried Wilhelm Leibniz, &lt;a href=&quot;https://books.google.it/books?id=H9wCnwEACAAJ&amp;amp;dq=The+Labyrinth+of+the+Continuum:+Writings+on+the+Continuum+Problem,+1672-1686&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=0ahUKEwiWvJL97oPMAhUESJoKHX_yDkIQ6AEIJjAB&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686&lt;/a&gt;, edited by Richard Arthur (2001), page 82: &lt;strong&gt;Infinite numbers&lt;/strong&gt; (ca.April 1676).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Consider the conclusion, page 99:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whenever it is said that a certain infinite series of numbers has a sum, I am of the opinion that all that is being said is that any finite series with the same rule has a sum, and that the error always diminishes as the series increases, so that it becomes as small as we would like. For numbers do not &lt;em&gt;in themselves&lt;/em&gt; go absolutely to infinity, since then there would be a greatest number [Nam ipsi &lt;em&gt;per se&lt;/em&gt; absolute numeri in infinitum non eunt, daretur enim numerus maximus]. [...] Therefore we conclude finally that there is no infinite multiplicity, from which it will follow that there is not an infinity of things, either. [...]&lt;/p&gt;&#xA;&lt;p&gt;Thus if you say that in an unbound [series] there exists no last finite number that can be written in, although there can exist an infinite one: I reply, not even this can exist, if there is no last number. The only other thing I would consider replying to this reasoning is that the number of terms is not always the last number of the series. That is, it is clear that even if finite numbers are increased to infinity, they never - unless eternity is finite, i.e.never - reach infinity. This consideration is extremely subtle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-04-10T10:56:05.743" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3642" PostTypeId="1" AcceptedAnswerId="3644" CreationDate="2016-04-11T12:56:02.703" Score="9" ViewCount="1448" Body="&lt;p&gt;I am just curious who was the first person to write down a differential equation? And what was this differential equation?&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="4251" LastEditDate="2020-06-21T19:49:52.613" LastActivityDate="2020-06-21T19:49:52.613" Title="Who wrote down the first differential equation?" Tags="|mathematics|differential-equations|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3643" PostTypeId="2" ParentId="3642" CreationDate="2016-04-11T15:00:14.043" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;But this will appear plainer by an Example or two. ...&quot; (Newton 1671) --- After outlining his general method for finding solutions of differential equations.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Newton obtained the solution of a differential equation satisfying a given initial condition in terms of infinite series. At each stage of his series solution, he inserted the series into his differential equation and integrated the resulting polynomial.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.phaser.com/modules/historic/newton/index.html&quot; rel=&quot;nofollow&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2016-04-11T15:06:46.723" LastActivityDate="2016-04-11T15:06:46.723" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3644" PostTypeId="2" ParentId="3642" CreationDate="2016-04-11T19:18:11.227" Score="10" Body="&lt;p&gt;It is impossible to tell &quot;who wrote down&quot; first. Leibniz PUBLISHED first (1684),&#xA;but some documents indicate that he knew how to solve some differential equations in 1666. Newton never published his &quot;Method of fluxions&quot;, but it is claimed that he discovered it in 1665 to 1667.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Method of fluxions of Newton was published after his death in 1736 but it was written in 1671. The first differential equation encountered in this book is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(3x^2-2ax+ay)dx+(ax-3y^2)dy=0$$&lt;/span&gt;&#xA;which by our modern classification is exact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the story is more complicated than this. Newton (and others) actually SOLVED&#xA;differential equations in many cases without WRITING them. Using some geometric arguments. For example, Newton in fact solved the differential equation which describes the two body problem in celestial mechanics, though he never wrote it as a differential equation, did not use the formal notion of derivative or fluxion, but argued by a purely geometric method. In the last volume of Principia he deals with very advanced questions (perturbation theory) which nowadays  belong to the theory of differential equations, but he did not state them in these terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The name &quot;differential equation&quot; was introduced by Leibniz. The earliest published paper, of which I could find an English translation, and where a differential equation is written and solved is this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.17centurymaths.com/contents/Leibniz/ae3a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.17centurymaths.com/contents/Leibniz/ae3a.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He solves the &quot;isochrone problem&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. In fact Galileo solved some simple differential equation long before Newton, when he was trying to figure out the path of a falling body. If &lt;span class=&quot;math-container&quot;&gt;$y(t)$&lt;/span&gt; is the height at time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y(0)=0$&lt;/span&gt;, his first guess was that &lt;span class=&quot;math-container&quot;&gt;$y'=ky$&lt;/span&gt; speed proportional to traveled distance. Then by complicated reasoning we concluded&#xA;that this cannot be the correct law. In modern language &lt;span class=&quot;math-container&quot;&gt;$y(t)=ce^{kt}$&lt;/span&gt;; if&#xA;you satisfy the initial condition you obtain &lt;span class=&quot;math-container&quot;&gt;$c=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y(t)=0$&lt;/span&gt;, so the motion never begins. Galileo's second guess was the correct &lt;span class=&quot;math-container&quot;&gt;$y'=kt$&lt;/span&gt;, which has a solution &lt;span class=&quot;math-container&quot;&gt;$y(t)=kt^2/2$&lt;/span&gt;, the Galileo law of the falling bodies. Of course, Galileo could not use these simple notations, and had to use some complicated arguments, but his arguments solving these differential equations were correct. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-02-06T12:44:57.390" LastActivityDate="2020-02-06T12:44:57.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3645" PostTypeId="1" AcceptedAnswerId="3647" CreationDate="2016-04-12T02:53:02.920" Score="15" ViewCount="419" Body="&lt;p&gt;It doesn't seem that he began his life in mathematics. Does anyone know how it came to be ? What did he read ?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="4251" LastEditDate="2020-05-13T15:00:06.917" LastActivityDate="2020-05-13T15:00:06.917" Title="What kind of mathematical education did Leibniz have?" Tags="|mathematicians|biographical-details|education|leibniz|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3647" PostTypeId="2" ParentId="3645" CreationDate="2016-04-12T07:01:21.377" Score="17" Body="&lt;p&gt;Mauro Allegranza's comment pretty much says it all but to elaborate a bit one could mention that Leibniz came to mathematics rather late in his intellectual career and was essentially a self-educated scholar. His older colleague Huygens encouraged him to pursue mathematics, and his encouragement (on many occasions) was instrumental in Leibniz's development. In a way being self-educated bestowed a certain advantage on Leibniz because it made it easier for him to question some of the reigning dogmas at the time. Specifically this applies to a traditional hostility to indivisibles and infinitesimals that had deep roots going back to the Council of Trent and connected to the dogma of the eucharist. Leibniz was able to shed these intellectual shackles in his work, giving us a mathematics that was original in both senses of the term.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-04-12T07:01:21.377" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3648" PostTypeId="1" CreationDate="2016-04-12T18:12:08.803" Score="1" ViewCount="148" Body="&lt;p&gt;I used to get confused when I first learned the electromagnetic field is said to have a momentum. A few years later, I came across an argument for why 0.999... = 1: When faced with the task of defining what a recurring decimal means, we as a community decided to make the definition consistent with the rules of algebra of &quot;normal&quot; numbers, paying the cost of this prima facie counter-intuitive result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When faced with a similar task in electrodynamics, we apparently chose to keep the theory consistent with the conservation laws of mechanics, at the cost of assigning energy and momentum to seemingly &quot;non-corporeal&quot; entities called fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why was this choice made? Why salvage conservation laws that seem &quot;wrong&quot;?&lt;/p&gt;&#xA;" OwnerDisplayName="user3971" LastEditorUserId="62" LastEditDate="2016-04-13T21:06:47.147" LastActivityDate="2017-06-06T17:57:10.207" Title="Why make fields have momentum and energy? (or, the importance of conservation laws)" Tags="|electromagnetism|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3649" PostTypeId="2" ParentId="3648" CreationDate="2016-04-12T21:35:12.770" Score="3" Body="&lt;p&gt;Keep in mind that &quot;intuitions&quot; are personal, 0.999... = 1 may be counterintuitive to you but natural to others, they also strongly depend on historical and cultural context. In 19th century first the wave optics and then Maxwell's electrodynamics strongly suggested existence of a material carrier for light and electromagnetic waves, the ether. Young conducted experiments which seemingly allowed one to &quot;see and touch&quot; the ether, he even constructed a &quot;ripple tank&quot; to demonstrate the analogy between water waves and light, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2629/how-did-young-perform-his-double-slit-experiment/2634#2634&quot;&gt;How did Young perform his double slit experiment?&lt;/a&gt; Faraday conducted experiments with magnetic fields demonstrating how his &quot;lines of force&quot; literally materialize when using metal filings, see his &lt;a href=&quot;https://books.google.com/books?id=yzgLAAAAYAAJ&amp;amp;pg=PA407&amp;amp;lpg=PA407&amp;amp;dq=%22On+the+physical+character+of+the+lines+of+magnetic+force%22&amp;amp;source=bl&amp;amp;ots=vuhrFRrROv&amp;amp;sig=Kpi1IOwE_8KRnZ1QnWokVgMgkHU&amp;amp;hl=en&amp;amp;ei=QJfzTKqLFI-6sQOKtqTCCw&amp;amp;sa=X&amp;amp;oi=book_result&amp;amp;ct=result&amp;amp;resnum=3&amp;amp;ved=0CCIQ6AEwAg#v=onepage&amp;amp;q=%22On%20the%20physical%20character%20of%20the%20lines%20of%20magnetic%20force%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Physical Character of the Lines of Magnetic Force&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even for gravity action at a distance was considered undesirable, so much so that a material intermediary was sought to explain it. Laplace in 1799 even researched a modification of Newtonian gravity with finite propagation speed, and Mossotti in 1830 proposed a hypothesis where gravity is the imbalance in the electromagnetic attraction and repulsion, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2388/what-19th-century-developments-contributed-to-the-general-theory-of-relativity/2395#2395&quot;&gt;What 19th century developments contributed to the General theory of Relativity?&lt;/a&gt; Many 19th century mathematicians (Cauchy, Navier, Green, Stokes, etc.) were constructing &lt;a href=&quot;http://www.ipme.ru/ipme/labs/dms/prive/ivanova/Home_page_Elena_Ivanova/Ether%20history%20ENG.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;elastic theories of ether&lt;/a&gt;, it was thought to be a special solid, and waves in it to be a case of elastic waves. In that context physicists did not see fields as &quot;incorporeal&quot;, elastic waves carry momentum and energy in a rather literal sense of the words, they can knock you off your feet in a particular direction. &lt;a href=&quot;http://people.seas.harvard.edu/~jones/cscie129/nu_lectures/lecture6/hertz/Hertz_exp.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hertz's 1887 experiments&lt;/a&gt; demonstrated that electromagnetic waves were as corporeal, they could also do mechanical work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the ether was just another mechanical system then it was only natural that it should obey mechanical conservation laws, and Hamiltonian reformulation of dynamics turned momentum and energy into fundamental notions of mechanics. Even after such a rationale was removed at the beginning of 20th century others took its place, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether&amp;#39;s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Noether theorem&lt;/a&gt; demonstrated a fundamental relation between conservation laws and symmetries, so conservation of momentum and energy came to be viewed as expressing fundamental symmetries of space and time.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-14T00:18:46.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3650" PostTypeId="2" ParentId="771" CreationDate="2016-04-13T00:02:58.243" Score="10" Body="&lt;p&gt;There's a very simple &lt;strong&gt;geometrical&lt;/strong&gt; response that provides a rounded solution which might reflect on this problem at &lt;a href=&quot;http://www.gjbath.com/SQR3.htm&quot; rel=&quot;noreferrer&quot;&gt;http://www.gjbath.com/SQR3.htm&lt;/a&gt;. It's so much easier seeing it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This involves circumscribing an equilateral triangle and taking a diameter from the apex. Then, applying $26/15$ as an approximation to $\sqrt 3$ ($\tan 30^\circ\approx 15/26$) and reiterating in the similar triangle on the base gives $15/(225/26)$. Thence, multiplying out by 26, and using the fact that the diameter is divided by the base in the ratio 3:1, and then combining, results in a further approximation of&#xA;$$\frac{3\cdot225+26\cdot26}{15\cdot2\cdot26}=\frac{1351}{780}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thereafter, subtracting the original approximation's numerator and denominator  $26/15$ gives $$\frac{1351-26}{780-15}=\frac{1325}{765}=\frac{265}{153}$$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is such straightforward geometry and reasoning that you wouldn't expect any comment from a Classical geometer. And it doesn't involve Pell!&lt;/p&gt;&#xA;" OwnerUserId="3976" LastEditorUserId="5020" LastEditDate="2018-09-03T15:21:52.143" LastActivityDate="2018-09-03T15:21:52.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3651" PostTypeId="1" AcceptedAnswerId="3653" CreationDate="2016-04-13T08:42:04.683" Score="2" ViewCount="305" Body="&lt;p&gt;Hilbert presented ten of his 23 problems at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne 1900.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In which of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert&amp;#39;s_problems&quot; rel=&quot;nofollow&quot;&gt;23&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_twenty-fourth_problem&quot; rel=&quot;nofollow&quot;&gt;24&lt;/a&gt; problems was rating (as he was speaking about actual mathematical problems he made estimations about their solvableness etc.) totally wrong and if it is possible to explain, why?&lt;/p&gt;&#xA;" OwnerUserId="2821" LastEditorUserId="4251" LastEditDate="2021-01-25T21:12:38.730" LastActivityDate="2021-01-25T21:12:38.730" Title="Hilbert's problems" Tags="|mathematics|mathematicians|hilbert|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3652" PostTypeId="2" ParentId="3648" CreationDate="2016-04-13T13:38:08.097" Score="1" Body="&lt;p&gt;Regarding 0.999... = 1, that's a just another representation of 1. Best advice: Get over it! There are lots of representations of $e$, for example $\sum_{n=0}^{\infty} \frac 1 {n!}$, or $\lim_{n-&amp;gt;0} (1+n)^{1/n}$, and a host of others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also lots of representations of 1 besides 0.999....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A couple of very simple tricks in mathematics are to add zero to an expression or multiply an expression by one (but zero is represented as $a-a$ rather than $0$, where $a$ is an expression of choice, or one is represented as $a/a$ rather than $1$, where $a$ is an expression known to be non-zero) and then rearrange and simplify. The mathematical world would be a much sadder place if there was only one representation of $\pi$, $e$, $0$, or $1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding electromagnetic fields carrying energy and momentum, what seems wrong about that? It should be intuitively obvious rather than confusing that electromagnetic fields carry energy. You can feel the heat of the Sun warming your skin thanks to the fact that the electromagnetic field carries energy. We can convert this energy to electricity, and use that to power the motors in the appliances in your house. Those motors would not work if electromagnetic field did not carry linear and angular momentum.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2016-04-13T13:38:08.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3653" PostTypeId="2" ParentId="3651" CreationDate="2016-04-13T14:07:41.473" Score="4" Body="&lt;p&gt;Contrary to what you say, Hilbert very rarely says anything which can be&#xA;interpreted as a &quot;rating&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no objective criterion to &quot;rate&quot; the problems except by the time&#xA;it took to solve them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some problems were solved even before Hilbert stated them, for example, problem 3. Problem 22 (uniformization) was solved (in dimension 1) in 1906. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Problem 7 was rated by Hilbert as very difficult. It was solved in 1935.&#xA;On the other hand, Hilbert was optimistic about problem 8 (Riemann hypothesis), which is understandable because there was a recent breakthrough at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some cases Hilbert was wrong on the answers: For example, problem 2 was&#xA;&quot;to prove that axioms of arithmetic are consistent&quot;. This is impossible to prove,&#xA;as Godel showed in 1936.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the problems were not specific questions but wide research programs. So one cannot really say whether they are solved or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I asked Gerry Myerson about his answer mentioned in KCd's comment. The information comes from the book of C. Reid, Hilbert. In it Hilbert's talk in 1920s is mentioned where he gave this assessment of the three problems.&#xA;His opinion about Fermat's problem was right. Only the comparison of the Riemann problem and transcendence problem turned out to be &quot;wrong&quot;. I suspect that majority of mathematicians would agree with Hilbert's rating of these two problems at that time. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-15T13:01:09.510" LastActivityDate="2016-04-15T13:01:09.510" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3655" PostTypeId="2" ParentId="3631" CreationDate="2016-04-14T02:02:00.367" Score="6" Body="&lt;p&gt;To a certain extend, the measurement problem (MP) is indeed philosophical, in the end, it still comes down to which interpretation you prefer. &#xA;However, there are also of course also certain physical results, which come into play here. These are also highly relevant to the various interpretations, which I will try to put in appropiate context. The keyword in order to understand the modern solutions of the measurement problem is &lt;em&gt;decoherence&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formally speaking the measurement problem is the obvious contradiction between two of the textbook axioms&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Time evolution of isolated systems is described by the Schrödinger equation $i\partial_t|\psi\rangle=H|\psi\rangle$. The important thing to notice is, that the time evolution operator $t_0\rightarrow t_1$ is unitary.&lt;/li&gt;&#xA;&lt;li&gt;After measuring the system in its eigenstate corresponding to the measurement outcome.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Formally speaking, the operator which describes the transition &quot;before measurement&quot; to &quot;after measurement&quot; is a strict projection (in case the system is not already in an eigenstate, which is generic), and therefore obviously not unitary, although it should be a time evolution in some sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some early versions of the Copenhagen Interpretation (CI), Bohr's views in particular, propose a different, classical set of physical rules as soon as macroscopic objects interact with the quantum objects, because the classical notions are inalienable in order to describe our world (&quot;&lt;em&gt;Unverzichtbarkeit klassischer Begriffe&lt;/em&gt;&quot;), the irreversible interaction between the quantum system and the measurement device are an integral part of our description. This point of view can be found in many of Bohr's essays, for example &quot;&lt;em&gt;Einheit des Wissens&lt;/em&gt; (1954) or his answer to EPR 1935 &quot;&lt;em&gt;Kann man die quantenmechanische Beschreibung der Wirklichkeit als vollständig erachten?&lt;/em&gt;&quot;. I should emphasize at this point that the Copenhagen Interpretation is not a clear-cut set of rules how to treat certain situations and how to interpret them physically, its more like an approach which works well in practice and is underpinned with various philosophical concepts, which differ between the various personalities. What I briefly sketched is mainly Bohr's point of view, von Neumann for example have a rather formal approach, while Heisenbergs emphasis was always on his uncertainty relations. However, the common denominator of what we call &quot;Copenhagen Interpretation(s)&quot; today is a somewhat nonunitary behavior at some classic-quantum boundary. For more details, see Jammer's book &quot;&lt;em&gt;The Philosophy of Quantum Mechanics&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Philosophically, CI, and in particular Bohr's views mimic Kant's concepts in some ways, however, it is surely not pleasing from a physicists point of view (at least not for me), as in this sense, QM is not &quot;more fundamental&quot; than the classical theories as we need the latter in order to formulate stuff like observables, as there is for example no quantum-mechanical notion of &quot;position&quot;.&#xA;One could say, that this anti-reductionist point of view is another different formulation of the measurement problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering a &quot;solution&quot; of the MP, as I already mentioned, the major breakthrough of the  &quot;second generation&quot; of QM-&quot;Philosophers&quot; was the development of Quantum decoherence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first step in this &quot;modern development&quot; was Everett '57 - &quot;The Theory of the Universal wave function&quot; or the (after getting massive flak from Bohr) shortened, first published version &quot;&quot;&lt;a href=&quot;http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29.454&quot; rel=&quot;noreferrer&quot;&gt;Relative-State&quot;-Formulation of Quantum Mechanics&lt;/a&gt;&quot;. The content of these famous publications is today known as &quot;Many-Worlds-Interpretation&quot; (MWI). While the ontological content of this work is most certainly debatable, the formal part of this work is what is mostly overlooked, although I personally think that Everett formulated the key idea for the modern approaches to the MP. In short, he re-formulated the known QM from a smaller set of axioms, in particular leaving out the &quot;collapse&quot; axiom and Born's rule, granting the Schrödinger equation universal validity, he then rederived Born's rule and something similar to the collapse from the reduced set of axioms. His description of the measurement process was basically &quot;System Observer+Experiment, where the state of the observer changes with the state of the experiment, i.e. he can determine the state of the system from his own state.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key concept of relative states he used (or rather his perspective on these) was probably crucial for the later development of decoherence. The mathematical observation is rather simple: Take a system consisting of two correlated systems $S=S_1\otimes S_2$ and put it in some state, then for a fixed state of the first system $S_1$, you can put a relative state by expanding in some basis of the second system. In this sense, you cannot independently fix states for $S_1$ or $S_2$ (especially as this works in any basis) - if you live in $S_1$, it only makes sense to talk about $S_2$ relative to your own state. Of course, this has been one of the well-known key features of quantum mechanics, entanglement - yet, Everetts key observation was, loosely speaking, that we can reproduce QM by entangling the observer to the system in our formal description. Notice, that this still does not solve the key problem &quot;Where do the classical states come from?&quot; - The basis we expand the relative system in is still arbitrary, so why should we expand in some &quot;classical bases&quot; from our point of view? This is in essence what is called &quot;preferred basis problem&quot; in the literature. In the end, it is basically again the measurement problem, stated in slightly different terms. However, these terms proved to be much more approachable than the philosophical discussion around Einstein&amp;amp;Bohr.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When Everett's work was first published, most people either either didn't  notice or didn't care. Everett's work was revitalized in the 70's by Graham and DeWitt, and it was no big surprise, that decoherence theory emerged shortly after and had a prime in the 80's and early 90's. Notable figures were Zeh, Joos and others. Decoherence is basically the process, where a quantum system interacts with a thermodynamic environment and loses its &quot;quantumness&quot; i.e. its ability to be in superposition states. Mathematically speaking, the offdiagonals of the corresponding density matrices are &quot;traced out&quot; by the environment. In this sense, the quantum irreversibility arises in a similar way as the irreversibility in thermodynamics from the classical theories - information gets lost in the heat of the environment. One key insight considering the &quot;preferred basis problem&quot; was made by Zurek , I think the first publication on this was &quot;&lt;em&gt;&lt;a href=&quot;http://journals.aps.org/prd/abstract/10.1103/PhysRevD.26.1862&quot; rel=&quot;noreferrer&quot;&gt;Environment-induced superselection rules&lt;/a&gt;&lt;/em&gt;&quot; in 1982 (Phys. Rev. D 26): The coupling to the environment induces a certain selection of quantum states which will not appear in superposition, this process is called &quot;environment-induced superselection&quot; or short &quot;Einselection&quot;, the basis &quot;chosen&quot; by this proces is also called &quot;pointer basis&quot;. In particular, it gives you a preferred basis in which it interacts with the environment, depending on the structure of the Hamiltionian. Details to this can be found in Zurek's publications, in particular&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://arxiv.org/abs/quant-ph/0306072&quot; rel=&quot;noreferrer&quot;&gt;Decoherence and the transition from quantum to classical&lt;/a&gt; (2003) &lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://arxiv.org/abs/quant-ph/0105127&quot; rel=&quot;noreferrer&quot;&gt;Decoherence, einselection, and the quantum origins of the classical&lt;/a&gt;  (2001)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now this sounds pretty good, does this solve our measurement problem, or at least the preferred basis problem? To some extend, yes. It explains, why the observer has a preferred basis, but it cannot exactly explain where this basis come from, or how an observer exactly has to look like to produce a classical basis. For example, einselection cannot explain why the stuff around here tends to be localized. An argument often brought up in this context is the so-called &quot;Quantum Darwinism&quot;: Observable states are states which are decently stable - if some state or respectively some basis was highly unstable, it would immediately trace out when interacting with the environment and therefore not be observable in a macroscopic context. In this sense, the observables which we perceive as classical are just the &quot;fittest&quot; states. As far as I know, this is still subject of current research. However, I know that some people claim that MWI/Consistent Histories+Decoherence+Einselection+Darwinism solve the measurement problem. The main argument to this arises from the anthropic principle (AP): If it was different, we wouldn't be there to observe it. Not a very satisfying answer if you ask me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to summarize my massive textwall:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Most people agree, that the measurement problem is &lt;em&gt;partially&lt;/em&gt;(!) solved by invoking a relative-state formulation for Observer+System+Environment and combining it with a suitable interpretation, mostly many worlds or consistent histories. The observer measures the system in his einselected basis, and the irreversibility comes from the information-loss in the environment.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most people also agree, that the quantum-to-classical transition (&quot;Collapse of the wave function&quot;) is a quantum-mechanical process which can be interpreted in a statistical sense. In particular, it is not some instantaneous collapse happening at some quantum-to-classical boundary as for example Bohr imagined it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sometimes, people claim that the measurement problem is solved by decoherence without further interpretation. This is clearly wrong, as decoherence only works in a &lt;em&gt;interpreted&lt;/em&gt; formalism. Also, decoherence is no magic addition to quantum theory, it arises naturally from &quot;normal&quot; quantum mechanics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Few people claim that Quantum Darwinism together with the AP solves the measurement problem. This is very controversial.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Most people agree, that the measurement problem is not fully solved by the decoherence program. Some people think, that in principle, it &lt;em&gt;can&lt;/em&gt; be solved, I imagine by deriving the preferred basis from some fundamental properties of our universe, such as the form of certain fundamental Hamiltonians and relations between the fundamental constants.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In conclusion, it still comes down more or less to which interpretation you prefer. However, my impression is that most people working in this area tend to Everett-ish interpretations (Many-Worlds, Consistent Histories, Existential Interpretation), in particular because the Schrödinger equation holds everywhere.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Most/Some/Few is of course my personal impression, I don't have stats to back this up. However, I hope I could give a decent overview over the historic development of this question and its current state. For a more detailed discussion about the current state, I can also recommend the overview article by Schlosshauer &lt;a href=&quot;http://arxiv.org/abs/quant-ph/0312059v4&quot; rel=&quot;noreferrer&quot;&gt;Decoherence, the measurement problem, and interpretations of quantum mechanics&lt;/a&gt;. There are also many, many other articles by different people with different opinions out there, if you're interested, you can actually find a lot of stuff out there. There are for example some very clearly written articles by one of the fathers of the decoherence program, H.D. Zeh, many of them are however in german, and he is very biased towards Many Worlds.&lt;/p&gt;&#xA;" OwnerUserId="354" LastEditorUserId="354" LastEditDate="2016-04-14T02:44:09.420" LastActivityDate="2016-04-14T02:44:09.420" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3656" PostTypeId="2" ParentId="3620" CreationDate="2016-04-14T15:32:23.020" Score="3" Body="&lt;p&gt;Confusingly, a &lt;strong&gt;Maxwell wheel&lt;/strong&gt; used to refer to a totally different invention: a color top invented by Maxwell as a student and described in &quot;&lt;em&gt;&lt;a href=&quot;http://www.jimworthey.com/archive/Maxwell_1855_OCRtext.pdf&quot; rel=&quot;nofollow&quot;&gt;Experiments On Colour, As Perceived By The Eye&lt;/a&gt;&lt;/em&gt;&quot; (1855). There are references to this as a 'Maxwell wheel' on Google Book search starting from 1882, though these days it's more commonly called &lt;strong&gt;Maxwell's disc&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell also invented a dynamical top to illustrate the classical laws of rotation, as well as various other pedagogical experiments involving spinning and axles. However, his papers don't seem to include a description of the modern Maxwell wheel, and it is not among his apparatus displayed at the Cavendish Laboratory museum. The first unambiguous mention that I can find of it is from a seminar report on the &quot;&lt;em&gt;Implementation of Curricula in Science Education with Special Regard to the Teaching of Physics&lt;/em&gt;&quot; (1974), though this doesn't go into its history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it's therefore likely that the wheel wasn't invented by Maxwell, though its history remains frustratingly murky.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastActivityDate="2016-04-14T15:32:23.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3659" PostTypeId="1" CreationDate="2016-04-15T23:19:06.213" Score="8" ViewCount="313" Body="&lt;p&gt;I'm having a discussion elsewhere about this. I suspect that it was Perl, with the hash maps, but that is because I don't know much about older languages rather than any exhaustive research on the matter. The wikipedia article says that the data structure was invented in 1953, so it is plausible that there are earlier examples.&lt;/p&gt;&#xA;" OwnerUserId="3958" LastActivityDate="2016-06-13T01:16:23.430" Title="What was the first programming language that implemented hash maps / dictionaries as a base type?" Tags="|computer-science|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3660" PostTypeId="1" CreationDate="2016-04-16T00:40:43.577" Score="7" ViewCount="93" Body="&lt;p&gt;Who introduced it, when, where, and with what if any rationale?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note that I am not asking about the origin of the nabla symbol, which is covered &lt;a href=&quot;https://en.wikipedia.org/wiki/Nabla_symbol&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="3669" LastActivityDate="2016-04-16T00:40:43.577" Title="Whence originates the use of the nabla (∇) for a connection or covariant derivative?" Tags="|notation|differential-geometry|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3661" PostTypeId="1" CreationDate="2016-04-16T15:08:40.370" Score="1" ViewCount="4544" Body="&lt;p&gt;In the Feynman lectures, it is mentioned that [Vol 1; Gravitation]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Galileo discovered a very remarkable fact about motion, which was&#xA;  essential for understanding these laws. That is the principle of&#xA;  inertia—&lt;strong&gt;if something is moving, with nothing touching it and&#xA;  completely undisturbed, it will go on forever, coasting at a uniform&#xA;  speed in a straight line. (Why does it keep on coasting?&lt;/strong&gt; We do not&#xA;  know, but that is the way it is.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How Gallileo discovered the law of inertia ? In those time, there were probably no frictionless area so that he could test his hypothesis. And uncontrolled &quot;thought experiment&quot; can lead you astray like Aristotle (who belived that continuous force is required to get a particle moving all the time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, I do find Feynman's quotation (of Tycho Brahe's &quot;idea&quot; ) relevant, but I can't make the connection:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This [&lt;em&gt;debates about the nature of the motions of the planets would best be resolved if the actual positions of the planets in the sky were measured sufficiently accurately&lt;/em&gt;] was a tremendous idea—that to find something out, it is better to&#xA;  perform some careful experiments than to carry on deep philosophical&#xA;  arguments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A quote by Paul Dirac is also relevant here (not directly, the bold words are only relevant):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;. For example, rough experiments about the relation between the&#xA;  pressure and volume of a gas at a fixed temperature give results&#xA;  fitting in with a law of inverse proportionality, &lt;strong&gt;but it would be&#xA;  wrong to infer that more accurate experiments would confirm this law&#xA;  with greater accuracy&lt;/strong&gt;, as one is here dealing with a phenomenon&#xA;  which is not connected in any very direct way with the fundamental&#xA;  laws of motion.&quot;,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How you can be sure that the less friction you use, the more accurate it becomes ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: Since most of the person are focusing on galileo did that (my ill written title misled them), I actually wanted to know how scientists get rid of their Aristotlian intuition and developed the idea (notion) of inertia.&lt;/p&gt;&#xA;" OwnerDisplayName="user3997" LastEditorDisplayName="user3997" LastEditDate="2016-04-17T03:37:39.630" LastActivityDate="2016-08-31T19:57:08.477" Title="How the law of inertia got discovered?" Tags="|physics|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3662" PostTypeId="2" ParentId="3661" CreationDate="2016-04-16T17:28:27.060" Score="2" Body="&lt;p&gt;Galileo' formulation of &lt;em&gt;inertia&lt;/em&gt; is not &quot;perfectly&quot; newtonian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the discussion into his &lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;nofollow&quot;&gt;Dialogo&lt;/a&gt; (1632), page 128 of &lt;a href=&quot;http://archimedes2.mpiwg-berlin.mpg.de/archimedes_templates/biography.html?-table=archimedes_authors&amp;amp;author=Salusbury,%20Thomas&amp;amp;-find&quot; rel=&quot;nofollow&quot;&gt;Th.Salusbury&lt;/a&gt;'s English translation (1661):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[&lt;a href=&quot;http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?step=thumb&amp;amp;dir=galil_syste_065_en_1661&quot; rel=&quot;nofollow&quot;&gt;page 128&lt;/a&gt; ] SALV. Now tell me, what would befall the same moveable upon a superficies that had neither acclivity nor &#xA;  declivity?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SIMPL. There being no declivity, there can be no natural incli­nation to motion: and there being no acclivity, there can be no resistance to being moved; so that there would arise an indifference between propension and resistance of motion; therefore, methinks it ought naturally to stand still. [...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SALV. Well: but if there be no cause of retardation, much less ought there to be any cause of rest. How long therefore would you have the moveable to move?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SIMP. As long as that superficies, neither inclined nor decli­ned shall last.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SALV. Therefore if such a space were interminate, the motion upon the same would likewise have no termination, that is, would be perpetual.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SALV. That hath been already supposed, when it was said, that all external and accidental impediments were removed, and the brittlenesse of the moveable in this our case, is one of those impediments accidental. Tell me now, what do you think is the cause that that same Ball moveth spontaneously upon the inclining &#xA;  plane, and not without violence upon the erected?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SIMP. Because the inclination of grave bodies is to move to­wards the centre of the Earth, and onely by violence upwards to­wards the circumference; and the inclining superficies is that which acquireth vicinity to the centre, and the ascending one, remotenesse.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SALV. Therefore a superficies, which should be neither de­clining nor ascending, ought in all its parts to be equally di­stant from the centre. But is there any such superficies in the World?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SIMP. There is no want thereof: Such is our Terrestrial Globe, if it were more even, and not as it is rough and montai­nous; but you have that of the Water, at such time as it is calm and still.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;SALV. Then a ship which moveth in a calm at Sea, is one of those moveables, which run along one of those superficies that are neither declining nor ascending, and therefore disposed, in case all obstacles external and accidental were removed, to move with the impulse once imparted incessantly and uniformly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The newtonian formulation is basically due to Descartes' &lt;a href=&quot;https://books.google.it/books?id=lHpbAAAAQAAJ&amp;amp;pg=PA55&quot; rel=&quot;nofollow&quot;&gt;Principia philosophiae&lt;/a&gt; (1644), art.XXXIX: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Altera lex naturae&lt;/em&gt;: quod omnis motus ex se ipso sit rectus; et ideo quae circulariter mouentur, tendere semper ut recedant a centro circuli quem describunt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-04-17T10:39:43.523" LastActivityDate="2016-04-17T10:39:43.523" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3663" PostTypeId="2" ParentId="771" CreationDate="2016-04-16T17:52:43.580" Score="6" Body="&lt;p&gt;Some information can be found here &lt;a href=&quot;https://math.stackexchange.com/questions/894862/archimedes-approximation-of-square-roots/1740660&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the secant method on the parabola $y = x^2$, a new estimate $x_{n+m}$ for $\sqrt{3}$ can be obtained from $x_n$ and $x_m$ as $$x_{n+m} = \frac{3 + x_n x_m}{x_n + x_m}$$ where if we start with $x_1 = \frac{5}{3}$ then $x_{2n} &amp;gt; \sqrt{3}$ and $x_{2n+1} &amp;lt; \sqrt{3}$.&lt;br&gt;&#xA;We therefore obtain $$x_1 &amp;lt; x_3 &amp;lt; \sqrt{3} &amp;lt; x_4 &amp;lt; x_2$$&#xA;Where $$x_1 = \frac{5}{3}$$&#xA;$$x_2 = \frac{3 + x_1 x_1}{x_1 + x_1} = \frac{26}{15}$$&#xA;$$x_3 = \frac{3 + x_2 x1}{x_2 + x_1} = \frac{265}{153}$$&#xA;$$x_4 = \frac{3 + x_3 x_1}{x_3 + x_1} = \frac{3 + x_2 x_2}{x_2 + x_2} = \frac{1351}{780}$$&lt;/p&gt;&#xA;" OwnerUserId="3999" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-04-16T17:52:43.580" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3664" PostTypeId="1" CreationDate="2016-04-16T14:34:51.197" Score="0" ViewCount="108" Body="&lt;p&gt;In the Feynman lectures, it is mentioned that [Vol 1; Gravitation]:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Galileo discovered a very remarkable fact about motion, which was&#xA;  essential for understanding these laws. That is the principle of&#xA;  inertia—&lt;strong&gt;if something is moving, with nothing touching it and&#xA;  completely undisturbed, it will go on forever, coasting at a uniform&#xA;  speed in a straight line. (Why does it keep on coasting?&lt;/strong&gt; We do not&#xA;  know, but that is the way it is.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How Gallileo discovered the law of inertia ? In those time, there were probably no frictionless area so that he could test his hypothesis. And uncontrolled &quot;thought experiment&quot; can lead you astray like Aristotle (who belived that continuous force is required to get a particle moving all the time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, I do find Feynman's quotation (of Tycho Brahe's &quot;idea&quot; ) relevant, but I can't make the connection:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This [&lt;em&gt;debates about the nature of the motions of the planets would best be resolved if the actual positions of the planets in the sky were measured sufficiently accurately&lt;/em&gt;] was a tremendous idea—that to find something out, it is better to&#xA;  perform some careful experiments than to carry on deep philosophical&#xA;  arguments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3997" LastActivityDate="2016-04-16T18:35:05.473" Title="How did Galileo discover the law of inertia?" Tags="|untagged|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2016-04-16T22:15:06.563" ContentLicense="CC BY-SA 3.0" />
  <row Id="3665" PostTypeId="2" ParentId="3661" CreationDate="2016-04-16T17:54:31.260" Score="0" Body="&lt;p&gt;I don't have any references, but I recall reading that he used inclined planes and marbles. Since the speed of a rolling marble is far less than a free falling one, air drag is much less noticeable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He then realized that if you let a marble go down a slope it will gain velocity, and if then it is forced to go up another slope, ir will stop at exactly the same height where it started, no matter the angles of the slopes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar experiment can be made with a pendulum: no matter if you bend the thead, the top height of the pendulum at both sides of the oscillation is always the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then Galileo reasoned: if the angle of the second up slide is very small, the distance needed to reach the original height will be very large. And if the angle is exactly 0, then the distance will be infinite and the marble will roll forever!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The failure in Galileo theory is that it only worked horizontally. You have to wait until Newton to get a really general law of conservation of movement.&lt;/p&gt;&#xA;" OwnerDisplayName="rodrigo" LastActivityDate="2016-04-16T17:54:31.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3666" PostTypeId="2" ParentId="3659" CreationDate="2016-04-16T19:43:54.107" Score="5" Body="&lt;p&gt;Not sure if there is a definitive answer but Smalltalk certainly had dictionaries (hash maps) built into the language (as such it can be given an OO language) and Smalltalk predates Perl.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2016-04-16T19:43:54.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3668" PostTypeId="2" ParentId="26" CreationDate="2016-04-17T09:55:49.960" Score="0" Body="&lt;p&gt;To respond to the requested &lt;em&gt;example where Occam’s razor favoured a theory A over another theory B, but theory B turned out to be a better description of reality later&lt;/em&gt; I would mention the history of real analysis which has been based since the 1870s on theory A (for Archimedean), which involves the Archimedean complete ordered field. An older/newer approach involves a theory B (for Bernoullian), working with infinitesimals as Johann Bernoulli did.  It turns out that while the background continuum is easier to describe in the A-track, the &lt;em&gt;procedures&lt;/em&gt; are easier to work with in the B-track. For example, instead of defining continuity of a function by requiring that for every epsilon greater than zero there should be a delta greater than zero such that the students already are falling asleep or taking calming pills, you can just follow Cauchy (1821) in requiring that every infinitesimal change $\alpha$ in input must produce an infinitesimal change in output: $f(x+\alpha)-f(x)$ is infinitesimal.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-04-17T09:55:49.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3669" PostTypeId="2" ParentId="2805" CreationDate="2016-04-17T15:19:22.573" Score="2" Body="&lt;p&gt;For the sake of taking this question off the unanswered list, these comments seem themselves to constitute answers:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Number-theoretical&quot; was criticized by many, not only by Hardy because it just sounds awful. The only reason why some people prefer it, is that &quot;Arithmetic&quot; as a book title can be confused with an elementary school textbook. Otherwise, &quot;arithmetic&quot;, &quot;arithmetical&quot; must be preferred. – Alexandre Eremenko Sep 16 '15 at 17:50&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Greek arithmos means counting number, and arithmetic was historically associated with counting and other elementary uses of numbers. Modern number theory is much broader, including analytic number theory for example, so authors that favor &quot;arithmetical&quot; go against the colloquial usage. There is nothing wrong with &quot;number theoretic&quot; grammatically, &quot;set theoretic&quot; is used the same way, and it makes the meaning more precise. – Conifold Sep 16 '15 at 20:00&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2016-04-17T15:19:22.573" CommentCount="2" CommunityOwnedDate="2016-04-17T15:19:22.573" ContentLicense="CC BY-SA 3.0" />
  <row Id="3670" PostTypeId="1" CreationDate="2016-04-17T15:26:40.757" Score="1" ViewCount="119" Body="&lt;p&gt;I have heard many people talk about this, is there any proof available?&lt;/p&gt;&#xA;" OwnerUserId="3249" LastEditorUserId="26" LastEditDate="2016-04-18T10:01:27.127" LastActivityDate="2016-05-01T22:26:09.870" Title="Was metallurgy of Zinc first started in India?" Tags="|india|materials-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3671" PostTypeId="1" AcceptedAnswerId="3673" CreationDate="2016-04-20T01:16:50.667" Score="7" ViewCount="1385" Body="&lt;p&gt;I'm writing a paper on the history of Feynman diagrams and am having trouble finding the first paper published by Feynman which features a calculation done with Feynman Diagrams.&lt;/p&gt;&#xA;" OwnerUserId="3219" LastEditorUserId="55" LastEditDate="2016-05-05T23:58:26.000" LastActivityDate="2016-05-05T23:58:26.000" Title="What was the first journal to have Feynman Diagrams?" Tags="|physics|quantum-mechanics|quantum-field-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3672" PostTypeId="2" ParentId="3671" CreationDate="2016-04-20T11:52:54.430" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://www.edition-open-access.de/proceedings/5/13/index.html&quot; rel=&quot;nofollow&quot;&gt;This paper&lt;/a&gt; is an analysis of Feynman's early work with Feynman diagrams, and should help you find the answer. Also look at Freeman Dyson's early papers, where he explains Feynman's techniques.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-04-20T11:52:54.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3673" PostTypeId="2" ParentId="3671" CreationDate="2016-04-20T19:06:56.573" Score="4" Body="&lt;p&gt;Comprehensive sources on the history of renormalization are &lt;a href=&quot;http://www.jstor.org/stable/20117830&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cao-Schweber survey paper&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=d0wS0EJHZ3MC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cao's book&lt;/a&gt;. On the diagrams specifically the source &lt;a href=&quot;http://www.press.uchicago.edu/ucp/books/book/chicago/D/bo3534300.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kaiser's Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics&lt;/a&gt;, see also his &lt;a href=&quot;http://www.americanscientist.org/issues/pub/richard-feynmans-diagrams&quot; rel=&quot;nofollow noreferrer&quot;&gt;interview to American Scientist&lt;/a&gt;. According to Kaiser, one of the first published diagrams (reproduced below) appears on p.772 of &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman, R. P.: 1949 &lt;a href=&quot;http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Feynman_QED_1949.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Space-Time Approach to Quantum Electrodynamics&lt;/a&gt;, Physical Review 76, 769-89.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stueckelberg might have used them earlier in an unpublished manuscript, the topic is controversial, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3531/did-feynman-develop-qed-based-on-stueckelbergs-manuscript/3538#3538&quot;&gt;Did Feynman develop QED based on Stueckelberg&amp;#39;s manuscript?&lt;/a&gt; The diagrams were a computational tool for the procedure developed in Feynman's seminal renormalization papers published a year earlier. Below are the citations in the order of publication with direct links.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Feynman, R. P.: 1948, &lt;a href=&quot;http://web.ihep.su/dbserv/compas/src/feynman48c/eng.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Space-time Approach to Non-relativistic Quantum Mechanics&lt;/a&gt;, Reviews of Modern Physics 20, 367-87.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Feynman, R. P.: 1948, &lt;a href=&quot;http://authors.library.caltech.edu/3516/1/FEYpr48a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Relativistic Cut-off for Classical Electrodynamics&lt;/a&gt;, Physical Review  74, 939-46.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Feynman, R. P.: 1948, &lt;a href=&quot;http://authors.library.caltech.edu/3517/1/FEYpr48b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Relativistic Cut-off for Quantum Electrodynamics&lt;/a&gt;, Physical Review 74, 1430-38.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of them had &quot;Feynman diagrams&quot; however, although there is a multiple path picture in the second one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PDgKa.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PDgKa.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-30T20:26:09.163" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3674" PostTypeId="1" AcceptedAnswerId="3676" CreationDate="2016-04-21T00:16:30.200" Score="4" ViewCount="376" Body="&lt;p&gt;For example in the early days, when it was the only way to learn Newtonian mechanics? Was it good as a textbook?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="55" LastEditDate="2016-05-05T23:56:39.757" LastActivityDate="2016-05-05T23:56:39.757" Title="Was the Principia ever used as a textbook of physics for the people wanting to learn?" Tags="|physics|biographical-details|social-context|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3675" PostTypeId="1" CreationDate="2016-04-21T01:09:54.017" Score="0" ViewCount="1070" Body="&lt;p&gt;I would like to know when Europe obtained the ability to predict lunar and solar eclipses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember reading about some story, where Columbus or someone like that used an eclipse chart to convince the native Indians that he was a god. This would imply that Europe had the ability somewhere around the year 1500, which seems pretty early to me. I would've thought It would have to wait until Keplers Laws at least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I want to be more exact than that. When did they start reliably making predictions without any misses? It would be nice to know which European nation did this first, and how fast it spread around Europe and the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If some other region did this first, I would love to know about it too.&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2016-04-23T01:53:10.413" Title="When did Europe start accurately predicting solar and lunar eclipses?" Tags="|astronomy|" AnswerCount="2" CommentCount="1" ClosedDate="2016-04-28T22:02:47.647" ContentLicense="CC BY-SA 3.0" />
  <row Id="3676" PostTypeId="2" ParentId="3674" CreationDate="2016-04-21T01:35:11.883" Score="5" Body="&lt;p&gt;At the time Principia (1687) came out it was not something to be learned, it was something to make sense of. Many consider Principia a tough read not only because of new ideas, but also because of obscuring effect of Euclidean geometric language in which it was coached, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2362/why-is-calculus-missing-from-newtons-principia/2366#2366&quot;&gt;Why is calculus missing from Newton&amp;#39;s Principia?&lt;/a&gt; But many ideas were already known in some narrow circles from works of Huygens, etc., and Newton's correspondence, and those close to him, like one of his early promoters Clarke, had extra insights. There was also competing Cartesian mechanics at the time (mostly qualitative but more intuitive) so it was not so much about teaching and learning in the early years as arguing and advocating between partisans on both sides. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's derivation of Kepler's laws was impressive however, and the new mechanics soon gained wide recognition. &lt;a href=&quot;http://www4.ncsu.edu/~kimler/hi322/textbooks.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;One of the first textbooks, Gravesande's&lt;/a&gt;, appeared in 1720-21, and others soon followed. Even then many concepts (force momentum, etc.), and their role in mechanics, remained unclear and controversial. The so-called vis viva controversy, that started with Leibniz criticizing Descartes's unconserved &quot;quantity of motion&quot; and turned into a mechanical off-shoot of Newton-Leibniz priority dispute, engulfed most prominent physicists in the first half of 18th century, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt; One positive outcome of it was clarification of the basic notions by Euler and D'alembert. It was reformulations in Euler's Mechanica (1736) and D'alembert's Traite de Dynamique (1743/58) that allowed new mechanics to be broadly understood, appreciated and learned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is something of a modern movement of teaching with original sources, especially in mathematics, here is a &lt;a href=&quot;https://www.math.nmsu.edu/~history&quot; rel=&quot;nofollow noreferrer&quot;&gt;programmatic website&lt;/a&gt;. It is not so popular in physics apparently, see &lt;a href=&quot;https://www.math.nmsu.edu/~history&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why don't most physics programs study the primary sources?&lt;/a&gt; on Physcis SE, which explicitly involves Newton's Principia. Still, one of the users wrote &quot;&lt;em&gt;I think it is possible to assign the Principia as an elementary physics book, if the translation is authentic and modernized. It requires rewriting the whole thing from scratch, but without throwing away any of Newton's insights. These insights are mostly contained in the special problems he solves&lt;/em&gt;&quot;. I have my doubts, but Bressoud uses (modernized) parts of Principia as a motivational springboard for teaching advanced calculus in his &lt;a href=&quot;https://books.google.com/books/about/Second_Year_Calculus.html?id=aeg48f5xk8EC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Second Year Calculus: From Celestial Mechanics to Special Relativity&lt;/a&gt;. In particular, he presents both Newton's and modern derivation of the inverse square law from Kepler's laws in the opening chapter.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-23T21:41:37.607" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3677" PostTypeId="2" ParentId="3674" CreationDate="2016-04-21T01:41:49.217" Score="0" Body="&lt;p&gt;Most definitely! Newton was hailed as a mathematical genius, especially in England. &lt;em&gt;Principia&lt;/em&gt; was the textbook on a variety of subjects, including physics and mathematics. I know for a fact that his fluxions (what Newton called his calculus) was exclusively used at Cambridge for a century after he died, which comes from his &lt;em&gt;Principia&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources:&#xA;&quot;Isaac Newton: Man, Myth, and Mathematics&quot; by V. Frederick Rickey; &#xA;&quot;The Newton Leibniz Controversy Concerning the Discovery of the Calculus&quot; by Dorothy Schrader&lt;/p&gt;&#xA;" OwnerUserId="4009" LastEditorUserId="26" LastEditDate="2016-04-21T12:22:54.200" LastActivityDate="2016-04-21T12:22:54.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3678" PostTypeId="2" ParentId="3675" CreationDate="2016-04-21T01:50:19.360" Score="4" Body="&lt;p&gt;It did not happen quite the way you imagined. The first accurate theories of solar and lunar motion in Europe are due to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hipparchus#Orbit_of_the_Sun&quot; rel=&quot;nofollow noreferrer&quot;&gt;father of astronomy Hipparchus&lt;/a&gt; (c. 190 – 120 BC), over millenium and a half before Columbus. They were geocentric and did not require Kepler's laws or any laws of dynamics, only geometry. But Hipparchus got his data from Babylonians, who could make accurate predictions of eclipses for centuries before that without even kinematics. What it required was careful observations and discerning subtle periodic patterns in the movements of the Sun and the Moon. Later Maya managed to do the same independently by a similar approach, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2783/how-accurate-are-mayan-astronomical-ephemeridestechniquese&quot;&gt;How accurate are Mayan astronomical &amp;quot;ephemerides&amp;quot;?&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-04-21T02:06:25.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3679" PostTypeId="1" CreationDate="2016-04-21T08:57:38.007" Score="7" ViewCount="1313" Body="&lt;p&gt;I searched about GMM (&lt;a href=&quot;https://en.wikipedia.org/wiki/Mixture_model#Gaussian_mixture_model&quot;&gt;Gaussian mixture model&lt;/a&gt;), but only found that normal distribution was invented by Carl Friedrich Gauss. I'd like to know who contributed to the development of GMM itself, and to using it for machine learning.&lt;/p&gt;&#xA;" OwnerUserId="4021" LastEditorUserId="55" LastEditDate="2016-04-21T20:52:44.057" LastActivityDate="2016-07-30T23:36:22.793" Title="Who developed Gaussian Mixture Model (GMM) and applied it to machine learning?" Tags="|mathematicians|computer-science|statistics|probability|machine-learning|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3680" PostTypeId="2" ParentId="3675" CreationDate="2016-04-21T12:05:24.050" Score="2" Body="&lt;p&gt;This question makes no sense as stated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) There is a huge difference between solar and lunar eclipses in this respect.&#xA;(Lunar eclipses occur everywhere, while Solar eclipses only on a small area on the Earth). For this reason, a solar eclipse is much harder to predict for a given location.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) The accuracy of prediction improved slowly with the development of astronomy.&#xA;So &quot;Predict accurately&quot; needs much further specification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c) Predict how much in advance?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;d) Most solar eclipses are incomplete: only a part of the Sun is obscured. This can be a very small part on a given location. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prediction of Lunar eclipses was possible already with Babylonian astronomy. Reliable prediction of solar eclipses for a given location few years in advance became possible only in the middle of 18th century when Lunar theory was fully developed. But there are reliably recorded cases when a lunar eclipse was predicted using ancient Babylonian methods. These cases happened in 18s and 19s century when predictions were made by Tamil (in India) astronomers using Babylonian methods, and recorded by French and English travelers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But one cannot predict a solar eclipse reliably (for a given location) with these methods for the reasons I explained above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Kepler laws have little to do with the problem. For the Moon, Kepler laws give only a crude approximation, comparable in accuracy to what Ptolemy already knew. The reason is that the system Earth-Moon-Sun is a three body problem, while Kepler's law is a solution of a two body problem.&#xA;It is good for planets. But gives a very rough approximation for the Moon.&#xA;Good theory of the Moon motion (sufficient for reliable solar eclipse prediction) was developed only in the middle of 18th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-23T01:53:10.413" LastActivityDate="2016-04-23T01:53:10.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3681" PostTypeId="1" AcceptedAnswerId="3682" CreationDate="2016-04-21T19:07:30.283" Score="16" ViewCount="503" Body="&lt;p&gt;What is the origin of calling a vector space over a field $F$ endowed with multiplication &lt;em&gt;an&lt;/em&gt; algebra? Tried searching, but not surprisingly Google likes to drop the article and just bring me to the etymology of algebra sans article. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="2448" LastEditorUserId="55" LastEditDate="2016-05-05T23:59:16.770" LastActivityDate="2016-11-11T20:18:41.130" Title="What is the origin of &quot;an algebra&quot; as in vector space with multiplication?" Tags="|mathematics|terminology|abstract-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3682" PostTypeId="2" ParentId="3681" CreationDate="2016-04-21T20:47:39.293" Score="13" Body="&lt;p&gt;Actually, it happened in the reverse order, algebras came first, and vector spaces only later. For the vector space story see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3214/when-did-people-start-viewing-a-matrix-as-a-linear-transformation-between-two-ve/3221#3221&quot;&gt;When did people start viewing a matrix as a linear transformation between two vector spaces?&lt;/a&gt; Peano gave the modern axiomatization of them only in 1888, and he called them linear systems. But the use of &quot;an algebra&quot; in essentially modern sense goes back to 1870, when Benjamin Peirce (the father of C.S. Peirce known for inventing the material conditional and giving Boolean algebra its modern form) introduced it in his &lt;a href=&quot;http://www.jstor.org/stable/2369153&quot; rel=&quot;nofollow noreferrer&quot;&gt;memoir Linear Associative Algebra&lt;/a&gt; read to the US National Academy (published in 1881). He explained it thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations. &lt;strong&gt;The symbols of an algebra&lt;/strong&gt;, with the laws of combination, constitute its language; the methods of using the symbols in the drawing of inferences is its art; and their interpretation is its scientific application. This three-fold analysis of algebra is adopted from President Hill, of Harvard University, and is made the basis of a division into books&lt;/em&gt;&quot; (boldface mine).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As the quote indicates the use of &quot;an algebra&quot; more broadly for a system of objects with &quot;addition&quot; and &quot;multiplication&quot; predates Peirce. &quot;Linear associative algebra&quot; specifically is what we today call an algebra over $\mathbb{C}$ in the narrow sense. An alternative term used at the time was &quot;hypercomplex number system&quot; and the field was a vibrant area of research since Hamilton's discovery of quaternions and Grassman's &quot;geometric algebra&quot;. Cayley, Sylvester, Frobenius, Veblen, Wedderburn, etc., were working in it in the second half of 19th early 20th century. The famous classification of division algebras over $\mathbb{R}$ (&lt;a href=&quot;https://en.wikipedia.org/wiki/Frobenius_theorem_%28real_division_algebras%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frobenius theorem&lt;/a&gt;) belongs to this period. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Various connections to geometry, already present in Hamilton and Grassman, were also explored at length, especially after Klein's promotion of Erlangen Program, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3230/what-was-the-motivation-for-minkowski-spacetime-before-special-relativity/3231#3231&quot;&gt;What was the motivation for Minkowski spacetime before special relativity?&lt;/a&gt; In addition to contemporary algebraic geometry the field was an important incubator of ideas and techniques that were later shaped into modern abstract algebra by Emmy Noether and van der Waerden in 1920s. That is when the definition was extended beyond $\mathbb{C}$ to algebras over arbitrary fields. For more on this later story see  &lt;a href=&quot;https://hsm.stackexchange.com/questions/2991/what-was-the-evolution-of-basis-and-generating-set-in-algebra/2999#2999&quot;&gt;What was the evolution of &amp;quot;basis&amp;quot; and &amp;quot;generating set&amp;quot; in algebra?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An authoritative historical work on the early days of linear associative algebras is &lt;a href=&quot;http://www.jstor.org/stable/41133752&quot; rel=&quot;nofollow noreferrer&quot;&gt;Parshall's Joseph H. M. Wedderburn and the Structure Theory of Algebras&lt;/a&gt;. A very insightful and nicely written (albeit less historically rigorous) account of 19th century work on hypercomplex numbers and Kleinian geometries is &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0817633162&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yaglom's Felix Klein and Sophus Lie&lt;/a&gt;. Its scope is much broader than its title, it explores the interplay of algebraic and geometric ideas also in works of Galois, Poncelet, Hamilton, Grassmann, Cayley, Peirce, Clifford, etc., and gives biographical details and extensive references to original and secondary sources.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-07-22T03:32:07.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3684" PostTypeId="1" CreationDate="2016-04-22T16:23:08.003" Score="3" ViewCount="390" Body="&lt;p&gt;There is this old adage that says something like: if you have not made your mark in Physics by the age of 30 then you are not going to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, I presume that other areas of study could be substituted for &quot;Physics&quot;.  But, when I was an undergrad Physics major in the latter half of the 1960s, this was often heard among students thinking about their future in Physics and wondering if they will &quot;make it by 30&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, where did this come from?  That is my question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In looking at the major players of the first part of the 20th century physics scene who are known for major breakthroughs we have:  Planck age 30 in 1900 when he published his work, Einstein age 26 in his miraculous year of 1905, Bohr was 28 when he published his atomic &quot;Bohr Model&quot; in 1913, de Broglie was 31 in 1923 when he published his work on particle-wave duality, Heisenberg was 24 when he published his &quot;Matrix Mechanics&quot; in 1925, Schrodinger was 39 during his 1926 publication of wave mechanics, Max Born was 45 when he published about &quot;Probability Waves&quot; in 1927 and Paul Dirac was 26 when he published his Electron Theory in 1928.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, a list of 8 well known physicists who definitely made a mark for themselves in Physics but only half of them were 30 or under in age.  And, two of them were in their 40s (old men!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am curious -- who says you need to be younger than 30.  I am hoping there is still hope for this 69-year old.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2016-05-22T20:35:18.963" Title="Making your mark by age 30?" Tags="|physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3685" PostTypeId="2" ParentId="3684" CreationDate="2016-04-22T17:29:23.350" Score="1" Body="&lt;p&gt;This opinion is quite common, especially in physics and mathematics, though everyone knows that there are exceptions. As a mathematician, I give examples from mathematics. G. H. Hardy, in his Mathematician's Apology wrote that mathematics is a sport for young people. I. M. Gelfand, when a 3-d year undergraduate approached him asking for a research topic, replied that s/he is&#xA;too old to begin research (Soviet 3-d year undergraduates usually were 20-21 years old). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples that you give show that usually the best achievement of top physicists are obtained before or around 30, and that there are exceptions.&#xA;In mathematics, the picture is the same.&#xA;Examples of mathematicians who wrote their best paper at much later age exist, of course. Roger Apery proved his most famous (by far) result (irrationality of&#xA;$\zeta(3)$) at the age of 63. One can argue that Antoni Zygmund made his most important contribution after 50, though he also did very important work before.&#xA;de Branges proved Bieberbach's conjecture at 52; this is his most famous result, but he had first rate results in his youth as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In these examples mathematicians made their top contributions at an advanced age, but they were established mathematicians. Examples when a mathematician&#xA;BEGAN his career late, and achieved top results, are much more rare, though they also exist. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my own experience I can tell that opinions of Hardy and Gelfand reflect general trend: most outstanding mathematicians obtain their best results between 30 and 40. But there are exceptions. Rare but very precious. For this reason I think that the attitude expressed in the legend about Gelfand is wrong. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-22T20:22:31.820" LastActivityDate="2016-04-22T20:22:31.820" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3689" PostTypeId="2" ParentId="2520" CreationDate="2016-04-23T22:20:55.833" Score="2" Body="&lt;p&gt;Apparently not.  On page 463 of Daniel Gottlieb's 1996 article &lt;a href=&quot;https://www.math.purdue.edu/~gottlieb/Bibliography/gauss-bonnet.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;All the Way with Gauss-Bonnet and the Sociology of Mathematics&lt;/em&gt;&lt;/a&gt; (The American Mathematical Monthly, Vol. 103, No. 6), he writes (according to Morris Hirsch) that Walter Dyck in 1888 was the first to connect the degree of the Gauss map with the Euler-Poincaré characteristic.  A few lines later in the same article, Gottlieb writes that Hans Samelson was unable to find a statement of the global Gauss-Bonnet theorem in Gauss' works.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2016-04-23T22:20:55.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3690" PostTypeId="2" ParentId="2362" CreationDate="2016-04-24T18:10:01.360" Score="2" Body="&lt;p&gt;Newton wanted to lay out his theory of gravity so that people would accept it. But his new methods of calculus were not yet widely accepted or known, and would therefore cast doubt on his physics. For that reason, Newton used classical geometry, the mathematical language of the ancient masters, to avoid having people attack his ideas based on his use of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton was well aware of the logical difficulties of his calculus, and struggled without success over the course of his career to explain the &quot;ultimate ratio&quot; of the difference quotient. He did not want to expose his theory of gravity to attacks based on those issues. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's some more background &lt;a href=&quot;https://blogs.otago.ac.nz/emxphi/2011/10/newton-and-the-case-of-the-missing-calculus/&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, supporting the idea that Newton understood the logical issues with his calculus and wanted to base his physics on math that everyone would believe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other reason was to deliberately make his work more difficult. As Newton said: &lt;em&gt;To avoid being baited by little smatterers in mathematics, I designedly made the Principia abstract; but yet as to be understood by able mathematicians ...&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We note in passing that were Newton to come back today and be introduced to the Internet, he'd be right at home. He suffered no fools, gladly or otherwise, and could flame with the best of them. &lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="340" LastEditDate="2016-04-24T18:18:07.573" LastActivityDate="2016-04-24T18:18:07.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3691" PostTypeId="1" AcceptedAnswerId="3698" CreationDate="2016-04-26T20:02:42.223" Score="1" ViewCount="253" Body="&lt;p&gt;I've seen and read that calculations before Feynman diagrams involved solving differential equations, but I can't find sample calculations like these. I've read that they would take months and pages to do, so I'm curious what the equations were that were being solved. Does anyone know an old textbook or papers where these are shown?&lt;/p&gt;&#xA;" OwnerUserId="3219" LastEditorUserId="4251" LastEditDate="2020-06-06T00:39:57.090" LastActivityDate="2020-06-06T00:39:57.090" Title="How were particle physics calculations done before Feynman diagrams?" Tags="|physics|theoretical-physics|particle-physics|quantum-field-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3692" PostTypeId="1" AcceptedAnswerId="3699" CreationDate="2016-04-27T08:38:18.470" Score="15" ViewCount="17991" Body="&lt;p&gt;In a lot of places I find this quote from Feynman:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I would rather have ques­tions that can’t be an­swered than an­swers that can’t be ques­tioned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, I cannot find any source of where it appeared written by him or reported by someone who heard him saying it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this quote authentic? Can you give a reference or debunk it? If not Feynman, who said it?&lt;/p&gt;&#xA;" OwnerUserId="4046" LastEditorUserId="4251" LastEditDate="2020-07-06T07:40:53.020" LastActivityDate="2020-07-06T07:40:53.020" Title="Is the &quot;ques­tions that can’t be an­swered over an­swers that can’t be ques­tioned&quot; quote by Feynman authentic?" Tags="|debunking|quote|feynman|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3693" PostTypeId="1" CreationDate="2016-04-27T11:29:26.033" Score="7" ViewCount="2757" Body="&lt;p&gt;Are there any English language (or perhaps German, Dutch or French, but preferably English) translations of &lt;a href=&quot;https://en.wikipedia.org/wiki/Luca_Pacioli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luca Pacioli&lt;/a&gt;'s 1494 (or later edition) &amp;quot;&lt;a href=&quot;http://www.e-rara.ch/zut/content/titleinfo/2683230&quot; rel=&quot;nofollow noreferrer&quot;&gt;Summa de arithmetica&lt;/a&gt;&amp;quot;? They do not have to be in print or of recent times, but just anyone in existence.&lt;/p&gt;&#xA;&lt;p&gt;Beware that there are a lot of translations of only the double-entry bookkeeping part of the &lt;em&gt;Summa&lt;/em&gt; which do not include the rest of the 600-page work. I can only guess why this is.&lt;/p&gt;&#xA;" OwnerUserId="4048" LastEditorUserId="4251" LastEditDate="2021-09-07T21:05:08.717" LastActivityDate="2021-09-07T21:05:08.717" Title="Translations of Luca Pacioli's Summa de arithmetica" Tags="|mathematics|reference-request|translations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3695" PostTypeId="2" ParentId="3693" CreationDate="2016-04-27T20:04:20.277" Score="3" Body="&lt;p&gt;If you read French, you will probably have no big difficulty with Latin.&#xA;Here is a recent edition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summa de arithmetica, geometria, proportioni et proportionalità, hg. Enrico Giusti, Abrizzi, Venedig 1994, ISBN 88-317-6008-4 (Nachdruck der Erstausgabe Paganinis, Venedig 1494).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a Russian translation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Пачоли Л. Трактат о счетах и записях / Под ред. Я. В. Соколова. — М.: Финансы и статистика, 1994.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another book of Pacioli, English, Italian and Russian translations seem to exist: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Engl. Divine Proportion, Norwalk, CT 2001; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ital. Übers. Andrea Masini, in: Arnaldo Bruschi u.a. (Hgg.): Scritti rinascimentali di architettura, Mailand 1978, S. 23-244.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rus. Лука Пачоли. О божественной пропорции. Репринт изд. 1508. С приложением перевода А. И. Щетникова. М.: Фонд «Русский авангард», 2007.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-04-27T20:10:20.473" LastActivityDate="2016-04-27T20:10:20.473" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3696" PostTypeId="2" ParentId="3692" CreationDate="2016-04-28T11:34:16.230" Score="11" Body="&lt;p&gt;This is all I have found for now:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“You see, one thing is, I can live with doubt, and uncertainty, and not knowing. I think it’s much more interesting to live, not knowing, than to have answers which might be wrong.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=I1tKEvN3DF0&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=I1tKEvN3DF0&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as discussed here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiquote.org/wiki/Talk:Richard_Feynman#.22I_would_rather_have_questions_that_cannot_be_answered.22&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikiquote.org/wiki/Talk:Richard_Feynman#.22I_would_rather_have_questions_that_cannot_be_answered.22&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4058" LastActivityDate="2016-04-28T11:34:16.230" CommentCount="0" CommunityOwnedDate="2016-04-28T11:34:16.230" ContentLicense="CC BY-SA 3.0" />
  <row Id="3697" PostTypeId="2" ParentId="2472" CreationDate="2016-04-28T13:48:49.243" Score="1" Body="&lt;p&gt;As to the first part of your question, conservation of momentum comes directly from Newton's Laws, so technically Newton himself came up with conservation of momentum, and others followed Newton's logic to notice that in certain scenarios that other conservation conditions exist (energy, angular momentum, mass, etc).  &lt;/p&gt;&#xA;" OwnerUserId="4060" LastActivityDate="2016-04-28T13:48:49.243" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3698" PostTypeId="2" ParentId="3691" CreationDate="2016-04-28T17:38:09.620" Score="1" Body="&lt;p&gt;Original renormalization papers that came out in 1947-48 by Bethe, Lewis, Schwinger and Tomonaga did not use Feynman diagrams, not even Feynman's own 1948 paper did. Here is &lt;a href=&quot;http://www.jstor.org/stable/20117830?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;Cao-Schweber's&lt;/a&gt; description and comparison of their approaches:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Neither Lewis nor Schwinger nor Tomonaga made explicit use of a cutoff. They directly identified the divergent terms with corrections to the mass and the charge, and removed them from the expressions for real processes by redefining the masses and charges. By contrast, Feynman's efficient calculational algorithm...  is based on the explicit use of a relativistic cutoff. The latter consists of a set of rules for regularization, which makes it possible to calculate physical quantities in a relativistically and gauge invariant manner, but still results in divergent expressions in the limit as the cutoff mass goes to infinity. With a finite cutoff, this artifice transforms essentially purely formal manipulations of divergent quantities, i.e., the redefinition of parameters, into quasi respectable mathematical operations. If, after the redefinition of mass and charge, other processes are insensitive to the value of the cutoff, then a renormalized theory can be defined by letting the cutoff go to infinity.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some of these papers are available online, Bethe's is pretty short.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bethe, H. A.: 1947, &lt;a href=&quot;http://web.ihep.su/dbserv/compas/src/bethe47/eng.PDF&quot; rel=&quot;nofollow&quot;&gt;The Electromagnetic Shift of Energy Levels&lt;/a&gt;, Physical Review 72, 339-341.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman, R. P.: 1948, &lt;a href=&quot;http://authors.library.caltech.edu/3517/1/FEYpr48b.pdf&quot; rel=&quot;nofollow&quot;&gt;Relativistic Cut-off for Quantum Electrodynamics&lt;/a&gt;, Physical Review 74, 1430-38.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schwinger, J.: 1948 &lt;a href=&quot;https://www.researchgate.net/publication/243692776_Quantum_Electrodynamics_I_A_Covariant_Formulation&quot; rel=&quot;nofollow&quot;&gt;Quantum Electrodynamics. I. A Covariant Formulation&lt;/a&gt;', Physical Review 74, 1439-61.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-04-28T17:38:09.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3699" PostTypeId="2" ParentId="3692" CreationDate="2016-04-28T21:46:12.550" Score="7" Body="&lt;p&gt;I have my doubts, although Feynman, like many others, expressed similar sentiments. &lt;a href=&quot;http://bookcritics.org/blog/archive/daniel_c._dennetts_breaking_the_spell1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dennett in the first chapter of Breaking the Spell (2006)&lt;/a&gt; writes:&quot;&lt;em&gt;Philosophy is questions that may never be answered. Religion is answers that may never be questioned&lt;/em&gt;&quot;. He attributes the quote to Anonymous, and does not mention Feynman. It appears that the quote's author, Feynman or not, prefers philosophy to religion in this allusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what &lt;a href=&quot;http://scitation.aip.org/content/aip/magazine/physicstoday/article/57/5/10.1063/1.1768652&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mermin writes&lt;/a&gt; about another &quot;Feynman's&quot; famous quote:&quot;&lt;em&gt;I noticed that not a single one of the Web sites attributing the phrase to Feynman cited a source or hinted at the circumstances under which he had said it... I suspect that it is only Feynman’s habitual irreverence that has linked him in the minds of many to the phrase “shut up and calculate”... Merton has taught us that it is only among the high and mighty that people tend to look... Broaden the search to embrace the low and powerless&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As in Mermin's case, not a single place sporting this quote cites anyone, let alone Feynman, or even tells an anecdote as to how it was said. They do not date too far back in time either.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-19T20:22:39.820" LastActivityDate="2019-06-19T20:22:39.820" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3700" PostTypeId="1" AcceptedAnswerId="6129" CreationDate="2016-04-29T05:33:42.277" Score="2" ViewCount="283" Body="&lt;p&gt;I was watching this lecture series by Dr. Richard Hamming on &quot;Learning to Learn&quot; and in &lt;a href=&quot;https://www.youtube.com/watch?v=2wO8ZHnn3Q4&amp;amp;index=14&amp;amp;list=PL2FF649D0C4407B30&quot; rel=&quot;nofollow&quot;&gt;this link&lt;/a&gt;, he describes &lt;a href=&quot;https://en.wikipedia.org/wiki/Harry_Nyquist&quot; rel=&quot;nofollow&quot;&gt;Harry Nyquist&lt;/a&gt; as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;a great man who could do two things, you dont meet very often. He was a very nice guy, a very smart guy, but...he had his goofy ideas too!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, I am interested in knowing what were Nyquist's &quot;goofy&quot; ideas? What was he trying to solve with those ideas and How were they disproved or why were they disregarded? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you&lt;/p&gt;&#xA;" OwnerUserId="3034" LastActivityDate="2017-06-07T18:25:49.603" Title="&quot;Goofy ideas&quot; of Harry Nyquist?" Tags="|biographical-details|quote|engineering|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3701" PostTypeId="2" ParentId="2472" CreationDate="2016-04-29T10:11:56.057" Score="2" Body="&lt;p&gt;Descartes wrote in *Le traité du monde et de la lumière *:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;…when one of these bodies pushes another, it cannot give the 2nd any motion, except by losing as much of its own motion at the same time…  (AT, XI, 41)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Motion being &quot;le transport d'une partie de la matière, ou d'un corps&quot; this is probably  the first enonciation of the conservation law for linear momentum. The book was published in 1664, some 30 years after it was written, and meanwhile the &lt;em&gt;Principles of philosophy&lt;/em&gt; published in 1644 stated:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lorsqu'une partie de la matière se meut deux fois plus vite qu'une autre, et que cette autre est deux fois plus grande que la première, nous devons penser qu'il y a tout autant de mouvement dans la plus petite que dans la plus grande ; et que toutes fois et quantes que le mouvement d'une partie diminue, celui de quelque autre partie augmente à proportion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Introducing a more refined &quot;quantity of movement&quot; the book however stated some erroneous propositions.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2016-04-29T10:11:56.057" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3702" PostTypeId="1" AcceptedAnswerId="3705" CreationDate="2016-04-29T12:05:20.067" Score="10" ViewCount="733" Body="&lt;p&gt;In any layman level description of &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_fluctuation#Quantum_fluctuations_of_a_field&quot;&gt;vacuum fluctuations in quantum field theory&lt;/a&gt; the fluctuations are described as a pair of virtual particles spontaneously appearing then disappearing within some short time determined by the particle energies and the uncertainty principle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However this appears to be unrelated to vacuum fluctuation as understood by all the quantum field theorists I know. They regard fluctuations in the vacuum expectation value of a field as simply the phenomenon that although the VEV may be zero the variance is not. The popular metaphor of the vacuum as a boiling sea of shortlived virtual particles appears nowhere in this description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is where did the &lt;em&gt;virtual particle pair&lt;/em&gt; metaphor come from? Does it derive from a treatment of the QFT vacuum that has now fallen out of favour, is it an analogy introduced by a particular author, or is it just a baseless notion arising from the fevered mind of some popular science writer?&lt;/p&gt;&#xA;" OwnerUserId="2082" LastEditorUserId="55" LastEditDate="2016-05-05T23:51:38.653" LastActivityDate="2016-05-05T23:51:38.653" Title="What is the origin of the &quot;virtual particle pair&quot; metaphor for vacuum fluctuations?" Tags="|terminology|theoretical-physics|quantum-mechanics|quantum-field-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3704" PostTypeId="1" CreationDate="2016-04-30T21:34:27.423" Score="0" ViewCount="160" Body="&lt;p&gt;Do we know if he knew of the previous works done on this subject? Do we have any idea of his thought process? Do we even know anything about this?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="2442" LastEditDate="2017-06-08T18:41:17.437" LastActivityDate="2017-06-08T18:41:17.437" Title="Do we have any idea on how Newton developed his method of fluxions?" Tags="|mathematics|calculus|newton|" AnswerCount="1" CommentCount="2" ClosedDate="2016-05-02T19:14:09.223" ContentLicense="CC BY-SA 3.0" />
  <row Id="3705" PostTypeId="2" ParentId="3702" CreationDate="2016-04-30T22:02:25.363" Score="2" Body="&lt;p&gt;The question is closely tied to the existence of &quot;virtual particles&quot;, elusive intermediaries in particle collisions whose &quot;paths&quot; appear as wavy lines on Feynman diagrams. The current majority view of them seems to be as expressed e.g. on &lt;a href=&quot;https://profmattstrassler.com/articles-and-posts/particle-physics-basics/virtual-particles-what-are-they&quot; rel=&quot;nofollow noreferrer&quot;&gt;Strassler's blog&lt;/a&gt;, &quot;&lt;em&gt;the Feynman diagram is actually a calculational tool, not a picture of the physical phenomenon&lt;/em&gt;&quot;, see also &lt;a href=&quot;https://physics.stackexchange.com/questions/221513/why-do-we-need-virtual-particles/221524#221524&quot;&gt;Do we need virtual particles?&lt;/a&gt; on Physics SE. This was not always the case, in his first diagram paper Feynman clearly reifies &quot;&lt;em&gt;the effect of exchange of one [virtual] quantum between two electrons&lt;/em&gt;&quot; (see his drawing reproduced and referenced under &lt;a href=&quot;https://hsm.stackexchange.com/questions/3671/what-was-the-first-journal-to-have-feynman-diagrams/3673#3673&quot;&gt;What was the first journal to have Feynman Diagrams?&lt;/a&gt;). It is a small step from that to interpreting diagrams with two wavy edges connecting two vertices as &quot;virtual pair production in a vacuum&quot; (since outgoing edges representing observable particles are missing).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, they idea predates Feynman. &quot;Virtual oscillations&quot; appeared in the &lt;a href=&quot;http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/BKS%20%281924%29.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1924 Bohr-Kramers-Slater paper&lt;/a&gt; even before the modern quantum mechanics emerged, and apparently inspired Heisenberg's uncertainty principle. In a 1933 letter to Bohr Dirac predicted the vacuum polarization within his negative sea with holes (positrons) theory, a precursor of QED. As &lt;a href=&quot;http://www.scholarpedia.org/article/User:Riccardo_Guida/Proposed/Quantum_field_theory:_origins&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scholarpedia's article&lt;/a&gt; (sponsored by Schweber, a leading expert on QFT history) writes:&quot;&lt;em&gt;In his letter Dirac had succinctly stated the physics entailed by vacuum polarization. The physical picture he sketched... was elaborated by Uehling (1935) and by Weisskopf (1936). In quantum electrodynamics the vacuum is not an empty medium but zero point fluctuations produce virtual electron-positron pairs. These are responsible for making the vacuum a dielectric medium that can be characterized by a dielectric constant, $\epsilon$. This dielectric constant is distance dependent: When a bare charge $e(0)$ is introduced into the vacuum the charge that is observed at a distance R is given by $e(0)/\epsilon(R)$, that is, the observed charge is reduced by an amount $\epsilon(R)$. Uehling determined that the larger R the more screening occurred...&lt;/em&gt;&quot; This picture of &quot;dressing up&quot; a bare charge was among key insights leading to the discovery of renormalization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the virtual pair production, this time of photons, reappeared in connection with the Hawking radiation from black holes. Although in his &lt;a href=&quot;http://www.nature.com/nature/journal/v248/n5443/abs/248030a0.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;original 1974 paper&lt;/a&gt; Hawking does not mention them explicitly, its 1976 follow up &lt;a href=&quot;http://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.2460&quot; rel=&quot;nofollow noreferrer&quot;&gt;Breakdown of Predictability in Gravitational Collapse&lt;/a&gt; gives the metaphorical picture reproduced in many popular accounts:&quot;&lt;em&gt;It is shown that the ignorance principle holds for the quantum-mechanical evaporation of black holes: The black hole creates particles in pairs, with one particle always falling into the hole and the other possibly escaping to infinity&lt;/em&gt;&quot;. The idea remains current today despite the well known ways to derive Hawking's result without using virtual pairs. For example, Parentani's paper titled &lt;a href=&quot;http://arxiv.org/abs/1003.3625&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Vacuum Fluctuations across an Event Horizon to Long Distance Correlations&lt;/a&gt; was &lt;a href=&quot;http://journals.aps.org/prd/abstract/10.1103/PhysRevD.82.025008&quot; rel=&quot;nofollow noreferrer&quot;&gt;published by Physical Review&lt;/a&gt;, a leading journal, in 2010.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me draw a historical analogy. There is no doubt that early development of electrodynamics was driven by vivid pictures involving ether. Young designed his double slit experiment to mimic waves in a water &quot;ripple tank&quot; he built (see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2629/how-did-young-perform-his-double-slit-experiment/2634#2634&quot;&gt;How did Young perform his double slit experiment?&lt;/a&gt;), Faraday viewed his &quot;lines of force&quot; as real entities, Maxwell set out to &quot;&lt;em&gt;explain electromagnetic phenomena by means mechanical action transmitted from one body to another by means of a medium&lt;/em&gt;&quot;, and Hertz believed that he was capturing ether vibrations on his antennas. Then came the Michelson-Morley experiment that failed to produce the expected ether wind. In response, Fitzgerald, Larmor and Lorentz came up with ad hoc contrivances of length contraction and time dilation, which turned ether from almost seen and touched into a forever elusive ghost. The ease with which Einstein dispatched it was largely due to the fact that by his time ether ceased to be the useful heuristic that it was in the 19th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar dynamic may be at work with virtual particles (and symbolically the not quite empty QED vacuum is the apparent successor to the luminiferous ether), but if the Hawking radiation is their Michelson-Morley experiment then its outcome went the other way. However, unlike in the Rutherford experiments for atoms, or Friedman-Kendall-Taylor's for quarks (see &lt;a href=&quot;https://physics.stackexchange.com/questions/201990/can-quarks-be-considered-elementary/201999#201999&quot;&gt;Can quarks be considered real and elementary?&lt;/a&gt;) event horizons of black holes do not exactly make for controlled laboratory conditions. Virtual particles are unlikely to be accepted unless more direct and less extravagant ways of &quot;actualizing&quot; them are found, and the jury may be out on that yet.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-05-02T19:31:12.217" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3706" PostTypeId="2" ParentId="3704" CreationDate="2016-04-30T23:23:34.297" Score="1" Body="&lt;p&gt;Here's a short answer $-$ others may be able to elaborate in more detail.  On page 98 of George Simmons' book &lt;em&gt;Calculus Gems: Brief Lives and Memorable Mathematics&lt;/em&gt; (published by the Mathematical Association of America), he summarizes the matter by stating that Newton wrote in a letter that his own early ideas about calculus came from &quot;Fermat's way of drawing tangents.&quot;  A more direct quote from the same letter can be found in the page 144 footnote of Sabra's &lt;em&gt;Theories of Light: From Descartes to Newton&lt;/em&gt;, which reads &quot;&lt;em&gt;I had the hint of this method [of fluxions] from Fermat's way of drawing tangents, and by applying it to abstract equations, directly and invertedly, I made it general.&lt;/em&gt;&quot;  &lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2016-04-30T23:23:34.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3707" PostTypeId="1" CreationDate="2016-05-01T03:17:59.713" Score="4" ViewCount="593" Body="&lt;p&gt;Historically, Special Relativity was motivated by apparent inconsistencies between Maxwell's Electrodynamics and Newtonian Mechanics. In Einstein's well known paper &quot;On the electrodynamics of moving bodies&quot; he explains quite well his motivations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Central objects of the theory are the Lorentz transformations. If one forgets motivations, history and intuition the Lorentz transformations are formally &lt;strong&gt;defined&lt;/strong&gt; as the linear transformations &#xA;$\Lambda:\mathbb{R}^4\to \mathbb{R}^4$ such that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\eta(\Lambda v,\Lambda w)=\eta(v,w),$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $\eta = \operatorname{diag}(-1,1,1,1)$. Furthermost, it seems that before this definition they were defined as the transformations which keep the speed of light the same in all frames.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: how did Lorentz transformations get this modern definition?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How were they first defined, how did they relate to Einstein's paper, and how did they get the modern definition as &quot;transformations which preserve the spacetime inner product&quot;? Specifically I'm interested in knowing how from the motivations for relativity physicists got to the definition of Lorentz transformations as the transformations $\Lambda$ such that $\eta(\Lambda x,\Lambda y) = \eta(x,y)$&lt;/p&gt;&#xA;" OwnerUserId="4072" LastEditorUserId="12699" LastEditDate="2017-06-05T14:44:19.007" LastActivityDate="2024-03-21T16:57:00.087" Title="How did Lorentz transformations get their modern definition?" Tags="|physics|theoretical-physics|relativity-theory|electromagnetism|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3709" PostTypeId="1" CreationDate="2016-05-01T06:08:30.623" Score="-1" ViewCount="177" Body="&lt;p&gt;I was curious to know if there are any resources similar to &lt;a href=&quot;http://www.earlymoderntexts.com/texts&quot; rel=&quot;nofollow noreferrer&quot;&gt;Early Modern Texts&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What the author of the site has done is taken older and also famous philosophy papers and translated them with more modern terminology. Not necessarily watering down the topic but making it more accessible to contemporary readers. Anyways, I was wondering if there are any such texts like that about mathematics and science.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, is there any book or such that collects many math papers in a single volume?&lt;/p&gt;&#xA;" OwnerUserId="2462" LastEditorUserId="10106" LastEditDate="2020-06-07T18:35:37.320" LastActivityDate="2020-06-07T18:35:37.320" Title="Any modern translations of famous math papers?" Tags="|mathematicians|reference-request|translations|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3710" PostTypeId="2" ParentId="3707" CreationDate="2016-05-01T11:05:07.953" Score="2" Body="&lt;p&gt;Transformations must preserve the structure of Maxwell's equations,  which then automatically preserves the speed of light. &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Woldemar_Voigt&quot; rel=&quot;nofollow&quot;&gt;Voigt, 1887, was the first&lt;/a&gt;, but there were several independent derivations. Note that Voight's transformations are not quite the same as those of Lorentz. The latter are consistent with the Principle of Relativity. &lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-05-01T11:05:07.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3711" PostTypeId="2" ParentId="3670" CreationDate="2016-05-01T22:26:09.870" Score="1" Body="&lt;p&gt;Zinc is principally known anciently from the production of brass which was widespread in the ancient world. The way that brass is made is that copper is combined with calamine (zinc oxide, a common mineral) and then reduced in situ with charcoal. Most ancient civilizations that had well-developed copper metallurgy readily discovered how to make brass as well, which involves the utilization of zinc. For example, among the ancient Greeks bronze (copper and tin) was known as &quot;kalchon&quot; and brass was known as &quot;orikalchon&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In India, there has always been a strong knowledge of copper and its uses, which of course involves the production of brass. However, to say that zinc was first used in India or was exclusively discovered in India is probably incorrect.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2016-05-01T22:26:09.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3712" PostTypeId="2" ParentId="3679" CreationDate="2016-05-01T22:35:52.383" Score="3" Body="&lt;p&gt;The idea of using Gaussian mixtures was popularized by Duda and Hart in their seminal 1973 text, &lt;em&gt;Pattern Classification and Scene Analysis&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2016-05-01T22:35:52.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3713" PostTypeId="2" ParentId="368" CreationDate="2016-05-01T22:54:10.027" Score="1" Body="&lt;p&gt;First of all, I would recommend reading the books of Robert Newton of Johns Hopkins University, especially his 1970 book. Newton understood this subject in far greater detail than practically anyone and his books have extremely clear and detailed discussions of the issues involved. Other books on the same subject are almost childish in comparison with Newton's books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In ancient times, the Greeks attributed to Hipparchus the first ability to predict eclipses, but it is likely that the Babylonians and Chaldeans had the capability as well and Newton's research suggests it is likely that Hipparchus was simply continuing a tradition that had been established by the Babylonians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In principle it is easy to predict an eclipse by projecting forwards from previously known eclipses. The farther ahead (or backward) the projection is, however, the more inaccurate it becomes because the motion of the moon is irregular in various ways. Beyond a distance of 150 years or so it becomes increasingly difficult to retroactively predict an eclipse accurately unless contemporary observations are known. And, of course, since we do not know what will happen in the future, predicting an eclipse even using modern methods beyond a few hundred years will have a high probability of error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the ancient astronomers the ability to predict eclipses 50 years in advance or less, based on contemporary observations was easily available.&lt;/p&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2016-05-01T22:54:10.027" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3714" PostTypeId="2" ParentId="3709" CreationDate="2016-05-02T16:55:41.897" Score="1" Body="&lt;p&gt;Off-hand, I know of three such books. You can probably find more such books by googling their titles and looking for any mention of other books that are like them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Birkhoff's &lt;a href=&quot;https://www.amazon.ca/dp/toc/0674822455&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;A Source Book in Classical Analysis&lt;/strong&gt;&lt;/a&gt; (1973)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Midonick's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/B000B09U50&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;The Treasury of Mathematics&lt;/strong&gt;&lt;/a&gt; (1965)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486646904&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;A Source Book in Mathematics&lt;/strong&gt;&lt;/a&gt; (1929)&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2016-05-02T16:55:41.897" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3718" PostTypeId="2" ParentId="3707" CreationDate="2016-05-03T20:26:04.453" Score="4" Body="&lt;p&gt;Wikipedia has a very adequate and well-sourced article on &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_Lorentz_transformations&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Lorentz transformations&lt;/a&gt;. Voigt formulated the not quite modern ones back in 1887, of which Lorentz was unaware, and had to work out his own version independently. This might have been just as well since he later said he would have used Voigt's if he knew about them, but he presented them partially (without the time dilation) in 1895, the first complete version is due to Larmor (1897). Lorentz was apparently unaware of that either, and supplied his own full version in 1899, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3594/what-made-einstein-believe-or-know-that-time-was-affected-by-speed-and-gravity/3600#3600&quot;&gt;What made Einstein believe (or know) that time was affected by speed and gravity?&lt;/a&gt;. None of them viewed the transformations algebraically or kinematically, they were seen as describing dynamic effects on bodies moving at high speeds. Larmor even supplemented a hypothesis that molecular forces are of electromagnetic nature, which would explain the effects. But as Poincaré showed in 1905 purely electromagnetic forces could not account for the stability of the electron. He had to conjecture an additional stabilizing non-electromagnetic force that nonetheless obeyed the same transformation laws, which made it ad hoc.&lt;/p&gt;&#xA;&lt;p&gt;The first algebraic observation, that the transformations form a group, was made by Poincaré in his 1904-1906 papers on the dynamics of the electron, but according to Weinstein &amp;quot;&lt;em&gt;Poincaré did not associate this quadratic form with propagation of light in order to define a null interval like Einstein or a metric like Minkowski&lt;/em&gt;&amp;quot;. This is particularly surprising because groups involved in Kleinian geometries are usually obtained by considering all transformations that preserve a quadratic form, as Poincaré well knew. As alluded to by Weinstein, it was Einstein in 1905 who first characterized them kinematically, as the transformations that preserve the speed of light in all frames (i.e. preserve the null interval), and only Minkowski, inspired by Einstein's paper, gave the modern geometric formulation of them as the transformations that preserve a (pseudo) metric in 1907-1909, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3230/what-was-the-motivation-for-minkowski-spacetime-before-special-relativity/3231#3231&quot;&gt;What was the motivation for Minkowski spacetime before special relativity?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10608" LastEditDate="2024-03-21T16:57:00.087" LastActivityDate="2024-03-21T16:57:00.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3719" PostTypeId="1" AcceptedAnswerId="3721" CreationDate="2016-05-03T21:44:08.237" Score="6" ViewCount="147" Body="&lt;p&gt;The second part of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_sixteenth_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert's 16th problem&lt;/a&gt; (determination of the upper bound on the number of limit cycles for two-dimensional polynomial vector fields of given degree), proposed in 1900, is still open and a bit mysterious. Its varìous aspects have been frequently revisited, &amp;quot;proven&amp;quot; and disproven! In particular, Petrovskiĭ and Landis claimed that they solved this problem in 1950s, but their &amp;quot;proof&amp;quot; turned out to have a gap. There is a &lt;a href=&quot;https://mathoverflow.net/questions/171988/the-error-in-petrovski-and-landis-proof-of-the-16th-hilbert-problem&quot;&gt;related discussion on Mathematics Overflow&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Here is my question&lt;/strong&gt;: I&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Does  there exist a paper which explicitly describe the essential mistake in that famous paper of Petrovski and Landis?&#xA;Am I correct to think that there is no universal agreement as to existence of such a paper?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4085" LastEditorUserId="4085" LastEditDate="2021-06-10T13:18:53.403" LastActivityDate="2021-06-10T13:18:53.403" Title="Is there an explicit description of Landis-Petrovskiĭ's mistake in their &quot;solution&quot; to Hilbert's 16th problem?" Tags="|mathematicians|philosophy-of-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3721" PostTypeId="2" ParentId="3719" CreationDate="2016-05-03T22:35:27.237" Score="4" Body="&lt;p&gt;Noah Schweber gives an excellent account of the context on MO, so I will only add the missing piece concerning the error Ilyashenko and Novikov found in the original Landis-Petrovskiĭ's argument. They never published it, but Landis and Petrovskiĭ wrote a &lt;a href=&quot;http://iopscience.iop.org/article/10.1070/SM1967v002n01ABEH002332/meta;jsessionid=4A8CE72595F5362E12226E0704CA049C.c4.iopscience.cld.iop.org&quot; rel=&quot;nofollow&quot;&gt;letter to the editors (1967)&lt;/a&gt; of Mathematics of the USSR-Sbornik, which published their &lt;a href=&quot;http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;amp;jrnid=sm&amp;amp;paperid=5216&amp;amp;option_lang=rus&quot; rel=&quot;nofollow&quot;&gt;original paper in 1955&lt;/a&gt;. It reads as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In our paper On the number of Limit Cycles of the Equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are Polynomials of the Second Degree in the proof of lemma 12 (p.242) there is an error pointed out to us by S.P. Novikov. A reference to this lemma is made in our&lt;/em&gt; [follow up] &lt;em&gt;paper On the number of Limit Cycles of the Equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are Polynomials. Therefore, one can only claim that the results of those works hold under the additional assumption of that lemma's validity&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, both original papers are from before 1967, when AMS started making English translations of the Sbornik, but they link to five (!) papers by Ilyashenko starting in 1969 that cite them, meaning their translations are available. At least to those who have the IOP Science subscription... unlike myself. Three of them are from 1969, one is titled &lt;a href=&quot;http://iopscience.iop.org/article/10.1070/SM1969v009n03ABEH001288/meta&quot; rel=&quot;nofollow&quot;&gt;An Example of Equations $dw/dz=P_n(z,w)Q_n(z,w)$ having a Countable Number of limit Cycles and Arbitrarily Large Petrovskii–Landis Genus&lt;/a&gt;, so they may have more details on the error. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-05-03T23:01:28.453" LastActivityDate="2016-05-03T23:01:28.453" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3722" PostTypeId="2" ParentId="3709" CreationDate="2016-05-04T01:06:56.453" Score="1" Body="&lt;p&gt;There is one book where you can find many remarkable papers by famous mathematicians and their biography too. It is complied by Stephen Hawking and called &lt;strong&gt;God created the integers&lt;/strong&gt;. It contains some of the landmark papers in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2016-05-04T01:06:56.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3723" PostTypeId="1" AcceptedAnswerId="3724" CreationDate="2016-05-04T01:54:24.760" Score="18" ViewCount="985" Body="&lt;p&gt;I found this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are good reasons to believe that nonstandard analysis, in some version or other, will be the analysis of the future.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What exactly did Kurt Gödel mean?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="4251" LastEditDate="2022-09-23T06:04:18.320" LastActivityDate="2024-01-12T14:08:11.673" Title="What did Kurt Gödel mean by nonstandard analysis being &quot;the analysis of the future&quot;?" Tags="|mathematics|real-analysis|quote|goedel|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3724" PostTypeId="2" ParentId="3723" CreationDate="2016-05-04T04:47:41.503" Score="14" Body="&lt;p&gt;The quote is from the remark Gödel made after Robinson's talk at the Institute for Advanced Study in Princeton in March 1973. It is reproduced in the preface to the second edition of &lt;a href=&quot;https://books.google.com/books?id=OkONWa4ToH4C&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Robinson’s Non-Standard Analysis (1974)&lt;/a&gt;. Here is the full text of the remark (boldface mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I would like to point out a fact that was not explicitly mentioned by Professor Robinson, but seems quite important to me; namely that non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be further from the truth. &lt;strong&gt;Rather there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future&lt;/strong&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;One reason is the just mentioned simplification of proofs, since simplifications facilitates discovery. Another, even more convincing reason, is the following: Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. I am inclined to believe that this oddity has something to do with another oddity relating to the same span of time, namely the fact that such problems as Fermat’s, which can be written down in ten symbols of elementary arithmetic, are still unsolved 300 years after they have been posed. Perhaps the omission mentioned is largely responsible for the fact that, compared to the enormous development of abstract mathematics, the solution of concrete numerical problems was left far behind.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In a short note &lt;a href=&quot;http://www.ux1.eiu.edu/%7Eyigordon/papers/about%20NSA.pdf&quot; rel=&quot;noreferrer&quot;&gt;About Perspectives of Nonstandard Analysis&lt;/a&gt; Gordon quotes Zeilberger suggesting a particular way of how Gödel's prediction might come true:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Continuous analysis and geometry are just degenerate approximations to the discrete world... While discrete analysis is conceptually simpler... technically it is usually much more difficult... continuous mathematics is an approximation of the discrete one in contraposition to the traditional point of view. Under this approach the notion of a very big finite set is very important and the definition of a hyperfinite set in NSA is an appropriate formalization of this notion that satisfies the modern requirements to mathematical rigor.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The NSA prediction comes from the same place as another Gödel's prediction,  about the continuum hypothesis. After Cohen proved it independent of ZFC Gödel suggested that new axioms will be &amp;quot;discovered&amp;quot; to resolve it. The root is his mathematical realism (often confused with Platonism), which dictates that there is one &amp;quot;true&amp;quot; mathematics, working like physics of the ideal:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://dialnet.unirioja.es/descarga/articulo/3974057.pdf&quot; rel=&quot;noreferrer&quot;&gt;da Costa's Gödel Incompleteness Theorems and Physics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-07-22T03:30:48.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3725" PostTypeId="2" ParentId="3723" CreationDate="2016-05-04T07:25:37.560" Score="8" Body="&lt;p&gt;In the 1970s Gödel recommended Robinson for membership in the British Academy of Sciences.  His recommendation was based in part on Robinson's framework for analysis with infinitesimals.  From the viewpoint of Gödel's realism, the significance of Robinson's work was to extend further the ordered number system, beyond the natural, integer, rational, and real numbers, so as to include the hyperreal numbers (e.g., infinitesimals). Robinson for his part did not subscribe to realist views about set theory, and wrote explicitly that he did not view set theory as a &quot;foundation&quot; of mathematics. In private correspondence, Gödel chastised Robinson for expressing views contrary to Gödel's realist ideas about number systems!  To Gödel, the significance of Robinson's work was to be able to give precise mathematical formulation in currently acceptable foundational frameworks to Leibniz's ideas about infinitesimal and infinite numbers.  Robinson and Gödel agreed in their admiration for Leibniz.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="2442" LastEditDate="2016-07-26T12:07:11.133" LastActivityDate="2016-07-26T12:07:11.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3726" PostTypeId="1" AcceptedAnswerId="3727" CreationDate="2016-05-04T13:09:08.103" Score="2" ViewCount="506" Body="&lt;p&gt;Martin Gardener was a great recreational mathematics expert and his column &amp;quot;Mathematical Games&amp;quot; is an all time hit. But is there any archive available online consisting of his articles (preferably in PDF format)? I searched online but I got only one or two articles in PDF format.&lt;/p&gt;&#xA;" OwnerUserId="2891" LastEditorUserId="2277" LastEditDate="2020-07-08T06:56:31.853" LastActivityDate="2020-07-09T20:05:13.053" Title="Is there a Martin Gardner's article archive available online?" Tags="|mathematics|reference-request|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="3727" PostTypeId="2" ParentId="3726" CreationDate="2016-05-04T16:58:33.547" Score="3" Body="&lt;p&gt;About 20 of Martin Gardner's books are available.  In print, in iBooks, in Kindle, and probably other sources.  Is there some reason you cannot pay for them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;added&lt;/strong&gt;&lt;br&gt;&#xA;All the &quot;Mathematical Games&quot; columns from Scientific American: you can get them on one &lt;a href=&quot;http://www.maa.org/press/books/martin-gardner-s-mathematical-games-the-entire-collection-of-his-scientific-american-columns-on-one&quot; rel=&quot;nofollow&quot;&gt;CD&lt;/a&gt;  ...about $51 from Amazon.com, cheaper from the MAA if you are a member.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2016-05-05T13:03:21.667" LastActivityDate="2016-05-05T13:03:21.667" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3728" PostTypeId="1" AcceptedAnswerId="3729" CreationDate="2016-05-05T16:26:16.103" Score="3" ViewCount="337" Body="&lt;p&gt;I would like to cite the original source for the Banach-Mazur Theorem that the space $\mathcal{C}([0,1])$ of continuous functions on the unit interval is universal for separable Banach spaces, i.e., that for every separable Banach space $X$ there is an isometric embedding $i\colon X\to\mathcal{C}([0,1])$. In the books on functional analysis I have there is no reference to the original proof of this result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In short my question is: where was the Banach-Mazur Theorem published originally?&lt;/p&gt;&#xA;" OwnerUserId="4096" LastEditorUserId="4096" LastEditDate="2016-05-05T17:10:43.043" LastActivityDate="2016-05-06T17:43:51.847" Title="Where was the Banach-Mazur Theorem originally published?" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3729" PostTypeId="2" ParentId="3728" CreationDate="2016-05-05T20:26:39.103" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ1_02%2FS0004972700041411a.pdf&amp;amp;code=ca2fbbcbc66494157005a5003ed21b36&quot; rel=&quot;nofollow&quot;&gt;Kleiber and Pervin (1969)&lt;/a&gt; cite Banach's monograph Théorie des Opérations Linéaires (1932) for the &quot;&lt;em&gt;classical theorem of Banach and Mazur&lt;/em&gt;&quot;. Here is a &lt;a href=&quot;http://kielich.amu.edu.pl/Stefan_Banach/e-operations.html&quot; rel=&quot;nofollow&quot;&gt;scanned copy of the original&lt;/a&gt; edition (in French) on Banach's home page. There is an &lt;a href=&quot;http://store.doverpublications.com/0486469832.html&quot; rel=&quot;nofollow&quot;&gt;English translation&lt;/a&gt; available from Dover.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-05-06T17:43:51.847" LastActivityDate="2016-05-06T17:43:51.847" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3732" PostTypeId="1" CreationDate="2016-05-06T08:51:25.620" Score="0" ViewCount="213" Body="&lt;p&gt;quote:&lt;br&gt;&#xA;every book I've seen on matrix algebra or linear algebra seem[s] to just define the matrix operations without providing any historical background&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Talk:Matrix multiplication - Wikipedia, the free encyclopedia&lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Talk%3AMatrix_multiplication&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Talk%3AMatrix_multiplication&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;relevant material:&lt;br&gt;&#xA;When was Matrix Multiplication invented?&lt;br&gt;&#xA;&lt;a href=&quot;http://www.math.harvard.edu/~knill/history/matrix/&quot; rel=&quot;nofollow&quot;&gt;http://www.math.harvard.edu/~knill/history/matrix/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Matrices and determinants&lt;br&gt;&#xA;&lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html&quot; rel=&quot;nofollow&quot;&gt;http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT:&lt;br&gt;&#xA;I have now found many links discussing the rationale for matrix multiplication,&#xA;but in particular I am looking for primary sources for Cayley other than:&lt;br&gt;&#xA;A Memoir on the Theory of Matrices on JSTOR&lt;br&gt;&#xA;&lt;a href=&quot;http://www.jstor.org/stable/108649?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow&quot;&gt;http://www.jstor.org/stable/108649?seq=1#page_scan_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4101" LastEditorUserId="4101" LastEditDate="2016-05-08T03:40:24.447" LastActivityDate="2016-05-08T03:40:24.447" Title="What questions led Cayley to the definition of matrix multiplication?" Tags="|mathematics|mathematicians|linear-algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3733" PostTypeId="1" CreationDate="2016-05-06T09:47:29.333" Score="3" ViewCount="96" Body="&lt;p&gt;More specifically:&lt;br&gt;&#xA;When were the symmetries of polygons/solids first presented as groups in Cayley tables?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Textbooks often use the symmetries of polygons/solids to introduce group theory,&#xA;however, I have found little historical material on&#xA;the early discoveries relating groups and geometry.&lt;/p&gt;&#xA;" OwnerUserId="4101" LastActivityDate="2016-05-08T02:56:12.663" Title="Early discoveries combining groups and geometry?" Tags="|mathematics|mathematicians|group-theory|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3734" PostTypeId="1" CreationDate="2016-05-06T10:33:23.923" Score="2" ViewCount="354" Body="&lt;p&gt;What questions led to the invention of complex differentiation/integration? How were their definitions agreed upon?&lt;br&gt;&#xA;Real differentiation/integration has an obvious meaning. To extend calculus to the complex numbers,&#xA;why would this be done, is it even meaningful to call it 'differentiation'/'integration'?&lt;/p&gt;&#xA;" OwnerUserId="4101" LastActivityDate="2016-05-06T12:45:59.800" Title="The origins of complex differentiation/integration" Tags="|mathematics|mathematicians|complex-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3735" PostTypeId="2" ParentId="3734" CreationDate="2016-05-06T12:38:05.633" Score="5" Body="&lt;p&gt;There is no difference in differentiation. Derivative of a complex function of a complex variable is defined by the same formula $$f'(z)=\lim_{h\to 0} (f(z+h)-f(z))/h$$&#xA;as for the real variable. Concerning integration, one reason was the desire to investigate integrals of real functions. One of the consequences (and motivations) of  Cauchy theory is that it evaluates integrals of real functions on real intervals by manipulating with integrals in the complex domain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is a more important and more profound reason. Many equations (differential and functional) can be solved in the form of power series. This was essentially discovered by Newton, and he considered this his main discovery in Calculus. A power series, if it converges anywhere except $x=0$, converges in a disk in the complex plane, and it is impossible to understand its properties while staying in the real domain only. This is the most important motivation of extending Calculus to complex domain. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a simple example: we have a power series expansion&#xA;$$f(x)=\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots.$$&#xA;It converges in the real domain for $-1&amp;lt;x&amp;lt;1$ only. But the left hand side $f(x)=1/(1+x^2)$ is a nice function on the whole real line. By looking at this function on the real line only it is impossible to understand what happens at $\pm 1$, why the series suddenly stops to converge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. I did not like the sentence &quot;definition agreed upon&quot;. These definitions are essentially forced on us by the nature of things. There is nothing here to agree or disagree upon:-) Most of these things were discovered by Cauchy in the beginning of 19th century, with important contributions of Gauss and Abel. But they were discovered, not &quot;agreed upon&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-05-06T12:45:59.800" LastActivityDate="2016-05-06T12:45:59.800" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3736" PostTypeId="1" CreationDate="2016-05-07T08:50:31.840" Score="3" ViewCount="60" Body="&lt;p&gt;In the recent movie about Stephen Hawking it is displayed that it is a coincidence that he got a computer with his wheelchair: Some friend was a computer engineer and installed the computer for him. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is that more like a fact or just fiction?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think today that it is normal with computers that help people in wheelchair and that Stephen Hawking would get that help today that in the movie was displayed a a coincidence that he had a friend who was a computer engineer. &lt;/p&gt;&#xA;" OwnerUserId="4103" LastEditorUserId="55" LastEditDate="2016-05-08T02:00:51.240" LastActivityDate="2016-05-08T02:00:51.240" Title="Is it true that Stephen Hawking got the computer with his wheelchair by coincidence?" Tags="|computers|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3737" PostTypeId="2" ParentId="3045" CreationDate="2016-05-07T18:02:53.883" Score="4" Body="&lt;p&gt;I originally answered this on a different forum*. BK's solution (with some minor improvements by GW) is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x=\sum_{i\geq 0}\frac{(-1)^{i}}{i! (i m+1)}\prod_{j=0}^{i-1}(\frac{i m+1}{n}-j).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears to be the first discovered general formula for real roots of the polynomial  $x^n + x^m - 1 = 0$. It is not clear whether other radical roots can also be obtained using hypergeometric series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*I would be interested in a discussion of mathematics forums and how they could be improved, and also how they could coordinate to address different kinds of questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did find a &quot;meta&quot; question here about questions like: &quot;Who first discovered X?&quot;. I couldn't participate in that discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason for deleting BK's original answer (according to comment) is because it was not on topic of history of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="4106" LastEditorUserId="4106" LastEditDate="2016-05-10T02:20:07.187" LastActivityDate="2016-05-10T02:20:07.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3738" PostTypeId="1" AcceptedAnswerId="3739" CreationDate="2016-05-08T01:07:12.333" Score="6" ViewCount="2473" Body="&lt;p&gt;I'm interested in the history behind Plato's formula $2m,m^2-1,m^2+1$ for generating pythagorean triples. Was Plato the first mathematician to come up with such a formula?&lt;/p&gt;&#xA;" OwnerUserId="2866" LastEditorUserId="2866" LastEditDate="2016-05-08T01:21:51.513" LastActivityDate="2019-04-03T15:40:48.240" Title="History of Plato's formula for generating pythagorean triples" Tags="|mathematics|mathematicians|pythagorean-proposition|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3739" PostTypeId="2" ParentId="3738" CreationDate="2016-05-08T02:42:45.117" Score="3" Body="&lt;p&gt;Plato was not a mathematician and it is unlikely that he invented anything in mathematics. This particular formula is credited to Pythagoras by ancient mathematician and historian of mathematics Proclus; he says this in his commentaries on Euclid.&#xA;Some special cases come from Babylonian mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did not check with Proclus. My source is the article of I. Bashmakova, Diophantus of Alexandria and his &quot;Atithmetic&quot;, published as the introduction to&#xA;the Russian translation of Diophantus (Moscow, Nauka, 1974).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only conclusion that can be made from Proclus's statement is that some unknown&#xA;Pythagoreans knew this formula, they were a secret society, and tended to attribute everything to their semi-legendary founder.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-05-08T02:42:45.117" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3740" PostTypeId="2" ParentId="3733" CreationDate="2016-05-08T02:56:12.663" Score="1" Body="&lt;p&gt;The earliest applications of groups in geometry are dealing with continuous groups&#xA;(Lie, Klein 1872). You seem to be talking about discrete groups. Their earliest application in a geometric subject was in crystallography in the 1879-1900. See &#xA;&quot;Crystallographic groups&quot; in Wikipedia.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-05-08T02:56:12.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3741" PostTypeId="1" CreationDate="2016-05-08T08:37:42.213" Score="5" ViewCount="243" Body="&lt;p&gt;AFAIK there's rare evidence to show that  $\sum$ has the meaning of &quot;sum&quot; in Greek, so why mathematician chose  $\sum$ to be the summation operator?&lt;/p&gt;&#xA;" OwnerUserId="4108" LastEditorUserId="2442" LastEditDate="2016-06-15T12:40:01.820" LastActivityDate="2016-06-15T12:40:01.820" Title="Why choose symbol $\sum$ to be the summation operator in maths?" Tags="|mathematics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3742" PostTypeId="2" ParentId="3741" CreationDate="2016-05-08T10:53:21.490" Score="8" Body="&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Florian_Cajori&quot; rel=&quot;noreferrer&quot;&gt;Forian Cajori&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp&quot; rel=&quot;noreferrer&quot;&gt;A History of Mathematical Notations&lt;/a&gt; (1928, also Dover reprint), Vol.II, page 61:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;438&lt;/strong&gt;. The sign &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; for summation is due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=sYE_AAAAcAAJ&amp;amp;pg=PA27&quot; rel=&quot;noreferrer&quot;&gt;Institutiones calculi differentialis&lt;/a&gt; (St.Petersburg, 1755), Cap.I,§26, page 27, who says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;summam indicabimus signo &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This symbol was used by Lagrange but otherwise received little attention during the eighteenth century. [...] The &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; to express &amp;quot;sum&amp;quot; occurs in 1829 in Fourier's &lt;em&gt;Theory of Heat&lt;/em&gt;, published in 1822, and in C.G.J. Jacobi's &lt;em&gt;Fundamenta nova theoriae functionum ellipticarum&lt;/em&gt; (1829).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt; We have to take into account that the latin &lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot; rel=&quot;noreferrer&quot;&gt;sum&lt;/a&gt; was already the source for Leibniz's symbol &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; for integral:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Utile erit scribi &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; pro &lt;em&gt;omnia&lt;/em&gt;, ut &lt;span class=&quot;math-container&quot;&gt;$\int l = \text {omn.} l$&lt;/span&gt;, id est summa ipsorum &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-05-08T15:29:13.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3743" PostTypeId="2" ParentId="3741" CreationDate="2016-05-08T11:41:13.323" Score="1" Body="&lt;p&gt;This is a typical example of mixing Greek with Latin in terminology and notation.&#xA;Greek letter which is equivalent to S stands for a Latin word summa which begins with S. Such things are very common. Some purists object this kind of mixture for&#xA;aesthetic and logical reasons, but this kind of confusion is so common that it is impossible to &quot;correct&quot; it. The reason is that scientists who introduce notation usually do not know much of Greek. (And modern ones also of Latin). I don't know whether Euler had a working knowledge of Greek. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is another example: from the Greek noun &quot;cone&quot;, one can make an adjective&#xA;&quot;conic&quot; using the Greek suffix -ic. In Latin the same role is played by the suffix -al. So the modern English adjective &quot;conical&quot; has two suffixes from different languages which mean exactly the same. From the point of a linguist this is a nonsense. But scientists do not mind.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-05-08T11:41:13.323" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3745" PostTypeId="1" CreationDate="2016-05-08T16:28:55.660" Score="0" ViewCount="36" Body="&lt;p&gt;I have a wild guess that science &lt;em&gt;flew&lt;/em&gt; when we started to take the concepts, fragment it and examine it piece by piece, fragmenting it again when needed for a better understanding. It seems that this process of modularization was pivotal to the scientific progress but where did it come from?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember of having read that for centuries, euclidean geometry was a sort of perfect scientific model in which we grab the idea of certain spaces and abstract them in order to obtain their intrinsic nature and hence, they &lt;em&gt;drilled a hole&lt;/em&gt; right to the axioms and confirmed that the axioms do indeed represent that idealization of space. Could this be the source?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe there is no &lt;em&gt;perfect&lt;/em&gt; answer to this question. But I guess we could at least trace where this idea was used and have a rough estimate of its beginning. Also, I'm interested in It's existence and am looking at references on both existential and historical aspects of the concept.&lt;/p&gt;&#xA;" OwnerUserId="332" LastActivityDate="2016-05-08T16:28:55.660" Title="Where did the idea of modularization come from?" Tags="|philosophy-of-science|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3746" PostTypeId="1" AcceptedAnswerId="3755" CreationDate="2016-05-08T20:18:01.147" Score="3" ViewCount="403" Body="&lt;p&gt;Apart from the location, brightness and color of the stars themselves, astronomy obviously originated from an interest in transient phenomena against the fixed but rotating star sky. The rise and set of objects, phases of the Moon, eclipses, movement of planets, passing by of comets, appearance of novas. They were all recorded in ancient times and there was astronomy or astrology made out of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what about meteor showers? Falling stars. Was any ancient astronomy done to predict or explain them? Meteorites falling from the sky were believed to come from volcanoes about 200 years ago. Did e.g. the ancient medetierraneans assume the same things since they knew about volcanoes? And bolides, fireballs crashing through the sky, are there ancient records of that? Can they have been confused with comets?&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorUserId="55" LastEditDate="2016-05-12T07:58:14.287" LastActivityDate="2016-05-12T07:58:14.287" Title="What role did meteors and comets play in early astronomy?" Tags="|astronomy|archaeoastronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3747" PostTypeId="1" AcceptedAnswerId="3748" CreationDate="2016-05-08T21:51:01.150" Score="0" ViewCount="1659" Body="&lt;p&gt;As is well known, Dirac introduced the creation and annihilation operators for the quantum harmonic oscillator ($\hat{a}^\dagger$ and $\hat{a}$ respectively), which are now part of every first course in quantum mechanics due to their far-reaching importance and simplicity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\hat{a}^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\left[\hat{x}-\frac{i}{m\omega}\hat{p}\right],\,\,\,\hat{a}=\sqrt{\frac{m\omega}{2\hbar}}\left[\hat{x}+\frac{i}{m\omega}\hat{p}\right]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in English we say &lt;em&gt;&quot;a-dagger&quot;&lt;/em&gt; for the creation operator, and the word &quot;dagger&quot; is certainly more associated with destruction (killing, stabbing, etc.) than it is with &lt;em&gt;&quot;creation&quot;&lt;/em&gt;. This is almost always an unspoken source of confusion for native English-speaking students of physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that this was originally done by an Englishman, how come we haven't simply swapped the definitions for $\hat{a}^\dagger$ and $\hat{a}$, and then called $\hat{a}^\dagger$ the annihilation operator and $\hat{a}$ the creation operator, a redefinition that would have no effect on the physics but have a substantial effect on a student's ability to remember these operators? Is there a reason why Dirac chose this language-awkward definition?&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2016-05-09T22:11:23.587" Title="Why is harmonic oscillator creation operator called &quot;a-dagger&quot;, $a^\dagger$?" Tags="|physics|quantum-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3748" PostTypeId="2" ParentId="3747" CreationDate="2016-05-09T02:29:06.080" Score="3" Body="&lt;p&gt;$\dagger$ is the transpose conjugate of a linear operator; once $a$ is defined as the annihilation operator (presumably because annihilation starts with the letter &lt;em&gt;a&lt;/em&gt;), Dirac showed that the corresponding creation operator corresponds to it's transpose conjugate, hence the creation operator is $a^{\dagger} $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus the &lt;em&gt;a&lt;/em&gt; comes first, and then the $\dagger $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wouldn't assume that Dirac is responsible for the $\dagger $ notation for transpose conjugate; it is not the only way it is written in mathematical texts, though it is certainly the favorite among physicists today.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastEditorUserId="3821" LastEditDate="2016-05-09T22:11:23.587" LastActivityDate="2016-05-09T22:11:23.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3752" PostTypeId="1" AcceptedAnswerId="3762" CreationDate="2016-05-10T17:38:55.643" Score="14" ViewCount="1317" Body="&lt;p&gt;Galileo used balls rolling down ramps to study the relationship between time and distance traveled. However, without any knowledge of physics, it doesn't seem immediately obvious that the time-distance relationship of an object rolling down a ramp is the same as if it were free falling. Why did Galileo assume this? Did he have sufficient physics knowledge to conclude this? Was this just a hypothesis which he confirmed by actual free fall experiments?   &lt;/p&gt;&#xA;" OwnerUserId="4123" LastActivityDate="2016-05-24T16:35:06.297" Title="How did Galileo know that objects rolling down a ramp was an accurate model for free fall?" Tags="|physics|experimental-physics|gravity|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3753" PostTypeId="2" ParentId="3741" CreationDate="2016-05-10T20:12:19.463" Score="2" Body="&lt;p&gt;The exact &lt;em&gt;mathematical&lt;/em&gt; Greek word for &quot;sum&quot; is written &quot;Αθροισμα&quot;. &lt;em&gt;But&lt;/em&gt; the &quot;everyday&quot; word for &quot;sum&quot; is written &quot;&lt;strong&gt;Σ&lt;/strong&gt;ύνολο&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The interesting thing is that, as a mathematical term,  in Greek the word &quot;&lt;strong&gt;Σ&lt;/strong&gt;ύνολο&quot; is used to mean &quot;set&quot;.&lt;/p&gt;&#xA;" OwnerUserId="244" LastActivityDate="2016-05-10T20:12:19.463" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3754" PostTypeId="1" AcceptedAnswerId="3756" CreationDate="2016-05-10T21:28:47.977" Score="9" ViewCount="616" Body="&lt;p&gt;I recently learned about irreducible and prime elements in a commutative ring. However, my professor was not quite sure who was the first to make this distinction, or give an example of an irreducible element which is not prime. Was it Dedekind? If so, where did he make this distinction? Could you also give some references regarding the history of commutative rings, UFD, and the evolution of these concepts?&lt;/p&gt;&#xA;" OwnerUserId="4126" LastEditorUserId="55" LastEditDate="2016-05-12T07:53:34.010" LastActivityDate="2016-05-12T07:53:34.010" Title="Who was first to differentiate between prime and irreducible elements?" Tags="|mathematics|mathematicians|terminology|abstract-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3755" PostTypeId="2" ParentId="3746" CreationDate="2016-05-11T00:09:08.120" Score="3" Body="&lt;p&gt;None. Ancient Greeks subdivided the world into and the superlunar the sublunar spheres, the heavens and the Earth. The former a place of perfection and order, the latter, not so much. Since meteors and comets did not display the reliable regularity of the fixed stars, or even of the &quot;wandering&quot; planets they clearly did not belong to the heavens. Greeks therefore saw them as the (upper) atmospheric phenomena, where air and fire had their &quot;natural place&quot;, along with  thunder and lightning, clouds, aurora, and rainbows. &quot;Meteorology&quot; and &quot;meteors&quot; got their names from Greek &quot;meteoros&quot;, raised up high. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle's view prevailed in both Europe and the Islamic East throughout the  middle ages. Comets acquired their celestial status first after Halley convincingly argued that the comet seen in 1705 (and now named after him) was the same comet seen in 1531, 1607 and 1682, and predicted that it will be seen again in 1758. Displaying such reliable periodicity quickly moved the comets to the realm of astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took longer for meteors, Kepler still believed that they were atmospheric phenomena, and Newton hypothesized that heavens were filled with stars, planets, and comets only. However, already in 1676 Wallis wrote sceptically:&quot;&lt;em&gt;that which makes it to me the more surprising is this; that I find the same [meteor] to have been seen in most parts of England&lt;/em&gt;&quot;. And in 1714 Halley wrote three papers, where he reasoned that atmosphere barely extended as high (45m/70km) as most of the &quot;fiery meteors&quot; appeared. But.. history is rarely linear, in 1719 Halley changed his mind and went back to Aristotle, and it took the good part of 18th century to finally move meteors into astronomy. See &lt;a href=&quot;http://adsabs.harvard.edu/full/1993JIMO...21..259B&quot; rel=&quot;nofollow&quot;&gt;Beech's Makings of Meteor Astronomy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the opening of &lt;a href=&quot;http://classics.mit.edu/Aristotle/meteorology.1.i.html&quot; rel=&quot;nofollow&quot;&gt;Aristotle's Meteorology&lt;/a&gt; (boldface mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;We have already discussed the first causes of nature, and all natural motion, also the stars ordered in the motion of the heavens, and the physical element-enumerating and specifying them and showing how they change into one another-and becoming and perishing in general. There remains for consideration a part of this inquiry which all our predecessors called meteorology. &lt;strong&gt;It is concerned with events that are natural, though their order is less perfect than that of the first of the elements of bodies. They take place in the region nearest to the motion of the stars. Such are the milky way, and comets, and the movements of meteors.&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And here is his theory of the shooting stars:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;We must think of what we just called fire as being spread round the terrestrial sphere on the outside like a kind of fuel, so that a little motion often makes it burst into flame just as smoke does: for flame is the ebullition of a dry exhalation. So whenever the circular motion stirs this stuff up in any way, it catches fire at the point at which it is most inflammable. The result differs according to the disposition and quantity of the combustible material...  if the whole length of the exhalation is scattered in small parts and in many directions and in breadth and depth alike, we get what are called shooting-stars... Shooting-stars further suffix to prove that the celestial sphere is not hot or fiery: for they do not occur in that upper region but below: yet the more and the faster a thing moves, the more apt it is to take fire.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-05-12T07:57:23.763" LastActivityDate="2016-05-12T07:57:23.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3756" PostTypeId="2" ParentId="3754" CreationDate="2016-05-11T02:15:27.403" Score="7" Body="&lt;p&gt;The phenomenon of nonunique factorization appears to have been first explicitly articulated in the setting of cyclotomic fields, by Kummer in the 1830s and 1840s: $\mathbf Z[\zeta_{23}]$ is the first cyclotomic ring that fails to have unique factorization. One of Kummer's first papers (in 1847) where he discusses how to save unique factorization in cyclotomic rings $\mathbf Z[\zeta_\lambda]$ for prime $\lambda$, using &quot;ideal numbers,&quot; is &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002146061&amp;amp;physid=PHYS_0340&quot;&gt;http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002146061&amp;amp;physid=PHYS_0340&lt;/a&gt;. (For readers who know algebraic number theory, on the last page of this paper at &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=PPN243919689_0035|LOG_0003&amp;amp;physid=PHYS_0380&quot;&gt;http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=PPN243919689_0035|LOG_0003&amp;amp;physid=PHYS_0380&lt;/a&gt; Kummer lists some examples of &lt;em&gt;exponents&lt;/em&gt; $n$ of class groups of these cyclotomic rings, which are not always equal to the class number $h$ of these rings. For example, when $\lambda$ is $29$ and $41$ he finds $n$ is $2$ and $11$, but $h$ is $8$ and $121$, respectively.)  See also Lenstra's article &lt;a href=&quot;http://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1979a/art.pdf&quot;&gt;http://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1979a/art.pdf&lt;/a&gt;. I do not believe at this time that anyone created special terminology to distinguish primes from irreducibles like we do today. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moving a few decades later, in Dedekind's famous XI-th Supplement on algebraic number theory in Dirichlet's Lectures on Number Theory (Vorlesungen über Zahlentheorie), which is from the 1870s, there is the example $3 \cdot 7 = (1+2\sqrt{-5})(1-2\sqrt{-5})$ from $\mathbf Z[\sqrt{-5}]$ in Section 159, where Dedekind says each of the four factors is unzerlegbare Zahl, which means an irreducible number. Dedekind's goal was to rescue unique factorization by using ideals instead of elements, and he does not dwell on the phenomenon of non-UFDs long enough to introduce new vocabulary like prime vs. irreducible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks to Franz Lemmermeyer, I can point you to a reference from 1903 that makes the distinction between primes and irreducibles. On the bottom of page 12 of the book Einleitung in die allgemeine Theorie der algebraischen Groessen, by Gyulya König, (see &lt;a href=&quot;https://books.google.com/books?id=3LhUAAAAYAAJ&amp;amp;pg=PA12&amp;amp;lpg=PA12&amp;amp;dq=irreduzibel+holoiden&amp;amp;source=bl&amp;amp;ots=aVbTTDuTi6&amp;amp;sig=oOb2Gna_A0T_e8ONlmwsSYy7tEM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjGreu8_dLMAhWIQSYKHYlxCY0Q6AEIHDAA#v=onepage&amp;amp;q=irreduzibel%20holoiden&amp;amp;f=false&quot;&gt;https://books.google.com/books?id=3LhUAAAAYAAJ&amp;amp;pg=PA12&amp;amp;lpg=PA12&amp;amp;dq=irreduzibel+holoiden&amp;amp;source=bl&amp;amp;ots=aVbTTDuTi6&amp;amp;sig=oOb2Gna_A0T_e8ONlmwsSYy7tEM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjGreu8_dLMAhWIQSYKHYlxCY0Q6AEIHDAA#v=onepage&amp;amp;q=irreduzibel%20holoiden&amp;amp;f=false&lt;/a&gt;) the terms Prim-größe and irreduzibel are introduced, where the first one means prime element and the second means irreducible irreducible element. At the top of page 13 he writes that each prime element is irreducible but not conversely, and a counterexample is presented over the course of pp. 20-23 in $\mathbf Z[\sqrt{-5}]$, where $2-\sqrt{-5}$ is shown to be irreducible but not prime. That it's not prime follows from $3 \cdot 3 = (2+\sqrt{-5})(2-\sqrt{-5})$.  (König was writing before the general concept of a ring was created. He always works in an integral domain, which he calls a Bereich, and he limits his definitions of prime and irreducible to what he calls holoid domains, which are integral domains of characteristic $0$, even though we'd recognize a restriction on the characteristic as an irrelevant hypothesis today. His notation for $\mathbf Z$ is $[1]$, for $\mathbf Q$ is $(1)$, and for $\mathbf Z[\sqrt{d}]$ is $[\sqrt{d}]$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since König is not well-known for his work in algebra, I'll mention another early 20-th century reference: on p. 11 of Krull's 1937 book Idealtheorie &#xA;(see &lt;a href=&quot;https://books.google.com/books?id=jCvKBgAAQBAJ&amp;amp;pg=PA11&amp;amp;lpg=PA11&amp;amp;dq=unzerlegbare+element&amp;amp;source=bl&amp;amp;ots=y_Y9BXubOy&amp;amp;sig=ZVQsi9-A-xYKb1TE5UAHE4-p1wk&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi71Z2NrdHMAhVDVD4KHfjsDUYQ6AEILTAD#v=onepage&amp;amp;q=unzerlegbare%20element&amp;amp;f=false&quot;&gt;https://books.google.com/books?id=jCvKBgAAQBAJ&amp;amp;pg=PA11&amp;amp;lpg=PA11&amp;amp;dq=unzerlegbare+element&amp;amp;source=bl&amp;amp;ots=y_Y9BXubOy&amp;amp;sig=ZVQsi9-A-xYKb1TE5UAHE4-p1wk&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi71Z2NrdHMAhVDVD4KHfjsDUYQ6AEILTAD#v=onepage&amp;amp;q=unzerlegbare%20element&amp;amp;f=false&lt;/a&gt;) he defines unzerlegbar and Primelement the way you'd define irreducible and prime, and he also cautions that while every prime element is irreducible, not every irreducible element is prime.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2016-05-11T23:22:32.800" LastActivityDate="2016-05-11T23:22:32.800" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3757" PostTypeId="1" AcceptedAnswerId="3758" CreationDate="2016-05-11T03:27:44.370" Score="4" ViewCount="163" Body="&lt;p&gt;The first paragraph of Albert Einstein's paper &lt;em&gt;On the Electrodynamics of Moving Bodies&lt;/em&gt;, written in 1905, mentions the failure of attempts to detect the earth's motion through the luminiferous aether.  Page 48 of Walter Isaacson's biography of Einstein says that as a graduate student Einstein read a paper by Wilhelm Wien that described the failures of thirteen different experiments that tried to measure motion through the aether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there were so many, then why is only one of them so famous?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;PS:&lt;/b&gt; &lt;em&gt;Possibly&lt;/em&gt; the paper of Wien is &lt;a href=&quot;https://de.wikisource.org/wiki/Translatorische_Bewegung_des_Licht%C3%A4thers&quot; rel=&quot;nofollow&quot;&gt;this one&lt;/a&gt;.  I haven't looked at it for more than a few seconds yet.  Isaacson's biography of Einstein actually doesn't cite the paper he says Einstein read.  The item I linked to here mentions the Michelson–Morley experiment.&lt;/p&gt;&#xA;" OwnerUserId="2193" LastEditorUserId="2193" LastEditDate="2016-05-12T19:25:03.353" LastActivityDate="2016-05-14T23:49:18.573" Title="Why is the Michelson–Morley experiment so much more famous than its rivals?" Tags="|experimental-physics|light|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3758" PostTypeId="2" ParentId="3757" CreationDate="2016-05-11T04:46:52.233" Score="6" Body="&lt;p&gt;Because it was the most accurate. The use of interferometers to measure length deviations eliminated the concern that the ether wind didn't register because the instruments were too blunt, or because the experimenters were too clutz. Remember that everything else up to that point confirmed ether, and Hertz confirmed it some more in 1887 by detecting electromagnetic waves. Extraordinary claims require extraordinary evidence. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concern remained even after &lt;a href=&quot;https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment#1881_and_1887_experiments&quot; rel=&quot;nofollow&quot;&gt;Michelson's first 1881 try&lt;/a&gt;, which is why he redid it with Morley in 1887 with sterling precision. Only after that Fitzgerald, Hertz, Lorentz and others started to take the outcome seriously. Fitzgerald suggested &lt;a href=&quot;https://en.wikipedia.org/wiki/Length_contraction#History&quot; rel=&quot;nofollow&quot;&gt;length contraction&lt;/a&gt; in 1889, Hertz gave his fix in Principles of Mechanics of 1894, and Lorentz followed up in 1895, see &lt;a href=&quot;http://file.scirp.org/pdf/AHS_2016021615305682.pdf&quot; rel=&quot;nofollow&quot;&gt;D’Agostino's Overview of the Nineteenth and Twentieth Centuries’ Theories of Light, Ether, and Electromagnetic Waves&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-05-14T23:49:18.573" LastActivityDate="2016-05-14T23:49:18.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3759" PostTypeId="1" CreationDate="2016-05-11T08:17:53.903" Score="11" ViewCount="2750" Body="&lt;p&gt;If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more detailed account) routinely get credit for the real numbers? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stevin did some detailed work (rather than vague general ideas) with unending decimals, including a proof of the intermediate value theorem for polynomials. Newton was in fact inspired by infinite decimals to introduce his general theory of power series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting point was raised in an answer by Peter Diehr. The so-called &lt;em&gt;Archimedean property&lt;/em&gt; (which is one of the defining characteristics of the real number field; though of course it does not suffice to characterize them as the rationals also satisfy it) was considered by authors like Euclid (&lt;em&gt;Elements V.4&lt;/em&gt;) considerably earlier. However as far as giving an actual construction (rather than axiomatic definition) Stevin seems to have been the first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1. To clarify, Stevin developed specific notation for decimals (more complicated than the one we use today) and did actual technical work with them rather than merely envisioning their possibility, unlike some of his predecessors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 2. One useful source for this is Malet, Antoni. &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086004000928&quot; rel=&quot;nofollow noreferrer&quot;&gt;Renaissance notions of number and magnitude&lt;/a&gt;. Historia Math.  33  (2006),  no. 1, 63–81. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 3. As Malet notes, &quot;Stevin does not justify his definition&quot; which identifies number and &quot;quantity of anything&quot; because to him the identification is obvious, and the implementation of number is his unending decimals. This was an appropriate move indeed since we know today that the Cantor-Dedekind postulate identifying the number line and the line in physical space is untenable based on what modern physics teaches us; similar remarks apply to magnitude/quantity. Stevin of course was not aware of &quot;transcendental&quot; numbers but no such knowledge is required in order to define the real numbers by means of unending decimals; namely this could have been done even if Liouville did not prove the existence of transcendental numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 4. I should clarify that Stevin dealt with unending decimals in his book &lt;em&gt;l&quot;Arithmetique&lt;/em&gt; rather than the more practically-oriented &lt;em&gt;De Thiende&lt;/em&gt; meant to teach students to work with decimals (of course, finite ones).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 5. As far as using the term &lt;em&gt;real&lt;/em&gt; to describe the numbers Stevin was concerned with, it should be clarified that the first one to describe the common numbers as &lt;em&gt;real&lt;/em&gt; may have been Descartes and at any rate this usage is later than Stevin. On the other hand, if we talk about representing &lt;em&gt;common&lt;/em&gt; numbers (including both rational and not so), Stevin not only speculated about the possibility of a representation scheme using decimals, but (unlike some of his predecessors) developed a specific notation (though different from what we use today) and moreover did work with this notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 6. Cantor thought that Cauchy Completeness (CC) was sufficient to characterize the real numbers axiomatically. Today we know this is not the case, as one also needs the Archimedean property. I found out recently that Dedekind was convinced he had a proof of the existence of an infinite set; see &lt;a href=&quot;https://mathoverflow.net/questions/240276/has-dedekinds-proof-of-existence-of-infinite-sets-been-analyzed-by-historians&quot;&gt;here&lt;/a&gt;. Do these misconceptions by Cantor and Dedekind indicate a shortcoming of the constructions of the real numbers they proposed? Hardly so. Stevin's approach to representing all common numbers by unending decimals similarly could not be held at fault because Stevin was not aware of certain future developments.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-09-02T20:36:05.270" Title="Who gets credit for the real numbers?" Tags="|mathematics|mathematicians|notation|debunking|real-analysis|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="3760" PostTypeId="2" ParentId="2740" CreationDate="2016-05-11T08:21:01.640" Score="1" Body="&lt;p&gt;Simon Stevin understood this at the end of the 16th century in his work &lt;em&gt;L'Arithmetique&lt;/em&gt;, where unending decimals are first spoken of explicitly. The fact that the decimal system was accepted in Europe may have been due to him but this involves only finite decimals obviously since those are useful in trade.  The &lt;em&gt;mathematical&lt;/em&gt; significance of his work of introducing unending decimal representation of numbers is often overlooked.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-05-11T08:21:01.640" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3761" PostTypeId="1" AcceptedAnswerId="3790" CreationDate="2016-05-11T09:07:23.447" Score="5" ViewCount="308" Body="&lt;p&gt;I'm convinced that there is no such a mathematician whose name is &quot;kernel&quot;. The wiki article about &lt;a href=&quot;https://en.wikipedia.org/wiki/Kernel_%28algebra%29&quot; rel=&quot;nofollow&quot;&gt;kernel&lt;/a&gt; doesn't include history in its content. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I wonder, who is the first mathematician to use kernel in his work and on which paper.     &lt;/p&gt;&#xA;" OwnerUserId="368" LastEditorUserId="26" LastEditDate="2016-05-18T06:34:05.953" LastActivityDate="2016-05-18T06:34:05.953" Title="Who coined kernel in mathematics?" Tags="|mathematicians|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3762" PostTypeId="2" ParentId="3752" CreationDate="2016-05-11T09:58:37.813" Score="7" Body="&lt;p&gt;From: &lt;a href=&quot;https://en.wikipedia.org/wiki/Two_New_Sciences#The_law_of_falling_bodies&quot; rel=&quot;noreferrer&quot;&gt;Discourses and Mathematical Demonstrations Relating to Two New Sciences&lt;/a&gt; (Italian: &lt;em&gt;Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze&lt;/em&gt;), published in 1638.&lt;/p&gt;&#xA;&lt;p&gt;See: &lt;a href=&quot;http://galileoandeinstein.physics.virginia.edu/tns_draft/tns_160to243.html&quot; rel=&quot;noreferrer&quot;&gt;Engl.transaltion by Henry Crew and Alfonso de Salvio&lt;/a&gt; (1914):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum (&lt;em&gt;celeritatis momenta&lt;/em&gt;) receives equal increments in equal times.&lt;/p&gt;&#xA;&lt;p&gt;This definition established, the Author makes a single assumption, namely,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The speeds acquired by one and the same body moving down planes of different nclinations are equal when the heights of these planes are equal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By the height of an inclined plane we mean the perpendicular let fall from the upper end of the plane upon the horizontal line drawn through the lower end of the same plane. [...]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;THEOREM II, PROPOSITION II&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances. [...] considering the total space traversed, that covered in double time will be quadruple that covered during unit time; in triple time, the space is nine times as great as in unit time.&lt;/p&gt;&#xA;&lt;p&gt;And in general the spaces traversed are in the duplicate ratio of the times, i.e., in the ratio of the squares of the times. [...]&lt;/p&gt;&#xA;&lt;p&gt;I have attempted in the following manner to assure myself that the acceleration actually experienced by falling bodies is that above described.&lt;/p&gt;&#xA;&lt;p&gt;A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent. We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i. e., of the channel, along which we rolled the ball. We also observed that the times of descent, for various inclinations of the plane, bore to one another precisely that ratio which, as we shall see later, the Author had predicted and demonstrated for them.&lt;/p&gt;&#xA;&lt;p&gt;For the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;level of accuracy&amp;quot; of Galileo's measurements is still highly debated, but a lot of hand-written sheets full of drawings and calculations are still extant.&lt;/p&gt;&#xA;&lt;p&gt;Thus, it is quite clear that Galileo makes assumptions and tested them experimentally (as usual for any good scientist).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-05-11T10:07:46.083" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3763" PostTypeId="1" CreationDate="2016-05-11T12:20:56.873" Score="0" ViewCount="65" Body="&lt;p&gt;Could you suggest a good biographic book about the life and work of the physicist Ludwig Boltzmann?&lt;/p&gt;&#xA;" OwnerUserId="3006" LastEditorUserId="4251" LastEditDate="2020-06-07T18:35:15.530" LastActivityDate="2020-06-07T18:35:15.530" Title="Ludwig Boltzmann: life and work" Tags="|physics|biographical-details|book-recommendation|physicists|boltzmann|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3764" PostTypeId="2" ParentId="3759" CreationDate="2016-05-11T13:41:44.363" Score="6" Body="&lt;p&gt;Many people get credit, because this was a long story beginning in the ancient Greece. Euclid has a theory of proportions (based on earlier research) which is equivalent to modern theory of real numbers. Infinite decimal expansions were gradually introduced since 17th century (Napier, Stevin), and the modern theories are due to Cantor and Dedekind. So the development took 2000 years, and it is impossible to credit one person.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-05-11T13:41:44.363" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3765" PostTypeId="2" ParentId="3759" CreationDate="2016-05-11T17:45:20.653" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedean_property&quot; rel=&quot;noreferrer&quot;&gt;The Archimedean property&lt;/a&gt; as it is called, was used as an axiom by Archimedes, and he credited &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedean_property#History_and_origin_of_the_name_of_the_Archimedean_property&quot; rel=&quot;noreferrer&quot;&gt;Eudoxus of Cnidus, who predates Euclid&lt;/a&gt;; also &lt;a href=&quot;http://www.mlahanas.de/Greeks/Eudoxus.htm&quot; rel=&quot;noreferrer&quot;&gt;see this.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086004000928&quot; rel=&quot;noreferrer&quot;&gt;Section 7: Stevin, Malet says: &lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;In fact Stevin does not justify his first definition (“Number is that by which one can tell the quantity of anything”)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it appears that, like Archimedes, Simon Stevin assumes that every point of a line corresponds to a distance from its origin; that is, magnitudes correspond to points of the line.  The nice mathematical distinctions that appear in the 19th century which sort out the details of the Real numbers, are not important to Stevin; what is important is that the decimal notation provides a convenient method for recording these magnitudes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His work was intended to teach students how to work with decimal numbers. Since even the concept of transcendental numbers does not appear until the 19th century, I don't see how any earlier work could be cited as referring to the Real numbers, except as an axiom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For reference: &lt;a href=&quot;http://mathworld.wolfram.com/ArchimedesAxiom.html&quot; rel=&quot;noreferrer&quot;&gt;Archimedes' Axiom&lt;/a&gt;  and &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Archimedean_axiom&quot; rel=&quot;noreferrer&quot;&gt;Archimedean axiom&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3821" LastEditorUserId="3821" LastEditDate="2016-05-15T21:26:39.570" LastActivityDate="2016-05-15T21:26:39.570" CommentCount="25" ContentLicense="CC BY-SA 3.0" />
  <row Id="3766" PostTypeId="2" ParentId="3759" CreationDate="2016-05-11T18:41:51.667" Score="2" Body="&lt;p&gt;Man, as a collective, is credited in a famous quote:  &quot;God made the integers; all else is the work of man&quot; (or &quot;&lt;a href=&quot;https://de.wikiquote.org/wiki/Leopold_Kronecker&quot; rel=&quot;nofollow&quot;&gt;Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk&lt;/a&gt;&quot;). The quote is often attributed to Leopold Kronecker, see for instance &quot;Philosophies of Mathematics&quot;, p.  13, Alexander George, Daniel J. Velleman, 2001. Apparently, the authenticity of the quote is disputed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book also kind of credits Dedekind:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Of particular note in this  connection is the accomplishment, due&#xA;  primarily to the German mathematician Richard Dedekind (1831-1916), of&#xA;  defining the integers, rationals and reals, taking only the system of&#xA;  natural numbers for granted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From my education, with a western twist, the  &lt;a href=&quot;https://en.wikipedia.org/wiki/Dedekind_cut&quot; rel=&quot;nofollow&quot;&gt;Dedekind cut&lt;/a&gt;  was a construction of real numbers that allowed me to get a instantaneous (perhaps faulty) inner picture of reals, based on (natural) rationals, which I did not grasp before (with unending decimal representations). I learned about it while studying rational (Diophantine) approximations of polynomial roots in the real field and in finite fields (continuing fractions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the history of science, the person who gets credits is not always the first. In the West, C. Columbus often gets credits for discovering America, which is probably unfair. Should the first one proving that there were at least one irrational number be credited too?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I begin (after your comments) to think that answers depend of &quot;what kind of real numbers?&quot;, in other words, which which structure? As points on a line, as a succession of figures, as a ring or  field structure, as a vector space or an an algebra, as a &quot;sense&quot; of continuity? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From my second hand knowledge, some say that arabic/muslim (in a wide sense) mathematicians were the first to treat irrational numbers as algebraic objects (possibly only surds), and indian ones developed trigonometric series (&lt;a href=&quot;http://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall14/128/CalcIslamIndia.pdf&quot; rel=&quot;nofollow&quot;&gt;Ideas of Calculus in Islam and India&lt;/a&gt;, Katz, 1995). And the first time I heard about (one instance of) the &lt;a href=&quot;https://en.wikipedia.org/wiki/Basis_(linear_algebra)#Hamel_basis&quot; rel=&quot;nofollow&quot;&gt;Hamel basis&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a basis for the real numbers \mathbb{R} as a vector space over the field \mathbb{Q} of&#xA;  rational numbers&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I understood that my level in mathematics was too narrow to understand what real numbers really  were. Since Heron of Alexandria is sometimes credited with the first (western) notion of imaginary numbers, can we expect the real was discovered after the complex?&lt;/p&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="4130" LastEditDate="2016-05-12T19:26:29.077" LastActivityDate="2016-05-12T19:26:29.077" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3767" PostTypeId="1" AcceptedAnswerId="3783" CreationDate="2016-05-11T18:46:10.150" Score="5" ViewCount="1752" Body="&lt;p&gt;Now set theory is taught even to kids and it is the foundation of mathematics. Can we say that Cantor won?&lt;/p&gt;&#xA;" OwnerUserId="4133" LastEditorUserId="4251" LastEditDate="2022-04-27T05:34:05.267" LastActivityDate="2022-04-27T05:34:05.267" Title="Kronecker vs Cantor — who won?" Tags="|mathematics|mathematicians|set-theory|cantor|kronecker|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="3768" PostTypeId="1" CreationDate="2016-05-12T10:33:03.713" Score="3" ViewCount="291" Body="&lt;p&gt;There are certain explanations on how integers might have evolved, like for example &quot;the wings of a bird to symbolize the number two, clover--leaves three, the legs of an animal four, the fingers on his own hand five.&quot;$_1$ Seeing all these, and making experience short--abstract, can be said to have given the numbers$_2$. Egyptian integer number notation can itself be traced back to their notion of marks on tally (for the sign of one), hobble for cattle (for ten), measuring rope (for hundred).$_3$   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly what are the explanations given for the evolution of fractions in Egypt, by any of the found evidence? Did division discovery gave raise to fractions or fractions discovery gave raise to division, in Egypt?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Book and journal reading suggestions would also be helpful.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$_1$Tobias Dnatzig, &lt;em&gt;Number: The Language of Science&lt;/em&gt;&lt;br&gt;&#xA;$_2$This expression can be found in Hamilton's letter of Sep.16, 1828, in the book &lt;em&gt;Life of Sir William Rowan Hamilton&lt;/em&gt;.&lt;br&gt;&#xA;$_3$Annette Imhausen, &lt;em&gt;Mathematics in Ancient Egypt: A Contextual History&lt;/em&gt; Page No.18-21.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2020-03-13T22:48:47.460" Title="How did egyptian mathematical fractions evolve in antiquity?" Tags="|mathematics|ancient-egypt|linguistics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3769" PostTypeId="1" CreationDate="2016-05-12T11:10:54.760" Score="0" ViewCount="34" Body="&lt;p&gt;I know for example that one of the reasons why our cellular phones need to be turned off in airplanes is that the navigation system of the plane (GPS and other systems) works on frequencies close to that of cellular phones, and that can effect the reading of the navigation equipment (the communication with the satelite goes bad). So my question is what is the technic reason for the wide-spread use of certain frequencies?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2016-05-12T11:10:54.760" Title="Why do so many electronic equipment work on the same or close frequencies?" Tags="|electromagnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3770" PostTypeId="1" AcceptedAnswerId="3774" CreationDate="2016-05-12T12:54:09.130" Score="4" ViewCount="786" Body="&lt;p&gt;The familiar quote is often incorrectly attributed to Kronecker directly. Actually a colleague of his named Weber claimed after Kronecker's death that Kronecker said this. I have doubts about this because Kronecker would not have used the term &quot;integer&quot;.  He was almost as suspicious of the negative numbers as he was of transcendental numbers. Furthermore he specifically wrote that the numbers are a creation of the human mind, implying a measure of contingency not conveyed by Weber's quote at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note. Thanks to Colin McLarty for providing the appropriate quote, this one literally from Kronecker.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="4251" LastEditDate="2022-04-27T05:29:50.133" LastActivityDate="2022-04-27T05:29:50.133" Title="Did Kronecker attribute immutable origin to the integers?" Tags="|mathematics|mathematicians|biographical-details|debunking|kronecker|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3771" PostTypeId="2" ParentId="3770" CreationDate="2016-05-12T14:46:51.293" Score="3" Body="&lt;p&gt;It seems Kronecker actually said&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and did not mention &quot;integers&quot; or any other English word.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.todayinsci.com/K/Kronecker_Leopold/KroneckerLeopold-Quotations.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;citation&lt;/a&gt; ... Speech at the Berlin meeting of the Society of German Scientists and Doctors in 1886, quoted by Weber in &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=PPN37721857X_0002|log6&amp;amp;physid=phys24&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung &lt;strong&gt;2&lt;/strong&gt; (1893), p. 19&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="658" LastEditDate="2017-06-04T20:28:20.750" LastActivityDate="2017-06-04T20:28:20.750" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3772" PostTypeId="2" ParentId="3768" CreationDate="2016-05-13T00:37:19.383" Score="3" Body="&lt;p&gt;According to Carl Boyer, in his text &lt;em&gt;A History of Mathematics&lt;/em&gt;, the earliest record we have regarding fractions in ancient Egypt comes from the Ahmes Papyrus, popularly known as the Rhind Papyrus, circa 1650 BCE.  The scribe, Ahmes, tells us that it is derived from a prototype of the Middle Kingdom, circa 2000-1800 BCE, and it is possible that some of this knowledge may be handed down from as early as 3000 BCE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Boyer, the general concept of fraction appears to have been an enigma to the Egyptians.  They did not regard the general rational fraction $\frac{n}{m}$ as an elementary &quot;thing&quot;, but as an uncompleted process. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where today we think of $\frac{3}{5}$ as a single irreducible fraction, Egyptian scribes thought of it as reducible to sum of the three unit fractions $\frac13$ and $\frac15$ and $\frac1{15}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;With this in mind, the papyrus begins with two tables; the first expressing fractions of the form $\frac{2}{n}$ for odd $n$ from $5$ to $101$ expressed as the sum unit factions ($\frac{1}{m}$). A second, shorter table follows giving a similar treatment for fractions of the form $\frac{n}{10}$ for $n = 1$ to $9$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(To complicate things, in addition to unit fractions, the Egyptians also appear to have been comforatble with the fraction $\frac23$, which appears in the tables. For example $\frac9{10}$ is given as $\frac{1}{30} + \frac15 + \frac23$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following the two tables is a list of eighty-four assorted problems which may provide some motivation for the &lt;strong&gt;evolution&lt;/strong&gt; of fractions in Egypt.  Typical of these problems is the &lt;em&gt;division&lt;/em&gt; of a given number of loaves of bread amongst ten men.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Boyes :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The fundamental arithmetic operation in Egypt was addition, and our operations of multiplication and division were performed in Ahmes' day through successive doubling, or &quot;duplication&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This suggests that the concept of fraction preceded the &lt;em&gt;formal&lt;/em&gt; concept of division.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-05-13T00:37:19.383" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3773" PostTypeId="1" AcceptedAnswerId="3775" CreationDate="2016-05-13T01:39:05.067" Score="1" ViewCount="90" Body="&lt;p&gt;Has anyone classified (or attempted to classify) all the experimentally-determined laws of physics, such as with a simple diagram or even in a full-length book?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing an &lt;a href=&quot;https://www.worldcat.org/search?q=su%3A%22experimental%20physics%22&amp;amp;qt=results_page&quot; rel=&quot;nofollow&quot;&gt;&quot;experimental physics&quot; subject search on WorldCat&lt;/a&gt; doesn't reveal much.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-05-13T16:27:06.637" LastActivityDate="2016-05-13T18:49:30.970" Title="Classification of experimentally-determined physics laws?" Tags="|physics|experimental-physics|resource-recommendation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3774" PostTypeId="2" ParentId="3770" CreationDate="2016-05-13T13:19:27.487" Score="9" Body="&lt;p&gt;Weber was a pretty reliable witness.  Probably Kronecker did tell the 1886 Berliner Naturforscher-Versammlung something like &quot;the whole numbers were made by dear God (der liebe Gott), the rest is the work of man.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Kronecker also published his endorsement of Gauss saying the opposite:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The principal difference between geometry and mechanics on one hand,&#xA;  and the other mathematical disciplines we comprehend under the name of&#xA;  \arithmetic, consists according to Gauss in this: the object of the&#xA;  latter, number, is a pure product of our mind, while space as well as&#xA;  time has reality also outside of our mind which we cannot fully&#xA;  prescribe a priori. (Kronecker, Uber den Zahlbegriff, 1887, p. 339)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is extremely unlikely that Kronecker or Weber intended the remark as anything serious about theology.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The late Walter Felscher has pointed out on the list HistoriaMatematica (26 May 1999)  archived at mathforum.org that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Lieber Gott&quot; is a colloquial phrase usually used only when speaking&#xA;  to children or illiterati.  Addressing grown-ups with it contains a&#xA;  taste of being unserious, if not condescending. . . ; no priest,&#xA;  pastor, theologian or philosopher would use it when expressing himself&#xA;  seriously. There is the well known joke of Helmut Hasse who, having&#xA;  quoted Kronecker's dictum on page 1 of his yellow Vorlesungen uber&#xA;  Zahlentheorie (1950), added to the index of names at the book's end&#xA;  under the letter L the entry&quot; Lieber Gott p. 1.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kronecker did seriously aim to replace the algebra of irrational numbers with the &quot;pure arithmetic&quot; of natural number polynomials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to mutability and contingency, there seem to be two questions: Would numbers exist if people (or God) had not thought of them?  Could some other version of numbers have been created instead of the ones we have?  I know no evidence that Kronecker ever considered either question -- or that he did not!  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the second, compare Dedekind.  He was explicit that the natural and real numbers are both our thought-creations yet he never breathed a word of any idea that we could have created them differently than we did. &lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2016-05-15T19:30:21.467" LastActivityDate="2016-05-15T19:30:21.467" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3775" PostTypeId="2" ParentId="3773" CreationDate="2016-05-13T18:49:30.970" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://store.doverpublications.com/0486656322.html&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;A History of Mechanics&lt;/em&gt;&lt;/a&gt;, by Dugas (1955) is available from Dover; Part V is an introduction to Quantum Theory.  Some experiments are discussed, but it is mostly a scholarly discussion of ideas and how they evolved over time, with proper references, as explained by a trained engineer and historian of science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The scope is well described by the preface and table of contents. Dugas references earlier compilers, and more detailed works of scholarship.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_electromagnetic_theory&quot; rel=&quot;nofollow&quot;&gt;History of electromagnetism&lt;/a&gt; is a Wikipedia resource, providing references. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scholarpedia provides an open-access, peer reviewed &lt;a href=&quot;http://www.scholarpedia.org/article/Encyclopedia:Physics&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Encyclopedia: Physics&lt;/em&gt;&lt;/a&gt;, which could keep you busy for a few years, or decades.  The content is intended to be &quot;a living review&quot; of each topic; it is not intended to provide a history of physics, but rather, the current state of it.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-05-13T18:49:30.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3776" PostTypeId="1" CreationDate="2016-05-14T01:28:43.243" Score="0" ViewCount="94" Body="&lt;p&gt;Were there physics teaching aids used in the past? Or did professors just basically follow the textbook the students read?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words: How was physics was taught in the past (e.g., in the 19th century)?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-05-14T01:28:43.243" Title="How was physics taught in the past (e.g., in the 19th century)?" Tags="|physics|education|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3777" PostTypeId="1" AcceptedAnswerId="3778" CreationDate="2016-05-14T01:46:50.133" Score="3" ViewCount="355" Body="&lt;p&gt;I know that G.H. Hardy hated applied mathematics, and I think I remember he was 'tricked' into discovering the Hardy-Weinberg equilibrium. How was he tricked into doing this, and what did Weinberg have to do with it, if he was all related to this?&lt;/p&gt;&#xA;" OwnerUserId="140" LastActivityDate="2016-05-14T03:25:44.550" Title="How was Hardy tricked into creating the Hardy-Weinberg Equilibrium" Tags="|mathematics|biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3778" PostTypeId="2" ParentId="3777" CreationDate="2016-05-14T03:08:42.443" Score="4" Body="&lt;p&gt;Hardy's personal friend and fellow cricket enthusiast, &lt;a href=&quot;https://en.wikipedia.org/wiki/Reginald_Punnett&quot; rel=&quot;nofollow&quot;&gt;Reginald Punnett&lt;/a&gt;, put a special case of the problem to Hardy.  They were both fellows of Trinity College and Gonville and Caius College, Cambridge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According the &lt;a href=&quot;http://www.genetics.org/content/179/3/1143&quot; rel=&quot;nofollow&quot;&gt;this article on the genetics.org website&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... Punnett was perplexed as to why the dominant gene did not continually increase in frequency, and this is the problem he put to Hardy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;em&gt;Later in the article, Punnett claims the problem was framed as a mathematical one.  See below.&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Social life revolved around the common meals, especially the evening “High-Table” dinner with its opportunities for relaxed conversation across the barriers between subjects. It has sometimes been said that Punnett put his genetic question to Hardy at one such dinner, but there is no evidence for this. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Punnett (1950) simply wrote that on his return to Cambridge from his lecture in London “I at once sought out G. H. Hardy with whom I was then very friendly.” He went on to explain that this was not only because they used to play cricket together, but also because they had acted as joint secretaries to a committee that had been lobbying for the retention of Greek in the Cambridge entrance examination. Nothing was more natural than that the wholly unmathematical Punnett should rub shoulders with the leading mathematician of the day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“I put my problem to him as a mathematical one. He replied that it was quite simple and soon handed to me the now well-known formula $pr = q^2$. Naturally pleased at getting so neat and prompt an answer I promised him that it should be known as ‘Hardy’s Law'—a promise fulfilled in the next edition of my Mendelism (Punnett 1950).”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is nothing here to suggest that any trickery was involved other than Punnett's claim that he posed the problem as a mathematical one.  Hardy simply appears to be responding to a request from a friend. In Hardy's letter to &lt;em&gt;Science&lt;/em&gt;, he comments on the simplicity of the problem : &quot;Moreover, I should have expected the very simple point which I wish to make to have been familiar to biologists.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, despite Punnett's claim, the article also includes the following :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hardy did not use the word “equilibrium” (as Pearson had), nor indeed did Punnett actually use the phrase “Hardy's law” in his book. But Hardy's law it became, and then the Hardy–Weinberg law (Stern 1943).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding your question on the role of Weinberg, again according to the genetics.org article :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Weinberg's] solution in fact preceded Hardy's (which was obtained independently).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Since Hardy was the first published solution, his name is attached to the result.  Since the biological community recognise Weinberg's solution as having preceded Hardy's, the result is now known as Hardy-Weinberg.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-05-14T03:25:44.550" LastActivityDate="2016-05-14T03:25:44.550" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3779" PostTypeId="1" CreationDate="2016-05-14T13:43:28.953" Score="1" ViewCount="63" Body="&lt;p&gt;I have read casual statements without reference suggesting that Maupertuis posed a question similar to Feynman's (in &lt;a href=&quot;http://www.feynmanlectures.caltech.edu/II_19.html&quot; rel=&quot;nofollow&quot;&gt;this lecture, for instance&lt;/a&gt;) as to how a particle &quot;knows&quot; to follow the path of stationary [but probably Maupertuis would have said least] action.  Does anyone know of a primary source where either Maupertuis or some other early proponent of action principles considered such a question?  Thank you very much in advance!&lt;/p&gt;&#xA;" OwnerUserId="4141" LastActivityDate="2016-06-13T17:42:51.837" Title="Did Maupertuis ask how a particle knows to follow the path of stationary action?" Tags="|theoretical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3780" PostTypeId="2" ParentId="3779" CreationDate="2016-05-14T17:01:49.817" Score="1" Body="&lt;p&gt;Maupertuis' consideration seems different...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pierre Louis Maupertuis, &lt;a href=&quot;https://books.google.it/books?id=cEdJAAAAcAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow&quot;&gt;Accord de différentes loix de la nature qui avoient jusqu'ici paru incompatibles&lt;/a&gt; (1744):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[page 12] &lt;em&gt;la Nature, dans la production de ses effets , agit toujours par les moyens les plus simples&lt;/em&gt;. Si un corps doit aller d'un point à un autre sans rencontrer nul obstacle , ou s'il n'y doit aller qu'après avoir rencontré un obstacle invincible , la Nature l'y conduit par le chemin le plus court,&#xA;  et par le temps le plus prompt.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 15] &lt;em&gt;la lumiere se meut le plus vite dans les milieux les plus denses&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 16] En méditant profondément sur cette matiere, j'ai pensé que la lumiere, lorsqu'elle passe d'un milieu dans un autre, abandonnant déjà le chemin le plus court, qui est celui de la ligne droite, pouvoit bien aussi ne pas suivre celui du temps le plus prompt. &lt;em&gt;En effet, quelle préférence devroit-il y avoir ici du temps sur l'espace ? la lumiere ne pouvant plus aller tout à la fois par le chemin le plus court , et par celui du temps le plus prompt,&#xA;  pourquoi iroit-elle plutôt par l'un de ces chemins que par l'autre ?&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;] &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Aussi ne suit-elle aucun des deux: elle prend une route qui a un avantage plus réel: &lt;em&gt;le chemin qu'elle tient est celui par lequel la quantité d'action est la moindre&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 19] Tous les phénomenes de la réfraction s'accordent maintenant avec le&#xA;  grand principe, que &lt;em&gt;la Nature, dans la production de ses effets, agit toujours par les voies les plus simples&lt;/em&gt;. De ce principe suit, que &lt;em&gt;lorsque la lumiere passe d'un milieu dans un autre, le sinus de son angle de refraction est au sinus de son angle d'incidence en raison inverse des vitesses qu'a la lumiere dans chaque milieu&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-05-14T17:01:49.817" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3781" PostTypeId="1" CreationDate="2016-05-14T18:15:08.633" Score="20" ViewCount="1844" Body="&lt;p&gt;A similar question is found below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/19/scientific-progress-claimed-to-be-caused-by-dreaming&quot;&gt;Scientific progress claimed to be caused by dreaming&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone ever made similar claims regarding other altered states of consciousness, besides dreaming?  I know similar claims have been found in the sciences, but what about mathematics specifically?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us exclude disease, as there are cases of mathematicians with mental illnesses that have certainly given much!&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2020-07-14T06:00:05.650" Title="Claims of mathematical breakthroughs while in an altered state of consciousness?" Tags="|mathematicians|biographical-details|" AnswerCount="6" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3782" PostTypeId="2" ParentId="3781" CreationDate="2016-05-14T18:59:10.573" Score="20" Body="&lt;p&gt;Scientists and mathematicians rarely self-report on psychological circumstances of their creative process, at best presenting a rationalization of their path to discovery. Kekule's dream and Kepler's vivid struggles described in Astronomia Nova are exceptions (and there are doubts that Kepler's descriptions are factual, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/1948/how-did-kepler-guess-his-third-law-from-data/1950#1950&quot;&gt;How did Kepler &amp;quot;guess&amp;quot; his third law from data?&lt;/a&gt;). Third person accounts often take on anecdotal character, like Eureka or &lt;a href=&quot;http://www.todayifoundout.com/index.php/2013/10/origin-apple-falling-newtons-head-story&quot; rel=&quot;noreferrer&quot;&gt;Newton's apple&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Two first hand works on psychology of discovery by mathematicians are a chapter in &lt;a href=&quot;https://archive.org/details/sciencemethod00poinuoft&quot; rel=&quot;noreferrer&quot;&gt;Poincare's Science and Method&lt;/a&gt; and &lt;a href=&quot;http://worrydream.com/refs/Hadamard%20-%20The%20psychology%20of%20invention%20in%20the%20mathematical%20field.pdf&quot; rel=&quot;noreferrer&quot;&gt;Hadamard's Psychology of Invention in the Mathematical Field&lt;/a&gt;. In particular, Poincare gives a famous report of his &amp;quot;altered state&amp;quot; when discovering what he called Fuchsian functions (now &lt;a href=&quot;https://en.wikipedia.org/wiki/Automorphic_form&quot; rel=&quot;noreferrer&quot;&gt;automorphic forms&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is time to penetrate further and to see what goes on in the very soul of the mathematician. For this, I believe, I can do best by recalling memories of my own. But I shall limit myself to telling how I wrote my first memoir on Fuchsian functions. I beg the reader's pardon; I am about to use some technical expressions, but they need not frighten him, for he is not obliged to understand them. I shall say, for example, that I have found the demonstration of such a theorem under such circumstances. This theorem will have a barbarous name, unfamiliar to many, but that is unimportant; what is of interest for the psychologist is not the theorem but the circumstances.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which came from the hypergeometric series; I had only to write out the results, which took but a few hours.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-07-22T01:10:10.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3783" PostTypeId="2" ParentId="3767" CreationDate="2016-05-14T21:43:42.077" Score="9" Body="&lt;p&gt;No, Cantor did not &quot;win&quot;, for a very simple reason: the race is not over.  Cantor may be in the lead, but there is no reason to think that Kronecker ( or somebody else ) will not be in the lead 100 or 200 years from now.  Also, it is incorrect to say that X is &lt;em&gt;the&lt;/em&gt; foundation for mathematics.  There are multiple competitors for that title, and the best bet ( IMO) is that there is no one true &quot;foundation&quot; for mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.  The major competitors today are Category Theory and (Homotopy) Type Theory.  There are probably others, but those two are pretty well-known.&lt;/p&gt;&#xA;" OwnerUserId="4062" LastEditorUserId="4062" LastEditDate="2017-06-05T19:23:07.813" LastActivityDate="2017-06-05T19:23:07.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3784" PostTypeId="2" ParentId="3781" CreationDate="2016-05-15T01:39:08.873" Score="15" Body="&lt;p&gt;Paul Erdős was a mathematician noted for his prolific contributions to the science of mathematics as well as a proclivity for the use of amphetamines. His friend Ronald Graham famously bet him $500 (in the 1970's) that he could not stop taking amphetamine for a month. He successfully completed the challenge, however begrudgingly, saying, &quot;Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Erdős resumed his amphetamine use after the month was over, allegedly asserting that the science of mathematics had been set back by a month due to his abstinence. While it is not immediately clear which of his mathematical works were completed under the influence of amphetamines and which were not, it is quite likely that at least some of his work, especially in the last quarter-century of his life, was created under the influence of stimulants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A mention should also be given to a commonly-related story about Drs. Watson and Crick that they discovered the structure of DNA under the influence of LSD; while although they have been historically credited for the discovery, credit has been more recently given to Dr. Rosalind Franklin, whose unpublished work on DNA structure was evidently shared with Watson and Crick some months before their historic article was published in Nature. Therefore, the story that they first imagined the double-helix structure of DNA molecules while under the influence of the psychedelic must be at least somewhat apocryphal.&lt;/p&gt;&#xA;" OwnerUserId="4142" LastEditorUserId="4142" LastEditDate="2016-05-15T04:54:24.017" LastActivityDate="2016-05-15T04:54:24.017" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3785" PostTypeId="2" ParentId="3781" CreationDate="2016-05-15T03:20:51.877" Score="-1" Body="&lt;p&gt;I recently watched a Ted-Ed video, and it talked about how Van Gogh supposedly painted turbulence in a mathematically correct way when he was in a period of intense suffering.  This is a feat because turbulence had not been discovered yet, and to this day is not fully understood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=PMerSm2ToFY&quot; rel=&quot;nofollow&quot;&gt;Link&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4143" LastActivityDate="2016-05-15T03:20:51.877" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3786" PostTypeId="1" CreationDate="2016-05-15T08:51:06.263" Score="1" ViewCount="310" Body="&lt;p&gt;Why did it take so long for Jensen's inequality?&lt;br&gt;&#xA;Jensen's inequality (compared with Lagrange multipliers, etc.) is based on comparatively elementary part of calculus. Still it was proven in 1906.&lt;br&gt;&#xA;Is it at all possible that someone might have gotten it but did not publish it ? &lt;/p&gt;&#xA;" OwnerUserId="4146" LastEditorUserId="2442" LastEditDate="2016-05-15T21:49:43.263" LastActivityDate="2016-05-15T21:49:43.263" Title="Why did it take so long for Jensen's inequality?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3787" PostTypeId="2" ParentId="3786" CreationDate="2016-05-15T13:39:07.903" Score="2" Body="&lt;p&gt;The history behind Jensen's inequality is actually quite a long and difficult story. There is even a &lt;a href=&quot;http://prr.hec.gov.pk/Thesis/2306S.pdf&quot; rel=&quot;nofollow&quot;&gt;thesis&lt;/a&gt; written on the subject by Rabia Bibi (2013). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1875 the following inequality was proven by J. Grolous:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f\left(\frac{x_1+x_2+\cdots+x_n}{n}\right) \leq \frac1n\left(f(x_1)+f(x_2)+\cdots+f(x_n)\right)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for all convex functions $f$ and $x_1, x_2, \cdots, x_n$ real numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1889 the following inequality was proven by O. Hölder:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f\left(p_1x_1+p_2x_2+\cdots+p_nx_n\right) \leq p_1f(x_1)+p_2f(x_2)+\cdots+p_nf(x_n)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for all convex functions $f$ and $x_1, x_2, \cdots, x_n$ real numbers and $p_1, p_2, \cdots, p_n$ positive real numbers such that $p_1+p_2+\cdots+p_n=1$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.cs.cmu.edu/~suvrit/teach/papers/1906_jensen_sur_les_fonctions_convexes_et_les_inegalites_entre_valeurs_moyennes.pdf&quot; rel=&quot;nofollow&quot;&gt;1906 paper of Jensen&lt;/a&gt; shows another result on integrals, in addition to the result on sums. This states, according to Bibi (2013):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$g\left(\frac{\int_a^b f(t) \mathrm{d}t}{b-a} \right) \leq \frac{\int_a^b g(f(t)) \mathrm{d}t}{b-a}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;with $a,b \in \mathbb R$ and $g$ an integrable convex function and $f$ an integrable function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these are sometimes called the Jensen inequality. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2016-05-15T13:39:07.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3789" PostTypeId="1" CreationDate="2016-05-16T11:51:49.583" Score="2" ViewCount="159" Body="&lt;p&gt;I read in several sites that Gauss made several contributions to technology, and that his main contributions were: the heliotrope, improvements of several optical devices (telescopes, introduction of Gauss-lenses,...), the photometer, magnetometer (and maybe several other electromagnetic instruments), and the electromagnetic telegraph. Actually, pages 186-189 of Clemens Schaefer's treatise on Gauss's contributions to physics deal with Gauss's small contribution to photometry. According to them, Gauss tried to establish a reliable method for photomety of stars - i.e for a quantitive measurement of the brightness of star. At his time, such a method didn't exist yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schaefer gives additional details about Gauss's own description of his design of photometer, according to which it was a comparative photometer that was able to &quot;tune&quot; the intesity of light from two stars, until the stars appeared &quot;equally bright&quot; to the eye (and that was something that had to be determined visually by the eye). Then the stars's relative brightness could be established by knowledge of the brightness magnification factor applied by the instrument. Gauss also described very clearly the difficulties in the problem of heterochromatic photometry (the problem of visually deciding whetever two light sources of different colours are equally bright). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, in Gauss: Titan of Science, Dunnington mentions that Gerling constructed a photometer according to principles outlined by Gauss. But despite of this, i coundn't find any information about the photometer which Gauss invented in the english literature. I'll specify what i want to know:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What was the internal structure of Gauss's photometer? what were his principles of operation? as far as i know, a comparative photometer must have two essential abilities: bringing the image of the measured star to the proximity of a standard source (even if the star might be angulary distant from the known source), and magnification of the brightness of one star. Gauss's comparative photometer must be judged by these criterions. &lt;/li&gt;&#xA;&lt;li&gt;Did Gauss's photometer have any importance  for the later developement of comparative photometers?    &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2018-10-12T10:51:29.177" LastActivityDate="2018-10-12T10:51:29.177" Title="What were the scientific principles of Gauss's photometer?" Tags="|biographical-details|optics|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="3790" PostTypeId="2" ParentId="3761" CreationDate="2016-05-16T14:40:29.037" Score="4" Body="&lt;p&gt;See the entry for kernel in Jeff Miller's &lt;a href=&quot;http://jeff560.tripod.com/k.html&quot; rel=&quot;nofollow&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt; web pages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, not mentioned in Jeff Miller's web pages is the &lt;a href=&quot;https://www.google.com/search?q=%22perfect+kernel%22+%22Cantor-Bendixson%22&quot; rel=&quot;nofollow&quot;&gt;&quot;perfect kernel&quot; in the Cantor-Bendixson theorem sense&lt;/a&gt; (&quot;nucleus&quot; and &quot;core&quot; are alternative terms that are also used). My initial thought was that this may predate most or all of the other uses at Jeff Miller's web pages, but after looking through some of my references on the Cantor-Bendixson theorem, I believe &quot;kernel&quot; was not used in this way until 1914 1st edition of Hausdorff's &lt;a href=&quot;https://en.wikipedia.org/wiki/Grundz%C3%BCge_der_Mengenlehre&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Grundzüge der Mengenlehre&lt;/strong&gt;&lt;/a&gt;. See pp. 220 &amp;amp; 226 in &lt;a href=&quot;https://archive.org/details/grundzgedermen00hausuoft&quot; rel=&quot;nofollow&quot;&gt;this copy of Hausdorff's 1914 book&lt;/a&gt;, where &quot;Kern&quot; -- the German version of the word -- is used.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2016-05-16T14:40:29.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3791" PostTypeId="1" AcceptedAnswerId="3796" CreationDate="2016-05-16T21:41:24.733" Score="2" ViewCount="1060" Body="&lt;p&gt;What are epicycles, deferents, eccentrics, equants, etc.? Who first introduced them, and when? Why were they introduced? How do they work? Are there visualizations of this?&#xA;&lt;hr&gt;&#xA;&lt;sup&gt;e.g., what is St. Thomas Aquinas describing here in his commentary on Aristotle's &lt;em&gt;De Cœlo&lt;/em&gt; (&lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#2-17&quot; rel=&quot;nofollow noreferrer&quot;&gt;lib. 2 l. 17&lt;/a&gt;)?&lt;blockquote&gt;&lt;sup&gt;To each of these he [Eudoxus or Aristotle?] assigned four movements, according to the four spheres that revolved the stellar body fixed in the lowest of them. Thus the first sphere moves the stellar body from east to west according to the diurnal motion; the second moves the stellar body in the opposite direction of west to east in the Zodiac — and this is called longitudinal motion; the third sphere moves a stellar body latitudinally, according to which a star is now in a more southerly, now a more northerly, position in the Zodiac. Now he placed the poles of this third sphere in the Zodiac; hence it followed that a major circle, equidistant from the poles, would go through the poles of the Zodiac. From this it seemed to follow that the planets in their latitudinal motion would sometimes reach the very poles of the Zodiac — a situation that never appears. Hence, he posited a fourth sphere that would move a star in an opposite direction to this movement and thus prevent it from ever reaching the poles of the Zodiac. He did not, however, attribute the motion of this fourth sphere to the sun and the moon, but tried to save their appearances by positing only three spheres, proportional to the first three of the other planets, but in such a way that the latitudinal motion of the moon would be greater than the sun's, as is explained in Metaphysics XII.&lt;/sup&gt;&lt;/blockquote&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-01-07T19:34:57.660" LastActivityDate="2020-01-07T19:34:57.660" Title="What are epicycles, deferents, eccentrics, equants, etc.?" Tags="|astronomy|celestial-mechanics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="3792" PostTypeId="1" AcceptedAnswerId="3793" CreationDate="2016-05-16T22:26:46.573" Score="9" ViewCount="1762" Body="&lt;p&gt;Who invented using $e^{\lambda t}$ as ansatz for solving linear differential equations with constant coefficients?&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="55" LastEditDate="2016-05-17T00:17:25.273" LastActivityDate="2016-07-22T01:07:28.977" Title="Who invented the exponential ansatz for linear differential equations with constant coefficients?" Tags="|mathematics|discoveries|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3793" PostTypeId="2" ParentId="3792" CreationDate="2016-05-16T23:36:23.997" Score="14" Body="&lt;p&gt;The short answer is Euler. Some details are given in the following long quote from &lt;a href=&quot;https://books.google.com/books?id=pOQy6-qnVx8C&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow&quot;&gt;Hald's History of Probability and Statistics and Their Applications before 1750&lt;/a&gt; (p.438):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;In 1743 Euler solved the homogeneous linear differential equation of the&#xA;  m-th order with constant coefficients (using the same idea as de Moivre) by&#xA;  guessing at a particular solution of the form $y = c\,e^{rx}$ and thus deriving the characteristic equation as an algebraic equation of the $m$-th degree. He pointed out that the general solution will be a linear combination of $m$ independent particular solutions. The year before he had proved (incompletely) that a polynomial with real coefficients can be decomposed into linear and quadratic factors with real coefficients, and he could therefore assert that the characteristic equation always has $m$ roots. He found that the particular solution for a single real root has the form $y = c\,e^{rx}$, for a real root of multiplicity $k$ the form  $e^{rx}$ times a polynomial in $x$ of degree $k$, and for a pair of conjugate complex roots the form $y = e^{ax}(c_1 \cos bx + c_2 \sin bx)$, where $c_1$ and $c_2$ are arbitrary constants.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In 1753 Euler solved the nonhomogeneous differential equation of the&#xA;  $m$-th order with constant coefficients by devising a method for reducing the&#xA;  order by unity and thus successively reducing the problem to the solution&#xA;  of a nonhomogeneous differential equation of the first order...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In the 1760s Lagrange proved that Euler's 1753 theorem also holds for a&#xA;  nonhomogeneous differential equation of the $m$-th order with variable&#xA;  coefficients so that the general solution may be expressed by means of the&#xA;  solution of the homogeneous equation and the solution of an adjoint equation&#xA;  of the first order.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The justification obviously depends on the Fundamental Theorem of Algebra, and according to the conventional wisdom Euler's reasoning for it had an irreparable gap, which was filled by Gauss. Depending on one's strictures for rigor even Gauss may not be beyond reproach, see &lt;a href=&quot;http://www.maa.org/programs/maa-awards/writing-awards/euler-and-the-fundamental-theorem-of-algebra&quot; rel=&quot;nofollow&quot;&gt;Dunham's Euler and the Fundamental Theorem of Algebra&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Euler-Lagrange presentation of a general nonhomogeneous solution as a sum of general homogeneous and particular solutions is recognized today as a direct consequence of linearity, this is a general form of solution to underdetermined linear systems. But of course this insight was not yet available in their time. The method for finding a particular solution which came to be known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Variation_of_parameters#History&quot; rel=&quot;nofollow&quot;&gt;the variation of parameters&lt;/a&gt; was used by Euler in a particular case already back in 1748, but was only given its final form by Lagrange in 1808-1810. Both Euler and Lagrange were solving equations related to perturbations of elliptic orbits of the planets and the Moon.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-07-22T01:07:28.977" LastActivityDate="2016-07-22T01:07:28.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3794" PostTypeId="2" ParentId="3752" CreationDate="2016-05-17T18:08:22.183" Score="3" Body="&lt;p&gt;Galileo knew that, if a stream of water is ejected from a pipe held horizontally, the stream would follow the shape of a parabola.  Mathematically, that means that the motion of the water is linear (uniform) horizontally, and quadratic vertically.  And that leads to the realization that the motion horizontally and the motion vertically can be treated independently.  He could thus use the inclined plane to stretch the vertical motion out so that the times were long enough to measure easily (probably using his pulse as a timer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did he know that the stream of water took the shape of a parabola?  Well, he was a man, and he often walked in the woods.&lt;/p&gt;&#xA;" OwnerUserId="2993" LastActivityDate="2016-05-17T18:08:22.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3795" PostTypeId="1" AcceptedAnswerId="3799" CreationDate="2016-05-17T19:41:49.720" Score="7" ViewCount="178" Body="&lt;p&gt;The following advertisement recently appeared in the sidebar on math.se:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kHGW4.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kHGW4.png&quot; alt=&quot;On the left is handwritten Greek script; on the right are the words “Does your question fit in this margin? History of Science &amp;amp; Math”&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the Greek script on the left actually from a mathematical manuscript?  What is its source?&lt;/p&gt;&#xA;" OwnerUserId="134" LastEditorUserId="55" LastEditDate="2016-05-17T22:30:47.647" LastActivityDate="2016-05-18T13:04:22.200" Title="What manuscript is depicted in the HSM advertisement?" Tags="|mathematics|reference-request|ancient-greece|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3796" PostTypeId="2" ParentId="3791" CreationDate="2016-05-17T21:44:20.727" Score="4" Body="&lt;p&gt;All these words are from astronomy, used to describe the motion of the Sun, Moon and planets.&#xA;The simplest model is this. Suppose that Earth is the center of a circle. This circle is called deferent. A point moves on&#xA;the deferent with constant angular velocity, this point is called the center of the epicycle, and epycycle itself is a circle centered at this point. The planet moves on the epicycle with constant angular speed. In another equivalent model,&#xA;earth is not in the center. But the center of the deferent moves on another  circle which is called the eccenter. In more comlicated models the motion of the center of the epicycle is not uniform with respect to the center of the epicycle  but uniform as seen&#xA;from another point which is called equant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope I explained all words. The two simplest models that I described in the beginning were introduced by Apollonius of Perga (262–190 bc) and he proved that they are equivalent. (In modern language this is equivalent to commutativity of addition of vectors). Hipparchus built first models for Sun and Moon, and possibly of planets&#xA;using this machinery. Equant was introduced by Ptolemy (100-170 ad) who built a satisfactory model for the whole Solar system. This model, with minor variations was used successfully  until 17th century, when it was replaces by Kepler's laws, where the motion is on ellipses and the angular velocity is not constant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However the model with circles and uniform motion is a very good approximation that worked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more detail, see Ptolemy's Almagest. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-05-17T21:57:32.693" LastActivityDate="2016-05-17T21:57:32.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3797" PostTypeId="2" ParentId="3795" CreationDate="2016-05-17T22:19:44.303" Score="3" Body="&lt;p&gt;This is an allusion to Fermat's comment in his copy of Diophantus's Arithmetica, whose page I assume is reproduced in the photo:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It  is  impossible  to  separate  a  cube  into  two  cubes,  or  a  biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof of it, but this margin is too small to contain it&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The comment was next to the Problem II.8:&quot;&lt;em&gt;partition a given square into two squares&lt;/em&gt;&quot;, the problem of finding the Pythagorean triples, triples of integers that can serve as side lengths of a right angle triangle. Diophantus does not mention the geometric connection however, see more under &lt;a href=&quot;https://hsm.stackexchange.com/questions/3291/who-discovered-integer-triangles-with-one-angle-trisecting-another/3304#3304&quot;&gt;Who discovered integer triangles with one angle trisecting another?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-05-17T23:31:30.637" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3798" PostTypeId="1" CreationDate="2016-05-18T08:44:30.717" Score="2" ViewCount="103" Body="&lt;p&gt;We just published an article making what we feel is a plausible case in favor of an affirmative answer in &lt;em&gt;Foundations of Science&lt;/em&gt;, see preprint &lt;a href=&quot;http://arxiv.org/abs/1605.03501&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. The basic argument is that while such a requirement may seem very limitative, such an embedding seems possible with a small number of additional ingredients like a black box for returing the sum of a series, and I was curious how well-supported this appears and if there are aspects that may have been overlooked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Crosslisted at &lt;a href=&quot;https://math.stackexchange.com/questions/1791361&quot;&gt;MSE&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-05-20T07:17:56.460" Title="Is Leibnizian calculus embeddable in first order logic?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3799" PostTypeId="2" ParentId="3795" CreationDate="2016-05-18T10:05:18.907" Score="11" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/users/84/plannapus&quot;&gt;User plannapus&lt;/a&gt; &lt;a href=&quot;https://hsm.stackexchange.com/questions/3795/what-manuscript-is-depicted-in-the-hsm-advertisement#comment7463_3797&quot;&gt;points out&lt;/a&gt; that the proposer of the ad links to the original source, which is the first page of  Diophantus’s &lt;em&gt;Arithmetica&lt;/em&gt;, specifically  the 1621 translation by &lt;a href=&quot;https://en.wikipedia.org/wiki/Claude_Gaspard_Bachet_de_M%C3%A9ziriac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Claude Gaspard Bachet de Méziriac&lt;/a&gt;.  The  right-hand side is Greek, and the left-hand side a Latin translation; the bottom seems to be the translator's commentary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FTKRQ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FTKRQ.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://books.google.com/books?id=zF1jAAAAcAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books has a scan&lt;/a&gt; of the British Museum copy of the 1621 version; it matches exactly.)&lt;/p&gt;&#xA;" OwnerUserId="134" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-05-18T13:04:22.200" CommentCount="0" CommunityOwnedDate="2016-05-18T10:05:18.907" ContentLicense="CC BY-SA 3.0" />
  <row Id="3800" PostTypeId="1" CreationDate="2016-05-18T10:56:09.073" Score="1" ViewCount="466" Body="&lt;p&gt;Groups, Rings and Fields are often referred to by the set involved without mention of the operation(s). For example, the &quot;group (G,+)&quot; may be called the &quot;group G&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did this practice originate and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems the labour saved in writing &quot;G&quot; instead of &quot;(G,+)&quot; (for example) is trivial compared with the loss of clarity and precision so I wonder what other reasons there could be? &lt;/p&gt;&#xA;" OwnerUserId="535" LastEditorUserId="26" LastEditDate="2016-05-19T15:49:31.660" LastActivityDate="2016-05-19T15:49:31.660" Title="Abbreviated Notation for Groups, Rings and Fields" Tags="|mathematics|notation|abstract-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3801" PostTypeId="2" ParentId="3800" CreationDate="2016-05-18T15:31:45.287" Score="3" Body="&lt;p&gt;The shift seems quite recent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the &quot;short&quot; writing, see some &quot;old&quot; Algebra's textbook, like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Bartel van der Waerden, &lt;a href=&quot;https://en.wikipedia.org/wiki/Moderne_Algebra&quot; rel=&quot;nofollow&quot;&gt;Modern Algebra&lt;/a&gt;, Vol.I (1949 - or.ed.1930-31): page 11 for &lt;em&gt;groups&lt;/em&gt; and page 38 for &lt;em&gt;rings&lt;/em&gt;;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nicolas Bourbaki, &lt;a href=&quot;https://books.google.it/books?id=STS9aZ6F204C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Elements of Mathematics. Algebra I: Chapters 1-3&lt;/a&gt; (1998 - or.ed.1970), page 30:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definition 1&lt;/strong&gt;. A set with an associative law of composition, possessing an identity element and under which every element is invertible, is called a group. &#xA;  [...] If $G$ and $H$ are two groups, ... &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Garrett Birkhoff &amp;amp; Saunders Mac Lane, &lt;a href=&quot;https://books.google.it/books/about/A_Survey_of_Modern_Algebra.html?id=FnP7sHxjt6gC&amp;amp;redir_esc=y&quot; rel=&quot;nofollow&quot;&gt;A Survey of Modern Algebra&lt;/a&gt; (4th ed - 1977), page 133:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definition&lt;/strong&gt;. A group $G$ is a system of elements with a binary operation &#xA;  which ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From the '90s, we can see the &quot;new wave&quot; emerging: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Saunders Mac Lane &amp;amp; Garrett Birkhoff, &lt;a href=&quot;https://books.google.it/books?id=L6FENd8GHIUC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Algebra&lt;/a&gt; (3rd ed 1991), page 43:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A &lt;em&gt;group&lt;/em&gt; $G$ is a set $G$ together with a binary operation...,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but see page 85:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A &lt;em&gt;ring&lt;/em&gt; $R=(K,+,∙,1)$ is a set $R$ with two binary operations...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;David Dummit &amp;amp; Richard Foote, &lt;a href=&quot;https://books.google.it/books?id=QkAxJgAACAAJ&quot; rel=&quot;nofollow&quot;&gt;Abstract algebra&lt;/a&gt; (3ed - 2004), page 16:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A &lt;em&gt;group&lt;/em&gt; is an ordered pair $(G,∙)$ where $G$ is a set ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-05-18T15:31:45.287" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3802" PostTypeId="1" AcceptedAnswerId="3803" CreationDate="2016-05-19T17:15:58.493" Score="7" ViewCount="400" Body="&lt;p&gt;There are many examples where the notation of $\partial{U}$ is used to represent the surface boundary of some volume $U$.  For example, the following representation of the Divergence Theorem:&#xA;$$&#xA;\int_U{\nabla \cdot \mathbf{F}}\,dV = \oint_{\partial{U}}\,\mathbf{F}\cdot\mathbf{n}\,dS&#xA;$$&#xA;What is the history and reasoning for the notation of using what appears to be a partial derivative of the volume for the surface.  I am guessing that the derivative represents the tangent at the surface and thus refers to surface boundary but is there more then that?&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2016-05-20T00:10:55.100" Title="What is rationale or history for using the symbol $\partial{U}$ to represent surface boundary?" Tags="|notation|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3803" PostTypeId="2" ParentId="3802" CreationDate="2016-05-20T00:02:08.590" Score="10" Body="&lt;p&gt;There seem to be two different rationales, one for how it first appeared, and the other for why it stuck, see &lt;a href=&quot;https://mathoverflow.net/questions/46252/is-the-boundary-partial-s-analogous-to-a-derivative/46410#46410&quot;&gt;MO thread&lt;/a&gt;. Francois Ziegler dug up the following remark in Michèle Audin's &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=3014192&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Henri Cartan &amp;amp; André Weil: du vingtième siècle et de la topologie&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;$d$ is for differential, no doubt, but what is $\partial$ for? Neither boundary nor frontier, in neither French nor German (nor even English), so? well, Rand, in German, written in gothic and abbreviated $\mathfrak{Rd}$, then $\mathfrak d$ (two letters, that was too much), then $\partial$!&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He also found the corresponding use in §16 of &lt;a href=&quot;https://books.google.com/books?id=rsb8zjP0XHoC&amp;amp;pg=PA61&quot; rel=&quot;nofollow noreferrer&quot;&gt;Seifert and Threlfall's &lt;em&gt;Lehrbuch der Topologie&lt;/em&gt; (1934)&lt;/a&gt;:&quot;&lt;em&gt;Der Rand wird mit dem Symbole $\mathscr R\partial$ bezeichnet&lt;/em&gt;&quot; [the boundary is denoted by the symbol $\mathscr R\partial$]. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems unlikely that this convoluted justification would have impressed many in adopting the symbol. Fortunately, it has other things to recommend itself. E.g. it dovetails nicely with the dual use of $d$ in cohomology and calculus on manifolds, e.g. in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Stokes&amp;#39;_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stokes formula&lt;/a&gt;, $\int_M d\omega = \int_{\partial M} \omega$, of which all integral formulas of vector analysis, like the OP one,  are particular cases. And it has some derivative like properties: $\partial(A\cap B)=(\partial A\cap B)\cup(A\cap\partial B)$ and $\partial(A\times B) =(\partial A\times B) \cup (A \times \partial B)$.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-05-20T00:10:55.100" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3804" PostTypeId="1" CreationDate="2016-05-20T08:02:22.103" Score="2" ViewCount="146" Body="&lt;p&gt;How do you think Euler felt when he discovered/invented $e^{ix}=\cos(x)+i\sin(x)$. Do you think he doubted his calculations for a length of time? Was it Readily accepted by the mathematical community quickly or did they object at first? If so when was it finally accepted and why?&lt;/p&gt;&#xA;" OwnerUserId="4167" LastActivityDate="2016-05-20T10:24:25.043" Title="Eulers identity history" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3805" PostTypeId="2" ParentId="3804" CreationDate="2016-05-20T10:24:25.043" Score="1" Body="&lt;p&gt;He proved it; see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Leonhard Euler, &lt;a href=&quot;https://books.google.it/books?id=_zpRAAAAYAAJ&amp;amp;pg=PA104&quot; rel=&quot;nofollow&quot;&gt;Introductio in analysin infinitorum. Tomus I&lt;/a&gt; (1748), §139, page 104.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And compare with: William Dunham, &lt;a href=&quot;https://books.google.it/books?id=x7p4tCPPuXoC&amp;amp;pg=PA93&quot; rel=&quot;nofollow&quot;&gt;Euler: The Master of Us All&lt;/a&gt; (1999), page 93, for a &quot;modern&quot; formulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why do you think that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was it Readily accepted by the mathematical community quickly or did they object at first? If so when was it finally accepted and why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The result has been proved by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow&quot;&gt;leading mathematician&lt;/a&gt; of his times and it is contained into the &lt;a href=&quot;https://en.wikipedia.org/wiki/Introductio_in_analysin_infinitorum&quot; rel=&quot;nofollow&quot;&gt;fundamental mathematical analysis textbook&lt;/a&gt; of his time. &lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-05-20T10:24:25.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3806" PostTypeId="1" CreationDate="2016-05-20T10:44:47.310" Score="-1" ViewCount="109" Body="&lt;p&gt;Just out of curiosity &lt;/p&gt;&#xA;&#xA;&lt;p&gt;if I have the function F and G. So that G is the left inverse of F. Because G o F = Idx. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is why is it named &quot;left&quot; inverse if it is like that? Is it because of the position of G in the composition notation in relation to F (G is at the left of F in the G o F = Idx equation)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possible guess would be  :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it's called a &quot;left &quot; inverse of F because to do G o F = Idx means that it goes from the domain to the codomain, then back to the domain again (the domain is usually illustrated at the left) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's called a &quot;right&quot; inverse F because to do F o H = Idy means that it goes from the codomain to the domain, then back to the codomain again (the codomain is usually illustrated at the right side). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or is it named &quot;left/right&quot; inverse because of some other reason? Does anybody knows the answer? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards.&lt;/p&gt;&#xA;" OwnerUserId="4168" LastActivityDate="2016-05-20T19:26:01.633" Title="About left / right Inverses" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3809" PostTypeId="2" ParentId="3806" CreationDate="2016-05-20T19:26:01.633" Score="0" Body="&lt;p&gt;Functions $F$ and $G$ have different domains, say $D_F=A$ and $D_G=B$.&#xA;So there are two identity maps $I_A$ and $I_B$. Thus the relations&#xA;$FG=I_B$ and $GF=I_A$ are very different things. In the first case we say that $G$&#xA;is the right inverse of $F$ and in the second case it is the left inverse.&#xA;Of course the terminology has the evident origin: on which side of $F$ $G$ stands&#xA;in the identity. The properties of $F$ to be right-invertible and left-invertible&#xA;are different properties.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-05-20T19:26:01.633" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="3810" PostTypeId="1" AcceptedAnswerId="3811" CreationDate="2016-05-20T21:53:16.180" Score="3" ViewCount="483" Body="&lt;p&gt;There are many examples where our senses are based off of log scales such as volume of a noise, ability to guess (i.e.) plus or minus a power of 10 with Fermi, and even when we measure pain on 1 to 10 scale most of us subconsciously do it as a log scale. There are even evolutionary advantages such as its much more for survival sake to know the log of a predator. (i.e.) it matters more if theirs 3 vs 1 lions but not if theirs 23 vs 21 lions. &lt;strong&gt;Are there any human senses that truly work off of a linear scale to a stimulus?&lt;/strong&gt; This leads me to wonder the following.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today we have a convention that is based off of linear scales based off of 1 and successors. From young we are taught linear scales and do everything in school under linear scales. &lt;strong&gt;Has this convention ever been reversed?&lt;/strong&gt; It clearly was used for a time with the slide rule but that's more of a way of switching between the two.&lt;/p&gt;&#xA;" OwnerUserId="4167" LastEditorUserId="55" LastEditDate="2016-05-20T23:03:57.513" LastActivityDate="2016-07-20T03:11:08.167" Title="What is the history of linear vs logarithmic scales?" Tags="|mathematics|biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3811" PostTypeId="2" ParentId="3810" CreationDate="2016-05-20T23:02:38.437" Score="6" Body="&lt;p&gt;According to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weber–Fechner law&lt;/a&gt; in vision and hearing perception is close to proportional to the logarithm of the stimulus. This however has no direct influence on the use of scales since they are used for correlating two different observables (&quot;stimuli&quot;), and the preference for linear or logarithmic scale depends on the behavior of what is correllated, not on our perception of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For exponential laws log-linear scales are most advantageous, and for power laws log-log scales are. Since Napier's invention of logarithms in early 1600s these insights were taken advantage of, first by brute conversion as with Kepler, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/1948/how-did-kepler-guess-his-third-law-from-data/1950#1950&quot;&gt;How did Kepler &amp;quot;guess&amp;quot; his third law from data?&lt;/a&gt; Then with the slide rule, and later logarithmic graph paper invented by Lallane in 1844, and logarithmic plotting. In fact, Lallane considered far more general scales than just linear and logarithmic, and in 1880s d'Ocagne developed his ideas into a whole discipline of nomography, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3580/when-was-the-earliest-use-of-log-log-plots-to-demonstrate-power-law-behavior/3584#3584&quot;&gt;When was the earliest use of log-log plots to demonstrate power-law behavior?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today there is no need for the slide rule or logarithmic paper because we have computers and calculators, but log plotting and transformations (along with many other types of rescaling) are used in modeling and statistical analysis of data even more broadly than before. In particular, logarithmizing and other nonlinear transformations are applied to reduce non-linear dependecies to linear ones, so that the linear regression (least squares fitting) can be applied to them directly, see e.g. &lt;a href=&quot;http://www.real-statistics.com/regression/exponential-regression-models/exponential-regression/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exponential Regression using a Linear Model&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-07-20T03:11:08.167" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3812" PostTypeId="1" AcceptedAnswerId="3813" CreationDate="2016-05-20T23:32:20.347" Score="15" ViewCount="721" Body="&lt;p&gt;The Wiki article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor's diagonal argument&lt;/a&gt; mentions that the first use of a diagonal argument was in the work of Paul du Bois-Reymond in 1875. This would be one year after Cantor's first proof of the uncountability of the reals and 16 years before Cantor's own diagonal argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was quite surprised to learn this. A Google search turned up &lt;a href=&quot;https://mathoverflow.net/questions/23953/earliest-diagonal-proof-of-the-uncountability-of-the-reals&quot;&gt;this MathOverflow question&lt;/a&gt;. John Stillwell's answer says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A more clearcut use of diagonalization before Cantor's 1891 proof, in&#xA;  my opinion, is in this 1875 paper by Paul du Bois-Reymond. Given a&#xA;  sequence of positive integer valued functions $f_1,f_2,f_3,\dots$, du&#xA;  Bois-Reymond constructs a function $f$ that grows faster than each&#xA;  $f_i$. In particular, $f$ differs from $f_i$ on the value $i$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The link in this response goes to a paper in German that comes up only intermittently in my browser; and in any event my German's not sufficient to understand it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know more about du Bois-Reymond's proof and how it relates to Cantor's work? Was Cantor already aware of the idea of a diagonal proof? Does Cantor incorrectly get credit for this idea? And was uncountability perhaps &quot;in the air&quot; among at least one of Cantor's contemporaries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(ps) Also found &lt;a href=&quot;https://books.google.com.mx/books?id=nMfAdizANuYC&amp;amp;pg=PA522&amp;amp;lpg=PA522&amp;amp;dq=Paul%20du%20Bois-Reymond%20diagonal%20argument&amp;amp;source=bl&amp;amp;ots=iGWJTqogqt&amp;amp;sig=50CdZZLEh4RwwpauUZWwwiQRfYs&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Paul%20du%20Bois-Reymond%20diagonal%20argument&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;. Turns out du Bois-Reymond is also responsible for recognizing the importance of lim inf and lim sup, among many other contributions. This article vagely refers (without detail) to du Bois-Reymond initiating a priority dispute with Cantor regarding the discovery of dense sets.&lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="4251" LastEditDate="2020-08-01T22:31:17.243" LastActivityDate="2020-08-01T22:31:17.243" Title="Did du Bois-Reymond invent the diagonal argument before Cantor?" Tags="|mathematics|mathematicians|discoveries|set-theory|diagonal-argument|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3813" PostTypeId="2" ParentId="3812" CreationDate="2016-05-21T00:50:17.850" Score="12" Body="&lt;p&gt;This paper is one in a series of papers in which du Bois Reymond studied functions on the positive real line ordered by the &amp;quot;order of infinity&amp;quot; (order of growth at infinity), what he later called infinitary pantachy. This was motivated by attempts to find &amp;quot;ideal boundary&amp;quot; between converging and diverging series in terms of the growth of their terms by analogy to filling in Dedekind cuts to get real numbers. Du Bois Reymond more or less established that such completion would not work. Later Hausdorff developed modern theory in Cantorian terms which explains why, the &amp;quot;gaps&amp;quot; in the orders of growth are more severe, beyond what countable limits can fill in. These are now called &lt;a href=&quot;https://en.wikipedia.org/wiki/Hausdorff_gap&quot; rel=&quot;noreferrer&quot;&gt;Hausdorff gaps&lt;/a&gt;, but du Bois Reymond did not work in cardinality terms.&lt;/p&gt;&#xA;&lt;p&gt;Fisher gives a detailed account in &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&amp;amp;type=pdf&amp;amp;doi=10.1.1.116.8707&quot; rel=&quot;noreferrer&quot;&gt;Infinite and Infinitesimal Quantities of du Bois-Reymond and their Reception&lt;/a&gt;, here is the reference with direct link:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://link.springer.com/article/10.1007/BF01443187&quot; rel=&quot;noreferrer&quot;&gt;Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre&#xA;Auflösung yon Gleichungen&lt;/a&gt;, &lt;em&gt;Mathematische Annalen&lt;/em&gt; 8 (1875) 363-414 (Nachtrage zur Abhandlung: Ueber asymptotische Werthe etc., 574-576).&lt;/p&gt;&#xA;&lt;p&gt;Du Bois-Reymond defines one order of infinity to be larger than another if the limit of the quotient of their representing functions is infinite. He is not very specific about the class of functions considered, and apparently overlooks the possibility of incomparable orders, an issue later investigated by Borel and Hausdorff. He then analogizes, and contrasts, this continuum of orders to the linear continuum of real numbers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Between the two domains there are many analogies... Instead of numbers as fixed signs in the domain of numbers, one has in the infinitary domain of quantities an unlimited number of simple functions: the exponential functions, the powers, the logarithmic functions, that likewise form fixed points of comparisons, and between whose arbitrarily close infinities a limitless number of infinities different from each other can still be inserted.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;These functions serving as numbers stretch, in accordance with the present&#xA;state of analysis, from exponential functions stacked arbitrarily high... down to logarithms repeated arbitrarily, often to infinity... The question which arises here, whether one actually includes in this interval of infinity the entire domain of infinity, in such a way that one can enclose any given infinity between two infinities of functions of that interval, as is possible for any number in the number series: This question I have already answered in substance and in fact negatively...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He refers to 1873 paper where only a particular case was considered. This time he gives a more general version, which Borel highly praised and termed&#xA;du Bois-Reymond's theorem. A consequence of it, and the motivation, is the non-existence of the &amp;quot;ideal boundary&amp;quot; that can be specified by a sequence of converging/diverging series, as Bertrand earlier proposed (he was thinking of reciprocals to products of powers and powers of iterated logarithms as terms). The du Bois-Reymond's theorem would be the &amp;quot;diagonal argument&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...if an unlimited family of more and more slowly increasing functions &lt;span class=&quot;math-container&quot;&gt;$\lambda_1(x), \lambda_2(x), \lambda_3(x)...$&lt;/span&gt; is given which for each &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; satisfies the condition &lt;span class=&quot;math-container&quot;&gt;$\lim\lambda_r(X)/\lambda_{r+1}(X) = \infty$&lt;/span&gt;, one can&#xA;always specify a function &lt;span class=&quot;math-container&quot;&gt;$\psi(x)$&lt;/span&gt; which becomes infinite with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, but more slowly than any function of that family&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The construction of &lt;span class=&quot;math-container&quot;&gt;$\psi(x)$&lt;/span&gt; is in a footnote on p. 365, and does show some &amp;quot;diagonality&amp;quot; if one looks hard. There is however no cardinality involved in du Bois-Reymond's setting (Hausdorff will relate gaps to cardinalities only later), so making it into the diagonal argument takes some reading in. In his 1882 book Die allgemeine Functionentheorie (The General Theory of Functions) du Bois-Reymond touches base with Cantorian set theory, and mentions that Cantor showed &amp;quot;continuum of the idealists&amp;quot; to be uncountable. He does not however point out any affinity between the diagonal argument by which it was shown and his earlier construction, let alone lay any claim to it. So whatever the relation between the two it was not apparent to du Bois-Reymond.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-07-20T03:10:12.277" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="3814" PostTypeId="1" AcceptedAnswerId="3815" CreationDate="2016-05-21T00:54:13.677" Score="14" ViewCount="1377" Body="&lt;p&gt;I read in an offhand comment in Amir Alexander's 2014 book &lt;em&gt;Infinitesimal&lt;/em&gt; (p.280), that John Wallis introduced the symbol $\infty$ for infinity.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was there any logic, reason, or precedent for this symbol, or did he &#xA;  (as far as we know) just make it up?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="606" LastEditorUserId="4703" LastEditDate="2017-05-30T15:41:27.230" LastActivityDate="2017-05-31T09:20:26.920" Title="Where did John Wallis get the idea for $\infty$?" Tags="|mathematics|mathematicians|notation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3815" PostTypeId="2" ParentId="3814" CreationDate="2016-05-21T01:03:07.647" Score="11" Body="&lt;p&gt;from &lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Earliest Uses of Various Mathematical Symbols&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Infinity. The &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; symbol  was introduced by John Wallis (1616-1703) in 1655 in his &lt;a href=&quot;https://books.google.it/books?id=03M_AAAAcAAJ&amp;amp;pg=PA4&quot; rel=&quot;noreferrer&quot;&gt;De sectionibus conicis (On Conic Sections)&lt;/a&gt; as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suppono in limine (juxta Bonaventurae Cavallerii &lt;em&gt;Geometriam Indivisibilium&lt;/em&gt;) Planum quodlibet quasi ex infinitis lineis parallelis conflari: Vel potius (quod ego mallem) ex infinitis Prallelogrammis [sic] aeque altis; quorum quidem singulorum altitudo sit totius altitudinis &lt;span class=&quot;math-container&quot;&gt;$\frac 1 \infty$&lt;/span&gt;, sive aliquota pars infinite parva; (esto enim &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; nota numeri infiniti;) adeoq; omnium simul altitude aequalis altitudini figurae.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Wallis also used the infinity symbol in various passages of his &lt;a href=&quot;https://books.google.it/books?id=Z5w_AAAAcAAJ&amp;amp;pg=PA70&quot; rel=&quot;noreferrer&quot;&gt;Arithmetica infinitorum (Arithmetic of Infinites)&lt;/a&gt; (1655 or 1656). For instance, he wrote (p. 70):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cum enim primus terminus in serie Primanorum sit &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, primus terminus in serie reciproca erit &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; vel infinitus: (sicut, in divisione, si diviso sit &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, quotiens erit infinitus.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;em&gt;Zero to Lazy Eight&lt;/em&gt;, Alexander Humez, Nicholas Humez, and Joseph Maguire write: &amp;quot;Wallis was a classical scholar and it is possible that he derived &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; [the infinity symbol] from the old Roman sign for 1,000, CD, also written M--though it is also possible that he got the idea from the lowercase omega, omega being the last letter of the Greek alphabet and thus a metaphor of long standing for the upper limit, the end.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Cajori (vol. 2, p 44) says the conjecture has been made that Wallis adopted this symbol from the late Roman symbol &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; for 1,000. He attributes the conjecture to Wilhelm Wattenbach (1819-1897), Anleitung zur lateinischen Paläographie 2. Aufl., Leipzig: S. Hirzel, 1872. Appendix: p. 41.&lt;/p&gt;&#xA;&lt;p&gt;This conjecture is lent credence by the labels inscribed on a Roman hand abacus stored at the Bibliothèque Nationale in Paris. A plaster cast of this abacus is shown in a photo on page 305 of the English translation of Karl Menninger’s &lt;em&gt;Number Words and Number Symbols&lt;/em&gt;; at the time, the cast was held in the Cabinet des Médailles in Paris. The photo reveals that the column devoted to 1000 on this abacus is inscribed with a symbol quite close in shape to the lemniscate symbol, and which Menninger shows would easily have evolved into the symbol M, the eventual Roman symbol for 1000 [Randy K. Schwartz].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Note,  see the actual link for things that don't come out after copy and paste.)&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-02-27T07:59:50.067" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3817" PostTypeId="1" CreationDate="2016-05-23T15:49:09.310" Score="0" ViewCount="76" Body="&lt;p&gt;I have to write an essay concerning philosophy of mathematics until the end of XIX century. I've heard that the reason why the Cauchy's theorem (if continuous functions $fn→f$ then $f$ is continuous too) is wrong nowadays is the fact that his real numbers line (number system) was different then the modern one. It is indeed a very interesting topic but I couldn't find anything specific about it. Could you direct me to some sources or suggest a different topic?&lt;/p&gt;&#xA;" OwnerUserId="4183" LastActivityDate="2016-05-23T15:49:09.310" Title="Cauchy's real line and math philosophy till XIX" Tags="|mathematics|mathematicians|philosophy-of-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3819" PostTypeId="1" CreationDate="2016-05-22T12:02:30.270" Score="4" ViewCount="62" Body="&lt;p&gt;Which concept was first introduced: the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagoras_number&quot; rel=&quot;nofollow&quot;&gt;pythagoras number of a field&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_field&quot; rel=&quot;nofollow&quot;&gt;pythagorean fields&lt;/a&gt;? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the idea of pythagorean fields to Pythagoras' theorem, since the equation $c^2 = a^2 + b^2$ has a solution $c$ for any $a, b \in k$ if and only if $k$ is pythagorean. More generally, the concept has geometric implications (see the linked Wikipedia article).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The pythagoras number of a field (or commutative ring), which I'm studying, does not seem directly related to anything Pythagoras did, so my best guess is that the pythagorean fields came first and that the pythagoras number of a field was a generalisation of this. A field is pythagorean if and only if its pythagoras number is one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any information (preferably with references) on this you might have is welcome. More broadly, I'd be interested in any information regarding the origin of the term pythagoras number.&lt;/p&gt;&#xA;" OwnerUserId="4184" OwnerDisplayName="Bib-lost" LastActivityDate="2016-05-23T17:16:29.913" Title="What came first: pythagoras number or pythagorean fields?" Tags="|reference-request|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3823" PostTypeId="2" ParentId="3752" CreationDate="2016-05-24T16:35:06.297" Score="2" Body="&lt;p&gt;According to Carl Boyer, in his text &lt;em&gt;A History of Mathematics&lt;/em&gt;, it is Simon Stevin who is credited with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Inclined_plane&quot; rel=&quot;nofollow&quot;&gt;Law of the Inclined Plane&lt;/a&gt;, as justified by his familiar &quot;wreath of spheres&quot; diagram.  The law itself had been stated previously by Jordanus Nemorarius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting to note that, again according to Boyer, Stevin observed and published Galileo's result some 50 years before the publication of Two New Sciences.  Boyer even goes so far to claim that it is doubtful that Galileo ever performed the experiment.  Quoting Boyer :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Stevin] and a friend dropped two spheres of lead, one ten times the weight of the other, from a height of 30 feet onto a board and found the sounds of their striking the board to be almost simultaneous.  But Stevin's published report (in Flemish in 1586) of the experiment has received far less notice than the similar and later experiment attributed, on very doubtful evidence, to Galileo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-05-24T16:35:06.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3824" PostTypeId="1" AcceptedAnswerId="4886" CreationDate="2016-05-25T19:22:24.983" Score="12" ViewCount="5046" Body="&lt;p&gt;It seems Einstein's most original contribution to physics was General Relativity Theory, as Lorentz and Poincaré already laid the foundations of Special Relativity. So, why didn't his &lt;a href=&quot;http://www.nobelprize.org/nobel_prizes/physics/laureates/1921/einstein-facts.html&quot; rel=&quot;noreferrer&quot;&gt;1921 Nobel Prize&lt;/a&gt; mention Special or General Relativity Theory at all?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, his prize motivation said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Especially?&quot; Einstein didn't make greater &quot;services to Theoretical Physics&quot; than &quot;the law of the photoelectric effect&quot;? (This was late: 1921, not pre-1918.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, two years later &lt;a href=&quot;http://www.nobelprize.org/nobel_prizes/physics/laureates/1923/&quot; rel=&quot;noreferrer&quot;&gt;Millikan's Nobel Prize&lt;/a&gt; was awarded &quot;for his work on the elementary charge of electricity and on the photoelectric effect&quot;. Did the Nobel Committee just have a liking for the photoelectric effect?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-06-21T23:27:45.480" LastActivityDate="2018-09-23T21:09:23.383" Title="Why didn't Einstein's Nobel Prize mention relativity theory?" Tags="|relativity-theory|nobel-prize|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3825" PostTypeId="1" CreationDate="2016-05-25T19:36:48.240" Score="2" ViewCount="88" Body="&lt;p&gt;Planck's &lt;a href=&quot;http://www.nobelprize.org/nobel_prizes/physics/laureates/1918/planck-facts.html&quot; rel=&quot;nofollow&quot;&gt;1918 Nobel Prize&lt;/a&gt; was awarded&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in 1900.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, why do I sometimes see Einstein attributed for this discovery?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-05-25T21:29:02.990" Title="Who discovered quanta first, Planck or Einstein?" Tags="|quantum-mechanics|nobel-prize|" AnswerCount="0" CommentCount="1" ClosedDate="2016-05-25T22:50:30.667" ContentLicense="CC BY-SA 3.0" />
  <row Id="3827" PostTypeId="1" CreationDate="2016-05-26T00:05:05.413" Score="3" ViewCount="119" Body="&lt;p&gt;When in our recorded history did first nonlinear optics appear?&lt;/p&gt;&#xA;&lt;p&gt;I am more interested in the first recorded use of nonlinear crystals.&lt;/p&gt;&#xA;" OwnerUserId="4196" LastEditorUserId="10106" LastEditDate="2020-06-21T15:25:22.090" LastActivityDate="2020-06-21T15:25:22.090" Title="When is considered the birth of nonlinear optics?" Tags="|physics|experimental-physics|optics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="3829" PostTypeId="2" ParentId="3824" CreationDate="2016-05-26T08:16:18.333" Score="3" Body="&lt;p&gt;It &lt;a href=&quot;https://en.wikipedia.org/wiki/Nobel_Prize_in_Physics&quot; rel=&quot;nofollow&quot;&gt;seems that&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The rules for the Nobel Prize in Physics require that the significance of achievements being recognized has been &quot;tested by time&quot;. In practice it means that the lag between the discovery and the award is typically on the order of 20 years and can be much longer. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If we browse the list of &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Nobel_laureates_in_Physics&quot; rel=&quot;nofollow&quot;&gt;Nobel laureates in Physics&lt;/a&gt;, we can &quot;inductively&quot; conclude that the prize is awarded mainly for &quot;experimental&quot; discoveries: new effects or devices, and hardly for pure theories.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-05-26T08:16:18.333" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3830" PostTypeId="2" ParentId="3827" CreationDate="2016-05-26T10:55:01.793" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://leccap.engin.umich.edu/leccap/site/qosbcytrdws33ghva0n&quot; rel=&quot;nofollow&quot;&gt;University of Michigan Physics Department -Nonlinear Optics 50th Anniversary Symposium&lt;/a&gt; was held on Octber 26, 2011. &lt;a href=&quot;http://course.ee.ust.hk/elec509/readings/nonlinear%20optics.pdf&quot; rel=&quot;nofollow&quot;&gt; Peter Franken had focused a ruby laser on a quartz crystal in 1961&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the first case of second harmonic generation in the optical regime, a material response to high intensity light. However, important nonlinear effects had been discovered much earlier: the Kerr effect in 1874.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polarization effects were known to prior to Newton's work on optics, based on &lt;a href=&quot;https://en.wikipedia.org/wiki/Iceland_spar&quot; rel=&quot;nofollow&quot;&gt;Iceland spar&lt;/a&gt;, in the mid-1600s, but these are explainable with linear optics. There is no intensity dependence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the research field of nonlinear optics begins the year after lasers: 1961.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastEditorUserId="3821" LastEditDate="2016-05-26T18:55:44.690" LastActivityDate="2016-05-26T18:55:44.690" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3832" PostTypeId="2" ParentId="3759" CreationDate="2016-05-26T17:20:38.903" Score="5" Body="&lt;p&gt;I've only recently begun reading on the subject of the history of mathematics, and my readings are currently limited to a single text; Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt;.  However, according to Boyer, and supported by the wikipedia entry for &lt;a href=&quot;https://en.wikipedia.org/wiki/Simon_Stevin#Decimal_fractions&quot; rel=&quot;noreferrer&quot;&gt;Simon Stevin&lt;/a&gt;, I believe your claim that Stevin dealt with &quot;all&quot; real numbers, &quot;(rational, surd, etc.)&quot;, is an overreach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quoting Boyer :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Viète, ... , in 1579 had urged the replacement of sexagesimal fractions by decimal fractions.  In 1585 an even stronger plea for the use of ten-scale fractions, as well as integers, was made by the leading mathematician in the Low Countries, Simon Stevin of Bruges.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This appears to stop short of claim that Stevin's work was conceptually embracing all real numbers.  The linked paper by Malet makes the clear claim that Stevin also considered (some) irrational numbers :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;That “any root whatsoever is number” [Stevin, 1585, 8] is also a consequence of identifying numbers and measures&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, according to Malet, Stevin does consider algebraic numbers, but again this stops short of claiming that Stevin was in possession of the correct notion of &quot;all real numbers&quot;.  In other words, although we know now that all real numbers can be represented in this way, it is not clear that Stevin was aware of the true and correct nature of the real numbers and their different types.  Perhaps this provides some explanation for why Stevin does not get full credit for the real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As final point, it may also be worth mentioning that Boyer notes :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is clear that Stevin was in no sense the &lt;em&gt;inventor&lt;/em&gt; of decimal fractions, nor was he the first systematic user of them.  More than incidental use of decimal fractions is found in ancient China, in medieval Arabia, and in Renaissance Europe; by the time of Viète's forthright advocacy of decimal fractions in 1579 they were generally accepted by mathematicians on the frontiers of research.  Among the common people, however, and even among mathematical practitioners, decimal fractions became widely known only when Stevin undertook to explain the system in full elementary detail.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-05-26T17:41:12.720" LastActivityDate="2016-05-26T17:41:12.720" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3834" PostTypeId="1" AcceptedAnswerId="5104" CreationDate="2016-05-27T17:49:54.940" Score="2" ViewCount="368" Body="&lt;p&gt;According to Boyer, Salviati introduces the idea of a higher order infinitesimal on the “third day” in Galileo’s &lt;em&gt;Two Chief Systems&lt;/em&gt; of 1632.  They are introduced in order to counter Simplicio’s argument that an object on a rotating earth should be thrown off tangentially.  Boyer includes a diagram intended to illustrate the role of the different orders of infinitesimals in the argument.  Obviously it is impossible to &lt;em&gt;draw&lt;/em&gt; an infinitesimal.  However, one would naturally think that the different orders of infinitesimals would be represented in different proportion to one another, but they are show in equal proportion.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boyer includes a quote in which Galileo appears to identify infinitesimals as &lt;em&gt;indivisibles&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On another occasion, however, Galileo has Salviati assert that infinites and indivisibles “transcend our finite understanding, the former on account of their magnitude, the latter on account of their smallness; Imagine what they are when combined.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If Galileo intends that an infinitesimal is an indivisible, then it is hard to imagine what is meant by higher orders of infinitesimals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boyer also notes that Galileo’s work implies that in an equation involving infinitesimals, those of higher order can be ignored since they have no effect on the final result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If find all of this very confusing.  How can indivisibility accommodate higher orders? Surely a higher order infinitesimal is not an infinitesimal portion of an infinitesimal measure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question : Is there a generally accepted description of what Galileo meant by higher order infinitesimals?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-08-16T11:05:10.233" Title="What is meant by higher order infinitesimals in the works of Galileo and Cavalieri" Tags="|physics|geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="3836" PostTypeId="1" AcceptedAnswerId="3844" CreationDate="2016-05-28T05:19:29.710" Score="2" ViewCount="3808" Body="&lt;p&gt;Hertz's experiments are said to have proven Maxwell's theory correct, but how exactly did his experiment do this?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-05-28T18:41:46.763" Title="How exactly did Hertz's experiments confirm Maxwell's theory?" Tags="|electromagnetism|light|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3837" PostTypeId="1" AcceptedAnswerId="3872" CreationDate="2016-05-28T05:36:30.280" Score="4" ViewCount="384" Body="&lt;p&gt;What did Einstein learn in his university electricity and magnetism courses? Did he learn about such prominent results as &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;nofollow&quot;&gt;Ampère's force law&lt;/a&gt;, Weber's force law derived from it (cf. Maxwell's &lt;a href=&quot;https://archive.org/stream/electricandmag02maxwrich#page/n457/mode/2up&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Treatise&lt;/em&gt; §§846&lt;/a&gt;-851), and &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002154420&quot; rel=&quot;nofollow&quot;&gt;Helmholtz's theory&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Doing a search through &lt;a href=&quot;http://einsteinpapers.press.princeton.edu/papers&quot; rel=&quot;nofollow&quot;&gt;Einstein's collected papers&lt;/a&gt;, I could not find him mentioning any of these things. Did he? If not, why?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-08-08T22:39:58.207" LastActivityDate="2022-08-28T14:41:48.040" Title="What did Einstein learn in his university electricity and magnetism courses?" Tags="|electromagnetism|electricity|magnetism|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="3838" PostTypeId="1" AcceptedAnswerId="3841" CreationDate="2016-05-28T06:15:04.663" Score="4" ViewCount="210" Body="&lt;p&gt;What topics were taught in late-19th century electricity and/or magnetism courses in Europe?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-05-28T18:47:41.920" LastActivityDate="2016-05-28T18:47:41.920" Title="What was taught in 19th century European electricity and/or magnetism courses?" Tags="|electromagnetism|education|electricity|magnetism|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3839" PostTypeId="1" AcceptedAnswerId="3840" CreationDate="2016-05-28T07:00:34.347" Score="0" ViewCount="189" Body="&lt;p&gt;I am looking for biographical works about Georg Cantor's life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I have found right now are short encyclopedia-style articles that skim the important points of his life or works mainly focused on his mathematical contributions; ideally I want something more thorough that has a greater emphasis on his life in general rather than just his mathematical contributions. I came across this book, 'Georg Cantor: Leben, Werk und Wirkung' by Herbert Meschkowski, that seems to be what I'm looking for, but unfortunately I haven't found a copy yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would prefer books, but any other references (papers, websites, etc.) that you'd think would be interesting or useful are very much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="4203" LastEditorUserId="4251" LastEditDate="2020-06-06T13:02:07.660" LastActivityDate="2020-06-06T13:02:07.660" Title="Biography on Georg Cantor" Tags="|mathematics|mathematicians|biographical-details|reference-request|cantor|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3840" PostTypeId="2" ParentId="3839" CreationDate="2016-05-28T08:19:33.687" Score="2" Body="&lt;p&gt;As pointed out in the comments on Math Stack Exchange, Dauben's book is highly regarded.  For reference, here's a link to the &lt;a href=&quot;http://www.ams.org/journals/bull/1980-02-01/S0273-0979-1980-14725-4/S0273-0979-1980-14725-4.pdf&quot; rel=&quot;nofollow&quot;&gt;Dauben book review&lt;/a&gt; published in the Bulletin of the AMS.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2016-05-28T08:19:33.687" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3841" PostTypeId="2" ParentId="3838" CreationDate="2016-05-28T15:40:53.993" Score="3" Body="&lt;p&gt;&lt;em&gt;Some data&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;The plan for the famous 1867 textbook &lt;a href=&quot;https://en.wikipedia.org/wiki/Treatise_on_Natural_Philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Treatise on Natural Philosophy&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Thomson&lt;/a&gt; (Professor of &lt;em&gt;Natural Philosophy&lt;/em&gt; at the University of Glasgow for over 50 years) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Peter_Tait_(physicist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peter Guthrie Tait&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;contemplated four volumes; the printing of the first volume began in 1862 and was completed in 1867. The other three volumes never appeared.&lt;/p&gt;&#xA;&lt;p&gt;It was intended to be an all-comprehensive treatise on physical science, the foundations being laid in kinematics and dynamics, and the structure completed with the properties of matter, heat, light, electricity and magnetism. But the literary partnership ceased in about eighteen years, when only the first portion of the plan had been completed, because each of the members felt he could work to better advantage separately than jointly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Tait, who held the chair of &lt;em&gt;Natural Pilosophy&lt;/em&gt; at Edinburgh from 1860 until shortly before his death in 1901, on his own produced books on:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sketch of Thermodynamics (1877);&lt;/p&gt;&#xA;&lt;p&gt;Heat (1884);&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/lighttaitpetergu00taitrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Light&lt;/a&gt; (1884);&lt;/p&gt;&#xA;&lt;p&gt;Properties of Matter (1885).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oliver_Heaviside&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oliver Heaviside&lt;/a&gt;'s papers on electromagnetism was collected into the three-volumes &lt;a href=&quot;https://archive.org/details/electromagnetict01heavrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Electromagnetic Theory&lt;/a&gt; (1893-1912).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations. The four re-formulated Maxwell's equations describe the nature of electric charges (both static and moving), magnetic fields, and the relationship between the two, namely electromagnetic fields.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/J._J._Thomson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph John Thomson&lt;/a&gt;, that in 1884 was chosen to become &lt;em&gt;Cavendish Professor of Physics&lt;/em&gt; at the University of Cambridge, published in 1895 his &lt;a href=&quot;https://books.google.it/books?id=w9kEAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements of the Mathematical Theory of Electricity and Magnetism&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[it] was a readable introduction to a wide variety of subjects, and achieved considerable popularity as a textbook.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-05-28T15:40:53.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3843" PostTypeId="1" CreationDate="2016-05-28T17:15:40.493" Score="3" ViewCount="170" Body="&lt;p&gt;As the biographer of the physiologist Emil du Bois-Reymond I'm delighted by the discussion of his brother Paul du Bois-Reymond's contribution to the invention of the diagonal argument.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know if Emil's 1872 address on &quot;The Limits of Science&quot; encouraged his brother Paul to think along those lines?&lt;/p&gt;&#xA;" OwnerUserId="4205" LastActivityDate="2016-05-29T07:06:17.867" Title="Did Emil du Bois-Reymond's 1872 lecture on the limits of science influence his brother Paul du Bois-Reymond's 1875 invention of the diagonal argument?" Tags="|diagonal-argument|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3844" PostTypeId="2" ParentId="3836" CreationDate="2016-05-28T18:41:46.763" Score="2" Body="&lt;p&gt;This article on the &lt;a href=&quot;http://spectrum.ieee.org/telecom/wireless/the-long-road-to-maxwells-equations&quot; rel=&quot;nofollow&quot;&gt;Institute for Electrical and Electronics Engineers&lt;/a&gt; site gives a detailed account of the history involved.  For those not wishing to read the entire article, I provide the relevant points here.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hertz’s experimental work on the subject began at the Technische Hochschule (now the Karlsruhe Institute of Technology) in Karlsruhe, Germany, in 1886. He noticed that something curious happened when he discharged a capacitor through a loop of wire. An identical loop a short distance away developed arcs across its unconnected terminals. Hertz recognized that the sparks in the unconnected loop were caused by the reception of electromagnetic waves that had been generated by the loop with the discharging capacitor.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Inspired, Hertz used sparks in such loops to detect unseen radio-frequency waves. He went on to conduct experiments to verify that electromagnetic waves exhibit lightlike behaviors of reflection, refraction, diffraction, and polarization. He performed a host of experiments both in free space and along wires. He molded a meter-long prism made of asphalt that was transparent to radio waves and used it to observe relatively large-scale examples of reflection and refraction. He launched radio waves toward a grid of parallel wires and showed that they would reflect or pass through the grid depending on the grid’s orientation. This demonstrated that electromagnetic waves were transverse: They oscillate, just as light does, in a direction perpendicular to the direction of their propagation. Hertz also reflected radio waves off a large sheet of zinc, measuring the distance between canceled-out nulls in the resulting standing waves in order to determine their wavelengths.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;With this data—along with the frequency of the radiation, which he calculated by measuring the capacitance and inductance of his circuitlike transmitting antenna—Hertz was able to calculate the speed of his invisible waves, which was quite close to that known for visible light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;....&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Maxwell had postulated that light was an electromagnetic wave. Hertz showed that there was likely an entire universe of invisible electromagnetic waves that behave just as visible light does and that move through space at the same speed. This revelation was enough, by inference, for many to accept that light itself is an electromagnetic wave.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.....&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eventually scientists accepted that waves could travel through nothing at all. And the concept of a field, at first distasteful because it lacked any mechanical parts to make it work, became central to much of modern physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-05-28T18:41:46.763" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3845" PostTypeId="2" ParentId="3838" CreationDate="2016-05-28T18:42:00.853" Score="1" Body="&lt;p&gt;These 2 volumes discuss research and academic courses in physics in the 19th century at German universities:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Christa  Jungnickel  and Russell McCormmach. Intellectual Mastery of Nature. Theoretical Physics from Ohm to Einstein, Volume 1: The Torch of Mathematics, 1800 to 1870. University of Chicago Press, paper cover, 1990a. ISBN 0-226-41582-1.&#xA;     Volume 2: The Now Mighty Theoretical Physics, 1870 to 1925. University of Chicago Press, Paper cover, 1990b. ISBN 0-226-41585-6.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in the years 1890-1893 Ludwig Boltzmann held the Chair of Theoretical Physics at the University of Munich and taught the theory of electrodynamics there. His lecture notes were written up by his students and published in two parts in 1891 and 1892. Quoting from the Volume 2 mentioned above, here is how he treated Maxwell's theory (which he supported):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;In the second volume of his lectures on Maxwell's theory, Boltzmann proceeded from the hypothesis that the ether communicates electric and magnetic actions from volume element to the neighboring volume element. Each element supposedly contains an unknown motion causing a displacement, which Boltzmann called the&lt;/code&gt;electrotonic state', following Faraday's terminology. As a starting point for the imagination, Boltzmann conceived of the displacement as the turning angle of a nucleus within each element, any change of which gives rise to a kinetic energy. To represent th e potential energy of an element, Boltzmann fell back on Maxwell's idea that between two rotating nuclei are particles, which serve as friction rollers; the potential energy is then proportional to the work done by the displacement of these particles. By substituting the terms for the kinetic and potential energies into the equation that express Hamilton's principle, Boltzmann derived one set of Maxwell's equations; the other set simply states  the assumptions of the derivation.&#xA;It was part of Boltzmann intention in his lectures to show how ideas from the distance-action theories relate to Maxwell's theory and to treat special problems by Maxwell's theory as thoroughly as they were treated by the other theories, making it unnecessary for students to go to the older textbooks any longer. In the language of potentials, Boltzmann compared Maxwell's equations with those of the older theories. As the potentials are expressed as integrals &#xA;over all space rather than in terms of states of neighboring volume elements, Boltzmann referred to the integral equations as ``Maxwell's distance-action equations' in contrast to his `contact-action' partial differential equations. Boltzmann went on to derive Neumann's induction law, Helmholtz's general theory, and other parts of the older electrodynamics.&quot;&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2016-05-28T18:42:00.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3846" PostTypeId="2" ParentId="3843" CreationDate="2016-05-29T07:06:17.867" Score="2" Body="&lt;p&gt;The main reference for this is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;McCarty, D. C.&#xA;Problems and riddles: Hilbert and the du Bois-Reymonds. &#xA;Synthese  147  (2005),  no. 1, 63–79. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both Paul du Bois-Reymond and Hilbert were profoundly influenced by Emil's lecture. Paul pretty much accepted the implications for mathematics, whereas Hilbert rejected them.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-05-29T07:06:17.867" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3847" PostTypeId="1" AcceptedAnswerId="3849" CreationDate="2016-05-29T13:13:09.477" Score="0" ViewCount="133" Body="&lt;p&gt;In Richard  Courant and Fritz  John's book&#xA;&lt;a href=&quot;https://books.google.com.hk/books?id=TiWsBAAAQBAJ&amp;amp;pg=PA6&amp;amp;lpg=PA6&amp;amp;dq=In%20modern%20times%20mathematics%20was%20recreated%20and%20vastly%20expanded%20on%20a%20foundation%20of%20number%20concepts%20rather%20than%20geometrical%20ones.&amp;amp;source=bl&amp;amp;ots=iXCek0rvpH&amp;amp;sig=OXqSEQTiIfFQn0D3Ga6YVfJEq_I&amp;amp;hl=zh-CN&amp;amp;sa=X&amp;amp;ved=0ahUKEwiY65G_zfzMAhURfiYKHcN1AlsQ6AEIHTAA#v=onepage&amp;amp;q=In%20modern%20times%20mathematics%20was%20recreated%20and%20vastly%20expanded%20on%20a%20foundation%20of%20number%20concepts%20rather%20than%20geometrical%20ones.&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt; Introduction  to  Calculus  and  Analysis Volume  I&lt;/a&gt;, says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In modern times mathematics was recreated and vastly expanded on a&#xA;foundation of number concepts rather than geometrical ones.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why such recreation happened ? For getting rid of the limitations of geometrical methods ?&lt;/p&gt;&#xA;" OwnerUserId="4210" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-05-30T12:57:59.460" Title="mathematics was recreated on a foundation of number concepts rather than geometrical ones" Tags="|calculus|number-theory|real-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3849" PostTypeId="2" ParentId="3847" CreationDate="2016-05-29T18:04:23.630" Score="1" Body="&lt;p&gt;The &amp;quot;reasons&amp;quot; behind the history of the &lt;em&gt;calculus&lt;/em&gt; are many.&lt;/p&gt;&#xA;&lt;p&gt;But a possible source of the &amp;quot;project&amp;quot; of&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;founding mathematics on number concepts rather than geometrical ones&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;may be searched in the development of algebraic symbolism and tools in the 16th and 17th Centuries.&lt;/p&gt;&#xA;&lt;p&gt;As an example, consider Leibniz's own words in his &lt;em&gt;Historia et origo calculi differentialis&lt;/em&gt; (ca.1714), translated into:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gottfried Wilhelm Leibniz, &lt;a href=&quot;https://books.google.it/books?id=tCmp_c3Q9S8C&amp;amp;pg=PA25&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Early Mathematical Manuscripts&lt;/a&gt; (J.M.Child editor: 1920), page 22-on:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;it certainly never entered the mind of anyone else before Leibniz [the memoir is written by Leibniz in third person] to institute the notation peculiar to the new calculus by which the &lt;em&gt;imagination is freed from a perpetual reference to diagrams&lt;/em&gt; [emphasis added], as was made by Vieta and Descartes in their ordinary or Apollonian geometry [i.e. &lt;em&gt;conic sections&lt;/em&gt;]; moreover, the more advanced parts pertaining to Archimedean geometry, and to lines which were called &amp;quot;mechanical&amp;quot; [i.e. &lt;em&gt;transcendent&lt;/em&gt; curves] by Descartes, were excluded by the latter in his calculus.&lt;/p&gt;&#xA;&lt;p&gt;But now by the calculus of Leibniz &lt;em&gt;the whole of geometry is subjected to analytical computation&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;], and those transcendent lines that Descartes called mechanical are also reduced to equations chosen to suit them, by considering the differences &lt;span class=&quot;math-container&quot;&gt;$dx, ddx$&lt;/span&gt;, etc., and the sums that are the inverses of these differences, as functions of the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;'s; and this, by merely introducing the calculus, whereas before this no other functions were admissible but [...] powers and roots.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Consider &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_Elements&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's Elements&lt;/a&gt; (c.300 BC): &lt;strong&gt;Book V&lt;/strong&gt; deals with &lt;em&gt;magnitudes&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookV/propV1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Prop.V.1&lt;/strong&gt;&lt;/a&gt;: If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The proposition is illustrated with &lt;em&gt;segments&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In 1638 Galileo Galilei, in the &lt;a href=&quot;https://books.google.it/books?id=n5cbjKmaZBwC&amp;amp;pg=PA155&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze&lt;/a&gt;, still illustrates his theorems on &lt;em&gt;motion&lt;/em&gt; with &lt;em&gt;ratio&lt;/em&gt; between segments.&lt;/p&gt;&#xA;&lt;p&gt;Then we have the invention of the &lt;em&gt;calculus&lt;/em&gt; and the language began to change.&lt;/p&gt;&#xA;&lt;p&gt;The first calculus textbook from &lt;a href=&quot;https://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guillaume de l'Hôpital&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.it/books?id=si8VAAAAQAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes&lt;/a&gt; (1696) still consider &lt;em&gt;magnitueds&lt;/em&gt;:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition I&lt;/strong&gt;. Those quantities [&lt;em&gt;quantités&lt;/em&gt;] are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while others change.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;With &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.it/books?id=_zpRAAAAYAAJ&amp;amp;pg=PA3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introductio in analysin infinitorum&lt;/a&gt; (1748), page 3:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Caput I: De Functionibus in Genere&lt;/strong&gt;. &lt;em&gt;Quantitas constans est quantitas determinata, perpetuo eumdem valorem servans&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ejusmodi &lt;em&gt;quantitates sunt numeri cujusvis generis&lt;/em&gt; [emphasis added], quippe qui eumdem, quem semel obtinuerunt, valorem constanter conservant [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally we may consider &lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Augustin-Louis Cauchy&lt;/a&gt;'s &lt;a href=&quot;https://books.google.it/books?id=M0or-HGe7D0C&amp;amp;pg=PA5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cours d'Analyse&lt;/a&gt; (1821), &lt;strong&gt;Preliminaries&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;First of all, we will indicate what idea will be appropriate to attach to the two words &lt;em&gt;number&lt;/em&gt; and &lt;em&gt;quantity&lt;/em&gt;. We always take the meaning of &lt;em&gt;numbers&lt;/em&gt; in the sense that is used in arithmetic, where numbers arise from the absolute measure of magnitudes, and we will only apply the term quantities to real positive or negative quantities, that is to say to numbers preceded by the signs &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$−$&lt;/span&gt;. Furthermore, we regard these quantities as intended to express increase and decrease, so that a given magnitude will simply be represented by a number if we only mean to compare it to another magnitude of the same type taken as a unit, and by the same number preceded by the sign &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; or the sign &lt;span class=&quot;math-container&quot;&gt;$−$&lt;/span&gt;, if we consider it as being capable of increasing or decreasing a given magnitude of the same kind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Conclusion&lt;/em&gt;: whith the development of the main new branch of mathematics: &lt;em&gt;analysis&lt;/em&gt;, the concepts of &lt;em&gt;number&lt;/em&gt; and &lt;em&gt;function&lt;/em&gt;, rather than geometrical concepts, became the fundamental ones.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-05-30T12:57:59.460" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3850" PostTypeId="1" CreationDate="2016-05-30T04:39:31.150" Score="2" ViewCount="109" Body="&lt;p&gt;Was there any name for electrical sparks before 'electric' and its related terms were invented as a result of &lt;a href=&quot;https://en.wikipedia.org/wiki/Etymology_of_electricity&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Magnete (1600)&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like, if people were in a dark room, and saw a &lt;a href=&quot;https://en.wikipedia.org/wiki/Triboelectric_effect#Sparks%20triboelectric%20spark&quot; rel=&quot;nofollow noreferrer&quot;&gt;triboelectric spark&lt;/a&gt;, what would they have called it, if anything?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly connected to &lt;a href=&quot;https://history.stackexchange.com/questions/19211/when-did-static-electricity-become-a-common-occurrence&quot;&gt;When did static electricity become a common occurrence?&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/questions/2214/since-when-do-humans-know-about-electric-charge&quot;&gt;Since when do humans know about electric charge?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://electricity.how-old-is-this.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Possibly&lt;/a&gt;, sparks were unknown before the discovery of electricity?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No... There's &lt;a href=&quot;https://en.wiktionary.org/wiki/scintilla&quot; rel=&quot;nofollow noreferrer&quot;&gt;scintilla&lt;/a&gt;, Latin for Spark, but that's only appeared in English from 1700s... Too new? But then again, English wasn't used for scientific writing at first. It was  Latin that ruled, initially.&lt;/p&gt;&#xA;" OwnerUserId="2313" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-08-01T23:31:17.220" Title="Was there any name for electrical sparks before 'electric' and its related terms were invented?" Tags="|electricity|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3851" PostTypeId="1" CreationDate="2016-05-30T05:03:51.563" Score="4" ViewCount="281" Body="&lt;p&gt;As we all know, Fermat has stated &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Fermat&amp;#39;s_Last_Theorem&quot; rel=&quot;nofollow&quot;&gt;his theorem&lt;/a&gt; at the corner of a book. That is far from the standard way of representing a theorem. He said he had a proof which we never saw. So before it was proven true by Andrew Wiles was it assumed that Fermat actually had a proof and thus its a THEOREM or it was called a CONJECTURE till it was proved by Andrew Wiles.&lt;/p&gt;&#xA;" OwnerUserId="4214" LastEditorUserId="4130" LastEditDate="2016-05-31T21:26:54.407" LastActivityDate="2016-05-31T21:26:54.407" Title="Was Fermat's last theorem ever a Conjecture?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ClosedDate="2016-05-30T17:49:08.200" ContentLicense="CC BY-SA 3.0" />
  <row Id="3852" PostTypeId="2" ParentId="3851" CreationDate="2016-05-30T07:39:00.283" Score="4" Body="&lt;p&gt;Mathematicians speculate that Fermat may have made a subtle error in a proposed &quot;proof&quot; such as for example the error of assuming unique factorisation of algebraic integers which was the basis for a 19th century incorrect proof apparently by Lamé.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is considered unlikely that Fermat had a correct proof.  The odds are against it, given the number of people who have worked on the problem and continue to do so :-)&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-05-31T11:10:21.333" LastActivityDate="2016-05-31T11:10:21.333" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3853" PostTypeId="1" CreationDate="2016-05-30T11:58:09.220" Score="6" ViewCount="481" Body="&lt;p&gt;I am in the process of preparing a lecture and remember that some of the Gods (highly esteemed researchers of the past) said, “Either you speak maths or you&#xA;speak nonsense.” Perhaps is was “Either you talk maths or you&#xA;talk nonsense.”&#xA;Any idea of the exact quotation and its original source, including the author?&lt;/p&gt;&#xA;" OwnerDisplayName="user4215" LastActivityDate="2016-06-01T08:24:26.330" Title="Who said “Either you speak maths or you speak nonsense?”" Tags="|mathematics|mathematicians|computer-science|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3854" PostTypeId="2" ParentId="2618" CreationDate="2016-05-31T04:28:29.630" Score="1" Body="&lt;p&gt;See Stigler history of Statistics, p. 92 in which the binomial comes from, Pascal, Newton, Bernoulli, and De Moivre in that order.&#xA;Hth&#xA;Best!&lt;/p&gt;&#xA;" OwnerUserId="4217" LastActivityDate="2016-05-31T04:28:29.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3855" PostTypeId="1" CreationDate="2016-05-31T15:35:43.990" Score="3" ViewCount="435" Body="&lt;p&gt;The time keeping system I've been taught and use has the following (cumbersome) units:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Months, Days of the month, hours of the day, minutes of the hour, and seconds of the minute... above and below these units base 10 units are used (e.g. years and milliseconds)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realize that the awkward calendar units are intended to align the year with the seasons, and the irregular clock units were &quot;designed&quot; for the convenience of dividing the hour (e.g. 1/2 hour, 1/4 hour, 1/5 hour, 1/6 hour) but these days it would seem that a reasonable approach would be Metric Time (e.g. base 10 for all units)...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is/was any such system ever proposed... or in use? If so, how does it work?&lt;/p&gt;&#xA;" OwnerUserId="4222" LastActivityDate="2016-06-02T08:48:31.160" Title="Was there ever a proposal for Metric time" Tags="|time|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3858" PostTypeId="2" ParentId="3853" CreationDate="2016-06-01T00:10:59.293" Score="0" Body="&lt;p&gt;Are you sure it was a short quote?  The page &lt;a href=&quot;http://zapatopi.net/kelvin/quotes/&quot; rel=&quot;nofollow&quot;&gt;http://zapatopi.net/kelvin/quotes/&lt;/a&gt; has the following quote from Kelvin in 1883 (more precise information is on the web page): &quot;I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, you'll see at that link that Kelvin said several times that flying machines other than balloons were impossible and the idea of commercial aviation was inconceivable, even as late as 1902 (the year before the Wright Brothers first flew their aircraft).&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2016-06-01T00:10:59.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3859" PostTypeId="2" ParentId="3855" CreationDate="2016-06-01T00:14:40.673" Score="10" Body="&lt;p&gt;The French Revolution introduced a decimal clock, where the day was divided into 10 &quot;hours&quot; each, these hours into 100 minutes, and these minutes into 100 seconds. The clock looked like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Decimal_time#/media/File:Clock-french-republic.jpg&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Decimal_time#/media/File:Clock-french-republic.jpg&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-06-02T08:48:31.160" LastActivityDate="2016-06-02T08:48:31.160" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3860" PostTypeId="2" ParentId="3853" CreationDate="2016-06-01T08:24:26.330" Score="4" Body="&lt;p&gt;Googling around, I have found this quote somewhere on facebook:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;John McCarthy:&#xA;Those who don't speak math are doomed to speak nonsense.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;However, this apparently are not the actual words. &lt;a href=&quot;https://en.wikiquote.org/wiki/John_McCarthy_(computer_scientist)&quot; rel=&quot;nofollow&quot;&gt;Wiki quote claims&lt;/a&gt; that the actual wording was:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;He who refuses to do arithmetic is doomed to talk nonsense&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;dated ~1995, so.. it is possible that somebody said it earlier too&lt;/p&gt;&#xA;" OwnerUserId="4226" LastActivityDate="2016-06-01T08:24:26.330" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3861" PostTypeId="1" AcceptedAnswerId="3865" CreationDate="2016-06-01T08:49:37.160" Score="7" ViewCount="566" Body="&lt;p&gt;According to Internet (actually, &lt;a href=&quot;https://en.wikipedia.org/wiki/E_(mathematical_constant)&quot;&gt;Wikipedia&lt;/a&gt; and &lt;a href=&quot;http://mathworld.wolfram.com/e.html&quot;&gt;Wolfram MathWorld&lt;/a&gt;), I have two information:  It was Euler who first introduced the symbol $e$ (before people used $b$);  the symbol is to honor Euler.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find this very strange.  My theory is that $e$ is for &quot;exponential&quot; not for Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what is the evidence that Euler was using the symbol to &quot;honor&quot; himself?&lt;/p&gt;&#xA;" OwnerUserId="629" LastEditorUserId="629" LastEditDate="2016-06-01T20:27:14.770" LastActivityDate="2016-06-02T05:38:41.033" Title="Is the symbol $e$ for the base of natural logarithm honoring Euler?" Tags="|notation|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3862" PostTypeId="2" ParentId="3861" CreationDate="2016-06-01T09:04:20.257" Score="2" Body="&lt;p&gt;It was common among mathematicians of the 17th century to use vowels; thus Fermat's writing on adequality is full of A's and E's.  Since A is used much too often the choice of the next vowel, E, was all too natural. I don't think Euler used the symbol to honor himself. Possibly once he chose E, the symbol was retained by later mathematicians so as to honor Euler.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-06-01T09:04:20.257" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3863" PostTypeId="2" ParentId="3861" CreationDate="2016-06-01T19:16:51.840" Score="5" Body="&lt;p&gt;You are right and your Internet source is wrong. First, Euler introduced this notation himself. And clearly this stood for &quot;exponential&quot;. Euler was a modest person, and I cannot imagine him naming anything for himself. Second, if someone wanted to name something after Euler, s/he would choose capital E.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-06-01T19:16:51.840" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3864" PostTypeId="2" ParentId="3855" CreationDate="2016-06-01T19:28:47.657" Score="1" Body="&lt;p&gt;EDITED.&#xA;To add some more information to fdb's answer. They divided the day+night period into 10 hours, so their hour was more than 2 of our hours. They also introduced decimal units for angles (which is closely related to time). 100 decimal degrees in the right angle, so all circumference makes 400 degrees. Every decimal degree is divided in 100 decimal minutes. So one decimal minute is 1/40000 of the circle,&#xA;and this explains the original meaning of the kilometer: it is one decimal minute, similar to nautical mile which is one usual minute of the Earth circumference. Clocks, watches and angle measuring devices made in that period can be sometimes seen on e-bay.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-06-02T05:51:59.520" LastActivityDate="2016-06-02T05:51:59.520" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="3865" PostTypeId="2" ParentId="3861" CreationDate="2016-06-02T05:38:41.033" Score="8" Body="&lt;p&gt;On the &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/HistTopics/e.html&quot;&gt;The number e&lt;/a&gt; page, one can find the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As far as we know the first time the number e appears in its own right&#xA;  is in 1690. In that year Leibniz wrote a letter to Huygens and in this&#xA;  he used the notation b for what we now call e.  [...] Retrospectively,&#xA;  the early developments on the logarithm became part of an&#xA;  understanding of the number e.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So much of our mathematical notation is due to Euler that it will come&#xA;  as no surprise to find that the notation e for this number is due to&#xA;  him. The claim which has sometimes been made, however, that Euler used&#xA;  the letter e because it was the first letter of his name is&#xA;  ridiculous. It is probably not even the case that the e comes from&#xA;  &quot;exponential&quot;, but it may have just be the next vowel after &quot;a&quot; and&#xA;  Euler was already using the notation &quot;a&quot; in his work. Whatever the&#xA;  reason, the notation e made its first appearance in a letter Euler&#xA;  wrote to Goldbach in 1731&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some other details are given &lt;a href=&quot;https://www.quora.com/What-is-the-origin-story-of-e-base-of-natural-exponential-function&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Euler started to use the letter e for the constant in 1727 or 1728, in&#xA;  an unpublished paper on explosive forces in cannons, and the first&#xA;  appearance of e in a publication was Euler's Mechanica (1736). While&#xA;  in the subsequent years some researchers used the letter c, e was more&#xA;  common and eventually became the standard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4130" LastActivityDate="2016-06-02T05:38:41.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3866" PostTypeId="1" CreationDate="2016-06-02T11:25:21.427" Score="2" ViewCount="152" Body="&lt;p&gt;It might look like not a serious question, since Leibniz discovered Cramer's rule (1684), so it makes much sense that he interpreted the determinant as an expansion factor for a volumetric element. I'm only asking if someone can give sources, references to works by Leibniz in which he gave this interpretation of the determinant concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Leibniz interpret his determinant as a volume function?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="62" LastEditDate="2016-06-02T23:08:45.547" LastActivityDate="2016-06-02T23:08:45.547" Title="Did Leibniz interpret his determinant as a volume function?" Tags="|mathematics|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="3867" PostTypeId="1" AcceptedAnswerId="3868" CreationDate="2016-06-03T12:35:01.170" Score="4" ViewCount="274" Body="&lt;p&gt;My question is based on the information on pages 108-109 of the book &lt;a href=&quot;https://doi.org/10.1142/8413&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Tangled Origins of the Leibnizian Calculus&lt;/a&gt;. I know that Leibniz&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;invented the stepped drum and used it to build the stepped reckoner — the first mechanical calculator capable of doing all four arithmetical operations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;sketched a design for a different machine‚ the pinwheel calculator.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;designed a cipher machine.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;developed binary arithmetic (based on &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;) and established its importance for computers, and even described in detail some of the fundamental principles of the modern computer (in his treatise &lt;em&gt;De progressione Dyadica&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;With such achievements in hand, it's not an exageration to believe the statement in this book.&lt;/p&gt;&#xA;&lt;p&gt;Can someone give a reference for Leibniz's supposed machine? Can someone give a reference for a drawing of Leibniz machine (how does it look like?)?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2021-09-22T05:52:32.730" LastActivityDate="2021-09-22T05:52:32.730" Title="Did Leibniz sketch a design for a machine capable of solving a system of linear equations?" Tags="|mathematicians|biographical-details|linear-algebra|computers|leibniz|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="3868" PostTypeId="2" ParentId="3867" CreationDate="2016-06-03T20:45:02.950" Score="3" Body="&lt;p&gt;I've found an indirect reference into:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Ehrenfried_Hofmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Ehrenfried Hofmann&lt;/a&gt;, &lt;a href=&quot;http://www.cambridge.org/us/academic/subjects/history/history-science-and-technology/leibniz-paris-1672-1676-his-growth-mathematical-maturity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz in Paris 1672-1676&lt;/a&gt; (1974, German ed.1949), page 30 and footnote 27:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;still in 1674 he [Leibniz] has plans for constructing an &amp;quot;analytical&amp;quot; counterpart to his calculating machine, an instrument for determining the solutions of equations, and he actually succeeds in doing so [&lt;em&gt;footnote 27&lt;/em&gt;]&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Footnote 27&lt;/em&gt; for details see Hofmann (1970) [&lt;a href=&quot;http://www.jstor.org/stable/40693589?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Uber fruhe mathematische Studien von G.W.Leibniz&amp;quot;, &lt;em&gt;Studia Leibnitiana&lt;/em&gt;, 2, 81-114&lt;/a&gt;] : 101-4. The instrument is mentioned in letters to Oldenburg: June, July, Dec.1675; also in L-Huygens, mid-Sept.1675. From Huygens' rather cautious reply, 30 Sept.1675 we may conclude that he has seen a sketch of the &amp;quot;Compas d'equation&amp;quot;, as had Tschirnhaus according to Leibniz's letter, 8 Jan.1694. [...] A similar instrument is described in the &lt;a href=&quot;https://books.google.it/books?id=zGREAAAAcAAJ&amp;amp;pg=PA834&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopédie, Suppl.II (1776): 834-5&lt;/a&gt; (with an illustration in the &lt;a href=&quot;http://www.biodiversitylibrary.org/item/156186#page/361/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planche suppl.(1777)&lt;/a&gt;) under &amp;quot;Algèbre&amp;quot;. This instrument works for quadratic equations only, but it is justly claimed as a prototype capable of adaptation to higher equations. The inventor is not named: he might indeed have been inspired by the hint in G.W.Leibniz, &lt;a href=&quot;https://archive.org/stream/derbriefwechselv00leibuoft#page/146/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Der Briefwechsel mit Mathematikern I (Berlin 1899): 146&lt;/a&gt; since the passage was available in print, in John Wallis, &lt;a href=&quot;https://books.google.it/books?id=TSEtJ31s0w4C&amp;amp;pg=PA620&quot; rel=&quot;nofollow noreferrer&quot;&gt;Opera mathematica III (Oxford 1699): 620-2&lt;/a&gt; where the letter in question is reproduced, and in &lt;a href=&quot;https://books.google.it/books?id=YDgPAAAAQAAJ&amp;amp;pg=PA45&quot; rel=&quot;nofollow noreferrer&quot;&gt;Commercium Epistolicum (1712): 45&lt;/a&gt; where an excerpt is given [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-06-04T16:16:43.277" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3869" PostTypeId="1" AcceptedAnswerId="3874" CreationDate="2016-06-03T20:50:18.937" Score="1" ViewCount="279" Body="&lt;p&gt;When and where did Riemann and Lebesgue give the well known &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann%E2%80%93Lebesgue_lemma&quot; rel=&quot;nofollow&quot;&gt;Riemann-Lebesgue lemma&lt;/a&gt;? A &quot;lemma&quot;  is usually used as a stepping stone to a larger result rather than as a statement of interest by itself. Did Riemann and Lebesgue intend to prove a larger result when they gave this work?&lt;/p&gt;&#xA;" OwnerDisplayName="user4178" LastActivityDate="2016-06-05T01:19:34.380" Title="Origin of the Riemann-Lebesgue lemma" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3870" PostTypeId="1" AcceptedAnswerId="3871" CreationDate="2016-06-04T04:27:07.943" Score="8" ViewCount="523" Body="&lt;p&gt;In the letter Ramanujan wrote to Hardy did he say that $1+2+3+\cdots=-\frac{1}{12}$. I have been hearing this ridiculous statement for some time now. And now people say that Ramanujan wrote it. Is it true? &lt;/p&gt;&#xA;" OwnerDisplayName="user4237" LastEditorUserId="2913" LastEditDate="2016-06-20T15:46:12.923" LastActivityDate="2017-08-22T00:50:46.220" Title="Was $1+2+3+\cdots=-\frac{1}{12}$ in Ramanujan's letter?" Tags="|mathematics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="3871" PostTypeId="2" ParentId="3870" CreationDate="2016-06-04T07:28:50.337" Score="6" Body="&lt;p&gt;Ramanujan did write down the expression in his first notebook as shown &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg&quot;&gt;here&lt;/a&gt;.  For a quick overview of this series, John Baez's &lt;a href=&quot;http://math.ucr.edu/home/baez/numbers/24.pdf&quot;&gt;lecture notes&lt;/a&gt; are a good starting point. These notes suggest that Ramanujan's observation goes back to Euler, though I have only seen the related alternating series $1 - 2 + 3 - 4 \dots = \frac{1}{4}$ mentioned in Euler's publications (see &lt;a href=&quot;https://math.dartmouth.edu/~euler/docs/originals/E352.pdf&quot;&gt;E352&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2016-06-04T08:16:11.327" LastActivityDate="2016-06-04T08:16:11.327" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="3872" PostTypeId="2" ParentId="3837" CreationDate="2016-06-04T23:18:35.220" Score="6" Body="&lt;p&gt;Einstein's physics teacher, H. F. Weber, apparently did not teach him any Helmholtz, as Einstein wrote in a 10 August 1899 letter to Mileva Marić:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I returned the Helmholtz volume* and am at present studying again in depth Hertz's propagation of electric force.** The reason for it was that [I] didn't understand Helmholtz's treatise on the principle of least action in electrodynamics. I am more and more convinced that the electrodynamics of moving bodies, as presented today, is not correct, and that it should be possible to present it in a simpler way. The introduction of the term &quot;ether&quot; into the theories of electricity led to the notion of a medium of whose motion one can speak without being able, I believe, to associate a physical meaning with this statement. I think that the electric forces can be directly defined only for empty space, [which is] also emphasized by Hertz…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;—&lt;a href=&quot;http://einsteinpapers.press.princeton.edu/vol1-trans/153&quot; rel=&quot;noreferrer&quot;&gt;Volume 1: The Early Years, 1879-1902 (English translation supplement) Page 131&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*Helmholtz's 1882 &lt;a href=&quot;http://books.google.com/books?id=Xus0AQAAMAAJ&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Wissenschaftliche Abhandlungen&lt;/em&gt;&lt;/a&gt; (not translated into English)&#xA;&lt;br&gt;**Hertz's &lt;a href=&quot;http://ncco.galegroup.com/gdc/ncco/MonographsDetailsPage/MonographsDetailsWindow?disableHighlighting=false&amp;amp;prodId=NCCO&amp;amp;action=1&amp;amp;activityType=BasicSearch&amp;amp;javax.portlet.action=viewPortletAction&amp;amp;documentId=GALE%7CBCIOCC304684907&amp;amp;dviSelectedPage=1&amp;amp;userGroupName=&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Electric waves being researches on the propagation of electric action with finite velocity through space&lt;/em&gt;&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's surprising Einstein was not formally taught the 19th century's leading way of doing E&amp;amp;M: deriving electric potentials from energetics first principles. This was taught in, e.g., Duhem's 1891-92 &lt;a href=&quot;https://archive.org/search.php?query=duhem%20Le%C3%A7ons%20sur%20l%27%C3%A9lectricit%C3%A9%20et%20le%20magn%C3%A9tisme&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Leçons sur l'électricité et le magnétisme&lt;/em&gt;&lt;/a&gt; course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's collected papers nowhere mention the derivation of Wilhelm Weber's law from Ampère's force law, which by the time Einstein was a physics student was at least ⅓ century old. Perhaps 19&lt;sup&gt;th&lt;/sup&gt; century physics instruction lagged behind cutting-edge research developments, much as today the vast majority of what is taught in physics courses were cutting-edge developments at least a century ago. Perhaps this is why Einstein was frustrated and rarely attended class.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-06-04T23:34:38.847" LastActivityDate="2016-06-04T23:34:38.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="3873" PostTypeId="2" ParentId="3759" CreationDate="2016-06-04T23:27:08.137" Score="-1" Body="&lt;p&gt;The history of decimal point is much older than Simon Stevin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC. But the oldest Chinese book introducing a decimal mark equivalent, is from the thirteenth century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around the middle of the tenth century, Al-Uqlidisi wrote &quot;Kitab al-fusul fi al-hisab al-Hindi&quot; (Chapter's book of Indian Arithmetic), which is the earliest surviving book that presents the Indian system.(Survived in a copy of the original which was made in 1157).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the fourth part of this book Al-Uqlidisi showed how to modify the methods of calculating with Indian symbols, which had required a dust board, to methods which could be carried out with pen and paper. This requirement of a dust board had been an obstacle to the Indian system's acceptance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Al-Uqlidisi's book is also historically important as it is the earliest known text offering a direct treatment of decimal fractions. Al-Uqlidisi used a decimal dash, above the first digit of the fractional part of the decimal number.( Very simple if compared to Stevin's notation. But similar to the decimal point used by Bartholomaeus Pitiscus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the Persian mathematician Jamshīd Al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that he was mistaken, as decimal fractions were first used five centuries before him by Al-Uqlidisi as early as the 10th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rashed puts Al-Kashi's important contribution into perspective. He shows that the main advances brought in by Al-Kashi are:-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Sorry, I am not permitted to post more than two links.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://vedicsciences.net/articles/history-of-numbers-part-2.html&quot; rel=&quot;nofollow&quot;&gt;http://vedicsciences.net/articles/history-of-numbers-part-2.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Uqlidisi.html&quot; rel=&quot;nofollow&quot;&gt;http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Uqlidisi.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4244" LastEditorUserId="4244" LastEditDate="2016-09-02T20:36:05.270" LastActivityDate="2016-09-02T20:36:05.270" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="3874" PostTypeId="2" ParentId="3869" CreationDate="2016-06-05T01:19:34.380" Score="1" Body="&lt;p&gt;Here's a partial answer regarding the when and where with links to the papers.  Riemann proved the Riemann-Lebesgue result for bounded Riemann integrable functions in his habilitation paper &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/Trig.pdf&quot; rel=&quot;nofollow&quot;&gt;&quot;&lt;em&gt;Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe&lt;/em&gt;&quot;&lt;/a&gt;.  He wrote the paper in 1853 (presented in 1854), but it was only published posthumously in 1867.  Lebesgue proved the result for $L^1$ functions in his 1903 paper &lt;a href=&quot;http://proba.jussieu.fr/~mazliak/Lebesgue_1903.pdf&quot; rel=&quot;nofollow&quot;&gt;&quot;&lt;em&gt;Sur les séries trigonométriques&lt;/em&gt;&quot;&lt;/a&gt;, which was published in Annales scientifiques de l'École Normale Supérieure.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2016-06-05T01:19:34.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4874" PostTypeId="1" CreationDate="2016-06-05T15:16:20.293" Score="1" ViewCount="227" Body="&lt;p&gt;My question refers to the following problem in electromagnetic theory which Gauss was concerned with (see &lt;strong&gt;Zur theorie der electrodynamik, volume 5 of Gauss's werke&lt;/strong&gt;): Determine the work done on a magnetic monopole of strengh &lt;strong&gt;m&lt;/strong&gt; when it is moved along a closed loop L_1 in the magnetic field produced by a closed current loop L_2 with current &lt;strong&gt;I&lt;/strong&gt;. This problem led Gauss to the introduction of the &lt;strong&gt;Gauss's linking integral&lt;/strong&gt; for computing the linking number of two interwinged loops. According to several books, this integral is constant under derformations of the loops as long as their number of interwingings is constant (so it's a measure of the degree of entaglement of the two loops). I'm trying to understand what was his result (what was his topological constant) and also what is his meaning, because according to several books is has an advanced physical meaning (the Gauss's linking integral is an advanced analytic tool).&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2016-06-05T19:14:10.420" LastActivityDate="2016-06-05T21:14:06.317" Title="What was the topological constant which Gauss derived in the context of his investigations in electromagnetism and knot theory?" Tags="|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4875" PostTypeId="2" ParentId="4874" CreationDate="2016-06-05T21:14:06.317" Score="3" Body="&lt;p&gt;I would suggest the following paper:&#xA;&lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/ricca.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.maths.ed.ac.uk/~aar/papers/ricca.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it contains a quite detailed analysis of Gauss original works in modern terms.&#xA;The linking number appears in computation of so-called helicity of vector fields, and in this form it is often applied to magnetic vector field.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2016-06-05T21:14:06.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4876" PostTypeId="1" CreationDate="2016-06-06T11:50:12.787" Score="3" ViewCount="961" Body="&lt;p&gt;I have recently been reading about serendipitous discoveries in science and I found them quite inspiring. Most of those discoveries are in Chemistry. I'm looking for examples of these kinds of discoveries in Mathematics and Computer Science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples of serendipitous discoveries include the discoveries of penicillin, saccharine and superglue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Poincare's accidental discovery of Chaos Theory qualifies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for the element of surprise and accident. A wonderful proof which consists of an extremely enlightened step does not qualify because the person was still searching for a proof. But, if a person was working in area X and suddenly realized that it's a way of solving Y, then that discovery would qualify.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note : There is a question on this already on the site. My question is different for two reasons. Firstly, I'm not asking for important discoveries. Secondly, I'm not asking for recency either. Any discovery in the history of Mathematics qualifies.&lt;/p&gt;&#xA;" OwnerUserId="3754" LastActivityDate="2018-11-03T02:26:15.160" Title="Serendipitous discoveries in Mathematics and Computer Science" Tags="|mathematics|discoveries|computer-science|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="4877" PostTypeId="1" AcceptedAnswerId="4878" CreationDate="2016-06-06T14:58:35.110" Score="4" ViewCount="185" Body="&lt;p&gt;Can cubit from Galileo Galilei books &quot; Dialogue Concerning the Two Chief World Systems&quot;, &quot;Two New Sciences&quot;, Newtons &quot;Philosophiae Naturalis Principia Mathematica&quot; or Mikołaj Kopernik books convert to other units like feet, meter, inch or does 1 cubit own different value nowadays? &lt;/p&gt;&#xA;" OwnerUserId="4249" LastActivityDate="2016-06-08T09:29:48.843" Title="History of cubit - Galilei, Kopernik, Newton" Tags="|units|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4878" PostTypeId="2" ParentId="4877" CreationDate="2016-06-06T15:54:23.833" Score="4" Body="&lt;p&gt;&lt;em&gt;Issues regarding ancient measure systems are not easy to manage&lt;/em&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo Galilei, in &lt;a href=&quot;https://en.wikipedia.org/wiki/Two_New_Sciences&quot; rel=&quot;nofollow&quot;&gt;Discorsi&lt;/a&gt;, uses &lt;em&gt;braccia&lt;/em&gt;; the &quot;old&quot; translation by Crew &amp;amp; Di Salvio uses: &quot;cubits&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://it.wikipedia.org/wiki/Braccio_(unit%C3%A0_di_misura)&quot; rel=&quot;nofollow&quot;&gt;braccio&lt;/a&gt; in Florence was: 583 mm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In ancient Rome:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;according to Vitruvius, a &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubit#Ancient_Rome&quot; rel=&quot;nofollow&quot;&gt;cubit&lt;/a&gt; was equal to $1 \frac 1 2$ Roman feet or 6 palm widths, which is 443.8 mm (17.47 in).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://books.google.it/books?id=c-nIrKjBqOwC&quot; rel=&quot;nofollow&quot;&gt;Stillman Drake&lt;/a&gt; [footnote page 469], the &lt;em&gt;braccio&lt;/em&gt; used by Galileo was 21 to 22 inches, i.e. between 533 and 558 mm; he translates it with &lt;a href=&quot;https://en.wikipedia.org/wiki/Yard&quot; rel=&quot;nofollow&quot;&gt;yard&lt;/a&gt;, which is 914 mm (36 in).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion, we may say that a good approximation for Galilei's &lt;em&gt;braccio&lt;/em&gt; is an half-meter.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Regarding Copernicus, the evaluation of ancient astronomers regarding the dimension of planets' orbits was expressed in &lt;em&gt;earth radii&lt;/em&gt;, i.e. with a &quot;relative&quot; measure; thus, I think that you can hardly find in his works measures expressed in &lt;em&gt;cubits&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Isaac Newton in the &lt;em&gt;Principia&lt;/em&gt; uses &lt;a href=&quot;https://en.wikipedia.org/wiki/Units_of_measurement_in_France_before_the_French_Revolution#Length&quot; rel=&quot;nofollow&quot;&gt;(Paris) feet&lt;/a&gt; [about 325 mm] and &lt;em&gt;inches&lt;/em&gt;; see &lt;strong&gt;Book 3&lt;/strong&gt;, &lt;em&gt;Prop.17, Th.15&lt;/em&gt; for :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And from these measures the circumference of the earth is found to be $123,249,600$ Paris feet, and its semidiameter $19,615,800$ feet, on the hypothesis that the earth is spherical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-06-08T09:29:48.843" LastActivityDate="2016-06-08T09:29:48.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4879" PostTypeId="1" AcceptedAnswerId="4885" CreationDate="2016-06-08T16:51:29.693" Score="5" ViewCount="398" Body="&lt;p&gt;Problems concerning tangents and quadrature have a long history predating the Newton/Leibniz formulation of calculus; indeed, they are amongst the oldest problems in mathematics.  It seems reasonable to assume that the relationship between the two problems was “well” understood for some time prior to the calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Carl Boyer, in his &lt;em&gt;A History of Mathematics&lt;/em&gt;, Isaac Barrow’s &lt;em&gt;Lectiones geometriae&lt;/em&gt; (1670) prominently feature both problems, and it is “likely” that he understood the relationship between the two.  However, Boyer also points out that Barrow’s dislike of analytic methods were “scarcely conducive to analytic discoveries” and that it was at Newton’s insistence that an analytic treatment of this material was included in Barrow’s text - Newton was acting as Barrow’s editor.  Boyes also notes that Barrow’s conservative adherence to geometric methods kept him from making effective use of this relationship.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many figures prior to Barrow appear to have been in position of all the necessary theory to positively identify the relationship - Fermat, Huygens, Wallis, Gregory, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was the inverse relationship between tangents and quadrature/area first identified?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-06-08T16:57:34.833" LastActivityDate="2016-06-09T11:36:49.383" Title="When was the inverse relationship between tangents and quadrature/area first identified?" Tags="|calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4880" PostTypeId="1" AcceptedAnswerId="4887" CreationDate="2016-06-08T21:46:21.600" Score="6" ViewCount="568" Body="&lt;p&gt;I want to know who derived&#xA; $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$ In school, our book mentioned that Euler proved this result. But on Math Stack Exchange, some people say that Laplace was the first person to derive this result. In Euler's book &quot;Zur Theorie der komplexen Funktionen&quot; he derives a more general integral and I think he knew how to use it to calculate the given integral. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be glad if someone could give a proper answer so I know who to give credit to.&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="62" LastEditDate="2016-12-29T22:20:39.347" LastActivityDate="2016-12-29T22:20:39.347" Title="Who derived $\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$?" Tags="|mathematics|discoveries|calculus|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4881" PostTypeId="2" ParentId="4880" CreationDate="2016-06-08T23:11:41.603" Score="1" Body="&lt;p&gt;According to Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt;, Abraham De Moivre was the first to &quot;work&quot; with this formula, having obtained it in a privately printed pamphlet.  Quoting Boyer :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;De Moivre apparently was the first one to work with the probability formula $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$&#xA;  as a result that appeared unobtrusively in a privately printed pamphlet of 1733 entitled &lt;em&gt;Approximatio ad summam terminorum binomii $(a+b)^n$ in seriem expansi&lt;/em&gt;.  This work, representing the first appearance of the law of errors or the distribution curve, was translated by De Moirvre and included in the second edition (1738) of his &lt;em&gt;Doctrine of Chances&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If there is a lack of clarity in Boyer's wording, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Abraham_de_Moivre&quot; rel=&quot;nofollow&quot;&gt;wikipedia page on De Moivre&lt;/a&gt; credits De Moivre as the author of the pamphlet.  One assumes that De Moivre was therefore the first to derive it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;http://www.york.ac.uk/depts/maths/histstat/demoivre.pdf&quot; rel=&quot;nofollow&quot;&gt;this paper on the York University site.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-06-08T23:32:34.977" LastActivityDate="2016-06-08T23:32:34.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4882" PostTypeId="1" CreationDate="2016-06-09T06:57:08.490" Score="1" ViewCount="126" Body="&lt;p&gt;This is a reference request prompted by some intriguing comments made by Felix Klein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be successful. Namely, one must be able to prove a mean value theorem (MVT) for arbitrary intervals, including infinitesimal ones: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The question naturally arises whether ... it would be possible to modify the traditional foundations of infinitesimal calculus, so as to include actually infinitely small quantities in a way that would satisfy modern demands as to rigor; in other words, to construct a non-Archimedean system. The first and chief problem of this analysis would be to prove the mean-value theorem $$ f(x+h)-f(x)=h \cdot f'(x+\vartheta h) $$ from the assumed axioms. I will not say that progress in this direction is impossible, but it is true that none of the investigators have achieved anything positive.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This comment appears on page 219 in the book (Klein, Felix Elementary mathematics from an advanced standpoint. Arithmetic, algebra, analysis) originally published in 1908 in German. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 1. When Klein writes that none of the current investigators have achieved, etc., who is he referring to? There were a number of people working &quot;in this direction&quot; at the time, and it would be interesting to know whose work Klein had in mind: Stolz, Paul du Bois-Raymond (somewhat earlier), Hahn, Hilbert, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 2. Did Klein elaborate in this direction in other works of his? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Question 3. Why is Klein focusing specifically on the mean value theorem?  Was this theorem a preoccupation of other mathematicians at the time?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-06-13T12:26:54.237" LastActivityDate="2016-06-13T12:26:54.237" Title="Felix Klein and the mean value theorem" Tags="|mathematicians|real-analysis|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4885" PostTypeId="2" ParentId="4879" CreationDate="2016-06-09T07:36:54.023" Score="3" Body="&lt;p&gt;For this complex historical process, you can see:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Margaret Baron, &lt;a href=&quot;https://books.google.it/books?id=S0L9Njm3xp8C&amp;amp;pg=PA177&quot; rel=&quot;nofollow noreferrer&quot;&gt;The origins of the infinitesimal calculus&lt;/a&gt;, (1969 - also Dover reprint).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;See &lt;em&gt;Introduction&lt;/em&gt;, page 4:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The infinitesimal calculus emerged in the form of two major classes of problem, the first concerned with the determination of arc, area, surface, volume and centre of gravity, the second with angle, chord, tangent, curvature, turning point and inflexion. [...] The inverse nature of the two classes of problem was approached in terms of geometric model by Torricelli, Gregory and Barrow but only with Newton did the relation emerge as central and general.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[page 177] Fermat: for one of the earliest link between differential and integral processes;&lt;/p&gt;&#xA;&lt;p&gt;[page 179] for Roberval relating tangent and area properties of curves;&lt;/p&gt;&#xA;&lt;p&gt;[page 189] for Torricelli clear understanding (?) of the inverse relation between integration and differentiation;&lt;/p&gt;&#xA;&lt;p&gt;[page 243] Isaac Barrow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-06-09T11:36:49.383" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4886" PostTypeId="2" ParentId="3824" CreationDate="2016-06-09T14:49:34.483" Score="16" Body="&lt;p&gt;It is often said that the reason why Einstein's Nobel Prize didn't mention Relativity Theory is the lack of sufficient evidence to the theory of relativity by 1922. But actually, by 1922, the special theory of relativity had been tested for almost all its major and pivotal predictions. The general theory of relativity had passed many highly significant tests with extreme precision. Like the deflection of light rays or the precession of the perihelion of the Mercury. It should be noted that although the precession of the perihelion of the Mercury was known before the formulation of the theory, GR was not an &lt;em&gt;ad hoc&lt;/em&gt; explanation for the same--and thus, the fact of precession should be regarded as a verified prediction of GR. So, the absence of the mention of the Relativity Theory remains a curious question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many believe that the actual reason has something to do with the views of Bergson, one of the most prominent philosophers of the 20th century, who was also more famous for his brilliance than Einstein was at the time, on Einstein's theory of relativity. Bergson considered Einstein's theory to be metaphysical and had declared his problem with the theory publicly. The writer of an account of the famous debate between Einstein and Bergson, Canales, has noted: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When the Nobel Prize was awarded to Einstein a few months later, it&#xA;  was not given for the theory that had made the physicist famous:&#xA;  relativity. Instead, it was given “for his discovery of the law of the&#xA;  photoelectric effect” — an area of science that hardly jolted the&#xA;  public’s imagination to the degree that relativity did. The reasons&#xA;  behind the decision to focus on this work other than relativity were&#xA;  directly traced to what Bergson said that day in Paris.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The president of the Nobel Committee explained that although “most&#xA;  discussion centers on his theory of relativity,” it did not merit the&#xA;  prize. Why not? The reasons were surely varied and complex, but the&#xA;  culprit mentioned that evening was clear: “It will be no secret that&#xA;  the famous philosopher Bergson in Paris has challenged this theory.”&#xA;  Bergson had shown that relativity “pertains to epistemology” rather&#xA;  than to physics — and so it “has therefore been the subject of lively&#xA;  debate in philosophical circles.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't read Walter Isaacson's biography of Einstein but the NatGeo show &quot;Genius&quot; which is based on the same indicates that Phillip Lennard had a major influence on the Nobel committee which he used to persuade them against Einstein. This seems plausible and it also explains why Einstein was given Nobel for the explanation of the photoelectric effect. Because Lennard's experiments were crucial in our experimental understanding of the phenomenon and thus, when the Nobel committee decided to give the Nobel to Einstein anyway, they tried to placate Lennard to some extent by giving it Einstein not for relativity but for the photoelectric effect.&lt;/p&gt;&#xA;" OwnerUserId="4257" LastEditorUserId="4257" LastEditDate="2018-09-19T07:57:42.247" LastActivityDate="2018-09-19T07:57:42.247" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4887" PostTypeId="2" ParentId="4880" CreationDate="2016-06-09T15:43:59.643" Score="7" Body="&lt;p&gt;While Nick R assumes in his answer that from Boyer's writing the formula is due to De Moivre, I think that is misleading and the first derivation of the formula you ask about really should be credited to Laplace. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;De Moivre was working on the binomial approximation to the normal distribution, and his approximation involved a constant $B$ (I am taking that notation from De Moivre's paper at the York University site that Nick R links to in his answer). De Moivre could express $B$ as a series but he was not able to express $B$ in closed form. Stirling told him that $B = \sqrt{2\pi}$, without proof, and with this clue De Moivre was later able to derive that nice formula for $B$ using the Wallis product for $\pi$. The key point I want to make is that in none of this work (from the 1720s) was De Moivre directly using anything like an integral of $e^{-x^2}$ in order to make any calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A better paper to look at on the York University website is &lt;a href=&quot;http://www.york.ac.uk/depts/maths/histstat/normal_history.pdf&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first proof of the integral formula you ask about where the author used  integrals was given by Laplace in the 1770s. He wrote the formula  (after the change of variables $y = e^{-x^2}$) as &#xA;$$&#xA;\int_0^1 \frac{dy}{\sqrt{-\log y}}\,dy = \sqrt{\pi}&#xA;$$&#xA;and derived this from a formula for integrals due to Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf&quot;&gt;here&lt;/a&gt; for many proofs of the integral formula you ask about. The eighth proof is  Laplace's argument I allude to above and explains how it uses a result of Euler, and the tenth proof shows that this formula is equivalent to identifying the constant in Stirling's asymptotic estimate for $n!$ as $\sqrt{2\pi}$. The second proof is a much nicer alternate proof by Laplace that is a forerunner of the standard proof via squaring the integral and passing to polar coordinates; Laplace's alternate argument squares the integral and makes a different change of variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the end of &lt;a href=&quot;http://www.math.uconn.edu/~kconrad/blurbs/analysis/stirling.pdf&quot;&gt;this&lt;/a&gt; for a discussion of how De Moivre's work on the binomial approximation to the normal distribution intersected with Stirling's work on Stirling's formula, and how Stirling found the constant $\sqrt{2\pi}$ in his formula.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2016-06-09T15:50:13.753" LastActivityDate="2016-06-09T15:50:13.753" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4890" PostTypeId="1" AcceptedAnswerId="4891" CreationDate="2016-06-10T10:59:07.520" Score="10" ViewCount="1347" Body="&lt;p&gt;It is quite well-known that Planck was one of the first people to realize the significance and importance of Einstein's theory of relativity. But I have heard a few times that when Einstein sent his manuscript of the paper to Planck for publication, while Planck was reading it, he had first thought that the paper was not fit to be published. But as he finished reading it, he had realized that Einstein was onto something big. Is this story true? Is there any record of any statement by Planck (or his friends) regarding the same? &lt;/p&gt;&#xA;" OwnerUserId="4257" LastEditorUserId="4251" LastEditDate="2020-06-21T13:32:11.770" LastActivityDate="2020-06-21T13:32:11.770" Title="Max Planck's reaction to Einstein's 1905 STR Paper" Tags="|theoretical-physics|relativity-theory|einstein|physicists|planck|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4891" PostTypeId="2" ParentId="4890" CreationDate="2016-06-10T13:11:54.383" Score="13" Body="&lt;p&gt;It seems not. See:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A.Douglas Stone, &lt;a href=&quot;https://books.google.it/books?id=4rxKCAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Einstein and the quantum: The quest of the valiant Swabian&lt;/a&gt; (2013):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[page 6] Planck was the first major figure to recognize Einstein’s seminal 1905 work on relativity theory, and he became Einstein’s greatest champion in the world of science and one of his closest personal friends.&lt;/p&gt;&#xA;&lt;p&gt;[page 83] [Einstein's 1905 paper on &lt;em&gt;Annalen&lt;/em&gt;] is clearly written and would have been relatively easy to understand for an expert like Planck (who was theory editor for the journal).&lt;/p&gt;&#xA;&lt;p&gt;[page 85] Despite his reservations about Planck’s method for deriving his radiation law, Einstein treats Planck quite delicately in his paper. [...] [ From] a cryptic comment by Besso much later, in 1928, it appears that Besso prevailed on Einstein to revise an earlier version of this paper in which more pointed comments were made about the correctness of Planck’s derivation.&lt;/p&gt;&#xA;&lt;p&gt;[page 93] “I do not seek the meaning of the quantum of action (light quantum) in the vacuum but at the sites of absorption and emission, and assume that processes in vacuum are described exactly by Maxwell’s equations.” This was Max Planck’s first known response to Einstein’s heuristic theory of light quanta, sent to Einstein in a letter of July 6, 1907, more than two years after the publication of the “revolutionary” paper of 1905. Planck must surely have known of Einstein’s ideas much earlier, since he was the theory editor of the journal in which they were published, &lt;em&gt;Annalen der Physik&lt;/em&gt;. Unfortunately none of the referee or editorial comments on Einstein’s papers of 1905 has survived, so we don’t know how direct a role Planck had in approving them for publication. Planck was known for being open to publishing scientific contributions with which he disagreed, as long as they did not contain clear errors, and this tolerance likely came into play with Einstein’s paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-06-10T13:11:54.383" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4892" PostTypeId="1" CreationDate="2016-06-10T14:02:37.623" Score="2" ViewCount="161" Body="&lt;p&gt;I read at many places that Leibniz made a lot of contributions to biology, anatomy, medicine, paleontology and other related life sciences, but I couldn't find concrete contributions of him to these fields. I'm not looking for philosophical remarks that have far reaching meaning (for example, his distinction between organic bodies and &quot;artificial&quot; machines) but rather for more practical contributions. I'm far from being an expert to Leibniz's writings and that is why i'm asking this question. I'm very curious to discover Leibniz's wisdom so if someone has references, please write them down.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="205" LastEditDate="2016-07-12T08:07:01.907" LastActivityDate="2016-08-11T08:56:26.673" Title="What were Leibniz's concrete contributions to the life sciences?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4893" PostTypeId="2" ParentId="3512" CreationDate="2016-06-10T14:06:08.600" Score="-2" Body="&lt;p&gt;The reason a clock uses 12 hours is actually simple: use a compass/forked stick in the sand to make a circle. then, naturally one would place one leg of the compass at some point on the circle, and draw a second circle.  then, one would naturally place the leg on one of the intersections just made, and draw a third one. 3 more times, and you have gone all around and drawn a pattern of 1 circle in the middle (the 7th)  and 6 circles around.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that, division in 6/12 is the easiest, and also most accurate way of dividing a circle in the past. Dividing a circle in 5/10 parts is really hard to do accurate with only the 'euclidian tools'(pencil, ruler, and a pair of compasses).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also that a number system based 60 is easy to make calculations with. Imagine a dollar having 60 cents, makes it much easier to share this dollar with 2,3,4,5,6 people. Also note that you can count 60 on your fingers, when counting 12 digits on your one hand, and 5 fingers on the other hand (5x12=60). Doing this, you even do not need one thumb, so you can hold your merchandise while counting :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since your question is about accuracy, i need to add that while we think of an hour now as a fixed unit, before the invention of the clock, hours were 'variable' in the sense that a day is shorter in the winter, so while there was a division in 12 parts, the parts were not always equal. &lt;/p&gt;&#xA;" OwnerUserId="4266" LastActivityDate="2016-06-10T14:06:08.600" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="4894" PostTypeId="2" ParentId="4892" CreationDate="2016-06-10T14:21:17.610" Score="1" Body="&lt;p&gt;See: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Justin Smith,&lt;a href=&quot;https://books.google.it/books?id=IcqhWpzRJ9MC&amp;amp;pg=PA16&quot; rel=&quot;nofollow&quot;&gt;Divine Machines: Leibniz and the Sciences of Life&lt;/a&gt; (2011), page 16:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Looking at his career as a whole, we may say that Leibniz’s biological interests lie at the center of at least three core issues in his philosophy:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;ol&gt;&#xA;    &lt;li&gt;&lt;p&gt;The problem of the structure and motion of the physical world, both of living entities as normally understood as well as of apparently lifeless things.&lt;/p&gt;&lt;/li&gt;&#xA;    &lt;li&gt;&lt;p&gt;The problem of coming into being, or generation.&lt;/p&gt;&lt;/li&gt;&#xA;    &lt;li&gt;&lt;p&gt;Species.&lt;/p&gt;&lt;/li&gt;&#xA;    &lt;/ol&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, no &quot;practical&quot; contribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also Leibniz's contribution to paleontology, with his principal work on this subject, &lt;a href=&quot;https://books.google.it/books?id=_QFdQJznrQsC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Protogaea&lt;/a&gt;, unpublished in his lifetime.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-06-10T14:21:17.610" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4895" PostTypeId="2" ParentId="381" CreationDate="2016-06-10T14:24:53.803" Score="0" Body="&lt;p&gt;The reason a clock uses 12 hours is actually simple: use a forked stick in the sand to make a circle. then, naturally one would place one leg of the compass at some point on the circle, and draw a second circle.  then, one would naturally place the leg on one of the intersections just made, and draw a third one. 3 more times, and you have gone all around and drawn a pattern of 1 circle in the middle (the 7th) and 6 circles around.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means that, division in 6/12 is the easiest, and also most accurate way of dividing a circle in the past. You can easily demonstrate the same effect by using 7 coins of the same size, 6 coins fit exactly around the 7th in the middle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dividing a circle in 5/10 parts is really hard to do accurate with only the 'euclidian tools' (pencil, ruler, and a pair of compasses). Same with 4 parts, since there is no way of knowing (without drawing a lot more circles) if your 2 lines making the division are perpendicular. And division in 7 Parts is nearly impossible to do. So 6/12 is basically the only logical option.&lt;/p&gt;&#xA;" OwnerUserId="4266" LastActivityDate="2016-06-10T14:24:53.803" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4896" PostTypeId="1" AcceptedAnswerId="4897" CreationDate="2016-06-12T07:48:59.050" Score="0" ViewCount="321" Body="&lt;p&gt;We have heard about the name of the famous German company Siemens.My friend said that this is because they were the first to introduce conductance bridges and hence named after the unit of conductance Siemens.&#xA;But in the Wikipedia article I saw that Siemens was founded by Werner von Siemens and thus the name.&#xA;So,is my friend wrong?What is the real reason behind the name?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any help!!&lt;/p&gt;&#xA;" OwnerUserId="2955" LastActivityDate="2016-06-12T12:51:03.137" Title="Is the name of German company Siemens related to the unit of conductance?" Tags="|physics|units|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4897" PostTypeId="2" ParentId="4896" CreationDate="2016-06-12T10:51:08.537" Score="7" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Siemens_(unit)&quot; rel=&quot;noreferrer&quot;&gt;unit &quot;Siemens&quot;&lt;/a&gt; was only introduced in 1971 whereas the &lt;a href=&quot;https://en.wikipedia.org/wiki/Siemens&quot; rel=&quot;noreferrer&quot;&gt;company&lt;/a&gt; was founded the mid-19th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both get their name from the same person, &lt;a href=&quot;https://en.wikipedia.org/wiki/Werner_von_Siemens&quot; rel=&quot;noreferrer&quot;&gt;Werner von Siemens&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4272" LastEditorUserId="4272" LastEditDate="2016-06-12T12:51:03.137" LastActivityDate="2016-06-12T12:51:03.137" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4898" PostTypeId="2" ParentId="3659" CreationDate="2016-06-13T01:16:23.430" Score="2" Body="&lt;p&gt;AWK had associative arrays from day one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Associative arrays seem to have been introduced in SNOBOL4. [1]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] &lt;a href=&quot;https://en.wikipedia.org/wiki/Associative_array#Language_support&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Associative_array#Language_support&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="186" LastActivityDate="2016-06-13T01:16:23.430" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4899" PostTypeId="1" CreationDate="2016-06-13T06:47:52.957" Score="0" ViewCount="38" Body="&lt;p&gt;Recently somebody said in answer (at PSE) that magnetic force violates 3rd law of motion and that Poincaré solved that anomaly introducing the magnetic field that compensates missing force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you tell if this is true and tell me where I can find more details on the issue, and most of all why 3rd law is violated without the field? I had never heard about that. Thanks&lt;/p&gt;&#xA;" OwnerDisplayName="user646" LastActivityDate="2016-06-13T06:47:52.957" Title="Poincaré and magnetic field" Tags="|magnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4900" PostTypeId="1" AcceptedAnswerId="4901" CreationDate="2016-06-13T08:14:04.707" Score="12" ViewCount="725" Body="&lt;p&gt;Did Turing have any knowledge of Babbage's work? Is it known if he ever mentioned Babbage?&lt;/p&gt;&#xA;" OwnerUserId="4277" LastEditorUserId="4251" LastEditDate="2020-07-08T17:06:25.187" LastActivityDate="2020-07-08T17:06:25.187" Title="Did Turing know of Babbage's work?" Tags="|computer-science|computers|turing|computer-scientists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4901" PostTypeId="2" ParentId="4900" CreationDate="2016-06-13T09:14:53.883" Score="16" Body="&lt;p&gt;See:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alan Turing, &lt;a href=&quot;https://books.google.it/books?id=5G4Eya27gCgC&amp;amp;pg=PA10&quot;&gt;Collected works: Mathematical Logic&lt;/a&gt; (R.O. Gandy &amp;amp; C.E.M. Yates editors, 2001).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[page 10, regarding Turing's paper &lt;em&gt;On computable numbers&lt;/em&gt;... (1937)] It should be remarked that there is no evidence that Turing had read any of the scanty and sporadic literature concerned with the general theory of mechanical computation. In particular, one can be sure that if Turing had read either account by Babbage of the Analytic Engine (Chapter VIII of '&lt;em&gt;Pages from the life of a philosopher&lt;/em&gt;' [1864]) or the account of Menabrea [1842] translated by the Countess of Lovelace [1843] he would have mentioned Babbage's ideas. He might well have read the article on Calculating Machines in the 11th edition of the &lt;em&gt;Encyclopaedia Britannica&lt;/em&gt;. In later years he often consulted the copy of the Encyclopaedia which he inherited from his father - but that article contains only a short and rather dismissive reference to the Analytic Engine: 'a much more powerful machine... intended to perform any series of possible arithmetical operations'. This would hardly have suggested to Turing that Babbage had in fact conceived a universal machine. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 15, regarding Turing machines and electronic computers] According to Randell [1980], Turing and Newman and a group of mathematicians and engineers at Bletchley Park discussed in 1942-43, out of office hours, Turing's universal machine, Babbage's plans for the Analytic Engine, and the possibilities of artificial intelligence. Turing played some part in the design of the first machines built under Newman's direction. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-06-13T09:14:53.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4902" PostTypeId="1" AcceptedAnswerId="4903" CreationDate="2016-06-13T16:21:52.307" Score="15" ViewCount="685" Body="&lt;p&gt;I have heard that the book &quot;Men of Mathematics&quot; by E. Bell is a very entertaining text composed of several biographies of a number of influential mathematicians, and is in fact one of the most popular popular mathematics books to date. I have also heard that E. Bell received more than a little criticism for his writings, due to the large number of embellishments and historical inaccuracies, as well as the less than inclusive title and contents. Up until now, I have avoided &quot;Men of Mathematics&quot; for this reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am quite interested in the history of mathematics, though, and would like to read a book with a similar subject matter to &quot;Men of Mathematics&quot;, except one written less to entertain and more to educate, without such criticisms. I have already read around a bit on the subject, for example I have read Boyer's textbook, but haven't found a history book exclusively focused on studying the lives of influential figures, yet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand that mathematics is collaborative, so the impact of these figures is less than the average casual followers of the subject would think, but I am interested in learning about them nonetheless.&lt;/p&gt;&#xA;" OwnerUserId="4279" OwnerDisplayName="Nethesis" LastEditorUserId="10106" LastEditDate="2022-02-22T06:59:07.443" LastActivityDate="2022-02-22T06:59:07.443" Title="Historically accurate alternatives to &quot;Men of Mathematics&quot;?" Tags="|mathematics|reference-request|book-recommendation|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4903" PostTypeId="2" ParentId="4902" CreationDate="2016-06-13T16:26:06.807" Score="8" Body="&lt;p&gt;Try &lt;em&gt;Journey Through Genius&lt;/em&gt;, by William Dunham.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every chapter mixes historical account with a mathematical discussion of a landmark theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an account of contemporary mathematicians, try&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Mathematical Lives:&#xA;Protagonists of the Twentieth Century From Hilbert to Wiles&lt;/em&gt;, by Bartocci et al.&lt;/p&gt;&#xA;" OwnerUserId="186" OwnerDisplayName="lhf" LastEditorUserId="186" LastEditDate="2016-06-13T16:54:53.123" LastActivityDate="2016-06-13T16:54:53.123" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4904" PostTypeId="2" ParentId="4902" CreationDate="2016-06-13T17:14:01.603" Score="1" Body="&lt;p&gt;The following detailed article counts as an alternative to Bell as it paints a more accurate picture of the history of analysis and also analyzes Bell's fantasy about George (Berkeley) battling the dragon of inconsistent infinitesimals: &lt;a href=&quot;http://dx.doi.org/10.1007/s10670-012-9370-y&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-06-14T12:16:51.187" LastActivityDate="2016-06-14T12:16:51.187" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4905" PostTypeId="1" CreationDate="2016-06-14T15:04:54.153" Score="3" ViewCount="228" Body="&lt;p&gt;My question is concerned with two hardly mentioned yet very ingenious inventions of Leibniz. These two inventions represent technological inventiveness and visionary insight (about wind energy technology). My reference is &lt;a href=&quot;http://www.gwlb.de/Leibniz/Leibnizarchiv/english/Leibniz_Life/windmill.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;this website&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A yaw system for the windmill is an automatic control mechanism that enables the windmill to capture maximum amount of wind energy by continuously turning the sails of mills to the direction of the wind (so the interaction with the wind will be maximal). A braking mechanism is a device that regulates the speed of the shaft to enable the wind to make mechanical work on the sails (by preventing the shaft from going into maximal speed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is mainly about getting explanations. I didn't find any English website to explain those inventions of Leibniz, and I'll be glad to see one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: I found a reference to those instruments of Leibniz in &quot;The oxford handbook of Leibniz&quot;, page 538. The author of the handbook mentions an instrument called &quot;&lt;strong&gt;inventum memorabile&lt;/strong&gt;&quot;, &quot;for the timely and regular braking of the wind shaft to avoid excessive damaging strain on the machinery resulting from excessively rapid rotation of the sails&quot;. The author also refers to the article &quot;Leibniz' Inventum Memorabile: Die Konzeption einer Drehzahlregelung vom März 1686&quot; by Hans Peter Munzenmayer. According to the abstract of the article, Leibniz's mechanism for speed regulation is in fact a feedback control system in the modern sense. Unfortunately, the article is in German, so I'm unable to read it.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10106" LastEditDate="2020-06-14T04:16:37.107" LastActivityDate="2020-06-14T04:16:37.107" Title="How did Leibniz's prototypes of a yaw system and a braking mechanism (for windmills) work?" Tags="|engineering|leibniz|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4906" PostTypeId="1" CreationDate="2016-06-14T18:36:20.713" Score="1" ViewCount="104" Body="&lt;p&gt;Because of the following relation,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation*} &#xA;\inf(S) = -\sup(-S),&#xA;\end{equation*}&#xA;minimization and maximization is essentially the same thing. However, take any optimization routine in R for example (optim, nlm, etc.), and the default setting is to minimize.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even for something named &lt;em&gt;maximum&lt;/em&gt; likelihood estimation, it is normal to &lt;em&gt;minimize&lt;/em&gt; the (negative) log-likelihood i.e. implementation is done as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{equation*}&#xA;\widehat \theta_n = \operatorname*{arg min}_{\theta \in \Theta} - l(\theta;x_1,\ldots, x_n)&#xA;\end{equation*}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and I have been taught to do this even when I was searching analytically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, what is the reasoning behind this more frequent use of minimization compared to maximization? I have a slight suspicion that it has something to do with how convex and concave functions historically have been analyzed but that may just come from thin air.&lt;/p&gt;&#xA;" OwnerUserId="4288" LastEditorUserId="4251" LastEditDate="2020-05-12T08:15:59.660" LastActivityDate="2020-05-12T08:15:59.660" Title="Why do we often minimize in optimization?" Tags="|mathematics|statistics|optimization|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4907" PostTypeId="1" AcceptedAnswerId="4908" CreationDate="2016-06-14T21:46:28.860" Score="12" ViewCount="13340" Body="&lt;p&gt;What was Euler's first proof of his famous formula?&lt;/p&gt;&#xA;&lt;p&gt;In Euler's book on complex functions he used the following proof. But was this his first proof?&lt;/p&gt;&#xA;&lt;p&gt;Euler starts with writing down De Moivre's Formula (can be proven by simple induction using some basic trig identities).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\cos(nx)+i\sin(nx)=\left( \cos(x)+i\sin(x)\right)^n$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;He says that &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is very large (&lt;span class=&quot;math-container&quot;&gt;$n \to \infty$&lt;/span&gt;) and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is very small (&lt;span class=&quot;math-container&quot;&gt;$x\to 0$&lt;/span&gt;). The product of both will be a finite number called &lt;span class=&quot;math-container&quot;&gt;$\omega =nx$&lt;/span&gt;. Then he applies this as substitution for De Moivres Formula:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cos(\omega)+i\sin(\omega)=\left( \cos(\frac{\omega}{n})+i\sin(\frac{\omega}{n})\right)^n$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Euler now applies the limit &lt;span class=&quot;math-container&quot;&gt;$n\to \infty$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cos(\omega)+i\sin(\omega)=\lim_{n\to \infty}\left( \cos(\frac{\omega}{n})+i\sin(\frac{\omega}{n})\right)^n$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;using small angle aproximations &lt;span class=&quot;math-container&quot;&gt;$\cos(x)\approx 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(x)\approx x$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cos(\omega)+i\sin(\omega)=\lim_{n\to \infty}\left( 1+\frac{i\omega}{n}\right)^n=e^{i\omega}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the last line he applied the limit representation &lt;span class=&quot;math-container&quot;&gt;$e^x=\lim\limits_{n\to \infty}(1+\frac{x}{n})^n$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="4251" LastEditDate="2022-11-11T19:36:26.710" LastActivityDate="2022-11-11T19:36:26.710" Title="Euler's first proof of $e^{ix}=\cos(x)+i\sin(x)$" Tags="|mathematics|complex-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4908" PostTypeId="2" ParentId="4907" CreationDate="2016-06-14T23:54:07.710" Score="12" Body="&lt;p&gt;According to Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt;, the identity first appeared in Euler's &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Introductio_in_analysin_infinitorum&quot; rel=&quot;nofollow&quot;&gt;Introductio in analysin infinitorum&lt;/a&gt;&lt;/em&gt; of 1748.  Euler started from the infinite series for $e^x$, $\sin x$, and $\cos x$, from which the identity quickly follows.  Thus from:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$&#xA;$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$$&#xA;$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;substituting $ix$ into the series for $e^x$ and rearranging terms quickly leads to the result. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boyer also notes that Roger Cotes had derived this identity, given by him in a &lt;em&gt;Philosophical Transactions&lt;/em&gt; article of 1714 in the equivalent form $\ln(\cos \theta + i \sin \theta) = i\theta$.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="205" LastEditDate="2016-07-11T14:27:06.063" LastActivityDate="2016-07-11T14:27:06.063" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4909" PostTypeId="2" ParentId="4902" CreationDate="2016-06-15T01:51:26.467" Score="5" Body="&lt;p&gt;Ioan James' &quot;&lt;em&gt;Remarkable Mathematicians: From Euler to von Neumann&lt;/em&gt;&quot; complements Bartocci et al's book (mentioned in another answer) by providing fairly good biographical sketches of mathematicians (60 different profiles) over a time period from roughly 1700 - 1950.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ronald Calinger's &quot;&lt;em&gt;Classics of Mathematics&lt;/em&gt;&quot; contains short (1-2 page) profiles of many mathematicians going back to Proclus and on into the 20th century.  In my opinion, the quality of the coverage tapers off noticeably as it moves beyond about 1850 or so, but the essays covering the development of mathematics prior to this time are quite well written.  The excerpts of original published works (translated to English) are the distinguishing feature of this book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;MacTutor History of Mathematics archive&lt;/em&gt;&lt;/a&gt;, a comprehensive, well-written, and reliable on-line resource, can be used as a supplement to the biographical sketches given in these books.  &lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2016-06-15T01:51:26.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4911" PostTypeId="1" CreationDate="2016-06-16T02:41:42.210" Score="1" ViewCount="98" Body="&lt;p&gt;Einstein said that incorporating the cosmological constant to his equations of general relativity was the greatest blunder of his life. But as it turns out, cosmological constant does indeed have application in understanding the evolution of the universe. So Einstein's greatest blunder indeed became a new way to reveal the secrets of the universe. So it was a mistake, considered by Einstein that revolutionized physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what are some other great blunders from great scientific minds that changed the way of science as we know it? &lt;/p&gt;&#xA;" OwnerUserId="2891" LastActivityDate="2016-06-16T02:41:42.210" Title="What are some greatest blunders from the great minds from science?" Tags="|physics|discoveries|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4913" PostTypeId="1" CreationDate="2016-06-16T10:05:32.707" Score="8" ViewCount="334" Body="&lt;p&gt;I am interested in the way scientific and mathematical subject areas developed (and are still developing). One of the great visual tools that can help us gain insight in how these areas developed is by creating a &quot;genealogy&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the case of mathematics: I already asked about it &lt;a href=&quot;https://www.quora.com/What-are-the-most-complete-genealogies-of-mathematical-subject-areas&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (on Quora). Although there is a genealogy of &lt;em&gt;mathematicians&lt;/em&gt;, (take a look over &lt;a href=&quot;https://genealogy.math.ndsu.nodak.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), to my knowledge there is no such thing for mathematical subject areas. There are a few so-called &quot;mind-maps&quot; that &lt;a href=&quot;https://math.stackexchange.com/questions/124709/mind-maps-of-advanced-mathematics-and-various-branches-thereof&quot;&gt;show&lt;/a&gt; how some (certainly not all) mathematical areas are connected, but they don't take into account the historical developments that are relevant (another example is the Mathematical Atlas made by Dave Rusin, which unfortunately currently can't be viewed online). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to come back to the question: do you know of any (annotated) genealogies of scientific and/or mathematical subject areas? I think those can be great to help one gain a broad perspective on the history of science and mathematics, however crude it may be. &lt;/p&gt;&#xA;" OwnerUserId="4296" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2017-12-20T17:36:45.360" Title="What are some of the most complete genealogies of scientific and mathematical subject areas?" Tags="|mathematics|physics|reference-request|resource-recommendation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4914" PostTypeId="1" CreationDate="2016-06-16T16:56:05.473" Score="1" ViewCount="152" Body="&lt;p&gt;In some texts about the simple pendulum we use to see references about some &quot;experiments&quot; Galileo Galilei did realize and whereby he found some important results, including that the period of the pendulum is proportional to the square root of the lenght of the (stem of the) pendulum. I would like to know if there is any book ou document wrote by Galileo (on the Dialogues or the Discourses) where I can find the explanation about these experiments realized by Galileo.&lt;/p&gt;&#xA;" OwnerUserId="4298" LastActivityDate="2016-06-16T16:56:05.473" Title="Galileo's pendulum and any references" Tags="|experimental-physics|philosophy-of-science|scientific-method|depiction-of-scientists|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4916" PostTypeId="2" ParentId="4913" CreationDate="2016-06-17T01:58:33.393" Score="4" Body="&lt;h1&gt;History of Mathematics Classification&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;To see how the classification of mathematics has developed historically, compare &lt;a href=&quot;https://philosophy.stackexchange.com/a/34544/2014&quot;&gt;the ancient classification&lt;/a&gt; of mathematics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Arithmetic (discrete math)&lt;/li&gt;&#xA;&lt;li&gt;Geometry (continuous math)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;to that of Descartes' era:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Analytic Geometry&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Arithmetic (discrete math)&lt;/li&gt;&#xA;&lt;li&gt;Geometry (continuous math)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;to the 1868 classification in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=I1fj_B60fyoC&amp;amp;pg=PA29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jahrbuch über die Fortschritte der Mathematik&lt;/a&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;History and Philosophy&lt;/li&gt;&#xA;&lt;li&gt;Algebra&lt;/li&gt;&#xA;&lt;li&gt;Number Theory&lt;/li&gt;&#xA;&lt;li&gt;Probability&lt;/li&gt;&#xA;&lt;li&gt;Series&lt;/li&gt;&#xA;&lt;li&gt;Differential and Integral Calculus&lt;/li&gt;&#xA;&lt;li&gt;Theory of Functions&lt;/li&gt;&#xA;&lt;li&gt;Analytic Geometry&lt;/li&gt;&#xA;&lt;li&gt;Synthetic Geometry&lt;/li&gt;&#xA;&lt;li&gt;Mechanics&lt;/li&gt;&#xA;&lt;li&gt;Mathematical Physics&lt;/li&gt;&#xA;&lt;li&gt;Geodesy and Astronomy&lt;/li&gt;&#xA;&lt;li&gt;(+ 38 subcategories)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;to the modern &lt;a href=&quot;https://zbmath.org/classification/&quot; rel=&quot;nofollow noreferrer&quot;&gt;zbMATH&lt;/a&gt; or &lt;a href=&quot;https://mathscinet.ams.org/mathscinet/msc/pdfs/classifications2010.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;MSC2010&lt;/a&gt; classifications of mathematics.&#xA;&lt;hr&gt;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;History of Physics Classification&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;For physics, cf. my question &quot;&lt;a href=&quot;https://hsm.stackexchange.com/q/3773/232&quot;&gt;Classification of experimentally-determined physics laws?&lt;/a&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2017-12-20T17:36:45.360" LastActivityDate="2017-12-20T17:36:45.360" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4917" PostTypeId="1" AcceptedAnswerId="4922" CreationDate="2016-06-17T13:39:50.030" Score="10" ViewCount="1498" Body="&lt;p&gt;Which mathematician did introduce the Method of Characteristics for solving linear Partial differential equations? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I some papers I saw that Saint Venant, Lagrange, Cauchy or Riemann were attributed with this method. But who was the first mathematician who introduced the method, which is taught in universities?&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="4251" LastEditDate="2020-06-21T19:49:38.583" LastActivityDate="2024-01-11T09:54:45.917" Title="Who did first use the Method of Characteristics?" Tags="|mathematics|mathematicians|calculus|differential-equations|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4919" PostTypeId="2" ParentId="4917" CreationDate="2016-06-18T09:20:43.870" Score="0" Body="&lt;p&gt;It is worth noting that Lagrange was born in 1736 (died in 1813), Cauchy in 1789, St. Venant in 1797, and Riemann in 1826.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So between those four, it would &quot;have&quot; to be Lagrange, who published his work on this subject in 1786, before the others were born.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also worth noting that the method of characteristics is tied to the Euler equations, and that Euler's lifespan was 1707-1783 (he died before Lagrange published his relevant work in 1786).&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="31" LastEditDate="2016-06-18T09:29:27.143" LastActivityDate="2016-06-18T09:29:27.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4920" PostTypeId="1" CreationDate="2016-06-18T12:41:32.053" Score="7" ViewCount="403" Body="&lt;p&gt;From &lt;a href=&quot;https://espresso.economist.com/a600bd172fcabd688500dac58ebda3a0&quot;&gt;Darkest matter: gravitational waves 2016-06-19 | Espresso&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But black-hole pairs orbiting one another emit something else: gravitational waves. These ripples spread outward like sound waves, ever so slightly stretching and squashing spacetime itself as they pass. The first ever to be caught by Earthly detectors, in September, proved Albert Einstein, who predicted their existence a century earlier, was right.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4302" LastEditorUserId="26" LastEditDate="2016-06-21T18:46:30.247" LastActivityDate="2016-06-21T18:46:30.247" Title="How did Albert Einstein predict the existence of gravitational waves?" Tags="|theoretical-physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4921" PostTypeId="2" ParentId="4920" CreationDate="2016-06-18T17:29:34.637" Score="1" Body="&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Gravitational_wave&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Predicted in 1916[2][3] by Albert Einstein on the basis of his theory of general relativity,[4][5]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Assuming you understand general relativity, why not consult the papers given there?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) Einstein, A (June 1916). &amp;quot;Näherungsweise Integration der Feldgleichungen der Gravitation&amp;quot;. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 688–696.&lt;/p&gt;&#xA;&lt;p&gt;(3) Einstein, A (1918). &amp;quot;Über Gravitationswellen&amp;quot;. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 154–167.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-06-18T17:29:34.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4922" PostTypeId="2" ParentId="4917" CreationDate="2016-06-18T20:51:47.730" Score="12" Body="&lt;p&gt;[EDIT: check the (*) for a potential earlier reference in Lagrange] The first mathematician (who used the methods of characteristics for differential equations) seems to be &lt;a href=&quot;https://fr.wikipedia.org/wiki/Paul_Charpit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Charpit de Villecourt&lt;/a&gt;  (?-1784). According to I. Grattan-Guinness et S. Engelsman, &lt;a href=&quot;http://dx.doi.org/10.1016/0315-0860(82)90140-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;The manuscripts of Paul Charpit&lt;/a&gt;, Historia Mathematica, vol. 9, 1982, p. 65-75:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He presented his paper on partial differential equations to the&#xA;Académie on June 30, 1784, but he died in Paris on December 28 of that&#xA;year.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The memoir was lost, then found again. Lagrange learnt about it in 1793 only (* see below for a comment). There is a series of papers on that topic  by N. Saltykow, such as &lt;a href=&quot;https://eudml.org/doc/192955&quot; rel=&quot;nofollow noreferrer&quot;&gt;Méthodes classiques d'intégration des équations aux dérivées partielles du premier ordre à une fonction inconnue&lt;/a&gt;, 1931. Page 7:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Charpit a eu, d'abord, la chance de formuler, le premier, les&#xA;équations différentielles ordinaires des caractéristiques, que l'on&#xA;attribue fréquemment à Lagrange&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;with the poor translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Charpit was lucky enough to be the first to express the ordinary differential&#xA;equations of characteristics, which are often attributed to Lagrange.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Full proof (in a modern sense) was apparently given by Jacobi.&lt;/p&gt;&#xA;&lt;p&gt;(*) &lt;a href=&quot;https://hsm.stackexchange.com/questions/4917/who-did-first-use-the-method-of-characteristics/4922?noredirect=1#comment24551_4922&quot;&gt;From a comment by Gerald&lt;/a&gt;, on page 198 of Mathematik für Physiker (Band 2: Gewöhnliche und partielle Differentialgleichungen, mathematische Grundlagen der Quantenmechanik), 2014 by Helmut Fischer and Helmut Kaul, it is said that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Charakteristikenmethode geht auf Lagrange zuruck. Nachdem er 1772&#xA;die nichtlineare DG &lt;span class=&quot;math-container&quot;&gt;$\frac{∂u}{∂y}  = f(x, y, u, \frac{∂u}{∂x})$&lt;/span&gt; auf&#xA;eine quasilineare DG zurückgeführt und 1774 eine Theorie für die&#xA;allgemeine Lösung aufgestellt hatte, gab er 1779&#xA;Differentialgleichungen für die Charakteristiken quasilinearer&#xA;Probleme an. Eine geometrische Begründung der Charakteristikenmethode&#xA;fand 1784 Gaspard Monge. Pfaff 1815 und Cauchy 1819 erweiterten diese&#xA;für &lt;span class=&quot;math-container&quot;&gt;$n &amp;gt; 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="4130" LastEditDate="2024-01-11T09:54:45.917" LastActivityDate="2024-01-11T09:54:45.917" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="4923" PostTypeId="2" ParentId="4920" CreationDate="2016-06-19T00:04:23.187" Score="6" Body="&lt;p&gt;General relativity is a classical field theory. In such theories, disturbances in the fields propagate as waves. The basic equation of GR is the Einstein field equation, which is a wave equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;GR replaced Newtonian gravity, which was a theory in which gravity acted instantaneously at a distance. Even as early as 1905, long before GR, it would have been clear that the Newtonian picture was inconsistent with relativity, which doesn't allow cause and effect to propagate at speeds greater than $c$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a funny twist in the story, which is that in 1936, Einstein and Rosen published a paper claiming that gravitational waves did not actually exist. The paper was seen to be incorrect by a referee, but Einstein an Rosen did not initially accept the criticism.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2016-06-19T00:04:23.187" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4925" PostTypeId="1" CreationDate="2016-06-19T10:26:57.547" Score="4" ViewCount="148" Body="&lt;p&gt;As is obvious from my previous questions, i'm very curius to learn as much as i can about Leibniz's inventions, simply because i think his inventions contain that kind of technological ingenuity that enabled later technical developements to emerge. This queustion asks what were the original ideas that Leibniz introduced in the design of the pumps of this fountain - i read that in his time this fountain was the highest in europe, with it's water reaching a height of 40 meters. &lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2017-06-04T10:44:49.663" Title="What innovations did Leibniz introduce in his design of the fountain of the Herrenhausen Gardens?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4926" PostTypeId="2" ParentId="54" CreationDate="2016-06-19T11:55:48.090" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;On the other hand, the original definition of the term &quot;science&quot;, as per its etymology, more or less just means &quot;knowledge&quot;. The term began to mean &quot;empirical science&quot; (in the sense of Bacon) later, perhaps in the 17th and 18th centuries, but it seems to me that Mathematics was still implicitly included by use of the phrase &quot;the sciences&quot; until significantly later. This was a source of some debate on our Area 51 proposal, and as Conifold suggested, I'm moving it here so that we can get an authoritative answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;English was not yet the dominant language of science in the 17th and 18th centuries. The changed meaning of the term &quot;science&quot; only became important after English had advanced to be the dominant language of science during the 20th century. This coincides well with the time when the term &quot;the sciences&quot; stopped to implicitly include mathematics.&lt;/p&gt;&#xA;" OwnerUserId="3935" LastActivityDate="2016-06-19T11:55:48.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4928" PostTypeId="2" ParentId="54" CreationDate="2016-06-19T17:50:38.930" Score="3" Body="&lt;p&gt;So we are using Wikipedia?  If we consult &lt;a href=&quot;https://en.wikipedia.org/wiki/Branches_of_science&quot; rel=&quot;nofollow&quot;&gt;Branches of Science&lt;/a&gt; we will find Mathematics listed in there.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-06-19T17:50:38.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4929" PostTypeId="1" CreationDate="2016-06-20T11:46:45.583" Score="9" ViewCount="2051" Body="&lt;p&gt;I'm a college student, just beginning to study Elementary Set Theory. In studying about the definition of injective and surjective function, out of curiosity, it came to my mind a question about the definition these functions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) How did they came up with such definition? How did it came about &amp;amp; why must it be defined that way? What is so special about injective &amp;amp; surjective function that makes them has to be defined in such a way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make clear  the context of my question, here are the conditions of this question:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;In short&quot; injective functions are defined as: for every element in the codomain, there is at &quot;most&quot; one element that maps to it from the domain. For Surjective functions: for every element in the codomain, there is at &quot;least&quot; one element that maps to it from the domain.&lt;/li&gt;&#xA;&lt;li&gt;As you can see, i'm not seeking about what exactly the definition of an Injective or Surjective function is (a lot of sites provide that information just from googling), but rather about why is it defined that way? What makes it so special that it has to be given it's own terminology (injective &amp;amp; Surjective functions)?&lt;/li&gt;&#xA;&lt;li&gt;Lastly, i'm just a human so if something is not clear about this post, just let me know in the comment &amp;amp; i'll try to edit the post to make it clearer. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I don't know how the real story goes and this is just a rough speculation in trying to guess how it came about (it is possible to be something along this line but don't see it as the real truth though):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The person who first coined these terms (surjective &amp;amp; injective functions) was, at first, trying to study about functions (in terms of set theory) &amp;amp; what conditions made them invertible. He observed that some functions are easily invertible (&quot;bijective function&quot;) while some are not possible to have any kind of inverse. As he continue to study, he discovers that some functions can have inverses if certain conditions are met, as if they were &quot;semi&quot; invertible type of functions. He discovers that these conditions can be distuingish into two types: right invertible or left invertible. Later on he can distuingish that, only those function who maps with a specific type of pattern can have a left inverse (which he called the pattern &quot;injective&quot;). And a similar thing happened as he discovered the pattern for surjective functions. So in short my opinion is: is not like he wanted to define injective &amp;amp; surjective function like so, but it  just happened that he &quot;discovered it&quot; in &quot;that form&quot;. Hence, the definition of injective &amp;amp; Surjective function. But, again, this is just me trying to be Mr. Know it all. I hope I'm not being too rude. If anyone do know the real story, do share.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Regards,&lt;/p&gt;&#xA;" OwnerUserId="4168" LastEditorUserId="4168" LastEditDate="2016-06-21T21:00:38.503" LastActivityDate="2020-06-05T21:52:00.723" Title="History of the definition of Injective &amp; Surjective Function" Tags="|mathematics|terminology|set-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4931" PostTypeId="2" ParentId="4929" CreationDate="2016-06-20T19:10:06.210" Score="8" Body="&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/i.html&quot;&gt;this page on &quot;earliest know uses of some mathematical words&quot;&lt;/a&gt;, the terms injective, surjective, and bijective were first introduced in Bourbaki's &lt;a href=&quot;http://tomlr.free.fr/Math%E9matiques/Bourbaki/Theorie%20Des%20Ensembles.pdf&quot;&gt;Théorie des ensembles&lt;/a&gt;, of 1954, page 80. The authors' motivations were to standardise terminology, stating : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Standard terms are badly needed for “one-to-one,” “onto” and “one-to-one onto”; will Bourbaki’s “injection,” “surjection” and “bijection” prove acceptable?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Note how the Bourbaki group refer to themselves in the third person.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The title of the Bourbaki text supports your guess that these terms were first introduced in connection with the study of set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same site resource attributes the first usage of one-to-one correspondence (i.e., injection) to the Danish mathematician Zeuthen, in his &lt;a href=&quot;http://jeff560.tripod.com/o.html&quot;&gt;&lt;em&gt;Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un&lt;/em&gt;,&lt;/a&gt; of 1870.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first use of &quot;Onto&quot;, i.e., surjection, is credited to C. C. MacDuffee in his &lt;em&gt;Introd. Abstract Algebra&lt;/em&gt; of 1940, shortly before Bourbaki's retreatment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to what makes these functions worthy of special attention, bijective functions are used to define equinumerousity in set theory.  (A &quot;two way&quot; injection argument can also be used to identify equinumerous sets.) Bijective functions which are structure preserving are a used to identify isomorphic structures in algebra. And a host of other applications.  In addition, injections, surjections, and bijections have interesting properties and are worth of study in their own right.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-06-20T19:15:38.330" LastActivityDate="2016-06-20T19:15:38.330" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4932" PostTypeId="1" CreationDate="2016-06-21T07:07:36.020" Score="5" ViewCount="1636" Body="&lt;p&gt;I'm not simply referring to the notion that Einstein treated the discrete emission and transference of energy (and matter) as &quot;real&quot; physical phenomena, but rather his major continuous role in the development of QM all the way up to Heisenberg Schrodinger's formulation of it. In short, this isn't simply a query of &quot;how Einstein's derivation of light quanta is different than Planck's&quot; but rather why - until recently - people only associate Einstein with the explanation of the photoelectric effect and the Bohr debates when he did SO much more in the field of quantum theory. Even the Nobel committee's acknowledgement of his 1905 paper is a bit of a misnomer: he doesn't merely &quot;explain the photoelectric effect&quot; (although that is one its consequences) but he's the first to actually quantize the radiation field. As physicists, this isn't a small quibble but a substantive difference. It suggests that the committee hadn't fully understand the development of QM from genesis to its 1926 formalism - and with the benefit of hindsight, it's quite clear. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a small exmaple of what I mean. In 1909 Einstein was the first to show that statistical fluctuations in thermal radiation fields display both particlelike and wavelike behaviour; his was the first demonstration of what would later become the principle of complementarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later in 1916, after completing General Relativity, he turned to the interplay of matter and radiation to create a quantum theory of radiation, he once again based his arguments on statistics and fluctuations. Bohr introduced a crucial new concept called stationary states in his 1913 paper on hydrogen but major features of Bohr's model could be interpreted as absolute nonsense because, according to electromagnetic theory, the electron would radiate intensely, emitting a broad spectrum as it crashed into the nucleus. Here we see contradictions in classical laws, and yet major properties of Bohr's hydrogen model rested on those laws. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein, always the original thinker, didn't take as his starting point the well-known field for thermal radiation given by the Planck radiation law. Instead, he assumed that the atoms are in thermal equilibrium and then deduced the properties of the radiation field required to maintain the equilibrium. Guess what? The field turned out to be given precisely by the Planck radiation law. He manages to create quantum effects (stimulated and spontaneous emission) from most classical principles. He uses Wien's displacement law, the canonical Boltzmann distribution, Poynting's theorem, and microscopic reversibility - all classical. The sole quantum idea was the concept of stationary states. And yet from these elements, he's the first to create a complete description of the basic radiation processes and a full description of the general properties of the photon. In his 1917 paper, he creates novel and elegant derivations of Planck's radiation law as well as a proof of Bohr's frequency rule. In it, among many other things, he answers the question of how a gas of atoms maintain the populations of its stationary states in equilibrium with a radiation field. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*The aforementioned novel concept of spontaneous emission, which embodies the FUNDAMENTAL interaction of matter with the vacuum, is a brilliant, Nobel-Prize worthy achievement. Why? Spontaneous emission sets the scale for ALL radiative interactions. The rates of absorption and stimulated emission, for instance, are proportional to the rate for spontaneous emission. Spontaneous emission can be viewed as the ultimate irreversible process and the fundamental source of noise in all of nature. With the development of cavity quantum electrodynamics - the study of atomic systems in close-to-ideal cavities - in the 1980s, the phystical situation was profoundly altered. In such cavities, spontaneous emission evolves into spontaneous-cavity oscillations. Although the dynamical behavior is totally altered, the atom-vacuum interaction that causes spontaneous emission sets the time scale for that evolution. It is first in Einstein's 1917 paper that the photon is demonstrated to possess all the properties of a fundamental excitation, and therefore it is quite clear that his radiation paper played a seminal role in the eventual creation of quantum electrodynamics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Apropos the second brilliant creation of his 1917 paper, stimulated emission of radiation, we see the first genesis of the laser. Stimulated emission underlies the basic mechanism of the laser and, by extension, laser cooling; his analysis of momentum transfer in a thermal radiation field can be immediately applied to atomic motion in a laser field. If the spectral width of a thermal field is replaced by the natural linewidth of the atom, Einstein's viscous damping force would give rise to the phenomenon known as optical molasses. This fundamental process of laser cooling was rediscovered by the atomic community in the 80's. Of course, you need Quantum Mechanics for a full realization of all the mechanisms of radiation, but papers like this are seminal contributions to what would eventually become QM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's theory of radiaton provided a complete characterization of the particlelike properties of the light quantum and, in retrospect, he was within an arms grasp of working out the statistical mechanics of these particles. Given that his 1905 proposal for the energy quantization of radiation was based on the analogy between entropies of thermal radiation and a system of particles, it is surprising that Einstein didn't extend his method of reasoning to derive the Planck law by treating photons as indistinguishable particles. He was VERY close, and it is quite apparent that Bose himself did not realize he had done anything novel. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fast forward to 1924 and Einstein, not Bose, applied the reasoning in Bose's treatment of photons as indistinguishable particle to a gas of indistuishable atoms thereby creating Bose-Einstein statistics. Subsequently, Einstein theorized Bose-Einstein condensation, a work for which 6 Nobel Prizes have been given. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just finished reading A. Douglas Stone's book (Head of Applied Physics at Yale Univ.) - a great theorist in his own right - titled &quot;Einstein and the Quantum: the Quest of the Valiant Swabian.&quot; It is a thrilling historical odyssey through the development of quantum mechanics, the best I've read since T.S. Kuhn's brilliant &quot;Black Body Problem and the Quantum Discontinuity.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stone convincingly argues that Einstein, not Planck, should be regarded as the true father of quantum theory. In the book Stone argues that: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein is well known for his rejection of quantum mechanics in the form it emerged from the work of Heisenberg, Born and Schrodinger in 1926. Much less appreciated are the many seminal contributions he made to quantum theory prior to his final scientific verdict, that the theory was at best incomplete. Einstein’s many conceptual breakthroughs need to be placed in historical context.&#xA;Einstein, much more than Planck, introduced the concept of quantization of energy in atomic mechanics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein proposed the photon, the first force-carrying particle discovered for a fundamental interaction, and put forward the notion of wave-particle duality, based on sound statistical arguments 14 years before De Broglie’s work. In fact, after reading the literature, De Broglie's contributions are a bit overrated in my estimation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I've pointed out above, Einstein was the first to recognize the intrinsic randomness in atomic processes, and introduced the notion of transition probabilities, embodied in the A and B coefficients for atomic emission and absorption.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein also preceded Born in suggesting the interpretation of wave fields as probability densities for particles, photons, in the case of the electromagnetic field. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I've already stated above, Einstein, stimulated by Bose, introduced the notion of indistinguishable particles in the quantum sense and derived the condensed phase of bosons, which is one of the fundamental states of matter at low temperatures. His work on quantum statistics in turn directly stimulated Schrodinger towards his discovery of the wave equation of quantum mechanics - a fact that Schrodinger always acknowledged. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein, in the EPR Paradox paper, and intending of course to poke holes in the modern iteration of QM (Schrodinger-Heisenberg-Born-Dirac), was the first to imply the phenomenon of quantum entanglement. Yes, Schrodinger fleshed it out after the fact, but it was the EPR paper which really highlighted that entanglement as the truly bizarre property of QM. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stone cogently argues that it was only due to his rejection of the final theory that he is not generally recognized as the most central figure in this historic achievement of human civilization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can check out the interview on youtube here:&#xA; &lt;a href=&quot;https://www.youtube.com/watch?v=anP54GxKgEQ&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=anP54GxKgEQ&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the Einstein paper Stone stumbled upon that precipitated his investigations: &#xA;lptms.u-psud.fr/nicolas_pavloff/files/2010/03/Stone-phys_today1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the book's summary (from his interview on NPR with Ira Flatow): &lt;a href=&quot;http://www.npr.org/2013/11/01/242356997/einsteins-real-breakthrough-quantum-theory&quot; rel=&quot;noreferrer&quot;&gt;http://www.npr.org/2013/11/01/242356997/einsteins-real-breakthrough-quantum-theory&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Check this out, it's a great book and it reveals with historical veracity how integral Einstein was in the development of QM. It wasn't just the photoelectric effect, he clearly did much more in QM theory than most people know. His 1906 work on the Specific Heat of Solids is also quite impressive. &lt;/p&gt;&#xA;" OwnerUserId="4315" LastEditorUserId="4315" LastEditDate="2016-06-21T18:53:46.140" LastActivityDate="2016-06-21T18:53:46.140" Title="Why Doesn't Einstein Get More Credit for Being the Father of Quantum Mechanics?" Tags="|mathematics|physics|theoretical-physics|quantum-mechanics|einstein|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4933" PostTypeId="2" ParentId="686" CreationDate="2016-06-21T07:21:20.977" Score="0" Body="&lt;p&gt;Grossmann himself always stated his role quite clearly, &quot;the theory is Einstein's.&quot; And it was Einstein, and Einstein alone, that came up with generally covariant field equations for General Relativity (to beat Hilbert of all people). There are some equations from the failed Entwurf theory that Grossmann came up with, but virtually all of GR's structure as a conceptual beauty (geodesics and so forth) was Einstein. (And the eponymously named &quot;Einstein Equations&quot; and &quot;Einstein metric&quot; tells you who conceived of them). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Martin Klein and Abraham Pais, the latter a rigorous theoretical physicist himself, downplayed Grossmann's role in the construction of GR, essentially stating that Grossman helped ensure the math was correct but did not contribute to the &quot;most abstract elements of the theory&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope that helps. &lt;/p&gt;&#xA;" OwnerUserId="4315" LastActivityDate="2016-06-21T07:21:20.977" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4934" PostTypeId="1" AcceptedAnswerId="4943" CreationDate="2016-06-21T15:22:56.957" Score="12" ViewCount="2260" Body="&lt;p&gt;Who was the first to prove the mean value theorem, i.e., the existence of an intermediate point where the slope equals the average slope over the interval?&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-05-01T12:10:01.193" Title="Who was the first to prove the mean value theorem?" Tags="|real-analysis|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="4935" PostTypeId="2" ParentId="2257" CreationDate="2016-06-21T17:03:17.453" Score="4" Body="&lt;p&gt;You should really read Douglas Stone's &quot;Einstein and the Quantum: The Quest of the Valiant Swabian.&quot; It's a brilliant work insofar as Prof. Stone shows precisely how Planck went about deriving the eponymously named Planck's law and why the quantum revolution did not beging with Planck but with Einstein. In the book Prof. Stone (Head of Applied Physics at Yale), having read T.S. Kuhn's work on the Black Body Problem and Quantum Discontinuity, as well as all of Planck's original papers and correspondence on the issue, in addition to all of Einstein's original papers and correspondence between 1879 and 1929, essentially argues that Planck's law arose as a mathematical fudgefactor. On the one end you had Rayleigh's law in the lower limit, and on the other Wien's law in  the  higher limit; Planck, having asserted to the Prussian Academy of Science that he had derived a formula directly from the second law of thermodynamics found himself in a dire quandary. His formula did not fit Kurlbaum's data and Planck, a very capable mathematician, simply guessed at an interpolation formula that fit Kurlbaum's new data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One Problem, however, he couldn't explain it and buried it for 5 years - in fact, no scientist in Europe brought it up again for 5 years - until Einstein launched quantum theory in earnest in 1905 and AGAIN in 1906 with one of the most underrated papers (as Walther Nernst later acknowledged) in the history of Quantum Mechanics titled &quot;On the Specific Heat of Solids.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The irony, of course, is that Bohr, his later sparring partner in the interpretation of QM, didn't actually believe in quantum theory for years, maintaining that &quot;the wave model of electromagnetism is tried and tested and any theory to the contrary cannot possibly fit experiment&quot; (or something to that effect, I'l' have to dig up the quote but Bohr was definitely not an instant believer).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After completing General Relativity, in 1917 Einstein rederives Planck's Law as well as Bohr's Frequency Rule in a completely novel way (crystallized in the A &amp;amp; B coefficients) and in so doing conjectures inherent randomness in quantum theory, a conceptual postulate that has remained with the theory ever since. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Watch a lecture Stone gives on the topic. Quite enlightening: &lt;a href=&quot;http://www.grassrootstv.org/view?showID=12681&quot; rel=&quot;nofollow&quot;&gt;http://www.grassrootstv.org/view?showID=12681&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One last word on the topic. Contrary to popular opinion, Einstein's foray into revolutionizing physics didn't begin in 1905 but rather between 1902 and 1903. This is where you see the genesis of his unconventional genius. He wrote a brilliant trilogy of papers in those years deriving the foundations of statistical mechanics from first principles (a work that would make J.W. Gibbs a legend - and Gibbs work, Einstein noted, was superior but also, in a way, less daring). Einstein didn't know of Gibbs work at the time as it hadn't been properly translated into German. But even so you can see the rudiments of Einstein's thinking on light quanta and how he, and not Planck, made the leap in 1905 that light, and indeed all energy and matter, is quantized. His complete mastery of statistical mechanics and thermodynamics comes to the fore again and again throughout his development on quantum theory.&#xA;It's interesting: &lt;a href=&quot;http://faculty.poly.edu/~jbain/heat/readings/98Navarro.pdf&quot; rel=&quot;nofollow&quot;&gt;http://faculty.poly.edu/~jbain/heat/readings/98Navarro.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4315" LastActivityDate="2016-06-21T17:03:17.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4936" PostTypeId="2" ParentId="64" CreationDate="2016-06-21T19:11:01.043" Score="2" Body="&lt;p&gt;Michael Weiss's answer is excellent. However, some clarificaiton is needed. Historians as late as 1997 went back to reevaluate the priority dispute and, the same way Newton created differential and integral calculus before Leibniz did (albeit independently), Einstein arrived at the correct field equations before Hilbert did (weeks before, not five days): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nytimes.com/1997/11/18/science/findings-back-einstein-in-a-plagiarism-dispute.html&quot; rel=&quot;nofollow&quot;&gt;http://www.nytimes.com/1997/11/18/science/findings-back-einstein-in-a-plagiarism-dispute.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hilbert's published equations were not generally covariant, Einstein's were, and it took him a few weeks to sort it out. Excellent point on the brilliant Emmy Noether (the most under-appreciated scientific genius of the 20th century). I also think the question is a bit silly given the fact that without Einstein's lectures at the University of Gottingen in 1914 and his numerous letters to Hilbert, Hilbert doesn't even come close to finding the final field equations. &#xA;In a way, Hilbert's behavior during the entire episode was a bit unethical. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, Hilbert's general theory of gravitation (he created his own) is riddled with errors - pointing to the fact that one can be a mathematical genius without necessarily being a scientific one. His conceptual understanding of physics was somewhat lacking. &lt;/p&gt;&#xA;" OwnerUserId="4315" LastActivityDate="2016-06-21T19:11:01.043" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4937" PostTypeId="2" ParentId="3824" CreationDate="2016-06-21T19:31:09.830" Score="3" Body="&lt;p&gt;@ Geremia, you should really abstain from citing a well-known anti-semite Bjerknes when trying to be objective about Einstein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And no, Special Relativity doesn't &quot;belong&quot; to Poincare and Lorentz but to Einstein. (And it's ludicrous to talk about SR without acknowledging Fitzgerald and Maxwell). Your argument is tantamount to crediting Newton with the full invention of calculus when the heavy lifting had already been done by Descartes and Fermat on analytical geometry - a curiousa that explains how Leibniz was able to derive calculus almost simultaneously with the great Sir Isaac. In fact, the Bernoullis - and even Lagrange - always acknowledged calculus as the birth child of Fermat. So if we aren't going to split hairs on an issue such as that, then why not simply give Einstein his due for Special Relativity (which in a real epistemic sense is different than both Lorentz and Poincare's conceptions of 'relativity'). In theory, Einstein didn't actually need his scientific predecessors to create SR given that the Lorentz transformations naturally arise out of Maxwell's equations. If we're being super picky, then let's give Lorentz, Poincare, Fitzgerald, and Maxwell 50% of the credit, and Einstein the other 50%. But why stop there? Give Laplace 30% of the credit for celestial mechanics. Max Born should get 40% of the credit for Matrix Mechanics too. Robert Hooke should get 50% of the credit for coming up with the inverse square law which Newton subsequently stole and then mathematized. (I hope you get my point - you run into a fair bit of troubling when delineating attributions of scientific discoveries).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At any rate, regarding your question, there are a few reasons. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The nobel committee didn't fully understand General Relativity nor was there an immediate realization of how important GR was to the modelling of the universe until Wheeler and company advocated for it later in the century. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The nobel committee didn't fully realize the universality of SR until Dirac. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Einstein really didn't care for the Nobel prize and the Nobel committee took offense. On the week he was about to be awarded the prize, he received a message suggesting he stay around &quot;for something big in Stockholm&quot; (wink, wink, hint, hint). However, he had already planned a trip to Japan that week and didn't care enough about the Nobel to delay his trip (the Nobel committee certainly took offense). Not only did he skip his own Nobel ceremony, but he didn't even WRITE his reception of the award in his journal the day he received it. Later quotes by journalists asking why he never received more than one Nobel mirror his general indiffernce to the award. His only use of the Nobel prize was to complete a promise to his then ex-wife Mileva Maric that, in exchange for a divorce, he would give her the Nobel Prize money once he inevitably got it for one of the many brilliant things he did. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Much of Einstein's work was so revolutionary that there was a sense in which the committee members believed them to be too speculative. Plus, the committee has generally prejudiced experimentalists over theorists - but even that isn't enough to account for why they've given out multiple nobels to other scientists whose work were nowhere near as fundamental as Einstein's.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;There's a great argument made by a plethora of science historians than Einstein deserved anywhere from 4 to 12 Nobel Prizes (klien wrote a great article on this a few years back). Stone: &lt;a href=&quot;http://www.huffingtonpost.com/a-douglas-ston/einstein-fantasy-physics_b_4948045.html&quot; rel=&quot;nofollow&quot;&gt;http://www.huffingtonpost.com/a-douglas-ston/einstein-fantasy-physics_b_4948045.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Count the amount of Nobels given for Brownian Motion. Bose-Einstein Condensates. Special Relativity. General Relativity. His 1909 and, later, his 1917 Paper on the Spontaneous and Stimulated Emission, and many others. Heck, even the EPR paper would be worthy contender. &lt;/p&gt;&#xA;" OwnerUserId="4315" LastEditorUserId="4315" LastEditDate="2016-06-21T21:22:37.170" LastActivityDate="2016-06-21T21:22:37.170" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4938" PostTypeId="1" CreationDate="2016-06-21T22:59:42.377" Score="1" ViewCount="66" Body="&lt;p&gt;Why are Bose-Einstein Condensates (BECs) named after Bose and Einstein? What did Einstein contribute to BECs?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="26" LastEditDate="2017-01-15T12:18:03.163" LastActivityDate="2017-01-15T12:18:03.163" Title="Why are Bose-Einstein Condensates (BECs) named after Bose and Einstein?" Tags="|theoretical-physics|einstein|statistical-mechanics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4939" PostTypeId="1" AcceptedAnswerId="4940" CreationDate="2016-06-22T01:04:48.177" Score="3" ViewCount="190" Body="&lt;p&gt;I seem to recall reading about a mathematician from antiquity whose view was, roughly, that to &quot;apply&quot; math was to do it violence (or indeed, to expedite violence) and so looked down on applications. Who/what might I be thinking of?&lt;/p&gt;&#xA;" OwnerUserId="4305" LastActivityDate="2016-06-22T03:52:42.420" Title="Which ancient mathematician was opposed to &quot;application&quot; of math?" Tags="|mathematicians|ancient-greece|ancient|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4940" PostTypeId="2" ParentId="4939" CreationDate="2016-06-22T03:52:42.420" Score="2" Body="&lt;p&gt;The sentiments you are describing are most likely those attributed to Plato.  Although not primarily a mathematician, Plato is the figure who is credited with drawing the distinction between &lt;em&gt;pure&lt;/em&gt; and &lt;em&gt;applied&lt;/em&gt; mathematics - drawing a line between the theoretical and the computational aspects of mathematics, and instilling this in his followers.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Boyer's &lt;em&gt;History of Mathematics&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[I]n geometry, he espoused the cause of pure mathematics as against the materialistic views of the artisan or technician.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Plutarch, in his &lt;em&gt;Life of Marcellus&lt;/em&gt;, speaks of Plato's &quot;indignation at the use of mechanical contrivances in geometry.&quot;  According to Boyer, Plato may largely be responsible for the restriction in Greek geometric constructions to those that can be effected by straightedge and compass alone.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The reason for this restriction is not likely to have been only the simplicity of the instruments used in constructing lines and circles, but rather the symmetry of the configurations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In arithmetic, Plato draws the line between those aspects of arithmetic that we now call &lt;em&gt;number theory&lt;/em&gt;, and &lt;em&gt;logistic&lt;/em&gt; as mere computation.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He regarded &quot;logistic&quot; as appropriate for business men and men of war who must learn the art of numbers or he will not know how to array his troops, while &quot;arithmetic&quot; was appropriate for philosophers because he has to arise out of the sea of change and lay hold of true being.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the &lt;em&gt;Republic&lt;/em&gt;, Plato says &quot;Arithmetic has a very great and elevating effect, compelling the mind to reason about abstract number.&quot;  Quoting Boyer :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So elevating are Plato's thoughts concerning number that they reach the realm of mysticism and apparent fantasy. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-06-22T03:52:42.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4941" PostTypeId="1" AcceptedAnswerId="5075" CreationDate="2016-06-22T14:49:32.450" Score="25" ViewCount="3460" Body="&lt;p&gt;What did Einstein contribute to what is now called Special Relativity theory that hadn't already been done by&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Lorentz in his 1904 &quot;&lt;a href=&quot;https://en.wikisource.org/wiki/Electromagnetic_phenomena&quot;&gt;Electromagnetic Phenomena in a System Moving with any Velocity Smaller than that of Light&lt;/a&gt;&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Poincaré in his &lt;em&gt;Rendiconti del Circolo Matematico di Palermo&lt;/em&gt; paper, &quot;&lt;em&gt;&lt;a href=&quot;https://fr.wikisource.org/wiki/Sur_la_dynamique_de_l%E2%80%99%C3%A9lectron#.C2.A7_9._.E2.80.94_Hypoth.C3.A8ses_sur_la_Gravitation.&quot;&gt;Sur la dynamique de l'électron&lt;/a&gt;&lt;/em&gt;&quot; (&lt;a href=&quot;https://en.wikisource.org/wiki/Translation:On_the_Dynamics_of_the_Electron_%28July%29&quot;&gt;free English translation&lt;/a&gt;; &lt;a href=&quot;https://dx.doi.org/10.1119/1.1976641&quot;&gt;paywalled English trans. part 1&lt;/a&gt;, &lt;a href=&quot;https://dx.doi.org/10.1119/1.11053&quot;&gt;part 2&lt;/a&gt;, &lt;a href=&quot;https://dx.doi.org/10.1119/1.1986815&quot;&gt;part 3&lt;/a&gt;), accepted in July 1905&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-06-23T03:55:07.057" LastActivityDate="2024-03-18T08:26:03.593" Title="What did Einstein contribute to Special Relativity that hadn't already been done by Lorentz in 1904 and Poincaré in 1905?" Tags="|relativity-theory|einstein|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4942" PostTypeId="2" ParentId="1838" CreationDate="2016-06-22T18:00:47.410" Score="0" Body="&lt;p&gt;As a general and high level overview, I enjoyed &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/039331071X&quot; rel=&quot;nofollow&quot;&gt;Mathematics for the Million&lt;/a&gt; by Lancelot Hogben. It's not very in-depth or modern, but it does put many mathematical developments in their historical context.&lt;/p&gt;&#xA;" OwnerUserId="4233" LastActivityDate="2016-06-22T18:00:47.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4943" PostTypeId="2" ParentId="4934" CreationDate="2016-06-22T18:02:56.453" Score="7" Body="&lt;p&gt;As per your comment requesting details of the linked paper :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://abesenyei.web.elte.hu/publications/meanvalue.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Besenyei's paper&lt;/a&gt; begins with a history of the development of Rolle's Theorem into its general form.  It follows this with the history of various generalisations of Rolle's Theorem equivalent to MVT and attributed to the likes of Cauchy, Bonnet, Serret, Dini, and Harnack.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at Besenyei's paper, at first it would appear that Joseph Alfred Serret is the first to state and prove the result in its modern form, however this is not the case - please see below.  From Serret's &lt;em&gt;Cours de calcu infinitesimal&lt;/em&gt; of 1868 as Theorem I (translation) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $f(x)$ be a function of $x$ which remains continuous for values of $x$ between two given limits, and which, for these values, has a well-determined derivative $f'(x)$. If $x_0$ and $X$ denote two values of $x$ between these same limits, the following $$\frac{f(X) - f(x_0)}{X-x_0} = f'(x_1)$$ will hold, with $x_1$ being a value between $x_0$ and $X$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Besenyei's paper does not include the relevant proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This appears on the page denoted 11/15 (lower right hand corner) - page 83/152 of the PDF document. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, according to &lt;a href=&quot;http://cerme8.metu.edu.tr/wgpapers/WG1/WG1_Chorlay.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper by Renaud Chorlay&lt;/a&gt;, Serret is here simply stating Bonnet's proof. (See page 7 of 16 of the PDF document.)  Thus, the correct answer as to who first proved MVT in its modern form appears to be &lt;strong&gt;Pierre-Ossian Bonnet&lt;/strong&gt;. The Renaud Chorlay paper does appear to give Bonnet's proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This conclusion appears to be contrary to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mean_value_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia entry for the MVT&lt;/a&gt; which credits Cauchy as the first to state it in its modern form.  However, Cauchy's statement of MVT is actually and extension of MVT to deal with two different continuous functions $f$ and $g$ which states $$\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}$$ and which predates the Serret/Bonnet version of MVT.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Mikhail has pointed out in his comment, the correct answer is in fact Cauchy, as is evident in Besenyei's paper.  Cauchy gives both forms of stating the MVT.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2018-04-25T03:38:09.883" LastActivityDate="2018-04-25T03:38:09.883" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="4944" PostTypeId="2" ParentId="4941" CreationDate="2016-06-22T23:48:46.090" Score="5" Body="&lt;p&gt;Both Einstein and Poincaré postulated the principal of relativity as a basis for their theories. Where Einstein's and Poincaré's theories differ is that Einstein postulated the one-way constancy of the speed of light in inertial frames and derived the Lorentz transformation as a consequence, while Poincaré postulated the Lorentz transformation and derived the round-trip constancy of the speed of light in inertial frames as a consequence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Poincaré's theory retained a metaphysical, unobservable ether frame. There is no need for an ether frame in Einstein's special relativity, and even Poincaré admitted that the ether frame is ultimately a useless concept. This concept of some special ether frame in which a body can deemed to be in a state of absolute rest prevented Poincaré from discovering the relativity of simultaneity, which Einstein did discover.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no test that can distinguish between Einstein's theory of special relativity and Lorentz's and Poincaré's Lorentz ether theory. The two theories are mathematically equivalent. Both theories have untestable metaphysical postulates. The one-way speed of light is unobservable, but then again, so is Poincaré's ether frame. The reason we now see relativity as Einstein's invention rather than that of Lorentz and Poincaré is quantum mechanics, which completely removes the need for an ether frame.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could someone have come up with Einstein's theory of special relativity had Einstein not been born? Of course. But it would have taken some time.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2016-06-22T23:48:46.090" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4945" PostTypeId="1" CreationDate="2016-06-23T04:07:46.660" Score="3" ViewCount="300" Body="&lt;p&gt;I'm searching a trusted historical sources about primitive Pythagoras triplets as being powerful integers (numerical examples), or a notable work of impossibility of such a triples, but couldn't find despite asking this question here at MSE and MOF&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do expect some old references for this issue, as when and who stated this issue, and what are the consequences of this problem on the important unsolved problems in mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please if any information available, provide exact dates with references.&lt;/p&gt;&#xA;" OwnerUserId="4075" LastActivityDate="2016-09-08T09:29:18.083" Title="Are there primitive Pythagoras triplets (in integers), being with all the terms as powerful numbers?" Tags="|mathematics|biographical-details|number-theory|pythagorean-proposition|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="4946" PostTypeId="1" AcceptedAnswerId="4947" CreationDate="2016-06-22T12:50:04.523" Score="59" ViewCount="2471" Body="&lt;p&gt;I would like to know the name of the device Ptolemy is holding in the following picture.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8kfu8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8kfu8.png&quot; alt=&quot;Public domain image of Ptolemy holding a cross-shaped device&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://es.wikipedia.org/wiki/Archivo:PSM_V78_D326_Ptolemy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;[Image Source]&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4325" OwnerDisplayName="hb20007" LastEditorUserId="4325" LastEditDate="2021-11-18T09:56:18.767" LastActivityDate="2021-11-18T09:56:18.767" Title="What is Ptolemy holding in this picture?" Tags="|geometry|ancient-greece|antiquity|ptolemy|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4947" PostTypeId="2" ParentId="4946" CreationDate="2016-06-22T12:59:47.337" Score="61" Body="&lt;p&gt;It is called &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jacob%27s_staff&quot;&gt;Jacob's staff&lt;/a&gt;&quot;. It was an old astronomical tool used for trigonometric purposes.&lt;/p&gt;&#xA;" OwnerUserId="4324" OwnerDisplayName="Euler_Salter" LastActivityDate="2016-06-23T02:01:09.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4948" PostTypeId="2" ParentId="4946" CreationDate="2016-06-22T13:07:48.127" Score="67" Body="&lt;p&gt;This device was invented by a Jewish Rabbi, &lt;a href=&quot;https://en.wikipedia.org/wiki/Gersonides&quot; rel=&quot;noreferrer&quot;&gt;Levi Ben Gershon&lt;/a&gt;. It was used to measure the angular distance between two stars or, in general, any pair of celestial bodies. Ptolemy lived 1000 years before the invention, so this is only the product of the artist’s fantasy.&lt;/p&gt;&#xA;" OwnerUserId="4329" OwnerDisplayName="Riccardo.Alestra" LastEditorUserId="9603" LastEditDate="2019-02-12T17:28:00.543" LastActivityDate="2019-02-12T17:28:00.543" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="4949" PostTypeId="2" ParentId="4946" CreationDate="2016-06-22T18:35:00.843" Score="19" Body="&lt;p&gt;And this tool has been known under many other latin names than &lt;em&gt;baculus Jacob&lt;/em&gt; (or Jacob's staff): &lt;em&gt;radius astronomicus&lt;/em&gt; (astronomic ray), &lt;em&gt;crux geometrica&lt;/em&gt; (geometrical cross), &lt;em&gt;revelatorem secretorum&lt;/em&gt; (secret decipherer).  This one is the hand-held version, a mounted version exists. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A collection of studies  on &lt;a href=&quot;https://en.wikipedia.org/wiki/Gersonides&quot; rel=&quot;noreferrer&quot;&gt;Gersonides (or Levi ben Gershon)&lt;/a&gt; is found in G. Freudenthal (ed.), &lt;a href=&quot;https://books.google.fr/books?id=DXSUpHqPiMgC&amp;amp;pg=PA267&amp;amp;lpg=PA267&amp;amp;dq=%22Studies%20on%20Gersonides%22&amp;amp;source=bl&amp;amp;ots=qqLfVDZo_L&amp;amp;sig=uFPvQDbFSY0XxdTmZmnn8_o9CEs&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=%22Studies%20on%20Gersonides%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Studies on Gersonides. A Fourteenth Century Jewish Philosopher&lt;/a&gt;. And in the more accessible 1981 paper by J. Roche, &lt;a href=&quot;http://dx.doi.org/10.1080/00033798100200101&quot; rel=&quot;noreferrer&quot;&gt;The radius astronomicus in England&lt;/a&gt;, Annals of Science, vol. 38:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abstract: This survey traces the history of the astronomer's cross staff on the&#xA;  Continent from Levi ben Gerson to Gemma Frisius, in England from John&#xA;  Dee to John Greaves, and again on the Continent from Tycho Brahe to&#xA;  Adrian Metius. The emphasis throughout is on sources and influences,&#xA;  on distinguishing the various kinds of cross staff, and on clarifying&#xA;  terminology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It has been used in navigation, perhaps as an upgraded version of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kamal_(navigation)&quot; rel=&quot;noreferrer&quot;&gt;kamal&lt;/a&gt;, a rectangular wooden card combined with a string equiped with several equally spaced knots, a navigation device that determines latitude.&lt;/p&gt;&#xA;" OwnerUserId="4130" OwnerDisplayName="Laurent Duval" LastEditorUserId="4130" LastEditDate="2019-10-05T15:22:04.130" LastActivityDate="2019-10-05T15:22:04.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4950" PostTypeId="1" CreationDate="2016-06-23T18:54:19.600" Score="6" ViewCount="121" Body="&lt;p&gt;The demarcation problem, i.e. the problem of differentiating science from pseudo science has been on my mind recently. This might be considered a philosophy topic for philoSE but since my question has mostly historical character and related significantly to the development of science, I thought this was a better place than philoSE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is: Does anybody have literature references discussing science vs pseudo science (or science versus religion) that date back further than the time of Hume or Descartes? It doesn't necessarily need to be under the name of science vs non-science, anything that pertains to this discussion that is earlier than Hume would be much appreciated.&lt;/p&gt;&#xA;" OwnerUserId="3652" LastActivityDate="2016-06-26T11:03:45.017" Title="What are the earliest known accounts of the demarcation problem (science versus pseudo science)?" Tags="|reference-request|philosophy-of-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4951" PostTypeId="1" CreationDate="2016-06-24T01:19:41.880" Score="6" ViewCount="269" Body="&lt;p&gt;Has anybody read or have access to&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alex Andrew&#xA;    Significance Feedback in Neural Nets&#xA;Report of Biological Computer Laboratory, University of Illinois, Urbana, IL&#xA;    GM-10718-03&#xA;TR No 5&#xA;September 1965&#xA;pages 1--12&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and can confirm the contribution this made to backprop?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the basic routine is done by economists since the late 19th century for inferring value to factors of factors of  ... production, with the chain rule, but for neural networks, this is cited by cybernetics oriented persons as the very first application. Is it?&#xA;Margaret Boden cites Paul Werbos as first but that I have read (it was republished in his book) but Werbos's dissertation came a decade after Andrew's report.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen this cited as the first modern backpropagation in cybernetics paper, and not surprisingly it is from the place where McCulloch, Ashby, et al worked. But I can't access it to confirm.&#xA;Alexander passed away recently so can't contact him either.&#xA;Anybody have this thing, or read it, or have it scanned?&#xA;Archive.org has some of the other reports (for example, on asynchronous computers) but not this one ... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;UPDATE: Interlibrary loan cannot get ahold of a copy of the first edition. They gave up on all requests! Does anybody have this paper, or the reprint of it from the 1990s?&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2016-10-29T06:57:17.270" LastActivityDate="2016-10-29T06:57:17.270" Title="history of backpropagation" Tags="|mathematics|computer-science|artificial-intelligence|cybernetics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4952" PostTypeId="2" ParentId="3870" CreationDate="2016-06-24T18:53:02.677" Score="3" Body="&lt;p&gt;Here is the relevant text from Ramanujan's second letter to G. H. Hardy, dated 27 February 1913 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source : &lt;a href=&quot;https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Ramanujan_summation&quot; rel=&quot;nofollow&quot;&gt;wikipedia page on $\Sigma n$, anchor at &quot;Ramanujan summation&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-06-24T18:53:02.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4953" PostTypeId="1" CreationDate="2016-06-24T19:07:42.917" Score="2" ViewCount="188" Body="&lt;p&gt;There are many ways to evaluate $\sum_{n \geq 1} 1/n^2$ as $\pi^2/6$, including multiple solutions using Fourier series. A colleague asked me who was the &lt;em&gt;first&lt;/em&gt; person to use Fourier series (or Fourier analysis) to make this computation. I don't know and was unable to figure this out. Any concrete evidence that a particular person was the first to do this would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="4251" LastEditDate="2020-06-21T15:16:44.267" LastActivityDate="2020-06-21T15:16:44.267" Title="First evaluation of $\sum_{n \geq 1} 1/n^2$ by Fourier series" Tags="|mathematics|discoveries|real-analysis|harmonic-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4954" PostTypeId="1" CreationDate="2016-06-25T11:47:09.310" Score="4" ViewCount="391" Body="&lt;p&gt;We like to repeat that &lt;a href=&quot;https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation&quot; rel=&quot;nofollow&quot;&gt;correlation does not imply casuation&lt;/a&gt;. My gut feeling is however that (source: &lt;a href=&quot;https://xkcd.com/552/&quot; rel=&quot;nofollow&quot;&gt;xkcd&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Correlation doesn't imply causation, but it does waggle its eyebrows&#xA;  suggestively and gesture furtively while mouthing 'look over there'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Were important discoveries rather the effect of &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;mathematical explorations (&lt;em&gt;&quot;I will take this and that equation, combine them together and test if this is true in the real world&quot;&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;or physical experiments (&lt;em&gt;&quot;The more I pull on this spring, the more resistance there is. Oh look, the relationship is linear - I will look at physics to understand why&quot;&lt;/em&gt;) - in that case correlation was the lead mechanism.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is obviously oversimplified, my question is about the predominance of each of these techniques in history. I guess that the second case was much more common initially, then with a more elaborate mathematical apparatus purely theoretical explorations were made possible (or easier).&lt;/p&gt;&#xA;" OwnerUserId="4337" LastActivityDate="2016-07-02T19:11:38.237" Title="Has correlation led to major discoveries?" Tags="|mathematics|theoretical-physics|experimental-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4955" PostTypeId="2" ParentId="4950" CreationDate="2016-06-26T11:03:45.017" Score="1" Body="&lt;p&gt;Galileo was way before Hume and even before Descartes.  His &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Assayer&quot; rel=&quot;nofollow&quot;&gt;The Assayer&lt;/a&gt; was the foundational text of modern science and simultaneously also a devastating critique of what he saw as pseudosience. The funny thing is that Galileo was actually wrong about the specific point of the nature of the comets :-)&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-06-26T11:03:45.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4956" PostTypeId="1" AcceptedAnswerId="4957" CreationDate="2016-06-27T00:36:14.597" Score="4" ViewCount="403" Body="&lt;p&gt;Boyer's &lt;em&gt;History of Mathematics&lt;/em&gt; includes a brief account of Bolzano's contributions in the chapter on Gauss and Cauchy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Building on Galileo's &quot;paradox&quot; on the one-to-one correspondence between positive integers and perfect squares, Bolzano was able to show that similar correspondences between infinite sets and a proper subset are commonplace, including examples of sets of real numbers.  Boyer then states :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bolzano seems even to have recognized, by about 1840, that the infinity of real numbers is of a type different from the infinity of the integers, being nondenumberable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bolzano's contributions to our understanding of infinite sets are featured in his posthumously published work of 1850, &lt;em&gt;Paradoxien des Unendlichen&lt;/em&gt;.  Online copies of this text are available in the original German, but I do not read German.  I am unable to locate any explanation as to how Bolzano reached this conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Bolzano conclude that the infinity of real numbers is of a different type from the infinity of integers, and if so, how did he reach this conclusion?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="26" LastEditDate="2016-06-27T13:38:32.840" LastActivityDate="2017-06-03T18:07:51.507" Title="Did Bolzano conclude that $| \mathbb R | \ne | \mathbb N|$?" Tags="|mathematics|set-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4957" PostTypeId="2" ParentId="4956" CreationDate="2016-06-27T09:21:25.850" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Did Bolzano conclude that the infinity of real numbers is of a different type from the infinity of integers ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Yes and no&lt;/em&gt; ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Guillermina Waldegg, &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs11191-004-2014-0&quot; rel=&quot;noreferrer&quot;&gt;Bolzano‘s Approach to the Paradoxes of Infinity: Implications for Teaching&lt;/a&gt; (2005):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[page 569] The comparison of infinite sets and the creation of an ‘order’ between them was one of Bolzano’s main concerns. In this respect, Bolzano identifies another of the paradoxes that constitutes ‘one of the most noteworthy characteristics of infinite sets’:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;When two sets are infinite, they can stand in such a relation to one another that (1) it is possible to couple each member of the first set with some member of the second in such wise that, on the one hand, no member of either set fails to occur in one of the couples; and on the other hand, not one of them occurs in two or more of the couples.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;However, Bolzano notes that these sets may simultaneously present another&#xA;  type of relationship, which is the decisive difference between Bolzano’s and&#xA;  Cantor’s criteria to compare two infinite sets.&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;(2) one of the two sets can comprise the other as a mere part of itself, in such a wise that the multiplicities to which they are reduced, when we regard all their members as interchangeable individuals, can stand in the more varied relationships to one another (Bolzano, &lt;em&gt;Paradoxes&lt;/em&gt;, 1851, p. 96)&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;For Bolzano, the obvious criterion for comparing two sets ought to be based&#xA;  on the second type of relationship, i.e., on the part-whole relationship between&#xA;  sets.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 570] Note Bolzano’s crucial idea that two abstract sets may present a one-to-one relationship, but this does not imply that they are equinumerous, because one of them may be a subset of the other. The possibility of establishing a bijection seems to be a (paradoxical) characteristic of infinite sets, since they may nevertheless have a different number of elements. This position clearly separates the works of Bolzano and Cantor. For Cantor (as well as for Dedekind), the part-whole relationship defines infinite sets and the ‘size’ (and order) of infinite sets is based on cardinality, a measure of the ‘number of elements’ of a set. If it is possible to establish a one-to-one correspondence between the members of two sets, the sets have the same cardinality, and are hence equivalent.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The paradoxical situation that arises from the comparison of infinite sets is&#xA;  a central concern in Bolzano’s work, and can be observed in the following  paragraph:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;As I am far from denying, an air of paradox clings to these assertions; but its sole origin is to be sought in the circumstance that the above and oft-mentioned relation between two sets, as specified in terms of couples, really does suffice, in the case of finite sets, to establish their perfect equimultiplicity in members. […] The illusion is therefore created that this ought to hold when the sets are no longer finite, but infinite instead (Bolzano, &lt;em&gt;Paradoxes&lt;/em&gt;, 1851, p. 98).&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;[Regarding Galileo's paradox] Bolzano did not accept to extend the bijection criterion so as to establish the equinumerosity between two infinite sets. Nevertheless, as with finite sets, the part-whole relationship did allow him to assert that one of the two infinite sets is larger than the other.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 571] Bolzano attempts to define the &lt;em&gt;arithmetic of infinity&lt;/em&gt;, by analyzing the possibility of introducing a certain ‘operability’ with infinity. However, as there existed no concept that would allow him to ‘quantify’ infinite sets, such as Cantor’s cardinality, Bolzano approached the problem of arithmetization of infinity in terms that are reminiscent of the theory of proportions:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;I confess that the mere idea of a &lt;em&gt;calculation with the infinite&lt;/em&gt; has the appearance of contradicting itself. For to try to calculate anything means, after all, to attempt a &lt;em&gt;determination&lt;/em&gt; of it in terms of number […] But this scruple vanishes when we reflect that a correctly conducted calculation with the infinite is not a numerical determination of what is therein not numerically determinable […] but only aims at determining the ratio [or relationship] between one infinite and another… (Bolzano, &lt;em&gt;Paradoxes&lt;/em&gt;, 1851, p. 107)&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Two observations stem from this paragraph. Firstly, when Bolzano talks&#xA;  about establishing the ratio between one infinity and another, he accepts that&#xA;  different infinities exist; this position is essentially different from his predecessors’, in particular from Galileo’s. [...] Secondly, Bolzano abandons the idea of equinumerous sets being defined in terms of a one-to-one relationship, a notion that would be fundamental to Cantor, in order to state the problem of arithmetization of infinity in terms of a theory of proportions.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[page 573] When Bolzano analyzes infinite sets of real numbers, he establishes a ‘metric’ comparison criterion by associating real numbers to points on the&#xA;  straight line. [...] The sets of points are full of geometric meaning for Bolzano. The perception of geometrical characteristics of the regions associated with a given set is a determining factor for establishing the order of magnitude that Bolzano attributes to such a set. The following paragraph illustrates this:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;We must attribute to a bilaterally indeterminated straight line an infinite length and a set of points infinite many times as great as the set of point in the unite straight line (Bolzano, &lt;em&gt;Paradoxes&lt;/em&gt;, 1851, p. 151).&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-06-27T09:21:25.850" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4959" PostTypeId="1" CreationDate="2016-06-27T14:55:37.407" Score="0" ViewCount="60" Body="&lt;p&gt;If so how did he managed to see this as molecules weren't known in his days?&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="131" LastEditDate="2016-06-27T15:18:13.653" LastActivityDate="2016-06-27T15:18:13.653" Title="Was Anaximander the first who described that the clouds and rain were formed by evaporation of the oceans?" Tags="|physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4960" PostTypeId="1" CreationDate="2016-06-27T21:19:38.257" Score="9" ViewCount="788" Body="&lt;p&gt;According to &lt;a href=&quot;https://web.archive.org/web/20141226210511/https://robert.accettura.com/wp-content/uploads/2010/10/Morgenstern_onGoedelcitizenship.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this document&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_G%C3%B6del&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Gödel&lt;/a&gt; had discovered logical inconsistiencies in the U.S. constitution that could theoretically allow it to become a dictatorship. &lt;a href=&quot;https://en.wikipedia.org/wiki/Oskar_Morgenstern&quot; rel=&quot;nofollow noreferrer&quot;&gt;Morgenstern&lt;/a&gt; told him not to bring it up during his U.S. citizenship exam, but:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And then [the examiner] turned to Gödel and said, &amp;quot;Now Mr. Gödel, where do you come from?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Gödel: &amp;quot;Where I come from? Austria.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The Examinor: &amp;quot;What kind of government did you have in Austria?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Gödel: &amp;quot;It was a republic, but the constitution was such that it finally changed into a dictatorship.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The Examinor: &amp;quot;Oh! This is very bad. This could not happen in this country.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Gödel: &amp;quot;Oh, yes, I can &lt;em&gt;prove&lt;/em&gt; it.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Luckily, the examiner changed the subject and Gödel could become a U.S. citizen.&lt;/p&gt;&#xA;&lt;p&gt;What was his proof?&lt;/p&gt;&#xA;" OwnerUserId="4349" LastEditorUserId="4251" LastEditDate="2022-09-23T06:05:06.163" LastActivityDate="2022-10-23T13:02:00.217" Title="What was Gödel's proof that errors in U.S. constitution could allow the U.S. to become a fascist dictatorship?" Tags="|mathematicians|biographical-details|logicians|goedel|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="4961" PostTypeId="1" CreationDate="2016-06-28T21:22:48.043" Score="7" ViewCount="685" Body="&lt;p&gt;I am reading through the Dover publication of Gauss' work on curves and curved surfaces called &quot;General Investigations of Curved Surfaces&quot;.  This Dover publication presents two major papers authored by Gauss in 1827 and 1825 (the 1825 paper was actually published later than the 1827 paper).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subject matter is not much different than taught in a typical Advanced Calculus class that presents the beginnings of Differential Geometry.  However, the book is still interesting reading although a tough travel due to the more crude notation and the total lack of any diagrams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This whole topic was made for the use of graphic diagrams to help with the presentation and yet nothing of that sort appears in the entire book.  The only diagrams that appear are in the end notes that briefly introduce the reader to Spherical Geometry.  These end notes are not part of Gauss' original work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I submit that this whole paper could be condensed by half or maybe to a third of its current size by the use of well appointed diagrams to help illustrate the mathematical development.  Indeed, reading through this is like reading a thousand words instead of the one well placed &quot;picture&quot;.  And, the very first section (aka paragraph) could be replaced with a diagram of the Unit Sphere with radial vectors displayed to denote points (x,y,z) on the surface of the sphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which brings me to my question.  Obviously these writers did not have LaTeX and TikZ at their disposal for writing and the idea of printing papers with diagrams was likely difficult, costly, or maybe impossible (without hiring an artist) in those earlier years.  At what point did diagrams become more prevalent in published papers.  I am referring of course to papers where a diagram is well suited to illustrating the concepts being described.  For the papers contained in this Dover book, I would guess almost every section could be augmented by a diagram.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am guessing not before 1900 and maybe even later did diagrams become more common.  I am probably older than most of the readers of this forum but my textbooks (physics &amp;amp; math) were loaded with diagrams (mid-1960 but many textbooks published in mid to late 1950s) and this was before Knuth, TeX, and of course, TikZ.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2021-01-05T07:25:59.043" Title="Diagrams in early Mathematical Paper Publication" Tags="|mathematics|publishing|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4962" PostTypeId="2" ParentId="4961" CreationDate="2016-06-28T21:46:14.690" Score="9" Body="&lt;p&gt;Probably not what you are really asking.  But it seems interesting anyway.&lt;/p&gt;&#xA;&lt;p&gt;This page of Euclid is from the Ninth Century.  There is a diagram.  It has other modern innovations, such as spaces between words.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mlMYg.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mlMYg.gif&quot; alt=&quot;Euclid&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mat.ucm.es/cosasmdg/cdsmdg/05edumat/geometriahoy/euclideselementos/Sample%20Page%20from%20Euclid%27s%20Elements.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The papyrus fragment below (also Euclid) is from the First Century, one of the earliest known instances of a geometric diagram.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/i207P.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i207P.jpg&quot; alt=&quot;EUclid&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.gearfuse.com/wonderful-gallery-of-science-euclids-elements/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A parchment from the Ninth Century with a diagram representing three dimensions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5r9Kl.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5r9Kl.jpg&quot; alt=&quot;Euclid&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/Math_extra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="11204" LastEditDate="2021-01-05T07:25:59.043" LastActivityDate="2021-01-05T07:25:59.043" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="4965" PostTypeId="1" CreationDate="2016-06-28T19:23:41.123" Score="0" ViewCount="98" Body="&lt;p&gt;It seems more intuitive to put price on the x-axis, since that is a variable that a producer can set, or is otherwise set for the consumer (a consumer walks into the store and sees the price, determining the quantity he is going to buy). There is a tendency to do this across undergraduate study, and I am curious as to why this seems to be the convention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there is a reason for this, or my intuition just incorrect? &lt;/p&gt;&#xA;" OwnerUserId="3822" OwnerDisplayName="Brandyn Murillo" LastEditorUserId="2442" LastEditDate="2016-08-12T15:45:02.040" LastActivityDate="2016-08-12T15:45:02.040" Title="Why do undergraduate texts insist on graphing price as a function of demand?" Tags="|economics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4966" PostTypeId="2" ParentId="4965" CreationDate="2016-06-28T19:49:50.497" Score="8" Body="&lt;p&gt;It isn't just something in undergraduate study, it's something common across economics in general.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consumers and producers can set quantity, not price. For example, when you enter a store, you can only choose the amount of stuff you buy, not the price at which you buy it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This logic is a little more confusing for the producer's side of things. One of the assumptions of microeconomics (at least at the beginning of most undergraduate studies) is perfect competition, which means that both individual producers and consumers cannot affect prices by their own actions. This isn't true in some real-world applications (Apple sets the prices on their laptops, for example) but is true in some others (restaurants in competitive neighborhoods generally have similar prices for similar lunch specials). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The price is also set as the y-axis because of the demand function's connection with utility. &lt;/p&gt;&#xA;" OwnerUserId="4360" OwnerDisplayName="Kontorus" LastActivityDate="2016-06-28T19:49:50.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4968" PostTypeId="1" CreationDate="2016-06-30T02:27:35.190" Score="4" ViewCount="186" Body="&lt;p&gt;I am looking for an online source that gives the original statement and proof of Chebyshev’s asymptotic relation $\vartheta(x)\sim x$ where $\vartheta(x)=\sum_{p\leq x}\log p$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have tried searching online but the multiple spellings of Chebyshev's name in Latin characters is a significant impediment. I have also looked at secondary sources but have not found one yet that gives a reference to the original paper by Chebyshev. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any reference of the type described would be appreciated.  &lt;/p&gt;&#xA;" OwnerUserId="535" LastEditorDisplayName="user459" LastEditDate="2016-07-02T22:47:31.603" LastActivityDate="2022-04-09T22:40:10.623" Title="Original Statement and Proof of Chebyshev’s $\vartheta(x) \sim x$" Tags="|number-theory|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4969" PostTypeId="1" AcceptedAnswerId="4970" CreationDate="2016-06-30T05:07:28.653" Score="1" ViewCount="214" Body="&lt;p&gt;The first few Fermat numbers are $F_{0}=2^{2^0}+1=3$, $F_{1}=2^{2^1}+1=5$, $F_{2}=2^{2^2}+1=17$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read that $F_5$ through $F_{32}$ have been shown to be composite. And right now $F_{33}$ is the smallest Fermat number whose primality is unknown.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: When was $F_{32}$ shown to be composite?&lt;/p&gt;&#xA;" OwnerDisplayName="user67" LastActivityDate="2016-06-30T09:40:51.807" Title="When was the Fermat number $F_{32}$ shown to be composite?" Tags="|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4970" PostTypeId="2" ParentId="4969" CreationDate="2016-06-30T09:40:51.807" Score="3" Body="&lt;p&gt;A short Google search yields the page &lt;a href=&quot;http://www.prothsearch.net/fermat.html&quot; rel=&quot;nofollow&quot;&gt;http://www.prothsearch.net/fermat.html&lt;/a&gt;, which says a factor of the 32nd Fermat number was found in 1963 by Wrathall and that only one prime factor is known. &lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2016-06-30T09:40:51.807" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4971" PostTypeId="2" ParentId="4961" CreationDate="2016-06-30T12:33:32.717" Score="3" Body="&lt;p&gt;I'm not a specialist in history of science and math but I have recently read something about your question accidentely in the book by MacLane &quot;Categories for the Working Mathematicians&quot; in the Notes to the Chapter I.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cUR0A.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cUR0A.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1833" LastActivityDate="2016-06-30T12:33:32.717" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4972" PostTypeId="1" CreationDate="2016-06-30T12:36:39.550" Score="2" ViewCount="110" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Marcel_Riesz&quot; rel=&quot;nofollow&quot;&gt;Marcel Riesz&lt;/a&gt; spent most of his career in &lt;a href=&quot;https://en.wikipedia.org/wiki/Lund&quot; rel=&quot;nofollow&quot;&gt;Lund&lt;/a&gt; (Sweden).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there any memorial of Marcel Riesz in Lund?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A plaque or a sculpture? I was looking for information about it, but I didn't find any. I know that this question is not fully in the profile of this site, but maybe someone could help.&lt;/p&gt;&#xA;" OwnerUserId="100" LastEditorUserId="100" LastEditDate="2016-07-03T22:57:25.917" LastActivityDate="2016-07-04T18:01:42.967" Title="Are there any memorial of Marcel Riesz in Lund?" Tags="|mathematicians|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4974" PostTypeId="1" CreationDate="2016-07-01T16:42:50.700" Score="7" ViewCount="959" Body="&lt;p&gt;We often think of mathematics as being clean-cut, with questions being settled with theorems, and not much room for disagreement. This isn't true though of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going back to ancient history, we hear the story of how Pythagoras &lt;em&gt;murdered&lt;/em&gt; the man that discovered irrational numbers. Jumping to modern mathematics, Cantor's ideas about infinity invoked such harsh criticism, he literally &lt;em&gt;went insane&lt;/em&gt;$^1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a good website or other resource recounting some of the great controversies and debates of mathematics?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$^{^{^1\text{Although I'm sure going insane isn't fun, you have to admit that's a pretty awesome reason, if one had the choice.}}}$&lt;/p&gt;&#xA;" OwnerUserId="4349" LastActivityDate="2016-07-05T16:00:44.927" Title="What's a good resource for some of mathematics greatest controversies and debates?" Tags="|mathematics|resource-recommendation|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4975" PostTypeId="2" ParentId="4974" CreationDate="2016-07-02T05:34:44.493" Score="0" Body="&lt;p&gt;Pythagoreas didn't murder the person that found irrational numbers. Actually, we know very little about the person Pythagoreas. There was a sect of men who followed the teachings of a mystic named Pythagoreas. It was a cult with strange rules like not eating beans. Men who joined were given knowledge of mathematics. To remove jealousy, any discovery made by anyone was attributed to Pythagoreas. It's just mathematical apocrypha that the man who discovered irrational numbers was thrown off the ship because it conflicted with their believe that the natural numbers describe everything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Howard Eves wrote a series of books called Mathematical Circles - Quadrants I, II, III, IV in which he tells apocryphal stories about Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="3754" LastActivityDate="2016-07-02T05:34:44.493" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4977" PostTypeId="2" ParentId="4974" CreationDate="2016-07-02T18:16:39.927" Score="4" Body="&lt;p&gt;You might be interested in &quot;Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever&quot; by Hal Hellman. Personally, I don't think very highly of such &#xA;books, however.  &lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-07-02T18:16:39.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4978" PostTypeId="2" ParentId="4954" CreationDate="2016-07-02T18:27:06.840" Score="3" Body="&lt;p&gt;I agree with David Hammen's comment that most science is based on correlation rather than causation. Also agree that it's awfully broad.  So broad I'm not sure of any definitive way to answer it. But there are many fun examples that illustrate the point.  So here are a few.  (Incidentially, I'm aware of the irony that I'm trying to answer the question with examples, which itself is correlation data.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edward Tufte, in his 1997 &lt;em&gt;Visual Explanations&lt;/em&gt;, describes how John Snow used correlation between cholera incidence and location to deduce that the London cholera epidemic of 1854 was explained by drinking water from a single contaminated well.  From this correlation data, he then designed an experiment to test his theory: He had the handle of the well pump removed, and cholera incidence dropped. Tufte's books are not available online, but this story is well-known. A google search on John Snow and cholera will give you dozens of different versions to read.  For example: &lt;a href=&quot;http://www.ph.ucla.edu/epi/snow/snowcricketarticle.html&quot; rel=&quot;nofollow&quot;&gt;http://www.ph.ucla.edu/epi/snow/snowcricketarticle.html&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, the entire field of epidemiology is fundamentally concerned with looking for statistical patterns.  (See &lt;a href=&quot;https://en.wikipedia.org/wiki/Epidemiology#Causal_inference&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Epidemiology#Causal_inference&lt;/a&gt;.) In medicine, clinical trials often compare different treatment options. These clinical trials may eventually lead to research that uncovers details of drugs mechanism of action, and to how the body often fails to respond to them.  But most of modern medicine is based on correlation.  The statistuical tests answer the question, &quot;If the data is truly random, what is the probability that we would see the patterns that we do?&quot;  If the probability is low enough, the studies assume the results are not random. But that's about all they truly say.   Much of the Evidence Based Medicine curriculum has to do with understanding the statistical tests that are used to evaluate the results of different studies. (See &lt;a href=&quot;https://en.wikipedia.org/wiki/Evidence-based_medicine&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Evidence-based_medicine&lt;/a&gt;.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The recent discovery of gravity waves is an example of your first bullet.  While I'm far from an expert, my understanding is the theoretical physicists (Einstein in particular) built a mathematical model of what gravity waves would look like, if they did actually exists.  That gave experimental physicists some idea of what to look for.  See &lt;a href=&quot;http://www.space.com/31922-gravitational-waves-detection-what-it-means.html&quot; rel=&quot;nofollow&quot;&gt;http://www.space.com/31922-gravitational-waves-detection-what-it-means.html&lt;/a&gt;.  The discovery of the Higgs Boson followed the same pattern. Researchers knew they had found it, only because the signals their equipment gave them matched those predicated by theoretical mathematical models.  See &lt;a href=&quot;https://en.wikipedia.org/wiki/Higgs_boson&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Higgs_boson&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm going to make a broad generalization that I'll leave it to others to judge: When there is a large body of well-tested theory, and mechanisms of action are fairly well known (such as in physics), then research is driven by mathematical theories (because they are relatively cheap to produce and to weed out the inconsistent ones) and confirmed by experiment (because they are relatively expensive to perform).  When mechanisms of action are not that well understood, and even the best theories can't explain the many, real-world exceptions (such as biology &amp;amp; medicine), then research tends to run physical experiments (because the theories are too weak for good predictive power), and them mathematics is used to find patterns in the data (that can help to refine the theory).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The field of chemistry is a nice example of the way these two approaches complement each other. In the days of alchemy, there was no coherent theory of chemistry. So research progressed mostly through experimentation. But that experimentation eventually led to the atomic theory of matter, and statistical mechanics, and for a while research was guided at proving the predictions made by those theories. But atoms can combine in very complicated ways that were mathematically too complicated to compute.  So even though the theories seemed correct in simple situations, for real-world applications, research was mostly guided by experimentation.  But now that computing power is relatively cheap and plentiful, we do see chemical research being guided by theory again: for example, &quot;designer drugs&quot; and computer modeling of macro-molecules.  Now, instead of weeding out ineffective drugs through experiment, many can be eliminated based on computer modeling alone.  See &lt;a href=&quot;http://wyss.harvard.edu/viewpressrelease/179/computer-model-sets-new-precedent-in-drug-discovery&quot; rel=&quot;nofollow&quot;&gt;http://wyss.harvard.edu/viewpressrelease/179/computer-model-sets-new-precedent-in-drug-discovery&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book &lt;em&gt;The Structure of Scientific Revolutions&lt;/em&gt; (Thomas Kuhn, 1962) introduced the term &quot;paradigm shift.&quot; It was a struggle for me to read (and I should probably read it again), but it describes this progression from immature sciences (that are guided by correlation because they have no coherent theory), to more mature sciences that are guided by theory and mathematical modeling.  Then, as experimental techniques get more sophisticated and accurate, shortcomings in the accepted theory become discovered.  This leads to a period of research led by experimentation again, and researchers look for patterns in the exceptions that are found to the original theory. Eventually, a &quot;paradigm shift&quot; happens, the old theory is replaced by a new one that can account for the exceptions, and the field once again moves to a period of research guided by mathematics and theory.&lt;/p&gt;&#xA;" OwnerUserId="3789" LastEditorUserId="3789" LastEditDate="2016-07-02T19:11:38.237" LastActivityDate="2016-07-02T19:11:38.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4979" PostTypeId="2" ParentId="4953" CreationDate="2016-07-02T18:35:02.527" Score="2" Body="&lt;p&gt;Kummer's interest in this question is due to the problem of the location of cubic Gauss sums, analogous to Gauss's determination of the sign of quadratic Gauss sums. In &lt;em&gt;De residuis cubicis disquisitiones nonnullae analyticae&lt;/em&gt; (Some analytic investigations on cubic residues), Crelle 32 (1846), Kummer studies cubic Gauss sums and connects these sums with L-series of cubic characters. He accomplishes this by using Fourier series and a very clever trick - I'm still trying to decipher his ideas, and I don't think this has been taken up by anyone at all. Kummer does not look at all at the simpler case of quadratic Gauss sums, where I think his methods yields a clever evaluation of $L(1,\chi)$ for quadratic characters,  let alone at the (rather trivial) case of the zeta function, but it is rather obvious that Kummer regarded these special cases &#xA;(Leibniz series or Euler's $\zeta(2)$) as less interesting or at least as not novel - at the time he was trying to impress Jacobi and Dirichlet. Kummer does not give any sources, but at the end he remarks that some of his formulas for cubic residues may be proved directly without the analytic methods that he regards as being invented by himself. &lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-07-02T18:35:02.527" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="4980" PostTypeId="2" ParentId="4974" CreationDate="2016-07-03T08:46:41.703" Score="5" Body="&lt;p&gt;One of the greatest mathematical controversies arguably is the one concerning infinitesimals: are they contradictory? are they useful? did Leibniz really use them? is there a connection between Leibniz's infinitesimals and Robinson's? are Euler's infinitesimal procedures sheer madness? are they &lt;em&gt;chimeras&lt;/em&gt;, as alleged by authors ranging from Berkeley to Moigno to Connes? Why did Cantor described them as &quot;abominations&quot;, &quot;paper numbers&quot;, and self-contradictory?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Published&lt;/em&gt; related materials can be found &lt;a href=&quot;http://u.cs.biu.ac.il/~katzmik/infinitesimals.html&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-07-03T12:24:17.157" LastActivityDate="2016-07-03T12:24:17.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4981" PostTypeId="2" ParentId="4974" CreationDate="2016-07-04T18:07:52.377" Score="3" Body="&lt;p&gt;This is not a list of controversies, but you might be interested in Imre Lakatos' book &lt;a href=&quot;https://en.wikipedia.org/wiki/Proofs_and_Refutations&quot; rel=&quot;nofollow&quot; title=&quot;Proofs and Refutations&quot;&gt;Proofs and Refutations&lt;/a&gt; which describes the mathematical progress. For example it provides a discussion of Euler's Polyhedron Formula as a discussion between a teacher and some students. The arguments the students use in this discussion are arguments which have been brought forward by different mathematicians. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book also describes different &quot;solutions&quot; which have been used in the history of mathematics if someone found a counterexample to a &quot;Theorem&quot;.&lt;/p&gt;&#xA;" OwnerUserId="4096" LastEditorUserId="4096" LastEditDate="2016-07-05T16:00:44.927" LastActivityDate="2016-07-05T16:00:44.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4983" PostTypeId="1" AcceptedAnswerId="4984" CreationDate="2016-07-05T12:43:14.560" Score="4" ViewCount="340" Body="&lt;p&gt;Which is Leonard Euler's Paper where he proves this central result? The term 'complex number' does not seem to have been used by him. I'm hoping someone could link me to some page in the Euler archive.&lt;/p&gt;&#xA;" OwnerUserId="4381" LastEditorUserId="205" LastEditDate="2016-07-12T08:06:57.313" LastActivityDate="2016-07-12T08:06:57.313" Title="Euler's paper for the theorem $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $?" Tags="|mathematicians|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4984" PostTypeId="2" ParentId="4983" CreationDate="2016-07-05T13:15:11.000" Score="5" Body="&lt;p&gt;This appears at the end of $\S\,138$ in &lt;em&gt;Introductio&lt;/em&gt; (1748).  In Blanton's translation this appears on page 112, line 5.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-07-05T14:43:56.707" LastActivityDate="2016-07-05T14:43:56.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4985" PostTypeId="1" CreationDate="2016-07-04T09:06:47.623" Score="4" ViewCount="110" Body="&lt;p&gt;I am trying to research how new species have been recorded over time, but specifically in the period before the 1930s.  Where would an academic who had discovered a new species get their discovery ratified and recorded for posterity over that period?  Were there multiple registries in different countries?  How centralized was the process?  I am particularly interested in marine species discovered in the USA as I may have had a relative involved at the time.&lt;/p&gt;&#xA;" OwnerDisplayName="Darius West" LastEditorUserId="84" LastEditDate="2016-07-07T06:52:55.717" LastActivityDate="2016-07-07T06:52:55.717" Title="History of species identification before 1930" Tags="|biology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4986" PostTypeId="2" ParentId="4968" CreationDate="2016-07-05T22:45:15.907" Score="2" Body="&lt;p&gt;@Vicache - Thanks for pointing me in the right direction. I believe I can now answer my own question. The source of my answer is Hardy's book &amp;quot;Ramanujan&amp;quot; (reference below).&lt;/p&gt;&#xA;&lt;p&gt;Chebyshev proved that the Prime Number Theorem  &lt;span class=&quot;math-container&quot;&gt;$\pi (x)$&lt;/span&gt; ~ &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{ln(x)}$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$\vartheta(x)$&lt;/span&gt; ~ &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and he proved that if the limit of  &lt;span class=&quot;math-container&quot;&gt;$\frac{\vartheta(x)}{x}$&lt;/span&gt; exists as &lt;span class=&quot;math-container&quot;&gt;$x\rightarrow\infty$&lt;/span&gt; then the limit is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. However Chebyshev could not prove the existence of the limit.&lt;/p&gt;&#xA;&lt;p&gt;The existence of the limit and hence the Prime Number Theorem and hence &lt;span class=&quot;math-container&quot;&gt;$\vartheta(x)$&lt;/span&gt; ~ &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; was finally proved by Hadamard and de la Vallée Poussin independently in 1896.&lt;/p&gt;&#xA;&lt;p&gt;Reference:&#xA;Hardy, G., &amp;quot;Ramanujan&amp;quot;, pp. 27-28, Chelsea,New York, 3e, 1978.&lt;/p&gt;&#xA;" OwnerUserId="535" LastEditorUserId="136" LastEditDate="2022-04-02T21:34:17.210" LastActivityDate="2022-04-02T21:34:17.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="4987" PostTypeId="2" ParentId="4985" CreationDate="2016-07-06T07:13:36.543" Score="4" Body="&lt;p&gt;Today, to describe a new species, you need to publish an article containing a description of the new species in a scientific journal that adhere to the rules of the International Code of Nomenclature relevant to your species (as there is a different one for animals, plants, bacteria, etc.), you need to name it according to the rules (and if possible recommandations) of said Code of Nomenclature, and most importantly, you need to deposit the name-bearing specimen (a. k. a. the &lt;strong&gt;holotype&lt;/strong&gt;) in a collection (most often in a museum but not necessarily, as long as the collection is correctly curated, and accessible to any other scientists, on request) where it is given an accession number.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, the 1st edition of &lt;a href=&quot;http://www.nhm.ac.uk/hosted-sites/iczn/code/index.jsp?nfv=true&amp;amp;booksection=contents&quot; rel=&quot;nofollow&quot;&gt;International Code of Zoological Nomenclature&lt;/a&gt; (in your case, I suppose) was only published in 1961. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Species described before that date, often had an holotype (as the concept was older than the ICZN) but not necessarily: all that was necessary was the publication process, with a clear illustration if possible or an unambiguous description (note that pre-1900 species descriptions rarely are but the name can be conserved if it is commonly used by the scientific community). Often, if necessary, a holotype can be designated a posteriori either from a series of specimen known to have been used by the original author to describe the species (in which case the new name-bearing specimen is a &lt;strong&gt;lectotype&lt;/strong&gt;) or a brand new specimen if none of the original specimens have been conserved (in which case it is a &lt;strong&gt;neotype&lt;/strong&gt;). Neotypes are designated only when deemed absolutely necessary, as they are by nature more dubious than lectotypes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What could make the species name &lt;a href=&quot;http://www.nhm.ac.uk/hosted-sites/iczn/code/index.jsp?article=11&amp;amp;nfv=true&quot; rel=&quot;nofollow&quot;&gt;unavailable&lt;/a&gt; is if the name was not published in accordance to &lt;a href=&quot;http://www.nhm.ac.uk/hosted-sites/iczn/code/index.jsp?nfv=true&amp;amp;article=8&quot; rel=&quot;nofollow&quot;&gt;Article 8&lt;/a&gt; of ICZN, if the name use another alphabet that the latin alphabet (diacritics, hyphen, apostrophe are not a problem though, but should be corrected a posteriori), if the work that contains the new name didn't make a consistent use of binomial nomenclature, if the name was already considered a synonym at time of publication or if the species name was not published unambiguously as being combined with a genus name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, species names published before 1931 are subjects to the previous rules of name availability (known as Art. 11) and to this requirement (&lt;a href=&quot;http://www.nhm.ac.uk/hosted-sites/iczn/code/index.jsp?article=12&amp;amp;nfv=true&quot; rel=&quot;nofollow&quot;&gt;Art. 12&lt;/a&gt;): &quot;To be available, every new name published before 1931 must satisfy the provisions of Article 11 and must be accompanied by a description or a definition of the taxon that it denotes, or by an indication.&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer one of your secondary questions: the process was not and still is not centralized. There is no global registry of new species names (there is Zoobank, though, which is a registry that is mandatory but only for species published in online-only journals, see Art. 8.5.3 of the ICZN).&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2016-07-06T08:03:43.280" LastActivityDate="2016-07-06T08:03:43.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4988" PostTypeId="2" ParentId="402" CreationDate="2016-07-07T06:03:49.913" Score="0" Body="&lt;p&gt;first 0 was invented by Brahmagupta. many people were not aware of it as he was not so famous as aryabatta and other mathematicians.even he invented the rules and method of cubes and cube root, square and square roots ,integers .&lt;/p&gt;&#xA;" OwnerUserId="4390" LastActivityDate="2016-07-07T06:03:49.913" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4989" PostTypeId="1" AcceptedAnswerId="4991" CreationDate="2016-07-07T18:51:04.273" Score="7" ViewCount="845" Body="&lt;p&gt;Were matrix theory and analysis well-known to physicists circa 1920-1925?  Did physicists make extended use of this theory in that period? The question is related to the discussion in &lt;a href=&quot;https://physics.stackexchange.com/questions/266173/how-did-heisenberg-come-up-with-matrix-mechanics/266185#comment591829_266185.&quot;&gt;How did Heisenberg come up with matrix mechanics? thread on Physics SE&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4376" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2017-02-15T16:54:55.990" Title="Were matrix theory and functional analysis well-known to physicists before the invention of matrix mechanics?" Tags="|quantum-mechanics|mathematical-physics|linear-algebra|" AnswerCount="5" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4990" PostTypeId="2" ParentId="4989" CreationDate="2016-07-07T21:00:04.327" Score="4" Body="&lt;p&gt;I heard a lecture by Heisenberg once, long ago.  (A public lecture at MIT in the early 1970's.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He remarked that he came up with a new, strange, sort of multiplication (that was not commutative).  But then found from his colleagues that mathematicians had already been using it for 100 years.  If Heisenberg wrote memoirs, presumably this is in there, too.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this is support for the side saying matrix multiplication was not well-known to Heisenberg.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-07-07T21:00:04.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4991" PostTypeId="2" ParentId="4989" CreationDate="2016-07-08T04:26:52.193" Score="6" Body="&lt;p&gt;One can probably say that the relevant parts of algebra were &quot;known to experts&quot;, rather than &quot;well-known&quot;, and the relevant parts of functional analysis did not exist at the time, see &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086085710257&quot; rel=&quot;nofollow&quot;&gt;Moore's Axiomatization of Linear Algebra: 1875-1940.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even finite dimensional matrices were not exactly standard teaching item yet, although Cayley gave the definition of matrix multiplication and developed some spectral theory back in 1850s, and Burali-Forti and Marcolongo published a book called Transformations Lineaires in 1912, which opens with:“&lt;em&gt;We briefly set forth the foundations of the general theory of linear systems and linear operators. Generally, these matters are familiar in large part&lt;/em&gt;”. The ideas started percolating among physicists after the use of tensors in Einstein's general relativity, and Weyl's book on it Space, Time and Matter (1918) even introduces axiomatic vector spaces, inner product and congruence-preserving transformations in them. That Born, who in 1904 studied in Göttingen under Hilbert and Minkowski and returned there in 1921, was familiar with matrices and linear transformations is not therefore surprising. Neither rotations nor Lorentz transformations commute. But connecting the idea to infinite matrices was more analogy and physical intuition than applying an established mathematical theory. It is equally not surprising that young Heisenberg was not familiar with it, his 1925 paper does not even mention matrices, see &lt;a href=&quot;https://arxiv.org/pdf/quant-ph/0404009v1.pdf&quot; rel=&quot;nofollow&quot;&gt;Understanding Heisenberg’s ‘Magical’ Paper&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hilbert introduced the &quot;Hilbert space&quot; in connection with integral equations beginning in 1904, but without treating them geometrically. Schmidt in a chapter &quot;Geometry in a Function Space&quot; (1908) wrote &quot;&lt;em&gt;For the geometric significance of the concepts and theorems developed in this chapter I am grateful to Kowalewski. It stands out even more clearly if $A(x)$ is defined, not as a function, but as a vector in a space of infinitely many dimensions&lt;/em&gt;&quot;. Riesz comes closest in his book (1913) on systems of linear equations with infinitely many variables, where he introduces the notion of orthogonal basis. In 1920-1922 Hahn, Banach and Wiener introduced normed linear spaces. However, this work did not study operators on infinite-dimensional spaces much, let alone presented them as infinite matrices. Such themes only emerge in von Neumann's works after 1927, and were motivated by quantum mechanics.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-07-26T04:06:40.837" LastActivityDate="2016-07-26T04:06:40.837" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="4992" PostTypeId="2" ParentId="4989" CreationDate="2016-07-09T01:24:43.060" Score="3" Body="&lt;p&gt;While their reception was rather slow, between Cayley's works from 1840s and 1850s and the much later development of vector spaces and functional analysis, matrices were considered by mathematicians of the late 19th and early 20th centuries in relations to complex numbers, quaternions, bilinear forms, systems of linear equations and determinants. Elements of matrix theory appeared in advanced textbooks and monographs. The book ``Z historie linearni algebry&quot; by Jind\v rich Be\v cva\v r (Matfyzpress, Prague, 2007) mentions e.g. the following which appeared before 1925: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ernesto Pascal: I determinanti. teoria ed applicazioni (1897).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eugen Otto Erwin Netto: Vorlesungen \&quot;uber Algebra (1896); Elementare Algebra. Akademische Vorlesungen f\&quot;ur Studierende der ersten Semester (1904); Die Determinanten (1910).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heinrich Weber: Lehrbuch der Algebra (second edition, 1898-99).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alfred North Whitehead: Treatise on universal algebra (1898).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leopold Kronecker: Vorlesungen \&quot;uber die Theorie der Determinanten (1903).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Salvatore Pincherle: Lezioni di algebra complementare (1906-1909).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxime B\^ocher: Introduction to higher algebra (1907).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cuthbert Edmund Cullis: Matrices and Determinoids (1913, 1918, 1925).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leonard Eugen Dickson: Algebras and their Arithmetics (1923). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Polish, there was an academic  textbook by W\l adys\l aw   Zaj\c aczkowski``The Principles of Higher Algebra” (1884), which,among other things,presented the theory of determinants and of algebraic equations.  There was also a wealth of  material in the monograph of J\'ozef Puzyna on analytic functions (1898, 1900), including resultants  and discriminants, binary forms,  and  the modular group. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was not necessary to go to Goettingen to be exposed to matrix theory before 1925 (apparently, it was not sufficient either- Heisenberg studied there). Indeed, in &#xA; his Nobel lecture in 1954, &quot;The Statistical Interpretation of Quantum Mechanics &quot;, Max Born stated explicitly:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;``This was in the summer of 1925. Heisenberg, plagued by hay fever took&#xA;leave for a course of treatment by the sea and gave me his paper for publication  if  I  thought  I  could  do  something  with  it.&#xA;The significance of the idea was at once clear to me and I sent the manuscript to the &#xA;Zeitschrift f\&quot;ur Physik. &#xA;I could not take my mind off Heisenberg’s&#xA;multiplication  rule,  and  after  a  week  of  intensive  thought  and  trial  I  suddenly&#xA;remembered  an  algebraic  theory  which  I  had  learned  from  my  teacher,&#xA;Professor [Jakob] Rosanes, in Breslau. Such square arrays are well known to mathematicians  and,  in  conjunction  with  a  specific  rule  for  multiplication,  are&#xA;called  matrices. &quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.pdf&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="3434" LastEditDate="2016-07-09T01:53:53.833" LastActivityDate="2016-07-09T01:53:53.833" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="4994" PostTypeId="1" CreationDate="2016-07-11T09:01:32.407" Score="4" ViewCount="369" Body="&lt;p&gt;Well, the question arose in the &lt;a href=&quot;http://chat.stackexchange.com/rooms/71/the-h-bar&quot;&gt;physics chat&lt;/a&gt; when &lt;a href=&quot;https://hsm.stackexchange.com/users/2628/slereah&quot;&gt;slereah&lt;/a&gt; mentioned about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, was there any prominent model that came to light after Bohr and before the actual beginning of Quantum Mechanics in early 20s?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Slereah mentions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;History of physics texts rarely mention models of that era&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What were the developments in this topic during that specific period?&lt;/p&gt;&#xA;" OwnerDisplayName="user1709" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-07-16T18:26:24.397" Title="Was there any atomic model(s) that came between Bohr's and the actual beginning of Quantum Mechanics in early 20s?" Tags="|physics|atomic-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4995" PostTypeId="1" CreationDate="2016-07-11T09:03:45.973" Score="4" ViewCount="73" Body="&lt;p&gt;The Rutherford model that invalidated the plum pudding model occured in 1909, while the first model for quantum mechanics was around 1924, leaving a period of 15 years without an adequate model for the atom, as the Rutherford planetary model did not work out due to the emission of radiations. That period of history for the atomic model is usually skipped in most history of science texts. What models did exist back then to explain the atomic structure? I vaguely recall something about ring-like electrons, but that's about it.&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2016-07-11T09:03:45.973" Title="Atomic models in the 1910's and early 1920's" Tags="|physics|quantum-mechanics|atomic-theory|" AnswerCount="0" CommentCount="3" ClosedDate="2016-07-24T18:56:10.773" ContentLicense="CC BY-SA 3.0" />
  <row Id="4996" PostTypeId="1" CreationDate="2016-07-11T19:57:57.613" Score="1" ViewCount="130" Body="&lt;p&gt;This question pursues the history of this noun and its etymons; compare with &lt;a href=&quot;https://english.stackexchange.com/a/222580/50720&quot;&gt;ELU&lt;/a&gt; and &lt;a href=&quot;https://french.stackexchange.com/q/16816/1995&quot;&gt;FSE&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.etymonline.com/index.php?term=physic&amp;amp;allowed_in_frame=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;physic (n.)&lt;/a&gt;&lt;br /&gt;&#xA;[1.]    c. 1300, &lt;em&gt;fysike&lt;/em&gt;, &amp;quot;art of healing, &lt;strong&gt;medical science&lt;/strong&gt;,&amp;quot; also &amp;quot;natural science&amp;quot; (c. 1300),&lt;br /&gt;&#xA;[2.] from Old French &lt;em&gt;fisike&lt;/em&gt; &lt;strong&gt;&amp;quot;natural science, art of healing&amp;quot;&lt;/strong&gt; (12c.)&lt;br /&gt;&#xA;[3.] and directly from Latin &lt;em&gt;physica&lt;/em&gt; (fem. singular of &lt;em&gt;physicus&lt;/em&gt;) &lt;strong&gt;&amp;quot;study of nature,&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;[4.] from Greek &lt;em&gt;physike&lt;/em&gt; (&lt;em&gt;episteme&lt;/em&gt;) &lt;strong&gt;&amp;quot;(knowledge) of nature,&amp;quot;&lt;/strong&gt;&lt;br /&gt;&#xA;from fem. of &lt;em&gt;physikos&lt;/em&gt; &amp;quot;pertaining to nature,&amp;quot; from &lt;em&gt;physis&lt;/em&gt; &amp;quot;nature,&amp;quot;&lt;br /&gt;&#xA;from &lt;em&gt;phyein&lt;/em&gt; &amp;quot;to bring forth, produce, make to grow&amp;quot; (related to phyton &amp;quot;growth, plant,&amp;quot; phyle &amp;quot;tribe, race,&amp;quot; phyma &amp;quot;a growth, tumor&amp;quot;)&lt;br /&gt;&#xA;from PIE root &lt;em&gt;*bheue-&lt;/em&gt; &amp;quot;to be exist, grow&amp;quot; (see &lt;a href=&quot;http://www.etymonline.com/index.php?term=be&amp;amp;allowed_in_frame=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;be&lt;/a&gt;).&lt;br /&gt;&#xA;[…]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Historically, how was 4 connected to 1-3?&lt;/p&gt;&#xA;&lt;p&gt;I do not understand the underlying notions because I (in 2016) cannot empathise with the perspective of the Ancient Greeks, Romans, or Old French of the equivalence of (knowledge of) nature, natural science, healing, medical science, and modern physics.&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-07-13T22:28:02.760" Title="Why were nature, natural science, physics, medicine all considered the same subject and so defined with the same noun?" Tags="|terminology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4997" PostTypeId="1" CreationDate="2016-07-12T12:20:06.953" Score="0" ViewCount="159" Body="&lt;p&gt;I read somewhere that, but unfortunately I cannot retrace where, can you help me find a reliable source?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually, they said a sad story, hardly believable: They stated that Einstein had circulated a different version of the formula, with a 1/2 factor, but someone accused him of plagiarism, because that formula had already been put forward by someone else. &#xA;Later, in the printed version, he changed the formula to the current form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you tell me if that is true and who had suggested the 1/2 formula and who accused him of plagiarism?&lt;/p&gt;&#xA;" OwnerDisplayName="user646" LastEditorUserId="26" LastEditDate="2016-07-12T16:04:16.147" LastActivityDate="2016-07-12T16:04:16.147" Title="Was Einstein original formula E =1/2 m c^2?" Tags="|theoretical-physics|relativity-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="4998" PostTypeId="1" AcceptedAnswerId="4999" CreationDate="2016-07-12T18:48:05.960" Score="2" ViewCount="910" Body="&lt;p&gt;Was Newton aware of the fact that his theory of gravity did not account for the motion of Mercury?&lt;/p&gt;&#xA;" OwnerUserId="4413" LastEditorUserId="4413" LastEditDate="2016-07-12T22:21:25.427" LastActivityDate="2016-07-13T22:14:25.383" Title="Was Newton aware his theory of gravity had flaws?" Tags="|physics|experimental-physics|gravity|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="4999" PostTypeId="2" ParentId="4998" CreationDate="2016-07-12T19:39:20.677" Score="4" Body="&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury&quot; rel=&quot;nofollow&quot;&gt;Perihelion precession of Mercury&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mercury deviates from the precession predicted from these Newtonian effects. This anomalous rate of precession of the perihelion of Mercury's orbit was first recognized in 1859 as a problem in celestial mechanics, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Urbain_Le_Verrier&quot; rel=&quot;nofollow&quot;&gt;Urbain Le Verrier&lt;/a&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-07-12T19:39:20.677" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5000" PostTypeId="2" ParentId="402" CreationDate="2016-07-13T13:06:13.007" Score="3" Body="&lt;p&gt;One should distinguish between the &lt;em&gt;notational&lt;/em&gt; zero (i.e., as placeholder in a positional system for representing numbers) and the &lt;em&gt;algebraic&lt;/em&gt; zero (i.e., as the neutral element of addition). We use the same symbol for both (and with good reason), but they were introduced independently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, the first mention of the algebraic zero is in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta&quot; rel=&quot;nofollow&quot;&gt;Brāhmasphuṭasiddhānta&lt;/a&gt; (ca. CE 628) of the Indian mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Brahmagupta&quot; rel=&quot;nofollow&quot;&gt;Brahmagupta&lt;/a&gt;, specifically in the section &lt;em&gt;Kuttaka&lt;/em&gt; (&quot;pulverizer&quot;), Rule §19. Quoting from &lt;a href=&quot;https://archive.org/details/algebrawitharith00brahuoft&quot; rel=&quot;nofollow&quot;&gt;Thomas Colebrooke's translation of 1817&lt;/a&gt; (Section II, 31 on page 339):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] The sum of two affirmative quantities is affirmative; of two negative&#xA;  is negative; of an affirmative and a negative is their difference; or, if they be equal, nought. The sum of cipher and negative is negative; of affirmative and nought is positive; of two ciphers is cipher.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Brahmagupta uses the two terms &lt;em&gt;nought&lt;/em&gt; (Sanskrit खम्, /kham/) and &lt;em&gt;cipher&lt;/em&gt; (Sanskrit शून्यम्, /śūnyam/) indiscriminately.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For those interested, the two sentences &lt;a href=&quot;http://gretil.sub.uni-goettingen.de/gretil/1_sanskr/6_sastra/8_jyot/brsphutu.htm&quot; rel=&quot;nofollow&quot;&gt;in transliteration&lt;/a&gt; read&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;dhanayos dhanam ṛṇam ṛṇayos dhana-ṛṇayos antaram sama-aikyam #kham/&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;ṛṇam aikyam ca dhanam ṛṇa-dhana-#śūnyayos #śūnyayos #śūnyam//&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4416" LastActivityDate="2016-07-13T13:06:13.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5001" PostTypeId="1" AcceptedAnswerId="5004" CreationDate="2016-07-13T17:09:21.337" Score="2" ViewCount="182" Body="&lt;p&gt;Hippocrates was able to construct the quadrature of three different lunes.  Euler found two more squarable lunes.  Tschebatorew and Durodnow showed that these five are the only squarable lunes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although Archimedes quadrature of the parabolic segment uses the method of exhaustion, the resulting formula, $\frac43 T$, does enable the finite construction of the square, so I will assume that this is a legitimate Euclidean quadrature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: In antiquity, did anyone manage to square any other curved figures in the plane using only a finite straightedge and compass construction?  &lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-07-22T03:16:05.623" Title="History of the quadrature of curvelinear figures prior to the middle ages" Tags="|euclidean-geometry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5002" PostTypeId="2" ParentId="4998" CreationDate="2016-07-13T22:08:06.693" Score="3" Body="&lt;p&gt;Newton probably believed that his theory is correct and exact. He could not know of the very small anomaly of Mercury. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there were some much more serous problems. Gravity law and the laws of motion only give a law (a differential equation) of motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ultimate test of the theory&#xA;would be solving this equation and comparing the result with observations.&#xA;But the equation can be explicitly solved only for the two-bodies case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When solving for two bodies (Sun and a planet, or a planet and a satellite) one neglects the influence of all other bodies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most conspicuous system which is&#xA;not adequately described by the two body equation is Sun-Earth-Moon. Newton worked very hard to explain the observed motion of the Moon by his gravitation theory but could explain only some gross features. (Technically: the so-called second inequality, but not the third). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This mathematical problem (to explain the visible motion of the Moon on the basis of the law of gravity) occupied &#xA;many best mathematicians of XVIII century, and some serious doubts in the law of gravitation were expressed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally a solution which predicts the motion of the Moon with sufficient precision (sufficient for the needs of navigation) was achieved in the second half of XVIII century. (And two people, Tobias Mayer and Leonard Euler&#xA;got their share in the Longitude Prize:-) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was one of the first great successes of Newton's &#xA;theory of gravitation, but it happened long after his death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next great success came when the orbits of small planets were computed, and&#xA;the most famous achievement: the discovery of a new planet (Neptune) from the&#xA;perturbations its gravity causes to Uranus's orbit. &#xA;Only at that point all doubts disappeared and one could say that&#xA;the gravity law was firmly established.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another great test was the shape of the Earth. Newton's gave a rough, essentially qualitative prediction. It was verified by very precise measurements in XVIII century. But a complete solution of the related equation (equation of equilibrium of rotated fluid) was not achieved until 20th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-07-13T22:14:25.383" LastActivityDate="2016-07-13T22:14:25.383" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5003" PostTypeId="2" ParentId="4996" CreationDate="2016-07-13T22:28:02.760" Score="6" Body="&lt;p&gt;As the knowledge grows various disciplines separate from each other.&#xA;In the beginning there was only one &quot;science&quot; and it was called philosophy.&#xA;Mathematics and astronomy separated very early. All the rest was &quot;physics&quot;, the study of nature. Medicine also separated early, but not completely, because very little specifically medical knowledge was available. Until the modern era many scientists also practiced medicine, and such things as astrology were considered indispensable for medical education. We see a lot of traces of this situation in the language: even in our time a mathematician or a physicist is called&#xA;&quot;Doctor of Philosophy (PhD)&quot;. But if you hear just &quot;Doctor&quot; you may assume that the person probably practices medicine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you noticed the word &quot;physics&quot; in Greek means just the knowledge of the nature. But nature includes not only the nature of stars and the laws of motion, but also the human and animal and plant body.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-07-13T22:28:02.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5004" PostTypeId="2" ParentId="5001" CreationDate="2016-07-14T04:02:52.073" Score="5" Body="&lt;p&gt;Pappus in &lt;a href=&quot;http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Pappus/Bookiv/Pappus.iv.21-25/Pappus.iv.21_25.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Collection, book IV&lt;/a&gt; &quot;squared&quot; the Archimedean spiral, the spiral is to the circle enclosing it as 1:3. The curve is obtained by composing uniform linear and circular motions in the plane, Archimedes himself in On Spirals attributes the discovery of it to his friend Conon. Some care is needed with calling this &quot;squaring with straightedge and compass&quot;. Neither parabola nor Archimedean spiral can be produced with straightedge and compass, so they would have to be drawn in the plane as given. However, &lt;a href=&quot;https://en.wikipedia.org/wiki/On_Spirals#Squaring_the_circle&quot; rel=&quot;nofollow noreferrer&quot;&gt;once the spiral is drawn&lt;/a&gt; it becomes possible to square &lt;em&gt;any&lt;/em&gt; circle with straightedge and compass as well, as Archimedes proved in On Spirals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first such curve was known even earlier, the quadratrix. Originally it was devised to trisect angles, and termed the trisectrix, by Hippias (c. 450 BC), a sophist derided by Plato in his dialogues for excessive vanity. About a century later Dinostratus adapted it to squaring circles, and renamed it into the quadratrix, that is all we know of him. Since the &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratrix_of_Hippias#Squaring_of_the_circle&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadratrix can be used to square circles&lt;/a&gt;, by design there are parts attached to it that can also be &quot;squared with straightedge and compass&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, &lt;a href=&quot;https://books.google.com/books?id=xUicBgAAQBAJ&amp;amp;pg=PA537&amp;amp;lpg=PA537&amp;amp;dq=area+of+ellipse+antiquity&amp;amp;source=bl&amp;amp;ots=BkC31TyKcZ&amp;amp;sig=igTw7lAdHJtuHsbwKfzaeUU-eJ8&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwifupz6hvLNAhVmwYMKHZGDCnoQ6AEIITAB#v=onepage&amp;amp;q=area%20of%20ellipse%20antiquity&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes states in On Conoids and Spheroids&lt;/a&gt; that an ellipse is to the circle with its major axis for diameter, as its minor axis is to the diameter, which gives $\pi ab$ for semi-axes $a,b$ in formulaic modernization. But we would have to say that ellipse is not &quot;squared with straightedge and compass&quot;, because the said circle can't be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we have a kind of hierarchy: lunes are both produced and squared by straightedge and compass, parabola must be drawn but can then be squared directly, quadratures of ellipse and spiral directly reduce only to the circle quadrature, but in the case of the spiral the circle can then be squared, if we are allowed to use it one more time. Parabola is a degenerate conic, circle and ellipse are non-degenerate ones, and Archimedean spiral and quadratrix are transcendental. But the pattern breaks down with hyperbola, areas under it require logarithms, and therefore do not reduce to the circle quadrature, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2981/how-did-saint-vincent-prove-the-logarithmic-property-of-areas-under-hyperbolas/3000#3000&quot;&gt;How did Saint Vincent prove the logarithmic property of areas under hyperbolas?&lt;/a&gt; Still, St. Vincent does prove the &quot;logarithmic property&quot; by Archimedean methods. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Greeks also knew some algebraic curves of degree higher than two, e.g. cubic cissoid of Diocles and quartic conchoid of Nicomedes, Newton's favorite, but those were only &quot;squared&quot; in modern times. For example, the area under the &lt;a href=&quot;http://mathworld.wolfram.com/CissoidofDiocles.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;cissoid by Huygens and Wallis in 1658&lt;/a&gt;, even though like ellipse it reduces to circle quadrature. In modern terms, the &quot;real&quot; pattern is complicated not only by the structure of anti-derivatives of algebraic functions, but also because some definite integrals are expressible in square radicals even when anti-derivatives are not.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-07-22T03:16:05.623" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5005" PostTypeId="1" CreationDate="2016-07-14T11:09:34.337" Score="2" ViewCount="96" Body="&lt;p&gt;The &lt;em&gt;Chinese Remainder Theorem&lt;/em&gt; is one of the fundamental theorems in modular arithmetic. As far as I know, this terminology for the theorem is due to the fact that the Chinese mathematicians were the first in developing it (as suggested by remaining works of the 3rd century of Sun Zia and the book &lt;em&gt;Mathematical Treatise in Nine Sections&lt;/em&gt; of 1247 of Qin Jiushao).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In parallel, the theorem was rediscovered by western mathematicians in the modern age. As far as I read, Gauss doesn't refer to it as the Chinese Remainder Theorem in his &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt;, so I am curious about when this terminology became commonplace in mathematics as it is today. Hence the questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Which is the first reference using the terminology &quot;Chinese Remainder Theorem&quot; for this theorem?&lt;/li&gt;&#xA;&lt;li&gt;At which point can we consider that this was the common place terminology of this theorem?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4422" LastActivityDate="2016-07-14T11:09:34.337" Title="Which is the first reference using the terminology &quot;Chinese Remainder Theorem&quot; for this theorem?" Tags="|mathematics|terminology|number-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5006" PostTypeId="1" AcceptedAnswerId="5010" CreationDate="2016-07-14T13:28:08.760" Score="4" ViewCount="1526" Body="&lt;p&gt;I just want to know the history of the derivative. Whenever I Google for it, I find the history of calculus or the tangent of a curve. However, they barely touch upon what happened before Leibniz and Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I cannot find is:&#xA;How was the tangent of a curve defined before the derivative? I am looking for a mathematical definition.&lt;/p&gt;&#xA;" OwnerUserId="2666" LastEditorUserId="2666" LastEditDate="2016-07-20T12:02:29.410" LastActivityDate="2016-08-09T13:00:54.850" Title="History of the derivative/tangent of a curve" Tags="|mathematics|calculus|differential-geometry|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5007" PostTypeId="2" ParentId="5006" CreationDate="2016-07-14T14:33:23.673" Score="4" Body="&lt;p&gt;Carl B. Boyer's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486605094&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;The History of the Calculus and Its Conceptual Development&lt;/strong&gt;&lt;/a&gt; is especially detailed on early work relating to calculus (it pretty much fizzles out in the mid 1800s, though), and for curves you'll also want to look at Boyer's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0486438325&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;History of Analytic Geometry&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2016-07-14T14:33:23.673" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5008" PostTypeId="1" AcceptedAnswerId="5063" CreationDate="2016-07-14T16:43:11.933" Score="4" ViewCount="510" Body="&lt;p&gt;Where in Gauss's works does he derive &quot;Gauss's Law&quot;? Or is &quot;Gauss's Law&quot; named after Gauss for a different reason?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-07-14T16:59:55.763" LastActivityDate="2016-08-03T17:31:06.323" Title="Where in Gauss's works does he derive &quot;Gauss's Law&quot;?" Tags="|physics|electromagnetism|mathematical-physics|electricity|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5010" PostTypeId="2" ParentId="5006" CreationDate="2016-07-15T05:36:24.417" Score="6" Body="&lt;p&gt;Keep in mind that the repertoire of curves available before calculus was very limited. The earliest known definition is given for circles by Euclid in &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookIII/bookIII.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;book III of Elements&lt;/a&gt;:&quot;&lt;em&gt;A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle&lt;/em&gt;&quot;. In other words, the tangent is a line that meets a curve but stays &quot;on one side&quot; of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may seem narrow, but it works for convex curves like conic sections that Greeks focused on, and even the Archimedean spiral. It was good enough for Euclid to &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/elements/bookIII/propIII16.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;demonstrate Proposition III.16&lt;/a&gt; whose Corollary states:&quot;&lt;em&gt;From this it is clear that the straight line drawn at right angles to the diameter of a circle from its end touches the circle&lt;/em&gt;&quot;. The proposition itself, and reasoning in its demonstration, are quite interesting:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What Euclid says is that the angle between a circle and its tangent is smaller than any rectilinear angle, i.e. it is &quot;infinitesimal&quot; relative to them. Such angles came to be called horn angles, and they are the first example of magnitudes violating the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedean_property&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedean axiom (of Eudoxus)&lt;/a&gt;, and Euclid's own treatment of &lt;a href=&quot;https://mathoverflow.net/questions/56286/what-is-the-state-of-research-on-horn-angles&quot;&gt;&quot;magnitudes&quot; as always having ratios&lt;/a&gt; to each other. &lt;a href=&quot;https://mathoverflow.net/questions/56286/what-is-the-state-of-research-on-horn-angles&quot;&gt;Horn angles&lt;/a&gt; and their relation to tangency were widely discussed during  Renaissance and in 17th century, motivating some calculus ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time of Archimedes Greeks already knew how to draw tangents to all conic sections, and Apollonius's demonstrations in Conica are along the lines of Elements III.16. But &quot;touches&quot; is good enough informally as long as one wants to just &lt;em&gt;find&lt;/em&gt; tangents without defining them. Archimedes possibly first used kinematic reasoning to find tangents to his spiral, and the idea was developed in 17th century, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3024/who-discovered-the-power-rule-for-derivatives&quot;&gt;Who discovered the power rule for derivatives?&lt;/a&gt; Descartes and others also had an algebraic way of finding tangents before Newton and Leibniz, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/1963/is-there-a-lost-calculus/1968#1968&quot;&gt;Is there a &amp;#39;lost calculus&amp;#39;?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-07-18T22:25:18.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5011" PostTypeId="1" AcceptedAnswerId="5013" CreationDate="2016-07-15T08:35:54.800" Score="0" ViewCount="225" Body="&lt;p&gt;I am already aware of the notation differences. But is it the only criterion?&lt;/p&gt;&#xA;" OwnerUserId="4431" LastActivityDate="2020-11-13T03:59:18.420" Title="Based on what criteria one could say Leibniz &quot;invented&quot; the differential calculus?" Tags="|calculus|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5012" PostTypeId="1" AcceptedAnswerId="5016" CreationDate="2016-07-15T09:45:51.767" Score="9" ViewCount="2701" Body="&lt;p&gt;I recently learned that &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama&quot; rel=&quot;nofollow&quot;&gt;Madhava of Kerala&lt;/a&gt; (&lt;em&gt;c&lt;/em&gt;.1340–&lt;em&gt;c&lt;/em&gt;.1425) was the first to discover &lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80&quot; rel=&quot;nofollow&quot;&gt;the following formula for $\pi$&lt;/a&gt;:&#xA;$$\frac{\pi}4\ =\ 1 - \frac13+\frac15 - \frac17 + \frac19 - \frac1{11} + \cdots$$&#xA;The formula was rediscovered by James Gregory (1671) and Gottfried Leibniz (1673) and is named after both of them. But Madhava discovered not one but two formulas for $\pi$, the following converging much more quickly than the one above:&#xA;$$\pi\ =\ \sqrt{12}\left(1 - \frac1{3\cdot3} +\frac1{5\cdot3^2}  - \frac1{7\cdot3^3}  +  \cdots\right)$$&#xA;In fact we now know that&#xA;$$\tan^{-1}\theta\ =\ \theta - \frac{\theta^3}3 + \frac{\theta^5}5 - \frac{\theta^7}7 + \cdots$$&#xA;and the first and second formulas can be obtained by putting $\theta=1$ and $\theta=\frac1{\sqrt3}$ respectively into the above Maclaurin series expansion of $\arctan\theta$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I would like to know is how Madhava discovered his formulas for $\pi$ without knowing anything about Maclaurin series. It is surely inconceivable that he knew as much calculus as Newton or Leibniz, the two men credited with the invention of calculus as a mathematical tool two and a half centuries after his death. What, then, inspired him to his discoveries?&lt;/p&gt;&#xA;" OwnerUserId="4433" LastEditorUserId="4433" LastEditDate="2016-07-17T00:41:34.717" LastActivityDate="2021-11-04T02:13:42.957" Title="Madhava and $\pi$" Tags="|mathematics|discoveries|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5013" PostTypeId="2" ParentId="5011" CreationDate="2016-07-15T10:24:24.127" Score="1" Body="&lt;p&gt;Based on written/published records.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz was &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_calculus#Newton_and_Leibniz&quot; rel=&quot;nofollow&quot;&gt;one of the two&lt;/a&gt; that discovered &lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy&quot; rel=&quot;nofollow&quot;&gt;independently&lt;/a&gt; the infinitesimal calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First written record from Newton: &lt;a href=&quot;https://en.wikipedia.org/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas&quot; rel=&quot;nofollow&quot;&gt;De analysi per aequationes numero terminorum infinitas&lt;/a&gt; (written: 1669; published: 1711).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First published record from Newton: &lt;a href=&quot;https://books.google.it/books?id=mxhfAAAAcAAJ&amp;amp;pg=PA163&quot; rel=&quot;nofollow&quot;&gt;Tractatus de Quadratura Curvarum&lt;/a&gt; (1704).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First published paper from Leibniz: &lt;a href=&quot;https://en.wikipedia.org/wiki/Nova_Methodus_pro_Maximis_et_Minimis&quot; rel=&quot;nofollow&quot;&gt;Nova Methodus pro Maximis et Minimis&lt;/a&gt; (1684).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-07-15T12:08:20.860" LastActivityDate="2016-07-15T12:08:20.860" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5014" PostTypeId="2" ParentId="5012" CreationDate="2016-07-15T13:49:04.027" Score="6" Body="&lt;p&gt;This may be of interest...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Ranjan Roy&lt;/em&gt;, MR 1081274 &lt;a href=&quot;http://dx.doi.org/10.2307/2690896&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The discovery of the series formula for $\pi$ by Leibniz, Gregory and Nilakantha&lt;/strong&gt;&lt;/a&gt;, &lt;em&gt;Math. Mag.&lt;/em&gt; &lt;strong&gt;63&lt;/strong&gt; (1990), no. 5, 291--306.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The series referred to is&#xA;$$&#xA;\frac{\pi}4\ =\ 1 - \frac13+\frac15 - \frac17 + \frac19 - \frac1{11} + \cdots&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He says the Indian discovery (author not definitely known, sometimes attributed to Nilakantha) was the result of an effort to rectify the circle. &lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2016-07-15T13:54:26.147" LastActivityDate="2016-07-15T13:54:26.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5015" PostTypeId="1" AcceptedAnswerId="5021" CreationDate="2016-07-15T17:31:34.550" Score="8" ViewCount="3855" Body="&lt;p&gt;Do we have any idea of how Newton proved the generalised binomial theorem?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastEditorUserId="55" LastEditDate="2016-09-13T01:20:37.253" LastActivityDate="2016-09-13T01:20:37.253" Title="How did Newton prove the generalised binomial theorem?" Tags="|mathematicians|calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5016" PostTypeId="2" ParentId="5012" CreationDate="2016-07-15T17:32:41.230" Score="8" Body="&lt;p&gt;According to &lt;a href=&quot;http://www.storyofmathematics.com/indian_madhava.html&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although almost all of Madhava's original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, you appear to be mistaken in your assumption that Madhava lacked knowledge of what we now call the Maclaurin series for trigonometric functions.  The &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_series&quot; rel=&quot;noreferrer&quot;&gt;wikipedia page on Madhava series&lt;/a&gt; gives details of Madhava's interesting derivation of the series expansions for the sine, cosine, and arctangent functions as well as the two formula for $\pi$ you have noted.  An account in Madhava's &quot;own words&quot; is provided together with a rendering in modern notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80&quot; rel=&quot;noreferrer&quot;&gt;page for $\frac{\pi}{4} = 1 - \frac13 + \frac15 - \dots$&lt;/a&gt; also notes that this expansion has more recently become referred to as the Madhava-Leibniz series.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-07-15T17:32:41.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5017" PostTypeId="2" ParentId="5001" CreationDate="2016-07-15T19:10:24.417" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zjk2F.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zjk2F.jpg&quot; alt=&quot;Andean squaring of circle&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Andeans used this system since 5,000 y.a. diagonals&#xA;were taken as geometric integers. It is present in &#xA;Ancient Egypt and Caral.&lt;/p&gt;&#xA;" OwnerUserId="4436" LastActivityDate="2016-07-15T19:10:24.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5018" PostTypeId="2" ParentId="4994" CreationDate="2016-07-15T21:34:29.197" Score="3" Body="&lt;p&gt;To my knowledge the first documented atomic model considering Planck’s action quantum was formulated by the Austrian Arthur Erich Haas (1884-1941). He hypothesized in 1910 a relation between the action quantum and the size of atoms. &#xA;A substantial initiation progress was postulated by John William Nicholson (1881-1955) (Cambridge) who in 1912 formulated the quantization of angular momentum and conjectured an explanation of Ritz’s series rule of hydrogen spectra. (shortly before Niels Bohr’s model).&#xA;In conclusion, Niels Bohr wasn’t the first to formulate a relationship between the quantum of action and the atomic structure. However Bohr soon became friend of Nicholson who must have inspired him. The whole story of the many try-and-error attempts of Bohr’s atomic model are documented in a trilogy published in “The Philosophical magazine” July 26, 1913.&lt;/p&gt;&#xA;" OwnerUserId="4438" LastActivityDate="2016-07-15T21:34:29.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5019" PostTypeId="1" AcceptedAnswerId="5020" CreationDate="2016-07-16T04:53:24.540" Score="0" ViewCount="78" Body="&lt;p&gt;I remember that not long ago wikipedia had somewhere info regarding the discovery of that phenomenon, but apparently all reference hasbeen removed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Searching the web I found a serbian &lt;a href=&quot;http://www.doiserbia.nb.rs/img/doi/0354-7310/2007/0354-73100704100B.pdf&quot; rel=&quot;nofollow&quot;&gt;site&lt;/a&gt; claiming that  Tesla discovered it in 1892, and published his finding in 1897. Tha is confirmend by Miles Mathis, but apparently he is not a reliable source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you have any precise and reliable details on the issue?&lt;/p&gt;&#xA;" OwnerDisplayName="user646" LastActivityDate="2016-07-16T06:19:39.840" Title="When was brehmsstrahlung discovered?" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5020" PostTypeId="2" ParentId="5019" CreationDate="2016-07-16T06:19:39.840" Score="1" Body="&lt;p&gt;Seems around 1896 by Tesla from here &lt;a href=&quot;http://pubs.rsna.org/doi/full/10.1148/rg.284075206&quot; rel=&quot;nofollow&quot;&gt;http://pubs.rsna.org/doi/full/10.1148/rg.284075206&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4442" LastActivityDate="2016-07-16T06:19:39.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5021" PostTypeId="2" ParentId="5015" CreationDate="2016-07-16T17:39:58.403" Score="11" Body="&lt;p&gt;According to William Dunham in his &lt;em&gt;Journey Through Genius&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Newton's Binomial Theorem is] not a &quot;theorem&quot; in the sense of Euclid or Archimedes in that Newton did not furnish a complete proof. Yet his insight and intuition served him well enough to devise a germane formula... Further, his inherent belief in the persistence of patterns suggested to him that the formula that correctly generated coefficients for binomial powers like $(a + b)^2$ or $(a + b)^3$ should work just as well for powers like $(a + b)^{\frac12}$ or $(a + b)^{-3}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although Newton did not prove this result, we do have some idea of how he arrived at it.  Newton's explanation is presented in a pair of letters sent to Leibniz dated 1676 and delivered via Henry Oldenburg of the Royal Society. In it, he explains how he arrived at his result by a somewhat indirect route.  He was investigating the areas under the curves $y = (1-x^2)^{\frac{n}{2}}$ from $0$ to $x$ for $n = 0, 1, 2, 3,\dots$. These are easy to calculate for even $n$.  By observing patterns and interpolating, Newton was able to guess the answer for odd values of $n$.  Then he realised that he could get the same answers by expressing $(1-x^2)^{\frac{n}{2}}$ as an infinite series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton wrote :&#xA;$$(P + PQ)^{\frac{m}{n}} = P^{\frac{m}{n}} + \frac{m}{n}AQ + \frac{m-n}{2n}BQ + \frac{m-2n}{3n}CQ + \dots$$ &#xA;where $P+PQ$ is the binomial to be considered; where $\frac{m}{n}$ is the power to which we shall raise the binomial &quot;whether that power is integral or (so to speak) fractional, whether positive or negative&quot;; and where $A,B,C, \dots$ represent the immediate preceding term in the expansion. (See TL;DR)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may look unfamiliar when compared to the modern form.  When you factor out the $P^{\frac{m}{n}}$ term from the right hand side and cancel you arrive at the more familiar form :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(1+Q)^{\frac{m}{n}} = 1 + \frac{m}{n}Q + \frac{(m/n)(m/n-1)}{2}Q^2 + \frac{(m/n)(m/n-1)(m/n-2)}{2\times3}Q^3 + \frac{(m/n)(m/n-1)(m/n-2)(m/n-3)}{2\times3\times4}Q^4 + \dots$$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, in Newton's notation, &#xA;$$A = P^{\frac{m}{n}}$$&#xA;$$B = \frac{m}{n}AQ = \frac{m}{n}P^{\frac{m}{n}}Q$$ &#xA;$$C = \frac{(m-n)}{2n} BQ = {\frac{(m-n)m}{(2n)n}}P^{\frac{m}{n}}Q^2 = {\frac{(m/n)(m/n-1)}{2}}P^{\frac{m}{n}}Q^2$$&#xA;etc.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-07-17T16:49:59.573" LastActivityDate="2016-07-17T16:49:59.573" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5022" PostTypeId="1" CreationDate="2016-07-17T03:48:22.847" Score="6" ViewCount="398" Body="&lt;p&gt;I would like to know some references (if any) for the claim that Grothendieck didn't like the idea of elementary topos.&lt;/p&gt;&#xA;" OwnerUserId="3857" LastEditorUserId="4251" LastEditDate="2021-09-11T09:44:18.350" LastActivityDate="2021-09-11T09:44:18.350" Title="Grothendieck and elementary topos" Tags="|mathematics|mathematicians|reference-request|algebraic-geometry|grothendieck|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5023" PostTypeId="2" ParentId="2656" CreationDate="2016-07-17T04:22:47.977" Score="8" Body="&lt;p&gt;“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”&lt;/p&gt;&#xA;&lt;p&gt;The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry.&lt;/p&gt;&#xA;&lt;p&gt;The lecturer was Claude Chevalley, and the date was december 12th 1955.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Sc7OP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Sc7OP.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.neverendingbooks.org/grothendiecks-cafe&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.neverendingbooks.org/grothendiecks-cafe&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.landsburg.com/grothendieck/mclarty1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The rising sea, Mclarty&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3857" LastEditorUserId="5031" LastEditDate="2023-01-22T08:02:16.007" LastActivityDate="2023-01-22T08:02:16.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5024" PostTypeId="1" CreationDate="2016-07-17T17:38:19.790" Score="0" ViewCount="115" Body="&lt;p&gt;After reading this &lt;a href=&quot;https://mathoverflow.net/questions/89103/who-first-cared-about-singular-points&quot;&gt;question&lt;/a&gt; and the answers to it, I am interested o know who were the first mathematicians who started classifying singular points of curves: i.e. different kind of nodes, of cusps etc., and when it reached the stage of the classification we use today? For example, a classification already appears in &quot;The elements of the differential calculus&quot; from 1839 of &lt;a href=&quot;https://archive.org/details/elementsofdiffer00youn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Young&lt;/a&gt;, but it seems to draw its conclusions from an existing investigation. If so, where did it appear earlier?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you, &#xA;David&lt;/p&gt;&#xA;" OwnerUserId="3232" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-07-22T23:05:38.147" Title="on the classification of singular points" Tags="|terminology|reference-request|calculus|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5025" PostTypeId="1" AcceptedAnswerId="5029" CreationDate="2016-07-17T18:12:26.000" Score="7" ViewCount="865" Body="&lt;p&gt;I am reading parts of Euclid’s &lt;em&gt;Elements&lt;/em&gt; and I am surprised, rightly or wrongly, to see that Euclid did not recognize that a conditional is logically equivalent to its contrapositive form.  Indeed, one often sees one proposition immediately followed by another proposition giving its contrapositive and a separate proof (usually by a &lt;em&gt;reductio&lt;/em&gt; argument on the original statement) .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stoicism&quot;&gt;The Stoics&lt;/a&gt;, who came just after Euclid, were the first to explicitly recognise &lt;a href=&quot;https://en.wikipedia.org/wiki/Modus_tollens&quot;&gt;&lt;em&gt;modus tollens&lt;/em&gt;&lt;/a&gt;  as valid argument form, and the equivalence of a conditional and its contrapositive follows from the application of &lt;em&gt;modus tollens&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the equivalence of the two forms is evident from their truth tables, I was equally surprised to read that truth tables have their origins in the late 19th century.  (According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Truth_table#History&quot;&gt;the wiki entry on truth tables&lt;/a&gt; it was the logician C.S. Pierce who is credited with first formulating truth tables, including them in an unpublished manuscript dated 1893.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So that “narrows” down the recognition of the equivalence of the two forms to sometime between Euclid and Pierce - a period of about 2200 years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt; : When did mathematicians first use the contrapositive form to prove a conditional?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-01-08T05:44:23.223" Title="When did mathematicians first use the contrapositive form to prove a conditional statement?" Tags="|mathematics|mathematical-logic|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5026" PostTypeId="1" CreationDate="2016-07-17T18:29:15.937" Score="-1" ViewCount="276" Body="&lt;p&gt;Who discovered this method of finding the number sequence of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Enter &lt;span class=&quot;math-container&quot;&gt;$\sin(0.72)$&lt;/span&gt; into a pocket calculator and divide it by 4:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0.012566039/4=0.003141509\tag{1}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is even better if you do &lt;span class=&quot;math-container&quot;&gt;$\sin(0.0072)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1763" LastEditorUserId="8052" LastEditDate="2021-08-26T11:58:36.733" LastActivityDate="2021-08-26T11:58:36.733" Title="Sine of 0.72 and pi" Tags="|mathematics|" AnswerCount="1" CommentCount="6" ClosedDate="2021-08-28T13:46:08.037" ContentLicense="CC BY-SA 4.0" />
  <row Id="5027" PostTypeId="2" ParentId="5026" CreationDate="2016-07-17T19:05:57.877" Score="7" Body="&lt;p&gt;This is&#xA;$$\lim_{x\to 0}\frac{\sin x}{x}=1&#xA;\tag{1}$$&#xA;so that is what we need to know the history of.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Explanation: $0.72=\frac{72}{100}=\frac{18}{25}$, so&#xA;  $$0.72\;\text{degrees}=\frac{18}{25}\cdot\frac{\pi}{180}=\frac{\pi}{250}$$&#xA;  and therefore by $(1)$,&#xA;  $$\sin(0.72\;\text{degrees})=\sin\left(\frac{\pi}{250}\right)\approx\frac{\pi}{250}\\&#xA;\frac{\sin(0.72\;\text{degrees})}{4}\approx\frac{\pi}{1000}$$&#xA;  And, as noted, for a better approximation, take $x$ closer to $0$, for example&#xA;  $$\frac{\sin(0.0072\;\text{degrees})}{4}\approx\frac{\pi}{100000}$$  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note: $(1)$ is one of the many reasons that, when a mathematician writes the sine function, he assumes it is computed in radians.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="72" LastEditDate="2017-12-16T13:16:51.010" LastActivityDate="2017-12-16T13:16:51.010" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5028" PostTypeId="2" ParentId="5025" CreationDate="2016-07-18T23:51:53.963" Score="1" Body="&lt;p&gt;I don't have enough reputation to comment. Are you reading Euclid from the original Ancient Greek text or are you reading it from a translation? And can you give an example of two (not necessarily consecutive) propositions where you observed this? I also noticed that you observed that an equivalence is proved in four successive propositions in some cases. Can you give an example of this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many instances, Euclid proves and states one conditional and immediately proves the &lt;em&gt;converse&lt;/em&gt;, not the contrapositive. But this might be difficult to catch because of the word order. I tried to translate from Ancient Greek earlier and I was surprised myself. But as an example you can see the &lt;a href=&quot;http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI18.html&quot; rel=&quot;nofollow&quot;&gt;guide for Proposition 1.18 and 1.19&lt;/a&gt; by David E. Joyce.&lt;/p&gt;&#xA;" OwnerUserId="4453" LastEditorUserId="4453" LastEditDate="2016-07-18T23:56:59.923" LastActivityDate="2016-07-18T23:56:59.923" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5029" PostTypeId="2" ParentId="5025" CreationDate="2016-07-19T00:59:53.817" Score="8" Body="&lt;p&gt;This is one of those cases where it is easy to take modern ideas for granted, and wonder how something so &amp;quot;obvious&amp;quot; could be missed. Modern formal logic with its array of logical operations, Boolean algebra and predicates is the creation of late 19th century, only finalized after much labor by Russell and Whitehead around 1910. Logical analysis that translates natural sentences into formally manipulable ones was also a 19th century invention (especially Frege's), and so was the axiomatic method (especially Pasch's, Peano's and Hilbert's). Stoics did introduce some propositional forms in the 3rd century BC, but not in relation to mathematics, and they never took. Between them and 19th century the only person to pursue propositional logic was Leibniz, who does list contraposition as a valid form in &lt;a href=&quot;http://www.nyu.edu/gsas/dept/philo/faculty/malink/The%20Logic%20of%20Leibnizs%20Generales%20Inquisitiones.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Generales Inquisitiones (1686)&lt;/a&gt;. But like most of his logical works it was not published until 1902.&lt;/p&gt;&#xA;&lt;p&gt;For everybody else formal logic was identical to Aristotle's syllogistic, where one could get from &amp;quot;all humans are mortal&amp;quot; to &amp;quot;all immortals are non-human&amp;quot;, but which did not have expressive means for intricate mathematical reasoning. &lt;a href=&quot;https://www.jstor.org/stable/41134289&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedman discusses in detail&lt;/a&gt; (pp.475-477) how expressive poverty of syllogistic impeded the rigorization of calculus. Since it was the only established formal logic flipping conditionals &amp;quot;blindly&amp;quot; would not have been kosher. And in the natural reasoning the difference between proof by contraposition and proof by contradiction is highly artificial, in both the conclusion is negated and the negation of something in the premise is derived. So  discerning the use of &amp;quot;contraposition&amp;quot; before Leibniz is a matter of one's willingness to read between the lines. This said, &lt;a href=&quot;http://arxiv.org/pdf/0902.3623.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to Wirth, Fermat&lt;/a&gt; was particularly self-conscious:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Fermat was the first who — instead of just proving a theorem — analyzed the method of proof search. Moreover, in his letter for Huygens and in [Diophantus, 1670], Fermat was also the first to provide a correct verbalization of proofs by Descente Infinie and to overcome the presentation of induction proofs as “generalizable examples”&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For instance, in Euclid’s Elements, Proposition VIII.7 is just the contrapositive of Proposition VIII.6, and this is just one of several cases that we find a proposition with a proof in the Elements, where today we just see a corollary. Moreover, even Fermat reported in his letter for Huygens that he had had problems to apply the Method of Descente Infinie to positive mathematical statements... Because of the work of Frege and Peano, these logical differences may be considered trivial today. Nevertheless, they were not trivial before, and to understand the history of mathematics and the fine structure in which mathematicians reasoned, the distinction between affirmative and negative theorems and between direct and apagogic methods of demonstration is important&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; Despite the common reading of Hilbert into Euclid, Euclid was not interested in building a deductive system from axioms and logic, and his &amp;quot;deviations&amp;quot; from this &amp;quot;task&amp;quot; are not oversights. He was interested in systematizing geometry, which he saw as an idealized representation of the &amp;quot;sensibles&amp;quot;, a kind of &amp;quot;theoretical physics&amp;quot;, and explicating common assumptions and types of reasoning used by geometers. These included reasoning from construction of diagrams, which was considered legitimate as long as they were used &amp;quot;indeterminately&amp;quot;, see &lt;a href=&quot;https://www.jstor.org/stable/41134289&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi's Euclid's Pseudaria&lt;/a&gt; (4.1). &lt;a href=&quot;https://math.stackexchange.com/questions/838184/contraposition-in-intuitionistic-logic&quot;&gt;When contraposition remains valid&lt;/a&gt; in such reasoning is not a priori obvious, in general it fails in constructivism.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-07-19T17:53:36.967" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5030" PostTypeId="1" AcceptedAnswerId="5032" CreationDate="2016-07-21T14:51:04.907" Score="5" ViewCount="2428" Body="&lt;p&gt;Who first used the term &amp;quot;Ordinary Differential Equation (ODE)&amp;quot;? Is it known why the word &amp;quot;ordinary&amp;quot; is used here? What makes an ODE &amp;quot;ordinary&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="1772" LastEditorUserId="10106" LastEditDate="2022-07-26T16:06:43.760" LastActivityDate="2022-07-26T16:06:43.760" Title="What is the origin of the term &quot;Ordinary Differential Equation&quot;?" Tags="|mathematics|terminology|reference-request|differential-equations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5031" PostTypeId="1" AcceptedAnswerId="5049" CreationDate="2016-07-21T16:33:08.093" Score="-1" ViewCount="123" Body="&lt;p&gt;I am curious, when and by whom it was proved that straight line is the shortest of measurable curves connecting two given points.&lt;/p&gt;&#xA;" OwnerUserId="4469" LastActivityDate="2016-07-25T17:27:10.963" Title="Straight line is the shortest of curves, who proved?" Tags="|mathematics|mathematicians|real-analysis|euclidean-geometry|geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5032" PostTypeId="2" ParentId="5030" CreationDate="2016-07-21T16:40:37.720" Score="8" Body="&lt;p&gt;The name &quot;Ordinary Differential Equation&quot;, together with an explanation of why the term &quot;ordinary&quot; is used,  is found in 1828 in An Elementary Treatise on the Differential and Integral Calculus by Jean-Louis Boucharlat and Ralph Blakelock: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If x, instead of being a function of two variables x and y, should contain only x, this would be no more than an ordinary differential equation, which, being integrated, would give...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source : &lt;a href=&quot;http://jeff560.tripod.com/o.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest know uses of some words of mathematics site&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting to note that the name &quot;Partial Differential Equation&quot; appears to predate the first use of ODE by almost 60 years.  From the same site, we have the first use of &quot;Partial Differential Equation&quot; given as :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Partial differential equation was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in the title &quot;Memoire sur les Equations aux différence partielles,&quot; which was published in Histoire de L'Academie Royale des Sciences, pp. 151-178, Annee M. DCCL&amp;amp;ldquo:III (1773).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first English use is given as :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Partial differential equation appears in English in 1809 in a letter from “Mr. Thomas Knight, Of Papcastle, near Cockermouth” in The Mathematical Repository, New Series, Volume III (1809). The same issue of The Mathematical Repository contains the expression partial fluxion. [James A. Landau]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-07-21T16:40:37.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5034" PostTypeId="2" ParentId="5001" CreationDate="2016-07-21T19:59:03.960" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XOca0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XOca0.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The feline was used in the Andeans and Egypt.&#xA;Pythagoras used the same...&lt;/p&gt;&#xA;" OwnerUserId="4436" LastActivityDate="2016-07-21T19:59:03.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5035" PostTypeId="1" CreationDate="2016-07-22T08:00:20.400" Score="1" ViewCount="188" Body="&lt;p&gt;Did he realise this as soon as he thought of light curving due to gravity or was this something that took him a while to realise?&lt;/p&gt;&#xA;" OwnerUserId="1849" LastEditorDisplayName="user466" LastEditDate="2016-08-04T22:45:28.797" LastActivityDate="2016-08-04T22:45:28.797" Title="At what point did Einstein realise that gravity was curvature of spacetime?" Tags="|physics|astronomy|einstein|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5036" PostTypeId="2" ParentId="5024" CreationDate="2016-07-22T23:05:38.147" Score="1" Body="&lt;p&gt;It's Julius Plücker who is at the origin of studies involving invariants relating the number of different singular points of a curve. He used duality as a mean of finding and proving his results, especialy the recently found &quot;reciprocal polars transformation&quot;. See the (not fully satisfying) article: &lt;a href=&quot;https://en.wikipedia.org/wiki/Pl%C3%BCcker_formula?oldid=360847971&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Pl%C3%BCcker_formula?oldid=360847971&lt;/a&gt; (its french counterpart is slightly richer). It was around 1831 that he made his first publication on the subject. See also the interesting document &lt;a href=&quot;https://tel.archives-ouvertes.fr/tel-01158895/document&quot; rel=&quot;nofollow&quot;&gt;https://tel.archives-ouvertes.fr/tel-01158895/document&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3730" LastActivityDate="2016-07-22T23:05:38.147" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5037" PostTypeId="2" ParentId="4989" CreationDate="2016-07-23T11:18:39.703" Score="2" Body="&lt;p&gt;In France, before the 1958 reform of (national curriculum) university studies, linear algebra wasn't taught for the &quot;licence&quot; degree.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://fr.wikipedia.org/wiki/Licence_(France)#1958-1966:_des_licences_moins_g.C3.A9n.C3.A9ralistes&quot; rel=&quot;nofollow&quot;&gt;https://fr.wikipedia.org/wiki/Licence_(France)#1958-1966:_des_licences_moins_g.C3.A9n.C3.A9ralistes&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3730" LastActivityDate="2016-07-23T11:18:39.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5038" PostTypeId="1" CreationDate="2016-07-23T12:29:35.413" Score="6" ViewCount="440" Body="&lt;p&gt;Today Klein's $j$-invariant is used in various context's, the most famous one being maybe &quot;Monstrous Moonshine&quot;. But what was the original motivation for the study of the $j$-invariant? &lt;/p&gt;&#xA;" OwnerUserId="3825" LastActivityDate="2017-01-05T14:28:50.693" Title="Origin of Klein's $j$-invariant" Tags="|mathematics|complex-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5039" PostTypeId="2" ParentId="4961" CreationDate="2016-07-23T19:57:46.997" Score="10" Body="&lt;p&gt;Two points:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;@Nick R: I don't think geometers doing pure or analytic geometry wanted to avoid diagrams. They were educated with Euclides, reasoning on figures/diagrams. This trend (doing geometry almost without figures) has flowed like a tsunami especially in the 1970s... &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Something that has not been underlined enough is the technical limitations of the time. From early 18th century,  and in particular at the time of Gauss, &lt;strong&gt;copper engraving&lt;/strong&gt; was dominant.  This process is time-consuming, costly, and last but not least, necessitates a very able &quot;artist&quot; (figures carved on wood can be found in mathematical texts till the early 18th, but this technique hadn't the precision that can be obtained with copper). For this reason, when figures were present in a book, they were generally gathered at the end of the book. Many new imaging processes were invented, mainly in the second half of the 19th century, under the influence of the huge development of massive (newspaper) printing techniques,  lithography, photoengraving, offset printing etc... providing mathematical treatises or textbooks with very good overall quality (printing, binding, ...) and especially good figures &lt;strong&gt;inserted in the text&lt;/strong&gt; at affordable prices everywhere in the 1900s.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3730" LastEditorUserId="2277" LastEditDate="2018-05-22T21:43:54.773" LastActivityDate="2018-05-22T21:43:54.773" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5040" PostTypeId="1" CreationDate="2016-07-23T21:45:42.857" Score="10" ViewCount="755" Body="&lt;p&gt;Many years ago I read an interview with a physics professor, where he recounted a funny situation when he was a graduate student at Harvard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When a first year, he was supposed to take the quantum mechanics course, but he could not stand the lecture and begged to be able to defer it for a year, hoping he'd get a better lecturer next year, and somehow he got his request approved. This left him with an open slot, however, so he, not knowing much about quantum mechanics, ended up taking a &lt;em&gt;quantum field theory&lt;/em&gt; course, because he had heard that the lecturer, Julian Schwinger, was very good.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When the time came for oral exams, each student chose the topic they would be interviewed on. The student chose electromagnetism, but most of the others chose quantum mechanics, so when due to some mix-up the examiners began asking him questions on quantum mechanics, and he answered them using his knowledge from taking quantum field theory. Eventually the examiners realized their mistake, asked him to write Maxwell's equations in differential form, which he did, and then passed him and encouraged him to go into theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the life of me, I cannot remember who this physics professor was. I do remember that he mentioned that Wendall Furry was one of his examiners though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is basically my desperate hope that someone out there has read that interview too and can dig it up again.&lt;/p&gt;&#xA;" OwnerUserId="4478" LastEditorUserId="26" LastEditDate="2016-07-24T09:06:27.840" LastActivityDate="2016-07-26T01:13:54.843" Title="Does anyone know the physics student who passed a quantum mechanics oral exam without taking quantum mechanics?" Tags="|biographical-details|reference-request|quantum-mechanics|quantum-field-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5041" PostTypeId="1" AcceptedAnswerId="5043" CreationDate="2016-07-24T19:26:39.673" Score="8" ViewCount="1819" Body="&lt;p&gt;I was studying interpolation by Splines in numerical analysis and started to wonder the word's origin. I've found that it was a system used in technical drawings using weights but couldn't find why they where called like that in that context. Any idea?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regards&lt;/p&gt;&#xA;" OwnerUserId="4481" LastActivityDate="2018-07-20T19:50:03.200" Title="Origin of &quot;Spline&quot; word" Tags="|terminology|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5042" PostTypeId="2" ParentId="4929" CreationDate="2016-07-24T20:09:53.920" Score="2" Body="&lt;p&gt;Some complementary information :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;About prefixes &quot;in&quot; and &quot;sur&quot; : although &quot;in&quot; is very neutral and does not evoke especialy &quot;one-to-one&quot;, &quot;sur&quot; means in French &quot;onto&quot; (&quot;le chat est sur la table&quot; = &quot;the cat is on the table&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The previous names in French were &quot;univoque&quot; (Middle age origin, contrary of &quot;équivoque&quot;) used by mathematicians around 1900 for &quot;injective&quot; (&lt;a href=&quot;https://www.cnrtl.fr/etymologie/univoque&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cnrtl.fr/etymologie/univoque&lt;/a&gt;) and &quot;bi-univoque&quot; for &quot;bijective&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It must be said that injectivity and surjectivity are dual (in a certain sense) one of the other as examplified by &lt;a href=&quot;https://en.wikipedia.org/wiki/Exact_sequence#Short_exact_sequence&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;short exact sequences&quot;&lt;/a&gt;:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\displaystyle 0\to A\;{\xrightarrow {\ f\ }}\;B\;{\xrightarrow {\ g\ }}\;C\to 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is injective and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is surjective, but in fact more than that : in the framework of categories, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is a &quot;monomorphism&quot; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; is an &quot;epimorphism&quot; : &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; operate on sets that have an algebraic/topological/geometrical structure, for example groups or vector spaces, or topological vector spaces, etc. Please note that prefixes &quot;mono&quot; and &quot;epi&quot; are the greek equivalent to &quot;single/one&quot; and &quot;onto&quot; resp.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;People working on &lt;a href=&quot;https://en.wikipedia.org/wiki/One-to-many_(data_model)&quot; rel=&quot;nofollow noreferrer&quot;&gt;relational databases&lt;/a&gt; have a current need for expressions like  &quot;one-to-one&quot;, &quot;one-to-many&quot; and &quot;many-to-many&quot;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Have a look at &lt;a href=&quot;https://math.stackexchange.com/q/63154&quot;&gt;this discussion&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3730" LastEditorUserId="3730" LastEditDate="2020-06-05T21:52:00.723" LastActivityDate="2020-06-05T21:52:00.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5043" PostTypeId="2" ParentId="5041" CreationDate="2016-07-24T23:21:16.863" Score="4" Body="&lt;p&gt;My dictionary has this as origin of the word:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;East Anglian dialect, perhaps related to &lt;em&gt;splinter&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-07-24T23:21:16.863" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5044" PostTypeId="2" ParentId="3035" CreationDate="2016-07-25T06:10:45.583" Score="2" Body="&lt;p&gt;It is clear now that nobody later was influenced or aware of either. The abstract was too vague. It was written also prior to the completed thesis.&#xA;The two citations are: &lt;strong&gt;MacLane, Saunders&lt;/strong&gt;, (Nov 1933) Jan 1934, Abbreviated Proofs in Logic Calculus, &lt;em&gt;Bulletin of the American Mathematical Society&lt;/em&gt; 40(1):37--38.&#xA;&lt;strong&gt;MacLane, Saunders&lt;/strong&gt;, 1934, &lt;em&gt;Abgekuerzte Beweise in Logikkalkul&lt;/em&gt;, Goettingen: Hubert.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thesis had insignificant distribution, mostly to the author's few dozen local friends (who had not interest in the subject). Weyl and Bernays saw it as the advisors, of course but didn't consider it much mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It goes: an input and the repetitive processes operating on it, a program, if worked, is equivalent all intermediate results of program execution and the output. Let this be a proof; if so we have two identical proofs, one often much shorter than the other but equally simple. They have same mathematical symbolic content, the same information we would say if it were binary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We record the symbolic repetitive inference procedures as the axiom symbolic inference functions, to be recursively applied as needed rather than recording any results of their use. This works there is proof that these functions give valid inferences and can recursively operated with, that they can compress symbols systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a compression scheme for standard logical symbolic proofs, similar to sets of shorter sequences and chunks having same information as the symbols sequences they chunked but requiring less storage, eliminating the storage problem (Miller 1956, Pribram 1970), and considering as equivalent bits strings and fewer bit programs that output these measurements sequences and so normalizing to common units the strings by the programs (Solomonoff 1960, Chaitin 1966).&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2016-07-25T06:10:45.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5045" PostTypeId="1" AcceptedAnswerId="15230" CreationDate="2016-07-25T11:09:08.460" Score="10" ViewCount="299" Body="&lt;p&gt;Did Jacobi prove the product formula &lt;span class=&quot;math-container&quot;&gt;$\Delta(\tau) = (2\pi)^{12}q\prod_{n \geq 1} (1 - q^n)^{24}$&lt;/span&gt; for the modular discriminant function, and if so where?&lt;/p&gt;&#xA;&lt;p&gt;I have tried without success to track down references (other than looking through Jacobi's collected works), but when I search for terms like &amp;quot;Jacobi product&amp;quot; I get lots of hits on his triple product formula, and inserting &amp;quot;discriminant&amp;quot; into the search is not helping on isolating a reference that would discuss Jacobi's own work on this.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-03-15T17:27:09.610" LastActivityDate="2023-03-16T16:05:00.290" Title="Jacobi's product for the discriminant" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5046" PostTypeId="1" AcceptedAnswerId="5048" CreationDate="2016-07-25T15:33:15.757" Score="9" ViewCount="231" Body="&lt;p&gt;Is there any reason why Hilbert's problems are ordered in the way they are?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, from whom does this ordering come?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that Hilbert talked about several problems on his list, namely 1, 2, 6, 7, 8, 13, 16, 19, 21 and 22, so I don't think that his talk in 1900 affected the order. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastEditorUserId="4251" LastEditDate="2021-01-25T21:12:33.113" LastActivityDate="2021-01-25T21:12:33.113" Title="Is there any reason why Hilbert's problems are ordered in the way they are?" Tags="|mathematics|hilbert|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5047" PostTypeId="1" CreationDate="2016-07-25T16:00:40.470" Score="3" ViewCount="136" Body="&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Following Mauro's comment, I have altered my question to ask only about any restrictions that may have been considered concerning the suitability of the incumbent's choice of questions for the challenger.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Renaissance mathematicians were well known for keeping their discoveries secret.  According to William Dunham in &lt;em&gt;Journey Through Genius&lt;/em&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To understand this behaviour - almost incomprehensible in the “publish or peril” world of today - we must consider the nature of the Renaissance university.  There, academic appointments were by no means secure.  Along with patronage and political influence, continued service depended on the ability to prevail in public challenges that could be issued by any quarter at any time. … a public humiliation could be disastrous to one’s career.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dunham then gives a colourful account of one such challenge.  It starts with Scipione del Ferro’s discovery of a method for solving a certain class of “depressed” cubic equations; those which lack a term of the second degree - i.e., those of the form &lt;span class=&quot;math-container&quot;&gt;$ax^3 + cx + d = 0$&lt;/span&gt;.  Ferro, on his deathbed,  revealed the formula to his student, Antonio Fior.  Fior then (rashly) issued a challenge to Tartaglia (Niccolo Fontana).  Tartaglia had boasted that he could solve depressed cubics missing a linear term - i.e., those of the form  &lt;span class=&quot;math-container&quot;&gt;$ax^3 + bx^2 + c = 0$&lt;/span&gt;.  Tartaglia won the challenge, having hastily worked hard in the allotted time to obtain a method for solving the depressed cubics of the form presented by Fior, while Fior lacked the ability to meet Tartaglia’s challenge.&lt;/p&gt;&#xA;&lt;p&gt;One assumes that a challenge is born of an “I know something you don’t know” claim.  Those conventions governing the number of problems and the time limit seem easy enough to decide. Dunham tells that each of the participants issued their opponent with a list of 30 problems requiring solution.  Apparently the challenge lasted over a period of some days since Tartaglia spent some days working to solve the alternative diminished cubic before providing his solutions.  Curiously, Dunham states that even if the challenger was stumped by the incumbent’s problems,  he could be confident that his secret knowledge, supposedly known only to himself, would guarantee the failure of his opponent, resulting in a stalemate.&lt;/p&gt;&#xA;&lt;p&gt;What is less clear is how to fairly judge the incumbent's subject choice.  Presumably the incumbent must present a set of problems on the same general subject.  If the incumbent assumes that the challenger has some “secret” knowledge, then obviously there is no point in asking the same type of problems as the challenger is posing.  In this example, there is no point in Tartaglia asking questions about cubics with no second degree term, so he asks questions about cubics with no linear term.  Since the challenger is making the challenge, obviously he cannot choose the type of problem that the incumbent may ask since that would give the challenger an unfair advantage.&lt;/p&gt;&#xA;&lt;p&gt;However, what if Fior possessed Cardano’s method for solving the general cubic.  What then would be Tartaglia’s options for subject choice.  He could not be required to also pose questions requiring the solution of cubics since that would give Firo an unfair advantage.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: &lt;strike&gt;What rules governed the scholastic challenges of the Renaissance?  What rules determine the incumbent’s choice of questions?&lt;/strike&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt; : What restrictions were considered appropriate regarding the incumbent's choice of subject, and who judged the fairness of the questions posed?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="4251" LastEditDate="2023-06-26T09:42:17.323" LastActivityDate="2023-06-26T09:42:17.323" Title="In the scholastic challenges of renaissance Italy, what restrictions were considered appropriate regarding the incumbent's choice of subject?" Tags="|mathematics|social-context|renaissance|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5048" PostTypeId="2" ParentId="5046" CreationDate="2016-07-25T17:22:03.767" Score="9" Body="&lt;p&gt;Yes, they are ordered by the broad areas of mathematics. If you read his paper, you will see that he hints this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1,2 - foundations, set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3,4, and 5 to some extent - foundations of geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6 - foundations of physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7-12 - number theory,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;13,14 -algebra,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;15,16,17 algebraic geometry,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;18 - metric geometry,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;19-23 - analysis and differential equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to be a natural order of any exposition of the whole mathematics&#xA;(compare with French Courses d'Analyse, or early 20th century Encyclopedia of mathematics, or even Bourbaki).  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-07-25T17:22:03.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5049" PostTypeId="2" ParentId="5031" CreationDate="2016-07-25T17:27:10.963" Score="2" Body="&lt;p&gt;Essentially Euclid. Exact statement depends on the exact notion of a curve and length. Euclid considers first broken lines, and proves the statement for them. Then he defines the length of other curves&#xA;(for example a circle) essentially as the limit of lengths of inscribed broken lines. By the way, the modern definition of length is the same. The statement easily follows, because the broken lines can be inscribed in such a way that their length increases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Of course, we do not know exactly what Euclid discovered himself, and what&#xA;was known before him, because his Elements is the only primary source for mathematics before him, and he does not give references. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-07-25T17:27:10.963" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5050" PostTypeId="2" ParentId="5040" CreationDate="2016-07-26T01:13:54.843" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://authors.library.caltech.edu/5456/1/hrst.mit.edu/hrs/renormalization/Stanley/index.html&quot; rel=&quot;noreferrer&quot;&gt;H.E. Stanley's 2002 interview&lt;/a&gt; has Harvard, taking quantum field theory ahead of quantum mechanics, Schwinger and a tough lecturer (Coleman), Wendell Furry, the electromagnetism course, and an exam. But the anecdotal details do not match up: Coleman &lt;em&gt;dis&lt;/em&gt;couraged him from going into QFT (he encouraged Jeff Mandula instead), the exam was take home, Furry was not the examiner, and there were no Maxwell's equations in differential form. So perhaps it was not Stanley but Jeff Mandula, or some other Harvard student from 1950s. American Institute of Physics conducted a number of interviews with physicists who studied at Harvard around that time, see e.g. &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/23362-1&quot; rel=&quot;noreferrer&quot;&gt;Anderson's&lt;/a&gt;, &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/33928&quot; rel=&quot;noreferrer&quot;&gt;Layzer's&lt;/a&gt; and &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/30248&quot; rel=&quot;noreferrer&quot;&gt;Schlihter's&lt;/a&gt;, but I could not find anything closer to your story than Stanley's. Here are the relevant passages:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;I made a lot of mistakes when I came because I lost a year on the Fulbright. I thought I could place out of some courses and Harvard is a little bit relaxed on how they do things. They don't force you to do much, and therefore I did not take a course in say quantum mechanics, because I went to take the [Julian] Schwinger's lecture on field theory and [Sidney] Coleman taught it also, and the bottom line is that I created holes in my physics education which I then was never able to fill because I never had the time to sort of go back and learn a lot of these things. So, it's made me a good teacher, because I tell students not to do this.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Yeah, yeah I did take Coleman's course and that's why I got out of field theory, I think that when I entered Harvard I wanted to be a Schwinger student.... In Germany I dreamed of Schwinger: he was a big monstrous guy but when I met him he was a short guy. So I took a reading course like so many students did the first semester, and had the exactly the same experience that everyone had of meeting him: once in the fall you saw him to sign a card and at which time he told me to come back in January at the end of the semester with the term paper. Which I did.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;So that didn't help me make a decision, but Coleman's course did, because Coleman's course was very hard... In particular, he gave a take home exam which was so tough that no one in the entire class could do it and you had 4-5 days. Except, the day before it was due Jeff Mandula said 'I can solve it,' and everyone didn't know if he was bluffing but he could, so Jeff got an A, the rest of us got B's and, basically, the message from Coleman was you shouldn't think of field theory if you couldn't get an A. Jeff Mandula could think of it if he wanted to and most of us really should not and I took that lesson to heart.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;At Lincoln Lab. John van Vleck taught a very nice course in magnetism, which I took. And taught it in a way which is also valuable to historians. His way of teaching is a way that I've never been able to quite teach but I've always wanted to. He would come in as if he were totally unprepared, you know and everyone would wonder whether he was or not, and he would say 'Today we're supposed to talk about paramagnetism and does anybody know about paramagnetism?' And we would say, 'yeah the spins don't interact.'&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;No he [Wendell Furry] then ... he didn't teach graduate courses. And he was regarded as a man who had fallen afoul of the [Joseph] McCarthy people...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-07-26T01:13:54.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5051" PostTypeId="2" ParentId="5022" CreationDate="2016-07-26T08:57:02.370" Score="1" Body="&lt;p&gt;The main evidence of this seems to be that whenever Grothendieck says topos, he means Grothendieck topos (unless someone can find a contradicting reference). However, it doesn't seem to be the case that he didn't like elementary toposes, so much as he didn't think he needed them for what he was doing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to page 6 of this interview with Lawvere &lt;a href=&quot;http://www.mat.uc.pt/~picado/lawvere/interview.pdf&quot; rel=&quot;nofollow&quot;&gt;http://www.mat.uc.pt/~picado/lawvere/interview.pdf&lt;/a&gt;, Grothendieck did call Lawvere the &quot;main contradictor&quot;, but this was with reference to politics rather than mathematics. Grothendieck does use the subobject classifier in a topos in &lt;a href=&quot;https://thescrivener.github.io/PursuingStacks/ps-online.pdf&quot; rel=&quot;nofollow&quot;&gt;Pursuing Stacks&lt;/a&gt;, and even calls it the &quot;Lawvere element&quot; of a topos. &lt;/p&gt;&#xA;" OwnerUserId="4490" LastActivityDate="2016-07-26T08:57:02.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5052" PostTypeId="2" ParentId="5041" CreationDate="2016-07-26T18:16:40.340" Score="8" Body="&lt;p&gt;The term comes from shipbuilding in the old days: A long thin flat spline which was fixed at discrete points by newts, flexes just like a cubic spline with natural boundary condition. In this case, the strain energy becomes a minimum. &#xA;The technique is still in use in experimental ship building.&lt;/p&gt;&#xA;" OwnerUserId="2000" LastActivityDate="2016-07-26T18:16:40.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5053" PostTypeId="2" ParentId="212" CreationDate="2016-07-27T11:28:04.100" Score="1" Body="&lt;p&gt;In 1928, at may 11 he was diagnosed yellow fever. He felt strange because he had immunity of yellow fever. However after he died, and his blood was injected to a monkey. The monkey was diagnosed yellow fever. Then also Dr.young died with yellow fever after about one week.&lt;/p&gt;&#xA;" OwnerUserId="4211" LastEditorUserId="4211" LastEditDate="2016-07-29T10:13:57.470" LastActivityDate="2016-07-29T10:13:57.470" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5054" PostTypeId="2" ParentId="205" CreationDate="2016-07-28T12:53:58.310" Score="6" Body="&lt;p&gt;MacLane's article &lt;a href=&quot;http://www.ams.org/notices/199510/maclane.pdf&quot; rel=&quot;nofollow&quot;&gt;&quot;Mathematics at Göttingen under the Nazis&quot;&lt;/a&gt; is a very vivid first hand account of the time 1931-33. MacLane writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... in 1933 eighteen mathematicians left or were driven out from the faculty at the Mathematical Institute in Göttingen&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which &#xA;shows how quickly the department disintegrated after the Nazis came to power.&lt;/p&gt;&#xA;" OwnerUserId="4498" LastEditorUserId="2442" LastEditDate="2016-08-12T17:05:13.733" LastActivityDate="2016-08-12T17:05:13.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5055" PostTypeId="1" AcceptedAnswerId="5076" CreationDate="2016-07-28T15:48:02.440" Score="3" ViewCount="868" Body="&lt;p&gt;Ampère, a half century before Maxwell, theorized that magnetism was caused by electrical currents. So, why is Maxwell and not Ampère credited for unifying electricity and magnetism?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;(cf. the question &quot;&lt;a href=&quot;https://hsm.stackexchange.com/q/5059/232&quot;&gt;Why is one of Maxwell's equations named after Ampère?&lt;/a&gt;&quot;)&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="4251" LastEditDate="2020-06-05T15:46:11.733" LastActivityDate="2020-06-05T15:46:11.733" Title="Why is Maxwell and not Ampère credited for unifying electricity and magnetism?" Tags="|physics|electromagnetism|electricity|magnetism|maxwell|" AnswerCount="2" CommentCount="6" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5056" PostTypeId="1" CreationDate="2016-07-29T14:00:48.590" Score="5" ViewCount="102" Body="&lt;p&gt;Historically, &lt;a href=&quot;https://en.wikipedia.org/wiki/Incandescent_light_bulb&quot; rel=&quot;noreferrer&quot;&gt;incandestent light bulbs&lt;/a&gt; were marketed primarily by their power consumption in watt.  I don't know if &lt;a href=&quot;https://en.wikipedia.org/wiki/Lumen_(unit)&quot; rel=&quot;noreferrer&quot;&gt;light output in lumen&lt;/a&gt; was specified at all.  I'm not aware of any other electrical good sold primarily marketing power consumption: it's listed in the specifications, but it's never the primary selling characteristic.  Loudseakers may be sold based on power in watt, but I believe that refers to useful power &lt;em&gt;output&lt;/em&gt; rather than power consumption.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that lights are mostly based on &lt;a href=&quot;https://en.wikipedia.org/wiki/Compact_fluorescent_lamp&quot; rel=&quot;noreferrer&quot;&gt;CFL&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/LED_lamp&quot; rel=&quot;noreferrer&quot;&gt;LED&lt;/a&gt;, and other techniques, it has become standard to specify light output in lumen.  Considering that light output is the obvious primary characteristic of any light source, why wasn't lumen used from the start when characterising any kind of bulb?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(There are even packages that show the hypothetical power consumption for an incandestent light bulb with the same light output!)&lt;/p&gt;&#xA;" OwnerUserId="111" LastEditorUserId="111" LastEditDate="2016-07-29T14:16:56.310" LastActivityDate="2016-07-29T14:16:56.310" Title="Why were light bulbs traditionally marketed based on power consumption rather than light output?" Tags="|light|technology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5057" PostTypeId="2" ParentId="5041" CreationDate="2016-07-29T19:55:35.077" Score="5" Body="&lt;p&gt;In his 1946 paper, &quot;&lt;a href=&quot;http://dx.doi.org/10.1007/978-1-4899-0433-1_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions&lt;/a&gt;&quot;. Quart. Appl. Math., mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Isaac_Jacob_Schoenberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isaac Jacob Schoenberg&lt;/a&gt; explains that:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A spline is a simple mechanical device for drawing smooth curves. It&#xA;  is a slender flexible bar made of wood or some other elastic material.&#xA;  The spline is place on the sheet of graph paper and held in place at&#xA;  various points by means of certain heavy objects (called &quot;dogs&quot; or&#xA;  &quot;rats&quot;) such as to take the shape of the curve we wish to draw.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is also called a &lt;a href=&quot;https://en.wikipedia.org/wiki/Flat_spline&quot; rel=&quot;nofollow noreferrer&quot;&gt;flat spline&lt;/a&gt; or a flexible  curve:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a long strip fixed in position at a number of points that relaxes to&#xA;  form and hold a smooth curve passing through those points for the&#xA;  purpose of transferring that curve to another material&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;depicted below:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PrV2E.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PrV2E.png&quot; alt=&quot;Flat spline or flexible curve&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="4130" LastEditDate="2018-07-20T19:50:03.200" LastActivityDate="2018-07-20T19:50:03.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5058" PostTypeId="2" ParentId="2935" CreationDate="2016-07-30T20:38:10.573" Score="5" Body="&lt;p&gt;Ampère never wrote down what is confusingly called &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re&amp;#39;s_circuital_law&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Ampère&lt;/em&gt;'s circuital law&lt;/a&gt;,&quot; not even the form without the displacement current term, as Ampère never dealt with the field concept.* Maxwell derived&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\nabla \times \mathbf{B} = \mu_0\mathbf{J}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in his 1855 paper &lt;a href=&quot;https://archive.org/stream/scientificpapers01maxw#page/n193/mode/2up&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;On Faraday's Lines of Force&lt;/em&gt;&lt;/a&gt;, based on analogies to hydrodynamics, which he corrected to be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}} {\partial t} \right)\qquad(2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in his 1861 paper &lt;a href=&quot;https://archive.org/stream/scientificpapers01maxw#page/450/mode/2up&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;On Physical Lines of Force&lt;/em&gt;&lt;/a&gt;; he never wrote down &lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re&amp;#39;s_force_law&quot; rel=&quot;nofollow&quot;&gt;Ampère's force law&lt;/a&gt; in either paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re&amp;#39;s_force_law&quot; rel=&quot;nofollow&quot;&gt;Ampère's force law&lt;/a&gt; is completely different from any of Maxwell's equations. It gives the force that current elements $I_1 d\vec {\ell }_1$ and&#xA;$I_2 d\vec {\ell }_2$ exert on one another to be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, it is appropriate that Equation (1) is one of &lt;em&gt;Maxwell&lt;/em&gt;'s equations. Gauss and Faraday utilized the field concept, so Equation (1) is the most &quot;Maxwellian&quot; of the four Maxwell's equations.&#xA;&lt;hr&gt;&#xA;&lt;sup&gt;*cf. Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015). &lt;em&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;nofollow&quot;&gt;Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience&lt;/em&gt; (PDF)&lt;/a&gt;. Montreal: Apeiron. ISBN 978-1-987980-03-5. ch. 15 pp. 221ff.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-07-30T23:41:30.713" LastActivityDate="2016-07-30T23:41:30.713" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5059" PostTypeId="1" AcceptedAnswerId="7609" CreationDate="2016-07-30T20:41:11.120" Score="9" ViewCount="697" Body="&lt;p&gt;Ampère never wrote down what is confusingly called &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re&amp;#39;s_circuital_law&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Ampère&lt;/em&gt;'s circuital law&lt;/a&gt;,&quot; not even the form without the displacement current term, as Ampère never dealt with the field concept.* Maxwell derived&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\nabla \times \mathbf{B} = \mu_0\mathbf{J}\qquad(1)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in his 1855 paper &lt;a href=&quot;https://archive.org/stream/scientificpapers01maxw#page/n193/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;On Faraday's Lines of Force&lt;/em&gt;&lt;/a&gt;, based on analogies to hydrodynamics, which he corrected to be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}} {\partial t} \right)\qquad(2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in his 1861 paper &lt;a href=&quot;https://archive.org/stream/scientificpapers01maxw#page/450/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;On Physical Lines of Force&lt;/em&gt;&lt;/a&gt;; he never wrote down &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_force_law&amp;amp;oldid=758353198#Equation&quot; rel=&quot;noreferrer&quot;&gt;Ampère's force law&lt;/a&gt; in either paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_force_law&amp;amp;oldid=758353198#Equation&quot; rel=&quot;noreferrer&quot;&gt;Ampère's force law&lt;/a&gt; is completely different from any of Maxwell's equations. It gives the force that current elements $I_1 d\vec {\ell }_1$ and&#xA;$I_2 d\vec {\ell }_2$ exert on one another to be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, it is appropriate that Equation (2) is one of &lt;em&gt;Maxwell&lt;/em&gt;'s equations. Gauss and Faraday utilized the field concept, thus Equation (2) is the most &quot;Maxwellian&quot; of the four Maxwell's equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, &lt;strong&gt;why are Equations (1) &amp;amp; (2) above named after Ampère? Who first named them after Ampère?&lt;/strong&gt;&#xA;&lt;hr&gt;&#xA;&lt;sup&gt;*cf. Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015). &lt;em&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;noreferrer&quot;&gt;Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience&lt;/em&gt; (PDF)&lt;/a&gt;. Montreal: Apeiron. ISBN 978-1-987980-03-5. ch. 15 pp. 221ff.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2017-01-06T01:40:03.663" LastActivityDate="2018-08-16T17:12:07.377" Title="Why is one of Maxwell's equations named after Ampère? Who first named it after Ampère?" Tags="|electromagnetism|electricity|terminology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5060" PostTypeId="1" CreationDate="2016-07-31T18:25:16.473" Score="3" ViewCount="117" Body="&lt;p&gt;I always was amazed by how disciplines which can give very different grades of accuracy (exact sciences in contrast to others which fails in its predictions more times than they are successful) can all be considered science by part of the people. So digging into this I've read that in a modern sense science requires empiricism and scientific method, and by part of the people in a broader interpretivist sense these properties aren't required. So my question is, was &lt;a href=&quot;https://en.wikipedia.org/wiki/Antipositivism&quot; rel=&quot;nofollow&quot;&gt;interpretivism, or antipositivism,&lt;/a&gt; always considered science by part of the people, or this conception came into play later on?&lt;/p&gt;&#xA;" OwnerUserId="4512" LastEditorUserId="55" LastEditDate="2016-08-13T20:39:16.663" LastActivityDate="2019-05-08T04:23:18.183" Title="Was interpretivism always considered scientific by part of the people?" Tags="|philosophy-of-science|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5061" PostTypeId="2" ParentId="635" CreationDate="2016-07-31T21:40:59.210" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Napier&quot; rel=&quot;noreferrer&quot;&gt;John Napier (1550-1617)&lt;/a&gt; published his table of logarithms &lt;a href=&quot;https://books.google.com/books?id=J19dAAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mirifici Logarithmorum Canonis Descriptio&lt;/em&gt;&lt;/a&gt; in 1614 after some twenty years of work and described his method of construction in &lt;a href=&quot;https://books.google.com/books?id=VukHAQAAIAAJ&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mirifici Logarithmorum Canonis Constructio&lt;/em&gt;&lt;/a&gt;, published posthumously in 1619 (Edinburgh) by his son Robert, with appendices by Napier and &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Briggs_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Henry Briggs (1561-1630)&lt;/a&gt;.  Briggs worked with Napier on improving the methods of calculation in the summers following the publication of the table until Napier's death and published his own method in 1624 (&lt;a href=&quot;https://books.google.com/books?id=-XR-2oDEb9oC&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;Vlacq's 1628 edition&lt;/a&gt;).  The following is based on the &lt;em&gt;Constructio&lt;/em&gt;, &lt;a href=&quot;https://books.google.com/books?id=Zlu4AAAAIAAJ&quot; rel=&quot;noreferrer&quot;&gt;Macdonald's translation&lt;/a&gt;, and &lt;a href=&quot;https://books.google.com/books?id=r7ffBwAAQBAJ&amp;amp;pg=PA2&quot; rel=&quot;noreferrer&quot;&gt;Goldstine's summary&lt;/a&gt; in &lt;em&gt;A History of Numerical Analysis from the 16th through the 19th Century&lt;/em&gt; (1977), 2-13.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Properties of geometric and arithmetic progressions were well-known by Napier's time, and the connection between a sequence of powers and its corresponding sequence of exponents that we call the law of exponents has roots in ancient mathematics (cf. Euclid IX.11 and Archimedes, &lt;em&gt;The Sand Reckoner&lt;/em&gt;). I do not know what enabled Napier to make the key connection that led to logarithms (or on the other hand, what prevented the discovery earlier). My take on Napier is that calculating was thought primarily in terms of integers or ratios of integers (or fractions), and Napier is at pains to make his calculation of logarithms in terms of integers accurate.  His break-through idea is quite ingenious and surprising (to me, at least).  It is to consider two points moving continuously, one (representing the logarithm) increasing &quot;arithmetically&quot; while the other (representing what we would now call the argument) decreasing &quot;geometrically.&quot;  Here arithmetic motion is the same as uniform motion while for geometric motion, Napier proves that the point moving geometrically toward a fixed point &quot;has its velocities proportional to its distances from the fixed one.&quot; (Prop. 25)  The notion is in modern terms that of the continuous interpolation of arithmetic and geometric sequences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like many first tries, this led to some inconveniences.  By the time his table was published, Napier had in mind some improvements, which are included in an appendix to the &lt;em&gt;Constructio&lt;/em&gt;. One of them is that the logarithm of 1 should be 0 and the logarithm of 10 should be some nice number, like $10^{10}$. (It is large, it appears, so that one can express logarithms accurately in terms of integers.)  Briggs too saw some room for improvement, and this led him to seek out Napier.  Briggs published a table in &lt;em&gt;Logarithmorum Chilias Prima&lt;/em&gt; (1617), the first published table of base-10 logarithms, and contributed some Notes in another appendix to the &lt;em&gt;Constructio&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Napier deals with velocities and distances as easily as we deal with derivatives and integrals.  In fact, from a modern point of view, there's is no real difference.  In our modern terms, we can represent Napier's definition of a logarithm (Prop. 26):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &quot;artificial number&quot; (logarithm) of a given sine is that which has increased Arithmetically always with the same velocity as the total sine [began] to decrease Geometrically and in the same time as the total sine decreased to that given sine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Throughout the &lt;em&gt;Constructio&lt;/em&gt;, Napier calls the logarithm &lt;em&gt;numerus artificialis&lt;/em&gt; and the given number either &lt;em&gt;sinus&lt;/em&gt; or &lt;em&gt;numerus naturalis&lt;/em&gt;.  The &quot;total sine&quot; is a fixed length through which the geometrically moving point can move.  It can be translated as &quot;radius&quot; and a given sine will be less than or equal to this; however, trigonometry does not figure in the construction, so the names are somewhat meaningless.  So a point begins moving along the total sine $r = TS$ from $T$ toward $S$.  Let $x$ be the distance to $S$, so that $r-x$ is the distance moved.  Then by Prop. 25, we have&#xA;$${d \over dt} (r-x) = x$$&#xA;(where we can pick the unit of time so that the constant of proportionality is $1$.)&#xA;If we let $y$ denote the distance the arithmetically moving point has covered, we have $dy/dt = r$.  Thus Napier's definition in modern terms is equivalent to &#xA;the initial value problem&#xA;$${d \over dt} (r-x) = x,\ {d \over dt} y = r,\ x(0) = r,\ y(0)=0$$&#xA;since at $x = r$ the two velocities equal.  (The notation here is the same as Goldstine's.)&#xA;Napier takes $r=10^7$, and it follows that Napier's logarithm is &#xA;$$y = 10^r \log 10^r/x = 10^7 \log 10^7/x\,.$$&#xA;To calculate his logarithm $y$, he uses his definition to prove some inequalities&#xA;$$\frac{r (r-x)}{x}&amp;lt;y&amp;lt;r-x, \quad \frac{r (x_1-x_2)}{x_1}&amp;lt;y_2-y_1&amp;lt;\frac{r (x_1-x_2)}{x_2}$$&#xA;where $x_1&amp;gt;x_2$ and $y_1,y_2$ are the corresponding logarithms.&#xA;It is easy to prove these inequalities from properties of the integral, but Napier did in terms of velocities and distances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Napier's first step in constructing his table is to approximate the logarithm of $x=9\,999\,999$, one less than the total sine $10^7$.  From the first inequality, he has&#xA;$$1 &amp;lt; y &amp;lt; 1.00000010000001$$&#xA;He splits the difference and takes $y = 1.00000005$ (which represents a error of less than $10^{-14}$).  With this he can fill out the logarithms of the geometric sequence with $r$ and this $x$ as the first terms.  He does this down to $9999900.0004950$. The he starts a new geometric sequence with $r$ and $x=9999900$ down to nearly $9995001$.  He then constructs a more extensive table with 69 columns so that the columns and rows were each in geometric progressions, starting with $r=10^7$ in one corner and ending with $x=4998609.4034$, or roughly $r/2$, in the opposite corner.  With these, he constructed his final table, the &quot;Canon of Logarithms.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that if $y=l(x)$ is Napier's logarithm, then&#xA;$$l(ab) = l(a)+l(b)-l(1)$$&#xA;From this he realized that if $l(1)=0$, logarithms would be easier to use.  But he had already constructed his table.  Some ask about the &quot;base&quot; of Napier's logarithm. Given that it is not a true modern logarithm, that's a bit difficult to answer.  If you think that $10^7 \log x$ is the basic function underlying $l(x)$, you can think of it either as a scaled natural logarithm or a logarithm with the odd base of $\exp(10^{-7})$. Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Napierian_logarithm&quot; rel=&quot;noreferrer&quot;&gt;Napierian logarithm&lt;/a&gt; effectively has the base as $10^7/(10^7 - 1)$; but that would imply the logarithm of $x=9\,999\,999$ is exactly $y=1$, not $y=1.00000005$ as Napier calculated.  Briggs' base-10 logarithm mentioned above is really $10^9 \log_{10} x$.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2016-07-31T21:40:59.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5062" PostTypeId="1" CreationDate="2016-08-01T02:27:37.540" Score="2" ViewCount="267" Body="&lt;p&gt;I was researching a little about the hollow earth theory, I know that many Scientist's back up this theory, namely Euler and Edmond Halley.&#xA;I would like to know what was the theory that Johannes Kepler had about it and what is his idea of the structure of the earth in his book called Somnium, the problem is that the book was written in Latin, and it is a little difficult to understand for that I would like to appreciate to know the opinion of a expert in this area.&#xA;An also to know if is that true that Kepler in this particular book claimed that the earth is hollow.&lt;/p&gt;&#xA;" OwnerUserId="4479" LastEditorUserId="4251" LastEditDate="2021-02-28T08:08:59.470" LastActivityDate="2021-02-28T08:08:59.470" Title="Is it true that Johannes Kepler thought that the earth was hollow?" Tags="|physics|17th-century|kepler|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5063" PostTypeId="2" ParentId="5008" CreationDate="2016-08-03T17:31:06.323" Score="2" Body="&lt;p&gt;The law was first¹ formulated by Joseph Louis Lagrange in 1773,² followed by Carl Friedrich Gauss in 1813,³ both in the context of the attraction of ellipsoids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;Notes&lt;br&gt;¹ &lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx#1&quot; rel=&quot;nofollow&quot;&gt;Pierre Duhem&lt;/a&gt;'s &lt;a href=&quot;https://archive.org/stream/leonssurllec01duheuoft#page/22/mode/2up&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Leçons sur l'électricité et le magnétisme&lt;/em&gt; (liv. 1, chap. 4, p. 22-23)&lt;/a&gt; shows that Lagrange has priority over Gauss. Others after Gauss discovered &quot;Gauss's Law,&quot; too.&lt;br&gt;² Lagrange, &quot;&lt;a href=&quot;https://books.google.com/books?id=4XkAAAAAMAAJ&amp;amp;pg=PA619&quot; rel=&quot;nofollow&quot;&gt;Sur l'attraction des sphéroïdes elliptiques&lt;/a&gt;&quot; , &lt;em&gt;Mémoires de l'Académie de Berlin&lt;/em&gt;, p. 125; 1773.&lt;br&gt;³ Gauss (C.-F.) , &lt;a href=&quot;https://books.google.com/books?id=0TxeAAAAcAAJ&amp;amp;pg=PA3&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Theoria attractionis corporum sphœroidicorum ellipticorum homogeneorum methodo nova tractata&lt;/em&gt;&lt;/a&gt; (Gauss, &lt;em&gt;Werke&lt;/em&gt;, t. V, p. 1). Gauss mentions Newton's related &lt;a href=&quot;https://archive.org/stream/newtonspmathema00newtrich#page/n243/mode/2up&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Principia&lt;/em&gt; proposition XCI&lt;/a&gt; regarding finding the force exerted by a sphere on a point anywhere along an axis passing through the spehere.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;(source: &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Gauss%27s_law&amp;amp;type=revision&amp;amp;diff=732846510&amp;amp;oldid=726736706&quot; rel=&quot;nofollow&quot;&gt;my recent edits on the &quot;'Gauss's' Law&quot; page&lt;/a&gt;)&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-08-03T17:31:06.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5064" PostTypeId="2" ParentId="3256" CreationDate="2016-08-03T17:44:12.677" Score="5" Body="&lt;p&gt;In 1964 Källén wrote a textbook with the title &quot;Elementary Particle Physics&quot;, which introduces the symbol with the words &quot;The $\lambda(x,y,z)$ is a symbol which occurs frequently in this book.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Wightman's obituary for Gunnar Källén (&lt;a href=&quot;http://projecteuclid.org/euclid.cmp/1103841217&quot; rel=&quot;nofollow&quot;&gt;http://projecteuclid.org/euclid.cmp/1103841217&lt;/a&gt;), he states that &quot;A typical remark about&#xA;the book was: That is the book on elementary particles the experimentalists really find helpful&quot;, which might indicate a certain popularity of the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives one possible answer to your question.&lt;/p&gt;&#xA;" OwnerUserId="4531" LastEditorUserId="2442" LastEditDate="2016-08-04T22:45:42.203" LastActivityDate="2016-08-04T22:45:42.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5065" PostTypeId="1" AcceptedAnswerId="5071" CreationDate="2016-08-04T10:02:00.740" Score="7" ViewCount="3135" Body="&lt;p&gt;Who discovered the &lt;a href=&quot;https://en.wikipedia.org/wiki/L&amp;#39;H%C3%B4pital&amp;#39;s_rule&quot;&gt;rule of L'Hôpital&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Wikipedia article, it is mentioned that it was not l'Hôpital who proved the rule but that it might have been Johann Bernoulli.&#xA;I even heard something about l'Hôpital paid Bernoulli for the privilege to publish the proof instead of him.&#xA;Is this true? Are there any proofs for this theory? Or is it totally nonsense?&lt;/p&gt;&#xA;" OwnerUserId="4535" LastActivityDate="2017-06-03T15:54:06.153" Title="Who discovered L'Hôpital's rule?" Tags="|mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5066" PostTypeId="1" AcceptedAnswerId="7550" CreationDate="2016-08-04T10:10:19.037" Score="3" ViewCount="284" Body="&lt;p&gt;Let's get this out of the way- I am aware that this is nonsense today. However, historically it would have been considered science, so I'm asking here rather than in the History SE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Classically, foods were associated with the four humors. But while I will hear people occasionally say things like &quot;apples were phlegmatic,&quot; I have no idea where they came by this information. Apparently applesauce and pork were combined with this in mind, and most classical recipes had these balances in mind while being made. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I would like recommendations for good sources/resources to find the humor associations of different foodstuffs. I like experimenting in my kitchen and I thought it would be a fun theme to try, what can I say?&lt;/p&gt;&#xA;" OwnerUserId="4537" LastActivityDate="2018-07-27T11:30:40.160" Title="Looking for source of foods associated with specific humors" Tags="|ancient-greece|medicine|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5067" PostTypeId="1" AcceptedAnswerId="5087" CreationDate="2016-08-04T14:21:51.133" Score="3" ViewCount="144" Body="&lt;p&gt;I am reading a &lt;a href=&quot;http://www.nytimes.com/interactive/projects/cp/obituaries/archives/otto-hahn&quot; rel=&quot;nofollow noreferrer&quot;&gt;NYTimes obituary of Otto Hahn&lt;/a&gt; and the discovery of fission, and &lt;a href=&quot;https://physics.stackexchange.com/q/272581/83380&quot;&gt;I've asked in physics SE&lt;/a&gt; about the function of the apparatus shown in that article.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The system shown demonstrated the ability to &lt;em&gt;induced&lt;/em&gt; fission by neutrons. &lt;a href=&quot;https://physics.stackexchange.com/a/272587/83380&quot;&gt;This answer&lt;/a&gt; to the linked question links to this &lt;a href=&quot;https://www.iaea.org/newscenter/news/pioneering-nuclear-science-discovery-nuclear-fission&quot; rel=&quot;nofollow noreferrer&quot;&gt;IAEA article&lt;/a&gt; which explains much more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my question I'm not certain how to attribute the development of the apparatus itself. I'd like to get the facts right so I can word the attribution correctly. I understand that Lise Meitner was involved generally, and dismissed for (most likely) being Jewish. What are the details here? How should I attribute this experimental apparatus and effort?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;note:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/N1SWg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/N1SWg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Above:&lt;/strong&gt; Image from the &lt;a href=&quot;http://www.nytimes.com/interactive/projects/cp/obituaries/archives/otto-hahn&quot; rel=&quot;nofollow noreferrer&quot;&gt;NYTimes Obituary&lt;/a&gt; credited there to &lt;em&gt;Hulton Archive/Getty Images&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="-1" LastEditDate="2017-04-13T12:40:15.403" LastActivityDate="2016-08-11T04:15:08.797" Title="Lise Meitner's contribution to this experimental apparatus and research effort?" Tags="|discoveries|experimental-physics|measurement|nuclear-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5068" PostTypeId="1" CreationDate="2016-08-04T14:28:34.757" Score="5" ViewCount="221" Body="&lt;p&gt;Anybody who has boiled water knows that a positive pressure builds up when steam is produced. Indeed the first conceptual design of a steam engine (the Aeolipile) was a &quot;positive pressure&quot; engine. The fact that condensing steam produces a vacuum, however, is much less intuitive. Why, then, were the first industrial steam engines (Newcomen engines) atmospheric engines that produced work as the result of the vacuum created by condensing steam? Why did they not use the more &quot;obvious&quot; solution of steam pushing a piston? Were it technical concerns (high pressure leading to boiler explosions)?&lt;/p&gt;&#xA;" OwnerUserId="4539" LastEditorUserId="8" LastEditDate="2016-08-04T19:43:35.387" LastActivityDate="2018-01-09T18:38:54.820" Title="Why were the first steam engines “atmospheric engines”" Tags="|technology|steam-engines|18th-century|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5069" PostTypeId="2" ParentId="5066" CreationDate="2016-08-04T16:44:04.607" Score="1" Body="&lt;p&gt;For those curious about the origins and legacy of theory of humors, the Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Humorism&quot; rel=&quot;nofollow&quot;&gt;https://en.wikipedia.org/wiki/Humorism&lt;/a&gt; offers a good description, even mentioning some food and wine associations. But it does not say much about  how and why specific foodstuffs were associated with certain humors.  The correspondence was apparently not always obvious: according to the above article,  ``when the chili pepper was first introduced to Europe in the sixteenth century, dieticians disputed whether it was hot or cold.&quot; &lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2016-08-04T16:44:04.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5070" PostTypeId="1" CreationDate="2016-08-04T20:41:25.160" Score="0" ViewCount="265" Body="&lt;p&gt;In Topology courses one learns&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An even dimensional sphere does not possess any continuous field of unit vectors&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is the importance of this result? I can't think of any applications off the top of my head.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more explanation see J Milnor &lt;a href=&quot;https://www.jstor.org/stable/2320860&quot; rel=&quot;nofollow&quot;&gt;&lt;strong&gt;Analytic Proofs of the &quot;Hairy Ball Theorem&quot; and the Brouwer Fixed Point Theorem&lt;/strong&gt;&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="193" LastActivityDate="2016-08-05T14:13:39.857" Title="why was the hairy ball theorem important" Tags="|differential-geometry|topology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5071" PostTypeId="2" ParentId="5065" CreationDate="2016-08-04T22:52:20.963" Score="5" Body="&lt;p&gt;According to Carl Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt;, this is the case.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Boyer's account, Jean Bernoulli's father Nicolaus had intended Jean to become a merchant or physician, and Jean did in fact write his dissertation in 1690 on effervescence and fermentation...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;but the following year he [Jean Bernoulli] became so deeply interested in the calculus that during 1691-1692 he composed two little textbooks on the differential and integral calculus, although neither was published until long afterward.  While in Paris in 1692, he instructed a young Marquis, G.F.A. de L'Hospital (1661-1704) in the new Leibnizian discipline; and Jean Bernoulli signed a pact under which, in return for a regular salary, he agreed to send L'Hospital his discoveries in mathematics, to be used as the marquis might wish.  The result was that one of Bernoulli's chief contributions, dating from 1694, has ever since been know as L'Hospital's rule of indeterminate forms.  ....  This well-known rule was incorporated by L'Hospital in the first textbook on differential calculus to appear in print - &lt;em&gt;Analyse des infiniment petits&lt;/em&gt;, published in Paris in 1696.  This book, the influence of which dominated most of the eighteenth century, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Boyer goes on to say of L'Hospital :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;L'Hospital was an exceptionally effective writer, for his &lt;em&gt;Traite analytique des sections coniques&lt;/em&gt;, published posthumously in 1707, did for analytic geometry of the eighteenth century what the &lt;em&gt;Analyse&lt;/em&gt; did for calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-08-05T14:28:16.793" LastActivityDate="2016-08-05T14:28:16.793" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5072" PostTypeId="2" ParentId="5070" CreationDate="2016-08-05T04:07:21.020" Score="1" Body="&lt;p&gt;At every time, there is &lt;a href=&quot;http://elevr.com/elevrant-the-hairy-ball-theorem-in-vr-video/&quot; rel=&quot;nofollow&quot;&gt;a place on earth where there is no wind&lt;/a&gt;. But you could be as well in the eye of the tornado.&lt;/p&gt;&#xA;" OwnerUserId="4130" LastActivityDate="2016-08-05T04:07:21.020" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5073" PostTypeId="2" ParentId="5070" CreationDate="2016-08-05T14:13:39.857" Score="2" Body="&lt;p&gt;One important fact is that it shows that vector fields on the sphere are not the same as vector-valued functions, &lt;em&gt;i.e.&lt;/em&gt; the tangent bundle of the sphere is not trivial. The tangent bundle of the circle is trivial. This is why we need sections of bundles to describe vector fields on manifolds, a more complicated notion than the vector-valued functions we can use to describe vector fields on $\mathbb{R}^n$.&lt;/p&gt;&#xA;" OwnerUserId="4490" LastActivityDate="2016-08-05T14:13:39.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5074" PostTypeId="2" ParentId="57" CreationDate="2016-08-05T22:48:09.697" Score="2" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;His Law of the Photoelectric Effect (a misnomer - it should REALLY be called, the Quantization of the Radiation Field).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;His Paper on the Specific Heat of Solids (1906)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;His Paper on Quantum Vibrations (1907) ...Mind you he was the ONLY physicist in the world seriously working on quantum theory - even Bohr thought his idea of quantized energy (i.e. photons) was silly, given how great thinkers like Poissant and Maxwell had &quot;proven&quot; that light was a wave. Not a SINGLE notable scientist believed in Einstein's 1905 paper until at least the First Solvay Conference of 1911, and even then the vast majority were 'quantum skeptics.'&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;4.In 1909 Einstein was the first to show that statistical fluctuations in thermal radiation fields display both particlelike and wavelike behaviour; his was the first demonstration of what would later become the principle of complementarity.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;1916/1917 marks Einstein's most underrated paper. After finishing his magnum opus, General Relativity, he turned to the interplay of matter and radiation to create a quantum theory of radiation. He once again based his arguments on statistics and fluctuations. Bohr introduced a crucial new concept called stationary states in his 1913 paper on hydrogen but major features of Bohr's model could be interpreted as absolute nonsense because, according to electromagnetic theory, the electron would radiate intensely, emitting a broad spectrum as it crashed into the nucleus. Here we see contradictions in classical laws, and yet major properties of Bohr's hydrogen model rested on those laws.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Einstein, always the original thinker, didn't take as his starting point the well-known field for thermal radiation given by the Planck radiation law. Instead, he assumed that the atoms are in thermal equilibrium and then deduced the properties of the radiation field required to maintain the equilibrium. Guess what? The field turned out to be given precisely by the Planck radiation law. He manages to create quantum effects (stimulated and spontaneous emission) from most classical principles. He uses Wien's displacement law, the canonical Boltzmann distribution, Poynting's theorem, and microscopic reversibility - all classical. The sole quantum idea was the concept of stationary states. And yet from these elements, he's the first to create a complete description of the basic radiation processes and a full description of the general properties of the photon. In his 1917 paper, he creates novel and elegant derivations of Planck's radiation law as well as a proof of Bohr's frequency rule. In it, among many other things, he answers the question of how a gas of atoms maintain the populations of its stationary states in equilibrium with a radiation field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The aforementioned novel concept of spontaneous emission, which embodies the FUNDAMENTAL interaction of matter with the vacuum, is a brilliant, Nobel-Prize worthy achievement. Why? Spontaneous emission sets the scale for ALL radiative interactions. The rates of absorption and stimulated emission, for instance, are proportional to the rate for spontaneous emission. Spontaneous emission can be viewed as the ultimate irreversible process and the fundamental source of noise in all of nature. With the development of cavity quantum electrodynamics - the study of atomic systems in close-to-ideal cavities - in the 1980s, the phystical situation was profoundly altered. In such cavities, spontaneous emission evolves into spontaneous-cavity oscillations. Although the dynamical behavior is totally altered, the atom-vacuum interaction that causes spontaneous emission sets the time scale for that evolution. It is first in Einstein's 1917 paper that the photon is demonstrated to possess all the properties of a fundamental excitation, and therefore it is quite clear that his radiation paper played a seminal role in the eventual creation of quantum electrodynamics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apropos the second brilliant creation of his 1917 paper, stimulated emission of radiation, we see the first genesis of the laser. Stimulated emission underlies the basic mechanism of the laser and, by extension, laser cooling; his analysis of momentum transfer in a thermal radiation field can be immediately applied to atomic motion in a laser field. If the spectral width of a thermal field is replaced by the natural linewidth of the atom, Einstein's viscous damping force would give rise to the phenomenon known as optical molasses. This fundamental process of laser cooling was rediscovered by the atomic community in the 80's. Of course, you need Quantum Mechanics for a full realization of all the mechanisms of radiation, but papers like this are seminal contributions to what would eventually become QM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's theory of radiation provided a complete characterization of the particlelike properties of the light quantum and, in retrospect, he was within an arms grasp of working out the statistical mechanics of these particles. Given that his 1905 proposal for the energy quantization of radiation was based on the analogy between entropies of thermal radiation and a system of particles, it is surprising that Einstein didn't extend his method of reasoning to derive the Planck law by treating photons as indistinguishable particles. He was VERY close, and it is quite apparent that Bose himself did not realize he had done anything novel.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;On the Quantization of Chaos (1919): In it, Einstein was the first to point out the fundamental problems that arise when one applies classical chaos theory to quantum states (a paper 50 years ahead of its time as this is a problem we have only now began to fully grasp): &lt;a href=&quot;http://boulderschool.yale.edu/sites/default/files/files/Einstein_chaos.pdf&quot; rel=&quot;nofollow&quot;&gt;http://boulderschool.yale.edu/sites/default/files/files/Einstein_chaos.pdf&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Fast forward to 1924 and Einstein, not Bose, applied the reasoning in Bose's treatment of photons as indistinguishable particle to a gas of indistinguishable atoms thereby creating Bose-Einstein statistics, and later, Bose-Einstein Condensation. Subsequently, Einstein theorized Bose-Einstein condensation, a work for which 6 Nobel Prizes have been given. Einstein was 45% of the way to the Schrodinger Equation. It was only after Schrodinger had read Einstein's paper that he derived his equations governing the wave function. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Einstein was the first to conceive of ghost fields as probability densities, a concept which he applied to a gas of photons (i.e. probability waves). Max Born essentially took the idea verbatim and applied it to electrons. Born always acknowledged this. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;EPR Paradox Paper: the first paper to show how quantum entanglement emerges from the QM's equations. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;*Einstein's work on wave-particle duality led directly to De Broglie's thesis on matter waves, and it seems unlikely De Broglie would have conceived of it without Einstein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein is pretty much the father of early quantum theory, and is one of the co-founders of modern Quantum Mechanics. The three major statistical systems governing the microscopic realm are: Fermi-Dirac Statistics, Einstein-Bose Statistics, and the Boltzmann Statistics. He would rightly be regarded as a legend for his work on BEC alone, and yet he contributed massively to Quantum Mechanics. Kindly check out his paper on the quantization of chaos, it's absolutely brilliant and shows how indispensable his thinking was to the development of QM. &lt;/p&gt;&#xA;" OwnerUserId="4315" LastEditorUserId="4315" LastEditDate="2016-08-05T22:58:50.670" LastActivityDate="2016-08-05T22:58:50.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5075" PostTypeId="2" ParentId="4941" CreationDate="2016-08-05T23:42:51.313" Score="41" Body="&lt;p&gt;To begin, let's start with a very apropos anecdote from Lorentz himself. At a conference on the Michelson–Morley experiment in 1927 at which Lorentz and Michelson were present, Michelson suggested that Lorentz was the initiator of the theory of relativity. Lorentz then replied: “I considered my time transformation only as a heuristic working hypothesis. So the theory of relativity is really solely Einstein's work. And there can be no doubt that he would have conceived it even if the work of all his predecessors in the theory of this field had not been done at all. His work is in this respect independent of the previous theories.” Lorentz, replying to Michelson at the Solvay Conference.&lt;/p&gt;&#xA;&lt;p&gt;Poincaré was confused on several points. (See the discussion on Wikipedia regarding “mass energy equivalence”.) He could never get the mechanical relations straight, since he could not figure out that &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt;. Einstein was aware of Poincaré's work, but he derived the theory simply as a geometric symmetry, and made a complete system from first principles using one, and only postulate: the speed of light is invariant and is the same for all frames of reference.&lt;/p&gt;&#xA;&lt;p&gt;Einstein did share the credit with Lorentz and Poincaré for special relativity for a while, and Fitzgerald and Maxwell deserve credit for their contributions as well. However, it was Einstein, and Einstein alone, that was able to derive SR from first principles without &lt;em&gt;ad hoc&lt;/em&gt; solutions to the transformation problem.  The great physicist Wolfgang Pauli in the Encyclopædia Britannica article famously credits Einstein alone for formulating the relativity principle, as did Lorentz. Poincaré was less accommodating but that is probably because he never fully understood Einsteinian relativity and was mired in the Galilean conception of relativity in which there is always a privileged frame of reference. Both Lorentz and Poincaré said some variation of “Einstein just assumed that which we were all trying to prove” (namely the principle of relativity) - Lorentz meant it affectionately whereas Poincaré mean it as an insult. Poincaré was still trying to get relativity from Maxwell's equations, rather than making a new postulate, and this is a huge difference. See below.&lt;/p&gt;&#xA;&lt;p&gt;Special relativity was ripe for discovery in 1905, but it was Einstein, and Einstein alone, who derived a complete system from first principles (using one fundamental postulate) and only he correctly derived &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt; from said postulate. Without his derivation of &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt;, nothing makes sense. Poincaré, Lorentz, Fitzgerald, and Maxwell deserve 50% of the credit (as Einstein himself accepted), but that's like splitting hairs and pointing out that Descartes and Fermat deserve 70% of the credit for calculus when Newton and Leibniz were the first to integrate the various derivations of analytical geometry.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Lorentz actually misinterpreted his own transformations as applying to the ether (in fact I think he derived them purposely to describe how the ether “reacts” in such a way as to be in accord with experiments (Michelson-Morley). So Einstein gets credit for being the first to correctly interpret the equations and do away with the ether concept. Both Poincaré and Lorentz believed in the ether - a non-trivial fact - which represents a privileged frame of reference (Galilean relativity) as opposed to a coordinate system that has NO preferred reference frame (Einsteinian relativity).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Einstein derived &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt; from first principle, notably that energy carries inertia from emitter to absorber, and the resulting rest frame that the separation “mass” and “energy” entails. This derivation of &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt; as a consequence of special relativity is epistemically critical to Dirac's later reconciliation of special relativity and quantum mechanics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Einstein had the correct electrodynamic transformations in 1905, and the correct energy density and momentum density expressions, and the right relation between mass and energy. Poincaré had all but the last, and this got him confused on several important points. Poincaré did not have spacetime geometry before Minkowski because: A) he did not define it through non-galilean relativity, B) Poincaré did not express particle motion in terms of a worldline, or define proper time as a worldline parameter. So inasmuch as spacetime geometry includes worldlines and proper time, Poincaré did not discover spacetime geometry. Einstein's complete systemization of the Lorentz transformations and the second order transformations essentially wove space and time together in a way that is non-Galilean (Poincaré never got that far). Minkowski, took Einstein's interval relation and mathematized it in 4-d vectors (but again, it was already implied in Einstein's 1905 paper).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Poincaré noted a preference against using Minkowski spacetime. Another curiosa, Moskowzki sat in the audience at his Berlin lecture where Poincaré had talked about Einstein's work both in the positive and negative. The speech-transcript does not include any explicit mention of Einstein by name though, so we are left to assume he did a bit of adlib when talking about Einstein specifically. From Moskowzki's impression, it seems Poincaré did indeed regard Einstein's work as not only different to his but… too revolutionary and daring. See &lt;a href=&quot;http://mathpages.com/home/kmath630/kmath630.htm&quot; rel=&quot;noreferrer&quot;&gt;this link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Einstein argued that a light pulse which is spherical in one inertial frame, is spherical in every inertial frame. According to Poincaré, a light pulse that is spherical in the above mentioned privileged frame is an elongated ellipsoid in every other inertial frame. The difference in description is due to that fact that Einstein recognized the relativity of spatio-temporal coordinates, when Poincaré did not. And, the aberration constant, Poincaré didn't derive it, Einstein did.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Although Poincaré understood independently of Einstein how the Lorentz transformations give rise to non-Galilean transformation rules for velocities (indeed Poincaré derived the correct relativistic rules), it is clear that he did not have a full appreciation of the modern operational significance attached to coordinate transformations. He did not seem to understand the role played by the second-order terms in the transformation. Compared with the cases of Lorentz and Larmor, it is clear that Poincaré did not fully understand either length contraction or time dilation to be a consequence of the coordinate transformation. What Poincaré was holding out for was no less than a new theory of ether and matter — something far more far-fetched than what appeared in Einstein's 1905 relativity paper. Einstein's rejection of the aether is important, but is not the pivotal difference between his conception of relativity and Lorentz/Poincaré conceptions. Why? Because it is always possible to add for whatever reason the notion of a privileged frame to special relativity, as long as one accepts that it will remain unobservable. However, in addition to the examples given above, there are other new features in Einstein's work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;*The full meaning of relativistic kinematics was simply not properly understood before Einstein. Nor was the 'theory of relativity' as Einstein articulated it in 1905 anticipated even in its programmatic form. It is impossible to understand the full implications of Einstein's discovery of special relativity without taking on board the impacts of the quantum in physics (see Paul Dirac). In respect to the conventional nature of distant simultaneity, Einstein was doing little more than expanding on a theme that Poincaré had already introduced. Where Einstein goes well beyond the great mathematician is in his treatment of the coordinate transformations. In particular, the extraction of the phenomena of length contraction and time dilation directly from the Lorentz transformations in section 4 of the 1905 paper is completely original. The genius of Einstein's 1905 paper is that the modern, dynamical interpretation of special relativity — as opposed to the kinematical approach of Einstein's 1905 paper — is ALREADY contained in Einstein's 1905-paper “masqueraded in the language of kinematics” (Physicist Harvey Brown) and the modern understanding of space-time.&lt;/p&gt;&#xA;&lt;p&gt;You cannot formulate General Relativity using Poincaré/Lorentz's conceptualization of relativity. You can, and Einstein did, using Einsteinian SR. Differences, for instance, in how one obtains the conservation of mass in GR vary widely depending on whether you use Einstein's approach or Poincaré/Lorentz. I hope this settles the debate once and for all. Until his death, Poicaré never fully grasped the full implications of what Einstein had done. Lorentz, a man Einstein called “the smartest man I have ever known,” eventually came around to it once General Relativity proved so successful.&lt;/p&gt;&#xA;" OwnerUserId="4315" LastEditorUserId="10608" LastEditDate="2024-03-18T08:26:03.593" LastActivityDate="2024-03-18T08:26:03.593" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5076" PostTypeId="2" ParentId="5055" CreationDate="2016-08-06T10:03:02.797" Score="3" Body="&lt;p&gt;On my opinion, the following citation from Poincare answers this question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“At the time, when Maxwell initiated his studies,&#xA;the laws of electrodynamics adopted before him explained&#xA;all known phenomena. He started his work&#xA;not because some new experiment limited the importance&#xA;of these laws. But, considering them from a&#xA;new standpoint, Maxwell noticed that the equations&#xA;became more symmetric, when a certain term was introduced&#xA;into them, although, on the other hand, this&#xA;term was too small to give rise to phenomena, that&#xA;could be estimated by the previous methods.&#xA;A priori ideas of Maxwell are known to have waited&#xA;for their experimental confirmation for twenty years;&#xA;if you prefer another expression, — Maxwell anticipated&#xA;the experiment by twenty years&quot;. (H. Poincare, The value of Science).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell's contribution was introduction of the so-called &quot;displacement current&quot;.&#xA;Then his equations predicted the existence of electromagnetic waves, the thing experimentally confirmed by Hertz.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-08-06T10:03:02.797" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5077" PostTypeId="1" AcceptedAnswerId="5079" CreationDate="2016-08-06T22:56:39.743" Score="3" ViewCount="112" Body="&lt;p&gt;Coefficient matching feels rather natural when solving equations and checking dimensions, however in footnote 2 to &lt;a href=&quot;http://web.ist.utl.pt/ist12219/data/60.pdf&quot; rel=&quot;nofollow&quot;&gt;&quot;Two alternative derivations of Bridgman's theorem&quot;&lt;/a&gt; (Berberan-Santos M N, Pogliani L, J. Math Chem 1999, 26, 255-261; p256) it's mentioned that Descartes shifted from line/area/volume ideas to the basic number line view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this where the idea of coefficient matching between the terms when solving polynomials began, or was it already well established for pure number polynomials?&lt;/p&gt;&#xA;" OwnerUserId="4547" LastActivityDate="2016-08-07T16:52:46.120" Title="when did polynomial coefficient matching start for solving equations?" Tags="|mathematical-physics|units|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5078" PostTypeId="1" CreationDate="2016-08-07T13:34:43.137" Score="8" ViewCount="1044" Body="&lt;p&gt;Neither Maxwell’s fundamental differential equations on electromagnetism nor Einstein’s first papers considered the magnetic vector potential A. &#xA;So who discovered, formulated or used A for the first time?&lt;/p&gt;&#xA;" OwnerUserId="4438" LastEditorUserId="232" LastEditDate="2017-03-27T23:17:16.350" LastActivityDate="2021-01-29T04:25:59.697" Title="Who discovered the magnetic vector potential, $\vec{A}$?" Tags="|electromagnetism|potential|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5079" PostTypeId="2" ParentId="5077" CreationDate="2016-08-07T16:52:46.120" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/sums-of-powers-of-positive-integers-thomas-harriot-c-1560-1621-england&quot; rel=&quot;nofollow&quot;&gt; Thomas Harriot, 1560-1621&lt;/a&gt; matched coefficients when generating formulas for sums of powers of positve integers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Gathering like terms&quot;, is, of course, an obvious technique, and was also used in the geometric derivations of these formulas. The link provided goes to the middle of a long article on the history of these formulas; Harriot appears to be the first to use symbolic notation, which allowed him to focus on the coefficients. Descarte, Fermat, and Bernoulli are all later.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-08-07T16:52:46.120" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5080" PostTypeId="2" ParentId="5078" CreationDate="2016-08-08T21:40:16.797" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Neumann_Franz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Franz Ernst Neumann&lt;/a&gt; was the first¹ to write down the magnetic vector potential in his 1845 paper &quot;General laws of induced electrical currents.&quot; He used it to write the equation summarizing Faraday's induction experiment (Faraday's law).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original paper:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;F. E. Neumann, “&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Allgemeine Gesetze Der Inducirten Elektrischen Ströme&lt;/a&gt;,” &lt;em&gt;Annalen Der Physik&lt;/em&gt; 143, no. 1 (January 1, 1846): 31–44, doi:&lt;a href=&quot;https://dx.doi.org/10.1002/andp.18461430103.&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1002/andp.18461430103&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Biography/Neumann__Franz_Ernst.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neumann's &lt;em&gt;Dictionary of Scientific Biography&lt;/em&gt;&lt;/a&gt; entry says this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Neumann and his contemporary &lt;a href=&quot;http://www.encyclopedia.com/topic/Wilhelm_Eduard_Weber.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilhelm Weber&lt;/a&gt; were the founders of the electrodynamic school in Germany, which later included, among others, Riemann, Betti, Carl Neumann, and Lorenz. The investigations and analyses of this group were guided by the assumption, held originally by Ampère, that electromagnetic phenomena resulted from direct action at a distance rather than through the mediation of a field. Neumann’s major contributions were contained in two papers published in 1845 and 1848, in which he established mathematically the laws of induction of electric currents. The papers, transmitted to the Berlin Academy, were entitled “Allgemeine Gesetze der inducirten elektrischen Ströme” and “Ober ein allgemeines Princip der mathematischen Theorie inducirter elektrischer Ströme.”&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;As a starting point Neumann took the proposition, formulated in 1834 by F. E. Lenz after Faraday’s discovery of induction, that the current induced in a conductor moving in the vicinity of a galvanic current or a magnet will flow in the direction that tends to oppose the motion. In his mathematical analysis Neumann arrived at the formula $E.Ds =—ν C.Ds$, where $Ds$ is an element of the moving conductor, $E.Ds$ is the elementary induced electromotive force, $v$ is the velocity of the motion, $C.Ds$ is the component of the inducing current, and $∊$ is a constant coefficient. With this formula Neumann was able to calculate the induced current in numerous particular instances. At present a common formulation is $E = — dN/dt$, where $E$ is the electromotive force generated in the circuit through which the number of magnetic lines of force is changing at the rate of $dN/dt$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Continuing his analysis Neumann noticed a way in which the treatment of currents induced in closed circuits moving in what is now termed a magnetic field might be generalized. He saw that the induced current depends only on the alteration, caused by the motion, in the value of a particular function. Considering &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_force_law&amp;amp;oldid=758353198#Equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère’s [force] equation&lt;/a&gt;s for a closed circuit, Neumann arrived at what is known as the mutual potential of two circuits, that is, the amount of mechanical work that must be performed against the electromagnetic forces in order to separate the two circuits to an infinite distance apart, when the current strengths are maintained unchanged. In modern notation the potential function, $Vii'$, is written:$$Vii'=-ii'\iint\frac{\mathbf{ds}\cdot\mathbf{ds'}}{r}.$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Continuing his analysis Neumann noticed a way in which the treatment $\mathbf{ds}.\mathbf{ds'}$ is the scalar product of the two vectors $\mathbf{ds}$ and $\mathbf{ds'}$, and $r$ their distance apart. If a fixed element $\mathbf{ds'}$ is taken and integrated with respect to $\mathbf{ds}$, the vector potential of the first circuit at the point occupied by $\mathbf{ds}$ is obtained. Maxwell arrived at the concept of vector potentials by another method and interpreted them as analytical measures of Faraday’s electrotonic state.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&lt;sup&gt;References&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;André Koch Torres Assis&lt;/a&gt;, &lt;a href=&quot;https://isidore.co/calibre/browse/book/4302&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Weber’s Electrodynamics&lt;/em&gt;&lt;/a&gt; (Dordrecht; Boston: Kluwer Academic, 1994), §5.2 (&quot;Franz Neumann&quot;), pp. 122-5.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maxwell used the &quot;magnetic vector potential&quot; (what he called the &quot;electromagnetic momentum&quot;) in his &lt;em&gt;A Dynamical Theory of the Electromagnetic Field&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/scientificpapers01maxw#page/554/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Scientific Works&lt;/em&gt; vol. 1 p. 555&lt;/a&gt;-556). $\mathbf{E}$ is $(P,Q,R)$ and $\mathbf{A}$ is $(F,G,H)$ in Maxwell's notation.&lt;/sup&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2017-02-12T00:42:15.723" LastActivityDate="2017-02-12T00:42:15.723" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5081" PostTypeId="2" ParentId="5006" CreationDate="2016-08-09T13:00:54.850" Score="0" Body="&lt;p&gt;The study of tangents goes back at least to Apollonius but was given a strong impetus through the work of Pierre de Fermat.  Fermat developed a technique called adequality or &quot;approximate equality&quot; which would amount in modern terminology to dropping second-order terms in an equation involving a small $E$.  Fermat did not have the notion of derivative but he made great strides toward developing the techniques that would later be unified by Leibniz and Newton.  For a recent discussion of his work at MO see &lt;a href=&quot;https://mathoverflow.net/questions/247065/fermats-enemies&quot;&gt;https://mathoverflow.net/questions/247065/fermats-enemies&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-08-09T13:00:54.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5082" PostTypeId="1" AcceptedAnswerId="5086" CreationDate="2016-08-10T08:26:44.863" Score="3" ViewCount="658" Body="&lt;p&gt;The imaginary unit is generally denoted $i$ or $\imath$. I have learned that the term imaginary (&quot;imaginaires&quot;) was coined by R. Descartes in 1637, and the &quot;i&quot; notation was introduced by L. Euler (cf. &lt;a href=&quot;http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Short History of Complex Numbers&lt;/a&gt;). In engineering and physics, the notation $j$ or $\jmath$ is often used. Some say it was used to limit confusions with the current, often denoted $i$ or &quot;I&quot;, as noted at &lt;a href=&quot;https://en.wikipedia.org/wiki/Electric_current#Symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;Electric current&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The conventional symbol for current is I, which originates from the&#xA;  French phrase intensité de courant, meaning current intensity .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Was the $\imath$ (or $i$) notation used before Euler?&lt;/li&gt;&#xA;&lt;li&gt;Who introduced the $\jmath$ (or $j$) notation?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;EDIT: I just found, answering &lt;a href=&quot;https://dsp.stackexchange.com/questions/33593/what-is-the-real-meaning-of-frequency-for-a-non-periodic-signal&quot;&gt;on a different topic about frequency&lt;/a&gt;, a mention of the word &quot;cisoid&quot; (abbreviated as $\mathrm{cis}$), I just found &lt;a href=&quot;http://dx.doi.org/10.1109/PAIEE.1911.6659711&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cisoidal Oscillations&lt;/a&gt;, 1911, by George A. Campbell, where he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The use of $i$ (or Greek $\imath$) for the imaginary symbol is nearly&#xA;  universal in mathematical work, which is a very strong reason for&#xA;  retaining it in the applications of mathematics in electrical&#xA;  engineering. Aside, however, from the matter of established&#xA;  conventions and facility of reference to mathematical literature, the&#xA;  substitution of the symbol $j$ is objectionable because of the vector&#xA;  terminology with which it has become associated in engineering&#xA;  literature, and also because of the confusion resulting from the&#xA;  divided practice of engineering writers, some using $j$ for $+i$ and&#xA;  others using $j$ for $- i$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Orezp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Orezp.jpg&quot; alt=&quot;against the j&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the story is not fully new, and we could date the $j$ before 1911.&lt;/p&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="-1" LastEditDate="2017-04-13T12:47:29.640" LastActivityDate="2016-08-29T18:43:10.650" Title="Introduction of $\imath$ and $\jmath$ notations for the imaginary unit" Tags="|mathematics|terminology|notation|complex-analysis|engineering|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5083" PostTypeId="2" ParentId="5082" CreationDate="2016-08-10T08:44:27.217" Score="2" Body="&lt;p&gt;I doubt it that the $i$ notation would have been used for $\sqrt{-1}$ before Euler because even Euler himself did not start using it until a rather late date, and moreover used $i$ in a different sense namely for an infinite integer, in his &lt;em&gt;Introductio&lt;/em&gt; and &lt;em&gt;Institutiones&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-08-10T08:44:27.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5084" PostTypeId="1" AcceptedAnswerId="5091" CreationDate="2016-08-10T17:02:57.827" Score="5" ViewCount="1360" Body="&lt;p&gt;Symmetry has become a central concept in mathematics.  The Euclidean concept of similarity is an example of symmetry, but similarity was not a subject of study in itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: How did symmetry come to take centre stage and become the subject of study in itself?  In particular, what results led mathematicians to focus their attention on the concept of symmetry.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-08-29T08:30:23.840" Title="What are the origins of the study of symmetry as a subject in itself?" Tags="|mathematics|group-theory|geometry|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5085" PostTypeId="1" CreationDate="2016-08-10T20:40:55.327" Score="8" ViewCount="375" Body="&lt;p&gt;Did there exist and does there still exist a debate over which school of mathematical thought (i.e. formalism, logicism, intuitionism, etc.) had the most affinity or application for physics?  In particular, I am looking at if a case can be made linking one these schools of mathematical thought can be linked to physics discoveries during the first half of the 20th century such as Einstein's discoveries of general relativity.&lt;/p&gt;&#xA;" OwnerUserId="4560" LastEditorUserId="12724" LastEditDate="2022-05-17T21:25:48.720" LastActivityDate="2022-05-17T21:25:48.720" Title="Has there been debate between relationship of philosophy of mathematics and physics?" Tags="|philosophy-of-science|relativity-theory|mathematical-physics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5086" PostTypeId="2" ParentId="5082" CreationDate="2016-08-11T02:44:37.163" Score="4" Body="&lt;p&gt;In answer to your second part of the question regarding $j$ for $\sqrt{-1}$, this was introduced into text books describing Power System Analysis of AC power circuits in the early 1900s by Charles P. Steinmetz.  I am not sure of the earliest date but my guess is between late 1890s and 1920s but certainly no later than 1923 as Steinmetz died in 1923.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Steinmetz is to Power Systems Engineers (EEs) as Einstein and $E=m c²$ is to physicists (and everyone else for that matter).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can read all about Charles Steinmetz &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Proteus_Steinmetz&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2016-08-11T02:44:37.163" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5087" PostTypeId="2" ParentId="5067" CreationDate="2016-08-11T04:15:08.797" Score="2" Body="&lt;p&gt;The above photo was reproduced in the biography: Lise Meitner, A life in Physics, by Ruth Lewin Sime; University Of California Press,1996, ISBN 0-520-20860-9.&#xA;A brief description of the parts is given on page  371, bottom and 372 top. The attached photo, describing some of the components, is from those included with the text. &#xA;&lt;a href=&quot;https://i.stack.imgur.com/dOuzU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dOuzU.jpg&quot; alt=&quot;Lise Meitner&amp;#39;s apparatus&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4561" LastActivityDate="2016-08-11T04:15:08.797" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5088" PostTypeId="1" AcceptedAnswerId="5089" CreationDate="2016-08-11T05:14:17.970" Score="3" ViewCount="97" Body="&lt;p&gt;In machine learning, people have been more and more letting go of the idea that they should understand how a particular algorithm works, and accepting it on the basis that it &quot;just works&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there other examples in the history of science where people started to achieve success by transitioning to methods that are less clearly understood than the previously used methods?&lt;/p&gt;&#xA;" OwnerUserId="4556" LastEditorUserId="2442" LastEditDate="2016-08-12T15:46:10.217" LastActivityDate="2016-08-12T15:46:10.217" Title="Are there fields of science that became more successful after becoming less clearly understood?" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5089" PostTypeId="2" ParentId="5088" CreationDate="2016-08-11T08:27:10.227" Score="4" Body="&lt;p&gt;The emergence of infinitesimal methods in the 17th century led to just such a situation.  The old-fashioned methods sometimes called &quot;method by exhaustion&quot; worked to calculate areas and solve other geometric problems, but they were cumbersome and less effective than the new infinitesimal methods.  On the other hand, it was less clear how exactly the infinitesimal methods work.  Moreover infinitesimals posed something of a philosophical if not theological challenge, in that they were entities you can't pin down the way you can pin down ordinary numbers.  A major battle raged that had talented mathematicians on both sides, and it took decades before the new techniques were accepted.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-08-11T08:27:10.227" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5090" PostTypeId="2" ParentId="5085" CreationDate="2016-08-11T08:34:22.303" Score="1" Body="&lt;p&gt;One such debate opposed the historian Geoffrey Hellman and constructivists like Douglas Bridges; see e.g., the article&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hellman, Geoffrey.&#xA;Quantum mechanical unbounded operators and constructive mathematics—a rejoinder to Bridges. &#xA;J. Philos. Logic  26  (1997),  no. 2, 121–127. &lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2016-10-02T08:47:47.003" LastActivityDate="2016-10-02T08:47:47.003" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5091" PostTypeId="2" ParentId="5084" CreationDate="2016-08-11T08:53:39.227" Score="6" Body="&lt;p&gt;The traditional answer to questions on symmetry would be to point to Felix Klein's &lt;a href=&quot;https://en.wikipedia.org/wiki/Erlangen_program&quot;&gt;&lt;em&gt;Erlangen Program&lt;/em&gt;&lt;/a&gt; as a way of systematizing the study of symmetry by focusing on the symmetries of a manifold as the essential feature thereof.  Earlier sources include of course Galois' group theory, which can always be interpreted as a study of symmetry.  In a multicultural vein, you would find the Mayas and the symmetries on their clay pots; modern historians are good at this sort of thing though I personally am sceptical; see &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=3245163&quot;&gt;this review&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-08-11T08:53:39.227" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5092" PostTypeId="2" ParentId="5084" CreationDate="2016-08-11T19:03:45.093" Score="6" Body="&lt;p&gt;In my opinion, there are two separate questions being asked here. The expectation that the answer should concern 18th-19th century was not formulated in the statement of the question(s), although one can say that this is when mathematicians started to `` focus their attention on the concept of symmetry.&quot;  At any rate, imposing such a time frame leaves out many important developments. So let me supplement Mikhail's answer with earlier examples and possible motivations (from traditional Western culture). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &quot;How did symmetry come to take centre stage and become the subject of study in itself?&quot; Two motivations emerged in antiquity: the study of nature and aesthetics, sometimes intertwined. In Plato's dialogue &quot;Timaeus&quot; (to which I linked in an earlier comment) an early cosmological model (so to speak) is offered. Plato seeks to explain the order and beauty observed in the nature. His explanation is that humans, animals, heavenly bodies etc. are orderly and beautiful because they are made out of orderly and beautiful `&quot;particles&quot; by the divine Craftsman (Demiurge). These &quot;particles&quot; are four Platonic solids: tetrahedron, octahedron, icosahedron and cube, corresponding tho the four &quot;elements&quot; considered by earlier philosophers. The dodecahedron can be thought to approximate the shape of the universe, which is a sphere that moves in a circular motion (most symmetric, hence most ``perfect&quot;). Formation of physical matter and phase transitions reduce to the composition, decomposition and exchange of triangles into which the faces of the Platonic solids can be divided. Hence the   study of nature and universe can be based on the study of symmetric solids and triangles (I am just summarizing Plato, not formulating  my own credo, in case someone wonders).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Roman architect Vitruvius &lt;a href=&quot;https://en.wikipedia.org/wiki/Vitruvius&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Vitruvius&lt;/a&gt; emphasized the importance of symmetry and proportion in architecture, which to him was imitation of nature and therefore should reflect the patterns of the cosmic order. Moreover, architecture should be useful to the human, whose body is the ultimate work of art. Vitruvius's  ten books on architecture inspired early modern architects,especially   Leonardo da Vinci and Andrea Palladio. The picture of the Vitruvian Man  by da Vinci is a well-known image, &lt;a href=&quot;https://en.wikipedia.org/wiki/Vitruvian_Man&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Vitruvian_Man&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And long before Galois theory there was substantial interest in symmetric functions, especially in relation with roots of algebraic equations. The names and results of Vieta, Girard and Newton are best known. Many more details can be found in the following article:&#xA;H. Gray Funkhouser:A short account of the history of symmetric functions of roots of equations, American Mathematical Monthly, 37 (7) (1930), 357–365: &#xA;&lt;a href=&quot;http://www.jstor.org/stable/2299273?origin=crossref&amp;amp;seq=5#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;http://www.jstor.org/stable/2299273?origin=crossref&amp;amp;seq=5#page_scan_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="3434" LastEditDate="2016-08-11T19:15:43.137" LastActivityDate="2016-08-11T19:15:43.137" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5093" PostTypeId="1" AcceptedAnswerId="5114" CreationDate="2016-08-11T22:45:18.457" Score="13" ViewCount="1451" Body="&lt;p&gt;It seems that the focus of mathematical research moves on every so often, and some areas are not proven wrong, but have just become uninteresting in the current mathematical culture.  I was under the impression that solutions to quartic (and high power) equations might be one of these abandoned area, but are there others? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &lt;a href=&quot;https://hsm.stackexchange.com/questions/4913/what-are-some-of-the-most-complete-genealogies-of-scientific-and-mathematical-su&quot;&gt;question&lt;/a&gt; has the current areas of mathematics mapped out, but what I'm looking for would be more the changes in the map over time, and which node are excluded from the current mapping.&lt;/p&gt;&#xA;" OwnerUserId="4565" LastActivityDate="2022-11-09T11:47:58.857" Title="What major areas of mathematics have been abandoned?" Tags="|mathematics|reference-request|academia|social-context|resource-recommendation|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5094" PostTypeId="2" ParentId="5093" CreationDate="2016-08-12T07:43:22.737" Score="9" Body="&lt;p&gt;&lt;strong&gt;Geometry&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure you can really call geometry abandoned, but it certainly was much more popular a few hundred years ago (discovery of spherical geometry and hyperbolical geometry, parallel axiom debate) and a few thousand years ago (the old Greeks developed a lot of geometry). Nowadays, there are very few papers about just plain Euclidean geometry (or spherical geometry, or hyperbolical geometry, for that matter). &lt;a href=&quot;http://arxiv.org/&quot;&gt;arXiv&lt;/a&gt; doesn't even have a specific category for it. They do have Algebraic Geometry, Metric Geometry and Differential Geometry, but these fields are no where near just studying the properties of triangles, circles, etc, while I think there are still things to discover. It might be partially a result of having only very few open problems. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2016-08-12T07:43:22.737" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5095" PostTypeId="2" ParentId="5060" CreationDate="2016-08-12T15:28:51.213" Score="1" Body="&lt;p&gt;According to the Wiki article you've linked to, Antipositivism is a relatively new concept which Newton would never have heard of. The article also explains that it only really applies to Social Sciences, according to it's proponents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I think it came about and is applied by who it's applied by because that can't, in fact, prove anything empirically and rigorously about their areas of science.&lt;/p&gt;&#xA;" OwnerUserId="1750" LastActivityDate="2016-08-12T15:28:51.213" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5096" PostTypeId="2" ParentId="3567" CreationDate="2016-08-13T00:04:33.697" Score="2" Body="&lt;p&gt;From this good review:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ricardo Karam, Debora Coimbra, and Maurício Pietrocola, “&lt;strong&gt;&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/7Q29PUT4/Karam%20et%20al.%20-%202014%20-%20Comparing%20Teaching%20Approaches%20About%20Maxwell%e2%80%99s%20Disp.pdf&quot; rel=&quot;nofollow&quot;&gt;Comparing Teaching Approaches About Maxwell’s Displacement Current&lt;/a&gt;&lt;/strong&gt;,” &lt;em&gt;Science &amp;amp; Education&lt;/em&gt; 23, no. 8 (August 1, 2014): 1637–61, doi:&lt;a href=&quot;https://dx.doi.org/10.1007/s11191-013-9624-3&quot; rel=&quot;nofollow&quot;&gt;10.1007/s11191-013-9624-3&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;on p. 1646 they quote this passage of Maxwell's &lt;a href=&quot;https://archive.org/stream/electricandmag02maxwrich#page/n173/mode/2up&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;Treatise on Electricity &amp;amp; Magnetism&lt;/em&gt; (vol. 2, pp. 142-144)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The current produces magnetic phenomena in its neighborhood. If any closed curve be drawn, and the line-integral of the magnetic force taken completely round it, then, if the closed curve is not linked with the circuit, the line-integral is zero, but if it is linked with the circuit, so that the current &lt;em&gt;i&lt;/em&gt; flows through the closed curve, the line-integral is &lt;em&gt;4πi&lt;/em&gt; […] Note—The line-integral &lt;em&gt;4πi&lt;/em&gt; depends solely on the quantity of the current, and not on any other thing whatever. It does not depend on the nature of the conductor through which the current is passing, as, for instance, whether it be a metal or an electrolyte, or an imperfect conductor. &lt;strong&gt;We have reason for believing that even when there is no proper conduction, but merely a variation of electric displacement, as in the glass of a Leyden jar during charge or discharge, the magnetic effect of the electric movement is precisely the same.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;(their emphasis)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;in the glass of a Leyden jar,&quot; Maxwell means &quot;in the space between the plates of a capacitor.&quot; (Leyden jars were an early form of a capacitor.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They also note (&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/7Q29PUT4/Karam%20et%20al.%20-%202014%20-%20Comparing%20Teaching%20Approaches%20About%20Maxwell%e2%80%99s%20Disp.pdf&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;ibid.&lt;/em&gt;&lt;/a&gt; fn12):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first record of such experiment[s &quot;that aimed at providing evidence for the existence of the displacement current by measuring the magnetic effects between the plates of a charging (or discharging) capacitor&quot;] goes back to 1899 in the report &quot;&lt;em&gt;On the Magnetic Action of Displacement-currents in a Dielectric&lt;/em&gt;&quot; written by Silvanus P. Thomson.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;As an aside, &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_circuital_law&amp;amp;oldid=732179087#cite_note-1&quot; rel=&quot;nofollow&quot;&gt;Ampère never wrote down &quot;Ampère's law&quot;&lt;/a&gt;, even the form without Maxwell's added displacement current term, as Ampère never dealt with the field concept. &lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law&quot; rel=&quot;nofollow&quot;&gt;Ampère's force law&lt;/a&gt; ≠ &lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law&quot; rel=&quot;nofollow&quot;&gt;4&lt;sup&gt;th&lt;/sup&gt; Maxwell equation called &quot;Ampère's&quot; circuital law&lt;/a&gt;.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-08-13T00:04:33.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5097" PostTypeId="1" CreationDate="2016-08-13T09:53:34.580" Score="5" ViewCount="191" Body="&lt;p&gt;During my research on branched coverings of the projective plane, I am interested to know the origins and history of branched coverings of the projective plane and the projective line, together with the rise of the concept of &quot;branch point&quot; or &quot;branch curve&quot; (resp. &quot;ramification points&quot; or &quot;ramification curve&quot;). It seems that the modern conception was developed by Zariski and Segre in the 20s and 30s of the 20th century, and the example of Zariski-Segre of the branch curve of degree 6 (having 6 cusps on a conic) of a cubic surface projected generically to the projective plane, is well known. But were there other mathematicians before them who helped conceptualize this concept? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you, Thomas&lt;/p&gt;&#xA;" OwnerUserId="4568" LastEditorUserId="4568" LastEditDate="2016-08-13T18:39:19.467" LastActivityDate="2016-08-13T20:43:22.503" Title="Origins and history of branched covering" Tags="|reference-request|terminology|topology|algebraic-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5098" PostTypeId="2" ParentId="5097" CreationDate="2016-08-13T20:43:22.503" Score="10" Body="&lt;p&gt;The theory of branched (or ramified) coverings  has its origins in continuation of analytic functions and the attempts to find maximal analytic continuations of a given function. However, certain complex functions, e.g. $f(z) =z^{1/2}$ are multi-valued in certain subdomains of the complex plane, so when trying to continue along the closed curve one might arrive at another branch of the multivalued function, not the original one, so the valued do not match. The idea of Riemann surfaces offers a geometric way to deal with this problem by introducing a Riemann surface as the natural domain of $z^{1/2}$ and similarly for other troublesome functions. One can picture the Riemann surface for $z^{1/2}$ as two sheets coming together at $z=0$. The Riemann surfaces appeared in Bernhard Riemann's inaugural thesis in 1851, &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/Grund.pdf&quot;&gt;http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/Grund.pdf&lt;/a&gt;,  and are considered the first examples of covering spaces.  &lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2016-08-13T20:43:22.503" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5099" PostTypeId="2" ParentId="2940" CreationDate="2016-08-13T22:04:25.840" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://ece.duke.edu/faculty/ingrid-daubechies&quot; rel=&quot;noreferrer&quot;&gt;Ingrid Daubechies&lt;/a&gt; did pioneering work in harmonic analysis, which led for instance to the development of finite support wavelets (orthogonal and biorthogonal). This enabled wavelet theory to enter the domain digital signal processing, perhaps similarly to the invention of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fast_Fourier_transform&quot; rel=&quot;noreferrer&quot;&gt;Fast Fourier Transform&lt;/a&gt; with respect to the mathematical Fourier transform.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two of these wavelets, called CDF 5/3 or &lt;a href=&quot;http://www.getreuer.info/home/waveletcdf97&quot; rel=&quot;noreferrer&quot;&gt;CDF 9/7&lt;/a&gt; for &lt;a href=&quot;https://en.wikipedia.org/wiki/Cohen-Daubechies-Feauveau_wavelet&quot; rel=&quot;noreferrer&quot;&gt;Cohen-&lt;strong&gt;Daubechies&lt;/strong&gt;-Fauveau&lt;/a&gt;, are at the core of the image compression algorithm JPEG 2000, and the &lt;a href=&quot;http://www.eetimes.com/document.asp?doc_id=1201549&quot; rel=&quot;noreferrer&quot;&gt;Motion JPEG 2000&lt;/a&gt; used in the Motion Picture industry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;She was the first woman to be president of the International Mathematical Union. She was a &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether_Lecture&quot; rel=&quot;noreferrer&quot;&gt;Noether lecturer&lt;/a&gt;, where you can find other influential mathematician women. And she is a baroness now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Her mathematical results not only made the path through industrial applications, but strongly modified the way people analyse data, in a multiscale (zoom-in/zoom-out) fashion.&lt;/p&gt;&#xA;" OwnerUserId="4130" LastActivityDate="2016-08-13T22:04:25.840" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5100" PostTypeId="2" ParentId="2578" CreationDate="2016-08-14T21:07:31.763" Score="6" Body="&lt;p&gt;I hope a version in French may shed some light on this question. In Le Séminaire de Royaumont 1959-1979, here is apparently a verbatim of the words of Jean Dieudonné. The previous link &lt;code&gt;http://www.ssrdm.ch/mathecole/wa_files/Mathecole_90.pdf&lt;/code&gt; is not available anymore. However, here is anotehr source: Calame,  A.  (1979).  L’enseignement  de  la  géométrie, Math-école, 90 Le séminaire de Royaumont 1959-1979. Novembre 1979, 11-16. Here is &lt;a href=&quot;https://www.revue-mathematiques.ch/files/4514/6288/8131/Mathecole_90.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math-école, volume 90, Le Séminaire de Royaumont 1959-1979&lt;/a&gt; in pdf, the full quote is on page 6.&#xA;Part of it is mentioned in  &lt;a href=&quot;https://limo.libis.be/primo-explore/fulldisplay?docid=LIRIAS1679371&amp;amp;context=L&amp;amp;vid=Lirias&amp;amp;search_scope=Lirias&amp;amp;tab=default_tab&amp;amp;lang=en_US&amp;amp;fromSitemap=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;« À bas Euclide ! » Mathématique Moderne au Séminaire OECE de Royaumont&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;C'est à Royaumont que Jean Dieudonné lança son cri de guerre « À bas&#xA;Euclide ! » et qu'il y définit, avec la collaboration de conférenciers&#xA;et de délégués issus de 16 Pays européens, des Etats-Unis et du&#xA;Canada, les contours d'un nouveau curriculum pour les mathématiques&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and quoted in &lt;a href=&quot;https://books.google.fr/books?id=zuxXDwAAQBAJ&amp;amp;pg=PT228&amp;amp;lpg=PT228&amp;amp;dq=%22maintenant%20envisager%20une%20r%C3%A9forme%20beaucoup%20plus%20profonde%22&amp;amp;source=bl&amp;amp;ots=SCzkPMxDPU&amp;amp;sig=ACfU3U3Iz1aaINPVTXe98SI6vT6wF8GWEw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwj917vaxN3xAhVNOBoKHU3TC-wQ6AEwAHoECAIQAw#v=onepage&amp;amp;q=%22maintenant%20envisager%20une%20r%C3%A9forme%20beaucoup%20plus%20profonde%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le sujet et l'objet, confrontations&lt;/a&gt;:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/AKpEJ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AKpEJ.jpg&quot; alt=&quot;Jean Dieudonné on Euclid&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Jean Dideudonné was apparently in favor of the teaching of 2D and 3D vector spaces, instead of mere geometry:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Je pense que le temps de ce &amp;quot;ravaudage&amp;quot; est dépassé et que nous devons&#xA;maintenant envisager une réforme beaucoup plus profonde, à moins que&#xA;nous n'acceptions de laisser la situation empirer au point d'entraver&#xA;sérieusement tout progrès scientifique ultérieur. Si je voulais&#xA;résumer en une phrase tout le programme que j'ai dans l'esprit, ce&#xA;serait par le ·slogan: &amp;quot;A bas Euclide&amp;quot;. Cette affirmation paraîtra&#xA;peut-être choquante à certains d'entre vous; je voudrais vous exposer&#xA;en détail les puissants arguments qui militent en sa faveur.&#xA;Permettez-moi de dire d'abord que j'ai la plus profonde admiration&#xA;pour les· résultat obtenus par les Grecs en mathématiques. Je&#xA;considère que leur création de la géométrie a peut-être été dans le&#xA;domaine intellectuel le progrès le plus extraordinaire que  l'humanité&#xA;ait jamais réalisé. C'est grâce aux Grecs que l'imposant édifice de la&#xA;science moderne a pu être bâti. Mais, au cours de cette évolution même,&#xA;on a repris à la base, surtout depuis le milieu du XIXe siècle, les&#xA;notions fondamentales de la géométrie: on a pu ainsi réorganiser la&#xA;géométrie euclidienne, lui redonner des fondements simples et solides&#xA;et réévaluer son importance par rapport aux mathématiques modernes, en&#xA;séparant ce qui est essentiel d'un amas chaotique de résultats qui ne&#xA;sont que les reliques éparses de méthodes maladroites ou de points de&#xA;vues désuets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, more a mere &amp;quot;Down with Euclid&amp;quot;, no mention of triangles so far.&lt;/p&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="4130" LastEditDate="2021-07-12T12:52:23.087" LastActivityDate="2021-07-12T12:52:23.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5101" PostTypeId="2" ParentId="162" CreationDate="2016-08-15T04:17:51.493" Score="4" Body="&lt;p&gt;Related question: &quot;&lt;a href=&quot;https://latin.stackexchange.com/q/1372/576&quot;&gt;What is the history of scientific Latin?&lt;/a&gt;&quot; &lt;a href=&quot;https://latin.stackexchange.com/a/1379/576&quot;&gt;This answer to that question&lt;/a&gt; has a very nice quote by Gauss discussing his Latin:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From G. Waldo Dunnington's 2004 biography of Gauss, &lt;a href=&quot;https://www.worldcat.org/oclc/53933110&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Carl Friedrich&#xA;  Gauss: Titan of Science&lt;/em&gt;&lt;/a&gt;, p. 37-8:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;… Of unusual interest is the part which Meyerhoff⁶ took in this book [&lt;em&gt;sc.&lt;/em&gt; Gauss's most important mathematical work: the &lt;em&gt;Disquisitiones&#xA;    arithmeticæ&lt;/em&gt;]—the correction of the Latin.&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;&lt;sup&gt;⁶Johann Heinrich Jakob Meyerhoff (1770-1812) became in 1794 collaborator, and in 1802 director, of the gymnasium in Holzminden. He&#xA;    was thoroughly grounded and trained in the ancient and modern&#xA;    languages. As a Göttingen student he had won a golden prize medal for&#xA;    a Latin dissertationon the Phoenicians. Yet mathematics was rather&#xA;    foreign to him.&lt;/sup&gt;&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;The above is striking enough, if one considers how little Gauss needed to mistrust his own proficiency in this respect. According to&#xA;    Moritz Cantor, Gauss wrote a classical Latin, giving rise to the&#xA;    expression that Cicero, if he could understand the mathematics of it,&#xA;    would have censured nothing in the Gaussian Latinity, except perhaps&#xA;    several customary incorrect modes of expression which Gauss used&#xA;    purposely. But it was Latin just the same and therefore attractive and&#xA;    stimulating to only a narrow circle of readers. Referring to&#xA;    Meyerhoff's work, Gauss wrote:&lt;/p&gt;&#xA;    &#xA;    &lt;blockquote&gt;&#xA;      &lt;p&gt;Of course I understand that it cannot be an especially attractive work for Mr. Meyerhoff, since he does not seem to be sufficiently&#xA;      acquainted with mathematics, in order to look on it just as reading.&#xA;      Thus the word &lt;em&gt;algorithmus&lt;/em&gt; was unknown to him. Only on a single point&#xA;      must I take the liberty of disagreeing with him. I well know that &lt;em&gt;si&lt;/em&gt;&#xA;      with the subjunctive is not good Latin; but modern mathematicians seem&#xA;      to have made for themselves the rule of constantly using the&#xA;      subjunctive in hypotheses and definitions; I do not remember an&#xA;      example of the opposite, and in &lt;a href=&quot;http://www.encyclopedia.com/topic/Christiaan_Huygens.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huyghens&lt;/a&gt;, who according to my&#xA;      notion writes the most elegant Latin and whom I purposely, therefore,&#xA;      have imitated, I find the subjunctive continually in these cases. I&#xA;      open at random and find &lt;em&gt;Opera&lt;/em&gt;, p. 156, &lt;em&gt;Quodsi fuerit&lt;/em&gt;; p. 157, &lt;em&gt;Si&#xA;      sit, si fiat, si agitetur&lt;/em&gt;; p. 158, &lt;em&gt;si suspendatur&lt;/em&gt;; pp. 188 &lt;em&gt;seqq.&lt;/em&gt;&#xA;      are examples by the dozen. Therefore, since in this instance the&#xA;      desire to be a genuine Roman would be only purism (which as far as I&#xA;      am concerned would be less allowable, because I am well minded not to&#xA;      be so in any case) and the thing is not at all absurd in itself, I&#xA;      went with the current. I hope Mr. M. will not take offense at me. What&#xA;      was incomprehensible to him in the &lt;em&gt;accedere possunt&lt;/em&gt;, p. 5, I have&#xA;      not been able able to guess; I have therefore let it stand. The&#xA;      passage p. 7, which previously ran thus: &lt;em&gt;Si numeri decadice expressi&#xA;      figuræ singulæ sine respectu loci quem occupant addantur&lt;/em&gt;, Mr. M.&#xA;      misunderstood, because he probably didn't know that &lt;em&gt;figuræ&lt;/em&gt; means&#xA;      numbers; he took &lt;em&gt;numeri&lt;/em&gt; for the nominative plural and &lt;em&gt;figuræ&lt;/em&gt; for&#xA;      the dative singular and on that account suggested to me that&#xA;      &lt;em&gt;singulus&lt;/em&gt; is not wrong; but just for this reason a mathematician will probably not construe it incorrectly, chiefly because it doesn't make&#xA;      sense; nevertheless I have now arranged the words somewhat differently&#xA;      by this time.&lt;/p&gt;&#xA;    &lt;/blockquote&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;Thus, Gauss seems to have worked with a &lt;em&gt;mathematical Latin&lt;/em&gt; based on&#xA;  classical Latin. Gauss also wasn't afraid to innovate the Latin. It&#xA;  seemed to be a truly living language to him.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Carl_Friedrich_Gauss.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss's &lt;em&gt;Dictionary of Scientific Biography&lt;/em&gt; entry&lt;/a&gt; says:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;He published in Latin not from internationalist sentiments but at the demands of his publishers.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, it seems that the publishers played a role in phasing out Latin and thus allowing the adoption of German.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wilhelm Weber, Gauss's student, never wrote any scientific Latin, to my knowledge. His famous works are in German.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="658" LastEditDate="2018-09-21T16:47:01.860" LastActivityDate="2018-09-21T16:47:01.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5102" PostTypeId="1" CreationDate="2016-08-15T04:32:40.433" Score="6" ViewCount="1148" Body="&lt;p&gt;What is the last physics paper or book written in Latin? I know Carl Neumann, for example, wrote papers in Latin in the 19th century. Are there any more recently than that?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2016-10-02T12:48:46.180" Title="What is the last physics paper or book written in Latin?" Tags="|physics|language|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5103" PostTypeId="1" CreationDate="2016-08-16T10:25:27.577" Score="16" ViewCount="953" Body="&lt;p&gt;&lt;a href=&quot;http://www.jstor.org/stable/23632817?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Redondi&lt;/a&gt; argued that Galileo's trial on heliocentrism was merely a show trial concealing the real objection against Galileo among the catholic establishment, which was his atomism thought to be at variance with canon 13.2 of the council of trent having to do with the eucharist.  Redondi based himself on an anonymous denunciation labeled G3 as well as similar objections expressed by Grassi.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other historians of Galileo disagreed.  Thus, &lt;a href=&quot;http://philpapers.org/rec/EQIGH&quot; rel=&quot;noreferrer&quot;&gt;Festa&lt;/a&gt; argued that the timing of the trial is off and that Scheiner is the probable source of the denunciation that led to the trial, rather than Grassi.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the state of the art on this debate?  Have there been additional arguments and counterarguments with regard to the thesis concerning Galileo's atomism?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1. In response to a question posed in a &lt;em&gt;comment&lt;/em&gt;, it may be useful to recall that Galileo was a protege of the pope Urban 8 (to the extent that the pope had a book of his read to him at mealtime). The pope was forced by the turn of political events to reorient his alliances in 1632 and could no longer ignore anti-Galileo pressure emanating from the jesuits, and therefore initiated the trial. On the other hand, to protect his (former?) protege, the pope channeled the trial in the direction of a relatively minor issue of heliocentricism resulting in a conviction leading to house arrest for Galileo. A trial on charges related to more serious canonical matters like Trent 13.2 could have led galileo to the stake which was something the pope wanted to avoid.  Again this is a summary of Redondi's argument (not my argument).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 2. See &lt;a href=&quot;https://mathoverflow.net/questions/247065/fermats-opponents&quot;&gt;this related thread&lt;/a&gt; on Fermat and Galileo.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 3. The names of the ten judges can be found &lt;a href=&quot;https://astronomy.stackexchange.com/questions/18107/how-many-judges-sat-on-galileos-trial-over-heliocentrism-and-who-were-they-exac&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 4. To summarize the relevant points of the &lt;em&gt;answer&lt;/em&gt; below, the &lt;em&gt;state-of-the-art&lt;/em&gt; since Redondi is the discovery of an additional document labeled EE291 which is a eucharist-based denunciation of Galileo (dating from a few months before Galileo was issued summons to stand trial) penned by jesuit Inchofer.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2017-09-05T13:36:48.687" LastActivityDate="2020-01-07T19:32:11.580" Title="Hidden agenda of the Galileo trial?" Tags="|physics|astronomy|philosophy-of-science|atomic-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5104" PostTypeId="2" ParentId="3834" CreationDate="2016-08-16T11:05:10.233" Score="2" Body="&lt;p&gt;Galileo used the term &lt;em&gt;non-quanta&lt;/em&gt; for infinitesimals (and &lt;em&gt;quanta&lt;/em&gt; for ordinary quantities).  A recent study on Galileo's infinitesimals is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bascelli, Tiziana&#xA;Galileo's quanti: understanding infinitesimal magnitudes. &#xA;Arch. Hist. Exact Sci.  68  (2014),  no. 2, 121–136. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cavalieri's indivisibles were different from Galileo's infinitesimals. In particular, it seems that it would be difficult to express the idea of second-order infinitesimal in Cavalieri's scheme of things as far as I can tell, but perhaps he did.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-08-16T11:05:10.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5105" PostTypeId="2" ParentId="2578" CreationDate="2016-08-16T11:22:49.997" Score="3" Body="&lt;p&gt;The book&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hersh, Reuben; John-Steiner, Vera Loving + hating mathematics. Challenging the myths of mathematical life. Princeton University Press, Princeton, NJ, 2011&lt;/p&gt;&#xA;&#xA;&lt;p&gt;on page 185 reports that in november 1959 at Centre Culturel de Royaumont, Dieudonne rose to his feet with the cry: &quot;A bas Euclide! Mort aux triangles!&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an &lt;a href=&quot;https://mathoverflow.net/questions/152352/is-euclid-dead&quot;&gt;MO thread&lt;/a&gt; dedicated to this.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-08-16T11:22:49.997" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5106" PostTypeId="2" ParentId="5103" CreationDate="2016-08-16T23:11:07.960" Score="10" Body="&lt;h1&gt;&lt;em&gt;The Assayer&lt;/em&gt; &amp;amp; Redondi's &amp;quot;G3&amp;quot;&lt;/h1&gt;&#xA;&lt;p&gt;Michele Camerota's 2008 &lt;a href=&quot;https://www.encyclopedia.com/people/science-and-technology/astronomy-biographies/galileo&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Complete Dictionary of Scientific Biography&lt;/em&gt; entry on Galileo&lt;/a&gt; describes this theory in a section entitled &amp;quot;Atomism and the Eucharist&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.encyclopedia.com/people/science-and-technology/astronomy-biographies/galileo#proper-ad-encyclopedia_content_4-2&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Atomism and the Eucharist&lt;/strong&gt;&lt;/a&gt;. In section 48 of &lt;em&gt;The Assayer&lt;/em&gt; (1623), Galileo set forth a theory of knowledge based on a sharp distinction between “objective” and “subjective” qualities. According to this view, whereas features such as shape, size, position, motion, and number are qualities intrinsic to real things, impressions such as colors, tastes, smells, or tactile properties do not exist in the objects themselves but only in the sentient subject experiencing them. For this reason, sensible qualities were characterized by Galileo as “mere names,” qualities that “reside only in the consciousness” and that would be “wiped away and annihilated” once human sensibility is removed. Behind sensible qualities are the true components of the real world, atoms, whose impinging on the sense organs produces sensory impressions. Hence, for example, the sensation of heat stems from the motion of a “multitude of minute particles” that penetrate human bodies; “their touch as felt by us when they pass through our substance is the sensation we call ‘heat.’” (trans. in Drake, &lt;em&gt;Discoveries and Opinions of Galileo&lt;/em&gt;, New York: Anchor Books, 1957, p. 277). Galileo’s stance was clearly rooted in the tradition of ancient atomism, whose most distinguished representatives, such as Democritus and Lucretius, had already stated similar views.&lt;/p&gt;&#xA;&lt;p&gt;Two documents discovered in the Archives of the Congregation for the Doctrine of the Faith (formerly Holy Office) show that Galileo’s atomistic theory was brought to the attention of the Inquisitorial authorities, most likely before the trial of 1633. The first document was found in 1982 by Pietro Redondi and is usually referred to as “G3,” from the code appearing on the top of its first page. G3 is a denunciation of the atomism of &lt;em&gt;The Assayer&lt;/em&gt;. The anonymous author protested that Galileo’s interpretation of sensible qualities clashed with the Catholic doctrine of the Eucharist, according to which, after consecration in the Mass, bread and wine become the body and blood of Jesus Christ. This transformation is understood as transubstantiation because it concerns the substances of bread and wine, whereas their “accidents,” or apparent qualities (color, odor, exterior shape), remain unchanged by virtue of a divine miracle. The author of G3 remarked that, according to the terms of Galileo’s argument, it would be impossible to separate the accidental properties of bread and wine from their own substances. Indeed, because those accidental properties are regarded as “mere names” and as nonexistent outside human sensory perception, on the basis of Galileo’s theory one would be obliged to conclude that “in the Sacrament there are substantial elements of the bread and the wine, which is an error condemned by the Sacred Council of Trent.”&lt;/p&gt;&#xA;&lt;p&gt;Redondi dated G3 to 1624 and attributed it to Orazio Grassi, the Jesuit mathematician against whom Galileo had written &lt;em&gt;The Assayer&lt;/em&gt;. Redondi also connected the document to the trial of 1633, suggesting that the charge of Copernicanism that motivated the trial was a stratagem devised by Pope Urban VIII (a former friend of Galileo) in order to avoid having the scientist face the more serious accusation of Eucharistic heresy.&lt;/p&gt;&#xA;&lt;p&gt;Redondi’s ascription of G3 to Grassi has been proved to be mistaken, and his thesis concerning the “true” (although disguised) reasons of the trial has been generally rejected by scholars. Nevertheless, Redondi’s book triggered a fresh wave of interest in the Galileo affair and renewed investigations into its cultural and political context.&lt;/p&gt;&#xA;&lt;p&gt;Another document, similar to G3, was discovered by Mariano Artigas in 1999 and has been carefully studied by Artigas himself along with Rafael Martinez and William Shea. This document is placed in the same volume as G3, the volume EE of the collection “Acta et Documenta,” where it occupies sheet 291. For this reason it has been called “EE 291.”&lt;/p&gt;&#xA;&lt;p&gt;Like G3, EE 291 is anonymous (it is in Latin while the former is in Italian), and it equally develops a criticism of the theory of sensible qualities expounded in &lt;em&gt;The Assayer&lt;/em&gt;, which it deems incompatible with the doctrine of the Eucharist. The author of this document has been identified as the Jesuit Melchior Inchofer, who probably was a member of the commission appointed by the pope in the summer of 1632 to examine Galileo’s &lt;em&gt;Dialogue&lt;/em&gt;. Inchofer, a firm opponent of Copernicanism, could have written EE 291 in order to worsen Galileo’s position by adding a further charge against him. Thus, the discoveries of G3 and EE 291, besides providing valuable pieces of information on previously unknown episodes of Galileo’s life, also opened a new chapter of investigation concerning the difficult relationship between atomism and Eucharistic doctrine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;The offensive quote from &lt;em&gt;The Assayer&lt;/em&gt;&lt;/h1&gt;&#xA;&lt;p&gt;This (from &lt;a href=&quot;https://sententiaedeo.blogspot.com/2009/12/science-and-modernism.html#feather&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;) is the offensive quote from &lt;em&gt;They Assayer&lt;/em&gt;, in which he denies that the nature of &amp;quot;feather,&amp;quot; something beyond the senses, can be known; for him, all knowledge begins and ends in the senses (empiricism):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I move my hand first over a marble statue and then over a living man. To the effect flowing from my hand, this is the same with regard to both objects and my hand; it consists of the primary phenomena of motion and touch, for which we have no further names. But the live body which receives these operations feels different sensations according to the various places touched. When touched upon the soles of the feet, for example, or under the knee or armpit, it feels in addition to the common sensation of touch a sensation on which we have imposed a special name, &amp;quot;tickling.&amp;quot; This sensation belongs to us and not to the hand. Anyone would make a serious error if he said that the hand, in addition to the properties of moving and touching, possessed another faculty of &amp;quot;tickling,&amp;quot; as if tickling were a phenomenon that resided in the hand that tickled. A piece of paper or a feather drawn lightly over any part of our bodies performs intrinsically the same operations of moving and touching, but by touching the eye, the nose, or the upper lip it excites in us an almost intolerable titillation, even though elsewhere it is scarcely felt. This titillation belongs entirely to us and not to the feather; if the live and sensitive body were removed it would remain no more than a mere word. I believe that no more solid an existence belongs to many qualities which we have come to attribute to physical bodies-tastes, odors, colors, and many more. I believe that no more solid an existence belongs to many qualities which we have come to attribute to physical bodies—tastes, odors, colors, and many more.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h1&gt;Catholic philosophical refutation of the &lt;em&gt;Assayer&lt;/em&gt; quote&lt;/h1&gt;&#xA;&lt;p&gt;The Catholic philosopher and historian of science &lt;a href=&quot;http://www.morec.com/nature.htm&quot; rel=&quot;noreferrer&quot;&gt;Fr. William A. Wallace, O.P.&lt;/a&gt; refutes, in &lt;a href=&quot;https://isidore.co/calibre/browse/book/5302&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Modeling of Nature&lt;/em&gt;&lt;/a&gt; (pgs. 148-149), the above &lt;em&gt;Assayer&lt;/em&gt; quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Those who hold for the subjectivity of sensible qualities maintain that such qualities have no existence independently of the sensing subject, and on this ground effectively deny the very existence of objective intentions for such qualities. They find convincing Galileo's example of the movement of a feather across the skin to explain the tickle. So they introduce a distinction between primary qualities such as movement and secondary qualities such as the sensed tickle, and hold that the primary qualities have objective existence whereas secondary qualities do not. As a result they populate the universe with particles in motion and attempt to explain all sensations by the various kinds of movement these particles undergo, meanwhile denuding the objective world of sensible qualities in their traditional understanding.&lt;/p&gt;&#xA;&lt;p&gt;The source of the difficulty here is an improper grasp of the role of the mental representation in the knowledge act. To think of the concept as &lt;em&gt;what&lt;/em&gt; is known, rather than seeing that the nature is what is known, though &lt;em&gt;by means of&lt;/em&gt; the concept, is to cut oneself off from intellectual knowledge of the real, for one is always left wondering about any extra-mental reality to which the concept might correspond. Similarly, to think of the sensation or the percept as itself &lt;em&gt;what&lt;/em&gt; is known, rather than seeing the sensible quality as what is known, though &lt;em&gt;by means of&lt;/em&gt; the sensation or percept, is to be imprisoned within one's sense organs and brain. The result is a radical solipsism that prohibits individuals from ever making statements about the objects of experience, leaving them to dwell in a world of their own imaginings.&lt;/p&gt;&#xA;&lt;p&gt;The tickle may be something sensed on the surface of the skin, but that admission surely does not permit the inference that there is no movement there, or extending the argument further to hold that there is no heat in boiling water, no color in a ruby or a rose, no sound in the cry of a bird, or no odor or taste in an onion. All of these are accidents or accidental modifications of the subjects in which they are sensed. Just as those subjects have natures (inorganic, plan or animal in kind), so accidents may be said to have natures in an analogous sense. And even if we cannot know &lt;em&gt;precisely&lt;/em&gt; the nature of heat, of color, and so on, we can at least &lt;em&gt;model&lt;/em&gt; those natures in terms of the modalities they introduce in the components of the substantial natures in which they exist, namely the electrons, atoms, and molecules […]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, the issue is not atomism &lt;em&gt;per se&lt;/em&gt; but nominalism/idealism vs. realism—or, specially in the Galileo case, it seems: &lt;a href=&quot;https://philosophy.stackexchange.com/q/9940/2014&quot;&gt;structural realism vs. scientific formalism&lt;/a&gt;, where Galileo, Kepler, et al. were &lt;a href=&quot;http://plato.stanford.edu/entries/structural-realism/&quot; rel=&quot;noreferrer&quot;&gt;structural realists&lt;/a&gt; and &lt;a href=&quot;http://www.encyclopedia.com/topic/Robert_Bellarmine.aspx#1&quot; rel=&quot;noreferrer&quot;&gt;Bellarmine&lt;/a&gt;, &lt;a href=&quot;http://www.encyclopedia.com/topic/Andreas_Osiander.aspx#1&quot; rel=&quot;noreferrer&quot;&gt;Osiander&lt;/a&gt;, et al. were scientific formalists, the latter thinking empirical science merely &amp;quot;&lt;a href=&quot;https://books.google.com/books?id=UofBybolmREC&amp;amp;lpg=PP1&amp;amp;pg=PA131#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;saves the appearances&lt;/a&gt;&amp;quot; rather than, as the former thought, uncovering nature's underlying reality. This is fundamentally the same debate for which Galileo was tried regarding (without sufficient proof at the time) his insistence on the reality of his hypothesis of the earth's motion. It was a philosophical debate, which is why he was condemned for holding a view that was &amp;quot;absurd and false &lt;em&gt;philosophically&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;h1&gt;Galileo trial not directly about atomism vs. anti-atomism&lt;/h1&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Saint_Thomas_Aquinas.aspx#1&quot; rel=&quot;noreferrer&quot;&gt;St. Thomas Aquinas&lt;/a&gt;, the greatest Doctor of the Catholic Church, describing the corruption of a consecrated Host, makes an allusion to atomism in &lt;a href=&quot;https://isidore.co/aquinas/summa/TP/TP077.html#TPQ77A4THEP1&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Summa Theologica&lt;/em&gt; III q. 77 a. 4 (&amp;quot;Whether the sacramental species can be corrupted?&amp;quot;)&lt;/a&gt; c. (my emphasis; cf. &lt;a href=&quot;https://christianity.stackexchange.com/a/52073/1787&quot;&gt;this answer&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;…if the change be so great that the substance of the bread or wine would have been corrupted, then Christ's body and blood do not remain under this sacrament; and this either on the part of the qualities, as when the color, savor, and other qualities of the bread and wine are so altered as to be incompatible with the nature of bread or of wine; or else on the part of the quantity, as, for instance, if the bread be reduced to &lt;strong&gt;fine particles&lt;/strong&gt; [atoms?], or the wine divided into &lt;strong&gt;such tiny drops&lt;/strong&gt; [atoms?] that the species of bread or wine no longer remain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Fr. &lt;a href=&quot;http://www.encyclopedia.com/doc/1G2-2830900549.html&quot; rel=&quot;noreferrer&quot;&gt;Rudjer J. Bošković&lt;/a&gt; (1711-1787) was a Jesuit priest and physicist, the &amp;quot;father of modern atomic theory,&amp;quot; and he was not condemned for his atomism as Galileo was for his thinking scientific hypotheses can contradict Holy Scripture.&lt;/p&gt;&#xA;&lt;h1&gt;Atomism vs. Dynamism&lt;/h1&gt;&#xA;&lt;p&gt;There was a philosophical debate in Catholic circles regarding atomism vs. dynamism in the 19&lt;sup&gt;th&lt;/sup&gt; and early 20&lt;sup&gt;th&lt;/sup&gt; centuries. Dynamism held that energy is the ultimate reality. Neither doctrine is held by Catholic scholastic philosophers to be a true natural philosophy.&lt;/p&gt;&#xA;&lt;p&gt;For a good overview of this debate, see:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4244&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Cosmology&lt;/em&gt;&lt;/a&gt; by Édouard Hugon, O.P. (treatise II, second &amp;amp; third articles, pp. 145-162)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/amanualofmoderns01mercuoft&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Modern Scholastic Philosophy&lt;/em&gt;&lt;/a&gt; (vol. 1) by Cdl. Desiré Joseph Mercier, 1851-1926, pp. 49-70, 131-144&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4133&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Physical System of St. Thomas&lt;/em&gt;&lt;/a&gt; by Giovanni Maria Cornoldi&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-07T19:32:11.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5108" PostTypeId="1" CreationDate="2016-08-17T14:42:53.260" Score="4" ViewCount="301" Body="&lt;p&gt;Dedekind in 1872 wrote that equations like  $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ ``to the best of my knowledge have never been established before'' (&lt;em&gt;Continuity and Irrational Numbers&lt;/em&gt;, p. 22).  This was not a crazy thing to say but it makes me wonder what were the standard textbook treatments at the time.  A book like Rouse Ball &lt;em&gt;History of the Study of Mathematics at Cambridge&lt;/em&gt; would be ideal for this question, except that it's account of instruction ends with the 18th century and it covers the 19th century only in  connection with the Tripos.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2018-01-16T15:11:44.153" Title="What were standard university texts in algebra in the mid 19th century?" Tags="|reference-request|teaching|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5109" PostTypeId="1" AcceptedAnswerId="5111" CreationDate="2016-08-17T15:07:47.670" Score="1" ViewCount="100" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Collins_(mathematician)&quot; rel=&quot;nofollow&quot;&gt;John Collins&lt;/a&gt; corresponded with numerous mathematicians in Europe in the 17th century. One of his correspondents mentioned in several letters is one Father Bertet.  Judging from the letters Bertet was in Italy or perhaps more specifically in Venice.  Beyond this I can't find any information on the fellow.  Who was he?&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-08-17T19:10:06.930" Title="Who is Father Bertet (correspondent of John Collins)?" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5110" PostTypeId="2" ParentId="3767" CreationDate="2016-08-17T15:23:14.920" Score="8" Body="&lt;p&gt;Very much so.  Today someone arguing Kronecker's position would be regarded as a crank (sort of like a finitist).  Kronecker was arguing for way more than just &quot;constructive mathematics&quot;; he believed things like:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;There is not set containing all the subsets of an infinite set (like the integers or the rationals).&lt;/li&gt;&#xA;&lt;li&gt;The nested interval property (that all infinite sequences of nested closed intervals have a common point) may fail (even in constructive math, this property holds for sequences given by an algorithm).&lt;/li&gt;&#xA;&lt;li&gt;Consequently, he believed that irrational and transcendental numbers do not exist (there are constructive proofs that numbers like $\pi$ and $e$ exist, and any classical proof whose result is a negation is also valid in constructive logic, so they are transcendental).&lt;/li&gt;&#xA;&lt;li&gt;For similar reasons, Kronecker's argument would imply that there is no set of real numbers (even if individual irrational or transcendental numbers were granted to exist).&lt;/li&gt;&#xA;&lt;li&gt;And, in general, Kronecker would deny the existence of uncountable seets (Brouwer, for example, took for granted that the real numbers were uncountable).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So there is a long list of propositions Kronecker held to that modern constructivists don't believe; they might criticize aspects of Cantor's set theory but not the whole concept of infinite sets the way Kronecker did.&lt;/p&gt;&#xA;" OwnerUserId="4434" LastActivityDate="2016-08-17T15:23:14.920" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5111" PostTypeId="2" ParentId="5109" CreationDate="2016-08-17T16:38:34.423" Score="5" Body="&lt;p&gt;Googling, I found ---&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Philip Beeley and Christoph J. Scriba (editors), &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0198569483&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Correspondence of John Wallis (1616-1703)&lt;/strong&gt;&lt;/a&gt;, Volume IV (1672-April 1675), Oxford University Press, 2014.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following is from p. 506:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bertet, Jean (1622-92).--- Born in Tarascon. Admitted S.J. 1637. Studied under Honoré Fabri at Collège de la Trinité in Lyon. Taught philosophy and mathematics at University of Grenoble. By 1659 professor of mathematics at Aix. In 1689 conspired with Leibniz and Antonio Baldigiani S.J. in Rome to have ban on Copernicanism lifted. Died in Paris on 29 June 1692 (new style).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wonder if Jean is the father or a grandfather of Laurent Dominique Bertet (1671-1739), who seems to be better known?&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2016-08-17T19:10:06.930" LastActivityDate="2016-08-17T19:10:06.930" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5112" PostTypeId="2" ParentId="5108" CreationDate="2016-08-17T16:38:36.750" Score="5" Body="&lt;p&gt;Here's an example of early 19&lt;sup&gt;th&lt;/sup&gt; century textbooks (from &lt;a href=&quot;https://math.stackexchange.com/a/986254/128568&quot;&gt;my answer here&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cauchy entered the &lt;em&gt;lycée&lt;/em&gt; (&quot;high school&quot;) &lt;em&gt;École Centrale du Panthéon&lt;/em&gt; in 1802, studying humanities. Then, in 1805 he entered the &lt;em&gt;École Polytechnique&lt;/em&gt; at age 15.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;core&quot; mathematics curriculum at the &lt;em&gt;École Polytechnique&lt;/em&gt; was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Analysis instruction&lt;/strong&gt;: the &lt;em&gt;Cours d'Analyse Algébrique&lt;/em&gt; by Garnier and&#xA;  the &lt;em&gt;&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k6213792j&quot; rel=&quot;nofollow noreferrer&quot;&gt;Traité Élémentaire de Calcul Différentiel et Intégral&lt;/a&gt;&lt;/em&gt; by&#xA;  Lacroix;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Mechanics instruction&lt;/strong&gt;: the &lt;em&gt;Traité de Mécanique&lt;/em&gt;, using the methods of Prony and edited by Francoeur and the &lt;em&gt;Plan Raisonne du Cours de Prony&lt;/em&gt;;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Descriptive geometry instruction&lt;/strong&gt;: the &lt;em&gt;&lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=nyp.33433057782298;view=1up;seq=13&quot; rel=&quot;nofollow noreferrer&quot;&gt;Géométrie Descriptive&lt;/a&gt;&lt;/em&gt; by Monge;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Applied analysis instruction&lt;/strong&gt;: the &lt;em&gt;Feuilles d'Analyse Appliquée a la&#xA;  Géométrie&lt;/em&gt; by Monge and the &lt;em&gt;Application de l’Algèbre à la Géométrie&lt;/em&gt;&#xA;  by Monge and Hachette.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Belhoste, Bruno. &lt;em&gt;&lt;a href=&quot;http://link.springer.com/book/10.1007/978-1-4612-2996-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Augustin-Louis Cauchy: A Biography&lt;/a&gt;&lt;/em&gt;. New York: Springer-Verlag, 1991. pp. 7-10. Appendix II is Cauchy's outlines of his analysis courses he taught at the &lt;em&gt;École Polytechnique&lt;/em&gt; from 1816-1819.&#xA;&lt;br&gt;See &lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=hvd.32044079809943;view=1up;seq=400&quot; rel=&quot;nofollow noreferrer&quot;&gt;the distribution of courses at the École Polytechnique when Cauchy was a student there&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-01-16T15:11:44.153" LastActivityDate="2018-01-16T15:11:44.153" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5113" PostTypeId="2" ParentId="5102" CreationDate="2016-08-18T15:18:11.250" Score="5" Body="&lt;p&gt;Here is a modern mathematical paper in Latin,&#xA;&lt;a href=&quot;http://www.math.univ-toulouse.fr/~schechtman/defin-nova-preprint.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.math.univ-toulouse.fr/~schechtman/defin-nova-preprint.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not sure about physicists:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-08-18T15:18:11.250" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5114" PostTypeId="2" ParentId="5093" CreationDate="2016-08-18T15:30:58.210" Score="17" Body="&lt;p&gt;I would say that no area of mathematics has ever been completely abandoned. The areas go in and out of fashion, but nothing seems to be completely abandoned.&#xA;For example, approximately in 1940's most mainstream mathematical journals stopped to consider papers on elementary geometry. But the area is not abandoned in any way. First of all, there are &quot;non-mainstream&quot; journals, second, new results in elementary geometry can be parts of the research dedicated to other subjects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give my own paper as an example:&#xA;Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann. of Math. (2) 152 (2000), no. 2, 551-592,  arXiv:math/0009251.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is published in a mainstream journal, and its main subject belongs to analysis. However the core argument in this paper is elementary-geometric, and it contains new results in elementary geometry. And in the reference list of this paper you can see some mid 19 century elementary geometry books.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-08-18T15:30:58.210" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5115" PostTypeId="2" ParentId="5084" CreationDate="2016-08-18T15:40:55.977" Score="3" Body="&lt;p&gt;On my opinion, there are two things which triggered this process in 19th cetury:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Galois theory (and the work of his predecessors, like Lagrange and Cauchy). They introduced the notion of group.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work on crystallography (second part of 19th century) where groups are explicitly used to develop a physical theory.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Of course the study of symmetry is much older than that but only in 19th century,&#xA;because of the evolution of the abstract notion of group it became a central part of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Then in the early 20th century group theory was applied with the great success to&#xA;understanding quantum mechanics and elementary particles, and from that time it was clearly realized  that symmetry also plays the fundamental role in the physical laws. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-08-18T16:23:28.653" LastActivityDate="2016-08-18T16:23:28.653" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5116" PostTypeId="1" AcceptedAnswerId="5117" CreationDate="2016-08-18T20:49:29.283" Score="8" ViewCount="1348" Body="&lt;p&gt;Where can I find this paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J.W. Strutt, Verh. d. deutsch. phys. Ges. &lt;strong&gt;2&lt;/strong&gt;, 65 (1900).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is presumably where Rayleigh derived the black-body radiation formula (the incorrect one that has ultraviolet infinity).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the fact that this paper is, at least historically, so important, I don't understand why it is not available anywhere. I had a look here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/verhandlungende01goog#page/n270/mode/2up&quot; rel=&quot;nofollow&quot;&gt;https://archive.org/stream/verhandlungende01goog#page/n270/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but I cannot see it. I also had a look at his collected papers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/scientificpapers04rayliala&quot; rel=&quot;nofollow&quot;&gt;https://archive.org/details/scientificpapers04rayliala&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user2831" LastEditorUserId="55" LastEditDate="2017-02-02T00:51:09.787" LastActivityDate="2020-04-11T01:53:21.203" Title="Where did Rayleigh derive the ultraviolet catastrophe?" Tags="|physics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5117" PostTypeId="2" ParentId="5116" CreationDate="2016-08-19T05:56:04.717" Score="12" Body="&lt;p&gt;The short answer is that you can not find it because it does not exist, Rayleigh never derived the &quot;ultraviolet catastrophe&quot;. Chapter VI of &lt;a href=&quot;https://books.google.com/books?id=zYUhVHzlr2IC&amp;amp;pg=PA291&amp;amp;lpg=PA291&amp;amp;dq=remarks+upon+the+law+of+complete+radiation.+Phil.+Mag.+49,+539-540&amp;amp;source=bl&amp;amp;ots=ePIyAXcv80&amp;amp;sig=Ioi65j7Zlo8JX5lqHJl3zJM9fT8&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjEz7C32czOAhUI5WMKHWm3DeQQ6AEIHDAA#v=onepage&amp;amp;q=remarks%20upon%20the%20law%20of%20complete%20radiation.%20Phil.%20Mag.%2049%2C%20539-540&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kuhn's book on the history of quantum mechanics&lt;/a&gt; reads: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The claim that black-body radiation should conform to the distribution law that has since been variously attributed to Rayleigh and Jeans was not made until 1905. But the main conceptual foundations for that claim can be found in a two-page note published by Rayleigh in the June 1900 issue of the Philosophical Magazine&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Notes to the chapter give the reference as Lord Rayleigh,&quot;Remarks upon the Law of Complete Radiation&quot;, The Philosophy Magazine, 49 (1900), 539-540; reprinted in John William Strutt, Baron Rayleigh, Scientific Papers, Vol. 4 (Cambridge, Eng., 1903), pp. 483-485. Here is the &lt;a href=&quot;https://archive.org/details/londonedinburgh549190lond&quot; rel=&quot;nofollow noreferrer&quot;&gt;direct link to the Philosophy Magazine&lt;/a&gt;. I suspect that your citation is garbled. Kuhn's note adds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This paper has been carefully discussed by Hans Kangro in his Vorgeschichte des Planckschen Strahlungsgesetzes (Wiesbaden, 1970), pp. 189-192, and also by M. J. Klein in his &quot;Max Planck and the Beginnings of the Quantum Theory,&quot; Archive for History of Exact Sciences, 1 (1962), 459-479, esp. 465-468. Both emphasize the importance of recognizing that it does not contain the Rayleigh-Jeans law&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here is more from the chapter:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Rubens and Kurlbaum compared a number of proposed radiation formulas with their data and concluded that Rayleigh's was satisfactory only in the limit where it coincided with&#xA;  Planck's. Since the law, as proposed, was almost totally ad hoc, there was no further reason to take it seriously. Less than six months after it had been suggested, it was set aside&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The historical importance of the paper, and of the &quot;ultraviolet catastrophe&quot; itself, has been greatly exaggerated by the &quot;rational reconstruction&quot; of history that turned it into the motivation for Planck's work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was no such motivation, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2257/what-was-different-about-plancks-quantization-of-light-compared-to-einsteins/2259#2259&quot;&gt;What was different about Planck&amp;#39;s quantization of light compared to Einstein&amp;#39;s?&lt;/a&gt; Planck derived his formula even before the Rayleigh's note, Rayleigh's two page argument was &quot;partly theoretical and partly ad hoc&quot;, and &quot;both cryptic and incomplete&quot;, according to Kuhn. Nobody knew of the &quot;catastrophe&quot; before 1905, although Lorentz already had some misgivings about the compatibility of classical electrodynamics with Planck's quanta. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And even after Jeans's 1905 derivation it was still mostly Lorentz who cared about the mismatch (Planck and others dismissed it because they rejected the equipartition theorem on which Jeans relied). Only after Lorentz gave an airtight derivation of the Jeans law from the most general classical principles in 1908 did the issue enter the public consciousness, and even then the anxiety was more about the incompatibility of Planck's law with the classical physics than about the &quot;ultraviolet catastrophe&quot;. The nickname was first circulated by Ehrenfest during the first Solvay Congress in 1911. For more references see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3404/how-did-planck-derive-the-black-body-radiation-formula-without-using-the-bose-st/3408#3408&quot;&gt;How did Planck derive the black body radiation formula without using the Bose statistics?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-11T01:53:21.203" LastActivityDate="2020-04-11T01:53:21.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5119" PostTypeId="2" ParentId="3681" CreationDate="2016-08-19T16:35:21.047" Score="3" Body="&lt;p&gt;Between algebras and vector spaces there was (and still is, as taught in some physics courses or in those  known as Vector Calculus at many American universities) the algebra of vectors developed by Oliver Heaviside to serve the needs of physics. It appeared in his paper &quot;On the forces, stresses, and fluxes of energy in the electromagnetic field&quot;, Philosophical Transactions of the Royal Society (1892), pp. 521-574 and treats rigorously the now-familiar operations on vectors such as scalar or vector product, also divergence and curl.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The commentary by Ido Yavetz in Chapter 49 (on Heaviside's electrical papers) in &quot;Landmark writings in Western Mathematics 1640-1940&quot;, edited by Ivor Grattan-Guinness, Elsevier, 2005, explicates the algebraic origins of Heaviside's work:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The remarkable thing about vector algebra as we currently know it, is that it did not emerge from a direct attempt to formalize such widely familiar applications. Instead, it made its first appearance in the context of Hamilton's highly innovative and idiosyncratic invention of quaternions. (...) In their quaternionic context, however, vectors possessed formal properties that proved awkward for application to physical problems. it was then left for Gibbs and Heaviside to extract the vector from its quaternionic foundations and establish it on its own formal grounds.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added on November 11, 2016: A good reference is: Michael J. Crowe, A history of vector analysis: the evolution of the idea of a vectorial system. Originally published by Notre Dame University Press, 1962; Dover reprints 1985 and 1994.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="3434" LastEditDate="2016-11-11T20:18:41.130" LastActivityDate="2016-11-11T20:18:41.130" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5120" PostTypeId="1" CreationDate="2016-08-20T14:36:45.590" Score="-5" ViewCount="241" Body="&lt;p&gt;i mean Bohr's model and Schroedinger equation are based on simple mathematics so why couldn't Einstein or another mathematician to discover them ?&lt;/p&gt;&#xA;" OwnerUserId="3786" LastActivityDate="2016-08-20T20:43:09.023" Title="If Einstein was a genious why didn't he discover the BOhr's atom model and Schroedinger equation" Tags="|physics|" AnswerCount="1" CommentCount="2" ClosedDate="2016-08-20T23:39:15.430" ContentLicense="CC BY-SA 3.0" />
  <row Id="5121" PostTypeId="2" ParentId="5120" CreationDate="2016-08-20T20:43:09.023" Score="2" Body="&lt;p&gt;First came the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rutherford_model&quot; rel=&quot;nofollow&quot;&gt;Rutherford model (1911)&lt;/a&gt;, based on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rutherford_scattering&quot; rel=&quot;nofollow&quot;&gt;Rutherford scattering experiments (1909)&lt;/a&gt; and the discovery of the nucleus.  The Rutherford model had several defects, but most importantly, how were the electrons kept in orbit about the nucleus, whose positive charge was constantly drawing them in?  Maxwell's equations require that an accelerating charge must radiate energy, and so the orbits couldn't be stable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pbs.org/wgbh/aso/databank/entries/dp13at.html&quot; rel=&quot;nofollow&quot;&gt;Neils Bohr (1912)&lt;/a&gt; came to work with Rutherford; he analyzed the defects of the model, and applied &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck%E2%80%93Einstein_relation&quot; rel=&quot;nofollow&quot;&gt;Planck's quantum of energy (1900)&lt;/a&gt; to the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result, after much careful pondering of the situation, including the experimental results, was the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr_model&quot; rel=&quot;nofollow&quot;&gt;Bohr model (1913)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This model works quite well for the hydrogen atom, but was soon shown to have its own defects.  These were resolved, in part, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Matter_wave&quot; rel=&quot;nofollow&quot;&gt;de Broglie's quantum waves (1924)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Putting together the Planck relation, $E=hf$ with the de Broglie relation (in wave mechanics form), $P=hk$, one can write the equation for a plane wave, here to be interpreted as de Broglie's matter wave, as $\psi = exp(i(kx-ft))=exp(i/h(px-Et))$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This &quot;wave equation&quot; was then solved by &lt;a href=&quot;https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation&quot; rel=&quot;nofollow&quot;&gt;Schrodinger (1925)&lt;/a&gt; by applying Hamiltonian mechanics to the problem.  Here the Hamiltonian is not the expression for the energy, but instead is the operator which extracts the energy from the wave function.  This is not the place to go into all the details of how he came to his solution, but he was in part inspired by &lt;a href=&quot;https://en.wikipedia.org/wiki/Heisenberg_picture&quot; rel=&quot;nofollow&quot;&gt;Heisenberg's slightly earlier results&lt;/a&gt;, as well as the de Broglie wave.  The two methods are complementary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that Bohr's reputation is not based solely upon his erroneous atomic model.  If Einstein had taken an interest in Rutherford's results, and had put aside his work on gravitation for half a year, he may have come up with something similar - but probably would not have published for the model only works for Hydrogen.  Why would a successful scientist want to publish something that is obviously wrong from the beginning?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By 1924, Einstein was already past his prime, and had completed major works on statistical mechanics, special relativity, general relativity, as well as several works on condensed matter; his most recent work was on light (1916) and Bose-Einstein statistics (1924). He had made major contributions to the &quot;old quantum theory&quot;, and simply missed the boat as it sailed by.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After all, one cannot do everything, and even if the equations look simple to you, there are many subtle issues underlying their development.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-08-20T20:43:09.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5124" PostTypeId="1" CreationDate="2016-08-22T18:36:20.503" Score="1" ViewCount="652" Body="&lt;p&gt;What physicist first discovered or produced an alternating current and recognized it as such?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-08-22T20:07:05.763" LastActivityDate="2016-08-22T20:07:05.763" Title="Who first discovered and produced an alternating current?" Tags="|physics|experimental-physics|electromagnetism|electricity|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5126" PostTypeId="1" AcceptedAnswerId="5127" CreationDate="2016-08-23T19:17:48.127" Score="7" ViewCount="372" Body="&lt;p&gt;In Europe, in the 20th century, $\int\frac{1}{x}dx$ equalled $\ln{x}+C$. (I have references from Poland for 1930-1947 and the UK for the 1960s and 1970s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, if one mentions $\int\frac{1}{x}dx=\ln{x}+C$ in Mathematics Stack Exchange, one is lynched. The doctrine is now that $\int\frac{1}{x}dx=\ln|x|+C$, and any suggestion to the contrary is a crime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't want to discuss the pros and cons of this alteration, but I am interested to know: who made this change, and when? Was it intentional, or just an influential textbook? Or is this less of an &quot;old millennium/new millennium&quot; and more of an &quot;Old World/New World&quot; kind of thing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;There is an argument for posting this to Maths Stack Exchange rather than here, and I will if asked; but it &lt;strong&gt;is&lt;/strong&gt; a historical question and so this seems a reasonable forum for it.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="4209" LastEditorUserId="55" LastEditDate="2016-08-23T21:02:36.197" LastActivityDate="2016-08-30T11:16:40.637" Title="When was the convention for the indefinite integral $\int\frac{1}{x}dx$ changed?" Tags="|mathematics|terminology|calculus|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5127" PostTypeId="2" ParentId="5126" CreationDate="2016-08-24T00:09:07.827" Score="5" Body="&lt;p&gt;Mathematicians say $\int \frac{1}{x}\;dx = \log x + C$.  It works even for the complex case.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Calculus instructors say $\int \frac{1}{x}\;dx = \ln|x|+C$ for some reason, but it is WRONG in the complex case.  (Perhaps that &quot;ln&quot; in there gives us a hint that they are writing for engineers and physicists rather than mathematicians.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my guess is that the change came around 1950 or so, when calculus textbook writers were no longer research mathematicians first, and textbook writers later.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-08-24T00:09:07.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5129" PostTypeId="1" AcceptedAnswerId="5134" CreationDate="2016-08-26T16:56:37.947" Score="15" ViewCount="2045" Body="&lt;p&gt;&lt;a href=&quot;http://lesswrong.com/lw/2v1/great_mathematicians_on_math_competitions_and&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; I found the following quote, attributed to the great &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Grothendieck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grothendieck&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.&lt;/p&gt;&#xA;&lt;p&gt;In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Did he &lt;em&gt;really&lt;/em&gt; said this? It seems very ironical, since Grothendieck is generally known as a genius, who was one of the greatest mathematicians of the 20th century.&lt;/p&gt;&#xA;&lt;p&gt;Could you attest the authenticity of this quote? The website I linked to where I found this quote gives as a source &lt;em&gt;Récoltes et Semailles&lt;/em&gt;. I can't understand french, and would be really thankful if someone of you could check if Grothendieck really wrote what is quoted above.&lt;/p&gt;&#xA;" OwnerUserId="4612" LastEditorUserId="4251" LastEditDate="2021-09-13T11:01:18.940" LastActivityDate="2021-09-13T11:01:18.940" Title="Did Grothendieck really say that he felt &quot;clumsy, even oafish, wandering painfully up an arduous track&quot;?" Tags="|mathematicians|biographical-details|quote|social-context|grothendieck|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5130" PostTypeId="1" AcceptedAnswerId="5136" CreationDate="2016-08-26T00:30:43.980" Score="3" ViewCount="303" Body="&lt;p&gt;Without a &lt;a href=&quot;https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae&quot; rel=&quot;nofollow&quot;&gt;Disquisitiones Arithmeticae&lt;/a&gt; at hand, I may ask... When the unique factorization theorem was first called &lt;em&gt;the&lt;/em&gt; Fundamental Theorem of Arithmetic?&lt;/p&gt;&#xA;" OwnerUserId="4617" OwnerDisplayName="André LFS Bacci" LastEditorUserId="98" LastEditDate="2016-08-26T18:08:02.057" LastActivityDate="2016-08-28T10:06:43.637" Title="First mention of Fundamental Theorem of Arithmetic" Tags="|mathematics|terminology|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5131" PostTypeId="2" ParentId="5130" CreationDate="2016-08-26T01:19:35.043" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/f.html&quot; rel=&quot;nofollow&quot;&gt;&quot;Earliest Known Uses of Some of the Words of Mathematics&quot;&lt;/a&gt; it's:G. H. Hardy and E. M. Wright 'An Introduction to the Theory of Numbers' first published in 1938 that first used the terminology Fundamental Theorem of Arithmetic to refer to this result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more detail it says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;FUNDAMENTAL THEOREM OF ARITHMETIC is another name for the unique factorization theorem, that any positive integer can be represented in exactly one way as a product of primes. The name was used by G. H. Hardy and E. M. Wright An Introduction to the Theory of Numbers (1938, p. 3) who remark that the theorem &quot;does not seem to have been stated explicitly before Gauss (Disquisitiones arithmeticae (1801, §16)). It was, of course, familiar to earlier mathematicians; but Gauss was the first to develop arithmetic as a systematic science.&quot; See also A. G. Ağargün &amp;amp; E. M. Özkan &quot;A Historical Survey of the Fundamental Theorem of Arithmetic,&quot; Historia Mathematica, 28, (2001), 207-214.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The term appears with another meaning in 1895 in the Century Dictionary: &quot;the proposition that any lot of things the count of which in any order can be terminated is such that the count in every order can be terminated, and ends with the same number.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="98" OwnerDisplayName="quid" LastActivityDate="2016-08-26T01:19:35.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5134" PostTypeId="2" ParentId="5129" CreationDate="2016-08-26T19:10:04.993" Score="17" Body="&lt;p&gt;Yes, the quote is essentially authentic. This is from a typeset version &quot;Récoltes et Semailles&quot;, specfically from &quot;2.2 L’importance d’être seul.&quot; (To find the document online should be possible without much trouble, I do not link it here, as I am not aware of a stable location.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Par la suite, j’ai eu l’occasion, dans ce monde des mathématiciens qui m’accueillait, de rencontrer bien des&#xA;  gens, aussi bien des aînés que des jeunes gens plus ou moins de mon âge, qui visiblement étaient beaucoup plus&#xA;  brillants, beaucoup plus &quot;doués&quot; que moi. Je les admirais pour la facilité avec laquelle ils apprenaient, comme&#xA;  en se jouant, des notions nouvelles, et jonglaient avec comme s’ils les connaissaient depuis leur berceau&#xA;  - alors que je me sentais lourd et pataud, me frayant un chemin péniblement, comme une taupe, à travers&#xA;  une montagne informe de choses qu’il était important (m’assurait-on) que j’apprenne, et dont je me sentais&#xA;  incapable de saisir les tenants et les aboutissants. En fait, je n’avais rien de l’étudiant brillant, passant haut la&#xA;  main les concours prestigieux, assimilant en un tournemain des programmes prohibitifs.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;La plupart de mes camarades plus brillants sont d’ailleurs devenus des mathématiciens compétents et réputés.&#xA;  Pourtant, avec le recul de trente ou trente-cinq ans, je vois qu’ils n’ont pas laissé sur la mathématique de notre temps une empreinte vraiment profonde. Ils ont fait des choses, des belles choses parfois, dans un contexte déjà tout fait, auquel ils n’auraient pas songé à toucher. Ils sont restés prisonniers sans le savoir de&#xA;  ces cercles invisibles et impérieux, qui délimitent un Univers dans un milieu et à une époque donnée. Pour les franchir, il aurait fallu qu’ils retrouvent en eux cette capacité qui était leur à leur naissance, tout comme elle était mienne : la capacité d’être seul.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As far as I can tell the English text is pretty much just a translation of this. There are some parts were it deviates in a minor way. Either there are slightly different versions, or the translation is just a bit free. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular the &quot;dump ox&quot; comes from &quot;une taupe&quot;, which is a mole not an ox, and &quot;despotic&quot; comes from &quot;impérieux,&quot; and the translation seems a bit on the negative end to me, I think &quot;ruling&quot;, &quot;domineering&quot;, or &quot;imperious&quot; could also have been used.   &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2016-08-26T19:10:04.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5136" PostTypeId="2" ParentId="5130" CreationDate="2016-08-27T14:21:32.903" Score="2" Body="&lt;p&gt;This is difficult to answer since the first time it was called fundamental theorem it might have been in German or Russian. In any case Eric Temple Bell used it in 1915, on p.~2 of &lt;a href=&quot;https://archive.org/details/ariththeorycernu00bellrich&quot; rel=&quot;nofollow&quot;&gt;An Arithmetical Theory of Certain Numerical Functions&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-08-27T14:21:32.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5138" PostTypeId="2" ParentId="5130" CreationDate="2016-08-28T09:56:44.480" Score="2" Body="&lt;p&gt;It is true that Gauss (1801) did not call his theorem no. 16 “theorema fundamentale arithmeticae”, but he does include it under the heading “theoremata praeliminaria de numeris primis, factoribus etc.” (page 14), which is not very much different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/pdfcache/PPN235993352/PPN235993352___LOG_0007.pdf&quot; rel=&quot;nofollow&quot;&gt;http://gdz.sub.uni-goettingen.de/pdfcache/PPN235993352/PPN235993352___LOG_0007.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-08-28T10:06:43.637" LastActivityDate="2016-08-28T10:06:43.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5140" PostTypeId="1" AcceptedAnswerId="5150" CreationDate="2016-08-28T15:03:03.320" Score="8" ViewCount="1715" Body="&lt;p&gt;Thomas Kuhn writes in The Structure of Scientific Revolutions&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Part of what the acceptance of Ohm’s Law demanded was a redefinition of both ‘current’ and ‘resistance’; if those terms had continued to mean what they had meant before, Ohm’s Law could not have been right; that is why it was so strenuously opposed as, say, the Joule-Lenz Law was not.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What where the historical definitions of the terms before Ohm's Law?&lt;/p&gt;&#xA;" OwnerUserId="4623" LastEditorUserId="55" LastEditDate="2017-03-29T00:55:57.013" LastActivityDate="2019-11-30T01:43:04.997" Title="What is the history of electric current and resistance?" Tags="|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5141" PostTypeId="1" CreationDate="2016-08-28T19:05:19.903" Score="8" ViewCount="1452" Body="&lt;p&gt;Throughout the centuries the meter has been redefined a number of times. At one point it was defined in relation to the distance from the north pole to the equator. Seeing how that is a bit of an awkward standard to use, a set of platinum bars was made. Each bar as close to a meter as was possible. On the 22nd of June 1799 the most accurate bar was selected and placed in the french national achieves. It is now known as the &quot;mètre des Archives&quot;. In 1889 the polar definition was abandoned and a new definition was to be based on a bar called the International Prototype Meter. The IPM was to be minted to the same length as the mètre des Archives. In total 30 bars were created, 1 was made the IPM and the others given out to serve as national standards. This new standard was ultimately abandoned in 1960 and replaced with the (now abandoned) krypton standard. So, my question is, what happened to the bars? Where are they now? I'm pretty sure the IPM and mètre des Archives are still maintained in France by the International Bureau of Weights and Measures. But what of the copies?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Here's what I've learned so far about the current state of the bars (not including an apparent failed production run that produced bars of unacceptable quality were still internationally distributed)&lt;br&gt;&#xA;Country assignment is given by &lt;a href=&quot;http://www.bipm.org/fr/CGPM/db/1/1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the resolution of the 1st CGPM&lt;/a&gt; (French only)&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;(Italy)&lt;/li&gt;&#xA;&lt;li&gt;(Switzerland)&lt;/li&gt;&#xA;&lt;li&gt;(Norway)&lt;/li&gt;&#xA;&lt;li&gt;(France)&lt;/li&gt;&#xA;&lt;li&gt;(Not on the table, I don't know why)&lt;/li&gt;&#xA;&lt;li&gt;Declared the IPM, maintained by the International Bureau of Weights and Measures &lt;/li&gt;&#xA;&lt;li&gt;(Bavaria/Germany now)&lt;/li&gt;&#xA;&lt;li&gt;(France)&lt;/li&gt;&#xA;&lt;li&gt;(Italy)&lt;/li&gt;&#xA;&lt;li&gt;(Portugal)&lt;/li&gt;&#xA;&lt;li&gt;(Russian Academy of Sciences)&lt;/li&gt;&#xA;&lt;li&gt;(Belgium)&lt;/li&gt;&#xA;&lt;li&gt;(Not on the table, Referred to as a witness meter? (machine translated))&lt;/li&gt;&#xA;&lt;li&gt;(Hungary)&lt;/li&gt;&#xA;&lt;li&gt;(Austria)&lt;/li&gt;&#xA;&lt;li&gt;(UK) Held by the National Physical Laboratory &lt;a href=&quot;http://www.npl.co.uk/reference/faqs/do-you-still-use-the-&amp;#39;metre-bar&amp;#39;-as-the-length-standard-(faq-length)&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt; (thanks Stuart Hall)&lt;/li&gt;&#xA;&lt;li&gt;(Spain)&lt;/li&gt;&#xA;&lt;li&gt;(Germany)&lt;/li&gt;&#xA;&lt;li&gt;(Austria)&lt;/li&gt;&#xA;&lt;li&gt;(France)&lt;/li&gt;&#xA;&lt;li&gt;(USA) With 27 &lt;a href=&quot;https://www.gpo.gov/fdsys/pkg/GOVPUB-C13-1e6fcfef0ffc7545cea4792e9aa6f78b/pdf/GOVPUB-C13-1e6fcfef0ffc7545cea4792e9aa6f78b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;source, see page 13&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;(Japan) Kept in the Paris Observatory (thanks Erik)&lt;/li&gt;&#xA;&lt;li&gt;(Belgium)&lt;/li&gt;&#xA;&lt;li&gt;(Spain)&lt;/li&gt;&#xA;&lt;li&gt;(Royal Observatory of Belgium)&lt;/li&gt;&#xA;&lt;li&gt;(International Bureau of Weights and Measures)&lt;/li&gt;&#xA;&lt;li&gt;(USA) Primary national standard, Kept in the NIST museum &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:US_National_Length_Meter.JPG&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt; &lt;a href=&quot;http://museum.nist.gov/object.asp?ObjID=37&quot; rel=&quot;nofollow noreferrer&quot;&gt;source 2&lt;/a&gt; (thanks robert bristow-johnson)&lt;/li&gt;&#xA;&lt;li&gt;(Russia)&lt;/li&gt;&#xA;&lt;li&gt;(Sweden)&lt;/li&gt;&#xA;&lt;li&gt;(Serbia)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4625" LastEditorUserId="4625" LastEditDate="2018-07-15T14:37:03.343" LastActivityDate="2018-07-15T14:37:03.343" Title="What happened to the meter bars?" Tags="|units|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="5142" PostTypeId="2" ParentId="5084" CreationDate="2016-08-29T08:30:23.840" Score="2" Body="&lt;p&gt;As a side remark: I recommend warmly the book: &quot;From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept&quot; by Hon\Goldstein, which deals with the emergence of the concept of symmetry till the beginning of the 19th cent.&lt;/p&gt;&#xA;" OwnerUserId="4126" LastActivityDate="2016-08-29T08:30:23.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5143" PostTypeId="1" CreationDate="2016-08-22T15:23:43.563" Score="5" ViewCount="153" Body="&lt;p&gt;The key of this question is not simply stating the content of his theory, but telling what results or principles he had at that time and how could they lead to the theory. He was as such the first person to measure relative atomic weights. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically what idea of technique was used and how to measure relative atomic weights using macroscopic measurements? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: Both wiki states that it was based on the principles of mass conservation and definite proportion. How?&lt;/p&gt;&#xA;" OwnerDisplayName="Kevin Kwok" LastEditorUserId="26" LastEditDate="2016-08-29T17:32:21.433" LastActivityDate="2016-08-29T17:32:21.433" Title="How could John Dalton hypothesised his atomic theory based on his experimental observations?" Tags="|experimental-physics|chemistry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5144" PostTypeId="2" ParentId="5141" CreationDate="2016-08-29T17:49:32.610" Score="6" Body="&lt;p&gt;The UK copies are still stored at the National Physical Laboratory.&#xA;See these links:&#xA;&lt;a href=&quot;http://www.npl.co.uk/reference/faqs/do-you-still-use-the-&amp;#39;metre-bar&amp;#39;-as-the-length-standard-(faq-length)&quot;&gt;Metre&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.npl.co.uk/educate-explore/redefining-the-kilogram/&quot;&gt;Kilogram&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4629" LastActivityDate="2016-08-29T17:49:32.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5145" PostTypeId="2" ParentId="5126" CreationDate="2016-08-30T08:52:26.757" Score="-3" Body="&lt;p&gt;I think that´s because now we use a special integral to integrate over the complex plane. It´s called &lt;a href=&quot;https://en.wikipedia.org/wiki/Methods_of_contour_integration&quot; rel=&quot;nofollow&quot;&gt;Contour Integral&lt;/a&gt;. I think then when we´d want to integrate over complex plane we can write a Contour integral and then it´s clear that the answer must be ln x and not the absolute value. But with this integral there is a problem. When you evaluate that integral by contour integration (over complex plane) the answer is then &lt;a href=&quot;https://wikimedia.org/api/rest_v1/media/math/render/svg/688f8412a8a0d422faf76fb1c9b07148d0cffab1&quot; rel=&quot;nofollow&quot;&gt;2pii&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4630" LastEditorUserId="4630" LastEditDate="2016-08-30T11:16:40.637" LastActivityDate="2016-08-30T11:16:40.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5147" PostTypeId="1" CreationDate="2016-08-30T19:17:00.890" Score="7" ViewCount="575" Body="&lt;p&gt;Let me quote from Terence Tao &amp;quot;Analysis 1&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Historically, the realization that numbers could&#xA;be treated axiomatically is very recent, not much more than a&#xA;hundred years old.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then, how could the people who lived before the axiomatization of real numbers be sure that, for example, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; always equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;? Because, back then they did have a set of axioms from which they could prove things, and thus they also didn't have a notion of rigorous proof. Does this mean that they just observed the pattern that if they took two concrete numbers, it didn't matter if they said &amp;quot;the first times the second&amp;quot; or &amp;quot;the second times the first&amp;quot;; and because of this observation they assumed &lt;span class=&quot;math-container&quot;&gt;$a \times b = b \times a$&lt;/span&gt; without a proof?&lt;/p&gt;&#xA;" OwnerUserId="4633" LastEditorUserId="6609" LastEditDate="2023-08-03T09:28:58.143" LastActivityDate="2023-08-03T09:28:58.143" Title="How could the people of the past be sure that $a \times b = b \times a$?" Tags="|mathematics|number-theory|mathematical-logic|" AnswerCount="3" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="5148" PostTypeId="1" AcceptedAnswerId="5154" CreationDate="2016-08-31T12:26:29.933" Score="9" ViewCount="394" Body="&lt;p&gt;I understand almost nothing about sphere packings and kissing numbers, but I am very curious to learn about the Newton - Gregory discussion of the problem. In particular, I wonder if Newton's guess was a &quot;blind guess&quot; or an educated guess with a certain logic behind it. So if someone knows a good article that focuses on the Newton - Gregory discussion please give a link.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="55" LastEditDate="2016-09-01T03:08:13.130" LastActivityDate="2016-09-04T03:31:56.283" Title="How did Newton guess that the answer to the 3-dimensional &quot;kissing problem&quot; is 12?" Tags="|mathematics|debunking|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5149" PostTypeId="2" ParentId="5147" CreationDate="2016-08-31T14:51:02.967" Score="9" Body="&lt;p&gt;Multiplication, before the invention of modern (axiomatic) algebra, was &lt;em&gt;defined&lt;/em&gt; as the operation giving the area of a rectangle with sides of a particular length.&lt;a href=&quot;http://www.themathpage.com/abooki/propI-39-41.htm&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt;  Commutativity of multiplication then follows from two axioms:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Congruent geometrical figures have equal areas&lt;/li&gt;&#xA;&lt;li&gt;Any geometrical figure reflected through a line is congruent to the original figure&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and the observation that reflecting a rectangle through a line through a corner at a 45 degree angle from the two sides flips the roles of the sides of the rectangle, so the side that formerly corresponded to $a$ in the figure now corresponds to $b$ and vice-versa.  Since the first rectangle corresponds to $a*b$ and the second rectangle corresponds to $b*a$, and they have the same area, $a*b = b*a$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: the geometrical intuition behind this proof is still used today, in the proof in set theory that $|A\times B| = |B \times A|$.  (The regular inductive proof you may have seen for integers would work for finite sets, but for infinite sets it's easier to do a direct 'geometrical' proof).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.themathpage.com/abooki/propI-39-41.htm&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt; For example, &lt;a href=&quot;http://www.themathpage.com/abooki/propI-39-41.htm&quot; rel=&quot;noreferrer&quot;&gt;Euclid states the theorem&lt;/a&gt; which today we would state as &quot;the area of a triangle with height $h$ and base $b$ is $\frac{1}{2}bh$ as &quot;If a parallelogram and a triangle are on same base and in the same parallels, the parallelogram is double the triangle&quot;, i.e., &quot;the area of a triangle with height $h$ and base $b$ is half the area of a parallelogram with the same height and same base&quot;.  &lt;a href=&quot;http://self.gutenberg.org/articles/eng/measurement_of_the_circle&quot; rel=&quot;noreferrer&quot;&gt;Archimedes goes a step further&lt;/a&gt; and states the theorem &quot;the area of a circle with radius $r$ is $\pi r^2$&quot; (which he was the first to prove) as &quot;The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle&quot;, i.e., &quot;the area of a circle is $\frac{1}{2}rC$&quot;.  Note that $\frac{1}{2}rC = \frac{1}{2}r\pi D = \frac{1}{2}r\pi 2r = \pi r^2$&quot;, but Archimedes evidently lacks the language to express his result in that form.&lt;/p&gt;&#xA;" OwnerUserId="4434" LastEditorUserId="4434" LastEditDate="2016-08-31T19:51:43.090" LastActivityDate="2016-08-31T19:51:43.090" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5150" PostTypeId="2" ParentId="5140" CreationDate="2016-08-31T17:13:29.767" Score="6" Body="&lt;p&gt;TL;DR At the time of Ohm's papers (1825-28) current and resistance were very young notions, and Ampere's recent theories shifted their understanding significantly, while Ohm himself completed the shift. Unfortunately, there is no straightforward way to map prior notions to familiar modern concepts. I suppose Kuhn chose this example for a reason, perhaps this is an example of his &quot;incommensurability&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was no notion of current and resistance to speak of before 1800, the only kinds of current encountered were the electrostatic discharges. Even so, du Fay and Franklin proposed &lt;a href=&quot;https://en.wikipedia.org/wiki/Fluid_theory_of_electricity&quot; rel=&quot;nofollow noreferrer&quot;&gt;two fluid (positive and negative) and one fluid theories of electricity&lt;/a&gt;, which made its flow imaginable. In 1800 Volta invented the first source of continuous current, his famous &quot;pile&quot; of paired copper-zinc plates interspaced with moistened cardboard. &lt;a href=&quot;https://www.jstor.org/stable/27757295&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brown's Electric Current in Early Nineteenth-Century French Physics&lt;/a&gt; gives a comprehensive survey of the state (or rather evolution) of the art between Volta and Ohm. According to Volta, the electric fluid was propelled into &quot;perpetual impulsion&quot; by an &quot;electromotive contact force&quot; between copper and zinc, multiplied by the pile's layers. However, at the time the only developed theory was electrostatic, and all measurements and measuring instruments were based on it. So Volta's &quot;courant&quot; was an empty word, and his &quot;theory&quot; was soon massaged beyond recognition by an authoritative commission of the French Academy consisting of Laplace, Coulomb, Fourcroy, Guyton, and Biot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The report reflecting the commission's views was written by Biot, and unsurprisingly described the pile not as a current source but as a sequencer of electrostatic discharges. But with it, unlike with Volta's theory, one could do measurements of the &quot;quantity of electricity&quot; (charge) and the &quot;exciting force&quot;/&quot;intensity&quot;/&quot;tension&quot; (roughly voltage between the terminals of the pile) with electrostatic instruments, like Coulomb's electroscope. To the extent that resistance figured there Biot identified it with &quot;isolating lengths&quot; in the pile. Biot's electric quasi-statics remained the orthodoxy until 1820s, and within it there was no way to even measure the current with any precision. The winds of change came with Oersted's discovery of the current affecting magnetic needles, and the invention of the galvanometer based on it. But the change maker was Ampere. As an outsider to the French scientific establishment he was less reserved about dismissing Biot's quai-static picture, and going back to Volta's original intuitions, but with better instruments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ampere introduced the idea of a perpetual circular movement of the two electricities in the wire somehow started by the pile, &quot;reunions in the wire&quot; as he called them. It is interesting that he first came up with his famous &lt;a href=&quot;https://commons.princeton.edu/josephhenry/amperes-theory/&quot; rel=&quot;nofollow noreferrer&quot;&gt;hypothesis of molecular currents&lt;/a&gt; in a magnet, the reunions in infinitesimal circles around each molecule, where it was obvious that the motion has to be continuous to prevent the immediate recombination, and only then transferred the idea to voltaic circuits. Ampere's theory was a hybrid though, as it sharply separated electrostatic effects in the pile, and electrodynamic ones in the wire. The tension of the pile was seen as of electrostatic nature, associated with attraction/repulsion and sparks, and when the pile's terminals were connected by a wire it was seen (and measured) to vanish. The current was seen to produce chemical and magnetic effects. A well known experiment, where several piles are placed together in a series, produced magnified tension, but essentially the same current as measured by the magnetic needle. Never the twain shall meet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was this Ampere's distinction that Ohm disturbed in 1827 with his law. To his fellow physicists he appeared to be suggesting something like a formula linking the intensity of light to its frequency, two clearly independent variables. Here is from &lt;a href=&quot;http://scitation.aip.org/content/aapt/journal/ajp/31/7/10.1119/1.1969620&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schagrin's Resistance to Ohm's Law&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...in having the magnetic action of the electric stream depend on the exciting force of the source, Ohm appeared to be confounding the well recognized distinction between tension electricity and current electricity... Ohm of course was able to explain away the observations that led to the distinction between tension and current electricity. But a scientist who had worked with this distinction, who had obtained fruitful results on the basis of it, could scarcely be expected to shift his viewpoint so radically. To this scientist Ohm's experimental results could not confirm the law; Ohm's remarks were not so much false as they were irrelevant (or even worse, a misuse of language).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a sense, one has to presuppose something like the Ohm's law to get the modern definitions of current, and especially voltage and resistance, their modern meanings in part depend on its form.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Ohm initiated another, more serious, conceptual innovation, of which the first signs can be found in these earlier papers. This was a further revision of the notion of Spannung such that each point in a closed circuit&#xA;  had a particular Spannung&lt;/em&gt; [tension]. &lt;em&gt;The Spannung of a source was, in these terms, the difference in Spannung between the terminals - a notion similar to the concept in the older theory. But that each point of a closed circuit had a Spannung was a notion entirely foreign to the older theory. With this revision of the meaning of &quot;Spannung&quot; Ohm was able to use his law on segments of a circuit rather than on the entire circuit at once.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-30T01:43:04.997" LastActivityDate="2019-11-30T01:43:04.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5151" PostTypeId="1" CreationDate="2016-08-31T18:04:44.387" Score="7" ViewCount="391" Body="&lt;p&gt;In the old days Aristotels argued that object needs a force to keep going in 'space'. Some philosophers, Philoponus and Buridanus (?), later argued that there was a need for some kind of 'impetus'. Did that still imply some kind of active force? Who was the first to suggest that objects only need a force to stop them, but not to keep them moving? Was it Newton or already someone before him?&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="55" LastEditDate="2016-09-01T01:55:37.730" LastActivityDate="2016-09-03T20:35:19.210" Title="Who was the first scientist to suggest that objects can keep moving without applied force?" Tags="|physics|philosophy-of-science|classical-mechanics|energy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5152" PostTypeId="2" ParentId="3661" CreationDate="2016-08-31T19:57:08.477" Score="2" Body="&lt;p&gt;Galileo, Newton, or even the medieval physicist &lt;a href=&quot;http://www.encyclopedia.com/topic/Jean_Buridan.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Buridan&lt;/a&gt; (1295-1358), who developed the notion of &lt;em&gt;impetus&lt;/em&gt;, were not the first to discover the law of inertia.&lt;/p&gt;&#xA;&lt;p&gt;The first was &lt;a href=&quot;http://www.encyclopedia.com/topic/John_Philoponus.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;John Philoponus&lt;/strong&gt;&lt;/a&gt; (&amp;quot;The Grammarian&amp;quot;), who lived in the late 5&lt;sup&gt;th&lt;/sup&gt; and 2&lt;sup&gt;nd&lt;/sup&gt; ½ of 6&lt;sup&gt;th&lt;/sup&gt; century A.D.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.encyclopedia.com/topic/John_Philoponus.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;his &lt;em&gt;Dictionary of Scientific Biography&lt;/em&gt; entry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Philoponus’ main significance for the history of science lies in his being, at the close of antiquity, the first thinker to undertake a comprehensive and massive attack on the principal tenets of Aristotle’s physics and cosmology, an attack unequaled in thoroughness until Galileo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://sententiaedeo.blogspot.com/2013/05/john-philoponus-facts-information.html?m=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, Philoponus&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;argued that the sun is fire and of terrestrial-like, corruptible matter. He devised a precursor to the notion of impetus which &lt;a href=&quot;http://sententiaedeo.blogspot.com/2012/01/medieval-scholars-applied-math-to.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Buridan&lt;/a&gt; later developed, that which keeps moving bodies in motion even after the mover ceases being in contact with them; air does not keep projectiles in motion. He discovered that light rays travel the same both backwards and forwards. He invented functions of variables and their &amp;quot;courses&amp;quot; (what we'd call &amp;quot;first derivatives&amp;quot; in modern calculus). &lt;strong&gt;He discovered the law of inertia, that bodies in motion remain in motion unless something impedes their movement&lt;/strong&gt;, literally a thousand years before Galileo, Newton, et al.!&lt;/p&gt;&#xA;&lt;p&gt;He's certainly one of the &amp;quot;&lt;em&gt;grands génies de l'Antiquité&lt;/em&gt;&amp;quot; (&amp;quot;great geniuses of Antiquity&amp;quot;) and &amp;quot;&lt;em&gt;principaux précurseurs de la Science moderne&lt;/em&gt;&amp;quot; (&amp;quot;principle precursers to modern Science&amp;quot;), as [the historian of medieval physics] &lt;a href=&quot;https://archive.org/stream/lesystmedumond01duhe#page/398/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre Duhem wrote&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-08-31T19:57:08.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5153" PostTypeId="2" ParentId="5151" CreationDate="2016-09-01T01:49:53.027" Score="6" Body="&lt;p&gt;This depends on which objects you have in mind and who you would call a scientist. There was a broad consensus in the ancient Greek natural philosophy that superlunar objects, like the stars and planets, were moving on their own steam, pardon, divine nature, and uniformly along circles at that. Early mathematical astronomers accepted this conviction to such an extent that when it ostensibly did not match the facts Eudoxus (c. 390–337 BC) came up with extremely clever mathematical construction of &lt;a href=&quot;http://www.jstor.org/stable/41134047&quot; rel=&quot;nofollow&quot;&gt;homocentric spheres&lt;/a&gt; to &quot;save the appearances&quot; this way. To the sublunar matter many natural philosophers ascribed self-motion as well, e.g. Empedoclus, Heraclitus, or the atomists, Leucippus and Democritus, and shortly after Aristotle, Stoics and Epicurus. Even Aristotle himself in addition to &quot;violent motions&quot; maintained by an active force had natural motions to natural places (down for earth and water, up for air and fire), which did not require an acting force and could only be stopped by one, see &lt;a href=&quot;https://arxiv.org/abs/1312.4057&quot; rel=&quot;nofollow&quot;&gt;Rovelli's Aristotle's Physics: a Physicist's Look&lt;/a&gt;. It is believed that Newton saw the idea of inertial motion as a modification of Aristotle's natural motions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On a somewhat more restrictive definition of a scientist, and restricting to the &quot;unnatural&quot; motions of the Earthly objects perhaps the first kind of theory that envisioned self-motion was a modification of Aristotle's theory of the projectile motion, where the &quot;impressed force&quot; was not a prototype of force but a prototype of inertia. It was &quot;impressed&quot; at the time of shooting, and  maintained the arrow's flight without the air behind constantly pushing it, as Aristotle contended. The theory is now called &lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_impetus&quot; rel=&quot;nofollow&quot;&gt;impetus theory&lt;/a&gt;, and the impetus was a precursor of both notions of momentum (Descartes) and kinetic energy (Leibniz) in 17th century. Here is &lt;a href=&quot;https://www.jstor.org/stable/3654219&quot; rel=&quot;nofollow&quot;&gt;Franco's Avempace, Projectile Motion, and Impetus Theory&lt;/a&gt; on  who came up with that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;While there is almost unanimous agreement about the theory that challenged and eventually supplanted the Aristotelian explanation, there is no such agreement about who first articulated the impetus theory according to which the arrow is kept in motion by a force impressed upon it by the projector. Samuel Sambursky argues that the idea was due to Hipparchus of Nicaea (second century BC), Shlomo Pines ascribes it to Alexander of Aphrodisias (third century AD), Henri Carteron has argued that is to be found first in the pseudo-Aristotelian treatise De Mechanica, McGuire has seen its origins, more generally, in the Stoic influenced John Philoponus (ca. 490-570), whereas Emil Duhem, and, more recently, Richard Sorabji, Michael Wolff, and others trace the idea of impetus to John Philoponus himself&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hipparchus of Nicea, the father of astronomy, is also the choice of &lt;a href=&quot;https://books.google.com/books/about/The_Forgotten_Revolution.html?id=MOTpnfz7ZuYC&quot; rel=&quot;nofollow&quot;&gt;Russo in his Forgotten Revolution&lt;/a&gt;, and his analysis of sources is interesting, but he is prone to some passionate exaggerations. Philoponus's mocking suggestion to make the arrow fly by waiving hands behind it was rhetorically effective against the Aristotle's theory, and he was certainly the source of Avicenna, Avempace, and European scholastics like Buridan and Oresme. The latter two developed a somewhat mathematical kinematics that eventually influenced Galileo (how much is controversial), and his idea of inertia, see section V in &lt;a href=&quot;http://sites.uci.edu/trabing/files/2013/11/Buridan-copy.pdf&quot; rel=&quot;nofollow&quot;&gt;Skrenes's dissertation Buridan’s Impetus Hypothesis&lt;/a&gt;:&quot;&lt;em&gt;Buridan’s impetus is permanent, and in this respect a forerunner of Newtonian inertia... I discuss whether Buridan anticipated Galileo’s law of acceleration.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But even if we restrict to modern scientists, Galileo and Descartes certainly recognized inertial motion before Newton.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-03T20:35:19.210" LastActivityDate="2016-09-03T20:35:19.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5154" PostTypeId="2" ParentId="5148" CreationDate="2016-09-01T02:58:02.193" Score="8" Body="&lt;p&gt;&quot;&lt;em&gt;In 1694, a famous discussion between two of the leading scientists of the day - Isaac Newton and David Gregory - took place on the campus of Cambridge University. Their dispute concerned the &quot;kissing problem.&quot;&lt;/em&gt;&quot; So starts &lt;a href=&quot;https://plus.maths.org/content/newton-and-kissing-problem&quot; rel=&quot;nofollow&quot;&gt;Newton and the Kissing Problem&lt;/a&gt; chapter from Szpiro's book Kepler's Conjecture. Pretty much so start (and quickly end) most descriptions of the incident in modern papers on the kissing problem. Not one of them cites a contemporary source, many cite... Szpiro. One exception, to its credit, is &lt;a href=&quot;https://books.google.com/books?id=Oo7lZl3Ru6AC&amp;amp;pg=PP77&amp;amp;lpg=PP77&amp;amp;dq=there+is+no+record+of+so+much+as+a+gentleman&amp;#39;s+wager+on+the+subject+and+certainly+no+possibility+of+anyone+collecting,+as+the+...&amp;amp;source=bl&amp;amp;ots=v2ecUw3O4p&amp;amp;sig=5Uvp7V66X68_6tMrUxKb6rvG2OY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiZrZ28j-3OAhVMRyYKHVt8DcwQ6AEIHDAA#v=onepage&amp;amp;q=there%20is%20no%20record%20of%20so%20much%20as%20a%20gentleman&amp;#39;s%20wager%20on%20the%20subject%20and%20certainly%20no%20possibility%20of%20anyone%20collecting%2C%20as%20the%20...&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Niederman's Number Freak&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story is an embellished tell-tale. The &quot;famous discussion&quot; was a passing mention in a conversation about astronomy, there was no &quot;dispute&quot;, and we are not even sure if Newton guessed right (but it is curious how everyone who retells the story assumes that he was). I suppose inserting the names of Newton and Gregory adds to the &quot;historical respectability&quot; of the kissing problem. Here is from &lt;a href=&quot;http://www.ams.org/notices/200408/comm-cass.pdf&quot; rel=&quot;nofollow&quot;&gt;Casselman's Difficulties of Kissing in Three Dimensions&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The history of the problem is obscure. It is commonly said that in a discussion that took place in Cambridge the Scottish astronomer and mathe-&#xA;  matician David Gregory asserted that 13 spheres could be placed in contact with a central sphere, while Isaac Newton claimed that only 12 were possible. evidence for exactly what was said in this discussion is murky. The first published reference to it that I know of is in the third volume of Newton’s&#xA;  correspondence, edited by H. W. Turnbull, which came out in 1961. There is an entry for May 4, 1694, one of several Latin memoranda written about that day by Gregory, summarizing a conversation with Newton on the distribution of stars&#xA;  of various magnitudes. On the question of 12 versus 13, the entry does not support what is commonly said. Two distinct possibilities are not mentioned, and the most plausible reading is that Newton himself thought that &#xA;  13 spheres surrounding a fourteenth was a possibility! More likely, some would think, is that Gregory didn’t understand what Newton had said in an apparently rather rapid discourse.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Turnbull refers to a more elaborate entry in a notebook of Gregory kept at Christ Church, Oxford, but at least one person’s attempt to locate that entry where Turnbull said it should be was unsuccessful. In any event, Turnbull’s paraphrase suggests that there is nothing important there not already mentioned in the published memorandum. Other puzzling features of this story are that the 1953 paper by Schütte and van der Waerden refers to a Newton-Gregory discussion,&#xA;  and in 1956 John Leech referred in more detail to the Christ Church notebook. These both appeared several years before the correspondence of Newton appeared in print. What was the source of their information? To paraphrase a familiar dictum of the Mattel Toy Company, history is hard.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-04T03:31:56.283" LastActivityDate="2016-09-04T03:31:56.283" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5157" PostTypeId="2" ParentId="5151" CreationDate="2016-09-02T11:24:41.967" Score="1" Body="&lt;p&gt;Actually, Aristotle argued that all bodies have a natural movement and do not need to be pushed to keep moving. The elements earth and water have a natural movement towards the centre of the cosmos, air and fire have a natural upward movement, and aether (the substance of the heavenly bodies) has a natural circular movement. This is all explained in his treatises “On generation and corruption” and “On the heavens”.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2016-09-02T11:24:41.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5158" PostTypeId="2" ParentId="5147" CreationDate="2016-09-02T21:39:11.740" Score="1" Body="&lt;p&gt;Form a series of conferences I heard about &quot;arabic/islamic&quot; mathematics (especially &lt;a href=&quot;http://www.storyofmathematics.com/islamic_alkhwarizmi.html&quot; rel=&quot;nofollow&quot;&gt;Al Khawarizmi&lt;/a&gt;), a practical motivation for completing the square (that yielded the algorithm to solve 2nd degree equation) could have been (very putative) the estimation of the quantity of tiles needed to extend a palace with a certain length of walls. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least for integers, placing square tiles on a rectangular floor provide a natural intuition than the product commutes. You can place tiles from East to West, or South to North, and get the same floor.&lt;/p&gt;&#xA;" OwnerUserId="4130" LastActivityDate="2016-09-02T21:39:11.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5159" PostTypeId="1" CreationDate="2016-09-03T04:57:56.207" Score="11" ViewCount="1245" Body="&lt;p&gt;A lot has been said about Grothendieck's capacity of work, but what we know?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(with references if possible) how was the work-day of Grothendieck, how many hours to day, what about his sleep?&lt;/p&gt;&#xA;&lt;p&gt;What is know about Grothendieck's habits of work (in manage routines, on time management)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm specially interested in the &amp;quot;active&amp;quot; years (50's, 60's)&lt;/p&gt;&#xA;" OwnerUserId="3857" LastEditorUserId="4251" LastEditDate="2021-09-22T05:45:08.560" LastActivityDate="2021-09-22T05:45:08.560" Title="What is known about Grothendieck's capacity of work?" Tags="|mathematicians|biographical-details|grothendieck|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5161" PostTypeId="1" AcceptedAnswerId="5163" CreationDate="2016-09-03T17:04:59.440" Score="6" ViewCount="424" Body="&lt;p&gt;I'm very confused by contradicting accounts of a supposed solution by Newton to the problem of finding the trajectory of a projectile moving under uniform gravity against resistance that is proportional to the square of the projectile's speed. Some sources state that he did solve it in the second edition of the Principia (not the first!), but did not solve the more general problem, when an arbitary power law is assumed for the resistance. According to them, this remarkably general problem was solved by John Bernoulli. Other sources claim that Newton did not even solve the quadratic case. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So did Newton find the quadratic law trajectory or not? And if he didn't, what kind of propositions are there in Section 2 of Book 2 of the Principia? I assume that Book 2 of the Principia marks the beginning of modern ballistics; that's why I consider this question important.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, i have another question, which is too concerned with the later part of Newton's life. The question is about the problem of orthogonal trajectories which newton solved in 1716 by deriving the general differential equation which the orthogonal family satisfies. I want to ask what is so important about this question ; it doesn't look hard - seriously, even i can solve it. So why so many books attach importance to this solution?, i understand the important connection with potential theory (field lines and equipotenial lines), but why several books declare that Newton's solution to this problem was one of the ingenius achievements of his later life?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2016-09-05T10:40:23.473" LastActivityDate="2016-09-05T10:40:23.473" Title="Did Newton find the trajectory of a body moving in uniform gravity under the quadratic resistance law (the ballistic problem)?" Tags="|biographical-details|classical-mechanics|debunking|differential-equations|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5162" PostTypeId="2" ParentId="5159" CreationDate="2016-09-03T17:28:42.570" Score="6" Body="&lt;p&gt;&lt;strong&gt;&quot;At the time, he had the capacity to be able to sleep when he wanted to, and for the number of hours he wanted to, in order to take up his work all the better afterward. In fact this capacity for work was to me something miraculous.&quot;&lt;/strong&gt; Memories of Schourik, Poénaru.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&quot;Schwartz came to realize that Grothendieck seemed to function on a biological schedule of 26 or 27 hours, which led him periodically to live on the opposite schedule from everyone else.&quot;&lt;/strong&gt; Mathematics III, Schneps and Scharlau&lt;/p&gt;&#xA;" OwnerUserId="3857" LastEditorUserId="3857" LastEditDate="2016-09-03T20:06:36.360" LastActivityDate="2016-09-03T20:06:36.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5163" PostTypeId="2" ParentId="5161" CreationDate="2016-09-03T22:42:00.147" Score="5" Body="&lt;p&gt;No, he did not. Not in the first (1687), not in the second (1713), and not in the third (1726) edition, below I quote from &lt;a href=&quot;https://books.google.com/books/about/Sir_Isaac_Newton_s_Mathematical_Principl.html?id=ySYULc7VEwsC&quot; rel=&quot;nofollow&quot;&gt;Cajori's edition of Motte's translation&lt;/a&gt; of the latter. Since he was unable to find the trajectory  analytically he tried to approximate it by pre-chosen curves. In the process he &#xA;apparently messed up with the semi-circles in the first edition of Principia, which Johann (=John) Bernoulli pointed out in 1712. Newton fixed the error in the next year's second edition, without mentioning Bernoulli. The problem was offered as a challenge in 1717, when Bernoulli solved it (in a parametric form) as a particular case of the problem with resistance proportional to an arbitrary power of speed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton did find the trajectory for the linear resistance law, suggested by Galileo, in Section 1 of Book II. But he argued that the square law is correct for rare media like the air, and even that it can be derived from his laws of motion! Section 2 examines this case, first without gravity, and then for the fall in a straight line. But the closest Newton comes to the general trajectory is when he studies the inverse problem under Proposition X of Book II:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Suppose the uniform force of gravity to tend directly to the plane of the&#xA;  horizon, and the resistance to be as the product of the density of the medium&#xA;  and the square of the velocity: it is proposed to find the density of&#xA;  the medium in each place, which shall make the body move in any given&#xA;  curved line, the velocity of the body, and the resistance of the medium in&#xA;  each place.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He considers four types of trajectories - semi-circle, parabola, hyperbola with oblique and vertical asymptote, and generalized hyperbola with similar asymptotes - and reasons what kind of medium distribution would produce them. His heuristic conclusion for the uniformly dense medium is this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;...the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant&#xA;  from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described.&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Newton's consideration of the semi-circle and hyperbolic cases appears to be influenced by &lt;a href=&quot;http://scindeks-clanci.ceon.rs/data/pdf/0352-2334/2014/0352-23341403114E.pdf&quot; rel=&quot;nofollow&quot;&gt;Tartaglia's Nova Scientia (1537)&lt;/a&gt;, one of the first books applying mathematics to ballistics. &lt;a href=&quot;http://scitation.aip.org/content/aapt/journal/ajp/84/2/10.1119/1.4935896&quot; rel=&quot;nofollow&quot;&gt;Hackborn's On Motion in a Resisting Medium: a Historical Perspective&lt;/a&gt; details the Bernoulli affair:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A “serious error” in Newton's analysis for the semi-circular trajectory in Proposition 10 of the first edition of Principia had unexpected consequences. Johann Bernoulli discovered the error and communicated it to Newton through his nephew, Nikolaus Bernoulli, who was visiting London in September 1712. Newton then tailored a long paste-up correction for the second (1713) edition, already printed, without citing Bernoulli's help... Eventually, responding to a 1717 challenge from Oxford professor John Keill to “[f]ind the curve which a projectile describes” subject to gravity and fluid drag varying as the square of its speed, Johann Bernoulli found an expression for this curve in the case of drag varying as an arbitrary power of speed&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-04T03:36:11.567" LastActivityDate="2016-09-04T03:36:11.567" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5164" PostTypeId="1" AcceptedAnswerId="5169" CreationDate="2016-09-04T14:18:12.893" Score="6" ViewCount="489" Body="&lt;p&gt;Two patterns in the structure of the Ptolemaic model make the transformation of coordinates to the Copernican model seem &quot;natural&quot; to modern eyes: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the alignment of the radii of the (second) epicycles for outer parents and the deferents of inner planets with the Earth-Sun line, and &lt;/li&gt;&#xA;&lt;li&gt;the equivalence of the outer epicycle and inner different radii with the Earth-Sun distance.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The former pattern was established since ancient times, and the latter was clearly established by Copernicus's time (something like $0.951R_\mathrm{\tiny{Sun}}$,  $0.997R_\mathrm{\tiny{Sun}}$, $0.989R_\mathrm{\tiny{Sun}}$, $0.998R_\mathrm{\tiny{Sun}}$, and $0.962R_\mathrm{\tiny{Sun}}$, for the known planets in order of modern distance). Earlier values for the latter were however far off (Ptolemy's values for the outer planets were something like $2.75R_\mathrm{\tiny{Sun}}$, $1.83R_\mathrm{\tiny{Sun}}$, and $1.48R_\mathrm{\tiny{Sun}}$; while his value for inner planets were off by a factor of ~10). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These values follow from the geometric structure of the model (which fixes the ratio of deferent and epicycle radii for each planet) and the estimated values of the deferent radii (or &quot;orbit sizes&quot;), expressed relative to Earth-Sun distance (which need not be — and was not — accurately known).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So while estimates relative &quot;orbit sizes&quot; were far off in ancient times, they had become sufficiently accurate by Copernican times to make his transformation of coordinates possible. But what happened in the intervening centuries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What were the estimates of the deferent radii (relative to the Eath-Sun distance) for the planets in the years between Ptolemy and Copernicus? At what point did they come close to producing the pattern necessary for Copernicus's coordinate transformation?&lt;/p&gt;&#xA;" OwnerUserId="2027" LastEditorUserId="55" LastEditDate="2016-09-06T08:24:12.137" LastActivityDate="2016-09-06T08:24:12.137" Title="When did the estimates of planetary distances made between Ptolemy and Copernicus produce the pattern suggesting heliocentrism?" Tags="|astronomy|cosmology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5166" PostTypeId="1" AcceptedAnswerId="13458" CreationDate="2016-09-04T16:32:54.740" Score="31" ViewCount="50816" Body="&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$\triangleq$&lt;/span&gt; is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the &lt;a href=&quot;https://www.fileformat.info/info/unicode/char/225c/index.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Unicode Character 'DELTA EQUAL TO' (U+225C)&lt;/a&gt;.&#xA;The notation  &lt;span class=&quot;math-container&quot;&gt;$t \triangleq m$&lt;/span&gt; (generally) means: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is defined to be &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;&amp;quot; or &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is equal by definition to &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;&amp;quot; (often under certain conditions).&lt;/p&gt;&#xA;&lt;p&gt;In a similar sense, some use &lt;span class=&quot;math-container&quot;&gt;$:=$&lt;/span&gt; or  &lt;span class=&quot;math-container&quot;&gt;$=:$&lt;/span&gt; (see for instance &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_mathematical_symbols#Symbols_based_on_equality&quot; rel=&quot;nofollow noreferrer&quot;&gt;Symbols based on equality&lt;/a&gt;). Yet, this &lt;em&gt;Delta variant&lt;/em&gt; is more important to me.&lt;/p&gt;&#xA;&lt;p&gt;The SE. Maths post &lt;a href=&quot;https://math.stackexchange.com/questions/1289339/what-is-meant-by-the-delta-equivalent-sign&quot;&gt;What is meant by the delta equivalent sign?&lt;/a&gt; proposes a slight distinction (not crystal-clear to me) between the above similar senses:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sometimes it is used with the slightly different meaning of &amp;quot;equal by definition&amp;quot;, to underline the difference w.r.t. &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$:=$&lt;/span&gt; &amp;quot; which is the definition itself.&#xA;i.e.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;a:=3;\\&#xA;5+a \triangleq 5 + 3 = 8 &#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I always took for granted that the &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; stood for letter &amp;quot;D&amp;quot;, i.e. for the  initial of &amp;quot;&lt;strong&gt;definition&lt;/strong&gt;&amp;quot;. Indeed, one sometimes finds &lt;span class=&quot;math-container&quot;&gt;$\overset{\mathrm{def}}{=}$&lt;/span&gt; too. In German apparently, one also uses &lt;span class=&quot;math-container&quot;&gt;$≙$&lt;/span&gt; (&lt;em&gt;Entspricht-Zeichen&lt;/em&gt;, with Unicode U+2259).&lt;/p&gt;&#xA;&lt;p&gt;Based on these prior hints, my questions are as follows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Who introduced this dual symbol first in science, and where (which source)?&lt;/li&gt;&#xA;&lt;li&gt;What motivated the Greek &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; notation? The abbreviation of some word, a symbol (why not a latin notation)?&lt;/li&gt;&#xA;&lt;li&gt;Why not merge the lower bar of the Delta with the upper bar of the equal sign, to save some ink, and create a lighter symbol?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;References: the symbol itself was already discussed in StackExchange:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;SE.math: &lt;a href=&quot;https://math.stackexchange.com/questions/1289339/what-is-meant-by-the-delta-equivalent-sign&quot;&gt;What is meant by the delta equivalent sign?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;SE.tex: &lt;a href=&quot;https://tex.stackexchange.com/questions/163829/delta-equal-to-symbol&quot;&gt;Delta-equal to symbol&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="12330" LastEditDate="2021-11-10T00:00:41.203" LastActivityDate="2021-11-10T00:00:41.203" Title="Who first defined the &quot;equal-delta&quot; or &quot;delta over equal&quot; ($\triangleq$) symbol?" Tags="|mathematics|reference-request|terminology|notation|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="5167" PostTypeId="2" ParentId="5161" CreationDate="2016-09-04T16:38:19.587" Score="2" Body="&lt;p&gt;According to Whittaker this problem was reduced to quadrature by D'Alembert in 1744: Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, p. 229 &lt;a href=&quot;http://archive.org/details/treatisanalytdyn00whitrich&quot; rel=&quot;nofollow&quot;&gt;http://archive.org/details/treatisanalytdyn00whitrich&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conifold's answer says Bernoulli solved it in a more general form in 1717. Presumably there is some difference between the types of solutions achieved by D'Alembert and Bernoulli.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be surprised if anyone could have accomplished anything on this problem as early as Newton's lifetime, given the primitive state of the art in math and physics at the time. Today, we describe the solution in terms of conserved quantities (there is a constant of the motion in terms of $v_x$ and $v_y$) and write out the analytic solution for the one-dimensional case using logarithms. Doing calculus with logarithms was a brand new thing at that time, and  I don't know if the concept of a conserved quantity even existed.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2016-09-04T16:38:19.587" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5168" PostTypeId="1" CreationDate="2016-09-05T03:38:44.480" Score="5" ViewCount="315" Body="&lt;p&gt;Was Hensel's lemma originally used for proving some other theorem? Or is it meant to be a standalone result? Why is it a &quot;lemma&quot; and not a theorem?&lt;/p&gt;&#xA;" OwnerDisplayName="user2084" LastActivityDate="2016-09-05T03:38:44.480" Title="What is Hensel's lemma a lemma for?" Tags="|mathematics|number-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5169" PostTypeId="2" ParentId="5164" CreationDate="2016-09-06T04:46:52.603" Score="3" Body="&lt;p&gt;The process was not gradual. There were only two astronomers between Ptolemy and Copernicus who independently estimated planetary distances, and it is uncertain if Copernicus was aware of their results, although there are indications that he was of al-Shāṭir's. Some of al-Shāṭir's estimates &#xA;can be found in &lt;a href=&quot;https://www.jstor.org/stable/227308&quot; rel=&quot;nofollow&quot;&gt;Planetary Theory of Ibn al-Shāṭir by Kennedy and Roberts&lt;/a&gt;, and there is a consensus that some of his planetary models are largely identical to Copernicus's (except for minor variations in parameters, and heliocentric reversal of the last vector in epicyclic linkages), see e.g. &lt;a href=&quot;http://www.americanscientist.org/issues/feature/greek-astronomy-and-the-medieval-arabic-tradition/1&quot; rel=&quot;nofollow&quot;&gt;Saliba's Greek Astronomy and the Medieval Arabic Tradition&lt;/a&gt;. So &lt;a href=&quot;http://www.jstor.org/stable/986463&quot; rel=&quot;nofollow&quot;&gt;Ibn al-Shāṭir (1304-1375)&lt;/a&gt; may be the unsung hero of the Copernican revolution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, some historians argue that Copernicus might have come to his estimates and models independently by working from the same Ptolemaic base, with its observartional problems, and sharing the methodological aspirations of Islamic astronomers to eliminate eccentrics and equants, see &lt;a href=&quot;http://adsabs.harvard.edu/full/1995JHA....26..133D&quot; rel=&quot;nofollow&quot;&gt;di Bono's Copernicus, Amico, Fracastro and Tusi's Device&lt;/a&gt;. Copernicus's method of estimating mean distances based on planetary parallaxes was explained to Kepler by his teacher Maestlin. It is nicely presented in the appendix he wrote, reproduced in &lt;a href=&quot;http://www.jstor.org/stable/986463&quot; rel=&quot;nofollow&quot;&gt;Grafton's paper&lt;/a&gt;, to his pupil's Mysterium Cosmographicum, where Kepler tried to derive said distances from dimensions of nested Platonic solids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Ptolemaic models only the epicycle/deferent ratio matters for planetary positions, if both sizes are say simultaneously doubled the observed trajectory on the celestial sphere remains exactly the same. Ptolemy deliberately avoids the discussion of mean distance ratios for different planets in the Almagest, but in Planetary Hypotheses he notes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The distances of the five planets are not as easy to determine as those of the two luminaries, for the distances of the two luminaries were determined, mostly, on the basis of combinations of eclipses. A similar proof cannot be invoked for the five planets, because no phenomenon allows us to fix their parallax with certainty.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://www.jstor.org/stable/229192&quot; rel=&quot;nofollow&quot;&gt;Goldstein's Theory and Observation in Medieval Astronomy&lt;/a&gt;, Ptolemy turns to the broader theory to do the task, the &quot;nesting hypothesis&quot; of Aristotelian cosmology: the maximum distance to each planet (mean plus epicycle radius) is set equal to the minimum distance (mean minus epicycle radius) to the next one. Even Ptolemy's solar and lunar distances (estimated according to &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Sizes_and_Distances_(Aristarchus)&quot; rel=&quot;nofollow&quot;&gt;Aristarchus's methodology&lt;/a&gt;) were far off, e.g. the Moon should have appeared twice as large at quadrature than at opposition, and for Venus say the size variation would have been nearly sevenfold, making it look two fifths the size of the Sun at its largest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy says nothing about these discrepancies. Silence is golden, and Ptolemy's silence doubly so. Most medieval astronomers followed his lead on it, with two exceptions. One was Jewish Talmudist and polymath &lt;a href=&quot;https://en.wikipedia.org/wiki/Gersonides#Estimation_of_stellar_distances_and_refutation_of_Ptolemy.27s_model&quot; rel=&quot;nofollow&quot;&gt;Gersonides (Levi ben Gerson, 1288–1344)&lt;/a&gt;, who pointed out Ptolemy's visible size absurdities, and used a pinhole camera to measure variations in sizes of the Moon and the planets. He discovered no correlations between them and epicyclic periods, and introduced a fluid layer between the neighboring spheres to account for his own estimates of the distances that violated the nesting hypothesis, see &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0498.1986.tb00860.x/full&quot; rel=&quot;nofollow&quot;&gt;Gerson’s Theory of Planetary Distances by Goldstein&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other was the aforementioned late representative of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gersonides#Estimation_of_stellar_distances_and_refutation_of_Ptolemy.27s_model&quot; rel=&quot;nofollow&quot;&gt;Maragha school of Islamic astronomers&lt;/a&gt; Ibn al-Shāṭir, who, according to Goldstein, &quot;&lt;em&gt;wished to find a new model that agreed with them as closely as possible - that is, he accepted Ptolemy's lunar positions in longitude and latitude and merely wished to repair the distance&lt;/em&gt;&quot;. Unlike Gersonides, al-Shāṭir's models were better than Ptolemy's even for longitudes, they also eliminated Ptolemy's Aristotelian transgressions of eccentrics and equants by replacing them with epicyclic linkages, a culmination of Maragha school's efforts. Neugebauer and Swerdlow argued that Maraghian astronomy was known in Italy in early 1500s, when Copernicus studied there. Neugebauer even found a Greek translation of an Arabic manuscript describing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tusi_couple&quot; rel=&quot;nofollow&quot;&gt;Tusi couple&lt;/a&gt;, a Maraghian device Copernicus employs in both Commentariolus and De Revolutionibus. But the manuscript does not mention al-Shāṭir, and so far as we know Copernicus did not read Arabic (although some of his Italian contacts might have).&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-06T07:37:27.633" LastActivityDate="2016-09-06T07:37:27.633" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5171" PostTypeId="1" AcceptedAnswerId="5173" CreationDate="2016-09-06T18:09:47.087" Score="10" ViewCount="927" Body="&lt;p&gt;I find it amazing that the geometric construction that underlies &lt;a href=&quot;https://en.wikipedia.org/wiki/Elliptic_curve#The_group_law&quot;&gt;the group law for elliptic curves&lt;/a&gt; gives rise to a group law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt;  Who was the first to identify the group law for elliptic curves and, if different, who was first to prove that it is associative? &lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-09-06T20:31:47.820" Title="Who first identified the group structure of an elliptic curve?" Tags="|abstract-algebra|group-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5172" PostTypeId="1" AcceptedAnswerId="5177" CreationDate="2016-09-06T20:18:54.307" Score="6" ViewCount="2544" Body="&lt;p&gt;What is the first or most ancient civilization to use a base-16, hexadecimal number system?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="55" LastEditDate="2016-10-15T03:01:25.507" LastActivityDate="2016-10-15T03:01:25.507" Title="What is the most ancient civilization that used base-16 (hexadecimal) number system?" Tags="|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5173" PostTypeId="2" ParentId="5171" CreationDate="2016-09-06T20:31:47.820" Score="5" Body="&lt;p&gt;In the 17th century Bachet and Fermat gave algebraic formulas for doubling a point on a cubic, and Newton showed how to do it in terms of chords and tangents. But that is as far as geometry progressed on its own, from there the path did not go from geometry to the group law, but the other way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1834 Jacobi pointed out a possible connection between cubic curves and elliptic functions, which Eisenstein proved in 1847. Clebsch made it more concrete in 1864 by suggesting parametrizations of cubic curves using elliptic functions (this where the term &quot;elliptic curve&quot; comes from). Weierstrass explicitly linked his addition formula for elliptic functions to the addition of points on cubic curves, and the geometry was studied by Juel in 1890s. Poincare's 1901 paper Sur les Proprietes Arithmetiques des Courbes Algebriques is considered to be the first systematization of the subject, where the chord-and-tangent addition (including the point at infinity as identity) is first shown to be a group law rigorously by modern standards, including associativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.jstor.org/stable/10.4169/math.mag.85.3.163&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why Ellipses Are Not Elliptic Curves by Rice and Brown&lt;/a&gt;, &lt;a href=&quot;http://search.proquest.com/docview/203726122&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elliptic curves from Mordell to Diophantus and back by Brown&lt;/a&gt;, and &lt;a href=&quot;https://math.stackexchange.com/questions/156650/history-of-elliptic-curves&quot;&gt;History of elliptic curves thread on Math SE&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-09-06T20:31:47.820" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5175" PostTypeId="1" CreationDate="2016-09-06T21:18:39.453" Score="2" ViewCount="142" Body="&lt;p&gt;To find - How did Newton derive the general binomial theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I know he approximated the area under functions.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;em&gt;But how did he approximate the area under CURVES like ( 1/((1-x^2)^(1/2))) or ( 1/((1-x^2)^(n/2))) , where n is odd.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I know he moved forward from here deriving sine inverse series and then sine series. But how did he approximate the area under the above function.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Kindly look to it. I have changed the title and the question a bit and now I don't think it's duplicate&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="4666" LastEditorUserId="4666" LastEditDate="2017-04-23T17:47:23.170" LastActivityDate="2017-04-23T17:47:23.170" Title="How did newton APPROXIMATE THE AREA UNDER THESE PARTICULAR CURVES" Tags="|calculus|" AnswerCount="0" CommentCount="4" ClosedDate="2016-09-15T19:01:54.980" ContentLicense="CC BY-SA 3.0" />
  <row Id="5177" PostTypeId="2" ParentId="5172" CreationDate="2016-09-07T01:00:06.723" Score="7" Body="&lt;p&gt;The only traditional use of hexadecimals that I know of, and that one is a bit of a stretch, is in Chinese weight units, e.g. one jīn (斤) was equal sixteen liǎng (兩), etc., see &lt;a href=&quot;http://forums.xkcd.com/viewtopic.php?t=68091&quot; rel=&quot;nofollow noreferrer&quot;&gt;Non Base-10 Number Systems in Languages&lt;/a&gt; forum discussion. When &lt;a href=&quot;https://en.wikipedia.org/wiki/Suanpan&quot; rel=&quot;nofollow noreferrer&quot;&gt;suan pan (abacus)&lt;/a&gt; was introduced in China c. 200 AD it accomodated both decimals and hexadecimals, but presumably the use of weight units predates that by a lot. Apparently, &lt;a href=&quot;https://en.wikipedia.org/wiki/Abacus#Japanese&quot; rel=&quot;nofollow noreferrer&quot;&gt;Japanese adopted the abacus&lt;/a&gt; in 14th century, and the weight hexadecimals were still in use there and in China as late as in 1930s. Other than that, hexadecimals owe their relevance to the rise of the computers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And yes, you read right about decimals on suan pan long before Indians invented the decimal notation. Chinese proto-decimal system was not fully positional, but that makes no difference for abacus calculations, see  &lt;a href=&quot;https://hsm.stackexchange.com/questions/2181/has-a-digit-ever-been-used-to-represent-the-number-10/2186#2186&quot;&gt;Has a digit ever been used to represent the number &amp;quot;10&amp;quot;?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-09-08T21:19:30.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5178" PostTypeId="1" CreationDate="2016-09-07T11:59:41.007" Score="3" ViewCount="133" Body="&lt;p&gt;The question is about the problem of orthogonal trajectories which newton solved in 1716 (in response to Leibniz's challenge) by deriving the general differential equation which the orthogonal family satisfies. I want to ask what is so important about this question ; it doesn't look hard - seriously, even i can solve it. The differential equation follows imediately from the condition for perpendicularity of two straight lines - that the products of their slopes is -1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why so many books attach importance to this solution? i.e i understand the connection with potential theory (field lines and equipotenial lines). Several sources describe Newton's solution to this problem as one of his most ingenius achievements in his later life - a par with his solution to the brachistochrone problem. So what do i miss?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2016-09-07T11:59:41.007" Title="Why do so many books attach such importance to Newton's solution to the problem of orthogonal trajectories?" Tags="|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5182" PostTypeId="1" AcceptedAnswerId="5185" CreationDate="2016-09-09T13:14:34.337" Score="11" ViewCount="305" Body="&lt;p&gt;Saunders Mac Lane says he attended Hilbert's weekly lectures on &quot;matters of general intellectual interest&quot; around 1931 or 1933 (&lt;em&gt;Saunders Mac Lane: A Mathematical Autobiography&lt;/em&gt; p. 44).  I believe these are the same as Mac Lane elsewhere called Hilbert's &quot;general lecture course on the philosophy of science which I heard him give in late 1931'' (1995 &quot;A matter of temperament&quot;, letter, with reply by Freeman J. Dyson. &lt;em&gt;New York Review of Books&lt;/em&gt;, 42(15):56.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any published descriptions of these lectures, beyond Mac Lane's remark that the topics included &quot;the overall importance of the discovery of the Americas&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hel Braun probably means these same lectures in her autobiography when she refers to Hilbert's legendary crepe de chine lectures (sagenhaften Crepe de Chine-Vorlesungen) (&lt;em&gt;Eine Frau und die Mathematik 1933–1940&lt;/em&gt;).  That is lectures given to a rather posh audience from beyond the university.  But she says nothing about content.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="4251" LastEditDate="2021-09-14T10:06:10.633" LastActivityDate="2021-09-14T10:06:10.633" Title="What were Hilbert's weekly 1933 lectures on &quot;matters of general intellectual interest&quot; about?" Tags="|mathematicians|biographical-details|social-context|hilbert|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5183" PostTypeId="1" CreationDate="2016-09-10T02:37:33.400" Score="1" ViewCount="64" Body="&lt;p&gt;So, for a couple years now I've been quoting someone that said that it would take about 150 years of education to learn everything it'd take to build an integrated circuit, start to finish.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thing is, now I can't find my original source. Does this figure sound familiar to anyone?&lt;/p&gt;&#xA;" OwnerUserId="4685" LastActivityDate="2016-09-10T02:37:33.400" Title="Years to Learn to Build an IC" Tags="|reference-request|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5184" PostTypeId="1" CreationDate="2016-09-10T06:20:20.507" Score="8" ViewCount="576" Body="&lt;p&gt;Does anyone know of a good book on the history of imaginary numbers and complex analysis and its role in physics?&lt;/p&gt;&#xA;" OwnerUserId="4686" LastEditorUserId="10106" LastEditDate="2020-06-22T17:44:17.110" LastActivityDate="2020-06-22T17:44:17.110" Title="History of complex analysis" Tags="|mathematical-physics|complex-analysis|book-recommendation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5185" PostTypeId="2" ParentId="5182" CreationDate="2016-09-11T01:15:57.773" Score="11" Body="&lt;p&gt;Here is MacLane's sentence about the lectures in its entirety:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Although the leading mathematician, David Hilbert, had retired and came only once a week to lecture on matters of general intellectual interest - for instance on the overall importance of the discovery of the Americas - many young stars had appeared&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is unclear if he actually attended a lecture or is referring a second hand account, or, if he did attend, whether he attended more than once. His shift of focus at the end of the sentence, and switching to Weyl, &quot;Hilbert's successor&quot;, in the next one suggest only passing familiarity with them. MacLane also mentions winter of 1933 on the same page as the sentence, it seems likely but not entirely certain that the lectures referred to are contemporaneous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the &lt;a href=&quot;http://www-gap.dcs.st-and.ac.uk/history/Biographies/Hilbert.html&quot; rel=&quot;nofollow&quot;&gt;McTutor biography&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1930 Hilbert retired but only a few years later, in 1933, life in Göttingen changed completely when the Nazis came to power and Jewish lecturers were dismissed. By the autumn of 1933 most had left or were dismissed. Hilbert, although retired, had still been giving a few lectures. In the winter semester of 1933-34 he gave one lecture a week on the foundations of geometry&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, McTutor's O'Connor and Robertson do not disclose their source of information on the content of the lectures either. But in 1932 Springer published Hilbert's and Cohn-Vossen's Intuitive Geometry (Geometry and the Imagination in English translation), which Nuno and Romao in &lt;a href=&quot;https://www.uni-goettingen.de/de/document/download/2a4f28e006bb2e7afcbf0fdbda8ce9b4.pdf/Abstract_Romao.pdf.&quot; rel=&quot;nofollow&quot;&gt;Hilbert as Outreacher&lt;/a&gt; describe as &quot;&lt;em&gt;widely acknowledged as one of the highest achievements in mathematical exposition to the general public, and it provides the layperson with an accurate (and timeless) impression of what the spirit of mathematics is about&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is confirmed by the list of Hilbert's lecture courses printed in &lt;a href=&quot;http://download.springer.com/static/pdf/966/bfm%253A978-3-540-69444-1%252F1.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fbook%2Fbfm%3A978-3-540-69444-1%2F1&amp;amp;token2=exp=1473634286~acl=%2Fstatic%2Fpdf%2F966%2Fbfm%25253A978-3-540-69444-1%25252F1.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fbook%252Fbfm%253A978-3-540-69444-1%252F1*~hmac=3860a56464016ee01e217b2cf5f009cbde910f3a8a229fdf17a9e358fa1e19fb&quot; rel=&quot;nofollow&quot;&gt;David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917-1933&lt;/a&gt;, see below. However, the only lecture in the volume from 1933 is called On the Infinite, which I suppose also could be of &quot;general intellectual interest&quot;. The editors' comments in the introduction on the general state of Hilbert's Nachlaß suggest that they prioritized mathematics and science related lecture notes for publication. So if there was something on the discovery of the Americas it would still sit handwritten in the archives:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The present Volume is the third in a series of six presenting a selection from Hilbert’s previously unpublished lecture notes on the foundations of mathematics and physics during the period 1890 to 1933. The Hilbert Nachlaß contains approximately eighty lecture notebooks, covering all aspects of his mathematical activity, and spanning almost the entirety of his teaching career; some are in his own hand, others were carefully worked out by his assistants as official protocols of his lectures... The structure of this Edition, the nature, location, and condition of the Hilbert lecture notes, their provenance, and what we have been able to reconstruct of their history, are all described in the general ‘Introduction to the Edition’, which is to be found at the beginning of Volume 1.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Googlebooks displays the relevant part of the &lt;a href=&quot;https://books.google.com/books?id=NxENTTHJCz4C&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;volume 1 introduction&lt;/a&gt;, with few exceptions the manuscripts are kept in the library of the Mathematics Institute, and the manuscript division of the university, both in Göttingen.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;List of Hilbert's lecture courses in 1931-34&lt;/strong&gt; (from David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917-1933, p.991)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SS 1931 Grundlagen der Mathematik, 1st.67 [Foundations of Mathematics]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;WS 1931/32 Einleitung in die Philosophie auf Grund moderner Naturwissenschaft (für Hörer aller Fakultäten), 1st. [Introduction to Philosophy of Modern Natural Science (for students of all departments)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SS 1932 Grundlagenfragen der Geometrie, 1st. [Foundational Questions of Geometry]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;WS 1932/33 Grundlagen der Logik, allgemeinverständlich (für Hörer aller Fakultäten), 1st. [Foundations of Logic, intelligibly (for students of all departments)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SS 1933 Wissen und Denken (für Hörer aller Fakultäten), 1st. [Knowledge and Thinking (for students of all departments)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;WS 1933/34 Grundlagen der Geometrie, mit Schmidt 2st. [Foundations of Geometry]&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-13T17:58:36.443" LastActivityDate="2016-09-13T17:58:36.443" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5186" PostTypeId="2" ParentId="5184" CreationDate="2016-09-12T18:20:24.867" Score="7" Body="&lt;p&gt;I would say that there is no good book which satisfies your description.&#xA;The Book of Bottazzini and Grey mentioned in the comments is OK, but it certainly does not cover the role of complex analysis in physics.&#xA;A better book is J. Dieudonne, Abrégé d'histoire des mathématiques 1700–1900,&#xA;Hermann, Paris, 1978, in 2 volumes, both volumes have chapters on complex analysis. Unlike Grey and Bottazzini,&#xA;Dieudonne was an outstanding mathematician himself, so he better understands the subject and its history. But it also does not cover interaction with physics. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-09-12T18:20:24.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5187" PostTypeId="1" CreationDate="2016-09-13T08:24:54.157" Score="5" ViewCount="96" Body="&lt;p&gt;The Vigenère cipher was broken by Kasiski in 1863 but I read novels and romans from older time, for example during the French Revolution where a student broke Vigenère cipher. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it likely that somebody have broken the code before Kasiski or is it not likely?&lt;/p&gt;&#xA;" OwnerUserId="4698" LastEditorDisplayName="user6215" LastEditDate="2019-02-03T11:48:26.083" LastActivityDate="2019-02-03T11:48:26.083" Title="Was the Vigenère cipher broken many years before Kasiski?" Tags="|computer-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5188" PostTypeId="1" AcceptedAnswerId="5195" CreationDate="2016-09-13T17:55:25.903" Score="24" ViewCount="3782" Body="&lt;p&gt;General Relativity was developed with Differential Geometry as the tool. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity? Who suggested him to use Differential Geometry?&lt;/p&gt;&#xA;" OwnerUserId="2903" LastActivityDate="2019-07-28T13:14:37.540" Title="How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?" Tags="|physics|theoretical-physics|relativity-theory|differential-geometry|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5190" PostTypeId="1" AcceptedAnswerId="5191" CreationDate="2016-09-14T03:42:57.137" Score="13" ViewCount="2732" Body="&lt;p&gt;(This question was first asked &lt;a href=&quot;https://math.stackexchange.com/questions/1926055/how-to-understand-the-word-exponent&quot;&gt;here&lt;/a&gt;. I modified it and moved it here based on some suggestions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have some difficulty in understanding the complex exponential function. So I decide to review the good old exponent function which I learned long ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I look up the word &lt;code&gt;exponent&lt;/code&gt;, the &lt;a href=&quot;http://www.dictionary.com/browse/exponent?s=t&quot; rel=&quot;noreferrer&quot;&gt;online dictionary&lt;/a&gt; says this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;noun&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;a person or thing that expounds, explains, or interprets: an exponent of modern theory in the arts.&lt;/li&gt;&#xA;  &lt;li&gt;a person or thing that is a representative, advocate, type, or symbol of something: Lincoln is an exponent of American democracy.&lt;/li&gt;&#xA;  &lt;li&gt;Mathematics. a symbol or number placed above and after another symbol or number to denote the power to which the latter is to be&#xA;  raised: The exponents of the quantities x&lt;sup&gt;n&lt;/sup&gt;, 2&lt;sup&gt;m&lt;/sup&gt;, y&lt;sup&gt;4&lt;/sup&gt; , and 3&lt;sup&gt;5&lt;/sup&gt; are,&#xA;  respectively, n, m, 4, and 5.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not a native English speaker. To me, these 3 meanings are so distinct. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when and why did powers come to be called 'exponents'? &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;ADD 1&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://www.mathsisfun.com/definitions/power.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, the exponent part is also called index, or power. I think the latter ones &lt;code&gt;index&lt;/code&gt; or &lt;code&gt;power&lt;/code&gt; are more acceptable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/csgD9.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/csgD9.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="592" LastEditorUserId="592" LastEditDate="2018-07-09T00:59:16.607" LastActivityDate="2019-06-14T13:47:21.937" Title="When and why did powers come to be called 'exponents'?" Tags="|mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5191" PostTypeId="2" ParentId="5190" CreationDate="2016-09-14T07:55:48.453" Score="11" Body="&lt;p&gt;The first use of the Latin &lt;em&gt;exponentem&lt;/em&gt; seems to be [thnaks to @fdb's answer]:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Michael_Stifel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Stifel&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=fndPsRv08R0C&amp;amp;pg=PA236&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arithmetica integra (1544)&lt;/a&gt;, page 236,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;with the latin source: &lt;a href=&quot;https://en.wiktionary.org/wiki/expono#Latin&quot; rel=&quot;nofollow noreferrer&quot;&gt;expono&lt;/a&gt; - to display.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt; uses it into his: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.it/books?id=_zpRAAAAYAAJ&amp;amp;pg=PA6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introductio in analysin infinitorum (1748)&lt;/a&gt;, page 6.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For the &quot;definition&quot;, see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Elements_of_Algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements of Algebra&lt;/a&gt;, &lt;a href=&quot;https://www.math.uni-bielefeld.de/~sieben/Euler_Algebra.ocr.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1st German ed.: 1765&lt;/a&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;§ 172 &lt;a href=&quot;https://books.google.it/books?id=X8yv0sj4_1YC&amp;amp;pg=PA50&quot; rel=&quot;nofollow noreferrer&quot;&gt;English transl. by John Hewlett (1822)&lt;/a&gt; : To avoid this inconvenience [that of writing e.g.: &lt;span class=&quot;math-container&quot;&gt;$aaaaa$&lt;/span&gt;], a much more commodious method of expressing such powers has been devised, which, from its extensive use, deserves to be carefully  explained. Thus, for example, to express the hundredth power, we simply write the number &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; above the quantity, whose hundredth power we would express, and a little towards the right-hand; thus &lt;span class=&quot;math-container&quot;&gt;$a^{100}$&lt;/span&gt; represents &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; raised to the &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;th power, or the hundredth power of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. It must be observed, also, that the name &lt;em&gt;exponent&lt;/em&gt; [German: der &lt;em&gt;Exponent&lt;/em&gt;] is given to the number written above that whose power, or degree, it represents, which, in the present instance, is &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A corresponding early English textbook uses instead &quot;index&quot; :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nicholas_Saunderson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicholas Saunderson&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=xq8rAQAAMAAJ&amp;amp;pg=PA60&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Elements of Algebra in Ten Books (1740)&lt;/a&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;instead of &lt;span class=&quot;math-container&quot;&gt;$xx$&lt;/span&gt; we write &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt;. [...] These products are called powers of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;; the figures representing the number of repetitions, are called the &lt;em&gt;indexes&lt;/em&gt; of those powers [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Exponent&lt;/em&gt; and &lt;em&gt;index&lt;/em&gt; are used into:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Colin_Maclaurin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Colin MacLaurin&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=QoAHAAAAcAAJ&amp;amp;pg=PA16&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Treatise of Algebra (1748)&lt;/a&gt;, page 16:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These figures which express the number of factors that produce powers are called their &lt;em&gt;indices&lt;/em&gt; or &lt;em&gt;exponents&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They are used as synonyms also later; see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/George_Peacock&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Peacock&lt;/a&gt;, &lt;a href=&quot;https://books.google.it/books?id=mcVc5v8_kyoC&amp;amp;pg=PA5&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Treatise on Algebra (1830)&lt;/a&gt;, page 5:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;if the symbol &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; be repeated any number of times (&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;) in the expression &lt;span class=&quot;math-container&quot;&gt;$aaa \ldots$&lt;/span&gt; &amp;amp;c it is written &lt;span class=&quot;math-container&quot;&gt;$a^n$&lt;/span&gt;, and is called the &lt;span class=&quot;math-container&quot;&gt;$n^{th}$&lt;/span&gt; power of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and the number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is called the &lt;em&gt;index&lt;/em&gt; or &lt;em&gt;exponent&lt;/em&gt; of the power.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Thus, I'm prone to think that Euler's &quot;revival&quot; of Stifel's term was the source of modern usage.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="9499" LastEditDate="2019-06-14T13:47:21.937" LastActivityDate="2019-06-14T13:47:21.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5192" PostTypeId="2" ParentId="5190" CreationDate="2016-09-14T08:29:38.363" Score="7" Body="&lt;p&gt;This is a question about etymology and the answer should be sought in the standard etymological dictionaries. The DWDS says of „Exponent“ „als mathematischer Terminus 1544 bei M. Stifel nachweisbar“. That is a good two centuries before Euler. &lt;a href=&quot;http://www.dwds.de/wb/Exponent&quot;&gt;http://www.dwds.de/wb/Exponent&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The logic is that the index numbers are “placed outside” the line of writing.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2016-09-14T08:29:38.363" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5193" PostTypeId="1" AcceptedAnswerId="5194" CreationDate="2016-09-14T11:53:46.287" Score="8" ViewCount="639" Body="&lt;p&gt;In complex geometry, there is the a lemma, analogous to the Poincaré lemma in (real) differential geometry, which states that a $(p,q)$-form that is $\bar\partial$-closed is locally $\bar\partial$-exact. In the book &lt;em&gt;Complex Geometry&lt;/em&gt; by Huybrechts, we find the following remark:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The [...] proposition and its corollary are known as the Grothendieck-Poincaré lemma. The first proof of it is due to Grothendieck and was presented by Serre in the Séminaire Cartan in 1958.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The claim that this lemma was (first) proven by Grothendieck is also backed up by &lt;a href=&quot;https://www.staff.science.uu.nl/~looij101/kahler.pdf&quot; rel=&quot;noreferrer&quot;&gt;these notes&lt;/a&gt;, in section 5. Now, I would like to see the original presentation of the proof, so I tried to find the source that Huybrechts refers to. The obvious thing to do is go through the &lt;a href=&quot;http://www.numdam.org/numdam-bin/browse?j=SHC&quot; rel=&quot;noreferrer&quot;&gt;Séminaire Cartan&lt;/a&gt; scripts. However, in the volumes corresponding to the year 1958 there is no contribution by Serre. &lt;s&gt;In fact, I don't think there is anything related to the Grothendieck-Poincaré lemma in the entire collection (I convinced myself of this by skipping through all of Serre's contributions, which can be found with a little bit of effort, e.g. on &lt;a href=&quot;http://www.numdam.org/numdam-bin/search?h=aur&amp;amp;aur=Serre,+Jean-Pierre&amp;amp;format=short&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;.&lt;/s&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, I would like to know the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Did Serre present this proof by Grothendieck anywhere (else)? Where, and when?&lt;/li&gt;&#xA;&lt;li&gt;If not, where and when was the proof due to Grothendieck first published or otherwise made public?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2016-09-14T19:52:22.520" LastActivityDate="2016-09-14T19:55:17.340" Title="Where can I find the original presentation of the proof, due to Grothendieck, of the $\bar\partial$-Poincaré lemma?" Tags="|mathematics|reference-request|differential-geometry|" AnswerCount="1" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5194" PostTypeId="2" ParentId="5193" CreationDate="2016-09-14T19:38:08.670" Score="7" Body="&lt;p&gt;An alternative name is the Dolbeault-Grothendieck lemma. &lt;a href=&quot;http://pierre.dolbeault.free.fr/Conferences/bedlewo0711-slides.pdf&quot;&gt;Dolbeault himself&lt;/a&gt; writes the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It is proved by P. Dolbeault in the $C^\omega$ case, by homotopy, as can&#xA;  been&lt;/em&gt; [sic!] &lt;em&gt;the Poincare lemma. H. Cartan brings the proof to the $C^\omega$&#xA;  case by a potential theoritical method [Do 53]. Simultanously, the lemma has been proved by A. Grothendieck, by induction on the dimension, from the case&#xA;  $n = 1$ a consequence of the non homogeneous Cauchy formula, see [Ca 53], expose 18.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[Ca 53] are the notes of  Séminaire Cartan 1953/54, &lt;a href=&quot;http://archive.numdam.org/ARCHIVE/SHC/SHC_1953-1954__6_/SHC_1953-1954__6__A18_0/SHC_1953-1954__6__A18_0.pdf&quot;&gt;expose 18&lt;/a&gt; is indeed by Serre, and the result is given by Proposition 1. It is likely that 1958 was a typo. [Do 53] are two short notes in Comptes Rendus:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P. Dolbeault, Sur la cohomologie des varietes analytiques complexes, C.R. Acad. Sci. Paris 236 (1953), 175-177.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P. Dolbeault, Sur la cohomologie des varietes,analytiques&#xA;complexes, II, C.R. Acad. Sci. Paris 236 (1953), 2203-2205.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-14T19:55:17.340" LastActivityDate="2016-09-14T19:55:17.340" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5195" PostTypeId="2" ParentId="5188" CreationDate="2016-09-15T03:11:48.237" Score="29" Body="&lt;p&gt;Einstein himself told the story in his Kyoto address of 1922, which I quote from &lt;a href=&quot;https://books.google.com/books?id=KdaiZDaB7TgC&quot; rel=&quot;noreferrer&quot;&gt;Pais's biography titled Subtle is the Lord&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If all systems are equivalent, then Euclidean geometry cannot hold in all of them. To throw out geometry and keep laws is equivalent to describing thoughts without words. We must search for words before we can express thoughts. What must we search for at this point? This problem remained insoluble to me until 1912, when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss's theory was contained in the geometry course given by Geiser when I was a student...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;I realized that the foundations of geometry have physical significance. My dear friend the mathematician Grossmann was there when I returned from Prague to Zurich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann's theory, namely, whether the invariants of the line element could completely determine the quantities I had been looking for&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1923 Einstein added:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I had the decisive idea of the analogy between the mathematical problem of the theory and the Gaussian theory of surfaces only in 1912, however, after my return to Zurich, without being aware at that time of the work of Riemann, Ricci, and Levi-Civita. This was first brought to my attention&#xA;  by my friend Grossmann when I posed to him the problem of looking for generally&#xA;  covariant tensors whose components depend only on derivatives of the coefficients &lt;span class=&quot;math-container&quot;&gt;$[g_{\mu\nu}]$&lt;/span&gt; of the quadratic fundamental invariant &lt;span class=&quot;math-container&quot;&gt;$[g_{\mu\nu}dx^\mu dx^\nu]$&lt;/span&gt;&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pais, who personally interviewed Einstein for the book, gives further details. Grossman and Einstein studied together at the ETH (Swiss Federal Institute of Technology in Zurich) from 1896 to 1900. After seven years of teaching high school Grossman was hired as a full professor of geometry at ETH in 1907. Interestingly enough, that was the same year that Einstein had &quot;&lt;em&gt;the happiest thought of my life&lt;/em&gt;&quot; that &quot;&lt;em&gt;the relativity postulate has to be extended to coordinate systems which, relative to each other, are in nonuniform motion&lt;/em&gt;&quot;, i.e. the idea of &lt;a href=&quot;https://en.wikipedia.org/wiki/General_covariance&quot; rel=&quot;noreferrer&quot;&gt;general covariance&lt;/a&gt;. But Einstein was in Prague at the time, and his original attempts at relativistic gravity involved a scalar &quot;c-field&quot;. Grossman was lobbying ETH for offering Einstein a position. He succeeded in 1912, and Einstein accepted, over Lorentz backed invitation to Utrecht. Here is what happened next, in Pais's words:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In order to appreciate what happened in August 1912, it is essential to know&#xA;  that before his arrival in Zurich Einstein had already concluded that the description of gravitation in terms of the single scalar c-field of the Prague days had to go and that a new geometry of physical space-time was needed. I am convinced that he arrived in Zurich with the knowledge that not just one but ten gravitational potentials were needed... It must have been at that time that he said, 'Grossmann, Du musst mir helfen, sonst werd' ich verrückt!' G., you must help me or else I'll go crazy! With Grossmann's help the great transition to Riemannian geometry must have taken place during the week prior to August 16, as is indicated by Einstein's letter to Hopf.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;These conclusions are in harmony with my own recollections of a discussion with Einstein in which I asked him how the collaboration with Grossmann began. I have a vivid though not verbatim memory of Einstein's reply: he told Grossmann of his problems and asked him to please go to the library and see if there existed an appropriate geometry to handle such questions. The next day Grossmann returned (Einstein told me) and said that there indeed was such a geometry, Riemannian geometry. It is quite plausible that Grossmann needed to consult the literature since, as we have seen, his own field of research was removed from differential geometry.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1912-13 Einstein and Grossmann worked on the first theory where gravity is represented by the metric tensor, the Entwurf (sketch in German), but they mistakenly concluded that the gravitational field could not be generally covariant. &lt;a href=&quot;http://mathpages.com/rr/s8-08/8-08.htm&quot; rel=&quot;noreferrer&quot;&gt;Brown in Reflections on Relativity&lt;/a&gt; opines that &quot;&lt;em&gt;this accident may have been all that prevented Grossmann from being perceived as a co-creator of general relativity&lt;/em&gt;&quot;, but quotes Einstein's contrary opinion from a 1915 letter to Sommerfeld: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Grossmann will never lay claim to being co-discoverer. He only helped in guiding me through the mathematical literature but contributed nothing of substance to the results.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="26" LastEditDate="2019-07-28T13:14:37.540" LastActivityDate="2019-07-28T13:14:37.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5196" PostTypeId="1" CreationDate="2016-09-15T04:11:53.713" Score="6" ViewCount="190" Body="&lt;p&gt;In Riemannian geometry we have two very important things named after Jacobi: the Jacobi equation $J''=R(\gamma',J)\gamma'$ and Jacobi's theorem which states geodesics never minimize past conjugate points. Why is Jacobi's name attached to this equation and theorem? &lt;/p&gt;&#xA;" OwnerUserId="3558" LastEditorUserId="55" LastEditDate="2016-09-16T00:58:41.627" LastActivityDate="2016-09-16T00:58:41.627" Title="What did Jacobi, who lived before Riemann, have to do with the equation and theorem named after him in Riemannian geometry?" Tags="|mathematics|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5197" PostTypeId="1" AcceptedAnswerId="6501" CreationDate="2016-09-15T10:10:26.590" Score="2" ViewCount="294" Body="&lt;p&gt;Max Abraham is well known for his electromagnetic mass theory, and his book &lt;em&gt;Theorie der Elektrizität&lt;/em&gt; is said to be read widely around 1900. I am trying to find this book, but the editions look like a hell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a two volume book. Its first volume was a separate book written by Föppl, and it was already published. Then, when Abraham's book was published, Föppl's book was merged as the volume 1 of that two-volume book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weirdly, it looks like these books have more than 15 editions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And more weirdly, although it was very popular once upon a time, it has never been translated into English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And more weirdly, most editions cannot be found in catalogs, such as worldcat.org.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And more weirdly, I found another book, which is one volume only, written by Max Abraham and Richard Becker, and translated by John Dougal. I don't know who is Becker, but in the preface it says this edition is published after Max Abraham's death, and he says &quot;although I kept many passages, some fairly extensive alterations have been made...&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am really lost. How dare he can take a book, after the author dies, make extensive alterations, etc. What is the point?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know the weird story of these books?&lt;/p&gt;&#xA;" OwnerDisplayName="user2831" LastEditorDisplayName="user2831" LastEditDate="2016-09-16T20:05:11.697" LastActivityDate="2017-09-11T13:02:54.703" Title="Abraham's electromagnetism book and its story" Tags="|physics|reference-request|electromagnetism|electricity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5198" PostTypeId="2" ParentId="5196" CreationDate="2016-09-16T00:57:29.013" Score="5" Body="&lt;p&gt;This is a natural question given the modern exposition, where Riemannian geometry is often the only place where some parts of the calculus of variations are encountered. And a reminder that historical order rarely follows the order of modern expositions. Jacobi's equation, condition, fields, etc., are not specific to Riemannian geometry, they relate to the second variation of functionals in the classical calculus of variations like $\int_a^b f(x,y,y')\,dx$, the arclength functional of the geodesic problem is just a particular case. Those were in the sights of the leading mathematicians since the &lt;a href=&quot;https://en.wikipedia.org/wiki/Brachistochrone_curve#History&quot; rel=&quot;noreferrer&quot;&gt;brachistochrone&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://arxiv.org/ftp/math/papers/0402/0402357.pdf&quot; rel=&quot;noreferrer&quot;&gt;Ferguson's Brief Survey of the History of the Calculus  of Variations and its Applications&lt;/a&gt;, Legendre in 1786 presented a memoir On  the  Method  of  Distinguishing  Maxima  from  Minima  in  the  Calculus  of  Variations, where he showed that necessarily $f_{y',y'}\geq0$ for minimizers, and $\leq0$ for maximizers. Half a century later Jacobi &quot;&lt;em&gt;in  a  paper  remarkable  for  its  brevity  and  obscurity&lt;/em&gt;&quot; gave a complementary sufficient condition, $f_{y',y'}&amp;gt;0$ and $b$ is closer to $a$ than the first conjugate point of $a$. The paper, On the Calculus of Variations and the Theory of Differential Equations (1836), &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;was so terse that rigourous  proofs were not  given but instead were hinted at. Perhaps, as one mathematical historian has suggested, Jacobi was in a rush to publish his results first to ensure intellectual priority. It is difficult to agree with such a theory since  progress  in  this  field  had  stagnated  for  half  a  century!&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, after another half a century Riemannian geometry became a natural place for applying Jacobi's result. The &quot;one historian&quot; is Goldstine, and more details can be found in his &lt;a href=&quot;http://www.springer.com/us/book/9781461381082&quot; rel=&quot;noreferrer&quot;&gt;History of the Calculus of Variations from the 17th through the 19th Century&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-09-16T00:57:29.013" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5200" PostTypeId="1" AcceptedAnswerId="5218" CreationDate="2016-09-16T16:46:47.747" Score="5" ViewCount="257" Body="&lt;p&gt;My question is only about a certain hystorical gem. The purpose of this question is to explore how vast were Archimedes's contributions to geometry that are not connected to calculus (i.e calculation of areas and volumes of curved figures), i.e to pure geometry. I know that Archimedes introduced his &quot;twin circles&quot; in proposition 5 of his book of lemmas, but the wiki article about the book of lemmas states that he also made use of Pappus's chain of tangential circles in this book, but i coudn't find where. So if anyone know the answer, please write it down.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-05-13T14:02:40.507" LastActivityDate="2020-05-13T14:02:40.507" Title="Did Archimedes make use of a Pappus chain according to the Book of Lemmas?" Tags="|biographical-details|archimedes|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5202" PostTypeId="1" AcceptedAnswerId="5204" CreationDate="2016-09-17T18:13:03.920" Score="4" ViewCount="442" Body="&lt;p&gt;I know that a function is called &quot;function&quot; because it's an &quot;execution&quot; of operations. Abbreviated notation is f. But why do we write &lt;code&gt;f(x)&lt;/code&gt; and not &lt;code&gt;(x)f&lt;/code&gt; or &lt;code&gt;f_x&lt;/code&gt; or &lt;code&gt;f-x-&lt;/code&gt; etc. ? &lt;/p&gt;&#xA;" OwnerUserId="4721" LastActivityDate="2016-09-18T10:24:17.230" Title="Why do we use brackets for function parameters?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5203" PostTypeId="1" AcceptedAnswerId="5206" CreationDate="2016-09-17T18:17:25.470" Score="3" ViewCount="4474" Body="&lt;p&gt;Albert Einstein is an example of someone knowing multiple and being significantly gifted at mathematics more specifically physics.&lt;/p&gt;&#xA;&lt;p&gt;How many languages did historically well-known mathematicians master?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;This was &lt;a href=&quot;https://math.stackexchange.com/q/1928115&quot;&gt;closed&lt;/a&gt; on the sister site Mathematics SE.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="4714" LastEditorUserId="4251" LastEditDate="2021-12-15T13:36:45.377" LastActivityDate="2021-12-15T13:36:45.377" Title="How many languages did historically well-known mathematicians master?" Tags="|mathematics|mathematicians|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5204" PostTypeId="2" ParentId="5202" CreationDate="2016-09-17T20:01:55.940" Score="7" Body="&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;E045&lt;/strong&gt; : &lt;a href=&quot;http://eulerarchive.maa.org/&quot; rel=&quot;noreferrer&quot;&gt;Additamentum ad dissertationem de infinitis curvis eiusdem generis (Addendum to the dissertation on infinite(ly many) curves of the same type)&lt;/a&gt;, originally published in &lt;em&gt;Commentarii academiae scientiarum Petropolitanae&lt;/em&gt;; 7, 1740, pp.184-200:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;si $f(\frac x a + c)$ denotet functionem quamcunque ipsius $\frac x a + c$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The definition of &lt;em&gt;function&lt;/em&gt; was already present into:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Bernoulli&quot; rel=&quot;noreferrer&quot;&gt;Johann Bernoulli&lt;/a&gt;, &lt;em&gt;Remarques sur ce qu'on a donne jusqu'ici de solutions des problemes sur les isopdrimitres&lt;/em&gt;, published in &lt;em&gt;Mem.Acad.roy.sci&lt;/em&gt;, Paris, 1718. See &lt;a href=&quot;https://books.google.it/books?id=Ku_1PxvPHPEC&amp;amp;pg=PA241&quot; rel=&quot;noreferrer&quot;&gt;Opera omnia, Tomus II&lt;/a&gt;, page 241:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Definition&lt;/em&gt;. On appelle ici &lt;em&gt;Fonction&lt;/em&gt; d'une grandeur variable, une quantité&#xA;  composée de quelque maniére que ce soit de cette grandeur variable et de constantes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the same &lt;a href=&quot;https://en.wikipedia.org/wiki/M%C3%A9moire&quot; rel=&quot;noreferrer&quot;&gt;mémoire&lt;/a&gt; [&lt;a href=&quot;https://books.google.it/books?id=Ku_1PxvPHPEC&amp;amp;pg=PA243&quot; rel=&quot;noreferrer&quot;&gt;page 243&lt;/a&gt;] Bernoulli proposed the Greek letter $\phi$ as a notation for the &lt;em&gt;caractéristique&lt;/em&gt; of a function, writing the argument without brackets: $\phi Pb$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I presume that the improved symbolism due to Euler was devised in order to avoid the mistake of interpreting justaxposition as multiplication.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-09-18T10:24:17.230" LastActivityDate="2016-09-18T10:24:17.230" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5205" PostTypeId="2" ParentId="5203" CreationDate="2016-09-18T02:18:36.003" Score="7" Body="&lt;p&gt;Einstein would not call himself a mathematician:-)&#xA;&amp;quot;Historically well-known&amp;quot; is an ill defined description. It depends on place and time.&lt;/p&gt;&#xA;&lt;p&gt;Until 18th century only one language was necessary to do mathematics: Latin.&lt;/p&gt;&#xA;&lt;p&gt;Euler wrote in German, French and Latin (mathematics in Latin, other things&#xA;in French and German). He lived and worked in Russia for a large part of his life, but the sources I know do not tell whether he spoke Russian:-)&#xA;Newton wrote mathematics in Latin, other things in English.&lt;/p&gt;&#xA;&lt;p&gt;Proceedings of St Petersburg Academy founded in 1740s had two official languages: Latin for mathematics and French for all other sciences.&#xA;Latin was a part of any high school and university curriculum in Europe until late 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Until the middle of 20th century most mathematicians could read mathematics in German, French and English. (Mathematical papers were rarely, almost never translated. And a serious mathematician had to be aware of the literature in his area. This was not possible without working knowledge of these three languages)).&lt;/p&gt;&#xA;&lt;p&gt;When I first met my future adviser (in 1971) he asked what languages I can read.&#xA;I was a first year student, and I replied with some pride: English and French.&#xA;He was not impressed and told me: a mathematician has to read in ALL languages:-) So we are returning to the situation which existed at the times&#xA;of Euler.&lt;/p&gt;&#xA;&lt;p&gt;Knowledge of foreign languages declined in the second half of 20th century: English gradually displaced other languages in mathematics, and at this time a mathematician can survive if s/he knows only English.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-09-30T01:46:09.647" LastActivityDate="2020-09-30T01:46:09.647" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="5206" PostTypeId="2" ParentId="5203" CreationDate="2016-09-18T02:37:44.433" Score="7" Body="&lt;p&gt;There are too many &quot;historically well-known mathematicians&quot; to list them all next to the number of languages they knew. As for those who knew particularly many, perhaps the best known is &lt;a href=&quot;https://en.wikipedia.org/wiki/John_von_Neumann&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Neumann&lt;/a&gt; (Hungarian-American, 1903-1957), who was fluent in French, German, Latin, Greek, English, Yiddish, and Hungarian. &lt;a href=&quot;https://en.wikipedia.org/wiki/P%C3%A9ter_Frankl&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frankl&lt;/a&gt; (Hungarian combinatorialist, Erdös' co-author, born 1953) is less well-known but is fluent in more: English, Russian, Swedish, French, Spanish, Polish, German, Japanese, Chinese, Thai, Korean. However, they both pale in comparison to &lt;a href=&quot;https://en.wikipedia.org/wiki/Al-Farabi&quot; rel=&quot;nofollow noreferrer&quot;&gt;al-Farabi&lt;/a&gt; (medieval Persian polymath, 872-950 AD), who is not exactly known as a mathematician (he did some work in logic), but allegedly mastered over 70(!) languages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From late middle ages to 18th century many Islamic and European mathematicians and scientists knew Greek and Latin in addition to their native tongue, Gauss still had to publish in Latin at the beginning of 19th. Since 17th it was not uncommon for them to at least be able to make out technical texts in English, French and German. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia has a long &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_polyglots&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of polyglots&lt;/a&gt;, dead and living. In a different vein, see &lt;a href=&quot;https://latin.stackexchange.com/questions/1372/what-is-the-history-of-scientific-latin&quot;&gt;What is the history of scientific Latin?&lt;/a&gt; and &lt;a href=&quot;http://www.bbc.com/news/magazine-29543708&quot; rel=&quot;nofollow noreferrer&quot;&gt;Porzucki's How English beat German as language of science&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T13:01:09.190" LastActivityDate="2016-09-18T02:46:32.403" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5209" PostTypeId="1" CreationDate="2016-09-19T11:16:41.793" Score="10" ViewCount="577" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_G%C3%B6del&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Gödel&lt;/a&gt; is one of the &quot;best&quot; logicians of the 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/270654&quot;&gt;Here&lt;/a&gt;, the user &quot;Jeffrey Kegler&quot; states that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Kurt Gödel considered himself a philosopher who did mathematics, rather than as a mathematician who did philosophy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Are there any sources where Gödel said this? Did Gödel really consider himself primarily a philosopher who is interested in mathematics (instead of the other way round)?&lt;/p&gt;&#xA;" OwnerUserId="4728" LastEditorUserId="4251" LastEditDate="2022-09-23T05:54:28.740" LastActivityDate="2022-09-23T05:54:28.740" Title="Did Gödel consider himself primarily a philosopher who is interested in mathematics (instead of the other way round)?" Tags="|mathematicians|biographical-details|mathematical-logic|logicians|goedel|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5210" PostTypeId="2" ParentId="5209" CreationDate="2016-09-19T13:31:10.810" Score="6" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/goedel/&quot; rel=&quot;noreferrer&quot;&gt;Kurt Gödel&lt;/a&gt; enrolled in the University of Vienna, attending lectures on physics, his initial field of interest, lectures on philosophy given by Heinrich Gomperz, and lectures on mathematics. [...]  eventually graduating under [the mathematician Hans] &lt;a href=&quot;https://en.wikipedia.org/wiki/Hans_Hahn_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Hahn&lt;/a&gt; with a Dr.phil. in mathematics in 1929. The main theorem of his dissertation was the completeness theorem for first order logic.&lt;/p&gt;&#xA;&lt;p&gt;Gödel's university years also marked the beginning of his attendance at meetings of the Vienna Circle, a group around Moritz Schlick that quickly became known as “logical positivists”.&lt;/p&gt;&#xA;&lt;p&gt;the decade [1940s] saw the beginning of Gödel's intensive study of Leibniz, which, Gödel reports, occupied the period from 1943 to 1946.&lt;/p&gt;&#xA;&lt;p&gt;Gödel's philosophical views can be broadly characterized by two points of focus, or, in modern parlance, commitments. These are: realism, namely the belief that mathematics is a descriptive science in the way that the empirical sciences are. The second commitment is to a form of Leibnizian rationalism in philosophy; and in fact Gödel's principal philosophical influences, in this regard particularly but also many others, were Leibniz, Kant and Husserl.&lt;/p&gt;&#xA;&lt;p&gt;By the decade [1950s]'s close Gödel developed a serious interest in phenomenology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Having said that, Gödel's main contributions are :&lt;/p&gt;&#xA;&lt;p&gt;A) mathematical ones:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the completeness theorem for the first order predicate calculus,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the incompleteness theorems,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the results in set theory regarding the consistency of the    Continuum Hypothesis and the Axiom of Choice,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;work in Intuitionistic Logic and Arithmetic;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;B) contribution to the &amp;quot;foundational debate&amp;quot; in the field of philosophy of mathematics, with the papers:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;“Russell's Mathematical Logic” (1944),&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“What is Cantor's Continuum Hypothesis?” (1947);&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;C) work on relativity, published in the article: “An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation” (1949);&lt;/p&gt;&#xA;&lt;p&gt;D) the unpublished &lt;a href=&quot;https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof&quot; rel=&quot;noreferrer&quot;&gt;ontological proof&lt;/a&gt;,  dated around 1941, based on modal logic.&lt;/p&gt;&#xA;&lt;p&gt;Having said that, the long G's involvement with philosophy produced a lot of discussion and manuscripits but no completed work, while his contributions to mathematical logic and set theory :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;are among the handful of landmark theorems in twentieth century mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-09-19T13:31:10.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5212" PostTypeId="1" CreationDate="2016-09-19T18:10:48.770" Score="4" ViewCount="701" Body="&lt;p&gt;There are several instances in history that are considered to be good candidates as discovery and first use of zero. Examples includes Ancient Sumarian and Babylonian civilizations, Mayans, medieval India, China and Cambodia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the mathematical concepts and ideas that can be logically considered as precursors to discovery of zero. For example, familiarity and extensive  use of a positional number system within a civilization could be considered as one such precursor. Another is evidence of experimenting with different symbols like empty space, dot, etc. to indicate absence of a number and need to deal with relatively large numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly what are the ideas that can be anticipated to follow discovery of zero?&#xA;Here are some examples: negative numbers, new rules of arithmetic to handle zero and negative numbers in addition to natural numbers, etc.&lt;/p&gt;&#xA;" OwnerUserId="4730" LastEditorUserId="4730" LastEditDate="2016-09-19T19:05:28.883" LastActivityDate="2020-01-26T22:51:41.340" Title="What are the precursors and postcursors to discovery/invention of zero?" Tags="|zero|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5213" PostTypeId="2" ParentId="5212" CreationDate="2016-09-19T20:31:18.933" Score="1" Body="&lt;p&gt;in the case of Arabic the &quot;precursors&quot; of zero we entirely practical, not really mathematical. you need zero to balance the books; algebra emerged from the need to keep the books, esp. in cases of inheritance based on Islamic law.  so zero effectively means even-steven: I don't owe you, you don't owe me, everything is fairly balanced.  (The term &quot;algebra&quot; comes from the title of al-khwarizmi's &quot;concise book of making whole and balancing&quot;,  where &quot;making whole&quot; = al-jabr, literally bonesetting.)&lt;/p&gt;&#xA;" OwnerUserId="4062" LastActivityDate="2016-09-19T20:31:18.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5214" PostTypeId="2" ParentId="5212" CreationDate="2016-09-19T21:17:38.777" Score="6" Body="&lt;p&gt;The main role of zero is not to express &quot;nothing&quot;, but to serve as a place holder&#xA;in a positional system. The need in a place holder begins when people start&#xA;comlicated calculations, that is in Babylonian astronomy. First they used a space&#xA;instead of zero, then invented a secial cuneiform sign (which I cannot reproduce here), see Van-der-Waerden, Science Awakening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Greeks used letter notation for numbers, but they also needed zero. Ptolemy uses&#xA;the Greek letter $\omicron$ (omicron).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus for example the number written in modern sexagesimal notation as&#xA;373; 10,0,58 (which means 373 degrees 10 minutes 0 seconds and 58/60 of a second) in Ptolemy would look like this: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\overline{\tau\omicron\gamma}\;\overline{\iota}\; \omicron\; \overline{\nu\eta}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Horisontal overline is used to show that letters under it are digits of one number. Here $\tau$ is 300, $\gamma$ is 3, $\iota$ is 10, $\nu$ is 50 and&#xA;$\eta$ is 8. The $\omicron$ has a double meaning here: overlined it stands for the 70&#xA;in 373, and not overlined for zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only difference with modern notation is that Ptolemy used decimal systems&#xA;for the integer part and sexagesimal for the fractional part. But &quot;zero&quot; he had. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Van der Waerden writes that the use of the letter $\omicron$ to denote zero&#xA;was known to some Iamblichus, a Pythagorean, but this cannot be the Iamblichus&#xA;of Wikipedia (245-345, a Syrian philosopher) because he lived after Ptolemy,&#xA;and van der Waerden is well aware that Ptolemy used $\omicron$ as zero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Indians and Arabs knew Ptolemy, crediting them with &quot;invention of zero&quot; is an&#xA;exaggeration. But of course they perfected the positional system. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-09-19T21:40:49.993" LastActivityDate="2016-09-19T21:40:49.993" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5215" PostTypeId="1" CreationDate="2016-09-20T01:31:54.677" Score="1" ViewCount="464" Body="&lt;p&gt;I know this is probably a precise and odd question, but I'm curious to know if any famous Mathematicians or Scientist had pets. I'm really looking for Mathematicians and dogs but I thought it better to widen my search. &lt;/p&gt;&#xA;" OwnerUserId="4731" LastActivityDate="2016-09-20T11:20:43.473" Title="Are there any accounts of famous Mathematicians/Scientists and their dogs, or any pets in general?" Tags="|mathematicians|biographical-details|depiction-of-scientists|" AnswerCount="1" CommentCount="4" ClosedDate="2016-09-20T15:21:22.633" ContentLicense="CC BY-SA 3.0" />
  <row Id="5216" PostTypeId="2" ParentId="5215" CreationDate="2016-09-20T11:20:43.473" Score="4" Body="&lt;p&gt;F. D. C. Willard co-authored a paper with Jack Hetherington. Willard was his cat. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/F.D.C._Willard&quot; rel=&quot;nofollow&quot;&gt;https://en.m.wikipedia.org/wiki/F.D.C._Willard&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2016-09-20T11:20:43.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5217" PostTypeId="1" CreationDate="2016-09-20T11:31:43.157" Score="4" ViewCount="174" Body="&lt;p&gt;My question refers to proposition 15 in book 2 of Isaac Newton's principia. In this proposition Newton stated sevaral corollaries on the orbital motion of bodies under a certain law of central force and under medium resistance that is proportional to the medium density and the velocity squared. His main result is that if the density of the medium varies inversely with the distance from the center than the orbit will be an &quot;equiangular spiral&quot;- or logarithmic spiral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question asks if there are any uses for this theorem, for example to calculate the decay of satellite orbits. As we know, the density in the atmosphere follows an exponential profile, so it seems this theorem is only a mathematical gem and not of any use. So is there any way to make use of htis thoerem?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2016-10-21T04:24:22.107" Title="Is there any significance to Newton's theorem on decaying orbits in book 2 of his Principia?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5218" PostTypeId="2" ParentId="5200" CreationDate="2016-09-20T22:28:58.607" Score="3" Body="&lt;p&gt;No, he did not. Wikipedia's editor was most likely misled by somewhat clumsy arrangement of the text in the Heath's 1897 translation of the &lt;a href=&quot;https://www.stmarys-ca.edu/sites/default/files/attachments/files/Book_of_Lemmas.pdf&quot; rel=&quot;nofollow&quot;&gt;Book of Lemmas&lt;/a&gt;. Right under the statement of Proposition 8 there is a long passage in small font describing the construction of the Pappus chain. This passage however is the continuation of the footnote from the previous page, which starts as:&quot;&lt;em&gt;It may be interesting to add the enunciation of the &quot;ancient proposition&quot; stated by Pappus (p. 208) and proved by him after several auxiliary lemmas&lt;/em&gt;&quot;. In other words, the only trace of Pappus chains in the Book of Lemmas is in... Heath's retelling of the chain described in Pappus's Mathematical Collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What we find in the book is the construction of the arbelos, a plane figure bounded by three semi-circles resting on the same segment, with the diameters of the smaller ones adding up to the largest one. The footnote refers to Proposition 6, which describes how to find the diameter of the circle  inscribed into the arbelos, i.e. the one touching all three semi-circles from the inside. In hindsight, this is the first (the largest) circle in the Pappus chain. But this is as far as it goes, the book does not inscribe any further circles into the arbelos, and the &quot;twin circles&quot; are unrelated to Pappus chains. Everything beyond that (iterating the inscribed circle construction, demonstrating that the centers of the circles all lie on an ellipse, etc.) is found only in Pappus's Collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only reason we know that the chains are not due to Pappus is because he himself calls the result about it an &quot;ancient lemma&quot;. Since Pappus is usually meticulous about giving credit when he knows the source (we know of many Hellenistic mathematicians' work largely due to Pappus) this probably means that in this case the lemma was already folklore. This was highly unlikely to happen to a result of Archimedes. It is worth mentioning that even the Book of Lemmas itself is not a work of Archimedes. &lt;a href=&quot;http://www.mlahanas.de/Greeks/Arbelos.htm&quot; rel=&quot;nofollow&quot;&gt;Here is Heath&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;... we have a collection of Lemmas which has reached us through the Arabic. [...] The Lemmas cannot, however, have been written by Archimedes in their present form, because his name is quoted in them more than once. ...though it is quite likely that some of the propositions were of Archimedean origin, e.g. those concerning the geometrical figures called respectively arbelos (literally 'shoemaker's knife') and salinon (probably 'salt-cellar')...&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pappus's Collection dates to c. 340 AD. But according to &lt;a href=&quot;https://www.cambridge.org/core/journals/science-in-context/article/two-approaches-to-foundations-in-greek-mathematics-apollonius-and-geminus/6A5FFBA7537E8B0D660C8BB4B29D7A4A&quot; rel=&quot;nofollow&quot;&gt;Acerbi's Two Approaches to Foundations in Greek Mathematics&lt;/a&gt;, Pappus's list of &quot;moderns&quot; in Collection IV.58 includes Demetrius of Alexandria, Phylo of Tiana, and Menelaus of Alexandria, Proclus also mentions Geminus among &quot;moderns&quot;. These characters lived as far back as the 1st century BC or after, so the &quot;ancient lemma&quot; most likely originated in the late third, or second century BC, near the end of Archimedes's lifetime, or shortly after his death in 212 BC. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-23T22:03:23.827" LastActivityDate="2016-09-23T22:03:23.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5219" PostTypeId="2" ParentId="5217" CreationDate="2016-09-21T03:47:11.247" Score="1" Body="&lt;p&gt;This theorem has no astronomical applications (or applications to satellites).&#xA;Newton worried that the space may not be empty (filled with some substance), and wanted&#xA;to know how this assumption would reflect on the motion of planets and satellites.&#xA;The theorem could have some implications for bodies moving in a rarefied atmosphere, if the law of density were correct. But it was incorrect.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-09-21T03:47:11.247" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5220" PostTypeId="2" ParentId="5209" CreationDate="2016-09-21T22:14:39.380" Score="5" Body="&lt;p&gt;Since there is no record of Gödel's view, any response to this question is a matter of opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the few philosophers who has published opinions on the merits of Gödel's philosophical views, as well as biographical details on role of philosophy in Gödel's professional life, is Palle Yourgrau of Brandeis University.  It is his opinion, expressed in this book &lt;em&gt;The Disappearance of Time: Kurt Gödel and the Idealistic Tradition in Philosophy&lt;/em&gt;, that Gödel did indeed consider himself to be primarily a philosopher.  I've skimmed the text a number of times but I am unable to locate the paragraph where Yourgrau explicitly states that he believes this is Gödel's view of his own work, which I am certain is there.  The best I can do is the following quote :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To philosophers, then and now, he was simply a logician trying to pass as a philosopher.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;TL/DR;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Yourgrau notes, in general philosophers appear to hold a different view.  Quoting Yourgrau:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gödel, throughout his academic life, was exceptionally anxious to avoid being considered a philosophical dilettante or crank, and a &quot;pious&quot; one at that.  In this he failed completely.  His &quot;special caution&quot; succeeded only in keeping him from contributing to important fields of research.  His refusal to publish his ontological argument for the existence of God fooled no-one.  He could not hide the fact that he was a kind of believer and that his argument, like Leibniz's before him, was anything but a mere intellectual exercise.  What Bertrand Russell called Gödel's' &quot;unadulterated Platonism&quot; marked him for some as a throwback to &quot;precritical&quot; times, before Kant launched is &quot;critique&quot; of pure reason . . . . .  Even Gödel's writings on Einstein succeeded only in convincing cosmologists that strange things happen when a logician pays too much attention to the equations of relativity, while forgetting their physical meaning.  His absence, in turn, from the Wittgenstein revolution . . . . . combined with his refusal to pay homage to the preeminent figure in contemporary analytic philosophy, W.V.O. Quine, marked him as a philosophical castaway. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In February of 1995, at Boston University, a symposium was held entitled &quot;Gödel's General Philosophical Significance&quot;. The two main speakers, Warren Goldfarb and Burton Dreben, both Harvard philosophers, were unrelenting in their ridicule.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following Goldfarb's speech, a single question was asked :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Do I understand you correctly, professor Goldfarb, that in your judgement Gödel, though a great logician, was a philosophical fool?&quot;  A polite smile was Goldfarb's only answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Dreben appears to have particularly irked that Gödel had the &quot;audacity&quot; to engage in rational theology - no doubt a reference to Gödel's recasting of the ontological argument. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For Dreben, it seemed, this was a kind of scandal.  He was moved to deliver a sermon on the harm that is done when people who are good at purely formal thinking get the idea that they are qualified to contribute to philosophy.  That Dreben's own position in the philosophical world owed much to his reputation in formal logic was an irony that appeared to be lost on him. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the years since the Boston symposium, philosophers' assessment of Gödel's philosophical work has been more polite, but no less dismissive. Yourgrau appears to be a lone voice in support of Gödel's philosophical views. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The case for Gödel as philosopher is unassailable.  Though he published few essays that could be considered explicit contributions to philosophy, they suffice to establish him as an important philosopher of mathematics and time and space.  The posthumous publication of several more of his philosophical studies, including the Gibbs lecture, his contribution to the Schilpp volume on Carnap, and the longer version of his contribution to the Schilpp volume on Einstein, confirm this assessment.  The essay he wrote for the Schilpp volume on Russell, which contained new and insightful discussions of Frege as well as Russell on the question of meaning, including an illuminating and prescient comparison of Russell on &quot;denoting&quot; with Frege on &quot;sense and reference&quot;, leaves little room for doubt that in the philosophy of language, too, his abilities were striking. He had clearly mastered the writings of most of the seminal figures in twentieth-century analytic philosophy ... His grasp of Hegel astonished the philosopher Georg Kreisel, a man not easy to impress. ( &lt;em&gt;paraphrasing&lt;/em&gt;: He also demonstrated extensive knowledge of Kant and Husserl.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-09-23T00:15:12.683" LastActivityDate="2016-09-23T00:15:12.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5221" PostTypeId="1" AcceptedAnswerId="5222" CreationDate="2016-09-23T11:18:26.240" Score="15" ViewCount="1493" Body="&lt;p&gt;Apparently the ﬁrst mathematical publication of Grothendieck where he uses the terms “functor” and “category” in the technical sense is the Kansas report.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From where Grothendieck had knowledge of categorial language?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3857" LastEditorUserId="15350" LastEditDate="2022-06-04T18:43:01.090" LastActivityDate="2022-06-04T18:43:01.090" Title="How did Grothendieck encounter and adopt the categorical language?" Tags="|mathematics|biographical-details|algebraic-geometry|grothendieck|category-theory|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5222" PostTypeId="2" ParentId="5221" CreationDate="2016-09-23T19:33:38.167" Score="16" Body="&lt;p&gt;Grothendieck's familiarity with the categories predates Kansas. In 1948-1949 he attended &lt;a href=&quot;http://www.numdam.org/numdam-bin/browse?id=SHC_1948-1949__1_&quot; rel=&quot;noreferrer&quot;&gt;Séminaire Cartan&lt;/a&gt; at École Normale Supérieure, where he &amp;quot;&lt;em&gt;took the liberty of speaking to Cartan, as if to his equal&lt;/em&gt;&amp;quot; (&lt;a href=&quot;https://www.ams.org/journals/notices/200409/200409FullIssue.pdf&quot; rel=&quot;noreferrer&quot;&gt;Cerf's obituary, quoted in Jackson's &lt;em&gt;Comme Appele du Neant&lt;/em&gt;&lt;/a&gt;). That would be  Henri Cartan, the agebraic topologist, he is the son of Élie Cartan known for his contributions to Lie group theory and differential geometry (e.g. the differential forms). The theme of the year was simplicial algebraic topology and sheaf theory, developed by Leray since 1945. Grothendieck also attended  a course on locally convex spaces, given by Leray himself at the Collège de France, so he probably heard the buzzwords. However, at the time Grothendieck felt out of his depth among people playing with the new algebraic toys &amp;quot;&lt;em&gt;like they have known them from the cradle&lt;/em&gt;&amp;quot; (&lt;a href=&quot;http://uberty.org/wp-content/uploads/2015/12/RS-grothendeick1.pdf&quot; rel=&quot;noreferrer&quot;&gt;Récoltes et Semailles&lt;/a&gt;). In 1949-1953 he went to study at the University of Nancy under Delsarte, Dieudonné, and Schwartz among others. As Dieudonné wrote in &lt;a href=&quot;http://docslide.us/documents/a-grothendiecks-early-work-19501960.html&quot; rel=&quot;noreferrer&quot;&gt;Grothendieck's early work (1950–1960)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;When, in 1953, it was time to grant him a doctor's degree, it was necessary to choose from among six papers he had written, any one of which was at the level of a good dissertation. Of course, his fundamental paper on tensor products and nuclear spaces is the one that was chosen and it rapidly became a landmark in functional analysis.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to Effros, this paper, Produits Tensoriels Topologiques  et&#xA;Espaces  Nucléaires, already employed categorical methods:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He was arguably the first to  realize  that  the  algebraic/categorical  methods that flourished after the  Second  World  War could be used in this highly analytic branch of functional analysis.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For 1953-54 Schwartz arranged for his brilliant pupil (who refused to become a French citizen because it entailed military service) a position in Brazil, and then Grothendieck spent a year (1955) at the University of Kansas, probably at the invitation of Aronszajn. It is there that his interests shifted from functional analysis to homological algebra. According to Malgrange, Grothendieck told him concerning topological vector spaces:&amp;quot;&lt;em&gt;There is nothing more to do, the subject is dead.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And it was there, in Lawrence, Kansas, that Grothendieck wrote the &amp;quot;Tôhoku paper&amp;quot; Sur Quelques Points D’algèbre Homologique (published in 1956), extending the work of Cartan and Eilenberg on modules, where the categories appear in their birthplace environment. He also began to correspond with Serre, whom he knew from Paris and Nancy and who pulled him towards algebraic geometry, where Grothendieck will employ sheaf-theoretic and categorical methods to make his principal mark. The &lt;a href=&quot;https://mattbakerblog.wordpress.com/2014/11/15/excerpts-from-the-grothendieck-serre-correspondence&quot; rel=&quot;noreferrer&quot;&gt;Grothendieck-Serre correspondence&lt;/a&gt; is translated into English. As Serre related much later,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;His knowledge of classical algebraic geometry was practically zero [...] I tried to help him with that. But... there were so many open questions that it didn’t matter.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Grothendieck confirms in Récoltes et Semailles that most of what he learned in geometry he learned from Serre, or taught  himself, and calls Serre the &amp;quot;&lt;em&gt;detonator&lt;/em&gt;&amp;quot; that provided the spark to ignite his explosion of ideas. In particular, it was Serre who described to Grothendieck the famous Weil conjectures in the cohomological language, a categorical bridge from homological algebra to algebraic geometry.&lt;/p&gt;&#xA;&lt;p&gt;I found &lt;a href=&quot;http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf&quot; rel=&quot;noreferrer&quot;&gt;Jackson's As If Summoned from the Void&lt;/a&gt; biography sketch that tracks the origins of Grothendieck's ideas very helpful.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-01-31T18:39:46.890" LastActivityDate="2022-01-31T18:39:46.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5223" PostTypeId="1" AcceptedAnswerId="5235" CreationDate="2016-09-25T17:13:12.617" Score="0" ViewCount="1162" Body="&lt;p&gt;I am curious whether or not Albert Einstein wrote a computer program. Did he write a computer program?&lt;/p&gt;&#xA;" OwnerUserId="2903" LastEditorUserId="62" LastEditDate="2016-09-25T23:22:40.100" LastActivityDate="2016-09-29T19:45:29.843" Title="Did Albert Einstein write a computer program?" Tags="|computer-science|einstein|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5224" PostTypeId="1" AcceptedAnswerId="5225" CreationDate="2016-09-25T18:02:13.147" Score="8" ViewCount="3998" Body="&lt;p&gt;According to what I have read, using Newton’s methods there are 78 different families of cubic curves.  Newton discovered 72 of them while “subsequent research identified another 6”.  &lt;a href=&quot;http://projecteuclid.org/download/pdf_1/euclid.rmjm/1250126126&quot; rel=&quot;nofollow noreferrer&quot;&gt;This paper&lt;/a&gt; suggests that those overlooked by Newton are attributable to Plucker and Cayley.  However, &lt;a href=&quot;https://math.stackexchange.com/questions/14921/how-many-cubic-curves-are-there&quot;&gt;this thread on MSE&lt;/a&gt; suggests that Plucker identified 219 additional types.  Obviously this is inconsistent with a total of 78.  Perhaps Plucker’s criteria were different from those of Newton.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt; : When, and by who, were the additional 6 families of cubic curves identified, and why might Newton have overlooked them? &lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2019-12-02T03:19:32.910" Title="How was Newton's classification of cubic curves completed?" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5225" PostTypeId="2" ParentId="5224" CreationDate="2016-09-25T23:04:01.100" Score="7" Body="&lt;p&gt;Stillwell gives some details in &lt;a href=&quot;https://books.google.com/books?id=WNjRrqTm62QC&amp;amp;pg=PA110&amp;amp;lpg=PA110&amp;amp;dq=stirling+cubic+curves&amp;amp;source=bl&amp;amp;ots=RTteT6C5EH&amp;amp;sig=Dizin3swNdrj_Q7Nhhtl_Fa7Xe4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjK6OX5xavPAhXElx4KHQn4AQcQ6AEIJjAC#v=onepage&amp;amp;q=stirling%20cubic%20curves&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Mathematics and its History&lt;/a&gt;. In modern terms, Newton made a general transformation of axes reducing the general cubic to four equation types, and then classified them into &quot;species&quot; according to the roots of the polynomials on the right hand side. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;His paper does not contain detailed proofs; these were supplied by Stirling (1717), along with four of the species Newton had missed. Newton’s classification was criticized by some later mathematicians, such as Euler, for lacking a general principle. A unifying principle was certainly desirable, to reduce the complexity of the classification. And such a principle was already implicit in one of Newton’s passing remarks, Section 29, “On the Genesis of Curves by Shadows.”&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The principle Stillwell has in mind is the projective classification over the complex field, but that is &quot;implicit&quot; indeed, no wonder Euler missed it.&#xA;&lt;a href=&quot;https://books.google.com/books?id=U4I82SJKqAIC&amp;amp;pg=PA130&amp;amp;lpg=PA130&amp;amp;dq=stirling+cubic+curves&amp;amp;source=bl&amp;amp;ots=L_hdcIeFub&amp;amp;sig=DsEuTO_IPMTBP7NCK6lMka5MXjI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjK6OX5xavPAhXElx4KHQn4AQcQ6AEIIzAB#v=onepage&amp;amp;q=stirling%20cubic%20curves&amp;amp;f=false&quot; rel=&quot;nofollow&quot;&gt;Guicciardini in Isaac Newton on Mathematical Certainty and Method&lt;/a&gt; remarks that &quot;the reader is not informed on the methods Newton deployed&quot; and gives examples of Newton's claims &quot;with no trace of demonstration&quot;, which Stirling's 1717 commentary fills in. I suppose this explains the gaps. Cayley gives a more detailed and less hindsighted description, as well as comparison to Plücker, in his On Classification of Cubic Curves (1864) reproduced in &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1440044910&quot; rel=&quot;nofollow&quot;&gt;volume 5 of his Collected Mathematical Papers&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The classification is according to the nature of the infinite branches; there are fourteen genera containing together seventy-two species, but four species were added by Stirling in his Linere Tertii Ordinis Newtoniare; sive Illustratio &amp;amp;c (1717), and two more by Murdoch or Cramer&lt;/em&gt; [Cayley's footnote: &lt;em&gt;The two additional species are, I believe, first mentioned in Murdoch's Genesis Curvarum per Umbras (1746), but one of them is there ascribed to Cramer], making in all seventy-eight species. A new classification was made by Plücker in his System der Analytischen Geometrie, 1835; this is likewise according to the nature of the infinite branches, but after his six head divisions, and some subordinate divisions thereof, Plücker establishes the divisions called Groups, which have nothing analogous to them in the Newtonian theory; there are sixty-one groups, and the total number of species is 219.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If you really want to delve into it there is a very detailed &lt;a href=&quot;http://plms.oxfordjournals.org/content/s1-22/1/104.extract&quot; rel=&quot;nofollow&quot;&gt;commentary by Rouse Ball in On Newton's Classification of Cubic Curves (1890)&lt;/a&gt;, who details how &quot;&lt;em&gt;four canonical forms or cases are divided into seven (or six) classes. These classes are subdivided into fourteen (or thirteen) genera, which contain the seventy-eight species, from which however the degenerate forms of a conic and a straight line and of three straight lines are excluded&lt;/em&gt;&quot;, and also tells us that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Cramer, in his Introduction a l'Analyse des Lignes Courbes Algebriques, published at Geneva in 1750, and Euler, in his Analysis Infinitorum, published at Lausanne in 1748, proposed to classify curves solely by reference to the character of their infinitely distant points. They both refused to recognise the various curves formed by the degeneration of an oval into an acnode, or its coalescence with other branches of the curve, as constituting distinct species.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-09-29T18:17:11.807" LastActivityDate="2016-09-29T18:17:11.807" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5227" PostTypeId="2" ParentId="666" CreationDate="2016-09-26T23:19:45.713" Score="9" Body="&lt;p&gt;&lt;p&gt;Given the historical circumstances the German education system might or might not have played a significant role in the formation of these scientists. In his collection of essays entitled &quot;Brocas Brain&quot;, Carl Sagan gives some excellent insight into Einsteins biography: (I think &lt;a href=&quot;https://newrepublic.com/article/117028/world-beckons&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; an accurate reprint of the original piece)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein dropped out of school before gaining qualifications to enter university and was in general critical of the Prussian education system of his time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Browsing through Charles Enz &lt;a href=&quot;https://global.oup.com/academic/product/no-time-to-be-brief-9780198564799?cc=gb&amp;amp;lang=en&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;biography&lt;/a&gt; of Wolfgang Pauli, I got the impression that he had some excellent teachers but didn't need much education as he had mastered the whole physics curriculum soon after entering university. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I'm not very familiar with the biographies of Planck, Schrödinger, Heisenberg &amp;amp; Born. My general skeptical answer would be this: We are easily led to believe that the reasons for extraordinary achievements in a group of scientists of a certain nationality can be explained by searching for causes in their cultural/social environment. While this might evidently be part of the story it is far from trivial to specify causal chains in this domain. We can fall prey to hindsight bias(we always knew how great the german education system was!) and might find that the grouping of these people by nationality/origin might be too arbitrary to deserve a clear-cut explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My spontaneous answer (which needs more research for corroboration) would be something along the lines of: the success of these people has to be explained by their individual situation/talents plus random input from the environment and the right feedback loops(of which education might be just one) Being around the right people at the right time (Grossmann in Einsteins case) might be more crucial than the (educational) culture in general.  &lt;/p&gt;&#xA;" OwnerUserId="3348" LastActivityDate="2016-09-26T23:19:45.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5228" PostTypeId="1" AcceptedAnswerId="5229" CreationDate="2016-09-27T02:50:31.300" Score="4" ViewCount="607" Body="&lt;p&gt;As described &lt;a href=&quot;https://en.wikipedia.org/wiki/Russell%27s_paradox&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt;, we know that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the foundations of mathematics, Russell's paradox (also known as&#xA;  Russell's antinomy), discovered by Bertrand Russell in 1901, showed&#xA;  that some attempted formalizations of the naive set theory created by&#xA;  Georg Cantor led to a contradiction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And &lt;a href=&quot;https://en.wikipedia.org/wiki/Russell%27s_paradox#Formal_presentation&quot; rel=&quot;nofollow&quot;&gt;here&lt;/a&gt; is the formal presentation which lead us &lt;code&gt;NST&lt;/code&gt; is inconsistent. But the inference used by Russel is a bit strange. So, I was wondering, how did he come up with that formal presentation? Do we have any ideas of how it came to be? For example was that built on the result of earlier mathematicians/logicians thinking? Or thinking this way in about axioms, definitions, properties and concepts is regular and usual?&lt;/p&gt;&#xA;" OwnerUserId="2965" LastEditorUserId="55" LastEditDate="2016-09-29T01:29:24.983" LastActivityDate="2023-03-22T05:07:09.220" Title="How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?" Tags="|mathematics|mathematical-logic|set-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5229" PostTypeId="2" ParentId="5228" CreationDate="2016-09-27T08:02:54.917" Score="5" Body="&lt;p&gt;Regarding:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;how did Russell come up with that formal presentation? Do we have any ideas of how it came to be? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;you can see &lt;a href=&quot;http://www.iep.utm.edu/par-russ/&quot; rel=&quot;noreferrer&quot;&gt;Russell's Paradox&lt;/a&gt; and &lt;a href=&quot;http://plato.stanford.edu/entries/russell-paradox/&quot; rel=&quot;noreferrer&quot;&gt;Russell's Paradox&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For details, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Ivor Grattan-Guinness, &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086078900460&quot; rel=&quot;noreferrer&quot;&gt;How Bertrand Russell discovered his paradox&lt;/a&gt; (1978)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Ivor Grattan-Guinness, &lt;a href=&quot;https://www.jstor.org/stable/30226029?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;Bertrand Russell on his paradox and the multiplicative axiom&lt;/a&gt; (1972)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Alejandro Garciadiego, &lt;a href=&quot;https://books.google.it/books?id=Yt4FCAAAQBAJ&amp;amp;pg=PR3&quot; rel=&quot;noreferrer&quot;&gt;Bertrand Russell and the Origins of the Set-theoretic &quot;Paradoxes&quot;&lt;/a&gt; (1992).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;You can see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Jean van Heijenoort (editor), &lt;a href=&quot;https://books.google.it/books?id=v4tBTBlU05sC&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;From Frege to Gödel : A Source Book in Mathematical Logic. 1879-1931&lt;/a&gt;, (1967)and &lt;a href=&quot;https://books.google.it/books?id=v4tBTBlU05sC&amp;amp;pg=PA124&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for the original &quot;formal presentation&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-09-27T13:43:28.160" LastActivityDate="2016-09-27T13:43:28.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5230" PostTypeId="1" AcceptedAnswerId="5232" CreationDate="2016-09-28T02:16:13.513" Score="4" ViewCount="1213" Body="&lt;p&gt;In the paper by A. J. Coleman, &quot;The greatest mathematical paper of all time&quot;&#xA;(Math Intelligencer, 11, no. 3 (1989), 29-39), on page 30 there is a passing remark&#xA;that the &quot;Jordan form is due to Weierstrass&quot;. Can anyone give a reference&#xA;to Weierstrass's publications or to a history source that discusses this? &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-30T00:13:42.430" LastActivityDate="2019-05-30T00:13:42.430" Title="Who discovered the Jordan normal form?" Tags="|mathematics|linear-algebra|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5231" PostTypeId="1" AcceptedAnswerId="5233" CreationDate="2016-09-28T03:07:30.897" Score="5" ViewCount="541" Body="&lt;p&gt;“At the age of 12 to 16 I familiarized myself with the elements of mathematics together with the principles of differential and integral calculus. In doing so I had the good fortune of finding books which were not too particular in their logical rigor, but which made up for this by permitting the main thoughts to stand out clearly and synoptically... I also had the good fortune of getting to know the essential results and methods of the entire field of the natural sciences in an excellent popular exposition, which limited itself almost throughout to qualitative aspects. …a work which I read with breathless attention.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://newrepublic.com/article/117028/world-beckons&quot;&gt;https://newrepublic.com/article/117028/world-beckons&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know what is this book ? Written by who ? Of what subject did it deal with (physics only?) What would be modern day equivalents to it ?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastActivityDate="2016-09-29T04:00:50.167" Title="Einstein's readings of popular sciences as a kid" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5232" PostTypeId="2" ParentId="5230" CreationDate="2016-09-28T04:26:11.667" Score="11" Body="&lt;p&gt;According to the following linked text, Weierstrass defined his equivalent form in a memoir presented to the Berlin Academy in 1868, two years prior to Jordan's Traite being published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://books.google.ca/books?id=W6DwCAAAQBAJ&amp;amp;pg=PA122&amp;amp;lpg=PA122&amp;amp;dq=weierstrass%20form%20jordan%20normal%20form&amp;amp;source=bl&amp;amp;ots=hJRS7Cp643&amp;amp;sig=WXbYcYqeV9-r_BNPuNmGApB61Qg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=weierstrass%20form%20jordan%20normal%20form&amp;amp;f=false&quot;&gt;Bartel L van der Waerden's &lt;em&gt;A History of Algebra&lt;/em&gt;&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On the history of Jordan Normal Forms, see Thomas Hawkins &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF02464978&quot;&gt;&quot;Weierstrass and the Theory of Matrices&quot;&lt;/a&gt; Archive for History of Exact Sciences 17, p 119-163 (1977).  In section four of this paper, Hawkins shows that Weierstrass, in his theory of Elementary Divisors, has already defined a normal form equivalent to that of Jordan.  Weierstrass presented his fundamental memoir &quot;Zur Theorie der bilinearen und quadratischen Formen&quot; to the Berlin Academy in 1868 (Monatsberichte, p. 311-338 = Werke 2, p.19-44) two years before the publication of Jordan's Traite. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="131" LastEditDate="2016-09-28T13:12:16.873" LastActivityDate="2016-09-28T13:12:16.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5233" PostTypeId="2" ParentId="5231" CreationDate="2016-09-29T00:08:23.280" Score="6" Body="&lt;p&gt;From &lt;a href=&quot;http://english4success.ru/Upload/books/1640.pdf&quot; rel=&quot;nofollow&quot;&gt;Walter Isaacson's &lt;em&gt;Einstein&lt;/em&gt;&lt;/a&gt; (page 116 of the linked pdf) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein’s greatest intellectual stimulation came from a poor student who dined with his family once a week. It was an old Jewish custom to take in a needy religious scholar to share the Sabbath meal; the Einsteins modified the tradition by hosting instead a medical student on Thursdays. His name was Max Talmud, and he began his weekly visits when he was 21 and Einstein was 10. .....Talmud brought Einstein science books, including a popular illustrated series called &lt;strong&gt;People’s Books on Natural Science&lt;/strong&gt;, “a work which I read with breathless attention,” said Einstein. The 21  little volumes were written by Aaron Bernstein, who stressed the interrelations between biology and physics, and reported in great detail the experiments being done at the time, especially in Germany. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the opening section of the first volume, Bernstein dealt with the speed of light, a topic that obviously fascinated him.  Indeed, he returned to it repeatedly in the subsequent volumes, including eleven essays on the topic in volume 8.  Judging from the thought experiments Einstein later used in constructing his theory of relativity, Bernstein's books appear to have been influential.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;For example, Bernstein asked readers to imagine being on a speeding train. If a bullet is shot through the window, it would seem to have been shot at an angle because the train would have moved between the time the bullet entered one window and exited the other window. ... (&lt;em&gt;and Bernstein continues with additional subject matter familiar to the Einsteinian way of thinking&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Talmud also provided non-popular reading material to the young Einstein :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Talmud also helped Einstein explore the wonders of mathematics by giving him a textbook on geometry two years before he was scheduled to learn that subject in school. When Talmud arrived each Thursday, Einstein delighted in showing him the problems he had solved that week. Initially, Talmud was able to help him, but he was soon surpassed by his pupil. “After a short time, a few months, he had worked through the whole book,” Talmud recalled. “Soon the flight of his mathematical genius was so high that I could no longer follow.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but this is outside the scope of your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&#xA;Note that Isaacson, or the translator, misnames the series as &quot;People's Books on Natural Science&quot;.  The correct name is &quot;Popular Books on Natural Science.  For practical use in every household, for readers of all classes&quot;.  Here is a &lt;a href=&quot;https://archive.org/details/popularbooksonna00bernrich&quot; rel=&quot;nofollow&quot;&gt;link to a PDF&lt;/a&gt;.  Click on the text to turn the page.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-09-29T04:00:50.167" LastActivityDate="2016-09-29T04:00:50.167" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5234" PostTypeId="2" ParentId="5228" CreationDate="2016-09-29T01:27:33.047" Score="17" Body="&lt;p&gt;Russell was not the first to discover &amp;quot;his&amp;quot; paradox. By June 1901 when he arrived at it (it was not published until the first edition of Principia in 1903), it was already known for a while to the Hilbert's circle at Göttingen. Russell did not belong to that circle, so his rediscovery was independent, but he was familiar with Schröder's Algebra of Logic (1890), the precursor to Principia, where Schröder argued that admitting &amp;quot;all objects of thought&amp;quot; into a class leads to contradictions. &lt;a href=&quot;http://rancho.pancho.pagesperso-orange.fr/Apprenti.htm&quot; rel=&quot;noreferrer&quot;&gt;Here is Ortiz Hill&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In a November 7, 1903 letter, Hilbert told Frege that Russell's antinomy was already known to them in Göttingen. Hilbert added that he believed that Ernst Zermelo had found it three or four years earlier after having learned of other, even more convincing, contradictions from Hilbert himself as many as four or five years before. Hilbert further commented that the idea that &amp;quot;a concept is already there if one can state of any object whether or not it falls under it&amp;quot; did not seem adequate to him. What would be decisive, he states, &amp;quot;is the recognition that the axioms that define the concept are free from contradiction.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Concrete evidence corroborating Zermelo's finding of the paradox is to be found in a note that he sent to Husserl in April 1902, in which Zermelo conveyed his proof... In Zermelo's opinion... Schröder had been basically right, but his reasoning had been faulty. According to Zermelo's argument as recorded by Husserl: given a set M which contains each of its sub-sets m, m'… as elements, and a set M0 which is the set of all sub-sets of M, which do not contain themselves as elements, it can then be shown that M0 both does and does not contain itself.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The principle Hilbert rejected (roughly, that every non-vague property defines a set of objects having it), now known as the axiom of (unrestricted) comprehension, was broadly used by 19th century logicians and mathematicians. It appears explicitly as the &amp;quot;third generation principle&amp;quot; of ordinals in Cantor's mathematico-philosophical opus Foundations of the General Theory of Sets (1883), and as the Basic Law V in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic&quot; rel=&quot;noreferrer&quot;&gt;Frege's Foundations of Arithmetic (1884)&lt;/a&gt;, another of Russell's points of departure. The unraveling that apparently led both Russell and Zermelo to the paradox started with the Schröder's monograph, and the Cantor's diagonal argument published in 1891. Russell commented that it was studying Cantor's theories that led him to the antinomy that ended the &amp;quot;&lt;em&gt;logical honeymoon&lt;/em&gt;&amp;quot; of the early work on Principia. In the diagonal argument a list of &amp;quot;all&amp;quot; sequences is first &amp;quot;compiled&amp;quot;, and then the list itself is manipulated to produce a sequence that does not belong to it, while it is supposed to.&lt;/p&gt;&#xA;&lt;p&gt;By 1895 Cantor already realized that his argument implies another paradox, now called after Burali-Forti who published it in 1897, that of the ordinal of all ordinals both having and not having a successor (or more precisely, its version for the alefs). As a result, according to &lt;a href=&quot;https://books.google.com/books/about/Gesammelte_Abhandlungen_mathematischen_u.html?id=PAJVAAAAYAAJ&quot; rel=&quot;noreferrer&quot;&gt;Zermelo's remarks in Cantor's Collected Works&lt;/a&gt;, in a far cry from the broad sweep of 1883 Cantor excluded  non-countable ordinals from his work:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The reason for this omission was apparently, first, the still absent proof that every cardinality is an alef, and second - that Cantor already had some skeptical doubts concerning &amp;quot;all&amp;quot; ordinal or cardinal numbers based on the already known to him &amp;quot;antinomy of Burali-Forti&amp;quot;...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The difference between the &amp;quot;Burali-Forti&amp;quot; paradox and the &amp;quot;Russell&amp;quot;'s, is that the latter does not require complicated notions like ordinals or cardinality. But they both rely on the unrestricted comprehension for definition, and arrive at the contradiction through the same kind of self-referential exploit. Russell's &lt;a href=&quot;https://en.wikipedia.org/wiki/Barber_paradox&quot; rel=&quot;noreferrer&quot;&gt;paradox of the barber&lt;/a&gt; (who shaves those and only those who do not shave themselves) stands in the same relation to &amp;quot;his&amp;quot; paradox, as the diagonal argument does to &amp;quot;Burali-Forti&amp;quot;'s. In the former there are presuppositions that can be rejected (existence of the barber, countability of the continuum), thus disarming the paradox, but in the latter the way out is blocked by the axiom of comprehension.&lt;/p&gt;&#xA;&lt;p&gt;One could say that the paradoxes of self-reference were &amp;quot;in the air&amp;quot;, and the particular type of reasoning Zermelo and Russell distilled was &amp;quot;in the air&amp;quot; too. But, as Ortiz Hill notes, Russell was &amp;quot;&lt;em&gt;the first to publicize Frege's errors and to bring the point home to Frege himself&lt;/em&gt;&amp;quot;, indeed he devised the paradox of the barber to popularize the issue in non-technical terms.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-10-02T00:11:28.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5235" PostTypeId="2" ParentId="5223" CreationDate="2016-09-29T19:45:29.843" Score="4" Body="&lt;p&gt;No. Very few people had written computer programs at the time of his death, 1955, and most had backgrounds in numerical analysis, and computational techniques. Programs were written in machine code in those days, or a few newer machines had an assembly language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;John von Neumann is a much better candidate for computer programming.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2016-09-29T19:45:29.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5236" PostTypeId="1" AcceptedAnswerId="5237" CreationDate="2016-09-30T03:47:48.610" Score="5" ViewCount="593" Body="&lt;p&gt;I'm wondering who is the first person in the history who came up with an idea to integrate Newton's $F=ma$ to obtain the law of mechanical energy conservation? And When did it happen?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also I have the same questions for the law of momentum conservation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that these laws were known by experience and intuitively formulated before someone realized that these are integrated from $F=ma$. But I want to know who found the relationship between the laws and $F=ma$.&lt;/p&gt;&#xA;" OwnerUserId="4772" LastEditorUserId="55" LastEditDate="2016-09-30T16:19:38.103" LastActivityDate="2016-09-30T17:19:00.253" Title="Who was first to integrate Newton's equations of motion to derive the conservation laws for mechanical energy and momentum?" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5237" PostTypeId="2" ParentId="5236" CreationDate="2016-09-30T16:18:32.703" Score="4" Body="&lt;p&gt;It is not enough to integrate the equations of motion in order to obtain the conservation laws. Newton derived conservation of momentum components in the absence of external forces from the third law of motion &lt;a href=&quot;https://archive.org/stream/100878576#page/86/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;already in Principia&lt;/a&gt;. The idea to integrate forces to obtain changes in momentum and kinetic energy (called &quot;vis viva&quot; at the time) first appears in Boscovich's De Viribus Vivis (Rome, 1745), a fifty page contribution in Latin to the then raging vis viva controversy, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt; for details. According &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://nature.berkeley.edu/departments/espm/env-hist/articles/1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Iltis's d'Alembert and the Vis Viva Controversy&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Employing both the ancient scholastic categories and the new mathematical&#xA;  methods of his time, Boscovich discussed the graphical representation&#xA;  of a pressure applied through a time and a force applied over a&#xA;  distance... Without taking a position on the definition of force, Boscovich measured the velocity acquired as a ratio composed of the pressure and its duration. A geometrical image is generated by the line representing the pressure&#xA;  with time as the second dimension of the diagram. The pressure is thus a&#xA;  function of time. Interpreting this in modern terminology, the momentum&#xA;  mv would be represented as the integral of these instantaneous pressures&#xA;  (or impulses) over a time, or $\int mdv = \int pdt$.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Boscovich suggested that, if the time coordinate is replaced by the space traversed and the pressure coordinate by the force which at any instant produces the velocity proportional to it, a second aspect of the phenomenon is represented... The new term 'force' must be interpreted as an entity proportional to the velocity engendered at any instant. If the pressure coordinate is changed to the force and the time coordinate to the space then the new geometrical image producing the velocity would be represented in modern notation as $\int Fds$. We would then interpret vis viva as $\int mvdv = \int Fds$ (where $ds = vdt$). Boscovich does not bring the mass into this analysis.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;D'Alembert included similar interpretations into the second edition of his Traite de Dynamique (1758), a milestone monograph on mechanics after Newton's Principia, which was widely read. It is unclear if he was aware of Boscovich's work. However, it was only &lt;a href=&quot;https://archive.org/details/mcaniqueanalyt02lagr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange in Mecanique Analytique (1788)&lt;/a&gt;, who  introduced the notion of potential energy (the name was proposed by Rankine only in 1853), and showed that for potential forces the sum of kinetic and potential energies is an &quot;integral of motion&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-09-30T17:19:00.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5238" PostTypeId="1" CreationDate="2016-10-01T15:10:07.663" Score="1" ViewCount="175" Body="&lt;p&gt;Who is the author of the term &quot;Coulomb logarithm&quot;? In fact, Coulomb logarithm was computed by Langmuir in his paper of 1928 where the term &quot;plasma&quot; was introduced into physics, but the term &quot;Coulomb logarithm&quot; seemed to appear later. The earliest reference I found is the paper Relativistic kinetic equation by Belyaev &amp;amp; Budker (1956).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;UPDATE: I found the term &quot;Coulomn logarithm&quot; in several papers by S.I.Braginsky, V.I. Kogan, and G.I.Budker dated as early as 1951. They are published in a volume of declassified works issued in 1958. &lt;/p&gt;&#xA;" OwnerUserId="3407" LastEditorUserId="3407" LastEditDate="2016-10-04T14:25:46.313" LastActivityDate="2016-10-04T14:25:46.313" Title="Who invented the term “Coulomb logarithm”?" Tags="|physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5240" PostTypeId="1" CreationDate="2016-10-02T09:50:52.023" Score="6" ViewCount="380" Body="&lt;p&gt;Looking in the past we can easily see that it was Europe who made most of the discoveries in STEM but today USA triumphs in this area. So, how did this shift took place and what were the reasons for the emergence of USA as the pioneer in these fields leaving behind Europe and other countries?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, is there a possibility of similar shift from USA to other country say China and India?&lt;/p&gt;&#xA;" OwnerUserId="2012" LastEditorUserId="2012" LastEditDate="2020-07-20T02:21:35.883" LastActivityDate="2020-07-20T05:12:31.877" Title="How did USA become world leader in STEM?" Tags="|discoveries|depiction-of-scientists|academic-awards|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5241" PostTypeId="1" CreationDate="2016-10-02T11:18:05.460" Score="7" ViewCount="409" Body="&lt;p&gt;Jost's recent &lt;a href=&quot;http://link.springer.com/book/10.1007%2F978-3-319-26042-6&quot;&gt;book on Riemann's famous 1854 essay&lt;/a&gt; claims in the introduction that Bertrand Russell misjudged Riemann's work.  However, Jost does not elaborate.  The index at the end of the book does not even contain an entry for Russell.  What is known about the reception of Riemann by Russell?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="55" LastEditDate="2016-10-02T21:01:26.933" LastActivityDate="2016-10-06T17:51:20.010" Title="How was Riemann's essay on the foundations of geometry misjudged by Russell?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5243" PostTypeId="2" ParentId="5102" CreationDate="2016-10-02T12:48:46.180" Score="5" Body="&lt;p&gt;Dissertations in The Netherlands were traditionally written in Latin, until the second half of the 19th century. At Leiden University the last physics thesis in Latin is from 1854, &lt;A HREF=&quot;http://ilorentz.org/history/proefschriften/sources/Bosscha_1854.pdf&quot; rel=&quot;noreferrer&quot;&gt;De galvanometra differentiali&lt;/A&gt; by &lt;A HREF=&quot;https://en.wikipedia.org/wiki/Johannes_Bosscha&quot; rel=&quot;noreferrer&quot;&gt;Johannes Bosscha.&lt;/A&gt;&lt;/p&gt;&#xA;" OwnerUserId="1697" LastActivityDate="2016-10-02T12:48:46.180" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5244" PostTypeId="2" ParentId="5240" CreationDate="2016-10-02T19:36:56.587" Score="1" Body="&lt;p&gt;A big boost in activity for WWII, rolling over into massive government support in the cold war and then the concentration of US venture capital in Silicon Valley today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not necessarily clear that it is still as much a leader today or tomorrow. Don't have the statistics at hand but it may not be at the forefront of funding / numbers of students / number of papers etc today.&lt;/p&gt;&#xA;" OwnerUserId="2838" LastActivityDate="2016-10-02T19:36:56.587" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5245" PostTypeId="2" ParentId="5241" CreationDate="2016-10-02T21:00:43.887" Score="9" Body="&lt;p&gt;Russell wrote a historico-philosophical &lt;a href=&quot;https://archive.org/details/117723764&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essay on the Foundations of Geometry (1897)&lt;/a&gt;, which &lt;a href=&quot;http://download.springer.com/static/pdf/197/chp%253A10.1007%252F978-3-319-26042-6_5.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-3-319-26042-6_5&amp;amp;token2=exp=1475439092~acl=%2Fstatic%2Fpdf%2F197%2Fchp%25253A10.1007%25252F978-3-319-26042-6_5.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Fchapter%252F10.1007%252F978-3-319-26042-6_5*~hmac=a7a539ad60183438d52915eaa86e6727279543068a49248af252cda83b0f429c&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jost cites on p.127&lt;/a&gt; and which is freely available on Gutenberg. Russell's reception of Riemann is already clear from the section titles concerning him:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;60&quot;&gt;&#xA;  &lt;li&gt;&lt;em&gt;Riemann regarded space as a particular kind of manifold, i.e.&#xA;  wholly quantitatively&lt;/em&gt; &lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;He therefore unduly neglected the qualitative adjectives of&#xA;  space&lt;/em&gt; &lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;His philosophy rests on a vicious disjunction&lt;/em&gt; &lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;His definition of a manifold is obscure&lt;/em&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;And his definition of measurement applies only to space&lt;/em&gt; &lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;Though mathematically invaluable, his view of space as a&#xA;  manifold is philosophically misleading&lt;/em&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Aside from young Russell being a little too hard on Riemann, what this makes clear is that his qualms were more philosophical than mathematical. It helps to remember that Riemann was a Kantian, he explicitly names Kant's successor at Königsberg, Herbart, as an influence in his lecture, the only other named influence is Gauss, see more in &lt;a href=&quot;https://hsm.stackexchange.com/questions/657/which-school-of-philosophy-motivated-thinking-about-spaces-of-higher-dimension/673#673&quot;&gt;Which school of philosophy motivated thinking about spaces of higher dimension?&lt;/a&gt; And, &lt;a href=&quot;http://www.iep.utm.edu/analytic/#H1&quot; rel=&quot;nofollow noreferrer&quot;&gt;as we know from Russell&lt;/a&gt;, &quot;&lt;em&gt;it was towards the end of 1898 that Moore and I rebelled against both Kant and Hegel&quot;&lt;/em&gt;, to found analytic philosophy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Riemann's &quot;&lt;em&gt;vicious disjunction&lt;/em&gt;&quot; is that &quot;&lt;em&gt;either the axioms must be consequences of general conceptions of magnitude, he thinks, or else they can only be proved by experience&lt;/em&gt;&quot;. In Russell's opinion, we must ask instead &quot;&lt;em&gt;What axioms, i.e. what adjectives of space, must be presupposed, in order that quantitative comparison of the parts of space may be possible at all?&lt;/em&gt;&quot; And not asking that, to Russell, was Riemann's cardinal sin:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;His philosophy is chiefly vitiated, to my mind, by this fallacy, and by the uncritical assumption that a metrical coordinate systern can be set up independently of any axioms as to space-measurement. Riemann has failed to observe, what I have endeavoured to prove in the next chapter, that, unless space had a strictly constant measure of curvature, Geometry would become impossible; also that the absence of constant measure of curvature involves absolute position, which is an absurdity.&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is most ironic about this is that the soon to be anti-Kantian Russell is criticizing a Kantian Riemann for... not being Kantian enough. Namely, overlooking the constitutive  principles that have to be adopted a priori for empirical measurements to become possible. The reason Russell puts such a premium on constant curvature is that only with that is transitive metric-preserving action possible, and hence measurement in his 1897 opinion. And that of course is a replication of Kant's similar thesis about the Euclidean geometry. Einstein will dispel such narrow construal of physical geometry soon enough, and credit Riemann's study of the foundations of geometry &quot;&lt;em&gt;in an even more profound way&lt;/em&gt;&quot; for providing a language of general covariance in 1912, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/5188/how-was-einstein-led-to-make-a-contact-with-differential-geometry-for-his-theory/5195#5195&quot;&gt;How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jost recommneded commentary on Russell on Riemann is &lt;a href=&quot;https://books.google.com/books?id=EcLrCAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toretti's Philosophy of Geometry from Riemann to Poincaré&lt;/a&gt;, see also this year's &lt;a href=&quot;https://books.google.com/books?id=dSPkDAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Space, Number, and Geometry from Helmholtz to Cassirer by Biagioli&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-10-06T17:51:20.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5246" PostTypeId="2" ParentId="5240" CreationDate="2016-10-02T23:01:28.513" Score="4" Body="&lt;p&gt;interesting question for historians,  a bit like &quot;why did Rome fall?&quot;  my guess:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;public money.  for many years the us govt  has funded scientific research to the tune of many billions of dollars &lt;em&gt;per year&lt;/em&gt;.  that's very attractive to scientists. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;politics. if you submit a crazy or only subversive proposal in the us the worst that can happen is you end up driving a cab.  you will not be arrested, declared mentally ill, and forced into a psychiatric ward or worse.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;law and econ..  it's relatively easy to start a company in the us, legally, and get funding.  the legal system is very stable and relatively non-corrupt.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;as for the future, who knows?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;one other historical factor:  immigration.  US STEM benefitted enormously from WWII.  Einstein, Gödel, von Braun, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;one more point: the cold war.  the 50s and 60s saw a huge govt investment in universities and tech generally, mostly military-driven.  today it's easy to overlook just how serious the cold war was back in the day.  Think &quot;Cuban missile crisis&quot;.  The space program that put a man on the moon was largely driven by cold war fears of Soviet technical domination.&lt;/p&gt;&#xA;" OwnerUserId="4062" LastEditorUserId="4062" LastEditDate="2016-10-05T20:47:12.587" LastActivityDate="2016-10-05T20:47:12.587" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5247" PostTypeId="1" AcceptedAnswerId="5249" CreationDate="2016-10-03T19:30:22.443" Score="4" ViewCount="210" Body="&lt;p&gt;There are numerous online sources giving biographical sketches of Ramsey's life; E.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Frank_P._Ramsey&quot; rel=&quot;nofollow&quot;&gt;Wikipedia&lt;/a&gt;, &lt;a href=&quot;https://www.repository.cam.ac.uk/bitstream/handle/1810/3484/RamseyText.html?sequence=5&quot; rel=&quot;nofollow&quot;&gt;Cambridge respository&lt;/a&gt;, &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Ramsey.html&quot; rel=&quot;nofollow&quot;&gt;St. Andrews MacTutor&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We learn that Ramsey died just before his 27th birthday in January of 1930, almost seven years after receiving his bachelor's degree in mathematics from Trinity, Cambridge.  No other degrees are cited as being conferred by Ramsey.  However, Ramsey seems to have been employed by King's College, Cambridge in various academic capacities - fellow and lecturer in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summarising :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1923 confers bachelor's degree in maths at Cambridge.&lt;/li&gt;&#xA;&lt;li&gt;1924 becomes a fellow of Kings College with the support of Keynes.&lt;/li&gt;&#xA;&lt;li&gt;1926 becomes a lecturer in maths at Kings College.&lt;/li&gt;&#xA;&lt;li&gt;1929 becomes Wittgenstein's &quot;nominal&quot; supervisor at Cambridge.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;He could not have done his graduate studies in 1924 since he was in Vienna undergoing psychoanalysis and getting chummy with Wittgenstein. None of the online sources I have consulted mention any graduate degree.  Certainly from a modern perspective it is hard to imagine someone becoming a university lecturer at Cambridge on the back of an undergraduate degree, even in the case of a senior Wrangler graduating with high honours. In the years between his bachelor's degree and his death he produced a number of papers on a wide variety of subjects - maths, philosophy, psychology, and economics - so he appears to have been busy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;:  Did Ramsey confer a graduate degree, and if so in what subject and what was the subject matter of his thesis?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="4251" LastEditDate="2022-09-23T05:55:55.643" LastActivityDate="2022-09-23T05:55:55.643" Title="Did F. P. Ramsey complete a graduate degree?" Tags="|mathematicians|biographical-details|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5248" PostTypeId="2" ParentId="5247" CreationDate="2016-10-03T20:34:14.117" Score="2" Body="&lt;p&gt;In Cambridge and Oxford a BA degree can be automatically upgraded to an MA after a short interval. There is a long tradition at these two &quot;ancient&quot; universities of appointing holders of no higher degree to senior teaching positions.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2016-10-03T20:34:14.117" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5249" PostTypeId="2" ParentId="5247" CreationDate="2016-10-04T11:28:11.917" Score="4" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nils-Eric Sahlin, &lt;a href=&quot;https://books.google.it/books?id=_QwGuE2UD9IC&amp;amp;pg=PA222&quot; rel=&quot;nofollow&quot;&gt;The Philosophy of F.P. Ramsey&lt;/a&gt; (1990), &lt;strong&gt;Ch.10 Biographical glimpses&lt;/strong&gt;, page 221-on :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;taking a degree in mathematics at Trinity College, Cambridge, in the summer of 1923. He got firstclass honours and became the most outstanding mathematics&#xA;  graduate at Cambridge. Ramsey never went on to write a Ph.D. thesis and I do not know whether the idea had even occurred to him.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In 1921 Ramsey became acquainted with J. M. Keynes through his friend Richard Braithwaite and thanks to Keynes's influence Ramsey became a Fellow of King's in 1924 at the age of only 21. Two years later he was made a lecturer in mathematics at Cambridge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-10-04T11:28:11.917" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5250" PostTypeId="2" ParentId="3012" CreationDate="2016-10-04T15:31:03.897" Score="2" Body="&lt;p&gt;I have written a report in question, though it turned out to be more on the history side than on the philosophy side, listing the important technological advances and breakthroughs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best introduction to the history of lithium-ion batteries is &lt;a href=&quot;https://en.wikipedia.org/wiki/Lithium-ion_battery#History&quot; rel=&quot;nofollow&quot;&gt;the corresponding section of the Wikipedia article&lt;/a&gt;, which gives a good timeline of achievements. Also, &lt;a href=&quot;http://www.nature.com/nature/journal/v407/n6803/full/407496a0.html&quot; rel=&quot;nofollow&quot;&gt;this &lt;em&gt;Nature&lt;/em&gt; article&lt;/a&gt; is important because it is the first mention of nanostructured materials for lithium-ion batteries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My report is &lt;a href=&quot;https://svavil.wordpress.com/2016/03/26/technological-advances-in-lithium-ion-battery-research/&quot; rel=&quot;nofollow&quot;&gt;posted here&lt;/a&gt; for anyone interested.&lt;/p&gt;&#xA;" OwnerUserId="3166" LastActivityDate="2016-10-04T15:31:03.897" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5252" PostTypeId="1" AcceptedAnswerId="5253" CreationDate="2016-10-04T23:47:40.603" Score="0" ViewCount="1458" Body="&lt;p&gt;In thermodynamics, one primarily studies systems that are in so-called &lt;em&gt;&quot;thermal equilibrium&quot;&lt;/em&gt;, a state in which certain prescribed &lt;em&gt;&quot;thermodynamic&quot;&lt;/em&gt; quantities such as temperature, heat, chemical potential, etc. are considered static in time. In fact, most modern textbooks admit that the study of systems &lt;strong&gt;not&lt;/strong&gt; in thermal equilibrium is extremely complicated, and is the primary focus of fields such as nonlinear dynamics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With this in mind, why isn't &lt;em&gt;thermodynamics&lt;/em&gt; called &lt;em&gt;thermostatics&lt;/em&gt;? If this were done, then we would have a proper word for the study of systems that are out of thermal equilibrium, thermodynamics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that, historically, thermodynamics was the study of heat and engines and such, so I can see where it got its name from. However, I don't see why it hasn't changed its name since then. Thermodynamics as is currently taught has been around for over 100 years!&lt;/p&gt;&#xA;" OwnerUserId="238" LastActivityDate="2024-03-17T00:28:28.017" Title="Why isn't &quot;thermodynamics&quot; replaced with &quot;thermostatics&quot;?" Tags="|physics|thermodynamics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5253" PostTypeId="2" ParentId="5252" CreationDate="2016-10-05T00:36:26.123" Score="5" Body="&lt;p&gt;Thermodynamics really studies CHANGES of states, not just the states themselves.&#xA;These changes are slow. The technical term is &quot;adiabatic&quot;. It started with&#xA;Carnot cycle, describing the action of a heat engine. A heat engine cycle passes through several different states.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-10-05T00:36:26.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5255" PostTypeId="2" ParentId="140" CreationDate="2016-10-05T15:13:05.463" Score="6" Body="&lt;p&gt;This question &lt;a href=&quot;https://economics.stackexchange.com/questions/13644/why-in-supply-and-demand-curves-does-price-go-on-the-y-axis/13649?noredirect=1#comment21505_13649&quot;&gt;cropped up on Economics.SE&lt;/a&gt;, where I posted this answer:&lt;/p&gt;&#xA;&lt;p&gt;This objection never made too much sense to me. In the standard model of perfect competition, firms take the price as given and respond by choosing their quantity. So you have a model in which a bunch of actors choose quantity and the market price emerges as a consequence of all of those decisions. This makes it sound awfully like price is the &amp;quot;dependent&amp;quot; variable, which by convention is always placed on the vertical access.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, this seems to be how Alfred Marshall (who originated the modern form of the Demand-Supply diagram) thought about things. Here's a quote from &lt;em&gt;An Introduction to Postitive Economics&lt;/em&gt;, Seventh ed. by Richard G. Lipsey (as quoted &lt;a href=&quot;https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Humanities/2009_January_24#why_does_my_economics_book_put_the_price_as_a_Y_axis_when_the_quantity_is_a_function_of_it.3F&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Readers trained in other disciplines often wonder why economists plot demand curves with price on the vertical axis. The normal convention is to put the independent variable on the X axis and the dependent variable on the Y axis. This convention calls for price to be plotted on the horizontal axis and quantity on the vertical axis.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The axis reversal - now enshrined by nearly a century of usage - arose as follows. The analysis of the competitive market that we use today stems from Leon Walras, in whose theory quantity was the dependent variable. Graphical analysis in economics, however, was popularized by Alfred Marshall, in whose theory price was the dependent variable. Economists continue to use Walras' theory and Marshall's graphical representation and thus draw the diagram with the independent and dependent variables reversed - to the everlasting confusion of readers trained in other disciplines. In virtually every other graph in economics the axes are labelled conventionally, with the dependent variable on the vertical axis.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="291" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-10-05T15:13:05.463" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5256" PostTypeId="1" AcceptedAnswerId="5257" CreationDate="2016-10-05T18:13:02.527" Score="5" ViewCount="154" Body="&lt;p&gt;In his book, &lt;em&gt;Category Theory&lt;/em&gt;&lt;sup&gt;[1]&lt;/sup&gt;, Awodey writes the following about it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Next, we are going to see the categorical definition of a product of two objects in a category. This was first given by Mac Lane in 1950, and it is probably the earliest example of category theory being used to define a fundamental mathematical notion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the last editions of &lt;em&gt;Algebra&lt;/em&gt;&lt;sup&gt;[2]&lt;/sup&gt; and &lt;em&gt;Categories for the Working Mathematician&lt;/em&gt;&lt;sup&gt;[3]&lt;/sup&gt;, the notion of product in a category is presented (of course). I was wondering how did MacLane first arrive at and where did he publish this notion?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;[1] Awodey, Steve: &lt;em&gt;Category Theory&lt;/em&gt;. Oxford University Press, 2010.&lt;br /&gt;&#xA;[2] Mac Lane, Saunders and Birkhoff, Garrett: &lt;em&gt;Algebra&lt;/em&gt;. AMS Chealsea Publishing, 1999.&lt;br /&gt;&#xA;[3] Mac Lane, Saunders: &lt;em&gt;Categories for the Working Mathematician&lt;/em&gt;. Springer, 1998.&lt;/p&gt;&#xA;" OwnerUserId="4776" LastEditorUserId="4251" LastEditDate="2021-09-08T10:56:38.187" LastActivityDate="2021-09-08T10:56:38.187" Title="Where did the notion of the product in a category first appear?" Tags="|mathematics|abstract-algebra|category-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5257" PostTypeId="2" ParentId="5256" CreationDate="2016-10-06T00:11:33.433" Score="4" Body="&lt;p&gt;It appears in &lt;a href=&quot;http://www.ams.org/journals/bull/1950-56-06/S0002-9904-1950-09427-0&quot; rel=&quot;nofollow&quot;&gt;MacLane's 1950 paper Duality for Groups&lt;/a&gt;, published in the Bulletin of AMS. Of course, he is dealing specifically with the category of groups, but the definition is categorical. Section 3, titled &quot;Free  products  and  direct  products&quot; starts with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Let $A\times B$ be  the  direct  (or Cartesian) product of the groups &#xA;  $A$ and  $B$,  defined  as  the  group  of pairs  $(a, b)$ for $a\in A$ and $b\in B$. Let $\alpha$ and $\beta$ denote the natural homomorphisms $\alpha(a, b)=a$, $\beta(a, b)=b$ of  the  direct  product  onto  its  respective  factors.  The  direct  product  may  then  be  described  conceptually in  terms of $\alpha$ and $\beta$ and the diagram&lt;/em&gt; [the now familiar one, labeled (3.1)] &lt;em&gt;as follows. Given any group $C$, and any homomorphisms $\alpha'$ and $\beta'$ of&#xA;  $C$ into $A$ and $B$ respectively, there exists one and only one homomorphism&#xA;  $\gamma: C\to A$ with $\alpha\gamma=\alpha'$ and $\beta\gamma=\beta'$...&#xA;  This property of the diagram (3.1) determines the direct product $A\times B$ and its mappings $\alpha$ and $\beta$ up to an isomorphism; hence it may serve as a definition of the direct product.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similar conceptualization of the free product is given next. However, the idea already appears two years earlier in &lt;a href=&quot;http://www.pnas.org/content/34/6/263.citation&quot; rel=&quot;nofollow&quot;&gt;MacLane's 1948 note Groups, Categories and Duality&lt;/a&gt; published in Proceedings of the National Academy of Sciences of the U. S. A. According to Hall's review on MathSciNet:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The direct product and the free product of two groups are defined abstractly in terms of homomorphisms, the two definitions being formally deducible one from the other by applying the following &quot;duality rules'': invert the direction of each homomorphism, invert the order of all products of homomorphisms, interchange homomorphisms onto with isomorphisms into.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-10-06T00:11:33.433" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5258" PostTypeId="1" CreationDate="2016-10-06T04:34:32.010" Score="23" ViewCount="17178" Body="&lt;p&gt;I heard somewhere that it was actually a mistake in translation. What's the correct story?&lt;/p&gt;&#xA;" OwnerDisplayName="user4795" LastEditorUserId="26" LastEditDate="2016-10-06T12:43:45.247" LastActivityDate="2021-12-02T17:34:02.513" Title="What is the etymology behind sine, cosine, tangent, etc.?" Tags="|mathematics|terminology|geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5259" PostTypeId="2" ParentId="5258" CreationDate="2016-10-06T06:12:20.327" Score="16" Body="&lt;p&gt;For &lt;a href=&quot;https://en.wikipedia.org/wiki/Sine&quot;&gt;sinus&lt;/a&gt;, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Victor Katz, &lt;a href=&quot;https://books.google.it/books?id=h50fAQAAIAAJ&quot;&gt;A History of Mathematics&lt;/a&gt; (3rd edition, 2008), apge 253 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The English word “sine” comes from a series of mistranslations of the Sanskrit &lt;em&gt;jya-ardha&lt;/em&gt; (chord-half). Aryabhata frequently abbreviated this term to &lt;em&gt;jya&lt;/em&gt; or its synonym &lt;em&gt;jiva&lt;/em&gt;. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise&#xA;  meaningless Arabicword &lt;em&gt;jiba&lt;/em&gt;. But since Arabic is written without vowels, later writers interpreted the consonants &lt;em&gt;jb&lt;/em&gt; as &lt;em&gt;jaib&lt;/em&gt;, which means bosom or breast. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word &lt;em&gt;sinus&lt;/em&gt;, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf. This Latin word has now become our English “sine.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For &lt;a href=&quot;https://en.wikipedia.org/wiki/Trigonometric_functions#Etymology&quot;&gt;cosine&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The prefix &quot;co-&quot; (in &quot;cosine&quot;, &quot;cotangent&quot;, &quot;cosecant&quot;) is found in &lt;a href=&quot;https://en.wikipedia.org/wiki/Edmund_Gunter&quot;&gt;Edmund Gunter&lt;/a&gt;'s &lt;a href=&quot;https://hal.inria.fr/inria-00543938/document&quot;&gt;Canon triangulorum&lt;/a&gt; (1620), which defines the &lt;em&gt;cosinus&lt;/em&gt; as an abbreviation for the &lt;em&gt;sinus complementi&lt;/em&gt; (sine of the complementary angle) and proceeds to define the &lt;em&gt;cotangens&lt;/em&gt; similarly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;co.sinus&lt;/em&gt; was suggested by the English mathematician Edmund Gunter (1581-1626) in his &lt;em&gt;Canon triangulorum, sive, Tabulae sinuum et tangentium artificialium ad radium 100000.0000. &amp;amp; ad scrupula prima quadrantis&lt;/em&gt; (1620). According to Smith (page 619), the term was &quot;soon modified by &lt;a href=&quot;https://en.wikisource.org/wiki/Newton,_John_(1622-1678)_(DNB00)&quot;&gt;John Newton&lt;/a&gt; (&lt;a href=&quot;https://books.google.it/books?id=xFlRAAAAcAAJ&amp;amp;pg=PA10&quot;&gt;Trigonometria Britanica&lt;/a&gt;, 1658) into &lt;em&gt;co-sines&lt;/em&gt;, a word which was thereafter received with general favor.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-10-06T13:56:16.283" LastActivityDate="2016-10-06T13:56:16.283" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5260" PostTypeId="2" ParentId="140" CreationDate="2016-10-06T06:17:28.487" Score="33" Body="&lt;p&gt;We have price on the vertical axis &lt;strong&gt;because that's how Alfred Marshall (&lt;a href=&quot;http://www.econlib.org/library/Marshall/marP.html&quot; rel=&quot;noreferrer&quot;&gt;1890&lt;/a&gt;) drew his graphs in &lt;em&gt;Principles of Economics&lt;/em&gt;. For better or worse, &lt;em&gt;Principles&lt;/em&gt; was hugely influential. And so the present-day convention is Marshall's convention.&lt;/strong&gt; As Humphrey (&lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.362.5766&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;1992&lt;/a&gt;) writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Marshallian cross diagram bears Marshall's name because he gave it&#xA;its most complete, systematic, and persuasive statement, not because&#xA;he was the first to invent it. His account was definitive, not&#xA;pathbreaking. For this he received — and deserved — credit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The above answer is not completely satisfactory. It merely pushes the question back one level: &lt;strong&gt;Why did Marshall put price on the vertical axis?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Short answer&lt;/em&gt;: Marshall first introduced a demand curve in &lt;a href=&quot;https://drive.google.com/open?id=0B6d27XgslBsBX2pDR1JGU0E3Wjg&quot; rel=&quot;noreferrer&quot;&gt;1879&lt;/a&gt;. There, &lt;strong&gt;he thought of quantity as the independent variable, with prices adjusting to clear the market.&lt;/strong&gt; It was thus perfectly logical for him to have quantity on the horizontal axis.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Long answer&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Marshall (1879) was not the first to draw demand or supply curves. According to Humphrey, we have:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Augustin Cournot (&lt;a href=&quot;https://books.google.com.sg/books?id=K2VHAAAAYAAJ&amp;amp;pg=PA201&quot; rel=&quot;noreferrer&quot;&gt;1838&lt;/a&gt;). &lt;strong&gt;Price on horizontal axis.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Karl Rau (&lt;a href=&quot;https://archive.org/stream/grundstzedervol00raugoog#page/n621/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;1841&lt;/a&gt;).  &lt;strong&gt;Price on vertical axis.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Jules Dupuit (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k408461p/f399.image&quot; rel=&quot;noreferrer&quot;&gt;1844&lt;/a&gt;). &lt;strong&gt;Price on horizontal axis.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Hans von Mangoldt (&lt;a href=&quot;https://archive.org/details/grundrissdervolk00manguoft&quot; rel=&quot;noreferrer&quot;&gt;1863&lt;/a&gt;). &lt;strong&gt;Price on vertical axis.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Fleeming Jenkin (&lt;a href=&quot;https://archive.org/details/papersliterarysc02jenk&quot; rel=&quot;noreferrer&quot;&gt;1870&lt;/a&gt;). &lt;strong&gt;Price on horizontal axis.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I'd also add&lt;/p&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;William Stanley Jevons (1871, &lt;a href=&quot;http://www.econlib.org/library/YPDBooks/Jevons/LF-BK0237-0126-0001.gif&quot; rel=&quot;noreferrer&quot;&gt;figure&lt;/a&gt;). &lt;strong&gt;Price on vertical axis.&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Did any of the above influence Marshall?&lt;/p&gt;&#xA;&lt;p&gt;Prior to his 1879 publication, Marshall may not have been aware of Rau or Mangoldt's work, who were less well-known. But Marshall was aware of Cournot, Dupuit, Jenkin, and Jevons's work (see e.g. Whitaker, &lt;a href=&quot;https://books.google.com/books/about/The_Early_Economic_Writings_of_Alfred_Ma.html&quot; rel=&quot;noreferrer&quot;&gt;1975&lt;/a&gt;). But of these four, only Jevons had price on the vertical axis. Cournot, Dupuit, and Jenkin all had price on the horizontal axis.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, according to the man himself, &lt;strong&gt;Marshall's main influence was Cournot&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;following the lead of Cournot I had anticipated all the central points of Jevons book and had in many respects gone beyond him,&lt;/p&gt;&#xA;&lt;p&gt;my obligations are solely to Cournot; not to Fleeming Jenkin or Dupuit. (Quoted in Whitaker.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But &lt;strong&gt;Cournot had price on the horizontal axis!&lt;/strong&gt; I can find in Marshall's writings no explicit explanation for why he chose to deviate from Cournot, who was his greatest influence (at least in this matter).&lt;/p&gt;&#xA;&lt;p&gt;However, when introducing for his very first demand curve (1879), he does write the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;we may draw what may be called &amp;quot;the Demand curve,&amp;quot; thus: Let &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; be&#xA;any point on &lt;span class=&quot;math-container&quot;&gt;$Ox$&lt;/span&gt; (fig. 20), and let the price at which it is possible to&#xA;dispose of &lt;span class=&quot;math-container&quot;&gt;$OM_1$&lt;/span&gt; coals annually be estimated and found to be equal to &lt;span class=&quot;math-container&quot;&gt;$ON_1$&lt;/span&gt;. &lt;a href=&quot;https://i.stack.imgur.com/4vIJM.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4vIJM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is, Marshall first takes &lt;strong&gt;the independent variable to be the quantity demanded&lt;/strong&gt; (the &lt;span class=&quot;math-container&quot;&gt;$OM_1$&lt;/span&gt; coals to be disposed of). Marshall's &lt;strong&gt;demand curve then tells us about the dependent variable, i.e. the price at which such coal can be disposed&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I thus conclude that &lt;strong&gt;Marshall thought of quantity as the independent variable, with prices adjusting to clear the market&lt;/strong&gt;. It was thus perfectly sensible for him to put quantity on the horizontal axis and price on the vertical.&lt;/p&gt;&#xA;&lt;p&gt;This way of thinking seems slightly unnatural to me (and many others). But perhaps this is only because we've been trained, since our first economics classes, to think of price as the independent variable.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, as some here have pointed out, it's not really all that absurd to think of quantity as the independent variable, as Marshall did.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;P.S. There's also another cynical explanation I once came across: &lt;strong&gt;Marshall simply wanted to differentiate himself from Cournot et al. So he deliberately switched the axes.&lt;/strong&gt; In my opinion this can plausibly serve as part of the explanation. After all, Marshall felt that he had done a lot of his own original work and was somewhat peeved that others, such as Jenkin, had anticipated his work (see e.g. Marshall's reaction to Jenkin's 1870 article, quoted in Whitaker, &lt;a href=&quot;https://books.google.com/books?id=K2Ru9J_2mWAC&amp;amp;pg=PA250&amp;amp;lpg=PA250&amp;amp;dq=%22I%20happened%20to%20come%20across%20%5BJenkin%27s%201870%20article%5D%20in%20the%20Easter%20vacation%20of%201870,%20when%20I%20was%20attending%20Marshall%27s%20lectures%20on%20diagrammatic%20economics,%20%26%20I%20shall%20never%20forget%20his%20chagrin%20as%20he%20glanced%20through%20the%20article%20when%20I%20showed%20it%20to%20him.%20There%20was%20nothing%20in%20Cournot%20which%20so%20closely%20agreed%20with%20Marshall%27s%20general%20approach%20to%20the%20Theory%20of%20Value%20%26%20particularly%20to%20his%20statement%20of%20the%20equation%20of%20supply%20%26%20demand%22&amp;amp;source=bl&amp;amp;ots=63whfqYbaU&amp;amp;sig=ChMWhlDy_0pVbCxprl8anuuPYkU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwic_oztxMXPAhXI6RQKHSXdDxsQ6AEIHDAA#v=onepage&amp;amp;q=%22I%20happened%20to%20come%20across%20%5BJenkin%27s%201870%20article%5D%20in%20the%20Easter%20vacation%20of%201870%2C%20when%20I%20was%20attending%20Marshall%27s%20lectures%20on%20diagrammatic%20economics%2C%20%26%20I%20shall%20never%20forget%20his%20chagrin%20as%20he%20glanced%20through%20the%20article%20when%20I%20showed%20it%20to%20him.%20There%20was%20nothing%20in%20Cournot%20which%20so%20closely%20agreed%20with%20Marshall%27s%20general%20approach%20to%20the%20Theory%20of%20Value%20%26%20particularly%20to%20his%20statement%20of%20the%20equation%20of%20supply%20%26%20demand%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;1975&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h2&gt;Supply and Demand Diagrams before Marshall (1879)&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DfHMX.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DfHMX.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oYaWs.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oYaWs.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2Jywy.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2Jywy.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RJI6g.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RJI6g.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pCiFE.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pCiFE.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qq5EK.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qq5EK.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user4795" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-10-06T06:49:54.853" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5261" PostTypeId="2" ParentId="3176" CreationDate="2016-10-06T14:52:59.340" Score="3" Body="&lt;p&gt;Please allow a correction: I think your statement &quot;he says that one should sincerely try to disproof hypotheses – and I am quite certain that he didn’t mean the null hypothesis that Fisher formulated but rather the hypothesis that is of critical importance to us&quot; is not really correct. Actually, this is exactly how Fisher would have described &quot;null hypothesis&quot;. For Fisher, there is only one type of hypothesis: the hypothesis to be tested, a.k.a. the null hypothesis. The distinction between null and alternative hypothesis, introduced by Neyman and Pearson, has never been accepted by Fisher (also Conifold's statement &quot;Instead of testing the actual hypothesis directly Fisher suggested testing its 'negation', the null hypothesis, if the null hypothesis comes out as unlikely after the test then our actual hypothesis 'survived the test'&quot; does not describe Fisher's approach correctly)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you read Fisher carefully, the commonly used version of &quot;null hypothesis&quot; (nil, no effect, &quot;just random variation&quot; and similar) is not what Fisher was referring to. The central meaning of a test for statistical significance is its ability to create results which can be, following Fisher (&quot;The Design of Experiments&quot;, 2nd ed., 1937), divided into two classes with different interpretations: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;“those which show a significant discrepancy from a certain hypothesis; […] and […] results which show no significant discrepancy from this hypothesis. This hypothesis […] is again characteristic of all experimentation. […] [W]e may speak of this hypothesis as the “null hypothesis”, and it should be noted that [it] is never proved or established, but is possibly disproved, in the course of experimentation. […] &lt;strong&gt;If it were asserted that the subject would never be wrong in her judgements we should again have an exact hypothesis, and it is easy to see that this hypothesis could be disproved by a single failure, but could never be proved by any finite amount of experimentation. It is evident that the null hypothesis must be exact, that is free from vagueness and ambiguity, because it must supply the basis of the ‘problem of distribution,’ of which the test of significance is the solution&lt;/strong&gt;. A null hypothesis may, indeed, contain arbitrary elements, and in more complicated cases often does so.” (p.18-20; emphasis added) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please note that the only distinctive feature of a ‘null hypothesis’, as it is characterized by Fisher, is its ability to partition the results of an experiment into two mutually exclusive classes (supporting and contradicting cases), and to do so, it needs to be exact. That's all. It is not assumed that a null hypothesis has to state the absence of any effect, as most statistical resources – websites, articles and textbooks – claim. It is also important that, as already mentioned, in Fisher's approach there is no &quot;alternative&quot; hypothesis, which is frequently equated with the &quot;research hypothesis&quot; to be supported by rejecting the null. In Fisher's approach, the &quot;null hypothesis&quot; &lt;strong&gt;is&lt;/strong&gt; the hypothesis the researcher wants to test. It can state the absence of an effect or its existence, and as long as it is exact, it can be tested. However, to test a non-nil null hypothesis appropriately, the usually applied tests (e.g. t-Tests) have to be replaced by versions that reflect the effect (size). In the case of the t-Test, any effect could be tested using a particular noncentral version of this test (which uses the same distribution that is used for power calculations in the Neyman-Pearson approach). A significant &lt;em&gt;p&lt;/em&gt; would - as usual - indicate that the data obviously do not correspond with the prediction based on the null hypothesis (our research hypothesis!), which would commonly be considered as rejection of the null. Interpreted this way, at least the majority (if not all) of the usually discussed flaws of &quot;NHST&quot; disappear. Moreover, this interpretation of significance testing looks like a statistical version of Popper's falsification principle, or at least as a statistical argument closely related to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The confusion of many users of statistical methods (which has also driven the - still ongoing - discussion about NHST, &quot;null hypothesis significance testing&quot; in the &quot;soft&quot; or - better - weak &quot;sciences&quot; [like Psychology]) is probably due to mixing up two distinct approaches to testing hypotheses - Fisher's significance test on one side and Neyman-Pearson's theory of statistical decision on the other - into an &quot;inconsistent hybrid that every decent statistician would reject&quot; (Gigerenzer, 1993). A prototypical study, at least in Psychology, works like this: The researcher has an assumption, call it A, that there is some effect. He/she assumes a medium effect size (Cohen's d=0.5; poor theory, probably, but anyway...) and calculates the sample size for this assumed effect to be indicated sensitively, say with Power=0.8 (this is a kind of Neyman-Pearson). Then he/she collects data, runs a standard (central) t- or F-Test, setting a strawman null hypothesis of &quot;no effect&quot; and if &lt;em&gt;p&lt;/em&gt;&amp;lt;0.05, he/she rejects the null (which is ok; this &lt;em&gt;is&lt;/em&gt; Fisher, but not testing the actual hypothesis) and accepts A (&lt;strong&gt;which is not ok&lt;/strong&gt;, as any effect size different from null gets support from rejecting the null, unless all other alternatives can be ruled out). This latter conclusion is neither Fisher, nor Neyman-Pearson, it is simply not correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, I think that the abstract you are referring to is rather one of the rare cases in which the essence of testing hypotheses (in a way that would have been accepted by Fisher) has been extracted more or less correctly. Still, there is some inexactness in the abstract, since it assumes that the null and the research hypothesis are different things. But actually, for both, Popper and Fisher, it is the research hypothesis that needs to be tested and, if necessary, rejected. Which actually &lt;em&gt;is&lt;/em&gt; Fisher's null hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With regard to your third question: &#xA;In &quot;Logik der Forschung&quot; Popper sometimes refers to Fisher's likelihood concept, but not to significance testing (or to the Neyman-Pearson theory). In Bennett's (1990) &quot;Statistical Inference and Analysis - Selected Correspondence of R.A. Fisher&quot;, Popper is not listed as correspondent. &lt;/p&gt;&#xA;" OwnerUserId="4797" LastEditorUserId="4797" LastEditDate="2016-10-11T11:02:43.400" LastActivityDate="2016-10-11T11:02:43.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5262" PostTypeId="1" AcceptedAnswerId="5270" CreationDate="2016-10-06T18:58:45.870" Score="3" ViewCount="284" Body="&lt;p&gt;My question reffers to section X  of book 1 of Isaac Newton's Principia, that deals with motion of particles on curved surfaces. I read that in proposition 56 Newton found the trajectory of a particle on a curved surface in the case where the centre of centripetal force lies on the axis of the surface. This proposition reads so: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Granting the quadrature of curvilinear figures, and both the law of centripetal force toward a given center and a curved surface whose axis passes through that center, to find the trajectory a body will describe on that surface, given its initial position and velocity.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By taking the centre of centripetal force to be at the infinite distance (on the axis of course) one gets a uniform gravity field. This is a very interesting point that I found little material about, since determining the motion of a particle on a curved surface involves complications and I think it's even somehow concerned with geodesics on the curved surface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm asking this question since I was unable to find material about this proposition, so if someone knows a good reference that analyses this point, I'll be glad to read it.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="55" LastEditDate="2016-10-09T03:09:38.017" LastActivityDate="2016-10-09T22:31:27.920" Title="Did Newton find geodesics on curved surfaces in Principia?" Tags="|biographical-details|classical-mechanics|gravity|celestial-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5263" PostTypeId="1" CreationDate="2016-10-06T22:58:38.733" Score="4" ViewCount="406" Body="&lt;p&gt;I have &lt;a href=&quot;https://mathoverflow.net/questions/251570/what-color-were-emmy-noethers-eyes?noredirect=1#comment618682_251570&quot;&gt;reposted&lt;/a&gt; this question from MathOverflow because it is questionably relevant there.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I am looking for a reference that definitively says what the color of Emmy Noether's eyes were. More generally, I would like to find a color photograph of her. Do any exist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The shading in &lt;a href=&quot;https://www.britannica.com/biography/Emmy-Noether&quot; rel=&quot;nofollow noreferrer&quot;&gt;this photo&lt;/a&gt; seems to suggest they are blue, but I am not an expert.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my search I also found &lt;a href=&quot;http://www.rzuser.uni-heidelberg.de/~ci3/noetherphoto-engl.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this document&lt;/a&gt; which discusses the difficulty of finding good photographs of Noether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not been able to find this information by (programmatically) searching the texts of existing biographies for the string &quot;eye.&quot;&lt;/p&gt;&#xA;" OwnerUserId="4798" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-10-07T14:48:25.870" Title="What color were Emmy Noether's eyes?" Tags="|mathematics|biographical-details|abstract-algebra|" AnswerCount="0" CommentCount="4" ClosedDate="2016-10-07T14:51:37.290" ContentLicense="CC BY-SA 3.0" />
  <row Id="5264" PostTypeId="1" CreationDate="2016-10-07T02:18:15.807" Score="3" ViewCount="470" Body="&lt;p&gt;G.E. Andrews writes, &quot;Leibniz was apparently the first person to ask about partitions. In a 1674 letter he asked J. Bernoulli about the number of “divulsions” of integers. In modern terminology, he was asking the first question about the number of partitions of integers. He observed that there are three partitions of 3 (3, 2 + 1, and 1 + 1 + 1) as well as five of 4 (4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find this assertion of priority incredibly difficult to believe as well as the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peter Luschny, an editor for the OEIS commenting on simply an allusion to a listing of integer partitions, certainly not an attribution as to the first publication with that order, &quot;I strongly appeal against the terms A&amp;amp;S-order or A-St-order. Whoever (...?) introduced these terms into the OEIS made a disservice. It is well known that this widely used order was introduced by C. F. Hindenburg in his 1779 dissertation and the proper way to reference it is either 'Hindenburg order' or a mathematical description in general terms. Neither Abramowitz nor Stegun have ever written a single line about monomial orders, not even in their Handbook. A detailed analysis of the Hindenburg way to enumerate the partitions can be found Knuth's chapter 7 in TAOCP 4. The term A&amp;amp;S does not appear a single time in this work. Let us base our terminology on serious literature rather than introduce private jargon into OEIS.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would assume that the ancient Babylonians and Chinese had addresssed the topic long before either of these two mathematicians (in diverse, reasonable, and exhaustive orderings), and that this OCD necessitating attributing priority to particular, long-deceased individuals is both gratuitous and naive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Accordingly, to support a definitive rebuttal of these somewhat trivial assertions and much more importantly to discourage such scholasticism in general, can anyone provide references on earlier investigations of the integer partitions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Some indication of interests of the ancients in integer partitions: The oldest known magic square dates to 2200 BCE (Cf: &lt;a href=&quot;http://www.math.ucsd.edu/~ronspubs/mis_10_encyclopedia.pdf&quot; rel=&quot;nofollow&quot;&gt;A brief survey of combinatorics)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2016-10-28T17:16:21.507" LastActivityDate="2016-10-28T17:16:21.507" Title="First to enumerate integer partitions?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="5265" PostTypeId="1" CreationDate="2016-10-07T08:01:14.080" Score="1" ViewCount="73" Body="&lt;p&gt;In 1777, &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Priestley&quot; rel=&quot;nofollow&quot;&gt;Joseph Priestley&lt;/a&gt;&lt;/strong&gt; wrote a book called &lt;strong&gt;&lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Disquisitions_relating_to_Matter_and_Spirit&quot; rel=&quot;nofollow&quot;&gt;Disquisitions relating to Matter and Spirit&lt;/a&gt;&lt;/em&gt;&lt;/strong&gt;, in which he says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is maintained in this treatise, that neither matter nor spirit (meaning by the latter the subject of sense and thought) correspond to the definitions above mentioned. &lt;strong&gt;For that matter is not that inert substance that it has been supposed to be&lt;/strong&gt;; that powers of attraction or repulsion are necessary to its very being, and that no part of it appears to be impenetrable to other parts. I therefore, &lt;strong&gt;define it to be a substance possessed of the property of extension, and of powers of attraction or repulsion&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From what I understand, both Descartes and Newton believed matter is &quot;inert&quot;, or has &quot;no power of its own&quot; until an outside force causes it to go into motion. Priestley talks about both men in his essay. Was Priestley describing fundamental interactions the only way he knew how?&lt;/p&gt;&#xA;" OwnerUserId="4801" LastEditorUserId="4801" LastEditDate="2016-10-09T22:13:21.437" LastActivityDate="2016-10-09T22:13:21.437" Title="Was Joseph Priestley describing fundamental interactions in the 18th century?" Tags="|philosophy-of-science|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5266" PostTypeId="2" ParentId="5258" CreationDate="2016-10-07T16:09:38.787" Score="20" Body="&lt;p&gt;Victor Katz is not a linguist and a lot of what he says in the quoted extract is wrong: for example that “Arabic is written without vowels” and that the word in question is spelt “jb”. In fact it is written jyb جيب  (as mobileink has pointed out). But the decisive error from the viewpoint of the history of science is his failure to remark that Sanskrit jyā and jīva mean “bow string” and are thus translations of the Greek khordē “gut string”, which was borrowed into Latin as chorda (which in turn is the source of both “chord” and “cord” in English). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a mathematical term khordē (and chord) means a line connecting two points on an arc A and B; if we draw lines from A and B to the centre of the circle (O) the length of the chord AB is defined as the chord of the angle AOB. The Indians borrowed Greek trigonometry and translated khordē literally as jyā, but they used this word not (usually) for a chord but for a half-chord, what we call a sine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Arabic astronomers inherited both the Greek and the Indian systems of trigonometry and the Sanstrit word jyā (jīva) was transcribed (not translated) as jīb. In the 12th century the famous Arabic-Latin translator Gerard of Cremona misread the latter as jayb “breast pocket” and rendered this with its Latin equivalent sinus. In Arabic script jīb and jayb are both written جيب .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: this is a Greek word, first translated into Sanskrit, then transcribed into Arabic, then mistranslated into Latin.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-10-07T16:36:15.143" LastActivityDate="2016-10-07T16:36:15.143" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5267" PostTypeId="2" ParentId="5240" CreationDate="2016-10-07T19:55:46.060" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;So, how did this shift took place and what were the reasons for the emergence of USA as the pioneer in these fields leaving behind Europe and other countries?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Other answers point to World War 2. The US's rise in science, technology, engineering, and mathematics started prior to WW2. I'll start with the pre-WW2 developments follow through to the end of the 20th century.&lt;/p&gt;&#xA;&lt;p&gt;The US's rise in engineering started with the birth of the country. At the very start, American interests saw the advantage in interchangeable parts and its subsequent offspring, mechanization and the assembly line. This came to forefront in the late 1860s in the Chicago meatpacking industry and in multiple places that manufactured arms for the American Civil War. This dominance became apparent in 1913 with Henry Ford's Model T.&lt;/p&gt;&#xA;&lt;p&gt;The US remained a backwater in the sciences until the mid 1930s. (The US did however produce some remarkable astronomers prior to this time.) What happened to the US in the 1930s was fascism in Europe. Germany was the world leader in much of the physical sciences throughout the 19th century and up until the mid 1930s. Fascist laws forced many of those European scientists to emigrate to the US. The list includes Hans Bethe, Albert Einstein, Enrico Fermi, John von Neumann, Leo Szilard, Edward Teller, Stanislaw Ulam, Eugene Wigner, and many others. It's dubious whether the US would have been able to build the bomb without those European scientists.&lt;/p&gt;&#xA;&lt;p&gt;World War II was a great impetus to US dominance. American engineering helped build the massive amounts of machinery needed to sustain that war effort, and the European scientists helped build the two machines that ended that war. In the immediate aftermath of that war, the US extracted 1500 German scientists, engineers, and technicians via Operation Paperclip, including Wernher von Braun. It's dubious whether the US would have been able to send men to the Moon in the 1960s without the help of those German emigres.&lt;/p&gt;&#xA;&lt;p&gt;In the meanwhile, WW2 devastated much of Europe. The US built machinery and technology while Europe struggled to rebuild its basic infrastructure. This advantage continued and grew for decades after WW2 ended.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Also, is there a possibility of similar shift from USA to other country say China and India?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The US developed the concept of the assembly line automobile manufacturing, enabling the US to dominate the automotive market for over half a century. That early dominance set the US back later on in the 20th century. Late 20th century American cars were nowhere close to best of class. Asian and European automobile manufacturers became better at making economical cars, luxury cars, and everything in-between compared to their American counterparts. So, yes, there's a precedence for a shift from the USA to elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-10-07T19:55:46.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5270" PostTypeId="2" ParentId="5262" CreationDate="2016-10-09T03:05:12.200" Score="2" Body="&lt;p&gt;You are on to something here. &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookI-X&quot; rel=&quot;nofollow&quot;&gt;Section X of Book 1 of Principia&lt;/a&gt; discusses the motion of particles constrained to curves or surfaces under the action of central forces (that is forces directed towards a fixed point, the center, with the magnitude depending only on the distance to it). This is an example of constrained dynamics, now associated with &lt;a href=&quot;https://en.wikipedia.org/wiki/Virtual_work#History&quot; rel=&quot;nofollow&quot;&gt;&quot;D'Alembert's&quot; principle of virtual work&lt;/a&gt; (formulated by John Bernoulli, see &lt;a href=&quot;http://link.springer.com/chapter/10.1007%2F978-3-319-05374-5_2&quot; rel=&quot;nofollow&quot;&gt;Maugin's Glimpse at the Eighteenth Century&lt;/a&gt;), and if we take the special case of zero central force, the trajectories will indeed be the geodesics on the surface. Taking the center to infinity while increasing the intensity can produce the constant force field in the limit, Newton considers this simpler case for curves but not surfaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us translate Proposition 56 into modern terms. &quot;Centripetal force toward a given center&quot; is a central force, &quot;curved surface whose axis passes through that center&quot; is a surface of revolution whose axis passes through the center of force, and &quot;granting the quadrature of curvilinear figures&quot; means that Newton is content to reduce problems to finding areas of curved regions, to quadratures. Thus, the proposition says that the initial value problem for motion on a surface of revolution under the action of a central force reduces to quadrature. The proof relies on the preceding Proposition 55, where Newton proves that if the particles trajectory on the surface is projected on a plane perpendicular to the axis its radius-vector will sweep equal areas in equal times. This is a generalization of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Second_law&quot; rel=&quot;nofollow&quot;&gt;Kepler's second law&lt;/a&gt; to arbitrary central forces and motions constrained to surfaces of revolution rather than just planes. In calculus terms, this gives a &lt;a href=&quot;https://en.wikipedia.org/wiki/Constant_of_motion#Integral_of_motion&quot; rel=&quot;nofollow&quot;&gt;first integral of motion&lt;/a&gt;, which explains the reduction to quadrature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The geodesic problem is a particular case of Newton's. This is not exactly &quot;solving&quot; it (as was finding ellipses, parabolas and hyperbolas for the inverse square law), but still. Struik gives some early history of geodesics in his &#xA;&lt;a href=&quot;https://books.google.com/books?id=TIXg4hSXmFAC&quot; rel=&quot;nofollow&quot;&gt;Lectures on Classical Differential Geometry (4-2)&lt;/a&gt; without mentioning Newton and Principia, even as he states that the geodesic problem on a surface of revolution reduces to quadrature:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The history of geodesic lines begins with John Bernoulli's solution of the&#xA;  problem of the shortest distance between two points on a convex surface (1697-&#xA;  1698). His answer was that the osculating plane (&quot;the plane passing through&#xA;  three points `quolibet proxima &quot;') must always be perpendicular to the tangent&#xA;  plane. For the further history see P. Stackel, Bemerkungen zur Geschichte der&#xA;  geodatischen Linien, Berichte sachs. Akad. Wiss., Leipzig, 45, 1893, pp. 444-467. The name &quot;geodesic line&quot; in its present meaning is, according to Stackel, due to J. Liouville, Journal de mat hem. 9, 1844, p. 401; the equation of the geodesics was first obtained by Euler in his article De linea brevissima in superficie quacumque duo quaelibet puncta jungente, Comment. Acad. Petropol. 3 (ad annum 1728), 1732; Euler's equation refers to a surface given by F(x, y, z) = 0.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The earlier propositions of Section X concerning motion constrained to curves are also of interest. Propositions 50-53 deal with the motion constrained to a cycloid and cast a new light on its famous relation to &lt;a href=&quot;https://en.wikipedia.org/wiki/Tautochrone_curve&quot; rel=&quot;nofollow&quot;&gt;pendulum clocks discovered by Huygens&lt;/a&gt; . This is discussed in detail in &lt;a href=&quot;http://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0020.php&quot; rel=&quot;nofollow&quot;&gt;Nauenberg's Huygens and Newton on Curvature and its applications to Dynamics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The question concerns the underlying reason for the isochronous property of the oscillations of a body moving on a cycloidal path under the action of the constant force of gravity. They found that the component of the force of gravity tangent to a cycloidal path is proportional to the distance along this path. This result implies that other physical systems where the force is linear with the distance leads to isochronous or harmonic oscillations. The best known example is the restoring force of a spring.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;You may find &lt;a href=&quot;https://books.google.com/books?id=qomP58txKQwC&quot; rel=&quot;nofollow&quot;&gt;Chandrasekhar's Newton's Principia for the Common Reader&lt;/a&gt; of interest. He systematically translates Newton's Euclidese into modern calculus, section by section, and comments on them, but he conspicuously skips Section X. &lt;a href=&quot;http://link.springer.com/chapter/10.1007%2F978-3-319-05374-5_2&quot; rel=&quot;nofollow&quot;&gt;Bressoud's Second Year Calculus: From Celestial Mechanics to Special Relativity&lt;/a&gt; deals with the Kepler's second law for gravity both in Newton's and in the calculus notation.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-10-09T22:31:27.920" LastActivityDate="2016-10-09T22:31:27.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5273" PostTypeId="1" AcceptedAnswerId="5274" CreationDate="2016-10-09T20:14:18.263" Score="3" ViewCount="135" Body="&lt;p&gt;&lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/D&amp;#39;Alembert.html&quot; rel=&quot;nofollow&quot;&gt;This source&lt;/a&gt; claims that &quot;d'Alembert won the prize of the Prussian Academy of Sciences with his essay on winds&quot;, and &lt;a href=&quot;http://biography.yourdictionary.com/jean-le-rond-d-alembert&quot; rel=&quot;nofollow&quot;&gt;this source&lt;/a&gt; claims that &quot;A prize essay on the theory of winds in 1746 led to membership in the Berlin Academy of Sciences&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't really know where to look (and searches involving 1746 haven't been fruitful) to find d'Almbert's essay on the theory of winds. What can I find it?&lt;/p&gt;&#xA;" OwnerUserId="3732" LastEditorUserId="26" LastEditDate="2016-10-10T06:14:26.073" LastActivityDate="2016-10-13T18:38:06.897" Title="Which paper/book did d'Alembert publish his essay on the theory of winds in?" Tags="|mathematics|physics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5274" PostTypeId="2" ParentId="5273" CreationDate="2016-10-09T21:23:16.713" Score="3" Body="&lt;p&gt;In 1740, the French Royal Academy of Science lauched a competition regarding the nature of tides, of which Colin MacLaurin was joint winner (I can find no record of who authored the other winning entry) for his essay “De Causa physica fluxus et refluxus maris” (On the cause of tides). In his essay, MacLaurin argues that the sea currents were affected by “the uneven velocity of a body carried by the earth in its daily motion around its axis”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is thought that the Berlin Academy of Sciences may have been inspired, in 1746, to announce a prize for anybody who could determine “the nature and the law” which the wind ought to obey in case the earth was covered by an ocean. The solution had to be presented in a form that allowed predictions. The winning contribution by d’Alembert was published in 1746 under the title “Reflexions sur la cause generale&#xA;des vents.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A copy of the essay can be found on the National Library of France site here : &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k62565n/f5.image&quot; rel=&quot;nofollow&quot;&gt;Réflexions sur la cause générale des vents&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can also buy a copy from &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/1166191788&quot; rel=&quot;nofollow&quot;&gt;Amazon&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clairaut attacked d'Alembert's methods :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In order to avoid delicate experiments or long tedious calculations, in order to substitute analytical methods which cost them less trouble, they often make hypotheses which have no place in nature; they pursue theories that are foreign to their object, whereas a little constancy in the execution of a perfectly simple method would have surely brought them to their goal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A heated argument between d'Alembert and Clairaut resulted in the two fine mathematicians trading insults in the scientific journals of the day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The source for my answer and additional details can be found here : &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/D&amp;#39;Alembert.html&quot; rel=&quot;nofollow&quot;&gt;St. Andrews MacTutor page on d'Alembert&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If inadvertently missed including the main source for my answer : &lt;a href=&quot;http://www.meteohistory.org/2006historyofmeteorology3/2persson_hadley.pdf&quot; rel=&quot;nofollow&quot;&gt;History of Meteorology Site PDF&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-10-10T20:04:43.217" LastActivityDate="2016-10-10T20:04:43.217" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5275" PostTypeId="1" CreationDate="2016-10-10T09:16:41.377" Score="3" ViewCount="1471" Body="&lt;p&gt;Newton is frequently seen as the founder of Western science. Was he the first person to explicitly articulate the scientific method?&lt;/p&gt;&#xA;" OwnerUserId="4623" LastEditorUserId="-1" LastEditDate="2016-10-10T14:59:30.370" LastActivityDate="2017-10-17T04:19:41.097" Title="Was Isaac Newton the first person to articulate the scientific method in Europe?" Tags="|scientific-method|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5276" PostTypeId="2" ParentId="5275" CreationDate="2016-10-10T10:30:49.583" Score="17" Body="&lt;p&gt;NO. See at least :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Gilbert_(astronomer)&quot;&gt;William Gilbert&lt;/a&gt; with &lt;a href=&quot;https://en.wikipedia.org/wiki/De_Magnete&quot;&gt;De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure&lt;/a&gt; (1600)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_Galilei&quot;&gt;Galileo Galilei&lt;/a&gt; with e.g. the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sidereus_Nuncius&quot;&gt;Sidereus Nuncius&lt;/a&gt; (1610)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Harvey&quot;&gt;William Harvey&lt;/a&gt; with &lt;a href=&quot;https://en.wikipedia.org/wiki/Exercitatio_Anatomica_de_Motu_Cordis_et_Sanguinis_in_Animalibus&quot;&gt;Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus&lt;/a&gt; (1628)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/descartes-physics/&quot;&gt;René Descartes&lt;/a&gt; with &lt;a href=&quot;https://en.wikipedia.org/wiki/Discourse_on_the_Method&quot;&gt;Discours de la méthode. Pour bien conduire sa raison, et chercher la vérité dans les sciences&lt;/a&gt; (Discourse on the Method. An introduction to the Essais, which include the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dioptrique&quot;&gt;Dioptrique&lt;/a&gt;, the Météores and the &lt;a href=&quot;https://en.wikipedia.org/wiki/La_G%C3%A9om%C3%A9trie&quot;&gt;Géométrie&lt;/a&gt;) (1637).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt; : &quot;scientific method&quot; is a vague concept: it is a &quot;recipe&quot; with various ingredients: observation, experiments, hypotheses, mathematization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Observations, for example, are as old as mankind: for Europe, they date back at least to Ancient Greece : see &lt;a href=&quot;http://plato.stanford.edu/entries/presocratics/#Mil&quot;&gt;Presocratic Philosophy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The peculiar mixture of the ingredients is an Early Modern invention: we can find it fully mature into Newton's works, but Newton relies heavily on Galileo and Descartes.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-10-10T11:43:40.610" LastActivityDate="2016-10-10T11:43:40.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5277" PostTypeId="2" ParentId="5275" CreationDate="2016-10-10T13:12:03.883" Score="9" Body="&lt;p&gt;Usually Francis Bacon (1561-1626) is credited with &quot;articulating the scientific method&quot; in general, and not only &quot;first in Europe&quot;, but just the first.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-17T04:19:41.097" LastActivityDate="2017-10-17T04:19:41.097" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5278" PostTypeId="2" ParentId="5275" CreationDate="2016-10-10T20:58:43.310" Score="2" Body="&lt;p&gt;No not in Europe. But scientific method has its roots in the Islamic World.  The origins of the scientific method hearken back to the Islamic World, not the Western one. Around 250 years before Roger Bacon expounded on the need for experimental confirmation of his findings, an Arab scientist named Ibn al-Haytham was saying the exact same thing. He not only was a precursor of the scientific method (both theoretically and in practice), but he was well-known in Europe (as Alhazen or Alhacen) since the Middle Ages and influenced many European thinkers such as Bacon himself, Galileo, Kepler, Descartes and others. In other words, he was not an isolated antecedent, but an influential precursor.&lt;/p&gt;&#xA;" OwnerUserId="4823" LastEditorUserId="4823" LastEditDate="2016-10-14T13:18:52.273" LastActivityDate="2016-10-14T13:18:52.273" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5281" PostTypeId="1" AcceptedAnswerId="5282" CreationDate="2016-10-11T18:11:09.953" Score="2" ViewCount="203" Body="&lt;p&gt;To my eye, the contents of Crelle's Journal (&lt;em&gt;Journal für die reine und angewandte Mathematik&lt;/em&gt;) around 1908 (not by chance the year Crelle's published Emmy Noether's dissertation) look very old-fashioned compared to those of the &lt;em&gt;Mathematische Annalen&lt;/em&gt; in the same year.  There are articles by currently well known mathematicians of course but not like in the &lt;em&gt;Annalen&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any authoritative statements or reasonably objective measures I could use to check my impression? &lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2016-10-13T00:15:34.943" Title="Are there objective measures for comparing Crelle's Journal to the Annalen circa 1908?" Tags="|mathematicians|publishing|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5282" PostTypeId="2" ParentId="5281" CreationDate="2016-10-12T19:13:18.760" Score="4" Body="&lt;p&gt;MathSciNet has &lt;a href=&quot;http://gethelp.library.upenn.edu/guides/scitech/databases/mathscinet/citationsearching.html&quot; rel=&quot;nofollow&quot;&gt;Journal Citation Search&lt;/a&gt; feature that might be useful. It lets you select a citing year (from 2000 to 2015 currently) and returns the histogram of total citations listed in MathSciNet from that year to past years (back to 1900), among other things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With 2015 as the citing year it shows for &lt;a href=&quot;http://www.ams.org/mathscinet/mrcit/journal.html?issn=0075-4102&amp;amp;eissn=1435-5345&amp;amp;max_citing_year=2015&amp;amp;min_citing_year=2015&quot; rel=&quot;nofollow&quot;&gt;Journal für die Reine und Angewandte Mathematik&lt;/a&gt;  13 citations to 1901-1904, 29 to 1905-1909, and 26 to 1910-1914. For &lt;a href=&quot;http://www.ams.org/mathscinet/mrcit/journal.html?issn=0025-5831&amp;amp;eissn=1432-1807&amp;amp;max_citing_year=2015&amp;amp;min_citing_year=2015&quot; rel=&quot;nofollow&quot;&gt;Mathematische Annalen&lt;/a&gt; it shows 34, 45, and 111, respectively. For the citing year 2010 it is even more definitive: for &lt;a href=&quot;http://www.ams.org/mathscinet/mrcit/journal.html?issn=0075-4102&amp;amp;eissn=1435-5345&amp;amp;max_citing_year=2010&amp;amp;min_citing_year=2010&quot; rel=&quot;nofollow&quot;&gt;Journal für die Reine und Angewandte Mathematik&lt;/a&gt; there is nothing to 1901-1908, 7 to 1909, and 1 to 1910-1914; for &lt;a href=&quot;http://www.ams.org/mathscinet/mrcit/journal.html?issn=0025-5831&amp;amp;eissn=1432-1807&amp;amp;max_citing_year=2010&amp;amp;min_citing_year=2010&quot; rel=&quot;nofollow&quot;&gt;Mathematische Annalen&lt;/a&gt; there are 18 to 1901-1904, 52 to 1905-1909, and 71 to 1910-1914.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Mathematische Annalen does appear increasingly more &quot;modern&quot; in those years, at least by these measures. You can try different citation years, and with extra work probably narrow it down to 1908 specifically, although I suspect that would require tracking individual citations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. Search links require MathSciNet access, and may be slow even with that.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-10-13T00:15:34.943" LastActivityDate="2016-10-13T00:15:34.943" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5286" PostTypeId="1" AcceptedAnswerId="5290" CreationDate="2016-10-13T21:54:49.730" Score="5" ViewCount="196" Body="&lt;p&gt;Heres the google ngram for the word syphilis. I was wondering what these three peaks could be. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6BYH5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6BYH5.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peak 1 - around 1870s - Repeal of the Contagious diseases act&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peak 2 - around 1910s - Discovery of &lt;em&gt;Treponema&lt;/em&gt; and Salvarsan&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peak 3 - around 1940s - World War and Penicillin&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is just a speculation and I don't know for sure. What are the correct explanations?&lt;/p&gt;&#xA;" OwnerUserId="4841" LastActivityDate="2016-10-14T16:28:46.693" Title="Explanation of the three peaks in Google ngram for 'syphilis'" Tags="|biology|medicine|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5287" PostTypeId="1" AcceptedAnswerId="5288" CreationDate="2016-10-14T06:38:23.803" Score="5" ViewCount="2830" Body="&lt;p&gt;Who is responsible for coming up with the gradient and why did they do so? In which work was it first described?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have Googled this extensively, to no avail, and &lt;a href=&quot;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&quot; rel=&quot;noreferrer&quot;&gt;Boyer's History of Calculus&lt;/a&gt; does not appear to contain the answer to this.&lt;/p&gt;&#xA;" OwnerUserId="4842" LastEditorUserId="4842" LastEditDate="2016-10-14T14:32:37.290" LastActivityDate="2017-08-31T21:56:44.717" Title="Who invented the gradient?" Tags="|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5288" PostTypeId="2" ParentId="5287" CreationDate="2016-10-14T07:33:41.123" Score="5" Body="&lt;p&gt;You can see &lt;a href=&quot;https://en.wikipedia.org/wiki/Nabla_symbol#History&quot; rel=&quot;noreferrer&quot;&gt;Nabla symbol&lt;/a&gt; as well as &lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot; rel=&quot;noreferrer&quot;&gt;VECTOR CALCULUS SYMBOLS&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The vector differential operator, now written &lt;span class=&quot;math-container&quot;&gt;$\nabla$&lt;/span&gt; and called &lt;em&gt;nabla&lt;/em&gt; or &lt;em&gt;del&lt;/em&gt;, was introduced by &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Rowan_Hamilton&quot; rel=&quot;noreferrer&quot;&gt;William Rowan Hamilton&lt;/a&gt; (1805-1865). Hamilton wrote the operator as a [90°] rotated nabla and it was &lt;a href=&quot;https://en.wikipedia.org/wiki/Peter_Tait_(physicist)&quot; rel=&quot;noreferrer&quot;&gt;P.G.Tait&lt;/a&gt; who established &lt;span class=&quot;math-container&quot;&gt;$\nabla$&lt;/span&gt; as the conventional symbol; see his &lt;a href=&quot;https://archive.org/details/anelementarytre03taitgoog&quot; rel=&quot;noreferrer&quot;&gt;An Elementary Treatise on Quaternions&lt;/a&gt; (1867). Tait was also responsible for establishing the term &lt;em&gt;nabla&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs&quot; rel=&quot;noreferrer&quot;&gt;Josiah Willard Gibbs&lt;/a&gt; &amp;amp; &lt;a href=&quot;https://en.wikipedia.org/wiki/Edwin_Bidwell_Wilson&quot; rel=&quot;noreferrer&quot;&gt;Edwin Bidwell Wilson&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_Analysis&quot; rel=&quot;noreferrer&quot;&gt;Vector analysis; a text-book for the use of students of mathematics and physics&lt;/a&gt; (1901), e.g &lt;a href=&quot;https://archive.org/details/vectoranalysiste00gibbiala&quot; rel=&quot;noreferrer&quot;&gt;page 138&lt;/a&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The vector sum which is the resultant rate of increase of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is denoted by &lt;span class=&quot;math-container&quot;&gt;$\nabla V$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\nabla V$&lt;/span&gt; represents a directed rate of change of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; - a directed or vector derivative of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, so to speak. For this reason &lt;span class=&quot;math-container&quot;&gt;$\nabla V$&lt;/span&gt; will be called the &lt;em&gt;derivative&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;; and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, the &lt;em&gt;primitive&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$\nabla V$&lt;/span&gt;. The terms &lt;em&gt;gradient&lt;/em&gt; and &lt;em&gt;slope&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; are also used for &lt;span class=&quot;math-container&quot;&gt;$\nabla V$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-10-14T07:33:41.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5289" PostTypeId="1" AcceptedAnswerId="5293" CreationDate="2016-10-14T09:47:59.860" Score="4" ViewCount="125" Body="&lt;p&gt;For some reason I recall hearing that around the time Euler wrote his Calculus books (1768-1770), or even before then, what we call integrals now were called solution cuvres (or even possibly the other way around since my memory on this is seriously hazy). Can I get some confirmation or refutation on this?&lt;/p&gt;&#xA;" OwnerUserId="3732" LastActivityDate="2016-10-15T01:44:22.503" Title="Were integrals really called solution curves (or vice versa)?" Tags="|mathematics|terminology|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5290" PostTypeId="2" ParentId="5286" CreationDate="2016-10-14T15:11:17.030" Score="5" Body="&lt;p&gt;I don't know if anyone can give the &quot;correct rationale&quot; for the peaks, but we can make some educated guesses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I agree with your explanations for the rise and peaks in the 1870s/80s and 1940s. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Contagious_Diseases_Acts&quot;&gt;Contagious Diseases Acts&lt;/a&gt; were quite controversial since their passage, and the press for their repeal gained quite a lot of publicity. Additionally, penicillin was a major breakthrough in treating a number of different diseases. I'd also argue that Nelson and Mayer's creation of the &lt;em&gt;Treponema pallidum&lt;/em&gt; immobilization test (TPI) in 1949 may have played a role in the 1940s peak (see &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3956094/&quot;&gt;Tampa et al. (2014)&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also agree that the early developments regarding &lt;em&gt;Treponema pallidum&lt;/em&gt; partially caused the early peak in the 1910s, first with its discovery in patients with syphilis and then the link between it and neurosyphilis, as put forth by &lt;a href=&quot;https://en.wikipedia.org/wiki/Hideyo_Noguchi&quot;&gt;Noguchi&lt;/a&gt; in 1913. There were other major breakthroughs around this time. You mentioned that Salvarsan and Neosalvarsan were major treatment options, thanks to Ehrlich. However, its application to war may have been what resulted in the peak. &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1053920/pdf/brjvendis00151-0005.pdf&quot;&gt;LW Harrison&lt;/a&gt; wrote quite a lot on this. Furthermore, developments in the use of bismuth (see &lt;a href=&quot;http://jmvh.org/article/syphilis-its-early-history-and-treatment-until-penicillin-and-the-debate-on-its-origins/&quot;&gt;here&lt;/a&gt;) in 1916 and then 1921 made it a second option, replacing arsenic. Without World War I, Ehrlich's discovery might not have gained quite as much attention. Also, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wassermann_test&quot;&gt;Wasserman test&lt;/a&gt;, developed in 1906 contributed to the diagnosis of syphilis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's one more amusing piece of information that may have led to a data bias in the late 19th and early 20th centuries. Syphilis cases rose in the United States between 1900 and 1920, and it was around this time that &lt;a href=&quot;http://www.essaysinhistory.com/articles/2012/103&quot;&gt;newspapers began using the term &quot;syphilis&quot;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Indeed, discussion of the disease only took place in specialized books and in medical journals, not in publications that most people would have read. Viewed as a subject beyond the &quot;boundaries of decency,&quot; syphilis was thought to be a disorder that affected only the immoral. In fact, the American press, yielding to the desires of the common people, was so unwilling to deal with the matter that the Reader's Guide to Periodical Literature did not include &quot;syphilis&quot; as a heading until 1907, and the New York Times Index avoided the term until late 1917.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This general trend should also account for a strong increase and then peak in the late 1910s/early 1920s.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-10-14T15:11:17.030" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5291" PostTypeId="2" ParentId="5286" CreationDate="2016-10-14T16:09:23.083" Score="0" Body="&lt;p&gt;I think this question is mainly about word usage. The English name for this disease is “pox”. You can find it dozens of times in Shakespeare, Swift etc. “Syphilis” started off as the personal name of a fictitious person (supposedly the first victim of the disease), the protagonist of a Latin poem from the 16th century. Later it became the pseudo-Latin medical designation for the disease. The surge in the usage of “syphilis” in English between 1850 and 1880 is presumably the result of Victorian prudery, which replaced “pox” by a more neutral sounding Latinate term. After 1880 we see a second wave of prudery, where “syphilis” itself becomes a dirty word and gets replaced by “infectious diseases” and the like. It comes back into favour at the time of the two world wars, when the dangers posed to British and American soldiers by women in the war zones is a frequent topic of the press.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-10-14T16:28:46.693" LastActivityDate="2016-10-14T16:28:46.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5292" PostTypeId="1" AcceptedAnswerId="5294" CreationDate="2016-10-14T21:13:21.450" Score="7" ViewCount="322" Body="&lt;p&gt;I would post this on Quora, since it is more of a &quot;just wondering&quot; sort of question, except that I much prefer StackExchange's platform:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As weird as Imperial units generally are, English &lt;strong&gt;volume&lt;/strong&gt; units surprisingly make a lot of sense.  I'm thinking there must be a reason for this... is there?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1 cup = 8 fluid ounces&lt;/li&gt;&#xA;&lt;li&gt;1 pint = 2 cups (16 fluid ounces)&lt;/li&gt;&#xA;&lt;li&gt;1 quart = 2 pints (32 fluid ounces)&lt;/li&gt;&#xA;&lt;li&gt;1 gallon = 4 quarts (128 fluid ounces)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Compared to some other craziness in the world of non-SI units, these make a ton of sense:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They are all base-2 multiples, which is &lt;strong&gt;really easy&lt;/strong&gt; for converting one unit to another, and for easily covering the &quot;space of interest&quot; (i.e., you'll never need to say 0.031 gallons, since you can instead say 4 fl. oz).&lt;/li&gt;&#xA;&lt;li&gt;Even the names are suggestive / helpful:&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a quart is a quarter of a gallon.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Was there a royal scientists or some-such academic responsible for this lack of insanity?&lt;/p&gt;&#xA;" OwnerUserId="4847" LastActivityDate="2016-10-15T03:16:00.170" Title="Why do English volume units use base 2?" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5293" PostTypeId="2" ParentId="5289" CreationDate="2016-10-15T01:44:22.503" Score="4" Body="&lt;p&gt;They are still so called in some contexts, although &lt;a href=&quot;https://en.wikipedia.org/wiki/Integral_curve&quot; rel=&quot;nofollow&quot;&gt;&quot;integral curves&quot;&lt;/a&gt; is more common. If $dy/dx=X(x)$ then $y$ is both an integral of $X$ and a solution to the differential equation. Such dual use does already appear in Euler's Institutionum Calculi Integralis. For example, he writes in §11 (in &lt;a href=&quot;http://www.17centurymaths.com/contents/euler/intcalvol1/preliminary.pdf&quot; rel=&quot;nofollow&quot;&gt;Bruce's translation&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Now even if here  only a single variable  quantity $x$  appears, yet in fact two are considered; for the other is that function itself, the  differential of which we take to be  $Xdx$; which if we designate with the letter $y$, becomes $dy=Xdx$ or $dy/dx=X$, thus as here generally is the ratio of the differentials $dy:dx$ proposed, which is equal to $X$, and thus there becomes $y=\int Xdx$.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And later in §§36-39 he talks about what is now called general and particular solutions to a differential equations, but in the context of (ordinary) integrals:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Hence the complete integral embraces all the particular integrals within itself and from that all these are able to be formed easily. But in turn from the particular integrals the general integral cannot become known. But many times, as henceforth it becomes apparent, a method of finding the complete integral from the particular integral is obtained.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He continues this dual use when he gets to solving the differential equations proper, e.g. in &lt;a href=&quot;http://www.17centurymaths.com/contents/euler/intcalvol1/part2ch1.pdf&quot; rel=&quot;nofollow&quot;&gt;§§411-415&lt;/a&gt; the problem of solving various examples is stated as:&quot;&lt;em&gt;To find the integral for this proposed differential equation&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-10-15T01:44:22.503" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5294" PostTypeId="2" ParentId="5292" CreationDate="2016-10-15T03:02:13.923" Score="8" Body="&lt;p&gt;The introduction of these units precedes the Royal Society by a long time, and the story of their origins is murky. Pints, pecks, gallons and quarts appeared in the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Winchester_measure&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Winchester Standards&quot;&lt;/a&gt; instituted in England by Henry VII in 1495. A standard bushel, gallon and quart made of bronze, issued in 1497 and stamped with the mark of Henry VII are still preserved in the British Museum. Later they became the basis of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Imperial_units#Volume&quot; rel=&quot;nofollow noreferrer&quot;&gt;Imperial Units&lt;/a&gt; defined in the British Weights and Measures Act of 1824. One of the earliest documents defining the gallon, bushel and quarter, is the Tractatus de Ponderibus et Mensuris, officially listed as an &quot;ancient statute of uncertain date&quot;, but estimated as  c. 1250 - 1305 and sometimes attributed to Henry III or Edward I: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;By Consent of the whole Realm the King’s Measure was made, so that... Twenty-pence make an Ounce; and Twelve Ounces make a Pound, and Eight Pounds make a Gallon of Wine; and Eight Gallons of Wine make a Bushel of London; which is the Eighth Part of a Quarter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But even in the Imperial Units the pint was still 20 fluid ounces, so it was not fully binarized even in 1824. There was no unit for 4 pints, I guess half-gallon worked well enough, but 1/4 pint was called gill.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason the &quot;Winchester Standards&quot; are so called is that c. 970, when the capital of England was Winchester, the then king &lt;a href=&quot;https://en.wikipedia.org/wiki/Edgar_the_Peaceful&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edgar the Peaceful&lt;/a&gt; allegedly instituted the said units already. Alas, no traces of them are found before the Norman Conquest of 1066, a century later, and most unit names (quarter, bushel, peck, gallon, pottle, quart, pint) have French roots. The units in use prior to the Norman Conquest, and even as late as 1196, were different and had Latin derived names (sester, amber, mitta, coomb, seam), although seams and sesters can be roughly equated to quarters and pints. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a side curiosity, the ancient Chinese units of weight went in multiples of 16, and we also do not know if some sage was behind it, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/5172/what-is-the-most-ancient-civilization-that-used-base-16-hexadecimal-number-sys/5177#5177&quot;&gt;What is the most ancient civilization that used base-16 (hexadecimal) number system?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-10-15T03:16:00.170" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5295" PostTypeId="1" CreationDate="2016-10-15T23:35:02.113" Score="5" ViewCount="289" Body="&lt;p&gt;What is the history of influential definitions of objective existence --- This Is Real, It Exists --- in physics? Where did they appear in the literature and in what context were they put forward?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please focus on the operational definitions of physicists, but you may also discuss arguments coming from mathematicians and natural philosophers. Who were the major individuals arguing in favor of each one? What were their reasons?&#xA;And what were the corresponding definitions or arguments regarding non-existence, un-reality, epiphenomenality, and of virtual objects or processes (virtual particles, virtual work, etc)?&lt;/p&gt;&#xA;" OwnerUserId="2713" LastEditorUserId="55" LastEditDate="2016-10-17T23:18:55.590" LastActivityDate="2016-10-22T02:58:43.937" Title="What did it historically mean in physics for something to &quot;exist&quot;?" Tags="|physics|philosophy-of-science|experimental-physics|theoretical-physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5296" PostTypeId="2" ParentId="5295" CreationDate="2016-10-18T23:53:15.677" Score="8" Body="&lt;p&gt;To not be measured is not to have any behavior. No dynamic is induced on &lt;em&gt;any&lt;/em&gt; other system in the universe by this object (Rosen 1978). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Objects without behavior do not exist. Their behavior &lt;em&gt;is&lt;/em&gt; that which makes some perceptions objective not subjective things (Hutton 1794).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is measured but not found different is double counted; it does not exist in addition to the other objects which exist (Leibniz 1686). Its whole behavior &lt;em&gt;is&lt;/em&gt; the object, however. This is what makes it different, something other than and partly independent of everything else including its observers and the empty state (Condillac 1756). So what is not measured ... (Hutton 1794, Bohr 1928; 1934, Rosen 1978, Wheeler 1994)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anticipating existence is not so much whether a symbol appears in a system of equations but whether the presence of one symbol somewhere being other than identity changes the result of applying one operator to another operator applied to ... in a way that builds up another operator in the system.*  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some formulations are equivalent but different in their virtual aspects. See the letters exchange between George Boole and Stanley Jevons regarding whether every object and operation arising during algebraic manipulation of an equation has and must have natural interpretation. Jevons: yes. Boole: no.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That measurement absence is nonexistence is quite noticeable at the smallest scale, because we can and do control for any other observers at that scale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads to some interesting results at the small scale regarding counterfactuals. (E.g. &lt;a href=&quot;http://rsta.royalsocietypublishing.org/content/374/2068/20150242&quot; rel=&quot;nofollow&quot;&gt;http://rsta.royalsocietypublishing.org/content/374/2068/20150242&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of this should not be confused with positivism, contrary Weinberg's (1992) allegation; see Wheeler's (1994) account. Positivism or operationalism are &lt;em&gt;further&lt;/em&gt; demands: that each specialized field of science only anticipates the existence of those objects and properties which can be measured by operations feasible or possible in that field without relying at all on any tools from any other fields. (So positivism and operationalism are just laziness codified into a method. Which is also why positivism and operationalism are sometimes popular ...)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;*Yes, people cannot directly measure primary scale particles. They are too small; but as distances get smaller ($10^{-15}$ etc) the frequencies and so the energies ($E = hv$) increase respectively. Wheeler (1971) suggested that larger particles are perturbations in the patterns of primary events in the quantum foam. This calculates out more or less correctly the lower energies we observe for the larger particles. However, measurement does not mean necessarily people measuring anything. Small objects measure other small objects, if one induces a dynamic in another. If by such constructive action we can get any of the objects in the group we suspect existence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;REFERENCES&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bohr&lt;/strong&gt;, &lt;strong&gt;Niels&lt;/strong&gt;, &lt;strong&gt;1928(4)&lt;/strong&gt;, The Quantum Postulate and the Recent Development of Atomic Theory, &lt;em&gt;Nature&lt;/em&gt; 121(15):580--590.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bohr&lt;/strong&gt;, &lt;strong&gt;Niels&lt;/strong&gt;, &lt;strong&gt;1934&lt;/strong&gt;, &lt;em&gt;Atomic Theory and the Description of Nature&lt;/em&gt;. Cambridge: Cambridge University Press.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bonnot de Condillac&lt;/strong&gt;, &lt;strong&gt;Etienne&lt;/strong&gt;, &lt;strong&gt;1754&lt;/strong&gt;, &lt;em&gt;Traite Des Sensations&lt;/em&gt;, I. London Paris: De Bure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Boole&lt;/strong&gt;, &lt;strong&gt;George&lt;/strong&gt;; &lt;strong&gt;Jevons&lt;/strong&gt;, &lt;strong&gt;Stanley&lt;/strong&gt;. &lt;strong&gt;1863&lt;/strong&gt;. Letters. In &#xA;1990(6), The Correspondence Between George Boole and Stanley Jevons, &lt;em&gt;History and Philosophy of Logic&lt;/em&gt; 12(1):15--35.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Hutton&lt;/strong&gt;, &lt;strong&gt;James&lt;/strong&gt;, &lt;strong&gt;1794&lt;/strong&gt;, &lt;em&gt;An Investigation of the Principles of Knowledge&lt;/em&gt;, I. Edinburgh London: Strahan Cadell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Leibnitz&lt;/strong&gt;, &lt;strong&gt;Gottfried&lt;/strong&gt;, &lt;strong&gt;1686&lt;/strong&gt;. In 1951, &lt;em&gt;Selections&lt;/em&gt;, Philip Wiener, trans. New York, Scribner. And 1970, &lt;em&gt;Philosophical Papers and Letters&lt;/em&gt;, Leroy Loemker, trans. Dordrecht: Reidel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Rosen&lt;/strong&gt;, &lt;strong&gt;Robert&lt;/strong&gt;, &lt;strong&gt;1978&lt;/strong&gt;, &lt;em&gt;Fundamentals of Measurement and Representation of Natural Systems&lt;/em&gt;. New York: North Holland.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Weinberg&lt;/strong&gt;, &lt;strong&gt;Steven&lt;/strong&gt;, &lt;strong&gt;1992&lt;/strong&gt;, &lt;em&gt;Dreams of a Final Theory&lt;/em&gt;. New York: Pantheon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Wheeler&lt;/strong&gt;, &lt;strong&gt;John&lt;/strong&gt;, &lt;strong&gt;1971&lt;/strong&gt;, From Atom To The Collapsing Star, &lt;em&gt;Periodicita e Simmetrie nella Struttura Elementare della Materia&lt;/em&gt;. Torino, Bona.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Wheeler&lt;/strong&gt;, &lt;strong&gt;John&lt;/strong&gt;, &lt;strong&gt;1994&lt;/strong&gt;, &lt;em&gt;At Home In The Universe&lt;/em&gt;. New York: American Institute of Physics.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2016-10-22T02:58:43.937" LastActivityDate="2016-10-22T02:58:43.937" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5297" PostTypeId="2" ParentId="5295" CreationDate="2016-10-19T09:37:51.630" Score="1" Body="&lt;p&gt;Guido's answer sounds fascinating but I am not sure I follow all of it so I will provide what seems to me a more down-to-earth analysis. In some branches of modern physics like string theory, the experiment and the idea of measurability have been left in the dust, and the field is approaching mathematics in its attitude toward the existence of objects.  If so, then the OP's question concerning modern theoretical physics is no different from a similar, and usual, question concerning the existence of entities in mathematics. The familiar joke cracked at the expense of these modern theories is that you would need an accelerator the size of the whole universe to confirm these theories.  I am actually not even sure of that being possible :-) This does not undermine the field; it simply ranks it alongside with mathematics.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2016-10-19T09:37:51.630" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5298" PostTypeId="2" ParentId="5264" CreationDate="2016-10-19T11:09:41.040" Score="2" Body="&lt;p&gt;Leibniz published in a book regarding this problem prior any of those letters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibnuetzio, Gottfredo, 1666, &lt;em&gt;Dissertatio De Arte Combinatoria&lt;/em&gt;, Lipsiae: Ficki, Seubold, p. 59. (Problem 3)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that this is listed as a problem he is trying to solve suggests the problem is much older. Which it probably is ... because of gambling. Likely this is one of the earlier publications however.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler published (as M.A. pointed out) his book somewhat later but gave a general formula and the modern statement (and name) of the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eulero, Leonhardo, 1748, &lt;em&gt;Introductio In Analysin Infinitorum&lt;/em&gt;, I, Lausannae: Bousquet, p. 255.&lt;/p&gt;&#xA;" OwnerUserId="163" LastActivityDate="2016-10-19T11:09:41.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5299" PostTypeId="1" AcceptedAnswerId="7657" CreationDate="2016-10-19T17:45:32.720" Score="4" ViewCount="443" Body="&lt;p&gt;I remember reading a historical overview of particle accelerators, which had a section devoted to the ill-fated &lt;a href=&quot;https://en.wikipedia.org/wiki/Superconducting_Super_Collider&quot; rel=&quot;nofollow&quot;&gt;Superconducting Super Collider&lt;/a&gt; (SSC). The SSC was partially constructed, in the early 1990s, but was cancelled after Congress stopped funding the project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The section contained an anecdote from congressional hearings about whether or not to continue funding the SSC, where a physicist involved in the project defended the project by talking about how a nation should ask itself if it is a nation of great artists and great poets, and that science should be just as important a cultural asset as art and poetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regrettably, I can't remember the quote or the physicist who said it, although I remember that it was the most eloquent defense of science funding I've ever read. Who was the physicist, and what was the quote?&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-08-31T04:25:10.140" Title="Which physicist defended the Superconducting Super Collider?" Tags="|experimental-physics|quote|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5300" PostTypeId="1" CreationDate="2016-10-19T19:58:08.247" Score="0" ViewCount="91" Body="&lt;p&gt;When were waves discovered/treated mathematically? Ex: basic sine wave&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I searched google for any information regarding the invention/establishment of a mathematical description of waves. I did not find any results. I know that there was an understanding of SHM in springs at around 1700 but I do not know if there exists a time period before then in which waves were abstractly constructed. &lt;/p&gt;&#xA;" OwnerUserId="4864" LastEditorUserId="4864" LastEditDate="2016-10-20T01:37:18.843" LastActivityDate="2016-10-20T01:37:18.843" Title="What time era were waves introduced a mathematical treatment?" Tags="|mathematics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5303" PostTypeId="1" AcceptedAnswerId="5304" CreationDate="2016-10-20T18:09:51.847" Score="11" ViewCount="421" Body="&lt;p&gt;I read that Zermelo formulated AC in 1904 in order to formally prove the well-ordering theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vitali’s 1905 proof of the existence of a non-measurable set of real numbers appears to the first “unpleasant” consequence of AC that is commonly cited.  &lt;strong&gt;Q&lt;/strong&gt;: Is this the first such example?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recall reading that someone has written a whole book on the unpleasant consequences of both AC and $\lnot$AC.  &lt;strong&gt;Q&lt;/strong&gt;: What is the first example obtained of an unpleasant consequence of $\lnot$AC, and what is the book whose name and author I cannot recall?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2016-10-21T15:50:03.043" LastActivityDate="2016-10-21T18:56:27.140" Title="What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?" Tags="|real-analysis|mathematical-logic|set-theory|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5304" PostTypeId="2" ParentId="5303" CreationDate="2016-10-20T21:56:34.717" Score="11" Body="&lt;p&gt;Vitali's construction is probably the first example &quot;unpleasant&quot; in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting that the continuum can be dissociated into points (contra Aristotle) was recent and hard enough, that it could be well-ordered strained credulity further. Borel and Lebesgue rejected it right away, and Poincare openly mocked the idea, Peirce and Weyl rejected even that the continuum is a set. Grothendieck topos would probably better fit what they had in mind than a set, but it is interesting that it was not proved until 1975 that &lt;a href=&quot;https://en.wikipedia.org/wiki/Diaconescu%27s_theorem&quot; rel=&quot;nofollow&quot;&gt;AC implies the law of excluded middle&lt;/a&gt;, and hence is automatically out for intuitionists. To Peano the axiom of choice itself looked like an unpleasant principle, not to be admitted. Interestingly enough, Poincare accepted the axiom of choice itself but rejected Zermelo's proof on the grounds of &lt;a href=&quot;https://en.wikipedia.org/wiki/Impredicativity&quot; rel=&quot;nofollow&quot;&gt;&quot;impredicativity&quot;&lt;/a&gt; of the well-ordering, a self-referential aspect in its definition. The same issue earlier led Russell to develop the theory of types. Zermelo's response (to both) was that without impredicativity many classical constructions (like the least upper bound) fall apart as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest compendium of axiom of choice zoo of pathologies is perhaps Sierpinski's long 1918 paper in French L'axiome de M. Zermelo et son róle dans la théorie des ensembles et l'analyse [Bulletin international de l'Académie des Sciences de Cracovie, Classe des sciences mathématiques et naturelles, Séie A, Sciences mathématiques, année 1918, pp. 97–152.] I do not think it was translated into English, but it started the enterprise of sifting through the classical mathematical theorems to determine whether or not they depend on AC, and looking for curiosities that follow from it or its negation (Sierpinski also did similar work for the continuum hypothesis). Polish mathematicians did a lot of work in these directions in 1918-1940, which is reflected in &lt;a href=&quot;https://books.google.com/books/about/Cardinal_and_ordinal_numbers.html?id=V4xsAAAAMAAJ&quot; rel=&quot;nofollow&quot;&gt;Sierpinski's Cardinal and Ordinal Numbers&lt;/a&gt;. In particular, the Banach-Tarski paradox came out of it, although it was a dramatization of the earlier Hausdorff's one from 1914. Another book not yet mentioned is &lt;a href=&quot;https://www.elsevier.com/books/equivalents-of-the-axiom-of-choice-ii/rubin/978-0-444-87708-6&quot; rel=&quot;nofollow&quot;&gt;Rubin-Rubin's Equivalents of the Axiom of Choice&lt;/a&gt;. Medvedev's Early History of the Axiom of Choice has a lot of early material with tracing of what does and does not depend on AC, but again it is not translated into English. Moore's book is perhaps the most accessible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the negation of AC, Sierpinski pointed out in 1918 that the measurability of the countable union of measurable sets depends on (countable) AC. That was a nice complement to Vitali, nonmeasurable sets may be unpleasant but without any AC there will be no Lebesgue measure theory at all. But already Zermelo's proof showed that AC is equivalent to well-orderability of all sets, so without it cardinalities would be thrown into &quot;chaos&quot;. That every cardinality is an alef (and in any case that continuum is) was Cantor's long-term conviction. He did not waver from it even when König at the 1904 International Congress of Mathematicians, in front of Cantor, presented his proof that the continuum could not be well-ordered. Cantor was sure that there must have been a mistake, then and there. Hausdorff and Zermelo soon realized that König made no mistakes, but there was a subtle gap in the proof of Bernstein's lemma which König used. So to Cantor not-AC would have been quite unpleasant. The eye of the beholder...&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-10-21T18:56:27.140" LastActivityDate="2016-10-21T18:56:27.140" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5306" PostTypeId="2" ParentId="2927" CreationDate="2016-10-22T19:22:56.033" Score="5" Body="&lt;p&gt;I would like to mention:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;J. L. Berggren, Mathematik im mittelalterlichen Islam, 2010&lt;/li&gt;&#xA;&lt;li&gt;M. Paty, Rationalités comparées des contenus mathématiques, 2002, Colloque des sciences arabes&lt;/li&gt;&#xA;&lt;li&gt;Michael Morgan: Lost History: The Enduring Legacy of Muslim Scientists, Thinkers, and Artists, 2008&lt;/li&gt;&#xA;&lt;li&gt;and especially the &lt;a href=&quot;https://en.wikipedia.org/wiki/Encyclopedia_of_the_History_of_Arabic_Science&quot; rel=&quot;nofollow&quot;&gt;Encyclopedia of the History of Arabic Science&lt;/a&gt;, edited by R. Rashed, with three volumes&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Volume 1: &quot;Astronomy--Theoretical and applied&quot;&lt;/li&gt;&#xA;&lt;li&gt;Volume 2: &quot;Mathematics and the Physical Sciences&quot;&lt;/li&gt;&#xA;&lt;li&gt;Volume 3: &quot;Technology, Alchemy, and the Life Sciences&quot;. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The works of &lt;a href=&quot;https://en.wikipedia.org/wiki/Roshdi_Rashed&quot; rel=&quot;nofollow&quot;&gt;Roshdi Rashed&lt;/a&gt; deserve a lot of attention. I recently heard from him through a collection of talks by François Nicolas, who draw relationships between IXth-XIIth century &quot;arabic&quot; mathematics toward modern category theory. Some of his works might be in French. Yet, some of his books are in arabic on one page, and translated (at least in French) on the other pages, which is quite  interesting.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Al-Bahir in Algebra As-Samaw'al&quot;, with S. Ahmad, University Press of Damascus, Damascus, 1972.&lt;/li&gt;&#xA;&lt;li&gt;&quot;The Work algebraic al-Khayyam&quot;, in collaboration with A. Djebbar, University Press of Aleppo, Aleppo, 1981.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Between Arithmetic and Algebra. Research on the History of Arabic Mathematics&quot;, Collection &quot;Arabic Sciences and Philosophy. Studies and included&quot;, Les Belles Lettres, Paris 1984. (ISBN 2251355316)&#xA;&quot;Studies on Avicenna&quot;, directed by J. Jolivet and R. Rashed, collection &quot;Arabic Sciences and Philosophy - Studies and included&quot;, Les Belles Lettres, Paris, 1984.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Mathematics of Sharaf al-Din al-Tusi. Algebra and Geometry in the twelfth century&quot;, Vol. II Collection &quot;Arabic Sciences and Philosophy - Texts and Studies&quot;, Les Belles Lettres, Paris, 1986.&#xA;Arabic translation: Beirut, 1998 .&lt;/li&gt;&#xA;&lt;li&gt;&quot;Optics and Mathematics: Research on the history of scientific thought in Arabic,&quot; Variorum reprints, Aldershot, 1992.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Geometry and Dioptrics the tenth century: Ibn Sahl al-Quhi and Ibn al-Haytham,&quot; Les Belles Lettres, Paris, 1993.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Al-Khayyam mathematician&quot;, in collaboration with B. Vahabzadeh, Bookstore Blanchard, Paris, 1999.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="4130" LastEditDate="2016-10-22T19:41:00.020" LastActivityDate="2016-10-22T19:41:00.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5307" PostTypeId="1" CreationDate="2016-10-23T16:50:15.280" Score="3" ViewCount="851" Body="&lt;p&gt;I'm looking for the way things were connected. It seems me to that it was like this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Spontaneous symmetry breakdown was detected in condensed matter physics. (I'm not sure of facts here)&lt;/li&gt;&#xA;&lt;li&gt;Later it turned out to be a problem in particle physics too.&lt;/li&gt;&#xA;&lt;li&gt;(1962) Backed by Weinberg and Salam, Goldstone published his theory where for continuous, exact and global symmetry breakdown there should be a massless scalar boson.&lt;/li&gt;&#xA;&lt;li&gt;(1964) Higgs and others worked on mechanisms that tried to solve the local gauge cause of spontaneous symmetry breakdown. In the Higgs mechanism three massive gauge bosons were predicted, the two W and Z, besides, of course, the Higgs boson.&lt;/li&gt;&#xA;&lt;li&gt;(1983) W and Z were detected at CERN.&#xA;(The Higgs boson is not really an issue in this question)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;What I'm trying to understand is specially the connection between points 2, 3 and 4. So here are my questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When, why and how did the spontaneous symmetry breakdown become a puzzle for particle physics? What was the event or theories that predicted the fact?&lt;/li&gt;&#xA;&lt;li&gt;It seems to me that Goldtone, Salam and Weinberg focused on the wrong problem. Am I right? How could have they simply not consider that the local gauge case was the place to look for the solution? &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="4580" LastEditorUserId="55" LastEditDate="2016-10-23T21:54:41.113" LastActivityDate="2016-10-24T20:25:19.027" Title="How was the spontaneous symmetry breaking introduced and how did it lead to the Higgs mechanism?" Tags="|physics|experimental-physics|theoretical-physics|particle-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5309" PostTypeId="2" ParentId="5307" CreationDate="2016-10-23T21:52:52.987" Score="2" Body="&lt;p&gt;Spontaneous symmetry breaking (SSB or SBS) refers to the fact that the lowest energy state, the vacuum, may not be invariant under all symmetries of the&#xA;theory, in other words, several vacua are possible. While it was always a logical possibility, before 1950s physicists generally regarded it as unphysical. First examples of SBS did indeed appear in condensed&#xA;matter physics, and the idea was transplanted into particle physics by analogy (as Nambu put it, &amp;quot;&lt;em&gt;I must say that my early exposure to condensed matter physics has been quite beneficial to me&lt;/em&gt;&amp;quot;). Jona-Lasinio gives some historical details in &lt;a href=&quot;http://www.gakushuin.ac.jp/univ/sci/phys/notice/JonaLasinioNov09.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spontaneous symmetry breaking:  some history and some variations on the theme:&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Heisenberg, Z. Naturforsch.14, 441 (1959), Proceedings of the 1960 Rochester Conference p. 851, was the first to consider SBS as a possibly relevant concept in particle physics... The theory of superconductivity of Bardeen, Cooper and Schrieffer which appeared in 1957 provided the key paradigm for the introduction of SBS in relativistic quantum field theory on the basis of an analogy proposed by Nambu, Phys. Rev. Lett. 4, 380 (1960)... The BCS theory of superconductivity was reformulated and developed by various authors including Bogolubov, Valatin, Anderson, Ricayzen and Nambu. The paper of Nambu, Phys. Rev.&#xA;117, 648 (1960), used a language akin to quantum field theory, that is Green’s functions formalism.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for &amp;quot;right&amp;quot; and &amp;quot;wrong&amp;quot; problems, it is always easy to make such judgements with a benefit of hindsight. Yang and Mills only introduced their non-Abelian gauge theories in 1956, much about them was still obscure and explored in the early 60s. The generality of the Goldstone &amp;quot;theorem&amp;quot;, and of the massless &lt;a href=&quot;https://en.wikipedia.org/wiki/Goldstone_boson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nambu-Goldstone bosons&lt;/a&gt; it predicted, was by no means clear, and neither was the &amp;quot;right&amp;quot; way to avoid them. Giving up the gauge principle altogether was considered as an option, along with questioning &amp;quot;theorem's&amp;quot; generality. In fact, it is these &amp;quot;wrong&amp;quot; bosons that led to the invention of the &amp;quot;right&amp;quot; Higgs mechanism. &lt;a href=&quot;http://philsci-archive.pitt.edu/9247/1/Karaca-Koray-Higgs.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karaca's Construction of the Higgs Mechanism and the Emergence of the Electroweak Theory&lt;/a&gt; gives a detailed account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This led, among the quantum field theorists, to the supposition that massless scalar bosons would be an inevitable consequence of all dynamical models of the Yang-Mills theory; thereby presenting a supposed dilemma which can be expressed as follows . The only way to solve the  zero - mass problem and give mass to gauge bosons is through the SSB of the original gauge symmetry of the Lagrangian. However, in this way, one encounters the difficulty posed by the Goldstone theorem — often referred to as the “Goldstone zero - mass difficulty” — namely that,  the existence of massless scalar bosons for which there is no experimental evidence...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The first explicit objection against the general validity of the Goldstone theorem came in 1963 from a solid-state physicist, namely, Philip Anderson... However, Anderson’s argument was an analogy argument from solid-state physics, and the demonstration of its validity within the theoretical framework of the Yang-Mills theory awaited contributions by a number of physicists... Higgs showed that the Goldstone theorem would break down in a class of non-manifestly Lorentz-covariant gauge theories that use the “radiation gauge”...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-10-24T20:25:19.027" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5311" PostTypeId="1" AcceptedAnswerId="5312" CreationDate="2016-10-24T16:31:55.527" Score="5" ViewCount="354" Body="&lt;p&gt;Dedekind’s 1858 introduction of Dedekind Cuts as a method of defining real numbers is a bona fide use of sets as mathematical objects.  He is using sets to define mathematical objects (real numbers) as a pair of sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prior to Dedekind, Boole worked with sets in his works &lt;em&gt;Mathematical Analysis of Logic&lt;/em&gt; of 1847 and &lt;em&gt;Investigations of the Laws of Thought&lt;/em&gt; of 1854.  Here, although Boole is using sets and defining operations on sets, it is with the aim of defining logical operations.  Since logical operators are not mathematical objects it is not clear to me that Boole’s works qualifies as a bona fide use of sets as mathematical objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prior to Boole, the waters become murkier. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;:  Who was first to introduce sets as bona fide mathematical objects?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2016-10-25T17:39:24.927" LastActivityDate="2022-09-27T21:04:45.700" Title="Who first treated sets of objects as mathematical objects in their own right?" Tags="|mathematics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5312" PostTypeId="2" ParentId="5311" CreationDate="2016-10-24T19:01:56.503" Score="6" Body="&lt;p&gt;The reason it gets murky is that the kind of use of which Dedekind cuts are prototypical became characteristic only after the arithmetization of analysis at the end of 19th century. Dedekind did not publish his 1858 work until 1872, by which time Cantor already started publishing his papers on point sets with similar ideas, including his own construction of real numbers as classes of rational sequences. This was a key difference between them and their closest precursor, Bolzano, who could not prove existence of say least upper bounds &amp;quot;rigorosly&amp;quot; because he lacked such constructions. We can bootstrap some prior uses into this but then what qualifies as &amp;quot;bona fide&amp;quot; becomes a judgement call. This is not uncommon, modern concepts have their precise meanings only in the modern context, when we try to apply them before that we are stretching it. But of course the modern context does not emerge out of nothing. Paradoxically, in history there are often no &amp;quot;firsts&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The tradition of &lt;a href=&quot;https://web.archive.org/web/20180302093615/http://mathforum.org/cgraph/history/planeloci.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;geometric loci&lt;/a&gt; predates Euclid. A locus is a figure singled out by a property (&amp;quot;symptom&amp;quot;, as Apollonius called it), e.g. the locus from which exactly two normals to a parabola can be drawn, considered by Apollonius. Aristheus (c.350 BC) wrote a book Plane Loci, not extant but mentioned by Pappus, which served as a stepping stone to the work of Euclid and Apollonius on conic sections. We now identify loci with &amp;quot;sets of points&amp;quot;, but this is not exactly how Greeks saw it, lines and planes were not dissociated into points then. Descartes brought these ideas closer to modern use in La Geometrie (1637), when he coordinatized the plane and identified loci with &amp;quot;zero sets&amp;quot; of polynomials, but even he does not treat them as &amp;quot;sets of coordinate pairs&amp;quot; as we do today.&lt;/p&gt;&#xA;&lt;p&gt;First Euler (1750, unpublished until 1849) and then more explicitly Gauss in Disquisitiones Arithmeticae (1798, published 1801) introduced congruence classes (Euler called them residues and preferred to think of them as remainders), and algebraic manipulations with them, the modular arithmetic, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3340/is-gauss-the-first-who-introduced-congruences/3346#3346&quot;&gt;Is Gauss the first who introduced congruences?&lt;/a&gt; These are perhaps closer in spirit to Dedekind cuts, indeed the whole idea of arithmetizing analysis was in part motivated by importing &amp;quot;Gauss's rigor&amp;quot; from arithmetic.&lt;/p&gt;&#xA;&lt;p&gt;The tradition Boole was following can be traced at least as far back as Leibniz, and with some charity to &lt;a href=&quot;https://en.wikipedia.org/wiki/Ramon_Llull#Llull.27s_Art&quot; rel=&quot;nofollow noreferrer&quot;&gt;Raymundus Lullius&lt;/a&gt; or even Aristotle. This is complicated by the conflation of intensions with extensions which persisted until the early 20th century, so in the early sources what is represented are more concepts than classes or sets they define. In any case, they were treated as &amp;quot;objects&amp;quot; with extensional connotation due to representation by circles or other figures in diagrams, what we now call &amp;quot;Venn diagrams&amp;quot;. The use of diagrams was popularized by Euler, but as &lt;a href=&quot;https://www.jstor.org/stable/3614533&quot; rel=&quot;nofollow noreferrer&quot;&gt;Baron writes in A Note on the Historical Development of Logic Diagrams&lt;/a&gt;:&amp;quot;&lt;em&gt;Despite Euler's stature as a mathematician he apparently made no  attempt to develop these relations algebraically. Nonetheless, through him, knowledge of the diagrams became widespread and they had some considerable influence in the nineteenth century&lt;/em&gt;&amp;quot;. Boole was one of the first to start thinking of what the circles represent as &amp;quot;classes&amp;quot; (extensions) rather than concepts, Venn, who gave the most comprehensive account of the diagrams and their manipulation, followed him. But the conflation is still present even in Russell and Zermelo, the purely extensional use is clearly articulated only in Hausdorff's Grundzuge der Mengenlehre (1914), see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2512/how-did-mathematicians-notate-the-empty-set-before-varnothing/2516#2516&quot;&gt;How did mathematicians notate the empty set before $\varnothing$?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="2277" LastEditDate="2022-09-27T21:04:45.700" LastActivityDate="2022-09-27T21:04:45.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5313" PostTypeId="2" ParentId="3468" CreationDate="2016-10-24T22:20:06.493" Score="6" Body="&lt;p&gt;Wilhelm  Körner between 1866-1874 was the first to use it to differentiate isomers of benzene rings. Since the Greek suffixes are fairly vague he chose them arbitrary. Its interesting he used orth- for 1,4 isomer, meta- for the 1,2 isomer , and pera-  for the 1,3 isomer. While later chemists used orth- for 1,2 isomer, meta- for the 1,3 isomer , and pera- for the 1,4 isomer. Then in 1879 the  Chemical Society of London officially adopted the backwards notation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So yea its a arbitrary and backwards system. I got all my info from the one link below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;ved=0ahUKEwjl6drurvTPAhXpxlQKHffWAysQFggeMAA&amp;amp;url=http%3A%2F%2Fwww.che.uc.edu%2Fjensen%2FW.%2520B.%2520Jensen%2FReprints%2F126.%2520Othro%2C%2520Meta%2C%2520Para.pdf&amp;amp;usg=AFQjCNGt4Kb4NAwOBG-x9FBKvsVSzM-Faw&amp;amp;sig2=t0Obb8qgxV8CfumN1wOa6A&amp;amp;cad=rja&quot; rel=&quot;noreferrer&quot;&gt;The Origins of the Ortho-, Meta-, and Para- Nomenclature&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="4880" LastActivityDate="2016-10-24T22:20:06.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5315" PostTypeId="2" ParentId="5311" CreationDate="2016-10-25T05:24:37.397" Score="2" Body="&lt;p&gt;Leibniz (1679) in an appendix to a letter sent to Huygens used sets of points to define geometric objects. For example: A(X) 8 AV is a constraint on the list of all points a,b,c, ... called {X} such that Aa, Ab, ... is congruent to line segment AV. They share an element A. So {X} a sphere. And so on.&#xA;He uses () not {} but ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was republished about three times in opera omnia type collections prior Dedekind and Cantor (and part of a question posed by Moebius to which Grassmann wrote his second monograph introducing inner products. Leibniz did nothing with this because Huygens was not interested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It was also the first abstract geometry. Without defining a metric as Riemann would later, any triangle is congruent to any other in this notation. Indeed, this confused Grassmann, and he suggested his own system winning the essay prize. Grassmann (1847) summarized the system at the start of his own book. This might have spread the notation.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Leibniz&lt;/strong&gt;, &lt;strong&gt;Gottfried&lt;/strong&gt;, &lt;strong&gt;1679&lt;/strong&gt;. In &lt;strong&gt;1969&lt;/strong&gt;, &lt;em&gt;Philosophical Papers and Letters&lt;/em&gt;, Leroy Loemker, trans., Dordrecht: Reidel, pp. 251--253.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Grassmann&lt;/strong&gt;, &lt;strong&gt;Hermann&lt;/strong&gt;, &lt;strong&gt;1847&lt;/strong&gt;, &lt;em&gt;Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik&lt;/em&gt;, Leipzig: Weidmann, pp. 4--5.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2016-10-25T15:29:31.393" LastActivityDate="2016-10-25T15:29:31.393" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5316" PostTypeId="1" CreationDate="2016-10-26T05:43:10.747" Score="4" ViewCount="255" Body="&lt;p&gt;&lt;sup&gt;&lt;strong&gt;Note&lt;/strong&gt;: This was first posted at Mathematics Stack Exchange, from which I was redirected here. Forgive me if this violates any crossposting rules I'm not currently aware of.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Currently, I'm starting work on a project about Evariste Galois. While I'm familiar with the basic details of his life — French, revolutionary, genius, headstrong, dead — and those details paint an amazing picture of him, I have not been able to find a comprehensive biography of him in English.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, my command of French is  poor and, thus, reading any biographies in French would be out of the question at this time. I was hoping you could point me to an authoritative biography in English, if such a thing exists, and if not, to compilations of his personal correspondence or the like.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your help.&lt;/p&gt;&#xA;" OwnerUserId="4890" LastEditorUserId="4251" LastEditDate="2021-06-22T08:56:37.277" LastActivityDate="2022-09-07T20:15:36.307" Title="Biography of Galois in English" Tags="|mathematicians|biographical-details|reference-request|19th-century|galois|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5317" PostTypeId="2" ParentId="5316" CreationDate="2016-10-26T05:58:19.683" Score="5" Body="&lt;p&gt;You can see : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Laura Toti Rigatelli, &lt;a href=&quot;https://books.google.it/books?id=zCjyBwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;Evariste Galois 1811–1832&lt;/a&gt; (1996).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Peter M. Neumann (editor), &lt;a href=&quot;https://books.google.it/books?id=fMh4KDMD_68C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow&quot;&gt;The Mathematical Writings of Évariste Galois&lt;/a&gt; (2011); there is a general Introduction as well as specific introductions to the  original papers : all in English.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-10-28T11:22:46.263" LastActivityDate="2016-10-28T11:22:46.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5319" PostTypeId="2" ParentId="5316" CreationDate="2016-10-26T19:37:01.653" Score="3" Body="&lt;p&gt;The classical source is Leopold Infeld, Whom the gods love,&#xA;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0873531256&quot; rel=&quot;nofollow&quot;&gt;https://www.amazon.com/Whom-Gods-Love-mathematics-education/dp/0873531256&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-10-26T19:37:01.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5320" PostTypeId="1" CreationDate="2016-10-26T06:12:11.107" Score="5" ViewCount="1502" Body="&lt;p&gt;Why is the thermoelectric figure of merit denoted by &lt;span class=&quot;math-container&quot;&gt;$Z T$&lt;/span&gt;? Does &lt;span class=&quot;math-container&quot;&gt;$Z T$&lt;/span&gt; come from the abbreviation of words in some language?&lt;/p&gt;&#xA;&lt;p&gt;Update: So far, &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; has been figured out — it is the temperature, to make the whole quantity dimensionless. So, the question is left only for &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4897" OwnerDisplayName="Alexander Zeng" LastEditorUserId="4251" LastEditDate="2020-07-08T06:55:22.253" LastActivityDate="2020-07-08T06:55:22.253" Title="Why is the thermoelectric figure of merit denoted by $ZT$?" Tags="|physics|terminology|notation|thermodynamics|solid-state-physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5321" PostTypeId="2" ParentId="5320" CreationDate="2016-10-26T07:36:05.887" Score="2" Body="&lt;p&gt;The concept of the thermoelectric figure of merit and the abbreviation ZT were introduced by the Russian physicist Abram Fedorovich Ioffe in 1949, so I assume the acronym originates from Russian. However all the references I can find are behind paywalls so frustratingly I cannot tell you how ZT is derived.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If your university has access to it, Ioffe's book where all this is described is Ioffe AF (1957) Semiconductor thermoelements, and Thermoelectric cooling.&lt;/p&gt;&#xA;" OwnerUserId="2082" OwnerDisplayName="John Rennie" LastActivityDate="2016-10-26T07:36:05.887" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5322" PostTypeId="1" CreationDate="2016-10-27T13:49:34.137" Score="3" ViewCount="4987" Body="&lt;p&gt;Otto Hahn (a german) found out about nuclear fission in 1938, so that should have given the Germans a head start in developing a nuclear bomb. However they did not succeed in doing so during WWII. I've heard a lot of reasons why they couldn't do it:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;allied bombing&lt;/li&gt;&#xA;&lt;li&gt;shortages of heavy water and other important raw material&lt;/li&gt;&#xA;&lt;li&gt;miscalculations (they thought graphite can't be used as moderator)  &lt;/li&gt;&#xA;&lt;li&gt;moral scruples by leading scientists&lt;/li&gt;&#xA;&lt;li&gt;German intelligence didn't know about the Manhattan project&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But does anybody know, which of these (or any others) have really been relevant? Why didn't the Germans succeed in building a nuclear bomb in second world war?&lt;/p&gt;&#xA;" OwnerUserId="3841" LastEditorUserId="5989" LastEditDate="2017-12-28T09:38:08.763" LastActivityDate="2018-01-02T18:57:44.217" Title="Why didn't Germany succeed in building a nuclear bomb in second world war?" Tags="|nuclear-physics|nazi-germany|world-war-2|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5323" PostTypeId="2" ParentId="401" CreationDate="2016-10-27T17:35:17.973" Score="0" Body="&lt;p&gt;Mantell v/s Owen, UK, 1830's/1840's (which of them deserved the credit for discovering the dinosaurs in the 1830's? - Owen was the first to name them - but Mantell had laid the groundwork: see Debbie Cadbury 'The Dinosaur Hunters')&lt;/p&gt;&#xA;" OwnerUserId="4899" LastActivityDate="2016-10-27T17:35:17.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5324" PostTypeId="2" ParentId="3257" CreationDate="2016-10-27T22:10:21.450" Score="3" Body="&lt;p&gt;Since $x^r+y^r=z^r$ can be transformed into a trinomial equation, we&lt;br&gt;&#xA;could give credit to Lambert, since he developed functions for it already&lt;br&gt;&#xA;in 1758, see for example this paper &lt;a href=&quot;https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; by Corless, et.al.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We quickly have $x^r*(1+(y/x)^r)=x^r*(z/x)^r$.&lt;br&gt;&#xA;Set $p=ln(y/x)$, $q=ln(z/x)$ and $s=e^r$. Then we get $1+s^p = s^q$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By further setting $w=s^q$ and $t=p/q$, we get the trinomial equation&lt;br&gt;&#xA;$-1+w=w^t$, which we have to solve for $w$ for a given $t$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit I:&lt;/strong&gt;&lt;br&gt;&#xA;The aforementioned paper mentions a series development by Euler,&lt;br&gt;&#xA;which is based on Lambert, which reads as follows:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$w^n = 1 + n*v + \frac{1}{2}*n*(n+α+β)*v^2$&lt;br&gt;&#xA;$\quad \quad + \frac{1}{6}*n*(n+α+2*β)*(n+2*α+β)*v^3$&lt;br&gt;&#xA;$\quad \quad + \frac{1}{24}*n*(n+α+3*β)*(n+2*α+2*β)*(n+3*α+β)*v^4$&lt;br&gt;&#xA;$\quad \quad + \, etc..$  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rough form matches the one as given by the OP. But concerning&lt;br&gt;&#xA;a full verification, i.e. matching the parameters α,β,v and positioning&lt;br&gt;&#xA;it correctly I was too busy so far and made a side question &lt;a href=&quot;https://math.stackexchange.com/questions/1989034/what-common-developments-are-there-to-solve-1x-xa&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit II:&lt;/strong&gt;&lt;br&gt;&#xA;The equation that Euler solves is $w^α-w^β = (α-β)*v*w^{α+β}$. We can&lt;br&gt;&#xA;choose $α=t-1$, $β=t$, $v=1$ and we get $w^{t-1}-w^t=(-1)*w^{2*t-1}$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we multiply both sides by $(-1)*w^{-t+1}$ we get $-1+w=w^t$ as required.&lt;br&gt;&#xA;We can choose $n=1$ and will get a development for $w$ which is on par&lt;br&gt;&#xA;with the function $f$ from the OP.&lt;/p&gt;&#xA;" OwnerDisplayName="user4900" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-11-06T18:43:24.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5325" PostTypeId="2" ParentId="5322" CreationDate="2016-10-27T22:24:39.577" Score="7" Body="&lt;p&gt;On my opinion, there are two important reasons:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) German government did not press the matter. Nobody convinced it that the thing is plausible and important. (In the US there was a lobby who could convince the US president in this. Szilard, V. Bush and Einstein played an important role). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) I think even if they wanted, they could not do it. They simply had no enough resources for a project of such scale. I mean resources of all kinds, including human, industrial, economic etc. Look how many top scientists were employed in the Manhattan project. And how many of them were immigrants from Germany, Austria, Italy. There were also problems with all other resources, and bombing played some role too. Look at the Oak Ridge uranium enrichment plant. Where could the Germans build anything comparable? And even if built this could not be kept secret, and would be certainly bombed.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-27T21:21:39.057" LastActivityDate="2017-12-27T21:21:39.057" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5326" PostTypeId="1" AcceptedAnswerId="5328" CreationDate="2016-10-28T20:51:22.893" Score="7" ViewCount="607" Body="&lt;p&gt;In the mathematics of the ancient Greek 1 has not been considered to be a number. Nevertheless 1 counted as a divisor of perfect numbers like 6 = 1 + 2 + 3. Is there any explanation for this inconsistency?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2016-10-29T07:44:27.097" LastActivityDate="2016-10-29T21:00:53.653" Title="Why did the ancient Greek count 1 in case of perfect numbers but not otherwise" Tags="|mathematics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5327" PostTypeId="2" ParentId="5311" CreationDate="2016-10-28T20:59:46.237" Score="2" Body="&lt;p&gt;The first to introduce the &lt;em&gt;terminus set&lt;/em&gt; (in German: Menge) in its mathematical meaning was Bernard Bolzano in &quot;Paradoxien des Unendlichen&quot;, Reclam, Leipzig, 1851, p. 2.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid's &quot;Therefore, prime numbers are more than any assigned multitude of prime numbers.&quot; certainly belongs to the first uses of the idea of set.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2016-10-29T07:40:24.340" LastActivityDate="2016-10-29T07:40:24.340" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5328" PostTypeId="2" ParentId="5326" CreationDate="2016-10-28T22:40:50.747" Score="9" Body="&lt;p&gt;Euclid (and this is not &quot;the Greeks&quot;, although the same definition can be found in other places as well) defined a number as a multitude of ones. Since 1 is not a multitude of ones, 1 was not a number. 6, on the other hand, is a multitude of ones, and for this reason has 1 as a divisor. And since 6 is not a multiple of 6, its only divisors were 1, 2 and 3. In particular, there is no inconsistency whatsoever in your example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Occasionally Euclid includes 1 as a number, in particular in a couple of propositions that are believed to go back to the Pythagoreans.&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2016-10-28T22:40:50.747" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5329" PostTypeId="1" AcceptedAnswerId="5342" CreationDate="2016-10-29T03:57:58.747" Score="5" ViewCount="2084" Body="&lt;p&gt;&lt;a href=&quot;http://www.organic-chemistry.org/namedreactions/saytzeff-rule.shtm&quot; rel=&quot;noreferrer&quot;&gt;Saytzeff&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Zaitsev%27s_rulehttps://en.wikipedia.org/wiki/Zaitsev%27s_rule&quot; rel=&quot;noreferrer&quot;&gt;Zaitsev&lt;/a&gt; rules are the same thing.Then why are they named differently?Some books favour to use Saytzeff while some prefer Zaitsev.Why are different names used for the same thing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks for any help!!&lt;/p&gt;&#xA;" OwnerUserId="2955" LastActivityDate="2018-10-11T20:26:15.033" Title="Why Saytzeff and Zaitsev Rules are named differently?" Tags="|chemistry|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5330" PostTypeId="2" ParentId="5320" CreationDate="2016-10-29T14:57:22.500" Score="2" Body="&lt;p&gt;My guess for the &lt;strong&gt;Z&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://tureng.com/en/german-english/figure%20of%20merit&quot; rel=&quot;nofollow&quot;&gt;THIS&lt;/a&gt; Germain-English dictionary, we see that &quot;Figure of merit&quot; in German could be&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Gütezahl&lt;/strong&gt;  $\qquad$  or  $\qquad$  &lt;strong&gt;Leistungskennzahl&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in electrical engineering.  So probably the letter &lt;strong&gt;Z&lt;/strong&gt; is for &lt;strong&gt;Zahl&lt;/strong&gt;, the German word for &quot;number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[moral: use Google translate with a grain of salt]&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-10-29T14:57:22.500" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5331" PostTypeId="2" ParentId="5329" CreationDate="2016-10-29T16:42:38.193" Score="6" Body="&lt;p&gt;The rule is named after the Russian chemist Александр Михайлович Зайцев. He studied in Germany and wrote most of his papers in German, spelling his name consistently as “Saytzeff”. Zaytsev or Zaitsev are English spellings of the same name. Actually, there is a case for spelling Russian names the way their bearers did themselves when they used Western languages. &lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-10-29T16:59:57.517" LastActivityDate="2016-10-29T16:59:57.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5332" PostTypeId="2" ParentId="5326" CreationDate="2016-10-29T18:25:09.957" Score="4" Body="&lt;p&gt;I am not convinced that there is any inconsistency. In El. VII, def. 2 Euclid writes “A number is a multitude composed of units.”  ( ̓Αριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος.) In Def. 22 he says: “A perfect number is that which is equal to its own parts.” (Τέλειος ἀριθμός ἐστιν ὁ τοῖς ἑαυτοῦ μέρεσιν ἴσος ὤν.) He does not say that the “parts” of perfect numbers are themselves “numbers”. The parts are a “unit” (that is: 1) and one or more “numbers”.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-10-29T21:00:53.653" LastActivityDate="2016-10-29T21:00:53.653" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5333" PostTypeId="2" ParentId="3540" CreationDate="2016-10-29T20:31:48.610" Score="1" Body="&lt;p&gt;I have heard the rumor that Carl Friedrich Gauss (1777 - 1855) called $9^{9^{9^9}}$ &quot;the measurable infinity&quot; (die messbare Unendlichkeit).&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2016-10-29T20:31:48.610" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5334" PostTypeId="1" AcceptedAnswerId="5349" CreationDate="2016-10-30T04:27:10.133" Score="2" ViewCount="107" Body="&lt;p&gt;I was telling a friend about a pioneer researcher in genetics who did his work in Northern California around the late 1800s. I was thinking this was in Marin, Sonoma, or Napa County.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I tried to remember the name I drew a blank. My first thought was, &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gregor_Mendel&quot; rel=&quot;nofollow&quot;&gt;Gregor Mendel&lt;/a&gt;?&quot;, but of course that's not right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I may be confused and it &lt;em&gt;was&lt;/em&gt; Mendel I was thinking of, and I just mixed up California with eastern Europe. But is there someone who fits the description I mentioned above?&lt;/p&gt;&#xA;" OwnerUserId="4910" LastEditorUserId="4251" LastEditDate="2020-07-13T01:55:17.853" LastActivityDate="2020-07-13T01:55:17.853" Title="Who was the Northern California pioneer of genetics?" Tags="|biology|genetics|biologists|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5335" PostTypeId="1" CreationDate="2016-10-30T14:53:53.873" Score="10" ViewCount="900" Body="&lt;p&gt;In &lt;strong&gt;Ramanujan's Notebooks Volume IV pg. 31&lt;/strong&gt; by Bruce C. Berndt, he describes an easy way to solve the general quartic by starting with the system$$x^2+ay=b\\y^2+cx=d\tag1$$&#xA;And solving for $x$; which gives you the depressed form of the quartic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I'm wondering whether Ramanujan had his own methods for solving the general cubic, quintic, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And according to &lt;a href=&quot;https://math.stackexchange.com/questions/540964/how-to-solve-fifth-degree-equations-by-elliptic-functions&quot;&gt;this MSE post&lt;/a&gt;, Ramanujan had his own method for solving a solvable quintics.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What was Ramanujan's method for solving the solvable quintic and quartic?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; I mean the sort of sneaky system (like $(1)$) employed by Ramanujan, and some substitution.&lt;/p&gt;&#xA;" OwnerUserId="4913" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-11-21T23:41:29.340" Title="Ramanujan's Method for solving cubic, quartic, quintic" Tags="|mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5336" PostTypeId="1" CreationDate="2016-10-30T19:44:23.010" Score="10" ViewCount="350" Body="&lt;p&gt;I have lost the reference to a story of someone (I believe it was Emil Artin) who found a strategy for understanding Emmy Noether's rapid delivery of new ideas during walks.  It was to walk fast enough and long enough that Noether would get winded, and  her delivery would slow down a bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone direct me to a source?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it is worth, Noether's Bryn Mawr student Ruth McKee tells us at age 52 Noether could climb between the rails of a rail fence without slowing her discussion. (in Srinivasan and Sally eds. &lt;em&gt;Emmy Noether in Bryn Mawr&lt;/em&gt; Springer 1983, p. 144.)  But still, I guess enough walking could slow her down a bit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since I have published several articles on Noether it is not feasible for  me to list here all the sources I have looked at for this.  So please let me know if you know a source but do not suggest sources that you just think might have it.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2016-10-31T13:48:17.093" LastActivityDate="2016-10-31T15:23:04.300" Title="Seeking anecdote on Emmy Noether and possibly Emil Artin" Tags="|biographical-details|mathematicians|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5338" PostTypeId="2" ParentId="5336" CreationDate="2016-10-31T09:13:16.287" Score="2" Body="&lt;p&gt;You can try with :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Auguste Dick, ,&lt;a href=&quot;https://books.google.it/books?id=3bjSBwAAQBAJ&amp;amp;pg=PP4&quot; rel=&quot;nofollow&quot;&gt;Emmy Noether 1882–1935&lt;/a&gt; (1981). &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-10-31T09:13:16.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5339" PostTypeId="2" ParentId="5336" CreationDate="2016-10-31T15:23:04.300" Score="10" Body="&lt;p&gt;I got it. The key was to search Natascha Artin. It is Emil Artin's strategy and it is on page 34 of the other premier source on Noether (along with Auguste Dick's book) C. Kimberling, &quot;Emmy Noether,&quot; in James W. Brewer and Martha K. Smith, editors, &lt;em&gt;Emmy Noether, A Tribute to Her Life and Work&lt;/em&gt;, Marcel Dekker, New York, 1981, Chapter 1, pages 1-61.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2016-10-31T15:23:04.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5340" PostTypeId="1" CreationDate="2016-10-31T23:29:58.690" Score="2" ViewCount="569" Body="&lt;p&gt;I'm taking a class in language acquisition called &quot;Nature vs Nurture&quot;. I'm not particularly fond of that framing, because the divide seems overly dichotomous. In addition, the N-vs-N debate has been used as a tool of oppression throughout history, namely in eugenics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm wondering whether there are any cases in linguistics (or related fields like Cognitive Sciences) where this debate is being used for harm?&lt;/p&gt;&#xA;" OwnerUserId="4907" LastEditorUserId="2442" LastEditDate="2016-11-03T16:37:23.680" LastActivityDate="2022-01-29T07:40:07.530" Title="Harm of nature vs nurture debate?" Tags="|social-context|neuroscience|linguistics|psychology|social-sciences|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5341" PostTypeId="1" AcceptedAnswerId="5343" CreationDate="2016-11-01T10:08:28.293" Score="12" ViewCount="1362" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy#Professor_at_.C3.89cole_Polytechnique&quot;&gt;Augustin-Louis Cauchy&lt;/a&gt; was a prolific writer, his writings range widely in mathematics and mathematical physics. As a professor at &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89cole_Polytechnique&quot;&gt;École Polytechnique&lt;/a&gt; he came in contact and reviewed &lt;a href=&quot;https://en.wikipedia.org/wiki/Niels_Henrik_Abel&quot;&gt;Abel's&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89variste_Galois&quot;&gt;Galois'&lt;/a&gt; early mathematical papers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to his busy work shedule, in the former (Abel's) case, it said that Cauchy&lt;sup&gt;1&lt;/sup&gt;, put the works aside and they were forgotten and the originals even lost, in the latter case (Galois') he refused to accept them for publication for reasons that still remain unclear. In both cases the consequences for the authors were devastating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any other &quot;victims&quot; of Cauchy's negligence and slopiness? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1. E.T. Bell - Men of Mathematics&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="2351" LastEditorUserId="4251" LastEditDate="2021-06-21T20:59:06.183" LastActivityDate="2021-06-21T20:59:06.183" Title="Did Cauchy forget or lose mathematical papers aside from Abel's and Galois's?" Tags="|mathematicians|biographical-details|debunking|19th-century|cauchy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5342" PostTypeId="2" ParentId="5329" CreationDate="2016-11-01T22:05:42.413" Score="7" Body="&lt;p&gt;In German language, &quot;Z&quot; is pronounced like English &quot;ts&quot;, and &quot;S&quot; is pronounced like English &quot;z&quot;. So a German reader would pronounce “Saytzeff” close to his Russian name, Зайцев.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So “Saytzeff” is close to German phonetic to his (Russian) name, while Zaitsev would be transliteration of it, or writing it in a way so English-speaking reader can pronounce it close to the sound of his Russian name, Зайцев.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;IOW: “Saytzeff” is a version for audience speaking German, &quot;Zaitsev&quot; is for  English-speaking one.&lt;/p&gt;&#xA;" OwnerUserId="4386" LastEditorUserId="4386" LastEditDate="2018-10-11T20:26:15.033" LastActivityDate="2018-10-11T20:26:15.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5343" PostTypeId="2" ParentId="5341" CreationDate="2016-11-02T01:02:37.650" Score="11" Body="&lt;p&gt;Apparently, Cauchy did not even forget or lose at least the papers of Galois. This is yet another example of how E.T. Bell and his Men of Mathematics, for all its literary virtues, are unreliable historical sources. See &lt;a href=&quot;https://hsm.stackexchange.com/questions/2397/what-resources-are-available-for-lives-of-recent-mathematicians-besides-e-t-bel/2400#2400&quot;&gt;What resources are available for lives of recent mathematicians besides E.T. Bell&amp;#39;s Men of Mathematics?&lt;/a&gt; for more on E.T. Bell and his writing style. As for Cauchy and Galois, Rothman has an informative chapter in his &lt;a href=&quot;https://books.google.com/books?id=rgkABAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Science a la Mode&lt;/a&gt;, and the article it is based on is freely available online, courtesy of MAA: &lt;a href=&quot;http://www.maa.org/programs/maa-awards/writing-awards/genius-and-biographers-the-fictionalization-of-evariste-galois&quot; rel=&quot;noreferrer&quot;&gt;Genius and Biographers: The Fictionalization of Evariste Galois&lt;/a&gt;. Not only was it not Cauchy but Fourier who &amp;quot;lost&amp;quot; Galois's memoir (and he had a pretty good excuse), but the Galois-theory-on-the-eve-of-the-duel story is also a piece of Bell's fiction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We now encounter a major myth which evidently had its origin in the very first writings on Galois and which had been perpetuated by virtually all writiers since. This myth is the assertion that Cauchy either forgot or lost the papers (Dupuy, Bell) or intentionally threw them out (Infeld). Recently, however, Rene Taton has discovered a letter of Cauchy's in the Academy archives, which conclusively proves that he did not lose Galois' memoirs but had planned to present them to the Academy in January 1830. There is even some evidence that Cauchy encouraged Galois...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;This letter makes it clear that, six months after their receipt, Cauchy was still in possession of Galois's manuscripts, had read them and was very likely aware of their importance. At the following session on 25 January, however, Cauchy, while presenting his own memoir, did not present Galois's work. Taton hypothesizes that between January 18 and January 25, Cauchy persuaded Galois to combine his researches into a single memoir to be submitted for the Grand Prize in Mathematics, for which the deadline was March 1. Whether or not Cauchy actually made the suggestion cannot yet be proved, but in February Galois did submit such an entry to Fourier in his capacity of perpetual secretary of mathematics and physics for the Academy.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The misfortune referred to above was the death of Fourier on May 16, 1830. Galois's entry could not be found among Fourier's papers and in Galois's eyes this could not be an accident. &amp;quot;The loss of my memoir is a very simple matter,&amp;quot; he wrote. &amp;quot;It was with M. Fourier, who was supposed to have read it and, at the death of this savant, the memoir was lost.&amp;quot; In addition to propagating the legend that Cauchy lost the manuscripts, Bell, curiously, does not mention Fourier by name in the preceding misadventure, although Dupuy is explicit on the identity of the Academy's Perpetual Secretary. Perhaps Bell felt it a little too much to &amp;quot;expose&amp;quot; Cauchy, Fourier and later Poisson as incompetents. Bell also does not make it clear that the papers listed above (plus a later memoir) constitute what is now called Galois theory. If this point had been clarified, the claim that Galois had written down the theory on the eve of the duel would be difficult to substantiate or even to suggest.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Rothman details Bell's other &amp;quot;creative contributions&amp;quot; to Galois's biography, like scrambled timeline, omitted events and manufactured quotes. Galois's early biographer, Dupuy, and Infeld share some of the &amp;quot;credit&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Taton's paper is in French, but it is not that recent (Rothman wrote back in 1982): &lt;a href=&quot;https://www.jstor.org/stable/24468266&quot; rel=&quot;noreferrer&quot;&gt;Sur les Relations Scientifiques d'Augustin Cauchy et d'Evariste Galois&lt;/a&gt;, Revue d'Histoire des Sciences 24 (2):123-148 (1971)&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-11-02T18:38:09.583" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5344" PostTypeId="1" AcceptedAnswerId="5345" CreationDate="2016-11-02T21:04:07.343" Score="3" ViewCount="267" Body="&lt;p&gt;What is the name of this indentity and which mathematician did derive this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="2442" LastEditDate="2016-11-03T16:27:09.647" LastActivityDate="2016-11-03T19:18:30.877" Title="What is the name of the identity $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ and who derived it?" Tags="|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5345" PostTypeId="2" ParentId="5344" CreationDate="2016-11-02T23:42:24.727" Score="6" Body="&lt;p&gt;Vector analysis identities typically do not have special names and appear as exercises in textbooks, they can be derived in too routine a manner to take notice who was first to do it. The first of such textbooks, Vector Analysis, a Text-book for the Use of Students of Mathematics and Physics has a &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_Analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; all to itself and was written by the co-founder (with Heaviside) of vector analysis Gibbs together with Wilson in 1901. The book went through seven editions afterwards, but it was based on Gibbs's 1881-1884 lectures Elements of Vector Analysis that were printed as a short pamphlet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://archive.org/stream/117714283#page/160/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;last edition of the text is available on Gutenberg&lt;/a&gt;, where an even more general identity appears as formula (46)′ on p.161:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\mathbf{\nabla (u \cdot v)_u = (u \cdot \nabla)v+ u \times (\nabla \times v )}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earlier on p. 159 the authors explain: &quot;&lt;em&gt;The notation $\bf\nabla (u \cdot v)_u$&#xA;will be used to denote that in applying the operator $\bf\nabla$ to the product $\bf(u\cdot v)$ the quantity $\bf u$ is to be regarded as &lt;strong&gt;constant&lt;/strong&gt;. That is, the operation $\bf\nabla$ is carried out only &lt;strong&gt;partially&lt;/strong&gt; upon the product $\bf(u\cdot v)$&lt;/em&gt;&quot;. Apparently, Gibbs and Wilson felt very strongly about this notation since they promote it in a footnote on the same page:&quot;&lt;em&gt;This idea and notation of a partial $\bf\nabla$ so to speak may be avoided by means of the formula (41). But a certain amount of compactness and simplicity is lost thereby. The idea of $\bf\nabla (u \cdot v)_u$ is surely no more complicated than $\bf(u \cdot \nabla)v$ or $\bf v\times(\nabla \times u)$&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By inspection from the coordinate formulas on the next page, $\bf\nabla (u \cdot v)_u$ is equal to $\frac{1}{2}\mathbf{\nabla (u \cdot u)}$ when $\bf v=u$. There is a lot of applying this identity on the subsequent pages, but I am afraid no name for it other than &quot;(46)′ &quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-11-03T19:18:30.877" LastActivityDate="2016-11-03T19:18:30.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5346" PostTypeId="2" ParentId="264" CreationDate="2016-11-03T07:24:08.313" Score="41" Body="&lt;p&gt;If anyone's still reading this thread, here's a few more data points that appear to back Feynmann's interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nature.com/nature/journal/v123/n3098/pdf/123409a0.pdf&quot; rel=&quot;noreferrer&quot;&gt;Erik Bäcklin, Nature vol 123&lt;/a&gt;, no. 3098, p. 409 (1929): $1.59875 \cdot 10^{-19} \pm 0.004796 \cdot 10^{-19} $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wide error bar overlaps Millikan's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://journals.aps.org/pr/pdf/10.1103/PhysRev.50.190&quot; rel=&quot;noreferrer&quot;&gt;Gunnar Kellstrom, Phys. Rev. 1935&lt;/a&gt;: $1.60709  \cdot 10^{-19} \pm 0.011 \cdot 10^{-19} $.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Revises Millikan's value based on updated viscosity of air. Wide error bars!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://iopscience.iop.org/0034-4885/4/1/312&quot; rel=&quot;noreferrer&quot;&gt;H.R. Robinson, 1937 Rep. Prog. Phys. 4 212&lt;/a&gt;: $4.803 \cdot 10^{-10} \, \text{esu} = 1.602 \cdot 10^{-19} \, \text{C}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concludes with a note added at the proof stage: &quot;Backlin and Flemberg’s result, however, comes down to $4.778\cdot10^{-10}$--or very nearly to Millikan's 1917 value! This is a most unfortunate note on which to end the Report.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WtmUj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WtmUj.png&quot; alt=&quot;Oil drop charge--historical reports&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4926" LastEditorUserId="2442" LastEditDate="2016-11-03T16:42:28.627" LastActivityDate="2016-11-03T16:42:28.627" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5347" PostTypeId="1" CreationDate="2016-11-03T13:11:26.637" Score="9" ViewCount="510" Body="&lt;p&gt;When a variable $y$ depends on other variables, say $y=c x^3$, one often writes &#xA;$$y|_{x=2}$$ &#xA;to say &quot;$y$ when $x$ has value $2$&quot;. This might be more familiar in the context of derivatives where we find $\left. \frac{dy}{dx}\right|_{x=2}=12c$. It also appears in other areas of mathematics. For example the fiber of a bundle $E$ over some point $p$ is often denoted with $E|_p$. If we interpret a fiber bundle as a space depending on points of the base this seems to be analogous to the previous example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice that in the example $y=cx^3$ it would be wrong to write $y(2)=8c$ since $y$ is not a map of type $\mathbb{R} \to \mathbb{R}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; Who introduced this bar notation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; How did people in the early days express &quot;$y$ under the condition that $x=2$&quot;? I'm guessing they just used words, but was there an alternative notation? Or did they abuse the notation for function application, as in $y(2)$, once Euler/Bernoulli had introduced it?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2017-08-16T15:26:39.740" LastActivityDate="2018-05-18T22:39:54.463" Title="Who introduced the notation $y|_{x=a}$?" Tags="|mathematics|notation|calculus|mathematical-logic|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5348" PostTypeId="2" ParentId="5347" CreationDate="2016-11-03T15:03:04.873" Score="10" Body="&lt;p&gt;You can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Giuseppe_Peano&quot; rel=&quot;noreferrer&quot;&gt;Giuseppe Peano&lt;/a&gt; , &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN585496110|LOG_0006&amp;amp;physid=PHYS_0024&quot; rel=&quot;noreferrer&quot;&gt;Lezioni di Analisi Infinitesimale&lt;/a&gt;, 2 vols., 1893, page 17 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$[f(x)]_{x=a}=f(a).$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not sure it is the earliest... but Peano was a prolific &quot;inventor of notations&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Regarding : how they express &quot;$y$ under the condition that $x=2$&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;see e.g. &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN585496110|LOG_0006&amp;amp;physid=PHYS_0039&quot; rel=&quot;noreferrer&quot;&gt;page 34&lt;/a&gt; [shortened] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;let $y$ the natural logarithm of $x$ [...] and $f(x) = \text {Log} x$. Compute $\text {Log} 1001$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-11-03T15:46:43.263" LastActivityDate="2016-11-03T15:46:43.263" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5349" PostTypeId="2" ParentId="5334" CreationDate="2016-11-04T23:57:29.963" Score="6" Body="&lt;p&gt;This was almost certainly &lt;a href=&quot;https://en.wikipedia.org/wiki/Luther_Burbank&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luther Burbank&lt;/a&gt;. He was a horticulturist who started in Lunenburg, Mass and moved to Santa Rosa, CA in 1875. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Burbank developed hundreds of fruit, vegetable, and flower cultivars . His most famous contribution is the &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Russet_Burbank&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russet Burbank&lt;/a&gt;&quot; potato, which is the most commonly grown potato variety in the US today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although his methods weren't particularly &quot;scientific&quot; (poor record-keeping) and he didn't practice &quot;genetics&quot; per se (the term hadn't been invented yet), his experiments were inspired by Darwin.&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="62" LastEditDate="2016-11-05T21:48:56.767" LastActivityDate="2016-11-05T21:48:56.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5351" PostTypeId="1" CreationDate="2016-11-08T10:32:06.027" Score="4" ViewCount="572" Body="&lt;p&gt;Why are the sign conventions of work done in thermodynamics different for physics and chemistry?&lt;/p&gt;&#xA;" OwnerUserId="4940" LastActivityDate="2017-05-15T02:12:46.757" Title="Different sign conventions in thermodynamics different for physics and chemistry" Tags="|physics|chemistry|thermodynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5352" PostTypeId="2" ParentId="5351" CreationDate="2016-11-08T10:41:28.350" Score="1" Body="&lt;p&gt;The difference can be understood by looking at the history of development of Thermodynamics . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During the initial stages  of industrial revolution the main focus was to get the work done by a machine the more the work done by the machine the more beneficial it was thought to be . Thus it assumes a positive sign for work done by the system as we are measuring the work done by the system (Machine). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand chemistry deals more with changes in the composition of the working fluid or composition of the system-surroundings more in general. Thus, work done on the system which brings about changes in the system is considered positive&lt;/p&gt;&#xA;" OwnerUserId="4940" LastActivityDate="2016-11-08T10:41:28.350" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5353" PostTypeId="1" CreationDate="2016-11-08T12:33:22.427" Score="7" ViewCount="76" Body="&lt;p&gt;I was wondering if there was a standard system, at some point in history, for academic positions in Latin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I mean is that today each country has a standard system that names each academic position, for example in England if you are the head of a university you will be called a &quot;chancellor&quot;, regardless which university you head, as long as it is in England.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Different countries today use different words for different positions, some are derived from Latin, some are not. I was wondering if there ever was a standard of Latin-language academic position titles that have been used in the past at some point by several or more countries? Because I'd wager some countries had Latin titles but they were the only ones to use them.&lt;/p&gt;&#xA;" OwnerUserId="4941" LastEditorUserId="4941" LastEditDate="2016-11-08T15:22:25.063" LastActivityDate="2016-11-08T15:22:25.063" Title="Latin academic positions" Tags="|academia|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5354" PostTypeId="2" ParentId="5353" CreationDate="2016-11-08T13:15:09.647" Score="3" Body="&lt;p&gt;Possible sources :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alan Cobban, &lt;a href=&quot;https://books.google.it/books?id=oz0OAAAAQAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Medieval Universities: Their Development and Organization&lt;/a&gt; (1975)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Alan Cobban, &lt;a href=&quot;https://books.google.it/books?id=nIiNAgAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;English University Life In The Middle Ages&lt;/a&gt; (1999).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;More general : &lt;a href=&quot;https://global.oup.com/academic/content/series/h/history-of-universities-series-hou/?lang=en&amp;amp;cc=us&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Universities Series&lt;/a&gt; directed by &lt;a href=&quot;https://www.hss.caltech.edu/content/mordechai-feingold&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mordechai Feingold&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-11-08T13:42:47.157" LastActivityDate="2016-11-08T13:42:47.157" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5355" PostTypeId="1" CreationDate="2016-11-08T17:31:16.393" Score="1" ViewCount="137" Body="&lt;p&gt;I am curious to know who first used the term &quot;scale&quot; in graphing, and what exactly did they mean.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Depending on which dictionary you use, a scale can mean &quot;a coating&quot; or a scale can mean a device for weighing (or measuring).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way scales are described in a &lt;a href=&quot;https://www.datacamp.com/courses/data-visualization-with-ggplot2-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Data Visualization Course I am taking&lt;/a&gt;, &quot;scales&quot; are a kind of aesthetic onto which we map our data (like &quot;scales&quot; on a fish); part of a theory of the &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387245448&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Grammar of Graphics&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, in other classes, it has always seemed like scales are a kind of measure of weight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who first used this term (&quot;scale&quot; or &quot;scales&quot;)?&lt;/li&gt;&#xA;&lt;li&gt;Which meaning (or analogy) of &quot;scale&quot; were they trying to convey?&lt;/li&gt;&#xA;&lt;li&gt;Also, has the meaning of &quot;scale&quot; (or &quot;scales&quot;) in graphing changed over time (e.g., with the &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387245448&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Grammar of Graphics&quot;&lt;/a&gt;, and, if so, by whom?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="2458" LastEditorUserId="2458" LastEditDate="2016-11-08T17:44:49.347" LastActivityDate="2016-11-11T13:01:15.403" Title="History of the word &quot;scales&quot; in graphing" Tags="|mathematics|mathematicians|measurement|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5356" PostTypeId="1" AcceptedAnswerId="5358" CreationDate="2016-11-08T19:03:37.700" Score="2" ViewCount="72" Body="&lt;p&gt;From G.H.Moore’s &lt;em&gt;Zermelo’s Axiom of Choice&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1890 Schröder had already introduced a notion of consistent and inconsistent classes in his &lt;em&gt;Algebra der Logik&lt;/em&gt;, well before the modern discovery of paradoxes in set theory and logic, and had utilized such inconsistent classes under certain conditions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;1890 is seven years before Cantor’s first published treatment of inconsistent classes in the second installment of &lt;em&gt;Beiträge&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since Schröder is working with and extending the methods of algebraic logic developed by Boole, I am guessing that these inconsistent classes are collections such as “the collection of all thoughts”, or somesuch.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an original German language PDF of Schröder’s &lt;em&gt;Algebra of Logik&lt;/em&gt; online at &lt;a href=&quot;https://archive.org/details/vorlesungenberd03mlgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;archive.org&lt;/a&gt;, however a search inside this PDF for &lt;em&gt;inkonsistent&lt;/em&gt; does not produce any matches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;:  What were  Schröder’s inconsistent classes and how did he demonstrate their inconsistency?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-11-08T19:08:44.473" LastActivityDate="2016-11-08T19:48:41.593" Title="What were the inconsistent classes identified by Ernst Schröder?" Tags="|set-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5357" PostTypeId="2" ParentId="5355" CreationDate="2016-11-08T19:11:34.000" Score="1" Body="&lt;p&gt;Scale  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;code&gt;3&lt;/code&gt;. a graduated range of values forming a standard system for measuring or grading something  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From Latin &lt;strong&gt;scala&lt;/strong&gt;, ladder&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in Chaucer, 1391:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Next the forseide cercle of the A. b. c., vnder the cros-lyne, is Marked the skale, in Maner of 2 Squyres or elles in Manere of laddres.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2016-11-08T19:11:34.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5358" PostTypeId="2" ParentId="5356" CreationDate="2016-11-08T19:23:13.577" Score="1" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bernard Linsky, &lt;a href=&quot;https://papyrus.bib.umontreal.ca/xmlui/bitstream/handle/1866/13299/Lepage_Fradet_linksy.pdf?sequence=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ernst Schroeder and Zermelo's Anticipation of Russell’s Paradox&lt;/a&gt; (2013).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Schroeder's argument involves the &lt;em&gt;universe of discourse&lt;/em&gt; and is similar to the argument regarding the universal class.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the argument as recounted by Frege using Boole's 0, 1 notation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As we have laid down, 0 would have to be contained in&#xA;  every class that can be got out of the manifold 1; … 0 would&#xA;  have to be the subject of every predicate. Now suppose we&#xA;  took a to be the class of those classes of the manifold that are equal&#xA;  to 1 (which would certainly be permissible if we could&#xA;  bring everything thinkable into the manifold 1), then this&#xA;  class of its very nature contains just one class, viz. the&#xA;  symbol 1 itself, or alternatively the whole of the manifold,&#xA;  which constitutes the reference of the symbol; but therefore&#xA;  besides this it would contain “nothing,” i.e. 0. Hence 1 and 0&#xA;  would make up the class of the objects that are to be equal&#xA;  to 1; and so we should have to admit not only: 1 = 1 but&#xA;  also: 0 = 1. For a predicate that applies to a class -- in our&#xA;  case, the predicate: to be identically equal to 1 -- must also apply to every individual in the class, by Principle II.&#xA;  (Frege, 1895, p.91) &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="3643" LastEditDate="2016-11-08T19:48:41.593" LastActivityDate="2016-11-08T19:48:41.593" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5359" PostTypeId="2" ParentId="5340" CreationDate="2016-11-08T23:35:36.187" Score="1" Body="&lt;p&gt;No, at least not in the academic sector.  It it now commonly held knowledge that it is not one or the other, but the interaction between nature and nurture that shapes an individual.  The &quot;debate&quot; therefore is currently how much of each contributes to any given developmental mechanism.  However, I cannot guarantee that some independent special interest group is not using a perversion of the &quot;original debate&quot; for their own agenda.  All this being said, it is important to note that there are some aspects of maturation (e.g., language or sight, for example) that have either sensitive or critical periods for development.  With that being said, it IS important, and extremely useful, to discuss nature/nurture within these contexts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Key individuals in this area of study include Robert Plomin, Arnold Buss, Michael Rutter, Urie Bronfenbrenner, Jay Belsky, Sandra Scarr, and Michael Pluess (to name a few).  If you are interested in additional information, many of these individuals have published works on this topic within the last 10 years.&lt;/p&gt;&#xA;" OwnerUserId="4938" LastEditorUserId="4938" LastEditDate="2016-11-10T05:39:36.120" LastActivityDate="2016-11-10T05:39:36.120" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5360" PostTypeId="1" CreationDate="2016-11-09T07:59:22.547" Score="3" ViewCount="360" Body="&lt;p&gt;Many had tried in vain to prove Euclid's parallel postulate using the existing axioms and theorems. But my question is that what is it about the parallel postulate that made it seem so much like a theorem that mathematicians just felt uncomfortable to accept it as an unprovable assumption?&lt;/p&gt;&#xA;" OwnerUserId="4946" LastActivityDate="2016-11-09T09:43:54.637" Title="Why did mathematicians have a hard time accepting Euclid's 5th postulate as a postulate?" Tags="|mathematics|euclidean-geometry|" AnswerCount="1" CommentCount="0" ClosedDate="2016-11-10T04:11:47.860" ContentLicense="CC BY-SA 3.0" />
  <row Id="5361" PostTypeId="1" CreationDate="2016-11-09T08:44:04.860" Score="4" ViewCount="1737" Body="&lt;p&gt;Is there any source which has the complete works of Abel, Lagrange, Jacobi and Gauss which has been translated into English? I'm asking for separate books for these mathematicians not necessarily a combined one for all of them.&lt;/p&gt;&#xA;" OwnerUserId="1888" LastEditorUserId="4251" LastEditDate="2020-06-07T20:16:59.000" LastActivityDate="2023-05-30T16:41:07.763" Title="Is there any complete works of Abel, Lagrange, Jacobi and Gauss translated into English?" Tags="|mathematics|mathematicians|reference-request|translations|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5362" PostTypeId="2" ParentId="5360" CreationDate="2016-11-09T09:43:54.637" Score="2" Body="&lt;p&gt;Because it was considerd (probably well before Euclid) as not as &quot;intuitive&quot; as the other postulates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See e.g. :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Boris Rosenfeld, &lt;a href=&quot;https://books.google.it/books?id=3wzSBwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;A history of non-euclidean geometry&lt;/a&gt; (1988; original ed.,1976)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;as well as the introduction to :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gerolamo Saccheri, &lt;a href=&quot;https://books.google.it/books?id=DPA-BAAAQBAJ&amp;amp;pg=PA6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid Vindicated (1733) : Edited and Annotated by Vincenzo De Risi&lt;/a&gt; (2014) :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We know of early attempts to prove this postulate in Classical Antiquity. In fact, these attempts probably preceded the composition of the &lt;em&gt;Elements&lt;/em&gt;, suggesting that Euclid perhaps assumed his Fifth Postulate unwillingly, because he could not devise the proof for it that he sought.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-11-09T09:43:54.637" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5363" PostTypeId="2" ParentId="5340" CreationDate="2016-11-09T10:38:53.830" Score="-1" Body="&lt;p&gt;There was, long ago, a confusion in social science circles. Alfred Marshall, Karl Pearson, Bernard Shaw, Ronald Fisher, etc, framed the question in terms of nature versus nature. However, this framing was later critically addressed, by prominent figures at the time within biology and neuroscience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1956 (2), &lt;a href=&quot;http://www.scholarpedia.org/article/Donald_Olding_Hebb&quot; rel=&quot;nofollow noreferrer&quot;&gt;Donald Hebb&lt;/a&gt; wrote that it is 100 percent both, that the nature versus nurture is misleading and incorrect, that there is no nature part and no nurture part. They cannot be separated or calculated as fractions. Environment determined how genes are expressed and while absence of one (now called developmental resource) causes some behavior not to be learned, no behavior is attributable to some environment or to some genetic component, because mere presence of that independent of the whole complex system does not produce that behavior. He was later elected president of the American Psychological Association. In his earlier famous coalitional neural networks monograph (1), he wrote that attributing substantial differences in human learning behavior specifically to different genetics is very misleading. Most significant genetic differences in a species, he argued, do not themselves cause any behavior difference when small and cause obvious physical illness when sufficiently great.&#xA;Before him Raymond Wheeler wrote much the same thing in places like &lt;em&gt;Psychological Review&lt;/em&gt;, and Conrad Waddington wrote the same thing on numerous occasions in his well known books in 1949, 1954, 1957, etc, and in places like &lt;em&gt;Nature&lt;/em&gt;. Today we talk about whole developmental systems and developmental resources that must be present at a certain time for something to occur but none of which by itself &lt;em&gt;codes&lt;/em&gt; or causes anything. So much for the &lt;em&gt;scientific&lt;/em&gt; history of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &lt;strong&gt;HEBB&lt;/strong&gt;, &lt;strong&gt;DONALD&lt;/strong&gt;. &lt;strong&gt;1949&lt;/strong&gt;. &lt;em&gt;The Organization of Behavior&lt;/em&gt;, New York: Wiley.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;strong&gt;HEBB&lt;/strong&gt;, &lt;strong&gt;DONALD&lt;/strong&gt;. &lt;strong&gt;1982&lt;/strong&gt;. &lt;em&gt;The Conceptual Nervous System&lt;/em&gt;, Oxford: Pergamon.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2016-11-09T10:47:55.517" LastActivityDate="2016-11-09T10:47:55.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5364" PostTypeId="1" CreationDate="2016-11-09T10:48:12.783" Score="3" ViewCount="132" Body="&lt;p&gt;My questions refer to propositions 44-46 of section 8 of book 2 of Isaac Newton's principia, where Newton draws analogy between oscillations of water in a U - shaped tube and water waves in deep water. Newton reasons from this analogy an important early result on water waves - the speed of water waves in the limit of deep water (i.e when the depth of water is much higher than the wavelength) is proportional to the squared root of their wavelength. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now comes my question. &lt;a href=&quot;https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chandrasekhar&lt;/a&gt;'s book &quot;Newton's principia for the common reader&quot; states that in the conclusion remarks of these propositions Newton also stated without proof the result for the speed of shallow water waves - that their speed is the squared root of gH. But i was unable to find it in the principia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: did Newton derive also the expression for shallow water waves or not?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="131" LastEditDate="2016-11-09T11:24:01.197" LastActivityDate="2016-11-09T15:59:01.363" Title="Did Isaac Newton derive the expression for the speed of water waves in both shallow water and deep water approximations?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5365" PostTypeId="2" ParentId="5364" CreationDate="2016-11-09T15:59:01.363" Score="2" Body="&lt;p&gt;It seems not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton states in Prop. XLV that the velocity of waves is as the square root of the wavelength, and again in Corol. 2 of Prop. XLVI. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton defines wavelength as &quot;the transverse distance either between bottoms of troughs or between tops of crests.&quot; Newton notes that he has considered that &quot;the parts of the water go straight up or straight down,&quot; but since &quot;this ascent and descent takes place more truly in a circle,&quot; he admits that &quot;in this proposition the time has been determined only approximately.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, it seems that there is no ref to the depth of water...&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-11-09T15:59:01.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5366" PostTypeId="2" ParentId="5361" CreationDate="2016-11-09T19:26:37.160" Score="3" Body="&lt;p&gt;The simple answer is &amp;quot;no&amp;quot;. They generally do not translate mathematical works from French and German to English, and rarely from Latin. Some separate works have been  translated. For example Gauss, Disquisitiones generales circa superficies curvas,&#xA;but very few of them. Much more is translated into European languages, for example into Russian, but even in Russian there are no translations of complete works of the mathematicians you listed.&lt;/p&gt;&#xA;&lt;p&gt;I address the comment. To my understanding, the reason is that translation is not profitable. On the one hand, the audience is relatively small, and many mathematicians can read mathematics in principal European languages. On the other hand, it is extremely difficult to make a good translation: mathematicians of highest qualification have to be employed. And those few who are able to do this would rather spend their time on doing their own mathematics.&lt;/p&gt;&#xA;&lt;p&gt;There are many examples of translations of mathematics where the translators did not really understand what they translated, and these are of very limited use, if any.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-05-30T16:41:07.763" LastActivityDate="2023-05-30T16:41:07.763" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5367" PostTypeId="2" ParentId="5361" CreationDate="2016-11-09T19:27:03.243" Score="4" Body="&lt;p&gt;&quot;Complete works&quot; usually refers to a collection of all author's publications and manuscripts, and that is almost never translated (one exception is Euler). Even individual books/papers are rarely translated from French or German into English, although some particularly famous ones were. For Abel see references under &lt;a href=&quot;https://hsm.stackexchange.com/questions/3087/where-i-can-find-the-translated-manuscript-of-abel&quot;&gt;Where I can find the translated manuscript of Abel?&lt;/a&gt; Bruce made English translations of some &lt;a href=&quot;http://www.17centurymaths.com/contents/lagrangecontents.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange's papers&lt;/a&gt;, Gauss's &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0387962549&quot; rel=&quot;nofollow noreferrer&quot;&gt;Disquisitiones Arithmeticae is translated&lt;/a&gt; and so are &lt;a href=&quot;https://books.google.com/books?id=SYJsAAAAMAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;General Investigations of Curved Surfaces of 1827 and 1825&lt;/a&gt;. See also relevant discussions on &lt;a href=&quot;https://mathoverflow.net/questions/151317/english-translation-of-gauss-principia-generalia-theoriae-figurae-fluidorum-in&quot;&gt;Principia generalia theoriae figurae fluidorum in statu aequilibri&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/questions/2008/translations-of-gauss-work-on-gaussian-integers&quot;&gt;Translations of Gauss&amp;#39; work on Gaussian integers&lt;/a&gt;. Also, &lt;a href=&quot;http://www.worldcat.org/title/jacobis-lectures-on-dynamics-delivered-at-the-university-of-konigsberg-in-the-winter-semester-1842-1843-and-according-to-the-notes-prepared-by-c-w-brockardt/oclc/730004039&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacobi's Lectures on Dynamics&lt;/a&gt; were recently translated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are probably more, but they are not conveniently gathered in one place, you'd have to look case by case. Bruce did collect some of his translations of other mathematicians under &lt;a href=&quot;http://www.17centurymaths.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some Mathematical Works of the 17th &amp;amp; 18th Centuries&lt;/a&gt;, and the &lt;a href=&quot;http://eulerarchive.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler Archive&lt;/a&gt; does translations methodically.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2016-11-10T01:55:24.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5368" PostTypeId="2" ParentId="2656" CreationDate="2016-11-10T05:37:16.643" Score="4" Body="&lt;p&gt;&quot;The term itself was coined by Chevalley, although accepted in a more restrictive sense than the term as used by Grothendieck. In &lt;em&gt;Foundations of Algebraic Geometry&lt;/em&gt;, André Weil had introduced into algebraic geometry the methods used by his mentor, Élie Cartan, in differential geometry (following Carl Friedrich Gauss and Jean Darboux). But Weil’s method was by no means intrinsic, and Chevalley wondered what was invariant, in Weil’s sense of variety. The answer, inspired by Zariski’s work, was simple and elegant: the scheme of an algebraic variety is the collection of local rings of the sub-varieties found inside the rational function field. There is no need for an explicit topology, a point of distinction between Chevalley and Serre, who at roughly the same time introduced his algebraic varieties using Zariski topologies and sheaves. Each of the two approaches had advantages, but also limitations: Serre had an algebraically closed base field; Chevalley had to work only with irreducible varieties. In both cases, the two fundamental problems of products of varieties and base change could only be approached indirectly. All the same, Chevalley’s point of view was better suited to future extensions to arithmetic, as Nagata soon observed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Évariste Galois was certainly the first to notice the polarity between equations and their solutions. One must distinguish between the domain in which coefficients of the algebraic equation are chosen and the domain in which solutions are sought. Grothendieck created a synthesis out of these ideas, based in essence on the conceptual presentation of Zariski-Chevalley-Nagata. Schemes are thus a way of encoding systems of equations as well as the transformations to which one may subject them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grothendieck presented the Galois problem in the following manner: a scheme is an absolute object, X, say; the choice of a field of constants (or a field of definition) corresponds to the choice of another scheme S and a morphism πX from X to S. In the theory of schemes, a commutative ring is identified with a scheme, its spectrum. A homomorphism from ring A to ring B likewise maps, inversely, the spectrum of B into the spectrum of A. Moreover, the spectrum of a field has a single underlying point (even though many different points exist, in this sense); consequently, giving the field of definition as being included in the universal domain corresponds to giving a scheme morphism πT from T to S. A solution of the system of equations X, with the domain of constants S, with values in the universal domain T, corresponds to a morphism φ from T to X such that πT is the composition of φ and of πT.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://inference-review.com/article/a-country-known-only-by-name&quot; rel=&quot;nofollow noreferrer&quot;&gt;A country known only by name, Cartier&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3857" LastActivityDate="2016-11-10T05:37:16.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5371" PostTypeId="1" CreationDate="2016-11-10T15:45:34.493" Score="1" ViewCount="44" Body="&lt;p&gt;What new technology enabled us to construct VR recently? Was it just the resolution which makes believable image?&lt;/p&gt;&#xA;" OwnerUserId="3630" LastActivityDate="2016-11-10T15:45:34.493" Title="What enabled the spread of virtual reality systems?" Tags="|discoveries|technology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5372" PostTypeId="1" AcceptedAnswerId="5376" CreationDate="2016-11-10T16:39:18.477" Score="1" ViewCount="101" Body="&lt;p&gt;I've read in &lt;a href=&quot;http://www.academia.cz/jak-letaji.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;How do they fly&quot; by Rudolf Dvořák&lt;/a&gt; that the main constraint from inventing some kind of remote-controlled flying machine that could go all directions was the lack of theoretical background that had been significantly improved with the continuing research on insect fly technology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this true or were there much bigger constrains that were making it hard to construct drones apart from the lack of knowledge on aerodynamics?&lt;/p&gt;&#xA;" OwnerUserId="3630" LastEditorUserId="7931" LastEditDate="2021-01-04T21:56:43.187" LastActivityDate="2021-01-04T21:56:43.187" Title="How much had the rise of drones been connected with the study of birds?" Tags="|biology|technology|aerodynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5374" PostTypeId="1" AcceptedAnswerId="5378" CreationDate="2016-11-10T18:56:29.860" Score="5" ViewCount="1316" Body="&lt;p&gt;Huygens' principle seemed quite complete in this description if applied correctly. I noticed that Fresnel was credited in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huygens-Fresnel principle&lt;/a&gt;, as well. I think it is because he was able to describe diffraction with the principle correctly, something Huygens could not do in his lifetime. The correct description of diffraction using Huygens' principle took ~120 years after Huygens' death. It is quite curious to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why couldn't Huygens explain diffraction using his principle?&lt;/p&gt;&#xA;" OwnerUserId="4864" LastEditorUserId="62" LastEditDate="2016-11-12T15:57:07.777" LastActivityDate="2016-11-12T15:57:07.777" Title="Why couldn't Huygens explain diffraction using his principle?" Tags="|physics|biographical-details|optics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5375" PostTypeId="2" ParentId="5374" CreationDate="2016-11-10T23:36:02.620" Score="5" Body="&lt;p&gt;Are you sure he did not try? Have you read Huygens? His contemporary Newton did try to explain it in his Optics. Anyway, if Huygens never addressed this question,&#xA;it is hard to tell why. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is easier to explain why it took 120 years before the correct explanation was given. The reason is that the wave theory was not popular until the end of 18th century. The prevalent theory was corpuscular, and it is hard to give a correct explanation of diffraction within corpuscular theory. (Newton tried).&#xA;One of the reasons corpuscular theory theory was almost universally accepted for so long was the enormous authority of Newton, though reading Newton indicates that he was not so sure.   &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-11-10T23:36:02.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5376" PostTypeId="2" ParentId="5372" CreationDate="2016-11-10T23:44:18.610" Score="3" Body="&lt;p&gt;Drones fly on the same principles as the ordinary airplanes or helicopters. So the main difficulty in developing of drones was the problem how to control them, not how to make them fly. And this problem of control has nothing to do with birds or insects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually airplanes have also little to do with birds and insects because they&#xA;use different aerodynamic principles. Airplanes (gliders) can be only compared with some large birds which glide for long periods of time. Attempts to imitate&#xA;birds and insects with artificial devices were made but they were not very successful. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-11-10T23:44:18.610" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5377" PostTypeId="1" AcceptedAnswerId="5379" CreationDate="2016-11-11T01:09:02.613" Score="9" ViewCount="472" Body="&lt;p&gt;Two Bible verses seem to indicate that ancients believed germination was the death of a seed, and a resurrection or rebirth of that seed into a plant:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Unless a grain of wheat falls into the earth and dies, it remains just a single grain; but if it dies, it bears much fruit. (John 12:34)&lt;/p&gt;&#xA;&lt;p&gt;What you sow does not come to life unless it dies. And as for what you sow, you do not sow the body that is to be, but a bare seed, perhaps of wheat or of some other grain. (1 Corinthians 15:36-37)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Context: In the first instance, Jesus is using germination as an image for his upcoming resurrection, and in the second, Paul is using it as a metaphor for the post-death resurrection Christians experience as a consequence of Jesus' sacrificial death.&lt;/p&gt;&#xA;&lt;p&gt;Textual background: The two writings are the Gospel of John and Paul's first epistle to the Corinthians. Both of them are written in Koine Greek and in the city of Ephesus, probably in the first century. John was likely written between 90 and 110, and 1 Corinthians between 50 and 55. The author of John is traditionally said to be the apostle John, a Jew from Galilee, though scholars generally deny that much is known of the actual author, besides that he is quite literate. 1 Corinthians was written by Paul, a learned Pharisaic Jew from Tarsus with training in Greek rhetoric.&lt;/p&gt;&#xA;&lt;p&gt;I only mention this background because in my mind there are three potential sources of their common germination illustration: a Jewish precedent, a Greek academic precedent (or a more local Turkish or Palestinian academic precedent or something), or something in the oral Christian tradition. The second, and maybe the first, would be on-topic to mention here, if I understand guidelines correctly.&lt;/p&gt;&#xA;&lt;p&gt;Is there an academic, protoscientific precedent within the Greek academy that describes germination in terms of death and rebirth?&lt;/p&gt;&#xA;" OwnerUserId="1814" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-11-17T02:56:19.280" Title="What did ancient near eastern protoscience believe about germination?" Tags="|ancient-greece|biology|roman-empire|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5378" PostTypeId="2" ParentId="5374" CreationDate="2016-11-11T01:20:45.497" Score="4" Body="&lt;p&gt;Because he did not think of wave fronts as &quot;fronts of amplitudes&quot; that can go both ways, and cancel each other when superimposed. Sometimes it is hard to imagine how simple ideas could be missed in retrospect, but perhaps they are &quot;simple&quot; to us only because we already know to pay attention. Huygens thought from what later came to be called the optico-mechanical analogy, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3486/did-wave-optics-anticipate-quantum-mechanics/3493#3493&quot;&gt;Did wave optics anticipate quantum mechanics?&lt;/a&gt; So he paid attention to the parallels between the spreading of waves and light (or particle) rays rather than the differences between them, on reproducing the effects of geometric optics. The time of focusing on wave effects came only later, &lt;a href=&quot;http://www.mathpages.com/home/kmath242/kmath242.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to Mathpages&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Subsequently, Augustin Fresnel (1788-1827) elaborated on Huygens' Principle by stating that the amplitude of the wave at any given point equals the superposition of the amplitudes of all the secondary wavelets at that point (with the understanding that the wavelets have the same frequency as the original wave).&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-11-11T01:20:45.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5379" PostTypeId="2" ParentId="5377" CreationDate="2016-11-11T03:36:25.417" Score="9" Body="&lt;p&gt;&lt;em&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0851997236&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Encyclopedia of Seeds: Science, Technology and Uses&lt;/a&gt;&lt;/em&gt;, edited by J. Derek Bewley, Michael Black, Peter Halmer, CABI International 2006 (Entry: &lt;a href=&quot;https://books.google.com/books?id=aE414KuXu4gC&amp;amp;lpg=PR1&amp;amp;pg=PA330&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of seed research&lt;/a&gt;) cites some ancient descriptions along similar lines, both mythical and proto-scientific:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For example, to the Greeks, parsley was associated with death: the notorious slow germination rate of the seeds was because they had to journey to the underworld and back several times before they could begin to germinate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They also cite &lt;a href=&quot;https://en.wikipedia.org/wiki/Pliny_the_Elder&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pliny the Elder&lt;/a&gt; (32–79 AD), who, following earlier observations by &lt;a href=&quot;https://en.wikipedia.org/wiki/Theophrastus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theophrastus of Eresos&lt;/a&gt; (371–287 BC), commented on the germination of mistletoe in &lt;a href=&quot;http://www.masseiana.org/pliny.htm#BOOK%20XVI&quot; rel=&quot;nofollow noreferrer&quot;&gt;book XVI&lt;/a&gt; of his &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Natural_History_(Pliny)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Natural History&lt;/a&gt;&lt;/em&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But universally when mistletoe seed is sown it never sprouts at all, and only when passed in the excrement of birds, particularly the pigeon and the thrush; its nature is such that it will not shoot unless it has been ripened in the stomach of birds. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, it seems that Theophrastus himself—considered the father of botany and particularly of seed physiology—did not support the death-and-birth theory of germination. He is cited saying &lt;a href=&quot;https://www.loebclassics.com/view/theophrastus-de_causis_plantarum/1976/pb_LCL471.55.xml?readMode=recto&quot; rel=&quot;nofollow noreferrer&quot;&gt;in his &lt;em&gt;De Causis Plantarum&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Every seed contains in itself a certain amount of food. This is why they are able to survive for some time, and do not, like the seed (semen) of animals, perish directly on separation from the parent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="1814" LastEditDate="2016-11-17T02:56:19.280" LastActivityDate="2016-11-17T02:56:19.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5380" PostTypeId="2" ParentId="5355" CreationDate="2016-11-11T12:46:48.050" Score="2" Body="&lt;p&gt;There are (at least) three different English words written and pronounced “scale”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The “scale” of a fish or reptile is a borrowing from an Old French word adopted from a Germanic form cognate with the English word “shell”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &quot;scale&quot; for weighing is originally a &quot;bowl&quot;, borrowed from Old Norse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Scale” in its various mathematical usages is borrowed from Latin scala “ladder”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are totally different words; homonyms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematical usage goes back a long time. Just to stay with English, the Oxford English Dictionary quotes the sentence from Chaucer’s Treatise on the Astrolabe from about 1400 referenced in Gerald's answer. This is the earliest recorded use of the word in English, but it follows the example of earlier usage in French and Latin.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-11-11T13:01:15.403" LastActivityDate="2016-11-11T13:01:15.403" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5381" PostTypeId="2" ParentId="666" CreationDate="2016-11-11T23:18:52.577" Score="4" Body="&lt;p&gt;The German/Prussian  education system was based on two pillars:  The first was a mandatory comprehensive primary  public school education with the aim to provide literacy, numeracy and good educational background for everybody (primary school = Volksschule).  Even the most deserted villages got their “Volksschule”. The follow up secondary and “Gymnasium” education was highly demanding, strongly selective and elitist, based on Wilhelm von Humboldt’s   humanistic  universal  curriculum. In addition, there were established  very ambitious natural science and mathematical “Real-Gymnasiums”.  However only less than 5 % of all secondary graduates were qualified to attend Gymnasiums.  The second pillar were the Universities being even more demanding and selective.  The renowned German “Professor”  was among the most respected personalities.  Austria and Hungary also adopted this elitist system.  Notably the “Fasori Evangélikus Gimnázium” in Budapest was a spring of later Nobel Laureates like Eugene Wigner and  John von Neumann and other distinguished scientists&lt;/p&gt;&#xA;" OwnerUserId="4438" LastActivityDate="2016-11-11T23:18:52.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5382" PostTypeId="1" AcceptedAnswerId="5383" CreationDate="2016-11-12T04:38:43.943" Score="7" ViewCount="203" Body="&lt;p&gt;It is well-known that Newton's Principia was published by Halley at his own expense, and the story of writing, editing, printing, correction and proofreading is well described in Newton's biographies. However I cannot find some important details:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How many copies of the first edition were printed?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For what price they were sold?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Did Halley get the proceeds from the sales? If not, who did? Newton? Royal Society?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Can anyone help me to find this information?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Since I asked this question, I found some relevant information in the article of B. Cohen, supplementing my own copy of Principia (1999 English translation:-)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A. N. Munby estimated the size of the first edition at some 300 or 400 copies, but the number has recently been increased to perhaps 500. Whiteside recons that&#xA;  granting this larger size of the edition, Halley would not have suffered financially by paying the printing costs and would even have made not less than 10 pounds in pocket for all his time and trouble.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;D. T. Whiteside, The prehistory of Principia from 1664 to 1786, Notes and records of the Royal Society, 45 (1991) 11-61. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-11-12T16:04:28.617" LastActivityDate="2016-11-12T16:04:28.617" Title="Publishing the Principia" Tags="|publishing|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5383" PostTypeId="2" ParentId="5382" CreationDate="2016-11-12T09:28:25.417" Score="8" Body="&lt;p&gt;Some info from :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I.B. Cohen, &lt;em&gt;Introduction to Newton's Principia&lt;/em&gt;, 1978, page 138 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;No one knows for certain just how large the edition of the &lt;em&gt;Principia&lt;/em&gt; was. Rouse Ball reckoned it to be some 250 copies [Ball (1893), &lt;em&gt;Essay&lt;/em&gt;, p.67], but A.N.L.Munby held it to be more nearly 300-400 [Munby (1952), &amp;quot;Distribution of the first edition&amp;quot;, p.37].&lt;/p&gt;&#xA;&lt;p&gt;The price to the trade was to be 6 shillings in quires (or sheets) (reduced to 5 shillings if payment was in cash or &amp;quot;at some short time&amp;quot;), but 9 shillings &amp;quot;bound in Calves leather or lettred&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-11-12T09:28:25.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5384" PostTypeId="1" AcceptedAnswerId="5385" CreationDate="2016-11-14T20:07:45.390" Score="5" ViewCount="98" Body="&lt;p&gt;Where did this terminology first appear, and what was the motivation behind using the word &quot;characteristic&quot; to refer to the property that $1 + 1 + ...+1$ a certain number of times gives us the additive identity?&lt;/p&gt;&#xA;" OwnerUserId="3129" LastActivityDate="2016-11-15T02:37:51.297" Title="Where did the terminology &quot;characteristic&quot; of a field originate?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5385" PostTypeId="2" ParentId="5384" CreationDate="2016-11-15T02:37:51.297" Score="5" Body="&lt;p&gt;The notion and the term were introduced by Steinitz in his seminal paper  Algebraische Theorie der Körper (Algebraic Theory of Fields), Crelle's Journal (1910), 167–309. He also defined such concepts as prime field and the transcendence degree there. In his own words, his goal was &quot;&lt;em&gt;to obtain a general overview of all possible types of fields and to determine their relations with each other&lt;/em&gt;&quot;. According to &lt;a href=&quot;https://www.rzuser.uni-heidelberg.de/~ci3/STEINITZ.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roquette's In Memoriam&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In any abstract  field, Steinitz showed that there is a unique smallest sub eld which he called the prime field; this is either in nite (and then isomorphic to the rationals) or its cardinality is a prime number p (and then it is isomorphic to the integers modulo p ). Accordingly he defined the characteristic of a  field to be either 0 or p respectively.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To appreciate the paper's significance it helps to quote van der Waerden, whose Modern Algebra text shaped the field:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In  earlier  treatises,  number   fields,  and   fields  of  algebraic  functions were usually treated in separate chapters, and  finite  fields in&#xA;  still  another  chapter.  The   first  to  give  a  unified  treatment,  starting  with  an  abstract  defnition  of  field,  was  E. Steinitz  in  his&#xA;  1910 paper. In my Chapter 5, called &quot;Korpertheorie&quot;, I essentially&#xA;  followed Steinitz...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Van der Waerden is a bit too generous, the first to give an abstract definition of a field was Weber (whom Steinitz credits) in Die Allgemeinen   Grundlagen der Galois'schen Gleichungstheorie, Math. Ann., 43:521-549, 1893. But he focused on Galois theory rather than studying general fields. As for the motivation for &quot;characteristic&quot;, it is a pretty common name in mathematics for a simpler object that &quot;characterizes&quot; something more complex, compare to characteristic of a logarithm, characteristic determinant, polynomial, function, etc. Steinitz may also have been influenced by Dedekind's, Frobenius's and Weber's earlier use of &quot;group character&quot; (German Gruppencharakter). According to Shapiro, (quoted from &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;However, in Dedekind’s edition of Dirichlet’s Vorlesungen ueber Zahlentheorie in 1894, Dedekind included a footnote in which he singled out the notion of &quot;character,&quot; defined it explicitly, and denoted it by chi(n). However, he did not give the function a name. Weber’s Lehrbuch der Algebra, II, 1899, defined the function chi(A) as a &quot;Gruppencharakter,&quot; and developed some of its elementary properties...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-11-15T02:37:51.297" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5386" PostTypeId="1" CreationDate="2016-11-15T03:32:49.260" Score="0" ViewCount="328" Body="&lt;p&gt;From &lt;em&gt;A Synopsis of Elementary Results and Applied Mathematics:&lt;/em&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/u01fq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/u01fq.png&quot; alt=&quot;Part 1&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Vcj0D.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Vcj0D.png&quot; alt=&quot;Part 2&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; I am confused on three main parts:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What is happening in &lt;strong&gt;35&lt;/strong&gt;? Why does the author get all of the numbers?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why multiply &lt;span class=&quot;math-container&quot;&gt;$3\times 2,3\cdot 2^2, 3\cdot 2^3$&lt;/span&gt; in the second line of the explanation?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How would you interpret the notation used?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I have spent an ample amount of time on this, and although I can understand * a bit* of finding the cube root, I'm still puzzled on the square root. I'm assuming that they &lt;em&gt;are&lt;/em&gt; supposed to be similar...&lt;/p&gt;&#xA;" OwnerUserId="4913" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-11-15T03:32:49.260" Title="Extracting a Square Root and Cube Root Notation" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5387" PostTypeId="1" CreationDate="2016-11-15T15:20:31.127" Score="1" ViewCount="116" Body="&lt;p&gt;I am exploring the topic of constructivism from discreet math, and think it is related to object oriented programming. Can anyone confirm or deny the two are related?&lt;/p&gt;&#xA;" OwnerUserId="4975" LastActivityDate="2016-11-15T15:20:31.127" Title="What is the relationship between constructivism and object oriented programming?" Tags="|computer-science|mathematical-logic|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5388" PostTypeId="1" AcceptedAnswerId="5390" CreationDate="2016-11-15T17:21:34.263" Score="7" ViewCount="304" Body="&lt;p&gt;When we introduce the Einstein equations in courses on General Relativity we use either the Bianchi indentity or the the variational principle to motivate the appearance of the Einstein tensor&#xA;$$&#xA;G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R.&#xA;$$&#xA;But, according to Pais (&quot;Subtle is the Lord&quot;), Einstein did not know the Bianchi identity, and he did not use the variational principle (introduced by Hilbert). But then: How did he get to the correct field equations? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pais seems to suggest that Einstein initially wrote down the field equations in trace reversed form&#xA;$$&#xA;R_{\alpha\beta}=8\pi G (T_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}T),&#xA;$$&#xA;but if that is the case, how did he know that you should consider the trace reverse of the stress tensor? &lt;/p&gt;&#xA;" OwnerUserId="4978" LastEditorUserId="55" LastEditDate="2016-11-16T01:46:43.317" LastActivityDate="2016-11-16T02:29:32.130" Title="How did Einstein arrive at his field equations without the Bianchi identities or variational formulation?" Tags="|relativity-theory|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5390" PostTypeId="2" ParentId="5388" CreationDate="2016-11-15T23:02:10.397" Score="3" Body="&lt;p&gt;Pais does not &quot;suggest&quot; that Einstein wrote the equation in this form, he reproduces on p.256 what Einstein wrote on November 25, 1915 with reference to his presentation to the Prussian Academy: $R^{\mu\nu}=-\kappa(T^{\mu\nu}-\frac{1}{2}g^{\mu\nu}T)$. This follows a detailed account on pp.250-255 of Einstein's struggles to implement, starting on November 4, his new/old guiding light:&quot;&lt;em&gt;I was led back to a more general covariance of the field equations, a requirement which I had abandoned only with a heavy heart in the course of my collaboration&lt;/em&gt; [on Entwurf] &lt;em&gt;with my friend Grossmann three years earlier&lt;/em&gt;&quot;. These struggles included considering and discarding several prior versions of the equation, including the penultimate version $R^{\mu\nu}=-\kappa T^{\mu\nu}$, trying to get right the conservation laws and figure out the status of the unimodularity condition $\sqrt{g}=1$. It started off as a principled requirement, but ended up being a selector of convenient coordinate systems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, it is because Einstein did not know of Bianchi identities in 1915 that he could use the conservation laws $T^{\mu\nu}_{;\nu}=0$ as an additional constraint on the theory, helping him to narrow down the form of equations. As a result, he could only assert that the Bianchi identities $(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R)_{;\nu}=0$ hold in the unimodular coordinate systems, where $\sqrt{g}=1$. It would be odd if conversely, Einstein arrived at his equation in terms of the $G$-tensor, which is artificially introduced for technical convenience, instead of the Ricci and stress-energy tensors that come up naturally. Einstein was working on extending relativity to gravity and implementing general covariance since 1907, went through a number of unsuccessful attempts, including Entwurf, looked at alternative theories, like Nordström's, lost and then regained confidence in the general covariance, and plowed through candidate equations even after that. Groundbreaking theories are often forged through trial and error, not through &quot;motivated&quot; textbook tricks, that get one to the &quot;right&quot; answer fast.  &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-11-16T02:29:32.130" LastActivityDate="2016-11-16T02:29:32.130" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5391" PostTypeId="1" CreationDate="2016-11-15T23:27:58.847" Score="2" ViewCount="439" Body="&lt;p&gt;I've been told personally by a well-known (even famous) geologist that Alfred Wegener doesn't get credit for continental drift because he didn't elucidate a correct mechanism, or even a mechanism.  This despite his understanding and description of isostasy and the less dense continents floating on the heavy basaltic crust, and despite his measurements of the yearly separation of Greenland from continental Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A day later I thought of a reply.  Under that logic Isaac Newton does not deserve credit for identifying gravity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen the same argument on popular educational programs.  This leads me to believe that it is a taught opinion.  A cynic would say that it is taught so that the identifiers of the Atlantic separation zone could get credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 1920's the British sought to punish German science after World War One.  Perhaps this action enabled geologists to &quot;forget.&quot;  Albert Einstein attempted at that time to stop this unprofessional kind of action. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my own experience, I was taught about the Mendeleev Table while a schoolboy in California.  In the US we hardly ever see that term, only &quot;Periodic Table.&quot;  On the other hand, my Russian book on solid state physics refers to J. Willard Gibbs as Canadian.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of schoolboy, I believed the Fifth-Grade teacher when she discounted another student's observation that the continents seem to fit together.  It was a lesson I have not forgotten.&lt;/p&gt;&#xA;" OwnerUserId="4979" LastActivityDate="2016-11-17T17:01:22.640" Title="Why don't geologists adopt Alfred Wegener as the founder of Continental Drift" Tags="|earth-sciences|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5392" PostTypeId="1" CreationDate="2016-11-16T12:36:34.790" Score="5" ViewCount="123" Body="&lt;p&gt;Who first wrote down the spin-orbit coupling term in the atom? Was it Dirac himself? The term should be derived from the Dirac equation. &lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="62" LastEditDate="2016-11-16T15:42:30.010" LastActivityDate="2016-11-16T16:16:49.703" Title="Who first wrote down the spin-orbit coupling term in the atom?" Tags="|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5393" PostTypeId="2" ParentId="5392" CreationDate="2016-11-16T16:16:49.703" Score="4" Body="&lt;p&gt;We can attribute the term to L. H. Thomas's &lt;a href=&quot;http://dieumsnh.qfb.umich.mx/archivoshistoricosMQ/ModernaHist/Thomas1927.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Kinematics of an Electron with an Axis&lt;/em&gt;&lt;/a&gt; (1927), derived a year before the Dirac equation was discovered. He writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\mathbf{K}=m[\mathbf{r}\times\mathbf{v}]$ is the angular momentum of revolution, and can be supposed nearly constant during one revolution, so the secular change in the direction of the spin axis is&#xA;  $$\frac{e}{mc}\mathbf{H}+\frac{1}{2}\frac{e^2}{mc^2}\frac{Z}{r^3}\mathbf{K}\tag{6.2}$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is for the case of, as Thomas puts it, a &quot;Coulomb field&quot;; $\mathbf{H}$ is an external magnetic field, and should not be confused with the Hamiltonian. The precession in this case is then&#xA;$$\mathbf{\sigma}=\frac{1}{2}\frac{e^2}{mc^2}\frac{Z}{r^3}\mathbf{K}$$&#xA;The change in the Hamiltonian from the precession is then just&#xA;$$\Delta H_T=\mathbf{\sigma}\cdot\mathbf{S}$$&#xA;although we could also apply this to $\text{(6.2)}$ in its entirety to get the full $\Delta H$.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2016-11-16T16:16:49.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5394" PostTypeId="1" AcceptedAnswerId="5396" CreationDate="2016-11-16T17:08:07.773" Score="2" ViewCount="1995" Body="&lt;p&gt;In this quote of his:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;“When I wrote this, only God and I understood what I was doing. Now, God only knows.”&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which piece of Mathematical work was this in reference to?&lt;/p&gt;&#xA;" OwnerUserId="4983" LastActivityDate="2016-11-16T17:29:54.637" Title="What was Karl Weierstrass referring to?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5395" PostTypeId="1" AcceptedAnswerId="5397" CreationDate="2016-11-16T17:23:12.627" Score="7" ViewCount="563" Body="&lt;p&gt;The theorem is that any continuous function on a compact is uniformly continuous. &#xA;It is called &quot;Heine&quot;, and sometimes also &quot;Heine-Cantor&quot; theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: what is the contribution of Cantor to this theorem, did Heine proved it after Cantor, or following Cantor's ideas? Was it proved before both Heine and Cantor?&lt;/p&gt;&#xA;" OwnerUserId="4984" LastEditorUserId="55" LastEditDate="2016-11-16T21:53:53.883" LastActivityDate="2016-11-16T21:53:53.883" Title="Who first proved the &quot;Cantor-Heine theorem&quot; on uniform continuity?" Tags="|mathematics|real-analysis|debunking|topology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5396" PostTypeId="2" ParentId="5394" CreationDate="2016-11-16T17:29:54.637" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;http://quoteinvestigator.com/2013/09/24/god-knows/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quote Investigator&lt;/a&gt;, the first written record of the quote is attributed to the German romantic writer &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Paul&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johann Richter&lt;/a&gt;, also known as Jean Paul, shortly after his death at a time when Weierstrass would have been just eleven years old. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The earliest instance known to QI appeared in a London newspaper in 1826 and featured the German writer Johann Paul Friedrich Richter who died shortly before in 1825. The anecdote used the alternate appellation John Paul Richter. The capitalization is in the original text. Boldface has been added to excerpts below:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The works of JOHN PAUL RICHTER are almost uninteresting to any but Germans, and even to some of them. A worthy German, just before RICHTER’S death, edited a complete edition of his works, in which one particular passage puzzled him. Determined to have it explained at the source, he went to JOHN PAUL himself, and asked him what was the meaning of the mysterious passage. JOHN PAUL’S reply was very German and characteristic. “My good friend,” said he, “when I wrote that passage, God and I knew what it meant. It is possible that God knows it still; but as for me, I have totally forgotten.”&#xA;  This story can be expressed in many ways and instances before 1826 may exist. Early examples of the anecdote typically feature German intellectuals, and the tale may have appeared previously in a German language book or periodical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As noted on Quote Investigator, a number of additional attributions followed shortly after the Richter attribution.  The author also notes :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Special thanks to Daniel Gackle who asked QI to investigate this topic after presenting the yarn and connecting it to the philosopher Hegel and the mathematician Weierstrass.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-11-16T17:29:54.637" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5397" PostTypeId="2" ParentId="5395" CreationDate="2016-11-16T20:16:36.897" Score="7" Body="&lt;p&gt;An explicit definition of uniform continuity was first published by Heine in Über Trigonometrische Reihen (On Trigonometric Series), Journal für die Reine und Angewandte Mathematik, 71 (1870), pp. 353–365. And two years later he published a proof that a function continuous on a closed interval is uniformly continuous there in Die Elemente der Functionenlehre (Elelements of Function Theory), Journal für die Reine und Angewandte Mathematik, 74 (1872), pp. 172–188. However, as &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086004000849&quot; rel=&quot;noreferrer&quot;&gt;Rusnock and Kerr-Lawson point out in Bolzano and Uniform Continuity&lt;/a&gt;: &quot;&lt;em&gt;He claimed no originality in these papers, however, and as it turns out his proof is an almost verbatim transcription of one given by Dirichlet in his lectures on definite integrals in 1854&lt;/em&gt;&quot;. They further argue:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The concern of this note, however, is to establish that Bolzano has a legitimate claim to priority. We intend to show, in particular, that he not only grasped the notion of uniform continuity but also gave an adequate characterization of the concept, stated and proved Theorem 2&lt;/em&gt; [a function can be continuous on an open interval &lt;em&gt;without being uniformly continuous there], and stated Theorem 1&lt;/em&gt; [&quot;Heine's&quot;] &lt;em&gt;in addition to providing a useful fragment of its proof.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sinkevich offers more details on Cantor's role in the genesis of the concept and the theorem in &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1502/1502.06942.pdf&quot; rel=&quot;noreferrer&quot;&gt;On the History of Epsilontics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1872,  Eduard  Heine  in  «Die  Elemente  der  Functionenlehre»  gave  a  definition  of  the  limit  function  using  Cantor’s  fundamental sequences. Every convergent sequence was represented as the sum of its limit and the  elementary  (decreasing)  sequence. On  this  basis,  Heine formulates  the  condition  of  continuity,  the  definition  of  uniform &#xA;  continuity in terms of $\epsilon-\delta$, the theorem of uniformly continuous functions and as a method  of  proof  of  there  was  cover  lemma. The  theorem of  uniformly  continuity was necessary for  intervals  between  irrational  number  and  its  limit, the rational number. (It was formulated by Cantor, as Heine wrote. Now it is Cantor-Heine theorem).&lt;/em&gt;&quot;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-11-16T21:53:24.473" LastActivityDate="2016-11-16T21:53:24.473" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5398" PostTypeId="2" ParentId="524" CreationDate="2016-11-17T03:26:54.503" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;Cajori, Florian. &quot;&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Philosophy%20&amp;amp;%20History/Origin%20of%20the%20Name%20Mathematical%20Induction%20(Cajori).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Origin of the Name &quot;Mathematical Induction&quot;&lt;/a&gt;&quot; &lt;em&gt;The&#xA;American Mathematical Monthly&lt;/em&gt; 25, no. 5 (1918): 197-201&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The process of reasoning called &quot;mathematical induction&quot; has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchmen B. Pascal and P. Fermat, and the Italian F. Maurolycus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;It seems &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Fermat.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermat&lt;/a&gt; (1601-1665) might have been the first.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://plato.stanford.edu/entries/peirce/&quot; rel=&quot;nofollow noreferrer&quot;&gt;C. S. Peirce&lt;/a&gt; says &quot;mathematical induction&quot; is an improper term for &quot;Fermatian inference&quot; (&lt;a href=&quot;https://isidore.co/calibre/browse/book/3989&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;CP&lt;/em&gt; 6&lt;/a&gt;.116):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In truth, of infinite collections there are but two grades of magnitude, the &lt;em&gt;endless&lt;/em&gt; and the &lt;em&gt;innumerable&lt;/em&gt;. Just as a finite collection is distinguished from an infinite one by the applicability to it of a special mode of reasoning, the syllogism of transposed quantity, so, as I showed in the paper last referred to, a numerable collection is distinguished from an innumerable one by the applicability to it of a certain mode of reasoning, the Fermatian inference, or, as it is sometimes improperly termed, “mathematical induction.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He cites Fermat, &lt;em&gt;Opera Omnia&lt;/em&gt; (Leipzig, 1911), vol. 1, §§340-351.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;Also, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Boyer, Carl B., and Uta C. Merzbach. 1991. &lt;a href=&quot;https://www.worldcat.org/oclc/23823042&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History of Mathematics&lt;/em&gt;&lt;/a&gt;. New York: Wiley.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;write, (ed. 2) xvii. 355:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In fact, mathematical induction, or reasoning by recurrence, sometimes is referred to as ‘Fermatian induction’, to distinguish it from scientific, or ‘Baconian,’ induction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-11-19T01:28:01.937" LastActivityDate="2016-11-19T01:28:01.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5399" PostTypeId="2" ParentId="5391" CreationDate="2016-11-17T17:01:22.640" Score="3" Body="&lt;p&gt;First, a correction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This despite his understanding and description of isostasy and the less dense continents floating on the heavy basaltic crust, and despite his measurements of the yearly separation of Greenland from continental Europe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The less dense continents do not float on the heavy basaltic crust. Instead, both the continental crust and oceanic crust float atop the upper mantle, which in general is denser than either continental crust or oceanic crust. While Wegener did measure the separation of Greenland from Europe, his measurements were flawed. His reported separation rate of 250 cm/year was 100 times greater than the very slow 2.5 cm/year that is observed by a number of modern techniques.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why don't geologists adopt Alfred Wegener as the founder of Continental Drift?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For a number of reasons.&lt;/p&gt;&#xA;&lt;p&gt;Wegener proposed that the continents somehow plowed through the denser oceanic crust. This is contrary to what plate tectonics says happens. Wegener's concept of continental drift and plate tectonics are distinct theories.&lt;/p&gt;&#xA;&lt;p&gt;Another issue is that Wegener was far from the first to propose that the east coast of the Americas and west coasts of Europe and Africa fit very nicely together. The basic concept predates Wegener by over 300 years. Those early proposals regarding continental drift were intimately tied with catastrophism (vaguely scientific explanations for Noah's flood). Nineteenth century geology was in part a war between catastrophism and uniformitarianism. Geologists of the late 19th / early 20th centuries who proposed ideas similar to Wegener's were told in no uncertain terms that they needed to stop lest they never publish or get a job in geology.&lt;/p&gt;&#xA;&lt;p&gt;Continental drift was not Wegener's idea. What set Wegener apart from his predecessors who proposed similar ideas was his dogged persistence. This persistence made many of those who later developed the theory of plate tectonics see a grain of truth in Wegener's concepts, even though much of what Wegener proposed was incorrect.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-11-17T17:01:22.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5400" PostTypeId="1" CreationDate="2016-11-19T14:40:09.497" Score="3" ViewCount="77" Body="&lt;p&gt;As far as I know, in the old quantum theory, a system can only reside in a stationary state. There was no Hilbert space, and no linear superposition. Or, a system cannot be simultaneously in two stationary states. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what is the first experimental observation showing that a system can be in a non-stationary state (or a generic state)?&lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2016-11-19T14:40:09.497" Title="What is the first effect demonstrating the importance (or existence) of quantum coherence?" Tags="|physics|quantum-mechanics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5401" PostTypeId="1" CreationDate="2016-11-19T15:57:57.840" Score="3" ViewCount="143" Body="&lt;p&gt;My question refers to the problem of the figure of the earth, and is intended to establish if Isaac Newton was or was not the first to discover the equation for the variation of gravity with latitude on an oblate spheroid.  As far as I know, Newton calculated a value of  $\frac54m$ for the eccentricity of earth, where $m$ is the equatorial ratio of centrifugal to gravitational force, and calculated $f = \frac{1}{230}$ for the ellipticity of earth. Later in book 3 of Principia, Newton stated that the gravitational force varies as the square of the sine of latitude, a statement that agrees in form with Clairaut's formula. My only question is whether Newton and Clairaut formulae are identical up to constants in the equation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An answer to this question will help me clarify the historical context surrounding the problem of the figure of the earth, so I'll be glad to get an answer.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="3643" LastEditDate="2016-11-19T20:23:25.723" LastActivityDate="2016-11-19T20:23:25.723" Title="Was Alexis Clairaut the first to discover Clairaut's theorem for the variation of gravity on spheroids?" Tags="|biographical-details|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5402" PostTypeId="1" AcceptedAnswerId="5404" CreationDate="2016-11-21T15:06:42.793" Score="6" ViewCount="1369" Body="&lt;p&gt;Can someone help me here? the language is archaic.  This is (translation of) Galileo&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If two particles carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed  The inveerse ratio of the time-intervals required&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think this is just $d = vt$ phrased in an elaborate way.  Here is another one from the same section:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the case of unequal speeds, the time-intervals required to traverse a given space are to each other inversely as the speeds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and in the proof he draw a diagram&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;A |----------|&#xA;C |-----------------|E|-------------| D&#xA;B |--------|&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;How did these diagrams help Gaileo reason about uniform motion?  Some of these propositions follow readily from his definition of uniform motion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By steady or uniform motion I mean one in which the distances traveersed by the moving particle during any equal intervals of time, are themselves equal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a bit of an idealization, no?  Even a car starts and stops and exhibits small amounts of acceleration driving up a hill or down a road.  Perhaps on a very smooth highway and for short periods of time.  One could argue hardly anything moves in the way he is decribing.&lt;/p&gt;&#xA;" OwnerUserId="193" OwnerDisplayName="john mangual" LastEditorUserId="62" LastEditDate="2016-11-22T14:19:02.100" LastActivityDate="2020-10-22T06:52:32.617" Title="Galileo's Discussion of Uniform Motion" Tags="|physics|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5403" PostTypeId="2" ParentId="5402" CreationDate="2016-11-21T17:36:59.860" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;If two particles carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed The inverse ratio of the time-intervals required&#xA;  I think this is just d=vt phrased in an elaborate way. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I would agree.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the case of unequal speeds, the time-intervals required to traverse a given space are to each other inversely as the speeds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think ( or hope:) this is self explanatory, the faster &quot;object&quot; gets to the end point in a time inversely proportionate to its velocity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The diagrams don't mean much to me without a scale, he seems to have taken one object to moving roughly twice as fast as the other.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By steady or uniform motion I mean one in which the distances traveersed by the moving particle during any equal intervals of time, are themselves equal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a bit of an idealization, no? Even a car starts and stops and exhibits small amounts of acceleration driving up a hill or down a road. Perhaps on a very smooth highway and for short periods of time. One could argue hardly anything moves in the way he is decribing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me this is simply a definition of constant velocity, nothing more than that. Obviously &lt;strong&gt;constant&lt;/strong&gt; anything depends on your level of measurement accuracy, but that's about as much as I can contribute. &lt;/p&gt;&#xA;" OwnerDisplayName="CountTo10" LastActivityDate="2016-11-21T17:36:59.860" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5404" PostTypeId="2" ParentId="5402" CreationDate="2016-11-22T08:27:44.250" Score="6" Body="&lt;p&gt;Is not &quot;phrased in an elaborate way&quot;; it is expressed in the current (at that point in time) &lt;a href=&quot;https://en.wikipedia.org/wiki/Ratio#History_and_etymology&quot; rel=&quot;noreferrer&quot;&gt;theory of ratios&lt;/a&gt;, when the symbolic algebra was still in his infancy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Uniform speed&lt;/em&gt; is defined in &lt;a href=&quot;http://galileoandeinstein.physics.virginia.edu/tns_draft/tns_153to160.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Two new sciences&lt;/em&gt;, Third Day&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is exactly the current definition of uniform motion (i.e. constant speed) : if $t_1=t_2$ then $s_1=s_2$, that implies : $v= \dfrac s t = \text {const}$ and yes, it is an &quot;idealization&quot; as anything in science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some &quot;obvious&quot; axioms are set forth and then some theorems are proved :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Th.I&lt;/strong&gt; If a moving particle, carried uniformly at a constant speed, traverses two distances the time-intervals required are to each other in the ratio of these distances.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I.e. with $v= \text {const}$ :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac {t_1} {t_2} = \dfrac {s_1} {s_2}$, that amounts to : $\dfrac {s_1} {t_1} = \dfrac {s_2} {t_2}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Th.III&lt;/strong&gt; In the case of unequal speeds, the time-intervals required to traverse a given space are to each other inversely as the speeds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I.e., given space $s$ :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac {t_1} {t_2} = \dfrac {v_1} {v_1}$, which amount to : $v_1 \times t_1 = v_2 \times t_2$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And also :   &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Th.IV&lt;/strong&gt; If two particles are carried with uniform motion, but each with a different speed, the distances covered by them during unequal intervals of time bear to each other the compound ratio of the speeds and time intervals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I.e.:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac {v_1} {v_2} = \dfrac {s_1} {s_2} \times \dfrac {t_2} {t_1}$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As you can easily verify, it is :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\dfrac {v_1} {v_2} = \dfrac {s_1} {t_1} \times \dfrac {t_2} {s_2}$. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-11-22T08:57:28.810" LastActivityDate="2016-11-22T08:57:28.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5405" PostTypeId="2" ParentId="5402" CreationDate="2016-11-22T14:29:09.620" Score="5" Body="&lt;p&gt;Yes, all these examples are the old fashioned ways to say that $d=vt$ and &#xA;$\Delta d=v\Delta t$. The reason why they did not speak of numbers and but always spoke of proportions is two-fold:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) The notion of real number is a modern concept. For ancients a &quot;number&quot; was an integer. Instead of the modern theory of real numbers they had an equivalent (but very cumbersome) &quot;theory of proportions&quot;, explained in Euclid. &#xA;That's why, even a century after Galileo Newton wrote: that &quot;attraction is inverse proportional to square of the distance&quot; instead of just saying $F=cm/d^2$&quot;. Even if I was in high school, they taught Newton's laws this way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) Until the end of 18th century, there was no universal units. Every physical quantity was measured in different units in different places. Writing $d=vt$ assumes that we use some units for $d,v,t$. Instead they said is that&#xA;&quot;distances&quot; are proportional to times, or that ratios or the distances are like&#xA;rations of the times etc. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-11-22T14:29:09.620" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5406" PostTypeId="1" AcceptedAnswerId="5407" CreationDate="2016-11-23T03:36:15.747" Score="11" ViewCount="1644" Body="&lt;p&gt;Hyperbolic numbers have the form $a+bj$ where $a,b \in \mathbb{R}$ and $j^2=1$. These are also known as a the Lorentz numbers or double numbers by some authors. I suppose these are a particular type of the Tessarines introduced by James Cockle in 1848, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Bicomplex_number#Tessarines&quot; rel=&quot;noreferrer&quot;&gt;this wikipedia article on the Tessarines&lt;/a&gt; for details. However, the hyperbolic numbers themselves are not so complicated and I wonder if they were known to other mathematicians before Cockle's Tessarines.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; was Cockle the first to introduce hyperbolic numbers ? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I happened to discover some of the hyperbolic numbers interesting features working with some students for fun back in 2012. We found,&#xA;$$ e^{j\theta} = \cosh (\theta) + j \sinh (\theta) $$&#xA;and a few other fun things before we stumbled upon the Wikipedia article which had much of what we found. We're not alone in our experience. The current article &lt;a href=&quot;https://en.wikipedia.org/wiki/Split-complex_number&quot; rel=&quot;noreferrer&quot;&gt;on split-complex numbers&lt;/a&gt; lists the following synonyms:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(real) tessarines, James Cockle (1848)&lt;/li&gt;&#xA;&lt;li&gt;(algebraic) motors, W.K. Clifford (1882)&lt;/li&gt;&#xA;&lt;li&gt;hyperbolic complex numbers, J.C. Vignaux (1935)&lt;/li&gt;&#xA;&lt;li&gt;bireal numbers, U. Bencivenga (1946)&lt;/li&gt;&#xA;&lt;li&gt;approximate numbers, Warmus (1956), for use in interval analysis&lt;/li&gt;&#xA;&lt;li&gt;countercomplex or hyperbolic numbers from Musean hypernumbers&lt;/li&gt;&#xA;&lt;li&gt;double numbers, I.M. Yaglom (1968), Kantor and Solodovnikov (1989), Hazewinkel (1990), Rooney (2014)&lt;/li&gt;&#xA;&lt;li&gt;anormal-complex numbers, W. Benz (1973)&lt;/li&gt;&#xA;&lt;li&gt;perplex numbers, P. Fjelstad (1986) and Poodiack &amp;amp; LeClair (2009)&lt;/li&gt;&#xA;&lt;li&gt;Lorentz numbers, F.R. Harvey (1990)&lt;/li&gt;&#xA;&lt;li&gt;hyperbolic numbers, G. Sobczyk (1995)&lt;/li&gt;&#xA;&lt;li&gt;paracomplex numbers, Cruceanu, Fortuny &amp;amp; Gadea (1996)&lt;/li&gt;&#xA;&lt;li&gt;semi-complex numbers, F. Antonuccio (1994)&lt;/li&gt;&#xA;&lt;li&gt;split binarions, K. McCrimmon (2004)&lt;/li&gt;&#xA;&lt;li&gt;split-complex numbers, B. Rosenfeld (1997)[13]&lt;/li&gt;&#xA;&lt;li&gt;spacetime numbers, N. Borota (2000)&lt;/li&gt;&#xA;&lt;li&gt;Study numbers, P. Lounesto (2001)&lt;/li&gt;&#xA;&lt;li&gt;twocomplex numbers, S. Olariu (2002)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The multitude of names reflects the fact that for the most part each author starts anew without much guidance from the previous generation. This seems to be the one uniting theme in the majority of the literature on hypercomplex analysis (if I could call it that, but, I really need to call it by many names, see how horrible the situation is even for the particular example of the hyperbolic numbers!)&lt;/p&gt;&#xA;" OwnerUserId="327" LastEditorUserId="26" LastEditDate="2016-11-23T13:46:25.513" LastActivityDate="2016-11-23T20:06:48.380" Title="Who invented or discovered the hyperbolic numbers?" Tags="|mathematics|discoveries|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5407" PostTypeId="2" ParentId="5406" CreationDate="2016-11-23T20:06:48.380" Score="7" Body="&lt;p&gt;According to the text &lt;a href=&quot;https://books.google.ca/books?id=68IvCwAAQBAJ&amp;amp;pg=PA2&amp;amp;lpg=PA2&amp;amp;dq=cockle%20tessarines%20hyperbolic&amp;amp;source=bl&amp;amp;ots=UAcFoBxJzg&amp;amp;sig=9QRc8iSZCrXLFiGuOKhuCoTMfcA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=cockle%20tessarines%20hyperbolic&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Bicomplex Holomorphic Funtions&lt;/em&gt;&lt;/a&gt; by &lt;em&gt;Luna-Elizarraras, et al&lt;/em&gt;, as far as we know Cockle was the first to introduce and study these objects, and their introduction was almost contemporary to Hamilton's introduction of the quaternions and certainly stimulated by it. From this text (using a sightly different notation - $k$ in place of $j$) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...it is not unreasonable to consider whether a four-dimensional algebra, containing $\mathbb C$ as a subalgebra, can be introduced in a way that perserves commutativity.  Not surprisingly, this can be done by simply considering two imaginary units $i, j$, introducing $k = ij$ (as in the quaternionic case), but now imposing that $ij = ji$.  This turns $k$ into what is known as a &lt;em&gt;hyperbolic imaginary unit&lt;/em&gt;, i.e., and element such that $k^2 = 1$.  As far as we know, the first time these objects were introduced was almost contemporary with Hamilton's construction, and in fact J. Cockle wrote, in 1848, a series of papers in which he introduced a new algebra that he called the algebra of &lt;em&gt;tessarines&lt;/em&gt;. Cockle's work was certainly stimulated by Hamilton's and he was the first to use tessarines to isolate the hyperboic trigonometric series as components of the exponential series. Not surprisingly, Cockle immediately realised that there was a price to pay for commutativity in four dimensions, and the price was the existence of zero-divisors.  This discovery led him to call such numbers &lt;em&gt;impossibles&lt;/em&gt;, and the theory has no further significant development for a while.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It was only in 1892 that the mathematician Corrado Segre, inspired by the work of Hamilton and Clifford, introduced what he called &lt;em&gt;bicomplex numbers&lt;/em&gt; ....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So Luna-Elizarraras history marks Segre's 1892 work as the next instance of the development of hyperbolic numbers, rather than that of Clifford, while noting Clifford and Hamilton's inspiration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cockle's original papers are available on the Biodiversity Heritage Library site as :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biodiversitylibrary.org/item/20157#page/449/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;On Certain Functions Resembling Quaternions ... (1848)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biodiversitylibrary.org/item/20121#page/51/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;On a New Imaginary in Algebra (1849)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biodiversitylibrary.org/item/20121#page/420/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;On the Symbols of Algebra and the Theory of Tessarines (1849)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biodiversitylibrary.org/item/20083#page/305/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;On the True Amplitude of Tessarines (1850)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biodiversitylibrary.org/item/120166#page/295/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;On Impossible Equations, on Impossible Quantities, and on Tessarines (1850) &lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2016-11-23T20:06:48.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5408" PostTypeId="1" CreationDate="2016-11-23T22:55:03.840" Score="4" ViewCount="409" Body="&lt;p&gt;When was the unit &quot;electronvolt&quot; (eV) first introduced, and by whom? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition of the eV requires the knowledge of the electron charge $e$, so it must have been introduced after 1909, the date of the famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Oil_drop_experiment&quot; rel=&quot;nofollow&quot;&gt;oil drop experiment&lt;/a&gt; conducted by R. A. Millikan and H. Fletcher, in which, as far as I know, $e$ was measured for the first time.&lt;/p&gt;&#xA;" OwnerUserId="5003" LastEditorUserId="5003" LastEditDate="2016-11-24T06:03:15.307" LastActivityDate="2016-11-24T19:32:06.560" Title="When was the electronvolt first introduced?" Tags="|physics|electromagnetism|units|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5409" PostTypeId="2" ParentId="5408" CreationDate="2016-11-24T03:48:42.750" Score="3" Body="&lt;p&gt;The &lt;em&gt;OED&lt;/em&gt;'s &quot;&lt;a href=&quot;http://www.oed.com/view/Entry/270197&quot; rel=&quot;nofollow noreferrer&quot;&gt;electronvolt&lt;/a&gt;&quot; entry gives this as the earliest usage in English:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.jstor.org/stable/94475?seq=20&quot; rel=&quot;nofollow noreferrer&quot;&gt;1925 &lt;em&gt;Proc. Royal Soc.&lt;/em&gt; A. &lt;strong&gt;109&lt;/strong&gt; 559&lt;/a&gt;   &#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Now one electron-volt = 1·59 .10&lt;sup&gt;−12&lt;/sup&gt; ergs, and 4·18 .10&lt;sup&gt;−17&lt;/sup&gt; ergs = 1 gram-calorie.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/94475?seq=20&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mlosT.gif&quot; alt=&quot;ibid., p. 549&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it defines what its measure is, not what it is.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2016-11-24T19:32:06.560" LastActivityDate="2016-11-24T19:32:06.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5411" PostTypeId="1" AcceptedAnswerId="5413" CreationDate="2016-11-25T13:35:03.327" Score="11" ViewCount="1233" Body="&lt;p&gt;The title is pretty self-explanatory: A &lt;a href=&quot;http://mathworld.wolfram.com/Pencil.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;pencil&lt;/em&gt;&lt;/a&gt; in projective (or algebraic) geometry is the family of all lines through a point. The above-linked website tells me that Cremona, &lt;a href=&quot;https://archive.org/details/elementsproject02cremgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;on page x of &lt;em&gt;Elements of Projective Geometry&lt;/em&gt;&lt;/a&gt;, asserts that Desargues was the first to make use of this geometric concept. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, inspection of the page shows that Cremona does not claim that Desargues came up with the name (in fact, the page says: &lt;em&gt;&quot;The geometric forms [...] are found, names excepted, in Desargues [...]&quot;&lt;/em&gt;). Therefore, I'm left wondering about, firstly, who &lt;em&gt;did&lt;/em&gt; come up with this name and where. Secondly, I'm interested in the reason why this concept got the name &lt;em&gt;&quot;pencil&quot;&lt;/em&gt;. The name is surprising enough for me to suspect that whoever came up with it had a good reason for it. I tried to look the term up in Lo Bello's book on mathematical terminology, but there is no relevant information to be found under &quot;pencil&quot; in that book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Has anyone ascertained whether Desargues (or Pascal) &lt;em&gt;actually wrote&lt;/em&gt; “pencil” (or “pinceau”) — or else, who introduced the word?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="26" LastEditorUserId="658" LastEditDate="2018-06-06T13:14:41.487" LastActivityDate="2018-06-06T13:14:41.487" Title="Where was the word &quot;pencil&quot; first used in (projective) geometry and what is the reason behind this curious name?" Tags="|mathematics|geometry|algebraic-geometry|projective-geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5412" PostTypeId="2" ParentId="5322" CreationDate="2016-11-25T17:54:51.397" Score="2" Body="&lt;p&gt;By accident I stumbled upon a recent article about the topic in a german science journal: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.spektrum.de/magazin/warum-es-hitlers-atombombe-nie-gab/1427403&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.spektrum.de/magazin/warum-es-hitlers-atombombe-nie-gab/1427403&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It is based on a the following publication:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1002/bewi.201601794&quot; rel=&quot;nofollow noreferrer&quot;&gt;Popp. M.: Misinterpreted Documents and Ignored Physical Facts: The History of &quot;Hitler’s Atomic Bomb&quot; Needs to be Corrected. In: Berichte zur Wissenschaftsgeschichte 39, S. 265 – 282, 2016&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is shown that until the end of the war the German physicists did&#xA;  not know that an atomic bomb can only be made with fast neutrons,&#xA;  except Heisenberg, who, however, discovered it rather late, did not&#xA;  communicate it clearly and did not study any bomb physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the article of the german science journal (spektrum.de) the author mentiones another important reason:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The physicists knew, that if they show the german government that it is possible to build a nuclear bomb they will all be forced to work on it. However if they would not succeed in two years, they would all be doomed to end in on of the infamous concentration camps. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the german scientists tried to avoid a big project. Instead there were a lot of small projects, which were not striving for success. The scientists just wanted to avoid being sent to the battlefront. The small projects on nuclear physics were more seen as a life insurance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the author concludes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The most important reason [for not developing the bomb] was fear of&#xA;  the Nazi regime. Already &lt;a href=&quot;https://en.wikipedia.org/wiki/Samuel_Goudsmit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Goudsmit&lt;/a&gt; had written, that &quot;science&#xA;  unser fascism was not, and in all probability could never be, the&#xA;  equal of science in a democracy&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3841" LastActivityDate="2016-11-25T17:54:51.397" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5413" PostTypeId="2" ParentId="5411" CreationDate="2016-11-25T18:09:07.713" Score="6" Body="&lt;p&gt;From the &lt;a href=&quot;http://jeff560.tripod.com/p.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt; site :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;PENCIL OF LINES. Desargues coined the term ordonnance de lignes, which is translated an order of lines or a pencil of lines [James A. Landau].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt; &lt;a href=&quot;https://archive.org/stream/AHistoryOfMathematics#page/n412&quot; rel=&quot;nofollow noreferrer&quot;&gt;(1968, p. 396)&lt;/a&gt; also attributes this terminology to Desargues.  From the section on Blaise Pascal, when discussing his &lt;a href=&quot;https://en.wikipedia.org/wiki/Pascal&amp;#39;s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Hexagrammum Mysticum Theorem&lt;/em&gt;&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... he followed the special language of Desargues, saying that if $A, B, C, D, E,$ and $F$ are successive vertices of a hexagon in a conic, and if $P$ is the intersection point of $AB$ and $DE$ and $Q$ is the intersection point of $BC$ and $EF$, then $PQ$ and $CD$ and $FA$ are lines &quot;of the same order&quot; (or, as we should say, the lines are members of a pencil, whether a pencil point or a parallel pencil). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The intuition behind the choice of &lt;em&gt;pencil&lt;/em&gt; in the English translation is not entirely clear.  Perhaps &quot;order&quot;, as in the translation &quot;an order of lines&quot;, was already too strongly associated with &lt;em&gt;order&lt;/em&gt;.  Boyer hints at this translated usage, writing : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Desargues assumes] that the parabola has a focus &quot;at infinity&quot; and that parallel lines meet at &quot;a point at infinity&quot;. The theory of perspective makes such ideas plausible, for light from the sun ordinarily is considered to be made up of rays that are parallel - comprising a cylinder or a parallel pencil of rays - whereas rays from a terrestrial source are treated as a cone or a pencil point.  The cylinder is merely a cone the vertex of which is infinitely distant, and a parallel pencil of lines is simply a family of lines all of which go through the same point at infinity. Desargues similarly studied a sheaf or bundle of planes through a point, finite or infinite.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reading further, it appears that Desargues was fond of colourful terminology.  For example, he calls a conic section a &quot;coup de rouleau&quot; - i.e., incidence with a rolling pin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also worth considering is that Desargues introduced these ideas and terms in a work not translated for at least two hundred years after its first appearance. Desargues' &lt;em&gt;Brouillon Projet d'une atteinte aux événements des rencontres du Cône avec un Plan&lt;/em&gt; (trans: Rough Draft of an Attempt to Deal with the Outcome of a Meeting of a Cone with a Plane) first appeared in Paris, 1639.  According to Boyer, Desargues published his works not to sell but to distribute to friends.  The work was completely lost until in 1847 a hand written copy made by Phillipe de Lahire, one of Desargues' few admirers, was found in a Paris library.  &lt;em&gt;Brouillon Projet&lt;/em&gt; is described by Boyer as one of &quot;the most unsuccessful great books ever produced&quot;. &lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="658" LastEditDate="2018-06-06T13:07:04.390" LastActivityDate="2018-06-06T13:07:04.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5414" PostTypeId="1" AcceptedAnswerId="5418" CreationDate="2016-11-26T06:09:55.737" Score="6" ViewCount="4163" Body="&lt;p&gt;I am reading &quot;Advice for young investigators&quot; by Santiago Ramón y Cajal , in which the author suggested Francis Bacon had made no impact on the scientific development*:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;It would not be wise in discussing general principles of research to overlook those panaceas of scientific method so highly recommended by Claude Bernard, which are to be found in Bacon’s Novum Organum and Descartes’s Book of Methods. They are exceptionally good at stimulating thought, but are much less effective in teaching one how to discover. After confessing that reading them may suggest a fruitful idea or two, I must further confess an inclination to share De Maistre’s view of the Novum Organum: “Those who have made the greatest discoveries in science never read it, and Bacon himself failed to make a single discovery based on his own rules.” Liebig appears even more harsh in his celebrated Academic Discourse when he states that Bacon was a scientific dilettante whose writings contain nothing of the processes leading to discovery, regardless of inflated praise from jurists, historians, and others far removed from science&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, I have also found that Thomas Young have said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Bacon first taught the world the true method of the study of nature, and rescued science from that barbarism in which the followers of Aristotle, by a too servile imitation of their master, had involved it.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, Charles Darwin has written that he proceeded&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“on true Baconian [inductive] principles and without any theory collected facts on a wholesale scale.” &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“How odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service!”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In order to solve the seemingly contradiction, I have read what Francis Bacon exactly had written, but the language was too hard for me to understand; while in general, I have read from other introductory texts claiming that Bacon has introduced the induction method into science; hence, so far I failed to see justification for Cajal's criticisms. Nevertheless, as I can't agree more with mostly everything else he wrote, I am left wondering:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What was the major influence of Francis Bacon on the development of modern science?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Besides, I remember Richard Feynman once said, Bacon's idea was great, but too impractical. However, I couldn't find the exact words.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Personally, I guess Beacon got the right idea, but wrong (sterile) rules. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2713" LastEditorUserId="2713" LastEditDate="2016-11-28T04:14:10.747" LastActivityDate="2016-11-30T07:15:13.767" Title="What was the major influence of Francis Bacon on the development of modern science?" Tags="|philosophy-of-science|scientific-method|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5415" PostTypeId="2" ParentId="5414" CreationDate="2016-11-26T17:28:19.733" Score="2" Body="&lt;p&gt;It is difficult to measure the influence of a specific book (or a writer). For example, I dare to say that very few scientists since the beginning of 18th century ever read anything by Newton (I have abundant evidence of this). I am speaking about Principia, not mentioning his mathematical papers. Most of his mathematical papers were not even published until they became &quot;history&quot;. And still the influence of Newton is enormous, and he invented calculus, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His ideas spread very quickly, and very soon more digestible expositions of them were published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even more this applies to Descartes. But there is hard evidence that Newton read Descartes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose this applies to Bacon as well. There is evidence of his large influence in the early 17th century (see Wikipedia article on Bacon). So his ideas spread somehow, even if few later physicists read him. And they spread so quickly, that &#xA;soon the need to read Bacon to learn these ideas disappeared: this just became &quot;common knowledge&quot;. This almost always happens with truly great discoveries. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that Bacon made no specific scientific discoveries. But his ideas had great influence nevertheless. Other philosophers who made no scientific discoveries were Aristotle, Voltaire, Kant. But their influence on the general intellectual atmosphere (in which scientific inquiry was possible at all) was enormous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Michael H. Hast wrote a book 100 most influential people in the world.&#xA;Of course one can argue with his list, but on my opinion it is essentially sound in its main features, and reflects common opinion (except some evident bias towards English-speaking authors). In this list the people I mentioned have these ranks:&#xA;Newton-2, Aristotle-13, Descartes-49, Voltaire-74, Bacon-90, and Kant is not included. (Any German list like this would have Kant). Here is the list:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.biographyonline.net/people/100-most-influential.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.biographyonline.net/people/100-most-influential.html&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT2: There is an evidence that Galileo either read Bacon or learned about his writings from an intermediary:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.unipune.ac.in/snc/cssh/ipq/english/IPQ/1-5%20volumes/01-3/1-3-3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.unipune.ac.in/snc/cssh/ipq/english/IPQ/1-5%20volumes/01-3/1-3-3.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and this paper also discusses the direct influence of Bacon on the Royal Society&#xA;in 17th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-11-26T17:58:34.400" LastActivityDate="2016-11-26T17:58:34.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5416" PostTypeId="1" AcceptedAnswerId="5417" CreationDate="2016-11-27T00:26:13.460" Score="8" ViewCount="557" Body="&lt;p&gt;I'm reading Colin Pask's book &lt;em&gt;Magnificent Principia&lt;/em&gt; and in 16.7.2 he states that the difficulty of the 3 body problem is in part tied to the lack of additional conservation laws at our disposal. In particular he says this with respect to Heinrich Bruns:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In fact, in 1887 mathematician and astronomer Ernst Heinrich Bruns proved that there are no further [beyond conservation of energy, momentum, and angular momentum] algebraic integrals or conservation laws to help us. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No reference is directly provided in the work for this result of Bruns. I am interested in the following:&#xA;1) What is the specific result of Bruns being referenced and where can it be found?&#xA;2) More generally, what did Bruns contribute to our knowledge of the 3 body problem. &lt;/p&gt;&#xA;" OwnerUserId="3273" LastEditorUserId="55" LastEditDate="2016-11-27T02:32:32.063" LastActivityDate="2016-11-29T22:46:04.877" Title="Did Bruns establish that the 3 body problem has no non-trivial conservation laws?" Tags="|physics|reference-request|astronomy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5417" PostTypeId="2" ParentId="5416" CreationDate="2016-11-27T02:29:03.720" Score="7" Body="&lt;p&gt;Poincaré made essential use of Bruns's results in his famous memoir Le Probléme des Trois Corps [The Three Body Problem], Revue générale des sciences pures et appliquees 2, 1-5 (1891). Here is Poincare's summary of Bruns's results, and his own, translated by &lt;a href=&quot;http://www.bourbaphy.fr/chenciner.pdf&quot; rel=&quot;noreferrer&quot;&gt;Chenciner in Poincaré and the Three-Body Problem&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The  differential  equations  of  the  Three-Body  Problem  possess  a  number  of  integrals  which  have  long  been familiar; these are those of the motion of the center of mass, the area integrals, the energy. It was extremely unlikely that they could have other algebraic integrals; it is, however only in the recent years that Mr. Bruns has proved this rigorously. But we can go further; apart from the known integrals, the Three-Body Problem admits no analytic and uniform integral; a careful study of the properties of periodic and asymptotic solutions is enough to establish this. It can be concluded that the various developments proposed so far are divergent; for their convergence would imply the existence of a uniform integral.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Chenciner gives a comprehensive discussion, and references Bruns's original paper, H. Bruns, Über  die  Integrale  des  Vielkörper-problems [On the Integrals of the Multi-body Problems], Acta  Mathematica,&#xA;volume 11 (1887). Also relevant is Poincare's paper Sur la Méthode de Bruns [On Bruns's method], C.R.A.S. 1896, t. 123, 1224-1228. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Curiously enough, despite all this there exists a convergent power series solution to the 3-body problem, which was found by Sundman in 1913. Saari gives an accessible account of Sundman's construction in &lt;a href=&quot;http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Saari.pdf&quot; rel=&quot;noreferrer&quot;&gt;A Visit to the Newtonian N-body Problem via Elementary Complex Variables&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Ironically, one of his major conclusions killed interest in a line of inquiry, so this particular result is not very well known. It should be; it is where Sundman &quot;solved&quot; the three-body problem according to accepted standards of the late 1800s and early 1900s.Indeed, in the late 1800s the King of Sweden and Norway established a prize for anyone who could find the solution of the N-body problem. The prize was awarded to Poincare in 1889 even though he hadn't solved the original problem. (On the other hand, Poincare's prizewinning paper contains a wealth of ideas that remain influential.) The originally stated problem finally was solved in 1913 by Sundman [16] when he found a converging series solution for the three-body problem. Unfortunately, his series converges so slowly that, essentially, it is useless for any practical purpose.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is a rare example of an explicitly constructive solution coming so far apart from practically useful. Nonetheless, there is a renewed interest recently in generalizing Sundman's methods to the $N$-body problems with $N&amp;gt;3$.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2016-11-27T02:29:03.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5418" PostTypeId="2" ParentId="5414" CreationDate="2016-11-27T03:47:12.693" Score="3" Body="&lt;p&gt;Francis Bacon did not himself do science, his chief contribution is to what is now called the methodology of science. But De Maistre only comments on one side of &lt;a href=&quot;https://en.wikipedia.org/wiki/Novum_Organum&quot; rel=&quot;nofollow noreferrer&quot;&gt;Novum Organum Scientiarum&lt;/a&gt;, its &quot;method&quot; of discovery by &quot;&lt;em&gt;inductio per enumerationem simplicem ubi non reperitur instant &amp;amp; contradictoria&lt;/em&gt;&quot; [induction by simple enumeration, excluding contradictory instances]. That is indeed not the most influential part of Bacon's methodological legacy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To better appreciate the New Organon we have to look at what it replaced, the Aristotle's old Organon, which reduced &quot;toolkit of science&quot; to mastering syllogisms (in fairness, Aristotle's view of the syllogistic was far richer than the redux scholastics turned it into). Bacon's view of induction might have been naive, but his takedown of the syllogistic in favor of it reoriented scientific inquiry in the right direction, from juggling empty generalities to observing and experimenting with nature, and gave it a ground to stand on in opposing &quot;old wisdom&quot; backed by ages and Aristotle's authority. His identification of major threats to successful scientific methodology and ethics was also highly liberating and influential, and his call to watch out for them is up to date even today. They are: the &lt;em&gt;Idols of the Tribe&lt;/em&gt; (general stereotypes of human nature), the &lt;em&gt;Idols of the Cave&lt;/em&gt; (personal preconceptions), the &lt;em&gt;Idols of the Market&lt;/em&gt; (linguistic stereotypes) and the &lt;em&gt;Idols of the Theatre&lt;/em&gt; (&quot;the dogmas of the philosophers&quot;, conceptual obstacles). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is one thing to imitate the masters of early science, but another to express what motivated and animated them generally and explicitly, if crudely. Bacon made the idea of science intellectually respectable, he made it make sense to outsiders, and something the scientists could rally to, he gave the popularizers, like Voltaire and the Encyclopedists, arguments and mottos to work with in promoting it to the public. Bacon was not alone in it, of course, but his was the most focused and comprehensive of the early accounts, and it became a landmark and a point of departure. To generations of scientists Bacon was so &quot;morally right&quot; that as late as mid 19th century his inductive naivete was mostly overlooked, and even partly reproduced, by &quot;professional&quot; methodologists like John Stewart Mill. This also shows that while the &quot;scientific method&quot; is &quot;clear&quot; to us in the works of Kepler, Galileo and Newton it took two centuries to finally spell out what it was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C.S. Peirce was one of the first to do it with the recognizable contours of the &quot;hypothetico-deductive method&quot;. Here is his somewhat irreverent assessment of both Bacon's strengths and weaknesses in &lt;a href=&quot;http://www.peirce.org/writings/p107.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fixation of Belief (1877)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;To Roger Bacon, that remarkable mind who in the middle of the thirteenth century was almost a scientific man, the schoolmen's conception of&#xA;  reasoning appeared only an obstacle to truth... Four centuries later, the more celebrated Bacon, in the first book of his &lt;strong&gt;Novum Organum&lt;/strong&gt;, gave his clear account of experience as something which must be open to verification and reexamination. But, superior as Lord Bacon's conception is to earlier notions, a modern reader who is not in awe of his grandiloquence is chiefly struck by the inadequacy of his view of scientific procedure.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;That we have only to make some crude experiments, to draw up briefs of the results in certain blank forms, to go through these by rule, checking off everything disproved and setting down the alternatives, and that thus in a few years physical science would be finished up -- what an idea! &quot;He wrote on science like a Lord Chancellor,&quot; indeed, as &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Harvey&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harvey&lt;/a&gt;, a genuine man of science said. The early scientists, Copernicus, Tycho Brahe, Kepler, Galileo, Harvey, and Gilbert, had methods more like those of their modern brethren.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is easy to say that &lt;em&gt;enumerationem simplicem&lt;/em&gt; can't do the trick, it turned out to be incredibly hard to say what does. Bacon's naivete can be all the more excused considering that the more recent methodologists of science, logical positivists and Popper, had even less to say on how scientific hypotheses are formed. In the ironically named Logic of Scientific Discovery Popper wrote:&quot;&lt;em&gt;The initial stage, the act of conceiving or inventing a theory, seems to me neither to call for logical analysis nor to be susceptible of it.&lt;/em&gt;&quot; The hypothetico-deductive method (H-D) is supposed to apply to the &quot;justification&quot; stage only. Hanson in &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/00048406085200111&quot; rel=&quot;nofollow noreferrer&quot;&gt;Is there a Logic of Scientific Discovery? (1960)&lt;/a&gt; points out that while Bacon's account is simplistic the hypothetico-deductive story is deficient without it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;There is something wrong. It is false. Scientists do not always discover every feature of a law by enumerating and summarizing observables... H-D accounts do not, however, tell us anything about the context in which laws are proposed in the first place; nor, perhaps, were they even intended to. The induction-by-enumeration story did intend to do this... But this does not strengthen the H-D account as against the inductive view. There is no H-D account of how &quot;sophisticated generalizations&quot; are derived. On his own principles, the H-D theorist's lips are sealed on this matter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Peirce did offer a more sophisticated analysis of hypothesis formation based on his theory of abductive inference, but unlike his &quot;fallibilism&quot; picked up by Popper, it was forgotten for a while. It became a topic of active research in the last decades however, see &lt;a href=&quot;http://www.springer.com/la/book/9783642036309&quot; rel=&quot;nofollow noreferrer&quot;&gt;Magnani's Abductive Cognition&lt;/a&gt; for a recent overview.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-11-29T01:08:30.567" LastActivityDate="2016-11-29T01:08:30.567" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5419" PostTypeId="2" ParentId="5411" CreationDate="2016-11-28T20:53:34.473" Score="3" Body="&lt;p&gt;This is a question about English terminology. As others on here have pointed out, the French terminology is different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original meaning of the English word “pencil” is a fine brush; this is also the meaning of French pinceau (as opposed to French &quot;crayon&quot; = English &quot;pencil&quot;). According to the Oxford English Dictionary the earliest English attestation of “pencil” in its mathematical sense is in a book from as late as 1840. The underlying image is evidently that of lines converging at a single point in the same way that the fine strands on a small pointed brush converge at the tip.&lt;/p&gt;&#xA;" OwnerUserId="364" LastEditorUserId="364" LastEditDate="2016-11-28T21:01:18.007" LastActivityDate="2016-11-28T21:01:18.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5420" PostTypeId="1" AcceptedAnswerId="5424" CreationDate="2016-11-29T05:02:12.710" Score="3" ViewCount="541" Body="&lt;p&gt;A previous &lt;a href=&quot;https://hsm.stackexchange.com/questions/56/when-was-the-method-of-getting-square-roots-invented-by-vi%c3%a8te-in-1610-and-devel&quot;&gt;question&lt;/a&gt; credits &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;François Viète&lt;/a&gt; with introducing the well-known longhand method for the computation of square roots in digit-by-digit manner. This method is related to the binomial theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in a German source (&quot;Kleine Enzylopädie Mathematik&quot;. Leipzig, Bibliographisches Institut 1965) I came across a remark that the German mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Michael_Stifel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Stifel&lt;/a&gt; was already aware of this method in the 16&lt;sup&gt;th&lt;/sup&gt; century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google provides a full scan of &lt;a href=&quot;https://books.google.com/books?id=7sVZAAAAcAAJ&amp;amp;dq&quot; rel=&quot;nofollow noreferrer&quot;&gt;the book&lt;/a&gt; &quot;Die Coss Christoffs Rudolffs, Mit schönen Exempeln der Cosz. Durch Michael Stifel Gebessert vnd sehr gemehrt&quot; of 1571. This is a book on algebra written originally by &lt;a href=&quot;https://en.wikipedia.org/wiki/Christoph_Rudolff&quot; rel=&quot;nofollow noreferrer&quot;&gt;Christoph Rudolff&lt;/a&gt; that was improved and expanded by Stifel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fourth chapter of the book describes the computation of square roots and cube roots. Starting at &lt;a href=&quot;https://books.google.com/books?id=7sVZAAAAcAAJ&amp;amp;pg=PA42#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fol. 42&lt;/a&gt; the procedure for taking square roots is demonstrated, using the computation of $\sqrt{72352036}$ as an example. What is described definitely appears to be the &lt;a href=&quot;https://xlinux.nist.gov/dads/HTML/squareRoot.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;longhand square root&lt;/a&gt;  algorithm taught in school until a few decades ago, before the advent of affordable electronic calculators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia does not have much on Rudolff, but notes that &quot;He introduced the radical symbol (√) for the square root.&quot; Since Stifel's additions to Rudolff's &quot;Coss&quot; of 1515 seem to be carefully indicated in the 1571 edition, it would seem that the longhand square root method was already described in Rudolff's original, but I have not managed to track down a scan of that yet. It is also not clear whether Rudolff invented the method. As Stifel's edition makes no such claim, I think it is possible that the algorithm predates Rudolff and possibly the 16th century.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-11-30T20:58:27.167" Title="Who first introduced the longhand square-rooting method into European mathematics?" Tags="|mathematics|discoveries|elementary-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5421" PostTypeId="2" ParentId="5416" CreationDate="2016-11-29T22:46:04.877" Score="0" Body="&lt;p&gt;I do not have enough reputation points to post this question as a comment which was my original intention but I can post it as an answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given Sundman's convergent power series solution what exactly does Chenciner's translation of Poincare mean by &quot;the Three-Body Problem admits no analytic and uniform integral&quot; ?&lt;/p&gt;&#xA;" OwnerUserId="535" LastActivityDate="2016-11-29T22:46:04.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5422" PostTypeId="2" ParentId="5414" CreationDate="2016-11-29T22:48:59.820" Score="1" Body="&lt;p&gt;I had not the time to write something shorter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Stanley Jevons&lt;/strong&gt;, &lt;strong&gt;1874&lt;/strong&gt;, &lt;em&gt;Principles of Science&lt;/em&gt;, 2, wrote: Bacon's method is strictly impossible, and if were possible, is the least productive method one could imagine: observe everything, write it all down, and wait for it to digest itself by eliminating disagreements into theories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem with scholastics was not that they did not base their universals on experience. They did not believe in laws of nature, or even principles, universals. The Nominalists based all on the concrete, much like Hutton (1794) and Joachim (1948) and Wheeler (1990), but significantly unlike them in that Hutton and Joachim and Wheeler based their universals ontologically on the concrete. The Nominalists rejected universals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They believed all was special intervention of God and God could always change his mind (even Jevons superstitiously inserted that clause in his book at late as 1874). They never bothered to make sense of their experience and Bacon recommends against it; he suggests it must be accumulated to digest itself. If it fails to give a theory, you just need more experience. But this method always fails. No amount of instances can prove a universal. At best we can assume an experiment is well controlled, so the sequence falsifies all other theories, because it is the same everywhere and everywhen, since all is measurable and nothing measurable can be inconsistent (Hutton) or simply that it at least falsified a competing conjecture because its predictions failed, regardless of how controlled the experiment was, at most leaving fewer explanations than before (Popper). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem with the Nominalists's methods is that they did not take theories or conjectures seriously. They did not believe in universals. For example: Buridan writes: an ass is a horse is true. Huh? &lt;em&gt;What did you say&lt;/em&gt;? Buridan explains: it is written on a piece of paper, the words are objective now (in a sense Popper (1972) reintroduced), therefore an instance of this exists, and truth is correspondence with experience. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The scholastics and nominalists did not take any predictions seriously, did not believe in universals, in laws of nature, all was special arbitrary choice of god, so they alleged, and they collected experience and tested theories, but never admitted that the theory applies to any other instances. No wonder science made no progress prior the seventeenth and eighteenth centuries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That had to wait for others to champion the idea of simple laws of nature explicitly. James Hutton (1794, 1795) and Ralph Gerard (1940) and then Stephen Wolfram (2002) write that all the universe complexity and complication is by repeated operation of extremely simple constant laws which have enormous scope in time to work out. Wheeler (1994) suggests the laws may self modify over time as part of their operation but that is it. Uniformitarianism. Conjectured universals. That was the big idea. First tacitly then in public, finally, by Leibniz and Hume and Hutton.&#xA;Newton writes that explanations should be as simple as possible to explain appearances. (Menger later added: Yes, but not simpler.) Newton also postulated rules that Euler revised into the Newtonian classical mechanics. The explicit fact that however even simpler rules over a long time can create complexity and all natural forms, instead of merely describing interaction of complex or complicated natural forms, had to wait.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jevons agrees with Hume and Popper that induction, in sense of learning universals from experience, does not exist at all, and there is no problem of induction, except strict inverse deduction, enumerating all combinations and solving a very complex combinatorics to find out all combinations of objects that could combine to construct those instances according to their know behavior. Enumerating all finite instances and eliminating the contradictions is not a method but proves nothing about universals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Joachim, Eddington, Hayek, Wheeler, Popper, Harris write that there are no foundations and no justifications only falsifications and elimination of free conjectures. The precedence of the conjectures allows us to know and use the truth without having to prove it and let us survive for often had we to prove it be some inductive procedure we would know what exists in our environment to late to survive (Hayek 1952, Popper 1962). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think some misunderstand Popper. He does not say anything about how conjectures are formed. True; but that is only because the conjectures are purely unfounded, free constructions. They can be aesthetic as Dirac, Eccles, Popper, Atiyah, Penrose, Poincare, Wheeler, etc, think. Based on nothing more than this would be pretty if it were true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no justification stage. Only falsification. We cannot justify a conjecture, only falsify competing ones. Popper's (and Hayek's), Wheeler's, and Errol Harris's are basically the first completely without foundations methods of science. Because foundations, justification of justification, leads to infinite regress, foundations that have foundations and yet in the end no foundations, no towers of turtles ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Joachim, Blanshard, Wheeler, Eddington, Hayek (but not Popper), Harris, etc, also point out that all is without foundations but circular and consistent in epistemology as in ontology. The conjecture not only defines what falsifies it (and this is what it means) but also what and how to measure. Fodor (1968) writes: different conjectures not only predict different results but perform different experiments and measure different things in different ways. Deutsch (2011) writes: most conjectures are not even falsified but cease to be entertain or become unentertainable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is all consistent with some empiricists (Hume, Condillac, Hutton, Bailey, modern physicists) and inconsistent with others (Bacon, Mill, Russell, Wittgenstein, logical positivists).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some further thoughts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Boden&lt;/strong&gt;, &lt;strong&gt;2006&lt;/strong&gt;, &lt;em&gt;Mind As Machine&lt;/em&gt;, 1, 2, is an empiricist who correctly emphasizes the framing problem for empiricism. You need a conjecture ready to decide what is relevant to then use to build bottom-up associations for further empirical theory generation. &lt;strong&gt;Chomsky&lt;/strong&gt; basically alleges this frame is genetically inherited. That is because he tacitly assumes conjectures can be falsified only a finite number at a time so only a finite number must be posited for an experience sieve to leave a finite filter. &lt;strong&gt;I&lt;/strong&gt; disagree with him; genes do not code for reasoning nor how to reason. Rather they are developmental resources. They only code for proteins, which have enormous degrees of freedom; the framing problem is still unsolved after five decades.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are hundreds of different algorithms, all possible for generating conjectures from experience, that is why Popper and all but Harris avoided talking about how conjectures are made. That is the field of artificial cognition and not methodology. Dozens of journals with new algorithms each year.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2016-11-30T07:15:13.767" LastActivityDate="2016-11-30T07:15:13.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5424" PostTypeId="2" ParentId="5420" CreationDate="2016-11-30T01:30:12.373" Score="3" Body="&lt;p&gt;Whenever you wonder about who first introduced something mathematical into Western Europe, Leonardo of Pisa aka &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fibonacci&lt;/a&gt;&lt;/strong&gt; should be the first person you think of. Fibonacci's 1202 work &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Liber_Abaci&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liber Abaci&lt;/a&gt;&lt;/em&gt; has a chapter dedicated to the finding of square and cube roots, with an introduction referencing Euclid and Al-Khwarizmi, and then launching into a demonstration of finding the square root of 10. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google Books has a preview of L. E. Sigler's translation, with the relevant section &lt;a href=&quot;https://books.google.com/books?id=PilhoGJeKBUC&amp;amp;pg=PA490&amp;amp;lpg=PA490&amp;amp;dq=&amp;quot;therefore+in+the+treatment+of+the+finding+of+roots&amp;quot;&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It can be difficult to parse Fibonacci's notation-free verbiage (F. calls &quot;first&quot; what we would call the &quot;last&quot; digit and vice versa), but if you read the chapter introduction you see it explained that it's based on the fact that $(a+b)^2 = a^2 +2ab+b^2$ (just as the modern method is) and as you go through the chapter (I used my printed copy) there are several examples of square root extractions with diagrams that extract the roots digit by digit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, Fibonacci does not instruct the reader to &quot;mark off pairs of digits starting from the right&quot; as an initial setup step. Instead, he tells us to &lt;em&gt;count&lt;/em&gt; the digits (his examples are all of integers) and understand that when a number has $2n$ or $2n-1$ digits, the square root has $n$ digits (in its integer part).  And if $2n-1$ digits, the leftmost group has only one digit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That isn't an actual difference in the method; it's is just another way of wording the initial step, and the difference in wording is not integral to the method.&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2016-11-30T20:58:27.167" LastActivityDate="2016-11-30T20:58:27.167" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5425" PostTypeId="1" CreationDate="2016-11-30T11:57:03.607" Score="10" ViewCount="2003" Body="&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Babylonian_mathematics&quot; rel=&quot;noreferrer&quot;&gt;Babylonian mathematics&lt;/a&gt; says that the ancient Babylonians usually used a round value for $\pi$ (3). But they knew a more precise value:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Babylonian texts usually approximated π≈3, sufficient for the architectural projects of the time. The Babylonians were aware that this was an  approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of π as 25/8=3.125, about 0.5 percent below the exact value.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question is: is this tablet the only one mentioning a better value for π than 3, or are there other known Babylonian sources with more precise values  before the Hellenistic conquest in 4th century BCE? &lt;/p&gt;&#xA;" OwnerUserId="3780" LastEditorUserId="55" LastEditDate="2016-12-02T01:50:39.583" LastActivityDate="2020-08-09T06:47:27.273" Title="What was the best approximation of π known to ancient Babylonians?" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="5426" PostTypeId="1" AcceptedAnswerId="5428" CreationDate="2016-11-30T15:12:17.503" Score="-6" ViewCount="290" Body="&lt;p&gt;Atheists claim Creation Science isn't really science while creationists claim it is. Historically, which is correct according to the scientific community and why do they say this? I hope this is not off-topic, but I don't know where else to ask this.&lt;/p&gt;&#xA;" OwnerUserId="5030" LastActivityDate="2016-11-30T19:27:34.393" Title="Creation Science" Tags="|debunking|" AnswerCount="1" CommentCount="7" ClosedDate="2016-12-03T11:02:40.303" ContentLicense="CC BY-SA 3.0" />
  <row Id="5427" PostTypeId="1" CreationDate="2016-11-30T18:06:24.293" Score="4" ViewCount="726" Body="&lt;p&gt;I'm in my first year of Mathematics at the University. Recently, we've learnt about &lt;a href=&quot;https://math.stackexchange.com/questions/265952/grassman-formula-for-vector-space-dimensions&quot;&gt;Grassmann Formula&lt;/a&gt; and when I was making a little research on the internet, I couldn't find a single reference confirming whether it is named that way because Hermann Grassmann was the one who discovered it or it is just named after him by another person. I was just very curious to know this, but I don't know where to look it up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know something about the explanation for the name of the formula?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;thanks in advance :)&lt;/p&gt;&#xA;" OwnerUserId="5031" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-12-03T23:34:55.750" Title="Grassmann Formula" Tags="|mathematics|terminology|linear-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5428" PostTypeId="2" ParentId="5426" CreationDate="2016-11-30T19:27:34.393" Score="3" Body="&lt;p&gt;The &quot;Scientific Community&quot; is an abstraction, rather than an organized group with stated positions on things, so a strictly factual response on what they believe is probably impossible.  No doubt there are scientists who believe Creation Science is valid science.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But several groups and organizations have addressed the question and found the negative.  In the Court case Kitsmiller v. Dover Area school district, the court found ID wasn't a science, based in part on measures of the scientific community as a whole.  And most of the relevant professional organizations for biologists have released statements against the idea that ID and Creation Science are sciences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just as an example, heres part of the American Institute of Biological Sciences statement on a Bush-era attempt to encourage ID in the classroom:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Intelligent design is not a scientific theory and must not be taught in science classes,&quot; said AIBS president Dr. Marvalee Wake, a perspective shared by President Bush's science advisor, Dr. John Marburger III. On Tuesday, August 2, Marburger stated in an interview that &quot;evolution is the cornerstone of modern biology&quot; and &quot;intelligent design is not a scientific concept.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Googling, you can find many other similar statements from both the AIBS and basically every other similar organization in the US.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to the extent that the &quot;Scientific Community&quot; has a position on the subject, I think its pretty clearly that Creation Science is not a science.  &lt;/p&gt;&#xA;" OwnerUserId="1731" LastActivityDate="2016-11-30T19:27:34.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5429" PostTypeId="2" ParentId="5427" CreationDate="2016-11-30T20:10:25.620" Score="4" Body="&lt;p&gt;For a detailed discussion of Grassmann's work see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hans-Joachim Petsche, &lt;a href=&quot;https://books.google.it/books?id=nWHeVXhUT2oC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Grassmann Biography&lt;/a&gt; (2009),&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and see &lt;a href=&quot;https://books.google.it/books?id=nWHeVXhUT2oC&amp;amp;pg=PA186&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 186&lt;/a&gt; for the discussion of the formula :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After basically applying the conceptual constructions from the first section of &lt;em&gt;Extension Theory&lt;/em&gt; in a more general way to elementary magnitudes in the second section, Graßmann introduces, after dealing with the problem of coordinates, a new form of generating products with these vector magnitudes. In quite difficult constructions, characterized by an excessive striving for extreme abstraction, he obtains a conjunctive product which is contrasted as a dual form of the exterior product and which he calls “eingewandtes Produkt” (a conjunction which later, in the 1861 revised edition of &lt;em&gt;Extension Theory&lt;/em&gt;, is termed the “regressive product” and comprehensively developed there).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Graßmann connects his reflections to dimensional considerations on the generating system of two multivectors $A$ and $B$ entering into a multiplicative conjunction. Thus he obtains the fundamental relation: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$dim(A) + dim(B) = dim(A + B) + dim(A ∩ B)$, &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;that is, the sum of the dimensions of the generating systems of $A$ and $B$ is equal to the sum of the dimension of the sum-space (“umfassendes Gebiet”) and the dimension of the intersection space (“gemeinschaftliches Gebiet”) of the generating systems of $A$ and $B$ [ ref to : Linear Extension Theory. In: A new branch of mathematics. The &lt;em&gt;Ausdehnungslehre&lt;/em&gt; of 1844 and Other Works by Hermann Graßmann. Translated by Lloyd C. Kannenberg. Chicago and La Salle: Open&#xA;  Court 1995, p. 202 ]. If one now equates the dimension of the initial space (“Hauptgebiet”) to $n$, then the exterior product of the multivectors $A$ and $B$ will only be unequal to zero if &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$dim(A ∩ B) = 0$ and $dim(A) + dim(B) = dim(A + B) ≤ n$, &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;that is, when “$A$ is completely outside the domain of $B$” (hence the term “exterior product”). For the product $AB$ we can then say: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$dim(AB) = dim(A) + dim(B) = dim(A + B)$.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course, we have to check on the original &lt;a href=&quot;https://books.google.it/books?id=92PzuM08o6UC&quot; rel=&quot;nofollow noreferrer&quot;&gt;German text&lt;/a&gt; : &lt;strong&gt;Drittes Kapitel&lt;/strong&gt; : Das eingewandte Produkt, §125-on, to find the &quot;best approximation&quot; to the said formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See footnote page 185, with the discussion of the example of two lines (&lt;em&gt;zweiter Stufe&lt;/em&gt; = second order):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;System zweier Linien, so wird, da jene als Elementarsystem von dritter, diese von zweiter Stufe sind, das gemeinschaftliche System von $(2+2-3)$ter, d.h. von erster Stufe sein, und somit entweder durch einen Punkt [...]&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-12-01T12:46:27.830" LastActivityDate="2016-12-01T12:46:27.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5430" PostTypeId="2" ParentId="5427" CreationDate="2016-11-30T20:24:48.367" Score="2" Body="&lt;p&gt;Grassman identified the concept of &quot;an algebra&quot; in his book &lt;em&gt;Die lineale Ausdehnungslehre...&lt;/em&gt; (The Theory of Linear Extensions).  In this book he introduced the concepts used in the referenced formula - those of subspace and dimensions - as well as proving this formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.39.1387&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper from the Penn State website&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He defines the notions of subspace, independence, span, dimension, join and&#xA;  meet of subspaces, and projections of elements onto subspaces. He is aware of&#xA;  the need to prove invariance of dimension under change of basis, and does so. He proves the Steinitz Exchange Theorem, named for the man who published it&#xA;  in 1913 (and who, incidentally, defined a linear space in terms of &quot;units&quot; in the same way Grassmann did). Among other such results, he shows that any finite set has an independent subset with the same span and that any independent set extends to a basis, and he proves the important identity $$\dim(U + V) = \dim(U) + \dim(V) - \dim(U \cap V).$$ He obtains the formula for change of coordinates under change of basis, defines elementary transformation of bases, and proves that every change of basis (equivalently, in modern terms, every invertible linear transformation), is a product of elementaries.    &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-12-03T23:34:55.750" LastActivityDate="2016-12-03T23:34:55.750" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5433" PostTypeId="1" AcceptedAnswerId="5438" CreationDate="2016-12-01T01:34:54.313" Score="2" ViewCount="3419" Body="&lt;p&gt;I am a freshman, mathematics. I have a presentation assignment for a class. I am expected to talk about the necessity of abstract sciences. The thesis I need to argue is the following: ''Some people are against some researches like space explorations because they find those just a waste of time and money.'' Indeed, even in movies we can see this, for instance ''Interstellar''. I have already prepared a speech in which I mention ''A Mathematician's Apology'' by Hardy. There is also some information from sources like NASA about why we should explore space. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, I believe that humankind has much more to do than just surviving, and so we need science that is not necessarily &quot;applicable&quot; in the narrow sense. But I feel like I need some historical sources discussing the role of &quot;pure&quot; mathematics and science, like the Hardy's book. Are there other such sources? Were there historical episodes when society or a subset of society turned against scientists because their science was &quot;inapplicable&quot;? How did scientists respond? Any help would be appreciated. &lt;/p&gt;&#xA;" OwnerDisplayName="user4962" LastEditorUserId="55" LastEditDate="2016-12-01T22:39:35.830" LastActivityDate="2016-12-02T15:36:23.677" Title="Why is pure mathematics important? More generally, why do some scientists deal with inaplicable notions?" Tags="|mathematicians|philosophy-of-science|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5434" PostTypeId="1" AcceptedAnswerId="5458" CreationDate="2016-11-29T15:32:38.480" Score="1" ViewCount="72" Body="&lt;p&gt;I want to know how different stages of human development were elucidated. What experiments were done? Do they involve taking out zygotes, embryos, and fetuses at different stages? How could one study all these stages in development of one individual? Does this mean that many developing humans were examined?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think MRI could have helped because it was discovered a bit later.&lt;/p&gt;&#xA;" OwnerUserId="5035" OwnerDisplayName="JM97" LastEditorUserId="62" LastEditDate="2016-12-08T17:59:08.313" LastActivityDate="2016-12-08T17:59:08.313" Title="Study of embryos" Tags="|biology|medicine|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5435" PostTypeId="1" CreationDate="2016-12-01T22:18:34.010" Score="0" ViewCount="145" Body="&lt;p&gt;I have heard that Galileo Ferraris and Nikola Tesla invented the induction motor independently.  Does anyone know who was the first and were they similar in power output and design?   This may be a duplicate question but the search does not seem to come up with anything for me. I am also interested in knowing why two people are credited with the same discovery if they worked independently. (Unless of course they both did it at the exact moment.:-)) &lt;/p&gt;&#xA;" OwnerUserId="5039" LastActivityDate="2016-12-01T22:18:34.010" Title="Who was the first to invent the induction motor?" Tags="|electromagnetism|electricity|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5436" PostTypeId="2" ParentId="5425" CreationDate="2016-12-02T04:49:02.417" Score="11" Body="&lt;p&gt;No additional or more precise approximations to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; seem to have been found in Babylonian records up till now.&lt;/p&gt;&#xA;&lt;p&gt;Herman C. Schepler, &amp;quot;The Chronology of PI&amp;quot;, &lt;em&gt;Mathematics Magazine&lt;/em&gt;, Vol. 23, No. 4, Mar. / Apr. 1950, pp. 216-228 (as reprinted in L. Berggren, J. Borwein, and P. Borwein, &amp;quot;Pi: A Source Book, 3rd Edition&amp;quot;, Springer 2003) summed up the extent of our knowledge of Babylonian &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;-approximations at that time:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Babylonians, Hindus, and Chinese used the value 3. It is probable that the Hebrews adopted this value from the Semites (Babylonian predecessors). No definite statement for the value of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; has yet been found on the Babylonian cylinders (1600 to 2300 B. C.).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Petr Beckmann, &amp;quot;A History of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&amp;quot;, New York: St. Martin's Press 1971, states (p. 21) that a translation of the tablet found in 1936 about 200 miles from Babylon was not published until 1950, and that it derives the value of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; from the circumscribed circle of a hexagon, giving the value in &lt;a href=&quot;https://en.wikipedia.org/wiki/Sexagesimal&quot; rel=&quot;noreferrer&quot;&gt;sexagesimal&lt;/a&gt; fractions as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{3}{\pi} = \frac{57}{60} + \frac{36}{(60)^{2}} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which yields &lt;span class=&quot;math-container&quot;&gt;$\pi={3{\frac{1}{8}}}$&lt;/span&gt;. Alfred S. Posamentier and Ingmar Lehmann, &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;: A Biography of the World's Most Mysterious Number&amp;quot;, New York: Prometheus Books 2004, likewise states (p. 44):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We now take a big leap in time to the Babylonians, which spans from 2000 BCE to about 600 BCE. In 1936 some mathematical tablets were unearthed at Susa (not far from Babylon). One of these compares the perimeter of a regular hexagon to the circumference of its circumscribed circle. The way they did this led today's mathematicians to deduce that the Babylonians used &lt;span class=&quot;math-container&quot;&gt;${3{\frac{1}{8}}} = 3.125$&lt;/span&gt; as their approximation for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This seems to be a pretty good indication that no new information on Babylonian approximations to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; has come to light in recent years. Kazuo Muroi, &amp;quot;The oldest example of &lt;span class=&quot;math-container&quot;&gt;$\pi \approx {3\frac{1}{8}}$&lt;/span&gt; in Sumer: Calculation of the area of a circular plot&amp;quot;, &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1610/1610.03380.pdf&quot; rel=&quot;noreferrer&quot;&gt;arXiv preprint&lt;/a&gt;, 2016, has specific examples from Babylonian sources for the &lt;em&gt;implicit&lt;/em&gt; use of both approximations to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; in the context of computing the area of a circle &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; from its circumference &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$A = \frac{c^{2}}{4\pi} \approx 0;5 \ c^{2}$&lt;/span&gt;, implying &lt;span class=&quot;math-container&quot;&gt;$\pi \approx 3$&lt;/span&gt;; &lt;span class=&quot;math-container&quot;&gt;$A = \frac{c^{2}}{4\pi} \approx 0;4,48 \ c^{2}$&lt;/span&gt;, implying &lt;span class=&quot;math-container&quot;&gt;$\pi \approx {3{\frac{1}{8}}}$&lt;/span&gt;. He remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematical problems the first formula frequently occurs but the second has only been found once so far, in a problem which concerns the volume of a cylindrical log.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Muroi &lt;em&gt;speculates&lt;/em&gt; that the Babylonians may have known more accurate approximations for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, which they did not use because they could not be conveniently represented as sexagesimal fractions.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2020-08-09T06:47:27.273" LastActivityDate="2020-08-09T06:47:27.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5438" PostTypeId="2" ParentId="5433" CreationDate="2016-12-02T14:45:19.570" Score="9" Body="&lt;p&gt;Historical experience shows that pure mathematics is one of the most useful parts of science. Pure mathematicians discover things which find applications later. Without pure mathematics, most of the &quot;applied mathematics&quot; and other sciences would be simply impossible. Let me give just one example. Ancient Greeks intensively studied curves called conic sections. There are various speculations of historians, about what motivated them, and it seems that nothing but pure curiosity, and other related pure mathematics problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1000 years later it was discovered by Kepler that planets and satellites move on conic sections. This was an application to the &quot;real world&quot;, astronomy, but still astronomy is a &quot;pure science&quot;. But discovery of Kepler triggered discoveries of Newton: to explain this motion of planets the whole science of mechanics was developed. Which is a basis of all modern engineering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Discoveries of Kepler and Newton would be impossible without that work of the Greeks 1000 years ago. And this pattern is essentially omnipresent, if one traces the development of science. In the modern times, the period between a discovery in pure mathematics and its influence on applied sciences is usually shorter.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not saying that EVERY mathematical discovery found or will find applications, but Mathematics is like a living organism: you cannot remove parts of it, all parts are necessary. And you never know in advance which part will&#xA;play a role in applications.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-02T14:45:19.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5439" PostTypeId="2" ParentId="5433" CreationDate="2016-12-02T15:36:23.677" Score="3" Body="&lt;p&gt;Basic cryptography used everyday by millions of people (security of email/text communications or e-commerce) depends on ideas from pure mathematics that were developed from pure intellectual interest rather than practical applications:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Fermat's little theorem is the backbone to RSA and is attributed to Fermat in the 1600s. Its standard formulation in terms of modular arithmetic goes back to Gauss in the early 1800s.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Elliptic curve cryptography depends on elliptic curves over finite fields. Elliptic curves were first studied over the complex numbers in the 19th century, developing from the study of elliptic functions (more pure math). Finite fields, in general, were created by Galois in the early 19th century as a topic in pure math. The consideration of elliptic curves over finite fields first came about in pure math and dates back to the first decades of the 20th century.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="643" LastActivityDate="2016-12-02T15:36:23.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5440" PostTypeId="1" AcceptedAnswerId="5442" CreationDate="2016-12-02T19:22:42.473" Score="1" ViewCount="5018" Body="&lt;p&gt;Basically, why did Newton write : &quot;Plato is my friend — Aristotle is my friend — but my greatest friend is truth.&quot; ? What were the influences of these two on him ? &lt;/p&gt;&#xA;" OwnerUserId="2310" LastActivityDate="2018-04-29T13:18:49.400" Title="Why does Newton claim : &quot;Plato is my friend — Aristotle is my friend — but my greatest friend is truth.&quot;?" Tags="|mathematics|physics|biographical-details|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5441" PostTypeId="2" ParentId="5440" CreationDate="2016-12-02T23:41:47.337" Score="0" Body="&lt;p&gt;This he said when he was a student in Cambridge. The cornerstone of &quot;science&quot; education at that time was reading Aristotle. So he started reading Aristotle. &#xA;But the new ideas were already &quot;in the air&quot;. Accidentally Newton looked into Euclid... then Descartes... and started to think about &quot;new science&quot;, that is real one, discarding Aristotle as he deserves. Source: Westfall, Never at rest (a standard scientific biography of Newton).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-02T23:41:47.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5442" PostTypeId="2" ParentId="5440" CreationDate="2016-12-03T15:43:54.947" Score="7" Body="&lt;p&gt;Newton (I assume the attribution is correct) was merely paraphrasing a well-known Latin phrase traditionally ascribed to Plato (though not found in any of Plato's authentic works). Roger Bacon quoted it as: Nam Plato dicit: &quot;Amicus est Socrates, magister meus, sed magis est amica veritas&quot;, and Bacon's contemporary Thomas Aquinas cites it in a similar form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More here:&lt;a href=&quot;https://en.wikipedia.org/wiki/Amicus_Plato,_sed_magis_amica_veritas&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Amicus_Plato,_sed_magis_amica_veritas&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2016-12-03T15:43:54.947" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5443" PostTypeId="1" CreationDate="2016-12-03T19:11:22.357" Score="3" ViewCount="319" Body="&lt;p&gt;I know that studying the evolution and history of a certain subject is a way of resolving the complexity of that subject. So I want to ask about references that describe the history and evolution of the foundation of mathematics as well as its philosophy and milestones of each part. For example, set-theory may be naive of axiomatic; and axiomatic have many versions. Each version is sufficient for explaining certain branch in mathematics. I want another reference emphasizing the application of each version in mathematics itself.&lt;/p&gt;&#xA;" OwnerUserId="579" LastEditorUserId="14422" LastEditDate="2021-05-14T07:03:24.553" LastActivityDate="2021-05-14T07:03:24.553" Title="History of Foundation of Mathematics" Tags="|mathematics|reference-request|mathematical-logic|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5444" PostTypeId="2" ParentId="5443" CreationDate="2016-12-03T23:31:58.380" Score="3" Body="&lt;p&gt;There is a good book on foundations with emphasis on history:&#xA;Abraham Fraenkel and Yehoshua Bar-Hillel, &#xA;Foundations of set theory. Original is in German. But a second revised edition,&#xA;which has one more author, Azriel Levy, has been translated into English. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-03T23:31:58.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5445" PostTypeId="1" AcceptedAnswerId="5450" CreationDate="2016-12-04T06:51:48.497" Score="5" ViewCount="773" Body="&lt;p&gt;I'm interested in particular in knowing about when $\iota$ began to be used as the imaginary unit/who began to use it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A majority of all text books that I have seen tend to just use $i$ as the imaginary unit. However, I remember being taught back in high school (in India) that the standard usage for the imaginary unit was $\iota$, which is most often just replaced with i everywhere. I can't find any source to back this up though --- for example wiki just says that $\iota$ is used when $i$ is being used for something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Am I remembering incorrectly? Was iota used initially historically maybe? Or is using iota just some notation that someone else tried to introduce? It definitely was/is considered valid notation if people are doubting the premise of the question (for example a quick search &quot;imaginary unit iota&quot; on Google comes up with several people using this notation. Even for example in the text that OP mentions &lt;a href=&quot;https://hsm.stackexchange.com/questions/5082/introduction-of-imath-and-jmath-notations-for-the-imaginary-unit&quot;&gt;here&lt;/a&gt; also talks about $\iota$ as valid notation.)&lt;/p&gt;&#xA;" OwnerUserId="5047" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-12-05T14:59:36.320" Title="What is the history of using $i$/$\iota$ as the imaginary unit?" Tags="|mathematics|notation|complex-analysis|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5446" PostTypeId="2" ParentId="5443" CreationDate="2016-12-04T08:23:24.930" Score="6" Body="&lt;p&gt;Some references :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Morris Kline, &lt;a href=&quot;https://books.google.it/books?id=RNwnUL33epsC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Mathematics: The Loss of Certainty&lt;/a&gt; (1982)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Marcus Giaquinto, &lt;a href=&quot;https://books.google.it/books?id=gyNbrpYXzxoC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;The Search for Certainty : A Philosophical Account of Foundations of Mathematics&lt;/a&gt; (2002) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;José Ferreiròs, &lt;a href=&quot;https://books.google.it/books?id=-WL0BwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics&lt;/a&gt; (1999)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-12-04T08:23:24.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5447" PostTypeId="2" ParentId="5443" CreationDate="2016-12-04T12:47:30.690" Score="4" Body="&lt;p&gt;I recently read a &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0691136106&quot; rel=&quot;nofollow noreferrer&quot;&gt;scholarly book&lt;/a&gt; relevant to this question.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jeremy Gray, &lt;em&gt;Plato's Ghost: The Modernist Transformation of Mathematics&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The transformation in question goes from 1880 to 1910 (roughly speaking).  Gray discusses how mathematics was done before, the turmoil surrounding the change, and how it was done after.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recommended.  But not for the faint of heart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Post copied from &lt;a href=&quot;https://math.stackexchange.com/a/2000433/442&quot;&gt;https://math.stackexchange.com/a/2000433/442&lt;/a&gt; ]&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2016-12-04T12:47:30.690" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5448" PostTypeId="1" AcceptedAnswerId="5451" CreationDate="2016-12-04T13:37:00.780" Score="4" ViewCount="168" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_Brown_(Scottish_botanist_from_Montrose)&quot; rel=&quot;nofollow noreferrer&quot;&gt;It is said&lt;/a&gt; that Robert Brown discovered the Brownian Motion by looking at pollen grains with a microscope in 1827. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder, did he look at pollen grains from a flower? And what kind of microscope was it that he used? I guess technology wasn't so good around 1827, right? Was it comparable to a nowadays reading lens?&lt;/p&gt;&#xA;" OwnerUserId="1706" LastEditorUserId="55" LastEditDate="2016-12-05T19:37:32.410" LastActivityDate="2017-05-31T22:19:34.883" Title="With what microscope and pollen was the Browinian motion discovered?" Tags="|mathematics|biology|einstein|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5449" PostTypeId="1" AcceptedAnswerId="5459" CreationDate="2016-12-04T20:51:03.800" Score="2" ViewCount="125" Body="&lt;p&gt;One would think that when DNA biology uses the word &lt;em&gt;palindrome&lt;/em&gt; it would mean approximately the same thing as &lt;em&gt;palindrome&lt;/em&gt; in other contexts.  As I understand it, this is not true.  Whereas, a normal palindrome built on CAT would be CATTAC, in DNA/RNA biology a palindrome built on CAT would be CATATG (!).  (I would have guessed they would instead have used a neologism like &lt;em&gt;complindrome&lt;/em&gt; for this pattern structure, in which the second half is the reverse of the nucleic-acid complement of the first half.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What person or paper introduced this nonstandard use of &lt;em&gt;palindrome&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A much less frequent term in molecular biology is &lt;em&gt;pseudopalindrome&lt;/em&gt;.  It's not clear if there is any consensus among biologists as to what the term means, although mostly it seems to refer to some form of almost-bio-palindrome, like CATATT.  When and where did &lt;em&gt;pseudopalindrome&lt;/em&gt; for particular DNA sequence patterns start?&lt;/p&gt;&#xA;" OwnerUserId="1723" LastEditorUserId="1723" LastEditDate="2016-12-05T19:00:59.983" LastActivityDate="2016-12-08T14:34:07.033" Title="Who originated the biological senses of palindrome and pseudopalindrome?" Tags="|terminology|biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5450" PostTypeId="2" ParentId="5445" CreationDate="2016-12-05T10:37:19.727" Score="6" Body="&lt;p&gt;For an early occurrence of &quot;iota&quot;, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alfred_Cardew_Dixon&quot; rel=&quot;noreferrer&quot;&gt;Alfred Cardew Dixon&lt;/a&gt; (English mathematician, 1865-1936), &lt;a href=&quot;http://quod.lib.umich.edu/u/umhistmath/acq6963.0001.001/15?page=root;rgn=full+text;size=100;view=pdf&quot; rel=&quot;noreferrer&quot;&gt;The elementary properties of the elliptic functions. With examples&lt;/a&gt; (London, 1894), p. 5 :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;§10&lt;/strong&gt;. writing $\iota$ for $\sqrt {-1}$, [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-12-05T14:59:36.320" LastActivityDate="2016-12-05T14:59:36.320" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5451" PostTypeId="2" ParentId="5448" CreationDate="2016-12-05T19:02:23.187" Score="6" Body="&lt;p&gt;Details of Brown's observations, concerning both the pollen and the microscope, are given by Pearle et al. in &lt;a href=&quot;https://arxiv.org/abs/1008.0039&quot; rel=&quot;nofollow noreferrer&quot;&gt;What Brown saw and you can too&lt;/a&gt;. The pollen was from the flower &lt;a href=&quot;https://en.wikipedia.org/wiki/Clarkia_pulchella&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clarkia pulchella&lt;/a&gt;, a.k.a. pinkfairy, ragged robin, or deerhorn clarkia, discovered on the return  trip  of  the  Lewis  and  Clark  expedition  by Lewis on June 1, 1806. He wrote in his journal: &quot;&lt;em&gt;I met with a singular plant today in blume, of which I preserved a specemine.  It grows on the steep sides of the fertile hills near  this  place...&lt;/em&gt;&quot;. To England seeds of Clarkia were brought only in 1826 by the expedition to the American northwest led by David Douglas, that begun on July 25, 1824 and was sponsored by the Horticultural Society of London and the Hudson's Bay Company. The society grew the flowers in its garden in Chiswick.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Brown's Slips Catalogue, dated June 12, 1827 and June 13, 1827 some sheets are labeled Clarkia with a line in Brown's handwriting having &quot;Hort Soc (Horticultural  Soc) Horticult (illegible) Chiswick (illegible)&quot;. The next line reads &quot;occident (western) Amer (illegible) by D Douglas&quot;. The results of his observations were reported in the paper &lt;strong&gt;R. Brown, A brief account of microscopical observations made  in  the  months  of  June,  July,  and  August,  1827, on  the  Particles  contained  in  the  Pollen  of  Plants;  and&#xA;on  the  general  existence  of  active  molecules  in  organic and inorganic bodies&lt;/strong&gt;, Edinburgh New Philos. J. 5 358-371  (1828);  reprinted  in  Philos. Mag. 4 161-173 (1828). The first paragraph described the microscope:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The observations, of which it is my object to give a summary in the following  pages,  have  all  been  made  with  a  simple  microscope,  and  indeed  with  one  and  the  same  lens,  the focal length of which is about 1/32 of an inch&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By the 10 inch rule of thumb (the near object is best seen from 10 inches away, so the magnification is roughly 10/focal length) the magnification of Brown's microscope was then about $\times 320$, Brown's own estimate was  $\times 370$. The two preserved Brown's microscopes, one at Kew Gardens and the other at the Linnean Society, have $\times 170$ lenses, so it probably was not one of them. The basic design goes back to Leeuwenhoek (1674), who already achieved magnification of $\times 300$ and observed bacteria through his microscope, see &lt;a href=&quot;http://www.history-of-the-microscope.org/microscope-history-timeline-design-development-inventions.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Microscopes timeline&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="12699" LastEditDate="2017-05-31T22:19:34.883" LastActivityDate="2017-05-31T22:19:34.883" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5452" PostTypeId="1" AcceptedAnswerId="5457" CreationDate="2016-12-06T01:01:46.450" Score="17" ViewCount="2173" Body="&lt;h2&gt;Background&lt;/h2&gt;&#xA;&lt;p&gt;When introducing Planck's switch to looking at black-body radiation, a number of sources -- like &lt;a href=&quot;https://www.youtube.com/watch?v=i1TVZIBj7UA&quot; rel=&quot;noreferrer&quot;&gt;MinutePhysics&lt;/a&gt;, &lt;a href=&quot;http://www.economist.com/node/443258&quot; rel=&quot;noreferrer&quot;&gt;the Economist&lt;/a&gt;, random &lt;a href=&quot;http://www.newworldencyclopedia.org/entry/Max_Planck&quot; rel=&quot;noreferrer&quot;&gt;online encyclopaedias&lt;/a&gt; and even here on &lt;a href=&quot;https://hsm.stackexchange.com/a/2259/3147&quot;&gt;HSM.SE&lt;/a&gt; (plus many popular history accounts: &lt;a href=&quot;https://books.google.co.uk/books?id=7XE7CQAAQBAJ&amp;amp;pg=PT48&amp;amp;lpg=PT48&amp;amp;dq=blackbody+radiation+planck+commissioned&amp;amp;source=bl&amp;amp;ots=ZOI51uboca&amp;amp;sig=mz1vYbTt_zZMYPmBP2SCwb6UB1U&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwim7_D6hd7QAhWeHsAKHTRaCUEQ6AEIRzAI#v=onepage&amp;amp;q=blackbody%20radiation%20planck%20commissioned&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://books.google.co.uk/books?id=qD_RF4P9HSkC&amp;amp;pg=PA31&amp;amp;lpg=PA31&amp;amp;dq=blackbody+radiation+planck+commissioned&amp;amp;source=bl&amp;amp;ots=ibk4xJKA51&amp;amp;sig=470NCLe2x15J6-wTQkA_4kWwzLI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwim7_D6hd7QAhWeHsAKHTRaCUEQ6AEISjAJ#v=onepage&amp;amp;q=blackbody%20radiation%20planck%20commissioned&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;2&lt;/a&gt;, &lt;a href=&quot;https://books.google.co.uk/books?id=LyZoAgAAQBAJ&amp;amp;pg=PA73&amp;amp;lpg=PA73&amp;amp;dq=blackbody+radiation+planck+commissioned&amp;amp;source=bl&amp;amp;ots=wBo7Juzgv5&amp;amp;sig=e2artYSKzxgQwNDh8CEl_AZAZOo&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiGgqyTh97QAhXsLsAKHer6B-84ChDoAQgfMAE#v=onepage&amp;amp;q=blackbody%20radiation%20planck%20commissioned&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;3&lt;/a&gt;, &lt;a href=&quot;https://books.google.co.uk/books?id=rzFoCwAAQBAJ&amp;amp;pg=PT430&amp;amp;lpg=PT430&amp;amp;dq=blackbody+radiation+planck+commissioned&amp;amp;source=bl&amp;amp;ots=fJJJPyjQJX&amp;amp;sig=CeKXEU8h6aLHV2cGOqA0p3zQTu4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiGgqyTh97QAhXsLsAKHer6B-84ChDoAQgvMAQ#v=onepage&amp;amp;q=blackbody%20radiation%20planck%20commissioned&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;4&lt;/a&gt;, etc.) -- propose motivations analogous to the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1894, Planck had been commissioned by electricity companies to discover how to generate the greatest luminosity from light bulbs with the minimum energy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Depending on the source, it is a single company, a local company, a consortium, or the government. But I haven't traced any of these mentions to an actual historical reference or document. Nor have I seen mentions of particular companies.&lt;/p&gt;&#xA;&lt;p&gt;The above story also rings several alarm bells:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why commission a theoretical physicist for an engineering problem if there are several other well-qualified experimentalists (say Lummer or others at the Physikalisch-Technische Bundesanstalt) in Berlin that could be commissioned instead?&lt;/li&gt;&#xA;&lt;li&gt;More reputable accounts (like &lt;a href=&quot;http://www.math.lsa.umich.edu/%7Ekrasny/math156_article_planck.pdf&quot; rel=&quot;noreferrer&quot;&gt;Helge Kragh's (pdf)&lt;/a&gt;) do not mention light bulb companies but suggest that Planck was motivated by the universality of Kirchhoff's law of black body radiation. That law was introduced in 1859, before incandescent light-bulbs were important. This suggests that the motivation for studying black body radiation, in general, had little to do with light bulbs (although we can use those theories now to think about them).&lt;/li&gt;&#xA;&lt;li&gt;Tying the previous two points together: Planck seems to have been much more concerned about universal, abstract, and foundational questions and doesn't seem like the kind of character that would take on a clearly practical chore like making light bulbs more energy efficient (even if they were all the rage at that time).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;Question&lt;/h2&gt;&#xA;&lt;p&gt;&lt;strong&gt;Did light bulbs directly motivate Planck?&lt;/strong&gt; Was he provided funding in 1894-1900 by anybody related to light bulbs? Were researchers close to him (like Lummer, Rubens, or Wien) motivated by making incandescent light bulbs more efficient? Or even by measuring their efficiency?&lt;/p&gt;&#xA;&lt;p&gt;Or is this simply one of the many myths made up after-the-fact by history textbooks? (If so, bonus question: who originated it?)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Notes&lt;/h2&gt;&#xA;&lt;p&gt;The most convincing story that I've found so far in this direction is on the wikipedia page for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Physikalisch-Technische_Bundesanstalt#Birth_of_quantum_physics&quot; rel=&quot;noreferrer&quot;&gt;Physikalisch-Technische Bundesanstalt&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first outstanding scientific achievement at the PTR was also closely connected with Max Planck. To decide whether electricity or gas would be more economic for street lighting in Berlin, the PTR was to develop a more precise standard for luminous intensity. For this purpose, in 1895, Otto Lummer and Wilhelm Wien developed the first cavity radiator for the practical generation of thermal radiation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But this is again stated without a reliable reference. Jorg Hollandt's article in &lt;a href=&quot;https://www.ptb.de/cms/fileadmin/internet/publikationen/ptb_mitteilungen/mitt2012/Heft2/PTB-Mitteilungen_2012_Heft_2.pdf#page=13&quot; rel=&quot;noreferrer&quot;&gt;this collection (pdf)&lt;/a&gt; might be able to corraborate the above story, but I cannot read German to verify.&lt;/p&gt;&#xA;" OwnerUserId="3147" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-24T01:32:39.980" Title="Did light bulb companies commission Planck to study black body radiation?" Tags="|physics|biographical-details|quantum-mechanics|debunking|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5453" PostTypeId="1" AcceptedAnswerId="5455" CreationDate="2016-12-06T01:55:32.190" Score="2" ViewCount="427" Body="&lt;p&gt;Poincaré wrote (English translation of the French) in 1903:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;``A very small cause that escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of  the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the&#xA;natural laws had no longer any secret for us, we could still only know the&#xA;initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation,  that is all we require, and we should&#xA;say that the phenomenon had been predicted, that it is governed by laws.&#xA;But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the  final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes&#xA;impossible, and we have the fortuitous phenomenon.''&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Where?* Poincaré wrote a lot of small papers, and gave many talks, and traveled.&lt;/p&gt;&#xA;&lt;p&gt;*Incidentally, Leibniz and Lichtenberg wrote similar things earlier, so this is a not a priority dispute question regarding instability. Later, but before modern citations, there was also, A. Compton, A. Lotka, E. Milne, J. Wheeler (a cigarette smoke causes a hurricane here and not elsewhere, about a year prior E. Lorenz's bird and butterfly examples), etc., so definitely not a priority question.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="10608" LastEditDate="2024-03-12T09:22:06.410" LastActivityDate="2024-03-12T09:22:06.410" Title="Poincaré quote regarding instability" Tags="|mathematics|physics|reference-request|quote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5454" PostTypeId="2" ParentId="5452" CreationDate="2016-12-06T02:32:12.527" Score="8" Body="&lt;p&gt;After looking into more reputable sources it seems that the &amp;quot;commissioned by electricity companies&amp;quot; is a confabulation, and the &amp;quot;commissioned by the German Bureau of Standards&amp;quot; is closer to the truth but still a huge stretch.&lt;/p&gt;&#xA;&lt;p&gt;In Kuhn's &lt;a href=&quot;https://books.google.com/books/about/Black_Body_Theory_and_the_Quantum_Discon.html?id=zYUhVHzlr2IC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Black-Body Theory and the Quantum Discontinuity, 1894-1912&lt;/a&gt;, which devotes two chapters to Planck's motivations, the light bulbs are mentioned only once:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;But quantitatively it&lt;/em&gt; [Michelson's law] &lt;em&gt;was not very satisfactory, a fact soon emphasized by H. F. Weber (1843-1912) of the Zurich Technische Hochschule, a physicist currently engaged in measuring the emission spectrum from &lt;a href=&quot;https://en.wikipedia.org/wiki/Incandescent_light_bulb#Dominance_of_carbon_filament_and_vacuum&quot; rel=&quot;nofollow noreferrer&quot;&gt;carbon filament lamps&lt;/a&gt;. After criticizing the theoretical basis of Michelson's derivation (including its reliance on the Stefan-Boltzmann law), Weber proposed an alternate formula based on his own and other experiments.&lt;/em&gt; (...) &lt;em&gt;When Wien, five years later, published the displacement law, his only reference to experiment was through Weber's law.&lt;/em&gt; (...) &lt;em&gt;As a product of theory, the Wien distribution law had, of course, little authority until Planck rederived it by a very different route in 1899.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But &lt;a href=&quot;https://books.google.com/books?id=IrqyBwAAQBAJ&amp;amp;pg=PA48&amp;amp;lpg=PA48&amp;amp;dq=Physikalisch-Technische+Reichsanstalt+planck&amp;amp;source=bl&amp;amp;ots=L5Z4619tG_&amp;amp;sig=K7WW6btT0uC-F0zYdhyb1qn1n-o&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjsyNm4t97QAhVCsBQKHRoWCj0Q6AEIVjAJ#v=onepage&amp;amp;q=Physikalisch-Technische%20Reichsanstalt%20planck&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brown's Planck: Driven by Vision, Broken by War&lt;/a&gt; gives an earlier connection.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Beyond the University of Berlin physicists, the nearby &lt;a href=&quot;https://en.wikipedia.org/wiki/Physikalisch-Technische_Bundesanstalt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physikalisch-Technische Reichsanstalt&lt;/a&gt; hosted a number of experimenters busily studying thermal radiation. The PTR, or Reichsanstalt in more commonplace parlance essentially functioned as a bureau of standards like today's National Institute of Standards and Technology in America. The focus there was applied technology, so why on earth would their lab scientists look into obscure issues of thermal radiation? They simply hoped to design a better electric light.&lt;/em&gt; (...)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Berlin, and particularly the Reichsanstalt, happened to house the world's foremost experts in measuring thermal radiation. These were men such as Lummer, Pringsheim, Rubens, Paschen, and Kurlbaum. Although Planck was much more of a pencil and equation physicist than these colleagues, he shared a great deal with them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If I had to guess the anecdote might have developed something like this: first Planck got &amp;quot;absorbed&amp;quot; by association into the Reichsanstalt crowd, which was perhaps motivated by improving light bulbs, then the Reichsanstalt was renamed into the better sounding German Bureau of Standards, which after all it essentially was, and finally somebody speculated that electric companies leaned on the Reichsanstalt concerning the bulbs (which may or may not be true), and decided to short-circuit it all the way back to Planck, for greater effect.&lt;/p&gt;&#xA;&lt;p&gt;The reality is vaguer and messier. The interests of the Reichsanstalt experimenters regarding light bulbs apparently influenced Planck's choice of topic in 1894, although it also dovetailed nicely with his interest in thermodynamics, and Wien's law of 1893, which rederived by Planck in 1899 became one of the major stepping stones to his law, was in its turn motivated by Weber's study of light bulbs in 1888. So the light bulbs were somewhat in the mix, albeit indirectly, and we can at best credit the industrial revolution with creating a research environment that led to a major discovery in fundamental physics.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-24T01:32:39.980" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5455" PostTypeId="2" ParentId="5453" CreationDate="2016-12-06T02:43:29.180" Score="5" Body="&lt;p&gt;This quote is taken from Poincaré's &lt;a href=&quot;https://archive.org/details/sciencemethod00poinuoft&quot; rel=&quot;noreferrer&quot;&gt;Science and Method&lt;/a&gt;, first pubished in 1908.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked webpage from archive.org is an English language translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote appears in section IV (titled Chance) of chapter I (titled The Scientist and Science).  In the linked archive.org facsimile, the quote begins at the bottom of page 66 and continues onto the top of page 67.  To view it, first click on the enlarge icon then use the horizontal scroll bar located at the bottom of the screen to advance to page 66.  Alternatively, just search for &quot;small cause&quot;, then click on the location pointer above the scroll bar at the bottom of the screen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Science and Method is one of a series of lectures given by Poincaré before the Société de Psychologie in Paris (1903?) and published as &lt;a href=&quot;https://archive.org/details/scienceandhypoth00poinuoft&quot; rel=&quot;noreferrer&quot;&gt;Science and Hypothesis&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Value_of_Science&quot; rel=&quot;noreferrer&quot;&gt;The Value of Science&lt;/a&gt;, and Science and Method.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2016-12-06T02:55:32.107" LastActivityDate="2016-12-06T02:55:32.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5457" PostTypeId="2" ParentId="5452" CreationDate="2016-12-07T05:06:46.830" Score="13" Body="&lt;p&gt;Sounds like we're all on the same page. But FWIW: In all my research for that Planck book (2015), I found no evidence that he was commissioned, contracted or paid by light bulb (or similar) companies in the 1890's for his theoretical work. Please note I'm not a proper historian, but I did, for instance, go through a number of German historical works on the topic (at embarrassingly glacial pace), including those of Dr. Dieter Hoffmann. Unless I missed something in those sources (always possible), Planck was simply working as a theoretical physicist exploiting ample state-of-the-art data arising in his own backyard. His switch in 1894, as far as I can tell, had more to do with wanting to explore something new, (after success in physical chemistry), especially after writing pieces to commemorate the recently departed titans Helmholtz and Hertz, (and especially after rereading Hertz's work on EM radiation). He opted for thermal radiation. In any case, I'm not sure light bulb companies would have cared much for Planck's work or really any pure theoretical work. Barely any physicists cared about his work at the time, from what I could tell. :-) He did later claim the obscurity of his 1890's work a godsend in that he had freedom and time without anyone competing or looking over his shoulder to make sense of it. Cheers.&lt;/p&gt;&#xA;" OwnerUserId="5058" LastActivityDate="2016-12-07T05:06:46.830" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5458" PostTypeId="2" ParentId="5434" CreationDate="2016-12-07T15:51:23.167" Score="3" Body="&lt;p&gt;Some references; for Ancient science, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;James Wilberding, &lt;em&gt;Embryology&lt;/em&gt;, &lt;strong&gt;Ch.20&lt;/strong&gt; of Georgia Irby (editor), &lt;a href=&quot;https://books.google.it/books?id=Y_JvCwAAQBAJ&amp;amp;pg=PA329&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Companion to Science Technology and Medicine in Ancient Greece and Rome&lt;/a&gt; (2016); see Aristotle (&lt;a href=&quot;https://en.wikipedia.org/wiki/Generation_of_Animals&quot; rel=&quot;nofollow noreferrer&quot;&gt;GenAnim&lt;/a&gt; 773a1–2) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Galen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galen&lt;/a&gt; (&lt;em&gt;On the Formation of the Fetus&lt;/em&gt; 54.13–14). &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For Early Modern science, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Paula Findlen, &lt;em&gt;Anatomy Theaters, Botanical Gardens, and Natural History Collections&lt;/em&gt;, &lt;strong&gt;Ch.12&lt;/strong&gt; of Katharine Park (editor), &lt;a href=&quot;https://books.google.it/books?id=2kvjAp92zcYC&amp;amp;pg=PA272&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cambridge History of Science. Vol.3 : Early Modern Science&lt;/a&gt;; see &lt;a href=&quot;https://en.wikipedia.org/wiki/Andreas_Vesalius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vesalius&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Harvey#On_Animal_Generation&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Harvey&lt;/a&gt;, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Walter_Pagel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Walter Pagel&lt;/a&gt;'s studies :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://books.google.it/books?id=m0cQQlZK43cC&amp;amp;pg=PA7&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Harvey's Biological Ideas : Selected Aspects and Historical Background&lt;/a&gt; (1967),&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://books.google.it/books?id=YZzkaAevzAoC&amp;amp;pg=PR5&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Light on William Harvey&lt;/a&gt; (1976).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Needham&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Needham&lt;/a&gt; (with Arthur Hughes), &lt;a href=&quot;https://books.google.it/books?id=rRZEBgAAQBAJ&amp;amp;pg=PA7&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Embryology&lt;/a&gt; (1st ed, 1934).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-12-07T16:00:43.423" LastActivityDate="2016-12-07T16:00:43.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5459" PostTypeId="2" ParentId="5449" CreationDate="2016-12-08T09:10:51.600" Score="4" Body="&lt;p&gt;It seems the term was coined in &lt;a href=&quot;http://dx.doi.org/10.1016/0022-2836(74)90216-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilson &amp;amp; Thomas (1974) Palindromes in Chromosomes. (Journal of Molecular Biology)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given the antiparallel character of the DNA double helix, only two kinds of repeating sequences are expected to have structural significance: ordinary repetitions (A B C.. . A B C) and inverted repetitions (A B C . . . C’B’A’) .&lt;br&gt;&#xA;  [...]&lt;br&gt;&#xA;  When inverted repetitions are separated by only a few intervening nucleotides, their similarity with a palindrome is sufficient to give them this name&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A', B' and C' being the complementaries of A, B and C.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact is, an actual sensu stricto palindrome (e. g. ACTTCA) would be an amusing trivia but bare no structural significance, therefore there is no real need to give it a name.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2016-12-08T14:34:07.033" LastActivityDate="2016-12-08T14:34:07.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5460" PostTypeId="1" AcceptedAnswerId="5461" CreationDate="2016-12-09T13:42:06.007" Score="10" ViewCount="6893" Body="&lt;p&gt;I know that Roman Empire built some of the most beautiful things for the epoch and also had an incredible culture (law, poetry, ...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my question is: Why there are there no known Roman mathematicans from the Roman Empire? All the famous roman were emperors, soldiers, lawyers, but not mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="3786" LastEditorUserId="4251" LastEditDate="2020-05-13T14:58:25.153" LastActivityDate="2021-08-12T19:14:18.020" Title="Why are there no known Roman mathematicans from the Roman Empire?" Tags="|mathematicians|roman-empire|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5461" PostTypeId="2" ParentId="5460" CreationDate="2016-12-09T14:29:20.047" Score="22" Body="&lt;p&gt;The Roman Empire did not consist of Latin-speaking &amp;quot;Romans&amp;quot; only. It included Greece, North Africa and parts of what we now call the Middle East. Probably, a majority of the inhabitants of the Roman Empire were Greek-speaking. And there were mathematicians among them. Menelaus, Ptolemy and Diophantus were living in the Roman Empire. Some of them could be Roman citizens, some not. We know practically nothing about their lives. Most likely all were Roman citizens after the citizenship was granted to all residents of the Empire.&lt;/p&gt;&#xA;&lt;p&gt;So the correct questions would be:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Why did Roman mathematicians write in Greek?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why did most of them prefer Northern Africa for their residence?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And the answer is that there was a tradition among the Greeks, in Greek language, especially in the Middle East, especially in Alexandria. Tradition means schools, libraries, scientific environment. This environment was created by the Greeks in Hellenistic times, and partially survived almost to the end of the Roman Empire.&lt;/p&gt;&#xA;&lt;p&gt;By the way, many Roman works in engineering, applied science (including military science!) were also mostly written in Greek. See a related discussion here: &lt;a href=&quot;https://hsm.stackexchange.com/questions/2043/roman-engineers&quot;&gt;Roman engineers&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Greek was the language of science in the Roman Empire. Similar to how Latin was the language of Science in Europe in the 16th to 18th centuries.&lt;/p&gt;&#xA;&lt;p&gt;EDIT. We really know nothing about Diophantus, except the books signed with this Greek-sounding name. Where did he live? When did he live? His masterpiece looks completely isolated (nothing remotely similar before or after or around his time). Could he create all this out of nothing? Very improbable.&lt;/p&gt;&#xA;&lt;p&gt;EDIT2. The question implicitly assumes that science and mathematics (in the modern sense) are necessary (or common, or desirable) activities of a civilization.&lt;/p&gt;&#xA;&lt;p&gt;Far from it. There were no mathematicians in Rome. But there were also no mathematicians in the Parthian Empire, India or China at that time. Mathematics (in the modern narrow sense) was invented by the Greeks, flourished in the Hellenistic world, then declined and vanished, as an activity, and very little of it spread elsewhere. Why should one expect it in Rome, or in Londinium or in the Parthian Empire? This was just some marginal activity of the Greek culture.&lt;/p&gt;&#xA;&lt;p&gt;It is only recently (since the 17th century) that mathematics became important and mathematical research spread widely.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-12T19:14:18.020" LastActivityDate="2021-08-12T19:14:18.020" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="5463" PostTypeId="1" AcceptedAnswerId="5468" CreationDate="2016-12-10T08:57:03.330" Score="6" ViewCount="438" Body="&lt;p&gt;Coupled oscillators can be broken down into a superposition of normal mode oscillations. Who was the first person to solve for this system in this way?&lt;/p&gt;&#xA;" OwnerUserId="5059" LastActivityDate="2016-12-17T17:20:36.363" Title="Who discovered the normal mode decomposition of coupled oscillators?" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5464" PostTypeId="1" CreationDate="2016-12-10T08:59:16.217" Score="4" ViewCount="202" Body="&lt;p&gt;After reading the article History of Entropy on Wikipedia, I noticed that it mentioned Carnot running into problems with entropy. I understand that entropy was fleshed out later by Clausius, but with respect to Carnot, what experiment did he perform that gave him reason to believe there was something else (although he didn't actually figure it out). I'm not looking for any answer that includes Boltzmann and probability, or any answer that includes the word &quot;chaos&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;TLDR: What did Carnot do or see that made him think there was something missing? What happened, possibly through an experiment or even thought experiment that made him realize things weren't adding up? &lt;/p&gt;&#xA;" OwnerUserId="5072" LastActivityDate="2016-12-20T19:53:23.960" Title="Carnot and Entropy" Tags="|thermodynamics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5465" PostTypeId="1" CreationDate="2016-12-10T09:40:05.800" Score="2" ViewCount="124" Body="&lt;p&gt;When was the fact that a spin could only have values S = n/2 derived? I cannot find any source that says when.&lt;/p&gt;&#xA;" OwnerUserId="5073" LastActivityDate="2016-12-11T03:35:27.773" Title="When was the quantization of spin discovered?" Tags="|physics|quantum-mechanics|particle-physics|" AnswerCount="1" CommentCount="2" ClosedDate="2017-11-09T15:33:27.993" ContentLicense="CC BY-SA 3.0" />
  <row Id="5466" PostTypeId="2" ParentId="5465" CreationDate="2016-12-11T03:35:27.773" Score="2" Body="&lt;p&gt;Quantization of spin was discovered experimentaly by Stern and Gerlach in 1922.&#xA;See Wikipedia article &quot;Stern-Gerlach experiment&quot;. The correct theoretical model for spin was constructed by Pauli and Dirac in 1928.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-11T03:35:27.773" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5467" PostTypeId="2" ParentId="5464" CreationDate="2016-12-11T03:44:31.880" Score="1" Body="&lt;p&gt;Carnot did not make any experiments, his work was purely theoretical. He constructed an idealized model of a heat engine, called &quot;Carnot's cycle&quot;, and deducted&#xA;a formula for maximal theoretical efficiency. Second Law of Thermodynamics was not formulated&#xA;yet at that time. Wikipedia article on Nicolas Leonard Sadi Carnot explains his work pretty well, and the work itself is available in English.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-11T03:44:31.880" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5468" PostTypeId="2" ParentId="5463" CreationDate="2016-12-11T04:37:16.080" Score="7" Body="&lt;p&gt;Edited. The story is long and complicated. From the mathematical point of view,&#xA;three statements are involved:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Eigenvalues of a self-adjoint operator are real and simple,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. There exists an orthogonal basis consisting of eigenvectors,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. Every quadratic form can be reduced to principal axes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1715 Brook Taylor found that functions&#xA;$$\sin\frac{\pi nx}{a}\cos\frac{\pi nct}{a}$$&#xA;describe the vibrations of a homogeneous string of length $a$&#xA;(normal modes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1750 Daniel Bernoulli proposed that the general solution of&#xA;the equation of a homogeneous string is a linear combination&#xA;(possibly infinite) of these normal modes. He used the previous work&#xA;of d'Alembert on solution of linear ODE with constant coefficients.&#xA;This conjecture of Bernoulli triggered a long discussion between&#xA;Euler, d'Alembert, Bernoulli on representability of an arbitrary&#xA;function by trigonometric series and on the general notion of function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1759-61 Joseph Louis Lagrange proved Bernoulli's conjecture by&#xA;considering an approximation of the string by a thread with beads of&#xA;equidistant equal masses, solving the corresponding linear ODE explicitly, and passing to the limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1748 Leonard Euler  showed that an arbitrary quadratic form&#xA;in dimension 3 can be reduced to principal axes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1787 Pierre Simon Laplace stated that all eigenvalues of a symmetric&#xA;matrix are real, but his proof involved physical arguments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1829 Augustin-Louis Cauchy proved that all eigenvalues of a symmetric&#xA;matrix are real, and simple. He writes that Charles Sturm has a simpler&#xA;proof to be published soon. I was unable to locate this proof. Cauchy is&#xA;aware of the significance of this theorem for mechanics and for&#xA;reduction of quadratic forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1813 Simeon Denis Poisson found the modern proof of reality of eigenvalues&#xA;for self-adjoint operators. (He spoke of differential operators, but his proof&#xA;applies to the general case. This is the proof given in modern linear algebra and functional analysis books).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1836 Charles Sturm proved A and B for ordinary linear differential operators&#xA;of second order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One had to wait till 20th century for a complete proof of B (the Spectral Theorem) in&#xA;full generality in infinite dimensional space. The final form is due to von Neumann.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References. I was unable to find a precise reference on B. Taylor, d'Alembert,&#xA;D. Bernoulli&#xA;and on the work of Sturm which Cauchy mentions.&#xA;Description of the work of Lagrange with references to his Oeuvres is here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/pdf/1305.3438.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/pdf/1305.3438.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The work of Euler, Laplace and Cauchy is described in the book&#xA;Emerging mathematics, by J. Stedall. The reference on Cauchy is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sur l'equation a l'aide de laquelle on determine les inegalites&#xA;seculaires de movment des planetes,&#xA;Exercises de mathematiques 1829, 140-142, 159-160.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very good reference for the whole story is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dieudonne, History of functional analysis, North Holland 1981.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-12-17T17:20:36.363" LastActivityDate="2016-12-17T17:20:36.363" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5469" PostTypeId="1" AcceptedAnswerId="5474" CreationDate="2016-12-11T11:09:22.723" Score="1" ViewCount="99" Body="&lt;p&gt;I'd like to know, why &lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Turner_Bennett&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward Turner Bennett&lt;/a&gt;, English zoologist, died at that early age (39 years) and was it in any connection to his work (like killed by some animal or as an effect of some disease, while on one of his scientific journeys)?&lt;/p&gt;&#xA;" OwnerUserId="2590" LastActivityDate="2016-12-12T22:42:36.707" Title="Why Edward Turner Bennett died early" Tags="|biographical-details|biology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5471" PostTypeId="1" AcceptedAnswerId="5472" CreationDate="2016-12-12T09:15:21.037" Score="0" ViewCount="92" Body="&lt;p&gt;There is are basically two versions of the St Petersburg paradox. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They differ only on a minor point, namely in the amount that is paid. In one version, $n$ heads and then tails gives a payout of $2^{n+1}$ (minus the price that was paid to play the game). In another version, $n$ heads and then tails gives a payout of $2^n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to know which version was originally used. I think the letters wherein the paradox was introduced are present in the second version of Ars conjectandi, 1713. However, I haven't been able to find the paradox or those letters in the versions of Ars conjectandi I found. &lt;/p&gt;&#xA;" OwnerUserId="2442" LastActivityDate="2016-12-12T15:18:06.077" Title="Which version of the St Petersburg paradox is the original one?" Tags="|mathematics|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5472" PostTypeId="2" ParentId="5471" CreationDate="2016-12-12T15:18:06.077" Score="3" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Raymond_de_Montmort&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre Remond de Montmort&lt;/a&gt; (1713), &lt;a href=&quot;https://books.google.fr/books?id=DIM_AAAAcAAJ&amp;amp;pg=PA402&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essay d'analyse sur les jeux de hazard&lt;/a&gt;, &lt;em&gt;Extrait d'une Lettre de M.N.Bernoulli à M.de M... du 9 Septembre 1713&lt;/em&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Cinquiéme Problème&lt;/em&gt;. On demande la meme chose si A promet à B de lui donner des écus en cette progression $1, 2, 4, 8, 16$ etc. [&lt;em&gt;Fifth Problem&lt;/em&gt;. One asks the same thing [see: &lt;em&gt;Fourth Problem&lt;/em&gt;] if A promises to B to give him&#xA;  some coins in this progression $1, 2, 4, 8, 16$ etc. ]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2016-12-12T15:18:06.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5473" PostTypeId="1" CreationDate="2016-12-12T19:02:37.580" Score="4" ViewCount="357" Body="&lt;p&gt;In modern parlance we talk about two different speeds of light in general relativity. We distinguish between the local speed of light, which is always $c$, and the coordinate speed of light, which can have any value from zero upwards without limit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Einstein's early papers he simply refers to &lt;em&gt;the speed of light&lt;/em&gt; and by this he means what we would today call the coordinate speed of light. I'm curious to know if at any point in his published work he started making the same distinction that we do today, and if so whether he used the modern terminology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've attempted to investigate this using the Einstein Digital Archive, but with 5,000 documents it's proving a bit impenetrable. I wonder if someone more familiar with Einstein's key papers would be able to answer this more easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it is of interest there is a related question in the Physics Stack Exchange: &lt;a href=&quot;https://physics.stackexchange.com/questions/297468/gr-einsteins-1911-paper-on-the-influence-of-gravitation-on-the-propagation-of&quot;&gt;GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2082" LastEditorUserId="4251" LastEditDate="2020-07-07T08:24:15.393" LastActivityDate="2020-07-07T08:24:15.393" Title="Did Einstein ever refer to the coordinate speed/velocity of light?" Tags="|physics|biographical-details|relativity-theory|einstein|physicists|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5474" PostTypeId="2" ParentId="5469" CreationDate="2016-12-12T22:02:35.047" Score="3" Body="&lt;p&gt;Reading biographies of famous people makes an impression that until the late 19th century lifespan around 40 was not unusual:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Byron (poet) 1788-1824,&#xA;Shelley (poet) 1792-1822,&#xA;Abel (mathematician) 1902-1829,&#xA;Eisenstein (mathematician) 1823-1852,&#xA;Sadi Carnot (physicist) 1796–1832,&#xA;Mozart (composer) 1756-1791,......&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and you can continue this list as long as desired. I intentionally do not use&#xA;statistical life expectancy (which is distorted by the child mortality and by people of low social status whose life expectancy was even shorter). I also do not include women who frequently died in childbirth. And certainly I do not include those who died in wars or duels or were murdered. So I am talking of middle or upper class males only who died of &quot;natural&quot; causes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;natural&quot; causes of death were pneumonia, tuberculosis, medicine (the main &quot;cure&quot; of everything was bloodletting, and many drugs contained lead, mercury and arsenic), sepsis (antiseptics were introduced only in the middle 19th century).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-12-12T22:42:36.707" LastActivityDate="2016-12-12T22:42:36.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5475" PostTypeId="1" CreationDate="2016-12-12T22:50:02.197" Score="1" ViewCount="146" Body="&lt;p&gt;I know that starting from the 19th century the style of mathematical proof was more and more formal and inductive. However, I am trying to find reference (article or books) that analyze the way which mathematicians proved their claims during the 16th-18th century. I assume these methods were pretty different from what we know from the 19th and 20th century....&#xA;any reference would be appreciated!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best, David &lt;/p&gt;&#xA;" OwnerUserId="3232" LastActivityDate="2016-12-13T03:13:48.107" Title="proofs of mathematical claims in the 16th, 17th and 18th cent" Tags="|mathematicians|reference-request|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5476" PostTypeId="2" ParentId="5475" CreationDate="2016-12-13T03:08:33.170" Score="5" Body="&lt;p&gt;You oversimplify what really happened to the standards of rigor. Rigorous proofs&#xA;were invented by ancient Greeks. (According to their own tradition, by Thales, but his writings do not survive). At the mature stage of Greek mathematics (Euclid, Archimedes, Apollonius) a very high standard of rigor was established.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It changed little since then to the modern time, if one speaks of the mainstream&#xA;mathematics (excluding the &quot;new&quot; areas of mathematical logic, set theory and foundations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the invention of calculus, this standard declined, because of the lack of foundations of calculus. Still at the time of Newton, best mathematicians perfectly understood this ancient standard; they just could not expose calculus on the same high standard as the Greek geometry. &#xA;And I suppose this is the main reason why Newton preferred to use&#xA;classical geometric methods (rather than calculus) in his Principia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 18th century people were more concerned with discovery of new things and in applications, then in the standards of rigor. So the standards declined. But in 19th century, by a long and difficult gradual process, this standard was finally achieved again. Rigorous foundation of calculus emerged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end of 19th century, the attention to foundations increased. This followed by a crisis in foundations in the beginning of 20th century, until some satisfactory situation was finally established. Again, this crisis in foundation&#xA;did not really affect much the work of most mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Those mathematicians who are not interested in foundations write on approximately the same standard of rigor which was established by the Greeks, leaving the care about foundations to the specialists. (This is somewhat oversimplified picture: there is some important interaction between foundations&#xA;and the rest of mathematics, but I cannot go into detail here).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all epochs, mathematicians were sometimes more concerned in discovery of new things, then in rigorous justifications. Archimedes himself wrote a famous book&#xA;with heuristic discoveries. He perfectly understood that the rigorous proofs were lacking. In the modern times, there is also plenty of such kind of research, I mean of the boundary of mathematics and physics. Where people use&#xA;&quot;semi-mathematical&quot; reasoning to discover new things, and don't know how to&#xA;prove these things rigorously.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-12-13T03:13:48.107" LastActivityDate="2016-12-13T03:13:48.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5477" PostTypeId="2" ParentId="5473" CreationDate="2016-12-13T07:17:55.420" Score="-1" Body="&lt;p&gt;I think the concept of &quot;local speed of light&quot; is (deliberately perhaps) confused in today's literature. &quot;Local&quot; should mean &quot;as measured by a freely falling observer&quot;, and some authors do define it in this way, but if the definition is not explicit, &quot;local&quot; becomes misleading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of &quot;coordinate speed of light&quot; is also confused. &quot;Coordinate&quot; should mean &quot;as measured by an observer at rest relative to the source of gravity&quot; but authors introduce some measuring of the speed of light from a distance, which is absurd, to say the least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a third confusion. Einstein's 1911 formula for the coordinate speed of light, c'=c(1+φ/c^2), is similar to the formula predicted by Newton's emission theory of light, and one sometimes teaches that they are identical. Actually Newton's formula is different: c'=c(1-φ/c^2). According to Einstein, the speed of light DECREASES as photons approach the source of gravity while according to Newton it INCREASES, like the speed of ordinary falling bodies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the final version of general relativity Einstein changed his formula - it became c'=c(1+2φ/c^2) - but the change was not essential. The sign remained &quot;plus&quot; and the speed of light continued to DECREASE as the source of gravity was approached. &lt;/p&gt;&#xA;" OwnerUserId="394" LastActivityDate="2016-12-13T07:17:55.420" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5479" PostTypeId="1" CreationDate="2016-12-13T21:35:44.930" Score="3" ViewCount="304" Body="&lt;p&gt;Which problem in celestial mechanics led Poincaré to his conjecture about fixed points of area preserving maps of the annulus to itself? I believe he was working on some differential equations. What was the problem and what was the differential equations he was working on?&lt;/p&gt;&#xA;" OwnerUserId="5090" LastEditorUserId="6609" LastEditDate="2024-03-12T11:13:35.463" LastActivityDate="2024-03-12T11:13:35.463" Title="Poincaré's last geometric theorem" Tags="|celestial-mechanics|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5480" PostTypeId="2" ParentId="5479" CreationDate="2016-12-14T00:57:49.097" Score="3" Body="&lt;p&gt;Poincaré was studying &lt;a href=&quot;https://en.wikipedia.org/wiki/Three-body_problem#Circular_restricted_three-body_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;the circular restricted three-body problem&lt;/a&gt;, where two bodies in a binary system revolve around their barycenter and a third test mass moves in some orbit around them. Birkhoff acknowledges this in his proof of the theorem, &lt;a href=&quot;http://www.ams.org/journals/tran/1913-014-01/S0002-9947-1913-1500933-9/S0002-9947-1913-1500933-9.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Proof of Poincaré's Geometric Theorem&lt;/em&gt;&lt;/a&gt;. Specifically, Poincaré was looking to prove that this third body may have periodic orbits.&lt;/p&gt;&#xA;&lt;p&gt;John Franks is quoted in Steve Nadis' &lt;a href=&quot;https://books.google.com/books?id=4e29AAAAQBAJ&amp;amp;pg=PA61&amp;amp;lpg=PA61&amp;amp;dq=poincare+last+geometric+theorem+three-body+problem&amp;amp;source=bl&amp;amp;ots=oY4qbR6eF-&amp;amp;sig=qUN_-waF9TInAp8jv0Hr-PwDH6E&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwj0so7buvLQAhVjJsAKHX88BGQQ6AEITzAJ#v=onepage&amp;amp;q=poincare%20last%20geometric%20theorem%20three-body%20problem&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History in Sum&lt;/em&gt;&lt;/a&gt; as describing the annulus as &amp;quot;a two-dimensional slice of three-dimensional space&amp;quot;. If a planet moves from a point on the annulus to the rest of the space and then back, this must be a fixed point, and there must therefore be periodicity. In fact, Poincaré's theorem states that if there is one such point there must be &lt;em&gt;a second&lt;/em&gt; point; one of the two describes the third body.&lt;/p&gt;&#xA;&lt;p&gt;Poincaré was not considering any differential equations, but instead the &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacobi_integral&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Jacobi integral&lt;/em&gt;&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$C_J$&lt;/span&gt;, which here &lt;a href=&quot;http://www.math.cornell.edu/%7Etemplier/junior/final_paper/Christopher_Schlom-Poincare_map_restricted.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;takes the form&lt;/a&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$C_J=-\frac{2H}{m}=-\left(\dot{x}^2+\dot{y}^2\right)+\left(x^2+y^2\right)+2\left(\frac{1-\mu}{r_1}+\frac{\mu}{r_2}\right)$$&lt;/span&gt;&#xA;for Hamiltonian &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;, test mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, coordinates &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;, masses of main bodies &lt;span class=&quot;math-container&quot;&gt;$1-\mu$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt;, and distances to those bodies &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="10608" LastEditDate="2024-03-12T09:21:37.210" LastActivityDate="2024-03-12T09:21:37.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5481" PostTypeId="2" ParentId="5460" CreationDate="2016-12-14T01:48:48.243" Score="5" Body="&lt;p&gt;In ancient Rome doing calculations with Roman numerals was a very cumbersome procedure which impeded the development of mathematics. Interestingly, ancient Rome also had no prominent scientists or philosophers as compared to ancient Greece. Probably science, mathematics and philosophy were not highly regarded in Roman society similarly to ancient China. These activities were left to Greek (slaves), who were also frequently used as teachers. &lt;/p&gt;&#xA;" OwnerUserId="5010" LastActivityDate="2016-12-14T01:48:48.243" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5482" PostTypeId="1" CreationDate="2016-12-14T13:46:49.550" Score="3" ViewCount="384" Body="&lt;p&gt;It is often said that he used Tycho's data. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the apparent motion of a planet is chaotic, right? The reason is simply that we are living on and observing from the earth, but not the sun. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how did he manage to recover the elliptic closed regular orbits of the planets from the original data of Tycho? Any good reference? &lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="4251" LastEditDate="2021-03-01T07:18:35.243" LastActivityDate="2021-03-01T07:18:35.243" Title="How did Kepler make his discovery?" Tags="|physics|reference-request|astronomy|celestial-mechanics|kepler|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5483" PostTypeId="2" ParentId="5482" CreationDate="2016-12-14T15:08:40.947" Score="5" Body="&lt;p&gt;The movement of planets is not &quot;chaotic&quot;. Otherwise prediction would be impossible. The planets move according to Kepler Laws, with very good accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Strictly speaking the description of the motion as regular or chaotic depends on&#xA;the time scale. On the scale of few thousand years, it is very regular,&#xA;obeying the Kepler laws. Deviations from this regular motion are small.&#xA;At the time of Kepler they were not observed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 18-19 centuries these deviations were observed, explained on the basis of Newton's laws, and permitted the discovery of new planets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But they may accumulate with time. There was a long discussion whether they actually accumulate or not (that is whether the mathematical model of the Solar system based on the Newton laws is stable or not. This discussion is not finished).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we look at the scale of billions years instead,&#xA;it may look much more complicated. But whether it is really chaotic or not&#xA;(on the scale of billions years) we do not really know, and the question has little practical meaning, because our solar system lifespan is only few billion years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question how Kepler derived his laws based on Brahe's observations was&#xA;discussed on this site:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/35/what-data-did-kepler-work-out-his-laws-from/196#196&quot;&gt;What data did Kepler work out his laws from?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best source for this is Kepler's Astronomia Nova itself. There is an English translation. In it Kepler described in the great detail, how after VERY many trials he figured the correct law. Unlike ost other authors, he describes the whole process, including all his mistakes, not only the final result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Explaining what and how Kepler did, even very roughly, needs some background in history of astronomy, and a bit more space than is allowed here. So I link a file with a non-technical explanation with essential background:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.purdue.edu/~eremenko/dvi/kepler.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.purdue.edu/~eremenko/dvi/kepler.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-12-16T21:53:25.830" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5484" PostTypeId="2" ParentId="534" CreationDate="2016-12-15T10:59:21.870" Score="9" Body="&lt;p&gt;Our recent edition of Christian Goldbach's correspondence with Leonhard Euler (&lt;em&gt;Leonhardi Euleri Opera Omnia&lt;/em&gt;, series IVA, vol.4, Springer: Basel, 2015) has a short biography of Goldbach (Introduction 1.1, p.3-12); thus I had reason to check again whether an authentic portrait might exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still haven't found one, but besides the Grassmann photograph - where the confusion seems to be due to an alphabetic list of mathematicians -, another spurious one has been published in the meantime: M.B.W. Tent's nice book on &lt;em&gt;Leonhard Euler and the Bernoullis&lt;/em&gt; (A.K.Peters: Natick, Mass., 2009) shows on p.151 the reproduction of an engraving. However, the caption &quot;Duque da Terceira&quot; is even discernible on this image: indeed, the person depicted is the Portuguese general and statesman Antonio José de Sousa Manuel de Meneses Severim de Noronha, Duke of Terceira (1792-1860). In this case the misinterpretation is due to a Brazilian memorial website listing both persons under their common birthday, March 18th.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A little &lt;em&gt;causerie&lt;/em&gt; I gave on the 250th anniversary of Goldbach's death - proposing, with a wink, yet another dubious likeness of the scholar from Königsberg - can be found (in German) on the website of the Bernoulli-Euler Center at Basel (&lt;a href=&quot;https://bez.unibas.ch/dokumente/Mattmueller-Goldbach.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://bez.unibas.ch/dokumente/Mattmueller-Goldbach.pdf&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a caution to anybody wishing to offer a prize for an authentic Goldbach portrait, I'd mention that a painting from 1776 portraying his colleague Daniel Bernoulli, thought to be lost for a long time, has recently been rediscovered at the Archangel'sk Museum of Fine Arts and used for the frontispiece of vol. IVA 3 of the Euler &lt;em&gt;Opera&lt;/em&gt; which has just been published. There is currently also a rumour that a portrait of Euler's wife Katharina Gsell (1707-1773) has been identified. So there's reason never to be too positive about the non-existence of images or documents!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Martin Mattmüller&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Bernoulli-Euler Center, University of Basel, Switzerland)&lt;/p&gt;&#xA;" OwnerUserId="5097" LastEditorUserId="5097" LastEditDate="2016-12-15T11:24:25.877" LastActivityDate="2016-12-15T11:24:25.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5485" PostTypeId="1" AcceptedAnswerId="6158" CreationDate="2016-12-15T20:07:03.453" Score="5" ViewCount="524" Body="&lt;p&gt;&quot;Gauss' planetary equations&quot; are a set of equations that describe the rate of change of the six elements of a planetary orbit in the presence of a disturbing force. I just intend to ask in which of his works did Gauss first write his planetary equations. Did he write them down in the &lt;em&gt;Theoria motus&lt;/em&gt; (1809) or perhaps in some of his shorter papers? &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10106" LastEditDate="2020-05-16T14:16:49.800" LastActivityDate="2020-05-16T14:16:49.800" Title="When did Gauss introduce &quot;Gauss' planetary equations&quot;?" Tags="|biographical-details|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5486" PostTypeId="1" CreationDate="2016-12-16T01:55:44.163" Score="1" ViewCount="82" Body="&lt;p&gt;Grey Walter writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Even in the very simplest system, with two active elements, multiple interconnection between elements give several modes for which simple observation is useless. The study of mentality, a product of interlaced patterns far too numerous to even write down, cannot therefore be defined, let alone solved, by contemplation of behavior, however patient the experiments, however sensitive his instruments. But nor is mentality made any clearer or more tractable by glib assertions of studying the organism as a whole. A holistic approach is not a practical strategy [either]. Norbert Wiener very properly stigmatizes holism as a bogey: `If a phenomenon can only be grasped as a whole and is completely unresponsive to analysis, there is no suitable material for any scientific description of it, for the whole is never at our disposal.' If, and in so far as, mentality must be defined as a function of the whole assembly of possible brain conditions, then, and to that extent, attempts to describe mentality in physiological terms are vain. But, mental phenomena are accessible to analysis, can be studied in groups of manageable size, and can be predicted from experimental observations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sub&gt;(&lt;strong&gt;WALTER&lt;/strong&gt;, &lt;strong&gt;GREY&lt;/strong&gt;. &lt;strong&gt;1953&lt;/strong&gt;. &lt;em&gt;The Living Brain&lt;/em&gt;. LONDON: DUCKWORTH. &lt;strong&gt;WIENER&lt;/strong&gt;, &lt;strong&gt;NORBERT&lt;/strong&gt;. &lt;strong&gt;1950(9)&lt;/strong&gt;. Maxims for Biologists and Psychologists. &lt;em&gt;Dialectica&lt;/em&gt; 4(3):186--191.)&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;He is against both behaviorism and holistic phenomenalism; his approach is basically cognitive: one at a time experimentation with specific but whole parts of the brain and construction of purposive models at the level of brain processes and their functions, not of stimuli and responses of neurons or entire organisms which is all that is directly and easily measurable (behaviorism), nor of minds as units with social histories that must all be known (holistic phenomenalism). Only behaviorist criticism of holism and holistic criticism of behaviorism are common in the literature 1920--1950.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are plenty of old behaviorist criticisms of holism. (They drop out with behaviorism.) Also there are many criticisms of behaviorism along the lines of absence of any one to one property in brain function and input and behavior. (This is one of them.) But what are the references prior this one that argue the same way against too broad holism in the context of brain information processing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="163" LastEditorUserId="4251" LastEditDate="2021-09-08T03:30:37.010" LastActivityDate="2021-09-08T03:30:37.010" Title="Grey Walter and Norbert Wiener regarding holism" Tags="|reference-request|computer-science|psychology|physiology|wiener|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5487" PostTypeId="1" CreationDate="2016-12-16T05:45:37.700" Score="6" ViewCount="440" Body="&lt;p&gt;I watched the movie 'The Man Who Knew Infinity' and read the biography of J.E. Littlewood in Wikipedia. He worked on Ballistics in WW1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were those works were stored somewhere? I have checked on the internet; but haven't found any.&lt;/p&gt;&#xA;" OwnerUserId="5099" LastActivityDate="2016-12-22T17:34:36.843" Title="Are the works of J.E. Littlewood in ballistics stored somewhere?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5489" PostTypeId="2" ParentId="5487" CreationDate="2016-12-16T20:01:58.340" Score="7" Body="&lt;p&gt;Littlewood described his WW1 work in Adventures in Ballistics, 1915-1918 published by Mathematical Spectrum magazine in two parts in 1971-72. Unfortunately, they store only &lt;a href=&quot;http://ms.appliedprobability.org/content.aspx?Group=home&amp;amp;Page=Back_Issues_4&quot; rel=&quot;nofollow noreferrer&quot;&gt;first pages online&lt;/a&gt;, so you'd have to get paper copies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Littlewood's contemporaneous papers, Formulae for Direct Fire and Anti-Aircraft Trajectories and Range Tables were published by War Office in London, 1917, which would be hard to get, I imagine. But they are discussed in Barrow-Green's &lt;a href=&quot;http://webcache.googleusercontent.com/search?q=cache:ysZKMe-FIn0J:smf4.emath.fr/Publications/RevueHistoireMath/21/pdf/smf_rhm_21_111-150.pdf+&amp;amp;cd=1&amp;amp;hl=en&amp;amp;ct=clnk&amp;amp;gl=us&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Anti-aircraft guns all day long”:Karl Pearson and computing for the Ministry of Munitions&lt;/a&gt; in Revue d’histoire des mathématiques 21 (2015), p. 111–150, which is conveniently cached by Google, and apparently reprinted in Pearson Papers (&lt;a href=&quot;https://books.google.com/books?id=8By8BAAAQBAJ&amp;amp;pg=PA123&amp;amp;lpg=PA123&amp;amp;dq=Littlewood,+Adventures+in+ballistics,+1915-1918&amp;amp;source=bl&amp;amp;ots=hoJODPhe6P&amp;amp;sig=7TuBFTcRfYuGCzy3zFvzAQ7WzSw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjPvb_0u_nQAhVJCiwKHWfMBzgQ6AEIIzAD#v=onepage&amp;amp;q=Littlewood%2C%20Adventures%20in%20ballistics%2C%201915-1918&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;cited here&lt;/a&gt;), which might be easier to find.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2016-12-16T21:03:14.393" LastActivityDate="2016-12-16T21:03:14.393" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5490" PostTypeId="2" ParentId="5485" CreationDate="2016-12-18T03:28:25.307" Score="1" Body="&lt;p&gt;Gauss was not quite so explicit, but it does seem that his &quot;planetary equations&quot; first appear in Theoria Motus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://arxiv.org/abs/1210.6658&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planetary Orbital Equations in Externally-Perturbed Systems: Position and Velocity-Dependent Forces by Veras and Evans&lt;/a&gt; we read: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The work of Burns (1976), subsequently popularised by Murray &amp;amp; Dermott (1999, pp. 54-57), provides a mechanism to obtain analytic equations for $da/dt$ , $de/dt$ , $di/dt$ , and $d\Omega/dt$ arising from a small perturbative force with a given prescription for radial, tangential and normal components. This line of attack can be traced back ultimately to Gauss (see e.g., section 9.13 of Brouwer &amp;amp; Clemence 1961).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This apparently refers to four of the &lt;a href=&quot;https://farside.ph.utexas.edu/teaching/celestial/Celestial/node164.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss planetary equations&lt;/a&gt;. The cited book &lt;a href=&quot;http://www.sciencedirect.com/science/book/9781483200750&quot; rel=&quot;nofollow noreferrer&quot;&gt;Methods of Celestial Mechanics by Brouwer and Clemence&lt;/a&gt; does not have a section IX.13. It does, however, have section XI.13, titled Gauss's Form, which reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Equations in this form were first derived by Gauss and applied to the calculation of first-order perturbations by Jupiter on Pallas. Gauss made use of these equations also for deriving secular perturbations in the elements. Finally, the equations have been used extensively for computing by numerical integration perturbations in the elements of comets and minor planets.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss and Ceres&lt;/a&gt; informs us that Gauss published his methods in &quot;&lt;em&gt;Theoria motus corporum coelestium in sectionibus conicus solem ambientium&lt;/em&gt;&quot; of 1809 (apparently, Gauss had to translate the book from German into Latin at publisher's request... for greater accessibility). But it has a reproduction of Gauss's sketch of orbits of Ceres, Pallas and Vesta referenced to &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://webdoc.sub.gwdg.de/ebook/e/2005/gausscd/html/kapitel_astro_ceres.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Astronomische Untersuchungen und Rechnungen vornehmlich über die Ceres Ferdinandea&lt;/a&gt; [Astronomical investigations and calculations mainly on Ceres Ferdinandea], 1802, SUB Göttingen: Cod. Ms. Gauß Handbuch 4, Bl.1 &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The accompanying text reads:&quot;&lt;em&gt;Gauss was able to calculate the orbit of the planetoid. On December 7, 1801, the planetoid Ceres reappeared exactly at the location predicted by Gauss&lt;/em&gt;&quot;. Apparently, Pallas and Vesta were less cooperative. The linked biography page adds that &quot;&lt;em&gt;at the time&lt;/em&gt;&quot; of Theoria Motus &quot;&lt;em&gt;he was able to calculate the exact path of the first three dwarf planets Ceres, Pallas and Juno as well as the to discover the first resonance relation in the asteroid belt.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is Davis's English translation of Theoria Motus, &lt;a href=&quot;https://books.google.com/books?id=LHE21JnAS5cC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections&lt;/a&gt;. Perturbations of Pallas by Jupiter are mentioned at the very end, on p.277. On the resonance  see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2323/calculation-of-gauss-leading-to-187-resonance-between-orbits-of-jupiter-and-pal&quot;&gt;Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2016-12-18T04:38:56.103" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5491" PostTypeId="1" CreationDate="2016-12-19T16:26:14.827" Score="11" ViewCount="643" Body="&lt;p&gt;What was the first written account about snowflake shapes and when did physicists start to study the reasons why the form appears and how snowflakes form in the air?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any old book that mentions these aspects?&lt;/p&gt;&#xA;" OwnerUserId="5109" OwnerDisplayName="yoyo_fun" LastEditorUserId="55" LastEditDate="2016-12-20T05:51:01.567" LastActivityDate="2016-12-20T23:33:37.220" Title="When did people notice the special shapes of snowflakes, and when did they start to study them?" Tags="|physics|thermodynamics|materials-science|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5492" PostTypeId="2" ParentId="5491" CreationDate="2016-12-20T04:34:38.413" Score="7" Body="&lt;p&gt;There is a very nice site &lt;a href=&quot;http://snowcrystals.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;SnowCrystals.com&lt;/a&gt; run by Kenneth Libbrecht, a professor of physics at Caltech, which covers both physics and history, and hosts many captivating images, videos and more.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://www.snowcrystals.com/history/history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Libbrecht's Snowflake History&lt;/a&gt;, the first written mention is by Han Yin c. 135-150 BC: &amp;quot;&lt;em&gt;Flowers of plants and trees are generally five-pointed, but those of snow, which are called ying, are always six-pointed&lt;/em&gt;&amp;quot;. In the 6th century AD Chinese poet Hsiao Tung seconded the observation:“&lt;em&gt;The ruddy clouds float in the four quarters of the cerulean sky. And the white snowflakes show forth their six-petaled flowers.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;p&gt;Kepler was not quite the first in Europe. Scandinavian bishop Olaus Magnus described snowflakes in 1555, although according to him their shapes included crescents, arrows, and even a human hand (Libbrecht has an image). English astronomer Thomas Harriot identified the characteristic hexagonal shape symmetry in 1591. But Kepler's Six-Cornered Snowflake (1611) was much more systematic, and regarding snowflake shapes, as in astronomy, he rejected the medieval mode of explanation a la &lt;a href=&quot;https://en.wiktionary.org/wiki/dormitive_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Molieres &amp;quot;opium puts you to sleep because it has sleep-inducing power&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Each single plant has a single animating principle of its own, since each instance of a plant exists separately, and there is no cause to wonder that each should be equipped with its own peculiar shape. But to imagine an individual soul for each and any starlet of snow is utterly absurd, and therefore the shapes of snowflakes are by no means to be deduced from the operation of soul in the same way as with plants.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But even Kepler was still mostly contemplative and speculative. On the other hand, &lt;a href=&quot;http://classiques.uqac.ca/classiques/Descartes/meteores/meteores.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes in Les Météores (1637)&lt;/a&gt; reported some &amp;quot;remarkably thorough&amp;quot; empirical naked-eye observations of snow crystals, including the rarer forms like capped columns.&lt;/p&gt;&#xA;&lt;p&gt;According to Libbrecht, &amp;quot;&lt;em&gt;not all snowflakes look like six pointed stars. Many do, but there are also quite a few odd-looking crystals falling from the winter clouds&lt;/em&gt;&amp;quot;. In particular, there are dodecagonal crystals, but there are no human handed or octogonal ones, Olaus Magnus and &lt;a href=&quot;http://www.pcmag.com/article2/0,2817,2398039,00.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google's Doodle&lt;/a&gt; notwithstanding. The hexagonal shape is predominant, and &lt;a href=&quot;http://www.doublexscience.org/why-are-snowflakes-always-six-sided&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francis' Why Are Snowflakes Always Six-Sided?&lt;/a&gt; explains the formation process:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;The hydrogen bonding dictates the shape of the ice crystals. You’ve learned that each water molecule is linked to four other water molecules in a tetrahedral arrangement. As the water freezes, these tetrahedrons come closer together and crystallize into a six-ring or hexagonal structure... Each point on the hexagon is an oxygen atom, and each side is a hydrogen bonded to one oxygen. As the water approaches freezing temperature, the water molecules continue to crystallize in this tetrahedral arrangement.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;But water does something unlike most substances. As it nears freezing, instead of continuing to contract, it expands slightly from about 4 degrees to 0 degrees Celsius as the motion of the molecules slows with the cold, and the hydrogen bonds extend the molecules to their fullest distance from each other. It’s like a ring of people holding hands, elbows bent, and then gradually straightening their arms to the fullest extension so that they’re at the greatest distance from each other. When water molecules do this, the hexagonal structure expands into a larger and larger hexagonal structure. The snowflake, with its six sides, is what results from this process...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2016-12-20T23:33:37.220" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5493" PostTypeId="1" AcceptedAnswerId="5494" CreationDate="2016-12-20T12:57:57.997" Score="3" ViewCount="103" Body="&lt;p&gt;In 1634 Descartes &lt;a href=&quot;http://www.earlymoderntexts.com/assets/pdfs/descartes1619.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;wrote&lt;/a&gt; to Mersenne :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;let me tell you about an experiment that was published not long ago in Leurechon’s &lt;em&gt;Mathematical Games&lt;/em&gt;. It involves a large cannon placed on flat ground, pointing straight up at the sky, and fired.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is Leurechon’s original description of this experiment still available to us?&lt;/p&gt;&#xA;" OwnerUserId="2331" LastEditorUserId="131" LastEditDate="2024-03-06T14:25:42.183" LastActivityDate="2024-03-06T14:25:42.183" Title="Is reference cited by Rene Descartes available?" Tags="|physics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5494" PostTypeId="2" ParentId="5493" CreationDate="2016-12-20T14:25:30.047" Score="6" Body="&lt;p&gt;Yes, it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Leurechon&quot; rel=&quot;noreferrer&quot;&gt;Jean Leurechon (1591–1670)&lt;/a&gt;, jesuit, professor of Mathematics from 1614 to 1627 and then professor of theology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He published : &lt;a href=&quot;https://books.google.fr/books?id=QsY5AAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;Récréation mathématique: Composée de plusieurs problèmes plaisants et facétieux : En feict d'Arithméticque, Géométrie, Méchanicque, Opticque, et autres parties de ces belles sciences&lt;/a&gt;, Pont-à-Mousson, 1624 (reprinted 1626).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See : &lt;strong&gt;86. Probleme des Canons&lt;/strong&gt;, page 108&amp;on; &lt;a href=&quot;https://books.google.fr/books?id=QsY5AAAAcAAJ&amp;amp;pg=PA110&quot; rel=&quot;noreferrer&quot;&gt;page 110&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A ce compte, dira quelqu'un, le Canon pointé droict au zenith deburoit tirer pòus fort, qu'un toute autre posture. [...] Mais pour moy qui trouve de la difficulté à croire cette experience, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The English translation was published in 1633 under the name of Laurechon's student : &lt;a href=&quot;https://books.google.it/books/about/Mathematical_Recreations_Or_A_Collection.html?id=7zsVvgAACAAJ&amp;amp;redir_esc=y&quot; rel=&quot;noreferrer&quot;&gt;Hendrick Van Etten&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Descartes reference is in his letter to Mersenne, dated April 1634.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Descartes discuss the movement of Earth, with reference to Galileo's condemnation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The issue is about possible experimental verification of Earth's movement; specifically about the fall on ground of a stone thrown vertically. Descartes asks to Mersenne about &quot;an experiment that was published not long ago in Leurechon’s book&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The author of the book says that the experiment has already been performed many times, and the cannon-ball didn’t once fall back to the ground. Many might think this quite incredible, but I don’t judge it to be impossible, and I think it’s well worth looking into.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-12-20T14:43:07.073" LastActivityDate="2016-12-20T14:43:07.073" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5495" PostTypeId="2" ParentId="5464" CreationDate="2016-12-20T19:53:23.960" Score="4" Body="&lt;p&gt;Carnot was not talking about thermodynamics in the same way that Clausius was. Carnot saw the fundamental action as a fluid substance (caloric) being manipulated into motion between a hot place and a cold place. He saw fire as a motive power that could push caloric from one place to another and thus perform work. It was all about &quot;la grande chute du calorique&quot; for Carnot, (the great fall of caloric). Thermodynamics as imagined by Clausius, Maxwell, and others, would abandon the idea of caloric and instead put into place a kinetic theory of heat that called into question the nature of heat itself.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest we might come to saying that Carnot &quot;saw&quot; entropy would be if he saw some of his caloric being lost, or if he saw some breakdown in the ability of heat to move it. And it is true that on p. 114 of his &quot;Reflexions&quot; when he calculates the motive power of burning coal, he adjusts (by half) his calculation in order to be conservative. It is not heavily theorized beyond accounting for the difference between theory and practice.  But this is really not the same thing as entropy as Clausius framed it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is all in his Reflexions, available in Google books.&lt;/p&gt;&#xA;" OwnerUserId="2819" LastActivityDate="2016-12-20T19:53:23.960" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5496" PostTypeId="1" CreationDate="2016-12-21T11:20:33.070" Score="3" ViewCount="1153" Body="&lt;p&gt;I was working on the area of &lt;a href=&quot;https://en.wikipedia.org/wiki/Fixed-point_iteration&quot; rel=&quot;nofollow noreferrer&quot;&gt;fixed point iteration&lt;/a&gt;. I couldn't find any historical account regarding the subject. Who was the first to use the this method?&lt;/p&gt;&#xA;" OwnerUserId="368" LastActivityDate="2016-12-22T20:01:19.777" Title="Who discovered the fixed point iteration method?" Tags="|mathematicians|discoveries|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5497" PostTypeId="2" ParentId="5496" CreationDate="2016-12-22T06:19:57.500" Score="3" Body="&lt;p&gt;There are several &quot;fixpoint iteration methods&quot; and several fixed point theorems underlying them. One of the earliest uses was &quot;Picard's iteration method&quot; for proving existence of solutions of ODE. It is based on the &quot;Banach fixed point theorem&quot;,&#xA;though Banach was not born yet when Picard discovered it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. Thanks to those who recalled Newton's method, which was discovered in a special case in Babylon thousands years ago. There are indeed very many &quot;fixpoint methods&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2016-12-22T20:01:19.777" LastActivityDate="2016-12-22T20:01:19.777" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5499" PostTypeId="1" CreationDate="2016-12-23T00:06:13.890" Score="7" ViewCount="398" Body="&lt;p&gt;I've been trying to find the source of the name of the DE modelling population growth known as logistic growth, for some time: why &quot;Logistic&quot; ? So far all my attempts to research it have hit dead ends - very curious if any math/ French language buffs have any idea. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only suggestion I have found is a rather tenuous tie to the French &quot;logis&quot; (for dwelling/home) but this doesn't seem very plausible given it doesn't have a corresponding adjectival form.&lt;/p&gt;&#xA;" OwnerUserId="5127" LastActivityDate="2018-11-22T03:40:09.170" Title="source of &quot;logistic growth&quot;?" Tags="|mathematics|terminology|differential-equations|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5500" PostTypeId="2" ParentId="5499" CreationDate="2016-12-23T09:43:10.173" Score="7" Body="&lt;p&gt;According to some on-line resources for historical French lexicon, around 1770 [see : &lt;a href=&quot;https://fr.wikipedia.org/wiki/Fortunato_Bartolomeo_De_Felice&quot; rel=&quot;noreferrer&quot;&gt;Fortunato Bartolomeo De Felice&lt;/a&gt;, vol.26 of: &lt;a href=&quot;https://books.google.it/books?id=f04VAAAAQAAJ&amp;amp;pg=PA513&amp;amp;lpg=PA513&quot; rel=&quot;noreferrer&quot;&gt;Encyclopédie ou Dictionnaire universel raisonné des connoissances humaines&lt;/a&gt; (1773)] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Adjectivement] &lt;strong&gt;Logarithmes logistiques&lt;/strong&gt;, logarithmes dans lesquels zéro est le logarithme correspondant au nombre 3600. Ces logarithmes sont commodes pour les calculs astronomiques [see &lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%A9r%C3%B4me_Lalande&quot; rel=&quot;noreferrer&quot;&gt;Jérôme Lalande&lt;/a&gt;, &lt;em&gt;Astronomie&lt;/em&gt;, par M. de La Lande, Tome premier (1764, enlarged edition, 4 vols, 1771–1781), &lt;strong&gt;Table LXXVII&lt;/strong&gt;, &lt;a href=&quot;https://books.google.it/books?hl=it&amp;amp;id=otEN5iMke6kC&amp;amp;q=logistique&quot; rel=&quot;noreferrer&quot;&gt;page 89&lt;/a&gt; ].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This, it seems, must be the source of the name: &lt;em&gt;courbe logistique&lt;/em&gt; [logistic curve] used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Fran%C3%A7ois_Verhulst&quot; rel=&quot;noreferrer&quot;&gt;Pierre François Verhulst&lt;/a&gt; in his 1845 work: &quot;&lt;em&gt;Recherches mathématiques sur la loi d'accroissement de la population&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See  &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN129323640_0018|LOG_0009&amp;amp;physid=PHYS_0021&quot; rel=&quot;noreferrer&quot;&gt;page 8&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2016-12-25T10:57:59.230" LastActivityDate="2016-12-25T10:57:59.230" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5502" PostTypeId="1" CreationDate="2016-12-25T21:43:06.923" Score="3" ViewCount="246" Body="&lt;p&gt;New here, had a question: Were there any famous Asian philosophers who were also mathematicians? Happy Xmas!&lt;/p&gt;&#xA;" OwnerUserId="5135" LastActivityDate="2016-12-28T12:43:35.193" Title="Were there any famous Asian philosophers who were also mathematicians?" Tags="|philosophy-of-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5503" PostTypeId="1" CreationDate="2016-12-25T22:42:20.757" Score="1" ViewCount="97" Body="&lt;p&gt;In textbooks on differential geometry, the first fundamental form looks like &lt;span class=&quot;math-container&quot;&gt;$E^2+2FG+H^2$&lt;/span&gt;, and its length is calculated through the help of the dot product. However, the inner product did not exist. The analytic formula was known that &lt;span class=&quot;math-container&quot;&gt;$x_1x_2+y_1y_2+z_1z_2$&lt;/span&gt; . Nor did the vector exist. So how did he prove it?&#xA;Same story with the unit normal: in textbooks they use the cross product &lt;span class=&quot;math-container&quot;&gt;$r(u) \times r(v)$&lt;/span&gt; But this did not exist either.  Again, how did he manage to calculate it? &lt;/p&gt;&#xA;" OwnerUserId="5136" LastEditorUserId="9968" LastEditDate="2019-06-11T13:23:18.017" LastActivityDate="2019-06-11T13:23:18.017" Title="Gauss fundamental form in differential geometry : use of dot products" Tags="|differential-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5504" PostTypeId="2" ParentId="5502" CreationDate="2016-12-26T03:30:24.140" Score="2" Body="&lt;p&gt;Kātyāyana, Brahmagupta, Jing Fang and many other from India, China and other regions.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Indian_mathematicians&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_Indian_mathematicians&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Chinese_mathematicians&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_Chinese_mathematicians&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At that time there wasn't a clear cut between disciplines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Babylonia and Egypt also had mathematicians but we don't know their names.&lt;/p&gt;&#xA;" OwnerUserId="4998" LastEditorUserId="4998" LastEditDate="2016-12-28T12:43:35.193" LastActivityDate="2016-12-28T12:43:35.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5505" PostTypeId="2" ParentId="5502" CreationDate="2016-12-26T06:12:44.900" Score="1" Body="&lt;p&gt;Most of famous mathematicians/astronomers of the Muslim countries were also philosophers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Science_in_the_medieval_Islamic_world&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Science_in_the_medieval_Islamic_world&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-26T06:12:44.900" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5506" PostTypeId="1" CreationDate="2016-12-26T14:14:13.630" Score="2" ViewCount="151" Body="&lt;p&gt;My question refers to Gauss's hyperbolic sine expression for a circle circumference in hyperbolic geometry with constant negative curvature. In particular, what specific conceptual developements it anticipated? I ask that because all derivations of this result that i found are based on something called the hyperbolic plane model, while in other places i read that the hyperbolic plane was an innovation of Poincare and Beltrami, not Gauss. So how did Gauss originally found his result? and did he anticipate much later notions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And by the way, i'll be glad if i'll get an explanation too for Gauss's result in his letter to Gerling from 1819, which gives a formula for the upper bound of area of a triangle in terms of a certain &quot;constant&quot; - the upper bound of the height of isocele triangle in constant negative curvature hyperbolic geometry.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2016-12-30T16:37:01.653" LastActivityDate="2016-12-30T16:37:01.653" Title="How did Gauss arrived at the formula for the circumference of circle in hyperbolic geometry?" Tags="|biographical-details|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5507" PostTypeId="1" AcceptedAnswerId="5664" CreationDate="2016-12-26T16:08:31.997" Score="10" ViewCount="2158" Body="&lt;p&gt;I had an interesting thought where I wondered that if I had a grandparent who had been a mathematician (or at least studied mathematics), how much of modern undergraduate mathematics would they understand?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think in other sciences this gap would be much greater than in mathematics. So I guess my question is, how much does mathematics, that is taught to students, change in a generation? In particular, how much has it changed in the last generation?&lt;/p&gt;&#xA;" OwnerDisplayName="user5140" LastEditorDisplayName="user5140" LastEditDate="2016-12-26T20:42:05.410" LastActivityDate="2021-07-11T20:42:04.803" Title="How much does Mathematics change in a generation?" Tags="|mathematics|mathematicians|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5508" PostTypeId="1" AcceptedAnswerId="5536" CreationDate="2016-11-19T12:01:39.573" Score="11" ViewCount="535" Body="&lt;p&gt;I am making a presentation about &lt;a href=&quot;https://en.wikipedia.org/wiki/Millennium_Prize_Problems&quot;&gt;Millennium Problems&lt;/a&gt;. There are 7 Millennium Problems but a lot of unsolved mathematics problems are also waiting to be solved. I know that these problems are much more important than many others but I need a more detailed explanation.  Why these 7? Who decided? Are they really the most important ones in mathematics? For example, one question comes to mind is why 3n+1 conjecture is not in this list, maybe solving it will provide us great mathematical devices like FLT. I will be really glad if you can provide some reliable links or a detailed explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;TT&lt;/p&gt;&#xA;" OwnerDisplayName="user4962" LastActivityDate="2017-01-23T17:05:32.657" Title="How were Millennium Problems chosen? (according to what criteria)" Tags="|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5511" PostTypeId="1" CreationDate="2016-12-26T22:54:22.143" Score="1" ViewCount="105" Body="&lt;p&gt;While I was looking for a source about Kronecker's Jugendtraum, I found &lt;a href=&quot;http://www.e-rara.ch/zut/content/pageview/5667300&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; document: Auszug aus einem Briefe von L. Kronecker an R. Dedekind vom 15. März 1880.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bottom of the given page has a note: Jacobi, Werke, Bd. 1, S. 266. I found &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k90209g/f279.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;, but it contains apparently no information about this letter from Kronecker to Dedekind. Does anyone know how this reference to Jacobi's works should be understood?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, we have only here an &quot;Auszug&quot; (= excerpt) of the letter. Is the full version available anywhere?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="3454" LastActivityDate="2016-12-27T11:40:42.947" Title="Reference for &quot;Auszug aus einem Briefe von L. Kronecker an R. Dedekind&quot;" Tags="|mathematicians|reference-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5512" PostTypeId="2" ParentId="5511" CreationDate="2016-12-27T11:40:42.947" Score="1" Body="&lt;p&gt;There seems to be a general problem with numbers here. The footnote 2) that links to Jacobi just is the reference for the paper cited in the letter, near the bottom of the page. The footnote numbered 1) is the one you need, which references &quot;Zusatz 35&quot; at the end of the volume. However what they probably meant was &quot;Zusatz 34&quot; (p. 510-515), as this is a longer note about the letter and the Jugendtraum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also there is a title page for the letter (p. 453), which gives the &quot;Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin&quot; as a source. However this seems to be the same excerpt.&lt;/p&gt;&#xA;" OwnerUserId="2785" LastActivityDate="2016-12-27T11:40:42.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5514" PostTypeId="1" AcceptedAnswerId="5515" CreationDate="2016-12-28T06:17:12.150" Score="3" ViewCount="229" Body="&lt;p&gt;I know that when solving geometric problem, Descartes used variables $x,y$ and derived equation such as $y^2=cy-\frac{cxy}{b}+ay-ac$. Conversely, in algebraic geometry, an arbitrary polynomial $F(X_1,...,X_n)$ is thought to define a geometric object as its zero set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was the first step in this direction taken?&lt;/p&gt;&#xA;" OwnerDisplayName="user5146" LastActivityDate="2016-12-31T21:12:55.923" Title="When did mathematician start to draw figures from equation?" Tags="|mathematics|geometry|algebraic-geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5515" PostTypeId="2" ParentId="5514" CreationDate="2016-12-28T18:14:09.733" Score="9" Body="&lt;p&gt;The coordinate method may be traced to antiquity, specifically to the works of Apollonius of Perga (c. 262 – c. 190 BC) The following quotation from &#xA; Carl B. Boyer,&quot;Apollonius of Perga&quot; (1991).   A History of Mathematics (Second ed.). John Wiley &amp;amp; Sons, Inc. pp. 156–157. ISBN 0-471-54397-7 (as referenced in the Wikipedia article on Apollonius) explains his approach: &quot;The method of Apollonius in the Conics in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use of a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed a posteriori upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down a priori for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation; [...] That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases.&quot;&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2016-12-28T18:14:09.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5516" PostTypeId="1" CreationDate="2016-12-29T08:30:06.677" Score="4" ViewCount="3747" Body="&lt;p&gt;I read this story a while ago, and I'm wondering whether there's any proof that it is true or whether it's just made up?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One day Shizuo Kakutani…was teaching a class at Yale. He wrote down a&#xA;  lemma on the blackboard and announced that the proof was obvious. One&#xA;  student timidly raised his hand and said that it wasn’t obvious to&#xA;  him. Could Kakutani explain? After several moments’ thought, Kakutani&#xA;  realized that he could not himself prove the lemma. He apologized, and&#xA;  said that he would report back at their next class meeting.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;After class, Kakutani, went straight to his office. He labored for&#xA;  quite a time and found that he could not prove the pesky lemma. He&#xA;  skipped lunch and went to the library to track down the lemma. After&#xA;  much work, he finally found the original paper. The lemma was stated&#xA;  clearly and succinctly. For the proof, the author had written,&#xA;  “Exercise for the reader.” The author of this 1941 paper was Kakutani.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user5140" LastActivityDate="2016-12-29T08:30:06.677" Title="Kakutani's Lemma" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5519" PostTypeId="1" CreationDate="2016-12-30T18:13:27.147" Score="4" ViewCount="661" Body="&lt;p&gt;Was it known before Maxwell's time that the vacuum permitivity was inversely related to the permeability by a factor of $c^2$?&lt;/p&gt;&#xA;" OwnerUserId="5178" OwnerDisplayName="Rascalniikov" LastActivityDate="2018-08-09T19:31:37.180" Title="Electromagnetic constants and the speed of light" Tags="|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5520" PostTypeId="1" AcceptedAnswerId="5529" CreationDate="2016-12-31T07:02:12.890" Score="3" ViewCount="452" Body="&lt;p&gt;They were contemporaries. Kepler lived from 1571 to 1630, and Galileo lived from 1564 to 1642. The former's life span was contained in that of the latter.&lt;/p&gt;&#xA;&lt;p&gt;So, did Galileo hear of Kepler's work and give some comment?&lt;/p&gt;&#xA;" OwnerUserId="4079" LastEditorUserId="4251" LastEditDate="2021-02-28T08:09:19.970" LastActivityDate="2021-02-28T08:09:19.970" Title="What was Galileo's opinion of Kepler's work?" Tags="|physics|astronomy|galileo|17th-century|kepler|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5521" PostTypeId="1" AcceptedAnswerId="5531" CreationDate="2016-12-31T10:21:01.057" Score="1" ViewCount="148" Body="&lt;p&gt;In &quot;natural life&quot; &quot;axis&quot; is often used in terms of an axis of rotation. But in the mathematical sense, it's more used like a ruler. One could say an axis in &lt;strong&gt;&quot;natural life&quot; sense has something to do with rotation around the axis&lt;/strong&gt; and &lt;strong&gt;in math, it's more like a translation (physical sense) in direction of the axis&lt;/strong&gt; (position on the axis).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In German it's even more difficult to distinguigh this meanings as &quot;axle&quot; and &quot;axis&quot; are both &quot;Achse&quot; in german.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that &lt;a href=&quot;https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes&quot; rel=&quot;nofollow noreferrer&quot;&gt;René Descartes&lt;/a&gt; deliberately chose the name &quot;axis&quot; for coordinate axis?&lt;/p&gt;&#xA;" OwnerUserId="4721" LastActivityDate="2017-01-01T17:57:21.853" Title="Coordinate axis - Why the name &quot;axis&quot;?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5523" PostTypeId="1" AcceptedAnswerId="5527" CreationDate="2016-12-31T16:44:59.730" Score="2" ViewCount="68" Body="&lt;p&gt;The scientific name of the Mediterranean Mytilus is &lt;a href=&quot;https://en.wikipedia.org/wiki/Mediterranean_mussel&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mytilus galloprovincialis&lt;/em&gt;&lt;/a&gt;, with &lt;a href=&quot;http://www.marinespecies.org/aphia.php?p=taxdetails&amp;amp;id=140481&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lamarck being reported as the creator&lt;/a&gt;. I wonder where this name comes from, in particular what is the (supposed) connection to the Gallia province? &lt;/p&gt;&#xA;" OwnerUserId="5165" LastEditorUserId="55" LastEditDate="2017-01-01T02:04:14.337" LastActivityDate="2017-01-01T07:31:51.513" Title="What is the connection between Lamarck's Mediterranean mussel and the province of Gallia?" Tags="|terminology|biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5524" PostTypeId="2" ParentId="5520" CreationDate="2016-12-31T17:35:07.060" Score="1" Body="&lt;p&gt;Galileo did not publish any opinion on Kepler's work, though they corresponded with Kepler. Why this is so, subject to speculation. See, for example Koestler, Sleepwalkers (this author is unfriendly to Galileo). One possible reason was that&#xA;Galileo simply did not appreciate enough Kepler's contributions, which was mostly mathematical, and very different in the spirit from Galileo's principal interests. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2016-12-31T17:35:07.060" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5525" PostTypeId="2" ParentId="5519" CreationDate="2016-12-31T18:44:40.003" Score="9" Body="&lt;p&gt;The relation of the speed of light $c$ to electrodynamics was known before Maxwell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1846, Weber derived his force law between point charges:&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$F=\frac{ee'}{r^2}\left[1-\frac{1}{2c^2}\left(\frac{dr}{dt}\right)^2+\frac{1}{c^2}r\frac{d^2r}{dt^2}\right]$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;from &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_force_law&amp;amp;oldid=758353198#Equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère's force law&lt;/a&gt;&lt;sup&gt;2&lt;/sup&gt; (not to be confused with one of Maxwell's equations, the &lt;a href=&quot;https://hsm.stackexchange.com/q/5059/232&quot;&gt;Ampère circuital law&lt;/a&gt;) between current elements  (written in modern vector notations and with modern units):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the form Weber wrote his law, it involved a constant $a$ that is related to the speed of light $c$ by a factor of $\sqrt{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1856, Kohlrausch &amp;amp; Weber experimentally determined the constant.&lt;sup&gt;3&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1857, Kirchoff explicitly tied this constant to the propagation of electricity in a wire:&lt;sup&gt;4&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The velocity of propagation of an electric wave is here equal to $\frac{c}{\sqrt{2}}$; it is therefore independent of the cross-section of the wire, of its conductivity, and, finally, of the electric density; […] it is thus very near the speed of light in empty space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/books.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Assis&lt;/a&gt;'s &lt;a href=&quot;https://isidore.co/calibre/browse/book/4302&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Weber's Electrodynamics&lt;/em&gt;&lt;/a&gt; and ch. 8 of Duhem's &lt;a href=&quot;https://isidore.co/calibre/browse/book/4976&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Electric Theories of J. Clerk Maxwell: A Historical and Critical Study&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, this is a good historical overview:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Rosenfeld, L. “&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/ACQ5SN9J/Rosenfeld%20-%201956%20-%20The%20velocity%20of%20light%20and%20the%20evolution%20of%20electro.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Velocity of Light and the Evolution of Electrodynamics&lt;/a&gt;.” &lt;em&gt;Il Nuovo Cimento&lt;/em&gt; 4, no. 5 (September 1, 1956): 1630–69. doi:&lt;a href=&quot;https://dx.doi.org/10.1007/BF02745315&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1007/BF02745315&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&#xA;References&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;W. Weber, &lt;a href=&quot;https://isidore.co/calibre/browse/book/4654&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Elektrodynamische Maassbestimmungen&lt;/em&gt;&lt;/a&gt; [&lt;a href=&quot;https://isidore.co/calibre/browse/book/4310&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Determinations of Electrodynamic Measure&lt;/em&gt;&lt;/a&gt;], Leipzig, 1846: p. 142 ff. of Weber's &lt;a href=&quot;https://isidore.co/calibre/browse/book/4654&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Werke&lt;/em&gt; vol. 3&lt;/a&gt; or p. 93 (§21 or PDF p. 94) ff. of &lt;a href=&quot;https://isidore.co/calibre/browse/book/4310&quot; rel=&quot;nofollow noreferrer&quot;&gt;this translation&lt;/a&gt; (also &lt;a href=&quot;https://www.21stcenturysciencetech.com/Articles%202007/Weber_1846.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). In 1848, Weber wrote a shorter paper, &quot;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4861&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Measurement of Electro-dynamic Forces&lt;/a&gt;&quot; (also &lt;a href=&quot;http://www.sizes.com/library/classics/Weber1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;); see spec. pp. 32-43 for the derivation of his law.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;First published English translation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Assis, André Koch Torres, J. P. M. C. Chaib, André-Marie Ampère. 2015. &lt;a href=&quot;https://isidore.co/calibre/browse/book/5464&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience&lt;/em&gt;&lt;/a&gt;. (also &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;R. Kohlrausch and W. Weber, &lt;em&gt;Elektrodynamische Maassbestimmungen, insbesondere Zurückführung der Stromintensitäts-Messungen auf mechanische Maass&lt;/em&gt;, Leipzig, 1856. [English translation: &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Weber-Kohlrausch(2003).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weber and Kohlrauch (2003)&lt;/a&gt;].&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;G. Kirchhoff, &lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/6TG7ABBC/Kirchhoff%20-%201857%20-%20Ueber%20die%20Bewegung%20der%20Elektricit%C3%A4t%20in%20Dr%C3%A4hten.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Ueber die Bewegung der Elektricität in Drähten&lt;/em&gt;&lt;/a&gt; [&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/335PCK2X/Kirchhoff%20-%201857%20-%20LIV.%20On%20the%20motion%20of%20electricity%20in%20wires.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the motion of electricity in wires&lt;/a&gt;] (&lt;em&gt;Poggendorff’s Annalen&lt;/em&gt;), Bd., 1857. [English translation: &lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/335PCK2X/Kirchhoff%20-%201857%20-%20LIV.%20On%20the%20motion%20of%20electricity%20in%20wires.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kirchhoff (1857a)&lt;/a&gt;].&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-08-09T19:31:37.180" LastActivityDate="2018-08-09T19:31:37.180" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="5526" PostTypeId="2" ParentId="5514" CreationDate="2016-12-31T21:12:55.923" Score="0" Body="&lt;p&gt;Probably Al-Khwarizmi.  His book on algebra (the first one, from which algebra gets its name), contains diagrams, but only to illustrate  (or geometrically butress) the &quot;equations&quot; (which are written in natural language, not symbols).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Greeks never really got to algebra.  Some of their stuff looks a bit like algebra, but it lacked the conceptual apparatus essential to genuine algebraic thinking, so it's historically inaccurate to call it algebra.&lt;/p&gt;&#xA;" OwnerUserId="4062" LastActivityDate="2016-12-31T21:12:55.923" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5527" PostTypeId="2" ParentId="5523" CreationDate="2017-01-01T07:09:39.527" Score="6" Body="&lt;p&gt;This mussel, indigenous to the Mediterranean but since spread across the globe by human activity, was first described in:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;M. Chevalier de Lamarck, &lt;em&gt;Histoire naturelle des animaux sans vertèbres&lt;/em&gt;, Tome sixième, 1&lt;sup&gt;re&lt;/sup&gt; partie, (Paris: February - June 1819), p. 126 (&lt;a href=&quot;http://www.biodiversitylibrary.org/item/47441#page/138/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There the French name is given as &quot;Moule de Provence&quot;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Provence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Provence&lt;/a&gt; is a region in the south of France, bordering the Mediterranean. In Roman times it was part of a province called first Gallia Transalpina, then &lt;a href=&quot;https://en.wikipedia.org/wiki/Gallia_Narbonensis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gallia Narbonensis&lt;/a&gt;, but commonly known simply as &quot;Provincia&quot; (= province), from which we get the French name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;galliaprovincialis&lt;/em&gt; is simply the Latin version of &quot;de Provence&quot;, referring to the place where the mussel was originally found according to Lamarck's description: near &lt;a href=&quot;https://en.wikipedia.org/wiki/Martigues&quot; rel=&quot;nofollow noreferrer&quot;&gt;Martigues&lt;/a&gt; in Provence, a small town in the vicinity of Marseille.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2017-01-01T07:31:51.513" LastActivityDate="2017-01-01T07:31:51.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5528" PostTypeId="1" CreationDate="2017-01-01T11:39:50.760" Score="2" ViewCount="114" Body="&lt;p&gt;Im trying to connect my study to a breif history of prime numbers . Also im interesting in reading history of numbers how they come and how they developed .&#xA;So can any one suggest for this question a good book or&#xA;any helpful reference for the history of prime numbers from Creek till 19 century and thank you in advanced .&lt;/p&gt;&#xA;" OwnerUserId="5168" LastActivityDate="2017-01-01T17:10:34.753" Title="reference need about History of prime number development" Tags="|reference-request|notation|number-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5529" PostTypeId="2" ParentId="5520" CreationDate="2017-01-01T16:12:32.033" Score="3" Body="&lt;p&gt;Yes, Galileo knowed Kepler's, as well as Tycho's, works.&lt;/p&gt;&#xA;&lt;p&gt;There is an extant copy of Tycho's &lt;a href=&quot;https://books.google.it/books/about/Astronomiae_instauratae_progymnasmata.html?id=0VpRAAAAcAAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Progymnasmata&lt;/em&gt;&lt;/a&gt; with Galileo's annotations, and we have an annotation from Galileo on Kepler's &lt;a href=&quot;https://books.google.it/books?id=Bk5RAAAAcAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De stella nova&lt;/em&gt;&lt;/a&gt;: the main interest showed by Galileo's annotations regards &lt;em&gt;novae&lt;/em&gt; and comets.&lt;/p&gt;&#xA;&lt;p&gt;Kepler published two booklets in defence of Galileo, after the pubblication in 1610 of &lt;a href=&quot;https://en.wikipedia.org/wiki/Sidereus_Nuncius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sidereus Nuncius&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.it/books?id=qXWc3fJVue4C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dissertatio cum Nuncio Sidereo&lt;/a&gt; (1611),&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;followed the same year by :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.it/books?id=vnBxmAEACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Narratio de Jovis Satellitibus&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The two corresponded and were &amp;quot;in touch&amp;quot;, although they did not meet.&lt;/p&gt;&#xA;&lt;p&gt;And yes, Galileo gave comments on Kepler.&lt;/p&gt;&#xA;&lt;p&gt;There are some references to Kepler in Galileo's &lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dialogue&lt;/a&gt;, ending with a critical judgement.&lt;/p&gt;&#xA;&lt;p&gt;The context is Galileo's &amp;quot;master argument&amp;quot; for the rotation of Earth: his pure &amp;quot;cinematical explanation of the tides, without need to &amp;quot;occult&amp;quot; attractions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[ &lt;a href=&quot;https://it.wikisource.org/wiki/Dialogo_sopra_i_due_massimi_sistemi_del_mondo_tolemaico_e_copernicano/Giornata_quarta&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giornata quarta, page 486&lt;/a&gt; ]  Ma tra tutti gli uomini grandi che sopra tal mirabile effetto di natura hanno filosofato, piú mi meraviglio del &lt;strong&gt;Keplero&lt;/strong&gt; che di altri, il quale, d’ingegno libero ed acuto, e che aveva in mano i moti attribuiti alla Terra, abbia poi dato orecchio ed assenso a predominii della Luna sopra l’acqua, ed a proprietà occulte, e simili fanciullezze.&lt;/p&gt;&#xA;&lt;p&gt;But among all the great men who have philosophized about this remarkable effect, I am more astonished at Kepler than at any other. Despite his open and acute mind, and though he has at his fingertips the motions attributed to the earth, he has nevertheless lent his ear and his assent to the moon’s dominion over the waters, to occult properties, and to such puerilities [&lt;em&gt;S.Drake's translation&lt;/em&gt;].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-01-01T16:30:07.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5530" PostTypeId="2" ParentId="5528" CreationDate="2017-01-01T17:10:34.753" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Oystein Ore, &lt;a href=&quot;https://archive.org/details/NumberTheoryItsHistory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Theory &amp;amp; its History&lt;/a&gt; (1948)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;André Weil, &lt;a href=&quot;https://books.google.it/books?id=Ar7gBwAAQBAJ&amp;amp;pg=PR7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Theory: An approach through history From Hammurapi to Legendre&lt;/a&gt; (1984).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-01-01T17:10:34.753" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5531" PostTypeId="2" ParentId="5521" CreationDate="2017-01-01T17:57:21.853" Score="2" Body="&lt;p&gt;The source of the use in analytic geometry seems to be from &lt;a href=&quot;https://en.wikipedia.org/wiki/Conic_section#Eccentricity.2C_focus_and_directrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;conics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Accoding to &lt;a href=&quot;http://jeff560.tripod.com/a.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tripod&lt;/a&gt; : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Axis&lt;/strong&gt; occurs in English in the phrase &quot;the Axis or Altitude of the Cone&quot; in 1571 in &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/mathematical-treasures-thomas-digges-pantometria&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Geometricall Practise named Pantometria&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonard_Digges_(scientist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonard&lt;/a&gt; (c.1515 – c.1559)and &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Digges&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Digges&lt;/a&gt; (1546?-1595).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As you can see into the first textbook on analytic geometry : Jan de Witt’s &lt;a href=&quot;https://books.google.it/books?id=6W1h-sHLGIkC&amp;amp;pg=PA123&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elementa Curvarum Linearum Liber Secundus&lt;/a&gt; (1659), page 123, the latin term &quot;axis&quot; is always referred to conics.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-01-01T17:57:21.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5532" PostTypeId="1" CreationDate="2017-01-02T10:38:36.600" Score="5" ViewCount="334" Body="&lt;p&gt;When I read our differential geometry book, I saw two theorema: &quot;Theorema Egregium&quot; and &quot;Theorema Elegantissimum&quot;. Mathematically, they are not the same. On wikipedia, there is nothing about Elegantissmum. What is the story behind it?&lt;/p&gt;&#xA;" OwnerUserId="5175" LastActivityDate="2017-01-02T13:52:33.720" Title="Does Gauss own two “Theorema”?" Tags="|differential-geometry|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5533" PostTypeId="2" ParentId="5532" CreationDate="2017-01-02T12:05:48.030" Score="5" Body="&lt;p&gt;In his &lt;a href=&quot;https://books.google.it/books?id=bX0AAAAAMAAJ&amp;amp;printsec=frontcover&amp;amp;hl=it#v=onepage&amp;amp;q=egregium&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Disquisitiones generales circa superficies curvas&lt;/a&gt; (1827), §12, page 24, Gauss called &lt;em&gt;egregium&lt;/em&gt; [sponte perducit ad egregium, i.e. &lt;em&gt;spontaneously leads to excellent&lt;/em&gt;] the following Theorem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Si superficies curva in quamcumque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet. [&lt;em&gt;If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged&lt;/em&gt;.]  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And see §20, &lt;a href=&quot;https://books.google.it/books?id=bX0AAAAAMAAJ&amp;amp;printsec=frontcover&amp;amp;hl=it#v=onepage&amp;amp;q=elegantissima&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;page 36&lt;/a&gt; for a result regarding the theory of curved surfaces called &lt;em&gt;elegantissima&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Excessus summae angolorum trianguli&lt;/em&gt;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to: John McCleary, &lt;a href=&quot;https://books.google.it/books?id=zfJEIU7YIioC&amp;amp;pg=PA205&amp;amp;lpg=PA205&amp;amp;dq=gauss+Theorema+Elegantissimum&amp;amp;source=bl&amp;amp;ots=QgWsC50roM&amp;amp;sig=YqrgRbkeWw-T5kNsxov2yZMj0Ms&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=0ahUKEwjmiuyouaPRAhWjLsAKHXHOBwo4ChDoAQgaMAA#v=onepage&amp;amp;q=gauss%20Theorema%20Elegantissimum&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Geometry from a Differentiable Viewpoint&lt;/a&gt; (2nd ed, 2013), the Gauss-Bonnet Theorem is linked to Gauss' &quot;Theorema Elegantissimum&quot;, referring to Gauss' &lt;em&gt;Disquisitiones&lt;/em&gt;, (1825, §20).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See also into &lt;a href=&quot;https://books.google.it/books?id=bX0AAAAAMAAJ&amp;amp;pg=PA30#v=onepage&amp;amp;q=elegantissimum&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Theoria residuorum biquadraticorum&lt;/a&gt; (1825), page 30, he calls &lt;em&gt;elegantissimum&lt;/em&gt; the result:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$P \equiv 2a (\text {mod} p)$.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-01-02T13:52:33.720" LastActivityDate="2017-01-02T13:52:33.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5534" PostTypeId="1" CreationDate="2017-01-02T14:05:21.597" Score="2" ViewCount="82" Body="&lt;p&gt;How did we come to know that, for example, the quantity  Work=Force*distance is conserved? &lt;/p&gt;&#xA;" OwnerUserId="5177" LastActivityDate="2017-01-02T14:05:21.597" Title="How exactly was the principle of conservation of energy discovered?" Tags="|energy|" AnswerCount="0" CommentCount="3" FavoriteCount="0" ClosedDate="2017-01-18T16:21:02.420" ContentLicense="CC BY-SA 3.0" />
  <row Id="5535" PostTypeId="1" CreationDate="2017-01-02T11:58:50.110" Score="0" ViewCount="33" Body="&lt;p&gt;How did we come to know that, for example, the quantity  Work=Force*distance is conserved? &lt;/p&gt;&#xA;" OwnerUserId="5177" OwnerDisplayName="Khalid T. Salem" LastActivityDate="2017-01-02T19:31:25.440" Title="How exactly was the principle of conservation of energy discovered?" Tags="|untagged|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2017-01-02T21:21:51.883" ContentLicense="CC BY-SA 3.0" />
  <row Id="5536" PostTypeId="2" ParentId="5508" CreationDate="2017-01-03T22:38:01.370" Score="7" Body="&lt;p&gt;The &lt;a href=&quot;http://www.claymath.org/millennium-problems/millennium-prize-problems&quot; rel=&quot;noreferrer&quot;&gt;website of the Clay Mathematical Institute&lt;/a&gt; writes the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The focus [...] was on important classic questions that have resisted solution for many years.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Arthur M. Jaffe, once director of the Clay Mathematical Insitute, wrote on the history of conceiving the Millennium Prize Problems (emphasis mine). &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As a first step, I requested that each SAB member submit a personal list of top questions. Each&#xA;  of these questions should be &lt;strong&gt;difficult and important — a  time-tested  challenge  on  which  mathematicians had worked without success&lt;/strong&gt;. [...] We  added  questions to the list one by one. With each new question we asked whether the list should be expanded&#xA;  or whether it might be improved by substituting a&#xA;  new question. [...] While each problem on the list was central and&#xA;  important, I want to stress that &lt;strong&gt;the SAB did not envisage making a definitive list, nor even a representative set of famous unsolved problems.&#xA;  Rather, personal taste entered our choices&lt;/strong&gt;; a different scientific advisory board undoubtedly would&#xA;  have come up with a different list. The persons we&#xA;  consulted were experts, but they were chosen under&#xA;  pressure of time. However, the spirit of the selection transcends these decisions: the resulting list&#xA;  represents &lt;strong&gt;an honest attempt to convey some excitement about mathematics&lt;/strong&gt;. We do not wish to address the question, “Why is Problem A not on your&#xA;  list?” Rather we say that the list highlights &lt;strong&gt;seven&#xA;  historic, important, and difficult open questions&lt;/strong&gt; in&#xA;  mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.ams.org/notices/200606/fea-jaffe.pdf&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt; of the above article also describes in more detail the actual process of creating the final list of $7$ problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For reference, the seven problems are:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;The Birch-Swinnerton-Dyer conjecture&#xA;  (Andrew Wiles)&lt;/li&gt;&#xA;  &lt;li&gt;The Hodge conjecture (Pierre Deligne)&lt;/li&gt;&#xA;  &lt;li&gt;The Navier-Stokes equation has&#xA;  smooth solutions (Charles Fefferman)&lt;/li&gt;&#xA;  &lt;li&gt;P is not NP (Stephen Cook)&lt;/li&gt;&#xA;  &lt;li&gt;The Poincaré conjecture (John Milnor)&lt;/li&gt;&#xA;  &lt;li&gt;Quantum Yang-Mills theory exists&#xA;  with a mass gap (Arthur Jaffe and Edward&#xA;  Witten)&lt;/li&gt;&#xA;  &lt;li&gt;The Riemann hypothesis (Enrico&#xA;  Bombieri)&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5134" LastEditorUserId="26" LastEditDate="2017-01-23T17:05:32.657" LastActivityDate="2017-01-23T17:05:32.657" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5537" PostTypeId="1" AcceptedAnswerId="5538" CreationDate="2017-01-04T00:44:08.867" Score="11" ViewCount="4445" Body="&lt;p&gt;In debates with climate change deniers and creationists, it is often claimed that consensus of scientists is not enough to establish a position beyond dispute.  Examples of the problem with consensus are geocentrism, species saltation, ambiogenesis, eugenics, racists theories, etc.  These were all theories believed by a large majority of scientists at the time, that were overturned by a few scientists, and often did not become widespread until long after their introduction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is this a fair claim by deniers and creationists, that scientific consensus by itself is not a good standard for considering a position beyond dispute?&lt;/p&gt;&#xA;" OwnerUserId="2617" LastActivityDate="2023-06-01T23:12:50.147" Title="Is scientific consensus ever significantly wrong?" Tags="|social-context|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5538" PostTypeId="2" ParentId="5537" CreationDate="2017-01-04T01:50:59.500" Score="14" Body="&lt;p&gt;There are all kinds of scientific claims, and I don't think they can necessarily be lumped together. One example of an incorrect scientific consensus is the theory of the luminiferous aether. Another false claim was that margarine was healthier than butter. I don't think there is any conceptual tent big enough to cover both of these examples and allow us to say anything meaningful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples like geocentrism going to heliocentrism and aether theories being killed off by relativity were huge revolutions in the foundations of entire sciences. If modern cosmology were to be completely overturned, as I suppose some creationists would like, it would probably require an equally momentous revolution. Historians of science such as Popper and Kuhn have had very different ideas about examples like these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Popper's description, scientific theories are falsifiable, meaning that at least in principle, there is some experiment or observation that could disprove it. Popper was trained as a psychologist in the 1920's, when Freudianism was still taken seriously. He would have considered Freudianism not to be a scientific theory, since it was not falsifiable. According to Popper, when an old theory is falsified, what usually happens in the physical sciences is that we find the old theory to have been a good approximation under certain circumstances, which were the circumstances under which it was tested, but a bad approximation under other circumstances. For instance, Newton's laws are an excellent description of the trajectory of a baseball, but a bad description of a subatomic particle moving at half the speed of light. Sometimes this is called the correspondence principle: the old theory wasn't wrong, within the circumstances where it had been accurate, and the new, broader theory has to be backward-compatible with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Kuhn's description, we have paradigm shifts, in which an accumulation of anomalies ends up causing a certain scientific world-view to break down and be replaced by a new one. According to Kuhn, there are periods of normal science, when things are boring and you just go around collecting beetles, and then there are revolutionary periods. Critics seem to see Kuhn as attacking scientific realism, the notion of scientific truth, and the authority of science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When it comes to climate change, I don't think it would require a scientific revolution to disprove it. It simply looks very unlikely right now that it's wrong. In that sense, it might be more like the margarine-butter example, although probably much less inherently uncertain because the physical sciences don't tend as much toward fads and junk science as does medical science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't think either Popper or Kuhn would have suggested that a consensus among scientists is the right way for &lt;em&gt;scientists&lt;/em&gt; to determine what is true in their own field of expertise. I think both were clearly aware of examples like geocentrism and aether theories where the consensus was wrong. If scientific consensus has a place, it's in guiding people who are &lt;em&gt;not&lt;/em&gt; scientists, and who therefore can't weigh the published evidence directly or do experiments and observations of their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some questions to ask a creationist or climate change denier:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) If they don't want to be guided by scientific consensus in matters of science, what alternative guide would they like to use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) What standard of proof do they think is appropriate, and why? In the case of climate change, we're talking about a global crisis, so even if it's only a 99% chance that it's right, it seems to me to make sense to take strong action to keep from driving our planet off a cliff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) On the issue being discussed, is there an alternative theory that fits the evidence? In cosmology, for example, we really don't have a viable alternative right now to a hot big bang.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2017-01-04T02:14:03.030" LastActivityDate="2017-01-04T02:14:03.030" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5539" PostTypeId="1" CreationDate="2017-01-04T15:37:00.080" Score="2" ViewCount="190" Body="&lt;p&gt;I'm searching for a book or an online copy of complete works of the mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;François Viète&lt;/a&gt;, preferably in English.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any help will be appreciated.&lt;br&gt;&#xA;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="5189" LastActivityDate="2017-01-04T15:37:00.080" Title="Works of mathematician François Viète" Tags="|mathematics|mathematicians|reference-request|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5541" PostTypeId="1" CreationDate="2016-12-22T04:55:13.267" Score="5" ViewCount="425" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Polyakov remembers the development of Renormalization Group and Conformal Bootstrap as &quot;With the use of the ingenious technique, developed by Gribov and Migdal [1] in the problem of reggeons, I found connections between phenomenological theory and “bootstrap” equations (Polyakov [2]). Sasha Migdal did very similar work independently. There was also something new–I formulated “fusion rules” for correlations, which we now would call operator product expansion [3]. I had mixed feelings when I found out later that the same rules at the same time and in more generality have been found by L. Kadanoff [4] and K. Wilson [5].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While for the Renormalization Group, we frequently also mentioned the names of K. Wilson and Michael Fisher together (e.g. Wilson-Fisher fixed point).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;question: &lt;strong&gt;What roles had Gribov, Migdal, Polyakov, L. Kadanoff, K. Wilson and M. Fisher play in the Renormalization Group?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. It seems that K. Wilson gets the most credit for the development of Renormalization Group (a single recipient for &lt;a href=&quot;https://en.wikipedia.org/wiki/Kenneth_G._Wilson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nobel Prize in Physics (1982)&lt;/a&gt;). But M. Fisher, L. Kadanoff and K. Wilson share credits for the earlier &lt;a href=&quot;https://en.wikipedia.org/wiki/Wolf_Prize_in_Physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolf Prize in Physics (1980), for &quot;for pathbreaking developments culminating in the general theory of the critical behavior at transitions between the different thermodynamic phases of matter.&lt;/a&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A side question: Why does K Wilson get the most credits for this subject?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] V. Gribov and A. Migdal, ZHETF 55, 1498 (1968).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] A. Polyakov, ZHETF 55, 1026 (1968)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[3] A. Polyakov, ZHETF 57, 271 (1969).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[4] L. Kadanoff, Phys. Rev. Lett. 23, 1430 (1969).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[5] K. Wilson, Phys. Rev. 179, 1499 (1969).&lt;/p&gt;&#xA;" OwnerUserId="2162" OwnerDisplayName="wonderich" LastActivityDate="2017-01-08T12:49:55.280" Title="The origin of Renormalization Group" Tags="|quantum-field-theory|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="5542" PostTypeId="2" ParentId="5541" CreationDate="2017-01-04T14:28:48.873" Score="3" Body="&lt;p&gt;I think the quote (a reference would be good) is more about the OPE than the RG although they are very closely related and must have germinated in Wilson's mind concurrently. About Wilson's priority regarding the OPE, the correct date for his paper is 1964 rather than 1969 which seems to make his contribution posterior to that of the Soviet School. They didn't have arXiv at the time and&#xA;Wilson's &lt;a href=&quot;http://inspirehep.net/record/1101781&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cornell preprint&lt;/a&gt; did not get a wide circulation (perhaps also because it had a mistake regarding anomalous dimensions). See section 1.11 of &lt;a href=&quot;https://arxiv.org/abs/1604.05259&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; for more references and in particular a &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0003491667901777&quot; rel=&quot;nofollow noreferrer&quot;&gt;1967 Ann. Phys. paper by Brandt&lt;/a&gt; with an account of Wilson's OPE.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Edit: Regarding Wilson's error about anomalous dimensions in the original OPE preprint, see Section 4 of &lt;a href=&quot;https://arxiv.org/abs/1405.7086&quot; rel=&quot;nofollow noreferrer&quot;&gt;this recent article&lt;/a&gt; by Peskin dedicated to Wilson's work.&lt;/p&gt;&#xA;" OwnerUserId="5190" OwnerDisplayName="Abdelmalek Abdesselam" LastEditorUserId="5190" LastEditDate="2017-01-08T12:49:55.280" LastActivityDate="2017-01-08T12:49:55.280" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5543" PostTypeId="2" ParentId="5230" CreationDate="2017-01-05T13:52:34.927" Score="2" Body="&lt;p&gt;Another nice source relative to this controversy between Weierstrass, Kronecker and Jordan is the review &quot;Sur les progr&amp;egrave;s de la th&amp;eacute;orie des invariants projectifs&quot; by Franz Meyer from 1897. As the link to Google books  put earlier on &lt;a href=&quot;https://mathoverflow.net/questions/96149/invariants-for-the-exceptional-complex-simple-lie-algebra-f-4/96519#96519&quot;&gt;MathOverflow&lt;/a&gt; does not seem to give access to the full text anymore, another link one can use is &lt;a href=&quot;https://archive.org/details/surlesprogrsdel00ocaggoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;.&#xA;The relevant discussion is between pages 27 and 35. &lt;/p&gt;&#xA;" OwnerUserId="5190" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2017-01-13T00:15:05.197" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5544" PostTypeId="2" ParentId="5093" CreationDate="2017-01-05T14:05:55.870" Score="4" Body="&lt;p&gt;I suppose the prime example would be classical invariant theory (CIT), although agreeing with Alexander, and despite exaggerated news of its death it had some bursts of revival recently. It is interesting to note that in the 19th century this subject was almost inseparable from classical elimination theory (CET). There has been a revival of CIT without CET (e.g., works of Dixmier, the recent book by Olver, etc.) as well as a revival CET without CIT (e.g., the book by Gelfand, Kapranov and Zelevinsky) but not really a revival of the combination CIT &amp;amp; CET. &lt;/p&gt;&#xA;" OwnerUserId="5190" LastActivityDate="2017-01-05T14:05:55.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5545" PostTypeId="2" ParentId="5038" CreationDate="2017-01-05T14:28:50.693" Score="3" Body="&lt;p&gt;This is an example of absolute invariant in the context of classical invariant theory. Of course it is of paramount importance in the theory of elliptic curves but it is not just about them. Such a curve can be written as &#xA;$y^2=f(x)$&#xA;where $f$ is a nonhomogenous polynomial of degree four (or three).&#xA;By homogenization this corresponds to a binary quartic&#xA;$$&#xA;F(x_1,x_2)=x_2^4 f\left(\frac{x_1}{x_2}\right)=&#xA;a_0x_1^4+4 a_1 x_1^3 x_2+ 6 a_2 x_1^2 x_2^2+4 a_3 x_1 x_2^3+a_4 x_2^4\ .&#xA;$$&#xA;The ring of $SL_2$ invariants is generated by&#xA;$$&#xA;S= a_0a_4 - 4a_1 a_3 + 3a_2^2&#xA;$$ &#xA;and&#xA;$$&#xA;T=a_0a_2a_4 - a_0a_3^2 - a_1^2a_4 + 2a_1a_2a_3 - a_2^3&#xA;$$&#xA;while the $j$ invariant is given by&#xA;$$&#xA;j=\frac{S^3}{S^3-27T^2}\ .&#xA;$$&#xA;It gives a way of parametrizing $SL_2$ orbits of such binary quartics (see &lt;a href=&quot;https://mathoverflow.net/questions/49571/quotient-space-of-mathbbc5-under-the-action-of-sl2-mathbbc/50171#50171&quot;&gt;this MO post&lt;/a&gt; for details) and thus of isomorphism classes of elliptic curves. &#xA;Early work on this is in papers by Eisenstein, Boole and Cayley from the first half of the 19th century. &lt;/p&gt;&#xA;" OwnerUserId="5190" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2017-01-05T14:28:50.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5547" PostTypeId="1" CreationDate="2017-01-05T17:35:56.460" Score="9" ViewCount="585" Body="&lt;p&gt;Over time, the science on the issue of dietary fat and heart disease has undergone drastic changes, and in particular, advice to eat margarine rather than butter has been reversed. What is the history of this subject?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-06-03T18:08:49.643" Title="What is the history of the science about butter, margarine, and health?" Tags="|medicine|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5548" PostTypeId="2" ParentId="5547" CreationDate="2017-01-05T17:35:56.460" Score="11" Body="&lt;p&gt;Throughout most of the 19th century, the commonly held scientific belief about the health impact of dietary fat was that fat, along with protein, tended to prevent obesity, by causing satiety. The best known proponent of this hypothesis was Jean Anthelme Brillat-Savarin, who wrote The Physiology of Taste in 1825. In this period, cardiovascular disease was poorly understood, and public health concerns centered on lack of nutrition among the poor. Because the poor in France often could not afford butter, Napoleon III offered a prize for developing a substitute, and this led to the invention of margarine, which was commercialized ca. 1870.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The early 20th century saw increased scientific and medical interest in cardiovascular disease, which was deadly and untreatable. Coronary artery disease was identified in 1912 by James B. Herrick, who coined the term “heart attack.” In the period before World War II, the focus was on diagnosis and treatment. Diet had not been implicated as a factor in heart disease, and food safety was seen as revolving around preventing the addition of adulterants. Dairy farmers in the U.S. successfully lobbied for the passage of legislation in 30 states to require that margarine be taxed and either sold with its natural gray color or dyed pink as a warning. These restrictions were only gradually eliminated. During World War II, food shortages caused many people to eat margarine rather than butter, and some people continued these eating habits after the war. But margarine continued to carry a stigma, and as late as 1959, a margarine company felt compelled to hire Eleanor Roosevelt to do a &lt;a href=&quot;https://www.youtube.com/watch?v=6HY8vxYX78s&quot; rel=&quot;noreferrer&quot;&gt;TV commercial&lt;/a&gt; reducing the stigma associated with margarine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The central figure in this story, starting around World War II, is the University of Minnesota physiologist Ancel Keys, a colorful and self-promoting figure who developed K rations (the “K” probably stood for “Keys”) and carried out a study, ethically dubious by modern standards, in which conscientious objectors were scientifically studied while they underwent starvation and were then nursed back to health. In 1952, he published the hypothesis that a diet rich in saturated fat caused an increase in blood cholesterol, which in turn led to atherosclerosis. From 1956 to 1978, he carried out his best known project, the Seven Countries Study, which searched for correlations between heart disease and lifestyle factors in the US, Finland, Netherlands, Italy, Yugoslavia, Greece, and Japan. He found that high blood pressure and smoking were correlated with heart disease, and also appeared to verify his hypothesis about saturated fat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What followed was a rush to judgment by public health officials to discourage the consumption of saturated fat, and in particular to convince people to switch from butter to margarine. In the US, the 1977 McGovern Report recommended that people shift their diets away from fat and toward complex carbohydrates. The only immediate blowback was from farmers whose economic interests would be harmed, as well as from doctors who believed that it was not the proper role of government to advise people about their health. The food industry responded in a variety of ways, including marketing carbohydrate-based foods as low in fat, making reduced-fat versions of dairy products, and producing fat substitutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Over the years, it became clear that the evidence on a link between diet and heart disease had always been weak, that the interpretations had been overblown, and that the scientific reality was both difficult to probe methodologically and extremely complicated (e.g., it took decades before “good” and “bad” cholesterol were distinguished, and for researchers to realize that bad cholesterol comes in a variety of particle sizes). The Seven Countries Study was criticized on a variety of methodological grounds, such as possible cherry-picking of the countries. Even studies that support a link between fat consumption and heart disease don't show a clear effect on mortality. In fact, one of Keys's own large studies, from institutionalized mental patients in the 1970s, was never published until 2016, probably because it appeared to show that patients put on a cholesterol-lowering diet were &lt;em&gt;more&lt;/em&gt; likely to die from heart disease. One very large study, the Women's Health Initiative, found no difference in heart disease among women who changed their diets in ways that lowered their “bad” cholesterol.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In retrospect, this episode in scientific history appears to be a demonstration of confirmation bias and the formation of alliances between scientists, government, and industry to promote a premature scientific consensus. Researcher Dariush Mozaffarian says, &quot;In the nutrition field, it's very difficult to get something published that goes against  established dogma. The dogma says that saturated fat is harmful, but that is not based, to me, on unequivocal evidence.&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="12699" LastEditDate="2017-06-03T18:08:49.643" LastActivityDate="2017-06-03T18:08:49.643" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5551" PostTypeId="1" CreationDate="2017-01-07T16:25:43.870" Score="0" ViewCount="90" Body="&lt;p&gt;I read a book a while ago about the history of physics in the renaissance, which treated it as an alternation between physical-causal descriptions (eg. inertia obtains because air molecules are pushed around an object and circle back around to push it in turn) and law-based descriptions (Newton's first law). It was a fairly thin volume.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is my description sufficient to locate this specific book? Thanks.&lt;/p&gt;&#xA;" OwnerUserId="5206" LastEditorUserId="4251" LastEditDate="2023-06-26T09:37:37.937" LastActivityDate="2023-06-26T09:37:37.937" Title="Looking for specific book about renaissance physics" Tags="|physics|reference-request|renaissance|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="5552" PostTypeId="1" AcceptedAnswerId="5554" CreationDate="2017-01-07T20:48:33.937" Score="4" ViewCount="2636" Body="&lt;p&gt;Before circa 1970, it was more common to say &quot;kilocycles&quot; than &quot;kilohertz&quot; (and analogously for &quot;mega-,&quot; &quot;giga-,&quot; etc.). What contributed to &quot;Hertz&quot; becoming more common than &quot;cycles&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;cf. &lt;a href=&quot;https://books.google.com/ngrams/graph?content=megahertz%2Cmegacycles%2Ckilocycles%2Ckilohertz%2Cgigacycles%2Cgigahertz&amp;amp;case_insensitive=on&amp;amp;year_start=1920&amp;amp;year_end=2008&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t4%3B%2Cmegahertz%3B%2Cc0%3B%2Cs0%3B%3Bmegahertz%3B%2Cc0%3B%3BMegahertz%3B%2Cc0%3B%3BmegaHertz%3B%2Cc0%3B%3BMegaHertz%3B%2Cc0%3B%3BMEGAHERTZ%3B%2Cc0%3B.t4%3B%2Cmegacycles%3B%2Cc0%3B%2Cs0%3B%3Bmegacycles%3B%2Cc0%3B%3BMegacycles%3B%2Cc0%3B%3BMEGACYCLES%3B%2Cc0%3B.t4%3B%2Ckilocycles%3B%2Cc0%3B%2Cs0%3B%3Bkilocycles%3B%2Cc0%3B%3BKilocycles%3B%2Cc0%3B%3BKILOCYCLES%3B%2Cc0%3B.t4%3B%2Ckilohertz%3B%2Cc0%3B%2Cs0%3B%3Bkilohertz%3B%2Cc0%3B%3BKilohertz%3B%2Cc0%3B%3BkiloHertz%3B%2Cc0%3B%3BKILOHERTZ%3B%2Cc0%3B.t4%3B%2Cgigacycles%3B%2Cc0%3B%2Cs0%3B%3Bgigacycles%3B%2Cc0%3B%3BGigacycles%3B%2Cc0%3B.t4%3B%2Cgigahertz%3B%2Cc0%3B%2Cs0%3B%3Bgigahertz%3B%2Cc0%3B%3BGigahertz%3B%2Cc0%3B%3BGigaHertz%3B%2Cc0%3B%3BGIGAHERTZ%3B%2Cc0%3B%3BgigaHertz%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;the Google Ngram graph for &quot;megahertz,megacycles,kilocycles,kilohertz,gigacycles,gigahertz&quot;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2021-11-12T23:57:54.873" Title="&quot;Cycles&quot; to &quot;Hertz&quot;: Why the shift circa 1970?" Tags="|physics|units|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5553" PostTypeId="2" ParentId="5552" CreationDate="2017-01-07T21:28:59.997" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/General_Conference_on_Weights_and_Measures&quot; rel=&quot;nofollow noreferrer&quot;&gt;General Conference on Weights and Measures&lt;/a&gt; (1960) adopted the &lt;strong&gt;herz&lt;/strong&gt; unit.  So you have found that it took about 10 years for English-speaking books to catch up with that change.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-01-07T21:28:59.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5554" PostTypeId="2" ParentId="5552" CreationDate="2017-01-07T21:29:01.723" Score="4" Body="&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/International_System_of_Units&quot; rel=&quot;nofollow noreferrer&quot;&gt;SI units&lt;/a&gt; started to be used.&lt;/h3&gt;&#xA;&lt;p&gt;In 1960 (not 1970!), the &lt;em&gt;Système international d'unités&lt;/em&gt; was established at the &lt;a href=&quot;https://en.wikipedia.org/wiki/General_Conference_on_Weights_and_Measures&quot; rel=&quot;nofollow noreferrer&quot;&gt;General Conference on Weights and Measures&lt;/a&gt;. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Hertz&quot; rel=&quot;nofollow noreferrer&quot;&gt;hertz&lt;/a&gt; was officially recognized as a unit derived from &lt;a href=&quot;https://en.wikipedia.org/wiki/Metric_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;metric system&lt;/a&gt; units (namely, the second). It was already in partial use (the &lt;a href=&quot;https://en.wikipedia.org/wiki/International_Electrotechnical_Commission&quot; rel=&quot;nofollow noreferrer&quot;&gt;International Electrotechnical Commission&lt;/a&gt; began using it &lt;a href=&quot;http://www.iec.ch/about/history/overview/&quot; rel=&quot;nofollow noreferrer&quot;&gt;in 1930&lt;/a&gt;). With the rise of the hertz, the phrase &amp;quot;cycles&amp;quot; in &amp;quot;cycles per second&amp;quot; fell out of fashion, leading to the general trend we see on the Google Ngram.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly enough, &lt;a href=&quot;https://books.google.com/ngrams/graph?content=megahertz%2Cmegacycles%2Ckilocycles%2Ckilohertz%2Cgigacycles%2Cgigahertz%2Chertz&amp;amp;case_insensitive=on&amp;amp;year_start=1920&amp;amp;year_end=2008&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t4%3B%2Cmegahertz%3B%2Cc0%3B%2Cs0%3B%3Bmegahertz%3B%2Cc0%3B%3BMegahertz%3B%2Cc0%3B%3BmegaHertz%3B%2Cc0%3B%3BMegaHertz%3B%2Cc0%3B%3BMEGAHERTZ%3B%2Cc0%3B.t4%3B%2Cmegacycles%3B%2Cc0%3B%2Cs0%3B%3Bmegacycles%3B%2Cc0%3B%3BMegacycles%3B%2Cc0%3B%3BMEGACYCLES%3B%2Cc0%3B.t4%3B%2Ckilocycles%3B%2Cc0%3B%2Cs0%3B%3Bkilocycles%3B%2Cc0%3B%3BKilocycles%3B%2Cc0%3B%3BKILOCYCLES%3B%2Cc0%3B.t4%3B%2Ckilohertz%3B%2Cc0%3B%2Cs0%3B%3Bkilohertz%3B%2Cc0%3B%3BKilohertz%3B%2Cc0%3B%3BkiloHertz%3B%2Cc0%3B%3BKILOHERTZ%3B%2Cc0%3B.t4%3B%2Cgigacycles%3B%2Cc0%3B%2Cs0%3B%3Bgigacycles%3B%2Cc0%3B%3BGigacycles%3B%2Cc0%3B.t4%3B%2Cgigahertz%3B%2Cc0%3B%2Cs0%3B%3Bgigahertz%3B%2Cc0%3B%3BGigahertz%3B%2Cc0%3B%3BGigaHertz%3B%2Cc0%3B%3BGIGAHERTZ%3B%2Cc0%3B%3BgigaHertz%3B%2Cc0%3B.t4%3B%2Chertz%3B%2Cc0%3B%2Cs0%3B%3BHertz%3B%2Cc0%3B%3Bhertz%3B%2Cc0%3B%3BHERTZ%3B%2Cc0&quot; rel=&quot;nofollow noreferrer&quot;&gt;adding &amp;quot;hertz&amp;quot; to the Ngram search&lt;/a&gt; shows that its usage, too, took off a bit after 1960, though it was already fairly popular.&lt;/p&gt;&#xA;" OwnerUserId="62" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-01-07T21:29:01.723" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5555" PostTypeId="2" ParentId="5552" CreationDate="2017-01-07T21:42:04.867" Score="1" Body="&lt;p&gt;There was indeed some resistance to adopting the new name in the United States. In 1967, &lt;a href=&quot;http://www.americanradiohistory.com/Archive-Electronics-Illustrated/Electronics-Illustrated-1967-03.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rufus Cartwright wrote about it in the magazine ``Electronic Illustrated&quot;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is the short, illustrious life of Heinrich Hertz any reason to change the English language? A quick answer is no, especially since hertzian waves and antennas already honor him. But that may be the short view. Perhaps we must take the long view and see life as a complete hertz. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What might have helped in popularizing the new unit is that saying 'cycle' is potentially confusing when 'cycle per second' is really meant. &lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="232" LastEditDate="2017-01-07T22:30:31.553" LastActivityDate="2017-01-07T22:30:31.553" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5556" PostTypeId="1" AcceptedAnswerId="5566" CreationDate="2017-01-08T03:13:29.217" Score="2" ViewCount="282" Body="&lt;p&gt;I am reading the book by Berggren, 'Episodes in the mathematics of medieval islam'. An important problem is determing the direction of Mecca with respect to a local city. The book introduced a method invented by Al-Biruni. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is based on the assumption that both the coordinates (latitudes and longitudes) of Mecca and the locality are known. While it is relatively easy to determine the latitude of a point, I cannot see how to determine the difference in longitude between two points. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what is the ancient trick? &lt;/p&gt;&#xA;" OwnerUserId="4079" LastActivityDate="2017-05-31T19:05:47.143" Title="How did the Arabs determine the longitudes of cities?" Tags="|mathematics|astronomy|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5557" PostTypeId="1" CreationDate="2017-01-08T14:10:06.027" Score="0" ViewCount="35" Body="&lt;p&gt;In high school students are taught the formula that describes the universal gravitational force $F=G\frac{m_1 m_2}{r^2}$. However it is not taught how and why Newton came up with it. Does Newton give a demonstration of the formula?&lt;/p&gt;&#xA;" OwnerUserId="5211" LastActivityDate="2017-01-08T14:10:06.027" Title="How did Newton come up with the formula for gravity?" Tags="|gravity|" AnswerCount="0" CommentCount="0" FavoriteCount="0" ClosedDate="2017-01-08T15:59:03.143" ContentLicense="CC BY-SA 3.0" />
  <row Id="5558" PostTypeId="1" CreationDate="2017-01-08T14:05:22.817" Score="8" ViewCount="5908" Body="&lt;p&gt;In high school students are taught the formula that describes the universal gravitational force $F=G\frac{m_1 m_2}{r^2}$. However it is not taught how and why Newton came up with it. Does Newton give a demonstration of the formula?&lt;/p&gt;&#xA;" OwnerUserId="5211" OwnerDisplayName="Peterix" LastEditorUserId="1815" LastEditDate="2023-12-03T22:25:51.840" LastActivityDate="2023-12-03T22:25:51.840" Title="How did Newton come up with his formula for gravitational force?" Tags="|physics|classical-mechanics|gravity|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="5559" PostTypeId="2" ParentId="5558" CreationDate="2017-01-08T15:48:08.750" Score="3" Body="&lt;p&gt;As Newton said he stood on the shoulders of giants. One of those giants was Kepler who found that the periodicity of a planetary orbit was related by&#xA;$$&#xA;T^2~\propto~r^3.&#xA;$$&#xA;This is Kepler's third law. Newton realized with the second law $\vec F~=~m\vec a$ of motion that centripetal force is&#xA;$$&#xA;\vec F~=~m\omega^2\vec r.&#xA;$$&#xA;Newton hypothesized there was some universal force between all masses. This is the basis of the semi-mythic story of the apple falling. Newton then said that with the second law the centripetal force is the $m\vec a$ and this force must be of the form $\vec F~=~r^n\vec r$ and so&#xA;$$&#xA;m\omega^2\vec r~=~Kr^n\vec r.&#xA;$$&#xA;It is clear that $n~=~3$ by Kepler. Also Kepler's third law has not dependency on the mass of the orbiting satellite. Newton did realize that the magnitude of this force must scale with the mass of the primary body. Hence $K~=~GMm$.&lt;/p&gt;&#xA;" OwnerUserId="5212" LastActivityDate="2017-01-08T15:48:08.750" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5560" PostTypeId="2" ParentId="5551" CreationDate="2017-01-08T18:07:50.717" Score="1" Body="&lt;p&gt;Found it. The Construction of Modern Science by Richard Westfall.&lt;/p&gt;&#xA;" OwnerUserId="5206" LastActivityDate="2017-01-08T18:07:50.717" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5561" PostTypeId="2" ParentId="5558" CreationDate="2017-01-08T20:29:32.030" Score="8" Body="&lt;p&gt;The formula was a commonly discussed hypothesis at that time (Ch. Wren, Hooke, Halley). First attempt to test the formula was made when Newton was a young student in Cambridge: he compared acceleration from gravity on the Earth surface (easy to measure by observing falling apples for example:-) with acceleration of the Moon on&#xA;its orbit (easy to compute). And compared these accelerations with the Earth radius and the Moon orbit radius. Agreement was poor and Newton abandoned the topic.&#xA;(The reason for the poor agreement was the wrong value of the Earth radius which Newton was apparently taught in his university).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second attempt was triggered by a letter from Hooke, many years later. Hooke literally proposed to&#xA;derive Kepler's laws (mainly the first law) assuming this formula. When Newton did this, this looked a convincing proof to him. Next time, when Edmund Halley asked him the same question, Newton was able to show him a proof.&#xA;It is still discussed whether Newton's proof was really a proof of what Halley asked (that the inverse square law implies elliptic orbits),&#xA;or he proved only the converse statement: that the motion on ellipses implies the inverse square law. In any case, any of the two implications constitute a convincing argument in favor of the inverse square law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the time Newton was writing the book, the precise Earth radius was already known to him, so his early argument was justified as well.&#xA;Another early successful test of the inverse square law was the prediction of return of the Halley comet, made by Halley. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this was a strong evidence for the universal gravitation but more testing was desired. (Only some principal features of the planet motions were explained, &#xA;but one wanted to make sure that the law is exact, not approximate).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were two crucial tests in 18th century.&#xA; First the prediction of the shape of the Earth (tested in several expeditions in 18th century by precise measurement of&#xA;the arcs of meridians). Second, and most important was quantitative explanation of irregularities of the Moon motion (due to the disturbance arising from the Sun, Moon does not exactly obey Kepler's laws). Here Newton himself had only partial success, explaining the order of magnitude of the so-called &quot;first inequality&quot;. Hard work of the best mathematicians of 18th century (Euler, Clairault, Lagrange and several others) finally achieved a success with quantitative prediction of the Moon motion. This was the decisive step in the proof of the Universal law of gravitation. Since then it was universally believed that this law is sufficient for explanation all observable features of the motion of planets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More testing was performed in 19th century, the most famous one was the prediction of existence of Neptune and calculation of its orbit before it was observed. After that nobody had any doubts anymore.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-01-08T20:59:01.197" LastActivityDate="2017-01-08T20:59:01.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5562" PostTypeId="1" CreationDate="2017-01-09T03:57:07.137" Score="1" ViewCount="109" Body="&lt;p&gt;I'm interested in examining the efficiency of the scientific process. Part of this involves examining what actually goes into making a discovery. One common objection is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Role_of_chance_in_scientific_discoverie&quot; rel=&quot;nofollow noreferrer&quot;&gt;Half of all important scientific findings were chance discoveries&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The implication is that these things can't be analysed and we should just throw money and minds into a pot and let them do science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a suspicion that due to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Availability_heuristic&quot; rel=&quot;nofollow noreferrer&quot;&gt;availability bias&lt;/a&gt; we tend to discuss surprise discoveries more and therefore overestimate their importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone published data on how frequently unexpected (chance) discoveries actually occur? Ideally, it would include a weighting on how important the discovery was.&lt;/p&gt;&#xA;" OwnerUserId="5219" LastActivityDate="2017-01-09T03:57:07.137" Title="How important were serendipitous scientific discoveries (objectively)?" Tags="|discoveries|philosophy-of-science|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5563" PostTypeId="1" AcceptedAnswerId="5564" CreationDate="2017-01-09T09:32:29.670" Score="9" ViewCount="6689" Body="&lt;p&gt;Who introduced the concept of eigenvalues and eigenvectors and where does the name come from? Is there a connection with the German word &quot;eigen&quot;?&lt;/p&gt;&#xA;" OwnerUserId="4096" LastActivityDate="2017-01-09T10:21:16.373" Title="Where does the name eigenvalue come from?" Tags="|mathematics|linear-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5564" PostTypeId="2" ParentId="5563" CreationDate="2017-01-09T10:08:01.390" Score="12" Body="&lt;p&gt;Exactly; see &lt;a href=&quot;https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Overview&quot;&gt;Eigenvalues&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The prefix &lt;a href=&quot;https://en.wiktionary.org/wiki/eigen-&quot;&gt;eigen-&lt;/a&gt; is adopted from the German word eigen for &quot;proper&quot;, &quot;inherent&quot;; &quot;own&quot;, &quot;individual&quot;, &quot;special&quot;; &quot;specific&quot;, &quot;peculiar&quot;, or &quot;characteristic&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It was David Hilbert who introduced the terms &lt;em&gt;Eigenwert&lt;/em&gt; and &lt;em&gt;Eigenfunktion&lt;/em&gt;; see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;David Hilbert, &lt;a href=&quot;http://quod.lib.umich.edu/u/umhistmath/acu8916.0003.001/10?page=root;rgn=full+text;size=100;view=pdf;q1=eigenwert&quot;&gt;Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen&lt;/a&gt; (1904); see e.g. page X for &lt;em&gt;Eigenwert&lt;/em&gt; and &lt;em&gt;Eigenfunktion&lt;/em&gt; [see &lt;a href=&quot;http://www.umiacs.umd.edu/~stewart/FHS.pdf&quot;&gt;here&lt;/a&gt; for the English translation].&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-01-09T10:21:16.373" LastActivityDate="2017-01-09T10:21:16.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5565" PostTypeId="1" CreationDate="2017-01-09T14:10:20.093" Score="12" ViewCount="12783" Body="&lt;p&gt;The ancient Greeks knew this fact. How about the chinese? If not, when did they realize that the earth is a sphere? By themselves or from other people? &lt;/p&gt;&#xA;" OwnerUserId="4079" LastActivityDate="2018-03-28T03:46:59.540" Title="Did the ancient Chinese know the earth is a sphere?" Tags="|astronomy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5566" PostTypeId="2" ParentId="5556" CreationDate="2017-01-09T18:21:54.317" Score="7" Body="&lt;p&gt;Terrestrial longitudes were derived from the astronomical longitudes given in the &quot;Hakimite Tables&quot; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ibn_Yunus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ibn Yunus&lt;/a&gt; (d. 1009).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially, during a lunar eclipse (predicted by the astronomical tables) two observers located some distance apart make measurements of when they first observe the shadow of the earth entering the disk of the moon.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source - &lt;a href=&quot;http://articles.adsabs.harvard.edu//full/1931PA.....39..527R/0000527.000.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harvard site article&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the determination of the difference in longitude between two places [on earth] from an eclipse of the moon it is necessary that for the measurement one be equipped with two very accurate instruments which have been tested by two men of learning and experience.... It is also necessary that the two know accurately the place where the earth's shadow enters upon the edge of the moon's disk in order that the gaze may be directed unflinchingly at the place of contact,...  And in regard to that which lies between the two moments of this contact and the moment when this eclipse becomes evident, certain of the earlier astronomers have said that in the interval between these two moments $1^{\circ} 49'$ of the equator turns and this makes $7$ minutes $16$ seconds of equivalent time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="12699" LastEditDate="2017-05-31T19:05:47.143" LastActivityDate="2017-05-31T19:05:47.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5567" PostTypeId="1" AcceptedAnswerId="5571" CreationDate="2017-01-09T20:36:45.003" Score="4" ViewCount="136" Body="&lt;p&gt;For my work on the history of hypertext systems and the WWW I cam across one sentence on the development of HTTP which I don't understand and where I could not find any information about:&lt;br&gt;&#xA;Page 55: (bold marking from me)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A new search and retrieve (SR) protocol, known as HTTP, was found to&#xA;  be necessary. Faster than FTP for document retrieval, &lt;strong&gt;HTTP also allows&#xA;  index search&lt;/strong&gt;. HTTP is similar in implementation to the Internet&#xA;  protocols above and similar in functionality to the WAIS protocol.&#xA;  Some differences are discussed below.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The paper can be accessed here for free:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.emeraldgrouppublishing.com/products/backfiles/pdf/backfiles_sample_5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.emeraldgrouppublishing.com/products/backfiles/pdf/backfiles_sample_5.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5222" LastActivityDate="2017-01-11T16:57:39.663" Title="What did Tim-Berners Lee mean with &quot;HTTP also allows index search&quot; in his paper &quot;World wide web: The Information Univesrse&quot;?" Tags="|computer-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5568" PostTypeId="2" ParentId="5565" CreationDate="2017-01-10T01:51:50.080" Score="12" Body="&lt;p&gt;There appear to be conflicting accounts of when the Chinese came to view the earth as spherical.  Chinese sources &lt;em&gt;might suggest&lt;/em&gt; the 11th century, while western sources suggest the 17th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Christopher_Cullen&quot;&gt;sinologist Christopher Cullen&lt;/a&gt;, the Chinese believed the earth to be flat and square while the heavens were round until the introduction of European astronomy in the 17th century. Quoting Cullen's 1976 text  &quot;A Chinese Eratosthenes of the Flat Earth: A Study of a Fragment of Cosmology in Huai Nan tzu 淮 南 子&quot; : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chinese thought on the form of the earth remained almost unchanged from early times until the first contacts with modern science through the medium of Jesuit missionaries in the seventeenth century. While the heavens were variously described as being like an umbrella covering the earth (the Kai Tian theory), or like a sphere surrounding it (the Hun Tian theory), or as being without substance while the heavenly bodies float freely (the Hsüan yeh theory), the earth was at all times flat, although perhaps bulging up slightly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The model of the egg was used by Chinese astronomers to describe the heavens : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The heavens are like a hen's egg and as round as a crossbow bullet; the earth is like the yolk of the egg, and lies in the centre.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Further details are provided at &lt;a href=&quot;https://en.wikipedia.org/wiki/Flat_Earth#Ancient_China&quot;&gt;wikipedia article on Flat Earth&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Needham&quot;&gt;Joseph Needham&lt;/a&gt;, in his more recent 1993 text &lt;em&gt;Chinese Cosmology&lt;/em&gt; reports that &lt;a href=&quot;https://en.wikipedia.org/wiki/Shen_Kuo&quot;&gt;Shen Kuo&lt;/a&gt; (1031-1095) used models of lunar eclipse and solar eclipse to conclude that &quot;celestial bodies&quot; are round, not flat.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If they were like balls they would surely obstruct each other when they met. I replied that these celestial bodies were certainly like balls. How do we know this? By the waxing and waning of the moon. The moon itself gives forth no light, but is like a ball of silver; the light is the light of the sun (reflected). When the brightness is first seen, the sun (-light passes almost) alongside, so the side only is illuminated and looks like a crescent. When the sun gradually gets further away, the light shines slanting, and the moon is full, round like a bullet. If half of a sphere is covered with (white) powder and looked at from the side, the covered part will look like a crescent; if looked at from the front, it will appear round. Thus we know that the celestial bodies are spherical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What is not clear here is whether or not Shen Kuo considered earth to be a &quot;celestial body&quot;.  If not, then perhaps Cullen's claim is the current view.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-01-10T02:10:47.747" LastActivityDate="2017-01-10T02:10:47.747" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5569" PostTypeId="1" AcceptedAnswerId="5570" CreationDate="2017-01-10T16:48:30.627" Score="6" ViewCount="731" Body="&lt;p&gt;While reading Chandrashrkhar's edition of Principia , I came to know that &lt;em&gt;Newton's Method of Divided Differences can be used to prove Taylor's Theorem. Could some one help me in knowing how this is possible.&lt;/em&gt; I found this &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S037704279900360X&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.sciencedirect.com/science/article/pii/S037704279900360X&lt;/a&gt; but it didn't help .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note- I want this&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Could someone give a reference to a book or a link where I can study the method of divided differences in sense in which Newton developed it&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="4666" LastEditorUserId="4666" LastEditDate="2017-05-03T11:44:03.277" LastActivityDate="2017-05-03T11:44:03.277" Title="Taylor's Theorem and Newton's Method of Divided Differences" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5570" PostTypeId="2" ParentId="5569" CreationDate="2017-01-11T01:34:33.857" Score="7" Body="&lt;p&gt;Not only can it be done, it is how Taylor did it: &lt;a href=&quot;https://books.google.com/books?id=r-Gq9YyZYXYC&amp;amp;pg=PA21&quot; rel=&quot;noreferrer&quot;&gt;B. Taylor, &lt;em&gt;Methodus incrementorum directa et inversa&lt;/em&gt; (1715), 21--23,&#xA; Prop. VII Theor. III &amp;amp; Coroll. II.&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taylor developed Taylor series through uniformly spaced nodes at $t_j=t_0+j\,\Delta t$, interpolating $x=x(t)$ between $t_0$ (Taylor uses $z$ equally for $t$ and $t_0$) and $t_0+v$, for some increment $v$ in $n$ steps. From Newton's interpolation formula, Taylor derived the formula (Prop. VII)&#xA;$$&#xA;\sum_{k=0}^n {(t-t_0) \strut\over\strut 1}{(t-t_0-\Delta t) \strut\over\strut 2}&#xA;  {(t-t_0-2\Delta t) \strut\over\strut 3}\cdots{(t-t_0-(k-1)\Delta t) \strut\over\strut k}&#xA;  {\Delta^k x \over \Delta t^k}\,.&#xA;$$&#xA;He wrote the proposition in terms of forward differences, but it is, of course, equivalent to divided differences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Letting $\Delta t \rightarrow 0$ (&lt;em&gt;Si pro Incrementis evanescentibus scribantur fluxiones...&lt;/em&gt;), &#xA;Taylor obtained that ${\Delta^k y / \Delta t^k} \rightarrow {d^ky /dt^k}$ and the other factors approach $(t-t_0)^k/k!$, which yields the Taylor series, since $n$ is determined by $v$ and $\Delta t$.&#xA;Klein called it a &quot;a limit-taking of unheard-of boldness&quot;&#xA;(&lt;em&gt;&quot;Hierin liegt nun tatsächlich Grenzübergang von unerhörter Kühnheit&quot;&lt;/em&gt;), &#xA;&lt;em&gt;Elementarmathematik vom höheren Standpunkte aus: Arithmetik, Algebra, Analysis&lt;/em&gt; (1908).&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2017-01-11T17:33:03.237" LastActivityDate="2017-01-11T17:33:03.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5571" PostTypeId="2" ParentId="5567" CreationDate="2017-01-11T16:57:39.663" Score="4" Body="&lt;p&gt;It's probably a reference to the &lt;code&gt;&amp;lt;ISINDEX&amp;gt;&lt;/code&gt; HTML element, which was like a &lt;code&gt;&amp;lt;FORM&amp;gt;&lt;/code&gt; with a single input element, used for constructing queries like &lt;code&gt;http://foo/bar?baz&lt;/code&gt; from the page &lt;code&gt;http://foo/bar&lt;/code&gt; when &lt;code&gt;baz&lt;/code&gt; was entered by the user. The purpose was for the page containing the &lt;code&gt;&amp;lt;ISINDEX&amp;gt;&lt;/code&gt; element to serve as a &quot;search engine&quot; for some collection of documents that could be returned in response to the queries with parameter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An explicit reference to search engines of the &quot;search the contents of everything everywhere&quot; type - and not as an idea but an implemented feature! - at this stage seems much less likely to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MDN has some historical links related to &lt;code&gt;&amp;lt;ISINDEX&amp;gt;&lt;/code&gt; that may be of interest: &lt;a href=&quot;https://developer.mozilla.org/en-US/docs/Web/HTML/Element/isindex&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://developer.mozilla.org/en-US/docs/Web/HTML/Element/isindex&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user5230" LastActivityDate="2017-01-11T16:57:39.663" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5572" PostTypeId="1" AcceptedAnswerId="5581" CreationDate="2017-01-12T15:30:28.563" Score="1" ViewCount="89" Body="&lt;p&gt;Some authors talk about &lt;em&gt;Gamma Sagittarii&lt;/em&gt; as a single star, while others use &lt;em&gt;Gamma1 Sgr&lt;/em&gt; for W Sgr and &lt;em&gt;Gamma2 Sgr&lt;/em&gt; for 10 Sgr (Gamma Sgr being then ambiguous). SIMBAD database uses Gamma1/Gamma2 as of 2017, for example, while Richard Hinckley Allen only talks about Gamma (1899).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the original Bayer designation &lt;a href=&quot;http://stars.astro.illinois.edu/sow/sgr-b.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;did not include superscript&lt;/a&gt;, and was referred only to what is today known as Gamma2 Sgr.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some authors did introduce superscripts to some double stars; some stayed (as for example Beta1 Sgr and Beta2 Sgr), while others did not (such as Gamma2 UMi, or Pherkad, and Gamma1 UMi, mostly known as 11 UMi; SIMBAD displays only Gamma UMi for Pherkad).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did the change for the case of Gamma Sgr happen? Was there a specific event when this was standardised? (Such as when Eugène Delporte published the &lt;em&gt;Atlas Céleste&lt;/em&gt; in 1930, accepted by the IAU to delimit the constellations, and hence getting rid of the Delta Pegasi designation in favour of Alpha Andromedae).&lt;/p&gt;&#xA;" OwnerUserId="2115" LastActivityDate="2017-01-15T21:03:02.160" Title="When did the Bayer designation Gamma Sagittarii become Gamma1 &amp; Gamma2 Sagittarii?" Tags="|astronomy|naming-conventions|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5573" PostTypeId="1" AcceptedAnswerId="5586" CreationDate="2017-01-13T00:37:48.187" Score="4" ViewCount="232" Body="&lt;p&gt;Does anyone know what the inscriptions, written presumably in Latin, are on this 3-D permutahedron sundial by Stefano Buonsignori (16th century) in the Medici collection presented by &lt;a href=&quot;http://brunelleschi.imss.fi.it/museum/esim.asp?c=403055&quot; rel=&quot;nofollow noreferrer&quot;&gt;Museo Galileo&lt;/a&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/bn2Z7.jpg&quot; alt=&quot;Sundial&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Added Mar. 9, 2024:&lt;/p&gt;&#xA;&lt;p&gt;The refined Euler characteristic polynomials / refined, signed, face polynomials of the permutahedra / permutohedra are the coefficients of the multiplicative inverse, or reciprocal, of a Taylor series with constant term unity (see &lt;a href=&quot;https://oeis.org/A133314&quot; rel=&quot;nofollow noreferrer&quot;&gt;OEIS A133314&lt;/a&gt;).&#xA;The 3-dimensional permutohedron is one of the 13 Archimedean solids--&lt;a href=&quot;https://en.wikipedia.org/wiki/Truncated_octahedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;the truncated octahedron&lt;/a&gt;. Here's some history on illustrations of these solids from a Wikipedia post on the truncated icosahedron:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The truncated icosahedron was known to Archimedes, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from Pappus of Alexandria, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by Piero della Francesca, in his 15th-century book De quinque corporibus regularibus, which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by Leonardo da Vinci, in his illustrations for Luca Pacioli's plagiarism of della Francesca's book in 1509. Although Albrecht Dürer omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, Underweysung der Messung, a description of it was found in his posthumous papers, published in 1538. Johannes Kepler later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, Harmonices Mundi.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2024-03-09T17:49:53.070" LastActivityDate="2024-03-09T17:49:53.070" Title="Inscriptions on a 16th century 3-dimensional permutahedron sundial?" Tags="|mathematics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5576" PostTypeId="1" CreationDate="2017-01-13T22:39:29.607" Score="2" ViewCount="152" Body="&lt;p&gt;My question refers to Gottfried Leibniz's &quot;instrumentum longuitidimum&quot; mentioned in some sites — a device that could record the course of a ship in sea and display its recent position directly on a map. I ask this because this statement inspired me to invent a mechanical device that is able to do so, and i'm very curius if this device was known to Leibniz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The device which came to my mind is based on naval odometer with two gears as its output. One gear is attached to and rotates a cylinder which represents a map of earth, and the other is connected through a complicated mechanism to a compass and a screw. The compass is attached to a kind of leibniz cylinder which is itself attached to a kind of screw which rotates and moved verticaly as a result of its rotation. The detailed structure of the device is hard to tell, but it functions as a kind of &quot;seismograph&quot; — when the compass needle is perpendicular to the ship body (i.e its lattitude remains unchanged), only the map cylinder rotates, otherwise both the map cylinder and the screw rotate. In this way when the ship moves north or south the route recorded on the cylinder map goes up and down, correspondly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the point here wasn't only to get compliment for the device - but to know if it's already known. So is there already such mechanical device? was this device known to Leibniz?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-05-13T12:21:35.773" LastActivityDate="2020-05-13T12:21:35.773" Title="What navigation instrument did Leibniz develop?" Tags="|biographical-details|leibniz|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5577" PostTypeId="2" ParentId="5573" CreationDate="2017-01-13T22:44:19.533" Score="3" Body="&lt;p&gt;From the &lt;a href=&quot;http://www.museumsinflorence.com/musei/History_of_Science_museum.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Museums in Florence site History of Science page&lt;/a&gt;, referring to a pair of panels comprising the Horary Quadrant (see below for description) of the Buonsignori sundial:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The front of this finely decorated quadrant carries (1) a brass disk with markings for days, months, and zodiac signs, and (2) a moon dial. There is a magnetic compass for orientation. On the back of the instrument are a shadow square, a degree scale, and a sundial with unequal hour lines set for latitude 43°30' (Florence). Made by Stefano Buonsignori, as &#xA;  indicated by the initials - D.STEP. B.F.F.- [Stefano Buonsignori Florentinus Fecit]. Provenance: Medici collections.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Presumably the mounted compass allows the user to correctly orient the sundial and thus read the applicable gnomon (the brass piece protruding from each panel used to cast a shadow).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, we are using the term &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadrant_(instrument)&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadrant&lt;/a&gt; to refer to an instrument used to measure angles up to ninety degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The four different quadrants are :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The sine quadrant, used for solving trigonometric problems and taking astronomical observations.&lt;/li&gt;&#xA;&lt;li&gt;The universal quadrant, used for solving astronomical problems for any latitude.&lt;/li&gt;&#xA;&lt;li&gt;The horary quadrant, used for finding the time with the sun.&lt;/li&gt;&#xA;&lt;li&gt;The astrolabe quandrant, a quadrant developed from the astrolabe used to represent the position of celestial bodies.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-01-13T23:03:16.203" LastActivityDate="2017-01-13T23:03:16.203" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5578" PostTypeId="2" ParentId="5576" CreationDate="2017-01-13T23:17:38.370" Score="1" Body="&lt;p&gt;It is an interesting question what Leibniz proposed, but I suppose his proposal was not practical: it is not mentioned in the histories of navigation, which probably means that it was never implemented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your proposal is also not practical for many reasons. Determination of ship's position by tracking the speed and direction was a principal method of navigation&#xA;even when astronomical methods became available. It has a strange name in English: &quot;dead reckoning&quot;.&#xA; It is very simply done by hand;&#xA;you just plot the course and distance traveled on a map and obtain your position.&#xA;The speed was measured with a log, as frequently as necessary, or in Leibniz times,&#xA;it was just estimated by eye. &#xA;The time was measured by hourglass and the course by magnetic compass corrected for deviation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This method is not very accurate because it does not take into account currents,&#xA;and drift, and because the usual ship logs do not measure the speed very accurately.&#xA;(There is no such thing as &quot;ship odometer&quot;. Odometer is used on wheeled vehicles). &#xA;Mechanization of the procedure which you propose requires a complicated mechanical device which will only introduce additional errors and have no real advantages. A person has to stand on watch all the time anyway, and it is not&#xA;difficult for this person to record the speed and course. Computation and plotting is a trivial procedure. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-01-13T23:17:38.370" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5579" PostTypeId="1" CreationDate="2017-01-14T20:08:50.720" Score="11" ViewCount="1477" Body="&lt;p&gt;Let me briefly recall the &amp;quot;Age of the Earth controversy&amp;quot;. Based on the hypothesis that Earth was originally hot and cools down ever since, and using the data on the gradient of temperature currently observed near the surface (about 1 degree per 100 meters), Willian Thomson (Lord Kelvin) derived that it must take about 100 million years to cool to the present state, starting from the melting temperature of rocks). Later he revised his estimate to a smaller number.&lt;/p&gt;&#xA;&lt;p&gt;This theory, though completely mathematically sound gives the wrong answer because it does not take into account the heat source inside (radioactivity, which was not known at that time). Mathematics of Kelvin's theory is completely clear. But my question is different:&lt;/p&gt;&#xA;&lt;p&gt;I know from many popular science histories that Kelvin's estimate was hotly disputed by geologists and evolutionists, who all thought that Earth must be much older &amp;quot;to allow sufficient time&amp;quot; for geological changes and for evolution.&lt;/p&gt;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How exactly did they estimate the time necessary for geological changes and evolution? How can you arrive with ANY numerical estimate of this sort?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For example Darwin himself presumably had estimated the time necessary for evolution as&#xA;300 million years. But he removed this computation from the third edition of his book (Origin of species). As I only can find later editions, I cannot see what arguments he used to obtain his estimate.&lt;/p&gt;&#xA;&lt;p&gt;There were many other such estimates,&#xA;for example, Samuel Haughton, a geologist, came with the lower estimate of&#xA;2300 million years. What sort of arguments could all these scientists&#xA;(I mean geologists and evolutionists) use to obtain their numerical estimates from below.&lt;/p&gt;&#xA;&lt;p&gt;Added later: I sort of understand the arguments of geologists referred in the comments. But how 19 century evolutionists estimated the rate of evolution is still completely unclear to me.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-02-04T02:07:42.343" LastActivityDate="2022-02-04T02:07:42.343" Title="How did 19th century scientists estimate the age of the Earth" Tags="|evolution|geology|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="5580" PostTypeId="1" AcceptedAnswerId="5588" CreationDate="2017-01-15T14:43:53.533" Score="5" ViewCount="187" Body="&lt;p&gt;&lt;a href=&quot;https://www.brainpickings.org/2014/11/13/van-gogh-starry-night-fluid-dynamics-animation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; I found that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sixty years later, Russian mathematician Andrey Kolmogorov furthered&#xA;our mathematical understanding of turbulence when he proposed that&#xA;energy in a turbulent fluid at length &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; varies in proportion to the&#xA;five-thirds power of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Was he the first one to describe the turbulence rigorously? I don't define the term &amp;quot;rigorously&amp;quot; on purpose, because I'd like to hear different views.&lt;/p&gt;&#xA;" OwnerUserId="5243" LastEditorUserId="4251" LastEditDate="2020-07-06T07:41:01.230" LastActivityDate="2020-07-06T07:41:01.230" Title="Who was the first person to describe turbulence in mathematical terms?" Tags="|physics|mathematical-physics|fluid-dynamics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5581" PostTypeId="2" ParentId="5572" CreationDate="2017-01-15T21:03:02.160" Score="1" Body="&lt;p&gt;I will self-answer my question: it seems the event I was wondering about (in which Gamma Sgr became officially Gamma2 Sgr) was the 20th IAU General Assembly in 1988, Baltimore (Maryland, United States). It was decided there that Bayer and Flamsteed designations accepted would be those found in the 4th edition of the &lt;em&gt;The Bright Star Catalogue&lt;/em&gt; (1982). By the way, this catalogue is the same introducing the HR (for &quot;Harvard Revised&quot;) codes for stars. Thanks to Eric Mamajek for his email clarification on this.&lt;/p&gt;&#xA;" OwnerUserId="2115" LastActivityDate="2017-01-15T21:03:02.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5582" PostTypeId="1" AcceptedAnswerId="5590" CreationDate="2017-01-16T00:08:34.400" Score="5" ViewCount="383" Body="&lt;p&gt;I have read that a lost book by Fibonacci (a commentary on Book X of Euclid's &lt;em&gt;Elements&lt;/em&gt;) gives a numerical treatment of incommensurable magnitudes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that Fibonacci grew up in North Africa and travelled extensively throughout the Middle East where he learned the Hindu-Arabic number system and became familiar with the contemporary developments in mathematics, it seems reasonable to wonder if he may have encountered a numerical treatment of incommensurable magnitudes during these travels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; Is there any record of Arab or Persian mathematicians developing a numerical treatment of incommensurables, or is Fibonacci’s lost commentary the first known treatment? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(N.B. A number of questions have already asked here regarding the Greek's discovery of incommensurable magintudes, but none appear to address this question.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Fibonacci.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor page on Fibonacci&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;His book on commercial arithmetic &lt;em&gt;Di minor guisa&lt;/em&gt; is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We know that the fourth section of &lt;em&gt;Liber Abaci&lt;/em&gt; has Fibonacci giving numerical approximations of surds, thereby removing them from geometry.  Thus, the missing text must go further than simple numerical approximations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, in the 1225 work, &lt;em&gt;Flos&lt;/em&gt;, Fibonacci details his solutions to the three questions posed to him by Johannes of Palermo during a meeting at the court of Fredrick II.  In the second of these problems he finds the only real root of the equation $x^3 + 2x^2 + 10x = 20$.  His argument establishes that such a root cannot be integral or rational, and that it cannot take any of the forms from Book X of &lt;em&gt;Elements&lt;/em&gt;. As such, this represented a new type of irrational number - one not capable of construction by straight edge and compass.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then, without justification, Fibonacci states that root is approximately &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$1; 22, 07, 42, 33, 04, 40$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in sexidecimal notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these know facts suggest that the missing commentary of Book X represented a significant advance in our understanding of irrational numbers.  I was hoping that any existing Middle Eastern sources might give a hint as to what this treatment may have included.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2017-01-21T21:39:00.020" LastActivityDate="2019-12-28T17:45:03.147" Title="Were ratios of incommensurable magnitudes interpreted as irrational numbers prior to Fibonacci?" Tags="|mathematics|mathematicians|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5583" PostTypeId="1" CreationDate="2016-12-10T08:53:47.270" Score="4" ViewCount="86" Body="&lt;p&gt;When was the fact that a spin could only have values $S = n/2$? I cannot find any source that says when.&lt;/p&gt;&#xA;" OwnerUserId="5073" OwnerDisplayName="The Bosco" LastActivityDate="2017-01-20T10:16:23.587" Title="When was the quantization of spin discovered?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5584" PostTypeId="2" ParentId="5583" CreationDate="2017-01-16T00:52:29.437" Score="2" Body="&lt;p&gt;The experiment of Stern and Gerlach showed that the spin of atomic silver was 1/2.  This experiment was &quot;simplified&quot; by Phipps and Taylor to hydrogen atoms so as to eliminate any possible ambiguities on the spin-1/2 nature of the electron.&lt;/p&gt;&#xA;" OwnerUserId="7093" OwnerDisplayName="user6552" LastActivityDate="2017-01-16T00:52:29.437" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5585" PostTypeId="2" ParentId="5582" CreationDate="2017-01-16T08:10:34.810" Score="3" Body="&lt;p&gt;It is highly unlikely that Fibonacci could give a rigorous treatment of incommensurables. Decimal fractions for representation of what we call now real numbers were introduced by John Napier in the end of 16th century, but the rigorous theory of incommensurable magnitudes (on the same level of rigor as the Greek treatment of them) had to wait until Dedekind and Cantor (second half of 19th century).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the answer to your question &quot;Who was to replace&quot; is this: Napier for practical purposes, and Dedekind/Cantor rigorously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Of course it was understood since the times immemorial that incommesurable quantities can be APPROXIMATED by rationals. But approximate is not the same as represent. As I understand, nobody was talking about infinite (decimal) expansions before Napier, which indeed represent the irrational numbers.&#xA;So there was no novelty when Fibonacci approximated his root with a sexagesimal fraction. Archimedes approximated pi with an ordinary fraction, and perfectly understood that such approximation can be obtained with any precision.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-01-17T08:59:11.590" LastActivityDate="2017-01-17T08:59:11.590" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5586" PostTypeId="2" ParentId="5573" CreationDate="2017-01-16T10:52:12.727" Score="3" Body="&lt;p&gt;The following is an e-mail kindly sent to me by Giorgio Strano, a curator at the museum:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides the usual &quot;Tropicus Cancri&quot;, &quot;AEquator&quot;, etc. the sundial contains the following sentences:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Ad Latitudinem Graduum Quadragintatrium Minutorum Triginta&quot; (For the latitude of 43° 30')&#xA;&quot;Sol idem semper, [nec non (?)] semper currit et ue[rtit ('?)]&quot; (The sun is always the same, [however ?] it always runs and [turns around ?])&#xA;&quot;Non sic antiq[ui]s notus, nunc tempora monstro&quot; (I show the time of today, not as it was known to the ancients [i.e. the dial shows equal, not unequal hours])&#xA;&quot;Surgentem ostendo solem, ostendoq[ue] cadentem&quot; (I show the rising and the setting Sun)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[ (?)] indates plausible integrations of a sentence where the words in the sundial are scratched away&#xA;[ ] indicates resolved Latin abbreviations&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this information is what you were looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Best regards,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Giorgio Strano&#xA;Curator &lt;/p&gt;&#xA;" OwnerUserId="1830" LastActivityDate="2017-01-16T10:52:12.727" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5587" PostTypeId="2" ParentId="5573" CreationDate="2017-01-16T18:56:15.100" Score="2" Body="&lt;p&gt;From &lt;a href=&quot;https://www.mhs.ox.ac.uk/epact/catalogue.php?ENumber=29766&amp;amp;Level=Detail&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mhs.ox.ac.uk/epact/catalogue.php?ENumber=29766&amp;amp;Level=Detail&lt;/a&gt;, which has &lt;a href=&quot;https://www.mhs.ox.ac.uk/epact/picturel.php?ENumber=29766&quot; rel=&quot;nofollow noreferrer&quot;&gt;an image showing two of the inscriptions&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The dials are drawn for Italian hours with pin gnomons for latitude 43? 30' as is stated in an inscription on the north inclining face 'Ad latitudinem Graduum Quadragintarium Minutorum Triginta'....Three of the four reclining dials carry a motto in an horizontal band at the top of the dial. These are, 'Sol idem semper n{o}s [?] semper currit e [illegibile]'; 'Non sic antiqs notus numc tempora monstro'; 'Surgentem ostendo solem, ostendoq cad tez.' (Anthony J. Turner)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2017-01-16T18:56:15.100" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5588" PostTypeId="2" ParentId="5580" CreationDate="2017-01-16T21:52:23.900" Score="7" Body="&lt;p&gt;Systematic study of turbulence originates with a series of experiments conducted by Osborne Reynolds starting in 1870s. His mathematical theory was developed in &lt;a href=&quot;http://phys.lsu.edu/tohline/IGERT/XDI/Notes.MT.day05.pdf&quot;&gt;On the dynamical theory of incompressible viscous fluids and the determination of the criterion (1895)&lt;/a&gt;. However, already in 1883 he described a classical experiment with a jet of dyed water at the center of a flow in a glass pipe that demonstrated the transition to turbulence. He also introduced what is now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reynolds_number&quot;&gt;Reynolds number&lt;/a&gt; as the parameter that governs the &lt;a href=&quot;https://en.wikipedia.org/wiki/Laminar-turbulent_transition&quot;&gt;transition from laminar to turbulent flows&lt;/a&gt;. Between the values of 2000 and 13000, depending on entry conditions (if extreme care was taken the threshold could even be increased to 40000), the dyed layer which remained distinct at smaller values, broke up and spread out across the entire cross-section of the pipe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some excerpts from the 1895 paper describing the relation of Reynolds's work to Stokes's before him. The &quot;assumption&quot; referred to below is the assumption shown by St. Venant and Stokes in 1845 to underlie the Navier-Stokes equations: that  the stresses, other than pressure, which is uniform in all directions, are linear functions of the rates of distortion, with a coefficient depending on the fluid:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;By obtaining a singular solution of these equations as applied to the cases of&#xA;  pendulums in steady periodic motion, Sir G. Stokes was able to compare the&#xA;  theoretical results with the numerous experiments that had been recorded... these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;em&gt;These experimental results&lt;/em&gt; [Reynolds's from 1883] &lt;em&gt;completely removed the discrepancy previously noticed, showing that, whatever may be the cause, in those cases in which the experimental results do not accord with those obtained by the singular solution of the equations, the actual motions of the water are different. But in this there is only a partial explanation, for there remains the mechanical or physical significance of the existence of the criterion to be explained.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;My object in this paper is to show that the theoretical existence of an inferior limit to the criterion follows from the equations of motion as a consequence: --&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(1) &lt;em&gt;Of a more rigorous examination and definition of the geometrical basis on&#xA;  which the analytical method of distinguishing between molar-motions and heat motions in the kinetic theory of matter is founded ; and&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(2) &lt;em&gt;Of the application of the same method of analysis, thus definitely founded, to distinguish between mean-molar-motions and relative-molar-motions...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-01-17T01:58:22.200" LastActivityDate="2017-01-17T01:58:22.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5589" PostTypeId="2" ParentId="5576" CreationDate="2017-01-16T22:43:22.063" Score="2" Body="&lt;p&gt;Leibniz bragged as early as 1671 about solving &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_longitude#Problem_of_longitude&quot; rel=&quot;nofollow noreferrer&quot;&gt;the problem of longitude&lt;/a&gt;, among a long list of other major problems, with the help of his &quot;combinatory art&quot;  At the time he was euphoric about the sweeping capabilities of his &quot;characteristica universalis&quot; to solve any intellectual problems. Unfortunately, his letter does not give many details:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I will demonstrate how to find longitudes completely, and provide a way for a person on a ship to know what his location is without the help of the sun, moon and stars which can not always be observed (yet Huygens's famous invention depends on them alone)&lt;/em&gt;&quot;. [quoted from &lt;a href=&quot;https://books.google.com/books?id=GmNDBAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz and the Kabbalah by Coudert&lt;/a&gt;, p. 154]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He published one solution in the Journal des Scavans in 1675, it was based on a clockwork with a new escapement mechanism. It is described, and pictured, in &lt;a href=&quot;https://books.google.com/books?id=JrcqBgAAQBAJ&amp;amp;pg=PA170&amp;amp;lpg=PA170&amp;amp;dq=leibniz+problem+of+longitude&amp;amp;source=bl&amp;amp;ots=YkhMu37ac5&amp;amp;sig=fKYi4HR_MSowapk84nezvmF9DLU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiTt7PwxMfRAhVB1mMKHUbZBHgQ6AEIHTAB#v=onepage&amp;amp;q=leibniz%20problem%20of%20longitude&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matthews's Time for Science Education&lt;/a&gt;, p.170: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He proposed using two balances and two spiral springs. Each spring was wound in tum by the train and then allowed to run down freely, controlled only by the inertia of the balance, the other spring being meanwhile wound. When the first spring had run down, it unlocked the second and so the cycle was to be repeated. He was oblivious to the fact, well known to Hooke, that as the spring unwound its motive power decreased.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The clockwork idea for the measurement of longitude was of course suggested already by Huygens. Leibniz's own comments show that he was a big idea man, and not a good fit for the problem of longitude which was largely practical. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;all these defects, that proceed from the imperfection of the matter, may be&#xA;surmounted by a general remedy, without examining them here in particular. And that is, that for executing it in great, we may make use of massy springs, as are those of cross-bows, we being the masters of them, not wanting force or place in a ship to govern a great weight that may serve to bend them continually again... And it is easy to demonstrate, that by augmenting the size of the engine, and the force of the massy springs, we may make the error as small as we please...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;It is easy to demonstrate&quot; will have a long life in mathematical literature, and as Hooke put it on another occasion &quot;&lt;em&gt;as a clockmaker, Leibniz made a great mathematician.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later Leibniz did get involved with something more perhaps in the spirit of the OP, but as usual with practical matters he did not pursue it too far. It was longitude measurement based on measuring magnetic inclination. In 1676 a British instrument maker and teacher of mathematics Henry Bond had claimed that magnetic inclinators can be used for measuring longitude. The Royal Society got interested, and Oldenbourg, its secretary, sent a copy of Bond’s book to Leibniz:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As Leibniz explained, magnets were also “inclined toward the horizon,” and since these inclinations were also inconsistent when measured in a single location over time, it made sense to study this phenomenon too. Yet this required the use of a different kind of instrument, an “Instrumentum inclinatorium,” Leibniz wrote, “so that Inclinati Magnetis, which is different from declination, could be observed.”&lt;/em&gt;&quot; [from &lt;a href=&quot;https://books.google.com/books?id=MLdYCAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Halle Orphanage as Scientific Community by Whitmer&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But it is possible that &quot;instrumentum longitudinum&quot; mentioned in &lt;a href=&quot;https://books.google.co.il/books?id=f8aADQAAQBAJ&amp;amp;pg=PA336&amp;amp;lpg=PA336&amp;amp;dq=leibniz+navigation+instrument&amp;amp;source=bl&amp;amp;ots=uWzFpF3NBB&amp;amp;sig=kAz_fjPRhbDy6ZM8WWZ8KE-z84M&amp;amp;hl=iw&amp;amp;sa=X&amp;amp;ved=0ahUKEwjM-vrI5MHRAhUENxQKHY7KCnAQ6AEIPTAF#v=onepage&amp;amp;q=leibniz%20navigation%20instrument&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kempe's book&lt;/a&gt;, that &quot;&lt;em&gt;could record the course of a ship and automatically transmit these records to the map&lt;/em&gt;&quot;, was yet another of Leibniz's big ideas from 1700s. Indeed it sounds like an automated version of the oldest method in history, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dead_reckoning#Marine_navigation&quot; rel=&quot;nofollow noreferrer&quot;&gt;dead reckoning&lt;/a&gt;, still in use until Harrison finally solved the problem of longitude with his marine chronometer in 1730s.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2017-01-16T22:43:22.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5590" PostTypeId="2" ParentId="5582" CreationDate="2017-01-17T01:56:11.877" Score="6" Body="&lt;p&gt;Transition from Eudoxian ratios of magnitudes to irrational &lt;em&gt;numbers&lt;/em&gt; was a long incremental process with no sharp thresholds. Some argue that already Archimedes deviates from Euclidean strictures about magnitudes when he &amp;quot;approximates&amp;quot; π in the &lt;a href=&quot;http://www.cs.xu.edu/math/math300/07f/archcircle.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Measurement of the Circle&lt;/a&gt; (c. 250 BC), and that his works show some  &amp;quot;intuitive idea&amp;quot; of irrational numbers. Heron described a popular iteration process for numerically &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Biographies/Heron.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;approximating square roots in his Metrica&lt;/a&gt;, he also had a version for cube roots. &lt;a href=&quot;https://books.google.com/books/about/Episodes_from_the_Early_History_of_Mathe.html?id=5wGzF0wPFYgC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy in Almagest&lt;/a&gt; (and likely already Hipparchus c. 150 BC), despite heavily Euclidean language, has no problem with expressing chords and angles in sexagesimals and even describes how to interpolate their tabulated values.&lt;/p&gt;&#xA;&lt;p&gt;This was somewhat opportunistic and sporadic. Medieval Islamic mathematicians became more systematic after the introduction of the decimal system and algebra. According to &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0321387007&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Katz's History of Mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The process of relating arithmetic to algebra, begun by al-Khwarizmi and Abu Kamil, continued in the Islamic world with the work of Abu Bakr al-Karaji (d. 1019) and al-Samaw’al over the next two centuries. These latter mathematicians were instrumental in showing that the techniques of arithmetic could be fruitfully applied in algebra and, reciprocally, that ideas originally developed in algebra could also be important in dealing with numbers...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;[...]Al-Karaji was more successful in continuing the work of Abu Kamil in applying arithmetic operations to irrational quantities. In particular, he explicitly interpreted the various classes of incommensurables in Elements X as classes of “numbers” on which the various operations of arithmetic were defined, but then noted that there were indefinitely many other classes composed of three or more surds. Like Abu Kamil, he gave no definition of “number,” but just dealt with the various surd quantities using numerical rather than geometrical techniques. As part of this process, he developed various formulas involving surds...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;[...]Given that al-Samaw’al thought of extending division of polynomials into polynomials in &lt;span class=&quot;math-container&quot;&gt;$1/x$&lt;/span&gt;, and thought of partial results as approximations, it is not surprising that he would divide whole numbers by simply replacing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;. As already noted, al-Samaw’al was the first to explicitly recognize that one could approximate fractions more and more closely by calculating more and more decimal places.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For more on al-Samawal see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2504/what-is-the-origin-of-polynomials-and-notation-for-them/2509#2509&quot;&gt;What is the origin of polynomials and notation for them?&lt;/a&gt; By the way, not taking a form of Book X irrationals does not amount to being non-constructible by straightedge and compass, Book X irrationals are quadratic surds, they do not cover iterated square roots. Not even Gauss could &lt;em&gt;prove&lt;/em&gt; roots of cubics non-constructible by straightedge and compass many centuries later, it had to wait until Wantzel, see &lt;a href=&quot;http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/Suzuki.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Suzuki's Brief History of Impossibility&lt;/a&gt;. Just how much trust medieval mathematicians put into irrationals as &amp;quot;numbers&amp;quot; is attested to by the &lt;a href=&quot;http://www.etymonline.com/index.php?term=surd&quot; rel=&quot;nofollow noreferrer&quot;&gt;etymology of &amp;quot;surd&amp;quot;&lt;/a&gt;: &lt;em&gt;surdus&lt;/em&gt; &amp;quot;deaf, dumb&amp;quot; is a Latin translation of Arabic &lt;em&gt;aṣamm&lt;/em&gt; &amp;quot;deaf&amp;quot;, itself apparently derived from Euclid's Greek &lt;em&gt;alogos&lt;/em&gt; &amp;quot;speechless, without reason&amp;quot;. Latin &lt;em&gt;surd&lt;/em&gt; shares the root with &amp;quot;absurd&amp;quot;. Even Cardano in Ars Magna (1545) treats them with gloves.&lt;/p&gt;&#xA;&lt;p&gt;But even if we are charitable, calling surds &amp;quot;numbers&amp;quot; does not amount to treating them as &amp;quot;irrational numbers&amp;quot; as understood today. That assumes familiarity with a whole class of &amp;quot;numbers&amp;quot; (say our real numbers) in which they belong to the complement of rationals. Neither Heron, nor Al-Karaji, nor Fibonacci, nor Cardano came close to that. We had a long debate in &lt;a href=&quot;https://hsm.stackexchange.com/questions/3759/who-gets-credit-for-the-real-numbers/3873&quot;&gt;Who gets credit for the real numbers?&lt;/a&gt; thread on whether Stevin's treatment of arbitrary unending decimals in Arithmetic (1585), amounts to such an insight. For subsequent developments up to Cantor and Dedekind see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2740/when-did-it-become-understood-that-irrational-numbers-have-non-repeating-decimal/2743#2743&quot;&gt;When did it become understood that irrational numbers have non-repeating decimal representations?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-28T17:45:03.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5593" PostTypeId="1" CreationDate="2017-01-17T17:55:03.710" Score="5" ViewCount="3495" Body="&lt;p&gt;&lt;strong&gt;When was the mechanism behind seasons on Earth discovered?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm talking about the angle between the Earth's rotational axis and its orbital axis. I read on Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Axial_tilt#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Earth's obliquity may have been reasonably accurately measured as early as 1100 BC in India and China.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But did they know this is related to the cycle of seasons? &lt;/p&gt;&#xA;" OwnerUserId="5249" LastEditorUserId="1843" LastEditDate="2017-01-20T12:22:40.640" LastActivityDate="2022-12-06T23:13:23.880" Title="When was the mechanism behind seasons on Earth discovered" Tags="|astronomy|earth-sciences|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5596" PostTypeId="2" ParentId="5593" CreationDate="2017-01-18T18:08:36.190" Score="5" Body="&lt;p&gt;To explain seasons you do not need a precise measurement of the inclination of the ecliptic, it is sufficient to know the fact. I suppose that this fact was known to the first people who started agriculture, that is more than 12 thousand years ago.&#xA;Indeed, agriculture strongly depends on the change of seasons. You have to know when to plant your plants for optimal results. 12 millennia ago, people did not have  calendars (not speaking of internet), and did not even know the precise length of&#xA;the year, and they did not count days (from which day will you begin your count anyway?). So how did they determine when seasons begin and end, and when&#xA;it is appropriate to do seasonal agricultural works by observing the sky?&#xA;Only by astronomical observations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear that you will get poor results if you rely on &quot;today's weather&quot; for&#xA;sowing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is confirmed by the earliest texts on agriculture that we possess, like Hesiod. He does not write &quot;On March 27 do this and this&quot; but he writes&#xA;&quot;When Sun is in such and such position on the sky (with respect to such and such star, do this and this&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We do not know whether peasants did astronomical observations individually or were advised by&#xA;their rulers or priests, perhaps the practice varied from one culture to another.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The beginnings of the seasons are determined by simple astronomical events:&#xA;equinoxes and solstices. So any ancient agricultural society must have been aware of these events, and on the motion of the Sun on the ecliptic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Position of the ecliptic in the sky determines that the days are longer and the Sun&#xA;is higher in summer than in winter. So what can be more natural than to assume&#xA;that this is exactly what CAUSES summer and winter? It is clear that we obtain all heat from the Sun (I am not counting firewood). Thus the warm season happens&#xA;when the Sun stays higher and the days are longer.&#xA;And this inequality happens exactly because the ecliptic is inclined under certain angle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course we do not have written sources from 12 millennia ago to confirm all this, but it seems very plausible that people observing the sky were aware&#xA;about connection between seasons and inclination of the ecliptic, since these observations began, that is since the beginning of agriculture, the latest.&#xA;And the earliest known manual for agriculture (Hesiod) confirms this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-01-19T20:30:24.463" LastActivityDate="2017-01-19T20:30:24.463" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="5597" PostTypeId="2" ParentId="3291" CreationDate="2017-01-18T18:12:15.293" Score="0" Body="&lt;p&gt;Its the same triangle as here: &lt;a href=&quot;https://math.stackexchange.com/a/2035612/4414&quot;&gt;https://math.stackexchange.com/a/2035612/4414&lt;/a&gt;&#xA;Where I used $y=(x-1)*\sqrt{x+1}$. Which came up by a similar question posted by the same OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the additional benefit of a rational parametrization. For $x=t^2−1$ with $t$ rational we get indeed $y=t*(t^2-2)$. And from this parametrization it should give dense rational points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I don't see any classical constructive method behind it, which shouldn't exist for ruler and compass. Problem is how do we get from a triangle with cos(γ) to a triangle with x?&lt;/p&gt;&#xA;" OwnerDisplayName="user4900" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2017-01-18T18:26:28.057" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5605" PostTypeId="2" ParentId="55" CreationDate="2017-01-21T11:40:39.767" Score="4" Body="&lt;p&gt;From Loemker's translation, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Leibniz's reasoning, though it strives for a broader application of&#xA;  the law of inverse squares than to gravity alone, is less general than&#xA;  Newton's (Principia, Book I, Propositions I, 2, 14), since it&#xA;  presupposes harmonic motion.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Leibniz, Gottfried Wilhelm &lt;em&gt;Philosophical Papers and Letters : A Selection&lt;/em&gt; / Translated and Edited, with an Introduction by Leroy E. Loemker. 2d ed. Dordrecht : D. Reidel, 1970. p.362&lt;/p&gt;&#xA;" OwnerUserId="5267" LastActivityDate="2017-01-21T11:40:39.767" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5606" PostTypeId="1" CreationDate="2017-01-21T14:26:37.133" Score="1" ViewCount="176" Body="&lt;p&gt;Who was the first mathematician to define and investigate measurable cardinals?&#xA;And how long did it take for the fundamental theorem of measurable cardinals to be hypothesized and proven?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be very grateful for some background information on these questions.&lt;/p&gt;&#xA;" OwnerUserId="5268" LastActivityDate="2017-02-22T15:04:13.750" Title="Measurable cardinals and the fundamental theorem" Tags="|mathematical-logic|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5608" PostTypeId="2" ParentId="1827" CreationDate="2017-01-22T20:40:46.667" Score="7" Body="&lt;p&gt;The book &amp;quot;&lt;a href=&quot;http://press.princeton.edu/titles/10557.html&quot; rel=&quot;noreferrer&quot;&gt;An Einstein Encyclopedia&lt;/a&gt;&amp;quot; (Princeton University Press, 2015) has a chapter about assistants. Indeed, some of them never published jointly with Einstein.&lt;/p&gt;&#xA;&lt;p&gt;For the record, they are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ludwig Hopf (Zürich then Prague)&lt;/li&gt;&#xA;&lt;li&gt;Emil Nohel (Prague)&lt;/li&gt;&#xA;&lt;li&gt;Otto Stern (Prague then Zürich)&lt;br&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Then comes a period with no assistant.&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Jakob Grommer (Berlin)&lt;/li&gt;&#xA;&lt;li&gt;Cornelius Lanczos (Berlin)&lt;/li&gt;&#xA;&lt;li&gt;Hermann Muentz, overlapping w/ Lanczos (Berlin)&lt;/li&gt;&#xA;&lt;li&gt;Walther Mayer (&amp;quot;Einstein's calculator&amp;quot;) (Berlin, then Princeton)&lt;/li&gt;&#xA;&lt;li&gt;Nathan Rosen (Princeton)&lt;/li&gt;&#xA;&lt;li&gt;Leopold Infeld (Princeton)&lt;/li&gt;&#xA;&lt;li&gt;Banesh Hoffmann (Princeton)&lt;/li&gt;&#xA;&lt;li&gt;Peter Bergmann, overlapping w/ Hofmann (Princeton)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The book lists later assistants (1940-1955) with no claim of exhaustivity (all in Princeton):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Venlentin Bargmann&lt;/li&gt;&#xA;&lt;li&gt;Ernst Strauss&lt;/li&gt;&#xA;&lt;li&gt;John Kemeny&lt;/li&gt;&#xA;&lt;li&gt;Robert Kraichnan&lt;/li&gt;&#xA;&lt;li&gt;Bruria Kaufman&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The book also contains a chapter about collaborators (and tons of other things).&lt;/p&gt;&#xA;" OwnerDisplayName="user2441" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-01-22T20:40:46.667" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5609" PostTypeId="2" ParentId="5606" CreationDate="2017-01-23T13:57:51.363" Score="2" Body="&lt;p&gt;You can see : Dov Gabbay &amp;amp; Akihiro Kanamori &amp;amp; John Woods (editors), &lt;a href=&quot;https://books.google.it/books?id=ZF_QckMFy-oC&amp;amp;pg=PR5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Handbook of the History of Logic. Volume 6: Sets and Extensions in the Twentieth Century&lt;/a&gt; (2012), especially :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ch.1&lt;/strong&gt; : Set Theory from Cantor to Cohen, By A.Kanamori,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ch.2&lt;/strong&gt; : History of the Continuum in the 20th Century, by J.Steprans,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ch.4&lt;/strong&gt; : Large Cardinals with Fprcing, by A.Kanamori.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also : Akihiro Kanamori, &lt;a href=&quot;https://books.google.it/books?id=FH_n84ocuSMC&amp;amp;pg=PR9&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings&lt;/a&gt; (2nd ed., 2009) : &lt;strong&gt;Introduction&lt;/strong&gt; and &lt;strong&gt;Chapter 1. Beginnings&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe useful also : Matthew Foreman &amp;amp; Akihiro Kanamori (editors), &lt;a href=&quot;https://books.google.it/books?id=DLCyehuI0i0C&amp;amp;pg=PR11&quot; rel=&quot;nofollow noreferrer&quot;&gt;Handbook of Set Theory&lt;/a&gt; (2010) : &lt;strong&gt;Introduction&lt;/strong&gt; by A.Kanamori.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For historical context, you can see also : Gregory Moore, &lt;a href=&quot;https://books.google.it/books?id=3RLGKcEjVIoC&amp;amp;pg=PA196&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zermelo's Axiom of Choice : Its Origin Development and Influence&lt;/a&gt; (1982): &lt;strong&gt;Ch.4 The Warsaw School&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-01-23T14:08:35.633" LastActivityDate="2017-01-23T14:08:35.633" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5610" PostTypeId="1" CreationDate="2017-01-23T15:16:28.800" Score="3" ViewCount="335" Body="&lt;p&gt;Why was it necessary for the velocity of light to be invariant between different inertial frames and hence what was the need for Lorentz transformation when it was believed that velocity of light was constant with respect to aether. Was Lorentz transformation build in order to explain the null result of Michelson-Morley experiment?&lt;/p&gt;&#xA;" OwnerUserId="5273" LastActivityDate="2019-07-21T23:41:51.337" Title="What was the need for Lorentz transformation in pre-relativity period?" Tags="|physics|light|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5611" PostTypeId="1" AcceptedAnswerId="6499" CreationDate="2017-01-23T21:29:35.097" Score="1" ViewCount="241" Body="&lt;p&gt;To me, one of the most exciting mathematical achievements of antiquity is Archimedes' &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Method&lt;/a&gt;. It is crazy to think that, had it not been for its miraculous recovery in the early 20th century, it would perhaps have been lost forever. In translation, it can be found online &lt;a href=&quot;http://wilbourhall.org/index.html#archimedes&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. The original text, however, seems as hard to find as ever, at least in terms of on-line availability. While I can find the Greek version of most of his other works online, I can't find it for the &quot;Method&quot;. Can someone help me?&lt;/p&gt;&#xA;" OwnerUserId="2353" LastEditorUserId="4251" LastEditDate="2020-06-03T08:12:57.387" LastActivityDate="2020-06-03T08:12:57.387" Title="Looking for an online version of Archimedes' &quot;The Method&quot; (in Greek)" Tags="|calculus|archimedes|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5612" PostTypeId="1" AcceptedAnswerId="5615" CreationDate="2017-01-23T22:26:48.460" Score="9" ViewCount="1445" Body="&lt;p&gt;I'm currently doing a numerical analysis course, and it seems many of the techniques were developed by mathematicians, such as Euler and Newton, who lived before the age of the computer. Why were these mathematicians interested in numerical analysis and what use did it have before computers?&lt;/p&gt;&#xA;&lt;p&gt;I have a vague idea of some uses it may have, but I still struggle to understand why such great mathematicians would choose to research numerical analysis instead of remaining focused on pure mathematics.&lt;/p&gt;&#xA;" OwnerDisplayName="user5140" LastEditorUserId="10106" LastEditDate="2020-06-23T16:04:49.807" LastActivityDate="2021-03-12T01:50:18.593" Title="What was the appeal of numerical analysis before computers?" Tags="|mathematics|mathematicians|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="5613" PostTypeId="2" ParentId="5093" CreationDate="2017-01-23T22:32:53.847" Score="5" Body="&lt;p&gt;Another example is the study of configurations; its history is given in §1.2 of Branko Grünbaum's &lt;em&gt;Configurations of Points and Lines&lt;/em&gt;, Graduate Studies in Mathematics volume 103, American Mathematical Society, 2009.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Broadly, this area of combinatorics, though not defined in full generality until 1876 by Theodor Reye, encompasses work of Pappus and Desargues.  Other names associated with work up to 1910 include Möbius, Cayley, Burnside, and Steinitz.  Then there was a &quot;dark ages&quot; until 1990 when Grünbaum and others reinvigorated the field.  (I was fortunate to attend a topics class he taught on this material at the University of Washington in the early 1990s.)  It is now an active area of research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, there is a statement about configurations from Hilbert &amp;amp; Cohn-Vossen's &lt;em&gt;Geometry and the Imagination&lt;/em&gt; that Grünbaum considers an overstatement:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;H &amp;amp; C-V: &quot;...there was a time when the study of configurations was considered the most important branch of geometry.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G: &quot;The author would like to conjecture that this is the greatest exaggeration of the truth that can be found in any of Hilbert's writings.  While it is a fact that---as mentioned above---in the &quot;classical period&quot; of the history of configurations there were quite a few people interested in the topic, configurations were never a central topic of mathematical (or geometric) research.&quot;&lt;/p&gt;&#xA;" OwnerUserId="3429" LastActivityDate="2017-01-23T22:32:53.847" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5614" PostTypeId="1" CreationDate="2017-01-24T02:04:53.070" Score="3" ViewCount="947" Body="&lt;p&gt;My understanding is that alchemy was a precursor to modern chemistry. Some might say that numerology, similarly, was an earlier form of what is now known as number theory, but I feel like it's a more complicated proposition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At it's heart, numerology is about playing games with and finding relationships between numbers, which seems very similar to number theory. The key distinction may be that number theory is scientific where numerology is mystical, with perhaps some overlap prior to the Enlightenment, &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagoreanism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pythagoreanism&lt;/a&gt; being an example.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My feeling is that arithmetic is very old indeed, and numerology cannot be said to precede arithmetic because it requires arithmetic.&lt;/strong&gt;    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I also know there is a distinction between classical arithmetic and modern number theory, so I'd welcome any additional insight, correction, or related information in support or against this idea. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note: The impetus for this question came out of a recent question on Stack Mythology regarding the origin of &lt;a href=&quot;https://mythology.stackexchange.com/questions/146/why-were-there-exactly-12-main-olympian-gods&quot;&gt;12 Olympians specifically&lt;/a&gt;.  In a mythological context such as this, numerological ideas can hold some validity.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Although contemporary numerology is largely concerned with non-scientific prediction and non-scientific attempts to influence reality, I have also read convincing arguments that it may have been used as method of encoding esoteric knowledge, indicating a possible, practical use.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The most satisfactory answer to the question in question was related to the practical aspects of the integer 12.&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="5197" LastEditorUserId="-1" LastEditDate="2017-04-13T13:00:19.830" LastActivityDate="2019-10-03T15:40:15.757" Title="Can Numerology be said to be the precursor of Number Theory?" Tags="|number-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5615" PostTypeId="2" ParentId="5612" CreationDate="2017-01-24T02:37:36.173" Score="10" Body="&lt;p&gt;I am not a historian but I propose first &quot;Astronomy&quot; as the answer to your question.  The study of planetary motions and seasons was essential for a variety of reasons (agriculture and religion come to mind).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, &quot;Navigation&quot;: the use of accurate tables (particularly multiplication tables) was essential for good navigation, and this entailed laborious calculations (mostly using spherical geometry).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, &quot;Money&quot;.  The calculation of interest and mortgages is rumoured to be the motivation for Euler introducing &quot;e&quot; as the limit of receiving interest over an ever-larger numbers of ever-smaller intervals.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorDisplayName="user5245" LastEditDate="2017-01-25T13:37:17.120" LastActivityDate="2017-01-25T13:37:17.120" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5616" PostTypeId="1" CreationDate="2017-01-24T15:10:13.090" Score="8" ViewCount="567" Body="&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/en/4/4f/The_official_poster_for_the_film_Hidden_Figures%2C_2016.jpg&quot; alt=&quot;Official poster for the film Hidden Figures&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was fascinated by the film &lt;a href=&quot;https://en.wikipedia.org/wiki/Hidden_Figures&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hidden Figures&lt;/a&gt;, and a related article from New Scientist magazine &lt;em&gt;&lt;a href=&quot;https://www.newscientist.com/article/mg23331090-400-gifted-and-black-the-incredible-woman-who-got-the-us-into-space/?utm_source=NSNS&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gifted and black: The brilliant woman who got the US into space&lt;/a&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to understand more about what it was like, before electronic computers, for these “human computers” to do “rocket science”. What would an everyday problem and calculation look like? What roles would be involved, and how would the workers accomplish them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the article I found &lt;a href=&quot;https://crgis.ndc.nasa.gov/crgis/images/b/b7/CompressibleFlow.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;an example&lt;/a&gt; of the sort of calculations that these amazing human computers carried out. It is signed at the bottom &quot;nmg - 2-3-53&quot;. I wonder who that was, and how this fit in to the work at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, I put together a modern re-working of it, with a &lt;a href=&quot;http://jupyter.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jupyter&lt;/a&gt; notebook and &lt;a href=&quot;https://www.python.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Python&lt;/a&gt; code. A bit of it is below, and you can see the whole thing, and the IPython notebook behind it, at my gist:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://gist.github.com/nealmcb/a48efd1c4bc1276865e39d22ea69be0b&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hidden Figures Example: Compressible Flow Equations and Table in Jupyter, Python&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;A bit of googling turned up some relevant background material on the math itself, which includes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/304440/flow-through-a-nozzle&quot;&gt;Flow through a nozzle - Physics Stack Exchange&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ntrs.nasa.gov/search.jsp?R=19930091974&quot; rel=&quot;nofollow noreferrer&quot;&gt;Equations for the Design of Two-Dimensional Supersonic Nozzles - Report No. 907&lt;/a&gt; from 1948&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But I'm still curious to know what the day-to-day work of the human computers looked like.  I'm guessing some of them would fill out tables like the one shown in the example, and in my gist, mostly by looking up logarithms and doing addition and subtraction with them. They would then use these new tables to do the calculations for specific problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But were there other clever tricks they used? Are there any more detailed descriptions of the everyday life of a human computer? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$M = { V \over a} $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;${H - p \over q}&#xA; = {F_c}&#xA; = {1 + \eta} &#xA; = { {2 \over {\gamma M^2}} { \left[{( {1 + {{\gamma - 1} \over 2}} M^2)}^{\gamma \over {\gamma - 1} } - 1\right]} } &#xA; = {1.42857 \over M^2} \left[(1 + 0.2 M^2) ^ {3.5} - 1\right] $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;&#xA;M       H−p/q   T/T0    A1/A    rho/rho0 Psonic     p/H     q/H &#xA;0.001   1.0000  1.0000  0.0017  1.0000  -673870.092 1.0000  0.00000&#xA;0.002   1.0000  1.0000  0.0035  1.0000  -168467.127 1.0000  0.00000&#xA;0.003   1.0000  1.0000  0.0052  1.0000  -74873.985  1.0000  0.00001&#xA;...&#xA;&lt;/code&gt;&lt;/p&gt;&#xA;" OwnerUserId="5279" LastEditorUserId="13" LastEditDate="2018-08-16T13:24:05.360" LastActivityDate="2018-08-16T20:42:30.813" Title="Day-to-day tasks of human computers, à la Hidden Figures movie" Tags="|classical-mechanics|depiction-of-scientists|computation|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="5617" PostTypeId="1" CreationDate="2017-01-24T16:13:31.513" Score="2" ViewCount="240" Body="&lt;p&gt;Can we imagine that scientific can still discover some basic simple but essential geometric rules such as the Pythagorean theorem in a near of far future:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^2 + b^2 = c^2$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or do we consider that the future of discovery in mathematic, geometry and physic only rely on heavy computational models ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean, is it possible to imagine that a man with some books and a brain (no computer for simluation, or heavy equipement) can still discover &lt;em&gt;essential&lt;/em&gt; stuffs that have never been discovered, such as Euclide, Thales or Pythogore did in the past ?&lt;/p&gt;&#xA;" OwnerUserId="5280" LastActivityDate="2017-02-06T21:02:58.277" Title="Essential basic geometrical discoveries still possible in our era?" Tags="|discoveries|computers|pythagorean-proposition|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5618" PostTypeId="2" ParentId="5617" CreationDate="2017-01-24T19:53:21.000" Score="6" Body="&lt;p&gt;This is exactly what is happening in modern mathematics: people with some books and brain discover essential things that have never been discovered, such as Euclid, Thales and Pythagoras did in the past. In Geometry as well as in&#xA;other areas of mathematics. And most of these people use computers only to write e-mails and to check their bank accounts). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Type &quot;Poincare conjecture&quot;, &quot;Perelman&quot; on Google and you find plenty of popular texts explaining one of the recent top achievements in geometry. If you want very recent (2016) type &quot;Marina Vyazovska&quot;, &quot;sphere packing&quot;. Average well-educated person can understand what has been done (not how was it done), from these popular explanations.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-01-25T07:18:57.843" LastActivityDate="2017-01-25T07:18:57.843" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5619" PostTypeId="2" ParentId="5612" CreationDate="2017-01-24T20:13:10.913" Score="13" Body="&lt;p&gt;When you apply mathematics to the real world this requires a lot of calculation.&#xA;Astronomy is the oldest science which applied mathematics on a large scale.&#xA;Then comes physics, chemistry, material sciences and all other sciences and engineering applications. They always required a lot of numerical analysis.&lt;/p&gt;&#xA;&lt;p&gt;The question is actually very strange: it was evidently MORE reasons for mathematicians to work on numerical methods BEFORE the computers spread than after their spread. Because all calculations had to be made by hand. So this was very time and labor consuming. It is exactly for this reason people were working hard to invent effective algorithms.&lt;/p&gt;&#xA;&lt;p&gt;For example, when Kepler learned about logarithms he said that &amp;quot;this invention&#xA;increased the life time of astronomers by a factor of 100&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As late as in 1940 Cornelius Lanczos invented the famous algorithm which is called the &amp;quot;Fast Fourier Transform&amp;quot;. In the introduction to his paper&#xA;(joint with his CHEMISTRY PhD student Danielson) he says: &amp;quot;One could use&#xA;an expensive harmonic analyser, but we found a method which permits to do this calculation by hand...&amp;quot; (Harmonic analyser is a very sophisticated analog computer designed to do just one task: Fourier expansion. These things were custom made and indeed enormously expensive).&lt;/p&gt;&#xA;&lt;p&gt;Until the late 1970s there was a profession &amp;quot;navigator&amp;quot;. Each long distance airplane and each ocean going ship had (at least) one. The main job of this person was calculations (by hand, using tables, and other aids). Invention of&#xA;electronic calculator did not eliminate the profession but changed it dramatically. Only the spread of satellite navigation systems+computers eliminated it.&lt;/p&gt;&#xA;&lt;p&gt;I did not even mention military applications, which were second only to astronomy and navigation in importance.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6229" LastEditDate="2021-03-12T01:50:18.593" LastActivityDate="2021-03-12T01:50:18.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5620" PostTypeId="1" CreationDate="2017-01-24T21:42:08.823" Score="1" ViewCount="47" Body="&lt;p&gt;The exponential-log transformation of exponential generating functions (see OEIS &lt;a href=&quot;http://oeis.org/A036040&quot; rel=&quot;nofollow noreferrer&quot;&gt;A036040&lt;/a&gt; and &lt;a href=&quot;http://oeis.org/A127671&quot; rel=&quot;nofollow noreferrer&quot;&gt;A127671&lt;/a&gt;) relate the &lt;a href=&quot;http://en.wikipedia.org/wiki/Cumulant&quot; rel=&quot;nofollow noreferrer&quot;&gt;classical cumulants&lt;/a&gt; to their associated moments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who were some of the first to introduce diagrammatic illustrations of the relationship between the classical cumulants and moments?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(A excellent discussion of the connections can be found in &quot;&lt;a href=&quot;http://arxiv.org/abs/1205.2097&quot; rel=&quot;nofollow noreferrer&quot;&gt;Three lectures on free probability&lt;/a&gt;&quot; by Novak and LaCroix.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Agata Fronczak in the &quot;&lt;a href=&quot;http://arxiv.org/abs/1209.4231&quot; rel=&quot;nofollow noreferrer&quot;&gt;The microscopic meaning of the grand potential&lt;/a&gt; ...&quot; mentions Mayer, Riddel, and Uhlenbeck in relation to the cluster expansion theorem.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastActivityDate="2017-01-24T21:42:08.823" Title="Origin of diagrammatics illustrating the relation between cumulants and moments?" Tags="|mathematical-physics|statistics|statistical-mechanics|combinatorics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5621" PostTypeId="2" ParentId="5616" CreationDate="2017-01-25T01:30:53.220" Score="10" Body="&lt;p&gt;If you are interested in descriptions of “everyday life” of human computers, here is an excerpt from Stan Ulam’s autobiography, &lt;em&gt;Adventures of a Mathematician&lt;/em&gt; (University of California Press, 1991) concerning the years 1949–1952, when he was a part of the team working in Los Alamos on thermonuclear explosion. The  account of the mathematical problems involved is rather general, but it mentions the female human computers – and shows some chauvinistic attitude on part of the team members.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My calculations with Everett  concerned the first phase of the explosion, the problem of the initial ignition. (...) The numerical work was done on desk computers with the assistance of a number of programmers from the laboratory's computing group, managed by a good-humored New Yorker, Max Goldstein. Much to Max's annoyance, Fermi wanted to encourage the girls to use slide rules; the machine precision was not really warranted because of our simplifications. But Max insisted on the usual routine with desk calculators. Reading from slide rules and using logarithms as Fermi did was much less accurate, but with his marvelous sense he had the ability to judge the right amount of accuracy which would be meaningful. The girls, who were merely making calculations without knowing the physics or general mathematics behind them could not do that, of course, so in a way Max was quite right in insisting on the standard routines.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I particularly remember one of the programmers who was really beautiful and well endowed. She would come to my office with the results of the daily computation. Large sheets of paper were filled with numbers. She would unfold them in front of her low-cut Spanish blouse and ask, ‘How do they look?’ and I would exclaim, ‘They look marvelous!’ to the entertainment of Fermi and others in the office at that time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ulam's wife, Francoise, also worked as a “computer” for some time.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="13" LastEditDate="2018-08-16T20:42:30.813" LastActivityDate="2018-08-16T20:42:30.813" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5622" PostTypeId="2" ParentId="2033" CreationDate="2017-01-25T02:12:50.117" Score="2" Body="&lt;p&gt;&quot;Good Will Hunting&quot; is a reasonable portrayal of modern mathematics; not so much the boy genius portrayal but the Gerald Lambeau portrayal, his office and general demeanour.  Also, there's another graduate student of Lambeau as part of the supporting cast, whom I thought gave a really good performance.  Bit of a pity these two roles have not been recognized for their authencity.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastActivityDate="2017-01-25T02:12:50.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5623" PostTypeId="1" CreationDate="2017-01-25T18:13:03.373" Score="4" ViewCount="183" Body="&lt;p&gt;Most specifically I wonder if there is any chance that he kept the notebooks he refers to in his autobiography.  Probably he did not.  But Gauss kept his diary....&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="4251" LastEditDate="2021-09-23T12:35:37.543" LastActivityDate="2021-09-23T12:35:37.543" Title="Is there an archive of Andre Weil's papers?" Tags="|mathematicians|weil|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5624" PostTypeId="1" AcceptedAnswerId="5625" CreationDate="2017-01-26T12:48:05.827" Score="0" ViewCount="94" Body="&lt;p&gt;In the second episode of &quot;Super Science Friends&quot; Tesla is telling of his fight with Edison that, in the comic, leads him to carry all his patents in the interior pockets of his jacket.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Freud &lt;a href=&quot;https://youtu.be/dmWvb1UXiZY?t=3m30s&quot; rel=&quot;nofollow noreferrer&quot;&gt;then&lt;/a&gt; comments &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;No chances that could ever backfire&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;while looking at Marie Curie, that reacts quite angrily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is being referenced here?&lt;/p&gt;&#xA;" OwnerUserId="5290" LastActivityDate="2017-01-27T00:44:05.207" Title="Which event involving Marie Curie is being referenced here?" Tags="|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5625" PostTypeId="2" ParentId="5624" CreationDate="2017-01-26T18:04:37.173" Score="2" Body="&lt;p&gt;Marie Curie died as a result of exposure to radiation.  This was, in part, the result of her carrying test tubes of radium in her pockets while doing her research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source &lt;a href=&quot;https://www.ilab.org/eng/documentation/608-marie_curie_-_a_woman_of_firsts.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;International League of Antiquarian Booksellers site &lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One inadvertent and tragic discovery of hers was the effects of radium.  Much of her work was done in a shed, without the safety measures that are used today. She had remarked on the pretty blue-green light that the substances gave off in the dark, and kept test tubes in her pocket and desk drawer.  In 1934 she died of aplastic anemia from exposure to radiation. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-01-26T18:04:37.173" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5627" PostTypeId="1" AcceptedAnswerId="5628" CreationDate="2017-01-28T00:25:59.237" Score="9" ViewCount="1439" Body="&lt;p&gt;We know the Earth isn't a sphere: that is, the equatorial circumference isn't equal to the polar circumference.  When (and how) was this discovered?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can put a lower bound of around the 3rd century BC, when Greek astronomy showed that the Earth was round rather than flat, and an upper bound of 1791, when the French Academy of Sciences specified a quadrant of &lt;em&gt;meridian&lt;/em&gt; in the definition of the meter.  An uncertainty of 2000 years is rather large, though.&lt;/p&gt;&#xA;" OwnerUserId="3186" LastEditorUserId="4251" LastEditDate="2023-06-26T09:51:50.387" LastActivityDate="2023-06-26T09:51:50.387" Title="When was it discovered that the Earth wasn't round?" Tags="|geography|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5628" PostTypeId="2" ParentId="5627" CreationDate="2017-01-28T08:21:53.170" Score="11" Body="&lt;p&gt;Discarding the ungrounded speculations (for example, Columbus claimed that Earth has a &quot;bulge&quot;),&#xA;scientific theories which predicted deviation of Earth's shape from the sphere were proposed in he late 17th century. The correct theory was due to Newton. There was a competing theory predicting that equator is shorter than meridian (either by Descartes himself, or by his followers). To decide which theory is correct, several expeditions were sent by the French Academy to measure arcs of meridians. The first expedition did this in Peru (modern Equador) in 1735-45. Before they finished in Peru, the second expedition (to Lapland, Sweden, 1736)&#xA;returned with a conclusive result that Newton's theory was correct.&#xA;The results were published in the book of Maupertuis, La figure de la Terre&#xA;in 1738. It was translated into English in the same year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This result was one of the first major tests of the theory of universal gravitation. (See &lt;a href=&quot;https://hsm.stackexchange.com/questions/5558/how-did-newton-come-up-with-his-formula/5561#5561&quot;&gt;How did Newton come up with his formula?&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Of course this was only the beginning. Since then geodesists made many more precise measurements, and mathematicians also investigated the question in great depth. Mathematically, the question is about equilibrium of rotating liquid body; it was finally solved only in 20th century (by Chandrasekhar in 1970).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-01-28T12:35:57.047" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5629" PostTypeId="1" CreationDate="2017-01-28T18:24:22.583" Score="4" ViewCount="1350" Body="&lt;p&gt;I am trying to understand the difference in &quot;analysis&quot; and &quot;synthesis&quot; as used by the ancient Greek mathematicians. Most sources characterize synthesis as working from givens to a desired conclusion, and analysis as starting from a conclusion and working back to its cause. What would be an example of the Greeks doing this? It seems that all arguments I have read in Euclid's &lt;em&gt;Elements&lt;/em&gt; would be synthetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, what is the connection between the historical and modern usages of these terms? Why did calculus come to be called &quot;real analysis&quot;?&lt;/p&gt;&#xA;" OwnerDisplayName="user3339" LastActivityDate="2017-02-01T21:26:08.130" Title="Analysis vs Synthesis in Greek Mathematics" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5630" PostTypeId="2" ParentId="5629" CreationDate="2017-01-28T19:40:39.137" Score="3" Body="&lt;p&gt;See &lt;a href=&quot;https://plato.stanford.edu/entries/analysis/s2.html#2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anaysis in Ancient Greek Geometry&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What analysis involves is the finding of appropriate principles, previously proved theorems, and constructional moves by means of which the problem can be solved (the desired figure constructed or the relevant theorem proved).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Synthesis instead, as you say, is the &quot;standard&quot; deductive approach: from postulates to theorems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With Early Modern mathematics, the new algebraic techniques was often called &quot;analytical&quot;; see e.g &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;François Viète &lt;/a&gt; and his :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/New_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;In Artem Analyticien Isagoge (Introduction to the art of analysis)&lt;/a&gt; (1591), &lt;strong&gt;Chapter I&lt;/strong&gt;, &lt;a href=&quot;https://books.google.it/books?id=omHqrYBzMpAC&amp;amp;pg=PA11&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation&lt;/a&gt;, page 11:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is a certain way for searching for the truth in mathematics that Plato is said first to have discovered. Theon called it analysis...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, &lt;em&gt;analysis&lt;/em&gt; became a sort of catchword for any new mathematical &quot;tools&quot; and methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Jaakko Hintikka &amp;amp; Unto Remes, &lt;a href=&quot;https://books.google.it/books?id=XwrvCAAAQBAJ&amp;amp;pg=PR9&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Method of Analysis: Its Geometrical Origin and Its General Significance&lt;/a&gt; (1974)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Michael Otte and Marco Panza (editors), &lt;a href=&quot;https://books.google.it/books?id=WFav-N0tv7AC&amp;amp;pg=PA3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analysis and Synthesis in Mathematics: History and Philosophy&lt;/a&gt; (1997).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Regarding Plato, see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Stephen Menn, &lt;a href=&quot;https://www.philosophie.hu-berlin.de/de/lehrbereiche/antike/mitarbeiter/menn/plato--analysis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plato and the Method of Analysis&lt;/a&gt; (2002).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-01-30T21:40:44.567" LastActivityDate="2017-01-30T21:40:44.567" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5631" PostTypeId="1" CreationDate="2017-01-29T17:32:06.677" Score="14" ViewCount="428" Body="&lt;p&gt;Can you think of examples of influential scientific articles that at first were rejected by the reviewers but where later recognized as a great and influential contribution? &lt;/p&gt;&#xA;" OwnerUserId="3983" LastActivityDate="2018-05-02T08:26:34.357" Title="Are there examples of influential scientific articles first rejected and later proved to be valid?" Tags="|publishing|scientific-method|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5632" PostTypeId="2" ParentId="5612" CreationDate="2017-01-29T17:58:34.383" Score="5" Body="&lt;p&gt;I find the question a bit bizarre but realize that it is likely due to the culture difference of those who have grown up in a society where computers are used everywhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was not that long ago in past history (~1965) when I was a undergrad physics major faced with a non-linear (transcendental) equation to solve.  No analytic solutions possible yet I had to find a value for 'x' (the unknown) up to 3 decimal places.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution in my case was Newton's method and a slide rule (which I still have and sits on the bookcase next to me right now).  We students of this class (Classical Mechanics) all had to solve the problem as part of a homework assignment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my fellow students though used the campus computer (CDC 3300) and the Fortran language to implement Newton's method and easily come up with an answer to 5 or 6 decimal digits.  And, his program was punched into the so-called IBM punched card.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That single event of learning the power of the computer over the drudgery of the slide rule turned me on to computer programming.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As others have said, computers help but they were not the reason for the development of numerical methods.  In physics, most real life problems must be solved numerically -- only text book assignments (usually) of limited scope have analytic solutions available to Algebra or Calculus. &lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2017-01-29T17:58:34.383" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5633" PostTypeId="2" ParentId="5631" CreationDate="2017-01-29T19:44:02.457" Score="11" Body="&lt;p&gt;As user njuffa has commented, my original answer was not strictly to the letter of the OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some other examples :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fermi's 1933 paper on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermi&amp;#39;s_interaction&quot; rel=&quot;nofollow noreferrer&quot;&gt;weak interaction&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Gell-Mann's 1953 paper on the classification of elementary particles.&lt;/li&gt;&#xA;&lt;li&gt;Higgs' 1964 paper introducing the Higgs Boson/Field - described as &lt;a href=&quot;https://en.wikipedia.org/wiki/Higgs_boson&quot; rel=&quot;nofollow noreferrer&quot;&gt;unimaginative by the publisher&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Ernst's 1966 paper on nuclear magnetic resonance. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Each of these authors went on to win the Nobel Prize.  Details of these cases together with an additional four Nobel Prize winners whose original papers were originally rejected are detailed in &lt;a href=&quot;http://www.sciencealert.com/these-8-papers-were-rejected-before-going-on-to-win-the-nobel-prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;ORIGINAL ANSWER&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My father suffered for many years with stomach ulcers.  The medical advice he had been given was to avoid stress - not an easy task in most families - and to eat a sensible diet, including drinking a lot of milk and consuming other dairy products and avoiding spicy foods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 1980s, &lt;a href=&quot;https://en.wikipedia.org/wiki/Barry_Marshall&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barry Marshall&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Robin_Warren&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robin Warren&lt;/a&gt; proposed that stomach ulcers were caused by bacteria.  The scientific community rejected, and even ridiculed this idea, arguing that stomach acids would kill any dangerous bacteria.  In the end, Marshall convinced the doubters by drinking the contents of a petri-dish full of the dangerous bacteria (&lt;em&gt;H Plyori&lt;/em&gt;) in order to obtain the &quot;desired result&quot;.  Although he expected it to take years for the ulcers to develop, in fact it only took three days.  See the research section of the Marshall's wiki article for further details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2005, Marshall and Warren were awarded the Nobel Prize for medicine for their discovery.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This page on the &lt;a href=&quot;https://mathoverflow.net/questions/13896/what-are-some-famous-rejections-of-correct-mathematics&quot;&gt;Math Overflow&lt;/a&gt; site gives a number of cases in mathematics.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt; Marshall and Warren first published their theory in &lt;em&gt;The Lancet&lt;/em&gt; on June 16, 1984.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2017-01-30T04:20:13.143" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5636" PostTypeId="2" ParentId="5631" CreationDate="2017-01-31T08:59:53.653" Score="5" Body="&lt;p&gt;As mentioned in his &lt;a href=&quot;http://www.nytimes.com/2010/10/31/us/31valen.html?_r=1&amp;amp;scp=1&amp;amp;sq=valen&amp;amp;st=cse&quot;&gt;obituary&lt;/a&gt;, Leigh Van Valen's &lt;a href=&quot;https://dl.dropboxusercontent.com/u/18310184/evolutionary-theory/vol-01/Vol.1%2CNo.1%2C1-30%2CL.%20Van%20Valen%2C%20A%20new%20evolutionary%20law..pdf&quot;&gt;evolutionary law (1973)&lt;/a&gt;, referred to more commonly as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Red_Queen_hypothesis&quot;&gt;Red Queen Hypothesis&lt;/a&gt;, has been repeatedly rejected. So much in fact that he decided to create his own journal (&lt;a href=&quot;http://leighvanvalen.com/evolutionary-theory/&quot;&gt;Evolutionary Theory&lt;/a&gt;) to publish it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For those not familiar with it, the 'Red Queen Hypothesis' is an evolutionary model driven by competition between species. It is indeed a process frequently invoked to explain coevolution, and is one of the most influential evolutionary theory of the last 50 years.&lt;/p&gt;&#xA;" OwnerUserId="84" LastEditorUserId="84" LastEditDate="2017-01-31T09:07:33.710" LastActivityDate="2017-01-31T09:07:33.710" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5637" PostTypeId="1" AcceptedAnswerId="5640" CreationDate="2017-02-01T01:22:03.637" Score="12" ViewCount="888" Body="&lt;p&gt;We know that a body is red because it absorbs (visible) light of all wavelengths except that for the red. But before Newton showed that white light was actually made up of seven colors how did people explain a body being red in color?&lt;/p&gt;&#xA;" OwnerUserId="5315" LastEditorUserId="55" LastEditDate="2017-02-02T18:42:12.600" LastActivityDate="2017-02-16T22:41:25.290" Title="How did people explain color before the splitting of white light was discovered?" Tags="|light|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5638" PostTypeId="2" ParentId="5631" CreationDate="2017-02-01T04:10:46.180" Score="9" Body="&lt;p&gt;Klaus von Klitzing's original submission on the integer quantum Hall effet was rejected (the story says by Physical Review Letters).  He won the Nobel in 1985 for this discovery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;George Uhlenbeck and Samuel Goudsmit proposed an erroneous theory of the electron spin.  They sent the manuscript while their thesis director Paul Ehrenfest was absent; Ehrenfest rapidly debunked his student's paper, but it had already been accepted.  Ehrenfest later commented that Uhlenbeck and Goudsmit were &quot;young enough&quot; to make such mistakes.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorDisplayName="user5245" LastEditDate="2017-02-01T22:29:06.040" LastActivityDate="2017-02-01T22:29:06.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5639" PostTypeId="2" ParentId="5612" CreationDate="2017-02-01T10:53:44.580" Score="8" Body="&lt;p&gt;It is striking to me that someone in the 21st century questions why people who had to calculate by hand were motivated to discover numerical algorithms. Let's see, imagine it is the early 1700s and you want to find the value of the sum of &lt;span class=&quot;math-container&quot;&gt;$1/n^2$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n = 1,2,3,\ldots$&lt;/span&gt; (this was an unsolved problem for about 90 years). Of course it helps to get an idea of what the answer might be before trying to derive it carefully. So you want a numerical estimate, say to 5 decimal places. You can't do that by hand just from summing the series: it takes over 5000 terms in the series to get within .00001 of the value of the full series. But if you are Euler, you develop (what came to be called) the Euler-Maclaurin summation formula that allows you to estimate the series to over 10 decimal places using just the first 10 terms of the series plus a few correction terms in that summation formula. Can you appreciate that this might be a useful achievement?&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2021-03-10T21:19:56.047" LastActivityDate="2021-03-10T21:19:56.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5640" PostTypeId="2" ParentId="5637" CreationDate="2017-02-01T19:19:55.367" Score="10" Body="&lt;p&gt;On a common sense view color requires no &quot;explanation&quot;: the world is full of objects, they have qualities (properties) — size, shape, weight, texture, etc. — color is one of them. This was systematized in Aristotle's qualitative physics, qualities were the constituents of &quot;substances&quot; forming objects, and our perceptions arise from their mixtures. Ancient theories of vision mostly focused on explaining how those qualities reach our eyes. By the way, &lt;a href=&quot;http://www.thestargarden.co.uk/Newtons-theory-of-light.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's optics&lt;/a&gt; was corpuscular, so he did not associate colors with wavelengths. The wave optics was proposed around 1678 by Huygens, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3486/did-wave-optics-anticipate-quantum-mechanics/3493#3493&quot;&gt;Did wave optics anticipate quantum mechanics?&lt;/a&gt;, but did not become dominant until &lt;a href=&quot;https://hsm.stackexchange.com/questions/2629/how-did-young-perform-his-double-slit-experiment/2634#2634&quot;&gt;Young's double slit experiments&lt;/a&gt; and Fresnel's theory of diffraction in 19th century. Newton in Opticks (1704) did introduce the term &quot;color spectrum&quot;, and concluded from experiments that color is a property of light reflected from objects, rather than of the objects themselves. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That white light splits into colored light is an interesting fact about light, but it has little to do with explaining color perception. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Young%E2%80%93Helmholtz_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;trichromatic (red-green-blue) theory of color vision&lt;/a&gt; was only proposed by Young in 1802, and postulated the existence of three types of retinal cone cells sensitive to RGB colors, which Helmholtz related to light wavelengths only in 1850. The existence of three cone types was experimentally confirmed by Svaetichin in 1956 (although the relation of their sensitivity ranges to green, blue, and especially red colors is rather loose). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is only in the context of singling out certain qualities as more &quot;fundamental&quot; and reducing the rest to them somehow that one needs to &quot;explain&quot; how color is so reduced. This point of view became prominent only with the rise of modern science in 17th century, when the distinction between &lt;a href=&quot;https://en.wikipedia.org/wiki/Primary/secondary_quality_distinction&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;primary and secondary qualities&quot;&lt;/a&gt; was made in the context of mechanistic natural philosophy. The term itself only appeared in &lt;a href=&quot;https://en.wikipedia.org/wiki/An_Essay_Concerning_Human_Understanding&quot; rel=&quot;nofollow noreferrer&quot;&gt;Locke's Essay Concerning Human Understanding (1689)&lt;/a&gt;, but the distinction is usually traced back to Galileo, and appears in the works of other early mechanists, like Descartes and Hobbes. The &quot;primary&quot; qualities were, naturally, the mechanical ones. According to the &lt;a href=&quot;http://galileo.rice.edu/sci/theories/atomism.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo Project&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In his Assayer of 1623, Galileo explained his notion of the difference between those qualities, mostly found by touch, that are inherent in bodies (weight, roughness, smoothness, etc.) and those that are in the mind of the observer (taste, color, etc.)--in other words, the difference between what we call primary and secondary qualities. In this discussion he referred to bodies that &quot;continually dissolve into minute particles&quot; and stated his opinion that &quot;for exciting in us tastes, odors, and sounds there are required in external bodies anything but sizes, shapes, numbers, and slow or fast movements.&quot; An anonymous cleric filed a report with the Inquisition in which he claimed the first citation to show that Galileo was an atomist and the second to be in conflict with the Council of Trent's pronunciations on the Eucharist. The report did not lead to any action against Galileo.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The picture of the world as &quot;particles in motion&quot;, and the mechanistic mode of explanation of course date further back to ancient atomists, the one natural philosophy in antiquity where an &quot;explanation&quot; of color was called for. Their list of primary qualities was even shorter, according to Leucippus and Democritus, only shapes and sizes of atoms were &quot;primary&quot;, while smells, tastes, and colors were our perceptions of those. For example, &quot;sweet atoms&quot; were round, and &quot;green atoms&quot; were thin and elongated, so they could pass through glass, etc. Epicurus added weight to the &quot;primary&quot; list, and &lt;a href=&quot;http://dpuadweb.depauw.edu/jeremyanderson_web/research/Epicurus.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;disagreed on sensory qualities like color&lt;/a&gt;. According to him, they were properties not of atoms themselves but of their &lt;em&gt;composites&lt;/em&gt;, which means that we perceive patterns in arrangement and orientation of atoms as colors, rather than the shapes of atoms themselves. The Epicurean physics is poetically described in &lt;a href=&quot;https://en.wikipedia.org/wiki/De_rerum_natura&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lucretius's De Rerum Natura (On the Nature of Things, c. 60 BC)&lt;/a&gt;, but like all of ancient atomism it was mostly natural philosophic speculation with only loose connection to observations. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-02-16T22:41:25.290" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5641" PostTypeId="2" ParentId="5637" CreationDate="2017-02-01T20:09:00.413" Score="6" Body="&lt;p&gt;Aristotle' s explanation of color was as follows: &quot;Firstly, white and black may be juxtaposed in such a way that by the minuteness of of the division of its parts each is invisible while their product is visible, and thus colour may be produced. This product can appear neither white nor black, but, since it must have some colour and can have neithe rof the above two,, it must be a sort of compound and a fresh kind of tint. In this way, then, we may conceive that numbers of colours over and above black and white may be produced, and that their multiplicity is due to differences in the proportion of their composition. (...)&quot; (20-30; Aristotle De Sensu and De Memoria, text and translation with introduction and commentary by G. R. T. Ross, D. Phil. (Edin.), Cambridge: at the University Press, 1906)&#xA;&lt;a href=&quot;https://books.google.com/books?id=aZGwq6KPmfAC&amp;amp;pg=PA57&amp;amp;dq=aristotle+%2B+%22de+sensu%22+%2B+white+and+black+may+be+juxtaposed%22&amp;amp;lr=&amp;amp;as_brr=1&amp;amp;cd=1#v=onepage&amp;amp;q&amp;amp;f=false&quot;&gt;https://books.google.com/books?id=aZGwq6KPmfAC&amp;amp;pg=PA57&amp;amp;dq=aristotle+%2B+%22de+sensu%22+%2B+white+and+black+may+be+juxtaposed%22&amp;amp;lr=&amp;amp;as_brr=1&amp;amp;cd=1#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Aristotelian theory of color as being produced by mixture of darkness and light found a notable adherent in Johann Wolfgang Goethe, who in 1810 published his ``Theory of Colours&quot;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_Colours&quot;&gt;https://en.wikipedia.org/wiki/Theory_of_Colours&lt;/a&gt; Since this work appeared many years after Newton and Huyghens and enjoyed some popularity (the English translation appeared in 1840), this may mean that the optical theories have not been widely accepted for a  long time. In particular, Goethe proposed an alternative explanation for Newton's observations of multiple colors appearing from white light through the prism. According to the above Wikipedia page: &quot;For Newton, the prism is immaterial to the existence of colour, as all the colours already exist in white light, and the prism merely fans them out according to their refrangibility. Goethe sought to show that, as a turbid medium, the prism was an integral factor in the arising of colour.&quot; While the underlying &quot;physical&quot; theory is rather naive, the work of Goethe contains a lot of  valid information about color perception and had influence on a number of artists.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2017-02-01T20:09:00.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5642" PostTypeId="2" ParentId="5629" CreationDate="2017-02-01T21:26:08.130" Score="0" Body="&lt;p&gt;Most problems in Diophantus' &lt;em&gt;Arithmetica&lt;/em&gt; are solved in an analytical manner (which is why his style of algebra and its offshoots were described as &lt;em&gt;ars analytica&lt;/em&gt; in later times): one starts off assuming a solution exists, and one works one's way towards finding them explicitly. This is why Diophantus has to verify that his solutions indeed satisfy the equation (or &quot;make the problem&quot;, as he words it), because he started out with the unwarranted assumption that a solution existed at all.&lt;/p&gt;&#xA;" OwnerUserId="2353" LastActivityDate="2017-02-01T21:26:08.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5644" PostTypeId="1" AcceptedAnswerId="5663" CreationDate="2017-02-03T12:52:17.573" Score="28" ViewCount="1776" Body="&lt;p&gt;It only occurred to me recently, in connection with this &lt;A HREF=&quot;http://mathoverflow.net/questions/261162/chudnovsky-algorithm-and-pi-precision/261220&quot;&gt;MO posting&lt;/A&gt;, that the way I write the base of the logarithm is not shared by the rest of the world. I am Dutch, and I learned at school to write the base as a superscript, $^{a}\!\log b$, rather than as a subscript, $\log_a b$. Searching the web, I found one &lt;A HREF=&quot;http://tex.stackexchange.com/questions/63526/customized-log-notation&quot;&gt;posting&lt;/A&gt; that indicates they write it the same way in Indonesia, and I also notice that the Dutch &lt;A HREF=&quot;https://nl.wikipedia.org/wiki/Logaritme&quot;&gt;Wikipedia page&lt;/A&gt; gives both superscript and subscript notations as alternative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my conjecture is that the superscript notation is peculiar for The Netherlands and its (former) colonies. Is this correct, and if it is, how and when did this difference originate? Was there some influential text book that promoted this?&lt;/p&gt;&#xA;" OwnerUserId="1697" LastActivityDate="2022-06-21T18:35:14.990" Title="Why did I learn to write the base of the logarithm differently from the rest of the world?" Tags="|notation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5645" PostTypeId="1" CreationDate="2017-02-05T10:12:49.760" Score="24" ViewCount="4815" Body="&lt;p&gt;At a first glance, Mathematics and Literature look like two completely unrelated subjects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder whether there are examples of acclaimed mathematicians which wrote novels, poems, or other literary works not related to mathematics.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT.&lt;/strong&gt; Following a request by  HDE 226868, I invite all the answerers to give some more details about the author and the cited literary works, whenever possible. Thanks to all of you. &lt;/p&gt;&#xA;" OwnerUserId="3715" LastEditorUserId="3715" LastEditDate="2018-09-08T18:58:20.430" LastActivityDate="2021-10-25T09:00:56.153" Title="Literary works authored by mathematicians" Tags="|mathematics|mathematicians|biographical-details|examples|" AnswerCount="22" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5646" PostTypeId="2" ParentId="5645" CreationDate="2017-02-05T16:02:38.737" Score="8" Body="&lt;p&gt;Several mathematicians wrote excellent recollections about their own life. Some of them are of high quality as literature. My favorite ones are by Weyl, Rudin and L. Schwartz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;David Ruelle (a famous mathematical physicist) wrote some philosophical fiction.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-02-05T16:02:38.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5647" PostTypeId="2" ParentId="5645" CreationDate="2017-02-05T17:40:21.780" Score="13" Body="&lt;p&gt;Many mathematicians wrote their autobiographies, see the answers to &#xA;&lt;a href=&quot;https://mathoverflow.net/questions/102597/history-question-autobiography-of-mathematicians/102676#102676&quot;&gt;https://mathoverflow.net/questions/102597/history-question-autobiography-of-mathematicians/102676#102676&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quite a few also have engaged in writing literary fiction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/Sofia_Kovalevskaya&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sofya Kovalevskaya&lt;/a&gt; (apart from memories about her childhood) wrote a novel: &lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0873527909&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nihilist Girl&lt;/a&gt;, translated by Natasha Kolchevska with Mary Zirin ; introduction by Natasha Kolchevska. Modern Language Association of America (2001) ISBN 0-87352-790-9.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kovalevskaya also collaborated with Anne Charlotte Leffler on a play (according to the Wikipedia page of Leffler, the title is &lt;em&gt;The struggle for happiness&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In our times, &lt;a href=&quot;https://en.wikipedia.org/wiki/Mich%C3%A8le_Audin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michèle Audin&lt;/a&gt;, known for her achievements in symplectic geometry, is a member of &lt;a href=&quot;https://en.wikipedia.org/wiki/Oulipo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oulipo&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She recently published a novel: &lt;a href=&quot;https://www.amazon.fr/formule-Stokes-roman-Mich%C3%A8le-Audin/dp/2842252063&quot; rel=&quot;nofollow noreferrer&quot;&gt;La formule de Stokes&lt;/a&gt;, roman, Cassini, 2016, 297 p. (ISBN 9782842252069) However, in view of the preferred mode of creation by Oulipo, this work may be not entirely unrelated to mathematics. It is a novel, though, not a  scholarly or popularizing work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some other examples can be found among answers to this question:&lt;a href=&quot;https://mathoverflow.net/questions/45185/pseudonyms-of-famous-mathematicians/62966#62966&quot;&gt;https://mathoverflow.net/questions/45185/pseudonyms-of-famous-mathematicians/62966#62966&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2017-02-05T18:22:32.937" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5648" PostTypeId="2" ParentId="5645" CreationDate="2017-02-05T17:53:20.910" Score="7" Body="&lt;p&gt;The Romanian mathematician Dan Barbilian  was better known as the modernist poet &lt;a href=&quot;https://en.wikipedia.org/wiki/Ion_Barbu&quot; rel=&quot;noreferrer&quot;&gt;Ion Barbu&lt;/a&gt;. It's interesting that the English wikipedia page deals mostly with his mathematical research. Also it is worth mentioning that he was the first to introduce Modern Algebra (a la Van der Waerden) in Romania.&lt;/p&gt;&#xA;" OwnerUserId="1764" LastActivityDate="2017-02-05T17:53:20.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5650" PostTypeId="1" CreationDate="2017-02-05T20:36:33.743" Score="4" ViewCount="5745" Body="&lt;p&gt;How did people count seconds? How did people count minutes? How come that every second of a clock takes exactly as much time as needed to be 86400 to fulfill 24 hours and 31536000 to fulfill a year? How was time divided when people didn't have any electricity or devices to count time with? I've heard of the Roman clock (is that what it's called) and hourglasses, but those don't explain how someone could actually divide time. How did people divide an exact day to 24 hours, then 60 minutes and then 60 seconds to get exactly as much as they needed? And when they counted it correctly, how did they redo the count? Did musicians maybe remember how long a second lasts as they have rhythmical talent? I really don't understand how people actually made clocks. I don't even know if this question exists and where on StackExchange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any idea?&lt;/p&gt;&#xA;" OwnerUserId="5332" LastActivityDate="2021-11-25T04:15:23.490" Title="How did people count seconds before clocks were invented?" Tags="|time|measurement|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5651" PostTypeId="2" ParentId="5645" CreationDate="2017-02-05T21:45:53.407" Score="16" Body="&lt;p&gt;Hausdorff wrote many literary and philosophical works, including poems and a play, under the pseudonym &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Hausdorff#Hausdorff_as_philosopher_and_writer_.28Paul_Mongr.C3.A9.29&quot;&gt;Paul Mongré&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2033" LastActivityDate="2017-02-05T21:45:53.407" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5652" PostTypeId="2" ParentId="5645" CreationDate="2017-02-06T02:37:14.820" Score="7" Body="&lt;p&gt;The mathematician Jordan Ellenberg (&lt;a href=&quot;https://www.math.wisc.edu/~ellenber/&quot; rel=&quot;noreferrer&quot;&gt;University page&lt;/a&gt;, &lt;a href=&quot;http://www.jordanellenberg.com/&quot; rel=&quot;noreferrer&quot;&gt;personal website&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Jordan_Ellenberg&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;, &lt;a href=&quot;https://quomodocumque.wordpress.com/&quot; rel=&quot;noreferrer&quot;&gt;blog&lt;/a&gt;) wrote a novel called &lt;a href=&quot;http://www.jordanellenberg.com/the-grasshopper-king/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Grasshopper King&lt;/em&gt;&lt;/a&gt; in 2003. Does this satisfy your criteria?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Acclaimed mathematician&quot;: child prodigy, two-time Putnam fellow, Guggenheim Fellowship (2015)&lt;/li&gt;&#xA;&lt;li&gt;&quot;novels, poems, or other literary works not related to mathematics&quot;: &lt;em&gt;The Grasshopper King&lt;/em&gt; is a novel, and not related to mathematics in any way. The setting of the (very funny) book is that of a university professor (of languages/literature) who has stopped speaking, so it is related to academia in some sense, but I don't remember any mathematics in the book.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="432" LastActivityDate="2017-02-06T02:37:14.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5653" PostTypeId="2" ParentId="5645" CreationDate="2017-02-06T02:38:12.203" Score="17" Body="&lt;p&gt;Bertrand Russell wrote many books and essays on philosophy and social justice, and was awarded a Nobel Prize in Literature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And the Persian mathematician Omar Khayyám wrote poetry: quatrains that have been translated into English as the &lt;em&gt;Rubáiyát of Omar Khayyám&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5333" LastActivityDate="2017-02-06T02:38:12.203" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5654" PostTypeId="2" ParentId="5645" CreationDate="2017-02-06T03:01:30.013" Score="11" Body="&lt;p&gt;The mathematician E. T. Bell (known for Bell numbers, and the first stirrings of umbral calculus) wrote many science-fiction novels under the name John Taine. Arthur C. Clarke considered John Taine one of his heroes. (According to Constance Reid's &lt;em&gt;The Search for E. T. Bell: also known as John Taine&lt;/em&gt;, which is a wonderful work of history/biography.)&lt;/p&gt;&#xA;" OwnerUserId="432" LastActivityDate="2017-02-06T03:01:30.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5655" PostTypeId="2" ParentId="5650" CreationDate="2017-02-06T07:33:55.663" Score="14" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Electricity has nothing to do with the question. Clocks which could count minutes and seconds were purely mechanical and precise mechanical clocks (which could count seconds reliably) were invented in 18th century. Until 1970s most&#xA;wristwatches were mechanical. They are available nowadays as well but good ones are expensive.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;People did not count minutes and seconds before the wide spread of mechanical clocks. They counted only hours (using halves and quarters of an hour when necessary). In the rare cases when short intervals had to be measured they used heartbeats. (Galileo in his early experiments with pendulum used his heart to measure short time intervals). Galileo actually invented the pendulum clock which made it possible to measure minutes. It took about a century after that to increase the accuracy to seconds.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nevertheless the notion of minute and second existed. It was used by astronomers. For example they could measure the length of the year (and other astronomical intervals) with high precision. But this was not done with clocks.&#xA;This was done by very long observations. Suppose you know that 100 years=N days,&#xA;then 1 year is approximately N/100 days, with accuracy 1/100 day.&#xA;As it is possible to measure the moment of beginning of the year (say spring equinox) within one hour, without any clocks, if you have two measurements 100 years apart, you can compute the length of the year with 1/100&#xA;of an hour accuracy. This is why astronomers invented minutes and seconds, long before they came into common use.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-11-11T03:25:38.273" LastActivityDate="2021-11-11T03:25:38.273" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5656" PostTypeId="1" AcceptedAnswerId="5662" CreationDate="2017-02-06T07:59:44.590" Score="14" ViewCount="5491" Body="&lt;p&gt;When we want to perform division, we write e.g. $8/2$ (this is what we already learn at school). But when we want to express that $2$ is a divisor of $8$, we write: $2\mid 8$. What the heck?? I do find this very counterintuitive, I would have expected $8\mid 2$ instead. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is there a good reason to write $2\mid 8$ instead of $8\mid 2$, and who invented that notation? &lt;/p&gt;&#xA;" OwnerUserId="4721" LastEditorUserId="2193" LastEditDate="2017-04-09T14:58:42.610" LastActivityDate="2022-12-08T05:13:36.283" Title="Who invented the divisibility symbol and why is it backwards?" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5657" PostTypeId="2" ParentId="5623" CreationDate="2017-02-06T15:54:48.737" Score="3" Body="&lt;p&gt;Casey Westerman, archivist at the IAS in Princeton, advises that André Weil's papers are held at the Institut de France Académie des Sciences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.academie-sciences.fr/fr/Transmettre-les-connaissances/inventaires-des-fonds-d-archives-personnelles.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.academie-sciences.fr/fr/Transmettre-les-connaissances/inventaires-des-fonds-d-archives-personnelles.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the Bibliothèque nationale de France holds Simone Weil's papers including her correspondence with André.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2017-02-06T15:54:48.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5660" PostTypeId="2" ParentId="5617" CreationDate="2017-02-06T21:02:58.277" Score="2" Body="&lt;p&gt;One incredible discovery of the 1800s was that when you speak of geometry, you need to specify &quot;where your geometry is located&quot; and specifically what constitutes points and lines. At the time this was applied to obtain &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperbolic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyperbolic Geometry&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Elliptic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elliptic Geometry&lt;/a&gt;. Recently, the field of &lt;a href=&quot;https://en.wikipedia.org/wiki/Finite_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Finite Geometry&lt;/a&gt; has become the subject of much foundational research. Finite Geometries arise when you define the concept of points and lines in such a way as to give rise to only finitely many points and lines, and has fundamental connections to theoretical computer science, in particular my field of linear algebra and combinatorics. &lt;a href=&quot;http://digitalcommons.mtu.edu/cgi/viewcontent.cgi?article=1198&amp;amp;context=etds&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; you can see a 140 page masters' thesis on the applications of finite geometry to coding theory. There are a number of open questions that are fundamentally important in finite geometry, particularly the classification of finite projective planes. Although finite projective planes are very important in finite geometry, for $n\geq12$ we don't know very much about the finite projective planes of order $n$, and don't even know how to prove that any exist for most $n$! See &lt;a href=&quot;https://en.wikipedia.org/wiki/Projective_plane#Finite_projective_planes&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; for more on finite projective planes.&lt;/p&gt;&#xA;" OwnerUserId="5340" LastActivityDate="2017-02-06T21:02:58.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5662" PostTypeId="2" ParentId="5656" CreationDate="2017-02-06T21:17:51.377" Score="14" Body="&lt;p&gt;In mathematics, we often write relations between $a$ and $b$ in the form $aRb$. I mean this both in the sense that we write that string to represent an abstract relation, as well as using that form to write expressions with particular relations. In almost every case, these are read as &quot;$a$ [relation] $b$.&quot; For a few examples, we have&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;$a:=b$, &quot;is defined to be&quot;&lt;/li&gt;&#xA;&lt;li&gt;$a\geq b$, &quot;is greater than or equal to&quot;&lt;/li&gt;&#xA;&lt;li&gt;$a\in b$, &quot;in / is an element of&quot;&lt;/li&gt;&#xA;&lt;li&gt;$a\subseteq$ &quot;is a subset of&quot;&lt;/li&gt;&#xA;&lt;li&gt;$a\to b$, &quot;maps to / is mapped to&quot;&lt;/li&gt;&#xA;&lt;li&gt;$a=O(b)$, &quot;is big-O of&quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Notably, every relation on this list is antisymmetric, so the ordering of $a$ first and then $b$ is important. This list is extremely incomplete, and there are dozens more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The correct reading of the symbol $|$ is &quot;divides / is a divisor of.&quot; When interpreted in this way, $a|b$ aka &quot;$a$ divides $b$&quot; fits this very well established pattern perfectly. Although it might be counter-intuitive to someone who has more experience with arithmetic than mathematics, it's actually a manifestation of a highly standardized pattern.&lt;/p&gt;&#xA;" OwnerUserId="5340" LastEditorUserId="5340" LastEditDate="2017-02-10T14:27:16.710" LastActivityDate="2017-02-10T14:27:16.710" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5663" PostTypeId="2" ParentId="5644" CreationDate="2017-02-06T22:03:04.373" Score="12" Body="&lt;p&gt;It helps to remember that there was no consensus notation for logarithms well into 20th century, with &lt;span class=&quot;math-container&quot;&gt;$\mathrm{l}\,x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\log x$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Log}\,x$&lt;/span&gt; often used by different authors and in different senses. The superscript notation was not peculiar to the Netherlands, at least not originally. According to &lt;a href=&quot;https://books.google.com/books/about/A_History_of_Mathematical_Notations.html?id=bT5suOONXlgC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History of Mathematical Notations&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Another notation was proposed by Crelle, in which the base is written above and to the left of the logarithm. This notation is sometimes encountered in more recent books. Thus, Stringham denotes a logarithm to the base &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; by &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\,^b\!\log$&lt;/span&gt;&amp;quot;; he denotes also a natural logarithm by &amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;&amp;quot; and a logarithm to the complex modulul&lt;/em&gt; [sic!] &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, by &amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\log_k$&lt;/span&gt;&amp;quot;. Stolz and Gmeiner signify by &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\,^a\log.b$&lt;/span&gt;&amp;quot; the &amp;quot;logarithm of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; to the base &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&amp;quot;. Crelle in 1831 and Martin Ohm in 1846 write the base above the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\,\log$&lt;/span&gt;&amp;quot;. This symbolism is found in many texts, for instance in Kimbly's Elementar-Mathematik.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Crelle founded and edited in 1826-1855 a very influential Journal für die Reine und Angewandte Mathematik, commonly nicknamed &lt;a href=&quot;https://en.wikipedia.org/wiki/Crelle%27s_Journal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crelle's Journal&lt;/a&gt;, or just Crelle. &lt;a href=&quot;https://archive.org/details/uniplanaralgebra00stri&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stringham's Uniplanar Algebra&lt;/a&gt; (San Francisco, 1893) was apparently influential in the adoption of &lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt; notation for natural logarithms, see &lt;a href=&quot;https://math.stackexchange.com/questions/1694/how-did-the-notation-ln-for-log-base-e-become-so-pervasive&quot;&gt;How did the notation “&lt;span class=&quot;math-container&quot;&gt;$\ln$&lt;/span&gt;” for “log base e” become so pervasive?&lt;/a&gt; Stolz and Gmeiner promoted their notation in the monograph &lt;a href=&quot;https://archive.org/details/theoretischearit00stoluoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theoretische Arithmetik&lt;/a&gt; (Leipzig, 1902), their &lt;a href=&quot;https://books.google.com/books?id=s3NRAQAAMAAJ&amp;amp;pg=PA119&amp;amp;lpg=PA119&amp;amp;dq=stolz+gmeiner&amp;amp;source=bl&amp;amp;ots=M2An0mNjUs&amp;amp;sig=HqZxFaYbFE7Bqk-cSD9_teWliOc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiX5vnDufzRAhVIjFQKHX0DDxcQ6AEIMjAH#v=onepage&amp;amp;q=stolz%20gmeiner&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Function Theory&lt;/a&gt; was also influential. But Crelle was a German, Stringham was an American, and Stolz and Gmeiner were Austrians. It is possible that a Dutch textbook author took a liking to their notation, but I would expect such a text to be in Dutch.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-06-21T18:35:14.990" LastActivityDate="2022-06-21T18:35:14.990" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5664" PostTypeId="2" ParentId="5507" CreationDate="2017-02-07T04:28:45.977" Score="1" Body="&lt;p&gt;Mathematics advances rapidly. A generation ago, ultrafiltres and category theory were pretty new and niche, now they are pretty close to foundational as concepts in algebra and topology. Theoretical Computer Science has changed massively in the past 30 years. Some of the greatest results in the field are recent discoveries. PCP Theorem, the Regularity Lemma, and the Triangle Removal Lemma are all pretty new. The classification of finite simple groups is another thing that has massively changed mathematics. Almost all of these things caused massive changes at the undergraduate level, and a few are first year graduate topics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By comparison, I think quantum computing and M-theory are the main changed in physics that would percolate to the undergraduate level (physicists please double check me though!).&lt;/p&gt;&#xA;" OwnerUserId="5340" LastActivityDate="2017-02-07T04:28:45.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5665" PostTypeId="2" ParentId="5645" CreationDate="2017-02-07T10:03:39.500" Score="7" Body="&lt;p&gt;Rudy Rucker writes science fiction. Some of his novels explore mathematical concepts such as infinity.&#xA;Ian Stewart, who is best known for his popular science and mathematics books (including the Science of Discworld books), has also written a couple of science fiction novels with Jack Cohen.&lt;/p&gt;&#xA;" OwnerUserId="4629" LastActivityDate="2017-02-07T10:03:39.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5666" PostTypeId="1" AcceptedAnswerId="5683" CreationDate="2017-02-07T10:40:15.817" Score="7" ViewCount="4187" Body="&lt;p&gt;Canonical coordinates are &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;coordinates $q_i$ and $p_i$ in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations:&#xA;  \begin{equation}&#xA;\{q_i,q_j\} = 0, \quad \{p_i,p_j\} = 0, \quad \{q_i,p_j\} = \delta_{ij},&#xA;\end{equation}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-Wikipedia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From this it is clear that we get canonical transformations and, with quantum mechanics, canonical commutation relations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did these coordinates start being called canonical and what was the reasoning?&lt;/p&gt;&#xA;" OwnerUserId="5059" LastEditorUserId="658" LastEditDate="2017-02-15T06:04:14.993" LastActivityDate="2018-01-25T19:51:54.413" Title="Why are canonical coordinates canonical?" Tags="|physics|terminology|classical-mechanics|symplectic-geometry|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5667" PostTypeId="2" ParentId="5645" CreationDate="2017-02-07T17:37:32.987" Score="6" Body="&lt;p&gt;Donald Knuth wrote a novel about surreal numbers (and he coined the term): &quot;Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3090" LastActivityDate="2017-02-07T17:37:32.987" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5668" PostTypeId="2" ParentId="5645" CreationDate="2017-02-07T17:44:52.153" Score="5" Body="&lt;p&gt;Greg Egan is a hard science fiction writer (Permutation City, Oceanic, Distress etc.), he also does some interesting computer science programming/research and he has collaborated with John Baez on many projects.&lt;/p&gt;&#xA;&lt;p&gt;Fiction - &lt;a href=&quot;https://www.gregegan.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gregegan.net&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Math research - &lt;a href=&quot;https://www.gregegan.net/BIBLIOGRAPHY/Online.html#NonFiction&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gregegan.net/BIBLIOGRAPHY/Online.html#NonFiction&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3090" LastEditorUserId="3090" LastEditDate="2021-10-25T09:00:56.153" LastActivityDate="2021-10-25T09:00:56.153" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5669" PostTypeId="2" ParentId="5645" CreationDate="2017-02-07T20:40:20.597" Score="6" Body="&lt;p&gt;Needless to say, &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Grothendieck&quot; rel=&quot;noreferrer&quot;&gt;Alexander Grothendieck&lt;/a&gt; (1928-2014) was one of the greatest (and perhaps the most visionary) mathematician of the XX century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1987-88 he wrote a manuscript entitled “&lt;em&gt;La Clef des Songes - ou Dialogue avec le Bon Dieu&lt;/em&gt;” (which can be translated as “The key of dreams - or conversation with the good Lord”). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The manuscript consist of 6 chapters, 315 pages, plus hundreds of pages of added notes. &#xA;It describes Grothendieck's arrival at the conviction of the existence of God, through his discovery of the meaning and significance of dreams. It also contains a wealth of personal biographical details on his early childhood, on the period spent in a French internment camp for undesirables during World War II and then in the Protestant foyer for Jewish children in Le Chambon sur Lignon, and on his parents. [&lt;a href=&quot;https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/recoltesetc.php&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt;]&lt;/p&gt;&#xA;" OwnerUserId="3715" LastActivityDate="2017-02-07T20:40:20.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5670" PostTypeId="2" ParentId="5645" CreationDate="2017-02-07T21:39:44.783" Score="7" Body="&lt;p&gt;An obscure one, perhaps.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Emanuel Lasker cowrote Vom Menschen die Geschichte (&quot;History of Mankind&quot;), a play, with his brother in 1925.  He has other works, although at least part of those were related to mathematics and game theory with his prediction in &quot;Kampf&quot; (Struggle) of a social science of the future.  In many ways this foresaw the application of game theory in the social sciences.  &lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2017-02-07T21:53:44.823" LastActivityDate="2017-02-07T21:53:44.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5671" PostTypeId="2" ParentId="5650" CreationDate="2017-02-07T22:39:58.373" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How did people count seconds before clocks were invented?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In general, they didn't, because there was no need to to so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The need for accuracy to the level of a second or less is a very recent need. Before the invention of trains and telegraphs, there was no need for even hour level accuracy, let alone minute or second level accuracy. Whether the roads were dry or muddy could spell the difference between being days early versus days late (or sometimes weeks late) for an appointment.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2017-02-07T22:39:58.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5672" PostTypeId="1" CreationDate="2017-02-08T05:02:39.650" Score="6" ViewCount="439" Body="&lt;p&gt;The height of medical science used to be applying leeches and administering useless poisons, like mercury. Undoubtedly many patients were killed by harmful treatments, and few if any were helped. Have historians of medicine estimated the year when calling a doctor first had a net positive impact on your chances of survival?&lt;/p&gt;&#xA;" OwnerUserId="5348" LastActivityDate="2017-02-08T12:20:57.693" Title="At what point in the history of medicine did doctors start having a net positive effect?" Tags="|medicine|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5673" PostTypeId="1" CreationDate="2017-02-08T05:52:04.167" Score="4" ViewCount="1163" Body="&lt;p&gt;Is it by Gibbs?  Or Boltzmann? I do not expect Maxwell. The point is, how was the textbook introduction of partition function today developed. &lt;/p&gt;&#xA;" OwnerUserId="4079" LastEditorUserId="4251" LastEditDate="2020-06-06T13:02:14.997" LastActivityDate="2023-05-06T11:17:17.037" Title="Who introduced the partition function?" Tags="|physics|statistical-mechanics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5674" PostTypeId="1" AcceptedAnswerId="5675" CreationDate="2017-02-08T07:26:13.980" Score="4" ViewCount="666" Body="&lt;p&gt;I was wondering when was the sine function (I suppose cosine too) first introduced or defined and what was the need for?&lt;/p&gt;&#xA;" OwnerUserId="5343" LastEditorUserId="5343" LastEditDate="2017-02-09T15:25:15.083" LastActivityDate="2017-02-12T07:26:50.687" Title="First appearance of the sine function" Tags="|discoveries|calculus|" AnswerCount="2" CommentCount="0" ClosedDate="2017-02-11T23:44:22.070" ContentLicense="CC BY-SA 3.0" />
  <row Id="5675" PostTypeId="2" ParentId="5674" CreationDate="2017-02-08T07:56:19.557" Score="3" Body="&lt;p&gt;Trigonometry was invented by the Greeks of Hellenistic period. The only trigonometric function they used was the chord related to the sine by&#xA;$$\mathrm{chd}\,x=2\sin(x/2).$$&#xA;The first known tables of the chord were composed by Ptolemy.&#xA;Sine was introduced in the Indian treatease Surya Siddhanta, see&#xA; &lt;a href=&quot;https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81&lt;/a&gt;&#xA;whose exact date is unknown (6th century), and the first table of sines was in Ariabhata (6th century). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The name sine is due to Gerard of Cremona who translated Arab text to the Latin in 12th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The use of sine vs chord is a question of convenience in a particular problem.&#xA;(There are even modern papers in which the chord function is used). The main application of trigonometry was in astronomy. See also &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_trigonometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/History_of_trigonometry&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-02-12T07:26:50.687" LastActivityDate="2017-02-12T07:26:50.687" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5676" PostTypeId="2" ParentId="5674" CreationDate="2017-02-08T11:26:56.970" Score="3" Body="&lt;p&gt;To add more details to Alexandre Eremenko's good answer, here is what I've found &lt;a href=&quot;http://mapage.noos.fr/r.ferreol/langage/notations/notations.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (in French):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First time: introduced by the indian astronomer Âryabhata (6th century) who used the word &quot;jîva&quot; meaning &lt;em&gt;rope or chord&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Then: the arabic mathematician Al-Fazzârî (8th century) translated this word to arabic as &quot;jîba&quot;, a word that doesn't necessarily have an arabic meaning.&lt;/li&gt;&#xA;&lt;li&gt;After that: the italian writer and translator Gerard of Cremona (12th century) confused this word with the arabic word &quot;jaîb&quot; meaning &lt;em&gt;pocket, sinus or cavity&lt;/em&gt; and hence translated it to latin as &lt;strong&gt;sinus&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5343" LastActivityDate="2017-02-08T11:26:56.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5677" PostTypeId="2" ParentId="5672" CreationDate="2017-02-08T12:15:43.733" Score="3" Body="&lt;p&gt;The problem with this question is that nobody can measure this &quot;net impact&quot;, especially in the past. Very ancient (stone age) remains show that surgeries were performed, and not only on the limbs but also on the scull. Perhaps it helped in some cases and harmed in other cases, but how can one perform any statistics of this sort?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in the recent time, we know of course that average life expectancy increased&#xA;dramatically, but how much of this can be attributed to medicine, and how much to&#xA;the general improvement of life conditions (hygiene etc.)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless some milestones can be marked of course. Invention of smallpox vaccine in the late 18th century certainly had enormous effect. Also the discovery of germs as causes of many diseases in 19th century, antiseptic,&#xA;and discovery of antibiotics in 20th century. There is no doubt that these discoveries saved many millions of lives.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-02-08T12:20:57.693" LastActivityDate="2017-02-08T12:20:57.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5678" PostTypeId="2" ParentId="5645" CreationDate="2017-02-09T17:55:15.090" Score="5" Body="&lt;p&gt;Christos Papadimitriou collaborated with Apostolos Doxiadis (who had written Uncle Petros and Goldbach's Conjecture (2000)) to write the graphic novel Logicomix: An epic search for truth (2009). As the Amazon page describes it,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This story is at the same time a historical novel and an accessible explication of some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a highly satisfying tale.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2822" LastEditorUserId="4251" LastEditDate="2021-10-21T16:36:41.210" LastActivityDate="2021-10-21T16:36:41.210" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5679" PostTypeId="2" ParentId="5666" CreationDate="2017-02-10T01:29:07.940" Score="1" Body="&lt;p&gt;I was told they are called &quot;canonical&quot; because of their prevalence and comparative simplicity.  I believe the etymology is from the Greek &quot;kanon&quot;, meaning &quot;standard&quot; or &quot;model&quot;, or &quot;usual&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastActivityDate="2017-02-10T01:29:07.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5680" PostTypeId="1" AcceptedAnswerId="5681" CreationDate="2017-02-10T16:30:38.453" Score="8" ViewCount="270" Body="&lt;p&gt;In the theory of boolean functions Sheffer's stroke is well-known. It is generalized for k-valued logic and known as Webb function:&#xA;$$&#xA;\mathrm{Webb}(x,y)=\max(x,y)+1;&#xA;$$&#xA;I want to find some information about Donald L. Webb -- the possible creator of this function. But I have gound nothing on the internet, except his name and the fact that he wrote an article in 1935.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would really appreciate any information on this topic.&lt;/p&gt;&#xA;" OwnerUserId="4228" LastActivityDate="2018-05-24T16:57:05.220" Title="Who is Donald L. Webb" Tags="|mathematicians|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5681" PostTypeId="2" ParentId="5680" CreationDate="2017-02-10T17:19:09.750" Score="15" Body="&lt;p&gt;Because &lt;strong&gt;HOW&lt;/strong&gt; I got this information might be as useful as the information itself (the &quot;teach a person how to fish&quot; viewpoint), I'll describe exactly the process I went through, which took about 2 or 3 minutes (helps to have the websites I mention bookmarked).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I began with a search at the &lt;a href=&quot;https://zbmath.org/&quot;&gt;Zentralblatt Math search page&lt;/a&gt; (because it's freely available, unlike Math. Reviews) and found &lt;a href=&quot;https://zbmath.org/?t=&amp;amp;s=0&amp;amp;r=xx&amp;amp;any=&amp;amp;au=Webb%2C+Donald&quot;&gt;these 3 papers by him&lt;/a&gt;. Then I went to the &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/search.php&quot;&gt;Mathematics Genealogy Project&lt;/a&gt; webpages, where I found that he &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=10389&quot;&gt;got his Ph.D.&lt;/a&gt; in 1936 under Eric Temple Bell at Caltech. The Math. Genealogy site doesn't give his middle name, so I then went to the &lt;a href=&quot;http://dissexpress.umi.com/dxweb/search.html&quot;&gt;ProQuest Search page&lt;/a&gt; and entered &quot;Webb, Donald&quot; (quotes included) in the author window and &quot;logics&quot; (quotes not included) in the title window, but I didn't learn anything new. FYI, the ProQuest page is a good place to try if the Math. Genealogy site doesn't give you a middle name. Then I tried a basic google books search, using &lt;a href=&quot;https://www.google.com/search?tbm=bks&amp;amp;q=%22webb%2C+Donald+L%22+math&quot;&gt;&quot;Webb, Donald L&quot; + math&lt;/a&gt;, where under the short google-excerpts in the &quot;Who's Who in Philosophy&quot; items I learned that he was born on 17 November 1907 in Wakita, Oklahoma; and he got his BS and MS degrees at University of Arizona in 1931 and 1933. Knowing his birth year allowed me to refine the google-books search to &lt;a href=&quot;https://www.google.com/search?tbm=bks&amp;amp;q=%22webb%2C+Donald%22+math+1907&quot;&gt;&quot;Webb, Donald&quot; + math + 1907&lt;/a&gt;, where I intentionally left off his middle initial so that I could possibly determine his middle name, and I was able to see among the excerpts of the hits that his middle name is Loomis (in the &quot;Who's Who in American Education&quot; item). Finally, knowing his middle name, I did a basic google search for &quot;Donald Loomis Webb&quot; (not using &quot;1907&quot;, but prepared to include &quot;1907&quot; if there seemed to be more than one person with that name or too many hits to be useful), and I found &lt;a href=&quot;https://www.findagrave.com/cgi-bin/fg.cgi/%3Cbr%3E..http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&amp;amp;GRid=169569985&quot;&gt;this gravesite page&lt;/a&gt;, where I learned that he died on 1 June 1971 in Tucson, Arizona.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Knowing these details about his life, you should now be able to conduct a variety of searches to find out more about him.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2017-02-10T17:59:01.290" LastActivityDate="2017-02-10T17:59:01.290" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5682" PostTypeId="1" CreationDate="2017-02-11T05:01:27.150" Score="6" ViewCount="5599" Body="&lt;p&gt;Can anyone give a list of famous scientists in the Nazi Party? Is there a complete list? &lt;/p&gt;&#xA;" OwnerUserId="4079" LastEditorUserId="4251" LastEditDate="2020-07-11T19:08:30.430" LastActivityDate="2021-09-08T01:33:53.757" Title="Famous scientists in the Nazi party" Tags="|mathematicians|physicists|nazi-germany|scientists|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5683" PostTypeId="2" ParentId="5666" CreationDate="2017-02-11T06:06:40.187" Score="15" Body="&lt;p&gt;Such coordinates were called canonical because they are those in which equations of motion (or, of the hamiltonian flow of a function $H$) take the “canonical form”&#xA;$$&#xA;\frac{dq_i}{dt}=\frac{\partial H}{\partial p_i},&#xA;\qquad&#xA;\frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i}&#xA;$$&#xA;first written by Poisson (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k4336710/f271&quot; rel=&quot;noreferrer&quot;&gt;1809, pp. 272, 313&lt;/a&gt;), Lagrange (&lt;a href=&quot;https://archive.org/stream/mmoiresdelinst10inst#page/343&quot; rel=&quot;noreferrer&quot;&gt;1810, p. 350&lt;/a&gt;), and Hamilton (&lt;a href=&quot;https://archive.org/stream/philtrans04757858/04757858&quot; rel=&quot;noreferrer&quot;&gt;1835, p. 98&lt;/a&gt;). &#xA;Actually, none of them called that form &lt;em&gt;canonical&lt;/em&gt; yet: according to the &lt;em&gt;Encyklopädie&lt;/em&gt; (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN360617913&amp;amp;DMDID=DMDLOG_0325&quot; rel=&quot;noreferrer&quot;&gt;1935, p. 573&lt;/a&gt;) the first person to do so was Jacobi (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k2964f/f61&quot; rel=&quot;noreferrer&quot;&gt;1837, pp. 65-66&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On trouve au moyen de ce&#xA;  théorème, par le calcul même, des éléments dont les valeurs différentielles, dans&#xA;  le mouvement troublé, prennent la forme simple qu'elles ont dans le théorème,&#xA;  forme que je désigne dans mon mémoire sous le nom de &lt;strong&gt;&lt;em&gt;canonique&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... and Thomson-Tait (&lt;a href=&quot;https://books.google.com/books?id=6odDAQAAMAAJ&amp;amp;pg=PA254&quot; rel=&quot;noreferrer&quot;&gt;1867, p. 254&lt;/a&gt;) commented:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This is the celebrated “canonical&#xA;  form” of the equations of motion of a system, though why it&#xA;  has been so called it would be hard to say.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As to the practice of calling the &lt;em&gt;coordinates&lt;/em&gt; or “elements” $p_i, q_i$ themselves “canonical”, it seems to originate also with Jacobi in the aforementioned memoir, completed in 1838 but only published posthumously (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002151529&quot; rel=&quot;noreferrer&quot;&gt;1862, p. 128&lt;/a&gt;; German translation: &lt;a href=&quot;https://archive.org/stream/neuemethodezuri00kowagoog#page/n158&quot; rel=&quot;noreferrer&quot;&gt;1906, p. 153&lt;/a&gt;): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Systema&#xA;  elementorum, quae in modum praecedentium per aequationes differentiales canonicas determinantur, et ipsum dicere convenit &lt;strong&gt;&lt;em&gt;canonicum elementorum&lt;/em&gt;&lt;/strong&gt; systema.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While this practice did not universally catch on at first (Thomson-Tait, Poincaré (&lt;a href=&quot;http://www.archive.org/details/lesmthodesnouv002poin&quot; rel=&quot;noreferrer&quot;&gt;1893&lt;/a&gt;), Whittaker (&lt;a href=&quot;https://archive.org/stream/treatisanalytdyn00whitrich#page/n281&quot; rel=&quot;noreferrer&quot;&gt;1917&lt;/a&gt;) or the &lt;em&gt;Encyklopädie&lt;/em&gt; call the equations canonical, but not the variables), it was adopted by e.g. Donkin (&lt;a href=&quot;https://archive.org/stream/atreatiseoninfi12pricgoog#page/n577&quot; rel=&quot;noreferrer&quot;&gt;1862, pp. v, 550&lt;/a&gt;), Tisserand (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k16412q.f263&quot; rel=&quot;noreferrer&quot;&gt;1868, p. 258&lt;/a&gt;; &lt;a href=&quot;https://archive.org/stream/traitemecanique00tissrich#page/164&quot; rel=&quot;noreferrer&quot;&gt;1889, p. 164&lt;/a&gt;), Schering (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002019558&quot; rel=&quot;noreferrer&quot;&gt;1873, p. 23&lt;/a&gt;), Lie (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002243008&quot; rel=&quot;noreferrer&quot;&gt;1874, p. 258&lt;/a&gt;), Routh (&lt;a href=&quot;https://archive.org/stream/advanceparttreat00routrich#page/n321&quot; rel=&quot;noreferrer&quot;&gt;1892, pp. 304-306&lt;/a&gt;), Dziobek (&lt;a href=&quot;https://archive.org/stream/mathematicaltheo00dziorich#page/102&quot; rel=&quot;noreferrer&quot;&gt;1892, pp. 102-103&lt;/a&gt;), Charlier (&lt;a href=&quot;https://archive.org/stream/diemechanikdesh04chargoog#page/n70&quot; rel=&quot;noreferrer&quot;&gt;1902, pp. 56-58&lt;/a&gt;), Dirac (&lt;a href=&quot;http://dx.doi.org/10.1098/rspa.1925.0150&quot; rel=&quot;noreferrer&quot;&gt;1925, p. 651&lt;/a&gt;), etc. In fact it must have leaked &lt;em&gt;somehow&lt;/em&gt; between 1838 and 1862, for Cayley has it already in (&lt;a href=&quot;https://archive.org/stream/reportofbritisha57brit/#page/n71&quot; rel=&quot;noreferrer&quot;&gt;1858, p. 9&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;18. There is, however, one important point which requires to be adverted to. Lagrange, in the memoir of 1810, and the second edition of the ‘Mécanique Analytique,’&#xA;  remarks, that for a particular system of arbitrary constants, viz., if $\alpha,...$ denote the&#xA;  initial values of the coordinates $\xi,..$ and $\lambda,..$ denote the initial values of $\smash[b]{\frac{dT}{d\xi'}},...$ then&#xA;  the equations for the variations of the elements take the very simple form&#xA;  $$&#xA;\frac{d\alpha}{dt}=-\frac{d\Omega}{d\lambda}..,\quad&#xA;\frac{d\lambda}{dt}=\frac{d\Omega}{d\alpha},...&#xA;$$&#xA;  This is, in fact, the original idea and simplest example of a system of &lt;strong&gt;&lt;em&gt;canonical&#xA;  elements&lt;/em&gt;&lt;/strong&gt;; viz. of a system composed of pairs of elements, $\alpha, \lambda$, the variations of&#xA;  which are given in the form just mentioned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-08-25T22:00:32.900" LastActivityDate="2017-08-25T22:00:32.900" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5684" PostTypeId="2" ParentId="5682" CreationDate="2017-02-11T08:11:41.553" Score="15" Body="&lt;p&gt;The most famous mathematicians who were the members of NSDAP are Ludwig Bieberbach and Oswald Teichmüller.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2021-09-07T19:28:22.687" LastActivityDate="2021-09-07T19:28:22.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5685" PostTypeId="2" ParentId="5645" CreationDate="2017-02-11T08:51:00.613" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Dixmier.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacques Dixmier&lt;/a&gt; published two books of short stories in science fiction, &lt;em&gt;L'Aurore des dieux&lt;/em&gt; (1993) and &lt;em&gt;Le Septième arrhe&lt;/em&gt; (1995).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And one more co-authored with &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Connes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alain Connes&lt;/a&gt; and Danye Chéreau: &lt;a href=&quot;https://www.odilejacob.fr/catalogue/art-et-litterature/romans-scientifiques/spectre-d-atacama_9782738141361.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Le Spectre d’Atacama&lt;/em&gt;&lt;/a&gt; (2018); recently &lt;a href=&quot;http://www.madore.org/~david/weblog/d.2018-06-18.2528.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;reviewed&lt;/a&gt; by @Gro-Tsen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/mathscinet/MRAuthorID/621986&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Bessis&lt;/a&gt; published a novel (&lt;a href=&quot;https://www.editions-allia.com/fr/livre/394/sprats&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sprats&lt;/em&gt;&lt;/a&gt;, 2005) and a &quot;sentimental atlas&quot; (&lt;a href=&quot;https://www.editions-allia.com/fr/livre/395/ars-grammatica&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Ars grammatica&lt;/em&gt;&lt;/a&gt;, 2006).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Recommended:&lt;/strong&gt; Michèle Audin, &lt;a href=&quot;http://store.cassini.fr/documents-essais-culture-scientifique/2-la-formule-de-stokes-roman.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;La Formule de Stokes, roman&lt;/em&gt;&lt;/a&gt; (2016), already mentioned in this thread; gave me several fits of the giggles in the past hour.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-07-29T18:40:30.063" LastActivityDate="2018-07-29T18:40:30.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5686" PostTypeId="2" ParentId="5682" CreationDate="2017-02-11T14:20:31.990" Score="8" Body="&lt;p&gt;Philip Lenard, who with Stark founded the &lt;em&gt;Deutsche Physik&lt;/em&gt; movement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also examples of scientists who claimed to be &quot;nominal members only&quot;: Wernher von Braun comes to mind (even if he admitted knowing about the slave labor used to construct the V-2 rockets); he was not only a member of the Nazi party but also of the SS.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorUserId="4251" LastEditDate="2020-05-12T17:00:15.300" LastActivityDate="2020-05-12T17:00:15.300" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5687" PostTypeId="1" AcceptedAnswerId="5688" CreationDate="2017-02-11T18:30:17.877" Score="1" ViewCount="974" Body="&lt;p&gt;After 2 month long search on the net, I was lucky enough to find a pdf on how Newton found the series for sine. It was a beautiful derivation mostly geometrical. But he used the Binomial Series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Now what I look for is Newton's proof (sort of proof) of the Binomial Series . I have read that he interpolated areas under circles and hyperbolas but a succinct answer jumps in terms of what he actually did without is what I am looking for. I know he made tables then followed the pattern. But there must be a lot of geometrical work too which would have led him to this point. All sorts of interpolation.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's what I want to understand. &lt;strong&gt;What did he actually do ? What was his geometrical work regarding it as well as what was his sort Interpolation of areas&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="4666" LastEditorUserId="5788" LastEditDate="2017-09-24T09:00:42.803" LastActivityDate="2017-09-24T09:00:42.803" Title="Newton , Binomial Series and Power Series , and James Gregory" Tags="|calculus|newton|combinatorics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5688" PostTypeId="2" ParentId="5687" CreationDate="2017-02-11T19:00:40.870" Score="2" Body="&lt;p&gt;See : Derek T. Whiteside, &lt;a href=&quot;https://www.jstor.org/stable/3612767?Search=yes&amp;amp;resultItemClick=true&amp;amp;searchText=whiteside&amp;amp;searchUri=%2Faction%2FdoBasicSearch%3Ffilter%3Djid%253A10.2307%252Fj100271%26Query%3Dwhiteside%26amp%3D&amp;amp;seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's Discovery of the General Binomial Theorem&lt;/a&gt; , &lt;em&gt;Math. Gaz.&lt;/em&gt; 45 (Oct.1961).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an English transaltion of &lt;a href=&quot;https://en.wikipedia.org/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas&quot; rel=&quot;nofollow noreferrer&quot;&gt;De analysi per aequationes numero terminorum infinitas&lt;/a&gt; (1669) see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.it/books?id=z-RJAAAAMAAJ&amp;amp;pg=PA321&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir Isaac Newton's Two Treatises: Of the Quadrature of Curves and Analysis by Equations of an Infinite Number of Terms, Explained : Containing the Treatises Themselves, Translated Into English, with a Large Commentary ; in which the Demonstrations are Supplied where Wanting, the Doctrine Illustrated, and the Whole Accommodated to the Capacities of Beginners, for Whom it is Chiefly Designed&lt;/a&gt; (1745), page 321-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-02-12T08:40:30.540" LastActivityDate="2017-02-12T08:40:30.540" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5689" PostTypeId="1" AcceptedAnswerId="5694" CreationDate="2017-02-12T09:22:01.543" Score="2" ViewCount="717" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Clifford_torus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clifford torus&lt;/a&gt; shows up a lot in differential geometry in connection with minimal surfaces, for example in the &lt;a href=&quot;https://arxiv.org/abs/1203.6597&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lawson's conjecture&lt;/a&gt;, the &lt;a href=&quot;https://eudml.org/doc/174480&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oh's Conjecture&lt;/a&gt;, etc. It can be described as the following surface in $\mathbb C^2$: &#xA;$$T = \left\{ (z_1, z_2) \in \mathbb C^2: |z_1| = |z_2| = \frac{1}{\sqrt 2}\right\}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd like to know in what context it was introduced, and why it is called the Clifford torus. From the Wikipedia page, Clifford &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Kingdon_Clifford#Mathematician&quot; rel=&quot;nofollow noreferrer&quot;&gt;is a geometer&lt;/a&gt;, but it does not mention if he used this torus in any way. &lt;/p&gt;&#xA;" OwnerUserId="5363" LastEditorUserId="55" LastEditDate="2017-02-12T23:21:06.947" LastActivityDate="2017-02-13T19:32:30.877" Title="Did Clifford introduce the &quot;Clifford torus&quot;, and for what purpose?" Tags="|mathematics|differential-geometry|riemannian-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5690" PostTypeId="2" ParentId="5682" CreationDate="2017-02-12T15:01:46.680" Score="6" Body="&lt;p&gt;After the Second World War, &lt;a href=&quot;https://en.wikipedia.org/wiki/Operation_Paperclip&quot; rel=&quot;nofollow noreferrer&quot;&gt;Operation Paperclip&lt;/a&gt; was started to bring certain top German scientists to America. Many of these had worked on military applications for the Nazis (developing rockets etc.), and some may have been involved with the Nazis in other capacities as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A short list of the more famous scientists includes&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Wernher_von_Braun&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wernher von Braun&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_Tank&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Tank&lt;/a&gt; - designed many German aircraft during WWII&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Eberhard_Rees&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eberhard Rees&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;There are others listed &lt;a href=&quot;https://en.wikipedia.org/wiki/Operation_Paperclip#Key_scientists_.28incomplete_list.29&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note added by &lt;a href=&quot;https://hsm.stackexchange.com/users/26/danu&quot;&gt;Danu&lt;/a&gt; to remove confusion caused by the initial wording of this post&lt;/strong&gt;: After the war, Operation Paperclip continued for many years, as an effort to bring German scientists to the USA to prevent them from being recruited by the Soviets in the Cold War. In this context, many others were brought to the USA, though there is no direct tie to any possible World War II activities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, there was a list called the Osenberg List. During the war, the Germans needed to recall all PhDs, scientists, etc, from active military service as they needed help developing new weaponry. The identified intellectuals were first politically cleared and then listed on the Osenberg List. I looked around and couldn't seem to find a copy of the Osenberg List, but it seems like that list is what you want.&lt;/p&gt;&#xA;" OwnerUserId="5202" LastEditorUserId="26" LastEditDate="2017-08-25T13:34:38.263" LastActivityDate="2017-08-25T13:34:38.263" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5691" PostTypeId="2" ParentId="3" CreationDate="2017-02-12T16:49:01.163" Score="6" Body="&lt;p&gt;From the evidence that we have, it is most likely that Fermat never even &lt;strong&gt;claimed&lt;/strong&gt; to have a proof of the FLT, see the extensive discussion at Mathoverflow &lt;a href=&quot;https://mathoverflow.net/questions/33265/how-do-we-know-that-fermat-wrote-his-famous-note-in-1637/33289#33289&quot;&gt;here&lt;/a&gt;. Quoting from the accepted answer:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Not only do we not know the date, we don't even know whether he wrote the remark at all. For all we know it might have been invented by his son Samuel, who published his father's comments.&lt;/p&gt;&#xA;&lt;p&gt;In his letters, Fermat never mentioned the general case at all, but quite often posed the problem of solving the cases n=3 and n=4. I am almost certain that Fermat discovered infinite descent around 1640, which means that in 1637 he did not have any chance of proving FLT for exponent 4 (let alone in general).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5362" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-02-12T16:49:01.163" CommentCount="2" CommunityOwnedDate="2017-02-12T16:49:01.163" ContentLicense="CC BY-SA 3.0" />
  <row Id="5693" PostTypeId="2" ParentId="5682" CreationDate="2017-02-12T23:02:49.967" Score="-3" Body="&lt;p&gt;To be really honest, all university professors who remained in office in Germany after 1933 collaborated in one way or another with the regime. This goes for big shots like Planck, Heisenberg, Schrödinger, and others.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-02-12T23:02:49.967" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5694" PostTypeId="2" ParentId="5689" CreationDate="2017-02-12T23:13:31.260" Score="7" Body="&lt;p&gt;The Clifford torus was introduced by Clifford in 1873, not as embedded into &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^4$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^2$&lt;/span&gt;, but first projectively and then intrinsically, by identifying the opposite sides of a flat parallelogram, a commonplace method in modern topology textbooks. The embedding into &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^4$&lt;/span&gt; first appeared in Killing's Die Nichteuklidischen Raumformen in Analytischer Behandlung (Non-Euclidean Space Forms in Analytic Treatment, 1885), where he also pointed out that the embedded torus is contained in a &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-sphere. The name &amp;quot;Clifford torus&amp;quot; was given by Klein in Zur Nicht-Euklidischen Geometrie (On Non-Euclidean Geometry, 1890), in this book he also used Clifford's identification method to construct the &lt;a href=&quot;https://en.wikipedia.org/wiki/Klein_bottle&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Klein bottle&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The Clifford torus was one of the first non-trivial examples of &lt;a href=&quot;https://en.wikipedia.org/wiki/Space_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;space forms&amp;quot;&lt;/a&gt;, manifolds locally isometric to a classical geometry (flat, spherical, hyperbolic, etc.). The term &amp;quot;space form&amp;quot; ( &amp;quot;Raumformen&amp;quot; in German) was coined by Killing in 1876-8, in 1891 he called them Clifford-Klein space forms, and showed how to construct them as &lt;a href=&quot;https://en.wikipedia.org/wiki/Homogeneous_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;homogeneous spaces&lt;/a&gt; (quotients of classical geometries by discrete group actions). Weyl proved in 1916 that all locally flat space forms are homogeneous spaces.&lt;/p&gt;&#xA;&lt;p&gt;Volkert gives a very nice account of the &lt;a href=&quot;http://www.boma.mpim-bonn.mpg.de/data/42print.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of space forms from 1873 to 1925&lt;/a&gt;  with references to the original sources. The watershed of 1925 is when Hopf's Zum Clifford-Kleinschen Raumproblem (On the Clifford-Klein Space Form Problem) came out, where he gave a complete list of flat surfaces and three-dimensional spherical space forms (which eventually led him to the discovery of Hopf fibration). Here is the passage concerning Clifford's flat torus:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Clifford-Klein space forms entered the history of mathematics in 1873 during a talk which was delivered by W. K. Clifford at the meeting of the British Association for the Advancement of Sciences (Bradford, in September 1873) and via an article he published in June 1873. The title of Clifford’s talk was ‘On a surface of zero curvature and finite extension’, the proceedings of the meeting only provide this title. But we know a bit more about it from F. Klein, who attended Clifford’s talk and who described it on several occasions.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In the context of elliptic geometry - which Clifford conceived in Klein’s way as the geometry of the part of projective space limited by a purely imaginary quartic - Clifford described a closed surface which is locally flat, the today so-called Clifford surface (this name was introduced by Klein). This surface is constructed by using (today so called) Clifford parallels; Bianchi later provided a description by moving a circle along an elliptic straight line in such a way that it is always orthogonal to the straight line. So Clifford’s surface is the analogue of a cylinder; but - since it closed - it is often called a torus.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;To show its local flatness, Clifford used a dissection into parallelograms which is defined in a natural way by the two sets of parallels (or generators) contained in the surface. By considering the angles of such a parallelogram one sees that their sum equals four right angles so it is a common flat parallelogram. Clifford concluded: ‘The geometry of this surface is the same as that of a finite parallelogram whose opposite sides are regarded as identical.’  This is a very early occurrence of the identification scheme for the torus!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-02-13T19:32:30.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5695" PostTypeId="1" CreationDate="2017-02-13T13:16:49.617" Score="3" ViewCount="190" Body="&lt;p&gt;I copied the question from &lt;a href=&quot;https://math.stackexchange.com/q/61497/378968&quot;&gt;https://math.stackexchange.com/q/61497/378968&lt;/a&gt; because I think it is more suitable for this site and I think an answer to this question here could do better than: Hilbert uses the term &lt;em&gt;Zahlring&lt;/em&gt; just because.&lt;/p&gt;&#xA;" OwnerUserId="5143" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2017-02-15T06:03:25.883" Title="Why are rings called rings?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5697" PostTypeId="2" ParentId="3498" CreationDate="2017-02-14T20:35:19.933" Score="9" Body="&lt;p&gt;Maybe it is interesting to note that the term &quot;l’inégalité de Schwarz&quot; was coined by Poincaré in an 1896 paper in Acta Mathematica 20, p. 73, and was used in the French and German literature for the integral inequality until well into the 20th century.&#xA;&lt;a href=&quot;https://archive.org/stream/actamathematica20upps#page/73/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/stream/actamathematica20upps#page/73/mode/1up&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term &quot;Cauchy-Schwarz inequality&quot; was used for the algebraic inequality by Hardy and Littlewood in a paper in Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1920, pp. 38-39.&#xA;&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002505649&amp;amp;physid=PHYS_0045&quot; rel=&quot;noreferrer&quot;&gt;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002505649&amp;amp;physid=PHYS_0045&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-02-15T08:12:26.317" LastActivityDate="2017-02-15T08:12:26.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5698" PostTypeId="2" ParentId="5682" CreationDate="2017-02-15T06:29:34.150" Score="9" Body="&lt;p&gt;The mathematician who would have been famous had he not been a member of the SS / Nazi Party and died in a POW camp is, I think, &lt;a href=&quot;https://en.wikipedia.org/wiki/Gerhard_Gentzen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerhard Gentzen&lt;/a&gt;. He invented/rigourously described &lt;a href=&quot;https://en.wikipedia.org/wiki/Natural_deduction&quot; rel=&quot;nofollow noreferrer&quot;&gt;natural deduction&lt;/a&gt; and sequent calculus but is well written out of the history&lt;/p&gt;&#xA;&lt;p&gt;In philosophy circles, in debates between analytic philosophers (logic) and continental philosophers, sometimes there are discussions about who was worse — Heidegger or Frege. Frege died before WW2 so Heidegger is worse. These questions are related to which philosophy — analytical or continental — is the easiest misused for bad goals&lt;/p&gt;&#xA;&lt;p&gt;But, if Gentzen was known, the question would be between Heidegger or Gentzen and the question could have a different answer.&lt;/p&gt;&#xA;&lt;p&gt;I have a book on natural deduction (Logic by &lt;em&gt;Paul Tomassi&lt;/em&gt;) it mentions lot of people involved in its development, but Gentzen is missing.&lt;/p&gt;&#xA;" OwnerUserId="557" LastEditorUserId="5989" LastEditDate="2021-09-08T01:33:53.757" LastActivityDate="2021-09-08T01:33:53.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5699" PostTypeId="2" ParentId="5682" CreationDate="2017-02-15T08:40:19.327" Score="4" Body="&lt;p&gt;He introduced groupoids and did some work on quadratic forms and quaternions, but I don't think he is considered a famous scientist: &lt;a href=&quot;https://en.wikipedia.org/wiki/Heinrich_Brandt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heinrich Brandt&lt;/a&gt;. However, he was a very active Nazi, being a member of some Nazi organisations and most notably, he was a &lt;a href=&quot;https://en.wikipedia.org/wiki/F%C3%B6rderndes_Mitglied_der_SS&quot; rel=&quot;nofollow noreferrer&quot;&gt;Förderndes Mitglied der SS&lt;/a&gt;. (According to German Wikipedia, referencing Harry Waibel, Diener vieler Herren: Ehemalige NS-Funktionäre in der SBZ/DDR (2011) p. 51.)&lt;/p&gt;&#xA;" OwnerUserId="79" LastActivityDate="2017-02-15T08:40:19.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5700" PostTypeId="2" ParentId="4989" CreationDate="2017-02-15T16:54:55.990" Score="3" Body="&lt;p&gt;At that time, physics students were normally not taught linear algebra in Germany, except when their professors had an affinity to this subject. Max Born's teacher had been Jacob Rosanes, an algebraist, among whose students were Steinitz and Toeplitz. Heisenberg, on the other hand, had studied in Munich with Lindemann. Perron only came to Munich in 1922, when Heisenberg had almost finished his studies. Göttingen was an exception in Germany, and Heisenberg later stated that he had learned &quot;optimism from Sommerfeld, physics from Bohr, and mathematics in Göttingen.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an analysis of Heisenberg's breakthrough paper from July 1925:&#xA;&lt;a href=&quot;https://arxiv.org/pdf/quant-ph/0404009.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/pdf/quant-ph/0404009.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here are interviews with Heisenberg and Born about this time: &#xA;&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4661-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4661-7&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4522-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4522-3&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2017-02-15T16:54:55.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5701" PostTypeId="1" CreationDate="2017-02-15T16:59:43.310" Score="1" ViewCount="522" Body="&lt;p&gt;Galileo was a real physicist and mathematician besides being an astronomer. But what were his contributions to mechanics like throw and fall and collision? Did he perhaps already know Newton's first law a little bit?&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="10106" LastEditDate="2020-05-16T18:45:04.627" LastActivityDate="2020-05-16T18:45:04.627" Title="What were the most important discoveries of Galileo in mechanics?" Tags="|physics|discoveries|classical-mechanics|galileo|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5702" PostTypeId="2" ParentId="198" CreationDate="2017-02-16T19:20:12.620" Score="7" Body="&lt;p&gt;I would say that in Germany there was a gradual development towards the matrix notation of linear equation systems from the 1920s onwards. Courant certainly was a pioneer in this development as he tells in this &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4562&quot; rel=&quot;noreferrer&quot;&gt;interview&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This textbook from 1927 on &lt;a href=&quot;https://books.google.de/books?hl=de&amp;amp;id=GEYgAAAAIAAJ&amp;amp;dq=wittwer+eisenbetonbau&amp;amp;focus=searchwithinvolume&amp;amp;q=matrix&quot; rel=&quot;noreferrer&quot;&gt;Statik im Eisenbetonbau&lt;/a&gt;, i.e. statics of concrete structures, features the term &quot;matrix&quot; 65 times and was surely not inspired by quantum mechanics, but by the simplicity of matrix notation of the large linear equation systems that occur in structural mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From 1950 onwards matrices were taught in all technical and scientific disciplines at German universities as can be seen from this textbook by &lt;a href=&quot;https://books.google.de/books?id=mDuHBwAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=matrizen+zurm%C3%BChl&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjgzt-dp5XSAhWLshQKHR-bDlAQ6AEIHDAA#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Zurmühl&lt;/a&gt; that went through three editions within 10 years.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-02-17T15:42:14.553" LastActivityDate="2017-02-17T15:42:14.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5703" PostTypeId="2" ParentId="2968" CreationDate="2017-02-16T20:58:27.167" Score="8" Body="&lt;p&gt;In 1887 a committee of the British Association was appointed for the purpose of &quot;considering the desirability of introducing uniform nomenclature for the fundamental units of mechanics of co-operating with other bodies engaged in similar work.” The committee issued a series of questions to members, and collected their replies. The result was that in 1888, when the committee met at Bath, they were able, amid much difference of opinion, to agree as to the desirability of introducing names for the C. G. S. units of velocity, &lt;strong&gt;momentum&lt;/strong&gt;, and pressure; the names suggested being, &lt;a href=&quot;https://books.google.com/books?id=_KvmAAAAMAAJ&amp;amp;pg=PA223&quot; rel=&quot;noreferrer&quot;&gt;kine, &lt;strong&gt;bole&lt;/strong&gt;, and barad&lt;/a&gt; respectively. &lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="658" LastEditDate="2017-02-16T22:43:01.507" LastActivityDate="2017-02-16T22:43:01.507" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5704" PostTypeId="1" CreationDate="2017-02-16T22:14:28.070" Score="0" ViewCount="42" Body="&lt;p&gt;I do understand that electrical resistance describes somehow the material resistance to electrons flow but I don't see the meaning behind the names capacitors and inductors (if there is any and it should I guess!)&lt;/p&gt;&#xA;" OwnerUserId="5343" LastActivityDate="2017-02-16T22:14:28.070" Title="Why are capacitors and inductors called that way?" Tags="|electromagnetism|electricity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5705" PostTypeId="1" CreationDate="2017-02-17T09:44:09.767" Score="2" ViewCount="44" Body="&lt;p&gt;The Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Cuckold&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cuckold&lt;/a&gt; indicates that the biological phenomenon of brood parasitism was known in the middle ages. (I suspect it may have been known in &lt;a href=&quot;https://youtu.be/QlQNJxxIw6s?t=3m20s&quot; rel=&quot;nofollow noreferrer&quot;&gt;other cultures&lt;/a&gt; as well, but I can't say for sure)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although observing such behaviour doesn't require much in the way of advanced technology (though binoculars can help), I assume observing this behaviour would take a lot of time, and the birds (both hosts and parasites) are of little economic significance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried searching about how brood parasitism was discovered, but I most of the hits about discovery were about whether or not the hosts discover the parasitism occurring.&lt;/p&gt;&#xA;" OwnerUserId="508" LastEditorUserId="508" LastEditDate="2017-02-17T09:52:55.870" LastActivityDate="2017-02-17T09:52:55.870" Title="Why was brood parasitism known about in the middle ages?" Tags="|biology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5706" PostTypeId="1" AcceptedAnswerId="5876" CreationDate="2017-02-17T10:19:40.003" Score="6" ViewCount="476" Body="&lt;p&gt;From Dieudonné's &quot;&lt;a href=&quot;https://books.google.de/books?id=mg7r4acKgq0C&amp;amp;pg=PA55&amp;amp;dq=dieudonne+picard+simple+pole&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjn-oHJ6pbSAhVKchQKHV5ZD8oQ6AEIHTAA#v=onepage&amp;amp;q=dieudonne%20picard%20simple%20pole&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Functional Analysis&lt;/a&gt;&quot; I learned that Picard in &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3073f/f504&quot; rel=&quot;nofollow noreferrer&quot;&gt;1893&lt;/a&gt; gave a characterization of an eigenvalue of the Laplacian as the simple pole of a meromorphic function.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an earlier source that makes this link?    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And who named this meromorphic function the resolvent? I have read somewhere that it was Hilbert. &lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="658" LastEditDate="2017-04-16T23:27:50.467" LastActivityDate="2018-10-07T03:45:59.230" Title="Who came up with the link between the spectrum of an operator and the poles of a meromorphic function?" Tags="|terminology|complex-analysis|differential-equations|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5707" PostTypeId="2" ParentId="4917" CreationDate="2017-02-17T15:30:46.967" Score="2" Body="&lt;p&gt;As a graphical method for the integration of hyperbolic PDEs the &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Method_of_characteristics&quot; rel=&quot;nofollow noreferrer&quot;&gt;method of characteristics&lt;/a&gt; was proposed by the Belgian engineer Massau in the late 19th century.&lt;br&gt;&#xA;&lt;a href=&quot;https://archive.org/stream/mmoiresurlintgr05massgoog#page/n4/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/stream/mmoiresurlintgr05massgoog#page/n4/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2017-02-17T15:30:46.967" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5708" PostTypeId="1" CreationDate="2017-02-18T23:28:34.133" Score="1" ViewCount="62" Body="&lt;p&gt;I'm interested in an English translation of the Alfonsine tables published in Paris in the 1400s. I know there is an English translation of what survived of earlier Alfonsine tables from Castile, but that's not what I'm interested in. If you know of software that implements the algorithms implied by the table, that would be good too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link to the book review of what I'm &lt;em&gt;not&lt;/em&gt; interested is &lt;a href=&quot;http://adsabs.harvard.edu/full/2004JHA....35..479S&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link to the Latin version of what I want is &lt;a href=&quot;http://books.google.com/books?id=4ppKAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5389" LastEditorUserId="10106" LastEditDate="2020-06-07T15:04:50.163" LastActivityDate="2020-06-07T15:04:50.163" Title="English translation of Parisian Alfonsine tables (the 1400s)" Tags="|astronomy|translations|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5709" PostTypeId="1" AcceptedAnswerId="5711" CreationDate="2017-02-19T04:40:43.760" Score="11" ViewCount="418" Body="&lt;p&gt;Today it has been shown that a 1-dimensional object in 4-dimensional space cannot be tied into a knot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I would like to know who first conjectured this and when? And when was it proven?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(P.S., is there a hyper-dimension tag by some other synonym?)&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="658" LastEditDate="2017-02-20T08:55:21.313" LastActivityDate="2018-10-23T13:33:04.240" Title="A knot cannot be tied in 4-dimensions, but when was this conjectured and proven?" Tags="|geometry|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5710" PostTypeId="1" CreationDate="2017-02-19T08:04:26.967" Score="1" ViewCount="92" Body="&lt;p&gt;Jacques Heyman in his article &lt;a href=&quot;https://books.google.gr/books?id=6LO_U6T-HvsC&amp;amp;pg=PA15#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Truesdell and the History of the Theory of Structures&lt;/a&gt;, mentions the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first full-scale tests on building frames were made in the 1920s&#xA;  in London, and the results were, in the fullest sense of the word,&#xA;  astounding. The internal stress resultants (in this case, bending&#xA;  stresses in beams) that were measured in practice bore almost no&#xA;  relation to those confidently calculated by the elastic (Navier)&#xA;  designer. The investigators quickly found what was wrong; while the&#xA;  equilibrium equations were obeyed, and the elastic equations of&#xA;  deformation were found to be almost exact, the boundary conditions&#xA;  used in the analysis did not represent reality. A small error in&#xA;  manufacture, a tiny dimensional mismatch, an inevitable forcing&#xA;  together of members during construction - all these led to a great&#xA;  change in the values of internal stresses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Where can I find more information about these astounding results?&lt;/p&gt;&#xA;" OwnerUserId="2906" LastEditorUserId="2906" LastEditDate="2017-02-23T07:18:43.953" LastActivityDate="2017-02-23T07:18:43.953" Title="Verification of Navier's theory of structures" Tags="|reference-request|engineering|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5711" PostTypeId="2" ParentId="5709" CreationDate="2017-02-19T12:04:38.037" Score="11" Body="&lt;p&gt;A nice account is found in a note to R. Steiner's &lt;em&gt;Die vierte Dimension&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/rudolf-steiner-ga-324a/page/n235&quot; rel=&quot;nofollow noreferrer&quot;&gt;1995&lt;/a&gt;; &lt;a href=&quot;https://books.google.com/books?id=SzsKPk2PgQ0C&amp;amp;pg=PT228&quot; rel=&quot;nofollow noreferrer&quot;&gt;translation&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Felix Klein&lt;/em&gt;&lt;/strong&gt; (1845–1925) seems to have been the first mathematician to draw attention to this phenomenon in the early 1870s. According to an account by Zöllner (&lt;a href=&quot;https://books.google.com/books?id=qKoPAAAAYAAJ&amp;amp;pg=PA276&quot; rel=&quot;nofollow noreferrer&quot;&gt;1878a&lt;/a&gt;), Klein spoke with him during a scientific conference on this subject shortly before publishing a treatise (&lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002243350&amp;amp;physid=phys485&quot; rel=&quot;nofollow noreferrer&quot;&gt;1876&lt;/a&gt;) in which he discussed this theme in passing. Klein also reported on their meeting (...) (&lt;a href=&quot;https://books.google.com/books?id=XtunBgAAQBAJ&amp;amp;pg=PA170&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926, pp. 169ff&lt;/a&gt;). While Klein (&lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002243350&amp;amp;physid=phys487&quot; rel=&quot;nofollow noreferrer&quot;&gt;1876, p. 478&lt;/a&gt;) discusses the subject only in general terms, Hoppe (&lt;a href=&quot;https://books.google.com/books?id=CLIEAAAAQAAJ&amp;amp;pg=RA1-PA224&quot; rel=&quot;nofollow noreferrer&quot;&gt;1879&lt;/a&gt;) uses an analytically formulated example to untie concretely a simple three-dimensional knot in four-dimensional space (see also Durège (&lt;a href=&quot;https://books.google.com/books?id=EIU6AQAAIAAJ&amp;amp;pg=RA1-PA135&quot; rel=&quot;nofollow noreferrer&quot;&gt;1880&lt;/a&gt;) and Hoppe (&lt;a href=&quot;https://books.google.com/books?id=VagEAAAAQAAJ&amp;amp;pg=PA423&quot; rel=&quot;nofollow noreferrer&quot;&gt;1880&lt;/a&gt;)).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Klein is also credited by Tait (&lt;a href=&quot;https://books.google.com/books?id=Uq49AQAAMAAJ&amp;amp;pg=PA190&quot; rel=&quot;nofollow noreferrer&quot;&gt;1877&lt;/a&gt;; &lt;a href=&quot;https://books.google.com/books?id=xT5IPDMtSJYC&amp;amp;pg=PA129&quot; rel=&quot;nofollow noreferrer&quot;&gt;1882&lt;/a&gt;), Dehn-Heegaard (&lt;a href=&quot;https://archive.org/stream/encyklomath1103encyrich#page/n195&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907&lt;/a&gt;), and himself (&lt;a href=&quot;https://archive.org/stream/gesammeltema02kleirich#page/n80&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922, p. 67&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt; To your additional question: Klein clearly convinced the experts that it wasn't just a conjecture. While he apparently never published his proof, I would bet he used the now-standard argument that one can eliminate one crossing at a time, as in e.g. Seifert-Threlfall (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?60.0496.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1934&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=rsb8zjP0XHoC&amp;amp;pg=PA328&quot; rel=&quot;nofollow noreferrer&quot;&gt;Remark 1&lt;/a&gt;), Weitzenböck (&lt;a href=&quot;http://www.emis.de/cgi-bin/JFM-item?55.0382.08&quot; rel=&quot;nofollow noreferrer&quot;&gt;1929&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=KBScBgAAQBAJ&amp;amp;pg=PA71&quot; rel=&quot;nofollow noreferrer&quot;&gt;§5&lt;/a&gt;) or — maybe a first? — the last four paragraphs of Durège (&lt;a href=&quot;https://books.google.com/books?id=IyAXAAAAYAAJ&amp;amp;pg=PA175&quot; rel=&quot;nofollow noreferrer&quot;&gt;1880&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-23T13:33:04.240" LastActivityDate="2018-10-23T13:33:04.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5712" PostTypeId="1" AcceptedAnswerId="5718" CreationDate="2017-02-19T21:32:34.307" Score="9" ViewCount="1278" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo&amp;#39;s_paradox&quot; rel=&quot;noreferrer&quot;&gt;Galileo's paradox&lt;/a&gt; is the observation that the natural numbers can be put into one to one correspondence with the square numbers, showing that an infinite set can be bijected to a proper subset of itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember reading that this was already known to Arab mathematicians in the 12th or 13th century, but now I can't find any references via Google. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there online references to the origin of this discovery?&lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="4251" LastEditDate="2020-07-08T01:28:17.217" LastActivityDate="2020-07-08T01:28:17.217" Title="What are the origins of Galileo's paradox?" Tags="|mathematics|set-theory|galileo|infinity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5713" PostTypeId="1" AcceptedAnswerId="5716" CreationDate="2017-02-20T11:54:29.433" Score="4" ViewCount="1391" Body="&lt;p&gt;In many text books on linear algebra one of the most important applications is geometry in Euclidean spaces and therefore there seems to be the impression that linear algebra was developed in order to consider geometrical problems in Euclidean spaces. On the other hand the axiomatic definition of a vector space seems to be a bit of an overkill for these problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which problems actually led to the development of modern linear algebra?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although there is a nontrivial intersection with the question &lt;a href=&quot;https://hsm.stackexchange.com/questions/3214&quot;&gt;on when people started viewing matrices as linear transformations&lt;/a&gt;, I am not only interested in the early development of linear algebra but more in the questions and problems which motivated the axiomatization.&lt;/p&gt;&#xA;" OwnerUserId="4096" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-02-21T11:44:15.653" Title="Motivation for the development of modern linear algebra" Tags="|mathematics|linear-algebra|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5714" PostTypeId="2" ParentId="5612" CreationDate="2017-02-20T13:17:30.577" Score="2" Body="&lt;p&gt;The following text is from the introduction to Carl Runge's &quot;&lt;a href=&quot;https://archive.org/stream/graphicalmethod01runggoog#page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graphical Methods&lt;/a&gt;&quot;, in which the author, well before the invention of computers, justifies the existence of numerical analysis: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A great many if not all of the problems in mathematics may be so&#xA;formulated that they consist in finding from given data the values of certain unknown quantities subject to certain conditions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We may distinguish different stages in the solution of a problem. The first stage we might say is the proof that the quantities sought for really exist. In many ... cases the first stage of the solution may be so easy, that we immediately pass on to the second stage of finding methods to calculate the unknown quantities sought for. Or even if the first stage of the solution is not so easy, it may be expedient to pass on to the second stage. For if we succeed in finding methods of calculation that determine the unknown quantities, the proof of their existence is&#xA;included.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are not a small number of men who believe the task of the&#xA;mathematician to end here. This, I think, is due to the fact that the pure mathematician as a rule is not in the habit of pushing his investigation so far as to find something out about the real things of this world. He leaves that to the astronomer, to the physicist, to the engineer. These men, on the other hand, take the greatest interest in the actual numerical values that are the outcome of the mathematical methods of calculation. Suppose the mathematician gives them a method of calculation, perfectly logical and conclusive but taking 200 years of incessant numerical work to complete. They would be justified in thinking that this is not much better than no method at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So there arises a third stage of the solution of a mathematical problem in which the object is to develop methods for finding the result with as little trouble as possible. I maintain that this third stage is just as much a chapter of mathematics as the first two stages and it will not do to leave it to the astronomer, to the physicist, to the engineer or whoever applies mathematical methods, for this reason that these men are bent on the results and therefore they will be apt to overlook the full generality of the methods they happen to hit on, while in the hands of the mathematician the methods would be developed from a higher standpoint and their bearing on other problems in other scientific inquiries would be more likely to receive the proper attention.&quot; &lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-02-21T17:09:30.000" LastActivityDate="2017-02-21T17:09:30.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5715" PostTypeId="2" ParentId="5713" CreationDate="2017-02-20T14:35:58.027" Score="4" Body="&lt;p&gt;Linear algebra originally developed because of applications to differential equations and mechanics. Usually a system of differential equations can be approximated with a linear system, and a linear system requires linear algebra to solve it. I mean the period 18-19 centuries, and mathematicians like Lagrange, Cauchy, Jacobi. Later in 19th century it found many more applications, and gradually was formalized by Cayley, Sylvester, and others; formal multiplication of matrices was introduced etc. Still this knowledge was restricted to some narrow circles of pure mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When Heisenberg invented modern quantum mechanics in 1925, he has never heard of matrices and their multiplication and had to invent these things himself. &#xA;(These things were not a part of a normal university curriculum).&#xA;After the invention of quantum mechanics, the knowledge of linear algebra spread enormously, and nowadays it is taught as a core undergraduate subject, as important as calculus. An important role played applications such as linear programming invented shortly after WWII, and fast Fourier transform.&#xA;It was understood that basic computationl algorithms of applied mathematics are&#xA;based on linear algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I omitted here the &quot;pre-history&quot; of linear algebra before 18th century. Mainly because I do not have sufficient knowledge of early applications. But people were solving linear systems since the antiquity, and studied determinants since 17th century at least. I am not sure what the main motivation was then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing however seems clear: linear algebra was NOT invented for the needs of&#xA;Euclidean geometry.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-02-20T14:35:58.027" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5716" PostTypeId="2" ParentId="5713" CreationDate="2017-02-20T17:18:55.773" Score="7" Body="&lt;p&gt;I think that by the end of the 19th and the beginning of the 20th century many mathematicians felt the need for a unified theory of &quot;linear things&quot; because they realized that they were doing the same things over and over again in different fields, e.g. Maxime Bocher writes in his &lt;a href=&quot;https://archive.org/stream/introductiontost00bcuoft#page/52/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Introduction to the study of Integral Equations&lt;/a&gt; that the notion of orthogonal functions was presented to him by Felix Klein in a lecture on the differential equations of mathematical physics in 1889.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jean-Luc Dorier writes in the conclusion to his paper on &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086085710245&quot; rel=&quot;noreferrer&quot;&gt;A general outline of the genesis of vector space theory&lt;/a&gt;:&#xA;&quot;Many examples of implicit linear methods in different contexts in mathematics or physics can be traced from Antiquity, but, until the 19th century at least, they remained isolated methods; the same author could use the same idea twice (in terms of the theory of linear algebra) in different contexts without noticing the similarity of the methods. In this sense, linear algebra long remained implicit because of a lack of&#xA;unification. Nevertheless, its origins can be found in various contexts, and &#xA;linearity has always been a question which has penetrated nearly every branch of mathematics.&quot; &lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-02-20T17:24:45.533" LastActivityDate="2017-02-20T17:24:45.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5717" PostTypeId="2" ParentId="5713" CreationDate="2017-02-20T21:54:09.890" Score="2" Body="&lt;p&gt;Some of the most significant steps in the development of matrix algebra were indeed taken in connection with problems of analytic geometry, e.g. Cramer's rule arose out of the problem of finding a curve that passes through a number of points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.gr/books?id=9tUrarQYhKMC&amp;amp;pg=PA2070#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; you can find a table with the &quot;Milestones in the History of Matrices and Determinants&quot;&lt;/p&gt;&#xA;" OwnerUserId="2906" LastActivityDate="2017-02-20T21:54:09.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5718" PostTypeId="2" ParentId="5712" CreationDate="2017-02-21T01:24:48.767" Score="9" Body="&lt;p&gt;The answer depends on what &quot;this&quot; means. According to &lt;a href=&quot;https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/measuring-the-size-of-infinite-collections-of-natural-numbers-was-cantors-theory-of-infinite-number-inevitable/325464FFF1E318E3E51A80E652FA1C5B&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mancosu's Measuring the Size of Infinite Collections of Natural Numbers&lt;/a&gt; (reprinted in his book &lt;a href=&quot;https://books.google.com/books?id=Pw2DDQAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abstraction and Infinity&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;It is actually unclear when the paradox, in the numerical form I just gave, appears. In the Greek tradition we have paradoxes that are related, but are not identical, to it; in this tradition what is claimed to be paradoxical is &#xA;  the existence of different sizes of infinity.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An early geometric example appears in Proclus's commentary on the Book I of Elements:&quot;&lt;em&gt;But if from one diameter two semicircles are produced, and if an indefinite number of diameters can be drawn through the center, it will follow that the number of semicircles is twice infinity&lt;/em&gt;&quot;. Proclus expresses the Aristotelian view that infinities have no &quot;comparative sizes&quot; (because actual infinities do not even exist), which, ironically, is what &quot;Galileo's paradox&quot; confirms. The &quot;Arab mathematicians&quot; is most likely &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/Thabit.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thābit ibn Qurra al-Ḥarrānī (826-901)&lt;/a&gt;, best known for his work on the parallel postulate (anticipating elements of hyperbolic geometry), and amicable numbers. Thābit was apparently the first known source to challenge the Aristotelian view and discuss &quot;comparative sizes&quot; of infinities more in the spirit of Cantor. In Questions asked of Thābit ibn Qurra (narrated by his student) we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We questioned him also regarding a proposition put into service by many revered commentators, namely that an infinite cannot be greater than an infinite. – He pointed out to us the falsity of this (proposition) also by reference to numbers. For (the totality of) numbers itself is infinite, and the even numbers alone are infinite, and so are the odd numbers, and these two classes are equal, and each is half the totality of numbers. That they are equal is manifest from the fact that in every two consecutive numbers one will be even and the other odd; that the (totality of) numbers is twice each of the two [other classes] is due to their equality and the fact that they (together) exhaust (that totality), leaving out no other division in it, and therefore each of them is half (the totality) of numbers.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the view that the paradox challenged. According to Murdoch's entry in &lt;a href=&quot;https://books.google.com/books/about/The_Cambridge_History_of_Later_Medieval.html?id=l821N_9GSusC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cambridge History of Later Medieval Philosophy&lt;/a&gt;, there was a lively discussion of the issue in the medieval Middle East (but not with numbers as examples), with Al-Shahrastani, Averroes and Maimonides participating. All of them are from 12th century, none is a mathematician, and only Averroes is an Arab. Murdoch further remarks: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Although the paradox was treated in other ancient sources they were either not available in the Latin Middle Ages or never cited; the same must be said of the Islamic discussions of the paradox. On the basis of presently available evidence, then, philosophers and theologians of the Latin West appear to have realized the importance of the paradox on their own&lt;/em&gt;”. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the West &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Bradwardine&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bradwardine (1300 – 1349)&lt;/a&gt; was apparently first to apply one-to-one correspondence to show that an infinity can be equal to one of which it is a proper part. But his argument in Geometria Speculativa is part of a reductio to argue against eternity of the world, a side of a contradiction, not a valid conclusion. &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicole_Oresme&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oresme (c. 1325 – 1382)&lt;/a&gt; comes closest to phrasing the paradox as it is phrased today in his commentary on Aristotle's Physics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;All infinite multitudes are simply infinite, therefore there is none that is greater than or less than another... consider the multitude of odd numbers and [perfect] squares; then, when there is the first, the second, the third and fourth, and thus, according to the order, without end all the numbers, it follows that there is no less than the multitude of odd numbers which lasted for the multitude of all numbers, and thus arguably the multitude of the parts of the even and the odd numbers is not less than the whole, or the whole multitude is not more than its parts.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By the way, Bradwardine and Oresme also anticipated Galileo's work on uniformly accelerated motion, including a proof of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mean_speed_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;mean speed theorem&lt;/a&gt;, although they did not apply it to falling bodies (or anything else physical), see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2317/was-galileo-a-plagiarizer/2319#2319&quot;&gt;Was Galileo a plagiarizer?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-02-22T04:39:45.953" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5720" PostTypeId="2" ParentId="3027" CreationDate="2017-02-21T19:54:17.413" Score="2" Body="&lt;p&gt;Two scientists got a unit named after them while alive: Joule and Weber.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Joule: the second International Electrical Congress in 1889 adopted the joule as the practical unit of electrical work. But time flies, and Joule died a few weeks after the event.(&lt;a href=&quot;https://arxiv.org/pdf/1512.07306.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jayson&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Weber. Around 1871, Latimer Clark introduced the weber as a name for the practical unit of current, equal to 1 volt per ohm, although the name was not adopted by all (&lt;a href=&quot;https://books.google.nl/books?id=vpobAAAAMAAJ&amp;amp;pg=PA70&amp;amp;dq=%22sometimes%20called%20the%20weber%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjOtIDb9qHSAhUNahoKHfY5BbcQ6AEIGjAA#v=onepage&amp;amp;q=%22sometimes%20called%20the%20weber%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Everett&lt;/a&gt;). In 1881 the International Congress of electrical engineers agreed on the name ampere instead of weber.(&lt;a href=&quot;https://web.archive.org/web/20160909183227/http://seaus.free.fr/spip.php?article964&quot; rel=&quot;nofollow noreferrer&quot;&gt;Borvon&lt;/a&gt;) &#xA;In 1891 Weber died. In 1893 the International Electrical Conference in Chicago officially adopted the name ampere. In 1933, the weber was reintroduced as another unit name, for magnetic flux.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no official guideline, but in 1891 the electrical engineer W.H. Preece wrote he thought it would be unwise to use the name of the living, he preferred the names of the dead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About temperature units: initially, the personal names Fahrenheit and Celsius were not inseparable parts of their temperature units. The unit was 'degree', and mentioning the scale was optional. Today they are inseparable. May be it would help to determine when writers started inserting a space before the degree symbol, for example 40 °C instead of 40° C.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2017-02-24T14:34:45.630" LastActivityDate="2017-02-24T14:34:45.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5721" PostTypeId="1" AcceptedAnswerId="5722" CreationDate="2017-02-22T03:52:09.113" Score="3" ViewCount="346" Body="&lt;p&gt;I have a follow up question from my previous question on math.SE, where I asked about the &lt;a href=&quot;https://math.stackexchange.com/questions/2148219/first-use-of-little-o-notation&quot;&gt;First use of little $o$ notation&lt;/a&gt; - for those who want to check - the answer goes back to Landau ($1909$), this could really be one of the earliest occurrences (the definition is motivated by the big $O$ notation and there is no reference to any other work, so it seems genuin).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I wonder when the slightly different notation of &lt;a href=&quot;https://en.wikipedia.org/wiki/Big_O_in_probability_notation#Small_O:_convergence_in_probability&quot; rel=&quot;nofollow noreferrer&quot;&gt;$o_p(\cdot)$ notation&lt;/a&gt; was first introduced. Clearly later but unfortunately there is no historically reference on Wikipedia. Does anyone have a clue or reference where I should look into?&lt;/p&gt;&#xA;" OwnerUserId="5404" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2017-02-22T18:53:59.963" Title="First use of litte $o_p$ (little $o$ in probability) notation?" Tags="|mathematics|terminology|notation|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5722" PostTypeId="2" ParentId="5721" CreationDate="2017-02-22T18:53:59.963" Score="4" Body="&lt;p&gt;From &lt;a href=&quot;http://jeff560.tripod.com/stat.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest use of mathematical symbols&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The convergence in probability symbol &lt;em&gt;plim&lt;/em&gt; was introduced by H. B. Mann and A. Wald &quot;On Stochastic Limit and Order Relationships,&quot; Annals of Mathematical Statistics, 14, (1943), 217-226. The stochastic order symbols $O_p$ and $o_p$, modelled on the $O$ and $o$, or Landau symbols, were introduced in the same paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-02-22T18:53:59.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5723" PostTypeId="1" AcceptedAnswerId="5724" CreationDate="2017-02-22T20:45:02.280" Score="2" ViewCount="391" Body="&lt;p&gt;Visible colors are caused due to the discreteness of energy levels in molecules and complexes, and scattering.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am not sure how physicists explained color before they understood this (before 1900). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what I know, they were able to understand the concept of wavelength. I also know that they knew that white light consisted of different &quot;colored light.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what was the predominant theory in explaining color? &lt;/p&gt;&#xA;" OwnerUserId="5426" OwnerDisplayName="Quantum AMERICCINO" LastActivityDate="2017-02-23T23:47:47.223" Title="How did physicists explain colors before quantum mechanics?" Tags="|optics|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="5724" PostTypeId="2" ParentId="5723" CreationDate="2017-02-23T01:46:21.520" Score="5" Body="&lt;p&gt;I am sure that there are some good and fairly complete histories of the thinking over time of what light is, and specifically the colors. The following site is a simple and quick summary of the highlights: &lt;a href=&quot;http://photonterrace.net/en/photon/history/&quot; rel=&quot;noreferrer&quot;&gt;http://photonterrace.net/en/photon/history/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It goes from Aristotle (white and black combinations for the colors), to Newton identifying refraction and how it splits light into its colors (his theory: particles with different refractivity), to Hyguins and Young determining that it is a wave and explaining refraction due to it and wavelengths, to Maxwell proving that it is electromagnetic and color was freq and equivalently wavelength, to Einstein who identified it as photons in the photoelectric effect - the start of anything quantum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there was also the study and theories of how the eye saw colors, that different colors affected different retina parts -- that took some time, and it actually was Goethe who first theorized that each person may see colors somewhat differently. Since that's less about physics than about biology I won't add more on that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I saw elsewhere a spectrum from different substances back before Maxwell, with sharp lines. See it in the link at &lt;a href=&quot;http://library.si.edu/exhibition/color-in-a-new-light/science&quot; rel=&quot;noreferrer&quot;&gt;http://library.si.edu/exhibition/color-in-a-new-light/science&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's from the Smithsonian, see the spectrum shown for Spectrum Analysis, Six Lectures, in 1868. Don't know what the thinking was on sharp lines in the spectrum, maybe you can get that from the Smithsonian and read. What they said in 1868 on the spectral lines. Somewhere there it says it allows one to identify different materials. Not too shabby for 1868. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after Bohr atom was there some explanation of discrete lines and jumps and orbitals, leading of course to QM. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the theories changed over time, and became more scientific really after Newton.&lt;/p&gt;&#xA;" OwnerUserId="5408" OwnerDisplayName="Bob Bee" LastActivityDate="2017-02-23T02:17:01.173" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5725" PostTypeId="2" ParentId="5723" CreationDate="2017-02-23T23:47:47.223" Score="0" Body="&lt;p&gt;Actually Goethe had a very interesting and ususual &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Theory_of_Colours#Goethe.27s_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;theory&lt;/a&gt; to rival Newton's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Unlike his contemporaries, Goethe didn't see darkness as an absence of light, but rather as polar to and interacting with light; colour resulted from this interaction of light and shadow. For Goethe, light is &quot;the simplest most undivided most homogenous being that we know. Confronting it is the darkness&quot; (Letter to Jacobi).&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastActivityDate="2017-02-23T23:47:47.223" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5726" PostTypeId="1" AcceptedAnswerId="5727" CreationDate="2017-02-24T02:18:45.993" Score="3" ViewCount="1585" Body="&lt;p&gt;In high school we were taught that the formula for pH is the negative of the common logarithm of hydrogen ion concentration: pH = -log[H+].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It wasn't until I took organic chemistry that the &quot;acid dissociation constant&quot; (pKa) was introduced.  It was at that point that the textbook explained that &quot;p&quot; was used as a mathematical operator, meaning the negative of the common logarithm.  (See &lt;a href=&quot;https://web.mst.edu/~gbert/logs/pH.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.mst.edu/~gbert/logs/pH.html&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why didn't they teach us that in high school?  There would have been no rote memorization of the pH formula: the equation is right there in the name!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this operator restricted to chemistry?  I've never seen it used in a (purely) mathematical context and have always wondered how the usage originated.  Does anyone know its etymology or the first time it appeared in a publication?&lt;/p&gt;&#xA;" OwnerUserId="3789" LastActivityDate="2017-12-09T19:58:51.837" Title="What is the etymology of lower case p as the operator for the negative of the common logarithm?" Tags="|mathematics|terminology|chemistry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5727" PostTypeId="2" ParentId="5726" CreationDate="2017-02-24T22:37:15.697" Score="4" Body="&lt;p&gt;How about this...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;pH&lt;/strong&gt;&lt;br&gt;&#xA;  1909, from German PH, introduced by S.P.L. Sörensen, from P, for German Potenz &quot;potency, power&quot; + H, symbol for the hydrogen ion that determines acidity or alkalinity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.etymonline.com/index.php?term=ph&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, yes, it is used only in chemistry.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-02-24T22:37:15.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5728" PostTypeId="1" AcceptedAnswerId="5729" CreationDate="2017-02-25T11:45:32.053" Score="2" ViewCount="157" Body="&lt;p&gt;I am going to teach history of physics next semester. Can anyone name some old textbooks on statistical mechanics? By old, I mean written around 1930 or earlier. I would like to compare the old books and the books nowadays. &lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2017-02-26T02:31:26.203" Title="Old textbooks on statistical mechanics" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5729" PostTypeId="2" ParentId="5728" CreationDate="2017-02-25T15:46:10.127" Score="5" Body="&lt;p&gt;Arnold Sommerfeld's last volume of his Lectures on Theoretical Physics, written in the 1940s and published posthumously, downloadable as &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.166906&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf file&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Planck's &lt;a href=&quot;https://archive.org/stream/theoryofheatbein007775mbp#page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theory of&#xA;Heat&lt;/a&gt; from 1932&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 1911 article by Paul and Tatiana Ehrenfest in Felix Klein's &lt;a href=&quot;https://archive.org/stream/p4encyklopdieder04akaduoft#page/n791/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyklopädie&lt;/a&gt;, of which there is also an English &lt;a href=&quot;https://books.google.de/books?id=g0xoBQAAQBAJ&amp;amp;pg=PT10&amp;amp;dq=ehrenfest+%22encyklop%C3%A4die+article%22&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjVpv3A2qzSAhVIK8AKHVgpAkwQ6AEIHzAA#v=onepage&amp;amp;q=ehrenfest%20%22encyklop%C3%A4die%20article%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;translation&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And here is a host of other books:&#xA;&lt;a href=&quot;https://archive.org/search.php?query=subject%3A%22Kinetic+theory+of+gases%22&amp;amp;sort=-date&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/search.php?query=subject%3A%22Kinetic+theory+of+gases%22&amp;amp;sort=-date&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-02-26T02:31:26.203" LastActivityDate="2017-02-26T02:31:26.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5731" PostTypeId="1" AcceptedAnswerId="5732" CreationDate="2017-02-26T07:15:57.000" Score="1" ViewCount="208" Body="&lt;p&gt;How many professors in EU can still read Latin in 1950? And how about 2000? I know many scientific literature were written in Latin in the past. &lt;/p&gt;&#xA;" OwnerUserId="2641" LastActivityDate="2017-02-27T00:17:18.790" Title="How many professors can read Latin in 1950 and 2000?" Tags="|language|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5732" PostTypeId="2" ParentId="5731" CreationDate="2017-02-26T07:31:18.923" Score="8" Body="&lt;p&gt;First of all &quot;can read&quot; must be defined more precisely. Many math professors in Europe whom I know had some Latin in school and can read a page from Euler, when necessary. But they certainly cannot read and enjoy Latin poetry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I never had any formal course of Latin, but I studied French, and also can understand 18 century mathematical texts in Latin with some efforts and perhaps with the help of a dictionary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that much scientific literature, especially in mathematics was written&#xA;in Latin until 19th century. But this does not mean that an average modern mathematician reads it. Those who need to read much of it just learn Latin.&#xA;I knew one mathematician who learned Latin with the special purpose to read Jacobi's works. But this is a rare exception. Most modern mathematicians read&#xA;only modern literature. Or translations to modern languages.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-02-26T07:31:18.923" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5735" PostTypeId="2" ParentId="5731" CreationDate="2017-02-27T00:17:18.790" Score="6" Body="&lt;p&gt;I agree with Alexandre: professors of &lt;strong&gt;science and mathematics&lt;/strong&gt; read mostly the current literature, the vast majority of which is in English.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in this forum, when the OP says &quot;professors&quot;, he may mean professors of &lt;strong&gt;History of Science&lt;/strong&gt; or &lt;strong&gt;History of Mathematics&lt;/strong&gt;.  In that case, depending on what era and what region of the world they study, other languages may be needed.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I heard a talk a few years ago by someone studying some topic in the history of geometry.  But he had come to an impasse because he did not read Arabic.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-02-27T00:17:18.790" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5736" PostTypeId="1" CreationDate="2017-02-27T10:24:56.417" Score="3" ViewCount="72" Body="&lt;p&gt;I find that papers reference Raymond Cattell suggesting 16 or 22, etc, traits, by factor analysis (basically regression), including all five of the modern reproducible traits (openness to experience, conscientiousness, extraversion, agreeableness --- he called it Independence --- neuroticism).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there is such a flood of publications by Cattell and other workers in this field, most of them applied, not theoretical, that it is is not obvious what the earliest publication containing all five of these factors was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tupes and Christal were the first, it seems, to be able to reproduce with a larger data set the only five factors that are reproducible, but &lt;em&gt;what is the earliest publication by CATTELL that includes all of these&lt;/em&gt; including the other dozen or so nonreproducible ones?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I found by Cattell is a 35 trait paper from 1947 that has all of these. Is there anything earlier? He worked in the 30's and 40's.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2017-02-27T10:57:36.060" LastActivityDate="2017-02-27T10:57:36.060" Title="Raymond Cattell and History of Personality Traits Prior 1947" Tags="|reference-request|biology|statistics|psychology|social-sciences|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5737" PostTypeId="1" AcceptedAnswerId="5741" CreationDate="2017-02-28T02:34:39.787" Score="5" ViewCount="546" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Herschel&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Herschel&lt;/a&gt; built a big telescope and observed Uranus in 1781. I'm trying to figure out who else had a comparable telescope at the time, who could verify these observations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently Johan Bode could but I could not find much data about his telescope. Another possibility is Anders Johan Lexell but I don't even know if he used a telescope or just computed the orbit via supplied observation recordings from someone else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Apparently, the telescope was &lt;a href=&quot;https://en.wikipedia.org/wiki/File:HerschelTelescope.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; one. It's described as 7 feet long, 6 inch diameter, f14 reflecting telescope. One way of investigating the question is to ask who had the second-largest telescope? Cannot find this on Google either.&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="2546" LastEditDate="2017-03-01T02:23:53.870" LastActivityDate="2017-03-01T02:23:53.870" Title="When William Herschel discovered Uranus, who else at the time had a comparable telescope?" Tags="|astronomy|telescope|18th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5738" PostTypeId="1" AcceptedAnswerId="5740" CreationDate="2017-02-28T16:27:53.887" Score="5" ViewCount="355" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Liouville&amp;#39;s_theorem_(Hamiltonian)#cite_note-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liouville's theorem&lt;/a&gt; asserts that the volume of phase space is incompressible. This was first shown by Gibbs 1903 but his name is mostly forgotten. The Wikipedia article says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is referred to as the Liouville equation because its derivation for&#xA;  non-canonical systems utilises an identity first derived by Liouville&#xA;  in 1838.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What identity does this refer to?&lt;/p&gt;&#xA;" OwnerUserId="5436" LastEditorUserId="658" LastEditDate="2017-02-28T20:41:56.513" LastActivityDate="2017-02-28T21:45:26.593" Title="What was Liouville's contribution to Liouville's theorem?" Tags="|biographical-details|theoretical-physics|classical-mechanics|symplectic-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5739" PostTypeId="1" CreationDate="2017-02-28T19:20:45.067" Score="3" ViewCount="151" Body="&lt;p&gt;After acquainting myself with the works of Richard Feynman in Physics: &lt;em&gt;&quot; The Feynman Lectures on Physics&quot;&lt;/em&gt; and partially with the works of Donald Knuth in Programming: &lt;em&gt;&quot;The Art of Computer Programming&quot;&lt;/em&gt;, both amazingly interesting, inspiring and attempting a holistic approach in the introduction and explanation of the respective field, I am wondering are there other authors with similar style in other fields of Science, e.g Mathematics, Chemistry, Biology and what are the titles of their works.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;" OwnerUserId="2351" LastActivityDate="2017-04-21T03:34:00.033" Title="Most known teachers of specific branch of Science" Tags="|mathematics|chemistry|biology|book-recommendation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5740" PostTypeId="2" ParentId="5738" CreationDate="2017-02-28T19:49:55.603" Score="7" Body="&lt;p&gt;Liouville shows (&lt;a href=&quot;http://archive.org/stream/s1journaldemat03liou#page/346&quot; rel=&quot;nofollow noreferrer&quot;&gt;1838, pp. 347-349&lt;/a&gt;) that if $x=\phi_t(a)$ is the &quot;complete integral&quot; of an ODE&#xA;$$&#xA;\frac{dx}{dt}=P(t,x)&#xA;\tag{*}&#xA;$$&#xA;on $\mathbf R^n$ (meaning that $a\in\mathbf R^n$ are $n$ otherwise arbitrary parameters for the general solution), then the jacobian $u=\det\bigl(\frac{\partial x_i}{\partial a_j}\bigr)$ satisfies (the identity you are after):&#xA;$$&#xA;\frac{du}{dt}=\operatorname{div}(P)u&#xA;\tag{**}&#xA;$$&#xA;where $\operatorname{div}(P)=\sum\frac{\partial P_i}{\partial x_i}$. In particular $u$ is constant if $\operatorname{div}(P)=0$, e.g. for $\frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i}$, $\frac{dq_i}{dt}=\frac{\partial H}{\partial p_i}$ (Hamilton equations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as observed by Prange (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN360617913&amp;amp;physid=PHYS_0687&quot; rel=&quot;nofollow noreferrer&quot;&gt;1935, p. 674&lt;/a&gt;) or Lützen (&lt;a href=&quot;https://books.google.com/books?id=px_vBwAAQBAJ&amp;amp;pg=PA51&quot; rel=&quot;nofollow noreferrer&quot;&gt;1990, pp. 51, 661&lt;/a&gt;), Liouville did &lt;em&gt;not&lt;/em&gt; apply his result to Hamilton equations, nor indeed speak of &quot;phase space volume&quot;, as his arbitrary constants $a$ need not be the initial conditions $\phi_0(a)$. That application came with Jacobi (&lt;a href=&quot;https://archive.org/stream/gesammeltewerkeh04jacouoft#page/451&quot; rel=&quot;nofollow noreferrer&quot;&gt;1844, §24&lt;/a&gt;), and the statistical or volumic interpretation with (before Gibbs) &lt;a href=&quot;https://books.google.com/books?id=px_vBwAAQBAJ&amp;amp;pg=PA665&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxwell, Boltzmann, and Kirchhoff&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Liouville's paper had fallen into oblivion when Maxwell and Boltzmann began their work, and therefore they rediscovered Liouville's theorem independently. In special cases, it was derived by Maxwell in his second discussion of the statistical distribution of the velocities of monatomic gas molecules in thermal equilibrium [1868] and by Boltzmann for other physical systems [1868].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(...) later the same year [1871b], Boltzmann derived the general theorem (**), and, moreover, showed that it was equivalent to Jacobi's principle of the last multiplier.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(...) as soon as the connection between Liouville's theorem and Jacobi's last multiplier had been established, the realization of Liouville's priority was not far away. Indeed, in Kirchhoff's &lt;em&gt;Vorlesungen über die Theorie der Wärme&lt;/em&gt;, held five times during the years 1876-1884 in Berlin, and published posthumously by Max Planck in 1894, it is remarked that Jacobi in his &lt;em&gt;Vorlesungen über Dynamik&lt;/em&gt; attributed the theorem to Liouville (cf. §23). Four years later, in his second part of the &lt;em&gt;Vorlesungen über Gastheorie&lt;/em&gt;, Boltzmann [1896-1898] baptized the theorem &quot;Liouville's theorem.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-02-28T21:45:26.593" LastActivityDate="2017-02-28T21:45:26.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5741" PostTypeId="2" ParentId="5737" CreationDate="2017-02-28T20:53:07.367" Score="4" Body="&lt;p&gt;(Partial answer - and a bit long winded)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding comparable telescopes - source : &lt;a href=&quot;https://books.google.ca/books?id=HFg2sOjrFtsC&amp;amp;pg=PA10&amp;amp;lpg=PA10&amp;amp;dq=Herschel%E2%80%99s%20telescopes%20far%20surpassed%20in%20both%20quality%20and%20size%20any%20other%20telescope%20in%20the%20world%20at%20that%20time.&amp;amp;source=bl&amp;amp;ots=jgmdi8Oc5h&amp;amp;sig=ElSxyFHQgTdq4SA0Vk0wwjB-5rU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Herschel%E2%80%99s%20telescopes%20far%20surpassed%20in%20both%20quality%20and%20size%20any%20other%20telescope%20in%20the%20world%20at%20that%20time.&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Herschel Objects and How to Observe Them&lt;/em&gt;, by James Mullaney&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Herschel's telescopes far surpassed in both quality and size any other telescope in the world at the time.  After comparative trials at a number of observatories in England including Greenwich, he stated with confidence &quot;I can now say that I absolutely have the best telescopes that were ever made.&quot;&#xA;  His fame as a telescope maker spread rapidly and soon he was flooded with requests from both other observes and obervatories to make instruments for them. While he was not in&#xA;  the telescope-making business as such, his allowance from the King – while freeing&#xA;  him from his musical duties – did not entirely meet his expenses, and so he began&#xA;  to make and sell telescopes privately. In addition to at least 60 complete instruments&#xA;  (most of them 7- and 10-feet in size), he also made several hundred mirrors upon&#xA;  order in addition to those for his own telescopes!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Curiously, Herschel thought he was observing a comet rather than a planet.  From &lt;a href=&quot;http://adsabs.harvard.edu/full/1944ASPL....4..254S&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harvard site&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Herschel's observations of the outline of the image on April 6 (perhaps a night with exceptionally good seeing) led him to remark, &quot;the comet appeared perfectly sharp upon the edges and extremely well-defined without the least appearance of any beard or tail.&quot;  This observation, however, seems not to have made him doubt that his object was a comet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Two years after his initial observations, Herschel wrote in &lt;em&gt;Philosophical Transactions, 73, 1, 1783&lt;/em&gt; (quoting from the same Harvard article) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sir - By the observations of the most eminent astronomers in Europe, it appears that the star which I had the honour of pointing out to them in March 1781 is a Primary Planet of our Solar System.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately Herschel does not name these astronomers, nor does he describe how they made their observations. However, Bode, who ultimately named Uranus, may not be amongst them.  According to &lt;a href=&quot;http://messier.seds.org/xtra/Bios/bode.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article on the Messier.org site&lt;/a&gt;, Bode's role may have been that of collecting the observations of other astronomers.  Bode names Mayer and Flamsteed as having observed Uranus prior to Herschel : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bode was greatly interested in the new planet discovered by William Herschel in March 1781. While Herschel always referred to this planet as &quot;Georgium Sidus&quot; to honor King George III of England, and Messier called it &quot;Herschel&quot; or &quot;Herschel's Planet,&quot; and Peitnet de Sevoy, &quot;Cybele,&quot; Bode proposed the name &quot;Uranus,&quot; which was soon adopted by the rest of the world. Bode collected virtually all observations of this planet by various astronomers, published many of them in the Astronomisches Jahrbuch, and found that Uranus had been observed before its discovery on a number of occasions, among them an observation of Tobias Mayer from 1756, and earliest by Flamsteed, in December 1690, cataloged as &quot;star&quot; 34 Tauri.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding Lexell, you appear to be correct in stating that he did not perform direct observations.  From the &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lexell.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;St. Andrew's Site&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When William Herschel discovered a new body in the solar system on 13 March 1781, Lexell computed its orbit which showed that it was a planet (later named Uranus) twice as far from the sun as Saturn, rather than a comet as had been thought at first. Although he did not predict the position of Neptune, as did Adams and Le Verrier, Lexell's initial calculations of the orbit of Uranus showed that it was being perturbed and he deduced that the perturbations were due to another more distant planet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-02-28T20:53:07.367" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5742" PostTypeId="1" CreationDate="2017-02-28T20:55:23.497" Score="0" ViewCount="132" Body="&lt;p&gt;I tried accesing the original article without much success. Can someone fill in the details ?&lt;/p&gt;&#xA;" OwnerUserId="2310" LastActivityDate="2017-02-28T20:55:23.497" Title="Why did Nikolai Luzin almost commit suicide?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5743" PostTypeId="1" CreationDate="2017-02-28T22:25:48.760" Score="3" ViewCount="245" Body="&lt;p&gt;The first term is used by Hilbert in his 1928 work, but in Gödel's later work, the same thing is referred to as Unvollständigkeitssatz (&quot;incompleteness theorem&quot;). For today's German CS researchers, it seems Unvollständigkeitssatz is more commonly used, and Entscheidungsproblem (&quot;decision problem&quot;) is still understood, but not necessarily associated with das Halteproblem (which seems to be more common after Turing's work on automata). On the other hand, for English CS researchers, Entscheidungsproblem is usually the only word they are familiar with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, when looking at the German Wikipedia, there is no entry for Entscheidungsproblem, but there is one for Gödelscher Unvollständigkeitssatz, and the entry about Hilbert uses Gödelscher Unvollständigkeitssatz. When looking at the English Wikipedia, one readily finds an entry for Entscheidungsproblem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How come Entscheidungsproblem is no longer used?&lt;/p&gt;&#xA;" OwnerUserId="5438" LastEditorUserId="5438" LastEditDate="2017-03-01T03:09:11.403" LastActivityDate="2017-03-02T14:45:16.123" Title="Entscheidungsproblem vs. Unvollständigkeitssatz" Tags="|mathematics|computer-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5744" PostTypeId="1" CreationDate="2017-03-01T14:41:00.047" Score="1" ViewCount="404" Body="&lt;p&gt;In the 1950s in The Netherlands, it was common for people to hang a little rope through the mailbox, so that children (which apparently did not have a copy of the key) could let themselves in at any time.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the outside, it looked like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.bottercourant.nl/images/nieuws/2016/12_December/SFA007002142.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;outside of door with rope through mailbox http://www.bottercourant.nl/images/nieuws/2016/12_December/SFA007002142.jpg&lt;/a&gt;&lt;br&gt;&#xA;&lt;em&gt;Source: &lt;a href=&quot;http://www.bottercourant.nl/nieuws/algemeen/11379-touwtje-aan-de-brievenbus-tijdens-kerst-in-de-vesting&quot; rel=&quot;nofollow noreferrer&quot;&gt;Botter Courant&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And on the inside:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://cdn.thingiverse.com/renders/8c/ef/d2/da/b8/IMG_1586_preview_featured.jpg&quot; alt=&quot;inside of the same&quot;&gt;&lt;br&gt;&#xA;&lt;em&gt;Source: &lt;a href=&quot;http://www.thingiverse.com/thing:738142&quot; rel=&quot;nofollow noreferrer&quot;&gt;user bjorntm, thingyverse&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly, this is actually a work-around for people who have a self-locking door but rather wouldn't.  I've visited areas where people don't normally lock their doors, and even where the cultural expectation for a visitor (stranger or not) is to simply enter and shout &lt;em&gt;Hello!&lt;/em&gt;.  Not locking the door in the first place would seem a rather more convenient solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the history of locks and door locks in particular, when were self-locking doors invented and when did they become commonplace?&lt;/p&gt;&#xA;" OwnerUserId="111" LastActivityDate="2017-05-29T13:35:02.547" Title="When were self-locking doors invented and when did they become commonplace?" Tags="|engineering|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5745" PostTypeId="2" ParentId="5743" CreationDate="2017-03-01T15:24:42.233" Score="4" Body="&lt;p&gt;The two problems of &lt;em&gt;completeness&lt;/em&gt; (&lt;em&gt;Vollstandigkeit&lt;/em&gt;) and &lt;em&gt;decidability&lt;/em&gt; of the predicate calculus were clearly defined in 1928 into the first &quot;modern&quot; mathematical logic textbook:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;David Hilbert and Wilhelm Ackermann, &lt;a href=&quot;https://en.wikipedia.org/wiki/Principles_of_Mathematical_Logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grundzüge der theoretischen Logik&lt;/a&gt; (see also the &lt;a href=&quot;https://books.google.it/books?id=45ZGMjV9vfcC&amp;amp;pg=PR1&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation&lt;/a&gt; of 1938 second edition: §10 and §12):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The 1928 edition included a clear statement of the &lt;em&gt;Entscheidungsproblem&lt;/em&gt; for FOL, and also asked whether that logic was complete (i.e., whether all semantic truths of FOL [= &lt;em&gt;universally valid&lt;/em&gt; (&lt;em&gt;Allgemeingultigkeit&lt;/em&gt;) formulas] were theorems derivable from the FOL axioms and rules). The first problem was answered in the negative by &lt;a href=&quot;https://en.wikipedia.org/wiki/Alonzo_Church#Mathematical_work&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alonzo Church in 1936&lt;/a&gt;. The second was answered affirmatively by &lt;a href=&quot;https://plato.stanford.edu/entries/goedel/#ComThe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Gödel in 1929&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First-order logic is &lt;em&gt;complete&lt;/em&gt; but &lt;em&gt;undecidable&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Entscheidungsproblem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Entscheidungsproblem&lt;/a&gt; = decision problem: &quot;the problem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers &quot;Yes&quot; or &quot;No&quot; according to whether the statement is universally valid&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The fact that every &lt;em&gt;valid&lt;/em&gt; formula can be derived (according to completeness) does not means that we have an algorithm for answering the above question : for first-order logic there is no algorithm like the truth table method for propositional logic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First-order arithmetic is &lt;em&gt;incomplete&lt;/em&gt; [ &lt;a href=&quot;https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Unvollständigkeit&lt;/em&gt;&lt;/a&gt; ].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two senses of &quot;completeness&quot; are both linked to Gödel, but they are not the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second fundamental result proved by &lt;a href=&quot;https://plato.stanford.edu/entries/goedel/#IncThe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gödel in 1931&lt;/a&gt; regards the &quot;capability&quot; of the first-order axiomatic theory of &lt;em&gt;arithmetic&lt;/em&gt; (FOL calculus plus non-logical axioms) to prove all the true arithmetical sentences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He found it working on &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_second_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert's second problem&lt;/a&gt; in the attempt to prove &lt;em&gt;consistency&lt;/em&gt; of arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The underlying calculus is complete but in this case we ask more: the true arithmetical sentences are not universally valid: they are true only in the structures that are models of the axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a nutshell, Gödel intuited that if a truth predicate $\text {Tr}_F(x)$ could be defined for a certain formal system $F$, then we could derive a version of the &lt;a href=&quot;https://plato.stanford.edu/entries/liar-paradox/#TarsHierLang&quot; rel=&quot;nofollow noreferrer&quot;&gt;liar antinomy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, such a truth predicate cannot be defined if $F$ is consistent, a result Gödel communicated in letters but never published (it was formally proved by Tarski; see Roman Murawski, &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/01445349808837306?journalCode=thpl20&quot; rel=&quot;nofollow noreferrer&quot;&gt;Undefinability of Truth. The Problem of Priority: Tarski vs Gödel&lt;/a&gt;, &lt;em&gt;History and Philosophy of Logic&lt;/em&gt;, Vol.19 (1998)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gödel realized that the antinomy can be avoided replacing &lt;em&gt;truth&lt;/em&gt; with &lt;em&gt;provability&lt;/em&gt;, but this implies that the two notions do not necessarily coincide: $\text{Tr} \ne \text {Pr}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for a &lt;em&gt;sound&lt;/em&gt; formal system (i.e. a system that proves only true arithmetical sentences) the only case when truth and provability differ is when $\text{Pr} \subsetneq \text {Tr}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, there will be true but formally unprovable sentences.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It is worth noting that not all FOL theories are incomplete and undecidable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Presburger_arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Presburger arithmetic&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Skolem_arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Skolem arithmetic&lt;/a&gt;, the arithmetical theories of &quot;$+$&quot; (plus) and &quot;$\times$&quot; (times) respectively, are both decidable and complete.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-03-02T14:45:16.123" LastActivityDate="2017-03-02T14:45:16.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5746" PostTypeId="1" AcceptedAnswerId="5775" CreationDate="2017-03-01T18:25:23.780" Score="5" ViewCount="358" Body="&lt;p&gt;Historian H. Mehrtens hypothesized an opposition between moderns and countermoderns in early 20th century mathematics, with the former led by Hilbert and the latter by Klein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hilbert's lecture at the ICM in Paris in 1900 presented 10 of the famous 23 open problems. It is well known that the idea of the lecture came from Hermann Minkowski. Hilbert was at Göttingen at the time where he was hired through untiring efforts of Felix Klein. As detailed by historian David Rowe and others, both Hilbert and Klein were involved in a battle against the Berliners at the time. The Berlin school dominated by followers of Kummer, Weierstrass, and Kronecker was known for its focus on arithmetized analysis. Hilbert's 23 open problems sought to broaden the scope of mathematics beyond such narrow focus. It seems as though it would have been natural for Hilbert to have discussed the 23 problems with Klein. Is there any evidence of such discussions in published work or private correspondence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what Minkowski wrote in a letter to Hilbert:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Most alluring would be the attempt to look into the future, in other words, a characterization of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now. Of course, prophecy is indeed a difficult thing&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Minkowski 1973, 5 January 1900; see &lt;a href=&quot;https://books.google.co.il/books?id=Hc-iBgAAQBAJ&amp;amp;pg=PA120&amp;amp;source=gbs_selected_pages&amp;amp;cad=3#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;German original&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reference is: Hermann Minkowski, Briefe an David Hilbert, Hg. L. Rüdenberg und H. Zassenhaus, New York: Springer-Verlag, 1973. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This information comes from page 16 of Rowe's article: Rowe, D. &quot;Mathematics made in Germany: on the background to Hilbert's Paris lecture.&quot; Math. Intelligencer 35 (2013), no. 3, 9--20. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beyond the issue of possible correspondence concerning Hilbert's Paris lecture, Frei's book on the Klein-Hilbert correspondence may contain further evidence that Klein and Hilbert were, first of all, allied against the Berliners, and second of all both moderns contrary to the thrust of the Mehrtens hypothesis on Klein being allegedly countermodern: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Der Briefwechsel David Hilbert-Felix Klein (1886-1918). [The correspondence between David Hilbert and Felix Klein (1886-1918)] Edited, with comments, by Guether Frei. Arbeiten aus der Niedersachsischen Staats- und Universitatsbibliothek Göttingen [Publications of the Lower Saxony State and University Library in Göttingen], 19. Vandenhoeck &amp;amp; Ruprecht, Göttingen, 1985.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question was posted at &lt;a href=&quot;https://mathoverflow.net/questions/261105&quot;&gt;MO&lt;/a&gt; a month ago without generating much input.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1. As Jan Peter Schäfermeyer pointed out, Klein not only published numerous papers by Cantor in &lt;em&gt;Mathematische Annalen&lt;/em&gt; but also used Cantor as a referee for the journal. From the modern perspective this would indicate a progressive attitude on Klein's part. Any further details would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2017-04-28T06:55:42.913" LastActivityDate="2018-03-20T11:15:57.193" Title="Evaluating the Mehrtens hypothesis concerning Felix Klein" Tags="|mathematics|mathematicians|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5747" PostTypeId="1" AcceptedAnswerId="5748" CreationDate="2017-03-01T19:00:43.323" Score="2" ViewCount="207" Body="&lt;p&gt;My mathematics are still quite rudimentary, but am I correct in assuming this is a reference to the &quot;finite&quot; state of closed manifolds as opposed to a potential, &quot;infinite&quot; state of the non-compacted case?&lt;/p&gt;&#xA;" OwnerUserId="5197" LastActivityDate="2017-03-01T20:40:48.950" Title="How is &quot;soul&quot; meant to be understood in the context of the &quot;Soul Theorem&quot;" Tags="|terminology|riemannian-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5748" PostTypeId="2" ParentId="5747" CreationDate="2017-03-01T20:40:48.950" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://www.nytimes.com/2008/06/19/science/19gromoll.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.nytimes.com/2008/06/19/science/19gromoll.html&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the soul theorem, published in 1972, Dr. Gromoll and Dr. Cheeger were studying the properties of certain surfaces that could have flat regions or curves like the outside of a sphere but not regions shaped liked saddles. They found that the properties of such surfaces, infinite in extent and existing in any number of dimensions, could be deduced from a finite central core region.&lt;/p&gt;&#xA;&lt;p&gt;Dr. Cheeger said it was Dr. Gromoll who suggested calling this finite region the “soul” of the object, because it captured the essence of the infinite expanse around it. “Just like inside a person,” Dr. Cheeger said.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-01T20:40:48.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5750" PostTypeId="2" ParentId="5746" CreationDate="2017-03-03T11:12:28.160" Score="3" Body="&lt;p&gt;I think it is safe to say that Klein was a more traditional mathematician than Hilbert. Yet Courant praises Klein's modern approach in teaching analysis in the preface to his &lt;a href=&quot;https://archive.org/stream/DifferentialIntegralCalculusVolI/Courant-DifferentialIntegralCalculusVolI#page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculus book&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are interviews with some people at Göttingen (&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4522-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Born&lt;/a&gt;,&#xA;&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4562&quot; rel=&quot;nofollow noreferrer&quot;&gt;Courant&lt;/a&gt;, and &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4935&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Karman&lt;/a&gt;) which show Klein's and Hilbert's shared deep interest in applying mathematics to physics. In this respect there was not much of a difference between the two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the antagonism between Berlin and Göttingen, I think it had more to do with personal animosities, as can be seen from the Frobenius biography by &lt;a href=&quot;https://books.google.de/books?id=UzTBBAAAQBAJ&amp;amp;pg=PA64&amp;amp;dq=frobenius+klein+%22low+esteem%22&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwiEn7-_krrSAhUrLZoKHZZHAwIQ6AEIHzAA#v=onepage&amp;amp;q=frobenius%20klein%20%22low%20esteem%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hawkins&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-03-10T13:37:56.377" LastActivityDate="2017-03-10T13:37:56.377" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5751" PostTypeId="2" ParentId="5739" CreationDate="2017-03-04T02:10:58.297" Score="2" Body="&lt;p&gt;The whole series &lt;em&gt;&quot;A course on Theoretical Physics.&quot;&lt;/em&gt; (in 10 volumes) by L. Landau and E. Lifshitz partially fits the bill, although the style is terse and entirely hypnotic. The lectures are certainly aspirational and certainly holistic, but well beyond elementary level.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorUserId="2351" LastEditDate="2017-04-20T18:55:23.517" LastActivityDate="2017-04-20T18:55:23.517" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5752" PostTypeId="2" ParentId="5739" CreationDate="2017-03-05T01:11:56.353" Score="1" Body="&lt;p&gt;John Maynard Smith in Evolutionary Biology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Richard Dawkins and Stephen Jay Gould are/were probably the most famous writers on Evolutionary Biology, but both's famous writings were for popular audiences, and nontechnical.)&lt;/p&gt;&#xA;" OwnerUserId="4791" LastEditorUserId="2351" LastEditDate="2017-04-21T03:34:00.033" LastActivityDate="2017-04-21T03:34:00.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5753" PostTypeId="2" ParentId="1816" CreationDate="2017-03-05T03:01:23.763" Score="3" Body="&lt;p&gt;Aristotle concluded in his law of motion that the speed of an object depends on the viscosity of the medium it is in.  In keeping with this line of thinking, since a perfect vacuum has zero viscosity, the speed of a falling object should approach infinity, as viscosity approaches zero.  Galileo in his incline plane experiment identified the role of gravity, explaining it to be Aristotle's attractive force pulling toward the &quot; natural place&quot;.  In addition, Galileo's Leaning Tower of Pisa experiment addressed the question: do falling masses of different sizes fall at the same speed?  He concluded that they will fall at the same speed.  Newton corrected Galileo's conclusion with his own gravitational theory which states that the the force of gravity exert the same acceleration on objects regardless of size; an idea that has gained acceptance over the years.   Just as Aristotle law of gravitation, ignoring the role of mathematics available to him at the time, was was the first approximation of the role of gravity on falling objects, Galileo, utilizing the same math that was available to Aristotle, came up with a second approximation of the role of gravity.  Newton with more advance mathematics came up with his law of gravitation, a third approximation to the law of gravitational motion; and of course, Einstein made a radical departure from all existing theories of the time with his upgrade. &lt;/p&gt;&#xA;" OwnerUserId="5458" LastActivityDate="2017-03-05T03:01:23.763" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5754" PostTypeId="1" AcceptedAnswerId="5758" CreationDate="2017-03-05T10:01:10.950" Score="6" ViewCount="784" Body="&lt;p&gt;Kepler apparently arrived at his first two laws based on the Tycho's data for  Mars. But Mars has the largest eccentricity except for Mercury, so it is easier to tell the difference between a circle and an ellipse. What about other planets, like Venus? It seems difficult, its eccentricity is only 0.006772. Wouldn't it be very hard to tell a circle from an ellipse with Tycho's data? Did Kepler really establish the two laws based on a single orbit of Mars? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="4251" LastEditDate="2021-02-28T08:10:38.030" LastActivityDate="2022-04-19T09:30:47.103" Title="Did Kepler arrive at his planetary laws based on Mars's orbit alone?" Tags="|physics|astronomy|kepler|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5755" PostTypeId="1" CreationDate="2017-03-05T20:43:33.940" Score="3" ViewCount="389" Body="&lt;p&gt;The theory of General Relativity as usually presented currently defines the &lt;em&gt;relativistic spacetime&lt;/em&gt; as a tuple $(M,g,\nabla, T)$ where $(M,g)$ is a four dimensional smooth lorentzian manifold, $\nabla$ is the Levi-Civita connection of the metric $g$ and $T$ is a time orientation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, $T$ is a nowhere vanishing vector field on $M$ such that $g(T,T)&amp;gt;0$ always.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, by what I know about the history of General Relativity, since its begining it invoked the idea of curvature of spacetime. More than that, Einstein's equations &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$R_{\mu\nu}-\dfrac{1}{2}{g_{\mu\nu}}R=\dfrac{8\pi G}{c^4}T_{\mu\nu}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;already requires the smooth manifold structure, the lorentzian metric and the Levi-Civita connection in order to have the curvature tensor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the structure $(M,g,\nabla)$ for spacetime was already included in Einstein's original formulation of the theory, even if in a different language than the one used today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However the time orientation is something that I still don't know where it historically came from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Einstein already needed and used somehow the idea of time orientation in his work on General Relativity? If not, who introduced time orientation as a part of the structure of spacetime and why he did so?&lt;/p&gt;&#xA;" OwnerUserId="4072" LastActivityDate="2017-12-22T15:15:03.687" Title="Einstein already used the idea of time orientation when formulating General Relativity?" Tags="|theoretical-physics|differential-geometry|relativity-theory|einstein|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5756" PostTypeId="2" ParentId="5755" CreationDate="2017-03-06T11:28:28.760" Score="3" Body="&lt;p&gt;I believe the question started being studied in earnest in the 1950s, with formal definition and first results usually attributed to Markus &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=73169&quot; rel=&quot;nofollow noreferrer&quot;&gt;(1955, p. 412)&lt;/a&gt; — by, e.g., Hawking in his famous &lt;a href=&quot;http://epjh.epj.org/epjh-news/875-epjh-highlight-10-1140-epjh-e2014-50014-x&quot; rel=&quot;nofollow noreferrer&quot;&gt;1966 prize essay&lt;/a&gt; (emphasis added):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;5.2 Time orientability&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In our own neighbourhood, there is a clear division of non-spacelike vectors into future- and past-directed. However, it is not so obvious that we could extend this to give a continuous division at all points of $M$. This problem has been dealt with by &lt;strong&gt;&lt;em&gt;Markus&lt;/em&gt;&lt;/strong&gt; &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=73169&quot; rel=&quot;nofollow noreferrer&quot;&gt;(1955)&lt;/a&gt;: as there is a Lorentz metric on $M$, we can find a continuous line-element field $(X,−X)$, where $X$ is a timelike vector. The manifold $\smash{\tilde M}$ is defined as the set of all pairs $\smash{(p,X|_p)}$ and $\smash{(p,-X|_p)}$ with the natural structure. Clearly, $\smash{\tilde M}$ covers $M$ twice. There are two possibilities: either $\smash{\tilde M}$ is connected, in which case it is impossible to introduce a continuous division into future- and past-directed vectors in $M$, but one exists in $\smash{\tilde M}$, or $\smash{\tilde M}$ consists of two disconnected components, in which case one can find a continuous division in $M$. Physically it would seem very reasonable to expect $M$ to be time orientable, but even if it was not, we could apply all the following theorems to the time orientable covering manifold $\smash{\tilde M}$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As for Einstein himself, Gödel’s model containing closed timelike curves &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=31841&quot; rel=&quot;nofollow noreferrer&quot;&gt;(1949)&lt;/a&gt; prompted him to at least some memories in &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=35229&quot; rel=&quot;nofollow noreferrer&quot;&gt;(1949, pp. 687-688)&lt;/a&gt; — where he is careful to distinguish the local and global question (emphasis added):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The problem here involved&#xA;  &lt;strong&gt;&lt;em&gt;disturbed me already&lt;/em&gt;&lt;/strong&gt; at the time of the building up of the &#xA;  general theory of relativity, without my having succeeded in&#xA;  clarifying it (...):&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If $P$ is a world-point, a “light-cone”&#xA;  ($\smash{ds^2 = 0}$) belongs to it. We draw&#xA;  a “time-like” world-line through $P$&#xA;  and on this line observe the close&#xA;  world-points $B$ and $A$, separated by&#xA;  $P$. Does it make any sense to provide&#xA;  the world-line with an arrow, and to&#xA;  assert that &lt;em&gt;$B$ is before $P$, $A$ after $P$&lt;/em&gt;?&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(...) If, therefore, $B$ and $A$ are two, sufficiently neighboring,&#xA;  world-points, which can be connected by a time-like line, then&#xA;  the assertion: “$B$ is before $A$,” makes physical sense. But does&#xA;  this assertion still make sense, if the points, which are connectable &#xA;  by the time-like line, are arbitrarily far separated from&#xA;  each other? Certainly not, if (...)&#xA;  those paradoxes, regarding the direction of the causal &#xA;  connection, arise, of which Mr. Gödel has spoken.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-12-22T15:15:03.687" LastActivityDate="2017-12-22T15:15:03.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5757" PostTypeId="1" CreationDate="2017-03-06T22:57:45.670" Score="4" ViewCount="592" Body="&lt;p&gt;In the SI unit prefixes, there's a general pattern of using uppercase prefixes for multipliers larger than 1 and lower case for prefixes that are smaller than one.  However, this is not a universal trend.  k-, in particular, is lower case, as are h- and da-.  In a system with very clear capitalization rules, these stand out as outliers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the rationale behind this capitalization?&lt;/p&gt;&#xA;" OwnerUserId="5470" LastActivityDate="2017-03-11T23:18:27.230" Title="Why is the SI prefix k- lower case?" Tags="|terminology|units|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5758" PostTypeId="2" ParentId="5754" CreationDate="2017-03-07T01:28:11.233" Score="7" Body="&lt;p&gt;This is a natural question given Kepler's own account of his discoveries in his most famous astronomical work, Astronomia Nova. It is a fascinating story of of how he arrived at the idea of the first two planetary laws (orbits are ellipses with the Sun at a focus, radius vector sweeps equal areas in equal times) by trying to fit Tycho's data on Mars first to epicycles, then to ovals, and finally to ellipses. The specific difficulty in the OP would not have been an issue: even if it was impossible to tell a circle from an ellipse for Venus and Mercury circle is a particular case of ellipse, so both laws would hold for a planet uniformly moving along a circle with the sun at its center. However, anomalies of the inner planets were known already at the time of Ptolemy, let alone Tycho, see Duke's &lt;a href=&quot;https://people.sc.fsu.edu/%7Edduke/inner.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Ancient Values of the Planetary Parameters of Venus&#xA;and Mercury&lt;/a&gt;. More importantly, modern scholarship indicates that Astronomia Nova is not a realistic description of how Kepler made his discoveries, it is a carefully crafted rhetorical fiction.&lt;/p&gt;&#xA;&lt;p&gt;Kepler was a devoted heliocentrist long before 1609 when Astronomia Nova was published, and metaphysical and mystical speculations played as much (if not more) of a role in his conviction than empirical data. He was intimately familiar with the mechanics of the Copernican system and its relation to the geocentric ones. Copernicus had circular orbits, and therefore still required epicycles, although they were much smaller than Ptolemy's (and sometimes referred to as epicyclets), see &lt;a href=&quot;https://www.aub.edu.lb/fas/cvsp/Documents/copernicus.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Astronomy and Cosmology of Copernicus&lt;/a&gt;. Kepler was discussing the fine points of this mechanics along with his own improvements for various planets with his like-minded teacher from Tübingen, Maestlin, before even working for Tycho. See &lt;a href=&quot;https://www.jstor.org/stable/986463&quot; rel=&quot;noreferrer&quot;&gt;Michael Maestlin's Account of Copernican Planetary Theory by Grafton&lt;/a&gt; for his correspondence with Kepler on various planets at the time Kepler's Mysterium Cosmographicum was published (1596).&lt;/p&gt;&#xA;&lt;p&gt;The area law is implicitly present in the epicyclic astronomy (to the first order of approximation) through the artifice of &lt;a href=&quot;https://en.wikipedia.org/wiki/Equant&quot; rel=&quot;noreferrer&quot;&gt;Ptolemy's equant&lt;/a&gt;, placing the point from which a planet's motion looks uniform off-center. Copernicus replaced the equant with a pair of epicyclets, and Kepler realized early on that it closely approximates &amp;quot;equal areas in equal times&amp;quot; when plotting the data for various planets, long before he came up with ellipses. In other words, his basis for the two laws was not just one orbit of Mars, although that alone provided quite a bit of data points to explain by two simple principles, but the entirety of epicyclic astronomy, including Copernicus, with all the known anomalies, and technical devices used to deal with them. As Voelkel writes in &lt;a href=&quot;http://press.princeton.edu/chapters/i7187.html&quot; rel=&quot;noreferrer&quot;&gt;The Composition of Kepler's Astronomia Nova&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Kepler named as one of the works he would produce his Commentaries on Mars--that is, the Astronomia nova. He was thus forced to conceive the book as a preliminary announcement of the fruits of his physical astronomy as applied to the orbit of Mars. It would contain his important finding regarding the bisection of the earth's eccentricity, which vindicated his physical account of the cause of the equant as well as clearing up certain problems in the orbit of Mars (and the orbits of Mercury and Venus as well). At that time, however, he had no clear idea of what the eventual solution to Mars's orbit would be. Although he was employing a form of the area law, the discovery of Mars's elliptical orbit was still two and a half years away.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Although the orbit of Mars was a catalyst for many developments they were accompanied with looking at the data for other planets, if for no other reason than to answer to Kepler's various correspondents and critics. Nor was empirical data his sole guide, his physical, metaphysical and mystical arguments for heliocentrism were deliberately pushed into the background by the circumstances at the time of the writing. Here is Voelkel again:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Recent research, especially that of William H. Donahue, has shown that the account Kepler offers his readers is not a true history of the course of his research--something Kepler never claimed--but is rather a didactic or rhetorical pseudohistory... Astronomia Nova is only accidentally modern--that is, that the particular context in which the book was composed forced Kepler to rein in his broader arguments for heliocentrism, leaving only a subset of his physical reasoning that appears distinctly modern in retrospect.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;[...] the argument of the book was a response to the various criticisms he had encountered during the course of his research. To the charge that his physical astronomy was an unjustified aberration, he responded by constructing his argument to make it appear as though he resorted to a physical approach to planetary theory only after a comprehensive failure of the most general kind of model in the classical form (which he presented in part 2, even though he actually completed the research only after parts of the research presented in part 3). He countered the charge that his radical innovations were themselves the source of the difficulties he had encountered by repeating many of the demonstrations in the book (as with the repeated demonstrations involving the true and the mean sun)... Thus, many features of the Astronomia Nova become comprehensible only when they are viewed in the context of Kepler's experience in writing the book as elements of an elaborate and purposefully-constructed rhetorical argument.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-07T01:28:11.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5759" PostTypeId="1" AcceptedAnswerId="5760" CreationDate="2017-03-07T13:32:18.493" Score="10" ViewCount="11559" Body="&lt;p&gt;At quite the end of the movie &lt;a href=&quot;https://www.imdb.com/title/tt0268978&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Beautiful Mind&lt;/a&gt;, John Nash tells a student &amp;quot;I am making progress&amp;quot; (towards proving the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann hypothesis&lt;/a&gt; (RH)). Actually, how much did Nash contribute to the proof of the RH?&lt;/p&gt;&#xA;" OwnerUserId="5473" LastEditorUserId="4251" LastEditDate="2022-04-26T16:10:49.617" LastActivityDate="2022-04-26T16:10:49.617" Title="How much did John Nash contribute to proving the Riemann hypothesis?" Tags="|mathematics|mathematicians|discoveries|number-theory|john-nash|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5760" PostTypeId="2" ParentId="5759" CreationDate="2017-03-07T15:55:07.840" Score="17" Body="&lt;p&gt;Nash was known to have been captivated by RH at an early age after reading E.T. Bell's &lt;em&gt;Men of Mathematics&lt;/em&gt;.  He had confided in some friends and colleagues that he had an idea that might work involving &lt;a href=&quot;https://en.wikipedia.org/wiki/Pseudoprime&quot; rel=&quot;noreferrer&quot;&gt;pseudoprimes&lt;/a&gt;, so there was a great deal of anticipation surrounding the announcement of his 1959 lecture at Columbia University sponsored by the American Mathematical Society.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Karl Sabbagh's &lt;em&gt;Dr Riemann's Zeros&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There have been several occasions in the past hundred years when mathematicians have announced that they have proved the Riemann Hypothesis.  But none can be stranger than the events that took place in New York in 1959 ....  So when two-hundred and fifty or so mathematicians gathered to listen to Nash, there was an air of expectation in the lecture hall.  Perhaps he really had done it?  But as Nash got deeper and deeper into his 'proof', the audience of mathematicians became more and more aghast.  It wasn't just wrong - it was nonsense. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;According to Donald Newman, 'One word didn't fit with the other.  Everyone knew something was wrong.  It was his chatter.  The maths was just lunatic. ... Nash's talk wasn't good or bad.  It was horrible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is no record of Nash demonstrating any significant progress toward a proof prior to the onset of his mental illness.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-03-07T17:56:43.347" LastActivityDate="2017-03-07T17:56:43.347" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5761" PostTypeId="1" CreationDate="2017-03-07T16:58:16.297" Score="1" ViewCount="98" Body="&lt;p&gt;I believe there is a story about a 20th century sciencist (either Neumann or Feynmann I think) who, at some dinner or some sort of gathering, made the following claim to his fellow colleagues: He asserted that in less than 1 minute he can find the answer (with 10% error) of any mathematical problem which could be stated in less than 10 seconds (I'm not 100% sure if I got the parameters exactly right). Is there any source of this story?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way I remember coming across the story is in a youtube video, where the author mentioned that the collegues should've asked for the value of $\tan(10^{1000})$, which would've required the sciencist to know $\pi$ to very many decimal places.&lt;/p&gt;&#xA;" OwnerUserId="4123" LastActivityDate="2017-03-07T16:58:16.297" Title="Taking less than a minute to solve a problem within 10% which was stated in less than 10 seconds?" Tags="|mathematics|physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5762" PostTypeId="1" AcceptedAnswerId="5766" CreationDate="2017-03-07T17:03:43.000" Score="3" ViewCount="224" Body="&lt;p&gt;On page 326 of his 2008 book, Jeremy Gray writes matter-of-factly that Albert Einstein &amp;quot;found he got a better hearing from Hilbert and Klein in Göttingen than he did from his colleagues in Berlin&amp;quot;. What is the background that would explain such a difference of attitude toward Einstein's work?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="4251" LastEditDate="2021-09-15T19:33:30.553" LastActivityDate="2021-09-15T19:33:30.553" Title="Why did Einstein get a better hearing in Göttingen than Berlin?" Tags="|physics|mathematicians|relativity-theory|einstein|hilbert|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="5763" PostTypeId="1" AcceptedAnswerId="5765" CreationDate="2017-03-07T18:38:10.450" Score="6" ViewCount="412" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy_Binet_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy-Binet formula&lt;/a&gt; in linear algebra gives the determinant of a square matrix &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; for rectangular matrices &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt; of appropriate shape in terms of the determinants of the submatrices of &lt;span class=&quot;math-container&quot;&gt;$A,B$&lt;/span&gt;. (See the linked Wikipedia page for the specifics.) In the special case of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; with two columns, this reduces to a summation identity which Wikipedia in turn refers to as the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Binet%E2%80%93Cauchy_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Binet-Cauchy identity&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why Cauchy-Binet for the determinant formula and Binet-Cauchy for the special case?&lt;/p&gt;&#xA;" OwnerUserId="107" LastEditorUserId="107" LastEditDate="2019-12-28T19:37:01.137" LastActivityDate="2019-12-28T19:37:01.137" Title="Binet-Cauchy or Cauchy-Binet?" Tags="|mathematics|naming-conventions|linear-algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5764" PostTypeId="1" CreationDate="2017-03-07T21:34:58.867" Score="0" ViewCount="155" Body="&lt;p&gt;One friend of mine has a cup with a lot of mathematical expressions printed on it. I would like to know more about one of these.&lt;/p&gt;&#xA;&lt;p&gt;What is the history behind the following inequality, and what is the related concept? Where would this inequality be used, or what natural phenomena does it represent?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sqrt {a\times b}\le\frac {a+b}{2}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="5474" LastEditorUserId="1901" LastEditDate="2022-05-05T06:19:23.480" LastActivityDate="2022-05-05T06:19:23.480" Title="Related concept for this equation?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5765" PostTypeId="2" ParentId="5763" CreationDate="2017-03-08T05:57:14.233" Score="5" Body="&lt;p&gt;I am not sure what the editors of Wikipedia had in mind when arranging the names (if anything). Linear algebra textbook authors have them arranged every which way for both the formula and the identity, see e.g. &lt;a href=&quot;https://books.google.com/books?id=6Pp2-DTOKWIC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shafarevich-Remizov&lt;/a&gt; (p.69),  &lt;a href=&quot;https://books.google.com/books?id=TRKDAgAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dym&lt;/a&gt; (p.103) and &lt;a href=&quot;https://books.google.com/books?id=TRKDAgAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lancaster&lt;/a&gt; (p.39). The formula was discovered independently but almost simultaneously in 1812 by both Binet and Cauchy.&lt;/p&gt;&#xA;&lt;p&gt;Oliver Knill reports that, according to Sternberg, Binet not only derived the formula but was also first to derive the rule for multiplying matrices about the same time, and presented it in a lecture on November 30, 1812 (the usual attribution is to Cayley's 1857 papers). Unfortunately, this is unconfirmed:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;My own digging on Binet did not get far. It would be especially interesting to get a hand on the publication or talk of Binet of 1812 and compare it with Cauchy who would prove at the same time the Cauchy-Binet formula. Cauchy and Binet lived at around the same time, went to the same school, competed for the same jobs. Its not surprising that they discovered the Cauchy-Binet formula at around the same time. They might even have communicated about it (even so this is only speculation).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;[...] It is interesting to me that determinants have appeared before matrix algebra or even matrices and that the multiplication rule for determinants predates the discovery of matrix multiplication. But in this case one can understand the reason: Cauchy-Binet is useful when trying to understand solutions of linear equations. The later can be understood also without matrix algebra, as it happened historically.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.math.harvard.edu/%7Eknill/history/matrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knill's webpage&lt;/a&gt; also has references, timeline of the early history of determinants, and links to the current research on generalizations of the formula.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-08T19:30:43.903" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5766" PostTypeId="2" ParentId="5762" CreationDate="2017-03-08T14:04:22.307" Score="4" Body="&lt;p&gt;There were a number of chemists in the Prussian Academy of Sciences, most notably Nernst and Fischer. They had hoped that Einstein would engage in atomic theory and bring about a better understanding of chemical processes. However, according to Jürgen Neffe's Einstein &lt;a href=&quot;https://books.google.de/books?id=B8K6n177ZwcC&amp;amp;pg=PA165#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography&lt;/a&gt;, nothing came of it. A new theory of gravitation was the least thing they were interested in.&#xA;Quite contrary to David Hilbert, who from 1912 on worked on a unified theory of gravitation and electromagnetism: &lt;a href=&quot;http://www.tau.ac.il/~corry/publications/articles/pdf/mie.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.tau.ac.il/~corry/publications/articles/pdf/mie.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-03-08T14:07:55.363" LastActivityDate="2017-03-08T14:07:55.363" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5767" PostTypeId="1" CreationDate="2017-03-08T19:22:02.410" Score="11" ViewCount="4824" Body="&lt;p&gt;When was it first discovered (or comprehended) that air has weight?&#xA;Did the ancients know of this, or did they think that air is weightless?            &lt;/p&gt;&#xA;" OwnerUserId="3780" LastEditorUserId="55" LastEditDate="2017-03-09T01:04:54.093" LastActivityDate="2017-03-17T01:19:40.577" Title="When was it first discovered (or comprehended) that air has weight?" Tags="|physics|discoveries|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5768" PostTypeId="2" ParentId="5767" CreationDate="2017-03-08T19:51:57.617" Score="3" Body="&lt;p&gt;Since ancient times, air was considered one of the four elements (air, water, fire, earth). As Aristotle said in &lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#2-22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Cœlo&lt;/em&gt; II, ch. 13&lt;/a&gt; (294b [359]):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;air is lighter than water, so is water than earth&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Cœlo&lt;/em&gt;&lt;/a&gt; IV, ch. 4 (311b), he says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;air is heavy when not in water or earth&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, he certainly believes air has some gravity.&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2017-03-08T19:51:57.617" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5769" PostTypeId="2" ParentId="5754" CreationDate="2017-03-08T23:02:21.137" Score="4" Body="&lt;p&gt;Let me add a couple of remarks to Conifold's response, based on Voelkel's book (mentioned by Conifold), and the earlier book &lt;em&gt;Kepler's Physical Astronomy&lt;/em&gt; by Bruce Stephenson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler did indeed come up with his first two laws based just on the orbit of Mars. The discovery was convoluted and not purely empirical, and Stephenson and Voelkel give a detailed account. However, once Kepler had his first two laws, he immediately generalized them to all the planets, first in his &lt;em&gt;Epitome of Copernican Astronomy&lt;/em&gt;, and then in his &lt;em&gt;Rudolphine Tables&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the &lt;em&gt;empirical&lt;/em&gt; case for elliptical orbits for the other planets was weak, Kepler belief in harmony and uniformity led him to extend his laws to all the planets. In addition, the &lt;em&gt;Astronomia Nova&lt;/em&gt; presents physical explanations for the first two laws; naturally, he assumed that the same physics would lead to the same results. (His physical explanations convinced almost nobody and rapidly disappeard from history.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even for Mars, the data was not that strong. As Curtis Wilson says in &quot;Predictive astronomy in the century after Kepler&quot; (in &lt;em&gt;The General History of Astronomy (vol. 2): Planetary astronomy from the Renaissance to the rise of astrophysics&lt;/em&gt;, ed. René Taton and Curtis Wilson):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Because of the level of unavoidable error in&#xA;  observations of position, and the near circularity of&#xA;  the orbits, the departure from circularity could be&#xA;  detected observationally only in the orbits of Mars&#xA;  and Mercury, where the eccentricity was exceptionally&#xA;  large; and even in these cases the choice&#xA;  between ellipse and other oval shapes was, as far as&#xA;  the observations could show, a matter for conjecture.&#xA;  Kepler, of course, had reasons for his choice: a&#xA;  causal [i.e., physical] account which led both to the elliptical orbit&#xA;  and to the planet's motion on that ellipse, with&#xA;  close agreement between the theoretical prediction&#xA;  and observation in the particular case of Mars.&#xA;  Ironically for Kepler, it was the ellipse that caught&#xA;  on most quickly and widely, as a simple curve that&#xA;  could be produced by a combination of circular&#xA;  motions, and as able to account for the bisection of&#xA;  the eccentricity and for the observably oval paths of&#xA;  Mars and Mercury; while Kepler's rationale of the&#xA;  planetary motions was all but universally rejected. The Keplerian ellipse became emblematic of the&#xA;  new astronomy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As Wilson notes, several other Keplerian innovations (e.g., his treatment of the &quot;mean sun&quot;) made a much bigger impact purely in terms of &lt;em&gt;accuracy&lt;/em&gt;, although &lt;em&gt;conceptually&lt;/em&gt; they were less important.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2017-03-08T23:02:21.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5770" PostTypeId="2" ParentId="5767" CreationDate="2017-03-09T00:54:43.877" Score="5" Body="&lt;p&gt;There is a subtle difference between realizing that air is a substance (as opposed to a void) and establishing that is has weight, see also &lt;a href=&quot;https://hsm.stackexchange.com/questions/2389/who-was-the-first-to-postulate-that-space-was-a-vacuum/2394#2394&quot;&gt;Who was the first to postulate that space was a vacuum?&lt;/a&gt; for a related issue. The former was demonstrated by Empedocles (c. 490–430 BC) by using perhaps one of the first experimental apparatuses in history, a glass tube called hydra. Here is from &lt;a href=&quot;https://books.google.com/books?id=oabrCbbMx6oC&amp;amp;pg=PT16&amp;amp;lpg=PT16&amp;amp;dq=empedocles+hydra+pressure+vacuum&amp;amp;source=bl&amp;amp;ots=bYeVIBI7YV&amp;amp;sig=pYqqU6hzt8YymVXDmhKw6v2cFWo&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ei=uo1nVYa2DYnvtQW5g4HYDw&amp;amp;ved=0CC8Q6AEwBQ#v=onepage&amp;amp;q=empedocles%20hydra%20pressure%20vacuum&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Close's The Void&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If you place your finger over it no water flows. If you empty the water from the hydra and then submerge it water will pour in and refill it as long as the open end remains open. However, if the end is covered with your finger no water enters the holes, and no air escapes either. This demonstrated that air and water coexist in the same space; no water can enter until the air leaves; air is a substance and not empty space.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, fire was also considered a substance, but a weightless one. Aristotle describes another experiment, with an inflated bladder, upon which his belief in air's weight was based in &lt;a href=&quot;http://classics.mit.edu/Aristotle/heavens.4.iv.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Caelo [On the Heavens] Book IV.4&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Earth, then, and bodies in which earth preponderates, must needs have weight everywhere, while water is heavy anywhere but in earth, and air is heavy when not in water or earth. In its own place each of these bodies has weight excepting fire, even air. Of this we have evidence in the fact that a bladder when inflated weighs more than when empty&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Aristotle's view was more or less accepted during the middle ages. But this might have been a case of believing the right thing for the wrong reasons, a historical instance of what is now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gettier_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gettier problem of knowledge&lt;/a&gt;. Galileo was skeptical:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I am inclined to believe that the increase of weight observed in the leather bottle or bladder arises, not from the gravity of the air, but from the many thick vapors mingled with it in these lower regions. To this I would attribute the increase in weight in the leather bottle.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Modern considerations also cast doubt on the validity of the bladder experiment, so if we exclude Gettierfied &amp;quot;knowledge&amp;quot; Galileo might have been the first to establish that air has weight, and measure it. A good historical survey of this topic is &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768090&quot; rel=&quot;nofollow noreferrer&quot;&gt;West's Torricelli and the Ocean of Air&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is interesting that, today, teachers of elementary physics often use an experiment with an inflated balloon to make the point that air has weight. There are several examples of these on the internet. In a typical demonstration, two inflated toy balloons are suspended at the ends of a long stick such as a meter rule, and this is supported in the middle so that it is balanced. One balloon is then burst by putting a match under it, and the other balloon tilts the balance downward.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In fact, this is a misleading demonstration because the reason why the intact balloon falls is that it contains air under pressure. If the two balloons are inflated with air at normal atmospheric pressure, deflating one balloon will not change the balance. The reason is that a balloon at normal pressure, such as a thin plastic bag that has been partly inflated, receives support by buoyancy from the air around it, which cancels the weight of the air inside it.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Galileo presumably understood this, although he does not appear to state it. What he does do is describe in detail a method for measuring the weight of the air. This is included in &lt;a href=&quot;https://en.wikipedia.org/wiki/Two_New_Sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discourses Concerning Two New Sciences&lt;/a&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-14T23:57:51.947" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5772" PostTypeId="1" AcceptedAnswerId="5777" CreationDate="2017-03-09T18:02:27.570" Score="11" ViewCount="1566" Body="&lt;p&gt;The notation $x\mapsto \sin x&#xA;$&#xA;and its meaning are well-known to most mathematicians. Less well-known seems to be the fact that $x \mapsto y$ means the same as Church's $\lambda x.y$ and Frege's $\grave x y$ (found in &lt;em&gt;Funktion und Begriff&lt;/em&gt;, 1891). At least at an informal level these all seem to denote the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both Frege's and Church's work are considered mayor milestones in the development of the notion of functions, but I've never heard of the person who came up with the notation $x\mapsto y$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By whom and when was $x\mapsto y$ first used?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="55" LastEditDate="2017-03-09T22:30:22.280" LastActivityDate="2017-11-03T12:16:33.843" Title="When was the function arrow notation $x \mapsto y$ first used?" Tags="|mathematics|notation|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5773" PostTypeId="2" ParentId="5772" CreationDate="2017-03-09T21:00:16.970" Score="5" Body="&lt;p&gt;A good source for these types of questions is &lt;a href=&quot;http://jeff560.tripod.com/operation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's website on earliest uses of mathematical symbols&lt;/a&gt;. This notation is recent (on the other hand, use of arrows for limits goes as far back as Riemann). The original version was introduced by Ore in  L'Agebre Abstraite (1936), but without the vertical bar:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We shall say that two algebraic systems $S$ and $S'$ are homomorphic (with respect to addition and multiplication) if there exists a correspondence $a\to a'$ between the elements of $S$ and $S'$ giving to each element $a$ of $S$ a unique image $a'$ in $S'$ such that each element of $S'$ is the image of at least one element of $S$ and further such that if $a\to a'$, $b\to b'$ we have $a+b\to a'+b'$ and $ab\to a'b'$&lt;/em&gt;&quot;. [translation mine]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1939 Bourbaki started using &quot;&lt;em&gt;la application $x\to f(x)$&lt;/em&gt;&quot;, and it spread far and wide along with their volumes. The earliest version of the $f: X\to Y$ notation seems to be in a Hurewicz-Steenrod paper of 1940. Algebraic topologists took a liking to it, and it was a blessing for the emerging homological algebra according to MacLane:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;At first the vivid arrow notation $f: X\to Y$ for a map was not available, and homomorphisms of homology groups (or rings) were always expressed in terms of the corresponding quotient group or rings. Thus the familiar long exact sequence of the homotopy groups of a fibration was originally described in terms of subgroups and quotient groups; this is the style used by all three discoveries of the sequence and of the covering homotopy theorem [...] The occurrence of exact sequences of homology groups (though not the name 'exact') was first noted by W. Hurewicz in 1941 [...] The practice of using an arrow to represent a map $f: X\to Y$ arose at the same time. I have not been able to determine who first introduced this convenient notation; it may well have appeared first on the blackboard, perhaps in lectures by Hurewicz and it is used in the Hurewicz-Steenrod paper, submitted November 1940...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The pedantic distinction between $\to$ and $\mapsto$ is the work of later textbook authors. &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=1179395&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cabillon comments on the NCTM Math Forum&lt;/a&gt; (credit to Francois Ziegler for linking the thread):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The notation &quot;a barred arrow f(a)&quot;&lt;/em&gt; [i.e. $a\mapsto f(a)$] &lt;em&gt;is fairly modern, I would dare to say. I have not been able to determine who first introduced this notation, but my guess is that it came with the &quot;New Math&quot; vogue of the early 60's. The first time I encountered this notation was in French books.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-03-09T22:20:00.183" LastActivityDate="2017-03-09T22:20:00.183" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5774" PostTypeId="2" ParentId="5772" CreationDate="2017-03-10T11:55:11.677" Score="4" Body="&lt;p&gt;Thanks for the useful answers so far. I find the &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=1179397&quot; rel=&quot;nofollow noreferrer&quot;&gt;answer&lt;/a&gt; of William Waterhouse in the discussion linked to by Francois Ziegler very interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He gives evidence for the following&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1) The &quot;barred arrow&quot; $\mapsto$ almost certainly was introduced between&#xA;  1960 and 1965.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2) It arose after a period of at least a decade (probably more) in&#xA;  which the plain arrow $\to$ was used ambiguously [to denote both the type of a function (domain/codomain), as well its element-to-element description].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;3) I do not know who invented it, but it definitely did not come  from&#xA;  the &quot;New Math&quot; school reform movement in the United States. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;4) My guess (though I cannot fully support this) is that need for  a&#xA;  distinct symbol became felt around this time just because  there were&#xA;  more situations arising where the ambiguity was a  possible source of&#xA;  confusion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and he also writes the following&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is one additional piece of evidence  that supports both 1) and&#xA;  4) and is probably unfamiliar:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the very early 1960's, one sometimes found the notation&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;/\/\/\--&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(i.e., a wiggle at the start of the arrow) in the meaning that was&#xA;  soon taken over by |--&gt;. (I must say this looks very much like a&#xA;  suitable place to recall Victor Borge's &quot;phonetic punctuation&quot;&#xA;  routine.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I don't know whether this appeared in &quot;formal&quot; publications. (Checking&#xA;  back, I see that I was using |--&gt; already as a graduate student in&#xA;  1966.) I have it in my notes from an algebra course taught by John Tate in Spring 1962, and it occurs in a semi-formal publication that&#xA;  should be in some libraries:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Dieudonne, Jean Alexandre &lt;em&gt;Algebraic geometry&lt;/em&gt;  [College Park], 1962 &#xA;  Maryland. University. Dept. of Mathematics / Lecture notes ; no.1 &#xA;  (Library of Congress number) QA564.D5&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then there is also an interesting &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=1179399&quot; rel=&quot;nofollow noreferrer&quot;&gt;answer&lt;/a&gt; by Emili Bifet about Riemann using the plain arrow $x\to y$ to denote the element-to-element action of a function, even before Ore (1936). (And not to denote the passage to a limit as in $\lim_{x\to 0}$.) Here is my translation of what Riemann &lt;a href=&quot;http://www.e-rara.ch/zut/content/pageview/5620190&quot; rel=&quot;nofollow noreferrer&quot;&gt;wrote&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;II. The integrals of a linear differential equation of second order at a branching point&#xA;  (From a lecture winter semester 1856/57.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If $a$ is a branching point of the solution to a linear differential equation of second order and, as $x$ moves clockwise around $a$, $z_1$&#xA;  goes over to $z_3$ while $z_2$ goes over to $z_4$, what shall be&#xA;  denoted with $z_1 \to z_3$ and $z_2 \to z_4$ for short, then $$ z_3 = t z_1 + u z_2  $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$z_4 = r z_1 + s z_2.$$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If $\epsilon$ is any constant, then $$ z_1 + \epsilon z_2 \to z_3 + \epsilon z_4. $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Since now $$ z_1 (x - a)^{-\alpha} \to z_3 (x - a)^{-\alpha} e^{-2 \alpha \pi i} $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;it must hold&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$ z_1 (x - a)^{-\alpha} \to z_1 (x - a)^{-\alpha}  + (z_1 + \epsilon z_2)k(x - a)^{-\alpha}. $$&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Since further&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$$ \frac{k}{2 \pi i} (x - a)^{-\alpha}(z_1 + \epsilon z_2)l(x - a) \to  \frac{k}{2 \pi i}  (x - a)^{-\alpha}(z_1 + \epsilon z_2)l(x - a) + k (x - a)^{-\alpha}(z_1 + \epsilon z_2),   $$ ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In modern terminology he is apparently talking about the monodromy map along a path in a complex domain. Bifet also writes that the arrow notation in analysis (limits) ist due to Leathem (around 1905) which agrees whit what Jeff Millers website &lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;says&lt;/a&gt; on this.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2017-03-12T17:59:36.580" LastActivityDate="2017-03-12T17:59:36.580" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5775" PostTypeId="2" ParentId="5746" CreationDate="2017-03-10T13:45:27.677" Score="5" Body="&lt;p&gt;Mehrtens describes in his book &quot;Moderne, Sprache, Mathematik&quot; the trend towards abstraction and axiomatization in mathematics from the late 19th century onwards, whose main exponents to him were Hilbert, Cantor, Zermelo and Hausdorff and whose main adversaries were Kronecker, Klein, Poincaré and Brouwer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While it is certainly true that there were controversies between Cantor and &lt;a href=&quot;http://mcps.umn.edu/philosophy/11_5Edwards.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kronecker&lt;/a&gt; and between Hilbert and &lt;a href=&quot;https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brouwer&lt;/a&gt;, which Poincaré participated in, I am not convinced that Klein's name belongs here. To begin with, Klein employed Cantor as a reviewer for the &quot;Mathematische Annalen&quot; (see Mehrtens, p. 207) and also published his main papers &lt;a href=&quot;https://glossar.hs-augsburg.de/Mathematische_Annalen&quot; rel=&quot;nofollow noreferrer&quot;&gt;there&lt;/a&gt;. Then he brought Hilbert and Zermelo to Göttingen and never interfered in their scientific work. So, Mehrtens has to stretch historical facts a bit to make his point and he consequently calls Klein a proponent of the &quot;kooperative Gegenmoderne&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He mainly bases his assertion on Klein's 1895 paper on &lt;a href=&quot;http://www.ams.org/journals/bull/1896-02-08/S0002-9904-1896-00340-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Arithmetizing of Mathematics&lt;/a&gt; where Klein criticizes Weierstrass and Kronecker who in turn considered him a &lt;a href=&quot;https://books.google.de/books?id=PlmjAzAOTpoC&amp;amp;pg=PA109&amp;amp;dq=%22geometrisch+anschaulichen%22+scharlatan&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwiJ-9W51cvSAhXJC8AKHTvJCj0Q6AEIPjAH#v=onepage&amp;amp;q=%22geometrisch%20anschaulichen%22%20scharlatan&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;charlatan&lt;/a&gt;. To Mehrtens, Klein's emphasis on geometric intuition, which not only shows up here, but also in a lecture he had given in &lt;a href=&quot;https://archive.org/stream/gesammeltema02kleirich#page/n237/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evanston&lt;/a&gt; and in  his &lt;a href=&quot;https://archive.org/stream/arxiv-0807.3161/0807.3161#page/n19/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erlangen program&lt;/a&gt;, makes him a counter-modern, while Gray's book features the latter as the first item in the chapter &lt;a href=&quot;https://books.google.de/books?id=ldzseiuZbsIC&amp;amp;pg=PA114&amp;amp;dq=jeremy+gray+%22erlangen+programm%22&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjp27PW9cvSAhXqI8AKHW1LCFgQ6AEIGjAA#v=onepage&amp;amp;q=jeremy%20gray%20%22erlangen%20programm%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical modernism arrives&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, Klein was - except for his Erlangen program - not a modern mathematician himself, but he was an enabler of modern mathematics, e.g. an early champion of Lie and Grassmann, and certainly not counter-modern.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-03-12T18:00:34.360" LastActivityDate="2017-03-12T18:00:34.360" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5776" PostTypeId="2" ParentId="5757" CreationDate="2017-03-11T23:04:17.083" Score="4" Body="&lt;p&gt;The decision that k is lowercase was made in the early days of the cgs system, long before the SI system. The available prefixes were milli up to myria.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.bipm.org/en/committees/cipm/publications-cipm.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1879&lt;/a&gt; the International Committee for Weights and Measures, CIPM, adopted the proposal to abbreviate only in lowercase, because a mixture with capitals would be sensitive to mistakes and reduce the ease of writing. The least used prefixes deca/deka and myria, having the same initial letters as deci and milli, were not abbreviated because they would require capitals or an extra letter. The member states of CIPM started using these official abbreviations. &#xA;The new prefix mega (1874) was not considered because it was rarely used and unimportant for commerce and trade. Higher prefixes did not exist. &lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2017-03-11T23:18:27.230" LastActivityDate="2017-03-11T23:18:27.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5777" PostTypeId="2" ParentId="5772" CreationDate="2017-03-11T23:33:29.990" Score="15" Body="&lt;p&gt;I believe the &lt;em&gt;when&lt;/em&gt; can be narrowed down to 1963-64, and &lt;em&gt;by whom&lt;/em&gt; to, likely, a Bourbaki member. In the treatise, a lower and upper bound are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=179291&quot; rel=&quot;noreferrer&quot;&gt;Intégration, Chap. 7-8 (&lt;strong&gt;1963&lt;/strong&gt;)&lt;/a&gt;: still uses $\to$ (e.g. pp. 10, &lt;a href=&quot;http://dx.doi.org/10.1007/978-3-540-35325-6&quot; rel=&quot;noreferrer&quot;&gt;121&lt;/a&gt;; &lt;em&gt;that&lt;/em&gt; goes back to &lt;a href=&quot;https://mathoverflow.net/questions/194377/when-was-the-arrow-notation-for-functions-first-introduced/272968#272968&quot;&gt;&lt;strong&gt;1891&lt;/strong&gt; or &lt;strong&gt;1905&lt;/strong&gt;&lt;/a&gt;!)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=219684&quot; rel=&quot;noreferrer&quot;&gt;Intégration, Chap. 1-4, 2nd ed. (&lt;strong&gt;1965&lt;/strong&gt;)&lt;/a&gt;: now uses $\mapsto$ (e.g. &lt;a href=&quot;https://books.google.com/books?id=zLHOy-4VIXoC&amp;amp;pg=PP4&quot; rel=&quot;noreferrer&quot;&gt;unaltered reprint&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=zLHOy-4VIXoC&amp;amp;pg=PA4&quot; rel=&quot;noreferrer&quot;&gt;p. 4&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of the &lt;a href=&quot;http://archives-bourbaki.ahp-numerique.fr/publications#elements-math&quot; rel=&quot;noreferrer&quot;&gt;volumes published in between&lt;/a&gt;, I could only view this relevant snippet (similar to, but different from M. Allegranza's) in &lt;a href=&quot;https://books.google.com/books?id=S1QgAQAAIAAJ&amp;amp;q=%C3%A9dition&quot; rel=&quot;noreferrer&quot;&gt;Théorie des ensembles, Fascicule de résultats, 4th ed.&lt;/a&gt; (&lt;a href=&quot;https://books.google.com/books?id=S1QgAQAAIAAJ&amp;amp;q=1964&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;1964&lt;/strong&gt;&lt;/a&gt;), pp. &lt;a href=&quot;https://books.google.com/books?id=S1QgAQAAIAAJ&amp;amp;q=%22sens+de+la+fl%C3%A8che%22&quot; rel=&quot;noreferrer&quot;&gt;13&lt;/a&gt;-&lt;a href=&quot;https://books.google.com/books?id=S1QgAQAAIAAJ&amp;amp;q=%22notation+l%C3%A9g%C3%A8rement+diff%C3%A9rente%22&quot; rel=&quot;noreferrer&quot;&gt;14&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lorsqu'une relation de la forme $y=\langle x\rangle$ (où $\langle x\rangle$ désigne une combinaison de signes dans laquelle peut figurer $x$) est une relation fonctionnelle en $y$, on désignera parfois la fonction qu'elle détermine par la notation $x\to\langle x\rangle$, ou même simplement par $\langle x\rangle$, ce qui est un abus de langage très fréquent (par exemple, on parlera de la fonction $\sin x$ dans $\mathbf R$). On notera que le sens de la flèche $\to$ est alors tout à fait différent de celui qui lui a été attribué ci-dessus; pour cette raison, &lt;strong&gt;&lt;em&gt;on remplace souvent la notation $x\to\langle x\rangle$ par la notation légèrement différente $x\mapsto\langle x\rangle$&lt;/em&gt;&lt;/strong&gt; pour éviter toute ambiguïté. Par exemple, si $\mathrm X$, $\mathrm Y$ sont deux parties d'un ensemble $\mathrm E$, la relation $\mathrm Y=\complement\mathrm X$ est fonctionnelle en $\mathrm Y$, et il y a lieu de désigner par $\mathrm X\mapsto\complement\mathrm X$ l'application de $\mathfrak P(\mathrm E)$ dans lui-même qu'elle détermine, pour éviter de la confondre avec une application de $\mathrm X$ dans $\complement\mathrm X$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Other early publications of $\mapsto$ by bourbakists are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Borel, Serre, Bruhat, and Lang, &lt;a href=&quot;http://www.numdam.org/issues/SB_1962-1964__8_&quot; rel=&quot;noreferrer&quot;&gt;séminaire Bourbaki nº 265, 270 (February &lt;strong&gt;1964&lt;/strong&gt;) and 271, 274 (May &lt;strong&gt;1964&lt;/strong&gt;)&lt;/a&gt; (already in the &lt;a href=&quot;https://books.google.com/books?id=ZebuAAAAMAAJ&amp;amp;q=Secr%C3%A9tariat&quot; rel=&quot;noreferrer&quot;&gt;1964 edition&lt;/a&gt;: pp. &lt;a href=&quot;https://books.google.com/books?id=ZebuAAAAMAAJ&amp;amp;q=expriment&quot; rel=&quot;noreferrer&quot;&gt;265-01&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=ZebuAAAAMAAJ&amp;amp;q=%22une+alg%C3%A8bre+sur+le+corps+premier%22&quot; rel=&quot;noreferrer&quot;&gt;270-03&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=ZebuAAAAMAAJ&amp;amp;q=271-02&quot; rel=&quot;noreferrer&quot;&gt;271-02&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=ZebuAAAAMAAJ&amp;amp;q=%22groupe+des+fonctions%22&quot; rel=&quot;noreferrer&quot;&gt;274-04&lt;/a&gt;);&lt;/li&gt;&#xA;&lt;li&gt;Borel and Tits, &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=207712&quot; rel=&quot;noreferrer&quot;&gt;Groupes réductifs (&lt;strong&gt;1965&lt;/strong&gt;), p. 57&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;Demazure's thesis, &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=197467&quot; rel=&quot;noreferrer&quot;&gt;Schémas en groupes réductifs (&lt;strong&gt;1965&lt;/strong&gt;), p. 370&lt;/a&gt;: &lt;strong&gt;&lt;em&gt;“Nous employons dans cet article le signe $\mapsto$ en lieu et place du signe $\sim\!\rightarrow$ utilisé antérieurement.”&lt;/em&gt;&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;Demazure's &lt;a href=&quot;https://books.google.com/books?id=yh9aAAAAYAAJ&amp;amp;q=%C3%A9dition&quot; rel=&quot;noreferrer&quot;&gt;SGA 3 talks (&lt;strong&gt;1962/64&lt;/strong&gt;)&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=yh9aAAAAYAAJ&amp;amp;q=ais%C3%A9ment&quot; rel=&quot;noreferrer&quot;&gt;p. 5&lt;/a&gt;, but only in the Springer Lecture Notes version (&lt;a href=&quot;https://books.google.com/books?id=_Ib2DAAAQBAJ&amp;amp;pg=PA5&quot; rel=&quot;noreferrer&quot;&gt;1970&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, the earlier switch from an overloaded $\to$ to the proto-mapsto $\sim\!\rightarrow$ (or $\rightsquigarrow$) seems to have been made by Grothendieck between Bourbaki seminars &lt;a href=&quot;http://www.numdam.org/item/SB_1958-1960__5__193_0&quot; rel=&quot;noreferrer&quot;&gt;nº 182&lt;/a&gt; and &lt;a href=&quot;http://www.numdam.org/item/SB_1958-1960__5__299_0&quot; rel=&quot;noreferrer&quot;&gt;nº 190&lt;/a&gt; (1959), and in print between EGA &lt;a href=&quot;http://www.numdam.org/item/PMIHES_1961__8__5_0&quot; rel=&quot;noreferrer&quot;&gt;Chap. II&lt;/a&gt; and &lt;a href=&quot;http://www.numdam.org/item/PMIHES_1961__11__5_0&quot; rel=&quot;noreferrer&quot;&gt;Chap. III&lt;/a&gt; (1961). (He stuck with $\rightsquigarrow$ much longer, until at least &lt;a href=&quot;http://www.numdam.org/item/SB_1964-1966__9__199_0&quot; rel=&quot;noreferrer&quot;&gt;nº 290&lt;/a&gt; (1965, p. 202) and &lt;a href=&quot;http://www.numdam.org/item/PMIHES_1967__32__5_0&quot; rel=&quot;noreferrer&quot;&gt;Chap. IV&lt;/a&gt; (1967).) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note added.&lt;/strong&gt; I asked Michel Demazure, who kindly allows me to share these recollections:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;In his mind, the innovation (and standardization) came from Bourbaki. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He thinks he learned it from them through Grothendieck before joining, not vice versa.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He seems to recall Serre using $\rightsquigarrow$ in his Collège de France (blackboard) lectures before 1959.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They called $\rightsquigarrow$ “tortillon” and $\mapsto$ “poussoir”.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In adopting any symbol, a significant consideration was its availability in the &lt;a href=&quot;http://dx.doi.org/10.1021/ac60182a857&quot; rel=&quot;noreferrer&quot;&gt;Typit&lt;/a&gt; system.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(One can indeed see this system used for $\rightsquigarrow$ and $\mapsto$ in e.g. the &lt;a href=&quot;http://www.numdam.org/issues/SB_1962-1964__8_&quot; rel=&quot;noreferrer&quot;&gt;1964 seminars&lt;/a&gt;, pp. 368, 403, 418.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Further note.&lt;/strong&gt; Bourbaki ($x\mapsto y$), Church ($\lambda x.y$) and Frege ($\grave xy$) have other antecedents: $\newcommand{SS}{\mathrel{\style{display:inline-block; transform:rotate(.25turn);}{\mathrm S}}}$&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Eduard Study apparently originated $x\to y$, initially typeset $x\rightarrowtail y$ (&lt;a href=&quot;https://mathoverflow.net/questions/194377/when-was-the-arrow-notation-for-functions-first-introduced/272968#272968&quot;&gt;1905, 1891&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Peano used $y|x$ and earlier $y\,\overline x$:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?31.0069.02&quot; rel=&quot;noreferrer&quot;&gt;1900&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=kVZLAAAAMAAJ&amp;amp;pg=PA38&quot; rel=&quot;noreferrer&quot;&gt;p.38&lt;/a&gt;): let $A$ be an expression containing the variable letter $x$; by $A|x$, which one can read « the expression $A$ regarded as a function of $x$ », we indicate the function sign $u$ which, written in front of $x$, produces the given formula $A$. (...) E. g.: $a^n/n!\,|n$ represents the function sign which for value $n$ of the variable has value $a^n/n!$.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;(&lt;a href=&quot;https://archive.org/stream/formulairedemat02peangoog#page/n83&quot; rel=&quot;noreferrer&quot;&gt;1898, p. VII&lt;/a&gt;): The sign $|$ is the inversion sign. In earlier work it had the shape $\overline{\phantom{m}}$.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;(&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?28.0063.03&quot; rel=&quot;noreferrer&quot;&gt;1897&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/formulairedemat02peangoog#page/n66&quot; rel=&quot;noreferrer&quot;&gt;pp. 2, 58&lt;/a&gt;): let $a$ be an expression containing the variable letter $x$. We indicate by $a\,\overline x$ the function sign $f$, such that one has $fx=a$. One has therefore&#xA;  $$&#xA;(fx)\,\overline x = f.&#xA;$$&#xA;  E. g. $x^2-3x$ is an expression containing $x$; if we put&#xA;  $$&#xA;f = (x^2-3x)\,\overline x&#xA;$$&#xA;  we will have $fx=x^2-3x$. The value of $fx$ for $x=5$, or $f5$, will therefore be indicated by&#xA;  $$&#xA;(x^2-3x)\,\overline x\,5=10.&#xA;$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Eisenstein used $x \SS y$, which Peano et al. (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?32.0070.02&quot; rel=&quot;noreferrer&quot;&gt;1901&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=kVZLAAAAMAAJ&amp;amp;pg=PA173&quot; rel=&quot;noreferrer&quot;&gt;p. 173&lt;/a&gt;) quote as a forerunner of $y|x$ even though its usage isn’t &lt;em&gt;quite&lt;/em&gt; the same:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?035.0987cj&quot; rel=&quot;noreferrer&quot;&gt;1847&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=RpxGAAAAcAAJ&amp;amp;pg=PA211&quot; rel=&quot;noreferrer&quot;&gt;p. 211&lt;/a&gt;): To be able to express myself more briefly, I will represent the concept of &lt;em&gt;Substitution&lt;/em&gt; by a special sign, say by the sign $\,\SS$, whose introduction is urgently needed in many investigations, whenever one deals with the values that some expressions assume when other expressions, of which the former are functions, change their form or value. This sign $\,\SS$, which saves a lot of words, reads in a premise (hypothesis): “If the expression on the sign’s left is replaced by the one on the right,” and in the ensuing conclusion (thesis): “Then the left-hand side goes over into the right-hand side.” E. g. the proposition, if $A\SS B$, then $C\SS D$ reads: if one replaces $A$ by $B$, or lets $A$ go over into $B$, then $C$ goes over into $D$, or $D$ replaces $C$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-11-03T12:16:33.843" LastActivityDate="2017-11-03T12:16:33.843" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5779" PostTypeId="2" ParentId="5616" CreationDate="2017-03-12T14:50:52.787" Score="4" Body="&lt;p&gt;The first memo, on orbit determination, that &lt;a href=&quot;https://en.wikipedia.org/wiki/Katherine_Johnson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Katherine Johnson&lt;/a&gt; did with a co-worker is &lt;a href=&quot;https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19980227091.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Determination of Azimuth Angle at Burnout for Placing a Satellite over a Selected Earth Position 1960. T.H. Skopinski, Katherine G. Johnson, NASA TN D-233&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This example from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hidden_Figures_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hidden Figures book&lt;/a&gt; provides some details of the actual computations that she did, checking the &lt;a href=&quot;https://en.wikipedia.org/wiki/IBM_7090&quot; rel=&quot;nofollow noreferrer&quot;&gt;IBM 7090&lt;/a&gt; trajectory calculations for the first US orbital mission. E.g. she was working to 8 significant digits:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Katherine Johnson] worked thru every minute of what was programmed to be [John Glenn's] three-orbit mission, coming up with numbers for eleven different output variables, each computed to eight significant digits.  It took a day and a half.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The book also notes that the human computers used mechanical calculators. Some great detail on their working environment comes from a 1942 memo reproduced in &lt;a href=&quot;https://airandspace.si.edu/stories/editorial/hidden-figures-and-human-computers&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hidden Figures and Human Computers | National Air and Space Museum&lt;/a&gt;, including these brief excerpts:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... the work of a computer required skill and judgment. Computers gathered data by reading pressure values from manometers placed in the wind tunnel. Depending on the application, the data were smoothed, plotted, and interpolated. Data reduction and analysis were carried out with the help of &lt;a href=&quot;https://en.wikipedia.org/wiki/Mechanical_calculator#1900s_to_1970s&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculators&lt;/a&gt;, slide rules, &lt;a href=&quot;https://en.wikipedia.org/wiki/Planimeter&quot; rel=&quot;nofollow noreferrer&quot;&gt;planimeters&lt;/a&gt;, drafting tools, and other instruments. The women in these roles knew how to organize computational work and how to do so quickly without making mistakes. This knowledge was unique to them....&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Equipment&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The automatic computing machines and &lt;a href=&quot;https://en.wikipedia.org/wiki/Comptometer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comptometers&lt;/a&gt; cost over $500.00 each....&lt;/p&gt;&#xA;&lt;p&gt;The automatic calculator is usually the Friden or Marchant.... The computers were also furnished with 20 inch (log-log duplex) slide rules....&lt;/p&gt;&#xA;&lt;p&gt;... for the type of computer who has some mathematical background the type of work required of her could probably be better done on the calculating machine....&lt;/p&gt;&#xA;&lt;p&gt;Each computing section has one light-table for tracing purposes.... A set of five-place logarithmic tables and trigonometric tables is also provided, together with a few scale, triangle and French curves....&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Type of Work&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is a large amount of simple calculation required in the work here at NACA. Most [wind] tunnels have means for taking photographs of banks of nearly a hundred manometers at a time. The computers read off the liquid levels and complete the analysis. On the other hand, some of the calculations are sent to the computers in the form of complicated formulas which necessitate a knowledge of trigonometry and sometimes of mathematics involving the calculus. In general, however, the group head would reduce this more complicated work down to tabular form requiring rather routine operations before it would  be given to the machine operator. Most of the work coming from the engineers is accompanied by a memo of calculating instructions or word-of-mouth explanations. The computers in any one section soon learn what the usual type of calculation required of them would be.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I wish the film had had just a bit more on those mechanical calculators. Here is a video of one of the models they used in action, a &lt;a href=&quot;https://www.youtube.com/watch?v=qKm9eM2BuM0&quot; rel=&quot;nofollow noreferrer&quot;&gt;1956 Friden STW 10&lt;/a&gt; taking a square root.&lt;/p&gt;&#xA;" OwnerUserId="5279" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-29T14:33:34.190" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5780" PostTypeId="1" AcceptedAnswerId="5781" CreationDate="2017-03-12T20:28:53.140" Score="6" ViewCount="988" Body="&lt;p&gt;I had a professor point out that it is odd we refer to more than one chain complex as &quot;complexes.&quot; It seems that in most other definitions in math we stick to the typical latin plural, i.e. we say vertices instead of &quot;vertexes&quot;, matrices instead of &quot;matrixes&quot;. So why do we say &quot;complexes&quot; instead of complices when referring to more than one chain complex?&lt;/p&gt;&#xA;" OwnerUserId="3129" LastEditorUserId="55" LastEditDate="2017-03-12T23:08:37.670" LastActivityDate="2017-03-12T23:11:15.773" Title="Why do we say &quot;Matrices&quot; and &quot;Vertices&quot;, but &quot;Complexes&quot; rather than &quot;Complices&quot;?" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5781" PostTypeId="2" ParentId="5780" CreationDate="2017-03-12T23:05:42.453" Score="12" Body="&lt;p&gt;Because unlike vertex, matrix or simplex, that came directly from Latin and have primarily mathematical uses, complex was &lt;a href=&quot;http://www.etymonline.com/index.php?term=complex&quot; rel=&quot;nofollow noreferrer&quot;&gt;borrowed through French&lt;/a&gt; around 1650s, with the meaning &quot;a whole comprised of parts&quot;. By the time of entering mathematics as a noun it already had colloquially established plural in English, complexes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is similar with apexes, annexes, and most ironically &lt;a href=&quot;https://english.stackexchange.com/questions/65541/plural-of-syntax&quot;&gt;syntaxes&lt;/a&gt;, regular speakers tend to ignore the pedantries of etymology. Appendices are an exception, possibly because appendixes is awkward to pronounce. &lt;a href=&quot;https://en.wiktionary.org/wiki/complex#Noun&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiktionary&lt;/a&gt; does list complices as archaic plural. Here is from &lt;a href=&quot;https://en.wikipedia.org/wiki/English_plurals#Irregular_plurals_from_Latin_and_Greek&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's English plurals&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The general trend with loanwords is toward what is called Anglicization or naturalization, that is, the re-formation of the word and its inflections as normal English words. Many nouns (particularly ones from Latin) have retained their original plurals for some time after they are introduced. Other nouns have become Anglicized, taking on the normal &quot;s&quot; ending. In some cases, both forms are still competing&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps Latin being the language of science and mathematics for a long time made them more resistant to Anglicization than the general public.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:38:16.393" LastActivityDate="2017-03-12T23:11:15.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5782" PostTypeId="1" AcceptedAnswerId="13336" CreationDate="2017-03-13T13:44:52.180" Score="4" ViewCount="287" Body="&lt;p&gt;It is often referred to as 'anomalous electron configurations' or 'unusual order of filling of orbitals' of a few elements e.g. Cr, Cu, Nb, Mo, Ru, Rh, Pd, Au etc. &lt;/p&gt;&#xA;" OwnerUserId="5496" LastActivityDate="2021-07-04T16:26:16.980" Title="Who first noticed violation of aufbau principle of elements like Cr, Cu, Pd, Au, and how?" Tags="|physics|theoretical-physics|experimental-physics|atomic-theory|nuclear-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5784" PostTypeId="1" CreationDate="2017-03-14T04:39:57.443" Score="4" ViewCount="585" Body="&lt;p&gt;As a middle-school student, I encountered equations like &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$  x^2 - 92 y^2  = 1 .$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I remember that I was not interested or motivated at all. Why should I care about this kind of problem? For what kind of problems do I need to study it? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, why did the ancients study it? Purely out of curiosity? &lt;/p&gt;&#xA;" OwnerUserId="4079" LastActivityDate="2017-03-17T06:35:21.817" Title="What was the motivation of studying Diophantine equations?" Tags="|mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5785" PostTypeId="1" CreationDate="2017-03-14T05:02:34.830" Score="2" ViewCount="107" Body="&lt;p&gt;When formulating General Relatvity Einstein proposed that freely falling particles, that is those subject to no forces other than gravity, follows geodesics through spacetime. In other words, those particles follow the straightest path possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question here is: what was Einstein's reasoning to conclude this? How did he conclude that the worldline of a freely falling particle is a geodesic? Was it with some thought experiment?&lt;/p&gt;&#xA;" OwnerUserId="4072" LastActivityDate="2017-03-14T05:02:34.830" Title="What was Einstein's reasoning to conclude particles follows geodesics through spacetime?" Tags="|theoretical-physics|relativity-theory|einstein|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5786" PostTypeId="1" CreationDate="2017-03-14T08:50:17.183" Score="1" ViewCount="66" Body="&lt;p&gt;According to Bar-Hillel, Carnap's coauthor in a 1952 report on probability, Carnap had, as of 1956 an unpublished but circulated theory distinguishing &quot;random&quot; refers to methods of production of sequences, rather than outcome, while &quot;disorderly&quot; refers to the sequence, independent of how it was produced.&#xA;So 1111000011110000 is low in disorder, and does not look &quot;random&quot; as in unpredictable in the usual sense of random, because it is highly predictable. Highly internally autocorrelated. But all the same, it might be random, if it were produced by a machine whose output we cannot predict. So a sequence might be random and disorderly, random and not disorderly, disorderly but not random, or neither disorderly nor random, in this view. It might appear random and be random, might be random but appear unlikely, might appear random but be nonrandom, and might be nonrandom and appear obviously thus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where was this first published by Carnap (beside Bar-Hillel's 1956 comment, which is the first publication)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: The 1956 note is a comment by Bar-Hillel in the end of a discussion at an Information Theory conference proceedings, edited by Cherry for publication. Presumably the Carnap work was a paper and still unpublished in 1954 or 1955 - when the conference occurred.&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="163" LastEditDate="2017-03-14T20:00:56.957" LastActivityDate="2017-03-14T20:00:56.957" Title="Carnap's last theory Of probability" Tags="|reference-request|philosophy-of-science|mathematical-logic|probability|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5787" PostTypeId="1" CreationDate="2017-03-14T09:21:26.150" Score="2" ViewCount="68" Body="&lt;p&gt;The journals kept by the likes of Galileo, Kepler and Herschel provide details about what objects they discovered and what kinds of optical instruments they used to discover them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taken together, this should allow placing a lower bound on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Visual_acuity&quot; rel=&quot;nofollow noreferrer&quot;&gt;visual acuity&lt;/a&gt; of the discoverer, perhaps even the deterioration of the same over their lifetime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, we can take for granted that most astronomy pioneers were individuals with exceptionally sharp eyesight, but has any systematic study been conducted with results of the type &lt;em&gt;&quot;to see Jupiter's moons using his 20x telescope, Galilei must have had a VA of at least 20/16&quot;&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5502" LastActivityDate="2017-03-14T09:21:26.150" Title="What can be inferred about the eyesight of early astronomers from their recorded observations?" Tags="|astronomy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5788" PostTypeId="2" ParentId="5645" CreationDate="2017-03-14T09:33:29.473" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_de_Fermat&quot; rel=&quot;noreferrer&quot;&gt;Pierre de Fermat&lt;/a&gt; wrote a 101-line hexameter poem in Latin that was recently &lt;a href=&quot;http://link.springer.com/article/10.1007/s00283-012-9295-y&quot; rel=&quot;noreferrer&quot;&gt;translated into English&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-03-14T09:33:29.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5790" PostTypeId="1" AcceptedAnswerId="5799" CreationDate="2017-03-14T23:15:10.800" Score="2" ViewCount="427" Body="&lt;p&gt;In an old book of Prescott (died in 1859), The conquest of Mexico, the author described Aztec calendar with a year of 365 days. To compensate for the difference, every 52 years&#xA;an additional 12.5 days were added (so that every other 52-years period started at noon,&#xA;and the rest at midnight). This gives the average length of the year 365.24038 day&#xA;which is only 2 minutes short, and is more precise than any European calendar before the Gregorian one. He refers to the research by Gama, Descripcion historica y chronologica de las Dos Piedras, (Mexico 1832).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I was unable to find a confirmation of this information in modern books,&#xA;for example, in Thurston, Early astronomy, Springer, 1994. Modern sources &#xA; such as Wikipedia, Thurston and others say that Aztec calendar is the same as Maya calendar, and none of them addresses the intercalary days and length of the year. Thurston cites some modern works which try to estimate the average length of the year in Maya calendar and concludes that the claims on high accuracy are unfounded. But nobody mentions the ingenious intercalation described by Prescott and Gama. Does this mean that Gama's research was somehow disproved or is it simply forgotten? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Following the suggestion in the comments I looked in the book by M. Coe, The Maya. It indeed contains a sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We know that none of the Maya&#xA;  intercalated days on leap years or the like, as we do, and it has&#xA;  been shown that more sophisticated corrections thought to&#xA;  have been made by them are a figment of the imagination.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No explanation, no references. He even does not mention Gama, the founding father of Maya studies. The question remains how the Maya did agriculture with a calendar not tied to seasons.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2018-04-22T14:15:00.567" LastActivityDate="2022-08-14T19:43:56.307" Title="Length of the year in Aztec/Maya calendar" Tags="|archaeoastronomy|calendar|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5791" PostTypeId="2" ParentId="5784" CreationDate="2017-03-14T23:47:23.813" Score="5" Body="&lt;p&gt;For the &quot;ancients&quot;, &quot;numbers&quot; were integers or fractions. They did not have a concept of a real number. (They had an equivalent concept of length in geometry, but analytic geometry did not exist, and they did not think of geometric problems&#xA;as problems about equations. Geometric problems had to be solved by construction). Interest to the theory of whole numbers in Greece can be traced back to Pythagoreans who defined the main objects (primes, squares, etc.) and &#xA;started to study their properties. According to the tradition, Pythagoras claimed that &quot;numbers rule the world&quot; (he meant integers). This sweeping claim&#xA;was based on his discovery that musical intervals correspond to certain fractions. (In the long run he was right as we know: the basic laws of nature have mathematical formulation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The early stages of development of Diophantine equations are unknown to us.&#xA;They were usually formulated as word problems. See, for example&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes%27_cattle_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes' cattle problem&lt;/a&gt; on Wikipedia as one especially famous problem. It is about an equation of the same type you give, it is called Pell's equation nowadays. This is an isolated example, and we do not really know why exactly people were interested in such problems. Perhaps indeed out of curiosity. Most of the important mathematics was invented like this, by the way, out of curiosity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, 3 centuries after Archimedes, we have the book of Diophantus which treats &#xA;a lot of much more complicated problems like that. The modern development starts essentially with Fermat who read Diophantus...a millenium later. Nowadays this is one of the most&#xA;fashionable areas of mathematics. There are some applications, but mostly it is&#xA;driven by curiosity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ancient Babylonians also had taste to these problems, for example they knew how to find all solutions of $a^2+b^2=c^2$ in positive integers. Whether the Greeks&#xA;started the subject independently or it came from Babylonia, it is not known.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="3643" LastEditDate="2017-03-15T00:16:44.417" LastActivityDate="2017-03-15T00:16:44.417" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5792" PostTypeId="1" CreationDate="2017-03-15T00:38:47.657" Score="0" ViewCount="72" Body="&lt;p&gt;Can someone please suggest me some books which treat and teach Newtonian Mechanics ( Statics and Dynamics ) without the use of vectors? I was wondering how can we carry out the same treatments with a scalar based method like it was done before the adoption of vector methods.&lt;/p&gt;&#xA;" OwnerUserId="3564" LastActivityDate="2017-03-15T00:38:47.657" Title="Scalar mechanics" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5793" PostTypeId="1" CreationDate="2017-03-15T20:55:47.770" Score="1" ViewCount="240" Body="&lt;p&gt;In addition to the general topical interest indicated in the title, I am also curious about British mathematics of the period viewed through the lens of competition with the rest of Europe (and perhaps even the rest of the world). For example, I have often heard it argued (informally) that the whole Newton-Leibniz-calculus kerfuffle can only be properly viewed in terms of a more general conflict between British and European mathematicians and scientists in that time period. Are there any books which address the existence and historical importance of such a conflict?&lt;/p&gt;&#xA;" OwnerUserId="5505" LastActivityDate="2017-04-17T19:26:18.607" Title="What are some good (and reasonably academic) books on 18th and 19th century British mathematics?" Tags="|mathematics|reference-request|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5794" PostTypeId="1" AcceptedAnswerId="5802" CreationDate="2017-03-16T17:41:52.893" Score="6" ViewCount="601" Body="&lt;p&gt;I would like to find out more about the history of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_XY_model&quot; rel=&quot;noreferrer&quot;&gt;XY model&lt;/a&gt;. While for the Ising and Potts model it is easy to find information, it's quite difficult for me to understand when and where the XY model (or the plane rotator model) was first presented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Kosterlitz%E2%80%93Thouless_transition&quot; rel=&quot;noreferrer&quot;&gt;KT transition&lt;/a&gt; is quite famous (1972), but even in the original &lt;a href=&quot;https://www.physics.uci.edu/~taborek/publications/other/jcv6i7p1181.pdf&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt; by Kosterlitz and Thouless there's nothing about history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was the XY model formulated for the first time? Who was the first scientist to introduce it?&lt;/p&gt;&#xA;" OwnerUserId="5509" OwnerDisplayName="fdesmond" LastActivityDate="2017-04-09T09:02:02.607" Title="History of XY model - Plane Rotator model" Tags="|statistical-mechanics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5795" PostTypeId="2" ParentId="5794" CreationDate="2017-03-16T19:26:24.950" Score="3" Body="&lt;p&gt;As far as I know, this model (well, the quantum version of it) was introduced in the following paper:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Two soluble models of an antiferromagnetic chain&lt;/em&gt;,&#xA;Elliott Lieb, Theodore Schultz and Daniel Mattis,&#xA;Annals of Physics, Volume 16, Issue 3, December 1961, Pages 407--466.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The paper can be found &lt;a href=&quot;https://www.math.ucdavis.edu/~bxn/lieb1961.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. The model is introduced in Section II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let us also address your comment below, concerning the other $O(N)$-models (the XY model being the $O(2)$-model). The $O(3)$-model, often called the Heisenberg model, was introduced much earlier, in the following paper:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Zur Theorie des Ferromagnetismus&lt;/em&gt;, W. Heisenberg, Zeitschrift für Physik, September 1928, Volume 49, Issue 9, pp 619--636.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Finally, the general $O(N)$-models seem to have been introduced in&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Dependence of Critical Properties on Dimensionality of Spins&lt;/em&gt;, H. E, Stanley, Phys. Rev. Lett. 20:589 (1968).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;However, I wouldn't be surprised if it was independently introduced by several people in different contexts...&lt;/p&gt;&#xA;" OwnerUserId="3401" LastEditorUserId="3401" LastEditDate="2017-03-17T12:02:10.827" LastActivityDate="2017-03-17T12:02:10.827" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5796" PostTypeId="2" ParentId="5793" CreationDate="2017-03-16T19:27:19.170" Score="2" Body="&lt;p&gt;The books on history of mathematics are not normally divided by national boundaries. I do not know of any book addressing specially the history of British mathematics, or German mathematics, or French mathematics. (Exception is Russian mathematics:-) Mathematics is not really divided on national lines, this is an international enterprise, at least in Europe, at least since 18th century. So 19 &#xA;and 20 century European mathematics is a single thing, with no national divisions. Mathematical journals, from the very beginning were international,&#xA;and published papers from all countries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good book on 19 century mathematics is Klein, History of mathematics in 19th century (there is an English translation). It addresses German, British, French and Italian mathematics, and of course Norwegian too.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-03-16T19:27:19.170" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5797" PostTypeId="2" ParentId="5767" CreationDate="2017-03-17T01:19:40.577" Score="1" Body="&lt;p&gt;The fact that air has weight was first well comprehended by Torricelli. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In ancient times people used to explain water suction through a pipe by assuming Nature has horror to vacuum. There is however two objections to this idea. &lt;a href=&quot;https://en.wikipedia.org/wiki/Gasparo_Berti&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gasparo Berti&lt;/a&gt; in 1640's created a stable vacuum by submerging the open end of a tube previously filled with water (a barometer), showing that Nature does not have horror to vacuum. Second, there was a practical problem that drew the attention of many people at that time: How to suck water through a pipe to a height greater than $10\, m$? Galileo himself proposed a solution by simply saying that the water column breaks down under its own weight when it reaches $10\, m$ tall. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Evangelista Torricelli was the first to explain both phenomena above. He assumed that air has weight and we live at the &quot;bottom of an ocean&quot;. The pressure due to air weight matches the pressure due to the liquid in the pipe. Therefore the maximum pressure the atmosphere can generate equals the pressure due to a $10\, m$ tall column of water. Torricelli then predicted that tallest column of Mercury would be $0.76\, m$, since Mercury is $13.6$ times denser than water, and that was later confirmed by his colleague Vicenzo Viviani.&lt;/p&gt;&#xA;" OwnerUserId="5436" LastActivityDate="2017-03-17T01:19:40.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5798" PostTypeId="2" ParentId="5784" CreationDate="2017-03-17T06:35:21.817" Score="2" Body="&lt;p&gt;One ancient mathematical topic that gives rise to Pell-type equations is polygonal numbers. Look at the list of triangular numbers (1, 3, 6, 19, 15, 21, 28, 36,...) and square numbers (1, 4, 9, 16, 25, 36, 49,...), and aside from the trivial common value 1 you will find another common value: 36. It is  natural to ask if there are other &quot;triangular-square numbers.&quot; There are in fact infinitely many, but the next example is greater than 1000. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asking more generally if there are integers that are both $m$-gonal and $n$-gonal for different $m, n \geq 3$ leads to a Pell-type Diophantine equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not suggesting polygonal numbers are an important topic anymore, but the level of math the ancient mathematicians were thinking about could not be expected to be too high.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2017-03-17T06:35:21.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5799" PostTypeId="2" ParentId="5790" CreationDate="2017-03-18T12:43:11.777" Score="2" Body="&lt;p&gt;The standard literature on the Mayan calendar is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. Eric S. Thompson, Maya Hieroglyphic Writing, Washington : Carnegie Institution, 1950.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lis Brack-Bernsen, 'Die Basler Mayatafeln', Verhandlungen der Naturforschenden Gesellschaft in Basel, Vol.86 (1977), 1-76.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Floyd G. Lounsbury, 'Maya numeration, computation, and calendrical astronomy', Dictionary of Scientific Biography, Volume XV, Supplement I, 1978.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All of these agree that the year had 18 “months” of 20 days each and one “month” of 5 days, making a year of 365 days. There is no evidence for intercalation.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-03-18T12:43:11.777" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5800" PostTypeId="2" ParentId="5793" CreationDate="2017-03-18T18:52:24.550" Score="2" Body="&lt;p&gt;Here are two references, from a broader perspective:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Mathematical Visions: The Pursuit of Geometry in Victorian England&quot; by Richards.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Mathematics in Victorian Britain&quot;, ed. by Raymond Flood, Adrian Rice, Robin Wilson.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3232" LastActivityDate="2017-03-18T18:52:24.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5801" PostTypeId="1" AcceptedAnswerId="5803" CreationDate="2017-03-18T19:08:00.927" Score="3" ViewCount="1192" Body="&lt;p&gt;I was reading lately that the &lt;a href=&quot;http://www.langorigami.com/article/angle-quintisection&quot; rel=&quot;nofollow noreferrer&quot;&gt;quintisection of an angle is possible with paper folding (origami)&lt;/a&gt;. Now, in contrast to the trisection of an angle, a problem which was discussed  historically, and was actually successfully solved already by the Greeks (although not with a compass and a straghtedge, but with other means), the history of quintisection is less discussed. Arthur Baragar shows &lt;a href=&quot;https://faculty.unlv.edu/baragar/papers/monthly151-164.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that if one can construct a complex number $a$ with a compass and twice-notched straightedge then it belongs to a field $K$ that lies in a tower of fields having degrees of extensions 2,3 or 5. I assume hence that one can quintisect an angle with these apparatuses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, what was the history of the quintisection problem? &lt;/p&gt;&#xA;" OwnerUserId="3232" LastEditorUserId="55" LastEditDate="2017-03-18T22:46:11.623" LastActivityDate="2024-02-16T04:28:10.517" Title="What is the history of angle quintisection (division into five equal parts)?" Tags="|mathematics|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5802" PostTypeId="2" ParentId="5794" CreationDate="2017-03-18T20:40:01.860" Score="6" Body="&lt;p&gt;Finally found!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This evening I was reading an article by Silver et al:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Silver, H., N. E. Frankel, and B. W. Ninham. “A Class of Mean Field&#xA;Models.” Journal of Mathematical Physics 13, no. 4 (April 1, 1972):&#xA;468–74. doi:10.1063/1.1666002,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;where you can find an explicit formula for the free energy of a class of mean field models (including XY model). The only references concerning the hamiltonian $H_N^{(\nu)}=-j \sum_{i,j} \boldsymbol{S}_i \cdot \boldsymbol{S}_j$, with $ \boldsymbol{S}_i=\left (S_i(1),\dots,S_i(\nu) \right)$ $\nu$-dimensional vectors, are a few papers by Stanley:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Stanley, H. E. “Spherical Model as the Limit of Infinite Spin&#xA;Dimensionality.” Physical Review 176, no. 2 (December 10, 1968):&#xA;718–22. doi:10.1103/PhysRev.176.718,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Stanley, H. E. “Exact Solution for a Linear Chain of&#xA;Isotropically Interacting Classical Spins of Arbitrary&#xA;Dimensionality.” Physical Review 179, no. 2 (March 10, 1969): 570–77.&#xA;doi:10.1103/PhysRev.179.570.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The article of 1968 is more about the Berlin-Kac model. But thanks to it, I found an interesting historical paper written by Kac himself, where he tells the story of his model. I guess it's the first one with continuous spins (isn't it?)! The &lt;em&gt;discovery&lt;/em&gt; is dated 1947 and the article I'm referring to is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kac, M. “The Work of T.H. Berlin in Statistical Mechanics …a Personal&#xA;Reminiscence.” Physics Today 17, no. 10 (October 1964).&#xA;doi:10.1063/1.3051173.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The article of 1969 written by Stanley talks about the XY model and it refers to it as &lt;em&gt;invoked in connection with certain magnetic insulators&lt;/em&gt;. It gives two references, in particular one by Betts and Lee:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Betts, D. D., and M. H. Lee. “Critical Properties of the XY Model.”&#xA;Physical Review Letters 20, no. 26 (June 24, 1968): 1507–10.&#xA;doi:10.1103/PhysRevLett.20.1507.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here there was no explicit date for the XY model, but a hint: &lt;em&gt;Matsubara and Matsuda introduced in 1956 a lattice model for liquid helium which reduces to the XY model for a hard-core molecular-inter-22 action potential&lt;/em&gt;. The article is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Matsubara, Takeo, and Hirotsugu Matsuda. “A Lattice Model of Liquid Helium, I.” Progress of Theoretical Physics 16, no. 6 (December 1, 1956): 569–82. doi:10.1143/PTP.16.569.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But I wasn't done. In Stanley1969 I found another trail to follow: &lt;em&gt;For a comprehensive introduction to exactly soluble models of interacting particles in one-dimension, see E. H. Lieb and D. C. Mattis, Mathematical Physics in One Dimension (Academic Press Inc., New York, 1966)&lt;/em&gt;. The same Lieb and Mattis @Yvan Velenik reported in his answer. So I looked for the book and I finally found the answer (cap. 6, around pag 463). Actually the model was firstly considered by Yoichiro Nambu in a paper dated 1950:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nambu, Yôichirô. “A Note on the Eigenvalue Problem in Crystal Statistics.” Progress of Theoretical Physics 5, no. 1 (February 1, 1950): 1–13. doi:10.1143/ptp/5.1.1,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;but it was denoted &quot;XY model&quot; for the first time in the article reported by Yvan Velenik:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Two soluble models of an antiferromagnetic chain, Elliott Lieb, Theodore Schultz and Daniel Mattis, Annals of Physics, Volume 16, Issue 3, December 1961, Pages 407--466.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It seems that the XY model was firstly introduced for two different reasons (magnetic insulators and quantum mechanics) not before 50's and had its name in 1961. I hope the (hi)story is concluded!&lt;/p&gt;&#xA;" OwnerUserId="5509" LastEditorUserId="5509" LastEditDate="2017-04-09T09:02:02.607" LastActivityDate="2017-04-09T09:02:02.607" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5803" PostTypeId="2" ParentId="5801" CreationDate="2017-03-18T22:41:47.353" Score="8" Body="&lt;p&gt;Not much history to it, I am afraid. It seems that methods of trisection rather obviously (to those who considered them) applied to quintisection as well, so the problem was of little theoretical interest in itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without restriction on tools the problem of angle multisection is easily solvable. For instance, the quadratrix of Dinostratus, originally the &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratrix_of_Hippias&quot; rel=&quot;noreferrer&quot;&gt;trisectrix of Hippias&lt;/a&gt; (c. 420 BC), can be used to multisect any angle into any number of parts (and square the circle to boot, as discovered by Dinostratus), although it was originally meant to trisect. The &lt;a href=&quot;https://en.wikipedia.org/wiki/On_Spirals#Trisecting_an_angle&quot; rel=&quot;noreferrer&quot;&gt;Archimedean spiral&lt;/a&gt; (invented by his friend Conon) can also be used for the same purposes, which would have been obvious to Archimedes, if not Hippias, although he also only discusses trisection. Euclid in Elements of course describes a construction of regular pentagon with straightedge and compass, which is a case of quintisection for a special angle. In a discussion on the &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=56677&quot; rel=&quot;noreferrer&quot;&gt;NCTM Math Forum&lt;/a&gt; another ancient case of quintisection is mentioned:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It would appear that at Cydonia such quintisection is DONE for a very&#xA;  specific reason (which by now I cannot disclose or Dr. Lahoz would&#xA;  harm me): but the environment SUGGESTS a very SIMPLE construction (???)&#xA;  which so far escapes our analysis: I guess that it may be related to&#xA;  the Archimedean trisection.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kydonia&quot; rel=&quot;noreferrer&quot;&gt;Cydonia or Kydonia&lt;/a&gt; was an ancient city-state on the northwest coast of the island of Crete, that lasted from archaic to Byzantine periods, and is a site of modern archeological excavations. I was unable to confirm or find specifics on this mention. &lt;a href=&quot;https://groups.google.com/forum/#!topic/sci.math/KlkV3i12UqA&quot; rel=&quot;noreferrer&quot;&gt;Angel Garcia suggested&lt;/a&gt; how the quintisection could be performed in the style of Archimedes with a marked straightedge (and avoiding towers of fields, etc.), see also &lt;a href=&quot;http://athang1504.blogspot.com/2011/05/trisection-and-pentasection-of-line.html&quot; rel=&quot;noreferrer&quot;&gt;Trisection and Pentasection of Line Segments and Angles&lt;/a&gt;, but we have no ancient sources describing such a construction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In modern times quintisection was considered at least as early as Viète (1540 – 1603) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Jost_B%C3%BCrgi&quot; rel=&quot;noreferrer&quot;&gt;Bürgi (1552 - 1632)&lt;/a&gt;,  Briggs used it to construct his trigonometric tables. &lt;a href=&quot;http://quod.lib.umich.edu/e/eebo/A67419.0001.001/1:2.4?rgn=div2;view=fulltext&quot; rel=&quot;noreferrer&quot;&gt;Wallis's Treatise of Angular Sections (1648, published 1685)&lt;/a&gt; also discusses the &quot;quinquisection&quot; explicitly. Here is from &lt;a href=&quot;http://locomat.loria.fr/briggs1633/briggs1633doc.pdf&quot; rel=&quot;noreferrer&quot;&gt;Roegel's Reconstruction of the Tables of Briggs and Gellibrand’s&#xA;Trigonometria Britannica (1633)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;After having considered the triplication and quintuplication, Briggs considered again the equations, but now in order to divide an arc into $3$ (chapter 4 [11, pp. 5–10]), $5$ (chapter 6 [11, pp. 12–18]), and $7$ (chapter 7 [11, pp. 19–20]) parts. For instance, if p is the chord of an angle a, we have seen that the chord of 3a is $c(3a) = 3p-p^3$, and trisecting an angle amounts to solve a cubic equation [13, p. 461]. For a division by $5$, the equation is &#xA;  $c(5a) = 5p -5p^3 + p^5$. Then we have $c(7a) = 7p - 14p^3 + 7p^5 - p^7$. And so on. Even sections lead to the equations $c(2a) = \sqrt{4p^2-p^4}$, $c(4a) = \sqrt{16p^2-20p^4 + 8p^6-p^8}$, and so on. The general case is considered in chapter 8 [11, pp. 20–28] and the coefficients of all these equations can be obtained from a table given by Briggs [11, p. 23]. This table can easily be extended.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Briggs’ work is certainly partly inspired from &lt;a href=&quot;http://www.17centurymaths.com/contents/Angular%20Sections.pdf&quot; rel=&quot;noreferrer&quot;&gt;François Viète’s Ad Angulares Sectiones&lt;/a&gt; which has such a table [83, p. 295]. Viète is in particular explicitly quoted on the cover of the Trigonometria Britannica. It is interesting to observe that Jost Bürgi also obtained another similar table for the same purpose, certainly independently, and described it in his “Coss,” probably around 1598 [50, pp. 33–35] [58, p. 77]... Like Bürgi before him [50], Briggs develops a method to find some roots of these equations by iteration... This happens to be exactly the so-called Newton-Raphson method, with the constraint that only one new digit is obtained at a time. The Newton-Raphson method actually goes back at least to Viète... The sixth chapter of the Trigonometria Britannica is devoted to the quintisection of arcs and expounds the same method. Briggs considers the equation $x^5-5x^3 + 5x = a$, a first approximation $b$ of a root, and he obtains a new approximation $L = b + c$ with $c=\frac{a-b^5+5b^3-5b}{&#xA;5b^4+15b^2+5}$.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even in the recent recreational literature the quintisection problem received little attention, especially compared to the endless recycling of the trisection. &lt;a href=&quot;https://books.google.com/books?id=0HYbAAAAMAAJ&amp;amp;pg=PA192-IA5&amp;amp;lpg=PA192-IA5&amp;amp;dq=angle+quinquisection&amp;amp;source=bl&amp;amp;ots=v7NIGeAc3Y&amp;amp;sig=rkVQc4xv4XIOfHMuRE_1VIy7mOk&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjq3vmkp-HSAhUGZCYKHYb6ALgQ6AEIMTAF#v=onepage&amp;amp;q=angle%20quinquisection&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Application of Miller's Trisector to Quinquisection of any Angle&lt;/a&gt; appeared in Proceedings of the Royal Society of Edinburgh in 1902. Duncan and Barnier published one page note &lt;a href=&quot;https://www.jstor.org/stable/2975657&quot; rel=&quot;noreferrer&quot;&gt;On Trisection, Quintisection,...,Etc.&lt;/a&gt; in the Monthly in 1982, their only reference is an algebra textbook. &lt;a href=&quot;https://www.tib.eu/en/search/id/BLSE%3ARN021224704/Trisection-and-pentasection-of-any-angle-by-dividers&quot; rel=&quot;noreferrer&quot;&gt;Balo-Roy's Trisection and Pentasection of any Angle by Dividers&lt;/a&gt; is not easily accessible, I am afraid. There is a bit more on the origami constructions. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-03-19T01:05:58.133" LastActivityDate="2017-03-19T01:05:58.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5805" PostTypeId="1" AcceptedAnswerId="5806" CreationDate="2017-03-20T01:16:48.423" Score="5" ViewCount="549" Body="&lt;p&gt;Apropos of the murder of Gaius Julius Caesar, the late V. I. Arnold told the following amusing story in pages 89-90 of his &quot;Yesterday and Long Ago&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;CAESAR AND GAULS: THE DEFENSE OF ROME FROM THE GERMANS&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Long ago, Caesar gave a vivid description of the Gaul's national character: affectation and striving for theatricality, big promises, and ineffectuality. He stated that it was not possible to rely on the Gaul's promises to defend Rome from the Germans: if a German regiment crossed the Rhine, the Gauls would hide in bushes and would never fight.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;According to Caesar's words, their troops were not able to fight, and they were also not able to cross the Alps: for this they would need good food and fine wines, but even with these they would be frozen before reaching a pass. The Gauls were always ready to sign any treaty profitable to them, but they would never honour it. Thus, there was only one way to defend Rome: to conquer Gaul.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Caesar developed a plan which could ensure a peaceful future for Europe. For this it was necessary to create a coalition with tribes that occupied a territory which is now in Russia, and then to attack the Germans from the East. Caesar had to inform the leaders of his army, who had fought with him in the Gallic war, about his plans for a new campaign; but the veterans preferred to grow cabbage, grapes, and apples on lands they had received from the Empire. Therefore, to avoid a new war and to save their gardens they had to kill Caesar when he came to the Roman Forum to obtain money for this campaign. Thus, Caesar's plan to ensure peace in Europe failed...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does anybody here know if the previous account was the &lt;em&gt;official&lt;/em&gt; version of Julius Caesar's murder in the schools of the Soviet Union once? Outrageous as you may find it, I have heard tell that, at some point of time, the famous Soviet mathematician Anatoly Fomenko put forward some historical claims which were even wilder than those by Arnold which one can find in the aforementioned book (yes, the one on Caesar's stabbing in the Ides of March of 44 BC is not the only outrageousness therein).&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2019-12-11T16:48:15.837" LastActivityDate="2019-12-11T16:48:15.837" Title="Beware the Ides of March!" Tags="|debunking|roman-empire|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5806" PostTypeId="2" ParentId="5805" CreationDate="2017-03-20T01:59:37.900" Score="9" Body="&lt;p&gt;No, I assure you, this nonsense was not taught at schools, at least in Soviet Union (I am not sure what is possible in Russia now). This is just a typical Russian &quot;pop historian&quot; interpretation, a genre pseudo-history quite popular with some Russian circles. In general V.I.Arnold's writing on history (including history of mathematics) cannot be taken seriously, though you are right, they are far from the outrageous delirium of Fomenko.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Perhaps &quot;popular in some circles&quot; is an understatement. Since the break up&#xA;of the Soviet Union a huge wave of this sort of pop history flooded Russian bookshops and later Internet. There is nothing like this in European countries or US. Books by Fomenko and his followers occupy whole shelves in the book stores&#xA;and apparently sell well. It is actually a strange sociological phenomenon, which needs a scientific explanation.   &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-03-20T02:06:48.627" LastActivityDate="2017-03-20T02:06:48.627" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5807" PostTypeId="1" AcceptedAnswerId="5808" CreationDate="2017-03-20T17:29:47.757" Score="2" ViewCount="411" Body="&lt;p&gt;What is the correct spelling of the name of the brother to the Swiss mathematician Johann Bernoulli? I've seen it spelt both ways, which one is the correct one?&lt;/p&gt;&#xA;" OwnerUserId="4948" LastActivityDate="2017-03-21T17:51:57.880" Title="Jakob or Jacob Bernoulli?" Tags="|mathematics|mathematicians|" AnswerCount="2" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5808" PostTypeId="2" ParentId="5807" CreationDate="2017-03-20T19:32:08.297" Score="4" Body="&lt;p&gt;It is simply the difference between German and Latin spelling&#xA;(German Wikipedia calls him Jakob).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The correct spelling (by definition) is how the person wrote his name himself. And here is how:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.kubkou.se/pdf/mh/jacobB.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.kubkou.se/pdf/mh/jacobB.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Jacobi in Latin means &quot;by Jacob&quot;). Of course it is possible that it was spelled Jacob in Latin and Jakob in German. But all mathematics was written in Latin at that time. (He could probably write Jakob in his financial or legal documents).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-03-20T19:32:08.297" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5809" PostTypeId="1" CreationDate="2017-03-21T09:06:48.500" Score="-1" ViewCount="134" Body="&lt;p&gt;It is actually quite strange that the unit of frequency was defined so later. How did people like Newton, Lagrange describe the frequency of a harmonic oscillator?  &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2017-03-21T14:11:25.983" Title="How did people describe the frequency of a wave before Hertz?" Tags="|physics|" AnswerCount="1" CommentCount="2" ClosedDate="2017-03-27T02:49:45.210" ContentLicense="CC BY-SA 3.0" />
  <row Id="5810" PostTypeId="1" AcceptedAnswerId="5814" CreationDate="2017-03-21T11:19:08.213" Score="5" ViewCount="973" Body="&lt;p&gt;The short story is that based on Kepler's and Newton's physics, in 1846 Le Verrier mathematically predicted the existence and current location of Neptune within a single angular degree, and it was observed at first attempt. It not only demonstrated the power of science, it taught us something about heaven and the planets, both historically the domain of religion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was the successful prediction of Neptune generally perceived as such a triumph for science that it caused a paradigm shift as to the role of science in society? Was astronomy until then maybe widely viewed as a leftover from astrology, and since the 17th century reduced to only meaninglessly predicting the movements of the planets without it having any significance for our destiny.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did it substantially help the science side against the Church during the debate about evolution that began 14 years later? Could Le Verrier's prediction even have stimulated the evolution debate to have come about?&lt;/p&gt;&#xA;" OwnerUserId="3663" LastActivityDate="2017-03-22T13:17:29.883" Title="How important was the prediction of Neptune in 1846 for the Oxford evolution debate?" Tags="|discoveries|astronomy|scientific-method|evolution|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5811" PostTypeId="2" ParentId="5810" CreationDate="2017-03-21T13:16:52.470" Score="2" Body="&lt;p&gt;Discovery of Neptune was indeed a significant step which increased the prestige of science but it was only one of a long chain of such events starting in 17th century. Let me give a short sample: discovery of Jupiter satellites with the newly discovered telescope (unlike the discovery of Neptune, this had a great practical application!), Newton's gravitation theory and celestial mechanics,&#xA;it predicted the correct shape of the Earth, verified by the French expeditions,&#xA;it gave the first working theory of tides, and it permitted Halley to predict the return if the Halley comet. Mathematical theory of Moon's motion solved the centuries old practical problem of finding longitude at sea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery of spectral lines made it possible to determine chemical composition of stars and the Sun.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This short list only mentions some discoveries in physics and astronomy. But remarkable discoveries were made in other areas as well, let me only mention smallpox vaccination which saved millions of lives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So by 1846, science had already sufficiently high prestige among educated public. All discoveries that I listed had more influence on people's lives than the discovery of Neptune. By the way, discovery of new planets (large asteroids,&#xA;Ceres, Pallas, Juno and Vesta)&#xA;before Neptune, also had enormous public resonance, now forgotten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. To address the comments, let me explain more about Jupiter satellites for example. The main thing about this discovery was not that they just exist, but that they obey Kepler laws. Kepler laws had to be discovered in advance to determine this. Now the fact that they obey Kepler's laws permitted to predict their motion. And this gave the first practical method of determining longitude, one of the most pressing scientific problems of 17th century. Jupiter satellites made possible determination of longitude only on land, because you need a telescope to see them, and a telescope cannot be installed on a ship.&#xA;Solution of longitude problem at sea had to wait for another century, and was&#xA;obtained in two ways: a) mathematicians managed to predict Moon's motion, using Newton's laws and advances in mathematics, and b) chronometer was invented.&#xA;(A lot of mathematics was used in both cases, especially for the Moon motion).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now about asteroids. You look at the sky through a telescope, and you see a new star. But you do not know what is this (a Nova, a comet, etc.) When you look again and again, you notice that it moves. So a Nova is excluded. But then you have to compute its orbit, to be sure that when you look in a year from now you see the same object. If the orbit turns to be elliptic with small excentricity, this is a new planet. In the case of Ceres, computation was performed by Gauss,&#xA;the greatest mathematician of that time. He had to develop new methods which enriched mathematics. This discovery has an enormous public resonance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in no way these milestones in the development of science can be called &quot;stamp collecting&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-03-22T13:17:29.883" LastActivityDate="2017-03-22T13:17:29.883" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5812" PostTypeId="2" ParentId="5809" CreationDate="2017-03-21T13:46:50.190" Score="2" Body="&lt;p&gt;In 1687 in Principia, Newton mentioned a pendulum &quot;vibrating four times in a second&quot;.(&lt;a href=&quot;https://books.google.nl/books?id=lSoJ2tJKfIEC&amp;amp;pg=PA363&amp;amp;lpg=PA363&amp;amp;dq=newton%20vibrating-four-times-in-a-second&amp;amp;source=bl&amp;amp;ots=PxtsezQRLh&amp;amp;sig=XAdjYG9-yjS874hHtFYrX6wWwHk&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwj0taC92efSAhXFPBQKHfuhDpAQ6AEIHTAB#v=onepage&amp;amp;q=newton%20vibrating-four-times-in-a-second&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;) Fractional values were allowed: in 1686, Molyneux mentioned that a &quot;pendulum 39.0 inches long vibrates 60 times in a minute, and 15 hundreds of a vibration more than 60 times&quot;.(&lt;a href=&quot;https://books.google.nl/books?id=YYsPAAAAQAAJ&amp;amp;pg=PP65&amp;amp;lpg=PP65&amp;amp;dq=%2215%20hundreds%20of%20a%20vibration%20more%20than%2060%22&amp;amp;source=bl&amp;amp;ots=1lDnQrXdnX&amp;amp;sig=47GtICZf00i8y6GjRIJXILLt_iE&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjUjfjs3efSAhXB8RQKHausDo8Q6AEIGjAA#v=onepage&amp;amp;q=%2215%20hundreds%20of%20a%20vibration%20more%20than%2060%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2017-03-21T14:11:25.983" LastActivityDate="2017-03-21T14:11:25.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5813" PostTypeId="2" ParentId="5807" CreationDate="2017-03-21T17:33:18.310" Score="5" Body="&lt;p&gt;There is an extensive Latin &lt;a href=&quot;http://www.springer.com/de/book/9783764329501&quot; rel=&quot;noreferrer&quot;&gt;correspondence&lt;/a&gt; between Leibniz (addressed, by the way, as &quot;Leibnitz&quot;) and Jacob Bernoulli, in which Bernoulli writes his name &quot;Jacobus&quot; or declensions thereof. A letter to Clüver written in French is signed with &quot;Jacques Bernoulli&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An &lt;a href=&quot;http://www.deutschestextarchiv.de/book/view/bernoulli_comet_1681?p=1&quot; rel=&quot;noreferrer&quot;&gt;astronomical treatise&lt;/a&gt; published in German in 1681 bears the name Jacob.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Johann signed his Latin letters as &lt;a href=&quot;http://www.ub.unibas.ch/bernoulli/index.php/1710-03-00_Bernoulli_Johann_I-Mencke_Johann_Burckhard&quot; rel=&quot;noreferrer&quot;&gt;Johannes&lt;/a&gt; and his French letters as &lt;a href=&quot;http://www.ub.unibas.ch/bernoulli/index.php/1743-01-02_Bernoulli_Johann_I-Bousquet_Marc-Michel&quot; rel=&quot;noreferrer&quot;&gt;Jean&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-03-21T17:51:57.880" LastActivityDate="2017-03-21T17:51:57.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5814" PostTypeId="2" ParentId="5810" CreationDate="2017-03-21T21:42:20.597" Score="6" Body="&lt;p&gt;First, the prediction of Neptune was a big win for science in the eyes of the general public. It was not exactly spotless though, especially in the eyes of the scientists. Here is from &lt;a href=&quot;https://hubpages.com/education/The-Drama-of-Neptunes-Discovery&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kelley's How was Neptune Discovered?&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The world was excited by the find, for never before had mathematics predicted a natural object. This confidence in the results was lessened, however, when discrepancies were noticed in the calculated values and the actual ones (Lyttleton 227). For example, Adams calculated an orbital period of 227 years and Le Verrier found it to be 218 years using Kepler’s Third Law (Period squared is proportional to average distance cubed). The actual value of the orbit is 165 years. This discrepancy was not a result of using Kepler’s Third Law but because of using Bode’s Law for the average distance (229). The only actual value they were close to, if one looks at the table, is the location in the sky it would be found. It is possible that both men were simply lucky with this. We shall never truly know (233).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Second, debate over evolution did not begin 14 years later, it was going on for a while. Lamarck proposed his version of evolution in 1801, and even the dominant in the early 19-th century &lt;a href=&quot;https://en.wikipedia.org/wiki/Georges_Cuvier#Catastrophism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cuvier's catastrophism&lt;/a&gt; posited a kind of evolution, although in discrete leaps. Lyell's gradualistic geology defended in &lt;a href=&quot;https://en.wikipedia.org/wiki/Principles_of_Geology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principles of Geology (1830-33)&lt;/a&gt; that displaced it was far more influential in the acceptance of Darwin's ideas than astronomical events (and it was a direct inspiration for Darwin himself). In 1860 Leverrier himself moved on to a new prediction, Vulcan, blamed for the anomalous precession of the perihelion of Mercury. A curious &lt;a href=&quot;http://www.pbs.org/wgbh/nova/next/physics/hunt-for-vulcan&quot; rel=&quot;nofollow noreferrer&quot;&gt;report about the annual meeting of the British Association for the Advancement of Science&lt;/a&gt; in Manchester, 1861 (soon after the Oxford debate) indicates that Neptune was not much of a factor:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1861, the city showcased both its wealth and brains as it hosted Britain’s largest celebration  of knowledge, the annual meeting of the British Association for the Advancement of Science. Charles Darwin had published The Origin of Species less than two years before, and that explosion continued to reverberate through every gathering of the learned. At the Manchester meeting Darwin’s defenders prepared to battle religious doubters. One speaker, the “blind economist” Henry Fawcett, made the ultimate claim: Darwin was a true scientific hero, one who solved his problem by the same methods, the same approach to experiment, observation, and generalization that the great Isaac Newton himself had used in his physics. Much else was discussed, of course — advances in dredging engineering, a report on birds of New Zealand, news from the balloon committee. The astronomy section was relatively quiet...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, in contrast to 17th and 18th centuries the role of the church in scientific debates was much diminished, and its reaction to Darwin (and more specifically to human origins, not evolution in general), was ambivalent. The nature and significance of the &lt;a href=&quot;https://en.wikipedia.org/wiki/1860_Oxford_evolution_debate&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford evolution debate of 1860&lt;/a&gt; might also have been glossed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Though the debate is frequently depicted as a clash between religion and science, the British Association at the time had a number of clergymen occupying high positions (including Presidents of two of its seven sections)... a case could be made for saying that for the many clerics in the audience, the underlying conflict was between traditional Anglicanism (Wilberforce) and liberal Anglicanism (Essays and Reviews)...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Many of the opponents of Darwin's theory were respected men of science: Owen was one of the most influential British biologists of his generation; Adam Sedgwick was a leading geologist; Wilberforce was a Fellow of the Royal Society (though at that time about half of the Fellows were well-placed amateurs)... The debate has been called &quot;one of the great stories of the history of science&quot; and it is often regarded as a key moment in the acceptance of evolution. However, at the time it received only a few passing references in newspapers, and Brooke argues that &quot;the event almost completely disappeared from public awareness until it was resurrected in the 1890s as an appropriate tribute to a recently deceased hero of scientific education&quot;&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-03-22T01:26:37.660" LastActivityDate="2017-03-22T01:26:37.660" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5815" PostTypeId="1" CreationDate="2017-03-22T00:36:50.437" Score="0" ViewCount="141" Body="&lt;p&gt;Could anyone please suggest me a good book which explains the history of mechanics clearly in a way that the modern reader can comprehend without knowing a breadth of historical facts? Also a book of similar nature with regards to the development of the energy concept in mechanics would be a great help. Thanks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;( I have tried out books by Mach and Dugas but they seem to be covered in obscure references or author's opinions )&lt;/p&gt;&#xA;" OwnerUserId="3564" LastActivityDate="2017-03-22T00:36:50.437" Title="History of mechanics" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5816" PostTypeId="2" ParentId="5645" CreationDate="2017-03-22T16:18:38.357" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Errett_Bishop&quot; rel=&quot;nofollow noreferrer&quot;&gt;Errett Bishop&lt;/a&gt; penned the following intriguing poem, cited &lt;a href=&quot;http://link.springer.com/article/10.1007%2Fs10699-013-9340-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Formalism.&#xA;The devil is very neat. It is his pride//&#xA;To keep his house in order. Every bit//&#xA;Of trivia has its place. He takes great pains//&#xA;To see that nothing ever does not fit.//&#xA;And yet his guests are queasy. All their food,//&#xA;Served with a flair and pleasant to the eye,//&#xA;Goes through like sawdust. Pity the perfect host!//&#xA;The devil thinks and thinks and he cannot cry.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-03-22T16:18:38.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5817" PostTypeId="1" AcceptedAnswerId="5818" CreationDate="2017-03-22T18:12:37.267" Score="12" ViewCount="1801" Body="&lt;p&gt;Title self explains: Is the symbol for set membership $\in$ derived from greek letter $\epsilon$? What is their historical relationship? Obviously the letter must be older, since greek alphabet is extremely old. &#xA;Paul Halmos Naive Set theory, page 2, second paragraph (equivalently page 2 lines 8-11)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This version of the Greek letter epsilon is so often used to denote&#xA;  belonging that its use to denote anything else is almost prohibited.&#xA;  Most authors relegate $\in$ to its set-theoretic use forever and use $\epsilon$&#xA;  when they need the fifth letter of the Greek alphabet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the comments of &lt;a href=&quot;https://math.stackexchange.com/questions/176783/what-does-the-letter-epsilon-signify-in-mathematics/176784#176784&quot;&gt;this answer&lt;/a&gt; is found a discussion, but not in the form of answer. Also this site seems more appropriate than Math.SE.&lt;/p&gt;&#xA;" OwnerUserId="5536" LastEditorUserId="-1" LastEditDate="2017-04-13T12:19:37.660" LastActivityDate="2020-01-16T12:05:26.943" Title="Is the symbol for set membership $\in$ derived from greek letter $\epsilon$?" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5818" PostTypeId="2" ParentId="5817" CreationDate="2017-03-22T19:19:47.707" Score="13" Body="&lt;p&gt;Yes and no. Peano originally used &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; in &lt;em&gt;Arithmetices Prinicipia Nova Methodo Exposita&lt;/em&gt; (1889), and stated that the symbol was an abbreviation for Latin &lt;em&gt;est&lt;/em&gt; (is), apparently using a Greek letter for a Latin word. However, as Mauro Allegranza pointed out, in &lt;em&gt;Principi di Logica Matematica&lt;/em&gt; (1891) he changes the script, and explains the use of &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; by reference to the Greek &lt;em&gt;ἐστι&lt;/em&gt; (is). In &lt;em&gt;Formulaire de Mathematiques&lt;/em&gt; (1895) Peano goes back to &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; (possibly the choice between the two depended on typographers). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the first edition of &lt;em&gt;Principia Mathematica&lt;/em&gt; (1903) Russell explicitly adopted Peano’s symbol &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;, along with the Greek lineage. Extensional (sets) and intensional (classes) notions were not separated at the time, so Russell-Peano's &lt;span class=&quot;math-container&quot;&gt;$a\,\varepsilon P$&lt;/span&gt; is more of intensional &quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;&quot;, with &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; as a class defining property, than modern extensional &quot;&lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is an element of the set &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;&quot;. Fully extensional interpretation of sets only appears in Hausdorff's Grundzuge der Mengenlehre (1914), see &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.9839&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kanamori's The empty set, the singleton, and the ordered pair&lt;/a&gt;. The stylized &lt;span class=&quot;math-container&quot;&gt;$\in$&lt;/span&gt; and its crossed version &lt;span class=&quot;math-container&quot;&gt;$\not\in$&lt;/span&gt; appear in Bourbaki's Theorie des Ensembles (1939), likely responsible for their widespread adoption, but they might have been used earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's site Earliest Uses of Symbols of Set Theory and Logic&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=bT5suOONXlgC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History of Mathematical Notations&lt;/a&gt; for these and similar questions.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-01T00:09:45.520" LastActivityDate="2020-01-01T00:09:45.520" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5823" PostTypeId="1" CreationDate="2017-03-23T08:13:59.327" Score="2" ViewCount="135" Body="&lt;p&gt;In Rene Dugas' History of Mechanics, the author starts off the book with this passage with details with regards to hellenistic mechanics.&#xA;He comments, and I paraphrase &quot; the author of the apocryphal treatise Problems in Mechanics defines the &quot;power&quot; of agency of motion equals to the product of the weight or mass of the body and the velocity of the body. For it is with this definition that the law of equilibrium of the a lever with two unequal arms carrying unequal weights at their ends can be formulated &quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author then talks about how the author of this book relates these properties to the magical properties of the circle and so on and so forth.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could someone kindly explain what does Rene mean when he says this?&lt;/p&gt;&#xA;" OwnerUserId="3564" LastEditorUserId="64" LastEditDate="2017-03-23T18:40:08.593" LastActivityDate="2017-03-23T18:40:08.593" Title="Hellenistic mechanics" Tags="|classical-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5824" PostTypeId="2" ParentId="5823" CreationDate="2017-03-23T10:24:30.693" Score="2" Body="&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Mechanics_(Aristotle)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle's &lt;em&gt;Mechanica&lt;/em&gt; (or &lt;em&gt;Mechanical Problems&lt;/em&gt;; Greek: &lt;em&gt;Μηχανικά&lt;/em&gt;)&lt;/a&gt;, traditionally attributed to Aristotle, though his authorship of it is disputed, 847b16: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The original cause of all such phenomena [lever, balance] is the circle. It is quite natural that this should be so; for there is nothing strange in a lesser marvel being caused by a greater marvel, and it is a very great marvel that contraries should be present together, and the circle is made up of contraries. For to begin with, it is formed by motion and rest, things which are by nature opposed to one another. Hence in examining the circle we need not be much astonished at the contradictions which occur in connexion with it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;marvel&quot; discussed here is specifically the power of a less weight to rise a greater one through the balance (with uequal arms), see 848a11:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Therefore, as has already been remarked, there is nothing strange in the circle being the origin of any and every marvel. The phenomena observed in the balance&#xA;  can be referred to the circle, and those observed in the lever to the balance; while practically all the other phenomena of mechanical motion are connected with the lever. Furthermore, since no two points on one and the same radius travel with the same rapidity, but of two points that which is further from the fixed centre travels more quickly, many marvellous phenomena occur in the motions of circles, which will be demonstrated in the following problems. [...] Mechanicians seizing on this inherent peculiarity of the circle, and hiding the principle, construct an instrument so as to exhibit the marvellous character of the device, while they obscure the cause of it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, in the end, the &quot;magical power of the circle&quot; is nothing more that the ability of the skileld mechanician to master the &quot;law of the circle&quot; that explain the functioning of the device: the balance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also: &lt;a href=&quot;http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1067&amp;amp;context=classicsfacpub&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mechanical Problems in the Corpus of Aristotle&lt;/a&gt; by Thomas Nelson Winter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also useful: Edoardo Benvenuto, &lt;a href=&quot;https://books.google.it/books?id=gfn2BwAAQBAJ&amp;amp;pg=PA34&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Introduction to the History of Structural Mechanics. Part I&lt;/a&gt;, Springer (1991), §1.8 The &quot;Mechanical Problems&quot;: The Peripatetic Explanation of the Law of the Lever and the Parallelogram Rule, page 34.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-03-23T13:01:07.473" LastActivityDate="2017-03-23T13:01:07.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5825" PostTypeId="2" ParentId="5817" CreationDate="2017-03-23T10:51:55.187" Score="17" Body="&lt;p&gt;Yes, it is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Giuseppe_Peano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giuseppe Peano&lt;/a&gt;'s &lt;a href=&quot;https://archive.org/stream/arithmeticespri00peangoog#page/n16/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Arithmetices Principia&lt;/em&gt;&lt;/a&gt; (1889), the &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; symbol is explained as follows (page x): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Signum &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; significat &lt;em&gt;est&lt;/em&gt;.  &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;[The sign &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; means &lt;em&gt;is&lt;/em&gt;.]&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In his &lt;a href=&quot;https://books.google.it/books?hl=it&amp;amp;id=MyxPAAAAYAAJ&amp;amp;dq=peano+logica+matematica&amp;amp;focus=searchwithinvolume&amp;amp;q=%288%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Principi di Logica matematica&lt;/em&gt;&lt;/a&gt; (1891), Peano gives the full explanation (page 3) : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Per indicare la proposizione singolare « &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; è un individuo della classe &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; » scriveremo (&lt;sup&gt;8&lt;/sup&gt;)&lt;br&gt;&#xA;  [To mean the proposition &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is an individual of the class &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;&quot; we will write]&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$x \varepsilon s,$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;e il segno &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;  si potrà leggere &lt;a href=&quot;https://en.wiktionary.org/wiki/essere#Italian&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;è&lt;/em&gt;&lt;/a&gt;, o &lt;em&gt;è un&lt;/em&gt;, o &lt;em&gt;fu&lt;/em&gt;, o &lt;em&gt;sarà&lt;/em&gt;, a seconda delle regole grammaticali; però il suo significato è sempre quello spiegato.&lt;br&gt;&#xA;  [and the sign &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; will be read &lt;em&gt;is&lt;/em&gt;, or &lt;em&gt;is a&lt;/em&gt;, ...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(&lt;sup&gt;8&lt;/sup&gt;): Il segno &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; è l'iniziale di &lt;a href=&quot;https://en.wiktionary.org/wiki/%E1%BC%94%CF%83%CF%84%CE%B9%CE%BD&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;ἐστι&lt;/em&gt;&lt;/a&gt;.&lt;br&gt;&#xA;  [The sign &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; is the initial of &lt;em&gt;ἐστι&lt;/em&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also: Hubert Kenendy, &lt;a href=&quot;https://projecteuclid.org/euclid.ndjfl/1093891001&quot; rel=&quot;nofollow noreferrer&quot;&gt;What Russell learned from Peano&lt;/a&gt;, NDJFL (1973).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And see: Alfred North Whitehead and Bertrand Russell, &lt;a href=&quot;https://books.google.it/books?id=ke9yGmFy24sC&amp;amp;pg=PA25&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Principia Mathematica&lt;/em&gt;&lt;/a&gt; (1910-1913), page 25:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The propositional function &quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a member of the class &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;&quot; will be expressed, following Peano, by the notation &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x \epsilon \alpha.$$&lt;/span&gt;&#xA;  Here &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; is chosen as the initial of the word &lt;em&gt;ἐστι&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1888, &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Dedekind&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard Dedekind&lt;/a&gt;, in &lt;a href=&quot;https://books.google.it/books?id=TK3OpvuABOMC&amp;amp;pg=PA2&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Was Sind und was Sollen Die Zahlen?&lt;/em&gt;&lt;/a&gt; (art. 3), used a &quot;reversed&quot; &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$A ∋ S$&lt;/span&gt;) to mean &quot;is part of&quot;, where this relation slid between membership and improper inclusion.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="10974" LastEditDate="2020-01-16T12:05:26.943" LastActivityDate="2020-01-16T12:05:26.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5828" PostTypeId="1" CreationDate="2017-03-23T22:51:21.947" Score="8" ViewCount="693" Body="&lt;p&gt;In theoretical physics, a Hamiltonian is traditionally denoted by some variant of $H$, a Lagrangian is some variant of $L$, but why is an action always some variant of S (as opposed to, say, $A$)? Does the letter refer to a particular person, place, or concept? In similar examples ($\mathbb{Z}$ for the Integers (Zahlen), $\mathbb{K}$ for a Field (Körper)), the letter traditionally used for the concept comes from the word of that concept in a language other than English. Is that the case here? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ideally, I'm looking for the history (perhaps etymology is the correct word) of the usage of the letter S being used to describe the action functional in theoretical physics.&lt;/p&gt;&#xA;" OwnerDisplayName="Mark B" LastEditorUserId="55" LastEditDate="2017-03-24T22:52:28.470" LastActivityDate="2017-03-24T23:32:15.680" Title="Why is the action from the principle of least action traditionally denoted $S$?" Tags="|notation|theoretical-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5829" PostTypeId="1" AcceptedAnswerId="7962" CreationDate="2017-03-24T00:20:51.127" Score="2" ViewCount="185" Body="&lt;p&gt;It's my understanding that the convention of using letters from the end of the alphabet ($x$, $y$, $z$) to represent $variables$, and letters from the start of the alphabet ($a$, $b$, $c$) to represent $constants$ came to us from F. Vieta (who proposed vowels and consonants) by way of Descartes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I was curious to know if there was ever any debate around subscript notation for the same distinction, i.e. $x$ vs. $x_o$, where the first is considered a variable and the second, an unknown constant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can imagine many reasons for $x_a$ being more popular than $a_x$; but, &lt;strong&gt;does anyone know where this notation got its start? &amp;amp;nd was there any contention?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="5127" LastEditorUserId="5127" LastEditDate="2017-04-04T17:06:00.960" LastActivityDate="2018-11-15T09:08:12.337" Title="Why $x_a$ (or $x_o$) and not $a_x$? (conventions for algebraic quantities)" Tags="|notation|elementary-algebra|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5830" PostTypeId="2" ParentId="5701" CreationDate="2017-03-24T02:13:15.037" Score="2" Body="&lt;p&gt;His most important contribution was the Principle of Galilean Relativity. His second most important contribution was his &lt;a href=&quot;https://m.youtube.com/watch?v=eghdN-GFuqo&quot; rel=&quot;nofollow noreferrer&quot;&gt;inclined plane experiment&lt;/a&gt; which showed all bodies accelerate at the same rate independent of their masses.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorUserId="10106" LastEditDate="2020-05-16T18:44:37.257" LastActivityDate="2020-05-16T18:44:37.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5831" PostTypeId="1" CreationDate="2017-03-24T02:35:06.937" Score="1" ViewCount="923" Body="&lt;p&gt;We know that Millikan ended up proving Einstein's photoelectric equation instead of &lt;em&gt;disproving&lt;/em&gt; it. Since he aimed at &lt;em&gt;disproving&lt;/em&gt; it, what did he have in mind? In other words, what did he expect to see? &lt;/p&gt;&#xA;" OwnerUserId="5558" OwnerDisplayName="Kunal Pawar" LastEditorUserId="55" LastEditDate="2017-03-24T23:21:42.023" LastActivityDate="2017-03-24T23:21:42.023" Title="What did Millikan expect to find when he tried to disprove the photoelectric effect, but proved it instead?" Tags="|physics|experimental-physics|debunking|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5832" PostTypeId="2" ParentId="5828" CreationDate="2017-03-24T21:41:34.103" Score="10" Body="&lt;p&gt;The notation was introduced by Hamilton in 1834, who shifted the focus from the original Maupertuis-Euler version of the least action principle. The tradition followed him probably because his exposition was definitive and systematic. It is also possible that thinking of &quot;stationary action&quot; makes the notation &quot;make sense&quot; to people, although this name is much more recent, and certainly not due to Hamilton.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neither Maupertuis in &lt;a href=&quot;https://en.wikisource.org/?curid=56506&quot; rel=&quot;noreferrer&quot;&gt;Derivation of the laws of motion and equilibrium from a metaphysical principle (1746)&lt;/a&gt;, nor Euler in &lt;a href=&quot;https://en.wikisource.org/wiki/Translation:Methodus_inveniendi/Additamentum_II&quot; rel=&quot;noreferrer&quot;&gt;Methodus inveniendi (1744)&lt;/a&gt; use any letter for action. Euler simply writes $\int vds$, and Maupertuis does not even write a formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamilton in &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics&quot; rel=&quot;noreferrer&quot;&gt;On a General Method in Dynamics (1834)&lt;/a&gt; uses $V$ (for &quot;&lt;em&gt;varying action&lt;/em&gt;&quot;), which he calls the &quot;&lt;em&gt;characteristic function&lt;/em&gt;&quot;, for the Maupertuis style action (see the &lt;a href=&quot;http://www.scholarpedia.org/article/Principle_of_least_action#Statements_of_Hamilton_and_Maupertuis_Principles&quot; rel=&quot;noreferrer&quot;&gt;Scholarpedia article&lt;/a&gt;  on the difference). But near the end he remarks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Before we conclude this sketch of our general method in dynamics, it will be proper to notice briefly a transformation of the characteristic function, which may be used in all applications. This transformation consists in putting, generally, $V=tH+S$, and considering the part $S$, namely, the definite integral&#xA;  $\int_0^t (T+U)dt$ as a function of the initial and final coordinates and of the time...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This $S$ is just called an &quot;&lt;em&gt;auxilliary function&lt;/em&gt;&quot; and no explanation for the choice of $S$ is given. It could have been as simple as destylizing $\int$ back into $S$. In the &lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics&quot; rel=&quot;noreferrer&quot;&gt;Second Essay on a General Method in Dynamics (1835)&lt;/a&gt; $S$ is renamed into the &quot;&lt;em&gt;Principal Function&lt;/em&gt;&quot;, and takes the center stage. This is where much of familiar Hamiltonian dynamics is developed in the familiar form. Still, we get no explanation as to the choice of $S$:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The former Essay contained a general method for reducing all the most important problems of dynamics to the study of one characteristic function, one central or radical relation. It was remarked at the close of that Essay, that many eliminations required by this method in its first conception, might be avoided by a general transformation, introducing the time explicitly into a part&#xA;  $S$ of the whole characteristic function $V$; and it is now proposed to fix&#xA;  the attention chiefly on this part $S$, and to call it the &lt;strong&gt;Principal Function&lt;/strong&gt;. The properties of this part or function $S$, which were noticed briefly in the former Essay, are now more fully set forth; and especially its uses in questions of perturbation, in which it dispenses with many laborious and circuitous processes, and enables us to express accurately the disturbed configuration of a system by the rules of undisturbed motion, if only the initial components of velocities be changed in a suitable manner.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://homes.chass.utoronto.ca/~cfraser/HJ.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Early History of Hamilton-Jacobi Dynamics 1834-1837 by Nakane and Fraser&lt;/a&gt; for more details.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-03-24T23:32:15.680" LastActivityDate="2017-03-24T23:32:15.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5833" PostTypeId="2" ParentId="5831" CreationDate="2017-03-24T23:20:07.663" Score="2" Body="&lt;p&gt;Millikan indeed &amp;quot;ended up&amp;quot; confirming Einstein's photoeffect formula, but that he was &lt;em&gt;aiming&lt;/em&gt; to disprove it in his 1916 experiment seems to be a later gloss from his Nobel lecture:&lt;/p&gt;&#xA;&lt;p&gt;“&lt;em&gt;This work resulted, contrary to my own expectation, in the first direct experimental proof… of the Einstein equation and the first direct photo-electric determination of Planck’s &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;p&gt;In an earlier paper from January 1916 he wrote something else:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Einstein’s photoelectric equation… cannot in my judgment be looked upon at present as resting upon any sort of a satisfactory theoretical foundation...&lt;/em&gt; [even though] &lt;em&gt;it actually represents very accurately the behavior.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Millikan could hardly have been aiming to experimentally disprove a &amp;quot;very accurate&amp;quot; equation. It appears that his goal was simply to measure the Planck constant. Here is from &lt;a href=&quot;http://physics.aps.org/story/v3/st23&quot; rel=&quot;nofollow noreferrer&quot;&gt;Holton's Centennial Focus&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Ironically, it had been Millikan’s experiment which convinced the experimentalist-inclined committee in Stockholm to admit Einstein to that select circle in 1922. One final irony: In 1950, at age 82, Millikan published his Autobiography, with Chapter 9 entitled simply “The Experimental Proof of the Existence of the Photon–Einstein’s Photoelectric Equation.” By then, Millikan had of course come to terms with the photon. Moreover, he had evidently changed his mind about what he had done around 1916, for now he wrote that as the experimental data became clear in his lab, they “proved simply and irrefutably, I thought, that the emitted electron that escapes with the energy hf gets that energy by the direct transfer of hf units of energy from the light to the electron, and hence scarcely permits of any other interpretation than that which Einstein had originally suggested, namely that of the semi-corpuscular or photon theory of light itself.”&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In the end, Millikan re-imagined the complex personal history of his splendid experiment to fit the simple story told in so many of our physics textbooks.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This was not Millikan's first time &amp;quot;re-imagining&amp;quot; his experiments, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2756/is-millikans-famous-oil-drop-experiment-a-fraud&quot;&gt;Is Millikan&amp;#39;s famous oil drop experiment a fraud?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-24T23:20:07.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5834" PostTypeId="1" CreationDate="2017-03-25T11:06:03.213" Score="2" ViewCount="544" Body="&lt;p&gt;I am reading &lt;em&gt;Big Bang&lt;/em&gt; by Simon Singh. Here is the short story (as I understood it):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein gets a contradiction by assuming that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1- The speed of light $c$ is constant with respect the ether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2- Galilean relativity is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;because if you were traveling at the speed of light, you wouldn't be able to see yourself in a mirror, since you would be moving with respect the ether at the same speed at the light leaving your face. So you would be able to tell that you are moving, contradicting galilean relativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here seems logical to deduce (assuming galilean relativity) that you will see your face on the mirror, therefore the speed of light can't be constant with respect the ether. And here comes where I am stuck, to me the most obvious way of solving this problem is thinking that light will act as any other thing we know, if you are moving at the speed of light and you throw a ball in the same direction of the movement at a speed $v$, you will see the ball leaving you at $v$ and someone that doesn't move with respect to you will see it at $c+v$. But according to the book, Einstein's first thought was to say that light has a constant speed for every observer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So finally here comes my question: Did Einstein (or anyone before him) consider the first argument I gave here and, if so, why did he reject it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.D. In case it is useful to know something about my background, I am a mathematician with very little knowledge of physics, just the things I read in these kind of divulgative books. That said, sorry about incongruences or inaccuracys, I would very much apreciate corrections.&lt;/p&gt;&#xA;" OwnerUserId="5560" LastActivityDate="2022-08-07T18:31:09.360" Title="How Einstein got the first idea for special relativity?" Tags="|physics|theoretical-physics|relativity-theory|einstein|light|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5835" PostTypeId="2" ParentId="5834" CreationDate="2017-03-25T12:42:39.543" Score="8" Body="&lt;p&gt;Einstein was reading Maxwell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Axiom a) follows from Maxwell's electromagnetic theory of light. Axiom b) is stated imprecisely in your message. The actual axiom is that you cannot detect the absolute motion of the observer (or that the laws of nature must be independent of the motion of the observer (rectilinear and with constant speed)).&#xA;This is the generalization of Galileo's relativity principle which Einstein&#xA;stated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In more precise mathematical language, Maxwell's equations which describe light&#xA;are not invariant under Galilean transformation. This is why &quot;Galilean relativity&quot;&#xA;has to be replaced by &quot;special relativity&quot;. This was understood by several people at approximately the same time (Einstein, Lorentz and Poincaré). They found the correct transformations which preserve Maxwell's equations. The group of these transformations is called the Poincaré group and the subgroup consisting of homogeneous transformations is called the Lorentz group.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="658" LastEditDate="2017-03-26T16:50:30.507" LastActivityDate="2017-03-26T16:50:30.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5836" PostTypeId="1" CreationDate="2017-03-27T09:11:05.843" Score="6" ViewCount="1469" Body="&lt;p&gt;Because regular physical exercise is in theory linked to better brain function and is also recommended in another question here on stack exchange, I wonder if there are any famous mathematicians who reputedly adhered to a strict exercise routine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I already asked this question on Mathematics Stack Exchange but it has been closed for being off topic. So far the following mathematicians were mentioned:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Leopold_Vietoris&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Leopold_Vietoris&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Urschel&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/John_Urschel&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagoras&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Pythagoras&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alan_Turing&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Alan_Turing&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ronald_Graham&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ronald_Graham&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Arnold&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Vladimir_Arnold&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/186726/math-and-mental-fatigue&quot;&gt;Math and mental fatigue&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S1364661307001581&quot; rel=&quot;noreferrer&quot;&gt;Capitalizing on cortical plasticity: influence of physical activity on cognition and brain function&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.nature.com/nrn/journal/v9/n1/abs/nrn2298.html&quot; rel=&quot;noreferrer&quot;&gt;Be smart, exercise your heart: exercise effects on brain and cognition&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0197458005002745&quot; rel=&quot;noreferrer&quot;&gt;Fitness, aging and neurocognitive function&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5483" LastEditorUserId="2277" LastEditDate="2019-04-18T11:32:14.057" LastActivityDate="2019-04-20T18:44:09.490" Title="Are there any famous mathematicians who did regular physical workouts?" Tags="|mathematicians|biographical-details|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="5837" PostTypeId="2" ParentId="5836" CreationDate="2017-03-27T13:32:27.240" Score="5" Body="&lt;p&gt;Mathematicians do not differ from other people in all respects except their professional activity. So fraction of mathematicians who regularly exercise and&#xA;involved with some sport is the same as for other people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, all sorts of physical exercises were very popular among the&#xA;Soviet mathematicians of 20 century, as we can see from their memoirs.&#xA;(Kolmogorov, Aleksandrov, Uhryson, Vinogradov etc.)&#xA;Long kayaking and mountaineering trips were especially  popular. Most Soviet mathematicians &#xA;of the middle 20th century participated in them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides kayaking and mointaineering the most popular activities were&#xA;rowing, swimming, skiing, soccer, volleyball,  even weight lifting. There is even a Russian book on this: Sadovskii, Mathematics and sport,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://math.ru/lib/bmkvant/44&quot; rel=&quot;noreferrer&quot;&gt;http://math.ru/lib/bmkvant/44&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-03-27T13:32:27.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5838" PostTypeId="2" ParentId="5836" CreationDate="2017-03-27T17:16:49.810" Score="7" Body="&lt;p&gt;Niels Bohr's brother, &lt;a href=&quot;https://en.wikipedia.org/wiki/Harald_Bohr&quot; rel=&quot;noreferrer&quot;&gt;Harald Bohr&lt;/a&gt;, was a notable mathematician and football (soccer) player.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Harald August Bohr (22 April 1887 – 22 January 1951) was a Danish mathematician and football player. After receiving his doctorate in 1910, Bohr became an eminent mathematician, founding the field of almost periodic functions. His brother was the Nobel Prize-winning physicist Niels Bohr. He was a member of the Danish national football team for the 1908 Summer Olympics, where he won a silver medal.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Bohr was also an excellent football player. He had a long playing career with Akademisk Boldklub, making his debut as a 16-year-old in 1903. During the 1905 season he played alongside his brother Niels, who was a goalkeeper. Harald was selected to play for the Danish national football team in the 1908 Summer Olympics, where football was an official event for the first time. Though a Danish side had played at the 1906 Intercalated Games, the opening match of the 1908 Olympic tournament was Denmark's first official international football match. Bohr scored two goals as Denmark beat the French &quot;B&quot; team 9–0. In the next match, the semi-final, Bohr played in a 17–1 win against France, which remains an Olympic record to this day. Denmark faced hosts Great Britain in the final, but lost 2–0, and Bohr won a silver medal. After the Olympics he made one further appearance for the national team, in a 2–1 victory against an England amateur team in 1910. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;His popularity as a footballer was such that when he defended his doctoral thesis the audience was reported as having more football fans than mathematicians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-03-27T17:26:55.613" LastActivityDate="2017-03-27T17:26:55.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5839" PostTypeId="1" CreationDate="2017-03-27T20:42:46.027" Score="1" ViewCount="1148" Body="&lt;p&gt;I am trying to trace (as far as known) the first example of a digital computing device. I am discounting (no pun) an abacus because I regard that as a device for representing, storing and manipulating discrete, not binary digital, data values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read that in 1937 Church and Turing published theoretical papers on computing. My understanding is that the (universal?) Turing machine does not explicitly require a digital representation of data and operations (maybe that is wrong).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to track down the point that it became clear that both data and operations could be represented by digital codes (maybe it was Church-Turing?). I have read that the earliest binary digital computers were produced by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1938: Konrad Zuse: Konrad Zuse creates the Z1 Computer a binary digital computer using punch tape    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1939    George Stibitz: George Stibitz develops the Complex Number Calculator - a foundation for digital computers&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was this the time when the relationship between high-level operations (such as integer multiplication) and manipulating binary digits became explicit?&lt;/p&gt;&#xA;" OwnerUserId="5568" LastActivityDate="2019-01-02T01:47:01.023" Title="Earliest concrete example of a binary / digital computing device" Tags="|computers|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5840" PostTypeId="2" ParentId="5839" CreationDate="2017-03-27T21:36:20.790" Score="5" Body="&lt;p&gt;Digital devices were quite common since 19th century, they were called cash registers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Cash_register&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cash_register&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many shops were equipped with them.&#xA;Similar device for scientific/engineering purposes was called arithmometer,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmometer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Arithmometer&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1970s I was solving PDE's with this arithmometer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%A4%D0%B5%D0%BB%D0%B8%D0%BA%D1%81_(%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%BE%D0%BC%D0%B5%D1%82%D1%80)#/media/File:Felix_kurskiy.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ru.wikipedia.org/wiki/%D0%A4%D0%B5%D0%BB%D0%B8%D0%BA%D1%81_(%D0%B0%D1%80%D0%B8%D1%84%D0%BC%D0%BE%D0%BC%D0%B5%D1%82%D1%80)#/media/File:Felix_kurskiy.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or adding machine. The earliest ones appear in 17th century (Pascal is credited&#xA;with invention of one.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. The main feature of modern computers which sets them apart is that they are programmable, not that they are digital. But programmable computers also existed in 19th century (Babbage machine, for example).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Babbage#/media/File:Opening_George_III_museum.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Charles_Babbage#/media/File:Opening_George_III_museum.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were also quite sophisticated analog computers (as opposite to digital)&#xA;and they were also in use since 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-01-01T14:58:10.290" LastActivityDate="2019-01-01T14:58:10.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5841" PostTypeId="1" AcceptedAnswerId="5842" CreationDate="2017-03-27T23:17:01.717" Score="0" ViewCount="219" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/a/5080/232&quot;&gt;This answer&lt;/a&gt; mentions Franz Ernst Neumann as the inventor of the vector potential, but who invented or discovered the scalar potential?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2019-07-30T19:58:51.957" Title="Who invented/discovered the scalar potential?" Tags="|potential|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5842" PostTypeId="2" ParentId="5841" CreationDate="2017-03-27T23:17:37.257" Score="0" Body="&lt;p&gt;Lagrange (1736-1813) in 1777, followed by Laplace (1749-1827) in 1782, was the first to introduce the scalar gravitational potential.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange's paper, &lt;em&gt;&lt;a href=&quot;https://archive.org/stream/oeuvresdelagrang04lagr#page/396/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Remarques générales sur lemouvement de plusieurs corps qui s'attirent mutuellement enraison inverse des carrés des distances&lt;/a&gt;&lt;/em&gt;, was read at the Academy of Berlin on 20 October 1777.&lt;sup&gt;2&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;References&lt;br&gt;1. &lt;a href=&quot;https://www.ifi.unicamp.br/~assis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Assis, André K. T.&lt;/a&gt; &lt;a href=&quot;https://www.ifi.unicamp.br/~assis/Relational-Mechanics-Mach-Weber.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Relational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force&lt;/em&gt;&lt;/a&gt;. Montréal: Apeiron, 2014, p. 475.&lt;br&gt;2. &lt;a href=&quot;https://www.encyclopedia.com/people/history/historians-miscellaneous-biographies/pierre-maurice-marie-duhem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;a href=&quot;https://archive.org/stream/leonssurllec01duheuoft#page/12/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Leçons sur l’électricité et le magnétisme&lt;/em&gt;&lt;/a&gt;. vol. 1. Paris: Gauthier-Villars, 1891, &lt;a href=&quot;https://archive.org/stream/leonssurllec01duheuoft#page/12/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 13&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-07-30T19:58:51.957" LastActivityDate="2019-07-30T19:58:51.957" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5843" PostTypeId="1" CreationDate="2017-03-27T18:40:47.963" Score="3" ViewCount="107" Body="&lt;p&gt;In his paper &lt;em&gt;The search for unity: Notes for a history of quantum field theory&lt;/em&gt; [Daedalus 4, 106 (1977)], Steven Weinberg writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first successful classical field theory was based on Newton's theory of gravitation. Newton himself did not speak of fields - for him, gravitation was a force which acts between every pair of material particles in the universe, &quot;according to the quantity of solid matter which they contain and propagates on all sides to immense distances, decreasing always as the inverse square of the distances.&quot; &lt;strong&gt;It was the mathematical physicists of the eighteenth century&lt;/strong&gt; who found it convenient to replace this mutual action at a distance with a gravitational field, a numerical quantity (strictly speaking, a vector) which is defined at every point in space, which determines the gravitational force acting on any particle at that point, and which receives contributions from all the material particles at every other point.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which mathematical physicists is he referring to?&lt;/p&gt;&#xA;" OwnerDisplayName="Funzies" LastEditorUserId="62" LastEditDate="2017-03-28T17:12:40.767" LastActivityDate="2017-03-28T17:12:40.767" Title="Who wrote down the equations governing gravity in a field language for the first time?" Tags="|mathematical-physics|gravity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5844" PostTypeId="2" ParentId="5843" CreationDate="2017-03-27T23:08:28.930" Score="5" Body="&lt;p&gt;Lagrange (1736-1813) in 1777, followed by Laplace (1749-1827) in 1782, was the first to introduce the scalar gravitational potential.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange's paper, &lt;em&gt;&lt;a href=&quot;https://archive.org/stream/oeuvresdelagrang04lagr#page/396/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Remarques générales sur lemouvement de plusieurs corps qui s'attirent mutuellement enraison inverse des carrés des distances&lt;/a&gt;&lt;/em&gt;, was read at the Academy of Berlin on 20 October 1777.&lt;sup&gt;2&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Weinberg is correct in saying &quot;eighteenth century&quot; (and not 19&lt;sup&gt;th&lt;/sup&gt; century).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;References&lt;br&gt;1. &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/&quot; rel=&quot;noreferrer&quot;&gt;Assis, André K. T.&lt;/a&gt; &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Relational-Mechanics-Mach-Weber.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Relational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force&lt;/em&gt;&lt;/a&gt;. Montréal: Apeiron, 2014, p. 475.&lt;br&gt;2. &lt;a href=&quot;http://www.encyclopedia.com/people/history/historians-miscellaneous-biographies/pierre-maurice-marie-duhem&quot; rel=&quot;noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;a href=&quot;https://archive.org/stream/leonssurllec01duheuoft#page/12/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Leçons sur l’électricité et le magnétisme&lt;/em&gt;&lt;/a&gt;. vol. 1. Paris: Gauthier-Villars, 1891, &lt;a href=&quot;https://archive.org/stream/leonssurllec01duheuoft#page/12/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;p. 13&lt;/a&gt;.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" OwnerDisplayName="Geremia" LastActivityDate="2017-03-27T23:13:44.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5845" PostTypeId="1" AcceptedAnswerId="5846" CreationDate="2017-03-28T17:15:16.353" Score="5" ViewCount="1645" Body="&lt;p&gt;An hour has been divided in sixty minutes &lt;a href=&quot;https://en.wikipedia.org/wiki/Minute&quot; rel=&quot;nofollow noreferrer&quot;&gt;since medieval times&lt;/a&gt;.  During the 16th and 17th century, clocks measuring &lt;a href=&quot;https://en.wikipedia.org/wiki/Second#Based_on_mechanical_clocks&quot; rel=&quot;nofollow noreferrer&quot;&gt;a &lt;em&gt;second&lt;/em&gt; subdivision&lt;/a&gt; emerged.  Today we still use the sexagesimal system for dividing hours into minutes and seconds, but below that we use the decimal system (and the second is the only SI unit for time). &lt;a href=&quot;https://en.wikipedia.org/wiki/Minute&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia notes&lt;/a&gt; that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For even further refinement, the term &quot;third&quot; ( 1⁄60 of a second) remains in some languages, for example Polish (tercja) and Turkish (salise), although most modern usage subdivides seconds by using decimals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The choice of the word &lt;em&gt;remains&lt;/em&gt; leave open the possibility that there was such a subdivision in the past.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has the English Language ever had a sexagesimal subdivision of the second?&lt;/p&gt;&#xA;" OwnerUserId="111" LastEditorUserId="55" LastEditDate="2017-03-28T17:56:46.407" LastActivityDate="2017-03-30T12:52:00.010" Title="Did English ever use a third (1/60 of a second) for measuring time?" Tags="|units|time|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5846" PostTypeId="2" ParentId="5845" CreationDate="2017-03-28T17:55:21.877" Score="5" Body="&lt;p&gt;Right after your quote Wikipedia has &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1267, the medieval scientist Roger Bacon, writing in Latin, defined the division of time between full moons as a number of hours, minutes, seconds, thirds, and fourths (horae, minuta, secunda, tertia, and quarta) after noon on specified calendar dates&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This use was retained in astronomy but not colloquially, &lt;a href=&quot;https://books.google.com/books?id=JnpmAAAAcAAJ&amp;amp;pg=PA46&amp;amp;lpg=PA46&amp;amp;dq=english+astronomy+thirds+and+fourths&amp;amp;source=bl&amp;amp;ots=Ce3x8ZOQAo&amp;amp;sig=OuwXG_9hABApyWkuWYcqEGM8wTI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiS2-7F4PnSAhUKPiYKHe0BDxIQ6AEIOzAI#v=onepage&amp;amp;q=english%20astronomy%20thirds%20and%20fourths&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moxon's Tutor to Astronomy and Geography. Or, An Easie and Speedy Way to Know the Use of Both the Globes (1698)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Hours are vulgarly divided into halves, quarters and half quarters, but mathematically into minutes, seconds, thirds, fourths, etc.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Arithmetical tables in the second edition of &lt;a href=&quot;https://books.google.com/books?id=EOoIAAAAQAAJ&amp;amp;pg=PA138&amp;amp;lpg=PA138&amp;amp;dq=english+astronomy+thirds+and+fourths&amp;amp;source=bl&amp;amp;ots=BXMSk8qN0P&amp;amp;sig=JGQUBbJ7gO9cSmsmqCYerB-aHyY&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiS2-7F4PnSAhUKPiYKHe0BDxIQ6AEIOTAH#v=onepage&amp;amp;q=english%20astronomy%20thirds%20and%20fourths&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rusher's English Spelling Book (1851)&lt;/a&gt; also list a third as 1/60 of a second.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-03-28T19:08:57.423" LastActivityDate="2017-03-28T19:08:57.423" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5847" PostTypeId="1" CreationDate="2017-03-29T00:05:24.500" Score="2" ViewCount="137" Body="&lt;p&gt;In the study of electrical engineering circuit diagrams it is usually the norm to show the + ( positive ) polarity as the direction of motion.  However in reality the electron is the elementary particle that is actually moving so the diagram is actually in reverse.  After some research the only explanation I can find on the internet is that when Benjamin Franklin first conceived of a positive and negative aspect and he just guessed at direction and got it wrong and engineers have been reluctant to undo all the legacy diagrams.  I don't know if I believe this story. I don't see how someone can guess the &quot;wrong&quot; direction since Franklin did not know about the electron. Wouldn't the better question be why didn't J.J Thompson name the electron to be + since that was the direction of motion in all circuit diagrams?  Obviously I am in error somewhere but need help sorting the historical details. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2017-03-29T00:05:24.500" Title="What is the history of why electrical circuit diagrams list positive as the direction of electron flow?" Tags="|biographical-details|terminology|electricity|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5848" PostTypeId="1" AcceptedAnswerId="5849" CreationDate="2017-03-29T03:31:11.533" Score="9" ViewCount="1172" Body="&lt;p&gt;I was researching the mathematics of this puzzle and wondered where it is be thought to originate. Any information, including context and precedents, would be greatly appreciated. (I'm particularly interested in puzzles that are thought of as combinatorial.)&lt;/p&gt;&#xA;" OwnerUserId="5197" LastEditorUserId="55" LastEditDate="2020-11-25T07:46:34.023" LastActivityDate="2020-11-25T07:49:16.633" Title="What was the origin of the Seven Bridges of Königsberg problem before Euler?" Tags="|discoveries|graph-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5849" PostTypeId="2" ParentId="5848" CreationDate="2017-03-29T19:13:20.647" Score="9" Body="&lt;p&gt;MAA has a very nice presentation of the &lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;problem's history and solution&lt;/a&gt; authored by Paoletti. For what happened after Euler see &lt;a href=&quot;https://hsm.stackexchange.com/a/12507/55&quot;&gt;Was there a continuation to Euler's work on the Seven Bridges of Königsberg problem between Euler and Hamilton?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem did not originate with Euler, although he was first to formalize it as a problem of existence of what is now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eulerian_path&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eulerian path&lt;/a&gt; in a graph, and the one who gave it its historical significance. Apparently, Euler was asked about the Königsberg bridge problem by &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Gottlieb_Ehler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Gottlieb Ehler&lt;/a&gt;, an astronomer, mathematician, and later a mayor of Danzig. In his reply to Ehler in 1736 Euler dismissed the problem as not even mathematical:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...Thus you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This despite the fact that a year earlier, on August 26, 1735, Euler presented a paper &lt;a href=&quot;http://eulerarchive.maa.org/pages/E053.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solutio Problematis ad Geometriam Situs Pertinentis (Solution of a Problem Relating to the Geometry of Position)&lt;/a&gt; to the St. Petersburg Academy devoted to solving just this problem. It was published in 1741, and is considered to be the first publication that started (mathematical) graph theory. Paoletti mentions that the puzzle was urban folklore in Königsberg:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;According to lore, the citizens of Königsberg used to spend Sunday afternoons walking around their beautiful city.  While walking, the people of the city decided to create a game for themselves, their goal being to devise a way in which they could walk around the city, crossing each of the seven bridges only once.  Even though none of the citizens of Königsberg could invent a route that would allow them to cross each of the bridges only once, still they could not prove that it was impossible.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, he cites no sources. But this is partially confirmed by Euler himself, who writes in his paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The problem, which I am told is widely known, is as follows: in Königsberg in&#xA;Prussia, there is an island &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; called the &lt;span class=&quot;math-container&quot;&gt;$Kneipho\!f$&lt;/span&gt;; the river which surrounds it is divided into two branches, as can be seen in Fig. [1.2], and these branches are crossed by seven bridges, a, b, c, d, e, f and g. Concerning these bridges it was asked whether anyone could arrange a route in such a way that he would cross each bridge once and only once. I was told that some people asserted that this was impossible, while others were in doubt; but nobody would actually assert that it could be done. From this, I have formulated the general problem: whatever be the arrangement and division of the river into branches, and however many bridges there be, can one find out whether or not it is possible to cross each bridge exactly once?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[Translation from &lt;a href=&quot;https://books.google.com/books?id=XqYTk0sXmpoC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graph Theory, 1736-1936 by Biggs, Lloyd and Wilson&lt;/a&gt;]&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-25T07:49:16.633" LastActivityDate="2020-11-25T07:49:16.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5850" PostTypeId="2" ParentId="5845" CreationDate="2017-03-30T12:52:00.010" Score="3" Body="&lt;p&gt;&lt;em&gt;Oxford English Dictionary&lt;/em&gt;&lt;br /&gt;&#xA;&lt;strong&gt;third,&lt;/strong&gt; &lt;em&gt;adj. (and adv.) and n.&lt;/em&gt;&lt;br /&gt;&#xA;Definition B.7.a.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The third of the subdivisions of any standard measure or dimension which is successively subdivided in a constant ratio; the subdivision next below seconds: see prime &lt;em&gt;n.&lt;/em&gt;&lt;sup&gt;2&lt;/sup&gt; 2   ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;with some quotes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1595   J. Davis &lt;em&gt;Seamans Secrets&lt;/em&gt; i. sig. B4,   Euery degree..doth containe 60 minutes, and euery minute 60 seconds, and euery second 60 thirds, &amp;amp;c.&lt;/p&gt;&#xA;&lt;p&gt;1694   W. Holder &lt;em&gt;Disc. Time&lt;/em&gt; ii. 32   To divide..an Hour into 60′ (Minutes), a Minute into 60″ (Second Minutes), a Second Minute into 60‴ (Thirds).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-03-30T12:52:00.010" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5851" PostTypeId="1" CreationDate="2017-03-30T21:37:30.383" Score="0" ViewCount="206" Body="&lt;p&gt;As I understand it, the first true algorithmic intelligences were designed to engage humans in combinatorial &lt;a href=&quot;https://en.wikipedia.org/wiki/Combinatorial_game_theory#cite_note-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;playgames&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a patent for &lt;a href=&quot;https://www.google.com/patents/US2215544?dq=2,215,544&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjn4bGzhbfTAhXMRiYKHWhVC04Q6AEIIzAA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nimatron&lt;/a&gt; filed by Westinghouse Electric in 1940, which was ultimately granted.  (&lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Condon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward Condon&lt;/a&gt; is one of the inventors.)  It was also reportedly displayed at the &lt;a href=&quot;http://www.goodeveca.net/nimrod/nimatron.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1940 World's Fair&lt;/a&gt;, and there is a &lt;a href=&quot;https://fr.wikipedia.org/wiki/Nimatron&quot; rel=&quot;nofollow noreferrer&quot;&gt;French Wikipedia article about it&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this timeline of &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_artificial_intelligence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Artificial Intelligence&lt;/a&gt; lists 1951 as the first working AI, again, notably, in reference to combinatorial playgames, in this case of the partisan variety.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There isn't a great deal of information about this on the web, so I wanted to put it before this Stack to get some clarity.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also throwing uncertainty into the mix, the period between 1870 and 1910 is referred to as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Automaton#Modern&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;The Golden Age of Automata&quot;&lt;/a&gt; and I wonder if it's possible that a mechanical game AI might have existed, in the spirit of Babbage.  &lt;/p&gt;&#xA;" OwnerUserId="5197" LastEditorUserId="5197" LastEditDate="2019-03-14T16:44:20.993" LastActivityDate="2019-03-14T16:44:20.993" Title="Was Nimatron the first AI?" Tags="|artificial-intelligence|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5852" PostTypeId="1" CreationDate="2017-04-01T11:32:34.770" Score="4" ViewCount="186" Body="&lt;p&gt;The first few big scientists like Copernicus, Tycho Brahe, Kepler, Galileo were not British. Then came Newton. And, since then, it seems that the British started dominating science.&lt;/p&gt;&#xA;&lt;p&gt;What was the social and political reason for this change?&lt;/p&gt;&#xA;" OwnerUserId="4079" LastEditorUserId="4251" LastEditDate="2021-03-01T00:40:52.550" LastActivityDate="2021-03-12T05:50:29.310" Title="When did British science take off?" Tags="|physics|social-context|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="5853" PostTypeId="2" ParentId="5852" CreationDate="2017-04-01T13:22:40.090" Score="2" Body="&lt;p&gt;&quot;Then comes Newton&quot;. You forget John Napier, a.k.a.  Marvellous Merchiston, who invented logarithms and decimal representation of real numbers. He was a  contemporary of Brahe, Galileo and Kepler. His inventions revolutionized mathematics and astronomy. And he was not alone: Henry Brigs was another British mathematician who cooperated with Napier. There is no doubt that their discovery&#xA;was the greatest mathematical discovery of that time.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-04-01T13:28:48.167" LastActivityDate="2017-04-01T13:28:48.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5854" PostTypeId="2" ParentId="5852" CreationDate="2017-04-01T15:56:35.720" Score="2" Body="&lt;p&gt;Before Newton, French physics was dominant. The medieval physicists before Galileo and Newton who made the biggest contributions to physics (e.g., &lt;a href=&quot;https://hsm.stackexchange.com/a/2147/232&quot;&gt;Oresme and Buridan&lt;/a&gt;) were Parisian, although there there were British ones like &lt;a href=&quot;https://hsm.stackexchange.com/a/2147/232&quot;&gt;Bradwardine&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the relationship between French and English physics, see &lt;a href=&quot;https://books.google.com/books?id=5mVPK7QBdTkC&amp;amp;pg=PA55&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 55ff. of Duhem's &lt;em&gt;Aim &amp;amp; Structure of Physical Theory&lt;/em&gt;&lt;/a&gt; or his article &quot;&lt;a href=&quot;https://books.google.com/books?id=UofBybolmREC&amp;amp;pg=PA50#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The English School and Physical Theory&lt;/a&gt;.&quot; &lt;a href=&quot;http://www.thomist.org/jourl/1968%2C%20vol.%2032/Jan/1968%20Jan%20A%20Wallace.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;This article&lt;/a&gt; discusses the transition of physics manuals before Newton, which focused more on Cartesian physics (with his vortex theory, etc.), to those that discussed Newtonian physics.&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2017-04-01T15:56:35.720" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5857" PostTypeId="1" AcceptedAnswerId="5858" CreationDate="2017-04-02T14:34:59.397" Score="1" ViewCount="216" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Titius-Bode rule&lt;/a&gt;'s fit to the solar system was a bit clunky at best, and it was not really testable in its day. It could not have been used to predict something else, and then that prediction tested. In that sense the whole thing was not in keeping with modern standards of what might be expected to refer to something as a law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And yet now it is very often called &quot;Bode's Law&quot; by many people. I remember reading about it in several (now considered 'old') introductory astronomy books, possibly even grade school text books. Even if it wasn't given any scientific weight, I always remember it being called a &quot;law&quot;, rather than a rule.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the Rydberg formula had some predictive ability, and yet it's not commonly referred to as a law or a rule, just a formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How and when did the Titius-Bode rule first become commonly known as a &quot;law&quot;? Is there a first instance of it being promoted to &quot;law&quot; status?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;note:&lt;/em&gt; As an aside, HSMSE has one instance of reference, and &lt;a href=&quot;https://hsm.stackexchange.com/a/2062/3869&quot;&gt;Titius-Bode rule&lt;/a&gt; is used. &lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-04-02T15:04:52.270" Title="How and when did the Titius-Bode rule first become known as a &quot;law&quot;?" Tags="|astronomy|scientific-method|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5858" PostTypeId="2" ParentId="5857" CreationDate="2017-04-02T15:04:52.270" Score="2" Body="&lt;p&gt;If you read carefully the Wikipedia article on which you refer, you find that&#xA;Titius-Bode law predicted the size of the orbit of Ceres and Uranus. This explains why it was considered a law, before Neptune (which does not obey the &quot;law&quot;) was discovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yes, it is a purely empirical &quot;law&quot;, which eventually turned out to be incorrect.&#xA;But the same applies to Kepler's 3-d law, which was also a purely empirical law, just a pattern, until satellites of Jupiter were discovered. Same applies to many other things like Balmer's series, which in the beginning was a pure numerical pattern which just 4 numbers obey. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-04-02T15:04:52.270" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5859" PostTypeId="1" CreationDate="2017-04-02T23:24:13.317" Score="4" ViewCount="259" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/British_flag_theorem&quot; rel=&quot;noreferrer&quot;&gt;British Flag Theorem&lt;/a&gt; is a fancy name for a law relating distances from the corners of a rectangle to an arbitrary point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The wikipedia article is small and has no history section. Could not find it on google either. When was this theorem first discovered or proven, and by whom?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2017-04-02T23:24:13.317" Title="When was the British Flag Theorem discovered or proven?" Tags="|mathematics|euclidean-geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5860" PostTypeId="1" AcceptedAnswerId="5863" CreationDate="2017-04-03T01:45:42.430" Score="22" ViewCount="3499" Body="&lt;p&gt;Alexander Grothendieck developed the theory of &quot;locally trivial coverings spaces for rings/schemes&quot; in SGAI as an analog to the theory of covering spaces in algebraic topology. He called such coverings étale morphisms; does anyone know why? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Étale translates to something like; spread out, or slack. But there are also a number of other possible translations. None of which seem to suit the apparent properties of étale. Is there a manner in which we can think of étale morphisms which make the name &lt;em&gt;obvious&lt;/em&gt;? &lt;/p&gt;&#xA;" OwnerUserId="2958" LastEditorUserId="4251" LastEditDate="2021-03-02T16:21:32.070" LastActivityDate="2021-03-08T05:50:52.463" Title="Why are étale morphisms called &quot;étale&quot;?" Tags="|mathematics|terminology|algebraic-geometry|grothendieck|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5861" PostTypeId="2" ParentId="3824" CreationDate="2017-04-03T05:44:33.063" Score="1" Body="&lt;p&gt;Taken literally, the answer is that it actually does, i.e. the handcrafted diploma given to Einstein mentions the theory general of relativity in a bizarre caveat: …independent of the value that (after eventual confirmation) may be credited to the relativity and gravitation theory.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be interpreted in different ways. One is that the committee points out that the theory (or perhaps theories – both the special and general versions of relativity) was a possibility for a future prize. However, according to the “Nobel prize official web site”, there are 62 nominations up to and including 1922, not a single one after. &lt;/p&gt;&#xA;" OwnerUserId="5602" LastActivityDate="2017-04-03T05:44:33.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5863" PostTypeId="2" ParentId="5860" CreationDate="2017-04-04T00:57:09.497" Score="9" Body="&lt;p&gt;The use of étale predates SGA, and &quot;spread out&quot; fits Grothendieck’s idea of all-encompassing topos, &quot;vast&quot; and &quot;slack&quot;, better than usual, as these things go. The name of étale morphisms derives from that. Also, French étaler comes from old French &lt;a href=&quot;https://en.wiktionary.org/wiki/estal#Old_French&quot; rel=&quot;noreferrer&quot;&gt;estal&lt;/a&gt;, which meant position/place, same as Greek &lt;a href=&quot;https://en.wiktionary.org/wiki/topos&quot; rel=&quot;noreferrer&quot;&gt;topos&lt;/a&gt;, although it is unclear if that was intended too. See &lt;a href=&quot;https://mathoverflow.net/questions/96264/what-are-the-benefits-of-viewing-a-sheaf-from-the-espace-%C3%A9tal%C3%A9-perspective&quot;&gt;What are the benefits of viewing a sheaf from the “espace étalé” perspective?&lt;/a&gt; thread on Math Overflow for a more mathematical discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is Grothendieck’s own explanation from &lt;a href=&quot;http://www.fermentmagazine.org/rands/recoltes1.html&quot; rel=&quot;noreferrer&quot;&gt;Récoltes et Semailles&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The crucial thing here, from the viewpoint of the Weil conjectures, is that the new notion&lt;/em&gt; [of space] &lt;em&gt;is vast enough, that we can associate to each scheme a “generalized space” or “topos” (called the “étale topos” of the scheme in question). Certain “cohomology invariants” of this topos (“childish” in their simplicity!) seemed to have a good chance of offering “what it takes” to give the conjectures their full meaning, and (who knows!) perhaps to give the&#xA;  means of proving them.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The idea was part of Grothendieck's general strategy for proving the Weil conjectures, which Serre explained to him in cohomological terms in 1955. Étale covers were inspired by Serre's &quot;isotrivial covers&quot;. As &lt;a href=&quot;http://www.landsburg.com/grothendieck/mclarty1.pdf&quot; rel=&quot;noreferrer&quot;&gt;McLarty's comments in The Rising Sea&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Cohomology gives algebraic invariants of a topos, just as it used to give invariants of a topological space. Each topological space determines a topos with the sheaf cohomology. Each group determines a topos with the group cohomology. The same, Grothendieck knew, would work for cases yet unimagined... For the Weil conjectures it only remained to find the natural topos for each arithmetic space—recalling that up to 1956 or so the spaces themselves were not adequately defined. In fact this conception of “toposes” came to Grothendieck as the way to combine his theory of schemes with Serre’s idea of isotrivial covers and produce the cohomology.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This strategy was put in motion in collaboration with Artin in the early 1960-s, the time of SGA, see &lt;a href=&quot;http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf&quot; rel=&quot;noreferrer&quot;&gt;Jackson's As If Summoned from the Void&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;When Grothendieck came to Harvard in 1961, “I asked him to tell me the definition of étale cohomology,” Artin recalled with a laugh. The definition had not yet been formulated precisely. Said Artin, “Actually we argued about the &#xA;  definition for the whole fall”. After moving to the Massachusetts Institute of&#xA;  Technology in 1962, Artin gave a seminar on étale cohomology. He spent much of the following two years at the IHÉS working with Grothendieck.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2017-04-13T12:58:31.857" LastActivityDate="2017-04-05T19:22:54.160" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5864" PostTypeId="1" CreationDate="2017-04-04T02:28:36.760" Score="1" ViewCount="228" Body="&lt;p&gt;In almost all Quantum Field Theories textbooks the same approach to quantization is presented as the first example: one considers the scalar real Klein-Gordon field $\phi$ and just write it as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\phi(x) =\int_{} \dfrac{d^3 p}{(2\pi^3)\sqrt{2\omega_p}} (a(p) e^{-i p_\mu x^\mu}+a^\dagger(p) e^{i p_\mu x^\mu})$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;being $a(p),a^\dagger(p)$ operators satisfying $[a(p),a^\dagger(q)]=(2\pi)^3\delta(p-q)$ and $[a(p),a(q)]=[a^\dagger(p),a^\dagger(q)]=0$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then textbooks continue by stating that $a(p),a^\dagger(q)$ are annihilation and creation operators in a Fock Space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Putting this all into a more organized way, what one has done here is: considering one wants to make classical fields turn to quantum fields obeying the canonical commutation relations, one possible approach is this one where the Hilbert space turns out to be a Fock Space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It turns out that this is not the only way. Unlike QM, there are infinitely many non equivalent representations of the CCR.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question here is: what is the history of &lt;em&gt;this specific approach based on the Fock Space?&lt;/em&gt; How Physicists first derived it? How it was discovered that one approach to achieve a representation of the CCR for a system of uncountably many degrees of freedom, i.e., a field, was this one based on the Fock Space? What was the first derivations of the Fock Space itself that appears in this aproach?&lt;/p&gt;&#xA;" OwnerUserId="4072" LastEditorUserId="4072" LastEditDate="2017-04-11T19:57:10.463" LastActivityDate="2018-04-04T11:12:07.223" Title="History of the fock space based quantization of fields" Tags="|theoretical-physics|quantum-mechanics|quantum-field-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5865" PostTypeId="1" CreationDate="2017-04-04T13:39:21.700" Score="3" ViewCount="250" Body="&lt;p&gt;What was the motivation to use the word &quot;kernel&quot; in algebra to denote the set of all arguments which are mapped to the idendity element (by a homomorphism)? Who introduced it?&lt;/p&gt;&#xA;" OwnerUserId="1772" LastActivityDate="2017-04-05T13:28:56.800" Title="Why is the term &quot;kernel&quot; used in algebra?" Tags="|terminology|abstract-algebra|linear-algebra|" AnswerCount="1" CommentCount="0" ClosedDate="2017-04-21T19:36:54.767" ContentLicense="CC BY-SA 3.0" />
  <row Id="5866" PostTypeId="2" ParentId="5865" CreationDate="2017-04-04T15:04:51.877" Score="2" Body="&lt;p&gt;Source: &lt;a href=&quot;http://jeff560.tripod.com/k.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathwords&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;KERNEL ...&#xA;  The use of kernel in algebra appears to be unrelated to its use in integral equations and Fourier analysis. The OED gives the following quotation from Pontrjagin’s Topological Groups i. 11 (translated by E. Lehmer 1946) &quot;The set of all the elements of the group G which go into the identity of the group G* under the homomorphism g is called the kernel of this homomorphism.&quot;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Kern&lt;/strong&gt; used this way (the set of elements of a group mapped to the identity by a homomorphism) is in my German-language copy of van der Waerden &lt;em&gt;Algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ziegler points out some earlier uses &lt;a href=&quot;https://hsm.stackexchange.com/a/5867/229&quot;&gt;HERE&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2017-04-13T12:51:33.527" LastActivityDate="2017-04-05T13:28:56.800" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5867" PostTypeId="2" ParentId="2149" CreationDate="2017-04-05T09:41:41.787" Score="9" Body="&lt;p&gt;The quoted 1946 English edition of Pontryagin's 1938 book is not the first appearance of &lt;em&gt;kernel&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=gl4nAQAAIAAJ&amp;amp;q=kernel&quot; rel=&quot;nofollow noreferrer&quot;&gt;It's already&lt;/a&gt; in the first English edition (&lt;a href=&quot;https://books.google.com/books?id=gl4nAQAAIAAJ&amp;amp;q=1939&quot; rel=&quot;nofollow noreferrer&quot;&gt;1939&lt;/a&gt;). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's earlier in e.g. 1938 papers of &lt;a href=&quot;http://www.jstor.org/stable/43412288&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. H. C. Whitehead (p. 703)&lt;/a&gt; and &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1546057&quot; rel=&quot;nofollow noreferrer&quot;&gt;Montgomery-Zippin (p. 364)&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The original German &lt;em&gt;Kern&lt;/em&gt; is in P. Alexandroff and H. Hopf, &lt;a href=&quot;https://books.google.com/books?id=8WWtBgAAQBAJ&amp;amp;pg=PA557&amp;amp;q=Kern#v=onepage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Topologie I (1935, p. 557)&lt;/a&gt;, and as Jan Peter comments below, in &lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?57.0717.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pontryagin (1931, pp. 186, 189, 202)&lt;/a&gt; – with p. 186 sounding like he’s introducing the word:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;sup&gt;28&lt;/sup&gt;) Wenn eine Gruppe $A$ auf eine Gruppe $B$ homomorph abgebildet ist, so heißt die Untergruppe von $A$, die aus allen Elementen besteht, welche auf das Einheits- (Null-) element von $B$ abgebildet werden, der &lt;strong&gt;&lt;em&gt;Kern&lt;/em&gt;&lt;/strong&gt; der homomorphen Abbildung.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;(Still, it seems to have come up in group theory first, and only later in linear algebra or ring theory.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-08-26T09:03:57.947" LastActivityDate="2017-08-26T09:03:57.947" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5868" PostTypeId="1" AcceptedAnswerId="5870" CreationDate="2017-04-05T22:05:23.830" Score="2" ViewCount="1253" Body="&lt;p&gt;I am setting out to write a paper, the thesis of which only needs to have the property that it addresses &quot;how was mathematics used in Ancient Greece&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have begun my own investigation (i.e. excessive googling) to see what I can find, but was curious if any users of this community are already familiar with a source or sources that address interesting aspects $X$ of Ancient Greece with the property that one could argue $X$ was an application of mathematical ways of thinking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some first thoughts I had were that if any form of trade and barter was occurring, an argument could be presented that this required some informal notion of solving elementary algebraic equations or arithmetic. Or perhaps if any form of architecture was taking place, an argument could be presented that some informal notions of geometry / spatial relations may have been being applied. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As you can see, there are many stances one could take, and a lot of leeway in how I argue that aspect $X$ of Ancient Greece was 'using mathematics'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to the fact that I do not know much about the history of Ancient Greece, I really don't know what types of things may have been occurring in their culture. Nor do I know where their notion or understanding of mathematics may have been during that time period. Of course, learning and researching both of these is part of the journey of writing this paper - so I am not expecting anyone to answer these for me, I am only hoping or inquiring as to whether anyone can point me towards some niche areas of the internet that address these topics, perhaps someone has something that may not come up in my google searching. If anyone has any sources or insight relating to the use of mathematics in Ancient Greece, that would be most appreciated! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Addendum: I did not even know enough history when I wrote the question to know how ambiguous the term 'Ancient Greece' was. The more specific time period I need to address  is Classical Greece, or approximately around 500 - 200 BCE. My instructor is most likely is intending us to address instances before Euclid's formalization. &lt;/p&gt;&#xA;" OwnerUserId="2469" LastEditorUserId="2469" LastEditDate="2017-04-07T19:50:20.983" LastActivityDate="2017-04-07T19:50:20.983" Title="Instances of the use and application of mathematics in Ancient Greece" Tags="|ancient-greece|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5869" PostTypeId="1" CreationDate="2017-04-06T03:07:07.757" Score="0" ViewCount="384" Body="&lt;p&gt;The question is in the title. I'm obviously refering to the 17/18 th century somewhere around this era.&lt;/p&gt;&#xA;" OwnerUserId="2310" LastActivityDate="2017-04-06T18:36:45.193" Title="Why was the question of the nature of light limited to the wave theory or the particle one ?" Tags="|physics|theoretical-physics|light|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5870" PostTypeId="2" ParentId="5868" CreationDate="2017-04-06T04:24:38.983" Score="2" Body="&lt;p&gt;You should specify more precisely what you mean by &quot;Ancient Greece&quot;.&#xA;Serious applications begin in the Hellenistic period (after Alexander's conquests).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main use of mathematics was in astronomy. Hellenistic astronomy was based on Euclid's geometry, and trigonometry was invented specially for the use in astronomy. Mathematics was also widely used in engineering: in particular in constructions of artillery, several Greek treatises on artillery survive,&#xA;and in geodesy (surveying).&#xA;Also in geography (I mean cartographic projections, determination of the size&#xA;of the Earth, determination of geographical coordinates).&#xA;Construction of sundials and water clocks also should be mentioned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hellenistic Greeks also had developed physics (statics, hydraulics, pneumatics, optics)&#xA;which used sophisticated mathematics. This physics was used in engineering applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A sophisticated mechanical computer for astronomical calculations exists in a museum in Athens, probably of &#xA;the beginning of the 2-nd century b.c. It is called Antikythera Mechanism (see Wikipedia).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of the earlier period (proper &quot;Ancient Greece&quot;) the evidence of mathematics and its applications is not so abundant, but there are some remarkable&#xA;pieces of indirect evidence of the use of mathematics in engineering. I mean the tunnel of Samos built in 530 bc. The tunnel exists: it was excavated in 1882. The remarkable feature is that is goes under the mountain (Castro mountain), and it was dug simultaneously from both ends! You cannot see the beginning of the tunnel from its end; the view is obscured by the mountain. Such an engineering feat requires quite sophisticated measurement which would be impossible without mathematics. The tunnel is described, for example in van der Waerden's book, Science Awakening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Since of all examples you seem to be interested only in this tunnel, I cite van er Waerden. It is 1 km long, and in the middle, where the diggers met, it is seen&#xA;how much error did they make: it is 3 meters in vertical direction and 10 meters&#xA;in horizontal direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar tunnels (dug from both ends) exist in other parts of the world, for example in Jerusalem (700 bc): but the technology is much more primitive: they just made frequent vertical wells to the surface to see where they are! And the resulting tunnel has a zig-zag shape. In the Samos case this method was impractical because of the mountain above the tunnel. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-04-06T16:21:20.317" LastActivityDate="2017-04-06T16:21:20.317" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5871" PostTypeId="2" ParentId="5869" CreationDate="2017-04-06T18:36:45.193" Score="1" Body="&lt;p&gt;A possible simple answer is that experiments on light manifested only the wave aspect of light behaviour or the particle one. These two aspects were not compatible, in the sense that light behaved as waves or behaved like particle depending on the nature of the experiment done. Heisemberg attributed this duality to the insufficience of our language to capture the real sense of the phenomena. By his point of view both aspects of light behaviour have to be treated as two different &quot;representation&quot; of the same phenomenon. More explicitly He remind us that already the category to which the terms &quot;particle&quot; and &quot;wave&quot; belong is the category of &quot;a model&quot;! He underlines that the products of radioactive emission going trough a Wilson Chamber shows themselves via a hyperfine-line of micro-droplets of condensend water supporting the &quot;saying&quot; that the radiation manifest itself as made of small particles moving at high velocity. On the contrary in the experiment of Davisson, Germer, Rupp, where one assume the particulate nature of radiation, since they also are products of redioactive decay, as a matter of fact they behave like waves giving rise to interference shapes similar to the one already known made by the X-Ray. In this case is appropriate &quot;saying&quot; that particulate radiation (see the source: radioactive decay) are indeed wave propagations. .... Heisember concludes that: &quot;Matter cannot behave at once as particles and as waves and that the two &quot;representations&quot; are deeply diferent. The solution of this problem must be found in the sense that both are &quot;models&quot; i.e., they are simply similarities, that sometimes seams to validly apply, and in other occasions just fail. ... For instance, about the electron is simply proven in certain experiments that it behaves like a particle, but there is nothing saying that the electron possess all properties of the corpuscular model. &quot;Mutatis mutandis&quot;, one can say the same thing about the undulatory model.&quot;. Up to now we dealt whith what we can say intuitively and using ordinary language. Heisemberg will take this, in accord with Bohr, as a starting point of the criticism of such level of knolwdge in order to found a proper logic reduction of the model theory and of the introduced analogous concepts.This in order to gain a mathematical coherent axiomatic from wich build the description of the new and unique nature of these phenomena. (from &quot;Die physikalischen Prinzipien der Quantentheorie&quot; 1930, in Italian &quot;I principi fisici della teoria dei quanti&quot; the text I used, and translate the best I could). I hope it can be of some use :-) ... &lt;/p&gt;&#xA;" OwnerUserId="5617" LastActivityDate="2017-04-06T18:36:45.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5873" PostTypeId="2" ParentId="5860" CreationDate="2017-04-07T00:15:33.670" Score="21" Body="&lt;p&gt;From &lt;a href=&quot;https://www.jmilne.org/math/CourseNotes/lec.html&quot; rel=&quot;noreferrer&quot;&gt;Milne's site&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are two different words in French, &amp;quot;étaler&amp;quot;, which means spread out or displayed and is used in&#xA;&amp;quot;éspace étalé&amp;quot;, and &amp;quot;étale&amp;quot;, which is rare except in poetry.&#xA;According to Illusie, it is the second that Grothendieck chose for étale morphism.&#xA;The Petit Larousse defines &amp;quot;mer étale&amp;quot; as &amp;quot;mer qui ne monte ni ne descend&amp;quot;, i.e., the sea at the point of high or low&#xA;tide.  For example, there is the quote from Hugo which I included in my book&#xA;&amp;quot;La mer était étale, mais le reflux commencait a se sentir&amp;quot;.&#xA;I think Grothendieck chose the word because the way he pictured étale&#xA;morphisms reminded him of a calm sea at high tide under a full moon&#xA;(locally almost parallel bands of light, but not globally). I find&#xA;this image beautiful. A footnote in Mumford's Red Book on Algebraic&#xA;Geometry says: &amp;quot;The word apparently refers to the appearance of the&#xA;sea at high tide under a full moon in certain types of weather.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Milne also linked this &lt;a href=&quot;https://yoganorthvancouver.com/wp-content/uploads/2018/06/transform-heal.jpg&quot; rel=&quot;noreferrer&quot;&gt;image&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5619" LastEditorUserId="6545" LastEditDate="2020-08-23T18:01:25.553" LastActivityDate="2020-08-23T18:01:25.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5874" PostTypeId="1" CreationDate="2017-04-08T13:34:25.133" Score="4" ViewCount="1811" Body="&lt;p&gt;In part 6 of his lecture series &quot;Character of Physical Law&quot;, Richard Feynman remarks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A philosopher once said, &quot;It is necessary for the very existence of science that the same conditions always produce the same results.&quot; Well, they don't!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;source: &lt;a href=&quot;https://youtu.be/aAgcqgDc-YM?t=3114&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who is the philosopher he refers to / cites?&lt;/p&gt;&#xA;" OwnerDisplayName="delete000" LastEditorUserId="55" LastEditDate="2017-04-11T05:08:22.593" LastActivityDate="2023-08-23T09:35:03.043" Title="Who is the philosopher Feynman cites as saying that existence of science requires the same conditions to produce the same results?" Tags="|physics|philosophy-of-science|quote|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5875" PostTypeId="1" CreationDate="2017-04-08T13:56:41.397" Score="0" ViewCount="325" Body="&lt;p&gt;This is not really a physical question, but it relates to notation in QED.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ${\gamma}^5$ matrix is defined as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$${\gamma}^5=i{\gamma}^0 {\gamma}^1{\gamma}^2{\gamma}^3$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wouldn't it be more logical to write ${\gamma}^5$ as ${\gamma}^4$?&lt;/p&gt;&#xA;" OwnerUserId="4120" OwnerDisplayName="descheleschilder" LastEditorUserId="4251" LastEditDate="2021-08-20T03:45:43.703" LastActivityDate="2021-09-19T04:07:30.690" Title="Why isn't the ${\gamma}^5$ matrix called ${\gamma}^4$?" Tags="|physics|terminology|quantum-field-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="5876" PostTypeId="2" ParentId="5706" CreationDate="2017-04-08T18:33:16.103" Score="3" Body="&lt;p&gt;A paper of Kneser (&lt;a href=&quot;https://archive.org/stream/mathematischean21behngoog#page/n95&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;) strongly suggests that the idea does (indeed) go back to &lt;strong&gt;Cauchy&lt;/strong&gt;, in connection with Sturm-Liouville problems (i.e. ordinary differential operators, as opposed to the Laplacian in the body of your question). Given functions &lt;span class=&quot;math-container&quot;&gt;$g,k,l$&lt;/span&gt; and writing &lt;span class=&quot;math-container&quot;&gt;$\smash{L=\frac d{dx}\left(k\frac{d}{dx}\,\cdot\right) - l},$&lt;/span&gt; Kneser considers the [for us: “eigenvalue”] problem&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;LV + rgV=0&#xA;$$&lt;/span&gt;&#xA;with boundary conditions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left[k\frac{dV}{dx}-hV\right](0)=0,\qquad&#xA;\left[k\frac{dV}{dx}+HV\right](X)=0.&#xA;\tag1&#xA;$$&lt;/span&gt;&#xA;Addressing the Sturm-Liouville (&lt;a href=&quot;http://www.archive.org/stream/s1journaldemat02liou#page/220&quot; rel=&quot;nofollow noreferrer&quot;&gt;1837&lt;/a&gt;) question whether any &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; can be expanded into a series of solutions &lt;span class=&quot;math-container&quot;&gt;$V_\nu$&lt;/span&gt; belonging to [“eigenvalues”] &lt;span class=&quot;math-container&quot;&gt;$r_\nu$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) = A_1V_1+A_2V_2+\cdots,&#xA;\tag3&#xA;$$&lt;/span&gt;&#xA;Kneser writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The particular analytic developments I use for this are inspired or drawn from relevant work of Dini (&lt;a href=&quot;https://archive.org/details/seriedifourierea00diniuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;1880&lt;/a&gt;), Harnack (&lt;a href=&quot;http://www.archive.org/stream/berichteberdiev31klasgoog#page/n658/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;1887&lt;/a&gt;), Poincaré (&lt;a href=&quot;https://archive.org/stream/rendicontidelci11palegoog#page/n71&quot; rel=&quot;nofollow noreferrer&quot;&gt;1894&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/thorieanalytiq00poin&quot; rel=&quot;nofollow noreferrer&quot;&gt;1895&lt;/a&gt;) and Stekloff (&lt;a href=&quot;http://www.numdam.org/item?id=AFST_1901_2_3_3_281_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901&lt;/a&gt;); but the basic idea can be explained as follows.&lt;/p&gt;&#xA;&lt;p&gt;These recent authors all use a device introduced by Cauchy (&lt;a href=&quot;https://books.google.com/books?id=w-741ZwVBnAC&amp;amp;pg=PA341&quot; rel=&quot;nofollow noreferrer&quot;&gt;1827&lt;/a&gt;) in his study of Fourier series: they build &lt;em&gt;&lt;strong&gt;a function of a complex variable &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; containing &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as a parameter, having poles at &lt;span class=&quot;math-container&quot;&gt;$r=r_\nu$&lt;/span&gt; as its only singularities&lt;/strong&gt;&lt;/em&gt;, and producing as residues the corresponding terms of the series &lt;span class=&quot;math-container&quot;&gt;$(3)$&lt;/span&gt;. Poincaré, apparently, first pointed out [I guess here: (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3074r/f385&quot; rel=&quot;nofollow noreferrer&quot;&gt;1894&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/thorieanalytiq00poin#page/309&quot; rel=&quot;nofollow noreferrer&quot;&gt;1895&lt;/a&gt;)] that Cauchy's auxiliary function is the solution of the equation [of the “resolvent” &lt;span class=&quot;math-container&quot;&gt;$\smash{(L+rg)^{-1}}$&lt;/span&gt;]&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;LV+rgV+f(x)=0&#xA;$$&lt;/span&gt;&#xA;satisfying conditions &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While this early literature is no easy reading, the treatises of Picard (&lt;a href=&quot;https://archive.org/stream/traitdanalyse11picagoog#page/n187&quot; rel=&quot;nofollow noreferrer&quot;&gt;1893&lt;/a&gt;, pp. 167-183), Poincaré (&lt;a href=&quot;https://archive.org/stream/thorieanalytiq00poin#page/210&quot; rel=&quot;nofollow noreferrer&quot;&gt;1895&lt;/a&gt;, pp. 210-223) and Watson (&lt;a href=&quot;https://archive.org/stream/treatiseontheory00watsuoft#page/576&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;, pp. 576-617) have chapter-long expositions of what may be the first three cases historically:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(g,k,l)=(1,1,0)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[0,\pi]$&lt;/span&gt; with Dirichlet boundary conditions. Then &lt;span class=&quot;math-container&quot;&gt;$V_\nu=\sin(\nu x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\smash{r_\nu=\nu^2}$&lt;/span&gt;, and (3) is a Fourier sine series. Or Neumann conditions, &lt;span class=&quot;math-container&quot;&gt;$V_\nu=\cos(\nu x)$&lt;/span&gt;, and cosine series; which Picard (p. 177) and Poincaré (p. 220) attribute to Cauchy (&lt;a href=&quot;https://books.google.com/books?id=w-741ZwVBnAC&amp;amp;pg=PA364&quot; rel=&quot;nofollow noreferrer&quot;&gt;1827&lt;/a&gt;, pp. 364-365).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(g,k,l)=(1,1,0)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; with Fourier's sphere cooling conditions (&lt;a href=&quot;https://archive.org/stream/thorieanalytiq00four#page/340&quot; rel=&quot;nofollow noreferrer&quot;&gt;1822&lt;/a&gt;, pp. 340-342): &lt;span class=&quot;math-container&quot;&gt;$V(0)=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$V(1)=AV'(1)$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$A&amp;gt;1$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$V_\nu=\sin(k_\nu x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\smash{r_\nu=k_\nu^2}$&lt;/span&gt; where the &lt;span class=&quot;math-container&quot;&gt;$k_\nu$&lt;/span&gt; are the positive solutions of &lt;span class=&quot;math-container&quot;&gt;$\tan(k)=Ak$&lt;/span&gt;. So (3) is a “nonharmonic Fourier series,” which Picard (pp. 178-183) and Poincaré (pp. 168-179, 220-223) attribute to Cauchy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(g,k,l)=(x,x,a^2/x)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$V(1)=0$&lt;/span&gt;, and no condition at the singular endpoint &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$V_\nu=J_a(k_\nu x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\smash{r_\nu=k_\nu^2}$&lt;/span&gt; where the &lt;span class=&quot;math-container&quot;&gt;$k_\nu$&lt;/span&gt; are the roots of the Bessel function &lt;span class=&quot;math-container&quot;&gt;$J_a$&lt;/span&gt;. So (3) is a “Fourier-Bessel series,”  which Watson (pp. 582-591) attributes to Schläfli (&lt;a href=&quot;https://books.google.com/books?id=faYEAAAAQAAJ&amp;amp;pg=PA137&quot; rel=&quot;nofollow noreferrer&quot;&gt;1876&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note added:&lt;/strong&gt; An earlier (and perhaps clearest) statement by Cauchy occurs in his &lt;em&gt;Application du calcul des résidus à l’intégration des équations différentielles linéaires et à coefficients constants&lt;/em&gt; (Exercices de mathématiques &lt;strong&gt;1&lt;/strong&gt; (&lt;a href=&quot;https://archive.org/stream/bub_gb_BmOo00FGCuAC#page/n209&quot; rel=&quot;nofollow noreferrer&quot;&gt;1826&lt;/a&gt;) 202-204 = Œuvres (2) &lt;strong&gt;6&lt;/strong&gt; (&lt;a href=&quot;https://archive.org/stream/oeuvresdaugusti206caucrich#page/252&quot; rel=&quot;nofollow noreferrer&quot;&gt;1887&lt;/a&gt;) 252-255):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider first the task of integrating the differential equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d^ny}{dx^n} + a_1&#xA;\frac{d^{n-1}y}{dx^{n-1}} + a_2&#xA;\frac{d^{n-2}y}{dx^{n-2}} + \ldots + a_{n-1}&#xA;\frac{dy}{dx} + a_ny=0,&#xA;\tag1&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$a_1, a_2,\dots a_{n-1}, a_n$&lt;/span&gt; denote constant coefficients; and let, for short&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;F(r) = r^n + a_1r^{n-1} + a_2r^{n-2}+\dots+ a_{n-1}r + a_n.&#xA;\tag2&#xA;$$&lt;/span&gt;&#xA;It is clear that, to satisfy the equation (1), it will suffice to take&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;y = \raise{-1ex}{\huge{\mathcal E}}\,\frac{\varphi(r)\,e^{rx}}{((\,F(r)\,))},&#xA;\tag3&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\varphi(r)$&lt;/span&gt; denotes any function of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; which does not become infinite for values of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; that verify the formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;F(r)=0.&#xA;\tag4&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Of course, this isn’t yet framed in terms of &lt;em&gt;eigenvalues&lt;/em&gt; of the differential operator. It becomes so if we replace &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$a_n -\lambda$&lt;/span&gt;, but Cauchy didn’t call (4) “characteristic equation” until &lt;a href=&quot;https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;1839&lt;/a&gt;, and names for its roots seem to have come even later — I’m not sure when.)&lt;/p&gt;&#xA;&lt;p&gt;Also, for symmetric operators (or quadratic forms) on &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^n$&lt;/span&gt; this is all in Weierstrass (&lt;a href=&quot;https://archive.org/details/berichtberdiez58berl/page/207&quot; rel=&quot;nofollow noreferrer&quot;&gt;1859&lt;/a&gt;), cf. p. 219.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-07T03:45:59.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="5879" PostTypeId="1" AcceptedAnswerId="5880" CreationDate="2017-04-09T16:34:53.943" Score="0" ViewCount="526" Body="&lt;p&gt;What role did astronomy play for Isaac Newton? He was a good friend with Edmond Halley the astronomer. Kepler's laws were obviously the foundation for his physics. And he revolutionized optics too, but did he make any systematical astronomical observations, as in for example checking the planet's movements or mapping the surface of the Moon? He isn't famous as an astronomer, but that is maybe just because it is overshadowed by his other accomplishments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KVX4Z.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KVX4Z.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Newton's telescope (replica).&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="3663" LastActivityDate="2017-04-09T22:52:58.613" Title="Did Isaac Newton make astronomical observations?" Tags="|astronomy|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5880" PostTypeId="2" ParentId="5879" CreationDate="2017-04-09T22:38:09.420" Score="2" Body="&lt;p&gt;Yes, he did. See Westfall, Never at rest: a biography of Issac Newton.&#xA;Many Newton's observations of comets are mentioned, including the one which will be later called Halley's comet. He also made some observations to test his telescope. But as an astronomical observer he did not make any great discoveries.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-04-09T22:52:58.613" LastActivityDate="2017-04-09T22:52:58.613" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5881" PostTypeId="2" ParentId="5874" CreationDate="2017-04-10T11:30:52.843" Score="6" Body="&lt;p&gt;Maybe a paraphrase of: David Hume, &lt;a href=&quot;https://books.google.it/books?id=vasuwvLnSYUC&amp;amp;pg=PA38&quot; rel=&quot;noreferrer&quot;&gt;An Enquiry Concerning Human Understanding&lt;/a&gt; (1748), II.21, pages 37-38:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;all inferences from  experience suppose, as their foundation, that the&#xA;  future will resemble the past, and that similar powers will be conjoined&#xA;  with similar sensible qualities. If there be any suspicion that the course of&#xA;  nature may change, and that the past may be no rule for the future, all&#xA;  experience becomes useless, and can give rise to no inference or conclusion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-04-10T11:30:52.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5882" PostTypeId="1" CreationDate="2017-04-10T15:33:26.493" Score="1" ViewCount="63" Body="&lt;p&gt;I am readying on Arago's experiment of 1810 in a book on the history of physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author writes that Arago conducted the experiment assuming that light consisted of particles and that faster light should have a lower refraction angle than slower light. The book states that in that regard, Arago was a follower of Netwon's light theories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can see (assumption of mine), up to this point in time, it had never really been proven that slower light refracts more (let alone that light consisted of particles).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If my assumption is right, why did Arago not just establish or propose a new law that states that the refraction angle of light is independant of the (incoming) light speed?&lt;/p&gt;&#xA;" OwnerUserId="2328" LastActivityDate="2017-04-10T15:33:26.493" Title="Why did Arago in 1810 not assume that the refraction angle is independant on the light speed?" Tags="|light|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5884" PostTypeId="2" ParentId="5874" CreationDate="2017-04-11T05:06:49.777" Score="7" Body="&lt;p&gt;Nobody in particular, it is what is called collective image in literature. Feynman's attitude towards philosophy is expressed by another of his quotes, &lt;a href=&quot;https://themultidisciplinarian.com/2013/11/02/youre-so-wrong-richard-feynman&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;&lt;em&gt;philosophy of science is about as useful to scientists as ornithology is to birds&lt;/em&gt;&quot;&lt;/a&gt;, or &quot;&lt;em&gt;we get into that paralysis of thought that comes to philosophers… one saying to the other: you don’t know what you are talking about! The second one says: what do you mean by ‘talking’? What do you mean by ‘you’? What do you mean by ‘know’?”&lt;/em&gt;&quot;. It seems rather unlikely that he read much of them. The irony is that &lt;a href=&quot;https://philosophynow.org/issues/114/Richard_Feynmans_Philosophy_of_Science&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feynman himself philosophized about science&lt;/a&gt; quite a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;A philosopher once said&quot; is a figure of speech Feynman uses (similar to &quot;a wise man once said&quot;, but with negative connotation) to express what he imagines philosophers might have said. The OP quote is from &lt;a href=&quot;http://www.informationphilosopher.com/solutions/scientists/feynman/probability_and_uncertainty.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Character of Physical Law, Chapter 6&lt;/a&gt;. In &lt;a href=&quot;http://evolvingthoughts.net/2011/09/more-feynman-on-philosophers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lectures on Physics, volume I, chapter 2&lt;/a&gt; the quote is attributed to &quot;philosophers&quot; in the plural, and &quot;some or other&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This is a horrible thing; in fact, philosophers have said before that one of the fundamental requisites of science is that whenever you set up the same conditions, the same thing must happen. This is simply not true, it is not a fundamental condition of science. The fact is that the same thing does not happen, that we can find only an average, statistically, as to what happens. Nevertheless, science has not completely collapsed. Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong. For example, some philosopher or other said it is fundamental to the scientific effort that if an experiment is performed in, say, Stockholm, and then the same experiment is done in, say, Quito, the same results must occur. That is quite false. It is not necessary that science do that; it may be a fact of experience, but it is not necessary.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In fairness, as the context of quantum mechanics makes clear, Feynman refers to the popularity of &lt;a href=&quot;https://plato.stanford.edu/entries/determinism-causal&quot; rel=&quot;nofollow noreferrer&quot;&gt;causal determinism&lt;/a&gt; among philosophers. It goes as far back as Democritus and Stoics, and was supported more recently by Hobbes, Spinoza, Leibniz, Kant, etc. Kant comes to mind in particular because the language of causality, etc., as &lt;a href=&quot;https://plato.stanford.edu/entries/kant-hume-causality&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;conditions of possibility of knowledge&quot;&lt;/a&gt; was his signature way of justifying science. Of course, a great deal of scientists, like Laplace and late 19th century Laplacian determinists, were just as culpable, as are many contemporary scientists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another well known quote of Feynman's is:&quot;&lt;em&gt;A poet once said, &quot;The whole universe is in a glass of wine&quot;&lt;/em&gt;&quot;. There was no such poet either, see &lt;a href=&quot;https://dobianchi.com/2013/07/16/universe-glass-wine-feynman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Do Bianchi's post&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I believe that the imagery comes from a “scientific letter” by Italian philosopher Lorenzo Magalotti (1637-1712) who cites Galileo’s [attributed] maxim, wine is a compound [mixture] of moisture [humor] and light (il vino è un composto di umore e di luce)... This celebrated observation of the physical world was transmitted anecdotally by Galileo’s student Raffaello Magiotti (1597-1656), who is quoted by Magalotti in the letter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-04-11T19:31:01.857" LastActivityDate="2017-04-11T19:31:01.857" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5885" PostTypeId="1" AcceptedAnswerId="5886" CreationDate="2017-04-11T19:11:58.980" Score="17" ViewCount="1266" Body="&lt;p&gt;Einstein's early career is well-known for the lack of success he had applying for assistant lecturer positions with universities; he could not get a position, and he ended up working in a Bern patent office, in large part because his academic record didn't garner attention or respect from those in hiring positions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet in 1905, Einstein's &quot;Annus Mirabilis,&quot; he published 4 separate papers in the peer-reviewed &lt;em&gt;Annalen der Physik&lt;/em&gt; (&lt;em&gt;Annals of Physics&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On the photoelectric effect: received March 18 and published June 9&lt;/li&gt;&#xA;&lt;li&gt;On Brownian motion: received May 11 and published July 18&lt;/li&gt;&#xA;&lt;li&gt;On special relativity: received June 30 and published Sept. 26&lt;/li&gt;&#xA;&lt;li&gt;On mass-energy equivalence: received Sept. 27 and published Nov. 21&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It shocks me that someone with no active academic credentials or track record was able to publish four times in one year in such a prestigious journal.  I have difficulty imagining the same thing happening today.  While reviewers are supposed to review the work in and of itself, I'm sure there's some bias towards academic credentials and standing in the field; Einstein's papers were published regardless.  Even if the reviewers were blinded to paper author, getting beyond the review to actual publication (four times!) seems a challenge.  To me, this publication feat is nearly as impressive as the papers' subject matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;QUESTION 1A:&lt;/strong&gt; Has there been any research/writing on the reviewers of these papers, their initial thoughts, and the process that led to publication?  I would think &lt;em&gt;Annalen der Physik&lt;/em&gt; might still have records of the review process.  Apparently, the reviewers looked beyond Einstein's lack of credentials and radical thinking.  That's magnificent, yet surprising.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;QUESTION 1B:&lt;/strong&gt; Were the papers so cogent that non-publication was inconceivable?&#xA;The papers were groundbreaking proposals for the most part (Lorentz and Poincare had done work leading towards special relativity), and the mass-energy equivalence paper had no empirical (atomic bomb) evidence to substantiate it.&lt;/p&gt;&#xA;" OwnerUserId="5566" LastEditorUserId="55" LastEditDate="2017-04-11T20:57:16.390" LastActivityDate="2017-04-11T21:50:57.617" Title="How did the publication feat of Einstein's four 1905 Annus Mirabilis papers get through peer review?" Tags="|physics|publishing|einstein|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5886" PostTypeId="2" ParentId="5885" CreationDate="2017-04-11T20:55:07.983" Score="16" Body="&lt;p&gt;I am afraid, you are overprojecting the power and the glory of peer review. Although some instances can be cited as early as 17th century, it only became what it is today with the triumph of publish or perish in mid 20th, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/611/how-did-publish-or-perish-become-the-scientific-priority-rule&quot;&gt;How did &amp;quot;publish or perish&amp;quot; become the scientific priority rule?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Concerning Einstein's papers specifically see &lt;a href=&quot;http://theconversation.com/hate-the-peer-review-process-einstein-did-too-27405&quot; rel=&quot;noreferrer&quot;&gt;Hate the peer-review process? Einstein did too&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Academic review process was different in Einstein’s time. In his brilliant career, the only time his work was subjected to blind peer review – the authors don’t know the reviewers and vice versa – he showed contempt for what is now the gold standard of science.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;For instance, the Annalen der Physik, in which Einstein published his four famous papers in 1905, did not subject those papers to the same review process. The journal had a remarkably high acceptance rate (of about 90-95%). The identifiable editors were making the final decisions about what to publish. It is the storied editor Max Planck who described his editorial philosophy as:&#xA;&lt;strong&gt;To shun much more the reproach of having suppressed strange opinions than that of having been too gentle in evaluating them&lt;/strong&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;only time&amp;quot; was in 1935 in the US, which was ahead of the curve on peer review. When Physical Review tried to peer review an Einstein-Rosen paper on gravitational waves it got the following response from Einstein:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We (Mr. Rosen and I) had sent you our manuscript for publication and had not authorised you to show it to specialists before it is printed. I see no reason to address the – in any case erroneous – comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-04-11T21:50:57.617" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5888" PostTypeId="1" AcceptedAnswerId="5889" CreationDate="2017-04-13T02:25:16.487" Score="2" ViewCount="2320" Body="&lt;p&gt;In my multi-variable calculus class we have been learning how to calculate double integrals over regions. To accomplish this task we usually make use of Fubini's Theorem because it relates the double integral to iterated integrals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, upon some research, I found out that Fubini's Theorem is a rather recent result. Wikipedia states that it was formulated in 1907.&#xA;I was under the impression that calculus was formulated in completion at a much earlier date.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So were double integrals over a region even calculated before? If so, how was it accomplished? &lt;/p&gt;&#xA;" OwnerUserId="5645" LastActivityDate="2017-04-13T03:32:06.187" Title="How were double integrals calculated before Fubini's Theorem?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5889" PostTypeId="2" ParentId="5888" CreationDate="2017-04-13T03:00:17.467" Score="9" Body="&lt;p&gt;Double integrals were calculated in the 18th and 19th centuries in the same way they teach you in calculus now, using special cases of what later became Fubini's theorem. Most of these special cases had no special name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fubini's theorem is just a name in honor of a person who&#xA;proved a much more general statement than that which is taught in calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually double and triple integrals where calculated even before the notion of&#xA;integral was formalized, since antiquity, by people like Archimedes and Eudoxus,&#xA;but they had to invent a new argument for each particular integral.&#xA;(They essentially approximated integrals by finite sums and then tried to find the limit. The difficulty is in finding the limit explicitly.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 17th century, Cavalieri's Principle (see Wikipedia) was formulated which helps in evaluating some&#xA;multiple integrals, especially for areas and volumes. Cavalieri principle was still mentioned in high school in 1960s. But modern calculus books prefer to refer to a very general&#xA;Fubini theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you wish to see how triple integrals were evaluated before the notion of integral was formalized (I mean before Newton and Leibniz and Cauchy), I recommend to read the remarkable little book by Kepler, Stereometry of wine barrels. I am not sure whether there is an English translation, but there is&#xA;a detailed exposition in English:&#xA;&lt;a href=&quot;http://www.maa.org/press/periodicals/convergence/kepler-the-volume-of-a-wine-barrel-keplers-inova-stereometria-doliorum-vinariorumi&quot; rel=&quot;noreferrer&quot;&gt;http://www.maa.org/press/periodicals/convergence/kepler-the-volume-of-a-wine-barrel-keplers-inova-stereometria-doliorum-vinariorumi&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-04-13T03:32:06.187" LastActivityDate="2017-04-13T03:32:06.187" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5890" PostTypeId="1" AcceptedAnswerId="11238" CreationDate="2017-04-14T11:46:10.237" Score="12" ViewCount="1253" Body="&lt;p&gt;In his 1950 paper in &lt;em&gt;Mind&lt;/em&gt; titled &quot;Computer Machinery and Intelligence&quot; which introduces the test that now bears his name, Alan Turing starts by describing a game which he calls the &quot;Imitation Game&quot;, which is played between three players, a man (A), a woman (B), and an interrogator (C) of either sex: C's goal is to decide which is a man and which is a woman, A's goal is to fool C in trying to pass as a woman, and B's goal is to assist C in making the correct decision.  Turing then proceeds to discuss a variation (or possibly, variations) of this game in which A is replaced by a machine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question concerns this original game, played between man and woman, without any mention of machines:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Did Turing invent the &quot;imitation game&quot; or was it already around?  Secondarily, did he invent the name &quot;imitation game&quot;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Various pages (such as &lt;a href=&quot;http://www.bbc.com/news/technology-18475646&quot; rel=&quot;noreferrer&quot;&gt;this one&lt;/a&gt;, which describes it as &quot;a Victorian-style competition called the imitation game&quot;) suggest that the game predated Turing, but they could be reproducing the same erroneous source and I found little in way of convincing evidence (the release of a motion picture with that name does not, of course, help with the Googling).  &lt;a href=&quot;https://books.google.com/ngrams/graph?content=imitation+game&amp;amp;year_start=1800&amp;amp;year_end=2015&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Cimitation%20game%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;Google Ngrams suggests&lt;/a&gt; that the term only became common after publication of Turing's paper, but that doesn't prove that he invented it, and even if he did, the game might have been known in some other way.&lt;/p&gt;&#xA;" OwnerUserId="3616" LastEditorUserId="4251" LastEditDate="2020-07-12T09:00:57.360" LastActivityDate="2023-02-28T20:45:41.727" Title="Did Turing invent the imitation game? Did he name it?" Tags="|biographical-details|artificial-intelligence|turing|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5891" PostTypeId="2" ParentId="3014" CreationDate="2017-04-14T15:14:47.060" Score="5" Body="&lt;p&gt;&quot;&lt;em&gt;The way to understand a mathematical problem is to express it in the mathematical world natural to it -that is, in the topos natural to it. Each topos has a natural cohomology, simply taking the category of abelian groups in that topos as the category of sheaves. The cohomology of that topos may solve the problem. In outline:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;1) Find the natural world for the problem (for example, the Etale topos of an arithmetic scheme).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;2) Express the problem cohomologically (state Weil's conjectures as a Lefschetz fixed point theorem).&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;3) The cohomology of that world may solve your problem, like a ripe avocado bursts in your hands.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.landsburg.com/grothendieck/mclarty1.pdf&quot; rel=&quot;noreferrer&quot;&gt;The rising sea, Mclarty&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5613" LastActivityDate="2017-04-14T15:14:47.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5892" PostTypeId="1" CreationDate="2017-04-15T04:10:08.143" Score="3" ViewCount="165" Body="&lt;p&gt;Recently I became curious about when the following ideas came about, and I couldn't really find information about them with some google searches.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^0 = 1$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^{\frac pq} = \sqrt [q] {2^p}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$a^{-x} = \dfrac {1}{a^x}$.&lt;/p&gt;&#xA;" OwnerUserId="4123" LastActivityDate="2017-04-26T16:36:44.377" Title="When was the idea of exponents generalized from &quot;repeated multiplication&quot;?" Tags="|elementary-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5893" PostTypeId="2" ParentId="5892" CreationDate="2017-04-15T12:52:31.153" Score="3" Body="&lt;p&gt;This was essentially done by Napier, see&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/42/which-came-first-the-natural-logarithm-or-the-base-of-the-natural-logarithm/47#47&quot;&gt;Which came first, the natural logarithm or the base of the natural logarithm?&lt;/a&gt;&#xA;and&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/635/how-did-napier-come-to-invent-logarithms/636#636&quot;&gt;How did Napier come to invent logarithms?&lt;/a&gt;&#xA;Napier was talking about the inverse function, but the tables he made can (and have to) be used both ways.&#xA;The formal definition of the exponent for all numbers, even for complex ones is due to Euler, who used power series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Of course, as it always happens, Euler had predecessors.&#xA;It is possible that the first work which discussed real (irrational) exponents&#xA;is by Nicole Oresme, De proportionibus proportionum, written approximately in&#xA;1360. It seems that it has never been fully translated from the Latin,&#xA;but there is a detailed exposition:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;E. Grant,&#xA;  Nicole Oresme and His De Proportionibus Proportionum,&#xA;  Source: Isis, Vol. 51, No. 3 (Sep., 1960), pp. 293-314&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The journal is available on JSTOR.&#xA;Of course, Oresme did not have (and could not possibly have) a precise rigorous definition of irrational exponents, but nevertheless he discusses them, and some of his observations are very clever. In particular he understood that multiples&#xA;of an irrational number are dense modulo 1, I think he was the first to state this clearly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Of course, Euler perfectly understood real numbers, and exponents etc.,&#xA;but strictly speaking the formal definition of a real number was not available until the late 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-04-26T16:36:44.377" LastActivityDate="2017-04-26T16:36:44.377" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5894" PostTypeId="1" CreationDate="2017-04-15T12:55:29.727" Score="1" ViewCount="136" Body="&lt;p&gt;Spacetimes with causality violations go back a bit of a way (at least as far back as Herman Weyl's &quot;Space, Time and Matter&quot; in 1922, followed by the Gödel spacetime in 1949), but one of the simplest example, the timelike cylinder (Minkowski space with the surfaces $t = 0$ and $t = T$ identified) seems to appear fairly late. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first example I can find of it is Reichenbach's 1957 &quot;The Philosophy of Space and Time&quot;, which features the cylinder p. 272. Unfortunately the book isn't big on bibliographical reference so it is hard to determine if this is its first appearance. Does that spacetime appear in any earlier source?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="604" LastEditDate="2023-06-19T13:40:04.420" LastActivityDate="2023-06-19T13:40:04.420" Title="First appearance of the timelike cylinder spacetime" Tags="|physics|relativity-theory|weyl|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5896" PostTypeId="1" CreationDate="2017-04-16T12:35:44.420" Score="4" ViewCount="175" Body="&lt;p&gt;Who, besides Hamilton, was instrumental in discovering at least part of the so-called Hamilton principle?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\delta \int_{a}^{b}L(q,\dot q ,t )dt$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $ L=T-V$, the Lagrangian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What exactly, was Lagrange's role, especially in the part that bears his name?&lt;/p&gt;&#xA;" OwnerUserId="3786" LastEditorUserId="2351" LastEditDate="2017-04-21T11:51:10.077" LastActivityDate="2017-04-21T11:51:10.077" Title="who was the first to discover the Hamilton principle of classical physics?" Tags="|physics|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5897" PostTypeId="2" ParentId="5896" CreationDate="2017-04-16T19:51:09.413" Score="1" Body="&lt;p&gt;Although called Hamilton's Principle, there were a number of mathematical physicists of the day who contributed to the overall work.  Contributions by mathematicians such as Euler, Lagrange, and similar or related work by Maupertius should not be ignored.  Also, powerful add on work was done by Jacobi and Poisson.  Plus, the original work with Calculus of Variations began (I think) with Johann Bernoulli's solution to the Brachistochrone Problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I think that the pure Hamilton's Principle should be written as the stationary action:&#xA;$$&#xA;\frac{\delta S}{\delta q_i(t)} = 0&#xA;$$&#xA;Where the action $S$ is written as you have in your post with the time integral of the Lagrangian.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2017-04-16T19:51:09.413" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5898" PostTypeId="1" CreationDate="2017-04-16T20:38:08.067" Score="2" ViewCount="38" Body="&lt;p&gt;Long before radiometric dating, Earth's history was divided into a timescale with named units based on bands of rocks and fossils. Thus events could be placed in order by superposition. Some time later, we were able to be more quantitative, e.g. stating that the Cretaceous began 145 mya. When were these dates computed and accepted?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2017-04-16T20:38:08.067" Title="When was the geological timescale determined by radiometric dating?" Tags="|nuclear-physics|geology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5899" PostTypeId="2" ParentId="42" CreationDate="2017-04-16T22:47:26.130" Score="5" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Which came first, the natural logarithm or the base of the natural logarithm?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Quick answer: logarithms came before Euler's Number, $e$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Euler's Number, $e$, one of the most important mathematical constants is an irrational number closely &lt;strong&gt;related to growth and rate of change&lt;/strong&gt;. The earliest written observation of number approximate to $e$ was made by J. Bernoulli, around 17-th century, arising from experimenting with the length and number of intervals of compound interest over an initial investment, where he observed a pattern which was later identified by Euler (and Gauss) as we know it Today. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The logarithms were developed, a century earlier (early 1600s) by Napier, as a practical tool for Astronomical calculations &lt;strong&gt;related with the multiplication of large numbers&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around that time (middle 1600s) the concept of &lt;em&gt;function&lt;/em&gt; became relevant together with &lt;em&gt;Calculus&lt;/em&gt;, which is essentially the language of rate of change. Main part of that &lt;em&gt;&quot;language&quot;&lt;/em&gt; is played by $e$ which arises naturally in expressions and functions related to growth. Calculus provided the &lt;em&gt;&quot;platform&quot;&lt;/em&gt; that allowed $e$ to be associated and connected with another mathematical (already existing) branches - geometry (areas under a curve (hyperbole)), trigonometry, etc. leading to a culmination, which is named: &lt;em&gt;&quot;The most beautiful formula.&quot;&lt;/em&gt; (Euler's identity.): $$e^{i \pi} + 1 = 0$$ applicable and helpful in numerous areas in science.&lt;/p&gt;&#xA;" OwnerUserId="2351" LastActivityDate="2017-04-16T22:47:26.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5901" PostTypeId="1" AcceptedAnswerId="5902" CreationDate="2017-04-05T14:28:58.237" Score="3" ViewCount="108" Body="&lt;p&gt;In lecture 12 of Professor &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Searle&quot; rel=&quot;nofollow noreferrer&quot;&gt;John R. Searle&lt;/a&gt;'s U.C. Berkeley class, &amp;quot;Philosophy of Mind&amp;quot;, he remarks &lt;a href=&quot;https://youtu.be/P1wZZPoPfUE?t=18m22s&quot; rel=&quot;nofollow noreferrer&quot;&gt;[18m22s]&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We no longer talk mysteriously about how heredity works. We can actually identify the mechanisms in DNA and RNA molecules. Indeed, according to Francis Crick, we don't really have much use for the notion of a gene anymore because we mostly deal with the molecular biochemistry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have looked for (presumably his later) works and interviews which might support Professor Searle's claim about Crick, and cannot find anything. Can anyone shed some light on this remark? Or, provide a reference citing Crick?&lt;/p&gt;&#xA;" OwnerDisplayName="Mr. Kennedy" LastEditorUserId="4251" LastEditDate="2020-07-14T06:02:40.313" LastActivityDate="2020-07-14T06:02:40.313" Title="Francis Crick and the usefulness of &quot;genes&quot; as compared to molecular biochemistry" Tags="|biographical-details|biology|quote|genetics|biologists|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5902" PostTypeId="2" ParentId="5901" CreationDate="2017-04-05T16:46:36.240" Score="5" Body="&lt;p&gt;I don't know if Crick actually said that, but it seems unlikely given (1) his whole career was focused on elucidating the genetic basis of life and (2) the notion of a gene is still widely invoked in modern biology. It is more likely that he (or someone) said that &lt;em&gt;the notion of a gene has changed with the advent of modern biochemistry&lt;/em&gt;. This is certainly true. I refer you to this profile of Crick titled &lt;a href=&quot;https://profiles.nlm.nih.gov/SC/&quot; rel=&quot;noreferrer&quot;&gt;The Francis Crick Papers&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Researchers working on DNA in the early 1950s used the term &amp;quot;gene&amp;quot; to mean the smallest unit of genetic information, but they did not know what a gene actually looked like structurally and chemically, or how it was copied, with very few errors, generation after generation.&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;Crick and Watson recognized, at an early stage in their careers, that gaining a detailed knowledge of the three-dimensional configuration of the gene was the central problem in molecular biology. Without such knowledge, heredity and reproduction could not be understood.&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;[Watson and Crick] had shown that in DNA, form is function: the double-stranded molecule could both produce exact copies of itself and carry genetic instructions. During the following years, Crick elaborated on the implications of the double-helical model, advancing the hypothesis, revolutionary then but widely-accepted since, that the sequence of the bases in DNA forms a code by which genetic information can be stored and transmitted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Perhaps you could send Dr. Searle an email and get it straight from the horse's mouth.&lt;/p&gt;&#xA;" OwnerDisplayName="canadianer" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-04-05T16:46:36.240" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5903" PostTypeId="2" ParentId="3108" CreationDate="2017-04-17T10:12:58.107" Score="6" Body="&lt;p&gt;To quote from Jordan's Traité des substitutions et des équations algébriques (1870), p. 56 (&lt;a href=&quot;https://archive.org/stream/traitdessubstit00jordgoog#page/n79/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/stream/traitdessubstit00jordgoog#page/n79/mode/2up&lt;/a&gt;): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;§ 67.  A group Γ is called &quot;isomorphic&quot; to another group G if one can establish between their substitutions [i.e., elements] a correspondence such that:  (1)  each substitution of G corresponds to a single substitution of Γ, and each substitution of Γ [corresponds] to one or several substitutions of G ; (2) the product of any two substitutions of G corresponds to the product of their corresponding [substitutions in Γ ], respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The isomorphism will be called &quot;mériédrique&quot; if several substitutions of G correspond to only one substitution of Γ, &quot;holoédrique&quot; in the contrary case.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus an &quot;isomorphisme mériédrique&quot; corresponds to a modern &quot;homomorphism&quot;, whereas an &quot;isomorphisme holoédrique&quot; corresponds to a modern &quot;isomorphism&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The English equivalent of the French term &quot;holoédrique&quot; would be the crystallographic term &quot;holohedral&quot; (having all of the symmetry planes that are required by the symmetry of a crystal system).  The equivalent of &quot;mériédrique&quot; would be be the crystallographic term &quot;merohedral&quot; (having only a fraction of the symmetry planes that are required by the symmetry of a crystal system).  Thus, in the case of &quot;holoédrique&quot;, the correspondence between the elements of the groups Γ and G is one-to-one and thus isomorphic, whereas in the case of &quot;mériédrique&quot; the correspondence is not one-to-one and therefore homomorphic.&lt;/p&gt;&#xA;" OwnerUserId="5662" LastEditorUserId="5662" LastEditDate="2017-04-17T11:41:00.640" LastActivityDate="2017-04-17T11:41:00.640" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5904" PostTypeId="1" AcceptedAnswerId="5906" CreationDate="2017-04-18T13:58:13.920" Score="3" ViewCount="221" Body="&lt;p&gt;Were complex numbers ever considered to be of limited usefulness  that is not very useful in practice (unlike modern science where we strongly need complex numbers)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note my question is &lt;em&gt;not&lt;/em&gt; about philosophical consideration of complex numbers as something &quot;imaginary&quot; (existing only in imagination of people).&lt;/p&gt;&#xA;" OwnerUserId="4469" LastActivityDate="2017-04-26T16:50:39.327" Title="Were complex number first considered of limited usefulness?" Tags="|mathematics|complex-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5905" PostTypeId="1" CreationDate="2017-04-18T14:54:53.453" Score="3" ViewCount="259" Body="&lt;p&gt;I'm researching about fractals history and one of its main contributor and promoter Benoit Mandelbrot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I'm concerned, when he published his first book about this subject in 1975, he was severely criticized and questioned. Later, as he published more books and expanded this subject, he started to get accepted by most mathematicians.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where could I find proofs about the first?&#xA;For example, any book or science / math magazine of the time criticizing him (or similar). &lt;/p&gt;&#xA;" OwnerUserId="5666" LastEditorUserId="12699" LastEditDate="2017-06-28T16:25:37.707" LastActivityDate="2017-06-28T16:25:37.707" Title="Books or articles criticizing Benoit Mandelbrot? (fractals)" Tags="|geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5906" PostTypeId="2" ParentId="5904" CreationDate="2017-04-18T19:13:36.600" Score="7" Body="&lt;p&gt;Usefulness was never seriously questioned. Complex numbers are useful, for example, if you want to make Cardano's formula work. And this is how they were invented.&#xA;They are useful for other mathematical problems, and since the beginning of 19th century, in physics too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was questioned sometimes is their justification, existence. These questions gradually disappear when their geometric interpretation became a common knowledge,&#xA;which happened in the beginning of 19th century. But still some mathematicians of&#xA;19th century (Chebyshev, for example) distrusted them and tried to avoid them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read somewhere (don't remember where) that students of Ecole Polytechnique once in the beginning of 19th century made a riot protesting teaching complex numbers (on the grounds that they are useless in practical applications),&#xA;if this story is true, these were just poorly educated students. Serious mathematicians never doubted that they are useful. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-04-26T16:50:39.327" LastActivityDate="2017-04-26T16:50:39.327" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5907" PostTypeId="1" CreationDate="2017-04-19T20:52:53.750" Score="10" ViewCount="16600" Body="&lt;p&gt;As discussed in this answer &lt;a href=&quot;https://physics.stackexchange.com/a/77691/667&quot;&gt;https://physics.stackexchange.com/a/77691/667&lt;/a&gt; there are several common conventions for the notation $[q]$ of a physical quantity $q$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I often see people to put the square brackets around the (SI-) unit itself. For example $[\mathrm{kg}]$ (for kilogram). Most time the people using this are over 50 years old and often engineers. So, I got the impression that this convention was common some years ago.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what are the origins and reasons for this old convention?&lt;/p&gt;&#xA;" OwnerUserId="5670" LastEditorUserId="5670" LastEditDate="2017-04-20T19:41:26.620" LastActivityDate="2023-10-31T23:00:44.400" Title="Old square bracket notation for units" Tags="|physics|notation|units|engineering|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5908" PostTypeId="1" AcceptedAnswerId="5912" CreationDate="2017-04-19T23:26:28.793" Score="10" ViewCount="654" Body="&lt;p&gt;The term &quot;Adjoint&quot; appears in many different mathematical areas and for sometimes seemingly different kinds of things.  Wikipedia says --  &quot;In mathematics, the term &lt;em&gt;adjoint&lt;/em&gt; applies in several situations.  Several of these share a similar formalism:  if $A$ is adjoint to $B$, then there is typically some formula of the type&quot;&#xA;$$&#xA;\left( A x, y \right) = \left(x, B y\right)&#xA;$$&#xA;Also, Wikipedia goes on to give examples and a few of them I repeat here:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hermitian Adjoint of a Linear Operator&lt;/li&gt;&#xA;&lt;li&gt;Adjoint Equation (as used in Differential Operators)&lt;/li&gt;&#xA;&lt;li&gt;Adjoint Matrix&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;And, there are other examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is about how and where was this term chosen.  Better yet, who was first to start using it.  Every time I see this word in a text or paper I am always thinking &quot;There has to be a better more descriptive term to use&quot;.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2019-12-26T14:20:27.287" Title="What is the first usage of the term &quot;Adjoint&quot; and why was this word chosen?" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5909" PostTypeId="2" ParentId="5905" CreationDate="2017-04-20T00:49:52.907" Score="4" Body="&lt;p&gt;This may not be quite what you seek, but there is a good review of &lt;em&gt;The Fractalist&lt;/em&gt; by Brian Hayes in &lt;em&gt;The American Scientist&lt;/em&gt; (&lt;a href=&quot;http://www.americanscientist.org/bookshelf/pub/father-of-fractals&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;).&#xA;He mentions the &quot;feud&quot; with Herbert Simon:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What’s not so endearing is to see some of the supporting actors denied even a listing in the credits. For example, Mandelbrot carried on a bitter public feud with the economist and computer scientist Herbert Simon. For six years they traded critiques and replies, final notes and postscripts to final notes in the pages of &lt;em&gt;Information and Control&lt;/em&gt;. The incident is mentioned in &lt;em&gt;The Fractalist&lt;/em&gt;, but Simon goes unnamed; Mandelbrot refers to him as “a loud opponent of mine.” It could be worse, I suppose. Instead of ignoring Simon, Mandelbrot might have taken one last swipe at him—a postscript to the postscript.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="606" LastActivityDate="2017-04-20T00:49:52.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5911" PostTypeId="1" AcceptedAnswerId="5914" CreationDate="2017-04-18T11:55:52.973" Score="8" ViewCount="233" Body="&lt;p&gt;How did Eratosthenes come up with the latitude 36 line, also called &lt;a href=&quot;https://en.wikipedia.org/wiki/36th_parallel_north&quot; rel=&quot;noreferrer&quot;&gt;36th parallel north&lt;/a&gt;, in the Mediterranean world? Rhodes was one of the navel points in his calculations and even today N36.00 goes through the southern part of the Rhodes. So I'd like to know the history of the calculation of this latitude and the origin of the latitudes in general.&lt;/p&gt;&#xA;" OwnerUserId="5674" OwnerDisplayName="MarkokraM" LastActivityDate="2017-04-20T17:19:48.790" Title="Origin of the latitude 36 of Eratosthenes" Tags="|geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5912" PostTypeId="2" ParentId="5908" CreationDate="2017-04-20T10:26:39.593" Score="4" Body="&lt;p&gt;Jeff Miller's very valuable collection of the origins of mathematical expressions has the entries adjoint equation (Lagrange), adjoint linear form (Cayley) and adjoint matrix (Bocher): &lt;a href=&quot;http://jeff560.tripod.com/mathword.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jeff560.tripod.com/mathword.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-04-20T17:58:51.243" LastActivityDate="2017-04-20T17:58:51.243" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5913" PostTypeId="1" AcceptedAnswerId="5915" CreationDate="2017-04-20T14:28:17.310" Score="6" ViewCount="193" Body="&lt;p&gt;According to the book &lt;em&gt;&quot;Models and Inferences in Science&quot;&lt;/em&gt;, p.156 - &quot;the idea of a number-theoretic approach to knots goes to Gauss (1798), who &lt;strong&gt;used the analogy between primes and knots&lt;/strong&gt;, and it was put forward by Schubert (1949) and Mazur (1973)&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my question is where is this analogy stated in Gauss's nachlass and in addition to that, are there any indications that Gauss has foreseen the connection between his law of quadratic reciprocity and his linking integral from 1833?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2351" LastEditDate="2017-04-21T11:51:36.400" LastActivityDate="2017-04-26T23:06:47.977" Title="Did Gauss see the analogy between number-theoretic concepts and knots?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5914" PostTypeId="2" ParentId="5911" CreationDate="2017-04-20T17:19:48.790" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;https://books.google.ca/books?id=VYQDAAAAQAAJ&amp;amp;pg=PA27&amp;amp;lpg=PA27&amp;amp;dq=36th%20parallel%20history%20eratosthenes&amp;amp;source=bl&amp;amp;ots=Vm9rfkd_DU&amp;amp;sig=8SV0J2few1OsRSkX4zFBq7a-zXA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiAmZLIt7PTAhXIw1QKHRxOAkYQ6AEIKTAB#v=onepage&amp;amp;q=36th%20parallel%20history%20eratosthenes&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Short History of Natural Science&lt;/em&gt; by Arabella Fisher&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He laid down the parallel of latitude in the following manner.  He knew that at all places on the equator the day was exactly the same length all the year round, and that the lengths of the days and nights varied more and more as you went northward; therefore he reasoned that, if he could draw a line east and west through a number of places whose longest day was exactly the same length, those places would be exactly the same distance from the equator. He began at the Straights of Gibraltar, when the longest day was exactly $14\frac{1}{2}$ hours, and then observing all those places whose longest day was also $14\frac{1}{2}$ hours, he drew a line through the south coast of Sicily, across the south of the Peloponnesus, the island of Rhodes, the bay of Issus, and across the Euphrates and Tigris, out to the mountains of India.  If you follow this line on a map you will find it is the $36$&lt;em&gt;th&lt;/em&gt; parallel of north latitude and that Eratosthenes observation was perfectly correct.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, Eratosthenes would not have identified this parallel as $36$ degrees north.  (According to Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt;, the introduction of the $360$ degree circle came a little later.)  However, he could calculate its distance from the equator using the same techniques he employed in calculating the circumference of the Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Boyer, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is not known when the systematic use of the $360$ degree circle came into mathematics, but it seems to be largely due to Hipparchus in connection with his table of chords.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hipparchus of Nicaea (ca. 180 - ca. 125 BCE).&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-04-20T17:19:48.790" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5915" PostTypeId="2" ParentId="5913" CreationDate="2017-04-21T16:45:39.757" Score="4" Body="&lt;p&gt;The answer is no, I am afraid. Gauss's writings on knots are in scattered notes from 1794 to 1833 (some undated), none of them from 1798. On the other hand, 1798 is the year when &lt;a href=&quot;https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae&quot; rel=&quot;nofollow noreferrer&quot;&gt;Disquisitiones Arithmeticae&lt;/a&gt; was written. This, and the reference to Mazur, suggest that the book's claim is based on a casual (mis)reading of &lt;a href=&quot;https://arxiv.org/abs/0904.3399&quot; rel=&quot;nofollow noreferrer&quot;&gt;Morishita's Analogies between Knots and Primes, 3-Manifolds and Number Rings&lt;/a&gt; (p.2):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As we shall see in the following, the analogy between primes and knots is well-suited with the arithmetic (for example, classfield theory) originated from Gauss' Disquisitiones Arithmeticae. Furthermore, as Gauss' linking number was connected to the electro-magnetic theory from its origin, the analogy between number theory and knot theory seems to be extended to the connection involving quantum field theory, dynamics etc (cf. [8],[28],[42] etc). I would like to e-mail Gauss in heaven and ask “Is this circle of thoughts what you dreamed to explore?”.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While Morishita's enthusiasm is infectious, &quot;the following&quot; (pp. 6, 18) makes clear what is already apparent on the second reading of the quote: what Morishita says &quot;originated from&quot; Gauss was the classfield arithmetic, not the knots-are-primes analogy. Moreover, he is explicit that by the analogy he does not mean the elementary idea (developed by Schubert in 1949) that both integers and knots can be decomposed into (connected sums of) primes, but something far more sophisticated:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Namely, the classical classfield theory by T. Takagi - E. Artin is restated as a sort of 3-dimensional Poincare duality in the etale cohomology of a number ring ([44]). The analogy between knots and primes was firstly pointed out by B. Mazur from this homotopical viewpoint, during that time, in the middle of 1960’s (and also by Y. Manin).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The attribution of the knots-are-primes analogy to Mazur and Manin has since been rescinded, and as a result of a Math Overflow discussion no less! With a little help, Mazur found his long lost preprint from 1964, which opens with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Mumford has suggested a most elegant model as a geometric interpretation of the above situation: &#xA;  $spec(\mathbb{Z}/p\mathbb{Z})$ is like a one-dimensional knot in $spec(\mathbb{Z})$ which is like a simply connected three-manifold.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For the full story with twists and turns see &lt;a href=&quot;http://www.neverendingbooks.org/who-dreamed-up-the-primesknots-analogy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Who dreamed up the primes=knots analogy?&lt;/a&gt;, which also has photocopies of original sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss's notes on knots are discussed in &lt;a href=&quot;http://www.math.ucla.edu/~radko/191.1.05w/erin.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Colberg's Brief History of Knot Theory&lt;/a&gt; and &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/przytycki2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Przyticki's Classical Roots of Knot Theory&lt;/a&gt;. His main contributions are the &quot;Gauss code&quot; for representing knots, and the linking integral. He did not publish on the issue, but his work was inherited by his student Listing, who developed many of Gauss's ideas in undeservedly overlooked Vorstudien zur Topologie (1847), the first publication with the word &quot;topology&quot; in it. Tait, the foremost explorer of knots in the 19th century, rediscovered much of it independently before Maxwell pointed out the Listing's paper to him. Unfortunately, the paper is still not translated into English, but Tait gave a &lt;a href=&quot;https://books.google.com/books?id=CUwbAAAAMAAJ&amp;amp;pg=PA30&amp;amp;lpg=PA30&amp;amp;dq=Tait++Listing%E2%80%99s++Topologie++Philosophical++Magazine&amp;amp;source=bl&amp;amp;ots=21kzwExugb&amp;amp;sig=aeHNXzdQCJgCgUP7bn7BStsjmAI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjjluP01bTTAhWEllQKHTKsAWkQ6AEIKDAA#v=onepage&amp;amp;q=Tait%20%20Listing%E2%80%99s%20%20Topologie%20%20Philosophical%20%20Magazine&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;detailed summary of Vorstudien&lt;/a&gt;. It seems that even of the more elementary &quot;prime decomposition&quot; analogy there is no trace in either Gauss or Listing. Listing talks about &quot;reduced&quot; knots, and Tait about &quot;simple forms&quot;, but those refer to eliminating redundant crossings, not decompositions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the linking integral, Gauss did not do much with it. It appears as a brief note in his diary of 1833 (published in 1867), and what little he wrote has been subject to interpretative speculations. He did not derive the integral, or noted the connection between it and the number of crossings, which appear in his &quot;code&quot; (it was pointed out by Boeddicker in 1876). Here is from &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/ricca.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss’s Linking Number Revisited by Ricca and Nipoti&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The concept of linking number was introduced by Gauss in a brief note on his diary in 1833 (see Sec. 2 below), but no proof was given, neither of its derivation, nor of its topological meaning. Its derivation remained indeed a mystery. Nevertheless this concept was seminal... Reconstructions of possible derivations made by Gauss have been offered by historians, in particular Epple, and physicists. These reconstructions rely on ideas that can certainly be traced back to Gauss, even though direct derivation of the linking number from Gauss’s own work is left to speculation. Here we offer a plausible reconstruction entirely based on Gauss’ own work on terrestrial magnetism...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ranicki has a &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/knots&quot; rel=&quot;nofollow noreferrer&quot;&gt;knot theory webpage&lt;/a&gt;, which links to many original sources, including most of Tait's papers.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-04-26T23:06:47.977" LastActivityDate="2017-04-26T23:06:47.977" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5918" PostTypeId="1" CreationDate="2017-04-21T23:53:59.277" Score="6" ViewCount="703" Body="&lt;p&gt;My question goes from &lt;a href=&quot;https://physics.stackexchange.com/questions/328161/why-do-we-write-e-mc2-and-not-e-c2-m?noredirect=1#comment736084_328161&quot;&gt;Phys.SE&lt;/a&gt; where people advised me to ask my question here.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I always learn in maths and physics when something is a constant in an equation we have to put it before which varies. Some days ago I just thought about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you say a function is linear you say something like, let $f$ be a linear function such as $f(x)=ax$. ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, why, when we say the energy is a linear function of the mass we write, &#xA;$$E=mc^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead of, $$E=c^2 m$$ because light speed is a constant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this something aesthetic, does Einstein wrote it like that and then everybody does? Why don't we respect the convention when we do relativity?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;My question is more general because if we look closely to GR equations when there are product terms with the speed of light then the speed of light is always the second. For example we also write $$E^2=m^2c^4+p^2c^2$$ which should be written as $$E^2=c^4m^2+c^2p^2$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And there are many others. &lt;/p&gt;&#xA;" OwnerUserId="5679" LastActivityDate="2017-04-24T08:41:38.280" Title="Why do we write $E=mc^2$ and not $E=c^2 m$?" Tags="|physics|notation|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5919" PostTypeId="1" AcceptedAnswerId="5920" CreationDate="2017-04-22T14:18:42.997" Score="2" ViewCount="662" Body="&lt;p&gt;I am preparing a series of lectures on the history of mathematics. Of course my lessons contain already sections about Egyptian and Arabian mathematics. But I don't know about mathematics in Sub-Saharan Africa except the famous Ishango bone. Are there topics or hints to literature available? Modern achievements would also be welcome.&lt;/p&gt;&#xA;" OwnerDisplayName="user3292" LastEditorUserId="26" LastEditDate="2017-04-24T18:46:09.117" LastActivityDate="2017-04-24T18:46:09.117" Title="Is there evidence for mathematics in Sub-Saharan Africa before modern times?" Tags="|mathematics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5920" PostTypeId="2" ParentId="5919" CreationDate="2017-04-23T03:32:05.223" Score="1" Body="&lt;p&gt;I've seen mentioned before, and played only a couple of times, a game in the Gebet'a or &quot;Mancala&quot; family of games.  This family of games dates back at least until 700 BC.  You could go through a brief history of these games.  They seem like a good bet, and I could definitely see how they'd have enough depth to be studied mathematically.  That might be too focused though, I don't know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://taneter.org/math.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://taneter.org/math.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="75" LastActivityDate="2017-04-23T03:32:05.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5921" PostTypeId="1" AcceptedAnswerId="5922" CreationDate="2017-04-23T13:16:36.363" Score="1" ViewCount="92" Body="&lt;p&gt;At around the 3 min mark in this video &lt;a href=&quot;https://www.youtube.com/watch?v=C6gNeKjC9Cc&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=C6gNeKjC9Cc&lt;/a&gt;, Dr. Martin Cooper, the inventor of the first cell phone says that he made the first phone call on 4/3/73 to impress a journalist. Could anyone provide a link to the article that was written by this journalist?&lt;/p&gt;&#xA;" OwnerUserId="5684" LastActivityDate="2017-04-29T01:40:06.117" Title="Is the article reporting the first cell phone still around?" Tags="|engineering|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5922" PostTypeId="2" ParentId="5921" CreationDate="2017-04-23T15:54:58.613" Score="-1" Body="&lt;p&gt;rarenewspapers.com/view/596532 Smith, G. (1973, April 4). Motorola Introduces Wire-less Phone. The New York Times, pp. 1,57&lt;/p&gt;&#xA;" OwnerUserId="5684" LastEditorUserId="5684" LastEditDate="2017-04-29T01:40:06.117" LastActivityDate="2017-04-29T01:40:06.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5924" PostTypeId="2" ParentId="5918" CreationDate="2017-04-24T08:41:38.280" Score="4" Body="&lt;p&gt;In his 1905-1907 papers Einstein derived the equation $E = m c^2 \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ for a mass moving at speed $v$. So $mc^2$ is a constant, and the variable is $v$.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastActivityDate="2017-04-24T08:41:38.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5925" PostTypeId="2" ParentId="2605" CreationDate="2017-04-24T18:38:42.727" Score="1" Body="&lt;p&gt;In Larson/Falvo sixth ed. (2009) the adjoint is defined as the transpose of the cofactor matrix. The term &quot;adjugate&quot; is not used and does not show up in the index. So apparently some continue to use the term adjoint to mean classical adjoint or adjugate.&lt;/p&gt;&#xA;" OwnerUserId="5688" LastActivityDate="2017-04-24T18:38:42.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5926" PostTypeId="2" ParentId="2486" CreationDate="2017-04-24T19:05:15.420" Score="11" Body="&lt;p&gt;In my grandfather's book, lately translated from German:&#xA;&lt;a href=&quot;http://www.springer.com/us/book/9783319308456&quot; rel=&quot;nofollow noreferrer&quot;&gt;Recollections of a Jewish Mathematician in Germany&lt;/a&gt;, by Abraham A. Fraenkel, edited by Jiska Cohen-Mansfield, translated by Allison Brown.&lt;/p&gt;&#xA;&lt;p&gt;Hilbert’s response to a question of Bernhard Rust, the Nazi Reich Minister for&#xA;Science, Education, and Popular Culture, was typical. At a banquet in 1934 in&#xA;Göttingen, Rust asked: “Is it really true, Mr. Professor, that your institute suffered so much from the departure of the Jews and their friends?” to which Hilbert replied, in his characteristic East Prussian dialect: “Suffered? No, it hasn’t suffered, Mr. Minister. It simply doesn’t exist anymore!”&lt;/p&gt;&#xA;&lt;p&gt;It was originally published in 1967.&lt;/p&gt;&#xA;" OwnerUserId="5689" LastEditorUserId="12724" LastEditDate="2021-03-09T07:35:24.983" LastActivityDate="2021-03-09T07:35:24.983" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5927" PostTypeId="1" CreationDate="2017-04-25T07:59:42.253" Score="6" ViewCount="352" Body="&lt;p&gt;Abraham Fraenkel &lt;em&gt;grandpère&lt;/em&gt; writes on page 127 of his &lt;a href=&quot;http://www.springer.com/gp/book/9783319308456&quot; rel=&quot;noreferrer&quot;&gt;book recently translated into English&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My professional career began in March 1919 with... an invitation to Göttingen to Privy Counselor Felix Klein, almost 70 years of age, but still active as the &quot;foreign minister&quot; of German mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is this to be interpreted as saying that Klein sought out Fraenkel's appointment at Göttingen?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; As pointed out by @user37237, Klein actually invited Fraenkel to write an article on &quot;number&quot;. I assume Fraenkel was already known at the time as a rising star in the new abstract field of axiomatic set theory. The fact that Klein asked Fraenkel to write an article on &quot;number&quot; seems to indicate that Klein still adhered to the importance of the &lt;em&gt;arithmetisation of analysis&lt;/em&gt;, a term he coined in his 1895 Goettingen address. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it reasonable to assume that Klein invited Fraenkel to write an article on &quot;number&quot; because Klein considered set theory to be the foundation of such an arithmetisation?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="4251" LastEditDate="2021-03-09T07:35:44.630" LastActivityDate="2021-03-09T07:35:44.630" Title="Fraenkel's appointment at Göttingen" Tags="|mathematics|mathematicians|biographical-details|gottingen|germany|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5928" PostTypeId="1" CreationDate="2017-04-25T16:51:46.143" Score="2" ViewCount="366" Body="&lt;p&gt;Every now and then I have seen the following idea, but I don't know the original author. I am interested in order to give credit to him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Every object of thought that can be identified, i.e., distinguished from all other objects of thought, exists in at least one memory or processing unit, i.e., in the spatio-temporal physical universe. Therefore it has a lot of rational spatio-temporal coordinates exclusively of its own. Take one of them to enumerate the object. Since the set of rational spatio-temporal coordinates, even in an infinite and eternal universe, is countable and since the set of thought objects can be injected into that countable set, the set of all thought objects is countable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: Who was the first to recognize this fact?&lt;/p&gt;&#xA;" OwnerDisplayName="user5693" LastActivityDate="2022-03-07T11:43:13.707" Title="Who has invented this enumeration of all objects of thought?" Tags="|mathematics|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5929" PostTypeId="1" CreationDate="2017-04-25T19:07:25.380" Score="1" ViewCount="537" Body="&lt;p&gt;There has been a recent uprise in philosophy (and it seems to me to be a very popular topic there), which is called Speculative Realism or Object-Oriented Ontology. One of the founding texts there is Quentin Meillassoux's &quot;After Finitude&quot; (including &quot;Iteration, Reiteration, Repetition: A Speculative Analysis of the Meaningless Sign&quot;), which discusses mathematics very prominently, e. g.:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;one of the main points, very roughly, is that correlationism, &lt;em&gt;&quot; every&#xA;philosophy that maintains the impossibility of acceding through thought to a being&#xA;independent of thought. We never, according to this type of philosophy, have access to any intended thing (understood in the most general sense) that is not always already correlated to an act of thinking (understood, again, in the most general sense). Consequently, correlationism posits, against all materialism, that thought cannot escape from itself so as to accede to a world not yet affected by the modes of apprehension of our subjectivity.&quot;&lt;/em&gt;, could be overcomed by mathematics and mathematized physics: &lt;em&gt;&quot;My plan is thus as follows: I will try to exhibit a minimal condition, modest&#xA;yet fundamental, of various contemporary formal languages – logical as well as&#xA;mathematical. This minimal condition, as we shall see, has to do with our capacity to&#xA;think a meaningless sign. I will then derive this capacity to think a meaningless sign&#xA;from the principle of factiality, by showing that there is an essential link between this&#xA;sort of sign and absolutized contingency. I therefore will have to show in what way&#xA;this factial derivation of the meaningless sign allows us to argue that physics (or any&#xA;other science of nature) must be based upon this absoluteness of the void sign in order&#xA;to produce hypothetical (revisable) descriptions of the present world, capable, in turn,&#xA;of being true in an absolute sense – that is to say, independently of our existence&quot;&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;he makes logical proofs about things-in-itself. For references, you could read modal logic intepretation of some of the arguments in Jon Cogburn's blog: &lt;a href=&quot;http://drjon.typepad.com/jon_cogburns_blog/2012/01/meillassouxs-key-argument-is-formulizable-in-modal-logic-and-valid.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;http://drjon.typepad.com/jon_cogburns_blog/2012/01/more-modal-logic-and-meillassoux.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;http://drjon.typepad.com/jon_cogburns_blog/2012/01/more-development-of-meillassouxs-argument-in-modal-logic-plus-a-possible-dialtheist-problem.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://drjon.typepad.com/jon_cogburns_blog/2012/01/more-modal-logic-and-meillassoux.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;he also discusses possible applications of paraconsistent logic, but maybe Graham Priest would be more appropriate here (in terms of not-purely-analytical philosophers, who use or mention mathematical logic in their works).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In overall, I have a somewhat strange impression, that this work in theory should/could be read by (addressed to) serious mathematicians, but I haven't found any commentary from mathematicians about recent developments in philosophy. (As a related question on hsm, you could read &lt;a href=&quot;https://hsm.stackexchange.com/questions/2643/conflict-between-physics-and-philosophy&quot;&gt;comments from physicists about continental philosophy pre-2005&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is, &lt;strong&gt;has any natural scientists have commented on speculative realism, object-oriented ontology, or more specifically, on the works of Quentin Meillassoux?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have also found more understandable review of his work &lt;a href=&quot;https://www.academia.edu/477081/Meillassouxs_Speculative_Philosophy_of_Science_Contingency_and_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;by philosopher Fabio Gironi&#xA;&lt;/a&gt;: &lt;em&gt;These two opening passages summarise Quentin Meillassoux‘s&#xA;two-fold ambition: to denounce the restrained nature (and inconsistency)of the premises of post-Kantian continental philosophy and to propose anew,&#xA;speculative&#xA;way of philosophising, based on direct access to theexternal world obtained through mathematical reasoning. These two&#xA;moments in Meillassoux‘s project are both a&#xA;negative&#xA;critique of contemporary philosophy and a&#xA; positive&#xA;demonstration of a new principle on which philosophical speculation can be founded: the absolute necessity of contingency and the recognition of mathematical discourse as its expression.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="3090" LastActivityDate="2017-04-26T08:29:54.870" Title="Has mathematicians read/reacted to speculative realism, specifically to philosophy of Quentin Meillassoux?" Tags="|mathematics|philosophy-of-science|" AnswerCount="1" CommentCount="3" ClosedDate="2017-05-02T22:02:54.670" ContentLicense="CC BY-SA 3.0" />
  <row Id="5931" PostTypeId="2" ParentId="5928" CreationDate="2017-04-26T07:34:45.907" Score="1" Body="&lt;p&gt;While it is true that from the modern viewpoint something as vague as an &quot;object of thought&quot; can hardly be identified, as correctly pointed out in the &lt;em&gt;comments&lt;/em&gt;, such creatures were used by no less an authority than Richard Dedekind to &quot;prove&quot; the existence of an infinite set.  The idea is essentially a recursive construction involving thinking about the previous thought. This was discussed in detail &lt;a href=&quot;https://mathoverflow.net/q/240276/28128&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-04-26T07:34:45.907" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="5932" PostTypeId="2" ParentId="5929" CreationDate="2017-04-26T07:43:40.297" Score="1" Body="&lt;p&gt;I will slightly generalize the question: how mathematicians react to modern philosophers who (ab)use mathematical terms and metaphors:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.physics.nyu.edu/faculty/sokal/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.physics.nyu.edu/faculty/sokal/index.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure which specific names and directions in philosophy are mentioned&#xA;there but it gives you a general impression on how mathematicians react on such&#xA;philosophers. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="604" LastEditDate="2017-04-26T08:29:54.870" LastActivityDate="2017-04-26T08:29:54.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5933" PostTypeId="2" ParentId="5907" CreationDate="2017-04-26T13:47:41.733" Score="6" Body="&lt;p&gt;It is a habit of text books and handbooks (of engineering branches as well as of chemistry and even physics) to indicate the unit of numerical values given in tables by column headings like &quot;&lt;em&gt;m&lt;/em&gt; [kg]&quot; or &quot;&lt;em&gt;m&lt;/em&gt; in kg&quot;. The mathematically correct way is &quot;&lt;em&gt;m&lt;/em&gt;/kg&quot; since if the mass is &lt;em&gt;m&lt;/em&gt; = 3 kg, then the numerical value is clearly 3 = &lt;em&gt;m&lt;/em&gt;/kg. But even when students are told this, I see often the aformentioned habit show through. This might be the reason for further [ab]use of square brackets. (It is not distorting. We know what is meant.)&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-04-26T13:47:41.733" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5935" PostTypeId="1" CreationDate="2017-04-28T07:55:11.917" Score="5" ViewCount="732" Body="&lt;p&gt;The Fundamental Theorem of Calculus links the concepts of differentiation and integration together.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did mathematicians of the past see the link between these two concepts? Integration is used to compute areas while differentiation is used to compute speeds. How did they link them together? &lt;/p&gt;&#xA;" OwnerUserId="5076" LastActivityDate="2017-04-28T08:46:54.540" Title="History of the Fundamental Theorem of Calculus" Tags="|calculus|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5936" PostTypeId="2" ParentId="5935" CreationDate="2017-04-28T08:46:54.540" Score="2" Body="&lt;p&gt;They didn't really &quot;link them together&quot;.  As late as the first half of the 17th century the topics that we classify under &quot;calculus&quot; today were treated by using separate techniques. This applies not merely to area-finding as opposed to what we call &quot;differential calculus&quot;, but also within the latter the techniques of finding tangent lines were often different from the techniques of finding maxima and minima.  When Fermat was working on problems of maxima and minima he wasn't necessarily thinking of &quot;doing the same thing&quot; as when he was looking for tangents to curves, though in the end he solved both by various adaptations of a basic technique called &lt;a href=&quot;https://en.wikipedia.org/wiki/Adequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;adequality&lt;/a&gt;. It was the generation after Fermat, including Barrow, and as some modern scholars argued also James Gregory, that began to realize the importance of the relation between what we consider to be the two &quot;inverse&quot; procedures.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-04-28T08:46:54.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5937" PostTypeId="1" CreationDate="2017-04-28T13:33:36.187" Score="9" ViewCount="9144" Body="&lt;p&gt;Einstein always believed that everything is certain, and we can calculate everything. That's why he rejected quantum mechanics, due to its factor of uncertainty. But still quantum physics was right. So why did Einstein never accept the fact of uncertainty? Was there any valid reason that we know of? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explain it like a story, so it will be interesting. &lt;/p&gt;&#xA;" OwnerUserId="5703" LastEditorUserId="4251" LastEditDate="2021-09-16T06:36:28.120" LastActivityDate="2021-09-16T06:36:28.120" Title="Why did Einstein oppose quantum uncertainity?" Tags="|physics|biographical-details|quantum-mechanics|einstein|physicists|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5939" PostTypeId="2" ParentId="5937" CreationDate="2017-04-28T21:58:55.083" Score="15" Body="&lt;p&gt;I am not sure what &amp;quot;explain it like a story&amp;quot; means, but the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates&quot; rel=&quot;noreferrer&quot;&gt;Bohr–Einstein debates&lt;/a&gt; is perhaps the most dramatic story of Einstein making his dissatisfaction known. Pais's scientific biography of Einstein &lt;a href=&quot;https://books.google.com/books?id=KdaiZDaB7TgC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Subtle is the Lord&lt;/a&gt; offers an even more comprehensive story of his life and the genesis of his philosophical outlook.&lt;/p&gt;&#xA;&lt;p&gt;As for the main reason for his opposition, Einstein was pretty explicit about it. He even wrapped it into a quip in a 1926 letter to Born:&amp;quot;&lt;em&gt;God doesn't play dice with the world&lt;/em&gt;&amp;quot; (anecdotally, when Einstein repeated the quip during the Bohr–Einstein debates, Bohr's response was &amp;quot;&lt;em&gt;we shouldn't tell God what to do with his dice&lt;/em&gt;&amp;quot;). The &amp;quot;God&amp;quot; here is metaphorical, in other words, he felt that physics must provide deterministic and observer-free description of reality. Einstein accepted that quantum mechanics was an experimentally confirmed empirical theory, he simply believed that some of its structural features make it inherently incomplete, as opposed to the view, advanced by Bohr and others, that quantum mechanics is as complete an account of the microphysics as the nature has to offer.&lt;/p&gt;&#xA;&lt;p&gt;The analogy was to thermodynamics vs statistical mechanics, Einstein felt that quantum mechanics is based on averaged notions, that the uncertainty was an artifact of the description rather than fundamental, and that a more fundamental account of how reality works was in order. He was certainly right on the last part, we are now looking for quantum gravity, but probably not on the idea that it will eliminate the indeterminism he detested. &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1301/1301.1656.pdf&quot; rel=&quot;noreferrer&quot;&gt;Natarajan's note What Einstein Meant When He Said “God Does Not Play Dice...”&lt;/a&gt;, after recounting Einstein's scientific trajectory, gives additional reasons:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We are now in a position to understand Einstein’s opposition to Quantum Mechanics. This was not the knee-jerk reaction of a person unqualified to speak about physics, but the considered opinion of an eminent scientist based on what he felt were several undesirable features of the theory. First, he was averse to the idea of randomness as a fundamental feature of any theory. He believed that randomness could appear as some form of statistical behaviour but could not be a part of the law, just like a pack of cards that is shuffled according to deterministic laws still shows a  random  arrangement.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;But  this  was  not  its  only  undesirable feature. The theory was also inherently nonlocal, or had a kind of Newtonian action-at-a-distance  built  into it,  but  relativity had taught us that all interactions had to propagate at a finite speed. In a landmark paper published in 1935 (twenty years after when Pais considers him finished!), Einstein highlighted the nonlocal and incomplete nature of Quantum Mechanics by proposing the famous Einstein–Podolsky–Rosen (EPR) paradox. The EPR paradox was a gedanken or thought experiment that brought out these undesirable features of Quantum Mechanics, and we all know that Einstein was a past master at thought experiments. Finally, of course, Einstein was opposed to the linear formalism of Quantum Mechanics as an approach to understanding Nature because we have  seen  above  that  only a  nonlinear  theory can  contain  the equations of motion.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This second part explains why in 1952, when Bohm came up with what Einstein seemingly wanted, a reinterpretation of quantum mechanics which removed the dreaded indeterminism by introducing deterministic hidden variables, the Bohmian particles, Einstein was not taken by it. The problem was that &lt;a href=&quot;https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory&quot; rel=&quot;noreferrer&quot;&gt;Bohmian mechanics&lt;/a&gt; endorsed the &amp;quot;spooky action at a distance&amp;quot; through instantaneous changes in the so-called &amp;quot;quantum potential&amp;quot;, which &amp;quot;guides&amp;quot; the Bohmian particles like the de Brogliean pilot wave. Moreover, the Bohmian particles also share the most undesirable property of Lorentz's ether, for which Einstein dispatched it, they are unobservable in principle.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-06-18T21:46:18.333" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5940" PostTypeId="1" AcceptedAnswerId="5941" CreationDate="2017-04-29T08:48:08.263" Score="2" ViewCount="290" Body="&lt;p&gt;Why is eV used instead of proton V even if they would&#xA;give the same value?&#xA;I have this view: from electrochemistry's Faraday's law,&#xA;scientists first calculated the charge of an electron. Then,&#xA;they introduced a unit of eV for measuring really small&#xA;work in electrochemistry.&#xA;But, scientist didn't know that proton would be equal in&#xA;charge to electron. Actually, when they calculated it,&#xA;nothing special was known about the attributes of proton.&#xA;eV was already established as a unit. So, scientist didn't&#xA;change it.&#xA;Am I right?&lt;/p&gt;&#xA;" OwnerUserId="5473" LastActivityDate="2017-04-29T18:41:01.397" Title="Why not proton volt instead of $eV$?" Tags="|physics|chemistry|units|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5941" PostTypeId="2" ParentId="5940" CreationDate="2017-04-29T17:50:43.700" Score="3" Body="&lt;p&gt;In this context, the letter e stands for the quantum of charge, not for the particle called the electron. That's why eV is notated as a multiplication, and that's why it has a positive value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stoney introduced the term &quot;electron&quot; in 1891 to refer to hypothetical positive and negative charges that were permanently attached to atoms. The absolute value of the charge of these particles was known, albeit with poor precision, from electrolysis combined with approximate knowledge of the atomic mass scale. In 1897, Thomson measured the charge to mass ratio of cathode rays. Its value was negative and very large, and he interpreted this as an indication that there was a light, negatively charged particle that was a universal building block of matter. In Thomson's paper, he uses the notation $e$ for the charge of these particles:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let m be the mass of each of the particles, e the charge carried by it. Let N be the number of particles passing across any section of the beam in a given time; then Q the quantity of electricity carried by these particles is given by...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later, it was shown by Millikan that charge was quantized in units of $e$. The proton was discovered later still.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the concept of $e$ as a fundamental charge originated in 1891, and the notation $e$ was used for it by Thomson in 1897. It would have been at some point during that six-year period that the notation $e$ was first used. The choice of that letter of the alphabet could have been motivated by &quot;electron&quot; or by &quot;quantity of electricity.&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2017-04-29T18:41:01.397" LastActivityDate="2017-04-29T18:41:01.397" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5942" PostTypeId="2" ParentId="5927" CreationDate="2017-04-29T21:22:42.493" Score="4" Body="&lt;p&gt;Fraenkel writes: &gt;&gt; ... Felix Klein. Er bat mich, in den von ihm herausgegebenen „Materialien für eine Wissenschaftliche Biographie von Gauß“ das Thema Zahlbegriff und Algebra zu behandeln. Das von mir verfaßte Heft VIII der „Materialien“ erschien 1920.&amp;lt;&amp;lt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This contribution to a scientific biography of Gauß seems to be the reason of Klein's invitation. An intended appointment of Fraenkel to Göttingen by Klein appears improbable because there are records about planned appointments in Klein's Nachlass in the Niedersächsische Staats- und Universitätsbibliothek Göttingen &lt;a href=&quot;http://hans.sub.uni-goettingen.de/nachlaesse/Klein.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://hans.sub.uni-goettingen.de/nachlaesse/Klein.pdf&lt;/a&gt; but Fraenkels Name does not appear there. &lt;/p&gt;&#xA;" OwnerDisplayName="user5707" LastActivityDate="2017-04-29T21:22:42.493" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="5943" PostTypeId="1" CreationDate="2017-04-29T22:33:13.150" Score="1" ViewCount="1075" Body="&lt;p&gt;I've recently finished reading Helge Kragh's &lt;em&gt;Quantum Generations&lt;/em&gt; and am looking forward to something to read next. I am hoping to find biographies or more information about certain physicists such as Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Max Born, Max Planck, Enrico Fermi, Paul Dirac, Arnold Sommerfeld, Lev Landau, David Bohm, John Bell, Wolfgang Pauli, or other prominent physicists involved in the development of quantum mechanics. I think some of these people may have interesting lives and am interested in reading just the lives of a single person at once. Does anyone know good biographies on some of these people?&lt;/p&gt;&#xA;" OwnerUserId="3219" LastEditorUserId="4251" LastEditDate="2021-09-07T00:26:35.603" LastActivityDate="2021-09-07T00:26:35.603" Title="Biographies on 20th Century physicists" Tags="|physics|reference-request|physicists|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="5945" PostTypeId="1" AcceptedAnswerId="5946" CreationDate="2017-04-30T21:46:52.410" Score="4" ViewCount="248" Body="&lt;p&gt;It is well known that $S^6$ admits an almost complex structure, inherited from its manifestation as the space of unit imaginary octonions. This almost complex structure is also well-known not to be integrable, i.e. not derived from a genuine complex structure. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of these facts were known by the early 50's, as mentioned e.g. in &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=130698&quot; rel=&quot;nofollow noreferrer&quot;&gt;the introduction to this paper by Calabi&lt;/a&gt;, which contains references (numbers 8 and 23 for its existence and numbers 7, 9 for non-integrability). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that the existence might have been known for some decades by that time, since it already appeared in a textbook (the above-mentioned reference 23) and is a relatively easy linear algebra-type result. Perhaps Cartan wrote it down already? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding non-integrability, I suspect (for no particularly strong reasons) that it was only discovered around 1950 (the references Calabi gives are from 1951), but I'm not sure.&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="658" LastEditDate="2017-05-01T00:36:59.560" LastActivityDate="2017-05-18T22:25:13.520" Title="Who first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?" Tags="|mathematics|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5946" PostTypeId="2" ParentId="5945" CreationDate="2017-04-30T22:33:55.240" Score="5" Body="&lt;p&gt;According to &lt;a href=&quot;https://arxiv.org/pdf/1610.09366.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this arxiv paper by Atiyah&lt;/a&gt;, existence and construction dates from 1947; non-integrability from 1951.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is Atiyah's history: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ehresmann 1947: Introduced the notion of almost complex structure&#xA;and showed that the 6-sphere admits an almost complex structure, but&#xA;explicitly points out that he does not know whether it has a complex&#xA;structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hopf 1947: Proved that $S^4$ and $S^8$ do not admit almost complex structures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kirchhoff 1947: Uses octonions to construct an explicit almost complex&#xA;structure on $S^6$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eckmann-Frohlicher and Ehresmann-Liberman 1951: Independently&#xA;prove that Kirchhoff’s almost complex structure on $S^6$ is not integrable&#xA;to a complex structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Borel and Serre 1953: Prove that $S^{2n}$ admits an almost complex structure&#xA;if and only if $n = 1$ or $3$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hirzebruch 1954 and Liberman 1955: Remarks that it is still not known&#xA;whether $S^6$ has a complex structure.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding your request for sources, here are my “mixed results”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ehresmann’s 1947 introductory results can be found here : &lt;a href=&quot;http://www.mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0412.0419.ocr.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ehresmann, C., Sur la theorie des espaces fibres, Colloque de Topologie Algebrique, C.N.R.S., Paris (1947), pp. 3.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Borel and Serre’s non-existence result for even dimensional spheres can be found here : &lt;a href=&quot;http://www.maths.ed.ac.uk/~aar/papers/borelserre.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A. Borel, J.P. Serre &lt;em&gt;Groupes de Lie et puissances réduites de Steenrod&lt;/em&gt; Amer. J. Math., 75 (1953), pp. 409-448&#xA;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Hopf’s 1947 result, I cannot locate the original paper, however the source is identified in &lt;a href=&quot;https://books.google.ca/books?hl=en&amp;amp;lr=&amp;amp;id=AGQECAAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PA99&amp;amp;dq=Hopf%201947%20almost%20complex&amp;amp;ots=DY6RLOJbpu&amp;amp;sig=sCcEhkZ3v9JVFC12NR6dEVGQQwU&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Hopf%201947%20almost%20complex&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Green and Shing-Tung Yau’s &lt;em&gt;Differential Geometry&lt;/em&gt; &#xA;&lt;/a&gt; as H. Hopf, Sur les champs d’element de surface dans les varietes a 4 dimensions, Topologie Algebrique, Paris 1947, Editions CNRS. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original paper giving Kirchhoff's construction is not easy to locate.  Possibly from A. KIRCHHOFF, C. R. Acad. Sci. Paris vol. 225 (1947).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not able to source the non-integrable to a complex structure result.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-05-18T22:25:13.520" LastActivityDate="2017-05-18T22:25:13.520" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5947" PostTypeId="2" ParentId="2605" CreationDate="2017-05-01T16:05:50.640" Score="4" Body="&lt;p&gt;According to the Oxford English Dictionary, the first use of the word 'adjugate' is by CL Dodgson (Lewis Carroll) Proc. Roy Soc 15(1866) at p. 151. He says &quot;In doing so I shall use the word &quot;adjugate&quot; in the following sense:-if there be a square block, and is a new block be formed, such that each of its terms is the determinant of the complemental minor of the corresponding term of the first block, the second block is said to be &lt;em&gt;adjugate&lt;/em&gt; to the first&quot; (italics in original).  I take his use of inverted commas to be an admission that he is defining a new word. &lt;/p&gt;&#xA;" OwnerUserId="5713" LastActivityDate="2017-05-01T16:05:50.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5948" PostTypeId="1" AcceptedAnswerId="5949" CreationDate="2017-05-02T19:16:35.563" Score="6" ViewCount="1205" Body="&lt;p&gt;There are many sources which say that Euler produced, on average, one mathematical paper every week in the year $1775$. &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/Euler.html&quot; rel=&quot;noreferrer&quot;&gt;Some&lt;/a&gt; even say he produced almost half his total works despite the total blindness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't seem to find much works after $1770$, on the page linked it says for the square root of -1 (1777) and Cauchy-Riemann equations in $1777$ does anyone have some references to Euler's works after he got blind?&lt;/p&gt;&#xA;" OwnerUserId="5723" OwnerDisplayName="kingW3" LastActivityDate="2018-04-02T18:43:34.323" Title="Euler's works after blindness" Tags="|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5949" PostTypeId="2" ParentId="5948" CreationDate="2017-05-03T06:44:19.183" Score="6" Body="&lt;p&gt;Euler indeed produced much of his work when he was blind, as can be seen from the &lt;a href=&quot;http://eulerarchive.maa.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler archive&lt;/a&gt;. The blindness developed&#xA;gradually: first he lost sight in one eye, several years later in another.&#xA;Euler himself joked: when he became completely blind he said that this is even better for his research, because fewer things will distract him. Euler was assisted by several secretaries, including his own children and grandchildren.&#xA;Source: William Dunham, Euler: The Master of Us All (MAA, 1999).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="26" LastEditDate="2018-04-02T18:43:34.323" LastActivityDate="2018-04-02T18:43:34.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5950" PostTypeId="1" AcceptedAnswerId="5953" CreationDate="2017-05-03T08:59:20.317" Score="5" ViewCount="374" Body="&lt;p&gt;General relativity (1915), as I've heard it explained, describes a universe that is either shrinking or expanding. By adding a cosmological constant it can describe a universe in eternal steady state, which was favored at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were there published suggestions, or otherwise discussed, that the universe could be expanding before Edwin Hubble observed that more distant galaxies indeed have higher redshift (1929)? Was Hubble deliberately looking for an expanding universe in the intergalactic space he had discovered in 1923?&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorDisplayName="user466" LastEditDate="2017-05-03T15:13:52.613" LastActivityDate="2020-12-05T21:44:41.770" Title="Was an expanding universe proposed before Edwin Hubble's observations?" Tags="|astronomy|relativity-theory|cosmology|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5951" PostTypeId="1" CreationDate="2017-05-03T10:18:18.837" Score="6" ViewCount="1299" Body="&lt;p&gt;My understanding is that magnitudes to ancient Greeks meant the actual line segments and plane regions (not the size of the line segment or the area of the plane region), the concept of ratio was then used to talk about the 'size' of different magnitudes. But Euclid defines ratio as 'sort of a relation in respect of size between two magnitudes of the same kind'. It seems this definition attaches no numerical meaning to the concept of ratio.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Was there a particular reason that the magnitudes were defined this way?&lt;/li&gt;&#xA;&lt;li&gt;With no numerical meaning, how was this concept of ratio used to talk about size?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5727" LastEditorUserId="55" LastEditDate="2017-05-03T22:25:54.430" LastActivityDate="2024-01-13T00:03:23.880" Title="What did the ratio of two magnitudes mean to ancient Greek mathematicians?" Tags="|euclidean-geometry|real-analysis|measurement|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="5952" PostTypeId="1" CreationDate="2017-05-03T11:42:13.687" Score="2" ViewCount="373" Body="&lt;p&gt;I'm a Spanish Physics undergrad, and I was writing a report on an experiment we did on the lab, in which we had to measure a solid's density using the so-called Westphal balance (aka Mohr balance, and in Spanish, Mohr-Westphal balance). After doing a little research on the internet, in both Spanish and English languages, I've encountered no more in the references on the history of the balance than the mention of &lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Friedrich_Mohr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl Friedrich Mohr&lt;/a&gt; as the inventor of the instrument, &lt;a href=&quot;https://es.wikipedia.org/wiki/Balanza_de_Mohr-Westphal&quot; rel=&quot;nofollow noreferrer&quot;&gt;here, among other sites&lt;/a&gt;, and the fact that Westphal was one who improved the usage of the balance. I was capable to find the latter only &lt;a href=&quot;http://www.ugr.es/~museojtg/instrumento17/ficha_fundamentos2.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (second sentence):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Posteriormente introdujo Westphal ciertas modificaciones empleando unas pesas cuyas masas guardan una determinada relación entre sí y a su vez con la masa de un flotador cilíndrico de vidrio en cuyo interior hay colocado un termómetro.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and there's no references in the webpage of second link.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm missing a few things:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who is Westphal, because I was unable to find a person behind this name.&lt;/li&gt;&#xA;&lt;li&gt;How exactly did contributed Mohr and Westphal to the development of the balance? There's practically no data on the web.&lt;/li&gt;&#xA;&lt;li&gt;Does somebody know anything by chance about the history of the instrument? Or does someone know where should I try to do the research?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5031" LastEditorUserId="55" LastEditDate="2017-05-03T21:39:02.350" LastActivityDate="2017-12-22T15:03:48.413" Title="Who is the Westphal behind the Mohr-Westphal specific gravity balance? And how exactly did he improve on Mohr?" Tags="|physics|experimental-physics|measurement|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5953" PostTypeId="2" ParentId="5950" CreationDate="2017-05-03T15:18:14.783" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Were there published suggestions, or otherwise discussed, that the universe could be expanding before Edwin Hubble observed that more distant galaxies indeed have higher redshift (1929)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes. De Sitter published expanding cosmological models in March 1917 in which there was a cosmological constant. Friedman published his models in 1922. People didn't take this sort of stuff very seriously at the time. This kind of speculation was not considered mainstream science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first Doppler shift of a galaxy was discovered in 1912: a blueshift for the Andromeda galaxy. By 1914 Slipher had realized that most galaxies had red shifts. His work was greeted with extreme skepticism initially. The Shapley-Curtis debate on the nature of the spiral nebulae didn't take place until 1920. Mount Wilson undertook a large-scale campaign of observing spiral nebulae in 1919, and the picture of them as galaxies wasn't really solidly confirmed until about 1924.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Cepheid distance scale dates to about 1924, so de Sitter and Friedman's models considerably predated the Hubble law. Lemaitre synthesized the various ideas and predicted in the Hubble law in 1925. Hubble confirmed it observationally in 1927.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So basically the theory and observations were developed side by side over a period of about 15 years. Hubble certainly was aware of the relevant ideas before he undertook the program of observation which, as one of its results, demonstrated the Hubble law empirically.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-05-03T15:18:14.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5954" PostTypeId="2" ParentId="5951" CreationDate="2017-05-03T16:54:37.650" Score="-2" Body="&lt;p&gt;For the Greeks, the number (or ratio) was what we call rational number nowadays. &#xA;(They did not have the concept of real number which was rigorously defined only in the end of 19th century).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was a great discovery that some segments (constructed by ruler and compass)&#xA;do not have a ratio. To deal with this the Greeks invented proportions.&#xA;Proportion says that two ratios (in the modern sense of the word) are equal.&#xA;The Greeks discovered that one can develop the theory of proportions (without defining the ratios themselves!) which works even in the case when the ratios&#xA;are not rational numbers. This theory is essentially equivalent to our modern theory of real numbers. It was influential, and until the end of 18th&#xA;century physicists were talking about proportions instead of real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even when I was at school (in 1960th) all physical laws (like Hooke's law or gravitation law or Coulomb law) were stated in the form of proportions rather&#xA;then equalities between real numbers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-03T16:54:37.650" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5955" PostTypeId="2" ParentId="5950" CreationDate="2017-05-03T17:01:46.543" Score="4" Body="&lt;p&gt;This is an extended comment to Ben Crowell's answer. Friedmann proved that&#xA;without the &quot;cosmological term&quot; the universe cannot be stationary: has either to&#xA;expand or contract. As a result of this Einstein introduced the cosmological term to his equations, the thing which he later called &quot;the greatest blunder&quot; in his life. The correct logical conclusion from Friedmann's theorem was that the universe is not stationary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Wikipedia spells him as Friedmann, but the exact transliteration from Russian&#xA;will be Fridman.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-03T17:01:46.543" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5958" PostTypeId="2" ParentId="5951" CreationDate="2017-05-03T22:23:30.073" Score="8" Body="&lt;p&gt;&lt;a href=&quot;http://www.ams.org/journals/bull/1979-01-06/S0273-0979-1979-14684-6&quot; rel=&quot;noreferrer&quot;&gt;Fowler's Ratio in Early Greek Mathematics&lt;/a&gt; is a standard reference on the subject, see also his book &lt;a href=&quot;https://archive.org/stream/MathematicsOfPlatosAcademyFowler/Mathematics_of_Plato%27s_Academy-Fowler_djvu.txt&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Of Plato's Academy&lt;/a&gt; (both are freely available).&lt;/p&gt;&#xA;&lt;p&gt;Book V, Definition 3 of Euclid's Elements reads: &amp;quot;&lt;em&gt;A ratio (logos) is a sort of relation in respect of size between two magnitudes of the same kind&lt;/em&gt;&amp;quot;. Thus, ratios of magnitudes were indeed  non-numbers, but they could be related to measurements. In the simplest form, originally used by Pythagoreans, one could talk about a ratio of magnitudes &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; being equal to a ratio of numbers &lt;span class=&quot;math-container&quot;&gt;$m:n$&lt;/span&gt; if there was a &lt;em&gt;common measure&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, which reproduced &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; times equals &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and reproduced &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; times equals &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (this makes particularly simple sense for line segments). The common measure could be found through the &lt;em&gt;anthyphairesis&lt;/em&gt;, the Euclidean algorithm in geometric form, which produces the sequence of partial quotients of the continuous fraction for &lt;span class=&quot;math-container&quot;&gt;$m/n$&lt;/span&gt;. Of course, Greeks did not talk of fractions or anything like &amp;quot;rational numbers&amp;quot;. Here is Fowler:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Operations corresponding to the multiplication of ratios do occur within the Elements, with treatments that are worthy of note. First observe that although the idea of taking the product of two ratios is never defined — nor could it be since ratio itself is not defined, but only the equality of two ratios — the operation is used in VI, 23... There could not be a general proposition in Book V on the compound of two proportions between magnitudes, since the result might seem to involve the product of the magnitudes, which would not in general be defined. Particular cases do however occur... At no point in the Elements, or in the surviving corpus of classical Greek mathematics does anything remotely related to a formal treatment of addition of ratios of numbers occur.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The situation was different in practical use, and later in Hellenistic astronomy, where fractions and even sexagesimals, were arithmetically manipulated. Fowler believes that experimentation with the &lt;em&gt;anthyphairesis&lt;/em&gt; might have led to the discovery that it does not terminate for some pairs of segments (like the sides and diagonals in squares and pentagons), and hence to the discovery that some magnitudes are incommensurable. This led to the first, &amp;quot;constructive&amp;quot;, theory of incommensurable ratios developed by Theaetetus based on non-terminating &lt;em&gt;anthyphairesis&lt;/em&gt; sequences. It was later replaced by the technically more convenient device of Eudoxus, presented in Book V of Elements, which defines when two potentially incommensurable ratios are equal by cleverly abstracting and modifying the original Pythagorean definition. Some results of Theaetetus's theory, although established by different methods, are presented in Book X of Elements, where Euclid surprisingly does not use the general Eudoxian theory of Book V.&lt;/p&gt;&#xA;&lt;p&gt;The Eudoxian definition was the basis for the double reductio arguments later termed the &lt;a href=&quot;https://en.wikipedia.org/wiki/Method_of_exhaustion&quot; rel=&quot;noreferrer&quot;&gt;method of exhaustion&lt;/a&gt;, which go back to Eudoxus himself (volume of the pyramid), and were extensively used by Archimedes. In modern translation, it declares two ratios &lt;span class=&quot;math-container&quot;&gt;$a:b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c:d$&lt;/span&gt; to be equal if for any pair of numbers &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; (meaning positive integers, of course) the equality/inequality between &lt;span class=&quot;math-container&quot;&gt;$na$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$mb$&lt;/span&gt; always goes in the same direction as the one between &lt;span class=&quot;math-container&quot;&gt;$nc$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$md$&lt;/span&gt;. Archimedes's results also provide nice examples of how the theory of ratios was used to talk about sizes. Instead of expressing areas and volumes as numbers, the way we do today, he states in On Sphere and Cylinder that the volume of a sphere is to the volume of the circumscribed cylinder as &lt;span class=&quot;math-container&quot;&gt;$2:3$&lt;/span&gt;, see details in &lt;a href=&quot;https://hsm.stackexchange.com/questions/1863/who-calculated-for-the-first-time-the-volume-and-surface-area-of-the-sphere-ex/1872#1872&quot;&gt;Who calculated for the first time the volume (and surface area) of the sphere exactly?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;How much the attitudes have changed by the end of 17th century is attested to by &lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmetica_Universalis&quot; rel=&quot;noreferrer&quot;&gt;Newton's Arithmetica Universalis (1707)&lt;/a&gt;, where he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We understand a number not as a set of units, but as the abstract ratio of&#xA;one magnitude to another magnitude of the same kind taken for that unit.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-04T14:41:30.107" LastActivityDate="2020-11-04T14:41:30.107" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5959" PostTypeId="2" ParentId="5952" CreationDate="2017-05-04T01:32:36.957" Score="3" Body="&lt;p&gt;Quoth H. R. Jenemann, &lt;a href=&quot;http://dx.doi.org/10.1002/bewi.19970200211&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Die Geschichte der Dämpfung an der Laboratoriumswaage&lt;/em&gt;&lt;/a&gt; (1997, p. 240):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1879, Georg Westphal filed a patent in Celle for a scale which was distinguished by various innovations and featured a similar brush lock&lt;sup&gt;19&lt;/sup&gt;. With this special scale, the scale carrier had no connection to the beam of the scale; it could swing freely about its own axis, and had the braking brush at its lower end. (...)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;19. Georg Westphal: &lt;em&gt;Neuerungen an Waagen&lt;/em&gt;. Patentschrift des Kaiserlichen Patentamtes, No. 8975, patented in the German Reich from September 4, 1879. On Westphal’s history in Celle, see Hans R. Jenemann: &lt;em&gt;Die Waagenkonstruktionen von Georg Westphal&lt;/em&gt;. In: Westphal Mechanik-Augenoptik (Hrsg.): 125 Jahre 1860-1985 Westphal-Mechanik/Westphal-Augenoptik. Celle 1985, pp. 35-45.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-12-22T15:03:48.413" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5962" PostTypeId="1" AcceptedAnswerId="5963" CreationDate="2017-05-04T17:00:38.783" Score="6" ViewCount="361" Body="&lt;p&gt;A little over 50 years ago I took my first Calculus class and learned the conventional form of an integral as:&#xA;$$&#xA;\int f(x)\,\, \textrm{d}x&#xA;$$&#xA;That is, the integral sign (definite or indefinite) followed by the function being integrated followed by the integration variable as $\textrm{d}x$ (in this example).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, during retirement I decided a self-teaching adventure into Quantum Field Theory and also General Relativity and other topics of mathematical physics.  Reading many different papers and text books on these subjects I find a lot of authors using this variation of the integral:&#xA;$$&#xA;\int \textrm{d}x\,\,f(x) &#xA;$$&#xA;Where the integral sign is immediately followed by designating the variables to be integrated.  Other examples include:&#xA;$$&#xA;S = \int_{t_1}^{t_2}\textrm{d}^4q\,\,\mathcal{L}(\phi,\dot{\phi}\,;t)&#xA;$$&#xA;For the action integral of a field described by Lagrangian Density function or an example from General Relativity,&#xA;$$&#xA;S=\int \textrm{d}^4 x\sqrt{-g}\,\,g^{\mu \nu}\,\,R_{\mu \nu}\,(\,\Gamma\,)&#xA;$$&#xA;Now that I am used to seeing and using this notation, I much prefer it over the notation I learned in Calculus classes and have used ever since in work and play.  I like the idea of seeing the integral, its limits, and the variables of integration all described at once before considering the function to be integrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question, is this notation recognized by others, in particular mathematicians as I myself have only seen it used in various papers and texts of topics in mathematical physics.  Also, who used it first and were they motivated by the same idea that I personally find as an improvement -- that is, it just reads better.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastEditorUserId="2075" LastEditDate="2017-06-25T15:35:31.963" LastActivityDate="2017-06-25T15:35:31.963" Title="Notational change with Integrals" Tags="|notation|calculus|mathematical-physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5963" PostTypeId="2" ParentId="5962" CreationDate="2017-05-04T23:12:43.673" Score="8" Body="&lt;p&gt;Calculus was originally formulated in terms of infinitesimals. Hundreds of years later, a second formulation was found in terms of limits. There were originally some doubts about whether the version using infinitesimals was logically OK, but these doubts were cleared up by Robinson and others ca. 1961.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Leibniz notation $\int f(x) dx$ was invented in the earlier period, so in this notation, $dx$ a notation for an infinitesimal. You can imagine a Riemann sum with the narrow rectangles having an infinitesimal width $dx$. The thing inside the integral sign is the width of one such rectangle: its height $f(x)$ multiplied by its width $dx$. Since multiplication is commutative, we have literally $f(x)dx=dxf(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some math teachers today still tell their students that the $dx$ is just punctuation, or only functionsas a statement of what variable is being integrated with respect to. This could be because they are afraid their students will be confused by talk about infinitesimals, or because the teachers themselves don't know that any concerns about the logical issues with infinitesimals have been cleared up.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My question, is this notation recognized by others, in particular mathematicians as I myself have only seen it used in various papers and texts of topics in mathematical physics. Also, who used it first[...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The notation was first used by Leibniz, with the factors written in whichever  order was convenient. It is universally recognized, with the factors in either order, by people who understand the above historical and mathematical facts.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="604" LastEditDate="2017-05-05T07:01:59.473" LastActivityDate="2017-05-05T07:01:59.473" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="5964" PostTypeId="1" AcceptedAnswerId="5971" CreationDate="2017-05-05T08:28:26.687" Score="2" ViewCount="87" Body="&lt;p&gt;As a follow-up to &lt;a href=&quot;https://hsm.stackexchange.com/q/5927/604&quot;&gt;this earlier question on Fraenkel and Klein&lt;/a&gt;, I am interested in the actual contents of the article Fraenkel apparently wrote for Klein's volume on Gauss, which I don't have access to.  Did Fraenkel's article actually discuss set theory which was Fraenkel's specialty?&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-05-06T14:28:32.633" Title="Did Fraenkel's article on $Number$ for Klein's volume on Gauss discuss set theory?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5965" PostTypeId="1" AcceptedAnswerId="6596" CreationDate="2017-05-05T11:00:09.717" Score="7" ViewCount="355" Body="&lt;p&gt;In my research area one seminal reference is &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF01341708&quot; rel=&quot;nofollow noreferrer&quot;&gt;H. Bethe, ''Zur Theorie der Metalle'', Z. Phys. &lt;strong&gt;71&lt;/strong&gt; 205 (1931)&lt;/a&gt;, see also the &lt;a href=&quot;https://homepages.dias.ie/dorlas/Papers/Bethe.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation by T. C. Dorlas (2009)&lt;/a&gt;. On page 206 of the original reference we read&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Solange von der Wechselwirkung abgesehen wird, gibt es für jedes Atom&#xA;  zwei Zustände mit gleicher Energie: Der Spin des Leuchtelektrons kann&#xA;  nach rechts oder nach links zeigen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus it seems that at the time, at least in the German-speaking scientific community, the terminology spin 'left' and 'right', rather than 'up' and 'down, was used for the two basis vectors of the spin-1/2 irrep of $su(2)$. Was this indeed common practice? If so, was this the original terminology used? And why and by whom was it changed to 'up' and 'down'?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;NB. I can speculate that the following happened. 'Left' and 'right' may originate from the Stern--Gerlach experiment, of course depending on the direction of the external magnetic field in the original experiment. Later on, '$\leftarrow$' and '$\rightarrow$' might then have been replaced by the more compact '$\uparrow$' and '$\downarrow$' for typographical reasons.&lt;/p&gt;&#xA;" OwnerUserId="4703" LastEditorUserId="4703" LastEditDate="2017-10-13T14:04:53.603" LastActivityDate="2018-03-26T06:44:46.000" Title="What is the origin of the terminology 'spin up/down'?" Tags="|terminology|quantum-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5966" PostTypeId="1" CreationDate="2017-05-05T12:13:21.927" Score="0" ViewCount="3347" Body="&lt;p&gt;If nowaday an engineer needs to know high level of mathematics and physics to complete the constructions of bridges buildings and so on&lt;/p&gt;&#xA;&#xA;&lt;p&gt;what happened in ancient rome, greek middle ages ? they had no idea of physics so how could build they a temple for example or a cathedral or similar&lt;/p&gt;&#xA;&#xA;&lt;p&gt;even today there are failures and buildings and bridges collapse so why for example a church or temple or pyramid do not fall ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;how could anyone build something without knowing the laws of physics ?&lt;/p&gt;&#xA;" OwnerUserId="3786" LastActivityDate="2017-05-06T11:27:42.440" Title="how did people build temples and buildings in the ancient times without knowing newton's law?" Tags="|newton|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5967" PostTypeId="2" ParentId="5966" CreationDate="2017-05-05T18:53:03.687" Score="3" Body="&lt;p&gt;One does not need to know Newton's Laws to construct buildings. Newton's laws are about dynamics, and the parts of physics most relevant to buildings are statics and material science.&#xA;(Statics was a highly developed science in Ancient Greece but not before that.&#xA;Material science did not exist until 17th century).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But even this is irrelevant. We know that large buildings were built long before Ancient Greece,&#xA;for example in Egypt and Babylonia. And the builders of Mexican temples did not know Ancient Greek statics, and it is doubtful that medieval builders knew much of the Greek science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these buildings were possible to build by trial and error and experiments on&#xA;scale models. We can only guess the details how very ancient buildings were built, but it is known that medieval builders of cathedrals used scale models,&#xA;and very little mathematics. And they used common sense, first of all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Similarly mankind built very good ships without hydrodynamics, even without Archimedes law, and complicated irrigation systems without hydrostatics, and artillery without ballistics). Seaworthy ships were built even before writing was invented! People did not understand as we do now how and why these things work, but they could make working things by very long trial and error.&#xA;By the way, even now we do not have a complete theoretical understanding how such things as sails or airfoil work. They are also made mostly by trial and error.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. We all know that Greek science disappeared and was forgotten during the Dark Age in Europe (approximately starting from the fall of the Western Roman empire. But it disappeared also in the Eastern Roman empire).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes they cite an edict of Justinian which formally prohibited mathematics&#xA;along with all other &quot;pagan&quot; science. On the other hand, the same Justinian (an Eastern Roman emperor) sponsored the Hagia Sophia cathedral, which was arguably the most sophisticated building built in the early Middle Age.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-06T11:27:42.440" LastActivityDate="2017-05-06T11:27:42.440" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5968" PostTypeId="2" ParentId="5962" CreationDate="2017-05-05T20:45:25.457" Score="1" Body="&lt;p&gt;$f(x)dx$ is a product such that commutativity is preserved (like the differential quotient as the limit of a sequence of fractions maintains the notation as a fraction). The &quot;physical&quot; notation spares you some parentheses when integrating over different variables with different ranges: $\int_a^b dx\int_c^d dyf(x, y)$. In $\int_a^b \int_c^d f(x, y)dxdy$ the ranges are less clear. This notation is not only used in physics. See for instance p. 270 of this mathematics text book:&lt;a href=&quot;https://www.amazon.de/Mathematik-ersten-Semester-Gruyter-Studium/dp/3110377330/ref=pd_cp_14_1?ie=UTF8&amp;amp;refRID=1TB0TDD9MB9DAA735D6K&quot; rel=&quot;noreferrer&quot;&gt;https://www.amazon.de/Mathematik-ersten-Semester-Gruyter-Studium/dp/3110377330/ref=pd_cp_14_1?ie=UTF8&amp;amp;refRID=1TB0TDD9MB9DAA735D6K&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user5707" LastActivityDate="2017-05-05T20:45:25.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5969" PostTypeId="1" CreationDate="2017-05-06T08:59:43.403" Score="2" ViewCount="194" Body="&lt;p&gt;My question is based on Dunnington's biography of Gauss: Gauss: titan of science. In it Dunnington mentions that Gauss, in his response to Janos Bolyai's paper, sent him a synthetic proof of the angular deficit theorem in hyperbolic geometry, and in addition recommended him to find the hyperbolic volume of a tetrahedron in this new geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also in Dunnington biography, he mentions a note from 1841 by Gauss, again on the volume of the tetrahedron, found among the pages of Lobachevsky's paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what is the importance of the problem, and where are the relevant notes in Gauss's Nachlass?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="64" LastEditDate="2017-07-11T21:42:38.183" LastActivityDate="2017-07-11T21:43:52.420" Title="Why did Gauss recommend Janos Bolyai to find the volume of a tetrahedron in hyperbolic geometry?" Tags="|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="5970" PostTypeId="2" ParentId="5969" CreationDate="2017-05-06T11:10:52.030" Score="2" Body="&lt;p&gt;On the importance of the volume of a tetrahedron in hyperbolic geometry, see the survey article&lt;/p&gt;&#xA;&#xA;&lt;p&gt;John Milnor, &#xA;Hyperbolic geometry: the first 150 years. &#xA;Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 9–24. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;freely available &lt;a href=&quot;http://www.ams.org/journals/bull/1982-06-01/S0273-0979-1982-14958-8/&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;. He discusses the volume of the tetrahedron in detail, which shows that it is important. How Gauss was aware of its importance? Well, this is an established (empirical) fact that Gauss somehow anticipated (or maybe influenced, determined?) the development of mathematics for the next 200 years:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-11T21:43:52.420" LastActivityDate="2017-07-11T21:43:52.420" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5971" PostTypeId="2" ParentId="5964" CreationDate="2017-05-06T11:54:37.623" Score="4" Body="&lt;p&gt;Fraenkel's article does not discuss set theory but gives a review, based on Gauss' letters and notes, how Gauss understood and defined the numbers in comparison with Bolzano, Cauchy, Dedekind, and others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only on p. 5 Fraenkel asserts that Gauss' famous quote &quot;I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics&quot; does not mean what it appears to mean to the avearge reader and does not condemn the infinite in general. He claims that Gauss would have agreed with Cantor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the last pages 48-49, Fraenkel mentiones sytems with infinitely many dimensions, but not set theory.&lt;/p&gt;&#xA;" OwnerDisplayName="user5737" LastEditorDisplayName="user5737" LastEditDate="2017-05-06T14:28:32.633" LastActivityDate="2017-05-06T14:28:32.633" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5972" PostTypeId="1" AcceptedAnswerId="5976" CreationDate="2017-05-06T15:53:48.010" Score="25" ViewCount="65694" Body="&lt;p&gt;Both Tesla and Einstein were brilliant scientists. Tesla said the following on the theory of relativity in a &lt;a href=&quot;http://query.nytimes.com/gst/abstract.html?res=9A02EFDD1E39E33ABC4952DFB166838E629EDE&amp;amp;legacy=true&quot; rel=&quot;noreferrer&quot;&gt;1935 New York Times interview&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The theory, wraps all these errors and fallacies and clothes them in&#xA;  magnificent mathematical garb which fascinates, dazzles and makes&#xA;  people blind to the underlying errors. The theory is like a beggar&#xA;  clothed in purple whom ignorant people take for a king. Its exponents&#xA;  are very brilliant men, but they are metaphysicists rather than&#xA;  scientists. Not a single one of the relativity propositions has been&#xA;  proved.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How can this be? If Einstein's theory is falsifiable yet very reinforced by evidence (the balance for a good scientific theory) why did Tesla disavowed it so strongly?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An emphasis: I am not a physicist, just want to know how could such a venerated scientist go against general relativity if it describes reality so accurately and that if he is of the same field.&lt;/p&gt;&#xA;" OwnerDisplayName="user5738" LastEditorUserId="4251" LastEditorDisplayName="user466" LastEditDate="2021-12-24T21:12:56.543" LastActivityDate="2021-12-24T21:12:56.543" Title="Why did Tesla disagree with Einstein?" Tags="|physics|tesla|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="5973" PostTypeId="2" ParentId="451" CreationDate="2017-05-06T17:39:07.540" Score="4" Body="&lt;p&gt;Cantor was definitively &lt;em&gt;not&lt;/em&gt; influenced by Galileo since he believed that Galileo was an opponent of actual infinite numbers. Cantor quoted Galileo as such (but only 15 years after inventing set theory) in Mitteilungen zur Lehre vom Transfiniten 1888 (footnote on p. 417 of the collected works) and in a letter to Hilbert of January 27, 1900.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The footnote refers to a book of Moigno:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Moigno: Imposs. d. nombre act. inf. Paris 1884. Hier werden Galilei, Gerdil, Torricelli, Guldin, Cavalieri, Newton, Leibniz als solche angeführt, welche sogenannte Beweise gegen die Möglichkeit aktual unendlicher Zahlen geführt hätten.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The letter to Hilbert contains the paragraph: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mein Gegensatz zu &lt;em&gt;Gauss&lt;/em&gt; besteht hingegen darin, daß Gauss &lt;em&gt;alle Vielheiten, mit Ausnahme&lt;/em&gt; der &lt;em&gt;endlichen&lt;/em&gt;, für &lt;em&gt;inconsistent&lt;/em&gt; hält (ich meine &lt;em&gt;unbewusst&lt;/em&gt;, d. h. ohne den Begriff zu haben) und daher &lt;em&gt;kategorisch&lt;/em&gt; und &lt;em&gt;principiell&lt;/em&gt; dasjenige Actualunendliche, welches ich &lt;em&gt;Transfinitum&lt;/em&gt; nenne, verwirft, mithin auch die transfiniten Zahlen, deren Existenz ich begründet habe, für &lt;em&gt;unmöglich&lt;/em&gt; erklärt (V. Brief von &lt;em&gt;Gauss&lt;/em&gt; an &lt;em&gt;Schumacher&lt;/em&gt;, v. 12. Juli 1831). Dies ist übrigens auch der Standpunct der Aristoteliker, auch von &lt;em&gt;Galilei, Leibniz, Newton, d'Alembert, Cauchy&lt;/em&gt;, etc. bis auf &lt;em&gt;Kronecker&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user5739" LastEditorUserId="658" LastEditDate="2017-05-18T00:55:46.550" LastActivityDate="2017-05-18T00:55:46.550" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5976" PostTypeId="2" ParentId="5972" CreationDate="2017-05-06T20:12:30.577" Score="26" Body="&lt;p&gt;For context, Tesla was 79 at the time he gave this interview. He had spent his entire life working on electromagnetism within the framework of prerelativistic theories, such as aether theories and a concept according to which &quot;all space is filled with a gaseous substance.&quot; He accomplished some important things, but he was also somewhat of a kook and a charlatan. He had only two years of formal university education. In the New York Times interview referred to in the question, he promised an apparatus, which he never actually produced, &quot;by which mechanical energy can be transmitted to any part of the terrestrial globe.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can this be? If Einstein's theory is falsifiable yet very reinforced by evidence (the balance for a good scientific theory) why did Tesla disavowed it so strongly?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1935, there was actually not very much direct, strong evidence supporting relativity. Evidence like Michelson-Morley and Eddington's 1919 eclipse observation could have been considered amiguous or uncertain. Ives-Stilwell wouldn't happen until 1938, Rossi-Hall 1941.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tesla actually claimed to possess evidence that falsified special relativity. The same 1935 interview says: &quot;He has measured cosmic ray velocities from Antarus ... which he found to be fifty times greater than the speed of light...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, Tesla claimed to have simple thought experiments that disproved relativity, and he disliked relativity on aesthetic, philosophical, and religious grounds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tesla contradicts a part of the relativity theory emphatically, holding that mass is unalterable; otherwise, energy could be produced from nothing, since the kinetic energy acquired in the fall of a body would be greater than that necessary to lift it at a small velocity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-Hugo Gernsback, Science and Mechanics, November 1931&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is 'thought' in relativity, for example, is not science, but some kind of metaphysics based on abstract mathematical principles and conceptions which will be forever incomprehensible to beings like ourselves whose whole knowledge is derived from a three-dimensional world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I hold that space cannot be curved, for the simple reason that it can have no properties. It might as well be said that God has properties.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;New York Herald Tribune, 1932 Sep 11&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We read a great deal about matter being changed into force and force being changed into matter by the cosmic rays. This is absurd. It is the same as saying that the body can be changed into the mind, and the mind into the body. We know that the mind is a functioning of the body, and in the same manner force is a function of matter. Without the body there can be no mind, without matter there can be no force.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;1932&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="26" LastEditDate="2020-03-23T22:10:40.893" LastActivityDate="2020-03-23T22:10:40.893" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="5977" PostTypeId="1" CreationDate="2017-05-07T21:08:34.517" Score="6" ViewCount="1024" Body="&lt;p&gt;&lt;a href=&quot;http://www.jmilne.org/math/apocrypha.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This article&lt;/a&gt; goes something like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a discussion with Grothendieck, Messing mentioned the formula expressing the integral of &lt;span class=&quot;math-container&quot;&gt;$\exp(-x^2)$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, which is proved in every calculus course. Not only did Grothendieck not know the formula, but he thought that he had never seen it in his life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But how does a great mathematician like Grothendieck hear nothing about the Gaussian integral? How is it possible to explain this?&lt;/p&gt;&#xA;" OwnerUserId="5749" LastEditorUserId="4251" LastEditDate="2021-03-09T09:52:41.690" LastActivityDate="2021-03-12T06:49:53.837" Title="Grothendieck and the Gaussian integral" Tags="|mathematicians|grothendieck|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5978" PostTypeId="1" CreationDate="2017-05-08T20:37:41.180" Score="10" ViewCount="4979" Body="&lt;p&gt;It is commonly said (to children) that we have five senses: taste, sight, touch, smell, and hearing. The term &quot;sixth sense&quot; refers to something supernatural. But we do have more senses. Balance, for example, is as tangible as the other senses. More importantly we sense thoughts that organize other senses in space, time and meaning. And emotions. Is there a complete list of the senses?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does the idea of five senses have ancient roots? Or, since they are selected to be informative about the external objective world, are they the child of the breakthrough of science and engineering since Newton?&lt;/p&gt;&#xA;" OwnerUserId="3663" LastActivityDate="2020-04-23T21:20:46.227" Title="What's the origin of the concept of the five senses?" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5979" PostTypeId="2" ParentId="5978" CreationDate="2017-05-08T22:14:35.473" Score="11" Body="&lt;p&gt;It goes back at least to Aristotle's &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Soul&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;De Anima&lt;/em&gt;&lt;/a&gt;, Book II, ch. 7-11 (these five chapters being respectively devoted to sight, hearing, smell, taste, and touch). This is perhaps where it started, since Aristotle was an incorrigible cataloguer of all human experience, be it either sensory or intellectual.&lt;/p&gt;&#xA;&lt;p&gt;That there are no more than five senses was subsequently argued by Aristotle in the beginning of Book III.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Perhaps it is also interesting to notice that in Buddhist psychology, there are not the five senses, but there are the six &lt;em&gt;saḷāyatana&lt;/em&gt;, which comprise the five&#xA;'Aristotelian' senses as well as &amp;quot;mind&amp;quot;, whose sense objects include thoughts, feelings, etc. See the page for &lt;a href=&quot;https://en.wikipedia.org/wiki/Ayatana&quot; rel=&quot;noreferrer&quot;&gt;Ayatana&lt;/a&gt; on Wikipedia.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Since this question has been revived, I want to add an interesting perspective found in the book &lt;em&gt;Inner Christianity&lt;/em&gt; by Richard Smoley, a respected author on topics such as religion and esotericism. He mentions the possibility of interpreting the story of Jesus and the Samaritan woman at the well as an allusion to the five senses.&lt;/p&gt;&#xA;&lt;blockquote&gt;&lt;p&gt;&#xA;One episode in John’s Gospel illuminates with special clarity the relation between the “I” and the world. Christ meets a woman of Samaria at a well and asks her for water. She refuses on the grounds that he is a Jew, and Jews have no dealings with Samaritans. In response he offers her living water and tells her to go and bring her husband back with her. She replies: “Sir, I have no husband. Jesus said unto her, Thou hast well said, I have no husband: for thou hast had five husbands; and he whom thou hast is not thy husband: in that thou sayest truly” (John 4:17–18).&lt;/p&gt;&#xA;&lt;p&gt;&#xA;From earliest times the woman of Samaria has been understood as a&#xA;personification of the soul. She has had five husbands, meaning the five senses (the number five in esoteric terms often has this connotation). She is living with another man, who is “not her husband.” Maurice Nicoll suggests that this indicates a state where the soul has begun to detach itself from the external world; “at this point the ‘soul’ vaguely turns to other interests — perhaps to some sort of philosophy or to different forms of so-called occultism, to opinions, theories and imagination and so on, in an endeavour to satisfy its thirst with truth other than the truth of the senses.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &lt;em&gt;Inner Christianity&lt;/em&gt;, ch. 6.&lt;/p&gt;&#xA;" OwnerUserId="2353" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-04-23T21:20:46.227" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="5981" PostTypeId="1" AcceptedAnswerId="5982" CreationDate="2017-05-09T02:59:43.143" Score="3" ViewCount="418" Body="&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Partition_of_unity&quot; rel=&quot;nofollow noreferrer&quot;&gt;partition of unity&lt;/a&gt; is a mathematical concept in geometry. I want to know when and in what context this concept appeared.&lt;/p&gt;&#xA;" OwnerDisplayName="user5146" LastActivityDate="2017-05-09T10:01:11.070" Title="When was a partition of unity discovered?" Tags="|mathematics|terminology|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5982" PostTypeId="2" ParentId="5981" CreationDate="2017-05-09T10:01:11.070" Score="7" Body="&lt;p&gt;Partitions of unity were formally introduced by Dieudonne (C. R. 205 (1937) 593-595), and for some time they were even called &quot;Dieudonne decompositions&quot;.&#xA;However is some special cases they were used by Whitney (TAMS, 36 (1934) 63-89).&#xA;This information is taken from Hormander, Analysis of Partial differential operators, vol. I, comments to Chapter I.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-09T10:01:11.070" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5983" PostTypeId="1" CreationDate="2017-05-09T10:45:59.617" Score="4" ViewCount="145" Body="&lt;p&gt;Based on what I know of the discovery of positive charges, Thompson measured the $e/m$ ratio of various nuclei and their $e/m$ ratio (charge-to-mass ratio) was not constant. People did not know whether they contined a single common positively charged particle charge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What made them think that? Given different charge-to-mass ratios, why wouldn't one assume multiple types of positive charges in an atom? What motivated people to think that a single common particle could be the cause of their positive charges? &lt;/p&gt;&#xA;" OwnerUserId="5756" LastEditorUserId="55" LastEditDate="2017-05-10T20:32:29.327" LastActivityDate="2017-05-10T20:32:29.327" Title="How was proton determined to be a common component of all atoms?" Tags="|electricity|particle-physics|atomic-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5986" PostTypeId="1" CreationDate="2017-05-09T21:30:20.930" Score="1" ViewCount="72" Body="&lt;p&gt;We know that low-dimensional materials and topological materials are some of the current hot topics in Physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did this happen with the pre-special relativity discussions in the late nineteenth and early twentieth century, too? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I couldn’t get a conclusion reading &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_special_relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/History_of_special_relativity&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5763" OwnerDisplayName="Caetes" LastEditorUserId="5763" LastEditDate="2017-05-09T22:26:33.173" LastActivityDate="2017-05-10T13:19:49.467" Title="Was the pre-special relativity discussion a hot topic in the late nineteenth and early twentieth century?" Tags="|physics|relativity-theory|electromagnetism|einstein|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5987" PostTypeId="2" ParentId="5983" CreationDate="2017-05-10T00:35:53.603" Score="2" Body="&lt;p&gt;A great source on the history of particle physics is Pais's &lt;a href=&quot;https://www.amazon.com/exec/obidos/ISBN=0198519974/physlinktheultim&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inward Bound&lt;/a&gt;, see also references in &lt;a href=&quot;https://physics.stackexchange.com/questions/220482/in-which-experiments-the-charge-to-mass-ratio-of-proton-was-determined&quot;&gt;In which experiments the charge to mass ratio of proton was determined?&lt;/a&gt; thread on Physics SE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Charge to mass ratios for various ions were measured soon after Thomson's discovery of the electron, first for hydrogen apparently by Wien in 1898 based on canal (or anode) rays discovered by Goldstein in 1886. In 1902 Thomson measured it also for chlorine, nitrogen and neon, which led to the discovery of isotopes, see &lt;a href=&quot;http://www.chemteam.info/AtomicStructure/HO3-DiscoveryProton.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Atomic Structure&lt;/a&gt;. Of course, the ratios for ions depended on a gas, unlike the ones for electrons. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the idea of a &quot;fundamental&quot; atom goes all the way back to 1815-16 when Prout, an English chemist, observed that the atomic weights were whole multiples of the atomic weight of hydrogen, and hypothesized that other elements were groupings of hydrogen atoms, which he called &lt;em&gt;protyles&lt;/em&gt; (from Greek first matter). More careful measurements by Berzelius and Turner in 1828 and 1832 disproved the &lt;a href=&quot;https://en.wikipedia.org/wiki/Prout%27s_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prout hypothesis&lt;/a&gt;, but Rutherford was influenced by it. After the scattering experiments of 1911, when he conjectured existence of a nucleus surrounded by a large void, van den Broek proposed that the atomic number of an element is equal to its nuclear charge. This was experimentally confirmed by Moseley in 1913, and revived the Prout hypothesis, the Berzelius-Turner discrepancies could now be attributed to isotopes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1917 experiments (reported in 1919) Rutherford was able to produce hydrogen nuclei by bombarding nitrogen with alpha-particles, thus confirming the Prout hypothesis directly. And in August 1920, at a meeting of the British Association for the Advancement of Science in Cardiff he modified Prout's &quot;protyles&quot; into &quot;protons&quot; (apparently, the play on &quot;prouton&quot; was  involved) referring to hydrogen nuclei. The 1917 experiments are sometimes even billed as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Proton#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;discovery&quot; of proton&lt;/a&gt;. Chadwick discovered neutrons only in 1932, and the mystery of isotopes and deviations from Prout was finally solved. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proton did not enjoy the &quot;fundamental&quot; status for very long, and not just because of quarks. For a long time after their first appearance in 1964 quarks were considered speculative entities, and their confinement made protons at least &quot;unbreakable&quot;. But many grand unification schemes, starting with the $SU(5)$ one of Georgi-Glashow, predicted &lt;a href=&quot;https://en.wikipedia.org/wiki/Proton_decay&quot; rel=&quot;nofollow noreferrer&quot;&gt;proton decay&lt;/a&gt;, and experimenters were chasing after it ever since.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2017-05-10T00:35:53.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5989" PostTypeId="2" ParentId="58" CreationDate="2017-05-10T11:06:08.580" Score="1" Body="&lt;p&gt;According to Cantor himself it was his desire to replace the mechanical explanation of nature by a more complete theory. See several aspects in &lt;a href=&quot;https://hsm.stackexchange.com/questions/5988/what-of-cantors-claims-has-become-true&quot;&gt;What of Cantor&amp;#39;s claims has become true?&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user5737" LastActivityDate="2017-05-10T11:06:08.580" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5990" PostTypeId="2" ParentId="5986" CreationDate="2017-05-10T13:19:49.467" Score="-1" Body="&lt;p&gt;Once electromagnetism produced a lot of awards in that time and it were researched by top physics, I think not exactly special relativity but Electromagnetism were a hot topic. Thanks @Conifold. &lt;/p&gt;&#xA;" OwnerUserId="5763" LastActivityDate="2017-05-10T13:19:49.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5992" PostTypeId="1" CreationDate="2017-05-10T16:46:04.020" Score="2" ViewCount="136" Body="&lt;p&gt;As is well-known, Maxwell's equations were already consistent with Special Theory of Relativity while the Newtonian law of Gravitation wasn't. It can be shown by solving the equations of Maxwell for a dynamic system that the &lt;em&gt;information&lt;/em&gt; of something done on a charged particle over here reaches somewhere else in time no shorter than it would take for the light to travel between the two places - without explicitly using any of the Relativity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noticing the remarkable resemblance between the Coulomb's law and the Newtonian law of Gravitation, it is tempting to ask whether one can formulate something like Maxwell's equations (which contains the Newtonian law as a special case just like Maxwell's equations contain the Coloumb's law as a special case) for describing the Physics of Gravitation. I am curious as to whether such an attempt was ever made or not. There are chances that before General Relativity, people tried making the laws of Gravitation consistent with Special Relativity exactly along these lines. I know that General Relativity reduces to a set of four equations in the weak field limit (in some particularly chosen frames) which resembles Maxwell's equations to a great extent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I am asking if such attempts were made before the formulation of General Relativity and whether they had got the right equations - the equations that we call today the gravitomagnetic equations (the equations that look like Maxwell's equations and are the weak field approximations of General Relativity)? &lt;/p&gt;&#xA;" OwnerUserId="4257" LastEditorUserId="4251" LastEditDate="2020-06-06T08:10:12.117" LastActivityDate="2020-06-06T08:10:12.117" Title="Modification of Newtonian Gravity based on Maxwell's Theory" Tags="|physics|theoretical-physics|electromagnetism|relativity-theory|gravity|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="5993" PostTypeId="2" ParentId="4956" CreationDate="2017-05-10T20:53:49.820" Score="1" Body="&lt;p&gt;Let me add one paragraph to the answer of Mauro Allegranza (which is completely correct: Bolzano did not accept the bijection as a tool to prove equicardinality. He claimed that a short line has less points than a longer line. A bijection would make all lines appear as equal. But there are different infinities, for instance there are twice as many focal points of ellipses than centers.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bolzano mentiones on p. 102f of Paradoxien des Unendlichen that the number of points of a line, may it be even very short, is infinitely larger than the infinitely many points of an infinite (e.g., geometrical) sequence taken from (or projected into) that line because between two points of the sequence there are infinitely many points of the line. So while the word &quot;nondenumerable&quot; is in fact out of place, the infinite can have different sizes. Bolzano explains this by the example of the circle which has one center (or periphery) but an infinity of diameters. The ratio is infinite. (J. BERG (ed.): Bernard Bolzano, Wissenschaftslehre §§ 1-45, Friedrich Frommann Verlag, Stuttgart (1985), p. 31) &lt;/p&gt;&#xA;" OwnerDisplayName="user5737" LastEditorUserId="12699" LastEditDate="2017-06-03T18:07:51.507" LastActivityDate="2017-06-03T18:07:51.507" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5994" PostTypeId="1" CreationDate="2017-05-11T01:29:46.973" Score="3" ViewCount="838" Body="&lt;p&gt;Thomson's atomic model is a static model. Rutherfold's model is a dynamic one. This poses an apparent problem, i.e., the electrons orbiting the nucleus will radiate and fall into the nucleus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically, was his model immediately widely accepted or was it questioned seriously by people? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="55" LastEditDate="2017-05-11T04:22:05.240" LastActivityDate="2017-05-11T04:22:05.240" Title="Was Rutherford's atomic model questioned at the time?" Tags="|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5995" PostTypeId="2" ParentId="5994" CreationDate="2017-05-11T02:15:38.683" Score="4" Body="&lt;p&gt;It was not accepted immediately, but quickly, courtesy of Bohr. It was he who pointed out in 1911 that electromagnetic radiation would cause electrons in the Rutherford's atom to fall upon the nucleus in no time. But he was also the one who &amp;quot;solved&amp;quot; the problem by suggesting... to ignore classical electrodynamics for quantized stationary orbits in his 1913 model. It was Rutherford's turn to return the favor by asking:&amp;quot;&lt;em&gt;How does an electron decide what frequency it is going to vibrate at when it passes from one stationary state to another? It seems to me that you would have to assume that the electron knows beforehand where it is going to stop&lt;/em&gt;&amp;quot;, see &lt;a href=&quot;http://galileo.phys.virginia.edu/classes/252/Bohr_to_Waves/Bohr_to_Waves.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Bohr's Atom to Electron Waves&lt;/a&gt;. Bohr's mathematician brother Harald intimated that young physicists at Göttingen considered Bohr's model to be too &amp;quot;bold&amp;quot; and &amp;quot;fantastic&amp;quot;, and even Bohr himself admitted that it made no physical sense.&lt;/p&gt;&#xA;&lt;p&gt;Things got worse in 1914, when Fowler made precision measurements on enhanced spectral lines of helium. They allowed him to determine that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rydberg_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Rydberg constant&amp;quot;&lt;/a&gt; for helium was &lt;span class=&quot;math-container&quot;&gt;$4.0016$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, as Bohr's theory predicted. But what a a difference a small discrepancy can make:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Bohr's response to this news was conclusive. He pointed out that his analysis of the circular motion had neglected the finite mass of the nucleus. He should really have taken the electron to have an effective mass equal to mM/(m + M). It is easy to verify that if this correction is made for both hydrogen and helium, the ratio changes from 4 to 4.0016!&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;This discovery won over most physicists to the point of view that Bohr was definitely on to something. Sir James Jeans remarked that the only justification for Bohr's postulates was &amp;quot;the very weighty one of success&amp;quot;. According to Rosenfeld (Introduction to Niels Bohr Collected Works): &amp;quot;At Göttingen, that high place of mathematics and physics, where the sense for propriety was strong, the prevailing impression was one of scandal, or at least bewilderment, before the undeserved success of such high-handed disregard of the canons of formal logic.&amp;quot; But all Germans didn't feel that way. Bohr's friend George Hevesy told Einstein about Bohr's identification of the Pickering-Fowler lines with helium. &lt;strong&gt;Einstein said: &amp;quot;This is an enormous achievement. The theory of Bohr must then be right&lt;/strong&gt;&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-05-11T02:15:38.683" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="5996" PostTypeId="1" CreationDate="2017-05-11T07:01:29.633" Score="1" ViewCount="183" Body="&lt;p&gt;I really cannot see why the Brownian motion is linked with the being of atoms. Cannot it be explained without atoms? You just need some fluctuations. &lt;/p&gt;&#xA;" OwnerUserId="5769" LastActivityDate="2017-05-11T21:27:04.213" Title="Why can Perrin's study of Brownian motion demonstrate the existence of atoms" Tags="|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="5997" PostTypeId="1" AcceptedAnswerId="6001" CreationDate="2017-05-11T18:11:06.153" Score="1" ViewCount="218" Body="&lt;p&gt;Euler obtained the famous result $1+2+3+...= -1/12$. Is this the only result for $\Sigma_{n=0}^{\infty}n$ or have other results been derived, for instance in set theory?&lt;/p&gt;&#xA;" OwnerDisplayName="user5699" LastActivityDate="2017-05-12T13:20:47.543" Title="What results for $1 + 2 + 3 + ...$ are known?" Tags="|number-theory|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="5999" PostTypeId="2" ParentId="701" CreationDate="2017-05-11T20:47:47.280" Score="3" Body="&lt;p&gt;Set theory and the term set (German: Menge) has been invented by Bernard Bolzano (1781-1848). In his posthumous book Paradoxien des Unendlichen, Reclam, Leipzig (1851), but also in his Wissenschaftslehre, Friedrich Frommann Verlag, Stuttgart (1985), Bolzano-Gesamtausgabe, Series I Vol. 11,1, he considers sets and the properties of their elements. It is just another set theory than that invented by Cantor but certainly not worse. Bolzano does not accept the one-to-one correspondence as a tool to measure sets. But he accepts different infinities: There are infinitely many tetrahedrons but there are four times as many corners of tetrahedrons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main idea of set theory however, grouping certain objects with common properties, is much older than the term set / Menge. Euclid for instance considered &quot;every given number of prime numbers&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="user5699" LastActivityDate="2017-05-11T20:47:47.280" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6000" PostTypeId="2" ParentId="5996" CreationDate="2017-05-11T21:27:04.213" Score="2" Body="&lt;p&gt;More precisely it has to do with existence of molecules. The demonstration was obtained after Einstein and Smoluchowski developed a QUANTITATIVE theory (based on the molecular theory) whose &#xA;predictions could be measured. Measuring parameters of Brownian motion (by Perrin)&#xA;confirmed the theory. There was no better (or competing) explanation of these measured parameters.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-11T21:27:04.213" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6001" PostTypeId="2" ParentId="5997" CreationDate="2017-05-12T12:49:53.460" Score="2" Body="&lt;p&gt;First let me explain that Euler in fact obtained this result since some comments to the original question seem to doubt that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leonhard Euler applied the $\zeta$ series to obtain the result, $\zeta(-1) = -1/12$ but he knew that summation of divergent series is doubtful, namely &quot;the cause of great dissent among mathematicians of whom some deny and others affirm that such a sum can be found.&quot; See Ed Sandifer: How Euler did it. &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2006-06.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://eulerarchive.maa.org/hedi/HEDI-2006-06.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact Euler applied the series where it does not converge at all and hence does not converge to $-1/12$ either. But at his times a lot of undigested math was en vogue. So he assumed with Leibniz and the Bernoullis that $1-1+1-1+1-+... = 1/2$ and with Wallis that $1/3 &amp;lt; 1/2 &amp;lt; 1/1 &amp;lt; 1/0 &amp;lt; 1/-1$. All that is wrong. Obviously $1+2+3+...&amp;gt; 1$. And if something else is meant by &quot;$+$&quot;, then it should be clearly indicated. Unfortunately Ramanujan and many modern mathematicians accept this confusing notation. On the other hand, Euler has obtained correct results from wrong assumptions like $ log2\infty - log\infty= log2 $ or $(1 + x/\infty)^\infty = e^x$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bernard Bolzano in his Paradoxien des Unendlichen, p. 44f denotes the sum of all natural numbers by $S^0$. Further he introduces $N^0$ what we would call the cardinal number today, taking one unit for every natural number. He claims that some (without quoting names) have put $N^0(N^0+1)/2 = S^0$. But he does not endorse this result. He only claims that the sum $S^0$ must be by far larger than $N^0$. If I remember correctly, Euler somewhere wrote $i(i+1)/2$ with $i$ for &lt;em&gt;numerus infinitus&lt;/em&gt; as he often did. But I am not sure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cantor summed all natural numbers with the result $	1 + 2 + 3 + ... \nu... = \omega$ in a letter of 1883 to Mittag Leffler (i.e., before he had introduced cardinal numbers). He obtained the same sum for the harmonic series. He emphasized however the order of the terms: $2 + 3 + ... + \nu+ ... + 1 = \omega + 1$. Whether $2$ is necessary as first term of the sequence is left open but obviously it is not necessary. For another order he states $1 + 3 + 5 +  ... + (2\nu + 1) + ... + 2 + 4 + 6 + ... + (2\nu) + ... = 2\omega$&lt;/p&gt;&#xA;" OwnerDisplayName="user5737" LastEditorDisplayName="user5737" LastEditDate="2017-05-12T13:20:47.543" LastActivityDate="2017-05-12T13:20:47.543" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6002" PostTypeId="1" CreationDate="2017-05-13T12:44:40.943" Score="0" ViewCount="182" Body="&lt;p&gt;I've read Marcus Du Sautoy's &quot;The Music of Primes&quot; recently and, although I'm not so much interested in number theory, I really liked it for the global perspective on the subject, though at an &quot;educational&quot;, &quot;popular &quot; level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also goig to read Aczel and Singh's &lt;em&gt;Fermat Last Theorem&lt;/em&gt;. Now, I am wondering wether there are similar books that a fourth-year student of Mathematics like me could appreciate. In particular I would like to have an insight in the recent history of the subjects, let's say from 1940-50 until now. Of course Du Sautoy's level is a bit low, so a little &quot;higher-level&quot; references are also welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, are there such books for algebraic topology, modern set theory, or other abstract issues? I'm asking because there doesn't seem to be a way to have a perspective on recent history of some topics, on the important and challenging theoretical problems and approaches, also - I must say - in order to choose my career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope the question is not off-topic. (Perhaps I should have posted it in the Math StackExchange?) Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="4135" LastEditorUserId="4135" LastEditDate="2017-05-13T19:29:12.640" LastActivityDate="2017-05-14T21:03:42.837" Title="Books on recent developments of abstract mathematics" Tags="|mathematics|mathematicians|mathematical-logic|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6003" PostTypeId="2" ParentId="6002" CreationDate="2017-05-13T15:59:56.610" Score="4" Body="&lt;p&gt;Princeton companion to mathematics, Gowers, ed.&#xA;&lt;a href=&quot;http://press.princeton.edu/titles/8350.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://press.princeton.edu/titles/8350.html&lt;/a&gt;&#xA;is perhaps what you want.&#xA;It tries to cover the whole mathematics (and of course fails).&#xA;But parts of it it could be an interesting reading.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now in most areas of mathematics there are introductory undergraduate-level books. Not on &quot;history&quot; but just introducing to the area.&#xA;But you have to specify the area, of course because otherwise the list will be too long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Usually mathematics after 1950 is not qualified as &quot;history&quot;: the main participants can be still alive. What really belongs to &quot;history&quot; will be decided by next generations. So they do not write &quot;history books&quot; but write surveys on various level. There is a nice collection &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Development of mathematics 1900–1950 (Luxembourg, 1992), 369–384, Birkhäuser, Basel, 1994,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but this is probably the latest period&#xA;&quot;history of mathematics&quot; deals with.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-13T16:07:04.853" LastActivityDate="2017-05-13T16:07:04.853" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6004" PostTypeId="1" AcceptedAnswerId="6005" CreationDate="2017-05-13T17:00:06.883" Score="0" ViewCount="141" Body="&lt;p&gt;Many prominent mathematicians like Barrow, Cavalieri, Wallis, Moigno, Bolzano, Bayes, Venn, Gerdil were clergymen. Are there investigations in the social sciences about the percentage of clergymen in mathematics compared to the percentage of clergymen in the population? I expect an overproportionate share because of the high intellectual level of clergyman compared to the general population. &lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-05-13T18:33:10.787" Title="Are there overproportionately many clergymen in mathematics?" Tags="|social|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6005" PostTypeId="2" ParentId="6004" CreationDate="2017-05-13T18:33:10.787" Score="4" Body="&lt;p&gt;The answer depends of the period, of course. I do not think you will find many clergymen among mathematicians in 20th century, or even in the 19th century.&#xA;The reason is simple: most mathematicians need some income for living.&#xA;So what jobs were available to mathematicians in various periods?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since 18th century they mostly take positions at the universities. Until 17th centuries, most universities were essentially religious institutions, their most important function was preparation of clergymen, and many professors were clergymen. If one did not stay with a university, becoming a priest was a good alternative, and usually brought a good income.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were alternatives of course: many mathematicians until 17th century practiced medicine or astrology. I think these three professions were most popular among mathematicians until 17th century: clergyman, medical doctor or astrologer. Except those who had independent means and did not need additional income. But this was always a small minority.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Newton was pressed to become a clergyman: this was a condition for his position in Cambridge. Only with great efforts he obtained an exemption (this had to be granted by the king!) And the reason was that Newton had beliefs which strongly differed from mainstream Anglican dogma (he was a secret Arian: did not believe in the Trinity). &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-13T18:33:10.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6006" PostTypeId="2" ParentId="3013" CreationDate="2017-05-14T05:14:32.770" Score="3" Body="&lt;p&gt;One Google page says &quot;He was as Lucasian Professor of Mathematics from 1839, resigning because of ill-health in 1849 having given no lectures and published only one paper&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I remember reading 20  or 30 years ago that in times past, the only responsibility of a professor at Cambridge or Oxford was to prepare the yearly examination paper. If that is so, he may have performed his duties to a high standard.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are the names of some of the luminaries who won the Smith's Prize after King:&#xA;1823 Airy, 1830 Heaviside, 1841 Stokes, 1842 Cayley.&lt;/p&gt;&#xA;" OwnerUserId="5780" LastActivityDate="2017-05-14T05:14:32.770" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6007" PostTypeId="1" AcceptedAnswerId="6024" CreationDate="2017-05-14T08:10:27.447" Score="-3" ViewCount="160" Body="&lt;p&gt;From the only reason that two sets, $A$ and $B$, are corresponding to each other by the fact that for every part $a$ being in $A$ there is a part $b$ being in $B$ such that all pairs $(a + b)$ formed in this way contain every thing contained in $A$ or $B$ and each thing only once – only from this reason it is, as we see, not yet allowed to conclude that these two sets, if they are infinite, with respect to their number of things (disregarding the differences of their parts) are equal to each other. [...] Two finite sets, if they are of a kind such that we can find to every thing $a$ of the first set a thing $b$ of the second set and no thing is remaining and no thing appears in two or more pairs, have always the same number of things. It appears as if this property should be maintained for infinite sets too. It appears so, but on closer inspection we see that it is not at all necessary because the reason lies in the finiteness and therefore vanishes if the sets instead of being finite are infinite. [...] The conclusion becomes invalid as soon as the set of things in $A$ is infinite because, by the definition of an infinite set, we never encounter a last thing in $A$ – how many things we may have counted, there are always others to be counted. And although there is no lack of things in the set $B$ too which can be paired with the things of $A$, the reason becomes invalid to conclude that the multitudes of both sets are equal. [Bernard Bolzano: &quot;Paradoxien des Unendlichen&quot;, Reclam, Leipzig (1851) 31-33]&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-05-14T08:15:44.910" LastActivityDate="2017-05-17T13:59:18.797" Title="Has Bolzano's opinion about the irrelevance of an infinite bijection ever found supporters?" Tags="|set-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6008" PostTypeId="1" AcceptedAnswerId="6010" CreationDate="2017-05-14T11:18:50.157" Score="8" ViewCount="939" Body="&lt;p&gt;I have started to compile a list of instances of suppression of scientific ideas in history. Up to now I have collected the following points:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;According to Dioganes Laertius Anaxagoras was imprisoned and then exiled because of impiety. He had claimed that Helios is not a God but a heap of glowing stones.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The cases of Giordano Bruno and Galileo Galilei are well known without sources.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cavalieri and Torricelli (a pupil of Galilei), two protagonists of  infinitesimals, were harassed and finally silenced by the leading Jesuit mathematicians. &quot;Once the Revisors had issued their decision, a well-oiled machinery of enforcement sprang into action.&quot; (Amir Aleander: &quot;Infinitesimal&quot;, Oneworld, 2015)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Brouwer was fired as an editor of Mathematische Annalen by Hilbert in 1928 because of incompatibility of Hilbert's and Brouwer's ideas. In this case Hilbert acted immorally and went beyond his legal rights. The whole story is given by Mueckenheim &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/KB/&quot; rel=&quot;noreferrer&quot;&gt;https://www.hs-augsburg.de/~mueckenh/KB/&lt;/a&gt; from KB 548 to KB 658 under the title (coined by Einstein) &quot;Krieg der Frösche und der Mäuse&quot;. The text is in German but with many English documents, some of them published for the first time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Few years later the &quot;Jewish science&quot; was banned from Germany, among it the theory of relativity (100 Autoren gegen Einstein).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am interested to collect more instances of suppression of scientific ideas. (I know that in Soviet union every author of a scientific book had to praise Lenin as the best human ever or at least had to quote some important phrase of his, but that is not what I would call a suppression.)&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2019-11-21T16:57:20.997" Title="Instances of suppression of scientific ideas" Tags="|social-context|" AnswerCount="6" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="6009" PostTypeId="1" AcceptedAnswerId="6019" CreationDate="2017-05-14T15:48:50.143" Score="5" ViewCount="967" Body="&lt;p&gt;When talking about the inconsistencies in physics that led up to Einstein's discovery of relativity today's professors always say that Maxwell's discovery of the constant speed of light  $c$ created a huge contradiction in physics. (namely: was there a special frame relative to which the speed of light was $c$ OR were the laws of EM wrong). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is why didn't the scientists realize something was wrong right when the law saying that the magnetic force on a body was proportional to it's velocity was first proposed? someone &lt;em&gt;had&lt;/em&gt; to ask &quot;velocity relative to which observer&quot; right? if they did realize something was wrong, why did they continue to use it in mainstream physics?&lt;/p&gt;&#xA;" OwnerUserId="4779" LastEditorUserId="4779" LastEditDate="2017-05-14T16:56:59.857" LastActivityDate="2017-05-18T06:12:56.940" Title="&quot;Tension&quot; between Electromagnetism and Newton's laws" Tags="|physics|philosophy-of-science|electromagnetism|light|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6010" PostTypeId="2" ParentId="6008" CreationDate="2017-05-14T19:06:34.577" Score="5" Body="&lt;p&gt;There is an interesting &lt;a href=&quot;https://books.google.de/books?id=tgraBwAAQBAJ&amp;amp;pg=PA380&amp;amp;dq=%22establishment+stifles+genius%22&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwiR25HjifDTAhUDXSwKHZ2VCKEQ6AEIJjAA#v=onepage&amp;amp;q=%22establishment%20stifles%20genius%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;essay&lt;/a&gt; by Truesdell on the early history of the kinetic theory of gases called &quot;The establishment stifles genius: &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Herapath&quot; rel=&quot;noreferrer&quot;&gt;Herapath&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/John_James_Waterston&quot; rel=&quot;noreferrer&quot;&gt;Waterston&lt;/a&gt;&quot;, in which two cases of British scientists are reported whose visionary papers on the kinetic theory of gases did not get published (or only in minor journals) and remained in the vaults of the Royal Society in London until after the authors' deaths.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-05-16T09:53:50.060" LastActivityDate="2017-05-16T09:53:50.060" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6011" PostTypeId="2" ParentId="6002" CreationDate="2017-05-14T20:55:38.493" Score="1" Body="&lt;p&gt;Have a look at Zalamea, Synththetic Philosophy of Contemporary Mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rads.stackoverflow.com/amzn/click/0956775012&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Synthetic-Philosophy-Contemporary-Mathematics-Fernando/dp/0956775012&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a book of philosophy, since as others noted it's too soon to write the history of post-1950's math. Zalamea's book gives an overview of the Grothendieck revolution in math and its philosophical relevance. It brings the philosophy of math out of its Russell-Frege-Gödel phase and into the present. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Very well written and easy to read. If you're interested in the modern history of algebraic topology, you'll find plenty here to enjoy.&lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="340" LastEditDate="2017-05-14T21:03:42.837" LastActivityDate="2017-05-14T21:03:42.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6012" PostTypeId="2" ParentId="6009" CreationDate="2017-05-15T07:48:09.127" Score="0" Body="&lt;p&gt;Concerning Einstein's relativity, there was only one relevant contradiction in physics before 1905: The ether theory had (wrongly) predicted that the speed of light is independent of the speed of the light source while Newton's emission theory of light had (correctly) said that the speed of light does vary with the speed of the source. The Michelson-Morley experiment had confirmed the latter (correct) prediction but Einstein found it profitable to adopt the former (wrong) one as his 1905 second (&quot;light&quot;) postulate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://philsci-archive.pitt.edu/1743/2/Norton.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://philsci-archive.pitt.edu/1743/2/Norton.pdf&lt;/a&gt; &#xA; &quot;The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://books.google.com/books?id=JokgnS1JtmMC&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://books.google.com/books?id=JokgnS1JtmMC&lt;/a&gt; &#xA; Banesh Hoffmann, Relativity and Its Roots, p.92: &quot;There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that &quot;the introduction of a 'luminiferous ether' will prove to be superfluous.&quot;&lt;/p&gt;&#xA;" OwnerUserId="394" LastActivityDate="2017-05-15T07:48:09.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6014" PostTypeId="2" ParentId="6008" CreationDate="2017-05-15T16:26:41.140" Score="17" Body="&lt;p&gt;In Soviet Union, in 1948 genetics was officially banned. Researchers in genetics&#xA;had either co confess publicly that they were doing pseudo-science and switch to another activity, or were dismissed. Several were imprisoned and even killed&#xA;(Isaak Agol, Solomon Levit, Grigorii Levitskii, Georgii Karpechenko and Georgii Nadson were executed. Nikolai Vavilov died in prison).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the most famous episode of persecution of science in Soviet Union, when a whole large area of science was formally banned. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of individual scientists, there were just too many cases to mention here.&#xA;Some of this persecution was not related to their scientific activity, but in many cases it was: they were repressed because their work was found somehow inconsistent with &quot;Marxist-Leninist philosophy&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The punishment varied from a reprimand (and a public &quot;confession&quot;) to firing, to a term in a prison camp (some of them were real death camps), to execution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cybernetics (informatics, computer science) was also banned for some period, in 1960s but this was after Stalin's death, and nobody was killed for cybernetics at least.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-15T17:38:13.487" LastActivityDate="2017-05-15T17:38:13.487" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6015" PostTypeId="2" ParentId="6008" CreationDate="2017-05-15T16:38:17.740" Score="4" Body="&lt;p&gt;Michael Servetus (Miguel Serveto) was burnt alive on John Calvin's order in 1553 in Geneva. He discovered the pulmonary blood circulation, published the book, and the book was labeled heretical.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6222" LastEditDate="2019-11-20T13:13:20.550" LastActivityDate="2019-11-20T13:13:20.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6016" PostTypeId="2" ParentId="6008" CreationDate="2017-05-15T16:54:00.737" Score="4" Body="&lt;p&gt;In 415 a mob incited by a Christian bishop in Alexandria lynched the last Alexandrian mathematician/astronomer Hypatia, thus putting the end to the greatest scientific school of the ancient world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 528 emperor of the Eastern Roman empire prohibited all pagan science. Which at that time meant ALL science. Seven philosophers, including mathematician Simplicius were exiled (to Persia). In 533 they were allowed to return on the condition that they will not teach or publish anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just two episodes of the systematic suppression of science by Christianity&#xA;(and to a much lesser extent by Islam too). The suppression was not directly aimed at science but at any kind of &quot;free thinking&quot;. I suppose these rulers did not know anything about science, so they could not target science intentionally.&#xA;Their aim was &quot;pagan rites&quot;, &quot;pagan philosophy&quot;, astrology and magic. Science,&#xA;which was a marginal activity in comparison with those listed above suffered what is called nowadays &quot;collateral damage&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. It is hard to tell what Justinian really had in mind.&#xA;The word &quot;science&quot; did not exist (at least not in the modern meaning) and the word &quot;mathematicus&quot; meant &quot;astrologist&quot; at the time of Justinian. (Probably because astrology was the only way to make a living for a mathematician). His&#xA;law actually prohibited &quot;pagan rites&quot;, teaching &quot;pagan philosophy&quot;, the practice of &quot;mathematici&quot;, weather prediction etc. I suppose that Justinian simply had no idea about the activities which we call now &quot;science&quot; and &quot;mathematics&quot;.&#xA;In any case, his laws (and similar laws by Christian emperors before him, and the general atmosphere of intolerance after Christianity became the state religion) had the effect that all&#xA;scientific activities whatsoever stopped for almost a millenium. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-17T20:24:17.713" LastActivityDate="2017-05-17T20:24:17.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6017" PostTypeId="1" CreationDate="2017-05-15T18:17:46.430" Score="1" ViewCount="277" Body="&lt;p&gt;How did Newton come up with the three laws of motion?&#xA;Are these laws are true of newton, or they are of Galileo.&lt;/p&gt;&#xA;" OwnerUserId="5784" LastActivityDate="2017-05-18T05:20:09.690" Title="Which problems led Newton to three laws of motion?" Tags="|experimental-physics|classical-mechanics|newton|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6018" PostTypeId="2" ParentId="6017" CreationDate="2017-05-15T21:44:45.303" Score="1" Body="&lt;p&gt;The nature of the surviving evidence makes it hard to tell 'how Newton came up with his laws of motion':  much of the evidence is in the successive verbal formulations of Newton's successive drafts, which often don't reveal much of his background thinking. Also, in the 'Principia' the laws of motion have the alternative title 'axioms'. One of the curiosities about them is that even though in retrospect they have been recognised as a highly original compilation, Newton did not seem to claim much by way of authorship of them: thus he concluded the statement of the axioms or laws of motion with a 'scholium' beginning: 'The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds. ....' and he went on to acknowledge work or books by Galileo, Wren, Wallis, Huygens and Mariotte. There is a body of opinion tending to the view that Newton was being unduly modest about this, and it is worth looking at R S Westfall's &quot;Force in Newton's Physics : the science of dynamics in the seventeenth century&quot; (1971) to see why.  It devotes chapters to detailed studies of the dynamics (or kinematics) of Galileo, Descartes and Huygens among others, as well as to Newton.      &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This is the kind of question on which wikipedia is sometimes not at its strongest, better to trust the linked sources than the wp article itself, which in this case doesn't cite Westfall's work and begins by anachronistically mis-stating the laws themselves.) Better to read the laws as Newton wrote them, plus their explanatory passages, directly from 'Principia' iself, for which there's a good 1729 English translation free of copyright restrictions. The wikipedia article on the Principia (&lt;a href=&quot;https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica&lt;/a&gt;) has plenty of links to different parts of it, such as (&lt;a href=&quot;https://books.google.co.uk/books?id=Tm0FAAAAQAAJ&amp;amp;pg=PA19&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.co.uk/books?id=Tm0FAAAAQAAJ&amp;amp;pg=PA19&lt;/a&gt;) for the laws of motion themselves, but please read on to cover the references to 'accelerative force' etc.  &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-05-15T21:44:45.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6019" PostTypeId="2" ParentId="6009" CreationDate="2017-05-16T01:05:24.710" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;someone had to ask &quot;velocity relative to which observer&quot; right?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They thought it was relative to the frame of the aether. People like Lorentz also knew ca. 1880-1895 that any observable effect of motion relative to the aether would vanish until you got to at least order $(v/c)^4$. They understood that the electric and magnetic fields would transform when you changed frames. They just didn't know that it was possible to make the form of Maxwell's equations &lt;em&gt;exactly&lt;/em&gt; frame-independent by doing what Einstein did in his 1905 paper. It was a &lt;em&gt;very&lt;/em&gt; radical and surprising step that Einstein took. Before 1905, people like Lorentz conceived of things like the Lorentz transformation in terms of stresses and strains generated by the motion of matter through the aether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related: &lt;a href=&quot;https://physics.stackexchange.com/questions/7049/history-of-electromagnetic-field-tensor&quot;&gt;https://physics.stackexchange.com/questions/7049/history-of-electromagnetic-field-tensor&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-05-16T01:05:24.710" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6020" PostTypeId="1" CreationDate="2017-05-17T03:23:30.463" Score="2" ViewCount="310" Body="&lt;p&gt;At the beginning,how and using which principles scientists determined the mass and distance from sun of various planets?&lt;/p&gt;&#xA;" OwnerDisplayName="S.Kundu" LastActivityDate="2017-05-20T16:09:13.853" Title="Determination of planetary mass and radius" Tags="|astronomy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6021" PostTypeId="1" CreationDate="2017-05-17T10:46:24.077" Score="4" ViewCount="2208" Body="&lt;p&gt;I asked the following question on the Physics stackexchange, only to be notified that it was more suitable for this forum. The link to original post can be found at: &lt;a href=&quot;https://physics.stackexchange.com/questions/333125/alexandrian-knowledge-on-distance-of-sun-from-earth&quot;&gt;https://physics.stackexchange.com/questions/333125/alexandrian-knowledge-on-distance-of-sun-from-earth&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is as follows. In the book &quot;Cosmos&quot;, Carl Sagan describes the methods used by Eratosthenes in determining the radius of earth (Chapter 1). Going through the steps of the method, he makes the statement &quot;The Sun is so far away that its rays are parallel when they reach the Earth&quot;, which is crucial for Eratosthenes' conclusion. Sagan does not explain how Alexandrian scholars knew this quoted fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it true that the Alexandrian scholars already knew this fact? If yes then how did they determine this or how might they have determined this? Did they know the distance of Sun from Earth? If yes then how?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Following link was given as an answer in Physics stackexchange: &lt;a href=&quot;https://en.wikipedia.org/wiki/Astronomical_unit#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Astronomical_unit#History&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, this link does not describe the method used by Eratosthenes, rather only by Ptolemy, who came into the scene much later. I am interested in the method used by or before Eratosthenes. &lt;/p&gt;&#xA;" OwnerUserId="5796" LastActivityDate="2017-05-30T17:47:31.217" Title="Historical knowledge of Distance of Earth from Sun" Tags="|astronomy|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6022" PostTypeId="2" ParentId="6008" CreationDate="2017-05-17T12:36:24.437" Score="0" Body="&lt;p&gt;Well, if philosophy counts as a science, then the case of Kant belongs here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was in 1794 a decree muzzling freedom of speech from Prussian King Friedrich Wilhelm II. to Immanuel Kant. &lt;a href=&quot;https://korpora.zim.uni-duisburg-essen.de/kant/briefe/640.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://korpora.zim.uni-duisburg-essen.de/kant/briefe/640.html&lt;/a&gt; threatening with highest disgrace if Kant continued to dishonour and disparage the Holy Bible and Christianity. Kant had 1781 in &quot;Kritik der reinen Vernunft&quot; and 1793 in &quot;Die Religion innerhalb der Grenzen der bloßen Vernunft&quot; published some heretical opinions, for instance with respect to the ontological &quot;proof&quot; of God's existence.&lt;/p&gt;&#xA;" OwnerDisplayName="user5739" LastEditorDisplayName="user5739" LastEditDate="2017-05-17T17:55:03.107" LastActivityDate="2017-05-17T17:55:03.107" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6023" PostTypeId="2" ParentId="6021" CreationDate="2017-05-17T13:21:42.320" Score="2" Body="&lt;p&gt;Aristarchus showed how to determine the relative distances and sizes of earth, sun and moon based on eclipses:&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=ozuEb_qLNys&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=ozuEb_qLNys&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the fact the sun is so far away that its rays can be considered parallel is obvious already without such calculations. When we travel large distances, the change in appearance of the night sky is consistent with the earth being a tiny dot and the sun and stars infinitely far away. If the sun was close in relation to the distance we travelled we would instead see drastic parallax effects.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2017-05-17T13:21:42.320" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6024" PostTypeId="2" ParentId="6007" CreationDate="2017-05-17T13:59:18.797" Score="2" Body="&lt;p&gt;D.D. Spalt [1] acknowledges at least that Bolzano's theory, although different from Cantor's, is not per se self-contradictory or wrong. But he deplores the general lack of eagerness to scrutinize his theory. There seems no readiness to deal with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] D.D. Spalt: &quot;Die Unendlichkeit bei Bernard Bolzano&quot; in &quot;Konzepte des &#xA;mathematisch Unendlichen im 19. Jahrhundert&quot; (G. König, Hrsg.), &#xA;Vandenhoeck &amp;amp; Ruprecht, Göttingen (1990), 189-218.&lt;/p&gt;&#xA;" OwnerDisplayName="user5739" LastActivityDate="2017-05-17T13:59:18.797" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6025" PostTypeId="2" ParentId="6021" CreationDate="2017-05-17T19:55:22.797" Score="2" Body="&lt;p&gt;Aristarchus &quot;method&quot; mentioned in the answer of @Viktor Blasjo was impractical. It is clear from Aristarchus's book that he never tried to actually measure &#xA;anything. His book has to be considered a pure mathematical exercise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The simplest method of finding distances is measuring parallax (see Wikipedia).&#xA;The ancients knew this and they also knew that parallax of the Sun is zero to the degree of the accuracy they could measure. Therefore they could only very crudely estimate this distance from below. The parallax of the Sun is too small to be measured in antiquity, or even much later. So they knew essentially nothing about this distance. The story of measuring Sun's parallax is explained here, for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.eso.org/public/outreach/eduoff/aol/market/collaboration/solpar/#chap2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.eso.org/public/outreach/eduoff/aol/market/collaboration/solpar/#chap2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first meaningful measurement of the distance to the Sun was made by Cassini&#xA;in 1672. But even at that time it was impossible to measure Sun's parallax directly: he measured parallax of Mars instead (which is closer, and easier to measure) and used the known RELATIVE distances which were known correctly without the knowledge of absolute distances since Copernicus. So it is sufficient to measure just one distance in the solar system to calculate all the rest. A century later an ingenious method based on observation of transit of Venus was developed and more precise measurement was made as a result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. I tried myself to follow Aristarchus:-) With much better instruments than those available to him. And I confirm by my own experience the common opinion of serious historians of astronomy that his method is useless in practice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Aristarchus proposed several methods. One of his ideas is to use a very large base for the Sun parallax, namely the distance to the Moon. Consider the triangle SME. To find it up to similarity we need two angles. Angle E is measured directly, and or the angle M, Aristarchus had an ingenious ides: just wait until it equals 90 degrees. At this moment you will see exactly half of the Moon. There are two problems. First, how accurately you can fix the moment when exactly half of the Moon is visible? Second, the angle E you measure is close to 90 degrees, say $90-\epsilon$, where $\epsilon$ is small. It is this $\epsilon$&#xA;which you really need and have to measure. But it is much smaller than the accuracy of any instrument Aristarchus could possibly make. (Compute it yourself, using Wikipedia data!)&#xA;So any measurement will give some number which can be slightly greater than 90 or slightly smaller, and you can make no conclusion whatsoever from such a measurement. This is why this method is impracticable. His other methods are of the same nature.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-18T05:46:32.400" LastActivityDate="2017-05-18T05:46:32.400" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6026" PostTypeId="2" ParentId="6020" CreationDate="2017-05-17T20:11:52.950" Score="2" Body="&lt;p&gt;For the distances, between planets, see my answer to the next question,&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/6021/historical-knowledge-of-distance-of-earth-from-sun&quot;&gt;Historical knowledge of Distance of Earth from Sun&lt;/a&gt;&#xA;Once you know the distance and the apparent (angular) size you get the radius.&#xA;Apparent size was known only for the Sun and Moon before telescopes came in use.&#xA;But with a telescope, you see a planet as a disk, and can measure the angular&#xA;size of this disk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mass is trickier. The size of a planet can be measured with a telescope and knowing the distance. Then you make some guess about the density. This gives some estimate. It is as good as your guess of the density.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For planets having satellites the situation is better. You know the orbit of a &#xA;satellite, and can compute the force of gravitational attraction. Using Newton's&#xA;law of attraction, you find the mass. This is about all you could do in the era before the&#xA;space craft. See&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://curious.astro.cornell.edu/physics/57-our-solar-system/planets-and-dwarf-planets/orbits/245-how-do-you-measure-a-planet-s-mass-beginner&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://curious.astro.cornell.edu/physics/57-our-solar-system/planets-and-dwarf-planets/orbits/245-how-do-you-measure-a-planet-s-mass-beginner&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more subtle method involves the perturbation by a planet of the orbits of&#xA;other planets. For example when Neptune's existence was calculated (before it was discovered), this calculation came with a rough estimate of its mass.&#xA;This was long before its satellites were discovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(EDIT: See the very interesting reference in the comment of Franz Lemmermeyer,&#xA;where the results obtained with this method are described.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difficult case&#xA;is Mercury and Venus.&#xA;The mass of Venus was reliably measured only after a space craft was sent to it. Observing the motion of the spacecraft near the Venus, you can compute the mass using the same Newton's Law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A remarkable story is related with Sirius. It has a satellite (also a star, but invisible: too small). This star orbits Sirius and perturbs Sirius's orbit.&#xA;From this perturbation, the mass of the satellite could be found. So in the beginning of 20th century we had a paradoxial situation: we knew the mass of&#xA;the invisible satellite of Sirius (9 light years away!) but did not know the mass of our closest neighbor (Venus).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-20T16:09:13.853" LastActivityDate="2017-05-20T16:09:13.853" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6028" PostTypeId="2" ParentId="6017" CreationDate="2017-05-17T20:17:44.503" Score="1" Body="&lt;p&gt;The problem of the motion of the planets and satellites. Moon, first of all. Also comets. To justify the inverse square law, Newton had to develop mechanics. His three laws summarize in a short, clear and elegant form what was known on mechanics. Then he could combine them with the gravitation law to compute the orbits of the planets, satellites and comets.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-18T05:20:09.690" LastActivityDate="2017-05-18T05:20:09.690" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6029" PostTypeId="2" ParentId="6009" CreationDate="2017-05-18T06:06:31.547" Score="1" Body="&lt;p&gt;This is an extended comment on the answer of @Pentcho Valev. In his first paper on relativity Einstein does not mention the Micheson-Morley experiment. When asked later he said that this was not of crucial importance for him. As I understand, of prime importance was careful reading of Maxwell and thinking about what Maxwell's theory implies. Maxwell's theory is indeed inconsistent with classical (Galileo) relativity Principle. (In mathematical language, Mazwell's equations are not invariant under the group of Galileo transformations. They are invariant under Lorenz transformations instead.) This logically implies special relativity, if one only believes that Maxwell's equations are correct.(And Maxwell's theory was spectacularly confirmed by Hertz). But only the most profound thinkers clearly understood this (Lorenz, Poincare and Einstein). I mean that these people could PREDICT the outcome of the Michelson-Morley experiment even if it were not made.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my short answer to the question is: there was a tension, but very few people&#xA;noticed it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Maxwell's book is a real treasure chest and it was read by very few people: &lt;a href=&quot;http://www.ams.org.ezproxy.lib.purdue.edu/journals/bull/1972-78-05/S0002-9904-1972-12971-9/home.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.ams.org.ezproxy.lib.purdue.edu/journals/bull/1972-78-05/S0002-9904-1972-12971-9/home.html&lt;/a&gt; but Einstin was certainly among them.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-18T06:12:56.940" LastActivityDate="2017-05-18T06:12:56.940" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6031" PostTypeId="2" ParentId="419" CreationDate="2017-05-18T12:49:16.210" Score="1" Body="&lt;p&gt;I assume that addition of numbers other than $n+1$ is &lt;em&gt;not&lt;/em&gt; counting. The first evidence for this gives the famous Ishango bone.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qbaBr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qbaBr.png&quot; alt=&quot;Ishango bone, 20000 to 25000 years old&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has three rows of numbers in unary notation which by turning act like a slide rule for addition in that additions like the following additions are solved:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) + 3 + 6 = 11&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 + 6 + 4 = 13&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 + 6 + 4 + 8 = 21&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The 2 is not graved into the bone.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Fo5DL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Fo5DL.png&quot; alt=&quot;Three rows of the addition &amp;quot;slide rule&amp;quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Wikipedia it has even been speculated that it works as a lunar calender and that the first mathematician was a woman, but that is not common consensus. Remarkable ist the complete set of prime numbers between 10 and 20 in row b.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-05-18T12:49:16.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6032" PostTypeId="1" CreationDate="2017-05-18T13:45:24.817" Score="11" ViewCount="1175" Body="&lt;p&gt;Nowadays we know it is a consequence of the wave property of light. But ironically it was discovered by Newton who held the particle point-of-view of light. So how did he explain his discovery?  &lt;/p&gt;&#xA;" OwnerUserId="5769" LastEditorUserId="55" LastEditDate="2017-05-19T07:16:07.900" LastActivityDate="2017-05-21T20:30:26.930" Title="How did Newton explain his interference rings without wave optics?" Tags="|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6033" PostTypeId="2" ParentId="5666" CreationDate="2017-05-18T17:52:23.583" Score="0" Body="&lt;p&gt;A more appropriate translation of canon (kanon) is &quot;law&quot;. Consider for example the dictionary definition of &quot;canonical hours&quot;: &lt;em&gt;Canonical hours: certain stated times of the day, fixed by ecclesiastical laws, and appropriated to the offices of prayer and devotion;&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="2822" LastActivityDate="2017-05-18T17:52:23.583" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6034" PostTypeId="2" ParentId="6020" CreationDate="2017-05-19T03:12:04.463" Score="0" Body="&lt;p&gt;In case of the Earth's mass, it was measured by William Cavendish through his experiment with a spring pendulum (like Coulomb did for the determination of the &lt;em&gt;Coulomb's Constant&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.juliantrubin.com/bigten/cavendishg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Explanation of his experiment&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5798" LastActivityDate="2017-05-19T03:12:04.463" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6035" PostTypeId="2" ParentId="6032" CreationDate="2017-05-19T04:43:48.570" Score="11" Body="&lt;p&gt;&amp;quot;Newton rings&amp;quot; in thin plates were discovered not by Newton but by Hooke, and not in 1717 but in 1664 (Boyle described a similar phenomenon in soap bubbles at about the same time). Hooke reported his experiments in Micrographia (1665), from which Newton learned about them, &amp;quot;&lt;em&gt;in which book he hath also largely discoursed of this… and delivered many other excellent things concerning the colours of thin plates&lt;/em&gt;&amp;quot;, as he acknowledged in his Discourse of Observations (1676). But when the Discourse was converted into Opticks and published  after Hooke's death the mention was taken out, hence the successors learned of them as &amp;quot;Newton rings&amp;quot;, see &lt;a href=&quot;http://www.encyclopedia.com/people/science-and-technology/physics-biographies/sir-isaac-newton&quot; rel=&quot;noreferrer&quot;&gt;Encyclopedia's summary&lt;/a&gt;. In fairness, Hooke only reported the picture qualitatively, whereas Newton performed detailed measurements so precise that Young could later derive wavelengths of various hues from Newton's data to high accuracy.&lt;/p&gt;&#xA;&lt;p&gt;Despite the commonly repeated assertion that Newton's optics was corpuscular (and this was likely his initial intuition) his considered position was actually a hybrid of corpuscular and wave theories (reminiscent of the later &lt;a href=&quot;https://en.wikipedia.org/wiki/Pilot_wave&quot; rel=&quot;noreferrer&quot;&gt;de Broglie's pilot wave theory&lt;/a&gt;), although he presented the wave part as &amp;quot;hypothetical&amp;quot;. The hybridization already appears in his optical manuscripts of 1670-s, when he had to accomodate the partial reflection and refraction in thin plates and diffraction (discovered by Grimaldi in 1665), see &lt;a href=&quot;http://www.newtonproject.ox.ac.uk/view/texts/normalized/OTHE00101&quot; rel=&quot;noreferrer&quot;&gt;Brewster's survey at the Newton Project&lt;/a&gt;. As a result, Newton's corpuscles were not elementary but had what we now call internal degrees of freedom, vibration modes, which he called &amp;quot;fits&amp;quot; and which interacted with the all-pervading &amp;quot;luminous ether&amp;quot;. Here is a detailed explanation from &lt;a href=&quot;https://archive.org/details/theorylight01presgoog&quot; rel=&quot;noreferrer&quot;&gt;Preston's Theory of Light&lt;/a&gt; (freely available online):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The existence of both reflection and refraction at the surface of a transparent substance presents at first sight a great difficulty in the emission theory, for it is not eaay to conceh-e how the same surface may at one time reflect and at another refract an impinging molecule. To meet the difficulty Newton was led from his observations on the coloured rings of thin plates (chap. VIII) to endow the luminous corpuscles with periodic phases or fits, as he terms it, of easy reflection and easy transmission, so that sometimes they are in a condition to be reflected, and sometimes in a condition to be refracted at a transparent surface.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;To communicate these fits to the luminous corpuscles he imagined all space to be filled with an all-pervading medium or ether. The luminous corpuscles, on striking a reflecting or refracting surface, excite waves in this ether which overtake them at regular intervals, and assist or oppose their motion periodically, so that at any new surface they are refracted or reflected according as the wave assists or opposes the corpuscle. The element of periodicity thus so ingeniously introduced, and which is so fundamentally involved in a wave motion, we should naturally expect to be independent of the angle of incidence. However, to reconcile the theory with his observations on thin plates, Newton found it necessary to suppose the length of a fit to vary as the secant of the angle of incidence, and it does not appear easy to account for such a law.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And here is Newton himself in book II of Opticks applying his hybrid theory to explain the rings in thin plates:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Every ray of light in its passage through any refracting surface is put into a certain transient constitution or state, which in the progress of the ray returns at equal intervals and disposes the ray at every return to be easily refracted through the next refracting surface, and between the returns to be easily reflected by it... This is manifest by the 5th, 9th, 12th, and 15th observations (coloured ring). For by those observations it appears that one and the same sort of ray at equal angles of incidence on any thin transparent plate is alternately reflected and transmitted for many successions accordingly as the thickness of the plate increases in arithmetical progression of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, etc.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Strictly speaking, Newton only needs coordinated &amp;quot;fits&amp;quot; to explain the rings, and mathematically his explanation is essentially equivalent to that in wave optics. But the nature of this coordination remains mysterious without the ether. To cover his bases Newton in Opticks first professes the familiar &lt;a href=&quot;https://en.wikipedia.org/wiki/Hypotheses_non_fingo&quot; rel=&quot;noreferrer&quot;&gt;hypotheses non fingo&lt;/a&gt; pretense:&amp;quot;&lt;em&gt;What kind of action or disposition this is or whether it consists in a circulating or a vibrating motion of the ray, or of the medium, or something else, I do not here inquire&lt;/em&gt;&amp;quot;. But for those &amp;quot;&lt;em&gt;averse from assenting to any new discoveries but such as they can explain by a hypothesis&lt;/em&gt;&amp;quot; he provides the rest of his account phrased into rhetorical questions (the idea dates back to &lt;a href=&quot;http://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00002&quot; rel=&quot;noreferrer&quot;&gt;Hypothesis Explaining the Properties of Light, 1675&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;And the vibrations or tremors incited in the air by percussion continue a little time to move from the place of percussion in concentric spheres to great distances. And in like manner, when a ray of light falls upon the surface of any pellucid body, and is there refracted or reflected, may not waves of vibrations, or tremors, be thereby excited in the refracting or reflecting medium at the point of incidence and continue to arise there, and to be propagated from thence... and are not these vibrations propagated from the point of incidence to great distances? And do they not overtake the rays of light, and by overtaking them successively, do they not put them into the fits of easy reflexion and easy transmission described above?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-05-21T20:30:26.930" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6036" PostTypeId="1" CreationDate="2017-05-19T07:09:41.343" Score="6" ViewCount="174" Body="&lt;p&gt;Their papers were published on the same issue of Nature back to back. Moreover, helix was also mentioned in Franklin's paper. So, what important insight or contribution of Watson and Crick was missed by Franklin? &lt;/p&gt;&#xA;" OwnerUserId="5769" LastEditorUserId="4251" LastEditDate="2021-09-10T06:47:00.743" LastActivityDate="2021-09-10T06:47:00.743" Title="What insight of Watson and Crick was missed by Franklin?" Tags="|biology|20th-century|biochemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6037" PostTypeId="2" ParentId="6036" CreationDate="2017-05-19T07:42:59.533" Score="6" Body="&lt;p&gt;Franklin did realize that the structure was the double helix independently, but she did not have time to construct a detailed model based on the diffraction data by the time Watson and Crick constructed theirs. Prior to the publication there was a meeting, and the content of Nature papers, and the splitting of the credit, were agreed upon in advance by both groups. Here is the timeline from &lt;a href=&quot;https://www.theguardian.com/science/2015/jun/23/sexism-in-science-did-watson-and-crick-really-steal-rosalind-franklins-data&quot; rel=&quot;noreferrer&quot;&gt;Cobb's Life's Greatest Secret&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Franklin’s laboratory notebooks reveal that she initially found it difficult to interpret the outcome of the complex mathematics... but by 24 February, she had realised that DNA had a double helix structure and that the way the component nucleotides or bases on each strand were connected meant that the two strands were complementary, enabling the molecule to replicate.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;To prove her point, she would have to convert this insight into a precise, mathematically and chemically rigorous model. Watson and Crick had already crossed the finishing line – the Cambridge duo had rapidly interpreted the double helix structure in terms of precise spatial relationships and chemical bonds, through the construction of a physical model.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In the middle of March 1953, Wilkins and Franklin were invited to Cambridge to see the model, and they immediately agreed it must be right. It was agreed that the model would be published solely as the work of Watson and Crick, while the supporting data would be published by Wilkins and Franklin.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Curiously, Watson's sexism might have prevented him from having any competition altogether (although, naturally, he was more concerned about Pauling than Franklin, whom he called 'Rosy' in print):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Ironically, the data provided by Franklin to the MRC were virtually identical to those she presented at a small seminar in King’s in autumn 1951, when Jim Watson was in the audience. Had Watson bothered to take notes during her talk, instead of idly musing about her dress sense and her looks, he would have provided Crick with the vital numerical evidence 15 months before the breakthrough finally came.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-05-19T07:42:59.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6038" PostTypeId="1" CreationDate="2017-05-19T10:16:01.653" Score="9" ViewCount="2340" Body="&lt;p&gt;This question has been discussed on several sites including MathOverflow but with not definite result. Presumely HSE is best suited.&lt;/p&gt;&#xA;&lt;p&gt;Jeremy Gray denies that Poincare said, &amp;quot;Later generations will regard Mengenlehre as a disease from which one has recovered.&amp;quot; &lt;sup&gt;[1]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;Skolem however notes as a rumor &amp;quot;he is reported to have said in a talk at the international congress of mathematicians at Rome in 1908 that in future set theory would be considered as a disease from which one has recovered.&amp;quot;  &lt;sup&gt;[2]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is there further evidence in support of Skolem?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;[1] Jeremy Gray, &amp;quot;Did Poincare say 'set theory is a disease'?&amp;quot;, Math. Intelligencer 13 (1991) 19-22.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;[2] Thoralf Skolem: &amp;quot;Über die Grundlagendiskussionen in der Mathematik&amp;quot;, Den Syvende Skandinav. Matematikerkongr. Oslo (1929).&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user5699" LastEditorUserId="4251" LastEditDate="2021-03-01T20:08:33.810" LastActivityDate="2021-03-01T20:08:33.810" Title="Did Poincaré say that set theory is a disease?" Tags="|mathematics|biographical-details|set-theory|debunking|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="6039" PostTypeId="2" ParentId="6038" CreationDate="2017-05-19T14:19:10.163" Score="5" Body="&lt;p&gt;I cannot answer the question but only quote some circumstancial evidence.&lt;/p&gt;&#xA;&lt;p&gt;Firstly, I have seen related quotes which are with certainty by Poincaré:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For my part I think, and I am not alone in so thinking, that the important thing is never to introduce any entities but such as can be completely defined in a finite number of words. Whatever be the remedy adopted, we can promise ourselves the joy of the doctor called in to follow a fine pathological case. (Henri Poincaré: &amp;quot;Science and Method&amp;quot;, Nelson, London, 1914 p. 44)&lt;/p&gt;&#xA;&lt;p&gt;There is no actual infinity. The Cantorians forgot this, and so have fallen into contradiction. (Henri Poincaré: &amp;quot;Science and Method&amp;quot;, Nelson, London, 1914 p. 195)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Secondly, Cantor was well aware of Poincaré's opposition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So I understand quite well the opposition of Mons. Poincaré, by which I felt myself honoured, so he never had in his mind to honour me, as I am sure. (Cantor in a letter to Russell, 19 Sept 1911)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is a reference to Poincaré's talk in a letter of Cantor to Grace Chisholm Young of June 1908:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Erstaunt war ich über das, was Herr Henri Poincaré im April d. J. in seiner römischen Vorlesung über den „Cantorisme&amp;quot; sagte, doch kann ich nicht behaupten, dass es mich aus meinem Gleichmuth gebracht hat. Seit vielen Jahren sehe ich es kommen, dass sich die französische Akademie (aufgehetzt von der Berliner Akademie) in ihrem Zorn wider das unerhörte Transfinite resp. „Actuale Unendliche&amp;quot;, das von mir ausgegangen ist, einmal fürchterlich blamiren würde. Dies scheint in Rom zur Thatsache geworden zu sein. Wie bisher, rühre ich auch in Zukunft keinen Finger um, was ich brachte gegen den verblendeten Hochmut der Akademiker diesseits und jenseits des Rheins zu vertheidigen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-05-19T20:57:11.907" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6040" PostTypeId="2" ParentId="419" CreationDate="2017-05-19T21:16:51.447" Score="0" Body="&lt;p&gt;Because this question is also semantically interesting, semiotic aspect should be considered as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taken certain signs in the ancient human history were used for accounting, i.e. &lt;strong&gt;marking cycles&lt;/strong&gt; and &lt;strong&gt;quantities of objects&lt;/strong&gt;, same signs could have been used for &quot;anything&quot; else. There is nothing fundamental restricting that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can count or express quantity without a written sign, for example by fingers. This already suggests that any symbol one takes, it by default means also something else. A finger is one, one is a finger. When writing systems developed, that continued to apply. For example &lt;a href=&quot;https://en.wikipedia.org/wiki/Egyptian_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;Egyptian symbols&lt;/a&gt; (3000 BC) denoting numerals also meant an heal, a rope, a lily, a frog, etc. In many semitic languages (for example in Hebrew), and others like Greek and Latin, alphabet signs were also numerals. And they still continue to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can see by these examples that numbers, were they ordinal or cardinal, represented other things too from the very beginning. No matter if signs were simple notches made with a sharp object to the other object, or if signs were more sophisticated ideograms or pictograms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So semantically answering to the question &quot;When were numbers first used &lt;strong&gt;for anything other&lt;/strong&gt; than counting?&quot; I'd say that practice goes before writing systems were established. And it was other way around, abstract numbers came later. Special cases usually become before a general abstraction. But mental ability to abstract the usage of numbers purely as quantities probably evolved also before writing systems so it is hard to tell when it happened. On the other hand it is quite safe to assume that numbers as separate mental signs and entities were strongly evolved along with agriculture and colonization in the Near East between 10000 BC - 5000 BC. In the 4th and 3rd millennium BC Mesopotamia we can find a lot of &lt;a href=&quot;http://cdli.ox.ac.uk/wiki/doku.php?id=numbers_metrology&quot; rel=&quot;nofollow noreferrer&quot;&gt;metrological texts&lt;/a&gt; using numbers to purely denote quantities and measures like a length, a weight, and a capacity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.math.wichita.edu/history/topics/num-sys.html#babylonian&quot; rel=&quot;nofollow noreferrer&quot;&gt;Babylonian number systems&lt;/a&gt; two different strokes made with a wedge were strictly used to denote single digits and powers of ten. There might be an era, where accounting practice related to taxation forced to use numerals in a very strict sense. One can see a wavy tendence to use number symbols purely for quantities, closer we come to our age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But as noted on other answers, even in 500 BC, Pythagoras promoted the idea of quality over quantity in numbers. So the number abstraction in the sense of strictly meaning only quantity or magnitude never broke totally thru, at least before the modern era. And actually in the modern mathematical notation we hardly ever use numbers, but variables and other abstract symbols instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe the question could be elaborated a bit so that we would like to know, when numbers were either used operatively within formulas or grouped to characteristic sets rather than just as physical counter ticks similar to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lebombo_bone&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lebombo bone&lt;/a&gt; or the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ishango_bone&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ishango bone&lt;/a&gt;? We can try to deduce it from the cave paintings and built structures left from the ancient past. But surely we know it only from the written texts in the 4000 BC - 3000 BC that there is a hint of areas being calculated in Uruk [a]. Also big numbers exceeding tens of thousands and millions might suggest that some addition / multiplication operations has been done rather than counting objects one by one. Major corpus of mathematical texts, where numbers were used in complex equations, not just as counters, come from Mesopotamia and Egypt, 1800 BC [b].&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;a) Pyramids and Cones in Cuneiform and Other Mathematical Texts, Jöran Friberg (1997)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) History of Mathematics, Vagn Lundsgaard Hansen, Jeremy Gray (2010, page 5)&lt;/p&gt;&#xA;" OwnerUserId="5674" LastActivityDate="2017-05-19T21:16:51.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6041" PostTypeId="1" CreationDate="2017-05-19T21:44:51.587" Score="5" ViewCount="257" Body="&lt;p&gt;In the book &lt;a href=&quot;http://www.paultaylor.eu/prafm/html/s22.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;'Practical Foundations of Mathematics' (Paul Taylor)&lt;/a&gt;&lt;/em&gt;, available online, one reads:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As Emile Borel stressed in 1908, the important observation about $\mathbb{Q}$ [that there is a bijection from $\mathbb{Q}$ to $\mathbb{N}$] is that there is an effective coding, not anything to do with its &quot;size&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does anyone know where evidence for Borel stressing this can be found?&lt;/p&gt;&#xA;" OwnerUserId="5802" LastEditorUserId="55" LastEditDate="2017-05-23T01:55:55.667" LastActivityDate="2017-05-23T20:00:14.813" Title="Where did Borel stress that $\mathbb{Q}$ being effectively enumerable by $\mathbb{N}$ is not about its size?" Tags="|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6042" PostTypeId="2" ParentId="6038" CreationDate="2017-05-19T22:35:00.780" Score="8" Body="&lt;p&gt;I would say that the result is pretty definite. Gray does not simply deny that the quote is genuine, he quotes the scholarship of Moore (Zermelo' s Axiom of Choice), Cassinet-Guillemot (L'Axiome du choix) and Dauben (Georg Cantor), all of whom neglect to mention it. Moreover, the sources that do mention it, like Skolem and Kline, trace it to Poincare's remarks at the Fourth International Congress of Mathematicians in Rome, 1908, but Poincare's essay circulated there, &lt;a href=&quot;http://ricardoheras.com/wp-content/uploads/2013/11/Poincare-the-future-of-mathematics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Future of Mathematics&lt;/a&gt;, has nothing of the sort. The closest he comes to curing &quot;disease&quot; is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I think, and I am not the only one who does, that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of the physician called in to follow a beautiful pathological case [beau cas pathologique].&lt;/em&gt;&quot;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;not be completely defined by a finite number of words&quot; is the phrasing that French proto-constructivists (Borel, Baire, Lebesgue) and Poincare used since 1905 to criticize Zermelo's proof of the well-ordering theorem from the axiom of choice, and similar arguments, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/6041/where-did-borel-stress-that-mathbbq-being-effectively-enumerable-by-mathb/6058#6058&quot;&gt;Where did Borel stress that $\mathbb{Q}$ being effectively enumerable by $\mathbb{N}$ is not about its size?&lt;/a&gt; It was motivated by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard's paradox&lt;/a&gt; (of the smallest number not definable in finitely many words), and led to Poincare's predicativist &lt;a href=&quot;https://en.wikipedia.org/wiki/Vicious_circle_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;vicious circle principle&lt;/a&gt;. Thus, the &lt;em&gt;beau cas pathologique&lt;/em&gt; referred specifically to Zermelo-type &quot;constructions&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Thus, although I am favorably disposed to accept Zermelo's Axiom, I reject his proof, which for an instant had made me believe that aleph-one could actually exist.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Better yet, Skolem's sources are reported rumours, and Kline's is... E.T. Bell, whose reputation for making up and embellishing stories is legendary, for some of his handiwork see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2397/what-resources-are-available-for-lives-of-recent-mathematicians-besides-e-t-bel/2400#2400&quot;&gt;What resources are available for lives of recent mathematicians besides E.T. Bell&amp;#39;s Men of Mathematics?&lt;/a&gt; &lt;a href=&quot;https://hsm.stackexchange.com/questions/2105/whats-the-famous-story-about-a-mathematician-who-gave-a-talk-without-saying-a-w/2106#2106&quot;&gt;What&amp;#39;s the famous story about a mathematician who gave a talk without saying a word?&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/questions/5341/did-cauchy-forget-or-lose-mathematical-papers-aside-from-abels-and-galoiss/5343#5343&quot;&gt;Did Cauchy forget or lose mathematical papers aside from Abel&amp;#39;s and Galois&amp;#39;s?&lt;/a&gt; (hint: Cauchy did not even lose Galois', but why let facts get in the way of a good story). Bell quotes the above passage before adding the &quot;&lt;em&gt;later generations will regard Mengenlehre as a disease from which one has recovered&lt;/em&gt;&quot;, dates it to 1905, and Kline copies him despite knowing that the essay is from 1908. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this time the juicy story fell into Bell's lap. The &quot;quote&quot; first appears in Pierpont's address of 1928 (Pierpont at least got the date right), and his source was likely Hölder's 1924 book. One or both of them were likely the sources of Skolem's &quot;reportage&quot; too. Except Hölder offers the &quot;quote&quot; as a summary, and he was not at the Congress himself. There is no trace of the &quot;quote&quot; before 1924, so Gray's reconstruction of events seems very reasonable:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Hölder, who was not at the Rome Congress but knew that Poincare's address had caused a stir, offered at the end of his book a summary of what Poincare&#xA;  had said. The beau cas pathologique became a disease, a not unfair translation possibly coloured by the gossip that would also have spread news of Poincare's&#xA;  opinions. The optimism that Poincare conveyed became the suggestion that one will look back on the disease and see that one has recovered from it.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-05-23T02:06:59.060" LastActivityDate="2017-05-23T02:06:59.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6043" PostTypeId="1" CreationDate="2017-05-20T00:33:07.730" Score="3" ViewCount="162" Body="&lt;p&gt;Do we have any insight or document into theoretical development of the EPR papers both in terms of math, ideas and anything related to actual physical problem.&#xA;I know that Einstein didn't write the papers, but I couldn't find anything about how the work can be actually credited between Einstein, Podolsky and Rosen. Did Einstein contribute essential ideas or just supported the papers with his name and interest to give them prominence and take them to the attention of the scientific community?&lt;/p&gt;&#xA;" OwnerDisplayName="Claudio P" LastEditorUserId="55" LastEditDate="2017-05-21T21:25:19.427" LastActivityDate="2017-05-21T21:25:19.427" Title="What was Einstein's contribution to the Einstein-Podolsky-Rosen (EPR) paper?" Tags="|physics|quantum-mechanics|einstein|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6045" PostTypeId="2" ParentId="6021" CreationDate="2017-05-20T10:24:57.980" Score="2" Body="&lt;p&gt;The best answer I can think of is to refer you to a 1985 book by Albert van Helden, &quot;Measuring the Universe, Cosmic Dimensions from Aristarchus to Halley&quot; &#xA;(1985, Univ of Chicago Press). The overall history is intricate, it would not be useful for me to try to summarize anything of it: just read the book. Van Helden does refer to what little is known of Eratosthenes' work in this connection. It seems likely that assumptions as well as measurement figured in that work.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-05-20T10:24:57.980" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6049" PostTypeId="2" ParentId="674" CreationDate="2017-05-20T14:34:46.197" Score="2" Body="&lt;p&gt;Machgielis (Max) Euwe (1901-1981) was a Dutch mathematician. His teachers were Roland Weitzenböck and L.E.J. Brouwer. The latter was his friend and Euwe held his funeral oration. Euwe was a teacher of mathematics himself. He has published many (if not most) pieces of chess literature, among them &quot;Mengentheoretische Betrachtungen über das Schachspiel&quot; (set-theoretic reflections about the game of chess). Euwe was the fifth world champion (1935 he defeated Alexander Aljechin) and president of the FIDE (world chess unione) just in the difficult time when Bobby Fisher, author of the best sold chess book ever, encountered Boris Spassky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe authored among others the following books:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe: Feldherrnkunst im Schach: eine Studie über die Entwicklung des Schachdenkens vom Jahre 1600 bis heute. Joachim Beyer Verlag, Eltmann 3. Auflage 2015&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe: Schach von A-Z - Vollständige Anleitung zum Schachspiel. Joachim Beyer Verlag, Eltmann 8. Auflage 2012&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe: Theorie der Schacheröffnungen. 12 Bände. Siegfried Engelhardt Verlag, Berlin-Frohnau, 1957&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe, Walter Meiden: Meister gegen Amateur. Joachim Beyer Verlag, Eltmann 8. Auflage 2012&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe, Walter Meiden: Amateur wird Meister. Joachim Beyer Verlag, Eltmann 8. Auflage 2012&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe, Walter Meiden: Meister gegen Meister. Joachim Beyer Verlag, Eltmann 4. Auflage 2011&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe: Positions- und Kombinationsspiel. Joachim Beyer Verlag, Eltmann 6. Auflage 2010&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe: Urteil und Plan im Schach. 3. Aufl. de Gruyter, Berlin, 1968&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Max Euwe: Endspieltheorie und -praxis. Joachim Beyer Verlag, Eltmann 2. Auflage 2014&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(References from German Wikipedia)&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-05-20T14:34:46.197" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6050" PostTypeId="1" AcceptedAnswerId="6054" CreationDate="2017-05-20T17:42:23.843" Score="2" ViewCount="3918" Body="&lt;p&gt;I've heard it claimed Galileo said or wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All truths are easy to understand once they are discovered. The point is to discover them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Where did he say this?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2023-12-16T03:54:27.197" Title="Where did Galileo say &quot;All truths are easy to understand once they are discovered. The point is to discover them.&quot;?" Tags="|reference-request|quote|galileo|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6051" PostTypeId="2" ParentId="639" CreationDate="2017-05-20T17:44:12.730" Score="1" Body="&lt;p&gt;Dictionary sources including Merriam-Webster, Oxford, Cambridge, show 'invention' as connected by etymology with Latin 'invenire' (to find (out)): they offer what looks like essential agreement on a core of meaning relevant here:  something, typically a process or device, that has been invented, discovered or contrived, something never made before; as well as sometimes giving more special meanings having to do with patents (law).  (Not further considered here, of course, are  widely different applications of the word 'invention' such as 'fiction' or 'genre of musical composition'.)  Relevant dictionary definitions of 'innovation' often derive from the verb 'innovate': make changes in something established, introduce some new change or idea, &amp;amp;c.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems there is no generally-accepted and specific agreement about the boundaries of the concept 'innovation'. A recent (2016) survey of 15 'experts' gave rather widely various answers about what they thought of the meaning of 'innovation' -- results that appear to support an idea that the concept 'innovation' has somewhat broad fuzzy boundaries, not necessarily a bad thing if one wishes to have a word of wide application for a general discussion. ('Idea to Value, Community for Creativity and Innovation': at &#xA;&lt;a href=&quot;https://www.ideatovalue.com/inno/nickskillicorn/2016/03/innovation-15-experts-share-innovation-definition/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ideatovalue.com/inno/nickskillicorn/2016/03/innovation-15-experts-share-innovation-definition/&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practical usage a distinction between the two appears connected with the way in which the meaning of 'invention' has to some degree been side-tracked by the complicated laws of patents for inventions. Originally, an 'invention' in law was also something new-created and practical and specially useful -- special enough to deserve the grant of exclusive trade privileges for a set period, e.g. 14 years. Over long periods of time, the legal definitions of invention in many countries have effectively become much more complex [US Manual of Patent Examining Procedure, &lt;a href=&quot;https://www.uspto.gov/web/offices/pac/mpep/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.uspto.gov/web/offices/pac/mpep/&lt;/a&gt; ; Guidelines for Examination in the European Patent Office, &#xA;&lt;a href=&quot;https://www.epo.org/law-practice/legal-texts/guidelines.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.epo.org/law-practice/legal-texts/guidelines.html&lt;/a&gt; .]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Against this background, a significant practical use of the word 'innovation' is to allow discussions about technical progress without the discussions becoming side-tracked into legalistic arguments about what exactly constitutes an invention, whether in law or otherwise. This can be helpful for example in broad-based discussions about technical progress that either have nothing to do with questions of patents, or are not to be limited to that topic.&lt;/p&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;{Added 14 November:} Besides the views expressed in the 'expert survey' already cited, another (and official) example shows a usage of 'innovation' that is broader than 'invention'.  The website of the &lt;a href=&quot;https://www.uspto.gov/about-us/organizational-offices/office-commissioner-patents/office-deputy-commissioner-patent-administration/office-innovation-development&quot; rel=&quot;nofollow noreferrer&quot;&gt;Office of Innovation Development&lt;/a&gt;, part of the US Patent and Trademark Office, refers to &quot;our &lt;a href=&quot;https://www.uspto.gov/learning-and-resources/inventors-entrepreneurs-resources&quot; rel=&quot;nofollow noreferrer&quot;&gt;Innovation Development&lt;/a&gt; webpage&quot;, which turns out to give information for &quot;inventors and entrepreneurs&quot; that concerns not only inventions but also some quite different matters such as trade marks and protectable designs. (Acknowledgement to &lt;a href=&quot;https://hsm.stackexchange.com/users/1815/michael-e2&quot;&gt;Michael E2&lt;/a&gt; for drawing attention to the example). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;'Innovation' can thus be a broader and more general word than 'invention'. It remains up to any particular writer either to say, more specifically, what s/he means by 'innovation' in any discussion, or else to leave it accepted that the word is somewhat vague.  The same applies to whatever is meant by saying that some innovations are 'sustaining', 'evolutionary', 'revolutionary, or  'disruptive': these are vague words with no generally accepted specific definitions in the context.  Articles may be written about the way in which some particular author understands and uses such concepts: and the articles may sometimes imply an assumption that the usage that they employ amounts to a generally-recognized definition -- even where no such general recognition really exists.  Specific definitions may begin to be  needed in special circumstances -- for example if the status of 'innovation' ever brings with it some tangible benefit. But there we begin to come full circle on the reason why there are two words in this area rather than one.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2017-11-14T11:04:57.823" LastActivityDate="2017-11-14T11:04:57.823" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6054" PostTypeId="2" ParentId="6050" CreationDate="2017-05-21T18:44:58.563" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Dialogo sopra i due massimi sistemi del mondo&lt;/em&gt;&lt;/a&gt; (1632), Day Two:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;SAGR. Tali [facili da intendersi] sono tutte le cose vere, doppo che son trovete; ma il punto sta nel saperle trovare.&lt;/p&gt;&#xA;&lt;p&gt;[Stilmann Drake transl., p.225] So [easy to understand] are all truths, once they are discovered; the point is in being able to discover them.&lt;/p&gt;&#xA;&lt;p&gt;[ &lt;a href=&quot;http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?step=thumb&amp;amp;dir=galil_syste_065_en_1661&quot; rel=&quot;nofollow noreferrer&quot;&gt;Th.Salusbury transl.,209&lt;/a&gt;] So are all truths, when once they are found out, but the difficulty lyeth in finding them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="232" LastEditDate="2023-12-16T03:54:27.197" LastActivityDate="2023-12-16T03:54:27.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6056" PostTypeId="1" CreationDate="2017-05-22T14:20:25.990" Score="4" ViewCount="2248" Body="&lt;p&gt;I am somewhat familiar with the works of Karl Popper and his opposition to using past data to induce prospects of future events, however disclaimed as uncertain, AKA &lt;a href=&quot;https://en.wikipedia.org/wiki/Historicism&quot; rel=&quot;nofollow noreferrer&quot;&gt;historicism&lt;/a&gt;. He contributed to formulating the &lt;a href=&quot;https://en.wikipedia.org/wiki/Problem_of_induction&quot; rel=&quot;nofollow noreferrer&quot;&gt;problem of induction&lt;/a&gt;, which IMHO conflates the certainty of an outcome with a rational probability distribution of possible outcomes based on slow changing detectable patterns, invalidating the latter for the absence of the former, a form of &lt;a href=&quot;https://en.wikipedia.org/wiki/Straw_man&quot; rel=&quot;nofollow noreferrer&quot;&gt;strawman fallacy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure if it's been formally recognized as such but &lt;a href=&quot;https://en.wikipedia.org/wiki/Bayesian_inference&quot; rel=&quot;nofollow noreferrer&quot;&gt;Baysian inference&lt;/a&gt; can be thought of as realistically, however partially, historicist and &lt;a href=&quot;https://en.wikipedia.org/wiki/Inductive_reasoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;inductive&lt;/a&gt;. Does Popper oppose the general validity of Baysian thought on the grounds of being inductive and historicist, perhaps not exact enough for his standards, although it never lays such claims as it clearly deals in probabilistic (approximate) and not exact terms?&lt;/p&gt;&#xA;" OwnerUserId="5772" LastEditorUserId="55" LastEditDate="2017-05-24T03:55:32.780" LastActivityDate="2020-06-10T06:33:01.083" Title="Did Karl Popper argue against Bayesian inference?" Tags="|philosophy-of-science|probability|scientific-method|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6057" PostTypeId="2" ParentId="6056" CreationDate="2017-05-22T23:24:17.003" Score="3" Body="&lt;p&gt;The answer is not so straightforward. Of course, the thrust of Popper's position was against probabilistic induction in general, and Bayesianism is often put forth as the leading alternative to falsificationism, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3176/hypothesis-testing-fisher-vs-popper-vs-bayes/3186#3186&quot;&gt;Hypothesis testing: Fisher vs. Popper vs. Bayes&lt;/a&gt;. In 1983 Popper (with Miller) even offered a technical argument published as a note &lt;a href=&quot;https://www.nature.com/nature/journal/v302/n5910/abs/302687a0.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;A proof of the Impossibility of Inductive Probability in Nature&lt;/a&gt;, which concludes with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This result is completely devastating to the inductive interpretation&#xA;  of the calculus of probability. All probabilistic support&#xA;  is purely deductive: that part of a hypothesis that is not deductively entailed by the evidence is always strongly countersupported by the evidence-the more strongly the more the evidence asserts. This is completely general; it holds for every hypothesis h ; and it holds for every evidence e, whether it supports h, is independent of h, or countersupports h. There is such a thing as probabilistic support; there might even be such a thing as inductive support (though we hardly think so). But the calculus of probability reveals that probabilistic support cannot be inductive support.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Popper-Miller argument is controversial, and many consider it invalid, see &lt;a href=&quot;http://intellectualmathematics.com/blog/the-popper-miller-argument-against-probabilistic-induction&quot; rel=&quot;nofollow noreferrer&quot;&gt;commentary at Intellectual Mathematics&lt;/a&gt;, but the sentiment seems clear. However, in &lt;a href=&quot;http://www.ph.ucla.edu/epi/faculty/greenland/Epi204/Greenland1998_InductionvsPopper_IJE.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Induction versus Popper: substance versus semantics&lt;/a&gt; Greenland argues that &quot;&lt;em&gt;some of the controversy is semantic, and hence Popperian criticisms of induction must be translated carefully into ordinary language  to  be  appreciated  by  inductively  oriented  epidemiologists.&lt;/em&gt;&quot; In other words, Popper's sense of &quot;induction&quot; is not the same as colloquially used, or as understood in Bayesianism and other formalized approaches:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Nearly every type of statistical approach has at times been called ‘inductive’.  R. A. Fisher  referred  to  his  significance-testing  procedures as methods for ‘inductive inference’; Neyman referred to  his  hypothesis-testing  procedures  as  methods  for  ‘inductive behavior’; R. von  Mises  referred  to  his  objective  Bayesian approach as an ‘inductive science’; and DeFinetti referred to his  subjective  Bayesian  approach  as  ‘inductive  reasoning’. Despite the diversity of approaches represented by these authors, their usage is defensible.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This leaves room for some conciliation. The Fisher-Neyman significance testing is usually endorsed by falsificationists, and in his own quantitative accounts of corroboration Popper used Bayesian looking measures. Indeed, the controversy seems to be chiefly over &lt;em&gt;interpretation&lt;/em&gt; of formalisms, including the Bayesian one, rather than their potential usefulness in specific contexts.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-05-23T19:47:29.487" LastActivityDate="2017-05-23T19:47:29.487" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6058" PostTypeId="2" ParentId="6041" CreationDate="2017-05-23T00:35:15.497" Score="3" Body="&lt;p&gt;This statement is characteristic of Borel's change of attitude in 1908, around the Fourth International Congress of Mathematicians. Borel always had constructivist leanings, and considered all infinities to be only potential (unfinished). This position became more pronounced after Zermelo's controversial proof of the well-ordering theorem from the axiom of choice appeared in 1905. But as late as 1908 Borel still identified allowable infinities with denumerable (countable) infinities, i.e. by size:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...a non-denumerable infinity of choices (successive or simultaneous)... appears to me... entirely meaningless. As for a denumerable infinity of&#xA;  choices, clearly one cannot effect them all, but... one is assured that any given choice will be effected at the end of a finite time&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This changed in &lt;a href=&quot;http://archive.numdam.org/article/ASENS_1908_3_25__443_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Les «Paradoxes» de la Théorie des Ensembles (1908)&lt;/a&gt;. Denumerability was of course still necessary for meaningful infinities, but no longer sufficient. Borel introduces a stronger notion of &lt;em&gt;effective enumerability&lt;/em&gt;, which will come into prominence two decades later. Here is from &lt;a href=&quot;https://books.google.it/books?id=kJrhBwAAQBAJ&amp;amp;hl=it&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore's Zermelo's Axiom of Choice&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In one respect Borel refined his earlier position. No longer did he regard&#xA;  the relevant distinction as that between denumerable and non-denumerable&#xA;  sets. From a practical point of view, the principal distinction was now&#xA;  between those sets effectively enumerable and those not. According to&#xA;  Borel, a set was effectively enumerable if one could state, &quot;by means of a&#xA;  finite number of words, a definite process for attributing unambiguously a&#xA;  [unique natural number as] rank to each of its elements&quot; [1908a, 446-447].&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Some denumerable sets were not effectively enumerable, since to attribute&#xA;  a rank to each element might require an infinity of arbitrary choices. Moreover,&#xA;  an infinite subset of an effectively enumerable set need not be effectively&#xA;  enumerable. At last Borel had produced a positive concept, effective enumerability, in response to his qualms about set theory and Zermelo's Axiom.&#xA;  Related notions of effectivity for subsets of $\mathbb{N}$, such as recursive enumerability, were investigated by recursion theorists beginning in the 1930s.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I should note that Borel's attitude was rather pragmatic, he did not deny some indirect utility to non-constructivist mathematics, and even used Cantorian arguments in his work on complex functions and measure theory. But throughout his life he insisted on sharp separation between &quot;Zermeloan&quot; and non-&quot;Zermeloan&quot; mathematics, classing the former as only provisional. As he wrote to Hadamard in 1905:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I prefer not to write alephs. Nevertheless, I willingly state arguments equivalent to those which you mention, without many illusions about their intrinsic value, but intending them to suggest other more serious arguments...&#xA;  One may wonder what is the real value of these arguments that I do not regard as absolutely valid but that still lead ultimately to effective results. In fact, it seems that if they were completely devoid of value they could not lead to anything since they would be meaningless collections of words. This, I believe, would be too harsh. They have a value analogous to theories in mathematical physics through which we do not claim to express reality but rather to have a guide that aids us, by analogy, in predicting new phenomena, which must then be verified.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-05-23T20:00:14.813" LastActivityDate="2017-05-23T20:00:14.813" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6059" PostTypeId="1" AcceptedAnswerId="6066" CreationDate="2017-05-23T02:31:44.507" Score="3" ViewCount="255" Body="&lt;p&gt;This idea provides the foundations of algebraic geometry now; and they have certainly gone down the rabbit hole with it. As a student studying this subject, I have always found it such a great leap to think that some ring of functions could have such a strong influence on geometry. &lt;em&gt;Given&lt;/em&gt; the idea, it seems natural. But to be the one that first had the idea, that seems a great leap.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have any information about the history of this idea? Who first thought about? Perhaps why they thought about it? I believe the answer is Riemann, when studying what we now call Riemann surfaces. But, he doesn't seem to have &quot;gone down the rabbit hole&quot; with it, so far as I can uncover. Were there any ideas like this beforehand? &lt;/p&gt;&#xA;" OwnerUserId="2958" LastEditorUserId="55" LastEditDate="2017-05-23T22:56:40.677" LastActivityDate="2017-05-24T02:49:57.233" Title="Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?" Tags="|differential-geometry|geometry|algebraic-geometry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6060" PostTypeId="1" CreationDate="2017-05-19T18:22:01.293" Score="0" ViewCount="77" Body="&lt;p&gt;I'm interested in models used before the Hubbard model (free electron model, band model, etc, both itinerant and localized models). What were their strengths and weaknesses? Which model is currently the main topic of modern research? Did the Hubbard model overcome any obstacles that other models didn't? &lt;/p&gt;&#xA;" OwnerDisplayName="Emmi" LastEditorUserId="4703" LastEditDate="2017-06-03T18:08:54.580" LastActivityDate="2017-06-03T18:08:54.580" Title="History of Magnetism before the Hubbard model" Tags="|theoretical-physics|electromagnetism|magnetism|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6061" PostTypeId="2" ParentId="6059" CreationDate="2017-05-23T16:12:37.693" Score="1" Body="&lt;p&gt;I don't think this was Riemann, or that Riemann knew that any ring of functions determines the surface. In fact, Riemann studied compact surfaces on which the ring of regular functions is trivial, and he studied the field of meromorphic functions instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that a ring of functions determines the space is of much later origin.&#xA;It can be traced to Gelfand's theory of commutative Banach algebras, and was brought to algebraic geometry by Grothendieck.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-23T16:12:37.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6062" PostTypeId="1" CreationDate="2017-05-23T17:48:05.480" Score="2" ViewCount="139" Body="&lt;p&gt;It is known that during the last quarter of the 19th century there was a flourishing of the production of material models (from plaster, strings, card-board etc) of curves and surfaces in Germany (but obviously not only there, but also before that in France); see example the collection in Göttingen (&lt;a href=&quot;http://modellsammlung.uni-goettingen.de/index.php?lang=en&amp;amp;r=8&amp;amp;sr=31&amp;amp;m=257&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). My question concerns the models of &lt;a href=&quot;http://did2.mathematik.uni-halle.de/modell/modell.php?Nr=R-006&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann surfaces&lt;/a&gt;. Obviously Riemann defined these surfaces and the notion of the branch point already in 1851 in his thesis: &quot;Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse&quot;, and I assume the notion of a multi-valued function became therefore clearer mathematically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is the following: were these models considered, along them being an additional way to visualize the branch points and how the different sheets 'change their position', necessary and essential for the mathematical understanding of how the Riemann surfaces 'look like' / behave? Or was this (this = what happens in the neighborhood of the branch point) already clear mathematically and the models were just a secondary resp. additional way to help the teaching of these surfaces?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also hardly found any literature specifically about these models and their connection to concetualization of Riemann surfaces, so any help would be great. Thanks!&lt;/p&gt;&#xA;" OwnerUserId="3232" LastActivityDate="2017-06-22T18:26:55.787" Title="Material models of Riemann surfaces" Tags="|differential-geometry|topology|algebraic-geometry|riemannian-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6063" PostTypeId="2" ParentId="6059" CreationDate="2017-05-23T18:27:46.703" Score="4" Body="&lt;p&gt;The idea is usually attributed to Dedekind and Weber in &lt;em&gt;Theorie der algebraischen Functionen einer Veränderlichen&lt;/em&gt; (1882): [&lt;a href=&quot;https://books.google.com/books?id=Qxte2mhlEOYC&amp;amp;pg=PP1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://mathoverflow.net/a/56108&quot;&gt;2&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=Bb30CjGW7EAC&amp;amp;pg=PA591&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=RTLRBK-wj6wC&amp;amp;pg=PA55&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=a3_1BwAAQBAJ&amp;amp;pg=PA127&amp;amp;dq=Bourbaki&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt;,...].&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-05-23T18:27:46.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6064" PostTypeId="2" ParentId="6062" CreationDate="2017-05-23T18:55:43.147" Score="3" Body="&lt;p&gt;Kharkiv University (Ukraine) subscribed to all models made M. Schilling, who probably was a student of Klein, and who run a company making and selling these models. Currently they photograph them and place on internet:&#xA;&lt;a href=&quot;http://touch-geometry.karazin.ua/list&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://touch-geometry.karazin.ua/list&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This page also has Schilling's catalog of his production.&#xA;Recently I learned that they have some Riemann surfaces made of cardboard. The corresponding items are described in M. Shilling's catalog. But they are not sure how to photograph them properly. This summer I will visit Kharkiv and hope that I will be able to help them. According to the description, one set of cardboard models is based on the PhD thesis of W. Ihlenburg, Klein's student who was apparently killed in WWI.&#xA;I have Ihlenburg's thesis posted on my web page, including the pictures:&#xA;&lt;a href=&quot;http://www.math.purdue.edu/~eremenko/dvi/ihlenburg3p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.purdue.edu/~eremenko/dvi/ihlenburg3p.pdf&lt;/a&gt;&#xA;Presumably they have in Kharkiv these Riemann surfaces made of cardboard. (Schilling's catalog mentions them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. In 1970s, in Lviv University (Ukraine), prof. A. A. Goldberg&#xA;taught a course of Riemann surfaces where students were assigned to make these models as a part of their HW. I've seen many of them, but I don't think the models from this course are preserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Specifically, the Riemann surface models have numbers 303-310 p. 159-180 in Schilling's catalog.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT2. Here are some photos made in Kharkiv on June 22, 2017:&#xA; &lt;a href=&quot;http://www.math.purdue.edu/~eremenko/Ihlenb/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.purdue.edu/~eremenko/Ihlenb/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-06-22T18:26:55.787" LastActivityDate="2017-06-22T18:26:55.787" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6065" PostTypeId="2" ParentId="639" CreationDate="2017-05-23T20:41:55.040" Score="-1" Body="&lt;p&gt;The word invention is often tied to patents and so it picks up meanings from patent law. Innovation can be anything new, patentable or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long before the US Revolutionary War, the monarch of England could grant a patent for whatever reason, even on things that were already in general use such as playing cards. In contrast, the Framers of the US Constitution authorized Congress to promote the useful arts by granting an exclusive right (what would become parents) to &quot;inventors&quot; for their &quot;discoveries.&quot; This slowly led to the development of a requirement of non-obviousness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before the term disruptive innovation was coined, a similar concept existed that some inventions are &quot;pioneering.&quot; Pioneering invention are rare and have little or no prior art in the field. A pioneering invention may become a disruptive innovation, or it may become a complete flop.&lt;/p&gt;&#xA;" OwnerUserId="5818" LastActivityDate="2017-05-23T20:41:55.040" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6066" PostTypeId="2" ParentId="6059" CreationDate="2017-05-23T22:42:27.483" Score="6" Body="&lt;p&gt;A canonical reference on this is &lt;a href=&quot;https://books.google.com/books?id=_uhlf38jOrgC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dieudonne's History of Algebraic Geometry&lt;/a&gt;. An abridged version &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Dieudonne.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historical Development of Algebraic Geometry&lt;/a&gt; is freely available, see also &lt;a href=&quot;http://www.robertweaston.com/wp-content/uploads/documents/presentations/History_of_Algebraic_Geometry.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Easton's slides&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let me make a general comment first. When we wonder &amp;quot;however did someone first connect these two [modern ideas]?&amp;quot; we tacitly presuppose that they were always separately available, waiting to be connected. But the truth often is that they were &lt;em&gt;developed connected&lt;/em&gt; to each other. Riemann was indeed instrumental in creating the modern algebro-geometric framework, but he did not have the idea to study surfaces via rings of functions for the simple reason than in his time the (general) concept of Riemann surfaces, let alone of rings of functions, did not exist. He was studying Abelian integrals, this led him to consider surfaces on which holomorphic and meromorphic functions, such as Abelian integrals, are defined. And by the time Kronecker and Dedekind-Weber developed the suitable algebraic concepts they already had the connection on display in Riemann's work. So &lt;em&gt;nobody&lt;/em&gt; had such an idea first.&lt;/p&gt;&#xA;&lt;p&gt;Here are some details as described by  Dieudonne:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is quite a paradox that in the work of this prodigious genius , out of which algebraic geometry emerges entirely regenerated, there is almost no mention of algebraic curve, it is from his theory of &lt;strong&gt;algebraic functions&lt;/strong&gt; and their integrals that all of the birational geometry of the nineteenth and the beginning of the twentieth century issues.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Instead of starting (as would all his predecessors and most of his immediate successors) from an algebraic equation &lt;span class=&quot;math-container&quot;&gt;$F(s, z) = 0$&lt;/span&gt; and the Riemann&#xA;surface of the algebraic function &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; which it defines, his initial object is an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-sheeted Riemann surface without boundary and with a finite number of ramification points, given a priori without any reference to an algebraic equation... Thus, the abstract Riemann surface &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is, in fact, identical to that of algebraic function &lt;span class=&quot;math-container&quot;&gt;$s(z)$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$F(s, z) = 0$&lt;/span&gt;, and Riemann attaches to it what will, after Dedekind's time, be called the &lt;strong&gt;field of meromorphic (or rational) functions&lt;/strong&gt; on &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.&lt;/em&gt; [emphasis Dieudonne's]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Riemann's insights were absorbed in two foundational papers from 1882, by Kronecker (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN243919689_0092&amp;amp;DMDID=dmdlog4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grundzüge einer arithmetischen Theorie der algebraischen Grössen, Crelle's journal, 92, 1–122&lt;/a&gt;) and Dedekind-Weber (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002158280&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal für die reine und angewandte Mathematik, 92, 181-290&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first task to which each school of algebraic geometry addressed itself was therefore the systematization of the birational theory of algebraic plane curves, incorporating most of Riemann's results with proofs in conformity&#xA;with the principles of the school... just as Riemann had revealed the close relationship between algebraic varieties and the theory of complex manifolds, Kronecker and Dedekind-Weber brought to light for the first time the deep similarities between algebraic geometry and the burgeoning theory of algebraic numbers... this conception of algebraic geometry is for us the clearest and simplest one, due to our familiarity with abstract algebra.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Kronecker started defining varieties in terms of rings of polynomials vanishing on them, and developed the notions of subvariety and dimension in terms of ideals (which he called &lt;em&gt;Modulsystems&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The goal of Dedekind and Weber in their fundamental paper was quite different&#xA;and much more limited; namely, they gave purely algebraic proofs for all the algebraic results of Riemann. They start from the fact that, for Riemann, a class of isomorphic Riemann surfaces corresponds to a field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; of rational functions, which is a finite extension of the field &lt;span class=&quot;math-container&quot;&gt;$C(X)$&lt;/span&gt; of rational fractions in one indeterminate over the complex field; what they set out to do, conversely, if a finite extension &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; of the field &lt;span class=&quot;math-container&quot;&gt;$C(X)$&lt;/span&gt; is given &lt;strong&gt;abstractly&lt;/strong&gt;, is to reconstruct a Riemann surface &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; will be isomorphic to the field of rational functions on &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-05-24T02:49:57.233" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6067" PostTypeId="2" ParentId="6056" CreationDate="2017-05-24T07:27:52.837" Score="5" Body="&lt;p&gt;Bayesian inference seeks to believe in that which has a high conditional probability as computed with Bayes's theorem. The problem with using &lt;span class=&quot;math-container&quot;&gt;$P\left( A|B\right)=P\left( B|A\right)\frac{P\left( A\right)}{P\left( B\right)}$&lt;/span&gt; to compute the LHS is that, although the first factor on the RHS is often known, at least one of the RHS's other probabilities usually won't be known. (If &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; denotes a theory and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; data, &lt;span class=&quot;math-container&quot;&gt;$P\left( B\right)$&lt;/span&gt; requires another theory to compute (a null hypothesis), and &lt;span class=&quot;math-container&quot;&gt;$P\left( A\right)$&lt;/span&gt; is even harder.) If you want to see Popper's arguments that (a) the probability of a theory given data will, counter-intuitively, be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and (b) probability is the wrong way to think about the case for or against a theory, but (c) a non-zero measure expressible in terms of conditional probability called verisimilitude is more relevant, see &lt;em&gt;Objective Knowledge: An Evolutionary Approach&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historicism doesn't mean “induction works”. As you can see in the Wikipedia article you linked to, Popper took the term to mean political theories that imply human society will inevitably turn out a certain way. In &lt;em&gt;The Poverty of Historicism&lt;/em&gt;, Popper critiqued such theories (e.g. Marxism claiming capitalism will be overthrown) on the grounds that the destiny of society depends on its future discoveries, which by definition cannot be foreseen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Popper's main contribution regarding induction was not to critique it (though he did explain how the existing criticism had been misunderstood), but to argue that science didn't need it so was more justifiable than anti-inductivists had thought. See &lt;em&gt;The Logic of Scientific Discovery&lt;/em&gt;. Hume formulated the first problem of induction, though Goodman and Popper each cited additional reasons it's a problematic way to reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can claim the problem “conflates the certainty of an outcome with a rational probability distribution of possible outcomes”, but it's important to understand that, while inductivists of the time such as Carnap and Russell felt induction and the problem thereof were that straightforward, Popper argued they'd misunderstood the problem. I'll summarize it in the hope its beyond-mere-uncertainty implications will become clear. Hume argued that, because a principle of uniformity of nature is (a) required to find induction convincing and (b) unknowable &lt;em&gt;a priori&lt;/em&gt; because it's not a tautology, the only way it could be justified is empirically. But evidence that nature fits such a principle would by definition be presented in an inductive argument! As Popper noted in (if memory serves) &lt;em&gt;Conjectures and Refutations&lt;/em&gt;, this forces us to either (1) assume such a principle in the argument that justifies it (circularity) or (2) use a different principle that only pushes the problem back (infinite regress).&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2020-06-10T06:33:01.083" LastActivityDate="2020-06-10T06:33:01.083" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6069" PostTypeId="1" AcceptedAnswerId="6070" CreationDate="2017-05-25T03:15:38.627" Score="0" ViewCount="302" Body="&lt;p&gt;Reading Dedekind original manuscript (&lt;a href=&quot;https://archive.org/details/essaysintheoryof00dedeuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/essaysintheoryof00dedeuoft&lt;/a&gt;) it seems that he was after formalizing the completeness of the real numbers, which was known to him as the continuity of real numbers. Formalizing the completeness of real numbers then easily led him to construction of real numbers using cuts of rational numbers, which also provided a model for the real numbers(&lt;a href=&quot;https://en.wikipedia.org/wiki/Dedekind_cut&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Dedekind_cut&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I have seen some sources talking about Eudoxus ratios (&lt;a href=&quot;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus&lt;/a&gt;) being the predecessor of the Dedekind cuts, which sort of means that Dedekind knew how real numbers could be constructed using rational numbers. This in turn means that the Dedekind construction of real numbers suggested the completeness of real numbers to Dedekind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question is that how did Dedekind conclude the real numbers are complete? Was he convinced that the reals were complete because they could be constructed using cuts of rational numbers (was this an acceptable (logical) argument to him)? or he followed a different logic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, what made Dedekind accept real numbers are complete, was it the construction of the real numbers or was it something else?&lt;/p&gt;&#xA;" OwnerUserId="5727" LastActivityDate="2017-05-25T07:29:56.103" Title="How did Dedekind arrive at the completeness of real numbers?" Tags="|real-analysis|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6070" PostTypeId="2" ParentId="6069" CreationDate="2017-05-25T07:29:56.103" Score="1" Body="&lt;p&gt;That the set of &quot;real numbers&quot; must complete was intuitively understood already by the Greeks, though they had neither a formal concept of real number nor the formal definition of completeness. For them, the &quot;numbers&quot; were rational numbers, and the &quot;irrationality of $\sqrt{n}$&quot; meant that there are no enough numbers to measure the lengths of all segments in geometry. But the intuitive notion of numbers which allow to measure any length was desirable and it slowly evolved. By the time of Dedekind, it was clear what mathematicians want of real numbers, and the problem was to &quot;construct&quot; them from rationals. In particular it was known that what we call &quot;completeness&quot; is a desirable property. Dedekind found a way to formally construct from rational numbers the system of numbers with desired properties, including completeness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was not the first such construction. Eudoxus's theory of proportions also achieves this purpose. Its disadvantage is that it is more cumbersome and awkward to use. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-05-25T07:29:56.103" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6072" PostTypeId="1" AcceptedAnswerId="6074" CreationDate="2017-05-25T17:25:58.223" Score="8" ViewCount="10984" Body="&lt;p&gt;I was taught that the Arabs introduced zero in their Arabic numerals and it was depicted as a decimal point.  They got their number system from India in turn in Sanskrit where the zero digit was also used.&lt;/p&gt;&#xA;&lt;p&gt;Roman numerals, however, &lt;a href=&quot;https://hsm.stackexchange.com/questions/11312/why-didnt-the-number-zero-0-have-a-roman-numeral-of-its-own?&quot;&gt;didn't have a zero digit.&lt;/a&gt; &lt;a href=&quot;https://www.scientificamerican.com/article/history-of-zero/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This article&lt;/a&gt; says Europe did not get it until 1200AD.  If the ancient Greeks did not have zero, they would have had great difficulty with some of their calculations, such as the &lt;a href=&quot;https://www.windows2universe.org/?page=/citizen_science/myw/w2u_eratosthenes_calc_earth_size.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;circumference of the Earth&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, how did the Greeks perform these calculations without it, or did they have it back then?&lt;/p&gt;&#xA;" OwnerUserId="5829" LastEditorDisplayName="user10930" LastEditDate="2020-07-20T06:39:43.770" LastActivityDate="2020-08-01T02:36:11.360" Title="Did the ancient Greeks have zero in their number system?" Tags="|mathematics|ancient-greece|zero|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="6073" PostTypeId="2" ParentId="6072" CreationDate="2017-05-25T19:03:56.513" Score="7" Body="&lt;p&gt;It depends on the definition of the &amp;quot;Ancient Greeks&amp;quot;. Ptolemy, who lived in 2 century AD, lived on the territory of the Roman empire, but wrote in Greek, and probably worked in Alexandria (modern Egypt) most of the time. He is usually considered Greek.&#xA;He used zero with sexagesimal Babylonian system (base 60, and digits were Greek letters). Zero was used as a place holder (which is actually its main function)&#xA;and was denoted by Greek letter &lt;span class=&quot;math-container&quot;&gt;$\omicron$&lt;/span&gt; (omicron).&lt;/p&gt;&#xA;&lt;p&gt;The work of Ptolemy, or some related/derived work spread to the East eventually reaching India, as seen from its influence on the later Arab and Indian astronomy. But I do not know whether invention of zero in India was dependent of this or not.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-08-01T02:36:11.360" LastActivityDate="2020-08-01T02:36:11.360" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6074" PostTypeId="2" ParentId="6072" CreationDate="2017-05-25T23:14:03.710" Score="18" Body="&lt;p&gt;During the classical and early Hellenistic period (until 200 BC) Greeks did not use any positional system, they had their own which was decimal but not positional. The units from 1 to 9 are assigned to the first nine letters of the archaic Greek alphabet from $\alpha$ to $\theta$. Each multiple of ten from 10 to 90 was assigned its own separate letter from the next nine letters from $\iota$ to ϙ. And so on, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Greek_numerals&quot; rel=&quot;noreferrer&quot;&gt;Greek numerals&lt;/a&gt;. This worked fine for practical and even some scientific calculations. In the 3rd century BC Eratosthenes and Aristarchus, Archimedes's contemporaries, did calculations for estimating the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eratosthenes#Measurement_of_the_Earth.27s_circumference&quot; rel=&quot;noreferrer&quot;&gt;circumference of the Earth &lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Sizes_and_Distances_(Aristarchus)&quot; rel=&quot;noreferrer&quot;&gt;distances to the Moon and the Sun&lt;/a&gt; without the benefit of not only zero but even of a positional system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there weren't enough letters to express very large numbers (which is where placeholder zeros may come handy), no more than the myriad ($10^4$) could be expressed. When the need arose &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Sand_Reckoner&quot; rel=&quot;noreferrer&quot;&gt;Archimedes in Sand Reckoner&lt;/a&gt; invented his own number system capable of expressing them. He introduced &quot;orders&quot; based on the myriad, which was the &quot;unit of the first order&quot;. The myriad myriads ($10^8$) was unit of the second order, and so on. Archimedes did not have zero either, but one can manage without it even for something like a positional system. Archimedes's was akin to the system with base $10^8$, but it did not take. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After 200 BC Hellenistic astronomers, such as the father of astronomy Hipparchus, started using the sexagesimal system (with base $60$), imported from Babylon along with long term precise astronomical data Greeks lacked. This gave a great boost to the development of mathematical astronomy, and the sexagesimal system was sometimes used for doing astronomical calculations, in particular by &lt;a href=&quot;https://en.wikipedia.org/wiki/Almagest&quot; rel=&quot;noreferrer&quot;&gt;Ptolemy in Almagest&lt;/a&gt; in the second century AD (presumably he was a Greek ethnically, but he lived and worked in Alexandria, Egypt). However, the sexagesimal system made no advances beyond a very narrow circle of practitioners, not even all astronomers used it. Zero was used as a punctuation mark for purely notational purposes, not as a number, and the symbol for it, when not simply blank space, was $\omicron$ for &quot;obol&quot; (a coin of smallest value). For details and further developments see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3245/was-the-concept-of-zero-ever-developed-without-relation-to-positional-number-sys/3249#3249&quot;&gt;Was the concept of zero ever developed without relation to positional number systems?&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-05-26T02:13:20.160" LastActivityDate="2017-05-26T02:13:20.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6075" PostTypeId="1" AcceptedAnswerId="6078" CreationDate="2017-05-26T10:20:38.157" Score="1" ViewCount="129" Body="&lt;p&gt;During times when there was no spectroscopy of any kind, people did make guesses on the molecular constituency and structure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, John Dalton's original atomic hypothesis was that all elements were mono-atomic and that the atoms in compounds have the simplest possible ratios. However, this was proved to be wrong and later it was recognized that oxygen and various other gases were actually diatomic.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How? How were the constitution of various compounds, in terms of both number and structure discerned? [for example, in this case, the atomicity of gases].&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Note: &lt;a href=&quot;https://hsm.stackexchange.com/questions/2542/how-was-the-composition-and-structure-of-water-determined&quot;&gt;This question here&lt;/a&gt; also deals with a similar problem but for the specific case of water. But, my question is about how this is done generally, what are the techniques and experiments and hypotheses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; &lt;a href=&quot;https://hsm.stackexchange.com/questions/3398/how-did-chemists-figure-out-atomic-weights-when-hydrogen-etc-were-first-disc&quot;&gt;This question and answer therein&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/questions/6075/history-of-molecular-constituency-and-structure/6078#6078&quot;&gt;the current answer to this question&lt;/a&gt; are specific to gases and water. Is there anything more generally applicable to organic and inorganic compounds (other than only gases)?&lt;/p&gt;&#xA;" OwnerUserId="5832" LastEditorUserId="5832" LastEditDate="2017-05-29T08:10:51.673" LastActivityDate="2017-06-02T23:01:47.823" Title="History of molecular constituency and structure" Tags="|discoveries|atomic-theory|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6076" PostTypeId="2" ParentId="5928" CreationDate="2017-05-26T16:17:17.953" Score="-3" Body="&lt;p&gt;I have seen the argument of surjecting a countable set of spatio-temporal coordinates on the set of all ideas including all real numbers that ever have been or will be realized: Countability of the real numbers, p. 275 of &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&lt;/a&gt;. Since no reference to earlier work is given, I assume this is the first appearance of the idea unless earlier authors can be identified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT(1): In the writings of Borel, Lebesgue, Peirce, Russell, or Moore's &quot;Zermelo's Axiom of Choice&quot; I have not been able to find this idea. Hadamard was anyway in favour of Zermelo's approach to set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit(2): Here is a nice definition of thought-object:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At request of the referee who asked what is a thought-object let me add: I understand it to be a thought about an object which may exist or not. Thus it is an electrochemical event in the brain or/and its record in the memory. In particular it is a physical thing in space time. Of course it is difficult to characterise any physical phenomena. But we have the ability to recognize thoughts as identical or different, just as we have the ability to recognize a silent lightning from a thunderous one. Hence I understand Hilbert's words as follows: mathematicians imagine sets which do not exist, but their thoughts about sets do exist and they can arise prior to the thoughts of most elements in those sets. (J. Mycielski: &quot;Russell's paradox and Hilbert's (much forgotten) view of set theory&quot; in G. Link (ed.): &quot;One hundred years of Russell's paradox: mathematics, logic, philosophy&quot;, de Gruyter, Berlin (2004) p. 534)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-27T11:29:00.990" LastActivityDate="2017-06-27T11:29:00.990" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6078" PostTypeId="2" ParentId="6075" CreationDate="2017-05-27T06:29:45.277" Score="3" Body="&lt;p&gt;Avogadro's Law says that the number of molecules in a fixed volume of gas under fixed temperature and pressure does not depend on the gas. So the number of water molecules in one cubic meter of vapor under temperature T and pressure P  is the same as the number of molecules of pure oxygen in the same volume and under the same conditions. Now add hydrogen to this oxygen and burn them together,&#xA;then condense water and release the excess hydrogen. Then evaporate water,&#xA;and see how much volume the resulting water vapor will occupy under the same &#xA;temperature and pressure as before. It will be two cubic meters (experimental fact). Conclusion: one molecule of oxygen produces two molecules of water.&#xA;Repeating this experiment with hydrogen as a starting substance, we conclude that one molecule of hydrogen produces one molecule of water.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, if you try to combine hydrogen with oxigen so that nothing but water remains, you discover that they combine in proportion $2:1$ (volumes under the same pressure and temperature).&#xA;Therefore water composition must be $H_2O$ and&#xA;one molecule of oxygen must contain two atoms of oxygen.&lt;br&gt;&#xA;This is the general principle. Technical detail of the experiment are not addressed.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-27T07:18:18.457" LastActivityDate="2017-05-27T07:18:18.457" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6079" PostTypeId="2" ParentId="2643" CreationDate="2017-05-27T16:43:46.343" Score="2" Body="&lt;p&gt;Is it really the case that English speaking physicists are less philosophically inclined?? I know of some statements which let me doubt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Steven Weinberg said: &quot;Religion is an insult to human dignity. With or without it you would have good people doing good things and evil people doing evil things. But for good people to do evil things, that takes religion.&quot; That is certainly a strong philosophical statement. This one and many others can be found in &lt;a href=&quot;https://en.wikiquote.org/wiki/Steven_Weinberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikiquote&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Richard Feyman said in &quot;&lt;a href=&quot;https://books.google.de/books?id=EpU5DgAAQBAJ&amp;amp;pg=PT21&amp;amp;lpg=PT21&amp;amp;dq=%22the%20theory%20that%20it%20is%20all%20arranged%20as%20a%20stage%20for%20God%20to%20watch%20man%27s%20struggle%20for%20good%20and%20evil%20seems%20inadequate%22&amp;amp;source=bl&amp;amp;ots=6_BE6stRCb&amp;amp;sig=zKSliyroFZ133ZtmA7uuFkvepwY&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwi33__ovJDUAhUF8RQKHchNBxsQ6AEIMDAB#v=onepage&amp;amp;q=%22the%20theory%20that%20it%20is%20all%20arranged%20as%20a%20stage%20for%20God%20to%20watch%20man&amp;#39;s%20struggle%20for%20good%20and%20evil%20seems%20inadequate%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Meaning of It All&lt;/a&gt;&quot;: &quot;... the theory that it is all arranged as a stage for God to watch man's struggle for good and evil seems inadequate.&quot; And Conifold showed in an answer to the question &lt;a href=&quot;https://philosophynow.org/issues/114/Richard_Feynmans_Philosophy_of_Science&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard Feynman’s Philosophy of Science&lt;/a&gt; that Feynman philosophized quite a lot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or take J.A. Wheeler: &quot;It from bit symbolizes the idea that every item of the physical world has at bottom — at a very deep bottom, in most instances — an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-no questions and the resistering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe.&quot; (Information, Physics, Quantum: The Search for Links&quot; in &quot;Complexity, Entropy and the Physics of Information&quot; (1990)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that most &lt;em&gt;good&lt;/em&gt; physicists can be called philosophers too.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-05-27T16:49:01.193" LastActivityDate="2017-05-27T16:49:01.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6080" PostTypeId="1" AcceptedAnswerId="6081" CreationDate="2017-05-28T04:05:25.883" Score="0" ViewCount="43" Body="&lt;p&gt;I heard that there is a famous letter between Egorov and Lusin , which best reflects relation between the teacher student duo. Can anyone provide any link of that letter?&lt;/p&gt;&#xA;" OwnerUserId="5834" LastActivityDate="2017-05-28T11:39:39.140" Title="Letter of Egorov to Lusin" Tags="|teaching|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6081" PostTypeId="2" ParentId="6080" CreationDate="2017-05-28T11:33:04.500" Score="1" Body="&lt;p&gt;If you mean the letter of 4th August 1905, you can read it &lt;a href=&quot;https://books.google.de/books?id=BFsnWXkqZaMC&amp;amp;pg=PA63&amp;amp;lpg=PA63&amp;amp;dq=Egorov%20Lusin%20Letter&amp;amp;source=bl&amp;amp;ots=A2gkpyiLIH&amp;amp;sig=Ay-wmYEzvf-adJUBGyhI_uM2JaM&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjW9M--vZLUAhXHcRQKHWoJB0kQ6AEIODAC#v=onepage&amp;amp;q=Egorov%20Lusin%20Letter&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; in English translation (from Russian Mathematicians in the 20th Century by Yakov Sinai). There are also excerpts of other letters by Egorov.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-05-28T11:39:39.140" LastActivityDate="2017-05-28T11:39:39.140" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6082" PostTypeId="2" ParentId="5645" CreationDate="2017-05-28T13:47:25.697" Score="5" Body="&lt;p&gt;Georg Cantor wrote a religious text:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;EX ORIENTE LUX. Gespräche eines Meisters mit seinem Schüler über wesentliche Punkte des urkundlichen Christentums. Berichtet vom Schüler selbst Georg Jakob Aaron, cand. sacr. theol. Erstes Gespräch. Herausgegeben von G. Cantor, Halle 1905.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Light from the East. Talks of a master with his pupil about the essential points of the original Christianity. Reported by the pupil himself.&lt;/p&gt;&#xA;&lt;p&gt;The essay concludes with a refusal of Catholicism: &amp;quot;It remains until the end of all days resting on an unshakeable rock, Christ himself: the invisible church, which He has founded. He is the Lord who does not need a governor on earth.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;An Essay about the true author of Shakespeares works Cantor wanted to start with the following autobiographical peom as an introduction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Allen&lt;/em&gt; stets zu gefallen ist Loos nur den Kindern Fortunens,&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Wen'gen&lt;/em&gt; Erlesenen, ach, selten glückt's dem Verdienst;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Niemandem&lt;/em&gt; recht es zu machen, von Allen geschmäht und verhöhnt sein,&lt;/p&gt;&#xA;&lt;p&gt;Bitter trägt dies der Mann, schmerzvoll entsagend das Weib.&lt;/p&gt;&#xA;&lt;p&gt;Dennoch, verweigern die Götter grausam die Gunst mir des Mittlern,&lt;/p&gt;&#xA;&lt;p&gt;Steh' ich und habe die Wahl, nur von den Aeussersten Ein's:&lt;/p&gt;&#xA;&lt;p&gt;Einsam unentwegt folg ich den Spuren der Wahrheit,&lt;/p&gt;&#xA;&lt;p&gt;Freudig verzichtend hinfür jeglichem Beifall der Welt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[H. Meschkowski, W. Nilson (eds.): &lt;em&gt;Georg Cantor Briefe&lt;/em&gt;, Springer, Berlin  (1991), p. 418]&lt;/p&gt;&#xA;&lt;p&gt;By the way, Cantor was also an excellent painter. See section 4 &lt;a href=&quot;http://disk.mathematik.uni-halle.de/history/cantor/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-05-28T18:10:16.810" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6085" PostTypeId="2" ParentId="5744" CreationDate="2017-05-29T13:35:02.547" Score="1" Body="&lt;p&gt;One inventor of a self-locking door was &lt;a href=&quot;https://books.google.de/books?id=50eWBAAAQBAJ&amp;amp;pg=PR52&amp;amp;lpg=PR52&amp;amp;dq=%22selbstschlie%C3%9Fende%20T%C3%BCr%22%20erfindung&amp;amp;source=bl&amp;amp;ots=xJVPvchziP&amp;amp;sig=Zg3JtRmvwcddTSAiolXLWQKujjc&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwiKwcGxl5XUAhXCfywKHaN1AqAQ6AEIRTAH#v=onepage&amp;amp;q=%22selbstschlie%C3%9Fende%20T%C3%BCr%22%20erfindung&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonardo da Vinci&lt;/a&gt; (1452-1519). But certainly there were earlier applications, in particular by trappers for many centuries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second part of the question is not precisely enough formulated. What percentage is &quot;common place&quot;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-05-29T13:35:02.547" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6086" PostTypeId="2" ParentId="3814" CreationDate="2017-05-29T14:07:37.953" Score="3" Body="&lt;p&gt;A vague conjecture is that the symbol was taken from a Roman inscription containing the symbol for 100 millions &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4JJNG.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4JJNG.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Wallis never visited Italy. So he can hardly have seen it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Wallis was a cleric, and 8 is an important element of Christian symbolism, among others standing for eternity. It may have happened that the cleric Wallis just took this 8 and turned it in order to distinguish it from the ordinary 8. (W. Mückenheim: Script to the lecture &lt;em&gt;Die Geschichte des Unendlichen&lt;/em&gt;, 7th ed., Maro-Verlag, Augsburg (2011) p. 15)&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-05-31T09:20:26.920" LastActivityDate="2017-05-31T09:20:26.920" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6087" PostTypeId="2" ParentId="2643" CreationDate="2017-05-29T16:17:49.783" Score="1" Body="&lt;p&gt;This is of course not a question with a final answer, but from my point of view one of the main reasons is the increasing specialization of the different fields in physics combined with the difficult-to-grasp mathematical overhead. &#xA;As a result, traditional philosophers often have little to no idea about what is happening at the forefront of research in physics since this would involve concepts of enormous complexity that cannot be understood without the mathematical overhead (e.g. Quantum Field Theories, AdS/CFT-Correspondence or Supersymmetry just to name a few). &#xA;Popular explanations of the modern theories are almost always imprecise with respect to some crucial details. This makes any attempt of a philosophical discussion among non-experts very difficult - even if a non-expert makes a valid argument, it can be easily dismissed due to possible misinterpretations of the theory. In contrast to this, many of the great philosophers of the past had an extensive understanding of the physics they were talking about - Kant had a number of important publications in celestial mechanics, many of the philosophers around logical positivism were also accomplished scientists themselves (Mach, Hertz, Gödel or Russel just to name a few), Popper repeatedly demonstrated his solid knowledge of Quantum Mechanics. Those who did not have a background in physics or maths often had close ties to those who did - the Vienna Circle is a classic example of this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another point that is probably related to the one I just made is the increasing self-imposed distance to falsifiability in the philosophy of science. While there are certainly valid points to be made in this regard, this does not sit well with most physicists - the gold-standard for scientific theories is still considered to be their falsifiability, hence the philosophical trend towards &quot;anything goes&quot; in the sense of Feyerabend and others is either ignored or immediately dismissed since it does not relate to everyday scientific work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;tl;dr: Physics is too complicated and philosophy is completely disconnected from the forefront of research.&lt;/p&gt;&#xA;" OwnerUserId="354" LastActivityDate="2017-05-29T16:17:49.783" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6088" PostTypeId="2" ParentId="6060" CreationDate="2017-05-30T14:50:40.050" Score="1" Body="&lt;p&gt;I'm certainly no expert on the history of these models, and can only answer from my working knowledge about the topic. (In particular, I'm not quite the right person to give a detailed comparison with the Hubbard model.) Nevertheless, the text below is too long for a comment, and it might still be of some help. I do not claim completeness, but think that these are the bare minimum one should know about in this context. I'd be interested to see more detailed answers by people who know more about this than I do!&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The Hubbard model is from the 1960s. Thus the &lt;strong&gt;Ising model&lt;/strong&gt; (Lenz, 1920) is certainly older, as is the &lt;strong&gt;Heisenberg model&lt;/strong&gt; (Heisenberg, 1928) for (anti)ferromagnetism. See e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Magnetism#Quantum-mechanical_origin_of_magnetism&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Magnetism&quot; at Wikipedia&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ferromagnetism#Explanation&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Ferromagnetism&quot; at Wikipedia&lt;/a&gt; and references there for more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Ising model is a classical-physical model, in the sense that the &quot;spins&quot; are discrete, and as such can be thought of as describing the limit of highly anisotropic interactions between the magnetic dipoles of the atoms. The Ising model in 1d was already solved by Ising in his PhD thesis; the 2d Ising model (on a square lattice) was famously tackled by Onsager (1944); while in 3d and up it is still topic of active research -- see e.g. references at &lt;a href=&quot;https://en.wikipedia.org/wiki/Ising_model#Three_and_four_dimensions&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Ising model&quot; at Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Heisenberg model is already interesting in 1d (spatial), and shows up in current research, ranging from topics such as &quot;&lt;a href=&quot;https://arxiv.org/abs/1501.06805&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantum integrability&lt;/a&gt;&quot;, to &quot;quench dynamics&quot;, to &lt;a href=&quot;https://arxiv.org/abs/1104.0983&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematically rigorous probabilistic approaches&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4703" LastActivityDate="2017-05-30T14:50:40.050" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6089" PostTypeId="2" ParentId="6021" CreationDate="2017-05-30T17:47:31.217" Score="0" Body="&lt;p&gt;An increasing number of modern scholars agree that the Alexandrian scholars knew the distance of the Earth from the Sun. The knowledge is found in the dimensions of the Great Pyramid, but how they knew this remains a question for many. More on the ancient Egyptian methods, sources for Eratosthenes' method and a detailed critique of Sagan's analysis from Christian Irigaray's &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;a href=&quot;https://www.academia.edu/27928173/How_the_Ancient_Egyptians_had_Calculated_the_Earths_Circumference_between_3750-1500_BC_a_revision_of_the_method_used_by_Eratosthenes&quot; rel=&quot;nofollow noreferrer&quot;&gt;How the Ancient Egyptians had Calculated the Earth’s Circumference between 3750-1500 BC: a revision of the method used by Eratosthenes&lt;/a&gt;&quot;  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We want to make it clear that we have nothing against Eratosthenes: on the contrary, after reading from Strabo it is clear that he was a great geographer which gave the Greeks a great number of details concerning the dimensions of the known world. All that we know from Eratosthenes comes from second hand accounts, and most of them speak very highly of Eratosthenes. The problem is that&#xA;  modernists and many Greeks congratulate Eratosthenes as “the first” to measure&#xA;   the circumference of the Earth. Many academics and scholars consider this true, but the reality is that this congratulation is unwarranted and based on biased accounts like the one given by Sagan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eratosthenes (276-194 BC) is credited as &quot;the first&quot; person to have measured the circumference of the Earth in our modern history books, but a serious study of the circumstances shows that Eratosthenes did not actually measure anything, and that he simply copied an earlier method of the Egyptians. Eratosthenes was a Greco-Egyptian director of the Library of Alexandria at the time, and he had access to the ancient records of science collected during the Alexandrian era which contained not only Egyptian but Mesopotamian and Persian knowledge of a variety of kinds. Among these documents was one from which Eratosthenes copied a method of deriving the Earth’s circumference, but the method itself (as we will see) shows that the actual experiment and record of this in Egypt occurred as early as 3750 BC, making “the first” to actually measure a portion of the Earth’s circumference and calculating were most probably Predynastic Egyptians living up to 3500 years before Eratosthenes’ time. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Related to our answer on this question from the hsm forum here: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/2143/in-ancient-times-how-did-people-conclude-that-the-shape-of-earth-is-a-sphere&quot;&gt;In ancient times, how did people conclude that the shape of Earth is a sphere?&lt;/a&gt;  &lt;/p&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2017-05-30T17:47:31.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6090" PostTypeId="1" CreationDate="2017-05-31T02:28:32.497" Score="6" ViewCount="201" Body="&lt;p&gt;As is well known, the Hadamard inequality is trivial from this point of view. But the Hadamard inequality is discovered so late.  &lt;/p&gt;&#xA;" OwnerUserId="5675" LastActivityDate="2017-06-03T16:01:26.017" Title="When did people realize that the determinant of a matrix is actually the volume of a parallelepiped?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6091" PostTypeId="2" ParentId="6090" CreationDate="2017-05-31T10:41:07.957" Score="5" Body="&lt;p&gt;According to &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;O'Connor and Robertson&lt;/a&gt;, the first observation dates 250 years ago:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lagrange, in a paper of 1773, studied identities for 3 × 3 functional determinants. However this comment is made with hindsight since Lagrange himself saw no connection between his work and that of Laplace and Vandermonde. This 1773 paper on mechanics, however, contains what we now think of as the volume interpretation of a determinant for the first time. Lagrange showed that the tetrahedron formed by $O(0,0,0)$ and the three points $M(x,y,z)$, $M'(x',y',z')$, $M''(x'',y'',z'')$ has volume&#xA;  $$\frac16 [z(x'y'' - y'x'') + z'(yx'' - xy'') + z''(xy' - yx')].$$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="658" LastEditDate="2017-06-03T16:01:26.017" LastActivityDate="2017-06-03T16:01:26.017" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6092" PostTypeId="1" CreationDate="2017-05-31T13:16:46.587" Score="5" ViewCount="572" Body="&lt;p&gt;I think is well known that greek scientists and even founding fathers of modern science did not use error propagation in their calculations. Today, instead, is unacceptable to work out any prediction of a physical theory without giving an estimate of how errors on the input parameters propagate into the computed quantity. So my question is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;when the use of error propagation became prominent? Who has been the pioneer who brought this into physics common practices?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5863" LastActivityDate="2020-08-31T12:52:52.440" Title="When did error propagation become prominent in physics?" Tags="|physics|statistics|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6093" PostTypeId="1" CreationDate="2017-05-31T14:34:47.950" Score="4" ViewCount="758" Body="&lt;p&gt;In your opinion, what are some of the best books/papers on Newton and his work that accurately cover the connections between his geometric proofs in the Principia and his development of the calculus and his disputes with others on these topics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, some articles in The Cambridge Companion to Newton seem pretty good to me in addressing the interplay between Newton's calculus and his geometric proofs of physics in the Principia as well as the associated disputes on priority and philosophical approach (physical intuition versus pure algebraic manipulation). See also the intro of Needham's Visual Complex Analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit March 2020: This question is about Newton's mathematics, but I can't resist quoting Keynes on Newton, which aptly places Newton's diverse interests into an understandable perspective:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind that looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2020-03-11T19:39:37.750" LastActivityDate="2020-03-11T19:39:37.750" Title="Best books/papers on Newton and his mathematical physics" Tags="|biographical-details|mathematical-physics|scientific-method|depiction-of-scientists|newton|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6094" PostTypeId="2" ParentId="6092" CreationDate="2017-05-31T15:51:17.193" Score="3" Body="&lt;p&gt;One of the first pioneers was Gauß. He needed it, like the least squares law, for his astronomical calculations. In German it is called Gaußsches Fehlerfortpflanzungsgesetz. &lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-05-31T15:51:17.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6096" PostTypeId="2" ParentId="6093" CreationDate="2017-05-31T20:22:23.613" Score="5" Body="&lt;p&gt;The standard book about Newton's life is Never at Rest by Richard Westfall.&#xA;On my opinion it is a very good book, it covers his life in great detail, and gives a general overview of his activities (not only in physics) but in astronomy, history, theology, alchemy, and as the Mint administrator. On physics, the latest English translation of Principia by Cohen (UCB Press, blue cover, not the cheaper editions!), has very comprehensive introduction and commentaries, (Introduction is about 1/3 of the total size of the book!) which covers all aspects of Principia (and Principia contains Newton's most important contributions to physics). I do not know an equally good commentaries on his Optics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Newton's Principia for the common reader&quot; by S. Chandrasekhar, written by one of the famous physicists of 20th century. This is a commentary to the first part of Principia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot; Huygens and Barrow, Newton and Hooke&quot; by V. Arnold, is a very interesting (and very small) book  written by a prominent mathematician. It explains very much of Newton's physics for a book of such small size.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I listed only my favorite general books. There is an enormous specialized literature (books and articles) on various aspects of Newton's physics, for example, on his derivation of Kepler's laws, on his theory of the Moon, on the theory of motion in resisting medium, on tides, on optics etc. The books listed above give a good general overview.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-05-31T20:40:28.680" LastActivityDate="2017-05-31T20:40:28.680" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6097" PostTypeId="1" CreationDate="2017-05-31T21:42:26.443" Score="4" ViewCount="144" Body="&lt;p&gt;I have always wondered if the equations of quantum field theory were actually ever used in the production of some invention or device other than needed to make predictions about the Standard Model of Particle physics of which quantum field theory was actually necessary in order to complete the Standard Model. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the development of radio, television, MRI or possibly all the advances in the construction of nuclear power plants. I know Einsteins's field equations of General Relativity come in handy in GPS systems and quantum mechanics was needed to develop the transistor and the vast explosion in micro processors, superconductors, and solid state devices.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is quantum field theory a set of equations used as a guide line when these inventions are created or is this some other finely tuned set of equations used only by the theoretical elite.  I am not sure where to post the question since this isn't a specific question about a historical date but someone versed in  history of scientific inventions might like to comment. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-07-08T07:38:36.690" LastActivityDate="2020-07-08T07:38:36.690" Title="During the development of QFT was this theory actually used to build any invention like the MRI?" Tags="|physics|quantum-mechanics|electromagnetism|technology|quantum-field-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6098" PostTypeId="1" CreationDate="2017-06-01T08:50:00.740" Score="0" ViewCount="111" Body="&lt;p&gt;Assuming we have a Riemann surface $S$ of degree $n$ and we look at it as a covering of the projective line $\mathbb{P}^1$. If $B$ is the set of branch points of $S$ (when $B$ is a subset in $\mathbb{P}^1$), then there is an epimorphism from the free group $\pi_1(\mathbb{P}^1 - B)$ to $Sym_n$, determining this covering (birationally, I assume). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have therefore two questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Was Riemann already thinking in these terms (of epimorphism to the symmetric group), or was it a later development? I assume that Riemann thought about his surfaces otherwise (in terms of algebraic functions and Abelian integrals?), and not as explicitly covering surfaces with epimorphisms to this group.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Who actually proved first the &quot;reverse&quot; theorem, that given a set of points $B$, if there is an epimorphism $\pi_1(\mathbb{P}^1 - B)$ to $Sym_n$, then there is a Riemann surface of degree $n$ branched along $B$?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;===&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: as was remarked below: &quot;the ideas of coverings, the fundamental group, and the symmetric group, [...] [were] developed together&quot;; I think nevertheless the the idea of a symmetry group existed before Riemann, but the developpment of the ideas indeed happened same time. Dieudonne writes in his book &quot;A History of Algebraic and Differential Topology 1900-1960&quot;, p. 293, that: &quot;In modern mathematics three notions are interconnected in such a way that each one essentially determines the other two: fundamental group, covering space, and properly discontinuous group. Historically, they appeared in the reverse order.&quot; The chapter &quot;Fundamental Group and Covering Spaces&quot; (pp. 293-310) in this book gives the historical overview on these topics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems however, that it was Adolf Hurwitz who indeed connected these ideas together: &quot;In 1891, Adolf Hurwitz published a paper on (closed) Riemann surfaces, understood as branched coverings of the complex number sphere with finitely many sheets and a finite number $n$ of branch points.&quot; (from: &quot;History of Topology&quot;, ed. I. M. James, p. 327, from the paper of Moritz Epple, &quot;Geometric aspects in the development of knot theory&quot;). Hurwitz indeed refers there directly to the symmetric group. The 1891 paper of Hurwitz is called &quot;Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten&quot;. I assume that answers at least my first question.&lt;/p&gt;&#xA;" OwnerUserId="3232" LastEditorUserId="3232" LastEditDate="2017-06-07T14:36:04.290" LastActivityDate="2017-06-07T14:36:04.290" Title="Riemann surfaces and covering" Tags="|reference-request|differential-geometry|topology|riemannian-geometry|" AnswerCount="0" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6099" PostTypeId="1" CreationDate="2017-06-02T00:31:59.083" Score="0" ViewCount="97" Body="&lt;p&gt;In his book 'Chaos' he says, 'In theory the WWII atomic bomb project was a problem in nuclear physics. In reality the nuclear physics had been mostly solved before the project began, and the business that occupied the scientists assembled at Los Alamos was a problem in fluid dynamics.'&lt;/p&gt;&#xA;" OwnerUserId="5045" LastActivityDate="2017-06-02T00:31:59.083" Title="Is this quote from James Gleick about the Manhattan project accurate?" Tags="|atomic-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6100" PostTypeId="1" CreationDate="2017-06-02T13:35:41.483" Score="2" ViewCount="140" Body="&lt;p&gt;So many elements were discovered without a prize. What is special about Radium? Radioactivity is not that rare. &lt;/p&gt;&#xA;" OwnerUserId="5675" LastActivityDate="2017-06-02T14:22:38.200" Title="Why was the discovery of Radium worth a Nobel prize?" Tags="|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6101" PostTypeId="2" ParentId="6100" CreationDate="2017-06-02T14:22:38.200" Score="4" Body="&lt;p&gt;Before radium was isolated, the only available sources of alpha particles were uranium and thorium. The naturally occurring isotopes of uranium and thorium have half-lives of billions of years, which is on the same order of magnitude as the age of the earth, allowing them to occur in nature in significant quantities. But the long half-lives imply low rates of decay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Curies chemically isolated several products of the decay chains of uranium and thorium. They initially referred to them all as radium, and there were names like radium C, radium C', and radium C2 for what we would today describe as isotopes of bismuth, polonium, and thallium. Because radium 226 has a half-life of only 1620 years, its alpha radioactivity is approximately a million times more intense than that of uranium or thorium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The availability of an intense alpha source was crucial for further experimental breakthroughs in nuclear physics. Alphas from radium were used in the 1909 discovery of the nucleus by Rutherford, and also in the 1917-1919 &lt;a href=&quot;https://web.lemoyne.edu/giunta/rutherford.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;discovery&lt;/a&gt; of nuclear transmutation and the proton, also by Rutherford.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-06-02T14:22:38.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6102" PostTypeId="2" ParentId="6093" CreationDate="2017-06-02T21:38:24.740" Score="3" Body="&lt;p&gt;I agree with the citation of Westfall's biography of Newton, the 'Cambridge Companion' and the 1999 translation and introduction of the 'Principia'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, for the calculus and an account of the dispute about it I would suggest A R Hall's 'Philosophers at War'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also cite excellent books and papers by: &#xA;Niccolo Guicciardini (esp. books including 'Reading the Principia', 'Isaac Newton on Mathematical Certainty and Method', &amp;amp; papers); &#xA;William L Harper (esp. 'Isaac Newton's Scientific Method').    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also valuable is almost everything by I Bernard Cohen, e.g.&lt;br&gt;&#xA;his earlier (1972/78) 'Introduction to Newton's Principia'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book on the Principia by Chandrasekhar is a strange mixture. Some parts of it can deliver great clarity and insight, even though it tends to project modern ways of thinking backwards on to a time before the concepts had yet developed. Other parts are so wrong on verifiable facts they show Chandrasekhar was telling the truth when he claimed (in the prologue) to have made &quot;no serious attempt to enlarge my knowledge derived from the Principia by any significant collateral reading.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Newton and his work are subjects on which a number of myths have developed (i.e. not based on evidence of primary documents, especially of Newton &amp;amp; his contemporaries, even contradicting the sources). Some of the myths are longstanding, even some of the better books have not been quite free of them. It's worth looking at a range of authors and at original documents whenever you can, to help tell fact from speculation, to see directly in what sort of terms Newton and his contemporaries held their discussions and argued, and identify some of the uncertainties. The published volumes of the Correspondence of Isaac Newton (7 vols) and the Mathematical Papers of Isaac Newton (8 vols) are bulky but can be found in some libraries and are well worth the discovery.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-06-02T21:38:24.740" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6103" PostTypeId="2" ParentId="6075" CreationDate="2017-06-02T23:01:47.823" Score="1" Body="&lt;p&gt;'Before spectroscopy' there were some established chemical structures based on more than just guesswork. In the 19th century much of the structural evidence for a given compound usually lay in combining experimental results about the elemental composition and proportions, with results about the range and kinds of chemical reactions in which the considered compound would participate (which included consideration of the combining proportions by mass), as well as the range and kinds of products that were (or were not) produced in those reactions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For each compound investigated, the problem might be considered as a special detective exercise, with an individual pattern of relationships between the pieces of evidence in each case. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the best known examples of this was in the identification of the molecular ring-structure of benzene by August Kekulé, who also from the 1850s onwards contributed much to the theory of (esp. organic) chemical structures and their identification, e.g. in papers published in the 'Annalen der Chemie und Pharmacie'. A parallel contributor during the same period was Archibald Couper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much later on, W H Bragg's X-ray crystallography became another powerful non-spectroscopic tool for discovery of chemical structures provided the considered compound could be crystallized.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-06-02T23:01:47.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6104" PostTypeId="1" AcceptedAnswerId="6113" CreationDate="2017-06-03T00:36:56.557" Score="23" ViewCount="2480" Body="&lt;p&gt;The question is in the title, but allow me to provide some background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m aware that Leibniz introduced the word “function” into mathematics (around 1673) and that &lt;a href=&quot;https://hsm.stackexchange.com/a/5204/3462&quot;&gt;Johann Bernoulli or Euler&lt;/a&gt; introduced the notation $f(x)$ (sometimes between 1706-1740). I've also read the Wikipedia article on the &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_function_concept&quot; rel=&quot;noreferrer&quot;&gt;History of the function concept&lt;/a&gt; and the book &lt;a href=&quot;https://books.google.ch/books/about/Die_Analysis_im_Wandel_und_im_Widerstrei.html?id=0fzWrQEACAAJ&amp;amp;redir_esc=y&quot; rel=&quot;noreferrer&quot;&gt;Die Analysis im Wandel und im Widerstreit&lt;/a&gt; by Spalt, plus a few other sources on the history of functions. But I haven't found the following addressed:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at definitions* and uses of the word &lt;em&gt;function&lt;/em&gt; among mathematicians up until the beginning of the 20th century, I see that they consistently call $y$ the function. By $y$ I mean the one in $y=f(x)$. Equivalently they call $f(x)$ the function. Actually they call $y$ a function &lt;em&gt;of $x$&lt;/em&gt;, but they eventually drop the “of $x$” when it is clear from the context or irrelevant for the discussion. None of them ever consider $f$ as a mathematical object in itself.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is in stark contrast to the modern definition, which calls $f$ the function, describes it as a rule or the like, and officially calls $f(x)$ something else (like the output of the function at input $x$ etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first mathematician I could find to distinguish clearly between $f$ and $f(x)$, with a definition similar to modern ones, was Cantor in 1895. But to my surprise he did not call $f$ a function but a “Belegung” (allocation). Here's my translation of what he wrote in &lt;a href=&quot;https://link.springer.com/article/10.1007/BF02124929&quot; rel=&quot;noreferrer&quot;&gt;Beiträge zur Begründung der transfiniten Mengenlehre, I&lt;/a&gt;, 1895, p.486:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By an &lt;em&gt;,allocation of the set $N$ with elements of the set $M$'&lt;/em&gt;  or&#xA;  simply put, by an &lt;em&gt;,allocation of $N$ with $M$'&lt;/em&gt; we understand a law,&#xA;  through which with every element $n$ of $N$ is connected a certain&#xA;  element of $M$, where one and the same element of $M$ may be used&#xA;  repeatedly.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The with $n$ connected element of $M$ is to a certain extent a unique&#xA;  function of $n$ and may be denoted with $f(n)$; it shall be called the&#xA;  &lt;em&gt;,allocation function of $n$'&lt;/em&gt;; the corresponding allocation of $N$ shall be called $f(N)$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Two allocations $f_{1}(N)$ und $f_{2}(N)$ are called equal if and only&#xA;  if, for all elements $n$ of $N$ the equation &#xA;  $$  f_{1}(n) = f_{2}(n),$$ holds, so that if for a single fixed element $n=n_{0}$ the&#xA;  equation does not hold, $f_{1}(N)$ and $f_{2}(N)$ are characterised as&#xA;  different allocations of $N$. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[$\ldots$]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The totality of all different allocations of $N$ with $M$ determines a&#xA;  set with the elements $f(N)$; we call it the &lt;em&gt;,set of allocations of&#xA;  $N$ with $M$'&lt;/em&gt; and denote it with $(N\mid M)$. Hence: &#xA;  $$ (N\mid M)=\{ f(N) \}. $$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Notice that he keeps the traditional meaning of the word function, so it makes perfect sense to give $f$ (or as he writes $f(N)$) a new name.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Two questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Was Cantor the first person to treat $f$ as an object on its own and&#xA;distinguish it from $f(x)$? In other words, was Cantor the&#xA;first one to give the modern definition of a function?&lt;/li&gt;&#xA;&lt;li&gt;When and why did it become standard to call $f$ a function, despite the&#xA;established use of the word for $f(x)$ (and despite Cantor calling&#xA;$f$ something else)?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;* Several such historical definitions can be found at the end of the following MO question: &lt;a href=&quot;https://mathoverflow.net/questions/307947/formalizations-of-the-idea-that-something-is-a-function-of-something-else&quot;&gt;Formalizations of the idea that something is a function of something else?&lt;/a&gt;.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2018-08-30T08:01:19.650" LastActivityDate="2018-08-30T08:01:23.813" Title="Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?" Tags="|mathematics|mathematicians|terminology|calculus|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="6105" PostTypeId="1" AcceptedAnswerId="6112" CreationDate="2017-06-03T03:32:56.643" Score="3" ViewCount="345" Body="&lt;p&gt;This is from Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In May–June 1897, Thomson investigated whether or not the rays could&#xA;  be deflected by an electric field.[4] Previous experimenters had&#xA;  failed to observe this, but Thomson believed their experiments were&#xA;  flawed because their tubes contained too much gas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is surprising to know that initially people failed to deflect the cathode ray. The mass of the electron is so small. It should be easily deflected by an electric field. The initial velocity of the electrons is not so high, right? &lt;/p&gt;&#xA;" OwnerUserId="5675" LastActivityDate="2017-06-03T19:30:43.270" Title="Thomson's discovery of the electrical deflection of the cathode ray" Tags="|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6106" PostTypeId="1" CreationDate="2017-06-03T04:21:36.197" Score="10" ViewCount="10132" Body="&lt;p&gt;Einstein's failure to win a Nobel until 1921, and that prize's not being awarded for his work on relativity, is generally ascribed to these factors:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Lack of sufficient experimental proof for the theories at the time (some distrust in the early results on the precession of Mercury, and the Eddington starlight observations).&lt;/li&gt;&#xA;&lt;li&gt;Failure by some members of the committee to understand relativity&lt;/li&gt;&#xA;&lt;li&gt;anti-Semitism&lt;/li&gt;&#xA;&lt;li&gt;skepticism generally about the utility of relativity, and whether it was physics at all.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If these factors were in fact the reason that the relativity work did not garner a prize in 1921, why didn't the relativity work merit a second Nobel prize after, say 1945? Surely the work on cyclotrons and the bomb provided firm experimental support for the validity of the theory, and by then the overall utility would have been understood as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For that matter it's interesting to speculate how many Nobel Prizes Einstein should have won; but this question just asks: why only one?)&lt;/p&gt;&#xA;" OwnerUserId="5870" LastActivityDate="2019-09-30T15:15:44.840" Title="Why didn't Einstein win a second Nobel Prize, for relativity?" Tags="|relativity-theory|einstein|nobel-prize|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6107" PostTypeId="2" ParentId="6106" CreationDate="2017-06-03T05:32:53.557" Score="13" Body="&lt;p&gt;The time when experimental evidence supporting relativity began to appear coincided with the rise of antisemitism in Germany following Germany's defeat in WWI.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Antisemitism was on the rise in Germany; Jews were being scapegoated for the country's defeat in the war. As both Jew and pacifist, Einstein was an obvious target. The complexity of relativity did not help either. Opponents such as Ernst Gehrcke and Philipp Lenard found it easy to cast doubt upon its labyrinthine mathematics.&lt;/p&gt;&#xA;&lt;p&gt;The situation reached crisis point in 1921 when, paralysed by indecision, the Nobel Committee decided it was better not to award a prize at all than to give it to relativity. The arguments raged for another year until a compromise was reached.&lt;/p&gt;&#xA;&lt;p&gt;At the suggestion of Carl Wilhelm Oseen, Einstein would receive the deferred 1921 prize, but not for relativity. He would be given it for his explanation of the photoelectric effect, a phenomenon in which electrons are emitted from a metal sheet only under certain illuminations. The work had been published back in 1905.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, after considerable hand-wringing and even deferring the prize for a year, in 1922 the Swedish Academy made the decision to award Einstein the 1921 prize a year late, writing to Einstein on the 10th of November 1922 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... the Royal Academy of Sciences has decided to award you last year's Nobel Prize for physics, in consideration of your work in theoretical physics and in particular your discovery of the law of the photoelectric effect, but &lt;strong&gt;without taking into account the value which will be accorded your relativity and gravitation theories after these are confirmed in the future.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Source : &lt;a href=&quot;https://books.google.ca/books?id=OjgTS12V0VUC&amp;amp;pg=PA15&amp;amp;lpg=PA15&amp;amp;dq=without%20taking%20into%20account%20the%20value%20that%20will%20be%20accorded%20your%20relativity%20and%20gravitation%20theories%20after%20these%20are%20confirmed%20in%20the%20future&amp;amp;source=bl&amp;amp;ots=OHOy7o2pit&amp;amp;sig=f_A4eBxwr6QhTIpUrDXNmBhfzb8&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjV5uDj7aDUAhVK5mMKHQsQBUQQ6AEIODAD#v=onepage&amp;amp;q=without%20taking%20into%20account%20the%20value%20that%20will%20be%20accorded%20your%20relativity%20and%20gravitation%20theories%20after%20these%20are%20confirmed%20in%20the%20future&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mass and Energy in Relativity Theory&lt;/em&gt;, by Lev Borisovi.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;At this stage relativity was open to future consideration. However, Einstein chose not to attend the ceremony to receive the award in person. This decision may have made a future prize for relativity more difficult, if not impossible, since it gave the impression that Einstein did not value the award.&lt;/p&gt;&#xA;&lt;p&gt;Having said that, at the time Einstein received confirmation of the award from the Nobel committee, he was committed to a lecture tour of Japan.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;German foreign minister Walther Rathenau had been murdered by anti-Semites. In the subsequent investigation, the police had found Einstein's name on a list of targets. In the face of such a death threat, leaving Germany to spend months in the Far East, rather than a few days in Stockholm, must have seemed prudent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-06-06T04:47:37.370" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6109" PostTypeId="2" ParentId="5065" CreationDate="2017-06-03T15:54:06.153" Score="2" Body="&lt;p&gt;This is only a marginal notice but too long for a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are different opinions about the matter. It is indubitable that l'Hospital paid Johann Bernoulli for educating him in analysis. It is possible that the &quot;rule&quot; has been found by Johann Bernoulli. On the other hand it is fact that Johann Bernoulli was one of the first self-promoters with a fair amount of self-confidence and that he did not easily admit any negative features. (See the famous quarrel of the Bernoulli brothers about a problem posed by Jakob Bernoulli. Johann did not admit his mistake for many years and confessed it only after his brother's dead.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make a long story short: An author the name of whom I have forgotten wrote: The longer l'Hospital was dead, the louder sounded Johann's claim that he was the author of the rule. This aspect should not be completely overlooked.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-03T15:54:06.153" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6110" PostTypeId="2" ParentId="6093" CreationDate="2017-06-03T16:51:28.290" Score="2" Body="&lt;p&gt;&lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Whiteside.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;D. T. Whiteside&lt;/a&gt;'s abundant commentary in his &lt;em&gt;Mathematical Papers of Isaac Newton&lt;/em&gt; (8 volumes, Cambridge University Press, 1967-1981) is widely regarded as &lt;strong&gt;&lt;em&gt;the&lt;/em&gt;&lt;/strong&gt; authoritative work on Newton's mathematics. (See Whiteside's obituaries in &lt;a href=&quot;https://www.theguardian.com/education/2008/may/05/science.mainsection&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Guardian&lt;/a&gt;, &lt;a href=&quot;http://www.independent.co.uk/news/obituaries/professor-d-t-whiteside-historian-of-mathematics-whose-prodigious-work-on-newtons-papers-astonished-820387.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Independent&lt;/a&gt;, &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Obits2/Whiteside_Times.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Times&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=nbGcAQAAQBAJ&amp;amp;pg=PA1181&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford Dictionary of National Biography&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=oYZ-0PUrjBcC&amp;amp;pg=PR5&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Investigation of Difficult Things&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding your specific question (“connections between his geometric proofs in the Principia and his development of the calculus”), Whiteside later &lt;a href=&quot;http://dx.doi.org/10.1098/rsnr.1991.0002&quot; rel=&quot;nofollow noreferrer&quot;&gt;wrote&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How often am I still asked: ‘&lt;em&gt;Did&lt;/em&gt; Newton use calculus to obtain&#xA;  the theorems in his &lt;em&gt;Principia&lt;/em&gt;?’ How, without seeming to patronize, do you lay the groundwork on which you can reply that the question is ill-formed and&#xA;  therefore meaningless? I will not here go into the reasons why.$\smash{^4}$&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;ol start=&quot;4&quot;&gt;&#xA;  &lt;li&gt;I have done so in ‘&lt;a href=&quot;http://adsabs.harvard.edu/abs/1970JHA.....1..116W&quot; rel=&quot;nofollow noreferrer&quot;&gt;The mathematical principles underlying Newton's &lt;em&gt;Principia&#xA;  Mathematica&lt;/em&gt;&lt;/a&gt;’, J. Hist. Astron. &lt;strong&gt;1&lt;/strong&gt;, 116-38 (1970); and more generally in pertinent footnotes in vol. 6 of my edition of &lt;em&gt;The mathematical papers of Isaac Newton&lt;/em&gt; (Cambridge University Press, 1974).&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-06-03T17:19:13.853" LastActivityDate="2017-06-03T17:19:13.853" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6111" PostTypeId="2" ParentId="139" CreationDate="2017-06-03T17:50:42.920" Score="4" Body="&lt;p&gt;The question implicitly assumes that other theoretical explanations had been offered before the theory of symbiogenesis.  I believe the truth is that up to a certain point around the later 60s, those active in the field were still too busy identifying constituents and properties of the plastids to think that the time was ripe to offer separate (scientifically-testable) theories of their origin: it does not seem that there were yet any theories of that kind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of the analytical tools and methods of molecular biology that are well-known now did not even exist then. Much of the confidence in evolutionary pathways that exists now comes from nucleic acid sequences, which did not then yet exist. Classical taxonomy without genetic sequence information was an uncertain business.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References that support that the idea was old, but that the questions of evolutionary origin were for a long time untestable and undecidable, include: Schimper A F W (1883) &lt;em&gt;Über die Entwicklung der Chlorophyllkörner und Farbkörner&lt;/em&gt;, Botanische Zeitung 41: 105-114 (for a scientifically prudent and restrained mention in a footnote that if it became definitely established that the plastids did not arise anew in the egg-cells (but propagated independently) then 'their relationship to the organism containing them would somewhat remind one of a symbiosis');  Mereschkowsky, C. (1905) &lt;em&gt;Über Natur und Ursprung der Chromatophoren im Pflanzenreiche&lt;/em&gt;,  Biologisches Centralblatt 25 593-604 (pointing out the lack of evidence supporting the old idea that the green particles arose under control of the cell nucleus from the cytoplasm, but going further and proposing their separate origin); and the same author (1910), in Biologisches Centralblatt, Band 30 (1910), (pages 277-303, 321-347, 353-367) gave an extensive elaboration with his idea of a &lt;em&gt;Theorie der zwei Plasmaarten als Grundlage der Symbiogenesis&lt;/em&gt; (theory of two kinds of plasma as a basis of symbiogenesis), but still there was no feasible way to test it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heavy disapproval of the idea in the period from about 1925 to the later 1960s is shown first by E B Wilson, &lt;em&gt;The Cell in Development and Heredity&lt;/em&gt; (3rd ed. 1925 and 1928 -- described by J T Bonner (&lt;em&gt;The Ideas of Biology&lt;/em&gt;, 1962) as 'one of the great books of biology'); and its continuing disrepute can be seen confirmed &amp;amp; reiterated by J T Bonner (in 'The Ideas of Biology', 1962); and by R M Klein &amp;amp; A Cronquist (in &lt;em&gt;Quarterly Review of Biology&lt;/em&gt;, vol.42(2), 108-296 at p.167).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The unusually emphatic terms in which the idea was condemned are themselves &#xA;a noteworthy piece of background to the current question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For E B Wilson (1925 &amp;amp; 1928) there were already too many unprovable biological conjectures, and he expressed reluctance to create or to entertain any more of them: thus, his preface (p.viii) quoted a predecessor's report, that although a certain biologist had 'rejected two hundred and sixty-two' &quot;groundless hypotheses of development&quot;, &quot;nothing is more certain than that [his] own theory formed the two hundred and sixty-third.&quot; Wilson then wrote &quot;I have no wish to add another to this list&quot;.  When he then went on to review Mereschkowsky's theory, he called it an &quot;entertaining fantasy&quot; with &quot;flights of the imagination&quot; (Wilson, 3rd ed., pp.738-9). He qualified that somewhat, acknowledging that &quot;the ... speculation [about chloroplasts] cannot be considered totally baseless in view of the symbiotic union of unicellular green algae with fungi in the lichens&quot;, and added that while &quot;many&quot; would rate the speculations &quot;too fantastic for present mention in polite biological society; nevertheless it is within the range of possibility that they may some day call for more serious consideration&quot;.  In spite of these somewhat positive qualifications, the negative part of Wilson's appraisal seems to have made more impact and to have contributed to adding the endosymbiotic idea to the already-long list of unprovable and disreputable biological conjectures.  For a long time it remained out of favor: thus, as late as the 1960s the endosymbiotic idea was described as a &quot;never well-regarded hypothesis&quot; (by J T Bonner, in 'The Ideas of Biology', 1962), and even in 1967, R M Klein &amp;amp; A Cronquist offered reasons for likening the endosymbiotic idea to a &quot;bad penny&quot; (Qtrly Review of Biology, vol.42(2), 108-296 at p.167).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the face of this weight of professional disapproval it would clearly have been a bold and risky move by anybody to revive the idea without the backup of new and at least potentially weighty and decisive evidence.  When Lynn Margulis (previously Lynn Sagan) did reintroduce and revitalize the endosymbiotic idea in her publications of 1967 (L Sagan, &lt;em&gt;The Origin of Mitosing Cells&lt;/em&gt;, J Theoret. Biol. 14, 225-274) and 1970 (&lt;em&gt;Origin of Eukaryotic Cells&lt;/em&gt;, Yale Univ Press) she did not claim to introduce something quite new; rather, she was proposing ways to make out of the old idea a scientifically testable theory, thus distinguishing her contribution from the ideas that had gone before.  For example, the prominent subtitle of her book, with the words &quot;evidence and research implications&quot; emphasized the new and hopefully persuasive means of theory-testing that she proposed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is probably no need here to go into the details of those areas where modern techniques have provided decisive evidence and those where they have not, except for one point: at least until recently (2008) there still existed some questions on this topic that arguably still could not be effectively tested even by any of the modern types of evidence. Thus, a review from 2008 (C J Howe, et al., 'The Origins of Plastids', Philosophical Transactions of the Royal Society B 363, 2675-2685), which acknowledges, of course, that the general question whether chloroplasts first arose through symbiosis has been tested, with an affirmative answer now beyond serious doubt. But it also gives reasoned consideration to the question &quot;Can we actually prove (or disprove) anything?&quot; about a still-open problem, whether the different known plastids descend from a single line of organisms that acquired the common endosymbiotic ancestor of all (monophyly). (The alternative possibility would be that there was an independent endosymbiosis for at least one of the current plastid types, thus with different ancestry than the rest (polyphyly).) The limits on testability arise (in the reviewers' prudent and probable opinion) from unavoidable uncertainties, e.g. about limits to the comprehensiveness of achievable taxon sampling, and the narrow effect of any evidence that could be obtained to bear on the problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of the history has been described by Jan Sapp (1994), in 'Evolution by Association: A History of Symbiosis'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, to be more specific about the questions originally asked, the original old idea up to the 1880s was that the green particles of plant cells arose from the cytoplasm under control of the nucleus, and by implication shared the same ancestry as the cells in which they occurred. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know whether there is any evidence bearing decisively on the question of a single or multiple origin of photosynthesis in prokaryotes, but the 2008 review mentioned above shows that the question so far as it may be applied to single or multiple evolutionary origin of chloroplasts, actually remained arguably untestable at least as of 2008 (showing that some questions in this field may even indefinitely remain undecidable).  &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2017-07-04T12:55:18.740" LastActivityDate="2017-07-04T12:55:18.740" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6112" PostTypeId="2" ParentId="6105" CreationDate="2017-06-03T19:30:43.270" Score="3" Body="&lt;p&gt;From &lt;a href=&quot;http://www-outreach.phy.cam.ac.uk/camphy/electron/electron2_1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;CambridgePhysics.org&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The gas in Hertz's tube had become ionised by the electric field produced by the plates. The positive ions moved towards the negative plate and the negative ions towards the positive plate. The net charge on the plates was therefore reduced and so the strength of the electric field was reduced. The remaining electric field deflected the cathode rays very little or not at all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This explanation accounts for Hertz's failure to observe any deflection.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;From the same source :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The nature of the cathode rays was controversial. Although Thomson thought the rays must be particles, many Europeans thought they were an 'etherial disturbance', like light. In Germany Hertz had observed the rays passing through thin sheets of gold. It seemed impossible that particles could pass through solid matter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-06-03T19:30:43.270" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6113" PostTypeId="2" ParentId="6104" CreationDate="2017-06-03T20:03:27.977" Score="10" Body="&lt;p&gt;Cantor 1895 is predated at least by Dedekind in &lt;a href=&quot;https://archive.org/stream/wassindundwasso00dedegoog#page/n28&quot; rel=&quot;noreferrer&quot;&gt;§2&lt;/a&gt; of &lt;a href=&quot;https://books.google.com/books?id=zoxKAAAAYAAJ&amp;amp;pg=PA6&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Was sind und was sollen die Zahlen?&lt;/em&gt; (1888)&lt;/a&gt; (&lt;a href=&quot;https://archive.org/stream/cu31924001586282#page/n63&quot; rel=&quot;noreferrer&quot;&gt;translation&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;21. Erklärung *). Unter einer &lt;strong&gt;Abbildung&lt;/strong&gt; $\varphi$ eines Systems $S$ wird ein Gesetz verstanden, nach welchem zu jedem bestimmten Element $s$ von $S$ ein bestimmtes Ding &lt;strong&gt;gehört&lt;/strong&gt;, welches das &lt;strong&gt;Bild&lt;/strong&gt; von $s$ heißt und mit $\varphi(s)$ bezeichnet wird; wir sagen auch, daß $\varphi(s)$ dem Element $s$ &lt;strong&gt;entspricht&lt;/strong&gt;, daß $\varphi(s)$ durch die Abbildung $\varphi$ aus $s$ &lt;strong&gt;entsteht&lt;/strong&gt; oder &lt;strong&gt;erzeugt&lt;/strong&gt; wird.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;*) Vergl. Dirichlet's &lt;a href=&quot;https://archive.org/stream/vorlesungenberz00dirigoog#page/n490&quot; rel=&quot;noreferrer&quot;&gt;Vorlesungen über Zahlentheorie, dritte Auflage, 1879, §. 163&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The footnote refers to a supplement in &lt;em&gt;Dedekind's&lt;/em&gt; edition of Dirichlet.) &#xA;Also, Dedekind in &lt;a href=&quot;https://archive.org/stream/wassindundwasso00dedegoog#page/n65&quot; rel=&quot;noreferrer&quot;&gt;§11&lt;/a&gt; clearly distinguishes between a &lt;em&gt;map&lt;/em&gt; $\psi$ and what he still calls the &lt;em&gt;function $\psi(n)$ it determines&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;135. Erklärung. Es liegt nahe, die im Satze 126 dargestellte Definition einer&#xA;  Abbildung $\psi$ der Zahlenreihe $N$ oder der durch dieselbe&#xA;  bestimmten &lt;strong&gt;Funktion&lt;/strong&gt; $\psi(n)$ auf den Fall anzuwenden, wo ($\ldots$)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; To your two questions,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Attribution to Dedekind 1888 seems confirmed by W. Sieg and D. Schlimm,  &lt;a href=&quot;https://dx.doi.org/10.1093/philmat/nku021&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Dedekind's Abstract Concepts: Models and Mappings&lt;/em&gt;, Philosophia Mathematica nku021 (2014) 1-26&lt;/a&gt;. They also quote from an earlier draft of 1872-78, where Dedekind wrote $x\,|\,f$ instead of $f(x)$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't know who (rather unfortunately, I'd say) suppressed his careful distinction between &lt;strong&gt;Abbildung&lt;/strong&gt; (mapping, application, &quot;Maler&quot;, $f$) and &lt;strong&gt;Funktion&lt;/strong&gt; (function, fonction, &quot;Bild&quot;, $f(x)$). Perhaps von Neumann (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?51.0163.04&quot; rel=&quot;noreferrer&quot;&gt;1925&lt;/a&gt;, &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002169606&amp;amp;physid=phys227&quot; rel=&quot;noreferrer&quot;&gt;pp. 222, 223&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;($\ldots$) aus einer Funktion $f$ (die von ihren Werten $f(x)$ wohl zu unterscheiden ist!) und einem Argumente $x$ den Wert $f(x)$ der Funktion $f$ für das Argument $x$ zu bilden. ($\ldots$) $\mathfrak M$ sei eine Menge, $f(x)$ eine (in $\mathfrak M$ definierte) Funktion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or more influentially van der Waerden in &lt;em&gt;Moderne Algebra&lt;/em&gt; (&lt;a href=&quot;http://dx.doi.org/10.1007/978-3-662-41906-9&quot; rel=&quot;noreferrer&quot;&gt;1930, §2, §6, §9&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wenn durch irgend eine Vorschrift jedem Element $a$ einer Menge $\mathfrak M$ ein einziges neues Objekt $\varphi(a)$ zugeordnet wird, so nennen wir diese Zuordnung eine &lt;em&gt;Funktion&lt;/em&gt; ($\ldots$) Man nennt eine solche Zuordnung ($\ldots$) auch eine (eindeutige) &lt;em&gt;Abbildung&lt;/em&gt; der Menge $\mathfrak M$ auf die Menge ($\ldots$) Ist $\varphi$ eine eineindeutige Abbildung ($\ldots$) Eine solche Funktion $\varphi$ gibt es ($\ldots$)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-09-04T13:23:26.693" LastActivityDate="2017-09-04T13:23:26.693" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6114" PostTypeId="2" ParentId="459" CreationDate="2017-06-03T21:16:50.870" Score="7" Body="&lt;p&gt;The absence of air in the interplanetary spaces &#xA;(to a very high degree of exhaustion) was appreciated &#xA;by Newton: he may have been the first. His demonstrations &#xA;in the Principia are heavily bound up with the combined &#xA;effects of gravitation and atmospheric resistance on the &#xA;motion of bodies, a subject that attracted his interest &#xA;and study, from the earliest 'De Motu..' of 1684, through &#xA;Book 2  of the 'Principia'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In book 2 of the Principia, Newton made it a point &#xA;to study at what rate the combination of gravitational &#xA;and (de)compressive effects should cause the density of &#xA;the earth's atmosphere to decrease with distance above the earth's surface (esp. Book 3 Prop. 10, &#xA;[&lt;a href=&quot;https://books.google.com/books?id=6EqxPav3vIsC&amp;amp;pg=PA230]&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.com/books?id=6EqxPav3vIsC&amp;amp;pg=PA230]&lt;/a&gt;,&#xA;(applying Book 2 Prop.22 Scholium) &#xA;[&lt;a href=&quot;https://books.google.com/books?id=6EqxPav3vIsC&amp;amp;pg=PA73&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.com/books?id=6EqxPav3vIsC&amp;amp;pg=PA73&lt;/a&gt; up to PA76]). &#xA;He calculated that at an altitude of even 200 miles up, the air should be 'rarer' than at the earth's surface in a ratio of roughly 75 x 10^12 to 1.&#xA;This is practically a vacuum. In Book 3 Prop.10 Newton went on to conclude that there should be at most no appreciable atmospheric-type drag on the motion of Jupiter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the General Scholium Newton afterwards wrote:&#xA;[&lt;a href=&quot;https://books.google.com/books?id=6EqxPav3vIsC&amp;amp;pg=PA388]&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.com/books?id=6EqxPav3vIsC&amp;amp;pg=PA388]&lt;/a&gt;&#xA;(translation slightly amended here, closer to the Latin original):&#xA;&quot;Bodies, projected in our air, suffer no resistance but from the air. &#xA;Withdraw the air, as is done in Mr. Boyle's vacuum, and the resistance &#xA;ceases. For in this void a bit of fine down and a piece of solid gold &#xA;descend with equal velocity. And the same reasoning applies to the &#xA;celestial spaces which are above the Earth's atmosphere&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Near-contemporaries saw the interplanetary void as a feature &#xA;of Newton's theories (and sometimes as an attackable one, even &#xA;absurd). Voltaire, with characterstic dry humour, included it in his contrasts between Newton and Descartes (1733): &quot;A Frenchman arriving &#xA;in London finds things quite changed in philosophy as in everything else: &#xA;he left the world full, he finds it empty. In Paris one sees that the &#xA;universe is composed of vortices of subtle matter: in London one sees &#xA;nothing of that.&quot; &amp;amp;c. (Voltaire, 'Lettres Philosophiques', 1733, &#xA;Lettre XIV. 'Sur Descartes et Newton'.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-06-03T21:16:50.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6115" PostTypeId="1" CreationDate="2017-06-04T01:20:32.277" Score="0" ViewCount="60" Body="&lt;p&gt;When I read about Nikolai  Lobachevsky  it is said that the university and Kazan itself was not very important or well known but also when I check Kazan in the 1800 's it was in the top 10 of the biggest Russian city's  at that time so that made me wonder&lt;/p&gt;&#xA;&#xA;&lt;p&gt;was the university of Kazan a small University in Russian standards in his time and how well known was  it compared with other Russian  universities ?&lt;/p&gt;&#xA;" OwnerUserId="557" LastActivityDate="2017-06-04T14:20:24.270" Title="Lobachevsky and the University of Kazan" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6116" PostTypeId="2" ParentId="6115" CreationDate="2017-06-04T06:14:12.190" Score="1" Body="&lt;p&gt;There is no contradiction in what you read. Kazan university was not well known or important in comparison with Western European universities. But Kazan was one of the 10 biggest cities in Russia. Most of those cities had no universities at all.&#xA;IN the beginning of 19th century, Russian empire had only 5 universities (Moscow, Petersburg, Derpt, Kharkov and Kazan), none of them especially famous in the rest of Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even now, Kazan University is mostly known in the West because of Lobachevsky (see, for example the very beginning of the Wikipedia article on this university).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-06-04T14:20:24.270" LastActivityDate="2017-06-04T14:20:24.270" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6118" PostTypeId="2" ParentId="4925" CreationDate="2017-06-04T10:44:49.663" Score="3" Body="&lt;p&gt;In fact, Leibniz is the father of the fountain. He supplied the decisive ideas in 1696 but died before the fountain could be finished.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz first recognized that enough water has to be supplied. This was realized by a channel from the nearby river &lt;em&gt;Leine&lt;/em&gt; which had to be damed by a weir. The heart of the device is a two storied &lt;em&gt;Wasserkunst&lt;/em&gt; (something he also had improved for the mining industry in the Harz mountains) consisting of a water wheel and four water pumps, two of which had to pump the water into a 12 m high container. The other pumps had to pump the water into a completely new invention: the &lt;em&gt;Windkessel&lt;/em&gt;, a device to compress air such that finally the fountain should reach a hight of 35 m, according to Leibniz' calculations. (And we can state that he was better than Euler whose construction of the fountains in the garden of Sanssouci Frederick the Great always gave reason to complaints.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz ideas have been applied and even improved after his dead by Johann Christian Böhme, such that a hight of 67 m was reached. Today, with electric power, even 82 m are reached.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For facts and photos see &lt;a href=&quot;http://www.technikatlas.de/~th2/wasserleitungen.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;technikatlas&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-04T10:44:49.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6119" PostTypeId="2" ParentId="3767" CreationDate="2017-06-04T16:54:53.403" Score="-3" Body="&lt;p&gt;Kronecker won, Cantor lost. Plenty of material proving this case is given in &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Chapter V Mückenheim has collected a tremendous amount of over 250 critical voices, most of which are directly refusing set theory or its basic assumption, namely actual or completed infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further he applies mathematical limit theory of analysis in order to show that not all rational numbers can be mapped on natural numbers. The fact that &lt;em&gt;every&lt;/em&gt; rational can be mapped on its own natural is irrelevant, he says, because &lt;em&gt;every&lt;/em&gt; rational number belongs to a finite initial segment which is followed by potentially infinitely many rationals. The mathematical limit of the not mapped rationals is infinite. He gives convincing evidence and a formal proof which to my knowledge has never been refuted. So if set theory is taken as it is, it is in contradiction with analysis and therefore cannot be its foundation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then he has shown that an irrational number can never be defined by its digit expansion because there is no number defined unless the last digit is known. If only a string of digits is given - without a finite formula like &quot;0.111...&quot; defining the sequence - then no real number is fixed. Therefore Cantor's diagonal argument does not yield a fixed real number let alone a transcendental number. The finite formulas like those known for $e$ or $\pi$ however are countable. By the way, even Hessenberg, a strong supporter of set theory and author of the famous uncountability proof using all subsets of the natural numbers, has recognized that &quot;a statement about the number defined by that process, which concerned only that number, would be possible only after completion of the process – and this process cannot be completed.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, Mückenheim shows that, if countability of set theory is assumed (contrary to its being in contradicition with analysis), all objects of thought can be enumerated by their rational spatio-temporal coordinates even in an infinite and eternal universe. Therefore there is nothing uncountable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many other arguments. But I would recommend to read first the statements in chapter V. They show that by far not all mathematicians agree to Cantor's victory over Kronecker.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-05T10:30:05.367" LastActivityDate="2017-06-05T10:30:05.367" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6120" PostTypeId="2" ParentId="3770" CreationDate="2017-06-04T17:10:28.673" Score="1" Body="&lt;p&gt;The mathematician Adolf Kneser (1862 - 1930), who in 1884 did his doctorate with L. Kronecker and E. E. Kummer in Berlin, reported on the occasion of Kronecker's 100 birthday in a speech on 19. December 1923 at the Mathematical Society of Berlin &lt;em&gt;from his personal memory&lt;/em&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Da trat eines schönen Tages auch &lt;em&gt;Kronecker&lt;/em&gt; auf dieses Gebiet über und tat den Ausspruch: Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Adolf Kneser: Leopold Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung &lt;strong&gt;33&lt;/strong&gt; (1925) 210-227, &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002127431&amp;amp;physid=phys231&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 221&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="658" LastEditDate="2017-06-04T20:26:22.690" LastActivityDate="2017-06-04T20:26:22.690" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6121" PostTypeId="1" CreationDate="2017-06-04T02:49:42.970" Score="0" ViewCount="60" Body="&lt;p&gt;I was preparing some history notes for a few subjects next week and got sidetracked on some details of the Michelson and Morley experiment. Could someone clarify or answer a few questions?&#xA;(1) What wavelength of light is usually used in the interferometer?&#xA;(2) What were the expected results? (3) which arm was predicted to take longer? Some articles say the arm parallel with the direction of the medium would be longest time but other articles have it reversed.&#xA;(4) most articles say there was no difference for either direction but every now and then an article will say there were small discrepancies as the experiment was rotated. So I'm not sure what that means.&#xA;Thanks&lt;/p&gt;&#xA;" OwnerUserId="12387" OwnerDisplayName="Bill Alsept" LastActivityDate="2017-06-04T17:21:54.327" Title="Michelson and Morley questions" Tags="|optics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6123" PostTypeId="2" ParentId="6104" CreationDate="2017-06-05T10:59:38.193" Score="6" Body="&lt;p&gt;&lt;strong&gt;Answering 2:&lt;/strong&gt; Maybe the idea of calling $f$ the function goes back to Frege? In &lt;a href=&quot;https://archive.org/details/festschriftludw00meyegoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Was ist eine Funktion?&lt;/a&gt; Festschrift Ludwig Boltzmann (1904, pp. 656-666) we find on p. 663:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Und haben wir hier nicht das, was wir suchen, die Funktion? Also wird&#xA;  auch »$f$« genaugenommen eine Funktion andeuten.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My translation: &quot;And don't we have here what we are looking for, the function? Hence also »$f$« will strictly speaking signify a function.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole article is a very interesting read, since Frege also tries to answer the questions &quot;What is a variable quantity?&quot; (he concludes they don't belong in mathematics), &quot;What is an indeterminate number?&quot; (he concludes there are none) and addresses the distinction between $f(x)$ as a syntactic expression (a term), versus $f(x)$ as a semantic entity (the variable quantity). I don't agree with all of his arguments, but it's fascinating to see how he single-handedly demolishes concepts that had been an integral part of mathematics for 200 years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added later:&lt;/strong&gt; In the comments Francois makes the relevant observation, that Lagrange called $f$ the &lt;em&gt;caractéristique&lt;/em&gt; of the function $f(x)$. One also finds this in Johann Bernoulli (in &lt;a href=&quot;http://www.e-rara.ch/doi/10.3931/e-rara-3595&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Remarques sur ce qu'on a donné jusqu'ici de solutions des Problemes sur les isoperimetres&lt;/em&gt;&lt;/a&gt;, 1718, p. 243) and Euler (in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E140.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sur la vibration des cordes&lt;/em&gt;&lt;/a&gt;, 1748, p. 78).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I interpret this similarly to Francois: $f$ is being treated merely a sign, that might helps us distinguish two functions, say $f(x)$ and $g(x)$. But $f$ was not yet considered as a mathematical object on its own. &lt;em&gt;On the other hand&lt;/em&gt; in Peano's &lt;a href=&quot;https://archive.org/details/arithmeticespri00peangoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Arithmetices Principia&lt;/em&gt;&lt;/a&gt; (1889, p. XIII) we find:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let $\phi$ be a sign, or aggregate of signs, such that if $x$ is an&#xA;  entity of class $s$, the sign combination $\phi x$ determines a new&#xA;  entity; we also suppose that equality is defined between the entities&#xA;  $\phi x$; and if $x$ and $y$ are entities of the class $s$, and $x=y$,&#xA;  we suppose that it can be deduced that $\phi x = \phi y$. Then the&#xA;  sign $\phi$ is said to be a &lt;em&gt;prefix sign for a function in the&#xA;  class $s$&lt;/em&gt;, and we write $\phi \, \backepsilon \, F`s$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Translation adapted from Johan Granström's &lt;a href=&quot;https://intuitionistic.files.wordpress.com/2010/07/theses_published_uppsala.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PhD thesis&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My interpretation here is that Peano &lt;em&gt;does&lt;/em&gt; treat $f$ as a mathematical object on its own, distinct from $f(x)$, just like his contemporaries, Frege, Dedekind and Cantor, &lt;em&gt;yet&lt;/em&gt; he calls $f$ a &lt;em&gt;sign for a function&lt;/em&gt;, just like Bernoulli, Euler and Lagrange.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(So I'm left with the feeling, that I first need to define more precisely what it means to &lt;em&gt;treat something as a mathematical object&lt;/em&gt;, in order for my interpretations or even my question to make sense. But currently I'm not able to provide such a definition.)&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2018-08-30T08:01:23.813" LastActivityDate="2018-08-30T08:01:23.813" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="6124" PostTypeId="1" CreationDate="2017-06-05T14:08:38.127" Score="7" ViewCount="3259" Body="&lt;p&gt;What is the origin of the symbol for a battery cell, consisting of a long line and a shorter line? Was it derived from voltaic cells composed of unequal sized disks?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ibApK.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ibApK.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Example of symbol resembling Voltaic pile with differently sized electrolyte disks&lt;/em&gt;. (&lt;a href=&quot;https://books.google.nl/books?id=TncEAAAAQAAJ&amp;amp;pg=PA227&amp;amp;dq=voltaic-pile&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjo34vu2KbUAhVrCcAKHdDhBEM4ChDoAQgjMAA#v=snippet&amp;amp;q=%22contruction%20of%20the%20voltaic%20pile%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt;,&lt;a href=&quot;https://books.google.nl/books?id=wQFEAAAAYAAJ&amp;amp;pg=PA493&amp;amp;dq=electrical-engineer%20diagram&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi0052ss6bUAhXmLsAKHZjnAzkQ6AEIKDAA#v=snippet&amp;amp;q=%22obtained%20the%20proportions%20in%20rheostat%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;2&lt;/a&gt;,&lt;a href=&quot;https://books.google.nl/books?id=dyVbAAAAcAAJ&amp;amp;pg=PA259&amp;amp;dq=battery%20diagram%20electricity&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwju7cqrvKbUAhVKLMAKHXCWDaAQ6AEIOTAD#v=onepage&amp;amp;q=battery&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;3&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2017-06-05T15:56:46.470" LastActivityDate="2017-06-08T03:03:01.197" Title="Origin of battery symbol?" Tags="|electricity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6125" PostTypeId="1" AcceptedAnswerId="6126" CreationDate="2017-06-05T21:00:49.397" Score="1" ViewCount="812" Body="&lt;p&gt;I am a little confused on who first split the atom.  Wikipedia says that J.J Thomspon did with his cathode ray tube experiments.  In what sense does this split the atom?  But then it also says Rutherford did using alpha particles and nitrogen. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="131" LastEditDate="2017-06-06T12:06:28.837" LastActivityDate="2017-06-06T12:06:28.837" Title="In what sense did J.J. Thompson split the atom?" Tags="|atomic-theory|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6126" PostTypeId="2" ParentId="6125" CreationDate="2017-06-05T23:04:31.973" Score="4" Body="&lt;p&gt;The cathode ray experiments separated an electron from an atom. This is usually not considered splitting the atom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery of alpha radiation showed that the atom could be broken up, and that nature did it as a matter of course. To 'control' the energy of the alpha particle one used different source elements. The desire to vary and control the energy of alphas, as well as to use other particles such as protons, led to the technology for high voltage accelerators. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, the $^{14}N (\alpha,p) ^{17}O$ reaction was first observed in Rutherford's lab in March of 1919, as noted in notebooks with various observers including Marsden. Final confirmation awaited a paper by PMS Blackett, also from Rutherford's lab, in 1925, where he looked at 400,000 cloud chamber pictures to pull out the 8 cases. His results are in the Proceedings of the Royal Society, Volum 107, issue 742, pages 349-360 (Feb. 1925).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Meanwhile, by 1932 Cockroft and Walton had developed their (controllable) high voltage ion accelerator. One of the first papers from it was the $^{7}Li (p,\alpha) \alpha$ reaction, where an incident proton makes a $^{8}Be$ nucleus which splits in to two $\alpha$ particles. This was published in the Proceedings of the Royal Society, Vol 137, No. 831 229-242 (July 1, 1932). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chadwick's paper on the discovery of the neutron was published in the June 1st 1932 issue of the Proceedings. By 1935 Chadwick was publishing things like 'Disintegration by Slow Neutrons', Mathematical Proceedings of the Cambridge Philosophical Society 31(4) 612-616 (1935). There he discusses the $^{6}Li (n,T) \alpha$, $^{10}B(n,\alpha)^{7}Li$, and $^{14}N (n,\alpha) ^{11}B$ reactions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Otto Hahn's 1944 Chemistry Nobel Prize (which &lt;em&gt;did not include&lt;/em&gt; Lise Meitner) was 'for his discovery of the fission of heavy nuclei'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: J.J. Thompson had nothing to do with splitting the atom. Similarly, atoms were known to split well before Hahn and Meitner.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2017-06-05T23:32:43.993" LastActivityDate="2017-06-05T23:32:43.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6127" PostTypeId="2" ParentId="3648" CreationDate="2017-06-06T14:25:55.927" Score="0" Body="&lt;p&gt;1) We know since Einstein that mass contains or &lt;em&gt;is&lt;/em&gt; energy. So the energy of a field has mass. If the field moves then we get a moving mass, $mv$ which is defined as momentum. This is not a choice that has been made but a fact that has been explored. It is of practical importance for instance in case of satellites the orbits of which are measurably changed by light-pressure, i.e., the momentum of photons (in addition to the &quot;solar wind&quot; consisting of particles with rest mass).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) I am always happy to see that good students don't easily swallow the equation $0.999... = 1$. In fact it is not only counterintuitive but strictly speaking wrong and should not be accepted without the explanation (usually only supplied by good teachers) that according to common agreement by $0.999...$ the &lt;em&gt;limit&lt;/em&gt; 1 of the sequence $0.9 + 0.09 + 0.009 + ...$ has to be understood like we write $3.1415... = \pi$, knowing that $\pi$ cannot be represented by any string of digits and that $3.1415... \rightarrow \pi$ would be correct.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-06T17:57:10.207" LastActivityDate="2017-06-06T17:57:10.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6129" PostTypeId="2" ParentId="3700" CreationDate="2017-06-07T18:25:49.603" Score="0" Body="&lt;p&gt;The only &quot;goofy idea&quot; that I could find out is that he was ahead of his time. Here sis the quote from his &lt;a href=&quot;https://nitum.wordpress.com/2012/09/29/biography-of-harry-nyquist/&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Good old Harry Nyquist also recommended that the number of samples per second for a good representation of the signal has to be twice as big as the number of Hertz of the fastest sine wave contained in the analog signal. Since the telephone only allows 4 kHz through the phone line, sampling for voice is done 8000 times per second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Signal Sampling Theory was an exercise in frustration for Nyquist, since it needed 30,000 samples a second to make it work, and no system at that time could measure, record, store and reread that much information that quickly. He had to wait for computers, binary language, transistors and integrated circuits – 60 years of technological progress – to make digital recording and playback a reality.&lt;/p&gt;&#xA;" OwnerDisplayName="user5707" LastActivityDate="2017-06-07T18:25:49.603" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6130" PostTypeId="2" ParentId="6124" CreationDate="2017-06-08T03:03:01.197" Score="8" Body="&lt;p&gt;Yes, it was derived from the looks of voltaic batteries. More generally, the early symbols for circuit diagrams (cell, resistor, capacitor, etc.) were  based on the physical appearance of the devices they represented. This lasted until the 1950-s, when the advent of the transistor changed the scene, see &lt;a href=&quot;http://www.yorku.ca/imhotep/Publications/EJI-HSNS3803_03.pdf&quot; rel=&quot;noreferrer&quot;&gt;Icons and Electronics by Jones-Imhotep&lt;/a&gt; for the fascinating story of how it happened. I would also like to thank the author, who was kind enough to help me locate the primary sources cited below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.gutenberg.org/ebooks/15617?msg=welcome_stranger#_jmp0_&quot; rel=&quot;noreferrer&quot;&gt;Cyclopedia of Telephony and Telegraphy, Vol. 1 (1919)&lt;/a&gt;, available on Gutenberg, describes the interpretation of the battery symbol, even less removed in the visual appearance from the batteries, shown in the OP, than the modern schematization:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The conventional symbol for a cell, either of the primary or the secondary type, consists of a long thin line and a short heavy line side by side and parallel. A battery is represented by a number of pairs of such lines, as in Fig. 67. The two lines of each pair are supposed to represent the two electrodes of a cell. Where any significance is to be placed on the polarity of the cell or battery the long thin line is supposed to represent the positively charged plate and the short thick line the negatively charged plate. The number of pairs may indicate the number of cells in the battery. Frequently, however, a few pairs of such lines are employed merely for the purpose of indicating a battery without regard to its polarity or its number of cells. In Fig. 67 the representation at A is that of a battery of a number of cells connected in parallel; that at B of a battery with the cells connected in series; and that at C of a battery with one of its poles grounded.&lt;/em&gt;&quot;&#xA;  &lt;a href=&quot;https://i.stack.imgur.com/Ras4r.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ras4r.gif&quot; alt=&quot;Battery symbol&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An even earlier &lt;a href=&quot;http://www.gutenberg.org/files/27574/27574-h/27574-h.htm&quot; rel=&quot;noreferrer&quot;&gt;Le Téléphone, le Microphone et le Phonographe (1878)&lt;/a&gt; displays circuit diagrams for a phonograph with the familiar battery symbol explicitly referred to as &quot;&lt;em&gt;batterie voltaïque&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZQhIC.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZQhIC.jpg&quot; alt=&quot;Phonograph circuit diagram&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2017-06-08T03:03:01.197" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6131" PostTypeId="2" ParentId="3433" CreationDate="2017-06-08T13:50:20.293" Score="2" Body="&lt;p&gt;Here you can see how Leandro Panizzon and his wife Marguerite looked in their old age.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MHExA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MHExA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After, as was usual at that time, testing his invention himself with no apparent result, he let his wife have a try (who suffered from low blood pressure). Methylphenidate apparently improved her tennis capabilities. Today it is used not only as medicine, in cases of ADHS (&lt;a href=&quot;https://nebadonien.wordpress.com/2013/05/11/beichte-auf-dem-sterbebett-ich-habe-gelogen-und-betrogen/&quot; rel=&quot;nofollow noreferrer&quot;&gt;which seems to be a fabricated illness&lt;/a&gt;), but also as a drug (called Rita).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1950 Panizzon together with Max Hartmann obtained a US patent (number 2507631) for an improved manufacturing method of methylphenidate. In 1954 methylphenidate came on the market. In 2014 Novartis had a turnover of 492 million US-Dollar with Ritalin. In 2013 338 kg Ritalin were sold compared to 23 kg in 1998.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Panizzon got vice-director of his Swiss company Ciba (today Novartis)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He died in 2003.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The couple had four children. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources: &lt;a href=&quot;https://adhspedia.de/wiki/Leandro_Panizzon&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://adhspedia.de/wiki/Leandro_Panizzon&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.unternehmerzeitung.ch/ausserdem/schweizer-pioniere/panizzons-wunderpille/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.unternehmerzeitung.ch/ausserdem/schweizer-pioniere/panizzons-wunderpille/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-08T13:50:20.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6133" PostTypeId="2" ParentId="3448" CreationDate="2017-06-08T14:10:46.843" Score="1" Body="&lt;p&gt;Originally zero was not considered a natural number. In fact it is one of the most unnatural numbers. Since the word  &quot;progression naturelle&quot; was introduced by Chuquet, in 1484 and &quot;natural number&quot; by Emmerson in 1763 always the sequence starting with 1 has been addressed. &quot;Natural number, defined as the numbers 1, 2, 3, 4, 5, etc., appears in the 1771 Encyclopaedia Britannica in the Logarithm article.&quot; &lt;a href=&quot;http://jeff560.tripod.com/n.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jeff560.tripod.com/n.html&lt;/a&gt; Cantor for instance (although rarely using this wording but prefering &quot;integers&quot; (ganze Zahlen) mentiones the &quot;natural sequence of numbers&quot; (natürliche Zahlenreihe): $1, 2, 3, ...,\nu , ...$&quot; in &lt;em&gt;Über unendliche lineare Punktmannigfaltigkeiten&lt;/em&gt; Math. Annalen 15, (1879). Also Peano started with 1 in his first papers. Later he switched to 0 like many others. (See the answer by Mauro Allegranza)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where did these conventions come from? What was the reason for this change? Cantor explained it in a letter to Dedekind of Juli 28,1899:&#xA;The sequence of ordinal numbers $0, 1, 2, 3, ... ,\omega_0, \omega_0 + 1, ..., \gamma,..  $ has the property that every number is the &lt;em&gt;ordinal type&lt;/em&gt; of the preceding numbers. Without  including zero this property would not be satisfied for the finite part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that this is the reason.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-08T14:10:46.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6134" PostTypeId="2" ParentId="3388" CreationDate="2017-06-08T15:28:06.900" Score="3" Body="&lt;p&gt;The first preserved list of primes is row (b) of the famous Ishango bone: 11, 13, 17, 19. See my answer to the question &lt;a href=&quot;https://hsm.stackexchange.com/questions/419/when-were-numbers-first-used-for-anything-other-than-counting/6031#6031&quot;&gt;When were numbers first used for anything other than counting?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-08T15:28:06.900" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6135" PostTypeId="2" ParentId="210" CreationDate="2017-06-08T20:31:53.037" Score="2" Body="&lt;p&gt;Originally (in Greece) the word mathematician was used in the same sense as today.&#xA;Astrology penetrated Greece from Babylon during Alexander's conquests, and gradually became very popular. The use of the word &quot;mathematicus&quot; in the sense &quot;astrologer&quot; spread in the Roman empire, among the people who had no slightest idea about what mathematics is. In the late antiquity, when mathematics (in the proper sense of the word) completely disappeared, the only meaning of the word &quot;mathematicus&quot; was &quot;astrologer&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-06-08T20:31:53.037" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6138" PostTypeId="2" ParentId="3323" CreationDate="2017-06-09T14:06:48.833" Score="-2" Body="&lt;p&gt;$\frac{d^2}{dx^2}$ $y$ = $\frac{d^2y}{dx^2}$ is too obviously built from $\frac{d}{dx}$$\frac{d}{dx}$ $y$ = $(\frac{d}{dx})^2$ $y$ to deserve any further explanation.&lt;/p&gt;&#xA;" OwnerDisplayName="user5693" LastActivityDate="2017-06-09T14:06:48.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6139" PostTypeId="1" CreationDate="2017-06-09T16:22:18.770" Score="1" ViewCount="618" Body="&lt;p&gt;One thing that bugs me when learning and explaining special relativity concepts is that the time axis on space-time charts always on the vertical axis. While almost everywhere else time is going on the horizontal axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover space-time coordinates often written in the order of $(t, x, y, z)$. And in case of 1 dimensions $(t, x)$. Now when we speak of the event $(A, B)$, $A$ goes on the vertical axis $B$ on the horizontal and not vice versa as it usually goes on the coordinate charts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I always need to twist my mind a bit when working with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did we started putting time on the vertical axis and what's the reason behind it?&lt;/p&gt;&#xA;" OwnerUserId="2178" LastEditorUserId="2178" LastEditDate="2017-06-09T19:11:12.803" LastActivityDate="2017-06-17T12:49:14.237" Title="Why is time represented on the vertical axis in space-time diagrams?" Tags="|relativity-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6140" PostTypeId="1" CreationDate="2017-06-10T01:34:36.993" Score="1" ViewCount="38" Body="&lt;p&gt;It is mentioned in many textbooks on statistical mechanics. But generally there is no reference. &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2017-06-10T01:34:36.993" Title="Who first discovered that the specific heat of gas decreases with temperature?" Tags="|physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6142" PostTypeId="1" CreationDate="2017-06-10T15:28:04.067" Score="3" ViewCount="211" Body="&lt;p&gt;Are the great mirrors and lenses of historic telescopes, especially those of Herschel, Rosse and many others back from the 18th and 17th centuries, preserved? The wooden structures of those telescopes, I assume, rotted and became firewood, but did they preserve the mirrors and lenses?&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorUserId="10106" LastEditDate="2023-01-12T02:51:43.337" LastActivityDate="2023-01-17T17:32:53.843" Title="Where are the mirrors of historic telescopes today?" Tags="|physics|astronomy|telescope|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6143" PostTypeId="1" CreationDate="2017-06-10T20:37:44.043" Score="0" ViewCount="107" Body="&lt;p&gt;The chemical composition of water was discovered &lt;a href=&quot;https://hsm.stackexchange.com/a/2543&quot;&gt;about 1784&lt;/a&gt;, but when has it been discovered that it is also composed of acid and base? &lt;/p&gt;&#xA;" OwnerDisplayName="user5903" LastActivityDate="2017-06-10T20:37:44.043" Title="How has it been discovered that water consists of acid and base?" Tags="|discoveries|chemistry|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="6144" PostTypeId="1" CreationDate="2017-06-11T01:27:34.773" Score="2" ViewCount="282" Body="&lt;p&gt;In &lt;a href=&quot;https://physics.stackexchange.com/a/336500/25794&quot;&gt;this answer&lt;/a&gt; to a question of mine on the stackexchange physics site, I learned about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tautochrone_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;tautochrone problem&lt;/a&gt;. Apparently it was solved by Huygens in 1659, which is before Newton's work on mechanics. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What did he base his assumptions on? Were​ there phenomenological laws, analogous to Kepler's laws for planetary motion?&lt;/li&gt;&#xA;&lt;li&gt;What were his assumptions and how did he solve the problem?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Answers just addressing one of these points are very welcome.&lt;/p&gt;&#xA;" OwnerUserId="305" LastEditorDisplayName="user466" LastEditDate="2017-06-11T03:13:49.763" LastActivityDate="2017-07-11T11:09:05.120" Title="How could Huygens have solved the tautochrone problem before Newton's theory of gravity and equations of motion?" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6146" PostTypeId="2" ParentId="6142" CreationDate="2017-06-11T06:25:37.100" Score="8" Body="&lt;p&gt;Many of these objects are collected in museums. The German &amp;quot;Deutsches Museum&amp;quot; for instance keeps famous original instruments of Fraunhofer &lt;a href=&quot;http://www.deutsches-museum.de/sammlungen/meisterwerke/meisterwerke-ii/refraktor/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.deutsches-museum.de/sammlungen/meisterwerke/meisterwerke-ii/refraktor/&lt;/a&gt; Such institutions are scattered all over the globe.&lt;/p&gt;&#xA;&lt;p&gt;Further there are many museums devoted to individual astronomers like &lt;a href=&quot;https://en.wikipedia.org/wiki/Herschel_Museum_of_Astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Herschel_Museum_of_Astronomy&lt;/a&gt;&#xA;keeping original relics and replica.&lt;/p&gt;&#xA;&lt;p&gt;Most relics will have been preserved at their original places. The Göttingen Sternwarte keeps original instruments used by Gauss. A list of such institutions would get fairly long.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="6514" LastEditDate="2023-01-17T17:32:53.843" LastActivityDate="2023-01-17T17:32:53.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6147" PostTypeId="1" CreationDate="2017-06-11T09:51:17.587" Score="1" ViewCount="77" Body="&lt;p&gt;Did people try that? Or did they consider energy quantization for the translation motion of a free particle? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2017-06-11T10:12:53.090" Title="Did people consider energy quantization of nonlinear oscillators after Planck?" Tags="|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6148" PostTypeId="2" ParentId="6147" CreationDate="2017-06-11T10:04:54.223" Score="3" Body="&lt;p&gt;The Schrodinger equation is linear, so perhaps &quot;quantization of non-linear oscillators&quot; refers to non-linearity of the classical system that one quantizes. If this is what you mean, the answer is yes. In the very first paper on modern quantum mechanics, Heisenberg's paper of 1925, non-linear oscillators with potentials $x^2+cx^3$ and $x^2+cx^4$&#xA;play a crucial role. They are called anharmonic oscillators. They were also studied before and after Heisenberg in context of quantization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References: B. L. van der Waerden, Sources in quantum mechanics, Dover, 1968.&#xA;This contains English translations and commentaries of the most important papers on quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. Most people who studied this paper of Heisenberg agree that his logic is very obscure. After stating the rules of the matrix mechanics, he applies them&#xA;to quantization of cubic and quartic potentials, and somehow concludes from the result that his rules must be correct. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-06-11T10:12:53.090" LastActivityDate="2017-06-11T10:12:53.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6149" PostTypeId="2" ParentId="6144" CreationDate="2017-06-11T10:18:59.627" Score="2" Body="&lt;p&gt;Everything he had to know about gravity and motion to state and solve the Tautochrone problem was well-known since Galileo at least. This problem assumes the uniform gravity (everything happens near the Earth surface), so Newton's law of gravity has nothing to do with it. Newton's law is only relevant for motions at great distances from Earth, like planet and satellite motions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-06-11T10:18:59.627" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6150" PostTypeId="1" CreationDate="2017-06-11T21:46:57.123" Score="5" ViewCount="254" Body="&lt;p&gt;I’m a fan of C. F. Gauss but I have to be objective — I don’t understand why the classical theory of lenses is called “Gaussian optics”. I think so since it seems that almost all the results of Gauss’s treatise from 1840 were already known — the lens-maker formula was known already from the 17th century, C. Huygens clarified many aspects of geometrical optics, and L. Euler published an extensive study of design of optical instruments in his &lt;a href=&quot;//www.archive.org/details/leonhardieulerid027742mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Dioptrica&lt;/em&gt;&lt;/a&gt;. So what is special and groundbreaking about Gauss’s contributions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On this occasion, I’ll be glad if anyone can give a concise summary of Gauss’s contributions to optics — his published as well as unpublished manuscripts.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="658" LastEditDate="2018-04-14T18:57:28.010" LastActivityDate="2018-04-14T18:57:28.010" Title="What results did C. F. Gauss add to Euler’s dioptrics?" Tags="|biographical-details|optics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6151" PostTypeId="1" AcceptedAnswerId="6153" CreationDate="2017-06-11T22:21:50.983" Score="2" ViewCount="139" Body="&lt;p&gt;Has any of you ever read the paper in which Georg Pick  made public &lt;a href=&quot;https://en.wikipedia.org/wiki/Pick%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;his famous formula&lt;/a&gt;? If so, would you be so kind as to tell me what it is that one can find in the introduction of it?&lt;/p&gt;&#xA;&lt;p&gt;The complete bibliographical information of the paper in question is:&lt;/p&gt;&#xA;&lt;p&gt;G. Pick, &lt;em&gt;Geometrisches zur Zahlenlehre&lt;/em&gt;. Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen „Lotos“ in Prag. 47. Band (1899), S. 315–323.&lt;/p&gt;&#xA;&lt;p&gt;In case you wish to share with me an electronic copy of it, you can find my email account in my profile...&lt;/p&gt;&#xA;&lt;p&gt;Let me thank you in advance for your attention and support.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-06-12T03:10:58.520" Title="On a paper by Georg Pick" Tags="|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6152" PostTypeId="1" AcceptedAnswerId="11512" CreationDate="2017-06-11T22:28:20.380" Score="2" ViewCount="575" Body="&lt;p&gt;From What I understand the Wright brother never had university training and although never getting a high school diploma they did attend high school. Looking at the diagrams of their original work they needed to work out pressure differentials on the surface of the propeller using mathematical formula. Their work appears to have quadratic equations and cross products in the calculations complete with torque and power.  Does anyone know where they may have learned these skills ?  I don't believe any high school of that day in that area of the country would have had the ready math and physics needed to these tasks. Or was it the case that they were self taught in all the needed math,experimental, managerial, drafting and tooling skills needed for such an accomplishment? The diagrams are more along the line of advanced undergraduate but way beyond high school or at least those of today. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2020-03-01T16:58:55.110" Title="How or where did the Wright brothers develop the expertise to formulate quadratic equations to design their first propeller?" Tags="|mathematics|aerodynamics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6153" PostTypeId="2" ParentId="6151" CreationDate="2017-06-12T03:10:58.520" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://biodiversitylibrary.org/page/14334051&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://biodiversitylibrary.org/page/14334051&lt;/a&gt; $\ \ \ \ \ \ \ \ \ \ \ \ \ \ $&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-06-12T03:10:58.520" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6155" PostTypeId="2" ParentId="6150" CreationDate="2017-06-12T08:54:56.637" Score="7" Body="&lt;p&gt;I’d say that F. Jentzsch (&lt;a href=&quot;http://dx.doi.org/10.1007/BF01559118&quot; rel=&quot;nofollow noreferrer&quot;&gt;1925&lt;/a&gt;, p. 863) gives just the concise summary you want (emphasis added):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Secondly, one had to understand and sufficiently clarify certain general properties of optical systems, such as the concepts of &lt;a href=&quot;https://en.wikipedia.org/wiki/Focal_length&quot; rel=&quot;nofollow noreferrer&quot;&gt;focal distance&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Cardinal_point_(optics)#Focal_planes&quot; rel=&quot;nofollow noreferrer&quot;&gt;focal points&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Cardinal_point_(optics)#Principal_planes_and_points&quot; rel=&quot;nofollow noreferrer&quot;&gt;principal points&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Magnification#Magnification_as_a_number_.28optical_magnification.29&quot; rel=&quot;nofollow noreferrer&quot;&gt;magnification&lt;/a&gt;, etc. This was provided by &lt;strong&gt;&lt;em&gt;Gauss&lt;/em&gt;&lt;/strong&gt;. He was the first to bring his forerunners’ works to a certain closure, mainly by also treating systems with &lt;strong&gt;&lt;em&gt;arbitrary thickness&lt;/em&gt;&lt;/strong&gt;, and by showing that any (rotationally symmetric) system could be &lt;strong&gt;&lt;em&gt;replaced in thought&lt;/em&gt;&lt;/strong&gt; by a single refracting surface, or by a thin lens and a translation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(“Forerunners” acknowledged in Gauss (&lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN385030444_1841_1|LOG_0005&amp;amp;physid=PHYS_0083&quot; rel=&quot;nofollow noreferrer&quot;&gt;1841&lt;/a&gt;, &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002017954&quot; rel=&quot;nofollow noreferrer&quot;&gt;1843&lt;/a&gt;) are &lt;a href=&quot;https://archive.org/stream/treatiseongeomet00hermuoft#page/77&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cotes&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=sBEOAAAAQAAJ&amp;amp;pg=PP7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smith&lt;/a&gt;, &lt;a href=&quot;http://www.archive.org/details/leonhardieulerid027742mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler&lt;/a&gt;, &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1778&amp;amp;seite:int=260&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=jqIAAAAAMAAJ&amp;amp;pg=RA2-PA13&quot; rel=&quot;nofollow noreferrer&quot;&gt;Piola&lt;/a&gt;, and &lt;a href=&quot;http://www.archive.org/stream/gesammeltewerkeh04mbuoft#page/477&quot; rel=&quot;nofollow noreferrer&quot;&gt;Möbius&lt;/a&gt;.) A more detailed account, by C. Schaefer in Gauss’ &lt;em&gt;Werke&lt;/em&gt; (Vol. XI.2, Abh. 2, §III, &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN236059505|LOG_0032&quot; rel=&quot;nofollow noreferrer&quot;&gt;1929&lt;/a&gt;), mainly agrees and adds (pp. 189, 201; paraphrased translation from Dunnington, &lt;a href=&quot;https://books.google.com/books?id=4mwSrfxBSzkC&amp;amp;pg=PA171&quot; rel=&quot;nofollow noreferrer&quot;&gt;1955&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN236020595|LOG_0132&amp;amp;physid=PHYS_0167&quot; rel=&quot;nofollow noreferrer&quot;&gt;his own claim&lt;/a&gt; he had possessed the results for forty or forty-five years, but had always hesitated to publish such elementary meditations. A &lt;a href=&quot;http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1840AN.....17..289B&quot; rel=&quot;nofollow noreferrer&quot;&gt;work of Bessel&lt;/a&gt; on the determination of the focal distance of the Königsberg heliometer gave him the impetus necessary for publication. Bessel’s method assumed mistakenly that the usual lens formula&#xA;  $$&#xA;\frac1g+\frac1b=\frac1f&#xA;$$&#xA;  is correct for lenses of finite thickness. As a consequence of this mistake, Bessel greatly underestimated the possible error of his measurement. &lt;em&gt;Dioptrische Untersuchungen&lt;/em&gt; [gives] formulas for a simple lens of nonvanishing thickness [$\ldots$] While Bessel estimated the error of his result at 1/75,000, Gauss &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002017954&amp;amp;physid=PHYS_0576&quot; rel=&quot;nofollow noreferrer&quot;&gt;showed&lt;/a&gt; that it amounted to 1/1,300.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-12-22T16:02:53.710" LastActivityDate="2017-12-22T16:02:53.710" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6156" PostTypeId="1" CreationDate="2017-06-12T09:59:38.727" Score="5" ViewCount="333" Body="&lt;p&gt;In a paper about Plato's &lt;em&gt;Timaeus&lt;/em&gt; I came across a cryptic note saying: &quot;Academic research usually avoids noting that Plato's assertion about water consisting of two parts air and one part fire is strikingly similar to the modern chemical formula (H&lt;sub&gt;2&lt;/sub&gt;O).&quot; Most obviously this refers to the text found at the end of 56d:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;water.. is capable of becoming a compound of one corpuscle of fire&#xA;  with two of air&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;Timaeus&lt;/em&gt; famously linked the the four elements with regular solids, water being the icosahedron (20 faces), air - the octahedron (8), fire - the tetrahedron (4), so the count is good: $20= 2\times8 +4$. The modern formula for water H&lt;sub&gt;2&lt;/sub&gt;O has been established early in the 19th century. Who was the first to  note  this coincidence and what later comments are there?&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="2277" LastEditDate="2018-05-12T04:33:13.340" LastActivityDate="2018-05-12T04:33:13.340" Title="Plato's chemical formula for water" Tags="|ancient-greece|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="6157" PostTypeId="1" AcceptedAnswerId="6212" CreationDate="2017-06-12T12:15:18.880" Score="9" ViewCount="1717" Body="&lt;p&gt;V. I. Arnold asserted that mathematics is a natural science:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.uni-muenster.de/Physik.TP/%7Emunsteg/arnold.html&quot; rel=&quot;noreferrer&quot;&gt;V.I. Arnold: &amp;quot;On teaching mathematics&amp;quot; (1997)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;This opinion is rare. Many mathematicians will oppose, in particular pure mathematicians. But is it solitary?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there famous mathematicians who have shared in written form Arnold's view that mathematics is a natural science?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2021-07-24T09:01:01.657" Title="Are there well-known mathematicians who shared Arnold's view about mathematics as natural science?" Tags="|mathematicians|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="6158" PostTypeId="2" ParentId="5485" CreationDate="2017-06-12T12:19:16.933" Score="1" Body="&lt;p&gt;I agree with the conclusion that the planetary equations are not in &lt;em&gt;Theoria motus&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/motionofheavenly00gausrich#page/n8&quot; rel=&quot;nofollow noreferrer&quot;&gt;1809&lt;/a&gt;). I believe Gauss first wrote them down in &lt;a href=&quot;https://archive.org/stream/werkehrsgvonderg07gausuoft#page/457&quot; rel=&quot;nofollow noreferrer&quot;&gt;§14&lt;/a&gt; of the essay &lt;em&gt;Exposition d'une nouvelle méthode de calculer les perturbations planétaires avec l'application au calcul numérique des perturbations du mouvement de Pallas&lt;/em&gt;, drafted (c.1812) for a Paris prize competition but only published posthumously (&lt;a href=&quot;https://archive.org/stream/werkehrsgvonderg07gausuoft#page/439&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-06-12T12:19:16.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6159" PostTypeId="1" CreationDate="2017-06-12T12:40:49.927" Score="14" ViewCount="630" Body="&lt;p&gt;I have frequently come across Kronecker's statement about set theory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is quoted in Wikipedia. Google shows over 400 hits. But I have never seen a hint to the original source.&lt;/p&gt;&#xA;&lt;p&gt;It need not be explained that Kronecker had a very negative attitude toward set theory. Did Kronecker say so? And if yes, where and when?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="4251" LastEditDate="2022-04-27T05:32:36.213" LastActivityDate="2022-04-27T05:32:36.213" Title="Did Kronecker say that set theory is not mathematics?" Tags="|mathematics|reference-request|set-theory|quote|kronecker|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6160" PostTypeId="2" ParentId="3131" CreationDate="2017-06-12T13:41:24.670" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://de.wikipedia.org/wiki/Ludwig_Scheeffer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ludwig Scheeffer&lt;/a&gt; (1859-1885) was a gifted mathematician. At 25 he became a Privatdozent in Munich. He died at only 26. In 1884, only 10 years after the invention of set theory and way before its breaktrough in 1891 he wrote in an Acta Mathematica paper about sets of first cardinality. Georg Cantor wrote an obituary which is reprinted in his &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN237853094&amp;amp;DMDID=DMDLOG_0002&quot; rel=&quot;nofollow noreferrer&quot;&gt;collected papers&lt;/a&gt; pp. 368f.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="17963" LastEditDate="2024-02-23T16:26:38.090" LastActivityDate="2024-02-23T16:26:38.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6161" PostTypeId="2" ParentId="5347" CreationDate="2017-06-12T14:38:02.720" Score="6" Body="&lt;p&gt;Concerning alternative notations for $y|_{x=0}$: Lagrange in &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k86263h&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Théorie des fonctions analytiques&lt;/em&gt;&lt;/a&gt;, 1797,&#xA; &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k86263h/f68&quot; rel=&quot;noreferrer&quot;&gt;p.57&lt;/a&gt; writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(...) et si on désigne par $(y), (y'), (y''),$ etc. les valeurs de&#xA;  $y,y',y'',$ etc. lorsque $x=0$, on a en général $$ y = (y) + x\,(y') + \frac{x^2}{2}(y'') +\ldots $$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is developed further to allow other values than zero in 1825 in Martin Ohm's &lt;a href=&quot;https://books.google.ch/books?id=hcJZAAAAcAAJ&amp;amp;pg=PA25#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Die Lehre vom Grössten und Kleinsten&lt;/em&gt;&lt;/a&gt;, p.5:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $V$ is a function of $x$, and $a$ a value of $x$, then we denote with $V_a$ or $(V)_a$ that which becomes of $V$ when we write $a$ everywhere instead of $x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(my translation)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This notation allows for ambiguities if there are other variables involved one might want so substitute for. (Moreover it is very close to the abuse of notation $V(a)$.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest use of the &quot;modern&quot; notation, but without the bar, I've so far found in Todhunter's &lt;a href=&quot;https://archive.org/details/histroyofthecalc033379mbp&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A History of the Calculus of Variations&lt;/em&gt;&lt;/a&gt; 1861, p.23 ff.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(...) thus the first line in for example (2) will really when written at full become&#xA;  \begin{align}&#xA;&amp;amp;\left\{ I\left( P' -\frac{dQ'}{dx} + \frac{d^2R'}{dx^2} - \ldots \right)\omega  \right\}_{x=\beta} \\&#xA;-&amp;amp; \left\{ I\left( P' -\frac{dQ'}{dx} + \frac{d^2R'}{dx^2} - \ldots \right)\omega  \right\}_{x=\alpha}&#xA;\end{align}&#xA;  and as the value of $I$ when $x=\beta$ will not generally be the same as when $x=\alpha$, we cannot as Lacroix does put $A$ for $I$ in the terms include in (2).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although the notation is quite self explanatory, Todhunter does not define it explicitly. So I suspect it was already being used at that time by someone else. But I haven't yet checked all the sources cited by Todhunter.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2017-07-29T11:19:50.077" LastActivityDate="2017-07-29T11:19:50.077" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6162" PostTypeId="1" CreationDate="2017-06-12T12:04:19.967" Score="1" ViewCount="256" Body="&lt;p&gt;Why is so much emphasis put on the concepts of absolute space and time as the grand foundations of Newtonian mechanics, and not as foundations also of previous theories, as for example Kepler's laws of planetary motion, or even the ancient Ptolemaic system?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is not the concept of universal force Newton's main contribution, and not the concepts of absolute space and time?&lt;/p&gt;&#xA;" OwnerUserId="5909" OwnerDisplayName="J.Avaris" LastEditorUserId="62" LastEditDate="2017-06-12T17:29:51.053" LastActivityDate="2017-06-20T10:16:56.373" Title="Absolute space and time before Newton" Tags="|relativity-theory|newton|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6163" PostTypeId="2" ParentId="6162" CreationDate="2017-06-12T12:35:40.600" Score="1" Body="&lt;p&gt;No, in his &lt;strong&gt;Philosophia Naturalis Principia Mathematica&lt;/strong&gt;, Newton stated - &quot;&lt;em&gt;Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time&lt;/em&gt;&quot; According to him absolute space and time existed independently regardless of the state of motion of observer, the Law of Gravitation was not the only work he published in the Principia Mathematica, this concept of absolute space and time was later disproved by Albert Einstein in his Theory of Relativity. &lt;/p&gt;&#xA;" OwnerDisplayName="Ajinkya Naik" LastActivityDate="2017-06-12T12:35:40.600" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6164" PostTypeId="2" ParentId="3131" CreationDate="2017-06-12T21:33:19.003" Score="14" Body="&lt;p&gt;Raymond E. Paley (7 January 1907 – 7 April 1933) fulfills both conditions asked for in the question. He died young, with 26 years, just after having started his collaboration with Wiener (see Paley-Wiener integral), &lt;strong&gt;and&lt;/strong&gt; in a  tragic way. Namely, he was killed by an avalanche, while his death was witnessed by companions lower down the mountainside.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Raymond_Paley&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="217" LastEditDate="2020-05-16T18:44:49.823" LastActivityDate="2020-05-16T18:44:49.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6165" PostTypeId="1" CreationDate="2017-06-12T22:39:28.013" Score="1" ViewCount="135" Body="&lt;p&gt;More specifically, was Newton aware that given an inverse pair of functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; such that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(h(x)) = x = h(f(x))$$&lt;/span&gt; about the origin that, for&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x,y)=(h(y),f(x)),$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the derivatives satisfy&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f^{'}(x) = 1/h^{'}(y)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$dy/dx = 1/(dx/dy)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;near the origin?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heuristically, this follows symbolically from&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$dy = f^{'}(x)dx = f^{'}(x)h^{'}(y)dy. $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it follows geometrically for a function whose graph lies in the first quadrant by reflection through the bisector of the first quadrant, the line &lt;span class=&quot;math-container&quot;&gt;$y=x$&lt;/span&gt;. Clearly, the slope for any tangent line is inverted by the reflection just as displacements along the &lt;span class=&quot;math-container&quot;&gt;$x-$&lt;/span&gt;axis and the &lt;span class=&quot;math-container&quot;&gt;$y-$&lt;/span&gt;axis are interchanged.  d. In fact, it follows directly from the tangent line perspective since&#xA; &lt;span class=&quot;math-container&quot;&gt;$$ y = m \; x + b$$&lt;/span&gt; and &#xA; &lt;span class=&quot;math-container&quot;&gt;$$y = \frac{1}{m}(x-b)$$&lt;/span&gt; describe an inverse pair.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely, with Newton's mastery of geometric calculus, he was aware of these relationships. Is there evidence of this in Newton's work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related &lt;a href=&quot;https://mathoverflow.net/questions/196058/history-of-frac-ddt-tan-1t-frac-11t2#comment671951_196058&quot;&gt;MO-Q&lt;/a&gt; by Ziegler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cross-posted from this &lt;a href=&quot;https://mathoverflow.net/questions/271568/was-a-nascent-inverse-function-theorem-known-to-newton?noredirect=1&amp;amp;lq=1&quot;&gt;MO-Q&lt;/a&gt; and an identical question on MSE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit 6/12/17:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of a calculation incorporating the IFT that would have been obvious to Newton and plausible for him to have performed if only as a simple check of his general formulas:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was known well before Newton that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d\tan(x)}{dx} = 1+ \tan^2(x),$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;or, with &lt;span class=&quot;math-container&quot;&gt;$y = \tan(x)$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dy}{dx} = 1+ y^2.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of fluxions and fluents, this could be put in the form of &lt;a href=&quot;https://books.google.com/books?id=vLBJSmA9zgAC&amp;amp;pg=PA69#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's implicit function&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$g(x,y,\dot{x},\dot{y})=\dot{y}-(1+y^2)\dot{x}=0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\dot{x}}{\dot{y}}= \frac{1}{1+y^2}=\frac{dx}{dy}, $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and application of the binomial theorem and integration would give the series &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\arctan(y) = x = y - \frac{y^3}3+\frac{y^5}5-\frac{y^7}7+\dots.&#xA;\tag3&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton could then have derived a series expression for &lt;span class=&quot;math-container&quot;&gt;$\tan(x)$&lt;/span&gt; using his series reversion formula (see &lt;a href=&quot;https://books.google.com/books?id=vLBJSmA9zgAC&amp;amp;pg=PA69#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferraro&lt;/a&gt;) for finding the series for the compositional inverse of a function from its power series. In fact, the same procedure is applied to finding a series for &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; in Ferraro on pages 76-78 following an alleged reconstruction by Horsley of Newton's derivation of the series.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2018-11-04T18:07:47.697" LastActivityDate="2018-11-04T18:07:47.697" Title="Was Newton aware of a nascent inverse function theorem?" Tags="|mathematics|mathematicians|biographical-details|scientific-method|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6167" PostTypeId="2" ParentId="3131" CreationDate="2017-06-13T01:49:05.637" Score="23" Body="&lt;p&gt;Felix Hausdorff, his wife, and his wife's sister committed suicide in 1942 after being ordered to a Nazi death camp. He was 74 at the time but his death is certainly tragic. &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Hausdorff&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="340" LastEditDate="2017-07-16T19:07:55.640" LastActivityDate="2017-07-16T19:07:55.640" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6169" PostTypeId="1" CreationDate="2017-06-13T08:22:33.923" Score="2" ViewCount="163" Body="&lt;p&gt;How do you think it possible for the ancients to be satisfied with the concept of FINITE SPACE, with &quot;fixed stars&quot; at its limits? To our minds, the idea of infinite extension comes so naturally, without the slightest effort, even for the mathematically unsophisticated. So what is really hiding here? &lt;/p&gt;&#xA;" OwnerUserId="5909" LastActivityDate="2017-10-11T13:00:27.430" Title="The ancient concept of Finite Space" Tags="|infinity|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6171" PostTypeId="2" ParentId="6169" CreationDate="2017-06-13T11:35:15.890" Score="2" Body="&lt;p&gt;Who believed in a closed world? The fathers of science, the Old Greek, did not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anaxagoras 499-428 BC knew: Of the great there is no greatest but there is always a greater.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anaximander of Milet 611-547 BC already said: The infinite (apeiron) is not exhaustible. Wherever the warrior stays, he can stretch out his spear farther.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-13T11:35:15.890" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6172" PostTypeId="2" ParentId="3131" CreationDate="2017-06-13T16:46:04.460" Score="17" Body="&lt;p&gt;&lt;strong&gt;Jacques Herbrand&lt;/strong&gt; (1908-1931) died et 23 in an accident during a mountain excursion. He made fundamental works in logic.&lt;/p&gt;&#xA;" OwnerUserId="5911" LastActivityDate="2017-06-13T16:46:04.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6173" PostTypeId="2" ParentId="2993" CreationDate="2017-06-13T17:10:49.307" Score="1" Body="&lt;p&gt;Notes on the theory and application of Fourier transforms (with&#xA;R. E. A. C. Paley) Trans. Amer. Math. Soc. 35 (1933) 348-355 and 761-791.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-13T17:10:49.307" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6175" PostTypeId="2" ParentId="162" CreationDate="2017-06-14T02:07:08.180" Score="1" Body="&lt;p&gt;German physics and the Journal &amp;quot;Poggendorff's &lt;em&gt;Annalen&lt;/em&gt;&amp;quot; contributed to German becoming a scientific language in circa 1840s.&lt;/p&gt;&#xA;&lt;p&gt;From Jungnickel &amp;amp; McCormmach's &lt;a href=&quot;https://isidore.co/calibre/browse/book/5957&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Second Physicist: On the History of Theoretical Physics in Germany&lt;/em&gt;&lt;/a&gt; (2017), &amp;quot;Chapter 6: Physics Research in “Poggendorff’s &lt;em&gt;Annalen&lt;/em&gt;” in the 1840s,&amp;quot; &amp;quot;§6.1 Foreign Recognition of German Physics,&amp;quot; pp. 137-8:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In an important respect, Poggendorff’s &lt;i&gt;Annalen&lt;/i&gt; in the middle of the nineteenth century was a different journal than the one he began editing twenty some years before. New foreign physics in translations and reports still had a prominent place in it, but the German physics appearing there was no longer in its shadow. The &lt;i&gt;Annalen&lt;/i&gt; now regularly published work by German physicists that equaled, and occasionally surpassed, the best work by foreign physicists. At the same time, German work was gaining recognition abroad. This was a greater achievement than one might think, particularly in France. Humboldt had found in his many years abroad that the German language did not “flourish excessively in the great Babel,” and that at the Institut de France “almost everything is lost that is sent in in German without excerpt and explanation.”&lt;sup&gt;1&lt;/sup&gt; Even a paper by Gauss might get lost, the reason why Humboldt translated his paper on absolute measures before submitting it. The &lt;i&gt;Journal de mathématiques pures et appliquées&lt;/i&gt; published no German mathematical physics at all in the early years following its founding by Joseph Liouville in 1836. Through the 1830s, the &lt;i&gt;Comptes rendus&lt;/i&gt; and the &lt;i&gt;Annales de chemie et physique&lt;/i&gt; together published only about a dozen papers on German physics. In the 1840s these journals added nearly a dozen new German names, mainly those of experimentalists though including the mathematical physicists Neumann and his student Kirchhoff. It was not until the 1850s that German physics came to be published in the &lt;i&gt;Annales&lt;/i&gt; as copiously as foreign physics had long been published in the German &lt;i&gt;Annalen&lt;/i&gt;. Twenty-five German physicists and well over 100 of their papers appeared in the &lt;i&gt;Annales&lt;/i&gt; between 1850 and 1863, and German mathematical physics now received almost as much attention as German experimental physics. Experimentalists such as Magnus, Plücker, and Buff were represented by more papers than before, and the mathematical physicist Clausius appeared eight times and Kirchhoff thirteen. Liouville&lt;i&gt;’s Journal&lt;/i&gt;, however, continued to publish very little German mathematical physics: of the new physicists, only Clausius appeared there with a single paper on the mechanical theory of heat in 1855.&lt;sup&gt;2&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;In Britain in the 1830s and 1840s, German physics papers appeared very occasionally in the &lt;i&gt;Philosophical Magazine&lt;/i&gt; and the &lt;i&gt;Edinburgh New Philosophical Journal&lt;/i&gt;, but frequently in &lt;i&gt;Taylor’s Scientific Memoirs&lt;/i&gt;. The &lt;i&gt;Memoirs&lt;/i&gt; published the work of experimentalists, Magnus again, H. W. Dove, and Hermann Knoblauch, and also a good many theoretical works. The volume for 1841 contained a translation of Ohm’s 1827 theory, &lt;i&gt;Galvanic Circuit,&lt;/i&gt; as well as of ten other, more recent German physics papers, mainly those by Gauss and Weber dealing with Earth magnetism. The volume for 1853 was devoted almost entirely to German work, including Helmholtz’s memoir on the conservation of force and many papers by Clausius. This being the last volume of the &lt;i&gt;Memoirs&lt;/i&gt;, from then on the &lt;i&gt;Philosophical Magazine&lt;/i&gt; assumed the responsibility of frequently publishing German physics papers. By the 1850s the work of German physicists could be read in translation in Britain about as regularly as the work of British physicists could be read by German physicists in the &lt;i&gt;Annalen&lt;/i&gt;.&lt;hr&gt;&lt;sup&gt;Notes&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;sup&gt;A. v. Humboldt to Gauss, 17 February 1833, in Alexander von Humboldt, &lt;i&gt;Briefe zwischen A. v. Humboldt und Gauss. Zum hunderjährigen Geburtstage von Gauss am 30. April 1877,&lt;/i&gt; ed. Karl Bruhns (Leipzig, 1977), 23.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;sup&gt;Our account of German physics in foreign publications is based on our survey of the journals mentioned. The work by German physicists appearing in the &lt;i&gt;Comptes rendus&lt;/i&gt; of the Paris Academy of Sciences constituted a minute fraction of the journal’s contents. The German physicists who published there in the 1840s were Dove (2 papers), Holtzmann (1), Kirchhoff (2), G. Karsten (1), Magnus (2), J. R. Mayer (3), Moser (2), Plücker (3), Poggendorff (2), Reich (2), and Wiedemann (1). In the 1840s Liouville’s &lt;i&gt;Journal&lt;/i&gt; published only three papers on mathematical physics by Germans: by Gauss (1) and Neumann (2). In the same decade, the &lt;i&gt;Annales&lt;/i&gt; published papers by Buff (1), Dove (2), Magnus (3), Moser (1), Poggendorff (4), and A. Seebeck (1). In the subsequent period, 1850–63, the &lt;i&gt;Annales&lt;/i&gt; published work by many more German physicists: Beer (2 and 1 with Plücker) Beetz (2), Buff (9 and 1 with Wöhler), Clausius (8), Dove (2), Eisenlohr (2), Hankel (1), Helmholtz (8 including some on physiology), Holtzmann (1), Kirchhoff (13), Knoblauch (6) R. Kohlrausch (3), Magnus (13), J. R. Mayer (1), J. Müller (2), Neumann (1), J. F. Pfaff (2), Plücker (8), Poggendorff (4), Quincke (7), Reich (3), Riess (6), Weber (1 with Kohlrausch), Wiedemann (11), and Wüllner (3); Gauss appeared with a paper on mathematical physics.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;sup&gt;The survey of the &lt;i&gt;Annalen&lt;/i&gt; for these years, on which the figures and discussions of the research published there are based, is our own. In 1840–1845, the ordinary professors of physics publishing in the &lt;i&gt;Annalen&lt;/i&gt; were Buff (1 paper, 13 pages), Dove (8, 151), Fechner (6, 127), Magnus (4, 77), Moser (7, 143), J. Müller (1, 10), Muncke (1, 1), Neumann (1, 28), Osann (1, 25), C. H. Pfaff (4, 93), Pohl (1, 24), and Weber (5, 90). German physicists publishing there who later became ordinary professors were: Beetz (1, 18), Feilitzsch (3, 58), Hankel (6, 121), G. Karsten (2, 33), Kirchhoff (1, 18), Knoblauch (1, 12), Ohm (5, 98), and A. Seebeck (8, 169). Physicists at the Berlin Academy were: Poggendorff (18, 355) and Riess (10, 194).&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-21T16:37:13.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6176" PostTypeId="1" CreationDate="2017-06-14T10:22:03.123" Score="3" ViewCount="220" Body="&lt;p&gt;WARNIG: Contrafactual question&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is commonly known that Einstein was not fond of quantum mechanics -- formulated in the famous quote &quot;God does not play dice with the universe&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This quote comes from the time when the Copenhagen interpretation of quantum mechanics was the dominant one, and one of the main point was the idea of wave functions collapse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the modern treatment of quantum mechanics, we don't really need wave function collapse, since we have the mechanism of &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_decoherence&quot; rel=&quot;nofollow noreferrer&quot;&gt;decoherence&lt;/a&gt;.&#xA;In this picture there in no real randomness, but randomness emerges due to the interplay of lots microscopic degrees of freedom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now for the question:&#xA;If the ideas of decoherence would have been around a few decades earlier, is it possible that Einstein would have been won over of the quantum mechanics camp after all - given that the randomness he was so opposed of isn't really a problem.&lt;/p&gt;&#xA;" OwnerUserId="5915" LastActivityDate="2017-06-14T11:40:26.080" Title="Einstein and the concept of decoherence" Tags="|quantum-mechanics|einstein|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6177" PostTypeId="2" ParentId="6176" CreationDate="2017-06-14T11:40:26.080" Score="1" Body="&lt;p&gt;The ideas of decoherence cannot avoid the &quot;spukhafte Fernwirkungen&quot; as demonstrated in the Bell-type experiments and save the local realist view of causality that was advocated by Einstein, Podolsky, and Rosen. The problem is not randomness but the predictable behaviour of entangled particles and their violating Bell's inequalities. Entangled photons from a 0 to 0 transition show with certainty 100 % correlation of spins when measured with polarizers under 90° or 0°. This cannot be explained in a local realist model. Therefore the answer is no.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: This seems to be more a speculative and less a historic question.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-14T11:40:26.080" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6178" PostTypeId="2" ParentId="3824" CreationDate="2017-06-14T18:53:07.927" Score="2" Body="&lt;p&gt;Some good history is to be found in Pais A., How Einstein Got the Nobel Prize: Why did the Nobel Committee for Physics wait so long before giving Einstein the Prize, and why did they not award it for relativity?. American Scientist 70.4 (1982): 358-365. &#xA;Isaacson’s book (p.311 ff.) adds some further details. But Canales theory about Bergson is mostly a recent journalistic invention.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the years before WWI SR was also known as Lorentz-Einstein theory and people were aware of Poincare’s earlier work. Awarding the prize to the three has not really been an option as Lorentz has already got the 1902 Nobel and Poincare died in 1912. Around 1910 Lorentz  emphasized that most of Einstein’s interpretations are ‘epistemological’ (erkenntnistheoretish) and not strictly physical.  His view was authoritative and well known. However since Einstein’s name appeared on newspapers frontpages in 1919 there was pressure and recommendations to award him the prize. The decision to give it to him was taken without taking account of report submitted to the Nobel Committee Actually, it was written by … an opthalmologue (a Nobel in medicine) who somehow did not like Relativity. &#xA;Svante Arrhenius, who had opposed Einstein’s winning,  knew Lorentz’s judgment but most probably knew from second hand that Bergson view was more or less similar. During the announcement he chose to mention Bergson,  with the remark that discussions of relativity pertain to epistemology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Canales imagines some kind of personal inimity and writes “Historians have often puzzled why Lorentz, Poincare, and Michelson—the three men whose research was closest to Einstein’s—failed to embrace the theory of relativity wholeheartedly. The role of Bergson as an individual, colleague, mentor, friend, and confidant—in addition to the general role and impact of his philosophy—was key” p88. But Bergson became interested in Relativity after 1911 and  his views appeared in print (in French) barely a mont before the Nobel Committee decision).&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-06-14T18:53:07.927" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6179" PostTypeId="1" AcceptedAnswerId="6180" CreationDate="2017-06-15T00:38:53.020" Score="6" ViewCount="334" Body="&lt;p&gt;I've been reading John Stillwell's translation of the famous &lt;em&gt;Analysis Situs&lt;/em&gt; and have become confused about the exact meaning of 'simply connected' in Poincaré's language. On page 7 (in the introduction), Stillwell claims Poincaré defines it, in paragraph 14, as a manifold with trivial fundamental group (as is the modern sense of the term). However, in the actual text I find no such definition clearly stated. The term 'simply connected' is first used on page 65, seemingly with no definition given. On page 74 we have something suggestive:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thus we have three manifolds whose group are of finite order, but non-isomorphic, so the manifolds cannot be homeomorphic. Nevertheless, they have the same Betti numbers $$P_1 = P_2 = 1.$$&#xA;  It seems natural to restrict the meaning of the term &lt;em&gt;simply connected&lt;/em&gt; to manifolds whose group $G$ reduces to a single substitution. Then a closed manifold of more than two dimensions can have a group $G$ of finite order without being simply connected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I can only interpret this as meaning that, to Poincaré, simply connected meant trivial fundamental group &lt;em&gt;plus something else&lt;/em&gt;. It seems to me like this something else is all Betti numbers being equal to 1 (note that Poincaré's Betti numbers differ from the modern ones, being shifted up by 1).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, on the very last page of the fifth (and last) complement, Poincaré states his famous conjecture:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it possible for the fundamental group of $V$ to reduce to the identity without $V$ being simply connected?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Poincaré goes on to end the paper claiming, ironically enough, &quot;However, this question would carry us too far away.&quot;) This would seem like a silly question if simply connected were a synonym of trivial fundamental group (unless I misunderstood what he means by 'reduce to the identity' -- I am interpreting this as being a trivial group). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if understood as I suggested above, this is still a bit strange a question, as it is vastly different from what gets called the Poincaré conjecture nowadays -- in fact, it's &lt;em&gt;easy&lt;/em&gt; to show that a simply connected (in the modern understanding of the term) closed 3-manifold is a homology sphere (in particular, has the same Betti numbers as the sphere), and so is simply connected even in the more restrictive definition. Indeed, the proof is just Poincaré duality, which obviously Poincaré knew about, so he couldn't have missed it. Nevertheless, on a footnote Stillwell claims this is a correct statement of Poincaré's conjecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How to make light of this confusion? What did Poincaré mean by 'simply connected' and how did he interpret his own conjecture?&lt;/p&gt;&#xA;" OwnerUserId="5917" LastActivityDate="2024-03-14T12:33:18.417" Title="What exactly did Poincaré mean by 'simply connected'?" Tags="|mathematics|geometry|topology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6180" PostTypeId="2" ParentId="6179" CreationDate="2017-06-15T01:16:51.580" Score="5" Body="&lt;p&gt;In the setting of the conjecture (closed manifolds) he certainly meant “homeomorphic to the $n$-sphere” — see &lt;a href=&quot;https://books.google.com/books?id=IE40AQAAMAAJ&amp;amp;pg=RA1-PA46&quot; rel=&quot;nofollow noreferrer&quot;&gt;second page of that fifth complement&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;simplement connexe au sens propre du mot, c'est-à-dire homéomorphe à l'hypersphère.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Stillwell, p. 179. Same on pp. 141, 152, 169, 255. Stillwell apparently missed the subtlety you note that what is said on p. 74 is not a definition, so he calls Poincaré confused in his Introduction, p. 7.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;br&gt;&#xA;In general Poincaré’s meaning was “&lt;a href=&quot;http://analysis-situs.math.cnrs.fr/spip.php?mot17&quot; rel=&quot;nofollow noreferrer&quot;&gt;homeomorphic to a ball or a sphere&lt;/a&gt;”, according to &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=2768303&quot; rel=&quot;nofollow noreferrer&quot;&gt;H. P. de Saint-Gervais&lt;/a&gt;’ site &lt;a href=&quot;http://analysis-situs.math.cnrs.fr&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Analysis Situs&lt;/em&gt;&lt;/a&gt;, or &lt;a href=&quot;https://books.google.com/books?id=ISXqDAAAQBAJ&amp;amp;pg=PA120&amp;amp;dq=simply+connected&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt; — which however don’t  seem to source the claim.  &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-06-15T18:44:18.863" LastActivityDate="2017-06-15T18:44:18.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6181" PostTypeId="2" ParentId="6179" CreationDate="2017-06-15T11:17:22.493" Score="2" Body="&lt;p&gt;Poincaré really was confused in the beginning. He probably wanted &amp;quot;simply connected&amp;quot; to mean &amp;quot;homeomorphic to the sphere&amp;quot;, as F. Ziegler points in his answer, and tried to find a condition of this in terms of the homology groups and the fundamental group which he introduced. In the beginning he thought that one can characterize the sphere in terms of homology groups, and even had a wrong &amp;quot;proof&amp;quot; of this. Then he discovered&#xA;what is now called &amp;quot;homology spheres&amp;quot; or &amp;quot;Poincaré spheres&amp;quot; which have the same&#xA;homology as the spheres but not homeomorphic to them. Whether the trivial fundamental group implies the sphere is the celebrated Poincaré conjecture, now proved.&lt;/p&gt;&#xA;&lt;p&gt;When you read Poincaré, take into account that he lived long before Bourbaki, and his terms do not always have exact meaning. Or this meaning could change as&#xA;his knowledge expanded.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="10608" LastEditDate="2024-03-14T12:33:18.417" LastActivityDate="2024-03-14T12:33:18.417" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6183" PostTypeId="2" ParentId="2810" CreationDate="2017-06-15T14:20:01.223" Score="-3" Body="&lt;p&gt;Nothing of mathematics was disturbed or spoilt by Russell's paradox. Not even a simple lemma. The reason ist that set therory is not foundational to mathematics (1,2) because it even contradicts mathematics (3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1)  &quot;set theory is largely irrelevant to the practice of most mathematics. Most professional mathematicians never have occasion to use the Zermelo-Fraenkel axioms, while others do not even know them. [Saunders Mac Lane: &quot;&lt;a href=&quot;http://home.dei.polimi.it/schiaffo/TFIS/philofmaths.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical models: A sketch for the philosophy of mathematics&lt;/a&gt;&quot;, American Mathematical Monthly, Vol. 88,7 (1981) p. 467f]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2)  &quot;I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique.&quot; [T. Skolem: &quot;Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre&quot;, Akademiska Bokhandeln, Helsinki (1923) 217-232, reprinted as &quot;Some remarks on axiomatized set theory&quot; in J. van Heijenoort: &quot;From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931&quot;, Harvard University Press, Cambridge, Mass. (1967) 290-301]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Here is the gist of an argument made by &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mückenheim&lt;/a&gt;: Scrooge McDuck every day receives 10 \$ and issues 1 \$. If he issues always the dollars received first and if he applies modern set theory, then he will go bankrupt because each of the dollars received will be spent. The &lt;em&gt;set-limit&lt;/em&gt; is empty. According to &lt;em&gt;mathematics&lt;/em&gt; we have to take the limit of the cardinality of the dollars held. This limit is infinite and stands in clear contradiction to set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Downvotes will not change the facts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The actual infinite is not required for the mathematics of the physical world. [S. Feferman: &quot;Infinity in mathematics: Is Cantor necessary?&quot; in &quot;In the light of logic&quot;, Oxford Univ. Press (1998) p. 30]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part if not all of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, &quot;Is Cantor Necessary?&quot;, is answered with a resounding &quot;no&quot;. [S. Feferman, &quot;In the light of logic&quot;, Oxford Univ. Press (1998) description from the jacket flap]&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-17T05:03:19.227" LastActivityDate="2017-06-17T05:03:19.227" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6184" PostTypeId="1" AcceptedAnswerId="6203" CreationDate="2017-06-15T17:26:35.427" Score="7" ViewCount="453" Body="&lt;p&gt;There were in last several years papers in &lt;em&gt;Science&lt;/em&gt; regarding DNA that does not code for any protein (&quot;junk&quot; DNA) as having epigenetic aspects, namely, switching on and off other genes and thus affecting heredity without coding. In the twisted and furled state, these noncoding genes are sufficiently near to the genes they conditionally switch on and off. The size of &quot;junk&quot; DNA also correlates much better with organizational complexity than traditional coding genes. This has led to &quot;junk&quot; DNA being considered partly a selected for operating system for the other genes and physiological development.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who first speculated correctly about this, but in detail, before the evidence was in? Any mathematical biologists?&lt;/p&gt;&#xA;" OwnerUserId="163" LastEditorUserId="55" LastEditDate="2017-06-19T00:13:08.177" LastActivityDate="2017-06-22T17:21:41.787" Title="Who first suggested, contra the central dogma, that &quot;junk&quot; DNA may perform epigenetic functions?" Tags="|biology|evolution|genetics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6185" PostTypeId="2" ParentId="2776" CreationDate="2017-06-15T19:31:07.003" Score="4" Body="&lt;p&gt;Surprising, that nobody mentions Cantor. His proof of uncountable sets of 1874 (although not as difficult as many proofs mentioned in the other answers) was not going to tempt anybody but a handful specialists. His &lt;em&gt;diagonal argument&lt;/em&gt; of 1891 however was so easy and striking that even laymen could understand it and popularized set theory all at once.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-15T19:31:07.003" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6186" PostTypeId="1" AcceptedAnswerId="6192" CreationDate="2017-06-15T19:42:28.000" Score="5" ViewCount="108" Body="&lt;p&gt;What I am referring to is the Balmer formula as it appears in Wikipedia. To come up with this series by trial and error along with its constants is asking a little too much  but I can't understand how one would go about deriving this formula which involve such exquisite measurements of its constants.   Specifically was there some perhaps experimental methodology that makes the formula a little more easy to see how  he may have arrived at it? It appears to be an ad hoc truly amazing leap in intuition. Which may be the case. Certainly doesn't hurt to ask.  &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorDisplayName="user466" LastEditDate="2017-06-16T17:01:04.913" LastActivityDate="2017-06-18T11:53:29.570" Title="Is there any history into the methodology that Balmer used for the spectral line formula?" Tags="|mathematics|atomic-theory|nuclear-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6188" PostTypeId="2" ParentId="2741" CreationDate="2017-06-15T21:16:17.357" Score="0" Body="&lt;p&gt;The roots of falten, Falte, Zwiefalt, Sorgfalt and Mannichfaltigkeit (Cantor's original use) or Mannigfaltigkeit (modern use) are the same but the question has to be answered by &quot;definitely not&quot;. The original meaning of the verb &quot;falten&quot; (to fold) is no longer present in Zwiefalt = Zweifel (doubt) and Sorgfalt (care). Same holds for manifold = Mannigfaltigkeit = Vielfalt. The emphasize is on viel = many, multiple. Neither Riemann nor Cantor had a folded space in mind.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-15T21:16:17.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6189" PostTypeId="1" AcceptedAnswerId="6238" CreationDate="2017-06-15T22:10:46.223" Score="13" ViewCount="1377" Body="&lt;p&gt;In a historical description of the Stern-Gerlach experiment, [Friedrich 2003] says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein and Paul Ehrenfest, among others, struggled to understand how the atomic magnets could take up definite, preordained orientations in the field. Because the interaction energy of atoms with the field differs with their orientation, it remained a mystery how splitting could occur when atoms entered the field with random orientations and their density in the beam was so low that collisions did not occur to exchange energy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They give this quote from Einstein (from a March 1922 letter to Born):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The most interesting achievement at this point is the experiment of Stern and Gerlach. The alignment of the atoms without collisions via radiative [exchange] is not comprehensible based on the current [theoretical] methods; it should take more than 100 years for the atoms to align. I have done a little calculation about this with [Paul] Ehrenfest. [Heinrich] Rubens considers the experimental result to be absolutely certain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Wve8g.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Wve8g.jpg&quot; alt=&quot;Gerlach&amp;#39;s postcard to Bohr&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone explain this? Classically, if we send a beam of randomly oriented dipoles through a Stern-Gerlach spectrometer, we expect an ellipsoidal pattern that is not split into well-defined components. According to modern quantum mechanics, we expect what we get in the famous postcard Gerlach sent to Bohr, above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So apparently Einstein was working in a theoretical universe that was very primitive and contained only a subset of modern quantum mechanics. However, this subset must have contained &quot;space quantization,&quot; i.e., the quantization of angular momentum in units of hbar along a particular axis. I guess this is what people refer to as early quantum theory. At the time, electron spin was unknown, and the Stern-Gerlach result was interpreted as a triumphant proof that the orbital angular momentum of the odd electron in the silver atom had the two values $\pm\hbar$. (Zero was forbidden.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was the intermediate theory under which the splitting of the beam into two discrete components was interpreted as evidence of quantization, but Einstein's concern about time scales still appeared? What would have been the reasoning that would have led to this concern about time scales?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Friedrich and Herschbach, &quot;Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics,&quot; Physics Today, December 2003, p. 56&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorUserId="10106" LastEditDate="2022-05-26T04:11:29.957" LastActivityDate="2022-05-26T04:11:29.957" Title="Einstein's confusion about Stern-Gerlach" Tags="|physics|quantum-mechanics|einstein|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6192" PostTypeId="2" ParentId="6186" CreationDate="2017-06-16T16:48:25.593" Score="7" Body="&lt;p&gt;Here the German Wikipedia helps better than the English one: Balmer was interested in a very broad range of sciences and &quot;sciences&quot; including numerology and cabbala. For instance he calculated the number of steps of pyramids and the floor plan of Jewish temples, probably from the information supplied by the bible. Therefore it is not surprising that he was interested in the problem of spectral wavelengths which Eduard Hagenbach (Prof. of physics at Basel university), knowing Balmer's deciphering skill, had suggested to him. In 1885 Balmer found the well-known formula &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\lambda = \frac{m^2h}{m^2-n^2}$  where $n = 2$ and $m = 3, 4, 5, ...$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which in 1888 was generalized by Rydberg.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Balmer predicted the line for m = 7 which was confirmed by Angström. Other lines, i.e. higher quantum states, were found in the spectra of white stars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps also of interest, but only available in German from the &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002074761&amp;amp;physid=phys56#navi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Swiss Mathematical Society&lt;/a&gt;: Balmer also lectured as a Privatdozent at the university of Basel. Alas there were no students interested. Of about 70 announced courses 50 did not take place. Only very rarely he had more than 1 or 2 students. So his main job remained to teach girls how to add and multiply fractions and to express thoughts in written form, 17 hours a week.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his notes of 1891 he speculates about atomic movement, points of force in circular or elliptic motion and even something resembling waves. These (historically) interesting notes are partially available on &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=PPN378850199_0016%7Clog21&amp;amp;physid=phys65#navi&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 59&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-18T11:53:29.570" LastActivityDate="2017-06-18T11:53:29.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6193" PostTypeId="1" CreationDate="2017-06-17T07:45:28.053" Score="4" ViewCount="815" Body="&lt;p&gt;My question refers to Gauss's 1811 letter to Friedrich Bessel, which contains a statement of Cauchy integral theorem. I have access to the letter, but I'm unable to read it. I know that Gauss gives the example of the multivalued nature of the line integral of $1/x$ around $x = 0$, and gives the correct reside of $2\pi i$. This is a result of a straightforward integration along infinitesimal arcs (which are described as infinitesimal complex mumbers) which gives a differential of $e^{{-i\theta}}\cdot e^{{i(\theta + \pi/2)}}\,d\theta = i\,d\theta$. But this could also be understood from the fact that the integral of $1/x$ is $\log x$, which is of course a multivalued function. I don't understand enough the subject of complex analysis to infer what these facts imply about Gauss's understanding of the resides theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that the example which Gauss gives can be viewed another way; he gives the integral of $1/x$ because the key to evaluating resides of line integrals around singularities is the coefficient $a_1$ of the $-1$ power term in the Laurent series of the complex function. Therefore the question that remains is whether Gauss knew Laurent series? I saw in his letter to bessel the function $(e^x - 1)/x$ as well as certain power series , but what do these power series really mean?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-06-22T17:23:30.440" LastActivityDate="2020-06-22T17:23:30.440" Title="Did Gauss know the residues theorem in complex analysis in 1811?" Tags="|mathematics|gauss|complex-analysis|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6195" PostTypeId="2" ParentId="6139" CreationDate="2017-06-17T12:49:14.237" Score="1" Body="&lt;p&gt;Two reasons I can think of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Minkowski's paper &lt;em&gt;Space &amp;amp; Time&lt;/em&gt; (in Wikipedia).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geometry: a circle is x^2+y^2=R, a family of ever widening circles is x^2+y^2=R.t;&#xA;stacking the circles gives a cone which is the usual x^2+y^2=t^2.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-06-17T12:49:14.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6196" PostTypeId="1" AcceptedAnswerId="6199" CreationDate="2017-06-17T14:02:00.343" Score="4" ViewCount="271" Body="&lt;p&gt;I seem to remember that as late as in the XIX century there was a prominent mathematician who denied that the theory of probability was part of mathematics, since it does not deal with certainty. &#xA;Do you know who was that mathematician?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="4251" LastEditDate="2022-02-01T13:36:06.740" LastActivityDate="2022-02-01T13:36:06.740" Title="Who said that theory of probability was not mathematics?" Tags="|mathematics|mathematicians|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6197" PostTypeId="2" ParentId="2693" CreationDate="2017-06-17T16:25:51.663" Score="1" Body="&lt;p&gt;Ludolph van Ceulen (Köln / Cologne) had calculated (using the method of Archimedes) 20 digits of $\pi$ in 1596. Toward the end of his life he knew even 35 digits. Therefore in Germany $\pi$ has also often been called &quot;Ludolphs Zahl&quot;. Obviously this was always considered a number or constant although not being named $\pi$. To answer the question: It is highly probable that he knew enough digits in 1592.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But much earlier the Chinese Tsu Ch’ung-Chih (430-501) had found the very good approximation 355/113 which amounts to 3.14159292... (and &lt;em&gt;not&lt;/em&gt; so on). In possession of decimal digits and the inconsistent MM-DD-YYYY date format he could have been the first to satisfy the request.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This accomplishment however has been forgotten soon, and his compatriot Liu Hwuy used $157/50 = 3.14$ in the 7th century. The Indian Brahmagupta had $\sqrt{10} = 3.16...$ in the 7th century. Then followed a really &lt;strong&gt;dark age&lt;/strong&gt;. The Byzantine Michael Psellus, 11th century put $\sqrt{8} = 2.828...$, and Franco de Liège, also 11th  century, $(9/5)^2 = 3.24$, both worse than the Old Egypts. Only the Dutch Adriaan Metius in 1585 rediscovered $355/113 = 3.1415929...$. So he would have been able in 1593 to celebrate the pi-day.  &lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-17T16:25:51.663" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6198" PostTypeId="1" CreationDate="2017-06-17T22:29:54.160" Score="2" ViewCount="219" Body="&lt;p&gt;I have had difficulty locating the full name and story of the mathematician Puppe whose name adorns the beloved underlying long-exact sequence algebraic topology is built on. Does anyone know who they are?&lt;/p&gt;&#xA;" OwnerUserId="2787" LastActivityDate="2017-06-18T11:47:18.933" Title="Who was Puppe of the Puppe sequence?" Tags="|mathematicians|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6199" PostTypeId="2" ParentId="6196" CreationDate="2017-06-18T10:04:00.903" Score="9" Body="&lt;p&gt;It was Hardy and Littlewood. They said in a footnote to an article in the early 1920s (this was notably &lt;em&gt;before&lt;/em&gt; Kolmogorov's measure-theoretic foundations were developed) that &quot;Probability is not a notion of pure mathematics, but of philosophy or physics.&quot; See p. 37 of their paper &lt;em&gt;Some Problems of Partitio Numerorum III&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2017-06-18T10:11:55.600" LastActivityDate="2017-06-18T10:11:55.600" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6200" PostTypeId="2" ParentId="6198" CreationDate="2017-06-18T10:53:05.387" Score="2" Body="&lt;p&gt;Siegmund Dieter Puppe (16. Dezember 1930 - 13. August 2005) has already been identified by Francois Ziegler. But since the question asks also for the story which is not yet available in English language, some additional information may be useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He studied physics and mathematics at Göttingen and Heidelberg and became a professor in 1960 at Saarbrücken. 1968 he returned to Heidelberg and stayed there for the rest of his career until 1996. He was guest professor at Princeton 1957/58, Chicago 1961 and Minneapolis 1966/67. He gave in 1962 a talk at the International Mathematical Congress at Stockholm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://de.wikipedia.org/wiki/Dieter_Puppe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Information from the German Wikipedia&lt;/a&gt;. There is also a photograph.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-18T11:47:18.933" LastActivityDate="2017-06-18T11:47:18.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6201" PostTypeId="2" ParentId="3131" CreationDate="2017-06-18T15:08:13.373" Score="27" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Frank_Plumpton_Ramsey&quot; rel=&quot;noreferrer&quot;&gt;Frank Ramsey&lt;/a&gt; died before his 27th birthday. He started Ramsey theory, which is a very active field in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="81" LastActivityDate="2017-06-18T15:08:13.373" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6202" PostTypeId="2" ParentId="256" CreationDate="2017-06-18T19:49:01.553" Score="1" Body="&lt;p&gt;&quot;Has there ever been a major basic science result that did not lead to practical applications within the next couple of hundred years?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Transfinite set theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It can be considered basic science in the sense that it was not mainly about application.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This result was considered a breakthrough at its time by notable sources&#xA;like Hilbert and many other mathematicians.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neither this result nor its successors are considered relevant today for any practical application in sciences like physics, chemistry, biology, technology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no relevant technological application (nor has there ever been) and it does not appear in modern textbooks of any scientific discipline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result was not a negative one, but an invention of new notions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only the last condition is not satisfied.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-18T19:49:01.553" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6203" PostTypeId="2" ParentId="6184" CreationDate="2017-06-19T00:08:40.030" Score="7" Body="&lt;p&gt;This is a great question as it touches on the history-in-the-making, a revolution in molecular biology taking place in our lifetime. In the last two decades the &lt;a href=&quot;https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;central dogma&amp;quot; of molecular biology&lt;/a&gt;, formulated by Watson and Crick  after the discovery of DNA in 1952-1958, has been overthrown. The neat picture of genes, exclusive containers of hereditary traits, coding proteins and having the code transcribed by enzymes and then delivered by messenger RNA from the nucleus to the ribosome, where it is further transcribed into amino acids; has been replaced with a much more complex, and messier, picture of epigenetic suppression and modification of the code, directly influenced by environmental factors, and regulated by vast swaths of non-protein-coding &amp;quot;DNA elements&amp;quot; (new inclusive term, replacing &amp;quot;genes&amp;quot;) previously dismissed as evolutionary &amp;quot;junk&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The recent papers mentioned in the OP derive from the simultaneous publication of 30 (!) linked articles on September 5, 2012, in Nature and several other journals, by the Encyclopedia of DNA Elements (ENCODE) consortium authors. They mark a major advance in deciphering the &amp;quot;junk&amp;quot; DNA functionality. ENCODE is an offshoot of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Human_Genome_Project&quot; rel=&quot;nofollow noreferrer&quot;&gt;Human Genome Project&lt;/a&gt; launced after its completion in 2003. The information dump was so massive that &lt;a href=&quot;http://www.nature.com/encode/#/threads/epigenetic-regulation-of-rna-processing/landscape-of-transcription-in-human-cells&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nature created a special website&lt;/a&gt; meant to guide the readers through it, ENCODE also has &lt;a href=&quot;https://www.encodeproject.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;its own website&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But the idea of active DNA not involved in protein coding, which was its primary function under the &amp;quot;central dogma&amp;quot;, but producing non-coding RNA (ncRNA) is much older. A good historical survey on this topic is &lt;a href=&quot;http://www.nature.com/nrg/journal/v2/n12/full/nrg1201-919a.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eddy's Nature review Non–coding RNA Genes and the Modern RNA World (2001)&lt;/a&gt;. Eddy traces the idea back to &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0022283661800727&quot; rel=&quot;nofollow noreferrer&quot;&gt;Genetic Regulatory Mechanisms in the Synthesis of Proteins by Jacob and Monod (1961)&lt;/a&gt;. They were not mathematical biologists, and the paper is not very mathematical, but it is certainly very theoretical:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The idea that ncRNA would be well adapted for regulatory roles is not new. In the process of defining many of the concepts of molecular genetics, including mRNA and operons, François Jacob and Jacques Monod distinguished “structural genes” (such as lacZ) from “regulatory genes” (such as lacI). At that time, regulators such as lacI had only been defined genetically, and they were known to specifically interact with cis-acting sequences (such as lacO), either at the DNA or mRNA level. Jacob and Monod reasoned that base complementarity would allow RNA to interact highly specifically with other nucleic-acid sequences. They proposed that structural genes encoded proteins, and regulatory genes produced ncRNAs (FIG.5). Forty years later, their proposal is looking more relevant than ever.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One can see from the bibliography in Eddy's review that discoveries of ncRNA genes with epigenetic regulatory functions picked up significantly in the late 1990-s, leading up to an explosion after 2000, when the initial draft of the Human Genome Project was published. The final draft came out in 2003. Some of the sound and the fury of 2000-2003 is reflected in &lt;a href=&quot;http://www.detectingdesign.com/PDF%20Files/The%20Unseen%20Genome.doc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gibbs's popular review Unseen Genome: Gems among the Junk (2003)&lt;/a&gt;. But isolated examples were known much earlier. One of the first direct confirmations of Jacob-Monod's speculation came in 1980, due to Rosalind Lee and her colleagues. Eddy again:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;A canonical example of the identification of a ncRNA gene by genetics is the story of the lin-4 regulatory RNA in the nematode C. elegans. The lin-4 locus&#xA;was identified in a screen for mutations that affect the timing and sequence of postembryonic development (HETEROCHRONIC MUTATIONS) in C. elegans... The lin-4 RNA inhibits accumulation of the LIN-14 and LIN-28 proteins by an unknown mechanism.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1988 &lt;a href=&quot;http://www.annualreviews.org/doi/abs/10.1146/annurev.ge.22.120188.003031&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simons and Kleckner in Biological Regulation by Antisense RNA in Prokaryotes&lt;/a&gt; described a different type of regulation mechanism  effected by genes on the wrong side of the DNA ladder, usually considered to be idle backup copies. From Gibbs:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In some cases, however, the backup has its own agenda. While the gene is producing a sensible RNA message, its alter ego can churn out an &amp;quot;antisense&amp;quot; RNA that has a complementary sequence. Whenever matched sense and antisense RNAs meet, they mesh to form their own double-stranded ladders -- effectively interfering with the gene's ability to express its protein... These competing RNAs may suppress a gene just by tying up the gene's messenger RNA.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it is not that epigenetic functions of non-coding RNA genes were not known before 2012, or even before 2000, but rather that they were considered to be flukes, rare offshoots on the highway of evolutionary heredity revealed by the &amp;quot;central dogma&amp;quot;, represented by the protein coding by the &amp;quot;true&amp;quot; genes through the three major lanes of messenger, transfer and ribosomal RNA. It is only with the completion of the Human Genome Project in 2000-2003 that the evidence snowballed, especially in humans, and made this view untenable. Gibbs's review describes many other ncRNA regulatory mechanisms that came to light in late 1990-s – early 2000-s, like active pseudogenes, riboswitches, and microRNA. And Eddy sketches the emerging &amp;quot;modern RNA world&amp;quot; hypothesis, meant to replace the &amp;quot;central dogma&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Many of the ncRNAs we see in fact have roles in which RNA is a more optimal material than protein. Non-coding RNAs are often (though not always) found to have roles that involve sequence specific recognition of another nucleic acid. ... RNA, by its very nature, is an ideal material for this role. Base complementarity allows a very small RNA to be exquisitely sequence specific.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Many functional roles do not require the more sophisticated catalytic prowess of proteins and could be carried out by simple RNAs. Post-transcriptional regulation, in particular, can be achieved simply by steric occlusion of sites on a target pre-mRNA or mature RNA. In cases requiring more sophistication than simple steric blockage, necessary catalytic functions can be delegated to a small number of shared proteins, whereas specific sequence recognition functions are carried out by a horde of individual small RNAs that interact with&#xA;these proteins.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-06-22T17:21:41.787" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6204" PostTypeId="1" CreationDate="2017-06-19T02:30:52.907" Score="5" ViewCount="384" Body="&lt;p&gt;The Renaissance created a number of prominent mathematicians. However, later in the 18th and especially 19th century, Germany and France became the hot centers of mathematical thinking.&lt;/p&gt;&#xA;" OwnerUserId="5618" LastEditorUserId="4251" LastEditDate="2023-06-26T09:43:31.297" LastActivityDate="2023-06-26T09:43:31.297" Title="Why did pre-17th century mathematics mostly come from Italy but later mathematics came from France, Germany and England?" Tags="|mathematics|19th-century|18th-century|renaissance|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6206" PostTypeId="1" AcceptedAnswerId="7409" CreationDate="2017-06-19T08:04:48.327" Score="17" ViewCount="841" Body="&lt;p&gt;Cajori, the website on &lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Uses of Symbols of Calculus&lt;/a&gt; and many other sources claim that Lagrange introduced the notation $f'(x)$ for the derivative of $f(x)$ with respect to $x$. But I see Euler using it 1748 in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E140.html&quot; rel=&quot;noreferrer&quot;&gt;Sur la vibration des cordes&lt;/a&gt; p. 78-79&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[$\ldots$] pourvu que nous marquions le differentiel de la fonction&#xA;  $f{:}z$ par $dz \, f'{:}z$, &amp;amp; le differentiel de la fonction $f'{:}z$ par&#xA;  $dz \, f''{:}z$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(He's using the alternative notation for &quot;function application&quot; $f{:}x$, instead of the nowadays common $f(x)$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around this time Lagrange was 12 and, according to Wikipedia, showed no interest in mathematics prior to being 17. So it seems unlikely that Euler got it from Lagrange. On the other hand, Lagrange studied the works of Euler and it is reasonable to suspect that Lagrange took it from him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone confirm this state of affairs?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2017-06-19T11:26:28.623" LastActivityDate="2018-06-06T09:14:21.777" Title="Is the prime notation for derivatives $f'$ due to Euler?" Tags="|mathematics|notation|calculus|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6207" PostTypeId="2" ParentId="6193" CreationDate="2017-06-19T11:52:31.960" Score="5" Body="&lt;p&gt;Here are the relevant passages of Gauss' letter:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... What should one think of $\int \phi xdx$ for $x = a + bi$? Obviously ... $x$ has to go by infinitely small increments (each one of the form $\alpha + \beta i$) from the value where the integral shall be $0$ to $x = a + bi$, and then we sum up all $\phi xdx$. ... The continuous transition from one value of $x$ to another $a + bi$ is done by means of a line and therefore is possible in infinitely many ways. I claim now that the integral $\int \phi xdx$ after two different transitions always takes the same value if within the area between the lines representing these transitions nowhere $\phi x = \infty$. This is a beautiful theorem the not very difficult proof of which I will give at an appropriate occasion. It is connected with other nice truths concerning the expansion of series. Therefore I demand to evade points where obviously the original basic notion of $\phi x$ becomes unclear and leads to contradictions. By the way, it is clear how a function generated by $\int \phi xdx$ for one and the same $x$ can have several values, namely in that during the transition we can not or once or several times go around a point where $\phi x = \infty$. Defining, for example, $log x$ by $\int (1/x)dx$, to start at $x=1$, then one gets to $logx$ either without including the point $x=0$ or by going around one or several times. Every time the constant $+2\pi i$ or  $-2\pi i$ is added. This makes clear that every number has many logarithms. If $\phi x$ can never become infinite for a finite $x$ then the integral is always single-valued.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This makes clear that Gauss was in possession of the &quot;not difficult to prove&quot; theorem and that the logarithm is only an example for an application, not the basis of his rule.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-19T11:58:43.380" LastActivityDate="2017-06-19T11:58:43.380" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6208" PostTypeId="2" ParentId="6162" CreationDate="2017-06-19T15:18:20.510" Score="3" Body="&lt;p&gt;Newton was not the first to conceive of the notion of absolute space. The Stoics, &lt;a href=&quot;http://www.encyclopedia.com/people/history/historians-miscellaneous-biographies/john-philoponus&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Philoponus&lt;/a&gt; in particular, and the medieval physicist Bishop &lt;a href=&quot;http://www.encyclopedia.com/topic/Nicole_Oresme.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicole Oresme&lt;/a&gt; (1320-1382) conceived of it. For the quote from Bishop Oresme's &lt;em&gt;Traite du ciel et du monde&lt;/em&gt; illustrating this, see pp. 264-266 of Pierre Duhem's &lt;a href=&quot;https://isidore.co/calibre/browse/book/4757&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2017-06-19T15:18:20.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6209" PostTypeId="2" ParentId="6204" CreationDate="2017-06-19T17:55:44.210" Score="2" Body="&lt;p&gt;This does not apply only to mathematics, but to all ancient culture. It was almost completely forgotten in Europe, and then it was slowly recalled, in the process which is called Renaissance. So, the natural restatement of your question would be: why did the Renaissance happen in Italy first?&lt;/p&gt;&#xA;&lt;p&gt;It is hard to name one reason. Probably there were several:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First of all, some memories of the ancient culture were certainly preserved in Italy more than in other places in Europe, because Italy was the center of the ancient Roman Empire.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Second, and probably more important, was more intensive contacts with the Eastern Roman Empire, which still existed at the time of the Renaissance, and where some ancient heritage was preserved.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And, finally, Italy in general had more contact with the rest of the world because the international trade was more developed in Italy. This includes Mediterranean countries and the same Eastern Empire. (After all, Marco Polo and Columbus were Italian, and Fibonacci was a merchant.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To a much smaller extent, one can also see a similar process in the Iberian Peninsula, where Europeans had a long contact with Muslims, who also preserved a part of ancient heritage. However, in Spain and Portugal this process was arrested by religious intolerance after the &lt;em&gt;Reconquista&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So the process seems to start on the cultural border (penetration of &amp;quot;new&amp;quot; ideas) but later it develops mainly in the places where intellectual climate is more more appropriate (Germany, England, France).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2020-07-08T16:58:17.137" LastActivityDate="2020-07-08T16:58:17.137" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="6211" PostTypeId="2" ParentId="6162" CreationDate="2017-06-20T10:16:56.373" Score="3" Body="&lt;p&gt;Max Jammer has an original theory which should be treated cautiously. In his book &lt;em&gt;Concepts of Space&lt;/em&gt;, (Einstein wrote a preface for it) he argues that Newton's absolute space is a notion of religious origin. It is an attribute of the absolute deity, an argument that was made visible in the controversy Leibniz-Clarke about space being the &quot;sensorium dei&quot;. He traces back the idea to early theology but that is no more 'history of science'. Such more or less mystic ideas are not uncommon during the Renaissance  and Jammer also comments some of them.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-06-20T10:16:56.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6212" PostTypeId="2" ParentId="6157" CreationDate="2017-06-20T10:39:00.557" Score="8" Body="&lt;p&gt;Newton is a dubious example here, but it is well known that in  the preface of his &lt;em&gt;Principia&lt;/em&gt; he asserted (p.xiii):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;geometry is a founded on mechanical practice and is nothing but that&#xA;  part of universal mechanics which accurately proposes and demonstrates&#xA;  the art of measuring.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Leaving aside the etymology harking back to 'measuring', for Newton geometry &lt;em&gt;is&lt;/em&gt; mathematics, as he disliked cartesianism and its algebraic approach.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before the XXth.c most of mathematics has been done by natural scientists-physicists; Imre Lakatos, who was a philosopher, proposed in his &lt;em&gt;Proofs and refutations&lt;/em&gt; the view that mathematics is just like any other science; later Quinn and Jaffe outlined  the idea that mathematics is theoretical and makes hypotheses, for which proofs are offered and so it functions as a natural science. &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-06-20T10:39:00.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6213" PostTypeId="2" ParentId="6157" CreationDate="2017-06-20T11:32:49.863" Score="6" Body="&lt;p&gt;An explicit statement has been given by Kronecker:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematics is a natural science – not better, not more complete, and not simpler the phenomena can be described than mathematically. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;About the notion of number in mathematics&quot;, Public lecture in summer semester 1891 at Berlin – Kronecker's last lecture. &quot;Sur le concept de nombre en mathematique&quot;  Retranscrit et commenté par Jacqueline Boniface et Norbert Schappacher: Revue d'histoire des mathématiques 7 (2001).&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-20T11:32:49.863" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6214" PostTypeId="2" ParentId="6206" CreationDate="2017-06-20T13:46:18.343" Score="6" Body="&lt;p&gt;That's an interesting find but notice that Euler is still talking about differentials here.  He seems to use $f'$ as somewhat ad hoc notation for an auxiliary entity used in describing the differential, rather than having the &quot; $f\mapsto f'$ &quot; picture in mind.  Lagrange introduced the new concept of a &quot;derived function&quot; even if he borrowed this piece of notation from Euler.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-06-20T13:46:18.343" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6215" PostTypeId="2" ParentId="2576" CreationDate="2017-06-20T13:53:56.233" Score="3" Body="&lt;p&gt;If we understand by mathematics simple counting and addition, then the oldest known relicts are the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ishango_bone&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ishango bone&lt;/a&gt; (20000 years) and the even older &lt;a href=&quot;https://en.wikipedia.org/wiki/Lebombo_bone&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lebombo bone&lt;/a&gt; (44000 years). It is even speculated that the duodecimal system and the sexagesimalsystem have their roots there. The &quot;nations&quot; producing it are today named Kongo and South Africa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we understand by mathematics some more sophisticated calculating like geometric series and drawing rectangular triangles, then Egypt, and Mesopotamia usually come to mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When we include astronomical observation and calculation, then not only China should be mentioned (execution of the astronomers Hsi and Ho 2000 BC) but also the &lt;a href=&quot;https://en.wikipedia.org/wiki/Nebra_sky_disk&quot; rel=&quot;nofollow noreferrer&quot;&gt;sky disc of Nebra&lt;/a&gt;, recently found in Germany, stemming also from about 2000 BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the modern meaning of proof-based mathematics, see the answers already given: Greek, India, China, all several centuries BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally it should be noted that the word μάθημα máthēma has been created by the Pythagoreans.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-20T13:53:56.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6216" PostTypeId="2" ParentId="2465" CreationDate="2017-06-20T15:12:51.267" Score="2" Body="&lt;p&gt;Relativistic mass is &lt;em&gt;not&lt;/em&gt; outdated. &quot;Roche states that about 60% of modern authors just use rest mass and avoid relativistic mass.&quot; Who is Roche? &quot;Roche suggests that relativistic mass is only an artifact.&quot; That should say enough.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mass_in_special_relativity#Relativistic_mass&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mass_in_special_relativity#Relativistic_mass&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When a photon of relativistic mass &lt;em&gt;m&lt;/em&gt; is absorbed, then the mass of the absorber increases by &lt;em&gt;m&lt;/em&gt;, although the rest mass of the photon is zero. That should say all.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-06-20T15:12:51.267" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6217" PostTypeId="2" ParentId="6204" CreationDate="2017-06-20T16:12:48.273" Score="5" Body="&lt;p&gt;Analysis got a head start in the work of Italian indivisibilists &lt;a href=&quot;https://en.wikipedia.org/wiki/Bonaventura_Cavalieri&quot; rel=&quot;noreferrer&quot;&gt;Cavalieri&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Evangelista_Torricelli&quot; rel=&quot;noreferrer&quot;&gt;Torricelli&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Stefano_degli_Angeli&quot; rel=&quot;noreferrer&quot;&gt;degli Angeli&lt;/a&gt;, and others. However, the budding analysis was viewed with disfavor by some in the Catholic hierarchy, including many leading Jesuits. This led to a suppression of active math centers in Italy and a graduate decline of the Italian mathematical tradition. The reasons for the clerical opposition to indivisibles are the subject of dispute among scholars. &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9525-z&quot; rel=&quot;noreferrer&quot;&gt;This recent publication by Sherry&lt;/a&gt; attributes this to the association of indivisibles with ideas that were seen as contrary to the Catholic interpretation of the eucharist, while &lt;a href=&quot;https://link.springer.com/article/10.1007/s10699-017-9523-1&quot; rel=&quot;noreferrer&quot;&gt;this recent publication by Alexander&lt;/a&gt; attributes the opposition to an adherence to a Euclidean ideal of rigor.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-06-20T16:12:48.273" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6218" PostTypeId="1" CreationDate="2017-06-20T17:52:53.093" Score="1" ViewCount="57" Body="&lt;p&gt;Below is a picture of a  3-D permutahedron sundial by Stefano Buonsignori (16th century) in the Medici collection presented by &lt;a href=&quot;http://brunelleschi.imss.fi.it/museum/esim.asp?c=403055&quot; rel=&quot;nofollow noreferrer&quot;&gt;Museo Galileo&lt;/a&gt;. The permutahedra / permutohedra and the closely related associahedra are important families of convex polytopes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the earliest known presentation of the 3-Dim permutahedron?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/bn2Z7.jpg&quot; alt=&quot;Sundial&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="1830" LastActivityDate="2017-06-20T17:52:53.093" Title="Earliest presentation of a 3-D permutahedron?" Tags="|mathematics|geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6219" PostTypeId="1" AcceptedAnswerId="6220" CreationDate="2017-06-20T18:02:33.193" Score="2" ViewCount="2254" Body="&lt;p&gt;It's a simple enough result I would have been unsurprised if it weren't named for anyone at all. I certainly find it odd it's named for a relatively modern physicist rather than an early-calculus mathematician. I assume Feynman earned the moniker by popularising its use in physics, but even that is a shock since it has obvious applications to 19th century thermodynamics.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="4251" LastEditDate="2020-06-05T20:21:52.383" LastActivityDate="2020-06-05T20:21:52.383" Title="Why is differentiation under the integral sign named Feynman's trick?" Tags="|mathematics|physics|calculus|feynman|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6220" PostTypeId="2" ParentId="6219" CreationDate="2017-06-20T19:48:13.570" Score="5" Body="&lt;p&gt;The name comes from Feynman's evaluation of the integral&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_0^\infty\frac{\sin x}{x}dx.$$&lt;/span&gt;&#xA;This is a classical example on applications of residues (I think it is originally due to Hardy. In the US residues are not a standard part of the undergraduate curriculum).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman proposed the following elementary argument. Consider instead&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(y):=\int_0^\infty\frac{\sin x}{x}e^{-xy}dx,$$&lt;/span&gt;&#xA;so we need &lt;span class=&quot;math-container&quot;&gt;$f(0)$&lt;/span&gt;. Then differentiate under the integral sign:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f'(y)=-\int_0^\infty e^{-xy}\sin x dx,$$&lt;/span&gt;&#xA;and this is easily evaluated (integration by parts twice). As we know &lt;span class=&quot;math-container&quot;&gt;$f(y)\to 0$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$y\to+\infty$&lt;/span&gt; we find &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; and our integral is &lt;span class=&quot;math-container&quot;&gt;$f(0)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This simple argument impressed people so much that they started calling it Feynman's trick. Those for whom this was the first interesting example of differentiation under the integral sign extended this name to any differentiation under integral sign.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-01-04T06:39:10.690" LastActivityDate="2019-01-04T06:39:10.690" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="6221" PostTypeId="2" ParentId="2460" CreationDate="2017-06-20T20:10:38.407" Score="0" Body="&lt;p&gt;How did Aristotle make these mistakes? Probably the resason is the same in most if not all cases: There was not a single experiment performed before Archimed. From atomism of Leukippos to Aristotle's laws of motion there was always only pure imagination or philosophy (except explanations based on observation, for instance the wrong life cycle of bees or the not reproducing eels). Aristoteles stated for instance acoustic dispersion: signals of different pitch should spread out with different velocities. Small wonder that many errors arose.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-20T21:45:42.147" LastActivityDate="2017-06-20T21:45:42.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6222" PostTypeId="1" CreationDate="2017-06-21T04:26:33.793" Score="1" ViewCount="150" Body="&lt;p&gt;Specifically, the associative, distributive, and commutative laws of addition and multiplication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it Peano?&lt;/p&gt;&#xA;" OwnerUserId="4095" LastActivityDate="2017-06-21T04:26:33.793" Title="Which mathematician first proved the laws of arithmetic?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="7" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6223" PostTypeId="1" CreationDate="2017-06-21T19:44:35.990" Score="3" ViewCount="1727" Body="&lt;p&gt;How much did Arabic and Persian scientists contribute to physics and mathematics? Would it have made any difference for modern natural sciences and mathematics if they hadn't made their discoveries?&lt;/p&gt;&#xA;" OwnerUserId="5941" LastEditorUserId="12699" LastEditDate="2017-07-08T11:23:18.180" LastActivityDate="2017-07-16T01:25:21.733" Title="How much did Arabic and Persian scholars contribute to physics and mathematics?" Tags="|mathematics|physics|" AnswerCount="4" CommentCount="13" ClosedDate="2017-07-04T22:52:29.057" ContentLicense="CC BY-SA 3.0" />
  <row Id="6224" PostTypeId="1" CreationDate="2017-06-22T13:36:40.727" Score="1" ViewCount="83" Body="&lt;p&gt;My question is a continuation of this posted question: &lt;a href=&quot;https://hsm.stackexchange.com/questions/5969/why-did-gauss-recommend-janos-bolyai-to-find-the-volume-of-a-tetrahedron-in-hype?rq=1&quot;&gt;Why did Gauss recommend Janos Bolyai to find the volume of a tetrahedron in hyperbolic geometry?&lt;/a&gt;. The issue is that since i asked the previous question i found additional material in Gauss's works - i think the relevant note in Gauss's nachlass is in page 228 of volume 8 of his  work - &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN236010751|LOG_0072&amp;amp;physid=PHYS_0234&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN236010751|LOG_0072&amp;amp;physid=PHYS_0234&lt;/a&gt;. It's a little note entitled &quot;CUBIRUNG DER TETRAEDER&quot;.  it seems to me like an important note since Stackel carefullty comments on it. So can anyone tell something about that?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2017-06-23T10:13:58.523" LastActivityDate="2017-06-23T10:13:58.523" Title="What was C.F Gauss's work on the problem of the volume of tetrahedron in hyperbolic geometry?" Tags="|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6226" PostTypeId="2" ParentId="2110" CreationDate="2017-06-22T20:49:53.850" Score="-1" Body="&lt;p&gt;First because of Herschel, second because of the designers of pocket calculators. Unfortunately many pocket calculators give $sin^{-1}(x) = arcsin(x)$.&#xA;That is incorrect. Also &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_trigonometric_identities&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; has it wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$sin^{-1}(x)$ means $\frac{1}{sin(x)}$ like $sin^{2}(x)$ means $sin(x)\cdot sin(x)$. Otherwise it would be highly inconsistent to write $sin^{2}(x) + cos^{2}(x) = 1$, as also Wikipedia does: $sin^2(x)$ = $(sin(x))^2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The inconsistency of Herschel's mis-design is also acknowledged by this &lt;a href=&quot;https://en.wikipedia.org/wiki/Inverse_trigonometric_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further it would be confusing to write $sin(x)^{n}$, in particular $sinx^{n}$ if the parentheses are not written as is often done.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-22T20:55:32.563" LastActivityDate="2017-06-22T20:55:32.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6227" PostTypeId="1" AcceptedAnswerId="6228" CreationDate="2017-06-19T03:11:48.217" Score="7" ViewCount="337" Body="&lt;p&gt;This is not a question about chemistry as a science, but as a school discipline.&lt;br&gt;&#xA;Why is it that Mendeleev as a person is barely mentioned in chemistry courses compared to all the others, like Coulomb, Bohr, Planck, Heisenberg, Boyle, Avogadro, etc.? His story is quite inspiring and I think we should all learn it while in school, not by watching a YouTube video (as was my case). In my layman's view, Mendeleev performed a thought jump (can't even call that an insight) equivalent to that of Einstein's special relativity when coming up with the concept of the periodic table. Today, we take the table for granted, but how would someone even begin to think of such a concept back when we knew so little about chemistry?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this some left over from the Cold War, because Mendeleev was Russian? Or something similar to Newton vs Leibniz, or maybe even Edison vs Tesla?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not believe the question I have asked is completely subjective, because I would assume some chemistry teachers to be present. Mendeleev's contributions to chemistry are indisputable, yet, we learn so little about him. &lt;/p&gt;&#xA;" OwnerUserId="5950" OwnerDisplayName="FinnTheHuman" LastEditorUserId="4251" LastEditDate="2020-06-08T10:47:32.577" LastActivityDate="2020-06-08T10:47:32.577" Title="Why do we learn little about Mendeleev when compared to other science figures?" Tags="|chemistry|mendeleev|chemists|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6228" PostTypeId="2" ParentId="6227" CreationDate="2017-06-19T08:42:41.827" Score="6" Body="&lt;p&gt;I share the same sentiments as well. Actually &lt;strong&gt;Dimitri Mendeleev,&lt;/strong&gt; is still one of my favourite Chemists (actually the best).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leaving everything non-chemistry aside, he made one of the most astonishing discoveries in chemistry. As the &lt;a href=&quot;http://www.rsc.org/education/teachers/resources/periodictable/pre16/develop/mendeleev.htm&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; article states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Good scientists discover new information and make sense of it, linking&#xA;  it to other data. They may go further by giving an explanation of this&#xA;  linked data which, maybe not immediately, other scientists accept as a&#xA;  correct explanation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And goes on further..&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;However the outstanding scientist goes further in predicting&#xA;  consequences of his ideas which can be tested. This boldness&#xA;  identifies the great scientist if the predictions are later found to&#xA;  be accurate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which really agrees with idea that he is one of the greatest. This true prediction later formed the foundations of various branches of chemistry including periodicity, much of inorganic chemistry and physical chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess the reason memories of him quickly vanish, is because his concepts are &lt;strong&gt;taught at at very basic or entry of chemistry&lt;/strong&gt; and as humans we tend to forget and quickly become preoccupied with more difficult stuff as we further study chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes &lt;strong&gt;its just preference&lt;/strong&gt;, for example my chemistry teacher's favourite scientist/chemist was &lt;strong&gt;Henry Louis Le Châtelier&lt;/strong&gt; while my medicinal chemistry teacher's favourite was &lt;strong&gt;Kekulé&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus all scientist who made contribution to the field of chemistry all deserve equal recognition and honour as chemistry would never be the same without them&lt;/p&gt;&#xA;" OwnerDisplayName="xavier_fakerat" LastActivityDate="2017-06-19T08:42:41.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6229" PostTypeId="2" ParentId="2465" CreationDate="2017-06-23T09:19:36.843" Score="5" Body="&lt;p&gt;Interestingly, the shift in physics appears to have occured late in the 20th c. while  philosophers were already discussing it as an exemple. Max Jammer's book &lt;em&gt;Concepts of Mass&lt;/em&gt; remains an invaluable source here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In philosophy it is well known one word used by two authors may be taken either to mean the same  thing ('stuff') or the same role ('function'), (e.g. 'soul' used by Christian authors and Greek thinkers). That is what happened with the term mass and in order to sort out the confusion one has to distinguish: 'classical' mass(1) in Newton and 'rest'(2) or 'relativistic'(3)  mass after Einstein. It appears that (1) and (2) are the same thing while (1) and (3) play the same role.  Philosophical debates about paradigm shifts after Kuhn's book hinge on such issues.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A paper from 1976 (Eriksen and Voyenli, “The Classical and Relativistic Concepts of Mass,” Found. Phys. 6, 115–24 (Feb.1976)) pointed that there is an invariant mass in the old and new theories but only in the old theory a sum  is conserved. In Relativistic physics it is the sum of relativistic masses that is conserved and this is the new role.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In language use sometimes a shift in semantics is prefered to a neologism. In the 70s &lt;em&gt;a computer&lt;/em&gt; was a device filling half of your basement but later appeared &lt;em&gt;personal&lt;/em&gt; computers; today  an unqualified  computer is a PC while descriptions are supplied for the other kind. Inversely &lt;em&gt;relativistic&lt;/em&gt; mass did not become so widespread as to became simply mass and contemporary writers obviously prefer to preserve the semantics and to use the new desription when it is needed.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-06-23T09:19:36.843" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6230" PostTypeId="1" AcceptedAnswerId="6231" CreationDate="2017-06-23T09:25:08.057" Score="4" ViewCount="321" Body="&lt;p&gt;I am interested in a timeline for the history of measuring the mass of the Earth from the first estimates to the currently accepted value, and in references containing more details.&lt;/p&gt;&#xA;" OwnerUserId="3915" OwnerDisplayName="Arnold Neumaier" LastActivityDate="2017-06-23T15:22:48.510" Title="History of measuring the mass of the Earth" Tags="|experimental-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6231" PostTypeId="2" ParentId="6230" CreationDate="2017-06-23T15:17:09.177" Score="2" Body="&lt;p&gt;Newton's gravity law says that acceleration of gravity on the Earth surface&#xA;is $\gamma Mr^{-2}$, where $\gamma$ is the gravitational constant, $M$ is the mass of&#xA;the Earth, and $r$ is the radius of the Earth. Radius is easy to measure (by astronomical means) and the acceleration is also easy to measure. One can use &#xA;instead the distance to the Moon and Moon's acceleration, for which you only need astronomical observations, which are usually more precise than any measurement&#xA;on the Earth). This shows that&#xA;the problem of measuring the mass is EQUIVALENT to the problem of finding the gravitational constant: once you measure one of them you obtain the other. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The gravitational constant was measured for the first time directly by Cavendish (1798)&#xA;with the result: $6.754×10^{−11} m^3⋅kg^{−1}⋅s^{−2}$.&#xA;(Modern value is $6.672×10^{−11} m^3 kg^{−1} s^{−2}$. So Cavendish's error was $1.2\%$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually Cavendish actually stated his result in terms of the mass of the Earth,&#xA;(or the average density of the Earth) rather then in terms of the gravitational constant. There is no much to say about measurements between the time of Cavendish and present (see Wikipedia (&quot;gravitational constant&quot;) which explains why it is difficult to measure). I do not think there is another reliable method of accurate measurement of the  mass of the Earth, except the direct measurement of the gravity constant.&#xA;(Any method based on the knowledge of what is the Earth made of will be much less accurate. Actually it works in the opposite direction: we guess the composition of the Earth using the knowledge of its mass).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-06-23T15:22:48.510" LastActivityDate="2017-06-23T15:22:48.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6232" PostTypeId="1" AcceptedAnswerId="6236" CreationDate="2017-06-23T17:28:50.973" Score="3" ViewCount="255" Body="&lt;p&gt;The Wikipedia article for &lt;a href=&quot;https://en.wikipedia.org/wiki/Cassini_oval&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cassini ovals&lt;/a&gt; claims in the introduction that &quot;Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval.&quot; This claim doesn't have an associated citation, but it appears that Wikipedia may have gotten it from &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cassinian.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this website&lt;/a&gt;, which doesn't cite any sources. Thus, my question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does there exists a reliable source (i.e., one with citations tracing back to a primary source) which backs up that claim that Cassini attempted to prove that the Sun traveled around the Earth in a Cassini oval?&lt;/p&gt;&#xA;" OwnerUserId="5505" LastActivityDate="2022-08-01T10:47:25.850" Title="Is there any evidence supporting this claim about Cassini and his ovals?" Tags="|reference-request|astronomy|geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6233" PostTypeId="1" AcceptedAnswerId="6239" CreationDate="2017-06-23T19:41:09.133" Score="3" ViewCount="182" Body="&lt;p&gt;I've seen conflicting statements about whether the Stern-Gerlach experiment involved an &lt;em&gt;oven&lt;/em&gt; to evaporate a sample of silver or a &lt;em&gt;filament&lt;/em&gt;, perhaps a filament made of silver. Is it possible that the latter is the truth, but the former is discussed in pedagogical treatments because we often think of thermodynamic concepts like blackbody radiation in terms of thought experiments involving ovens?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I unfortunately haven't had any luck finding such details from the primary sources, both because I can't access all the original papers (there were more than one) and because I don't read German.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-06-25T19:09:56.097" Title="Stern-Gerlach: oven, or filament?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6234" PostTypeId="2" ParentId="6232" CreationDate="2017-06-24T08:20:24.533" Score="4" Body="&lt;p&gt;In his article &lt;em&gt;Traité de la comete, qui a paru en Decembre 1743&lt;/em&gt; (see &lt;a href=&quot;https://books.google.de/books?id=wdYq3blEl7cC&amp;amp;pg=PA9&amp;amp;dq=Giovanni%20Cassini%20%20ovale&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwihvfHe_9XUAhWD1xQKHUHHAtgQ6AEIJjAA#v=onepage&amp;amp;q=Giovanni%20Cassini%20%20ovale&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) G. Cassini writes that the orbits of comets and planets around the sun are ovals. At the end of this paragraphe he writes that the mathematicians call  these ovals ellipses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The claim that Giovanni Cassini used his ovals for describing the orbits of planets was made by his son Jean-Dominique on p. 149 of his &lt;em&gt;&lt;a href=&quot;http://www.e-rara.ch/doi/10.3931/e-rara-19698?lang=fr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements d'Astronomie&lt;/a&gt;&lt;/em&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Since the exact observations of the apparent size of the diameter of the sun, my father has found another curve different from the ellipse that may serve as representing rather precisely the true movement of the sun&quot; (Depuis l'observation exacte de la grandeur apparente des diamètres du soleil, mon père a trouvé une autre courbe différente de l'ellipse, qui sert à représenter fort exactement les mouvements vrais du Soleil).&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2017-06-24T08:20:24.533" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6235" PostTypeId="2" ParentId="1965" CreationDate="2017-06-24T14:18:30.483" Score="1" Body="&lt;p&gt;It will be rather difficult if not impossible to find the first one who discovered a difference between infinities. But it is clear that this one was not George Cantor (1845-1918). He came much later. (Cantor merely devised a certain, rather arbitrary, tool, namely the one-to-one correspondence or bijection, to base his theory upon it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very old source is Robert Grosseteste (1168-1253) who said that actual infinite is definite. There are more moments in a long time interval than in a short one. The number of points in a segment one ell long is its true measure.&#xA;&lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&lt;/a&gt; p. 106&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Blaise Pascal (1623-1662) taught the existence of the three orders: Infinitely small, finite, and infinitely large (and applied it to body, mind, and God).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gottfried Wilhelm Leibniz (1646-1716) also distinguished three grades of infinity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cantor himself mentions Bernard de Fontenelle (1657-1757) who invented actual infinite numbers. (G. Cantor, letter to A. Schmid, 26 March 1887, translated in &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&lt;/a&gt; p. 106)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We know that Leonhard Euler (1707-1783) accepted different infinities. $a/dx^2$ quantitas infinita infinities maior quam $a/dx$ (the first term is a quantity infinitely many infinities larger than the second). (W. Mückenheim: Die Geschichte des Unendlichen, 7. ed, Maro, Augsburg, p. 50)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even way before Cantor Bernard Bolzano (1781-1848) distinguished infinities, for instance there are twice as many foci of ellipses than centres of ellipses. There are infinitely many more diameters of circles that centres of circles.&#xA;(J. BERG (ed.): Bernard Bolzano, Wissenschaftslehre §§ 1-45, Friedrich Frommann Verlag, Stuttgart (1985), Bolzano-Gesamtausgabe, Reihe I Band 11,1, p. 31ff)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is merely a short list, by no means complete, but sufficient to show that Cantor was not the first to distinguish different infinities.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-06-24T22:12:55.477" LastActivityDate="2017-06-24T22:12:55.477" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6236" PostTypeId="2" ParentId="6232" CreationDate="2017-06-24T17:18:32.150" Score="5" Body="&lt;p&gt;The history of Cassini's curve is summarized by Curtis Wilson, along with citations to seventeenth-century sources, in his chapter &lt;em&gt;'Predictive Astronomy in the Century after Kepler&quot;&lt;/em&gt;, in &quot;General History of Astronomy&quot; vol. 2A, eds R Taton &amp;amp; C Wilson (Cambridge, 1989). Wilson's account includes this about the eldest Cassini and his curve: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;... around 1690 he [Cassini] invented the &lt;em&gt;cassinoid&lt;/em&gt;, with the aim of obtaining a possible orbit for the planets in which the superior focus would serve as equant point. The hypothesis was first mentioned in 1691 by Jacques Ozanam in his &lt;em&gt;Dictionnaire mathématique&lt;/em&gt;, and was desribed by Cassini in 1693 in his treatise &lt;em&gt;&quot;Sur l'origine et le progrès de l'astronomie&quot;&lt;/em&gt;. In the cassinoid the product of the distances from any point to the two foci is a constant.&quot; Curtis Wilson's chapter goes on to recount the reaction of astronomical contemporaries to the 'cassinoid' as a pattern for planetary orbits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cassini aimed to arrive at a model for the orbits from which the angles could be calculated directly for a given point in time, without guesswork or approximation. The background to this was one of some dissatisfaction, during the later 17th century, with Kepler's hypothesis of ellipses and constant areal-velocities. Such dissatisfaction may be hard to appreciate now, when much of the talk of Kepler for almost three centuries has been about &quot;Kepler's laws&quot;. An early source that sums up contemporary complaints about Kepler's hypotheses is a letter by Nicolas Mercator that appeared in &lt;em&gt;Philosophical Transactions&lt;/em&gt; (of the Royal Society) in 1670, vol.5, pp.1168-1175.  Mercator wrote (at p.1174,in Latin, translated here): &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;No-one has been found up to now who would deny that Kepler's areas can satisfy the appearances; but since neither he himself nor anyone after him could determine them by direct calculation, some have criticized Kepler for having parted ways with geometry while indulging too much in [speculation about] physical causes.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It appears that mathematical conservatives of that time required proofs that went directly to a final certain result by successive valid steps of demonstration, each leading to a true intermediate statement. Those who thought like this would not accept that truth could emerge from a succession of approximations, thus, a succession of errors or at best of uncertainties. Kepler himself had admitted that to find the orbital angles for a given time required either guesswork or approximation, e.g. a series of approximations: (Kepler, &lt;em&gt;'Epitome Astronomiae Copernicanae'&lt;/em&gt;, book 5 (1621), pt 2, at p.695, &quot;Hic via directa nulla est...&quot; ('Here there is no direct way....'), see also N M Swerdlow, 'Kepler's Iterative Solution to Kepler's Equation', Journal for the History of Astronomy v.31 (2000) pp.339-341.) More about the background can be found in another paper by Curtis Wilson: &quot;From Kepler's laws, so-called, to universal gravitation: Empirical factors&quot;, Archive for History of Exact Sciences 1970, Volume 6, Number 2, pp.89-170.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-06-24T17:18:32.150" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6237" PostTypeId="1" AcceptedAnswerId="6286" CreationDate="2017-06-24T19:13:01.687" Score="3" ViewCount="144" Body="&lt;p&gt;G. Waldo Dunnington in his biography of Gauss states (on p. 165):&quot;only a portion of his results in the field of curved surfaces was presented in the Disquisitiones generalis, and a second memoir on the subject was planned.&quot; I must mention in adddition that i found an enormous amount of material on these subjects in volumes 8 and 9 of his works - the titles of the various manuscripts range from geodesic lines on ellipsoid, the figure of the earth (&quot;erdellipsoid&quot;), and lots of kinds of conformal mappings. It came as a surprise to me that i didn't find any summaries of these writings, and as far as i know Gauss, the probability that the importance of these writings is neglible is pretty small. I didnt even find a concise evaluation of his work in theoretical geodesy (like his memoirs from 1843 and 1846). So my questions is mainly to get sources - if anyone knows good articles about the subjects mentioned, please add it. &lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2017-07-11T18:27:59.627" Title="Summary of Gauss's contributions to differential geometry, conformal mappings, and geodesy" Tags="|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6238" PostTypeId="2" ParentId="6189" CreationDate="2017-06-24T21:25:52.833" Score="7" Body="&lt;p&gt;I came across an explanation of this in a paper that Nick R pointed out, Schmidt-Boecking et al., &quot;The Stern-Gerlach Experiment Revisited,&quot; &lt;a href=&quot;https://arxiv.org/pdf/1609.09311.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://arxiv.org/pdf/1609.09311.pdf&lt;/a&gt; .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein and Ehrenfest apparently reasoned as follows. Before a particular silver atom enters the magnetic field, its magnetic moment $\mu$ is randomly oriented. Once it enters the magnetic field, it has an energy $\mu\cdot B$. Unless there is a mechanism for the transfer of energy in or out of the atom, this energy can't change, and therefore the magnetic moment can only precess about the $B$ vector, but it can't change its orientation with respect to it. One mechanism for energy loss is Larmor radiation, but this is very weak and would take 100 years to have enough of an effect to produce alignment -- many orders of magnitude too long for the times of flight of the atoms in the actual apparatus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what was missing conceptually was the idea that a single atom, before entering the field, could be in a superposition of two different states with the magnetic moment oriented parallel to or antiparallel to the field.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-06-24T21:25:52.833" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6239" PostTypeId="2" ParentId="6233" CreationDate="2017-06-25T19:09:56.097" Score="7" Body="&lt;p&gt;The following diagram of the original setup is from Stern's private slide collection and was included in Stern and Gerlach's 1924 paper &lt;em&gt;Über die Richtungsquantelung im Magnetfeld. Annalen der Physik, 74:673–699&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It shows an oven which is heated by a filament. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This image comes from page 12 of arXiv paper &lt;a href=&quot;https://arxiv.org/pdf/1609.09311.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Stern-Gerlach Experiment Revisited&lt;/em&gt;&lt;/a&gt; by Schmidt-Böcking, et al.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1Mgyl.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1Mgyl.jpg&quot; alt=&quot;SGE setup&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-06-25T19:09:56.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6240" PostTypeId="1" CreationDate="2017-06-25T22:07:03.020" Score="4" ViewCount="418" Body="&lt;p&gt;Hadamard's lemma, in one dimension, says for any smooth function $f \colon \mathbf R \rightarrow \mathbf R$ there is a first-order expansion of $f$ at $0$: $f(x) = f(0) + xg(x)$ where $g \colon \mathbf R \rightarrow \mathbf R$ is smooth. Explicitly, $g(x) = \int_0^1 f'(tx)\,dt$. (The analogue in higher dimensions is on the Wikipedia page for Hadamard's lemma.) &#xA;My question is: where/when did Hadamard first present this result, or is it not due to him at all (Stigler's law)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I looked through the four volumes of Hadamard's &lt;em&gt;Oeuvres&lt;/em&gt; and the only thing I could find resembling this result was in his 1892 PhD thesis on analytic functions. Sections 32-35 contain some integrals of the form $\int_0^1 V(t)f(tx)\,dt$ for different choices of $V(t)$.  Perhaps these motivated someone else (Whitney?) later on when the general foundations of manifolds were laid down in the 20th century.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2019-05-11T16:04:49.893" Title="First appearance of Hadamard's lemma on smooth functions" Tags="|differential-geometry|real-analysis|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6241" PostTypeId="1" AcceptedAnswerId="6243" CreationDate="2017-06-26T01:24:36.853" Score="0" ViewCount="264" Body="&lt;p&gt;Einstein began working  on unified field theory at a time when the weak and strong force were perhaps not known I believe...but as the years proceeded and the weak and strong force were being outlined why was it he missed these as necessary components to complete his field theory.  I guess what was known when is very difficult to ascertain?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4995" LastEditDate="2017-06-28T21:06:36.967" LastActivityDate="2017-06-28T21:06:36.967" Title="Why didn't Einstein try to include the strong and weak forces in his attempts at a unified theory?" Tags="|einstein|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6243" PostTypeId="2" ParentId="6241" CreationDate="2017-06-26T16:24:24.703" Score="3" Body="&lt;p&gt;You can unify some of the forces of nature without unifying others. That's what's happened historically. Maxwell unified electricity and magnetism. In the 1970s we unified the electromagnetic and electroweak forces. Einstein was trying to unify the electromagnetic and gravitational forces. The discovery of forces A and B doesn't imply that we shouldn't try to unify forces C and D.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also that Einstein was working on classical unified theories, but you aren't going to get a classical theory of the weak force because the W and Z are unstable.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2017-06-27T13:50:31.937" LastActivityDate="2017-06-27T13:50:31.937" CommentCount="0" CommunityOwnedDate="2017-06-27T13:48:03.583" ContentLicense="CC BY-SA 3.0" />
  <row Id="6244" PostTypeId="1" AcceptedAnswerId="6249" CreationDate="2017-06-26T20:11:56.663" Score="6" ViewCount="2593" Body="&lt;p&gt;The question &lt;a href=&quot;http://hsm.stackexchange.com/questions/6227/why-do-we-learn-little-about-mendeleev-when-compared-to-other-science-figures&quot;&gt;&quot;Why do we learn little about Mendeleev when compared to other science figures?&quot;&lt;/a&gt; let me ask: Why do we learn next to nothing about Julius &lt;a href=&quot;https://en.wikipedia.org/wiki/Julius_Lothar_Meyer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lothar Meyer (1830-1895)&lt;/a&gt; who in parallel to Mendeleev, found the periodic system?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="4251" LastEditDate="2020-06-21T13:48:29.140" LastActivityDate="2020-06-21T13:48:29.140" Title="Why is Mendeleev credited with the discovery of the periodic table much more often than Meyer?" Tags="|chemistry|mendeleev|chemists|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6245" PostTypeId="2" ParentId="5928" CreationDate="2017-06-27T06:30:51.493" Score="0" Body="&lt;p&gt;I believe the desired answer is &lt;a href=&quot;https://en.wikipedia.org/wiki/Jules_Richard&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Jules Richard&lt;/strong&gt;&lt;/a&gt; and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the &lt;strong&gt;&lt;em&gt;Richard&lt;/em&gt;&lt;/strong&gt; &lt;strong&gt;antinomy&lt;/strong&gt; (all things which form objects of thought are defined through finitely many words, therefore denumerable; on the other hand the set of all real numbers is already nondenumerable)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The above is a translation by W. Boos (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1746503&quot; rel=&quot;nofollow noreferrer&quot;&gt;1995&lt;/a&gt;, p. &lt;a href=&quot;https://books.google.com/books?id=kmKlBwAAQBAJ&amp;amp;pg=PT343&amp;amp;dq=%22Richard+antinomy%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;285&lt;/a&gt;) of the &lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?41.0089.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Jahrbuch review&lt;/em&gt;&lt;/a&gt; of Hermann Weyl’s &lt;em&gt;Habilitationsvortrag&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=U0k6AQAAMAAJ&amp;amp;dq=%22Richardschen+Antinomie%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;). It is an oversimplified paraphrase of both Weyl and Richard’s &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;original paradox&lt;/a&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Jules_Richard#Richard.27s_Paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;), which was technically precise and about “numbers”, not “things”. Also,  like  &lt;a href=&quot;https://hsm.stackexchange.com/questions/5928/who-has-invented-this-enumeration-of-all-objects-of-thought#comment9684_5928&quot;&gt;@DaveLRenfro&lt;/a&gt; it replaces the OP’s “rational spatio-temporal coordinates” by “verbal definitions”. Last twist, Boos &lt;a href=&quot;https://books.google.com/books?id=kmKlBwAAQBAJ&amp;amp;pg=PT347&amp;amp;dq=review&quot; rel=&quot;nofollow noreferrer&quot;&gt;attributes&lt;/a&gt; the quoted review to Skolem, but the online Jahrbuch says Salkowski! &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-06-27T06:51:55.693" LastActivityDate="2017-06-27T06:51:55.693" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6246" PostTypeId="1" CreationDate="2017-06-27T12:01:29.517" Score="3" ViewCount="716" Body="&lt;p&gt;I am aware that perhaps the earliest source concerning problems of maximum and minimum values occurs in Euclid's &lt;em&gt;Elements&lt;/em&gt;. After Euclid, Archimedes of Syracuse and Apollonius of Perga seem to consider such problems in their treatises, but not with an extensive study of them.&#xA;But after them, it seems that no mathematician gets interested in the realm till in the 17th Century A.D., Pierre de Fermat writes a paper to describe a method for solving problems of maxima and minima.[ See this paper: &lt;a href=&quot;http://science.larouchepac.com/fermat/fermat-maxmin.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://science.larouchepac.com/fermat/fermat-maxmin.pdf&lt;/a&gt; ]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this paper, written by Fermat, makes me believe that there is still some previous activity regarding the study of maxima and minima. The degree of sophistication and certitude of this &lt;em&gt;new style of problems&lt;/em&gt; concerning maximum and minimum properties of quantities is undeniably built on some Giant's shoulders.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore I am curious if there was any mathematician in the middle or mid-late Renaissance time who contributed to this field. Precisely, I should say I am searching for a &lt;strong&gt;timeline&lt;/strong&gt; of the study of maxima and minima, especially in the middle ages and in the Renaissance but before Fermat. However, I will still appreciate any related information regarding it.&lt;/p&gt;&#xA;" OwnerUserId="5359" LastActivityDate="2017-06-27T12:06:19.593" Title="History of the origins and development of problems of finding maximum and minimum values of quantities" Tags="|mathematicians|biographical-details|reference-request|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6247" PostTypeId="2" ParentId="6246" CreationDate="2017-06-27T12:06:19.593" Score="1" Body="&lt;p&gt;Pappus of Alexandria, Viete, and Diophantus.  See &lt;a href=&quot;http://dx.doi.org/10.1162/POSC_a_00101&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt; for details.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-06-27T12:06:19.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6249" PostTypeId="2" ParentId="6244" CreationDate="2017-06-29T20:43:01.503" Score="13" Body="&lt;p&gt;The timeline is well-known, even Wikipedia has a specific article on &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_periodic_table&quot; rel=&quot;noreferrer&quot;&gt;History of the Periodic Table&lt;/a&gt;. Several people suggested some version of the periodic law in 1860-s before Mendeleev, including Meyer (1864), Newlands (1863-1867), Odling (1864) and Hinrichs (1867). Meyer's original table only had 44 elements, Odling's had 57. Meyer's expanded table was only published in 1870, a year after Mendeleev's, and Mendeleev sent a copy of his to Meyer, among many others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the timeline is not the crux of the story. When the law of gravity is mentioned it is typically with Newton's name attached, even though Bullialdus, Hooke, Wren and Halley suggested the inverse square law before Newton, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/1973/who-was-first-to-explain-intuitively-the-inverse-square-law-of-gravity/1975#1975&quot;&gt;Who was first to explain intuitively the inverse square law of gravity?&lt;/a&gt; Not only was Meyer's original table incomplete but he was also reticent about it, talking of only a &quot;possibility of periodical law&quot;. Mendeleev, on the other hand, did not hesitate to change atomic weights when they did not fit (for which Meyer criticized him), and predict new elements when a spot was empty. Soon his changes and predictions were confirmed, as a result Mendeleev's table turned from a mere classification tool to tested predictive theory. So Mendeleev, like Newton, did not just suggest it, he made people believe it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-06-29T20:57:13.300" LastActivityDate="2017-06-29T20:57:13.300" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6251" PostTypeId="2" ParentId="6223" CreationDate="2017-06-29T23:20:12.737" Score="2" Body="&lt;p&gt;Modern physics without the contribution of scientists who happened to be muslims is absolutely inconceivable. The most obvious example: algebra was invented by a Persian who happened to be a muslim.  Same guy described the decimal number system.  Physics would be impossible without both - and it is a historical fact that Europe learned both from him.  There are many other examples of great contributions which were coopted by Europe without crediting the original source.  This is neither surprising nor nefarious, given the historical competition between Chrstiandom and the House of Islam - it's the way the world works.&lt;/p&gt;&#xA;" OwnerUserId="4062" LastEditorUserId="12699" LastEditDate="2017-06-30T10:59:16.137" LastActivityDate="2017-06-30T10:59:16.137" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="6252" PostTypeId="2" ParentId="6223" CreationDate="2017-07-01T20:43:59.663" Score="1" Body="&lt;p&gt;There were a number of &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_optics#Optics_and_vision_in_the_Islamic_world&quot; rel=&quot;nofollow noreferrer&quot;&gt;optics&lt;/a&gt; contributions, including Snell's law and curved mirrors and lenses.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2017-07-01T20:43:59.663" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6254" PostTypeId="2" ParentId="6223" CreationDate="2017-07-02T20:55:48.277" Score="8" Body="&lt;p&gt;Science is the result of endeavors of many nations and races and its very difficult to say how much each nation has contributed. So I just mention a few examples of what has been accomplished by scientists who happened to be Muslim. This obviously is not a comprehensive list, just a few examples.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;One example is the theory of evolution. We all know that Darwin did many observations and developed the theory in 18th century. However, Nasir al-Din Tusi, a Persian scientist, wrote a book with very similar ideas based on some of his observations 600 years before Darwin. The story is explained here: &lt;a href=&quot;http://www.thevintagenews.com/2016/08/27/priority-first-theory-evolution-600-years-older-darwin/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The first theory of evolution 600 years older than Darwin&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another example is what we usually know as: Pascal's triangle. The triangle was well known several centuries earlier by Persian mathematicians such as Al-Karaji (953–1029) and Omar Khayyám (1048–1131) and later by Chinese mathematician Yang Hui (1238–1298). See here:&lt;a href=&quot;https://en.wikipedia.org/wiki/Pascal%27s_triangle&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Pascal%27s_triangle&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Yet other example is the development of Algebra. For example in &lt;a href=&quot;https://en.wikipedia.org/wiki/Algebra#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; it says:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.[14]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He [Al-Khwarizmi] introduced the methods of &quot;reduction&quot; and &quot;balancing&quot; (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,[17] and that he gave an exhaustive explanation of solving quadratic equations,[18] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[19] His algebra was also no longer concerned &quot;with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study&quot;. He also studied an equation for its own sake and &quot;in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems&quot;.[20]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe.[21] Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.[22] He also developed the concept of a function.[23] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[24] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In Chemistry, as it is explained &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_chemistry&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;An early scientific method for chemistry began emerging among early Muslim chemists, beginning with the 9th century chemist Jābir ibn Hayyān (known as &quot;Geber&quot; in Europe), who is considered as &quot;the father of chemistry&quot;.[24][25][26][27] He introduced a systematic and experimental approach to scientific research based in the laboratory, in contrast to the ancient Greek and Egyptian alchemists whose works were largely allegorical and often unintelligble.[28] He also invented and named the alembic (al-anbiq), chemically analyzed many chemical substances, composed lapidaries, distinguished between alkalis and acids, and manufactured hundreds of drugs.[29]&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among other influential Muslim chemists, Abū al-Rayhān al-Bīrūnī,[31] Avicenna[32] and Al-Kindi refuted the theories of alchemy, particularly the theory of the transmutation of metals; and al-Tusi described a version of the conservation of mass, noting that a body of matter is able to change but is not able to disappear.[33] Rhazes refuted Aristotle's theory of four classical elements for the first time and set up the firm foundations of modern chemistry, using the laboratory in the modern sense, designing and describing more than twenty instruments, many parts of which are still in use today, such as a crucible, cucurbit or retort for distillation, and the head of a still with a delivery tube (ambiq, Latin alembic), and various types of furnace or stove.&quot;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As a more modern example: Iranian-American Mathematician and the first female winner of the Fields Medal (the highest honor in mathematics AKA &quot;Noble Prize of mathematics&quot;), Maryam Mirzakhani &quot;thrived in study of curved surfaces such as doughnut shapes and amoebas -- to a degree that other bright minds in the field dared not explore.... Her work could help advance understanding in physics, quantum mechanics and areas outside math, Stanford said in an online news article.&quot; Sadly she died at age 40 today as I'm editing this post. See here: &lt;a href=&quot;http://www.cnn.com/2017/07/15/us/mirzakhani-obituary-first-woman-win-math-prize/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maryam Mirzakhani&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5984" LastEditorUserId="5984" LastEditDate="2017-07-16T01:25:21.733" LastActivityDate="2017-07-16T01:25:21.733" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6255" PostTypeId="2" ParentId="1936" CreationDate="2017-07-02T21:50:57.113" Score="1" Body="&lt;p&gt;According to Bernstein, Dedekind imagined a set like a closed sack containing certain defined things. [Becker: Grundlagen, S. 316] Isn't that an example of pictorial thinking?&lt;/p&gt;&#xA;" OwnerDisplayName="user5739" LastActivityDate="2017-07-02T21:50:57.113" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6256" PostTypeId="1" CreationDate="2017-07-03T13:58:33.590" Score="8" ViewCount="686" Body="&lt;p&gt;In pp. 283–285 of volume 2 of Dickson's “history of the theory of numbers” appear several formulas of striking similarity: some of them are stated by Gauss (p. 283) and some are stated by Jacobi (p. 285); they are actually the same and only the notation differs (&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in Gauss's formula and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; in Jacobi's formula). Gauss's formulas are the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp; (1 + 2y + 2y^4 + 2y^9 + \cdots + 2y^{n^2})^4 \\[6pt]&#xA;= {} &amp;amp; (1 - 2y + 2y^4 - \cdots)^4 + (2y^{1/4} + 2y^{{9/4}} + 2y^{(2n - 1)^2/4})^4&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;&amp;amp; (1 + 2y + 2y^4 + 2y^9 + \cdots + 2y^{n^2})^4 \\[6pt]&#xA;= {} &amp;amp; 1 + 8\left( \frac y {1 - y} + \frac{2y^2}{1 + y^2} + \frac{3y^3}{1 - y^3} + \cdots + \frac{ny^n}{1 + (-1)^n y^n} \right)&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The point is that the last equality means that the coefficients of the &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;th power in the right side of the last equality must be equal to &lt;span class=&quot;math-container&quot;&gt;$r_4(k)$&lt;/span&gt; (number of representations of &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; as sum of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; squares), and an additional interpretation (by certain manipulations) of the right side of the equality gives the result of Jacobi: &lt;span class=&quot;math-container&quot;&gt;$r_4(k) = 8\sigma(k)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$24\sigma(k)$&lt;/span&gt;, depends if &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is odd or even.&lt;/p&gt;&#xA;&lt;p&gt;The only other reference for Gauss's result that i found in the english literature is in the &amp;quot;Sums of Squares&amp;quot; chapter of Ranjan Roy's book &amp;quot;&lt;strong&gt;Elliptic and Modular Functions from Gauss to Dedekind to Hecke&lt;/strong&gt;&amp;quot; (chapter 15, p.386) where the author writes that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Note that Gauss independently discovered (15.2) and (15.3); they were written without proof, and without explicit application to the sums of squares, in an undated manuscript published after his death.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(15.2) and (15.3) in this book are the identities mentioned before in this post. Reading what Ranjan Roy says, several questions arise:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If Gauss didn't write down a proof of Jacobi's identity, why Ranjan Roy states that Gauss &amp;quot;independently discovered&amp;quot; Jacobi's identity? i mean - this is a general question on the authenticity of posthomous papers; in this case, how can we be sure that Gauss didn't simply copy Jacobi's identity after he saw his Fundamenta Nove from 1828? how is authenticity of an historical document confirmed?&lt;/li&gt;&#xA;&lt;li&gt;Looking again into Gauss's manuscript, i found out that in the commentary on it, the editor Schering states it was found in a notebook dated approximately to 1808, while there are no dates on the actual manuscripts. So, along the same lines like the first question, i'd like to know what is the degree of certainty in such &amp;quot;evaluations of dates&amp;quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Additional evidence&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, when an already known result is discovered in a posthomous manuscript, and is not rigorously derived in this manuscript, the only way to argue in favour of an independent discovery of the author of the manuscript is to find new (previously unknown) results in the same direction in this manuscript.&lt;/p&gt;&#xA;&lt;p&gt;So, for the completeness of the discussion, i must add another relevant reference. In the chapter &amp;quot;sum of four squares&amp;quot; of volume 2 of Dickson's work, at page 300, he mentions that the czech mathematician Karel Petr proved two formulas by Gauss (Werke, III,p. 476) on theta functions by the method outlined by Gauss. The point is that K. petr used those identities of Gauss to derive relations giving the number of representations of a number N by three quaternary quadratic forms: &lt;span class=&quot;math-container&quot;&gt;$x^2 + y^2 + 9z^2 + 9u^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2+z^2+9u^2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^2+9y^2+9z^2+9u^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I mention this fact because in light of Karel Petr's results it seems that Gauss-Jacobi identity wasn't an isolated result by Gauss, but was part of a grand plan Gauss had for the subject of analysis, and theta functions, in particular. Unfortunately i'm far from being knowledgable enough in those matters, so i can't make a conclusion about how this additional result sheds light on Gauss's possible derivation of Jacobi's four sqaures theorem (and perhaps several additional arithmetical facts). So i'd like to know an expert opinion on this question.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="-1" LastEditDate="2020-12-06T07:56:15.803" LastActivityDate="2021-01-06T01:56:47.593" Title="Did Gauss know Jacobi's four squares theorem?" Tags="|mathematics|biographical-details|gauss|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6257" PostTypeId="2" ParentId="6223" CreationDate="2017-07-03T15:03:21.390" Score="7" Body="&lt;p&gt;Around the year 1000, &lt;a href=&quot;https://en.wikipedia.org/wiki/Al-Biruni&quot; rel=&quot;noreferrer&quot;&gt;Al-Biruni&lt;/a&gt; measured the side of the Earth. He did it in two steps. First he discovered a way of measuring the height of a mountain. Then, if a person climbed to the top of that mountain, that person could measure the &lt;a href=&quot;https://en.wikipedia.org/wiki/Strike_and_dip&quot; rel=&quot;noreferrer&quot;&gt;dip angle&lt;/a&gt;. Using this data and trigonometry, he obtained that the radius of the Earth is 6,339.6 km (the correct value is 6,356.75 km) and the error he made is less than $2\%$. You will find more details &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_geodesy#Al-Biruni&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will find &lt;a href=&quot;https://www.ijsciences.com/pub/pdf/V220131220.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; more about Al-Biruni's scientific output.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2017-07-04T17:20:00.813" LastActivityDate="2017-07-04T17:20:00.813" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6260" PostTypeId="1" CreationDate="2017-07-04T11:59:33.180" Score="0" ViewCount="1062" Body="&lt;p&gt;In Wikipedia in article about Rota's conjecture it is written: “ A proof of the conjecture has been announced by Geelen, Gerards, and Whittle” . In Quanta magazine in article about mathematician June Huh  by KEVIN HARTNETT “A Path Less Taken to the Peak of the Math World” ,it is declared that Rota’ conjecture is proved by Adiprasito, Huh and Katz&#xA;Can somebody in this community explain who has the priority &lt;/p&gt;&#xA;" OwnerUserId="5993" LastActivityDate="2017-07-04T11:59:33.180" Title="Who first proved Rota' conjecture" Tags="|mathematics|combinatorics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6261" PostTypeId="1" AcceptedAnswerId="6263" CreationDate="2017-07-06T01:15:05.337" Score="1" ViewCount="96" Body="&lt;p&gt;Has the modern logic negation $\lnot$ been adapted from &lt;a href=&quot;https://en.wikipedia.org/wiki/Begriffsschrift&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frege's Begriffsschrift&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5998" LastEditorUserId="131" LastEditDate="2017-07-06T12:54:06.433" LastActivityDate="2017-07-06T12:54:06.433" Title="Has the modern logic negation $\lnot$ been adapted from Frege's Begriffsschrift?" Tags="|notation|mathematical-logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6263" PostTypeId="2" ParentId="6261" CreationDate="2017-07-06T06:44:09.723" Score="1" Body="&lt;p&gt;No. See &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The main symbol for negation wich is used today is $¬$. It was first used in 1930 by &lt;a href=&quot;https://en.wikipedia.org/wiki/Arend_Heyting&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arend Heyting&lt;/a&gt; in “Die formalen Regeln der intuitionistischen Logik,” &lt;em&gt;Sitzungsberichte der preußischen Akademie der Wissenschaften, phys.-math.Klasse&lt;/em&gt;, 1930, p. 42-65 [see page 43].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-07-06T06:44:09.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6266" PostTypeId="1" AcceptedAnswerId="6274" CreationDate="2017-07-08T15:13:27.910" Score="2" ViewCount="268" Body="&lt;p&gt;Leonhard Euler is called the &lt;em&gt;Father of Graph Theory&lt;/em&gt;. He also started off the studies of &lt;em&gt;Calculus of Variations&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which all are the other mathematical/science disciplines that evolved as an independent field of study because of him?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quick Google search did not return any good documents on it, the closest I could find is this &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler#Notes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia Article&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Contributions_of_Leonhard_Euler_to_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;, but both of them doesn't explicitly mention disciplines that were started off by him, far as I read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Also &lt;a href=&quot;https://math.stackexchange.com/q/2350449/45937&quot;&gt;asked in Math.SE&lt;/a&gt;, moved it here since it was suggested that it's more appropriate to hsm.SE)&lt;/p&gt;&#xA;" OwnerUserId="6014" LastActivityDate="2017-07-11T21:57:33.677" Title="Fields of Study Introduced by Leonhard Euler" Tags="|mathematics|physics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6268" PostTypeId="1" AcceptedAnswerId="6273" CreationDate="2017-07-08T16:42:49.283" Score="-1" ViewCount="354" Body="&lt;p&gt;Did the church continue to prosecute scientists for heliocentrism? If not, why not? What changed? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why didn't the church go after Isaac Newton?&lt;/p&gt;&#xA;" OwnerUserId="4095" LastActivityDate="2017-08-14T01:47:30.123" Title="How did heliocentrism survive after Galileo's sentencing?" Tags="|newton|galileo|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6271" PostTypeId="2" ParentId="6268" CreationDate="2017-07-08T20:57:37.603" Score="0" Body="&lt;p&gt;It got...difficult.  At one point, there was a model where all the planets &lt;em&gt;except&lt;/em&gt; Earth revolved around the Sun, but Earth revolved around it's &lt;em&gt;own&lt;/em&gt; sun...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In all honesty, some people still don't believe it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: re: Newton ...  If you read &lt;em&gt;principia&lt;/em&gt; Newton goes out of his way to avoid religion.  As a weird historical sidenote, Newton knew Darwin's father, through the church.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT^2: It was his &lt;em&gt;grandfather&lt;/em&gt; Erasmus Darwin, &lt;em&gt;not&lt;/em&gt; his father Robert Darwin.&lt;/p&gt;&#xA;" OwnerUserId="6016" LastEditorUserId="6016" LastEditDate="2017-07-08T22:47:37.343" LastActivityDate="2017-07-08T22:47:37.343" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6272" PostTypeId="2" ParentId="6268" CreationDate="2017-07-08T21:21:47.133" Score="1" Body="&lt;p&gt;&quot;The church&quot;, as you call it, had nothing to say in Protestant England.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-07-08T21:21:47.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6273" PostTypeId="2" ParentId="6268" CreationDate="2017-07-08T21:52:27.100" Score="4" Body="&lt;p&gt;This will answer two out of three parts of the question: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) 'Why didn't the church go after Isaac Newton?' It was not at all the whole church that was involved in the Galileo affair: it was the establishment of the Roman Catholic church of the time. In much of (mostly northern) Europe, the Roman Catholic church had no authority at all: the reformed churches predominant there did not acknowledge the Pope (and usually disagreed with what was done in his name). The church of England was one of the reformed churches, roughly speaking, and was generally anti-Pope, so that Newton was in no danger of suffering the same thing as was done in the Galileo affair. Evidence to show how the reformed churches were completely independent of Rome and generally opposed to what the Pope did can be seen, for example, in sources that show how the reformed churches even refused for a long time to accept the (Roman, papal) Gregorian reform of the calendar: see M Hoskin, 'The reception of the calendar by other churches', pp.255-264 in 'Gregorian Reform of the Calendar', (conference proceedings ed. G V Coyne et al., 1983).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope others with better familiarity than mine with the Galileo affair will answer the part of the question referring to 'prosecution for heliocentrism', because from what I recall of careful historical accounts I believe it can be questioned whether what happened was simply a 'prosecution for heliocentrism'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) 'Did the church continue to prosecute scientists for heliocentrism?' In any event, whatever the details of the Galileo affair, there was little or no obstacle for those under Roman Catholic authority who wished to discuss heliocentrism or Newtonianism. Evidence of that can be seen in both the existence and the content of a well-known annotated edition of Newton's 'Principia' produced in 1740-42 by two Roman Catholic priests who were also teachers of mathematics: Fathers Thomas Le Seur and Francis Jacquier (members of the 'Minim' monastic order, but their edition was sometimes mistakenly referred to as the 'Jesuit edition'). At the start of the third book of the Principia they inserted a declaration, stating in effect that it was not possible to discuss the book without speaking with Newton's voice. For their own part they accepted the papal decrees against the motion of the earth. And that was it: it can be seen from the content of the book that after making their brief statement they just carried on giving the text and discussing it in the notes as if there was nothing against it. (Their declaration can be seen here, along with the rest of the volume, in a digital scan of an 1822 reprint of their edition -- their edition was valued for a long time for its annotations and commentaries: &lt;a href=&quot;https://archive.org/stream/philosophiaenatu03newtuoft#page/n16/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/stream/philosophiaenatu03newtuoft#page/n16/mode/1up&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2017-07-08T22:29:59.993" LastActivityDate="2017-07-08T22:29:59.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6274" PostTypeId="2" ParentId="6266" CreationDate="2017-07-08T22:02:11.950" Score="3" Body="&lt;p&gt;There are many areas of science in which Euler can be called a founding father, though in most cases some predecessors can be found:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics and mechanics: Calculus of variations (Newton and Bernoulli were predecessors), fluid dynamics (again Newton and Bernoulli can be counted as predecessors), rotation of a rigid body, elliptic integrals, use of generating functions in number theory, the theory of partitions, elasticity theory, complex function theory. He also discovered Gamma, Beta, zeta and hypergeometric functions, &#xA;and possibly Bessel functions, which qualifies him as the father of the Special function theory. (I do not even mention that he discovered the exponential function of a complex argument too). Euler is also a founding father of  holomorphic dynamics which is very fashionable nowadays: he was the first to&#xA;study systematically the iteration of the exponential functions (though here too he had a predecessor, marquis de Condorcet). 90% of the modern undergraduate course of Ordinary Differential Equations is also due to Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In physics and engineering: interior ballistics (no predecessors that I know),&#xA;perturbation theory in celestial mechanics (predecessor: Newton).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-11T21:57:33.677" LastActivityDate="2017-07-11T21:57:33.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6275" PostTypeId="1" CreationDate="2017-07-09T07:44:50.857" Score="1" ViewCount="152" Body="&lt;p&gt;My question is based on the information that appears in several websites, according to which I found three examples of a kind of instrument that was probably an analog computer. The first one mentioned, which is from the great soviet encyclopedia (so I suspect it might not be true), is a kind a integrator, i.e, a machine that can solve differential equations. The second instrument is from wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Tractrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and it's supposedly a drawing machine. The third one is reported by Couturat and was a kind of an algebraic calculator, i.e, a calculator that can perform algebraic operations. And I have to mention another &lt;a href=&quot;http://web.mit.edu/STS.035/www/PDFs/Newell.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt; which states:&#xA;&quot;Many other elementary analog devices were described before the&#xA;modern period: Differential gears (Figure 1). used for adding&#xA;or subtracting two variables. arc usually ascribed to&#xA;Leonardo da Vinci: and Leibniz is credited for the idea late&#xA;in the seventeenth century of a &lt;strong&gt;similar-triangles device for&#xA;equation solving or root solving&lt;/strong&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So can anyone offer explanation of the underlying principle of this instruments? and mayby even offer explanation of their inner-workings?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10106" LastEditDate="2020-05-12T19:15:41.510" LastActivityDate="2020-05-12T19:15:41.510" Title="Did Leibniz also design analog computers?" Tags="|biographical-details|computers|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6276" PostTypeId="1" CreationDate="2017-07-09T09:41:08.203" Score="3" ViewCount="309" Body="&lt;p&gt;Basically, the square root operation was discovered and proved rigorously from the Pythagorean theorem, it was denoted by square root of a rational number say $n$ as $\sqrt{n}$, but at a later stage, it used to be denoted by fractional power equals to $1/2$, also for multi-square root operation for say a prime number $p$ as $\sqrt[2^{n}]{p}$ or $p^{2^{- n}}$, for some positive integer $n$, and then this was so generalized to any fraction number &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, the power or exponent operation was basically defined for positive integer numbers, where the number of using the multiplication operation is basically a natural number such that it makes sense, and this was simply generalised to a ricipricical or the inverse of the integer number of power operation to be extended to negative integers &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To illustrate that simply in examples for say $n^3 = n*n*n$, so we used the multiplication operation twice which is an integer number of times&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And generally, we use the multiplication operation as ($k - 1$) times, to express a power integer $n^k$ for $k$ positive integer &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what does it mean when the exponent provided is a fraction number in accordance with the sensible basic definition of power integer?    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does that mean that an integer is multiplied by itself a fractional number of times? which seems meaningless&lt;/p&gt;&#xA;" OwnerUserId="2855" LastActivityDate="2017-07-09T11:18:35.827" Title="Why and who was the first to denote the square root operation in fractional form as $1/2$" Tags="|mathematics|physics|mathematicians|terminology|mathematical-logic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6277" PostTypeId="2" ParentId="6275" CreationDate="2017-07-09T10:57:12.297" Score="1" Body="&lt;p&gt;Yes, Leibniz did design a famous computing machine called &lt;strong&gt;the stepped-reckoner&lt;/strong&gt;. It was &lt;strong&gt;a (digital) computer, but not an analog computer&lt;/strong&gt;. Don't confuse digital with electronic. The difference is only that digital computers use discrete values for computation, whereas analog computers use continuous values, for example real numbers in case of Tractrix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you're mixing up different definitions, and also the question doesn't match the description. I'll try to explain the rest below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A computer can be defined (in an extremely generalized way) as &lt;strong&gt;&lt;em&gt;a machine that performs certain logical (or arithmetical) operations automatically (usually on a given input).&lt;/em&gt;&lt;/strong&gt; Guess what, the foundational proves of Euclidean Geometry, and a lot of geometrical algorithms were written by &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid&lt;/a&gt; for a computer that requires only a ruler and a compass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All the machines that you've mentioned can be described with the above definition, except that the details of their implementation will vary from case to case. Also, this is where the line between computer science and engineering starts arising. A computer scientist may just assume that there is some computer which can perform arithmetical operations. Now he/she might go on to write an algorithm, say to find square roots of a natural number. An engineer on the other hand, will put in his or her expertise to build a machine that can support the requirements for that machine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the &lt;strong&gt;underlying principle&lt;/strong&gt; behind all the computers you've mentioned? &#xA;It is the definition of computers itself- &lt;strong&gt;perform certain logical or arithmetical operations (on an input).&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Explanation of inner workings? Those are specific to different machines and you will have to look into each of them. But if you know what they are doing, it's not that difficult to figure out. For example, try figuring out how one can build a Leibniz's stepped-reckoner for simple addition and subtraction. Watch this animation &lt;a href=&quot;https://www.youtube.com/watch?v=klLB5k3LkwU&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=klLB5k3LkwU&lt;/a&gt; and see if you can explain its inner workings completely. &lt;/p&gt;&#xA;" OwnerUserId="6017" LastActivityDate="2017-07-09T10:57:12.297" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6278" PostTypeId="2" ParentId="6276" CreationDate="2017-07-09T11:10:56.267" Score="4" Body="&lt;p&gt;Nicole d'Oresme was the first to use fractional exponents. &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicole_Oresme&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nicole_Oresme&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Algorismus proportionum and De proportionibus proportionum, Oresme developed the first calculation-method of powers with fractional irrational exponents.&#xA;&lt;a href=&quot;http://www.nicole-oresme.com/seiten/oresme-biography.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.nicole-oresme.com/seiten/oresme-biography.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The exponent 1/2 means that the square root of $x$ has to be multiplied by itself in order to give $x$.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-07-09T11:18:35.827" LastActivityDate="2017-07-09T11:18:35.827" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6279" PostTypeId="1" CreationDate="2017-07-09T16:10:41.687" Score="5" ViewCount="241" Body="&lt;p&gt;As someone reputed among certain historians to have &lt;em&gt;given you the epsilon&lt;/em&gt; Cauchy startled me by using $\varepsilon$ to denote an infinitely small &lt;em&gt;number&lt;/em&gt; in his 1826 text on differential geometry; see page 98 &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN591098687&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[S]i l'on désigne par $\varepsilon$ un nombre infiniment petit, on aura $$ \frac{\sin\left(\frac{\pi}{2}\pm\varepsilon\right)}{r}= \frac{\sin(\pm\Delta\tau)}{\sqrt{\Delta x^2+\Delta y^2}} $$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Are there other instances of such nonweierstrassian usage by Cauchy?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2017-07-16T10:33:06.263" LastActivityDate="2017-07-16T10:33:06.263" Title="Who gave you infinitesimal epsilon?" Tags="|mathematics|mathematicians|reference-request|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6280" PostTypeId="1" AcceptedAnswerId="6281" CreationDate="2017-07-10T10:07:06.680" Score="1" ViewCount="171" Body="&lt;p&gt;For a very long time I've searched in the internet what was the actual integrating device which Leibniz designed in 1693. Yesterday I found the answer - the instrument which Leibniz devised is the earliest example of an &quot;integraph&quot; - a mechanical quadrature device for plotting the integral value of a continuous function. But I didn't succeed in understanding the way it works. I'm the editor which yesterday added the information in Wikipedia about Leibniz's integraph, so I don't need the wiki references. Also I don't need the wiki article &quot;integraph&quot;, since I tried to read it and I didn't understand that article.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question asks for a detailed explanation of the mechanism of Leinbiz's integraph. I'll also be glad to find out what was the influence of Leibniz's device in the 18th century, but that is not the main question.   &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2277" LastEditDate="2020-04-13T16:00:36.567" LastActivityDate="2020-04-13T16:00:36.567" Title="What was the mechanical principle of Leibniz's &quot;integraph&quot;?" Tags="|biographical-details|calculus|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6281" PostTypeId="2" ParentId="6280" CreationDate="2017-07-10T18:05:04.380" Score="3" Body="&lt;p&gt;Leibniz did indeed describe a mechanism for tracing the solution to any differential equation of the form dy/dx=f(x) using tractional motion. For a detailed explanation, see Viktor Blåsjö, The myth of Leibniz’s proof of the fundamental theorem of calculus, Nieuw Archief voor Wiskunde, ser. 5, 16(1), 2015, 46–50, especially Figure 6, available for free here: &lt;a href=&quot;http://www.nieuwarchief.nl/serie5/pdf/naw5-2015-16-1-046.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.nieuwarchief.nl/serie5/pdf/naw5-2015-16-1-046.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz's mechanism was intended to make a theoretical point. It is useless in practice. The influence of Leibniz's device in the 18th century was, I believe, negligible, although others continued making tractional integraphs (see Dominique Tournès, La construction tractionnelle des équations différentielles, Paris: Blanchard, 2009).&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2017-07-10T18:05:04.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6282" PostTypeId="1" CreationDate="2017-07-10T22:23:53.177" Score="3" ViewCount="354" Body="&lt;p&gt;Whilst several not-very-authoritative sources e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Qubit#Origin_of_the_concept_and_name&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; state that the word qubit was derived, partially, as a play on the word cubit (obviously it also stands for 'quantum bit'), is there any credible source for this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The coining of the term &quot;qubit&quot; is attributed to Benjamin Schumacher. In the acknowledgments of his paper, Schumacher states that the term qubit was invented in jest due to its phonological resemblance with an ancient unit of length called cubit, during a conversation with William Wootters.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, &lt;a href=&quot;https://dx.doi.org/10.1103/PhysRevA.51.2738&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schumacher (1995)&lt;/a&gt; (the above referenced paper) merely states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The term &quot;qubit&quot; was coined in jest during one of the&#xA;  author's many intriguing and valuable conversations with W. K. Wootters, and became the initial impetus for this work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;'In jest' would suggest it's perhaps not merely derived from 'quantum bit', but there's no further information. In fact the only source I can find that mentions qubit and cubit together in this context is Mermin (2007, p. 4) (whom I don't have the reputation to hyperlink but DOI: 10.1017/CBO9780511813870):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My guess is that “qubit” has gained acceptance because it visually resembles an obsolete English unit of distance, the homonymic cubit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which really doesn't tell us anything.&lt;/p&gt;&#xA;" OwnerUserId="6028" LastActivityDate="2017-07-10T22:23:53.177" Title="Etymology of 'qubit'; is there any relation to cubits?" Tags="|reference-request|quantum-mechanics|computer-science|terminology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6283" PostTypeId="1" CreationDate="2017-07-11T07:22:17.577" Score="4" ViewCount="105" Body="&lt;p&gt;I have a question related to Japan History and cryptography record. As you may know, Meiji is the period in Japan between 1868 and 1912, in which occidental reforming was performed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After (and through) this period I know there's some record of cryptography techniques (in war) such as the &lt;em&gt;purple machine&lt;/em&gt;. But I'm kind of curious about &lt;strong&gt;other cryptography techniques&lt;/strong&gt;, maybe more rustic, and older, &lt;strong&gt;not influenced&lt;/strong&gt; by occidental 'modern' war techniques.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you guys know anything about this? Thank you in advance. &lt;/p&gt;&#xA;" OwnerUserId="6023" LastEditorUserId="4251" LastEditDate="2020-05-03T05:59:59.640" LastActivityDate="2020-05-03T05:59:59.640" Title="Cryptography in Japan before Meiji" Tags="|mathematics|japan|cryptology|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6284" PostTypeId="2" ParentId="657" CreationDate="2017-07-11T11:05:38.197" Score="0" Body="&lt;p&gt;The question being rather vague I will propose another answer: The French school. Descartes in his algebraic approach was well aware that there was no necessary connection between powers and dimensions, and  values greater than 3 were not essentially different. After Newton the understanding that rational or analytic mechanics is geometry + time became rather common. At the end of the 18th. c. Lagrange states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“La mécanique peut être regardée comme une géométrie à quatre&#xA;  dimensions et l'analyse mécanique comme une extension de l'analyse&#xA;  géométrique /Mechanics can be considered to be a geometry with 4&#xA;  dimensions/&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He was of course aware of the formalism behind inverse functions and derivatives,or more generally the  symmetry x(t)&amp;lt;-&gt;t(x). However sometime earlier d'Alembert has written:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;on pourrait cependant regarder la durée comme une quatrième dimension,&#xA;  et que le produit du temps par la solidité serait en quelque manière&#xA;  un produit de quatre dimensions    /one could consider duration as a&#xA;  4th dimension/&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and rather clearly he is indebted to Descartes. Both quotes suggest to say that the French Encyclopedists, who were generally materialistic and mechanistic philosophers, offered the first views of higher dimensional thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly it was an opponent to Descartes, the Cambridge Platonist Henry More who tried to solve the mind-body problem in his &lt;em&gt;Enchiridion Metaphysicum&lt;/em&gt; by proposing a  4th dimension to which only pure spirits have access. His idea was never really forgotten and during the 19th. c. when spiritism was in fashion it is frequently mentioned and one can even find a joking allusion to it in Einstein’s popular writings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Refs to this post are to be found in Meyerson’s &lt;em&gt;La Deduction relativiste&lt;/em&gt;)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2017-07-12T08:31:26.787" LastActivityDate="2017-07-12T08:31:26.787" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6285" PostTypeId="2" ParentId="6268" CreationDate="2017-07-11T15:11:03.503" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/5103/604&quot;&gt;This question&lt;/a&gt; discusses the common assumption that the only issue of the Galileo trial was heliocentrism.  Briefly, some scholars have argued that an additional issue was atomism and its difficult relation with doctrinal issues. Even as far as heliocentrism is concerned, the opposition was not to using it as a technical hypothesis in scientific calculation so much as the more sweeping assumption that it describes the &quot;true&quot; state of affairs.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-07-11T15:11:03.503" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6286" PostTypeId="2" ParentId="6237" CreationDate="2017-07-11T18:27:59.627" Score="2" Body="&lt;p&gt;Here is a review of Vol. 8 of Gauss's works: &lt;a href=&quot;https://projecteuclid.org/euclid.bams/1183417390&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://projecteuclid.org/euclid.bams/1183417390&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for his works in earthly geodesy, i.e. the adjustment of triangulations, which forms a part of Vol. 4 and Vol. 9, the best introduction to it is in chapter 3 of &quot;Numerical analysis of symmetric matrices&quot; by Schwarz, Rutishauser and Stiefel: &lt;a href=&quot;https://books.google.de/books?id=j8O7AAAAIAAJ&amp;amp;q=adjustment+%22constrained+fitting%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.de/books?id=j8O7AAAAIAAJ&amp;amp;q=adjustment+%22constrained+fitting%22&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more thorough exposition is in chapters 5 and 6 of Wright/Hayford: &lt;a href=&quot;https://archive.org/stream/adjustmentobser01hayfgoog#page/n6/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/stream/adjustmentobser01hayfgoog#page/n6/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2017-07-11T18:27:59.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6287" PostTypeId="1" AcceptedAnswerId="6468" CreationDate="2017-07-11T20:00:41.020" Score="4" ViewCount="417" Body="&lt;p&gt;There are only four types of plane isometries: translations, rotations, reflections, and glide reflections. My question is: who was the first person who proved this? I asked this question personally to two historians (&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Stillwell&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Stillwell&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Jeremy_Gray&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeremy Gray&lt;/a&gt;) and none of them was able to provide an answer.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="10106" LastEditDate="2020-06-07T06:53:17.123" LastActivityDate="2020-06-07T06:53:17.123" Title="Who classified plane isometries first?" Tags="|mathematics|mathematicians|euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6288" PostTypeId="2" ParentId="3400" CreationDate="2017-07-12T01:15:57.577" Score="1" Body="&lt;p&gt;Max Cosyns, professor at the Free University of Brussels, survived Dachau.&lt;/p&gt;&#xA;" OwnerUserId="6035" LastActivityDate="2017-07-12T01:15:57.577" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6289" PostTypeId="2" ParentId="3400" CreationDate="2017-07-12T04:23:09.147" Score="8" Body="&lt;p&gt;“Cammaille” or “Camaille” sound like misspellings for &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Cavaill%C3%A8s&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Jean Cavaillès&lt;/strong&gt;&lt;/a&gt;. This seems confirmed by &lt;a href=&quot;http://cds.cern.ch/record/1727695&quot; rel=&quot;noreferrer&quot;&gt;CERN Bulletin Issue No. 18/1999&lt;/a&gt;, which on page 15 (numbered 2) quotes &lt;a href=&quot;https://fr.wikipedia.org/wiki/Catherine_Chevalley&quot; rel=&quot;noreferrer&quot;&gt;Catherine Chevalley&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Heisenberg the physicist who “apparently tried, from within Germany, to save Cavaillès”$^1$.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;$^1$Catherine Chevalley: in her preface to “Werner Heisenberg” (see footnote 3 below) page 12, note 1 (...)&lt;br&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;$^3$“Werner Heisenberg: Philosophie, manuscrit de 1942”, translation and introduction by C. Chevalley, Editions du Seuil, Paris, 1998 (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-07-12T04:23:09.147" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6290" PostTypeId="1" AcceptedAnswerId="6291" CreationDate="2017-07-12T14:08:58.750" Score="5" ViewCount="582" Body="&lt;p&gt;Does anybody know a paper or sorts in which the term &quot;colour&quot; charge in QCD was introduced first? Or any other source in which this label was proposed?&lt;/p&gt;&#xA;" OwnerUserId="6036" OwnerDisplayName="uitty400" LastEditorUserId="4251" LastEditDate="2020-07-06T15:39:36.697" LastActivityDate="2020-07-06T15:39:36.697" Title="Who first proposed the &quot;colour&quot; charge?" Tags="|physics|quantum-mechanics|theoretical-physics|physicists|quantum-field-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6291" PostTypeId="2" ParentId="6290" CreationDate="2017-07-12T14:17:00.390" Score="4" Body="&lt;p&gt;O.W. Greenberg is most often credited with the introduction of colour [1]. Note the date: it was almost immediately after Gell-Mann and Zweig introduced quarks. However his approach was formally completely different as he postulated that quarks do not obey the Fermi statistics but instead, a generalisation of it, the parafermion statistics of order 3. In fact, it turns out that Greenberg wrote an essay on the history of the discover of colour [2]. In this article Greenberg credits Han and Nambu :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Han and Nambu [13] were the first who explicitly introduced the color $SU(3)_c$ symmetry which is implicit in the parastatistics model. One of their motivations was to avoid fractional quark charges, so they arranged for their three flavor triplets to have different electric charges : (1, 0, 0); (1, 0, 0); (0, −1, −1) so that they are distinguishable. Averaging over the charges for each flavor gives the fractional charges of the original quark model. They proposed that the forces between these quarks would be mediated by the exchange of an $SU(3)_c$-octet of gauge vector mesons, thus giving the hidden color degree of freedom a dynamical role, and showed that such forces would make the $SU(3)_c$-singlets be the ground-state particles, to be identified with the known baryons and mesons. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As it should be clear, the charges are not right! But a few paragraphs later, Greenberg goes on to say that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While preparing for this session, I looked into the origin of the use of the word “color” for the gauged $SU(3)_c$ degree of freedom. Bram Pais [17], in a discussion in the Erice summer school of 1965, was the first to use color in this way. Donald Lichtenberg [18] also used color with this meaning in his book published in 1970. Color in this sense came into general use following the articles by Gell-Mann and by William Bardeen, Harald Fritzsch and Gell-Mann [19].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A work that H. Fritzsch, M. Gell-Mann And H. Leutwyler further consolidated [3] in the article I cited in an earlier version of this answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] O.W. Greenberg: Spin and Unitary Spin Independence in a Paraquark Model of Baryons and Mesons. Phys. Rev. Lett. 13, 598-602 (1964)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] Color, from baryon spectroscopy to QCD, &lt;a href=&quot;https://arxiv.org/abs/hep-ph/9301207v1&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv:hep-ph/9301207&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[13] Y. Nambu, in Preludes in Theoretical Physics, ed. A. de Shalit, H. Feshbach and L. Van Hove (North Holland, Amsterdam, 1966), p. 133; M.Y. Han and Y. Nambu, Phys. Rev. 139 (1965) B1006. A. Tavkhelidze, in High-Energy Physics and Elementary Particles (Int. Atomic Energy Agency, Vienna, 1965), also suggested three different triplets with integral charges, but did not suggest the gauge interaction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[17] A. Pais, in Recent Developments in Particle Symmetries, ed. A. Zichichi (Aca- demic, New York, 1966) p. 406.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[18] D.B. Lichtenberg, Unitary Symmetry and Elementary Particles (Academic, New York, 1970) p. 227.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[19] M. Gell-Mann, Elementary Particle Physics, ed. P. Urban (Springer, Vienna, 1972); Acta Phys. Austriaca Supp. 9 (1972) 733; W.A. Bardeen, H. Fritzsch and M. Gell-Mann, in Scale and Conformal Symmetry in Hadron Physics, ed. R. Gatto (Wiley, New York, 1973) p 139.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[3] H. Fritzsch, M. Gell-Mann, and H. Leutwyler. Advantages of the color octet gluon picture. Physics Letters B, 47(4):365 – 368, 1973.&lt;/p&gt;&#xA;" OwnerDisplayName="user6037" LastEditorDisplayName="user6037" LastEditDate="2017-07-12T19:45:51.093" LastActivityDate="2017-07-12T19:45:51.093" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6292" PostTypeId="1" CreationDate="2017-07-12T15:36:21.463" Score="-1" ViewCount="152" Body="&lt;p&gt;What are some famous and brilliant metaphors and analogies used by physicists and science writers and communicators to explain complex physical ideas?&lt;/p&gt;&#xA;" OwnerUserId="5236" LastActivityDate="2017-07-13T08:35:55.330" Title="Metaphors and analogies used by physicists and science communicators in history?" Tags="|physics|" AnswerCount="1" CommentCount="1" ClosedDate="2017-07-20T11:32:42.377" ContentLicense="CC BY-SA 3.0" />
  <row Id="6293" PostTypeId="2" ParentId="1865" CreationDate="2017-07-12T20:40:26.590" Score="7" Body="&lt;p&gt;There is a terrific summary &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Pi_chronology.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; which includes the degree of accuracy with chronology, if that is what you are interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Pre computer calculations of π&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Mathematician           Date     Places Comments    &#xA;1   Rhind papyrus       2000 BC  1      3.16045 (= 4(8/9)2)&#xA;2   Archimedes          250 BC   3      3.1418 (average of the bounds)&#xA;3   Vitruvius           20 BC    1      3.125 (= 25/8)&#xA;4   Chang Hong          130      1      3.1622 (= √10)&#xA;5   Ptolemy             150      3      3.14166&#xA;6   Wang Fan            250      1      3.155555 (= 142/45)&#xA;7   Liu Hui             263      5      3.14159&#xA;8   Zu Chongzhi         480      7      3.141592920 (= 355/113)&#xA;9   Aryabhata           499      4      3.1416 (= 62832/20000)&#xA;10  Brahmagupta         640      1      3.1622 (= √10)&#xA;11  Al-Khwarizmi        800      4      3.1416&#xA;12  Fibonacci           1220     3      3.141818&#xA;13  Madhava             1400    11      3.14159265359&#xA;14  Al-Kashi            1430    14      3.14159265358979&#xA;15  Otho                1573     6      3.1415929&#xA;16  Viète               1593     9      3.1415926536&#xA;17  Romanus             1593    15      3.141592653589793&#xA;18  Ludolph Van Ceulen  1596    20      3.14159265358979323846&#xA;19  Ludolph Van Ceulen  1596    35      3.1415926535897932384626433832795029&#xA;20  Newton              1665    16      3.1415926535897932&#xA;21  Sharp               1699    71  &#xA;22  Seki Kowa           1700    10 &#xA;23  Kamata              1730    25 &#xA;24  Machin              1706    100 &#xA;25  De Lagny            1719    127     Only 112 correct&#xA;26  Takebe              1723    41  &#xA;27  Matsunaga           1739    50  &#xA;28  von Vega            1794    140     Only 136 correct&#xA;29  Rutherford          1824    208     Only 152 correct&#xA;30  Strassnitzky        1844    200 &#xA;31  Clausen             1847    248 &#xA;32  Lehmann             1853    261 &#xA;33  Rutherford          1853    440 &#xA;34  Shanks              1874    707     Only 527 correct&#xA;35  Ferguson            1946    620&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Computer calculations of π&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Mathematician        Date      Places   Type of computer&#xA;Ferguson            1947       710      Desk calculator&#xA;Ferguson, Wrench    1947       808      Desk calculator&#xA;Smith, Wrench       1949      1120      Desk calculator&#xA;Reitwiesner et al.  1949      2037      ENIAC&#xA;Nicholson, Jeenel   1954      3092      NORAC&#xA;Felton              1957      7480      PEGASUS&#xA;Genuys  Jan         1958     10000      IBM 704 &#xA;Felton  May         1958     10021      PEGASUS&#xA;Guilloud            1959     16167      IBM 704&#xA;Shanks, Wrench      1961    100265      IBM 7090&#xA;Guilloud, Filliatre 1966    250000      IBM 7030&#xA;Guilloud, Dichampt  1967    500000      CDC 6600&#xA;Guilloud, Bouyer    1973    1001250     CDC 7600&#xA;Miyoshi, Kanada     1981    2000036     FACOM M-200&#xA;Guilloud            1982    2000050&#xA;Tamura              1982    2097144     MELCOM 900II&#xA;Tamura, Kanada      1982    4194288     HITACHI M-280H&#xA;Tamura, Kanada      1982    8388576     HITACHI M-280H&#xA;Kanada, Yoshino, &#xA;Tamura              1982    16777206    HITACHI M-280H&#xA;Ushiro, Kanada      1983    10013395    HITACHI S-810/20&#xA;Gosper              1985    17526200    SYMBOLICS 3670&#xA;Bailey  Jan         1986    29360111    CRAY-2&#xA;Kanada, Tamura      1986    33554414    HITACHI S-810/20&#xA;Kanada, Tamura      1986    67108839    HITACHI S-810/20&#xA;Kanada, Tamura, &#xA;Kubo                1987    134217700   NEC SX-2&#xA;Kanada, Tamura      1988    201326551   HITACHI S-820/80&#xA;Chudnovskys         1989    480000000&#xA;Chudnovskys         1989    525229270&#xA;Kanada, Tamura      1989    536870898&#xA;Chudnovskys         1989    1011196691&#xA;Kanada, Tamura      1989    1073741799&#xA;Chudnovskys         1991    2260000000&#xA;Chudnovskys         1994    4044000000&#xA;Kanada, Tamura      1995    3221225466&#xA;Kanada              1995    4294967286&#xA;Kanada              1995    6442450938&#xA;Kanada, Takahashi   1997    51539600000     HITACHI SR2201 &#xA;Kanada, Takahashi   1999    206158430000    HITACHI SR&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerDisplayName="user6039" LastEditorUserId="217" LastEditDate="2019-08-28T10:24:09.430" LastActivityDate="2019-08-28T10:24:09.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6294" PostTypeId="2" ParentId="6292" CreationDate="2017-07-13T08:35:55.330" Score="2" Body="&lt;p&gt;Assuming that you include physicists dabbling in mathematics in the scope of your question, I would mention that Leibniz used the comparison of a grain of sand and the Earth to motivate his notion of an infinitesimal number.  He also compared distances on Earth with distances to the Heavens to give an idea of an infinite number.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-07-13T08:35:55.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6295" PostTypeId="1" CreationDate="2017-07-13T13:16:48.557" Score="9" ViewCount="4473" Body="&lt;p&gt;Eratosthenes measured the radius of the Earth with an incredibly accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do it, you need to measure the length of the shadows from 2 different cities at the same time of the day. Then knowing the distance between the cities and a little bit of geometry he calculated the radius of the Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: How did he know when it was &quot;the same time of the day&quot;?&#xA;There was no long range communications to synchronize two people measuring the shadows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if you measure the shadow one day, then travel to the other city to measure the shadow next day, how can you know that you will do it at the same time of the day?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays with clocks or telephones the experiment is very easy, but back then I can not figure out how he managed to do it.&lt;/p&gt;&#xA;" OwnerUserId="6041" LastActivityDate="2024-03-20T15:34:02.563" Title="How did Eratosthenes knew the exact time of the day?" Tags="|geometry|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6296" PostTypeId="1" AcceptedAnswerId="6299" CreationDate="2017-07-14T15:18:17.250" Score="2" ViewCount="170" Body="&lt;p&gt;I can find many resources regarding Cantor's proof of the uncountability of the continuum. But was he the first to prove that $\mathfrak{c} = | \mathfrak{P} (\mathbf{N})|$? Which paper?&lt;/p&gt;&#xA;" OwnerUserId="6045" LastActivityDate="2017-07-14T21:36:43.647" Title="Where was it first proved that the cardinality of the continuum equals the cardinality of the power set of the naturals?" Tags="|set-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6297" PostTypeId="2" ParentId="6295" CreationDate="2017-07-14T15:19:37.883" Score="17" Body="&lt;p&gt;The first sentence of the question is not justified. Accuracy of Erathosphenes measurement was much discussed in the literature, and it is certainly not &quot;incredible&quot;. What the actual accuracy was is unknown, because we are not sure what his unit &quot;stadium&quot; exactly was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering time measurement, he did not need it. He used two cities on approximately the same longitude and measured shadows at noon. Noon is determined when the shadow is shortest, and one does not need any clock for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the most doubtful thing is how he measured the distance between the two cities, or whether he measured it at all.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-07-14T15:19:37.883" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6298" PostTypeId="1" AcceptedAnswerId="6308" CreationDate="2017-07-14T18:48:09.447" Score="1" ViewCount="98" Body="&lt;p&gt;I have read in several places that both the developed concept and the term for &lt;a href=&quot;https://en.wikipedia.org/wiki/Dynamic_pressure&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;dynamic pressure&lt;/em&gt;&lt;/a&gt; &lt;em&gt;as it pertain to fluid dynamics&lt;/em&gt; was first published in &lt;a href=&quot;https://en.wikipedia.org/wiki/Hydrodynamica&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hydrodynamica&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Daniel_Bernoulli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Daniel_Bernoulli&lt;/a&gt;, but I can not find any direct evidence of this. Assuming it is there, I assume that it would be in the original latin. Is there a linkable source where I could see it as originally written and defined? Either the original latin, or a scholarly translation into a different language if necessary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If not a linkable source, even a screen shot of the relevant passage would also be helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;below:&lt;/strong&gt; HYDRODYNAMICA, Danielis Bernoulli, &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:HYDRODYNAMICA,_Danielis_Bernoulli.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;from here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SCJZ2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SCJZ2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2017-07-17T07:07:29.097" LastActivityDate="2017-07-18T19:48:28.683" Title="First use of the term and principle of dynamic pressure — Daniel Bernoulli?" Tags="|mathematics|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6299" PostTypeId="2" ParentId="6296" CreationDate="2017-07-14T21:16:32.733" Score="5" Body="&lt;p&gt;I believe the credit mainly goes to Bertrand Russell and Émile Borel.&lt;/p&gt;&#xA;&lt;p&gt;José Ferreirós &lt;strong&gt;[2]&lt;/strong&gt; writes (see below) that Bertrand Russell &lt;strong&gt;[4]&lt;/strong&gt; (1903) was largely responsible for rephrasing Cantor's 1892 diagonalization result in terms of the collection of all subsets of a set, and then emphasizing the importance of this rephrasing. However, Russell was not the only person who was communicating Cantor's 1892 results in this manner. Émile Borel, in his influential 1898 book &lt;strong&gt;[1]&lt;/strong&gt;, includes Cantor's 1874 proof for the uncountability of any interval (pp. 14-15) and Cantor's 1892 diagonalization proof that given any set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; the set of all functions from &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; into the two element set &lt;span class=&quot;math-container&quot;&gt;$\{0, \, 1\}$&lt;/span&gt; has cardinality greater than the cardinality of &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; (pp. 107-108), and Borel explicitly makes the connection to the cardinality of collections of subsets of a set (pp. 109-110, translated excerpt below). Also, in an 1898 conference Charles Sanders Peirce &lt;strong&gt;[3]&lt;/strong&gt; (top of p. 159) said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\ldots$&lt;/span&gt; this collection shall consist of all finite multitudes [= positive integers] together with all possible collections of those multitudes, together with all possible collections of collections of those multitudes, together with all possible collections of collections of collections of those multitudes, and so on &lt;em&gt;ad infinitum&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Émile Félix Édouard Justin Borel, &lt;a href=&quot;https://archive.org/details/leconstheoriefon00borerich&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Leçons sur la Théorie des Fonctions&lt;/strong&gt;&lt;/a&gt; [Lessons on the Theory of Functions], Collection de Monographies sur la Théorie des Fonctions (Borel's Series) #1, Gauthier-Villars, 1898, x + 136 pages.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; José Ferreirós, &lt;strong&gt;Labyrinth of Thought. A History of Set Theory and Its Role in Modern Mathematics&lt;/strong&gt;, Science Networks / Historical Studies #23, Birkhäuser Verlag, 1999, xxii + 440 pages.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Charles Sanders Peirce, &lt;em&gt;Lecture Three: The Logic of Relatives&lt;/em&gt;, pp. 146-164 in &lt;strong&gt;Reasoning and the Logic of Things. The Cambridge Conferences Lectures of 1898&lt;/strong&gt;, edited by Kenneth Laine Ketner, Harvard University Press, 1992, xiv + 297 pages.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Bertrand Arthur William Russell, &lt;a href=&quot;https://archive.org/details/principlesmathe00russgoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Principles of Mathematics. Volume I&lt;/strong&gt;&lt;/a&gt;, Cambridge University Press, 1903, xxx + 534 pages. [Note: A second volume was never published.]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from p. 306 of Ferreirós [2])&lt;/strong&gt; It has to be noted that it was Russell, not Cantor in his published work, who focused on the Cantor Theorem as a central result of great importance. He seems to have been the first mathematician who presented it as showing that the set of all subsets of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; has always a greater cardinality than &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; itself [Russell (1903), §§ 346-347]. Thus, it was Russell who formulated it for the first time as a purely set-theoretical result. (In Cantor's version it showed that, given a set &lt;span class=&quot;math-container&quot;&gt;$S,$&lt;/span&gt; a certain set of &lt;em&gt;functions&lt;/em&gt; has greater cardinality, and functions were &lt;em&gt;not&lt;/em&gt; taken to be sets.) In the process, Russell was the first to emphasize something like the Power Set Axiom. All of this was in itself an important contribution, for until then the theorem lay rather forgotten in the first annual report of the &lt;em&gt;DMV&lt;/em&gt; and its significance had not been clearly grasped. Moreover, the Cantor Theorem led Russell to the discovery of his paradox. Thus Russell's early reformulations of previous work were important in the process of extensionalization of set theory, although he was never partisan of a purely extensional conception of classes (see Russell (1903), §§ 66-79), and he gradually became more and more a Fregean on this account.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from pp. 109-110 of Borel [1], translated)&lt;/strong&gt; It also seems difficult for us to introduce such functions for consideration and, in general, any class of functions that forms a set of power greater than that of the continuum, because one can use a function in calculus only if it is defined by means of countably many conditions. This is the case with continuous functions and also with functions that are discontinuous at only countably many values of the variable. Such functions can, in fact, be defined by means of countably many conditions and, if one wishes to be more precise, by means of countably many whole numbers &lt;span class=&quot;math-container&quot;&gt;$({ }^1)$&lt;/span&gt; [&lt;em&gt;footnote&lt;/em&gt;: These conditions are not arbitrary, by the way; we will return to this point in Note III.]&lt;/p&gt;&#xA;&lt;p&gt;The same applies to sets of points. One can see that each function &lt;span class=&quot;math-container&quot;&gt;$f(x),$&lt;/span&gt; equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; for every value of the variable, partitions the reals into two classes, and therefore defines two sets of points: the set of points at which the function has the value &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and the set of points at which the function has the value &lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt; Thus, the set &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; of all such functions is identical to the set of all possible sets (with the reals as elements). Therefore, the set &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; has a power greater than that of the continuum, and hence its &lt;em&gt;general&lt;/em&gt; element (that is, an &lt;em&gt;arbitrary&lt;/em&gt; set of reals) cannot be defined except by means of a set of conditions that has the power of the continuum. This is what makes reasoning about arbitrary sets so difficult and sometimes impossible. This is also what often makes such reasoning pointless in the present state of science, since it does not seem as if a set defined by uncountably many conditions can be easily introduced into a train of reasoning. On the other hand, we have seen (p. 50) that the collection of perfect sets has the power of the continuum. It is easy to see that the same applies to the collection of sets we have called &lt;em&gt;measurable&lt;/em&gt;; it is this aspect [&lt;em&gt;means&lt;/em&gt;: being definable by countably many conditions] that allows us to reason with more facility about these particular classes of sets and to discover their properties.&lt;/p&gt;&#xA;&lt;p&gt;In summary, there is the need to distinguish, both in the case of sets of points and in the case of discontinuous functions, two large classes: the sets and functions which cannot be defined by countably many conditions and those which can be so defined. It seems that only the latter class can be profitably investigated at present.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Regarding Borel's claim &amp;quot;the same applies to the collection of sets we have called &lt;em&gt;measurable&lt;/em&gt;&amp;quot;, Borel is talking about the collection of Borel sets. Because Borel did not give a mathematically precise definition for his collection of (Borel) measurable sets, Borel's claim is something that he probably felt was intuitively evident. The claimed result is true, but because the construction process which Borel incompletely described (elsewhere) actually involves uncountably many steps (as shown by Lebesgue in 1905, who also gave a rigorous proof of the claim), the situation is not as intuitively evident as I suspect Borel may have thought it to be.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-07-14T21:36:43.647" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6300" PostTypeId="1" CreationDate="2017-07-15T01:44:42.210" Score="0" ViewCount="52" Body="&lt;p&gt;Every now and then, I run across a rumor of the Soviets building a tri-nary processor, way back when.  But I can never find a source beyond the infamous guy who knows a guy...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea is interesting, but I suspect it's mostly fantasy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has anyone ever heard of this from a &lt;em&gt;credible&lt;/em&gt; source?  Or is this the Maths/Science/Engineering version of the majestic yeti?&lt;/p&gt;&#xA;" OwnerUserId="6016" LastActivityDate="2017-07-15T01:44:42.210" Title="Tri-nary processor?" Tags="|computer-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6301" PostTypeId="1" AcceptedAnswerId="6304" CreationDate="2017-07-15T05:49:11.433" Score="1" ViewCount="375" Body="&lt;p&gt;In &lt;a href=&quot;https://hsm.stackexchange.com/a/2275/3869&quot;&gt;this answer&lt;/a&gt; there is an interesting passage:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Also, Aristotle posited that the speed of sound was dependent on frequency. Gassendi had religious/philosophical issues with Aristotle's views, and one reason for the sound experiments was to prove Aristotle incorrect.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;followed by an interesting &lt;a href=&quot;https://hsm.stackexchange.com/questions/2271/#comment4124_2275&quot;&gt;comment&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Aristotle had no concept of sound &quot;frequency&quot;, let alone &quot;speed of sound&quot; depending on it. He talks of air &quot;falling upon and striking the air which is next to it&quot;, and is too vague to make sense of either the frequency or the speed. Those require concept of a wave, which came much later. And although Roemer was first to measure the speed of light successfully people suggested that light takes time to travel centuries before him. Aristotle mentions Empedocles, Avicenna and Alhazen also held such view.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I recommend going to the original question &lt;a href=&quot;https://hsm.stackexchange.com/q/2271/3869&quot;&gt;When was it first realized that sound travels with finite speed?&lt;/a&gt; and reading through it, and all of the answers there, as I am trying to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I almost asked my question in a comment, but then thought it might be interesting enough for a full answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the modern concept of frequency would not be appropriate, did Aristotle suggest that different &lt;em&gt;kinds&lt;/em&gt; of sound might travel at different speeds?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Pierre Gassendi later determine that different kinds of sound probably travelled &lt;em&gt;at the same&lt;/em&gt; speed?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2017-07-15T13:00:52.970" Title="Did Aristotle suggest that different kinds of sound might travel at different speeds?" Tags="|discoveries|experimental-physics|acoustics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6302" PostTypeId="1" AcceptedAnswerId="6303" CreationDate="2017-07-15T06:15:30.250" Score="2" ViewCount="275" Body="&lt;p&gt;I am not a student of thermodynamics, but I will reproduce some equations and discussion from Wikipedia to highlight the principles involved in obtaining a fairly accurate — one percent perhaps — value for the speed of sound in air.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the article &lt;a href=&quot;https://en.wikipedia.org/wiki/Speed_of_sound&quot; rel=&quot;nofollow noreferrer&quot;&gt;Speed of sound&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$c=\sqrt{\frac{K_s}{\rho}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $K_s$ is &quot;...a coefficient of stiffness, the isentropic &lt;a href=&quot;https://en.wikipedia.org/wiki/Bulk_modulus&quot; rel=&quot;nofollow noreferrer&quot;&gt;bulk modulus&lt;/a&gt; (or the modulus of bulk elasticity for gases)&quot; and the isentropic bulk modulus $K_s=\gamma p$ where $p$ is the pressure, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Heat_capacity_ratio&quot; rel=&quot;nofollow noreferrer&quot;&gt;heat capacity ratio&lt;/a&gt; $\gamma$ for a diatomic gas is equal to $1+2/5$ or 1.4. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article continues: &quot;For general &lt;a href=&quot;https://en.wikipedia.org/wiki/Equations_of_state&quot; rel=&quot;nofollow noreferrer&quot;&gt;equations of state&lt;/a&gt;, if &lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;classical mechanics&lt;/a&gt; is used, the speed of sound c is given by&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$c=\sqrt{\left( \frac{dp}{d\rho} \right)_s}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where again $p$ is the pressure, $\rho$ the density, and the derivative is taken &lt;em&gt;isentropically&lt;/em&gt;, that is, at constant &lt;a href=&quot;https://en.wikipedia.org/wiki/Entropy&quot; rel=&quot;nofollow noreferrer&quot;&gt;entropy&lt;/a&gt; $s$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm wondering if there is any well recognized &lt;em&gt;first accurate calculation&lt;/em&gt; of the speed of sound from modern principles? I realize the history of science is a continuum and it's likely concepts were added in stages, but it's possible there was a moment when a theory was first developed such that it produced a nearly-correct speed of sound, and there was an &quot;aha!&quot;  &lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2017-07-15T09:25:58.943" Title="When was the speed of sound first &quot;correctly&quot; calculated?" Tags="|physics|thermodynamics|acoustics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6303" PostTypeId="2" ParentId="6302" CreationDate="2017-07-15T08:55:36.780" Score="3" Body="&lt;p&gt;One of the first theories is due to Newton. He derived a formula for the speed of sound from his wave theory, and compared with experiment.&#xA;(The experiment was difficult at that time, because of the lack of exact clocks).&#xA;His theory had a reasonable agreement with the experiment but was not very precise because he did not take thermodynamics into account at all (it did not exist!).&#xA;With the development of thermodynamics, more and more precise theories were developed. In the last edition of Principia, Newton arrived at the number 1142 feet/sec, which was in good agreement with the latest measurements.&#xA;The details of the story are described in&#xA;Westfall, Never at rest, p. 734-735. But one cannot point the exact moment when &quot;accurate&quot; theory was created, while all previous theories were inaccurate or incorrect. The history after (and before!) Newton is described in detail in this article:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www3.nd.edu/~powers/ame.20231/finn1964.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www3.nd.edu/~powers/ame.20231/finn1964.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The paper is concentrated on Laplace's contribution but many people were involved, and one cannot point the precise moment when the theory &quot;became accurate&quot;. But this happened in the early 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-15T09:25:58.943" LastActivityDate="2017-07-15T09:25:58.943" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6304" PostTypeId="2" ParentId="6301" CreationDate="2017-07-15T11:09:50.263" Score="2" Body="&lt;p&gt;Yes, Aristotle claimed that sound velocity depends on pitch. For instance &lt;a href=&quot;https://books.google.de/books?id=8xS_ky1OhqAC&amp;amp;pg=PA98&amp;amp;lpg=PA98&amp;amp;dq=Aristotle%20sound%20velocity&amp;amp;source=bl&amp;amp;ots=wyUiOLsVTl&amp;amp;sig=lzHzdiOKGLhihDo88ZCfw6VKb-M&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjTvLiqiovVAhUKLsAKHZSVD7gQ6AEITjAF#v=onepage&amp;amp;q=Aristotle%20sound%20velocity&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle&lt;/a&gt; said:&quot;Hence it passes through more slowly, and so becomes lower-pitched&quot;. A detailled discussion can be found in the linked source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: The link directs to Google books: Greek Musical Writings: Volume 2, Harmonic and Acoustic Theory, Andrew Barker (ed.), Cambridge University Press&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;De Audibilibus&lt;/em&gt; is preserved only as a quotation in Porphyr's &lt;em&gt;Commentary on Ptolemy's Harmonics&lt;/em&gt;. ... Porphyr ascribes it to Aristotle, and its general approach and the theories it propounds are certainly close to his.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another quote from the text: &quot;pitch depends on the velocity of a sound transmission, a view that Aristotle shares&quot;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-07-15T13:00:52.970" LastActivityDate="2017-07-15T13:00:52.970" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6305" PostTypeId="2" ParentId="3131" CreationDate="2017-07-16T11:11:26.633" Score="58" Body="&lt;p&gt;Maryam Mirzakhani, the first Iranian &lt;em&gt;and&lt;/em&gt; first woman to win the Fields medal, died of breast cancer in July 2017. She was only 40 years of age.&lt;/p&gt;&#xA;" OwnerUserId="6049" LastEditorUserId="26" LastEditDate="2017-07-16T22:33:48.800" LastActivityDate="2017-07-16T22:33:48.800" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6306" PostTypeId="1" AcceptedAnswerId="7496" CreationDate="2017-07-16T22:19:05.517" Score="6" ViewCount="217" Body="&lt;p&gt;When describing the projective Hilbert space as the state space in quantum mechanics, physicists habitually refer to its elements as &quot;rays in Hilbert space&quot;, while the mathematical literature seems to favour calling the elements of projective spaces &quot;lines&quot; in the original space. Granted, I've not done a systematic literature research on this, but personal experience and corroborating evidence (like googling &quot;projective space ray&quot; and observing that most of the results are indeed about Hilbert spaces) seem to support this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the question is - who first referred to the elements of projective (Hilbert) space as &quot;rays&quot;? Can all later usage be traced back to this instance or did it arise in several instances separately?&lt;/p&gt;&#xA;" OwnerUserId="3797" LastActivityDate="2018-07-03T00:18:19.453" Title="Where does the habit of calling the elements of a projective Hilbert space &quot;rays&quot; originate from?" Tags="|physics|terminology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6307" PostTypeId="1" CreationDate="2017-07-17T13:09:14.453" Score="5" ViewCount="224" Body="&lt;p&gt;&lt;strong&gt;Redactional note&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like suggested by one of &lt;a href=&quot;https://mathoverflow.net/questions/273292/where-did-zermelo-first-model-the-natural-numbers-by-iterates-of-the-singleton-o/273419#273419&quot;&gt;this question&lt;/a&gt;'s (or so I understand the meta.discussion), &lt;a href=&quot;https://mathoverflow.net/questions/273292/where-did-zermelo-first-model-the-natural-numbers-by-iterates-of-the-singleton-o/273419#273419&quot;&gt;this question&lt;/a&gt; should be given some too-many-subquestions-redacting, in particular since the easiest sub-question (0) of it has been answered by myself after another pass over the original literature and its subquestion (3), which can be seen to not be covered by the main question's title has been kindly answered by one relevant literature reference in a comment by F. Ziegler. (I would find more references to systematically ordinal-comparing references useful, though.)  I therefore created this satellite question, taking the advice of M. Katz to post it in this forum, leaving the original subquestion in  &lt;a href=&quot;https://mathoverflow.net/questions/273292/where-did-zermelo-first-model-the-natural-numbers-by-iterates-of-the-singleton-o/273419#273419&quot;&gt;this question&lt;/a&gt; in place, but with an edit saying that these questions should be answered here, and repeating them here: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Are there scholarly references (or possibly, testimonies from people having attended those lectures) about &lt;em&gt;whether Zermelo defined the natural numbers this way in his lectures&lt;/em&gt; in G&amp;ouml;ttingen and Z&amp;uuml;rich? &#xA;(It of course seems unlikely that he did, having been much involved in axiomatic set theory, and having received what is now the standard definition from von Neumann.) NOTE: This question was first asked &lt;a href=&quot;https://mathoverflow.net/questions/273292/where-did-zermelo-first-model-the-natural-numbers-by-iterates-of-the-singleton-o&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Are there scholarly references about whether Zermelo ever publicly discussed, in lectures or in writing, von  Neumann's definition, possibly comparing his definition to his own? EDIT: This question was first asked &lt;a href=&quot;https://mathoverflow.net/questions/273292/where-did-zermelo-first-model-the-natural-numbers-by-iterates-of-the-singleton-o&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6060" LastActivityDate="2017-07-17T13:09:14.453" Title="What was Zermelo's _reception_ of von Neumann's definition of ordinal numbers?" Tags="|set-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6308" PostTypeId="2" ParentId="6298" CreationDate="2017-07-17T13:45:11.717" Score="3" Body="&lt;p&gt;The experiment described in &lt;a href=&quot;http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10871847_00274.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Section 12, § 7&lt;/a&gt; of the &quot;Hydrodynamica&quot; and depicted in the &lt;a href=&quot;http://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10871847_00339.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;appendix&lt;/a&gt; (second picture from above on the left part of the page), can be described as follows: water flows out of a perforated tube and rises to the height of the water in a nearby manometer, thus establishing the equality of hydrostatic and hydrodynamic pressure. See also &lt;a href=&quot;https://books.google.de/books?id=UdGBy8iLpocC&amp;amp;pg=PA134&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mikhailov&lt;/a&gt; in: Grattan-Guinness, &quot;Landmark Writings in Western Mathematics 1640-1940&quot;, Elsevier 2005.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Daniel's as well as Johann Bernoulli's treatises on hydraulics were translated into &lt;a href=&quot;https://books.google.de/books/about/Hydrodynamics_and_Hydraulics.html?id=sJggAQAAMAAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;English&lt;/a&gt; by Carmody and Kobus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also the introduction to &lt;a href=&quot;https://books.google.de/books?id=sT1cHAAACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonardo Euleri Opera Omnia, Ser. II, Vol. 12&lt;/a&gt;, written by Clifford Truesdell.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-07-18T19:48:28.683" LastActivityDate="2017-07-18T19:48:28.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6309" PostTypeId="1" CreationDate="2017-07-17T13:58:37.843" Score="5" ViewCount="157" Body="&lt;p&gt;I could give (but, not being a professional historian, nor a native Dutch speaker, only &lt;em&gt;few&lt;/em&gt;) references and background-remarks, but I will keep this short, to make more use of what a Q&amp;amp;A-sites offers, hoping for a summary by professional hiostorians and philophers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In &lt;em&gt;constructivism&lt;/em&gt; (in the technical sense), is &quot;mind&quot; more or less merely a &lt;em&gt;convenient&lt;/em&gt; monosyllabic and four-letter &lt;em&gt;synonym&lt;/em&gt; for &quot;algorithm/computer/lambda calculus/Turing machine&quot;, or is there more to the predilection of constructivists for using this word when describing what constructivism is about? (Such as in &quot;proofs are object constructed by our minds&quot;.) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How much weight did Brouwer himself attach to using this problematic concept/word &quot;mind&quot; in describing his philosophy? Is there anything left of his writings in which he discusses this very point (i.e., whether &quot;mind&quot; and &quot;algorithm&quot; are interchangeable synonyms).&#xA;Wouldn't he be so much better off by keeping to some objective metaphor, like calculating machines/clockworks/cogs/levers etc when it came to having his philosophy be adopted by others?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Part of my motivation for asking this is a censor-like &quot;Oh, cut out that mind-bit!&quot;-reaction that I often have when doing background reading on constructivism. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I could imagine that Brouwer held opinions in the direction of mind-is-the-most-fundamental-concept-of-all-something-like-a-schopenhauerian-will-and-any-model-of-algorithm-will-be-driven-by-mind-too, or something like that, and would be interested in whether there are writings on that.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Would it be comparing apples with oranges to a professional philosopher to argue that &quot; Brouwer's &quot;mind&quot; is Schopenhauer's &quot;will&quot; &quot; ? EDIT: this is a vague question of course; let me give it precise variant here, more in the direction of comparative literature: have Brouwer's ideas and Schopenhauers concept of &quot;will&quot; been (recognizably) discussed in a published philosophical article? I think this improbable.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It (&quot;mind&quot;) appears over and over again in writings on constructivism, also in very technical mathematical contexts, and is used by professional mathematicians/type theorists. My intuitive reaction to this always has been: Constructivism is just about equally &lt;em&gt;platonic&lt;/em&gt; as &lt;em&gt;platonism&lt;/em&gt;, and constructive proofs are an objective eternal reality as well, so why for goodness' sake do they always mix-in the m-word?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="6060" LastEditDate="2017-07-19T10:36:59.067" LastActivityDate="2017-07-19T10:36:59.067" Title="To constructivists, is &quot;mind&quot; more than a convenient synonym for &quot;algorithm&quot;?" Tags="|mathematics|philosophy-of-science|computers|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6312" PostTypeId="2" ParentId="6309" CreationDate="2017-07-17T20:51:20.760" Score="2" Body="&lt;p&gt;Regarding the algorithmic conception of the mind:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Von Neumann &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Computer_and_the_Brain&quot; rel=&quot;nofollow noreferrer&quot;&gt;definitely regarded human intelligence as algorithmic&lt;/a&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;He may not have known how the brain/computer worked exactly, but it's a reasonable assumption that it involves procedures and memory, which are analogous to the way computers function.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There has been much resistance to this idea, making the point that the human brain is not a &lt;em&gt;digital&lt;/em&gt; computer, but it is unclear to what degree detractors understand computing at a fundamental level, &lt;a href=&quot;http://www.plumbingengineer.com/content/how-mechanics-urinal-flush-inspired-early-computer&quot; rel=&quot;nofollow noreferrer&quot;&gt;which is not restricted to microprocessors&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I personally prefer the term &quot;algorithmic intelligence&quot; over &quot;artificial intelligence&quot; because it is more precise and descriptive, and doesn't carry the stigma of artificiality. (Philip Dick, the narrative philosopher, &lt;a href=&quot;https://en.wikipedia.org/wiki/Do_Androids_Dream_of_Electric_Sheep%3F&quot; rel=&quot;nofollow noreferrer&quot;&gt;has a very famous meditation&lt;/a&gt;, partly on this subject.)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is much we don't understand about consciousness or the workings of the human mind, but materialists would assume any functions to be ultimately rooted in nature. (&lt;a href=&quot;http://www.cs.princeton.edu/courses/archive/fall09/cos126/lectures/23AI-2x2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conway's Game of Life&lt;/a&gt; has theoretical implications regarding algorithmic intelligence/super intelligence, regarding such as merely functions of the complexity of the system.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't comment on Brouwer or Brouwer Schopenhauer, but hopefully this sheds some light on the &quot;proceduralist&quot; conception &lt;em&gt;(if I can rebrand that term to give it a computer science definition;)&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="5197" LastActivityDate="2017-07-17T20:51:20.760" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6313" PostTypeId="1" CreationDate="2017-07-18T06:27:47.277" Score="0" ViewCount="58" Body="&lt;p&gt;Several famous chemists were blown up, poisoned, blinded, mutilated or got lethally sick by radiation. Are these isolated anecdotes or was researching chemistry a truly dangerous job during the period of, say, 1750 to 1950? Since the casualty ratio among the most famous chemists was pretty high, one could imagine that it was even worse among the many who tried but never got that far. I wonder if there's any substance to this assumption (to quote what might've been for example Gustaf Dalén's last words).&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorUserId="3663" LastEditDate="2017-07-18T07:16:46.003" LastActivityDate="2017-07-18T07:16:46.003" Title="How common were death and mutilation among chemists 1750-1950?" Tags="|chemistry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6314" PostTypeId="2" ParentId="3131" CreationDate="2017-07-18T06:34:49.227" Score="26" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Yutaka_Taniyama#Depression_and_death&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yutaka Taniyama&lt;/a&gt; (12 November 1927 – 17 November 1958) was a notable Japanese mathematician known for the Taniyama–Shimura or Taniyama–Shimura–Weil conjecture now referred to as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Modularity_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Modularity theorem&lt;/a&gt; which &amp;quot;states that elliptic curves over the field of rational numbers are related to modular forms.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Proof of a significant special case of the modularity theorem (for semistable elliptic curves) was an essential component of the proof of &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermat's last theorem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Yutaka_Taniyama#Depression_and_death&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On 17 November 1958, Taniyama committed suicide. He left a note explaining how far he had gotten with his teaching duties, and apologizing to his colleagues for the trouble he was causing them. His mystifying suicide note read:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Until yesterday I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="1815" LastEditDate="2023-12-03T11:56:02.430" LastActivityDate="2023-12-03T11:56:02.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6315" PostTypeId="1" CreationDate="2017-07-18T13:40:13.630" Score="1" ViewCount="138" Body="&lt;p&gt;In a letter to Zeilberger the late Alexander Zenkin states that Brouwer called set theory &quot;a pathological incident in history of mathematics from which future generations will be horrified&quot;. (See Doron Zeilberger: &quot;Opinion 68&quot; (2005)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could not find the source. Where, if at all, did Brouwer say so?&lt;/p&gt;&#xA;" OwnerDisplayName="user6065" LastActivityDate="2017-07-27T13:05:51.917" Title="Citation of Brouwer needed" Tags="|biographical-details|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="6316" PostTypeId="2" ParentId="3131" CreationDate="2017-07-18T17:02:24.013" Score="11" Body="&lt;p&gt;Heinrich Kornblum proved an analogue to Dirichlet's theorem on arithmetic progressions in the setting of polynomial rings over finite fields for his PhD thesis. Sadly he was called to fight in WWI and died in battle. His advisor, Edmund Landau, was tasked with typing and publishing the work of his student. You can find a source in &lt;em&gt;Number Theory in Function Fields&lt;/em&gt; by Rosen.&lt;/p&gt;&#xA;" OwnerUserId="6067" LastEditorUserId="62" LastEditDate="2019-08-21T14:17:31.743" LastActivityDate="2019-08-21T14:17:31.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6317" PostTypeId="1" AcceptedAnswerId="6319" CreationDate="2017-07-18T02:15:32.360" Score="12" ViewCount="2539" Body="&lt;p&gt;I'm sorry if this is a dumb question but I've never heard a convincing explanation for why seemingly all of physics names the position vector &quot;$\vec{r}$&quot;. I've tried translating it into just about every language that would make sense, and I've asked several professors at my college, but I can't find a satisfactory answer. Was it just something somebody decided one day?&lt;/p&gt;&#xA;" OwnerUserId="7113" OwnerDisplayName="user2996957" LastActivityDate="2017-07-18T21:22:35.477" Title="Why is the letter $\vec{r}$ used for position?" Tags="|terminology|notation|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6318" PostTypeId="2" ParentId="6317" CreationDate="2017-07-18T02:19:53.007" Score="4" Body="&lt;p&gt;It can be motivated by considering central force motions, like around the sun. The force points against the &lt;em&gt;radial&lt;/em&gt; vector $\vec r$ from the sun to the heavenly body in question. &lt;/p&gt;&#xA;" OwnerUserId="5975" OwnerDisplayName="Enrique Mendez" LastActivityDate="2017-07-18T09:21:15.643" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6319" PostTypeId="2" ParentId="6317" CreationDate="2017-07-18T02:20:43.660" Score="14" Body="&lt;p&gt;The $r$ is for &quot;radius&quot;, and in particular, describes the radial vector from the origin to the location described by the vector.  This is sensible because some sort of polar or spherical coordinates are the most common for many physical applications, where the forces described have some sort of spherical symmetry, and point radially outward.&lt;/p&gt;&#xA;" OwnerUserId="6068" OwnerDisplayName="Jerry Schirmer" LastActivityDate="2017-07-18T02:20:43.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6320" PostTypeId="2" ParentId="6317" CreationDate="2017-07-18T02:23:34.420" Score="6" Body="&lt;p&gt;As far as I know, the letter $r$ comes from the word &quot;radius.&quot; This is best seen in polar coordinates, where the position vector of an object is given by $r\,\hat{r}$, where $r$ is the distance from the origin to the object (ie the &lt;em&gt;radius&lt;/em&gt; of the circle that passes through our object and is concentric with the origin) and $\hat{r}$ is the &lt;em&gt;radial&lt;/em&gt; basis vector in polar coordinates. Since we write $r\,\hat{r}$, we might as well use another $r$ to write the expression for the position of an object in polar coordinates as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\vec{r}=r\,\hat{r}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I know, this is the origin of this notation. Often people will write $\vec{x}$, but they don't want to confuse it with the $x$-direction in a particular problem (or they've already used that letter), and so they use the next thing they know is often used to describe a position, and thus we have $\vec{r}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps!&lt;/p&gt;&#xA;" OwnerDisplayName="Bob Knighton" LastActivityDate="2017-07-18T02:23:34.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6321" PostTypeId="2" ParentId="6317" CreationDate="2017-07-18T02:56:02.157" Score="2" Body="&lt;p&gt;$\vec{r}$ is actually a radius vector but if we consider center of our circle as origin of our x-y axes then the radius vector will also denote the position of particle performing circular motion that's why we can say that $\vec{r}$ is position vector but if we consider origin as any other point in space then $\vec{r}$ won't denote position vector.&lt;/p&gt;&#xA;" OwnerUserId="6446" OwnerDisplayName="Krishna Deshpande KD" LastActivityDate="2017-07-18T02:56:02.157" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6325" PostTypeId="1" CreationDate="2017-07-21T18:46:06.423" Score="0" ViewCount="55" Body="&lt;p&gt;Based on the period when Turing went to school, it is fairly inconceivable he was not well familiar with Protagoras, and the statement that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Man is the measure of all things&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;*&lt;a href=&quot;https://philosophy.stackexchange.com/questions/41237/is-protagoras-the-philosophical-root-of-the-turing-test/41277#41277&quot;&gt;I asked a question about this on philosophy&lt;/a&gt;, and garnered answers casting doubt on the ancient axiom's relevance to computing, but any such categorizations, as my own, are purely subjective, and one critique in general regards the &quot;Imitation Game&quot; as not a philosophical subject, where it is more properly a philosophical &quot;rabbit hole&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless, my question here is more direct: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Did Turing, or those commenting on the Imitation Game, mention the famous axiom attributed to Protagoras?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5197" LastEditorUserId="4251" LastEditDate="2020-07-13T00:33:24.673" LastActivityDate="2020-07-13T00:33:24.673" Title="Did Turing ever mention Protagoras regarding the Imitation Game?" Tags="|philosophy-of-science|artificial-intelligence|turing|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6326" PostTypeId="1" AcceptedAnswerId="6339" CreationDate="2017-07-22T02:39:24.597" Score="4" ViewCount="891" Body="&lt;p&gt;On this and other Stack Exchange website, there have been question about the so-called geometric series, and where its name comes from. My problem is that most answers follow one of two different ideas.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Either they show an ingenious, but obviously not ancient, picture that illustrates concrete examples of geometric series summation. &lt;/li&gt;&#xA;&lt;li&gt;Or they make some reference to the middle of three consecutive terms in the sequence being the &quot;geometric mean&quot; of the other two.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Well, I want to raise the question again, because neither of these prototype answers really explains where the use of &quot;geometric&quot; started, nor who (if known) came up with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any good explanation?&lt;/p&gt;&#xA;" OwnerUserId="1861" LastEditorUserId="26" LastEditDate="2017-07-22T12:02:01.070" LastActivityDate="2022-09-26T14:49:54.307" Title="Where does the name &quot;geometric sequence&quot; come from?" Tags="|terminology|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6327" PostTypeId="2" ParentId="6315" CreationDate="2017-07-22T13:06:03.287" Score="4" Body="&lt;p&gt;I have done a lot of research without any result. The quote is also given in &quot;How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics&quot; and asked the author by e-mail 3 days ago. There was no reply. I am convinced meanwhile that the quote is not authentic. If there will be any positive reaction I will edit this answer.&lt;/p&gt;&#xA;" OwnerDisplayName="user6065" LastActivityDate="2017-07-22T13:06:03.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6328" PostTypeId="2" ParentId="6326" CreationDate="2017-07-22T13:57:06.400" Score="3" Body="&lt;p&gt;In former times sequence and series have often been used for the same purpose.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;GEOMETRIC SERIES is found in 1723 in A System of the Mathematics James Hodgson [Google print search, James A. Landau].&lt;/p&gt;&#xA;&lt;p&gt;The term GEOMETRIC PROGRESSION was used by Michael Stifel in 1543: &amp;quot;Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis&amp;quot; [James A. Landau].&lt;/p&gt;&#xA;&lt;p&gt;Geometrical progression appears in English in 1557 in the Whetstone of Witte by Robert Recorde: &amp;quot;You can haue no progression Geometricalle, but it must be made either of square nombers, or els of like flattes&amp;quot; [OED].&lt;/p&gt;&#xA;&lt;p&gt;Geometric progression is found in English in 1696 in A new theory of the earth, from its original, to the consummation of all things by William Whiston: “Which to how immense a Sum it would arise, those who know any thing of the nature of Geometrick Progressions will easily pronounce.” [OED]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;http://jeff560.tripod.com/g.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jeff560.tripod.com/g.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user6080" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-07-22T13:57:06.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6329" PostTypeId="1" CreationDate="2017-07-23T15:40:00.600" Score="6" ViewCount="330" Body="&lt;p&gt;I often check biographies of physicists and at one point or another in the beginning of their careers they have the opportunity to be mentored by other and often also great scientists. But, I know for sure there are scientists who never needed mentors, such as Gauss (correct me if I'm wrong).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there famous examples of people who did not need mentors to become great physicists? I'm particularly interested of post WWI science.&lt;/p&gt;&#xA;" OwnerUserId="6083" LastActivityDate="2018-03-14T16:26:04.070" Title="Great physicists with no mentors" Tags="|physics|biographical-details|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6331" PostTypeId="1" AcceptedAnswerId="6333" CreationDate="2017-07-23T17:31:21.923" Score="-1" ViewCount="209" Body="&lt;p&gt;The spacetime metric of relativity &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(ds)^2 = - (cdt)^2 + (dx)^2 + (dy)^2 + (dz)^2 $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;attaches physical significance to $\sqrt{-1}$.  (In order to achieve invariance the time differential used must be $-(cdt)^2 = (c\sqrt{-1}dt)^2$ rather than $(cdt)^2$.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prior to Einstein, which physical theories treated $\sqrt{-1}$ as physically significant?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-07-23T17:56:37.303" LastActivityDate="2017-07-23T21:40:46.340" Title="The physical significance of $\sqrt{-1}$ prior to Einstein" Tags="|physics|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6332" PostTypeId="2" ParentId="6331" CreationDate="2017-07-23T21:34:21.767" Score="2" Body="&lt;p&gt;I must say that I don't understand in which way does the expression$$(ds)^2=-(c\,dt)^2+(dx)^2+(dy)^2+(dz)^2$$“attaches physical significance to $\sqrt{-1}$”. Nobody has to think in terms of $\sqrt{-1}$ in order to understand it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides, what has Einstein to do with it? This expression was introduced by Minkowski in his 1907 lecture &lt;em&gt;The fundamental equations for elecromagnetic processes in moving bodies&lt;/em&gt;. If I am wrong, please tell me where did Einstein mention it before.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-07-23T21:34:21.767" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6333" PostTypeId="2" ParentId="6331" CreationDate="2017-07-23T21:40:46.340" Score="3" Body="&lt;p&gt;Complex numbers enter physics in the work of Fresnel on wave theory of light, namely in his derivation of total reflection. This was in the early 19 century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fresnel_equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Fresnel_equations&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For more details, see  Whittaker, E. T., A History of the Theories of Aether and Electricity.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-07-23T21:40:46.340" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6334" PostTypeId="2" ParentId="6329" CreationDate="2017-07-23T21:52:39.837" Score="8" Body="&lt;p&gt;&quot;Correct me if I am wrong&quot;: Gauss's mentor was Johann Friedrich Pfaff.&#xA;See, for example &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18231&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18231&lt;/a&gt;&#xA;It is very rare to have no mentor, no teacher and no adviser. I doubt that you will have convincing 20 century examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One example which comes to my mind is George Green (Early 19 century). He can be equally qualified as a physicist and mathematician, and he made his main discoveries (Green's identity, Green's function, the notion of potential) before&#xA;he entered the university. It seems that he was completely self-taught. This is a very exceptional case, indeed. He had no formal education when he made his discoveries. He was a son of a miller and a miller himself.&#xA;Modern research shows that the only advanced mathematical book available to him was Laplace's Celestial Mechanics which happened to be in the local library in Nottingham. (It is difficult to imagine how a person without preparation can read this book!) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: D. M. Cannell, George Green: miller and mathematician,  City of Nottingham Arts Dept., 1988.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. English translation of Laplace did not exist at that time, so he could only use an original French copy. Where and how did he learn French? All this story looks very enigmatic.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-03-14T16:26:04.070" LastActivityDate="2018-03-14T16:26:04.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6335" PostTypeId="1" CreationDate="2017-07-24T02:23:26.907" Score="3" ViewCount="235" Body="&lt;p&gt;There were (I think) no number theoretic or relation to science that time, and the only impression I get from reading books is that they did it too one-up other persons on competitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That's not a mathematical Motivation. What was the mathematical Motivation ?&lt;/p&gt;&#xA;" OwnerUserId="6088" LastEditorUserId="31" LastEditDate="2017-08-13T01:52:27.297" LastActivityDate="2017-08-13T01:52:27.297" Title="Why were people in sixteenth century interested in solving the cubic?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6336" PostTypeId="1" AcceptedAnswerId="6337" CreationDate="2017-07-24T13:58:28.260" Score="8" ViewCount="706" Body="&lt;p&gt;In analytic number theory, the function&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;L(s,\chi_m) = \sum_{n=1}^\infty \frac{\chi_m(n)}{n^s}.&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is called the &lt;em&gt;Dirichlet L-function&lt;/em&gt; and has many important uses in the study of prime numbers. In particular, if $\chi_1$ is the trivial Dirichlet character, we have the identity&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;L(s,\chi_1)=\zeta(s).&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why were such functions called &quot;&lt;em&gt;L-functions&lt;/em&gt;? What is the functional significance of the letter &lt;em&gt;L&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="3990" LastActivityDate="2017-07-24T23:58:50.687" Title="Why is Dirichlet's L-function called &quot;L-function&quot;?" Tags="|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6337" PostTypeId="2" ParentId="6336" CreationDate="2017-07-24T23:58:50.687" Score="8" Body="&lt;p&gt;It's the original notation used by Dirichlet. The reason why he chose L, without commenting on the choice, rather than some other letter is not known. Chances are there is not much of a reason, and he could just as well have chosen another letter. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Answers to the MathOverflow question &lt;a href=&quot;https://mathoverflow.net/questions/21375/why-are-they-called-l-functions&quot;&gt;Why are they called L-functions?&lt;/a&gt; mainly also assert this, but also present some theories, including:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;It just fitted in naturally with the rest of the notation of a paper. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is for Legendre. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It is for Lejeune. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I rather doubt 3. The plausibility of 2. is based on the original context being closely linked to Legendre, but it's still a bit of a stretch (and presented rather tongue-in-cheek on MO). &lt;/p&gt;&#xA;" OwnerUserId="98" LastActivityDate="2017-07-24T23:58:50.687" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6338" PostTypeId="1" AcceptedAnswerId="6345" CreationDate="2017-07-25T08:26:37.087" Score="3" ViewCount="207" Body="&lt;p&gt;Most sources attribute the first use of &quot;function&quot; in the context of mathematics to Leibniz. But D'Alembert, Lacroix and Dini claim the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D'Alembert in &lt;a href=&quot;http://artflsrv02.uchicago.edu/cgi-bin/philologic/getobject.pl?c.6:47.encyclopedie0416&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Encyclopédie&lt;/em&gt;&lt;/a&gt; 1757:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;les anciens géometres, ou plûtôt les anciens analystes ont appellé &lt;em&gt;fonctions&lt;/em&gt; d'une quantité quelconque $x$ les différentes &lt;em&gt;puissances&lt;/em&gt; de cette quantité.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lacroix in &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k92729z&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Traité du calcul différentiel et du calcul intégral&lt;/em&gt;&lt;/a&gt; 1797, p.1:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Les anciens Analystes comprenoient en général sous la dénomination de &lt;em&gt;fonctions&lt;/em&gt; d'une quantité, toutes les puissances de cette quantité.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Dini in &lt;a href=&quot;https://archive.org/stream/fondamentiperla00dinigoog#page/n49/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Fondamenti per la teorica della funzioni di variabili reali&lt;/em&gt;&lt;/a&gt; 1878, p.35&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gli antichi usarono dapprima la parola funzione per esprimere le varie potenze du un stessa quantità, e solo da Leibnitz, dai Bernoulli e più specialmente poi da Eulero fu esteso il concetto di funzione [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;None of them explain who they mean by &lt;em&gt;ancients&lt;/em&gt;, but Dini makes it quite clear that he doesn't mean Leibniz. So who were these ancients? The greeks?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2017-07-26T09:48:46.560" LastActivityDate="2017-07-26T09:48:46.560" Title="Was the word &quot;function&quot; used in mathematics prior to Leibniz?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6339" PostTypeId="2" ParentId="6326" CreationDate="2017-07-25T17:44:47.530" Score="1" Body="&lt;p&gt;The &quot;why&quot; is not clear. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The origin is with the &lt;a href=&quot;https://plato.stanford.edu/entries/pythagoreanism/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pythagorean School&lt;/a&gt; (see also: &lt;a href=&quot;https://plato.stanford.edu/entries/archytas/#Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archytas&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The early extant souce seems to be &lt;a href=&quot;https://books.google.it/books?id=2ietQ7tX8TEC&amp;amp;pg=PA79&amp;amp;lpg=PA163#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fragment 2&lt;/a&gt; of the lost work of &lt;em&gt;On Music&lt;/em&gt; of Archytas. &lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-07-25T17:44:47.530" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6340" PostTypeId="1" CreationDate="2017-07-25T21:39:53.443" Score="2" ViewCount="103" Body="&lt;p&gt;In chapter 3 of &quot;The Revolutions of the Heavenly Spheres&quot; where Copernicus defines &quot;how the Earth forms a single sphere with water&quot;, the Edward Rosen translation of the book states this. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Moreover, there is no difference between the earth's centers of gravity and&#xA;  magnitude. This can be established by the fact that from the ocean inward the&#xA;  curvature of the land does not mount steadily in a continuous rise.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Based on this. I'm believe that the center of magnitude is referring to the center of mass. And that the figure that he is referring to with a difference between the center of gravity and mass, would be a hollow Earth. Assuming that this is correct, how would a hollow Earth have land that would continuously rise with no changes in elevation from that.&lt;/p&gt;&#xA;" OwnerUserId="6095" LastActivityDate="2018-02-23T22:04:52.327" Title="How would land &quot;mount steadily in a continuous rise&quot; if there was no difference between center of gravity and magnitude, according to Copernicus" Tags="|gravity|earth-sciences|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6341" PostTypeId="2" ParentId="6340" CreationDate="2017-07-25T23:15:28.797" Score="1" Body="&lt;p&gt;I think here &quot;center of magnitude&quot; simply means the geometric center (center of the sphere). If it did not coincide with the center of gravity, the ocean surface would be different from from the surface of the solid part.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The argument seems to be closely related to Archimedes, On floating bodies,&#xA;Section II, where Archimedes proves that the surface of the ocean must be spherical. As the surface of the land only slightly differs from the ocean surface, this indicates  sphericity of Earth.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-28T18:33:41.023" LastActivityDate="2017-07-28T18:33:41.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6342" PostTypeId="2" ParentId="6329" CreationDate="2017-07-25T23:31:55.577" Score="4" Body="&lt;p&gt;What of the great British experimentalists: Cavendish and Faraday?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to post WWI all Nobels were schooled in physics.&lt;/p&gt;&#xA;" OwnerDisplayName="user5245" LastEditorDisplayName="user5245" LastEditDate="2017-07-26T02:46:44.943" LastActivityDate="2017-07-26T02:46:44.943" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6344" PostTypeId="1" CreationDate="2017-07-26T03:18:24.450" Score="3" ViewCount="322" Body="&lt;p&gt;In chapter 3 of &quot;The Revolutions of the Heavenly Spheres&quot; where Copernicus defines &quot;how the Earth forms a single sphere with water&quot;, the Edward Rosen translation of the book states this.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For they do not realize that the water cannot be even seven times greater and still leave any part of the land dry, unless earth as a whole vacated the center of gravity and yielded that position to water, as if the latter were heavier than itself For, spheres are to each other as the cubes of their diameters. Therefore, if Earth were the eighth part to seven parts of water, earth's diameter could not be greater than the distance from [their joint] center to the circumference of the waters.&#xA;  For context, &quot;they&quot; is referring to Aristotelians which according to Copernicus, believe that there is ten times more water than land.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My main lack of understanding comes from how Copernicus uses the statement &quot;spheres are to each other as the cubes of their diameters.&quot; To prove that &quot;if Earth were the eighth part to seven parts of water, earth's diameter could not be greater than the distance from [their joint] center to the circumference of the waters.&quot; Thereby showing that a 1:7 ratio of land to water is the threshold where water submerges the entire Earth.&lt;/p&gt;&#xA;" OwnerUserId="6097" LastEditorUserId="6097" LastEditDate="2017-07-26T03:32:10.953" LastActivityDate="2017-07-31T17:55:25.440" Title="How Copernicus figured out that it took 7 parts of water and 1 part earth until the entire Earth was submersed in water" Tags="|mathematics|earth-sciences|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6345" PostTypeId="2" ParentId="6338" CreationDate="2017-07-26T06:36:28.060" Score="2" Body="&lt;p&gt;The &lt;em&gt;Encyklopädie&lt;/em&gt; article &quot;Principles of the theory of functions&quot; (&lt;a href=&quot;https://archive.org/stream/p1n1encyklopdied02akaduoft#page/3&quot; rel=&quot;nofollow noreferrer&quot;&gt;1899, p. 3&lt;/a&gt;) relays the assertion, quoting Lagrange and Lacroix, but notes it's unconfirmed.  In an &quot;Encyclopedia Chat Room&quot;, Archiv der Math. u. Phys. &lt;strong&gt;3&lt;/strong&gt; (1902) 317-319, G. Eneström &lt;a href=&quot;https://archive.org/stream/archivdermathem45unkngoog#page/n332/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;comments&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lagrange's assertion is no doubt literally false, presumably all he meant is that the first analysts generally dealt with no functions other than powers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The much-expanded translation in &lt;em&gt;Encyclopédie&lt;/em&gt; (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k2025807/f5&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912, p. 3&lt;/a&gt;) traces the quotes of D'Alembert and Leibniz we have above, considers the possibility that D'Alembert used &quot;function&quot; to translate the latin &quot;dignitas&quot; once used for powers (Tartaglia, Bombelli), but in the end, sides with Eneström:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This assertion of &lt;em&gt;J. d'Alembert&lt;/em&gt; is no doubt inaccurate (...) one can affirm that &lt;em&gt;J. d'Alembert&lt;/em&gt; likely didn't bother to find out whether early analysts actually used the word &quot;function&quot;, or not.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-07-26T06:36:28.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6346" PostTypeId="2" ParentId="6329" CreationDate="2017-07-26T13:00:51.840" Score="3" Body="&lt;p&gt;Perhaps it is not quite the sort of example you are looking for, but Lars Onsager seems not to have had a &quot;mentor&quot; in the classical sense, although early in his career he encountered people like Debye, Weyl and Schrödinger. He wrote his thesis some years after beginning to work and publishing now famous papers on irreversible processes. Bundled together they weren't accepted as a thesis in Norway. Later, after hired at Yale he wrote (entirely on his own) a thesis on Mathieu functions that apparently had to be evaluated by Einar Hille in the math department. See for instance this &lt;a href=&quot;http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/onsager-lars.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;NAS biographical memoir&lt;/a&gt; and the commentaries in his Collected Works.&lt;/p&gt;&#xA;" OwnerUserId="633" LastActivityDate="2017-07-26T13:00:51.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6347" PostTypeId="1" AcceptedAnswerId="6349" CreationDate="2017-07-27T00:17:07.107" Score="3" ViewCount="278" Body="&lt;p&gt;I've used the phrase more than once, but in &lt;a href=&quot;https://space.stackexchange.com/a/22170&quot;&gt;this answer&lt;/a&gt; I wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;closing thoughts:&lt;/strong&gt; Realistic orbits are not perfect conics, and so they and their Keplerian elements &lt;em&gt;do not represent realistic orbits&lt;/em&gt;. They are only approximations to reality, and so are not right even though they are close.&lt;/p&gt;&#xA;&lt;p&gt;Keplerian elements were used when people were writing with feathers using light from burning animal fat (if they were not busy being burned at the stake themselves). They are a mixed blessing in the 21st century when everything has so many more digits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I understand that I am probably jumbling a few different centuries together there, but I am wondering about the hardware more than the politics.&lt;/p&gt;&#xA;&lt;p&gt;In the fields of mathematics and orbital mechanics, say from Kepler to Lagrange, were mathematical derivations and computations done with pens made from feathers, and if an inspiration came after sundown, was light often as not produced by animal fat, whale oil, or the like?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-07-27T11:04:52.030" Title="Tools of the trade: were early scientists and mathematicians really &quot;writing with feathers using light from burning animal fat?&quot;" Tags="|mathematics|computation|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6348" PostTypeId="1" CreationDate="2017-07-27T08:03:57.003" Score="3" ViewCount="635" Body="&lt;p&gt;&lt;strong&gt;Archimedes principle:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Any object, wholly or partially immersed in a stationary fluid, is&#xA;  buoyed up by a force equal to the weight of the fluid displaced by the&#xA;  object.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I get that this can be shown with force calculations on an object in a fluid. But how come this fact was suddenly so obvious to someone taking a bath thousands of years ago?&lt;/p&gt;&#xA;" OwnerDisplayName="Immurse" LastEditorUserId="4251" LastEditDate="2021-12-14T20:28:50.513" LastActivityDate="2021-12-14T20:28:50.513" Title="How did Archimedes arrive at his principle in his time?" Tags="|physics|ancient-greece|archimedes|antiquity|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6349" PostTypeId="2" ParentId="6347" CreationDate="2017-07-27T10:45:28.893" Score="5" Body="&lt;p&gt;Quick Internet search says that feather quills were mainly used in 600-1800 AD.&#xA;After that people gradually switched to steel. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First true mathematicians (Babylonians) wrote on clay tablets.&#xA;In the Greek/Roman Antiquity they wrote on papyrus, presumably with reed pens or brushes.&#xA;Apparently they switched from brushes to pens in Ptolemaic Egypt, that is exactly when serious mathematics started:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.brooklynmuseum.org/community/blogosphere/2010/09/22/pigments-and-inks-typically-used-on-papyrus/&quot; rel=&quot;noreferrer&quot;&gt;https://www.brooklynmuseum.org/community/blogosphere/2010/09/22/pigments-and-inks-typically-used-on-papyrus/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Papyrus was used for &quot;publication&quot;, for books and perhaps for letters, &#xA;not for the actual work.&#xA;The work was done on wax tablets, since the Roman times, and on sand before that. Even in 18th century, paper was not cheap enough for use as &quot;scratch paper&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The data on when pencil became common diverge by 200 years, some say in 16th century, others in 18th. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of light sources, it does not have to be animal fat. The most common lamps used all kinds of vegetable oil. These lamps of various shapes were commonly used since Paleolith till 20th century in many places. &#xA;In many locations, vegetable oil is cheaper than animal fat or wax candles.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-27T11:04:52.030" LastActivityDate="2017-07-27T11:04:52.030" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6350" PostTypeId="1" AcceptedAnswerId="6352" CreationDate="2017-07-27T12:38:39.427" Score="6" ViewCount="85" Body="&lt;p&gt;When Kepler was an assistant of Tycho Brahe, did Kepler in public declare his support for Copernicus' system or Brahe's system (the earth at rest, but the planets orbiting the sun) or was he undecided? The latter is told in the novel by Max Brod: &quot;Tycho Brahe's path to God&quot;. But that's a novel only.&lt;/p&gt;&#xA;" OwnerDisplayName="user6027" LastEditorUserId="4251" LastEditDate="2021-02-28T08:08:44.177" LastActivityDate="2021-02-28T08:08:44.177" Title="Kepler's position with respect to Tycho Brahe's world system?" Tags="|astronomy|17th-century|kepler|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6351" PostTypeId="2" ParentId="6315" CreationDate="2017-07-27T13:05:51.917" Score="1" Body="&lt;p&gt;The quote is probably no authentic since Mückenheim, who is very careful in these matters, does not ascribe it to Brouwer in his Zenkin text quoted on p. 197 of &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Transfinity&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user6027" LastActivityDate="2017-07-27T13:05:51.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6352" PostTypeId="2" ParentId="6350" CreationDate="2017-07-27T16:02:27.597" Score="5" Body="&lt;p&gt;Kepler never supported Brahe's system in print. &#xA;In his first book (1596) he proposes&#xA;his own version of heliocentric system. Collaboration with Brahe is 1600-1612.&#xA;During this period he wrote Astronomia Nova (1609) based on heliocentric system.&#xA;And all his later books are based on heliocentric system as well.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-07-27T16:02:27.597" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6353" PostTypeId="2" ParentId="3131" CreationDate="2017-07-27T19:50:02.653" Score="7" Body="&lt;p&gt;Mukremin Nesheli.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was a young Turkish mathematician. He had such strong mathematical abilities that the Canadian mathematician Langlands immediately recognized him and Langlands wanted to invite him to Yale University. &#xA;But Mukremin was a leftist and was killed by a right-wing group in Turkey. This story always reminds me of Evariste Galois.&lt;/p&gt;&#xA;" OwnerUserId="6049" LastActivityDate="2017-07-27T19:50:02.653" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6355" PostTypeId="1" AcceptedAnswerId="6356" CreationDate="2017-07-28T01:56:21.163" Score="2" ViewCount="183" Body="&lt;p&gt;In the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&#xA; page about Cantor's diagonal argument&lt;/a&gt;, it says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Historically, the diagonal argument first appeared in the work of Paul du Bois-Reymond in 1875.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, the diagonal argument is usually associated with the name &lt;em&gt;Cantor&lt;/em&gt;. Thus I wonder: &lt;strong&gt;Did Cantor knew the work of Paul du Bois-Reymond?&lt;/strong&gt; Or did he discover the diagonal method independently of Paul du Bois-Reymond?&lt;/p&gt;&#xA;" OwnerDisplayName="user6103" LastActivityDate="2017-07-28T12:02:48.460" Title="Did Cantor knew the work of Paul du Bois-Reymond (the original inventor of the diagonal argument proof method in mathematics)?" Tags="|mathematics|mathematicians|set-theory|" AnswerCount="1" CommentCount="1" ClosedDate="2017-08-19T05:15:32.107" ContentLicense="CC BY-SA 3.0" />
  <row Id="6356" PostTypeId="2" ParentId="6355" CreationDate="2017-07-28T12:02:48.460" Score="2" Body="&lt;p&gt;It is very probable that Cantor knew the work of du Bois-Reymond.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Meschkowski and Nilson write in Georg Cantor - Briefe, p. 78 (my translation): &quot;With respect to the intensity with which Cantor used to study the papers of others, it is hardly possible that he did not notice this example.&quot; (Here they refer to the &quot;Cantor set&quot; which had been described by du Bois-Reymond earlier.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Meschkowski and Nilson write in Georg Cantor - Briefe, p. 37  that du Bois-Reymond has used the diagonal procedure 15 years before Cantor. With respect to their remark in (1) it is very probable that Cantor learnt this argument from du Bois-Reymond.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) This suspicion is supported by other plagiarisms of Cantor. For instance Dedekind proved that the algebraic numbers are countable and wrote it in a private letter to Cantor. Cantor used his argument and published this proof without giving credit to Dedekind. Dedekind himself has made a note of this case where he mentiones this incident including the fact that Cantor used the technical term &quot;Höhe&quot; which Dedekind had invented just for the use in his proof and had wrote it to Cantor. This note is published in E. Noether, J. Cavaillès: Briefwechsel Cantor-Dedekind, Hermann, Paris (1937), p. 18:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Hierauf habe ich umgehend geantwortet, dass ich die erste Frage nicht entscheiden könnte, zugleich aber den Satz ausgesprochen und vollständig bewiesen, dass sogar der Inbegriff aller algebraischen Zahlen sich dem Inbegriffe (n) der natürlichen Zahlen in der angegebenen Weise zuordnen lässt (&lt;em&gt;dieser Satz und Beweis ist bald darauf fast wörtlich, selbst mit dem Gebrauch des Kunstausdruckes Höhe&lt;/em&gt;, in die Abhandlung von Cantor in Crelle Bd. 77  übergegangen, nur mit der gegen meinen Rath festgehaltenen Abweichung, dass nur der Inbegriff aller reellen algebraischen Zahlen betrachtet wird).&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Italics are mine.&lt;/p&gt;&#xA;" OwnerDisplayName="user6027" LastActivityDate="2017-07-28T12:02:48.460" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6357" PostTypeId="2" ParentId="6348" CreationDate="2017-07-28T18:06:59.877" Score="6" Body="&lt;p&gt;The bath is a later legend. The principle can be discovered and proved by the following argument which requires no calculation. Suppose that a body is submerged by water.&#xA;The force with which water acts on it is the pressure on the surface of the body.&#xA;Now replace the inside of the body by water. Then we have just water in water and nothing moves (equilibrium). Which means that the weight of the water inside the surface is equal to the pressure of the water outside of the surface. But this last pressure evidently does not depend on what's inside the surface. This proves the Archimedes law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was Archimedes's own argument, given in his book &quot;On Floating bodies&quot;.&#xA;All stories about bath and crown and bull's sacrifice and Eureka were invented by later biographers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-28T18:19:32.577" LastActivityDate="2017-07-28T18:19:32.577" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6358" PostTypeId="1" CreationDate="2017-07-28T18:16:20.443" Score="33" ViewCount="38720" Body="&lt;p&gt;There is a story about Alexander Grothendieck and the &quot;Grothendieck Prime&quot;  57, which goes roughly as follows (cf. &lt;a href=&quot;https://en.wikipedia.org/wiki/57_(number)&quot; rel=&quot;noreferrer&quot;&gt;this wikipedia article&lt;/a&gt;): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a mathematical conversation, someone suggested to Grothendieck&#xA;  that they should consider a particular prime number.&#xA;  “You mean an actual number?” Grothendieck&#xA;  asked. The other person replied, yes, an actual&#xA;  prime number. Grothendieck suggested, “All right, take 57.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This quote is taken from Allyn Jackson's article &lt;a href=&quot;http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;Comme Appelé du Néant—&#xA;As If Summoned from the Void:&#xA;The Life of Alexandre&#xA;Grothendieck&quot;&lt;/a&gt;. Jackson refers to the story as a &quot;legend&quot;. One can argue that the story is quite believable given Grothendieck's way of thinking (David Mumford: &quot;He (Grothendieck) doesn’t think concretely&quot;).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; What (if any) is the factual basis of the story? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, when/where did it happen? (In different versions it is said to have happened during or after a Grothendieck's talk.) Did anybody hear this story from somebody present at Grothendieck's talk? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My guess is that the story is just a legend, but I could be mistaken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Just for the sake of completeness, here is what Georges Elencwajg had to say about this issue (extracted from my conversation with him in comments to an answer to &lt;a href=&quot;https://math.stackexchange.com/questions/2373333/do-most-mathematicians-know-most-topics-in-mathematics&quot;&gt;this&lt;/a&gt; math.stackexchange question): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The story is not made up: Grothendieck did make that silly blunder, in an exchange after a talk, after being asked to be more concrete by a member of the audience. Of course this doesn't change anything to the fact that Grothendieck was one of the most profound arithmeticians of the 20th century. And indeed 57 looks a bit prime for some psychological reason :-) . Conversely many mathematicians think I'm pulling their leg when I tell them that 4999 is prime! ... I've heard this story a long time ago. I think it is true but I can't prove it since, alas, most of the protagonists are dead. Anyway, this is just an amusing but quite meaningless anecdote: a genius made a lapsus linguae. So what? On the other hand I'm quite sure that Allyn Jackson can't disprove what he (strangely) calls a legend... &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Cross-posted at Mathoverflow: &lt;a href=&quot;https://mathoverflow.net/questions/326912/story-of-grothendiecks-prime-number&quot;&gt;https://mathoverflow.net/questions/326912/story-of-grothendiecks-prime-number&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="4251" LastEditDate="2021-09-25T15:10:06.253" LastActivityDate="2021-09-26T05:16:53.537" Title="Story of Grothendieck's Prime Number" Tags="|mathematicians|debunking|grothendieck|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="6359" PostTypeId="1" CreationDate="2017-07-28T22:01:46.677" Score="-1" ViewCount="123" Body="&lt;p&gt;I need the details please. I have been searching information about that but I can't find something useful.&lt;/p&gt;&#xA;" OwnerUserId="6108" LastActivityDate="2017-07-29T15:19:53.997" Title="who calculated for the first time the volume of a cylinder? How?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6360" PostTypeId="2" ParentId="6359" CreationDate="2017-07-29T15:13:34.243" Score="2" Body="&lt;p&gt;The question is not well defined. That the volume of the cylinder equals to the product of the area of the base and height is pretty evident, and once a rigorous theory of volume was stated, the proof is immediate. This is contained in Euclid, for example. But this only reduces the volume of the cylinder to the area of the disk. For the area of the disk it was known to Euclid that it is proportional to &#xA;the square of the radius. And also equals (1/2) times radius times circumference.&#xA;The coefficient of proportionality is called $\pi$. Same number $\pi$ you need to&#xA;calculate the circumference.&#xA;This number $\pi$ cannot be&#xA;&quot;calculated&quot;. One can only approximate it. For example, Archimedes approximated it to 5 digits, while the modern mathematicians with the help of a computer obtained about one billion digits. But that's about all one can do.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-07-29T15:19:53.997" LastActivityDate="2017-07-29T15:19:53.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6361" PostTypeId="2" ParentId="6344" CreationDate="2017-07-30T11:47:25.593" Score="2" Body="&lt;p&gt;I am not able to give a complete answer to your question and admit that I do not really understand what Copernicus is saying, but give this as background information: The notion that the proportion of water to earth in the cosmos is 7 : 1 goes back to Averoes (Ibn Rushd) in his Shorter Commentary to Aristotle’s Meteorology, with decidedly specious arguments. It seems that Copernicus accepted this bit of mediaeval speculation. To my recollection, Aristotle himself has nothing to say on this matter.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-07-30T11:47:25.593" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6362" PostTypeId="1" AcceptedAnswerId="6369" CreationDate="2017-07-30T12:17:07.020" Score="4" ViewCount="467" Body="&lt;p&gt;Until recently, i thought that the first to observe the connection between modular forms and the hyperbolic plane (a model of non-euclidean geometry) was Felix Klein. But in the last week i read the chapter on non-euclidean geometry in the book of John Stillwell - &quot;Mathematics and it's history&quot;, where it's mentioned that using his theory of modular forms Gauss discovered a tessellation of the Poincare disk model by equilateral triangles of angles: $$\pi/4,\pi/4,\pi/4$$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how Gauss could accomplish this without knowing the hyperbolic plane? i understand that this work was done in the context of elliptic functions and modular forms (not hyperbolic geometry), but the fact that he used triangles with angles sum less than pi, suggests that he might suspect that it's connected with hyperbolic geometry. Also, the fact that Klein studied very closely the writings of Gauss might suggest that this accomplishment has historical influence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So i'll be glad to clarify this &quot;historical corner&quot;.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10856" LastEditDate="2024-03-11T07:43:05.553" LastActivityDate="2024-03-11T10:19:14.577" Title="Was Gauss aware of the non-euclidean implications of his work on modular forms?" Tags="|biographical-details|gauss|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6363" PostTypeId="1" CreationDate="2017-07-30T14:56:32.950" Score="1" ViewCount="1921" Body="&lt;p&gt;When did Grace Murray Hopper say &quot;The wonderful thing about standards is that there are so many of them to choose from?&quot;  This quote is attributed to her in many places (including in one of my books), but I have never found the source of it. Does a source exist?&lt;/p&gt;&#xA;" OwnerUserId="6112" LastActivityDate="2017-08-29T17:03:31.637" Title="When did Grace Murray Hopper say &quot;The wonderful thing about standards is that there are so many of them to choose from?&quot;" Tags="|biographical-details|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6364" PostTypeId="2" ParentId="6363" CreationDate="2017-07-30T16:38:13.080" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikiquote.org/wiki/Andrew_S._Tanenbaum&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikiquote&lt;/a&gt;, it is due to Andrew S. Tanenbaum. It is taken from is book &lt;em&gt;Computer Networks&lt;/em&gt;. The first edition appeared in 1980 (I don't know if it contains that sentence) and I confirmed that the fifth edition (2001) has the sentence.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-07-30T16:38:13.080" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6367" PostTypeId="1" AcceptedAnswerId="6373" CreationDate="2017-07-31T12:53:33.680" Score="3" ViewCount="2607" Body="&lt;p&gt;After the Second World War Russia had a vastly larger number of students being trained in their universities compared to the United States around the time of the International Geophysical Year in 1957.  Previous to the discovery of the transistor a great deal of discoveries were made in the area of radio and radar which lead right into the quest to learn about semiconductors but the names I read about appear to be mainly American and British in the semiconductor initial research and development.  I am thinking did Russia have its own version of a semiconductor explosion fueled by it's own scientists ? As for example I believe but am not sure they credit their own scientists with powered flight and I believe do not recognize the Wright Brother's as being the first, as for example. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-07-08T06:56:05.493" LastActivityDate="2020-07-08T06:56:05.493" Title="Did the Soviet Union have their own discoveries in the development of semiconductors and the semiconductor industry?" Tags="|discoveries|technology|soviet-union|electronics|cold-war|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6369" PostTypeId="2" ParentId="6362" CreationDate="2017-08-01T15:06:06.637" Score="4" Body="&lt;p&gt;The best source on this is Klein's Lectures on mathematics in XIX century, vol. I.&#xA;It has a whole chapter on Gauss, with main focus on elliptic integrals and modular forms. Klein can be trusted because he himself worked on this and really read a lot of Gauss's writings. It seems to me from this book that Gauss did not know about the connection with non-Euclidean geometry, and Poincare's model is indeed&#xA;Poincare's invention.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-08-01T15:06:06.637" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6370" PostTypeId="1" CreationDate="2017-07-31T15:23:48.667" Score="7" ViewCount="353" Body="&lt;p&gt;How did people realize they could do logic with electronics? Are there anecdotes of the first realizations? I'm wondering about the first &quot;eureka&quot; moments.&lt;/p&gt;&#xA;" OwnerDisplayName="Kinnard Hockenhull" LastEditorUserId="4251" LastEditDate="2020-07-08T17:06:28.467" LastActivityDate="2020-07-08T17:06:28.467" Title="How did people realize they could do logic with electronics?" Tags="|technology|logic|electronics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6373" PostTypeId="2" ParentId="6367" CreationDate="2017-08-02T23:23:15.923" Score="8" Body="&lt;p&gt;Here is a partial translation of a &lt;a href=&quot;http://gete.ru/post_1172774080.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russian article&lt;/a&gt; about the history of semiconductors (part that addresses Soviet contributions):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1948-1950. Not only in the United States, but also in other countries there have been scientific research in the field of semiconductors. For instance, a Soviet physicist V.E. Loshkarev as early as in 1946 discovered the bipolar diffusion of nonequilibrium current carriers in semiconductors. A.V.Krasilov and his group designed germanium diodes for radar stations. [Presumably still classified research] in Fryazino (Moscow region) in the Research Institute-160 (Research Institute &amp;quot;Istok&amp;quot;). A.V.Krasilov and S.G.Madoyan first observed the transistor effect. The creators of the domestic transistor A.V.Krasilov and S.G.Madoyan published the first article in the USSR on transistors, &amp;quot;Crystal Triode.&amp;quot; Laboratory samples of germanium transistors were developed by B.M.Vul, A.V.Rzhanov, V.S.Vavilov, and others (FIAN), V.M.Tuchkevich, D.N.Nasledov (LPTI), S.G. Kalashnikov, N.A. Penin and others (IRE USSR AS)...&lt;/p&gt;&#xA;&lt;p&gt;1960s. In the USSR in 1963 the Center for Microelectronics was established in Zelenograd. A Soviet engineer F.A. Shchigol developed a planar transistor 2T312 and its open-ended analogue 2T319, which became the main active element of hybrid circuits. In 1964, the Angstrem plant at the Research Institute of Precision Technology created the first integrated circuit IC-&amp;quot;Path&amp;quot; with 20 elements on the crystal, performing the function of transistor logic with resistive connections. At NIIME in Zelenograd technology was created and the first planar transistors &amp;quot;Ploskost&amp;quot; were launched. Under the direction of B.V. Malin at NII-35 (nowadays the scientific research institute &amp;quot;Pulsar&amp;quot;), the first series of silicon integrated circuits TS-100 (degree of integration - 37 elements on the crystal) was created. In 1966, the first experimental station for the production of planar integrated circuits began operating at the Pulsar Research Institute. At the NIIME the group of Dr. Valieva,  started production of logical and linear integrated circuits. In 1968, the Pulsar Research Institute produced a batch of first hybrid thin-film ICs with planar, open-frame transistors of the types KD910, KD911, KT318 intended for television, radio broadcasting and communications. Digital and linear ICs of mass application (series 155) were developed in the Research Institute of ME. In 1969, a Soviet physicist Zh. I. Alferov formulated and practically realized his ideas for controlling electron and light fluxes in classical heterostructures based on the gallium arsenide arsenide system. Digital and linear ICs of mass application (series 155) were developed in the Research Institute of ME.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is a fascinating story of two Soviet spies, Barr and Sarant, who were at the origin of foundation of the  &amp;quot;Soviet Silicon Valley&amp;quot; in Zelinograd in 1963 (see above); it is discussed in &lt;a href=&quot;https://en.wikipedia.org/wiki/Joel_Barr&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; wikipedia article and in much more detail in Usdin's book that I liked quite a bit when I read it few years ago:&lt;/p&gt;&#xA;&lt;p&gt;Steven T. Usdin, &lt;em&gt;Engineering Communism: How Two Americans Spied for Stalin And Founded the Soviet Silicon Valley&lt;/em&gt;, Yale University Press, 2005, hardcover, ISBN 0-300-10874-5.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-08-06T00:46:25.830" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6374" PostTypeId="1" CreationDate="2017-08-03T01:16:43.680" Score="4" ViewCount="180" Body="&lt;p&gt;What are some examples of big results in mathematics and or physics that took a long time to be considered groundbreaking? What was the length of time from the original publication to the recognition?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some examples for which this process was really quick?&lt;/p&gt;&#xA;" OwnerUserId="6127" LastEditorDisplayName="user466" LastEditDate="2017-08-04T12:58:10.490" LastActivityDate="2017-08-04T13:06:13.783" Title="Time for big results to become widely recognized in the scientific community" Tags="|mathematicians|publishing|mathematical-physics|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6376" PostTypeId="2" ParentId="6374" CreationDate="2017-08-03T09:03:28.253" Score="7" Body="&lt;p&gt;There is no point in giving examples on the second question, because for most truly groundbreaking results this time is really short.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the first question, one can mention almost all results of Archimedes and Apollonius, which had almost no development in antiquity, and during the middle age there were no people who understood them. (People read and discussed more elementary books of Euclid, but not Archnimedes or Apollonius). True recognition only came in 16th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the modern times, the period is smaller. One can mention Sharkovskii's theorem (see Wikipedia) which was proved in 1964, but was not noticed by the large mathematics community until the paper by Li and Yorke (1975) who proved a special case. This special case created a lot of excitement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example is Fatou-Julia theory of iteration of rational functions&#xA;created around 1920. It was dormant and almost forgotten until new developments made it one of the most fashionable areas of mathematics in the middle 1980s.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-08-03T09:03:28.253" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6378" PostTypeId="2" ParentId="6374" CreationDate="2017-08-03T15:51:05.950" Score="6" Body="&lt;p&gt;From an &lt;a href=&quot;http://www.damtp.cam.ac.uk/user/tong/em/dyson.pdf&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; by Freeman Dyson:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the year 1865, Maxwell published his paper 'a dynamical theory of the electromagnetic field' ... we, with the advantage of hindsight can clearly see that his paper was the most important event in the 19th C in the history of the physical sciences ... but the importance of Maxwells work was not obvious to his contemporaries. For more than twenty years his work was largely ignored. Physicists found it hard to understand because the equations were complicated. Mathematicians found it hard to understand as he used physical language. It was regarded as an obscure speculation without much experimental support.&lt;/p&gt;&#xA;&lt;p&gt;The physicist Michael Pupin, in his autobiography 'from immigrant to inventor' describes how he travelled from America to Europe in search of someone who understood Maxwell ... Pupin went first to Cambridge hoping to learn the theory from Maxwell himself. He did not know that Maxwell had died four years earlier (!). After learning of his death he stayed on in Cambridge and was assigned a college tutor who knew less about Maxwell than he did and was only interested in teaching him to answer questions in the Mathematical Tripos.&lt;/p&gt;&#xA;&lt;p&gt;He was amazed to discover, as he said, 'How few the physicists were that had caught the meaning of the theory, even after twenty years after Maxwell had first stated it in 1865'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-08-03T15:51:05.950" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6379" PostTypeId="2" ParentId="6374" CreationDate="2017-08-03T17:15:05.077" Score="7" Body="&lt;p&gt;Discovery of non-Euclidean geometries in 19th century, it took about 40-50 years for them to get accepted.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2017-08-03T17:15:05.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6380" PostTypeId="2" ParentId="6374" CreationDate="2017-08-04T13:06:13.783" Score="4" Body="&lt;p&gt;The Michelson-Morley experiment took a very long time to be widely known, and it also took a very long time to be accepted, even by Michelson and Morley themselves.  The original experiment was in 1887, but Michelson continued doing different versions of the experiment until 1929, trying to get the &quot;right&quot; result. Einstein may or may not have known about it in 1905, and if he did, it played no major part in his reasoning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Special relativity also took a long time to be widely accepted. Einstein published it in 1905, but there was a lot of skepticism until around World War II. The early evidence was either uncertain or ambiguous. The more solid evidence came with Ives-Stilwell in 1938 and Rossi-Hall in 1941.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-08-04T13:06:13.783" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6381" PostTypeId="1" CreationDate="2017-08-04T15:24:03.600" Score="0" ViewCount="1187" Body="&lt;p&gt;What's the significance of date 0000? Why was this day chosen to be the start of the calendar?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What comes before 0000? I know they say 1BC. But then shouldn't current time be called AD?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Don't know if 0000 is termed as &lt;a href=&quot;https://en.wikipedia.org/wiki/Year_zero&quot; rel=&quot;nofollow noreferrer&quot;&gt;Year Zero&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6131" LastEditorDisplayName="user10930" LastEditDate="2020-05-05T21:29:10.087" LastActivityDate="2020-05-05T21:29:10.087" Title="Why was 0000 chosen to be the start of the calendar?" Tags="|calendar|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6382" PostTypeId="2" ParentId="6370" CreationDate="2017-08-04T23:21:53.527" Score="4" Body="&lt;p&gt;I think the resources listed above are good background references but if you want to know how logic was introduced to actual electronic equipment then consider Claude Shannon.  At the age of 21, writing his MIT Master's degree thesis, he laid out the methods of how electrical applications of Boolean logic can carry out any logical or numerical process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was in the year 1937 and although it is not the start of work on computing devices or unique, Shannon, more than any other single person, is responsible for answering your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Claude Shannon is considered the father of Information Theory and also wrote many papers on the entropy of information and using statistical methods that were not that different from statistical physics (e.g. Boltzmann Distribution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, a small extra -- Shannon wrote at least two papers on algorithms for a chess playing computer implementation.  I wrote a chess playing program back in the late 1960s and early 1970s when I completed all my work on it.  Most of my motivation came from Shannon's papers which I still have in some binder someplace.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2017-08-04T23:21:53.527" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6383" PostTypeId="1" CreationDate="2017-08-05T01:17:34.130" Score="1" ViewCount="84" Body="&lt;p&gt;Which journals were the most commonly used journals for the publication of research papers in physics in the late 19th century and the early 20th century, specifically from 1875 to 1925?&lt;/p&gt;&#xA;" OwnerUserId="6133" LastEditorUserId="26" LastEditDate="2017-08-05T13:39:59.793" LastActivityDate="2017-08-05T13:39:59.793" Title="Journals of the early 20th century" Tags="|physics|publishing|resource-recommendation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6384" PostTypeId="2" ParentId="6381" CreationDate="2017-08-05T10:52:59.543" Score="4" Body="&lt;p&gt;In the common counting (in history) there is no year zero. Our common calendar&#xA;begins with year 1 AD preceded by year 1 BC. See&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Calendar_era&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Calendar_era&lt;/a&gt;,&#xA;chapter Dionysian Common Era.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course this is inconvenient for astronomy. &#xA;So astronomers sometimes use year 0,&#xA;which is the same as 1 BC in the common system, and then use negative numbers&#xA;for the preceding years. See the same article, chapter Dionysian derived.&#xA;For other, more convenient systems used in astronomy, see the same article, and&#xA;the article &quot;Julian Day&quot; in Wikipedia, they place year 0 further in the past, so that all events in the recorded history have positive year numbers. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-08-05T11:01:56.773" LastActivityDate="2017-08-05T11:01:56.773" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6385" PostTypeId="2" ParentId="6381" CreationDate="2017-08-06T01:47:25.870" Score="3" Body="&lt;p&gt;I suspect that your notation 000 misses one extra zero and what you have in mind is the &lt;a href=&quot;https://en.wikipedia.org/wiki/ISO_8601&quot; rel=&quot;nofollow noreferrer&quot;&gt;ISO 8601&lt;/a&gt; calendar. This calendar does have the year 0000 (which corresponds to 1BC), dates prior to that have negative value (say, 2BC = -0001, etc). The date 0000 has no particular significance, according to Christian traditions, this is within a couple of years from the birth of Jesus. From the modern physics viewpoint, a more reasonable choice for &quot;year zero&quot; would be the time of the &quot;big bang&quot;. More information you can easily find by googling. &lt;/p&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2017-08-06T01:47:25.870" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6386" PostTypeId="1" CreationDate="2017-08-07T11:57:35.757" Score="3" ViewCount="107" Body="&lt;p&gt;In what connection did Zermelo say: &quot;Could not just this seemingly so fruitful hypothesis of the infinite have introduced straight contradictions into mathematics, thereby destroying the basic nature of this science that is so proud upon its consistency?&quot; (H.-D. Ebbinghaus, V. Peckhaus: &quot;Ernst Zermelo, an approach to his life and work&quot;, Springer (2007) p. 292) And what can he have meant by this sentence?&lt;/p&gt;&#xA;" OwnerDisplayName="user6050" LastActivityDate="2017-08-08T02:46:37.150" Title="Information about a statement of Zermelo required" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6387" PostTypeId="2" ParentId="6386" CreationDate="2017-08-07T19:45:33.513" Score="0" Body="&lt;p&gt;The answer to the second question is quite clear: he means that mathematics dealing with sets of infinite cardinality might be inconsistent (i.e. lead to contradictions). As a specific example , we do not know if ZF (the Zermelo-Frankel axiomatic set theory) is consistent, and, more worrisome,  the Peano arithmetic might be inconsistent. We still do not know. There are some recent attempts by &lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Nelson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ed Nelson&lt;/a&gt; to prove inconsistency of Peano's Axioms, but, as far as I know, they were unsuccessful (per wikipedia: &quot;An error was found in the proof, and he retracted the claim&quot;). See also &lt;a href=&quot;https://en.wikipedia.org/wiki/Finitism&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; and  &lt;a href=&quot;https://en.wikipedia.org/wiki/Ultrafinitism&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; wikipedia articles on finitism and ultrafinitism, which are attempts to avoid dealing with infinities in math.  &lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2017-08-07T22:58:44.207" LastActivityDate="2017-08-07T22:58:44.207" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6388" PostTypeId="2" ParentId="6386" CreationDate="2017-08-08T00:37:31.790" Score="3" Body="&lt;p&gt;The quote is given in section &lt;strong&gt;4.4.5&lt;/strong&gt; of &lt;a href=&quot;https://books.google.ca/books?id=0WB1CgAAQBAJ&amp;amp;pg=PA181&amp;amp;lpg=PA181&amp;amp;dq=Ebbinghaus%20Peckhaus%20Ernst%20Zermelo%20%22this%20seemingly%20so%20fruitful%20hypothesis%22&amp;amp;source=bl&amp;amp;ots=TIshZboD1q&amp;amp;sig=Qv9VeGESGfnMf3EEMP4VQw_ZUsU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwirs5Hmm8bVAhUX9WMKHU69BREQ6AEILjAB#v=onepage&amp;amp;q=Ebbinghaus%20Peckhaus%20Ernst%20Zermelo%20%22this%20seemingly%20so%20fruitful%20hypothesis%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ebbinghaus, Peckhaus&lt;/a&gt; titled &quot;&lt;strong&gt;The Justification Problem&lt;/strong&gt;&quot;, where The Justification Problem is defined as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the necessity of justifying the existence of (infinite) models&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to Ebbinghaus, Zermelo expresses the view that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;in general it is impossible to prove consistency, i.e., to exhibit a model.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Note that Zermelo is speaking here in 1929, two years before Gödel proved his incompleteness results, with its implications for the provability of consistency.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the &quot;fruitful hypothesis&quot; is the assumption of a consistent model of axiomatic set theory, and the contradictions are those which may be present in any axiomatization in the absence of a consistency proof.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;TL/DR;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Zermelo's reasoning is that in a mathematical proof of consistency, we could never be sure that we have taken into account all modes of inference used in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Such a demonstration, if it were possible, would have to rest on a thorough and complete formalisation of all the logic relevant to mathematics.  Any &quot;incompleteness&quot; in the underlying &quot;proof theory&quot; such as a neglected possible inference would jeopardise the entire proof.  But, since such &quot;completeness&quot; could obviously never be guaranteed, it is, in my opinion, not possible to furnish a formal proof of the consistency.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Zermelo goes on to note that a consistency proof would have to deal with an infinite set of sentences and therefore would simply shift the problem from one infinite domain to another. &lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-08-08T02:46:37.150" LastActivityDate="2017-08-08T02:46:37.150" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6389" PostTypeId="1" AcceptedAnswerId="6390" CreationDate="2017-08-08T13:43:35.417" Score="8" ViewCount="3637" Body="&lt;p&gt;It is believed, and often cited, that in relation to the Collatz conjecture Paul Erdos once said &quot;&lt;em&gt;Mathematics is not yet ready for such problems&lt;/em&gt;&quot;. However, I have not found any credible reference for this. Is it rumor, common misconception or the truth?&lt;/p&gt;&#xA;" OwnerUserId="3990" LastActivityDate="2021-08-22T02:43:41.403" Title="Paul Erdos' quote &quot;Mathematics is not yet ready for such problems&quot;" Tags="|mathematics|mathematicians|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6390" PostTypeId="2" ParentId="6389" CreationDate="2017-08-08T15:16:15.223" Score="9" Body="&lt;p&gt;An &lt;a href=&quot;https://arxiv.org/abs/math/0309224&quot; rel=&quot;noreferrer&quot;&gt;annotated bibliography by Lagarias&lt;/a&gt; on the 3x+1 problem shows a couple mentions of papers by Erdős that touch on related topics, but seems to trace the quote as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;77&quot;&gt;&#xA;&lt;li&gt;Richard K. Guy (1983a) &lt;em&gt;Don’t try to solve these problems!&lt;/em&gt;, Amer. Math. Monthly &lt;strong&gt;90&lt;/strong&gt; (1983), 35–41.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The article gives some brief history of work on the 3&lt;i&gt;x&lt;/i&gt; + 1 problem. It mentions at second hand a statement of P. Erdős regarding the 3&lt;i&gt;x&lt;/i&gt; + 1 problem: “Mathematics is not yet ripe enough for such questions.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The standard form of the quote, “Mathematics is not yet ready for such problems,” appears to come from an article by the same Lagarias, &lt;em&gt;The 3x+1 Problem and its Generalizations&lt;/em&gt; (1985, &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Lagarias3-23.pdf&quot; rel=&quot;noreferrer&quot;&gt;PDF from the publisher&lt;/a&gt;), and the dates are sufficient that he might be misquoting the above reference, so I would take his later bibliography as authoritative: I am guessing that he was somewhat bugged by this quote, too, and that this was the earliest reference he could find to it.&lt;/p&gt;&#xA;" OwnerUserId="3012" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-08-08T15:16:15.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6391" PostTypeId="1" CreationDate="2017-08-08T02:14:11.640" Score="4" ViewCount="223" Body="&lt;p&gt;How did Christiaan Huygens come across the fact that the quantity $mv^2$ is conserved during motion? I would like to know his thought process and how he derived or proved this result.&lt;/p&gt;&#xA;" OwnerUserId="3564" OwnerDisplayName="SaitamaSensei" LastEditorUserId="658" LastEditDate="2017-08-09T03:04:25.640" LastActivityDate="2017-08-18T04:35:14.933" Title="Equation for the conservation of vis viva" Tags="|physics|classical-mechanics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6392" PostTypeId="1" AcceptedAnswerId="6393" CreationDate="2017-08-08T21:56:01.943" Score="4" ViewCount="116" Body="&lt;p&gt;In December 1892, the french mathematician Charles Hermite had a jubilee celebration. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He received a number of letters and telegrams. One of them is from the members of the University of Berlin, see the following picture:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/KedAo.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KedAo.jpg&quot; alt=&quot;Letter from the University of Berlin&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can identify all but one name:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_von_Helmholtz&quot; rel=&quot;noreferrer&quot;&gt;von Helmholtz&lt;/a&gt; (physicist), &lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Weierstrass&quot; rel=&quot;noreferrer&quot;&gt;Weierstrass&lt;/a&gt; (mathematician), &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Kummer&quot; rel=&quot;noreferrer&quot;&gt;Kummer&lt;/a&gt; (mathematician), the missing one&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Wilhelm_von_Bezold&quot; rel=&quot;noreferrer&quot;&gt;von Bezold&lt;/a&gt; (physicist), &lt;a href=&quot;https://en.wikipedia.org/wiki/Emil_du_Bois-Reymond&quot; rel=&quot;noreferrer&quot;&gt;E. Dubois-Reymond&lt;/a&gt; (physician), &lt;a href=&quot;https://en.wikipedia.org/wiki/August_Kundt&quot; rel=&quot;noreferrer&quot;&gt;Kundt&lt;/a&gt; (physicist), &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Wilhelm_Vogel&quot; rel=&quot;noreferrer&quot;&gt;H.W. Vogel&lt;/a&gt; (chemist), &lt;a href=&quot;https://en.wikipedia.org/wiki/Lazarus_Fuchs&quot; rel=&quot;noreferrer&quot;&gt;Fuchs&lt;/a&gt; (mathematician).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Can you identify the missing one? (It seems that it is something like S. S...ers)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6151" LastActivityDate="2017-08-09T03:02:45.490" Title="Name of a XIXth-century German scientist from the University of Berlin" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6393" PostTypeId="2" ParentId="6392" CreationDate="2017-08-08T23:01:07.377" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Auwers&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Arthur_Auwers&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All signatories (and Hermite) were members of the &lt;em&gt;Prussian Academy&lt;/em&gt; : &lt;a href=&quot;https://archive.org/stream/sitzungsberichte1892deut#page/n10&quot; rel=&quot;noreferrer&quot;&gt;1/1/1892&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/sitzungsberichte1893deut#page/n10&quot; rel=&quot;noreferrer&quot;&gt;1/1/1893&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-08-09T03:02:45.490" LastActivityDate="2017-08-09T03:02:45.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6394" PostTypeId="1" CreationDate="2017-08-09T08:01:38.433" Score="1" ViewCount="58" Body="&lt;p&gt;The German historian of medieval thought, &lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_Flasch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Flasch&lt;/a&gt; claims in one of his books (&quot;Das philosophische Denken im Mittelalter&quot;, my translation &quot;philosophical thought in medieval times&quot;) the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;If there was any interest in natural history, in botany or mineralogy, for example, people looked for explanations in Pliny's Naturalis Historia.&quot;[^1]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm wondering if you can think of other works that were available and actually used by medieval natural historians/philosophers. Was aristotle (or at least his natural history) not available to people in the (european) middle ages?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[^1]: &lt;a href=&quot;https://books.google.de/books?id=321hDAAAQBAJ&amp;amp;pg=PT105&amp;amp;lpg=PT105&amp;amp;dq=Wenn+%C3%BCberhaupt+naturkundliche+Tatsachen,+z.+B.+der+Botanik+oder+der+Mineralogie,+Interesse+fanden,+so+suchte+man+die+Erkl%C3%A4rungen+auf,+die+in+Plinius%27+Naturgeschichte+gegeben+waren&amp;amp;source=bl&amp;amp;ots=EOdj14B873&amp;amp;sig=Rh77VfgT4vvrgpPSWSbR7wmYmZM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Wenn%20%C3%BCberhaupt%20naturkundliche%20Tatsachen%2C%20z.%20B.%20der%20Botanik%20oder%20der%20Mineralogie%2C%20Interesse%20fanden%2C%20so%20suchte%20man%20die%20Erkl%C3%A4rungen%20auf%2C%20die%20in%20Plinius&amp;#39;%20Naturgeschichte%20gegeben%20waren&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Wenn überhaupt naturkundliche Tatsachen, z. B. der Botanik oder der Mineralogie, Interesse fanden, so suchte man die Erklärungen auf, die in Plinius’ Naturgeschichte gegeben waren.&quot;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: &lt;a href=&quot;https://history.stackexchange.com/questions/35485/what-are-other-ancient-encyclopaedic-texts-like-plinys-naturalis-historia-wer&quot;&gt;I asked this question on the history SE as well.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6147" LastEditorUserId="6147" LastEditDate="2017-08-12T19:36:44.477" LastActivityDate="2017-08-12T19:36:44.477" Title="What are other ancient encyclopaedic texts like Pliny's “Naturalis Historia” that were available in medieval times?" Tags="|middle-ages|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6395" PostTypeId="1" CreationDate="2017-08-09T08:43:36.967" Score="1" ViewCount="63" Body="&lt;p&gt;Reprints of articles from scientific journals seem to have been an important part of scientific practice before the copying machine and the internet. Authors of articles were given a number of those reprints to share with their peers in times when going to a library and put it on a copying machine or simply downloading a pdf file were not an option.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in how reprints (&quot;Sonderdrucke&quot; as we call them in German) shaped scientific practice. Were they a social factor? (Probably) Who paid for them and why? What kind of impact had the relative scarcity of reprints in comparison to today?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be especially grateful for specific info on that topic around 1900 and for case studies from the history of biology (doesn't have to be from 1900).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: &lt;a href=&quot;https://history.stackexchange.com/questions/35609/how-important-were-reprints-of-scientific-articles-in-scientific-practice-and-co&quot;&gt;I asked this question on the history SE as well.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6147" LastEditorUserId="6147" LastEditDate="2017-08-09T14:55:11.427" LastActivityDate="2017-08-09T14:55:11.427" Title="How important were reprints of scientific articles in scientific practice and communications before the copier, the computer and the internet?" Tags="|biology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6396" PostTypeId="1" CreationDate="2017-08-09T09:41:12.587" Score="3" ViewCount="279" Body="&lt;p&gt;I'm always amazed by the the apparent amount of foreign languages that scientists in the 18th and 19th centuries seem to have possessed. With the end of Latin as the main scholarly language, researchers started to write in their mother tongue. I'm aware that sometimes papers got translated when they were reprinted in a foreign journal, which was pretty common back then (the reprinting, not the translation as far as I know…), but it seems obvious for example in the field of botany in Germany, that people had to have a command (at least to the level of a reading comprehension) of at least Latin, French, and English in order to participate in scholarly discourse.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering where and how scientists acquired those skills, or if this is a case where there is a nowadays unknown army of translators that facilitated the discourse and bridged the gap in that way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S.: &lt;a href=&quot;https://history.stackexchange.com/questions/38096/where-and-how-did-scientists-of-the-18th-and-19th-century-learn-foreign-language&quot;&gt;I asked this question on the history SE as well.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6147" LastEditorUserId="6147" LastEditDate="2017-08-09T10:04:00.600" LastActivityDate="2017-08-17T13:11:43.010" Title="Where and how did scientists of the 18th and 19th century learn foreign languages?" Tags="|biology|language|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6397" PostTypeId="1" CreationDate="2017-08-09T12:47:33.327" Score="1" ViewCount="531" Body="&lt;p&gt;The arithmetic mean of a set of points $\{x_1, x_2, ..., x_n\}$ is defined by &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac{1}{n}\sum_{i=1}^n x_i.$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is remarkable for its ubiquitous use and universal understanding. It represents a center of mass, a point of equilibrium, a fair ressource distribution, the center of a probability distribution, etc. It is also not only an elementary concept: its generalization by integrals is quite fundamental in higher mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the history of the arithmetic mean? When did it first arise and when did it become widely understood through its different interpretations?&lt;/p&gt;&#xA;" OwnerUserId="5555" LastEditorUserId="5555" LastEditDate="2017-08-09T18:17:29.757" LastActivityDate="2017-08-09T18:17:29.757" Title="History of the arithmetic mean" Tags="|mathematics|reference-request|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6398" PostTypeId="2" ParentId="6395" CreationDate="2017-08-09T14:23:41.590" Score="4" Body="&lt;p&gt;My experience (somewhat later than 1900, however)...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Publishers would provide a certain number of reprints to the author for free (and the author could order more by paying, which I never did).  Then the author could send (by post) these reprints to likely interested parties.  And others who hear of the paper (for example, seeing it in a library copy of the journal) could send the author a postcard and request a reprint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I still have reprints for most of my papers in a drawer here, but it is many years since I have been asked for one.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-08-09T14:23:41.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6400" PostTypeId="1" CreationDate="2017-08-10T03:11:40.567" Score="8" ViewCount="1129" Body="&lt;p&gt;There are famous quotes about what Farcas Bolyai wrote to his son Janos to persuade him not to study the &quot;theory of parallels &quot; or what is now known as hyperbolic geometry &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But not all translation of this quote are the same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was originally written (and for fun what is the most outrageous translation of it)&lt;/p&gt;&#xA;" OwnerUserId="557" LastEditorUserId="557" LastEditDate="2017-08-10T17:51:45.187" LastActivityDate="2017-09-03T14:08:19.373" Title="What did Farcas Bolyai write to his son?" Tags="|mathematicians|geometry|quote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6401" PostTypeId="2" ParentId="6400" CreationDate="2017-08-10T04:31:37.820" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikiquote.org/wiki/Farkas_Bolyai&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikiquote site&lt;/a&gt; gives the original quote from a letter of 4, April 1820:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A parallelákat azon az útan ne próbáld: tudom én azt az utat is mind végig — megmértem azt a feneketlen éjszakát én, és az életemnek minden világossága, minden öröme kialudt benne...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which it translates as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(It reads like a Morrissey song lyric.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bolyai_Farkas.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor site&lt;/a&gt; gives the same translation of the April 1820 letter, while also quoting an earlier letter as stating : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Detest it as lewd intercourse, it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;In addition to these two quotes from two separate letters, the &lt;a href=&quot;https://todayinsci.com/B/Bolyai_Farkas/BolyaiFarkas-Quotations.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;TodayInScience&lt;/a&gt; site quotes a third letter from 1820 stating :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For God’s sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So we have three different letters, each expressing similar sentiments.   Citations for the sources of each of the three translations can be found on the TodayInScience site (linked above).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Alternatively, &lt;a href=&quot;http://bolyai.mtak.hu/en/life.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Hungarian site&lt;/a&gt;, devoted to the life and work of János, simply states :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the spring of 1820 he informed his father about his experiences concerning parallels. Farkas in a long letter warned his son against this: “…don’t go any step further, or else you’re a lost person.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rereading my answer over my morning coffee it seems clear to me that the second and third quotes are different translations of the same text.  So it appears that we are dealing with two different letters expressing similar sentiments.  &lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-08-10T16:57:17.453" LastActivityDate="2017-08-10T16:57:17.453" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6403" PostTypeId="1" CreationDate="2017-08-11T05:54:14.430" Score="-3" ViewCount="222" Body="&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What early &lt;em&gt;published&lt;/em&gt; or &lt;em&gt;citably attested&lt;/em&gt; examples (preferably in the mathematical literature) can you give of the following convention?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let $\mathbb{S}$ denote some nonempty subset of some specified alphabet $\mathbb{A}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then the convention is made that&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;in any formula, any letter from $\mathbb{S}$ must be interpreted to a &lt;em&gt;strictly negative&lt;/em&gt; real number (i.e. $&amp;lt;0$), while any letter from $\mathbb{A}\setminus \mathbb{S}$ must be interpreted to a &lt;em&gt;strictly positive&lt;/em&gt; real number (i.e. $&amp;gt;0$).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;An example is $\mathbb{A}:=$Latin alphabet, $\mathbb{S}:=$consonants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, for example,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\mathrm{for}\ \mathrm{all}\ a,\mathrm{for}\ \mathrm{all}\ z,\qquad a\cdot z &amp;lt; 0\qquad (\mathrm{sen})$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is a true sentence,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;while, needless to say, with the usual convention of indifference to the typographical shape of variable symbols, (sen) is false, since &lt;em&gt;without the convention&lt;/em&gt; there are both $(a,z)$ with $a\cdot z&amp;lt;0$ and $(a,z)$ with $a\cdot z&amp;gt;0$.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;This is a &lt;strong&gt;&lt;em&gt;rare&lt;/em&gt;&lt;/strong&gt; convention, yet it &lt;em&gt;exists&lt;/em&gt; in the mathematical literature. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It can make formulas more concise. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give a &lt;em&gt;toy example&lt;/em&gt; (this is &lt;em&gt;not&lt;/em&gt; the application in the published article which made me ask this question): this convention is useful to more concisely realize the idea of &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;'necessarily false formula' &lt;/li&gt;&#xA;&lt;li&gt;'possibly yet not necessarily true formula'. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Needless to say, a formula (in the model-theoretic sense) is &lt;em&gt;never&lt;/em&gt; true or false, only after interpretation via a &lt;em&gt;specified satisfaction-relation&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Definition:  a 'necessarily false formula'$:=$shorthand for the longer term 'necessarily false formula over a specified signature $\Sigma$, and w.r.t. a specified satisfaction relation $\vDash$, and w.r.t. a specified form-formula $\Phi$' (in the usual sense of model theory)$:=$shorthand for quadruple ($\varphi,\Phi,\Sigma,\vDash$) wherein $\Sigma$ is a signature, $\varphi$ and $\Phi$ are formulas over $\Sigma$, $\vDash$ is a satisfaction relation for formulas over $\Sigma$ (note that this implies that there is a specified structure $\mathfrak{R}$ in the background), such that $\varphi$ is obtained from $\Phi$ by substitution of variables (in the usual technical sense, where as usual &lt;em&gt;substitution of variables is indifferent to sorts of variables; you may substitute any sort of variable for any other&lt;/em&gt;), and $\neg\ (\mathfrak{R}\vDash \varphi[\alpha])$ for every variable assignment $\mathfrak{\alpha}$ (variable assignment maps must respect sorts of variables).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the example, if the above convention is in force, then &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;$a&amp;lt;z$ is a 'necessarily false formula' w.r.t. the form-formula $x&amp;lt;y$, the signature $&amp;lt;,\mathrm{negvar},\mathrm{posvar}$ and the (standard satisfaction relation $\vDash$ of) the structure $(\mathbb{R},&amp;lt;)$,&lt;/li&gt;&#xA;&lt;li&gt;$z&amp;lt;a$ is a 'possibly yet not necessarily true' formula w.r.t. the form-formula $x&amp;lt;y$, the signature $&amp;lt;,\mathrm{negvar},\mathrm{posvar}$ and the (standard satisfaction relation $\vDash$ of) the structure $(\mathbb{R},&amp;lt;)$,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now where's the &lt;em&gt;difference&lt;/em&gt; to the usual situation in which variables are not &lt;em&gt;sorted&lt;/em&gt;? In this situation, &lt;em&gt;it is impossible to write a 'necessarily false formula' of the form '$x&amp;lt;y$' with $x,y$ arbitrary variables&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole question is related to what, in particular in categorical logic, are called &lt;em&gt;sorted signatures&lt;/em&gt;. In a sense, here, the consonants have sort 'strictly negative real number' and the non-consonants have sort 'strictly positive real number'.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Example citations I will not give, since I am hoping for examples far older than the one I could give. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="6060" LastEditDate="2017-08-12T05:44:45.600" LastActivityDate="2017-08-12T05:44:45.600" Title="What are early examples of the rare notational convention to make the sign of the real number represented by a letter depend on the typography?" Tags="|mathematics|terminology|notation|mathematical-logic|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="6404" PostTypeId="2" ParentId="6403" CreationDate="2017-08-11T13:05:25.967" Score="2" Body="&lt;p&gt;It is not completely clear to me what you mean... Would this be an example?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the FORTRAN programming language (1957)$^1$, variables beginning with letters &lt;code&gt;I,J,K,L,M,N&lt;/code&gt; represent integers, and other variables represent floating point values.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$^1$ Backus, J. W.; H. Stern, I. Ziller, R. A. Hughes, R. Nutt, R. J. Beeber, S. Best, R. Goldberg, L. M. Haibt, H. L. Herrick, R. A. Nelson, D. Sayre, P. B. Sheridan (1957). &quot;The FORTRAN Automatic Coding System&quot;. Western joint computer conference: Techniques for reliability. Los Angeles, California&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-08-11T13:05:25.967" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6405" PostTypeId="1" AcceptedAnswerId="6700" CreationDate="2017-08-11T22:53:44.623" Score="0" ViewCount="186" Body="&lt;p&gt;I'm writing a piece of paper and trying to understand, the meaning of word 'science' in the context of history. I'm not a native English speaker and I know I could look up in the dictionary, but my question is when the word 'science' is mentioned, does it usually mean all the scientific outcomes that we have now, or does it mean a general way of discovering and researching?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if one says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;modern science has several issues.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It could mean that the outcome of science has drawbacks, e.g. global warming, or it could mean that the way people do research has limitation, e.g. something has to be seen before it can be studied. Or it means both?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks a lot!&lt;/p&gt;&#xA;" OwnerUserId="6160" LastActivityDate="2017-11-13T18:49:56.380" Title="Does the word 'science' mean what's been found so far, or the way that human discovers unknown?" Tags="|discoveries|philosophy-of-science|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6406" PostTypeId="1" AcceptedAnswerId="6424" CreationDate="2017-08-12T07:41:58.717" Score="7" ViewCount="335" Body="&lt;p&gt;In a XVth century french manuscript on arithmetic and astrology, there is a description of a numeral system as follows (it starts &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/btv1b9059038x/f82.image&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; in the manuscript).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numbers between 1 and 9 are depicted by a vertical bar and a set of small bars on the upper-right (the conversion to arabic numerals is obviously using a slightly earlier version than current-day digits, but this is nothing difficult and not the topic of the question):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KtZHY.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KtZHY.png&quot; alt=&quot;digits 1 to 9&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then the mirror sign on the left is used for 10 to 90:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FnSJc.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FnSJc.png&quot; alt=&quot;numbers 10 to 90&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so that numbers 11 to 29 read:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2U9gn.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2U9gn.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then larger numbers are represented with more bars, for instance with:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jHZpX.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jHZpX.png&quot; alt=&quot;100 to 900&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;one can then write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fElLH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fElLH.png&quot; alt=&quot;111 to 169&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and with:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ddq5u.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ddq5u.png&quot; alt=&quot;1000 to 11000&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;one can write:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EqjH2.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EqjH2.png&quot; alt=&quot;numbers with 4 digits&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, very large numbers can be written by embedding symbols:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Lyvu2.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lyvu2.png&quot; alt=&quot;very large numbers&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I would like to know the name (if any) of this numeral system, who first came up with it (if this is known), and relevant references about it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6151" LastEditorUserId="6151" LastEditDate="2017-08-12T07:50:49.910" LastActivityDate="2017-08-16T17:42:18.553" Title="What is the name of this numeral system?" Tags="|mathematics|terminology|notation|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6407" PostTypeId="1" CreationDate="2017-08-12T08:22:50.160" Score="4" ViewCount="148" Body="&lt;p&gt;The Bessel function appears in many mathematics and physics problems. Their zeros are solutions of many problems. So how did people calculate them at 1900? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did they have some good series? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2017-08-12T08:22:50.160" Title="How did people calculate the zeros of the Bessel functions before the electronic computer?" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6408" PostTypeId="1" CreationDate="2017-08-12T18:13:55.650" Score="2" ViewCount="72" Body="&lt;p&gt;A thermometer is a &quot;scaled&quot; and therefore very accurate device in measuring body temperature, and thereby detecting fever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What happened before thermometers were introduced in their current form in the 17th century? For instance, what was the usefulness and limitations of  &lt;a href=&quot;https://en.wikipedia.org/wiki/Thermoscope&quot; rel=&quot;nofollow noreferrer&quot;&gt;thermoscopes&lt;/a&gt;? And what might have preceded them other than the &quot;feel&quot; test?&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2017-08-14T18:58:15.510" Title="How effective were pre-thermometer devices in e.g. detecting fever?" Tags="|medicine|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6409" PostTypeId="2" ParentId="6396" CreationDate="2017-08-12T19:34:55.473" Score="2" Body="&lt;p&gt;I can think of several reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all, in the 18th and 19th centuries, most science was conducted in Europe, within a fairly narrow (though linguistically diverse) area. People of the time that were wealthy and educated enough to be scientists traveled enough, or at least received enough travelers from nearby countries, to have a lot of it &quot;rub&quot; off on them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, the scope of scientific vocabulary is much narrower, and has much more in common with each other from one language to another than everyday language. That is, an intensive &quot;scientific English&quot; or &quot;scientific German&quot; course can teach most people what they need to know within a few hundred hours.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read anecdotally that when e.g. Albert Einstein first came to the U.S., he had little trouble with &quot;scientific&quot; English, but a lot of trouble going down to the corner grocery store, the kind of thing every ten year old native speaker knows. I have personally had the same issues with non-English languages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In botany, a lot of the terminology has Latin roots and most educated people in the 18th and 19th centuries studied Latin as well as their native language. They may or may not be so good at the &quot;English&quot; or &quot;French&quot; part of the discipline (if that wasn't their native tongue).Even so, a study of Latin would help with French, and a knowledge of German or English would help with the other.&lt;/p&gt;&#xA;" OwnerUserId="31" LastEditorUserId="31" LastEditDate="2017-08-15T11:20:06.807" LastActivityDate="2017-08-15T11:20:06.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6410" PostTypeId="2" ParentId="6396" CreationDate="2017-08-12T20:40:44.807" Score="3" Body="&lt;p&gt;A typcial gymnasium in Germany in the 19th century would offer courses in Latin, Ancient Greek and French, as well as in Hebrew (probably only for Jewish students). If you wanted to learn other languages (Gauss started learning Russian when he was quite old already) you needed a tutor. For scientific purposes, Latin, French and German covered more than 90 % of the important literature during much of the 19th century. In mathematics, there exist several translations of German texts into French, but I am not aware of a single translation of a French text into German during the 19th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way of learning languages was the &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Grand_Tour&quot; rel=&quot;nofollow noreferrer&quot;&gt;grand tour&lt;/a&gt;&quot;, which was fashionable up to the early 18th century.&lt;/p&gt;&#xA;" OwnerDisplayName="user2255" LastActivityDate="2017-08-12T20:40:44.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6411" PostTypeId="2" ParentId="6335" CreationDate="2017-08-13T01:50:45.987" Score="2" Body="&lt;p&gt;This appears to have arisen out of the &lt;a href=&quot;https://hsm.stackexchange.com/questions/347/why-is-cardanos-formula-wrongly-attributed-to-him&quot;&gt;interaction&lt;/a&gt; of two men, Tartaglia and Cardano. Of the two, Tartaglia was the first to address the problem, but Cardano's solutions probed the matter deeper, covered more cases, and was basically the more &quot;general&quot; method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is noteworthy that Cardano's lasting legacy is that one of his students, Ferrari solved the &lt;a href=&quot;https://en.wikipedia.org/wiki/Quartic_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;quartic&lt;/a&gt; shortly afterward.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2017-08-13T01:50:45.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6414" PostTypeId="2" ParentId="6408" CreationDate="2017-08-13T14:06:31.553" Score="1" Body="&lt;p&gt;Before the invention of thermometer, there was no notion of &quot;temperature&quot;,&#xA;as a physical quantity, a number. (To define a physical quantity one needs to specify a way to measure it). Only qualitative differences could be discussed:&#xA;&quot;hot&quot;, &quot;cold&quot;, &quot;fever&quot; etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fever is easily detected without any thermometer. By physical contact of the person who detects it (presumably having normal temperature). Even now, when thermometer is not available fever is easily detected by touching the person's forehead with one's hand or better with the lips.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mercury thermometers measured reliably with 0.1 degree Celsius accuracy, but this &#xA;accuracy is rarely required. Since the use of mercury was prohibited, electronic thermometers are used which is much less reliable and less accurate. But one does not really need to measure temperature with such high precision.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-08-14T18:58:15.510" LastActivityDate="2017-08-14T18:58:15.510" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6415" PostTypeId="2" ParentId="6268" CreationDate="2017-08-14T01:47:30.123" Score="0" Body="&lt;p&gt;To answer the last part of the question, Isaac Newton lived in England, and the Catholic Church had been supplanted as the country's highest religious authority by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Church_of_England&quot; rel=&quot;nofollow noreferrer&quot;&gt;Church of England&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason was that King Henry VIII wanted a divorce from his first wife, Catherine of Aragon, so that he could marry Anne Boleyn. The Pope refused him the divorce, so Henry &quot;disestablished&quot; the Catholic Church and set up the Church of England in its place, answerable to the king (or queen) of England.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other &quot;heliocentric&quot; scientists, e.g. Kepler, lived in Germany or other parts of northern Europe where Catholicism had been replaced by Lutheranism or some other Protestant faith.&lt;/p&gt;&#xA;" OwnerUserId="31" LastActivityDate="2017-08-14T01:47:30.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6417" PostTypeId="2" ParentId="6405" CreationDate="2017-08-14T12:11:14.097" Score="0" Body="&lt;p&gt;Science has the etymological rootes &quot;scire&quot; (know) and &quot;scientia&quot; in Latin. It came via old French to the middle English science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The present contents of the word &quot;science&quot; is the set of methods and techniques invented in the course of cultural development of mankind in order to gain knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A scientist is a person who has acquired, usually at a university, these techniques for a special field and endeavours to increase the knowledge in this field - contrary to engineers who apply the knowledge in order to construct reliably working useful (sometimes also useless or even detrimental) machines. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the results obtained by scientists are also part of science. Often they are extolled as &quot;scientifically proven&quot;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-08-14T12:11:14.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6418" PostTypeId="2" ParentId="6391" CreationDate="2017-08-14T12:54:10.420" Score="3" Body="&lt;p&gt;In &quot;Tractatus de motu corporum ex percussione&quot; Christiaan Huygens showed energy conservation during the elastic impact. His investigations were triggered by a question of the Royal Society. Also Wren and Wallis gave answers. Wallis treated the inelastic impact, Huygens and Wren the elastic impact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Huygens considered the impact in two different inertial systems, one of them being the centre-of-mass system. There he found that for both cases the vis viva was conserved. He wrote the equation already in modern form:&#xA;$m_1v_1^2 + m_2v_2^2 =m_1v_1'^2 + m_2v_2'^2 $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: Norbert Pucker: &lt;em&gt;Physikalische Grundlagen der Energietechnik&lt;/em&gt;, Springer (2013)&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2017-08-14T12:54:10.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6419" PostTypeId="1" AcceptedAnswerId="6420" CreationDate="2017-08-14T19:36:14.777" Score="4" ViewCount="169" Body="&lt;p&gt;I saw this quote in Richard Burton's &lt;em&gt;Physiology by the Numbers&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let us therefore take it that in a man the amount of blood pushed&#xA;  forward in the individual heartbeats is half an ounce, or three drams,&#xA;  or one dram, this being hindered by valves from re-entering the heart.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-William Harvey, De Motu Cordis, 1628 (from the Latin)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stroke_volume&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia gives the stroke volume of a healthy adult man as 94 mL&lt;/a&gt;, and Burton cites 70 mL in his own text below the quote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In modern terms, &quot;one ounce&quot; means about 30 mL, so it sounds to me like Harvey was saying the stroke volume was about 15 mL. But Burton doesn't point out any discrepancy. Also, half an ounce is either 4 drams or 8 drams, depending on what type of drams you're using (according to &lt;a href=&quot;https://en.wikipedia.org/wiki/Dram_(unit)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;), not three drams (but perhaps he was approximating for convenience?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What units was Harvey using, what did he think was the stroke volume of a human heart, and if he was wrong by about a factor of 5 or 6, why?&lt;/p&gt;&#xA;" OwnerUserId="2238" LastEditorUserId="2238" LastEditDate="2017-08-14T19:43:31.820" LastActivityDate="2017-08-14T23:20:21.117" Title="What did William Harvey think the stroke volume of a heart was?" Tags="|medicine|physiology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6420" PostTypeId="2" ParentId="6419" CreationDate="2017-08-14T23:20:21.117" Score="1" Body="&lt;p&gt;I now see that this was answered on the next page of the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Harvey's &quot;ounces&quot; were today's ounces, and his drams were &quot;fluid drams&quot;, each one eighth of an ounce. It isn't explained why he identified &quot;half an ounce&quot; with &quot;three drams&quot;, but possibly he found &quot;three eighths of an ounce&quot; cumbersome and unnecessarily precise, he was imprecise about other figures, too, such as assuming a human heart rate of around 30 bpm instead of the normal 70 bpm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I haven't found a precise description of Harvey's methodology for measuring stroke volume, but &lt;a href=&quot;https://sydney.edu.au/medicine/museum/mwmuseum/index.php/The_Circulation_of_the_Blood&quot; rel=&quot;nofollow noreferrer&quot;&gt;a museum webpage&lt;/a&gt; gives this description of his successor, Stephen Hales' method:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This scientifically interested layman undertook in Tordington (sic) in 1710, 53 years after the death of William Harvey (1578–1657), the first precise definition of the capacity of a heart. He bled a sheep to death and then led a gun-barrel from the neck vessels into the still-beating heart. Through this, he filled the hollow chambers with molten wax and then measured from the resultant cast the volume of the heartbeat and the minute-volume of the heart, which he calculated from the pulse-beat. Besides this, Stephen Hales was also the first, in 1727, to determine arterial blood pressure, when he measured the rise in a column of blood in a glass tube bound into an artery. (From the acceptance speech for the Nobel Prize in Medicine, given by Werner Forssmann in 1956.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding Harvey's methodology, I found is in &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2776239/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this commentary&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He isolated parts of the heart; he ligated and divided arteries; he exerted pressure on veins on either side of the valves. His observations of dissected hearts showed that the valves in the heart allowed blood to flow in only one direction. Harvey measured the volume of the left ventricle and calculated that the amount of blood that passes through the heart of a man in an half hour and established that it was greater than the amount contained in the whole body&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And &lt;a href=&quot;http://circ.ahajournals.org/content/circulationaha/23/2/286.full.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this discussion&lt;/a&gt; of Harvey's results:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To make his calculation Harvey had to measure the pulse rate and the amount of blood that the heart ejects with each beat. Here, he tackled a tough problem; even today there is no widely accepted figure for cardiac output. The determination of cardiac output is most difficult, and the various procedures&#xA;  for measuring it give results that vary as much as 25 per cent. The lowest figure, however-one dram, apothecaries weight-which Harvey used was only one-eighteenth of the lowest calculation accepted today. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it seems that Harvey wasn't overly concerned with precision, and he used only crude methods of inspection to estimate the heart volume, meaning he was off by quite a bit, although that didn't really matter much for his arguments.&lt;/p&gt;&#xA;" OwnerUserId="2238" LastActivityDate="2017-08-14T23:20:21.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6421" PostTypeId="1" AcceptedAnswerId="6791" CreationDate="2017-08-15T08:43:33.167" Score="2" ViewCount="183" Body="&lt;p&gt;On page 347 in his &quot;Mathematical Thought, volume 1&quot;, Morris Kline writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The work on the third class of problems, finding the maxima and minima of functions, may be said to begin with an observation by Kepler. He was interested in the shape of casks for wine; in his Stereometria Doliorum (1615) he showed that, of all right parallelepipeds inscribed in a sphere and having square bases, the cube is the largest. His method was to calculate the volumes for particular choices of dimensions. This in itself was not significant; but he noted that as the maximum volume was approached, the change in volume for a fixed change in dimensions grew smaller and smaller. Fermat in his Methodus ad Disquirendam gave his method, which he illustrated with the following example: Given a straight line (segment), it is required to find a point on it such that the rectangle contained by the two segments of the line is a maximum. He calls the whole segment B and lets one part of it be A. Then the rectangle is AB - A2. He now replaces A by A + E. The other part is then B - (A + E), and the rectangle becomes (A + E)(B - A - E). He equates the two areas because, he argues, at a maximum the two function values-that is, the two areas--should be equal, etc.&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not sure if Kline implies a connection between Kepler and Fermat here but it would certainly be interesting to explore the issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Question. Did d'Espagnet's library at Bordeaux that Fermat had access to, contain works by Kepler?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that Mahoney's book on Fermat does not discuss such a possible influence directly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1. &lt;a href=&quot;https://mathoverflow.net/q/278871/28128&quot;&gt;Related question at MO&lt;/a&gt; on d'Espagnet's library.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="4251" LastEditDate="2021-02-27T07:41:38.443" LastActivityDate="2021-02-27T07:41:38.443" Title="Did Kepler influence Fermat?" Tags="|mathematicians|biographical-details|17th-century|fermat|kepler|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6422" PostTypeId="2" ParentId="2042" CreationDate="2017-08-15T11:39:52.920" Score="3" Body="&lt;p&gt;The site &lt;a href=&quot;http://jeff560.tripod.com/d.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt; cites Katz, who in &lt;em&gt;&lt;a href=&quot;https://sites.math.washington.edu/~morrow/335_17/history%20of%20stokes%20thm.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The History of Stokes' Theorem&lt;/a&gt;&lt;/em&gt; (1979) attributes the first use of &quot;exterior differential form&quot; to Elie Cartan in 1922. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Katz writes p.154:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The mathematician chiefly responsible for clarifying the idea of a&#xA;  differential form was Elie  Cartan. In his fundamental paper of 1899&#xA;  [Sur certaines expressions diffrentielles et sur le probleme de Pfaff], he first defines an &quot;expression differentielle&quot; as a  symbolic&#xA;  expression given by a finite number of sums and products of the $n$&#xA;  differentials  $dx_1,dx_2,\ldots,dx_n$, and certain coefficient &#xA;  functions of the variables $x_1,x_2,\ldots,x_n$. A differential &#xA;  expression of the first degree, $A_1 dx_1 + A_2 dx_2 + \ldots + A_n&#xA;&amp;gt; dx_n$, he calls an &quot;expression de Pfaff&quot;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;By 1922, Cartan had extended his work on differential expressions in&#xA;  &lt;em&gt;&lt;a href=&quot;https://archive.org/details/leonssurlesinv00cartuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leçons sur les invariants intégraux&lt;/a&gt;&lt;/em&gt;. It is here that he  first uses the current terminology of&#xA;  &quot;exterior differential form&quot; and &quot;exterior derivative.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm not sure if &quot;differential form&quot; (without the exterior) was used prior to that. I randomly picked a work of Cartan from 1901 &lt;a href=&quot;http://www.numdam.org/item?id=ASENS_1901_3_18__241_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;L'intégration des systèmes d'équations aux différentielles totales&lt;/em&gt;&lt;/a&gt; and a search for &quot;forme&quot; didn't yield any &quot;forme differentielle&quot;. But I found the following in a footnote p.255:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Par example, si les équations de $E_p$ sont $$P_1 =P_2 = \ldots = P_{r-p}=0,$$ où les $P$ sont des formes linéaires en $dx_1,\ldots,dx_r$, les equations de $E_{p+1}$ sont [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This suggests, as you did in the question, that if there is a continuity between Grassmanns &quot;form&quot; and &quot;differential form&quot;, it might be through the use of &quot;linear form&quot;. But I don't know when the use of the latter startet.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2017-08-15T11:39:52.920" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6423" PostTypeId="1" CreationDate="2017-08-15T20:37:53.930" Score="5" ViewCount="267" Body="&lt;p&gt;A lot of sources attribute the definition to Arthur Samuel (1959), &quot;the field of study that gives computers the ability to learn without being explicitly programmed&quot;, but none of these sources directly point to where he actually said the quote. Just a bunch of people saying he said that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some give it as his 1959 paper on checkers, but I don't find any such definition there.&lt;/p&gt;&#xA;" OwnerUserId="6174" LastEditorUserId="10974" LastEditDate="2020-04-25T06:43:28.207" LastActivityDate="2020-04-25T18:08:53.743" Title="Who coined the term “machine learning”?" Tags="|terminology|computer-science|machine-learning|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6424" PostTypeId="2" ParentId="6406" CreationDate="2017-08-16T17:42:18.553" Score="2" Body="&lt;p&gt;According to S.Lamassé in &lt;em&gt;Scientific Sources and Teaching Contexts Throughout History: Problems and Perspectives&lt;/em&gt; (page 130, footnote 28. A.Bernard &amp;amp; C.Proust, Eds. Springer, 2014.), this a system that is close to the &lt;em&gt;Algorismus Linealis&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, &lt;a href=&quot;https://bildsuche.digitale-sammlungen.de/index.html?c=viewer&amp;amp;lv=1&amp;amp;bandnummer=bsb00024979&amp;amp;pimage=00024979&amp;amp;suchbegriff=&amp;amp;l=en#&quot; rel=&quot;nofollow noreferrer&quot;&gt;a copy of this short incunable&lt;/a&gt; is available online, but I must say I do not find the comparison too enlightening. Lamassé seems to wish to regroup them under the idea of &quot;a table with counters&quot;, but the differences are quite stark to me. Yet, since there doesn't seem to be a better answer in the literature, I'll leave it at that.&lt;/p&gt;&#xA;" OwnerUserId="6151" LastActivityDate="2017-08-16T17:42:18.553" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6425" PostTypeId="1" CreationDate="2017-08-17T03:34:10.063" Score="2" ViewCount="364" Body="&lt;p&gt;For absolute beginners in  physics, what are some good books in the history of physics (especially mechanics) ? (But it should not be full of mathematical hand waving - actual historical development wth mathematical background should be there)&lt;/p&gt;&#xA;" OwnerUserId="6088" LastActivityDate="2017-12-07T09:56:50.767" Title="Good book on history of mechanics" Tags="|physics|book-recommendation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6426" PostTypeId="2" ParentId="6425" CreationDate="2017-08-17T09:47:48.077" Score="5" Body="&lt;p&gt;&lt;em&gt;A History of Mechanics&lt;/em&gt;, by René Dugas, is a classic.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2017-12-07T09:56:50.767" LastActivityDate="2017-12-07T09:56:50.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6427" PostTypeId="1" AcceptedAnswerId="6428" CreationDate="2017-08-17T11:28:07.353" Score="1" ViewCount="430" Body="&lt;p&gt;In 1696 Johann Bernoulli posed aproblem to Europe's mathematicians to find the path of shortest time for a mass to descend from a point A to a point B. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apart from Bernoulli only a hand full of other people found the solution, one of which was Isaac Newton. &#xA;Newton claimed to have been unaware of this competition until he received a letter from Bernoulli in January of 1697. He pulled an all-nighter and found the solution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Brachistochrone_curve#Newton.E2.80.99s_Solution&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia article&lt;/a&gt; on this problem however raises doubts about Newtons story. The article also claims that Newton was &quot;not always truthful about his accomplishments, especially when it comes to his dealings with Leibnitz and his associates&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are no sources for this statement and a cursory google search didn't reveal any either. &#xA;Now I would like to know if there are any accounts of Newton embelishing or outright lying about his work.&lt;/p&gt;&#xA;" OwnerUserId="6179" LastEditorUserId="5788" LastEditDate="2017-08-19T08:11:12.393" LastActivityDate="2017-08-28T11:17:47.730" Title="Allegations of Newton's untruthfulness" Tags="|mathematics|physics|newton|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6428" PostTypeId="2" ParentId="6427" CreationDate="2017-08-17T12:16:39.710" Score="6" Body="&lt;p&gt;A comprehensive scientific biography of Newton is &quot;Never at rest&quot; by Westfall.&#xA;He tells the story, and expresses no doubt about it. The article in Wikipedia is an example of sloppy writing. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-08-17T12:16:39.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6429" PostTypeId="2" ParentId="6396" CreationDate="2017-08-17T13:11:43.010" Score="1" Body="&lt;p&gt;In the 18th century, the main characteristic was not to be a scientist but to be a scholar which means to be literate and be able to &lt;em&gt;write&lt;/em&gt; latim and &lt;em&gt;read&lt;/em&gt; latim, greek, hebrew and arabic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course for your day to day life where you also supposed to speak the language (or languages) of the country in which you lived, as well as the language of the court you were part of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Add that people were traveling a lot more than today, and that traveling was implying much longer stops of a month or two, between cities with different local dialect. At the time there was not a one German or an Italian but hundreds of dialects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To live in country with 3-4 languages did not seams to bother people very much, and to master 10-12 languages was certainly a distinctive mark of education.&lt;/p&gt;&#xA;" OwnerUserId="3475" LastActivityDate="2017-08-17T13:11:43.010" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6430" PostTypeId="1" AcceptedAnswerId="6432" CreationDate="2017-08-17T15:56:59.913" Score="2" ViewCount="464" Body="&lt;p&gt;Little thinking into basic Real Analysis results like &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Arbitrary union of open sets is open&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;made me wonder who could have possibly proved it first - do we have any historic data on it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, who defined open and closed sets etc (basic definitions in analysis/topology)?&lt;/p&gt;&#xA;" OwnerUserId="6014" LastEditorUserId="6014" LastEditDate="2017-08-17T16:17:46.517" LastActivityDate="2017-08-18T02:48:44.267" Title="Basic Theorems in Topology: Who proved them first?" Tags="|mathematics|real-analysis|topology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6432" PostTypeId="2" ParentId="6430" CreationDate="2017-08-17T21:10:27.073" Score="4" Body="&lt;p&gt;For first use of mathematical terms, see &lt;a href=&quot;http://jeff560.tripod.com/mathword.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jeff560.tripod.com/mathword.html&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;CLOSED SET&lt;/strong&gt;. Georg Cantor (1845-1918) in &quot;De la puissance des ensembles parfaits de points,&quot; &lt;em&gt;Acta Mathematica&lt;/em&gt; IV, March 4, 1884, introduced (in French) the concept and the term &quot;ensemble fermé&quot; [Udai Venedem].  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;OPEN SET&lt;/strong&gt;. René Baire, “Sur les fonctions de variables réelles,” &lt;em&gt;Annali di Matematica Pura ed Applicata&lt;/em&gt; (3) 3 (1899), 1-123 has:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Je dirai que $S$ est une sphère fermée, $S'$ une sphère ouverte. La différence essentielle entre ces deux ensembles réside dans la fait suivant:&#xA;  Etant donné un point quelconque de $S'$, il existe une sphère ayant ce point pour centre et de rayon positif, dont tous les points font partie de $S'$.   &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;J'appelle d'une manière générale &lt;em&gt;domaine ouvert à n dimensions&lt;/em&gt; tout ensemble de points possédant cette propriété.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A translation is:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I will say that $S$ is a closed sphere and $S'$ is an open sphere. The essential difference between these two sets resides in the following fact:&#xA;  Given any point in $S'$, there exists a sphere having this point for its center, and of positive radius, of which all its points are a part of $S'$. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;More generally, I will call an &lt;em&gt;open domain in n dimensions&lt;/em&gt; any set of points that possesses this property.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This citation and translation were provided by Dave L. Renfro, who writes, “I believe that, in addition to being the first appearance of the term ‘open set,’ this may also be the first appearance of the concept itself. Prior to this, I think everyone just talked about the interior of a set of points, or about the ‘sum’ (i.e. union) of the interiors of sets from some specified collection of sets (often intervals).”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-08-17T21:10:27.073" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6433" PostTypeId="2" ParentId="6430" CreationDate="2017-08-18T02:48:44.267" Score="3" Body="&lt;p&gt;“Them theorems” (+ definitions...) makes this really too broad, but G. H. Moore’s &lt;em&gt;The emergence of open sets, closed sets, and limit points in analysis and topology&lt;/em&gt; (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=2455202&quot; rel=&quot;nofollow noreferrer&quot;&gt;2008&lt;/a&gt;) has good answers for the specific three you ask about and more. E.g. you’ll see the union result attributed to Hausdorff, and the definitions of closed &amp;amp; open to several people under several names in several contexts.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Focusing on particular words can also be misleading: e.g. Weierstrass (&lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=07-abh/1876&amp;amp;seite:int=562&quot; rel=&quot;nofollow noreferrer&quot;&gt;1876&lt;/a&gt;, p. 13) didn’t use “closed” nor “open”, but did freely use “interior”, “boundary”, “neighborhood” in the complex plane. &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-08-18T02:48:44.267" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6434" PostTypeId="2" ParentId="6391" CreationDate="2017-08-18T04:35:14.933" Score="1" Body="&lt;p&gt;F. Chareix’s &lt;em&gt;La découverte des lois du choc par Christiaan Huygens&lt;/em&gt; (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=2007367&quot; rel=&quot;nofollow noreferrer&quot;&gt;2003&lt;/a&gt;) discusses this in great detail, on the basis of Huygens manuscripts (1652–). Generally his arguments were of relativistic nature, i.e. he drew conclusions from the possibility of viewing the collision “from the ground”, then “from the boat”, etc. But while Chareix sketches (pp. 18-19) the simple proof he may have had along such lines, Huygens does not seem to have written that: instead, he states conservation of &lt;em&gt;vis viva&lt;/em&gt; as an unproved “Axiom” (2, pp. 18, 52), “Proposition” (XI, p. 55), or “Rule” (VI, p. 57).  &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-08-18T04:35:14.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6435" PostTypeId="1" AcceptedAnswerId="6492" CreationDate="2017-08-19T12:34:29.040" Score="0" ViewCount="121" Body="&lt;p&gt;Alphones Pyramus de Candolle (1806-1893), the son of Augustin-Pyrame de Candolle (1778-1841), has been an important figure (as was his father) in the development of plant geography.&lt;/p&gt;&#xA;&lt;p&gt;The younger Candolle's work &amp;quot;Géographie botanique raisonnée&amp;quot; (1856) has been an especially  important text for the history of plant geography since it combines botanical arithmetic with hypothesis driven scientific research. The scholar Janet Browne call this text &amp;quot;the cornerstone of mid-century ideas on the subject&amp;quot;.[^1]&lt;/p&gt;&#xA;&lt;p&gt;I was wondering if this text (or any of his &lt;s&gt;for example his &amp;quot;Essai élémentaire de Géographie botanique&amp;quot; from 1820&lt;/s&gt;) has been translated in either German or English as these two languages were the other principal languages of biological distribution studies. It would also help myself, since my French is regrettably limited.&lt;/p&gt;&#xA;&lt;p&gt;[^1]: &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1038/npg.els.0003092/full&quot; rel=&quot;nofollow noreferrer&quot;&gt;Browne, J. 2001. History of Biogeography. eLS&lt;/a&gt;, p. 2&lt;/p&gt;&#xA;&lt;p&gt;EDIT: As noted in my answer the Essai élémentaire de Géographie botanique was written by Auguste de Candolle.&lt;/p&gt;&#xA;" OwnerUserId="6147" LastEditorUserId="10608" LastEditDate="2024-02-27T07:45:46.570" LastActivityDate="2024-02-27T07:46:12.810" Title="Has Alphonse Pyramus de Candolle's &quot;Géographie botanique raisonnée&quot; ever been translated into German or English?" Tags="|reference-request|biology|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6436" PostTypeId="2" ParentId="6435" CreationDate="2017-08-19T15:48:54.207" Score="3" Body="&lt;p&gt;Origin of cultivated plants / by Alphonse de Candolle.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Main author:Candolle, Alphonse de, 1806-1893.&#xA;Title:Origin of cultivated plants / by Alphonse de Candolle.&#xA;Other Entries:Darwin, Francis, Sir, 1848-1925, former owner.&#xA;University of Cambridge. Department of Plant Sciences, former owner.&#xA;Published:London : Kegan Paul, Trench &amp;amp; co, 1884.&#xA;Description:viii p., 1 l., 468 p ; 20 cm.&#xA;Series statement:International scientific series ; vol. xlix&#xA;Notes:Original French edition, 1883; first American edition, 1885.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;[From the Cambridge University Library catalogue]&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-08-19T15:48:54.207" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6437" PostTypeId="1" AcceptedAnswerId="6440" CreationDate="2017-08-19T17:12:56.890" Score="7" ViewCount="224" Body="&lt;p&gt;According to Melanie Mitchell's book, &lt;a href=&quot;http://waltersorrentino.com.br/wp-content/uploads/2012/02/melanie-mitchell-complexity_a-guided-tour-366-paginas.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Complexity: A Guided Tour&lt;/a&gt; [PDF], Mendel's model&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] was published in a rather obscure journal and was not appreciated as being of great importance until 1900, after which several scientists had obtained similar results in experiments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Experiments_on_Plant_Hybridization&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Wikipedia page for his paper&lt;/a&gt; describes his work as &amp;quot;being largely forgotten&amp;quot;. So, how was his work rediscovered again, given that 45 years are very long comparing to human lifetime?&lt;/p&gt;&#xA;&lt;p&gt;I hope that I am not exaggerating, but either someone had to read his paper and still remember it after 45 years (which contradicts the claim of its &amp;quot;being largely forgotten&amp;quot;), or re-read every old paper just to come across his (which has a very low chance with an obscure journal).&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="4251" LastEditDate="2020-07-05T14:50:50.537" LastActivityDate="2020-07-05T14:50:50.537" Title="How was Mendel's work rediscovered after being forgotten for 45 years?" Tags="|biology|evolution|genetics|mendel|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6438" PostTypeId="2" ParentId="3131" CreationDate="2017-08-20T07:09:30.933" Score="38" Body="&lt;p&gt;Bernhard Riemann (1826–1866) died at the age of 39. Paraphrasing the following from &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Bernhard_Riemann&amp;amp;oldid=791189307&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The son of a poor pastor, he enrolled at age 19 at the University of Göttingen to obtain a degree in Theology. There Gauss urged him to become a mathematician. When he was 31, there was an attempt to promote him to extraordinary professor status. When he was 33, Dirichlet died, and he became head of the department of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He is known for Riemannian geometry (Riemann surfaces etc.), the Riemann integral (the first rigorous formulation of the integral), &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=On_the_Number_of_Primes_Less_Than_a_Given_Magnitude&amp;amp;oldid=792864896&quot; rel=&quot;noreferrer&quot;&gt;one of the most influential papers in analytic number theory&lt;/a&gt;, in which he stated the Riemann hypothesis, and &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=List_of_things_named_after_Bernhard_Riemann&amp;amp;oldid=755899620&quot; rel=&quot;noreferrer&quot;&gt;a few other things&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1866, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Austro-Prussian_War&quot; rel=&quot;noreferrer&quot;&gt;Austro-Prussian War&lt;/a&gt; broke out. The armies of Prussia and Hanover clashed in Göttingen. Riemann fled the city. In Italy, he died of tuberculosis. (The war lasted just over two months, from June to August 1866.) He was 39. Much of his unpublished work was lost.&lt;/p&gt;&#xA;" OwnerUserId="432" LastActivityDate="2017-08-20T07:09:30.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6439" PostTypeId="1" AcceptedAnswerId="6442" CreationDate="2017-08-20T09:51:25.643" Score="4" ViewCount="281" Body="&lt;p&gt;I am curious about who first noticed that block matrices can be multiplied blockwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a section about matrices partitioned into submatrices that describes block matrix multiplication in &quot;An introduction to the theory of canonical matrices&quot; by Turnbull and Aitken, but there are earlier references that use block matrices and their multiplication and inverses, for example, Schur's dissertation &quot;Über eine Klasse von Matrizen...&quot;. &#xA;A natural guess for the origin of block matrices would be Jordan's &quot;Traité des substitutions et des équations algebriques&quot;, but it seems from the formulas that Jordan does not employ block matrices and I don't read French, so I cannot say if Jordan mentions something in text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a reference earlier that Turnbull–Aitken that talks explicitly about multiplication of block matrices or was it a folklore simple thing?&lt;/p&gt;&#xA;" OwnerUserId="2578" LastActivityDate="2017-08-21T21:01:16.630" Title="Where did block matrix multiplication appear?" Tags="|mathematics|linear-algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6440" PostTypeId="2" ParentId="6437" CreationDate="2017-08-20T16:36:43.613" Score="5" Body="&lt;p&gt;The story is told in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gregor_Mendel#Rediscovery_of_Mendel.27s_work&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Mendel's work was not &amp;quot;forgotten&amp;quot; but rather &amp;quot;ignored&amp;quot; because it did not fit in the mainstream science of that time. But some people go to libraries and read old papers. So when discoveries which were &amp;quot;much ahead of its time&amp;quot; are eventually rediscovered it is frequent that someone finds an older work. Such things happen regularly. The work published in established scientific journals usually does not disappear, as long as we have scientific libraries. Such things constantly happen in all sciences. For example, the Fast Fourier Transform (FFT) algorithm was rediscovered several times in the 19th and 20th centuries, until it became &amp;quot;common knowledge&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Opinions on which journals are &amp;quot;obscure&amp;quot; and which are not are always highly subjective.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2020-07-05T13:50:27.030" LastActivityDate="2020-07-05T13:50:27.030" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6441" PostTypeId="2" ParentId="6439" CreationDate="2017-08-21T10:11:35.867" Score="2" Body="&lt;p&gt;In section 20 of the supplement to his &quot;&lt;a href=&quot;http://www.archive.org/stream/abhandlungenmet00gausrich#page/n87/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theoria combinationis&lt;/a&gt;&quot; Gauss proposed the solution of the normal equations in groups, which is perhaps not the kind of block multiplication that you are looking for, but in modern terms this method would be called a Block Gauss-Seidel method. Here is an &lt;a href=&quot;https://archive.org/stream/adjustmentobser01hayfgoog#page/n184/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;English description&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alternatively, here is a &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN599415665_0003|LOG_0018&amp;amp;physid=PHYS_0302&quot; rel=&quot;nofollow noreferrer&quot;&gt;short note&lt;/a&gt; from 1858 by Zehfuss introducing the Kronecker product of two matrices.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2017-08-21T10:11:35.867" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6442" PostTypeId="2" ParentId="6439" CreationDate="2017-08-21T19:20:23.343" Score="2" Body="&lt;p&gt;Matrix multiplication was defined by Cayley (&lt;a href=&quot;https://archive.org/stream/collmathpapers02caylrich#page/n202&quot; rel=&quot;nofollow noreferrer&quot;&gt;1855&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/collmathpapers02caylrich#page/n492&quot; rel=&quot;nofollow noreferrer&quot;&gt;1858&lt;/a&gt;), so anything earlier can only involve products of determinants, not of the matrices themselves. According to Cayley (&lt;a href=&quot;https://archive.org/stream/collmathpapers02caylrich#page/n621&quot; rel=&quot;nofollow noreferrer&quot;&gt;1889&lt;/a&gt;),&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The next later memoir on the theory of Matrices, so far as I am aware is that by Laguerre, “Sur le Calcul des Systèmes Lineaires,” &lt;em&gt;Jour. Ec. Polyt.&lt;/em&gt; t. xxv. (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k433698b.f219&quot; rel=&quot;nofollow noreferrer&quot;&gt;1867&lt;/a&gt;), pp. 215–264. A “système lineaire” is what I called a matrix (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lo and behold, Laguerre’s remarkable paper does introduce blockwise multiplication, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k433698b/f259&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 255–256&lt;/a&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-08-21T21:01:16.630" LastActivityDate="2017-08-21T21:01:16.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6443" PostTypeId="2" ParentId="3870" CreationDate="2017-08-22T00:50:46.220" Score="-1" Body="&lt;p&gt;He did say that in his second letter to Hardy, and as it turned out, it is not so ridiculous after all. This sum is being used in physics like in string theory.&lt;/p&gt;&#xA;" OwnerUserId="6193" LastActivityDate="2017-08-22T00:50:46.220" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6445" PostTypeId="1" AcceptedAnswerId="6465" CreationDate="2017-08-22T08:09:18.580" Score="5" ViewCount="404" Body="&lt;p&gt;Recently I found a website with good historical information about the contributions of Gauss to Analysis situs (the old term for topology). The site is in German so I made a Google translate to English on it. The site contains the following remark by Möbius on a certain oral communication  of Gauss: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One can easily visualize a double ring by cutting out a sheet of paper in the form of a cross, and then the ends FH and F'H '(see figure) of the pair of opposing arms approximately above the initial plane Of the cross and the ends BD and B'D 'of the other pair below this plane, the latter still possesses the remarkable property (after an oral communication by Gauss; what led Gauss to consider this surface, is unknown to me), that from any four points P, Q, R, S following each other on their perimeters, the first one is connected to the third and the second to the fourth by two lines PTT'R, and QUU'S, which lie in the surface itself and yet do not intersect one another - as would always happen if the surface had a basic form of the first class (for example, a Cylinder, C.H.).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The site, whose link is here: &lt;a href=&quot;http://www.solidaritaet.com/ibykus/1997/1/gauss.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.solidaritaet.com/ibykus/1997/1/gauss.htm&lt;/a&gt;, also includes an a certain pic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is this surface a kind of Möbius strip?; I checked it and it's not the classical &quot;twisted&quot; Möbius strip - in fact I'm not sure at all this is a kind of Möbius strip (I checked it). But I certainly think this surface implies some basic topological ideas introduced by Gauss. So what are these topological ideas? &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-21T18:42:40.427" LastActivityDate="2019-10-21T18:42:40.427" Title="What topological ideas did Gauss introduce to his student Möbius?" Tags="|biographical-details|geometry|topology|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6446" PostTypeId="2" ParentId="6445" CreationDate="2017-08-22T08:58:01.430" Score="2" Body="&lt;p&gt;No, this surface is actually less interesting than the Moebius strip, it is just a torus (if you do not know what this means, think of the surface of a doughnut)  with a (rather large) hole in it. However, it does suggest a nontrivial (at the time of writing, but quite standard now) and useful mathematical idea, namely, constructing surfaces by identifying sides of some (not necessarily convex) polygons. &lt;/p&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2017-08-22T08:58:01.430" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6447" PostTypeId="1" CreationDate="2017-08-23T15:05:14.550" Score="3" ViewCount="1274" Body="&lt;p&gt;To quote &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Raphson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Raphson's most notable work... contains a method, now known as the Newton–Raphson method... Newton had developed a very similar formula in his Method of Fluxions, written in 1671, but this work would not be published until 1736, nearly 50 years after Raphson's Analysis. However, Raphson's version of the method is simpler than Newton's, and is therefore generally considered superior. For this reason, it is Raphson's version of the method, rather than Newton's, that is to be found in textbooks today.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's a shame we normally call it Newton's method, since if anything it should be named after the otherwise forgotten Raphson. But what was Newton's version of the formula? Raphson's is our modern formula,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x\mapsto x-\frac{f(x)}{f'(x)}.$$&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2021-12-17T07:22:18.827" LastActivityDate="2021-12-17T07:22:18.827" Title="How did Newton &amp; Raphson's versions of the N-R method differ?" Tags="|mathematicians|calculus|newton|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6448" PostTypeId="2" ParentId="6447" CreationDate="2017-08-23T17:37:30.510" Score="3" Body="&lt;p&gt;According to Peter Deuflhard's &lt;a href=&quot;https://www.math.uni-bielefeld.de/documenta//vol-ismp/13_deuflhard-peter.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Short History of Newton's Method&lt;/em&gt;&lt;/a&gt;, Newton began by familiarising himself with the methods of Vieta, which had been simplified by Oughtred.  (Vieta's method was already known to al-Kāshī, who published a method using Vieta's &lt;em&gt;perturbation technique&lt;/em&gt; in 1427, though Vieta appears to have been unaware of al-Kāshī's work.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As an example, he discussed the numerical solution of the cubic&#xA;  polynomial $$f(x) := x^3 - 2x - 5 = 0.$$ Newton first noted that the integer part of the root is $2$ setting $x_0 = 2$.&#xA;  Next, by means of $x = 2 + p$, he obtained the polynomial equation  $$p^3+6p^2+10p-1 = 0.$$ He neglected terms higher than first order setting $p ≈ 0.1$. Next, he&#xA;  inserted $p = 0.1 + q$ and constructed the polynomial equation $$q^3 + 6.3q^2 + 11.23q + 0.061 = 0.$$  Again neglecting higher order terms he found $q ≈ −0.0054$. Continuation of the process one further step led him to $r ≈ 0.00004853$ and therefore to the third iterate $$x_3 = x_0 + p + q + r = 2.09455147.$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Deuflhard continues &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1690, Joseph Raphson (1648–1715) managed to avoid the tedious computation&#xA;  of the successive polynomials, playing the computational scheme&#xA;  back to the original polynomial; in this now fully iterative scheme, he&#xA;  also kept all decimal places of the corrections. He had the feeling that&#xA;  his method differed from Newton’s method at least by its derivation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that neither Newton nor Raphson mention derivatives, even if, as Deuflhard notes :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Note that the relations $10p − 1 = 0$ and $11.23q + 0.061 = 0$ given above&#xA;  correspond precisely to $$p = x_1 − x_0 = −f(x_0)/f′(x_0)$$&#xA;  and to $$q = x_2 − x_1 = −f(x_1)/f′(x_1).$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1740, Thomas Simpson (1710–1761) actually introduced derivatives&#xA;  (‘fluxiones’) in his book ‘Essays on Several Curious and Useful Subjects in&#xA;  Speculative and Mix’d Mathematicks [No typo!], Illustrated by a Variety&#xA;  of Examples’. He wrote down the true iteration for one (nonpolynomial)&#xA;  equation and for a system of two equations in two unknowns thus making&#xA;  the correct extension to systems for the first time. His notation is already&#xA;  quite close to our present one (which seems to date back to J. Fourier).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also worth noting is that, according to &lt;a href=&quot;https://www.math.ubc.ca/~anstee/math104/104newtonmethod.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;UBC Math&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At first sight, the method Newton uses doesn’t look like the Newton&#xA;  Method we know. The derivative is not even mentioned, even though &lt;strong&gt;the&#xA;  same manuscript develops the Newtonian version of the derivative&lt;/strong&gt;!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(My emphasis.)&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-08-23T17:37:30.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6449" PostTypeId="2" ParentId="6447" CreationDate="2017-08-24T00:00:14.743" Score="2" Body="&lt;p&gt;Newton's method of successive approximation to roots of equations first appeared in print in 1685 in John  Wallis's  'A Treatise of Algebra, both Historical and Practical' at pp.338-347, Chap.XCIV &amp;amp; XCV, 'A new method of extracting roots in Simple and Affected Equations' -- containing also Wallis's attribution to Newton.  This presentation has been discussed by Niccolo Guicciardini, who has remarked that Newton's description 'is based on concrete examples ... a craft more than a theory':  N Guicciardini (2009), '&lt;a href=&quot;https://books.google.co.uk/books?id=U4I82SJKqAIC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isaac Newton on Mathematical Certainty and Method&lt;/a&gt;'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton published a further concrete example of the method in the 'Principia' (all editions, from 1687), at Book 1, Prop.31 (Scholium). This example was an application of the method, to solve an equation defining the orbital position of a planet, one version of Kepler's problem.  The example was discussed in 1882 by John Couch Adams (&lt;a href=&quot;http://adsabs.harvard.edu/abs/1882MNRAS..43...43A&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monthly Notices of the Royal Astronomical Society 43, pp.43-49&lt;/a&gt;), who pointed out that in the first edition of the Principia, Newton had tried to describe a refinement of the method, but had made a mistake (equivalent to forgetting to update one of the 'temporary variables' in the iteration); Newton corrected the description (as amended in the second and the third editions) with a simplified version of the quadratically-converging method.  In the 1999 translation of Newton's Principia, the method appears at pp.514-515 (Isaac Newton, 'The Principia ... A new Translation', I Bernard Cohen, Anne Whitman, 1999). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Raphson's method of successive approximations to roots of equations was first published in his 'Analysis Aequationum Universalis', 1690, which went through at least three editions. (Neither Newton nor Raphson used any notation close to the modern &quot; -f(x)/f'(x) &quot; for the iterative corrections.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Raphson's method has been discussed and compared with Newton's method by F Cajori among others (1911, American Mathematical Monthly, Vol.18, 29-32). Cajori showed how Raphson's method resembled that of Newton especially in that they both used a divisor for the next correction that evaluates, in effect, to (what in modern notation would be) f'(r), where r was the most recent corrected value. (But neither of them used a symbolic notation such as f'(r), they expressed each case of the quantity in an extended e.g. polynomial form.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But where Newton had derived each successive step of approach to the root from a newly formed equation derived from the original one, Raphson, closer in that respect to the modern procedure, found each successive corrected value by substitution in the original equation.  (Cajori also remarked that Newton's forerunners such as Vieta had used a different and generally non-equivalent principle for evaluating the relevant divisor. But in some specific example-cases, Vieta's value of the divisor had coincided with the value of Newton's divisor -- which had led some commentators mistakenly to assimilate Newton's method with Vieta's.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cajori gave a history of the methods, and in view of Raphson's important modification proposed the name 'Newton-Raphson' method as a more accurate attributive label than just the name 'Newton's method', which had been current.  His suggestion, made over a century ago now, does seem to have taken hold, 'Newton-Raphson' is widely used to denote the modern version of the method.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="104" LastEditDate="2017-09-24T09:00:39.113" LastActivityDate="2017-09-24T09:00:39.113" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6452" PostTypeId="1" CreationDate="2017-08-24T12:46:04.740" Score="7" ViewCount="155" Body="&lt;p&gt;In (the AMS Chelsea Publishing version of) what is perhaps the first genuine textbook on graph theory ever, Dénes Kőnig on p. 28 gives the illustration&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LthaF.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LthaF.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;and the footnote&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2Fgnh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2Fgnh.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;which when translated says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since the graph in Fig. 14 can be drawn in the plane (on the 2-sphere), this graph constitutes a simple counterexample to a false claim that is  frequently recurring in the literature on the Four Color Theorem, cf. e.g. as late as 1930: Jahresbericht der deutschen Mathematiker-Vereinigung, 39, S. &lt;span class=&quot;math-container&quot;&gt;$\textit{51}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Kőnig here is referencing the following text, which I take from &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN37721857X_0039&quot; rel=&quot;nofollow noreferrer&quot;&gt;this electronic version of the Jahresbericht der deutschen Mathematiker-Vereinigung&lt;/a&gt; (note that in those days this journal had what a model theorist may be tempted to call 'typed' or 'sorted' page numbers: there were upright page numbers, and there were pages numbers in &lt;em&gt;italics&lt;/em&gt;, the latter were used for the appendix.&lt;/p&gt;&#xA;&lt;p&gt;Schoblik's text is roughly of the genre 'research announcement' or rather 'extended abstract', and it was published as part of the proceedings of the annual meeting of the DMV in Prague, 16.-23. September 1929:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2x2KE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2x2KE.png&quot; alt=&quot;p. &lt;span class=&quot;math-container&quot;&gt;$\textit{51}$&lt;/span&gt;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uHOEP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uHOEP.png&quot; alt=&quot;p. &lt;span class=&quot;math-container&quot;&gt;$\textit{52}$&lt;/span&gt;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I now give a literal translation of Schoblik's extended abstract&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;F. Schoblik, Brünn: On the problem of coloring maps.&lt;/p&gt;&#xA;&lt;p&gt;The problem of coloring maps on a surface of genus zero (Four Color Problem) asserts that it is possible to color a map on the 2-sphere with four colors in such a way that no two like colors abut along any line.&#xA;One may assume without loss of generality that one is coloring a 3-regular graph without 'Blätter' [I do not know what Schoblik means here; literally 'Blätter' means 'leaves', which does not make sense &lt;em&gt;if&lt;/em&gt; one would understand this in the contemporary graph-theoretic sense of 'leaves', since a 'leaf' is just a vertex of degree precisely one, of which a 3-regular graph does not have any].&#xA;Every set &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; of vertices of the graph, together with all incident edges, which has the property that every vertex of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is incident with at most one 'free' edge [I am not sure what Schoblik means here; presumably he means that every vertex of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; has at most one neighbor &lt;em&gt;outside&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;], shall be called a &lt;em&gt;part&lt;/em&gt; of the graph; moreover, each single vertex together with the three incident edges, shall be subsumed under the term 'part', too. The 'free' edges of a 'part' are called its &lt;em&gt;accesses&lt;/em&gt;. We assume that the graph does not contain any 'part' with only two 'accesses' ('bridge'). [Schoblik's claim that 'bridgeless' were equivalent with non-existence of any part with two 'freie' edges seems wrong, unless I misunderstand his definition of 'freie' edges: a bridge in a graph may have any cardinality of 'freie' edges, if 'freie edges' is understood to mean 'edges incidence with the bridge]. The general case can be reduced to coloring a finite number of bridgless such graphs. For such special graphs, the problem is solved by the following theorem:&lt;/p&gt;&#xA;&lt;p&gt;Every bridgeless 3-regular graph on the 2-sphere contains a closed sequence of edges which contains all the vertices, and contains each vertex only once. [In contemporary language, Schoblik here claims that every bridgeless 3-regular graph on the 2-sphere contains a Hamilton circuit.]&lt;/p&gt;&#xA;&lt;p&gt;When trying to traverse such a sequence of edges it becomes apparent that every traversed edge, together with the preceding edges, requires-or rules out-certain edges among those which are still eligible (we call such edges 'successor edges') [I find it confusing that Schoblik seems to permit that a 'successor edge' may either be of type 'required edge' or of type 'ruled-out edge']. The traversal of an edge, together with all the accompanying 'successor edges' may briefly be called a 'walk'. We moreover say that a walk &lt;em&gt;obstructs&lt;/em&gt; a 'part' of the graph if it 'requires' the traversal of all heretofore not forbidden 'accesses' of the relevant part, and if moreover the number of those [required-yet-not-yet-forbidden] 'accesses' is odd. Likewise, we say that a walk 'blocks' a 'part' if it 'forbids' all but at most one of the 'accesses' of the part. Walks which 'obstruct' or 'block' are called &lt;em&gt;impermissible&lt;/em&gt;, otherwise, &lt;em&gt;permissible&lt;/em&gt;. If the notion 'successor edge' is made suitably precise, the one can show that if a graph of the above-mentioned kind is traversed, then of the two possible edges after a single-edge-traversal, at least one must be permitted. By concatenating permitted edge-traversals one obtains a closed sequence of edges of the kind claimed above. This immediately implies the theorem of Tait for 3-regular graphs embedding in the 2-sphere, according to which any such graph can be decomposed into three 1-regular graphs [=matchings].&lt;/p&gt;&#xA;&lt;p&gt;A detailed presentation will be published elsewhere.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;It seems clear that no proof of the Four Color Theorem can be expected to come of Schoblik's approach, and the 'theorem' he singles out is wrong, as correctly pointed out in Kőnig's 1936 footnote.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Yet my main question is: &lt;em&gt;Did Schoblik's announced 'ausführliche Darstellung' survive&lt;/em&gt;? Or even appear somewhere in print? If not, is there any reason to hope that it lies undiscovered somewhere? Whom to ask in the Czech Republic today?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am particularly interested if anything can be said with certainty about what stopped Schoblik's announced ''ausführliche Darstellung' from publication (and apparently it &lt;em&gt;was&lt;/em&gt; stopped). Is there anything surviving of the &lt;em&gt;refereeing process&lt;/em&gt; involved?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If Schoblik's proof did not appear, are there any surviving documents about what became of the [presumably once existant] manuscript of Schoblik's?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there any surviving documents on the &lt;em&gt;refereeing process&lt;/em&gt; for Schoblik's wrong proof?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;More generally, what are &lt;em&gt;other&lt;/em&gt; occurrences of the &amp;quot;in der Literatur des Vierfarbensatzes oft wiederkehrende falsche Behauptung&amp;quot;? This seems to be a question which has a unique answer, yet currently I do not have a single example. I would appreciate as many precise occurrences as possible. More precisely: who in print did claim, besides Schoblik, that any edge-2-connected 3-regular planar graph is Hamiltonian. Note that &lt;strong&gt;this is not Tait's conjecture&lt;/strong&gt;, as disproved by Tutte, it is another, much easier to disprove wrong statement. Currently, my count for the 'oft' stands at 1.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remarks.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;First and foremost, one should point out that what Schoblik announces in the extended abstract translated above is nothing less than a proof of a statement which if true would imply the Four Color Theorem. Of course, the statement about Hamiltonian-ness of any 2-edge-connected 3-regular finite graph embedded in the sphere is &lt;em&gt;wrong&lt;/em&gt;, like Kőnig correctly points out in his footnote (the graph represented by Fig. 14 is a counterexample: it is edge-2-connected&lt;sub&gt;1&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Apparently, the author of the extended abstract was &lt;a href=&quot;http://web.math.muni.cz/biografie/friedrich_schoblik.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedrich Schoblik&lt;/a&gt;, who published on real analysis. It seems (from what I can make of the Czech language) that he obtained a doctorate in Brno in 1933, related to the &lt;em&gt;other&lt;/em&gt; extended abstract in the Jahresbericht cited below (the one on differential forms), and that in 1939 he became an associate professor in Brno, and that in 1939 he finished his work on the book&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Friedrich Lösch, Fritz Schoblik: Die Fakultät (Gammafunktion) und verwandte Funktionen. Mit besonderer Berücksichtigung ihrer Anwendungen. B. G. Teubner, Leipzig 1951.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;whose publication was delayed until 1951 (when it was edited by Friedrich Lösch).&lt;/p&gt;&#xA;&lt;p&gt;Apparently Schoblik in 1944,went missing in what (a search engine translation of) the biographic article cited above calls &amp;quot;the Eastern Front&amp;quot;.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://web.math.muni.cz/biografie/bib/hlavni/schoblik.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This list of Schoblik's publications&lt;/a&gt;&#xA;lists the extented abstract I translated above, yet seems not to contain any trace of the announced &amp;quot;ausführliche Darstellung&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Besides announcing a proof of the Four Color Theorem, Schoblik in the same conference proceedings (loc. cit. &lt;span class=&quot;math-container&quot;&gt;$\textit{41}$&lt;/span&gt;) also published a long extended abstract on differential forms.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; &lt;sub&gt;Yet evidently the graph in Fig. 14 is not vertex-3-connected, hence is not a &lt;a href=&quot;http://mathworld.wolfram.com/PolyhedralGraph.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;polyhedral graph&lt;/a&gt;. In particular, what Schoblik claims is &lt;em&gt;stronger&lt;/em&gt; than the &lt;em&gt;also-wrong-but-disproved-ten-years-after-Kőnig's-footnote&lt;/em&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Tait%27s_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tait's conjecture&lt;/a&gt;.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-08-25T15:28:22.767" Title="F. Schoblik's announced ''ausführliche Darstellung&quot;: a lost wrong proof of the Four Color Theorem?" Tags="|mathematics|biographical-details|combinatorics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6453" PostTypeId="1" CreationDate="2017-08-24T21:51:11.103" Score="3" ViewCount="105" Body="&lt;p&gt;I am somewhat amazed at the depth of the theorem but I believe it involved his work with electricity and magnetism. Additionally I don't think he had a formal education.  I have not found much on this historical development in the context of electricity and magnetism.  Most work centers around Faraday and Maxwell's brilliant unification. Can someone shed more light on this?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2017-08-25T16:42:41.803" Title="What motivated Green to develop his theorem in order to calculate work for a non conservative vector field?" Tags="|mathematicians|electricity|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6454" PostTypeId="1" CreationDate="2017-08-24T22:01:59.517" Score="3" ViewCount="427" Body="&lt;p&gt;We say that two dynamical variables $f$ and $g$ are in involution if their Poisson bracket vanishes, i.e., $\{f,g\}=0$. Why is it called involution?&lt;/p&gt;&#xA;" OwnerUserId="5436" LastActivityDate="2017-08-24T23:21:29.277" Title="What is the origin of the term &quot;involution&quot; used in Hamiltonian mechanics" Tags="|physics|terminology|classical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6455" PostTypeId="2" ParentId="6454" CreationDate="2017-08-24T23:21:29.277" Score="2" Body="&lt;p&gt;An involution is, according to the &lt;a href=&quot;http://www.oed.com/view/Entry/99200#eid16688&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;OED&lt;/em&gt;&lt;/a&gt;, &quot;A function or transformation that is equal to its inverse.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whittaker discusses involution systems in his &lt;a href=&quot;https://isidore.co/calibre/browse/book/4482&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Treatise on the Analytic Dynamics of Particles and Rigid Bodies&lt;/em&gt;&lt;/a&gt; p. 322:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;147&lt;/strong&gt;. &lt;em&gt;Invoultion systems.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Let $(u_1,, u_2 , …, u_r)$ denote $r$ functions of $n$ independent variables $$(q_1, q_2 , …, q_n , p_1, …, p_n);$$if it is possible to express all the Poisson-brackets $(u_i,u_k)$ as functions of $(u_1,u_2,…,u_r)$, the functions $(u_1,u_2,…,u_r)$ are said to form a &lt;em&gt;function-group&lt;/em&gt;*. Any function of $(u_1,u_2,…,u_r)$ belongs to this group.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If the quantities $(u_i,u_k)$ are all zero, the functions $(u_1,u_2,…,u_r)$ are said to be &lt;em&gt;in involution&lt;/em&gt;, or to form an &lt;em&gt;involution-system&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;hr&gt;&lt;sup&gt;*Lie, &lt;em&gt;Math. Ann.&lt;/em&gt; VIII. (1875). p. 215.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2017-08-24T23:21:29.277" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6456" PostTypeId="1" CreationDate="2017-08-24T23:27:11.953" Score="0" ViewCount="124" Body="&lt;p&gt;Was it common knowledge around 600 AD or perhaps 30 BCE that 200 solar years for example were equal to 209 lunar years? In which period of history precisely can we say that such a thing was confirmed?&lt;/p&gt;&#xA;" OwnerUserId="6207" LastActivityDate="2017-08-25T23:58:05.170" Title="Were people in the ancient or medieval times aware of how exactly a solar year is equal to a lunar year?" Tags="|astronomy|time|calendar|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6457" PostTypeId="2" ParentId="6456" CreationDate="2017-08-25T13:05:42.917" Score="0" Body="&lt;p&gt;Such things were never &quot;common knowledge&quot; as they are not now. People who dealt with such things are called astronomers, and there were not many of them in any period. Precise relations of the sort you mention were first discovered in the ancient Babylonia, and they were known to the Greek astronomers of the Hellenistic period (probably this knowledge penetrated from Babylonia). &#xA;(Of course similar relations, but less precise were also discovered in other civilizations: everyone knows indeed that there are approximately 12 months in a year.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The period you mention (600 a.d.) was the &quot;Dark Age&quot; in Europe (and in the Middle East) when this knowledge was forgotten. There were no known astronomers, and in many places this occupation was even prohibited. Still, the work of Ptolemy somehow survived to our day, which means that it was at least copied. &#xA;And perhaps those who copied understood something of it. Most of the other Greek works on astronomy were not sufficiently copied in this period, and they are lost. Babylonian astronomy is known from the clay tables which are fortunately durable enough. They were discovered and read in 19 century. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-08-25T23:58:05.170" LastActivityDate="2017-08-25T23:58:05.170" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6458" PostTypeId="2" ParentId="6456" CreationDate="2017-08-25T15:44:40.043" Score="1" Body="&lt;p&gt;Hipparchus, who lived in the 2nd century BCE, calculated the length of the synodic lunar month as 29.53059 days, and that of the tropical year as 365.24667 days (though he expressed both values not in decimal notation but in sexagesimal notation). These values were adopted by Ptolemy and became “common knowledge” among people who were interested in astronomy, whether in the Greek world, or in the Islamic world, or even in Western Europe. The Jews used these values as the basis for the calculation of their luni-solar calendar. &lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-08-25T15:44:40.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6460" PostTypeId="1" CreationDate="2017-08-27T00:04:12.293" Score="9" ViewCount="457" Body="&lt;p&gt;The &lt;a href=&quot;https://hsm.stackexchange.com/questions/2149/what-came-first-the-kernel-from-vector-spaces-or-from-group-theory&quot;&gt;kernel&lt;/a&gt; question leads to another : Today, &lt;strong&gt;&lt;em&gt;homomorphism&lt;/em&gt;&lt;/strong&gt; (resp. &lt;em&gt;isomorphism&lt;/em&gt;) means what Jordan (&lt;a href=&quot;https://archive.org/stream/traitdessubsti00jorduoft#page/56&quot; rel=&quot;noreferrer&quot;&gt;1870&lt;/a&gt;) had called &lt;em&gt;isomorphism&lt;/em&gt; (resp. &lt;em&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/3108/jordan-called-isomorphisms-iso-and-homomorphisms-iso-holoedriques-and-iso&quot;&gt;holoedric&lt;/a&gt; isomorphism&lt;/em&gt;). How did the switch happen?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;“Homomorphic” appears e.g. in de Séguier (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?36.0187.02&quot; rel=&quot;noreferrer&quot;&gt;1904&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/thoriedesgroup00sguoft#page/65&quot; rel=&quot;noreferrer&quot;&gt;pp. 65–66&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(...) supposons entre les groupes $\mathrm A$ et $\mathrm B$ une correspondance telle qu’à chaque élément de $\mathrm A$ réponde un élément au moins de $\mathrm B$ et à chaque élément de $\mathrm B$ un élément au moins de $\mathrm A$ et que si $a_i$ de $\mathrm A$ et $b_i$ de $\mathrm B$ se correspondent, $a_ia_k$ et $b_ib_k$ se correspondent aussi. On dit que $\mathrm A$ et $\mathrm B$ sont &lt;em&gt;homomorphes&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“Homomorphism” (as &lt;em&gt;property&lt;/em&gt; of being homomorphic) appears e.g. in Loewy (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?43.0199.02&quot; rel=&quot;noreferrer&quot;&gt;1912&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/festschriftheinr06webeuoft#page/198&quot; rel=&quot;noreferrer&quot;&gt;p. 198&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;§1 (...) untersucht den Homomorphismus zweier abstrakter Gruppen (...).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“Homomorphism” (as &lt;em&gt;map&lt;/em&gt; between groups) appears e.g. in Hopf (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?54.0610.02&quot; rel=&quot;noreferrer&quot;&gt;1928&lt;/a&gt;, &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002507641&amp;amp;physid=phys139&quot; rel=&quot;noreferrer&quot;&gt;p. 128&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;4) Unter einem Homomorphismus der Gruppe $\mathfrak G$ in die Gruppe $\mathfrak H$ verstehen wir eine eindeutige Abbildung $f$ von $\mathfrak G$ auf einen echten oder unechten Teil von $\mathfrak H$, bei der stets $f(x+y)=f(x)+f(y)$ ist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Are these the earliest occurrences?&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-08-27T01:28:55.880" LastActivityDate="2017-08-31T18:32:57.913" Title="Whence “homomorphism”, “homomorphic”?" Tags="|terminology|abstract-algebra|group-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6461" PostTypeId="1" AcceptedAnswerId="6470" CreationDate="2017-08-27T07:28:47.243" Score="2" ViewCount="781" Body="&lt;p&gt;The American Physical Society's &lt;a href=&quot;https://www.aps.org/publications/apsnews/200606/history.cfm&quot; rel=&quot;nofollow noreferrer&quot;&gt;This Month in Physics History; June, ca. 240 B.C. Eratosthenes Measures the Earth&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eratosthenes &lt;strong&gt;had heard from travelers&lt;/strong&gt; about a well in Syene (now Aswan, Egypt) with an interesting property: at noon on the summer solstice, which occurs about June 21 every year, the sun illuminated the entire bottom of this well, without casting any shadows, indicating that the sun was directly overhead. Eratosthenes then measured the angle of a shadow cast by a stick at noon on the summer solstice in Alexandria, and found it made an angle of about 7.2 degrees, or about 1/50 of a complete circle. (emphasis added)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However in this small video clip titled &lt;a href=&quot;https://www.youtube.com/watch?v=3EspZtA7C3o&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Sagan - How We Know The Earth Is Round&lt;/a&gt;, Prof. Sagan describes it differently. Here is a transcription of a small bit after the &lt;code&gt;01:00&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He was also the chief librarian of the great library of Alexandria, and one day, &lt;strong&gt;while reading a papyrus book in the library&lt;/strong&gt;, he came upon a curious account: Far to the south, he read, at the military outpost of Syene, something notable could be seen, on the longest day of the year. (emphasis added)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I understand there might not be certainty, but is there some general agreement if it was the words of travelers, or a book in the library where Eratosthenes first heard of the &quot;well of  Syene&quot;? Of course, it's also possible that the book was written by travelers, and so the two descriptions are not necessarily contradictory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d0TAVm.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d0TAVm.jpg&quot; alt=&quot;Carl Sagan discusses Eratosthenes&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2017-08-27T12:13:27.387" LastActivityDate="2019-07-30T12:52:29.523" Title="Is it known where Eratosthenes first heard of the &quot;well of Syene&quot;?" Tags="|astronomy|measurement|ancient-egypt|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6462" PostTypeId="2" ParentId="6142" CreationDate="2017-08-27T12:34:53.907" Score="5" Body="&lt;p&gt;One of the most important mirrors historically is that of Sir Isaac Newton's telescope shown below. In the Periodic Video &lt;a href=&quot;https://www.youtube.com/watch?v=4ESW_NTIhBM&quot; rel=&quot;noreferrer&quot;&gt;Amazing piece of metal (speculum)&lt;/a&gt;, Professor &lt;a href=&quot;https://royalsociety.org/people/martyn-poliakoff-12107/&quot; rel=&quot;noreferrer&quot;&gt;Martyn Poliakoff&lt;/a&gt; (CBE, CChem, FRS, FRSC, FIChemE) describes the telescope, notes about building the telescope, and what is believed to be the speculum mirror used in the telescope, at the Royal Society in London. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EwC6n.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EwC6n.jpg&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5NlMZ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5NlMZ.jpg&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Lrz1V.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lrz1V.jpg&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2017-08-27T12:34:53.907" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6463" PostTypeId="1" AcceptedAnswerId="6493" CreationDate="2017-08-27T18:43:43.597" Score="2" ViewCount="390" Body="&lt;p&gt;The grudge (to say the least) of Philipp Lenard against Einstein is well-known. But he also held a grudge against the Curies. For instance at the book &lt;em&gt;The Man Who Stalked Einstein&lt;/em&gt;, by Birgit Ertl-Wagner, Bernd C. Wagner, and Bruce Hillman, there is this passage: “[&amp;hellip;] Lenard was responsible for providing the foundation from which Roentgen's and the Curies' research had sprung. Then he had looked on from the sidelines as each of those who had followed and benefited from his discoveries had been selected [as Nobel prize winners] before him.” Besides, Lenard's book &lt;em&gt;Great Men in Science, A History of Scientific Progress&lt;/em&gt; doesn't include the Curies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my question is: in which way was Lenard's work important for the Curies?&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="4251" LastEditDate="2020-05-13T06:06:19.050" LastActivityDate="2020-05-13T06:06:19.050" Title="Philipp Lenard and the Curies" Tags="|radiation|physicists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6464" PostTypeId="2" ParentId="6427" CreationDate="2017-08-28T11:17:47.730" Score="6" Body="&lt;p&gt;The article cited by the questioner incorrectly represents the limited amount of historical evidence that we have about the incident described.  When the article is compared with the evidence, it can be seen that its account has been extensively fictionalized. Thus, it is an unsuitable basis on which to question anyone's truthfulness -- except perhaps that of the author of the fiction, whoever that was. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quoting from the article cited by the questioner, &quot;Newton ... claimed he had been unaware of the challenge until he first saw it at 4 pm on 29 January, some 5 weeks after its publication.&quot; &quot;He further claims that he solved it by 4 am the following morning...&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of that is fictional embellishment. Thus, there are two different historical sources about the incident, but they tell us much less about the incident. Both are reported in the notes given in D T Whiteside's edition of Newton's 'Mathematical Papers', vol.8, pp.72-3. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that Newton is known to have said or written about the background was (original in Latin):  &quot;I received yesterday ... two copies (published at Groningen) of problems ...&quot;.  (Nothing more than that about the background, he then went on to quote the problems, and to give his solutions.)  The document is a letter from Newton dated 30 January 1696|97 (also given in full in the 'Correspondence of Isaac Newton', vol.4 from p.220) in which he sent the solution to his friend Montague, who was then President of the Royal Society. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a source for the 'late-night' aspect of the story, but the source was not Newton.  It makes a big difference to put a reminiscence from years after an event fictitiously into the mouth of a person other than its source.  The source here was Newton's niece, Catherine Barton, who lived with Newton after he moved to London to be Warden of the Mint. Newton's niece seems to have organised his domestic and social life until she married. Her recollection was recorded years later by her husband, Newton's friend John Conduitt, who noted that his source for the reminiscence was his wife. She also told her husband about Newton's characteristic behaviour (quoted too in Mathematical Papers vol.8 p.73): &quot;when [he was] come from the Tower&quot; [i.e. where the Mint was located, he] &quot;would stand still a great while if any book or paper [had been] sent him[,] before he would eat&quot;. (This kind of intense focus on Newton's part was also separately reported by his amanuensis of the 1680s, Humphrey Newton, whose reminiscences of Newton's forgetfulness to eat or sleep while concentrating on a problem has often been quoted, e.g. by R S Westfall in 'Never at rest', p.406.)  There doesn't seem to be anything implausible or out of Newton's character about the reminiscence given by Newton's niece to her husband. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What has happened with the story quoted by the questioner, as shown by its comparison with the sources, is that an incident that we can barely glimpse from proper historical sources has been given fictional embellishments (here by some person unknown), and then the fiction itself has been used to draw   conclusions about the incident.  This has nothing to do with history or historical investigation: it more closely resembles the peddling of false gossip or juicy scandal.  (It is to be hoped that the source from which the questioner cited the embellished story will be properly amended before too long.) &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-08-28T11:17:47.730" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6465" PostTypeId="2" ParentId="6445" CreationDate="2017-08-28T18:21:10.943" Score="9" Body="&lt;p&gt;While you already accepted an answer, &#xA;it seems not superfluous to add another one, &#xA;in particular since you are implicitly asking &#xA;for a better translation/understanding of the &#xA;passage you quoted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will give you a translation here, and some comments of mine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Beforehand, though, I think it is fair to say &#xA;hat you seem to have been &lt;em&gt;misled&lt;/em&gt; when you wrote &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;website with good historical information about the contributions of Gauss to Analysis situs&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think it is fair to say that &#xA;this website gives &lt;em&gt;good motivation&lt;/em&gt;, &#xA;yet one should not say it gives &#xA;&quot;good historical information&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It does not meet the standards of &#xA;professional historical writing. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Needless to say, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I am not blaming the author &#xA;of the text on the website, who might perhaps have &#xA;consciously written in a way that they just &#xA;thought fit for the purpose, without even aspiring &#xA;to produce a piece of professional historical writing,&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I myself do not claim that what comes below is of &#xA;professional quality; I also will focus more or less &#xA;on your mathematical question, and there are some &#xA;flaws in the below, too.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Yet to me it seems one should tell you &#xA;that the text you link to, while nice, has its flaws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The text you cite is nice, &#xA;yet it can hardly be considered a scientific source; &#xA;in parts, it is misleading, historically and mathematically, &#xA;sometimes it is quite loosely crafted, &#xA;and the German has stylistic flaws. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Literal copy-paste-quote of a superset&lt;sup&gt;1&lt;/sup&gt; &#xA;of the text from the &lt;a href=&quot;http://www.solidaritaet.com/ibykus/1997/1/gauss.htm&quot; rel=&quot;noreferrer&quot;&gt;website&lt;/a&gt; you cited: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] Damit warf er [Gauß] die gesamte Newtonsche Mechanik in hohem Bogen zum Fenster hinaus, denn die Bewegung eines Körpers wird durch seine Lage im Raum und durch die gesamten ihn beeinflussenden anderen Körper erklärt und nicht nur durch die Anziehungs- und Abstoßungskräfte einer Zentralkraft.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Darüber hinaus behandelt Gauß in seinen Arbeiten zur Theorie der krummen Flächen die Frage der kürzesten Linie, genau die gleiche Frage, die Bernoulli schon 1697 den Geometern gestellt hatte. Denn beide Fragen hängen sehr eng zusammen. Der sich bewegende Körper verfolgt erstens nach dem Gesetz der kleinsten Wirkung [comment: &lt;em&gt;this seems an inaccuracy in the text from the website&lt;/em&gt;: usually Gauß is credited with the &lt;a href=&quot;https://i.stack.imgur.com/lZqn0.png&quot; rel=&quot;noreferrer&quot;&gt;Prinzip des kleinsten Zwangs&lt;/a&gt; (principle of least constraint), which is similar to, yet clearly kept similar from &lt;a href=&quot;https://en.wikipedia.org/wiki/Principle_of_least_action&quot; rel=&quot;noreferrer&quot;&gt;the principle of least action&lt;/a&gt; ], den kürzesten Weg von einem Punkt zum anderen, wird aber dabei abgelenkt durch die &quot;Rahmenbedingungen&quot;, die ihm durch seine Lage im Raum gesetzt sind.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In diesem Zusammenhang ist es erwähnenswert, daß Gauß sehr früh begann, sich mit allen möglichen Verschlingungen oder Umschlingungen eines Fadens oder Verkettungen von Kurven im Raum zu beschäftigten. Schon 1794 macht er eine Liste von 13 sauber gezeichneten Ansichten von Knoten, die er in einem Buch gefunden hat. Diese werden später wichtig bei der Untersuchung elektrischer und magnetischer Ströme. Auch in der Astronomie finden sich bei der Untersuchung von Sternennebeln oder den Atmosphären von Planeten oft eigentümliche Kurven und Verkettungen. Erst in den letzten Jahren hat man aus Satellitenaufnahmen im Saturnring unerklärliche Verschlingungen entdeckt.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Gauß war sich im klaren darüber, daß die Euklidische Geometrie nur eine Geometrie von vielen möglichen darstellt. Wenn man nämlich einen gekrümmten Raum annimmt, so fallen die zeitlich kürzesten Bahnen mit den räumlich kürzesten zusammen. Er hat sich zum Beispiel immer mit der Frage beschäftigt, welche Gesetze für solche gekrümmten Bewegungen gelten, bzw. wie die Koordinatensysteme wohl aussehen, wenn Körper alle Arten von verschlungenen Bahnen annehmen. Auch bei den Planetoiden tritt ja mehrfach das kettenartige Ineinandergreifen von verschiedenen Bahnen auf!&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Als Beispiel dafür, wie sich die Gesetze bei anderen &quot;Rahmenbedingungen&quot; für die Möglichkeiten der Bewegung eines Körpers ändern, ist folgende Konstruktion, die Möbius nach Äußerungen von Gauß überliefert hat. Möbius hatte seit dem Herbst 1813 ein Semester lang unter Leitung von Gauß auf der Göttinger Sternwarte gearbeitet. In seinen Aufzeichnungen über die krummen Flächen bezieht er sich ausdrücklich auf eine mündliche Mitteilung von Gauß über die Eigenschaften eines Doppelringes:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Abbildung Doppelring] &quot;Man kann sich so einen Doppelring leicht zur Anschauung bringen, wenn man ein Blatt Papier in Form eines Kreuzes ausschneidet und hierauf die Enden FH und F'H' (siehe Abbildung) des einen Paares gegenüberliegender Arme etwa oberhalb der anfänglichen Ebene des Kreuzes und die Enden BD und B'D' des anderen Paares unterhalb dieser Ebene mit einander vereinigt. Es besitzt diese nur von einer Linie ABB' IHH' GD'D EF'FA begrenzte Fläche noch die merkwürdige Eigenschaft (nach einer mündlichen Mitteilung von Gauß; wodurch Gauß zur Betrachtung der Fläche geführt worden ist, ist mir unbekannt), daß man von irgend vier auf ihrem Perimeter auf einander folgenden Punkten P, Q, R, S den ersten mit dem dritten und den zweiten mit dem vierten durch zwei Linien PTT'R und QUU'S verbinden kann, welche in der Fläche selbst liegen und dennoch einander nicht schneiden - wie dies doch immer geschehn würde, wenn die Fläche eine Grundform der ersten Klasse (zum Beispiel ein Zylinder, C.H.) wäre.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Gauß hinterließ also nicht nur ein unermeßliches Werk von großen Entdeckungen und Erkenntnissen; vielmehr warfen seine Gedanken und Ideen noch einen Lichtstrahl einer erahnten Erkenntnis auf alle möglichen unerforschten Naturerscheinungen. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now my translation and comments: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In doing so [the author is referring to Gauss' &#xA;  formulation of what is nowadays called &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss%27s_principle_of_least_constraint&quot; rel=&quot;noreferrer&quot;&gt;Gauss's principle of least constraint&lt;/a&gt;&#xA;  he threw the entire Newtonian mechanics out of &#xA;  the window, [This is misleading hyperbole. &#xA;  I am not a professional historian, &#xA;  yet I am sure that one cannot justifiably say &#xA;  that Gauss threw out Newtonian mechanics out of the window. &#xA;  I have no time to adduce references, yet I have a faint recollection &#xA;  that there are original documents in which Gauss expresses &#xA;  his appreciation of Newton. One can now quibble &#xA;  that the author is not referring to Newton's &lt;em&gt;mathematics&lt;/em&gt; &#xA;  but to Newton's &lt;em&gt;mechanics&lt;/em&gt; only, yet still I think that &#xA;  the author here is misleadingly hyperbolic.] since the movement &#xA;  of a body is determined by its position in space, &#xA;  and by all other bodies which influence it---not by &#xA;  the attractive or repulsive forces of a central force.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Moreover, Gauss in his work on the theory of &#xA;  curved surfaces treated the question of geodetics, &#xA;  precisely the same question that already in 1697 &#xA;  had been posed by Bernoulli [there is more than &#xA;  one notable mathematician named 'Bernoulli'; &#xA;  the author should have made it clear which one was meant] &#xA;  to the geometers. For both questions are closely related.&#xA;  The moving body takes (according to the law of least action) &#xA;  the shortest path from one point to another, &#xA;  yet is deflected by the 'boundary conditions' &#xA;  which are imposed on it by its position in space.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In this respect it is worth mentioning that Gauss &#xA;  began very early on to occupy himself with all possible &#xA;  entanglings and loopings [I do not care much &#xA;  about translating the words 'Verschlingungen' and 'Umschlingungen' here: &#xA;  this is one of the worst parts of the text you linked to: &#xA;  while 'Verschlingung' is indeed the usual German translation &#xA;  of the technical term 'link' in contemporary knot theory, &#xA;  'Umschlingung' does not have any usual meaning, not even in the vernacular, &#xA;  and to juxtapose it with 'Verschlingug' is misleading; &#xA;  moreover, the 'allen moglichen Verschlingungen' is misleading, &#xA;  or at least, inappropriate German for a scientific text; &#xA;  it seems that the author has a penchant to let the &#xA;  German colloquialism 'alle möglichen' (roughly corresponding to the English: 'all sorts of linkings' or 'any old linking', with a dismissive connotation) slip into the text] &#xA;  of a thread, or linkages of curves in space. &#xA;  Already in 1794 he makes a list of 13 neatly drawn pictures of knots &#xA;  which he found in a book. These will become important later on, &#xA;  in investigations of electric and magnetic currents. &#xA;  Also in astronomy one often finds, in investigations &#xA;  of nebulas or the atmospheres of planets, peculiar curves or linkages. &#xA;  Only a few years ago one has discovered [in the original, saying &#xA;  'man hat aus... etwas entdeckt' is not correct German: one 'entdeckt' &#xA;  something 'in' something else, not 'aus] inexplicable links in photographs &#xA;  made by space probes of the rings of Saturn [the term 'Satellitenaufnahmen' &#xA;  in the text from the website is just &lt;em&gt;wrong&lt;/em&gt;:  in German one has to say 'Raumsonde' &#xA;  in this context: what made these photographs of the rings of Saturn &#xA;  were not 'satellites' but 'space probes'.]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Gauss was aware that the Euclidean geometry represents but one &#xA;  of many possible geometries. For if one assumes a curved space, &#xA;  then the timelike geodesics coincide with the spacelike geodesics. &#xA;  [I am not sure what the author means here, especially in view of &#xA;  the vaguely general hypothesis 'if one assumes a curved space'.]&#xA;  He has, for example, always occupied himself what laws hold for &#xA;  such curved movements, or rather, how the coordinate systems might look, if bodies follow all sorts of linked orbits. [The author is very vague and confusing when writing 'alle Arten von verschlugenen Bahnen annehmen'. This almost sounds as if the author would be thinking of Feynman path integrals... The universal quantifier 'alle Arten' seems misleading to me, it might be a German colloquialism having slipped into this text] Also in the case of minor planets the linking of distinct orbits occurs.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;An example for how the laws change. if different 'boundary conditions' are imposed on the possibilities of movement of a body [The German sentence I tranlsated here is grammatically incorrect: it starts with &quot;Als Beispiel dafur, wie ...&quot; which one cannot finish as the author tries to do with &quot;ist folgende Konstruktion&quot;. It just does not parse, while the meaning is clear of course.] is the following construction, which Möbius related after having heard it from Gauss. Möbius had worked since the autumn of 1813 for one semester under Gauss' direction in the observatory in Göttingen. In his notes on curved surfaces he explicitly refers to an oral communication of Gauss on the propertiese of a 'double ring': &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;: It is easily possible to visualize such a 'double ring' if one cuts out a cross-shaped piece of paper and then identifies the edges FH and FH' (see Figure) of one pair of opposing 'arms' above (say) the initial plane and  identifies the edges BD and B'D' of the other pair below that plane. The resulting surface, which has only one boundary ABB'IHH'GD'DEFFA the remarkable property (according to an oral communication of Gauss'; what led Gauss to consider this surface is not known to me) that it is possible to choose a sequence of four arbitrary distinct points P,Q,R,S on its boundary(-line), and then connect the third first in the sequence with the third in the sequence, and the second in the sequence with the fourth in the sequence by two lines PTT'R and QUU'S, the lines being contained in the surface itself, and not intersecting each other --- such an intersection would however always occur if the surface were a 'Grundform der ersten Klasse' [I do not know what Möbius means by 'Grundform der ersten Klasse'] (for example a cylinder , C.H.) [Presumably the C.H. are the initials of the author of the text on the website; C.H. should, I think, rather have written 'one-way-infinite cyclinder' or 'semi-infinite cylinder', i.e. the cartesian product $\{x\in\mathbb{R}\colon 0&amp;lt;x\}\times\mathbb{S}^1$, where $\mathbb{S}^1$ denotes the circle; reason:&#xA;   C.H.'s comment only makes sense if all the four vertices are on the &lt;em&gt;same&lt;/em&gt; boundary component of the cyclinder, and then refers to a topological property which is hinted at in the following quick sketch I just made for you &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lZqn0.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lZqn0.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and wherein I used the same letters as in Möbius' original paper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please note:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The most mathematically substantive part of Möbius' remarks is not the statement &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;daß man von irgend vier auf ihrem Perimeter auf einander folgenden Punkten P, Q, R, S den ersten mit dem dritten und den zweiten mit dem vierten durch zwei Linien PTT'R und QUU'S verbinden kann, welche in der Fläche selbst liegen und dennoch einander nicht schneiden&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but what comes afterwards, i.e., the only nontrivial statement (to my mind) is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;wie dies doch immer geschehn würde, wenn die Fläche eine Grundform der ersten Klasse (zum Beispiel ein Zylinder, C.H.) wäre.&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(My &lt;em&gt;hunch&lt;/em&gt; is that this is equivalent to the Jordan curve theorem, but please not that this is a &lt;em&gt;hunch&lt;/em&gt; and I have not looked closely into this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure what Möbius meant exactly with 'Grundform der ersten Klasse', yet the following, taken from his &lt;a href=&quot;https://books.google.de/books?id=7BARAwAAQBAJ&amp;amp;pg=PA550&amp;amp;lpg=PA550&amp;amp;dq=%22Grundform+der+ersten+Classe%22&amp;amp;source=bl&amp;amp;ots=VPBoszxQzR&amp;amp;sig=DLdF-rqfARUukjTEiRXozv8HKIo&amp;amp;hl=de&amp;amp;sa=X&amp;amp;ved=0ahUKEwjxs_7Kn_rVAhXLuhQKHckfD-gQ6AEIKDAA#v=onepage&amp;amp;q=%22Grundform%20der%20ersten%20Classe%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;collected works&lt;/a&gt;, comes close to a definition:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QMKdd.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QMKdd.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The highlighted passage reads&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Addendum. A piece of a 2-sphere which is bounded by a closed line --- or, in other words: a 2-sphere with a hole --- is a 'Grundform erster Klasse. Likewise, a 2-sphere with two, three, etc, $n$ holes is a 'Grundform der zweiten, dritten, etc, $n$ten Klasse. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is not modern language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What seems most ambiguous about Möbius's passage is &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;whether he regards a '2-sphere with a hole' to be a &lt;em&gt;punctured sphere&lt;/em&gt; (i.e. sphere with a point removed; this is homeomorphic via stereographic projection to $\mathbb{R}^2$) or rather a &lt;em&gt;sphere with an &lt;strong&gt;open disc&lt;/strong&gt; removed&lt;/em&gt; (which is itself homeomorphic to a closed disc, yet is &lt;em&gt;not&lt;/em&gt; a manifold, only a manifold-with-boundary). These two notions are not the same; more precisely, the two interpretations one can give Möbius's formulation 'Kugelfläche mit einer Öffnung' are not homeomorphic. However, his formulation &quot;Ein von einer geschlossenen Linie begrenztes Stück einer Kugelfläche&quot; rather sounds like he intends the boundary to be part of the 'Stück', yet is is simply ambiguous to contemporary eyes.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;E.g. in the &lt;a href=&quot;https://books.google.de/books?id=8OYxdADnhZoC&amp;amp;pg=PA571&amp;amp;dq=%22punctured+surface%22&amp;amp;hl=de&amp;amp;sa=X&amp;amp;redir_esc=y#v=snippet&amp;amp;q=%22moduli%20space%20of%20hyperbolic%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Handbook of Geometric Topology&lt;/a&gt; you find a usual definition of 'punctured surface': &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4B2Nm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4B2Nm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point of my above 'C.H. should' is that 'cylinder' is vague and makes people think of the ordinary finite cylinder surace which has &lt;em&gt;two boundary components&lt;/em&gt; and then the reader is perhaps confused what the correspondence to the one-boundary-component case of  Möbius' surface is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point that C.H. is making that on a semi-infinite cylinder, which by the way is homeomorphic to the 'plane with an &lt;em&gt;open&lt;/em&gt; disc removed', the intersection of the P-&gt;R arc with the Q-&gt;S arc cannot be avoided. And this is intuitively quite obvious (though this is not a proof, of course), for the 'semi-infinite cylinder with boundary' is homeomorphic to the 'plane with an &lt;em&gt;open&lt;/em&gt; disc removed', represented by the following picture: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xBLjC.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xBLjC.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 'open disc' is represented by the gray circle. &#xA;The intersection of the blue and red line cannot be avoided if these lines are supposed to be arcs in the complement of the 'open disc'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now let's go on with the translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So Gauß did not only leave an inestimable body of work, full of great discoveries and insights; his thoughts and ideas rather threw a ray of light of an anticipated understanding on all possible uninvestigated natural phenomena. [I think it is fair to say that the German in this last sentence, could be better, to say the least. &#xA;  I did not make much effort to render this into decent English. First of all, the German colloquialism 'alle möglichen' (roughly: 'all sorts of'; it has a pejorative ring to it), inappropriate in a scientific text, and irritating at least to mathematicians' sensitivities in its abuse of a universal quantifier (&quot;allen&quot;), appears again. &#xA;  Second, 'Lichtstrahl einer erahnten Erkenntnis', to my mind, is (an example of the strained-use-of-words-version-of-the-literary-concept-of) a &lt;a href=&quot;https://en.wikipedia.org/wiki/Catachresis&quot; rel=&quot;noreferrer&quot;&gt;catachresis&lt;/a&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now to the mathematics. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The object that Möbius was told about by Gauß to me essentially seems to be the &lt;strong&gt;punctured torus&lt;/strong&gt;. You can easily find much on this by searching. It is not completely standardized what precisely 'punctured torus'. For the present purposes, the following excert from H.B.Griffiths: &lt;em&gt;Surfaces&lt;/em&gt;, Cambridge University Press. 2nd edition (1981) should suffices (I suppose you do not know the usual notions of basic algebraic topology, which would be needed to give a definition of 'punctured torus', so I won't try to do so)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8riTD.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8riTD.png&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the topic of &lt;em&gt;why&lt;/em&gt; Möbius' paper-surface is homotopic to the punctured torus see also &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/187617/continuous-deformation-of-punctured-torus&quot;&gt;this thread&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Relevant pictures are &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ej2JR.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ej2JR.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/X2QmN.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/X2QmN.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7QnpH.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7QnpH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(source: &quot;Ian Mallet&quot;, cf. &lt;a href=&quot;https://www.youtube.com/watch?v=j2HxBUaoaPU&quot; rel=&quot;noreferrer&quot;&gt;https://www.youtube.com/watch?v=j2HxBUaoaPU&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While you might have found this yourself, it might still be useful to you to here cite the relevant passage from Möbius [I reproduce this from the Cambridge University Press 2011 edition of Gauss's works, Volume 10, Part 2]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QeR15.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QeR15.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the larger question which to me seems to be behind your question (and its title), it seems useful to cite from the Cambridge University Press edition of Gauss' collected works in greater detail: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8S4Tt.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8S4Tt.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jAWEe.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jAWEe.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I won't translate this; you can do so with a search engine if you are really interested in it. The author is in particular saying here that Gauss influenced each of Listing, Möbius and Riemann, and a letter of Gauss to Schumacher is cited in which Gauss (in free translation) says that he thought that Möbius was someone who wanted and could think for himself instead of being spoon-fed with knowledge, and should only be given some hint here and there. Moreover, it seems there are many speculations as to what Gauß told whom when, in particular in the context of Riemann's 1854 Habilitation lecture. I know little about this, yet much of this seems speculation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To end with addressing your follow-up question in your comment&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;O.k I accept your answer, since i noticed that the surface has two sides so it cannot be a mobius strip. But i still dont understand what is the &quot;remarkable property&quot; which Gauss refers to in the last part of the passage: – user2554 Aug 22 at 9:27&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and your comment&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;can you explain to me what the last part of the passage means? (that i cited in my last comment) - i need the answer to close the discussion and understand the property which Gauss describes. – user2554 Aug 22 at 10:28&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Let me try to summarize that: in modern terms, the &quot;remarkable property&quot; that Möbius wrote he was told by Gauss is this: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;whatever ordered sequence of four distinct points $P$, $Q$, $R$, $S$ you pick on the one boundary $\partial( \{x\in\mathbb{R}\colon 0&amp;lt;x\}\times \mathbb{S}^1  )$ of the surface $S$  Möbius is describing a construction of, there exist, for both $i\in2$, arcs $[0,1]\xrightarrow[]{A_i}S$ such that &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;$A_0(0)=P$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A_0(1)=R$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A_1(0)=Q$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$A_1(1)=S$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;for all $(t_0,t_1)\in[0,1]\times[0,1]$ we have $A_0(t_0)$ not equal to $A_1(t_1)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term 'arc' here means 'continuous and injective map $[0,1]\rightarrow$(the surface in question).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An illustration of this is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/54A5l.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/54A5l.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me also mention that that the  historical literature &lt;em&gt;seems&lt;/em&gt; (I am no expert on this historical matter) to be saying that no student-teacher-relationship in a conventional sense of these words existed between Möbius and Gauß. In particular, Möbius's doctoral advisor was &lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Friedrich_Pfaff&quot; rel=&quot;noreferrer&quot;&gt;Johann Friedrich Pfaff&lt;/a&gt;. And this Pfaff was also the doctoral advisor of, guess who, Gauß. So the title of your question seems not quite correct: formally-genealogically, Gauß and Möbius were &lt;em&gt;academic brothers&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please let me know briefly whether your question in your comments is sufficiently answered now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum up, if I would have to answer all your question in &lt;em&gt;one&lt;/em&gt; sentence, my answer would be: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Gauß gave his academic brother Möbius a hint that the punctured torus is not simply-connected.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 &lt;sub&gt;I am doing this to give you examples of properties of this text which, I think, you should be warned against.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update September 17, 2017.&lt;/strong&gt; By chance I found out that there is a new book by &lt;a href=&quot;http://perso.ens-lyon.fr/ghys/home/&quot; rel=&quot;noreferrer&quot;&gt;Étienne Ghys&lt;/a&gt;, entitled &lt;a href=&quot;http://perso.ens-lyon.fr/ghys/promenade/&quot; rel=&quot;noreferrer&quot;&gt;A singular mathematical promenade&lt;/a&gt;, which &lt;em&gt;contains a chapter relevant to your question&lt;/em&gt;. (The book is amazing by itself, too, something of a 'Book 2.0'.) For your convenience, I here reproduce a few parts of Ghys' book relevant to your question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gXAma.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gXAma.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QWvct.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QWvct.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ywZ3f.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ywZ3f.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hMAqf.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hMAqf.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="6060" LastEditDate="2017-09-17T07:52:23.960" LastActivityDate="2017-09-17T07:52:23.960" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6466" PostTypeId="1" AcceptedAnswerId="6471" CreationDate="2017-08-29T04:40:12.047" Score="0" ViewCount="705" Body="&lt;p&gt;Could anyone recommend books that introduce to a layman science and its method/spirit? In particular, it should talk about what is science, what is the scientific method and why it works. After reading, the reader should appreciate modern science and its methods (and know why it works better than most things we have tried, like other attempts at doing 'science'.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have look through the internet for some time and I have found some candidates&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Martin Goldstein -- How We Know: An Exploration Of The Scientific Process&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hugh G. Gauch Jr -- Scientific Method in Brief&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stephen Carey -- A Beginner's Guide to Scientific Method&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Barry Gower -- Scientific Method: A Historical and Philosophical Introduction&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am not sure if they are any good, or which of them are better. And I think I might have missed some of the good ones. In particular, none of the books I found seem particularly well-known, which is strange. So I am looking for some recommendations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry if I am asking this in the wrong place, there seems to be no general science thread/community in StackExhange, this thread looks like the closest. Anyway, a good such book should probably involve some discussion of the history of science, so I guess it's relevant here.&lt;/p&gt;&#xA;" OwnerUserId="6219" LastActivityDate="2017-08-29T19:59:27.457" Title="Books on scientific method for laymen" Tags="|philosophy-of-science|scientific-method|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6467" PostTypeId="2" ParentId="6466" CreationDate="2017-08-29T06:54:53.500" Score="0" Body="&lt;p&gt;The best way to appreciate the full impact of the emergence of the scientific method and its scope is to study the resistance and the controversy that it engendered.  Only by understanding the context of the early 17th century can you fathom what kind of change in thinking was involved.  One fine book that studies both the emergence of the method and the sinister resistance it engendered is the book by Redondi on the Galileo affair:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Redondi, Pietro. &lt;em&gt;Galileo: heretic&lt;/em&gt;. Translated from the Italian by Raymond Rosenthal. Princeton University Press, Princeton, NJ, 1987.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-08-29T06:54:53.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6468" PostTypeId="2" ParentId="6287" CreationDate="2017-08-29T07:41:42.950" Score="6" Body="&lt;p&gt;Coolidge (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=2769&quot; rel=&quot;nofollow noreferrer&quot;&gt;1940&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=QmkGAQAAIAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22The+earliest+proof+I+have+seen%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 273&lt;/a&gt;) gives a proof which ends:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(...) Hence any real transformation of the Euclidean plane&#xA;  which keeps distances invariant is either a rotation, a translation, or the&#xA;  product of one of these and a reflection in a line.†&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;† The earliest proof I have seen for this is Chasles (&lt;a href=&quot;https://archive.org/stream/bulletindesscie08unkngoog#page/n328&quot; rel=&quot;nofollow noreferrer&quot;&gt;1830, p. 321&lt;/a&gt;). I think, however, it must have been known earlier, as the corresponding problem in three dimensions, which I take up next, was solved by Euler [&lt;a href=&quot;http://www.biodiversitylibrary.org/item/38593#page/275&quot; rel=&quot;nofollow noreferrer&quot;&gt;1776&lt;/a&gt;].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I’ll let you judge if you think (1830) contains the theorem... To me it rather seems that Coolidge has in mind what Chasles says while &lt;em&gt;discussing&lt;/em&gt; (1830) in his later book (&lt;a href=&quot;https://archive.org/stream/bub_gb_SZIKAAAAYAAJ#page/n552&quot; rel=&quot;nofollow noreferrer&quot;&gt;1837, pp. 548–549&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Quand une figure plane éprouve un mouvement infiniment petit dans son plan, il existe toujours un point qui, pendant ce mouvement, reste fixe ;&lt;/em&gt; (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;En énonçant [ce théorème], il y a quelques années (voir &lt;a href=&quot;https://archive.org/stream/bulletindesscie08unkngoog#page/n328&quot; rel=&quot;nofollow noreferrer&quot;&gt;1830&lt;/a&gt;), nous l'avons présenté comme un cas particulier d'un théorème sur le déplacement fini quelconque d'une figure plane (...). Mais ces théorèmes dépendent eux-mêmes d'un principe encore plus général, que voici :&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Si l'on conçoit, dans un plan, deux figures qui ont été primitivement la perspective l'une de l'autre, et qui se trouvent actuellement placées d'une manière quelconque l'une par rapport à l'autre ;&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Chaque point de l'une des figures aura son homologue dans l'autre figure ;&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Il existera généralement trois points, dans l'une des figures, qui se trouveront superposés respectivement sur leurs homologues dans la seconde figure ;&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;L'un de ces trois points sera toujours réel ; les deux autres pourront être imaginaires.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Il résulte de là qu'il y a aussi trois droites, dans l'une des figures, qui se trouvent superposées sur leurs homologues dans la seconde figure : ce sont les droites qui joignent deux à deux les trois points.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;L'une de ces droites est toujours réelle, et les deux autres peuvent être imaginaires.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Quand les deux figures sont semblables, ce qui est un cas particulier de la perspective, deux des trois points et deux des trois droites sont toujours imaginaires ; le troisième point est réel ; la troisième droite est aussi réelle ; mais elle se trouve située à l'infini.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Cela a lieu pareillement quand les deux figures sont égales entre elles.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ces propriétés des figures planes ont leurs analogues dans les figures à trois dimensions, pour lesquelles j'ai déjà énoncé quelques théorèmes qui se rapportent à cette théorie. (Voir &lt;a href=&quot;https://archive.org/stream/bulletindesscie08unkngoog#page/n328&quot; rel=&quot;nofollow noreferrer&quot;&gt;1830, p. 321&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Indeed, Coolidge’s lines leading to the above quote are fleshing out this argument, which is about (what we would call) eigenvalues and eigenvectors of the isometry’s matrix&#xA;$$&#xA;\begin{pmatrix}&#xA;\cos\theta &amp;amp; \mp\sin\theta &amp;amp; a\\&#xA;\sin\theta &amp;amp; \pm\cos\theta &amp;amp; b\\&#xA;0&amp;amp;0&amp;amp;1&#xA;\end{pmatrix}:&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The characteristic equation here is of the third degree and, since it is real, it must have one real root at least, so that there is at least one real fixed point. As a circle goes into a circle, so the circular points at infinity are either invariant, or interchanged. Suppose that they are invariant and that we have a real finite fixed point [... then] we have a rotation about this point. If, however, the real fixed point is infinite [... then] we have a translation. When the circular points at infinity are interchanged and there is a finite fixed point, the transformation can be factored into a rotation about this point and a reflection in a line through it. If the fixed real point be infinite [... then] the transformation is the product of a translation and a reflection in a line through one of these points. Hence (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-12-14T20:04:07.087" LastActivityDate="2017-12-14T20:04:07.087" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6470" PostTypeId="2" ParentId="6461" CreationDate="2017-08-29T10:02:49.817" Score="4" Body="&lt;p&gt;The earliest extant description of Erathostenes' method is found in a book by Cleomedes written in the 1st century BC, and no wells are mentioned at all, if it's true that it's stated that whenever the Sun, 'being in Crab at the summer solstice, the pointers of the sundials in Syene throw no shadows...'&lt;/p&gt;&#xA;" OwnerUserId="6221" LastActivityDate="2017-08-29T10:02:49.817" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6471" PostTypeId="2" ParentId="6466" CreationDate="2017-08-29T19:59:27.457" Score="0" Body="&lt;p&gt;It is much harder than it might seem to answer this question and find recommendable &quot;books that introduce to a layman science and its method/spirit&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subject is immense: it covers all theories and all observations: making it difficult or impossible for books on scientific method to digest entirely and present their subject.  Some books concentrate on theory, others, perhaps a valuable minority, on the way theories are built from the ground upwards on observation and experiment. Some achieve a degree of brevity but pay a heavy price in accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In view of these difficulties, an author who can extract an important gist briefly from them has an enviable skill, and so I would recommend first, in spite of its age, a short essay by :  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fernando Sanford  (1899) -- 'The Scientific Method and Its Limitations'  ~ 20pp. (Stanford University Press) [&lt;a href=&quot;https://stacks.stanford.edu/file/druid:mp840zw9344/SC1020_1899.pdf]&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://stacks.stanford.edu/file/druid:mp840zw9344/SC1020_1899.pdf]&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sanford's brief essay dated the modern awakening of scientific investigation in Europe to about the year 1600, and especially to the works of William Gilbert on magnetism, and of Galileo on moving (falling) bodies. Sanford explained why he considered their innovations in method to be so fundamental, and described important features (and difficulties) of carrying out experimental science more generally. He did refer also to some (now) very dated and dead controversies, and the world has moved on unimaginably since 1899, but still much of what Sanford wrote about scientific principle and method remains of value.  Much more could of course be and have been said, but in the meantime the essay can still deliver briefly a core of material to &quot;introduce to a layman science and its method/spirit&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the almost perpetual puzzles about science is the question, what degree of confidence, or what certainty if any, can we place in a theory and belief, and what circumstances should we take to reinforce or undermine that confidence?  The spirit of science is one of critical assessment, and at its foundations ideas of probability. The history of what the idea of 'probability' has meant has shown radical change. Thus I would also recommend:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ian Hacking's 'The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference' (1975), [&lt;a href=&quot;https://books.google.com/books?id=Z2g0F1V5DywC]&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?id=Z2g0F1V5DywC]&lt;/a&gt;, &lt;/p&gt;&#xA;a book that insightfully shows important historical antecedents and pre-modern underpinnings of current attitudes towards indicative signs, facts, and evidence. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Isaac Newton's work for a long time had considerable influence in forming ideas of what scientific method is and should be. In spite of all that has happened since, there is much in what Newton did, and in what his successors made of it, that still illuminates &quot;science and its method/spirit&quot;. The questioner's &quot;laymen&quot; will find much of that method and spirit in I Bernard Cohen's historical study of the experimental side of Newton's legacy and its connection with that great experimentalist Benjamin Franklin:-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I Bernard Cohen, &quot;Franklin and Newton&quot;, Memoirs of the American Philosophical Society vol.43 (1956), repr. 1966; &lt;/p&gt;&#xA;and also &lt;/p&gt;&#xA;&quot;The Cambridge Companion to Newton&quot; (2002), eds. I.Bernard Cohen, George E. Smith. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Equally and still relevant to the method and spirit of science are questions posed and answered in: &lt;/p&gt;&#xA;(a) Steffen Ducheyne (2012) &quot;The Main Business of Natural Philosophy: Isaac Newton's Natural-Philosophical Methodology&quot;; &lt;/p&gt;&#xA;(b) William L. Harper (2011) &quot;Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology&quot;; and &lt;/p&gt;&#xA;(b) George E Smith's &quot;Closing the Loop: Testing Newtonian Gravity, then and now&quot;, which has appeared in several forms (essay and lecture transcript), most recently at pp.262-352 of &quot;Newton and Empiricism&quot; (2014), eds. Zvi Biener &amp;amp; Eric Schliesser.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To close, a word about one of the books given in the questioner's list:--&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Barry Gower's (1996) &quot;Scientific Method -- A Historical and Philosophical Introduction&quot; &lt;/p&gt; is a broad historical survey by a professor in the philosophy of science. It stretches in coverage from Galileo, Bacon and Newton to Reichenbach, Popper and Carnap.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many riches in this book, but it is worth mentioning that it shares one characteristic of numerous modern philosophical approaches to theories of science, a strong emphasis on theoretical problems such as skepticism and the status of induction.  The result can sometimes be a great distance between the theory of philosophy of science and the practice of science and of scientific theorists.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, I could not imagine that any metallurgist for thousands of years has ever had reason to doubt that copper can be melted: but the chosen remit of this author's book makes it relevant for him to consider at some length, for example, Reichenbach's assertion that &quot;because the conclusion that all copper is fusible is inductive and we have yet to establish whether such conclusions are acceptable, we cannot claim that it is true that all copper is fusible, or even that it is probable that all copper is fusible.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clearly it is up to each prospective reader to decide whether this distance from the real world of scientific theory and practice brings the book's subject-matter closer to or farther away from the field of his/her interest. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-08-29T19:59:27.457" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6472" PostTypeId="1" CreationDate="2017-08-30T03:50:33.040" Score="2" ViewCount="214" Body="&lt;p&gt;Grace Hopper is frequently credited with finding the first actual bug. In fact, she just popularized the story of how William (Bill) Burke found the bug in the Mark II computer and documented it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to find out when Bill Burke was born and died. Does anyone know?&lt;/p&gt;&#xA;" OwnerUserId="6112" LastEditorUserId="6112" LastEditDate="2017-08-31T17:25:37.997" LastActivityDate="2017-08-31T17:25:37.997" Title="When did William (Bill) Burke, who discovered the first actual bug, live and die?" Tags="|biographical-details|computers|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6473" PostTypeId="2" ParentId="3131" CreationDate="2017-08-30T04:46:32.123" Score="7" Body="&lt;p&gt;James Gregory also died very young. He discovered the basic ideas of Calc the same time Newton did...he died at 36 only 1 year after accepting a position at the University of Edinburgh&lt;/p&gt;&#xA;" OwnerUserId="6228" LastActivityDate="2017-08-30T04:46:32.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6475" PostTypeId="2" ParentId="5068" CreationDate="2017-08-31T12:24:51.293" Score="0" Body="&lt;p&gt;Thomas Savery used steam pressure in his engines to force water to a height beyond the 10.3m limit afforded by a vacuum but from what i've read he only managed a pressure of around 45psi and he had several incidents busting his boilers.  So i guess the materials at the time just weren't strong enough.&lt;/p&gt;&#xA;" OwnerUserId="6232" LastActivityDate="2017-08-31T12:24:51.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6476" PostTypeId="1" CreationDate="2017-08-31T12:31:35.327" Score="1" ViewCount="256" Body="&lt;p&gt;I've been reading about Newcomen's engine but i'm not clear on how he overcame the lift limit due to gravity on the water column overcoming the suction of the partial vacuum.  Did the engine somehow create a continual positive pressure and if so how?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="6233" LastActivityDate="2017-10-14T11:32:12.323" Title="How did Newcomen's Atmospheric Engine lift water above the 10.3m vacuum limit?" Tags="|engineering|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6479" PostTypeId="1" AcceptedAnswerId="6491" CreationDate="2017-09-01T07:59:51.893" Score="6" ViewCount="290" Body="&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(bibliographic) Are there recommendable, preferably &lt;em&gt;modern&lt;/em&gt;, &lt;em&gt;detailed&lt;/em&gt; and &lt;em&gt;technical&lt;/em&gt; discussions in the literature on the kind of consistency proofs that Otto Blumenthal mentions Hilbert proposed in 1904 in an address to the third ICM in Heidelberg? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(philosophical) Do you think it is meaningful to distinguish between the following three, given in chronological order, essentially different kinds of consistency proofs: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;semantic&lt;/em&gt; consistency-proofs (roughly: constructing a model of the axioms, the model preferably being perceived as reasonable by others)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Heidelberg-style&lt;/em&gt;&lt;sup&gt;1&lt;/sup&gt; consistency-proofs (this kind of consistency proof is the topic of this question; by these I mean the ones mentioned by Blumenthal, and they seem not to be discussed in the technical literature, probably &lt;em&gt;with a reason&lt;/em&gt;, for perhaps this is not known to work, which is why I am asking, yet extrapolating from the words this means: define a group of symmetries acting on all wellformed formulae over the relevant signature, then show that sets of sentences which are inconsistent have, in some precise sense, little symmetry, then turn toward the axioms whose consistency one would like to prove, and prove that these axiom &lt;em&gt;do&lt;/em&gt; have much symmetry, which implies consistency; this sounds like vague speculation and wishful thinking and I have not seen this made precise anywhere, which is perhaps explained by Blumenthal's dismissive words 'erwiesen sich bei genauerem Eingehen als  unzulänglich'.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;syntactic&lt;/em&gt; consistency-proofs (by this I mean consistency proofs more-or-less in the style of Gentzen's famous 1936 'Zur Widerspruchsfreiheit der reinen Zahlentheorie'; this is very well documented, not the topic of this question, and I won't say more about this here)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(mathematical) Do you know a specific mathematical reason for and/or discussion in the literature of Blumenthal's verdict &quot;unzulänglich&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Commentary.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In Otto Blumenthal's biographical essay about David Hilbert, written on the occasion of the 1935 Springer publication of the third volume of Hilbert's collected works, Blumenthal mentions  'misunderstood' what one might, perhaps distortingly, call a failed proposal of Hilbert's for consistencies proofs for formal systems. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IsICS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IsICS.png&quot; alt=&quot;O. Blumenthal: Lebensgeschichte, in third volume of 1935 edition of Hilbert&amp;#39;s collected works&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The highlighted passages can be translated thus&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] basic idea of his later proof theory: that one should examine the form of the formulae that can be deduced from a given system of axioms, and that then a certain invariance of this form would become apparent [...] formulas which are contradictory do not exhibit this invariance [...] the talk remained, for all I know, completely ununderstood, and the approaches proposed in the talk turned out to be unsuitable upon a closer examination&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Part of my motivation is working on a non-ahistorical answer to &lt;a href=&quot;https://mathoverflow.net/questions/275362/what-did-zermelo-say-he-was-hoping-for-on-the-consistency-of-set-theory&quot;&gt;this MO question&lt;/a&gt;. Hilbert's address predates Zermelo's comments by several years, so this seems relevant.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bibliographic remark.&lt;/strong&gt; The published version of Hilbert's 1904 ICM talk at Heidelberg was published in the proceedings of the conference: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;David Hilbert &lt;em&gt;Über die Grundlagen der Logik und der Arithmetik&lt;/em&gt;, Verhandlungen des 3. Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, pp. 174-185 (available e.g. &lt;a href=&quot;http://archiv.ub.uni-heidelberg.de/volltextserver/16037/1/hilbert-1904a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;from University of Heidelberg&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and there is at least one translated version: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;David Hilbert &lt;em&gt;On the foundations of logic and arithmetic&lt;/em&gt;, The Monist, Vol. 15, No. 3 (July 1905), pp. 338-352. Translated by George Bruce Halsted. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; This term is not attested and was invented for the present question. &lt;/p&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="6060" LastEditDate="2018-02-10T19:12:14.653" LastActivityDate="2018-02-10T19:12:14.653" Title="A 'third way' to consistency (besides Dedekind's and Gentzen's)? Hilbert's 'unverstanden' 1904 Heidelberg ICM proposal for consistency proofs" Tags="|mathematics|mathematical-logic|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6480" PostTypeId="1" CreationDate="2017-09-01T15:17:43.387" Score="6" ViewCount="375" Body="&lt;p&gt;It is often said that he discovered non-Euclidean geometry. But in which sense? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am reading the book 'geometry' by Brannan et al. They use the disk model as an example of hyperbolic geometry. Did Lobachevsky have a similar model? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2017-09-03T06:03:41.043" Title="What did Lobachevsky do?" Tags="|mathematics|geometry|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6481" PostTypeId="2" ParentId="6480" CreationDate="2017-09-01T17:29:11.157" Score="4" Body="&lt;p&gt;When developing his geometry, Lobachevsky worked exclusively in the (Lobachevskian) plane.  It was the Italian mathematician Beltrami who first showed that the geometry of (part of) the Lobachevskian plane coincided with the geometry of a certain surface - namely the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pseudosphere&quot; rel=&quot;nofollow noreferrer&quot;&gt;pseudosphere&lt;/a&gt;.  Beltrami's work came some forty-two years after Lobachevsky first formulated and completed his ideas: 1868 vs 1826.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Beltrami's interpretation shows that to every statement of Lobachevskian geometry referring to part of the plane there corresponds an immediate fact about the intrinsic geometry of the psuedosphere. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, not all of the Lobachevskian plane is realised on the pseudosphere, but only part of it.  It was Klein, in 1870, who first gave an actual interpretation of Lobachevskian geometry on the whole plane and, more generally, his geometry in space.  (See, for example, Klein's interpretation in the circle and the sphere.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0486409163&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;MATHEMATICS Its Contents, Methods, and Meaning&lt;/em&gt;&lt;/a&gt; by Aleksandrov, Kolmogorov, and Lavrent'ev.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-09-02T13:58:26.763" LastActivityDate="2017-09-02T13:58:26.763" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6482" PostTypeId="2" ParentId="6480" CreationDate="2017-09-01T18:06:05.543" Score="4" Body="&lt;p&gt;Yes, Lobachevsky was a discoverer of non-Euclidean (more precisely, hyperbolic) geometry. (It was independently discovered by few other people, but Lobachevsky's&#xA;treatment was by far the most complete). More precisely, he just accepted as an axiom the negation of the &quot;5-th postulate&quot;, and developed a comprehensive geometry based on these new axioms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But he did not have any model. It was developed axiomatically, in the same way as Euclid. The role of the later models (Beltrami, Poincare, Klein) was to show rigorously that IF there is no contradiction in Euclidean geometry, THEN these is no contradiction in the Lobachevsky's one. So they are equally &quot;true&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The question whether there is a contradition in Euclidean geometry, or in the rest of mathematics, does not belong to mathematics itself: this is unprovable by the usual mathematical methods).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-09-01T18:06:05.543" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6483" PostTypeId="1" CreationDate="2017-09-01T20:04:27.657" Score="2" ViewCount="254" Body="&lt;p&gt;Does anyone know how the definitions for &lt;em&gt;onto&lt;/em&gt; and &lt;em&gt;into&lt;/em&gt; map to the spoken language definitions of the words?  I compared the Bourbaki definitions to these words and have a suspicion that the German word &lt;em&gt;thrawn&lt;/em&gt;, meaning &quot;twisted&quot; may clarify this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;MATHMATICAL DEFINITIONS&lt;/strong&gt; (summarized)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Injective&lt;/em&gt; = every element of the co-domain is mapped to by at most one element of the domain.  1-1.  May be &lt;strong&gt;Into&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Surjective&lt;/em&gt; = every element of the co-domain is mapped to by at least one element of the domain.  Is &lt;strong&gt;Onto&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;DICTIONARY DEFINITIONS&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Into&lt;/strong&gt;: expressing movement or action with the result that something becomes enclosed or surrounding by something else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;In,En&lt;/strong&gt;: to restrict, on all sides completely.  Negative force &lt;em&gt;Latin&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;To&lt;/strong&gt;: proposition indicating relation to another word, to place, before, indicates direction towards something&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Onto&lt;/strong&gt;: moving to a location on (the surface of something), moving aboard with intention of traveling in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;On&lt;/strong&gt;: Being (&lt;em&gt;Greek&lt;/em&gt;), above and in contact, (&lt;em&gt;German&lt;/em&gt;) directly supported by another, by, near, close, next to, followed by another time point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Injective&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In: negative force&lt;/li&gt;&#xA;&lt;li&gt;jacere: throw (&lt;em&gt;Latin&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;ive: relating, belonging, tending to&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Surjective&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sur: over, above, up&lt;/li&gt;&#xA;&lt;li&gt;jacere: throw (&lt;em&gt;Latin&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;ive: relating, belonging, tending to&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm also wondering if the translation of &lt;em&gt;jacere&lt;/em&gt; really means &quot;twisted, distorted&quot; from the German word &lt;em&gt;thrawn&lt;/em&gt;.  If thats the case, I can see how the word &lt;em&gt;injection&lt;/em&gt; really means &quot;not a twisted relationship from above&quot; and &lt;em&gt;surjection&lt;/em&gt; means &quot;a twisted relationship from above&quot;.  Where from above represents the function F mapping domain to co-domain or the reverse.  This would imply injection is a &quot;1-1 mapping preserved by the function&quot;&quot; and surjection is a &quot;1-1 mapping not preserved by the function&quot;.  From this interpretation, it seems as if there is some topological relation.  &lt;em&gt;Into&lt;/em&gt; keeps the form of the domain.  &lt;em&gt;Onto&lt;/em&gt; has twisted form/surface because of the possible many to one relationship from domain to co-domain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Into Example&lt;/strong&gt;&#xA;$F$ maps $A$ &lt;em&gt;into&lt;/em&gt; $B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$F: A \rightarrow B$ and $F \subseteq B$ and $F=A$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Onto Example&lt;/strong&gt;&#xA;$F$ maps $A$ &lt;em&gt;onto&lt;/em&gt; $B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$F: A \rightarrow B$ and $F=B$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Appreciate anyone's insight here.  I'm trying to clear up this language because it makes the math easier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="6241" LastEditorUserId="131" LastEditDate="2017-09-13T08:27:20.373" LastActivityDate="2017-09-15T14:07:33.547" Title="Set Theory, onto and into their relation to spoken language definitions" Tags="|mathematics|terminology|set-theory|topology|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6484" PostTypeId="2" ParentId="6480" CreationDate="2017-09-02T06:26:15.197" Score="2" Body="&lt;p&gt;Another important contributions of Lobachevsky was the development of hyperbolic trigonometry (from the axioms: unlike the modern treatments, he did not use any models since he did not have any). An interesting, and often overlooked, application of this development is the &lt;strong&gt;uniqueness&lt;/strong&gt; of the Lobachevskian geometry:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theorem.&lt;/strong&gt; Suppose that $X_1, X_2$ are spaces satisfying Lobachevsky axioms with the same &quot;curvature normalization&quot;: Each ideal triangle has area $c$ in both geometries. (You can take $c=\pi$ if you like, this corresponds to the curvature $-1$.) Then $X_1$ is isometric to $X_2$.    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can explain how it follows if you are interested. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Proof.&lt;/strong&gt; Pick a base point $o_1\in X_1$ and a reference ray $\rho_1$ emanating from $o$. This defines the &quot;polar coordinates&quot; on $X_1$, namely $P(r,\theta)$, where $r$ is the distance from $P$ to $o_1$ and $\theta\in [0,2\pi)$, the (oriented) angle between $o_1P$ and $\rho_1$. Now, do the same in the &#xA;second space $X_2$: Pick a base-point $o_2$, a ray $\rho_2$, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, define a map $f: X_1\to X_2$ by sending $P(r,\theta)\in X_1$ to $Q(r,\theta)\in X_2$. It is clearly a bijection, let us prove that it is an isometry. Take two points $A, B\in X_1$. Then their distance in $X_1$ is given by the hyperbolic cosine law (due to Lobachevsky: I did not check, but it is possible that Bolyai also proved it):&lt;br&gt;&#xA;$$&#xA;cosh(|AB|)= cosh(|o_1A|)cosh(|o_1B|) - sinh(|o_1A|)sinh(|o_1B|) cos(\angle(Ao_1B)).&#xA;$$&lt;br&gt;&#xA;By the construction, the map $f$ preserves all the quantities on the right-hand side of this equation (the distance to the origin and the angle). Since the hyperbolic cosine formula also holds in $X_2$, it follows that $f$ is an isometry. &lt;strong&gt;qed&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This proof uses the notion of an oriented angle which can be easily avoided by working in one half-plane in $X_i$ (defined by the unique line through $\rho_i$ $i=1,2$) at a time. &lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2017-09-03T06:03:41.043" LastActivityDate="2017-09-03T06:03:41.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6485" PostTypeId="2" ParentId="6400" CreationDate="2017-09-03T14:08:19.373" Score="3" Body="&lt;p&gt;According to Jeremy Gray's &quot; Jonas Bolayai, Non-Euclidean geometry and the nature of space&quot; (2006) &#xA;and &quot;worlds out of nothing &quot; (2007 ) &#xA;it is all from just one  long quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;you must not attempt this approuch to the parallels.&#xA;  I know this way to the very end.&#xA;  I have transversed this bottomless night,&#xA;  which extinguished all light and  joy of my life.&#xA;  I entreat you, leave the science of parallels alone&#xA;  [...]&#xA;  I thought I would sacrifice myself for the sake of truth.&#xA;  I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind.&#xA;  I accomplished monstrous, enormous labours: my creations are far better than those of others and yet I have not achieved complete satisfaction&#xA;  [...]&#xA;  I turned back when I saw no man can reach the bottom of this night.&#xA;  I turned back unconsoled , pitying myself and all mankind.&#xA;  Learn from my example:&#xA;  I wanted to know about parallels.&#xA;  I remain ignorant, this has taken all the flowers of my life and all my time from me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And yet again:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I admit I expect nothing from the deviation of your lines.&#xA;  It seems to me that I have been in these regions;&#xA;  that I have travelled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail.&#xA;  The ruin of my disposition and my fall date to this time.&#xA;  I thoughtlessly risked my life and happiness&#xA;  -aut Ceasar aut nihil (either Ceasar or Nothing).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in &quot;Jonas Bolyai, Non-Euclidean geometry and the nature of space&quot; (2006) the reference is to&#xA;Stackel's &quot;Wolfgang and Jonas Bolyai, geometrische untersuchungen, ... &quot; Leipzig, Treubner 1913 page 81 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &quot;worlds out of nothing &quot; (2007 ) the reference is to&#xA;Menschkowski's  &quot;Non euclidean geometry &quot;New York, Academic press,1964  (tr. A Shenitzer) page 31-34&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be nice to have a complete translation of what was written &lt;/p&gt;&#xA;" OwnerUserId="557" LastActivityDate="2017-09-03T14:08:19.373" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6490" PostTypeId="2" ParentId="22" CreationDate="2017-09-05T07:53:49.437" Score="0" Body="&lt;p&gt;1) The Ozone layer was discovered by a French physicist- Charles Fabry.&#xA;2) The properties were explored in details by a British scientist- G. Dobson who developed an instrument [Dobsometer] which could be used to measure stratospheric ozone from ground level.&#xA;3) Dobson unit is thus a measure of the density of ozone overhead.&lt;/p&gt;&#xA;" OwnerUserId="6258" LastActivityDate="2017-09-05T07:53:49.437" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6491" PostTypeId="2" ParentId="6479" CreationDate="2017-09-05T17:47:03.573" Score="5" Body="&lt;p&gt;In regards to your bibliographic question, I suggest the highly informative essay by Craig Smoryński, &lt;a href=&quot;https://ir.cwi.nl/pub/18149&quot; rel=&quot;noreferrer&quot;&gt;Hilbert's programme&lt;/a&gt;; he gives a highly compressed account of Hilbert's 1904 proof on pp.7-8 of the pdf file (pp. 9-10 of the paper), including some commentary on its limitations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In regards to your &quot;philosophical&quot; question, first, note that, if Smoryński is right, then Hilbert's 1904 proof is not exactly like the idea you suggested, since there is no need to invoke group theoretic ideas. Rather, the &quot;invariant property&quot; claimed for the axioms and theorems is that of being &lt;em&gt;homogeneous equations&lt;/em&gt;, that is, equations whose equated terms have the same number of symbols. And the proof here is a simple induction on the length of &#xA;proofs: the axioms are all homogeneous equations, and the rules of inference also take homogeneous equations to homogeneous equations (this is immediate by inspection, if you look at the axioms and rules on p. 7 of the pdf). This shows that the the system is consistent because it's clear that, say, 0=S(0) is not homogeneous, so the system can't derive that particular equation. Although very primitive, this is essentially the idea behind Gentzen's consistency proof of the first-order sequent calculus: cut elimination shows that every canonical proof has the subformula property, and any purported derivation of a contradiction would not have this property (because a contradiction, in the sequent calculus, has no subformulas---unfortunately, as is well-known, this strategy won't work for Peano Arithmetic).&lt;/p&gt;&#xA;" OwnerUserId="4097" LastActivityDate="2017-09-05T17:47:03.573" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6492" PostTypeId="2" ParentId="6435" CreationDate="2017-09-06T15:48:46.347" Score="0" Body="&lt;p&gt;Further research has shown, that there exists a list of de Candolle's publications from 1824 until May 1883:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Liste complète des ouvrages et opuscules publiés par M. Alph. de Candolle dans le laps des soixante années de 1824 à mai 1883.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://catalog.hathitrust.org/Record/009173065&quot; rel=&quot;nofollow noreferrer&quot;&gt;It's available on Hathi Trust.&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On top of that, and even more comprehensive, is Frans Stafleu's Taxonomic Literature. Here's is &lt;a href=&quot;http://biodiversitylibrary.org/page/33120564&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volume 1&lt;/a&gt; and here's the &lt;a href=&quot;http://biodiversitylibrary.org/page/33266663&quot; rel=&quot;nofollow noreferrer&quot;&gt;supplementary Volume III&lt;/a&gt; on Alphonse de Candolle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Both do not list any translation of &amp;quot;Géographie botanique raisonnée&amp;quot;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;P.S.: I also realized, that the &amp;quot;Essai élémentaire de Géographie botanique&amp;quot; from 1820 was actually written by Auguste de Candolle (the father) and not by Alphonse. I have edited my question as regards to this.&lt;/p&gt;&#xA;" OwnerUserId="6147" LastEditorUserId="10608" LastEditDate="2024-02-27T07:46:12.810" LastActivityDate="2024-02-27T07:46:12.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6493" PostTypeId="2" ParentId="6463" CreationDate="2017-09-06T21:23:56.433" Score="4" Body="&lt;p&gt;There is an indirect connection between the works of Lenard and the Curies. This link is exactly the work of Röntgen and the influence it had. Let me expand on this a bit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lenard was unhappy that the public attributed the discovery of X-rays entirely to Röntgen. He believed that he should also be acknowledged, because Röntgen's discoveries were made by Crookes–Hittorf tube and Lenard tubes. After the war, during an interview made by an American colonel, V. Etter (who was a medical doctor, and would later a professor of radiology), Lenard said the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I am the mother of X-rays, Just as the mid-wife is not responsible for the mechanism of birth, so was Röntgen not responsible for the discovery of X-rays since all the groundwork had been prepared by me. Without me, the name of Röntgen would be unknown today.&quot;&#xA;(See, Etter, L. E. (1946) Some historical data relating to the discovery of the Roentgen rays. Am. J. Roent. Radium Ther. 56,220-231.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery of X-rays directly influenced the research of Becquerel and the Curies. Indeed, Becquerel was in the audience, when Poincaré (after getting a letter from Röntgen) gave an enthusiastic talk in 1896 at the Académie des Sciences in Paris about the discoveries of X-rays. Influenced by this talk, Becquerel wondered whether there was any connection between the X-ray images on photographic plates and the phenomena of fluorescence and phosphorescence that he was studying. The rest we know: He designed an experiment. He was exposing fluorescing material to the sun, and then placing it and a metal object over an unexposed photographic plate. This didn't really work. However, while doing this experiment during a rainy day, he wrapped a fluorescing uranium compound (potassium uranyl sulfate) in a black cloth along with a photographic plate and a copper Maltese cross - and put it in a drawer. Several days later, when removing the plate from the drawer, he discovered to his surprise that a distinct image of the cross appeared on the plate (despite it had never been exposed to sunlight). The image resembled Röntgen's X-ray images, and the radiation seemed to have the penetrating power of X-rays. The enthusiastic Becquerel wanted to know whether this was X-ray (turned out not to be), and which other materials could emit such radiation. His young student, Marie Sklodowska Curie decided to look into these &quot;uranium rays&quot; as a possible field of research for a thesis. And there followed a period of intense research into radioactivity by her and her husband Pierre Curie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, there was a clear line of influence Röntgen-Becquerel-the Curies with superb research results (with all of these people acknowledging each other). Lenard believed that he should be in this line, as his results directly influenced Röntgen.&#xA;(Personal note: Lenard was acknowledged and was regarded as a great scientist, but his ego couldn't tolerate that others, like Röntgen and the Curies, got even more fame.)&lt;/p&gt;&#xA;" OwnerUserId="6226" LastEditorUserId="6226" LastEditDate="2017-09-06T22:14:28.597" LastActivityDate="2017-09-06T22:14:28.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6494" PostTypeId="1" AcceptedAnswerId="6495" CreationDate="2017-09-07T16:41:05.623" Score="3" ViewCount="593" Body="&lt;p&gt;The Gamma function for positive arguments can be defined with the integral&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The function $ x^{\alpha-1} e^{-x} $ is called the Gamma distribution when normalized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How are the two names related?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Did the Gamma function get this name and the $\Gamma$ symbol because the distribution was already called Gamma distribution?  &lt;/li&gt;&#xA;&lt;li&gt;Or backwards, did the Gamma distribution get named from the Gamma function?  &lt;/li&gt;&#xA;&lt;li&gt;Or were the two named together at the same time?  &lt;/li&gt;&#xA;&lt;li&gt;Or are the two names independent and only accidentally the same letter, despite the apparent connection?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This question came up in a discussion about spelling mathematical terms, and while it's probably not actually relevant for that, I'm now curious about the origin of the names.&lt;/p&gt;&#xA;" OwnerUserId="2258" LastActivityDate="2017-09-07T17:39:30.063" Title="Name of the Gamma function" Tags="|mathematics|terminology|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6495" PostTypeId="2" ParentId="6494" CreationDate="2017-09-07T17:39:30.063" Score="6" Body="&lt;p&gt;According to Herbert A. David’s &lt;em&gt;First (?) Occurrence of Common Terms in Statistics and Probability&lt;/em&gt; (&lt;a href=&quot;https://web.archive.org/web/*/http://stat.iastate.edu/preprint/articles/2006-07.pdf&quot; rel=&quot;noreferrer&quot;&gt;2010 revision&lt;/a&gt;), “Gamma distribution” goes back to Paul G. Hoel, &lt;em&gt;A significance test for component analysis&lt;/em&gt;, Annals of Mathematical Statistics &lt;strong&gt;8&lt;/strong&gt; (&lt;a href=&quot;http://dx.doi.org/10.1214/aoms/1177732411&quot; rel=&quot;noreferrer&quot;&gt;1937&lt;/a&gt;) 149–158. That’s much later than Legendre’s naming of the Gamma function (see Wikipedia). &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-09-07T17:39:30.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6496" PostTypeId="1" CreationDate="2017-09-07T20:00:06.600" Score="1" ViewCount="64" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What was the first recorded use of someone calling exponential families (in probability/statistics) for which the dimension of the natural parameter space is strictly less than the dimension of the ambient space containing it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(When the inequality isn't strict, i.e. the affine hull of the natural parameter space is all of the parameter space's ambient space, then the family is called a full-rank exponential family.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See, e.g. here: &lt;a href=&quot;http://yaroslavvb.com/papers/notes/lauritzen-expfams.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://yaroslavvb.com/papers/notes/lauritzen-expfams.pdf&lt;/a&gt; , or Keener, &lt;em&gt;Theoretical Statistics: Topics for a Core Course&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5797" LastActivityDate="2017-09-07T22:40:45.033" Title="First use of the term/name &quot;curved exponential family&quot;?" Tags="|terminology|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6497" PostTypeId="2" ParentId="6496" CreationDate="2017-09-07T22:22:04.480" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.63.135&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper on the Penn State site&lt;/a&gt;, page 5 (PDF page 7/22):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The family of distributions&#xA;  $$P(Y=y) = exp\{ \eta(\theta)^tZ(y) - \psi[\eta(\theta)]\} , \theta \in R^q  $$&#xA;  is called the &lt;em&gt;curved exponential family&lt;/em&gt; in the terminology of Efron (1975).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Efron (1975) reference is then given as :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Efron, B. (1975), Defining the curvature of a statistical problem (with applications to second order&#xA;  efficiency) (with discussion), Annals of Statistics, 3: 1189–1242.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt; I've just noticed that the above link does not seem to load correctly.  If you have problems view this PDF, here is an alternative link : &lt;a href=&quot;http://sites.stat.psu.edu/reports/2004/tr0402.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Penn State site paper&lt;/a&gt; - here see page 7.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-09-07T22:40:45.033" LastActivityDate="2017-09-07T22:40:45.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6498" PostTypeId="1" AcceptedAnswerId="6500" CreationDate="2017-09-08T11:53:39.147" Score="2" ViewCount="356" Body="&lt;p&gt;One of Einstein's famous quotes is &lt;em&gt;&quot;God does not play dice&quot;&lt;/em&gt;, referring to his dislike of the probabilistic model of the universe, as found by quantum theory. However, in the book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/024100344X&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Quantum Mechanics: The Theoretical Minimum&quot;&lt;/a&gt;, the author notes that Einstein was actually the first to propose that quanta of light are governed by probabilistic rules.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what happened to Einstein that made him go completely against his earlier ideas, given that the majority of the scientific community were in favour of quantum theory?&lt;/p&gt;&#xA;" OwnerUserId="6106" LastActivityDate="2017-09-08T15:30:50.793" Title="Why did Einstein turn his back on a probabilistic universe?" Tags="|physics|quantum-mechanics|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6499" PostTypeId="2" ParentId="5611" CreationDate="2017-09-08T12:40:05.610" Score="1" Body="&lt;p&gt;The question was answered by Viktor Blasjo in the comments: the Greek text is included in &lt;a href=&quot;http://www.jstor.org/stable/4473078&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article by Heiberg.&lt;/p&gt;&#xA;" OwnerUserId="2353" LastActivityDate="2017-09-08T12:40:05.610" CommentCount="0" CommunityOwnedDate="2017-09-08T12:40:05.610" ContentLicense="CC BY-SA 3.0" />
  <row Id="6500" PostTypeId="2" ParentId="6498" CreationDate="2017-09-08T15:30:50.793" Score="6" Body="&lt;p&gt;Einstein did not go against his earlier ideas.  The probabilistic rules that the author may be referring to is the use of statistical methods of physics to account for the myriad of particles involved (e.g. as used in kinetic theory and other developments also used by Planck in his earlier work on black body radiation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather, from my brief review of the book, the author presents Einstein's objections to quantum entanglement as his major complaint of quantum mechanics.  Indeed, every index reference to Einstein in the OP's referenced book is on the topic of quantum entanglement.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2017-09-08T15:30:50.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6501" PostTypeId="2" ParentId="5197" CreationDate="2017-09-11T13:02:54.703" Score="1" Body="&lt;p&gt;I believe you are talking about this book:&#xA;&lt;a href=&quot;http://store.doverpublications.com/0486642909.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Becker: Electromagnetic Fields and Interactions&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the back side text on my own copy (from memory), it started - as you mentioned - as a textbook by Föppl, taken over by Abraham who edited it in several editions, then taken over by Becker who also edited it in several editions, then finally taken over by Sauter after Becker's death. This final edition is from 1964 and was published in English. (Phew!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The link above goes to the Dover reprint of this edition. It contains both volumes I and II.&lt;/p&gt;&#xA;" OwnerUserId="6284" LastActivityDate="2017-09-11T13:02:54.703" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6502" PostTypeId="1" CreationDate="2017-09-11T16:09:21.863" Score="2" ViewCount="265" Body="&lt;p&gt;I do not mean the &quot;debauch of indices&quot; discussed at &lt;a href=&quot;https://hsm.stackexchange.com/questions/3320/debauches-of-indices-translation-request&quot;&gt;Debauches of indices: Translation request&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a memory of someone talking about a &quot;plague&quot; of indices, or perhaps of tensors.  Maybe I am just confusing it with the &quot;gruppenpest,&quot; the image of group theory as a plague on physics, but I don't think I am. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gerald Edgar shows Dwight Neuenschwander used the phrase in a 2014 book on tensors but I suspect he got it from someone earlier.  I believe I heard it much earlier than that, and since I have read Neuenschwander's book on Noether's conservation theorem I would have remembered if was associated with him.  Sadly he passed away last spring and I cannot ask him.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="38" LastEditDate="2017-09-11T23:21:27.257" LastActivityDate="2017-09-11T23:21:27.257" Title="Did someone refer to a &quot;plague of indices&quot; in tensor calculus?" Tags="|mathematicians|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6503" PostTypeId="2" ParentId="6502" CreationDate="2017-09-11T19:30:50.987" Score="2" Body="&lt;p&gt;Given the topic I came across this recently in the short bio of &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Jan_Arnoldus_Schouten&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jan Arnoldus Schouten&lt;/a&gt; (of the Schouten tensor), which you might find interesting/entertaining:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schoutens dissertation applied his 'direct analysis' modelled on the vector analysis of Gibbs and Heaviside to higher order tensor like entities he called 'affinors' ... [and also] entities such as axiators, perversors and deviators appeared in this analysis ... just as vector analysis had the dot and cross product, so affinor analysis had different kinds of products for tensors at various levels. However, instead of two kinds of multiplication symbols, Schouten had at least &lt;em&gt;twenty&lt;/em&gt; [!!!]; this made his work a chore to read, although the conclusions were valid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schouten said later to Weyl&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;he'd like to throttle the man who wrote this book&amp;quot; [meaning himself!]&lt;/p&gt;&#xA;&lt;p&gt;... Weyl himself said that Schoutens early book&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;has orgies of formalism that threaten the peace of even the technical scientist&amp;quot;; and Weitzenbock wrote&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;of the terrible book he had committed&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Lucky for us that Einstein didn't learn the Ricci tensor calculus from Schouten!&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-09-11T19:30:50.987" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6504" PostTypeId="1" CreationDate="2017-09-11T20:16:58.057" Score="1" ViewCount="165" Body="&lt;p&gt;I have an approximation of a quote lodged in my brain, and I'm looking to source it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Whenever you don't understand something, there is a simpler thing that you also don't understand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am almost certain this quote was due to a mathematician or physicist, but I can't track it down.  Does it ring a bell for anyone?&lt;/p&gt;&#xA;" OwnerUserId="35" LastEditorUserId="62" LastEditDate="2017-09-12T03:02:31.867" LastActivityDate="2017-09-12T03:02:31.867" Title="Source: quote about problem solving" Tags="|quote|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6505" PostTypeId="2" ParentId="6504" CreationDate="2017-09-11T20:50:20.247" Score="1" Body="&lt;p&gt;Accordin to &lt;a href=&quot;https://math.berkeley.edu/~gmelvin/polya.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;THIS&lt;/a&gt;, Pólya included &quot;Solve a simpler problem&quot; as one strategy in his book &lt;em&gt;How to Solve It&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-09-11T20:50:20.247" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6506" PostTypeId="1" CreationDate="2017-09-12T10:46:00.700" Score="2" ViewCount="3635" Body="&lt;p&gt;How and when did we build the first ruler ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have an hard time figuring how to build something &quot;straight&quot; without having first a &quot;straight&quot; tool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sure you could just stretch a thread/wire and then cut a wood block to have something ressembling a ruler. But is that it ?&lt;/p&gt;&#xA;" OwnerUserId="6286" LastActivityDate="2017-09-12T21:01:23.440" Title="How does human design the first ruler?" Tags="|discoveries|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6507" PostTypeId="2" ParentId="6506" CreationDate="2017-09-12T12:21:07.150" Score="4" Body="&lt;p&gt;The ruler is made by an expert joiner with an instrument similar to the modern&#xA;plane: &lt;a href=&quot;https://en.wikipedia.org/wiki/Plane_(tool)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Plane_(tool)&lt;/a&gt;.&#xA;The straightness is checked by looking along the edge of the ruler (one end&#xA;of the ruler is close to the eye). That is the edge is&#xA;compared with the ray of light. In this way, the joiner detects the concavities, of the edge, and carefully removes them, until the edge looks straight.&#xA;The surface of the instrument itself does not have to be very straight for this&#xA;procedure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A more difficult question is how the divided ruler is made, if you do not have&#xA;a divided ruler for comparison. To divide the ruler, a construction with a compass&#xA;is used, essentially consecutive dissections. This is how they divided precision instruments until the second half of 18th century. This was very labor consuming and thus expensive, until a &quot;dividing engine&quot; was invented. It uses essentially a comparison with already existing very finely divided scale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. One of the most famous craftsmen of 18 century (John Bird, instrument maker) once said that there is no mechanical way to produce a straight line.&#xA;One can only produce an arc of a circle (perhaps of a very great diamter), by compass. Since then, a mechanical mean of producing a straight line was actually discovered (it is called an Inversor, discovered by Paucellier, with rigorous proof by Lipkin&#xA;(late 19 century)), &lt;a href=&quot;https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I've never heard of this device being used in practice.&#xA;Its original purpose was to convert the circular motion to rectilinear motion&#xA;(and back) in a steam engine. But the progress is lubricants made this invention redundant for its original purpose. It is easier to make a piston to move on&#xA;APPROXIMATE straight line, then to implement an inversor. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-09-12T21:01:23.440" LastActivityDate="2017-09-12T21:01:23.440" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6508" PostTypeId="1" CreationDate="2017-09-12T15:45:32.670" Score="2" ViewCount="156" Body="&lt;p&gt;I recently read the book &quot;The secular ark&quot; by Janet Browne on the history of plant geographical studies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the chapter &quot;A Science of Patterns&quot;, a subsection entitled &quot;Tabellenstatistik&quot; is included, that tries to answer, why people were so enamored with numerical surveys of plants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On page 75 in this subsection she writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Later, this preoccupation with figures was derisively called Tabellenstatistik, after a similar episode in German political economy where, for a short period, theory was disregarded in favour of numerical data alone.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does anybody know what Browne is referring to here? It would be very interesting to know the context a little bit better in which this was said and of course by whom it was said. So far my own searches didn't bring anything up. I have also wrote a message to Browne herself in August, but thought it won't hurt to ask here as well.&lt;/p&gt;&#xA;" OwnerUserId="6147" LastEditorUserId="6147" LastEditDate="2017-09-12T20:26:12.267" LastActivityDate="2017-12-22T14:46:05.147" Title="Who used &quot;Tabellenstatistik&quot; as a derogatory term in the history of biogeography?" Tags="|biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6509" PostTypeId="2" ParentId="6508" CreationDate="2017-09-12T21:08:27.963" Score="3" Body="&lt;p&gt;Histories by Twellmann (&lt;a href=&quot;http://nbn-resolving.de/urn:nbn:de:bsz:352-0-268290&quot; rel=&quot;nofollow noreferrer&quot;&gt;2014, pp. 148-154&lt;/a&gt;), Westergaard (&lt;a href=&quot;https://books.google.com/books?id=BBhRAQAAMAAJ&amp;amp;pg=PA223&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916, pp. 230-231&lt;/a&gt;; &lt;a href=&quot;https://archive.org/stream/diegrundzgedert00westgoog#page/n261&quot; rel=&quot;nofollow noreferrer&quot;&gt;1890, pp. 249-250&lt;/a&gt;), John (&lt;a href=&quot;https://archive.org/stream/geschichtedersta00johnuoft#page/88&quot; rel=&quot;nofollow noreferrer&quot;&gt;1884, pp. 88 sq, 128 sq&lt;/a&gt;) or Knies  (&lt;a href=&quot;https://archive.org/stream/diestatistikals00kniegoog#page/n35&quot; rel=&quot;nofollow noreferrer&quot;&gt;1850, pp. 23-24&lt;/a&gt;) all point to “the Göttingen school (1806-1811)”, with e.g. Lueder who wrote in (&lt;a href=&quot;https://books.google.com/books?id=pwlDAAAAcAAJ&amp;amp;pg=PA221&quot; rel=&quot;nofollow noreferrer&quot;&gt;1817, pp. 221-222&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After and next to the true statisticians, who described the State in words, there arose a new kind of statisticians, who described the State in numbers: &lt;em&gt;political arithmeticians&lt;/em&gt;, &lt;em&gt;table fabricators&lt;/em&gt;, or, as &lt;a href=&quot;https://de.wikipedia.org/wiki/Ernst_Brandes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brandes&lt;/a&gt; so aptly named them, &lt;em&gt;table serfs&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; As to similar controversies in biogeography, Rudwick (&lt;a href=&quot;https://archive.org/stream/palaeontology21121978pala#page/n245&quot; rel=&quot;nofollow noreferrer&quot;&gt;1978, p. 236&lt;/a&gt;) has a slightly earlier and more specific claim than Browne:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Heinrich_Georg_Bronn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bronn&lt;/a&gt; (&lt;a href=&quot;https://archive.org/details/italienstertirge00bron&quot; rel=&quot;nofollow noreferrer&quot;&gt;1831&lt;/a&gt;)’s tables are an outstanding example of the ‘statistical’ approach in paleontology at this time (remembering that ‘statistics’ meant simply a compilation of quantitative data). (...) Yet only incidentally in the midst of all this &lt;em&gt;‘Tabellenstatistik’&lt;/em&gt; (as its critics scornfully called it) did Bronn (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;By the way, Rudwick (p. 229) attributes the first (non-derisive) use of the word &lt;em&gt;statistics&lt;/em&gt; in botanical geography to de Candolle (&lt;a href=&quot;https://archive.org/stream/bub_gb_4uwTAAAAQAAJ#page/n366&quot; rel=&quot;nofollow noreferrer&quot;&gt;1820, p. 362&lt;/a&gt;; see also &lt;a href=&quot;https://books.google.com/books?id=UBIPAAAAQAAJ&amp;amp;pg=PA1&amp;amp;dq=statistique&quot; rel=&quot;nofollow noreferrer&quot;&gt;1838a&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=UBIPAAAAQAAJ&amp;amp;pg=PA58-IA31&amp;amp;dq=statistique&quot; rel=&quot;nofollow noreferrer&quot;&gt;1838b&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=0j84AAAAMAAJ&amp;amp;pg=PA287&amp;amp;dq=Statistik&quot; rel=&quot;nofollow noreferrer&quot;&gt;1840&lt;/a&gt;), but it is already (semi-derisively...) in von Humboldt (&lt;a href=&quot;https://archive.org/stream/b2170630x#page/18&quot; rel=&quot;nofollow noreferrer&quot;&gt;1817, p. 18&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-12-22T14:46:05.147" LastActivityDate="2017-12-22T14:46:05.147" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6510" PostTypeId="2" ParentId="6476" CreationDate="2017-09-14T11:15:55.627" Score="1" Body="&lt;p&gt;The engine in question does not &lt;em&gt;lift&lt;/em&gt; water; rather it &lt;em&gt;pushes&lt;/em&gt; water up from the bottom of the vessel.  As is clearly shown in the animated image &lt;a href=&quot;https://en.wikipedia.org/wiki/Newcomen_atmospheric_engine&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, condensed steam is forced into the bottom of the lifting pipe.  A freeze-frame from that page:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hfPwd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hfPwd.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So long as the combination of partial vacuum in the cylinder (as steam condenses into liquid) and the net mass (weight) of the piston-linkage assembly is sufficient to overcome the pressure on the outlet valve, the machine will pump water into the upper chamber.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2017-09-14T11:15:55.627" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6511" PostTypeId="1" AcceptedAnswerId="6515" CreationDate="2017-09-14T18:42:19.720" Score="5" ViewCount="1114" Body="&lt;p&gt;I found &lt;a href=&quot;https://hsm.stackexchange.com/questions/3422/example-of-abstract-math-theory-that-was-later-found-to-be-applicable-to-physica&quot;&gt;this question&lt;/a&gt; that discusses abstract theories that later found application. I am interested in accepted (at least at one point in time) abstract theories that:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;was contradicted by attempts to apply and/or observed phenomenon (as currently understood).&lt;/li&gt;&#xA;&lt;li&gt;are irreconcilable with another accepted and intersecting abstract theory&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I am a licensed engineer by profession; but turning to philosophy. Some individuals point to the &lt;em&gt;a priori&lt;/em&gt; or abstract nature of mathematics as justification that other a priori forms of knowledge exist. I &lt;em&gt;hope&lt;/em&gt; to demonstrate that this dependence is unreliable by showing that mathematics requires &lt;em&gt;a posteriori&lt;/em&gt; verification to acquire meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be happy with 2–3 examples, or if someone could refer any texts that might document the history of such errors that would be helpful.&lt;/p&gt;&#xA;" OwnerUserId="6291" LastEditorUserId="658" LastEditDate="2018-08-28T13:15:08.983" LastActivityDate="2018-09-02T11:02:58.723" Title="Examples of abstract proofs that turned out to be false" Tags="|mathematics|reference-request|" AnswerCount="6" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="6512" PostTypeId="2" ParentId="6511" CreationDate="2017-09-14T20:05:01.120" Score="3" Body="&lt;p&gt;Proofs (your title) in mathematics (your tag) don’t get invalidated. Some of Kant’s more exalted efforts, on the other hand... (See e.g. Prop. 4 or 7 in his &lt;a href=&quot;https://archive.org/stream/kantsprolegomen00kantuoft#page/169&quot; rel=&quot;nofollow noreferrer&quot;&gt;Metaphysical foundations of dynamics&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;A famous quote comes to mind [&lt;a href=&quot;https://archive.org/stream/geometrieunderf00einsgoog#page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;german&lt;/a&gt;, &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;english&lt;/a&gt;, &lt;a href=&quot;http://www.pitt.edu/%7Ejdnorton/teaching/HPS_0410/chapters/significance_GR_geometry/Einstein_on_Kant.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;context&lt;/a&gt;] (emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?&lt;/p&gt;&#xA;&lt;p&gt;In my opinion the answer to this question is, briefly, this: &lt;em&gt;&lt;strong&gt;As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality&lt;/strong&gt;&lt;/em&gt;. It seems to me that complete clearness as to this state of things first became common property through that new departure in mathematics which is known by the name of mathematical logic or “Axiomatics.” The progress achieved by axiomatics consists in its having neatly separated the logical-formal from its objective or intuitive content; according to axiomatics the logical-formal alone forms the subject-matter of mathematics, which is not concerned with the intuitive or other content (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-01T21:00:20.220" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6513" PostTypeId="2" ParentId="6511" CreationDate="2017-09-14T20:44:28.723" Score="3" Body="&lt;p&gt;There are some borderline cases.  Frege wrote a two volume treatise on set theory,  &lt;em&gt;Die Grundlagen der Arithmetik&lt;/em&gt; [&lt;em&gt;The Foundations of Arithmetic&lt;/em&gt;] (1884).  Russell found a contradiction in Frege's system (Russell's paradox).  So mathematicians came up with other systems for doing set theory.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2017-09-17T12:16:34.313" LastActivityDate="2017-09-17T12:16:34.313" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6514" PostTypeId="2" ParentId="6511" CreationDate="2017-09-16T08:34:09.893" Score="5" Body="&lt;p&gt;I'm not sure if this is what you are interested in, but two supposed proofs of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Four_color_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;four color theorem&lt;/a&gt; appeared in the XIX&lt;sup&gt;th&lt;/sup&gt; century and were accepted as correct proofs for eleven years, but both turned out to be wrong. Quoting form Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One alleged proof was given by Alfred Kempe in 1879, which was widely acclaimed; another was given by Peter Guthrie Tait in 1880. It was not until 1890 that Kempe's proof was shown incorrect by Percy Heawood, and in 1891, Tait's proof was shown incorrect by Julius Petersen—each false proof stood unchallenged for 11 years.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2017-09-22T08:39:50.103" LastActivityDate="2017-09-22T08:39:50.103" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6515" PostTypeId="2" ParentId="6511" CreationDate="2017-09-16T08:41:28.233" Score="3" Body="&lt;p&gt;You can see &lt;a href=&quot;https://en.wikipedia.org/wiki/Imre_Lakatos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Imre Lakatos'&lt;/a&gt; book: &lt;a href=&quot;https://en.wikipedia.org/wiki/Proofs_and_Refutations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs and Refutations&lt;/a&gt;, with a beautiful discussion of some case histories of theorems whose proofs had to be &quot;revised&quot; in order to take into account counterexamples. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See: I.Lakatos, &lt;a href=&quot;https://books.google.it/books?id=zb8qDgAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proofs and Refutations: The Logic of Mathematical Discovery&lt;/a&gt;, Cambridge UP, or.ed.1976.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-09-16T08:41:28.233" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6516" PostTypeId="2" ParentId="3131" CreationDate="2017-09-16T08:48:30.953" Score="14" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oswald_Teichm%C3%BCller&quot; rel=&quot;noreferrer&quot;&gt;Oswald Teichmüller&lt;/a&gt; (1913–1943), was a gifted German mathematician who worked on Riemann surfaces, among other subjects. He was also a dedicated Nazi, who volunteered for combat on the Eastern Front and was killed in action in September 1943.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2019-03-07T18:52:18.627" LastActivityDate="2019-03-07T18:52:18.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6520" PostTypeId="2" ParentId="3131" CreationDate="2017-09-18T23:44:28.400" Score="17" Body="&lt;p&gt;&quot;&lt;a href=&quot;http://www.kellenmyers.org/deaths.html&quot; rel=&quot;noreferrer&quot;&gt;This page&lt;/a&gt; features a collection of mathematicians who died under unfortunate or unfitting circumstances.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of those not already mentioned in previous answers are&lt;br&gt;&#xA;Vladimir Markov, 1871-1897 (25, tuberculosis)&lt;br&gt;&#xA;René Gâteaux, 1889-1914 (25, died in battle during WWI)&lt;br&gt;&#xA;Stanisław Saks, 1897-1942 (44, murdered by the Gestapo)&lt;br&gt;&#xA;Dénes Kőnig, 1884-1944 (60, suicide after Hungarian Nazi coup)&lt;br&gt;&#xA;Dmitri Egorov, 1869-1931 (61, starvation while on hunger strike)&lt;br&gt;&#xA;Archimedes, 287-212 (75, killed)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's add &lt;a href=&quot;https://en.wikipedia.org/wiki/Hypatia&quot; rel=&quot;noreferrer&quot;&gt;Hypatia&lt;/a&gt; and Hippasus (although we're straying into legend here). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a footnote on page 4 of Mathematicians Fleeing from Nazi Germany, author Reinhard Siegmund-Schultze names over two dozen mathematicians murdered by the Nazis. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: I should also mention Walter Koppelman, murdered by a graduate student at Penn in 1970. &lt;/p&gt;&#xA;" OwnerUserId="6308" LastEditorUserId="-1" LastEditDate="2017-09-19T09:59:53.543" LastActivityDate="2017-09-19T09:59:53.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6521" PostTypeId="1" AcceptedAnswerId="6527" CreationDate="2017-09-19T12:54:16.373" Score="7" ViewCount="736" Body="&lt;p&gt;I'm reading &lt;em&gt;&quot;Professor Stewart's Incredible Numbers,&quot;&lt;/em&gt; by Ian Stewart and in there it is claimed that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In early Greece, the list of numbers started $2, 3, 4,$ and so on: $1$ was special, not &quot;really&quot; a number. Later, when this convention started to look really silly, $1$ was deemed to be a number as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is new to me. What evidence is there of this and when did the change occur?&lt;/p&gt;&#xA;" OwnerUserId="6312" LastActivityDate="2017-09-21T08:36:03.650" Title="When $1$ wasn't really a number in Greece" Tags="|ancient-greece|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6523" PostTypeId="2" ParentId="6521" CreationDate="2017-09-19T16:05:20.857" Score="9" Body="&lt;p&gt;Euclid's &lt;a href=&quot;https://mathcs.clarku.edu/%7EDJoyce/java/elements/bookVII/bookVII.html#defs&quot; rel=&quot;noreferrer&quot;&gt;Elements&lt;/a&gt; &lt;em&gt;Book VII&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition 1&lt;/strong&gt;: A &lt;em&gt;unit&lt;/em&gt; is that by virtue of which each of the things that exist is called one.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition 2&lt;/strong&gt;: A &lt;em&gt;number&lt;/em&gt; is a multitude composed of units.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also: &lt;a href=&quot;https://plato.stanford.edu/entries/aristotle-mathematics/#10&quot; rel=&quot;noreferrer&quot;&gt;Aristotle on unit (&lt;em&gt;monas&lt;/em&gt;) and number (&lt;em&gt;arithmos&lt;/em&gt;)&lt;/a&gt; and  &lt;em&gt;Metaphysics&lt;/em&gt;, Book N, 1088:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;One&amp;quot; evidently means a measure. And in every case it is some underlying&#xA;thing with a distinct nature of its own [...] And this is reasonable; for the one means the measure of some plurality, and number means a measured plurality&#xA;and a plurality of measures. Thus it is natural that one is not a number; for the measure is not measures, but both the measure and the one are starting-points.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These speculations affect the &amp;quot;theory of numbers&amp;quot; of Pythagorean origin and not the everyday counting practice; see Plato, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Abook%3D7%3Asection%3D525a&quot; rel=&quot;noreferrer&quot;&gt;Republic, Book 7, 525a&lt;/a&gt;, with the distinction between the &amp;quot;science of arithmetic&amp;quot; (&lt;em&gt;ἀριθμητική&lt;/em&gt;) and the &amp;quot;art of calculation&amp;quot; (&lt;em&gt;λογιστική&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-09-21T06:28:12.293" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6526" PostTypeId="1" CreationDate="2017-09-20T10:38:13.177" Score="0" ViewCount="235" Body="&lt;p&gt;I was just wondering if there is a process or set of properties that exist to name a score, such as, Cohen’s &lt;strong&gt;&lt;em&gt;Kappa&lt;/em&gt;&lt;/strong&gt;, Fleiss’ &lt;strong&gt;&lt;em&gt;Kappa&lt;/em&gt;&lt;/strong&gt;, Krippendorff’s &lt;strong&gt;&lt;em&gt;Alpha&lt;/em&gt;&lt;/strong&gt;, or if it is just at the creators choice.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance &lt;/p&gt;&#xA;" OwnerUserId="6317" LastEditorUserId="658" LastEditDate="2017-09-20T14:16:22.570" LastActivityDate="2019-08-28T10:22:39.067" Title="What defines the 'name' of a score i.e. gamma, kappa etc" Tags="|statistics|naming-conventions|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6527" PostTypeId="2" ParentId="6521" CreationDate="2017-09-20T17:13:21.057" Score="5" Body="&lt;p&gt;The opening words of the Platonic dialogue &lt;em&gt;Timaeus&lt;/em&gt; are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;b&gt;Σωκράτης:&lt;/b&gt;&lt;p&gt;εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων, τὰ νῦν δὲ ἑστιατόρων;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;b&gt;Socrates:&lt;/b&gt;&lt;p&gt;&#xA;One, two, three, — but where, my dear Timaeus, is the fourth of our guests of yesterday, and our hosts of today?&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;clearly showing that the word εἷς &amp;quot;one&amp;quot; was used for counting in the same way as all other positive integers.&lt;/p&gt;&#xA;&lt;p&gt;If at some point later on (some) Greek mathematicians accorded a special status to the number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, I would be tempted to compare this to a modern discussion of whether &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; is or is not to be included in the natural numbers -- or whether the zero ring is an integral domain or not. It is a technicality, of the sort that only becomes meaningful once one has run into the necessity of &amp;quot;constructing&amp;quot; objects that have already been understood on a more intuitive level.&lt;/p&gt;&#xA;&lt;p&gt;To be sure, there seems to be a certain attraction to the statement as presented by Ian Stewart: it is intriguing to consider the possibility that the Greeks somehow thought &amp;quot;differently&amp;quot; about numbers than we do. It is a statement that borders on the unprovable. Heidegger is even more outrageous, writing somewhere (in &amp;quot;Modern Science, Metaphysics, and Mathematics&amp;quot;, which is reprinted in the &amp;quot;Basic Writings&amp;quot; collection) that the Greeks didn't even consider 2 to be a number: since 1 is the unit, and 2 is only ever a &amp;quot;pair&amp;quot;, and not a collection of &amp;quot;two&amp;quot; separate objects. Only when the number three is first encountered, does the concept of number really start to shine forth -- or so the the little magician from Messkirch would have it. It sounds good, but I have yet to be convinced that there is more to it than idle speculation.&lt;/p&gt;&#xA;" OwnerUserId="2353" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-09-21T08:36:03.650" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6528" PostTypeId="1" CreationDate="2017-09-21T19:14:48.883" Score="4" ViewCount="224" Body="&lt;p&gt;&lt;em&gt;Riddell's formula for unlabeled graphs&lt;/em&gt; is a generating function transformation $$1 + B(x) = \exp \sum_{k=1}^\infty \frac{A(x^k)}{k}$$ which gives the number of graphs whose connected components have a property P in terms of the number of connected graphs with property P. From a modern perspective it is &quot;just&quot; the application of Pólya's enumeration theorem to the natural action of the symmetric groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pólya published in 1937 (ten years after Redfield, but attracting much more attention); the theorem as I understand it is given in terms of subgroups of the symmetric groups, and $S_n$ is arguably the most obvious subgroup, and one of the most useful, of $S_n$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riddell's name is attached to the formula in consequence of his 1951 doctoral dissertation in physics, and Harary gives to understand in a 1955 paper&lt;sup&gt;1&lt;/sup&gt; that Riddell had priority; curiously he describes the formula as &quot;&lt;em&gt;a well known identity&lt;/em&gt;&quot; but states that &quot;&lt;em&gt;Riddell's combinatorial derivation ... is not readily available in the literature&lt;/em&gt;&quot; (presumably because doctoral dissertations didn't have as wide circulation as journals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I find it surprising that it should have taken so long for the enumeration theorem to be applied to the full $S_n$, and Harary's paper hints at the result being known more widely than is easily accounted for if Riddell truly had priority. Is there more to the story than this? Was the formula circulating as folklore but unpublished for up to 14 years?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; The Number of Linear, Directed, Rooted, and Connected Graphs, Frank Harary, &lt;em&gt;Transactions of the American Mathematical Society&lt;/em&gt; 78.2 (1955)&lt;/p&gt;&#xA;" OwnerUserId="104" LastActivityDate="2017-09-21T19:14:48.883" Title="Was Riddell's formula really so late?" Tags="|mathematics|naming-conventions|combinatorics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6529" PostTypeId="2" ParentId="6511" CreationDate="2017-09-22T11:37:16.870" Score="2" Body="&lt;p&gt;Read this link for a relatively recent example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.siobhanroberts.com/in-mathematics-mistakes-arent-what-they-used-to-be/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;In Mathematics, Mistakes Aren’t What They Used To Be&lt;/em&gt;&lt;/a&gt;&#xA;&lt;br&gt;by Siobhan Roberts&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&#xA;[Voevodsky] and Kapranov shared a passion for developing the mathematics of new higher dimensional objects and categories, and they published an important result in 1990. [...]&#xA;&lt;/p&gt;&#xA;&lt;p&gt;&#xA;But along the way he met a bump in the road. In 1998, the American mathematician Carlos Simpson published a paper indicating there might be a mistake in Voevodsky and Kapranov’s 1990 result. For years Voevodsky sifted through the details without making much progress. He remained convinced the result was right. Then, in the autumn of 2013, as the leaves changed color and summer gave way to autumn, he made a breakthrough. Of sorts. He confirmed the error. The important result was no longer quite so important.&#xA;&lt;/p&gt;&#xA;&lt;p&gt;&#xA;“It is plainly wrong. The main theorem is incorrect,” he says. “It’s not that there is some gap in the proof. It’s that the main theorem is plainly wrong.” The mistake, he explains, was in failing to question the obvious. “We had proved that an assertion was indeed true in all of the difficult cases, but it turned out to be false in the simple case. We never bothered to check.” In confirming the error, he added an addendum to the original citation in his official publications list—“Warning: The main theorem of this paper was shown by Carlos Simpson to be false.”&#xA;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2353" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-09-22T12:04:04.477" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6531" PostTypeId="1" CreationDate="2017-09-22T16:40:22.587" Score="0" ViewCount="87" Body="&lt;p&gt;A &lt;a href=&quot;http://solar-center.stanford.edu/AO/Ancient-Observatories.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2015 document from Stanford Solar Centre&lt;/a&gt; informs us that Stonehenge, The Goseck Circle and Majorville Medicine Wheel are on the same latitude (51° N.)  with the Newgrange monument  slightly to the north, and explains that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These sites lie on the exact latitude at which the midsummer sunrise&#xA;  and sunsets are at 90° to the Moon’s northerly setting and southerly&#xA;  rising. This particular phenomena is only possible within a band of&#xA;  less than one degree of which Stonehenge and Goseck lies in the&#xA;  middle-third. The sites also sit on one of two unique latitudes in the&#xA;  world where the full Moon passes directly overhead on its maximum&#xA;  zeniths.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which somehow I fail to understand. What  directions are meant  - by the builders, eventually,  and/or  by the authors of the document? Does this text describe an original discovery? &lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2017-09-24T08:55:18.530" LastActivityDate="2017-09-24T08:55:18.530" Title="What is the remarkable configuration of directions at 51° N?" Tags="|archaeoastronomy|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6532" PostTypeId="1" CreationDate="2017-09-23T13:57:39.583" Score="4" ViewCount="188" Body="&lt;p&gt;Telescope images of stars look like crosses, although they are point sources of light. Featureless dots. Diffraction along the vanes of the support structure that holds up their secondary mirror is what causes this optical distortion. And makes the stars look as if they have these beautiful spikes, like a painted compass on an ancient map.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But why would anyone before Newton, who invented the refracting telescope, have perceived a star as having spikes? Did anyone? The spikes aren't visible by the naked eye, are they?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were stars depicted as having spikes before Newton, or before any telescope?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did astronomers, like Tycho Brahe, mark the location of a star on the sky map with a cross (an asterisk) for clarity, and did that &quot;typographically&quot; lead to the star shape we all use to celebrate Christmas?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Not to go into the specifics of this here, but to exemplify: What about the pre-Newton &quot;star of David&quot;? Was it originally really intended to depict a &quot;star&quot; as in an observable star on the sky? Or was it just a geometric figure, later called &quot;a star&quot;? And what about the pentagram of the Devil in Catholicism, was that originally in any way a star related symbol? And not the least the islamic star next to the new moon. Why is that spiky? Did these symbols really refer to a star in the sky when they were created? And if so why their spikes?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorUserId="3663" LastEditDate="2017-09-23T14:24:14.320" LastActivityDate="2017-09-24T00:31:05.127" Title="When and why did the 'spiky star' appear as an image in art and science?" Tags="|optics|archaeoastronomy|ancient|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6534" PostTypeId="2" ParentId="6532" CreationDate="2017-09-24T00:31:05.127" Score="1" Body="&lt;p&gt;Here are some Roman Denari.  Moon and stars.  Looks like points on the stars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NgaWt.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NgaWt.jpg&quot; alt=&quot;image&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-09-24T00:31:05.127" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6535" PostTypeId="2" ParentId="2362" CreationDate="2017-09-24T15:40:57.210" Score="8" Body="&lt;p&gt;Although this question and the answers now have some age to them, I suggest that it's important not to overlook the mythical character of the assumption that underlies this question. The question explicitly supposes that 'calculus is missing from the &lt;em&gt;Principia&lt;/em&gt;'.  But that is not true: it is not missing, not only have skilled commentators from the 17th to the 21st centuries clearly recognised the content of calculus in the work, but also one can point directly to many arguments and demonstrations in the work itself that definitely employ principles belonging to the field of calculus. &#xA;&lt;/p&gt; &#xA;What is practically absent from the &lt;em&gt;Principia&lt;/em&gt; is a largely different matter: it is mainly about notation. As to this, it is worthwhile to bear in mind the late Clifford Truesdell's assessment that &quot;... a modern mathematician, according little respect to those who confuse notations with notions, finds the &lt;em&gt;Principia&lt;/em&gt; a book dense with the theory and application of the infinitesimal calculus.&quot; (Essays in the History of Mechanics, 1968, 99 (at n.4)). &#xA;&lt;/p&gt;&#xA;Truesdell was not alone in his view. Published for example already in 1696 was the book &quot;Analyse des infiniment petits&quot; (Infinitesimal analysis) by the Marquis de l'Hospital (or l'Hôpital): this was an exposition of Leibniz's form of the differential calculus. In its preface, after due praise of Leibniz, one reads (translating from the French): &quot;... credit is also due to the learned M. Neuwton, as M. Leibnis himself (in the Journal des Sçavans of 30 August 1694) has acknowledged: that he too&quot; [i.e. Newton] &quot;had found something similar to the differential Calculus, as appears by the excellent book entitled 'Philosophia naturalis principia Mathematica', which he gave us in 1687, of which almost all is of this calculus&quot; ['lequel est presque tout de ce calcul']. &quot;However, the notation of Mr. Leibnis makes his much easier and more expeditious ...&quot;.  (L'Hospital's last-quoted phrase, his assessment of the convenience and utility of Leibniz's notation, is of course an assessment echoed many times since 1696, applying also to modern descendant forms of Leibniz's notation, and understandably attracting wide agreement, as Newton's expository choices and forms have been regarded by many as a hindrance.)  As part of the background to the acknowledgement of Newton by Leibnizians it is worth recalling, that at the relevant period no quarrel had yet been fomented between Newton and Leibniz, and each of them had made courteous acknowledgement of the other.  As A.Rupert Hall put it, in 'Philosophers at War' (1980), for example &quot;the [earlier] triangular exchanges of letters between Wallis, Leibniz, and Newton consistently suggest esteem and amity&quot;. &#xA;&lt;/p&gt; &#xA;Coming now to more detailed mathematical considerations, a study by Bruce Pourciau (2001), in &lt;em&gt;Historia Mathematica&lt;/em&gt; 28, 18-30, investigates &quot;Newton’s understanding of the limit concept through a study of certain proofs appearing in the &lt;em&gt;Principia&lt;/em&gt;.&quot;  Pourciau finds &quot;that Newton, not Cauchy, was the first to present an epsilon argument, and that, in general, Newton’s understanding of limits was clearer than commonly thought.  We observe Newton’s distinction between two properties easily confused, namely f/g --&gt; 1 and f - g --&gt; 0, [and] we resolve a problem created by a spurious translation appearing in Cajori’s revision of Motte’s original translation, ...&quot;.  Pourciau specially points out three important Lemmas in the &lt;em&gt;Principia&lt;/em&gt;, &quot;Lemma I on the limit of a difference [&lt;em&gt;sic&lt;/em&gt;, this must be a slip for 'ratio'   which appears in Lemma 1], Lemma II on the existence of the integral, and Lemma XI on the second derivative&quot; and discusses how &quot;their statements and proofs most clearly reveal Newton’s grasp of the limit process.&quot; But &quot;to read these lemmas&quot;, says Pourciau, &quot;requires a double translation, not only a first translation from the original Latin into English (for which we rely on [I B Cohen's 1999 translation of the &lt;em&gt;Principia&lt;/em&gt;]), but then a second translation as well, for the lemmas come to us packed in the &lt;em&gt;Principia&lt;/em&gt;’s unique blend of Euclidean geometry and limits, a sort of geometric calculus, and we cannot sort out what the lemmas really say without doing some unpacking. But any translation disturbs meaning, and we must take great care to minimize that disturbance, to preserve as far as possible Newton’s original intent.&quot; Thus Pourciau begins to show that while the notation of calculus as we now know it was largely absent from the Principia, the content is indeed to be found, expressed in a geometric form of infinitesimal calculus, often based on limits of ratios of vanishing small quantities. &#xA;&lt;/p&gt;&#xA;One should also go directly to the source on such a question. Immediately after Newton's Definitions and Laws of Motion, the opening section 1 in &lt;em&gt;Principia&lt;/em&gt;'s Book 1 has a number of lemmas all about &quot;the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow.&quot; The 'first' and 'last' ratios are explained as the limits of ratios of quantities either growing from zero ('nascent') or decaying to zero ('evanescent'). These lemmas include those discussed by Pourciau (2001) already cited.  Then in the body of the work we find inter alia Propositions 1, 6, 10, 11: in Proposition 1 the argument proceeds by assembling a finite series of triangles expressing by their equal areas the increments of motion occurring after a finite series of discrete impulses, then Newton writes &quot;now let the number of those triangles be augmented and their breadth diminished &lt;em&gt;in infinitum&lt;/em&gt;&quot;, thus he expresses an argument of limits clearly belonging to the field of calculus, and draws his conclusion about a curved trajectory and its relation to a continuous force both expressed by reference to the results in the limit of a process involving an indefinitely increasing number of (individually) indefinitely diminishing elements. There are arguments of limits in the mentioned later propositions too, sometimes expressed more briefly, and it may be, loosely, as when Newton writes first of a geometrically-defined 'solid' expressed by a ratio in which one of the factors, in both numerator and denominator, depends on a distance PQ, and then of 'that magnitude which [the solid] ultimately acquires when the points P and Q coincide'. The context and the 'ultimately' both indicate that with the phrase 'when the points ... coincide', Newton is referring to the limit of the ratios developed as the points approach each other, in the way discussed in the opening section on the 'method of first and last ratios'.&#xA;&lt;/p&gt;&#xA;It is worth mentioning that the notation of fluxions amounted to only one notational form, essentially the latest, among Newton's various expressions of his ideas in this field. He seems to have held a view that any particular notation was relatively unimportant compared to the idea 'which may be without it'. One may certainly quarrel with Newton's assessment and choice of notation and exposition, but his ideas and work applying them are attested in the various sources cited and discussed in the references already given. &#xA;&lt;/p&gt;&#xA;In short, calculus is clearly not 'missing' from the &lt;em&gt;Principia&lt;/em&gt;:  and while this mistaken idea has become one of many myths about Newton, it tells more about the history of commentators and commentaries than about Newton's actual work.  &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-09-24T15:40:57.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6536" PostTypeId="1" CreationDate="2017-09-24T16:35:37.167" Score="2" ViewCount="377" Body="&lt;p&gt;It seems that almost every where that physics is taught, in particular teaching relativity or quantum phenomena, that examples or demonstrations using persons Bob and Alice is a common tradition.  Professor Susskind uses Bob and Alice quite a bit in his ER=EPR lectures.  Several text books I have on relativity, both special and general, use Bob and Alice in various diagrams usually highlighting their respective world lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am curious.  Who started this, is there a first example using Bob and Alice?&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2017-09-24T16:35:37.167" Title="The Pedagogy of Physics -- Bob and Alice?" Tags="|physics|examples|teaching|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6539" PostTypeId="2" ParentId="2557" CreationDate="2017-09-25T21:26:31.943" Score="1" Body="&lt;p&gt;The question is rather misleading. The different clestial matter was a successful theory devised by  Aristotle. The sources being what they are,  it is rather speculative to tell what the ancients believed. Obviously many presocratics, atomists, earlier and later, did not believe such a thing. Anaxagoras is a famous exemple. But he was tried for impiety and Aristotle apparently approved that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle presented his view as taditional piety but it allowed him to reject all variants of atomism and to explain why mathematics is unapplicable in the sublunar work. Seen thus &lt;em&gt;De Coelo&lt;/em&gt; is his alternative to Plato's &lt;em&gt;Timaeus&lt;/em&gt;, something scholars have noted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Refs: the Timaeus connection is a leading theme in Bowen, A. and Wildberg, C. (ed.). 2009. &lt;em&gt;New Perspectives on Aristotle’s De Caelo&lt;/em&gt;. Leiden/Boston: Brill; also see Solmsen, F. 1960. &lt;em&gt;Aristotle's System of the Physical World. A Comparison with his Predecessors&lt;/em&gt;, Cornell Studies in Classical Philology 33&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2017-09-25T21:36:38.033" LastActivityDate="2017-09-25T21:36:38.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6540" PostTypeId="2" ParentId="256" CreationDate="2017-09-26T02:37:40.700" Score="4" Body="&lt;p&gt;&lt;strong&gt;Rene Thom's Catastrophe Theory&lt;/strong&gt; After Thom classified them, there was a frenzy of claims for how catastrophes were a universal model for abrupt changes in &lt;em&gt;real life&lt;/em&gt; situations. The mathematical theorems are sound, but the prospect of applications died quickly, and today no one talks of catastrophes.&lt;/p&gt;&#xA;" OwnerUserId="1861" LastActivityDate="2017-09-26T02:37:40.700" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6541" PostTypeId="1" CreationDate="2017-09-26T04:37:26.667" Score="2" ViewCount="57" Body="&lt;p&gt;Its function was to expel the air that came inside the cylinder along with the steam to prevent the engine to become &quot;wind logged&quot;. Since the valve was non-return, how did Newcomen stopped steam from leaving after pushing out the air?&lt;/p&gt;&#xA;" OwnerUserId="6341" LastActivityDate="2017-09-26T04:37:26.667" Title="How did the snifting valve in Newcomen's steam engine work?" Tags="|engineering|steam-engines|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6542" PostTypeId="1" CreationDate="2017-09-26T18:32:03.700" Score="1" ViewCount="815" Body="&lt;p&gt;I tried to find up the history of (Rogers-)Hölder Inequality. In wikipedia, I found L. J. Rogers's origin paper, saying how to extent the &quot;well known&quot; AM-GM (arithmetic mean-geometric mean) inequality and get more another ones (&lt;a href=&quot;https://archive.org/stream/messengermathem01unkngoog#page/n183/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;An extension of a certain theorem in inequalities&lt;/a&gt;). I learned AM-GM  inequality just for doing math exercise or Math competition, not really understood its own origin. Although it is said that Greek mathematician had the geometric proof (like this &lt;a href=&quot;http://artofproblemsolving.com/wiki/index.php?title=Root-Mean_Square-Arithmetic_Mean-Geometric_Mean-Harmonic_mean_Inequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality&lt;/a&gt;) , yet I wonder:  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a) Why did they investigate this and did they know the general case $\frac{a_1+\cdots + a_n}{n}\geq \sqrt[n]{a_1\cdots a_n}$?  &lt;/li&gt;&#xA;&lt;li&gt;b) Is it used frequently analytically after the emergence of Calculus? &lt;/li&gt;&#xA;&lt;li&gt;c) How did it develop from ancient ages to Rogers's ages (1888)? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This would help me solve &quot;the mystery of invention of Hölder and Young inequality&quot; (the final step) in the course of Real-Analysis.&lt;/p&gt;&#xA;" OwnerUserId="6346" LastEditorUserId="4130" LastEditDate="2017-10-04T04:51:22.133" LastActivityDate="2017-10-04T04:51:22.133" Title="History of Arithmetic and Geometric Inequality" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6543" PostTypeId="1" CreationDate="2017-09-26T20:43:09.573" Score="3" ViewCount="2071" Body="&lt;p&gt;Explanations of stellar parallax that I have found involve examining the apparent motion of a nearer star relative to a background of more distant stars, as the Earth moves around the sun or as the Earth rotates.  We can see the apparent motion because we have telescopes.  In Tycho Brahe's time, telescopes were not available, and this kind of motion couldn't be seen.  The fact that stellar parallax couldn't be observed is supposed to be evidence that Tycho and others took as evidence that the Earth didn't move.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as I understand it, Tycho believed that the stars move together in a fixed sphere, so it seems that he would not even try to look for parallax using the near star/far star method.  He would have had to look for motion of the moon or a planet relative to the stars.  The moon and planets were in constant motion, however.  This doesn't mean that you can't test for parallax, but doing so would be more complicated.  How was this done?&lt;/p&gt;&#xA;" OwnerUserId="277" LastEditorUserId="4251" LastEditDate="2021-10-09T09:33:19.250" LastActivityDate="2021-10-09T09:33:19.250" Title="How was stellar parallax tested by Tycho Brahe?" Tags="|astronomy|16th-century|astronomers|brahe|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6544" PostTypeId="2" ParentId="6543" CreationDate="2017-09-26T20:58:47.407" Score="4" Body="&lt;p&gt;Parallax was always measured by measuring the angular distances between stars. If it changes during the year, you see the parallax due to the Earth motion.&#xA;The accuracy is limited by the accuracy of your angle-measuring device.&#xA;A typical such device at the time of Brahe could measure to 1' or perhaps 0.5'&#xA;(1' is the average resolution of the human eye). This is not enough to detect any&#xA;parallax of &quot;fixed stars&quot;. Of course they always attempted to measure it,&#xA;because this would decide whether the Earth moves or not (with respect to the stars). The negative result was actually one of the strongest arguments in favor&#xA;of geocentric theories. To deal with this argument against his theory, Copernicus&#xA;had to postulate really HUGE distance to the stars, the thing which did not look plausible to many.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parallax was actually detected only in 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The whole story is told in great detail in this book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parallax: The Race to Measure the Cosmos,&#xA;by Alan W. Hirshfeld&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-09-26T20:58:47.407" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="6545" PostTypeId="1" CreationDate="2017-09-27T06:25:02.333" Score="6" ViewCount="376" Body="&lt;p&gt;Sometimes, coming up with good mathematical notation is key to understanding parts of mathematics. For example, consider the quadratic formula. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brahmagupta formulated a version of the quadratic formula in 628 A.D., which goes as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Compare this to our modern notation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not only is the modern notation more succinct, it also makes certain properties of quadratic equations more apparent. For example, it's easy to determine the number of real roots of a quadratic equation using the modern formula. But Brahmagupta's formula is more opaque, and it's not immediately obvious how to find the number of real roots.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second example is the introduction of the digit 0 and the place value system. When doing complicated arithmetic, Roman numerals are difficult to manage. But number systems with place value systems can handle addition much more intuitively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I am looking for is specific examples of mathematical notation, where its introduction accelerated progress in that particular field.&lt;/strong&gt; I'm especially interested in more elementary mathematics--i.e. everything before the 1700s--but examples in more modern math are welcome too. &lt;/p&gt;&#xA;" OwnerUserId="6348" LastActivityDate="2017-10-08T00:24:47.720" Title="Examples of when the development of math notation accelerated progress in math research?" Tags="|mathematics|notation|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6546" PostTypeId="2" ParentId="34" CreationDate="2017-09-28T03:08:41.443" Score="5" Body="&lt;p&gt;Apart from the necessity in the calculation of roots of cubic polynomials, &#xA;there is another, more fundamental role complex numbers play in polynomial equations,&#xA;which was only beginning to be appreciated in the 17th century.&#xA;This role is expressed through the &lt;em&gt;fundamental theorem of algebra&lt;/em&gt;, &#xA;which says that any nonconstant polynomial equation has at least one root, if we allow complex numbers to be roots.&#xA;That is, if $a_0,a_1,\ldots,a_n$ are real numbers such that at least one of $a_1,a_2,\ldots,a_n$ is nonzero,&#xA;then the equation&#xA;\begin{equation}\label{e:polynomial-x-0}&#xA;p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 ,&#xA;\end{equation}&#xA;has a solution, provided $x$ may have complex values.&#xA;If $a_1=a_2=\ldots=a_n=0$, then the equation $p(x)=0$ becomes $a_0=0$, &#xA;which does not have any (complex) solution when $a_0\neq0$.&#xA;So the condition that at least one of $a_1,a_2,\ldots,a_n$ is nonzero&#xA;(i.e., $p(x)$ is nonconstant) is simply to rule out this trivial case.&#xA;The fundamental theorem of algebra is miraculous because complex numbers are designed to solve any quadratic equation,&#xA;and it is &lt;em&gt;a priori&lt;/em&gt; conceivable that we need to introduce a new kind of &quot;number&quot; every time we increase the degree of a polynomial equation.&#xA;The first formulation of the fundamental theorem of algebra was given by Albert Girard (1595-1632) in 1629,&#xA;although he did not attempt a proof.&#xA;Indeed, rigorous proofs of this theorem did not appear until the early 19th century,&#xA;which incidentally marks the beginning of an era when the existence and usefulness of complex numbers were widely accepted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any doubts on the existence and importance of complex numbers were completely disposed of after the development of &lt;em&gt;complex analysis&lt;/em&gt;,&#xA;which is also known as &lt;em&gt;function theory&lt;/em&gt;.&#xA;The initial motivation for studying functions of a complex variable was to use them to compute (or simplify) real definite integrals,&#xA;and the pioneering works in this direction were done by Euler and Joseph-Louis Lagrange (1736-1813)&#xA;around 1760-1780.&#xA;Their research was taken up later in the 1810's by Augustin Louis Cauchy (1789-1857), &#xA;who realized by 1821 that complex functions have a rich theory of their own.&#xA;Gauss reached the same understanding as early as 1811, and played a major role in popularizing complex numbers,&#xA;but he did not directly contribute to the development of complex analysis.&#xA;Thus roughly between 1820-1850, Cauchy singlehandedly developed all the basic results of complex analysis, perhaps with the exception of Laurent series,&#xA;which first appeared in a paper submitted by Pierre Alphonse Laurent (1813-1854) in 1843.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastActivityDate="2017-09-28T03:08:41.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6547" PostTypeId="1" CreationDate="2017-09-28T06:34:37.920" Score="14" ViewCount="732" Body="&lt;p&gt;The standard presentation of &quot;coordinatizing the plane&quot; in 19th century British textbooks on geometry (Salmon, Smith, Besant, and many more) take the plane as being rigorously (at the time) axiomatized by Euclid's axioms, and establishing a bijection between the plane and $\mathbf{R}^{2}$ by picking a point $O$, two distinct lines passing through $O$ meeting at right angles, and a suitable unit length. An arbitrary point of the plane $P$ can then be associated to a pair of real numbers $(a,b) \in \mathbf{R}^{2}$ by projection. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But towards the end of the 19th century, several distinct constructions of $\mathbf{R}$, discovered (mainly) by Continental mathematicians, were already in the air. Extensive work by Cantor on the theory of &quot;point sets&quot;, as well as the work by topologists on the dimension of $\mathbf{R}^{n}$ later on made $\mathbf{R}^{2}$ the &quot;standard definition of the plane&quot;, and the &quot;synthetic plane&quot; given by axioms fell by the wayside. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The situation then was completely the reverse of the way we do things now: lines, planes, angles, etc. were considered more rigorously established, and more trustworthy, than the real numbers. Thus the real numbers were only considered as &quot;a tool&quot; to study geometric phenomena. Nowadays we take it for granted that $\mathbf{R}^{2}$ is the &quot;Euclidean plane&quot;, and in differential geometry and topology, when we talk about &quot;Euclidean space&quot;, we really mean $\mathbf{R}^{n}$. The modern way of studying Euclidean space can also be confusing: in manifold theory, for example, $\mathbf{R}^{2}$ can be coordinatized in many different ways by different atlases, which means that we should really picture it as being &quot;featureless&quot; before the imposition of coordinate lines, but people usually talk about $\mathbf{R}^{2}$ as being the coordinate plane, which muddles things a bit. The older counterpart of giving different atlases of $\mathbf{R}^{2}$ can be thought of as the process I described in the first paragraph above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the story I proposed in the previous paragraph is only a vague sketch of (the way I see it) how things went; I haven't read proper accounts of why and when mathematicians switched completely to using $\mathbf{R}^{2}$ as the formalization of the &quot;infinitely thin flat plane&quot; of our intuition, and not the object defined by axioms. I would be interested to know more.&lt;/p&gt;&#xA;" OwnerUserId="6045" LastEditorUserId="6045" LastEditDate="2017-09-29T11:31:19.583" LastActivityDate="2019-04-12T19:02:10.357" Title="When did people start accepting $\mathbf{R}^{2}$ as &quot;the plane&quot;?" Tags="|mathematics|euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6548" PostTypeId="1" AcceptedAnswerId="6549" CreationDate="2017-09-28T17:18:57.663" Score="2" ViewCount="156" Body="&lt;p&gt;Of course, I could ask who conjectured the most, but conjectures can later be shown to be either true or false, so perhaps who has the most unresolved conjectures, in publication?  More than that though, also considered would be mathematicians with a reputation towards conjecture for discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another related question would be the person with the most disproven conjectures, in publication?  I'm also looking for interesting accounts of someone being shown to be wrong in conjecture while alive, yet still go on to write mathematics, hopefully of high quality, despite the setback.&lt;/p&gt;&#xA;" OwnerUserId="75" LastEditorUserId="75" LastEditDate="2017-09-28T18:01:29.373" LastActivityDate="2017-09-28T21:23:19.597" Title="Mathematicians with a history of conjecturing a lot" Tags="|mathematics|conjecture|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6549" PostTypeId="2" ParentId="6548" CreationDate="2017-09-28T21:23:19.597" Score="4" Body="&lt;p&gt;Probably Paul Erdos was the champion in the number of conjectures, but I do not&#xA;want to do the research needed to prove this. Another candidate that I know was Lee Rubel from the University of Illinois. But on my opinion, it is a waste of time to count such things. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-09-28T21:23:19.597" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6551" PostTypeId="1" CreationDate="2017-09-29T20:38:43.797" Score="4" ViewCount="105" Body="&lt;p&gt;What was the first radio navigation system capable of triangulating your position, and when was it built? How accurate was it and what was its range?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This does not have to be GPS or a satellite system. A series of 3 or more radio towers, broadcasting beacon signals, can be anywhere on Earth. A receiver in theory can calculate its position &lt;strong&gt;if it knows the distance to each one.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This would require two things: (1) Accurate chronometers so that the receiver and beacons have their time in sync. (2) Precise chronometers capable of resolving time down to the millisecond or finer. These two things are needed because the speed of light is so fast, a difference of 1 millisecond evaluates to an error of 30 km (19 mi). OTOH, 1 microsecond would be 30 meters, which is good enough accuracy for the old days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can those things even be done with old analogue tech from the WW2 era or even earlier? I'm hoping to find a system capable of resolving your position down to 1 km or so, which should be good enough for naval purposes in the old days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did look into this around the WW2 era. I found &lt;a href=&quot;https://en.wikipedia.org/wiki/Battle_of_the_Beams&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Battle of the Beams&lt;/a&gt;, which seems to revolve around a directional beam sent out towards London, and probably based on distance but only by measuring strength/weakness of the signal. I found &lt;a href=&quot;https://en.wikipedia.org/wiki/Sonne_(navigation)&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Sonne System&lt;/a&gt; and its predecessor, the Elektra, but again, those seemed to be based on finding your direction to an anchor beacon, not distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I couldn't find any that &lt;strong&gt;measure time delay from 3 or more anchor beacons.&lt;/strong&gt; The system doesn't actually have to calculate distance and then position. That might be asking too much from the 1940's. Instead, it just needs to measure time delay and display it on a screen. The crew of a ship or airplane could then do the calculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(P.S., I couldn't find any better tags. I saw &quot;radiation&quot; but not &quot;radio.&quot; Can't think of what else would be a good tag except electricity and computers.)&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2017-09-30T14:10:31.773" Title="What/When was the first radio nav system capable of triangulating your position?" Tags="|electromagnetism|electricity|computation|computers|radiation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6552" PostTypeId="1" CreationDate="2017-09-30T08:04:47.677" Score="4" ViewCount="73" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The question is in the title. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A pictorial representation of what this is about is the following:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/cgbzw.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cgbzw.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;(created with &lt;a href=&quot;https://en.wikipedia.org/wiki/GeoGebra&quot; rel=&quot;noreferrer&quot;&gt;GeoGebra&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/GIMP&quot; rel=&quot;noreferrer&quot;&gt;GIMP&lt;/a&gt;)&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Schl%C3%A4fli_orthoscheme&quot; rel=&quot;noreferrer&quot;&gt;orthoschemes&lt;/a&gt; named after &lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Schl%C3%A4fli&quot; rel=&quot;noreferrer&quot;&gt;Ludwig Schläfli&lt;/a&gt; are very relevant but more special, and I suspect that the above decomposition has been recognizably described much earlier than the mid 1800s. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I would also appreciate a view on whether this recognizably occurs in Euclid's elements. To me it seems it doesn't.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there attested extant non-European mentionings of this?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6060" LastActivityDate="2017-09-30T08:04:47.677" Title="What is the earliest attested mention of the fact that a parallelepiped in Euclidean 3-space can be decomposed into six tetrahedra?" Tags="|euclidean-geometry|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6554" PostTypeId="2" ParentId="6545" CreationDate="2017-09-30T12:59:05.033" Score="3" Body="&lt;p&gt;The classic example is Diophantus of Alexandria (ca. 250 AD), the first mathematician to use something that can reasonably be called algebraic notation. His work famously influenced Fermat, a millennium and a half later.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2017-09-30T12:59:05.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6555" PostTypeId="2" ParentId="6551" CreationDate="2017-09-30T14:10:31.773" Score="1" Body="&lt;p&gt;Radio navigation was used on a large scale during the &quot;Battle of Britain&quot; in WWII, to direct the fighter planes to the enemy bomber formations. Of course, they experimented much earlier than that, even during WWI, but practical use on a large scale started during WWII.&#xA;See Wikipedia article &quot;Radio navigation&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-09-30T14:10:31.773" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6556" PostTypeId="2" ParentId="6545" CreationDate="2017-09-30T14:29:46.510" Score="5" Body="&lt;p&gt;The most fundamental example the OP has already mentioned themselves: they called it 'place value system'. This is indeed a very important example. It seems to have taken humanity many millenia to realize that that makes it possible to harness the magic of the logarithm function to speed up specified tasks. Most of what you see around you is ultimately based on the 'place value system' (to use you words, which conveniently relieves me of having to pick one of the synonyms).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another, slightly more 'theoretical' example are:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;exponents in the form of Hindu-Arabic numerals&lt;/strong&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am sure I once saw an original copy of a book of Euler in which he still writes something like $xxx$ for $x^3$. It is not that long ago that the latter was an innovation. The 'acceleration' (to use the OP's word) roughly corresponds to the advantages of &lt;a href=&quot;https://en.wikipedia.org/wiki/Positional_notation&quot; rel=&quot;noreferrer&quot;&gt;Positional notation&lt;/a&gt; over &lt;a href=&quot;https://en.wikipedia.org/wiki/Unary_numeral_system&quot; rel=&quot;noreferrer&quot;&gt;unary number systems&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also have to say that your question is rather broad and somehow too unfocused.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very good advanced theoretical example, quite surprisingly not mentioned so far, is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Einstein summation convention&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;for which I refer you to the following recommendable summary:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Willie Wong: &lt;em&gt;Why use Einstein Summation Notation?&lt;/em&gt;, URL (version: 2017-04-13): &lt;a href=&quot;https://math.stackexchange.com/q/1926173&quot;&gt;https://math.stackexchange.com/q/1926173&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that Einstein Summation Notation &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;does not only have an 'acceleration' effect&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;but also &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;has one of those '&lt;em&gt;limit the language&lt;/em&gt;' effects.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="6060" LastEditDate="2017-10-02T10:56:08.380" LastActivityDate="2017-10-02T10:56:08.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6558" PostTypeId="1" AcceptedAnswerId="6568" CreationDate="2017-09-30T18:36:22.773" Score="3" ViewCount="353" Body="&lt;p&gt;As far as I understand, the Kaluza-Klein theory, despite its unprecedentedly profound and beautiful character, had a modest following in its early days. I guess that two of the many reasons might be the lack of additional experimental predictions by the theory and the reluctance towards accepting the possibility of additional spatial dimensions. Anyway, the idea of additional spatial dimensions, although in a somewhat different context, was revived by the rise of String Theory in the 1980s and today, we know the Kaluza-Klein theory is mentioned as Kaluza-Klein Miracle at many places (including my GR classes :P). I wonder if the rise of String Theory made the old Kaluza-Klein idea a sensation. Or was Kaluza-Klein theory equally appreciated and loved even before? &lt;/p&gt;&#xA;" OwnerUserId="4257" LastEditorUserId="4257" LastEditDate="2018-12-06T00:21:12.197" LastActivityDate="2018-12-06T00:21:12.197" Title="When Was Kaluza-Klein Theory Appreciated?" Tags="|theoretical-physics|relativity-theory|electromagnetism|mathematical-physics|string-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6559" PostTypeId="1" CreationDate="2017-10-01T16:44:56.457" Score="2" ViewCount="91" Body="&lt;p&gt;In lambda calculus there are three types of reduction,&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;$\alpha$-renaming&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\beta$-reduction&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;$\eta$-conversion&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The use of $\eta$ in $\eta$-conversion seems rather strange to me. Since they already used $\alpha$ and $\beta$ I would expect the pattern to be continued with $\gamma$ the third Greek letter, but $\eta$ is used instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is $\eta$ used in $\eta$-conversion?&lt;/p&gt;&#xA;" OwnerUserId="6372" LastActivityDate="2017-10-01T16:44:56.457" Title="Why is $\eta$ used in $\eta$-conversion?" Tags="|terminology|computer-science|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6560" PostTypeId="1" AcceptedAnswerId="6565" CreationDate="2017-10-02T13:21:41.847" Score="10" ViewCount="875" Body="&lt;p&gt;Galois theory occupies a rather central place in modern number theory, from class field theory, to the Langlands program, to the ideas found in Grothendieck's Esquisse d'un Programme. But the Wikipedia article on Galois theory says that the mathematical community of the 19th century was not immediately very receptive to Galois theory. The Kronecker-Weber theorem involves abelian extensions, so would I be correct in assuming that Galois theory had already been adopted by the time of Kronecker? But anyway, my main question is, when did Galois theory start to become so central to number theory?&lt;/p&gt;&#xA;" OwnerUserId="6373" LastActivityDate="2018-03-10T16:53:50.840" Title="History of Galois Theory after Galois" Tags="|number-theory|galois-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6561" PostTypeId="1" CreationDate="2017-10-02T15:18:42.870" Score="0" ViewCount="82" Body="&lt;p&gt;On &lt;a href=&quot;https://en.wikipedia.org/wiki/LaSalle&amp;#39;s_invariance_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; this article about the invariance principle and article states that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The general result was independently discovered by J.P. LaSalle (then&#xA;  at RIAS) and N.N. Krasovskii, who published in 1960 and 1959&#xA;  respectively. While LaSalle was the first author in the West to&#xA;  publish the general theorem in 1960, a special case of the theorem was&#xA;  in communicated in 1952 by Barbashin and Krasovskii, followed by a&#xA;  publication of the general result in 1959 by Krasovskii.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hence, I am wondering if this claim is correct (references please I couldn't find the reference form 1959).&lt;/p&gt;&#xA;" OwnerUserId="3966" LastEditorUserId="14365" LastEditDate="2022-03-20T13:46:27.677" LastActivityDate="2022-03-20T13:46:27.677" Title="Invariance principle for stability in the sense of Lyapunov" Tags="|mathematics|reference-request|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6562" PostTypeId="1" CreationDate="2017-09-07T13:37:55.413" Score="4" ViewCount="10952" Body="&lt;p&gt;Why were the &lt;a href=&quot;https://en.wikipedia.org/wiki/Metric_prefix&quot; rel=&quot;nofollow noreferrer&quot;&gt;SI unit prefixes&lt;/a&gt;, i.e.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;\begin{align}&#xA;\mathrm{giga}  &amp;amp;&amp;amp;   10^9 \\&#xA;\mathrm{mega}  &amp;amp;&amp;amp;  10^6 \\&#xA;\mathrm{kilo}  &amp;amp;&amp;amp;  10^3 \\&#xA;\mathrm{milli} &amp;amp;&amp;amp;  10^{-3} \\&#xA;\mathrm{micro} &amp;amp;&amp;amp;  10^{-6} \\&#xA;\mathrm{nano}  &amp;amp;&amp;amp;  10^{-9}&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;chosen to be a multiple power of 3?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: Although this question stemmed from a physics based discussion, I'm not sure which stack exchange would be better suited for this question.&lt;/p&gt;&#xA;" OwnerUserId="6267" OwnerDisplayName="crezefire" LastActivityDate="2022-12-21T18:29:36.637" Title="Why were the SI-unit prefixes chosen to be a multiple power of 3?" Tags="|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6563" PostTypeId="2" ParentId="6562" CreationDate="2017-10-02T23:31:12.120" Score="1" Body="&lt;p&gt;I'd have to say because the SI prefixes correspond to the standard number grouping, i.e.:&lt;ul&gt;&lt;li&gt;kilo = thousand&lt;/li&gt;&lt;li&gt;mega = million&lt;/li&gt;&lt;li&gt;giga = billion&lt;/li&gt;&lt;li&gt;tera = trillion&lt;/li&gt;&lt;/ul&gt;&lt;br&gt;Here's the complete list:&lt;br&gt;&lt;a href=&quot;https://i.stack.imgur.com/cRgP4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cRgP4.jpg&quot; alt=&quot;SI Prefixes&quot;&gt;&lt;/a&gt;&lt;br&gt;Not to mention, an SI prefix for every ten-fold increase would be hugely cumbersome. Can you imagine a prefix for 10,000, 100,000, 10,000,000; 100,000,000...?&lt;/p&gt;&#xA;" OwnerUserId="6376" LastEditorUserId="6376" LastEditDate="2017-10-03T18:46:24.357" LastActivityDate="2017-10-03T18:46:24.357" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6564" PostTypeId="2" ParentId="6561" CreationDate="2017-10-03T02:51:45.210" Score="1" Body="&lt;p&gt;MR0106313 Krasovskiĭ, N. N. {\cyr Nekotorye zadachi teorii ustoĭchivosti dvizheniya.} (Russian) [Certain problems in the theory of stability of motion] Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow 1959 211 pp.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0052616 Barbašin, E. A.; Krasovskiĭ, N. N. On stability of motion in the large. (Russian) Doklady Akad. Nauk SSSR (N.S.) 86, (1952). 453-456.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-03T02:51:45.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6565" PostTypeId="2" ParentId="6560" CreationDate="2017-10-03T07:49:53.090" Score="12" Body="&lt;p&gt;I suggest that you reed B. Melvin Kiernan's &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF00327219.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Development of Galois Theory from Lagrange to Artin&lt;/em&gt;&lt;/a&gt;. In particular, it says there that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the 1890's a few noteworthy expositions of GALOIS Theory were published, perhaps stimulated by the appearance of a German translation of GALOIS' works, in 1889, and the reprinting in book form, in 1897, of GALOlS' papers &#xA;  already published. Certainly the most innovative of these expositions were the two presentations by HEINRICI WEBER (1842–1913), an article, &amp;ldquo;Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie,&amp;rdquo; published in 1893, and the development in his &lt;em&gt;Lehrbueh der Algebra&lt;/em&gt;, published in 1895.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that Kronecker died in 1891. The expositions of Galois theory by Weber were published only after the death of Kronecker. Therefore, it is unlikely that “Galois theory had already been adopted by the time of Kronecker”&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="62" LastEditDate="2018-03-10T16:53:50.840" LastActivityDate="2018-03-10T16:53:50.840" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6566" PostTypeId="2" ParentId="6562" CreationDate="2017-10-03T16:56:27.850" Score="1" Body="&lt;p&gt;Because of a tradition that goes back to ancient Greece. The greek alphabet was also used to denote numerals with just a little tweaking. Three archaic letters were used to have a total of 27 signs, so the first nine leters were 1..9, the next nine 10,20..90 and the last 100 to 900. This is known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sampi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miletsian system&lt;/a&gt;. It is obvious that after 999 some kind of loop should be implemented but there was not a universally agreed way to do it. Apparently the archaic latter 'sampi' was often used as a factor for thousands and is this usage is called 'numeric sampi' (there  was some confusion as it could also be just 900).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For refs and details see the wikipedia article Sampi &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-10-03T16:56:27.850" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6567" PostTypeId="2" ParentId="6558" CreationDate="2017-10-04T17:22:38.540" Score="3" Body="&lt;p&gt;Here is a &lt;a href=&quot;https://arxiv.org/abs/hep-ph/9810524&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; that features the early history of the Kaluza-Klein theory. It seems that one of its earliest supporters was &lt;a href=&quot;http://onlinelibrary.wiley.com/wol1/doi/10.1002/andp.19334100307/abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pauli&lt;/a&gt;.&#xA;When Victor Weisskopf became Pauli's assistant in 1933, he told him that he would work on anything but the Kaluza-Klein theory, claiming he did not understand it: &lt;a href=&quot;https://books.google.de/books?id=LdMVAwAAQBAJ&amp;amp;pg=PA281&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.de/books?id=LdMVAwAAQBAJ&amp;amp;pg=PA281&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2017-10-04T17:22:38.540" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6568" PostTypeId="2" ParentId="6558" CreationDate="2017-10-04T22:29:53.833" Score="3" Body="&lt;p&gt;The answer depends on the meaning of &quot;appreciated&quot;. One could say that it was appreciated from the start, Einstein commented on the idea positively back in 1919, and in 1921 he presented Kaluza's paper “Zum Unit ̈atsproblem der Physik&quot; to the Prussian Academy. The reason Kaluza-Klein theory did not receive the kind of recognition it would have received in 19-th century is that in 1920s the focus was on quantum mechanics, in 1930-40s on quantum electrodynamics, and later on quantum gauge theories, emphasis on quantum. The classical unification programme, to which Kaluza-Klein was the most successful entry after Maxwell,  however dear to Einstein's heart, came to be seen as outdated. If there was a hope of unification it head to unify quantized theories, and Kaluza-Klein was not that. Pauli's evolution on the issue is characteristic, he supported Weyl's 1918 conformal unification theory, but by 1920-21 he sounded a different tune:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I would like to ask Einstein if he agrees that one can only expect a solution to the problem of matter by a modification of our ideas about space (and perhaps also time) and electric fields in the sense of atomism...or should one hold on to the rudiments of the continuum theories? [...] New elements foreign to the continuum must be added to the basic structure of the theory before one can arrive at a satisfactory solution of the problem of matter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is pretty much the majority position today. Sure enough, Einstein, given his attitude towards quantum mechanics, was unimpressed. In fact, he intensively studied Kaluza-Klein theory in 1938-1943 in hopes of finding an alternative to quantum mechanics, see &lt;a href=&quot;https://arxiv.org/abs/gr-qc/0009087&quot; rel=&quot;nofollow noreferrer&quot;&gt;van Dongen's Einstein and the Kaluza-Klein particle&lt;/a&gt;. That did not work out.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The revival of interest in the theory was due to Witt's 1964 observation that by adding more dimensions one could unify gravity with non-Abelian gauge theories. Of course, the unification was still classical, but the idea took off and was indeed boosted by extra dimensions in the nascent string theory and later supergravity. &lt;a href=&quot;https://inspirehep.net/record/199895/files/v13-n4-p171.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chodos's 1984 overview of Kaluza-Klein theories&lt;/a&gt; describes the situation in the early 80-s, which one could call their second appreciation: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In the late seventies, spurred by recent progress in higher-dimensional supergravity theories, and by the fact that the grand-unified scale was only a few orders of magnitude above the Planck length, the Kaluza-Klein idea suddenly acquired widespread notoriety, with the result that by now the diligent researcher who wishes to become acquainted with the field is faced with a reading list of several hundred papers, all written within the last few years.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-10-04T23:16:13.853" LastActivityDate="2017-10-04T23:16:13.853" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6569" PostTypeId="2" ParentId="6545" CreationDate="2017-10-05T09:28:39.427" Score="2" Body="&lt;p&gt;The introduction (in Europe) of decimal fractions by Simon Stevin in 1585 and the simplification in its notation introduced by John Neper a few decades later were very important, because sudenly a much larger number of people were able to do numerical computations. This had a large impact in scientific research.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-10-05T09:28:39.427" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6570" PostTypeId="1" CreationDate="2017-10-05T14:16:53.173" Score="1" ViewCount="231" Body="&lt;p&gt;It seems that functional analysis is greatly motivated by quantum mechanics, right? Is there any reference on the relation between the two? &lt;/p&gt;&#xA;" OwnerUserId="2641" LastEditorUserId="5989" LastEditDate="2017-10-10T18:03:22.183" LastActivityDate="2017-10-10T18:03:22.183" Title="Mathematics motivated by quantum mechanics" Tags="|mathematics|quantum-mechanics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6571" PostTypeId="2" ParentId="6570" CreationDate="2017-10-05T15:40:02.693" Score="5" Body="&lt;p&gt;Read Jean Dieudonné's &lt;em&gt;History of Functional Analysis&lt;/em&gt;. More precisely, read &lt;a href=&quot;https://books.google.de/books?id=mg7r4acKgq0C&amp;amp;pg=PA171&quot; rel=&quot;noreferrer&quot;&gt;§VII.4: &lt;em&gt;The spectral theory of von Neumann&lt;/em&gt;&lt;/a&gt;. Chapter I of von Neumann's &lt;em&gt;Mathematical foundations of Quantum Mechanics&lt;/em&gt; is very interesting too.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5376" LastEditDate="2017-10-05T15:53:21.303" LastActivityDate="2017-10-05T15:53:21.303" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6572" PostTypeId="2" ParentId="6570" CreationDate="2017-10-05T15:48:04.873" Score="5" Body="&lt;p&gt;The main contribution of quantum mechanics to functional analysis is the spectral theory of unbounded operators developed by &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002273535&quot; rel=&quot;noreferrer&quot;&gt;John von Neumann&lt;/a&gt; (with a little help from &lt;a href=&quot;https://books.google.de/books?id=neimAAAAIAAJ&amp;amp;dq=%22e+schmidt+und+der+verf%22+existenz&quot; rel=&quot;noreferrer&quot;&gt;Erhard Schmidt&lt;/a&gt;) and &lt;a href=&quot;http://bookstore.ams.org/coll-15&quot; rel=&quot;noreferrer&quot;&gt;Marshall Stone&lt;/a&gt;. See this article by &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/0315086084900363&quot; rel=&quot;noreferrer&quot;&gt;Kreyszig and Birkhoff&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2017-10-05T15:48:04.873" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6573" PostTypeId="2" ParentId="6570" CreationDate="2017-10-05T23:15:25.423" Score="5" Body="&lt;p&gt;I also recommend Dieudonne (following Jose Carlos Santos) for a brief historical account, and also von Neumann's own classic book Mathematical foundations of quantum mechanics. And volumes&#xA;by Reed and Simon, Methods of Mathematical Physics., if you want some later development. The influence was (and is) very strong, in particular the theory of unbounded operators was developed mostly to satisfy the demands of quantum mechanics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-05T23:15:25.423" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6577" PostTypeId="1" AcceptedAnswerId="6579" CreationDate="2017-10-06T23:52:48.257" Score="3" ViewCount="496" Body="&lt;p&gt;I noticed there are two varieties of Maxwell's equations and although I am not sure which form Maxwell originally worked with and perhaps the differential form came much latter, I was wondering if he made use of Green's Theorem. I ask because after reading about Green , I got the impression he was motivated to develop his theorem for trying to understand electrical behavior.  I looked at Wiki and James Clerk Maxwell lived from 1831 to 1879 and  George Green from 1793 to 1841. Certainly Maxwell was old enough to have read about Green's work before Green passed away. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-06-06T12:58:28.563" LastActivityDate="2020-06-06T12:58:28.563" Title="Is it thought that Maxwell used Green's Theorem to derive his equations?" Tags="|mathematics|physics|electromagnetism|maxwell|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6578" PostTypeId="2" ParentId="6545" CreationDate="2017-10-08T00:24:47.720" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Variable_(mathematics)&quot; rel=&quot;noreferrer&quot;&gt;Variables&lt;/a&gt;; &lt;a href=&quot;https://hsm.stackexchange.com/questions/6104/who-first-considered-the-f-in-fx-as-an-object-in-itself-and-who-decided-t&quot;&gt;function notation&lt;/a&gt;; &lt;a href=&quot;https://mathoverflow.net/questions/156075/history-of-calculus-of-several-variables/156133#156133&quot;&gt;partial derivatives&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_Analysis&quot; rel=&quot;noreferrer&quot;&gt;vector calculus&lt;/a&gt;; &lt;a href=&quot;https://mathoverflow.net/questions/264013/etymology-of-exterior-in-exterior-calculus&quot;&gt;exterior&lt;/a&gt; &lt;a href=&quot;https://mathoverflow.net/questions/37301/why-did-the-word-exterior-get-chosen-for-the-idea-of-exterior-derivative&quot;&gt;calculus&lt;/a&gt;; &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Matrices_and_determinants.html&quot; rel=&quot;noreferrer&quot;&gt;matrices&lt;/a&gt;; &lt;a href=&quot;https://mathoverflow.net/questions/75765/who-coined-the-name-tensor-and-why&quot;&gt;tensors&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Operational_calculus&quot; rel=&quot;noreferrer&quot;&gt;operational calculus&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirac_delta_function&quot; rel=&quot;noreferrer&quot;&gt;delta function&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmetices_principia,_nova_methodo_exposita&quot; rel=&quot;noreferrer&quot;&gt;set operations&lt;/a&gt;; &lt;a href=&quot;https://mathoverflow.net/questions/135347/who-introduced-the-terms-equivalence-relation-and-equivalence-class&quot;&gt;equivalence relations&lt;/a&gt;;&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/194377/when-was-the-arrow-notation-for-functions-first-introduced&quot;&gt;mapping arrows&lt;/a&gt;; &lt;a href=&quot;https://mathoverflow.net/questions/160183/origin-of-exact-sequences&quot;&gt;exact sequences&lt;/a&gt;; &lt;a href=&quot;https://mathoverflow.net/questions/59456/whence-commutative-diagrams&quot;&gt;commutative diagrams&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Feynman_diagram&quot; rel=&quot;noreferrer&quot;&gt;Feynman diagrams&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Abstract_index_notation&quot; rel=&quot;noreferrer&quot;&gt;abstract index notation&lt;/a&gt;; &lt;a href=&quot;https://en.wikipedia.org/wiki/Infinity_symbol&quot; rel=&quot;noreferrer&quot;&gt;$\infty$&lt;/a&gt;; &lt;a href=&quot;https://hsm.stackexchange.com/questions/546/what-was-eulers-motivation-for-introducing-i-for-sqrt-1&quot;&gt;$i$&lt;/a&gt;; ...too broad!&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-10-08T00:24:47.720" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6579" PostTypeId="2" ParentId="6577" CreationDate="2017-10-08T13:12:03.420" Score="8" Body="&lt;p&gt;Of course Maxwell knew Green's theorem, by the time he was writing this was the common knowledge. Maxwell's book has a mathematical preliminary chapter (chapter 2) where he explains mathematical tools he uses, and this contains Gauss, Green, Stokes theorems and much more. (In fact he anticipates what was later called Hodge theory). In the chapter where Maxwell equations are derived, (Chap 9, vol. 2) he freely passes from differential to integral formulations and back.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Green's book (1828) remained in obscurity for some time, until William Thomson&#xA;(future Lord Kelvin) found it in 1845 and made his results widely known.&#xA;And yes, Green's book was called  &quot;An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism&quot;, so it is about electricity and magnetism.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-08T13:12:03.420" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6580" PostTypeId="1" CreationDate="2017-10-11T16:54:10.167" Score="6" ViewCount="2889" Body="&lt;p&gt;I want to understand the extent to which scientists and technicians employed in the Manhattan project understood that they were building a bomb that would kill hundreds of thousands of people. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia has this to say:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;An example of compartmentalization was the Manhattan Project. Personnel at Oak Ridge constructed and operated centrifuges to isolate Uranium-235 from naturally occurring uranium, but most did not know what, exactly, they were doing. Those that did know, did not know why they were doing it. Parts of the weapon were separately designed by teams who did not know how the parts interacted.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;( &lt;a href=&quot;https://en.wikipedia.org/wiki/Compartmentalization_(information_security)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Compartmentalization_(information_security)&lt;/a&gt; )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously some top scientists knew that they were building a bomb. For example, I know (from hearsay) that von Neumann and Ulam (and some others?) developed Monte-Carlo methods in order to analyze when the reaction would reach a critical point, and I also know (from hearsay) that there was a general fear that Germans would build the atom bomb first. (Though it seems like this fear was unfounded.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also interested in learning about how they were propagandized. (I guess that the fear of Germans building a bomb is part of this.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone recommend some resources? &lt;/p&gt;&#xA;" OwnerUserId="3545" LastActivityDate="2017-10-13T21:52:53.877" Title="To what extent were the scientists involved in the Manhattan project aware of the goals?" Tags="|atomic-theory|nuclear-physics|world-war-2|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6581" PostTypeId="2" ParentId="6580" CreationDate="2017-10-11T22:26:44.563" Score="8" Body="&lt;p&gt;The sources are abundant and easy to find, for example&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leslie Groves, Now it can be told. A history of Manhattan project (written by a project leader from the military side),&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Robert Jungk, Brighter than a thousand suns, (written by a journalist).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides this there are literally hundreds of books, including many participants memoirs, describing in great detail all aspects of the program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You are essentially right in your conjectures: high level participants knew its goal (to build an atomic bomb). Lover level were not told the goal, but I suppose many could easily guess, people were talking to each other after all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking of the original motivation, you are also essentially right: the main motivation (at least for scientists) was the fear that Germany could do this.&#xA;(The project started with a letter of Einstein to Roosevelt citing exactly this motivation). Later it became clear that Germany is not likely to build a bomb but this was a classified information, and nobody could know for sure. Defeating the Japanese quickly was also an important motivation for many. When the bomb was finally made, the government and scientists still were not sure what to do with it, several options were discussed. All this information is easily available in many books, including the books I mentioned.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-11T22:26:44.563" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6582" PostTypeId="1" AcceptedAnswerId="6583" CreationDate="2017-10-11T23:34:36.180" Score="9" ViewCount="330" Body="&lt;p&gt;I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead you to homology. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But how does one motivate/understand the expansion into other areas of mathematics? From the geometric/topological point of view simplices seem to be of fundamental importance. Yet, in all other homological theories (groups, sheaves etc.) I don't see simplices. Groups are not broken down into simplices for instance. All I see are these chains of morphisms; how did they take center stage? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could someone please comment on how it might seem natural to arrive at these different homologies? Or point me in the direction of a reference which discusses how group homology arose from the more geometric simplicial/singular homology theories?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="2958" LastActivityDate="2017-10-12T08:46:54.033" Title="(Co)Homology: From topology to the rest of mathematics?" Tags="|topology|abstract-algebra|algebraic-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6583" PostTypeId="2" ParentId="6582" CreationDate="2017-10-12T01:41:05.743" Score="10" Body="&lt;p&gt;I'd recommend Weibel’s &lt;em&gt;History of homological algebra&lt;/em&gt; (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1721123&quot; rel=&quot;noreferrer&quot;&gt;1999&lt;/a&gt;)(&lt;a href=&quot;http://sites.math.rutgers.edu/~weibel/&quot; rel=&quot;noreferrer&quot;&gt;pdf&lt;/a&gt;). He describes many threads, such as roots of group cohomology in Hurewicz’s observation that cohomology of an aspherical space $Y$ depends only on (what we now call cohomology of) its fundamental group $\pi=\pi_1(Y)$:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Since] homology and cohomology groups of $Y$ (with coefficients in $A$) were independent of the choice of $Y$, Eilenberg and Mac Lane took them as the definition of $H_n(\pi; A)$ and $H^n(\pi; A)$. To perform computations, Eilenberg and Mac Lane chose a specific abstract simplicial complex $K(\pi)$ for the aspherical space $Y$. (...) Thus one way to calculate the cohomology groups of $\pi$ was to use the cellular cochain complex of $K(\pi)$, whose $n$-chains are functions $f : \pi^q \to A$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the simplices are not to be found in $\pi$ but in $K(\pi)$. Another crucial thread is well summarized in the first lines of Chevalley-Eilenberg (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=24908&quot; rel=&quot;noreferrer&quot;&gt;1948&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[We] give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions concerning Lie algebras. This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham’s theorems (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the consideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-10-12T08:38:47.430" LastActivityDate="2017-10-12T08:38:47.430" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6584" PostTypeId="2" ParentId="6582" CreationDate="2017-10-12T08:46:54.033" Score="7" Body="&lt;p&gt;As a complement to the answer provided by Francois Ziegler, I would add the first three paragraphs of &lt;em&gt;Homological Algebra&lt;/em&gt; (1956), by Henri Cartan and Samuel Eilenberg:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to present a unified account of these developments and to lay the foundations of a full-fledged theory.&lt;/p&gt;&#xA;&lt;p&gt;The invasion of algebra has occurred on three fronts through the&#xA;construction of cohomology theories for groups, Lie algebras, and&#xA;associative algebras. The three subjects have been given independent&#xA;but parallel developments. We present herein a single cohomology (and&#xA;also a homology) theory which embodies all three; each is obtained&#xA;from it by a suitable specialization.&lt;/p&gt;&#xA;&lt;p&gt;This unification possesses all the usual advantages. One proof replaces three. In addition an&#xA;interplay takes place among the three specializations; each enriches&#xA;the other two.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-10-12T08:46:54.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6585" PostTypeId="2" ParentId="6580" CreationDate="2017-10-12T22:24:01.883" Score="4" Body="&lt;p&gt;New scientific and engineering personal arriving at Los Alamos were given a series of lectures about the project, the relevant physics and the proposed designs of the weapons.  These were later published as the Los Alamos Primer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So at least in Los Alamos, the purpose of the bomb was understood by all the technical personnel.  Indeed, the ability to do so securely was part of the purpose of that site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;David Greenglass (the spy that would give information to the Rosenbergs) was just an enlisted machinist, and wasn't formally informed of what the project was on, but was able to figure it out from gossip.  So even amongst the lower level employees, who were in theory kept in the dark, the purpose of the project seems to have been an open secret.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman, who was sent to Oak Ridge to oversee safety concerns regarding uranium storage, claims in one of his books that even the higher-ups were only vaguely aware that they were working on some sort of bomb, and the rest of the staff new nothing.  Though his stories are obviously meant mainly as amusing anecdotes or morality tales, so I'm not sure how great a source he is for factual info.   &lt;/p&gt;&#xA;" OwnerUserId="1731" LastActivityDate="2017-10-12T22:24:01.883" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6586" PostTypeId="2" ParentId="6580" CreationDate="2017-10-13T12:33:22.107" Score="4" Body="&lt;p&gt;The website &lt;a href=&quot;http://manhattanprojectvoices.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://manhattanprojectvoices.org&lt;/a&gt; is full of interviews with participants in the Manhattan project. Some interviews address exactly the question asked here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, here are two interviews with members of what was called the &quot;Special Engineering Detachment&quot; in which the interviewed described quite specifically at what moment they were told they were working on a bomb:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://manhattanprojectvoices.org/oral-histories/robert-js-browns-interview&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://manhattanprojectvoices.org/oral-histories/robert-js-browns-interview&lt;/a&gt;&#xA;(he is asked &quot;When did you really know that you were working on an atomic bomb?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://manhattanprojectvoices.org/oral-histories/benjamin-bedersons-interview&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://manhattanprojectvoices.org/oral-histories/benjamin-bedersons-interview&lt;/a&gt;&#xA;(see where he is asked &quot;That's great. So, when did you work for George Kistiakowsky?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bederson states &quot;We'd hear all these stories about need to know and about how everything at Los Alamos was compartmentalized—that was nonsense! Within three months of my getting my clearance as a PFC [private first class] at the time, I was told this immense secret without any hesitation by Professor Kistiakowsky. Of course, I could never forget the feeling. But, he was a very interesting guy. He certainly put it across to these low level individuals that he spoke to.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(However, this is not a random sample. Both of these men worked in the group directed by George Kistiakowsky, and perhaps Kistiakowsky was more open than some.)&lt;/p&gt;&#xA;" OwnerUserId="633" LastEditorUserId="633" LastEditDate="2017-10-13T17:48:40.993" LastActivityDate="2017-10-13T17:48:40.993" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6587" PostTypeId="2" ParentId="3131" CreationDate="2017-10-13T16:18:52.847" Score="9" Body="&lt;p&gt;Vladimir Voevodsky died on September 30, 2017 at the age of 51. He won the Fields medal in 2002 (due to the proof of the Milnor conjecture)&lt;/p&gt;&#xA;" OwnerUserId="6049" LastActivityDate="2017-10-13T16:18:52.847" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6588" PostTypeId="2" ParentId="6580" CreationDate="2017-10-13T21:52:53.877" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I want to understand the extent to which scientists and technicians employed in the Manhattan project understood that they were building a bomb that would kill hundreds of thousands of people. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are two separate questions here: (1) whether they knew it was a bomb, and (2) whether they knew it would be used to kill large numbers of civilians. Even the scientists at the highest level had no way of knowing that the bombs would be dropped on cities full of civilians. The bombs could have been used as demonstrations, off shore, or to attack military targets rather than population centers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even Truman claimed, in his July 25, 1945 diary entry, that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I have told [Stimson] to use it so that military objectives and soldiers and sailors are the target and not women and children. [...] The target will be a purely military one ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And in his August 9 radio address, he said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The world will note that the first atomic bomb was dropped on Hiroshima, a military base. That was because we wished in this first attack to avoid, insofar as possible, the killing of civilians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's unclear (to me) whether Truman was really ignorant of the fact that Hiroshima was a city. But in any case, I think this shows pretty vividly that there was no way for scientists working on the project to know with any certainty that the bomb would be used to annihilate civilian populations. If Truman's statements are taken at face value, even he didn't know that.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-10-13T21:52:53.877" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6589" PostTypeId="1" AcceptedAnswerId="6594" CreationDate="2017-10-14T02:56:35.213" Score="6" ViewCount="898" Body="&lt;p&gt;The neutron was predicted in 1920 and discovered 12 years later. Before this time, how did scientists explain standard atomic weights?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Lithium has a standard atomic weight of about 6.9 because the natural abundance of the isotopes is about 92% Li-7 and 8% Li-6. Another example is Iron with 92% Fe-56, 6% Fe-54, and 2% Fe-57 for a SAW of 55.7.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But they did not know the isotopes. They only knew the SAWs, and that some of them were not integer multiples of Hydrogen (the proton). How did they explain this before the neutron was discovered?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I included the chemistry tag just in case this was considered a chemistry thing back in the early 1900's.)&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2017-10-15T19:57:37.317" Title="Before the discovery of the neutron, how did scientists explain standard atomic weights?" Tags="|chemistry|atomic-theory|nuclear-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6590" PostTypeId="1" AcceptedAnswerId="6595" CreationDate="2017-10-14T05:20:33.407" Score="4" ViewCount="203" Body="&lt;p&gt;I was reading Russell's &lt;em&gt;An Inquiry into Meaning and Truth&lt;/em&gt; (1940) and I noted the following passage in the Introduction,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finally, there is the question of the relation between truth and knowledge. &lt;strong&gt;Attempts have been made to define “truth” in terms of “knowledge”, or of concepts, such as “verifiability”, which involve “knowledge”. Such attempts, if carried out logically, lead to paradoxes which there is no reason to accept.&lt;/strong&gt; I conclude that “truth” is the fundamental concept, and that “ knowledge” must be defined in terms of “truth”, not vice versa. T his entails the consequence that a proposition may be true although we can see no way of obtaining evidence either for or against it. It involves also a partial abandonment of the complete metaphysical agnosticism that is favoured by the logical positivists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;My Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know which paradoxes Russell is referring to here? If so, can some references be cited?&lt;/p&gt;&#xA;" OwnerDisplayName="user459" LastEditorUserId="4251" LastEditDate="2021-09-10T14:46:51.087" LastActivityDate="2021-09-10T14:46:51.087" Title="What are the paradoxes of which Russell is talking about in the following?" Tags="|philosophy-of-science|logic|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6591" PostTypeId="1" CreationDate="2017-10-14T14:52:54.583" Score="2" ViewCount="2568" Body="&lt;p&gt;A 1929 paper of Chittenden contains the following sentence (about the derived set operator on a space $P$):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;ldquo;Thus the relation $E' = K(E)$ defines a single-valued set-valued set-function, whose range is the class of all subsets of $P,$ and whose values are also subsets of $P.$&amp;rdquo;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since he used &lt;em&gt;range&lt;/em&gt; to mean &lt;em&gt;domain&lt;/em&gt; (in current terminology), it made me wonder when the currently used terms became the norm.  It evidently wasn't before 1929.&lt;/p&gt;&#xA;" OwnerUserId="6269" LastActivityDate="2017-10-15T09:14:26.400" Title="origin of the terms &quot;domain&quot; and &quot;range&quot;?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6594" PostTypeId="2" ParentId="6589" CreationDate="2017-10-15T19:57:37.317" Score="5" Body="&lt;p&gt;Historically, the discovery of naturally-occurring isotopes of chemical elements by means of the mass-spectrograph provided a correct explanation for the fractional character of the  experimentally-measured values for the atomic weights of many chemical elements -- as compared with values that might have been expected if the atoms of a given element had been all alike as integer multiples of the atomic weight of hydrogen, or (later) as integers on a scale obtained by defining the atomic weight of oxygen O = 16. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, Francis W Aston (1877-1945) discovered, by means of his mass-spectrograph (1919), many &lt;em&gt;isotopes&lt;/em&gt; -- a designation coined with a Greek etymology for 'equal place', and signifying chemical elements occupying the &lt;em&gt;same place&lt;/em&gt; in the periodic table but with atoms that differ in atomic weight: Aston was later awarded a Nobel Prize for his discoveries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aston's book 'Isotopes' (1922) (&lt;a href=&quot;https://archive.org/details/isotopes00asto&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/isotopes00asto&lt;/a&gt;) gives a short history of the measurement of atomic weights and of attempted explanations of these weights, starting with the hypotheses of Dalton (1803) and of Prout (about ten years after). Dalton's far-reaching atomic theory or hypothesis included the idea that all atoms of a given element were alike: and Prout's hypothesis had been that all atoms are aggregations of atoms of hydrogen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on it became clear, with increasing accuracy of measurement, that many atomic weights could not possibly represent whole-number aggregates of hydrogen, and Dalton's hypothesis (so far as it related to the weights of atoms) and Prout's could not both be true. The weight of chemical opinion at that point, Aston tells us, was to give up Prout's idea and to prefer Dalton's.  In retrospect, and with awareness of the hindsight of discovery in respect of isotopes, Aston labeled that decision as &quot;as wise in principle as it was wrong in fact&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Aston (1922): &quot;The idea that atoms of the same element are are identical in weight could not be challenged by chemical methods, for the atoms are by definition chemically identical and numerical ratios were only to be obtained in such methods by the use of quantities of the element containing countless myriads of atoms. At the same time it is rather surprising, when we consider the complete absence of positive evidence in its support that no theoretical doubts were publicly expressed until late in the nineteenth century, first by Schutzenberger and then by Crookes, and that these doubts have been regarded, even up to the last few years, as speculative in the highest degree.&quot;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was a further attempt at explaining among other phenomena the diversity of atomic weights, William Crookes' idea (1886) of &lt;em&gt;meta-elements&lt;/em&gt;. Crookes reached the idea that chemically inseparable elements might consist of a variety of atoms. This idea seemed at first to apply to the results of Crookes' study of yttria (oxide of yttrium) -- &quot;But as more and more refined chemical methods were applied, the rare earths one after another yielded to analysis and the different spectra observed by Crookes were shown to be due to the fact that he was dealing with a mixture of real elements, each of which had a characteristic spectrum and a definite atomic weight. The theory of meta-elements was therefore abandoned and the problem of fractional atomic weight remained unsolved.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Accordingly, it seems that there was essentially no viable explanation of the well-known fractional atomic weights until the mass-spectrographic discovery of isotopes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the fact that many standard atomic weights are not close to integers on a scale where hydrogen=1, or any closely similar scale, was explained (satisfactorily), before knowledge of the neutron, by the discovery that isotopes of the same chemical element actually exist with different atomic weights and a variety of natural relative abundances.  It can also be seen from Aston's work that the neutron had essentially no role in this discovery.  For example, the existence of chlorine-35 and chlorine-37 with about 3:1 relative natural abundance was shown by the mass-spectrographic work, and it explains the accepted value of about 35.5 for the atomic weight of chlorine.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(There is naturally an onward question &lt;em&gt;how&lt;/em&gt; such isotopes could exist, but that matter is separate from the question answered by experimental demonstration that such isotopes exist in fact.  The onward question 'how' involves among other things the distinction between atomic weight and the atomic numbers discovered experimentally in 1913 by H G Moseley, as well as the evolution of theories about atomic structure, theories that came to include, of course, the neutron.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-10-15T19:57:37.317" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6595" PostTypeId="2" ParentId="6590" CreationDate="2017-10-15T21:26:03.957" Score="2" Body="&lt;p&gt;Eames discusses this issue in &lt;a href=&quot;https://books.google.com/books?id=h-J0e9TOI0YC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bertrand Russell's Theory of Knowledge, pp.151-154&lt;/a&gt;. The targets are Dewey's theory of warranted assertibility and Reichenbach's theory of verifiability, which both tie truth to what is or can be known. Russell's main objections are that truth must provide a goal for knowledge, and therefore must be conceived as wider than knowledge. What Russell calls &quot;paradoxes&quot; are the consequences that the law of excluded middle would have to be abandoned for a number of claims if one identifies truth with knowable truth. His example is &quot;&lt;em&gt;It snowed on Manhattan island in the year 1 AD&lt;/em&gt;&quot; being either true or false regardless of the fact that no evidence may ever be found one way or the other. Even the proposition &quot;&lt;em&gt;Caesar conquered Gaul&lt;/em&gt;&quot;, which we can verify based on historical documents, is not true in lieu of such documents but in lieu of the event that occurred in the past. To this Russell adds a moral criticism of pragmatists and positivists that by restricting truth they removed it as a goal from beyond inquiry towards which the search for knowledge is directed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Russell was of course aware that the law of excluded middle was rejected by intuitionists, hence his oblique admission that the &quot;paradoxes&quot; can be accepted. Their rejection was, however, largely restricted to mathematics, and logical positivists, at least, did not abandon classical logic overtly. But even Carnap, a positivist logicist, did not accept Frege-Russell's universalism about logic (no object/meta language separation), so they could accept it as a law of the object language while admitting existence of undecidable statements. An extension of intuitionism to philosophy of science was later developed by Dummett under the name of &lt;a href=&quot;https://en.wikipedia.org/wiki/Anti-realism&quot; rel=&quot;nofollow noreferrer&quot;&gt;anti-realism&lt;/a&gt;. Dummett calls the kinds of truths Russell was talking about &quot;verification-transcendent truths&quot;, and those who accept them &lt;a href=&quot;https://plato.stanford.edu/entries/realism-sem-challenge/#Wha&quot; rel=&quot;nofollow noreferrer&quot;&gt;metaphysical realists&lt;/a&gt;. Since metaphysical realism has well known problems with explaining meaningfulness of verification-transcendent claims (not yet prominent in Russell's time) Dummett replays some of Russellian arguments, but now as arguments &lt;em&gt;against&lt;/em&gt; the law of excluded middle.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-10-16T17:51:31.527" LastActivityDate="2017-10-16T17:51:31.527" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6596" PostTypeId="2" ParentId="5965" CreationDate="2017-10-15T22:25:12.917" Score="7" Body="&lt;p&gt;Bethe is quoting and following &lt;strong&gt;Felix Bloch&lt;/strong&gt;, who said left/right in (&lt;a href=&quot;https://doi.org/10.1007/BF01340281&quot; rel=&quot;nofollow noreferrer&quot;&gt;1929&lt;/a&gt;; &lt;a href=&quot;https://doi.org/10.1007/BF01339661&quot; rel=&quot;nofollow noreferrer&quot;&gt;1930, p. 207&lt;/a&gt;). But I don’t think this was ever “common practice”. Stern, in his original Stern–Gerlach paper, drew the magnetic field $\mathfrak H$ and quantized angular momentum component $\mathfrak m_z$ vertically (&lt;a href=&quot;https://doi.org/10.1007/BF01332793&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921, p. 251&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\hspace{11em}$&lt;img src=&quot;https://i.stack.imgur.com/GDrkQ.png&quot; width=&quot;198&quot; height=&quot;216&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In doing so, he followed (and quoted) Sommerfeld’s &lt;em&gt;Atombau und Spektrallinien&lt;/em&gt; (2nd ed., &lt;a href=&quot;https://archive.org/stream/atombauundspekt00sommgoog#page/n429&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921, pp. 408-411&lt;/a&gt;) and ultimately Sommerfeld’s original &lt;a href=&quot;https://en.wikipedia.org/wiki/Spatial_quantization&quot; rel=&quot;nofollow noreferrer&quot;&gt;“spatial quantization”&lt;/a&gt; paper (&lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1002/andp.19163561702/abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916, pp. 29-32&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ay0ZP.png&quot; width=&quot;247&quot; height=&quot;286&quot;&gt;&#xA;&lt;img src=&quot;https://i.stack.imgur.com/JCUtG.png&quot; width=&quot;396&quot; height=&quot;174&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If Fig. 6 looks like a horizontal component is being quantized, it’s only because the lines represent the orbital plane, not the angular momentum vector perpendicular to it; Sommerfeld didn’t switch to picturing the latter until &lt;em&gt;Atombau&lt;/em&gt;’s 5th edition (&lt;a href=&quot;http://edition-open-access.de/studies/2/7/#5&quot; rel=&quot;nofollow noreferrer&quot;&gt;1931; 1934 translation, p. 123&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\hspace{50em}$&lt;img src=&quot;https://i.stack.imgur.com/kY8He.png&quot; width=&quot;483&quot; height=&quot;197&quot;&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It’s true that the Stern–Gerlach &lt;em&gt;experimental&lt;/em&gt; magnetic field was apparently horizontal (&lt;a href=&quot;https://doi.org/10.1007/BF01326983&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;, &lt;a href=&quot;http://onlinelibrary.wiley.com/doi/10.1002/andp.19243791602/abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;1924&lt;/a&gt;). But they didn’t call it left/right any more than front/back or up/down; and the subsequent theories of Heisenberg–Jordan (&lt;a href=&quot;https://doi.org/10.1007/BF01397100&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926, p. 266&lt;/a&gt;), Pauli (&lt;a href=&quot;https://doi.org/10.1007/BF01397326&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927, p. 605&lt;/a&gt;), Dirac (&lt;a href=&quot;https://doi.org/10.1098/rspa.1928.0023&quot; rel=&quot;nofollow noreferrer&quot;&gt;1928, p. 620&lt;/a&gt;) and expositions by Weyl (&lt;a href=&quot;https://books.google.com/books?id=ebjPAAAAMAAJ&amp;amp;dq=Stern+Gerlach&quot; rel=&quot;nofollow noreferrer&quot;&gt;1928, p. 65&lt;/a&gt;), Dirac (&lt;a href=&quot;https://archive.org/stream/in.ernet.dli.2015.177580#page/n162&quot; rel=&quot;nofollow noreferrer&quot;&gt;1930, p. 152&lt;/a&gt;), Wigner (&lt;a href=&quot;https://books.google.com/books?id=X9XzBgAAQBAJ&amp;amp;pg=PA238&amp;amp;dq=Z-Richtung&quot; rel=&quot;nofollow noreferrer&quot;&gt;1931, p. 238&lt;/a&gt;) or Bethe himself (&lt;a href=&quot;https://books.google.com/books?id=pQOdBgAAQBAJ&amp;amp;pg=PA276&amp;amp;dq=z-Richtung&quot; rel=&quot;nofollow noreferrer&quot;&gt;1933, p. 276&lt;/a&gt;) all quantized in the $z$ direction!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;br&gt;As to the title question, my guess is that the standard terms were “positive spin” and “negative spin” from around the time of Sommerfeld–Bethe (&lt;a href=&quot;https://books.google.com/books?id=Z47KBgAAQBAJ&amp;amp;pg=PA588&amp;amp;dq=positiven&quot; rel=&quot;nofollow noreferrer&quot;&gt;1933&lt;/a&gt;) and until maybe the early 1950s, when “spin up” and “spin down” began to appear, with and then without quotes, in references like Blatt–Weisskopf (&lt;a href=&quot;https://books.google.com/books?id=CTrOAAAAMAAJ&amp;amp;dq=%22We+shall+refer+to+these+two+possibilities%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1952&lt;/a&gt;), Kittel (&lt;a href=&quot;https://books.google.com/books?id=Ag5RAAAAMAAJ&amp;amp;dq=%22spin+up%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1953&lt;/a&gt;), Bethe–Maximon (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=62020&quot; rel=&quot;nofollow noreferrer&quot;&gt;1954&lt;/a&gt;), Bethe–Salpeter (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=96526&quot; rel=&quot;nofollow noreferrer&quot;&gt;1957&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-03-26T06:44:46.000" LastActivityDate="2018-03-26T06:44:46.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6597" PostTypeId="1" CreationDate="2017-10-16T07:56:13.733" Score="1" ViewCount="32" Body="&lt;p&gt;It's common in machine learning to train on 70% of non-validation data, testing on the remaining 30%. I'm not sure whether the motive for this is theoretical or &quot;empirically this works well&quot;, although my reading suggests the latter. Where was this ratio first justified either way?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2017-10-16T07:56:13.733" Title="How was the &quot;train on 70% of data&quot; convention established?" Tags="|computer-science|machine-learning|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6598" PostTypeId="1" CreationDate="2017-10-16T14:43:23.683" Score="3" ViewCount="642" Body="&lt;p&gt;The pop-sci answer is that Lorenz characterised chaotic atmospheric dynamics with the hypothetical example of a butterfly's flapping wings changing whether a tornado results. However, since butterfly-shaped solutions in the Lorenz attractor are sometimes called &lt;a href=&quot;https://www.youtube.com/watch?v=aR16sQa5LUw&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorenz butterflies&lt;/a&gt;, my question is: was the butterfly effect named for the shape of the plot or the hypothetical scenario of a butterfly's effect?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2023-02-03T08:32:39.830" Title="Why is it called the butterfly effect?" Tags="|terminology|discoveries|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6599" PostTypeId="2" ParentId="6598" CreationDate="2017-10-16T16:50:47.187" Score="10" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Butterfly_effect&quot; rel=&quot;noreferrer&quot;&gt;SEE&lt;/a&gt; for more&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ed Lorenz in 1963 wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Following suggestions from colleagues, in later speeches and papers Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted &lt;em&gt;Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?&lt;/em&gt; as a title.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I believe both of these were earlier than the possibility of computer images of the Lorenz attractor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mUmy3.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mUmy3.jpg&quot; alt=&quot;image&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/File:TwoLorenzOrbits.jpg&quot; rel=&quot;noreferrer&quot;&gt;[source]&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-10-16T16:50:47.187" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6601" PostTypeId="2" ParentId="6590" CreationDate="2017-10-17T02:17:38.020" Score="0" Body="&lt;p&gt;There was a lot of use of the  &quot;Verification Principle&quot; back then, which went something like:  A sentence has meaning iff it can be verified.  This was accepted for a while, but stock in the Verification Principle went down when it came to light that the Verification Principle itself could not be verified. Thing is, though, I think it was Wittgenstein (rather than Russell) who actually pointed this out.&lt;/p&gt;&#xA;" OwnerUserId="6417" LastActivityDate="2017-10-17T02:17:38.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6602" PostTypeId="1" CreationDate="2017-10-17T02:20:36.597" Score="-1" ViewCount="158" Body="&lt;p&gt;Before World War Two, how did pipelines work for crude oil? What kind of metal or plastic were they made out of? What kind of sensors and monitoring did they need? Did they have &quot;pigs&quot; that travel up and down the pipe looking for problems? How were they kept warm so as not to freeze during winter?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2017-10-20T17:29:38.623" Title="How did crude oil pipelines work before WW2?" Tags="|thermodynamics|engineering|energy|technology|materials-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6603" PostTypeId="1" AcceptedAnswerId="6608" CreationDate="2017-10-17T15:08:04.120" Score="4" ViewCount="355" Body="&lt;p&gt;Important results in the theory of PDEs regarding boundary-value problems are trace and extension theorems. Since the trace operator (not to be confused with the trace from linear algebra) essentially acts by restriction to the boundary of the domain, I was wondering how it got the name &quot;trace&quot;: who came up with this name, when and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I asked this earlier on &lt;a href=&quot;https://mathoverflow.net/q/283162&quot;&gt;MathOverflow&lt;/a&gt; but did not get an answer.)&lt;/p&gt;&#xA;" OwnerUserId="4703" LastEditorUserId="4703" LastEditDate="2017-10-31T19:47:48.700" LastActivityDate="2017-10-31T19:47:48.700" Title="Origin of the terminology “trace operator” related to boundary-value problems for PDEs" Tags="|terminology|differential-equations|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6604" PostTypeId="2" ParentId="284" CreationDate="2017-10-17T19:32:45.540" Score="2" Body="&lt;p&gt;In point of fact, Euler and Lagrange were the first mathematicians that held titles of that type. According to A. Weil (&quot;Number theory: an approach through history&quot;, Birkhäuser Verlag, 1984): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(Chap. III, §III, p. 169): No mathematician ever attained such an undisputed position of leadership in all branches of mathematics, pure and applied, as Euler did for the best part of the eighteenth century. In 1745 his old teacher Johann Bernoulli, not a modest man as a rule, addressed him as &quot;&lt;strong&gt;&lt;em&gt;mathematicorum princeps&lt;/em&gt;&lt;/strong&gt;&quot; (&lt;a href=&quot;https://archive.org/stream/bub_gb_P2ThmkmB-UkC#page/n121&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Corr&lt;/em&gt;.II.88.92&lt;/a&gt;).&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;(Chap. IV, §I, p. 309): In 1745  Euler had been hailed by his old teacher Johann Bernoulli as&#xA;  &quot;&lt;strong&gt;&lt;em&gt;mathematicorum princeps&lt;/em&gt;&lt;/strong&gt;&quot;, the first of mathematicians (cf. Chap.III,&#xA;  §III). By 1775 he clearly felt ready to pass the title on to&#xA;  Lagrange. &quot;It is most flattering to me&quot;, he wrote to his younger&#xA;  colleague, &quot;to have as my successor in Berlin the most outstanding&#xA;  geometer of this century&quot;(&lt;em&gt;loc.cit.&lt;/em&gt; Chap.III, §IX). Such was indeed by then the universal&#xA;  verdict of the scientific world. &quot;&lt;strong&gt;&lt;em&gt;Le célèbre Lagrange, le premier des&#xA;  géomètres&lt;/em&gt;&lt;/strong&gt;&quot; is how Lavoisier referred to him in 1793 in an official&#xA;  request to the Convention on behalf of his friend (&lt;a href=&quot;https://archive.org/stream/uvresdelagrange14lagr#page/314&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Lag&lt;/em&gt;.XIV.314–315&lt;/a&gt;)&#xA;  at the onset of the Terror which was soon to claim Lavoisier himself&#xA;  as a victim. In the next century the title of &quot;&lt;strong&gt;&lt;em&gt;princeps&#xA;  mathematicorum&lt;/em&gt;&lt;/strong&gt;&quot; was bestowed upon Gauss by the unanimous consent of&#xA;  his countrymen. It has not been in use since.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2017-10-19T00:26:39.480" LastActivityDate="2017-10-19T00:26:39.480" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6605" PostTypeId="2" ParentId="6603" CreationDate="2017-10-17T20:46:11.660" Score="0" Body="&lt;p&gt;The name Trace here is related to the notion of the trace of function  in general which basically means the restriction.&lt;br&gt;&#xA;I other word, the Trace of a function in general its restriction (whenever it makes sense) on a &lt;strong&gt;subset&lt;/strong&gt; of ITS DOMAIN ( usually named as trace set of the original function) .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But be aware  the actual situation is much more delicate. Since given a function defined almost everywhere on  a bounded domain it is not always possible to take its &lt;strong&gt;TRACE&lt;/strong&gt; (&lt;strong&gt;or its restriction )&lt;/strong&gt; on the boundary. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather if we consider a CONTINUOUS function up the the boundary , it is possible to take its TRACE (or its restriction) on the boundary.  &lt;strong&gt;This means that the OPERATOR that maps a continuous function up the boundary to it TRACE (to its restored) on the boundary is WELL define .&lt;/strong&gt; Moreover, this OPERATOR is bounded if we endowe both depature and arrived spaces with suitable L.p. or Sobolev NORMS. &#xA;&lt;strong&gt;THIS THEN GIVES THE NAMES TRACE OPERATOR.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;FINALLY since the space of continuous function up to boundary is dense IN SOME SUITABLE L.p. space or SOBOLEV SPACE, we know that, OUR LATEST TRACE OPERATOR CAN BE CONTINUOUSLY EXTEND TO THE corresponding L.p. or Sobolev spaces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;AND IT PRESERVES THE NAME TRACE OPERATOR AS WELL&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: &#xA;1. There are some cases where   the density argument may not apply but One can still use HANH-BANACH theorem to get A CONTINUOUS EXTENTION.&#xA;2. One may chose other spaces instead of the space of continuous function up the boundary.  Like the space consisting of functions which are  restrictions of functions that are smooth (on a bigger domian ) on closure of the DOMIAN  (the we are working with ).&lt;/p&gt;&#xA;" OwnerUserId="6074" LastEditorUserId="6074" LastEditDate="2017-10-17T20:55:15.380" LastActivityDate="2017-10-17T20:55:15.380" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6606" PostTypeId="1" AcceptedAnswerId="6607" CreationDate="2017-10-18T10:16:03.043" Score="1" ViewCount="389" Body="&lt;p&gt;I assume that there was a single person who at a specific day in time was the very first to see and understand perfectly clearly that temperature is somehow &lt;em&gt;identical&lt;/em&gt; with the mean kinetic energy of a bunch of molecules or atoms, and who was able to write this done as a mathematical equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though this precise idea grew and shaped slowly starting from some rather vague ideas and over generations of physicists (and inside the mind of this one physicist) I believe it became clear to him somehow suddenly (maybe but not necessarily accompanied by an &lt;a href=&quot;https://en.wikipedia.org/wiki/Eureka_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;eureka experience&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Do we know who this person was, and when it happened (roughly)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or do you believe this kind question doesn't make sense or is unanswerable? But what about Einstein and his special theory of relativity, then?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Over there &lt;a href=&quot;https://physics.stackexchange.com/questions/363554/discovery-of-kinetic-gas-theory&quot;&gt;at Physics SE&lt;/a&gt;, the names of Waterston, Clausius, Boltzmann and Clark-Maxwell were mentioned.]&lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2017-10-18T17:06:21.680" LastActivityDate="2017-10-19T13:25:50.793" Title="Discovery of kinetic gas theory" Tags="|discoveries|thermodynamics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6607" PostTypeId="2" ParentId="6606" CreationDate="2017-10-18T23:25:35.310" Score="2" Body="&lt;p&gt;Those who have posted comments on this question have already suggested (and I agree) that the questioner's assumption is unsound where it supposes a &quot;single person&quot; who was the &quot;very first ... to see and understand perfectly clearly that temperature is somehow identical with the mean kinetic energy&quot; of a quantity of atoms or molecules in gaseous form.  One of the reasons why this 'single-person' explanatory format is arguably an unrealistic romantic idealization (and has actually been so criticized) is that the history shows a succession of important experimental contributions, on which the eventual mathematization fundamentally depended. It did not emerge from mathematical contemplation alone, and it would have lacked meaningful reference if not for the antecedent studies of the phenomena along with mathematical relations describing them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ideal gas laws to be explained by the kinetic theory are often expressed in summary as &lt;/p&gt;p.V = n.R.T ,&lt;/p&gt; where the successive symbols are for pressure, volume, number of moles or molecules, a constant R, and temperature on an absolute scale T. (e.g. among other sources W J Moore, 'Physical Chemistry', many editions from 1950 onwards with chapters on the kinetic theory and gas laws, giving (latterly well-known) physical/mathematical derivation of the kinetic theory as well as describing its relation with the basic observable phenomena and their laws).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The historical contributors whose results in aggregate enable this summary of gas laws are often given as:- &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Robert Boyle (1662), who found the reciprocal relationship between p and V ('Boyle's Law'), for a constant quantity of air at constant temperature -- noting that in Boyle's 1662 account (&quot;A Defense of the Doctrine Touching the Spring and Weight of the Air, Propos'd by Mr. R. Boyle in His New Physico-mechanical Experiments, Against the Objections of Franciscus Linus ...&quot;) the author acknowledged a number of precursor experiments by others;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jacques Charles (1780s), whose unpublished work was acknowledged and re-expressed by Gay-Lussac, showing that for a given quantity of [gas] at constant pressure the volume was proportional (or the density inversely proportional) to the temperature: there had been others including Francis Hauksbee (1708) who had arrived at comparable results; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and Amedeo Avogadro (1811), whose law or hypothesis stated that at a given/constant temperature and pressure, equal volumes of different gases contain an equal number of molecules. (The background of this contribution also included, of course, Dalton's atomic theory or hypothesis (1803-1810).)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One should also not forget the conceptual identification of heat with motion made long ago by Francis Bacon, limited though it was.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This naming of early contributors and their contributions makes no attempt to undermine the importance or results of the later 19th-c. scientists already cited, including e.g. Waterston, Clausius, Boltzmann and Clerk Maxwell. But the inclusion of the earlier history does show more of the perspective and context against which their later contributions fall to be appreciated.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the contribution of J J Waterston was to add to the gas laws comprising the constituent laws of Boyle, Charles/Gay-Lussac, Dalton and Avogadro, the important explanation of gas-temperature as proportional to molecular kinetic energy expressed as 'vis-viva'.  In view of the comments on Waterston already given here and on Physics SE, there may be interest in a link to an appreciation of Waterston and his contribution given by Lord Rayleigh in 1892, in a foreword to a belated posthumous publication of Waterston's original 1845 paper (&quot;On the physics of media that are composed of free and perfectly elastic molecules in a state of motion&quot;, J. J. Waterston, Lord Rayleigh, Sec. R. S., Phil Trans Roy Soc A (1892) v.183, pp.1-79  (&lt;a href=&quot;http://rsta.royalsocietypublishing.org/content/183/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://rsta.royalsocietypublishing.org/content/183/1&lt;/a&gt; )).  There is no need here to go into Rayleigh's appreciation of where Waterston's contribution advanced and where it stopped, except to mention (a) that Waterston's paper shows signs as might be expected of relying on previous experimental and theoretical work, which he cited as the laws of Mariotte (effectively Boyle's law) and of Dalton and Gay-Lussac (effectively including Charles' law), successfully modeling the effects described by those laws on the hypothesis that temperature is proportional to vis-viva i.e. to the square of molecular velocity, and (b) that one of Waterston's publications (as quoted by Rayleigh) expressed his contribution as a correction to previous work of about 1821 in which heat was supposed to represent motion but the motion was mistakenly measured there as effectively proportional to particle-momentum (i.e. to first instead of second power of velocity).   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem of appreciating respective contributions made by a succession of experimental and theoretical scientists is a substantial one. It is arguably not fairly addressed by opposing, as the question does, on one side, one who &quot;see[s] and understand[s] perfectly clearly&quot; and expresses that vision as a mathematical equation, against, on the other side, &quot;some rather vague ideas .. over generations of physicists&quot;. (This arrangement and expression of the question arguably does suggest an unrealistic and disparaging prejudice against physicists and their contributions). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is overlooked by the assumption made in the question can be seen as, first, the existence of a long succession of contributing researchers who made partial contributions to what eventually became the kinetic theory and its basis; and second, the crucial role of successive pieces of experimental evidence and of discovery of the phenomena of gases. Also the whole idea of theoretical discovery as something that arises from theoretical or intellectual consideration may be criticised by reference to historical facts, including those cited above,  showing how such discoveries with mathematical content can be firmly rooted in experimental evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In principle, it is perhaps conceivable before studying the history that -- as the question assumes -- one of the parties to a discovery indeed had some 'eureka' moment in which a physical perception and its mathematical expression could emerge intellectually &lt;em&gt;in toto&lt;/em&gt;.  But then again, at the other extreme, it is equally conceivable that a mathematical theory might have done no more than express, as an arguably routine translation into formulaic language, the clear physical insights arrived at by earlier experimental contributors.  The likelihood here, especially in view of the cited history, is that neither of these extremes is true of the kinetic theory, and that the respective contributions were partial, successive, occurring occasionally as parallel and independent near-duplicates, and that they depended both on mathematics and on experiments. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2017-10-18T23:53:13.487" LastActivityDate="2017-10-18T23:53:13.487" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6608" PostTypeId="2" ParentId="6603" CreationDate="2017-10-19T00:54:28.287" Score="3" Body="&lt;p&gt;Normally, I'd refer to &lt;a href=&quot;http://jeff560.tripod.com/mathword.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics site&lt;/a&gt; for these kinds of questions, but it is silent on this one, and generally &quot;trace&quot;'s history seems unexplored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first use of &quot;trace&quot; in the sense of restriction to the boundary on MathSciNet is (as far as I could find) in the review of Slobodeckii's 1958 note Sobolev's Spaces of Fractional Order and Their Application to Boundary Problems for Partial Differential Equations (Dokl. Akad. Nauk SSSR (N.S.) v. 118, 1958, pp. 243–246). The reviewer was Lions, who himself began to use the term afterwards, perhaps the first in the West, e.g. in the monograph Équations Différentielles Opérationnelles et Problèmes aux Limites (1961). The author of the first &quot;trace theorems&quot;, Sobolev, does not use the word in his well-known book, Some Applications of Functional Analysis in Mathematical Physics (1950), which summarized his work from 1930-s and brought him the world fame. But it does occur in his Varenna lectures Sur les Equations aux Dérivées Partielles Hyperboliques Non-linéaires published in Rome in 1961 as a monograph:&quot;&lt;em&gt;The first four lectures develop the properties of generalized derivatives, including important theorems about the traces of functions on varieties in $R^n$ (in particular, the hyperplanes $t=\,$const&lt;/em&gt;&quot;. So it might have been Slobodeckii's doing, but I wouldn't claim it without more direct evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the name's motivation, I can only speculate. &quot;The traces of functions on varieties&quot; is reminiscent of algebraic geometry, where intersections of surfaces with planes were traditionally called their &quot;traces&quot;, see e.g. &lt;a href=&quot;https://books.google.com/books?id=LVJxBvxVP5QC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lardner's 1831 Treatise on Algebraic Geometry&lt;/a&gt;, p.229. If one thinks of functions as graphs over the domain then graph's intersection with the cylinder over the boundary (in nice cases) will be exactly the &quot;trace&quot; in Slobodeckii's sense.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-10-19T22:36:27.707" LastActivityDate="2017-10-19T22:36:27.707" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6609" PostTypeId="2" ParentId="771" CreationDate="2017-10-19T06:36:19.383" Score="10" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Archimedes might, of course, have used a better method. (He also produces the approximation $\frac{1351}{780}$, for which the foregoing is not obviously practical...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I disagree on the foregoing not being practical: it's a trivial computation compared to some which were carried out by hand by later mathematicians. But I think that really is the nub: any explanation for how Archimedes produced the approximation $\frac{265}{153} &amp;lt; \sqrt{3} &amp;lt; \frac{1351}{780}$ should also explain why he &lt;em&gt;didn't&lt;/em&gt; produce the approximation $\frac{989}{571} &amp;lt; \sqrt{3} &amp;lt; \frac{1351}{780}$. Here both your approach and the continued fraction approach fall down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also the point that this tabulation method is rather ungeometrical, and so not really in the expected style. I suspect that Archimedes would have preferred not to publish a result than to face the embarrassment of explaining that it was calculated thus when someone asked him for the proof.&lt;/p&gt;&#xA;" OwnerUserId="104" LastActivityDate="2017-10-19T06:36:19.383" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6610" PostTypeId="2" ParentId="6606" CreationDate="2017-10-19T13:20:24.120" Score="1" Body="&lt;p&gt;To illustrate the intricateness of progress in science, I reproduce here the first paragraph of Clausius' seminal paper &lt;a href=&quot;http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Clausius%20(1857)_Nature%20motion%20we%20call%20heat.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The nature of the motion which we call heat&lt;/a&gt; and its German original &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k15185v/f371.image.langDE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Über die Art der Bewegung, welche wir Wärme nennen&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Nbww1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Nbww1.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lPcli.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lPcli.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;From the title of Clausius' paper, Brush derived the title of his outstanding monograph &lt;a href=&quot;https://isidore.co/calibre/get/PDF/Brush,%20Stephen%20G_-The%20Kind%20of%20Motion%20We%20Call%20Heat_%20A%20History%20of%20the%20Kinetic%20Theory%20of%20Gases%20in%20the%2019th%20Century_4438.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The kind of motion we call heat&lt;/a&gt; (1976).&lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2017-10-19T13:25:50.793" LastActivityDate="2017-10-19T13:25:50.793" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6611" PostTypeId="1" AcceptedAnswerId="9737" CreationDate="2017-10-19T14:01:48.260" Score="3" ViewCount="524" Body="&lt;p&gt;Gauss gave 6 published and 2 unpublished proofs of quadratic reciprocity (see, e.g., &lt;a href=&quot;https://www.rzuser.uni-heidelberg.de/~hb3/rchrono.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).  I suspect this was to try to understand the &quot;real reason&quot; quadratic reciprocity holds (though please correct me if you know otherwise), but I'd like to know what Gauss actually thought about his different proofs.  Do we know to what extent Gauss was satisfied with his proofs, or which proofs he regarded as better than others?  &lt;/p&gt;&#xA;" OwnerUserId="2529" LastActivityDate="2019-06-17T16:10:15.103" Title="Gauss's proofs of quadratic reciprocity" Tags="|number-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6612" PostTypeId="1" CreationDate="2017-10-19T23:35:42.577" Score="3" ViewCount="214" Body="&lt;p&gt;Both &lt;a href=&quot;https://en.wikipedia.org/wiki/Mechanical_explanations_of_gravitation#Vortex&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Rodolfo_Llin%C3%A1s#Selected_bibliography&quot; rel=&quot;nofollow noreferrer&quot;&gt;Llinas&lt;/a&gt; use the term of &lt;strong&gt;vortex&lt;/strong&gt; to illustrate their theories, but none of them does it really specifically, and none of them is mentioned in the Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Vortex&quot; rel=&quot;nofollow noreferrer&quot;&gt;vortices&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could anyone please give a somehow concise explanation of &quot;vortex&quot; that simultaneously helps understanding Descartes and Llinas better?&lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2017-10-20T10:38:36.330" LastActivityDate="2017-10-24T22:20:43.287" Title="What is a vortex?" Tags="|gravity|psychology|neuroscience|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6614" PostTypeId="1" AcceptedAnswerId="6621" CreationDate="2017-10-20T12:25:48.393" Score="7" ViewCount="374" Body="&lt;p&gt;This &lt;a href=&quot;https://matheducators.stackexchange.com/a/10584/590&quot;&gt;answer&lt;/a&gt;, to a question on teaching the chain rule, suggests writing something like this&#xA;$$&#xA;\frac{\mathrm{d}\, \mathrm{e}^\sqrt{s}}{\mathrm{d}\,s}=\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\sqrt{s}}\frac{\mathrm{d}\sqrt{s}}{\mathrm{d}\,s}&#xA;$$&#xA;where a composite term ($\sqrt{s}$) is substituted for the variable occurring in the denominator of a derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is still common today to substitute a term for $y$ in $\frac{\mathrm{d}y}{\mathrm{d}x}$, but I have rarely (or never) seen people substitute for $x$ in $\frac{\mathrm{d}y}{\mathrm{d}x}$. On the other hand it makes a lot of sense, when the &quot;differential coefficient&quot; $\frac{\mathrm{d}y}{\mathrm{d}x}$ is literally understood as &quot;the coefficient in front of the differential $\mathrm{d}x$ in $\mathrm{d}y=c\mathrm{d}x$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Question: Can one find expressions like $\frac{\mathrm{d} \,\mathrm{e}^\sqrt{s}}{\mathrm{d}\sqrt{s}}$ in classical works of differential calculus? Was it popular at some time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The motivation for this question comes from another question I asked on formalizing classical calculus: &lt;a href=&quot;https://cs.stackexchange.com/questions/82230&quot;&gt;https://cs.stackexchange.com/questions/82230&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2017-10-23T07:10:03.780" LastActivityDate="2020-02-11T00:41:51.240" Title="Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?" Tags="|mathematics|notation|calculus|real-analysis|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6615" PostTypeId="2" ParentId="6602" CreationDate="2017-10-20T17:29:38.623" Score="0" Body="&lt;p&gt;The implication of information on war construction of the &quot;Big Inch&quot; and &quot;Little Big Inch&quot; pipelines from Texas to the East Coast of the US (at &lt;a href=&quot;https://aoghs.org/petroleum-in-war/oil-pipelines/&quot; rel=&quot;nofollow noreferrer&quot;&gt;American Oil and Gas Historical Society&lt;/a&gt; is that there were not major long-distance pipelines before the war. Pictures at the indicated web site clearly show that the pipeline was welded steel. Buried 4 feet deep with moving product in it there was not an issue with freezing.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2017-10-20T17:29:38.623" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6616" PostTypeId="1" CreationDate="2017-10-20T20:05:13.707" Score="2" ViewCount="238" Body="&lt;p&gt;In this video is explained that in November 1584 the width of the Atlantic ocean was discovered by comparing the hour when the eclipse of the moon started in England with the hour when the eclipse occurred in Virginia USA. But doesn't also the Sun arise on particularly times in England and USA to calculate the distance? Why was the eclipse of the moon used?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See min 22:00 &lt;a href=&quot;https://www.youtube.com/watch?v=w6JZSzVcKgo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=w6JZSzVcKgo&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="3677" LastEditDate="2017-10-21T08:40:00.243" LastActivityDate="2017-10-23T13:09:52.753" Title="Why was in 1584 the eclipse of the moon used to calculate the width of the Atlantic ocean rather than just using the Sun itself?" Tags="|astronomy|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6617" PostTypeId="1" AcceptedAnswerId="6644" CreationDate="2017-10-21T02:17:42.423" Score="2" ViewCount="139" Body="&lt;p&gt;What's the first appearance of ultrastatic spacetimes (that is, spacetimes with a metric of the form $ds^2 = -dt^2 + h$, with $h$ a Riemannian metric)? The oldest reference I can find on the topic is Fulling's 1977 paper &quot;Alternative vacuum states in static space-times with horizons&quot;, but from the article it seems to not be the first appearance of the term, but I'm unable to find an older reference from there.&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2017-10-29T03:02:14.510" Title="First descriptions of ultrastatic spacetimes?" Tags="|physics|relativity-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6618" PostTypeId="2" ParentId="6616" CreationDate="2017-10-21T14:37:28.350" Score="1" Body="&lt;p&gt;Read the interesting book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/080271529X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time&lt;/a&gt; to find why measurement of longitude simply by observing the sun (or the stars) was impossible.  Until there were accurate clocks.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-10-21T14:37:28.350" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6619" PostTypeId="2" ParentId="6616" CreationDate="2017-10-21T17:36:11.877" Score="0" Body="&lt;p&gt;More precisely, they tried to determine the difference of longitude between two places. The difference of longitude is the difference in local times at the places. While local time is easy to determine by astronomical observations, there is no simple way of comparing time at different locations. Moon eclipse is one such mean. The principle was clear from the antiquity, and this was the principal method of practical determining of difference in longitude between distant places, until the discovery of Jupiter satellites, which have two advantages: their eclipses occur more frequently, and also their eclipse times are easier to predict with high accuracy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. One cannot use a Sun eclipse for this purpose. The difference is that Moon's eclipse occurs at a definite time, which can be predicted. So if one&#xA;observer observes it at his local time $T_1$, and another at his local time $T_2$ then the difference of longitude it $T_1-T_2$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sun eclipses occur at times which depend on locations.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-23T13:09:52.753" LastActivityDate="2017-10-23T13:09:52.753" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6620" PostTypeId="1" CreationDate="2017-10-21T21:40:15.513" Score="2" ViewCount="1524" Body="&lt;p&gt;I see in many essays about Newton's &lt;em&gt;Principia&lt;/em&gt; how it was very difficult to read and follow, so I was wondering if any of you know of any quotes from mathematicians of the time to that effect.&lt;/p&gt;&#xA;" OwnerUserId="6435" LastEditorUserId="1815" LastEditDate="2017-10-22T19:24:19.130" LastActivityDate="2017-10-22T19:24:19.130" Title="Difficulty in Understanding Newton's Principia" Tags="|mathematics|calculus|newton|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6621" PostTypeId="2" ParentId="6614" CreationDate="2017-10-22T10:21:35.600" Score="9" Body="&lt;p&gt;Gibbs (&lt;a href=&quot;https://archive.org/stream/americanjourna3371889newh#page/140&quot; rel=&quot;nofollow noreferrer&quot;&gt;1889, p. 140&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\qquad&#xA;\dfrac{d\,\log\mathrm V}{d\,\log p} = - &#xA;\dfrac{d\,\log n}{d\,\log\lambda}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riemann (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN250442582_0013?tify=%7B&amp;quot;pages&amp;quot;:[199]%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1868, p. 89&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\qquad&#xA;\dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=4\dfrac{d\smash[t]{\dfrac{dy}{d(x+\alpha t)}}}{d(x-\alpha t)}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hamilton (&lt;a href=&quot;https://archive.org/details/s03philosophicalmag23londuoft/page/362&quot; rel=&quot;nofollow noreferrer&quot;&gt;1843, p. 362&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\quad\ \dfrac d{d\log t}\,(1+t)^{-m}\,\mathrm T_m&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jacobi (&lt;a href=&quot;https://books.google.com/books?id=mgRCAAAAcAAJ&amp;amp;pg=PA336&quot; rel=&quot;nofollow noreferrer&quot;&gt;1841, p. 336&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\qquad&#xA;\dfrac{\partial R}{\partial a}= \sum\cdot \dfrac{\partial R}{\partial \frac{\partial f_i}{\partial x_k}}\cdot\dfrac{\partial^2 f_i}{\partial a\,\partial x_k}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Legendre (&lt;a href=&quot;https://archive.org/stream/bub_gb_c4M_AAAAcAAJ#page/n491&quot; rel=&quot;nofollow noreferrer&quot;&gt;1826, p. 466&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\quad&#xA;\dfrac{d\,l\,\Gamma a}{da} +&#xA;\dfrac{d\,l\,\Gamma(\frac12+ a)}{d(\frac12+ a)} - &#xA;\dfrac{2d\,l\,\Gamma(2a)}{d(2a)} =&#xA;-2l\,2&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss (&lt;a href=&quot;https://archive.org/details/bub_gb_ORUOAAAAQAAJ/page/n42&quot; rel=&quot;nofollow noreferrer&quot;&gt;1809, p. 27&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\qquad&#xA;\dfrac{\mathrm{d}\frac12v}{\mathrm{d}\log\operatorname{tang}\frac12v}=&#xA;\dfrac{\sin v}2&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;L’Huilier (&lt;a href=&quot;https://archive.org/stream/principiorumcal00lhugoog#page/n136&quot; rel=&quot;nofollow noreferrer&quot;&gt;1795, p. 96&lt;/a&gt;):&#xA;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\qquad&#xA;\dfrac{d.a^{\mathrm z}}{d.\log.z}= A.a^{\mathrm z}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-02-11T00:41:51.240" LastActivityDate="2020-02-11T00:41:51.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6622" PostTypeId="2" ParentId="6620" CreationDate="2017-10-22T17:22:07.980" Score="6" Body="&lt;p&gt;It would probably not have been easy for a contemporary mathematician to formulate a direct critique that Newton was difficult to understand without also 'reflecting' unwanted discredit on the skill or level of understanding of the critic. I can't recall seeing such a direct critique, but what can be found perhaps more easily are statements pointing more-or-less indirectly to an early reputation of the 'Principia' for difficulty.  Material of this kind from Newton's lifetime, or subsequent decades, includes statements of three sorts:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1 * A-B comparison with a conclusion about which presentation is 'easier':  Under this heading (and perhaps nearest to a direct statement of the kind sought by the question), there is the preface to the Marquis de l'Hospital's 1696 book &quot;Analyse des infiniment petits&quot; (Infinitesimal analysis). The focus here is on the content of calculus in the 'Principia', as compared with Leibniz's calculus. The writer of the 1696 preface says: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;the learned M. Neuwton&quot; ... &quot;had found something similar to the differential Calculus, as appears by the excellent book entitled 'Philosophia naturalis principia Mathematica', which he gave us in 1687, of which almost all is of &#xA;this calculus.&quot; ['lequel est presque tout de ce calcul']. &quot;However, the notation of Mr. Leibnis makes his much easier and more expeditious ...&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, by implication, Newton's Principia, at least in its presentation of matters related to the calculus, was being politely described as much more difficult.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2 * Anecdotes which might be enjoyably told by raconteurs without personally admitting any lack in their own understanding:  Under this heading there is an anecdote of a remark supposedly overheard in Cambridge while Newton was there: &quot;There goes the man that writt a book that neither he nor any one else understands.&quot; (recorded by John Conduitt, MSS at King's College, Cambridge, Keynes 130.6, 130.5, quoted by D T Whiteside  (1970), J Hist Astron 1: 116.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3 * Simplified explanations written ostensibly for students or 'for ladies', carrying indirect implication that some of the material was too difficult to include. Under this heading fall two books of the early 18th-c: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1720:  Willem 's Gravesande (1688-1742) wrote an introduction to Newton's physics in which some of the more difficult mathematical material was omitted and replaced by description of experiments to illustrate relevant points (&quot;Physices elementa mathematica, experimentis confirmata, sive introductio ad philosophiam Newtonianam&quot;; English translation by W T Desaguliers &quot;Mathematical Elements of Natural Philosophy, Confirmed by Experiments; or, an Introduction to Newtonian Philosophy&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1739: &quot;Il Newtonianismo per le Dame ... &quot; ('Newtonianism for Ladies, ... Dialogues about Light, Colours, and Attraction') by Francesco Algarotti (mostly about Newton's optics, and less about the supposedly more difficult gravitation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One should also mention a statement by Newton himself. His introduction to Book 3 of the Principia included often-quoted remarks that easily attract criticism that his presentation was deliberately made hard to understand: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I composed an earlier version of book 3 in popular form, so that it might be more widely read. But those who have not sufficiently grasped the principles set down here will certainly not perceive the force of the conclusions, nor will they lay aside the preconceptions to which they have become accustomed over many years; and therefore, to avoid lengthy disputations, I have translated the substance of the earlier version into propositions in a mathematical style, so that they may be read only by those who have first mastered the principles.&quot; (from I B Cohen's 1999 translation of the Principia, at page 793.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The early version 'in popular form' and presumably easier to understand had been written in 1685 by Newton: it was superseded but not destroyed, and was retrieved and published after his death, both in Latin, and also in English as Newton's 'System of the World' (1728, 1731, 1740).)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2017-10-22T17:22:07.980" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6623" PostTypeId="2" ParentId="6616" CreationDate="2017-10-23T12:34:21.940" Score="3" Body="&lt;p&gt;Perhaps the following &lt;a href=&quot;http://astroguyz.com/2008/02/14/detrimine-your-longitude-the-lunar-eclipse-method-part-i/&quot; rel=&quot;nofollow noreferrer&quot;&gt;quote&lt;/a&gt;will be more, ahem, illuminating.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As any Eclipse chaser can attest, total solar eclipses are extremely&#xA;  rare from any given location. A spot on Earth may go centuries before&#xA;  totality graces the skies. In theory, a partial would be much more&#xA;  common. And of course, this would require something such as smoked&#xA;  glass to observe first contact. That leaves us with lunar eclipses.&#xA;  Any given eclipse can easily span more than one full hemisphere of the&#xA;  Earth. Also, while contact with the diffuse outer cone of Earth’s&#xA;  shadow, or penumbra, is rather subtle, contact with the inner core, or&#xA;  umbra, is not. Such an event can be defined by eye to a span of time&#xA;  of about five minutes. Up until the time of the invention of the&#xA;  telescope, eclipse timing was the only semi reliable way to judge&#xA;  longitude. The idea is simple; observe the local time of first contact&#xA;  with the umbra, then compare it with the known time from a separate&#xA;  fixed location. The Earth rotates 15 degrees per hour, (360/24=15) or&#xA;  1 degree every 4 minutes. If umbral contact was said to begin at&#xA;  2:05AM at a given location and you observed it to begin at 1:05AM, you&#xA;  are 15 degrees west of the first location. The Greeks first noted this&#xA;  during an eclipse on September 20th, 331BC. Alexander the Great in&#xA;  Mesopotamia recorded the eclipse as beginning two hours after sunset,&#xA;  while on the north shores of Africa, the city of Carthage it was seen&#xA;  to start at sunset. Ptolemy correctly deduced that Alexander’s&#xA;  position was 30 degrees or 2 hours east of Carthage. None other than&#xA;  Christopher Columbus was a student of the works of Ptolmey. He&#xA;  attempted to use this method to judge his position on his second and&#xA;  third voyage to the new world. The eclipses concerned were on&#xA;  September 14th, 1494 (part of saros cycle #119) and February 29, 1504&#xA;  (saros cycle #109).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2017-10-23T12:34:21.940" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6624" PostTypeId="1" CreationDate="2017-10-23T18:30:51.270" Score="5" ViewCount="380" Body="&lt;p&gt;In the 17e century van Leeuwenhoek discovered with his microscope new kind of animals and cells. How was this discovery of van Leeuwenhoek received by the ordinary people when there seemed to be more animals than human kind could ever think of? Was it a bit like discovering extraterrestrial life would be nowadays,  or was his whole discovery only interesting for the scientists of those days?&lt;/p&gt;&#xA;" OwnerUserId="3677" LastEditorUserId="55" LastEditDate="2017-10-23T23:42:35.583" LastActivityDate="2017-10-24T17:53:17.713" Title="Was it a major cultural event when Van Leeuwenhoek discovered unseen animals under the microscope?" Tags="|discoveries|biology|social-context|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6625" PostTypeId="1" CreationDate="2017-10-23T18:20:57.780" Score="8" ViewCount="1118" Body="&lt;p&gt;In Euclid's elements, some of the theorems (e.g. SAA congruence) can be proven using the parallel postulate, much easier than without it. But it seems that Euclid has intentionally avoided using it, when possible.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Am I right?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the reason behind this choice?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6452" OwnerDisplayName="Behzad" LastActivityDate="2017-10-24T09:54:57.920" Title="Why did Euclid Avoid Using the 5th Postulate?" Tags="|euclidean-geometry|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6626" PostTypeId="2" ParentId="6625" CreationDate="2017-10-23T18:40:06.397" Score="3" Body="&lt;p&gt;If he avoided use of that postulate where he saw that it is possible to do like that he did it most likely in the spirit of proving results with minimal assumptions, so that result proven in such a way will hold for some axiom systems different than the one used by him in his books, and different in the sense that those alternative axiom systems will contain some of his axioms but not all, so, if he proved that something is true without the usage of parallel postulate then he proved also that it is true with the usage of parallel postulate, so he most likely did it all in the spirit of trying to be as general as possible, and aware of the fact that other axioms systems and geometries are also possible.&lt;/p&gt;&#xA;" OwnerDisplayName="user6444" LastActivityDate="2017-10-23T18:40:06.397" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6627" PostTypeId="2" ParentId="6625" CreationDate="2017-10-23T18:50:46.293" Score="2" Body="&lt;p&gt;I cannot answer with confidence as I havent extensively studied his work.  I can think of a number of possible reasons, however.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Maybe he didnt think of that solution. Genius or not, no one sees everything.&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maybe that solution required a theorem he had yet to prove, and he thought it unnecessary to go back and change it. Its easy to judge in hindsight bias when you already know the theorem.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Maybe unbeknownst to you, the theorem you would have to invoke on some level requires the theorem you would like to prove with it, in order to itself be proven, and it would be circular reasoning to do it your way. &lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;To avoid unnecessary assumption. As Antoine pointed out, the whole point of his work is to minimize assumption.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Maybe he foresaw that if one postulate were proven wrong at a later date, it would undermine your &quot;simpler proof&quot;, whereas taking an alternate approach prevented this possibility.&lt;/li&gt;&#xA;&lt;li&gt;Maybe he didnt have the utmost confidence in that postulate.&lt;/li&gt;&#xA;&lt;li&gt;Maybe he wanted to keep things as generalizable as possible so that it could be applicable to a different set of axioms and the like.&lt;/li&gt;&#xA;&lt;li&gt;Maybe he wanted to exercise his genius instead of taking the &quot;easy path&quot;.&lt;/li&gt;&#xA;&lt;li&gt;Maybe he wanted to demonstrate the approach/thinking/technique for later scholars benefit, rather than just arrive at a solution.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6454" OwnerDisplayName="CogitoErgoCogitoSum" LastActivityDate="2017-10-23T19:43:26.797" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="6628" PostTypeId="2" ParentId="6625" CreationDate="2017-10-23T19:01:51.623" Score="0" Body="&lt;p&gt;Yes. Somebody calls &quot;Absolute geometry&quot; the set of propositions that are provable without the $5$-th postulate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;IMHO this is an because it seems  less obvious than the others, at least as we learnt at school the version &quot;In a plane given a line and a point outside the line, there exists one and only one  line passing through the given point and parallel to the given line&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is honestly quite hard to understand because it implies an impossible infinite producing of the parallel line to &quot;verify&quot; it does not intersect the given one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Euclid does not use these words! He says, very smartly &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Let the following be postulated: [...] &lt;/p&gt;&#xA;  &#xA;  &lt;ol start=&quot;5&quot;&gt;&#xA;  &lt;li&gt;That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two&#xA;  straight lines, if produced indefinitely, meet on that side on which&#xA;  are the angles less than the two right angles.&quot;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is much easier to digest: at least the producing stops at a certain point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope it is useful&lt;/p&gt;&#xA;" OwnerUserId="6447" OwnerDisplayName="Raffaele" LastEditorUserId="6447" LastEditDate="2017-10-24T09:54:57.920" LastActivityDate="2017-10-24T09:54:57.920" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6629" PostTypeId="2" ParentId="6625" CreationDate="2017-10-23T20:38:59.047" Score="1" Body="&lt;p&gt;Euclid does not call on his fifth postulate until $I, 29$, where he cannot do without it. It is not needed until the treatment of parallels, which begins at $I, 27$.  The last of the triangle congruence theorems is $I, 26$. Euclid had some dramatic sense: it would be premature to bring postulate five onstage needlessly, and just moments before the scene that really requires it.    &lt;/p&gt;&#xA;" OwnerUserId="6453" OwnerDisplayName="Edward Porcella" LastActivityDate="2017-10-23T20:38:59.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6630" PostTypeId="2" ParentId="6624" CreationDate="2017-10-23T23:40:01.763" Score="3" Body="&lt;p&gt;Short answer: no. They did not have TV, internet, twitter, public schools or even scientific journals back then, so the group of people that got information about such things was very narrow to begin with, mostly scientists gathered in royal academies and societies. &quot;Ordinary people&quot; had more pressing concerns with surviving day to day. Gest describes the early days of microbilology in &lt;a href=&quot;http://rsnr.royalsocietypublishing.org/content/58/2/187.long&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Discovery of Microorganisms by Robert Hooke and Antoni van Leeuwenhoek&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leeuwenhoek in 1676 was not the first to observe microorganisms under the microscope, Hooke published his observations of microfungus about a decade earlier in Micrographia (1665):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;These spots appear’d, through a good Microscope, to be a very pretty shap’d&#xA;  Vegetative body, which, from almost the same part of the Leather, shot out multitudes of small long cylindrical and transparent stalks, not exactly streight, but a little bended with the weight of a round and white knob that grew on the top of each of them... The whole substance of these pretty bodies was of a very tender constitution, much like the substance of the softer kind of common white Mushroms, for by touching them with a Pin, I found them to be brused and torn; they seem’d each of them to have a distinct root of their own; for though they grew neer together in a cluster...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Leeuwenhoek's submitted his first letter to the Royal Society in 1673, and his descriptions of &quot;mould&quot; indicate that he was familiar with Micrographia. When he later (1676) observed &quot;very little animals of divers kind&quot; (bacteria) his letter was originally met with skepticism. Hooke could not observe the creatures through his microscope until he left the sample be for five to six days. He daoes express fascination and wonderment, but not of the kind associated today with extraterrestrials:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I examined again the said water; and then much to wonder I discovered vast multitudes of thoseexceeding small creatures, which Mr. Leeuwenhoeck had described; and upon making use of other lights and glasses, as I shall by and by shew, I not only magnified those I had thus discovered to a very great bigness, but I discovered many other sorts very much smaller than those I first saw, and some of these so exceeding small, that millions of millions might be contained in one drop of water. I was very much surprized at this so wonderful a spectacle, having never seen any living creature comparable to these for smallness: nor could I indeed imagine that nature had afforded instances of so exceedingly minute animal productions.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Modern scholarship suggests more generally that Hooke deserves more credit as a co-founder of microbiology, for instance what was later commonly called a (single-lens) ‘Leeuwenhoek microscope’ was already described by Hooke, and while Leeuwenhoek was notoriously secretive about how he polished his lenses, Hooke published detailed descriptions of his microscopes and their making. What &quot;ordinary people&quot; were not in a position to appreciate right away was still assimilated, but slower.  Boorstin in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Discoverers&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Discoverers&lt;/a&gt; even compares Hooke to Galileo:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;What Galileo’s &lt;a href=&quot;https://en.wikipedia.org/wiki/Sidereus_Nuncius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sidereus Nuncius&lt;/a&gt; had done for the telescope and its heavenly vistas, Hooke’s Micrographia now did for the microscope. Just as Galileo did not invent the telescope, neither did Hooke invent the microscope. But what he described seeing in his compound microscope awakened learned Europe to the wonderful world within. Fifty-seven amazing illustrations drawn by Hooke himself revealed for the first time the eye of a fly, the shape of a bee’s stinging organ,the anatomy of a flea and a louse, the structure of feathers, and the slantlike form of molds. When he discovered the honeycomb structure of cork, he said it was made of ‘cells.’ Frequently reprinted, Hooke’s illustrations remained in textbooks into the nineteenth century&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-10-24T17:53:17.713" LastActivityDate="2017-10-24T17:53:17.713" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6631" PostTypeId="1" AcceptedAnswerId="6632" CreationDate="2017-10-24T14:05:18.837" Score="3" ViewCount="154" Body="&lt;p&gt;Raymond Ayoub quoted Legendre in his paper on the early history of elliptic functions(&lt;em&gt;The Lemniscate and Fagnano's Contributions to Elliptic Integrals,1984&lt;/em&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...&lt;em&gt;lucky events which do not arise except to those who know how to give them birth.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;G. N. Watson also referred to a similar text in his paper &quot;The Marquis and the Land-agent; a Tale of the Eighteenth Century&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;...as Legendre once remarked in another connection, occur only to those who know how to produce them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is Legendre's original text of these quotations?&lt;/p&gt;&#xA;" OwnerUserId="6450" LastEditorUserId="4251" LastEditDate="2022-10-19T16:26:36.717" LastActivityDate="2022-10-19T16:26:36.717" Title="Legendre's original text on elliptic functions" Tags="|mathematics|legendre|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6632" PostTypeId="2" ParentId="6631" CreationDate="2017-10-24T15:02:02.000" Score="8" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Euler par une combinaison qu'on peut regarder comme fort heureuse,&#xA;  quoique  ces hazards n'arrivent qu'à ceux qui savent les faire naître,&#xA;  trouva cette integrale...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Adrien Marie Legendre, &lt;em&gt;Traité des fonctions elliptiques et des intégrales eulériennes: (1825) Théorie des fonctions elliptiques et son application à différens problèmes de géométrie et de mécanique&lt;/em&gt;, Imprimerie de Huzard-Courcier, page 2 (easily found in google books)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-10-24T15:02:02.000" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6633" PostTypeId="2" ParentId="6612" CreationDate="2017-10-24T22:20:43.287" Score="1" Body="&lt;p&gt;As far as I remember, there was some explanation in 'The History of the theories of aether and electricity&quot;, Vol. 1. &lt;/p&gt;&#xA;" OwnerUserId="3014" LastActivityDate="2017-10-24T22:20:43.287" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6634" PostTypeId="2" ParentId="2042" CreationDate="2017-10-25T05:09:28.690" Score="3" Body="&lt;p&gt;“Differential forms” seem to have also evolved from “differential formulas” — a terminology used by e.g. Cousin (1777):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://archive.org/stream/bub_gb_obF1teMTx6oC#page/n210&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 175&lt;/a&gt;): la quadrature et la rectification des lignes courbes (...) se réduiront visiblement à intégrer une différentielle de cette forme &lt;span class=&quot;math-container&quot;&gt;$X\,dx$&lt;/span&gt;, par &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; j'entens une fonction de &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; et de constantes: réciproquement, on pourra toujours faire dépendre l’intégrale d’une &lt;em&gt;&lt;strong&gt;formule différentielle&lt;/strong&gt;&lt;/em&gt; telle que &lt;span class=&quot;math-container&quot;&gt;$X\,dx$&lt;/span&gt;, de la quadrature ou de la rectification de quelque courbe (...)&lt;/p&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://archive.org/stream/bub_gb_obF1teMTx6oC#page/n322&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 287–289&lt;/a&gt;): On a la formule différentielle homogène &lt;span class=&quot;math-container&quot;&gt;$(hx+iy)dx-(kx+ly)dy$&lt;/span&gt; qu’il faut rendre exacte (...) Soit &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; un facteur propre à rendre intégrable la &lt;em&gt;&lt;strong&gt;formule différentielle&lt;/strong&gt;&lt;/em&gt; quelconque du premier ordre &lt;span class=&quot;math-container&quot;&gt;$\alpha\,dx + ϐ\,dy$&lt;/span&gt; (...)&lt;/p&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://archive.org/stream/bub_gb_O_f_nSIssuIC#page/n8&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 403&lt;/a&gt;): Chapitre VII: &lt;em&gt;De l'intégration des formules différentielles qui ne renferment qu’une seule variable&lt;/em&gt;. Il va être question de l’intégration de la &lt;em&gt;&lt;strong&gt;formule différentielle&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$X\,dx$&lt;/span&gt; (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Other examples: Agnesi (&lt;a href=&quot;https://books.google.com/books?id=XQ1bAAAAQAAJ&amp;amp;q=%22formule+diff%C3%A9rentielle%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1775&lt;/a&gt;), du Bourguet (&lt;a href=&quot;https://books.google.com/books?id=UlzSjYXJL1gC&amp;amp;pg=PA83&amp;amp;dq=%22formule+diff%C3%A9rentielle%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1810&lt;/a&gt;), Garnier (&lt;a href=&quot;https://books.google.com/books?id=h2tbAAAAQAAJ&amp;amp;q=%22formule+diff%C3%A9rentielle%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1812&lt;/a&gt;), Abel (&lt;a href=&quot;https://books.google.com/books?id=ZppGAAAAcAAJ&amp;amp;pg=PA185&amp;amp;dq=Differential+Formel&quot; rel=&quot;nofollow noreferrer&quot;&gt;1826&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=oZpGAAAAcAAJ&amp;amp;pg=PA264&amp;amp;dq=formules+diff%C3%A9rentielles&quot; rel=&quot;nofollow noreferrer&quot;&gt;1829&lt;/a&gt;), Liouville (&lt;a href=&quot;https://books.google.com/books?id=E6rxAAAAMAAJ&amp;amp;pg=PA95&amp;amp;dq=formule+diff%C3%A9rentielle&quot; rel=&quot;nofollow noreferrer&quot;&gt;1835&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=5c5GAAAAcAAJ&amp;amp;pg=PA6&amp;amp;dq=formule+diff%C3%A9rentielle&quot; rel=&quot;nofollow noreferrer&quot;&gt;1851&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=mMVDfrvRyb0C&amp;amp;pg=PA192&amp;amp;dq=formule+diff%C3%A9rentielle&quot; rel=&quot;nofollow noreferrer&quot;&gt;1856&lt;/a&gt;).&#xA;Initially some authors&lt;sup&gt;1&lt;/sup&gt; only used “differential formula” for &lt;em&gt;exact&lt;/em&gt; forms, calling the general &lt;span class=&quot;math-container&quot;&gt;$\sum A_i\,dx_i$&lt;/span&gt; another name such as “differential”&lt;sup&gt;2&lt;/sup&gt;, “formula”&lt;sup&gt;3&lt;/sup&gt;, or “differential expression”&lt;sup&gt;4&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Bernoulli (&lt;a href=&quot;https://books.google.com/books?id=Db9eAAAAcAAJ&amp;amp;pg=PP739&amp;amp;dq=differentielle&quot; rel=&quot;nofollow noreferrer&quot;&gt;1712&lt;/a&gt;), Euler (&lt;a href=&quot;https://books.google.com/books?id=XLQEAAAAQAAJ&amp;amp;pg=PA107&amp;amp;dq=%22formules%20diff%C3%A9rentielles%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1767&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/novaactaacademia07impe#page/n86&quot; rel=&quot;nofollow noreferrer&quot;&gt;1793&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/novaactaacademia07impe#page/99&quot; rel=&quot;nofollow noreferrer&quot;&gt;1793&lt;/a&gt;), Lagrange (&lt;a href=&quot;https://archive.org/stream/mmoiresdelacadm04turigoog#page/n229&quot; rel=&quot;nofollow noreferrer&quot;&gt;1786&lt;/a&gt;).&lt;br&gt;&#xA;&lt;sup&gt;2&lt;/sup&gt; Le Seur–Jacquier (&lt;a href=&quot;https://archive.org/stream/elementsducalcu00jacqgoog#page/n23&quot; rel=&quot;nofollow noreferrer&quot;&gt;1768&lt;/a&gt;), Sauri (&lt;a href=&quot;https://books.google.com/books?id=-GUVAAAAQAAJ&amp;amp;pg=PA291&amp;amp;dq=diff%C3%A9rentielle&quot; rel=&quot;nofollow noreferrer&quot;&gt;1774&lt;/a&gt;).&lt;br&gt;&#xA;&lt;sup&gt;3&lt;/sup&gt; Euler (&lt;a href=&quot;https://archive.org/details/leonhardieuleri02eulegoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;1770&lt;/a&gt;), Lexell (&lt;a href=&quot;https://www.biodiversitylibrary.org/item/38594#page/27&quot; rel=&quot;nofollow noreferrer&quot;&gt;1772&lt;/a&gt;), Bossut (&lt;a href=&quot;https://books.google.com/books?id=pro_AAAAcAAJ&amp;amp;pg=PA127&amp;amp;dq=formule&quot; rel=&quot;nofollow noreferrer&quot;&gt;1798&lt;/a&gt;), Poisson (&lt;a href=&quot;https://books.google.com/books?id=zt2wKlNovI8C&amp;amp;pg=PA291&amp;amp;dq=formule&quot; rel=&quot;nofollow noreferrer&quot;&gt;1811&lt;/a&gt;).&lt;br&gt;&#xA;&lt;sup&gt;4&lt;/sup&gt; Gauss (&lt;a href=&quot;https://archive.org/stream/werkecarlf03gausrich#page/n245&quot; rel=&quot;nofollow noreferrer&quot;&gt;1815&lt;/a&gt;), Jacobi (&lt;a href=&quot;https://books.google.com/books?id=EgVCAAAAcAAJ&amp;amp;pg=PA240&amp;amp;dq=expressio+differentialis&quot; rel=&quot;nofollow noreferrer&quot;&gt;1845&lt;/a&gt;), Sturm (&lt;a href=&quot;https://books.google.com/books?id=N-I3AAAAMAAJ&amp;amp;pg=PA38&amp;amp;dq=expression+diff%C3%A9rentielle&quot; rel=&quot;nofollow noreferrer&quot;&gt;1877&lt;/a&gt;), Frobenius (&lt;a href=&quot;https://books.google.com/books?id=ME40AQAAMAAJ&amp;amp;pg=PA231&amp;amp;dq=Differentialausdruck&quot; rel=&quot;nofollow noreferrer&quot;&gt;1877&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=85HxAAAAMAAJ&amp;amp;pg=PA1&amp;amp;dq=Differentialausdruck&quot; rel=&quot;nofollow noreferrer&quot;&gt;1879&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;As mentioned in comments to M. Bächtold’s answer, the switch to “differential form” (and merger with the multilinear terminology of Grassmann(?), Kronecker, Weierstrass) seems to have occurred in papers of Christoffel (&lt;a href=&quot;https://books.google.com/books?id=9D9bAAAAcAAJ&amp;amp;pg=PA56&amp;amp;q=%22lineare+Form%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1869&lt;/a&gt;), Lipschitz (&lt;a href=&quot;https://books.google.com/books?id=9D9bAAAAcAAJ&amp;amp;pg=PA74&amp;amp;q=%22Formen+von%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1869&lt;/a&gt;), Darboux (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3050z/f837&quot; rel=&quot;nofollow noreferrer&quot;&gt;1882&lt;/a&gt;, &lt;a href=&quot;http://www.numdam.org/item?id=BSMA_1882_2_6_1_14_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1882&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-10-27T01:46:31.807" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6635" PostTypeId="1" CreationDate="2017-10-25T05:34:39.250" Score="2" ViewCount="6370" Body="&lt;p&gt;Double-paned glass windows have a hollow space between two panes of glass, usually just filled with air. This can make a much better insulator than normal glass, according to &lt;a href=&quot;https://www.engineeringtoolbox.com/heat-loss-transmission-d_748.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Engineering Toolbox&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know when and where the first double-paned glass was made, and by whom? How was it constructed?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="2546" LastEditDate="2017-10-26T01:16:44.910" LastActivityDate="2017-10-26T01:16:44.910" Title="When was double-paned glass first invented and used for insulation?" Tags="|thermodynamics|engineering|technology|materials-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6636" PostTypeId="2" ParentId="6635" CreationDate="2017-10-25T12:32:03.450" Score="5" Body="&lt;p&gt;lmgtfy&lt;br&gt;&#xA;&lt;a href=&quot;http://webcache.googleusercontent.com/search?q=cache:CRrm-gkPvE0J:directory.mimivanderhaven.com/article/a-history-of-glass.html%20&amp;amp;cd=4&amp;amp;hl=en&amp;amp;ct=clnk&amp;amp;gl=us&quot; rel=&quot;noreferrer&quot;&gt;google cache of mimivanderhave.com&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1945  The first double-pane, insulated window was introduced&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's without attribution;   &lt;a href=&quot;http://www.doorandwindow.com/windows/types/double-pane/double-pane-windows-and-ancient-history.html&quot; rel=&quot;noreferrer&quot;&gt;door and window&lt;/a&gt; claims&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From single to double pane windows… The true evolution of double pane&#xA;  windows originated in 1913 when the Andersen Lumber Co. became the&#xA;  first to manufacture wood window frames. In 1932 they added the sash&#xA;  and hardware, and two years later, even the glazing. Making the entire&#xA;  factory-squared, tight-fitting assembly was the first step in laying&#xA;  the groundwork for what would become today's double pane windows for&#xA;  the home. In 1952, they introduced a Welded Insulated Glass window,&#xA;  and the double pane window was born.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An interesting note over at &lt;a href=&quot;http://www.ecobuildingpulse.com/products/efficient-window-technologies-a-history_s&quot; rel=&quot;noreferrer&quot;&gt;ecobuildingpulse&lt;/a&gt;  says: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Robert Struble, communications manager at PPG Industries (formerly&#xA;  Pittsburgh Plate Glass), says that the concept of double glazing&#xA;  originated in the 1860s. An inventor filed a claim with the U.S.&#xA;  Patent and Trademark Office for an insulating glass product consisting&#xA;  of two glass panes separated by rope and bound with tar. “Of course,&#xA;  nothing happened with [that product],” Struble says. “Nobody used it,&#xA;  and it was not necessarily commercially viable.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1940s and '50s: Double and Triple Glazing&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The east- and west facings of the United Nations' headquarters in New&#xA;  York, completed in 1953, are clad in insulated glazing. Muhammad&#xA;  Ghouri/Flickr The east- and west facings of the United Nations'&#xA;  headquarters in New York, completed in 1953, are clad in insulated&#xA;  glazing. In 1945, PPG developed one of the first commercially viable&#xA;  double-glazed IGUs in the U.S. Though they were originally used for&#xA;  Pullman railroad cars, the units found their way into buildings the&#xA;  following year, reducing U-factors from the 1-plus values of&#xA;  single-glazed units to 0.47.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2017-10-25T12:32:03.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6637" PostTypeId="1" CreationDate="2017-10-25T16:32:14.023" Score="4" ViewCount="164" Body="&lt;p&gt;Faraday, after drawing his lines of force in 19th century, is normally credited as the first to use the term field in physics. But...&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;... was not the term field used in the context of gravitational field already in the 17th or 18th century? &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Moreover, Euler in the 18th century studied fluid flows by looking at every point of the fluid and associating to that point a vector corresponding to the velocity of the particles passing through that point. This results in the velocity field which is actually a vector field. &lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Did Euler use any specific term for that velocity field? Did Euler draw field lines?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;So, related to the previous questions,&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Why does Faraday deserve all the credit?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5436" LastActivityDate="2017-10-25T16:32:14.023" Title="Who was the first to use the term field in physics?" Tags="|physics|terminology|classical-mechanics|electromagnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6638" PostTypeId="1" CreationDate="2017-10-26T06:27:16.847" Score="11" ViewCount="2908" Body="&lt;p&gt;Of course much of this can be debated on what you mean by the word &amp;ldquo;modern&amp;rdquo;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But most of us would agree that the Arabic World and places like India were the leading mathematical and scientific civilizations way beyond Europe for centuries, all throughout the middle ages and even beyond that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then suddenly that all completely shifted. Starting with Copernicus, and then Galileo and Kepler and culminating with Newton, Europe started to become the main powerhouse of science and mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why wasn't the Arabic World and India able to maintain their &amp;ldquo;scientific dominance&amp;rdquo; why couldn't they stay on level with Europe's increasing scientific hold? Why did someone like Newton emerge in the West and not the Arabic World or India&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cultural reasons? Cultural shifts? Was European society structured differently that allowed people to flourish in the sciences. what are the main reasons Europe become the leading scientific power?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastEditorUserId="5989" LastEditDate="2017-10-27T18:11:18.503" LastActivityDate="2019-08-18T18:29:38.280" Title="Why was modern science and mathematics a European phenomenon?" Tags="|mathematics|mathematicians|philosophy-of-science|" AnswerCount="6" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6640" PostTypeId="2" ParentId="6638" CreationDate="2017-10-26T17:06:12.620" Score="3" Body="&lt;p&gt;One cannot give a definitive answer to this question, because we do not have&#xA;an &quot;alternative history&quot; to compare. I can repeat what I answered on a somewhat similar question here:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/666/why-were-20th-century-german-scientists-so-impressive/672#672&quot;&gt;Why were 20th Century German scientists so impressive?&lt;/a&gt;&#xA;Development of science correlates with strong economic and social development.&#xA;I mean that since 16 or 17 century Europe rushed ahead not only in science but&#xA;in &quot;everything else&quot;.&#xA;But what is the cause and what is the outcome, is open to speculation.&#xA;Fast development of science, capitalism and technology start almost simultaneously,&#xA;and clearly they are interconnected. Probably the root of all of this lies in the organization of society which made this development possible. Possibly this is the general idea of individual freedom which evolved in Europe for historical reasons.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-10-26T17:06:12.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6641" PostTypeId="2" ParentId="6638" CreationDate="2017-10-26T20:04:18.387" Score="2" Body="&lt;p&gt;Leaving aside ambiguities and vagueness, let’s say that there are at least three scales at which to look for an answer. At the largest one are Jared Diamond’s books or  Ian Morris’ &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Why_the_West_Rules%E2%80%94For_Now&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why the West rules - for now&lt;/a&gt;&lt;/em&gt;. More modestly one can look through some books about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientific_revolution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scientific revolution &lt;/a&gt;which is still a fashionable topic among hsm writers. Lastly one can develop a narrative about some kind of cultural shift . Thus one could identify three major factors: in the economy, a revival brought about by Spanish Empire (see e.g. Braudel’s &lt;em&gt;Mediterrannee&lt;/em&gt;), a new effective circulation of information and ideas due to the printing press and in the European society the shattering of traditional religion caused by the Reformation. Perhaps a good word should be said here for Steven Toulmin’s books (&lt;em&gt;Cosmopolis&lt;/em&gt;, &lt;em&gt;Return to Reason&lt;/em&gt;) which offer a revisionist narrative, pointing that Renaissance humanism with its interest in art and letters has degenerated into a quest for certainty and effectiveness. According to him the SciR is actually a counter-revolution to the Renaissance, that is a relapse into a new single dimensional ideology without alternatives. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there are also different other accounts insisting more on ‘intrinsic factors’ and perhaps they look more satisfying. There is an analogy between the s.c. Greek miracle and The SciR: soon after inventing the phonetic alphabet, ancient Greeks were writing books on philosophy and other forms of theory; soon after the German cossists and the &lt;em&gt;logistica speciosa&lt;/em&gt; Europeans were developping theoretical physics. There is perceptible line from Viete’s logistica, which is algebra, to Descartes’ Geometrie, which unified algebra and geometry, and further to Leibniz’ extended work on mathematics and its symbols. In his own view what emerged was a kind of autonomous infallible reason, which he called sometimes &lt;em&gt;cognitio caeca&lt;/em&gt;, that is the ability to arrive at valid knowledge about the word from purely formal operations. Leibniz is famous (Toulmin also insists here) for stating that disputes could be resolved by calculations - which supposedly would avert disputes and even wars (in his time the 30 years war, most murderous in European history, was still present in memories).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So narrowing down the scope one could arrive at the figure of Rene Descartes who decided to believe nobody and published his &lt;em&gt;Discourse&lt;/em&gt; examplified by the &lt;em&gt;Geometrie&lt;/em&gt;. &#xA;Newton started by disagreeing with Descartes’ physics (and philosophy), just as Leibniz who never stopped criticizing him. Actually it was a badly shaken society which produced the resonance to such work. The Enligthment lasted as long as science was more comprehensible and/or convincing than the mysteries of religion, a balance than appears to have shifted lately. &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-10-26T20:04:18.387" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6642" PostTypeId="2" ParentId="6638" CreationDate="2017-10-27T08:38:06.600" Score="5" Body="&lt;p&gt;Europe had an advantage which is often overlooked: the fact that it was (and it is) divided into many countries. As far as intellectual pursuits are concerned, this is an advantage, because whenever a scholar was in an environment in which he was in risk of being punished because of his ideas, he could move to somewhere else. Or, at least, his texts could be published somewhere else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there are other factors, such as a level of economic prosperity which allowed some individuals to focus their minds in intelectual matters, but this happened also at another times and another places.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-10-27T08:38:06.600" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6643" PostTypeId="1" CreationDate="2017-10-27T19:45:08.703" Score="0" ViewCount="95" Body="&lt;p&gt;Or more broadly when did people start marking everything against a common level? Or say when was it known how many feet Rome was above the sea, or if New Delhi or Chengdu was higher?&lt;/p&gt;&#xA;" OwnerUserId="6468" LastActivityDate="2017-11-09T01:28:01.237" Title="When was sealevel discovered?" Tags="|measurement|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6644" PostTypeId="2" ParentId="6617" CreationDate="2017-10-27T20:22:33.027" Score="2" Body="&lt;p&gt;Both MathSciNet and APS do not even find Fulling'77 (probably because &quot;ultrastatic&quot; is not used in the abstract). The earliest metadata occurrences are Fulling, S. A. Narcowich, F. J. Wald, Robert M. Singularity Structure of the Two-point Function in Quantum Field Theory in Curved Spacetime, II. Ann. Physics 136 (1981), no. 2, 243–272; and  Page, Don N. Thermal Stress Tensors in Static Einstein Spaces, Phys. Rev. D (3) 25 (1982), no. 6, 1499–1509. 83C47 (81E20). This suggests that &quot;ultrastatic&quot; was not a term/keyword before 1980s. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, Fulling'77 does not credit or reference anybody when it defines &quot;ultrastatic&quot; on p. 919, and on p. 948 it even states: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This procedure and interpretation are presumably physically &#xA;  trustworthy when the orbits of the Killing vector are also the geodesics normal to each constant-time hypersurface, as in the class of models &lt;strong&gt;we have called   ‘ultrastatic’&lt;/em&gt;&lt;/strong&gt;&quot;. [emphasis mine]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, this could just mean &quot;we have called within this paper&quot;, but then it would be odd not to mention the source of definition. Moreover, according to Fulling, his inspiration was Unruh, specifically &lt;a href=&quot;https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.870&quot; rel=&quot;nofollow noreferrer&quot;&gt;Unruh's Notes on Black-Hole Evaporation, Phys. Rev. D 14, 870 (1976)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The key observation, due to Unruh (1976), is that the horizon can be embedded in a  flat or ultrastatic space-time; equivalently, that each of the two null surfaces making up the horizon is invariant under translation in its affine parameter, V or U.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But Unruh's paper does not use the term &quot;ultrastatic&quot; despite deriving results about ultrastatic spacetimes. So it seems entirely plausible to me that Fulling came up with the term himself.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-10-29T03:02:14.510" LastActivityDate="2017-10-29T03:02:14.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6645" PostTypeId="1" CreationDate="2017-10-28T03:23:52.213" Score="1" ViewCount="418" Body="&lt;p&gt;Rather it be within a single discipline and the sub fields within it. Or completely different scientific fields. People that had significant contributions in many fields?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who are some of the most prominent polymaths before the era of specialization truly kicked in?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2017-10-28T03:23:52.213" Title="Who are considered the last true polymaths of math and science" Tags="|polymath|" AnswerCount="0" CommentCount="5" ClosedDate="2017-11-09T15:32:57.127" ContentLicense="CC BY-SA 3.0" />
  <row Id="6646" PostTypeId="2" ParentId="6638" CreationDate="2017-10-28T10:42:43.457" Score="8" Body="&lt;p&gt;It's hard to put a precise date without generating debate, but I'd put forward that Europe took the lead at some point between the early 14th to late 15th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Several processes were in full motion around then:&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Medieval_technology&quot; rel=&quot;noreferrer&quot;&gt;Medieval Scientific Dynamism&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Where there was indeed a dearth of books and literate scholars in the Early Middle Ages, science and technology in the Mid- to Late Medieval Europe actually was a lot more dynamic than is often credited. It saw developments such as the Three Field System, plowing-related technologies, mill-related technologies, etc. - all of which contributed to significantly boost the agricultural output.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also worth highlighting during the period are &lt;strong&gt;universities&lt;/strong&gt;, which began to appear between the 11th and 13th century depending on the region.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My point here is that it's incorrect to assume Europe was roamed by uneducated peasants &lt;em&gt;only&lt;/em&gt;. And the Byzantine Empire to the East was far from backwards, and still interacting with Western Europe. Science never really died during the period.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;noreferrer&quot;&gt;Hindu-Arabic Numerals&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Fibonacci is known nowadays mostly for the sequence named after him, but the latter actually is a mere example problem he gave in passing within Liber Abaci, a book he published in 1202.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Liber Abaci introduced Hindu-Arabic numerals and place value to Europe, enjoyed a very wide success. It was transformative for both science and finance in Northern Italy and then Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being a savvy marketer in his time, he followed it with a shorter, more to the point version for use by merchants. One could argue that it laid the foundations that were necessary to lift Europe out of the middle ages.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Reconquista&quot; rel=&quot;noreferrer&quot;&gt;The Reconquest of Spain&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;When one thinks Renaissance, one usually thinks about the rediscovery of Classic arts and literature. But this is actually hiding the reality that preceded it in the late Middle Ages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numerous scholars went to Spain to study the very large collection of scientific literature that the Muslims had left behind - particularly after Seville fell in 1248. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Norman-Arab-Byzantine_culture&quot; rel=&quot;noreferrer&quot;&gt;Sicily&lt;/a&gt; arguably was a major cultural gateway before that.)&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Printing_press&quot; rel=&quot;noreferrer&quot;&gt;The Printing Press&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Gutenberg's invention of metal movable type around 1440 allowed to mass print books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bible was, of course, the top seller. Most titles below it were scientific in nature. (In particular works based on Fibonacci's book.)&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Greek_scholars_in_the_Renaissance&quot; rel=&quot;noreferrer&quot;&gt;Byzantine Refugees&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Waves of Byzantine scholars and émigrés came to Italy in the aftermath of the Turks conquering Constantinople and dismantling the Byzantium Empire in 1453.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They brought numerous manuscripts with them, ultimately setting the stage for the Renaissance.&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Islamic_attitudes_towards_science&quot; rel=&quot;noreferrer&quot;&gt;Change of Muslim Attitude Towards Science&lt;/a&gt;&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Depending on the historian you ask, the pivotal point occurred between the 14th and the 16th century. Whichever date it is, what's clear is that the period saw the &quot;rise of a clerical faction which froze this same science and withered its progress&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Taqi_ad-Din_Muhammad_ibn_Ma%27ruf&quot; rel=&quot;noreferrer&quot;&gt;Tarqi ad-Din&lt;/a&gt; observatory in Istanbul, for instance, was ultimately destroyed by a squad of Janissaries &quot;by order of the sultan, on the recommendation of the Chief Mufti&quot; around 1577.&lt;/p&gt;&#xA;" OwnerUserId="6471" LastActivityDate="2017-10-28T10:42:43.457" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6647" PostTypeId="1" CreationDate="2017-10-28T21:39:12.993" Score="2" ViewCount="170" Body="&lt;p&gt;Complex numbers were used in as early as the 16th century to solve cubic equations, but they didn't gain wide acceptance until the late 18th and early 19th century. What is the reason for the 200 year delay in accepting imaginary numbers? Were there any famous mathematicians (or philosophers) that renounced imaginary numbers, and why?&lt;/p&gt;&#xA;" OwnerUserId="6348" LastActivityDate="2017-10-28T21:39:12.993" Title="18th and 19th century skeptics of imaginary numbers?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6648" PostTypeId="1" CreationDate="2017-10-29T13:12:23.777" Score="9" ViewCount="1740" Body="&lt;p&gt;The Chinese Remainder Theorem says, in rough terms, that if you know the remainders of an integer $n$ modulo $m_1,m_2,\dots,m_r$, you also know $n$ modulo $\mathrm{lcm}(m_1,m_2,\dots,m_r)$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the mathematical folklore, I've often heard it associated to the problem of counting the size of an army. That comes in two flavours. The first (which seems pretty unlikely) is that instead of counting troops, the Chinese generals would order them to assemble in rows of different depths (hence obtaining the remainder of the number of soldiers modulo different integers) and then they would use the Chinese Remainder Theorem to compute the size of the army. Another version which I've heard says that the Chinese Remainder Theorem was used to guard against spies. Instead of recording the number of soldiers in an army, the Chinese would record remainders modulo various numbers (presumably in hope that the intended recipient would be able to apply the theorem to compute the original number, but the spy would not).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Has any of these things actually happened?&lt;/p&gt;&#xA;" OwnerUserId="3068" LastEditorUserId="55" LastEditDate="2017-10-31T19:57:27.583" LastActivityDate="2017-10-31T19:57:27.583" Title="Has Chinese Remainder Theorem ever been used by Chinese military?" Tags="|mathematics|debunking|ancient-china|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6649" PostTypeId="2" ParentId="6648" CreationDate="2017-10-29T22:40:52.960" Score="3" Body="&lt;p&gt;I am not able to prove that it was never used in the ways that you mentioned, but it's highly unlikely, to say the least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The oldest known appearence of that theorem is in a Chinese text, the &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sunzi_Suanjing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sunzi Suanjing&lt;/a&gt;&lt;/em&gt; (3&lt;sup&gt;rd&lt;/sup&gt; to 5&lt;sup&gt;th&lt;/sup&gt; centuries AD). It is stated as a purely mathematical problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are certain things whose number is unknown. A number is repeatedly divided by $3$, the remainder is $2$; divided by $5$, the remainder is $3$; and by $7$, the remainder is $2$. What will the number be?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As you can see, there is no reference to counting soldiers here. The theorem only reappears in a Chinese text in the 13&lt;sup&gt;th&lt;/sup&gt; century, in two books: the &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_Treatise_in_Nine_Sections&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Treatise in Nine Sections&lt;/a&gt;&lt;/em&gt; written by Qin Jiushao in 1247 and in a book written by Yang Hui around 1275. Again, there is no reference to applications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, again, I very much doubt that it has ever been used for counting soldiers in ancient China.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2017-10-29T22:54:43.970" LastActivityDate="2017-10-29T22:54:43.970" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6650" PostTypeId="2" ParentId="6648" CreationDate="2017-10-29T22:45:10.537" Score="8" Body="&lt;p&gt;Is it possible that &quot;Chinese remainder theorem&quot; was used by the Chinese military? Who knows. But we know that the proliferation of Chinese generals and soldiers in connection with it comes from late fables. The original source of the &quot;theorem&quot; is &lt;a href=&quot;https://en.wikipedia.org/wiki/Sunzi_Suanjing&quot; rel=&quot;noreferrer&quot;&gt;Sunzi Suanjing&lt;/a&gt;, which has nothing to say about counting soldiers. The problem is stated thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book is dated to 3-5th century AD, and nothing is known about the author, but the name Sun Zi is spelled the same way as &lt;a href=&quot;https://en.wikipedia.org/wiki/Sun_Tzu&quot; rel=&quot;noreferrer&quot;&gt;Sun Tzu&lt;/a&gt;, the famous Chinese general who wrote The Art of War. That Sun Tzu lived c. 500 BC does not stop some modern enthusiasts from attributing Sunzi Suanjing to him anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another Chinese general associated with the theorem is &lt;a href=&quot;https://en.wikipedia.org/wiki/Han_Xin&quot; rel=&quot;noreferrer&quot;&gt;Han Xin&lt;/a&gt;, who lived c. 200 BC. A &lt;a href=&quot;https://chinesetuition88.com/2015/04/25/chinese-remainder-theorem-history-%E9%9F%A9%E4%BF%A1%E7%82%B9%E5%85%B5&quot; rel=&quot;noreferrer&quot;&gt;Singaporean blogger&lt;/a&gt; tells an &quot;amazing&quot; story of him and his army of 1500 soldiers estimated 400-500 of which died in a battle:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;When the soldiers stood 3 in a row, there were 2 soldiers left over. When they lined up 5 in a row, there were 4 soldiers left over. When they lined up 7 in a row, there were 6 soldiers left over. Han Xin immediately said, “There are 1049 soldiers&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;No source for this amazing story is given (it is repeated or mentioned in many other places though), but from &lt;a href=&quot;https://books.google.com/books?id=AG2XBCmxYcUC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Volkov's paper in From China to Paris, p.402&lt;/a&gt; we learn that &lt;em&gt;Han Xin dian bing&lt;/em&gt; (Han Xin's way of counting soldiers) dates to 13th century AD, when Cheng Dawei mentions it as a name of a rhymed stanza. &lt;em&gt;Qin wang an dian bing&lt;/em&gt; (Prince of Qin's method of secretly counting soldiers) occurs somewhat earlier in &lt;a href=&quot;https://en.wikipedia.org/wiki/Yang_Hui&quot; rel=&quot;noreferrer&quot;&gt;Yang Hui's Xiangjie Jiuzhang Suanfa (c. 1265 AD)&lt;/a&gt;, better known for  magic squares, magic circles and Yang Hui's (a.k.a. Pascal's) triangle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This explosion of popularizations seems to follow &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_Treatise_in_Nine_Sections&quot; rel=&quot;noreferrer&quot;&gt;Qin Jiushao's Shùshū Jiǔzhāng (c.1245 AD)&lt;/a&gt;, which presented a complete solution to the remainder systems of equations with multiple examples. As usual with anecdotes, memorable illustrations were then invented and ascribed to famous historical figures, acquiring more and more details over the subsequent centuries. The name &quot;Chinese Remainder Theorem&quot; comes from 19th century Europe, see &lt;a href=&quot;https://mathoverflow.net/questions/11951/what-is-the-history-of-the-name-chinese-remainder-theorem&quot;&gt;What is the history of the name “Chinese remainder theorem”?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a more mathematically focused history see &lt;a href=&quot;http://www.math.harvard.edu/~knill/crt/lib/Kangsheng.pdf&quot; rel=&quot;noreferrer&quot;&gt;Kangsheng's Historical development of the Chinese remainder theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-10-30T00:55:30.653" LastActivityDate="2017-10-30T00:55:30.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6651" PostTypeId="1" CreationDate="2017-10-31T00:13:46.170" Score="5" ViewCount="1396" Body="&lt;p&gt;It seems like the focus always tends to be on the achievement of Greek math (which strikes of eurocentrism a little bit) while civilizations like the Egyptians and Babylonians are overlooked&lt;/p&gt;&#xA;&#xA;&lt;p&gt;why do we overlook their mathematical contributions?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2019-12-10T21:16:27.620" Title="how sophisticated was Egyptian and Babylonian mathematics compared to the Greeks" Tags="|ancient-greece|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6652" PostTypeId="1" CreationDate="2017-10-31T06:47:27.713" Score="2" ViewCount="677" Body="&lt;p&gt;In the history of science and technology, there are some cases when scientists did not believe something that later turned out to be real.&lt;/p&gt;&#xA;&lt;p&gt;For example, Bill Gates was saying 640K is sufficient to any computer :)&lt;/p&gt;&#xA;&lt;p&gt;Did such thing happen in the history of nuclear weapon? Did someone famous not believe nuclear weapons were possible?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Maybe some problems were regarded as unsolvable initially, for example obtaining pure enough fuel or handling undercritical masses?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE 2&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Urban legends also ok. For example, if it was popular opinion that A said something impossible.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE 3&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let's try to narrow the time. Let it be before the discovery of neutron and fission (1932). Wasn't it possible to understand that that the nucleus contain a lot of binding energy, but not knowing about fission to doubt that it can be extracted?&lt;/p&gt;&#xA;" OwnerUserId="6479" LastEditorUserId="6504" LastEditDate="2022-11-14T07:28:28.467" LastActivityDate="2022-11-14T07:28:28.467" Title="Did any famous physicists think nuclear weapons were impossible?" Tags="|nuclear-physics|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="6653" PostTypeId="2" ParentId="6652" CreationDate="2017-10-31T07:11:53.340" Score="3" Body="&lt;p&gt;I've never heard of a physicist denying the possibility of building atomic weapons. The closest thing that I am aware of is Werner Heisenberg's statement &lt;a href=&quot;https://www.atomicheritage.org/history/german-atomic-bomb-project&quot; rel=&quot;nofollow noreferrer&quot;&gt;“I don't believe a word of the whole thing,”&lt;/a&gt; concerning the use of atomic weapons in Japan. But he was not denying the possibility of building the weapons. He just did not believe that the Americans had actually been able to do it.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="229" LastEditDate="2017-10-31T12:45:07.017" LastActivityDate="2017-10-31T12:45:07.017" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6654" PostTypeId="2" ParentId="6643" CreationDate="2017-10-31T13:27:29.270" Score="1" Body="&lt;p&gt;You probably mean MSL, which is Mean Sea Level. In order to calculate the predicted height of a tide we work with amplitudes relative to a mean level as the tide approximates a sinusoidal behaviour. It was therefore necessary to measure the level of the sea surface relative to some arbitrary reference level, called Chart Datum (C.D.). As the tides vary in behaviour and amplitude throughout the world the MSL varies from place to place. By observing and recording the tidal heights over long periods of time it is possible to derive the height of MSL relative to this C.D. Unfortunately weather patterns like wind and atmospheric pressure influence tides as well, which also influences MSL and in various locations around the world the Tide Tables will indicate the variation in MSL during a year. As the need for this info arose with the advent of needing navigation charts it probably started in the early 1600's. I don't know whether the Romans were even interested in knowing how high Rome was above MSL.&lt;/p&gt;&#xA;" OwnerUserId="6481" LastActivityDate="2017-10-31T13:27:29.270" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6655" PostTypeId="2" ParentId="6651" CreationDate="2017-10-31T13:36:35.793" Score="1" Body="&lt;p&gt;It was incomparable to Greek mathematics. So much that many historians who studied the subject claimed that &#xA;mathematics was invented by the Greeks. Neither Egyptian nor Babylonian mathematics had a notion of proof. And real mathematics begins only with invention of proof (Of course this depends on how we define &quot;mathematics&quot;, but proof is what makes mathematics different from all other sciences). In any case there is no trace of mathematical proofs in any culture but Greek. In this exact sense the Greeks invented mathematics. See&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. L. van der Waerden, Science Awakening.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;O. Neugebauer, Exact sciences in antiquity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;L. Russo, Forgotten revolution.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Interestingly, Greek historians themselves claimed that the sources of Greek mathematics lie in Egypt. Modern research does not confirm this legend.&#xA;These Greek historians did not understand what they were writing about.&#xA;For example, they tell us that Thales traveled to Egypt, presumably to study.&#xA;And they also tell us how he surprised Egyptian priests by measuring the height of a pyramid. So the modern historians are puzzled: If he surprised them with such a minor thing, what could they possibly teach him in Egypt?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. On &quot;eurocentrism&quot; mentioned in a comment. Nowadays mathematics is cultivated in many places, and great contribution to it comes from places outside Europe, for example from China and Japan and other places. However, WHAT do they study in schools, universities, and what they develop? Greek mathematics, not Mesopotamian, not indigenous Chinese, not indigenous Japanese etc.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-10-31T19:46:08.197" LastActivityDate="2017-10-31T19:46:08.197" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6656" PostTypeId="2" ParentId="6651" CreationDate="2017-10-31T16:48:27.593" Score="6" Body="&lt;p&gt;The achievements of Babylonian mathematics are on a par with the Greek ones. Actually a rather significant part of Greek mathematics is a rewriting in a more modern style of Babylonian material. In contrast, Egypt which received even earlier a similar input, did not find interesting developments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much new material and ideas have been presented by Jöran Friberg in the two volumes of solid work (published unfortunately with bombastic titles):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Unexpected Links Between Egyptian and Babylonian Mathematics&lt;/em&gt;, Singapore: World Scientific (2005),&#xA;&lt;em&gt;Amazing Traces of a Babylonian Origin in Greek Mathematics&lt;/em&gt;, 2007, World Scientific &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Clay tablets &lt;a href=&quot;https://en.wikipedia.org/wiki/Plimpton_322&quot; rel=&quot;noreferrer&quot;&gt;Plimpton 322&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/YBC_7289&quot; rel=&quot;noreferrer&quot;&gt;YBC 7289 &lt;/a&gt;are famous and offer indubitable evidence of sophistication, e.g. pythagorean triples larger than 10000 and  srt(2) with six digits.&#xA;The crux of the history, of course, is the discovery of incommensurables by the Greeks, that is the failing of the old programme &quot;Everything is a Number&quot;; no satisfactory solution being available, a new programme was launched with the moto &quot;Geometry is Real&quot;. However in astronomy, during Hellenistic times, the Greeks were still using Babylonian computational techniques (see e.g. A Jones work). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The accessible sources are not equivalent: there are no traces what arguments or ideas did the Babylonian use, but what they have left are not the results of guesswork. In contrast the text known under the name of Euclides is obviously focused on method.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-10-31T16:48:27.593" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6657" PostTypeId="1" CreationDate="2017-11-01T02:39:03.060" Score="-1" ViewCount="218" Body="&lt;p&gt;Was Heliocentrism something that exploded with Copernicus seemingly out of nowhere, or were there predecessors, and what were those predecessors?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read somewhere that their might of been versions before Copernicus?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2017-11-06T03:42:16.227" Title="What were the precursor works to Copernicus Heliocentrism?" Tags="|discoveries|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6658" PostTypeId="1" AcceptedAnswerId="6665" CreationDate="2017-11-01T03:12:35.703" Score="5" ViewCount="463" Body="&lt;p&gt;It is said &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Beginnings_of_set_theory.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set theory was becoming the centre of controversy. Kronecker, who was on the editorial staff of Crelle's Journal, was unhappy about the revolutionary new ideas contained in Cantor's paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However the authors of the entry don't elaborate on reasons why Kronecker was so unhappy, which prompted my curiosity.&lt;/p&gt;&#xA;" OwnerUserId="6274" LastEditorUserId="4251" LastEditDate="2022-04-27T05:25:38.870" LastActivityDate="2022-04-27T05:25:38.870" Title="Why was Kronecker dissatisfied with Cantor's submitted paper?" Tags="|mathematics|mathematicians|set-theory|cantor|kronecker|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6659" PostTypeId="1" CreationDate="2017-11-01T09:46:27.573" Score="0" ViewCount="624" Body="&lt;p&gt;Strangely, many of Heisenberg's very important scientific papers are not translated into English. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF01397160&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt; seems not to have been translated.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where can I get the translated ones? &lt;/p&gt;&#xA;" OwnerUserId="2641" LastEditorUserId="4251" LastEditDate="2020-06-08T03:22:45.273" LastActivityDate="2020-10-25T01:12:53.343" Title="English translation of Heisenberg's papers" Tags="|physics|reference-request|physicists|translations|heisenberg|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6660" PostTypeId="2" ParentId="6659" CreationDate="2017-11-01T13:39:51.983" Score="2" Body="&lt;p&gt;It is not strange. They rarely translate papers in physics and mathematics from German or French into English. However two most important papers of Heisenberg and other founders of quantum mechanics are translated in the book&#xA;B. L. van der Waerden, Sources on quantum mechanics, (there is a cheap Dover edition).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-11-01T13:39:51.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6661" PostTypeId="2" ParentId="6657" CreationDate="2017-11-01T13:49:22.240" Score="1" Body="&lt;p&gt;There were ancient predecessors, among Hellenistic Greeks, and possibly even earlier, but their works did not survive. Best known author is Aristarchus of Somos. His work on heliocentric system is only known from a brief reference by Archimedes in his surviving book Psammite (Sand reckoner). Wikipedia's article &quot;heliocentrism&quot; gives a list of other ancient Greek authors about whom we know the references saying or hinting that they developed some heliocentric theory. But none of these works survived. Copernicus mentions these ancient authors, but he could not read them, because their work was already lost at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Interestingly, Newton also mentions obliquely some &quot;ancients&quot; in connection with the inverse square law. And there is even speculation on whether Newton possibly had some ancient texts which did not survive to our days.&#xA;Ref.: L. Russo, Forgotten revolution. This whole book discusses what Hellenistic scientists probably knew.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Wikipedia (article Al-Biruni) mentions that he discussed heliocentric system and apparently not only he alone, because he compares the views of different scholars on the subject, and concludes that it is plausible, but unclear how to prove. If this is indeed so (I have not read Biruni) than his point of view is even more reasonable than that of Galileo, because at the time of Galileo there was also no conclusive proof. (Biruni probably read Archimedes).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-06T03:42:16.227" LastActivityDate="2017-11-06T03:42:16.227" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6662" PostTypeId="1" CreationDate="2017-11-01T17:52:18.913" Score="0" ViewCount="182" Body="&lt;p&gt;Suppose I was living in his time. How would Kepler explain that the orbits are elliptical?&lt;/p&gt;&#xA;&lt;p&gt;Since none of his 3 laws explain why orbits are elliptical, I assume he must have had other reasons to believe why orbits are elliptical. Also, calculus had not yet been invented. So, how did he do that? How did he know that the distance to the Sun was changing, and that the velocity of the planet was changing to compensate for that? Was it solely because of the observational data provided by Tycho Brahe?&lt;/p&gt;&#xA;&lt;p&gt;Can someone provide the details of his thinking on this?&lt;/p&gt;&#xA;" OwnerUserId="6484" LastEditorUserId="4251" LastEditDate="2021-02-28T08:09:07.533" LastActivityDate="2021-02-28T08:09:07.533" Title="How did Kepler know that orbits are elliptical?" Tags="|physics|astronomy|celestial-mechanics|17th-century|kepler|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="6663" PostTypeId="2" ParentId="3006" CreationDate="2017-11-01T18:26:13.020" Score="1" Body="&lt;p&gt;This isn't a proper answer, but just a speculation: if I think about how the polar angle is defined in polar coordinates, it only makes sense that it's increasing in the counter clockwise direction: you want to start on the positive half of the $x$ axis, and then increase the angle staying in the $x&amp;gt;0, y&amp;gt;0$ quarter for the first $\pi/2$ radians. This of course was true before the formalization of polar coordinates, for instance in the geometric definitions of the trigonometric functions on the unitary circle: you want the angle to be defined that way so that $\sin\theta$ is positive for $0&amp;lt;\theta&amp;lt;\frac{\pi}{2}$. So my instinctive answer would be: because western writing is done left to right, so the axis of our graphs are oriented that way as we see the right direction as positive, thus we define angles a certain way.&lt;/p&gt;&#xA;" OwnerUserId="6485" LastActivityDate="2017-11-01T18:26:13.020" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6665" PostTypeId="2" ParentId="6658" CreationDate="2017-11-02T14:21:26.677" Score="7" Body="&lt;p&gt;About Kronecker's views on &quot;foundational&quot; issues, you can see:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Anne Troelstra &amp;amp; Dirk van Dalen, &lt;a href=&quot;https://books.google.it/books?id=-tc2qp0-2bsC&quot; rel=&quot;noreferrer&quot;&gt;Constructivism in mathematics: An Introduction. Volume 1&lt;/a&gt; (1988), page 17.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There is no &quot;explicit&quot; philosophy of mathematics in Kronecker's works.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He may be regarded as a &quot;constructivist&quot;, or perhaps as a precursor of the &lt;em&gt;finitist&lt;/em&gt; approach. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his essay &quot;Uber den Zahlbegriff&quot; (1887) he outlined the project of &quot;arithmetizing&quot; Algebra and Analysis; that is, to found these disciplines on the fundamental notion of number, avoiding thus geometrical intuition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his arithmetization project he considered a mathematical definition acceptable only if it could be checked in a finite number of steps, criticizing the &quot;pure&quot; existence proofs. He stated that an existence proof for a number could be considered correct only if it contained a method to find the number whose existence was proven.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of his remarks belong to mathematical folklore, like his widely reported statement that: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;the Lord made the natural numbers (&lt;em&gt;ganze Zahlen&lt;/em&gt;), everything else is the work of men&quot;; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;the same idea is reiterated in the following statement:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;I consider mathematics only as an abstraction of the arithmetical reality&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;: for some references to Kronecker's opposition to Cantorian set theory, we can see José Ferreirós, &lt;a href=&quot;https://books.google.it/books?id=vrQLbbxGNMsC&quot; rel=&quot;noreferrer&quot;&gt;Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics&lt;/a&gt;, Birkhäuser (2007), &lt;em&gt;sub voce&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In conclusion, it seems that Kronecker never published his critique of Cantor's theory, but we can conjecture from the sources above that he disliked Cantor's new theory of the &quot;arithmetics of the &lt;em&gt;infinite&lt;/em&gt;&quot; as well as Cantor's wide use of &quot;pure&quot; existence proofs.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-11-02T14:21:26.677" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6666" PostTypeId="1" CreationDate="2017-11-02T14:26:49.110" Score="1" ViewCount="110" Body="&lt;p&gt;Recently I &lt;a href=&quot;https://phys.org/news/2017-11-physicists-rapid-bounding-gravity.html#jCp&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;read this article&lt;/strong&gt;&lt;/a&gt; in Phys.org that states:&lt;/p&gt;&#xA;&lt;h3&gt;edit: providing previous precision value&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Over the past few months, physicists have made very rapid progress in&#xA;bounding the speed of gravity using gravitational wave observations.&#xA;Initially, the first LIGO detections of gravitational waves&#xA;constrained the speed of gravity to within 50% of the speed of light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Then,...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;...they were able to constrain the difference between the speed of gravity and the speed of light to between -3 x &lt;span class=&quot;math-container&quot;&gt;$10^{-15}$&lt;/span&gt; and 7 x &lt;span class=&quot;math-container&quot;&gt;$10^{-16}$&lt;/span&gt; times the speed of light. ...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So has there ever been such a large leap achieved in the precision of any Physics measurement? Also, why is there a minus sign on the three in &lt;strong&gt;-3&lt;/strong&gt; x &lt;span class=&quot;math-container&quot;&gt;$10^{-15}$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user6489" LastEditorUserId="-1" LastEditorDisplayName="user6489" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-11-03T14:58:44.913" Title="Is this the largest delta precision achieved in any Physics measurement?" Tags="|experimental-physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6667" PostTypeId="1" CreationDate="2017-11-02T18:50:42.247" Score="2" ViewCount="2377" Body="&lt;p&gt;When I talk about the western mathematical tradition, I think roughly of the mathematics done by the Greeks, and then take up by European countries and then countries with primarily European descendants. Also, I'm not thinking of the mathematicians in particular, but the methods they used, and the work they built upon and added to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What non-western and non-modern mathematical traditions contributed to modern mathematics? The biggest example I know of is Algebra. I also know some small things, like the Chinese remainder theorem, and I remember hearing that African mathematicians independently discovered Fibonacci numbers at some point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I would say that Ramanujan's work counts as western or modern mathematics, even though he himself is Indian, since he primarily worked with mathematicians using western techniques, and built upon that tradition.&lt;/p&gt;&#xA;" OwnerUserId="4349" LastEditorUserId="4349" LastEditDate="2017-11-02T18:58:46.130" LastActivityDate="2017-11-16T17:20:33.203" Title="Contribution of non-western mathematics to modern mathematics" Tags="|mathematics|big-list|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6668" PostTypeId="2" ParentId="6667" CreationDate="2017-11-02T19:24:04.767" Score="11" Body="&lt;p&gt;This is a meaningless question. First, you count medieval &quot;Arab&quot; contribution as &quot;non-western&quot;, while it is well-known that Arab mathematics comes from the Greek  and Mesopotamian sources. Arabs did not invent mathematics themselves but developed Greek and Mesopotamian mathematics.&#xA;Is Mesopotamian mathematics &quot;western&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Really, mathematics cannot be separated into &quot;western&quot; and &quot;non-western&quot;&#xA;until 17 century. Greek mathematics was influenced by Mesopotamian tradition.&#xA;Muslim peoples inherited Greek and Mesopotamian mathematics. Indians too,&#xA;to some extent. Then Europeans of 16 century learned a lot from Muslim (and Indian) mathematics, including many achievements of the Greeks. Much of trigonometry, and positional number system is of non-European origin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And why is Greek mathematics WESTERN? Greece is in Eastern Europe, by the way, and most&#xA;significant Greek mathematicians worked in Africa, namely in Alexandria. And Western European civilization is very different from the Greek one. Yes, we inherited Greek mathematics but so also the &quot;Arabs&quot; did. (I use quotation marks because many medieval mathematicians were not Arabs but Iranians and people from Middle Asia).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is indeed very hard to tell which tradition never experienced Greek influence.&#xA;(Perhaps American-Indian is the only clear case). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, you mention Chinese reminder theorem and African independent discovery&#xA;of Fibonacci numbers. Can we really say that these independent discoveries&#xA;CONTRIBUTED anything to Western Mathematics? Did Western mathematicians really learn them from Africans and Chinese? No. There is a difference between &quot;discover independently&quot; and &quot;contribute&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me give another clear example which is well documented. In 18th century Japanese mathematician&#xA;Seki Takakazu developed the theory of determinants. Few decades earlier&#xA;than his European counterparts. But did European mathematicians learn anything about determinants from him? Did Japanese mathematicians of 20th century study determinants from his work? No, they studied&#xA;them from &quot;western&quot; sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the Meiji revolution, the door for cultural exchange was wide open,&#xA;Japanese students learned &quot;western mathematics&quot; very quickly and started to&#xA;contribute to it on the highest level, since approximately 1900.&#xA;But they learned it from European sources, not from native Japanese sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did European mathematicians really learn anything new from pre-19 century Japanese&#xA;mathematics? I strongly doubt it. There was a high quality mathematics in Japan&#xA;before 19th century, but it did not contribute to modern mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I dare to claim that none of the famous mathematicians who made a substantial&#xA;contribution to modern mathematics did this by developing any &quot;non-western&quot;&#xA;tradition, no mater where this mathematician comes from. Even when some of them claim the contrary. One cannot contribute anything essential to modern mathematics without being educated in this &quot;western tradition&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-03T12:33:15.867" LastActivityDate="2017-11-03T12:33:15.867" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6669" PostTypeId="2" ParentId="6667" CreationDate="2017-11-02T22:22:45.477" Score="2" Body="&lt;p&gt;I was taught that the standard notation for representing integers was invented in India and transmitted to Europe by Arabs.  I suppose this is part of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2017-11-02T22:22:45.477" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6670" PostTypeId="1" CreationDate="2017-11-03T01:07:21.110" Score="3" ViewCount="262" Body="&lt;p&gt;I don't believe in the ridiculous conspiracy theories that she was the true founder of relativity and Einstein stole credit from her or any of that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but did she play a role in anyway helping him? someone said that before he met her he didn't have a single peer review publication?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;how much did she help Einstein?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2017-11-03T01:07:21.110" Title="So how much did Mileva Maric contribute to Einstein's work?" Tags="|einstein|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6671" PostTypeId="2" ParentId="6667" CreationDate="2017-11-03T05:02:56.200" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://quatr.us/africa/science/numbers.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Counting&lt;/a&gt; is non-western in origin:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Around 35,000 BC, somebody in South Africa made the earliest known counting stick, or tally stick, and left it in Lebombo Cave. Somebody cut 29 notches into the stick.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next advance in counting wasn't made until the early 20th C by Cantor in his cardinal arithmetic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As 29 is roughly the number of days between one full moon and another, we could also say that this is also the earliest evidence we have of astronomical observation and so one of the earliest hints of science. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also &lt;a href=&quot;http://it.stlawu.edu/~dmelvill/mesomath/obmetrology.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mensuration&lt;/a&gt; or the measurement of length, area and volume in ancient Sumeria, Babylonia and the Harrapan civilisation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first unification result in mathematics is likely to have been this - not only can we count things, we can count length, area and volume and then it developed into a distinct discipline - geometry - which as the name describes, is the measurement of the earth.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2017-11-16T17:20:33.203" LastActivityDate="2017-11-16T17:20:33.203" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6672" PostTypeId="2" ParentId="6667" CreationDate="2017-11-03T09:55:28.100" Score="4" Body="&lt;p&gt;The question as formulated 'What non-western and non-modern mathematical traditions contributed to modern mathematics?' does not leave much place for anything except a strictly negative answer. The nature of mathematics being what it is, it is possible to (re)discover results independently of preceding authors and traditions, e.g. we know that Leibniz discovered the expansion of pi later than Madhava and naming the series with the hyphenated &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava-Leibniz_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Madhava-Leibniz&lt;/a&gt; acknowledges priority, not influence or contribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The self reliance of mathematics exists along with an historical deficience of communications, so the West did not know anything of &lt;a href=&quot;https://en.wikipedia.org/wiki/Indian_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indian mathematics&lt;/a&gt; which developped independently to an impressive degree. It is an historical fact that Christianity deleted pratically all of the Greek mathematics and the Scholastic revival started with translation from Arab copies into Latin e.g. Adelard of Bath translation of Euclid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recent research has uncovered a remarkably high level of mathematics (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Plimpton_322&quot; rel=&quot;nofollow noreferrer&quot;&gt; Plimpton 322&lt;/a&gt; and  &lt;a href=&quot;https://en.wikipedia.org/wiki/YBC_7289&quot; rel=&quot;nofollow noreferrer&quot;&gt;YBC 7289&lt;/a&gt;) during the Old Babylonian  epoch, a millenium before the Greeks, but its later transmission remains unclear. Some of it went to Egypt  but mostly it was reconstructed by the Greeks as Joran Friberg demonstrated in &lt;em&gt;Amazing Traces of a Babylonian Origin in Greek Mathematics&lt;/em&gt;, 2007, World Scientific. &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-11-03T09:55:28.100" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6673" PostTypeId="1" AcceptedAnswerId="6676" CreationDate="2017-11-05T08:14:54.730" Score="0" ViewCount="851" Body="&lt;p&gt;Once, after a lecture, my professor of differential equations said, that Newton did not use derivatives in his work as we do today. He told us that Newton rather used some series expansions for his equations of motion because it helped him to get better approximation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunely I have not been able to find any written referense of it, is there any? Or maybe someone knows how that series looked like?&lt;/p&gt;&#xA;" OwnerUserId="6497" LastActivityDate="2017-11-06T00:50:29.470" Title="How did Newton write his equations?" Tags="|physics|notation|classical-mechanics|newton|differential-equations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6675" PostTypeId="1" CreationDate="2017-11-05T19:40:21.330" Score="2" ViewCount="103" Body="&lt;p&gt;Much of the advertised reason for scientific research is that there is a trickle-down effect from science to technology. For instance, here is a quote by Enrico Fermi that I grabbed from Wikipedia.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some of you may ask, what is the good of working so hard merely to collect a few facts which will bring no pleasure except to a few long-haired professors who love to collect such things and will be of no use to anybody because only few specialists at best will be able to understand them? In answer to such question[s] I may venture a fairly safe prediction. History of science and technology has consistently taught us that scientific advances in basic understanding have sooner or later led to technical and industrial applications that have revolutionized our way of life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In talking to people, I have more or less found two ideas for how scientific research affects technology:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Most scientific research projects result in tiny advances that primarily feed back into more research, but which also have the potential for small applications. Huge numbers of these projects eventually sum to tools and technologies that have an impact outside of just scientific research itself.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are many scientific research projects, most of which don't come to much outside (or sometimes even inside) of the scientific community. However, the ones that do have  an enormous impact on technology, and it is difficult to tell which ones will beforehand. The impact of the projects that do come to something is so profound that it justifies all of the research that is done.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here is my question: Does anyone know of any studies that attempts to verify, quantify, and understand these science-to-technology trickle-down ideas, beyond just assuming that one of the above statements is true?&lt;/p&gt;&#xA;&lt;p&gt;As an example, it could be something like surveying numerous patents in the last decade that are deemed to be very impactful to civilization, and looking at the papers cited in those patents, and then looking at papers that those papers cite in turn, etc. In this way, citations among papers could be used to get a rough idea of what technogical impact factor a paper has had, provided that one or more significant patents cite either that paper, or other papers which are linked to it via citations.&lt;/p&gt;&#xA;&lt;p&gt;I am asking this for several reasons. Some of them are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;A lot of research seems very obscure. It is not uncommon for papers to be aimed at only a handful of scientists other than the authors. Even for papers that are aimed at a much larger scientific audience, it is often very hard to connect the dots from their scientific interest to how they will affect life outside of scientific fields. And, even when these dots are connected, a lot of times it is in such a way that they hint at a hoped-for technology, and it is still unestablished whether or when that technology will be attained.&lt;/p&gt;&#xA;&lt;p&gt;It seems like it would be important to know to what degree things like this could actually be expected to broaden their relevance over time. How often are the hoped-for technologies attained? How often are other technologies, which are not the ones that were originally being aimed for, attained? What shape does the path from highly specialized papers, to more generally understandable papers, to technologies take? How does this vary by field? What percentage of papers are likely to be impactful outside of their field? Etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am curious as to what kind of time scales are typical to go from research to technologies, especially technologies that are relevant to every day life in an obvious way.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am curious as to how true arguments 1 and 2 above are. I have had conversations with people, both scientists and not, where they show at least a partial disdain for both arguments. So, the fact that they are both heavily used arguments, but also doubted, makes them seem like an interesting thing to understand better. Additionally, most applications for scientific funding rely more or less on arguments 1 and 2 above. It seems like we should understand as well as possible how true they are if we're going to keep using them.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Thanks for any help!&lt;/p&gt;&#xA;&lt;p&gt;P.S. I'm not positive that this is the correct Stack Exchange site to post this thread on - if it's not, could someone please point me to a better one?&lt;/p&gt;&#xA;" OwnerUserId="6500" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-11-06T04:03:13.340" Title="Study of path from science to technology" Tags="|technology|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6676" PostTypeId="2" ParentId="6673" CreationDate="2017-11-05T23:46:27.160" Score="2" Body="&lt;p&gt;This question involves two rather different points. It involves Newton's practices for notating not only derivatives, but also the functions (or whatever else) he used to indicate the values of those derivatives.  In cases where we might now write dy/dx = {&lt;em&gt;some-function&lt;/em&gt;} , the question involves Newton's practices for expressing both the left and the right sides.  'Equations', as we now notate them, were only one among several ways in which he wrote down relationships of this kind, as can be seen from the following examples and their sources.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The comment given by &lt;em&gt;Michael E2&lt;/em&gt; shows one of the ways (fluxional notation) in which Newton indicated a derivative for which we might now write dy/dx .  Beyond that, Newton also used other ways to indicate derivatives or rates of change, not all of them related to the notation of fluxions, and some of them not involving symbols at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One notable type of example can be found e.g. in Proposition 34 (Book 3) of the &lt;em&gt;Principia&lt;/em&gt;. Here, Newton undertook 'to find the horary variation of the inclination of the Moon's orbit to the plane of the ecliptic'. Newton gave a rather complicated geometrical specification for the quantity of this variation: it amounts, when translated, to a triple product of sines/cosines. Newton then went on (in Proposition 35) to integrate (a simplified version of) this quantity geometrically with respect to the time, to produce a result showing the periodic variation of the actual inclination of the orbit at any time (strictly, its difference from the mean inclination, which plays the role here of the arbitary constant in the integration). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From all of that, especially the integration, it is clear that Newton's 'horary variation of the inclination' meant what we could now  write as di/dt , where i is the inclination and t the time. P-S Laplace later treated it so. Thus, when Laplace considered this same passage in Newton, he 'translated' Newton's geometric expression for the value of the 'horary variation' into a trigonometrical formula (see P-S Laplace, Traite de Mecanique Celeste, tome 5 (1825), livre XVI, ch.2, art.3 at p.375, discussing the inclination from p.379 onwards; at (&lt;a href=&quot;https://books.google.com/books?&amp;amp;id=UdZGAQAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?&amp;amp;id=UdZGAQAAMAAJ&lt;/a&gt;)). Laplace then developed its integral (which he did with wider generality than Newton's treatment, which had involved some simplification before the integration, but that doesn't alter the main point of interest here). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example so far given shows Newton using, for the quantity of the rate of change in which he was interested, a geometric definition (which was translatable -- and later translated -- into trigonometric notation). In other examples he used algebraic series, but Newton's uses of algebraic series appear to stem primarily from an earlier part of his career.  When Newton did use series, his notation for the terms was sometimes not too different from what might be written now, except for details such as aaa for $a^3$ and so on. See for example &lt;em&gt;The Mathematical Papers of Isaac Newton&lt;/em&gt;, volume II: 1667-1670, (ed.) D T Whiteside (1968), (&lt;a href=&quot;https://books.google.com/books?id=AQ3tveOwseoC&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?id=AQ3tveOwseoC&lt;/a&gt;), where page 206 et seq. give Latin &amp;amp; English versions of Newton's 'On analysis by equations infinite in number of terms' and show some of his practices for writing down such series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 'Correspondence of Isaac Newton' vol.1 at p.52 (24 Dec 1670, John Collins to James Gregory) Collins reported, among other things obtained from Newton, Newton's series for the arcsine.  A longer description of how Newton obtained this (by term-by-term integration of the series obtained by binomial expansion of 1 / $\sqrt(1-x^2)$ ) is given for example (but with modernized notation) in 'The Calculus Gallery: Masterpieces from Newton to Lebesgue', by William Dunham (2005), at p.17.  (Newton's purpose in using the series does not seem, at least in this case, to have been &quot;to get better approximation&quot;: rather it was to convert an expression that was effectively intractable as it stood, into a readily-integrable form.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(John Collins was a well-known mathematical 'intelligencer' of Newton's time, who saw a number of Newton's (often algebraic) inventions and reported them to his correspondents. Collins later understood that the prospects seemed poor for getting any more mathematics, especially no doubt in algebraic form, from Newton, because Newton (and Barrow) were then &quot;beginning to thinke mathematicall speculations to grow at least nice and dry, if not somewhat barren&quot; (Isaac Newton Corresp vol 1 p.355 (19 Oct 1675, John Collins to James Gregory)). The letter turned out to be prophetic, in that Collins never afterwards heard anything more of mathematics from Newton, and Newton turned more towards the language of geometry to express his mathematical investigations. The reasons for Newton's (reportedly) growing disaffection from the kind of mathematics that he had engaged in up to that time perhaps deserve fuller investigation.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2017-11-06T00:50:29.470" LastActivityDate="2017-11-06T00:50:29.470" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6678" PostTypeId="2" ParentId="6638" CreationDate="2017-11-06T10:13:25.510" Score="1" Body="&lt;p&gt;To complement Denis de Bernardy's &lt;a href=&quot;https://hsm.stackexchange.com/a/6646/604&quot;&gt;excellent answer&lt;/a&gt; I would mention &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francois Vieta&lt;/a&gt;. Vieta's introduction of symbolic algebra was the decisive catalyst of the transformation of natural philosophy/science that took place in the 17th century.  Had Vieta lived in North Africa, who knows, the dominant role of Western Europe that we take for granted may have turned out to be the result of historical factors more contingent than we like to admit.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-11-06T10:13:25.510" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6679" PostTypeId="5" CreationDate="2017-11-06T15:34:33.370" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2017-11-06T15:34:33.370" LastActivityDate="2017-11-06T15:34:33.370" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6680" PostTypeId="4" CreationDate="2017-11-06T15:34:33.370" Score="0" Body="This is a special tag that is designed to be used only by the system for questions that have had all of their other tags removed. Do not add this tag to existing questions, it is reserved for use by the system to identify posts with no valid tags." OwnerUserId="2277" LastEditorUserId="2277" LastEditDate="2017-12-06T04:33:01.393" LastActivityDate="2017-12-06T04:33:01.393" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6681" PostTypeId="1" CreationDate="2017-11-06T19:47:14.730" Score="0" ViewCount="147" Body="&lt;p&gt;Historically steam engines got developed in England after the introduction of the hypothetico-deductive scientific method. Was the hypothetico-deductive method or knowledge about thermodynamics or metallurgy required for the development of commercial steam engines?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or could the technology that's needed for commercial steam engines also have been developed without scientists moving to the hypothetico-deductive method?&lt;/p&gt;&#xA;" OwnerUserId="4623" LastActivityDate="2017-11-07T22:44:18.960" Title="Was knowledge derived from the scientific method required to build commercial steam engines?" Tags="|steam-engines|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6682" PostTypeId="2" ParentId="6681" CreationDate="2017-11-07T08:41:28.910" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Humphry_Davy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Humphry Davy&lt;/a&gt; said of &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Watt&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Watt&lt;/a&gt;: &amp;ldquo;Those who consider James Watt only as a great practical mechanic form a very erroneous idea of his character; he was equally distinguished as a natural philosopher and a chemist, and his inventions demonstrate his profound knowledge of those sciences, and that peculiar characteristic of genius, the union of them for practical application.&amp;rdquo;&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-11-07T08:41:28.910" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6683" PostTypeId="1" CreationDate="2017-11-07T17:12:02.250" Score="3" ViewCount="939" Body="&lt;p&gt;I've been asked to write an essay on whether the work on PDE's in the 19th century belonged to applied or pure mathematics. Does anyone know of any useful sources I could use?&lt;/p&gt;&#xA;" OwnerUserId="6435" LastEditorUserId="4251" LastEditDate="2020-07-05T14:50:24.757" LastActivityDate="2020-07-05T14:50:24.757" Title="History of PDE's in the 19th Century" Tags="|mathematics|reference-request|mathematical-physics|differential-equations|19th-century|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6684" PostTypeId="2" ParentId="6683" CreationDate="2017-11-07T22:20:10.657" Score="5" Body="&lt;p&gt;It belonged to both. For example, the most important work on PDE in 19th century&#xA;was arguably Fourier's Analytic theory of heat. You don't have to read the book,&#xA;to conclude that this was applied mathematics, just from the title.&#xA;Other important work on PDE, comes from pure mathematics (differential geometry for example), or the work of Liouville on the Liouville's equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting example is Cauchy-Riemann equations which come from Complex Analysis (pure mathematics) but closely related to fluid dynamics (applied mathematics).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laplace's equation appeared first in the work on Saturn rings (applied mathematics) but later was used everywhere and its theory (the theory of harmonic functions) was pure mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, the distinction between pure and applied mathematics was never sharp and clear, especially in 19th century.&#xA;So the question you are assigned has actually little sense.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-11-07T22:20:10.657" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6685" PostTypeId="2" ParentId="6681" CreationDate="2017-11-07T22:44:18.960" Score="2" Body="&lt;p&gt;This is an extended set of comments.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To be sure, the first known steam engine was invented by Hero of Alexandria in&#xA;1-st century AD. It was not &quot;commercial&quot;, of course but this has nothing to do with scientific method: just &quot;commerce&quot; was not developed enough at that time.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Scientific method was not &quot;introduced&quot; in England in the modern times. It was just &quot;articulated&quot;, explicitly stated. Scientific method was used by many people&#xA;before modern era; they just did not know that this will be called &quot;scientific method&quot; sometimes. Some Hellenistic Greeks used scientific method in their work on physics and astronomy and medicine, there can be no doubt in it. Certainly Archimedes and Hero used it.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Thermodynamics had not been invented by the time of Watt invented his engine.&#xA;Usually its invention is credited to Sadi Carnot, and when he was working on it,&#xA;he certainly knew about steam engines.&#xA;However there is no doubt that Watt used scientific method. He was aware of&#xA;physics of his time, and he certainly experimented (experiment is one of the main features of scientific method). Which does not imply, of course that Watt read Francis Bacon. But he probably knew the Boyle-Mariotte law, and whatever was known about heat and steam at that time. And all this knowledge was obtained by scientific method.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-11-07T22:44:18.960" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6686" PostTypeId="2" ParentId="6643" CreationDate="2017-11-09T01:28:01.237" Score="0" Body="&lt;p&gt;My longish answer to the original question tells you how we go about establishing mean sea level, so I don't understand your additional question. If you want to know where this mean sea level is in your location get a tide table that has tidal info for your location and it will tell you where MSL is in your location. If your location is not contained in a Tide Table, get a chart for your area as it will also provide info on the reference level, i.e. Chart Datum, and the variation between various long term tidal heights, such as spring and neap high and low water levels so you can work out where MSL is. If you are lucky it may provide data on where the bench mark is in relation to chart datum so you could run a line of levels from it to the water's edge to see where that level is relative to CD at the time you do this. If you need more info buy a book on tides or navigation or ask a local harbour pilot or lecturer at a maritime college. I could recommend some books, but as I do not know anything about your skills in mathematics these could possibly be beyond your ability to cope with. As it is not efficient to keep on dealing through Stack Exchange you could contact me directly at ghgoldsteen@netspace.net.au&lt;/p&gt;&#xA;" OwnerUserId="6481" LastActivityDate="2017-11-09T01:28:01.237" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6687" PostTypeId="1" CreationDate="2017-11-09T03:27:43.567" Score="1" ViewCount="1780" Body="&lt;p&gt;Further, I believe that the idea to call it thus had to do with its resemblance to the &quot;torus&quot; in the base of some Greek columns of old:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/XKRTP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XKRTP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What do you think of this hypothesis of mine?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance for your knowledgeable replies.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2017-11-09T03:32:59.737" LastActivityDate="2017-11-10T19:07:17.703" Title="Who was the first individual that used the word &quot;torus&quot; to refer to $\mathbb{S}^{1} \times \mathbb{S}^{1}$?" Tags="|terminology|topology|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6688" PostTypeId="2" ParentId="6687" CreationDate="2017-11-09T12:22:12.200" Score="1" Body="&lt;p&gt;The OED gives an 1870 Cayley citation.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2017-11-09T12:22:12.200" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6689" PostTypeId="2" ParentId="6687" CreationDate="2017-11-09T13:56:54.193" Score="3" Body="&lt;p&gt;Answer from &lt;a href=&quot;http://jeff560.tripod.com/t.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathword&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;TORUS.&lt;/strong&gt; Hero mentions a mathematician named Dionysodorus as the author of &lt;em&gt;On the Tore&lt;/em&gt;, in which a formula for the volume of the torus is given [DSB].&lt;/p&gt;&#xA;&lt;p&gt;An early use of torus as a mathematical term in English is in 1860 in &lt;em&gt;The Practical Draughtsman’s Book of Industrial Design&lt;/em&gt; by William Johnson: “In geometry, the torus is a solid, generated by a circle, revolving about an axis, continuing constantly in the plane of this axis, in such a manner, that all sections made by planes passing through the axis are equal circles, and all sections by planes perpendicular to the axis will also be circles, but of variable diameters.” [Google print search]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, do you think the term used for Greek columns was named for the geometric term as in Dionysodorus?  Or the other way around?&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-11-09T13:56:54.193" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6690" PostTypeId="1" CreationDate="2017-11-09T16:34:20.567" Score="0" ViewCount="42" Body="&lt;p&gt;I'm trying to convince my institution to improve their internal Computer Science archives, and want one of my argument angles to be based on empirical value-adds of archiving. History, library science, etc are not my area of domain knowledge at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any empirical or quantified research in the economic, social, or other benefits of archives? STEM and Computational archives are of interest in particular. &lt;/p&gt;&#xA;" OwnerUserId="4907" LastActivityDate="2017-11-09T16:34:20.567" Title="What is the (economic, social) value of STEM archives?" Tags="|computation|social|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6691" PostTypeId="2" ParentId="6687" CreationDate="2017-11-10T09:11:30.500" Score="2" Body="&lt;p&gt;The mathematical usage comes from the usage in architecture. The word is of latin origin &quot;&lt;em&gt;torus&lt;/em&gt;&quot; and not of greek origin, and has a number of different meanings. It means &lt;em&gt;muscle&lt;/em&gt; but also &lt;em&gt;pillow&lt;/em&gt; (probably circular one) and it means also a &lt;em&gt;strong rope&lt;/em&gt;. In this last meaning it was used to describe circular decoration made of strong ropes and afterwards it started being used to describe the part of columns you're referring to. With this meaning it is used already by &lt;strong&gt;Vitruvio Pollione&lt;/strong&gt; (80ac-15 ac). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nothing strange that the word passed from architecture to  geometry, the two subjects being strongly related throught Renaissance and Baroque periods. I easily found Italian citations much pre-dating Cayley:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Vincenzo Antonio Rossi&lt;/strong&gt; &quot;Intorno ad una superficie anulare...&quot; &lt;strong&gt;1844&lt;/strong&gt;&#xA;(Architecture - &quot;About an annular surface...&quot; in which he explicitely discuss toric surface, with the usual $\mathbb R^3$ equations).&#xA;but also&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Francesco Paolo Tucci&lt;/strong&gt; Elementi di Calcolo Differenziale e Integrale &lt;strong&gt;1850&lt;/strong&gt;&#xA;which is a textbook for students of a Military school (and where you see the expression &quot;superficie torica o toro&quot; together with its equations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;while in&#xA;&lt;strong&gt;V. Gambardella&lt;/strong&gt; &quot;Istituzioni di Architettura secondo il metodo matematico&quot; - &lt;strong&gt;1798&lt;/strong&gt; I've found many reference to the torus surface but no equations.&#xA;This is just half an hour search on Googlebooks in Italian so I guess many more instances can be found easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess that the word easily migrated from architecture to Geometry and it will be difficult to trace back a &quot;first usage&quot; of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ancient origin of the word is unclear. It should be related to the suffix Tor-Tur that is connected to the idea of revolving around like in &quot;turning&quot; or in the Italian &quot;tornare&quot; (which means &quot;coming back&quot;) and appears in many Indo-European languages, so that it is supposed to be a very old lemma. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ADDED For what concerns &quot;modern&quot; usage it has to be remarked that &lt;strong&gt;Kepler&lt;/strong&gt; in his &lt;em&gt;Nova Stereometria&lt;/em&gt; &lt;strong&gt;1615&lt;/strong&gt; is credited for having computed the volume of a torus. Since the text is in Latin it is plausible he is using the word &lt;em&gt;torus&lt;/em&gt; but it would be nice to check it.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="1689" LastEditDate="2017-11-10T19:07:17.703" LastActivityDate="2017-11-10T19:07:17.703" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6692" PostTypeId="1" AcceptedAnswerId="6693" CreationDate="2017-11-10T14:16:00.313" Score="2" ViewCount="408" Body="&lt;p&gt;I asked this over on Math Stackexchange, and someone said it might be good to ask it over here too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some authors use different equals signs for different purposes. For the most part, they are &quot;$=$&quot;, &quot;$\equiv$&quot;, &quot;$:=$&quot;, and &quot;$:\equiv$'.  I read that &quot;$=$&quot; dates back to 1557, and is of mathematical origin.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What about the others?  I have read that &quot;$:=$&quot; appears in some programming languages from the $70$'s.  But, does it occur earlier? What about $\equiv$ and $:\equiv$, what are their origins?&lt;/p&gt;&#xA;" OwnerUserId="6213" LastActivityDate="2017-11-10T15:07:43.843" Title="Origins of the Equals Signs" Tags="|notation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6693" PostTypeId="2" ParentId="6692" CreationDate="2017-11-10T14:19:44.923" Score="6" Body="&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/relation.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Uses of Symbols of Relation&lt;/a&gt; for the first occurrence of the equality sign $=$, that was first used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_Recorde&quot; rel=&quot;noreferrer&quot;&gt;Robert Recorde (c.1510-1558)&lt;/a&gt; in 1557 in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Whetstone_of_Witte&quot; rel=&quot;noreferrer&quot;&gt;The Whetstone of Witte&lt;/a&gt;, with a link to the image of the original print.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the equivalence sign, see &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/a&gt;, with the origin of $\leftrightarrow$, that was apparently first used in 1933 by Albrecht Becker &lt;em&gt;Die Aristotelische Theorie der Möglichkeitsschlüsse&lt;/em&gt;, Berlin, 1933.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The double arrow $\Leftrightarrow$ was first used in 1954 by Nicholas Bourbaki, in: &lt;em&gt;Theorie des ensembles&lt;/em&gt;, 3rd edition, Paris, 1954 (see &lt;a href=&quot;https://books.google.it/books?id=VDGifaOQogcC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;page I.30&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The symbol $\equiv$ for the bi-conditional was firstly used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Gottlob_Frege&quot; rel=&quot;noreferrer&quot;&gt;Gottlob Frege (1848–1925)&lt;/a&gt; into his 1879 &lt;a href=&quot;https://en.wikipedia.org/wiki/Begriffsschrift&quot; rel=&quot;noreferrer&quot;&gt;Begriffsschrift&lt;/a&gt; and subsequently &quot;popularized&quot; by Alfred North Whitehead and Bertrand Russell's &lt;a href=&quot;https://books.google.it/books?id=ke9yGmFy24sC&amp;amp;pg=PA7&quot; rel=&quot;noreferrer&quot;&gt;Principia Mathematica (1910-1913)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Principia&lt;/em&gt; uses $=$ followed by Df. on the right-end of the formula for definitional identity.(see &lt;a href=&quot;https://books.google.it/books?id=ke9yGmFy24sC&amp;amp;pg=PA15&quot; rel=&quot;noreferrer&quot;&gt;page 11 and 15&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can find the usage of $=_{Def}$ already into &lt;a href=&quot;https://en.wikipedia.org/wiki/Cesare_Burali-Forti&quot; rel=&quot;noreferrer&quot;&gt;Cesare Burali-Forti&lt;/a&gt;, Logica Matematica (1894), that uses $=$ for (logical) equivalence (see page 26):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$(a=b) =_{Def} (a \supset b) . (b \supset a)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2017-11-10T15:07:43.843" LastActivityDate="2017-11-10T15:07:43.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6694" PostTypeId="2" ParentId="6687" CreationDate="2017-11-10T15:24:13.090" Score="4" Body="&lt;p&gt;Hero mentions the work of Dionysodorus in his &quot;Metrica&quot; this way: &quot;διονυσωδώρω έν τω περί τησ σπείρασ έπιγραφομένω&quot;, namely &quot;Dionysodorus in his work On σπείρασ&quot;. &#xA;In modern Greek σπείρασ stays for &quot;spirals&quot;, but was used in ancient Greek for the Latin torus.&#xA;You can find the original text &lt;a href=&quot;http://www.wilbourhall.org/pdfs/Heronis_Alexandrini_Opera_quae_SupersuntVOL_III.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; at page 128, line 3 and 4.&lt;/p&gt;&#xA;" OwnerUserId="6530" LastActivityDate="2017-11-10T15:24:13.090" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6695" PostTypeId="1" CreationDate="2017-11-10T19:12:04.547" Score="4" ViewCount="282" Body="&lt;p&gt;What is the earliest known written attempt at the definition of number? Is it Euclid's Elements?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A unit is that by virtue of which each of the things that exist is called one.&lt;/p&gt;&#xA;&lt;p&gt;A number is a multitude composed of units.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6532" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-07-18T23:14:29.197" Title="The earliest definition of number" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6696" PostTypeId="1" CreationDate="2017-11-11T01:10:55.720" Score="1" ViewCount="147" Body="&lt;p&gt;Why aren't turns used more often in math? It seems much more fundamental of a unit compared to degrees and radians. The value of a turn is definite, and other units can be derived from it. The problem with radians is we'll never know exactly what a radian is, unless you're putting it in terms of pi or tau. And degrees are quite arbitrary (360? Where did we get that from?)&#xA;Turns could be put in terms of transcendental numbers, using fractions. This doesn't need to be done as often, but it's not too inconvenient to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;360 degrees = 1 turn&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2π radians or  radians = 1 turn&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1/400 gradians = 1 turn&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just think it's more fundamental. The Planck unit of arc, if you will. Probably just my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is simply why radians are used far more than turns as a unit.&lt;/p&gt;&#xA;" OwnerUserId="6534" LastActivityDate="2017-11-11T14:29:58.803" Title="On Units of Arc" Tags="|units|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6697" PostTypeId="2" ParentId="6696" CreationDate="2017-11-11T14:29:58.803" Score="3" Body="&lt;p&gt;Turns are used in mathematics  when convenient. For example, some authors write Fourier transform as&#xA;$$\int e^{-2\pi i xz}f(x)dx,$$&#xA;while other authors as&#xA;$$\int e^{-ixz}f(x)dz.$$&#xA;In the first formula $x$ is in &quot;turns&quot; (1 turn=$2\pi$ radians). This is a question&#xA;of convenience. Using turns, the Taylor expansion look more complicated:&#xA;$$e^{2\pi x}=\sum_{n=1}^\infty \frac{(2\pi x)^n}{n!}.$$&#xA;In many mathematical formulas $\pi$ is unavoidable. So people choose units to minimize the number of occurrences of $\pi$ in the formulas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same in physics. There is angular frequency $\omega$ (radians per second) and there is frequency $\omega/(2\pi)$ (oscillations/turns per second).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-11-11T14:29:58.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6698" PostTypeId="1" CreationDate="2017-11-12T20:24:08.160" Score="16" ViewCount="611" Body="&lt;p&gt;Arthur Cayley's first paper on abstract groups, in 1854, can be found in his Collected Papers on the Internet Archive, starting at &lt;a href=&quot;https://archive.org/stream/collectedmathema02cayluoft#page/122/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/stream/collectedmathema02cayluoft#page/122/mode/2up&lt;/a&gt;. He states on the 3rd page that groups of prime order are cyclic and then goes on to classify the groups of order 4 and 6 up to isomorphism (he doesn't use that term and refers to the order of an element as its &quot;index,&quot; but to modern eyes it is clear what he is doing). If you go to the 5th page of the paper, at &lt;a href=&quot;https://archive.org/stream/collectedmathema02cayluoft#page/126/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/stream/collectedmathema02cayluoft#page/126/mode/2up&lt;/a&gt;, he is classifying groups of order 6 and writes near the bottom &quot;we have thus two, and only two, essentially distinct forms of a group of six [symbols].&quot; The next page shows the Cayley tables for these groups, corresponding to the cyclic group and $S_3$, respectively. So far so good. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moving ahead to 1878, in the first volume of the &lt;em&gt;American Journal of Mathematics&lt;/em&gt; Cayley has a short paper on groups. See &lt;a href=&quot;https://www.jstor.org/stable/2369433?seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;https://www.jstor.org/stable/2369433?seq=1#page_scan_tab_contents&lt;/a&gt;. On the bottom of the second page he writes &quot;The general problem is to find all the groups of a given order $n$.&quot; He goes on to say that when $n = 2$ and $3$ there is one group, when $n = 4$ there are two groups (a footnote points out when $n = 5$ there is one group), and then comes the big surprise: &quot;when $n = 6$, there are three groups&quot;. That is not only wrong, but contradicts what he wrote in 1854. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cayley describes the three groups of order 6 as $1, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5$ where $\alpha$ has order 6; and two groups of the form $1, \beta, \beta^2, \alpha, \alpha\beta, \alpha\beta^2$ where $\alpha$ has order 2 and $\beta$ has order 3, distinguished from each other by the rules $\alpha\beta = \beta\alpha$ &#xA;or $\alpha\beta = \beta^2\alpha$. (Strictly speaking, he lists that last relation together with $\alpha\beta^2 = \beta\alpha$, but it's redundant since it follows from $\alpha\beta = \beta^2\alpha$ since $\beta^3 = 1$.) The second group in his list is recognizable as a direct product of a group of order 2 and a group of order 3, so it appears that Cayley does not realize his second group is isomorphic to his first group.  The mistake is understandable; even today, students learning group theory who do not yet know the Chinese remainder theorem could easily believe at first that a direct product of cyclic groups of order 2 and 3 is not isomorphic to a cyclic group of order 6 since the groups don't &quot;look&quot; the same (confusing appearance with structure). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is this: did Cayley in print later acknowledge his mistake or explain why he got it right in 1854 and later got it wrong? In a 1913 Amer. Math. Monthly paper G. A. Miller writes about Cayley's mistake on &lt;a href=&quot;https://www.jstor.org/stable/2973510?seq=4#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;https://www.jstor.org/stable/2973510?seq=4#page_scan_tab_contents&lt;/a&gt;, but is silent about Cayley ever issuing a correction or explanation later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: Near the end of writing up this question I found Cayley's error mentioned by Richard Stanley in a Mathoverflow post about mathematical errors -- see &lt;a href=&quot;https://mathoverflow.net/questions/879/most-interesting-mathematics-mistake/921&quot;&gt;https://mathoverflow.net/questions/879/most-interesting-mathematics-mistake/921&lt;/a&gt; -- but my question still stands: did Cayley ever bring up his error after 1878?&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="2277" LastEditDate="2018-08-10T21:03:46.917" LastActivityDate="2023-09-29T17:11:25.257" Title="A basic mistake by Cayley" Tags="|group-theory|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6699" PostTypeId="1" CreationDate="2017-11-13T17:41:41.147" Score="3" ViewCount="252" Body="&lt;p&gt;We take it for granted these days that the stars are unimaginably bigger than the planets and the moons. But when you look at the sky, it does not appear this way. The moon looks bigger and brighter than the stars. This leads me to wonder when astronomy first discovered the relative size of the stars. Did people believe that the moon was bigger than the stars for a long time? If so, where was this believed, and up until when? What discoveries or observations led people to conclude that the stars are larger?&lt;/p&gt;&#xA;" OwnerUserId="6546" LastActivityDate="2017-11-13T21:29:27.333" Title="When did astronomy first discover that the stars are bigger than the moon?" Tags="|astronomy|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6700" PostTypeId="2" ParentId="6405" CreationDate="2017-11-13T18:49:56.380" Score="0" Body="&lt;p&gt;I think the term is even more ambiguous than the original question implies.  In addition to referring to the corpus of scientific facts (discoveries, theories, etc) obtained so far, and to the methods of discovering them, it can also refer to the social organization of the activity.  These three are interrelated and their relationship has a history.  A sentence like &quot;modern science has several issues&quot; can be read as a statement about what we think we know now, or as one about the &quot;inner&quot; methodology of science, or about (say) the politics or economics or sociology etc of science.  Does this government fund science wisely, is there corruption in some nation's Academy of Sciences, are students being inspired to become scientists, are all &quot;issues&quot; with modern sciences.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2017-11-13T18:49:56.380" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6701" PostTypeId="2" ParentId="6699" CreationDate="2017-11-13T21:15:02.737" Score="5" Body="&lt;p&gt;It was known since antiquity that stars have no parallax visible by naked eye (parallax due to the displacement&#xA;of the observer on the Earth). This indicates that they must be much further than the Moon (Moon has significant parallax which was known in&#xA;antiquity). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next step was heliocentric system of Copernicus. If one accepts the motion of the Earth around the Sun, the stars must have parallax due to this motion, unless they are VE-ERY far. Actually this was used as an important argument against &#xA;the heliocentric system. The argument was that they cannot be THAT far. The universe cannot consist of mostly empty space. Why would God create such a strange universe? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As they are very far, it follows that they are very much BRIGHTER than the Moon and some of them have to be brighter then the Sun, because we see them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Strictly speaking this does not imply that they are bigger (they can be brighter but smaller, in principle). But if we make a natural assumption that they are of similar nature to the Sun, then their brightness and distance indicate they they are&#xA;much bigger than the Moon, and some of them are bigger than the Sun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this could be made precise when the parallax (due to the Earth motion around the Sun) was eventually discovered (in 18th century), and when it was proved that the nature of the stars is indeed similar to the nature of the Sun (by spectral analysis, in 19th century). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Direct measurement of the size is still impossible (except perhaps of the closest ones), they all look like dots, even with the most powerful telescope. But we have a theory about their inner structure, this theory is well confirmed and it predicts their actual dimensions.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-13T21:29:27.333" LastActivityDate="2017-11-13T21:29:27.333" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6702" PostTypeId="2" ParentId="6683" CreationDate="2017-11-14T10:25:11.233" Score="1" Body="&lt;p&gt;I suggest to start with Morris Kline's &quot;&lt;a href=&quot;https://archive.org/stream/MathematicalThoughtFromAncientToModernTimes/Mathematical_Thought_from_Ancient_to_Modern_Times#page/n15/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Thought from Ancient to Modern Times&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A valuable source is also Felix Klein's &quot;&lt;a href=&quot;https://books.google.de/books/about/Development_of_Mathematics_in_the_19th_C.html?id=NM36hgqmOLkC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Development of Mathematics in the 19th Century&lt;/a&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-11-14T20:22:37.133" LastActivityDate="2017-11-14T20:22:37.133" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6703" PostTypeId="2" ParentId="5943" CreationDate="2017-11-15T00:53:18.007" Score="3" Body="&lt;p&gt;I would like to mention a biography I just finish reading: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0465022103&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Strangest Man - The hidden life of Paul Dirac, Mystic of the atom&lt;/a&gt;. It has turned out to be not just the best biography but also one of the best books I have ever read. It is a wonderful combination of a life description of a quite &quot;strange man&quot; and a very well written story. As a plus, it contain many adjacent stories about names that you mentioned such as Bohr, Heisenberg, Pauli, Landau, etc, and gives a nice description on how Quantum Mechanics was developed. &lt;/p&gt;&#xA;" OwnerUserId="5436" LastActivityDate="2017-11-15T00:53:18.007" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6704" PostTypeId="1" AcceptedAnswerId="6705" CreationDate="2017-11-14T17:29:23.743" Score="8" ViewCount="1223" Body="&lt;p&gt;Historically, where did the concept of operators in quantum mechanics come from?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How did people first understand that momentum operator should be of the form of &lt;span class=&quot;math-container&quot;&gt;$i \hbar \frac{{\rm d}}{{\rm d}x}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also how did they find the expression for kinetic energy operator?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6277" OwnerDisplayName="user157588" LastEditorUserId="4251" LastEditDate="2020-07-11T17:15:10.883" LastActivityDate="2020-07-11T17:15:10.883" Title="Origin of operators in quantum mechanics" Tags="|physics|quantum-mechanics|mathematical-physics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6705" PostTypeId="2" ParentId="6704" CreationDate="2017-11-14T18:57:54.963" Score="5" Body="&lt;p&gt;The idea of $p\to \frac{\partial }{\partial q}$ can be found in the canonical Hamilton-Jacobi theory, where &#xA;$$&#xA;p=\frac{\partial S}{\partial q}\, .&#xA;$$&#xA;This was (apparently) the inspiration for Schrodinger.  In fact, on the first page of his original paper, one can find the equations&#xA;\begin{align}&#xA;H\left(q,\frac{\partial S}{\partial q}\right)&amp;amp;=E\, ,\\&#xA;S&amp;amp;=K\log\psi\, ,\\&#xA;H\left(q,\frac{K}{\psi}\frac{\partial \psi}{\partial q}\right)&amp;amp;= E&#xA;\end{align}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It follows immediately that the kinetic energy is proportional to the second derivative of $\psi$ w/r to the position $q$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DNybC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DNybC.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Here's the translation of that first page:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QNkTe.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QNkTe.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7093" OwnerDisplayName="user6552" LastEditorDisplayName="user6552" LastEditDate="2017-11-15T21:16:16.193" LastActivityDate="2017-11-15T21:16:16.193" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6706" PostTypeId="2" ParentId="6704" CreationDate="2017-11-15T04:57:33.530" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;From where did the concept of operator in quantum mechanics came, historically?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;This was a gradual development started by Heisenberg's insight. He invented (infinite) matrices (without any prior knowledge of matrix multiplication).&#xA;This was followed by by Born, Jordan and Dirac. Dirac's book Principles of quantum mechanics (1930) explains in great detail where the operators come from. Two years later a rigorous mathematical theory of self-adjoint operators needed in quantum mechanics was developed by von Neumann. &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Momentum operator.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Representation of momentum operator comes from the analogy with classical Hamiltonian mechanics, as explained in the answer by ZeroTheHero.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Kinetic energy&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Comes from the general quantization rule: you take a classical expression and substitute the operators in it instead of coordinates and momenta.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. To address the question asked in a comment to the answer of ZeroTheHero:&#xA;There is a book by B. L. van der Waerden, Sources in quantum mechanics, where many papers of the early period are translated into English, with comments.&#xA;Unfortunately, Schroedinger's papers are not there. But the main papers by Heisenberg, Born, Jordan and Dirac of 1925-26 (and many earlier papers) are in the book.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-15T20:24:43.210" LastActivityDate="2017-11-15T20:24:43.210" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6707" PostTypeId="2" ParentId="6652" CreationDate="2017-11-15T10:23:52.600" Score="1" Body="&lt;p&gt;The question really hangs on when nuclear energy was thought &lt;a href=&quot;https://www.nobelprize.org/nobel_prizes/themes/physics/fusion/&quot; rel=&quot;nofollow noreferrer&quot;&gt;possible&lt;/a&gt;; because once it is thought possible it's easy enough to speculate whether it could be weaponised. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, in 1896 Wilhelm Rontgen and Henri Bequerel discovered radioactivity, and then in 1903 Pierre Curie announced that Radium salts radiated heat without cooling down and thus revealing a new source of energy; a year later, Ernest Rutherford showed that alpha particle radiation released enormous amounts of radiation, he wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The discovery of the radioactive elements which in their disintegration liberate enormous amounts of energy increases the possible limit of the duration of life on this planet, and allows the time claimed by the geologist and biologist for the process of evolution.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here, he was referring to a controversy between Darwin &amp;amp; Kelvin of the age of the earth; Darwin had come up with an estimate of 200 million years by considering the process of erosion on the earths surface, whereas Kelvin had estimated only 30 million years by supposing gravitational contraction powered the sun. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery of radioactivity freed theorists from considering gravity as the source of the suns energy, but it was shown that the sun had very little radioactive elements being mostly composed of hydrogen so some other process was required. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1905, Einstein had established his famous equation relating energy and mass; this fueled speculation that this could be the process that fueled the sun. Empirical evidence for such a claim came in 1919 when FW Aston demonstrated that four hydrogen atoms weighed more than a helium atom; then Sir Arthur Eddington, leveraging Einsteins insight, argued that this mass difference was what powered the sun. He said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If, indeed, the sub-atomic energy in the stars is being freely used to maintain their great furnaces it seems to bring a little nearer fulfillment our dream of controlling this latent power for the well-being of the human race - or it's suicide.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This must be one of the earliest acknowledgements that nuclear energy could be hugely destructive; this it's probable that a debate had been circulating amongst scientists between the discovery of a new energy source by Pierre Curie in 1903 and Eddingtons speech in 1920 to whether such a source could be weaponised. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2017-11-15T10:28:59.093" LastActivityDate="2017-11-15T10:28:59.093" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6708" PostTypeId="1" CreationDate="2017-11-15T19:24:41.173" Score="7" ViewCount="547" Body="&lt;p&gt;From what little I know of sociology, Karl Marx is considered one of the fathers of sociology, along with Emile Durkheim and Max Weber. Marx is most known for his political views, which don't have a lot of support in current Western thought; are his sociological views similarly treated, or are they still the basis for any substantial portion of modern sociology?&lt;/p&gt;&#xA;" OwnerUserId="6554" LastActivityDate="2019-02-09T20:18:11.357" Title="How much do Marx's theories of sociology influence modern Western sociology?" Tags="|social-sciences|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6709" PostTypeId="2" ParentId="6704" CreationDate="2017-11-16T10:04:30.260" Score="4" Body="&lt;p&gt;Max Jammer in &lt;em&gt;The conceptual development of Qm&lt;/em&gt;, 1967, (chap. 5.2) has a written a dozen page history about operators, an idea that was developped in the study of differential equations. He starts with a Leibniz paper from 1710 (note 87) &lt;em&gt;Symbolismus memorabilis&lt;/em&gt;, mentions Lagrange and piles up scores of obscure names. In 1903 Ludwig Silberstein published &lt;em&gt;Eine theorie der physikalischen operatoren&lt;/em&gt; in Otswalds Annalen that might have been known by later authors. As a careful historian Jammer refrains from pointing who has been the first, but it appears rather unambiguosly that he would accord priority to the tandem Born-Wiener. He writes that in work published 1925-6 &quot;Born and Wiener, in generalizing matrix mechanics, introduced the operator calculus&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-11-16T10:04:30.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6710" PostTypeId="1" AcceptedAnswerId="6713" CreationDate="2017-11-16T12:05:24.463" Score="1" ViewCount="3206" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Which scientists were known to be messy?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know of &lt;a href=&quot;https://en.wikipedia.org/wiki/Albert_Einstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;Albert Einstein&lt;/a&gt;:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/sMZ2e.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sMZ2e.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And of &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Piaget&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Piaget&lt;/a&gt;:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/diFyz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/diFyz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there other examples?&lt;/p&gt;&#xA;" OwnerUserId="6558" LastEditorUserId="4251" LastEditDate="2020-07-11T19:09:02.323" LastActivityDate="2020-07-11T19:09:02.323" Title="Famous messy scientists" Tags="|biographical-details|examples|scientists|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6712" PostTypeId="1" CreationDate="2017-11-17T00:50:43.243" Score="4" ViewCount="253" Body="&lt;p&gt;I've noticed the Doppler effect in the change of pitch of music as I ride past on my bicycle, so I assume in the old days people did this on horses too. Are there any ancient mentions or theories about this, maybe the ancient greeks?&lt;/p&gt;&#xA;" OwnerUserId="6560" LastActivityDate="2017-11-17T23:32:39.257" Title="How did ancient people describe the Doppler effect?" Tags="|physics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6713" PostTypeId="2" ParentId="6710" CreationDate="2017-11-17T09:24:52.757" Score="3" Body="&lt;p&gt;Including mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Horton_Conway&quot; rel=&quot;nofollow noreferrer&quot;&gt;John H Conway&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rVtsw.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rVtsw.jpg&quot; alt=&quot;John Horton Conway in his office at Princeton University in 1993.&quot;&gt;&lt;/a&gt;&#xA;John Horton Conway in his office at Princeton University in 1993.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conway was eventually crowded out of that office, writes biographer Siobhan Roberts, by the &quot;wanton mess that accumulated into a full-fledged tip, with multicoloured paper polyhedra hanging from the ceiling and gigantic spongy Escher puzzle pieces tiling the floor.&quot;&lt;/p&gt;&#xA;" OwnerUserId="6562" LastActivityDate="2017-11-17T09:24:52.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6714" PostTypeId="2" ParentId="6712" CreationDate="2017-11-17T13:46:25.153" Score="5" Body="&lt;p&gt;Have you really performed the experiment with riding a bicycle past a music source? I do not know with what speed you can ride your bicycle, but with a generous assumption of $10$m/sec, I arrive at the frequency distortion of $1.03$.&#xA;This is less than half-tone (the ratio of the frequencies of two neighboring piano keys (black and white). So I imagine that &quot;an ancient&quot; standing and listening how&#xA;a horseman riding past him at a full speed sounds a horn will not notice much difference. To have this $1.03$ frequency change, the horseman has to ride straight at you; if he passes you at some distance, the effect will not be sharp&#xA;but gradual. And the horseman has to produce the same note on his horn, continuously. So I doubt that this was noticed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way Doppler himself discovered the effect by observing light, rather than sound. (Railroads already existed in 1842 when the effect was discovered. One can easily hear the Doppler effect when a train on full speed passes you while sounding the signal). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-17T23:32:39.257" LastActivityDate="2017-11-17T23:32:39.257" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6715" PostTypeId="1" CreationDate="2017-11-17T05:02:21.400" Score="1" ViewCount="91" Body="&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein%E2%80%93Szil%C3%A1rd_letter&quot; rel=&quot;nofollow noreferrer&quot;&gt;the letter to FDR&lt;/a&gt; during WWII written by Szilárd and signed by Einstein, a rather massive (emphasis added) nuclear bomb is theorized:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This new phenomenon would also lead to the construction of bombs, and it is conceivable — though much less certain — that extremely powerful bombs of a new type may thus be constructed. A single bomb of this type, &lt;strong&gt;carried by boat&lt;/strong&gt; and exploded in a port, might very well destroy the whole port together with some of the surrounding territory. However, such bombs might very well prove to be &lt;strong&gt;too heavy for transportation by air&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suppose this was before the notion that U&lt;sub&gt;235&lt;/sub&gt; could be isolated from natural uranium bringing the critical mass to a wispy 52 kg.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regardless, what is the notional design of such a boat-delivered bomb? Is there a critical mass of natural uranium, or some mixture of natural uranium with a moderator? Was the theory to make something like the Chicago Pile, but through a to-be-determined method make it supercritical and blow up rather than melt down?&lt;/p&gt;&#xA;" OwnerDisplayName="Nick T" LastActivityDate="2017-11-17T15:55:19.217" Title="What was the boat-carried nuclear bomb envisaged by the Einstein–Szilárd letter?" Tags="|nuclear-physics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6716" PostTypeId="2" ParentId="6710" CreationDate="2017-11-17T15:40:32.773" Score="2" Body="&lt;p&gt;Just google &quot;messy professor&quot; images to find lots of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p7bJt.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p7bJt.jpg&quot; alt=&quot;Al Gore&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-11-17T15:40:32.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6717" PostTypeId="2" ParentId="6715" CreationDate="2017-11-17T15:55:19.217" Score="2" Body="&lt;p&gt;I think it's pretty obvious (at least in hindsight and having read Richard Rhode's excellent  &lt;a href=&quot;https://smile.amazon.com/Making-Atomic-Bomb-25th-Anniversary/dp/1451677618/&quot; rel=&quot;nofollow noreferrer&quot;&gt;history&lt;/a&gt; ) that they simply didn't know the total mass (fissile plus compression explosives plus shell) of the final device.&lt;br&gt;&#xA;Side note - the destructive force is much greater when a bomb is detonated at a calculated altitude rather than ground level. They probably hadn't talked with explosives experts yet, either. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2017-11-17T15:55:19.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6718" PostTypeId="2" ParentId="3169" CreationDate="2017-11-18T21:40:38.387" Score="-1" Body="&lt;p&gt;No matter how much I admire Weber's potential and force laws, these laws don't predict the physics of optics (for instance reflection and refraction, interference, etc ...). The Maxwell field theory predicts laws of optics as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Weber's force law in case of magnetostatics is exactly Ampere's force law which satisfies Newton's third principle of motion, where the Lorentz force violates this principle in case of magnetostatics. The violation by Grassmann's force law of momentum conservation is NOT compensated by the momentum of fields. Magnetostatica experiments are such that the momentum of fields do not compensate for the inbalance in the forces on massive current carrying objects (metalic conductors).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps it is wise to reconsider the Lorentz force law, and replace it for a better electromagnetic field based force law, which agrees with Newtonean classical mechanics. Whittaker was working on this problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The retardation of fields and the finite velocity of near fields and far fields can be measured, however, no experiment clearly confirms the Coulomb near fields also spread with TEM wave far field velocity. On the contrary, several Coulomb field speed measurements show super luminal velocity, for instance, Wheatstone's famous rotating mirror experiment shows super luminal velocity of 'electricity' (partly wired partly wireless path).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since Weber's laws relies only on the relative distance between two charges, it is reference frame independent, also independent from relative rotation of frames. I wonder if all electrodynamics in a rotating lab frame is the same as electrodynamics in a non-rotating lab frame. &lt;/p&gt;&#xA;" OwnerUserId="6572" LastActivityDate="2017-11-18T21:40:38.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6719" PostTypeId="1" CreationDate="2017-11-18T22:05:11.387" Score="4" ViewCount="211" Body="&lt;p&gt;What is the historical motivation of continuous functions? Also, does anyone know who first isolated the idea?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2018-05-27T10:26:35.013" Title="Motivation of Continuous Functions" Tags="|real-analysis|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6720" PostTypeId="1" CreationDate="2017-11-18T22:47:40.023" Score="1" ViewCount="118" Body="&lt;p&gt;What is the historical motivation of infinite series? According to Wikipedia, they are arose separately by Newton, Leibniz and Somayaji.&lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2017-11-19T01:11:50.710" Title="Motivation of Infinite Series" Tags="|real-analysis|euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6721" PostTypeId="1" AcceptedAnswerId="6732" CreationDate="2017-11-18T22:59:00.053" Score="5" ViewCount="1291" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Number, indeed, has two proper kinds (ιδια ειδη), odd and even, and a&#xA;  third mixed together from both, the even-odd(αρτιοπέριττον). Of each&#xA;  of the two kinds there are many shapes, of which each thing itself&#xA;  gives signs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Philolaus F5 = Stobaeus, Eclogae 1.21.7c; tr. Huffman 1993)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote is from Horky's &lt;em&gt;Plato and Pythagoreanism&lt;/em&gt;, 2013,  (p.141), it is commented twice further in text. On p185 the author writes &quot; the so-called even-odd&#xA;(αρτιοπέριττον), which appears to refer to the “one.” Aristotle, too, knew this fragment, and he is explicit in associating the “one” with the “even-odd” class and stating that it is derived from the “even” and the “odd.” On p190, disappointingly, he says &quot;however It is beyond the scope of this study to examine in more detail the significance of the&#xA;concept of “mixture” of Forms or classes&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The “one” is not the number preceding two which Ancient greeks apparently did not accept as a number. Philolaos quote might suggest that it is set apart as it has no shape but I am tempted to read it as suggesting that the mixed numbers have multiple shapes, while the numbers from the proper kinds have each just one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what are the 3 kinds of numbers?&#xA;The odd numbers are perhaps the ones that we also call odd, (starting with 3), while the even numbers are seen as, so to say, 'strictly even', the powers of 2, that is 2,4,8,16 and the rest, that is 6,10,12 etc. as even-odd. References and comments (or should this be moved to Philosophy SE)?&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="131" LastEditDate="2017-11-29T13:31:00.087" LastActivityDate="2022-06-11T07:05:27.297" Title="What are Philolaos' “even-odd” numbers?" Tags="|mathematicians|reference-request|ancient-greece|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6722" PostTypeId="2" ParentId="6720" CreationDate="2017-11-19T01:11:50.710" Score="2" Body="&lt;p&gt;I have not read Leibniz and Somayaji, but Newton himself says that his motivation is that one can solve &quot;any equation&quot; using infinite series, he means by this&#xA;algebraic equations, differential equations (integrals is a special case of them)&#xA;and many other types of functional equations. &#xA;He demonstrates this with examples in his two letters to Leibniz, and then in expanded form in his papers on Analysis. What he uses is called Taylor series nowadays, and some more general series with non-integer powers of the variable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also explicitly mentions the analogy between the infinite decimal expansions&#xA;and infinite series (decimal expansions are in fact a very special case).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear that other mathematicians were also thinking along these lines.&#xA;For example $\pi$ as a real number has a decimal expansion, in principle. But one cannot find all digits of this expansion explicitly. On the other hand&#xA;various representations of $\pi$ by other explicit infinite expansions (series or products) were found. These expansions are more explicit than decimal representation because the pattern is clear in them.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-11-19T01:11:50.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6723" PostTypeId="1" AcceptedAnswerId="6724" CreationDate="2017-11-19T01:11:51.937" Score="3" ViewCount="158" Body="&lt;p&gt;Nonrelativistic Quantum Mechanics (QM) relies in a number of postulates and although some authors may disagree about the exact set, there are a few which are quite indisputable:  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;States are represented by vectors in a Hilbert space.&lt;/li&gt;&#xA;&lt;li&gt;Observables are represented by Hermitian operators acting on the Hilbert space. &lt;/li&gt;&#xA;&lt;li&gt;If the system is in state $|\psi\rangle$, then the measurement of the observable $A$ yields one of the eigenvalue of the operator associated to $A$.&lt;/li&gt;&#xA;&lt;li&gt;(Born Postulate) The probability of measuring $A$ and obtaining $a$, given that the system is in the state $|\psi\rangle$, is $|\langle a|\psi\rangle |^2$. After the measurement, the system is found to be in state $|a\rangle$.&lt;/li&gt;&#xA;&lt;li&gt;The state vector of the system obeys the Schrodinger equation.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Who did formulate the first three of these principles (or any other you may regard as fundamental) as fundamental postulates of QM?&lt;/p&gt;&#xA;" OwnerUserId="5436" LastActivityDate="2017-11-19T02:56:13.503" Title="Who did enunciate each postulate of modern Quantum Mechanics" Tags="|physics|theoretical-physics|quantum-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6724" PostTypeId="2" ParentId="6723" CreationDate="2017-11-19T02:49:47.707" Score="5" Body="&lt;p&gt;All these principles gradually evolved, in a series of papers starting from Heisenberg, Schrodinger and followed by Born, Jordan and Dirac, some of these papers are joint, the contribution of each author is impossible to separate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the form that you state them all of them were formulated for the first time&#xA;in Dirac's book.&#xA;However these postulates lack mathematical rigor (and Dirac understood this). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean the use of the words &quot;Hermitian&quot; and &quot;eigenvalue&quot;. The operators representing observables must be self-adjoint. The exact meaning of these things was clarified for the first time by von Neumann.&#xA;(Technically speaking, observables may not have any eigenvalues or eigenstates, they may not be &quot;acting on the Hilbert space&quot; etc., in the case of unbounded observables and/or counituius spectrum).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-19T02:56:13.503" LastActivityDate="2017-11-19T02:56:13.503" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6725" PostTypeId="2" ParentId="6719" CreationDate="2017-11-19T04:06:04.753" Score="5" Body="&lt;p&gt;Isolation of the modern concept is &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_function#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;generally attributed&lt;/a&gt; to Bolzano (&lt;a href=&quot;https://books.google.com/books?id=uccGAAAAcAAJ&amp;amp;pg=PA11&amp;amp;dq=Stetigkeit&quot; rel=&quot;nofollow noreferrer&quot;&gt;1817&lt;/a&gt;; &lt;a href=&quot;//ams.org/mathscinet-getitem?mr=572277&quot; rel=&quot;nofollow noreferrer&quot;&gt;translation&lt;/a&gt;) and/or Cauchy (&lt;a href=&quot;https://archive.org/stream/coursdanalysedel00cauc#page/34&quot; rel=&quot;nofollow noreferrer&quot;&gt;1821&lt;/a&gt;), with some controversy on their independence: see Grattan-Guinness (&lt;a href=&quot;https://zbmath.org/?q=an:0198.00601&quot; rel=&quot;nofollow noreferrer&quot;&gt;1970&lt;/a&gt;), Freudenthal (&lt;a href=&quot;https://zbmath.org/?q=an:0246.01010&quot; rel=&quot;nofollow noreferrer&quot;&gt;1971&lt;/a&gt;). Kline (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=472307&quot; rel=&quot;nofollow noreferrer&quot;&gt;1972&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/MathematicalThoughtFromAncientToModernTimes#page/n968&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 950&lt;/a&gt;) says Bolzano’s motivation was to rigorously prove the fundamental theorem of algebra, and on his way, the intermediate value theorem — stated in his memoir’s title.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-05-27T10:26:35.013" LastActivityDate="2018-05-27T10:26:35.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6726" PostTypeId="1" CreationDate="2017-11-19T19:22:53.973" Score="8" ViewCount="2528" Body="&lt;p&gt;When was the use of \mathbb popularized as an alternative to \mathbf? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course there are the subjective preferences of certain authors, but when I read older articles, there appears to be an almost unanimous usage of what we would think of as \mathbf (possibly because of the use of typewriters before LaTeX), and now more recently it seems as if the use of \mathbb is becoming increasingly common. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a reason for this switch in notation?&lt;/p&gt;&#xA;" OwnerUserId="3129" LastActivityDate="2017-11-19T19:22:53.973" Title="\mathbb versus \mathbf" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6727" PostTypeId="1" AcceptedAnswerId="6736" CreationDate="2017-11-20T13:04:34.047" Score="11" ViewCount="917" Body="&lt;p&gt;When I first learned calculus a few decades ago, the books I read used italicized letter &quot;d&quot;s in derivatives (like this: $\frac{dy}{dx}$).  But a few years ago, I started seeing upright &quot;d&quot;s (like this: $\frac{{\mathrm d}y}{{\mathrm d}x}$).  When did this way of writing derivatives start?&lt;/p&gt;&#xA;" OwnerUserId="72" LastEditorUserId="658" LastEditDate="2018-03-01T03:55:07.930" LastActivityDate="2018-10-23T14:44:08.487" Title="$\frac{dy}{dx}$ versus $\frac{{\mathrm d}y}{{\mathrm d}x}$" Tags="|mathematics|notation|calculus|" AnswerCount="3" CommentCount="11" ContentLicense="CC BY-SA 3.0" />
  <row Id="6729" PostTypeId="2" ParentId="6708" CreationDate="2017-11-20T20:57:54.243" Score="5" Body="&lt;p&gt;Still the basis you ask, no, not currently; but...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you have asked this question at a very interesting time. When you see discussions of the far Right and neo-fascism, then you can count on the fact that there will be a rise of the Left; a reaction by a &quot;new&quot; New Left to present circumstances, if you will. These &quot;repetitions&quot; are never exactly the same as the first (or second, or third) time around, but they are suggestive of prior periods. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The particular theories and associated material you ask about in sociology is very rich, and I should remark on this rather peculiar feature about such studies:  the volume of output drops off a cliff after 1979, with relatively few books, articles etc. straggling along into the mid-'80s.  (This could be a fascinating study in itself, but let's just say if you wanted tenure, you might have found it wise to switch away from Marx beginning in the early 1980s in America).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You will see a characteristic overlap in this area too, between sociology, economics, and philosophy; but primarily between sociology and economics.  I have in my hands now this book:  &quot;Theories of Modern Capitalism&quot; by Tom Bottomore, Controversies in Sociology (Series) Vol. 17, George Allen &amp;amp; Unwin, London (1985), found at Dewey Decimal Classification (DDC) 330.122, the Economics(!) section of the library, and not in the Sociology section, even though it is a part of a sociology series. But of course you will find titles in the sociology section of the library also. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;T.B. Bottomore, you will see his name a lot, and this is probably his most important work:  Title:  Political sociology, Author: Bottomore, T. B., 1920-1992.   Publisher:Harper &amp;amp; Row,Pub date:1979.  A search of &quot;T.B. Bottomore&quot; at a used book website (or at a university library or even a very good public library) will yield even more titles related to your question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are interested in doing some reading, then I suggest these search terms: alienation, sociology of knowledge, class, class structure, critical theory. These subjects could be some discrete subjects you could research in sociology and the Marx or Marxian school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The subject of &quot;alienation&quot; might be more rewarding and interesting for someone new to this area. You may want to start here with:  Karl Marx, Selected Writings in Sociology &amp;amp; Social Philosophy, T.B. Bottomore &amp;amp; Maximilien Rubel Eds., McGraw-Hill, New York (1956), (1964), see particularly Fromm's forward, pg. xvii, &quot;....what we ourselves produce into an objective power above us....&quot;. Fromm does not use the word alienation here, but he is talking about a key part of the concept in that the system of objects that workers create through labor turns around to dominate the people who make it. This book is widely available on the used market, and libraries will also have it. This is just a suggestion, there are many more books on the subject of alienation/Marxian school. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sociology of Knowledge: Title:  Marx and Mead : contributions to a sociology of knowledge Author Goff, Tom W.   Publisher:Routledge &amp;amp; Kegan Paul,Pub date:1980.  (The Mead referred to here is George H Mead). Also, Karl Mannheim, the most important figure, who was influenced in part by Marx. &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Karl_Mannheim&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Karl_Mannheim&lt;/a&gt;  Title:  The concept of ideology Author Larraín, Jorge, 1942-  Publisher:University of Georgia Press,Pub date:c1979. Title:  Marx's theory of ideology Author Parekh, Bhikhu C.   Publisher:Johns Hopkins University Press,Pub date:1982.&lt;/p&gt;&#xA;" OwnerUserId="6586" LastEditorUserId="6586" LastEditDate="2017-11-22T02:46:30.677" LastActivityDate="2017-11-22T02:46:30.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6730" PostTypeId="1" CreationDate="2017-11-20T21:00:44.877" Score="0" ViewCount="282" Body="&lt;p&gt;I was born after the 1980s so I don't really know what it was like then. However, whenever I hear older folks reminisce about all the new and innovative technology that came out back then I have to wonder why there was such a boom in the industry. Most of the technology we use today was first introduced in the 1980s - the first cell phone, the first camcorder, the CD player and so on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was it about the 1980s that allowed for so many new tech devices and revolutionary ideas to be introduced in the matter of 10 years&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;GPS&lt;/li&gt;&#xA;&lt;li&gt;Video Gaming&lt;/li&gt;&#xA;&lt;li&gt;Cable TV&lt;/li&gt;&#xA;&lt;li&gt;Home video&lt;/li&gt;&#xA;&lt;li&gt;Cell phones&lt;/li&gt;&#xA;&lt;li&gt;Portable electronics (radios, audio players)&lt;/li&gt;&#xA;&lt;li&gt;Compact Disc&lt;/li&gt;&#xA;&lt;li&gt;Camcorder&lt;/li&gt;&#xA;&lt;li&gt;Personal computing&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6585" LastActivityDate="2017-11-20T21:00:44.877" Title="Why was there so much technological advancement in the 1980s?" Tags="|technology|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6731" PostTypeId="2" ParentId="2588" CreationDate="2017-11-20T22:57:00.953" Score="4" Body="&lt;p&gt;Mayan numerals go from top to bottom...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here on the left we see 8.5.16.9.7&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pBuZY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pBuZY.jpg&quot; alt=&quot;numeral&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Maya_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;(SOURCE)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2017-11-20T22:57:00.953" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6732" PostTypeId="2" ParentId="6721" CreationDate="2017-11-21T11:00:56.580" Score="4" Body="&lt;p&gt;Aristotle comments on the Pythagorean theory in &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0052%3Abook%3D1%3Asection%3D986a&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Met&lt;/em&gt;, Book I (A), 986a14-986a22&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;these thinkers also consider that number is the principle both as matter for things and as forming their modifications and states, and hold that the elements of number are the even and the odd, and of these the former is unlimited, and the latter limited; and the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; proceeds from both of these (for it is both even and odd), and number from the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;; and the whole heaven, as has been said, is numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But see also Thomas Heath, &lt;a href=&quot;https://archive.org/details/cu31924008704219&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Greek Mathematics. Volume I (1921)&lt;/a&gt;, page 71:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The explanation of this strange view might apparently be that the unit, being the principle of all number, even as well as odd, cannott itself be odd and must therefore be called &lt;em&gt;even-odd&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is, however, another explanation, attributed by Theon of Smyrne to Aristotle, to the effect that the unit when added to an even number makes an odd number, but when added to an odd number makes an even number: which could not be the case if it did not partake of both species.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Philolaus' Fr.5 is discussed at lenght in: Carl Huffman, &lt;a href=&quot;https://books.google.it/books?id=GtliSUQqeyoC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philolaus of Croton: Pythagorean and Presocratic&lt;/a&gt;, Cambridge UP (1993), page 178-on. There are various interpretations, including the possibility of an interpolation.&lt;/p&gt;&#xA;&lt;p&gt;See page 190:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In summary, I believe that the even-odd is a derived class of numbers&#xA;whose first member is, as the ancient tradition indicates, the one, but which&#xA;also includes numbers that consist of even and odd numbers combined in&#xA;ratios (e.g. &lt;span class=&quot;math-container&quot;&gt;$2 : 1 , 4 : 3$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$3 : 2$&lt;/span&gt; ) . This class of numbers corresponds to the third class of things in Fr.2, which consists of members that are harmonized from both limiting and unlimited constituents. The even-odd numbers are the numbers by which these harmonized things are known. This connection&#xA;of course remains conjectural, but I believe that it is a plausible way&#xA;to make sense of both Fr.2 and Fr.5 of Philolaus and Aristotle's testimony&#xA;that there was a connection between the even - odd dichotomy and the&#xA;unlimited - limited dichotomy, although my suggestion does not identify&#xA;the two as Aristotle does.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-11-21T13:17:29.030" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6733" PostTypeId="1" CreationDate="2017-11-21T13:38:54.817" Score="1" ViewCount="213" Body="&lt;p&gt;As it is globally known that set theory as a foundation of mathematics, although in the beginning we didn't call it &quot;Set&quot; rather group of elements. For example - set of [1(banana) + 2(apple)+1(cow)] =&gt; contains 4 elements. Of course, counting objects as 1,2,3,4,5,... was the basis and earliest example of set theory though at that time we didn't give it the name &quot;Sets&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So counting as sets was foundation of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After this developing basic &quot;operations&quot; like addition/subtraction seems like a logical development of foundation in mathematics. &lt;em&gt;I can be wrong also I would assume multiplication/division as complex operations&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;then &lt;strong&gt;what&lt;/strong&gt; did we developed after this and &lt;strong&gt;why&lt;/strong&gt; and &lt;strong&gt;how&lt;/strong&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did we invented geometry right after this foundation and then number system to make mathematics more concrete? Did we invented/discovered  negative number / decimal / fraction / rational / irrational / Integers after geometry as mathematical foundation? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In what order of mathematical foundation we invented/discovered trigonometry? In what order of mathematical foundation we invented/discovered Algebra? When did we started using complex mathematical operations (multiplication/division)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Few points: &#xA;1. I don't need dates as when it was invented rather &quot;after what&quot; it was discovered/invented.&#xA;2. What would be the logical timeline you will decide if just keep the history aside?&#xA;3. What is the actual timeline of mathematical foundations? And why did we discovered/invented them?    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;P.S:&lt;/strong&gt; Please don't mark it as broad question because it is a broad question and all kind of example, references,links, study material is welcomed.  &lt;/p&gt;&#xA;" OwnerUserId="6576" LastActivityDate="2017-11-21T13:38:54.817" Title="Timeline of mathematical foundation?" Tags="|mathematics|mathematicians|terminology|mathematical-logic|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6734" PostTypeId="2" ParentId="6727" CreationDate="2017-11-21T18:45:00.033" Score="5" Body="&lt;p&gt;Standards bodies recommend using roman font for mathematical operators and italic font for physical quantities or variables. The $\mathrm{d}$ in the derivative is an operator (&quot;differential of&quot;), therefore it has to be written in roman font as $\mathrm{d}x$, whereas a variable $d$ like in $y= ax^4 + bx^3 + cx^2 + dx$ is written in italic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This convention also applies to other variables/operators. For instance a delta in roman font usually denotes the operator of small variation, whereas a delta in italic font usually means the decay coefficient. I don't know when those recommendations did start.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.iupac.org/cms/wp-content/uploads/2016/01/ICTNS-On-the-use-of-italic-and-roman-fonts-for-symbols-in-scientific-text.pdf&quot; rel=&quot;noreferrer&quot;&gt;On the use of italic and roman fonts for symbols in scientific text&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://physics.nist.gov/cuu/pdf/typefaces.pdf&quot; rel=&quot;noreferrer&quot;&gt;Typefaces for Symbols in Scientific Manuscripts&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6580" LastActivityDate="2017-11-21T18:45:00.033" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6735" PostTypeId="1" CreationDate="2017-11-21T18:59:38.127" Score="4" ViewCount="84" Body="&lt;p&gt;It's standard nowadays for the real line to be horizontal (negative numbers on the left, positive numbers on the right) and for $i$ to be above (rather than below) 0 in the complex plane. Were these conventions universally adhered to in the past?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, I don't know how Argand and Wessel and Gauss actually drew the complex plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For that matter, did Oresme (who used something resembling Cartesian coordinates) always draw time as going from left to right, and time-varying quantities on a vertical axis?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(This question is also &lt;a href=&quot;https://mathoverflow.net/questions/286637/variants-in-graphical-presentation-of-real-and-complex-numbers&quot;&gt;https://mathoverflow.net/questions/286637/variants-in-graphical-presentation-of-real-and-complex-numbers&lt;/a&gt; at MathOverflow. I'm new to HSM, so please feel free to correct my tags if others would be more appropriate.)&lt;/p&gt;&#xA;" OwnerUserId="6587" LastActivityDate="2017-11-21T18:59:38.127" Title="Variants in graphical presentation of real and complex numbers" Tags="|real-analysis|complex-analysis|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6736" PostTypeId="2" ParentId="6727" CreationDate="2017-11-22T01:46:43.173" Score="8" Body="&lt;p&gt;The earliest use of upright d to indicate a differential or derivative operator as a standard practice I have found is from the International Union of Pure and Applied Physics (IUPAP), 1987:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://iupap.org/wp-content/uploads/2014/05/A4.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://iupap.org/wp-content/uploads/2014/05/A4.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This formed the basis for International Organization for Standardization's &lt;em&gt;ISO 31&lt;/em&gt; (1992) standard:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/ISO_31&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/ISO_31&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A discussion of this standard may be found in &lt;a href=&quot;http://www.tug.org/TUGboat/Articles/tb18-1/tb54becc.pdf&quot; rel=&quot;noreferrer&quot;&gt;this 1997 TUGboat article&lt;/a&gt;.  This standard has been superseded by &lt;a href=&quot;https://en.wikipedia.org/wiki/ISO_80000-2&quot; rel=&quot;noreferrer&quot;&gt;ISO 80000-2&lt;/a&gt;, which maintains the use of the upright d for the differential operator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite &lt;a href=&quot;https://en.wikipedia.org/wiki/ISO/IEC_80000&quot; rel=&quot;noreferrer&quot;&gt;the claim&lt;/a&gt; that &quot;the notations used in mathematics and science textbooks at schools and universities follow closely the guidelines in this standard,&quot; it is clear to me that the notation $dx$ is still widely followed in mathematics textbooks and journals in the US.  Perhaps it is different in physics, which seems to be a source for the change in notation.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2017-11-22T01:46:43.173" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6737" PostTypeId="2" ParentId="30" CreationDate="2017-11-22T20:15:03.877" Score="2" Body="&lt;p&gt;Davidovits' theory appears to be moving slowly from fringe toward mainstream.&#xA;Also  the possibility of a mixed technique i.e. some carved stones along&#xA;with cast blocks offers a compromise in the debate. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;why were the pyramids not made wholly of these cast stones? After all it is easier to cast a stone than to cut and hoist it into place. The answer, embedded in our finding, is simple: making geopolymeric stone was an expensive &#xA;  proposition. For a primitive society, the fabrication of lime is non-trivial, especially the millions of tons that would have been required. Along the same lines, crushing tons of limestone is non-trivial either. Under these constraints, the ancient builders apparently compromised [1]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The quote goes along with a  photograph (fig.6) showing distinctly the two types,  the leading author, M. W. Barsoum, being a highky cited scientist (h=83) and the journal is a most academic one (J. Am. Ceram. Soc.). However there is an acknowledgement that some of the samples were provided by Davidovits. A more recent and no less academic work about Senefru's &#xA;pyramid by Barsoum et al. [2] states &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;results suggest that the casing stones consist of limestone grains from the Tura quarry, cemented with an amorphous calcium-silicate gel formed by human intervention, by the addition of extra silica, possibly diatomaceous earth, from the Fayium area.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A 2015 Springer &lt;em&gt;Handbook of Materials Structures, Properties, Processing and &#xA;Performance&lt;/em&gt; [3] begins by mentioning the casting of blocks via ref[2].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Refs&#xA;[1]M. W. Barsoum et al, (2006) Microstructural Evidence of Reconstituted Limestone Blocks in the Great Pyramids of Egypt, J. Am. Ceram. Soc., 89 [12] 3788–96 &lt;a href=&quot;https://watershedmaterials.com/s/PyramidPaper_NSF.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;(pdf)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2]M. W. Barsoum et al,Were the casing stones of Senefru's Bent Pyramid &#xA;in Dahshour cast or carved?: Multinuclear NMR evidence, Mat. Lett.Vol. 65, Iss. 2, (2011),p 350-2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[3]L. E. Murr, (2015), Examples of Materials Science and Engineering in Antiquity, Handbook of Materials Structures...p.12 &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2017-11-22T20:15:03.877" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6738" PostTypeId="1" CreationDate="2017-11-23T23:34:24.373" Score="8" ViewCount="488" Body="&lt;p&gt;On chapter 9 of M. Spivak's book on calculus there is an exercise in which Spivak asks the reader to prove that Galileo &quot;got his facts wrong&quot;. More specifically, Spivak asks one to to show if a body falls a distance $d(t)$ in $t$ second and $d^{\prime}$ is proportional to $d$ then $d$ cannot be a function of the form $d(t) = ct^{2}$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Settling it is kind of a no-brainer: yet, did Galileo really claim what Spivak is attributing to him therein? Do you know if this &quot;mistake&quot; by Galileo had been noticed before? If I understand correctly, even Newton took for granted the claim by Galileo according to which &quot;the descent of bodies varied as the square of the time&quot; (cf. p. 21 of vol. I of the University of California Press edition of the &lt;em&gt;Principia&lt;/em&gt;)? What's going on here? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me thank you for your comments, suggestions, links, answers, etc.&lt;/p&gt;&#xA;" OwnerUserId="6596" LastEditorUserId="10106" LastEditDate="2022-03-27T15:59:03.580" LastActivityDate="2022-03-27T15:59:03.580" Title="Is Spivak right in what he says about Galileo?" Tags="|mathematics|reference-request|calculus|debunking|galileo|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6739" PostTypeId="2" ParentId="6738" CreationDate="2017-11-24T00:53:05.270" Score="13" Body="&lt;p&gt;Yes, indeed when trying to obtain the law of falling bodies, Galileo's first conjecture was that the speed is proportional to the distance traveled. After some contemplation, Galileo understood that this cannot be the case and eventually came with the correct law.&lt;/p&gt;&#xA;&lt;p&gt;Good source on Galileo: S. Drake, Galileo at work. (There are many editions).&lt;/p&gt;&#xA;&lt;p&gt;From the modern point of view, this is a good story to tell to students who begin to study differential equations: equation &lt;span class=&quot;math-container&quot;&gt;$y'=ky$&lt;/span&gt; and initial condition &lt;span class=&quot;math-container&quot;&gt;$y(0)=0$&lt;/span&gt; imply &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;, so the motion never begins. In Galileo's time this was non-trivial.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-11T00:36:27.767" LastActivityDate="2021-09-11T00:36:27.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6740" PostTypeId="2" ParentId="6738" CreationDate="2017-11-24T07:01:35.807" Score="8" Body="&lt;p&gt;Yes, Galileo made that error (and so did Descartes). Only later did he realise that the speed is proportional to the time ellapsed, not to the distance already covered. I suggest that you read &lt;em&gt;The new science of motion: A study of Galileo's De motu locali&lt;/em&gt;, by Winifred L. Wisan (Archive for History of Exact Sciences, June 1974, &lt;strong&gt;13&lt;/strong&gt;, Issue 2–3, pp 103–306).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-11-24T07:01:35.807" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6741" PostTypeId="1" CreationDate="2017-11-24T22:53:04.043" Score="2" ViewCount="73" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Frederick_W._Lanchester&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lanchester&lt;/a&gt; was arguably the first to somewhat accurately describe the physical mechanism for the generation of lift. In 1894, the Birmingham Natural History and Philosophical Society &lt;a href=&quot;https://books.google.com/books?id=iaQ_AQAAMAAJ&amp;amp;pg=PA106&quot; rel=&quot;nofollow noreferrer&quot;&gt;read his paper&lt;/a&gt;, but according to &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0521669553&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anderson&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he never published that paper, because at that time he had his eyes on the Royal Society. However, a member of the Royal Society suggested that the work was more appropriate for the Physical Society. When Lanchester submitted his paper to the Physical Society, it was rejected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Anderson goes on to explain that the reason for this rejection &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;was Lanchester's poor writing style. His explanations were not easy to follow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wikipedia, meanwhile, explains without citation that the paper was &quot;too advanced for its time.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neither of these explanations seems particularly convincing to me. To the first point, we can read a lot of Lanchester's &lt;a href=&quot;http://catalogue.lanchesterinteractive.org/records/LAN&quot; rel=&quot;nofollow noreferrer&quot;&gt;writings&lt;/a&gt; contemporary to the paper in question (which is not known to have survived), and some relevant thoughts in his &lt;em&gt;&lt;a href=&quot;https://archive.org/details/aerodynamicscons00lanc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aerodynamics&lt;/a&gt;&lt;/em&gt; (although this work was published some ten years later). While his explanations could be more lucid, they are not impenetrable to the modern reader, nor should they have been to the learned people of the day. To the second point, the unsourced claim on Wikipedia seems more of a fanciful cliché to me than a practical reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there other factors that could have influenced the society's rejection of the paper, or am I missing something in the given explanations?&lt;/p&gt;&#xA;" OwnerUserId="6011" LastEditorUserId="6011" LastEditDate="2017-11-25T06:44:15.087" LastActivityDate="2017-11-25T06:44:15.087" Title="Why was F. W. Lanchester's first aeronautical paper rejected by the Physical Society?" Tags="|physics|biographical-details|discoveries|aerodynamics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6743" PostTypeId="1" AcceptedAnswerId="6744" CreationDate="2017-11-25T10:20:41.727" Score="2" ViewCount="161" Body="&lt;p&gt;There are many textbooks which states mathematical theorems, but in them, by whom and when the theorems are discovered is not explained.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know good references for this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this scope is too broad, I limit the topic to elementary geometry.&lt;/p&gt;&#xA;" OwnerUserId="6603" LastActivityDate="2017-11-26T13:55:34.193" Title="Books that we can know when which theorem was discovered by whom" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6744" PostTypeId="2" ParentId="6743" CreationDate="2017-11-25T13:42:05.483" Score="4" Body="&lt;p&gt;Such books are called &quot;history of mathematics&quot;. Of course one book will not tell you the origin of every theorem, mathematics is too large, you have to search in many books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the subject of elementary geometry, one good book is &#xA;Florian Cajori, History of elementaty mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, most of the elementary geometry is not of ancient Greek origin.&#xA;Euclid only laid the foundation of the subject, and his immediate successors were more interested in the more advanced subjects (like conic sections, integration etc.)&#xA;Most of the existing body of &quot;elementary geometry&quot; was discovered in 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-11-26T13:55:34.193" LastActivityDate="2017-11-26T13:55:34.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6746" PostTypeId="1" CreationDate="2017-11-26T22:10:58.053" Score="0" ViewCount="133" Body="&lt;p&gt;Mesoscopic physics is a topic of major research as nanotechnology becomes an important hot topic. There doesn't seem to be a major writing about the history of mesoscopic physics. I know a little about Yoseph Imry and Rofl Landauer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there more important physicists recognized mainly by the work in mesoscopic physics and quantum transport?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to read or find a resume of the important experiments that led to the foundation of the field.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2017-11-28T19:06:12.460" LastActivityDate="2017-11-28T19:06:12.460" Title="History of mesoscopic physics" Tags="|physics|terminology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6747" PostTypeId="1" CreationDate="2017-11-28T17:08:51.463" Score="3" ViewCount="1138" Body="&lt;p&gt;I only found the following at the &lt;a href=&quot;https://www.nobelprize.org/nobel_prizes/facts/physics/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nobel Prize webpage&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why were the Nobel Prizes not awarded in those years? In the statutes&#xA;of the Nobel Foundation it says: &amp;quot;If none of the works under&#xA;consideration is found to be of the importance indicated in the first&#xA;paragraph, the prize money shall be reserved until the following year.&#xA;If, even then, the prize cannot be awarded, the amount shall be added&#xA;to the Foundation's restricted funds.&amp;quot; During World War I and II,&#xA;fewer Nobel Prizes were awarded.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which does not says much. In 1916, 1940-42 it is quite reasonable to assume that there was no awarding due to the world wars. But then, what about 31 and 34? Of course there were many contributions to physics which where achieved in the preceding years and that could have been awarded.&lt;/p&gt;&#xA;" OwnerUserId="5436" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-11-28T21:08:52.667" Title="Why were the Nobel Prize in Physics not awarded in 1916, 1931, 1934, 1940, 1941, and 1942?" Tags="|physics|nobel-prize|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6749" PostTypeId="1" CreationDate="2017-12-01T21:06:18.703" Score="2" ViewCount="1188" Body="&lt;p&gt;My question is about the length of the inch which is a subunit of the Imperial foot. &#xA;Is there any connection whatsoever between the Imperial system for units of measure and the dimensions of the earth? If so, what is the relationship?&lt;/p&gt;&#xA;" OwnerUserId="6676" LastEditorUserId="131" LastEditDate="2017-12-02T20:48:51.700" LastActivityDate="2018-01-22T13:06:53.750" Title="Why is an inch (in the English Imperial system of measure) as long as it is?" Tags="|terminology|astronomy|units|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6751" PostTypeId="1" CreationDate="2017-12-03T16:27:23.423" Score="1" ViewCount="130" Body="&lt;p&gt;Euler proved $n=641$ divides $2^{32}+1$ by noting $n=5^4+2^4=5\times 2^7+1$ so $$2^{32}\equiv-5^4\times 2^{28}=-(5\times 2^7)^4\equiv-1\,(\text{mod}\, n).$$How did he happen upon this realisation? One possibility is he tried long division until he saw $641|2^{32}+1$, then saw if $641$ could be expressed in terms of powers of $2$ in a way that explains why this would be so. Another is that he realised he could prove $2^{32}+1$ is composite if he found an equation of the right form, then solved some Diophantine equations. Is it known what happened historically?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2017-12-03T16:27:23.423" Title="How did Euler stumble on this proof?" Tags="|mathematics|mathematicians|discoveries|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6752" PostTypeId="1" AcceptedAnswerId="6754" CreationDate="2017-12-03T16:34:43.887" Score="6" ViewCount="529" Body="&lt;p&gt;I think it's  fuzzy as to whether or not this question is appropriate to ask on this site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason I ask it that the characteristics of Hilbert spaces are very much used in expressing quantum mechanics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering if Hilbert had this in mind all along or was it just a fortunate accident for the early pioneers of quantum mechanics that he developed this when he did? Thank you&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-07-12T02:03:10.167" LastActivityDate="2020-07-12T02:03:10.167" Title="Are there any records that show how Hilbert came to &quot;invent&quot; or &quot;discover&quot; Hilbert spaces?" Tags="|mathematics|mathematicians|quantum-mechanics|functional-analysis|hilbert|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6753" PostTypeId="1" CreationDate="2017-12-03T18:50:31.360" Score="3" ViewCount="175" Body="&lt;p&gt;I recall that I read---in a book by Constance Reid---of a named principle that guided the arithmetic conventions that applied to operations on newly discovered mathematical objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, when negative numbers were finally admitted into the fold of arithmetic, the product of two negative numbers was defined to be positive (even if it appeared on face value to be counter intuitive) to maintain logical consistency with all that went before it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought this idea was called the &quot;Principle of Arithmetic Consistency&quot; or something like that, but I have not been able to Google successfully for any such candidate expression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not believe my recollection to be false.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could someone please tell is if such a principle does indeed exist and is recognized, and if so, what its formal name and definition are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6710" LastEditorUserId="131" LastEditDate="2017-12-04T11:09:55.393" LastActivityDate="2017-12-04T11:09:55.393" Title="What is the name given to the principle that guides mathematical conventions like the product of two negative numbers is positive" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6754" PostTypeId="2" ParentId="6752" CreationDate="2017-12-03T23:13:53.097" Score="7" Body="&lt;p&gt;From Steen’s paper mentioned in a very similar &lt;a href=&quot;https://mathoverflow.net/questions/287517/reference-request-for-functional-analysis&quot;&gt;MO question&lt;/a&gt; just yesterday:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hilbert himself was astonished that the spectra of his quadratic forms should&#xA;  come to be interpreted as atomic spectra. “I developed my theory of infinitely many variables from purely mathematical interests, and even called it ‘spectral analysis’ without any presentiment that it would later find an application to the actual spectrum of physics.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;”Records of how he came to invent” his space (originally just $l^2$) are his papers, quoted by Steen.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-12-03T23:13:53.097" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6756" PostTypeId="1" CreationDate="2017-12-04T13:57:12.467" Score="3" ViewCount="109" Body="&lt;p&gt;I am looking for papers or books that describe the history of the development of braid theory, mainly during the 19th and the 20th century. I know Moritz Epple book on the history of the theory of knots (&quot;Die Entstehung der Knotentheorie&quot;), which gives already several surveys on Gauss, Hurwitz and Artin's works on braids, but I was wondering if there are other manuscripts which are dedicated only to the history of braids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="3232" LastActivityDate="2017-12-04T13:57:12.467" Title="History of Braids" Tags="|reference-request|topology|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6757" PostTypeId="2" ParentId="6752" CreationDate="2017-12-04T14:50:52.273" Score="9" Body="&lt;p&gt;The notion of Hilbert space comes from Hilbert's theory of integral equations. Of course, it was partially motivated by physics, by the theory of oscillations in classical mechanics, but this theory was developed much earlier than quantum mechanics:&#xA;the main work Grundzuge einer allgemeinen Theorie der linearen Intergalgleichungen (Foundations of the general theory of linear integral equations), in 6 parts was published in 1904-1910.&#xA;Hilbert's own exposition (besides original papers) is in Courant Hilbert, Methods of mathematical physics. The very title shows that he definitely was motivated by physics, while quantum mechanics was not discovered yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many anecdotes showing that Hilbert was surprised by applications to&#xA;quantum mechanics and by the very term &quot;Hilbert space&quot;. Once in a conference, he asked the speaker: &quot;What is a Hilbert space?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;W. Feller was fond of repeating: &quot;I could stare all my life at a symmetric matrix, I will never get Hilbert space out of it!&quot; (Cited from G-C Rota, Indiscrete thoughts p. 223).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are many other examples when mathematicians develop new tools just in time, shortly before they find new applications in physics (Riemannian geometry and tensor calculus, Hamiltonian formalism (wave-particle duality) etc.) On the other hand, this mathematical research is motivated by already existing physics,&#xA;at least partially.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-04T17:11:06.293" LastActivityDate="2017-12-04T17:11:06.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6758" PostTypeId="1" AcceptedAnswerId="6765" CreationDate="2017-12-05T11:20:36.937" Score="3" ViewCount="737" Body="&lt;p&gt;Freeman Dyson expresses the opinion in his &lt;a href=&quot;http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-12971-9/S0002-9904-1972-12971-9.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1972 essay titled &amp;quot;Missed Opportunities&amp;quot;&lt;/a&gt; that Maxwell's equations could have played a much bigger role, one that is comparable to classical mechanics, in advancing mathematics (and then in turn, advancing physics).&lt;/p&gt;&#xA;&lt;p&gt;He says &lt;em&gt;&lt;strong&gt;mathematicians&lt;/strong&gt;&lt;/em&gt; in the late 19th century failed to grasp the opportunity, and that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;If they had taken Maxwell's equations to heart as Euler took Newton's, they would have discovered, among other things, Einstein's theory of special relativity, the theory of topological groups and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A great part of twentieth century physics and mathematics could have been created in the nineteenth century&lt;/strong&gt;, simply by exploring to the end the mathematical concepts to which Maxwell's equations naturally lead.&amp;quot;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\quad$&lt;/span&gt;(emphasis mine)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dyson spends quite some paragraphs addressing this line of thought in &amp;quot;Missed Opportunities&amp;quot; (which I totally enjoyed). More discussions can be seen in another article of his titled &amp;quot;&lt;a href=&quot;http://www.clerkmaxwellfoundation.org/DysonFreemanArticle.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why is Maxwell's Theory So Hard To Understand&lt;/a&gt;&amp;quot; where he acknowledges that historically, even among the physicists, &amp;quot;It was regarded as an obscure speculation without much experimental evidence to support it... (quoting Pupin) how few were the physicists who had caught the meaning of the theory, even twenty years after...&amp;quot;&lt;/p&gt;&#xA;&lt;h1&gt;Here's My question:&lt;/h1&gt;&#xA;&lt;p&gt;Where do researchers stand on this? Have any authors remarked somewhere that Dyson's proposal seems more of a wishful thinking (if not naive) than a plausible suggestion of an alternative history?&lt;/p&gt;&#xA;&lt;p&gt;(Researchers/authors include: historians, mathematicians, physicists, as well as journalists, pop science writers, social commentators, etc)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Dyson's &amp;quot;Missed Opportunities&amp;quot; is a non-technical essay that is &lt;a href=&quot;https://www.google.com.tw/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=1&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=0ahUKEwiFmcSS1PLXAhVJxLwKHUszDm4QzgIIKjAA&amp;amp;url=https%3A%2F%2Fscholar.google.com.tw%2Fscholar%3Fum%3D1%26ie%3DUTF-8%26lr%26cites%3D9423466732591958050&amp;amp;usg=AOvVaw0O0GDYlKidjULy0wClEOMj&quot; rel=&quot;nofollow noreferrer&quot;&gt;often cited by mathematicians and physicists (it seems)&lt;/a&gt;. I'm not sure how to dig through the list to find useful information for my inquiry.&lt;/p&gt;&#xA;&lt;p&gt;This question requires a certain level of expertise in both history and math/phys. Currently within hsm StackExchange I can find only &lt;a href=&quot;https://hsm.stackexchange.com/questions/10&quot;&gt;one post that is remotely related&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure if will be a good idea to phrase this question sufficiently differently to cross post on physics StackExchange or MathOverflow, asking about the plausibility of Dyson's idea more in terms of the pure logic of math and physics.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;To put some comments in context, they involve the original title &amp;quot;&lt;em&gt;Did math lag by 40 years due to under appreciation of Maxwell's Equations?&lt;/em&gt;&amp;quot;, which was problematic thus replaced.&lt;/p&gt;&#xA;" OwnerUserId="4723" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2021-12-21T20:27:13.537" Title="Math development and under-appreciation of Maxwell's Equations" Tags="|mathematics|physics|relativity-theory|mathematical-physics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6759" PostTypeId="1" AcceptedAnswerId="6761" CreationDate="2017-12-05T12:54:07.033" Score="3" ViewCount="138" Body="&lt;p&gt;In a German book by Helmut Kracke I found Frank Irvin mentioned. Irvin proved in 1916 the convergence of the harmonic series when all terms containing a 9 were omitted. I would like to include Irvin in my lecture but could not find anything about him except a mentioning of his name in &lt;a href=&quot;https://archive.org/stream/B-001-004-281/The_American_Mathematical_Monthly_djvu.txt&quot; rel=&quot;nofollow noreferrer&quot;&gt;AMM&lt;/a&gt; 1915 concerning the solution of a problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would need the dates of birth and dead and possibly a picture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note. From the material contained in the answer supplied by Michael E2 I have seen that the original work is due to Kempner (1914) and already described in &lt;a href=&quot;https://en.wikipedia.org/wiki/Kempner_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-12-06T13:53:43.793" LastActivityDate="2017-12-06T13:53:43.793" Title="Who knows Frank Irvin?" Tags="|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6761" PostTypeId="2" ParentId="6759" CreationDate="2017-12-05T15:19:15.630" Score="8" Body="&lt;p&gt;It's &lt;em&gt;Frank Irwin&lt;/em&gt;: &lt;a href=&quot;https://www.jstor.org/stable/2974352&quot; rel=&quot;noreferrer&quot;&gt;https://www.jstor.org/stable/2974352&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obituary: &lt;a href=&quot;http://texts.cdlib.org/view?docId=hb9p300969;NAAN=13030&amp;amp;doc.view=frames&amp;amp;chunk.id=div00015&amp;amp;toc.depth=1&amp;amp;toc.id=&amp;amp;brand=calisphere&quot; rel=&quot;noreferrer&quot;&gt;http://texts.cdlib.org/view?docId=hb9p300969;NAAN=13030&amp;amp;doc.view=frames&amp;amp;chunk.id=div00015&amp;amp;toc.depth=1&amp;amp;toc.id=&amp;amp;brand=calisphere&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can't find a photo.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2017-12-05T15:19:15.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6762" PostTypeId="2" ParentId="6758" CreationDate="2017-12-05T17:26:59.293" Score="3" Body="&lt;p&gt;I think Maxwell's theory was well received by the physicists of the 19th century, see &lt;a href=&quot;https://archive.org/stream/EncyklopdieDerMathematischenWissenschaftennfterBandPhysik/PPN360709672_LOG_0001#page/n77/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list&lt;/a&gt; of references most of which date from the 1890s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The time lag of 20 years after the publication of Maxwell's equations in &lt;a href=&quot;https://archive.org/stream/atreatiseonelec03maxwgoog#page/n4/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;book form&lt;/a&gt; is certainly not longer than the time lag between Newton and Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true, however, that there is only one mathematician on this list, Poincare, whose &lt;a href=&quot;https://archive.org/stream/maxwellstheorya01poingoog#page/n17/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;treatise&lt;/a&gt; of Maxwell's theory leaves something to be desired, because, oddly for a mathematician, &quot;mathematics and abstruse reasoning are avoided&quot;.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2017-12-05T17:33:44.400" LastActivityDate="2017-12-05T17:33:44.400" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6763" PostTypeId="1" CreationDate="2017-12-05T19:35:18.283" Score="1" ViewCount="235" Body="&lt;p&gt;When did (cognates of) the word &lt;em&gt;polynomial&lt;/em&gt; become standard in mathematical writing for expressions like $x^2 -x + 1$ and $ax^2 + bxy + cy^2$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 3rd edition of an 1822 English translation by Rev. John Hewlett of a French edition of &lt;strong&gt;Euler's&lt;/strong&gt; &lt;em&gt;Elements of Algebra&lt;/em&gt;, &lt;strong&gt;1770&lt;/strong&gt;, one does not see the word polynomial for such expressions---they are usually called &quot;formulas&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Legendre's&lt;/strong&gt; &lt;strong&gt;1798&lt;/strong&gt; &lt;em&gt;Essai sur la Théorie des Nombres&lt;/em&gt; uses the word &lt;em&gt;polynôme&lt;/em&gt;. Various cognates of this word (e.g., Polynom, polynomial, etc.) appear regularly in publications from then till the present.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The title of &lt;strong&gt;Gauss's&lt;/strong&gt; PhD Thesis of &lt;strong&gt;1799&lt;/strong&gt; has the phrase &lt;em&gt;Omnem Functionem Algebraicam Rationalem Integram&lt;/em&gt;, which translates into English as &quot;all integral (whole) rational algebraic functions&quot;. Gauss continues using similar phrases in his famous &lt;em&gt;Disquitiones Arithmeticae&lt;/em&gt; (&lt;strong&gt;1801&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such a phrase appears in many German publications throughout the 19th century as &lt;em&gt;ganze rationale algebraische Funktion&lt;/em&gt;, often shortened to &lt;em&gt;ganze rationale Funktion&lt;/em&gt;, or even &lt;em&gt;ganze Funktion&lt;/em&gt;. It was used by &lt;strong&gt;Dedekind&lt;/strong&gt; in &lt;strong&gt;1857&lt;/strong&gt;, by &lt;strong&gt;Dedekind and Weber&lt;/strong&gt; in their &lt;strong&gt;1882&lt;/strong&gt; paper giving an algebraic treatment of Riemann surfaces, and in &lt;strong&gt;Weber's&lt;/strong&gt; influential &lt;em&gt;Lehrbuch der Algebra&lt;/em&gt; (&lt;strong&gt;1895&lt;/strong&gt;, &lt;strong&gt;1896&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Dirichlet's 1840&lt;/strong&gt; paper on number theory uses the word &lt;em&gt;polynôme&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;George Boole's&lt;/strong&gt; &lt;em&gt;Mathematical Analysis of Logic&lt;/em&gt;, &lt;strong&gt;1847&lt;/strong&gt;, uses simply &lt;em&gt;function&lt;/em&gt; or &lt;em&gt;expression&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Joseph Serret's&lt;/strong&gt; &lt;em&gt;Cours d'algèbre supérieure&lt;/em&gt;, &lt;strong&gt;1849&lt;/strong&gt;, uses both &lt;em&gt;polynôme&lt;/em&gt; and &lt;em&gt;fonction rationale entière&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;George Chrystal's&lt;/strong&gt; &lt;em&gt;Algebra&lt;/em&gt; of &lt;strong&gt;1885&lt;/strong&gt; uses English translations of Gauss's terminology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did not find the word &lt;em&gt;Polynom&lt;/em&gt; while browsing through &lt;strong&gt;Hilbert's&lt;/strong&gt; Collected Works. He does use Gauss's wording in some of his publications.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern algebra was introduced, in the German language, by the trio of &lt;strong&gt;E. Noether&lt;/strong&gt; (German), &lt;strong&gt;E. Artin&lt;/strong&gt; (Austrian) and &lt;strong&gt;B. van der Waerden&lt;/strong&gt; (Dutch). Noether was the oldest, van der Waerden the youngest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Noether&lt;/strong&gt; started out using Gauss's wording in &lt;strong&gt;1908&lt;/strong&gt;, then used both wordings in two papers in &lt;strong&gt;1915&lt;/strong&gt;, then just Gauss's wording in two more papers of &lt;strong&gt;1915&lt;/strong&gt; and in one paper in &lt;strong&gt;1916&lt;/strong&gt;. It seems that she permanently switched to &lt;em&gt;Polynom&lt;/em&gt; in &lt;strong&gt;1918&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Artin&lt;/strong&gt; started with a paper in &lt;strong&gt;1924&lt;/strong&gt; using Gauss's terminology, but then switched to using &lt;em&gt;Polynom&lt;/em&gt; in 4 papers of &lt;strong&gt;1927&lt;/strong&gt; (two with &lt;strong&gt;Schreier&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Van der Waerden&lt;/strong&gt; seems to have used only &lt;em&gt;Polynom&lt;/em&gt;, starting in &lt;strong&gt;1927&lt;/strong&gt;, and most importantly in his very influential two volumes (&lt;strong&gt;1930&lt;/strong&gt;, &lt;strong&gt;1931&lt;/strong&gt;) of &lt;em&gt;Moderne Algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems that the abandonment of Gauss's terminology by the leading German language algebraists was taking hold near the end of WWI. Had the French been responsible for the modern algebra, then of course one would expect the terminology to be &lt;em&gt;polynôme&lt;/em&gt;; but why did the German language leaders in algebra choose to go with &lt;em&gt;Polynom&lt;/em&gt; instead of Gauss's terminology? Was it connected to the passing of the old guard,  &lt;strong&gt;Weber&lt;/strong&gt; (1913) and &lt;strong&gt;Dedekind&lt;/strong&gt; (1916)?&lt;/p&gt;&#xA;" OwnerUserId="6738" LastEditorUserId="6738" LastEditDate="2017-12-29T18:10:14.407" LastActivityDate="2017-12-29T18:10:14.407" Title="When did usage of the word polynomial become standard?" Tags="|algebra|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6765" PostTypeId="2" ParentId="6758" CreationDate="2017-12-07T02:30:29.127" Score="2" Body="&lt;p&gt;Maxwell's equations as they are expressed today, with modern vector notation and not Maxwell's quaternion notation, were first written by &lt;a href=&quot;http://www.encyclopedia.com/people/science-and-technology/physics-biographies/oliver-heaviside&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oliver Heaviside&lt;/a&gt; in a few papers in the mid- to late 1880s, but it wasn't until 1893 that the first volume of his &lt;em&gt;Electromagnetic Theory&lt;/em&gt; appeared (and vol. 2 in 1899 and vol. 3 in 1912), so these delays in better notation could have stalled the mathematical appreciation of Maxwell's equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Crowe, Michael J. 1967. &lt;a href=&quot;https://isidore.co/calibre/browse/book/5497&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A history of vector analysis: the evolution of the idea of a vectorial system&lt;/em&gt;&lt;/a&gt;. New York: Dover. &lt;strong&gt;pp. 162-176&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastActivityDate="2017-12-07T02:30:29.127" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6766" PostTypeId="1" AcceptedAnswerId="6770" CreationDate="2017-12-07T18:40:15.350" Score="5" ViewCount="2530" Body="&lt;p&gt;I'm looking for good books on history of chemistry and chemical methodology. There appear to be surprisingly few available on the market based on my own 'research' (just googling and phrase-searching for different variations on 'history','chemistry'). Thanks a lot!&lt;/p&gt;&#xA;" OwnerUserId="6274" LastActivityDate="2021-09-10T07:12:56.690" Title="Looking for books on the history of chemistry?" Tags="|chemistry|resource-recommendation|book-recommendation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6767" PostTypeId="1" CreationDate="2017-12-07T21:48:02.733" Score="5" ViewCount="2587" Body="&lt;p&gt;In other words, how is Aristotle's logic represented in Euclid's &lt;em&gt;Elements&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read many articles where Euclid's &lt;em&gt;Elements&lt;/em&gt; is linked to Aristotle's logic, but I do not understand, and I can't find any examples explaining how deductive logic (i.e. syllogism) or laws of thought or any other work from Aristotle could have influenced the development of geometry and the concepts Euclid came up with when writing his book.&lt;/p&gt;&#xA;" OwnerUserId="6755" LastEditorUserId="4251" LastEditDate="2020-07-21T00:44:01.660" LastActivityDate="2020-07-21T00:44:01.660" Title="How did Aristotle influence Euclid?" Tags="|geometry|ancient-greece|mathematical-logic|euclidean-geometry|euclid|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6768" PostTypeId="2" ParentId="6767" CreationDate="2017-12-08T11:53:55.913" Score="3" Body="&lt;p&gt;You are right: syllogisms are not used by Euclid. More generally: “Although Aristotle emphasized the use of syllogisms as the building blocks of logical arguments, Greek mathematitians apparently never used them.” (I am quoting from &lt;em&gt;A History of Mathematics&lt;/em&gt;, by Victor J. Katz (3&lt;sup&gt;rd&lt;/sup&gt; edition)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The infuence of Aristotle on Euclid lies elsewhere. Quoting from the same source: “If one accepts the premises of a syllogism as true, then one must also accept the conclusion. One cannot, however, obtain every piece of knowledge as the conclusion of a syllogism. One has to begin somewhere with truths that are accepted without argument. Aristotle distinguishes between the basic truths that are peculiar to each particular science and the ones that are common to all. The former are often called &lt;strong&gt;postulates&lt;/strong&gt;, and the latter are known as &lt;strong&gt;axioms&lt;/strong&gt;.” I suppose that you recognise in this description the way Euclid's &lt;em&gt;Elements&lt;/em&gt; are written.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-12-08T11:53:55.913" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6769" PostTypeId="1" AcceptedAnswerId="6775" CreationDate="2017-12-08T12:23:29.320" Score="9" ViewCount="409" Body="&lt;p&gt;As noted on &lt;a href=&quot;https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Importance_and_range_of_validity&quot; rel=&quot;noreferrer&quot;&gt;this Wikipedia page&lt;/a&gt; Newton's laws of motion and of gravity were &quot;&lt;em&gt;verified by experiment and observation for over 200 years&lt;/em&gt;&quot; and found to be a &quot;&lt;em&gt;good approximation for macroscopic objects under everyday conditions&lt;/em&gt;&quot;. Now, of course, we know from Einstein and quantum mechanics that while not &lt;em&gt;incorrect&lt;/em&gt;, his laws &quot;only&quot; work for a certain range of inputs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While some of those early experiments were probably carried out with the &lt;em&gt;hope&lt;/em&gt; of proving Newton wrong, nobody &lt;em&gt;knew&lt;/em&gt; that there were limits to their applicability (very small scales; very high speeds; very strong gravitational fields) until much later (and, as far as I know, no one even &lt;em&gt;suspected&lt;/em&gt; that there might be such limits).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As noted in the answers to &lt;a href=&quot;https://hsm.stackexchange.com/q/2818/2552&quot;&gt;this HofS&amp;amp;M question&lt;/a&gt;, there were some known &quot;anomalies&quot;, such as the motion of the moon or Mercury, but these were more to do with the problem of applying Newton's laws to a three-body problem than problems with the laws themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is: if Newton, or scientists around his time somehow had an &quot;inkling&quot; that there actually &lt;em&gt;were&lt;/em&gt; limits to his laws, and set off on a &quot;directed search&quot; to explore those limits, would they have had the technology to expose such limits? In other words, would they have been able to explore a small-enough scale or move large-enough masses, at fast-enough speeds, &lt;em&gt;and&lt;/em&gt; measure things with enough precision to detect any discrepancies above experimental noise and be reasonably confident in concluding that Newton's laws don't hold under all circumstances?&lt;/p&gt;&#xA;" OwnerUserId="2552" LastActivityDate="2017-12-19T02:45:50.987" Title="Could scientists of Newton's time have explored the limits of his laws of motion?" Tags="|experimental-physics|relativity-theory|classical-mechanics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6770" PostTypeId="2" ParentId="6766" CreationDate="2017-12-08T17:10:11.893" Score="4" Body="&lt;p&gt;I agree that there are (surprisingly?) few books on the history of chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most succinct one is probably C. Cobb, H. Goldwhite; &quot;Creations of Fire - Chemistry's Lively History from Alchemy to the Atomic Age&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much more detailed, although somewhat old now, is A. Ladenburg; &quot;Lectures on the History of the Development of Chemistry Since the Time of Lavoisier&quot;. Is a very good book, even if it is difficult to read since chemistry in the 19th century lacked a lot of our concepts, but I think the author does a good job in not only presenting the material as well as analyzing the thoughts of our predecessors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the real of comprehensiveness you cannot do better than J. R. Partington, &quot;A History of Chemistry&quot;, in four massive volumes, covering a lot from the greeks until modern chemistry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a lighter reading I highly recommend S. Kean; &quot;The Disappearing Spoon&quot;. Is not so much a history book, but rather a collection of anecdotes from the history of chemistry. Each chapter selects a couple of elements from the periodic table and then recounts some historical events relating to these elements. Is not an academic view, but it is very entertaining, and surprisingly illuminating.  &lt;/p&gt;&#xA;" OwnerUserId="5904" LastActivityDate="2017-12-08T17:10:11.893" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6772" PostTypeId="1" AcceptedAnswerId="6774" CreationDate="2017-12-09T11:19:13.540" Score="3" ViewCount="577" Body="&lt;p&gt;My question is mainly about getting explanation to Gauss's third note (note III) in his writing &quot;Zur Astralgeometrie&quot;, (see &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/pdfcache/PPN236010751/PPN236010751___LOG_0038.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cubierung der Tetraeder&lt;/a&gt;, pages 228-229), and about placing it in the right historical context. In this note Gauss is writing the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In the tetrahedron 1234 (1,2,3,4 are it's vertexes), whose faces 124 and 134 are orthogonal, denote the volume as &lt;span class=&quot;math-container&quot;&gt;$$\Delta,$$&lt;/span&gt; then it holds that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\partial \Delta = -24.\partial341,$$&lt;/span&gt; and if the face angles at vertex 3 are constant, then the following also holds:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\alpha\alpha\cdot cotg^2341 - \beta\beta\cdot (tgi.24/i)^2 = 1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;when: &lt;span class=&quot;math-container&quot;&gt;$$\alpha = cotg431,\quad &#xA;\beta = cotg 234.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two things i didn't understand about Gauss's notation immediately came to my mind:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What does the notation 24 &lt;strong&gt;point&lt;/strong&gt; &quot;something&quot; mean? does the point mean product? or something else?&lt;/li&gt;&#xA;&lt;li&gt;What is the &quot;&lt;strong&gt;i&lt;/strong&gt;&quot; that appears in his formulas? how is it defined?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The questions i asked are of great interest to me since I think the problem of determining the volume of the non-euclidean tetrahedron really represents the peak of Gauss's work on hyperbolic geometry, and I really want to get a formal statement of his result, and to understand how is it compared with later results of Lobachevsky and Schlafli. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: recently i became aware of a certain interconnection between this fragment (CUBIRUNG DER TETRAEDER) and a second fragment of this kind - &quot;&lt;strong&gt;Volumenbestimmungen in der Nichteuklidischen Geometrie&lt;/strong&gt;&quot; (Gauss's werke, volume 8, p.232-233). This fragment was found among the pages of Gauss's copy of Lobachevsky's 1840 publication (these two fragments are the only ones that deal with the probelm of hyperbolic tetrahedron). As a matter of fact, Stackel mentions, in his commentary of Gauss's &quot;cubirung der tetraeder&quot;, that Gauss used the results of the second fragment implicitly when he derived his first displayed formula from p. 228 (&lt;span class=&quot;math-container&quot;&gt;$\partial \Delta = -\frac{{1}}{{2}}(24)d(341)$&lt;/span&gt;) - he used the two formulas:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$d(341) = -sinh(14)\cdot d(13)$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$\partial \Delta = \frac{{1}}{{2}}d(13)\cdot (24)\cdot sinh(14)$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;to set up this relation (&lt;span class=&quot;math-container&quot;&gt;$\partial \Delta = -\frac{{1}}{{2}}(24)d(341)$&lt;/span&gt;). The second of these formulas was also mentioned in his later fragment, and though this fragment was written later (Gauss wrote this since the publication of Lobachevsky captured his attention), it seems that he was aware of it earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I made this update because maybe dechipering Gauss's second fragment might serve as a breakthrough in understanding what Gauss knew on hyperbolic tetrahedrons. &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-22T07:21:17.587" LastActivityDate="2019-10-22T07:21:17.587" Title="Did Gauss's expression for the differential of the hyperbolic volume of the tetrahedron agree with later results?" Tags="|biographical-details|gauss|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6773" PostTypeId="2" ParentId="5726" CreationDate="2017-12-09T17:57:15.260" Score="1" Body="&lt;p&gt;The term first appeared&lt;sup&gt;&lt;a href=&quot;https://doi.org/10.1021%2Fed800002c&quot; rel=&quot;nofollow noreferrer&quot;&gt;[source]&lt;/a&gt;&lt;/sup&gt; in S. P. L. Sørensen's paper “Enzyme Studies II” (1909), which did not state what it stood for. Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/PH#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;claims&lt;/a&gt;&lt;sup&gt;&lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=PH&amp;amp;oldid=814222559&quot; rel=&quot;nofollow noreferrer&quot;&gt;[permalink]&lt;/a&gt;&lt;/sup&gt; that the meaning of p is disputed between a number of possible words beginning with p (emphasis mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The exact meaning of the &quot;p&quot; in &quot;pH&quot; is disputed, but according to the Carlsberg Foundation, pH stands for &quot;&lt;strong&gt;power&lt;/strong&gt; of hydrogen&quot;.&lt;sup&gt;[5]&lt;/sup&gt; It has also been suggested that the &quot;p&quot; stands for the German &lt;strong&gt;&lt;em&gt;Potenz&lt;/em&gt;&lt;/strong&gt; (meaning &quot;power&quot;), others refer to French &lt;strong&gt;&lt;em&gt;puissance&lt;/em&gt;&lt;/strong&gt; (also meaning &quot;power&quot;, based on the fact that the Carlsberg Laboratory was French-speaking). Another suggestion is that the &quot;p&quot; stands for the Latin terms &lt;strong&gt;&lt;em&gt;pondus&lt;/strong&gt; hydrogenii&lt;/em&gt; (quantity of hydrogen), &lt;strong&gt;&lt;em&gt;potentia&lt;/strong&gt; hydrogenii&lt;/em&gt; (capacity of hydrogen), or &lt;strong&gt;potential&lt;/strong&gt; hydrogen. It is also suggested that Sørensen used the letters &lt;strong&gt;&quot;p&quot; and &quot;q&quot;&lt;/strong&gt; (commonly paired letters in mathematics) simply to label the test solution (p) and the reference solution (q).&lt;sup&gt;[6]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The source [6] cited (&lt;a href=&quot;https://doi.org/10.1021%2Fed800002c&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1021/ed800002c&lt;/a&gt;) disputes &quot;power&quot; since the article was published in German, French and Danish, but not English. It prefers the &lt;em&gt;p&lt;/em&gt; and &lt;em&gt;q&lt;/em&gt; suggestion for the test and reference cells, citing &lt;a href=&quot;https://doi.org/10.1016/S0968-0004(99)01517-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1016/S0968-0004(99)01517-0&lt;/a&gt;. This final paper shows that there were equations in Sørensen's papers using the symbols $p_H^+, C_p, C_q, π_p$ and $π_q$.&lt;/p&gt;&#xA;" OwnerUserId="6764" LastEditorUserId="6764" LastEditDate="2017-12-09T19:58:51.837" LastActivityDate="2017-12-09T19:58:51.837" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6774" PostTypeId="2" ParentId="6772" CreationDate="2017-12-10T01:28:53.627" Score="1" Body="&lt;p&gt;(Edited). My German is not too good, but the dot clearly stands for multiplication, and $i=\sqrt{−1}$ so  $$\frac{\mathrm{tg}i.24}{i}$$ is what we call the hyperbolic tangent today, $\tanh(24)$. Symbol probably $\partial$ means the differential. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can read German, read the comments below the note: Stackel explains what is going on there. In particular he says that Gauss missed the multiple of $1/2$&#xA;in his variation formula (his first displayed formula). The formula describes the&#xA;variation of the volume under the conditions that the following angles 142, 342, 312, 314 are right. Under these conditions he derives Gauss's second displayed formula, (written by Stackel in the bottom line on p. 228), and then differentiates it to obtain Gauss's first formula,&#xA;with extra factor $1/2$.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-10T17:10:01.530" LastActivityDate="2017-12-10T17:10:01.530" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6775" PostTypeId="2" ParentId="6769" CreationDate="2017-12-10T02:10:01.690" Score="9" Body="&lt;p&gt;At the time of Newton, the scientists could NOT detect any deviation of the Newton's laws from reality. As we know now, the only visible effect of this deviation in the Solar system is the anomalous precession of Mercury's perihelion. It was detected only in the middle of 19th century; it makes about 43 seconds per century, and it was not known at Newton's times. The problem was not with accuracy of observations, in principle they could detect the phenomenon. The problem was comparison with the theory: equations resulting from&#xA;Newton's laws are notoriously difficult to solve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's theory of gravitation was not immediately generally accepted, and scientists indeed expressed doubts. Newton's theory was originally derived to explain Kepler laws, but Kepler's laws themselves only hold approximately, and Newton's main claim was that his gravity law explains not only Kepler's laws but also all observed DEVIATIONS from these laws.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took about a century and half to verify these claims. (The problem is very difficult: Newton's theory only tells you how to WRITE the equations, but another matter is how to SOLVE them). Newton tried hard to explain the motion of the Moon (I mean the small non-uniformities of this motion) and only partially succeeded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematicians who struggled with the theory of the Moon indeed expressed doubts&#xA;in the gravitation law, Clairaut, for example thought at some point that the gravity law is only an approximation to the &quot;true law&quot;, but then he changed his opinion. The motion of the Moon was finally explained to the accuracy sufficient for navigation only in the second part of 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the end of 19th century ALL motions in the Solar system were explained (or believed to be explainable to the accuracy of observations of that time) except this&#xA;Mercury perihelion precession. The fact that Newton's gravity law does not explain the orbit of Mercury was discovered by Leverrier in the middle of 19th century. They started searching a new planet between Mercury and Sun whose perturbation could explain the effect. The search lasted long time with some claims of discovery which were later refuted.&#xA;At some point even the name was given to this &quot;new planet&quot;: Vulcan. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Any scientific law is supposed to work only in certain range of phenomena.&#xA;After the law is discovered, the efforts are directed at the precise determination of this range. This is what mathematicians and astronomers were occupied with for the century after Newton.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. One can also think whether scientists at the time of Newton could detect the deviation of light rays from stars as they pass near the Sun. The effect was observed by Eddington in 1918 and this was a major check of general relativity.&#xA;The accuracy required was about 0.5-1''. Flamsteed star catalog has accuracy 4''. So this phenomenon also could not be observed in Newton's time. And the theory of light was not sufficiently developed to discuss this phenomenon.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-19T02:45:50.987" LastActivityDate="2017-12-19T02:45:50.987" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6776" PostTypeId="1" AcceptedAnswerId="6777" CreationDate="2017-12-10T19:08:11.163" Score="0" ViewCount="252" Body="&lt;p&gt;In what work did Jacobi introduce the jacobian, and what was his motivation for doing so?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2017-12-10T21:23:00.770" Title="The Origin of the Jacobian" Tags="|calculus|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6777" PostTypeId="2" ParentId="6776" CreationDate="2017-12-10T21:23:00.770" Score="2" Body="&lt;p&gt;I don't know about the motivation, but it was introduced in his article &lt;em&gt;&lt;a href=&quot;https://archive.org/download/ueberdiefunction00jacouoft/ueberdiefunction00jacouoft.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die Functionaldeterminanten&lt;/a&gt;&lt;/em&gt; (Journal für die reine und angewandte mathematik, bd. 22, 1841). However, Jacobians had already been studied by Cauchy in 1815.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2017-12-10T21:23:00.770" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6778" PostTypeId="1" AcceptedAnswerId="6788" CreationDate="2017-12-10T22:34:48.867" Score="14" ViewCount="3448" Body="&lt;p&gt;Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Flat_Earth&quot; rel=&quot;nofollow noreferrer&quot;&gt;notes&lt;/a&gt; that, while knowledge Earth is approximately spherical was obtained in ancient Greece, and became standard among educated people in Europe and the Middle East long before 1300 AD, China persisted in the flat-Earth model until a Jesuit influence in the 1600s, despite a 1267 gift of a Persian globe from Jamal ad-Din. China is also unusual in that its flat-Earth model was of a square rather than a circle, the latter common in other cultures.&lt;/p&gt;&#xA;&lt;p&gt;My guess is that knowledge elsewhere of Earth's shape derived from Greek texts (and replication of the experiments they described), and that it wasn't so much that China was &amp;quot;the odd one out&amp;quot; as that there were very few independent realisations of the Earth's sphericity. However, feel free to correct me on this.&lt;/p&gt;&#xA;&lt;p&gt;Why was Chinese astronomy so slow to adjust on this point? Why did Jamal ad-Din make so much less of an impression than the later Jesuits?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2023-08-27T13:08:16.537" LastActivityDate="2023-08-27T13:08:16.537" Title="Why was China slow to recognise the sphericity of Earth?" Tags="|discoveries|astronomy|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6779" PostTypeId="1" CreationDate="2017-12-11T12:25:35.843" Score="1" ViewCount="307" Body="&lt;p&gt;This is a follow up to this question: &lt;a href=&quot;https://hsm.stackexchange.com/q/6683/6435&quot;&gt;History of PDE&amp;#39;s in the 19th Century&lt;/a&gt;&#xA;The question I have been given to answer is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The history of partial differential equations in the 19th Century belongs to applied mathematics, not pure mathematics. To what extent do you agree, and why?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I plan to argue that the question is misconceived as there is far too much overlap between the definitions of pure and applied to definitively categorise mathematicians work as solely pure or applied and to some extent why does it matter? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematicians whose work I have been asked to look at are Cauchy, Kovalerskaya, Fourier, Thomson, Dirichlet, Schwarz, Poincare, Riemann, Hamilton-Jacobi, Hilbert and Hadamard. Clearly there are mathematicians who lie on either side of the pure/applied spectrum with Cauchy clearly focused on pure and Thomson focused on applied for example. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am really struggling on sources for the whole pure vs applied question, I can find more than enough sources giving detailed reviews of their work so it comes down to my classification which I do not entirely trust. I also need to consider obvious counter arguments so I would be interested to see other points of view on the question so I can deal with any refutations to my answer. So what I am asking for is any sources you may think relevant and your opinions on the question, thanks for any reply in advance.&lt;/p&gt;&#xA;" OwnerUserId="6435" LastEditorUserId="4251" LastEditDate="2020-07-06T06:59:35.340" LastActivityDate="2020-07-06T06:59:35.340" Title="History of PDE's in the 19th Century (part 2)" Tags="|mathematics|calculus|mathematical-physics|differential-equations|19th-century|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6780" PostTypeId="2" ParentId="6767" CreationDate="2017-12-11T13:30:00.473" Score="3" Body="&lt;p&gt;As Simpson clearly states in the section &quot;Aristotelian logic&quot; of his article, principle 3 is the law of excluded middle: $P \vee \neg P$ is true. Euclid's &lt;em&gt;Elements&lt;/em&gt; indeed rely on classical logic rather than intuitionistic logic.  There have been attempts recently to rewrite Euclid in an intuitionistic framework; see e.g., the work of Michael J. Beeson.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2017-12-11T13:30:00.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6781" PostTypeId="1" CreationDate="2017-12-11T14:13:30.420" Score="5" ViewCount="218" Body="&lt;p&gt;i'm interested in the history of the 2 Sobolev Inequalities. The Gagliardo-Nirenberg-Sobolev inequality and Morrey inequality as here &lt;a href=&quot;https://en.wikipedia.org/wiki/Sobolev_inequality&quot; rel=&quot;noreferrer&quot;&gt;Sobolev inequality&lt;/a&gt;  .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did these two inequalities arise? What is the context of its emergence? In what year were they discovered?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="6774" LastActivityDate="2018-03-26T19:40:17.840" Title="About Sobolev Inequalities" Tags="|mathematics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6783" PostTypeId="2" ParentId="6769" CreationDate="2017-12-11T21:07:37.717" Score="4" Body="&lt;p&gt;One of the main criticisms of Newtons theory of Gravitation was philosophical; in that his theory incorporated action at a distance; this criticism was something that Newton was very much aware of, and that he accepted; he could not see how any man of a 'philosophical' temper could not accept this principle; however, he saw no way forward through this and hence he 'framed no hypotheses' about it. &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Action_at_a_distance&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is how he put it himself in a letter to Bentley in 1692/3.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is inconceivable that inanimate matter should. without the mediation of something else which is not material, operate upon, and affect other matter &#xA;  without mutual contact...that gravity should be innate, inherent and essential to matter so that one body may act upon another thro' a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters a competent facility of thinking can ever fall into it. Gravity must be cause by an agent acting constantly according to certain laws. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This critique is of ancient provenance, Aristotle mentions it in one of his many books (I think it's in the &lt;em&gt;De Caelo&lt;/em&gt; - &lt;em&gt;On the Heavens&lt;/em&gt;) that it was valid is seen by the fact that the accepted theory of gravity now, that by Einstein - General Relativity - incorporates this principle. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2017-12-11T22:05:36.897" LastActivityDate="2017-12-11T22:05:36.897" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6784" PostTypeId="1" AcceptedAnswerId="6786" CreationDate="2017-12-12T01:25:41.383" Score="2" ViewCount="4045" Body="&lt;p&gt;Newton's laws of motion are for linear motion. Did he also discover the laws of motion for rotation - angular momentum, torque and the moment of inertia; or was this discovered by others?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorDisplayName="user466" LastEditDate="2017-12-16T01:23:56.100" LastActivityDate="2018-05-30T19:39:19.087" Title="Who invented/discovered angular momentum?" Tags="|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6785" PostTypeId="1" AcceptedAnswerId="6787" CreationDate="2017-12-13T01:10:44.413" Score="6" ViewCount="2098" Body="&lt;p&gt;According to Wikipedia,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis(&quot;as the extension, so the force&quot; or &quot;the extension is proportional to the force&quot;). Hooke states in the 1678 work that he was aware of the law already in 1660.&#xA;  The anagram was given in alphabetical order, ceiiinosssttuu, representing Ut tensio, sic vis – &quot;As the extension, so the force&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does anybody know an alternative view in which he is not inventor of the law that has his name? I found something about C. Huygens, but with no historical reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason of my question is because the mathematician M. Viana said that in one of his record lectures at IMPA (Instituto de Matemática Aplicada, at Brazil). Lecture in portuguese: &lt;a href=&quot;https://youtu.be/TOcWLGyrjjk?t=1176&quot; rel=&quot;noreferrer&quot;&gt;https://youtu.be/TOcWLGyrjjk?t=1176&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6480" LastEditorUserId="6480" LastEditDate="2017-12-13T01:18:03.150" LastActivityDate="2017-12-13T03:00:51.433" Title="Who really discovered/invented the Hooke's law?" Tags="|physics|experimental-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6786" PostTypeId="2" ParentId="6784" CreationDate="2017-12-13T01:34:38.340" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/people/philosophy-and-religion/philosophy-biographies/jean-buridan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Buridan&lt;/a&gt; (1295-1358), the discoverer/inventor of momentum (&lt;em&gt;impetus&lt;/em&gt;), also discussed angular momentum. From &lt;a href=&quot;http://www.encyclopedia.com/people/philosophy-and-religion/philosophy-biographies/jean-buridan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ernest A. Moody's &lt;em&gt;Complete Dictionary of Scientific Biography&lt;/em&gt; entry on Buridan&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Buridan’s concept of impetus is further distinguished from the modern inertial concept by the fact that he construes rotational motion at uniform angular velocity as due to a rotational impetus analogous to the rectilinear impetus involved in projectile motion. Galileo did likewise, and was in this respect nearer to Buridan than to Newton. But Buridan makes a striking use of his impetus concept, in its rotational sense [i.e., angular momentum], by arguing that since the celestial spheres posited by the astronomers encounter no external resistance to the rotational movements and have no internal tendency toward a place of rest (such as heavy and light bodies have), their uniform rotational motions are purely inertial and require no causes acting on them to maintain their motions. There is, therefore, no need to posit immaterial intelligences as unmoved movers of the heavenly spheres, in the manner that Aristotle and his commentators supposed. “For it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more.”&lt;sup&gt;5&lt;/sup&gt; It was in this way, Buridan adds, that God rested on the seventh day and committed the motions of the bodies he had created to those bodies themselves.&lt;hr&gt;&lt;sup&gt;5. &lt;em&gt;Qu. De caelo et mundo&lt;/em&gt; II, 22 (1942), 227.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;http://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/thomas-bradwardine&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Bradwardine&lt;/a&gt; (d. 1349) appears to have been the first to define angular velocity, which he called &lt;em&gt;velocitas circuitionis&lt;/em&gt;; cf. &lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem&lt;/a&gt;'s &lt;a href=&quot;https://archive.org/stream/tudessurlona03duhe#page/304/mode/2up/search/velocitas+circuitionis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Études sur Léonard de Vinci&lt;/em&gt; vol. 3, p. 305&lt;/a&gt;-306 (p. 219 of &lt;a href=&quot;https://dx.doi.org/10.13140/RG.2.2.23235.71201&quot; rel=&quot;nofollow noreferrer&quot;&gt;the English translation&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-05-30T19:39:19.087" LastActivityDate="2018-05-30T19:39:19.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6787" PostTypeId="2" ParentId="6785" CreationDate="2017-12-13T03:00:51.433" Score="7" Body="&lt;p&gt;Proportionality law for elastic forces is one discovery at which Hooke did arrive first. He describes it in De Potentia Restitutiva (1678):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;”&lt;em&gt;The Power of any Spring is in the same proportion with the Tension thereof: That is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward. Now as the Theory is very short, so the way of trying it is very easie. […] it is very evident that… in every springing body… the force or power thereof to restore itself to its natural position is always proportionate to the Distance or space it is removed therefrom.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, Hooke does not differentiate between Boyle’s law for gases and the law for elastic bodies, his “springing body” in the quote can represent either, and his surmise is clearly inspired by the former (he worked as Boyle’s assistant in 1659). And his theoretical justification for the elastic case specifically was erroneous. He modeled the spring as a row of boxes containing vibrating microscopic particles. At the equilibrium the tendency of the particles to recede from each other is balanced by the external pressure from the surrounding subtle matter. The rate of particles' collisions with their neighbors accounts for the restoring force of springs. But according to &lt;a href=&quot;http://www.journals.uchicago.edu/doi/abs/10.1086/351771&quot; rel=&quot;noreferrer&quot;&gt;Moyer’s Robert Hooke's Ambiguous Presentation of “Hooke's Law”&lt;/a&gt; his model produces an inverse square dependence, which he confuses with the observed proportionality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Something about Huygens” probably refers to Hooke’s dispute with him which did involve springs, it was over the priority for inventing balance-spring watches. The affair goes back to 1665 when Huygens got patent for his famous isochronous pendulum clock employing cycloidal arcs. Hooke disputed the originality of the invention and offered his own coiled spring watch as prior art:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“&lt;em&gt;I my self had an other way of continuing and equalling the vibrations of a pendulum by clock work long before I heard of Monsier Zulichems way, nay through equated with a Cycloeid yet I have not either cryd eureka or publisht it and yet I think I can produce a sufficient number of Credible witnesses that can testify for it about these 12 years. Soe that the argument that he soe much Relys upon to secure to him the Invention is not of soe great force as to preswaid all the World that he was the first &amp;amp; sole inventor of that first particular of applying a pendulum to a clock.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Fast forward to 1675 when Huygens published the description of his balance-spring watch in French Academy’s Journal des Sçavans. It appears that between 1665 and 1675 Royal Society’s Oldenburg and Moray shared Hooke’s balanced-spring designs with Huygens, conveniently off the record, and Hooke even accused Oldenburg of “spying” for Huygens. Nonetheless, the Royal Society ultimately sided with Huygens in 1676, formally rebuked Hooke, and commended &quot;Oldenburg's fidelity and honesty ‘in the management of the intelligence of the Royal Society'&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hooke was really good at making enemies, but one productive thing that came out of the affair was that it turned Hooke’s interest to the restoring force of springs, although it does not seem that he realized the relation between its proportionality and the isochrony of balance-spring watches. See &lt;a href=&quot;http://shipseducation.net/modules/phys/hooke/hooke.htm&quot; rel=&quot;noreferrer&quot;&gt;Horibe’s Robert Hooke, Hooke's Law &amp;amp; the Watch Spring&lt;/a&gt; for the full story and design pictures.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2017-12-13T03:00:51.433" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6788" PostTypeId="2" ParentId="6778" CreationDate="2017-12-13T05:53:20.120" Score="9" Body="&lt;p&gt;It is hard to say why history happened one way rather than another. But we can point at some differences between ancient Greece, Hellenistic Mediterranean or Renaissance Europe and ancient and medieval China that arguably contributed not only to the persistence of flat Earth, but to the cultural slow motion generally, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2344/55&quot;&gt;Is the Scientific Method uniquely Western?&lt;/a&gt; for a broader theme. The former were a motley of city-states or countries with high levels of diversity, mobility and cultural exchange, while China was a much more culturally uniform traditional society that favored isolationism. Traditional societies effect change differently than dynamic ones, their tools and procedures improve through accumulation of incremental glacial changes, generational trial and error, rather than through intentional innovation guided by scientific inquiry. Here is &lt;a href=&quot;http://strangebeautiful.com/other-texts/stein-eudoxos-dedekind.pdf&quot; rel=&quot;noreferrer&quot;&gt;Stein's characterization&lt;/a&gt; of Babylonian mathematics, that applies as well to Chinese astronomy:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...for the Babylonians, mathematics was not an enterprise, but a lore and a skill possessed by the priestly or scribal class, for use in essentially administrative functions, and passed on from generation to generation much as were the techniques of the craft guilds in the middle ages. Of course, the techniques had to have been discovered; and innovations might occur from time to time; but invention or discovery was no more the business of those trained in the lore, than it was the business of the medieval master of a craft.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In recent decades there has been some progress in evolutionary modeling of cultural development, see. e.g. &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/15058714&quot; rel=&quot;noreferrer&quot;&gt;Is human cultural evolution Darwinian? by Mesoudi et al. (2004)&lt;/a&gt;, although the approach is still controversial. From this perspective traditional societies suppress random drift, and do not provide enough opportunities for cultural &quot;mutations&quot; (so-called &quot;memes&quot;) to spread and gain a foothold even when they do arise. This is how endemic populations are maintained in stable isolated environments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Flat square Earth surrounded by round heaven was the traditional belief integrated with other mythical beliefs, the language used to think about the heavens was mythical and poetic, not geometric or precise. Zhang Heng (78-139 AD) writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The heavens are like a hen's egg and as round as a crossbow bullet; the earth is like the yolk of the egg, and lies in the centre... Heaven takes its body from the Yang, so it is round and in motion. Earth takes its body from the Yin, so it is flat and quiescent&lt;/em&gt;&quot;. [in case you thought that the &quot;yoke&quot; of Earth is spherical] &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Was there a compelling reason to change this belief? It depends on one's view of &quot;compelling&quot;. And we often forget how the same observations can be interpreted in many different ways, and reconciled with just about anything. Essentially the same measurements that Eratosthenes used to estimate the Earth's circumference &lt;a href=&quot;https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Flat_Earth.html&quot; rel=&quot;noreferrer&quot;&gt;Chinese astronomers who wrote Huainanzi&lt;/a&gt; in the 2nd century BC interpreted to calculate the height of the Sun above the earth. This is just geometric inversion. What about the shadow of the Earth on the Moon during the eclipses, which is one of Aristotle's main arguments for spherical Earth? Eclipses are rare events of little practical significance, and in a traditional society would not be valued as highly as in the cosmopolitan Hellenistic world. But for what it is worth some Chinese astronomers, like Li Ye in 13th century, Jamal-ad-Din's contemporary in Kublai Khan's time, did challenge the idea that the Earth was square, he thought it was still flat but round like the sky. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There was one earlier exception that confirms the rule. China underwent a remarkable period of &lt;a href=&quot;http://www.history.ac.uk/reviews/review/1091&quot; rel=&quot;noreferrer&quot;&gt;cultural innovation during the Song dynasty (960–1279)&lt;/a&gt;, sometimes compared to European Renaissance. The invention of gunpowder and widespread use of printing press and paper money in China date to this period. &lt;a href=&quot;https://en.wikipedia.org/wiki/Shen_Kuo#Astronomy_and_instruments&quot; rel=&quot;noreferrer&quot;&gt;Shen Kuo (1031–1095)&lt;/a&gt;, a polymath often compared to Leonardo, among other things invented nautical compass, measured the distance between the North star and the true north, improved designs of the armillary sphere, and, most importantly for us, wrote on solar and lunar eclipses explaining how phases indicated that the Sun and the Moon were spherical rather than flat like fans. The Sun and the Moon, yes, but not the Earth, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5568/55&quot;&gt;Did the ancient Chinese know the earth is a sphere?&lt;/a&gt; Close, but Song China did not develop systematic theoretical and empirical science just as Shen Kuo stopped short of spherical Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to why the Jesuits had more of an effect than &lt;a href=&quot;https://en.wikipedia.org/wiki/Jamal_ad-Din_(astronomer)&quot; rel=&quot;noreferrer&quot;&gt;Jamal ad-Din&lt;/a&gt; that is relatively easy to explain. Kublai Khan let him set up an Islamic Astronomical Bureau in Beijing, rival to the traditional Chinese bureau, and had his subordinates check calculations of the Chinese. This maintained the traditional bureaucracy but hardly inspired the spirit of scientific cooperation. And the surrounding traditional community would not have been affected much even if the Chinese bureau did change its mind. Jesuits, on the other hand, were preaching directly to the people.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good source on historical interaction of China with Western science is &lt;a href=&quot;http://www.hup.harvard.edu/catalog.php?isbn=9780674016859&quot; rel=&quot;noreferrer&quot;&gt;Elman's On Their Own Terms: Science in China, 1550-1900&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2017-12-15T03:54:54.560" LastActivityDate="2017-12-15T03:54:54.560" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6789" PostTypeId="1" CreationDate="2017-12-13T21:28:45.697" Score="0" ViewCount="87" Body="&lt;p&gt;My question is a direct continuation of my already posted question &lt;a href=&quot;https://hsm.stackexchange.com/questions/6772/did-gausss-expression-for-the-differential-of-the-hyperbolic-volume-of-the-tetr&quot;&gt;Did Gauss&amp;#39;s expression for the differential of the hyperbolic volume of the tetrahedron agree with later results?&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I simply didn't find any sources that say that Gauss's result was a nonsense; in his commentary on Gauss's relevant note, Stackel doesn't say Gauss was mistaken (apart from a factor of $1/2$ that was missing from his expression for the volume differential), and in particular, the book &quot;Mathematics and Its History&quot;, p379, emphasizes that Gauss &quot;&lt;em&gt;did have many of the results of non-euclidean geometry by this time, &lt;strong&gt;including&lt;/strong&gt; the answer to the volume problem he raised to test his young rival (see Gauss (1832 - Cubirung der Tetraeder))&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to place Gauss's result in historical context, but the conceptual framework of hyperbolic geometry is as new to me as it was to Gauss's contemporaries. My previous post helped me understand the Gauss's formula for the orthoscheme tetrahedron; it connects the length of the side 24 with the angle 341 by the formula:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\alpha^2\cdot \cot^2(341) - \beta^2\cdot \tanh^2(l_{24}) = 1$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;when: $$\alpha = \cot(431),\quad &#xA;\beta = \cot(234).$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I understand the method of exhaustion is universal and doesn't depend on type of geometry; whether the geometry is hyperbolic, elliptic or euclidean, one can find volumes by dividing it into slices and then integrate them. But I don't know how to move from the Gauss's expression for the differential:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$(1) \partial \Delta = - \tfrac12 l_{24}\cdot \partial A_{341},$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;to the volume function; in particular, I lack an expression for the area of the face 341 as a function of the length 24. If anybody can help me with that, then I believe I'll be able to derive an expression for the volume function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps I'm not completely appreciating the difficulty of the subject (I know three-dimensional hyperbolic geometry is a pretty advanced topic) and there are very few people who can answer my question, but I won't give up until I've exhausted all of my options.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="6780" LastEditDate="2017-12-14T20:01:44.663" LastActivityDate="2017-12-14T20:01:44.663" Title="How to derive from Gauss's result on the volume of orthoscheme tetrahedron the formulas of Lobachevsky and Bolay?" Tags="|biographical-details|differential-geometry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6790" PostTypeId="1" AcceptedAnswerId="6862" CreationDate="2017-11-25T01:50:10.820" Score="15" ViewCount="2107" Body="&lt;p&gt;My question is on the notation of the &lt;strong&gt;Digamma function&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The Factorial function $n!$ (which is met in secondary school), is conceptually seminal to the Digamma function.  The Factorial function is defined as:&#xA;    $$0!=1,\qquad n!=\frac{(n+1)!}{n+1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This concept is extended with Gauss's Pi function:&#xA;    $$\operatorname\Pi(z)=\frac{\operatorname\Pi(z+1)}{z+1}$$&#xA;and with a simple unit offset (&lt;a href=&quot;http://math.stackexchange.com/questions/267989/the-gamma-function-and-the-pi-function&quot;&gt;see here&lt;/a&gt;), the more familiar Euler Gamma function:&#xA;    $$\operatorname\Gamma(z)=\frac{\operatorname\Gamma(z+1)}z$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Digamma function is a further extension:&#xA;    $$\psi(z)=\dfrac{\operatorname\Gamma'(z)}{\Gamma(z)}$$&#xA;and can itself be extended with the Polygamma function:&#xA;    $$\psi^m(z)=\frac{\mathrm d^m}{\mathrm dz^m}\psi(z)&#xA;%=\frac{\mathrm d^{m+1}}{\mathrm d z^{m+1}}\ln \operatorname\Gamma(z)&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;All sources I have encountered that mention the Digamma (or Polygamma) denote it as above with the greek letter 'psi' $\psi$.  This befuddles and confuses me.  Archaic greek's uppercase form of &lt;a href=&quot;http://en.wikipedia.org/wiki/Digamma&quot; rel=&quot;noreferrer&quot;&gt;digamma&lt;/a&gt; resembles the glyph of capital gamma $\Gamma$, but with and additional horizontal cross bar, making it appear as a 'double gamma' or a latin $\mathrm F$.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The uppercase gamma, and lowercase digamma and psi characters are available in LaTeX/MathJax, where &lt;code&gt;$$\Gamma,\digamma,\psi$$&lt;/code&gt; produces: $$\huge\Gamma,\digamma,\psi$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It makes sense that 'Digamma' would be chosen as a symbol that is both visually and etymologically similar to the Gamma function, because these functions are closely related.&#xA;It does not make any sense to chooses the symbol $\psi$.  But what really gets me, is that the name of the function is a complete mismatch to the symbol of the function.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question(s):&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;If we are going to call it a 'Digamma' function why don't we use a 'Digamma' symbol?&lt;/em&gt;&lt;br&gt;&#xA;or conversely&lt;br&gt;&#xA;&lt;em&gt;If we are going to use a 'psi' symbol why don't we call it a 'psi' function?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some justification for the establishment or continuation of this entrenched convention?  Is this merely an historical artefact of idiotic humanity (a typographical error)?  Or is there more to the story?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My best guess&lt;/strong&gt; is that it many have something to do with typographical limitations, or overloading.  But it still seems to ridiculously confusing notation to learn, (let alone to teach and propagate).&lt;/p&gt;&#xA;" OwnerUserId="6780" OwnerDisplayName="Elements in Space" LastEditorUserId="6780" LastEditDate="2017-12-29T02:01:16.667" LastActivityDate="2018-10-12T12:42:45.033" Title="Why is the Digamma function always denoted with the letter &quot;psi&quot;?" Tags="|mathematics|notation|naming-conventions|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6791" PostTypeId="2" ParentId="6421" CreationDate="2017-12-14T13:17:48.753" Score="2" Body="&lt;p&gt;The question of Kepler's possible influence on Fermat is a subject of scholarly controversy, exhaustively analyzed by Cifoletti in her book on Fermat.  For details see &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9542-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 2018 publication in &lt;em&gt;Foundations of Science&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-01-07T12:22:57.543" LastActivityDate="2018-01-07T12:22:57.543" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6792" PostTypeId="1" CreationDate="2017-12-14T13:53:34.313" Score="0" ViewCount="290" Body="&lt;p&gt;I've been reading about Bell Labs in the USA and find the idea of private labs to be interesting. What are some other major private labs that have existed in the US that did/do physics research?&lt;/p&gt;&#xA;" OwnerUserId="3219" LastActivityDate="2017-12-14T20:44:30.900" Title="Major private labs that did physics research in the USA" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6793" PostTypeId="1" CreationDate="2017-12-11T13:30:06.823" Score="1" ViewCount="70" Body="&lt;p&gt;i'm interested in the history of the 2 Sobolev Inequalities. The Gagliardo-Nirenberg-Sobolev inequality and Morrey inequality as here &lt;a href=&quot;https://en.wikipedia.org/wiki/Sobolev_inequality#Gagliardo%E2%80%93Nirenberg%E2%80%93Sobolev_inequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sobolev inequality&#xA;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did these two inequalities arise? What is the context of its emergence? In what year were they discovered?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="6774" OwnerDisplayName="C. Junior" LastActivityDate="2017-12-14T16:51:35.620" Title="About Sobolev inequalities" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6794" PostTypeId="1" CreationDate="2017-12-14T17:10:56.677" Score="-1" ViewCount="353" Body="&lt;p&gt;My question is straight forward. How did Lord Kelvin accurately determine that -273.16°C is absolute zero. I guess this was not determined by experiments cause the equipment was not so developed. ( Please note that I am not sure that it was Lord Kelvin only who gave the concept of absolute zero.)&lt;/p&gt;&#xA;" OwnerDisplayName="user497826" LastEditorUserId="6784" LastEditDate="2017-12-16T02:37:52.213" LastActivityDate="2017-12-16T02:37:52.213" Title="Lord Kelvin's wonderful prediction" Tags="|thermodynamics|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6795" PostTypeId="2" ParentId="6794" CreationDate="2017-12-14T17:40:46.280" Score="0" Body="&lt;p&gt;He didn't accurately determine the value - just the principle that there was an absolute zero temperature scale.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although probably fairer to say he popularised it, or put it on a firmer theoretical foundation, since &lt;a href=&quot;https://en.wikipedia.org/wiki/Guillaume_Amontons&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guillaume Amontons&lt;/a&gt;  seems to have made a reasonable experimental estimate of absolute zero 150 years earlier.&lt;/p&gt;&#xA;" OwnerUserId="2838" LastActivityDate="2017-12-14T17:40:46.280" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6796" PostTypeId="2" ParentId="6794" CreationDate="2017-12-14T18:10:41.163" Score="2" Body="&lt;p&gt;It was observed that if you change the temperature of a gas at constant pressure, its volume changes linearly with the temperature (Charles' law), at least if the temperature is low enough.  The slope of the line turns out to be proportional to the volume of the gas at some reference temperature, say $T_0 = 0^\circ$C, with the proportionality constant $\alpha$:  $\Delta V / \Delta T = \alpha V_0$.  That means if you extrapolate the $V$ vs. $T$ line back to zero volume, you get a minimum possible temperature $T_\text{min}$, which is given by $T_\text{min} = T_0-1/\alpha$.  Furthermore, that minimum temperature turns out to be the same for different gases,  or for the same gas at different pressures, or for different initial volumes $V_0$. That is, it's a universal constant and not the property of a particular gas.  Gay-Lussac first measured it in 1802 to be $-267^\circ C$, and more recent measurements have put it at $-273.15^\circ C$.&lt;/p&gt;&#xA;" OwnerUserId="6783" LastEditorUserId="6783" LastEditDate="2017-12-14T18:16:21.447" LastActivityDate="2017-12-14T18:16:21.447" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6797" PostTypeId="1" CreationDate="2017-12-14T20:07:24.987" Score="1" ViewCount="69" Body="&lt;p&gt;I have heard some stuff about how Einstein got is ideas from third parties. I've read a biography about him by Walter Isaacson, and a lot of possibly wrong one-liners on the internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I'm wondering, are there specific texts that Einstein has written about the chronology of how he discovered the things he discovered? For example, did he directly write himself about his intellectual influences, what kind of things he read, and so forth?&lt;/p&gt;&#xA;" OwnerUserId="4556" LastEditorUserId="4556" LastEditDate="2017-12-15T04:38:15.947" LastActivityDate="2017-12-15T04:38:15.947" Title="What has Einstein actually written about the chronology of his discoveries?" Tags="|einstein|resource-recommendation|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6798" PostTypeId="2" ParentId="6794" CreationDate="2017-12-14T20:17:34.000" Score="5" Body="&lt;p&gt;Long ago I had to make a project about this topic. I disagree with other answers: Lord Kelvin, with the help of Joule, did actually &lt;strong&gt;find&lt;/strong&gt; the absolute scale, because there's much more you are missing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's true that it was well known that $T=-273 ºC$ was acually the temperature at which wolume predicted for ideal gases gets zero. You can see it in many books like Callen thermodynamics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, setting $T=ºC+273$ is NOT enough to consider it an &quot;absolute scale&quot;, because it still depends on the instrument you use. You must DEFINE temperature in such a way it doesn't depend on the substance used. You always measure a physical property related to T, but not directly T. What Kelvin did was finding a relationship between T and energies, heats and works, which are universal physical quantities that do not depend on the sustance. For this, he used the results of the ideal Carnot's engine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Liek this, it doesn't depend on the substance but on measurable energies/heats. This is now indeed an absouet scale.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the full development, check this book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gray, Andrew (1908). Lord Kelvin: an account of his scientific life&#xA;  and work&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which you can find here:&#xA;&lt;a href=&quot;https://archive.org/details/lordkelvinanacc01graygoog&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/lordkelvinanacc01graygoog&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6787" LastActivityDate="2017-12-14T20:17:34.000" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6799" PostTypeId="2" ParentId="6792" CreationDate="2017-12-14T20:44:30.900" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;http://www.geglobalresearch.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;GE research lab in Niskayuna&lt;/a&gt; (Schenectady) NY - variously called GE CR&amp;amp;D (corporate research and development) or GRC (global research center) has a long history of innovation - from the early invention of ductile tungsten (first for the light bulb, and then resulting in the Coolidge tube which is still the foundation of virtually all medical X-ray tubes today), through jet engines, magnetrons, laser diodes, superconductivity (for which &lt;a href=&quot;https://en.wikipedia.org/wiki/Ivar_Giaever#The_Nobel_Prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ivar Giaever shared the Nobel prize&lt;/a&gt;), thin films (for which &lt;a href=&quot;https://www.chemheritage.org/historical-profile/irving-langmuir-and-katharine-burr-blodgett&quot; rel=&quot;nofollow noreferrer&quot;&gt;Langmuir received the Nobel prize&lt;/a&gt;), amorphous silicon X-ray detectors, MRI, wind turbines, locomotives, remote sensing, robotics, plastics, ceramics, ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is one of the few large private labs that is &quot;still standing&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other labs included the Xerox &lt;a href=&quot;https://www.xerox.com/en-us/innovation/parc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Palo Alto Research Center (PARC)&lt;/a&gt;, &lt;a href=&quot;https://www.kodak.com/US/en/science/innovation/default.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kodak research center&lt;/a&gt;, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Philips_Natuurkundig_Laboratorium&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philips Natuurkundig Laboratorium&lt;/a&gt; (now High Tech Campus Eindhoven - no longer purely for Philips, and not in the US), Siemens (multiple locations - &lt;a href=&quot;https://www.siemens.com/content/dam/internet/siemens-com/global/company/innovation/innovation/corporate-technology/pdf/CT-Standardpresentation_E_March_2017.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;very diverse&lt;/a&gt;), and many others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of these are &quot;just physics&quot; labs - they usually thrive because they are cross-disciplinary.&lt;/p&gt;&#xA;" OwnerUserId="543" LastActivityDate="2017-12-14T20:44:30.900" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6800" PostTypeId="1" CreationDate="2017-12-14T22:48:14.083" Score="0" ViewCount="1258" Body="&lt;p&gt;The question in the title is clear enough to not to write additional details here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only to say, as far as I know, Goldbach´s conjecture and the problem of existence of odd perfect numbers are among the oldest unsolved problems that I know of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What problems do you know of?&lt;/p&gt;&#xA;" OwnerDisplayName="user6444" LastActivityDate="2017-12-15T01:17:32.250" Title="What are some of the oldest unsolved problems in mathematics?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6801" PostTypeId="2" ParentId="6800" CreationDate="2017-12-15T01:17:32.250" Score="3" Body="&lt;p&gt;The questions of existence of odd perfect numbers and are there infinitely many even perfect numbers go back to Euclid (although I'm sure Euclid did not explicitly state these problems). There are not many advanced mathematical texts before Euclid, so they have to be the oldest unsolved problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand if you count the questions like &quot;who wins with the optimal strategy in a particular game (e.g. Go)?&quot; as an open problem, then it could be even older.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2017-12-15T01:17:32.250" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6802" PostTypeId="1" AcceptedAnswerId="6990" CreationDate="2017-12-15T15:21:32.033" Score="6" ViewCount="789" Body="&lt;p&gt;The 43rd entry(Oct. 1796) of Gauss' mathematical diary &quot;&lt;strong&gt;Vicimus GEGAN&lt;/strong&gt;&quot; remained a mystery for a long time. K. R. Biermann found &lt;a href=&quot;https://mathoverflow.net/questions/86657/a-couple-of-questions-on-gausss-mathematical-diary&quot;&gt;evidence&lt;/a&gt; that GEGAN is related the famous arithmetic-geometric mean(AGM) of Gauss. He also referred this entry to the link &lt;strong&gt;between the AGM and general elliptic functions&lt;/strong&gt;. Since I do not have access to Biermann's article, I am still confused about Biermann's conclusion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--The 51st entry(Jan. 1797) of the diary shows that Gauss started his research on lemniscate;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--The 92nd entry(July. 1798) of the diary suggests that Gauss found the connection between Jacobi theta-functions and lemniscate functions;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--The 98th entry(May. 1799) of the diary shows that Gauss numerically computed the AGM between $1$ and $\sqrt{2}$ and verified the AGM is equal to a certain lemniscatic integral to 11 digital places;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--The 102nd entry(Dec. 1799) shows that Gauss proved his conjecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As is indicated in these later entries, Gauss was likely to be unaware of the link between elliptic functions and the AGM before 1799(i.e., he did not know as much of elliptic functions in 1796 as in 1799). What is the exact argument of Biermann in his 1997 paper?&lt;/p&gt;&#xA;" OwnerUserId="6450" LastEditorUserId="8" LastEditDate="2018-01-28T14:29:42.327" LastActivityDate="2018-01-30T08:08:25.693" Title="A question on Gauss' &quot;Vicimus GEGAN&quot;" Tags="|calculus|debunking|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6803" PostTypeId="2" ParentId="5943" CreationDate="2017-12-16T05:56:19.370" Score="4" Body="&lt;p&gt;Here are some books I've enjoyed that have a lot of biographical information and context. Most are &quot;popular&quot; biographies, not in-depth scientific studies, however, books by Abraham Pais are exceptional and include good technical material. I also list Richard Rhodes, &quot;The Making of the Atomic Bomb&quot; since it is so in-depth and relevant to your question, I think. Rhodes's book is the first in an excellent series. I also include a book on J.C. Maxwell, though he's from the 19th century, he's such an important figure and the book is enjoyable. And I list a terrific biography on Newton because, in addition to the biographical material, it contains lots of good technical information.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, I agree with the previous post, the book by Graham Farmelo on Dirac was excellent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Schrodinger: Life and Thought&lt;/em&gt;&quot;, by Walter J. Moore&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Planck: Driven by Vision, Broken by War&lt;/em&gt;&quot;, by Brandon R. Brow&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Beyond Uncertainty: Heisenberg, Quantum Physics, and The Bomb&lt;/em&gt;&quot;, by David Cassidy&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The Age of Entanglement: When Quantum Physics was Reborn&lt;/em&gt;&quot;, Louisa Gilder&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality&lt;/em&gt;&quot;, by Manjit Kumar&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The Pope of Physics: Enrico Fermi and the Birth of the Atomic Age&lt;/em&gt;&quot;,&#xA;by Gino Segrè and Bettina Hoerlin&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Genius: The Life and Science of Richard Feynman&lt;/em&gt;&quot;, by James Gleick&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Henri Poincare: A Scientific Biography&lt;/em&gt;&quot;, by Jeremy Gray&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Niels Bohr's Times, In Physics, Philosophy, and Polity&lt;/em&gt;&quot;, by Abraham Pais&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Subtle is the Lord&lt;/em&gt;&quot;, Abraham Pais&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Inward Bound&lt;/em&gt;&quot;, by Abraham Pais&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Uncertainty: Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science&lt;/em&gt;&quot;, by David Lindley&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt;&quot;, by Richard Rhodes&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The Man Who Changed Everything: The Life of James Clerk Maxwell&lt;/em&gt;&quot;, by Basil Mahon&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Never at Rest: A Biography of Isaac Newton&lt;/em&gt;&quot;, by Richard S. Westfall&lt;/p&gt;&#xA;" OwnerUserId="6792" LastEditorUserId="6792" LastEditDate="2017-12-17T04:32:55.737" LastActivityDate="2017-12-17T04:32:55.737" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6804" PostTypeId="1" CreationDate="2017-12-16T12:26:45.410" Score="1" ViewCount="294" Body="&lt;h2&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stigler's Law of Eponymy&lt;/a&gt;&lt;/em&gt;&lt;/strong&gt; states that: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Mathematical and Scientific laws/discoveries/inventions/&amp;amp;c. are simply &lt;strong&gt;not&lt;/strong&gt; named after their original discoverer.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Stigler's &quot;Law&quot; is a perfect example of itself, being that Stephen Stigler himself attributes the discovery of &lt;em&gt;this very law&lt;/em&gt; to a Robert K. Merton, but the idea itself is much older.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(While there are certainly counter-examples of this law) there are &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_examples_of_Stigler%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;hundreds of examples&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can &lt;em&gt;only&lt;/em&gt; be very confusing/misleading for students.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;&lt;strong&gt;Question(s)&lt;/strong&gt;&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Why is Stigler's Law of Eponymy so standard in Mathematics and the Sciences?&lt;/em&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Is there a way to avoid the confusion of Stigler's Law?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(Edit)&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Why are such laws/discoveries not given simple descriptive names?;  Why are they named after anyone at all?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6780" LastEditorUserId="55" LastEditDate="2017-12-19T00:53:54.857" LastActivityDate="2017-12-19T04:28:08.513" Title="How come so many laws were not discovered by people they are named after?" Tags="|discoveries|social-context|naming-conventions|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6805" PostTypeId="2" ParentId="6804" CreationDate="2017-12-16T23:58:29.963" Score="2" Body="&lt;p&gt;Why not? Science and mathematics are communal efforts, every discovery typically has multiple precursors (&quot;&lt;em&gt;standing on the shoulders of giants&lt;/em&gt;&quot;), and &quot;the discoverer&quot; is often not the one from whom subsequent scholars and/or the public learn about it because they did not explain or publicize it well enough. &quot;Stigler's law&quot; is an example of the latter. Sometimes it is due to historical obscurity of the &quot;original discovery&quot;, as with &quot;Pythagorean&quot; theorem or &quot;Heron's&quot; formula, sometimes due to countries or schools promoting &quot;their&quot; discoverer even if he/she was not the first, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The naming, upon which recognizable reference depends, has to deal with  pragmatic constraints of cultural change. Although renaming efforts are undertaken to correct some blatant misnomers they typically take a long time to take hold and are not a practical priority (US and UK still use feet and pounds for the same reason). And more generally, historical significance of &quot;original discoverers&quot; is overrated in the popular literature (as can be seen anecdotally from the number of &quot;who was first&quot; questions on our SE), which contributes to the &quot;&lt;em&gt;great leaps by great minds&lt;/em&gt;&quot; misconceptions about history. Here is &lt;a href=&quot;http://www.ub.edu/geocrit/geo84.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Renn's From the History of Science to the History of Knowledge – and Back&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The history of knowledge has traditionally been studied from a restricted perspective that favors innovation over implementation, transmission and transformation. In the past, historians of science and technology have often focused on the question of who was the first to discover a fact that later became a key innovation and when this took place. Much less attention has been paid to the question of what role these discoveries or inventions played in the contemporary context of knowledge and how they changed their meaning when transmitted to a different context.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2017-12-16T23:58:29.963" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6806" PostTypeId="2" ParentId="5943" CreationDate="2017-12-18T09:40:13.323" Score="2" Body="&lt;p&gt;I would suggest:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Born, Max&lt;/strong&gt;, &lt;em&gt;My Life: Recollections of a Nobel Laureate&lt;/em&gt;. Taylor &amp;amp; Francis, London, 1978.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2017-12-18T09:40:13.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6807" PostTypeId="2" ParentId="6804" CreationDate="2017-12-19T04:08:50.217" Score="2" Body="&lt;p&gt;&quot;X's Theorem&quot; is just a convenient label, not an indication of priority. And many researchers, including myself, will agree that &quot;Pythagoras Theorem&quot; is a more convenient label than the &quot;Right-angled Triangle Theorem&quot;. The name that you proposed also does not reflect the matter because there are several theorems about right-angled triangle. Anyway, mathematicians do not find such labels &quot;confusing and misleading&quot;, and students have to live with this:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides this, finding the first person who stated something can be very difficult, many results developed gradually, with contribution from several&#xA;authors,&#xA;and further research about history would lead to changing attributions. Changing names of established theorems is not desirable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also recommend this related discussion of the common naming practice in mathematics:&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/285627/how-are-constants-functions-named-after-their-discoverer/285644#285644&quot;&gt;https://mathoverflow.net/questions/285627/how-are-constants-functions-named-after-their-discoverer/285644#285644&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-19T04:28:08.513" LastActivityDate="2017-12-19T04:28:08.513" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6808" PostTypeId="1" AcceptedAnswerId="7128" CreationDate="2017-12-19T14:35:46.720" Score="3" ViewCount="120" Body="&lt;p&gt;The oldest appearances of causality troubles in modern theories are of course the Kurt Gödel paper (1949), Reichenbach's philosophy of time and space (1927), Weyl's &quot;Space, time and matter&quot; (1922) and then the Tolman paradox in &quot;The theory of the relativity of motion&quot; (1917). There's a few papers on the topic of advanced waves before that but none of them really seem to deal with any causality issues at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any older papers concerning hints at causality issues with modern theories or are those the oldest ones?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2018-03-08T12:53:39.497" Title="First appearance of causality issues" Tags="|physics|relativity-theory|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6809" PostTypeId="2" ParentId="3324" CreationDate="2017-12-19T14:42:03.073" Score="1" Body="&lt;p&gt;in his first paper on the calculus (1669) Newton used the series  to integrate the function $y=\frac1{1+x^2}$ where he used the binomial theorem and integrated term by term the produced the infinite polynomial.&lt;/p&gt;&#xA;" OwnerUserId="6805" LastEditorUserId="5989" LastEditDate="2017-12-19T22:26:19.937" LastActivityDate="2017-12-19T22:26:19.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6810" PostTypeId="1" CreationDate="2017-12-20T04:23:55.690" Score="1" ViewCount="133" Body="&lt;p&gt;I can think of no discovery bigger in terms of correction of a belief if the metric is, How much bigger is something really than believed? Of course that is not the only significance. It is a major change in &quot;world view&quot; -- it seems like it would be equivalent to overturning helio-centrism (although perhaps astronomers having once had such a belief overturned would be more open to new ideas) and if so, it would have had impact far beyond astronomy itself.&lt;/p&gt;&#xA;" OwnerUserId="6807" LastActivityDate="2017-12-21T18:30:58.297" Title="What was the impact on science and philosophy of the discovery that the Milkyway is not the whole Universe?" Tags="|astronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6811" PostTypeId="2" ParentId="6810" CreationDate="2017-12-20T23:17:17.120" Score="1" Body="&lt;p&gt;There was no such discovery. Atomists believed in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cosmic_pluralism&quot; rel=&quot;nofollow noreferrer&quot;&gt;plurality of worlds&lt;/a&gt; when mainstream astronomy was single world and geocentric. Although there is no a priori reason for the two to go together, historically the overturning of geocentrism went hand in hand with accepting the plurality of worlds. One reason was that to explain the absence of a visible &lt;a href=&quot;https://en.wikipedia.org/wiki/Stellar_parallax#Early_theory_and_attempts&quot; rel=&quot;nofollow noreferrer&quot;&gt;stellar parallax&lt;/a&gt; (change in position of stars due to rotation of the Earth around the Sun) heliocentrists had to postulate that the stars are extremely distant, and then it was natural to assume that they were analogous to the Sun. In the ancient times the absence of parallax was one of the main arguments against heliocentrism, which was proposed as a hypothesis by &lt;a href=&quot;https://en.wikipedia.org/wiki/Aristarchus_of_Samos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristarchus&lt;/a&gt; and some others, and Brahe's chief objection to Copernicus was that the void between the last planetary sphere and the sphere of fixed stars would have to be implausibly huge to explain no parallax. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More philosophically, once heliocentrism turned the Earth from the center of the universe into one planet among many, and the metaphysical difference between Aristotle's sublunar and superlunar spheres was erased, the natural next step to take was that the Sun too was just one star among many. After that the &quot;discovery&quot; that Milky Way was just one galaxy among many was rather routine and anti-climactic, it is hard to even pinpoint when it &quot;happened&quot;, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_knowledge_about_galaxies,_clusters_of_galaxies,_and_large-scale_structure&quot; rel=&quot;nofollow noreferrer&quot;&gt;Timeline of knowledge about galaxies, clusters of galaxies, and large-scale structure&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the closest thing to the &quot;event&quot; that the OP is looking for was the &lt;a href=&quot;https://en.wikipedia.org/wiki/Copernican_Revolution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Copernican Revolution&lt;/a&gt;. Copernicus vaguely entertained the plurality of worlds already, although not publicly due to the cultural context of the time, a bit later Giordano Bruno was burned for it (along with other heresies). But with the spread of telescopes by mid 16th century the plurality of worlds was a commonality among scientists, Newton assumed it as a matter of course in Principia even though the stellar parallax was only detected by Bessel in 1838. Still, even the impact of the Copernican Revolution was enhanced by the cultural context that it displaced, as Renn explains in &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-14553-2_16&quot; rel=&quot;nofollow noreferrer&quot;&gt;The History of Science and the Globalization of Knowledge&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In response to the encompassing religious worldview, the new knowledge accumulated by these scientist-engineers began to assume the character of an&#xA;  equally all-embracing interpretation of the world, as can be found in the great&#xA;  philosophical concepts of the early-modern period, for instance, in the works of&#xA;  Giordano Bruno or Rene Descartes. Science eventually became a kind of counter&#xA;  ideology by which the emerging bourgeoisie could defend its claims to power, not&#xA;  according to a transcendent, religious order, but according to immanent laws of&#xA;  nature and society. The new knowledge thus also assumed a normative dimension.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;This situation helps to explain why, in the sixteenth century, the reform of&#xA;  astronomy by Copernicus, placing the Sun rather than the Earth at the center of&#xA;  the universe, could have had such far-reaching ideological consequences: it occurred within a context of a socially dominant system of knowledge that claimed to be universal and exclusive. The geocentric worldview, placing&#xA;  the Earth at the center of the universe, was deeply anchored within this system of knowledge. Questioning this claim, even with good scientific reasons and without any intent of heretic provocation, still amounted to unhinging the whole system and thus causing an ideological revolution by means of an astronomical, and at the outset purely scientific innovation.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In contrast, there was no comparable revolution in seventeenth-century China when Jesuit missionaries introduced Copernican theory, or even Galileo’s telescope, which made the new view of the heavens so intuitively plausible. In Ming China, there was simply no combined religious and philosophical worldview that this new discovery could potentially provoke.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2017-12-20T23:17:17.120" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6812" PostTypeId="1" AcceptedAnswerId="6821" CreationDate="2017-12-21T11:18:52.683" Score="5" ViewCount="317" Body="&lt;p&gt;When and by who was it first observed that the &lt;a href=&quot;https://en.wikipedia.org/wiki/E_(mathematical_constant)&quot; rel=&quot;nofollow noreferrer&quot;&gt;transcendental number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&lt;/a&gt; is NOT a &lt;a href=&quot;https://en.wikipedia.org/wiki/Liouville_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liouville number&lt;/a&gt;?&lt;/p&gt;&#xA;&lt;p&gt;This fact is stated in a lot of web pages, and a proof can be found in the Mathematics Stack Exchange question &lt;a href=&quot;https://math.stackexchange.com/questions/1946535/why-is-e-not-a-liouville-number&quot;&gt;Why is &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; not a Liouville number?&lt;/a&gt;, but I have yet to find any nontrivial historical information about this fact.&lt;/p&gt;&#xA;&lt;p&gt;I suspect this can be found in Édmond Maillet’s 1906 book &lt;a href=&quot;https://archive.org/details/introductionlat00mailgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions&lt;/strong&gt;&lt;/a&gt; (where the phrase “Liouville number” was first coined) or in Oskar Perron’s 1913 book &lt;a href=&quot;https://archive.org/details/dielehrevondenk00perrgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Die Lehre von den Kettenbrüchen&lt;/strong&gt;&lt;/a&gt;, but my very poor French and my non-existent German stands in the way.&lt;/p&gt;&#xA;&lt;p&gt;I began wondering about this when I came across some remarks on this topic in the following 1916 Master of Arts thesis:&lt;/p&gt;&#xA;&lt;p&gt;Jewell Constance Hughes [&lt;em&gt;after 1935&lt;/em&gt;: Jewell Hughes Bushey] (1896-1989), &lt;a href=&quot;https://mospace.umsystem.edu/xmlui/handle/10355/49197&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Transcendentalism of Curves and of Numbers&lt;/strong&gt;&lt;/a&gt;, Master of Arts thesis (under Wilhelmus David Allen Westfall, 1879-1951), University of Missouri, 1916, 24 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(first paragraph on p. 17)&lt;/strong&gt; As we have already explained, in 1844 [&amp;quot;1873&amp;quot; intended] Hermite first showed that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is not algebraic by proving that it cannot satisfy an algebraic equation of degree &lt;span class=&quot;math-container&quot;&gt;$m.$&lt;/span&gt; Is it not possible to prove the same thing by showing that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; cannot satisfy the Liouville inequality? The following results show the efforts of the writer attacking the problem. This Chapter may be regarded as a preliminary report on an investigation of rather far reaching character which the writer hopes to carry further after this year.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(last paragraph on p. 24)&lt;/strong&gt; As we said in the beginning, this Chapter is a preliminary report of an investigation which the writer hopes to carry on later. We have not been able to prove that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; violates the Liouville inequality. Perhaps it does do so, perhaps it does not. Perhaps &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is one of an uncountable class of numbers which fail to violate Liouville's Theorem. At the present time we do not know.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2017-12-22T07:15:23.117" Title="When was $e$ first observed to not be a Liouville number?" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6814" PostTypeId="2" ParentId="6766" CreationDate="2017-12-21T18:48:23.260" Score="0" Body="&lt;p&gt;I found an exhausting volume that spends too long on earlier efforts at explanations of nature. It may have the range you are looking for.  I believe the title is &lt;em&gt;The Chemical Philosophy&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6814" LastActivityDate="2017-12-21T18:48:23.260" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6816" PostTypeId="2" ParentId="6749" CreationDate="2017-12-21T20:19:16.793" Score="2" Body="&lt;p&gt;At first an inch was the width of a man's thumb. In the 14th century, King Edward II of England ruled that 1 inch equalled 3 grains of barley placed end to end lengthwise. So no, no relationship to the dimensions of the Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Ref &lt;a href=&quot;https://www.britannica.com/science/inch&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.britannica.com/science/inch&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="6817" LastEditorUserId="6817" LastEditDate="2017-12-21T20:24:28.167" LastActivityDate="2017-12-21T20:24:28.167" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6817" PostTypeId="1" AcceptedAnswerId="6818" CreationDate="2017-12-20T14:20:06.663" Score="1" ViewCount="302" Body="&lt;p&gt;If differential equation theory was known and also studied by Galileo, so why he didn't manage to discover a normal distribution (its discovery had to wait for Laplace and Gauss)?&lt;/p&gt;&#xA;" OwnerUserId="6821" OwnerDisplayName="Lili" LastEditorDisplayName="user6769" LastEditDate="2017-12-30T16:10:48.567" LastActivityDate="2017-12-30T16:10:48.567" Title="Galileo and normal distribution discovery" Tags="|calculus|statistics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6818" PostTypeId="2" ParentId="6817" CreationDate="2017-12-21T01:22:53.110" Score="3" Body="&lt;p&gt;Even aside from the fact that Galileo knew nothing of differential equations, or derivatives for that matter (he lived before Newton and Leibniz), and that the normal distribution was not discovered by Laplace and Gauss but by De Moivre, why the connection? De Moivre discovered the bell curve not by solving differential equations but looking for a good approximation to binomial distributions with large n, Bernoulli's formula with binomial coefficients was not very practical for calculations. And Galileo could not do that either because even the simplest cases of the binomial distribution (or of any statistical distributions, for that matter) were not considered until the &lt;a href=&quot;https://en.wikipedia.org/wiki/Problem_of_points&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermat-Pascal correspondence&lt;/a&gt; 12 years after his death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laplace did try to derive the error curve using differential equations, but the curves he derived that way had cusps or vertical asymptotes at the origin and did not even look like the normal distribution, see &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stahl's Evolution of the Normal Distribution&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" OwnerDisplayName="Conifold" LastActivityDate="2017-12-21T01:22:53.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6819" PostTypeId="1" CreationDate="2017-12-21T22:23:51.813" Score="1" ViewCount="111" Body="&lt;p&gt;I've read this long transcript of a speech given by Richard Hamming:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cs.virginia.edu/~robins/YouAndYourResearch.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cs.virginia.edu/~robins/YouAndYourResearch.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was an amazingly good speech in my view, and extremely interesting. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: &lt;strong&gt;Do you know of more articles like this? Articles or transcripts of speeches by successful scientists giving their life's wisdom and advice in condensed form?&lt;/strong&gt; I've read a lot of philosophy and history of science, and while that is very interesting, supplementing it with more practical insights by scientists (rather than philosophers or historians of science) is helpful. &lt;/p&gt;&#xA;" OwnerUserId="4556" LastEditorUserId="4251" LastEditDate="2020-07-13T03:00:44.013" LastActivityDate="2020-07-13T03:00:44.013" Title="Canonical examples of successful scientists giving detailed advice about doing science/mathematics?" Tags="|scientists|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6820" PostTypeId="1" AcceptedAnswerId="6822" CreationDate="2017-12-21T22:27:01.627" Score="10" ViewCount="593" Body="&lt;p&gt;Sometimes, mathematical proofs accepted in the scientific community are found to be wrong. &#xA;What is  the disproved proof that was believed to be correct for the longest period of time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;clarification: Note that I'm asking only about proofs where the result itself turned out to be false.&lt;/p&gt;&#xA;" OwnerDisplayName="user6769" LastEditorDisplayName="user6769" LastEditDate="2017-12-22T10:56:53.163" LastActivityDate="2023-06-12T06:57:46.670" Title="Invalid mathematical proof that was considered valid for the longest period of time?" Tags="|mathematics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6821" PostTypeId="2" ParentId="6812" CreationDate="2017-12-21T23:11:27.140" Score="4" Body="&lt;p&gt;As you guessed, the observation is in Maillet’s 1906 book: Introduction (p. iii), footnote (p. 27) and Corollary II&lt;sub&gt;6&lt;/sub&gt; (&lt;a href=&quot;https://archive.org/stream/introductionlat00mailgoog#page/n148&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 125&lt;/a&gt;; &lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?37.0237.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jahrbuch credit&lt;/a&gt;). He announced it earlier in &lt;em&gt;Comptes Rendus&lt;/em&gt; (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?36.0444.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3095m/f419&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 419&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\smash{e^{\frac1q}}$ ($q$ integer $\geqslant 1$) is not a Liouville number&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and gave himself credit in the &lt;em&gt;Encyclopédie&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=QMdJAQAAMAAJ&amp;amp;pg=PA288-IA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=QMdJAQAAMAAJ&amp;amp;pg=PA383&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 383&lt;/a&gt;). The observation doesn't seem to be in Perron’s 1913 &lt;em&gt;Lehre von den Kettenbrüchen&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/dielehrevondenk00perrgoog#page/n155&quot; rel=&quot;nofollow noreferrer&quot;&gt;§33&lt;/a&gt;), but it is in his 1921 &lt;em&gt;Irrationalzahlen&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/irrationalzahlen00perruoft#page/174&quot; rel=&quot;nofollow noreferrer&quot;&gt;§47&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In any case, the number $e$ does not belong to the known types of transcendental numbers: it is not a Liouville number; for its continued fraction expansion, which we derived in §32, obviously does not satisfy the condition of Proposition 71. (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-12-22T07:15:23.117" LastActivityDate="2017-12-22T07:15:23.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6822" PostTypeId="2" ParentId="6820" CreationDate="2017-12-22T03:53:38.813" Score="11" Body="&lt;p&gt;In 1923 Henri Dulac published a solution of Hilbert problem 16, second part.&#xA;This was accepted, he received a prize of the French academy, and the problem was considered solved until a gap in the proof was found by Ilyashenko in 1982.&#xA;It took about 10 years to fill the gap (Ilyashenko and Ecalle, independently, 1992). 1982-1923=59, so for 59 years the fact was believed to be proven while it was not. This is the longest example that I know.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-22T15:36:09.723" LastActivityDate="2017-12-22T15:36:09.723" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6823" PostTypeId="2" ParentId="6819" CreationDate="2017-12-22T04:00:26.427" Score="2" Body="&lt;p&gt;Gian-Carlo Rota, Ten lessons I wish I had been taught, published in his book&#xA;Indiscrete thoughts, Springer 2008 (can be also found on Internet).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-12-22T04:00:26.427" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6824" PostTypeId="2" ParentId="6820" CreationDate="2017-12-22T07:20:33.917" Score="2" Body="&lt;p&gt;Are you asking about proofs of results where the result itself turns out to be false, or do you allow proofs with one part that was incomplete (a proof not covering all cases is strictly speaking not a valid proof of the claimed theorem) and the gap was filled in later? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The history of the Kronecker-Weber theorem provides an example of an incomplete proof that went unquestioned for a long time. Weber published a proof in 1886 and a gap was found in it and filled in 95 years later by Neumann. See the Wikipedia page about the Kronecker-Weber theorem. Arguably the reason nobody noticed the gap for so long is that, in the intervening years (starting with Hilbert ten years after Weber) other proofs and further conceptual generalizations of the Kronecker-Weber theorem were found, so there was not a strong incentive to recheck Weber's proof in order to have a proof of the result.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, if what you are really after is proofs that were fundamentally mistaken rather than being incomplete, this example I give does not fit. Please clarify your intentions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arguably, until a rigorous definition of real numbers and a proof of their completeness was found in the 1800s, lots of earlier work involving the real numbers was not fully justified. For example, Euclidean geometry needs further conditions related to continuity or completeness of the reals, which the ancient Greeks did not realize. Thus you could say there are 2000-year old examples in Euclidean geometry, and you'd be hard-pressed to find examples over a longer time period than that.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2017-12-22T07:30:15.393" LastActivityDate="2017-12-22T07:30:15.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6825" PostTypeId="2" ParentId="6819" CreationDate="2017-12-22T11:36:23.597" Score="1" Body="&lt;p&gt;Santiago Ramón y Cajal wrote &lt;a href=&quot;https://cvc.cervantes.es/ciencia/cajal/cajal_reglas/default.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Reglas y consejos sobre investigación científica&lt;/em&gt;&lt;/a&gt;, translated into English as &lt;em&gt;Advice for a Young Investigator&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="633" LastActivityDate="2017-12-22T11:36:23.597" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6826" PostTypeId="1" AcceptedAnswerId="6827" CreationDate="2017-12-22T18:03:14.653" Score="5" ViewCount="1096" Body="&lt;p&gt;I was calculating the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirichlet_integral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichlet Integral&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\displaystyle \int_{0}^{\infty}\frac{\sin x}{x} \text{d} x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and then I wondered about his name and history:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Why it has the name of &lt;a href=&quot;https://es.wikipedia.org/wiki/Peter_Gustav_Lejeune_Dirichlet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichlet&lt;/a&gt;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When was it first calculated?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In what context it showed up for the first time and from which interests it needed (if so) to be evaluated?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I don't know any resources (except for Stack Exchange) for researching this questions, and I haven't found anything related to them after surfing along the first 4 pages of results of &amp;quot;dirichlet integral history&amp;quot; in Google.&lt;/p&gt;&#xA;&lt;p&gt;Does anybody know anything about its history or where to find information about it?&lt;/p&gt;&#xA;" OwnerUserId="5031" LastEditorUserId="4251" LastEditDate="2020-10-01T06:56:17.387" LastActivityDate="2020-10-01T06:56:17.387" Title="Dirichlet integral's history" Tags="|mathematics|calculus|dirichlet|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6827" PostTypeId="2" ParentId="6826" CreationDate="2017-12-22T19:34:13.800" Score="11" Body="&lt;p&gt;It arose in &lt;a href=&quot;https://archive.org/stream/diepartiellendi04webegoog#page/n55&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichlet’s famous proof&lt;/a&gt; of the convergence of Fourier series (&lt;a href=&quot;https://books.google.com/books?id=oZpGAAAAcAAJ&amp;amp;pg=PA157&quot; rel=&quot;nofollow noreferrer&quot;&gt;1829&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=oZpGAAAAcAAJ&amp;amp;pg=PA161&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 161&lt;/a&gt;),&lt;sup&gt;1&lt;/sup&gt; then again in his “discontinuous factor” method to compute integrals (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k2967c/f160&quot; rel=&quot;nofollow noreferrer&quot;&gt;1839&lt;/a&gt;; &lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?35.0312.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;, pp. &lt;a href=&quot;https://archive.org/stream/glejeunedirichle00lejeuoft#page/193&quot; rel=&quot;nofollow noreferrer&quot;&gt;193-195&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/glejeunedirichle00lejeuoft#page/353&quot; rel=&quot;nofollow noreferrer&quot;&gt;353-385&lt;/a&gt;)&lt;sup&gt;2&lt;/sup&gt; and in his “jump function” proof of the central limit theorem (1846; &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=2743162&quot; rel=&quot;nofollow noreferrer&quot;&gt;2011&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=v7kTwafIiPsC&amp;amp;pg=PA70&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 69-74&lt;/a&gt;).&lt;sup&gt;3&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently first evaluated by Euler in &lt;em&gt;De valoribus integralium a termino variabilis $x = 0$ usque ad $x = \infty$ extensorum&lt;/em&gt; (1781; 1794, pp. 337–&lt;a href=&quot;https://books.google.com/books?id=LmwyliFqN7cC&amp;amp;pg=PA345&quot; rel=&quot;nofollow noreferrer&quot;&gt;345&lt;/a&gt;; 1932, pp. &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k69676/f65&quot; rel=&quot;nofollow noreferrer&quot;&gt;lxiii&lt;/a&gt;, 217–&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k69676/f297&quot; rel=&quot;nofollow noreferrer&quot;&gt;227&lt;/a&gt;);&lt;sup&gt;4&lt;/sup&gt; Dirichlet’s uses are also foreshadowed in Fourier (&lt;a href=&quot;https://archive.org/stream/oeuvresdefourier01four#page/510&quot; rel=&quot;nofollow noreferrer&quot;&gt;1822, pp. 510-512&lt;/a&gt;),&lt;sup&gt;5&lt;/sup&gt; Poisson (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433090910104&amp;amp;num=273&quot; rel=&quot;nofollow noreferrer&quot;&gt;1824, p. 276&lt;/a&gt;), Bessel (&lt;a href=&quot;http://adsabs.harvard.edu/abs/1838AN.....15..369B&quot; rel=&quot;nofollow noreferrer&quot;&gt;1838&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=eFY_AAAAcAAJ&amp;amp;pg=PA393&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 393&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Incidentally, I suspect that calling it &lt;strong&gt;&lt;em&gt;the&lt;/em&gt;&lt;/strong&gt; Dirichlet integral is a misnomer introduced by Wikipedia: so far as I can tell, none of the below-quoted authors do that. Rather, they call &lt;em&gt;any&lt;/em&gt; integral of the form $\smash{\int_0^\infty f(x)\frac{\sin kx}{x}dx}$ (including of course this one) &lt;strong&gt;&lt;em&gt;a&lt;/em&gt;&lt;/strong&gt; Dirichlet integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References in English:&lt;/strong&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&lt;a href=&quot;https://doi.org/10.11429/subutsukiji1885b.3.249&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichlet&lt;/a&gt; p. 255, &lt;a href=&quot;http://archive.org/stream/introductiontoth00carsrich#page/200&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carslaw&lt;/a&gt; pp. 200-212, &lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=860945&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bottazzini&lt;/a&gt; pp. 190-196, &lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=2807493&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roy&lt;/a&gt; pp. 417-421, &lt;a href=&quot;https://books.google.com/books?id=oUfBAQAAQBAJ&amp;amp;pg=PA238&amp;amp;lpg=PA233&amp;amp;vq=Dirichlet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Petrovic&lt;/a&gt; Prob. 9.2.14. &lt;sup&gt;2&lt;/sup&gt;&lt;a href=&quot;https://archive.org/stream/atreatiseoninfi06pricgoog#page/n442&quot; rel=&quot;nofollow noreferrer&quot;&gt;Price&lt;/a&gt; pp. 399-410. &lt;sup&gt;3&lt;/sup&gt;&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=2743162&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fischer&lt;/a&gt; pp. 44-52. &lt;sup&gt;4&lt;/sup&gt;&lt;a href=&quot;https://arxiv.org/abs/0705.4640&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler&lt;/a&gt; p. 7. &lt;sup&gt;5&lt;/sup&gt;&lt;a href=&quot;https://archive.org/stream/analyticaltheory00four#page/439&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier&lt;/a&gt; §423.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2017-12-28T14:13:56.160" LastActivityDate="2017-12-28T14:13:56.160" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6828" PostTypeId="1" CreationDate="2017-12-23T18:23:49.190" Score="1" ViewCount="102" Body="&lt;p&gt;There are many biographies that go into detail into the lives of a single, or maybe a couple of scientists. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering, is there a good book that does not focus on an individual, but explicitly takes theses individuals as examples, of the &quot;typical&quot; scientist, or looks at the various types of scientists? &lt;/p&gt;&#xA;" OwnerUserId="4556" LastActivityDate="2017-12-24T14:29:01.880" Title="Biographies comparing the personal and professional lives of various influential scientists/mathematicians?" Tags="|biographical-details|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6829" PostTypeId="1" CreationDate="2017-12-23T19:54:06.647" Score="4" ViewCount="857" Body="&lt;p&gt;In 1909, Cambridge undergraduate G.I. Taylor published a letter describing his observation of diffraction using light of an extremely low intensity. For years, I've been teaching my students the interpretation of this experiment as demonstrating certain core features of the quantum theory of light. The story was that Taylor did this to prove that a single quantum could be diffracted. I recently learned that this was wrong on both historical and physics grounds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.physics.wisc.edu/undergrads/courses/spring09/205/handouts/GITaylor_Interference.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;actual paper&lt;/a&gt; never even mentions quanta. It cites no references, but the only theoretical motivation given is from an obscure theoretical suggestion by Thomson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, it turns out that, for reasons that could not possibly have been anticipated in 1909, the experimental conditions did not really test this aspect of quantum mechanics; the first confirmation of such an effect that is now believed to have been valid was by Grangier et al. in 1986. (Grangier, Roger, and Aspect, &quot;Experimental evidence for a photon anticorrelation effect on a beamsplitter,&quot; Europhys. Lett. 1 (1986) 173. Paywalled.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There apparently was some point in history after 1909 and before 1986 when people did interpret Taylor's experiment as a correct test of quantum mechanics, with a positive result. When was this? Did it play any decisive role in the debate, in which Bohr's followers wanted to quantize the atom but keep the EM field classical?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2017-12-26T17:54:30.417" Title="Photon interpretation of G.I. Taylor's diffraction of feeble light" Tags="|quantum-mechanics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6830" PostTypeId="2" ParentId="6828" CreationDate="2017-12-23T22:40:33.653" Score="5" Body="&lt;p&gt;Not sure these exactly fit your needs, but I am reminded of:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Gindikin, Simon&lt;/em&gt;, &lt;a href=&quot;http://dx.doi.org/10.1007/978-0-387-48811-0&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Tales of mathematicians and physicists. Transl. from the Russian by Alan Shuchat&lt;/strong&gt;&lt;/a&gt;, New York, NY: Springer (ISBN 0-387-36026-3/pbk; 0-387-048811-1/ebook). xx, 382&amp;nbsp;p. (2007). &lt;a href=&quot;https://zbmath.org/?q=an:1105.01009&quot; rel=&quot;noreferrer&quot;&gt;ZBL1105.01009&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Mashaal, Maurice&lt;/em&gt;, Bourbaki: A secret society of mathematicians. Transl. from the French by Anna Pierrehumbert, Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3967-5). 168&amp;nbsp;p. (2006). &lt;a href=&quot;https://zbmath.org/?q=an:1099.01022&quot; rel=&quot;noreferrer&quot;&gt;ZBL1099.01022&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Gillispie, Charles C.&lt;/em&gt;, &lt;a href=&quot;https://books.google.com/books/about/Science_and_Polity_in_France.html?id=uNI3YgEACAAJ&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Science and Polity in France: The Revolutionary and Napoleonic Years&lt;/strong&gt;&lt;/a&gt;. Princeton University Press (2004).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Yandell, Benjamin H.&lt;/em&gt;, The honors class. Hilbert’s problems and their solvers, Natick, MA: A K Peters. ix, 486 p. (2002). &lt;a href=&quot;https://zbmath.org/?q=an:0993.01001&quot; rel=&quot;noreferrer&quot;&gt;ZBL0993.01001&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Schweber, Silvan S.&lt;/em&gt;, QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga, Princeton Series in Physics. Princeton, NJ: Princeton University Press,. xxviii, 732 p. (1994). &lt;a href=&quot;https://zbmath.org/?q=an:0815.01011&quot; rel=&quot;noreferrer&quot;&gt;ZBL0815.01011&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Hunt, Bruce J.&lt;/em&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Maxwellians&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Maxwellians&lt;/strong&gt;&lt;/a&gt;. Cornell University Press (1991). &lt;a href=&quot;https://www.ams.org/mathscinet-getitem?mr=1260666&quot; rel=&quot;noreferrer&quot;&gt;MR1260666&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Grattan-Guinness, Ivor&lt;/em&gt;, Convolutions in French mathematics, 1800-1840. Volume I: The settings. Volume II: The turns. Volume III: The data, Science Networks. Historical Studies. 2-4. Berlin: VEB Deutscher Verlag der Wissenschaften. Basel etc.: Birkhäuser, 1601 p. (1990). &lt;a href=&quot;https://zbmath.org/?q=an:0836.01013&quot; rel=&quot;noreferrer&quot;&gt;ZBL0836.01013&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Bell, E. T.&lt;/em&gt;, Men of mathematics, XXI + 593 p. New York, Simon and Schuster (1937). &lt;a href=&quot;https://zbmath.org/?q=an:63.0793.03&quot; rel=&quot;noreferrer&quot;&gt;ZBL63.0793.03&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Marie, Maximilien&lt;/em&gt;, &lt;a href=&quot;https://archive.org/search.php?query=Maximilien%20Marie%20Histoire%20des%20sciences%20math%C3%A9matiques%20et%20physiques&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Histoire des sciences mathématiques et physiques&lt;/strong&gt;&lt;/a&gt;. Paris, Gauthier-Villars, 12 vol. (1883–1888).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Lord Brougham&lt;/em&gt;, &lt;a href=&quot;https://archive.org/details/livesofphilosoph00brourich&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Lives of philosophers of the time of George III&lt;/strong&gt;&lt;/a&gt;. London, Glasgow, R. Griffin (1855).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="658" LastActivityDate="2017-12-23T22:40:33.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6831" PostTypeId="2" ParentId="6828" CreationDate="2017-12-24T14:21:03.200" Score="1" Body="&lt;p&gt;Let me add few items to the nice list of Francois: Koestler's biographies of Copernicus, Galileo and Kepler: Arthur Koestler, Sleepwalkers: History of Man's changing vision of the universe, multiple editions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and F. Klein, History of mathematics in 19th century. (It is mostly about mathematics, but includes short biographies and some comparison of them).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and Michele Audin, &quot;Fatou, Julia, Montel: the Great Prize of Mathematical Sciences of 1918 and beyond&quot;, also several editions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2017-12-24T14:29:01.880" LastActivityDate="2017-12-24T14:29:01.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6832" PostTypeId="2" ParentId="6829" CreationDate="2017-12-24T15:14:46.967" Score="1" Body="&lt;p&gt;Googling I came up with&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fields and Particles: An Introduction to Electromagnetic Wave Phenomena and Quantum Physics&#xA;  Francis Bitter, Heinrich Adolf Medicus&#xA;  American Elsevier, 1956 - Science - 688 pages&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On page 167 there is a discussion of quanta and a mention of Taylor's paper saying that it wasn't appreciated at the time. I can't see the whole page cleanly since I was using snippets from Goggle Books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quoting part...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For example, if we use a photographic plate which we must obviously expose for a long time because of the low light intensity, will we observe interference fringes on it? Such experiments have been performed, the first one(42) as early as 1909 which, however, did not make a great impact on the physicists of those days. The first experiment that was carried out under good conditions was published in 1927(43). Monochromatic light of very low intensity from a helium gas discharge passed through a certain type of interferometer and was registered on a film. It clearly produced interference fringes. Later experiments (44) also confirmed that single quanta ended up predominantly in those regions&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(42)G. I. Taylor, Proc. Cambridge Phil. Soc. 15, 114 (1909)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(43) &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.30.644&quot; rel=&quot;nofollow noreferrer&quot;&gt;A. J. Dempster and H. F. Batho, Phys. Rev. 30, 644 (1927)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/F191g.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/F191g.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3107" LastEditorUserId="3107" LastEditDate="2017-12-26T16:39:28.427" LastActivityDate="2017-12-26T16:39:28.427" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6833" PostTypeId="2" ParentId="2934" CreationDate="2017-12-24T18:06:04.623" Score="3" Body="&lt;p&gt;Dirac was a mathematician above all who liked the beauty of mathematics. This is one of the reasons why he objected the ugliness of renormalization theories.&#xA;You should read Pretty Mathematics International Journal of Theoretical Physics, Vol. 21, Nos. 8//9, 1982 , where he essentially explains that he was playing with 2X2 matrices, quoting him&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Such papers are largely ignored by physicists in the field, though.&lt;/p&gt;&#xA;" OwnerUserId="6826" LastActivityDate="2017-12-24T18:06:04.623" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6834" PostTypeId="2" ParentId="2087" CreationDate="2017-12-24T21:29:36.070" Score="4" Body="&lt;p&gt;The original 1864 paper of James Clerk Maxwell, &quot;&lt;em&gt;A dynamical theory of the electromagnetic field&lt;/em&gt;&quot; contained the whole electrodynamics in the form of 20 nonvectorial equations in 20 variables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following this paper, efforts were made for casting these equations into a more systematic form. At those times, algebraic quaternions were the unique mathematical instrument available to that aim. J.C. Maxwell tried to work  with quaternions, but was not very successful. Anyway, such efforts were merely considered for the beauty of the gest, because the complete theory was already available from Maxwell's original paper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventually, Josiah Gibbs and Oliver Heavyside came up with vector analysis, and what today is called &quot;Maxwell's Equations&quot; is in fact a reformulation by Oliver Heavyside of the original equations proposed by J.C. Maxwell. So far for the engineering part of the story.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it has rarely been noticed that quaternions are the &quot;square root&quot; of an algebraic number identity: Leonhard Euler's 4-squares identity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;(a_0 + ia_1 + ja_2 + ka_3)\times(b_0 + ib_1 + jb_2 + kb_3)&#xA;&amp;amp;=  (a_0b_0 - a_1b_1 - a_2b_2 - a_3b_3)\\&#xA;&amp;amp;+ i(a_0b_1 + a_1b_0 + a_2b_3 - a_3b_2)\\&#xA;&amp;amp;+ j(a_0b_2 - a_1b_3 + a_2b_0 + a_3b_1)\\&#xA;&amp;amp;+ k(a_0b_3 + a_1b_2 - a_2b_1 + a_3b_0)&#xA;\end{align}$$  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The square of this product is, &#xA;$$(a\times b)\times (a\times b)'' = a\times(b\times b'')\times a''$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and with the following&#xA;$$q\times q'' = q''\times q = (q_0^2 + q_1^2 + q_2^2 + q_3^2)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have Leonhard Euler's 4-squares-identity:&#xA;$$\begin{align}&#xA;(v_0^2 + v_1^2 + v_2^2 + v_3^2)(w_0^2 + w_1^2 + w_2^2 + w_3^2) &#xA;&amp;amp;= (v_0w_0 - v_1w_1 - v_2w_2 - v_3w_3)^2\\&#xA;&amp;amp;+ (v_0w_1 + v_1w_0 + v_2w_3 - v_3w_2)^2\\&#xA;&amp;amp;+ (v_0w_2 - v_1w_3 + v_2w_0 + v_3w_1)^2\\&#xA;&amp;amp;+ (v_0w_3 + v_1w_2 - v_2w_1 + v_3w_0)^2&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Electrodynamics using quaternions has thus the advantage of being&#xA;metaphysically backed by an algebraic number identity: everything&#xA;is conserved under such transformation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A further advantage is the easy and transparent formulation.&#xA;The gradient, $\delta\times A$, of the vector potential $A$ is not a 2-form, but a vector, &#xA;as expected for a physical quantity supposed to represent a force or a movement,&#xA;having a direction in space:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\begin{align}&#xA;(\delta_0 + i\delta_1 + j\delta_2 + k\delta_3)\times(A_0 + iA_1 + jA_2 + kA_3) &#xA;&amp;amp;= (\delta_0A_0 - \delta_1A_1 - \delta_2A_2 - \delta_3A_3)\\&#xA;&amp;amp;+ i(\delta_0A_1 + \delta_1A_0 + \delta_2A_3 - \delta_3A_2)\\&#xA;&amp;amp;+ j(\delta_0A_2 + \delta_2A_0 - \delta_1A_3 + \delta_3A_1)\\&#xA;&amp;amp;+ k(\delta_0A_3 + \delta_3A_0 + \delta_1A_2 - \delta_2A_1)&#xA;\end{align}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Under Lorenz gauge, the first line vanishes identically,&#xA;and the remaining lines yield the electromagnetic field in&#xA;source $(E/c)$ and $\operatorname{curl} (B)$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\delta\times A = i(E_1/c + B_1) + j(E_2/c + B_2) + k(E_3/c+ B_3)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source and curl are the two parts of a 4-dimensional isoclinic&#xA;double-rotation around a point; a quaternion being an &#xA;operator for performing general isoclinic double-rotation &amp;amp; stretching&#xA;operations in 4-dimensional space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The square $(\delta\times A) \times (\delta\times A)''$ is equal to $(E⁄c)^2 + B^2$, i.e. there are no mixed terms involving $E$ and $B$, because both parts of an isoclinic double rotation are orthogonal to each other.&lt;/p&gt;&#xA;" OwnerUserId="6827" LastEditorUserId="6780" LastEditDate="2018-05-22T21:43:05.120" LastActivityDate="2018-05-22T21:43:05.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6835" PostTypeId="1" AcceptedAnswerId="6839" CreationDate="2017-12-24T23:59:11.097" Score="7" ViewCount="977" Body="&lt;p&gt;I think most of us know about the construction of the first atomic bomb at &lt;a href=&quot;https://nl.wikipedia.org/wiki/Los_Alamos_National_Laboratory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Los Alamos&lt;/a&gt;, with &lt;a href=&quot;https://en.wikipedia.org/wiki/J._Robert_Oppenheimer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Oppenheimer&lt;/a&gt; (who said he became &quot;The destroyer of worlds&quot;, which goes to show he regretted his participation; nevertheless he &lt;strong&gt;did&lt;/strong&gt; participate) in charge of an enormous complex where many, many brilliant physicists (including Feynman) were offered (ordered for participating in?) a well-paid job, housing, food and drink, etc. The project was initiated by Einstein after sending a letter to Roosevelt (correct me if I'm wrong), which seems to contradict his pacifist attitude. But that's not that relevant to my question. Which is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did it take such a long time (2-3 years) for all these men (and some women), working on that enormous complex, to construct an actual working device [the first test (called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trinity_(nuclear_test)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trinity&lt;/a&gt; nuclear test) hit the jackpot], while in principle you &quot;just&quot; have to smash together two masses of Plutonium below the critical mass, which after the smash have a mass above that mass? That &lt;strong&gt;was&lt;/strong&gt; known (i.e. in theory) at the time. Was it because it was the beginning of the atomic era, and there was still much to learn? Was it to prevent failure? I've read many times the Nazis were on the verge of constructing one too, and I suppose the Americans knew that too. So why not hurry a bit more? &quot;Luckily&quot;, the Americans were first, though there were two dropped on Japan since Germany had already surrendered. Even a third was planned to be thrown because there could be used three different elements in the bomb, and the Americans wanted to see how all three exploded. The second one, dropped on Nagasaki, was i.m.o. totally superfluous. By the way, the only relevant tag I could find was &quot;nuclear physics&quot;.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="1930" LastEditDate="2018-05-08T02:38:34.643" LastActivityDate="2020-04-12T20:42:55.067" Title="Why did it take an army of physicists, working on a huge complex, so long to produce a working nuclear device, while it seems so easy in theory?" Tags="|nuclear-physics|world-war-2|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="6837" PostTypeId="2" ParentId="6835" CreationDate="2017-12-25T09:30:57.217" Score="14" Body="&lt;p&gt;One of the big issues was isotope separation. Extracting uranium 235 from uranite was a huge problem that took quite an effort to overcome. In fact, just before the war started Niels Bohr believed that extracting enough uranium 235 to build a bomb would be an impossible task (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Margaret_Gowing&quot; rel=&quot;noreferrer&quot;&gt;Margaret Gowing&lt;/a&gt;'s &lt;em&gt;Niels Bohr and Nuclear Weapons&lt;/em&gt;, in &lt;em&gt;Niels Bohr: A Centenary Volume&lt;/em&gt; (Harvard University Press, 1987)).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suggest that you read Richard Rhodes' &lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2018-01-06T09:39:08.190" LastActivityDate="2018-01-06T09:39:08.190" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6838" PostTypeId="2" ParentId="6835" CreationDate="2017-12-25T12:57:52.993" Score="8" Body="&lt;p&gt;You should do some preliminary research before asking such a question. It was not so easy, even in theory. They did not know anything about Plutonium when they started, it simply did not exist. They had to make it, and study its properties etc. This was a slow process. The first bomb was based on Urainum rather then Plutonium, and again there was a huge problem to obtain the correct isotope of Uranium. There were many other difficult problems, or example how exactly you compress a piece of Plutonium to achieve critical mass. This is done with ordinary explosives, and the technology which does this (called implosion) had to be developed both in theory and in practice. These are only some of the most difficult problems they had. And it is actually amazing that they were able to do all of this so quickly. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-12-25T12:57:52.993" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6839" PostTypeId="2" ParentId="6835" CreationDate="2017-12-26T04:17:22.303" Score="12" Body="&lt;p&gt;Some of the most critical numbers needed are the reaction cross sections.  There was basically no serious way of calculating those beyond orders of magnitude.  The cross-sections had to be measured, and this is a tedious process, even with good sources and pure samples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, understanding (in an age where all computers were human) the hydrodynamics of the shock wave propagation in plasma bounded by a medium that is about to be blown to bits was a big headache, never mind the reflection of the shock waves to produce spherical compression.   The precise timings of the various physical phenomena, and how one step in the sequence affected later steps had to be understood. The thermodynamics alone is not exactly equilibrium stuff.  Indeed, the actual compression design was rather controversial as it required machining precision that was, at that time, exquisite.  Working out the actual geometry of the lenses, and getting rid of the imperfections, were also necessary breakthroughs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside, you can get a good survey of the math involved in thermonuclear (i.e. fusion) devices from the book &lt;a href=&quot;http://www.wlym.com/archive/fusion/book/1981ThermoBombBook.pdf&quot; rel=&quot;noreferrer&quot;&gt;The physical principles of themonuclear explosive devices&lt;/a&gt; by F. Winterberg.  This is not what you’re asking for but it will give you a sense that the ideas might not be so complicated, but the math remains challenging and the engineering to make it work is even more sophisticated.&lt;/p&gt;&#xA;" OwnerDisplayName="user6552" LastActivityDate="2017-12-26T04:17:22.303" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6840" PostTypeId="2" ParentId="6829" CreationDate="2017-12-26T17:28:11.827" Score="0" Body="&lt;p&gt;Searching Google books with &lt;a href=&quot;https://www.google.com/search?q=Taylor+Dempster+Batho&amp;amp;client=firefox-b-1&amp;amp;biw=984&amp;amp;bih=513&amp;amp;source=lnt&amp;amp;tbs=cdr%3A1%2Ccd_min%3A1%2F1%2F1900%2Ccd_max%3A12%2F31%2F1950&amp;amp;tbm=bks&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Taylor Dempster Batho&quot; from 1901 through 1950&lt;/a&gt; I came up with&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Archives de philosophie - Volumes 17-18 - Page 31, Beauchesne, 1947&lt;/p&gt;&#xA;&#xA;&lt;p&gt;En 1909 G. I. Taylor, puis en 1937 Dempster et Batho, utilisant la propriété des plaques photographiques d'accumuler indéfiniment l'action de la lumière, réussirent à photographier- des franges d'interférence avec des sources lumineuses si faibles qu'en tenant compte de l'intensité totale et du temps de pose on est obligé d'admettre, dans la théorie corpusculaire de la lumière, que les photons ont passé séparément, l'un après l'autre,, dans l'interféromètre. En pareil cas la production ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were a number of such hits in the mid to late 1950s. &lt;/p&gt;&#xA;" OwnerUserId="3107" LastActivityDate="2017-12-26T17:28:11.827" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6841" PostTypeId="2" ParentId="6829" CreationDate="2017-12-26T17:41:00.413" Score="4" Body="&lt;p&gt;It looks like the date when people started (incorrectly) interpreting Taylor as single-photon interference is 1927.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The dominant Bohr school said the EM field wasn't quantized, and nobody really took light quanta seriously until the BKS theory was disproved by Bothe and Geiger in 1925-6. As late as 1925, Slater (the &quot;S&quot; in BKS) wrote up a paper, Phys. Rev. 25, 395, which sounds like the canonical reference on BKS, and which has the first reference to Taylor that I was able to find in a citation index. Although the Slater paper is paywalled, I imagine that he manages to dismiss Taylor's result somehow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the 5th Solvay conference in 1927, there was a consensus for photons. (The word was coined in 1926.) 1927 was also the year of an experiment by Dempster and Batho, which was essentially a redo of Taylor under better conditions. The paper is paywalled (90 years later!), but the abstract is available, and they explicitly interpret their result as single-photon interference. They cite Taylor, and this is only the second reference to Taylor that I could find in a citation index.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems that 1927 was when people started to notice Taylor as important and want to interpret it as single-photon interference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Re the related question of when it was realized that Taylor could &lt;em&gt;not&lt;/em&gt; correctly be interpreted as supporting single-photon interference, here's what I dug up. It appears that there was some controversy and confusion about the correct quantum interpretation of the Hanbury-Twiss interferometer. There was a paper on this in 1956, E.M. Purcell, Nature 178, 1449 (1956), in which he apparently correctly analyzed the Bose-Einstein correlations that cause bunching. But it doesn't seem to have been widely understood for another couple of decades, even among specialists, that these same correlations also invalidate the interpretation of Taylor as single-photon interference. There is a &lt;a href=&quot;https://archive.org/details/InterferenceOfPhotons&quot; rel=&quot;nofollow noreferrer&quot;&gt;1959 PSSC film&lt;/a&gt; that reenacts Taylor with a photomultiplier tube and presents it as single-photon interference. As late as a 1978 review article by Pipkin, even some experts don't seem to have correctly applied this to the historical experiments like Taylor. However, Clauser, Phys. Rev. D 9, 853–860 (1974) seems to have pointed out that bunching invalidated the interpretation. (I'm not 100% certain about all of this, because a lot of the papers are paywalled.)&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2017-12-26T17:54:30.417" LastActivityDate="2017-12-26T17:54:30.417" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6842" PostTypeId="2" ParentId="6835" CreationDate="2017-12-26T18:32:37.963" Score="5" Body="&lt;p&gt;I can only recommend R. Rhodes, &lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt; once more and very strongly. It is also instructive to note that although Leo Szilard [sp] was inspired to understand and fight for the chain reaction concept and also held the patent it was years before he or anyone else could figure out what material could be used to demonstrate the concept. Now, of course, there is a whole declension of what elements turn into other elements as a sun burn through its life. &lt;/p&gt;&#xA;" OwnerUserId="6834" LastActivityDate="2017-12-26T18:32:37.963" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6843" PostTypeId="2" ParentId="6835" CreationDate="2017-12-26T21:01:28.463" Score="11" Body="&lt;p&gt;&lt;strong&gt;(1) Weapon-grade uranium is hard to make&lt;/strong&gt; - it took almost a year to separate 64 kg of weapon-grade uranium for the bomb in the Y-12 Oak Ridge facility, so &quot;Little boy&quot; bomb was not even tested (they had no spare). And because of the &quot;Little boy&quot; bomb design, most of that expensive uranium was NOT involved in the nuclear explosion (bomb exploded into sub-critical mass before most of the uranium was involved in the chain reaction).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How hard is to make the weapon-grade uranium: in &lt;a href=&quot;https://en.wikipedia.org/wiki/Y-12_National_Security_Complex&quot; rel=&quot;nofollow noreferrer&quot;&gt;Y-12 complex in Oak Ridge&lt;/a&gt; electromagnetic isotope separation, they used 12 300 tonnes (1 tonne = 1000 kg = 2204 lb, $300 million worth) of silver (copper as strategic material was not available in necessary quantities) converted to wires for the coils. After manufacturing the wires, facility was burned down to extract spilled silver from the ashes. Less than 0.036% was lost. As you can see, Y-12 is extremely complex process, resulting in building 1 bomb per year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(2)&lt;/strong&gt; But to have &lt;strong&gt;more feasible nuclear bomb (&quot;Fat Man&quot;), you need plutonium&lt;/strong&gt;, which has to be first isolated (in 1940) and researched, then, &lt;strong&gt;industrial amounts of sufficiently pure specific isotope of plutonium&lt;/strong&gt; (Pu-239, because not all isotopes are equally good for chain reaction - one of the problems they need to figure out and to overcome) has to be  manufactured in a &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Breeder_reactor&quot; rel=&quot;nofollow noreferrer&quot;&gt;breeder nuclear reactor&lt;/a&gt;&lt;/strong&gt;, which is also an easy to bomb target (and extremely &lt;strong&gt;expensive to build&lt;/strong&gt;).           &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(3) Shaped explosion:&lt;/strong&gt; Scientist were well aware of the inefficiency of the &quot;Little boy&quot; design, therefore they used &lt;strong&gt;shaped explosion&lt;/strong&gt; to create superctitical mass of 11 kg of Plutonium - which has to be created from industry-grade (not weapon-grade) uranium in a breeder reactors, so it is much cheaper and faster process (it took about 1 month to breed enough Plutonium for 1 bomb). But you need to build also breeder reactors to manufacture the plutonium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because scientists were not sure if plutonium bomb would even work (decision could not be made based on the minute amount of the plutonium isotopes available), Manhattan Projects decided to invest in &lt;strong&gt;both&lt;/strong&gt; approaches to save time and halve the risk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(4) Calculating explosion-shaped charge&lt;/strong&gt; used in &lt;a href=&quot;https://en.wikipedia.org/wiki/Fat_Man&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fat Man&lt;/a&gt; design is substantially more complex design (as compared to just two pieces of metal hitting each other as in Little boy): requires &lt;strong&gt;precise synchronization of multiple explosions&lt;/strong&gt; (to shape explosion inward: hundred nanoseconds difference can make it or break it), so mathematical models of such explosions had to be developed, and calculated (by hand), therefore it was tested - &lt;a href=&quot;https://en.wikipedia.org/wiki/Trinity_(nuclear_test)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trinity&lt;/a&gt; - and they could not use computer simulation for explosion because computers were not invented yet, all calculation were made by hand. And wiring to deliver signals to start shaped explosions needed to be invented and tested.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;(5) Bomb delivery&lt;/strong&gt; is another big problem: Developing &lt;a href=&quot;https://en.wikipedia.org/wiki/Boeing_B-29_Superfortress&quot; rel=&quot;nofollow noreferrer&quot;&gt;B-29&lt;/a&gt; &quot;strategic&quot; bomber was &lt;strong&gt;even more expensive than the Manhattan project&lt;/strong&gt;. It was about twice heavier than &quot;standard&quot; heavy bomber, &lt;a href=&quot;https://en.wikipedia.org/wiki/Boeing_B-17_Flying_Fortress&quot; rel=&quot;nofollow noreferrer&quot;&gt;B-17&lt;/a&gt; and so substantially more complex. Getting enough power from just 4 engines, and &lt;strong&gt;fully pressurized cabin&lt;/strong&gt; (to be able of long flights in high altitude, where enemy fighters cannot reach it), was especially hard. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unexpected part that it at the end, &lt;strong&gt;nuclear bombing not only saved lives&lt;/strong&gt; (it prevented expected 500K US casualties and tens of millions expected Japanese casualties during the planned invasion &lt;a href=&quot;https://en.wikipedia.org/wiki/Operation_Downfall&quot; rel=&quot;nofollow noreferrer&quot;&gt;Operation Downfall&lt;/a&gt; - bigger than D-Day), but also &lt;strong&gt;saved money&lt;/strong&gt;: because it shortened war by several months, war cost were about 1 Billion per week, and cost of B-29 and Manhattan Project were in total only about $4B.&lt;/p&gt;&#xA;" OwnerUserId="4386" LastEditorUserId="4386" LastEditDate="2020-04-12T20:42:55.067" LastActivityDate="2020-04-12T20:42:55.067" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="6844" PostTypeId="1" CreationDate="2017-12-27T02:41:52.903" Score="1" ViewCount="247" Body="&lt;p&gt;It is stated that it is included in his book &lt;em&gt;Discourse on the Method&lt;/em&gt;. But I cannot even find the keyword 'rainbow' in this book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;so where is it?  &lt;/p&gt;&#xA;" OwnerUserId="2641" LastActivityDate="2017-12-28T09:33:28.717" Title="Where can I find Descartes' account of the rainbow?" Tags="|physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6845" PostTypeId="2" ParentId="6844" CreationDate="2017-12-27T06:07:55.727" Score="2" Body="&lt;p&gt;Descartes &lt;em&gt;Discourse on the Method&lt;/em&gt; was published with three additional essays: &lt;em&gt;Dioptrics, Meteorology, Geometry&lt;/em&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is difficult to find a PDF of the original meteorology essay complete with its many hand-drawn images,  but there are a number of summaries supplementing  the main body of &lt;em&gt;Discourse&lt;/em&gt;.  For example, see &lt;a href=&quot;http://www.rlwclarke.net/Theory/SourcesPrimary/DescartesDiscourseonMethod.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 61/161&lt;/a&gt; from &quot;rlwclarke.net&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://eo.ucar.edu/rainbows/&quot; rel=&quot;nofollow noreferrer&quot;&gt;UCAR Centre for Science Education&lt;/a&gt; includes the following details and image:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Descarte describes how he held up a large sphere in the sunlight and looked at the sunlight reflected in it. He wrote &quot;I found that if the sunlight came, for example, from the part of the sky which is marked AFZ and my eye was at the point E, when I put the globe in position BCD, its part D appeared all red, and much more brilliant than the rest of it; and that whether I approached it or receded from it, or put it on my right or my left, or even turned it round about my head, provided that the line DE always made an angle of about forty-two degrees with the line EM, which we are to think of as drawn from the center of the sun to the eye, the part D appeared always similarly red; but that as soon as I made this angle DEM even a little larger, the red color disappeared; and if I made the angle a little smaller, the color did not disappear all at once, but divided itself first as if into two parts, less brilliant, and in which I could see yellow, blue, and other colors ... When I examined more particularly, in the globe BCD, what it was which made the part D appear red, I found that it was the rays of the sun which, coming from A to B, bend on entering the water at the point B, and to pass to C, where they are reflected to D, and bending there again as they pass out of the water, proceed to the point&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0YfKP.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0YfKP.gif&quot; alt=&quot;Rainbow image from Meteorology&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This quotation illustrates how the shape of the rainbow is explained. To simplify the analysis, consider the path of a ray of monochromatic light through a single spherical raindrop. Imagine how light is refracted as it enters the raindrop, then how it is reflected by the internal, curved, mirror-like surface of the raindrop, and finally how it is refracted as it emerges from the drop. If we then apply the results for a single raindrop to a whole collection of raindrops in the sky, we can visualize the shape of the bow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lkjnZ.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lkjnZ.gif&quot; alt=&quot;Shape of Rainbow&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2017-12-27T06:07:55.727" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6846" PostTypeId="1" AcceptedAnswerId="6847" CreationDate="2017-12-27T07:45:36.180" Score="1" ViewCount="250" Body="&lt;p&gt;Wikipedia says: First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence ${\displaystyle \forall P\,\forall x(x\in P\lor x\notin P)} \forall P\,\forall x(x\in P\lor x\notin P)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who has first published this doctrin? And why has it been accepted?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2017-12-28T19:19:34.120" LastActivityDate="2017-12-28T19:19:34.120" Title="Why second-order logic?" Tags="|mathematical-logic|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6847" PostTypeId="2" ParentId="6846" CreationDate="2017-12-27T08:59:33.863" Score="3" Body="&lt;p&gt;Modern logic began with Boole, Peirce and &lt;a href=&quot;https://plato.stanford.edu/entries/frege/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frege&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Frege's system (1879) was the first &quot;fully formalized&quot; system of mathematical logic and was a &lt;a href=&quot;https://plato.stanford.edu/entries/logic-higher-order/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Second- and Higher-order system of Predicate Logic&lt;/a&gt;: quantification over predicate of any order was allowed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Subsequently, the first modern math lofic textbook: Hilbert &amp;amp; Ackermann's &lt;a href=&quot;https://en.wikipedia.org/wiki/Principles_of_Mathematical_Logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grundzüge der Theoretischen Logik&lt;/a&gt; (1928) codified the distinction between first-order: quantification over individuals but not predicates, and second- and higher-order.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2017-12-27T08:59:33.863" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6848" PostTypeId="1" CreationDate="2017-12-27T14:11:02.223" Score="0" ViewCount="43" Body="&lt;p&gt;(Apologies if this isn't the right place for such a post. I see lots of information out there for math/science students wanting to do research, but haven't seen anything about doing things on the &quot;soft&quot; side, like history and pedagogy.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm an undergrad math student (will have one more term left after the summer) and I'm hoping to get some advice on summer plans. While I really like math, and I'm considering grad school, I've always been really interested in science education, science in pop culture, and the history of science. I love the book &lt;em&gt;A Short History of Nearly Everything&lt;/em&gt; and Youtube channels like Vsauce. Working as a science writer or museum curator would probably be a dream job for me. I'm hoping to do something in that realm in the future. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of any good ways to get involved in this sort of thing? It seems like people in this arena come from a pretty wide range of backgrounds. When I think about the parts of math and science that I really like, I keep coming back to the historical, cultural, philosophical, and pedagogical aspects, so something along those lines would be awesome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm living in Canada right now, but I'd love to travel so I'm open to any location. I also have good grades (just under 4.0), if that matters for anything. That said, I'm really open to anything, academia or otherwise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;~~~&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If anyone has any suggestions I would be very grateful. Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="6837" LastActivityDate="2017-12-27T14:11:02.223" Title="Are there any undergraduate research/internship opportunities in math/science education/history?" Tags="|mathematics|philosophy-of-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6850" PostTypeId="2" ParentId="6844" CreationDate="2017-12-28T09:33:28.717" Score="3" Body="&lt;p&gt;Here is a link to the original French version, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/btv1b86069594/f334.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;De l'arc-en-ciel&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastActivityDate="2017-12-28T09:33:28.717" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6852" PostTypeId="2" ParentId="6790" CreationDate="2017-12-29T04:52:03.110" Score="2" Body="&lt;p&gt;According to the &lt;a href=&quot;https://fr.wikipedia.org/wiki/Fonction_digamma&quot; rel=&quot;nofollow noreferrer&quot;&gt;French wikipedia page &lt;em&gt;Fonction digamma&lt;/em&gt;&lt;/a&gt;, it was James Stirling (1730) who first introduced and studied the digamma function, denoting it with the Greek letter digamma (upper case) $\digamma$.  However, this claim is marked as requiring a citation. After noting its subsequent study by Legendre, Poisson and Gauss around 1810, the article then simply states &quot;Elle est désormais le plus souvent notée par la lettre $\psi$ (psi minuscule)&quot; - &quot;It is now most often denoted by the letter $\psi$&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the gamma and digamma functions were introduced they were generalised as the full sequence of polygamma functions.  In the context of the polygamma functions it is obviously desirable to have a single functional denotation - i.e., &#xA;$$\psi^{(n)}(z) := \frac{d^{m+1}}{dz^{(m+1)}} \text{ ln } \Gamma (z)$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $\psi^{(0)}$ denotes the digamma function, $\psi^{(1)}$ the trigamma function, etc...  In other words, in the context of the sequence of polygamma functions, there is not reason for the digamma function to have a special designation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If is not clear why &lt;em&gt;psi&lt;/em&gt; was chosen, but it seems reasonable to assume that this is why the special $\digamma$ &lt;em&gt;Digamma&lt;/em&gt; designation introduced by Stirling fell out of usage.  It was simply a matter of notational elegance in the general case.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2017-12-30T17:22:32.950" LastActivityDate="2017-12-30T17:22:32.950" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6853" PostTypeId="1" AcceptedAnswerId="6858" CreationDate="2017-12-29T09:35:02.807" Score="1" ViewCount="1472" Body="&lt;p&gt;Consider the three &quot;main&quot; trigonometric functions, sine cosine and tangent; whose notations are sin(x); cos(x); tan(x).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there instances of alternative notations being used for these particular functions?&#xA;The only one I could find so far was the &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_trigonometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;use of &quot;tang&quot; by Leonard Euler&lt;/a&gt; for even when referencing trigonometry from Indian Babylonian, Egyptian or Chinese antic cultures, wikipedia uses the standard notation.&lt;/p&gt;&#xA;" OwnerDisplayName="user6769" LastEditorDisplayName="user6769" LastEditDate="2017-12-29T12:26:24.383" LastActivityDate="2017-12-29T14:10:46.817" Title="Instances of alternative notation being used for the trigonometric functions?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="6854" PostTypeId="1" CreationDate="2017-12-29T10:33:39.280" Score="1" ViewCount="69" Body="&lt;p&gt;This is a cross-post from mathstack:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/2584003/gentzen-and-computer-science?noredirect=1#comment5333947_2584003&quot;&gt;https://math.stackexchange.com/questions/2584003/gentzen-and-computer-science?noredirect=1#comment5333947_2584003&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to learn a bit about the connections between Gentzen's discoveries in the 30's related to proof theory and, in particular, his sequent calculus, and the later development of computer languages. Could anyone elaborate a bit on that? I am particularly interested in knowning whether one can find in Gentzen some predecessor of a pointer to a memory address. If so, in which specific form?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="6845" LastEditorUserId="6845" LastEditDate="2017-12-29T12:50:16.983" LastActivityDate="2017-12-29T12:50:16.983" Title="Gentzen and computer science" Tags="|terminology|philosophy-of-science|mathematical-logic|computer-science|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6855" PostTypeId="5" CreationDate="2017-12-29T11:55:39.013" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2017-12-29T11:55:39.013" LastActivityDate="2017-12-29T11:55:39.013" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6856" PostTypeId="4" CreationDate="2017-12-29T11:55:39.013" Score="0" Body="Questions involving topology, the mathematical study of properties of spaces preserved by continuous maps." OwnerDisplayName="user6769" LastEditorUserId="26" LastEditDate="2018-01-13T13:06:23.340" LastActivityDate="2018-01-13T13:06:23.340" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6857" PostTypeId="1" AcceptedAnswerId="6859" CreationDate="2017-12-29T13:22:42.040" Score="3" ViewCount="508" Body="&lt;p&gt;My question refers to Gauss's heliotrope - an optical device for projecting sunlight to a distance by a reflecting mirror, that was used in geodetic surveys in order to mark reference points. The problem is (and Gauss was aware of this problem) that since the index of refraction changes with the alttitude above sea, than according to snell's law the rays of light are not straight but rather curved. So my question is  how did geodesits make a practical use of hte heliotrope? and did Gauss originally face the problem? &lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2018-03-30T22:54:43.213" Title="How did Gauss overcome the problem of curvature of sunlight rays when he developed his heliotrope?" Tags="|biographical-details|optics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6858" PostTypeId="2" ParentId="6853" CreationDate="2017-12-29T14:10:46.817" Score="4" Body="&lt;h2&gt;Notation in indian trigonometry&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The Indian mathematician and astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Aryabhata&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aryabhata&lt;/a&gt; (476-550 CE) was the author of first &lt;a href=&quot;https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table&quot; rel=&quot;nofollow noreferrer&quot;&gt;sine table&lt;/a&gt; ever.&#xA;It was a  stanza composed of sanscrit &lt;a href=&quot;https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa_numeration&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alphanumerals&lt;/a&gt; and words, which contained relations between arcs and half-chords of a circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So in this way of writing, notation for sine would just be it's name.&#xA;The sanscrit word for half-chord is &lt;em&gt;ardha-jyā&lt;/em&gt; what we would commonly see today as the sine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/40JNq.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/40JNq.jpg&quot; alt=&quot;litteral meaning&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the word &lt;em&gt;jyā&lt;/em&gt; ( ज्या in sanscrit ) came to mean &lt;em&gt;half-chord&lt;/em&gt; when &lt;em&gt;ardha-jyā&lt;/em&gt; was shortened :&#xA;&lt;a href=&quot;https://i.stack.imgur.com/5yLrB.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5yLrB.jpg&quot; alt=&quot;technical meaning&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the notations would be:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;   sine : ज्या (jyā)&#xA;   cosine : कोटिज्या (koti-jyā)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;jyā&lt;/em&gt; passed into Arabic as &lt;em&gt;jība&lt;/em&gt; and was corrupted into &lt;em&gt;jaib&lt;/em&gt; which means &quot;bosom&quot; and translates to &lt;em&gt;sine&lt;/em&gt; in Latin&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Feynman Notation&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Apparently, during his younger years, The physicist Richard Feynman invented his own notation for the trig functions (which was never really used afterwards) :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mC0rT.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mC0rT.png&quot; alt=&quot;Feynman trig notation&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;sources : &#xA;    &lt;a href=&quot;https://tex.stackexchange.com/questions/274463/feynman-trig-notation-creating-custom-characters&quot;&gt;https://tex.stackexchange.com/questions/274463/feynman-trig-notation-creating-custom-characters&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.physicsforums.com/threads/feynmans-trig-notations.78087/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.physicsforums.com/threads/feynmans-trig-notations.78087/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user6769" LastActivityDate="2017-12-29T14:10:46.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6859" PostTypeId="2" ParentId="6857" CreationDate="2017-12-29T14:25:08.253" Score="2" Body="&lt;p&gt;In geodesy, one usually looks horizontally, or almost horizontally, so the ray travels in the same layer of the atmosphere, where the temperature and pressure &#xA;are almost constant, thus refraction is negligible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sun only serves as the source of light in this case, so the refraction of the ray coming from&#xA;the Sun to the devise does not matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Refraction is not negligible for astronomic observations (which were also used in geodesy, for the measurements in which the Sun position is relevant), to take it into account, the tables of refraction are used, but the true refraction may depend on the state of the atmosphere and be different from the tables value. This is one of the major difficulties with astronomical measurements. (Gauss certainly was aware of this problem, it is well known since the antiquity, and the most precise tables of refraction in Gauss's time are due to his friend Bessel).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are several possible uses of the heliotrope, so one has to specify the exact setting to discuss whether refraction is relevant or not in this setting.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2017-12-29T14:25:08.253" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6860" PostTypeId="2" ParentId="6857" CreationDate="2017-12-29T15:38:48.257" Score="4" Body="&lt;p&gt;In 1825, Gauss introduced the coefficient of refraction &lt;strong&gt;k&lt;/strong&gt;, which represents a way to quantify terrestrial refraction. He defined it as the ratio of the radius of the earth &lt;strong&gt;R&lt;/strong&gt; and radius of curvature  &lt;strong&gt;r&lt;/strong&gt; of light ray (the line of sight). He found, on the basis of reciprocal vertical angle measurements near hannover, an average value of 0.13 for this coefficient. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, in geodesy there is a mathematical trick of dealing with the curvature of line of sight - it's called &lt;strong&gt;effective earth radius&lt;/strong&gt; convention, and it's very useful in calculating horizons seen from an heightened point. It's defined mathematicaly as:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ R_{eff} = \frac {{R_{earth}}}{{1 - k}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By replacing earth with an imaginary earth with greater radius, one can make calculations from a new reference system. In order to direct the heliotrope ray to a distant observer, one has to calculate angles on the basis of the horizontal distance &lt;strong&gt;x&lt;/strong&gt; between two points &lt;strong&gt;A&lt;/strong&gt; and &lt;strong&gt;B&lt;/strong&gt;, their heights $h_1$ and $h_2$, and the effective earth radius $R_{eff}$.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2017-12-29T19:17:45.803" LastActivityDate="2017-12-29T19:17:45.803" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6861" PostTypeId="2" ParentId="5322" CreationDate="2017-12-30T05:24:23.130" Score="1" Body="&lt;p&gt;Given enough time and resources, the Germans would have produced a bomb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What held them back was firstly that they were persecuting a large section of their existing scientific and engineering community (many literally going up the chimney) - either for their political views (socialists and communists) or because of their ethnic background (Jews). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Allied powers on the other hand were mopping up a lot of that German scientific community and were not generally expelling or persecuting anyone (and certainly not systematically killing).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The effect of the Nazi regime on scientific morale was also probably a factor. They had reasonably committed scientists like Von Braun, but not everybody was ideologically reconciled, and would have seen many friends and associates disappeared or forced to leave the country (the Nazi regime was promoted mainly to suppress growing socialist support amongst the populace in Germany).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whereas the Allies did not seem to have the same ideological problem at the time - the majority were reasonably unified in their opposition to the Nazis, moreso as time went on. That would have been accentuated in scientific circles, surely, by the fact of working alongside refugee German scientists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A third problem is that of resources. American industry was booming whilst German industry was being smashed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In places like Britain where industry was also being pummelled, the economy was extensively nationalised at the outbreak of war in 1939 in order to organise and control it efficiently for war, whereas the Germans continued with private sector production until well into the 40s (prompted only because they were falling onto the back foot economically by then).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By time they might have wanted a bomb, they had decimated their scientific community (in numbers and morale), their economy was under siege, and there were too many competing demands (against a speculative nuclear bomb project) for the remaining resources, in a war that by that time was going against them.&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2017-12-30T05:24:23.130" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6862" PostTypeId="2" ParentId="6790" CreationDate="2017-12-30T08:33:30.940" Score="11" Body="&lt;p&gt;I believe it’s because this function was used, and denoted “psi”, much before it got a name.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, it looks like &lt;span class=&quot;math-container&quot;&gt;$(\log\circ\,\Pi)'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\log\circ\,\Gamma)'$&lt;/span&gt; first occur in &lt;strong&gt;Euler&lt;/strong&gt; (&lt;a href=&quot;http://eulerarchive.maa.org/pages/E212.html&quot; rel=&quot;noreferrer&quot;&gt;1755&lt;/a&gt;, &lt;a href=&quot;http://www.e-rara.ch/zut/content/pageview/1174761&quot; rel=&quot;noreferrer&quot;&gt;pp. 797-801&lt;/a&gt;; &lt;a href=&quot;http://eulerarchive.maa.org/pages/E368.html&quot; rel=&quot;noreferrer&quot;&gt;1769&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=lAQZAAAAYAAJ&amp;amp;pg=RA1-PA17&quot; rel=&quot;noreferrer&quot;&gt;p. 17&lt;/a&gt;), resp. Legendre (&lt;a href=&quot;https://archive.org/stream/mmoiresdelinst10inst#page/n543&quot; rel=&quot;noreferrer&quot;&gt;1810, p. 502&lt;/a&gt;), with no special name or notation. Surveys like Brunel (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?18.0440.01&quot; rel=&quot;noreferrer&quot;&gt;1886&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/monographiedela00brungoog#page/n68&quot; rel=&quot;noreferrer&quot;&gt;p. 58&lt;/a&gt;; &lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?30.0272.04&quot; rel=&quot;noreferrer&quot;&gt;1899&lt;/a&gt;, &lt;a href=&quot;http://archive.org/stream/p1n1encyklopdied02akaduoft#page/162&quot; rel=&quot;noreferrer&quot;&gt;p. 162&lt;/a&gt;) or Jensen (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1503576&quot; rel=&quot;noreferrer&quot;&gt;1916, p. 140&lt;/a&gt;) agree that they are denoted &lt;span class=&quot;math-container&quot;&gt;$\varPsi$&lt;/span&gt; after &lt;strong&gt;Gauss&lt;/strong&gt; (&lt;a href=&quot;https://archive.org/stream/bub_gb_9Iyw570bQYQC#page/n220&quot; rel=&quot;noreferrer&quot;&gt;1813, p. 34&lt;/a&gt;), resp. &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; after — &lt;em&gt;unclear whom&lt;/em&gt;. (Clausen (&lt;a href=&quot;https://books.google.com/books?id=a59EAAAAcAAJ&amp;amp;pg=PA166&quot; rel=&quot;noreferrer&quot;&gt;1858, p. 169&lt;/a&gt;) or Bertrand (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?02.0298.01&quot; rel=&quot;noreferrer&quot;&gt;1870&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/traitecalculdiff02bertrich#page/n267&quot; rel=&quot;noreferrer&quot;&gt;p. 252&lt;/a&gt;) might be early examples.) In any case, they are still unnamed.&lt;/p&gt;&#xA;&lt;p&gt;Instead, as one learns in Pearson (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?48.0627.01&quot; rel=&quot;noreferrer&quot;&gt;1922&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/cu31924001444870#page/n11&quot; rel=&quot;noreferrer&quot;&gt;p. viii&lt;/a&gt;), Davis (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?59.1220.01&quot; rel=&quot;noreferrer&quot;&gt;1933&lt;/a&gt;, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015070366862&amp;amp;num=277&quot; rel=&quot;noreferrer&quot;&gt;p. 277&lt;/a&gt;; &lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?61.1337.03&quot; rel=&quot;noreferrer&quot;&gt;1935&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/in.ernet.dli.2015.212087#page/n23&quot; rel=&quot;noreferrer&quot;&gt;p. 9&lt;/a&gt;; &lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=1563067&quot; rel=&quot;noreferrer&quot;&gt;1935, p. 243&lt;/a&gt;), or Jordan (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=441&quot; rel=&quot;noreferrer&quot;&gt;1939&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=3RfZOsDAyQsC&amp;amp;pg=PA58&quot; rel=&quot;noreferrer&quot;&gt;p. 58&lt;/a&gt;), the name “digamma” and notation &lt;span class=&quot;math-container&quot;&gt;$\digamma$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\varPsi$&lt;/span&gt; arose for the first time on p. 5 of&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Pairman, Eleanor&lt;/em&gt;, &lt;a href=&quot;https://archive.org/details/cu31924001468416&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Tables of the digamma and trigamma functions. (Tracts for computers, edited by Karl Pearson, Nr. 1.)&lt;/strong&gt;&lt;/a&gt;, Cambridge: University Press, 19 S (1919). &lt;a href=&quot;https://zbmath.org/?q=an:47.0510.15&quot; rel=&quot;noreferrer&quot;&gt;ZBL47.0510.15&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/vNxyz.png&quot; alt=&quot;pairman&quot; /&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In short, I think &lt;strong&gt;Pairman&lt;/strong&gt; won her bet to name the yet-unnamed function — but failed to displace the existing “psi” notation. (The only adopters I could find are Jeffreys &amp;amp; Jeffreys (&lt;a href=&quot;http://www.ams.org/mathscinet-getitem?mr=19678&quot; rel=&quot;noreferrer&quot;&gt;1946&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/methodsofmathema031187mbp#page/n478&quot; rel=&quot;noreferrer&quot;&gt;§15.04&lt;/a&gt;).)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note added:&lt;/strong&gt; On the other hand, Legendre in his &lt;em&gt;Exercices&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/exercicescalculi01legerich/page/n24&quot; rel=&quot;noreferrer&quot;&gt;1811, p. 19&lt;/a&gt;) famously divided elliptic integrals into 3 kinds to be called &lt;em&gt;Nome, Epinome, Paranome&lt;/em&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;Ϝ=\int\frac{d\varphi}{\Delta},\qquad&#xA;\mathrm E = \int\Delta\,d\varphi,\qquad&#xA;\Pi= \int\frac{d\varphi}{(1+n\sin{\!}^2\varphi)\Delta}&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\Delta=\sqrt{1-c^2\sin{\!}^2\varphi}$&lt;/span&gt;. When these words didn’t catch on, Verhulst (&lt;a href=&quot;https://archive.org/details/bub_gb_WS8LAAAAYAAJ/page/n11&quot; rel=&quot;noreferrer&quot;&gt;1841, p. v&lt;/a&gt;) justified an attempt to rename them &lt;em&gt;digamma, epsilon, kappa&lt;/em&gt; (notation &lt;span class=&quot;math-container&quot;&gt;$\mathrm{dig}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{eps}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{kap}$&lt;/span&gt;) with&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1º The letter Ϝ, used by Legendre to denote the function &lt;span class=&quot;math-container&quot;&gt;$\int\frac{d\varphi}\Delta$&lt;/span&gt;, bears the Greek name &lt;em&gt;δίγαμμα&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Digamma#Computer_encodings&quot; rel=&quot;noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$$\Huge\style{font-family:Arial}{\text{Ϝ ϝ  }}$$&lt;/span&gt;&lt;/a&gt;&lt;/h2&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-12T12:42:45.033" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6863" PostTypeId="1" CreationDate="2017-12-30T09:30:00.713" Score="5" ViewCount="411" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law#Notable_derivations_of_Amp%C3%A8re%27s_force_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; credits this to Maxwell. This derivation can be found in Maxwell's &lt;a href=&quot;https://archive.org/stream/electricandmag02maxwrich#page/n177/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Treatise on Electricity and Magnetism&lt;/em&gt; vol. 2, part 4, ch. 2 (§§502-527)&lt;/a&gt;. I went through the derivation and found two self cancelling mistakes made by Maxwell. Those mistakes were corrected by J.J. Thomson in his edition of Maxwell's treatise.&lt;/p&gt;&#xA;&lt;p&gt;The fact that Maxwell made some mistakes and that the final result is unaffected is itself an indication that someone else before Maxwell originally made that derivation and Maxwell simply took this derivation from someone else (although he doesn't mention in his treatise from where he took this derivation).&lt;/p&gt;&#xA;&lt;p&gt;So was it really Maxwell who originally made this derivation of general force law equation or was it someone else?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT: (@Geremia)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Ampere derived in 1820s the force between current elements. He made an assumption that it was along the line joining the two elements. A few scientists after Ampere pointed out that current elements have direction as well and hence there is not a solid reason on why force between current elements would be like other forces in nature. In 1845, Hermann Grassmann derived another force law (which is taught today in schools). The force law of Grassmann doesn't obey Newton's third law even in the weak form.&lt;/p&gt;&#xA;&lt;p&gt;In the year 1873, Maxwell published his treatise. In that treatise, there is a derivation (Vol 2, Article 502-527) of general force law equation. The general force law equation says there can be infinite valid force laws as far as source circuit is closed. The general force law equation is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$d^{2}\vec{F}=kII'dsds'\left[\left(\frac {1}{r^{2}}\left({\frac {\partial r}{\partial s}}{\frac {\partial r}{\partial s'}}-2r{\frac {\partial ^{2}r}{\partial s\partial s'}}\right) +r{\frac {\partial ^{2}Q}{\partial s\partial s'}}\right)(\hat{r})-{\frac {\partial Q}{\partial s'}}(\hat{s})+{\frac {\partial Q}{\partial s}}(\hat{s'})\right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\hat{r}$&lt;/span&gt; is a unit vector pointing from &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$s'$&lt;/span&gt; (field circuit to source circuit)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is any function of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. But we must be careful while assuming the value of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; because we &lt;strong&gt;should&lt;/strong&gt; give same form for &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s'$&lt;/span&gt; coordinates.&lt;/p&gt;&#xA;&lt;p&gt;This general force law equation can be found in article 525 of Maxwell's treatise. Thus I am only asking who originally made this &amp;quot;general force law equation&amp;quot; (and &lt;strong&gt;not&lt;/strong&gt; Ampere's original force law).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Wikipedia call's it &lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law#Notable_derivations_of_Amp%C3%A8re%27s_force_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxwell's 1873 derivation&lt;/a&gt;. As explained in my original post, it doesn't seem to me that it was Maxwell who originally wrote it down.&lt;/p&gt;&#xA;&lt;p&gt;Please tell if it was really Maxwell or someone else?&lt;/p&gt;&#xA;" OwnerUserId="5313" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-01T13:03:39.007" Title="Who originally derived the general force law equation of force between current elements?" Tags="|physics|theoretical-physics|electromagnetism|mathematical-physics|electricity|" AnswerCount="1" CommentCount="6" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6865" PostTypeId="2" ParentId="6863" CreationDate="2017-12-30T20:07:20.557" Score="8" Body="&lt;p&gt;Ampère did. &lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/force/index-en.php&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Ampère's force law&lt;/strong&gt;&lt;/a&gt; (&lt;a href=&quot;https://hsm.stackexchange.com/q/5059/232&quot;&gt;not to be confused with one of Maxwell's equations, &lt;em&gt;&quot;Ampère&quot;'s circuital law&lt;/em&gt;, which Ampère never wrote down, as Ampère didn't deal with the field concept&lt;/a&gt;), written in modern vector notation, gives the force that current elements $I_1 d\vec {\ell }_1$ and&#xA;$I_2 d\vec {\ell }_2$ exert on one another to be:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$d^2\vec{F_{21}^A} = - \frac{\mu _0 }{4\pi }I_1 I_2 \frac{\hat {r}_{12} }{r_{12}^2 }\left[2(d\vec {\ell }_1 \cdot d\vec {\ell }_2) - 3({\hat {r}_{12} \cdot d\vec {\ell }_1 })({\hat {r}_{12} \cdot d\vec {\ell }_2 })\right] = - d^2\vec{F_{12}^A}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the English translation of Ampère's work where he derives his force law:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/&quot; rel=&quot;noreferrer&quot;&gt;Assis, André Koch Torres&lt;/a&gt;; Chaib, J. P. M. C; Ampère, André-Marie (2015). &lt;em&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;noreferrer&quot;&gt;Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience&lt;/em&gt; (PDF)&lt;/a&gt;. Montreal: Apeiron. ISBN 978-1-987980-03-5.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2017-12-30T20:13:37.890" LastActivityDate="2017-12-30T20:13:37.890" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6866" PostTypeId="1" CreationDate="2018-01-01T11:08:36.667" Score="2" ViewCount="65" Body="&lt;p&gt;What did Gauss mean when he said that you'd never be a good mathematician if you didn't think that De Moivre's theorem was obvious? I'm looking for a specifically historically correct answer with evidence based on his letters, remarks, etc. &amp;amp; not a mathematical guess based on the properties of Taylor series, etc., which would be based on calculation &amp;amp; nothing that could be regarded as &quot;obvious&quot;. Gauss was a deep guy; give me a deep answer. This may not be the right discussion forum for this question; please suggest another if you can.&lt;/p&gt;&#xA;" OwnerUserId="6857" LastActivityDate="2018-01-01T11:08:36.667" Title="gauss' opinion on de moivre's theorem" Tags="|complex-analysis|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6867" PostTypeId="1" AcceptedAnswerId="6875" CreationDate="2018-01-01T19:41:55.670" Score="6" ViewCount="1413" Body="&lt;p&gt;I read recently that &lt;em&gt;Cours&lt;/em&gt; includes a famous, or perhaps infamous, error in that Cauchy states and proves a false result concerning sequences of continuous functions.  (Here, obviously, &lt;em&gt;continuous function&lt;/em&gt; means continuous according to Cauchy’s intuitive definition of continuity and his notion of the continuum together with Cauchy’s concept of function.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There appear to be conflicting accounts of what Cauchy actually stated as a theorem.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some interpret the result as saying that the pointwise limit of a sequence of continuous functions is a continuous function.  Let’s call this the &quot;limit interpretation&quot;: $f(x) = \lim_{n \to \infty} f_{n}(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Others say, no, no, no - Cauchy is talking about the series whose terms are a sequence of functions, and that interpreted this way the result is much more “plausible”  according to Cauchy’s own definition of continuity - more plausible in the sense that counter-examples are not completely trivial.  Let’s call this the &quot;series interpretation&quot;: $f(x) = \sum_{n=1}^{\infty} f_{n}(x)$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A recent translation (see TL/DR below) suggests that the series interpretation is correct, however it is not clear if this translation is a statement of Cauchy’s original 1821 theorem or a statement of his 1853 clarification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; According to Cauchy’s notion of function and the continuum, what is the correct interpretation of Cauchy’s original 1821 theorem and his subsequent 1853 clarification.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;TL/DR&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia states only the limit interpretation, referring to “some” historians and “purported”  counter-examples.  The &lt;a href=&quot;https://en.wikipedia.org/wiki/Uniform_convergence#History&quot; rel=&quot;noreferrer&quot;&gt;wikipedia article on uniform convergence&lt;/a&gt; states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some historians claim that Augustin Louis Cauchy in 1921 published a false statement, but with a purported proof, that &lt;strong&gt;the pointwise limit of a sequence of continuous functions is always continuous&lt;/strong&gt;; however, Lakotos offers a re-assessment of Cauchy’s approach.  Niels Henrik Abel in 1826 found purported counterexamples to this statement in the context of Fourier series, arguing that Cauchy’s proof had to be incorrect.  Cauchy ultimately responded in 1853 with a clarification of his 1821 formulation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here on HSM, the limit interpretation is also assumed in the question &lt;a href=&quot;https://hsm.stackexchange.com/questions/774/why-did-mathematicians-not-see-that-f-nx-xn-is-a-counterexample-to-cauchys&quot;&gt;Why did mathematicians not see that $f_n(x) = x^n$ is a counter-example to Cauchy's &quot;theorem&quot; about limits of continuous functions&lt;/a&gt;, which begins with the sentence “In 1821 Cauchy claimed that the limit of a sequence of continuous functions is continuous.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the series interpretation, see, for example, the HSM user MJD’s blog posting &lt;a href=&quot;https://blog.plover.com/math/Cauchy.html&quot; rel=&quot;noreferrer&quot;&gt;Cauchy and the continuum&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The confusion about Cauchy’s controversial theorem arises from a perennially confusing piece of mathematical terminology: a convergent &lt;em&gt;sequence&lt;/em&gt; is not at all the same as a convergent &lt;em&gt;series&lt;/em&gt;.  Cauchy claimed that a convergent series of continuous functions had a continuous limit.  He did not ever claim that a convergent sequence of continuous functions had a continuous limit.  But I have often encountered claims that he did that, even though such claims are extremely implausible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cauchy’s words, according to a recent translation&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe that the theorem in question is given as &lt;a href=&quot;https://books.google.ca/books?id=M0or-HGe7D0C&amp;amp;pg=PA90&amp;amp;lpg=PA90&amp;amp;dq=When%20the%20various%20terms%20of%20series%20are%20functions%20of%20the%20same%20variable%20x,%20continuous%20with%20respect%20to%20this%20variable%20in%20the%20neighborhood%20of%20a%20particular%20value%20for%20which%20the%20series%20converges,%20the%20sum%20s%20of%20the%20series%20is%20also%20a%20continuous%20function%20of%20x%20in%20the%20neighborhood%20of%20this%20particular%20value.&amp;amp;source=bl&amp;amp;ots=FzGc-dOZLu&amp;amp;sig=wf1r7Z92rvS7519bCbLp-gAtq1o&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiB1eKTtrPYAhVNz2MKHTfFC6IQ6AEILzAB#v=onepage&amp;amp;q=When%20the%20various%20terms%20of%20series%20are%20functions%20of%20the%20same%20variable%20x%2C%20continuous%20with%20respect%20to%20this%20variable%20in%20the%20neighborhood%20of%20a%20particular%20value%20for%20which%20the%20series%20converges%2C%20the%20sum%20s%20of%20the%20series%20is%20also%20a%20continuous%20function%20of%20x%20in%20the%20neighborhood%20of%20this%20particular%20value.&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Theorem 1 on page 90 of Bradley and Sandifer’s &lt;em&gt;Cauchy’s Cours d’analyse An Annotated Translation&lt;/em&gt;, Springer 2009&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;THEOREM 1&lt;/strong&gt; - When the various terms of series (1) are functions of the same variable x, continuous with respect to this variable in the neighborhood of a particular value for which the series converges, the sum s of the series is also a continuous function of x in the neighborhood of this particular value.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As the note (1) indicates, it is important to consider exactly what Cauchy meant by &lt;em&gt;series&lt;/em&gt; and &lt;em&gt;convergent series&lt;/em&gt;.  According to the Bradley and Sandifer's translation, Cauchy uses the following definitions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We call a &lt;em&gt;series&lt;/em&gt; an indefinite sequence of quantities, $$u_0, u_1, u_2, u_3, \dots ,$$ which follow from one to another according to a determined law.  These quantities themselves are the various &lt;em&gt;terms&lt;/em&gt; of the series under consideration.  Let $$s_n = u_0 + u_1 + u_2 + \dots + u_{n-1}$$ be the sum of the first &lt;em&gt;n&lt;/em&gt; terms, where &lt;em&gt;n&lt;/em&gt; can be any integer number.  If, for ever increasing values of &lt;em&gt;n&lt;/em&gt;, the sum $s_n$ indefinitely approaches a certain limit &lt;em&gt;s&lt;/em&gt;, the series is said to be &lt;em&gt;convergent&lt;/em&gt; and the limit in question is called the sum of the series.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So Cauchy appears to be using the term &lt;em&gt;series&lt;/em&gt; to describe what we now call a &lt;em&gt;sequence&lt;/em&gt;.  However, when it comes to a &lt;em&gt;convergent series&lt;/em&gt;, Cauchy is not talking about what we now call  a convergent sequence. Rather he is using the term in the modern sense of a convergent series - albeit with a different underlying notion of function and the continuum.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This suggests that our &quot;series&quot; interpretation is the correct interpretation.   However, as noted above, it is not clear that this translation is stating the original 1821 theorem or the 1853 clarification.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2018-01-03T10:01:04.467" Title="What is the correct statement of Cauchy’s erroneous theorem on continuity?" Tags="|mathematics|mathematicians|real-analysis|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6868" PostTypeId="1" CreationDate="2018-01-01T22:03:19.940" Score="0" ViewCount="56" Body="&lt;p&gt;In physics, the capacitance of a conductor is (roughly) the net charge accumulated when it is grounded relative to a unit voltage at infinity.  The same quantity comes up in mathematics, but we call it capacity instead.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why don't both communities call this capacity?  I can't help but think engineers accustomed to &quot;resist&lt;em&gt;ance&lt;/em&gt; is what a resist&lt;em&gt;or&lt;/em&gt; has&quot; and the word &quot;capacit&lt;em&gt;or&lt;/em&gt;&quot; came along and invented &quot;capacit&lt;em&gt;ance&lt;/em&gt;&quot; by analogy...&lt;/p&gt;&#xA;" OwnerUserId="6862" LastActivityDate="2018-01-02T09:58:30.880" Title="Capacitance or capacity?" Tags="|mathematics|physics|electromagnetism|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6869" PostTypeId="1" CreationDate="2018-01-02T00:00:57.447" Score="2" ViewCount="2531" Body="&lt;p&gt;I've read somewhere that Hipparchus measured the distance to the Moon using the lunar and solar eclipse and obtained a value of around 67.3 Earth radii. It also says that soon after Ptolemy gave a more accurate value of 59.7 Earth radii, but I want to know how did he arrive at this?&lt;/p&gt;&#xA;" OwnerUserId="6865" LastEditorUserId="4251" LastEditDate="2021-09-15T05:12:43.780" LastActivityDate="2021-09-15T05:12:43.780" Title="How did Ptolemy calculate the distance to the Moon?" Tags="|astronomy|ancient-greece|archaeoastronomy|ptolemy|antiquity|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6870" PostTypeId="2" ParentId="6869" CreationDate="2018-01-02T01:04:31.750" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/On_Sizes_and_Distances&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On Sizes and Distances&lt;/em&gt;&lt;/a&gt;, describes the method of Hipparchus, with modern reconstructions.  As stated in the article, Ptolemy gives these results in &lt;em&gt;Almagest V, 11&lt;/em&gt;.  Ptolemy provides his own, improved estimate, and his procedure, in section &lt;em&gt;13&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a nice discussion in this article on &lt;em&gt;&lt;a href=&quot;http://www.etwright.org/astro/moonpar.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lunar Parallax&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3821" LastActivityDate="2018-01-02T01:04:31.750" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6871" PostTypeId="2" ParentId="6869" CreationDate="2018-01-02T01:14:39.620" Score="5" Body="&lt;p&gt;Ptolemy knew about the Moon's parallax (he explains it in section 11, Ch. V of Mathematical Syntaxis). To measure it he invented the &quot;parallactic instrument&quot; described in section 12. Section 13 is dedicated to determination of Moon's distance, where he explains his observation in great detail. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roughly speaking he computes the Moon's geocentric position in the sky from theory (which was very accurate for this purpose), and compares it&#xA;with an observation. He finds a parallax of 1.7 degrees.&#xA;(Of course he makes the necessary precaution that the theory describing the true&#xA;motion of the Moon is independent of the parallax. It describes the Moon's position as seen from the center of the Earth.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The parallax gives the distance in units of the Earth's radius: he finds this distance to be approximately 39 (at the time of observation). The average distance according to Ptolemy is 59. This is a good result for the&#xA;accuracy of observations at that time. He does not discuss the Earth radius in  other units, like stadii, probably he did not care much. He also discusses the parallax of the Sun, but here he is wrong by far: the Sun's parallax is too small to be reliably measured in Ptolemy's time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The existence of the Moon's parallax was already known to Hipparchus, but none of the technical writings of Hipparchus survives, so all our information is based&#xA;on Ptolemy's Syntaxis (&quot;Almagest&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: There are easily available translations of Ptolemy, but is a difficult reading. For the general idea, type various combinations of the words &quot;Ptolemy&quot; &quot;parallax&quot;, &quot;parallactic instrument&quot; and&#xA;&quot;triquetrum (astronomy)&quot; on Google. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="2277" LastEditDate="2020-03-22T13:28:58.737" LastActivityDate="2020-03-22T13:28:58.737" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="6872" PostTypeId="1" CreationDate="2018-01-02T07:58:28.287" Score="3" ViewCount="588" Body="&lt;p&gt;The Mayan's astronomical observations were outstanding given the tools they had, and investigations of the Dresden codex published by Harvey and Victoria Bricker in 1983 predicted the July 11th, 1991 eclipse according to the Mayan text.  Given that the Mayan's didn't have true model of the mechanics and scale of the solar system, could they have done this just with extrapolations from their observations?  Lunar eclipse, sure, but I'm not familiar enough with the variables you would need to know in order to predict a solar eclipse 500 years ahead of time to know if it would have actually been possible.  The publication says it is so, but in 1983 modern astronomers could predict the 1991 eclipse and it would have been easily accessible knowledge so you wonder if the Brickers could have set out looking so hard for something that they saw it.  There are certainly other debates over interpretations of the meanings in the codex so maybe, you know, are they are getting too much credit? But maybe it's totally legitimate interpretation, I can't read Mayan hieroglyphics and interested to know.  &lt;/p&gt;&#xA;" OwnerUserId="6863" LastActivityDate="2018-01-02T08:53:08.287" Title="Did the Mayan's really predict the solar eclipse?" Tags="|astronomy|archaeoastronomy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6873" PostTypeId="2" ParentId="6872" CreationDate="2018-01-02T08:53:08.287" Score="2" Body="&lt;p&gt;Yes, they really predicted the 1991 solar eclipse. I suggest that you read &lt;em&gt;Classic Maya Prediction of Solar Eclipses&lt;/em&gt;, by Harvey M. Bricker and Victoria R. Bricker, together with comments and replies (&lt;em&gt;Current Anthropology&lt;/em&gt;, Vol. 24, No. 1 (Feb., 1983), pp. 1&amp;ndash;23). &lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-01-02T08:53:08.287" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6874" PostTypeId="2" ParentId="6868" CreationDate="2018-01-02T09:58:30.880" Score="1" Body="&lt;p&gt;Because the word &quot;capacity&quot; has a very general meaning, whereas capacitance is a very specific property in the theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, resistance could be reformulated as &quot;the capacity to resist&quot;, and capacitance could be reformulated as &quot;the capacity to hold (charge)&quot;. We could also talk about &quot;current-carrying capacity&quot;, or &quot;battery capacity&quot;. But in each case the type or nature of the capacity being referred to is different.&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2018-01-02T09:58:30.880" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6875" PostTypeId="2" ParentId="6867" CreationDate="2018-01-02T10:54:18.483" Score="4" Body="&lt;p&gt;Our 2018 publication &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9534-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Cauchy's infinitesimals, his sum theorem, and foundational paradigms&quot;&lt;/a&gt; in &lt;em&gt;Foundations of Science&lt;/em&gt; (currently in &lt;em&gt;online first&lt;/em&gt; status) deals with this issue in detail. Cauchy's original formulation of the &quot;sum theorem&quot; in 1821 was ambiguous but his 1853 clarification, while also somewhat cryptic, admits a better interpretation in the context of a modern theory of infinitesimals and an enriched continuum than in the context of an Archimedean continuum.  Briefly, Cauchy required the condition of convergence also at infinitesimal and other points.  Such a pointwise condition for uniform convergence is indeed available in a hyperreal setting; the publication mentioned above provides the appropriate theorem from Abraham Robinson's book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a lot of resistance in the community to accepting the fact that Cauchy's theorem cannot be called &quot;erroneous&quot; (at most &quot;ambiguous&quot;) due to defective work of some historians (as evidenced also by the alternative &lt;em&gt;answer&lt;/em&gt; to this &lt;em&gt;question&lt;/em&gt;); see &lt;a href=&quot;https://mathoverflow.net/q/268163/28128&quot;&gt;this MO post&lt;/a&gt; for a lively discussion&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-01-03T10:01:04.467" LastActivityDate="2018-01-03T10:01:04.467" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6876" PostTypeId="2" ParentId="6867" CreationDate="2018-01-02T14:03:13.227" Score="-1" Body="&lt;p&gt;The correct statement is that is that uniform limit of continuous functions is continuous. Cauchy's proof proofs exactly this. The problem is that uniform convergence was not formalized at the time of Cauchy, so he confused pointwise convergence with uniform convergence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Neither was the notion of continuous function formalized. For example, Fourier, a contemporary of Cauchy, defined a continuous function as a function whose graph can be drawn without detaching your pencil from the paper, and according to his definition the &quot;function&quot; $f(x)=1, x&amp;gt;0, f(x)=-1, x&amp;lt;0$ with the graph completed by a vertical segment at $0$ was continuous. He obtained this function as a limit of Fourier series, and he apparently believed Cauchy's theorem. The modern notion of function was formalized by Dirichlet.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-01-02T14:10:54.587" LastActivityDate="2018-01-02T14:10:54.587" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6878" PostTypeId="2" ParentId="5322" CreationDate="2018-01-02T18:57:44.217" Score="1" Body="&lt;p&gt;As regards the Soviet's quick progress on the bomb please read &lt;em&gt;Dark Sun&lt;/em&gt; which is Richard Rhodes second book on the making of the atomic bomb. Early on he describes the wholesale shipping of plane loads of secrets to the soviets by their well respected U.S patriot spies here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reasons given are well considered but unmentioned is Hitler's impatience which I think was mentioned in &lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt;. The prospect of a multi year project for just a bomb would not interest him in the least. He turned down more development of the first working jet airplane for similar narrow minded reasons.&lt;/p&gt;&#xA;" OwnerUserId="6870" LastActivityDate="2018-01-02T18:57:44.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6879" PostTypeId="1" AcceptedAnswerId="6889" CreationDate="2018-01-02T23:25:01.677" Score="1" ViewCount="74" Body="&lt;p&gt;According to &lt;a href=&quot;https://books.google.com/ngrams/graph?content=willpower&amp;amp;case_insensitive=on&amp;amp;year_start=1800&amp;amp;year_end=2008&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t4%3B%2Cwillpower%3B%2Cc0%3B%2Cs0%3B%3Bwillpower%3B%2Cc0%3B%3BWillpower%3B%2Cc0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google ngrams&lt;/a&gt; the term &lt;code&gt;willpower&lt;/code&gt; wasn't used much before 1880. How did humans reason about motivation back then? What kind of conceptions of human motivation was displaced by adopting the new concept of &lt;code&gt;willpower&lt;/code&gt;?&lt;/p&gt;&#xA;" OwnerUserId="4623" LastActivityDate="2018-01-04T04:19:07.933" Title="How did scientists thought about human motivation before the term willpower came into fashion?" Tags="|psychology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6880" PostTypeId="2" ParentId="5055" CreationDate="2018-01-03T06:08:33.327" Score="2" Body="&lt;p&gt;There are two reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As explained by Alexandre Eremenko, the laws of electrodynamics before Maxwell (like Ampere's force law, Neumann's law of induction, Weber's law, etc.) explained all the known phenomena in an &lt;em&gt;action at a distance&lt;/em&gt; manner. When action at a distance theories fell out of favour, the pioneers of these theories were mostly discredited. Not to mention the fact that these pioneers had their name attached to some part of field theory. For eg &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; Ampere's circuital law, Gauss law, etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In the later accepted field theory, the pioneer was James Clerk Maxwell (Faraday did initiated field theory but it was Maxwell who mathematically expressed them). Not only he expressed them mathematically but also symmetrically.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From these two reasons it would be obvious that it was only Maxwell who would be eligible for being credited as &amp;quot;unifier of electricity and magnetism&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Here is a quote from Maxwell's treatise regarding field theory vs action at a distance theory:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In a philosophical point of view, moreover, it is exceedingly important that two methods should be compared, both of which have succeeded in explaining&#xA;the principal electromagnetic phenomena, and both of which have attempted to explain the propagation of light as an electromagnetic phenomenon, and have&#xA;actually calculated its velocity, while at the same time the fundamental conceptions of what actually takes&#xA;place, as well as most of the secondary conceptions of&#xA;the quantities concerned, are radically different.&#xA;I have therefore &lt;em&gt;taken the part of an advocate rather&#xA;than that of a judge&lt;/em&gt;, and have rather exemplified one method than attempted to give an impartial description of both. I have no doubt that the method which I&#xA;have called the German one (action at a distance) will also find its supporters, and will be expounded with a skill worthy of its ingenuity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5313" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-03T06:13:51.153" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6881" PostTypeId="1" AcceptedAnswerId="6884" CreationDate="2018-01-03T06:24:12.677" Score="3" ViewCount="248" Body="&lt;p&gt;Lakatos, in his &lt;em&gt;Proofs and Refutations&lt;/em&gt;, rejects the Euclidean methodology and exposition of mathematics: where axioms and definitions precede the proofs. In other words, a Euclidean mathematician makes clear the axioms and definitions, then proves novel theorems in an altogether linear fashion. This rigid methodology is not present in mathematical practice, Lakatos contests. &lt;em&gt;“This led to&quot;&lt;/em&gt; his &lt;em&gt;&quot;dialectical methodology, a ‘heuristic style’ that reveals the struggle and adventure of mathematical creation&quot;&lt;/em&gt; [Paolo Mancosu’s Preface in &lt;em&gt;Proofs and Refutations&lt;/em&gt;]. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lakatos seeks to convince us of this through case studies -- the historical contexts behind uniform convergence, bounded variation, and the Carathéodory definition of a measurable set, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are the cases that Lakatos discusses special cases? Or does most of post-Euclidean mathematics progress in this heuristic, dialectical style? Examples of some recent discoveries and how they came about would be interesting.&lt;/p&gt;&#xA;" OwnerUserId="6867" LastActivityDate="2018-01-04T11:02:47.110" Title="Does Lakatos' argument in favour of 'informal mathematics' hold up in most cases?" Tags="|mathematics|mathematicians|philosophy-of-science|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6882" PostTypeId="1" AcceptedAnswerId="6887" CreationDate="2018-01-03T11:30:30.070" Score="10" ViewCount="991" Body="&lt;h2&gt;Background&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;In the famous Boltzmann's entropy formula, &lt;a href=&quot;https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula#/media/File:Zentralfriedhof_Vienna_-_Boltzmann.JPG&quot; rel=&quot;noreferrer&quot;&gt;carved in the physicist's tombstone&lt;/a&gt;, a mysterious quantity $W$ appears (a):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$S=k_B \log W\label{1}\tag{1}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We often hear that $W$ represents &quot;the number of microstates corresponding to the macrostate of a system&quot;. However, this definition is problematic in classical physics, as noted for example by Fermi (bold is mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have seen that a thermodynamical state of a system is not a sharply defined state of the system, because it cor- responds to a large number of dynamical states. This consideration led to the Boltzmann relation:&#xA;  $$S = k \log \pi$$&#xA;  where $\pi$ is called the probability of the state. Strictly speaking, $\pi$ is not the probability of the state, but is actually the number of dynamical states that correspond to the given thermodynamical state. This seems at first sight to give rise to a serious difficulty, since &lt;strong&gt;a given thermodynamical state corresponds to an infinite number of dynamical states.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-E. Fermi, &lt;em&gt;Thermodynamics&lt;/em&gt; (1937), Chapter VIII&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Fermi then proceeds to propose a solution for this inconsistency in the framework of classical physics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The phase space is divided into a number of very small&#xA;  cells all of which have the same hyper-volume $\tau$; the state is then characterized by specifying the cell to which the point representing the state belongs. Thus, states whose representative points all lie in the same cell are not considered as being different. This representation of the state of a system would evidently become exact if the cells were made infinitesimal. The cell representation of the dynamical states of a system introduces a discontinuity in the concept of the state of a system which enables us to calculate $\pi$ by the methods of combinatory analysis, and, hence, with the aid of the Boltzmann relation, to give a statistical definition of the entropy.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-E. Fermi, &lt;em&gt;Thermodynamics&lt;/em&gt; (1937), Chapter VIII&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This solution also has a problem, however, since the arbitrariness in the choice of $\tau$ leads to an arbitrary additive constant in $S$. Fermi claims that this inconsistency can ultimately be solved only by quantum mechanics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] quantum theory introduces a discontinuity quite naturally into the definition of the dynamical state of a system (the discrete quantum states) without having to make use of the arbitrary division of the phase space into cells. It can be shown that this discontinuity is equivalent, for statistical purposes, to the division of the phase space into cells having a hyper-volume equal to $h^f$, where $h$ is Planck's constant ($h = 6.55 \cdot 10^{-27}$ cm$^2$ gm sec$^{-1}$) and $f$ is the number of degrees of freedom of the system.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-E. Fermi, &lt;em&gt;Thermodynamics&lt;/em&gt; (1937), Chapter VIII&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, in modern statistical mechanics, it seems that the problem of the infinity of the microstates is not present (or ignored?)(b): in the microcanonical ensemble, entropy is simply defined as (c)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$S= k_B \log \left(\int d^{3N}\mathbf p \ d^{3N} \mathbf q \ \rho({\{\mathbf p, \mathbf q\})} \right)\label{2}\tag{2}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $\rho(\{\mathbf p, \mathbf q\})$ is the microcanonical probability density:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;\rho(\{\mathbf p, \mathbf q\})=&#xA;\begin{cases}&#xA;\text{const.} &amp;amp; E&amp;lt;\mathcal H (\{\mathbf p, \mathbf q\}) &amp;lt;E+\Delta E\\&#xA;0 &amp;amp; \text{otherwise}&#xA;\end{cases}&#xA;\tag{3}\label{3}&#xA;$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Eq.\ref{3}, $\mathcal H$ is the system's hamiltonian, $E$ is its energy and $\Delta E$ is chosen such that $\Delta E / E \ll 1$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eq.\ref{2} does not present the problem discussed by Fermi, since the integral appearing in it is just an hypervolume in phase space, which is finite if $E$ is finite. It appears therefore that Fermi's problem only appears if we (loosely?) define $W$ of Eq.\ref{1} as &quot;the number of microstates of the system&quot; without having a precise notion of what a microsatate is in statistical mechanics. However, this inconsistency is avoided if a definition like \ref{2} is chosen, where there is no mention of &quot;microstates&quot;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;My question&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;What was the original interpretation Boltzmann gave of the $W$ appearing in Eq.\ref{1}? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did he see it as a &quot;number of microstates&quot;? If so, was he conscious of the ambiguity of this concept, as pointed out by Fermi? Or did he see it, in more modern terms, as an hypervolume in phase space?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;(a) Even if it appears that the equation was &lt;a href=&quot;https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula#History&quot; rel=&quot;noreferrer&quot;&gt;actually reformulated in this way later by Planck&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) But thr problem of the indeterminate additive constant remains. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(c) See for example K. Huang, &lt;em&gt;Statistical Mechanics&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5003" LastEditorUserId="4251" LastEditDate="2020-06-07T10:35:49.833" LastActivityDate="2020-11-20T16:44:08.317" Title="Interpretation of $W$ in Boltzmann's entropy formula" Tags="|physics|theoretical-physics|statistical-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6883" PostTypeId="1" CreationDate="2018-01-03T16:11:10.783" Score="6" ViewCount="863" Body="&lt;p&gt;I know that there are no standard definitions for pure and applied mathematics however I would like to know who first considered them as two separate entities, I have seen people mention it was around the late 18th early 19th century but I can't find any conclusive evidence to support this.&lt;/p&gt;&#xA;" OwnerUserId="6435" LastActivityDate="2018-03-29T21:21:53.000" Title="When were the concepts of pure and applied Mathematics introduced?" Tags="|mathematics|mathematicians|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6884" PostTypeId="2" ParentId="6881" CreationDate="2018-01-03T16:30:16.730" Score="12" Body="&lt;p&gt;I do not agree with the assumptions made in the title and in main text of this question, namely that Lakatos &lt;em&gt;rejects the Euclidean methodology and exposition of mathematics&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the way I read it, Lakatos is putting his fingers on a rather different topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people, while describing mathematics as a science, tend to confuse the &lt;em&gt;Euclidean exposition&lt;/em&gt; with what mathematics is. In passing let me say that I doubt any mathematician ever in the world thought that there was something like &lt;em&gt;an Euclidean methodology&lt;/em&gt;: a way of doing mathematics in the linear and terse style of an axiomatic theory. So Lakatos is pointing us towards understanding that math &lt;em&gt;in practice&lt;/em&gt; is very different from its final outcome and therefore, from a philosophical point of view we cannot describe it merely as an activity aiming at &lt;em&gt;proving theorems&lt;/em&gt;, which requires that you have a theorem to start with and a path to prove it afterwards.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics progresses in a heuristic, dialectical style, yes. But this was not Lakatos' thought. Just read first hand reports on Bourbaki method of work (I guess the Bourbaki team can be rightfully considered as exponent of contemporary &lt;em&gt;Euclidean exposition&lt;/em&gt;) and you will see that Cartan, Weil, Dieudonné, Grothendieck all knew very well that new math does not flow in the form of Definition-Theorem-Proof like in their books (and no surprise that Bourbaki's books have very interesting historical notes - something Lakatos would definitely have advocated). This was wide clear to many. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lakatos put his finger on one problem. Describing math from a philosophical viewpoint in those years it was common to forget this fact and to identify math as it was written with math as it was done. And this is not the case. &lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="1689" LastEditDate="2018-01-04T11:02:47.110" LastActivityDate="2018-01-04T11:02:47.110" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6885" PostTypeId="2" ParentId="6883" CreationDate="2018-01-03T17:54:26.693" Score="4" Body="&lt;p&gt;As Mauro Allegranza wrote in his comments the distinction can be traced to ancient Greece, more precisely to Hellenistic times, though it was not formalized. Subjects like music, mechanics, optics and astronomy were considered parts of mathematics. At the same time there was &quot;physics&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The distinction was that in &quot;mathematics&quot; they had axioms and theorems while in physics they started with &quot;phenomena&quot; (observations).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid wrote books on Optics and on harmony treating them as mathematical subjects. First Greek books on astronomy were pure mathematical, with no much regard to observations. Notice that Ptolemy's book (which is known now as Almagest) was called Mathematical Syntaxis, and Newton's book Principia Mathematica, though certainly observations played a fundamental role in both.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The distinction was probably formalized in 19th century when the first mathematical journals appeared:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Journal fur die reine und angewandte Mathematik, and&#xA;Journal de mathematiques pures et appliquee.   &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-01-03T17:54:26.693" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6886" PostTypeId="1" CreationDate="2018-01-03T22:33:40.090" Score="8" ViewCount="1671" Body="&lt;p&gt;I teach mathematics at a Catholic liberal arts college. This semester, I am teaching Real Analysis and I want students to complete a semester-long research project that extends beyond the course content and into relevant issues of history and philosophy. One idea I had is for students to investigate the relationship between the Catholic church and mathematics, perhaps including (but not limited to):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some historical figure from the church who also contributed significantly to the development of mathematics &lt;/li&gt;&#xA;&lt;li&gt;Examples of mathematical concepts or theorems that were interpreted divinely&lt;/li&gt;&#xA;&lt;li&gt;Some significant mathematician who was also a practicing Catholic and wrote about the relationship between these two aspects of their life&lt;/li&gt;&#xA;&lt;li&gt;Examples of the Catholic church commenting on, or intervening in, the development of mathematics&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Googling has not yielded anything particularly fruitful. I am not really interested in the interpretation of axiomatic mathematics as a reflection of divinity, because this is not especially unique to Catholicism. And I do want to stick to mathematics and not science, in general (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_affair&quot; rel=&quot;noreferrer&quot;&gt;the Galileo affair&lt;/a&gt;), because this is an upper-level course that is required for mathematics majors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Main question: Can anyone suggest some examples of sources that I can suggest to any of my students who want to work on such a project?&lt;/strong&gt; This may include books, films, blog posts, scholarly articles, any format and depth, really. What matters most is that it is highly relevant to the historical development of mathematics (not just interpreting mathematics retrospectively) and the Catholic church. (Bonus points if it's somehow related to the development of mathematical &lt;em&gt;analysis&lt;/em&gt; and calculus, specifically!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;em&gt;Meta comment:&lt;/em&gt; I would like to tag this with &quot;religion&quot; but do not yet have the reputation. I will leave it to more frequent site-users to determine if this should be a new, valid tag, or if something more specific like &quot;Catholicism&quot; would be warranted, as well.)&lt;/p&gt;&#xA;" OwnerUserId="125" LastActivityDate="2018-01-18T11:13:06.123" Title="Seeking sources: Catholic church and the development of mathematics" Tags="|mathematics|education|resource-recommendation|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6887" PostTypeId="2" ParentId="6882" CreationDate="2018-01-03T23:33:10.870" Score="6" Body="&lt;p&gt;One should remember the context of the time when Boltzmann wrote his 1877 paper. Weierstrass just created a rigorous version of mathematical analysis and probability theory was not even close to Kolmogorov's formalization. Kinetic theory of gases was highly controversial, with Mach and Ostwald rejecting it altogether. Boltzmann's earlier publication (1872) of the &quot;$H$-theorem&quot; about entropy was contravened by the Loshmidt's paradox. He could not be expected to have as many mathematical scruples as Fermi, and his focus was on demonstrating the viability and fruitfulness of the kinetic program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boltzmann did not write the formula as $S=k\ln W$ in his 1877 paper. In place of $W$ he had the &quot;probability of a state&quot; $P$, and $k$ was generally ignored in 19th century calculations starting with Maxwell, it was seen as merely a choice of units. Planck introduced it explicitly when it became clear that it was important for precision estimates of the Avogadro constant. Since probability is merely proportional to the number of microstates it is clear that Boltzmann did not much worry about the additive constant either. He did not put much weight even on the logarithm, in his verbal formulations it is not even mentioned:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The initial state of a system will be, in most cases, a not so probable state and the system will tend always towards more probable states, until it will reach the most probable state, i.e. the state of thermodynamic equilibrium. If we apply this to the second law of thermodynamics, we can identify the quantity that is usually called entropy with the probability of the corresponding state. [...] the total entropy of the system cannot but increase. In our present interpretation, this has no other meaning than the fact that the probability of the global state of the bodies of the system must continuously increase: the system cannot but pass from a state to a more probable state.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Boltzmann starts from a gas of $n$ particles with discrete energies $(0, e, 2e, ..., pe)$, &quot;&lt;em&gt;an unrealizable fiction&lt;/em&gt;&quot; by his own admission, and fixed total energy, and calls the collection of integers $n_k$, the number of particles with energy $ke$, &quot;&lt;em&gt;state distribution&lt;/em&gt;&quot;. Repartitions in which each particle has an assigned energy are called &quot;&lt;em&gt;complexions&lt;/em&gt;&quot; and the usual multifactorial formula for the probability of a complexion is then derived. Here is from &lt;a href=&quot;https://books.google.com/books/about/Ludwig_Boltzmann.html?id=iikSDAAAQBAJ&quot; rel=&quot;noreferrer&quot;&gt;Cercignani's Boltzmann's biography&lt;/a&gt;, which discusses this issue in detail:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Boltzmann took the number of complexions... as a measure of the likelihood of a given distribution, underlining that in this way he considered the complexions as a priori equiprobable, in the same way as the probability that the sequence 1, 2, 3, 4, 5 will come out of the hat in a game of lotto is no different from that of any preassigned sequence of five numbers. In this way, the problem is reduced to the search for a constrained maximum of P... The convenience of working with the logarithm of P is then clear.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To get the continuous case he discretizes it and passes to the limit. This is done quite heuristically, sums are replaced by integrals. It turns out that the analog of $-\log P$ is the $H$ from his 1872 paper, given by the integral of $f\log f$ and identified with entropy there, and this is how the Boltzmann relation is arrived at. It is interesting that already Einstein in 1905 criticized Boltzmann's handling of probabilities in his calculations with complexions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The word probability is used in a sense that does not conform to its definition as given in the theory of probability. In particular, &quot;cases of equal probability&quot; are often hypothetically defined in instances where the theoretical pictures used are sufficiently definite to give a deduction rather than a hypothetical assertion.&lt;/em&gt;&quot; [quoted from Cercignani]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The original paper is: L. Boltzmann (1877). Uber die Beziehung eines allgemeine mechanischen Satzes zum zweiten Hauptsatze der Warmetheorie. Sitzungsberichte der Akademie der Wissenschaften, Wien, II, 75, 67-73 [English translation in: S.G. Brush, Kinetic theory, Vol. 2, Irreversible processes, pp. 188-202, Pergamon Press, Oxford (1966)].&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-04T03:58:17.757" LastActivityDate="2018-01-04T03:58:17.757" CommentCount="13" ContentLicense="CC BY-SA 3.0" />
  <row Id="6888" PostTypeId="2" ParentId="6886" CreationDate="2018-01-04T00:27:40.107" Score="6" Body="&lt;p&gt;&lt;a href=&quot;http://www.newadvent.org/cathen/03457a.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy&lt;/a&gt;, who gave calculus its modern formalization (cf. Grabiner's &lt;a href=&quot;https://isidore.co/calibre#book_id=4178&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Origins of Cauchy's Rigorous Calculus&lt;/em&gt;&lt;/a&gt;), was a &quot;significant mathematician who was also a practicing Catholic.&quot; From &lt;a href=&quot;https://isidore.co/calibre/#book_id=4340&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;Belhoste's biography of him&lt;/a&gt;, p. viii:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Truth,&quot; he wrote in 1842,&#xA;  &quot;is a priceless treasure which, whenever we manage to acquire it, cannot bring&#xA;  us remorse and sorrow; it cannot disquiet and distress our soul. The mere&#xA;  thought of its heavenly attributes, of its divine beauty suffices to replenish us&#xA;  for all the sacrifices we may have made in discovering it. Indeed, the joy of&#xA;  heaven itself is but the full and complete possession of immortal truth.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/lavieetlestravau00vals#page/n197/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vol. 1, ch. 12 &amp;amp; 13 of Valson's biography&lt;/a&gt; of Cauchy are on his Christian thoughts and works. Cauchy's profession of faith is on &lt;a href=&quot;https://archive.org/stream/lavieetlestravau00vals#page/n201/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;vol. 1, pp. 173&lt;/a&gt;-4.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;Bishop &lt;a href=&quot;http://www.encyclopedia.com/people/philosophy-and-religion/other-religious-beliefs-biographies/nicole-oresme#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicole Oresme&lt;/a&gt; (1320-1382) (cf. &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/history/Biographies/Oresme.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;his MacTutor mathematical biography&lt;/a&gt;) studied fractional powers, invented &lt;em&gt;x-y&lt;/em&gt; coordinate geometry, and is part of the pre-history of calculus (cf. &quot;III. Medieval Contributions&quot; of Boyer's &lt;a href=&quot;https://isidore.co/calibre/#book_id=4179&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The History of the Calculus and Its Conceptual Development&lt;/em&gt;&lt;/a&gt; and ch. 2 &quot;The Dream of Oresme&quot; of Peter Pesic's &lt;a href=&quot;https://isidore.co/calibre/#book_id=5685&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Music and the Making of Modern Science&lt;/em&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;See also:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4757&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds&lt;/a&gt;&lt;/em&gt;. Edited and translated by Roger Ariew. Chicago: University of Chicago Press, 1985.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;Part I: The Two Infinities&quot; is more mathematically oriented.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hannam, James. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4182&quot; rel=&quot;nofollow noreferrer&quot;&gt;God’s Philosophers: How the Medieval World Laid the Foundations of Modern Science&lt;/a&gt;&lt;/em&gt;. London: Icon Books, 2009.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;references from &lt;a href=&quot;https://hsm.stackexchange.com/a/2147/232&quot;&gt;this answer&lt;/a&gt;; cf. &lt;a href=&quot;https://hsm.stackexchange.com/a/3611/232&quot;&gt;this answer&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://sententiaedeo.blogspot.com/2013/11/st-thomas-on-mathematicss-certitude.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;St. Thomas Aquinas on mathematics's certitude&lt;/a&gt; / co-naturality to the human intellect&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;https://archive.org/details/FilosofiaDeLosMatematicasEnSantoTomas&quot; rel=&quot;nofollow noreferrer&quot;&gt;Filosofía de las Matemáticas en Santo Tomás&lt;/a&gt;&lt;/em&gt; by José Alvarez Laso, C.M.F.&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;cf. its excellent &lt;a href=&quot;https://archive.org/stream/FilosofiaDeLosMatematicasEnSantoTomas/FilosofaDeLosMatemticasEnS.Thoms#page/n139/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Index locorum ubi S. Thomas de mathematicis agit&lt;/em&gt;&lt;/a&gt;, which collects all the references to St. Thomas where he treats of mathematics or uses mathematical examples&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&lt;a href=&quot;https://isidore.co/calibre/#book_id=5186&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bishop M.-L. Guérard des Lauriers, O.P.&lt;/a&gt;, was a 20&lt;sup&gt;th&lt;/sup&gt; century Dominican, mathematician, and philosopher of mathematics; he did his dissertation under Cartan and was classmates with Levi-Civita and André Weil.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-01-17T17:47:02.107" LastActivityDate="2018-01-17T17:47:02.107" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6889" PostTypeId="2" ParentId="6879" CreationDate="2018-01-04T00:56:10.270" Score="4" Body="&lt;p&gt;Scientific psychology only came into being in 19th century, at the end of 18th Kant, now considered one of its forefathers, even opined that psychology can never be a science. Before that it was mostly the domain of philosophers and theologians (much like physics before 17th), Brett's monumental History of Psychology, published in 1912-1921, devotes 90% of space to their writings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While &quot;willpower&quot; may be recent the &quot;will&quot; as a &quot;power&quot; or &quot;faculty&quot; of the soul is ancient. It is the English translation of Latin &quot;voluntas&quot;, which in turn translated Aristotle's &quot;boulesis&quot; (wish, desire). Even before Aristotle Plato in a famous allegory represented human soul as a &lt;a href=&quot;https://en.wikipedia.org/wiki/Chariot_Allegory&quot; rel=&quot;nofollow noreferrer&quot;&gt;chariot driven by the two horses, of noble and lowly passions&lt;/a&gt;, with the reason for the charioteer. Here is from &lt;a href=&quot;https://books.google.com/books/about/Aquinas_on_Mind.html?id=5vjJfL4BTjQC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kenny's Aquinas on Mind&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The discussion of the will derives from the philosophical tradition going back to Aristotle. The discussion of free choice harks back to the theological controversy between St. Augustine and his Pelagian opponents concerning the relation between human freedom and divine grace... Aristotle says that the origin of conduct is choice, and the origin of choice is appetition plus means-end reasoning; reasoning by itself moves nothing, only means-end reasoning concerned with good conduct which is the end of appetition. The passage concludes ‘Therefore choice is either appetitive intelligence or ratiocinative appetite’ (Nicomachean Ethics 6, 1139a30ff.)... In the Ethics Aristotle also makes a distinction between wish (boulesis) and choice (prohairesis): wish relates to the end, choice to the means to the end; we wish to be healthy, but we choose what will lead to health...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Aquinas and other scholastics developed this into a full blown theory of soul's cognitive vs appetitive powers. What was common to this tradition, and its 18th century derivatives like Wolffian &quot;rational psychology&quot; that Kant criticized, is the pervasive rationalism about the will and behavior. We were supposed to have a special power to act towards ends dictated by (mostly noble) passions and mediated by rational deliberation about the best means to achieve them. This picture was challenged in 18th century, first by British empiricists (Locke, Berkeley, and most notably Hume), who themselves contributed to what became empirical psychology, and later by German romantics and French vitalists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hume, who saw himself as the Newton of psychology, dispensed with the soul itself and its &quot;powers&quot; as fictions, and represented the mental as a bundle of ideas that attract according to the &quot;law of association&quot;. The passions he analyzed into &quot;affections&quot;, ideas associated to feelings, which sometimes include impulses to act (this is his &lt;a href=&quot;https://www.rep.routledge.com/articles/thematic/mind-bundle-theory-of/v-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;bundle theory of mind&lt;/a&gt;). Hume's mind has no driver, no horses, and is not even a chariot itself, reason and passions are fictions to organize ideas floating around and clustering according to the association laws modeled on Newton's gravity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While Hume dethroned both the reason and the will romantics and vitalists soon rehabilitated the latter without the former, see &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/19245109&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huneman's Montpellier Vitalism and the Emergence of Alienism in France (1750-1800)&lt;/a&gt; for a treatment of passions.&#xA;But it was Schopenhauer's and von Hartmann's irrationalist voluntarism (enabled by Kantian critique of reason, romantic elevation of &quot;animal spirits&quot; by Schelling and others, and the popularity of Oriental philosophies like Vedanta), which  paved the way for Freud's semi-empirical psychoanalysis, where deep unconscious urges are placed at the front and center of human motivation. Brett credits the rise of will psychology at the end of 19th century specifically to Schopenhauer's cultural influence, but notes that it was also supported by the early empirical work on psychophysiology of Cabanis and Bichat.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-04T04:19:07.933" LastActivityDate="2018-01-04T04:19:07.933" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6890" PostTypeId="2" ParentId="6886" CreationDate="2018-01-04T09:34:06.320" Score="3" Body="&lt;p&gt;Look up Bernard Bolzano, who should be known to all students of analysis through the Bolzano-Weierstrass theorem. His work became known largely after his death and after others had already rediscovered his results independently.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2018-01-04T09:48:12.567" LastActivityDate="2018-01-04T09:48:12.567" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6891" PostTypeId="2" ParentId="6886" CreationDate="2018-01-04T10:19:34.423" Score="7" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;&quot;Examples of the Catholic church commenting on, or intervening in, the development of mathematics:&quot;&lt;/em&gt;&lt;/strong&gt; Counter-reformation issues in the 17th century around transubstantiation/consubstantiation interpretation of the eucharist directly influenced attitudes toward the techniques of indivisibles that were seen as closely related to atomism and therefore opposed on doctrinal grounds by many catholic theologians based on canon 2 of the 13th Session of the Council of Trent in the preceding century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The said canon was a particularly doctrinaire endorsement of the pagan Aristotelian doctrine of &lt;a href=&quot;https://en.wikipedia.org/wiki/Hylomorphism&quot; rel=&quot;nofollow noreferrer&quot;&gt;hylomorphism&lt;/a&gt;.  The doctrine postulates a pseudoscientific substratum (&quot;hylo&quot;) underpinning all forms (that's the &quot;morph&quot; part), or a kind of primordial undifferentiated Play-Do.  Such a speculative scheme is somewhat analogous to ether, similarly rejected by scientists in the 19th century. Meanwhile, protestants envisioned less literal interpretations (such as &lt;em&gt;consubstantiation&lt;/em&gt;) that tolerated elements of &lt;em&gt;atomism&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The result was that in catholic Italy, developments in indivisibles and the techniques leading to the emerging infinitesimal calculus came to a virtual stop whereas mostly protestant lands tolerated such mathematical activity leading to the development of the infinitesimal analysis:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;en 1700, c'est le vide intégral en ce qui concerne la pratique des mathématiques nouvelles en Italie (page 183 in Robinet). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The reference is Robinet, A. (1991) La conqu\^ete de la chaire de mathématiques de Padoue par les leibniziens. Revue d'Histoire des Sciences, 44(2), 181--201. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;le grand nombre des mathématiciens de [l'Ordre] resta jusqu'`a la fin du XVIII$^e$ si`ecle profondément attaché aux méthodes euclidiennes (page 77 in Bosmans). &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The reference is Bosmans, H. (1927) André Tacquet (S. J.) et son traité d''Arithmétique théorique et pratique.' Isis, 9(1), 66--82. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;These and related issues are discussed in detail in &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9542-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 2018 publication in &lt;em&gt;Foundations of Science&lt;/em&gt;&lt;/a&gt; which is &lt;a href=&quot;https://arxiv.org/abs/1801.00427&quot; rel=&quot;nofollow noreferrer&quot;&gt;also available on the arxiv&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-01-18T11:13:06.123" LastActivityDate="2018-01-18T11:13:06.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6892" PostTypeId="2" ParentId="6886" CreationDate="2018-01-04T10:41:32.497" Score="6" Body="&lt;p&gt;You must tell them about the book &lt;em&gt;&lt;a href=&quot;https://books.google.pt/books/about/Infinitesimal_How_a_Dangerous_Mathematic.html?id=HwROAgAAQBAJ&amp;amp;redir_esc=y&quot; rel=&quot;noreferrer&quot;&gt;Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World&lt;/a&gt;&lt;/em&gt;, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Amir_Alexander&quot; rel=&quot;noreferrer&quot;&gt;Amir Alexander&lt;/a&gt;. It's about, among other things, how the Jesuit priests stopped the progress of Infinitesimal Calculus in Italy because the foundations of the subject were shaky at the time (to say the least).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps that you might also mention the fact that the Big Bang theory is due to a catholic priest, who was also a physicist, &lt;a href=&quot;https://fr.wikipedia.org/wiki/Georges_Lema%C3%AEtre&quot; rel=&quot;noreferrer&quot;&gt;Georges Lemaître&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-01-04T10:41:32.497" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6893" PostTypeId="2" ParentId="6886" CreationDate="2018-01-04T10:56:33.393" Score="6" Body="&lt;p&gt;I would add, as example, that can be a bit investigated the work of Padre &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Saccheri.html&quot; rel=&quot;noreferrer&quot;&gt;Girolamo Saccheri&lt;/a&gt;, a Jesuit priest, and his famous &quot;&lt;em&gt;Euclides ab omni naevo vindicatus&lt;/em&gt;&quot; (&lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5048&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Euclid Vindicated from Every Blemish&lt;/em&gt;&lt;/a&gt;) in which he first discovered many theorems of what will then be called hyperbolic geometry. He however ended his books by saying that such theorems were evidently wrong and concluded, erroneously, that hyperbolic geometry was not possible, thus failing to anticipate Non Euclidean Geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were many discussion whether this false statement reflected theological considerations (there was and should be only one geometry allowed by God) and whether it would have been possible for him to print his book if his conclusions would have been different (the book had to be approved by Inquisition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It certainly contributed the fact, correctly cited above by Mikhail Katz, that he rejected completely the theory of infinitesimals and the analytic approach to geometry. Not by chance his major mistake in the book appears when he tries to extend  at infinity some properties of hyperbolic lines.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="232" LastEditDate="2018-01-04T17:02:18.660" LastActivityDate="2018-01-04T17:02:18.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6894" PostTypeId="2" ParentId="343" CreationDate="2018-01-04T14:03:11.493" Score="6" Body="&lt;p&gt;Newton was certainly referring at least to Viete, who introduced the prototype of the symbolic notation which enabled the rapid development of tools leading to analysis.  Also surely Newton was standing on the shoulders of Simon Stevin, Pierre Fermat, Isaac Barrow, and John Wallis.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-01-04T14:03:11.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6895" PostTypeId="2" ParentId="2935" CreationDate="2018-01-04T14:13:47.930" Score="2" Body="&lt;p&gt;Let's consider each law seperatey.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;1) and 2): Gauss law for electricity and magnetism:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Its integral form was first formulated by Joseph-Louis Lagrange in 1773 &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss%27s_law#cite_note-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;(See here)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The next step, the Gauss divergence theorem, was also at first formulated by Joseph-Louis Lagrange in 1762 &lt;a href=&quot;https://en.wikipedia.org/wiki/Divergence_theorem#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;(See here)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;However, it may seem strange, that the credit to both the above equations is given to Gauss.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;3) Maxwell-Faraday equation:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This law is about electromagnetic induction which was discovered independently by both Faraday in 1831 and Joseph Henry in 1832. Faraday explained electromagnetic induction using a concept he called lines of force. However, all scientists at that time rejected him. Then Maxwell came along and mathematically expressed Faraday's theoretical ideas. &lt;em&gt;Thus this law of electromagnetic induction can be righteously called Maxwell-Faraday law.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It states that, whenever magnetic flux linked with a circuit changes then an induced electromotive force (emf) equal to the rate of change of magnetic flux is set up in the circuit.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;4) Ampere's circuital law:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Historically, there are two forms for this law. The one without displacement current term and the other with it. Both were introduced into physics for the first time by Maxwell in 1855 and 1861 respectively. Ampere has nothing to do with this law. I strongly believe if Ampere was alive at the time Maxwell presented this law, he will not accept it as it was opposed to his action at a distance Newtonian way of explaining electrodynamic phenomena. &lt;em&gt;Anyway, it is apparent that why Maxwell is credited for this law.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the above discussion we can only say that Maxwell could only be credited for two equations but not for all four. Most probably it was entirely due to Oliver Heaviside that these equations got their name. But there is a reason on why Maxwell is credited for these. In his 1865 paper &amp;quot;A Dynamical Theory of the Electromagnetic Field&amp;quot;, for the first time using field concept, he used these four equations to derive the electromagnetic wave equation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Thus these four equations bear and should bear Maxwell's name.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5313" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-04T14:13:47.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6897" PostTypeId="2" ParentId="6886" CreationDate="2018-01-04T18:11:19.450" Score="2" Body="&lt;p&gt;There are gigantic lists (Wikipedia) of men of the cloth in Mathematics and Science:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Catholic_clergy_scientists&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_Catholic_clergy_scientists&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Christians_in_science_and_technology&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_Christians_in_science_and_technology&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2141" LastActivityDate="2018-01-04T18:11:19.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6898" PostTypeId="2" ParentId="6883" CreationDate="2018-01-04T19:26:11.730" Score="4" Body="&lt;p&gt;There is, of course a wikipedia entry on pure mathematics and a ref to a &lt;a href=&quot;https://en.wikipedia.org/wiki/Sadleirian_Professor_of_Pure_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sadleirian Professor of Pure Mathematics&lt;/a&gt; since 1710. Arguably,  Plato may be the first to have emphasized the distinction number theory vs. computation ('arithmetic' vs. 'logistike')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is more or less obvious that pure maths somehow developed from 'impure' maths. The ultimate motives for the sexagesimal system adopted by Old Babylonian mathematics are its advantages for computation (especially fractions). But clay tablets as &lt;a href=&quot;https://en.wikipedia.org/wiki/YBC_7289&quot; rel=&quot;nofollow noreferrer&quot;&gt;YBC 7289&lt;/a&gt; or Plimpton 322 already present results which appear to be of purely academic interest. &#xA;The name geo-metry  adopted in Greece a thousand years later betrays also application as origin. The so called pythagoreans appear to be the first interested in numbers as such. Disregarding the current debate about their nonexistence (Burkert, Zhmud) they are a generation or two before Plato. &#xA;Greek astronomy originally was mostly 'calendrical' or observational  but presocratics thinkers already proposed models which came to be refined. Later Ptolemy's purely geometrical model generated realiable data, while the nested spheres adopted by peripatetics were used as an explanatory physical device. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton famously said in the preface of his &lt;em&gt;Principles&lt;/em&gt; that geometry is really mechanics. And geometry   as a natural science survived until the advent of non-Euclidean geometries which  decisively marked a branching of pure and applied forms of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2018-01-04T19:26:11.730" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6899" PostTypeId="1" AcceptedAnswerId="6900" CreationDate="2018-01-05T01:19:21.717" Score="4" ViewCount="528" Body="&lt;p&gt;The Runge-Kutta method is a milestone of computational mathematics. I can find the papepr by Runge, but not the one by Kutta. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By wikipedia, the title is &lt;strong&gt;Beitrag zur näherungweisen Integration totaler Differentialgleichungen&lt;/strong&gt; (1901)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone find it? &lt;/p&gt;&#xA;" OwnerUserId="2641" LastActivityDate="2018-01-26T23:50:31.337" Title="The original paper by Kutta" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6900" PostTypeId="2" ParentId="6899" CreationDate="2018-01-05T02:27:54.213" Score="8" Body="&lt;p&gt;Here is a facsimile of the journal: &lt;a href=&quot;https://archive.org/stream/zeitschriftfrma12runggoog#page/n449/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/stream/zeitschriftfrma12runggoog#page/n449/mode/2up&lt;/a&gt;.  See pp. 435f. &lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="5376" LastEditDate="2018-01-26T23:50:31.337" LastActivityDate="2018-01-26T23:50:31.337" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6902" PostTypeId="2" ParentId="6883" CreationDate="2018-01-05T11:38:58.937" Score="6" Body="&lt;p&gt;The earliest evidence we have of the distinction being made explicitly is in Plato's Republic, where he criticizes the manner in which geometers express themselves given the exalted (according to him) object of their study:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;They speak, I suppose, very laughably and perforce, for they mention squaring, applying, and adding, and state all their claims as if they are engaged in action and fashioning all their proofs for the sake of action; but the fact is, I presume, that the whole science is pursued for the sake of knowledge... That the knowledge pursued is of the eternally existent but not of&#xA;  what comes to be at any particular time and perishes.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Plato himself, of course, also encouraged geometers to &quot;save the phenomena&quot; of the heavens by reconstructing their apparent disorder from orderly circular motions. In answering his call Eudoxus created mathematical astronomy where rotating nested spheres were featured. But because epigones have to be holier than their authority Plato's successors at the Academy, notably his nephew Speusippus who took over after him, extended his disapproval from the manner of expression to entire parts of mathematics that used &quot;mechanical&quot; methods, such as rotations that produced curves Eudoxus and Archytas used to duplicate the cube. Proclus describes a debate over &quot;problems&quot; vs &quot;theorems&quot; in post-Plato's Academy, where Menaechmus (the inventor of conic sections) opposed Speusippus by defending geometers' methods, see &lt;a href=&quot;https://www.academia.edu/1985485/Menaechmus_versus_the_Platonists_Two_Theories_of_Science_in_the_Early_Academy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Menaechmus versus the Platonists by Bowen&lt;/a&gt;. In any case, by the time of Plutarch the distinction between mathematics of the intelligible (pure) and mathematics of the sensible (applied) already solidified. Plutarch writes in De vita Marcelli:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;For Eudoxus and Archytas, who embellished geometry with subtlety, initiated this highly regarded and famous art of mechanics, by supporting problems not easily solved with proof through discursive argument and diagram, by means of sensible and practical illustrations: for example, both reduced the problem&#xA;  concerning two mean proportionals, an element in many geometrical figures, to mechanical constructions in adapting certain mean proportionals from curved lines and sections.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;But, since Plato was displeased and opposed them on the ground that they destroyed and corrupted the good of geometry, which relapses from incorporeal, intelligible objects to sensibles and, moreover, uses bodies needing much vulgar manual labour, mechanics was thus distinguished as falling outside of geometry, and, since it was for a long time disregarded by philosophy, it has become one of the military arts.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Academy's influence, with its condemnation of applied as corrupted, was felt in antiquity despite the pushback from prominent mathematicians. Euclid deliberately avoids motion or any &quot;mechanics&quot; in the Elements, many of his proofs are more convoluted because he replaces use of congruence by constructing various auxiliary triangles. Hellenistic geometers, especially Archimedes and Apollonius, made a point of mixing mechanics and geometry, including the use of mechanical curves like spirals and helices. Apollonius even suggested some alternative proofs for Elements to relax Euclid's purity, see &lt;a href=&quot;https://www.academia.edu/8016013/Two_Approaches_to_Foundations_in_Greek_Mathematics_Apollonius_and_Geminus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi's Two Approaches to Foundations in Greek Mathematics:&#xA;Apollonius and Geminus&lt;/a&gt;. But even Archimedes shaped his work on the law of the lever and floating bodies into the Euclidean mold, and described his mechanical method for computing areas and volumes only in private letters, reproving them by Euclid-approved double reductio (&quot;method of exhaustion&quot;) in &quot;official&quot; works. That the Academic attitude survives to modern times can be seen from &lt;a href=&quot;https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hardy's Mathematician's Apology&lt;/a&gt;, where he exalts pure mathematics over applied on the familiar platonist grounds:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way. [...] A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. [...] The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. [...] Pure mathematics is on the whole distinctly more useful than applied. [...]  I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is often pointed out that, ironically, Hardy's number theoretic work was put to the amenity of the world very directly soon after his death, in public key cryptography.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-29T21:21:53.000" LastActivityDate="2018-03-29T21:21:53.000" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6904" PostTypeId="1" AcceptedAnswerId="6905" CreationDate="2018-01-06T01:37:17.290" Score="5" ViewCount="276" Body="&lt;p&gt;A much quoted passage from Gödel is the opening section of his &lt;em&gt;Remarks before the Princeton bicentennial conference on problems in mathematics (1946)&lt;/em&gt; where he praises Turing's Turing machine model of computation as being philosophically well founded:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://global.oup.com/academic/product/collected-works-9780195147216?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;Tarski has stressed in his lecture (and I think justly) the great importance of the concept of general recursiveness (or Turing's computability). It seems to me that this importance is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. In all other cases treated previously, such as demonstrability or definability, one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for. For the concept of computability, however, although it is merely a special kind of demonstrability or decidability, the situation is different. By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What strikes me as confusing is that Post published in 1944 his paper &lt;a href=&quot;http://www.ams.org/journals/bull/1944-50-05/S0002-9904-1944-08111-1/home.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Recursively enumerable sets of positive integers and their decision problems&lt;/em&gt;&lt;/a&gt; where he introduced the idea of Turing degrees which seem (to me) to distinguish exactly the type of orders of non-recursiveness that Gödel was arguing do not exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To compound my confusion even further, the introduction to Gödel's paper provided by Parsons in &lt;em&gt;Collected Works Vol. II&lt;/em&gt; seems to also ignore Turing degrees:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://global.oup.com/academic/product/collected-works-9780195147216?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;Gödel refers here not primarily to the equivalence of different formulations such as Turing computability, λ-definability and Herbrand-Gödel general recursiveness, but to the absence of the sort of relativity to a given language that leads to stratification of the notion, such as (in the case of definability in a formalized language) into definability in languages of greater and greater expressive power. Such stratification is driven by diagonal arguments. But, since a function enumerating the recursive functions is not recursive and there is no reason to think it computable, the diagonal function it gives rise to is simply non-recursive, rather than &quot;recursive at the next level&quot;.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The way Parsons phrases &quot;But, since a function enumerating the recursive functions is not recursive and there is no reason to think it computable, the diagonal function it gives rise to is simply non-recursive, rather than 'recursive at the next level'.&quot; sounds like a direct contradiction to the definition of Turing degree, and it surprises me greatly that he wrote this in (or at least it was published by) 1990. It also seems perfectly sensible to me to make a comparison to the stratification of Turing degrees and the stratification between $PA &amp;lt; ZFC &amp;lt; ZFC+Con(ZFC) &amp;lt; etc.$ which is what Gödel was originally talking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was Gödel not aware of Turing degrees in 1946? Or am I misunderstanding what Gödel and Parsons are arguing for?&lt;/p&gt;&#xA;" OwnerUserId="5664" LastEditorUserId="4251" LastEditDate="2022-09-23T06:02:56.407" LastActivityDate="2022-09-23T06:02:56.407" Title="Did Gödel know about Turing degrees in 1946?" Tags="|mathematical-logic|goedel|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6905" PostTypeId="2" ParentId="6904" CreationDate="2018-01-06T05:59:09.910" Score="7" Body="&lt;p&gt;I do not believe that degrees of computability were directly relevant to the topic of Gödel's 1946 lecture because they do not relativize computability in the relevant language dependent sense that Gödel stressed, and Parsons after him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stillwell gives a lucid explanation of the difference in this regard between Turing computability (solvability, recursiveness) and things like definability and decidability in his &lt;a href=&quot;https://www.jstor.org/stable/3219226&quot; rel=&quot;nofollow noreferrer&quot;&gt;Emil Post and His Anticipation of Gödel and Turing&lt;/a&gt;, and indicates that both Post and Gödel were aware of it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As we have seen, Post's starting point was the concept of computation, which he believed could be formalized and made subject to the diagonal argument. Diagonalization yields problems that are absolutely unsolvable, in the sense that no computation can solve them. In turn, this leads to relatively undecidable propositions, for example, propositions of the form $n\notin S_n$. No consistent formal system $F$ can prove all true propositions of this form, hence any such $F$ must fail to prove some true proposition $n_o\notin S_{n_0}$. But this proposition is only relatively undecidable, not absolutely, because&#xA;  $F$ can be consistently extended by adding it as an axiom.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Gödel did not at first believe in absolutely unsolvable problems, because he did not believe that computation is a mathematical concept. Instead, he proved the existence of relatively undecidable propositions directly, by constructing a kind of diagonal argument inside Principia Mathematica&quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What of &quot;degrees of computability&quot;? Turing hints at them in passing in a 1939 paper where he introduces oracles, but the first clear exposition is indeed Post's 1944 paper. However, it does not relativize recursiveness to a language, but introduces a new notion of &lt;em&gt;recursive reducibility&lt;/em&gt;, which can be formalized in terms of consulting Turing oracles. As Post explains in his paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;At certain times the otherwise machine determined process raises the question is a certain positive integer in a given recursively enumerable set $S_2$ of positive integers, and that the machine is so constructed that were the correct answer to this question supplied on every occasion that arises, the process would automatically continue to its eventual correct conclusion.  We could then say that the machine effectively reduces the decision problem of $S_1$&#xA;  to that of $S_2$.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But we do not get a larger class of recursive functions by extending the formalism, consulting an oracle is no such extension. Once we get into the Peano arithmetic the recursive is fixed, moving to ZFC or further changes nothing, non-recursive remains non-recursive. That some problems are recursively reducible to some other problems and not vice versa does establish a hierarchy, but it is not based on a hierarchy of languages. More expressive languages alter decidability and definability but they do not make uncomputable computable, that is absolute.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was Gödel aware of Post's paper in 1946 (not that it mattered for his praise of Turing)? Soare's &lt;a href=&quot;http://www.people.cs.uchicago.edu/~soare/Turing/shagrir.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turing-Post Relativized Computability&#xA;and Interactive Computing&lt;/a&gt; gives a detailed historical account of the developments from 1936 to 1954. Unfortunately, there is no direct answer. Post had a knack for being overlooked. In a sense, he proved the incompleteness theorem before Gödel, and described the Turing machine before Turing, but got little recognition for either (see Stillwell). Hao Wong, Gödel's closest fried and colleague who spent countless hours interviewing him, suggests that Gödel overlooked him too (unlike Church, he is not even mentioned):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Over  the  years  G  habitually  credited  A.M.  Turing’s  paper  of 1936 as the definitive work in capturing the intuitive concept [of computability], and did not mention Church or E. Post in this connection.  He must have felt that Turing was the only one who gave persuasive arguments to show the adequacy of the precise concept... In particular, he had probably been  aware  of  the&#xA;  arguments offered by Church for his “thesis” and decided that they were inadequate. It is clear that G and Turing (1912–1954) had great admiration for each other...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This would be particularly not surprising given the topic of 1946 lecture and the fact that earlier &quot;&lt;em&gt;Post did not attempt to prove that his formalism coincided with any  other  formalism,  such  as  general  recursiveness,  but  merely  expressed the expectation that this would turn out to be true&lt;/em&gt;&quot;. On the other hand, &quot;&lt;em&gt;it was only during the next phase from 1940 to 1954 that Post’s remarkable influence was fully felt. As Turing left the subject of pure computability theory in 1939, his mantle fell on the shoulders of Post&lt;/em&gt;&quot;. It is certainly likely that Gödel became aware of Post's and Kleene's work on recursive reducibility at least in 1950-s.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-07T20:45:22.063" LastActivityDate="2018-01-07T20:45:22.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6906" PostTypeId="1" AcceptedAnswerId="6907" CreationDate="2018-01-06T12:29:20.137" Score="4" ViewCount="1334" Body="&lt;p&gt;I heard that Newton used the Moon acceleration to come up with his universal law of gravity, but in order to calculate its acceleration you need the distance to the Moon. Whose measurement of this distance was used and how did they measure it?&lt;/p&gt;&#xA;" OwnerUserId="6865" LastEditorUserId="6884" LastEditDate="2018-01-08T14:26:55.403" LastActivityDate="2018-01-08T14:26:55.403" Title="Whose value of the distance to the Moon did Newton use?" Tags="|experimental-physics|gravity|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6907" PostTypeId="2" ParentId="6906" CreationDate="2018-01-06T13:50:16.697" Score="3" Body="&lt;p&gt;In Newton's time distance to the moon was not a problem: since the ancient times this distance (in terms of the Earth radius) was measured using parallax. Newton used the value 60 for the ratio of Moon's orbit to the Earth radius. This is close to the value 59 which can be found in Ptolemy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's problem was with Earth radius which was not known to young Newton with sufficient accuracy. For this reason his&#xA;first check of the gravitation law gave incorrect result, and Newton abandoned the subject. Only when a more precise measurement became available (by Picard), Newton saw that his gravitation theory is correct. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference. The article Newton's moon test by Douglas W. MacDougal in the book Newton's gravity, Springer 2012, gives the following values of the distance available in Newton's time: 59 (Ptolemy), 60 (Huygens), 60 1/3 (Copernicus), &#xA;60 2/5 (Street) and 60 1/2 (Tyho Brahe). Newton does not mention all these authors, just taking 60 as a &quot;well-known&quot; number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remarks. Since the question of size the the Earth is raised in the comments, let me comment on this too. The best measurement which Newton could use&#xA;is by Snell (6369 km), but Newton did not know this result&#xA;at the time of calculation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia claims that Biruni's measurement was accurate to 17 km (0.2%), which is hard to believe, given his method described in Wikipedia. I followed the links from Wikipedia article, and could not trace the reference for this&#xA;unbelievably precise number. Quick search hows various figures attributed to al-Biruni, and the ridiculously precise value cited in Wikipedia is certainly made up.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-01-06T22:34:51.847" LastActivityDate="2018-01-06T22:34:51.847" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6908" PostTypeId="2" ParentId="1796" CreationDate="2018-01-06T23:55:30.537" Score="-3" Body="&lt;p&gt;Just was at Pulkovo Astronomical Observatory last year, they definitely said that, first time the distance was measured, after the Lunokhod 1 delivered the Corner reflectors to the Moon. There are also atomic clocks, made with lamps in museum &lt;/p&gt;&#xA;" OwnerUserId="6876" LastActivityDate="2018-01-06T23:55:30.537" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6909" PostTypeId="1" AcceptedAnswerId="6917" CreationDate="2018-01-07T04:57:28.040" Score="2" ViewCount="181" Body="&lt;p&gt;I did manage to find technical drawings of the World's fair after considerable searching but probably I am not savvy enough to find the right items. In one of the drawings it does show what appears to be almost a geothermal structure of some type.  Other drawings clearly show the alternating current generation room , and in fact that was one of the main attractions at the fair, but I cannot make out where the drive shaft is going to make it spin. Does anyone happen to be a late 1800 historian of the &quot;alternating current wars&quot; will probably know off the top of his head.  Thank you &lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-01-08T12:57:09.087" Title="What did Westinghouse use for power to drive Tesla's generators during the 1892 World's Fair" Tags="|electricity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6912" PostTypeId="1" AcceptedAnswerId="6913" CreationDate="2018-01-07T17:37:50.923" Score="1" ViewCount="867" Body="&lt;p&gt;Did James Prescott Joule also design this thermodynamic workcycle, separate from George Brayton? I've not found any reference to Joule's involvement besides that from George Brayton. Could someone clarify?&lt;/p&gt;&#xA;" OwnerUserId="6880" LastEditorUserId="55" LastEditDate="2018-01-08T03:49:07.677" LastActivityDate="2018-01-08T03:49:07.677" Title="Why is the Brayton cycle also known as the Joule cycle? What was Joule's contribution?" Tags="|discoveries|thermodynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6913" PostTypeId="2" ParentId="6912" CreationDate="2018-01-07T21:54:57.337" Score="1" Body="&lt;p&gt;As is usual with technological inventions no one person invents the whole thing. Watt's steam engine was indebted to Newcommen's, and toy steam engines were known in antiquity and described by Heron. The same is true of the constant pressure cycle. Here is from &lt;a href=&quot;https://pdfs.semanticscholar.org/865e/5e53089bf4aad8e0acf39a40ce2bda71ace8.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comparative Study of Open and Closed Heat-Engines by Eames, Evans and Pickering&lt;/a&gt;, which includes references to Joule's and Brayton's original works:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The constant pressure cycle too has a long history. In 1791&#xA;  John Barber patented the first engine of this type, which incorporated an air compressor, a combustion chamber and an early type of turbine [4]. However, it was James Prescott Joule (1850) that provided the first theoretical description of the constant pressure cycle [5], more than 20 years before George Brayton&#xA;  invented his improvements [6]. It is interesting to note that both Joule’s and Brayton’s engines were reciprocating piston types and both the original work of Barber and the improvements by Brayton were based on internal combustion engines but that Joule’s own test engine was externally heated.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Eckardt, D. Gas Turbine Powerhouse; De Gruyter Oldenbourg: Berlin, Germany, 2014.&lt;/li&gt;&#xA;&lt;li&gt;Joule, J.P. On Air-Engines; Report of the British Association: Manchester, UK; 19; June; 1851.&lt;/li&gt;&#xA;&lt;li&gt;Brayton, G. Improvements in Gas-Engines. U.S. Patent 125166A, 2 April 1871.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/108534&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Air Engine&lt;/a&gt; was published in the Philosophical Transactions of the Royal Society of London the next year, 1852, as a joint work with William Thomson, a.k.a Lord Kelvin, just two years after Joule's most famous paper &lt;a href=&quot;https://www.jstor.org/stable/108427&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Mechanical Equivalent of Heat&lt;/a&gt;. Joule did not actually construct the engine and perform experiments on it, but he described it in detail, and gave a mathematical theory of its work developed with Thomson's help. For more on Joule and Thomson's joint work, which clarified the foundations of modern thermodynamics, see &lt;a href=&quot;http://rsnr.royalsocietypublishing.org/content/64/1/43&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Joule, William Thomson and the Concept of a Perfect Gas by Rowlinson&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-07T22:26:12.023" LastActivityDate="2018-01-07T22:26:12.023" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6914" PostTypeId="2" ParentId="6886" CreationDate="2018-01-08T10:23:54.183" Score="4" Body="&lt;p&gt;An interesting story is the one of &lt;a href=&quot;https://en.wikipedia.org/wiki/Matteo_Ricci&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matteo Ricci&lt;/a&gt;, a missionary jesuit in China. He wrote the first Chinese translation of Euclid's &lt;em&gt;Elements&lt;/em&gt;. He was also a cartographer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding real analysis, &lt;a href=&quot;https://en.wikipedia.org/wiki/Pietro_Mengoli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pietro Mengoli&lt;/a&gt; proposed one of the most influential problem in the early days of calculus: the Basel problem, famously solved by Euler. Also, &lt;a href=&quot;https://en.wikipedia.org/wiki/Francesco_Fa%C3%A0_di_Bruno&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francesco Faà di Bruno&lt;/a&gt; was an avid mathematical researcher as well as a devoted priest and theologian.&lt;/p&gt;&#xA;" OwnerUserId="6884" LastActivityDate="2018-01-08T10:23:54.183" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6915" PostTypeId="1" AcceptedAnswerId="6918" CreationDate="2018-01-08T10:32:27.460" Score="8" ViewCount="752" Body="&lt;p&gt;Euler is often credited with introducing the notation &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, and people  &lt;a href=&quot;https://hsm.stackexchange.com/a/5204/3462&quot;&gt;cite&lt;/a&gt; the example &lt;span class=&quot;math-container&quot;&gt;$f(\frac{x}{a}+c)$&lt;/span&gt;, where he &lt;em&gt;had&lt;/em&gt; to use parentheses around the function argument. On the other hand, when the argument was a single letter like &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, I have mainly seen Johann Bernoulli and Euler just write &lt;span class=&quot;math-container&quot;&gt;$f\, x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f\colon x$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$\phi\, x$&lt;/span&gt;), &lt;em&gt;without&lt;/em&gt; the parentheses. If I recall correctly even Lagrange in his lectures introduced the function notation without parentheses.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Did Euler (or Johann Bernoulli) ever write &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;In case the answer is no, the follow up question is: when did it become standard to put parentheses around &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="4251" LastEditDate="2021-10-07T13:05:45.277" LastActivityDate="2021-10-07T13:05:45.277" Title="Did Euler ever write $f(x)$, with parentheses?" Tags="|mathematics|mathematicians|notation|euler|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="6916" PostTypeId="2" ParentId="663" CreationDate="2018-01-08T11:44:56.870" Score="4" Body="&lt;p&gt;As to your question &quot;why did the university [at Göttingen] become so prevalent&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer is surely &quot;Felix Klein&quot;. A dynamic, versatile, and brilliant mathematician, he was also a talented organizer and administrator. The list of first-rate mathematicians he brought to Göttingen is probably too long to fit within the allowable limits of a SE &lt;em&gt;answer&lt;/em&gt;; some details can be found &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Klein&quot; rel=&quot;nofollow noreferrer&quot;&gt;at wiki&lt;/a&gt;. As editor he similarly transformed &lt;em&gt;Mathematische Annalen&lt;/em&gt; into a leading journal.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-01-08T12:02:01.653" LastActivityDate="2018-01-08T12:02:01.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6917" PostTypeId="2" ParentId="6909" CreationDate="2018-01-08T12:57:09.087" Score="3" Body="&lt;p&gt;The  &lt;a href=&quot;http://www.teslasociety.com/columbia_expo.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tesla Society&lt;/a&gt; has a photo of a &quot;Allis-Corliss Engine.&quot;&lt;br&gt;&#xA;Wikipedia identifies this as a large steam-powered engine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And over at &lt;a href=&quot;http://wiki.vintagemachinery.org/Steam-Engines-at-the-World-s-Fair-Parts-1-2.ashx&quot; rel=&quot;nofollow noreferrer&quot;&gt;vintagemachinery.org&lt;/a&gt;,  it is stated explicitly that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The duty of the World's Fair [Corliss] engine will, however, be of a different&#xA;  nature from that of the Corliss engine at the Centennial, which&#xA;  transmitted its power to the main jack shaft. The Allis engine is&#xA;  belted by two belts made by the Page Belting Company, of Concord, N.&#xA;  H., to two Westinghouse dynamos, each of a capacity of 10,000&#xA;  incandescent light. The speed necessary for these machines to produce&#xA;  20,000 light of sixteen candle-power is 200 revolutions. The&#xA;  illustration shows only one belt, but when completed the second belt&#xA;  will be directly over the f1rst connected to the second dynamo&#xA;  immediately beyond. The fly-wheel is thirty feet in diameter, and at&#xA;  sixty revolutions, will have a periphery speed of over 5500 feet per&#xA;  minute. The face of the wheel is seventy-six inches, not as large as&#xA;  many others the Allis Company have built, but in construction it is&#xA;  probably stronger than any of its size ever made. It has twelve arms&#xA;  and the rim is made up of the same number of segments. It weighs&#xA;  complete 135,900 pounds, the rim itself weighing 88,000 pounds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd add in passing that, having lived on Chicago's South Side only a few blocks from the Midway,  I can attest there's no chance of a Hydro-powered source anywhere near there :-) &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-01-08T12:57:09.087" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6918" PostTypeId="2" ParentId="6915" CreationDate="2018-01-08T14:01:42.490" Score="6" Body="&lt;p&gt;I’m guessing &lt;strong&gt;no&lt;/strong&gt;. But how does one make sure? (Maybe with &lt;a href=&quot;http://www.springer.com/birkhauser/history+of+science?SGWID=0-40295-6-121672-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;85+&lt;/a&gt; volumes of clean pdfs...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cajori, who started that $f(\frac xa+c)$ example, &lt;a href=&quot;https://archive.org/stream/AHistoryOfMathematicalNotationVolII#page/n292&quot; rel=&quot;nofollow noreferrer&quot;&gt;points out&lt;/a&gt; a $\varphi(z)$ in &lt;strong&gt;D’Alembert&lt;/strong&gt; (&lt;a href=&quot;https://books.google.com/books?id=Bw8OAAAAQAAJ&amp;amp;pg=PA50&quot; rel=&quot;nofollow noreferrer&quot;&gt;1754, p. 50&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For “standard”, I would say &lt;strong&gt;Lacroix&lt;/strong&gt; (&lt;a href=&quot;https://archive.org/stream/traitducalculdi02lacrgoog#page/n128&quot; rel=&quot;nofollow noreferrer&quot;&gt;1797, p. 87&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;4. Pour représenter une fonction sans indiquer, en aucune manière comment elle peut être composée, je me servirai de la caractéristique&#xA;  $\mathrm f$; et il faudra entendre, par l'expression $\mathrm f(x)$, une fonction&#xA;  quelconque de $x$, en comprenant sous cette dénomination tout ce  que&#xA;  comporte la définition du mot &lt;em&gt;fonction&lt;/em&gt; (&lt;em&gt;Intr.&lt;/em&gt; nº 1) : on  doit donc&#xA;  bien se garder de prendre la lettre $\mathrm f$ pour un coefficient de $x$.&#xA;  J’indiquerai la substitution de $x+k$ aulieu de $x$ dans $\mathrm f(x)$,  en&#xA;  écrivant $\mathrm f(x+k)$, et cela voudra dire que le résultat est composé en&#xA;  $x+k$, comme la fonction primitive l’est en $x$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Side remark tying into your &lt;a href=&quot;https://hsm.stackexchange.com/questions/6104/who-first-considered-the-f-in-fx-as-an-object-in-itself-and-who-decided-t&quot;&gt;other question&lt;/a&gt;: This book of Lacroix writes “the function $f$” very often; e.g. pp. 93, 212, 258, 483–496, 502, mainly when describing results of Monge who also did this a lot (but avoided unnecessary parentheses). I think “$f$” all started with solutions of PDEs depending on “arbitrary functions” — though only Dedekind, I would say, made them “objects” in the sense you want at the other question.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;br&gt;In E213 “Remarques sur les mémoires précedens de M. Bernoulli” (1755), just quoted &lt;a href=&quot;https://hsm.stackexchange.com/questions/6206/is-the-prime-notation-for-derivatives-f-due-to-euler&quot;&gt;elsewhere&lt;/a&gt;, you can see Euler “forget” his evaluation colon and slip into writing $\Phi'(x)$ (p. 215) and eventually $\Phi(x)$ (&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1753-pub1755#page/n218&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 216&lt;/a&gt;). Same thing in E441 (1773, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092611839&amp;amp;num=429&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 429&lt;/a&gt;). So in the end, &lt;strong&gt;yes&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-06-04T12:06:13.997" LastActivityDate="2018-06-04T12:06:13.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="6922" PostTypeId="1" CreationDate="2018-01-09T01:06:42.470" Score="6" ViewCount="458" Body="&lt;p&gt;Cauchy's Rigidity theorem says that if the corresponding faces of two convex polytopes are isometric (congruent)&#xA;then the polytopes are related by a (proper or improper) motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cauchy's biography (by Bruno Belhoste, Springer 1991) says that this statement&#xA;is in Euclid book &quot;9, Theorem 11&quot;. I checked: Proposition 11 in book 9 is not related to this. In general, book 9 is not about stereometry. English and French Wikipedia also say it is in Euclid, but do not say where exactly.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone tell where in Euclid can this (or similar, or related) statement be found?&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2020-07-20T12:13:08.750" LastActivityDate="2020-07-20T12:13:08.750" Title="Has Euclid stated Cauchy's theorem?" Tags="|mathematics|geometry|euclid|" AnswerCount="1" CommentCount="1" FavoriteCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6924" PostTypeId="2" ParentId="6922" CreationDate="2018-01-09T02:53:37.117" Score="8" Body="&lt;p&gt;I found it. It is Definition 10 in Book XI.&#xA;&lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/elements/bookXI/defXI9.html&quot; rel=&quot;noreferrer&quot;&gt;https://mathcs.clarku.edu/~djoyce/elements/bookXI/defXI9.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid takes the assumption of Cauchy's theorem as the definition of equality of&#xA;polytopes! Evidently he did not see the difficulty which is involved here.&#xA;(Cauchy's theorem is highly non-trivial, and far from evident).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-01-09T18:46:59.930" LastActivityDate="2018-01-09T18:46:59.930" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6925" PostTypeId="1" CreationDate="2018-01-09T07:37:25.080" Score="9" ViewCount="1903" Body="&lt;p&gt;This question is prompted by (comments at) &lt;a href=&quot;https://hsm.stackexchange.com/questions/6882/interpretation-of-w-in-boltzmanns-entropy-formula&quot;&gt;another one&lt;/a&gt;. There, I was surprised to find that despite traditional claims to the contrary, Boltzmann himself &lt;em&gt;did&lt;/em&gt; once write his formula $S=k\log W$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\hspace{11em}$&lt;img src=&quot;https://i.stack.imgur.com/lK2K2.png&quot; width=&quot;232&quot; height=&quot;73&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That’s in his book (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?30.0818.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1898&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/vorlesungenber02bolt#page/172&quot; rel=&quot;nofollow noreferrer&quot;&gt;§61, p. 172&lt;/a&gt;), with a pointer to (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?26.1056.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;1896&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/vorlesungenber01bolt#page/60&quot; rel=&quot;nofollow noreferrer&quot;&gt;§8, p. 60&lt;/a&gt;) where he  says the same thing in words, and emphasizes that $RM$ is &lt;em&gt;the same constant for all gases&lt;/em&gt;. In fact, if we note that his $R$ is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ideal_gas_law#Molar_form&quot; rel=&quot;nofollow noreferrer&quot;&gt;specific&lt;/a&gt; gas constant (equal to $P\ /\ \rho T$ by the gas law, p. 53) and his $M$ the molecular mass $\rho V\ /\ N$, we see on multiplying that his $RM$ is indeed our $PV\ /\ NT=k$. So all seems well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But now: there is another common story, which we get to hear again as 2018 is the year the kelvin should be &lt;a href=&quot;https://doi.org/10.1098/rsta.2015.0037&quot; rel=&quot;nofollow noreferrer&quot;&gt;redefined by freezing $k$&lt;/a&gt;. E.g. White and Fischer (&lt;a href=&quot;https://doi.org/10.1088/0026-1394/52/5/S213&quot; rel=&quot;nofollow noreferrer&quot;&gt;2015&lt;/a&gt;) (emphasis mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although Boltzmann published his famous definition of entropy in &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044093294908&amp;amp;num=373&quot; rel=&quot;nofollow noreferrer&quot;&gt;1877&lt;/a&gt;, the constant of proportionality in Boltzmann’s definition, was not identified as Boltzmann’s constant until 1900 when Planck published his analysis of blackbody radiation (&lt;a href=&quot;http://archive.org/stream/verhandlungende01goog#page/n212&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900a&lt;/a&gt;, &lt;a href=&quot;http://archive.org/stream/verhandlungende01goog#page/n247&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900b&lt;/a&gt;), where he identified the constant as $k$ and &lt;strong&gt;named it after Boltzmann&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And that, again, seems not true. Of course Planck does introduce $k$ — in (1900b, p. 241), after $h$, as “a second constant of nature” such that an entropy is $k\log\mathfrak R_0$. But he does not name it after Boltzmann there — nor, unless I missed it, in any of the obvious or oft-quoted places. Not in the follow-up articles (&lt;a href=&quot;https://books.google.com/books?id=KhtbAAAAYAAJ&amp;amp;pg=PA553&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901a&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=KhtbAAAAYAAJ&amp;amp;pg=PA564&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901b&lt;/a&gt;). Not in the Boltzmann &lt;em&gt;Festschrift&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/festschriftludw00meyegoog#page/n133&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;: no mention of $k$). Not in his books (&lt;a href=&quot;https://archive.org/stream/uberdietheoewarm00planrich#page/n152&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=fjrQAAAAMAAJ&amp;amp;pg=PA55&amp;amp;dq=%22Konstante%20k%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/vorlesungenberd01plangoog#page/n137&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=tMzPAAAAMAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22universelle+Faktor+k%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1930&lt;/a&gt;). Not in his Nobel lecture (&lt;a href=&quot;https://archive.org/stream/MaxPlanckVortraegeRedenErinnerungen#page/n26&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This constant is frequently termed Boltzmann’s constant, although to the best of my knowledge &lt;strong&gt;Boltzmann himself never introduced it&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not in his autobiography (&lt;a href=&quot;http://archive.org/stream/MaxPlanckVortraegeRedenErinnerungen#page/n69&quot; rel=&quot;nofollow noreferrer&quot;&gt;1948&lt;/a&gt;) where $k$ is “the so-called absolute gas constant”. And most remarkably — even if meant with self-deprecating irony — not in his recollections (&lt;a href=&quot;http://archive.org/stream/MaxPlanckVortraegeRedenErinnerungen#page/n24&quot; rel=&quot;nofollow noreferrer&quot;&gt;1943&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I (...) was not even taken seriously, in some places. But I did not let such doubts deter me from trusting &lt;strong&gt;my constant&lt;/strong&gt; $k$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The literature also seems to have settled slower than legend has it — see chronology in the CW answer below. (A name Planck &lt;em&gt;did&lt;/em&gt; propose in (1900b, p. 245) is “Boltzmann-Drude constant” for $\alpha=3k/2$, but few besides Abraham (&lt;a href=&quot;https://www.emis.de/cgi-bin/JFM-item?36.0906.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/142unkngoog#page/n375&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 284, 362&lt;/a&gt;) seem to have adopted it — e.g. Perrin (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.31822017841578&amp;amp;num=18&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;) calls $\alpha$ “la constante d’énergie moleculaire”.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: If not in 1900, when did $k$ get its name? Was there ever a debate (e.g. after the unveiling of Boltzmann’s famous tombstone)? Was there a concerted decision? Were the above-quoted pages of Boltzmann’s book ever invoked? Or was it no-one’s doing, just &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck%27s_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;resolution by attrition&lt;/a&gt;? Finally, if Planck did not &lt;em&gt;name the constant after Boltzmann&lt;/em&gt;, how did we end up with the tale that he did?&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-01-17T15:11:11.553" LastActivityDate="2018-02-26T18:09:31.873" Title="When did the name “Boltzmann constant” prevail, and how?" Tags="|statistical-mechanics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6926" PostTypeId="2" ParentId="6915" CreationDate="2018-01-09T09:51:40.950" Score="2" Body="&lt;p&gt;You can find all of Euler's original papers in the &lt;a href=&quot;http://eulerarchive.maa.org/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler archive&lt;/a&gt;. Glancing through his later papers does not yield a hit. But if you are interested enough you can probably exclude that notation for all his writings.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-01-09T09:51:40.950" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6927" PostTypeId="2" ParentId="5068" CreationDate="2018-01-09T18:38:54.820" Score="3" Body="&lt;p&gt;The best answer to that question is the following book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The steam engine of Thomas Newcomen&#xA;by L. T. C. Rolt, J. S. Allen&#xA;1977&#xA;Moorland Pub. Co. ; New York : Science History Publications, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize what I learned from this book.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The boilers did not generate enough steam pressure to move pistons.  You needed to use the weight of the atmosphere for the power stroke.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Even if the boiler could generate high steam pressures, the art of machining pistons and cylinders was still in its infancy&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Kevin Olsen&lt;/p&gt;&#xA;" OwnerUserId="6890" LastActivityDate="2018-01-09T18:38:54.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6928" PostTypeId="2" ParentId="252" CreationDate="2018-01-11T09:06:46.817" Score="1" Body="&lt;p&gt;I don't think it is quite so clear cut as the popular opinion (&quot;Euler thought only of symbolic expressions, while Dirichlet gave the modern definition&quot;) makes us believe. Consider for example this definition of functions from Eulers later work, &lt;a href=&quot;https://archive.org/details/bub_gb_sYE_AAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Institutiones calculi differentialis&lt;/em&gt;&lt;/a&gt;, 1755, Preface p.VI::&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thus when some quantities so depend on other quantities, that if the&#xA;  latter are changed the former undergo change, then the former&#xA;  quantities are called functions of the latter ; this definition&#xA;  applies rather widely, and all ways, in which one quantity could be&#xA;  determined by others, are contained in it. If therefore $x$ denotes a&#xA;  variable quantity, then all quantities, which depend upon $x$ in any&#xA;  way, or are determined by it, are called functions of it.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Examples are $x^{2}$, the square of $x$, or any other powers of $x$,&#xA;  and indeed, even quantities that are composed with these powers in any&#xA;  way, even transcendentals, in general, whatever depends on $x$ in such&#xA;  a way that when $x$ increases or decreases, the function changes. From&#xA;  this fact there arises a question; namely, if the quantity $x$ is&#xA;  increased or decreased, by how much is the function changed, whether&#xA;  it increases or decreases?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To my mind this is not substantially different from what Dirichlet said. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, Dirichlet never talked of sets or the domain or codomain of a map, nor did he call the &quot;rule&quot; the function, as the modern definition you find in all books does. See also &lt;a href=&quot;https://hsm.stackexchange.com/q/6104/3462&quot;&gt;&lt;em&gt;Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2018-01-11T09:06:46.817" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6929" PostTypeId="1" CreationDate="2018-01-11T10:28:02.253" Score="5" ViewCount="95" Body="&lt;p&gt;Fermat had a friend at Toulouse named Lalouvère.  Lalouvère  was censor, jesuit, and mathematician (in alphabetical order). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Antonella Romano writes on page 512 of her book &lt;em&gt;La Contre-Réforme Mathématique. Constitution et diffusion d'une culture mathématique jésuite à la Renaissance (1540-1640)&lt;/em&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Un premier constat regarde les jésuites confrontés à la censure : sur tous les cas du Fondo Gesuitico qui concernent la France, deux professeurs seulement appartiennent à la liste établie dans le cadre de cet ouvrage, B. Labarthe et V. Léotaud. Si d'autres mathématiciens y apparaissent, c'est tout aussi exceptionnellement, et au titre de censeur, comme Antoine Lalouvère 114. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Footnote 114 there reads: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;C'est lui qui porte un jugement négatif sur l'ouvrage de B. Labarthe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On on page 515 Romano writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Antoine Lalouvère, le mathématicien toulousain qui s'est illustré dans le débat sur la cycloïde128, émet un jugement négatif sur le &lt;em&gt;Hermetis Mathematici praeludium&lt;/em&gt; de B. Labarthe.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, Lalouvère in his capacity as a censor sank at least one mathematical book, namely that by his fellow jesuit Labarthe. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Questions 1.&lt;/em&gt;&lt;/strong&gt; Why did Lalouvère report negatively on Labarthe's book?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question 2.&lt;/em&gt;&lt;/strong&gt; Were there other books reviewed negatively by him?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be of particular interest to determine whether or not the grounds for Lalouvère's negative report on Labarthe's book had to do with possible use of indivisibles or the infinitely small.  See a &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9542-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;related 2018 publication in &lt;em&gt;Foundations of Science&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question was posted &lt;a href=&quot;https://mathoverflow.net/q/289685/28128&quot;&gt;at MO&lt;/a&gt; with limited success.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-01-17T11:39:07.490" LastActivityDate="2018-01-17T11:39:07.490" Title="Nature of Fermat's friend Lalouvère's activities as censor?" Tags="|mathematics|mathematicians|biographical-details|social-context|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6930" PostTypeId="2" ParentId="6925" CreationDate="2018-01-11T11:32:17.467" Score="4" Body="&lt;p&gt;&lt;strong&gt;Q: When did the name “Boltzmann constant” prevail, and how?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Herzfeld's review article from 1920 for the &lt;A HREF=&quot;https://doi.org/10.1007/978-3-663-16016-8_12&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyklopädie der Mathematischen Wissenschaften&lt;/A&gt; the symbol $k$ is called Boltzmann's constant without further discussion, so this must have been common usage at that time. This implies that there is no connection with Boltzmann's tombstone, which is from 1933 (as described &lt;A HREF=&quot;https://books.google.nl/books?id=iIyKTjZLEc8C&amp;amp;pg=PA231&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/A&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tetrode in his 1912 &lt;A HREF=&quot;http://dx.doi.org/10.1002/andp.19123430708&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/A&gt; still refers to Planck's constant (and makes a point that his $\ln$ is Planck's $\log$), so the formula $S=k\log W$ was still a novelty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would conclude that the transition from Planck's constant to Boltzmann's constant happened in the 1912-1920 time frame.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning the role of Lorentz in the naming of the constant, I note that in a 1917 paper entitled &lt;em&gt;Some remarks on the theory of monatomic gases&lt;/em&gt;, Lorentz refers to $k$ as &quot;Planck's well-known coefficient&quot; (&lt;A HREF=&quot;http://www.dwc.knaw.nl/DL/publications/PU00012398.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/A&gt;, page 47). I did not find a later reference from Lorentz, however, in the Ph.D. thesis of his student Van de Sande Bakhuyzen, from 1921, the constant $k$ is called &quot;Boltzmann's constant&quot; (&lt;A HREF=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/sources/vandeSandeBakhuyzen_1921.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/A&gt;, page 39).$^*$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this is consistent with the idea that the name Boltzmann constant emerged within a few years before 1920.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;One might ask what prompted the switch from Planck constant to Boltzmann constant. This screen shot from Van de Sande Bakhuyzen's Ph.D. thesis suggests the obvious answer: once quantum mechanics and statistical mechanics came together, a single formula would contain both $h$ and $k$ and different names were needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;IMG SRC=&quot;https://ilorentz.org/beenakker/MO/Bakhuyzen.png&quot;/&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;$^*$ I also searched for references to Boltzmann's constant in other Ph.D. theses of Lorentz's students (online &lt;A HREF=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/lorentz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/A&gt;):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;A HREF=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/sources/Fokker_1913.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fokker&lt;/A&gt;&#xA;(1913): calls $k$ the single-molecule gas constant (page 20) &lt;/li&gt;&#xA;&lt;li&gt;&lt;A HREF=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/sources/Bruins_1918.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bruins&lt;/A&gt;&#xA;(1918): uses the symbol $s$ instead of $k$ and calls it the gas constant of a molecule (page 92) &lt;/li&gt;&#xA;&lt;li&gt;&lt;A HREF=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/sources/vanSlingelandt_1919.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;van&#xA;Slingelandt&lt;/A&gt; (1919): on page 39 writes $\eta =\frac{R}{N}\log W$ and calls it&#xA;the &quot;principle of Boltzmann&quot;, but without introducing a separate&#xA;notation for $R/N$.&lt;/li&gt;&#xA;&lt;li&gt;&lt;A HREF=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/sources/vandeSandeBakhuyzen_1921.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;van de Sande Bakhuyzen&lt;/A&gt; (1921) finally calls $k$ the &quot;constant of Boltzmann&quot; (page 39).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So 1920 seems to be the &quot;phase transition&quot; year, at least for Lorentz's students.&lt;/p&gt;&#xA;" OwnerUserId="1697" LastEditorUserId="1697" LastEditDate="2018-01-19T14:52:36.717" LastActivityDate="2018-01-19T14:52:36.717" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6932" PostTypeId="1" CreationDate="2018-01-12T03:59:12.660" Score="2" ViewCount="1010" Body="&lt;p&gt;Why its true that the Egyptians did not see much developments. It has been said that the Babylonians were equal to the Greeks in mathematical achievement in terms of having an axiomatic, deductive system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would you agree with that claim? If not, why is there scholarly debate about, or more precisely what is the scholarly debate about in terms of this subject.&lt;/p&gt;&#xA;" OwnerUserId="6459" LastEditorUserId="6459" LastEditDate="2018-01-12T21:28:05.823" LastActivityDate="2018-01-21T17:18:02.060" Title="Was Babylonian Mathematics as sophisticated as Greek mathematics?" Tags="|mathematics|ancient-greece|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="6933" PostTypeId="2" ParentId="6932" CreationDate="2018-01-12T04:52:29.620" Score="4" Body="&lt;p&gt;The short answer is certainly no. Mathematics as it is practiced today has its own unique method of telling what is true and what is not. This method is called &quot;mathematical proof&quot;. This is what distinguishes mathematics from all other activities. No other culture practiced any form of mathematical proof before Greeks. Moreover, there is absolutely no indication that  that any culture invented it independently before or after. In this precise sense the Greeks invented mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course one can use a broader definition of mathematics, to include all activities related to counting and measuring something. Then every culture had some form of this of course. But even with this broad definition no other culture stands any comparison with the Greeks in its contribution to mathematics that we practice and study today.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-01-12T04:52:29.620" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6934" PostTypeId="1" CreationDate="2018-01-13T00:23:39.683" Score="5" ViewCount="277" Body="&lt;p&gt;A friend of mine pointed out that there were no graphs in Adam Smith's &lt;em&gt;The Wealth of Nations&lt;/em&gt;, which was published in 1776. This surprised me because René Descartes (1596-1650) is well known as being the inventor of the coordinate plane, but I understand now he never actually graphed an equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who was the first known mathematician to graph an equation? And if it differs, who was the first to “popularize” them?&lt;/p&gt;&#xA;" OwnerUserId="6904" LastEditorUserId="5989" LastEditDate="2018-01-13T13:06:31.787" LastActivityDate="2018-01-13T13:06:31.787" Title="Who was the first known mathematician to graph an equation?" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6936" PostTypeId="1" AcceptedAnswerId="6937" CreationDate="2018-01-13T12:30:33.973" Score="7" ViewCount="591" Body="&lt;p&gt;Who first proved the interchangeability of partial derivatives? I never see any reference in textbooks. This is not a trivial result. &lt;/p&gt;&#xA;" OwnerUserId="2641" LastActivityDate="2022-09-30T18:14:22.523" Title="Who first proved that $f_{xy} = f_{yx}$?" Tags="|mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6937" PostTypeId="2" ParentId="6936" CreationDate="2018-01-13T13:31:37.760" Score="9" Body="&lt;p&gt;It looks as if it was Euler who first proved it. See &lt;em&gt;&lt;a href=&quot;https://web.archive.org/web/20170923003138/http://mathforum.org/kb/thread.jspa?forumID=13&amp;amp;threadID=1580215&amp;amp;messageID=5765194&quot; rel=&quot;nofollow noreferrer&quot;&gt;A note on the history of mixed partial derivatives&lt;/a&gt;&lt;/em&gt;, by Thomas James Higgins (&lt;em&gt;Scripta Mathematica&lt;/em&gt; &lt;strong&gt;7&lt;/strong&gt; (1940), pp. 59–62).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="2277" LastEditDate="2022-09-30T18:14:22.523" LastActivityDate="2022-09-30T18:14:22.523" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6938" PostTypeId="2" ParentId="6936" CreationDate="2018-01-13T16:07:45.460" Score="5" Body="&lt;p&gt;This question could also be asked as: Who first found an example for the not-interchangeability of partial derivatives?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was H.A. Schwarz who proved the theorem: If a function $f:\mathbb{R}^n \rightarrow\mathbb{R}^n$ is $m$ times differentiable &lt;em&gt;and continuous&lt;/em&gt;, then the $m$th mixed derivatives are independent of the order.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-01-13T16:07:45.460" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6939" PostTypeId="1" CreationDate="2018-01-15T06:18:37.197" Score="1" ViewCount="6046" Body="&lt;p&gt;I wondered if back in the time of ancient Greeks mathematicians, 3D geometry was discovered as result of plane geometry? (Was there anything in the axioms of plane geometry that indicated existence of 3D geometry?) or was 3D geometry a familiar concept already and there were simply attempts to find its axioms?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Basically, at the time of ancient Greeks mathematicians was geometry discovered mathematically or was it familiar concepts for which axioms were derived? &lt;/p&gt;&#xA;" OwnerUserId="5866" LastActivityDate="2018-02-15T22:11:27.137" Title="How was geometry with 3 dimensions discovered/invented?" Tags="|mathematics|euclidean-geometry|geometry|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6940" PostTypeId="2" ParentId="6939" CreationDate="2018-01-15T08:34:20.097" Score="4" Body="&lt;p&gt;Three-dimensional Geometry is what the last three books of Euclid's &lt;em&gt;Elements&lt;/em&gt; are about. No, it was not discovered mathematically, at least not in sense that it was seen as a generalization of two-dimensional Geometry. Already in ancient Egypt and Babylon it was known how to compute the volume of a truncated quadrangular pyramed, for instance, long before axioms were introduced in Geometry. What Euclid added was an axiomatic approach to an already known subject, besides, possibly, adding a few new theorems of his own.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-01-15T08:34:20.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6941" PostTypeId="1" CreationDate="2018-01-15T16:48:36.593" Score="1" ViewCount="159" Body="&lt;p&gt;I read a lot of historical articles that doubt the possibility that Gauss saw in the pseudosphere the realization of hyperbolic geometry; that geodetic triangles on the pseudosphere obey the same trigonometric rules valid in hyperbolic geometry (that can be modelled by the hyperbolic plane). Even proponents of the view that Gauss was in fact at full possesion of non-euclidean geometry, like Felix Klein (see: &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.bams/1183417390&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://projecteuclid.org/download/pdf_1/euclid.bams/1183417390&lt;/a&gt;), doubt whether or not Gauss saw this connection, despite that to my opinion it's one of the more obvious conclusions of hyperbolic geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect he did see this connection, simply because he possesed all the essential elements needed in order to make this conclusion: &lt;strong&gt;he proved Gauss-Bonnet theorem&lt;/strong&gt; for geodetic triangles on any surface, from his differential geometry &lt;strong&gt;he developed the equations of the pseudosphere&lt;/strong&gt; (he called it &quot;counterpart of the sphere&quot;) - the first example of a surface with constant negative curvature, and in his 1832 letter to Janos Bolyai &lt;strong&gt;he provided a synthetic proof of the angular deficit theorem in the hypothetic non-euclidean geometry (hyperbolic geometry)&lt;/strong&gt;. So if he knew both the angular deficit theorem of hyperbolic geometry and the Gauss-Bonnet theorem of differential geometry, and also had in hand an example of a surface with constant negative curvature, why there is still a long way to make this cnclusion?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask this question simply because i want to gain additional understanding of the subject of hyperbolic geometry. It's a simple fact that people tend to believe what they want to believe, and i personally have huge admiration for Gauss (so i  preffer to believe he did make this conclusion), but i still want to gain better insight into this riddle. &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2018-01-15T19:25:39.637" LastActivityDate="2018-01-16T17:10:39.080" Title="Why is there some doubt whether or not Gauss saw the pseudosphere as the embodiment of hyperbolic geometry?" Tags="|biographical-details|geometry|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6942" PostTypeId="1" CreationDate="2018-01-15T13:31:20.503" Score="4" ViewCount="151" Body="&lt;p&gt;I am looking into the historical perspective of how the concept of work and energy came about: who coined the terms &quot;mechanical work&quot; and &quot;energy&quot;, and how the concept evolved over time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that James Joule developed a relation between mechanical work and the mechanical equivalent of heat, but was he the first to use this term, or was it in use before him?&lt;/p&gt;&#xA;" OwnerDisplayName="Rajneesh Singh" LastEditorUserId="62" LastEditDate="2018-01-16T01:50:22.260" LastActivityDate="2018-01-16T01:50:22.260" Title="Who assigned the name &quot;work&quot; to the quantity $\int F\,{\rm d}r$?" Tags="|physics|terminology|classical-mechanics|energy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6943" PostTypeId="2" ParentId="6942" CreationDate="2018-01-15T13:40:58.480" Score="2" Body="&lt;p&gt;In 1826, French mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaspard-Gustave_de_Coriolis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaspard-Gustave de Coriolis&lt;/a&gt; coined the term work as &quot;weight lifted through height&quot;.  Here, weight id the Force F, and height is distance d.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kinetic energy was defined by Gottfried Leibniz and Johann Bernoulli as $\frac{1}{2}mv^2$&lt;/p&gt;&#xA;" OwnerDisplayName="XcoderX" LastActivityDate="2018-01-15T13:46:12.223" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6945" PostTypeId="2" ParentId="6942" CreationDate="2018-01-15T14:32:48.140" Score="-1" Body="&lt;p&gt;According to the description of &lt;a href=&quot;http://www.springer.com/gp/book/9789400777040&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; book, &lt;em&gt;A History of the Concept of Work&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In modern times the concept of work is analyzed in the context of applied mechanics mainly in Lazare Carnot mechanics and the mechanics of the new generation of polytechnical engineers like Navier, Coriolis and Poncelet. In this context the word &amp;quot;work&amp;quot; is finally adopted.&lt;/p&gt;&#xA;&lt;p&gt;These engineers are also responsible for the incorporation of the concept of work into the discipline of economics when they endeavoured to combine the study of the work of machines and men together.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2324" OwnerDisplayName="Mozibur Ullah" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-15T14:32:48.140" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6946" PostTypeId="2" ParentId="6941" CreationDate="2018-01-16T17:10:39.080" Score="5" Body="&lt;p&gt;Pseudosphere is &lt;strong&gt;&lt;em&gt;not&lt;/em&gt;&lt;/strong&gt; a realisation of hyperbolic geometry for two reasons: (1) it is not simply-connected; (2) it is not complete.  Thus one cannot claim that there is a unique geodesic through any given pair of points, nor that any line can be indefinitely extended.  In fact the hyperbolic plane cannot be embedded in Euclidean 3-space.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-01-16T17:10:39.080" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="6947" PostTypeId="2" ParentId="6939" CreationDate="2018-01-16T21:22:23.107" Score="2" Body="&lt;p&gt;It is not known how two-dimensional or three-dimensional geometry were discovered. All we have are some legends recorded by the people who lived centuries after these discoveries. It is quite plausible that 2-dimensional and 3-dimensional geometry were discovered simultaneously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the legends says that Thales measured the height of an Egyptian pyramid.&#xA;It gives no detail about how exactly he did it, but it is quite possible that he had to use some geometry in space. (Thales was probably the earliest  mathematician on whom we have any record, but his work did not survive and details which are mentioned by later writers are unclear).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-01-18T22:13:12.390" LastActivityDate="2018-01-18T22:13:12.390" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6949" PostTypeId="2" ParentId="6925" CreationDate="2018-01-17T11:42:19.713" Score="7" Body="&lt;h3&gt;Community-wiki timeline of relevant literature:&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;By year written, sources where $k$ is named after...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/pjfGa.png&quot; alt=&quot;histogram&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;Author         |Dated    |Published |Denoted|Called &#xA; Boltzmann                 &lt;a href=&quot;//zbmath.org/?q=an:09.0760.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1877&lt;/a&gt;  &lt;a href=&quot;//doi.org/10.3390/e17041971&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  ––   S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044093294908&amp;amp;num=428&quot; rel=&quot;nofollow noreferrer&quot;&gt;ein constanter Factor&lt;/a&gt; &lt;sup&gt;1&lt;/sup&gt; &#xA; Boltzmann•      1895.09   &lt;a href=&quot;//zbmath.org/?q=an:26.1056.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;1896&lt;/a&gt;  &lt;a href=&quot;//google.com/books?id=AZrq7PcqhG0C&amp;amp;pg=PA74&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  RM   S  &lt;a href=&quot;//archive.org/stream/vorlesungenber01bolt#page/60&quot; rel=&quot;nofollow noreferrer&quot;&gt;für alle Gase gleiche Constante RM&lt;/a&gt; &lt;sup&gt;2&lt;/sup&gt; &#xA; Boltzmann•      1898.08   &lt;a href=&quot;//zbmath.org/?q=an:30.0818.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1898&lt;/a&gt;  &lt;a href=&quot;//google.com/books?id=AZrq7PcqhG0C&amp;amp;pg=PA371&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  RM   S  &lt;a href=&quot;//archive.org/stream/vorlesungenber02bolt#page/172&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &lt;sup&gt;3&lt;/sup&gt; &#xA; Drude           1900.02   &lt;a href=&quot;//zbmath.org/?q=an:31.0811.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900&lt;/a&gt;       2α/3 E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d00326673d&amp;amp;num=571&quot; rel=&quot;nofollow noreferrer&quot;&gt;–– (school using α = 3k/2 instead)&lt;/a&gt; &#xA; Planck          1900.12   &lt;a href=&quot;//zbmath.org/?q=an:31.0846.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900b&lt;/a&gt; &lt;a href=&quot;//archive.org/stream/PlancksOriginalPapersInQuantumPhysics#page/n27&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//archive.org/stream/verhandlungende01goog#page/n251&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine zweite Naturconstante&lt;/a&gt; &#xA; Planck          1901.01   &lt;a href=&quot;//zbmath.org/?q=an:32.0913.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901a&lt;/a&gt; &lt;a href=&quot;//google.com/books?id=CbkfAQAAQBAJ&amp;amp;pg=PA12&amp;amp;dq=universal+constants&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//google.com/books?id=KhtbAAAAYAAJ&amp;amp;pg=PA561&amp;amp;dq=universelle+Constante&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Constante&lt;/a&gt; &#xA; Planck          1901.01   &lt;a href=&quot;//zbmath.org/?q=an:32.0908.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901b&lt;/a&gt; &lt;a href=&quot;//digitalcollections.library.cmu.edu/awweb/awarchive?type=file&amp;amp;item=49373&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//google.com/books?id=KhtbAAAAYAAJ&amp;amp;pg=PA566&amp;amp;dq=Strahlungsconstanten&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Strahlungsconstante k&lt;/a&gt; &#xA; Planck          1901.05   &lt;a href=&quot;//zbmath.org/?q=an:32.0913.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901c&lt;/a&gt;      k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101066023571&amp;amp;num=547&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle positive Constante&lt;/a&gt; &#xA; Jeans           1901.06   &lt;a href=&quot;//zbmath.org/?q=an:33.0742.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1902&lt;/a&gt;       λ    R  &lt;a href=&quot;//google.com/books?id=F387AQAAMAAJ&amp;amp;pg=PA5&amp;amp;dq=%22gas+constant%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;the gas constant&lt;/a&gt; &#xA; Planck                    &lt;a href=&quot;//zbmath.org/?q=an:32.0892.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901d&lt;/a&gt;      k    S  &lt;a href=&quot;//archive.org/stream/archivesnerland261901laha#page/58&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Proportionalitätsconstante k&lt;/a&gt; &#xA; Einstein        1902.06   &lt;a href=&quot;//zbmath.org/?q=an:33.0911.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1902&lt;/a&gt;       2ϰ   E  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/56&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Constante&lt;/a&gt; &#xA; Lorentz         1903.04   &lt;a href=&quot;//zbmath.org/?q=an:34.0989.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1903&lt;/a&gt;       k    U  &lt;a href=&quot;//archive.org/stream/verslagvandegewo111902_11#page/788&quot; rel=&quot;nofollow noreferrer&quot;&gt;een universeele constante&lt;/a&gt; &#xA; Planck          1903.07   &lt;a href=&quot;//zbmath.org/?q=an:35.0918.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;       ––      &lt;a href=&quot;//archive.org/stream/festschriftludw02meyegoog#page/n134&quot; rel=&quot;nofollow noreferrer&quot;&gt;–– (Festschrift)&lt;/a&gt; &#xA; Einstein        1904.03   &lt;a href=&quot;//zbmath.org/?q=an:35.0918.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;       2ϰ   E  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/83&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine absolute Konstante&lt;/a&gt; &#xA; Jeans•          1904.11   &lt;a href=&quot;//zbmath.org/?q=an:36.0972.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;       R    E  &lt;a href=&quot;//archive.org/stream/dynamicaltheoryo00jeanrich#page/109&quot; rel=&quot;nofollow noreferrer&quot;&gt;an absolute constant&lt;/a&gt; &#xA; Abraham•        1905.03   &lt;a href=&quot;//zbmath.org/?q=an:36.0906.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101061579874&amp;amp;num=362&quot; rel=&quot;nofollow noreferrer&quot;&gt;die universelle Konstante k&lt;/a&gt; &#xA; Einstein        1905.03   &lt;a href=&quot;//zbmath.org/?q=an:36.0883.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;       C    S  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/109&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Konstante&lt;/a&gt; &lt;sup&gt;4&lt;/sup&gt; &#xA; Rayleigh        1905.05   &lt;a href=&quot;//zbmath.org/?q=an:36.0972.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c2754948&amp;amp;num=55&quot; rel=&quot;nofollow noreferrer&quot;&gt;a constant&lt;/a&gt; &#xA; Jeans           1905.06   &lt;a href=&quot;//zbmath.org/?q=an:36.0842.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905a&lt;/a&gt;      R    R  &lt;a href=&quot;//archive.org/stream/londonedinburghd610190lon#page/97&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Jeans           1905.07   &lt;a href=&quot;//zbmath.org/?q=an:36.0981.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905b&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c2754948&amp;amp;num=293&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Jeans           1905.10   &lt;a href=&quot;//zbmath.org/?q=an:36.0981.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905c&lt;/a&gt;      R    R  &lt;a href=&quot;//google.com/books?id=-Bk_AQAAMAAJ&amp;amp;pg=PA547&amp;amp;dq=%22constant+of+the+theory+of+gases%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;the constant of the theory of gases&lt;/a&gt; &#xA; Langevin                  &lt;a href=&quot;//zbmath.org/?q=an:36.0944.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905a&lt;/a&gt;      r    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.hx3e42&amp;amp;num=116&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante de l’équation des gaz&lt;/a&gt; &#xA; Langevin                  &lt;a href=&quot;//zbmath.org/?q=an:36.0945.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905b&lt;/a&gt;      r    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=chi.105148032&amp;amp;num=691&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante de l’équation des gaz&lt;/a&gt; &#xA; Einstein        1906.03   &lt;a href=&quot;//zbmath.org/?q=an:37.0856.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906a&lt;/a&gt;      R/N  R  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/208&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; von Laue        1906.04   &lt;a href=&quot;//zbmath.org/?q=an:37.0951.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;       k    U  &lt;a href=&quot;//google.com/books?id=MFFNAQAAIAAJ&amp;amp;pg=PA367&amp;amp;dq=%22Konstanten+des+Energieverteilungsgesetzes%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Konstante des Verteilungsgesetzes&lt;/a&gt; &#xA; Planck•         1906.04   &lt;a href=&quot;//zbmath.org/?q=an:37.0949.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;       k    S  &lt;a href=&quot;//archive.org/stream/uberdietheoewarm00planrich#page/n152&quot; rel=&quot;nofollow noreferrer&quot;&gt;universelle Integrationskonstante&lt;/a&gt; &#xA; Winkelmann•               &lt;a href=&quot;//zbmath.org/?q=an:37.0927.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;       k    S  &lt;a href=&quot;//archive.org/stream/handbuchderphys10unkngoog#page/n402&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine Konstante&lt;/a&gt; &#xA; Ehrenfest       1906.07   &lt;a href=&quot;//zbmath.org/?q=an:37.0950.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906a&lt;/a&gt;      k       &lt;a href=&quot;//google.com/books?id=RycKAAAAIAAJ&amp;amp;pg=PA530&amp;amp;dq=Boltzmannschen+Konstante&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmannsche&lt;/strong&gt; Konstante&lt;/a&gt; &lt;sup&gt;5&lt;/sup&gt; &#xA; Einstein        1906.08   1906b      k    S  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/226&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine (universelle) Konstante&lt;/a&gt; &#xA; Ehrenfest       1906.10   &lt;a href=&quot;//zbmath.org/?q=an:37.0027.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906b&lt;/a&gt;      ––      &lt;a href=&quot;//google.com/books?id=jLhnGaMbhjIC&amp;amp;pg=RA2-PA205&quot; rel=&quot;nofollow noreferrer&quot;&gt;–– (obituary)&lt;/a&gt; &#xA; Einstein        1906.12   &lt;a href=&quot;//zbmath.org/?q=an:38.0911.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907&lt;/a&gt;       R/N  R  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/239&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &lt;sup&gt;6&lt;/sup&gt; &#xA; von Laue        1907.03   &lt;a href=&quot;//zbmath.org/?q=an:38.0937.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907a&lt;/a&gt;      k    U  &lt;a href=&quot;//google.com/books?id=iMBGAQAAMAAJ&amp;amp;pg=PA18&amp;amp;dq=%22Konstanten+des+Planckschen+Energieverteilungsgesetzes%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Konstante des Planckschen Gesetzes&lt;/a&gt; &#xA; von Laue        1907.10   &lt;a href=&quot;//zbmath.org/?q=an:38.0938.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907b&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=chi.21403652&amp;amp;num=610&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Konstante&lt;/a&gt; &#xA; Reiche          1908.01   &lt;a href=&quot;//zbmath.org/?q=an:39.0974.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144294&amp;amp;num=534&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine gewisse allgemeine Konstante&lt;/a&gt; &#xA; Planck          1908.06   &lt;a href=&quot;//zbmath.org/?q=an:39.0958.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;       k    S  &lt;a href=&quot;//google.com/books?id=hPxFAQAAMAAJ&amp;amp;pg=PA638&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; von Laue        1908.09   &lt;a href=&quot;//zbmath.org/?q=an:39.0969.08&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015068320947&amp;amp;num=779&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Konstante&lt;/a&gt; &#xA; Kowalski        1908.10   1908       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b2832027&amp;amp;num=752&quot; rel=&quot;nofollow noreferrer&quot;&gt;Konstante des Planck’schen Gesetzes&lt;/a&gt; &#xA; Lorentz•        1909.01   &lt;a href=&quot;//zbmath.org/?q=an:40.0922.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;       k    U  &lt;a href=&quot;//archive.org/stream/theoryofelectron00loreuoft#page/79&quot; rel=&quot;nofollow noreferrer&quot;&gt;a universal physical constant&lt;/a&gt;&#xA; Einstein        1909.01   &lt;a href=&quot;//zbmath.org/?q=an:40.0985.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909a&lt;/a&gt;      R/N  R  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/376&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Reinganum       1909.04   &lt;a href=&quot;//zbmath.org/?q=an:40.0986.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433090815253&amp;amp;num=352&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Schwingungsenergie.. beträgt.. kT&lt;/a&gt; &#xA; Planck          1909.04   &lt;a href=&quot;//zbmath.org/?q=an:41.0868.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910a&lt;/a&gt; &lt;a href=&quot;//archive.org/stream/eightlecturesont00planuoft#page/55&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//google.com/books?id=fjrQAAAAMAAJ&amp;amp;pg=PA55&amp;amp;dq=%22Konstante%20k%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Konstante k&lt;/a&gt; &#xA; Wien•           1909.05   &lt;a href=&quot;//zbmath.org/?q=an:40.0984.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;       k    S  &lt;a href=&quot;//archive.org/stream/encyklomath305encyrich#page/n323&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Jahn            1909.05   1909       k       &lt;a href=&quot;//google.com/books?id=ne4tAQAAIAAJ&amp;amp;pg=RA1-PA233&amp;amp;dq=Plancksche+Konstante&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &lt;sup&gt;7&lt;/sup&gt;&#xA; Lorentz         1909.06   1909       2α/3 R  &lt;a href=&quot;//ilorentz.org/IL-publications/sources/Lorentz_CW_1909.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;α -- een werkelijk universeele constante&lt;/a&gt;&#xA; Einstein        1909.09   &lt;a href=&quot;//zbmath.org/?q=an:40.0984.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909b&lt;/a&gt;      k    S  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol2-trans/403&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Konstante&lt;/a&gt; &#xA; Larmor          1909.11   &lt;a href=&quot;//zbmath.org/?q=an:41.1001.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;       k    S  &lt;a href=&quot;//archive.org/stream/philtrans09937134#page/n3&quot; rel=&quot;nofollow noreferrer&quot;&gt;a universal constant&lt;/a&gt; &#xA; Kowalski        1909.12   &lt;a href=&quot;//zbmath.org/?q=an:41.0969.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910a&lt;/a&gt;      K       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924087713800&amp;amp;num=16&quot; rel=&quot;nofollow noreferrer&quot;&gt;die universelle &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Planck          1910.01   &lt;a href=&quot;//zbmath.org/?q=an:41.0997.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910b&lt;/a&gt;      k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3336735&amp;amp;num=759&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine Konstante&lt;/a&gt; &#xA; Haas            1910.03   &lt;a href=&quot;//zbmath.org/?q=an:41.0999.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015073682190&amp;amp;num=121&quot; rel=&quot;nofollow noreferrer&quot;&gt;2/3 der Boltzmann-Drude’schen Konstanten&lt;/a&gt; &#xA; Kowalski        1910.05   1910b      k       &lt;a href=&quot;//archive.org/stream/archivesdesscien429gen#page/524&quot; rel=&quot;nofollow noreferrer&quot;&gt;la constante universelle de &lt;strong&gt;Boltzmann&lt;/strong&gt;&lt;/a&gt; &#xA; Lindemann       1910.06   &lt;a href=&quot;//zbmath.org/?q=an:41.0996.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770845&amp;amp;num=609&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Debye           1910.10   &lt;a href=&quot;//zbmath.org/?q=an:41.0876.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910a&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770845&amp;amp;num=1117&quot; rel=&quot;nofollow noreferrer&quot;&gt;universelle Konstante, welche..&lt;/a&gt; &#xA; Einstein        1910.10   &lt;a href=&quot;//zbmath.org/?q=an:41.0927.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;       R/N  R  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol3-trans/245&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Debye           1910.10   &lt;a href=&quot;//zbmath.org/?q=an:41.1000.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910b&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144375&amp;amp;num=1431&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Einstein        1910.11   &lt;a href=&quot;//zbmath.org/?q=an:42.0855.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911&lt;/a&gt;       k    U  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol3-doc/451&quot; rel=&quot;nofollow noreferrer&quot;&gt;Konst. der Planckschen Strahlungsformel&lt;/a&gt; &#xA; Planck          1911.02   &lt;a href=&quot;//zbmath.org/?q=an:42.0984.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911a&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113495&amp;amp;num=140&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Planck          1911.04   &lt;a href=&quot;//zbmath.org/?q=an:42.0961.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911b&lt;/a&gt; &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433090815378&amp;amp;num=685&quot; rel=&quot;nofollow noreferrer&quot;&gt;(g)&lt;/a&gt;  k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.32106016223882&amp;amp;num=354&quot; rel=&quot;nofollow noreferrer&quot;&gt;la constante k&lt;/a&gt; &#xA; Sommerfeld      1911.07   2004       k       &lt;a href=&quot;//sommerfeld.userweb.mwn.de/KurzFass/01546.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Planck&lt;/strong&gt;’s.. universelle Constante k&lt;/a&gt; &#xA; Ehrenfest       1911.07   &lt;a href=&quot;//zbmath.org/?q=an:42.0911.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144391&amp;amp;num=110&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Planck          1911.07   &lt;a href=&quot;//zbmath.org/?q=an:42.0912.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911c&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=msu.31293030452266&amp;amp;num=729&quot; rel=&quot;nofollow noreferrer&quot;&gt;der universelle Faktor k&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; Debye           1911.08   1911       k    S  &lt;a href=&quot;//archive.org/stream/archivesdesscien432gen#page/152&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante de la loi de Boltzmann&lt;/a&gt; &#xA; Ornstein        1911.09   &lt;a href=&quot;//zbmath.org/?q=an:43.1054.09&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911&lt;/a&gt;  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.a0001473115&amp;amp;num=78&quot; rel=&quot;nofollow noreferrer&quot;&gt;(f)&lt;/a&gt;  R/N  R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101073434514&amp;amp;num=244&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Sommerfeld      1911.09   &lt;a href=&quot;//zbmath.org/?q=an:42.0850.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911&lt;/a&gt;       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015011417709&amp;amp;num=1058&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Perrin          1911.11   &lt;a href=&quot;//zbmath.org/?q=an:43.0983.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       r    R  &lt;a href=&quot;//archive.org/stream/lathoriedurayo00inst#page/246&quot; rel=&quot;nofollow noreferrer&quot;&gt;la constante universelle R/N&lt;/a&gt; &#xA; Langevin        1911.11   &lt;a href=&quot;//zbmath.org/?q=an:43.0983.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k       &lt;a href=&quot;//archive.org/stream/lathoriedurayo00inst#page/445&quot; rel=&quot;nofollow noreferrer&quot;&gt;le coefficient de &lt;strong&gt;Boltzmann&lt;/strong&gt;&lt;/a&gt; &#xA; Debye           1911.12   &lt;a href=&quot;//zbmath.org/?q=an:43.1002.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912a&lt;/a&gt;      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770878&amp;amp;num=100&quot; rel=&quot;nofollow noreferrer&quot;&gt;universelle Konstante, mit welcher..&lt;/a&gt; &#xA; K.Onnes-Keesom• 1911.12   &lt;a href=&quot;//zbmath.org/?q=an:43.1039.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k&lt;sub&gt;P&lt;/sub&gt;      &lt;a href=&quot;//gdz.sub.uni-goettingen.de/id/PPN360709532?tify=%7B%22pages%22:[648],%22view%22:%22toc%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Planck&lt;/strong&gt;’sche Konstante&lt;/a&gt; &#xA; Lorentz                   1912       k    E  &lt;a href=&quot;//ilorentz.org/IL-publications/sources/Lorentz_RS_1912.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;une constante universelle&lt;/a&gt; &#xA; Planck          1912.01   &lt;a href=&quot;//zbmath.org/?q=an:43.0972.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3336741&amp;amp;num=652&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Einstein        1912.01   &lt;a href=&quot;//zbmath.org/?q=an:43.0970.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       ϰ       &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol4-trans/105&quot; rel=&quot;nofollow noreferrer&quot;&gt;–– (ambiguous!)&lt;/a&gt; &#xA; Bohr            1912.02   &lt;a href=&quot;//zbmath.org/?q=an:43.1001.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k    R  &lt;a href=&quot;//archive.org/stream/londonedinburg6231912lond#page/985&quot; rel=&quot;nofollow noreferrer&quot;&gt;gas-constant referred to a molecule&lt;/a&gt; &#xA; Kroò            1912.02   &lt;a href=&quot;//zbmath.org/?q=an:43.1003.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770878&amp;amp;num=249&quot; rel=&quot;nofollow noreferrer&quot;&gt;die universelle &lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Warburg         1912.02   1912       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951p00527608t&amp;amp;num=225&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Planck&lt;/strong&gt;sche Strahlungskonstante&lt;/a&gt; &#xA; Debye           1912.03   1912b      k       &lt;a href=&quot;//archive.org/stream/archivesdesscien433gen#page/256&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante universelle de &lt;strong&gt;Boltzmann&lt;/strong&gt;&lt;/a&gt; &#xA; Tetrode         1912.03   &lt;a href=&quot;//zbmath.org/?q=an:43.0917.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3336742&amp;amp;num=434&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Frank           1912.04   &lt;a href=&quot;//zbmath.org/?q=an:43.0973.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770878&amp;amp;num=506&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Keesom          1912.04   1912       k&lt;sub&gt;P&lt;/sub&gt;      &lt;a href=&quot;http://www.dwc.knaw.nl/DL/publications/PU00012940.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Debye           1912.07   &lt;a href=&quot;//zbmath.org/?q=an:43.1037.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912c&lt;/a&gt;      k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3336743&amp;amp;num=791&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Drude-Boll•               1912       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=wu.89097503155&amp;amp;num=275&quot; rel=&quot;nofollow noreferrer&quot;&gt;une constante universelle&lt;/a&gt; &#xA; Schrödinger     1912.10   &lt;a href=&quot;//zbmath.org/?q=an:43.1008.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015051107277&amp;amp;num=1940&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;-Konstante&lt;/a&gt; &lt;sup&gt;8&lt;/sup&gt; &#xA; Schrödinger     1912.10   &lt;a href=&quot;//zbmath.org/?q=an:43.1008.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015051107277&amp;amp;num=1963&quot; rel=&quot;nofollow noreferrer&quot;&gt;die universelle &lt;strong&gt;Boltzmann&lt;/strong&gt;konstante&lt;/a&gt; &#xA; Westphal        1912.10   1912       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056112091&amp;amp;num=1012&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Konstante der Entropiegleichung&lt;/a&gt; &#xA; Nernst•         1912.11   &lt;a href=&quot;//zbmath.org/?q=an:44.1047.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;  &lt;a href=&quot;//archive.org/stream/chemitheoretical00nernrich#page/256&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  ––      &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101017959451&amp;amp;num=257&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &lt;a href=&quot;//google.com/books?id=3eqiBgAAQBAJ&amp;amp;pg=PP7&amp;amp;dq=%22Boltzmannsche+Konstante%22+%22Quantenkonstante+f%C3%BCr+Frequenzen%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;(school using β = h/k instead)&lt;/a&gt; &#xA; Planck•         1912.11   &lt;a href=&quot;//zbmath.org/?q=an:44.1051.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;  &lt;a href=&quot;//archive.org/stream/theoryofheatradi00planrich#page/119&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//archive.org/stream/vorlesungenberd01plangoog#page/n137&quot; rel=&quot;nofollow noreferrer&quot;&gt;universelle Integrationskonstante&lt;/a&gt; &#xA; Lorentz•        1912.11   1916       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044009941865&amp;amp;num=5&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante qui détermine l’énergie&lt;/a&gt;&#xA; Born-von Kármán 1912.11   &lt;a href=&quot;//zbmath.org/?q=an:44.1032.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770894&amp;amp;num=16&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Hasenöhrl       1912.12   1913       k    S  &lt;a href=&quot;//phaidra.univie.ac.at/o:170477&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Reiche          1912.12   &lt;a href=&quot;//zbmath.org/?q=an:44.0954.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//google.com/books?id=pjVLAAAAMAAJ&amp;amp;pg=PA16&amp;amp;dq=Planck&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Jellinek•       1913.01   1913       k    S  &lt;a href=&quot;//google.com/books?id=OWDQAAAAMAAJ&amp;amp;pg=PA168&quot; rel=&quot;nofollow noreferrer&quot;&gt;eine universelle Konstante&lt;/a&gt; &#xA; Einstein-Stern  1913.01   &lt;a href=&quot;//zbmath.org/?q=an:44.1037.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol4-trans/149&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Tetrode         1913.01   &lt;a href=&quot;//zbmath.org/?q=an:44.1051.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770894&amp;amp;num=213&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Sommerfeld      1913.02   &lt;a href=&quot;//zbmath.org/?q=an:44.0972.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951p003587298&amp;amp;num=770&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Born-Bolza &amp;amp; al 1913.02   &lt;a href=&quot;//zbmath.org/?q=an:44.1049.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b5381230&amp;amp;num=228&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Planck&lt;/strong&gt;sche universelle Konstante&lt;/a&gt; &#xA; Einstein        1913.03   &lt;a href=&quot;//zbmath.org/?q=an:44.1043.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       R/N  R  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol4-trans/161&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Born                      &lt;a href=&quot;//zbmath.org/?q=an:44.0982.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015000428766&amp;amp;num=504&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Planck          1913.04   &lt;a href=&quot;//zbmath.org/?q=an:44.0959.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.d0001333319&amp;amp;num=361&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Planck          1913.04   &lt;a href=&quot;//zbmath.org/?q=an:45.0955.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;       k    E  &lt;a href=&quot;//google.com/books?id=Y_lYAAAAYAAJ&amp;amp;pg=PA3&amp;amp;dq=absolute+Gaskonstante&quot; rel=&quot;nofollow noreferrer&quot;&gt;absolute Gaskonstante&lt;/a&gt; &#xA; Debye           1913.04   &lt;a href=&quot;//zbmath.org/?q=an:45.0955.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=Y_lYAAAAYAAJ&amp;amp;pg=PA24&amp;amp;dq=Boltzmannsche&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Smoluchowski    1913.04   &lt;a href=&quot;//zbmath.org/?q=an:45.0955.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;       H/N  R  &lt;a href=&quot;//google.com/books?id=Y_lYAAAAYAAJ&amp;amp;pg=PA99&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Sommerfeld      1913.04   &lt;a href=&quot;//zbmath.org/?q=an:45.0955.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=Y_lYAAAAYAAJ&amp;amp;pg=PA154&amp;amp;dq=Konstante&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Lorentz         1913.04   &lt;a href=&quot;//zbmath.org/?q=an:45.0955.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914a&lt;/a&gt;      k    E  &lt;a href=&quot;//google.com/books?id=Y_lYAAAAYAAJ&amp;amp;pg=PA169&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Stern           1913.05   &lt;a href=&quot;//zbmath.org/?q=an:44.1050.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770902&amp;amp;num=630&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Ehrenfest       1913.05   &lt;a href=&quot;//zbmath.org/?q=an:44.1034.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c003267959&amp;amp;num=451&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Ratnowsky       1913.05   &lt;a href=&quot;//zbmath.org/?q=an:44.0998.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c003267959&amp;amp;num=499&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Millikan        1913.06   1913  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770902&amp;amp;num=811&quot; rel=&quot;nofollow noreferrer&quot;&gt;(g)&lt;/a&gt;  k       &lt;a href=&quot;//google.com/books?id=kLsWAAAAYAAJ&amp;amp;pg=PA142&amp;amp;dq=entropy#v=twopage&quot; rel=&quot;nofollow noreferrer&quot;&gt;the &lt;strong&gt;Boltzmann&lt;/strong&gt; entropy constant&lt;/a&gt; &#xA; Bolza           1913.07   1913       k       &lt;a href=&quot;//archive.org/stream/anwendungdertheo00bolz#page/39&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Planck&lt;/strong&gt;sche universelle Konstante&lt;/a&gt; &#xA; Born-Courant    1913.07   &lt;a href=&quot;//zbmath.org/?q=an:44.0908.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101054770902&amp;amp;num=736&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Debye           1913.07   &lt;a href=&quot;//zbmath.org/?q=an:44.0975.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913a&lt;/a&gt;      k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c003267959&amp;amp;num=681&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Ortvay          1913.07   &lt;a href=&quot;//zbmath.org/?q=an:44.1033.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c003267959&amp;amp;num=774&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Debye           1913.08   &lt;a href=&quot;//zbmath.org/?q=an:44.0953.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913b&lt;/a&gt;      k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c003267959&amp;amp;num=781&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Kroò            1913.09   &lt;a href=&quot;//zbmath.org/?q=an:44.0994.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144456&amp;amp;num=1373&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Lorentz         1913.09   1914b      R/N  R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101076796703&amp;amp;num=374&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Einstein                  &lt;a href=&quot;//zbmath.org/?q=an:45.0093.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915&lt;/a&gt;       R/N  R  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol4-trans/255&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; von Laue        1913.10   &lt;a href=&quot;//zbmath.org/?q=an:44.1009.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       ϰ       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144456&amp;amp;num=1561&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Fokker          1913.10   1913       k    S  &lt;a href=&quot;//lorentz.leidenuniv.nl/IL-publications/dissertations/lorentz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;de gasconstante voor één molecuul&lt;/a&gt; &#xA; von Laue        1913.10   1921       k       &lt;a href=&quot;//archive.org/stream/lastructuredelam00inst#page/101&quot; rel=&quot;nofollow noreferrer&quot;&gt;la constante de &lt;strong&gt;Boltzmann&lt;/strong&gt;&lt;/a&gt; &#xA; Sommerfeld      1913.10   1921a      k    U  &lt;a href=&quot;//archive.org/stream/lastructuredelam00inst#page/129&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante du rayonnement&lt;/a&gt; &#xA; Lorentz         1913.10   1921       k       &lt;a href=&quot;//archive.org/stream/lastructuredelam00inst#page/283&quot; rel=&quot;nofollow noreferrer&quot;&gt;la constante k de M. &lt;strong&gt;Planck&lt;/strong&gt;&lt;/a&gt; &#xA; Einstein        1913.11   1913       k    S  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol4-trans/285&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Schrödinger     1913.11   &lt;a href=&quot;//zbmath.org/?q=an:44.1023.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1913&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c003267959&amp;amp;num=1168&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;-konstante&lt;/a&gt; &#xA; Schrödinger     1913.11   1914       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433090815022&amp;amp;num=81&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;konstante&lt;/a&gt; &#xA; Pockels                   1913       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uva.x000269589&amp;amp;num=1211&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Stern           1914.02   1914       k    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015068231193&amp;amp;num=500&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Bragg           1914.03   1914       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.%24b562674&amp;amp;num=390&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Richardson•     1914.05   1914a      k    S  &lt;a href=&quot;//archive.org/stream/electrontheoryof00richrich#page/400&quot; rel=&quot;nofollow noreferrer&quot;&gt;a universal constant&lt;/a&gt; &#xA; Griffiths       1914.06   1914       k       &lt;a href=&quot;//doi.org/10.1098/rsta.1914.0021&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;’s constant&lt;/a&gt; &#xA; Lorentz         1914.09   1917       k       &lt;a href=&quot;//archive.org/stream/proceedingsofsec191koni#page/737&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Planck&lt;/strong&gt;’s well known coefficient&lt;/a&gt; &#xA; Richardson                &lt;a href=&quot;//zbmath.org/?q=an:45.1009.10&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914b&lt;/a&gt;      R    R  &lt;a href=&quot;//archive.org/stream/londonedinburg6281914lond#page/n676&quot; rel=&quot;nofollow noreferrer&quot;&gt;the gas constant for one molecule&lt;/a&gt; &#xA; Lindemann       1914.12   1915       R/n  R  &lt;a href=&quot;//archive.org/stream/londonedinburg6291915lond#page/128&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Czukor          1915.02   1915       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113297&amp;amp;num=76&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Tetrode         1915.02   1915  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d000145025&amp;amp;num=1169&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uiug.30112109543097&amp;amp;num=1112&quot; rel=&quot;nofollow noreferrer&quot;&gt;de constante van &lt;strong&gt;Planck&lt;/strong&gt;&lt;/a&gt; &#xA; Richardson      1915.04   1915       k       &lt;a href=&quot;//doi.org/10.1098/rspa.1915.0046&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;’s constant&lt;/a&gt; &#xA; Richardson                1915       k       &lt;a href=&quot;//archive.org/stream/londonedinburg6301915lond#page/296&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;’s constant&lt;/a&gt; &#xA; Guillaume                 &lt;a href=&quot;//zbmath.org/?q=an:45.1271.13&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915&lt;/a&gt;       k    R  &lt;a href=&quot;//archive.org/stream/archivesdesscien439gen#page/304&quot; rel=&quot;nofollow noreferrer&quot;&gt;la constante d’énergie moléculaire&lt;/a&gt; &#xA; von Laue        1915.05   1915a      k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113297&amp;amp;num=198&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; von Laue•       1915.07   &lt;a href=&quot;//zbmath.org/?q=an:45.1042.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915b&lt;/a&gt;      k    S  &lt;a href=&quot;//archive.org/stream/encyklomath305encyrich#page/n428&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Born            1915.07   &lt;a href=&quot;//zbmath.org/?q=an:45.1043.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915&lt;/a&gt;       k       &lt;a href=&quot;//archive.org/stream/dynamikderkrist00borngoog#page/n84&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Rubens          1915.07   1915       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113297&amp;amp;num=330&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Planck          1915.11   1915       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113297&amp;amp;num=407&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Mandersloot     1915.11   1916       K       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433071592475&amp;amp;num=4&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Kroò            1915.11   1915       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113297&amp;amp;num=457&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Planck          1915.11   1915       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924056113297&amp;amp;num=418&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Jeans•          1916.01   &lt;a href=&quot;//zbmath.org/?q=an:46.1200.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916&lt;/a&gt;       R    S  &lt;a href=&quot;//archive.org/stream/dynamicaltheoryo00jeanuoft#page/79&quot; rel=&quot;nofollow noreferrer&quot;&gt;the universal gas-constant&lt;/a&gt; &#xA; Stern           1916.02   1916       k    R  &lt;a href=&quot;//doi.org/10.1002/andp.19163540703&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Gerlach         1916.04   1916       k    U  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144529&amp;amp;num=268&quot; rel=&quot;nofollow noreferrer&quot;&gt;die universelle Konstante k&lt;/a&gt; &#xA; Wolfke          1916.04   1916       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015010783705&amp;amp;num=199&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Stern           1916.08   1916       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433090771183&amp;amp;num=238&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Einstein        1916.08   1916       ϰ    S  &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol6-trans/230&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Debye                     1916       k       &lt;a href=&quot;//google.com/books?id=Eus2AQAAMAAJ&amp;amp;pg=RA5-PA22&amp;amp;dq=Boltzmann&amp;#39;sche&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;’sche Konstante&lt;/a&gt; &#xA; Einstein                  1916       k       &lt;a href=&quot;//einsteinpapers.press.princeton.edu/vol6-trans/234&quot; rel=&quot;nofollow noreferrer&quot;&gt;die bekannte &lt;strong&gt;Boltzmann&lt;/strong&gt;’sche Konstante&lt;/a&gt; &#xA; Hertz                     &lt;a href=&quot;//zbmath.org/?q=an:46.1505.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=GgVJAAAAIAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22Boltzmann-Plancksche+Konstante%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA;&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;•&lt;/code&gt; $=$ textbook or reference book. &lt;code&gt;(e|f|g)&lt;/code&gt; $=$ English | French | German translation.&lt;br&gt;&#xA;Letters &lt;code&gt;S|U|R|E&lt;/code&gt; tell the formula each article &lt;em&gt;not&lt;/em&gt; naming $k$ uses to first introduce it:&#xA;$$&#xA;S=k\log W\qquad\quad&#xA;U=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}\qquad\quad&#xA;R=Nk\qquad\quad&#xA;E=\tfrac32kT.&#xA;$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notes (in progress, very much open to debate / correction):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; First known occurrence of the principle. (Quoted by Planck 1900b.)&lt;br&gt;&#xA;&lt;sup&gt;2&lt;/sup&gt; First known occurrence of the constant. (Ever quoted?)&lt;br&gt;&#xA;&lt;sup&gt;3&lt;/sup&gt; First known occurrence of the formula. (Ever quoted?)&lt;br&gt;&#xA;&lt;sup&gt;4&lt;/sup&gt; Earli(est?) to call the principle Boltzmann’s.&lt;br&gt;&#xA;&lt;sup&gt;5&lt;/sup&gt; Earli(est?) to call the constant Boltzmann’s.&lt;br&gt;&#xA;&lt;sup&gt;6&lt;/sup&gt; Earli(est?) to call the formula Boltzmann’s.&lt;br&gt;&#xA;&lt;sup&gt;7&lt;/sup&gt; Earli(est?) to call the constant Planck’s.&lt;br&gt;&#xA;&lt;sup&gt;8&lt;/sup&gt; Earli(est?) to call the constant Boltzmann-Planck’s.&lt;br&gt;&#xA;&lt;sup&gt;9&lt;/sup&gt; Calls the constant &lt;strong&gt;&lt;em&gt;not&lt;/em&gt;&lt;/strong&gt; Boltzmann’s.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="1697" LastEditDate="2018-02-26T18:09:31.873" LastActivityDate="2018-02-26T18:09:31.873" CommentCount="8" CommunityOwnedDate="2018-01-17T11:42:19.713" ContentLicense="CC BY-SA 3.0" />
  <row Id="6950" PostTypeId="1" AcceptedAnswerId="6952" CreationDate="2018-01-17T15:05:54.377" Score="8" ViewCount="1598" Body="&lt;p&gt;Complex number are extremely useful in every branch of physics dealing with ondulatory phenomena. In electromagnetism, for example, they allow to write the solution of Maxwell's equations in a form which is particularly simple to manipulate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did the use of complex numbers become widespread in physics?&lt;/p&gt;&#xA;" OwnerUserId="5003" LastEditorUserId="5003" LastEditDate="2018-01-17T17:59:43.293" LastActivityDate="2022-02-23T04:17:30.173" Title="When did the use of complex numbers become widespread in physics?" Tags="|physics|theoretical-physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6951" PostTypeId="1" AcceptedAnswerId="6985" CreationDate="2018-01-17T15:08:47.790" Score="2" ViewCount="157" Body="&lt;p&gt;The (infinte dimensional) separable quotient problem asks whether every infinite dimensional Banach space has a separable infinite dimensional quotient. In the literature I have seen that is problem is attributed to Banach or to Mazur. Is it known which one of them posed the problem and when it was posed?&lt;/p&gt;&#xA;" OwnerUserId="4096" LastActivityDate="2018-01-26T21:40:22.693" Title="Who posed the separable quotient problem (and when)?" Tags="|mathematics|reference-request|functional-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6952" PostTypeId="2" ParentId="6950" CreationDate="2018-01-17T18:05:10.827" Score="9" Body="&lt;p&gt;It depends on the definition of &amp;quot;widespread&amp;quot;. Mathematicians certainly used them in 18th century. They play prominent role in the work of Cauchy and Fourier (which belongs to both physics and mathematics). Probably they entered physics with the work of Fresnel on the wave theory of light (1823). Maxwell uses them in his Treatise on Electricity and Magnetism (1873). The standard textbook on mathematical physics of the late 19th century (Kelvin and Tait) freely uses them.&lt;/p&gt;&#xA;&lt;p&gt;Remark. When searching the last mentioned book use the word &amp;quot;imaginary&amp;quot;, not &amp;quot;complex&amp;quot;. I found this word used many times without any special&#xA;explanation: the use of complex numbers by that time was routine. Another English word used early in 19th century was &amp;quot;impossible&amp;quot;. J. B. Airy talks about &amp;quot;impossible roots&amp;quot; meaning &amp;quot;complex roots&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Second remark (to address the issue raised in comments). The engineers started using complex numbers since the criteria of stability were discovered. (It is due to Maxwell (1868) that for stability of a linear system the roots of the characteristic polynomial must lie in the left half-plane.) Since then, all engineers designing steam engines use complex numbers. Maxwell was not the first to address stability in engineering from the mathematical point of view, but he gave the first general criterion. The first was J. B. Airy, an astronomer (1840). And any criterion of stability uses complex numbers. The names of 19 century engineers working on stability are still mentioned in the complex variables textbooks (Stodola, Vyshnegradski).&lt;/p&gt;&#xA;&lt;p&gt;Edit. I was right when I mentioned Fresnel first of all.&#xA;Here is a paper which describes the first essential instance when complex numbers appeared in physics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ricardo Karam, Fresnel's original interpretation of complex numbers in 19th century optics,&#xA;American Journal of Physics 86, 245 (2018).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By a correct interpretation of a complex amplitude, Fresnel discovered the phase shift of light after an internal reflection.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-02-23T04:17:30.173" LastActivityDate="2022-02-23T04:17:30.173" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="6954" PostTypeId="2" ParentId="6950" CreationDate="2018-01-17T22:34:13.410" Score="11" Body="&lt;p&gt;Roughly speaking, complex numbers seem to have become widely used in physics and engineering around 1940, after gradually gaining popularity starting around 1920.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the late 19th and early 20th centuries, there were three algebraic systems of numbers competing for attention: complex numbers, quaternions, and vectors. Vectors were late to the party. Maxwell used quaternions (1873), and they were enthusiastically promoted by their adherents, especially Hamilton, for purposes that today we would use vectors for. Quaternions can be viewed as a generalization of both vectors and complex numbers. Most physicists considered quaternions to be very abstruse. As late as 1905, Einstein's paper on special relativity states Maxwell's equations without the use of either vector or quaternion notation. The 1895 edition of Kelvin and Tait's massive  Treatise on Natural Philosophy has no index entries for &quot;complex,&quot; &quot;imaginary,&quot; &quot;vector,&quot; or &quot;quaternion,&quot; and complex numbers are used only infrequently. Tait wrote an 1890 book on quaternions, so it's not as though he didn't know about them. They just weren't the kind of thing that readers could be expected to know about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gibbs and Heaviside created the vector system as an easier alternative to quaternions, and it gradually gained popularity. Gibbs described it in an 1881 book, which he promoted vigorously and successfully.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kennelly introduced the notion of a complex impedance in 1893, and this was recognized as a big selling point of the use of complex numbers. A popular 1941 textbook by Stratton, Electromagnetic Theory, used complex numbers. So I think the widespread use of complex numbers in physics and engineering must date to approximately 1900-1940. To pin this down a little more, I think it's helpful to look at the google ngrams graphs for &quot;&lt;a href=&quot;https://books.google.com/ngrams/graph?content=complex%20impedance&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Ccomplex%20impedance%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;complex impedance&lt;/a&gt;&quot; and &quot;&lt;a href=&quot;https://books.google.com/ngrams/graph?content=complex%20exponential&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Ccomplex%20exponential%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;complex exponential&lt;/a&gt;,&quot; which both show nonnegligible usage starting around 1920, and an inflection point around 1940.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2018-01-18T15:43:23.300" LastActivityDate="2018-01-18T15:43:23.300" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6957" PostTypeId="1" CreationDate="2018-01-21T00:34:36.343" Score="10" ViewCount="319" Body="&lt;p&gt;Today it is common for mathematicians to formulate conjectures and publish/communicate them as a spur toward further research.&#xA;An example is &lt;a href=&quot;https://en.wikipedia.org/wiki/Goldbach%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Goldbach's conjecture&lt;/a&gt;, posed in a letter&#xA;from Goldbach to Euler in 1742. &#xA;Another is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Collatz_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Collatz conjecture&lt;/a&gt;, dating from about 1937.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. When were mathematical conjectures first posed as explicit conjectures?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Perfect numbers were defined in Euclid, so the question of whether or not&#xA;there are an infinite number of perfect numbers (still unresolved) could have been an explicit conjecture in the time of Euclid, but I don't know if it was ever formulated&#xA;in that manner.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The line between an explicit &lt;em&gt;open problem&lt;/em&gt; and a &lt;em&gt;conjecture&lt;/em&gt; is a fine one.&#xA;The former usually means the author is agnostic, whereas a conjecture usually &#xA;indicates the author has studied the question enough to suggest the likely answer,&#xA;and at least in modern times, to risk being disproved.&lt;sup&gt;1&lt;/sup&gt;&#xA;Early explicit acknowledgements that clearly formulated problems are &quot;open&quot; would&#xA;also be of interest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;Pisier, Gilles. &quot;Counterexamples to a conjecture of Grothendieck.&quot; &lt;em&gt;Acta Mathematica&lt;/em&gt; 151.1 (1983): 181-208.&lt;/p&gt;&#xA;" OwnerUserId="606" LastActivityDate="2019-04-11T19:36:54.043" Title="When were mathematical conjectures first formulated and communicated?" Tags="|mathematics|conjecture|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6958" PostTypeId="2" ParentId="6957" CreationDate="2018-01-21T01:45:05.780" Score="8" Body="&lt;p&gt;The oldest &quot;open problems&quot; are well known, duplication, trisection and quadrature were widely discussed in antiquity, the presumption is that by the time of Archimedes and Apollonius the experts realized that they are unsolvable with straightedge and compass (this is supported by Pappus's classification of problems into plane, solid and &quot;mechanical&quot;), but nobody could even frame an impossibility proof at the time, see &lt;a href=&quot;https://hal.archives-ouvertes.fr/tel-01098493/document&quot; rel=&quot;noreferrer&quot;&gt;Crippa's Impossibility Results&lt;/a&gt; on how this field emerged. Another old one is Dido's isoperimetric problem. Simplicius claims that circle was &quot;known&quot; to have the greatest area among figures with the same perimeter &quot;by the time of Aristotle&quot; and Archimedes and Zenodorus &quot;proved it&quot;. The surviving fragments of Zenodorus show that he only dealt with the simplified problem for polygons, and it is unclear how the isoperimetric property could have been &quot;proved&quot; by Greeks' methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for conjectures, they were not the business they are today. Confessing in writing one's inability to prove a suspected theorem was rarely done even in 19-th century, and when it was it was usually in a passing remark, not a Conjecture (as did Riemann with the zeta function zeros). In 17-18th centuries some conjectures were stated in private correspondence as &quot;theorems&quot;, e.g. Goldbach in a letter to Euler so reports Fermat's contention that $2^{2^n}+1$ is always prime. More often, conjectures were communicated to students privately (compare to Arnold's intimation in Geometrical Methods of ODE:&quot;&lt;em&gt;unmotivated definitions that hide fundamental ideas and methods are like parables only explained privately to students&lt;/em&gt;&quot;) or inferred by others from peculiarities of the public text. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prototypical example is Euclid avoiding the use of the parallel postulate in Elements for as long as he could, which was taken as a cue to prove it. In another instance, Apollonius boasted in the &lt;a href=&quot;http://sites.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html&quot; rel=&quot;noreferrer&quot;&gt;preface to Conics&lt;/a&gt;:&quot;&lt;em&gt;I observed that Euclid had not worked out the synthesis of the locus with respect to three and four lines, but only a chance portion of it and that not successfully: for it was not possible that the synthesis could have been completed without my additional discoveries&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An even earlier example is alluded to by Plato in Theaetetus, see &lt;a href=&quot;https://www.jstor.org/stable/2690123&quot; rel=&quot;noreferrer&quot;&gt;McCabe's Theodorus' Irrationality Proofs&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet; and at that he stopped.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Presumably, he did not stop because he thought that $\sqrt{19}$ was rational but because his even/odd method of proof faltered. The now familiar method from Euclid's Elements working for all primes was found by Theaetetus, his student, who presumably inherited the problem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Archimedes wrote a famous letter to Eratosthenes, opening with &quot;&lt;em&gt;I formerly sent you some theorems discovered by myself with the request that you find the proofs which I provisionally withheld&lt;/em&gt;&quot;, where he reported a device for generating conjectures, his method of balancing weights. He expressed hope that others would find new theorems using it, but the only results he listed specifically were those already reproved by the Eudoxian double reductio. According to Toomer's interpretation of the preface to Diocles's On Burning Mirrors, Dositheus, Archimedes's friend and frequent correspondent, discovered the focal property of the parabola &quot;by practical means&quot;, which Diocles undertakes to prove (he also discovered the spiral which Archimedes  studied). This interpretation is controversial, however, see &lt;a href=&quot;https://www.jstor.org/stable/41287704&quot; rel=&quot;noreferrer&quot;&gt;Acerbi's paper&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid does not state anything remotely like conjectures about the perfect numbers. Nichomachus of Gerasa (c. 100 AD), a neo-Pythagorean, has a passage in his Arithmetic where he declares:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Only one is found among the units, only one other among the tens, and a third in the rank of the hundreds, and a fourth within the limits of the thousands... And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nichomachus does not qualify these as conjectures nor offers even a hint of arguments for them. The first was taken to mean that there is one perfect in each order of magnitude, in particular that there are infinitely many of them. Al-Bagdhadi (c. 980 AD) commented on them thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He who affirms that there is only one perfect number in each power of 10 is wrong; there is no perfect number between ten thousand and one hundred thousand. He who affirms that all perfect numbers end with the figure 6 or 8 is right&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;He&quot; would be right only if the odd perfect numbers do not exist, of course. Mersenne followed suit in 1644 by publishing a list of &quot;Mersenne primes&quot; up to 257. He did not bill it as a conjecture either, but a couple were extraneous and some missing. Still, he was remarkably close, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2424/how-did-mersenne-discover-mersenne-primes&quot;&gt;How did Mersenne discover Mersenne primes?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Stepping back a bit, here is from &lt;a href=&quot;http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1998cBis_Dioph.pdf&quot; rel=&quot;noreferrer&quot;&gt;Schappacher's paper on Diophantus&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Around 940 al-Khazin refuted an argument that abu-M. al-Khujandi&#xA;  had proposed to show that the equation (in our notation) $x^3 + y^3 = z^3$ has&#xA;  no solution in positive integers, and this discussion was carried on further&#xA;  involving also Abdallah ben Ali! And it was also al-Khazin, and not Diophantus, who formulated the problem which has become over the past 15&#xA;  years among arithmetic algebraic geometers a favourite topic of lectures for&#xA;  a wider audience:&#xA;  &lt;strong&gt;Congruent Number Problem&lt;/strong&gt;. Decide whether a given squarefree integer is the area of a right triangle with rational sides. It is still unsolved, although we today have a very simple conjectural answer...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Omar Khayyám, after classifying all cases of the cubic and solving each by intersecting conic sections, wrote in Al-jabr w'al-Muqabala: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;When however the object of the problem is an absolute number neither we nor any of those who are concerned with algebra have been able to solve this equation - perhaps others who follow us will be able to fill the gap...&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Others tried in vain, and in 1494 Summa Arithmeticae Luca Pacioli went so far as to say that algebraically &quot;&lt;em&gt;solving the cubic is as impossible as the quadrature&lt;/em&gt;&quot;. This was 19 years before del Ferro did it... and withheld the method from all but his student Fiore, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2234/how-was-geometry-historically-used-to-solve-polynomial-equations&quot;&gt;How was geometry historically used to solve polynomial equations?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-22T06:36:33.653" LastActivityDate="2018-01-22T06:36:33.653" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6959" PostTypeId="1" CreationDate="2018-01-21T01:54:11.903" Score="12" ViewCount="243" Body="&lt;p&gt;A &lt;a href=&quot;https://physics.stackexchange.com/questions/381179/why-is-angular-momentum-sometimes-called-moment-of-momentum#comment854479_381203&quot;&gt;recent comment&lt;/a&gt; called my attention to &lt;a href=&quot;https://books.google.com/ngrams/graph?content=angular+momentum&amp;amp;year_start=1860&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&quot; rel=&quot;noreferrer&quot;&gt;the Google Ngram for 'angular momentum'&lt;/a&gt;, which shows a very strong and rather sharp peak in the usage of the phrase shortly after 1960, followed by a steady decline with a loss of ~60% of the original frequency in today's usage.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IuuEd.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IuuEd.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I find this extremely surprising, and I'm coming up short in terms of possible explanations for why this should be the case. Is there some specific aspect of physics that saw particularly strong research or dissemination then and which has then fallen into disuse? Did the focus shift from the concept into slightly adjacent ones? Was there a boom in the writing of textbooks? (None of those feel particularly satisfying or plausible to me.) Or is this just an artifact of the measuring tool?&lt;/p&gt;&#xA;" OwnerUserId="272" LastActivityDate="2018-01-22T21:59:39.610" Title="What explanations are there for this strong spike in the use of 'angular momentum' in the 1960s?" Tags="|physics|language|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6960" PostTypeId="2" ParentId="6957" CreationDate="2018-01-21T09:22:56.473" Score="6" Body="&lt;p&gt;The earliest known instance that I am aware of a mathematical problem being described as an open problem took place around the year 1080. Omar Khayyam wrote a treatise about cubic equations, in which he described how to solve them using Geometry. Concerning the problem of solving these equation algebraically he admitted that he did not know how to do it and added that &amp;ldquo;possibly someone else will come to know it after us&amp;rdquo;. This took about 500 years.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-01-21T09:22:56.473" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6961" PostTypeId="1" CreationDate="2018-01-21T16:08:30.840" Score="0" ViewCount="215" Body="&lt;p&gt;Clemens Schaefer says, in his treatise &quot;&lt;em&gt;Über Gauss' physikalische Arbeiten&lt;/em&gt;&quot;, that for a very short time in the years 1835/1836 Gauss showed interest in the new diffraction experiments of Fresnel and Fraunhoffer, and he wrote one note. After making Google Translate on the relevant passage from Schaefer's treatise, it reads like the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1835 the work of the Speyerer Professor F.M.Schwerd appeared: The diffraction defects were analytically developed and represented in pictures  from the fundamental laws of the theory of undulation, which had a very great significance for the fixation of the undulation theory. In the meantime Schwerd had put together a &quot;diffraction apparatus&quot; which would enable his relatives to repeat his experiments. Regarding the importance of the problem, it is not strange that Gauss was also busy on it; He wrote to Schumacher on June 7, 1836: &quot;In the last week I have been busy with optical, especially with the very essentive diffraction experiments. Schwerd has done quite a great deal in this field to deduce from a principle these highly varied phenomena, to which he first opened the way. In the meantime, however, there is still a great deal left until the theory can be regarded as complete and exhausting&quot;. On July 23, 1836, Gauss also informs Olbers that he has acquired a SHERWER apparatus in order to try diffractive experiments. &quot;These objects become almost as interesting to me as magnetism and galvanism. But now I've had to break it all by force, since I do not have the time&quot;. Unfortunately, the split was final: Gauss never had the leisure to return to the diffraction problem. From the estate we have received some notes on General formulas for the effect of a luminous point P on a point p, which apparently come from the same time (1835/36).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After this he says: &quot;The first of these notes is interesting in several respects&quot;, and examines the Gaussian attempt to give exact mathematical formulation to the Huygens-Fresnel principle. Gauss's integral formula describes a similar scenario like the one in Kirchhoff's description - wave disturbance of a spherical monochromatic wave passes through an opening in an opaque screen. His formula reflects the basic intuition that waves propagate in a certain direction, while the basic Huygens's principle describe points on a wavefront as sources of spherical waves (without the preferred direction). According to Schaefer, Gauss's formula is the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\int_S\frac {{ds}}{{Rr}}\frac {{e^{ik(R + r)}(\cos (R,v) - \cos(r,v))}}{{\sin (\omega)}}$$ &lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $S$ in the opening region, $ds$ an element of area, $R$ is the distance of point in the opening to the source of the wave, $r$ the distance from the point in the opening to the point $p$, $k$ the wavenumber, and $v$ is the normal vector to the opening ($(R,v)$ and $(r,v)$ are the angles between $R$ and $r$ to $v$), and $\omega$ the angle between $R$ and $r$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kirchhoff's formula is the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$-\frac {{iA}}{{2\lambda}}\int_S\frac {{ds}}{{Rr}} e^{ik(R + r)}(\cos (R,v) - \cos(r,v))$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where $A$ is the amplitude of the emanating wave at point $P_0$. The factor $\cos (R,v) - \cos(r,v)$ in both formulas is called the obliquity factor $K(\chi)$ and is responsible for the fact the waves don't travel backwards. The factor $i = \sqrt {{-1}}$ in Kirchhoff's formula means that the secondary waves are $\pi/2$ out of phase with the primary waves. The factor $\frac {{1}}{{\sin\omega}}$ is very strange and I don't know what Gauss meant in this factor. It seems to me that since there is much similarity in the two formulas, there is some stuff to discuss here. Schaefer also says that the whole Gaussian approach isn't complete since he eventually left these problems&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is just about getting explanation to Gauss's note. There isn't very much information in the internet about Gauss's less known contributions to optics. If anyone wants to read what Schaefer says in his treatise, the relevant pages are 209-211 on his treatise (that is in Gauss's werke, band XI1).&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="6780" LastEditDate="2018-09-14T02:20:14.363" LastActivityDate="2018-09-14T02:20:14.363" Title="Did Gauss anticipate Kirchhoff's diffraction formulation in 1836?" Tags="|biographical-details|optics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="6963" PostTypeId="2" ParentId="6932" CreationDate="2018-01-21T17:18:02.060" Score="0" Body="&lt;p&gt;Mathematics does not need axioms and does not need verbalized proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Applied mathematics answers questions like: if I have 3 apples and get 2 apples, how many apples do I have? 3A + 2A = 5A is a theorem. It can be proved and has been proved by simply doing the experiment. (Mathematics is physics where the experiments are cheap (V.I. Arnold).) No axioms are required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same holds for calculating the contents of granaries. Since mathematics is abstracted from reality it can best be checked by reality which is a better computer than all human productions of that kind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general axioms only are introduced to show post festum that also useful mathematics can be formalized. But mathematics without axioms is not of less value than the Greek mathematics and it has been pursued for thousands of years by Egypts and Babylonians in a much more sophisticated way than by the Greek.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Egypts were the first to solve a quadratic equation. That is mathematics, if the solution is correct as can be proved by doing a suitable experiment. Further proof is not required.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the high level of ancient Egyptian mathematics see &lt;a href=&quot;https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-01-21T17:18:02.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6964" PostTypeId="2" ParentId="6959" CreationDate="2018-01-22T05:22:14.530" Score="4" Body="&lt;p&gt;I looked at the contexts of the &quot;angular momentum&quot; cluster around the spike: angular momentum of electrons; angular momentum about the nucleus; anti-neutron differs from the neutron in that its intrinsic magnetic moment is directed parallel to the spin angular momentum; the quantum number of angular momentum, the spin Hamiltonian, and the angular momentum Hamiltonian; quantum theory of scattering, etc. Orbital dynamics contributed some, but much less, it seems. I would not be surprised, however, by many spurious correlations. The turn of 1960-s was a &quot;cultural moment&quot; that lifted all boats, the romance with nuclear physics and space exploration were only two aspects of it. So were high hopes for thinking computers, loosening of the iron curtain, and the rise of youth counterculture and rock-n-roll.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bulk seems to come from expositions of quantum theory, presumably prompted by its technologization that created a new audience of engineers (lasers, semiconductors, transistors, nuclear energy, etc.). The so-called magic-angle spinning in nuclear magnetic resonance was also discovered in 1958-59, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_quantum_mechanics#1950%E2%80%931959&quot; rel=&quot;nofollow noreferrer&quot;&gt;Timeline of quantum mechanics&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second contributing factor seems to be the &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_subatomic_physics#Strange_particles_and_mysteries_of_the_weak_interaction&quot; rel=&quot;nofollow noreferrer&quot;&gt;great search for the new subatomic particles&lt;/a&gt; in 1950-s after the large accelerators were built, followed by the efforts to make sense of the &quot;particle zoo&quot; in 1960-s. &quot;&lt;em&gt;Had I foreseen this, I would have gone into botany&lt;/em&gt;&quot;, Pauli quipped. Soviet movie Nine Days of One Year, that romanticized the endeavor, came out in 1962 and became a hit. The spin of course is the distinction between bosons and fermions so it came up a lot. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-22T21:59:39.610" LastActivityDate="2018-01-22T21:59:39.610" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6965" PostTypeId="1" AcceptedAnswerId="6968" CreationDate="2018-01-22T12:29:28.690" Score="4" ViewCount="167" Body="&lt;p&gt;We are sitting in class and I was wondering: who came up with the arithmetic mean? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who proposed it or who mentioned it first in his/her work? Google showed the oldest mentions in 1750-1800 but...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We looked at Wikipedia but didn't find the answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for any lead, as even the teacher doesn't know.&lt;/p&gt;&#xA;" OwnerUserId="6942" LastEditorUserId="2277" LastEditDate="2018-05-25T09:06:31.807" LastActivityDate="2018-05-25T09:06:31.807" Title="Arithmetic mean - who, when?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="6966" PostTypeId="2" ParentId="6959" CreationDate="2018-01-22T12:42:17.260" Score="4" Body="&lt;p&gt;First thing, we have to realize that the n-grams show the percentage of some sequence of characters in &lt;em&gt;all the available literature&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you take out some constant background signal, you will notice that the increase in the frequency of &quot;angular momentum&quot; takes place around 1900, when quantum theory is born.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MMM5m.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MMM5m.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can notice that &quot;angular momentum&quot; correlates extremely well with &quot;energy levels&quot; and &quot;hydrogen atom&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my guess is simply that the literature about atomic theory started to increase immediately after quantum mechanics was born in the first years of 1900 and eventually reached a plateau around 1960, with a consequent decrease in the relative frequency in the literature. In those years, quantum electrodynamics was developed, and not much was left to be said about atomic theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given the nature of physics, there is no real point in writing many new books about old theories, so it only makes sense that once atomic theory had reached its full development the production of new books about it decreased.&lt;/p&gt;&#xA;" OwnerUserId="5003" LastActivityDate="2018-01-22T12:42:17.260" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6967" PostTypeId="2" ParentId="6749" CreationDate="2018-01-22T13:01:15.500" Score="4" Body="&lt;p&gt;It goes back to the ancient Romans.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Romans, when working with fractions, often divided things into twelfths rather than tenths (since twelve divides nicely into thirds as well as quarters, whereas ten only nicely divides into tenths and fifths which aren't as useful as thirds and quarters). So the Roman foot was divided into twelve smaller units called &quot;unciae&quot; (singular: &quot;uncia&quot;), whence the English word &quot;inch&quot; (though the Roman uncia was slightly smaller than the Imperial inch).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In regard to measuring the earth, there are five Roman feet in a pace, and a thousand paces in a Roman mile (the English word &quot;mile&quot;, in fact, comes from the Latin word &quot;mille&quot; which means &quot;thousand&quot;). So, one Roman mile was 1000 paces, i.e. 5000 feet (again, this was slightly smaller than the Imperial unit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of these units are directly relative to the Earth's size, though. They're only relative to other units.&lt;/p&gt;&#xA;" OwnerUserId="6943" LastEditorUserId="6943" LastEditDate="2018-01-22T13:06:53.750" LastActivityDate="2018-01-22T13:06:53.750" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6968" PostTypeId="2" ParentId="6965" CreationDate="2018-01-22T13:12:50.773" Score="5" Body="&lt;p&gt;The source is ancient Greek mathematics: at least the Pythagorean &lt;a href=&quot;https://plato.stanford.edu/entries/archytas/#Math&quot; rel=&quot;noreferrer&quot;&gt;Archytas&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The early extant souce seems to be &lt;a href=&quot;https://books.google.it/books?id=2ietQ7tX8TEC&amp;amp;pg=PA163&quot; rel=&quot;noreferrer&quot;&gt;Fragment 2&lt;/a&gt; of the lost work of &lt;em&gt;On Music&lt;/em&gt; [cited by Porphyry, &lt;em&gt;On Ptolemy’s Harmonics&lt;/em&gt;, 1.5] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And Archytas speaking about the means writes these things:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;“There are three means in music: one is the arithmetic [&lt;em&gt;αριθμητικά&lt;/em&gt;], the second geometric [&lt;em&gt;γεωμετρικά&lt;/em&gt;] and the third sub-contrary [, which they call “harmonic”].&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-01-22T13:12:50.773" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6969" PostTypeId="1" AcceptedAnswerId="6974" CreationDate="2018-01-23T08:07:27.350" Score="5" ViewCount="201" Body="&lt;p&gt;I am sure many scientists would initially have attempted to explain Young's double slit experiment's results using the concept of light as a stream of particles. Can somebody tell me what these attempts were and how they failed?&lt;/p&gt;&#xA;" OwnerUserId="5832" LastActivityDate="2018-01-23T09:39:09.670" Title="Corpuscular theory of light and Double slit experiment" Tags="|physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6970" PostTypeId="5" CreationDate="2018-01-23T08:37:35.917" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2018-01-23T08:37:35.917" LastActivityDate="2018-01-23T08:37:35.917" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6971" PostTypeId="4" CreationDate="2018-01-23T08:37:35.917" Score="0" Body="Nuclear physics is the field of Physics that studies atomic nuclei and their constituents and interactions. It started with the discovery of radioactivity by Henri Becquerel in 1896, together with the discovery of the electron by J. J. Thomson a year later." OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2018-01-30T08:08:01.757" LastActivityDate="2018-01-30T08:08:01.757" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6972" PostTypeId="5" CreationDate="2018-01-23T08:43:10.480" Score="0" Body="&lt;p&gt;This tag should be used along with &lt;a href=&quot;/questions/tagged/physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;physics&lt;/a&gt; tag.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-02-23T08:33:08.787" LastActivityDate="2022-02-23T08:33:08.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6973" PostTypeId="4" CreationDate="2018-01-23T08:43:10.480" Score="0" Body="For questions about Electromagnetism - a branch of Physics involving the study of the electromagnetic force, a physical interaction that occurs between electrically charged particles" OwnerUserId="5989" LastEditorUserId="10106" LastEditDate="2022-02-23T08:33:35.013" LastActivityDate="2022-02-23T08:33:35.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="6974" PostTypeId="2" ParentId="6969" CreationDate="2018-01-23T09:39:09.670" Score="5" Body="&lt;p&gt;Young specifically designed the experiment to rule out corpuscular explanations, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2634/55&quot;&gt;How did Young perform his double slit experiment?&lt;/a&gt;, so the Newtonians' reactions were more of the nature of Brougham's &quot;&lt;em&gt;absurdity and one of the most incomprehensible suppositions that we remember to have met with in the history of human hypothesis&lt;/em&gt;&quot; than alternative explanations. Not even Newton himself could offer a corpuscular explanation of interference, he had to come up with a speculative hybrid of corpuscles with &quot;fits of easy reflection and easy transmission&quot; and a pilot wave. But neither he nor his successors offered a quantitative theory based on it, which wave optics provided, see &lt;a href=&quot;https://hsm.stackexchange.com/a/6035/55&quot;&gt;How did Newton explain his interference rings without wave optics?&lt;/a&gt; Here is the gist from &lt;a href=&quot;http://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00002&quot; rel=&quot;noreferrer&quot;&gt;Hypothesis Explaining the Properties of Light, 1675&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;And the vibrations or tremors incited in the air by percussion continue a little time to move from the place of percussion in concentric spheres to great distances. And in like manner, when a ray of light falls upon the surface of any pellucid body, and is there refracted or reflected, may not waves of vibrations, or tremors, be thereby excited in the refracting or reflecting medium at the point of incidence and continue to arise there, and to be propagated from thence... and are not these vibrations propagated from the point of incidence to great distances? And do they not overtake the rays of light, and by overtaking them successively, do they not put them into the fits of easy reflexion and easy transmission described above?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-01-23T09:39:09.670" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6975" PostTypeId="1" CreationDate="2018-01-23T12:35:29.237" Score="1" ViewCount="271" Body="&lt;p&gt;The concept of Lipschitz continuous mappings is probably at the present time the most important mathematical concept associated with Lipschitz's name. These mappings play an important role in the theory of differential equations and they also appear in many other branches of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was R. Lipschitz' original motivation for considering these mappings?&lt;/p&gt;&#xA;" OwnerUserId="4096" LastEditorUserId="264" LastEditDate="2018-01-23T17:08:21.453" LastActivityDate="2018-01-23T20:44:56.110" Title="What was Lipschitz's original motivation for the introduction of Lipschitz continuity?" Tags="|mathematics|real-analysis|functional-analysis|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="6976" PostTypeId="2" ParentId="6975" CreationDate="2018-01-23T20:44:56.110" Score="3" Body="&lt;p&gt;It is existence and uniqueness question for ordinary differential equations. When I was a student (in 1970s) Lipschitz functions were not omnipresent in Analysis. The only context where this name appeared (in undergraduate curriculum) was existence and uniqueness theorem for ODE. And this was apparently his original motivation, as the reference in the comment of @Michael E2 shows. Nowadays this condition has more wide use because of the development of non-smooth analysis in the recent decades.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-01-23T20:44:56.110" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6977" PostTypeId="1" CreationDate="2018-01-24T05:57:36.940" Score="1" ViewCount="119" Body="&lt;p&gt;In mathematics terminology, a function is defined over two sets. One is is input set and other one is output set and for a particular input element, we the following notation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$f(x) = y$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $x$ is the input element and $y$ is the output element.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Coming to conditional probability the notation is as follows&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$p(A/B) = \dfrac{p(A\cap B)}{p(B)}$$&#xA;The numerator and denominator are perfectly okay as per notations. But coming to the notation for conditional probability, the input is just $A$ under the assumption of $B$ occurrence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the reason behind that notation, since $A/B$ is not an eligible input where $A$ has to be?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastActivityDate="2018-01-24T10:27:56.997" Title="Notation for conditional probability" Tags="|notation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6978" PostTypeId="2" ParentId="6977" CreationDate="2018-01-24T07:10:16.917" Score="2" Body="&lt;p&gt;See: A.N. Kolmogorov, &lt;a href=&quot;https://books.google.it/books?id=a5ubnQEACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Foundations of the Theory of Probability&lt;/a&gt;, (1956: original ed.1933), page 3:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To each set $A$ in [the collection of &lt;em&gt;elementary events&lt;/em&gt;] $\mathfrak F$ is assigned a non-negative real number $\mathsf P(A)$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In symbols: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\mathsf P(A): \mathfrak F \to \mathbb R$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And see page 6:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If $\mathsf P(B) &amp;gt; 0$, then the quotient $\mathsf P_B(A)=\dfrac {\mathsf P(AB)}{\mathsf P(B)}$ is defined to be the &lt;em&gt;conditional probability&lt;/em&gt; of the event $A$ under the condition $B$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Formally the conditional probability is a function of two variables:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$\mathsf P_B(A): \mathfrak F \times \mathfrak F \to \mathbb R$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other terms, for each $B$, we have that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$\mathsf P_B(A): \mathfrak F \to \mathbb R$.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-01-24T10:27:56.997" LastActivityDate="2018-01-24T10:27:56.997" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6979" PostTypeId="1" AcceptedAnswerId="6981" CreationDate="2018-01-24T15:30:18.927" Score="4" ViewCount="220" Body="&lt;p&gt;What is the most number of digits of a transcendental mathematical constant (for example, $\pi$ or $\mathrm{e}$) that have been necessary for an actual computation?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Note that I am asking about the most extreme case, not usual practice. Usual practice is well documented; for example, &lt;a href=&quot;https://www.vox.com/2016/4/2/11350518/nasa-digits-pi&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; quotes a NASA engineer as saying that they use only 15 digits for their calculations.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;By &quot;actual computation&quot; I mean a computation that has been performed with some scientific or engineering purpose.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;By &quot;necessary&quot; I mean that if the value of the fundamental constant were less precise, then the computation could give a result so inaccurate as to be legitimately less useful for the intended scientific or engineering purpose.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6949" LastActivityDate="2018-01-25T22:58:05.557" Title="What is the most number of digits of a mathematical transcendental constant that have been required for a real computation?" Tags="|mathematics|computation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6980" PostTypeId="2" ParentId="6979" CreationDate="2018-01-24T18:24:30.000" Score="0" Body="&lt;p&gt;Of course many mathematical constants can be computed to billions of digits,&#xA;I suppose the question is about physics: how many digits can be actually tested,&#xA;what are the most precise measurements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Littlewood (who wrote this in 1948), the most exact computations used in science are time computations in astronomy (up to 15 decimal digits). &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Some of them involve time periods of thousands years which are computed to 0.001 second, and for control one usually&#xA;  keeps one-two more digits&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't think this changed since the time of Littlewood.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference J. E. Littlewood, A mathematian's Miscellany, London 1957. Chapter &quot;Large Numbers&quot;. Originally published in Math Gazette, JUly 1948, vol 32, N 300.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some computations in &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Precision_tests_of_QED&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantum electrodynamics&lt;/a&gt; (Wikipedia) have 12-13 significant digits which have been tested. So this comes close.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-01-24T18:31:32.030" LastActivityDate="2018-01-24T18:31:32.030" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="6981" PostTypeId="2" ParentId="6979" CreationDate="2018-01-25T00:53:22.990" Score="6" Body="&lt;p&gt;Lamb discusses the issue in &lt;a href=&quot;https://blogs.scientificamerican.com/observations/how-much-pi-do-you-need&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Much Pi Do You Need?&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I asked a NASA scientist how many digits of pi the agency uses for its calculations. Susan Gomez, manager of the International Space Station Guidance Navigation and Control (GNC) subsystem for NASA, said that calculations involving pi use 15 digits for GNC code and 16 for the Space Integrated Global Positioning System/Inertial Navigation System (SIGI). SIGI is the program that controls and stabilizes spacecraft during missions.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Believe it or not, there is a committee that makes recommendations about the values of these fundamental constants. The Committee on Data for Science and Technology, or CODATA, an interdisciplinary group from the International Council for Science, periodically publishes a set of accepted values of the fundamental physical constants... Peter Mohr, a physicist who works for the Fundamental Constants Data Center at the National Institute for Standards and Technology, which is involved in calculating and disseminating the accepted CODATA values, says that the institute uses 32 significant digits of pi in their computations.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need&quot; rel=&quot;nofollow noreferrer&quot;&gt;NASA's Marc Rayman&lt;/a&gt; explained why one would not need to go far beyond the CODATA precision even in the hypothetical future astronomy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places. If you think about how fantastically vast the universe is... and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One may worry that sensitivity effects in ill-posed problems, such as weather forecasts, would require extra high precision in the input. However, in those cases the imprecision of measured inputs makes high precision in mathematical constants pointless. Still, in 2002 Tucker employed a combination of interval arithmetic and 80-bit IEEE extended arithmetic (17-digit precision) to prove that the Lorenz equations support a “strange” attractor, which solved Smale's problem. Bailey and Borwein describe other examples in &lt;a href=&quot;http://www.mdpi.com/2227-7390/3/2/337/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;High-Precision Arithmetic in Mathematical Physics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But if &amp;quot;scientific or engineering purpose&amp;quot; includes cryptography then the threshold is raised dramatically. Ciphers based on transcendental numbers have been proposed, and they require exorbitant precision to maintain security. Here is a simple version from &lt;a href=&quot;http://www.m-hikari.com/ams/ams-2014/ams-173-176-2014/viswanathAMS173-176-2014.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Viswanath's Transcendental Numbers and Cryptography&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The decimal  expansion  of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is  known  for  more  than  1000  million  places of  decimals and is known as the mountain of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. We use this mountain of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; in the encryption of  messages.  One  can  use  any  other transcendental  number  for  which  the mountain is available! Assume that Bob is encrypting a message with he mountain of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. In this encryption, the Key &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is the position of the decimal place of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; which is used to begin  the  encryption.  The  number  at &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-th place,  say &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is  used  to  substitute  the beginning  letter  of  the  plaintext  by  shifting the  alphabet by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; units(mod 26). Afterwards the  process is continued with the next integer and the next alphabet in the  plaintext and  so  on,  till  the  entire  message  is  encrypted. One  can  raise  the  level  of  security  further  by  using  two  transcendental numbers one for encryption and the other for decryption and the key is chosen as before.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Actually, digits of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; are known to 10 trillion places now.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-25T22:58:05.557" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6982" PostTypeId="2" ParentId="6979" CreationDate="2018-01-25T17:08:17.047" Score="2" Body="&lt;p&gt;In cryptography, designs often have &quot;constants&quot;: effectively parameters which can use an arbitrary number (or an arbitrary number which doesn't have any simple patterns) but which must use the same number in all implementations. However, it's considered suspicious if the numbers are truly arbitrary: perhaps you calculated them in such a way as to create a &quot;back door&quot;. So it is now common to use &lt;a href=&quot;https://en.wikipedia.org/wiki/Nothing_up_my_sleeve_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;nothing up my sleeve&quot; numbers&lt;/a&gt;, and elementary transcendental numbers are the most common type of such numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Blowfish_(cipher)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blowfish cipher&lt;/a&gt; uses the first 704 bits of the fractional part of pi to initialise its key schedule. That's about 212 decimal digits.&lt;/p&gt;&#xA;" OwnerUserId="104" LastEditorUserId="104" LastEditDate="2018-01-25T19:13:07.697" LastActivityDate="2018-01-25T19:13:07.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6983" PostTypeId="2" ParentId="5666" CreationDate="2018-01-25T19:51:54.413" Score="1" Body="&lt;p&gt;I was told by the professor in a graduate physics course that the equations were called &quot;canonical&quot; because they were so perfect that they could be laws of the church, that is canon laws.  At the time of their creation, the church was the most powerful entity extant.  Thus, they were perfect enough to be church laws or &quot;canonical&quot; laws.&lt;/p&gt;&#xA;" OwnerUserId="6957" LastActivityDate="2018-01-25T19:51:54.413" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6984" PostTypeId="1" CreationDate="2018-01-26T13:02:02.683" Score="1" ViewCount="151" Body="&lt;p&gt;The following words concerning Carl Friedrich Gauss are attributed to Laplace in an article from the Mathematics magazine by Teets and Whitehead (&lt;a href=&quot;https://www.maa.org/programs/maa-awards/writing-awards/the-discovery-of-ceres-how-gauss-became-famous&quot; rel=&quot;nofollow noreferrer&quot;&gt;The discovery of Ceres: how Gauss became famous&lt;/a&gt;, first page.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Duke of Brunswick has discovered more in his country than a&#xA;  planet: a super-terrestrial spirit in a human body.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am looking for the original quote of Laplace, presumably in French, together with a reference.&lt;/p&gt;&#xA;" OwnerUserId="6959" LastEditorUserId="55" LastEditDate="2018-01-26T19:51:46.747" LastActivityDate="2018-01-26T19:51:46.747" Title="Laplace's characterization of Gauss as &quot;a super-terrestrial spirit in a human body&quot;" Tags="|biographical-details|reference-request|quote|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6985" PostTypeId="2" ParentId="6951" CreationDate="2018-01-26T21:26:38.437" Score="1" Body="&lt;p&gt;It seems the only firm fact about the origin of the problem is that it is discussed for the first time in print in 1969 in &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0022123669900111&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rosenthal's On quasi-complemented subspaces of Banach spaces&lt;/a&gt;, without attribution to Banach or Mazur. The attribution to Mazur appears in &lt;a href=&quot;https://arxiv.org/abs/1709.09646&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferrando--Kakol--Lopez-Pellicer--Sliwa paper On the separable quotient problem for Banach spaces&lt;/a&gt; with the date of 1932, but no details. &lt;a href=&quot;https://arxiv.org/abs/1707.09546&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leiderman-Morris-Tkachenko in On the separable quotient problem for Banach spaces&lt;/a&gt; write:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The Separable Quotient Problem for Banach Spaces has its roots in the 1930s and, according to a private communication from Wladyslaw Orlicz to Jerzy Kakol, is due to Stefan Banach and Stanislaw Mazur&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I emailed Jerzy Kakol to clarify, and he replied:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Formally this problem has been announced by Mazur. But I was told by prof. Orlicz, who was a student of Banach, that Banach  knew already this problem much earlier than mentioned by Mazur.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How exactly Mazur announced it I am not sure. The story of the Lvov mathematical school, to which both Banach and Mazur belonged, is told in &lt;a href=&quot;https://www.maa.org/press/maa-reviews/pearls-from-a-lost-city-the-lvov-school-of-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duda's prize-winning Pearls from a Lost City&lt;/a&gt;. Much of the interaction took place in Lvov's cafés, and none of it was recorded until Banach's wife introduced the now famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Scottish_Book&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scottish notebook&lt;/a&gt; in 1935 (by then the meetings regularly took place in the Scottish Café). It is there that a cousin problem 153 (with a live goose for a prize), which was later connected to Schauder bases and solved by Enflo, was entered by Mazur on November 1936. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If Mazur &quot;formally announced&quot; the separable quotient problem in a café in 1932 there would be no record of it, and we can at best hope for an eyewitness testimony. The same goes for Orlicz's attribution to Banach. After Soviet takeover of Poland in 1939 the Lvov school was essentially snuffed out and after World War II its members scattered to the winds. It is possible that by word of mouth the problem eventually reached Rosenthal, who was not, however, aware of its source.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-26T21:40:22.693" LastActivityDate="2018-01-26T21:40:22.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6986" PostTypeId="1" AcceptedAnswerId="7016" CreationDate="2018-01-27T01:24:32.410" Score="4" ViewCount="376" Body="&lt;p&gt;Is there a source collecting data on the numbers of Mathematics professors (I mean with rank of Ordinarius, or ordentlicher Professor) in German Universities around 1890?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see how clearly both the Ministry of Education and the faculty members distinguished mathematics from other subjets, and indeed different fields of mathematics from one another, see Renate Tobies &lt;em&gt;Zur Berufungspolitik Felix Kleins. - Grundsätzliche Ansichten&lt;/em&gt;, in: &lt;em&gt;Natur Technik, Medizin&lt;/em&gt;, 24 (1987) 2, S. 43-52.&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="4251" LastEditDate="2021-09-14T10:27:14.800" LastActivityDate="2021-09-14T10:27:14.800" Title="Seeking numbers of German mathematics professors in 1890" Tags="|education|19th-century|germany|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="6987" PostTypeId="1" AcceptedAnswerId="6995" CreationDate="2018-01-27T18:31:27.093" Score="3" ViewCount="158" Body="&lt;p&gt;Did Euclid formulate his definitions/postulates/common notions at the beginning of Book I of the &lt;em&gt;Elements&lt;/em&gt; before or after writing the 465 theorems of the &lt;em&gt;Elements&lt;/em&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;cf.: Michael J. Crowe, “&lt;a href=&quot;http://www.f.waseda.jp/sidoli/Crowe_10_Misconceptions.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ten Misconceptions about Mathematics and Its History&lt;/a&gt;,” &lt;em&gt;History and Philosophy of Modern Mathematics&lt;/em&gt; 11 (1988): 260.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-01-28T03:44:48.843" LastActivityDate="2018-01-29T05:03:40.040" Title="Did Euclid formulate his definitions/postulates/common before or after writing all his theorems?" Tags="|mathematics|mathematical-logic|euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6988" PostTypeId="2" ParentId="6767" CreationDate="2018-01-27T18:48:36.123" Score="3" Body="&lt;p&gt;Heath, the famous translator of the &lt;em&gt;Elements&lt;/em&gt;, concludes in his introduction to &lt;a href=&quot;https://isidore.co/calibre#book_id=6070&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;vol. 1 of his translation of the &lt;em&gt;Elements&lt;/em&gt;&lt;/a&gt;, §&amp;quot;3. First Principles: Definitions, Postulates, and Axioms&amp;quot;, that Euclid's usage of these terms aligns most closely to Aristotle's. Heath begins that § by quoting &lt;em&gt;in extenso&lt;/em&gt; from Aristotle's &lt;a href=&quot;https://isidore.co/aquinas/PostAnalytica.htm#18&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Posterior Analytics&lt;/em&gt; 1.10 (76&lt;sup&gt;a&lt;/sup&gt;5)&lt;/a&gt; (&amp;quot;Difference between principles and non-principles, common and proper principles&amp;quot;) and commenting upon it. Here's Heath's translation with his useful parenthetical remarks relating what Aristotle is saying to geometry:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“By first principles in each genus I mean those the truth of which it is not possible to prove. What is &lt;em&gt;denoted&lt;/em&gt; by the first (terms) and those derived from them is assumed; but, as regards their &lt;em&gt;existence&lt;/em&gt;, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers.&lt;/p&gt;&#xA;&lt;p&gt;“Now that which is &lt;em&gt;per se&lt;/em&gt; necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer’s hypotheses false, as some have said: I mean those who say that ’you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.’ The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is &lt;em&gt;illustrated&lt;/em&gt; by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither” (because the subject is of equal extent with what is predicated of it).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-27T18:48:36.123" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6989" PostTypeId="1" AcceptedAnswerId="6991" CreationDate="2018-01-28T05:44:55.230" Score="9" ViewCount="861" Body="&lt;p&gt;Fermat is easily one of the best known mathematicians of all time. We all know about Fermat's Last Theorem, Fermat's Little Theorem, his quadrature rule, his invention of probability theory, etc. Every introduction of Fermat, however, mentions that mathematics wasn't his &quot;day job&quot; and that he was a lawyer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As much as I search, I cannot seem to find anything on his work as a lawyer. Unless his job was divine mathematics as a lawyer, I get the feeling that his work as a lawyer was either unremarkable or just forgotten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have a reference that they can point me to that could describe his role in law? Or something that would tell me what his job would have been?&lt;/p&gt;&#xA;" OwnerUserId="2214" LastEditorUserId="4251" LastEditDate="2021-02-27T07:41:32.867" LastActivityDate="2021-02-27T07:41:32.867" Title="What did Fermat do as a lawyer?" Tags="|mathematicians|biographical-details|reference-request|fermat|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6990" PostTypeId="2" ParentId="6802" CreationDate="2018-01-28T06:39:07.760" Score="5" Body="&lt;p&gt;zy_:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two parts to Biermann's hypothesis on the meaning of &quot;Vicimus GEGAN&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i) GEGAN should be read &lt;em&gt;in inverse order&lt;/em&gt; as standing for (vicimus)&#xA;    &lt;strong&gt;N&lt;/strong&gt;[exum medii] &lt;strong&gt;A&lt;/strong&gt;[rithmetico-] &lt;strong&gt;G&lt;/strong&gt;[eometricum] &lt;strong&gt;E&lt;/strong&gt;[xpectationibus] &lt;strong&gt;G&lt;/strong&gt;[eneralibus];&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ii) With &quot;Vicimus GEGAN&quot; the great Gauß was alluding to his discoveries on the connection between the arithmetic-geometric means and the general theory of elliptic functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Biermann's paper in the &lt;em&gt;Mitt. Gauß-Ges. Göttt.&lt;/em&gt; does confirm the first part of the hypothesis or at least &lt;strong&gt;the portion of it wherein he asserts that GEGAN should be read backwards&lt;/strong&gt;. To be more precise, Biermann reported on a manuscript (by Gauß) found in the Göttingen astronomical observatory by Herr Dr. Hartmut Grosser sometime in 1997; on a certain sheet of the manuscript (there is a copy of it in the penultimate page of &lt;a href=&quot;https://drive.google.com/file/d/1-jCsmpwyEgKzVS392m6jpgOsd7Rcx-W2/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;the paper&lt;/a&gt;!), we can find the abbreviation NAGEG at least three times, the well-known GEGAN, and even a drawing of something which resembles a lemniscate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Despite the fact that Biermann wrote in &quot;das Fazit&quot; of his paper something like &quot;it remains to determine exactly what the E stands for&quot;, he wrapped it up expressing  his belief that the sketch of the lemniscate just below the &quot;GEGAN&quot; on that page of the manuscript also &lt;em&gt;validated&lt;/em&gt; the second part of his conjecture:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was nun aber der Fund so bedeutsam macht, sind die Majuskeln &lt;strong&gt;G E G A N&lt;/strong&gt; mit der Skizze einer &lt;strong&gt;Lemniskate&lt;/strong&gt; darunter. Das Wort GEGAN begegnet uns auf der Seite noch mehrfach, teils kunstvoll kalligraphisch verschnörkelt, auch in der Form gegan. Was nun aber allem die Krone aufsetzt, ist die Begegnung mit dem so lange vergeblich gesuchten Schlüsselwort &lt;strong&gt;NAGEG&lt;/strong&gt;, auch in der Form &lt;strong&gt;nageg&lt;/strong&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Welche Schlußfolgerungen können nun aus diesem Sachverhalt gezogen werden?&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;Das gemeinsame Auftreten des Schlüsselworts GEGAN und einer Lemniskate beweist, daß die Hypothese ... , wonach GEGAN mit dem agM in Verbindung steht, begründet war.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;Das nunmehr bei Gauß nachgewiesene Schlüsselwort NAGEG stützt entscheidend die obige Hypothese ..., wonach die Gaußschen Schlüsselwörter rückwärts zu lesen sind. &lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;Fazit: Die vom Verfasser bei seinen Dechiffrierungsversuchen Gaußscher Schlüsselwörter und Buchstabengruppen erzielten Resultate werden durch das von Herrn Dr. Grosser gefundene und freundlicherweise als Kopie zur Verfügung gestellte Gaußsche Notizblatt bestätigt. Mag unsicher bleiben, ob etwa E zu Exspectatio bzw. zu Expressio bzw. zu Explicatio zu ergänzen sei usw. – der Sinn der zunächst rätselhaften Schlüsselwörter dürfte nunmehr unstrittig sein: Die Chiffren bezogen sich auf die große Gaußsche Entdeckung des Zusammenhangs zwischen lemniskatischen Funktionen, agM und Potenzreihen und der daraus resultierenden Theorie der Modulfunktion und der elliptischen Transzendenten. Teils haben sie retrospektiven (&quot;vicimus&quot;, &quot;demonstravimus&quot;), teils programmatischen (&quot;revolve!&quot;) Charakter. Dergestalt ermöglicht uns das Studium der Gaußschen Verschlüsselungen einen Blick in die geistige Werkstatt eines Genies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What makes this discovery so significant are the capital letters &lt;strong&gt;G E G A N&lt;/strong&gt; with the sketch of a &lt;strong&gt;lemniscate&lt;/strong&gt; below them. We encounter the word GEGAN several additional times on this page, sometimes in ornate calligraphic flourishes, and also in the variant gegan. What tops it all is the encounter with the keyword &lt;strong&gt;NAGEG&lt;/strong&gt;, also in the variant &lt;strong&gt;nageg&lt;/strong&gt;, the search for which had so long proved elusive.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What conclusions can now be draw from this factual situation?&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;The joint appearance of the keyword GEGAN and a lemniscate proves that hypothesis ..., &#xA;  according to which GEGAN is linked to the AGM, was well founded.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;The keyword NAGEG now documented in Gauss's writings decisively supports hypothesis ..., &#xA;  according to which Gaussian keywords are to be read in reverse.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;In conclusion: The results achieved by the author in his attempts to decipher Gaussian keywords and letter groups are being confirmed by the sheet of scratch paper found and kindly supplied in copy by Dr. Grosser. While it may remain uncertain whether, for example, E should be expanded into Exspectatio, or Expressio, or Explicatio etc, the sense of the initially enigmatic keywords should now be beyond dispute:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;These ciphers referred to Gauss's major discovery of the connection between lemniscatic functions, the AGM, and power series and the resulting theory of the modular form and elliptic transcendentals. Their character is in part retrospective (&quot;vicimus&quot;, &quot;demonstravimus&quot;), in part programmatic (&quot;revolve!&quot;). In this way the study of Gaussian cipherings allows for a peek into the mental workshop of a genius.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If after reading Biermann's paper through and through, you still find hard to reconcile the second part of his hypothesis with the entries in Gauß's diary which you listed in your question, I think you can do no worse than to take a look at this paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;D. A. Cox, &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4757-3240-5_55&quot; rel=&quot;nofollow noreferrer&quot;&gt;The arithmetic-geometric mean of Gauß&lt;/a&gt;. &lt;em&gt;L'Enseignement Mathématique&lt;/em&gt;, vol. 30, 1984, pp. 275-330.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, on page 321 of Cox's paper you will find this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... how much did [Gauß] know about the agM [in May, 1799]? Unfortunately, this a very difficult question to answer. Only a few scattered fragments dealing with the agM can be dated before May 30, 1799 (see [12, X.1, pp. 172-173 and 260]). As for the date 1791 of his discovery of the agM, it comes from a letter he wrote in 1816 (see [12, X.1, p. 247]), and Gauß is known to have been wrong in his recollections of dates. The only other knowledge we have about the agM in this period is an oral tradition which holds that Gauß knew the relation between theta functions and the agM in 1794 (see [12, III, 493]). We will soon see that this claim is not as outrageous as one might suspect.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Interestingly enough, Cox did not mention Biermann's hypothesis on the &quot;Vicimus GEGAN&quot; entry at all in his article...&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="4201" LastEditDate="2018-01-30T08:08:25.693" LastActivityDate="2018-01-30T08:08:25.693" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6991" PostTypeId="2" ParentId="6989" CreationDate="2018-01-28T09:50:15.977" Score="18" Body="&lt;p&gt;Fermat wasn't so much a &quot;lawyer&quot; as a &lt;em&gt;magistrat&lt;/em&gt; which means that he sat on successively higher levels of the &lt;em&gt;Parlement&lt;/em&gt; of Toulouse, France.  This period (17th century) was before the emergence of the doctrine of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Separation_of_powers#Montesquieu&amp;#39;s_separation_of_powers_system&quot; rel=&quot;noreferrer&quot;&gt;separation of powers&lt;/a&gt; in Western thought, so that the &lt;em&gt;Parlement&lt;/em&gt; was not a &quot;Parliament&quot; in the modern English sense of the word, but rather the judicial, legislative, and executive branches rolled into one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fermat was extremely active as a &lt;em&gt;magistrat&lt;/em&gt; and penned numerous legal documents.  In fact he was busy writing one only a few days before he died.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the most controversial cases Fermat was involved in concerned the trial of the priest Delpoy and his eventual execution.  Fermat himself was unconvinced of Delpoy's guilt but was apparently overruled by his more powerful peers on the &lt;em&gt;Parlement&lt;/em&gt;, including the president of the &lt;em&gt;Parlement&lt;/em&gt; who is considered to have been rather hostile to Fermat.  Fermat is said to have been in shock over the verdict and execution to such an extent that his productivity at the court declined sharply.  He was similarly unable to work on mathematics for a period of a few months, as claimed by Digby.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More details on the background of Fermat's legal activity can be found in &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9542-y&quot; rel=&quot;noreferrer&quot;&gt;this 2018 publication in &lt;em&gt;Foundations of Science&lt;/em&gt;&lt;/a&gt; where you will also find additional references on Fermat's activity at the &lt;em&gt;Parlement&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-01-28T09:50:15.977" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="6992" PostTypeId="2" ParentId="6989" CreationDate="2018-01-28T09:55:27.973" Score="9" Body="&lt;p&gt;Read Klaus Barner's article &lt;em&gt;&lt;a href=&quot;http://www.ems-ph.org/journals/newsletter/pdf/2001-12-42.pdf&quot; rel=&quot;noreferrer&quot;&gt;Pierre de Fermat (1601? &amp;ndash; 1665): His life beside mathematics&lt;/a&gt;&lt;/em&gt; (pages 12&amp;ndash;16).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-01-28T09:55:27.973" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6993" PostTypeId="1" CreationDate="2018-01-28T11:58:00.217" Score="0" ViewCount="104" Body="&lt;p&gt;Why is $\textbf{J}$ called angular momentum operator? Can anyone explain why the expectation value of J is angular momentum?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how $J$ is defined: The rotation operator&#xA;$$&#xA;U(\alpha)=\exp(-i {\bf \alpha} \cdot \mathbf{J})&#xA;$$&#xA;Rotates the system around the direction of $\alpha$, through an angle $|\alpha |$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sorry that this is not quite research level, but I cannot find some intuitive explanation about this elsewhere.&lt;/p&gt;&#xA;" OwnerDisplayName="user6964" LastActivityDate="2018-01-28T11:58:00.217" Title="Why is the angular momentum written as JJ in quantum mechanics?" Tags="|reference-request|quantum-mechanics|mathematical-physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6994" PostTypeId="2" ParentId="6802" CreationDate="2018-01-28T22:12:08.793" Score="1" Body="&lt;p&gt;@ José Hdz. Stgo.:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you for this great answer. I think I should add a few comments on the related materials I collected from different books. (Since I am neither a native speaker of German language nor well-versed in it, sometimes I need google translation or deepL translation for a better understanding on all these mysteries)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe Biermann used some hypotheses originated from L. Schlesinger on Gauss' work of AGM and elliptic functions(see &lt;em&gt;Fragmente zur Theorie des arithmetrisch-geometrischen Mittels aus den Jahren 1797-1799&lt;/em&gt;, L. Schlesinger, 1911). Schlesinger made a great effort on dating some fragments on elliptic functions in Gauss' &lt;em&gt;Leistenotizen&lt;/em&gt;(in which most of the records are dated no later than 1798), which is extremely important for understanding the development of Gauss' theory on elliptic functions. Schlesinger pointed out that the fragments in the &lt;em&gt;Leistenotizen&lt;/em&gt; containing AGM are consisted of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a)AGM and the rectification of ellipses;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b)AGM and the series whose exponents are quadratic functions(i.e., theta functions);&lt;/p&gt;&#xA;&#xA;&lt;p&gt;c)Series expansion of elliptic integrals as well as linear differential equation for elliptic integral of the first kind and the second kind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1911 Schlesinger dated these materials no later than 1798 as well as other records in the &lt;em&gt;Leistenotizen&lt;/em&gt;. It is very likely that this hypothesis contradicts what Gauss wrote in his dairy on May, 30th 1799(which implies that Gauss did not know the relation between AGM and elliptic integral before 1799), so later Schlesinger changed it to the summer of 1799(Gauss's &lt;em&gt;Werke, X-1&lt;/em&gt;, pp. 273). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the same year(Nov. 1799), Gauss started his &lt;em&gt;Scheda Ac&lt;/em&gt;, another important record on Gauss' own development of elliptic function theory. In the &lt;em&gt;Scheda Ac&lt;/em&gt; one can see the encrypted word &lt;strong&gt;GALEN&lt;/strong&gt;(Gauss's &lt;em&gt;Werke, X-1&lt;/em&gt;, pp. 273). I trust Biermann's interpretation of the &lt;strong&gt;GALEN&lt;/strong&gt; that Gauss realized the importance of the reciprocal of AGM(which is equal to some elliptic integral of the first kind), and I do not reject Biermann's hypothesis that &quot;Vicimus &lt;strong&gt;GEGAN&lt;/strong&gt;&quot; is a discovery on AGM. but I do not think &quot;Vicimus &lt;strong&gt;GEGAN&lt;/strong&gt;&quot; is a discovery on the AGM and elliptic integrals/thetafunctions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I)The date of the record &quot;Vicimus &lt;strong&gt;GEGAN&lt;/strong&gt;&quot; is Oct. 21, 1796. Gauss started his research on lemniscate months after that(Jan. 1797). It is highly improbable that Gauss had already known the relation between the &lt;strong&gt;general&lt;/strong&gt; AGM and the elliptic integral of the first kind when he had not even started his research on a special case(the lemniscatic integral).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;II)I doubt whether the oral tradition that Schering recorded is reliable. Gauss already know that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\theta_4(e^{-\pi})=1-e^{-\pi}+e^{-4\pi}-\cdots=\sqrt{\frac{\varpi}{\pi}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\theta_2(e^{-\pi})=2e^{-\pi/4}+e^{-9\pi/4}+\cdots=\sqrt{\frac{\varpi}{\pi}}$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\theta_3(e^{-\pi})=1+e^{-\pi}+e^{-4\pi}+\cdots$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\theta_3^4=\theta_4^4+\theta_2^4$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in 1798(&lt;em&gt;Scheda Aa&lt;/em&gt;, see Gauss's &lt;em&gt;Werke, III&lt;/em&gt;, pp. 418), where $\varpi$ is the lemniscatic constant&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\varpi=2\int_{0}^1\frac{\mathrm{d}x}{\sqrt{1-x^4}}.$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If he did know the relation between AGM and the theta functions in 1794, he can prove that $$AGM((\theta_3(q))^2,(\theta_4(q))^2)=1$$ without any effort, which would definitely answer Gauss' question on May 30th, 1799. Pfaff's letter(Gauss's &lt;em&gt;Werke, X-1&lt;/em&gt;, pp. 273, which is quoted in &lt;em&gt;Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory&lt;/em&gt; by Jeremy Gray) suggests that the proof was elusive to Gauss even in Nov. 1799.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;III)If Gauss had discovered anything about AGM in 1796, it might be something else, e.g. the asymptotic formula for AGM in &lt;em&gt;Scheda Ac&lt;/em&gt;(Gauss's &lt;em&gt;Werke, X-1&lt;/em&gt;, pp. 186, Gauss also said he found that AGM can be written as the quotient of two transcendental functions &lt;strong&gt;long ago&lt;/strong&gt; in the 101st entry of his diary). But it is likely that we can never know what Gauss discovered on Oct. 21, 1796.  &lt;/p&gt;&#xA;" OwnerUserId="6450" LastEditorUserId="6450" LastEditDate="2018-01-29T15:03:06.453" LastActivityDate="2018-01-29T15:03:06.453" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6995" PostTypeId="2" ParentId="6987" CreationDate="2018-01-28T23:58:17.943" Score="3" Body="&lt;p&gt;Nothing is factually known not only about how Elements were written but even about Euclid himself. At one point even his identity as a single person was disputed, and some historians suggested that it was a name for a collective of Alexandrian mathematicians, something like an ancient Bourbaki, see &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/978-3-319-39649-1_10.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paumier-Aubin's Polycephalic Euclid&lt;/a&gt;. This idea is considered unlikely now. Recent studies of the diagrammatic method of Euclid's demonstrations suggest that postulates and common notions (a.k.a axioms) had to be formulated explicitly before the final version of demonstrations was written, because they are transparently alluded to in them, see &lt;a href=&quot;http://spiritual-minds.com/philosophy/assorted/Mancosu%20-%20Philosophy%20of%20Mathematical%20Practice%20(Oxford,%202008).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manders's classic Euclidean Diagram&lt;/a&gt; and &lt;a href=&quot;https://arxiv.org/abs/1109.4298&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rodin's Doing and Showing&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, it was felt by many that the opening definitions of Book I do not fit well with the rest of the Elements and are perhaps later insertions for didactic purposes, to accord with the Aristotelian idea of structuring science starting with definitions, eventually interpolated into the main text by copiers. Lucio Russo, of the &lt;a href=&quot;https://archive.org/details/springer_10.1007-978-3-642-18904-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Forgotten Revolution&lt;/a&gt; fame, makes a detailed case, based on textual and circumstantial evidence, that the definitions of Book I are excerpts from Definitions by Heron of Alexandria, an engineer and mathematician of 1st century AD known for his steam powered toys and a treatise on computational mathematics, Metrica, see &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/s004070050016.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Definitions of Fundamental Geometric Entities&#xA;Contained in Book I of Euclid’s Elements&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps the most interesting part of Russo's speculation is his suggested view of interpolation's motivation. The traditional critique of Book I definitions, that they are useless in demonstrations, has the same air of anachronism to it as the common practice of first ascribing to Euclid the modern axiomatic method and then detecting his &amp;quot;gaps&amp;quot;. Russo puts a less anachronistic face on it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the history of thought we can recognize two completely different views about the function of definitions. According to the first view definitions have the purpose of describing the true essence of the defined entities, whose actual existence is of course considered to be independent of their definitions. In the case of mathematics this concept, which was sustained by PLATO and was dominant from the Imperial period until (at least) the XVIII century, implies, of course, that mathematical entities actually exist, independently from the mathematicians describing and using them. We shall call this first view essentialist or Platonic. This view was substantially shared by ARISTOTLE.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;According to the second view, which we shall call nominalist, the function of definitions is the introduction of a short label for a long defining formula. The existence of the object so defined must of course be ascertained by other means. In the case of mathematics the existence of the defined objects can be warranted by means of their actual geometrical construction. This second concept, so completed, shall be called nominalist and constructivist... People sharing this second concept certainly realize the necessity of avoiding a regress ‘ad infinitum’ by assuming some undefined terms as a starting point...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That a dispute between essentialists and constructivists took place in Academy after Plato, with Speusippus, his nephew, representing the former, and Menaechmus, the inventor of conic sections, the latter, we know from Proclus, see &lt;a href=&quot;https://www.pdcnet.org//pdc/bvdb.nsf/purchase?openform&amp;amp;fp=ancientphil&amp;amp;id=ancientphil_1983_0003_0001_0012_0029&amp;amp;onlyautologin=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bowen's Menaechmus versus the Platonists: Two Theories of Science in the Early Academy&lt;/a&gt;. Russo himself mentions that according to ANTISTHENES’ school &amp;quot;&lt;em&gt;it is possible to give a definition of the composite kind of things or substances, whether they are sensible things or objects of intellectual intuition, but not of their primary parts (ARISTOTLE, Met., 1043b, 23–32)&lt;/em&gt;&amp;quot;. Moreover, in Archimedes's and Apollonius's writings Aristotelian pseudo-definitions do not appear, and Apollonius explicitly describes geometric entities as motivated by everyday experience (e.g. lines by walls and roads), see also &lt;a href=&quot;https://www.academia.edu/8016013/Two_Approaches_to_Foundations_in_Greek_Mathematics_Apollonius_and_Geminus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi Two Approaches to Foundations in Greek Mathematics: Apollonius and Geminus&lt;/a&gt;. That Euclid's text lends itself better to the constructivist interpretation is also confirmed by the already mentioned studies of the diagrammatic method.&lt;/p&gt;&#xA;&lt;p&gt;Russo is known for somewhat bold conjectures, but his command of the original sources is not in dispute, he is not the only one to suggest interpolation, and in this case the idea is not too far fetched considering that most of extant copies of Elements date to Theon's IV-th century AD recension, which is known to contain interpolations, including his own. Moreover, Sextus Empiricus explicitly distinguishes between describing (&lt;span class=&quot;math-container&quot;&gt;$\upsilon\rho o\gamma\rho\alpha\phi o\nu\tau\epsilon\varsigma$&lt;/span&gt;) and defining (&lt;span class=&quot;math-container&quot;&gt;$o\rho\iota\zeta o\mu\epsilon\nu o\iota$&lt;/span&gt;) geometric entities, with only the latter applied to Elements and the former to the Book I style definitions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The identification of the mathematician actually referred to by SEXTUS is not too difficult, since the description of the point reported by SEXTUS is nothing but the initial part of the first of Heron’s Definitions (HERON OF ALEXANDRIA, vol. IV)... HERON’s Definitions contain not only complemented and illustrated versions of EUCLID’s definitions (as in the case of the circle) but also definitions of geometrical entities which EUCLID had not defined at all, such as circumference and trapezium. Some of these entities (not all of them) are nevertheless defined in the manuscript tradition of the Elements&amp;quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-01-29T05:03:40.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="6996" PostTypeId="2" ParentId="6986" CreationDate="2018-01-29T11:14:36.450" Score="7" Body="&lt;p&gt;A relevant source to find answers to this questions is the book (out of print in Germany):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Ferber 1956] Christian von Ferber: &lt;em&gt;Die Entwicklung des Lehrkörpers der deutschen Universitäten und Hochschulen 1864-1954&lt;/em&gt;.&#xA;  Band 3 von Untersuchungen zur Lage der deutschen Hochschullehrer.&#xA;  Vandenhoeck &amp;amp; Ruprecht, 1956, 244 pages&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which is the declared source of the data displayed in the following table: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KWCTs.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KWCTs.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I did not check these data. And yet, I decided to trust these number. Strictly speaking the statements implicit in the above table are not quite defined, as the time-resolution is too coarse. As a programming exercise, I constructed the following histograms, and computed three usual statistics (mean, standard deviation, coefficient of variation), pathetic as that may be given such a small sample size:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Y83lZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Y83lZ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recognize that the OP is about mathematics only, and yet, while I was at it, I fed the numbers in [Mehrtens 1990, Tabelle 3] about the subjects physics and chemistry into my program as well, obtaining the following charts, which I reproduce here, too, for comparison:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Physics&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9Dix7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Dix7.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Chemistry&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TUaHH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TUaHH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A specialized reference.&lt;/strong&gt; The following is restricted to &lt;em&gt;one&lt;/em&gt; German university only. In the 19th century, there was but one university in Berlin; there was also the  &lt;a href=&quot;https://de.wikipedia.org/wiki/Gewerbeinstitut_Berlin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gewerbeinstitut/Gewerbeakademie&lt;/a&gt; (which roughly means &#xA;'industry/trade institute'), which had a more limited offering of courses, and was Weierstraß's first academic employer). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A standard source-book for the early history of University of Berlin is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;[Biermann 1988]&lt;/strong&gt; Kurt-Reinhard Biermann: &lt;em&gt;Die Mathematik und ihre Dozenten an der Berliner Universität 1810-1933.&lt;/em&gt; Akademie-Verlag Berlin 1988.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book is in German, and is out of print, even in Germany. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This book meticulously describes the history of University of Berlin in the interval given in the book's title. In particular, the book has voluminous 'backmatter' with reproductions of original documents (such as proposals to the ministry, or reviews of theses; these are &lt;em&gt;not&lt;/em&gt; given in facsimile though), and tables on professors and doctoral students; regarding your &lt;em&gt;question&lt;/em&gt;, let me simply describe the 'line of succession' in its entirety, taking the liberty to interpret your 'around 1890' as the interval [1810,1933]. The following is my literal translation from [Biermann 1988, p. 343-348]; I couldn't finish the translation yet. [Biermann 1988, p. 341] says that the source for the following list is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Johannes Asen: &lt;em&gt;Gesamtverzeichnis des Lehrkörpers der Universität Berlin&lt;/em&gt;. I. 1810-1945. Leipzig 1955&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and adds that the data have been 'checked, expanded by numerous additional data, and corrected in some cases'. Biermann says that the basis for these corrections were files in an archive of the 'Philosophische Fakultät' at Berlin, some personnel files, and some university calendars. [Biermann, p. 341] says that the entries are given in 'chronological order, according to the date of the start of the first employment at University Berlin, with ties broken by rank, and ties in both date and rank being broken by alphabetical order of last name'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1 &lt;a href=&quot;https://de.wikipedia.org/wiki/Johann_Georg_Tralles&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Tralles&lt;/strong&gt;&lt;/a&gt;, &lt;em&gt;Johann Georg&lt;/em&gt;.&#xA;  Born 15 October 1763 Hamburg. Died 19 November 1822 London. &#xA;  Ordinarius (for mathematics and physics) since 30 May 1810.&#xA;  Lectured from Summer 1811 till Summer 1822. Position terminated: by death. Successor: Dirksen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;2 &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=222309&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Fischer&lt;/strong&gt;&lt;/a&gt;, &lt;em&gt;Ernst Gottfried&lt;/em&gt;.&#xA;  Born 17 July 1754 Hoheneiche bei Saalfeld (Thüringen). Died  27 January 1831 Berlin. &#xA;  Extraordinarius (for mathematics and physics) since 30 May 1810.&#xA;  Lectured: no &lt;em&gt;mathematical&lt;/em&gt; lectures. Position terminated: by death.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;3 &lt;a href=&quot;https://de.wikipedia.org/wiki/Johann_Albert_Eytelwein&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Eytelwein&lt;/strong&gt;&lt;/a&gt;, &lt;em&gt;Johann Albert&lt;/em&gt;.&#xA;  Born  21 December 1764 Frankfurt am Main. Died 18 August 1849 Berlin. &#xA;  Extraordinarius (for mechanics and mathematics) since 1 October 1810.&#xA;  Lectured from Winter 1811/12 till Winter 1814/15. Position terminated: by changing to the position of 'Oberlandesbaudirektor' in Berlin on 4 January 1816.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;25 &lt;a href=&quot;https://de.wikipedia.org/wiki/Karl_Weierstra%C3%9F&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Weierstraß&lt;/strong&gt;&lt;/a&gt;, Karl.&#xA;  Born 31 October 1815 Ostenfelde bei Warendorf (Westfalen). Died 19 February 1897 Berlin. &#xA;  Extraordinarius (for mathematics) since 11 October 1856, before and simultaneously until 1864 Professor at the 'Gewerbeinstitut Berlin'. Ordinarius (for mathematics) since 2 July 1864. Position terminated: by death. Successor: Schwarz.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;26 &lt;a href=&quot;https://de.wikipedia.org/wiki/Alfred_Clebsch&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Clebsch&lt;/strong&gt;&lt;/a&gt;, Alfred.&#xA;  Born 19 January 1833 Königsberg/Prussia. Died 7 November 1872 Göttingen. &#xA;  Privatdozent (for mathematics) since 12 December 1872. Lectured in Summer 1858. Position terminated: by becoming professor of theoretical mechanics ato TH Karlsruhe on 22 September 1858 &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;27 &lt;a href=&quot;https://de.wikipedia.org/wiki/Elwin_Bruno_Christoffel&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Christoffel&lt;/strong&gt;&lt;/a&gt;, Elwin Bruno.&#xA;  Born 10 November 1829 Monschau/Eifel. Died 15 March 1900 Strasbourg. &#xA;  Privatdozent (for mathematics) at Berlin since 2 August 1859. Lectured at Berlin from Winter 1859/60 till Summer 1862. Position terminated: by becoming professor of mathematics at ETH Zurich on 5 September 1862. [not in Biermann 1988: Christoffel became Ordinarius for mathematics at the (then German) Strasbourg University in 1872]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;28 &lt;a href=&quot;https://de.wikipedia.org/wiki/Leopold_Kronecker&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Kronecker&lt;/strong&gt;&lt;/a&gt;, Leopold. Born 7 December 1823 Liegnitz/Legnica (Poland). Died 29 December 1891 Berlin. Lecturing member of the Berlin Academy since 1 October 1861. Predicate 'Professor' since 18 March 1864. Ordinarius (for mathematics) since 11 March 1883. Lectured from Winter 1861/62 to Winter 1891/92. Position ended: by death. Successor: Frobenius. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;29 &lt;a href=&quot;https://de.wikipedia.org/wiki/Lazarus_Fuchs&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Fuchs&lt;/strong&gt;&lt;/a&gt;, Lazarus. Born 5 May 1833 Mosina/Moschin (Poland). Died 26 April 1902 Berlin. Privatdozent (for mathematics) 27 October 1865. Extraordinarius (for mathematics) 7 December 1866, with predecessor being Arndt. Lectured at Berlin from Summer 1866 to Winter 1868/69. Position ended by becoming Ordinarius (for mathematics) at University of Greifswald on 31 March 1869. Successor: &lt;a href=&quot;https://genealogy.math.ndsu.nodak.edu/id.php?id=9724&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomé&lt;/a&gt;. Ordinarius (for mathematics) at Berlin since 7 April 1884; before that was Ordinarius at University Heidelberg, with predecessor being Kummer. Lectured from Winter 1884/85 to Winter 1901/02. Position ended: by death. Successor: Schottky.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;30 &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=222309&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Thomé&lt;/strong&gt;&lt;/a&gt;, Wilhelm. Born 13 March 1841 Oberdollendorf bei Königswinter/Rhein. Died 1 October 1910. Privatdozent at Berlin (for mathematics) 17 April 1869. Extraordinarius at Berlin (for mathematics) 16 April 1870, with predecessor being Fuchs. Lectured from Winter 1869/70 till Winter 1873/74. Position ended: 23 April 1874, by becoming Ordinarius at Greifswald. Successor: Bruns&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;31 &lt;a href=&quot;https://de.wikipedia.org/wiki/Leo_August_Pochhammer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Pochhammer&lt;/strong&gt;&lt;/a&gt;, Leo August. Born 25 August 1841 Stendal/Saxony-Anhalt. Died 24 March 1920 Kiel. Privatdozent (for mathematics) at Berlin since 25 January 1872. Lectured at Berlin from Summer 1872 to Winter 1873/74. Position ended 21 March 1874, by becoming Extraordinarius at University Kiel. Not in [Biermann 1988]: Pochhammer became Ordinarius at Kiel in 1877, and seems to not have retired until well into his 78th year. In particular, Pochhammer was Ordinarius in the requested year 1890. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;32 &lt;a href=&quot;https://de.wikipedia.org/wiki/Ferdinand_Georg_Frobenius&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Frobenius&lt;/strong&gt;&lt;/a&gt;, Ferdinand Georg. Born 26 October 1849. Died 3 August 1917 Berlin-Charlottenburg. Extraordinarius (for mathematics) 27 March 1874. Lectured at Berlin from Winter 1874/75 till Summer 1875. &#xA;  Position ended 30 September 1875, by becoming professur at ETH Zurich, succeeded by Wangerin. Ordinarius (for mathematics) at Berlin 16 March 1892, up until then professor at ETH Zurich; predecessor: Kronecker. Lectured at Berlin from Winter 1892/93 till Summer 1916. Position ended 3 July 1916, by retirement. Successor: Carathéodory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;33 &lt;a href=&quot;https://de.wikipedia.org/wiki/Heinrich_Bruns_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Bruns&lt;/strong&gt;&lt;/a&gt;, Ernst Heinrich. Born 4 September 1848 Berlin. Died 23 September 1919 Leipzig. Extraordinarius (for mathematics) 24 January 1876, before that Privatdozent at University Dorpat/Tartu (Estonia), preceded by Thomé. Lectured from Summer 1876 to Summer 1881. Position ended 31 March 1882 , by becoming Ordinarius at University of Leipzig. Successor: Hettner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;34 &lt;a href=&quot;https://de.wikipedia.org/wiki/Albert_Wangerin_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Wangerin&lt;/strong&gt;&lt;/a&gt;, Albert. Born 18 November 1844 Greifenberg(Pomerania). Died 25 October 1933 Halle (Saale). Extraordinarius (for mathematics) 2 March 1876, as successor of Frobenius. Lectured at Berlin from Summer 1876 till Summer 1882. Position ended 30 September 1882, by becoming Ordinarius (for mathematics) at &lt;a href=&quot;https://en.wikipedia.org/wiki/Martin_Luther_University_of_Halle-Wittenberg#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Halle&lt;/a&gt;, Successor: Netto. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;35 (not applicable: professor of astronomy)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;36 &lt;a href=&quot;https://de.wikipedia.org/wiki/Georg_Hettner&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Hettner&lt;/strong&gt;&lt;/a&gt;, Georg. Born 21 August 1854 Jena. Died 24 May 1914 Berlin. Extraordinarius (for mathematics) 11 January 1882, before: lecturer at Göttinge. Predecessor: Bruns. Ordinarius at TH Berlin snce 1 October 1894. Lectured at Berlin from Summer 1882 to Winter 1913/14. Position ended: by death.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;37 &lt;a href=&quot;https://en.wikipedia.org/wiki/Eugen_Netto&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Netto&lt;/strong&gt;&lt;/a&gt;, Eugen. Born 30 June 1846 Halle/Saale. Died 13 May 1919 Gießen. Extraordinarius (for mathematics) at Berlin 6 September 1882, before: Extraordinariu at Strasbourg; predecessor Wangerin. Lectured at Berlin from Winter 1882/83 to Winter 1887/88. Position ended 31 May 1888 by becoming Ordinarius at Gießen; Successor: Knoblauch&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;38 &lt;a href=&quot;https://de.wikipedia.org/wiki/Johannes_Knoblauch&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Knoblauch&lt;/strong&gt;&lt;/a&gt;, Johannes. Born 27 August 1855 Halle/Saale. Died 22 July 1915 Berlin. Lecturer (for mathematics) at Berlin 15 March 1883. Extraordinarius at Berlin 4 April 1889 (unsalaried); predecessor: Netto. Extraordinarius at Berlin 28 May 1890 (salaried); lectured at Berlin from Summer 1883 to Summer 1915. Position ended: by death. Successor: [Issai] Schur. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;39 &lt;a href=&quot;https://www.deutsche-biographie.de/gnd118803476.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Runge&lt;/strong&gt;&lt;/a&gt;, Carl. Born 30 August 1856 Bremen. Died 3 January 1927 Göttingen. Lecturer (for mathematics) at Berlin 9 June 1883. Lectured at Berlin from Winter 1883/84 to Summer 1886. Position ended 30 September 1886 by becoming professor at TH Hannover&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;40 &lt;a href=&quot;https://de.wikipedia.org/wiki/Kurt_Hensel&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Hensel&lt;/strong&gt;&lt;/a&gt;, Kurt. Born 29 December 1861 Königsberg/Prussia. Died 1 June 1941 Marburg. Lecturer (for mathematics) at Berlin 2 August 1886. Extraordinarius at Berlin 7 March 1892. Lectured from Winter 1886/87 to Summer 1902. Position ended 31 October 1902 (by becoming Ordinarius at Marburg)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;41 &lt;a href=&quot;https://de.wikipedia.org/wiki/Ernst_K%C3%B6tter&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Kötter&lt;/strong&gt;&lt;/a&gt;, Ernst. Born 7 August 1859 Berlin. Died 26 January 1922 Aachen. Lecturer (for mathematics) at Berlin 15 March 1887. Titular professor at Berlin 6 June 1894. Lectured at Berlin from Summer 1887 to Winter 1896/97. Position ended 31 March 1897 (by becoming professor at TH Aachen)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;42 &lt;a href=&quot;https://de.wikipedia.org/wiki/Ludwig_Schlesinger_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schlesinger&lt;/strong&gt;&lt;/a&gt;, Ludwig. Born 1 November 1864 Trnava (Tyrna). Died 16 December 1933 Gießen. Lecturer (for mathematics) at Berlin 7 June 1889. Titular professor at Berlin 6 June 1894. Lectured at Berlin from Winter 1889/90 till Winter 1896/97. Position ended 15 April 1897 (by becoming Extraordinarius at Bonn) &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;43 &lt;a href=&quot;https://de.wikipedia.org/wiki/Paul_G%C3%BCnther_(Mathematiker,_1867)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Günther&lt;/strong&gt;&lt;/a&gt;, Paul. Born 2 April 1867 Bernburg/Saale. Died 27 September 1891 Berlin. Lecturer (for mathematics) 6 August 1890. Lecture announced for Summer 1891, lecture not given for health reasons. Position ended: by death. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;44 &lt;a href=&quot;https://de.wikipedia.org/wiki/Hermann_Amandus_Schwarz&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schwarz&lt;/strong&gt;&lt;/a&gt;, Hermann Amandus. Born 25 January 1843 Sobieszów (Hermsdorf unterm Kynast). Died 30 November 1921 Berlin. Ordinarius (for mathematics) 1 April 1892, before: Ordinarius at Göttingen; predecessor: Weierstraß. Lectured at Berlin from Summer 1892 to Summer 1918. Retired 31 March 1917; Successor Schmidt&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;45 &lt;a href=&quot;https://de.wikipedia.org/wiki/Edmund_Landau&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Landau&lt;/strong&gt;&lt;/a&gt;, Edmund. Born 14 February 1877 Berlin. Died 19 February 1938 Berlin. Lecturer (for mathematics) at Berlin 27 June 1901. Titular professor at Berlin 29 September 1905. Lectured at Berlin from Winter 1901/02 till Winter 1908/09. Position ended 22 February 1909 (by becoming full professor at Göttingen)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;46 &lt;a href=&quot;https://de.wikipedia.org/wiki/Friedrich_Schottky&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schottky&lt;/strong&gt;&lt;/a&gt;, Friedrich. Born 24 July 1851 Breslau. Die 12 August 1935 Berlin. Ordinarius (for mathematics) at Berlin 1 October 1902, before: Ordinarius at Marburg; predecessor Fuchs. Lectured at Berlin from Winter 1902/03 till Summer 1926. Retired 31 March 1921. Successor Schur.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;47 &lt;a href=&quot;https://de.wikipedia.org/wiki/Issai_Schur&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schur&lt;/strong&gt;&lt;/a&gt;, Issai. Born 10 January 1875 Mogilew/Dnepr (Mogiljow,Mahiljou). Died 10 January 1941 Tel Aviv. Lecturer (for mathematics) at Berlin 5 January 1903. Titular professor at Berlin 23 December 1909. Lecured at Berlin from Summer 1903 till Winter 1912/13. Position ended 21 April 1913 (by becoming Extraordinarius at Bonn). Extraordinarius at Berlin 1 April 1916, predecessor Knoblauch. &quot;Persönliches Ordinariat&quot; 29 December 1919, with salary of Extraordinarius; successor Hamburger. Ordinarius 21 May 1921; predecessor Schottky. lectured at Berlin from Summer 1916. Retired 30 September 1935&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;48 &lt;a href=&quot;https://de.wikipedia.org/wiki/Konrad_Knopp&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Knopp&lt;/strong&gt;&lt;/a&gt;, Konrad. Born 22 July 1882 Friedenau bei Berlin. Died 30 April 1957 Annecy/Haute-Savoie/France. Lecturer (for mathematics) at Berlin 13 November 1911. Extraordinarius at Berlin 1 May 1915; predecessor &lt;a href=&quot;https://de.wikipedia.org/wiki/Rudolf_Lehmann-Filh%C3%A9s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lehmann-Filhés&lt;/a&gt;. Lectured at Berlin Summer 1912 till Summer 1919. Position ended 30 September 1919 (by becoming Ordinarius at Königsberg/Prussia. Successor: v. Mises&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;49 &lt;a href=&quot;https://de.wikipedia.org/wiki/Rudolf_Rothe&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Rothe&lt;/strong&gt;&lt;/a&gt;, Rudolf. Born 15 October 1873 Berlin. Died 26 October 1942 Berlin. Lectureship (&quot;Lehrauftrag&quot;) for mathematics at Berlin 1 April 1915 as Ordinarius at TH Berlin. [I don't quite understand Biermann's data here; it is unclear whether Biermann means 'lectureship as an Ordinarius', which seems oxymoronic, or whether Biermann means that Rothe had a lectureship at University of Berlin and was Ordinarius at TH Berlin, which seems more plausible] Lectured at Berlin from Summer 1915 till Summer 1919. Position ended 30 September 1919 (no reason given)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;50 &lt;a href=&quot;https://de.wikipedia.org/wiki/Robert_Jentzsch&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Jentzsch&lt;/strong&gt;&lt;/a&gt;, Robert. Born 4 November 1890 Königsberg/Prussia. Died 21 March 1918 (fallen). Lecturer (for mathematics) 11 December 1916. Lectured at Berlin in Summer 1917. Position ended: by death&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;51 &lt;a href=&quot;https://de.wikipedia.org/wiki/Erhard_Schmidt_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schmidt&lt;/strong&gt;&lt;/a&gt;, Erhard. Born 13 January 1876 Dorpat/Tartu (Estonia). Died 6 December 1959 Berlin. Ordinarius (for mathematics) 1 October 1917. Before this: Ordinarius at Breslau. Predecessor: Schwarz. Lectured at Berlin since Winter 1917/18. Retired 1 August 1950. Co-director of the 'I. Math. Inst.' until 22 October 1952&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;$\quad$&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;52 &lt;a href=&quot;https://de.wikipedia.org/wiki/Constantin_Carath%C3%A9odory&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Carathéodory&lt;/strong&gt;&lt;/a&gt;, Constantin. Born 13 September 1873 Berlin. Died 2 February 1950 Munich. Ordinarius (for mathematics) 1 October 1918, before this: Ordinarius at Göttingen. Predecessor: Frobenius. Lectured at Berlin from Winter 1918/19 till Winter 1919/20. Position ended: 31 December 1919 (by changing to Greece) [ Biermann probably means Carathéodory's change to İzmir/Smyrna to participate in the abortive attempt of founding the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ionian_University_of_Smyrna&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ionian University of Smyrna&lt;/a&gt; ]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[...] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(to be continued) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, the above is mainly about University of Berlin only. In total, [Biermann 1988] lists &lt;strong&gt;67&lt;/strong&gt; lecturers at University of Berlin from 1810 to 1933; of these, &lt;strong&gt;18&lt;/strong&gt; are listed as Ordinarius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks on the definedness of the question.&lt;/strong&gt; While the intuitive meaning is clear, strictly speaking, 'German universities' in the OP is not well-defined (more defined would be 'German speaking', or 'on the territory of the current Federal Republic of Germany', or, but that would be more restrictive, 'Prussian Universities'): in the 19th century, Germany was more united politically/administratively than in former centuries, but still much &lt;em&gt;less united&lt;/em&gt; than e.g. in the 20th-century: in particular, Prussia and the kingdom of Saxony had a difficult relationship throughout the 19th century (and went to war against one another as late as 1866 in the 'Battle of Königgrätz' in which Saxony was allied with Austria against Prussia). The kingdom of Bavaria was yet another separate administrative entity. And yet, the Saxon university of Leipzig, and the Bavarian universities of Erlangen and Munich were 'German universities', too, though completely independently administered. To give an anecdote: a letter of Weierstraß, dated 20 December 1885, has survived in which he expresses his outrage at Klein having helped Sophus Lie to a chair at Leipzig (which, Weierstraß implies, should have gone to a German, not to a foreigner, and a young foreigner at that), and then Weierstraß (professor at Berlin) explicitly says that if Leipzig was a Prussian university, he would protest with the ministry (and implies that the Prussian ministry does not have any say in matters of Leipzig University). Such administrative independence is perhaps hard to guess given the proximity of the cities of Leipzig and Berlin. And then there are the issues around University of Strasbourg and University of Königsberg, both of which were German speaking for some time (and the latter having a role in the history of mathematics) but aren't any more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Professors of mathematics with rank 'Ordinarius' at German-speaking universities in the requested year 1890&lt;/strong&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To answer the OP systematically, it is necessary to first enumerate all German-speaking universities in 1890. Not all German universities which currently exist already existed back then. According to the source [Ferber 1956, p. 37], in 1890 there were the following German-speaking universities (I list the polytechnics, too, though these are not universities in the usual sense, and accordingly were called 'Hochschule' in the 19th century):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A&lt;/strong&gt;achen (Technische Hochschule) : (...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;erlin: &lt;a href=&quot;https://www.deutsche-biographie.de/gnd116844329.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Fuchs&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1884&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;erlin: &lt;a href=&quot;https://de.wikipedia.org/wiki/Leopold_Kronecker&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Kronecker&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1883&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;erlin: &lt;a href=&quot;https://de.wikipedia.org/wiki/Karl_Weierstra%C3%9F&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Weierstraß&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1862&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;erlin (Technische Hochschule): &lt;a href=&quot;https://www.deutsche-biographie.de/pnd1012183734.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;du Bois-Reymond&lt;/strong&gt;&lt;/a&gt;: Etatmäßiger Professor (i.e. the analogue of 'Ordinarius' at the 'Hochschulen'; similar to the status of being a 'faculty member' mentioned in the OP) 1884 (cf. page 446 of &lt;a href=&quot;https://digital.zlb.de/viewer/image/14688302_1884/456/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; source)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;erlin (Technische Hochschule): &lt;a href=&quot;https://de.wikipedia.org/wiki/Julius_Weingarten&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Weingarten&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1871&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;onn: &lt;a href=&quot;https://en.wikipedia.org/wiki/Rudolf_Lipschitz&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Lipschitz&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1864&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;raunsberg (Lyceum Hosianum): &lt;a href=&quot;https://de.wikipedia.org/wiki/Wilhelm_Killing&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Killing&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.deutsche-biographie.de/gnd119097117.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; since 1882 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;raunschweig (Technische Hochschule): &lt;a href=&quot;https://www.deutsche-biographie.de/gnd118524259.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Dedekind&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1862 (Direktor 1872-75)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;B&lt;/strong&gt;reslau: &lt;a href=&quot;http://www.jewishencyclopedia.com/articles/12827-rosanes-jacob&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Rosanes&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1876&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;D&lt;/strong&gt;orpat (now &lt;a href=&quot;https://de.wikipedia.org/wiki/Universit%C3%A4t_Tartu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tartu&lt;/a&gt;; German-speaking throughout 19th century): &lt;a href=&quot;https://de.wikipedia.org/wiki/Adolf_Kneser&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Kneser&lt;/strong&gt;&lt;/a&gt;: Ordinarius since &lt;a href=&quot;https://www.uni-marburg.de/uniarchiv/pkat/einfache_suche/details?current=48&amp;amp;q=Mathematik&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1890&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;D&lt;/strong&gt;resden (Technische Hochschule): &lt;a href=&quot;https://de.wikipedia.org/wiki/Martin_Krause_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Krause&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1888&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;E&lt;/strong&gt;rlangen: &lt;a href=&quot;http://www.ub.uni-heidelberg.de/helios/fachinfo/www/math/homo-heid/noether-m.htm#top&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Noether&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1888&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;F&lt;/strong&gt;reiburg: (...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;ießen: &lt;a href=&quot;https://de.wikipedia.org/wiki/Moritz_Pasch&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Pasch&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1875 (dean 1885, rector 1893)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;ießen: &lt;a href=&quot;https://www.deutsche-biographie.de/gnd116944684.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Netto&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.deutsche-biographie.de/gnd116944684.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1888&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;öttingen: &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Klein&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Klein&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1886&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;öttingen: &lt;a href=&quot;https://www.deutsche-biographie.de/gnd117367028.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schwarz&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1875 (successor of Fuchs)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;öttingen: &lt;a href=&quot;http://www.mathematik.uni-muenchen.de/~fritsch/Schering.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schering&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1868 (with regard to the OP, Schering is a borderline case, as &lt;a href=&quot;http://www.mathematik.uni-muenchen.de/~fritsch/Schering.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; source explicitly says he was &quot;ohne spezielle Zuordnung zum Fach Mathematik&quot;; Schering is perhaps most remembered for his work as &lt;a href=&quot;https://www.jstor.org/stable/3027835?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;editor of Gauß's collected works&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;raz (Technische Hochschule): &lt;a href=&quot;https://austria-forum.org/af/Biographien/Mertens%2C_Franz&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mertens&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://austria-forum.org/af/Biographien/Mertens%2C_Franz&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1884 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;G&lt;/strong&gt;reifswald: (...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;H&lt;/strong&gt;alle: &lt;a href=&quot;https://de.wikipedia.org/wiki/Georg_Cantor&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Cantor&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.deutsche-biographie.de/gnd118518887.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1879&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;H&lt;/strong&gt;alle: &lt;a href=&quot;https://de.wikipedia.org/wiki/Albert_Wangerin_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Wangerin&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1882&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;H&lt;/strong&gt;annover (Technische Hochschule): &lt;a href=&quot;http://www.mathematik.uni-muenchen.de/~fritsch/Schering.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Runge&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.deutsche-biographie.de/gnd118803476.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1886&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;H&lt;/strong&gt;eidelberg: &lt;a href=&quot;https://www.deutsche-biographie.de/gnd116293802.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Koenigsberger&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1884 (retired 1914, resumed teaching in 1918)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;J&lt;/strong&gt;ena: &lt;a href=&quot;https://de.wikipedia.org/wiki/Carl_Johannes_Thomae&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Thomae&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1879&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;K&lt;/strong&gt;arlsruhe (Technische Hochschule): &lt;a href=&quot;https://de.wikipedia.org/wiki/Wilhelm_Schell&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schell&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1863 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;K&lt;/strong&gt;önigsberg: &lt;a href=&quot;https://de.wikipedia.org/wiki/Ferdinand_von_Lindemann&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Lindemann&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.deutsche-biographie.de/sfz51526.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1883 (changed to Münchenin 1893)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;L&lt;/strong&gt;eipzig: &lt;a href=&quot;http://research.uni-leipzig.de/catalogus-professorum-lipsiensium/leipzig/Neumann_109/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Neumann&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1868&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;L&lt;/strong&gt;eipzig: &lt;a href=&quot;http://research.uni-leipzig.de/catalogus-professorum-lipsiensium/leipzig/Scheibner_971/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Scheibner&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;http://research.uni-leipzig.de/catalogus-professorum-lipsiensium/leipzig/Scheibner_971/&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1868&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;M&lt;/strong&gt;arburg: &lt;a href=&quot;https://de.wikipedia.org/wiki/Heinrich_Weber_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Weber&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.uni-marburg.de/uniarchiv/pkat/gnd?id=1012808513&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1884 (dean at Marburg 1887, rector at Marburg 1891; changed to Göttingen in 1892)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;M&lt;/strong&gt;ünchen: &lt;a href=&quot;https://de.wikisource.org/wiki/ADB:Seidel,_Philipp_Ludwig&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Seidel&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1855&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;M&lt;/strong&gt;ünster: &lt;a href=&quot;https://de.wikipedia.org/wiki/Rudolf_Sturm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Sturm&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;http://wwwmath1.uni-muenster.de/historie/kapitel51.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1878&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;M&lt;/strong&gt;ünster: &lt;a href=&quot;https://de.wikipedia.org/wiki/Paul_Bachmann_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Bachmann&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;http://wwwmath1.uni-muenster.de/historie/kapitel51.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1868. (Bachmann is a borderline case with regard to the OP, since Bachmann left the position in the very year 1890. He &lt;em&gt;was&lt;/em&gt; Ordinarius at  &lt;strong&gt;M&lt;/strong&gt;ünster in 1890 though. This is an example of the 'time-resolution'-objection noted in the histograms; very strictly speaking, the case of Bachmann alone is enough reason to consider the table in [Mehrtens 1990] strictly-speaking ill-defined).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;R&lt;/strong&gt;ostock: (...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;S&lt;/strong&gt;trasbourg: (...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;T&lt;/strong&gt;übingen: (...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;W&lt;/strong&gt;ien: &lt;a href=&quot;https://de.wikipedia.org/wiki/Emil_Weyr&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Weyr&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1875&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;W&lt;/strong&gt;ien: &lt;a href=&quot;https://de.wikipedia.org/wiki/Josef_Stefan&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Stefan&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://de.wikisource.org/wiki/ADB:Stefan,_Josef&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1863&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;W&lt;/strong&gt;ien (Hochschule für Bodenkultur): &lt;a href=&quot;https://de.wikipedia.org/wiki/Josef_Schlesinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schlesinger&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1875&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;W&lt;/strong&gt;ürzburg: &lt;a href=&quot;https://www.deutsche-biographie.de/gnd119034727.html#ndbcontent&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Prym&lt;/strong&gt;&lt;/a&gt;: Ordinarius since 1869&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Z&lt;/strong&gt;ürich: &lt;a href=&quot;https://de.wikipedia.org/wiki/Carl_Friedrich_Geiser&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Geiser&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;http://www.ethistory.ethz.ch/materialien/professoren/listen/alle_profs_eintritt/&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1873&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Z&lt;/strong&gt;ürich: &lt;a href=&quot;https://en.wikipedia.org/wiki/Friedrich_Schottky&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Schottky&lt;/strong&gt;&lt;/a&gt;: Ordinarius &lt;a href=&quot;https://www.uni-marburg.de/uniarchiv/pkat/einfache_suche/details?current=76&amp;amp;q=Mathematik&quot; rel=&quot;nofollow noreferrer&quot;&gt;since&lt;/a&gt; 1882&lt;/p&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="6060" LastEditDate="2018-02-27T20:02:13.367" LastActivityDate="2018-02-27T20:02:13.367" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="6997" PostTypeId="1" AcceptedAnswerId="7001" CreationDate="2018-01-29T23:08:14.450" Score="8" ViewCount="325" Body="&lt;p&gt;What is the difference between the terms &quot;analysis&quot; and &quot;synthesis&quot; used in a mathematical context?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Hawkins's &lt;a href=&quot;https://isidore.co/calibre#book_id=4507&amp;amp;panel=book_details&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Emergence of the Theory of Lie Groups&lt;/em&gt;&lt;/a&gt; &lt;a href=&quot;https://books.google.com/books?id=lz3aBwAAQBAJ&amp;amp;pg=PA4#v=snippet&amp;amp;q=%22They%20were%20self-styled%20synthesists%20in%20the%20midst%20of%20analysts%20and%20arithmeticians.%22&quot; rel=&quot;noreferrer&quot;&gt;p. 3&lt;/a&gt; says that Klein and Lie&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;were self-styled &quot;synthesists&quot; in the midst of analysts and arithmeticians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What does this mean?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="55" LastEditDate="2018-01-30T01:21:17.230" LastActivityDate="2018-01-30T01:39:10.567" Title="Who are &quot;analysts&quot; and &quot;synthesists&quot; in mathematics?" Tags="|mathematics|terminology|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="6998" PostTypeId="2" ParentId="6925" CreationDate="2018-01-29T23:08:51.890" Score="1" Body="&lt;h3&gt;Community-wiki timeline (continued, 1917–):&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;After 1920, a vast majority of papers call $k$ Boltzmann’s constant. So (with few exceptions) mention is only made of sources calling it another name, plus some textbooks and reference books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;Author         |Dated    |Published |Denoted|Called &#xA; Millikan        1917.01   1917       k    S  &lt;a href=&quot;//archive.org/stream/londonedinburg6341917lond#page/14&quot; rel=&quot;nofollow noreferrer&quot;&gt;the entropy constant&lt;/a&gt; &#xA; Schrödinger               1917       R/N  R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015000428808&amp;amp;num=541&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Gerlach         1917.08   1918       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101049863556&amp;amp;num=156&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Konstante der Entropie&lt;/a&gt; &#xA; Flamm           1917.09   1917       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015009220800&amp;amp;num=519&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Entropiekonstante&lt;/a&gt; &#xA; Born            1917.10   1917       k       &lt;a href=&quot;//google.com/books?id=BixLAAAAMAAJ&amp;amp;pg=PA247&amp;amp;dq=%22Maxwell-Boltzmann&amp;#39;sche+Konstante%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Maxwell-Boltzmann&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Sommerfeld                1918       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015000428337&amp;amp;num=196&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Konstante k des Boltzmannschen Prinzips&lt;/a&gt; &#xA; Wien                      1918       k    S  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015000428337&amp;amp;num=205&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Entropiekonstante&lt;/a&gt; &#xA; von Laue        1918.07   1918       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101049863556&amp;amp;num=205&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Schrödinger     1918.07   1919       k    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.hxgq91&amp;amp;num=422&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Bruins          1918.07   1918       s    R  &lt;a href=&quot;//lorentz.leidenuniv.nl/IL-publications/dissertations/lorentz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;de gasconstante van een molecuul&lt;/a&gt; &#xA; von Laue        1918.08   1918       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101049863556&amp;amp;num=259&quot; rel=&quot;nofollow noreferrer&quot;&gt;der Boltzmannsche k&lt;/a&gt; &#xA; Stern           1918.11   1919       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101043122728&amp;amp;num=67&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Reiche          1918.12   1919       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.%24b562041&amp;amp;num=669&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; von Laue        1918.12   1919       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.%24b562041&amp;amp;num=695&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt;&#xA; Lorentz•                  &lt;a href=&quot;//zbmath.org/?q=an:47.0706.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1919a&lt;/a&gt;      k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=wu.89119127215&amp;amp;num=74&quot; rel=&quot;nofollow noreferrer&quot;&gt;2/3 der translatie-energie per graad&lt;/a&gt; &#xA; Lorentz•                  &lt;a href=&quot;//zbmath.org/?q=an:53.0812.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1919b&lt;/a&gt; &lt;a href=&quot;//google.com/books?id=xVkmAAAAMAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22Planck%27s+constant%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=wu.89119127215&amp;amp;num=134&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante van &lt;strong&gt;Planck&lt;/strong&gt;&lt;/a&gt; &#xA; Nernst-Wulf     1919.04   1919       k    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.c058625598&amp;amp;num=301&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaskonstante pro Molekül&lt;/a&gt; &#xA; Lindemann       1919.05   1919a      k       &lt;a href=&quot;//archive.org/stream/londonedinburg6381919lond#page/177&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmann’s constant&lt;/a&gt; &#xA; Lindemann       1919.07   1919b      k       &lt;a href=&quot;//archive.org/stream/londonedinburg6381919lond#page/676&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmann’s constant&lt;/a&gt; &#xA; Birge           1919.07   1919       k       &lt;a href=&quot;//google.com/books?id=4agOAAAAIAAJ&amp;amp;pg=RA1-PA362&amp;amp;dq=%22Boltzmann+entropy+constant%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Boltzmann entropy constant&lt;/a&gt; &#xA; Sommerfeld•     1919.09   1919       ?       ? &#xA; Lindemann       1919.09   1920       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uiug.30112053429541&amp;amp;num=22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmann’s constant&lt;/a&gt; &#xA; van Slingelandt 1919.12   1919       k    E  &lt;a href=&quot;//lorentz.leidenuniv.nl/IL-publications/dissertations/lorentz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante, die kinetische energie bepaalt&lt;/a&gt; &#xA; Stern           1919.12   1920       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=inu.30000089907038&amp;amp;num=149&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Silberstein     1920.02   &lt;a href=&quot;//zbmath.org/?q=an:47.0708.13&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920&lt;/a&gt;       k    U  &lt;a href=&quot;//archive.org/stream/reportonquantumt00silbrich#page/36&quot; rel=&quot;nofollow noreferrer&quot;&gt;constant in Planck’ formula&lt;/a&gt; &#xA; Lyon-Wolfram    1920.03   1920              &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.%24b562046&amp;amp;num=741&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Stern           1920.04   1920       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=inu.30000089907046&amp;amp;num=49&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Ladenburg       1920.05   1920       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=njp.32101049863531&amp;amp;num=98&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Entropiekonstante&lt;/a&gt; &#xA; Planck          1920.06   &lt;a href=&quot;//zbmath.org/?q=an:47.0708.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920&lt;/a&gt;  &lt;a href=&quot;//archive.org/stream/origindevelopmen00planrich#page/10&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    U  &lt;a href=&quot;//archive.org/stream/MaxPlanckVortraegeRedenErinnerungen#page/n31&quot; rel=&quot;nofollow noreferrer&quot;&gt;die erste Strahlungskonstante&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; Sommerfeld•     1920.09   1921b &lt;a href=&quot;//archive.org/stream/AtomicStructureAndSpectralLines#page/n228/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       &lt;a href=&quot;//archive.org/stream/atombauundspekt00sommgoog#page/n485&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Reiche•         1920.10   &lt;a href=&quot;//zbmath.org/?q=an:48.0876.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921&lt;/a&gt;  &lt;a href=&quot;//archive.org/stream/quantumtheory00reicrich#page/108&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       &lt;a href=&quot;//archive.org/stream/diequantentheor00reicgoog#page/n151&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Schaefer•       1920.10   &lt;a href=&quot;//zbmath.org/?q=an:48.0874.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921&lt;/a&gt;       k    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433084030653&amp;amp;num=350&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Herzfeld•       1920.12   &lt;a href=&quot;//zbmath.org/?q=an:48.0882.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920&lt;/a&gt;       k       &lt;a href=&quot;//gdz.sub.uni-goettingen.de/id/PPN360709532?tify=%7B%22pages%22:[985],%22view%22:%22toc%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Haas•           1921.03   &lt;a href=&quot;//zbmath.org/?q=an:48.0875.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921&lt;/a&gt;       k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433084030836&amp;amp;num=108&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Gerlach                   1921       k       &lt;a href=&quot;//archive.org/stream/dieexperimentel00gerlgoog#page/n149&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Planck&lt;/strong&gt;sche Entropiekonstante&lt;/a&gt; &#xA; V.d.S.Bakhuyzen 1921.12   1921       k       &lt;a href=&quot;//lorentz.leidenuniv.nl/IL-publications/dissertations/lorentz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante van Boltzmann&lt;/a&gt; &#xA; Gerlach-Stern   1922.04   1922       k    E  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=inu.30000089551091&amp;amp;num=354&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Hertz                     &lt;a href=&quot;//zbmath.org/?q=an:48.0036.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=2uzPAAAAMAAJ&amp;amp;pg=PA69&amp;amp;dq=Boltzmann+Planckschen&quot; rel=&quot;nofollow noreferrer&quot;&gt;die &lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Eucken                    &lt;a href=&quot;//zbmath.org/?q=an:48.0036.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=2uzPAAAAMAAJ&amp;amp;pg=PA146&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Henning                   &lt;a href=&quot;//zbmath.org/?q=an:48.0036.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=2uzPAAAAMAAJ&amp;amp;pg=PA166&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Born•           1922.09   &lt;a href=&quot;//zbmath.org/?q=an:49.0569.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1923&lt;/a&gt;       k       &lt;a href=&quot;//archive.org/stream/encyklomath305encyrich#page/n651&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Lorentz•                  &lt;a href=&quot;//zbmath.org/?q=an:53.0812.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1923&lt;/a&gt;  &lt;a href=&quot;//google.com/books?id=xVkmAAAAMAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22Planck%27s+constant%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=msu.31293001943376&amp;amp;num=39&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante van &lt;strong&gt;Planck&lt;/strong&gt;&lt;/a&gt; &#xA; Darwin•                   1923       k    E  &lt;a href=&quot;//archive.org/stream/dictionaryofappl00glazmiss#page/531&quot; rel=&quot;nofollow noreferrer&quot;&gt;the atomic gas constant&lt;/a&gt; &#xA; Coblentz•                 1923       k       &lt;a href=&quot;//archive.org/stream/dictionaryofappl00glazmiss#page/564&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmann gas constant&lt;/a&gt; &#xA; Berliner &amp;amp; al•  1924.04   &lt;a href=&quot;//zbmath.org/?q=an:50.0523.10&quot; rel=&quot;nofollow noreferrer&quot;&gt;1924&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=pI-NBwAAQBAJ&amp;amp;pg=PA105&amp;amp;dq=Boltzmannsche&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Einstein        1924.07   1924  &lt;a href=&quot;//lucaamendola.wordpress.com/other-stuff/&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  ϰ    S  &lt;a href=&quot;http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Schrödinger     1924.11   &lt;a href=&quot;//zbmath.org/?q=an:51.0755.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1924&lt;/a&gt;       k       &lt;a href=&quot;//doi.org/10.1007/BF01331851&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannkonstante&lt;/a&gt; &#xA; Einstein        1925.01   &lt;a href=&quot;//zbmath.org/?q=an:51.0755.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1925a&lt;/a&gt;      ϰ       &lt;a href=&quot;http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Einstein        1925.01   &lt;a href=&quot;//zbmath.org/?q=an:51.0755.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1925b&lt;/a&gt;      ϰ       &lt;a href=&quot;http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/sitzungsberichte&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Smekal•         1925.06   &lt;a href=&quot;//zbmath.org/?q=an:52.0952.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//archive.org/stream/encyklomath305encyrich#page/n883&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Smekal•         1925.06   &lt;a href=&quot;//zbmath.org/?q=an:52.0952.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//archive.org/stream/encyklomath305encyrich#page/n946&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Schrödinger     1925.12   &lt;a href=&quot;//zbmath.org/?q=an:52.0956.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;  &lt;a href=&quot;//homepages.dias.ie/dorlas/papers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       –– &#xA; Geiger-Scheel•            &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=3eqiBgAAQBAJ&amp;amp;pg=PP7&amp;amp;dq=%22Boltzmannsche+Konstante%22+%22Quantenkonstante+f%C3%BCr+Frequenzen%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Henning-Jaeger•           &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=3eqiBgAAQBAJ&amp;amp;pg=PA504&amp;amp;dq=Boltzmannsche&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Bennewitz•                &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=16eoBwAAQBAJ&amp;amp;pg=PA163&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Smekal•                   &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=16eoBwAAQBAJ&amp;amp;pg=PA224&quot; rel=&quot;nofollow noreferrer&quot;&gt;die sog. &lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Byk•                      &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=16eoBwAAQBAJ&amp;amp;pg=PA305&amp;amp;dq=multiplikative&quot; rel=&quot;nofollow noreferrer&quot;&gt;die molekulare Gaskonstante&lt;/a&gt; &#xA; Jäger•                    &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=16eoBwAAQBAJ&amp;amp;pg=PA374&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Henning•                  &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=16eoBwAAQBAJ&amp;amp;pg=PA530&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Henning•                  &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=16eoBwAAQBAJ&amp;amp;pg=PA611&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; Grüneisen•                &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=zlPQBgAAQBAJ&amp;amp;pg=PA18&amp;amp;dq=Planckschen&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Planck&lt;/strong&gt;sche Strahlungskonstante&lt;/a&gt; &#xA; van der Waals•            &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=zlPQBgAAQBAJ&amp;amp;pg=PA131&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Schrödinger•              &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=zlPQBgAAQBAJ&amp;amp;pg=PA277&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Simon•                    &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;//google.com/books?id=zlPQBgAAQBAJ&amp;amp;pg=PA363&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Gerlach•                  &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       R/N     &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k95109d/f46&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Pauli•                    &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k95111z/f87&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &#xA; Ladenburg•                &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k    U  &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k95111z/f294&quot; rel=&quot;nofollow noreferrer&quot;&gt;die universelle Entropiekonstante&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; Franck-Jordan•            &lt;a href=&quot;//zbmath.org/?q=an:52.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;       k       &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k95111z/f746&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmannsche Konstante&lt;/a&gt; &#xA; Guillaume &amp;amp; al•           1926       k    R  &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=wu.89048366553&amp;amp;num=18&quot; rel=&quot;nofollow noreferrer&quot;&gt;molecular gas constant&lt;/a&gt; &#xA; Tolman•         1926.08   &lt;a href=&quot;//zbmath.org/?q=an:53.0876.18&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;       k    R  &lt;a href=&quot;//archive.org/stream/statisticalmecha00tolm#page/57&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Boltzmann constant&lt;/a&gt; &#xA; Uhlenbeck       1927.07   1927       k       &lt;a href=&quot;//ilorentz.org/IL-publications/dissertations/ehrenfest.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;constante van Boltzmann&lt;/a&gt; &#xA; Planck•         1930.03   &lt;a href=&quot;//zbmath.org/?q=an:56.1265.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1930&lt;/a&gt;  &lt;a href=&quot;//archive.org/stream/theoryofheatbein007775mbp#page/n241&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//google.com/books?id=tMzPAAAAMAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22universelle+Faktor+k%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;der universelle Faktor k&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; Berliner &amp;amp; al•  1931.11   1932       k       &lt;a href=&quot;//google.com/books?id=8o7LBgAAQBAJ&amp;amp;pg=PA154&amp;amp;dq=Boltzmannsche&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Boltzmann&lt;/strong&gt;-&lt;strong&gt;Planck&lt;/strong&gt;sche Konstante&lt;/a&gt; &#xA; (tombstone)     1933.07   1933       k    S  &lt;a href=&quot;//de.wikipedia.org/wiki/Ludwig_Boltzmann#Posthume_Ehrungen&quot; rel=&quot;nofollow noreferrer&quot;&gt;––&lt;/a&gt; &#xA; Planck                    &lt;a href=&quot;//zbmath.org/?q=an:0028.27602&quot; rel=&quot;nofollow noreferrer&quot;&gt;1943&lt;/a&gt;       k    S  &lt;a href=&quot;//archive.org/stream/MaxPlanckVortraegeRedenErinnerungen#page/n24&quot; rel=&quot;nofollow noreferrer&quot;&gt;meine Konstante k&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; van Dantzig     1939.06   1939       k       &lt;a href=&quot;http://www.dwc.knaw.nl/DL/publications/PU00017347.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmann's number&lt;/a&gt;&#xA; Planck          1945.03   &lt;a href=&quot;//zbmath.org/?q=an:0031.09909&quot; rel=&quot;nofollow noreferrer&quot;&gt;1948&lt;/a&gt;  &lt;a href=&quot;//google.com/books?id=F3iMBAAAQBAJ&amp;amp;pg=PT23&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k    S  &lt;a href=&quot;//archive.org/stream/MaxPlanckVortraegeRedenErinnerungen#page/n69&quot; rel=&quot;nofollow noreferrer&quot;&gt;die sog. absolute Gaskonstante&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; Meissner                  1951       k       &lt;a href=&quot;//archive.org/stream/in.ernet.dli.2015.72541#page/n102&quot; rel=&quot;nofollow noreferrer&quot;&gt;the &lt;strong&gt;Boltzmann-Planck&lt;/strong&gt; constant&lt;/a&gt; &lt;sup&gt;9&lt;/sup&gt; &#xA; Sommerfeld•               &lt;a href=&quot;//ams.org/mathscinet-getitem?mr=62030&quot; rel=&quot;nofollow noreferrer&quot;&gt;1952&lt;/a&gt;  &lt;a href=&quot;//google.com/books?id=kkkvQAicWhkC&amp;amp;pg=PA213&quot; rel=&quot;nofollow noreferrer&quot;&gt;(e)&lt;/a&gt;  k       &lt;a href=&quot;//google.com/books?id=R1O4AAAAIAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22selbst+die+Formel+niemals%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;die Boltzmannsche Konstante&lt;/a&gt; &lt;sup&gt;9,10&lt;/sup&gt;&#xA;&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;•&lt;/code&gt; $=$ textbook or reference book. &lt;code&gt;(e)&lt;/code&gt; $=$ English translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;9&lt;/sup&gt; Calls the constant &lt;strong&gt;&lt;em&gt;not&lt;/em&gt;&lt;/strong&gt; Boltzmann’s.&lt;br&gt;&#xA;&lt;sup&gt;10&lt;/sup&gt; Calls the formula &lt;strong&gt;&lt;em&gt;not&lt;/em&gt;&lt;/strong&gt; Boltzmann’s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-02-02T11:47:36.593" LastActivityDate="2018-02-02T11:47:36.593" CommentCount="1" CommunityOwnedDate="2018-01-29T23:08:51.890" ContentLicense="CC BY-SA 3.0" />
  <row Id="7000" PostTypeId="2" ParentId="6997" CreationDate="2018-01-30T01:17:57.440" Score="7" Body="&lt;p&gt;There is a long history to analytic and synthetic. But after historical roots and philosophical battles were forgotten analytic and synthetic came to mean two different styles of reasoning, the more formal vs the more intuitive, the more discrete and reductive vs the more holistic, etc., and not even necessarily in mathematics. There was &lt;a href=&quot;http://www.iiuedu.eu/press/journals/sds/SDS-2012/SSc_Article1.pdf&quot; rel=&quot;noreferrer&quot;&gt;analytic and synthetic Cubism&lt;/a&gt;. In neuroscience the &quot;analytic-synthetic theory&quot; sometimes refers to the respective specializations of &lt;a href=&quot;http://psychology.wikia.com/wiki/Analytic-synthetic_theory&quot; rel=&quot;noreferrer&quot;&gt;the left and the right hemispheres&lt;/a&gt;, the more &quot;rational&quot; and the more &quot;creative&quot; one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 19th century mathematics the roughly Kantian usage was given a twist: &quot;analytic&quot; came to be associated with more formalization and arithmetization, while &quot;synthetic&quot; concerned the more intuitive and constructive approaches, especially in geometry (see e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Synthetic_geometry&quot; rel=&quot;noreferrer&quot;&gt;synthetic geometry&lt;/a&gt;). When Frege argued against mathematics being synthetic he meant to rid it of &quot;non-rigorous&quot; Kantian intuition, and construction that required it. It is this loosened Kantian meaning that Hawking appears to have in mind. He follows Klein himself, who in &lt;a href=&quot;https://books.google.com/books?id=3IqQDAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Elementary Mathematics from a Higher Standpoint: Volume II: Geometry, pp.66-67&lt;/a&gt; bemoaned the one-sidedness of the moderns:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In higher mathematics, however, these words have, curiously, taken on&#xA;  an entirely different meaning. Synthetic geometry is that which studies figures&#xA;  as such, without recourse to formulas, whereas analytic geometry consistently&#xA;  makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates. Rightly understood, there exists only a difference of gradation between these two kinds of geometry, according as one&#xA;  gives more prominence to the figures or to the formulas...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In mathematics, however, as everywhere else, men are inclined to form&#xA;  parties, so that there arose schools of pure synthesists and schools of pure&#xA;  analysts, who placed chief emphasis upon absolute &quot;purity of method,&quot;&#xA;  and who were thus more one-sided than the nature of the subject demanded.&#xA;  Thus the analytic geometricians often lost themselves in blind calculations devoid of any geometric representation. The synthesists, on the other hand,&#xA;  saw salvation in an artificial avoidance of all formulas, and thus they accomplished nothing more, finally, than to develop their own peculiar language&#xA;  formulas, different from ordinary formulas.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The original use is derived from &lt;a href=&quot;https://plato.stanford.edu/entries/analysis/s1.html#Pappus&quot; rel=&quot;noreferrer&quot;&gt;Pappus&lt;/a&gt;, who likely reports a consensus conception that emerged in late antiquity, see &lt;a href=&quot;http://www.springer.com/us/book/9789027705327&quot; rel=&quot;noreferrer&quot;&gt;The Method of Analysis by Hintikka and Remes&lt;/a&gt; for modern commentary. And he clearly talks about two &lt;em&gt;complementary&lt;/em&gt; aspects of solving a geometric problem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the Elder, and proceeds by the method of analysis and synthesis. Now analysis is the way from what is sought — as if it were admitted — through its concomitants [akolouthôn] in order to something admitted in synthesis. For in analysis we suppose that which is sought to be already done, and we inquire from what it results, and again what is the antecedent [proêgoumenon] of the latter, until we on our backward way light upon something already known and being first in order. And we call such a method analysis, as being a solution backwards [anapalin lysin].&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In synthesis, on the other hand, we suppose that which was reached last in analysis to be already done, and arranging in their natural order as consequents [epomena] the former antecedents [proêgoumena] and linking them one with another, we in the end arrive at the construction of the thing sought. And this we call synthesis.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But the later usage was influenced by Kant's distinction between analytic and synthetic (anticipated by Locke), which altered the ancient meaning. Where analytic relied on formal logical (to Kant, syllogistic) derivations, synthetic relied on construction in intuition. Nonetheless, Kant's intuitive construction was explicitly modeled on the synthetic side of Euclid's demonstrations, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s11229-012-0066-2&quot; rel=&quot;noreferrer&quot;&gt;Friedman's Kant on Geometry and Spatial Intuition&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It appears, in fact, that the proof-procedure of Euclid’s Elements is paradigmatic of construction in pure intuition throughout Kant’s discussion of mathematics in the first Critique — which includes a fairly complete presentation of the elementary Euclidean geometry of the triangle. In the Transcendental Aesthetic, for example, Kant presents the corresponding side-sum property of triangles — that two sides taken together are always greater than the third (Proposition I.20) — as an illustration of how geometrical propositions “are never derived from universal concepts of line and triangle, but rather from intuition, and, in fact, [are thereby derived] a priori with apodictic certainty” (A25/B39). And the Euclidean proof of this proposition proceeds, just like Proposition I.32, by auxiliary constructions and inferences starting from an arbitrary triangle ABC...&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-30T01:27:30.047" LastActivityDate="2018-01-30T01:27:30.047" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7001" PostTypeId="2" ParentId="6997" CreationDate="2018-01-30T01:38:22.653" Score="0" Body="&lt;p&gt;Grattan-Guinnesses's &lt;a href=&quot;https://isidore.co/calibre#book_id=6616&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Convolutions in French Mathematics, 1800–1840 From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics&lt;/em&gt;&lt;/a&gt; §3.2.5 &quot;'Analysis' and 'synthesis', and their mathematical connections&quot;, p. 135:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The basic distinction between analysis and synthesis in proof-methods in mathematics is that in analysis one starts out from the desired result and regresses until apparently impeccable principles are found, while synthesis begins with those principles and derives the result. However, especially during the 18th century mathematicians became accustomed to associate analysis with algebra and synthesis with geometry, although these connections were not clear, least of all in the development and use of the calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-01-30T01:38:22.653" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7002" PostTypeId="2" ParentId="6997" CreationDate="2018-01-30T01:39:10.567" Score="3" Body="&lt;p&gt;As Francois Ziegler pointed out in his comment, the distinction of  synthetic vs. analytic refers here to two approaches to geometry. In a nutshell, synthetic geometry is the study based solely on axioms, logical arguments  and classical constructions, while analytic geometry allows the use of coordinates or calculus. The Wikipedia page gives a pretty good account:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Synthetic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Synthetic_geometry&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Klein did not completely ignore the analytic approach, as he wrote the following in his Erlangen Program:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;``The distinction between modern synthesis and modern analytic geometry must no longer be re-&#xA;garded as essential, inasmuch as both subject-matter and methods of reasoning have gradually&#xA;taken a similar form in both. We choose therefore in the text as common designation of them both&#xA;the term&#xA;projective geometry&#xA;. Although the synthetic method has more to do with space-perception&#xA;and thereby imparts a rare charm to its first simple developments, the realm of space-perception&#xA;is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be&#xA;looked upon as a precise and perspicuous statement of geometrical relations. On the other hand,&#xA;the advantage to original research of a well formulated analysis should not be underestimated, -&#xA;an advantage due to its moving, so to speak, in advance of the thought. But it should always be in-&#xA;sisted that a mathematical subject is not to be considered exhausted until it has become intuitively&#xA;evident, and the progress made by the aid of analysis is only a first, though a very important, step.&quot; (translated by M. W. Haskell; available at &lt;a href=&quot;https://arxiv.org/abs/0807.3161&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/0807.3161&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might be wrong on this, but it seems that the coordinate geometry is nowadays absent from the American high school curriculum; some elements are taught in multivariable calculus and/or linear algebra, which are considered ``advanced&quot; courses (at the college level). Therefore the distinction between analytic and synthetic geometry appears mysterious. On the other hand, when I was in my junior high school grade in Poland in 1980s, I took a (compulsory)  course on analytic geometry, from a text with a matching title. It was preceded by 2 years of axiomatic geometry, so the distinction was clear.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2018-01-30T01:39:10.567" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7004" PostTypeId="1" AcceptedAnswerId="7008" CreationDate="2018-01-30T13:21:18.700" Score="2" ViewCount="237" Body="&lt;p&gt;In this article in Wikipedia about Katsumi Nomizu&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Katsumi_Nomizu&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Katsumi_Nomizu&lt;/a&gt;&#xA;it is written that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Over the course of his career, Katsumi Nomizu was influential in determining the course of differential geometry by stressing what he called the structural approach.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In what does  this &lt;em&gt;structural approach&lt;/em&gt; constist? Are there some papers written by Nomizu where this concept is clarified?&lt;/p&gt;&#xA;" OwnerUserId="6975" LastActivityDate="2018-01-30T21:51:36.640" Title="Nomizu's structural approach to differential geometry" Tags="|mathematicians|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7005" PostTypeId="1" AcceptedAnswerId="7009" CreationDate="2018-01-30T16:33:07.997" Score="3" ViewCount="254" Body="&lt;p&gt;When you study Newtonian physics in introductory classes, the equations don't include any statistical indications, such as error, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However modern physics articles indicate statistical properties of their theories and equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did the physics community recognize the need to include statistics in the mathematical description of reality?&lt;/p&gt;&#xA;" OwnerUserId="2708" LastActivityDate="2018-02-01T01:00:04.937" Title="When did statistics become an integral part of physics?" Tags="|physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7006" PostTypeId="1" AcceptedAnswerId="7015" CreationDate="2018-01-30T17:46:02.230" Score="1" ViewCount="441" Body="&lt;p&gt;Reading about the scientific method I found that the classical definition I got in school (observation - induction - hypoteshis - experimentation - demostration or refutation and tesis or scientific law)  was made by Francis Bacon who lived between 22 January 1561 – 9 April 1626. Was this the first definition of the scientific method or is there any older definition?&lt;/p&gt;&#xA;" OwnerUserId="4512" LastActivityDate="2018-01-31T16:46:14.710" Title="Who was the first person to give a definition of the scientific method?" Tags="|scientific-method|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7007" PostTypeId="2" ParentId="7005" CreationDate="2018-01-30T19:49:28.373" Score="2" Body="&lt;p&gt;Statistics entered physics in two ways: a) when working with experimental data, which always come with random errors. A mathematical theory of random errors, addressing the processing of experimental data was developed independently by Legendre and Gauss; it is called the Method of Least Squares. (This was in the end of 18th and the very beginning of 19th century). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) In a more substantial way, statistics entered with the development of statistical mechanics. This begins with D. Bernoulli, but real development happened in the middle of 19th century (Clausius, Maxwell, Boltzmann, Gibbs). &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-01-30T19:49:28.373" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7008" PostTypeId="2" ParentId="7004" CreationDate="2018-01-30T20:55:37.360" Score="3" Body="&lt;p&gt;The intended sense of &quot;structure&quot; comes from Bourbaki, who reformulated all of mathematics as a theory of structures founded on set theory in their multi-volume Elements of Mathematics (with the obvious allusion to Euclid), see &lt;a href=&quot;http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicolas Bourbaki: Theory of Structures&lt;/a&gt;. A structure is a set with operations and/or relations defined on it and/or additional objects attached to it. Topology, metric, group operation, partial order, etc., are examples of structures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In differential geometry the basic structure is the differential one, and one can put extras on top of it, affine connection, Riemannian metric, symplectic form, complex or Kähler structure, etc. Despite having prominent geometers among its members and decrying the &quot;debauchery of indices&quot; in classical expositions, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/599/what-was-the-motivation-for-the-development-of-modern-intrinsic-differential-g&quot;&gt;What was the motivation for the development of modern, intrinsic, differential geometry?&lt;/a&gt;, Bourbaki never quite got to writing its own volume on differential geometry. So &lt;a href=&quot;https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kobayashi-Nomizu's Foundations of Differential Geometry&lt;/a&gt; became the stand in for that. It promoted the systematic use of invariant notation and intrinsic descriptions of geometric objects through the systematic use of the language of Lie groups, homogeneous spaces, fiber bundles, connections on bundles, and other differential geometric structures. Nomizu particularly contributed to the revival of interest in affine structures in 1980-s with his Munster lecture What is Affine Differential Geometry? and subsequent work, see &lt;a href=&quot;https://books.google.com/books?id=OAUrDwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry And Topology Of Submanifolds VII, p.35&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-01-30T21:51:36.640" LastActivityDate="2018-01-30T21:51:36.640" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7009" PostTypeId="2" ParentId="7005" CreationDate="2018-01-30T21:50:31.920" Score="6" Body="&lt;p&gt;A theory of observational errors first appeared in Simpson's memoir of 1755, which postulated that positive and negative errors are equally probable, and discussed several possible distributions of error over the interval where they were supposed to fall, including uniform and triangular distribution. In 1765 Lambert added the semicircle distribution to the list, and Lagrange added parabolic in 1776 and raised cosine and logarithmic distributions 1781. The first systematic attempt to &lt;em&gt;derive&lt;/em&gt; the &quot;error curve&quot; was undertaken by Laplace in 1774, but the distribution he obtained had a cusp at the origin. He derived the &quot;correct&quot; error curve, the normal one, four years later, Gauss rediscovered it around 1795. The name is due to Pierce (1873), who studied measurement errors for an object dropped on a wooden base.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tobias Mayer, in his study of the libration of the moon (1750), suggested to reduce observational errors by averaging over groups of similar equations instead of just averaging observations under identical circumstances. In 1755 Boškovic in his work on the shape of the earth proposed that the true value of a series of observations would be that which minimizes the sum of absolute errors, i.e. what we call the median. The method of least squares to minimize measurement errors was published independently by Adrien-Marie Legendre (1805), Robert Adrain (1808), and Carl Friedrich Gauss (1809). Gauss earned European fame in 1801 by using it to predict the location of Ceres based on rather imprecise data, see &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_statistics#Origins_in_probability_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of statistics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, some basic statistical tools were developed by the beginning of the 19-th century and tried in astronomy. But after that the major developments and applications of error theory happened not in physics but in &quot;soft&quot; sciences like medicine, psychology and sociology. In particular, the notions of standard deviation, correlation and regression were only introduced by Galton (1889) in field studies of human anatomy. Galton's work inspired Pearson to promote statistics as a basic tool of science, but it only became standard practice after Fisher's textbooks Statistical Methods for Research Workers (1925) and The Design of Experiments (1935), see &lt;a href=&quot;https://en.wikipedia.org/wiki/Design_of_experiments&quot; rel=&quot;nofollow noreferrer&quot;&gt;Design of experiments&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Slow adoption of statistical methods in physics may have had something to do with the controversial status of statistical mechanics in the second half of 19th century. Bernoulli's 1738 idea of treating gases as large conglomerates of particles was picked up in 1859 by Maxwell, who derived the distribution of molecular velocities, and starting in 1870-s by Boltzmann, who became the greatest advocate of statistical methods in mechanics. But molecular theory was bitterly opposed by physical and chemical establishment (although, due to the success of the periodic table, chemists were much more receptive to the idea than physicists), led by Mach and Ostwald, as a metaphysical speculation, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2514/55&quot;&gt;Did anybody question the indivisibility of the atom because there were “too many” elements?&lt;/a&gt; Only after the success of Lorentz's theory of electrons, and especially Einstein's and Smoluchowski's theory of Brownian motion, which vindicated Boltzmann's ideas, was the barrier removed, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Statistical_mechanics#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Statistical mechanics: history&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-01T01:00:04.937" LastActivityDate="2018-02-01T01:00:04.937" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7011" PostTypeId="1" AcceptedAnswerId="7017" CreationDate="2018-01-31T10:13:37.997" Score="5" ViewCount="2661" Body="&lt;p&gt;French curves are a set of curvilinear rulers used in industrial design, before the advent of CAD, when everything still had to be drawn by hands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most popular set of such rulers is made up of 3 rulers chosen among a set of 28 that were described in a book by Burmester at the beginning of the XXth century. (Lexicon der Gesamten Technik (1904)). For this reason they are often known also as Burmester's curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the engineering US literature they are often mentioned also as irregular curves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have the following questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Does anyone knows which curves are describing their profiles? Either of the 3 or of the full set of 28.&lt;/li&gt;&#xA;&lt;li&gt;Connected to the previous: wikipedia (without source) and some others mention sometimes the fact that profile curves contain arcs of clothoid. Though quite reasonable from some points of view (linearly approximating curvature) I've found no explicit evidence for this. Any reference you may know for this &quot;factoid&quot;?&lt;/li&gt;&#xA;&lt;li&gt;Burmester certainly built up his 28 profiles for some reasons: is there any math reason for the 28 profiles? &lt;/li&gt;&#xA;&lt;li&gt;Burmester certainly was capable of physically buildings his models: how were their constructed? Since Burmester is known for his work on geometric linkages is there any evidence that such profiles can be described via a (simple?) planar linkage?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Any help on even one of these topics is welcome.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="55" LastEditDate="2018-02-01T00:55:07.043" LastActivityDate="2018-02-01T18:29:57.517" Title="What is the origin of French/Burmester's curves?" Tags="|classical-mechanics|differential-geometry|geometry|engineering|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7013" PostTypeId="1" AcceptedAnswerId="7019" CreationDate="2018-01-31T13:35:43.953" Score="0" ViewCount="154" Body="&lt;p&gt;Who first did use the symbol $\begin{bmatrix}   1 &amp;amp;   2 \\   3 &amp;amp;   4 \end{bmatrix}$ for a matrix and similarly $\begin{pmatrix}   1 &amp;amp;   2 \\   3 &amp;amp;   4 \end{pmatrix}$?&lt;br&gt;&#xA;Why do these two conventions exist independently  to the present day?    &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were whenever efforts made to unify the symbols?&lt;/p&gt;&#xA;" OwnerUserId="6955" LastEditorUserId="6955" LastEditDate="2018-02-01T17:40:50.223" LastActivityDate="2018-02-01T17:40:50.223" Title="Invention of matrix symbols" Tags="|mathematics|linear-algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7014" PostTypeId="1" CreationDate="2018-01-31T15:37:37.147" Score="5" ViewCount="429" Body="&lt;p&gt;There is an interesting cartoon in the book &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lillian Hoddeson, Ernst Braun, Jurgen Teichmann, Spencer Weart (Eds.) Out of the Crystal Maze: Chapters from The History of Solid State Physics. Oxford Univesity Press, 1992 (online &lt;a href=&quot;https://books.google.co.il/books?id=WCpPPHhMdRcC&amp;amp;printsec=frontcover&amp;amp;vq=cartoon&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q=cartoon&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The cartoon (see below) appears on page 15 and depicts recognizably professorial types exchanging books for cartoonish bags of coins brought by recognizably capitalist types. Hovering over the scene are a sun-like Felix Klein and an angelic-looking &lt;a href=&quot;https://fr.wikipedia.org/wiki/Friedrich_Althoff&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedrich Althoff&lt;/a&gt;, depicted as bestowing blessings upon the congregants. The book says that the cartoon dates from 1908 but does not provide any additional details. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question 1.&lt;/em&gt;&lt;/strong&gt; Where did the cartoon first appear in 1908? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question 2.&lt;/em&gt;&lt;/strong&gt; Who created the cartoon? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question 3.&lt;/em&gt;&lt;/strong&gt; Althoff died on 20 october 1908.  Did the cartoon appear before or after his death in 1908? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question 4.&lt;/em&gt;&lt;/strong&gt; There are several books that have reprinted this cartoon and they all seem to agree that the cartoon originates from 1908.  Where did they get this information from?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Question 5.&lt;/em&gt;&lt;/strong&gt; Armin Hermann writes in his 1991 book that the cartoon was distributed with the invitations to a february 1908 meeting at Goettingen, as pointed out in &lt;a href=&quot;https://hsm.stackexchange.com/a/7023/604&quot;&gt;this answer&lt;/a&gt;.  Where does Hermann's information come from?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gqcnW.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gqcnW.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-02-04T12:41:41.620" LastActivityDate="2018-02-04T12:41:41.620" Title="Source of cartoon lampooning Felix Klein" Tags="|physics|mathematicians|" AnswerCount="1" CommentCount="12" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7015" PostTypeId="2" ParentId="7006" CreationDate="2018-01-31T16:46:14.710" Score="1" Body="&lt;p&gt;The history of scientific method and its study (on which &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_scientific_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt; has many good points and references) shows a great variety of attempted conceptualizations. This variety in itself arguably indicates that a final definition, whether classical or otherwise, has not yet been written. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that proviso, an at least partial attempt can be made to answer the question &quot;is there any older definition?&quot; of scientific method (other than the quoted one attributed to Bacon). As &lt;em&gt;Conifold&lt;/em&gt; has pointed out in his comment, the substance of the quoted definition is rather a 19th-century development or refinement, often referred to as the 'hypothetico-deductive' definition, where Bacon's own ideas were less developed or more naive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bacon's discussion of method came out in 1620 in his book 'Novum Organum' (Latin) (an English translated text is &lt;a href=&quot;http://www.constitution.org/bacon/nov_org.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And yes, there is to some extent an older anticipation of Bacon, at least one from 20 years before. It is from William Gilbert (1544-1603) in 'De Magnete' (Latin) of 1600. (An English translation by P Fleury Mottelay (1893, New York (Wiley)) is &lt;a href=&quot;http://www.gutenberg.org/files/33810/33810-h/33810-h.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;found here&lt;/a&gt;.)  There were earlier writers whose works have been pointed out as examples of experimental method, but I do not know whether Gilbert was the first to discuss method itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gilbert did not expand as much as Bacon on the subject of methodology, his book was mainly a groundbreaking study of magnetism, which achieved wide reception and had considerable influence in Europe. But it did contain explicit discussion of method, identified a core of experimental philosophy that was put into practice by its application to magnetic phenomena and has been described as &quot;one of the finest examples of inductive philosophy&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Gilbert, &quot;stronger reasons are obtained from sure experiments and demonstrated arguments than from probable conjectures and the opinions of philosophical speculators of the common sort&quot;.   Gilbert wrote that he was dealing &quot;in hypotheses that are provable, the things that we have through a long experience discovered.&quot; He discussed and rejected as insufficient the reliance that he ascribed to the classical Greeks, a reliance on terminology and logical argumentation alone. One sees in germ in Gilbert's work the principle of confrontation between arguments/hypotheses/ideas and the reality of experiment and observation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(But Gilbert was also careful about the use of words, and in a paragraph that arguably still deserves attention from those who create jargon in modern times, he distinguished two kinds of motives for introducing new terms: &quot;we sometimes employ words new and unheard-of, not (as alchemists are wont to do) in order to veil things with a pedantic terminology and to make them dark and obscure, but in order that hidden things which have no name and that have never come into notice, may be plainly and fully published.&quot;) &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-01-31T16:46:14.710" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7016" PostTypeId="2" ParentId="6986" CreationDate="2018-01-31T23:47:12.873" Score="6" Body="&lt;p&gt;I recommend the authoritative volume &quot;Das Studium der Mathematik an den deutschen Universitäten seit Anfang des 19. Jahrhunderts&quot; by Wilhelm Lorey, published in 1916. This &lt;a href=&quot;http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Lorey%2C%20Wilhelm%2C%201873-&quot; rel=&quot;nofollow noreferrer&quot;&gt;digitalized edition&lt;/a&gt; can be accessed from inside the US.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the number of professors: to the best of my knowledge, German universities had two ordinary and one extraordinary professor, except Berlin and later Göttingen, which had three ordinary professors.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="5376" LastEditDate="2018-02-02T16:17:46.770" LastActivityDate="2018-02-02T16:17:46.770" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7017" PostTypeId="2" ParentId="7011" CreationDate="2018-02-01T00:23:06.073" Score="3" Body="&lt;p&gt;There seems to be little secondary literature on this so answering the OP questions fully would take some serious digging into the original sources. One promising secondary source that I was unable to locate is Ceccarelli and Koetsier, Burmester and Allevi: A Theory and its Application for Mechanism Design at the end of 19th Century, Proceedings of IDETC/CIE 2006 ASME 2006 International Design Engineering Technical Conferences &amp;amp; Computers and Information in Engineering Conference. I will only give some comments.&lt;/p&gt;&#xA;&lt;p&gt;There has been a revival of interest in the old drawing techniques recently due to their digitization, for touch-based input sketch-based techniques give a more natural feel and fluidity than conventional interfaces that employ control point manipulation of splines like Bezier curves. I suspect that the reference to clothoid comes from modern digital reconstructions of French curves, for instance &lt;a href=&quot;http://www.dgp.toronto.edu/%7Emccrae/projects/french/mccraeSBIM2011.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;McCrae and Singh&lt;/a&gt; stitch them from clothoid pieces. But there is no connection to Burmester's works, only reference to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lexikon_der_gesamten_Technik&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lexicon der gesamten Technik&lt;/a&gt;, and earlier Singh used &lt;a href=&quot;https://www.rhino3d.com/nurbs&quot; rel=&quot;nofollow noreferrer&quot;&gt;cubic NURBS&lt;/a&gt; (Non-Uniform Rational B-Splines) to the same end. By the way, Lexicon der gesamten Technik (Encyclopaedia of All Technology and its Auxiliary Sciences) is not a book by Burmester, it is a multi-volume encyclopedia first published by Lueger in 1894–1899 that had a picture of a complete Burmester set, it is unclear that Burmester directly contributed to it.&lt;/p&gt;&#xA;&lt;p&gt;Burmester's doctoral thesis was on the geometry of &lt;a href=&quot;https://en.wikipedia.org/wiki/Isophote&quot; rel=&quot;nofollow noreferrer&quot;&gt;isophotes&lt;/a&gt; (lines of equal brightness), but his opus magnum Lehrbuch der Kinematik (Textbook of Kinematics, 1888) developed kinematic synthesis of curve-drawing linkages, an elegant and sadly forgotten theory originated by Reuleaux (Gibson has a brief section on it in his &lt;a href=&quot;https://books.google.com/books/about/Elementary_Geometry_of_Algebraic_Curves.html?id=cKVkMP4mZSAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elementary Geometry of Algebraic Curves (1998)&lt;/a&gt;, which ironically does not even mention Reuleaux). Roughly, the idea was to sketch various curves by designing linkages and other mechanical devices whose points would trace them in motion, such as the classical &lt;a href=&quot;https://en.wikipedia.org/wiki/Four-bar_linkage&quot; rel=&quot;nofollow noreferrer&quot;&gt;four-bar linkage&lt;/a&gt; employed in Watt's engine. It seems likely that the curves Burmester used were linkage coupler curves (traced by a point on the coupler bar). The &lt;a href=&quot;http://s.goessner.net/articles/CubicOfStationaryCurvature.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;cubic of stationary curvature&lt;/a&gt; (an asymmetric strophoid, below), traced by the four-bar linkage, is reminiscent of the shapes in French curves. &lt;a href=&quot;http://ebooks.library.cornell.edu/k/kmoddl/toc_hartenberg1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hartenberg's Kinematic Synthesis of Linkages&lt;/a&gt;, published when the subject was still lively (1964), gives some history:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Modern kinematics had its beginning with Reuleaux. His now classical &amp;quot;Theoretische Kinematik&amp;quot; of 1875 presented many views finding general acceptance then that are current still... Reuleaux regarded a mechanism as a (kinematic) chain of connected links (or parts), one link being fixed... Through hindsight we saw in Watt's straight-line linkage the amorphous beginning of an ordered and advanced synthetic process. Nearly a century later Reuleaux identified synthesis as a concept, as an entity that might be and must be pursued to guide the designer through the maze of mechanisms. His views were limited to only type synthesis: by this is meant the determination of the type of mechanism for a given job... which lies ahead of the related and more recent fields of number and dimensional synthesis.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Geometers and algebraicists of the 1870s became interested in linkages as curve-drawing devices, not as hardware; their work has since been made part of the corpus of the kinematics of mechanisms. It was discovered that a link motion can be found to describe an algebraic curve of any order. In particular, the coupler-point motion of the four-bar linkage (then called a &amp;quot;three-bar motion&amp;quot;) was studied. Samuel Roberts showed that a coupler point describes a curve of the sixth order.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Concepts far beyond those of Reuleaux were added to the picture he had attempted to paint; the whole is now called synthesis of linkages... The third and last kind is dimensional synthesis, or the determination of the proportions (lengths) of the links needed to accomplish the specified motion transformation. In Germany, Burmester was in accord with Reuleaux's fundamental concepts and most of his nomenclature and definitions. Making extensive use of mathematical principles (mostly geometrical), and considering displacement, velocity, and acceleration, Burmester's &amp;quot;Lehrbuch der Kinematik&amp;quot; (1888) developed&#xA;geometric methods that furthered analysis and showed the way to synthesis.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By the way, Hartenberg uses &lt;a href=&quot;https://hsm.stackexchange.com/a/7000/55&quot;&gt;&amp;quot;analysis and synthesis&amp;quot;&lt;/a&gt; in their original ancient sense described by Pappus.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Cubic of stationary curvature traced by a four-bar linkage:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Dv2rm.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Dv2rm.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-01T18:29:57.517" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7018" PostTypeId="1" AcceptedAnswerId="7021" CreationDate="2018-02-01T04:48:02.910" Score="6" ViewCount="361" Body="&lt;p&gt;I was asked recently if the tensor product symbol $\otimes$ had been used before Bourbaki's publication on multilinear algebra in 1948 (a draft of this document can be seen at &lt;a href=&quot;http://sites.mathdoc.fr/archives-bourbaki/PDF/040_iecnr_049.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://sites.mathdoc.fr/archives-bourbaki/PDF/040_iecnr_049.pdf&lt;/a&gt; starting on p. 10). I was unable to find any earlier usage, so that leads to my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who introduced the symbol $\otimes$ for use in tensor products?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="643" LastEditorUserId="72" LastEditDate="2018-03-01T03:31:36.033" LastActivityDate="2018-03-01T03:31:36.033" Title="First appearance of tensor product symbol $\otimes$" Tags="|notation|linear-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7019" PostTypeId="2" ParentId="7013" CreationDate="2018-02-01T13:41:02.907" Score="1" Body="&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/matrices.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Uses of Symbols for Matrices and Vectors&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first similar symbol seems to be the &quot;mixed&quot; one used by Cayley into his &lt;a href=&quot;https://www.jstor.org/stable/pdf/108649.pdf?refreqid=excelsior:bbca73d3dad85de36b0239af2b5092b8&quot; rel=&quot;nofollow noreferrer&quot;&gt;“A Memoir on the Theory of Matrices”&lt;/a&gt; (1858).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A following reference is M.Bocher, &lt;a href=&quot;https://books.google.it/books?id=3XBQm4uWkiwC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Higher Algebra&lt;/a&gt; (1907), page 20-21.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;squared parenthesis&quot; notation is due to Cullis, &lt;a href=&quot;https://books.google.it/books?id=LqL5rrXcY7MC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matrices and Determinoids&lt;/a&gt;, 1st ed. 1913.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-02-01T13:41:02.907" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7020" PostTypeId="2" ParentId="2200" CreationDate="2018-02-01T14:05:34.690" Score="8" Body="&lt;p&gt;In Jataka, there's a story about Losaka&#xA;&lt;a href=&quot;http://www.sacred-texts.com/bud/j1/j1044.htm&quot; rel=&quot;noreferrer&quot;&gt;http://www.sacred-texts.com/bud/j1/j1044.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So in time it came to pass that the people fell into a wretched plight. Reflecting that such had not been their lot in former days, but that now they were going to rack and ruin, they concluded that there must be some breeder of misfortune among them, and resolved to divide into two bands. This they did; and there were then two bands of five hundred families each. Thence-forward, ruin dogged the band which included the parents of the future Losaka, whilst the other five hundred families throve apace. So the former resolved to go on halving their numbers, and did so, until this one family was parted from all the rest. Then they knew that the breeder of misfortune was in that family, and with blows drove them away&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6983" LastEditorUserId="-1" LastEditDate="2018-04-11T16:07:15.767" LastActivityDate="2018-04-11T16:07:15.767" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7021" PostTypeId="2" ParentId="7018" CreationDate="2018-02-01T14:54:43.453" Score="4" Body="&lt;p&gt;Keith:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest usage we have been able to find is in a 1936 paper by Francis J Murray and John von Neumann, entitled &quot;Rings of Operators&quot;(Annals of Mathematics, Vol. 37, pp. 116-229). Paul Halmos later used it in the earliest published version of his &quot;Finite Dimensional Vector Spaces,&quot; published by Princeton University Press in 1942 (see Appendix II-Direct Products).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jim Casey&lt;/p&gt;&#xA;" OwnerUserId="6984" LastEditorUserId="7064" LastEditDate="2018-02-22T00:46:21.530" LastActivityDate="2018-02-22T00:46:21.530" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7022" PostTypeId="2" ParentId="2200" CreationDate="2018-02-01T16:26:07.660" Score="3" Body="&lt;p&gt;The method of bisection was used over three hundred years before Bolzano, to prove the intermediate value theorem by Simon Stevin, in the case of a particular cubic polynomial (of course Stevin did not use the term &quot;binary search&quot;).  For details see &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-011-9228-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 2012 publication in &lt;em&gt;Foundations of Science&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-02-01T16:26:07.660" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7023" PostTypeId="2" ParentId="7014" CreationDate="2018-02-01T18:43:01.230" Score="8" Body="&lt;p&gt;Armin Hermann states in his book on &lt;a href=&quot;https://books.google.de/books?id=vrrPBgAAQBAJ&amp;amp;pg=PA355&amp;amp;lpg=PA355&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 355&lt;/a&gt; that this caricature was enclosed in the invitation letters. Since the anniversary meeting was held in February 1908, it appeared before Althoff's death. Here the conference proceedings can be downloaded: &lt;a href=&quot;https://link.springer.com/book/10.1007/978-3-663-15735-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/book/10.1007/978-3-663-15735-9&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5376" LastEditorUserId="604" LastEditDate="2018-02-02T08:15:43.727" LastActivityDate="2018-02-02T08:15:43.727" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7024" PostTypeId="1" CreationDate="2018-02-01T22:50:24.137" Score="4" ViewCount="606" Body="&lt;p&gt;I've been reading Courant's Integral and Differential Calculus for a bit, at some sections, there are problems which do not seem to be answerable with previously given material in the book.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Several days ago, I made &lt;a href=&quot;https://math.stackexchange.com/q/2381361/25805&quot;&gt;this question&lt;/a&gt; but although the answer is fair, it's a little bit vague. I don't know much about German educational system nor its history, so at the time of publication of Courant's book, I believe people studied it in university (I may be mistaken, it seems that in the past, calculus was taught at high-school and still is at some countries). What were the subjects people typically ought to know when facing Courant's book through formal education?  &lt;/p&gt;&#xA;" OwnerUserId="332" LastEditorUserId="12699" LastEditDate="2018-02-05T00:34:36.167" LastActivityDate="2018-02-05T00:34:36.167" Title="What did the typical German student know before reading/studying Courant's Calculus when it was published?" Tags="|calculus|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7025" PostTypeId="2" ParentId="7024" CreationDate="2018-02-02T01:13:20.667" Score="2" Body="&lt;p&gt;First of all, there was no such subject as &quot;calculus&quot; in Germany in 1920s.&#xA;The original title of the book is  Vorlesungen über Differential-und Integralrechnung (1927), and the subject is called Analysis. I have no first hand experience with mathematics education in Germany; I was educated in Soviet Union, but Soviet system was modeled on the German one, and all that I know about German system shows that they were similar, and the textbooks were similar. (A Russian translation of Courant was used sometimes in some universities).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analysis was one of the three principal subject taught to 1-st year students,&#xA;the other two were called Higher Algebra and Analytic Geometry. So the background was high school mathematics. (In Germany it is called Gimnasium). High school gave a&#xA;solid background in elementary algebra and geometry (rigorous, everything with proofs, including stereometry and trigonometry). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, European and American education systems were (and still are) very different. The main difference is that in Germany (and in Soviet Union) there is no analog of American &quot;undergraduate education&quot;. When you enroll to a university&#xA;you have to decide from the very beginning what you are going to study. If you choose Mathematics, then you are taught real (completely rigorous) Mathematics (not &quot;Calculus&quot;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope some German-educated mathematician will give more qualified answer.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-02-02T01:13:20.667" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7026" PostTypeId="2" ParentId="7024" CreationDate="2018-02-02T01:19:12.527" Score="5" Body="&lt;p&gt;By the time Courant's text was written, the German mathematics curriculum underwent a  reform spearheaded by Felix Klein around 1900. It was later adopted by most European countries (but not the US). Prior to the reform only algebra, elementary geometry and trigonometry were taught at secondary schools. Note that all three were far less &amp;quot;mechanical&amp;quot; than today and involved deriving non-trivial identities and making Euclidean demonstrations. Indeed in classical education the purpose of mathematics was seen as exercising &amp;quot;brain muscles&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Klein's reform added analytic geometry and calculus to the list, and made the notion of function central to the curriculum. Klein's original motivation was to satisfy the new demands that industrial revolution imposed on workers' mathematical backgrounds. In alliance with German industrialists he established in 1898 the Göttinger Vereinigung für angewandte Physik (Göttingen Association for Applied Physics), broadened in 1901 to &amp;quot;und Mathematik&amp;quot;. This was followed by reorienting the curriculum of secondary schools away from &amp;quot;brain muscles&amp;quot; and towards science and engineering. The reform was finalized in the then Austrian town Meran in 1905, the name &amp;quot;Meraner Reform&amp;quot; is sometimes used, its main document is known as the Meran syllabus. Fenker's textbook 'Arithmetische Aufgaben' gives an idea of what the syllabus looked like and what it replaced, see &lt;a href=&quot;https://books.google.com/books?id=sojkAwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bierman-Jahnke's chapter in Transformation - A Fundamental Idea of Mathematics Education, p.20&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is from &lt;a href=&quot;http://www.icmihistory.unito.it/portrait/klein.php#ag&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of the International Comission on Mathematical Instruction&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Klein deplored what he used to call the &amp;quot;double gap&amp;quot;: the discontinuity between school mathematics and university mathematics and the double forgetting of the respective knowledge: first one had to forget school mathematics upon beginning one's university studies and later as a teacher one had to forget university mathematics and return to school mathematics.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Klein proposed a radical reform. Since the mathematical courses at the technical colleges consisted of a basic preparatory &amp;quot;general&amp;quot; part and an advanced or higher part, he recommended that the basic studies be transferred to the preparatory schools, i.e. the secondary schools, and that only the advanced studies should remain as part of the college curriculum. However, they would be reformed and taught not as independent branches of knowledge, but rather &amp;quot;in permanent touch with the requirements of the students of mechanical and construction engineering.... Mathematics at the colleges can thus become once again what it ought to be: a specific power pervading the whole&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Transferring the preparatory courses to secondary school would help create a firm mathematical basis for the pursuit of college studies, due, in particular, to the &amp;quot;benevolent coercion of school,&amp;quot; as opposed to a university environment where academic freedom reigned. But which subjects were to be considered as part of the basic core of this new secondary mathematical curriculum? According to Klein, this core would include analytical geometry, but primarily differential and integral calculus! In private letters to close friends and co-workers, Klein in fact always stressed that the key to his reform plans was to solve the question of mathematical preparation for the technical colleges.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-04T22:30:22.940" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7028" PostTypeId="1" CreationDate="2018-02-02T13:28:57.843" Score="6" ViewCount="702" Body="&lt;p&gt;In modern set theory the difference between actual infinity and potential infinity is often not understood or even denied. Some decades back however mathematicians like Hilbert or Poincaré, let alone Cantor or Fraenkel were fully aware of the difference. My question: Does there exist a formal definition of potential infinity in contrast to actual infinity, and who was the first to give it?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="55" LastEditDate="2018-02-03T09:33:42.947" LastActivityDate="2018-02-06T03:40:59.537" Title="Is there a formal distinction between potential and actual infinities?" Tags="|mathematical-logic|set-theory|infinity|" AnswerCount="4" CommentCount="17" ContentLicense="CC BY-SA 3.0" />
  <row Id="7029" PostTypeId="2" ParentId="7028" CreationDate="2018-02-02T14:38:48.107" Score="3" Body="&lt;p&gt;Not &quot;formal&quot; but quite precise: &lt;a href=&quot;https://plato.stanford.edu/entries/aristotle-mathematics/index.html#12&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle and &lt;em&gt;apeiron&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;em&gt;Meta&lt;/em&gt;, Book IX ($\Theta$), &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0052%3Abook%3D9%3Asection%3D1048b&quot; rel=&quot;nofollow noreferrer&quot;&gt;1048b10&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The infinite and the void and all similar things are said to exist potentially&#xA;  and actually in a different sense from that in which many other things are said&#xA;  so to exist, e.g. that which sees or walks or is seen. [...] But the infinite does not exist potentially in the sense that it will ever actually have separate existence; its separateness is only in knowledge. For the fact that division never ceases to be possible gives the result that this actuality exists potentially, but not that it exists separately.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And &lt;em&gt;Phys&lt;/em&gt;, Book III, 4, 206b17:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By addition then, also, there is potentially an infinite, namely, what we have&#xA;  described as being in a sense the same as the infinite in respect of division. For it will always be possible to take something &lt;em&gt;ab extra&lt;/em&gt;. Yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed and there will always be a smaller part.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And 207a32:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is reasonable that there should not be held to be an infinite in respect&#xA;  of addition such as to surpass every magnitude.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-02-02T14:59:25.630" LastActivityDate="2018-02-02T14:59:25.630" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7030" PostTypeId="2" ParentId="7024" CreationDate="2018-02-02T15:54:03.697" Score="2" Body="&lt;p&gt;One must not forget that the Courant book was an edition of his lectures, see the description of its creation in Reid's biograpy of &lt;a href=&quot;https://books.google.de/books?id=TGn1BwAAQBAJ&amp;amp;pg=PA110&quot; rel=&quot;nofollow noreferrer&quot;&gt;Courant&lt;/a&gt;. The exercises stemmed from the university courses and were not intended to be solved without tuition from the assistants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two other German calculus books from the late 19th and early 20th century, by &lt;a href=&quot;https://archive.org/stream/grundrissderdif02steggoog#page/n6/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kiepert&lt;/a&gt; and &lt;a href=&quot;https://archive.org/stream/grundzgederdif00kowauoft#page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kowalewski&lt;/a&gt;.&#xA;The latter book attracted Stanley &lt;a href=&quot;https://books.google.de/books?id=U2_zEZOHdU4C&amp;amp;pg=PA20&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ulam&lt;/a&gt; to mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2018-02-02T15:54:03.697" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7031" PostTypeId="1" AcceptedAnswerId="7033" CreationDate="2018-02-03T01:01:10.070" Score="4" ViewCount="270" Body="&lt;p&gt;I am trying to decipher a portion of James Joseph Sylvester's 1869 address entitled &quot;&lt;a href=&quot;https://books.google.com/books?id=n8sSAwAAQBAJ&amp;amp;pg=PA1758&amp;amp;lpg=PA1758&amp;amp;dq=%22The%20Study%20That%20Knows%20Nothing%20of%20Observation%22&amp;amp;source=bl&amp;amp;ots=8OtVic-BSf&amp;amp;sig=KOJvcdubF_AZM3ilm5gwb-SFZX0&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjVy-WXxYjZAhUBWq0KHSCtBrEQ6AEIKjAB#v=onepage&amp;amp;q=%22The%20Study%20That%20Knows%20Nothing%20of%20Observation%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Study That Knows Nothing of Observation&lt;/a&gt;&quot;, which, among other things, surveys the landscape of 19th century mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In response to the self-posed question, &quot;how did algebraic forms originate?&quot;, he replies,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;[they originated] in the accidental observation by Eisenstein, some&#xA;  score or more years ago, of a single invariant (the Quadrivariant of a&#xA;  Binary Quartic) which he met with in the course of certain researches&#xA;  ...&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I can't make sense of this, because it seems many of these mathematical objects have since been renamed.  What are 'algebraic forms'?  What are 'quadrivariants'?  Who was Eisenstein, and what was the significance of his discovery?&lt;/p&gt;&#xA;" OwnerUserId="3811" LastEditorUserId="55" LastEditDate="2018-02-03T06:15:43.093" LastActivityDate="2022-03-15T00:55:31.280" Title="What was the significance of Eisenstein's discovery of invariants?" Tags="|mathematicians|terminology|discoveries|abstract-algebra|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7032" PostTypeId="2" ParentId="5860" CreationDate="2018-02-03T05:07:43.297" Score="5" Body="&lt;p&gt;The mathematical terminology &quot;étalé&quot; [spread out] was used by Grothendieck in his 1957 Tohoku paper, and was preexisting at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Grothendieck, A. (1959). Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. Séminaire Bourbaki, 5, 299-327.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;contains the passage&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...$f[:T \to S]$ est dit étale ou encore $f$ est appelé un étalement, ou $T$ est dit étalé sur $S$...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this, it appears that Grothendieck wanted an adjective for a map whose domain $T$ is étalé over $S$, and selected the adjective étale [calm, slack, still] due to its formal similarity to étalé. This is feasible since the meaning of étale happens to be reasonably applicable.&lt;/p&gt;&#xA;" OwnerUserId="6988" LastEditorUserId="6988" LastEditDate="2018-05-28T15:42:08.927" LastActivityDate="2018-05-28T15:42:08.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7033" PostTypeId="2" ParentId="7031" CreationDate="2018-02-03T06:14:45.290" Score="7" Body="&lt;p&gt;The terminology did not change that much.  Algebraic forms are more often called homogeneous polynomials, but for polynomials in two variables &quot;binary form&quot; is still often used. Invariant of a binary form is a polynomial in its coefficients that remains unchanged under the transformation of said coefficients induced by the action of the special linear group on form's variables. There is also a more general notion of covariant. The discriminant of a form is an invariant, and the determinnat of its Hessian is a covariant&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Invariants of a binary form form a ring, and the study of invariants was a flourishing industry in 19-th century that Eisenstein helped to initiate, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Invariant_of_a_binary_form#Covariants_of_a_binary_quartic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Invariant of a binary form&lt;/a&gt;. &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Eisenstein.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferdinand Gotthold Max Eisenstein (1823-1852)&lt;/a&gt; was a brilliant German mathematician, who became fascinated by number theory after buying a French translation of Gauss's Disquisitiones Arithmeticae when he was 19. He was picked out as a young genius by Crelle of the famous journal in 1844, and became a lifelong protege of Alexander von Humboldt. Despite staying on the margins of 1848 revolution he was arrested with much detriment to his already poor health. Like Galois he died young, only 4 years later. He is perhaps best known for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eisenstein%27s_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eisenstein criterion of irreducibility&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eisenstein_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eisenstein series&lt;/a&gt; in the theory of modular forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ring of invariants of a binary quartic is generated by invariants of degrees 2 and 3. This ring is naturally isomorphic to the ring of modular forms of level 1, and the two generators correspond to the Eisenstein series $E_4$ and $E_6$ (&quot;quadrivariant&quot; must be the couterpart of $E_4$). Sylvester himself developed the theory of invariants and covariants of a binary quintic, and contributed to the theories of binary septic, octavic and duodecimic. But the &quot;king of invariants&quot; was &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Gordan.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gordan&lt;/a&gt;. His signature achievement was to construct finite basis for invariants of binary forms of fixed order in 1868. He tried to extend the result to ternary and higher forms, but to no avail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It all came crushing down after Hilbert proved the finite basis theorem in 1888. Except he &quot;cheated&quot;, at least in the eyes of classical invariant theorists. Instead of constructing the basis Hilbert gave a pure existence proof based on ascending chain arguments. He submitted to Mathematische Annalen and Klein sent his paper to &quot;king&quot; Gordan, who opposed it and allegedly quipped &quot;This is not mathematics, this is theology&quot;. At least, this is the traditional story that McLarty calls the &quot;origin myth&quot;, propounded by Hilbert himself and others 30 years later, see &lt;a href=&quot;http://wayback.archive-it.org/all/20090116011956/http://people.math.jussieu.fr/~harris/theology.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;McLarty's Theology and Its Discontents The Origin Myth of Modern Mathematics&lt;/a&gt;. In truth, Gordan and Hilbert cooperated on the 1888 paper, and Gordan's objection was to its &quot;unfinished&quot; state, not the non-constructive proof, and Hilbert &quot;finished&quot; the theory in 1893 by adding Nullstellensatz, etc. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Gordan and most of his contemporaries were far too quick with their reasoning to notice the difference between a statement and its double negation. Without that distinction you cannot sharply distinguish proof by contradiction from direct proof... Apparently Gordan meant just what he said: the proof was not clear to him. When he published his own version he added that Hilbert’s ideas offered more help with calculating specific systems than Hilbert had bothered to use: &quot;The proof Hilbert has given is entirely correct in substance; yet&#xA;  I feel a gap in his explication as he is satisfied to prove the existence&#xA;  of [solutions] without discussing their properties. To repair this gap I give a somewhat different proof...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Klein ended up accepting Hilbert's 1888 paper without changes. In 1893 Gordan effectively developed what is now called Gröbner bases to construct invariants along Hilbert's lines, Emmy Noether merged both Hilbert's axiomatic method and Gordan's &lt;a href=&quot;https://en.wikipedia.org/wiki/Symbolic_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;symbolic method&lt;/a&gt; in her work. But be it as it may, the effect of Hilbert's work was the rise of abstract algebra and the decline of classical invariant theory. For a good part of 20-th century it was almost forgotten until plucked out of obscurity by Rota in 1960-s, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s00012-003-1827-z&quot; rel=&quot;nofollow noreferrer&quot;&gt;The work of Gian-Carlo Rota on invariant theory by Grosshans&lt;/a&gt;. Buchberger introduced &quot;Gröbner bases&quot; in 1965, and with the rise of computers &lt;a href=&quot;https://www.maa.org/press/maa-reviews/computational-invariant-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;computational/algorithmic invariant theory&lt;/a&gt; made something of a comeback. Mumford's &lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_invariant_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;geometric invariant theory&lt;/a&gt; recovers iinvariants in a more abstract guise derived from Noether.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-03T23:31:19.680" LastActivityDate="2018-02-03T23:31:19.680" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7034" PostTypeId="2" ParentId="7028" CreationDate="2018-02-03T09:34:02.247" Score="2" Body="&lt;p&gt;The distinction between potential and actual infinities was Aristotle's clever solution to Zeno's paradoxes. The idea was that while we can mentally divide segments in half indefinitely actualizing the resulting sequence, which is what Zeno does in Dichotomy, is in error. From Metaphysics VIII.8:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Though there are infinitely many halves in a continuum, these are potential and not actual... So the reply we have to make to the question whether it is possible to traverse infinitely many parts... is that there is a sense in which it is possible, and which it is not. If they exist actually, it is impossible; but if they exist potentially, it is possible.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, after the invention of calculus alternative resolutions became available, and Cantor's work convinced many mathematicians that other &quot;paradoxes of infinity&quot; could be dealt with as well, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2419/55&quot;&gt;Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?&lt;/a&gt;. Moreover, Cantor developed a theory of actualized infinities which was seen as fruitful (Hilbert's &quot;&lt;em&gt;no one shall expel us from the Paradise that Cantor has created&lt;/em&gt;&quot; is oft-quoted). The actual infinity is built into the standard axiomatizations of set theory, therefore it can not be distinguished from potential infinity &lt;em&gt;within&lt;/em&gt; them. To do the distinguishing one needs &lt;em&gt;alternative axiomatizations&lt;/em&gt; of mathematics. This underscores Hilbert's idea of axioms as implicit definitions of terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The underlying conceptions were developed by intuitionists, mainly Brouwer and Weyl, see &lt;a href=&quot;https://dspace.library.uu.nl/bitstream/handle/1874/26962/preprint211.pdf?sequence=1.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum by Atten et al.&lt;/a&gt; Poincaré, Borel, Baire, Lebesgue and other so-called proto-intuitionists anticipated these ideas earlier, see &lt;a href=&quot;https://hsm.stackexchange.com/a/6042/55&quot;&gt;Did Poincaré say that set theory is a disease?&lt;/a&gt; (not exactly).  Here is Weyl's informal description of his and Brouwer's conception in Philosophy of Mathematics and Natural Science (1949): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus remains in statu nascendi. This ‘becoming’ selective sequence represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number falling into the continuum. The continuum no longer appears, to use Leibniz’s language, as an aggregate of fixed elements but as a medium of free ‘becoming’&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://quantitability.files.wordpress.com/2013/04/hermann-weyl-the-continuum-a-critical-examination.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Das Kontinuum (1918)&lt;/a&gt; Weyl undertook an intuitionistic reconstruction of classical analysis based on these infinities in becoming. Weyl's reconstruction was first of many. In 1930 Heyting formalized  &lt;a href=&quot;https://en.wikipedia.org/wiki/Intuitionistic_logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;intuitionistic logic&lt;/a&gt;, which allowed to give full formal expression to Brouwer's and Weyl's potential infinities. It turned out that it is not enough to alter set theory axioms (in particular, axioms of infinity and choice have to be dropped), but to also drop the law of excluded middle, which allows reasoning by contradiction. Indeed, if the infinite is never &quot;completed&quot; certain statements about it can not have truth values one way or the other. More recent developments in this direction, like &lt;a href=&quot;https://en.wikipedia.org/wiki/Constructive_analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bishop's constructive analysis&lt;/a&gt;, usually go under the name of constructivism (to avoid Kantian associations), see also &lt;a href=&quot;https://en.wikipedia.org/wiki/Constructive_set_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Constructive set theory&lt;/a&gt;. While a minority position among mathematicians constructivism proved to be a lasting presence.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-03T09:42:23.807" LastActivityDate="2018-02-03T09:42:23.807" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7035" PostTypeId="2" ParentId="7031" CreationDate="2018-02-03T14:18:06.103" Score="2" Body="&lt;p&gt;I recommend that you read some history of mathematics if you are interested in 19th century mathematics. A good one is Klein, Lectures on development of mathematics in 19th century, Chapter IV, section Theory of Invariants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eisenstein is Gotthold Eisenstein, a famous German mathematician, you may look in Wikipedia for his biography.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-02-03T14:18:06.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7036" PostTypeId="2" ParentId="4917" CreationDate="2018-02-04T03:03:56.393" Score="2" Body="&lt;p&gt;Sophus Lie's 1895 paper &quot;&lt;em&gt;&lt;a href=&quot;https://archive.org/stream/gesammabhan04lierich#page/n337/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung&lt;/a&gt;&lt;/em&gt; [&lt;a href=&quot;https://isidore.co/calibre#book_id=6541&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;General Theory of Partial Differential Equations of Arbitrary Order&lt;/a&gt;]&quot; begins with a historical summary in which he seems to think Monge was the first (p. 4 of the &lt;a href=&quot;https://isidore.co/calibre#book_id=6541&amp;amp;panel=book_details&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;…theory of characteristics [&lt;em&gt;Theorie der Charakteristiken&lt;/em&gt;] developed by Monge, Laplace, Ampére and Darboux…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;…the notion of characteristics [&lt;em&gt;Begriff Charakteristik&lt;/em&gt;] introduced by Monge…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-02-04T03:03:56.393" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7037" PostTypeId="1" AcceptedAnswerId="7039" CreationDate="2018-02-04T18:56:03.333" Score="3" ViewCount="1499" Body="&lt;p&gt;Being Italian, I have a biased view on my homeland's mathematical impact in the world, so I would like to get some impartial opinion on the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would measure the mathematical relevance in terms, for example, of:&#xA;-contemporary importance&#xA;-historical importance&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Contemporary importance is about the mathematical productivity of new papers, the quantity and quality of resourceful and respected mathematicians of a nation, and the international prestige of math universities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historical importance is almost the same but in the broader landscape of history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would also like some thoughts about perks and defects of Italian mathematics, again, historically and contemporarily.&lt;/p&gt;&#xA;" OwnerUserId="6996" LastEditorUserId="55" LastEditDate="2018-02-05T04:35:00.007" LastActivityDate="2018-02-06T03:43:49.890" Title="What is the international standing of Italian mathematics?" Tags="|mathematics|mathematicians|social-context|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7038" PostTypeId="2" ParentId="7028" CreationDate="2018-02-05T01:59:01.043" Score="0" Body="&lt;p&gt;A continuous function can be characterized as a function where the output can be determined to any finite accuracy using only finite information about the input. (By finite information, I mean accuracy. Stupid English language!)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether an infinity is &quot;potential&quot; or &quot;actual&quot; depends exactly on what you do with it. A real number is an infinite object; it's got infinitely many digits. Certain operations on real numbers are feasible, whereas others are not. For instance, it's clearly &lt;em&gt;not&lt;/em&gt; feasible to decide if a real number is equal to $0$, because that would require knowing every digit of the number. It &lt;em&gt;is&lt;/em&gt; feasible to multiply a real number by $2$, because this can be determined to accuracy $\epsilon$ by reading only finitely many digits of the input.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether an operation is feasible can be captured either by computability or continuity. Topology and computability theory define the notions of continuous function and computable function. If a function is discontinuous or uncomputable then you've got an actual infinity. Also, continuity and computability are fairly similar in practice.&lt;/p&gt;&#xA;" OwnerUserId="6999" LastActivityDate="2018-02-05T01:59:01.043" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7039" PostTypeId="2" ParentId="7037" CreationDate="2018-02-05T04:31:41.433" Score="3" Body="&lt;p&gt;I do not believe that the question can be answered here in its sweeping generality so I will only offer some remarks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;US News and World Report has a list of &lt;a href=&quot;https://www.usnews.com/education/best-global-universities/mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Best Global Universities for Mathematics&lt;/a&gt;, although one can expect bias in their listings. Still, in the top 10 we have US, UK, France and Switzerland. The highest Italian one is #32, University of Milan, with Sapienza University of Rome close second, #36. The highest German one is University of Bonn, #26. Population of Italy is not far behind UK and France, but it is 20 million less than Germany's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for &quot;prestige&quot; of mathematical journals, one can get some idea using SJR index, a &quot;prestige&quot; sensitive version of the impact factor. Unlike the impact factor, SJR assigns different values to citations depending on the &quot;prestige&quot; of the journals they came from, citations from the more &quot;prestigious&quot; journals are valued higher. &lt;a href=&quot;http://www.scimagojr.com/journalrank.php?category=2601&quot; rel=&quot;nofollow noreferrer&quot;&gt;The top ten &quot;prestigious&quot; mathematics journals&lt;/a&gt; have SJR indices between 4.063 and 9.989 and come from the US (3), Germany (3), Sweden, UK, Russia and Switzerland (1 each). The highest ranking Italian journal is Rivista di Matematica, #33, with SJR index 1.708. Only two others have it above 1, Annali della Scuola normale superiore di Pisa (1.456, #41) and Atti della Accademia Nazionale dei Lincei (1.083, #68). These are the only three in the top 100. I am not sure how to measure output of mathematical papers from a country, but perhaps one can extract something from MathSciNet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia gives a &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Italian_mathematicians&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of Italian mathematicians by century&lt;/a&gt;, but&#xA;I am not quite sure how to measure &quot;prestige&quot;, &quot;importance&quot;, etc., in centuries past either. Here is a shorter list of &lt;a href=&quot;https://www.thefamouspeople.com/italian-mathematicians.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;famous&quot; Italian mathematicians&lt;/a&gt;. If one insists on subjective impressions, I'd say that Italy is not quite at the epicenter of world's mathematics, where Cardano and Ferrari, or Cavalieri and Toricelli, or Peano, Ricci and Levi-Civita were. Peano's axiomatization of arithmetic, Volterra's integral equations and Ricci-Curbastro's tensor calculus and the Ricci flow, date to the turn of the 20th century. Fewer names come to mind after 1950s, one is Calabi of the Calabi-Yau manifolds prominent in applications to string theory. &lt;a href=&quot;http://www.springer.com/cda/content/document/cda_downloaddocument/9783642136054-c1.pdf?SGWID=0-0-45-1000857-p174011171&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guerraggio writes&lt;/a&gt; that &quot;&lt;em&gt;Renato Caccioppoli is probably the most “storied” Italian mathematician, the one who has been most been talked and written about, even beyond the circle of specialists&lt;/em&gt;&quot;, but I am not sure that many US mathematicians would recognize the name. His student De Giorgi, who solved Hilbert's 19th problem, might be better known. Bombieri, who collaborated with him on the Bernstein's problem for minimal surfaces, is probably better known for winning the 1974 Fields medal for applications of the large sieve in number theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps the most recognizable (recent) Italian names, other than the aforementioned, come from algebraic geometry: Castelnuovo, Fano, Enriques, del Pezzo, Veronese, Segre, Severi. As for &quot;defects&quot;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Italian school of algebraic geometry&lt;/a&gt; again comes to mind, unfortunately. There was a progressive decline in rigor standards from Castelnuovo to Enriques to Severi, right at the time when the general trend went the other way. Eventually, the results became too uncertain to be usable and had to be reworked in Zariski-Weil framework, see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-642-18908-1_9&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Italian School of Algebraic Geometry and Abel’s Legacy by Brigaglia et al.&lt;/a&gt; Here is Mumford:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The Italian school, and notably Severi, Todd, Eger, and B. Segre, developed a general theory of Chern classes in the algebraic case... In the period 1935 to 1950, Severi published many papers on series of equivalence and its generalizations to higher dimensions. It is hard to untangle everywhere what he conjectured and what he proved and, unfortunately,&#xA;  some of his conclusions are incorrect.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of other relevant resources I'll mention &lt;a href=&quot;https://books.google.com/books?id=S5fpiWCghYoC&amp;amp;dq=history+of+italian+mathematics&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guerraggio-Nastasi's book Italian Mathematics Between the Two World Wars&lt;/a&gt; and proceedings of 2014 &lt;a href=&quot;https://www.ehu.eus/en/web/fjim2014/42&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Mathematics in Italy and Spain: State of the Art and Future Developments&lt;/a&gt;, abstracts are available online.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-06T03:43:49.890" LastActivityDate="2018-02-06T03:43:49.890" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7040" PostTypeId="2" ParentId="7028" CreationDate="2018-02-05T07:05:20.277" Score="0" Body="&lt;p&gt;This is not an answer but too long for a comment. It shows the direction where the answer should be looked for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A collection or class $C$ is potentially infinite if and only if for every subset $A$ there is a subset $B$ such that $A$ is a &lt;em&gt;proper&lt;/em&gt; subset of $B$. Otherwise $C$ is finite or actually infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$C$ is potentially infinite $\Leftrightarrow \forall A\subseteq C$ $\exists B \subseteq C:A\subset B$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: The complete collection or set of all natural numbers does not exist in potential infinity. The complete real interval $[0, 1]$ does not exist in potential infinity (for instance, because the complete collection of unit fractions does not exist).&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2018-02-06T03:40:59.537" LastActivityDate="2018-02-06T03:40:59.537" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7041" PostTypeId="2" ParentId="7037" CreationDate="2018-02-05T11:56:00.257" Score="0" Body="&lt;p&gt;The very top of the mathematics profession is shown by the &lt;a href=&quot;http://stats.areppim.com/listes/list_fieldsxmedal.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fields Medal&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Italian mathematician Enrico Bombieri received the Fields Medal in 1974.  So there is an Italian in that august list.  (He is the only one.)  Most countries have no recipients of the Fields Medal at all, so this is an argument that Italy has high international standing in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-02-05T11:56:00.257" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7042" PostTypeId="1" CreationDate="2018-02-04T07:47:41.950" Score="12" ViewCount="1738" Body="&lt;p&gt;In the book of &lt;em&gt;Experiment in Modern Physics&lt;/em&gt;, by Melissinos, at page 2, it is given that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;2. The Millikan Oil Drop Experiment&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In 1909, R. Millikan reported a reliable method for measuring ionic charges. It consists of observing the motion of small oil droplets under the influence of an electric field. Usually the drops acquire a few electron charges and thus conventional fields impart to them velocities that permit isolation of a drop and continuous observation for a considerable length of time; further, the mass of the oil droplet remains almost constant (there is very slight evaporation) during these long observation times.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But how did they know this in that time?&lt;/p&gt;&#xA;" OwnerUserId="6993" OwnerDisplayName="onurcanbektas" LastActivityDate="2018-08-07T17:23:30.970" Title="How did Milikan know that oil drops would acquire only few electron charges?" Tags="|electromagnetism|experimental-physics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7043" PostTypeId="1" AcceptedAnswerId="7048" CreationDate="2018-02-06T03:42:12.833" Score="13" ViewCount="566" Body="&lt;p&gt;The &lt;a href=&quot;http://www.oed.com/view/Entry/91163&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;OED&lt;/em&gt;'s entry for &quot;iff&quot;&lt;/a&gt; lists this as the earliest usage:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1955   J. L. Kelley &lt;em&gt;Gen. Topol.&lt;/em&gt; vii. 232:   &quot;F is equicontinuous at &lt;em&gt;x&lt;/em&gt; iff there is a neighborhood of &lt;em&gt;x&lt;/em&gt; whose image under every member of &lt;em&gt;F&lt;/em&gt; is small.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there an earlier usage?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-02-06T13:13:36.063" Title="Who coined the term &quot;iff&quot; for &quot;if and only if&quot;?" Tags="|mathematics|terminology|mathematical-logic|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7044" PostTypeId="2" ParentId="7042" CreationDate="2018-02-06T06:43:31.683" Score="3" Body="&lt;p&gt;One would think that in the second Melissinos and Napolitano would update on how &amp;quot;reliable&amp;quot; Millikan's method was. Even &lt;a href=&quot;https://en.wikipedia.org/wiki/Oil_drop_experiment#Fraud_allegations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia mentions&lt;/a&gt; that according to his own notes he &amp;quot;edited&amp;quot; the data he had to reduce the statistical error, see also &lt;a href=&quot;https://hsm.stackexchange.com/questions/2756/is-millikans-famous-oil-drop-experiment-a-fraud&quot;&gt;Is Millikan&amp;#39;s famous oil drop experiment a fraud?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In any case, the idea of electrons and ions was in circulation since 1890-s, when &lt;a href=&quot;https://en.wikipedia.org/wiki/Lorentz_ether_theory#Basic_concept&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorentz's electron theory&lt;/a&gt; explained magneto-optics, the  behavior of conductors and non-conductors, etc., using them. Thomson directly observed them in cathode (then Lenard) rays in 1897. Here is from &lt;a href=&quot;https://www.nobelprize.org/nobel_prizes/physics/laureates/1902/lorentz-lecture.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorentz's Nobel lecture (1902)&lt;/a&gt; devoted to the electron theory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The theory of which I am going to give an account represents the physical world as consisting of three separate things, composed of three types of building material: first ordinary tangible or ponderable matter, second electrons, and third ether... The covibrating particles must, we concluded, be electrically charged; so we can conveniently call them &amp;quot;electrons&amp;quot;, the name that was introduced later by Johnstone Stoney...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;At the same time, a start was made on a general theory which ascribed all electromagnetic processes taking place in ponderable substances to electrons. In this theory an electrical charge is conceived as being a surplus of positive or negative electrons, but a current in a metallic wire is considered to be a genuine progression of these particles, to which is ascribed a certain mobility in conductors, whereas in non-conductors they are bound to certain equilibrium positions, about which, as has already been said, they can vibrate.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If we combine the results to which Zeeman's experiments lead with those which can be deduced from the colour dispersion of gases, on the hypothesis that it is the same type of electrons which is under consideration in both cases, we come to the conclusion that the charge of an electron is of the same order of size as the charge of an electrolytic ion. The mass, however, is much smaller... the principal thing is that, as we have remarked before, the electron is very small compared with the atom. The latter is a composite structure, which can contain many electrons, some mobile, some fixed.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Do not think that Einstein's 1905 theory immediately abolished ether, in 1900-s even Einstein himself occasionally used the ether talk. In any case, one could simply replace &amp;quot;ether&amp;quot; with &amp;quot;electromagnetic field&amp;quot; in most places. Millikan’s and Fletcher's apparatus had a parallel pair of horizontal metal plates separated by an insulator with a uniform electric field created in between. A fine mist of oil droplets was sprayed into a chamber above the plates. Oil drops were charged either through friction with the spraying nozzle or by ionising radiation from an X-ray tube.&lt;/p&gt;&#xA;&lt;p&gt;According to Lorentz's theory, the charging was the result of a few extra electrons sticking to the droplet (too many would be repelled by the excess negative charge). By  changing the voltage across the plates Millikan could make the droplets rise, fall or stand still. From the latter (he claimed) he determined the charges to be small integer multiples of &lt;span class=&quot;math-container&quot;&gt;$1.5924(17)×10^{−19}\,{\rm C}$&lt;/span&gt;, about 0.6% below the current value. Feynman describes the comical aftermath when subsequent experimenters gradually increased the Millikan's value to &lt;span class=&quot;math-container&quot;&gt;$1.602176487(40)×10^{−19}\,{\rm C}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Why didn't they discover the new number was higher right away? It's a thing that scientists are ashamed of — this history — because it's apparent that people did things like this: When they got a number that was too high above Millikan's, they thought something must be wrong — and they would look for and find a reason why something might be wrong. When they got a number close to Millikan's value they didn't look so hard. And so they eliminated the numbers that were too far off, and did other things like that...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-06T06:43:31.683" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7045" PostTypeId="5" CreationDate="2018-02-06T10:02:39.970" Score="0" Body="&lt;p&gt;If the concerned question is about one of the works of Albert Einstein, this tag should be used along with &lt;a href=&quot;/questions/tagged/physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;physics&lt;/a&gt; and the tag of the specific area of that work (&lt;a href=&quot;/questions/tagged/relativity-theory&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;relativity-theory&amp;#39;&quot; rel=&quot;tag&quot;&gt;relativity-theory&lt;/a&gt; etc.).&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-19T00:37:40.823" LastActivityDate="2022-03-19T00:37:40.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7046" PostTypeId="4" CreationDate="2018-02-06T10:02:39.970" Score="0" Body="Albert Einstein (1879–1955) was one of the most important physicists of all time. His work touched many major areas of physics, although he is mostly known as the creator of the theory of relativity and the photoelectric effect. Use this tag for questions about Einstein's scientific life and his works." OwnerUserId="5989" LastEditorUserId="10106" LastEditDate="2022-03-18T07:33:10.393" LastActivityDate="2022-03-18T07:33:10.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7047" PostTypeId="2" ParentId="7042" CreationDate="2018-02-06T13:05:12.493" Score="4" Body="&lt;p&gt;I think the way Millikan deduced the number of electrons was, aside from the current state of knowledge as Conifold relates, by taking lots and lots of measurements (certainly that's what I did when repeating the experiment in undergrad).  When you plot the calculated charge on each droplet, you see the points on the graph are not a continuum but rather they show up at quantized separations.  Since the uncertainty in the electron count for a droplet containing, say, 100 electrons is likely to be several electrons' worth, concentrate on the points with the fewest quantized charge units. [edit: to clarify, I mean the error bars on such drops are several electron charge units wide] From that you can make a pretty good guess there's no &quot;fractional&quot; charge that would occur if your estimate of the electron charge were twice the truth.&lt;br&gt;&#xA;So, while the number of electrons in a given droplet may follow some statistical distribution, making use of the least ambiguous subset of all captured droplets leads to a more reliable answer.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-02-06T13:05:12.493" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7048" PostTypeId="2" ParentId="7043" CreationDate="2018-02-06T13:13:36.063" Score="10" Body="&lt;p&gt;As Francois Ziegler notes, Kelley attributes it to Halmos.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the past (before Halmos), definitions might be given in the form&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A group is called abelian if $xy=yx$ for all $x,y$.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...and every mathematics student would need to be told &quot;since this is a definition, &lt;strong&gt;if&lt;/strong&gt; means &lt;strong&gt;if and only if&lt;/strong&gt;.&quot;  I was one of the students told this: in the 1960s many textbooks (and of course research papers) still used &quot;if&quot; in this way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then Halmos proposed the new word &quot;iff&quot; ... it was to be used in definitions, where traditionally &quot;if&quot; appeared meaning &quot;if and only if&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[In &lt;em&gt;Measure Theory&lt;/em&gt;, Halmos still used &quot;if&quot; in definitions, so I do not know when (or even if) he used this in print.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kelley adopted Halmos's usage whole-heartedly.  But if you look carefully, you will see that &quot;iff&quot; appears only in definitions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, the word came to be used for &quot;if and only if&quot; sometimes also in other contexts.  And today every mathematics student needs to be told &quot;iff&quot; means &quot;if and only if&quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-02-06T13:13:36.063" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7051" PostTypeId="1" AcceptedAnswerId="7053" CreationDate="2018-02-07T20:55:17.980" Score="4" ViewCount="460" Body="&lt;p&gt;In some lecture notes I found online it was claimed that Maxwell discovered the Maxwell-Boltzmann distribution while in the course of solving a (then unsolved) examination problem given by Stokes. The reference is to &lt;em&gt;Experiment, Theory, Practice&lt;/em&gt; by Kapitsa. I was able to find a Russian copy. Indeed the anecdote is there:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Приведу вам еще четвертый пример. Происходило это в Кембридже, во второй половине прошлого века. Теоретическую физику тогда преподавал Стоке. К нему пришел сдавать аспирантский экзамен один молодой человек. Аспирантский экзамен в те времена был довольно трудный, потому что аспирантур тогда было очень мало — всего две-три, и состязание за право попасть в аспирантуру былр очень трудным. Стоке давал задачу, причем система была такая: давался десяток задач, и студент сам выбирал те, которые он хотел решить. Ему давалось определенное число часов, и Стоке, не стесняясь, ставил часто неразрешимые задачи, чтобы посмотреть, знает ли студент, что эта задача неразрешима. Он ставил, например, такую задачу (то были домаксвелловские времена): найти распределение скоростей в газе. Тогда это распределение скоростей не было известно. Бернулли и все остальные считали, что скорости примерно равны. Молодой человек, к удивлению Стокса, решил эту задачу, и решил правильно. Вы догадываетесь, что этот молодой человек был не кто иной, как Максвелл. Таким образом, открытие закона распределения скоростей молекул в газе было сделано Максвеллом на экзамене.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately Kapitsa doesn't cite a reference for this anecdote, nor does he appear to give enough detail for independent verification. The exam he refers to may be the &lt;a href=&quot;https://en.wikipedia.org/wiki/Smith&amp;#39;s_Prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smith's Prize exam&lt;/a&gt;, on which (in a more well-known anecdote) Stokes also set the problem of proving &lt;a href=&quot;https://en.wikipedia.org/wiki/Kelvin%E2%80%93Stokes_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;the theorem that bears his name&lt;/a&gt; and Maxwell tied for first with Routh. This anecdote is also mentioned in the book, so I don't think it's likely that Kapitsa confused the two. (The problem about the velocity distribution of gases does not appear on the &lt;a href=&quot;http://www.clerkmaxwellfoundation.org/SmithsPrizeExam_Stokes.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1854 exam&lt;/a&gt; that contained Stokes's theorem.)&lt;/p&gt;&#xA;" OwnerUserId="7017" OwnerDisplayName="Brian Bi" LastEditorUserId="4251" LastEditDate="2020-06-04T17:10:07.940" LastActivityDate="2020-06-04T17:10:07.940" Title="Did Maxwell discover the distribution of velocities in a gas during an exam set by Stokes?" Tags="|physics|biographical-details|statistical-mechanics|maxwell|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7052" PostTypeId="2" ParentId="7051" CreationDate="2018-02-07T22:50:24.040" Score="3" Body="&lt;p&gt;Who am I to cast doubt on what Kapitsa wrote? But I thought Maxwell's first work on kinetic theory, including his first argument for the Maxwellian distribution, was presented in 1859 and had been stimulated by his reading of Clausius. He'd graduated from Cambridge in 1854. &lt;/p&gt;&#xA;" OwnerUserId="15941" OwnerDisplayName="Philip Wood" LastActivityDate="2018-02-07T22:50:24.040" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7053" PostTypeId="2" ParentId="7051" CreationDate="2018-02-08T22:00:37.093" Score="7" Body="&lt;p&gt;No, he did not. Stokes could not have offered a problem of the distribution of velocities of gas particles for the simple reason that he never thought of fluids as consisting of particles. He was a typical 19-th century continualist, as a look at &lt;a href=&quot;https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet#Fluid_dynamics&quot; rel=&quot;nofollow noreferrer&quot;&gt;his works&lt;/a&gt; shows. After Bernoulli's derivation of the ideal gas law in 1738 kinetic theory was moribund until late 1850-s, it could not have been put on an entrance exam in 1854, especially by Stokes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Excerpts from the early works on statistical physics, and commentary, are collected in &lt;a href=&quot;http://philsci-archive.pitt.edu/2691/1/UffinkFinal.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Uffink's Compendium of the foundations of classical statistical physics&lt;/a&gt;, where we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Despite this initial success, no further results were obtained in kinetic&#xA;  gas theory during the next century. By contrast, the view that modeled a gas as a continuum proved much more fertile, since it allowed the use of powerful tools of calculus... Nevertheless, the kinetic view was revived in the 1850s, in works by Krönig and Clausius.  The main stimulus for this revival was the Joule-Mayer principle of the equivalence of heat and work, which led to the First Law of thermodynamics, and made it seem more plausible that heat itself was just a form of motion of gas particles.  (A point well-captured in the title of Clausius’ 1857 paper: “The kind of motion we call heat”, subsequently adopted by Stephen Brush (1976) for his work on the history of this period.)&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;It was Maxwell’s paper of 1860 that really marks the re-birth of kinetic theory.  Maxwell realized that if a gas consists of a great number N of moving particles, their velocities will suffer incessant change due to mutual collisions, and that a gas in a stationary state should therefore consist of a&#xA;  mixture of slower and faster particles. More importantly, for Maxwell this was not just an annoying complication to be replaced by simplifying assumptions, but the very feature that deserved further study.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Krönig's paper came out in 1856, &lt;a href=&quot;https://isidore.co/calibre/get/PDF/Brush,%20Stephen%20G_-The%20Kind%20of%20Motion%20We%20Call%20Heat_%20A%20History%20of%20the%20Kinetic%20Theory%20of%20Gases%20in%20the%2019th%20Century_4438.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brush's book&lt;/a&gt; is available online. On the mathematical side the origin of Maxwell's insight is somewhat speculative, but it seems that he mimicked Herschel's derivation of the error curve. According to Brush, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;While Maxwell was indirectly influenced by earlier writers on statistics such as Laplace, Poisson, Cournot, and Quetelet, the most immediate stimulus for Maxwell's first derivation of his velocity-distribution law was a review of Quetelet's works on probability by Sir John Herschel in the Edinburgh Review of July 1850... Aside from the striking similarity to Maxwell's own derivation (see below) there is only a letter from Maxwell to Lewis Campbell, which is undated but thought to be written around June 1850 by Campbell and Garnett, who published it in their Life of Maxwell. According to C. W. F. Everitt, who discusses this question in his forthcoming biography of Maxwell, the letter was&#xA;  probably written by Maxwell just after he read Herschel's review,&#xA;  since it repeats some of Herschel's remarks on probability theory in&#xA;  only slightly different words.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-09T19:15:58.450" LastActivityDate="2018-02-09T19:15:58.450" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7054" PostTypeId="1" AcceptedAnswerId="7059" CreationDate="2018-02-09T02:33:01.717" Score="1" ViewCount="156" Body="&lt;p&gt;I am playing with Ngram viewer and I plugged in several subatomic particles and saw this graph (see attached). The term &quot;proton&quot; was used starting the 1920 and neutron was discovered in 1932, which is consistent with the starting points in the graph. But neutron has a peak around the 1960s. Did anything significant happened that year about neutron? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JhWxx.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JhWxx.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7019" LastActivityDate="2018-02-14T04:17:58.590" Title="Did anything significant happened in 1960s that involve neutron?" Tags="|nuclear-physics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7055" PostTypeId="1" AcceptedAnswerId="7056" CreationDate="2018-02-10T05:07:28.450" Score="6" ViewCount="696" Body="&lt;p&gt;There's a well known treatise by Apollonius on conic sections, but these objects are absent in Euclid's Elements. Why? If I were to guess, I'd say that conic sections cannot be constructed using a compass and a straightedge, but I cannot be sure. &lt;/p&gt;&#xA;" OwnerUserId="6045" LastActivityDate="2018-02-13T17:05:32.143" Title="Why didn't Euclid's Elements treat conic sections?" Tags="|mathematics|euclidean-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7056" PostTypeId="2" ParentId="7055" CreationDate="2018-02-10T06:35:49.003" Score="11" Body="&lt;p&gt;While &lt;em&gt;Elements&lt;/em&gt; contains no reference to conic sections it does define &lt;em&gt;angled cones&lt;/em&gt;, given as &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookXI/defXI18.html&quot; rel=&quot;noreferrer&quot;&gt;definition 18 of Book XI&lt;/a&gt;, and it examines some of their properties in Book XII.&lt;/p&gt;&#xA;&lt;p&gt;History identifies Menaechmus as inventor of conic sections (350-360BC), obtained by considering these angled cones intersecting a fixed plane.&lt;/p&gt;&#xA;&lt;p&gt;In addition to &lt;em&gt;Elements&lt;/em&gt;, Euclid also wrote four books on conic sections, now lost. Indeed, Apollonius makes reference to Euclid's lost work in &lt;em&gt;Conics&lt;/em&gt;.  Quoting from &lt;a href=&quot;http://sites.math.rutgers.edu/%7Echerlin/History/Papers1999/schmarge.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Conic Sections in Ancient Greece&lt;/em&gt;&lt;/a&gt;, by Ken Schmarge of Rutgers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We next come to the (again, lost) works of Aristaeus `the elder' and of the celebrated Euclid on conic sections. Since we do not have the original works by these two men on conic sections, our knowledge of them is derived from the comments of Pappus, whose writings are discussed in Heath, using a translation by Hultsch:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The four books of Euclid's conics were completed by Apollonius, who added four more and produced eight books of conics. Aristaeus, who wrote the still extant five books of solid loci connected with the conics, called one of the conic sections the section of an acute-angled cone, another the section of a right-angled cone and the third the section of an obtuse-angled cone.... Apollonius says in his third book that the `locus with respect to three or four lines' had not been completely investigated by Euclid, and in fact neither Apollonius himself nor any one else could have added in the least to what was written by Euclid with the help of those properties of conics only which had been proved up to Euclid's time; Apollonius himself is evidence for this fact when he says that the theory of that locus could not be completed without the propositions which he had been obliged to work out for himself. Now Euclid-regarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system, being moreover in no wise contentious and, though exact, yet no braggart like the other-wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations ... (Heath, 1961, pp. xxi-xxii)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The introduction in Schmarge's paper sums things up pretty well :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The knowledge of conic sections can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C.; it is reported that he used them in his two solutions to the problem of &amp;quot;doubling the cube&amp;quot;. Following the work of Menaechmus, these curves were investigated by Aristaeus and of Euclid. The next major contribution to the growth of conic section theory was made by the great Archimedes. Though he obtained many theorems concerning the conics, it does not appear that he published any work devoted solely to them. Apollonius, on the other hand, is known as the &amp;quot;Great Geometer&amp;quot; on the basis of his text Conic Sections, an eight-&amp;quot;book&amp;quot; (or in modern terms, &amp;quot;chapter&amp;quot;) series on the subject. The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-13T17:05:32.143" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7057" PostTypeId="2" ParentId="3770" CreationDate="2018-02-10T12:35:27.747" Score="2" Body="&lt;p&gt;To me as a German and after reading all your posts and uncertainty I would explain it like this and I base this on the truth of what you have written:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On a meeting in a speech Weber tried to remember people about what somebody else said. It can be very well that Weber just &lt;strong&gt;added&lt;/strong&gt; the word &quot;lieber&quot; to &quot;Gott&quot;.&#xA;Remembering somebody on something, with an ironic vain (which a lot of mathematicians have), is somehow like to talking to a child. Here I would like to mention the citation of Walter Felscher in the first entry by Colin McLarty. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If it is the case that Kronecker said the sentence &lt;strong&gt;without&lt;/strong&gt; the above word, one, as a German native, cannot deduce with so much certainty which part of the sentence should be stressed. So then one could not deduce the intentions of Kronecker, but rather the intentions of Weber as to see in the next paragraph.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let us now analyze with an native linguistic feeling the above sentence:&#xA;&quot;Die ganzen Zahlen hat der lieber Gott gemacht, alles andere ist Menschenswerk.&quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;-My claim is that the stress lies on the second half &quot;the rest is done by human work&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Proof: Suppose the contrary. The stress was on the first part. The grammatical structure of the sentence is easy and consists of object, subject, predicate; in this ordering. The object, the &quot;whole&quot; numbers are not new to the listener; they were common knowledge; they are not a new invention. So the stress does not lie on the &quot;whole&quot; numbers. They are common knowledge, so if the stress was there and he meant something else he would had to redefine the term &quot;whole&quot; numbers by specifying it in a subordinate clause.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also with the aid of native language feeling if I wanted to stress the object I would put it to the end of the sentence. I would rather say: &quot;Der [liebe] Gott hat die ganzen Zahlen [gemacht]&quot;. But I would even then use the word &quot;erschaffen&quot; which is like &quot;create&quot; rather than &quot;made=[gemacht]&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we see that the information to be distributed if the stress lied on the first half of Kronecker's sentence would then be the predicate and subject. But the predicate is simple. It is only &quot;has made&quot;. It does not contain any information. I would use as said before the word &quot;create&quot; if the information shall be put in the predicate. &#xA;Hence, what rests is the subject &quot;[lieber] Gott&quot; - &quot;lovely god&quot;. But natives in German would then stress the subject by pronouncing it more so to make clear the information to be sent. But this sounds ridiculous if I say :&#xA;&quot;Die ganzen Zahlen hat der liieeeebe Gott gemacht&quot; where this is to be translated as &quot; the whole numbers were made by the looooveely god&quot;. &#xA;It does sound ridiculous because I would never talk about god seriously with the adjective &quot;liebe(r) - lovely&quot;. If I want to stress somebody in the sentence I would not try to harm the subject or make it childish by using the word &quot;liebe(r)&quot; to the subject. Here I again want to make reference to the citation of McLarty about somebody using the word &quot;liebe(r)&quot;. &#xA;Another argument, I would even change the ordering of the sentence as before so to put the subject in front and then say what he/she did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we see that the stress really lies on the second half of the sentence. -End of proof.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In the commentaries somebody said that it is known Kronecker to have approved/accepted the dictum by Gauss which contains the information that numbers are product of human mind. If this is so then it even becomes more clear that Kronecker really wanted to say that the &quot;rest is done by human&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The sentence has the structure: even ..... is true, we still have .....&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In German one puts the stress/ or main information normally at the end of the sentence. (The only exception I can think of is the subject which one puts in front if one wants to stress it - ** the second half of the sentence contains no subject because a general law or fact is stated. &quot;Something is done by...&quot;&#xA;The subject here has become an object &quot;by ...&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The end of the sentence is that 'the rest is done by human kind'.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As result I would say:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If Kronecker really said it like this then I see it in the following ironic manner: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My dear friends, remember shortly that we can be lucky that lovely god has given as at least some numbers to be existent, but the real thing is that the rest is done by human work! &#xA;(stress on the second part)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I don't believe the sentence to be about God.&#xA; It only uses an existence property related to God as an assumption.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The term &quot;ganze Zahlen&quot; (whole numbers):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As said before the stress might not lie in this part of the sentence. The &quot;whole numbers&quot; are to be seen as common knowledge. But common knowledge for whole numbers back then was integer numbers I guess.&#xA;In old texts people use the words &quot;ganze Zahlen&quot; as &quot;integer numbers&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, &quot;ganz&quot; can also be meant in German in the sense of &quot;vollkommen&quot; which is like &quot;perfect&quot; or &quot;not extendible&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I personally do not believe he meant only natural numbers in this sentence, but well, I actually don't know much about Kronecker. This is just a linguistic attempt to the meaning of sentences.&lt;/p&gt;&#xA;" OwnerUserId="7021" LastEditorUserId="2277" LastEditDate="2020-04-19T07:49:23.207" LastActivityDate="2020-04-19T07:49:23.207" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7059" PostTypeId="2" ParentId="7054" CreationDate="2018-02-12T02:15:09.673" Score="4" Body="&lt;p&gt;Google Ngrams are based on searching mentions in book hits from the corresponding time period. By looking at the titles and contexts where the word is mentioned one can usually figure out what the dominant reason for its occurence is. Here is the &lt;a href=&quot;https://books.google.com/ngrams/graph?content=neutron&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Cneutron%3B%2Cc0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ngram for neutron&lt;/a&gt; and here are the &lt;a href=&quot;https://www.google.com/search?q=%22neutron%22&amp;amp;lr=lang_en&amp;amp;tbs=lr:lang_1en,cdr:1,cd_min:1/1/1956,cd_max:12/31/1962&amp;amp;tbm=bks&amp;amp;ei=__KAWqHVEZjijwS-5YugBA&amp;amp;start=10&amp;amp;sa=N&amp;amp;biw=1173&amp;amp;bih=491&amp;amp;dpr=1.36&quot; rel=&quot;nofollow noreferrer&quot;&gt;book hits for the 6 year range&lt;/a&gt; around the spikes in 1957 and 1961. Radiochemistry, chemistry of radioactive materials, and other nuclear reactor related contexts seem to dominate:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;fissionable nuclides and for fast neutron fission&quot;, &quot;Reactors for University Research&quot;, &quot;Radiochemistry of Iron&quot;, &quot;A Glossary of Terms in Nuclear Science and Technology&quot;, &quot;Radiochemistry of the Transcurium Elements&quot;, &quot;Radiochemistry of Silicon&quot;, &quot;Radiochemistry of Osmium&quot;, &quot;condensate from the dissolution of neutron-activated uranium metal&quot;, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it seems the spikes are due to the rise of nuclear power in late 1950-s, which created the need for technological literature on how different materials behave in radioactive conditions, measurement standards, etc. All of that mentioned fission chain reactions and their &lt;a href=&quot;https://en.wikipedia.org/wiki/Nuclear_chain_reaction#Effective_neutron_multiplication_factor&quot; rel=&quot;nofollow noreferrer&quot;&gt;neutron multiplication factors&lt;/a&gt;, among other things. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1955 the &quot;First Geneva Conference&quot; of the UN met to discuss nuclear power, in 1957 EURATOM and the International Atomic Energy Agency (IAEA) were launched. Calder Hall, the first commercial nuclear power plant, opened in UK in 1956, the Shippingport in Pennsylvania, the first commercial one in the US, opened in 1957. The U.S. Army had a nuclear power program since 1954, its nuclear power plant at Fort Belvoir, Virginia, opened in 1957 even before Shippingport. Army's experimental nuclear reactor at the National Reactor Testing Station in Idaho had a steam explosion and meltdown, which killed its three operators, in January 1961, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Nuclear_power#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Nuclear Power&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-14T04:17:58.590" LastActivityDate="2018-02-14T04:17:58.590" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7060" PostTypeId="1" AcceptedAnswerId="7061" CreationDate="2018-02-12T11:01:11.507" Score="2" ViewCount="150" Body="&lt;p&gt;The word algorithm comes from al-Khwārizmī. From Wikipedia, I can read that he did a great work on algebra, but I could not find any algorithm attributed to him. Has he actually made any algorithm?&lt;/p&gt;&#xA;" OwnerUserId="7029" LastActivityDate="2018-02-12T12:31:12.673" Title="What is the work of al-Khwārizmī for algorithms?" Tags="|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7061" PostTypeId="2" ParentId="7060" CreationDate="2018-02-12T12:31:12.673" Score="3" Body="&lt;p&gt;Perhaps his most sophisticated contribution was &lt;a href=&quot;https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi#Algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;solving quadratic equations&lt;/a&gt;. However, he didn't write the roots of an arbitrary quadratic as functions of its coefficients in the way we do today, because that would require conceding negative numbers exist and can be the arguments of functions. in other words, he classified several types of quadratic, each solved a different way (if roots were conceded). If that sounds odd, remember that centuries later Europe's doubts about negative numbers motivated calling the square roots of negative numbers imaginary, and the different types of quadratic distinguished in the above link were still considered separately. (The eventual motive for taking imaginary/negative numbers seriously was &lt;a href=&quot;https://en.wikipedia.org/wiki/Casus_irreducibilis&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-02-12T12:31:12.673" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7062" PostTypeId="2" ParentId="384" CreationDate="2018-02-13T21:37:46.240" Score="3" Body="&lt;p&gt;Although it's an old question, but perhaps this insight would be beneficial to those stumbling on this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Gauss wasn't the first to derive this formula per se.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Algebraically, you'd see &quot;how&quot; to derive the formula by working through Gauss' derivation. Purely by observation you'd see that you can pair up the extreme values to get the &lt;em&gt;same sum&lt;/em&gt; and just count/multiply them out and divide the result by two.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this formula was well known in ancient Greece too. Pythagoreans had a habit of &quot;visualizing&quot; numbers by arranging discs/stones in particular patterns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A triangular number was that which could be arranged in a triangle:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;*   *      *&#xA;   * *   *   * &#xA;        *  *  * &#xA;1   3      6&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;A square number was that which could be arranged in a square (hence our usage of $x$ &lt;em&gt;squared&lt;/em&gt;):&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt; *  * *   * * *&#xA;    * *   * * *&#xA;          * * *&#xA; 1   4      9&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Now, if you see carefully, $6 = 1 + 2 + 3$ . By extension $10 = 1 + 2 + 3 + 4$ and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, you can easily see the relation between a triangular number and a corresponding square number, by doubling the triangle &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;* + + + +&#xA;* * + + +  =&amp;gt; 2 x T(10) = S(4) + 4 = 4 x 5&#xA;* * * + +&#xA;* * * * +&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Doubling gives you a rectangle - you can get back the triangle simply by &lt;em&gt;observing&lt;/em&gt; that&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$2 \cdot T(n) = n(n+1) =&amp;gt; T(n) = \frac{n(n+1)}{2}$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Triangular number just happen to arrange themselves in a simple pattern of an arithmetic progression of consecutive numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Gauss did was think of this algebraically. We credit Gauss with this story, but there isn't a reason to. I'd say we look at the story from the POV of a &quot;smart kid&quot; who may or may not have known about the Greek history of numbers but was able to &quot;sit back and think&quot; vs. attacking the problem in a brute force fashion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Side note: The formula for sum of squares took some work and isn't really immediately apparent. I believe Archimedes derived it successfully.)&lt;/p&gt;&#xA;" OwnerUserId="7035" LastEditorUserId="7035" LastEditDate="2018-02-14T19:28:48.157" LastActivityDate="2018-02-14T19:28:48.157" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7063" PostTypeId="1" CreationDate="2018-02-14T09:22:00.223" Score="0" ViewCount="111" Body="&lt;p&gt;Who first introduced the concept of magnetic vector potential and why?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it introduced only for ease of mathematical calculation or it was done  keeping in mind its physical significance like connection with momentum and all?&lt;/p&gt;&#xA;" OwnerUserId="6277" OwnerDisplayName="user157588" LastEditorUserId="3643" LastEditDate="2018-02-14T18:12:55.370" LastActivityDate="2018-02-14T18:12:55.370" Title="Introduction of magnetic vector potential" Tags="|physics|electromagnetism|potential|" AnswerCount="1" CommentCount="1" ClosedDate="2018-03-01T23:09:43.143" ContentLicense="CC BY-SA 3.0" />
  <row Id="7064" PostTypeId="2" ParentId="7063" CreationDate="2018-02-14T10:00:43.077" Score="2" Body="&lt;p&gt;It was the German mathematician F. E. Neumann.&#xA;I refer you to &lt;a href=&quot;https://hsm.stackexchange.com/questions/5078/who-discovered-the-magnetic-vector-potential-veca&quot;&gt;this&lt;/a&gt; discussion on the History of Science and Mathematics Stack Exchange, where you will find your answers in detail.&lt;/p&gt;&#xA;" OwnerDisplayName="jfeis" LastActivityDate="2018-02-14T10:00:43.077" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7065" PostTypeId="2" ParentId="6939" CreationDate="2018-02-15T22:11:27.137" Score="0" Body="&lt;p&gt;It wasn't really discovered in a mathematical sense of &quot;exploration of higher dimensions&quot; but more as a result of everyday experiences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclidean &quot;plane&quot; is classic 2D Geometry - studying what can be done with lines/triangles and circles, the &lt;em&gt;primitive&lt;/em&gt; geometric elements, and deriving various proofs. This &lt;em&gt;is not&lt;/em&gt; clear from everyday experiences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You don't find perfect triangles in nature. They're a geometrical construct that uses the concept of a line (which itself is not &quot;well defined&quot; by modern standards in The Elements). So, the study of 2D geometry does require creative thought and application of deductive logic/proofs, constructions with straight-edge and compass (for the most part).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, we &lt;em&gt;see the world in 3 dimensions&lt;/em&gt;. To give &quot;perfect meaning&quot; to shapes, they used the 2D analogs and built what we call &quot;surfaces of revolution&quot; to &lt;em&gt;define&lt;/em&gt; some basic shapes they were familiar with (mostly owing to their experiences with pottery perhaps):&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cone&lt;/strong&gt;: A right triangle is rotated about one of it's 'right' sides&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Cylinder&lt;/strong&gt;: A rectangle rotated around it's central axis or just around it's sides (sort of equivalent)&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Sphere&lt;/strong&gt;: Circle rotated around it's diameter.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;In theory you could just rotate any primitive shape and get a corresponding 3D shape. We've lost Euclid's book on Conics so we don't know if they ever rotated those to get some interesting shapes. The cube was just the most primitive shape. But I'm sure they made some vases in those days so I'd conjecture they did something :)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They studied some &quot;other shapes&quot; like the platonic solids, the various &quot;hedrons&quot; (tetra-, dodeca-, icosa- ...) since they combined symmetrical shapes to get interesting 3D objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, they just analyzed the shapes and explored the corresponding 2D counterparts i.e., areas in 2D implied volumes in 3D. They concluded that volumes are proportional to the &quot;cube&quot; of the &quot;variable&quot; i.e., length of radius/side of a cone/cube, since areas are proportional to the &quot;squares&quot; in the 2D world. Finding exact formulae was a separate thing (e.g., for a sphere)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eudoxus had conjectured that the volume of a pyramid is $\frac{1}{3}$rd that of the enclosed &quot;cylinder&quot; i.e., a cube. Euclid 'perfected' this method of exhaustion as we call it, and published a clean proof in his books.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you read the elements you'll see, that 3D analysis was a natural extension owing to everyday experiences. They proved what they could and what made most sense &quot;by analyzing&quot; shapes or cutting them (e.g., you'd always get an ellipse if you cut a cylinder diagonally across it's height).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The advantage of this &quot;study&quot; was applications to architecture I believe but nothing beyond that. The greeks firmly believed in a world they could &quot;perceive&quot; and hence they spent time analyzing that, even in their mathematical endeavors. But I don't think they did it as an exploration of &lt;em&gt;higher dimension math&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-02-15T22:11:27.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7066" PostTypeId="1" CreationDate="2018-02-16T07:05:46.237" Score="2" ViewCount="1150" Body="&lt;p&gt;Sines and cosines are commonly introduced as ratios of sides of a triangle with its hypotenuse and attributed to ancient Indian scholars. However, I've never actually thought of the &lt;em&gt;reason for introducing&lt;/em&gt; sine/cosine to mathematical analysis. What exactly led to the need for sines and cosines? What were the Indian astronomers trying to compute that necessitated the existence of such trigonometric functions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest I've come is this &lt;a href=&quot;https://www.youtube.com/watch?v=6QOJ_KQQWfg&quot; rel=&quot;nofollow noreferrer&quot;&gt;lecture&lt;/a&gt; by an Indian professor who explains it as follows: &#xA;&lt;a href=&quot;https://i.stack.imgur.com/xoZ0c.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xoZ0c.png&quot; alt=&quot;Epicycle Model&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The derivation below has a few odd symbols, not in the diagram above, but it's easy to verify this by hand. The basic premise is that the orbit of planets isn't purely circular and thus a correction needs to be applied to compute its true position. This computation necessitates the need for sines and cosines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tbgxU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tbgxU.png&quot; alt=&quot;Derivation&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The calculation above &lt;em&gt;depends&lt;/em&gt; on the existence of $sin$ and $sin^{-1}$ functions &lt;em&gt;&quot;proving&quot;&lt;/em&gt; their need, so to speak.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the first time I've come across something like this. Are there any other references anywhere in existing math literature? Is this accurate?&lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-02-16T22:55:37.823" Title="What astronomical reason led to the creation of the trigonometric sine and cosine?" Tags="|mathematics|astronomy|geometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7067" PostTypeId="2" ParentId="7066" CreationDate="2018-02-16T14:12:33.637" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Sine and cosine were not the first trigonometric functions introduced. The ancients used chord instead $\mathrm{chd}\, x=2\sin(x/2)$. All other trigonometric functions were introduced much later, in the Middle Age.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The main reason for their introduction has nothing to do with planetary systems, it is much simpler. The reason is that one has to transfer from one coordinate system in the sky to another. There are three natural coordinate systems. The first is related to the observer on the earth (zenith distance and azimuth). The second&#xA;is polar (declination and right ascention), it is important because the stars are fixed in these coordinates. The third one is the ecliptic system (longitude and latitude). The motion of the Sun, Moon and planets is best described in ecliptic coordinates. To pass from one to another one has to do spherical trigonometry (to solve a spherical triangle). This is why trigonometric functions are needed.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The simplest type of problems solved in astronomy (ancient and modern) is predicting the&#xA;time and place of rising and setting of a star. Or duration of the day light for a given date at a given location. This requires a transfer from polar (for stars) or ecliptic (for Sun, Moon and planets) coordinates to the&#xA;coordinates of the observer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(In the ancient times ll these coordinates had somewhat different names, and their definition slightly varied, but the essence is the same).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Literature: Ptolemy, Almagest, or any later textbook on Astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy's book begins with trigonometry, then passes to rising and setting phenomena, and only after that describes the motions of the Sun Moon and planets. All later astronomy books followed this pattern.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difference between the ancient trigonometric function (the chord) and the modern ones (sine, cosine, tangent etc.) is a question of convenience in specific calculations. (Other trigonometric functions, now forgotten, were also used, for example, versine). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-02-16T18:32:19.400" LastActivityDate="2018-02-16T18:32:19.400" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7068" PostTypeId="2" ParentId="7051" CreationDate="2018-02-16T17:25:34.230" Score="0" Body="&lt;p&gt;Here is a &lt;a href=&quot;https://archive.org/stream/memoirscientific02stok#page/8/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;letter&lt;/a&gt; that Maxwell wrote to Stokes on May 30, 1859, in which he states that he read a paper by Clausius in the Philosophical Magazine of &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/14786445908642626&quot; rel=&quot;nofollow noreferrer&quot;&gt;1859&lt;/a&gt;. Of course, this may not have been the first time he raised this issue, for there was another paper by Clausius in &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/14786445708642360&quot; rel=&quot;nofollow noreferrer&quot;&gt;1857&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5376" LastActivityDate="2018-02-16T17:25:34.230" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7069" PostTypeId="2" ParentId="7066" CreationDate="2018-02-16T22:55:37.823" Score="4" Body="&lt;p&gt;This is a supplement to Alexandre Eremenko's answer. There is useful context to the question and answers in David Eugene Smith's &lt;em&gt;History of Mathematics&lt;/em&gt; (originally published 1923-25), especially in &lt;a href=&quot;https://archive.org/details/historyofmathema031897mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;volume 2&lt;/a&gt;. This contains a topical history of trigonometry (in ch.8, pp.600-632), both in terms of named mathematicians and of particular functions, such as sines and the chords that preceded them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With reference to the parts of the question that refer to motivations for the development of these topics, it is worth noting that ancient historical evidence often shows what was done or used, but without necessarily giving direct evidence of a motivation for its development. Examples of this can be seen in the cited chapter 8. The motivations that impelled the ancients towards their work and results can sometimes thus be matters of secondary inference, less certain than the historical evidence for the procedures and functions themselves. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the context, e.g. in the Almagest, as already noted by Alexandre Eremenko, can well indicate a likely motivation, e.g. in this case to account mathematically for rising and setting and related phenomena, a set of problems prior to any analysis of planetary orbits. Thus, the opening theorems in Book 1 of the Almagest (see &lt;a href=&quot;https://books.google.com/books/about/Ptolemy_s_Almagest.html?id=43XvAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;G J Toomer's translation from 1984&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Almagest&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;), show &quot;that the heavens move like a sphere ... that the earth, too, taken as a whole, is sensibly spherical ... in the middle of the heavens&quot;. Then Ptolemy showed that there are two kinds of 'primary motion' in the heavens, the daily motions parallel to the equator and the slower motions in the opposite direction but parallel to a relatively oblique plane.  Ptolemy then went on to discuss the &quot;size of chords&quot; along with a table of chords, and spherical trigonometry, as a means to solve problems arising from the spherical phenomena initially introduced. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-02-16T22:55:37.823" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7071" PostTypeId="1" CreationDate="2018-02-18T01:45:48.270" Score="3" ViewCount="2076" Body="&lt;p&gt;Recently, I came across &lt;a href=&quot;https://web.archive.org/web/20150915045955/http://mathforum.org/kb/message.jspa?messageID=6623661&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;this article&lt;/strong&gt;&lt;/a&gt; and wondered if there really is a definitive answer to the question of who invented the number line?&lt;/p&gt;&#xA;" OwnerUserId="83" LastEditorUserId="83" LastEditDate="2022-09-18T06:09:36.820" LastActivityDate="2022-09-23T09:34:14.520" Title="Who Invented The Number Line?" Tags="|mathematics|mathematicians|terminology|geometry|arithmetic|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7072" PostTypeId="2" ParentId="1816" CreationDate="2018-02-18T04:02:04.183" Score="2" Body="&lt;p&gt;Aristotle was considering the max speed of an object being pulled through a resistant media - eg: an ox pulling a cart, or a feather falling through air.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is obvious to any idiot that two 1 lb weights falling side by side hit the ground at the same time as 1 2lb weight formed by gluing the two together.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is far less obvious how long it will take an identical shape and size weighing twice as much while falling through a resistant media.  It will certainly be faster, but since resistance is NOT directly proportionate to speed, it will not be twice as fast.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, give Aristotle some credit, he was NOT taking about objects falling in a vacuum.&lt;/p&gt;&#xA;" OwnerUserId="7052" LastActivityDate="2018-02-18T04:02:04.183" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7073" PostTypeId="2" ParentId="7071" CreationDate="2018-02-18T06:46:33.840" Score="6" Body="&lt;p&gt;Not mentioned in the linked thread is Bombelli, who although not widely published until after cited “candidates” Stevin and Wallis, came before them per Bourbaki’s &lt;a href=&quot;https://books.google.com/books?id=kTFSfmsjDM0C&amp;amp;pg=PA411&quot; rel=&quot;nofollow noreferrer&quot;&gt;historical notes&lt;/a&gt; (my emphasis):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(...) up to the end of the Middle Ages [the] “ratios” of Euclid were customarily described as “numbers”, and the rules for calculating with integers were applied to them without any attempt to analyse the reasons for the success of these methods.&lt;/p&gt;&#xA;&lt;p&gt;Nevertheless we see &lt;strong&gt;R. Bombelli&lt;/strong&gt;, as early as the middle of the 16th century, expounding a point of view on this subject in his &lt;em&gt;Algebra&lt;/em&gt; [1572] (*), which is essentially correct (provided that the results of Book V of Euclid are assumed to be known); having realized that once the unit of length has been chosen there is a one-to-one correspondence between lengths and ratios of magnitudes, he defines the various algebraic operations &lt;em&gt;on lengths&lt;/em&gt; (assuming of course that the unit has been fixed) and, &lt;strong&gt;representing numbers by lengths&lt;/strong&gt;, obtains the geometrical definition of the field of real numbers (a point of view which is usually credited to Descartes) and thus gives his algebra a solid geometrical foundation (**).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;(*) We are concerned here with Book IV of his &lt;em&gt;Algebra&lt;/em&gt;, which remained unpublished until modern times; for our purposes it matters little whether or not the ideas of Bombelli on this subject were known to his contemporaries.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;(**) (...) Bombelli, in the same context, gives with perfect clarity the purely formal definition (such as one would find in modern algebra) not only of negative numbers, but also of complex numbers.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-18T06:46:33.840" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7074" PostTypeId="1" CreationDate="2018-02-19T09:21:31.747" Score="4" ViewCount="376" Body="&lt;p&gt;There is the well-known quote of Hilbert: &amp;quot;No one shall drive us from the paradise which Cantor has created for us.&amp;quot; [D. Hilbert: &amp;quot;Über das Unendliche&amp;quot;, Mathematische Annalen 95 (1925) p. 167] On the other hand Hilbert concludes his paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finally we will return to our original topic and draw the conclusions of all our investigations about the infinite. On balance the result is this: The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking – a remarkable harmony of being and thinking. &lt;sub&gt;[loc cit, p. 190]&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Further Hilbert devised &amp;quot;Hilbert's hotel&amp;quot;: An infinite hotel is completely filled with guests. Another guest arrives. He gets room no. 1 after every resident guest has moved on by one number from &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n+1$&lt;/span&gt;. Even infinitely many guests can be accommmodated when every resident guest doubles his room number.&lt;/p&gt;&#xA;&lt;p&gt;Did Hilbert set up this example in order to counter Cantor's list?&lt;/p&gt;&#xA;&lt;p&gt;Hilberts infite hotel is really infinite, unfinished, extendable but Cantor's list is not. Two different interpretations of one and the same infinity.&lt;/p&gt;&#xA;&lt;p&gt;Only that allows to conclude that the antidiagonal, as a new guest differing from all resident guests or entries of the list, cannot be inserted, for instance into the first position when every other entry moves on by one &amp;quot;room number&amp;quot;. Without Cantor's arbitrary constraint even all infinitely many antidiagonals that ever could be constructed could be accommodated. Cantor's theorem would go up in smoke.&lt;/p&gt;&#xA;&lt;p&gt;My question: Is there evidence that Hilbert intended this counter argument? Or did he not realize that it is a counter argument? But then what was his purpose?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="4251" LastEditDate="2020-07-12T02:03:27.327" LastActivityDate="2020-07-12T02:03:27.327" Title="Was Hilbert ambivalent about set theory?" Tags="|mathematics|mathematicians|set-theory|hilbert|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7075" PostTypeId="2" ParentId="7074" CreationDate="2018-02-19T20:50:34.170" Score="9" Body="&lt;p&gt;No, he was not, as one can see from the &lt;em&gt;full&lt;/em&gt; passage from &lt;a href=&quot;https://math.dartmouth.edu/%7Ematc/Readers/HowManyAngels/Philosophy/Philosophy.html&quot; rel=&quot;noreferrer&quot;&gt;Hilbert's lecture On the Infinite&lt;/a&gt; delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, and published in Mathematische Annalen vol. 95 (1926):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought — a remarkable harmony between being and thought. In contrast to the earlier efforts of Frege and Dedekind, we are convinced that certain intuitive concepts and insights are necessary conditions of scientific knowledge, and logic alone is not sufficient. Operating with the infinite can be made certain only by the finitary.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The role that remains for the infinite to play is solely that of an idea — if one means by an idea, in Kant's terminology, a concept of reason which transcends all experience and which completes the concrete as a totality — that of an idea which we may unhesitatingly trust within the framework erected by our theory.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;An idea which we may unhesitatingly trust&lt;/em&gt;&amp;quot; does not exactly sound ambivalent. Hilbert was a formalist, not a platonist, he did not believe that &amp;quot;actually infinite&amp;quot; actually exists, or that it needs to exist to be talked about. Being a Kantian regulative idea (&amp;quot;noumenon&amp;quot;), finitely axiomatized into a formal theory, was more than enough for mathematics, according to him. The only requirement is that the said theory be consistent, and the whole lecture was to promote the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_program&quot; rel=&quot;noreferrer&quot;&gt;Hilbert programme&lt;/a&gt; of proving consistency of infinitary theories by finitary means.&lt;/p&gt;&#xA;&lt;p&gt;It should be clear from this that Hilbert's hotel is neither real, nor meant to counter Cantor. What Cantor did or did not assume in the diagonal argument is moot since it is derivable in finitely many steps from set-theoretic axioms. Inspection of Hilbert's earlier 1924 lecture, where it is described (along with a similarly minded infinite dance party and a quip &amp;quot;&lt;em&gt;in a world with an infinite number of houses and occupants there will be no homeless&lt;/em&gt;&amp;quot;), confirms it. It was a smalltalk introduction of the difference between the finite and the infinite aimed at lay audience, it is not connected to Cantor, discussed or even mentioned in that lecture, or any other, after being introduced. Here is from&#xA;&lt;a href=&quot;https://arxiv.org/abs/1403.0059&quot; rel=&quot;noreferrer&quot;&gt;The True (?) Story of Hilbert’s Infinite Hotel by Kragh&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It was merely an example and one that he attached no particular importance to. Nor did other people at the time find it important. Had the hotel not been&#xA;resuscitated by Gamow more than two decades later it might well be unknown today. The only allusion to it before 1947 that I know of is from a textbook on calculus published in 1938 and written by  Otto Haupt, a mathematician&#xA;at Erlangen University.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-19T21:02:08.890" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7076" PostTypeId="1" CreationDate="2018-02-20T20:41:35.727" Score="3" ViewCount="1254" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.etymonline.com/word/choleric&quot; rel=&quot;nofollow noreferrer&quot;&gt;Etymonline&lt;/a&gt;, &quot;choleric&quot; derives from from Greek kholera &quot;a type of disease characterized by diarrhea, supposedly caused by bile&quot; with bile being &quot;khole&quot;, so called for its color, greenish-yellow. From 1580s comes the meaning of &quot;easily angered, hot-tempered&quot; and &quot;pertaining to cholera&quot; from 1834. But why was yellow bile associated with anger?&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastEditorUserId="55" LastEditDate="2018-02-20T21:52:53.793" LastActivityDate="2018-02-21T02:25:52.093" Title="Why was yellow bile/choler believed to cause anger?" Tags="|physiology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7077" PostTypeId="2" ParentId="7076" CreationDate="2018-02-20T22:24:04.197" Score="2" Body="&lt;p&gt;We do not really know. The association comes from ancient medicine of humors, black bile, yellow bile, phlegm, and blood, four supposed bodily fluids. Considering that &quot;phlegm&quot; does not exist and black and yellow bile are the same substance in modern view it is hard to say why semi-fictional &quot;fluids&quot; were associated with observed phenomena, perhaps they were postulated just for the purpose of explaining them. Indeed, it is not the fluid associations that were the rational kernel of humoric explanations, but rather the interrelations between various ailments that they were used to express.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The doctrine of four humors can be traced back to Ancient Egypt and Mesopotamia, but it took its classical form in Greece around 400 BC and was canonized by Hippocrates of Kos. Greeks associated the four humors to four elements (earth, water, air and fire). In Roman times Galen (129–201 AD) further related them to  four temperaments (melancholic, phlegmatic, sanguinic and choleric). The yellow bile was associated with fire, which explains its later relation to the &quot;fiery&quot; choleric temperament, and, in excess, to anger, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Humorism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's History of Humorism&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A Swedish physician Fahraeus speculated that the idea of humors came from observing blood clotting. His speculation would explain why the lightest &quot;fluid&quot; (yellow bile) was associated with the lightest element (fire), but it is not entirely convincing. Here is from &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/11843802&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hart's Descriptions of Blood and Blood Disorders before the Advent of Laboratory Studies&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Fahraeus  (1921),  a  Swedish  physician  who  devised  the erythrocyte   sedimentation   rate,   suggested   that   the   four humours were based upon the observation of blood clotting in  a  transparent  container.  Following  the  clotting  of  the homogeneous  red  fluid,  it  separates  into  a  dark  red/black clot  at  the  bottom  with  a  thin  layer  of  red  cells  above  it. Above this is a pale green or whitish layer and the contents are  surrounded  by  clear   yellow  serum.  Certainly,  clotted blood reveals the humours of blood and yellow bile, but the appearance  of  black  bile  and  phlegm  is  not  so  apparent.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-02-21T02:25:52.093" LastActivityDate="2018-02-21T02:25:52.093" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7078" PostTypeId="1" CreationDate="2018-02-21T00:23:58.247" Score="-1" ViewCount="155" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;[ &lt;a href=&quot;https://www.etymonline.com/word/sanguine&quot; rel=&quot;nofollow noreferrer&quot;&gt;Etymonline :&lt;/a&gt; ] &quot;blood-red,&quot; late 14c. (late 12c. as a surname),&lt;br&gt;&#xA;  from Old French sanguin (fem. sanguine),&lt;br&gt;&#xA;  from Latin sanguineus &quot;of blood,&quot; also &quot;bloody, bloodthirsty,&quot;&lt;br&gt;&#xA;  from sanguis (genitive sanguinis) &quot;blood&quot; (see &lt;a href=&quot;https://www.etymonline.com/word/sanguinary?ref=etymonline_crossreference&quot; rel=&quot;nofollow noreferrer&quot;&gt;sanguinary&lt;/a&gt;).&lt;br&gt;&#xA;  &lt;strong&gt;Meaning &quot;cheerful, hopeful, confident&quot; first attested c. 1500, because these qualities were thought in medieval physiology to spring from an excess of blood as one of the four humors. [bold mine]&lt;/strong&gt; Also in Middle English as a noun, &quot;type of red cloth&quot; (early 14c.).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know that the Ancient Greeks and Romans lacked the knowledge of modern medicine, but wouldn't they have been able to observe the following symptoms from patients with excess of blood who obviously couldn't have been &quot;cheerful, hopeful, confident&quot;, e.g. if they fainted?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://newsnetwork.mayoclinic.org/discussion/blood-disorder-causes-body-to-make-too-many-red-blood-cells/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Due to the excess red blood cells, though, some people may eventually develop skin itchiness and redness, headaches, dizziness, shortness of breath, and numbness, tingling, burning or weakness in their hands, feet, arms or legs. Because PV causes blood to thicken and slow, it can also increase the risk of developing blood clots, which can cause a stroke.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2018-02-21T09:28:11.097" Title="Why was excess of blood believed to cause cheerfulness, hopefulness?" Tags="|physiology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7079" PostTypeId="1" CreationDate="2018-02-21T09:16:46.133" Score="3" ViewCount="158" Body="&lt;p&gt;Toward the end of his paper on the infinite Hilbert sketched his proof of the continuum hypothesis. [D. Hilbert: &quot;Über das Unendliche&quot;, Mathematische Annalen 95 (1925)] Can experts on Hilbert give more details of his proof than are published there?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-02-21T09:16:46.133" Title="Details of Hilbert's proof of the continuum hypothesis" Tags="|set-theory|" AnswerCount="0" CommentCount="2" ClosedDate="2018-03-03T05:17:27.813" ContentLicense="CC BY-SA 3.0" />
  <row Id="7080" PostTypeId="2" ParentId="7078" CreationDate="2018-02-21T09:28:11.097" Score="1" Body="&lt;p&gt;See e.g.: James Irwin, &lt;a href=&quot;http://www.tandfonline.com/doi/abs/10.1080/00221309.1947.9918106?journalCode=vgen20&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galen on the Temperaments&lt;/a&gt; (1947).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You have to follow the link from &lt;a href=&quot;https://en.wikipedia.org/wiki/Hippocrates&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hippocrates&lt;/a&gt;’ ideas of humors, or bodily fluids, as a part of character, though the modifications of the later physician, [Galen], into a doctrine of “temperaments” as a semi-scientific study of the larger field of character study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Greek theory of four elements has been adopted by Hippocrates:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Now all animals including man, are composed of two things, different in power but working together in their use, namely, fire and water.&#xA;  [...] These elements have severally the following attributes. Fire has the&#xA;  hot and the dry, water the cold and the moist. Mutually, too, fire has&#xA;  the moist from water, for in fire there is moisture, and water has the&#xA;  dry from fire, for there is dryness in water also.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He also asserts that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the constituents of the body to be blood, phlegm, yellow bile, and black bile.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In conclusion, we can summarize the theory with two schema:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;hot and dry equals fire,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;hot and moist equals air,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;cold and dry equals earth,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;cold and moist equals water.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;hot and dry equals yellow bile,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;hot and moist equals blood,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;cold and dry equals black bile,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;cold and moist equals phlegm.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When these four humors are properly mingled the body is in a state of health; when there is a defect or an irregularity in the mixture, disease arises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through Aristotle, this theory is adopted by Galen:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Diseases affecting the the four humors:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;are “dyscrasia,” and contrast with the complete and harmonious mixture of the humors or “eucrasia.” But this state of ideal health rarely occurs, and the predominance of the various humors gives rise to the “temperaments” or sanguine, phlegmatic, bilious or melancholic, which though partaking of the nature of the disease, are not actually to be so called unless they produce a perversion of&#xA;  function.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The specific temperaments arise when one of the four humors is in predominance in the body and from which they are specifically derived. Thus, when examined&#xA;  in this light the temperaments were actually more than just humors or bodily&#xA;  fluids for they were actual conditions of the body arising out of the preponderance of their immediate specific humor as the sanguine temperament&#xA;  from the preponderance of blood, to illustrate the point.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The sanguine temperament is associated to the humor of blood and to the quality of hot and moist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See Galen, &lt;a href=&quot;https://books.google.it/books?id=hx99p9L6ep4C&amp;amp;pg=PA154&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Diseases and Symptoms&lt;/a&gt; (Ian Johnston editor, 2006), XII.2, page 154:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;because of the aforementioned humours, even if in form they are all moist, in capacity they are not moist, for black bile is dry and cold, and yellow bile is dry and hot, whereas moist and cold is of phlegm, whilst the blood is moist and hot.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Much more difficult is to assess the relationship between temperaments (the body) and passions (the soul). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See R.J. Hankinson (editor), &lt;a href=&quot;https://books.google.it/books?id=ygBactKUkAMC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cambridge Companion to Galen&lt;/a&gt;, Cambridge University Press (2009), &lt;strong&gt;Ch.7&lt;/strong&gt;, P.Donini, &lt;em&gt;Psychology&lt;/em&gt;, page 184-on:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;‘the movements of the passions are always consequent upon the dispositions of the body’ and that ‘they are the product of temperaments of the body which are similar to them’.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Galen adopted the thesis of the Platonic origin of the tripartition of the soul into the rational (&lt;em&gt;logistikon&lt;/em&gt;), the spirited (&lt;em&gt;thumoeides&lt;/em&gt;) and&#xA;desiderative (&lt;em&gt;epithumetikon&lt;/em&gt;), a thesis which he also attributes to Hippocrates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galen maintains that the Platonic location of the three parts in three distinct organs: the rational part in the brain, spirited part in the heart and desiderative part in the liver.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galen adopt also an Aristotelic point of view: according to it the soul is the form of the body, and its parts are understood by Galen as being the forms of&#xA;the bodily organs in which each of them resides. But the form is&#xA;identical with the temperament, i.e. with the mixture of elementary&#xA;bodies or of the corresponding qualities which constitute the organ.&#xA;Thus the desiderative soul will be the form and temperament of the&#xA;liver, the spirited will be the form and temperament of the heart,&#xA;while the rational soul must consequently be the form and temperament&#xA;of the brain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conclusion: this seems to be the long road from &lt;em&gt;blood&lt;/em&gt; to the &lt;em&gt;sanguine&lt;/em&gt; temperament to a &quot;spirited&quot; behaviour. &lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-02-21T09:28:11.097" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7081" PostTypeId="1" CreationDate="2018-02-22T01:27:44.240" Score="5" ViewCount="514" Body="&lt;p&gt;All answers to questions like this dodge the question by saying it's a postulate of Matrix Mechanics, so let me rephrase it. Instead of how to derive the CCR, how does it follow from Heisenberg's matrices:&#xA;$$\hat X=\{X_{mn}e^{i\omega_{mn}t}\}\quad\hat P=\{P_{mn}e^{i\omega_{mn}t}\}$$&#xA;Where $\omega_{mn}=\frac{E_m-E_n}{\hbar}$, such that&#xA;$$\hat X(t)=\left(\begin{matrix}&#xA;X_{11}&amp;amp;X_{12}e^{i\omega_{12}t}&amp;amp;X_{13}e^{i\omega_{13}t}&amp;amp;\cdots\\&#xA;X_{21}e^{i\omega_{21}t}&amp;amp;X_{22}&amp;amp;X_{23}e^{i\omega_{23}t}&amp;amp;\cdots\\&#xA;X_{31}e^{i\omega_{31}t}&amp;amp;X_{32}e^{i\omega_{32}t}&amp;amp;X_{33}\cdots\\&#xA;\vdots&amp;amp;\vdots&amp;amp;\vdots&amp;amp;\ddots&#xA;\end{matrix}\right)$$&#xA;$$\hat P(t)=\left(\begin{matrix}&#xA;P_{11}&amp;amp;P_{12}e^{i\omega_{12}t}&amp;amp;P_{13}e^{i\omega_{13}t}&amp;amp;\cdots\\&#xA;P_{21}e^{i\omega_{21}t}&amp;amp;P_{22}&amp;amp;P_{23}e^{i\omega_{23}t}&amp;amp;\cdots\\&#xA;P_{31}e^{i\omega_{31}t}&amp;amp;P_{32}e^{i\omega_{32}t}&amp;amp;P_{33}\cdots\\&#xA;\vdots&amp;amp;\vdots&amp;amp;\vdots&amp;amp;\ddots&#xA;\end{matrix}\right)$$&#xA;And from the Old quantum condition:&#xA;$$\oint pdx=nh$$&#xA;I know Heisenberg came up with it through analogies between the matrices and the classical observables, but I'd like to know how he did it.&lt;/p&gt;&#xA;" OwnerUserId="7065" LastEditorUserId="7065" LastEditDate="2018-02-22T01:42:46.803" LastActivityDate="2018-08-04T12:04:03.450" Title="How did Born come up with the Canonical Commutation relation ($\hat X \hat P-\hat P\hat X=i\hbar$)?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7082" PostTypeId="2" ParentId="7081" CreationDate="2018-02-22T04:48:56.807" Score="5" Body="&lt;p&gt;Van der Waerden’s &lt;em&gt;Sources of Quantum Mechanics&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/SourcesOfQuantumMechanics#page/n44&quot; rel=&quot;nofollow noreferrer&quot;&gt;1967, pp. 36-39&lt;/a&gt;) gives a thorough account based on Born’s own recollections, later published in &lt;em&gt;Mein Leben&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=wBaLAwAAQBAJ&amp;amp;pg=PP5&quot; rel=&quot;nofollow noreferrer&quot;&gt;1975&lt;/a&gt;). It was guesswork! Briefly, Born interpreted a formula of Heisenberg (&lt;a href=&quot;https://archive.org/stream/SourcesOfQuantumMechanics#page/n267&quot; rel=&quot;nofollow noreferrer&quot;&gt;1925, eq. 16&lt;/a&gt;) as giving the diagonal entries of $pq-qp$, conjectured that all others vanish (“this was only a guess, and my attempts to prove it failed”), and then accepted Jordan’s argument that this must be true (“not an exact mathematical proof”).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whittaker has an interesting discussion and “simplified proof” in (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=61057&quot; rel=&quot;nofollow noreferrer&quot;&gt;1953&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/AHistoryOfTheTheoriesOfAetherAndElectricity#page/n713&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 258-260&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-08-04T12:04:03.450" LastActivityDate="2018-08-04T12:04:03.450" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7083" PostTypeId="1" AcceptedAnswerId="7084" CreationDate="2018-02-22T15:14:31.063" Score="4" ViewCount="386" Body="&lt;p&gt;Can Newton’s process for finding the derivative of y as a function of x resulting in $\frac{dy}{dx}$ be thought of as an application of the chain rule? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My thinking is:&#xA;If y is a (continuous?) function of x then as a particle moves along the curve traced out, the particle has a velocity $\dot y  (\frac{dy}{dt})$ in the y direction and a velocity $\dot x (\frac{dx}{dt})$ in the x direction. Hence the gradient function, $\frac{dy}{dx}$ is $\frac{\dot y}{\dot x} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dt}.\frac{dt}{dx} = \frac{dy}{dx}$ ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Separately, was the terminology “fluent” used because Newton was interested in using (differential) calculus to model the motion of objects?&lt;/p&gt;&#xA;" OwnerUserId="5568" LastEditorUserId="55" LastEditDate="2018-02-22T22:57:54.027" LastActivityDate="2023-07-06T15:26:32.603" Title="Was Newton's method of finding derivatives of his fluents based on applying the chain rule?" Tags="|calculus|newton|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7084" PostTypeId="2" ParentId="7083" CreationDate="2018-02-22T22:53:38.797" Score="9" Body="&lt;p&gt;The answer is more of a yes, but with many buts. Newton did not have the modern concept of function, it was introduced by Dirichlet in the 19th century, or even its predecessor as assignment of values according to a &amp;quot;law&amp;quot;, as Euler defined it in the 18th. Fluent was closer to what was later called  variable quantity, something like a temporally changing (&amp;quot;&lt;em&gt;evanescent&lt;/em&gt;&amp;quot;) magnitude, and it was conceived on a kinematic model based on Aristotelian &amp;quot;fluid&amp;quot; continuum. Newton, of course, did not use the modern Dedekind-Cantor notion of the continuum composed of points either, his geometric line was not assembled from &amp;quot;real numbers&amp;quot;. Here is from &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03962/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Quadrature of Curves&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I don't here consider Mathematical Quantities as composed of Parts extremely small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux, and so in the rest.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Fluxion was the &amp;quot;derivative&amp;quot; of a fluent only speaking anachronistically. Newton, of course, could not use predicates and nested quantifiers of modern analysis to define limits and then derivatives. He originally used infinitesimals in the definition, but grew dissatisfied with their haziness and unreliability, and replaced them in Principia with &amp;quot;first and last ratios&amp;quot;. These were kinematic substitutes for limits, but the best he could describe with was just an analogy, to velocities, see &lt;a href=&quot;https://hsm.stackexchange.com/a/9590/55&quot;&gt;What was the notion of limit that Newton used?&lt;/a&gt; Still, this allowed to connect them to geometry, which was on a much firmer footing than infinitesimals. Newton even uses the word &amp;quot;limit&amp;quot;, but fluxions of evanescent quantities are no limits of functions. Here is from &lt;a href=&quot;http://gravitee.tripod.com/booki1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book I of Principia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since Newton did not have functions he could not have had compositions and the chain rule either. But he did have fluents traversing curves and those curves did have Cartesian equations like &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt;, and their tangents had slopes. Of course, Newton did not &amp;quot;apply&amp;quot; the &amp;quot;chain rule&amp;quot;, that would have been circular even aside from the difference in concepts, but he did use a procedure somewhat like the one described in the OP to find slopes of curves using fluxions. The procedure was based on their analogy to velocities and geometry. Here is &lt;a href=&quot;https://www.jstor.org/stable/2185244&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedman's description&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We start with fluents or &amp;quot;flowing quantities&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt;, conceived as continuous functions of time. We can then form the fluxions or time-derivatives &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt;, because continuously changing quantities obviously have well-defined instantaneous velocities or rates of change. If we are then given a curve or figure &lt;span class=&quot;math-container&quot;&gt;$y = f(x)$&lt;/span&gt; generated by independent motions in rectangular coordinates of the fluents &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt; the derivative (slope of the tangent line) will be &lt;span class=&quot;math-container&quot;&gt;$dy/dx = \dot{y}/\dot{x}$&lt;/span&gt; by the &amp;quot;parallelogram of velocities&amp;quot;. Finally, we can recover the integral from the derivative via the Fundamental Theorem (which is also understood temporally). So all the basic notions of the calculus are explained without ever appealing to differentials or infinitely small quantities.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Moreover, although the kinematic interpretation of the calculus certainly does not meet modern standards of rigor, it is also not afflicted with the obvious problems about consistency and coherence facing an interpretation based on differentials, infinitesimals, and infinitely small quantities. Indeed, when the kinematic interpretation was explicitly criticized by mathematicians like D'Alembert and l'Huilier in the late eighteenth century, it was not on grounds of coherence and consistency but because it was thought to import a &amp;quot;foreign&amp;quot; or &amp;quot;physical&amp;quot; element into pure mathematics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-01T18:51:32.127" LastActivityDate="2021-02-01T18:51:32.127" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7086" PostTypeId="1" CreationDate="2018-02-21T17:27:14.067" Score="0" ViewCount="54" Body="&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Children%27s_Encyclop%C3%A6dia&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Children's Encyclopedia&lt;/a&gt;, edited by Arthur Mee which was in print from 1908 to 1964 (Vol 3, p 1614), there is an essay about Energy in which the following is asserted:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A man of science once said that there was enough energy in a glass of&#xA;  water to lift the whole of the Royal Navy into the air as high as St.&#xA;  Paul's Cathedral.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Can anyone help me locate the source of this quote? Who stated this? When? In what context? I figure the quote has to date roughly post 1697 when the building was consecrated and sufficiently before 1908 to allow for writing, editing and publication. I also assume it would have been an English physicist or scientist, probably London-based, given the reference point to the cathedral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas would be gratefully received.&lt;/p&gt;&#xA;" OwnerUserId="6676" OwnerDisplayName="Leon Conrad" LastEditorUserId="2277" LastEditDate="2018-05-12T04:34:05.333" LastActivityDate="2018-05-12T04:34:05.333" Title="Query re Physics-related quote about the amount of energy in water" Tags="|quote|energy|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="7087" PostTypeId="1" AcceptedAnswerId="7088" CreationDate="2018-02-23T23:06:18.060" Score="3" ViewCount="334" Body="&lt;p&gt;For the sake of curiosity, I'm interested in the &quot;list of problems&quot; that were laid out in the ancient Chinese text on Math. However, I haven't found a &quot;list&quot; in English anywhere. Only a few &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Nine_chapters.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;excerpts&lt;/a&gt; on a few problems. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an English version listing out the 200+ problems in the book?&lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-02-24T00:19:50.630" Title="Where can I find the list of problems from the (Chinese) &quot;Nine Chapters on Mathematical Art&quot;?" Tags="|reference-request|elementary-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7088" PostTypeId="2" ParentId="7087" CreationDate="2018-02-24T00:19:50.630" Score="4" Body="&lt;p&gt;From what I understand the &amp;quot;list of problems&amp;quot; is supposed to be the full English translation. Here is the bibliographic reference for the &lt;a href=&quot;https://global.oup.com/academic/product/the-nine-chapters-on-the-mathematical-art-9780198539360?cc=us&amp;amp;lang=en&amp;amp;#&quot; rel=&quot;nofollow noreferrer&quot;&gt;1999 translation&lt;/a&gt;, the only one so far:&lt;/p&gt;&#xA;&lt;p&gt;The Nine Chapters on the Mathematical Art: Companion and Commentary. By Shen Kangshen, John N. Crossley, and Anthony W.-C. Lun. Oxford and New York: Oxford University Press, 1999.&lt;/p&gt;&#xA;&lt;p&gt;There is no online version, unfortunately. The organization of the volume is controversial, here is from &lt;a href=&quot;https://www.jstor.org/stable/2695288?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Swetz's MAA review&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;A disappointing (frustrating!) aspect of this volume is its organizational format and  the over-abundance of information that it attempts to communicate beyond the translation itself. It falters in its role as a companion and commentary... The repeated comparisons of the Jiujiang's problems and techniques with those of other countries appear chauvinistic and become tedious. Comparisons certainly are warranted, but they would be more effective if they were limited in number and confined to one section or chapter devoted to the topic.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Historical commentaries, followed by the modern translators' comments and observations, are linked by a complex system of multireferencing. In some instances, a particular translated term or statement may be supported by as many&#xA;as three superscripts guiding the reader to further comments that may, in turn, be referenced to footnotes. The contemporary translators' extensive annotations, inserted  amongst the Jiujiang text, disrupt the flow of the original mathematical presentation. Once again, such material would be better placed in chapter summaries.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-02-24T00:19:50.630" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7089" PostTypeId="1" AcceptedAnswerId="7090" CreationDate="2018-02-24T01:58:40.737" Score="3" ViewCount="520" Body="&lt;p&gt;I'm looking for an English translation of Rafael Bombelli's L'Algebra.  From what I can tell after having searched the usual corners of the web, it doesn't exist, but I'm asking here just in case.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm particularly interested in his analysis of Cardano/Tartaglia/Ferro's methods for cubics and how they relate to complex numbers, so if there's a decently comprehensive history book covering that, that might be suitable. (For whatever it's worth, I currently have C. Boyer's &lt;em&gt;A History of Mathematics&lt;/em&gt; and P. Nahin's &lt;em&gt;An Imaginary Tale&lt;/em&gt;, but I'm looking for something a little more focused to accompany those.)&lt;/p&gt;&#xA;" OwnerUserId="7074" LastEditorUserId="10106" LastEditDate="2020-06-07T13:41:11.613" LastActivityDate="2020-06-07T13:41:11.613" Title="Does an English translation of Bombelli's L'Algebra exist?" Tags="|mathematics|reference-request|translations|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7090" PostTypeId="2" ParentId="7089" CreationDate="2018-02-24T04:20:52.800" Score="1" Body="&lt;p&gt;González-Velasco (&lt;a href=&quot;https://zbmath.org/?q=an:1243.01002&quot; rel=&quot;nofollow noreferrer&quot;&gt;2011&lt;/a&gt;) has a nice dozen pages on &lt;em&gt;L’Algebra&lt;/em&gt; (§3.3). His preface (p. x) says there has been no translation.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-02-24T04:20:52.800" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7092" PostTypeId="1" CreationDate="2018-02-27T10:23:31.307" Score="3" ViewCount="359" Body="&lt;p&gt;Between 1960 and 1983 the metre was defined as being equal to 1 650 763.73 wavelengths of an orange emission line of krypton-86. How did they actually use that definition? Was it possible to move the mirror of an interferometer slowly over a distance of 1 metre, perhaps by a very smooth screw, and count the appearance of 1.6 million fringes without mistakes? Was the coherence length of the krypton light more than 1 metre?&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2277" LastEditDate="2018-10-05T14:55:10.473" LastActivityDate="2018-10-05T14:55:10.473" Title="Wavelength definition of the metre" Tags="|measurement|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7093" PostTypeId="2" ParentId="7092" CreationDate="2018-02-27T12:51:04.447" Score="3" Body="&lt;p&gt;There are much easier ways to measure a wavelength than by &quot;tromboning&quot; an interferometer.  A simple &lt;a href=&quot;https://en.wikipedia.org/wiki/Fraunhofer_diffraction_equation#Two_slits&quot; rel=&quot;nofollow noreferrer&quot;&gt;two-slit grating&lt;/a&gt;, for example, will give you a pretty decent measurement.    You don't need to measure a &quot;meter's worth&quot; of light, just run an experiment where the output data (such as a diffraction pattern) can be used to calculate the wavelength to high precision and accuracy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, the definition was basically &quot;in reverse.&quot; Using existing measurement techniques, NIST or ISO or whomever took the current measurement of this narrow (no broadening, no fine-structure splitting) line and used that to define the meter.  &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-02-27T12:51:04.447" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7094" PostTypeId="1" AcceptedAnswerId="7116" CreationDate="2018-02-27T21:23:24.763" Score="5" ViewCount="432" Body="&lt;p&gt;I have studied that Gauss was one of the firsts mathematicians to defend this idea, about the Abstract Math and the conception of number, claiming that &quot;What is calculated (in the sense of things already counted) are not substances (thinkable objects for themselves), but relations between two objects counted two by two.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This thought may have been influenced (albeit indirectly) by a fellow-country philosopher of Gauss, Immanuel Kant, once in an article from a time not far from Gauss, Kant argued that the idea behind negative numbers should go beyond the traditional logical idea.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &quot;Attempt to introduce the conception of Negative Quantities into Philosophy&quot;, Kant puts in balance the logical-contradictory and real-oppositional ideas that the negative numbers have. As, for example, the logical idea of negative numbers attests to the fact that if a body is in motion, the negation of this movement would be rest. However, there would be no conditions for this body to be at rest and in motion simultaneously. On the other hand, the real-oppositional idea holds that the denial of a movement of a body would be the same movement in the opposite direction, and, finally, the simultaneous events of these two events would bring rest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is, the focus in the understanding of number is not about the things you are observing, but about the relationships between each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What mathematicians and philosophers between the 18th and 19th centuries were defenders or precursors of this kind of abstract mathematics and number as a relation?&lt;/p&gt;&#xA;" OwnerUserId="7089" LastEditorUserId="55" LastEditDate="2018-03-03T00:21:02.590" LastActivityDate="2018-03-03T00:23:06.017" Title="Who influenced Gauss in his abstract approach to mathematics?" Tags="|mathematicians|biographical-details|reference-request|philosophy-of-science|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7096" PostTypeId="1" AcceptedAnswerId="7097" CreationDate="2018-02-28T01:41:22.477" Score="0" ViewCount="2216" Body="&lt;p&gt;I wonder if anyone would know the origin of the &lt;a href=&quot;https://www.allaboutcircuits.com/textbook/reference/chpt-2/resistor-color-codes/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Resistor Color Code&lt;/a&gt;.  &lt;/p&gt;&#xA;" OwnerUserId="7093" LastActivityDate="2018-02-28T14:37:01.330" Title="Resistor color code" Tags="|physics|electricity|naming-conventions|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7097" PostTypeId="2" ParentId="7096" CreationDate="2018-02-28T09:36:29.323" Score="6" Body="&lt;p&gt;The color code was developed in the 1920's by the Radio Manufacturers Association (RMA) as a three band code for resistor values. The three bands were more compact than the number value because the third band represented the number of zeroes. For example, 250 000 Ω was reduced to three bands. In addition, color bands remained visible in whatever position a resistor was soldered, whereas a stamped number value could be out of sight. The fourth band, representing the tolerance, was added later.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initially the three colors were not three separate bands, and there was no neutral background color. Instead, they were the body color, the tip color and the dot color, as in the diagram. Identical adjacent colors were allowed.  &lt;a href=&quot;https://books.google.nl/books?id=VJALAQAAIAAJ&amp;amp;q=%22The%20code%20identifies%20resistors%20by%20means%20of%203%20colors,%20known%20as%20%22body,%22%20%22tip%22%20and%20%22dot%22%20colors%22&amp;amp;dq=%22The%20code%20identifies%20resistors%20by%20means%20of%203%20colors,%20known%20as%20%22body,%22%20%22tip%22%20and%20%22dot%22%20colors%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiGoq-m5cjZAhWCjqQKHQmgBi8Q6AEILTAB&quot; rel=&quot;noreferrer&quot;&gt;Radio Physics Course, Ghirardi, 1931&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The sequence of the central group of colors, ROYGBV, was chosen to match the rainbow mnemonic: red-orange-yellow-green-blue-violet, &quot;ROYGBIV&quot;,  without indigo because most people do not distinguish, with their eyes, a separate color between blue and violet. The rainbow group is preceded by two low brightness colors, black and brown, representing the lowest digits 0 and 1, and succeeded by two bright colors, grey and white, representing the highest digits 8 and 9. &lt;a href=&quot;https://books.google.nl/books?id=4tkiAAAAMAAJ&amp;amp;redir_esc=y&quot; rel=&quot;noreferrer&quot;&gt;Electrical Engineering Science, Preston, 1960, p.115&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SULygm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SULygm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2018-02-28T14:37:01.330" LastActivityDate="2018-02-28T14:37:01.330" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7098" PostTypeId="2" ParentId="6727" CreationDate="2018-02-28T14:41:31.923" Score="10" Body="&lt;p&gt;One already finds upright d's in Lacroix's &lt;a href=&quot;https://archive.org/details/traitlmentaired01lacrgoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Traité élémentaire de calcul différentiel et de calcul intégral&lt;/em&gt;&lt;/a&gt; (1802). I don't now if this is the earliest, but it is interesting to note that the first edition of his &quot;non-elementary&quot; &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k92729z/f130.image&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Traité du calcul différentiel et du calcul intégral&lt;/em&gt;&lt;/a&gt; (1797) has italicized d's, while the &lt;a href=&quot;https://archive.org/stream/traitducalculdi05lacrgoog#page/n226/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;second edition&lt;/a&gt; from 1810 has upright differentials.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; As noted in the comments by Francois Ziegler, Lacroix’s &lt;span class=&quot;math-container&quot;&gt;$\mathrm d$&lt;/span&gt;’s are predated by Lagrange (&lt;a href=&quot;https://archive.org/stream/mlangesdephilo02soci#page/230&quot; rel=&quot;noreferrer&quot;&gt;1762&lt;/a&gt;), L’Huilier (&lt;a href=&quot;https://archive.org/stream/principiorumcal00lhugoog#page/n76&quot; rel=&quot;noreferrer&quot;&gt;1795&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A somewhat tangential update:&lt;/strong&gt; I was reading Euler's &lt;em&gt;Institutiones Calculi Differentialis&lt;/em&gt; (&lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/1174066&quot; rel=&quot;noreferrer&quot;&gt;1755&lt;/a&gt;; &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/2351397&quot; rel=&quot;noreferrer&quot;&gt;1790&lt;/a&gt;) and found the following nice passage which could have influenced the change from &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathrm{d}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;   119. It should be kept in mind that the letter &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; here does not denote a quantity, but is used as a symbol to express the word &lt;em&gt;differential&lt;/em&gt;, in the same way that the letter &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; is used for the word logarithm when the theory of logarithms is being discussed or the symbol &lt;span class=&quot;math-container&quot;&gt;$\smash\surd$&lt;/span&gt; is used to denote the root. Hence &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; does not signify, as it usually does in analysis, the product of two quantities &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, but must be read as the differential of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. In a similar way, if we write &lt;span class=&quot;math-container&quot;&gt;$d^2 y$&lt;/span&gt;, this is not the square of a quantity &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, but it is simply a short and apt way of writing the &lt;em&gt;second differential&lt;/em&gt;. Due to this use of the letter &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; in differential calculus, one should — in order to avoid confusion — not use it to denote a quantity in calculations when many different quantities occur, just like we avoid the letter &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; to designate a quantity in calculations where logarithms occur. It would be desirable if the letters &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; were altered to a different appearance, lest they be confused with other letters of the alphabet that are used to designate quantities. This is what has happened to the letter &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, which first was used to indicate a root; before it was distorted to &lt;span class=&quot;math-container&quot;&gt;$\smash\surd$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And here's a quote from Cajori's &lt;em&gt;A history of mathematical notation&lt;/em&gt;  (1928) § 595.:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It looked indeed as if the different mathematical architects engaged in erecting a proud mathematical structure found themselves confronted with the curse of having their sign language confounded so that they could the less readily understand each other's speech. At this juncture certain writers on the calculus concluded that the inter­ests of their science could be best promoted by discarding the straight­ letter &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and introducing the rounded &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;. Accordingly they wrote &lt;span class=&quot;math-container&quot;&gt;$\partial y$&lt;/span&gt;&#xA;  for the total differential, &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial y}{\partial x}$&lt;/span&gt; for the total derivative, and &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{\partial y}{\partial x}\right)$&lt;/span&gt; for the&#xA;  partial derivative. G. S. Klügel refers to this movement when he says in 1803&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;: &quot;It is necessary to distinguish between the symbol for differential and that for a [finite] quantity by a special form of the letter. In France&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; and in the more recent memoirs of the Petersburg Academy, writers&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; have begun to designate the differential by the curved &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;.&quot; So Klügel himself adopts this symbolism from now on for his dictionary. Euler's Institutiones calculi integralis, which in its first edition of 1768-70 used the straight &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, appeared in the third edition of 1824 with the round &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;, both for the total differential and the total derivative. This same notation is found in J. A. Grunert's calculus&lt;span class=&quot;math-container&quot;&gt;$^4$&lt;/span&gt; of 1837. However, the movement failed of general adoption; for some years both &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt; were used by different writers for total differentiation.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;: G. S. Klügel, &lt;em&gt;Mathematisches. Wörterbuch&lt;/em&gt;, 1. Abth., 1. Theil (Leipzig, 1803), art. &quot;Differentiale.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;: In France, Le Marquis de Condorcet had used the &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt; in writing total deriva­tives in his &lt;em&gt;Probabilité des décisions&lt;/em&gt;, in 1785.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; The &lt;em&gt;Nova Acta Petropolitana&lt;/em&gt;, the first volume of which is for the year 1785, contain the rounded &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^4$&lt;/span&gt; Johann August Grunert, &lt;em&gt;Elemente der Differential und Integralrechnung&lt;/em&gt; (Leipzig, 1837).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But I couldn't find Cajori address the distinction &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; vs &lt;span class=&quot;math-container&quot;&gt;$\text{d}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2018-10-23T14:44:08.487" LastActivityDate="2018-10-23T14:44:08.487" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7099" PostTypeId="1" CreationDate="2018-02-28T17:16:56.200" Score="2" ViewCount="44" Body="&lt;p&gt;What was the &quot;doctrine of sterilazio magna&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example from 1912 article about the variability of drug effectiveness: &quot;Although the doctrine of the &lt;em&gt;sterilazio magna&lt;/em&gt; has only been urged against the pathogenic protozoa, its principle doubtless extends in many cases to the body cells...&quot; H. D'Arcy Power&lt;/p&gt;&#xA;" OwnerUserId="7096" LastActivityDate="2018-03-01T19:10:43.323" Title="Doctrine of the sterilazio magna" Tags="|terminology|medicine|cell-biology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7100" PostTypeId="1" AcceptedAnswerId="7104" CreationDate="2018-02-28T19:00:09.233" Score="10" ViewCount="902" Body="&lt;p&gt;Euclid's Elements could tout to have the longest and most famed publishing history of any book ever written. First written in 300 B.C., Euclid's Elements became the standard text from which aristocracy and the common people learned mathematics well into the 1800s, giving it a little over 2000 years of use. For just about any metric one could create for a book, the Elements would consistently place first or second (only to the Bible).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, it is rarely seen or used. Synthetic Geometry itself hasn't gone away (although it doesn't get the same amount of spotlight as it did). Geometry is still taught today in the grade schools, but this is frequently taught out of more modern textbooks that only live to see a few editions (and I would scarcely call them more &quot;readable&quot;).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why, or how, did the Elements wane from usage?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A couple of suspicions I have are:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Hilbert published his Grundlagen der Geometrie in 1899 in which he outlined different axioms which spotlighted the holes in Euclid's axiomatizations. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Enlightenment pushed Geometry away from the central role it had in mathematics for centuries in favor of the Calculus and subsequent branches of Analysis.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Although I'm sure these contribute in some manner, they don't really meld to create a story for the Elements decline. For such a work that had such great cultural significance to humanity, I feel like it deserves a far better eulogy.&lt;/p&gt;&#xA;" OwnerDisplayName="user6918" LastActivityDate="2020-05-23T14:02:52.367" Title="What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?" Tags="|mathematics|euclidean-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7101" PostTypeId="1" AcceptedAnswerId="7103" CreationDate="2018-02-28T22:25:22.707" Score="6" ViewCount="1718" Body="&lt;p&gt;I've always thought that defining conic sections by a locus of points w.r.t the ratio of the distance to the focus and directrix was always &quot;too artificial&quot; - how does one actually &lt;em&gt;discover&lt;/em&gt; this mysterious directrix, a line that isn't really lying anywhere when a double-cone is cut by a plane; or a focal point where all the &quot;rays reflecting the conic&quot; would meet (i.e., for a parabola)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some mathematical thinking/creativity must have gone into &quot;discovering&quot; this fact. I've scoured ancient Greek history but have failed to uncover anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best I have is along these lines (my best interpretation, from &lt;a href=&quot;https://archive.org/details/historyofgreekma029268mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Greek Math Vol 2 - Heath&lt;/a&gt;, Pg 119):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Aristaeus' Solid Loci considered loci which proved to be Conic&#xA;  sections. The theorems that Pappus of Alexandria alludes to are those&#xA;  put forth by the Solid Loci. There's a possibility that the&#xA;  focus-directrix property was known to Euclid but there's no conclusive&#xA;  evidence. However, the focus-directrix was probably &quot;defined&quot; as such&#xA;  by Pappus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;UPDATE:&lt;/strong&gt; The accepted answer provided a great direction and helped me dig up the Arabic Translation of &quot;&lt;a href=&quot;http://www.math.harvard.edu/archive/hist_206r_2009/Too_1976.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Burning Mirrors by Diocles&lt;/a&gt;&quot; that proves the existence of a focus (Proposition 1) as well as provides construction of the parabola via the focus-directrix property (Prop. 4-5). This is probably the &quot;first&quot; extant source that shines light on how this came about and Diocles attributed it to Dositheus, but the latter never furnished a geometric proof it seems.&lt;/p&gt;&#xA;" OwnerUserId="7035" LastEditorUserId="7035" LastEditDate="2018-03-02T01:14:51.693" LastActivityDate="2018-06-21T16:05:24.880" Title="How was the focus/directrix property of conic sections discovered?" Tags="|mathematics|reference-request|ancient-greece|geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7102" PostTypeId="2" ParentId="7101" CreationDate="2018-03-01T00:07:19.357" Score="2" Body="&lt;p&gt;I tend to agree with the sentiment expressed in the question, that the focus-directrix definition of conics seems artificial (and awkward). Yet it has had its adherents. To answer the question 'How could the student discover the focus/directrix properties on her own?' it might be useful to adopt an &lt;em&gt;a posteriori&lt;/em&gt; approach, by considering the material in a book of conics that works extensively with these properties.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of a book that did use this approach achieved some 19th-century popularity, at least it ran to many editions: &lt;em&gt;&lt;a href=&quot;https://archive.org/details/conicsectionstre00besarich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conic sections, treated geometrically&lt;/em&gt; by W H Besant&lt;/a&gt;. Its proofs may (with meditation) provide some heuristic clues!  &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="7035" LastEditDate="2018-03-02T18:17:48.387" LastActivityDate="2018-03-02T18:17:48.387" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7103" PostTypeId="2" ParentId="7101" CreationDate="2018-03-01T00:29:05.210" Score="5" Body="&lt;p&gt;The answer will be disappointing, I am afraid, but typical, &quot;natural&quot; and &quot;artificial&quot; are very relative notions. The &quot;discovery&quot; of the focus-directrix property was likely a technical lemma in investigations of solid loci, which was later also used in the study of curved mirrors. When it did appear Greeks did not make much of it, its modern prominence in the theory of conic sections is a late artifact. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest extant source we have is Diocles' On Burning Mirrors, and he is working off of Euclid's previous work on spherical mirrors. The point was of course to find a shape that would gather sun rays (assumed to be parallel) at a single point, the focus. As Knorr pointed out, reducing ray-focusing property to the focus-directrix property to &quot;discover&quot; that it is possessed by parabola would require something like solving a differential equation, which was beyond the devices of Greeks. He then speculated that someone (perhaps, Archimedes's friend Dositheus) made a &quot;lucky guess&quot;, and then proceeded to show that the curve constructed by the focus-directrix property is a parabola (similar demonstration for ellipse is described by Anthemius). Acerbi suggests a different reconstruction in &lt;a href=&quot;https://www.jstor.org/stable/41287704&quot; rel=&quot;noreferrer&quot;&gt;The Geometry of Burning Mirrors in Greek Antiquity&lt;/a&gt;, where one can find geometric details of how the property came up as a lemma:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;strong&gt;&lt;em&gt;Focal properties of the parabola were obtained as a by-product of an attempt at proving that the property of the subtangent is a sumptoma of the curve&lt;/strong&gt;. About the relevance of this property to the eyes of the ancient geometers it is not necessary to spend so many words. Recall in fact that the property of the subtangent was well known to Archimedes... As a consequence, the focal property proves crucial to employ the subtangent as a method to construct a parabola with given vertex and parameter (cf. Con. I.52, that, however, does not lead in an immediate way to a pointwise construction) and makes it possible to solve by reduction a particular case of the problem of tracing a conic section after some of its tangents are known.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;&lt;strong&gt;&lt;em&gt;[Assumption] is that a property so important (to us) as the focus-directrix could not be reduced to the status of mere functionality to a pointwise determination of a parabola&lt;/strong&gt;, as indisputably happens in Diocles’ approach and as Knorr himself has to recognize... &lt;strong&gt;Of this property, there is no trace in Apollonius’ Conica&lt;/strong&gt; as they are now extant (recall that it holds in reference to the axis only, and therefore it is not a sumptoma that can be referred to any diameter).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;&lt;em&gt;Toomer surmised that Diocles himself was the discoverer of the property in the case of the parabola, its validity being extended after him to the other&#xA;  conic sections. The few lemmas Pappus offers, at the very end of book VII of his&#xA;  Collectio... is the only attestation in the ancient corpus of the fact that the focus–directrix property is a sumptoma of the several conic sections (the formulation as a locus bears this out exactly).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Zeuthen (1886, pp. 210–215 and 367–371) observes that, if Pappus had to make such a lemma explicit, then the focus–directrix property should have been taken for granted in the Loci on a Surface. Zeuthen’s remark can be completed&#xA;  by the conjecture that the most “natural” place where the property was established, of course in the above form of a sumptoma formulated as a constraint identifying a locus, is in Aristaeus’ Solid Loci...  In sum, &lt;strong&gt;if one is not willing to venture in empty conjecturing, then a minimal hypothesis is that the property had been employed in the Loci on a Surface as an intermediate step in the analysis of some locus, but that it was simply considered as a sumptoma of a conic section&lt;/em&gt;&lt;/strong&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-03-01T00:29:05.210" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7104" PostTypeId="2" ParentId="7100" CreationDate="2018-03-01T01:42:39.827" Score="8" Body="&lt;p&gt;The contest between synthetic and analytic methods in geometry predates Hilbert and even calculus, one can trace its origins to Vieta's algebraic conversions of geometric problems that streamlined their solution, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2408/55&quot;&gt;Viète's Relevance and his Connection to Euler&lt;/a&gt; and their systematization in Descartes's analytic geometry. But the rise of calculus and eventually arithmetization and logicization of mathematics by Dedekind, Cantor, Frege, Pasch, Hilbert, etc., certainly sealed the deal. An early episode in the open rivalry between two styles of mathematical thinking, interesting due to its particular ferocity, is described by &lt;a href=&quot;https://www.jstor.org/stable/236738&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mazzotti in Geometers of God&lt;/a&gt;. The confrontation took place over the first half of 19th century in Naples. Analytics tended to be more modern and liberal and emphacized the role of mathematics in sciences and practical matters. Synthetics were conservative traditionalists who saw analytics as (sic!) &amp;quot;morally depraved&amp;quot;, &amp;quot;antiscientific&amp;quot;, and &amp;quot;corrupting&amp;quot; the minds of young students. As Ventura put it in 1824:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Among the sciences, the mathematical ones are those which have taken the more false and disastrous direction. They were the first to be included in the assault of the philosophers against Christianity... ; they have become deadly weapons in the hands of impiety and pride; they have broken every restraint; they have unchained all the passions; they have eroded the foundations of society and order&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is Mazzotti's more measured side by side comparison of the pros and cons of the two approaches:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The synthetic method is specific: every problem to be solved calls for a different construction; thus the geometer requires skill, knowledge, and experience, which can be gained only through long training. This method relies on the intuition of the geometer, who must choose the construction most suitable for a particular problem. Although demanding, the synthetic method was held by its supporters to be the only natural and sound method of reasoning in mathematics; they presented analysis as an artificial tool whose operations are not only epistemologically subordinate to geometry but also much more complex and counterintuitive.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The analytic method is general: every problem can be put in the form of an equation, and then solved, by following the same steps. Through solving the &amp;quot;equation of the problem&amp;quot; we obtain, mechanically, all the possible solutions. Nothing is left to the intuition of the geometer. The analytics stressed that this method is &amp;quot;easy,&amp;quot; &amp;quot;mechanical,&amp;quot; and easily learned. The fact that it often required great ingenuity to formulate and solve the equation was not considered an essential problem. The point was to show that an extension of the &amp;quot;empire&amp;quot; of analysis to the field of geometry is possible, and indeed desirable, because it&#xA;would render the problem-solving procedure entirely mechanical.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Despite the political headwinds, the popular triumph of the general and the mechanical over the old school ingenuity and subtlety is not surprising, nor is its replication elsewhere. The controversy was replayed, with less moralizing, between &amp;quot;synthesists&amp;quot; and &amp;quot;analysts&amp;quot; in the big leagues of mathematics in 19-th century, see &lt;a href=&quot;https://hsm.stackexchange.com/a/7000/55&quot;&gt;Who are “analysts” and “synthesists” in mathematics?&lt;/a&gt; One of the key figures in the Neapolitan debate, Padula, offered arguments that one can imagine being made today&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Thanks to Lagrangian &amp;quot;algebraic analysis,&amp;quot; Padula continues, &amp;quot;the mathematical sciences have finally been reduced to that unity of principles so sought by philosophers.&amp;quot; Padula stresses the importance of enabling students to read works written in the &amp;quot;new language&amp;quot; as early as possible. For this purpose, he supports the use of French and North Italian textbooks of analytic geometry; these should replace more traditional synthetic textbooks on conic sections - which, he says, in the &amp;quot;civilized&amp;quot; countries have almost disappeared. The study of &amp;quot;geometrical abstractions,&amp;quot; he claims, does not introduce the student to &amp;quot;the modem development of mathematics&amp;quot; and &amp;quot;to the enormous material advantages that society can obtain from the application of mathematics to the arts, manufactures, industries.&amp;quot;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As already mentioned, the stress on rigor at the end of 19th century only added to the dissatisfaction with synthetic methods. Recent studies of Euclid's method revealed, however, that it has resources (nondeductive use of diagrams, etc.) for maintaining reliability that are irreducible to Hilbert style axiomatic reasoning, and, in conjuction with heuristics, are far more effective for developing new results, as opposed to axiomatic organization of known ones. This makes it interesting as a model of mathematical practice and educational tool, aside from inspiring development of similarly minded approaches beyond classical geometry, see e.g. &lt;a href=&quot;https://arxiv.org/abs/1210.1478&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rodin's Axiomatic Method and Category Theory&lt;/a&gt;. A recent classic in the study of Euclid's synthetic method is Manders's Euclidean Diagram (published in &lt;a href=&quot;http://spiritual-minds.com/philosophy/assorted/Mancosu%20-%20Philosophy%20of%20Mathematical%20Practice%20(Oxford,%202008).pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mancosu edited volume&lt;/a&gt;, freely available). In a preface to it he addresses and evaluates perceived reasons for eschewing synthetics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;One student, apparently the beneficiary of enthusiastic instruction in the virtues of the modern logical account, recently compared the study of Euclidean demonstration as mathematical justifications to the study of the Flat Earth. While myself not in a position to judge the contributions of the flat earth tradition to modern plate tectonics, I believe this expresses at least two complaints... that diagram-based reasoning in Euclid is unreliable and justificationally inadequate, and that the study of a tradition of argument so obsolete cannot benefit the philosophy of mathematics. Let me take these in turn.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Assessments based on diagrams are held to be unreliable on several grounds. Drawn diagrams are imperfect in that, say, lines are not perfectly straight; regardless, human assessments of straightness or equality of line segments would be imperfect. Moreover, geometrical figures are individual, or at least atypical compared to the generality of geometrical conclusions. Next, there are different forms of geometry, which differ in their conclusions, and so a single diagram-based form of reasoning cannot serve them all; indeed there are forms (such as plane coordinate geometry restricted to rational coordinates) in which the two circles would not have an intersection point. Finally, there are geometricals such as space-filling curves which utterly defeat diagram-based reasoning.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;In research profile, only a corner of modern geometry; but in terms of the footprint of mathematics in the modern world, that is most of modern mathematics. Euclid, and Apollonius and Archimedes, are virtually without error: their every result has a counterpart in modern mathematics... Nor should we fool ourselves that modern logical reconstructions by Hilbert (1899/1902) and Tarski (1959), however great their interest in other respects, give such an account&lt;/em&gt; [of their justification]&lt;em&gt;: ancient geometers achieved their lasting, subtle and powerful results precisely by the means that philosophers dismiss so high-handedly today, without the benefit of modern logic and Hilbert’s refined control of coordinate domains... Euclidean diagram use forces us to confront mathematical demonstrative practice, in a much richer form than is implicit in the notions of mathematical theory and formal proof... and to confront rigorous demonstrative use of non-propositional representation.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-08T14:24:59.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7105" PostTypeId="1" CreationDate="2018-02-28T22:58:15.217" Score="3" ViewCount="347" Body="&lt;p&gt;I've been recently reading about the famous Michelson-Morley experimental attempt to detect the Earth motion with respect to the aether. According to Wikipedia, Michelson initially made a mistake calculating the time that the light takes to propagate in the transverse direction(perpendicular to the Earth velocity). In his calculation he assumed that the transverse travel time $T_t=2cL_t$, where $L_t$ is the length of the transverse channel. In this case, the speed of light in the aether frame is $\sqrt{c^2+v^2_{Earth}}$, since Michelson did not know about the relativity theory yet.Later, it was corrected by Lorenz, who said that the speed of light in the transverse direction must be $c_t=\sqrt{c^2-v_{Earth}^2}$, thus in the aether frame the speed of light remains $c$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was Lorenz' motivation to think that the speed of light is always $c$ in the  aether frame? Is it because they believed at the time that Maxwell's equations give the speed of light in aether? And how did they know that the light must acquire the longitudinal component $v_{Earth}$? Maybe when the light is radiated it does not have the longitudinal component in the aether frame.&lt;/p&gt;&#xA;" OwnerDisplayName="Kirill" LastEditorUserId="2277" LastEditDate="2018-04-22T12:34:32.857" LastActivityDate="2018-04-22T12:34:32.857" Title="Speed of light in aether" Tags="|physics|light|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7106" PostTypeId="2" ParentId="7092" CreationDate="2018-03-01T05:03:00.813" Score="2" Body="&lt;p&gt;It was actually done with a &quot;tromboning&quot; Michelson Interferometer&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FkBGi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FkBGi.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://www.bipm.org/en/measurement-units/history-si/former-prototype-metre.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.bipm.org/en/measurement-units/history-si/former-prototype-metre.html&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1892 by Michelson. He measured a distance of 10cm to an accuracy of  ± 0.00004 mm using a Cadmium lamp . The 1960 re-definition with Krypton used the same data - it was just a matter of measuring the ratio of the wavelengths of the Cadmium and Krypton lines &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Michelson's 1907 Noble prize speech describes it &#xA;&lt;a href=&quot;http://qsd.magnet.fsu.edu/course/exp3802/optics/AAMichelson1907.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://qsd.magnet.fsu.edu/course/exp3802/optics/AAMichelson1907.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2838" LastActivityDate="2018-03-01T05:03:00.813" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7107" PostTypeId="1" AcceptedAnswerId="7111" CreationDate="2018-03-01T06:06:20.140" Score="8" ViewCount="2754" Body="&lt;p&gt;The name &quot;azimuthal quantum number&quot; is often used for the total orbital angular momentum quantum number $\ell$ in an atom.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the origin of this name? It makes no sense to me, since the &lt;a href=&quot;https://en.wiktionary.org/wiki/azimuthal#English&quot; rel=&quot;noreferrer&quot;&gt;usual meaning&lt;/a&gt; of &quot;azimuthal&quot; is apparently &quot;of or pertaining to the azimuth; in a horizontal circle&quot;, but $\ell$ has no information about orientation.&lt;/p&gt;&#xA;" OwnerUserId="7017" LastActivityDate="2018-03-01T14:55:04.273" Title="Why is the azimuthal quantum number so named?" Tags="|terminology|quantum-mechanics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7108" PostTypeId="2" ParentId="7107" CreationDate="2018-03-01T07:54:44.620" Score="4" Body="&lt;p&gt;When the Schrödinger equation is solved in spherical coordinates by separation of variables it is split into three equations whose spectra produce the first three quantum numbers. One of them, ℓ, corresponds to the selection of spherical harmonics and determines the number of planar nodes going through the nucleus (planar node is the midpoint between crest and trough with zero magnitude). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The name &quot;azimuthal quantum number&quot; for ℓ was originally introduced by Sommerfeld, who refined Bohr's semi-classical model by replacing circular orbits with elliptic ones. The spherical orbitals were similar (in the lowest-energy state) to a rope oscillating in a large &quot;horizontal&quot; circle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Azimuthal_quantum_number#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Azimuthal quantum number: history&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-03-01T07:54:44.620" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7109" PostTypeId="1" CreationDate="2018-03-01T12:20:55.687" Score="2" ViewCount="1782" Body="&lt;p&gt;It has been reported that on 18th August 1913, in a game of roulette at the Monte Carlo Casino, the ball fell on black 26 times in a row. This anecdote is often mentioned in the context of the gambler's fallacy (see the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gambler%27s_fallacy&quot; rel=&quot;nofollow noreferrer&quot;&gt;wiki&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I am looking for information about the roulette wheel in use in Monte Carlo in 1913&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular: was it the &quot;European&quot; type with one zero or the &quot;American&quot; type with two zeros? I would also appreciate a description --- even better a photo or sketch --- of the pattern of the numbers and colors around the wheel. These patterns have changed over the years and I have no information about the one that would have been in use back in 1913 in Monte Carlo. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My uneducated guess is that it would be a &quot;European&quot; type. But I have not found a reliable source of information on the matter. I have gleaned a few bits and pieces: The &quot;American&quot; roulette was originally the &quot;French&quot; roulette (introduced into what is now the United States via the French-owned Louisiana territory), while the &quot;European&quot; roulette was originally the &quot;German&quot; roulette. At some point (presumably by the end of the nineteenth century), the European/German model (which offers a better deal to gamblers) had displaced the American/French model throughout Europe. This did not happen in the United States, where the American/French model remains the most prevalent to this day. Given my limited information I venture to guess that by 1913, the European/German roulette would have completely replaced the American/French roulette in Monte Carlo. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Monte Carlo being an atypical semi-enclave bordering France, perhaps their roulette was also atypical of the time and place? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I opted to post this question under &quot;History of Science and Mathematics,&quot; but that was a bit of a gamble and I welcome suggestions to move it if there is a better outlet! (moderators, feel free to move it if convenient)&lt;/p&gt;&#xA;" OwnerUserId="463" LastActivityDate="2018-03-03T07:29:51.830" Title="Description of the Monte Carlo roulette wheel in 1913" Tags="|probability|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7110" PostTypeId="2" ParentId="7099" CreationDate="2018-03-01T13:25:02.937" Score="2" Body="&lt;p&gt;My reading of that article suggests a slightly idiomatic translation as &amp;quot;Kill Them All.&amp;quot;  While that may be poor long-term strategy in a war amongst peoples, the point raised in this medical article is that failure to completely wipe out the enemy (pathogen, bacteria, whatever) can lead to the remaining organisms becoming resistant to the drug treatment.&lt;br /&gt;&#xA;This was a rather prescient article, I think, given how long it took for the medical community to realize the manner/ease with which pathogens can evolve to be drug-resistant.&lt;/p&gt;&#xA;&lt;h2&gt;edit&lt;/h2&gt;&#xA;&lt;p&gt;I spent some time surfing, and found a &lt;a href=&quot;https://books.google.ca/books?id=goNucnnnI5QC&amp;amp;pg=PA368&amp;amp;lpg=PA368&amp;amp;dq=sterilizzazione%20magna&amp;amp;source=bl&amp;amp;ots=dS75ZqUM4d&amp;amp;sig=afwiXODthiu6YZJEYuEw5ZOvS-w&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwj_gOup6MvZAhUFrlQKHTYtAvMQ6AEIZDAI#v=onepage&amp;amp;q=sterilizzazione%20magna&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; or &lt;a href=&quot;https://books.google.ca/books?id=Bbo4BQAAQBAJ&amp;amp;pg=PA420&amp;amp;lpg=PA420&amp;amp;dq=%22sterilisatio%20magna%22&amp;amp;source=bl&amp;amp;ots=zZDTKffs54&amp;amp;sig=z2pss2SkJkkzVtelcWMShQpiJjM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjNv5mq6cvZAhVhwFQKHdJVCvIQ6AEISDAH#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;two&lt;/a&gt; in which the term is spelled &amp;quot;Sterisilatio Magna&amp;quot; .  I can only take a hack at translating, but the words and the drawing there suggest this is the same medical subject.&lt;/p&gt;&#xA;&lt;p&gt;and dang me, there's even a &lt;a href=&quot;https://www.facebook.com/pages/F%C3%BCnf-F%C3%A4lle-von-konnataler-Tuberkulose-die-M%C3%B6glichkeit-der-Sterilisatio-magna-durch-intensive-Fr%C3%BChtherapie-in-einem-Falle-von-konnataler-Deglutinationstuberkulose/316232768481747&quot; rel=&quot;nofollow noreferrer&quot;&gt;Facebook page&lt;/a&gt; on this concept!&lt;/p&gt;&#xA;&lt;p&gt;So, with the preferred spelling,  references show up everywhere.&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-01T19:10:43.323" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7111" PostTypeId="2" ParentId="7107" CreationDate="2018-03-01T14:55:04.273" Score="5" Body="&lt;p&gt;I believe this is what happened. Sommerfeld (&lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=osu.32435031013592&amp;amp;num=962&quot; rel=&quot;noreferrer&quot;&gt;1915, p. 430&lt;/a&gt;) $=$ (&lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144537&amp;amp;num=12&quot; rel=&quot;noreferrer&quot;&gt;1916a, p. 12&lt;/a&gt;) originally introduced an “azimutal” quantum condition and number $n$ such that $\smash{\oint p_\varphi\,d\varphi=nh}$. This $n$ is what we now call the &lt;a href=&quot;https://en.wikipedia.org/wiki/Magnetic_quantum_number&quot; rel=&quot;noreferrer&quot;&gt;magnetic&lt;/a&gt; quantum number $m$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then in (&lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015010783705&amp;amp;num=491&quot; rel=&quot;noreferrer&quot;&gt;1916b&lt;/a&gt;) he implicitly switched his &lt;a href=&quot;//en.wikipedia.org/wiki/Azimuth&quot; rel=&quot;noreferrer&quot;&gt;azimuth&lt;/a&gt; from being $\alpha$ to $\gamma$ in this &lt;a href=&quot;//en.wikipedia.org/wiki/Euler_angles#Geometrical_definition&quot; rel=&quot;noreferrer&quot;&gt;figure&lt;/a&gt;, where the tilted plane would be his “Bahnebene”. As he writes, the resulting “azimutal quantum $n$ splits into quanta $n_1$ and $n_2$ belonging to the coordinates $\varphi$ and $\vartheta$”: $\smash{\oint p_\varphi\,d\varphi=n_1h}$ and $\smash{\oint p_\vartheta\,d\vartheta=n_2h}$. That new $n=n_1+n_2$ is what we &lt;a href=&quot;//hsm.stackexchange.com/questions/728/why-did-angular-momentum-get-the-letter-l&quot;&gt;now&lt;/a&gt; call $\ell$. It measures angular momentum in the direction normal to a putative “orbit’s plane” rather than in a fixed vertical direction.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-03-01T14:55:04.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7112" PostTypeId="1" CreationDate="2018-03-02T11:30:52.573" Score="7" ViewCount="438" Body="&lt;p&gt;Context: I have been thinking about Frege's &lt;em&gt;Begriffsschrift&lt;/em&gt;, where he introduces a version of what we now think of as the standard quantifier/variable notation. Philosophers who write on Frege tend to present this as a rather stunning advance in logic. But arguably Frege is just regimenting what he sees as existing practice with the use of variables to make general statements in mathematics (his novelty being the explicit scope-markers). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Query: Can you point me to some useful article(s) on how mid nineteenth century mathematicians, and perhaps earlier ones, used and thought of variables (especially in making general claims)? Preferably some piece(s) of limited size, rather a Big Book on the history of maths!&lt;/p&gt;&#xA;" OwnerUserId="343" LastEditorUserId="343" LastEditDate="2018-03-02T22:40:38.103" LastActivityDate="2018-03-02T22:40:38.103" Title="How were variables used and understood in (particularly) 19th century maths?" Tags="|mathematics|reference-request|notation|mathematical-logic|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="7114" PostTypeId="2" ParentId="7112" CreationDate="2018-03-02T21:16:17.777" Score="3" Body="&lt;p&gt;Frege did not see himself as regimenting existing practice. He explicitly says that he developed Begriffsschrift (concept-script) because he was dissatisfied with the vagueness of natural language for the &lt;em&gt;conceptual analysis&lt;/em&gt; of mathematics, the purpose of which was not to service practice but to remake it after exposing its logical foundations (logicist program). Notationally, Frege's version of the predicate calculus with quantifiers is sometimes described as &amp;quot;atrocious&amp;quot;, and it played virtually no role in the adoption of the variable/quantifier notation. &lt;a href=&quot;https://plato.stanford.edu/entries/algebra-logic-tradition/#DeMorPeiRelQuaAlgLog&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP's Algebra of Logic Tradition&lt;/a&gt;, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01445348408837062&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dipert's Peirce, Frege, the Logic of Relations, and Church's Theorem&lt;/a&gt; and &lt;a href=&quot;https://pdfs.semanticscholar.org/4d36/5e07c3104dd1bc5780620424ecdafa89979a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peckhaus's Calculus Ratiocinator vs. Characteristica Universalis? The Two Traditions in Logic, Revisited&lt;/a&gt; give detailed accounts of relevant developments and controversies.&lt;/p&gt;&#xA;&lt;p&gt;Two years after Begriffsschrift, in 1881, C.S. Peirce and his student Mitchell independently introduced their version of the quantifier calculus that evolved into what we have today, the original versions of &lt;span class=&quot;math-container&quot;&gt;$\exists$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\forall$&lt;/span&gt; were &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt;, respectively, introduced by Peirce in 1885. It spread after being enshrined in Schröder's monumental Vorlesungen über die Algebra der Logik, the three volumes published in 1890, 1895 and 1905, and picked up by Peano and Russell, among others. Vorlesungen became the notational blueprint for Russell's Principles of Mathematics (1903), and later his with Whitehead Principia Mathematica (1910-1913). Frege's role would likely have been forgotten if Russell did not pluck him from obscurity in his promotion of logicism.&lt;/p&gt;&#xA;&lt;p&gt;Peirce built on past work more than Frege but even he was not codifying existing practice. The use of variables as in quantifier logic was largely alien to pre-19th century mathematics, and even informal use of nested quantifiers does not really appear until the work of Dedekind and Weierstrass (one can detect some precursors to it in Gauss, Cauchy, Dirichlet and Riemann, among others, but their use is transitional from the older framework of &amp;quot;variable quantities&amp;quot;). Aristotelian logic, syllogistic, is roughly equivalent to the monadic (one-place) predicate calculus, and does not require variables, mathematical purposes met by nested quantifiers were served by iterated construction (as in Euclidean geometry) and infinitesimal/kinematic interpretations (as in calculus). For a comparison of pre and post quantifier practice see &lt;a href=&quot;https://www.jstor.org/stable/2185244&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedman's Kant's Theory of Geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The movement towards modern use originated in the rise of abstract algebra, the origins of which can perhaps be traced to Vieta, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2408/55&quot;&gt;Viète's Relevance and his Connection to Euler&lt;/a&gt;, and it was partly anticipated by Leibniz. However, Leibniz's logical ideas had very little influence, and much of his work remained unpublished until after they were rediscovered by others. An extension of algebra to logic was developed by Boole in The Mathematical Analysis of Logic (1847), where propositional variables were introduced, and de Morgan introduced polyadic predicates (relations) in On the syllogism, No. IV, and on the Logic of Relations (1859), thus setting the stage for quantifiers. Weierstrass's limits and Dedekind's cuts soon generated applications for them. Here is from Dipert:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Frege, apparently in response to the negative reviews attracted by his Begriffsschrift, especially one by Schroder, wrote several essays in the early 1880s in an attempt to show how his logic was superior to traditional and Boolean logic (see Frege 1880 and 1882). He shows no appreciation of the work by De Morgan and Peirce, primarily criticizes Boole's work from a quarter century earlier, and seizes mainly on issues and faults which are controversial, even today, or from which most Booleans had retreated or were then retreating... The advantages of his elegant, rigorous theory, and the implications it had for the theory of relations, became lost in this polemic for most 19th century readers.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;However, Frege's battle was not lost. His victory was merely postponed a generation. Unfortunately, the legitimate contributions of Peirce and Schröder, especially to the logic of the relations, did get lost in the ensuing fray. Neither Peirce nor Schröder had the services of such an excellent propagandist as Russell. The Peirce-Schröder calculus was portrayed as purely algebraic, without the variable-binding operators Peirce regarded as essential and to which Schröder usually resorted; its weaknesses were rhetorically exploited with the bon mot 'too complicated'; its subtlest achievements were ignored (e.g., clever theorems proven and Peirce's insights into the differences between the monadic and polyadic predicate logic); and, in a final injustice, the development of the theory of relations in Principia Mathematica owes most, especially in notation, to Schröder via the influence of Peano rather than to Frege, but it was presented without substantial acknowledgement. Alfred Tarski is one of the few logicians or historians writing in the 20th century who seems to realize the proportions of this injustice&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-02T21:31:18.907" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7115" PostTypeId="2" ParentId="1939" CreationDate="2018-03-02T21:32:46.537" Score="4" Body="&lt;p&gt;You make a fair amount of valid points however you need to see the progression of what was happening at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we assume that Indian math developed independently you'd see they were at the forefront of a lot of &quot;inventions&quot; and &quot;discoveries&quot; in math. Here are a few in as close chronological order as possible from Aryabhatta to Bhaskara II and Madhava in 1500 CE:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The base 10 numeric system and the creation of 0 that actually makes modern math &quot;easy&quot; in a sense as opposed to sexagesimal system of Babylonians or Greek/Roman numerals&lt;/li&gt;&#xA;&lt;li&gt;Creation of sines and cosines as opposed to Ptolemy's chord - literally creating the field of trigonometry&lt;/li&gt;&#xA;&lt;li&gt;Prediction of Heliocentricity w.r.t. computing the orbits of the planets (although this is still being debated since the computational models of Aryabhatta seemed to be based on epicycles. Wikipedia should provide necessary references on this and further reading)&lt;/li&gt;&#xA;&lt;li&gt;Discovery and acceptance of negative numbers as &quot;numbers&quot; and how to operate with them&lt;/li&gt;&#xA;&lt;li&gt;Discovery of the quadratic formula taught in schools&lt;/li&gt;&#xA;&lt;li&gt;Discovery of the various infinite series for sine, cosine and also $\pi$ - mostly by Madhava around 1500 CE, 200 years before Newton/Liebniz rediscovered them.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;These are just the &lt;em&gt;hallmark&lt;/em&gt; discoveries I've highlighted. After 1300 CE we see invasions by the Khilji Dynasty, followed by the Moghuls. &lt;a href=&quot;https://en.wikipedia.org/wiki/Taxila#Centre_of_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Takshila University&lt;/a&gt; predated Nalanda but was caught up in wars around the first 300 yrs CE. A large amount of knowledge was perhaps &lt;a href=&quot;https://2ndlook.wordpress.com/2009/08/04/destruction-of-takshashila-a-defining-moment/&quot; rel=&quot;nofollow noreferrer&quot;&gt;decimated&lt;/a&gt;. But the Kerela school where Madhava taught was perhaps safe from all this (my conjecture).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this brings us to the second case with Indian Math - AFAIK Indians were not into &quot;spreading&quot; their discoveries/knowledge. Culturally, people &quot;went to India&quot; to seek knowledge. In the European world, you see a lot of peer snubbing and arrogance to &quot;show each other&quot; how good they were and thus you see a lot of &quot;correspondence&quot; going back and forth (apologies for my strong opinion on this). A side effect of this was proliferation and sharing of knowledge and how our modern &quot;conferences&quot; were born.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;India lacked this so it took us this long to even know that discoveries we attribute to Newton, Liebniz and other prominent European mathematicians, Indians already had a good grasp on it. In fact, Newton wasn't even aware of the advances of Indian math!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, &quot;if it didn't come from Europe it wasn't any good&quot; was a common belief owing to the &quot;math superiority&quot; at the time. Hence there's a high chance that everything else was ignored.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the time, India was really struggling with too many invasions and would continue to do so until 1947 (India's freedom from the British rule). Some folks like Ramanujan were fortunate to be recognized but not many were that fortunate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Change of schooling system: The British rule put forth a new system of education as per the needs of the industrial revolution - it involved &quot;understanding the current state of affairs/knowledge&quot; without much regard to innovation/thinking (as well as not accepting Indian education as a &quot;good enough&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given all this, you can see that circumstances just weren't in India' favor and hence the feeling of &quot;lagging behind&quot;. It &quot;missed out&quot; because India was busy trying to just survive and focus on freeing itself from the various empires.&lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-03-02T21:32:46.537" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7116" PostTypeId="2" ParentId="7094" CreationDate="2018-03-03T00:13:57.393" Score="9" Body="&lt;p&gt;Gauss in &lt;a href=&quot;http://epsaleph.tripod.com/sitebuildercontent/sitebuilderfiles/disquisitionesarithmeticae.pdf&quot; rel=&quot;noreferrer&quot;&gt;Disquisitiones Arithmeticae (1799)&lt;/a&gt; does indeed express something close to what is now called mathematical formalism and structuralism. He writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;What is calculated (in the sense of things already counted) are not substances (thinkable objects for themselves), but relations between two objects counted two by two... The mathematician abstracts totally from the nature of the objects and the content of their relations; he is concerned solely with the counting and the comparison of the relations among themselves&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But it is hard to ascribe similar opinions to Kant, in fact his view was just the opposite. That concepts without intuitions are empty, and that (pure) intuitions for mathematical concepts are supplied by imaginative construction according to a priori schemata of space and time. In other words, relations are constructed only along with their intuitive objects. This led Kant to claim the a priori certainty for Euclidean geometry, something &lt;a href=&quot;https://anewdifferentworld.wordpress.com/2015/07/16/gauss-kastner-and-kant&quot; rel=&quot;noreferrer&quot;&gt;Gauss explicitly criticized him for&lt;/a&gt;:&quot;&lt;em&gt;Precisely the impossibility of deciding a priori between [Euclidean and non-Euclidean space] gives the clearest proof that Kant was not justified in asserting that space is just the form of our perception.&lt;/em&gt;&quot; Indeed, relationalization and formalization of mathematics in 19-th century went hand in hand with rejecting (notably by Frege and Hilbert) Kant's intuitive conception of it (which was retained more by Poincare and intuitionists).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss had plenty of earlier sources to build on though. The abstractization of mathematics can be traced back to Vieta's Isagoge (1591). Bos writes in &lt;a href=&quot;https://books.google.com/books?id=ApbfBwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Redefining Geometrical Exactness, Ch.8&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was Viète who first introduced and promoted the idea that algebra was proper method for analysis of problems both in geometry, and in the theory of numbers... Viète usually reserved the term specious logistics for that part of his new algebra that dealt with abstract magnitude and in which therefore no assumptions coud be made about the actual effectuation of algebraic operations... Viète did not see algebra as a technique concerning numbers... but as a method of symbolic calculation concerning abstract magnitudes.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For more see &lt;a href=&quot;https://philosophy.stackexchange.com/questions/49451/gauss-and-the-defense-of-abstract-mathematics&quot;&gt;Esteve's The Role of Symbolic Language in the transformation of Mathematics&lt;/a&gt;. The next thinker to advance relational/abstract view of mathematics, and Kant's chosen foil, was Leibniz. He is the one who called imaginary numbers and infinitesimals &quot;useful fictions&quot; and promoted the so-called &lt;a href=&quot;http://www.jstor.org/stable/41134263&quot; rel=&quot;noreferrer&quot;&gt;&quot;generality of algebra&quot;&lt;/a&gt; (the term was coined by Cauchy), treating algebraic idenities as purely formal rules that apply regardless of the nature of the quantities involved. Peckhaus describes in &lt;a href=&quot;https://pdfs.semanticscholar.org/4d36/5e07c3104dd1bc5780620424ecdafa89979a.pdf&quot; rel=&quot;noreferrer&quot;&gt;Calculus Ratiocinator vs. Characteristica&#xA;Universalis&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;‘Abbreviating’ means that as soon as a characteristic sign has been established for a complex object, memory can be relieved of the burden of retaining all the characteristic elements of this object. Natural languages are not sufficient for this job of designating objects unambiguously. Only in the language of arithmetic and algebra has this idea been partially realized. All reasoning in these branches consists in using characters. Errors in reasoning prove to be miscalculations... He uses arbitrarily chosen letters according to the model of mathematics. This notation allows ‘calculating with concepts’ according to sets of rules, each of them forming a calculus ratiocinator.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Frege specifically names Leibniz's calculus ratiocinator as inspiration for his concept-script (predicate calculus). Generality of algebra was later extensively used by Euler, Lagrange, and Gauss himself, in &quot;algebraic analysis&quot; that preceded Weierstrassian one, see &lt;a href=&quot;http://www.jstor.org/stable/41134263&quot; rel=&quot;noreferrer&quot;&gt;The Foundational Aspects of Gauss's Work by Ferraro&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Finally, eighteenth-century functions were characterised in an essential way by the use of a formal methodology that it made it possible to operate upon analytical expressions, independently of their meaning. This formal methodology was based upon two closely connected analogical principles, the generality of algebra and the extension of rules and procedures from the finite to the infinite. The generality of algebra consisted of the following assumption: (GA) if an analytical formula was derived by using the rules of algebra, then it was thought to be valid in general&quot;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Peacock later renamed this into &lt;a href=&quot;https://hsm.stackexchange.com/questions/606/why-does-the-principle-of-permanence-have-two-different-definitions&quot;&gt;Principle of Permanence of Form&lt;/a&gt; in his Symbolical Algebra (1831):&quot;&lt;em&gt;Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote.&lt;/em&gt;&quot; The abstractization of algebra was further promoted, before Hilbert, by Hankel and Dedekind.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-03T00:23:06.017" LastActivityDate="2018-03-03T00:23:06.017" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7117" PostTypeId="2" ParentId="7109" CreationDate="2018-03-03T07:29:51.830" Score="1" Body="&lt;p&gt;Please allow me to answer my own question by putting together information gleaned from &lt;a href=&quot;https://en.wikipedia.org/wiki/Roulette&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt; and &lt;a href=&quot;http://casinoreviews.co.uk/roulette/history&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, I am reasonably confident that the following is correct. If you have stronger evidence, such as a local newspaper photo/sketch of that time period, please do post an answer and I'll accept it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The layout of the wheel in use at the Monte Carlo casino on August 1913 was almost certainly the so-called &quot;European&quot; style with a single zero. By the end of the nineteenth century, the European layout (originating from Germany) had replaced the American layout (originating from France) at Monte Carlo. The layout has remained basically unchanged since (order of the number sequence and coloring scheme) except that the zero is now commonly colored green.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jqvFU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jqvFU.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:La_Roulette_de_Monte-Carlo_R%C3%A8gle_du_Jeu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://commons.wikimedia.org/wiki/File:La_Roulette_de_Monte-Carlo_R%C3%A8gle_du_Jeu.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks to users Carl Witthoft and sand1 for their comments.&lt;/p&gt;&#xA;" OwnerUserId="463" LastActivityDate="2018-03-03T07:29:51.830" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7118" PostTypeId="1" AcceptedAnswerId="7119" CreationDate="2018-03-03T17:24:16.763" Score="4" ViewCount="2949" Body="&lt;p&gt;Kinetic energy is often written as &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$KE$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$E_k$&lt;/span&gt;. Where does &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; come from in quantum mechanics? Why and how did it come to be different?&lt;/p&gt;&#xA;" OwnerUserId="7105" OwnerDisplayName="Arch Stanton" LastEditorUserId="7105" LastEditDate="2020-02-07T14:28:14.170" LastActivityDate="2020-02-07T14:28:14.170" Title="Why is kinetic energy denoted by the letter $T$ in quantum mechanics?" Tags="|energy|terminology|notation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7119" PostTypeId="2" ParentId="7118" CreationDate="2018-03-04T04:26:36.007" Score="5" Body="&lt;p&gt;I’m pretty sure Lagrange started this, in &lt;em&gt;Méchanique analitique&lt;/em&gt; (1788, &lt;a href=&quot;http://www.e-rara.ch/zut/content/pageview/9874626&quot; rel=&quot;noreferrer&quot;&gt;p. 224&lt;/a&gt;; 1809, &lt;a href=&quot;http://www.archive.org/stream/mmoiresdelinst09inst#page/263&quot; rel=&quot;noreferrer&quot;&gt;p. 263&lt;/a&gt;; 1811, &lt;a href=&quot;http://www.e-rara.ch/zut/content/pageview/6303351&quot; rel=&quot;noreferrer&quot;&gt;p. 311&lt;/a&gt;; 1815, &lt;a href=&quot;http://www.e-rara.ch/zut/content/pageview/6303472&quot; rel=&quot;noreferrer&quot;&gt;p. 2&lt;/a&gt;): his predecessors mostly worked with the &lt;em&gt;vis viva&lt;/em&gt; ($=2T$) instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(As to why he chose the letter  $T$, no idea. Words like kinetic, energy, work, appeared only &lt;a href=&quot;https://en.wikipedia.org/wiki/Kinetic_energy#History_and_etymology&quot; rel=&quot;noreferrer&quot;&gt;later&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-03-04T08:25:48.443" LastActivityDate="2018-03-04T08:25:48.443" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7120" PostTypeId="2" ParentId="7118" CreationDate="2018-03-04T06:55:39.837" Score="6" Body="&lt;p&gt;Gaspard-Gustave Coriolis and Jean-Victor Poncelet used the name &quot;quantité de travail&quot; (quantity of work) and &quot;travail mécanique&quot; (mechanical work) to denote the kinetic energy. I guess that this (the term &quot;travail&quot;) could be the origin of the symbol $T$. See at the end of &lt;a href=&quot;https://en.wikipedia.org/wiki/Vis_viva&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Vis_viva&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Be patient for my superficiality, I am very far from being a historian of science, I am just a mathematical physicist and this post migrated here from PSE.)&lt;/p&gt;&#xA;" OwnerUserId="7109" LastEditorUserId="7109" LastEditDate="2018-03-04T09:21:02.697" LastActivityDate="2018-03-04T09:21:02.697" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7122" PostTypeId="1" AcceptedAnswerId="7123" CreationDate="2018-03-05T02:00:52.867" Score="4" ViewCount="506" Body="&lt;p&gt;I have been reading about the history of Farey fractions and I am intrigued by the appearance of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mediant_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;mediant&lt;/a&gt; in the Farey tree and related objects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After a bit of searching, &lt;a href=&quot;https://fr.wikipedia.org/wiki/Nicolas_Chuquet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicolas Chuquet's&lt;/a&gt; &lt;em&gt;Triparty&lt;/em&gt; (drafted 1484, lost, rediscovered and published in 1880) is the earliest reference I can find featuring the mediant -  what Chuquet calls the “règle des nombres moyens”.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although such an &quot;operation&quot; appears &lt;em&gt;natural&lt;/em&gt; from the point of view of modern algebraic notations, it is not clear to me if this operation would have received any attention from previous generations of mathematicians.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: When did the mediant first appear in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2018-03-06T18:32:02.483" LastActivityDate="2020-12-04T15:16:19.857" Title="When did the mediant (&quot;freshman sum&quot;) of two fractions first appear in mathematics?" Tags="|mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7123" PostTypeId="2" ParentId="7122" CreationDate="2018-03-06T01:26:44.190" Score="4" Body="&lt;p&gt;Precursors of continued fractions in a geometric guise go back to ancient Greece, geometric version of producing successive terms of a continued fraction expansion was known as anthyphairesis. Fowler is perhaps the most thorough scholar of ratios and anthyphairesis. Here is from his &lt;a href=&quot;https://www.jstor.org/stable/41133888&quot; rel=&quot;nofollow noreferrer&quot;&gt;Approximation Technique, and its Use by Wallis and Taylor&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This procedure can be used to approximate a wide class of numbers whose values are not known in advance, like roots and logarithms (as in Taylor's calculation, below), or it can be used to find more convenient approximations to a number  for which some accurate but inconvenient approximation is given (as in Wallis' calculation).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;This mediant has a long history going back to classical Greece: a special case&#xA;of it is to be found at Plato's Parmenides 154b-d, and, while the inequality&#xA;does not occur in Euclid's Elements, the case of equality was treated as V 12&#xA;and VII 12. It was then stated and proved in Pappus, Collection VIII 8. Chuquet&#xA;rediscovered it and called it 'la rigle des nombres moyens&amp;quot; in his Triparty en&#xA;la Science des Nombres (1484). It was, we shall see, adduced by both Wallis&#xA;and, in a variant form, by Taylor. It was the first theorem enunciated in Cauchy's  Cours d'Analyse (1821). It is also, inter alia, the basic generating property of  Farey series. Thus we have an enduring motif of mathematics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This second paragraph inspired Guthery to write a monograph &lt;a href=&quot;http://www.maths.ed.ac.uk/%7Eaar/papers/farey.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Motif of Mathematics&lt;/a&gt; dedicated specifically to the mediant, its significance and history, from ancient roots to Farey series and the Riemann hypothesis.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-06T01:26:44.190" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7125" PostTypeId="1" CreationDate="2018-03-07T03:20:45.170" Score="5" ViewCount="336" Body="&lt;p&gt;As far as I can tell, the first occurrence of the sum notation (&lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt;) was in Euler's book &lt;em&gt;Institutiones calculi differentialis&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quemadmodum ad differentiam denotandam usi sumus signo &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;, ita summam indicabimus signo &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt;...&lt;/p&gt;&#xA;&lt;p&gt;(Just as we used the symbol &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; to signify a difference, so we use the symbol &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; to indicate a sum...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I can't seem to find any indication of where the product symbol (&lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt;) first occurred. Does anyone know the source?&lt;/p&gt;&#xA;" OwnerUserId="7118" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-08T01:04:46.550" Title="First appearance of the product symbol ($\Pi$)" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7126" PostTypeId="2" ParentId="7125" CreationDate="2018-03-07T18:17:35.480" Score="0" Body="&lt;p&gt;According to Cajori's &lt;em&gt;A History of Mathematical Notations Volume 2&lt;/em&gt; the notation first appeared in Gauss' &quot;Disquisitiones generales circa seriem infinitam $1+\frac{\alpha\beta}{1\cdot\gamma}x+\frac{\alpha(\alpha+1)\beta(\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}xx+\frac{\alpha(\alpha+1)(\alpha+2)\beta(\beta+1)(\beta+2)}{1\cdot2\cdot3\cdot\gamma(\gamma+1)(\gamma+2)}x^3+$ etc&quot;&#xA;which appears in Werke Vol III. &lt;/p&gt;&#xA;" OwnerUserId="3273" LastActivityDate="2018-03-07T18:17:35.480" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7127" PostTypeId="2" ParentId="7125" CreationDate="2018-03-07T20:12:59.140" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jan_Gullberg#Mathematics:_From_the_Birth_of_Numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gullberg's Mathematics: From the Birth of Numbers&lt;/a&gt; attributes first use to Descartes. This is not exactly a history book, and it comes with no source reference, so it is not very credible. If Descartes did use it it would have been a historical accident with no connection to modern use. Vieta before him and Wallis after him, along with a host of 17-18th century authors including Euler, just wrote several factors followed by some equivalent of &quot;etc.&quot;, a whole sentence for Vieta and just &quot;&amp;amp;c.&quot; for Wallis. Modern use of ${\small \Pi}$ tracks to 19-th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/AHistoryOfMathematicalNotationVolII&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History of Mathematical Notations, v.2, p.78&lt;/a&gt; cites Gauss's &lt;a href=&quot;https://www.cambridge.org/core/books/werke/disquisitiones-generales-circa-seriem-infinitam/067B5BE5A68C4DBBE44E9E4C42092FE1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Disquisitiones Generales circa Seriem Infinitam&lt;/a&gt; as the earliest use, but what Gauss introduces there is the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Gamma_function#Pi_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pi function&lt;/a&gt;, not the product symbol: &#xA;$$&#xA;{\small \Pi (k,z)={\frac {~1~~~\,\cdot\,~~~2~~~\,\cdot\,~~~3\,\,.\,.\,.\,.\,.\,\cdot\,k~~~}{(z+1)(z+2)(z+3)\,.\,.\,.\,.\,(z+k)}}k^{z}\,.}&#xA;$$&#xA;I suppose Cajori believed that this is from where the notation spread, eventually mutating in meaning. His reference to Jordan's Traite de Substitutionis (1870) is the earliest where the use is recognizably modern. There was a &lt;a href=&quot;https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2007_April_8&quot; rel=&quot;nofollow noreferrer&quot;&gt;discussion of big Pi by Wikipedia editors&lt;/a&gt; one of whom pointed to earlier occurrences:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Did a bit of digging myself, and found one use of $\Sigma$ for summation in Fourier's 1822 Théorie analytique de la chaleur and one in Cauchy's 1826 Leçons sur les applications de calcul infinitésimal. Both authors assume the reader is unfamiliar with the notation and provide an explanation. Some thirty years later, Riemann in his 1859 Über die Anzahl der Primzahlen unter einer gegebenen Größe and Dedekind in his 1863 Vorlesungen über Zahlentheorie are both using $\Sigma$ and $\Pi$ without further explanation. This indicates that the summation and product notations passed into general usage sometime between 1820 and 1860.&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems that one would have to pore over first half of 19th century papers and books to locate the &quot;earliest&quot; use.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-08T01:04:46.550" LastActivityDate="2018-03-08T01:04:46.550" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7128" PostTypeId="2" ParentId="6808" CreationDate="2018-03-08T12:53:39.497" Score="2" Body="&lt;p&gt;As far as general relativity is concerned, the issue seems to date back at least to 1914, before general relativity was properly done, in Einstein's paper &quot;Die formale Grundlage der allgemeinen Relativitätstheorie&quot; (formal foundations of relativity). It includes the following statement on the topic : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An even deeper-reaching question of fundamental significance shall now be brought up - and I am not able to answer it. In the ordinary theory of relativity, every line that describes the movement of a material point, i.e., every line consisting only of time-like elements, is necessarily nonclosed, the reason being that such a line never contains elements for which $dx_4$ vanishes. An analogous statement cannot be claimed for the theory developed here. It is therefore a priori possible to imagine a point movement where the four-dimensional curve of the point is almost a closed one. In this case &lt;em&gt;one and the same&lt;/em&gt; material point could exist in an arbitrarily small spacetime domain in &lt;em&gt;several seemingly mutually independant representations&lt;/em&gt;. This runs counter to my physical imagination in the most vivid manner. However, I am not able to demonstrate that the occurence of such curve can be excluded from the theory that has been developed here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2018-03-08T12:53:39.497" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7129" PostTypeId="1" CreationDate="2018-03-08T19:12:51.983" Score="3" ViewCount="85" Body="&lt;p&gt;I stumbled on this image from &lt;a href=&quot;https://math.berkeley.edu/~giventh/kepler_091615.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Io2yu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Io2yu.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's mentioned w.r.t. to the elliptical orbits of planets and how the focus-directrix property comes into play. It's an interesting POV but when was such a thing known about an ellipse's directrix? Does it predate Kepler or was it a modern discovery? Or a side effect of &lt;a href=&quot;https://en.wikipedia.org/wiki/Dandelin_spheres&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dandelin's spheres&lt;/a&gt; which are relatively modern? Or something discovered as a result of studying projective geometry?&lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-03-08T19:12:51.983" Title="When was this projective property of an ellipse's directrix known?" Tags="|geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7130" PostTypeId="1" AcceptedAnswerId="7131" CreationDate="2018-03-09T08:49:19.737" Score="5" ViewCount="353" Body="&lt;p&gt;There is a period of general relativity history famous for its lack of activity, where during the 1950's, almost no general relativity was done. Looking through various bibliographies, there were very little papers written on the specific topic of general relativity during that period, save for some papers, mostly by John Synge, L. Markus (mostly writing on the topic as a mathematician) and Bonnor (for the action of EM fields in GR). There's a handful of other authors I can find (Matte, Buchdahl, Nariai, Taub, Schwinger, Milner, Finkelstein, etc), but overall it's fairly rare to encounter a paper from the 1950's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then in the 1960's, the activity picks up, a lot of papers on causal structures, singularities and cosmology pop up. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a particular set of papers that triggered this renaissance? I can't really think of any really famous paper from the era that would trigger it (the famous papers tended to be more in the middle of this era). &lt;/p&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="4251" LastEditDate="2023-06-26T09:38:49.223" LastActivityDate="2023-06-26T09:38:49.223" Title="What triggered the general relativity renaissance?" Tags="|physics|relativity-theory|20th-century|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7131" PostTypeId="2" ParentId="7130" CreationDate="2018-03-09T16:53:32.207" Score="8" Body="&lt;p&gt;An influential figure in the later development of general relativity was John Archibald Wheeler (1911-2008). In the academic year 1952/53 he offered a yearlong course on special and general relativity at Princeton (the first such course there). According to Paul Halpern's book cited below, ``among the highlights of the spring session was a class trip on May 16 to Einstein's house (...). There, the eight graduate students in attendance had the unprecedented chance to ask the theory's designer any questions on their minds.&quot; Here are some of Wheeler's  papers on general relativity from mid- to late 1950s (the actual list may be longer if he published something in physics journals that were not covered by Mathematical Reviews):&#xA;MR0127375 Wheeler, J. A. Neutrinos, gravitation and geometry. 1959 Rend. Scuola Internaz. Fis. &quot;Enrico Fermi'', Corso XI pp. 67–196 Zanichelli, Bologna &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0100507  Weber, Joseph; Wheeler, John A. Reality of the cylindrical gravitational waves of Einstein and Rosen. Rev. Mod. Phys. 29 1957 509–515. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0094239  Lindquist, Richard W.; Wheeler, John A. Dynamics of a lattice universe by the Schwarzschild-cell method. Rev. Mod. Phys. 29 1957 432–443. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0093387  Misner, Charles W.; Wheeler, John A. Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space. Ann. Physics 2 (1957), 525–603.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0091832  Regge, Tullio; Wheeler, John A. Stability of a Schwarzschild singularity. Phys. Rev. (2) 108 (1957), 1063–1069. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0091828  Brill, Dieter R.; Wheeler, John A. Interaction of neutrinos and gravitational fields. Rev. Mod. Phys. 29 (1957), 465–479. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0090447  Power, Edwin A.; Wheeler, John A. Thermal geons. Rev. Mod. Phys. 29 (1957), 480–495. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR0067622 Wheeler, John Archibald Geons. Phys. Rev. (2) 97, (1955). 511–536. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Added:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Kruskal's paper mentioned in the comments was written up and submitted on his behalf by Wheeler.  &lt;/li&gt;&#xA;&lt;li&gt;While it might be difficult to attribute the increase in interest in general relativity to a single paper, there was an important event that influenced later developments. In 1957, a conference on general relativity later known as GR1 was organized by Bryce DeWitt and Cecile Morette DeWitt. According to the report by Peter Bergmann (one of participants) in Reviews of Modern Physics, &lt;a href=&quot;https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29.352&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29.352&lt;/a&gt;, one of main inspirations was a quest for developing quantum theory of gravity (still elusive). A readable account can be found in &quot;The quantum labyrinth. How Richard Feynman and John Wheeler  revolutionized time and reality.&quot; by Paul Halpern (Basic Books, New York, 2017) and in Feynman's own memoirs.  &lt;/li&gt;&#xA;&lt;li&gt;The history of general relativity  seems to have followed a more complex path than the `beginning-to-slump-to-golden age' scheme, according to Hubert Goenner in his paper &quot;A golden age of general relativity? Some remarks on the history of general relativity.&quot; It is available here: &lt;a href=&quot;https://arxiv.org/pdf/1607.03319.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/pdf/1607.03319.pdf&lt;/a&gt; Some developments and statistics from 1950s that he lists do not seem to support the thesis of there having been ``hardly any paper published&quot; in this decade.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="1930" LastEditDate="2018-03-17T19:26:18.790" LastActivityDate="2018-03-17T19:26:18.790" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7132" PostTypeId="1" AcceptedAnswerId="7133" CreationDate="2018-03-10T00:26:19.430" Score="8" ViewCount="224" Body="&lt;p&gt;In his article on &quot;Kant's Theory of Geometry&quot;, Michael Friedman claims that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(...) before Dedekind mathematicians would commonly give what we call the definition of denseness when explaining what they meant by &quot;continuity&quot;: namely, &quot;for every element there is a smaller&quot; or &quot;between every two elements there is a third&quot;. (p. 463/p. 60 of &lt;em&gt;Kant and the Exact Sciences&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I admit I was initially puzzled by &quot;for every element there is a smaller&quot;; I imagine he means something like &quot;every spatial region can be subdivided into parts&quot; or something)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, Friedman does not cite any of those pre-Dedekind mathematicians (I find interesting also that he cites Dedekind as the key figure; why not Bolzano, Cauchy or Weierstrass?). So here's my question: who were those mathematicians who thought of continuity in terms of denseness? Any clues?&lt;/p&gt;&#xA;" OwnerUserId="4097" LastActivityDate="2018-03-10T17:08:55.540" Title="Definitions of continuity pre-Dedekind" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7133" PostTypeId="2" ParentId="7132" CreationDate="2018-03-10T02:01:29.570" Score="5" Body="&lt;p&gt;Regrading the early consideration of &lt;em&gt;density&lt;/em&gt; as a means of defining the continuum :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;William of Ockham (c. 1280–1349) brought a considerable degree of dialectical subtlety to his analysis of continuity; it has been the subject of much scholarly dispute. For Ockham the principal difficulty presented by the continuous is the infinite divisibility of space, and in general, that of any continuum. The treatment of continuity in the first book of his Quodlibet of 1322–7 rests on the idea that between any two points on a line there is a third &lt;strong&gt;—perhaps the first explicit formulation of the property of density—&lt;/strong&gt; and on the distinction between a continuum “whose parts form a unity” from a contiguum of juxtaposed things. Ockham recognizes that it follows from the property of density that on arbitrarily small stretches of a line infinitely many points must lie, but resists the conclusion that lines, or indeed any continuum, consists of points. Concerned, rather, to determine “the sense in which the line may be said to consist or to be made up of anything.”, Ockham claims that “no part of the line is indivisible, nor is any part of a continuum indivisible.” While Ockham does not assert that a line is actually “composed” of points, he had the insight, startling in its prescience, that a punctate and yet continuous line becomes a possibility when conceived as a dense array of points, rather than as an assemblage of points in contiguous succession.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Subsequently, Bolzano, Cauchy, and Weierstrass all gave definitions of continuous functions and wrestled with the concept of the continuum.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dedekind appears to be the first to recognise that density is not a sufficient condition for continuity:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Following Weierstrass's efforts, another attack on the problem of formulating rigorous definitions of continuity and the real numbers was mounted by Richard Dedekind (1831–1916). Dedekind focussed attention on the question: exactly what is it that distinguishes a continuous domain from a discontinuous one? He seems to have been the first to recognize that the property of density, possessed by the ordered set of rational numbers, is insufficient to guarantee continuity. In Continuity and Irrational Numbers (1872) he remarks that when the rational numbers are associated to points on a straight line, “there are infinitely many points [on the line] to which no rational number corresponds” so that the rational numbers manifest “a gappiness, incompleteness, discontinuity”, in contrast with the straight line's “absence of gaps, completeness, continuity.” Dedekind regards this principle as being essentially indemonstrable; he ascribes to it, rather, the status of an axiom “by which we attribute to the line its continuity, by which we think continuity into the line.” It is not, Dedekind stresses, necessary for space to be continuous in this sense, for “many of its properties would remain the same even if it were discontinuous.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Dedekind's continuum is a point set.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is to be noted that Dedekind does not identify irrational numbers with cuts; rather, each irrational number is newly “created” by a mental act, and remains quite distinct from its associated cut. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://plato.stanford.edu/entries/continuity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stanford Encyclopedia of Philosophy&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the arXiv paper &lt;a href=&quot;https://arxiv.org/pdf/1503.03117.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the History of the Number Line&lt;/a&gt;, Bolzano's algebraic approach to the problem used the notions of equality, order, and density. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He introduced the notion of a measurable number, relations described as “equal to”, “greater than”, “less than”; asserted density (pantachisch) of a set of real numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The paper makes no mention of density with reference to Cauchy or earlier mathematicians - e.g., no mention of Ockham at all.  Also, Bolzano's term &lt;em&gt;pantachisch&lt;/em&gt; does not translate using google's translate program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding Leibniz's view of density (mentioned in the comments), &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One might be tempted to attribute to Leibniz the modern view that density is not a sufficient condition for continuity. (...) There is, however, ample evidence that for Leibniz, dense quantities, namely that between any two numbers there exists a third, are continuous. For Leibniz,&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;Continuous, is a whole such that its co-intergrating (cointegrands) parts, (i.e., such that taken together coincide with the whole) have something in common, and moreover such that if its parts are not redundant (i.e., they have no part in common, that is, there aggregate is equal to the whole) they have at least a term [terminum] in common.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source : &lt;a href=&quot;https://books.google.ca/books?id=UMnnCwAAQBAJ&amp;amp;pg=PA118&amp;amp;lpg=PA118&amp;amp;dq=leibniz%20density%20of%20rational%20numbers&amp;amp;source=bl&amp;amp;ots=ONXh1l9nuH&amp;amp;sig=XoG27bUctf5iiqjwU2GsO0u9Dfw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi8rJW2meLZAhVSzGMKHTiQAckQ6AEIKTAA#v=onepage&amp;amp;q=leibniz%20density%20of%20rational%20numbers&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz and Clarke - A Study of Their Correspondence&lt;/a&gt;, by Ezio Vailati.    &lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2018-03-10T17:08:55.540" LastActivityDate="2018-03-10T17:08:55.540" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7134" PostTypeId="1" CreationDate="2018-03-11T19:49:38.623" Score="5" ViewCount="3075" Body="&lt;p&gt;Why is it that, in ancient Greece, mathematicians tried to solve geometrical problems using compass and straightedge only and, apparently, only if that failed, they tried to use other tools? Note that this question is not about &lt;em&gt;what&lt;/em&gt; they were (and weren't) able to achive using compass and straightedge alone, but about &lt;em&gt;why&lt;/em&gt; that was their first choice.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-04-27T05:00:03.300" Title="Compass and straightedge: why?" Tags="|ancient-greece|euclidean-geometry|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7135" PostTypeId="2" ParentId="7134" CreationDate="2018-03-11T22:31:19.543" Score="7" Body="&lt;p&gt;The real reason is probably that straightedge and especially compass are the simplest, most primitive instruments, and also easy to make. At the same time they are quite accurate. (Straightedge is not so easy to make and is not so accurate, but certainly it stands next to the compass on these criteria).&#xA;A modern high school student tends to think that to double the cube it is easier to measure the side, then to calculate the cubic root approximately. But this is anachronism. Deprive the student of all modern tools, and prohibit to use the decimal system. How will s/he do the job?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you think of the tool which is needed to double the cube (a parabola, invented by Menaechmus, exactly for this purpose),&#xA;it is much more complicated than a straightedge and a compass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This real reason was disguised: these instruments were probably considered &quot;sacred&quot; at some time. There is a popular theory that in general, geometry has ritualistic origin:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/article/10.1007/BF00327767&quot; rel=&quot;nofollow noreferrer&quot;&gt;The ritual origin of geometry&quot;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the problem of doubling the cube clearly has some ritualistic origin: the original question was about doubling an Apollo altar. And it is natural to expect that only some sacred tools were permitted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a later time the sacred principle became a philosophical principle: the &#xA;straight line and circle are the only &quot;perfect&quot; figures because &quot;they look the same at all of their parts.&quot; (In modern language: isometry group acts transitively on them.) This was explicitly proclaimed by Plato and Aristotle.&#xA;In particular, Plato declared that the main goal of astronomy is to reduce&#xA;all celestial motions to uniform circular motions. (Why circular? Why not on ellipses/parabolas?) This philosophy actually slowed down the development of astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: the ancient Greeks were great theorists in mathematics but not so good&#xA;in practical applications. They probably thought a straightedge was a great&#xA;instrument without trying it in practice for precise constructions. That a straightedge is not really useful when high accuracy is required was only discovered by the 18th century by precision instrument makers. If the Greeks had discovered this, they would have limited their geometric constructions to &quot;compass only.&quot; In fact, one can do with just a compass everything which one can do with straightedge + compass.&#xA;That the straightedge is really redundant was proved in 19th century. See&#xA; &lt;a href=&quot;https://hsm.stackexchange.com/questions/287/division-of-the-circle-and-compass-constructions&quot;&gt;Division of the circle and compass constructions&lt;/a&gt; for a discussion of this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2019-05-08T23:19:13.227" LastActivityDate="2019-05-08T23:19:13.227" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7137" PostTypeId="1" CreationDate="2018-03-12T08:12:42.417" Score="-2" ViewCount="225" Body="&lt;p&gt;In the course of preparing a lecture for graduate students about errors of early set theory I have come across this nice example reported by Hausdorff which violates Peano's successor axiom but nevertheless seems to have been accepted for more than 40 years:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was usual to argue as follows: From the set $A$ to be well-ordered take by arbitrary choice an element and denote it as $a_0$, then from the set $A - \{a_0\}$ an element $a_1$, then an element from the set $A - \{a_0, a_1\}$ and so on. If the set $\{a_0, a_1, a_2, …\}$ is not yet the complete set $A$, we can choose from $A - \{a_0, a_1, a_2, …\}$ an element $a_{\omega}$, then an element $a_{\omega+1}$, and so on. This procedure must come to an end, because beyond the set $W$ of ordinal numbers which are mapped on elements of $A$, there are greater numbers; these obviously cannot be mapped on elements of $A$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is reported as late as in 1914 by Felix Hausdorff, obviously without reservations because he remarks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“We cannot share most of the doubts which have been raised against this method.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hausdorff only deplores the undesired impression of a temporal process but confirms that the element $a_{\omega}$ is fully determined in the sense of transfinite induction and claims that every single action of choosing an element as well as their order has to be understood as timeless. [F. Hausdorff: “Grundzüge der Mengenlehre”, Veit, Leipzig (1914); reprinted: Chelsea Publishing Company, New York (1965) p. 133f]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested to know who of our great colleagues has originally endorsed this method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark: This question had been asked in &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Hau.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathOverflow&lt;/a&gt; where it disappeared without an answer, receiving some startling comments though. History of Science and Math seems better suited anyway.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-03-26T12:18:59.373" Title="Endorsers of the method of well-ordering reported by Hausdorff?" Tags="|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7138" PostTypeId="1" AcceptedAnswerId="7141" CreationDate="2018-03-12T16:26:58.397" Score="4" ViewCount="3772" Body="&lt;p&gt;While distance in physical formulas is often abbreviated as &lt;strong&gt;&lt;em&gt;d&lt;/em&gt;&lt;/strong&gt; (which is pretty intuitive), another common abbreviation is &lt;strong&gt;&lt;em&gt;s&lt;/em&gt;&lt;/strong&gt;, as seen e.g. &lt;a href=&quot;https://physics.info/velocity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://www.calculatorsoup.com/calculators/physics/displacement-calculator-svt.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;https://physics.stackexchange.com/questions/259507/considering-velocity-distance-graph&quot;&gt;here&lt;/a&gt;. It also seems to be used in optics to represent focal distances.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone happen to know what &lt;em&gt;S&lt;/em&gt; stands for? A quick look at Latin and Greek terms for &quot;distance&quot; doesn't seem to reveal anything starting with an S.&lt;/p&gt;&#xA;" OwnerUserId="7138" LastEditorUserId="658" LastEditDate="2018-03-13T16:56:38.520" LastActivityDate="2018-03-13T16:56:38.520" Title="Why is distance sometimes abbreviated S?" Tags="|physics|notation|differential-geometry|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7139" PostTypeId="2" ParentId="7134" CreationDate="2018-03-12T19:22:09.820" Score="2" Body="&lt;p&gt;The Greek appear to have been deeply concerned with the problem of change-and- permanance, what we call today 'invariants'. The (straight)line and the circle are the two obvious cases when an object moves and stays at the same place. Actually mechanical physics relies on the two when it admits as basic translations and rotations. Greek geometry is just thinly disguised mechanics and it has already caught this fundamental idea. Every form can be reduced to its permanent constituents which are just straight lines and circles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mohr%E2%80%93Mascheroni_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mohr–Mascheroni theorem&lt;/a&gt; showed that straightedge and compass constructions are possible with just a compass. &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2018-03-12T19:22:09.820" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7140" PostTypeId="1" CreationDate="2018-03-12T21:26:01.077" Score="4" ViewCount="123" Body="&lt;p&gt;The numbering of the chapters of Bertrand Russell and Alfred Whitehead's Principia Mathematica is the following : &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 30, etc&lt;/p&gt;&#xA;&#xA;&lt;p&gt;overall, there's about 30 chapters missing from this numbering schemes. Is this some kind of numbering where he left gaps intentionally to not have to renumber them later on? (there are technically 10 chapters per sections) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was that common practice back then?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2018-03-12T21:26:01.077" Title="The Principia Mathematica's missing chapters" Tags="|mathematics|mathematical-logic|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7141" PostTypeId="2" ParentId="7138" CreationDate="2018-03-13T00:57:53.463" Score="9" Body="&lt;p&gt;I think it most likely stands for &lt;em&gt;spatium&lt;/em&gt;. E.g. Euler’s first book &lt;em&gt;Mechanica&lt;/em&gt; (&lt;a href=&quot;http://www.archive.org/details/mechanicasivemot01eule&quot; rel=&quot;noreferrer&quot;&gt;1736&lt;/a&gt;) uses &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; throughout and first introduces it as follows (p. 13):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorema.&lt;/strong&gt; (...) &lt;em&gt;oportet determinare tempus, quo arcus &lt;span class=&quot;math-container&quot;&gt;$\mathrm{AM}$&lt;/span&gt; absolvitur&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Solutio.&lt;/strong&gt; Sit spatium &lt;span class=&quot;math-container&quot;&gt;$\mathrm{AM}$&lt;/span&gt;, sive sit linea recta sive curva, &lt;span class=&quot;math-container&quot;&gt;$=s$&lt;/span&gt;, et celeritas, quam corpus habet in &lt;span class=&quot;math-container&quot;&gt;$\mathrm M$&lt;/span&gt; sit &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; (...). Integrando ergo habebitur tempus, quo totus arcus &lt;span class=&quot;math-container&quot;&gt;$\mathrm{AM}$&lt;/span&gt; absolvitur &lt;span class=&quot;math-container&quot;&gt;$=\int\frac{ds}c$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I don’t know who started this. A letter of Leibniz to Huygens (&lt;a href=&quot;https://archive.org/stream/leibnizensmathe08leibgoog#page/n276&quot; rel=&quot;noreferrer&quot;&gt;1690&lt;/a&gt;) has “resistence &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, vistesse &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, temps &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, espace &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;”. Johann Bernoulli’s work on caustics (&lt;a href=&quot;https://archive.org/stream/operaomnia00primgoog#page/n86&quot; rel=&quot;noreferrer&quot;&gt;1692&lt;/a&gt;) has &lt;em&gt;spatium curvilineum&lt;/em&gt; (p. 57) but apparently not the letter &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, which he uses in later papers (ibid. pp. 60, 309, 316, 409).)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-13T01:53:50.103" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7142" PostTypeId="2" ParentId="7134" CreationDate="2018-03-13T04:54:05.780" Score="2" Body="&lt;p&gt;The Greek historian &lt;a href=&quot;https://en.wikipedia.org/wiki/Herodotus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Herodotus&lt;/a&gt; writes of the origins of geometry in Egypt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They say that that king [Sesostri, ca. 2000 a. C.] distributed the land among all of the Egyptians, each one having an equal lot in the a square shape, and from these subdivisions obtained tribute, having imposed an annual payment. If the river bore away a part, the owner announced the loss, and officials were sent to observe the extent to which the plot had been diminished for the purpose of adjusting the tribute payment. It is my feeling that this indicates the invention of geometry here, prior to its passing to Greece.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Geometry - literally &lt;em&gt;earth measurement&lt;/em&gt; - was the responsibility of the &quot;arpedonapti&quot;, those who knot ropes.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is by tightening ropes that they drew the two simplest and most important lines in geometry: the straight line and the circle. The first, simply tightening a rope between two points, a kind of operation which image is still present in the expressions &quot;to draw a line&quot;, &quot;to draw a perpendicular&quot;; the second, making one of the two points turn around the other which is held fixed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the suggestion is that the &quot;earth measuring arpedonapti&quot; of ancient Egypt used only lines and circles and this was the art inherited by the Greeks. (I'm not entirely convinced.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid makes no reference to the edge and compass in his axioms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source : &lt;a href=&quot;http://php.math.unifi.it/users/archimede/archimede_NEW_inglese/curve/curve_giusti/prima.php?id=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Garden of Archimedes&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2018-03-13T04:59:15.360" LastActivityDate="2018-03-13T04:59:15.360" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7144" PostTypeId="1" AcceptedAnswerId="7145" CreationDate="2018-03-15T11:30:19.157" Score="8" ViewCount="2797" Body="&lt;p&gt;Mathematicians of the 18th century and the Swiss people are known to speak and write every language. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leonhard Euler belongs to both of these categories and wrote articles in any language, I am not able to track down any information on his native language. Is there any information on that? Or did he already speak French, German and Latin at the day of his birth?&lt;/p&gt;&#xA;" OwnerUserId="7153" LastActivityDate="2018-09-13T08:33:20.430" Title="What was Euler's first language?" Tags="|mathematicians|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7145" PostTypeId="2" ParentId="7144" CreationDate="2018-03-15T14:08:29.290" Score="17" Body="&lt;p&gt;I suppose that Euler's first language was German. Probably the &lt;a href=&quot;https://en.wikipedia.org/wiki/Swiss_German&quot; rel=&quot;noreferrer&quot;&gt;Swiss German&lt;/a&gt; dialect, which they speak in Basel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I say this because: a) he was born and went to school and to the university in the German-speaking part of Switzerland (Basel), b) His mother's (Brucker) and brother's (Johann Heinrich) names sound definitely German, and c) he had a German wife (Katharina Gsell whom he met in St. Petersburg), d) He was a Protestant and his father was a Protestant pastor, e) Euler is a German name. There is little doubt that he also learned&#xA;French in his childhood, (his thesis is written in French) as well as Latin, Greek and Hebrew at school. Later he learned Russian and English. There are indications that he preferred German in daily life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Following the custom of the time, he wrote his mathematical papers in Latin, papers on other subjects in German and French.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a reference that confirms my guess:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=tePqBgAAQBAJ&amp;amp;pg=PA210&amp;amp;lpg=PA210&amp;amp;dq=euler%27s+native+language&amp;amp;source=bl&amp;amp;ots=we-wRpYEuh&amp;amp;sig=cOwEtIeDTFzT13wV_A13GYtzpAw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjhmauyv-7ZAhUMRKwKHY-gCtoQ6AEIVTAF#v=onepage&amp;amp;q=euler%27s%20native%20language&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Leonhard Euler and the Bernoullis: Mathematicians from Basel&#xA;By M. B. W. Tent&lt;/a&gt;, p. 210.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-03-16T02:36:52.260" LastActivityDate="2018-03-16T02:36:52.260" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7147" PostTypeId="2" ParentId="7137" CreationDate="2018-03-18T14:31:33.360" Score="-1" Body="&lt;p&gt;Cantor claimed that it is always possible to transform a well-defined set into the form of a well-ordered set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a letter to Mittag-Leffler of Dec. 17, 1882 Cantor explains this well-ordering :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1) There is a first element $E_1$.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;2) For every element $E'$, except the last one, if a last one is existing, there exists an element $E''$ following next upon $E'$. (*)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;3) For every finite or infinite part $(E')$ of $(E)$ there is an element $E$* which is following next upon all $E'$ - except that an element following next upon  all $E'$ is not existing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Cantor applies this method of stepwise well-ordering also in his proof of theorem A that every transfinite set $T$ has a subset of cardinality $\aleph_0$:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After having removed according to some rule a finite number of elements $t_1, t_2, ...t_{\nu-1}$ from the set $T$ then there is always the possibility to remove another element $t_{\nu}$. The set $\{t_{\nu}\}$, where $\nu$ is an arbitrary finite cardinal number, is a partial set of ${T}$ having cardinal number $\aleph_0$ because $\{t_{\nu}\}$ ~ $\{\nu\}$. [E. Zermelo: &quot;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN237853094&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts&lt;/a&gt;&quot;, Springer, Berlin (1932) p. 293]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Zermelo comments:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &quot;proof&quot; of theorem A, which is purely intuitive and logically unsatisfactory is reminiscent of the well-known primitive attempt to obtain a &lt;em&gt;well-ordering&lt;/em&gt; of a given set by successively taking away arbitrary elements. [loc. cit. p. 352]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Therefore this &quot;primitive attempt&quot; must have been rather popular. However I could not find any mathematician of that time who in written form has accepted it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(*) Note the phrase &quot;following next upon all&quot; which Cantor often used, for instance denoting $\omega$ as following next upon all finite ordinals. It is in clear contradiction with the linear character of any well-ordering of natural numbers.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-03-18T14:31:33.360" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7148" PostTypeId="1" AcceptedAnswerId="7149" CreationDate="2018-03-18T20:46:21.413" Score="10" ViewCount="7811" Body="&lt;p&gt;I am trying to equate the famous Stephen Hawking to some of our other famous scientists and noted that the vast majority have an associated equation with their name.  As for example Einstein was the famous E=mc&lt;sup&gt;2&lt;/sup&gt; and his stress tensor field equation linking matter and energy to space and time. Schrödinger like his one famous Schrödinger's equation.   Boltzmann's entropy formula.  Heisenberg's uncertainty equation.  The Dirac equation. and so on and so forth. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am thinking he will be remembered for some formula that links quantum mechanics to general relativity and information. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone venture what this might be?   I assume there are other formulas that are perhaps more unclear in the sense that I saw a book by Leonard Susskind to the effect how he  made the world safe for quantum mechanics and this was related to Hawking's formulas I assume but frankly the book was over my level of physics training.  Maybe someone can comment on how this plays into Hawking's most notable &quot;formula&quot;.  Assuming such a thing exists.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="-1" LastEditDate="2018-03-25T23:21:57.390" LastActivityDate="2019-11-21T19:52:55.063" Title="What equation is Stephen Hawking most noted for?" Tags="|quantum-mechanics|cosmology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7149" PostTypeId="2" ParentId="7148" CreationDate="2018-03-18T21:46:52.610" Score="16" Body="&lt;p&gt;One can only speculate what Hawking will be remembered for, but &lt;a href=&quot;http://www.nytimes.com/2002/01/22/science/hawking-s-breakthrough-is-still-an-enigma.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;according to NYT&lt;/a&gt; in 2002 he apparently expressed a wish to have what he saw as his biggest accomplishment engraved on his tombstone (following the tradition that goes back at least to Archimedes):&quot;&lt;em&gt;Perhaps in imitation of Boltzmann, Dr. Hawking declared at the end of the meeting that he wanted the formula for black hole entropy engraved on his own tombstone&lt;/em&gt;&quot;. The formula in question is  $$S=\frac{kc^3}{4G\hbar}A,$$ where $A$ is the area of the event horizon, $c$ is the speed of light and $k$, $G$, and $\hbar$ are the Boltzmann, gravitational and reduced Planck constants, respectively. It is also often written in the form $S=\frac{k}{4l_P^2}A$, where $l_P:=\sqrt{\frac{G\hbar}{c^3}}$ is the Planck length, and called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Black_hole_thermodynamics#Overview&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bekenstein–Hawking formula&lt;/a&gt; (Bekenstein conjectured  the proportionality to $A$). The formula is related to Hawking's work on black hole evaporation (&quot;Hawking radiation&quot;) back in 1973-4. It led to the so-called black hole information loss paradox, see &lt;a href=&quot;https://arxiv.org/abs/1801.05923&quot; rel=&quot;nofollow noreferrer&quot;&gt;(Information) Paradox Regained? by Manchak and Weatherall&lt;/a&gt; for a recent status review, and was later linked to 't Hooft's 1993 holographic principle in string theory. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-20T20:21:20.250" LastActivityDate="2018-03-20T20:21:20.250" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7150" PostTypeId="1" CreationDate="2018-03-19T08:14:39.260" Score="3" ViewCount="77" Body="&lt;p&gt;I have a book from 1795 where the mixing of quantities is done by Alligation.  Depending on the supplied data this can be Partial, Alternate, Medial or Total.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I asked several teachers 'how would you calculate the mixture of quantities' &#xA;they answered 'that's no longer on the syllabus' and would go no further.  Much later I found that Alligation is still taught at University to Pharmacy students.  The given methods do not necessarily give 'best' answers but do give quick and easy answers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When and Why did Alligation leave the syllabus? &#xA;JK&lt;/p&gt;&#xA;" OwnerUserId="7169" LastActivityDate="2018-03-19T08:14:39.260" Title="Alligation - when and why did it disappear?" Tags="|mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7152" PostTypeId="1" AcceptedAnswerId="7153" CreationDate="2018-03-19T18:56:32.920" Score="5" ViewCount="192" Body="&lt;p&gt;I'm trying to find a good source for the definition of degenerate matter to differentiate it from Fermi gases. For my research a good section on history would be nice. This question is more specifically about stars composed of degenerate matter.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="55" LastEditDate="2018-03-19T20:59:50.290" LastActivityDate="2018-03-19T20:59:50.290" Title="Who coined the term &quot;degenerate star&quot;?" Tags="|physics|terminology|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7153" PostTypeId="2" ParentId="7152" CreationDate="2018-03-19T20:58:31.983" Score="4" Body="&lt;p&gt;Originally, &amp;quot;degenerate star&amp;quot; was just a regular phrase describing a star made of degenerate matter, which referred to the dense matter made of nearly completely ionized atoms. The &amp;quot;coining&amp;quot; was a joint effort by Eddington, Fowler and Milne in 1920-s, they developed a theory of degenerate stars, see &lt;a href=&quot;https://books.google.com/books?id=r47qBwAAQBAJ&amp;amp;pg=PA186&amp;amp;lpg=PA186&amp;amp;dq=degenerate+star+history&amp;amp;source=bl&amp;amp;ots=2qCtTQDoKr&amp;amp;sig=pX162HWrqr1zUwTt21p822wTbVI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwibv_r8n_nZAhVEcq0KHZQ4D9YQ6AEIgAEwCA#v=onepage&amp;amp;q=degenerate%20star%20history&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leverington's History of Astronomy (p.186)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Arthur Eddington had explained that in a star the radiation pressure and gas pressure balance the gravitational attraction. He suggested, however, that, as the radiation pressure is proportional to the fourth power of the temperature, a star could explode if the temperature became too high. In a parallel development, he had also explained in 1924 that white dwarfs, like Sirius B, are made of atoms that are almost completely ionised, which enabled the atoms to be packed very close together into what was called degenerate matter. In the following year, Nova Pictoris was seen to be caused by a star swelling up and exploding.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Ralph Fowler of Cambridge developed a theory of degenerate matter in 1926, in which he confirmed that a white dwarf is largely composed of degenerate matter, with only its outer layer remaining gaseous. As the white dwarf cools near the end of its life, more and more of this gaseous layer becomes degenerate and, eventually, the whole star consists of degenerate matter at a temperature of absolute zero. The star is then a black dwarf radiating no energy. Two years later, Arthur Milne of Oxford examined the properties of stars, not just white dwarfs, assuming that there was some degenerate matter in the centre of most of them.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-19T20:58:31.983" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7154" PostTypeId="2" ParentId="5746" CreationDate="2018-03-20T11:15:57.193" Score="2" Body="&lt;p&gt;Our analysis of Mehrtens' claims appears at &lt;a href=&quot;https://arxiv.org/abs/1803.02193&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/1803.02193&lt;/a&gt; and is forthcoming at &lt;em&gt;Mat. Stud.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). &#xA;We argue that Klein and Hilbert, both at Goettingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. &#xA;Klein's Goettingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. &#xA;Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. Mehrtens' claim that [the future SS-Brigadefuehrer] &quot;Theodor Vahlen ... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism&quot; fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position. &lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-03-20T11:15:57.193" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7156" PostTypeId="1" CreationDate="2018-03-21T14:45:14.000" Score="2" ViewCount="182" Body="&lt;p&gt;The index of refraction of air at STP is about 1.0003. When was this established, and to what precision?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an analogy, you can find historical summaries of values for the speed of light, showing its convergence on the current value. Is there a similar timeline available for the index of refraction of air? I'm particularly interested in early values and specific dates (year level, not days or months).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since the ability to measure optical angles was very advanced by the time vacuum technology became common, I'd assume that the early values (1800's) would be very close to the current value. Of course, the current value is actually changing (since the amount of CO2 is changing), but that's a different question. &lt;/p&gt;&#xA;" OwnerUserId="7183" OwnerDisplayName="WhatRoughBeast" LastActivityDate="2018-05-21T12:55:13.337" Title="Historical values of the index of refraction of air" Tags="|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7157" PostTypeId="1" CreationDate="2018-03-21T20:34:51.693" Score="0" ViewCount="44" Body="&lt;p&gt;One of the ones the make the infinite vacuum zero point energy finite is to use the &lt;a href=&quot;https://en.wikipedia.org/wiki/Zeta_function_regularization#Heat_kernel_regularization&quot; rel=&quot;nofollow noreferrer&quot;&gt;heat kernel regularization&lt;/a&gt;.&#xA;Does anyone know the origin of this method? Or the first paper published discussing such a model?&lt;/p&gt;&#xA;" OwnerUserId="7184" OwnerDisplayName="John" LastActivityDate="2018-03-21T21:37:28.703" Title="Origin of heat kernel regularization" Tags="|quantum-field-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7158" PostTypeId="1" CreationDate="2018-03-22T10:46:12.977" Score="1" ViewCount="154" Body="&lt;p&gt;When I look up&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;'that which is right-angled but not equilateral'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;there are translations that show the word before the above phrase to '&lt;a href=&quot;http://www2.clarku.edu/~djoyce/elements/booki/defI22.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;oblong&lt;/a&gt;', some that show '&lt;a href=&quot;http://euclid.trentu.ca/math/sb/3260H/Fall-2017/Euclids-axioms.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;rectangle&lt;/a&gt;' and some that show both with one term in brackets (&lt;a href=&quot;http://math.bu.edu/people/sr/549-10/handouts/Euclid.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; &lt;a href=&quot;https://knowledge.autodesk.com/support/autocad/learn-explore/caas/simplecontent/content/euclids-elements-definitions.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; &lt;a href=&quot;http://www.employees.csbsju.edu/tsibley/Appendix-A.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is this? Guesses:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Translation error&lt;/li&gt;&#xA;&lt;li&gt;Euclid didn't consider squares to be rectangles.&lt;/li&gt;&#xA;&lt;li&gt;Euclid made a mistake.&lt;/li&gt;&#xA;&lt;li&gt;Other&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Related:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://korean.stackexchange.com/questions/3394/in-korea-are-squares-considered-rectangles&quot;&gt;In Korea, are squares considered rectangles?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/13831/&quot;&gt;Are kindergartners supposed to be steered from squares being rectangles?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/13700/in-what-curricula-are-rectangles-defined-so-as-to-exclude-squares&quot;&gt;In what curricula are “rectangles” defined so as to exclude squares?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/2249741/why-do-we-have-circles-for-ellipses-squares-for-rectangles-but-nothing-for-tria&quot;&gt;Why do we have circles for ellipses, squares for rectangles but nothing for triangles?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/394471&quot;&gt;What are/should kids (be) taught about the colour of the sun?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3338" LastActivityDate="2018-03-22T18:36:39.467" Title="Whether Euclid considered squares to be rectangles" Tags="|reference-request|ancient-greece|euclidean-geometry|geometry|language|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7159" PostTypeId="2" ParentId="7156" CreationDate="2018-03-22T11:42:16.773" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;http://okina.univ-angers.fr/publications/ua13816/1/refrindexstor.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this lecture&lt;/a&gt;, Newton did measure $n_{air}$ compared with vacuum, tho' they give no details. Perhaps the original source, &quot;Newton, Opticks, The Fourth edition, Ed. W. Innys, London, 1730,&quot; provides more details.  It's available at Gutenberg.org .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peeking at Newton's book, I find &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Refraction of the Air in this Table is determin'd by that of the&#xA;  Atmosphere observed by Astronomers. For, if Light pass through many&#xA;  refracting Substances or Mediums gradually denser and denser, and&#xA;  terminated with parallel Surfaces, the Sum of all the Refractions will&#xA;  be equal to the single Refraction which it would have suffer'd in&#xA;  passing immediately out of the first Medium into the last. And this&#xA;  holds true, though the Number of the refracting Substances be&#xA;  increased to Infinity, and the Distances from one another as much&#xA;  decreased, so that the Light may be refracted in every Point of its&#xA;  Passage, and by continual Refractions bent into a Curve-Line. And&#xA;  therefore the whole Refraction of Light in passing through the&#xA;  Atmosphere from the highest and rarest Part thereof down to the lowest&#xA;  and densest Part, must be equal to the Refraction which it would&#xA;  suffer in passing at like Obliquity out of a Vacuum immediately into&#xA;  Air of equal Density with that in the lowest Part of the Atmosphere.&lt;br&gt;&#xA;  There may be more details elsewhere but I haven't read the entire chapter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There are plenty of papers dating back at least to the 1950s which provide empirical formulas for calculating the index based on gas ratios and relative humidity of the sample.    But to your first question: any published value for $n$ is published with the calculated uncertainty; if that's missing then the default is to assume $\pm 1$ in the last digit, or half that.  Since you quoted a number without reference to a specific sample, I suspect that's as precise as one can get for &quot;average atmosphere at sea level and $25^{\circ} C $ .&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="609" LastEditDate="2018-03-22T11:57:11.117" LastActivityDate="2018-03-22T11:57:11.117" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7160" PostTypeId="2" ParentId="7158" CreationDate="2018-03-22T12:42:23.803" Score="4" Body="&lt;p&gt;The word in Euclid is &quot;ἑτερομήκης&quot;.  Whether you translate it &quot;rectangle&quot; or &quot;oblong&quot; depends on what you think those two words mean.  It is clear from Euclid that &quot;ἑτερομήκης&quot; does not include squares.  If the translator thinks rectangles do not include squares, then he can use &quot;rectangle&quot; in the translation.  But if the translator thinks squares are rectangles, then he may use a different translation, such as &quot;oblong&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my answer is (2), with the proviso that we use &quot;ἑτερομήκης&quot;, and not the English word &quot;rectangle&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Note: word ἑτερομήκης added from Mauro's answer.]&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2018-03-22T18:36:39.467" LastActivityDate="2018-03-22T18:36:39.467" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7161" PostTypeId="1" CreationDate="2018-03-22T13:20:51.373" Score="0" ViewCount="126" Body="&lt;p&gt;As in what were the main factors, personalities, policies, etc involved in its decline  during the $16^{th}$ century to the present day?&lt;/p&gt;&#xA;" OwnerUserId="7186" LastActivityDate="2018-03-22T13:20:51.373" Title="What is the reason behind the decline of Indian Physics?" Tags="|ancient-india|india|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7162" PostTypeId="2" ParentId="7158" CreationDate="2018-03-22T15:25:47.240" Score="3" Body="&lt;p&gt;See &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DDef%3Anumber%3D22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's &lt;em&gt;Elements&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition 22&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of quadrilateral (&lt;a href=&quot;https://en.wiktionary.org/wiki/%CF%80%CE%BB%CE%B5%CF%85%CF%81%CE%AC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;τετράπλευρος&lt;/em&gt;&lt;/a&gt;) figures, a &lt;em&gt;square&lt;/em&gt; (&lt;a href=&quot;https://en.wiktionary.org/wiki/%CF%84%CE%B5%CF%84%CF%81%CE%AC%CE%B3%CF%89%CE%BD%CE%BF%CF%82&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;τετράγωνος&lt;/em&gt;&lt;/a&gt;) is that which is both equilateral and right-angled; an &lt;em&gt;oblong&lt;/em&gt; (&lt;a href=&quot;http://www.perseus.tufts.edu/hopper/morph?l=e%28tero%2Fmhkes&amp;amp;la=greek&amp;amp;can=e%28tero%2Fmhkes0&amp;amp;prior=orqogw/nion&amp;amp;d=Perseus:text:1999.01.0085:book=1:type=Def:number=22&amp;amp;i=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;ἑτερομήκης&lt;/em&gt;: with sides of uneven length&lt;/a&gt;) that which is right-angled but not equilateral; a &lt;em&gt;rhombus&lt;/em&gt; that which is equilateral but not right-angled; and a &lt;em&gt;rhomboid&lt;/em&gt; that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called &lt;em&gt;trapezia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And see &lt;a href=&quot;https://books.google.it/books?id=HxhEBgAAQBAJ&amp;amp;pg=PA188&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heath's commentary&lt;/a&gt;, page 188:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Tetragonon&lt;/em&gt; was already a square with the Pythagoreans, and it is so most commonly in Aristotle; but in &lt;em&gt;De anima&lt;/em&gt;, II.3, 414b31 it seems to be a quadrilateral, and in &lt;em&gt;Metaph&lt;/em&gt;., 1054b2, &amp;quot;equal and equiangular &lt;em&gt;tetragona&lt;/em&gt;,&amp;quot; it cannot be anything else but quadrilateral if &amp;quot;equiangular&amp;quot; is to have any sense. Though, by introducing &lt;em&gt;tetrapleuron&lt;/em&gt; for any quadrilateral, Euclid enabled ambiguity to be avoided, there seem to be traces of the older vague use of &lt;em&gt;tetragonon&lt;/em&gt; in much later writers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-03-22T15:25:47.240" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7163" PostTypeId="1" AcceptedAnswerId="7168" CreationDate="2018-03-23T00:46:41.427" Score="9" ViewCount="2570" Body="&lt;p&gt;AFAIK, Archimedes is credited with discovering the following formula for computing the sum of squares:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to have come up in his quest for finding the area of a parabolic segment. I believe the sum of the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; numbers was already known since it's easily derivable by observing the arrangement of stones as triangular numbers (although I haven't been able to find a reference yet, but remember reading about it in my trip down math history).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I wasn't able to find any credible reference to Archimedes' derivation of this formula. All derivations taught in school today have some &quot;modern&quot; variation attached to it. How was this derived by Archimedes given the state of math at his time? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest reference is this &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/sums-of-powers-in-discrete-mathematics-archimedes-sums-squares-in-the-sand&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA Page&lt;/a&gt; but it doesn't give any insight into this (or even state things clearly). Here's a gist on this &lt;a href=&quot;https://math.stackexchange.com/questions/2630208/how-to-interpret-this-visual-proof-for-archimedes-derivation-of-sum-of-squares&quot;&gt;Math.SE Question&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="7035" LastEditorUserId="4251" LastEditDate="2020-05-13T18:58:35.913" LastActivityDate="2020-05-13T18:58:35.913" Title="How was the sum of squares formula discovered by Archimedes?" Tags="|mathematics|mathematicians|elementary-algebra|archimedes|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7165" PostTypeId="2" ParentId="7163" CreationDate="2018-03-23T13:57:58.633" Score="4" Body="&lt;p&gt;You are right about the person (Archimedes), but not about the work. He stated it as proposition 10 of his text &lt;em&gt;On spirals&lt;/em&gt;. In modern language, the statement is$$(\forall n\in\mathbb{N}):(n+1)n^2+(1+2+\cdots+n)=3(1^2+2^2+\cdots+n^2).$$See page 162 of &lt;a href=&quot;https://archive.org/details/worksofarchimede00arch&quot; rel=&quot;nofollow noreferrer&quot;&gt;this edition&lt;/a&gt; of the works of Archimedes.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="658" LastEditDate="2018-03-23T19:47:12.340" LastActivityDate="2018-03-23T19:47:12.340" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7166" PostTypeId="1" AcceptedAnswerId="7167" CreationDate="2018-03-23T14:34:50.213" Score="4" ViewCount="428" Body="&lt;p&gt;It is kind of cute that q-analogues are used in physics (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Q-analog#Applications_in_the_physical_sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt; for example), but it is also kind of confusing because the 'q' does &lt;strong&gt;not&lt;/strong&gt; stand for 'quantum'. It predates that use!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, where does the 'q' really come from? Is it only that McMahon (or someone else) first used that particular letter?&lt;/p&gt;&#xA;" OwnerUserId="1861" LastEditorUserId="55" LastEditDate="2018-03-23T19:01:07.047" LastActivityDate="2018-03-23T19:09:44.763" Title="What is the origin of q-calculus notation?" Tags="|notation|calculus|combinatorics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7167" PostTypeId="2" ParentId="7166" CreationDate="2018-03-23T18:59:20.413" Score="3" Body="&lt;p&gt;A good review of the history of the subject is &lt;a href=&quot;http://www.springer.com/cda/content/document/cda_downloaddocument/9783034804301-c2.pdf?SGWID=0-0-45-1345006-p174314098.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ch. 2 of Ernst's Comprehensive Treatment of q-Calculus&lt;/a&gt;, which names Bernoulli's numbers as an early appearance, and characterizes the work on elliptic and theta functions since 1750-s as &quot;pre-$q$ mathematics&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Presumably, $q$ stands for &quot;quotient&quot; or rather its Latin counterpart &quot;quotus&quot;, it traditionally appears in the geometric series notation. Euler in 1738 studied generating functions of the Bernoulli polynomials, and already introduced the &lt;a href=&quot;https://en.wikipedia.org/wiki/Q-Pochhammer_symbol#q-series&quot; rel=&quot;nofollow noreferrer&quot;&gt;$q$-series&lt;/a&gt; (a.k.a the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler function&lt;/a&gt;) in the oft-cited chapter 16 of Introductio in Analysin Infinitorum (1740) called &lt;a href=&quot;https://www.math.psu.edu/vstein/alg/antheory/preprint/andrews/16.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Partitio Numerorum&lt;/a&gt;. Euler's work on partitions was prompted by questions about them raised to him by a French mathematician Naude, see &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0020739X.2015.1071441&quot; rel=&quot;nofollow noreferrer&quot;&gt;Debnath's Brief History of Partitions of Numbers&lt;/a&gt;. Jacobi continued to use the letter in his work on theta functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ernst divides the $q$-tradition into two major &quot;schools&quot;, Watson's and Austrian, and makes interesting remarks about its notational history:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The various Schools of $q$ to be outlined below have developed over a period of roughly 300 years since the Bernoullis and Euler. These Schools today use and communicate in such different languages, that they have problems understanding each other. The big School of Watson speaks—both literally and metaphorically—&#xA;  English. It has little exact consciousness of, indeed may not find it necessary, to study and know the roots of the vernacular... And we have in the q-area, in relation to other areas, a tendency to develop specific, new languages (e.g. notations), which surpasses this tendency in other fields of mathematics. This is either because practitioners find these more easy to speak, because they find them more beautiful or simply because they do not know and cannot pronounce the older forms.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Both Schools or traditions recognize the early legacies of Gauß and Euler. Only the Austrian School, however, represents and incorporates the entire historical background which includes the pre-$q$ mathematics, namely the Bernoulli and Euler numbers, the theta functions and the elliptic functions. The vast area of theta functions and elliptic functions... is in fact $q$-analysis before $q$ was really introduced.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;The Austrian School is little known in the English speaking world, e.g. the USA and the Commonwealth, for two main reasons: Immediately after the first world war, ca. 1920 until 1925, the German and Austrian mathematicians were barred from participation in the big mathematical conferences... The second and more important reason has to do with languages. Most of the mathematicians of the Austrian School wrote in either German or French and, as regards the oldest,&#xA;  namely Euler, Gauß and Jacobi, in Latin. Only few English-speaking mathematicians today master these languages... The Watson School is today the most widespread and influential of the two principal Schools/traditions. But again this is mainly due to the language.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-23T19:09:44.763" LastActivityDate="2018-03-23T19:09:44.763" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7168" PostTypeId="2" ParentId="7163" CreationDate="2018-03-23T22:40:58.180" Score="9" Body="&lt;p&gt;Although it is commonly said that Archimedes summed the geometric series in &lt;em&gt;Quadrature of the Parabola&lt;/em&gt;, and summed squares of integers in &lt;em&gt;On Conoids and Spheroids&lt;/em&gt; and &lt;em&gt;On Spirals&lt;/em&gt;, that is not what he did. He &quot;sums&quot; not even areas of figures (as in numbers associated to them) but figures themselves, and he certainly does not relate any of that to stacking pebbles into triangular numbers, let alone to any algebraic manipulations. Those date to neo-Pythagoreans at least three centuries after Archimedes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To see how different what Archimedes did is from a formula here is the relevant passage from &lt;em&gt;On Spirals&lt;/em&gt; in Dijksterhuis's translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If a series of any number of lines be given, which exceed one another by an equal amount, and the difference be equal to the least, and if other lines be given equal in number to these and in quantity to the greatest, the squares on the lines equal to the greatest, plus the square on the greatest and the rectangle contained by the least and the sum of all those exceeding one another by an equal amount will be the triplicate of all the squares on the lines exceeding one another by an equal amount.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Translated&quot; into a formula, nothing close to which could have been written until Vieta almost two millenia later, this becomes $$(n+1)L_n^2+L_1(L_1+L_2+\cdots+L_n)=3(L_1^2+L_2^2+\cdots+L_n^2),$$&#xA;where $L_{i+1}-L_i$ is constant. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How it was derived is discussed at length in section 1.2 of &lt;a href=&quot;https://www.math.nmsu.edu/~history/#Capstone&quot; rel=&quot;noreferrer&quot;&gt;Mathematical Masterpieces by Knoebel et al.&lt;/a&gt;, who remark that &quot;&lt;em&gt;while Archimedes expressed this as an equality of areas, today we tend to interpret it as just a formula about numbers, ignoring the dimensionality involved&lt;/em&gt;&quot;, and refer to Kanim's conjectural reconstruction of Archimedes’s geometric reasoning. The most detailed commentary on it with a step by step &quot;discovery&quot; guide is in &lt;a href=&quot;https://www.cs.nmsu.edu/historical-projects/Projects/sums-of-powers.pdf&quot; rel=&quot;noreferrer&quot;&gt;Pengelley's Sums of Numerical Powers in Discrete Mathematics (pp. 8-10)&lt;/a&gt;. Kanim, whose idea was published as a short note &lt;a href=&quot;https://www.maa.org/sites/default/files/Kanim.MathMag.2001.pdf&quot; rel=&quot;noreferrer&quot;&gt;How Did Archimedes Sum Squares in the Sand?&lt;/a&gt; (freely available from MAA) accompanied by a &quot;proof without words&quot; (below), writes for her part: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The modern transcriptions of his proof (Dijksterhuis, Heath, Heiberg) are completely algebraic and hard to follow. The geometry of Archimedes' proof is depicted visually below. Each step of his written proof is transparent in the geometry of the picture. One wonders if this is the picture Archimedes drew in the sand.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OO74R.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OO74R.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-23T23:36:34.293" LastActivityDate="2018-03-23T23:36:34.293" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7169" PostTypeId="1" AcceptedAnswerId="7171" CreationDate="2018-03-24T05:08:46.347" Score="3" ViewCount="299" Body="&lt;p&gt;I read &lt;a href=&quot;https://physics.stackexchange.com/q/338244/41634&quot;&gt;somewhere&lt;/a&gt; that projective geometry was an inspiration to Einstein to build the theory of relativity. However, the keyword &quot;projective geometry&quot; isn't mentioned in &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_general_relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of general relativity&lt;/a&gt; and only passingly in &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_special_relativity#Minkowski%27s_spacetime&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of special relativity&lt;/a&gt; that Clifford had contributions to the field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though the connection has been showed later by Penrose (1984), did it really inspire Einstein? If yes, how? If no, then what really intuitively inspired him?&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="65" LastEditDate="2018-03-24T11:24:07.270" LastActivityDate="2018-03-25T11:57:37.973" Title="Did Einstein get inspired by projective geometry?" Tags="|relativity-theory|einstein|projective-geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7170" PostTypeId="1" AcceptedAnswerId="7174" CreationDate="2018-03-24T17:59:11.483" Score="2" ViewCount="225" Body="&lt;p&gt;What did the Greeks call the &quot;trial and error&quot; reasoning process?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://ndpr.nd.edu/news/peirces-empiricism-its-roots-and-its-originality/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bruce Aune's review of Wilson's &lt;em&gt;Peirce's Empiricism: Its Roots and Its Originality&lt;/em&gt;&lt;/a&gt; claims &quot;The name '&lt;em&gt;empirici&lt;/em&gt;' is in fact traceable to a Greek verb meaning 'to make trial of.'&quot;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-03-25T23:38:51.563" Title="What did the Greeks call the “trial and error” reasoning process?" Tags="|ancient-greece|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7171" PostTypeId="2" ParentId="7169" CreationDate="2018-03-25T11:57:37.973" Score="3" Body="&lt;p&gt;It is hard to prove that the answer is negative, but have this in mind: a friend who helped Einstein a lot (and who co-wrote two articles with him) while Einstein was trying to formulate the General Theory of Relativity was Marcel Grossmann, who was a geometer (his PhD thesis was about Non-Euclidean Geometry and he wrote two textbooks on Descriptive Geometry). However, the Geometry that he used to help Einstein was the tensor calculus which was developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita, not Projective Geometry.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-03-25T11:57:37.973" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7174" PostTypeId="2" ParentId="7170" CreationDate="2018-03-25T19:57:27.040" Score="1" Body="&lt;p&gt;&quot;Trial and error&quot; applied not to a &quot;reasoning process&quot; but to medical practice, and the name of the practice was derived from Greek ἐμπειρία, experience. The inspiration for the approach apparently came from Greek Pyrrhonism, which recommended permanently suspending judgment on how things are in their own nature, what Pyrrho of Elis (c. 360 – 270 BC) called &lt;a href=&quot;https://en.wikipedia.org/wiki/Acatalepsy&quot; rel=&quot;nofollow noreferrer&quot;&gt;ακατάληψία&lt;/a&gt; in contrast to Stoic κατάληψις, perceptual grasping of knowledge. As Aune writes in the review: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He begins with the ancient empirici who were physicians. Unlike their competitors, the dogmatici, the empirici paid little attention to the supposed underlying causes of disease; their concern was principally directed to methods of treatment founded on trial and error... Later empiricists were less concerned specifically with trial and error practice; they instead focused on endorsing only factual claims with an evidential basis in sense experience&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wikipedia even has an article about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Empiric_school&quot; rel=&quot;nofollow noreferrer&quot;&gt;empirici&lt;/a&gt;. Perhaps the best known figure of the school is Sextus Empiricus (c. 160-210 AD), whose multi-volume Adversus Mathematicos is one of our major sources on the ancient science and philosophy. Literally, the title translates as &lt;a href=&quot;https://en.wikipedia.org/wiki/Sextus_Empiricus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Against the Mathematicians&lt;/a&gt;, but Greek &lt;a href=&quot;https://seansturm.wordpress.com/2010/07/12/ta-mathemata-we-can-only-learn-what-we-already-know&quot; rel=&quot;nofollow noreferrer&quot;&gt;tà mathémata&lt;/a&gt; meant what can be learned and taught, so the commonly used free translation Against the Professors is closer to the original meaning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Sextus belongs to the later, more philosophical, generation of empirici. The founders of the school in 3rd century BC, Serapion of Alexandria and Philinus of Cos, were physicians who opposed the &quot;dogmatic&quot; Hippocratic tradition. Their works have not survived, but some fragments are quoted by Galen, Pliny and others. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-03-25T23:38:51.563" LastActivityDate="2018-03-25T23:38:51.563" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7175" PostTypeId="1" CreationDate="2018-03-26T06:56:48.603" Score="1" ViewCount="59" Body="&lt;p&gt;Mendeleev is famous for having published his first periodic table in 1869. This was a short-form or 8-column table.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also published a number of 18 column periodic tables. I am asking for instances, dates, references for any such longer tables.&lt;/p&gt;&#xA;" OwnerUserId="7203" LastEditorUserId="4251" LastEditDate="2021-09-09T06:14:07.367" LastActivityDate="2021-09-09T06:14:07.367" Title="What is the first instance in which Mendeleev published a long-form table as compared with his 8-column table of 1869?" Tags="|reference-request|chemistry|mendeleev|chemists|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7176" PostTypeId="2" ParentId="6766" CreationDate="2018-03-26T07:01:33.530" Score="4" Body="&lt;p&gt;I recommend Norton's History of Chemistry by William Brock.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a history of the periodic table try my own:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eric Scerri, The Periodic Table, Its Story and Its Significance, OUP, 2007.&lt;/p&gt;&#xA;" OwnerUserId="7203" LastEditorUserId="2277" LastEditDate="2018-03-26T09:54:01.500" LastActivityDate="2018-03-26T09:54:01.500" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7177" PostTypeId="2" ParentId="557" CreationDate="2018-03-26T07:04:47.317" Score="1" Body="&lt;p&gt;The symbols that we now use for the elements were first introduced by Jacob Berzelius the Swedish chemist.  This would have been approximately in the 1820s if I recall correctly.&lt;/p&gt;&#xA;" OwnerUserId="7203" LastEditorUserId="2277" LastEditDate="2018-03-26T16:00:23.320" LastActivityDate="2018-03-26T16:00:23.320" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7178" PostTypeId="2" ParentId="6244" CreationDate="2018-03-26T07:18:32.693" Score="3" Body="&lt;p&gt;The traditional response to this question is that Mendeleev made successful predictions concerning missing elements/blank spaces in his periodic table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mendeleev successfully predicted Sc, Ga and Ge and a further 5 or so elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;see my book, Eric Scerri, The Periodic Table, Its Story and Its Significance, OUP, 2007.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some periodic table scholars, including myself, argue that Mendeleev's successful accommodations were as influential in the acceptance of his periodic table.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A further reason for Mendeleev's greater priority is due to the fact that Lothar Meyer was working in a highly formal German Chemical Society which discouraged speculations.  Meanwhile Mendeleev was working in Russia at the time when the Russian chemical society was being formed and was welcoming radical new ideas.&lt;/p&gt;&#xA;" OwnerUserId="7203" LastActivityDate="2018-03-26T07:18:32.693" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7179" PostTypeId="2" ParentId="6227" CreationDate="2018-03-26T07:23:15.030" Score="1" Body="&lt;p&gt;The answer to your question is that Mendeleev wrote mostly in Russian and there are few translations of his original papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case his name is synonymous with the discovery of the periodic table and I would argue that he is one of the most famous chemists of the modern era.  See my book on the periodic table.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Periodic Table, Its Story and Its Significance, OUP, 2007.&lt;/p&gt;&#xA;" OwnerUserId="7203" LastEditorUserId="5435" LastEditDate="2018-03-26T12:41:38.490" LastActivityDate="2018-03-26T12:41:38.490" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7180" PostTypeId="2" ParentId="6781" CreationDate="2018-03-26T19:40:17.840" Score="1" Body="&lt;p&gt;The Sobolev Inequality was proved up to a change of coordinates and only for radially symmetric functions by GA Bliss in 1930 (I believe). Sobolev actually did not publish his paper proving the so-called Sobolev Inequality (which does not require radial symmetry) until 1938. I don't know the context for why the inequality was proven. If you have more particular question about further developments of the Sobolev Inequality, let me know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyone know about the history of Morrey's Inequality?&lt;/p&gt;&#xA;" OwnerUserId="7207" LastActivityDate="2018-03-26T19:40:17.840" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7181" PostTypeId="1" CreationDate="2018-03-26T21:01:36.983" Score="9" ViewCount="176" Body="&lt;p&gt;Who was the first to use the &quot;does not exist&quot; sign ∄? I'm aware that Giuseppe Peano originated serifed ∃ and, moreover that Whitehead and Russell repurposed Peano's serifed ∃; I'm also aware that Gerhard Gentzen introduced the ∃ sans serif (and used it like Russell and Whitehead rather than like Peano). Yet, I cannot find who was the first to use ∄ as ~∃ to mean &quot;there does not exist&quot;. Any idea? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="5383" LastActivityDate="2018-04-04T19:07:47.070" Title="Who was the first to use the &quot;does not exist&quot; sign ∄?" Tags="|terminology|notation|mathematical-logic|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7182" PostTypeId="1" CreationDate="2018-03-26T22:24:23.120" Score="1" ViewCount="209" Body="&lt;p&gt;This term has been discussed on this forum, e.g. under &lt;a href=&quot;https://hsm.stackexchange.com/questions/187/examples-of-kuhn-loss/7172#comment12642_7172&quot;&gt;Examples of Kuhn loss?&lt;/a&gt;, and has been attributed to Kuhn himself. The term refers to the loss of explanations and predictions of the prior paradigm in the paradigm that replaced it. Cartesian mechanical explanations of gravity, and unification of various phenomena by the phlogiston theory are commonly given as examples.&lt;/p&gt;&#xA;" OwnerUserId="7208" LastEditorUserId="55" LastEditDate="2018-03-27T00:32:26.377" LastActivityDate="2018-04-26T01:34:49.823" Title="Who invented the term &quot;Kuhn loss&quot;?" Tags="|terminology|philosophy-of-science|kuhn|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7183" PostTypeId="2" ParentId="7182" CreationDate="2018-03-26T22:28:09.333" Score="2" Body="&lt;p&gt;The term was first suggested by the late Heinz Post from King's College in London in a 1971 article. Post claims that there is no Kuhn loss. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the reference: Post, Heinz R. (1971), &lt;a href=&quot;http://www.springer.com/gp/book/9780792320852&quot; rel=&quot;nofollow noreferrer&quot;&gt;Correspondence, Invariance and Heuristics: In Praise of Conservative Induction&lt;/a&gt;, Studies in History and Philosophy of Science, 2, 213-255. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It so happens that Post was my PhD advisor.)&lt;/p&gt;&#xA;" OwnerUserId="7208" LastEditorUserId="55" LastEditDate="2018-03-27T00:20:23.960" LastActivityDate="2018-03-27T00:20:23.960" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7184" PostTypeId="1" CreationDate="2018-03-27T12:43:12.453" Score="1" ViewCount="370" Body="&lt;p&gt;I am looking for some good physics book(s) which shows the development of physics ideas from antiquity since 1850 or 1900 or something like that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book should covers all &quot;elementary&quot; topics (Quantum Mechanics/more advanced topics etc are not needed) i.e those which are taught at high school senior/beginning college first year topic (ie mechanics, thermodynamics, introductory electrodynamics, optics and bunch of small topics) . Also, note that I will use the book to read alongside (and occasionally to supplement) Halliday and Resnick, so I am not looking for books with pop sci etc, books with actual mathematical contents are appreciated. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an example what I'm looking: History of mechanics by Ernst Mach almost fits, but I don't like two aspects of it (it's way too terse, and honestly I don't have the time to read ~600 page on just the history of mechanics, something like ~100 pages at max would be okay)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since this will be a community wiki soon (I have flagged it to make it), please recommend one book per post, so votes have some meaning. Also there is a bit of confusion below, I'm not looking for books just on the history of mechanics, I am looking for all other topics too. &lt;/p&gt;&#xA;" OwnerUserId="7209" LastEditorUserId="7209" LastEditDate="2018-03-28T04:55:27.060" LastActivityDate="2018-04-02T15:11:45.733" Title="Good book on the history of introductory physics (upto first year college)" Tags="|physics|resource-recommendation|" AnswerCount="6" CommentCount="8" CommunityOwnedDate="2018-04-02T13:40:26.213" ContentLicense="CC BY-SA 3.0" />
  <row Id="7185" PostTypeId="2" ParentId="7184" CreationDate="2018-03-27T17:34:50.640" Score="1" Body="&lt;p&gt;I would recommend&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;René Dugas, &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4429&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;A History of Mechanics&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;, trans. John Royden Maddox.&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.goodreads.com/book/isbn/0486656322&quot; rel=&quot;nofollow noreferrer&quot;&gt;GoodReads reviews&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;Monumental study traces the history of mechanical principles chronologically from their earliest roots in antiquity through the Middle Ages to the revolutions in relativistic mechanics and wave and quantum mechanics of the early 20th century. Contributions of ancient Greeks, Leonardo, Galileo, Kepler, Lagrange, many other important figures. 116 black-and-white illustrations.&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It starts with Hellenistic mechanics and ends with quantum mechanics.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-03-28T16:23:56.787" LastActivityDate="2018-03-28T16:23:56.787" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7188" PostTypeId="2" ParentId="5565" CreationDate="2018-03-28T03:46:59.540" Score="3" Body="&lt;p&gt;As a Chinese, I gotta say that our ancestors believed that the earth was flat while the heavens were round, just like a dome. But I am not sure whether the fact that the earth is spherical was found by themselves or introduced to them.  &lt;/p&gt;&#xA;" OwnerUserId="7214" LastActivityDate="2018-03-28T03:46:59.540" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7189" PostTypeId="2" ParentId="7184" CreationDate="2018-03-28T11:41:44.483" Score="0" Body="&lt;p&gt;I highly recommend the book &lt;em&gt;The mechanical universe: heat and mechanics, advanced edition&lt;/em&gt; written by &lt;em&gt;Steven Frautschi&lt;/em&gt;, &lt;em&gt;Tom Aposlot&lt;/em&gt;, &lt;em&gt;Richard Olenick&lt;/em&gt; and &lt;em&gt;David Goodstein&lt;/em&gt;. (Pick the advanced edition)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book uses a historical approach to teach you elementary physics and how to think scientifically, instead of hard core problem solving and the math. &lt;/p&gt;&#xA;" OwnerUserId="2713" LastActivityDate="2018-03-28T11:41:44.483" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7190" PostTypeId="2" ParentId="7184" CreationDate="2018-03-28T16:24:17.117" Score="1" Body="&lt;p&gt;For electromagnetism:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;E. T. Whittaker, &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=3705&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History of the Theories of Aether &amp;amp; Electricity&lt;/em&gt;&lt;/a&gt;, 1910.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-03-28T16:24:17.117" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7191" PostTypeId="2" ParentId="7184" CreationDate="2018-03-28T16:27:02.560" Score="1" Body="&lt;p&gt;For thermodynamics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Clifford Ambrose Truesdell, &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=4548&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Tragicomical History of Thermodynamics, 1822-1854&lt;/em&gt;&lt;/a&gt; (New York, NY: Springer New York, 1980).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-03-28T16:27:02.560" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7192" PostTypeId="2" ParentId="6857" CreationDate="2018-03-28T23:23:24.583" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;(Cannot comment, so answer to the comment of Alexandre Eremenko here.  However, this also contains an answer to the original question — if one takes it literally, not “in spirit”.)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to a &lt;a href=&quot;https://books.google.com/books?id=NmxNAAAAYAAJ&amp;amp;pg=PA456&amp;amp;lpg=PA456&amp;amp;dq=gauss%20hanover%20geodesy%20errors&amp;amp;source=bl&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiK99vo8I7aAhUH5oMKHZRMDBIQ6AEIjQEwDg#v=onepage&amp;amp;q=gauss%20hanover%20geodesy%20errors&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;text of 1875&lt;/a&gt;, the mean error of Gauss’ survey was 0.59″.  So an error of 5″ would be a killer.  (No, I have no clue how one would be able to average enough theodolite measurements to get down to 0.59″.  I would expect one needs thousands of them…)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also: I have no idea why precision of &lt;em&gt;heliotropes&lt;/em&gt; enters the equation.  With Sun’s angular size of 0.5°, you do not need precision much more than this.  I imagine the process as a heliotrope at one end, and a theodolite at the other end.  It is the precision of theodolite which matters, right?  &lt;/p&gt;&#xA;" OwnerUserId="7216" LastEditorUserId="7216" LastEditDate="2018-03-30T22:54:43.213" LastActivityDate="2018-03-30T22:54:43.213" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7193" PostTypeId="1" AcceptedAnswerId="7194" CreationDate="2018-03-29T09:30:01.163" Score="1" ViewCount="183" Body="&lt;p&gt;I want the answer to incorporate apparent magnitudes of those moons, power of his telescope, light pollution of that time, successful recreation of the scenario of watching Galilean moons using that telescope recently, possibilities of extraordinary eye sight, etc.&lt;/p&gt;&#xA;" OwnerUserId="7218" LastActivityDate="2018-03-29T12:07:05.403" Title="Did Galileo really see Galilean Moons?" Tags="|astronomy|optics|telescope|galileo|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7194" PostTypeId="2" ParentId="7193" CreationDate="2018-03-29T12:07:05.403" Score="8" Body="&lt;p&gt;There can be no doubt that he has seen them, for the simple reason that he determined their periods and configuration correctly, and published them. Therefore the other things (magnification of his telescope, light pollution etc.) are irrelevant for the answer. You can easily see them yourself using an 8x binocular.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-03-29T12:07:05.403" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7195" PostTypeId="1" AcceptedAnswerId="7198" CreationDate="2018-03-30T09:27:03.863" Score="0" ViewCount="353" Body="&lt;p&gt;I've heard of Nazi Germany developing many a technology that we still use today and that communications technology such as the telegraph came about  due to wartime needs. I was wondering to what extent war speeds up research and technological advancements and why we do not achieve such speeds during times of peace.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also would like to know some technology we use in our every day life that came about due to war.&lt;/p&gt;&#xA;" OwnerUserId="7223" LastEditorUserId="4251" LastEditDate="2020-05-13T06:06:56.953" LastActivityDate="2020-05-13T06:06:56.953" Title="Does war speed up technological advancements and research?" Tags="|technology|world-war-2|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7196" PostTypeId="2" ParentId="7195" CreationDate="2018-03-30T12:25:28.707" Score="0" Body="&lt;p&gt;It has heen argued that the development of electronic computers was speeded up by code-breaking efforts in World War II.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3uh89.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3uh89.jpg&quot; alt=&quot;colossus&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Colossus_computer&quot; rel=&quot;nofollow noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-03-30T12:25:28.707" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7197" PostTypeId="1" AcceptedAnswerId="7199" CreationDate="2018-03-30T12:37:08.933" Score="0" ViewCount="370" Body="&lt;p&gt;Were there PhD thesis in the time of Galileo and Newton?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Newton and Galileo make a PhD thesis?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Due to the poor level of science (before calculus and so on), were PhD thesis made in math and physics?&lt;/p&gt;&#xA;" OwnerUserId="3786" LastEditorUserId="5989" LastEditDate="2019-08-21T07:09:38.920" LastActivityDate="2019-08-21T07:09:38.920" Title="Which were the first PhD thesis in Mathematics and Physics?" Tags="|mathematical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7198" PostTypeId="2" ParentId="7195" CreationDate="2018-03-30T13:44:28.057" Score="3" Body="&lt;p&gt;Not only war but preparations to war, military-related research had very large influence on the development of science and  technology. Of the recent examples I mention GPS,&#xA;Internet, space exploration, not even speaking of nuclear energy. GPS and satellite navigation in general were originally developed for military purposes, and most importantly the research was financed by the military. Internet was developed by US military as a communication system which could withstand nuclear attack. Space exploration is a byproduct of the development of ballistic missiles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Gerald Edgar wrote, digital electronic computers were developed during WWII, but coding/decoding was not the only purpose. Other purposes were the nuclear bomb project, which involved enormous amount of computation, and the devices for control of anti-aircraft guns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can give very many examples from earlier history, beginning with Hellenistic Greece. (Tyrants of Syracuse employed people like Archimedes not because of their love of science: the main reason was their desire to develop advanced military technology).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Military considerations also have large influence on development of science education. Think of the French Grand Ecoles founded by Napoleon, or of a huge&#xA;boost in American science education after the launch of the first Soviet satellite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can conclude that military considerations is one of the MAIN driving forces&#xA;of development of applied science and technology.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-12T12:27:03.790" LastActivityDate="2020-05-12T12:27:03.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7199" PostTypeId="2" ParentId="7197" CreationDate="2018-03-30T18:00:01.733" Score="4" Body="&lt;p&gt;According to the &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo Galilei was a student of Ostilio Ricci&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The official tutor of Isaac Newton (M.A.) was Benjamin Pulleyn&#xA;; Isaac Barrow was NOT a formal advisor but probably had the greatest influence on the young Newton&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But neither of these held the actual title &quot;Ph.D.&quot;...&lt;a href=&quot;https://en.wikipedia.org/wiki/Doctor_of_Philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;History&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If Erhard Weigel (1652) is indeed the first Ph.D. (as some have suggested), then that would be mathematics.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2018-03-30T18:05:05.090" LastActivityDate="2018-03-30T18:05:05.090" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7200" PostTypeId="1" CreationDate="2018-03-31T02:00:07.000" Score="1" ViewCount="60" Body="&lt;p&gt;Maybe the wrong area...should be physics maybe...but there is some history here as well.  I became interested in knowing how Fermi came to pick graphite as a neutron moderator. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example beryllium may be the moderator in a radium beryllium neutron source . I am interested in knowing what  the B reactor at the  Hanford Site used as a neutron source so naturally a history of how moderators were chosen is helpful. I know  graphite was a moderator for the uranium oxide at Hanford reactor B but then what was the moderator for the neutron source and even more interesting what was the neutron source to first start the chain reation, it would have to be something other than uranium oxide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now you see why I have this in the history section.  I assume some moderators are better than others so a historical perspective may be the best approach.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="5040" LastEditDate="2018-03-31T02:06:42.997" LastActivityDate="2018-03-31T02:06:42.997" Title="What is the history of how a neutron moderator is chosen in nuclear chain reactions?" Tags="|nuclear-physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7201" PostTypeId="1" AcceptedAnswerId="7211" CreationDate="2018-03-31T16:53:38.410" Score="3" ViewCount="171" Body="&lt;p&gt;Riemann was the first to talk about elliptic geometry: Bolyai and Lobačevskij (even Gauss too) studied only hyperbolic geometry. But of course some theorems of (planar) elliptic geometry were known since more than fifteen centuries, since the surface of a sphere is a space of constant positive curvature even if it's not a real model of elliptic geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Riemann actually notice this? If not, who was the first to point out the equivalence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[EDIT: I tried to explain what I meant. The author of the approved answer read anyway into my mind!]&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="81" LastEditDate="2018-04-03T14:50:49.043" LastActivityDate="2018-04-03T14:50:49.043" Title="Who was first to recognize the link between (synthetic) elliptic geometry and geometry on the sphere?" Tags="|geometry|riemannian-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7202" PostTypeId="2" ParentId="7195" CreationDate="2018-04-01T16:51:09.293" Score="3" Body="&lt;p&gt;In physics, the four most important developments had nothing to with war or the preparation of war: Newtons calculus and his theory of gravitation, Maxwells equations, Einsteins theory of special and general relativity, Heisenbergs Matrix Mechanics in QM. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2018-04-01T16:51:09.293" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7203" PostTypeId="1" CreationDate="2018-04-01T19:37:40.807" Score="0" ViewCount="107" Body="&lt;p&gt;The book &quot;Differential Equations and Dynamical Systems&quot; by Lawrence Perko says that the first proofs of the Stable Manifold Theorem are from Hadamard in 1901, Perron in 1928, and Liapunov and Perron. I am trying to find these proofs. &lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2018-04-01T19:37:40.807" Title="The Original Proofs of The Stable Manifold Theorem" Tags="|reference-request|differential-equations|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7204" PostTypeId="1" AcceptedAnswerId="7205" CreationDate="2018-04-02T10:13:43.107" Score="2" ViewCount="119" Body="&lt;p&gt;I'm looking for references about how the concept of &lt;em&gt;mechanical work&lt;/em&gt; (&quot;$\boldsymbol{F}\cdot\mathrm{d}\boldsymbol{r}$&quot;) or the concept of mechanical power (&quot;$\boldsymbol{F}\cdot\boldsymbol{v}$&quot;) came about and developed. Please note that I'm not asking about &lt;em&gt;virtual&lt;/em&gt; work and its related principle, and that I'm asking for &lt;em&gt;references&lt;/em&gt; rather than direct reports as &lt;a href=&quot;https://hsm.stackexchange.com/q/6942/7231&quot;&gt;this question&lt;/a&gt; asks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I consulted Truesdell &amp;amp; Toupin's &lt;em&gt;The Classical Field Theories&lt;/em&gt; (Springer 1960), which usually provides many historical references, and Dugas's &lt;em&gt;A History of Mechanics&lt;/em&gt; (Griffon 1955), but I didn't find very much in either. I know of Jammer's &lt;em&gt;Concepts of Force&lt;/em&gt;, but I found so many nonsensical statements in that book that I don't trust it anymore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd be very grateful if you could share any other references you know. Thank you!&lt;/p&gt;&#xA;" OwnerUserId="7231" LastActivityDate="2018-04-02T11:21:54.843" Title="References about the the development of the concept of mechanical work" Tags="|terminology|classical-mechanics|energy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7205" PostTypeId="2" ParentId="7204" CreationDate="2018-04-02T11:21:54.843" Score="4" Body="&lt;p&gt;You can see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Agamenon Oliveira, &lt;a href=&quot;https://books.google.it/books?id=JcLEBAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of the Work Concept: From Physics to Economics&lt;/a&gt;, Springer (2014).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also interesting :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Danilo Capecchi, &lt;a href=&quot;https://books.google.it/books?id=8qUXV6lo8-wC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Virtual Work Laws: A History of Mechanics Prospective&lt;/a&gt;, Birkhauser (2012).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-04-02T11:21:54.843" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7206" PostTypeId="1" CreationDate="2018-04-02T14:40:37.417" Score="7" ViewCount="1071" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Catenary&quot; rel=&quot;noreferrer&quot;&gt;The Wikipedia-entry for catenaries&lt;/a&gt; lists Robert Hooke as the first to study catenaries mathematically, in the 1670s. However, the dome of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Florence_Cathedral#Dome&quot; rel=&quot;noreferrer&quot;&gt;Florence Cathedral&lt;/a&gt;, completed structurally in 1436, follows a catenary arch, which &lt;a href=&quot;http://schillerinstitute.org/educ/pedagogy/2013/vereycken-dome-1.html&quot; rel=&quot;noreferrer&quot;&gt;its builder seems to have been aware of&lt;/a&gt;, meaning that the principle of using a catenary as a good solution (although not at that time mathematically proven to be optimum) for self-supporting structures must have been known earlier than the time of Hooke. Many other examples exists, &lt;a href=&quot;https://en.wikipedia.org/wiki/Saqqara_ostracon&quot; rel=&quot;noreferrer&quot;&gt;some of which&lt;/a&gt; even dates back several thousands of years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;When did knowledge of the above-mentioned principle first become widespread?&lt;/strong&gt; &lt;strong&gt;Who discovered it first&lt;/strong&gt; (if possible to answer)&lt;strong&gt;?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I'm not only interested in Western architecture; for instance, did Persian architects building mosques pre-Renaissance know about it? &lt;/p&gt;&#xA;" OwnerUserId="3579" LastActivityDate="2018-04-05T19:49:10.767" Title="When did architects first become aware of the usefulness of the catenary arch?" Tags="|discoveries|architectural-engineering|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7207" PostTypeId="2" ParentId="7184" CreationDate="2018-04-02T15:10:45.437" Score="0" Body="&lt;p&gt;On optics, you can add to the list above : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Olivier Darrigol, &lt;a href=&quot;https://books.google.it/books?id=ImM62wvWE_cC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;A history of optics: From Greek antiquity to the nineteenth century&lt;/a&gt;, Oxford University Press (2012). &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-04-02T15:10:45.437" CommentCount="0" CommunityOwnedDate="2018-04-02T15:10:45.437" ContentLicense="CC BY-SA 3.0" />
  <row Id="7208" PostTypeId="2" ParentId="7184" CreationDate="2018-04-02T15:11:45.733" Score="0" Body="&lt;p&gt;On mechanics, a recent book is: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Danilo Capecchi, &lt;a href=&quot;https://books.google.it/books?id=kArvAwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Problem of the Motion of Bodies: A Historical View of the Development of Classical Mechanics&lt;/a&gt;, Springer (2016).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-04-02T15:11:45.733" CommentCount="0" CommunityOwnedDate="2018-04-02T15:11:45.733" ContentLicense="CC BY-SA 3.0" />
  <row Id="7209" PostTypeId="2" ParentId="5864" CreationDate="2018-04-02T16:09:51.523" Score="1" Body="&lt;p&gt;It's a long story, and students are constantly confused about the point even today. The basic appreciation that a quantized field is an infinite collection of quantum oscillators suitably organized is &lt;a href=&quot;https://en.wikipedia.org/wiki/Pascual_Jordan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pascual Jordan's&lt;/a&gt;, of course. But the technical innovations of 2nd quantization methods, the specific &quot;packaging&quot; of them is Dirac's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some chronological order, the epochal Dreimännerarbeit, &quot;Zur Quantenmechanic II&quot;,  &lt;em&gt;Zeitschrift for Physik&lt;/em&gt; (1926) (in German)  &lt;strong&gt;35&lt;/strong&gt; (8) &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF01379806&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1007/BF01379806&lt;/a&gt;, by Born, M.; Jordan, P.; Heisenberg, W., gets there on pp 606-615, Ch 4, §3, no &quot;if&quot;s and &quot;but&quot;s!! That section &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0199573263&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;is Entirely the work of Jordan&quot;&lt;/a&gt; reassures one Anthony Duncan  in &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0199573263&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Conceptual Framework of Quantum Field Theory&lt;/em&gt;&lt;/a&gt; p 19.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more familiar language suitably applied to the EM field is Dirac, P. A. M. (1927), &quot;The quantum theory of the emission and absorption of radiation&quot; &lt;em&gt;Proc. R. Soc. Lond. A.&lt;/em&gt; Royal Society Publishing &lt;strong&gt;114&lt;/strong&gt; (767): 243–65.   &lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/114/767/243&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1098/rspa.1927.0039&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But so many strands went into this tapestry that it is hard to parse them out and assess their detailed significance. &#xA;The most meticulous judgments vindicating Jordan's contributions have been detailed in Silvan S. Schweber's &lt;a href=&quot;https://books.google.com/books?id=61n5dE7FJQgC&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q=for%20Jordan%20on%20the%20other%20hand%20&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;QED and the Men who Made it: Dyson, Feynman, Schwinger, and Tomonaga&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I am skipping Fock space itself, as it is basically a technical formalism innovation, and @Conifold covers it in his comment: a well-meaning reader would recognize the logic of Fock space already at the level of the Dreimännerarbeit. The other realizations of field quantization (Wavefunctional quantization, phase-space field quantization, etc.) are mere mathematical curiosities by and large.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2018-04-04T11:12:07.223" LastActivityDate="2018-04-04T11:12:07.223" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7210" PostTypeId="2" ParentId="7206" CreationDate="2018-04-02T23:30:52.983" Score="6" Body="&lt;p&gt;What the builder Neri di Fioravanti was aware of is not catenary arch but &lt;em&gt;quinto acuto&lt;/em&gt; (pointed fifth) or Gothic arch, as the linked article explicitly states. The arc sits on top of a diameter composed of two halves drawn with 4/5 of this diameter spliced at the cusped top (sometimes smoothed out). It is hard to say if this is a better approximation to the catenary than parabola, which Galileo described as approximating hung chain in Two New Sciences (1638). That the chain curve is not a parabola was shown by  Joachim Jungius also before Hooke, but only published in 1669. It was Hooke who linked the catenary to architechture in connection with the rebuilding of St Paul's Cathedral , see &lt;a href=&quot;https://en.wikipedia.org/wiki/Catenary#History&quot; rel=&quot;noreferrer&quot;&gt;Catenary: History&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Although Romans still used spherical domes, &lt;a href=&quot;https://en.wikipedia.org/wiki/Taq_Kasra&quot; rel=&quot;noreferrer&quot;&gt;Taq Kasra, the Archway of Ctesiphon&lt;/a&gt; has the shape close to the &amp;quot;catenary&amp;quot; or pointed fifth, the method is conjectural. This seems to be the earliest known occurence, a c. 3-6th century AD Persian structure, presumably the main portico of the audience hall of the Sassanid shah's throne room, when Persia was still Zoroastrian. After the Islamic conquest it was converted into a mosque. Explicit uses of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gothic_architecture#Pointed_arch&quot; rel=&quot;noreferrer&quot;&gt;pointed arch&lt;/a&gt; also predate the Florentine dome and Gothic architecture like Notre-Dame de Paris.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The pointed arch is also a characteristic feature of Near Eastern pre-Islamic Sassanian architecture that was adopted in the 7th century by Islamic architecture and appears in structures like the Al-Ukhaidir Palace (775 AD), the Abbasid reconstruction of the Al-Aqsa mosque in 780 AD, Ramlah Cistern (789 AD), the Great Mosque of Samarra (851 AD), and the Mosque of Ibn Tulun (879 AD) in Cairo. It also appears in the Great Mosque of Kairouan, Mosque–Cathedral of Córdoba, and several structures of Norman Sicily...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The majority view of scholars however is the idea that the pointed arch was a simultaneous and natural evolution in Western Europe as a solution to the problem of vaulting spaces of irregular plan, or to bring transverse vaults to the same height as diagonal vaults, as evidenced by Durham Cathedral's nave aisles, built in 1093.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-04-02T23:30:52.983" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7211" PostTypeId="2" ParentId="7201" CreationDate="2018-04-03T00:00:12.687" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Elliptic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elliptic geometry&lt;/a&gt; is not equivalent to geometry on the sphere because there is non-unique line through antipodal points on the sphere, contrary to one of the axioms. One needs to identify the antipodal points, which gives the real projective plane. With induced metric it will be a model. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Kline, Riemann introduced manifolds of constant curvature in his 1861 paper, although there is some controversy as to what can be read into it. It is also unclear if he was aware that two different geometries were involved for constant positive curvature, it was later explicitly pointed out by Klein. Beltrami, who also showed that the pseudo-sphere is a (partial) model of hyperbolic geometry, pointed out that the sphere is a model for one of them. See &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=1380423&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jim Murdock's post on Math Forum&lt;/a&gt; for relevant quotes from Kline's Mathematical Thought from Ancient to Modern Times, the controversy is also discussed there.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-04-03T00:09:18.677" LastActivityDate="2018-04-03T00:09:18.677" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7212" PostTypeId="1" CreationDate="2018-04-03T22:09:14.963" Score="3" ViewCount="343" Body="&lt;p&gt;According to most discussions of Euclid's Elements, this work - and indeed, much of Ancient Greek geometry - should be seen as engaged in the game of figuring out what can be done with straightedge and (collapsible) compass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although Euclid's Elements can certainly be (mathematically) interpreted in this way, Euclid never says anything to explicitly support this. Nor does one find an explicit discussion of the idea that this is how we should view the ancient practice of geometry (with respect to Euclid or the Greek geometrical tradition in general) in Plato, Aristotle, Proclus and so on - as far as I can tell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any ancient source who specifically talks about straightedges and compasses and the idea that much of geometry revolves around questions of what can and can't be done with such tools? I am aware that much discussion of Greek mathematics simply assumes this, so I'm not looking for references to modern people who repeat this piece of folklore - there are examples of that everywhere. I'm rather looking for actual extant ancient texts that support this piece of folklore.&lt;/p&gt;&#xA;" OwnerUserId="7237" LastEditorUserId="7237" LastEditDate="2018-04-03T22:56:34.857" LastActivityDate="2018-04-03T23:52:53.980" Title="Straightedge and compass" Tags="|mathematics|ancient-greece|euclidean-geometry|geometry|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7213" PostTypeId="2" ParentId="7212" CreationDate="2018-04-03T23:52:53.980" Score="3" Body="&lt;p&gt;“Geometry … teaches us the use of the rule and compasses.” Vitruvius, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.02.0073%3Abook%3D1%3Achapter%3D1%3Asection%3D4&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De architectura&lt;/em&gt; I.1.4&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“I describe a circle with the compass. … With the straight ruler I set to work to inscribe a square within this circle.” Aristophanes, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0026:card=1000&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Birds&lt;/em&gt;, 1000&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“A circle drawn with geometrical compasses.” Plutarch, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2008.01.0321:section=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Maxime&lt;/em&gt; 1&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the importance of knowing what can and cannot be done by these tools, Pappus (&lt;a href=&quot;https://www.loebclassics.com/view/greek_mathematical_works_ix_special_problems/1939/pb_LCL335.349.xml&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Collection&lt;/em&gt; IV&lt;/a&gt;) says it is “no small error for geometers” to use more advanced means when a problem “can be solved by means of a straight line and a circumference of a circle,” that is, by ruler and compass.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2018-04-03T23:52:53.980" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7214" PostTypeId="1" AcceptedAnswerId="7215" CreationDate="2018-04-04T02:16:37.583" Score="2" ViewCount="236" Body="&lt;p&gt;My question is: Who is responsible for incorporating the initial conditions into the one dimensional wave equation solution? References or technical information would be appreciated, especially regarding the development of the integral term.&lt;/p&gt;&#xA;&lt;h2&gt;Background&lt;/h2&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$\phi(t,x)$&lt;/span&gt; is the general solution to the one dimensional wave equation and if the initial conditions &lt;span class=&quot;math-container&quot;&gt;$\phi(0,x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi_t(0,x)$&lt;/span&gt; are given, then the formula for the solution incorporating initial conditions is:&lt;/p&gt;&#xA;&lt;p&gt;Letting &lt;span class=&quot;math-container&quot;&gt;$g(x)=\phi(0,x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h(x)= \phi_t(0,x)$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$c=1$&lt;/span&gt;),&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\phi(t,x)= \frac 12[ g(x-t)+ g(x+t) ]+ \frac12 \int_{x-t}^{x+t} h(y)dy .$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;All of this is usually attributed to d'Alembert, however my &amp;quot;References for Euler / d'Alembert&amp;quot; imply that the solution with initial conditions were contributed by Euler.  If so, then the integral term was Euler's work?&lt;/p&gt;&#xA;&lt;h2&gt;References&lt;/h2&gt;&#xA;&lt;p&gt;References for Euler / d'Alembert:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Michela Massimi, &lt;em&gt;Kant and Philosophy of Science Today&lt;/em&gt;, p185&lt;/li&gt;&#xA;&lt;li&gt;Amir R. Alexander, &lt;em&gt;Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics&lt;/em&gt;, p63&lt;/li&gt;&#xA;&lt;li&gt;Manuel D. Salas, &lt;em&gt;A Shock-Fitting Primer&lt;/em&gt;, p3&lt;/li&gt;&#xA;&lt;li&gt;Hankins, &lt;em&gt;Jean d'Alembert-Science&lt;/em&gt; , p48&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Leonhard Euler: Life, Work and Legacy&lt;/em&gt;, edited by Robert E. Bradley, Ed Sandifer, p30&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;References for d'Alemberts formula:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathworld.wolfram.com/dAlembertsSolution.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;D'Alembert's Solution&lt;/a&gt;, MathWorld&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/D%27Alembert%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;D'Alembert's formula&lt;/a&gt;, English Wikipedia&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.jirka.org/diffyqs/htmlver/diffyqsse32.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;4.6 PDEs, separation of variables, and the heat equation&lt;/a&gt;, Jiří Lebl website&lt;/li&gt;&#xA;&lt;li&gt;R. L. Herman, &lt;a href=&quot;http://people.uncw.edu/hermanr/pde1/dAlembert/dAlembert.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;d'Alembert Solution of the Wave Equation&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Additional sources:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;English translation of what may be Euler's work on this: &lt;a href=&quot;https://download.uni-mainz.de/mathematik/Algebraische%20Geometrie/Euler-Kreis%20Mainz/E119Englisch.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Treatise on the vibration of chords&lt;/em&gt;&lt;/a&gt; (Server of the University of Mainz)&lt;/li&gt;&#xA;&lt;li&gt;Original in Latin: Euler, Leonhard, &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/119&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De vibratione chordarum exercitatio&lt;/em&gt;&lt;/a&gt;&#xA;(1749). Euler Archive - All Works. 119.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This question was first asked on:&#xA;&lt;a href=&quot;https://physics.stackexchange.com/q/302304/45664&quot;&gt;https://physics.stackexchange.com/q/302304/45664&lt;/a&gt;.&#xA;It was too old to migrate.&lt;/p&gt;&#xA;" OwnerUserId="7211" LastEditorUserId="6609" LastEditDate="2023-09-18T20:41:37.717" LastActivityDate="2023-09-18T20:41:37.717" Title="Did Euler or did d'Alembert incorporate initial conditions into the solution to the 1D wave equation?" Tags="|mathematical-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7215" PostTypeId="2" ParentId="7214" CreationDate="2018-04-04T07:34:01.883" Score="2" Body="&lt;p&gt;&lt;del&gt;I doubt&lt;/del&gt; D’Alembert had the integral term. Weber’s treatise (1901, &lt;a href=&quot;//archive.org/stream/diepartiellendi01riemgoog#page/n235&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 213&lt;/a&gt;) quotes the formula as “originating with” him, but I think only the general form $\Psi(x+ct)+\Phi(x-ct)$ is meant here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;del&gt;In fact&lt;/del&gt; Riemann (1867, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uiug.30112113441676&amp;amp;num=90&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 90&lt;/a&gt;; &lt;a href=&quot;//www.emis.de/cgi-bin/JFM-item?02.0175.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1869&lt;/a&gt;, pp. &lt;a href=&quot;//books.google.com/books?id=jRJTAAAAcAAJ&amp;amp;pg=PA111&quot; rel=&quot;nofollow noreferrer&quot;&gt;111-113&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=jRJTAAAAcAAJ&amp;amp;pg=PA188&quot; rel=&quot;nofollow noreferrer&quot;&gt;188-201&lt;/a&gt;) — relied on by both Weber (p. 205) and the &lt;em&gt;Encyklopädie&lt;/em&gt; (1900, &lt;a href=&quot;//archive.org/stream/p1n1encyklopdied02akaduoft#page/557&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 557&lt;/a&gt;) — attributed the integral term to Euler (1750, likely &lt;a href=&quot;//archive.org/stream/euler-e140#page/n9&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 78&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler was using rather different notation, still found in e.g. Forsyth (1929, &lt;a href=&quot;//archive.org/stream/treatiseondiffer032528mbp#page/n507&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 488&lt;/a&gt;). If you want the literal formula, Kline (1972, &lt;a href=&quot;//archive.org/stream/MathematicalThoughtFromAncientToModernTimes#page/n531&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 513&lt;/a&gt;) seems to say it was first written by Lagrange (1762, &lt;a href=&quot;//archive.org/stream/mlangesdephilo02soci#page/n262&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 27&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; It seems I put too much faith in Riemann. Here is from Burkhardt (&lt;a href=&quot;//zbmath.org/?q=an:32.0388.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/entwicklungenna01burkgoog#page/n29&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 12&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In an immediately subsequent second paper&lt;sup&gt; 44)&lt;/sup&gt;, D’Alembert first details the construction of the curve $y= \Psi(x+t)-\Psi(x-t)$ from the curve $y=\Psi(x)$. Then he turns to the more general assumption that at $t=0$ one arbitrarily prescribes not only the initial ordinate $y=\Sigma(x)$ but also the initial velocity $\partial y\,/ \,\partial t=\sigma(x)$ of each point of the string&lt;sup&gt; 45)&lt;/sup&gt;. In that case (...)&#xA;  $$&#xA;\Psi(x)-\Psi(-x)=\Sigma(x)&#xA;\tag9&#xA;$$&#xA;  and:&#xA;  $$&#xA;\Psi'(x)-\Psi'(-x)=\sigma(x)&#xA;$$&#xA;  or:&#xA;  $$&#xA;\Psi(x)+\Psi(-x)=\textstyle\int\sigma(x)\,dx.&#xA;\tag{10}&#xA;$$&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;44)&lt;/sup&gt; &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1747&amp;amp;seite:int=249&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 220&lt;/a&gt; [1749].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;45)&lt;/sup&gt; nº 23, &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=02-hist/1747&amp;amp;seite:int=259&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 230&lt;/a&gt;. &lt;strong&gt;&lt;em&gt;Riemann’s oft-repeated assertion that this case was first treated by Euler must be based on an oversight of this section&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-06-04T12:05:36.747" LastActivityDate="2018-06-04T12:05:36.747" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7216" PostTypeId="1" CreationDate="2018-04-04T15:19:10.560" Score="2" ViewCount="214" Body="&lt;p&gt;Who first noted the connection between Heisenberg's &lt;a href=&quot;https://en.wikipedia.org/wiki/Uncertainty_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;uncertainty principle&lt;/a&gt; and the Fourier transform?&lt;/p&gt;&#xA;" OwnerUserId="7241" LastEditorUserId="4251" LastEditDate="2023-12-05T14:55:06.367" LastActivityDate="2023-12-05T14:55:06.367" Title="Who first noted the connection between Heisenberg's uncertainty principle and the Fourier transform?" Tags="|physics|quantum-mechanics|harmonic-analysis|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7217" PostTypeId="1" AcceptedAnswerId="7220" CreationDate="2018-04-04T18:21:36.687" Score="7" ViewCount="1316" Body="&lt;p&gt;As far as I know, while Galileo's hypothesis about the tides was wrong, his discovery of the phases of Venus while passing the sun should have been sufficient to leave the geocentric crystal spheres shattered - figuratively and literally.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in the context of an answer to another question (&lt;a href=&quot;https://philosophy.stackexchange.com/a/50328&quot;&gt;https://philosophy.stackexchange.com/a/50328&lt;/a&gt;), comments claim the reasons Galileo gave for the heliocentric model were no good science at all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That seems suspicious to me but I thought I'd better get an expert's view on how valid Galileo's arguments were.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know the exact reasons for the criticism, maybe someone who shares it wants to elaborate here; otherwise I'd be happy with a short summary of the scientific &quot;canon&quot; view on the topic.&lt;/p&gt;&#xA;" OwnerUserId="7243" LastActivityDate="2018-04-05T03:10:05.513" Title="How scientifically valid were Galileo's heliocentric arguments?" Tags="|astronomy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7219" PostTypeId="2" ParentId="7206" CreationDate="2018-04-04T19:43:04.903" Score="4" Body="&lt;p&gt;The following is not an answer; it's a comment.&lt;br&gt;&#xA;Also, I'm not providing &lt;em&gt;historical&lt;/em&gt; information.&lt;br&gt;&#xA;However, I feel that the information that I present is interesting and useful to know.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Thinking that a catenary arch has general usefulness in architecture is actually problematic, I will argue. There is a wrong assumption at play. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A catenary is the mathematical solution to the problem of a structure (either hanging chain or standing arch) that has a &lt;em&gt;uniform weight per unit of length&lt;/em&gt;. As we know, the catenary is the solution with equilibrium everywhere along the length.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in the case of any large building you make the walls thick at the base, so they can carry the total load, and you make the walls thinner as you go higher. That means that near the base the structure will have &lt;em&gt;more weight per unit of arch length&lt;/em&gt; than the top section. And that means the mathematical solution will be somewhat different from a catenary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Domes&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With domes using a catenary cross-section is even less applicable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, take the case of a theoretical dome, a dome with a shell that has the same thickness everywhere. What can we say about its equilibrium shape? If you would take a catenary arch, and you would rotate that around its symmetry axis then the resulting dome that is &lt;em&gt;not&lt;/em&gt; an equilibrium shape. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Demonstration:&lt;br&gt;&#xA;Imagine two catenary arches, at right angles to each other, crossing each other at the top. That means the top stone is &lt;em&gt;shared&lt;/em&gt; between the two arches. It follows that that in order to be overall in equilibrum the cross-section of that structure must be somewhat flatter than a catenary. Imagine adding more and more crossing arches (adding up to a dome). At the base of that structure weight will be contributed by &lt;em&gt;each arch&lt;/em&gt;, but at the top most of the weight will be &lt;em&gt;shared&lt;/em&gt; weight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: even in the case of a dome with a uniform thickness of the shell the equilibrium shape will not be one that has a catenary-shape cross section.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's take one of the most famous domes: the Pantheon in Rome. Naturally the builders of that structure constructed it with a tapered thickness of the shell. Very thick at the base, gradually thinning out towards the top. (Also, the aggregate in the concrete varies, with pumice used at the very top.) The result: different weight per unit of surface from base to top.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in the case of a real world dome there are two independent factors that affect what the equilibrium shape will be, and both of them will shift the mathematical solution away from a catenary cross-section.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Of course, this doesn't necessarily mean that you cannot build a dome with a catenary-shape cross-section. If the structure is strong enough then you can. The structure has to be able to withstand wind loads anyway. That is, the structure must be able to withstand bending stresses anyway, so chances are that the bending stresses that arise from not being an equilibrium shape are well within the range of what the structure is able to handle.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2018-04-04T19:43:04.903" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7220" PostTypeId="2" ParentId="7217" CreationDate="2018-04-05T00:13:49.857" Score="8" Body="&lt;p&gt;I assume the doubts refer to the commentary on &lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;noreferrer&quot;&gt;Dialogue Concerning the Two Chief World Systems&lt;/a&gt; in &lt;a href=&quot;https://www.lesswrong.com/posts/rsAvZtTpTnHhmL5Ht/the-galileo-affair-who-was-on-the-side-of-rationality&quot; rel=&quot;noreferrer&quot;&gt;The Galileo affair: who was on the side of rationality?&lt;/a&gt; on Less Wrong. I do not believe that Galileo's case was as weak as the post suggests.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main objections to heliocentrism and rotating Earth were known since antiquity: the absence of stellar parallax or of fly-off (to which more subtle &quot;Coriolis effect&quot; was added later). There was little that either Copernicus or Galileo could say about the former, other than to speculate that the stars are too distant for the effect to be measurable. The parallax was only measured in 1838 by Bessel. As for the fly-off argument, Galileo's experiments illustrating the relativity of motion, such as descriptions of events on a moving ship, were directly meant to disarm it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo's arguments were also aimed at undermining geocentric Aristotelian physics generally, e.g. it explanation of falling bodies by their movement to their &quot;natural place&quot;, the center of the Earth, mentioned in the post. According to Aristotle, the &quot;degree&quot; of attraction depends on the body's weight, heavier bodies fall faster. This idea was mocked by Philoponus already in antiquity, and Galileo's famous thought experiment with tying two bodies together to show that their weights are moot was also plausible if not logically airtight, see &lt;a href=&quot;https://philosophy.stackexchange.com/q/18351/9148&quot;&gt;Is Galileo's argument about falling bodies logically flawed?&lt;/a&gt;. So, contrary to the psot's author, this was not a particularly strong argument for geocentrism at the time. It is true, however, that Galileo did not have an alternative comprehensive dynamical theory to replace Aristotle's, that would only come later, with Descartes, Huygens and ultimately Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Judging the tidal argument &quot;silly&quot; is anachronistic. Yes, Galileo thought that the tides are due to the rotating Earth rather than Moon's gravity, and that would not work under Newtonian dynamics. But there was no Newtonian dynamics at the time, and the implied effect is real, it explains Earth's bulged shape if not the tides. And the geocentric allusion to &quot;Moon's influence&quot; had no basis in Aristotelian physics of natural places. The connection was based on observed  correlations, not on any geocentric insight, and so provides no support either way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main contemporary arguments for heliocentrism are not even discussed there. We are not talking about the time a century earlier when Copernicus still had to use epicyclets and excenters to match his system to observations. After Kepler replaced circles with ellipses the match heliocentric system provided was unparalleled. One would need towers of epicycles to get the same precision geocentrically. And trying to incorporate ellipses into something like Tycho Brahe's hybrid system would upend Aristotelian physics anyway. In fairness, Galileo himself mentions neither Tychic system nor elliptic orbits in the Dialogue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Galileo's own discoveries of the moons of Jupiter and of the phases of Venus did appear, and suggested that the Earth was no special &quot;natural place&quot;. Kepler already speculated about gravity like effects, see &lt;a href=&quot;https://hsm.stackexchange.com/a/1975/55&quot;&gt;Who was first to explain intuitively the inverse square law of gravity?&lt;/a&gt;, his speculations were based on Gilbert's theory of magnetism, which Galileo explicitly invokes in the Dialogue.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-04-05T00:13:49.857" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7221" PostTypeId="1" AcceptedAnswerId="7222" CreationDate="2018-04-05T01:02:21.910" Score="2" ViewCount="673" Body="&lt;p&gt;I learned in school that Watson and Crick discovered DNA, but in fact DNA was discovered many years before, and it turns out that it was the &lt;strong&gt;structure&lt;/strong&gt; of DNA that Watson and Crick discovered (or proposed).&lt;/p&gt;&#xA;&lt;p&gt;However, there was also a woman whose experiments contributed to this, although she did not receive the Nobel Prize.&lt;/p&gt;&#xA;&lt;p&gt;Who was she, and what was her contribution?&lt;/p&gt;&#xA;" OwnerUserId="7247" LastEditorUserId="7247" LastEditDate="2024-03-19T02:36:48.680" LastActivityDate="2024-03-19T02:36:48.680" Title="Woman who contributed to Watson &amp; Crick's discovery of the double-helical structure of DNA" Tags="|biology|biologists|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7222" PostTypeId="2" ParentId="7221" CreationDate="2018-04-05T01:53:49.157" Score="17" Body="&lt;p&gt;The woman you're thinking of is, notably, &lt;a href=&quot;https://en.wikipedia.org/wiki/Rosalind_Franklin&quot; rel=&quot;noreferrer&quot;&gt;Rosalind Franklin&lt;/a&gt;. Her specialty was &lt;a href=&quot;https://en.wikipedia.org/wiki/X-ray_crystallography&quot; rel=&quot;noreferrer&quot;&gt;X-ray crystallography&lt;/a&gt;, and she worked with &lt;a href=&quot;https://en.wikipedia.org/wiki/Maurice_Wilkins&quot; rel=&quot;noreferrer&quot;&gt;Maurice Wilkins&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Raymond_Gosling&quot; rel=&quot;noreferrer&quot;&gt;Raymond Gosling&lt;/a&gt; in the early 1950s, continuing their previous work, to apply X-ray diffraction to produce images of DNA. Watson and Crick's models were built and confirmed by data gathered by Franklin, Wilkins and Gosling; the images allowed them to discount previous models and identify the double helix.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To say that Rosalind Franklin helped with the experiments is erroneous; she &lt;em&gt;performed&lt;/em&gt; the experiments, along with her colleagues (and yes, other work on the problem of structure had been done by other groups around the same time). It would perhaps be more correct to say that Franklin performed the experiments and Watson and Crick interpreted them in more detail (as Franklin's group also came to the conclusion that a double helix was responsible for some of DNA's shape).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's notable that the group Franklin was in was separate from Watson and Crick. She was at &lt;a href=&quot;https://en.wikipedia.org/wiki/King%27s_College_London&quot; rel=&quot;noreferrer&quot;&gt;King's College, London&lt;/a&gt;, while the latter two were at the University of Cambridge's &lt;a href=&quot;https://en.wikipedia.org/wiki/Cavendish_Laboratory&quot; rel=&quot;noreferrer&quot;&gt;Cavendish Laboratory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2018-04-05T01:53:49.157" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7223" PostTypeId="2" ParentId="7217" CreationDate="2018-04-05T02:55:51.790" Score="6" Body="&lt;p&gt;Galileo had no proof of heliocentrism, and no proof of the diurnal rotation of the Earth. His main argument was about tides, and this argument is incorrect. Moreover, it contradicts Galileo's own mechanics, namely the relativity principle.&#xA;(Contrary to what Conifold writes, the foundations of mechanics are due to Galileo, not to Newton. But when he became engaged in this Earth rotation polemics,&#xA;he somehow forgot his earlier discoveries). Of course, heliocentrism was an attractive hypothesis since Copernicus, but certainly there was no proof of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact this question does not deserve the attention which it attracts: it is a question of the choice of a coordinate system, and the choice of a coordinate system is actually a question of convenience. Geocentric co-ordinates are still&#xA;widely used in many problems, whenever convenient. So &quot;heliocentrism&quot; belongs more to philosophy than to science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earth diurnal rotation is another matter. This is something which can be proved &#xA;or disproved by experiment and various experiments were proposed. But as far as I know the first decisive and successful experiment was Foucalut pendulum, and this was much later than the time of Galileo. By the way, the main subject of &quot;discussion&quot; between Galileo and Catholic church was diurnal rotation, not&#xA;heliocentrism. And both sides used wrong arguments in this discussion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler's great discovery that the orbits are actually ellipses, not circles,&#xA;or combinations of circles&#xA;was truly revolutionary, and  it led to replacement of Ptolemy's system by the new one. But this has nothing to do with Galileo: Galileo never even mentioned Kepler's laws in his writings!   &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-05T03:10:05.513" LastActivityDate="2018-04-05T03:10:05.513" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7224" PostTypeId="1" CreationDate="2018-04-05T10:32:06.570" Score="9" ViewCount="3362" Body="&lt;p&gt;All in the title:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When did scientists realize that our eye functions like a lens/magnifying glass? &lt;/p&gt;&#xA;" OwnerUserId="3019" LastEditorUserId="-1" LastEditDate="2018-04-10T20:21:36.793" LastActivityDate="2019-05-06T01:23:22.367" Title="When did people realize that the eye was a lens?" Tags="|biology|optics|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7225" PostTypeId="1" CreationDate="2018-04-05T11:06:02.183" Score="5" ViewCount="645" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Spin_(physics)#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; states, that the spin degree of freedom was first formulated by Pauli in 1924:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1924 Wolfgang Pauli introduced what he called a &quot;two-valued quantum degree of freedom&quot; associated with the electron in the outermost shell.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, there is no reference to this quote. Does someone know the publication where Pauli introduced this quantum number for the first time? The only paper I know &lt;a href=&quot;http://www.springerlink.com/index/20W1M0VR050J033R.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;, where he formulates his exclusion principle. But this is from 1925, so one year later.&lt;/p&gt;&#xA;" OwnerUserId="7248" LastEditorUserId="131" LastEditDate="2018-12-04T12:30:15.707" LastActivityDate="2018-12-04T12:30:15.707" Title="Pauli's first paper about the spin" Tags="|physics|biographical-details|quantum-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7226" PostTypeId="2" ParentId="7224" CreationDate="2018-04-05T12:51:44.273" Score="9" Body="&lt;p&gt;Probably the Swiss physician &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Plater&quot; rel=&quot;noreferrer&quot;&gt;Felix Platter (1536 – 1614)&lt;/a&gt; that - according to Olivier Darrigol's &lt;a href=&quot;https://books.google.it/books?id=ImM62wvWE_cC&amp;amp;pg=PA23&quot; rel=&quot;noreferrer&quot;&gt;A history of optics : From Greek antiquity to the nineteenth century&lt;/a&gt;, Oxford University Press (2012), page 23 - in his :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;anatomic treatise of 1583 made the retina the sensor of the visual species, and the crystalline humor a magnifying glass (&lt;em&gt;perspicillum&lt;/em&gt;). [...] &lt;em&gt;Perspicillum&lt;/em&gt; was the Latin name then given to a growing popular device: the eyeglass.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See  Felix Platter, &lt;a href=&quot;https://books.google.it/books?id=4fQ6AAAAcAAJ&amp;amp;pg=PA186&quot; rel=&quot;noreferrer&quot;&gt;De partium corporis humani structura eius et usu Libri III (Basel, 1603 ed)&lt;/a&gt;, page 186. &lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-04-05T12:51:44.273" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7227" PostTypeId="2" ParentId="7225" CreationDate="2018-04-05T13:03:21.520" Score="3" Body="&lt;p&gt;According Charles Enz's &lt;a href=&quot;https://books.google.it/books?id=lSoRzxFye-4C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;W. Pauli's Scientific Work&lt;/a&gt;, into Jagdish Mehra (editor), The physicist's conception of nature, Reidel (1973), page 770-on (emphasis added):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The third paper on the anomalous Zeeman effect [&lt;em&gt;Über den Einfluß der Geschwindigkeitsabhängigkeit der Elektronenmasse auf den Zeemaneffekt&lt;/em&gt;, &lt;a href=&quot;//doi.org/10.1007/BF02980592&quot; rel=&quot;nofollow noreferrer&quot;&gt;Z. Phys. &lt;strong&gt;31&lt;/strong&gt; (1925) 373-385&lt;/a&gt;], &lt;strong&gt;submitted on 2 December 1924&lt;/strong&gt;, is of fundamental importance because it introduces, still in disguised form, the &#xA;  spin quantum number of the electron. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="658" LastEditDate="2018-04-05T14:31:40.233" LastActivityDate="2018-04-05T14:31:40.233" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7228" PostTypeId="2" ParentId="7224" CreationDate="2018-04-05T13:51:29.417" Score="7" Body="&lt;p&gt;Here's a very early bit of research.  While it identifies the lens as critical to vision, it doesn't seem to recognize  the lens as a focussing device  From some &lt;a href=&quot;https://web.stanford.edu/class/history13/earlysciencelab/body/eyespages/eye.html&quot; rel=&quot;noreferrer&quot;&gt;classroom notes&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the second century A. D., Galen had at least two different theories&#xA;  of the eye to choose from.  He chose the extramission theory because&#xA;  it corresponded well with his image of sight as a function of an&#xA;  optical pneuma, flowing forth from the brain to the eyes through&#xA;  hollow optic nerves.  Galen defined many of the fundamental features&#xA;  of the anatomy and physiology of the eye until the seventeenth&#xA;  century.  Benefiting from the work of the anatomists who dissected in&#xA;  Alexandria, such as Rufus of Ephesus, he described the retina, cornea,&#xA;  iris, uvea, tear ducts, and eyelids, as well as two fluids he called&#xA;  the vitreous and aqueous humors.  He noted some of the peculiar&#xA;  features of sight such as binocular vision.  Galen paid particular&#xA;  attention to the crystalline lens, which he described as a round lens&#xA;  in the middle of the eye.  He concluded:  &quot;the crystalline lens is the&#xA;  principal instrument of vision, a fact clearly proved by what&#xA;  physicians call cataracts, which lie between the crystalline humor and&#xA;  the cornea and interfere with vision until they are couched.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This same page goes on to credit Platter as recognizing the lens as an imager (and more or less putting to be the concept of the optic nerve as an emitter).&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-04-05T13:51:29.417" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7229" PostTypeId="2" ParentId="7224" CreationDate="2018-04-05T17:06:01.260" Score="11" Body="&lt;h2&gt;When Did Humans First Observe a Lens in an Eye?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;People probably realized that the eye contained a lens (at least the eye of a fish) as soon as they began fishing for dinner.  So, sadly the discovery of the hard lump that is a fish lens was probably prehistoric.&#xA;Here is a picture of a fish lens from an Instructable on fish dissection.&lt;a href=&quot;https://i.stack.imgur.com/JecxD.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JecxD.jpg&quot; alt=&quot;http://www.instructables.com/id/Fish-Dissection/&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.instructables.com/id/Fish-Dissection/&quot; rel=&quot;noreferrer&quot;&gt;http://www.instructables.com/id/Fish-Dissection/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, we have 3 things left to figure out:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When did humans figure out what this clear, &quot;lentil shaped&quot; (the word for lentil and the word for lens are the same in Greek and Latin) object in a fish eye was for?&lt;/li&gt;&#xA;&lt;li&gt;When did they find out that the human eye also had one?&lt;/li&gt;&#xA;&lt;li&gt;When are we sure they put all this together with a theory?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;h2&gt;1. When Did Humans Know What The Fish Lens Was For?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;We have a hard, clear lump that humans could have found in a fish's eye.  When did humans start to use similarly shaped items to see things better so that we have some idea that they knew what that lump in the fish eye was for?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This article (&lt;a href=&quot;http://www.wikiwand.com/en/History_of_optics&quot; rel=&quot;noreferrer&quot;&gt;http://www.wikiwand.com/en/History_of_optics&lt;/a&gt;) notes the Nimrud-Layard Lens dated from 750 BCE as the first evidence we have.  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;But, if you follow their link to the site and read the full description, there is evidence that this &quot;lens&quot; was decorative.&lt;/li&gt;&#xA;&lt;li&gt;Yet, the object has optical properties that have been measured such as its focal point&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Harry Thurston Peck, in Harpers Dictionary of Classical Antiquities (1898) assures us (without a citation) that the ancient Assyrians:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;&#xA;  &lt;p&gt;...were acquainted with glass; that they employed the arch in building; that they used the lens as a magnifying instrument; and had, among other mechanical appliances, the lever and the roller.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Euclid describes the geometry of optics in his work, &lt;em&gt;Optics&lt;/em&gt; in 300 BCE.  And, it is important for us to remember that Euclid was more of a textbook compiler of existing knowledge than someone who was creating and discovering the things he wrote about.  Much of his work has been traced back to Sumer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, his &lt;em&gt;Optics&lt;/em&gt; could be better titled &lt;em&gt;The Geometry of Perspective&lt;/em&gt; as they don't really relate to the refractive properties of a lens (although I haven't read them in depth).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his book, &lt;em&gt;QED: The Strange Theory of Light and Matter&lt;/em&gt;, Feynman points out that the evidence that prehistoric man understood refraction goes back to the first successful spearfishing.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/fN5K2.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fN5K2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;br&gt;&#xA;&lt;a href=&quot;http://legacy.sciencelearn.org.nz/Contexts/Light-and-Sight/Sci-Media/Video/Refraction/(quality)/hi&quot; rel=&quot;noreferrer&quot; title=&quot;Science Learning Hub&quot;&gt;Science Learning Hub&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For evidence of use of an optical lens from at least 68 AD we have:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In the writings of Pliny the Elder, we have a description of the use&#xA;of what was either an emerald of another green gemstone that was&#xA;classed as a smaragdus.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And then, besides, when viewed from a distance, these stones appear all the larger to the sight, reflecting as they do, their green hues&#xA;  upon the circumambient air...In form they are mostly concave, so as&#xA;  to re-unite the rays of light and the powers of vision: and hence it&#xA;  is, that it is so universally agreed upon among mankind to respect&#xA;  these stones, and to forbid their surface...When the surface of the&#xA;  smaragdus is flat, it reflects the image of objects in the same&#xA;  manner as a mirror.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;h2&gt;2. When did they find out that the human eye also had one?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;We have apparently lost the original works where Aristotle published his vivisection and dissection notes from his work on the Island of Lesbos between leaving Plato's Academy (when he was passed over to head it after Plato's death in 348 BCE) and a couple of years later when he returned to Macedonia to act as Alexander the Great's tutor.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, thankfully, the Arab scholars did not throw out the &quot;Pagan&quot; works of the Greeks.  And in Ibn al-Haytham's work (965–1039 AD), we have what is said to be a copy of one of Aristotle's diagrams of the eye.  It appears to show a lens.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure what animal this is from.  But, we know that Aristotle noted a difference between mammalian eyes and those of cuttlefish.  So, I would say it was almost certain that at this time is was known that the human eye contained a lens (even if no one admitted to human dissection).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am also not certain that this is actually a copy of Aristotle's drawings.  This may have been reported this way by those who wish to demonstrate that Arab cultures contributed nothing to science.  So, this drawing may have dated from  about 348 BCE or about 1400 years later...  Reading the Arabic would help.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, either this diagram predates Galen's 2nd Century AD work by a few centuries or it came a few centuries after...  Sorry!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lbbf7.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lbbf7.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.researchgate.net/publication/261744073_Ibn_al-Haytham_965-1039_AD_The_original_portrayal_of_the_modern_theory_of_vision&quot; rel=&quot;noreferrer&quot;&gt;Ibn al-Haytham (965–1039 AD), The original portrayal of the modern theory of vision&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We certainly have established that knowledge by the time Leonardo makes his scandalous drawings of cadaver dissections.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/goWCE.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/goWCE.jpg&quot; alt=&quot;One of Many Drawings of Human Eye Dissections made by da Vinci&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;3. When are we sure they put all this together with a theory?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;The excellent answer above from &lt;a href=&quot;https://hsm.stackexchange.com/users/609/carl-witthoft&quot;&gt;CarlW&lt;/a&gt; about Galen is earlier than what I knew about from the Islamic Golden Age.  Namely that Ibn al-Haytham (965–1039 AD) seems to have pulled this all together in his work.  But, that isn't to say that the investigators in Europe in the Scientific Revolution didn't have to rediscover this on their own and came up with additional nuances and questions.  (One of Newton's problems with the particle nature of corpuscles and light passing through lenses would not be answered until Quantum Mechanics-- as Feynman points out.)&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Final Note:&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;There are a lot of &quot;but they couldn't have &lt;em&gt;understood&lt;/em&gt; optics&quot; writers that comment on the early or ancient use of the lens.  But, we humans have a tendency to believe that all who came before us were far more ignorant and unimaginative than our own enlightened generation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thinkers who wrote about optics in the Scientific Revolution, for example, were wealthy and far removed from the act of fishing for their dinner and preparing that fish.  As a child, I fished with my father for dinner many times.  And, while preparing it for dinner, he showed me how to extract the lens from a fish's eye.  So, earlier thinkers had a ready-made example of how the lense should look (lentil-shaped and transparent) for them to copy and experiment with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps!  Oh-- and don't forget-- the Greeks used Camera Obscura techniques to observe the eclipse.  That's how Aristarchus estimated the distance from the Earth to the Sun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A huge help in tracking down sources for my answer was this web page.  After you search for a word in their database, there will be a place to the right of the page where you can limit the results by document where they are found-- so you can get closer to the time period you are interested in.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.perseus.tufts.edu/hopper/&quot; rel=&quot;noreferrer&quot;&gt;http://www.perseus.tufts.edu/hopper/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7005" LastEditorUserId="7005" LastEditDate="2018-04-06T15:07:00.157" LastActivityDate="2018-04-06T15:07:00.157" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7232" PostTypeId="1" CreationDate="2018-04-06T17:10:29.547" Score="1" ViewCount="105" Body="&lt;p&gt;My students just learned about Émilie du Châtelet's Salon with Voltaire in her husband's residence in Paris and in their country estates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to give them a homework assignment for extra credit of hosting one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suggestions?&lt;/p&gt;&#xA;" OwnerUserId="7005" LastEditorUserId="7005" LastEditDate="2018-04-06T18:19:13.760" LastActivityDate="2018-04-06T21:10:04.203" Title="Suggestions for hosting a Scientific Salon of Paris in Émilie du Châtelet's time" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7233" PostTypeId="2" ParentId="7232" CreationDate="2018-04-06T21:10:04.203" Score="3" Body="&lt;p&gt;One popular type of salon amusement was spectacular scientific experiments such as &lt;a href=&quot;https://www.amazon.com/review/RBE2JIEMPLXOG&quot; rel=&quot;nofollow noreferrer&quot;&gt;sending electric shocks through a line of people holding hands&lt;/a&gt;, &lt;a href=&quot;https://www.amazon.com/gp/customer-reviews/RNDC23BY7355L/&quot; rel=&quot;nofollow noreferrer&quot;&gt;tricking spiders into eating anything, making scorpions commit suicide&lt;/a&gt;, &lt;a href=&quot;https://www.amazon.com/review/R1H2QVGHRAEVWZ&quot; rel=&quot;nofollow noreferrer&quot;&gt;cross-species blood transfusions&lt;/a&gt;, and &lt;a href=&quot;https://www.amazon.com/review/R4PYQTQ6IMFZX&quot; rel=&quot;nofollow noreferrer&quot;&gt;giving opium to dogs&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some mathematical topics of documented salon origins are &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1600-0498.1988.tb00714.x&quot; rel=&quot;nofollow noreferrer&quot;&gt;Perrault's watch&lt;/a&gt; (the tractrix) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Buffon%27s_needle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Buffon's needle&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Salon conversation also included more philosophical matters such as &lt;a href=&quot;https://www.amazon.com/review/R1NSA74MQZ8T11&quot; rel=&quot;nofollow noreferrer&quot;&gt;whether nature is teleological&lt;/a&gt; and &lt;a href=&quot;https://www.amazon.com/gp/customer-reviews/R1013SPL074V6P/&quot; rel=&quot;nofollow noreferrer&quot;&gt;what a study of blind people tells us about the role of sight in our understanding of the world&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2018-04-06T21:10:04.203" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7234" PostTypeId="1" AcceptedAnswerId="7241" CreationDate="2018-04-07T17:02:03.950" Score="18" ViewCount="9375" Body="&lt;p&gt;So I found this on the internet the other day-&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ikyrM.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ikyrM.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is this fake?&#xA;  Are there any ways to prove that it's fake?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does there, if any, exist any real copy of such a rejection letter?(Was Einstein ever really rejected?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought that why should people use English and not German here. That's suspicious of the originality of this letter!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit:Interested people may also see this-&lt;a href=&quot;https://history.stackexchange.com/questions/30501/did-albert-einstein-really-receive-this-rejection-letter-from-the-university-of/30505&quot;&gt;https://history.stackexchange.com/questions/30501/did-albert-einstein-really-receive-this-rejection-letter-from-the-university-of/30505&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="2955" LastEditDate="2018-04-22T18:24:23.323" LastActivityDate="2019-07-18T22:14:09.863" Title="Is this Einstein rejection letter fake?" Tags="|physics|biographical-details|depiction-of-scientists|" AnswerCount="3" CommentCount="12" ContentLicense="CC BY-SA 3.0" />
  <row Id="7235" PostTypeId="2" ParentId="7234" CreationDate="2018-04-07T17:33:17.033" Score="19" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.snopes.com/fact-check/einstein-rejection-letter/&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;, it's a modern fabrication:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although Einstein’s initial application for a doctorate at the University of Bern (he had previously been awarded a PhD by the University of Zürich in 1905) was indeed rejected as insufficient in 1907, and it was not until the following year that he completed a new dissertation that resulted in his being awarded a doctorate by the University of Bern and given a position as a lecturer at that school, this image depicts a modern creation and not an actual letter sent to Einstein in 1907.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-04-07T17:33:17.033" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7236" PostTypeId="1" CreationDate="2018-04-07T20:55:45.263" Score="2" ViewCount="66" Body="&lt;p&gt;Wikipedia states that &lt;a href=&quot;https://en.wikipedia.org/wiki/R%C3%B8mer_scale&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rømer temperature scale&lt;/a&gt; initially set 0 °Rø for freezing brine and 60 °Rø for boiling water. (He then slightly changed the scale, but the difference is irrelevant for my purposes).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also &lt;a href=&quot;https://en.wikipedia.org/wiki/Fahrenheit#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;states that Fahrenheit&lt;/a&gt; started with this scale &lt;em&gt;where body temperature is 22.5 °Rø&lt;/em&gt;, enlarged it of a factor 4 (so that freezing water is 30 degrees and body temperature is 90 degrees) and then tweaked it so that the two values became 32 °F and 96 °F respectively; eventually some minor modification was made, and now body temperature is about 98 °F.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However this does not work: with such operations, water would be boiling over 240 °F (say, 250 °F). 212 °F would be compatible with 50 °Rø for boiling water, but I don't think that in 1701 Rømer thought to use such a value (60 °Rø made much sense, because it was easily divided in parts). Indeed, since a Rømer degree is more or less equal to 2 Celsius degrees, the value of 22.5 °Rø for the human body temperature is too low: 27.5 °Rø would make more sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know what really happened?&lt;/p&gt;&#xA;" OwnerUserId="81" LastActivityDate="2018-04-07T20:55:45.263" Title="discrepancy between Rømer and Fahrenheit scale" Tags="|measurement|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7237" PostTypeId="2" ParentId="7234" CreationDate="2018-04-08T15:12:57.067" Score="12" Body="&lt;p&gt;Its far more obvious than all of your assumptions, the supposed author of the letter was dead for more than 25 years by 1907. Case closed&lt;/p&gt;&#xA;" OwnerUserId="7272" LastActivityDate="2018-04-08T15:12:57.067" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7238" PostTypeId="1" AcceptedAnswerId="7239" CreationDate="2018-04-09T04:17:49.653" Score="2" ViewCount="644" Body="&lt;p&gt;Does any know the earliest known usage of the Greek letter gamma for showing a reducible representation of a group? This symbolism is commonly used in character tables in chemical applications of group theory, for example as shown here. Thank you in advance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oencB.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oencB.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-04-12T22:28:09.910" Title="Earliest known usage of letter gamma &quot;Γ&quot; for reducible representation in group theory" Tags="|mathematics|terminology|quantum-mechanics|chemistry|group-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7239" PostTypeId="2" ParentId="7238" CreationDate="2018-04-09T05:25:37.167" Score="2" Body="&lt;p&gt;Early examples are Burnside (&lt;a href=&quot;//zbmath.org/?q=an:41.0173.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=qAj_sgkzjvAC&amp;amp;pg=PA1266&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 324-325&lt;/a&gt;; &lt;a href=&quot;//zbmath.org/?q=an:42.0151.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=rGMGAQAAIAAJ&amp;amp;pg=PA271&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 271&lt;/a&gt;) where $\color{red}{\textrm{ir}}$reducible representations are called $\Gamma$, $\Gamma_1$, $\Gamma_2$, etc. (Earlier in (&lt;a href=&quot;//zbmath.org/?q=an:32.0140.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901&lt;/a&gt;) he had called them $G_1$, $G_2$, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speiser (&lt;a href=&quot;//zbmath.org/?q=an:49.0081.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1923&lt;/a&gt;, p. 104; &lt;a href=&quot;//zbmath.org/?q=an:53.0104.12&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=Ak59BwAAQBAJ&amp;amp;pg=PA151&amp;amp;dq=%22Ist%20nunmehr%20eine%20Darstellung%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 151&lt;/a&gt;) uses the same convention, but allows $\Gamma$ to be reducible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that for them a &lt;em&gt;representation&lt;/em&gt; is not a homomorphism to $\mathrm{GL}(n)$ (a map) but rather its image (a group). So your question morphs into: who first wrote $\Gamma$ for a group of linear substitutions? Now &lt;em&gt;that&lt;/em&gt; goes back to at least Jordan (&lt;a href=&quot;//zbmath.org/?q=an:03.0042.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1870&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/traitdessubsti00jorduoft#page/221&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 221 sq&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-04-10T23:06:58.573" LastActivityDate="2018-04-10T23:06:58.573" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7240" PostTypeId="1" CreationDate="2018-04-09T12:30:18.073" Score="1" ViewCount="137" Body="&lt;p&gt;I am trying to collect all available letters written or received by Cantor or written between his colleagues about Cantor. I have searched already the literature given below. But I would like to collect further ressources. Thank you.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E. Zermelo: &quot;Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts&quot;, Springer, Berlin (1932).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;E. Noether, J. Cavaillès: &quot;Briefwechsel Cantor-Dedekind&quot;, Hermann, Paris (1937).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jean van Heijenoort: &quot;From Frege to Gödel - A Source Book in Mathematical Logic, 1879-1931&quot;, Harvard University Press, Cambridge (1967).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I. Grattan-Guinness: &quot;The correspondence between Georg Cantor and Philip Jourdain&quot;, Jahresbericht der Deutschen Mathematiker-Vereinigung 73 (1971) 111-130.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I. Grattan-Guinness: &quot;The rediscovery of the Cantor-Dedekind correspondence&quot;, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 (1974) 104-139.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;H. Meschkowski: &quot;Georg Cantor: Leben, Werk und Wirkung&quot;, 2nd ed., Bibliographisches Institut, Mannheim (1983).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;W. Scharlau: &quot;Rudolf Lipschitz: Briefwechsel mit Cantor, Dedekind, Helmholtz, Kronecker, Weierstrass und anderen&quot;, Vieweg, Braunschweig (1986).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;W. Purkert, H.J. Ilgauds: &quot;Georg Cantor 1845-1918&quot;, Birkhäuser, Basel (1987).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J.W. Dauben: &quot;Georg Cantor: His mathematics and philosophy of the infinite&quot;, Princeton University Press, Princeton (1990).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;H. Meschkowski, W. Nilson: &quot; Georg Cantor – Briefe&quot;, Springer, Berlin (1991).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;R. Ziegler: &quot;Georg Cantor und Rudolf Steiner&quot;, Beiträge zur Rudolf Steiner Gesamtausgabe, Dornach, Nr. 114/115 (1995).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. Tapp: &quot;Kardinalität und Kardinäle&quot;, Franz Steiner Verlag, Stuttgart (2005).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. Ferreirós: &quot;Paradise Recovered? Some Thoughts on Mengenlehre and Modernism (2008).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;B. Russell: &quot;Autobiography&quot;, Routledge, London (2009).&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="4251" LastEditDate="2020-07-04T08:46:00.967" LastActivityDate="2020-07-04T08:46:00.967" Title="Looking for Cantor's correspondence" Tags="|mathematicians|reference-request|cantor|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7241" PostTypeId="2" ParentId="7234" CreationDate="2018-04-09T14:18:19.587" Score="13" Body="&lt;p&gt;&lt;a href=&quot;http://www.uniaktuell.unibe.ch/2016/die_einstein_faelschung/index_eng.html&quot; rel=&quot;noreferrer&quot;&gt;http://www.uniaktuell.unibe.ch/2016/die_einstein_faelschung/index_eng.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is the official link from The University of Bern itself which declares it as a fraud.&lt;/p&gt;&#xA;" OwnerUserId="2955" LastEditorUserId="2955" LastEditDate="2018-05-31T16:12:56.713" LastActivityDate="2018-05-31T16:12:56.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7243" PostTypeId="1" CreationDate="2018-04-10T18:21:54.663" Score="1" ViewCount="131" Body="&lt;p&gt;The &lt;em&gt;OED&lt;/em&gt; defines &quot;&lt;a href=&quot;http://www.oed.com/view/Entry/90707&quot; rel=&quot;nofollow noreferrer&quot;&gt;iatromathematics&lt;/a&gt;&quot; as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Practising medicine in conjunction with astrology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pre-17&lt;sup&gt;th&lt;/sup&gt; century, it seems most scientists (physicians included) believed in the influence of the stars on the terrestrial world, so it would seem ancient medicine took the influence of the stars into account. Is this true?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;OED&lt;/em&gt; also says iatromathematics was&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a school of physicians which arose in Italy in the 17th century, whose system of physiology and medicine was founded on the principles of mathematics and mechanics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Who founded this school?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;Explaining how astrology can make what today we call statistical associations, St. Thomas Aquinas writes (&lt;a href=&quot;https://isidore.co/aquinas/summa/SS/SS095.html#SSQ95A5THEP1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Summa Theologica&lt;/em&gt; II-II q. 95 a. 5&lt;/a&gt; ad 2):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;astrologers not unfrequently forecast the truth by observing the stars…because a great number of men follow their bodily passions, so that their actions are for the most part (&lt;em&gt;in pluribus&lt;/em&gt;) disposed in accordance with the inclination of the heavenly bodies: while there are few, namely, the wise (&lt;em&gt;sapientes&lt;/em&gt;) alone, who moderate these inclinations by their reason. The result is that astrologers in many cases foretell the truth, especially in public occurrences which depend on the multitude (&lt;em&gt;ex multitudine&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, it would seem that today's evidence-based medicine (EBM), which views &quot;disease as statistical associations at a population level&quot; (&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/6IYMPRG7/Chin%E2%80%90Yee%20-%202014%20-%20Underdetermination%20in%20evidence%E2%80%90based%20medicine.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chin-Yee 2014&lt;/a&gt; p. 921), is the modern version of iatromathematics or &quot;astrology&quot; (&quot;statistical associations at a population level&quot;) being practiced in conjunction with medicine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is EBM and its viewing &quot;disease as statistical associations at a population level&quot; the end-result of the historical development of iatromathematics (astrology applied to medicine)?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-04-06T20:46:25.350" LastActivityDate="2020-04-06T20:46:25.350" Title="Historical development of role of astrology in medicine?" Tags="|astronomy|medicine|statistics|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7245" PostTypeId="1" AcceptedAnswerId="7246" CreationDate="2018-04-13T01:53:22.497" Score="5" ViewCount="323" Body="&lt;p&gt;The variational principle is named after Hamilton, instead of Lagrange. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it seems that he did not derive his equation by the variational method. &lt;/p&gt;&#xA;" OwnerUserId="2641" LastActivityDate="2018-04-13T03:10:44.163" Title="How did Lagrange get his equation (the so-called Euler-Lagrange equation)" Tags="|physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7246" PostTypeId="2" ParentId="7245" CreationDate="2018-04-13T03:10:44.163" Score="4" Body="&lt;p&gt;Lagrange derived it in his &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4321&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Méchanique Analytique&lt;/em&gt;&lt;/a&gt;. Dugas summarizes it in his &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4429&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;History of Mechanics&lt;/em&gt;&lt;/a&gt; pp. 342-344:&#xA;&lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4429&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NvUIb.png&quot; alt=&quot;Dugas p. 342&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4429&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jLlOI.png&quot; alt=&quot;Dugas p. 343&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4429&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wdWHJ.png&quot; alt=&quot;Dugas p. 344&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-04-13T03:10:44.163" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7247" PostTypeId="1" AcceptedAnswerId="7252" CreationDate="2018-04-13T06:58:22.397" Score="28" ViewCount="6230" Body="&lt;p&gt;The joke is found on &lt;a href=&quot;https://www.reddit.com/r/math/comments/8bisfd/is_it_just_me_or_does_the_mathematicians_desire/dx72r0k/?st=jfxlds3v&amp;amp;sh=b13469f6&quot; rel=&quot;noreferrer&quot;&gt;this comment chain&lt;/a&gt; on Reddit. One user told the joke:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The version I heard is that Pauli was lecturing, and he said &quot;this is obvious&quot;. A student raises his hand and says &quot;sorry professor, I don't think that is obvious&quot;. Pauli stares at the board, back at the students. He thinks for a bit. He starts pacing in front of the class, thinking. He looks back at the board. Eventually he leaves the room, comes back 20 minutes later and says &quot;I've thought about it and yes, it is obvious&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The main characters are said to be Pauli, Wiener and Lax, Hardy, von Neumann and Einstein. So who were the real characters?&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="65" LastEditDate="2018-12-14T13:06:42.597" LastActivityDate="2022-01-11T21:33:17.070" Title="In a popular anecdote, who took 20 minutes to decide that a thing was obvious?" Tags="|biographical-details|debunking|anecdote|" AnswerCount="5" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7249" PostTypeId="2" ParentId="7247" CreationDate="2018-04-13T12:35:56.160" Score="6" Body="&lt;p&gt;I have reason to believe that's an apocryphal story, as it's been told to me by different professors at different universities with different (often local &quot;legends&quot; ) as the protagonists. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-04-13T12:35:56.160" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7250" PostTypeId="1" AcceptedAnswerId="7251" CreationDate="2018-04-13T20:51:38.087" Score="8" ViewCount="1542" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Wittgenstein&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; about Wittgenstein says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In Norway it was clear that Moore was expected to act as Wittgenstein's secretary, taking down his notes, with Wittgenstein falling into a rage when Moore got something wrong. When he returned to Cambridge, &lt;strong&gt;Moore asked the university to consider accepting Logik as sufficient for a bachelor's degree, but they refused&lt;/strong&gt;, saying it wasn't formatted properly: no footnotes, no preface. Wittgenstein was furious, writing to Moore in May 1914: &quot;If I am not worth your making an exception for me even in some STUPID details then I may as well go to Hell directly; and if I am worth it and you don't do it then—by God—you might go there.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Furthermore it says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Despite this fame, he could not initially work at Cambridge as he did not have a degree, so he applied as an advanced undergraduate. &lt;strong&gt;Russell noted that his previous residency was sufficient to fulfill eligibility requirements for a PhD, and urged him to offer the Tractatus as his thesis.&lt;/strong&gt; It was examined in 1929 by Russell and Moore; at the end of the thesis defence, Wittgenstein clapped the two examiners on the shoulder and said, &quot;Don't worry, I know you'll never understand it.&quot; Moore wrote in the examiner's report: &quot;I myself consider that this is a work of genius; but, even if I am completely mistaken and it is nothing of the sort, it is well above the standard required for the Ph.D. degree.&quot; Wittgenstein was appointed as a lecturer and was made a fellow of Trinity College.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How was it possible for Wittgenstein to get a PhD in philosophy without having a Bachelor's degree in philosophy? And how was it possible that his work satisfied requirements for a PhD but not for a Bachelor's degree?&lt;/p&gt;&#xA;" OwnerDisplayName="user6103" LastEditorUserId="4251" LastEditorDisplayName="user6103" LastEditDate="2022-09-23T05:52:31.093" LastActivityDate="2022-09-23T08:58:52.693" Title="How did Wittgenstein fulfill eligibility requirements for a PhD in philosophy without having a Bachelor's degree in philosophy?" Tags="|biographical-details|philosophy-of-science|academia|logicians|" AnswerCount="2" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7251" PostTypeId="2" ParentId="7250" CreationDate="2018-04-14T21:16:47.483" Score="4" Body="&lt;p&gt;In the early and mid-20th century in Anglo-American higher education, exceptions of this sort occurred for individuals who showed that they could hold their own in elite academic circles. An example in addition to Wittgenstein and Ramsey is &lt;a href=&quot;https://en.wikipedia.org/wiki/Garrett_Birkhoff&quot; rel=&quot;nofollow noreferrer&quot;&gt;Garrett Birkhoff&lt;/a&gt;. After receiving a BA at Harvard in 1932 and then spending some time at Cambridge, he returned to Harvard and became an instructor in 1936, eventually retiring from there in 1981, after a long and storied career, as George Putnam Professor of Pure and Applied Mathematics. The BA was his only formal degree.&lt;/p&gt;&#xA;" OwnerUserId="7298" LastEditorUserId="1930" LastEditDate="2018-04-24T12:56:27.527" LastActivityDate="2018-04-24T12:56:27.527" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7252" PostTypeId="2" ParentId="7247" CreationDate="2018-04-15T00:16:31.950" Score="17" Body="&lt;p&gt;The closest to an attribution that I could find is &lt;a href=&quot;https://math.stackexchange.com/a/28912/152568&quot;&gt;by Matt Calhoun on Math SE&lt;/a&gt;, who claims &quot;&lt;em&gt;it was personally told to me&lt;/em&gt;&quot; by &quot;&lt;em&gt;my mechanics professor (G. Horton) [who] took lectures from Pauli...&lt;/em&gt;&quot; G. Horton is presumably George Horton who held a physics post-doc in Zurich in 1949-51. Pauli stayed in Zurich from 1946 until his death in 1958, so they did intersect. Horton moved to Rutgers in 1960 where he taught until 2009. He was popular enough for students to dedicate a &lt;a href=&quot;http://george-horton.memory-of.com/&quot; rel=&quot;noreferrer&quot;&gt;memorial website&lt;/a&gt; to him. The anecdote is most often told about Pauli, but the story does show many signs of being made up, no first hand source, lack of detail, revolving names, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also reminiscent of another, older story, with a somewhat better credentials but the opposite ending. Reid reports it in her &lt;a href=&quot;https://books.google.com/books?id=-pTwBwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;book on Hilbert (p.93)&lt;/a&gt; (she also gives no direct attribution but implies that it was reported by Göttingen students whom she interviewed for the book):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;&quot;This theorem has not yet been proved, but that is because only mathematicians of the third rank have occupied themselves with it,&quot; Minkowski announced to the class in a rare burst of arrogance. &quot;I believe I can prove it.&quot; He began to work out his demonstration on the spot. By the end of the hour he had not finished. The project was carried over to the next meeting of the class. Several weeks passed this way. Finally, one rainy morning, Minkowski entered the class followed by a crash of thunder. At the rostrum he turned towards the class with a serious expression on his gentle round face. &quot;Heaven is angered by my arrogance,&quot; he announced. &quot;My proof of the Four Color Theorem is also defective. He then took up the lecture on topology at the point where he had dropped it several weeks before.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even assuming that this does have some basis in reality the &quot;several weeks&quot; and the melodramatic effects are most certainly embellishments.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-04-15T00:48:27.040" LastActivityDate="2018-04-15T00:48:27.040" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7253" PostTypeId="1" CreationDate="2018-04-15T15:27:04.927" Score="2" ViewCount="292" Body="&lt;p&gt;We know that the rules of relative number where laid down in India (a product of 2 debts is a fortune) and in Europe they were spread by Bombelli, who , again, only mentions the product of two minuses. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When/by whom was it first specified that the square of a negative is a positive?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;the question ha been misinterpreted and id is &lt;em&gt;not&lt;/em&gt; a duplicate: someone is taking for granted that the rule of minus times minus authomatically implies the realization that negative squares are impossible (besides the subtle fact that another conclusion was theoretically possible).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So , Bombelli mentioned the rules of multiplication, but did he explicitly state that the square of minus one (or any other negative) is plus one? When was the general public of scientists fully aware that the roots of negatives are missing on the number line on the left of zero?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surely not in the Middle ages before Bombelli even though (as the answer here and there imply) negatives where known from the 6th or 3rd century or even earlier. Is this clear now?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="7300" LastEditDate="2018-04-19T07:06:39.540" LastActivityDate="2018-04-22T01:43:07.823" Title="When was the square of negative numbers specified?" Tags="|number-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7255" PostTypeId="1" AcceptedAnswerId="7256" CreationDate="2018-04-16T09:42:57.793" Score="1" ViewCount="75" Body="&lt;p&gt;On a tour of the Royal Observatory at Greenwich I saw the following display on the history of mechanical clocks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JCSit.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JCSit.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PgbZH.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PgbZH.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/008B1.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/008B1.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The suggest that mechanical clocks and astronomy weren’t really combined in England until the 16th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I thought mechanical clocks &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Clock#Early_mechanical&quot; rel=&quot;nofollow noreferrer&quot;&gt;came into use&lt;/a&gt; in the 13th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;When did mechanical tower clocks start being used?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2018-04-19T23:09:02.230" LastActivityDate="2018-04-19T23:09:02.230" Title="When did mechanical tower clocks start being used?" Tags="|physics|astronomy|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7256" PostTypeId="2" ParentId="7255" CreationDate="2018-04-16T13:30:18.783" Score="2" Body="&lt;p&gt;In 14th century, but it is not known precisely, where and when was the first clock made. Salisbury cathedral has on display something they call the &quot;oldest surviving clock&quot;, made in 1386. But similar claims are made about some French and Italian cocks for which they give earlier dates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See   &lt;a href=&quot;https://en.wikipedia.org/wiki/Salisbury_cathedral_clock&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Salisbury_cathedral_clock&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is about mechanical (weight-driven) clocks. Water clocks are much older,&#xA;exist since antiquity, and there were also some large and complicated water clocks in China.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-16T18:38:51.110" LastActivityDate="2018-04-16T18:38:51.110" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7257" PostTypeId="2" ParentId="7253" CreationDate="2018-04-16T13:40:30.040" Score="1" Body="&lt;p&gt;The oldest surviving book on algebra is &quot;Arithmetic&quot; by Diophantus of Alexandria.&#xA;He defines negative numbers and arithmetic operations on them, including multiplication. So he knew that the product of negative numbers is positive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earlier sources on algebra did not survive, but there is little doubt that they existed: it is hard to imagine that such advanced mathematics appeared suddenly out of blue. Unfortunately, it is not known precisely when Diophantus lived. The accepted date is around 250 AD, but certainly he could not write after Theon (mid 300s) because Theon mentions his book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, in the comment you ask a different question: when and why did people start to discuss square ROOTS of negative numbers. This happened in Italy, in 16th century when they discovered a formula for solving the cubic equation.&#xA;When you apply this formula to a cubic equation which has 3 real roots,&#xA;some intermediate term which you obtain is a square root of a negative number.&#xA;So the question was how to interpret this root and to make the formula work.&#xA;For details, see   &lt;a href=&quot;http://www.math.purdue.edu/~eremenko/dvi/cardano.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.purdue.edu/~eremenko/dvi/cardano.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. I was asked for the exact citation of Diophantus. The copy that I have is in Russian, so I translate from the Russian to English as literally as I can:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arithmetic, book I, section IX (p. 40 of my Russian edition): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Deficiency multiplied on deficiency gives an asset; deficiency multiplied on an asset gives a deficiency; we use the following sign for a deficiency&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;sorry, I have no font to reproduce his sign for the minus. The words which I translated as &#xA;&quot;deficiency&quot; and &quot;asset&quot; can be also translated as &quot;lack&quot; and &quot;availability&quot;. &#xA;Section X:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After this explanation of multiplication the division must be clear; it is recommended to the beginner to exercise in addition, subtraction and multiplication of these kinds...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-19T18:17:29.543" LastActivityDate="2018-04-19T18:17:29.543" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7258" PostTypeId="1" AcceptedAnswerId="7274" CreationDate="2018-04-16T13:44:50.747" Score="0" ViewCount="520" Body="&lt;p&gt;On a tour of the Royal Observatory at Greenwich I saw the following display on the history of mechanical clocks:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TEPIP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TEPIP.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7J6JT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7J6JT.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LPzI2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LPzI2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They suggest that mechanical clocks and astronomy weren’t really combined in England until the 16th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I thought mechanical clocks &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Clock#Early_mechanical&quot; rel=&quot;nofollow noreferrer&quot;&gt;came into use&lt;/a&gt; in the 13th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;When did spring-driven clocks start being used?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastActivityDate="2018-04-18T19:47:05.410" Title="When did spring-driven clocks start being used?" Tags="|physics|astronomy|classical-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7259" PostTypeId="1" CreationDate="2018-04-16T14:55:28.890" Score="10" ViewCount="4160" Body="&lt;p&gt;Reading Wikipedia's articles on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Flat_Earth&quot; rel=&quot;noreferrer&quot;&gt;flat earth&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Spherical_Earth&quot; rel=&quot;noreferrer&quot;&gt;spherical earth&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_geodesy&quot; rel=&quot;noreferrer&quot;&gt;history of geodesy&lt;/a&gt; makes it clear that virtually every society recognizing the spheroidal shape of the earth today owes the observation to the ancient Greeks. But one possible exception is India. It's not clear to me, after reading the articles and the sources I could reach online, whether India came to this conclusion via cultural exchange with the Hellenistic world like others did or if their own astronomers reached it independently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect that the reason it wasn't clear to me is that historians aren't certain themselves. If that's the case, a good answer will lay out the evidence pro and con and explain the general views of historians.&lt;/p&gt;&#xA;" OwnerUserId="1814" LastActivityDate="2022-12-12T01:09:17.330" Title="Did Indian astronomers realize the sphericity of the earth independently of the Greeks?" Tags="|astronomy|ancient-greece|ancient-india|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7261" PostTypeId="2" ParentId="7258" CreationDate="2018-04-16T18:56:45.353" Score="1" Body="&lt;p&gt;Spring-driven clocks were invented in 15 century (as pocket clocks). Clocks had no use in astronomy in the beginning (neither weight-driven nor spring-driven) because they were not accurate enough. Accurate clocks require a pendulum or a balance, and these are 17th century inventions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nuremberg_eggs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nuremberg_eggs&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-17T03:25:39.737" LastActivityDate="2018-04-17T03:25:39.737" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7263" PostTypeId="1" AcceptedAnswerId="7270" CreationDate="2018-04-17T13:31:11.527" Score="1" ViewCount="116" Body="&lt;p&gt;The solution to the problem of geodesic lines on a &lt;strong&gt;biaxial&lt;/strong&gt; ellipsoid (when two of the axes are equal) is not very hard and can be solved by mathematical tools that existed prior to Gauss - i.e via Clairaut's relation for geodesics on bodies of revolution. However, the solution for &lt;strong&gt;triaxial&lt;/strong&gt; ellipsoids is very difficult, and was made by Jacobi in a remarkable paper fron 1839. My question is indeed intended to learn of Gauss's work in this field - i found, especially in volume 9 of his collected work (about geodesy) many unpublished titles bearing the subject &quot;geodesic lines on ellipsoid&quot; (for example, ERDELLIPSOID AND GEODETIC LINE, on p.65 - 105 in volume 9). I also read that in his treatise on higher geodesy (published at years 1843,1846), he dealt much with calculation with &quot;lines&quot; on ellipsoid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, were these calculations just about ellipsoid of revolution (biaxial ellipsoid)? or there was also work about lines in a triaxial ellipsoids (in his published or unpublished work)?  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2018-04-17T23:13:15.487" Title="Summary of Gauss's work on geodesic lines on ellipsoid" Tags="|biographical-details|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7264" PostTypeId="1" AcceptedAnswerId="7267" CreationDate="2018-04-17T13:41:42.587" Score="4" ViewCount="719" Body="&lt;p&gt;In the Royal Observatory in London they have a mechanical clock called the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Martin_Burgess#Notable_clocks&quot; rel=&quot;nofollow noreferrer&quot;&gt;Burgess B&lt;/a&gt;, based on a 200 year old design by &lt;a href=&quot;https://en.m.wikipedia.org/wiki/John_Harrison&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harrison&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EamYd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EamYd.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9Za2k.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Za2k.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CBowY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CBowY.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;Why couldn’t the Burgess B Clock built in 1974 be built in 1775 by Harrison?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastActivityDate="2020-09-26T18:06:53.010" Title="Why couldn’t the Burgess B Clock built in 1974 be built in 1775 by Harrison?" Tags="|physics|classical-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7265" PostTypeId="2" ParentId="7259" CreationDate="2018-04-17T17:44:19.703" Score="9" Body="&lt;p&gt;&lt;em&gt;(I started writing this answer from memory, but on a second look at some sources it turns out to answer the question for the statement of a &lt;strong&gt;rotating&lt;/strong&gt; earth rather than that of a spherical earth... but I'll leave it here; hope the other question is still interesting to you.)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most notable statement of a &lt;s&gt;spherical&lt;/s&gt; rotating earth, in Indian astronomy, is by &lt;a href=&quot;https://en.wikipedia.org/wiki/Aryabhata&quot; rel=&quot;noreferrer&quot;&gt;Āryabhaṭa&lt;/a&gt;. In his &lt;a href=&quot;https://en.wikipedia.org/wiki/Aryabhatiya&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Āryabhaṭīya&lt;/em&gt;&lt;/a&gt; (499 CE), he explains the familiar observation of the night sky, of the stars collectively appearing to go around the earth, as being because of the earth's rotation. (He gives the analogy of being in a boat, where things on the shore appear to be moving.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Āryabhaṭa was always universally revered in the Indian astronomical tradition, and had his followers (especially in Kerala) until pretty much modern times (in the 18th century, the French astronomer Guillaume Le Gentil was in India during a lunar eclipse, and found that the prediction of its length by local astronomers based on Āryabhaṭa's method was more accurate than the European one), but this particular statement about the earth's motion was rejected by many later Indian astronomers (even his own commentators kind of explained it away), because of various physical objections. For example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And if the earth turns, then how do birds reach their nests again after leaving them? Arrows released up toward the sky should fall down in the west if there is an eastward rotation of the earth, and clouds should move toward the west. Now, if you argue that these effects do not happen because the motion is very slow, then how can the earth complete a full rotation in one day?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Indian astronomy by this time there were many Greek sources known (starting with the influence of the Indo-Greek kingdoms after the breakup of Alexander's empire); consider works (either available or mentioned) like &lt;em&gt;Pauliśa-siddhānta&lt;/em&gt;, &lt;em&gt;Romaka-siddhānta&lt;/em&gt;, &lt;em&gt;Yavana-jātaka&lt;/em&gt;, etc. Astronomers  Varāhamihira and Brahmagupta, both later than Āryabhaṭa, mention these Greek sources and also praise them: “Though the Yavanas [Greeks] are barbarians (&lt;em&gt;mleccha&lt;/em&gt;), this science (astronomy) is good among them, and they are revered (&lt;em&gt;pūjyante&lt;/em&gt;) like sages — a scholar (div) in these sciences even more so.” However when Brahmagupta criticizes Āryabhaṭa's spinning-earth theory, he does not attribute it to a Greek source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I suppose that eventually in a few centuries the rotating earth was widely accepted, or maybe it was not, but in any case it doesn't matter much for the calculations.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to summarize, we see that neither Āryabhaṭa nor his commentators nor other astronomers attributed the theory of a rotating earth to Greek astronomy even though they were aware of it, nor was the theory widespread the way (some) ideas from Greek astronomy were, so it seems quite probable that the idea originated in India independently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Sources&lt;/strong&gt;: see Kim Plofker's &lt;em&gt;Mathematics in India&lt;/em&gt;, the course “Mathematics in India” by M.D.Srinivas, M.S.Sriram and K.Ramasubramanian (I can't find the link to the lecture notes and associated writing, but the &lt;a href=&quot;https://www.youtube.com/playlist?list=PLbMVogVj5nJThf31TNSQzuN7zqxe7HdRN&quot; rel=&quot;noreferrer&quot;&gt;videos are here&lt;/a&gt;), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(As I said, I realize this is not exactly the question you were asking. The idea of a spherical earth itself (even if possibly fixed instead of rotating) was strongly defended by both Āryabhaṭa's commentators and by other astronomers, and is older than Āryabhaṭa... it was universally preferred in Indian astronomy over any kind of flat earth model.)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="432" LastActivityDate="2018-04-17T17:44:19.703" CommentCount="21" ContentLicense="CC BY-SA 3.0" />
  <row Id="7266" PostTypeId="1" CreationDate="2018-04-17T19:52:45.520" Score="0" ViewCount="165" Body="&lt;p&gt;Did Zermelo in 1914 miss the fact, known in 1905 already, that there are only countably many finite strings, for instance polynomials with defined parameters? He wrote: Every real or complex number $\alpha$ satisfies at least one algebraic equation of the form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$g(x, \eta) = g(x, \eta_1, \eta_2, \eta_3, ..., \eta_t) = 0$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where $g$ is a polynomial with integer coefficients, and $\eta_1, \eta_2, \eta_3, ..., \eta_t$ are basis numbers (a Hamel basis), and  the largest power of $x$ is at least 1.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[E. Zermelo: &quot;Über ganze transzendente Zahlen&quot;, Math. Annalen 75 (1914) pp. 434-442]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1920 Skolem introduced his &quot;normal form &quot;, showed that every satisfiable well formed formula of first order predicate calculus has a satisfiable Skolem normal form (and vice versa), and improved and generalized the proof of Löwenheim's theorem: Every proposition in normal form either is a contradiction or is already satisfiable in a finite or denumerably infinite  domain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Zermelo regarded Skolem's position as a real danger for mathematics and, therefore, saw 'a particular duty' to fight against it. [...] His remedy consisted of infinitary languages. [...] Skolem had considered such a possibility, too, but had discarded it because of a vicious circle.&quot; [Heinz-Dieter Ebbinghaus: &quot;Ernst Zermelo, an approach to his life and work&quot;, Springer (2007) p. 200ff]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additional question relevant for the main question: What is the first paper where Zermelo explicitly considers uncountable alphabets?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2018-04-18T09:14:25.577" LastActivityDate="2018-04-18T11:16:07.867" Title="Did Zermelo in 1914 miss the fact that there are only countably many finite strings?" Tags="|set-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7267" PostTypeId="2" ParentId="7264" CreationDate="2018-04-17T20:27:55.537" Score="4" Body="&lt;p&gt;Perhaps it could be, in principle, given desire and unlimited time and money. But it took even Burgess to build a &amp;quot;nearly identical&amp;quot; clock &amp;quot;nearly 11 years&amp;quot; (Harrison died in 1776), and he used modern materials, mostly duraluminium and invar (a nickel-iron alloy), only the mechanism design was Harrison’s. The purpose of building Clock B was specifically to test Harrison's claim that his designs could remain within 1 second over 100 days, and apparently no one believed or perhaps even understood Harrison's claims in 1775. &lt;a href=&quot;https://www.independent.co.uk/news/science/john-harrisons-longitude-clock-sets-new-record-300-years-on-10187304.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Independent had Betts&lt;/a&gt;, a senior horologist at the Royal Observatory, comment on the matter:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In his later years, Harrison had left instructions on how to build the clock in an obscure book, which was so hard to read it became known as “the ramblings of superannuated dotage” by later horologists. “It was a claim that Harrison made and a claim nobody believed because the best clocks of the day couldn’t do better than about a second a week, if they were lucky,” Mr Betts said. “So the idea that somebody was going to keep time to an accuracy of a second in a 100 days was preposterous. It was only in the 20th century that people thought that Harrison may have been right.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;“This does not claim to be a replica of Harrison’s clock. It claims to be the use of Harrison’s design and concept. It’s important to realise his design goes against everything the establishment has claimed is the best throughout history,” Mr Betts said. Traditional clockmaking argued that a heavy pendulum bob and a short swing would make for good timekeeping. However, Harrison thought the opposite and designed a clock with a relatively light pendulum bob and wide swing. “That was such a radical difference it was another reason why they didn’t believe it. They thought it was rather heretical,” Mr Betts explained.”&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-04-17T20:27:55.537" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7268" PostTypeId="2" ParentId="7266" CreationDate="2018-04-17T21:23:11.293" Score="3" Body="&lt;p&gt;The assumption on which this question is based is incorrect, so, no, Zermelo did not miss this. The set of basis numbers, as defined by Zermelo, is itself uncountable, and the set of finite strings over an uncountable alphabet is of course uncountable as well.&lt;/p&gt;&#xA;" OwnerUserId="6265" LastEditorUserId="6265" LastEditDate="2018-04-18T05:19:06.507" LastActivityDate="2018-04-18T05:19:06.507" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7269" PostTypeId="2" ParentId="7264" CreationDate="2018-04-17T23:06:34.843" Score="3" Body="&lt;p&gt;Harrison spent most part of his later life trying to perfect the marine chronometer, and to win the Longitude prize (the prize was eventually collected by his widow). This clock is not a marine chronometer, as far as I can tell from the picture, and probably Harrison had no time and resources to build it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The demand for a stationary high precision clock at that time was not as urgent&#xA;as for a practical marine chronometer. Making a single high grade clock or watch&#xA;at the time of Harison consumed years and a lot of money because all parts (and tools!) had to be made by hand.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-04-17T23:06:34.843" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7270" PostTypeId="2" ParentId="7263" CreationDate="2018-04-17T23:13:15.487" Score="2" Body="&lt;p&gt;I do not read German, but I can tell what I know about this. Gauss' worked on a generic ellipsoid (with three different axes) but this work was motivated by geodesy (rather than pure mathematics). So he considered an ellipsoid with three different axes which is close to the sphere (the ratios of the axes are close to $1$), and derived approximate (asymptotic) formulas for small values of parameter responsible for excentricity. His formulas were to be used in geodesy, and their accuracy was higher than that required in geodesy.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-04-17T23:13:15.487" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7271" PostTypeId="1" AcceptedAnswerId="7281" CreationDate="2018-04-18T08:00:34.397" Score="6" ViewCount="390" Body="&lt;p&gt;Every cubic equation (with real coefficients) has a real root. Who was the first person who proved this? Is it already contained in Bombelli's &lt;em&gt;Algebra&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-04-21T23:32:21.557" Title="When it was discovered that cubic equations always have roots?" Tags="|mathematics|algebra|" AnswerCount="2" CommentCount="14" ContentLicense="CC BY-SA 3.0" />
  <row Id="7272" PostTypeId="2" ParentId="7266" CreationDate="2018-04-18T11:16:07.867" Score="2" Body="&lt;p&gt;The above equation has as parameters the so-called &lt;a href=&quot;https://books.google.it/books?id=XB2nd2ovakIC&amp;amp;pg=PA281&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;basis numbers&quot;&lt;/a&gt; $\eta$ belonging to $\Omega$, where $\Omega$ is an &lt;em&gt;arbitrary&lt;/em&gt; well-ordering of [the &lt;em&gt;continuum&lt;/em&gt;] $\mathfrak c$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding Zermelo's &quot;infinitary&quot; logic, see :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Über Stufen der Quantifikation und die Logik des Unendlichen&lt;/em&gt; [&lt;a href=&quot;https://books.google.it/books?id=XB2nd2ovakIC&amp;amp;pg=PA543&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On levels of quantification and the logic of the infinite&lt;/em&gt;&lt;/a&gt;] (1932) :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Proceeding from the assumption that it should be possible to represent all&#xA;  mathematical concepts and theorems by means of a fixed finite &lt;em&gt;system of&#xA;  signs&lt;/em&gt;, we inevitably run into the well-known “&lt;em&gt;Richard antinomy&lt;/em&gt;” already in&#xA;  the case of the arithmetical continuum. [...] a healthy “metamathematics”, a&#xA;  true “logic of the infinite”, will only become possible once we have &lt;em&gt;definitively renounced&lt;/em&gt; the assumption characterized above, which I would like to call the “&lt;em&gt;finitistic prejudice&lt;/em&gt;”. Mathematics, generally speaking, is not really concerned with “combinations of signs”, as some assume, but with &lt;em&gt;conceptually ideal relations&lt;/em&gt; among the elements of a conceptually posited &lt;em&gt;infinite manifold&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Dirk van Dalen &amp;amp; Heinz-Dieter Ebbinghaus, &lt;a href=&quot;https://www.jstor.org/stable/421203?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zermelo and the Skolem paradox&lt;/a&gt;, &lt;em&gt;Bull.Symb.Log&lt;/em&gt;. (2000).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-04-18T11:16:07.867" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7273" PostTypeId="2" ParentId="7271" CreationDate="2018-04-18T12:13:34.137" Score="1" Body="&lt;p&gt;According to an excerpt from The Heritage of Thales&#xA;By W.S. Anglin, J. Lambek, via GoogleBooks, a method was found:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DiBCm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DiBCm.png&quot; alt=&quot;quote&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this doesn't indicate a proof that there had to be a real root.  The Wikipedia page on cubic equations adds that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Cardano noticed that Tartaglia's method sometimes required him to&#xA;  extract the square root of a negative number. He even included a&#xA;  calculation with these complex numbers in Ars Magna, but he did not&#xA;  really understand it. Rafael Bombelli studied this issue in detail[21]&#xA;  and is therefore often considered as the discoverer of complex&#xA;  numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't know when someone demonstrated that complex roots had to come in pairs when all equation coefficients are real; presumably that's what will drive the full answer to the original question.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-04-18T12:13:34.137" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7274" PostTypeId="2" ParentId="7258" CreationDate="2018-04-18T19:47:05.410" Score="1" Body="&lt;p&gt;From the British Museum we can see one of the first spring driven clocks built in 1450 here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Z31Gk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z31Gk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ySQT3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ySQT3.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastActivityDate="2018-04-18T19:47:05.410" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7276" PostTypeId="1" CreationDate="2018-04-20T11:01:02.637" Score="1" ViewCount="474" Body="&lt;p&gt;I'm very curius to learn of the solution to the problem of the gravitational attraction of triaxial ellipsoid, both in his internal and external parts. From what i read, i understood that the solution for the attraction of spheroid (biaxial ellipsoid) in his &lt;strong&gt;internal part&lt;/strong&gt; was given implicitly by Newton in his Principia, and the mathematical formulas involved were made explicit by Maclaurin&#xA;(see Chandrasekhar: Newton's Principia for the common reader).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the solution for the &lt;strong&gt;external part&lt;/strong&gt; of the spheroid is complicated even for a homogenous ellipsoid, as is being said here &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S003206331730257X&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencedirect.com/science/article/pii/S003206331730257X&lt;/a&gt;, and according with this article, it was made for the first time by Gauss and later by Dirichlet. I also read that Gauss gave different expression for the internal part of ellipsoids, involving quadratic transfomation of the coordinates with coefficient values that are elliptic integrals (see &lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-01592829/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://hal.archives-ouvertes.fr/hal-01592829/document&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what was the method of potential theory used by Gauss? how did he tackle this unapproachable problem? in paricular, can anyone give an account of the results and the structure of Gauss's 1813 article?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-02-14T23:16:22.200" LastActivityDate="2020-02-15T13:47:34.450" Title="What were Gauss's results on the attraction of homogenous ellipsoids in his paper from 1813?" Tags="|biographical-details|gauss|potential|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 3.0" />
  <row Id="7277" PostTypeId="1" AcceptedAnswerId="7278" CreationDate="2018-04-20T12:45:32.220" Score="3" ViewCount="941" Body="&lt;p&gt;I would like to understand what are the ideas (or data) on which the famous scientist has based himself to arrive at such a fundamental principle for quantum mechanics.&lt;/p&gt;&#xA;" OwnerUserId="7326" LastEditorUserId="131" LastEditDate="2018-04-20T17:01:52.807" LastActivityDate="2018-04-21T00:10:31.217" Title="How did Pauli come to the Pauli exclusion principle?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 3.0" />
  <row Id="7278" PostTypeId="2" ParentId="7277" CreationDate="2018-04-20T20:20:27.530" Score="3" Body="&lt;p&gt;There is a whole book dedicated exactly to this question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ian Duck, E.C.G. Sudarshan, Pauli and the spin-statistics theorem, World Scientific, 1997.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Very roughly speaking, the story is the following. Bohr's model did not explain the&#xA;periodic table, namely the mysterious numbers 2, 8, 18 and 32...&#xA;To fit these numbers (and also anomalous Zeeman's effect, and spectra of alkali metals), Stoner proposed that there exists a new quantum number which takes two values. (An important contribution was also made by Lande).&#xA;Pauli's guess was that this number there must be a new quantum number for the electron which takes two values, and that two electrons in an atom cannot have all numbers the same. So originally this was just a guess. It explained the observed&#xA;spectral lines, their splitting in the Zeeman effect and the periods of the&#xA;periodic table. Later, with the development of relativistic quantum mechanics and field theory this guess was explained by the so-called spin-statistics theorem. Pauli acknowledged that his original guess was &quot;pure combinatorics&quot; and spent the next 30 years of his career to find deeper physical reasons.&#xA;In 1940 he gave a proof of the theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book that I cited reproduces all relevant papers (in English!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another excellent exposition of the same story is S. Tomonaga, The story of spin, U. Chicago Press, 1997.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-21T00:10:31.217" LastActivityDate="2018-04-21T00:10:31.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7279" PostTypeId="2" ParentId="7253" CreationDate="2018-04-21T01:01:51.807" Score="1" Body="&lt;p&gt;In his 1784 work &lt;em&gt;Algebra&lt;/em&gt;, Colin MacLaurin presents the following argument for why a negative number multiplied by a negative number is (or rather, must be) positive (see Chapter III case IV, &lt;a href=&quot;https://ia800304.us.archive.org/32/items/atreatisealgebr03maclgoog/atreatisealgebr03maclgoog.pdf&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;MacLaurin's Treatise on Algebra&quot;&gt;here&lt;/a&gt;; it's page 35 of the PDF).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-n(a-a)$&lt;/span&gt; must equal 0 (since &lt;span class=&quot;math-container&quot;&gt;$a-a = 0$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Using the distributive property, the first term &lt;span class=&quot;math-container&quot;&gt;$-n \times a$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$-na$&lt;/span&gt;.  The only way for the distributive property to still hold, and the statement to be true, is if &lt;span class=&quot;math-container&quot;&gt;$-n \times -a=na$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If we let &lt;span class=&quot;math-container&quot;&gt;$-n=-a$&lt;/span&gt; from our example above, then &lt;span class=&quot;math-container&quot;&gt;$-a \times -a$&lt;/span&gt; will of course be a positive number, &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From this it is clear that there is no way to square a real number and end up with a negative result, hence Cardano's (and everyone else's) confusion over what to do when confronted with something like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-n}$&lt;/span&gt;: such an operation was undefined, because there wasn't any number at the time that could be squared to get a negative result.&lt;/p&gt;&#xA;&lt;p&gt;In Book I of &lt;em&gt;L'Algebra&lt;/em&gt; (1572), Bombelli specifies that &amp;quot;minus times minus makes plus&amp;quot;, and even offers an example that is farily close to MacLaurin's:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Multiply &lt;span class=&quot;math-container&quot;&gt;$(6-4) \times (5-2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-2 \times -4 = 8$&lt;/span&gt;, and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$-2 \times 6 = -12$&lt;/span&gt;, and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5 \times -4 = -20$&lt;/span&gt;, and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5 \times 6 = 30$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;so &lt;span class=&quot;math-container&quot;&gt;$(6-4) \times (5-2) = 30-20-12+8$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bombelli does not take the extra step to explain that a negative times a negative must be positive for the calculation to work out properly, however from this example and his multiplication rules he would have realized that (a) the square of a negative number is positive, and (b) thus there was no way to square a number and get a negative result, rendering the square roots of negative numbers perplexing at best.  (A full version of &lt;em&gt;L'Algebra&lt;/em&gt; in Italian can be found &lt;a href=&quot;http://mathematica.sns.it/media/volumi/9/L%27algebra.pdf&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Bombelli's L'Algebra&quot;&gt;here&lt;/a&gt;.  The above excerpts are from Book I {Libro Primo}, pages 70 and 71 {127 and 128 of the PDF}. Without knowing you at all, I bet your Italian is better than mine...)&lt;/p&gt;&#xA;&lt;p&gt;I will say that I am not 100% certain that MacLaurin was the first one to actually demonstrate this &amp;quot;minus times minus is plus&amp;quot; rule (versus just stating it).  Bombelli gave an example, but MacLarin's &lt;em&gt;Treatise&lt;/em&gt; is the earliest publication I have found that offers something like a proof.  I whole-heartedly invite the pros here to fact check me.  I do hope that I have addressed the general spirit of your question, though.&lt;/p&gt;&#xA;" OwnerUserId="7074" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-04-22T01:43:07.823" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7280" PostTypeId="1" CreationDate="2018-04-21T01:28:29.590" Score="1" ViewCount="106" Body="&lt;p&gt;The Babylonian method for computing square roots is described (among other places) in &lt;a href=&quot;https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Wikipedia article&lt;/a&gt;.  What evidence is there that they actually used this method?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found several references that imply that they did &lt;em&gt;not&lt;/em&gt; use an iterative process, but used a closely related method that stopped after a single iteration (see, for instance, &lt;a href=&quot;https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Square  Root  Approximations  in  Old  Babylonian  Mathematics: YBC  7289  in  Context&lt;/a&gt;, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#cite_note-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia notes for Square Root of 2&lt;/a&gt;, Note 1).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet other references imply that they indeed used the iterative process.  In his &lt;em&gt;A History of Mathematics&lt;/em&gt;, C. Boyer happily explains the iterative process (which I find unusual because every other sentence in C. Boyer's works usually reads something like, &quot;...but it would be foolish to assume that these ancient mathematicians had any idea what they were doing...&quot;...but I digress). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Going back to the Yale tablet YBC 7289, the Babylonians evidently knew $\sqrt{2}$ to six decimal places. Is it safe to assume they &lt;em&gt;did&lt;/em&gt; have an iterative process, based on this and other circumstantial evidence?&lt;/p&gt;&#xA;" OwnerUserId="7074" LastActivityDate="2018-04-21T01:28:29.590" Title="What evidence is there that the Babylonians used the Babylonain method of estimating square roots?" Tags="|mathematics|ancient|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7281" PostTypeId="2" ParentId="7271" CreationDate="2018-04-21T07:28:04.693" Score="6" Body="&lt;p&gt;There is a problem with how the question is phrased, which is unfortunately common. Modern concepts pull together many different threads which come apart when projected into the past far enough. &quot;Cubic equation&quot; and &quot;real numbers&quot; are two such concepts. When they are used to ask questions about discoveries before 19th, or at least 17th, century there can be no answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When people talk about &quot;cubic equations&quot; in ancient Greece they mean geometric problems that can &lt;em&gt;today&lt;/em&gt; be converted into cubic equations. And their coefficients would be integers, or at least constructible numbers. Duplication and trisection are such problems, already Menaechmus used intersecting conic sections to solve duplication, we find more examples in Archimedes and Apollonius, Pappus generally describes problems solvable by intersecting conic sections as &quot;solid&quot; (as opposed to straightedge and compass &quot;plane&quot; ones). Netz in his book &lt;a href=&quot;https://books.google.com/books/about/The_Transformation_of_Mathematics_in_the.html?id=kTDRJwCCMecC&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Transformation of Mathematics in the Early Mediterranean World: From Problems to Equations&lt;/a&gt; tries to trace the morphing of geometry through Diophantus's arithmetic into Islamic algebra of equations, which culminates with Khayyam. Khayyam did look at cubic equations, with positive integer coefficients only, and painstakingly reduced each of the 14 cases, that appear without negative numbers, to intersecting two conics. He also only recognized positive roots, so to him some cubic equations had no roots, some had one or two, and he missed the possibility of three, even all positive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure if this answers &quot;“proof” based upon the observation that two certain lines must intersect&quot; version of the question because of the negatives and because I am not sure if Khaayyam's conics are the relevant lines. Even so, Netz's interpolation is controversial. &lt;a href=&quot;http://www.math.harvard.edu/archive/archimedes09/pdf/Acerbi_Phantom_Paths.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi in Archimedes and the Angel:&#xA;Phantom Paths from Problems to Equations&lt;/a&gt; charges him with &quot;&lt;em&gt;unfalsifiable conjectures about the aims and intentions of ancient mathematicians, unsatisfactory command and highly tendentious use of primary sources and of secondary literature, unwarranted resort to over-generalization and to modern mathematical concepts, taking conjectures as established facts about ancient mathematics — and treating the received texts and diagrams as if they were what their alleged author wrote in the first place&lt;/em&gt;&quot;. This should give some idea of the controversies over modernizations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll skip the separate story of Cardano's formula, its inability to produce the root in the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Casus_irreducibilis&quot; rel=&quot;nofollow noreferrer&quot;&gt;irreducible case&lt;/a&gt;, and &lt;a href=&quot;http://www.nickalls.org/dick/papers/maths/descartes2006.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vieta's trigonometric solution to it&lt;/a&gt; (published posthumously in 1615). But it means that Vieta knew how to express roots of a cubic either in radicals or in inverse trigonometric functions in all cases, including negative roots (he did not operate with a notion of function, of course). I have to mention that it was also Vieta who devised a general procedure for converting geometric problems into equations, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2408/55&quot;&gt;Viète's Relevance and his Connection to Euler&lt;/a&gt;. Let me now address Dave Renfro's variation on the question: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who/when was the first to observe, without getting ensnared in the possible methods of representing a root and without getting ensnared in what is meant by a real number, the simple fact that due to the intermediate value property (referred to in variously different terms back then), each odd-degree polynomial has at least one real root? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I do not believe we can entirely avoid getting ensnared in what is meant by a real number, but there is a school of thought (our Mikhail Katz is a member) that we can, informally, trace the notion to &lt;a href=&quot;https://en.wikipedia.org/wiki/Simon_Stevin#Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stevin's Arithmetic of 1594&lt;/a&gt;, which promoted the arithmetical use of infinite decimals without reservations. This too is controversial, see debates under &lt;a href=&quot;https://hsm.stackexchange.com/a/3765/55&quot;&gt;Who gets credit for the real numbers?&lt;/a&gt;&#xA;But modulo that, Stevin did prove the intermediate value theorem for cubic polynomials, and almost in the way modern textbooks do. Except instead of successive bisection he used successive decisection, subdivision of the interval into 10 parts, which produces an additional decimal digit of the root at each step, see &lt;a href=&quot;https://arxiv.org/abs/1107.3688&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stevin Numbers and Reality, p.3&lt;/a&gt;. Since cubics (and odd-degree polynomials more generally) cross from negative to positive or vice versa this would prove existence of a root by appealing to the observation that two certain lines, the graph and the $x$-axis, must intersect. But even here there is anachronism, Descartes only introduced coordinates in 1637.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-04-21T23:32:21.557" LastActivityDate="2018-04-21T23:32:21.557" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7282" PostTypeId="1" AcceptedAnswerId="7286" CreationDate="2018-04-21T07:58:49.187" Score="8" ViewCount="959" Body="&lt;p&gt;&lt;em&gt;I found an interesting &lt;a href=&quot;https://math.stackexchange.com/q/2396682/279515&quot;&gt;question&lt;/a&gt; on Math SE asked by &lt;a href=&quot;https://math.stackexchange.com/users/92456/dilemian&quot;&gt;@Dilemian&lt;/a&gt; that seems more on topic here, and since it lacks answers there I thought to post it here so that it can receive good answers here.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;There are several equivalent ways to define a measurable subset of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way is to start with the Lebesgue outer measure and then restrict it to the set of subsets satisfying the Caratheodory criterion, namely the set of all sets &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$m^*A = m^*(A \cap E) + m^*(A \cap E^c)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$A \subset \mathbb{R}$&lt;/span&gt;. This is the approach in H. L. Royden's &lt;em&gt;Real Analysis&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way is to first define a positive linear functional on &lt;span class=&quot;math-container&quot;&gt;$C_c(\mathbb{R})$&lt;/span&gt; and then apply the Riesz representation theorem on locally compact Hausdorff spaces. This is the approach in Rudin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another way is to start with a premeasure on an algebra of sets and then extend it to a measure on a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet another way mentioned is to first define a subset with a finite outer measure to be measurable if its outer measure equals its inner measure. Then, for a general subset we take intersections with measurable subsets and define things the way they're supposed to be. This approach is taken in Frank Jones' book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With all these different methods, it makes me wonder what Lebesgue's original approach was to define a measurable set. I haven't come across any textbook on Measure Theory/Real Analysis that points out what was Lebesgue's method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, which construction of measure did Lebesgue use, and what definition did he use for a measurable set?&lt;/p&gt;&#xA;" OwnerDisplayName="user3224" LastEditorDisplayName="user3224" LastEditDate="2020-03-20T10:56:07.203" LastActivityDate="2020-03-20T10:56:07.203" Title="What was Lebesgue's original definition of a measurable set?" Tags="|terminology|real-analysis|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7283" PostTypeId="1" AcceptedAnswerId="7287" CreationDate="2018-04-21T10:08:57.467" Score="6" ViewCount="184" Body="&lt;p&gt;Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Decimal_time#Fractional_days&quot; rel=&quot;noreferrer&quot;&gt;states&lt;/a&gt; that decimal time - where the time of the day is expressed as a decimal part of the day - &quot;have been used by astronomers ever since [Laplace introduced them]&quot;. An example by Herschel is also shown. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know if the use is still followed today, or it faded after the XIX century?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="55" LastEditDate="2018-04-21T22:41:29.080" LastActivityDate="2018-04-22T01:08:30.217" Title="Do astronomers still use decimal time?" Tags="|astronomy|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7284" PostTypeId="2" ParentId="7282" CreationDate="2018-04-21T10:12:51.977" Score="7" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Thomas Hawkins, &lt;a href=&quot;https://books.google.it/books?id=oV1aLqag6WwC&amp;amp;pg=PA122&quot; rel=&quot;noreferrer&quot;&gt;Lebesgue's theory of integration&lt;/a&gt;, AMS (1975), page 122, regarding Lebesgue's thesis &quot;Intégrale, longueur, aire,&quot; published in 1902:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$m(E)$ is a nonnegative measure on bounded sets $E$, such that: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;(1) $m(E) \ne 0$ for some set $E$;&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;(2) $m(E + a) = m(E)$ for every real number $a$; &lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;(3) if $E_n$ are pairwise disjoint, then $m(\bigcup E_n)=\sum m(E_N)$.&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also : G.T.Q. Hoare and N.J.Lord, &lt;a href=&quot;http://www.cnd.mcgill.ca/~ivan/Papers/3621569.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;Intégrale, longueur, aire&quot; the Centenary of the Lebesgue Integral&lt;/a&gt;, &lt;em&gt;The Mathematical Gazette&lt;/em&gt;, Vol. 86 (2002). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And see also : David Bressoud, &lt;a href=&quot;https://books.google.it/books?id=TxxMoGjXC-wC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;A radical approach to Lebesgue's theory of integration&lt;/a&gt;, Cambridge (2008), &lt;strong&gt;Ch.5 The Development of Measure Theory&lt;/strong&gt;, on Borel and Lebesgue. &lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-04-21T10:28:44.720" LastActivityDate="2018-04-21T10:28:44.720" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7285" PostTypeId="2" ParentId="4934" CreationDate="2018-04-21T19:43:45.450" Score="5" Body="&lt;p&gt;The theorem is black on white in &lt;strong&gt;Lagrange&lt;/strong&gt;’s &lt;em&gt;Théorie des fonctions analytiques&lt;/em&gt;, as the initial case of “Taylor’s theorem with &lt;a href=&quot;//en.wikipedia.org/wiki/Taylor%27s_theorem#Explicit_formulas_for_the_remainder&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt; remainder” (1797, &lt;a href=&quot;//archive.org/stream/theoriedesfoncti00lagr#page/49&quot; rel=&quot;nofollow noreferrer&quot;&gt;§§51–53&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hence the value of the quantity $P$ relative to $z=1$ [i.e., $\frac{f(x)\ -\ f(0)}x$] can be expressed as $f'u$, $u$ being a quantity between $0$ and $x$. Likewise (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;52. Whence there results finally this new theorem, remarkable in its simplicity and its generality, that denoting by $u$ a quantity unknown but comprised between the limits $0$ and $x$, one can successively develop any function of $x$ and of arbitrary other quantities according to the powers of $x$, thus (...)&#xA;  \begin{align}&#xA;f(z+x) &amp;amp;= fz + x\,f'(z+u),\\&#xA;&amp;amp;=fz + x\,f'z + \frac{x^2}2\,f''(z+u),\\&#xA;&amp;amp;=fz + x\,f'z + \frac{x^2}2\,f''z + \frac{x^3}{2.3}\,f'''(z+u),\\&#xA;&amp;amp;\ \ \&amp;amp;\mathrm c.&#xA;\end{align}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lagrange also spells out the “Rolle” case in his next book (1798, &lt;a href=&quot;//archive.org/stream/bub_gb_T5IQirvZebQC#page/n177&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 166, 175&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consideration of the &lt;em&gt;maxima&lt;/em&gt; and &lt;em&gt;minima&lt;/em&gt; of parabolic lines led Stirling to a method for counting and bounding real roots in degrees 3 and 4, which &lt;strong&gt;Euler&lt;/strong&gt; generalized in his Differential calculus [&lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/1174624&quot; rel=&quot;nofollow noreferrer&quot;&gt;§298&lt;/a&gt;]. This method boils down, basically, to Rolle’s (... :) &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;between two consecutive real roots of the equation $\mathrm F\,x=0$, there always falls a real root of the equation $\mathrm F'x=0$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So I question the accepted attribution to Cauchy (Wikipedia &lt;a href=&quot;//en.wikipedia.org/w/index.php?title=Mean_value_theorem&amp;amp;diff=127874760&amp;amp;oldid=127720730&quot; rel=&quot;nofollow noreferrer&quot;&gt;2007&lt;/a&gt;, &lt;a href=&quot;//en.wikipedia.org/w/index.php?title=Mean_value_theorem&amp;amp;diff=726867001&amp;amp;oldid=721633751&quot; rel=&quot;nofollow noreferrer&quot;&gt;2016&lt;/a&gt;) — though of course it all hinges on: “first to prove” &lt;em&gt;under what assumptions&lt;/em&gt;, and how rigorously? For discussion of that: e.g.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Dugac, Pierre&lt;/em&gt;, Histoire du théorème des accroissements finis, Arch. Int. Hist. Sci. 30, 86-101 (1980). &lt;a href=&quot;https://zbmath.org/?q=an:0447.26003&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL0447.26003&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Grabiner, Judith&lt;/em&gt;, &lt;a href=&quot;//doi.org/10.1016/0315-0860(81)90043-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Review of “Histoire du théorème des accroissements finis”&lt;/strong&gt;&lt;/a&gt; by Pierre Dugac. Paris (Université Pierre et Marie Curie). 1979. 60 pp. Historia Math. 8, 214-219 (1981).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Persson, Lars-Erik; Rafeiro, Humberto; Wall, Peter&lt;/em&gt;, &lt;a href=&quot;http://siba-ese.unisalento.it/index.php/notemat/article/view/18358&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Historical synopsis of the Taylor remainder&lt;/strong&gt;&lt;/a&gt;, Note Mat. 37, 1-21 (2017). &lt;a href=&quot;//mathscinet.ams.org/mathscinet-getitem?mr=3733800&quot; rel=&quot;nofollow noreferrer&quot;&gt;MR3733800&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit as requested:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Briefly, Dugac contends (and repeats in a later &lt;a href=&quot;//books.google.com/books?id=_K7iBQAAQBAJ&amp;amp;pg=PA14&amp;amp;dq=%22Th%C3%A9or%C3%A8me+des+accroissements+finis%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; and &lt;a href=&quot;//zbmath.org/?q=an:1060.01002&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt;) that 1º) Lagrange first singled out, stated, and proved — perhaps inconclusively — the mean-value property for analytic functions; 2º) he was to an extent foreshadowed by Cavalieri (1635, &lt;a href=&quot;//books.google.com/books?id=WCRqHkVzb80C&amp;amp;pg=PA19&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 19&lt;/a&gt;); 3º) the first conclusive or “rigorous” proof in print is by &lt;strong&gt;Dini&lt;/strong&gt; (1878, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924060184227&amp;amp;num=71&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 71&lt;/a&gt;). To me this is all rather noncontroversial: the &lt;a href=&quot;//books.google.com/books?id=iIkpAQAAMAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22Lagrange%27s+theorem%22+Rolle&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; I learned it from called it “Lagrange’s theorem”. &lt;a href=&quot;//books.google.com/books?id=vrBFAQAAMAAJ&amp;amp;pg=PA82&amp;amp;dq=%22Th%C3%A9or%C3%A8me+de+Lagrange%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;So&lt;/a&gt; &lt;a href=&quot;//books.google.com/books?id=V2hbAAAAcAAJ&amp;amp;pg=PA72&amp;amp;dq=Th%C3%A9or%C3%A8me+de+Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;do&lt;/a&gt; &lt;a href=&quot;//books.google.com/books?id=1yy_H-1NW8wC&amp;amp;pg=PA217&amp;amp;dq=Frequentemente+designado&quot; rel=&quot;nofollow noreferrer&quot;&gt;quite&lt;/a&gt; &lt;a href=&quot;//books.google.com/books?id=mMPCAgAAQBAJ&amp;amp;pg=PA72&amp;amp;dq=%22Lagrange&amp;#39;s+theorem%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;a&lt;/a&gt; &lt;a href=&quot;//books.google.com/books?id=LRrvAAAAMAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22th%C3%A9or%C3%A8me+de+Lagrange%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;few&lt;/a&gt; &lt;a href=&quot;//books.google.com/books?id=ncxGAAAAQBAJ&amp;amp;pg=PA240&amp;amp;dq=Lagrange&amp;#39;s&quot; rel=&quot;nofollow noreferrer&quot;&gt;others&lt;/a&gt;. Current textbooks (&lt;a href=&quot;//books.google.com/books?id=lVUZAQAAIAAJ&amp;amp;focus=searchwithinvolume&amp;amp;q=%22first+stated+by+Joseph-Louis+Lagrange%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=spiaBAAAQBAJ&amp;amp;ots=CXB_c79NrU&amp;amp;pg=PA216&amp;amp;dq=Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stewart&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=oUfBAQAAQBAJ&amp;amp;ots=YYBotBH3sq&amp;amp;pg=PA106&amp;amp;dq=Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;Petrovic&lt;/a&gt;,...) credit it to Lagrange. So do Bottazzini (&lt;a href=&quot;//zbmath.org/?q=an:0597.01011&quot; rel=&quot;nofollow noreferrer&quot;&gt;1986&lt;/a&gt;, p. 53), Grabiner (1981, &lt;a href=&quot;//books.google.com/books?id=XuFcx-laQmIC&amp;amp;ots=3UnmA98zu3&amp;amp;pg=PA122&amp;amp;dq=%22Cauchy+was+not+the+first%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 122&lt;/a&gt;), Edwards (1979, &lt;a href=&quot;//books.google.com/books?id=ilrlBwAAQBAJ&amp;amp;ots=GYUfO5TRpo&amp;amp;pg=PA313&amp;amp;dq=Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 313&lt;/a&gt;), Cajori (1910, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924001049513&amp;amp;num=308&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 308-311&lt;/a&gt;), the &lt;em&gt;Encyklopädie&lt;/em&gt; (1899, &lt;a href=&quot;//archive.org/stream/p1n1encyklopdied02akaduoft#page/65&quot; rel=&quot;nofollow noreferrer&quot;&gt;II A 2 §§7, 11&lt;/a&gt;), and so on all the way to Lacroix (1810: vol. 1, pp. &lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/15801244&quot; rel=&quot;nofollow noreferrer&quot;&gt;liv&lt;/a&gt;, &lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/15801632&quot; rel=&quot;nofollow noreferrer&quot;&gt;385-386&lt;/a&gt;; 1800: vol. 3, pp. &lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/2483142&quot; rel=&quot;nofollow noreferrer&quot;&gt;vii&lt;/a&gt;, &lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/2483527&quot; rel=&quot;nofollow noreferrer&quot;&gt;384&lt;/a&gt;). I am beginning to wonder if Besenyei might be the &lt;em&gt;only&lt;/em&gt; author who doesn’t attribute the theorem to Lagrange?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, it is true that a wide shift of emphasis happened between the way Lagrange thought of his theorem and the way we use (and prove) it now. Besides already containing most of the above story, the French &lt;em&gt;Encyclopédie&lt;/em&gt; article (1912, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k2025807/f272&quot; rel=&quot;nofollow noreferrer&quot;&gt;II 3 §§11, 22&lt;/a&gt;) says this well in footnote 116:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From Rolle’s theorem&lt;sup&gt;114)&lt;/sup&gt; one readily deduces&lt;sup&gt;115)&lt;/sup&gt; the &lt;em&gt;mean value formula&lt;/em&gt; which plays a basic role throughout Calculus&lt;sup&gt;116)&lt;/sup&gt; and is also, like Rolle’s theorem, basically a mere translation into precise language of the intuitive fact that &lt;em&gt;B. Cavalieri&lt;/em&gt; had brought to the geometers’ attention.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;114) On the first rigorous proofs of Rolle’s theorem, see &lt;em&gt;A. Pringsheim&lt;/em&gt; (1900, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015053327824&amp;amp;num=454&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 454&lt;/a&gt;); see also &lt;strong&gt;&lt;em&gt;U. Dini&lt;/em&gt;&lt;/strong&gt; (1878, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=coo.31924060184227&amp;amp;num=75&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 75&lt;/a&gt;); &lt;em&gt;A. Harnack&lt;/em&gt; (1881); &lt;em&gt;M. Pasch&lt;/em&gt; (1882); &lt;em&gt;P. Mansion&lt;/em&gt; (1887); &lt;em&gt;J. Tannery&lt;/em&gt; (1886); &lt;em&gt;A. Demoulin&lt;/em&gt; (1902).&lt;/sup&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;115) In his lectures at École polytechnique, &lt;em&gt;O. Bonnet&lt;/em&gt; fully rigorously demonstrated this formula on the basis of Rolle’s theorem. He also gave a proof of Rolle’s theorem which, at least as exposed in &lt;em&gt;J. A. Serret&lt;/em&gt; (1868, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=chi.092051445&amp;amp;num=17&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 17-19&lt;/a&gt;), is far from immune to objections. (...)&lt;/sup&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;116) One should note that this fundamental role stems mainly from the mode of exposition of Calculus generally adopted today. For &lt;strong&gt;&lt;em&gt;J. L. Lagrange&lt;/em&gt;&lt;/strong&gt;, who took a different point of view, this formula was merely a consequence of Taylor’s formula (1797, &lt;a href=&quot;//archive.org/stream/theoriedesfoncti00lagr#page/49&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 49&lt;/a&gt;).&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To summarize, my take for now (based on these references and subject to your corrections):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1635:  Cavalieri states MVT hence Rolle. (Proof rejected by Guldin, 1640.)&lt;br&gt;&#xA;1690:  Rolle proves Rolle for polynomials.&lt;br&gt;&#xA;1755:  Euler states Rolle for differentiable functions. (Sketch of proof accepted at the time.)&lt;br&gt;&#xA;1797:  Lagrange proves MVT (hence Rolle) for analytic functions. (Proof accepted at the time.)&lt;br&gt;&#xA;1823:  Cauchy proves MVT (hence Rolle) for C&lt;sup&gt;1&lt;/sup&gt; functions. (Proof essentially same as Lagrange.)&lt;br&gt;&#xA;1868:  Bonnet rearranges proof of MVT to rely on Rolle as a lemma.&lt;br&gt;&#xA;1878:  Dini proves Rolle (hence MVT now) for differentiable functions. (Our current proof.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-05-01T12:10:01.193" LastActivityDate="2018-05-01T12:10:01.193" CommentCount="10" ContentLicense="CC BY-SA 3.0" />
  <row Id="7286" PostTypeId="2" ParentId="7282" CreationDate="2018-04-21T21:48:40.747" Score="8" Body="&lt;p&gt;Lebesgue used what Caratheodory called the outer measure. Fermat's library has an annotated translation of &lt;a href=&quot;http://fermatslibrary.com/s/on-a-generalization-of-the-definite-integral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lebesgue's 1901 paper On a Generalization of the Deﬁnite Integral&lt;/a&gt;, he is much briefer than the subsequent elaborations: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Let us consider a set of points of $(a, b)$; one can enclose in an inﬁnite number of ways these points in an enumerably inﬁnite number of intervals; the inﬁmum of the sum of the lengths of the intervals is the measure of the set.&#xA;  A set $E$ is said to be measurable if its measure together with that of the set of points not forming $E$ gives the measure of $(a, b)$.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is it. The translator remarks that &quot;&lt;em&gt;Lebesgue is deﬁning $E$ to be measurable if $m^*(E) + m^*((a, b)\cap E^c)=b-a$&lt;/em&gt;&quot;, which is the &quot;Caratheodory criterion&quot; restricted to intervals.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-18T22:13:34.320" LastActivityDate="2018-05-18T22:13:34.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7287" PostTypeId="2" ParentId="7283" CreationDate="2018-04-21T22:39:55.010" Score="6" Body="&lt;p&gt;Wikipedia refers to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Julian_day#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julian Date system&lt;/a&gt;, which is used, primarily by astronomers, to this day. It is convenient due to irregularities in the standard calendar periods. The Julian days give a continuous count of days since the beginning of the Julian Period, introduced by Scaliger in 1583, i.e. since 1 January 4713 BC (Julian calendar), originally for the meridian of Alexandria. The Julian date (JD) is the Julian day number for the preceding mean noon in Universal Time plus the fraction of the day, the fraction is expressed in decimals. Here is an example from &lt;a href=&quot;https://books.google.com/books?id=O31j9UJ3U4oC&amp;amp;dq=decimal+part+of+the+day+astronomers&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ridpath's Oxford Dictionary of Astronomy (2012)&lt;/a&gt;: 1962 June 24 at 18h UT is JD 2,437,840.25, and here is online &lt;a href=&quot;http://www.onlineconversion.com/julian_date.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julian Date Converter&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Originally the Julian Period was used only to count years, and the Julian calendar was used to express days within years. Counting the days since the beginning was Herschel's innovation of 1849, in Outlines of Astronomy, the decimal fraction of the day addition was indeed Laplace's idea from Traité de Mécanique Céleste (1799). Another change, to the meridian of Greenwich instead of Alexandria, was made after its adoption as the Prime Meridian at the International Meridian Conference in Washington in 1884. Julian dates became standard practice after being adopted by Pickering, of the Harvard College Observatory, in 1890. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-04-22T01:08:30.217" LastActivityDate="2018-04-22T01:08:30.217" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7288" PostTypeId="1" CreationDate="2018-04-22T01:55:04.300" Score="5" ViewCount="611" Body="&lt;p&gt;This is a basic formula in mechanics, which determines the acceleration of a particle performing uniform circular motion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By who first derived it? In Newton's Principa, what one can find is that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$a \propto v^2/r  $$&lt;/p&gt;&#xA;" OwnerUserId="2641" LastActivityDate="2019-02-21T19:53:09.763" Title="Who first derived $a =v^2/r$" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7289" PostTypeId="2" ParentId="7288" CreationDate="2018-04-22T15:59:41.030" Score="9" Body="&lt;p&gt;(a) In answer, first a comment about proportionalities and equalities.  Many 17th-century writers, including Huygens and Newton, commonly used proportions rather than equalities, especially to avoid doing what would now be calculation of ratios between physical quantities of unlike kind, such ratios being considered somehow illegitimate. This old tradition and theory of proportion dating back to Euclid is explained by N Guicciardini in &quot;Mathematics and the New Sciences&quot;, chapter 8, (esp. sec. 8.3.2 (p.232-233) in 'Oxford Handbook of the History of Physics' ed J Buchwald et al., 2013).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) There is a chronology that seems to be widely accepted for the discoveries and publications of the &lt;span class=&quot;math-container&quot;&gt;$ v^2/r $&lt;/span&gt; relation between speed, curvature and bending force or acceleration: and it is usually concluded that Newton and Huygens made their findings independently. Here is the chronology along with references to documents: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) Huygens' manuscript of 'De vi centrifuga' carries a date of 1659, and includes the &lt;span class=&quot;math-container&quot;&gt;$ v^2/r $&lt;/span&gt; relation, stated verbally and in terms of proportion and centrifugal force, with its derivation: however, according to the 20th-c. editors of Huygens' &lt;a href=&quot;https://www.dbnl.org/tekst/huyg003oeuv16_01/huyg003oeuv16_01_0014.php&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Oeuvres complètes&lt;/em&gt;, Vol. XVI, at p.254&lt;/a&gt;, this work was not published until the appearance of the 'Opuscula posthuma' of 1703, with a re-impression in 1728.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) There is a Newton manuscript of about 1669 or before which derives the &lt;span class=&quot;math-container&quot;&gt;$ v^2/r $&lt;/span&gt; relation in terms of proportions and of 'conatus' which can amount to centrifugal force -- rather than in terms of (modern) force or acceleration, concepts which were only clarified later -- and in the same ms Newton then employed the result to derive (from the Kepler three-halves power relation between orbital radius and period) an inverse-square law of centrifugal force for the approximated case of assumedly circular planetary orbits.  The ms was not published by Newton, it was first published and discussed in modern literature by Rupert Hall in “Newton on the Calculation of Central Force,” Annals of Science 13 (1957), 62–71. The document was also reproduced, translated and commented in the Correspondence of Isaac Newton, vol.1 (1661-1675) (Cambridge, 1959), at pp.297-303 and plates facing pages 297,299. It also appeared in John Herivel's 'The Background to Newton's Principia' (Oxford, 1965), where pp.192-198 give commentary, Newton's diagram, text and an English translation from Latin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) Huygens' book 'Horologium oscillatorium' (1673) (available in Latin at &lt;a href=&quot;https://books.google.com/books?id=YgY8AAAAMAAJ&amp;amp;pg=PA159&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.com/books?id=YgY8AAAAMAAJ&amp;amp;pg=PA159&lt;/a&gt;) contains in part V, p.159 onwards, a number of 'theorems' out of 'De vi centrifuga' which Huygens set out as conclusions but without derivation/proof. The result relevant to the present question is stated verbally and in terms of proportion and centrifugal force. It can be gathered from p.160, theorems II, III and IV. To show the style, what Huygens wrote is transcribed and translated here as: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Original Latin): &#xA;&quot;II: Si duo mobilia aequalia, aequali celeritate ferantur, in circumferentiis inaequalibus; erunt eorum vires centrifugae in ratione contraria diametrorum.&#xA;&lt;p&gt;III: Si duo mobilia aequalia in circumferentiis aequalibus ferantur, celeritate inaequali, sed utraque motu aequabili, quarum in his omnibus intelligi volumus; erit vis centrifuga velocioris, ad vim tardioris, in ratione duplicata celeritatum.&#xA;&lt;p&gt;IV: Si mobilia duo aequalia, in circumferentiis inaequalibus circumlata, vim centrifugam aequalem habuerint; erit tempus circuitus in majori circumferentia, ad tempus circuitus in minori, in subdupla ratione diametrorum.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Translated: &#xA;&quot;II: If two equal movable objects are carried with equal speed around unequal circumferences, [then] their centrifugal forces will be in the inverse ratio of the diameters.&#xA;&lt;p&gt;III: If two equal movable objects are carried around equal circumferences with unequal speeds, but each with equable motion, which we wish to be understood in all these theorems, [then] the centrifugal force of the faster, to the force of the slower, will be in the duplicate ratio of the speeds.&#xA;&lt;p&gt;IV: If two equal movable bodies, carried around unequal circumferences, have equal centrifugal force, [then] the time of circuit around the greater circumference, to the time of circuit around the lesser, will be in the subduplicate ratio of the diameters.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(4) Newton's manuscript 'De Motu Corporum in gyrum' (1684) (more details are at (&lt;a href=&quot;https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum&lt;/a&gt;)) derived the result as its Theorem 2, showing that for any given time-segment, the centripetal force (directed towards the center of a circle, treated as a center of attraction) is proportional to the square of the arc-length traversed, and inversely proportional to the radius. This short precursor of the 'Principia' was made available to members of the Royal Society by its entry as a copy in late 1684/early 1685 into the Royal Society's register books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(5) The &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k3363w.image&quot; rel=&quot;noreferrer&quot;&gt;1687 edition&lt;/a&gt; of Newton's 'Principia' gave in Book 1, proposition 4, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k3363w/f51.image&quot; rel=&quot;noreferrer&quot;&gt;at pages 41-43&lt;/a&gt; essentially the same result as (4). This was the first appearance of a derivation in print. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(6) The 1703 posthumous edition of Huygens' works eventually printed &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/1681890&quot; rel=&quot;noreferrer&quot;&gt;'De vi centrifuga'&lt;/a&gt; with his derivation of the relationships quoted above (at p.409 onwards). &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-02-21T19:53:09.763" LastActivityDate="2019-02-21T19:53:09.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7290" PostTypeId="2" ParentId="3604" CreationDate="2018-04-22T19:16:03.060" Score="2" Body="&lt;p&gt;The National Institute of Mental Health Data Archive (NDA) makes available human subjects data collected from hundreds of research projects across many scientific domains. The NDA provides infrastructure for sharing research data, tools, methods, and analyses enabling collaborative science and discovery. De-identified human subjects data, harmonized to a common standard, are available to qualified researchers.  Summary data is available to all.&lt;/p&gt;&#xA;" OwnerUserId="7335" LastActivityDate="2018-04-22T19:16:03.060" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7291" PostTypeId="1" AcceptedAnswerId="7297" CreationDate="2018-04-22T23:12:02.847" Score="6" ViewCount="209" Body="&lt;p&gt;I'm looking for info on why E is commonly used for the total space of a fiber bundle. I understand F (fiber) and B (base), but there doesn't seem to be any particularly obvious reason for choosing E. &lt;/p&gt;&#xA;" OwnerUserId="7336" LastActivityDate="2018-04-23T21:02:12.143" Title="Notation for fiber bundles - why E for total space?" Tags="|notation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7292" PostTypeId="1" CreationDate="2018-04-23T08:57:25.217" Score="14" ViewCount="705" Body="&lt;p&gt;Has science fiction ever caused scientists to do real research?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Science fiction&lt;/em&gt; here means fiction that tries to explain things in the world rather than speculate about the future or unexplorable past (e.g. aliens having travelled to Earth 100 million years ago is unexplorable).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Real research&lt;/em&gt; means the one based on scientific methods, such as experimental analysis, statistics and so on.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am interested in natural science such as physics, biology, chemistry, etc, as well as history rather than human-created such as mathematics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Some examples of science fiction according to my definition may include the fiction describing &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Dark_matter_in_fiction&quot; rel=&quot;noreferrer&quot;&gt;dark matter&lt;/a&gt;, &#xA;human and animal behavior (e.g. fiction describing hunter-gatherer society where women can be hunters), historical speculations (say, ancient europeans travelled Americas), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No way I am asking for a big/complete list of such examples, just most remarkable ones if there even are any.&lt;/p&gt;&#xA;" OwnerUserId="7339" LastEditorUserId="7339" LastEditDate="2018-04-23T09:53:46.750" LastActivityDate="2022-12-28T15:18:56.287" Title="Has science fiction ever caused scientists to do real research?" Tags="|physics|astronomy|chemistry|biology|earth-sciences|" AnswerCount="6" CommentCount="9" ContentLicense="CC BY-SA 3.0" />
  <row Id="7293" PostTypeId="2" ParentId="7292" CreationDate="2018-04-23T12:30:54.293" Score="9" Body="&lt;p&gt;One such case comes to my mind: Jules Verne's novel Green ray (1882). Jules Verne popularized this rare phenomenon, and it seems that it has not been scientifically studied before. After Jules Verne novel, the green ray has been photographed and explained. See Wikipedia &quot;Green flash&quot; where there is a list of literature (Jules Verne included!).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-04-23T12:30:54.293" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7294" PostTypeId="2" ParentId="7292" CreationDate="2018-04-23T13:23:01.703" Score="16" Body="&lt;p&gt;Science fiction author Arthur C. Clarke is often credited with the idea of communication satellites.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_C._Clarke&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;  That was in 1945, long before any artificial satellite had been launched in reality.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-04-23T13:23:01.703" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7296" PostTypeId="2" ParentId="7292" CreationDate="2018-04-23T19:32:47.200" Score="8" Body="&lt;p&gt;While it may not be &lt;em&gt;research&lt;/em&gt; exactly, the fictional cases of Sherlock Holmes did actually influence the way real world police agencies and detectives approached forensic science.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sir Arthur Conan Doyle wrote about Holmes using fingerprint identification roughly 10 years before the police started to rely on it.  Also, Holmes did a lot of soil analysis before it was standard forensic practice. &lt;/p&gt;&#xA;" OwnerUserId="7346" LastActivityDate="2018-04-23T19:32:47.200" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7297" PostTypeId="2" ParentId="7291" CreationDate="2018-04-23T21:02:12.143" Score="7" Body="&lt;p&gt;Early contributors to the theory of fiber bundles, like Ehresmann who developed the geometric formalism of general connections, were French. In French total space is &lt;a href=&quot;https://ncatlab.org/nlab/show/%C3%A9tal%C3%A9+space&quot; rel=&quot;noreferrer&quot;&gt;espace étalé&lt;/a&gt;. In more refined terminology espace étalé is something more structured, which only coincides with the total space as a set, and applies to sheaves, which generalize fiber bundles. This generalization was also developed by the French, espace étalé definition was used in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sheaf_(mathematics)#History&quot; rel=&quot;noreferrer&quot;&gt;1950 Cartan seminar&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-04-23T21:02:12.143" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7298" PostTypeId="1" AcceptedAnswerId="7301" CreationDate="2018-04-24T06:35:36.867" Score="5" ViewCount="305" Body="&lt;p&gt;On a recent visit to the Royal Observatory at Greenwich I was struck by its proximity to the Naval Academy next door. The theme of the history of clocks and development of astronomy was driven by the need to stop ships crashing due to navigation failures. In theory you could solve this by better clocks and better astronomical models. Still they spent hundreds of years on the problem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FkvG3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FkvG3.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7ilGy.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7ilGy.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Included in the themed display was &lt;a href=&quot;https://science.nasa.gov/science-news/science-at-nasa/2004/28may_cook&quot; rel=&quot;nofollow noreferrer&quot;&gt;Captain Cook’s trip to Tahit&lt;/a&gt;i to &lt;a href=&quot;https://en.m.wikipedia.org/wiki/First_voyage_of_James_Cook&quot; rel=&quot;nofollow noreferrer&quot;&gt;observe the transit of Venus&lt;/a&gt;.  At the time Captain Cook &lt;a href=&quot;https://maas.museum/observations/2007/07/25/how-captain-cook-found-his-longitude/&quot; rel=&quot;nofollow noreferrer&quot;&gt;didn’t carry a chronometer&lt;/a&gt; because they were too rare at the time. (He carried them on subsequent voyages.) The problem he was trying to solve was the size of the solar system (which a good observation of Venus would solve.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xw5cM.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xw5cM.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The method of determining longitude was lunar distances, where you measure the angle of the moon and compare it to the angle of a star, compare this to a logbook and do a lot of calculations.&lt;/p&gt;&#xA;&lt;p&gt;Now it seems to me that one of the ‘stars’ referred to in the lunar distances calculation could have been a planet. This would explain why Captain Cook’s measurement would improve navigation.&lt;/p&gt;&#xA;&lt;p&gt;Alternatively there may have been an alternative method to lunar distances based on ‘planet distances ‘ that Captain Cooks measurement would assist with.&lt;/p&gt;&#xA;&lt;p&gt;But it’s not clear to me how Captain Cook’s trip was going to improve Naval Navigation. It seems improbable that his trip was funded on the basis of an academic question about the size of the solar system.&lt;/p&gt;&#xA;&lt;p&gt;My question is: &lt;strong&gt;Was Captain Cook’s voyage to observe the transit of Venus going to enable better ship navigation at the time?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="4251" LastEditDate="2020-07-10T18:52:26.400" LastActivityDate="2020-07-10T18:52:26.400" Title="Was Captain Cook’s voyage to observe the transit of Venus going to enable better ship navigation at the time?" Tags="|mathematics|physics|astronomy|navigation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7299" PostTypeId="1" AcceptedAnswerId="7300" CreationDate="2018-04-24T20:01:22.453" Score="12" ViewCount="1968" Body="&lt;p&gt;There is a story featuring Henri Poincaré and an unscrupuolous baker. Every day Poincaré bought a piece of bread which should have weighted 1 kg. After an year, the mathematician brought the baker to the judge: he weighted all the pieces of bread and found out that the average weight was just 950g. The baker was then fined. The story goes on: next year Poincaré sued the baker again. Now the average weight was more or less 1 Kg, but he showed that the distribution of weights was not a Gaussian curve but rather its tail. This meant, concluded Poincaré, that the baker chose the heaviest pieces of bread to give him, but he kept baking lighter pieces of bread.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The anecdote may be found in Leonard Mlodinov, &lt;em&gt;The Drunkard's Walk&lt;/em&gt;, who in turn cites Bart Holland, &lt;em&gt;What Are the Chances?&lt;/em&gt; pp. 41–42. I read both passages, but I think the story was made up. Poincaré could have done this, but would it have been able to convince a judge that his reasoning is correct? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody know some actual sources for the story?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="55" LastEditDate="2018-04-25T02:39:56.587" LastActivityDate="2022-08-22T17:59:59.667" Title="Poincaré and the baker: was the anecdote true?" Tags="|mathematicians|debunking|probability|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7300" PostTypeId="2" ParentId="7299" CreationDate="2018-04-25T02:39:37.253" Score="8" Body="&lt;p&gt;It is interesting that even many of those who retell the anecdote immediately disown it &quot;The following anecdote about him is probably fabricated, but it makes an interesting probability problem&quot;, says &lt;a href=&quot;http://www.actuarialoutpost.com/actuarial_discussion_forum/archive/index.php/t-253939.html&quot; rel=&quot;noreferrer&quot;&gt;Actuarial Outpost&lt;/a&gt;. Oloffson who like Mlodinov even put it into &lt;a href=&quot;https://books.google.com/books?id=g8DlBQAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;his book (p.203)&lt;/a&gt;, writes &quot;Although the anecdote illustrates a clever way to use probabilities to detect cheating, I doubt that it is true. What Frenchman who has just bought a loaf of bread would go home and weigh it?&quot; &lt;a href=&quot;https://jcdverha.home.xs4all.nl/scijokes/10.html#Poincare_6&quot; rel=&quot;noreferrer&quot;&gt;Laurie Snell from Dartmouth&lt;/a&gt; even mentions a themed exhibit at Boston Science Museum, but alas... no attribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story appears to be made up, and recently. The earliest mention I could find is from 2000, on &lt;a href=&quot;https://everything2.com/title/%2522true%2522+story+about+Poincar%25C3%25A9%2527s+baker&quot; rel=&quot;noreferrer&quot;&gt;Everything2's collection&lt;/a&gt; of equally &quot;credible&quot; anecdotes, like Gauss's fast addition of numbers from 1 to 100. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no  agreement even on whether the following normal distribution quote belongs to Poincare:&quot;&lt;em&gt;Everyone  believes  in  it:  experimentalists  believing  that  it  is  a  mathematical  theorem, mathematicians believing that it is an empirical fact&lt;/em&gt;&quot;. And that one at least occurs in his book. &lt;a href=&quot;http://aidanlyon.com/media/publications/Lyon-normal_distributions.pdf&quot; rel=&quot;noreferrer&quot;&gt;According to Cramer&lt;/a&gt;, it is Lippman's, and Poincare only quoted it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-04-25T02:39:37.253" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7301" PostTypeId="2" ParentId="7298" CreationDate="2018-04-25T03:07:12.170" Score="8" Body="&lt;p&gt;Cook's voyage had multiple purposes. One of them was observing the transit of Venus. Transit of Venus is a rare event, important for astronomy, but it has no direct bearing on navigation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Study of the new methods of navigation (in particular, determination of longitude by the method of Lunar distances)&#xA;was another, separate purpose of Cook's voyage. The method of Lunar distances just reached a degree of perfection at that time, so that it became practical, and this was a good opportunity to test it (because of the great length of the voyage, and because of the presence of a professional astronomer on board). As you say, in the later voyages he also tested the newly invented chronometers for determination of longitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Venus (like any other star or planet) can be used for navigation by Lunar distance (measuring the distance (angle) between Moon and Venus, instead of the more usual and more convenient distance between the Moon and the Sun),&#xA;but this has nothing to do with transits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem of determination of longitude at sea (which is evidently important for navigation) was solved for the first time in the middle of 18th century&#xA;with two different methods. One was the Lunar distances, and another the invention of chronometer. Both methods had advantages and disadvantages.&#xA;(Lunar distances required serious special education of navigators, this was a problem. Also Moon is not always available. Very accurate measurements with good instruments were required. Using a chronometer is much simpler, but chronometers were very expensive: each piece was made by hand and this took months or sometimes years of masters of highest qualifications. And what do you do when a chronometer breaks by some accident. Only mass machined production made them cheaper in the middle of 19th century. Lunar distances were practiced as an alternative method, frequently for backup, until late 19th century.&#xA;Chronometers were indispensable until the spread of satellite navigation in&#xA;the 1980s.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-04-26T22:21:43.970" LastActivityDate="2018-04-26T22:21:43.970" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7302" PostTypeId="2" ParentId="3615" CreationDate="2018-04-25T21:01:58.893" Score="12" Body="&lt;p&gt;As noted, Mermin was probably the first to utter the exact words “Shut up and calculate”.  However, the equivalent rallying cry of “Get the numbers out” has its origins some decades earlier.&lt;/p&gt;&#xA;&lt;p&gt;According to the &lt;em&gt;Nature&lt;/em&gt; article &lt;a href=&quot;https://www.nature.com/news/history-shut-up-and-calculate-1.14458#/b10&quot; rel=&quot;nofollow noreferrer&quot;&gt;History: Shut up and calculate&lt;/a&gt; , referring to the role of leading physicist in the war effort (WWII) and how their attitude to their art became largely pragmatic as a result:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This war-forged pragmatism produced enormously impressive research and influenced a generation of leading scientists. Their approach to basic research — and the institutions in which they pursued it — assumed an aura of inevitability. But the approach came with some trade-offs, largely unnoticed at the time. Important questions that resisted the powerful, phenomenological methods tended to get eclipsed. Anything that smacked of 'interpretation', or worse, 'philosophy', began to carry a taint for many scientists who had come through the wartime projects. Conceptual scrutiny of foundations struck many as a luxury. The wartime style was reinforced in the United States by exponentially rising university enrolments after the war. The new classroom realities left little space for informal discussion of philosophy or foundations. The Rad Lab rallying cry of “Get the numbers out” shaded into “Shut up and calculate!”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, “Rad Lab” refers to the Radiation Laboratory set up by the American government in 1940 to perform military research.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I was chatting with a friend last night and I was reminded of a story I had heard, most likely apocryphal, which perfectly exemplifies this attitude amongst physicists. The story was told by Sean Carroll during an interview in one of the many popular physics videos available on YouTube, and which I am unfortunately unable to relocate.&lt;/p&gt;&#xA;&lt;p&gt;The story goes that Murray Gell-Mann used to carry a note in his breast pocket which he claimed was a doctor's note. If anyone raised a philosophical issue, then Gell-Mann would produce this note and say &amp;quot;I'm sorry, I cannot talk about philosophy. My doctor said that it's bad for my blood pressure.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="6609" LastEditDate="2023-10-19T23:17:03.380" LastActivityDate="2023-10-19T23:17:03.380" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7304" PostTypeId="1" AcceptedAnswerId="7306" CreationDate="2018-04-26T23:55:45.163" Score="2" ViewCount="520" Body="&lt;p&gt;The D'Alembertian is a generalization of the Laplacian operator to a space of arbitrary dimension and metric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where does the D'Alembertian symbol $\Box$ come from?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/D%27Alembert_operator&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; it has to do with 4-dimensional spaces, but the authors don't cite sources and I don't see why the relation.&lt;/p&gt;&#xA;" OwnerUserId="5143" LastEditorUserId="-1" LastEditDate="2018-04-28T23:16:36.467" LastActivityDate="2018-04-30T13:48:25.070" Title="D'Alembertian symbol $\Box$" Tags="|notation|calculus|differential-equations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 3.0" />
  <row Id="7306" PostTypeId="2" ParentId="7304" CreationDate="2018-04-27T06:27:12.400" Score="6" Body="&lt;p&gt;The symbol seems due to &lt;strong&gt;Poincaré&lt;/strong&gt; (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?32.0818.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/lectricitetopti00poingoog#page/n481&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 456&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For brevity we will introduce the following symbol, by setting,&#xA;  $$&#xA;\Delta\mathrm U -\mathrm K_0\frac{d^2\mathrm U}{dt^2}=\Box\mathrm U&#xA;$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here $\mathrm K_0=1\,/\,c^2$ (&lt;a href=&quot;//archive.org/stream/lectricitetopti00poingoog#page/n361&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 336&lt;/a&gt;). It didn’t immediately catch on: it appears without comment in Poincaré’s (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?37.0886.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;, &lt;a href=&quot;//www.archive.org/stream/rendicontidelci09palegoog#page/n156&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 132&lt;/a&gt;) (cited by Wikipedia), but not in the book where &lt;strong&gt;Lorentz&lt;/strong&gt; named it (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?40.0922.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/electronstheory00lorerich#page/17&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 17&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the operation $\Delta-\frac1{c^2}\frac{\partial^2}{\partial t^2}$ may be given the name of the &lt;em&gt;Dalembertian&lt;/em&gt; (...) in commemoration of the fact that the mathematician d’Alembert was the first to solve a partial differential equation (...) which contains this operation, or rather the operation $\frac{\partial^2}{\partial x^2}-\frac1{c^2}\frac{\partial^2}{\partial t^2}$ &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?41.0143.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uva.x001677364&amp;amp;num=175&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 175&lt;/a&gt;) Lewis (of photon fame) was still suggesting the notation&#xA;$$&#xA;\diamondsuit^2=&#xA;\frac{\partial^2}{\partial x_1^2} +&#xA;\frac{\partial^2}{\partial x_2^2} +&#xA;\frac{\partial^2}{\partial x_3^2} +&#xA;\frac{\partial^2}{\partial x_4^2}\qquad(x_4=ict),&#xA;$$&#xA;but Sommerfeld (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?41.0759.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;, &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144375&amp;amp;num=661&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 661&lt;/a&gt;) and von Laue (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?42.0718.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/dasrelativittsp00lauegoog#page/n85&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 71&lt;/a&gt;) chose again $\Box$ (pronounced “Delta”). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Sommerfeld (ibid.) says the symbol was already introduced by &lt;strong&gt;Cauchy&lt;/strong&gt;. This is repeated by Silberstein (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?44.0778.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/londonedinburg6231912lond#page/805&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 805&lt;/a&gt;; &lt;a href=&quot;//emis.de/cgi-bin/JFM-item?45.1150.07&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/theoryofrelativi00silbrich#page/113&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 113&lt;/a&gt;), Bateman (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?45.1324.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/mathematicalanal00baterich#page/6&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 6&lt;/a&gt;) and Pauli (&lt;a href=&quot;//emis.de/cgi-bin/JFM-item?48.0977.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/encyklomath205encyrich#page/n618&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 607&lt;/a&gt;), but neither gives a citation. (Perhaps they mean &lt;a href=&quot;//archive.org/stream/exercicesdanaly02caucrich#page/n349&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;?)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-04-30T13:48:25.070" LastActivityDate="2018-04-30T13:48:25.070" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7307" PostTypeId="1" CreationDate="2018-04-27T08:26:12.337" Score="6" ViewCount="613" Body="&lt;p&gt;&quot;The general set theory [...] definitely belongs to metaphysics. You can easily convince yourself when examining the categories of cardinal numbers and the order type, these basic notions of set theory, on the degree of their generality. &#xA;    [...] and the fact that my presently written work is issued in mathematical journals does not modify the metaphysical contents and character of this work. &#xA;    [...] By me Christian philosophy is for the first time confronted with the true teachings of the infinite in its beginnings.&quot; [G. Cantor, letter to T. Esser (15 Feb 1896)] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I have to draw your attention to two facts: 1st to the untearable ties that connect metaphysics and theology, in that the latter is the lodestar according to which the former is adjusting itself. [...] There follows a threefold: a) In metaphysical discussions it is sometimes inevitable to have theology join in. b) Every real progress in metaphysics strengthens or multiplies the tools of theology; it may even happen that human reason (of course subjugating to the infallible decisions of the church) may obtain deeper and richer symbolic insights with respect to the mysteries of the religion than has been expected or foreseen.&quot; [G. Cantor, letter to T. Esser (15 Feb 1896)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;[...] it is clear that the theological considerations by which Cantor motivated his notion of the actual infinite, were metaphysical in nature.&quot; [A. Heyting: &quot;Technique versus metaphysic in the calculus&quot;, in Imre Lakatos (ed.): &quot;Problems in the philosophy of mathematics&quot;, North Holland, Amsterdam (1967) p. 43]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Cantor is probably the last exponent of the Newtonian attitude with respect to religion.&quot; [H. Meschkowski, W. Nilson: &quot;Georg Cantor Briefe&quot;, Springer, Berlin (1991) p. 15]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;[...] it was a certain satisfaction for me, how strange this may appear to you, to find in Exodus chapt. XV, verse 18 at least something reminiscent of transfinite numbers, namely the text: 'The Lord rules in infinity (eternity) and beyond.' I think this 'and beyond' hints to the fact that $\omega$ is not the end but that something is existing beyond.&quot; [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Compare the concurring perception of the whole sequence of numbers as an actually infinite quantum by St Augustin (De civitate Dei. lib. XII, chapt. 19) [...] While now St Augustin claims the total, intuitive perception of the set ($\nu$), 'quodam ineffabili modo', a parte Dei, he simultaneously acknowledges this set formally as an actual infinite entity, as a transfinitum, and we are forced to follow him in this matter.&quot; [G. Cantor, letter to A. Eulenburg (28 Feb 1886)]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So according to Cantor, the inventor of set theory, set theory $S$ by the following syllogism, is part of theology $T$:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$S\subseteq M \wedge M\subseteq T \Rightarrow S \subseteq T $&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It appears as if today set theorists do not like to remember the roots. But have they ever been cut, and if so when and where?&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="4251" LastEditDate="2021-09-14T17:34:35.453" LastActivityDate="2024-03-20T10:54:22.420" Title="When did set theory throw off theology?" Tags="|mathematics|set-theory|cantor|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 3.0" />
  <row Id="7308" PostTypeId="2" ParentId="7307" CreationDate="2018-04-27T13:42:59.733" Score="13" Body="&lt;p&gt;It was never really connected. Cantor's mathematical papers contain nothing of the sort you cite. That Cantor himself was interested in theology is no more relevant than Newton's interest in the same. But you do not ask &quot;when Mechanics threw of theology?&quot; They were never related. But one person can write on several different subjects.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-04-27T13:42:59.733" CommentCount="6" ContentLicense="CC BY-SA 3.0" />
  <row Id="7309" PostTypeId="1" CreationDate="2018-04-29T09:38:50.203" Score="9" ViewCount="315" Body="&lt;p&gt;Today there is no doubt that the empty set for the whole of mathematics is as reasonable and useful as zero for arithmetic. This however was not always the case, and surprisingly even Zermelo, who based his infinite set on the empty set, was unsure. The following quotes shed some light on this mostly forgotten development. My question is, when had these doubts become completely forgotten? I would also be interested in further quotes doubting the &quot;justifyability&quot; of the empty set.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named a nothing an empty set. In German the word set has the meaning of many or great quantity. Often we find in German texts the expression große (great or large) Menge, rarely the expression kleine (small) Menge. Therefore Bolzano apologizes for using this word in case of sets having only two elements: &quot;Allow me to call also a collection containing only two parts a set.&quot; [J. Berg (ed.): B. Bolzano, Einleitung zur Grössenlehre, Friedrich Frommann Verlag, Stuttgart (1975) p. 152] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the non-empty set of less than two elements: &quot;For the uniformity of the wording it is useful to permit also the special case that a system S consists of a single (of one and only one) element a, i.e., that the thing a is element of S but every thing different from a is not an element of S. The empty system, however, which does not contain any element, shall be excluded completely for certain reasons, although it might be convenient for other investigations to fabricate such.&quot; [R. Dedekind: &quot;Was sind und was sollen die Zahlen?&quot; Vieweg, Braunschweig (1887), 2nd ed. (1893) p. 2]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bertrand Russell considered an empty class as not existing: &quot;An existent class is a class having at least one member.&quot; [Bertrand Russell: &quot;On some difficulties in the theory of transfinite numbers and order types&quot;, Proc. London Math. Soc. (2) 4 (1906) p. 47]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Georg Cantor mentioned the empty set with some reservations and only once in all his work: &quot;Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O. P = O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such.&quot; [Cantor, p. 146] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And even Ernst Zermelo who made the &quot;Axiom II There is an (improper) set, the 'null-set' 0 which does not contain any element&quot; [E. Zermelo: &quot;Untersuchungen über die Grundlagen der Mengenlehre I&quot;, Mathematische Annalen 65 (1908) p. 263], this same Zermelo himself said in private correspondence: &quot;It is not a genuine set and was introduced by me only for formal reasons.&quot; [E. Zermelo, letter to A. Fraenkel (1 Mar 1921)] &quot;I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification.&quot; [E. Zermelo, letter to A. Fraenkel (9 May 1921)] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer and comments have shown me that it is hard if not impossible to answer this question. Zermelo for instance expressed his doubts only in private correspondence. What about McAtee or Peano or Young mentioned by Gerad Edgar? Therefore I will take the alternative: Who was the last recognized mathematician to utter doubts with respect to the existence of the empty set? Kanamori, mentioned by Dave L Renfro gives a hint to 1943.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="12699" LastEditDate="2018-04-29T18:02:01.567" LastActivityDate="2018-04-29T18:02:01.567" Title="Who was (were) the first mathematician(s) who did not doubt the empty set?" Tags="|set-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7310" PostTypeId="2" ParentId="7309" CreationDate="2018-04-29T11:46:33.120" Score="4" Body="&lt;p&gt;In &lt;a href=&quot;http://jeff560.tripod.com/e.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathWords&lt;/a&gt; we find&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;EMPTY SET&lt;/strong&gt;. A &lt;em&gt;JSTOR&lt;/em&gt; search found the term used--without explanation--in J. E. McAtee &quot;Modular Invariants of a Quadratic Form for a Prime Power Modulus,&quot; &lt;em&gt;American Journal of Mathematics&lt;/em&gt;, &lt;strong&gt;41&lt;/strong&gt;, (1919), p. 237. The term began to become common in the 1930s, although in these early days it was nothing like as common as &lt;em&gt;null set&lt;/em&gt;.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;NULL CLASS&lt;/strong&gt; is found in Bertrand Russell’s &lt;em&gt;Principles of Mathematics&lt;/em&gt; (1903). G. Peano’s &quot;Dizionario di Matematica,&quot; &lt;em&gt;Revue de mathématiques&lt;/em&gt;, &lt;strong&gt;7&lt;/strong&gt;, (1900-1), 160-172 has an entry under &lt;em&gt;Nulla:&lt;/em&gt; &quot;La classe nulle, classe non contenente individui ...&quot;&lt;br&gt;&#xA;  see NULL SET and EMPTY SET&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;NULL SET&lt;/strong&gt;. Null-set appears in 1906 in &lt;em&gt;Theory of Sets and Points&lt;/em&gt; by W. H. and G. C. Young &lt;em&gt;[OED]&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-04-29T11:46:33.120" CommentCount="3" ContentLicense="CC BY-SA 3.0" />
  <row Id="7311" PostTypeId="2" ParentId="5440" CreationDate="2018-04-29T13:18:49.400" Score="3" Body="&lt;p&gt;It may be perhaps worth to mention that the quote is strongly reminiscent of a passage in Aristotle as well. In the &lt;em&gt;Nicomachean Ethics&lt;/em&gt; (Bekker numbering 1096a), we read (transl. Roger Crisp):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;ἀμφοῖν γὰρ ὄντοιν φίλοιν ὅσιον προτιμᾶν τὴν ἀλήθειαν.&lt;br&gt;&lt;br&gt;&#xA;For one might love both, but it is nevertheless&#xA;a sacred duty to prefer the truth to one's friends.&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since a similar quote is ascribed to Plato, this may explain why Newton sees fit to reference both philosophers in his own version.&lt;/p&gt;&#xA;" OwnerUserId="2353" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-04-29T13:18:49.400" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7313" PostTypeId="2" ParentId="1824" CreationDate="2018-04-30T21:51:58.767" Score="0" Body="&lt;p&gt;The following screenshots (linked in a prior &lt;a href=&quot;https://hsm.stackexchange.com/a/1832&quot;&gt;answer&lt;/a&gt;) are taken directly from Volume 1 of &quot;The Collected Papers of Albert Einstein&quot; and are all the courses he was required to take during his undergraduate years at the Polytechnic of Zurich. The grades are also shown. As we can see, Einstein did not know about differential geometry, or to be more precise, about &lt;strong&gt;absolute differential calculus&lt;/strong&gt; which was the theory of tensors developed by Ricci Curbastro, Levi Civita and others in the decades before. He will learn these topics in a second time, while working on General Relativity. He will have many correspondences with Tullio Levi-Civita, who helped him with the math as well as his good friend mathematician Grossmann.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gPeh3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gPeh3.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;a href=&quot;https://i.stack.imgur.com/oRPnu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oRPnu.jpg&quot; alt=&quot;][1]&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/3nyM4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3nyM4.jpg&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/oSotL.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oSotL.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7369" LastEditorUserId="-1" LastEditDate="2018-05-01T07:48:23.843" LastActivityDate="2018-05-01T07:48:23.843" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7314" PostTypeId="1" AcceptedAnswerId="7316" CreationDate="2018-04-30T21:55:04.987" Score="7" ViewCount="423" Body="&lt;p&gt;I have studied General Relativity from various textbooks already, and the subject fascinated me immensely. I was wandering if there is any textbook that deals with the chronological &quot;steps&quot; that Einstein took (from 1906 to 1916) to complete his theory. I have read some passages from &quot;The Collected Papers of Albert Einstein&quot; and found them very interesting, especially all his correspondences with Tullio Levi-Civita. What I am looking for is something along the lines of this: &lt;a href=&quot;https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/index.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/index.html&lt;/a&gt; but in much greater mathematical detail. Is there any such textbook around?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="7370" LastActivityDate="2019-11-11T05:54:11.653" Title="Textbook on the History of General Relativity" Tags="|physics|relativity-theory|resource-recommendation|book-recommendation|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7315" PostTypeId="2" ParentId="7314" CreationDate="2018-04-30T22:14:17.420" Score="4" Body="&lt;p&gt;Edmund Whittaker's A history of the theories of aether and electricity, vol. II &#xA;(Harper Brothers, NY, 1960) has a chapter on the history of general relativity with a lot of mathematical details.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-04-30T22:14:17.420" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7316" PostTypeId="2" ParentId="7314" CreationDate="2018-04-30T23:01:59.443" Score="5" Body="&lt;p&gt;To have one man’s steps distilled into &lt;em&gt;textbook&lt;/em&gt; form is a lot to ask: who teaches a course on that? The next best thing might be the proceedings and collections in &lt;a href=&quot;//www.springer.com/series/4890?detailsPage=titles&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Einstein Studies&lt;/em&gt;&lt;/a&gt; (with contributions of the same John Norton):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;   1. &lt;a href=&quot;//zbmath.org/?q=an:0846.01007&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein and the history of general relativity (1989)&lt;/a&gt;,&lt;br&gt;&#xA;   3. &lt;a href=&quot;//zbmath.org/?q=an:0846.01010&quot; rel=&quot;nofollow noreferrer&quot;&gt;Studies in the history of general relativity (1992)&lt;/a&gt;,&lt;br&gt;&#xA;   5. &lt;a href=&quot;//zbmath.org/?q=an:0846.01009&quot; rel=&quot;nofollow noreferrer&quot;&gt;The attraction of gravitation: New studies in the history of general relativity (1993)&lt;/a&gt;,&lt;br&gt;&#xA;   7. &lt;a href=&quot;//zbmath.org/?q=an:0924.00026&quot; rel=&quot;nofollow noreferrer&quot;&gt;The expanding worlds of general relativity (1999)&lt;/a&gt;,&lt;br&gt;&#xA;   9. &lt;a href=&quot;//zbmath.org/?q=an:1029.01021&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein from &quot;B'' to &quot;Z'' (2002)&lt;/a&gt;, by J. Stachel,&lt;br&gt;&#xA; 11. &lt;a href=&quot;//zbmath.org/?q=an:1087.83001&quot; rel=&quot;nofollow noreferrer&quot;&gt;The universe of general relativity (2005)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-05-05T22:32:27.870" LastActivityDate="2018-05-05T22:32:27.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7317" PostTypeId="2" ParentId="7307" CreationDate="2018-05-01T09:59:59.277" Score="7" Body="&lt;p&gt;Cantor's metaphysical views are probably largely irrelevant compared to his lasting contribution to set-theoretic foundations commonly used today.  Nevertheless there may be an interesting issue here having to do with mathematical modernity.  Many historians today associate mathematical modernity with Cantorian set theory and Hilbert's formalism.  On this view, Cantor is one of the early moderns.  There is of course a tension here between Cantor's metaphysical views, on the one hand, and an attempt to portray him as a modern.  Attitudes of this sort among historians become particularly problematic when they are combined with attempts to portray other mathematicians as &lt;em&gt;countermodern&lt;/em&gt; and notably Felix Klein.  Our assessment of the work of such historians, including Herbert Mehrtens and Jeremy Gray, is due to appear in 2017 in &lt;em&gt;Mat.Stud.&lt;/em&gt; and is available &lt;a href=&quot;https://arxiv.org/abs/1803.02193&quot; rel=&quot;nofollow noreferrer&quot;&gt;on the arxiv&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note (added in 2024): A follow-up paper appeared in &lt;em&gt;Antiquitates Math&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Heinig, P.; Katz, M.; Kuhlemann, K.; Schaefermeyer, J.P.; Sherry, D. &amp;quot;Exploring Felix Klein's contested modernism.&amp;quot; Antiquitates Mathematicae 17 (2023), 101-137. &lt;a href=&quot;https://dx.doi.org/10.14708/am.v17i1.7245&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://dx.doi.org/10.14708/am.v17i1.7245&lt;/a&gt;, &lt;a href=&quot;https://arxiv.org/abs/2402.00122&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/2402.00122&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2024-03-20T10:54:22.420" LastActivityDate="2024-03-20T10:54:22.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7318" PostTypeId="1" CreationDate="2018-04-24T11:55:24.313" Score="11" ViewCount="1341" Body="&lt;p&gt;I propose this as a companion wiki page to the one about &lt;a href=&quot;https://mathoverflow.net/questions/298600/phd-dissertations-which-contain-a-final-solution-to-an-open-problem&quot;&gt;PhD dissertations which contain a solution to an open problem&lt;/a&gt; in the style of &lt;code&gt;big-list&lt;/code&gt; questions, thinking in terms of &lt;a href=&quot;https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf&quot; rel=&quot;noreferrer&quot;&gt;the well-known paradigm&lt;/a&gt; that splits mathematical research into &lt;em&gt;problem solving&lt;/em&gt; and &lt;em&gt;theory building&lt;/em&gt;. Theories are at times developed to solve famous open problems, but sometimes the concrete problems they solve are quickly dwarfed by the possibilities that a new theory opens.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you name modern mathematicians who already in their PhD theses (or earlier in their career) developed a substantial new theory or laid the foundations of a new field of research?&lt;/p&gt;&#xA;" OwnerUserId="7364" OwnerDisplayName="Bence Mélykúti" LastEditorUserId="4085" LastEditDate="2018-05-05T15:08:50.280" LastActivityDate="2020-02-11T13:01:46.397" Title="Mathematics PhD dissertations that opened a new field of research" Tags="|mathematics|mathematicians|philosophy-of-science|big-list|" AnswerCount="5" CommentCount="7" ContentLicense="CC BY-SA 3.0" />
  <row Id="7319" PostTypeId="2" ParentId="7318" CreationDate="2018-04-24T11:55:55.743" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Forbes_Nash_Jr.#Game_theory&quot; rel=&quot;noreferrer&quot;&gt;John Forbes Nash Jr.&lt;/a&gt; got a Nobel Prize for his.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nash earned a Ph.D. degree in 1950 with a 28-page dissertation on non-cooperative games.&lt;/p&gt;&#xA;&lt;p&gt;The thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash equilibrium, a crucial concept in non-cooperative games. It won Nash the Nobel Memorial Prize in Economic Sciences in 1994.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7364" OwnerDisplayName="Bence Mélykúti" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-04-24T11:55:55.743" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7320" PostTypeId="2" ParentId="7318" CreationDate="2018-04-24T12:33:30.180" Score="3" Body="&lt;p&gt;I’ll pick Philippe Delsarte’s 1973 thesis “An algebraic approach to the association schemes of coding theory” which basically expressed classical extremal problems in designs and codes as algebraic questions involving eigenspaces of related association schemes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a link to a talk on what is now known as “&lt;a href=&quot;http://www.math.is.tohoku.ac.jp/~combin/ac2014/docs/tanaka.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Delsarte Theory&lt;/a&gt;”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe not up to the Nash standard, but pretty good for a PhD!&lt;/p&gt;&#xA;" OwnerDisplayName="Gordon Royle" LastActivityDate="2018-04-24T12:33:30.180" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7321" PostTypeId="2" ParentId="7318" CreationDate="2018-04-24T15:13:49.063" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_G._Kurtz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas G. Kurtz&lt;/a&gt;' PhD &lt;a href=&quot;https://searchworks.stanford.edu/view/2189942&quot; rel=&quot;nofollow noreferrer&quot;&gt;dissertation&lt;/a&gt; at Stanford University (1967) was titled &lt;em&gt;Convergence of operator semigroups with applications to Markov processes&lt;/em&gt;. He went on to write with his PhD student Stewart N. Ethier the book &lt;em&gt;Markov Processes: Characterization and Convergence&lt;/em&gt; (John Wiley &amp;amp; Sons Inc., 1986), which is &quot;the standard reference for the advanced theory of Markov processes&quot;. Its first chapter is &lt;em&gt;Operator semigroups&lt;/em&gt;. He made a stellar research career on these foundations. Much of the modern theory of stochastic processes is variations of what he pioneered: &quot;establishing the convergence of Markov processes and characterising the limiting process&quot; (quotes from the Wikipedia page). I do not claim that he single-handedly created the area, but his contribution is immense.&lt;/p&gt;&#xA;" OwnerUserId="7364" OwnerDisplayName="Bence Mélykúti" LastActivityDate="2018-04-24T15:13:49.063" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7323" PostTypeId="1" AcceptedAnswerId="7324" CreationDate="2018-05-01T16:10:15.793" Score="8" ViewCount="400" Body="&lt;p&gt;Today it is well known that a regular heptagon cannot be constructed with straightedge and compass, since it would require to solve an equation of third degree which is not possible with the standard Euclidean tools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, a marked ruler or a compass which can carry a distance to a different point would allow us to solve such an equation; and Greeks exploited such tools to trisect an angle. Is there any reference of such an use to the problem of constructing an heptagon?&lt;/p&gt;&#xA;" OwnerUserId="81" LastEditorUserId="55" LastEditDate="2018-05-01T21:19:47.207" LastActivityDate="2018-05-01T21:27:21.570" Title="Was a regular heptagon ever constructed by ancient Greeks?" Tags="|ancient-greece|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 3.0" />
  <row Id="7324" PostTypeId="2" ParentId="7323" CreationDate="2018-05-01T21:14:44.373" Score="7" Body="&lt;p&gt;We have Archimedes's construction of regular heptagon due to Arab transcription by Thabit Ibn Qurra, see &lt;a href=&quot;http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/Regular%20Heptagon/ArchHept.html&quot; rel=&quot;noreferrer&quot;&gt;Mendell's translation&lt;/a&gt;. Aaboe gives modernized exposition and commentary in &lt;a href=&quot;https://books.google.com/books/about/Episodes_from_the_Early_History_of_Mathe.html?id=5wGzF0wPFYgC&quot; rel=&quot;noreferrer&quot;&gt;Episodes from the Early History of Mathematics&lt;/a&gt;, &lt;a href=&quot;http://mathworld.wolfram.com/Heptagon.html&quot; rel=&quot;noreferrer&quot;&gt;MathWorld's Heptagon&lt;/a&gt; has a shorter version along with modern variants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The construction uses neusis of sorts, i.e. marked straightedge as in the trisection of the angle, but the condition for marking the point is unusual, it requires two areas to become equal (in typical neusis one verges a line until it is tangent to something or some segments are in a given ratio). Neither Archimedes nor Thabit indicate how it is supposed to be met. According to Aaboe, this type of neusis is unique in all of extant Greek corpus, but he points out that one could determine when the areas are equal by intersecting two conic sections. The remaining straightedge and compass construction is itself much more intricate and relies on a &quot;miraculous&quot; coincidence at a key point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a broader perspective see &lt;a href=&quot;https://www.jstor.org/stable/41133724&quot; rel=&quot;noreferrer&quot;&gt;Hogendijk's Greek and Arabic Constructions of the Regular Heptagon&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-01T21:27:21.570" LastActivityDate="2018-05-01T21:27:21.570" CommentCount="0" ContentLicense="CC BY-SA 3.0" />
  <row Id="7325" PostTypeId="2" ParentId="7314" CreationDate="2018-05-02T00:17:26.253" Score="5" Body="&lt;p&gt;The closest I can think of is &lt;a href=&quot;https://books.google.com/books/about/Subtle_is_the_Lord_The_Science_and_the_L.html?id=KdaiZDaB7TgC&quot; rel=&quot;noreferrer&quot;&gt;Pais's Subtle is the Lord&lt;/a&gt;, although it is not a history textbook but rather Einstein's scientific biography, which literally retraces his steps based on the author's own interviews, among other things. He does also go into some mathematical details unlike the usual &quot;literary&quot; biographies. On  differential geometry and the role of Marcel Grossman in leading Einstein to adopt Ricci and Levi-Civita's calculus specifically see &lt;a href=&quot;https://hsm.stackexchange.com/a/5195/55&quot;&gt;How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-05-02T00:17:26.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7326" PostTypeId="1" CreationDate="2018-05-02T04:38:51.653" Score="5" ViewCount="218" Body="&lt;p&gt;The first thing a contemporary student of physics learns is the measurement error. As far as I understand, the idea of imprecision was totally foreign to natural philosophers at least until the end of 18th century. In 20th century the notion of experimental error became standard, and on top of that, a source of profound ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it known when, and by whom, and in which paper, an error margin of an observation was first published and discussed?&lt;/p&gt;&#xA;" OwnerUserId="1779" LastEditorUserId="55" LastEditDate="2018-05-02T22:05:41.060" LastActivityDate="2018-05-03T17:49:31.910" Title="How was the idea of observation error introduced?" Tags="|experimental-physics|scientific-method|statistics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7327" PostTypeId="2" ParentId="5631" CreationDate="2018-05-02T08:26:34.357" Score="2" Body="&lt;p&gt;Lous Mordell's paper &lt;em&gt;Indeterminate equations of the 3rd and 4th degrees&lt;/em&gt; was &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Extras/Mordell_reminiscences.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;rejected by the London Mathematical Society&lt;/a&gt;. Mordell wrote later that this paper “marks the greatest advance in the theory of indeterminate equations of the 3&lt;sup&gt;rd&lt;/sup&gt; and 4&lt;sup&gt;th&lt;/sup&gt; degrees since the time of Fermat; and it is all the more remarkable that it can be proved by quite elementary methods&amp;hellip; We trust that the author may be pardoned for speaking thus of his results. But the history of this paper has shown him that in his estimation, it has not been properly appreciated by English mathematicians.”&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-05-02T08:26:34.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7328" PostTypeId="2" ParentId="7326" CreationDate="2018-05-02T09:14:03.123" Score="2" Body="&lt;p&gt;In &lt;em&gt;The Oxford Guide to the History of Physics and Astronomy&lt;/em&gt;, under the heading &lt;em&gt;Error and the personal equation&lt;/em&gt;, Kathryn Olesko writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Since Greek times astronomers have recognized that observations were afflicted by errors, that results based on them might be only approximate, and that the quality of data varied. Astronomers in early modern Europe took the first steps toward giving reliable estimates of those errors. Johannes Kepler, who used Tycho Brahe's observations to derive the elliptical shape of planetary orbits, was probably the first to construct a correction term that assigned a magnitude to error, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That would seem to point to the answer being Kepler's &lt;em&gt;Astronomia nova&lt;/em&gt; (1609).&lt;/p&gt;&#xA;" OwnerUserId="104" LastActivityDate="2018-05-02T09:14:03.123" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7329" PostTypeId="2" ParentId="7326" CreationDate="2018-05-02T13:24:10.883" Score="1" Body="&lt;p&gt;First of all, the idea of experimental error was not &quot;introduced&quot; but was forced on scientists by the nature of experiments themselves. As soon as you start measuring something, you immediately see that the results differ. If you measure something several times and obtain slightly different results, it is natural to take the average, which scientists did routinely. Now, until 17th century, almost the only exact sciences (which involved precise measurements) were astronomy and geography/geodesy. (I am aware of only one description of a physical experiment &#xA;with measurements from antiquity, and it is very likely that this was really a &quot;thought experiment&quot;, that is it was never actually performed and the numbers were computed theoretically. I mean Ptolemy's table of refraction.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another matter is a mathematical theory of errors. It was developed in the end of 18th century independently by Legendre and Gauss (the method of least squares). Legendre was the first, but this theory is usually attributed to Gauss. Their main motivation was astronomy and geodesy. But the method quickly spread to other sciences.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-05-02T13:29:14.727" LastActivityDate="2018-05-02T13:29:14.727" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7330" PostTypeId="2" ParentId="7318" CreationDate="2018-05-02T15:55:31.173" Score="8" Body="&lt;p&gt;There are many examples. Here are a few that come to mind:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simon Donaldson's thesis &lt;a href=&quot;https://ora.ox.ac.uk/objects/uuid:4a728d07-8f2e-4c73-a756-90cd5972d4b1&quot; rel=&quot;noreferrer&quot;&gt;The Yang-Mills equations on Kahler manifolds&lt;/a&gt;&#xA;contains the first major steps in his work on the differential topology of four manifolds. The following paraphrases its abstract. He gave a new proof of a theorem of Narasimhan and Seshadri characterizing those holomorphic bundles over a projective curve that admit a flat connection and used it to prove the simplest interesting case of the conjecture of Hitchin and Kobayashi. He studied the moduli space of self-dual connections on a simply-connected four manifold and used it to deduce obstructions to the realization of a matrix as the intersection pairing on the second cohomology of such a manifold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;John Tate's thesis is another well known example, although I'm not competent even to summarize it. It has its own &lt;a href=&quot;https://en.wikipedia.org/wiki/Tate%27s_thesis&quot; rel=&quot;noreferrer&quot;&gt;wikipedia page&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mikio Sato's doctoral thesis (based on some already published work) introduced the theory of hyperfunctions as boundary values of holomorphic functions. See this &lt;a href=&quot;https://webusers.imj-prg.fr/~pierre.schapira/mispapers/Sato.pdf&quot; rel=&quot;noreferrer&quot;&gt;survey&lt;/a&gt; by P. Schapira and this &lt;a href=&quot;http://www.ams.org/notices/200702/fea-sato-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;interview&lt;/a&gt; with Sato. (Nothing about Sato's education is standard.)&lt;/p&gt;&#xA;" OwnerUserId="633" LastActivityDate="2018-05-02T15:55:31.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7331" PostTypeId="2" ParentId="7326" CreationDate="2018-05-02T21:53:59.567" Score="6" Body="&lt;p&gt;Astronomers had to deal with experimental errors to parametrize their geometric models at least as early as Hipparchus, and possibly earlier. There are some techniques and ad hoc methods that can be seen in hindsight as dealing with them in Ptolemy's Almagest, he discusses interpolation, for example. Ptolemy's &quot;massaging&quot; of Hipparchus's data even became a point of controversy recently,  he apparently passed certain interpolations for observed data, with accusations of fraud and plagiarism advanced by Newton and others, see &lt;a href=&quot;https://hsm.stackexchange.com/a/3079/55&quot;&gt;When did plagiarism become a major misconduct in academia?&lt;/a&gt; Here is a more charitable Gingerich's description in &lt;a href=&quot;https://www.journals.uchicago.edu/doi/10.1086/343243&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Trouble with Ptolemy&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Ptolemy clearly understood the geometry and realized that by stretching the period of greatest elongation he could get the relative placements of Venus and the earth (for him, the sun) much closer to the ideal positions he needed. Such approximations are characteristic of our most insightful scientists, who see them as a way to tackle otherwise intractable problems... It is clear that he deliberately moved away from the exact time of the greatest elongations in order to get the specific geometry he required... As Ptolemy wrestled with errors of measurement without any error theory, he was repeatedly forced into compromises to reconcile discordant observations.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In Optics V.2 Ptolemy proposes a series of experiments to substantiate his claim that &quot;&lt;em&gt;the angles [of refraction] do bear a certain consistent quantitative relation to one another with respect to the normals&lt;/em&gt;&quot;, and presents tabulations of relevant data. However, according to &lt;a href=&quot;https://www.jstor.org/stable/pdf/3185879.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smith's Ptolemy and the Foundations of Ancient Mathematical Optics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;There is, in fact, a specific mathematical law implicit in Ptolemy's tabulations, but its proper formulation in algebraic terms would have been beyond Ptolemy given the limitations of mathematical notation in his day.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Be it as it may, Ptolemy's Optics inspired a tradition, Islamic authors wrote elaborations that included new experimental data, e.g. Ibn al-Haytham's Book of Optics (1021) and al-Farisi's Optics (c. 1320). However, the first person to thematize experimental errors explicitly might be Ibn al-Haytham's contemporary &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_scientific_method#Al-Biruni&quot; rel=&quot;nofollow noreferrer&quot;&gt;al-Biruni&lt;/a&gt;. His interests included minerology, mechanics and even what we would call sociology. He talked of &quot;&lt;em&gt;errors caused by the use of small instruments and errors made by human observers&lt;/em&gt;&quot;, and of analysis (qualitative) of multiple observations to arrive at a &quot;&lt;em&gt;common-sense single value for the constant sought&lt;/em&gt;&quot;, to get a &quot;&lt;em&gt;reliable estimate&lt;/em&gt;&quot;, even suggesting the  arithmetic mean. This is similar, and more specific, than Bacon's later four idols of the mind. Rozhanskaya and Levinova write in Statics (see Rashed edited &lt;a href=&quot;https://archive.org/details/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia Of The History Of Arabic Science&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The phenomena of statics were studied by using the dynamic approach so that two trends – statics and dynamics – turned out to be inter-related within a single science, mechanics... Numerous fine experimental methods were developed for determining the specific weight, which were based, in particular, on the theory of balances and weighing. The classical works of al-Biruni and al-Khazini can by right be considered as the beginning of the application of experimental methods in medieval science.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Needless to say, experimentation and dealing with errors became widespread in 17th century Europe, and even earlier Copernicus, Tycho Brahe and Kepler were conscious of astronomical observation errors. But theory had to wait for some development of probability and statistics. Quantitative theory of observation errors only appears in Simpson's memoir of 1755, which discussed several possible error distributions, including uniform and triangular distribution. For the further story see &lt;a href=&quot;https://hsm.stackexchange.com/a/7009/55&quot;&gt;When did statistics become an integral part of physics?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-03T17:49:31.910" LastActivityDate="2018-05-03T17:49:31.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7332" PostTypeId="1" AcceptedAnswerId="7333" CreationDate="2018-05-04T02:27:57.873" Score="3" ViewCount="278" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The best result of mathematics is to be able to do without it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The above is a quote by Oliver Heaviside, an electrical engineer and mathematician. What does the quote really mean? &lt;/p&gt;&#xA;" OwnerUserId="7378" LastEditorUserId="4251" LastEditDate="2020-06-06T13:02:49.803" LastActivityDate="2020-06-06T13:02:49.803" Title="A peculiar quote from Oliver Heaviside" Tags="|physics|theoretical-physics|mathematical-physics|quote|heaviside|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7333" PostTypeId="2" ParentId="7332" CreationDate="2018-05-04T12:40:34.913" Score="5" Body="&lt;p&gt;Since I have no shame, I screen-shotted the reference that J.G.  found:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XK4ju.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XK4ju.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/lbrvm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lbrvm.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-05-04T12:40:34.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7334" PostTypeId="1" CreationDate="2018-05-04T13:35:27.260" Score="1" ViewCount="80" Body="&lt;p&gt;How actually people invented the quadratic equation? What practical problem did they solve?&#xA;There are some info about this exist in internet, but it is very abstract - I cant find the concrete task which they try to solve. Also I can`t find the way of they thought during solve of that concrete problem.&lt;/p&gt;&#xA;" OwnerUserId="7380" LastActivityDate="2018-05-04T13:35:27.260" Title="How was invented quadratic equation?" Tags="|elementary-algebra|algebra|" AnswerCount="0" CommentCount="5" ClosedDate="2018-05-06T13:52:09.680" ContentLicense="CC BY-SA 4.0" />
  <row Id="7336" PostTypeId="2" ParentId="7314" CreationDate="2018-05-05T16:15:00.200" Score="2" Body="&lt;p&gt;Although not a text book, a very enjoyable account that covers the history of general relativity from Einstein (and before) to the current research and understanding is &quot;The Perfect Theory: A Century of Geniuses and the Battle over General Relativity&quot; by Pedro G. Ferreira&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0544483863&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2018-05-05T16:15:00.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7337" PostTypeId="1" AcceptedAnswerId="7338" CreationDate="2018-05-05T18:11:20.967" Score="1" ViewCount="578" Body="&lt;p&gt;Who first understood that celestial bodies (stars, planets, and moons) are moving in a 3D space and not points of light moving on Earth's dome? Also, when and how did they realize that celestial bodies have actual masses just like the Earth?&lt;/p&gt;&#xA;" OwnerUserId="7384" LastEditorUserId="140" LastEditDate="2018-05-11T21:23:02.373" LastActivityDate="2018-05-24T20:44:53.717" Title="Who was the first to understand the concept of outer-space?" Tags="|astronomy|celestial-mechanics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7338" PostTypeId="2" ParentId="7337" CreationDate="2018-05-06T04:26:59.183" Score="1" Body="&lt;p&gt;As it almost always happens it is impossible to tell exactly &quot;who was the first&quot;. But Hellenistic scientists certainly knew this. The moon has visible parallax, so that they could prove that Moon is much closer than the Sun and planets. According to Ptolemy, Hipparchus knew about Moon's parallax. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They supposed (correctly but without proof) that stars are much more remote than Sun and planets. The standard model in the medieval times was Moon, then inner planets, then Sun, then outer planets and fixed stars. This order is essentially correct, and it is present in the works of Ptolemy (our principal source for Greek astronomy). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But how it evolved in the ancient Greece and Hellenistic states, we do not know because few other astronomical texts survive. Neither the books of astronomers themselves nor the Hellenistic books on history of astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We know that from the very beginning of the Greek astronomy, they tried to build&#xA;space models for planet motions (Eudoxus). Nothing like this was present in the&#xA;earlier, Babylonian astronomy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, they did not know the exact ratios of distances, and they knew nothing about fixed stars, except their positions on the celestial sphere. So they attached the fixed stars to one large sphere, but they had no way to check any conjectures about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The speculations that stars are actually like the Sun, and that they are on different distances started in 17th century but no proof could be given until  1838 when the stellar parallax was successfully measured by Bessel.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-05-06T04:26:59.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7339" PostTypeId="1" AcceptedAnswerId="7340" CreationDate="2018-05-06T13:33:59.957" Score="3" ViewCount="502" Body="&lt;p&gt;I am reading the &lt;a href=&quot;http://pubs.rsc.org/en/Content/ArticleLanding/1870/JS/JS8702300074#!divAbstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;classic paper&lt;/a&gt; by Thomas Andrews, in which he discovered the critical point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The gas he used in his experiments is called by him 'carbonic acid'. By its critical temperature being 31.1 Celsius, I infer that it is actually 'carbon dioxide'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do I get it right? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2018-05-06T17:33:14.797" Title="Was carbon dioxide also called carbonic acid historically?" Tags="|physics|chemistry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7340" PostTypeId="2" ParentId="7339" CreationDate="2018-05-06T17:33:14.797" Score="5" Body="&lt;p&gt;Yes.  The Oxford English Dictionary gives instances of the term &quot;Carbonic acid&quot; meaning CO${}_2$ as old as about 1800.  By 1900, the term &quot;carbonic acid gas&quot; was in common use for CO${}_2$ (my father called it that, 1950-2000, for instance).  More recently it turns out that there is a gaseous phase of H${}_2$CO${}_3$, so &quot;carbonic acid gas&quot; has become an ambiguous term.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2018-05-06T17:33:14.797" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7341" PostTypeId="1" AcceptedAnswerId="7488" CreationDate="2018-05-06T16:55:48.870" Score="12" ViewCount="688" Body="&lt;p&gt;Who assigned the term &lt;a href=&quot;https://en.wikipedia.org/wiki/Born_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;''Born's rule''&lt;/a&gt; to the statement that the measurement of a quantum observable is one of its eigenvalues, with a probability given by the square of the coefficient in the spectral expansion of the system state?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I checked the quantum mechanics books by von Neumann and by Landau and Lifschitz, and didn't find this term. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: In view of the answers and their discussion below, my precise question is when Born's name was first linked (as Born's rule or something similar) to an explicit statement about measuring &lt;strong&gt;arbitrary&lt;/strong&gt; observables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit (March 13,2019): Some of the early history of Born's rule (as I now see it) can now be found in Subsections 3.1 and 3.2 of my paper in &lt;a href=&quot;http://arxiv.org/abs/1902.10778&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv:1902.10778&lt;/a&gt;. I am grateful for the help of Francois Ziegler, who had answered below, in the early stages of the literature search.&lt;/p&gt;&#xA;" OwnerUserId="3915" OwnerDisplayName="Arnold Neumaier" LastEditorUserId="3915" LastEditDate="2019-03-13T14:09:36.180" LastActivityDate="2019-03-13T14:09:36.180" Title="Who coined the term ''Born's rule''?" Tags="|quantum-mechanics|terminology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7342" PostTypeId="2" ParentId="1816" CreationDate="2018-05-07T13:09:47.780" Score="3" Body="&lt;p&gt;Actually Carlo Rovelli considers the situation ignoring any viscosity/friction effects and points out that a body still falls slower in a denser medium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider a mass of specific gravity 2 falling through water. The net gravitation force is mg/2 but the mass is still m, hence the acceleration is halved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the specific gravity of air is 0.0013 so that effect is negligible in air. Aristotle thought that in a zero dense medium the mass would fall infinitely fast and he was obviously wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;if a thing moves through the thickest medium such and such distance in such and such time, it moves through the void with a speed beyond any ratio&quot;. (Physics 215b)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;People who think Aristotle's treatment was purely qualitative should try reading him. Look at Physics 215b and 216a in particular. There's plenty of calculation there. It seems to me that most of his problems arise from not understanding the concept of zero - hardly surprising since it wasn't introduced into maths for another thousand years (by the Indians). Maybe his horror of the void comes down to the same issue.&lt;/p&gt;&#xA;" OwnerUserId="7391" LastActivityDate="2018-05-07T13:09:47.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7343" PostTypeId="1" AcceptedAnswerId="7346" CreationDate="2018-05-09T12:22:52.147" Score="4" ViewCount="204" Body="&lt;p&gt;When, by whom and in which paper where moments of a random variable used first in probability theory?&lt;/p&gt;&#xA;" OwnerUserId="7309" LastActivityDate="2018-05-09T17:14:20.800" Title="Who introduced moments of a random variable first?" Tags="|mathematics|probability|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7344" PostTypeId="2" ParentId="7343" CreationDate="2018-05-09T12:39:57.877" Score="4" Body="&lt;p&gt;&lt;em&gt;My answer taken from the closed version of this question&lt;/em&gt; &lt;a href=&quot;https://math.stackexchange.com/q/2773382/442&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://jeff560.tripod.com/m.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathWords&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Moment&lt;/strong&gt; was taken into Statistics from Mechanics by Karl Pearson when he treated the frequency-curve (or observation curve) as the sheet enclosed by the curve and the horizontal axis. See his &quot;Asymmetrical Frequency Curves,&quot; &lt;em&gt;Nature&lt;/em&gt; October 26th 1893: &quot;Now the centre of gravity of the observation curve is found at once, also its area and its first four moments by easy calculation.&quot; [OED].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The phrase &lt;strong&gt;method of moments&lt;/strong&gt; was used in a statistics sense in the first of Karl Pearson’s &quot;Contributions to the Mathematical Theory of Evolution,&quot; (&lt;em&gt;Philosophical Transactions of the Royal Society&lt;/em&gt; A, 185, (1894), p. 75.). Pearson used the method to estimate the parameters of a mixture of normal distributions. For several years Pearson used the method on different problems but the name only gained general currency with the publication of his 1902 &lt;em&gt;Biometrika&lt;/em&gt; paper &quot;On the systematic fitting of curves to observations and measurements&quot; (David 1995). In &quot;On the Mathematical Foundations of Theoretical Statistics&quot; (&lt;em&gt;Phil. Trans. R. Soc.&lt;/em&gt; 1922), Fisher criticized the method for being inefficient compared to his own maximum likelihood method (Hald pp. 650 and 719).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Moment generating function&lt;/strong&gt;. R. A. Fisher seems to have brought this term into English in his &quot;Moments and Product Moments of Sampling Distributions.,&quot; &lt;em&gt;Proceedings of the London Mathematical Society&lt;/em&gt;, Series 2, 30, (1929), p. 238. He probably took the term from V. Romanovsky &quot;Sur Certaines Éspérances Mathématiques et sur l'Erreur Moyenenne du Coefficient de Corrélation, &lt;em&gt;Comptes Rendus&lt;/em&gt;, 180, (1925), 1897-1899. Romanovsky refers to &quot;la function génératrice des moments&quot; (p. 1898).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-05-09T12:39:57.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7346" PostTypeId="2" ParentId="7343" CreationDate="2018-05-09T16:54:55.480" Score="4" Body="&lt;p&gt;Long before Pearson, Chebyshev and his student A. A. Markov used moments to prove the Central Limit Theorem. The earliest paper of Chebyshev on this topic is dated 1887.&#xA;But I do not claim that Chebyshev &quot;was the first&quot;. On my opinion, a question of the type &quot;who was the first&quot; is meaningless and usually impossible to answer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.530.4669&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.530.4669&amp;amp;rep=rep1&amp;amp;type=pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chebyshev's original paper is called &quot;On two theorems in probability&quot;,&#xA;published in Russian In vol. 40 No 6 of the Imperial academy of sciences,&#xA;French translation in Acta Math 14, 305-315. It is freely available in French &#xA;in Chebyshev's Collected papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Chebyshev did not call them &quot;moments&quot; but used the name &quot;integral residues&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-05-09T17:14:20.800" LastActivityDate="2018-05-09T17:14:20.800" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7347" PostTypeId="1" AcceptedAnswerId="7348" CreationDate="2018-05-10T03:09:39.667" Score="10" ViewCount="544" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Indiana_Pi_Bill&quot; rel=&quot;noreferrer&quot;&gt;Wiki&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, one of the most notorious attempts to establish mathematical truth by legislative fiat.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, what are the other attempts?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Context: &lt;a href=&quot;https://matheducators.stackexchange.com/questions/13831/are-kindergartners-supposed-to-be-steered-from-squares-being-rectangles&quot;&gt;I'm meeting a member of the legislative branch next week&lt;/a&gt;, though technically this could be more &lt;a href=&quot;https://matheducators.stackexchange.com/revisions/13700/9&quot;&gt;mathematical definitions&lt;/a&gt; than mathematical truth.&lt;/p&gt;&#xA;" OwnerUserId="3338" LastEditorUserId="3338" LastEditDate="2018-05-10T03:18:46.483" LastActivityDate="2018-05-12T21:07:09.153" Title="Indiana Pi Bill: Other attempts to establish mathematical truth by legislative fiat?" Tags="|mathematics|terminology|geometry|social-context|naming-conventions|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7348" PostTypeId="2" ParentId="7347" CreationDate="2018-05-10T05:28:45.487" Score="8" Body="&lt;p&gt;It seems that this attempt made an impression, when one needs to make the point Indiana Pi itself is typically invoked. &lt;a href=&quot;http://www.nmsr.org/alabama.htm&quot; rel=&quot;noreferrer&quot;&gt;NMSR Reports&lt;/a&gt; modeled their 1998 April Fool's story on it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;NASA engineers and mathematicians in this high-tech city are stunned and infuriated after the Alabama state legislature narrowly passed a law yesterday [March 30, 1998] redefining pi, a mathematical constant used in the aerospace industry. The bill to change the value of pi to exactly three was introduced without fanfare by Leonard Lee Lawson (R, Crossville), and rapidly gained support after a letter-writing campaign by members of the Solomon Society, a traditional values group... &quot;I think that it is the mathematicians that are being irrational, and it is time for them to admit it,&quot; said Lawson. &quot;The Bible very clearly says in I Kings 7:23 that the altar font of Solomon's Temple was ten cubits across and thirty cubits in diameter, and that it was round in compass.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, upon closer examination Indiana Pi, officially  &quot;&lt;em&gt;A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897&lt;/em&gt;&quot; does not legislate (or even mention) the value of pi, it rather mandates an (erroneous) method of squaring the circle. The &quot;new mathematical truth&quot; is just a side effect. The bill was invoked again more recently in connection with the laws banning strong encryption. According to &lt;a href=&quot;https://www.michigandaily.com/section/columns/david-harris-legislating-math&quot; rel=&quot;noreferrer&quot;&gt;Michigan Daily&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;State legislatures in California and New York have introduced bills that would ban encryption that is unbreakable to law enforcement in personal devices. Taking it a step further, United Kingdom Prime Minister David Cameron is on record as wishing to ban strong encryption entirely, which elicited responses from the technical community that stated Cameron very literally “had no idea what he was proposing.” Because at its core, encryption is simply a widely available application of mathematics, such that Wikipedia co-founder Jimmy Wales directly compared that a ban on encryption would be like banning a form of mathematics itself.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On this issue Australian prime minister Malcolm Turnbull outdid even Cameron. After introducing a bill that would &quot;force technology companies to break into end-to-end encrypted messages&quot; he was asked by a reporter whether the laws of mathematics would trump the laws of Australia. &lt;a href=&quot;https://www.brookings.edu/blog/techtank/2017/07/17/no-the-laws-of-australia-dont-override-the-laws-of-mathematics&quot; rel=&quot;noreferrer&quot;&gt;According to Brookings&lt;/a&gt;, his response was:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“&lt;em&gt;Well, the laws of Australia prevail in Australia, I can assure you of that. The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Jokes aside, these bills would be harmless if they truly mandated the impossible, but that is not what they do. What they mandate is harmful but not impossible: Indiana Pi would have led to erroneous computational prescriptions, and &quot;banning end-to-end encryption&quot; means mandating so-called “back doors” to be built in into commercially available encryption systems. In the same spirit,  &lt;a href=&quot;https://en.wikipedia.org/wiki/Butler_Act&quot; rel=&quot;noreferrer&quot;&gt;Tennessee's Butler Act&lt;/a&gt; did not mandate that human evolution did not happen, but rather that it is not to be taught in state-funded schools.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And while you are at it, in the US and Canada &lt;a href=&quot;https://en.wikipedia.org/wiki/Trapezium&quot; rel=&quot;noreferrer&quot;&gt;trapezium&lt;/a&gt; is a quadrilateral with no parallel sides, while in the rest of the world, going at least as far back as Heron of Alexandria, it is a quadrilateral with at least one pair of parallel sides. Perhaps the legislator can do something about that. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-05-10T05:28:45.487" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7349" PostTypeId="1" AcceptedAnswerId="7350" CreationDate="2018-05-10T06:25:42.987" Score="3" ViewCount="282" Body="&lt;p&gt;The theorem often be called Lebesgue's MCT or Levi's theorem. Who did originally prove it or what is the contribution of Lebesgue and Levi respectively?&lt;/p&gt;&#xA;" OwnerUserId="7401" LastEditorUserId="55" LastEditDate="2018-05-10T08:30:19.913" LastActivityDate="2018-05-10T08:30:19.913" Title="Who proved the monotone convergence theorem for the Lebesgue integral?" Tags="|real-analysis|functional-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7350" PostTypeId="2" ParentId="7349" CreationDate="2018-05-10T08:29:29.310" Score="2" Body="&lt;p&gt;The original version of the dominated convergence, from which the monotone convergence trivially follows &lt;em&gt;assuming that the limit is Lebesgue integrable&lt;/em&gt;, was published by Lebesgue in &lt;a href=&quot;https://archive.org/details/leconegrarecher00leberich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leçons sur l'Intégration et la Recherche des Fonctions Primitives (1904)&lt;/a&gt;. This is a compilation of his lectures at Collège de France over the preceeding five years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://archive.org/stream/RendicontiIstitutoLombardoS2V39/Rendiconti_Istituto_Lombardo_S2V39#page/n805/mode/2up/search/775&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sopra l'Integrazione delle Serie&lt;/a&gt; (in Rendiconti - Reale Istituto lombardo di scienze e lettere, 39 (1906) 775-780) Beppo Levi quotes Lebesgue's result and notes that for positive monotone sequences one can drop the a priori integrability assumption for the limit. He then applies the result to sums of positive functional series, as the title indicates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://books.google.com/books?id=GffpCgAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kubrusly, Essentials of Measure Theory, p.63&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=Uzf0BwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chae, Lebesgue Integration, p.69&lt;/a&gt; for modern comments.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-05-10T08:29:29.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7351" PostTypeId="1" AcceptedAnswerId="7352" CreationDate="2018-05-10T17:55:05.990" Score="2" ViewCount="747" Body="&lt;p&gt;In statistical mechanics, why microcanonical, canonical and macrocanonical ensemble are called that way?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any reason according to the size of the system they can describe properly ( I don't think so) the order they are estudied or something similiar?&lt;/p&gt;&#xA;" OwnerUserId="7403" LastEditorUserId="55" LastEditDate="2018-05-11T21:40:02.020" LastActivityDate="2018-05-12T00:51:55.387" Title="Why are microcanonical, canonical and macrocanonical ensembles called that way?" Tags="|physics|terminology|statistical-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7352" PostTypeId="2" ParentId="7351" CreationDate="2018-05-11T21:39:04.090" Score="2" Body="&lt;p&gt;People often expect names and notations to have some deep meanings, or at least a story behind them, but in more cases than not all there is are some loose associations, if that, described as &quot;historical reasons&quot;, which is a polite way of saying &quot;just cuz&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gibbs introduced the names in &lt;a href=&quot;https://archive.org/details/elementaryprinci00gibbrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elementary Principles in Statistical Mechanics (1902)&lt;/a&gt;, and he did not exactly give explanations for them:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;When an ensemble of systems is distributed in phase in the manner described, i. e. when the index of probability is a linear function of the energy, we shall say that the ensemble is canonically distributed, and shall call the divisor of the energy ($\Theta$) the modulus of distribution.&lt;/em&gt;&quot; (p. 33-34)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Ensemble&quot; just means collection, of moving particles in this case. One can guess from the context that &quot;canonical&quot; might be due to the use of &quot;canonical coordinates&quot; in setting up the distribution. That name is due to Jacobi, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/5666/why-are-canonical-coordinates-canonical&quot;&gt;Why are canonical coordinates canonical?&lt;/a&gt;, who explained in &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k2964f/f61&quot; rel=&quot;nofollow noreferrer&quot;&gt;Note sur l'intégration des équations différentielles de la Dynamique(1837)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;By means of this theorem we find, through the calculation itself, elements whose differential values, in the disturbed motion, take the simple form which they have in the theorem, a form which I designate in my memory under the name of canonical.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Latin &lt;em&gt;canonicus&lt;/em&gt; means &quot;according to rule&quot;, and &quot;canon&quot; was traditionally used for ecclesiastical laws in medieval Church Latin, seems like Jacobi is talking about a &quot;head canon&quot;. Gibbs is slightly more forthcoming with the microcanonical ensemble:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Thus, the limiting distribution obtained from the part of a canonical ensemble between two limits of energy, when the difference of the limiting energies is indefinitely diminished, is independent of the modulus, being determined entirely by the energy, and is identical with the limiting distribution obtained from a uniform density between limits of energy approaching the same value. We shall call the limiting distribution at which we arrive by this process microcanonical.&lt;/em&gt;&quot; (p.115)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Micro&quot; might refer to the &quot;indefinitely diminished&quot; difference. Gibbs is explicit on the &quot;grand&quot;, although who knows why he did not go for &quot;macrocanonical&quot; instead. Maybe it sounded too close to &quot;microcanonical&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Instead of considering, as in the preceding chapters, ensembles of systems differing only in phase, we shall now suppose that the systems constituting an ensemble are composed of particles of various kinds, and that they differ not only in phase but also in the numbers of these particles which they contain. The external coordinates of all the systems in the ensemble are supposed, as heretofore, to have the same value, and when they vary, to vary together. For distinction, we may call such an ensemble a grand ensemble, and one in which the systems differ only in phase a petit ensemble. A grand ensemble is therefore composed of a multitude of petit ensembles. The ensembles which we have hitherto discussed are petit ensembles... If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distributed, we shall say that its distribution is canonical with respect to generic phases.&lt;/em&gt;&quot; (p. 189-191)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Gibbs never actually says &quot;&lt;em&gt;grand canonical ensemble&lt;/em&gt;&quot;, only &quot;&lt;em&gt;grand ensemble canonically distributed&lt;/em&gt;&quot;, and there is no real reason to why his &quot;&lt;em&gt;petit&lt;/em&gt;&quot; did not take.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-12T00:51:55.387" LastActivityDate="2018-05-12T00:51:55.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7353" PostTypeId="2" ParentId="6348" CreationDate="2018-05-12T12:08:49.593" Score="2" Body="&lt;p&gt;Fortunately there is some documentary evidence about Archimedes' discovery, in his work &quot;On Floating Bodies&quot;, which survives in manuscript forms (&lt;a href=&quot;https://en.wikipedia.org/wiki/On_Floating_Bodies&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/On_Floating_Bodies&lt;/a&gt;). From the available sources it also translated into English by Thomas L Heath in the 1890s. It is included in his 1897 commentary and set of translations 'Works of Archimedes' (&lt;a href=&quot;https://archive.org/details/worksofarchimede029517mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/worksofarchimede029517mbp&lt;/a&gt;), the translation can be read (with Heath's notes) at pages 253-300. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just as &lt;em&gt;Alexandre Eremenko&lt;/em&gt;'s earlier answer states, the treatment in the book is by 'argument which requires no calculation'.  The style is theoretical and geometrical, making no appeal to experiment.  On the other hand, that is a standard ancient Greek approach to presentation, and can not really conclude the question whether there may also have been some experimental background to the discovery.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The anecdotes about experiments that have been handed down may indeed be partly or wholly the inventions of later commentators. But there has always been an almost irresistible urge (and not an entirely unreasonable one), upon commentators from the ancient to the modern, to suppose that such a practical matter may very probably have been grounded in some practical experience. Unfortunately the evidence that has come down is too meager to allow evidence-based conclusions about what any such experiences by Archimedes may have been. The translator Heath's own ideas are contained in his notes, and each reader will judge the plausibility of Heath's and the other anecdotes and conjectures.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-05-12T12:08:49.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7354" PostTypeId="1" AcceptedAnswerId="7357" CreationDate="2018-05-12T22:57:45.800" Score="10" ViewCount="231" Body="&lt;p&gt;The concept of a moduli space goes back to Riemann's count of $3g-3$ (or $3p-3$, in older notation) coordinates to describe Riemann surfaces of genus $g$ when $g &amp;gt; 1$. See the bottom of p. 33 &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/AbelFn.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, where Riemann introduces the term Moduln and writes multiple times on later pages about Anzahl der Moduln (number of moduli/parameters/coordinates).&#xA;This was in 1857. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 19th century gave rise to other similar-sounding terms: modular functions, modular forms, and the modular group $\text{SL}_2(\mathbf Z)$. As far as I am aware, these terms were introduced in work of people like Dedekind, Fricke, and Klein in the 1870s-1890s, and is related to Dedekind's creation of the term Modul for a lattice (in Euclidean space). Lattices in $\mathbf C$, after suitable scaling normalization to the form $\mathbf Z + \mathbf Z\tau$, are closely related to $\text{SL}_2(\mathbf Z)$, and modular functions and modular forms are related to the action of this group on the upper half-plane (and modular forms can be thought of as certain homogeneous functions on the space of lattices). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question: is there any reason (through inspiration perhaps, if not for precise technical reasons) to expect Dedekind's term was inspired specifically by Riemann's term?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is an attempt to get some clarification on my answer to the MathOverflow question &lt;a href=&quot;https://mathoverflow.net/questions/300013/the-origin-of-modular-and-moduli&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2020-11-05T23:36:42.787" Title="Riemann's moduli and Dedekind's modules: any connection?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7355" PostTypeId="1" AcceptedAnswerId="7356" CreationDate="2018-05-14T18:53:40.310" Score="9" ViewCount="418" Body="&lt;p&gt;In &lt;em&gt;The Man Who Loved Only Numbers&lt;/em&gt;, a biography about Paul Erdős, by Paul Hoffman, the author claims that Paul Cohen was &quot;Gödel's former assistant&quot; (p 225). However, I can't find any other sources corroborating the claim, and while I don't know much about either of them, I was under the impression that Cohen's main work before his work on the Continuum Hypothesis was in analysis, and that his achievements in logic were especially impressive because he didn't have much prior experience in the field. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to wikipedia, Cohen was &quot;a fellow at the Institute for Advanced Study at Princeton [from 1959-61]&quot;, during which time Gödel was also there, so it's conceivable that they could have met, but it still seems surprising that he would have taken an analyst as his assistant. Are there any other sources that make this claim?&lt;/p&gt;&#xA;" OwnerUserId="7416" LastEditorUserId="4251" LastEditDate="2023-12-10T16:48:11.623" LastActivityDate="2023-12-10T16:48:11.623" Title="Was Paul Cohen a student or assistant of Gödel?" Tags="|mathematicians|biographical-details|mathematical-logic|set-theory|goedel|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7356" PostTypeId="2" ParentId="7355" CreationDate="2018-05-14T20:54:25.140" Score="13" Body="&lt;p&gt;No, he was not. Cohen wrote his own account of the history of forcing, &lt;a href=&quot;https://projecteuclid.org/euclid.rmjm/1181070010&quot; rel=&quot;noreferrer&quot;&gt;The Discovery of Forcing&lt;/a&gt; (Rocky Mountain J. Math. 32 (4) (2002), 1071-1100), where he addresses his relationship with Gödel in detail. In particular, he calls him &quot;&lt;em&gt;one of my heroes&lt;/em&gt;&quot; (along with Skolem), and mentions some conversations &quot;&lt;em&gt;years later, after my own proof&lt;/em&gt;&quot;. Nonetheless, it ends with the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Finally a personal remark. I cannot say that I was a friend of Kurt Gödel. We met relatively few times and there was a gulf of age and background that I found difficult to bridge. Yet, my meetings with him were charged with an emotion that was intense, yet difficult to describe. We each traversed journeys that had much in common. I would like to dedicate this talk to his memory.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Cohen was self-taught as a logician. His interest in independence came from discussions on foundations of mathematics with Feferman and Royden at Stanford in 1962. At some point he came to read &quot;Gödel's monograph&quot; &lt;a href=&quot;https://press.princeton.edu/titles/1034.html&quot; rel=&quot;noreferrer&quot;&gt;The Consistency of the Continuum Hypothesis&lt;/a&gt;, actually a compilation of lecture notes for a course given at the Institute for Advanced Study in 1938-39 (Cohen was there in 1959-61). He was even unaware of Shepherdson's follow-up work which showed that inner models could not prove the consistency of the negation of AC or CH, something he rediscovered in 1962.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-14T21:39:31.797" LastActivityDate="2018-05-14T21:39:31.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7357" PostTypeId="2" ParentId="7354" CreationDate="2018-05-14T21:38:20.163" Score="3" Body="&lt;p&gt;Dedekind explained his choice of &amp;quot;&lt;em&gt;modul&lt;/em&gt;&amp;quot; in a footnote with reference to Gauss's congruences, not to Riemann's work, see &lt;a href=&quot;https://books.google.com/books?id=8MK_AwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Episodes in the History of Modern Algebra (1800-1950), p. 79&lt;/a&gt;. In 1863 Dedekind published posthumously Dirichlet's Lectures on Number Theory, and Dirichlet defined for &lt;span class=&quot;math-container&quot;&gt;$a=sk+r$&lt;/span&gt;, , see &lt;a href=&quot;https://arxiv.org/abs/1211.6057&quot; rel=&quot;nofollow noreferrer&quot;&gt;Siebeneicher's Residues: The gateway to higher arithmetic&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In what follows we shall say that &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the remainder of the number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; relative to the &lt;strong&gt;modulus&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dedekind's point apparently was that the congruence relation is determined by a set, a subset of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;, just as his lattice &lt;em&gt;moduls&lt;/em&gt; are subsets (additive subgroups, in modern terms) of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Riemann's work on complex surfaces was clearly an inspiration for many algebraists including Kronecker and Dedekind-Weber. Dieudonné in &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.1972.11993142&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historical Development of Algebraic Geometry&lt;/a&gt; (American Mathematical Monthly, 79 (1972) no. 8, 827-866) writes, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The goal of Dedekind and Weber in their fundamental paper was quite different and much more limited; namely, they gave purely algebraic proofs for all the algebraic results of Riemann.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;fundamental paper&amp;quot; is Theory of Algebraic Functions of One Variable, and 1882 is after Dedekind already introduced his &lt;em&gt;modul&lt;/em&gt; and published Dirichlet's lectures with an independent use of &amp;quot;modulus&amp;quot;. If anything, Riemann's use of &amp;quot;modulus&amp;quot; might have been &amp;quot;inspired&amp;quot; by Dirichlet's, but in a purely linguistic sense, the word was around and its Latin root fit.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-05T23:36:42.787" LastActivityDate="2020-11-05T23:36:42.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7359" PostTypeId="2" ParentId="7341" CreationDate="2018-05-14T23:37:15.337" Score="3" Body="&lt;p&gt;Some general remarks first. The answer to the title question is likely &quot;nobody&quot;, &quot;coining&quot; often happens without a &quot;coiner&quot;. Born's authorship of the rule was acknowledged at least since von Neumann's book. Some might have used &quot;Born's rule&quot; in conversation or in print early meaning by it no more than a rule due to Born. After it was used this way long enough others thought of it as a &quot;term&quot;, but to them it was already &quot;coined&quot;. This is a typical way expressions spread, not on the authority of the original user (&quot;coiner&quot;) but because they seem handy to many, independently so. It makes no difference who happens to be first because there is often no causal link between them and subsequent users, and the original use is certainly not the reason for the eventual adoption, even if some later users are aware of it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an early occurence in Studia Philosophica, 4 (1949) p. 192, which seems to confirm it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This law is linked to the so-called statistical interpretation of quantum mechanics given for the first time for certain particular cases by Born, and afterwards generalized so as to apply to other cases. For brevity's sake we will call it Born's rule.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It comes up as the only match in &lt;a href=&quot;https://www.google.com/search?lr=lang_en&amp;amp;tbo=p&amp;amp;tbm=bks&amp;amp;q=%22born%27s+rule%22&amp;amp;tbs=,cdr:1,cd_min:Jan+1_2+1940,cd_max:Dec+31_2+1955&amp;amp;num=10&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google's advanced book search from 1940 to 1955&lt;/a&gt;, all other citations fed into Google's Ngram are spurious. However, it is possible that people used it orally and in papers before 1949, which book search may not find, although the context indicates that philosophers did not get it from there. MathSciNet and APS searches do not return anything promising either.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-05-14T23:37:15.337" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7360" PostTypeId="1" CreationDate="2018-05-15T13:33:24.677" Score="-1" ViewCount="1276" Body="&lt;p&gt;From &quot;Rhind Papyrus&quot; from 1600 BC we know that the Egyptians had an estimate for $\pi$, namely 3.16, meaning they knew only 2 digits of $\pi$. According to this article they knew more digits, at least 4 digits of $\pi$. Around 200 BC Archimedes estimated pi to 22/7 which is 3 digits of $\pi$. This indicates that the Egyptians knew more digits 2000 years before Archimedes, however, it's not clear to me how many digits they actually knew.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.bcamt.ca/wp-content/uploads/2015/03/Yochim.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.bcamt.ca/wp-content/uploads/2015/03/Yochim.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7419" LastEditorUserId="26" LastEditDate="2018-05-22T21:43:47.387" LastActivityDate="2018-10-08T03:00:32.170" Title="How many digits of Pi did the old Egyptians know?" Tags="|ancient-egypt|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7361" PostTypeId="2" ParentId="7360" CreationDate="2018-05-15T15:20:17.687" Score="2" Body="&lt;p&gt;The article does not say what you think it does. The article hypothesizes that the ancient Egyptians used a trundle wheel to accurately measure distance. An estimate of pi is not needed to make an accurate trundle wheel. What's needed is an accurate measure of the circumference. There's no reason to measure the diameter, which is harder to measure accurately than is the circumference. Furthermore the circumference will not be given by $\pi\cdot d$ if the wheel isn't perfectly round.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastActivityDate="2018-05-15T15:20:17.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7362" PostTypeId="2" ParentId="7360" CreationDate="2018-05-15T15:59:36.897" Score="0" Body="&lt;p&gt;Ancient Egyptians estimated the area of a diameter-$d$ circle &lt;a href=&quot;https://en.wikipedia.org/wiki/Egyptian_geometry#Area&quot; rel=&quot;nofollow noreferrer&quot;&gt;as that of&lt;/a&gt; a side-$8d/9$ square. This is equivalent to $\pi\approx\frac{256}{81}\approx 3.16$. This means they were wrong in the second DP, and wouldn't even truncate to 1DP accurately, as they'd round up to $3.2$ (not that they used base $10$ like that). Even by the standards of the ancient world, it's a quite inaccurate estimate of $\pi$ (although better than the &lt;a href=&quot;https://www.biblegateway.com/passage/?search=1+Kings+7%3A23&amp;amp;version=NRSV&quot; rel=&quot;nofollow noreferrer&quot;&gt;Biblical&lt;/a&gt; $\pi\approx 3$). However, it gives what seems to be the only ancient rational approximation of the fourth root of $\pi$ (namely as $4/3$, and this is indeed its first &lt;a href=&quot;https://en.wikipedia.org/wiki/Continued_fraction#Infinite_continued_fractions_and_convergents&quot; rel=&quot;nofollow noreferrer&quot;&gt;convergent&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-05-15T15:59:36.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7363" PostTypeId="1" AcceptedAnswerId="7364" CreationDate="2018-05-15T17:37:45.560" Score="2" ViewCount="174" Body="&lt;p&gt;Who first discovered the Virial Theorem? Who first wrote it down? When? Where?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;My guess: a 19&lt;sup&gt;th&lt;/sup&gt; century thermodynamicist&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-05-15T17:46:52.190" LastActivityDate="2018-05-15T18:57:23.410" Title="Who discovered the Virial Theorem?" Tags="|reference-request|thermodynamics|energy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7364" PostTypeId="2" ParentId="7363" CreationDate="2018-05-15T17:45:10.963" Score="4" Body="&lt;p&gt;According to the &lt;a href=&quot;http://www.oed.com/view/Entry/223773&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;OED&lt;/em&gt;&lt;/a&gt;, Clausius coined the German word &quot;virial&quot; (from &lt;em&gt;vīs&lt;/em&gt; force, strength):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a. In Clausius' kinetic theorem of gases: (see quots.). &lt;strong&gt;&lt;em&gt;virial theorem&lt;/em&gt;&lt;/strong&gt;, the theorem that for a steady-state system of particles obeying an inverse square law of force, the time-average of the kinetic energy equals the time-average of the virial; or equivalently, that the potential energy is twice the total energy and the kinetic energy is the negative of the total energy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;R. Clausius, “&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/XZVZN9VN/Clausius%20-%201870%20-%20XVI.%20On%20a%20mechanical%20theorem%20applicable%20to%20heat.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;XVI. On a Mechanical Theorem Applicable to Heat&lt;/a&gt;,” &lt;em&gt;The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science&lt;/em&gt; 40, no. 265 (August 1870): 122–27.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-05-15T18:57:23.410" LastActivityDate="2018-05-15T18:57:23.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7365" PostTypeId="2" ParentId="3131" CreationDate="2018-05-16T08:19:52.703" Score="4" Body="&lt;p&gt;&lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Cholesky.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;André-Louis Cholesky&lt;/a&gt; (1875-1918) was killed in action near the end of World War I:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cholesky died from wounds received on the battle field on 31 August&#xA;1918 at 5 o'clock in the morning in the North of France. After his&#xA;death one of his fellow officers, Commandant Benoit, published&#xA;Cholesky's method of computing solutions to the normal equations for&#xA;some least squares data fitting problems in &lt;em&gt;Note sur une méthode de&#xA;resolution des équations normales provenant de l'application de la&#xA;méthode des moindres carrés à un système d'équations lineaires en&#xA;nombre inferieure à celui des inconnues. Application de la méthode à&#xA;la resolution d'un système defini d'équations lineaires&lt;/em&gt;, published in the &lt;em&gt;Bulletin Géodesique&lt;/em&gt; in 1924.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;/questions/tagged/mathematicians&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;mathematicians&amp;#39;&quot; rel=&quot;tag&quot;&gt;mathematicians&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-02-28T11:53:14.530" LastActivityDate="2021-02-28T11:53:14.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7366" PostTypeId="1" AcceptedAnswerId="7368" CreationDate="2018-05-17T01:30:36.600" Score="5" ViewCount="135" Body="&lt;p&gt;I can't find any information about the history of mercury in fish as a public health concern. When was it first recognized that fish contain higher levels of mercury than land animals? How long have we had the technology to measure the amount of mercury in fish?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://chemistry.stackexchange.com/questions/96996/when-did-people-first-become-concerned-about-mercury-in-fish&quot;&gt;(moved from chemistry.stackexchange.com)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7422" LastActivityDate="2018-05-17T11:32:24.653" Title="When did people first become concerned about mercury in fish?" Tags="|chemistry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7368" PostTypeId="2" ParentId="7366" CreationDate="2018-05-17T11:32:24.653" Score="2" Body="&lt;p&gt;must... not... write  ....  L****Y....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://www.sciencedirect.com/topics/agricultural-and-biological-sciences/mercury-in-fish&quot; rel=&quot;nofollow noreferrer&quot;&gt;ScienceDirect&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Investigations into the presence of mercury in food have been carried&#xA;  out in a comparatively small number of foods in the UK since 1966.&lt;br&gt;&#xA;  There's considerably more info, several articles, at that page. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From  &lt;a href=&quot;https://www.nytimes.com/1971/01/14/archives/rise-in-mercury-is-found-in-eaters-of-certain-fish.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;TheOldGreyLady&lt;/a&gt; ,possibly paywalled, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Rise in Mercury Is Found In Eaters of Certain Fish By DAVID BIRDJAN.&#xA;  January 14, 1971, Page 74 The New York Times Archives The chemist&#xA;  whose findings last month led to the recall of mercury‐tainted tuna&#xA;  and swordfish reported yesterday that he had found as much as five&#xA;  times the mercury in per sons who ate those fish regu larly compared&#xA;  with those who did not. [...]His discovery was believed to be the first time &#xA;  that high levels of mercury had been found in ocean fish. Before that&#xA;  time, only fresh‐water fish in bodies of water polluted mainly by&#xA;  industrial waste had been, declared unfit for consumption.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;a href=&quot;https://emedicine.medscape.com/article/1175560-overview&quot; rel=&quot;nofollow noreferrer&quot;&gt;medscape&lt;/a&gt; wanders over to Japan, which had a terrible industrial pollution problem, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1961, researchers in Japan correlated elevated urinary mercury&#xA;  levels with the features of the previously mysterious Minamata&#xA;  disease. Before the etiology of Minamata disease was discovered, it&#xA;  plagued the residents around Minamata Bay in Japan with tremors,&#xA;  sensory loss, ataxia, and visual field constriction. (See&#xA;  Presentation.) &lt;a href=&quot;https://emedicine.medscape.com/article/1175560-overview&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Minamata disease is an example of organic toxicity. In Minamata Bay, a&#xA;  factory discharged inorganic mercury into the water. The mercury was&#xA;  methylated by bacteria and subsequently ingested by fish.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Finally, from a &lt;a href=&quot;http://apps.who.int/iris/bitstream/handle/10665/38082/9241571012_eng.pdf;jsessionid=7A6CFC1558F93EF2A980795F2A407A31?sequence=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;WHO pamphlet&lt;/a&gt;, suggesting a start-date for a reasonably reliable test methodology,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The methods summarized in Table I have been selected from a large&#xA;  number of publications. They are typical of the various methods&#xA;  available for analysis of total mercury and its inorganic or organic&#xA;  species. All represent a considerable improvement on the original&#xA;  &quot;dithizone&quot; method. This method was widely used up to the introduction&#xA;  of atomic absorption in the late 1960s. Basically it involved the&#xA;  formation of a coloured complex with dithizone after all the mercury&#xA;  in the sample had been converted to Hg++ compounds by oxidation in&#xA;  strong acids.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://watermark.silverchair.com/ajcpath23-0789.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAakwggGlBgkqhkiG9w0BBwagggGWMIIBkgIBADCCAYsGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQM2S-fdJfiA-eapE78AgEQgIIBXB_5ALMMQ1B5NYEPQpLsNf1xaoGzcWn8CUhkHUwDqvAKnbMNB_0Vfq_JpCnFVMb8KrVaszild1x96GYDDbQIwu3B5Prw0Rb7neBgkPKVdSqHvhw0tm_g2qTDZYQYuyb792NJfjYI87COVTpLLSBf7Ff9xX-iDfBGQV2-BSEIlDCBqIj8iUNdA9xl0eLgPGoBDbdITI2pe2KGu_01gyJVZ5WNkcf23h900tZ2KJg7b3_KBdEFi3abREYvc4UKDTEaCu0yQX18uD01i6apod4dlyvd6QoRthWWH9zvw-doMyEeu9eRxHRbzkbqpv3JuBLswzTN0191HggaFk2beW69yqqM99ufiWMxRVVkTPgn3eekVZYqkoS43XaYjkETvj-te03AC7eo-ReyMMSDU01G1-xt0IMv2Uvki9jVUuC3AWdQjUcGhcFH3ovKkOzv3TP57Y-3K06sIZYOm_FQrQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paper from 1953&lt;/a&gt;  describes that &quot;dithizone&quot; methodology.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-05-17T11:32:24.653" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7370" PostTypeId="1" CreationDate="2018-05-18T02:01:24.943" Score="1" ViewCount="2076" Body="&lt;p&gt;I read Charles Babbage invented the first mechanical computer (the Analytic Engine) and that Ada Lovelace devised the first computer programs.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also I realize John Von Neumann invented the first electronic programmable digital computer  and had actually invited Alan Turing to work with him on devising computer programs after reading about Turing's Universal Turing machine.  Yet another programmable computer? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are these all manifestations of the same principles in digital computers or is each a separate distinct invention building on the others.  Thank you.  &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-05-13T11:21:34.810" LastActivityDate="2020-05-15T14:59:52.527" Title="Who first invented the programmable digital computer?" Tags="|computer-science|computers|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7371" PostTypeId="1" AcceptedAnswerId="7376" CreationDate="2018-05-18T08:44:41.383" Score="6" ViewCount="924" Body="&lt;p&gt;During the late 16th century and early 17th century, published work about probability theory (e.g. &lt;em&gt;Liber de ludo aleae&lt;/em&gt; by J. Cardan published in 1663 but writen around 1564) studied dice games using &lt;em&gt;equiprobability&lt;/em&gt; on a finite sample space.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays, we use terms such as &lt;em&gt;equiprobability&lt;/em&gt; and &lt;em&gt;uniform distribution&lt;/em&gt; to refer to their work, but at the time those people didn't use these terms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who coined the terms &lt;em&gt;uniform&lt;/em&gt; and &lt;em&gt;equiprobability&lt;/em&gt; ? &lt;/p&gt;&#xA;" OwnerUserId="7309" LastEditorUserId="7309" LastEditDate="2018-05-18T09:00:57.430" LastActivityDate="2018-05-18T21:17:40.120" Title="Who coined the term &quot;uniform&quot; as in &quot;uniform distribution&quot;?" Tags="|mathematics|terminology|probability|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7372" PostTypeId="1" CreationDate="2018-05-16T07:18:59.937" Score="6" ViewCount="407" Body="&lt;p&gt;Looking at &lt;a href=&quot;https://en.wikipedia.org/wiki/Fat_Cantor_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, I see that fat Cantor sets are also called Smith-Volterra-Cantor sets. Another name which is sometimes associated with these sets is &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Hankel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Hankel&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suppose that Cantor's name appears there simply because this construction is similar to the construction &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor set&lt;/a&gt;. The construction of &lt;a href=&quot;https://en.wikipedia.org/wiki/Volterra%27s_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volterra's function&lt;/a&gt; might explain the connection with Volterra.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is this construction named after &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_John_Stephen_Smith&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henry Smith&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Vito_Volterra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vito Volterra&lt;/a&gt;? Did both of them discover this construction? Or are there also some other mathematicians who independently arrived at the same object? What is the history behind the discovery of fat Cantor sets?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be grateful both for references giving some historical background and also for answer giving brief account of this. (If it is short enough to fit a post on Stack Exchange.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I have seen the following in Barry Simon: &lt;em&gt;Real Analysis: A Comprehensive Course in Analysis, Part 1&lt;/em&gt;, &lt;a href=&quot;https://books.google.com/books?id=pkMACwAAQBAJ&amp;amp;pg=PA202&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 202&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The nearly simultaneous discovery of the Cantor set is connected with some work of Hermann Hankel (1839–73) in 1870 [391] which attempted to prove that certain functions were Riemann integrable (see the Notes to Section 4.4 for the definition of Riemann integrability of discontinuous functions). Hankel made the error of thinking nowhere dense sets are thin; they can have positive size as seen in the following variant of the Cantor set, called the Smith–Volterra–Cantor set, S.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[391] H. Hankel, &lt;em&gt;Untersuchungen über die unendlich oft oscillirenden und unstetigen Funktionen.&#xA;  Ein Beitrag zur Feststellung des Begriffes der Funktion überhaupt&lt;/em&gt;, Universitätspr. Tübingen, 1870.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I also found this &lt;a href=&quot;http://chat.stackexchange.com/transcript/message/4857978#4857978&quot;&gt;chat message by t.b.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;@robjohn I looked in one of my old history books (by Walter, no online version available): there the situation is presented as follows. Riemann's student Hankel fleshed out the passage I linked you to. Unfortunately he made some mistakes, which subsequently led Smith to find the fat Cantor sets (found independently by Volterra in his construction of the function named after him). The situation was clarified only Lebesgue, Vitali and Young who independently proved the &quot;Lebesgue criterion&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book then continues with description first of fat Cantor set and then Volterra's function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I happened to find also &lt;a href=&quot;https://chat.stackexchange.com/transcript/36?m=2240661#2240661&quot;&gt;chat message by robjohn&lt;/a&gt; which mentions the paper &lt;em&gt;A Historical Gem from Vito Volterra&lt;/em&gt; by William Dunham; &lt;a href=&quot;https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-historical-gem-from-vito-volterra&quot; rel=&quot;nofollow noreferrer&quot;&gt;MAA website&lt;/a&gt;, link to &lt;a href=&quot;https://www.maa.org/sites/default/files/0025570x02688.di021166.02p0055m.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The context is somewhat different, but this part seems to be related to the topic of this question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One such mathematician was Hermann Hankel, who in 1870 made a detailed examination of the &quot;punktirt unstetigen,&quot; i.e., &quot;pointwise discontinuous,&quot; functions.&#xA;  (This rather odd-sounding term was introduced as a contrast to the &quot;total unstetige,&quot; i.e., &quot;totally discontinuous,&quot; functions, whose points of continuity were not dense.) Hankel gave a proof [3, pp. 89-90] purporting to show that a function is (Riemann) integrable if and only if it is pointwise discontinuous. This conclusion certainly thrust pointwise discontinuous functions into the limelight, and, as Thomas Hawkins observes in his excellent book Lebesgue's Theory of Integration: Its Origins and Development [4, p. 301, seemed to identify them as precisely &quot;the functions amenable to mathematical analysis.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;However, Hankel's reasoning was flawed, and the error was exposed in 1875 by Oxford professor H. J. S. Smith [7, p. 150]. While Smith agreed that any integrable function must be pointwise discontinuous, his explicit example of a pointwise discontinuous but non-integrable function destroyed the &quot;if and only if' nature of Hankel's proof. Smith had established, of course, that the integrable functions form a proper subset of the pointwise discontinuous ones, but his paper was not widely read and its impact was minimal. Consequently, pointwise discontinuous functions occupied an important, although not necessarily well-understood, position in the research of the day. (Hawkins gives a detailed account of the situation in [4, Ch. 21.)  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This paper references the same book as already mentioned in &lt;a href=&quot;https://hsm.stackexchange.com/questions/7372/what-is-history-behind-smith-volterra-cantor-sets#comment13150_7372&quot;&gt;Hans Lundmark's comment&lt;/a&gt;: T. Hawkins, &lt;em&gt;Lebesgue's Theory of Integration: Its Origins and Development&lt;/em&gt;, Chelsea, 1975. &lt;/p&gt;&#xA;" OwnerUserId="205" OwnerDisplayName="Martin" LastEditorUserId="205" LastEditDate="2018-05-20T08:35:00.133" LastActivityDate="2018-05-20T08:35:00.133" Title="What is history behind Smith-Volterra-Cantor sets?" Tags="|mathematics|real-analysis|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7373" PostTypeId="2" ParentId="7372" CreationDate="2018-05-16T07:38:00.057" Score="6" Body="&lt;p&gt;See :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;David Bressoud, &lt;a href=&quot;https://books.google.it/books?id=TxxMoGjXC-wC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;A radical approach to Lebesgue's theory of integration&lt;/a&gt;, Cambridge UP (2008), &lt;strong&gt;4.1 The Smith-Volterra-Cantor Sets&lt;/strong&gt; :&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cantor would prove in 1883 that given any set S, its derived set is a union of a countable set and a perfect set, a set that is its own derived set.&lt;/p&gt;&#xA;&lt;p&gt;The first construction of a perfect, nowhere dense set was by the British mathematician Henry J. S. Smith (1826-1883) in 1875 [&amp;quot;On the integration of discontinuous functions&amp;quot;, &lt;em&gt;Proceedings of the London Mathematical Society&lt;/em&gt;].&lt;/p&gt;&#xA;&lt;p&gt;In 1881, Vito Volterra showed how to construct such a set, but Volterra was still a graduate student, and he published in an Italian journal that was not widely read [&amp;quot;Alcune osservazioni sulle funzioni punteggiate discontinue&amp;quot;, &lt;em&gt;G. Mat&lt;/em&gt;]. Again, little notice was paid. Finally, in 1883, Cantor rediscovered this construction for himself, and suddenly everyone knew about it. Cantor's example is known as the Cantor ternary set. We shall use&#xA;this term to refer to Cantor's specific example, but the family of examples of&#xA;perfect, nowhere dense sets exemplified by the work of Smith, Volterra, and Cantor will be referred to as the Smith-Volterra-Cantor sets, or SVC sets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" OwnerDisplayName="Mauro ALLEGRANZA" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-05-16T07:43:29.280" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7374" PostTypeId="2" ParentId="7371" CreationDate="2018-05-18T13:05:05.520" Score="2" Body="&lt;p&gt;Regarding English, I think that the first treatise was &lt;a href=&quot;https://en.wikipedia.org/wiki/Abraham_de_Moivre&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abraham De Moivre&lt;/a&gt;'s treatise &lt;a href=&quot;https://books.google.it/books?id=3EPac6QpbuMC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Doctrine of Chances (1718)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the discrete case, see &lt;a href=&quot;https://books.google.it/books?id=3EPac6QpbuMC&amp;amp;pg=PA7&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 7&lt;/a&gt; : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;If the Events in question are $n$ in number and are such as have the same number $a$ of Chances by whcih they may Happen...&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And see also &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Emerson_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Emerson (1701 – 1782)&lt;/a&gt;'s &lt;a href=&quot;https://books.google.it/books?id=C_k4AAAAMAAJ&amp;amp;pg=PA3&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Laws of Chance (1776), page 3&lt;/a&gt; : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;strong&gt;AXIOM I&lt;/strong&gt;. In computing the number of Chances, it is supposed that all Chances are equal, or made with equal facility.&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And page 4 : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;strong&gt;SCHOLIUM&lt;/strong&gt;. As $1$ represents a certainty, or when an event has an infinitely great probability of happening, so $\frac 1 2$ represents an &lt;em&gt;equal probability&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;] for happening or failing. For $1 − \frac 1 2 = \frac 1 2$.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For a good overview of the early history of probability in general and of &lt;em&gt;equipossibility&lt;/em&gt; in particular, see Ian Hacking, &lt;a href=&quot;https://books.google.it/books?id=Z2g0F1V5DywC&amp;amp;pg=PA122&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Emergence of Probability : A Philosophical Study&lt;/a&gt;, Cambridge UP (2nd ed 2006), Ch.14, page 122-on.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-05-18T13:05:05.520" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7375" PostTypeId="2" ParentId="7372" CreationDate="2018-05-18T14:01:44.320" Score="4" Body="&lt;p&gt;In his book &lt;em&gt;The Fractal Geometry of Nature&lt;/em&gt;, Mandelbrot discusses the Cantor set.  And says it is actually due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_John_Stephen_Smith&quot; rel=&quot;nofollow noreferrer&quot;&gt;H.J.S. Smith&lt;/a&gt; (1875).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I asked a graduate student (William Darnieder) to look into that.  He read Smith's paper carefully, and did his Master's thesis on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So:  Is the Cantor set due to Smith?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, maybe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Smith has no good way to talk about &quot;point sets&quot;, since that was pioneered later by Cantor.  Smith discusses certain functions in view of the question of whether or not they are Riemann integrable.  He has an example of a non-integrable function.  Nowadays, we would say &quot;The set of discontinuities of the function is a fat Cantor set&quot;.  And Smith's arguments clearly show this, at least to us, since we know the later notion of &quot;point set&quot;.  And the main point: Smith shows that (in our language) contrary to Hankel, measure zero Cantor set compared fat Cantor set determine whether his examples are integrable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A fat Cantor set:&lt;/em&gt; (It seems, from the quotes, that modern texts use the name &quot;Smith-Volterra-Cantor set&quot; for this:  An innovation since my schooldays.) Of course none of these guys: Hankel, Smith, Volterra, or Cantor, had a grasp of Lebesgue measure, since that came later.  But those of us who know Lebesgue measure can see that they were feeling their way along toward it.  Surely Lebesgue was a great man; but he didn't start from nothing.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="3643" LastEditDate="2018-05-18T15:46:10.733" LastActivityDate="2018-05-18T15:46:10.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7376" PostTypeId="2" ParentId="7371" CreationDate="2018-05-18T21:17:40.120" Score="3" Body="&lt;p&gt;The first uses of what we call &quot;uniform distribution&quot; occur very early, discrete arguably already in Cardano, and continuous in Simpson and Bayes. According to &lt;a href=&quot;http://fpfn24.wdfiles.com/local--files/suplementos-de-clase/The_Triangular_Distribution.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Handbook of Beta Distributions&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;One of the  first records that mentions  the continuos uniform distribution is the  famous  paper by the reverend Thomas Bayes (1763) (only a few years after Simpons' written records in 1757).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The paper is &lt;a href=&quot;https://en.wikipedia.org/wiki/An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essay towards solving a Problem in the Doctrine of Chances&lt;/a&gt;, where conditional probabilities and the Bayes's formula are introduced. Bayes considered independent experiments with the probability of success being some number $p$ between $0$ and $1$ and made the number itself to be random with the probability to fall into any interval between $0$ and $1$ equal to the length of that interval. However, the term &quot;uniform distribution&quot; itself is much more recent. According to James Landau's entry to &lt;a href=&quot;https://www.leidenuniv.nl/fsw/verduin/stathist/1stword.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;strong&gt;&lt;em&gt;Uniform distribution&lt;/strong&gt; appears in 1937 in Introduction to Mathematical Probability by J. V. Uspensky. Page 237 reads, &quot;A stochastic variable is said to have uniform distribution of probability if probabilities attached to two equal intervals are equal.&quot; This is a slight variant of the modern terminology, which would be &quot;a variable is said to be uniformly distributed&quot; or &quot;a variable from the uniform distribution&quot;. &lt;strong&gt;Uniformly distributed&lt;/strong&gt; is found in H. Sakamoto, &quot;On the distributions of the product and the quotient of the independent and uniformly distributed random variables,&quot; Tohoku Math. J. 49 (1943).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-05-18T21:17:40.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7378" PostTypeId="1" AcceptedAnswerId="7380" CreationDate="2018-05-19T14:52:37.377" Score="3" ViewCount="101" Body="&lt;p&gt;I am curious if anyone has seen any records on how this alloy was first developed or perhaps accidentally  discovered?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One could ask a similar question for any alloy but since this is peculiar in that it has to have a small amount of arsenic relative to copper and tin, I am baffled as to how it would have been originally discovered.  Issac Newton made use of this in making his and the first reflecting telescope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found in &quot;Henley's Twentieth Century Book of Recipes&quot; the formula 66.22 parts copper ; tin 33.11 parts ; arsenic 0.67.  Since that is such a small amount of arsenic it appears to be &quot;finely tuned&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only logical explanation I can think of is by accident and experimentation , exhausting all possibilities but somehow this does not sit well with me.  My question:  Any documentation on this development or ideas?  &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="5040" LastEditDate="2018-05-19T14:58:34.423" LastActivityDate="2018-05-19T22:30:29.333" Title="Are there any records of the history for the development of speculum alloy?" Tags="|chemistry|telescope|alchemy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7379" PostTypeId="1" CreationDate="2018-05-19T16:49:30.163" Score="1" ViewCount="447" Body="&lt;p&gt;It is sometimes said that Einstein didn't accept quantum mechanics. Some stronger claims are that he viewed it as wrong altogether, considering it not a viable description of nature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, I want to know the historical truth about this. Did Einstein really despise quantum mechanics to the point of dequalifying it as a correct theory of nature? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or he simply didn't agree to the interpretations people gave to it? Or even he just considered it to be &quot;right, but not the end&quot;, so that he believed one day something more fundamental could be found giving a better explanation to it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might be wrong, but I think Einstein didn't really totally viewed quantum mechanics as wrong, after all if I'm not mistaken (and I can certainly be), he was one of the first to work with something related to entanglment (the so-called EPR paradox).&lt;/p&gt;&#xA;" OwnerUserId="4072" LastActivityDate="2018-05-21T05:40:07.370" Title="Einstein really didn't &quot;accept&quot; quantum mechanics?" Tags="|quantum-mechanics|theoretical-physics|einstein|" AnswerCount="2" CommentCount="3" ClosedDate="2018-05-22T21:42:02.767" ContentLicense="CC BY-SA 4.0" />
  <row Id="7380" PostTypeId="2" ParentId="7378" CreationDate="2018-05-19T20:29:56.153" Score="2" Body="&lt;p&gt;The only descriptions I have seen seem to show by implication how experimental (and perhaps accidental) the subject was.  Newton was a long way from being the first to investigate speculum metal, but for the following reasons his descriptions can form a useful reference-point: (a) Some of the difficulties of accounting for early developments in 'chymistry' or any other proto-chemistry arise because for many materials, techniques for achieving and assessing chemical purity were in short supply, and for end-products gained by use of such materials, the effects of procedure and impurities are specially hard to trace; (b) Newton's descriptions had a strong practical content and focus on desired end-product qualities in relation to his experiences of methods for achieving them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Newton's view on the purpose and effect of including small proportions of arsenic in the formulations was clearly given. He described the danger of getting a product &quot;full of small pores onely discoverable by a Microscope&quot;, resulting in irregularity of the (seemingly) polished reflecting surface. His remedy and explanation was that &quot;white Arsenick both blanches the Metall and leaves it solid without any such pores, especially if the fusion hath not been too violent&quot; (18 January 1671|2, letter to Oldenburg, in 'Correspondence of Isaac Newton', ( vol.1, Cambridge UP 1959, ed. H W Turnbull) p.82). (His descriptions give many more practical details than that (ibid., esp. p.82-88), including alternative compositions that did not include arsenic at all.) At the same time the matter is not simple and caution is needed before accepting his ingredient-lists as an indication of the final compositions, for example, as a later writer put it (in Journal of the Franklin Institute, vol.5 (1828), 39-46 at p.41):  &quot;Sir Isaac Newton melted the copper first, then added the arsenic, and lastly the tin; as, without doubt, he knew that the tin should remain in a fluid state the shortest time possible. It is true that he added the arsenic to the melted copper, but as he well knew that a great part of it would be rendered volatile, he therefore added a very large quantity of it, viz. arsenic one, to copper six.&quot; So according to that account, much of the arsenic added in Newton's processes was volatilized away.  In these and other ways the subject for a long time remained highly experimental and practical, systematization was hard to achieve and seems to remain so in historical retrospect. An interesting view of speculum metals in the hands of Newton and some contemporaries is given by A M Roos, &quot;A speculum of chymical practice&quot;, Notes &amp;amp; Records of the Royal Society (2009) [&lt;a href=&quot;http://rsnr.royalsocietypublishing.org/content/roynotesrec/early/2009/12/15/rsnr.2009.0067.full.pdf]&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://rsnr.royalsocietypublishing.org/content/roynotesrec/early/2009/12/15/rsnr.2009.0067.full.pdf]&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2018-05-19T22:30:29.333" LastActivityDate="2018-05-19T22:30:29.333" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7381" PostTypeId="2" ParentId="1816" CreationDate="2018-05-20T14:01:16.770" Score="3" Body="&lt;p&gt;Physicists now tend to think in terms of &lt;em&gt;effective&lt;/em&gt; theories; that is a theory which is accurate in a certain energy regime but fails in another. Thus, for example we get super-gravity as a low energy effective (ie approximate) theory of string theory.&lt;/p&gt;&#xA;&lt;p&gt;It's not quite fair to compare Newton and Aristotle after all a span of two millenia separates them. This is a hefty amount of time by any measure; and the earlier work should be evaluated in its own terms, in the context of its time, as well as it's influence.&lt;/p&gt;&#xA;&lt;p&gt;Julian Barbour in his &lt;em&gt;History of Dynamics&lt;/em&gt; clearly points out that Aristotles critical approach to the problem of change, space, time &amp;amp; motion was a clear precursor to later thinking about motion; and I think, if I recall rightly, that he said that Newtons laws were in some sense, in an embryonic form in his works.&lt;/p&gt;&#xA;&lt;p&gt;It was the later infatuation with Aristotle beginning in the Renaissance after Averroes rediscovery of Aristotles works and his commentaries on them; and also the close association of the Church with Aristotle and then then the rise of anti-clericalism in Europe after the bourgeois revolutions that our modern dislike of Aristotle stems from; the popular literature is littered with disdain for his works; this is not an accurate view of his importance; and nor, to be honest of the Greek work on dynamics of which Aristotle is the most prominent representative.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's not that Aristotle himself who held back scientific thought; after all, he himself was not backward in critiquing Plato, Parmenides or Democritus when he felt their ideas hadn't sufficient justification. He was not over-awed by them, perhaps due to the closeness in time to these giant figures, after all, he himself was a student of Plato.&lt;/p&gt;&#xA;&lt;p&gt;Had the early to late medivals understood this aspect of his critical activity and took it upon themselves then perhaps scientific thought might have been recovered more rapidly; instead, they found themselves over-awed  by their achievements and on the whole, could do no more than tinker with them, and it took the time to the early moderns before general scientific culture in Europe had risen to a sufficient height and had become fluid and vigorous that they could begin to develop where the Ancient world had stopped.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;After Einstein, we see gravity as the curvature of spacetime. This means that spacetime, far from being the passive theatre in which events happen, also has dynamical properties. Did anyone before Einstein forsee this possibility? Well, Clifford did when he stated that this curvature was all that there was in terms of physical change in the world.&lt;/p&gt;&#xA;&lt;p&gt;Whilst Aristotle didn't forsee this, he was capable of asking the question: &amp;quot;does place itself have a place&amp;quot;? That question, if pushed further, suggests that space is dynamical. After all, a stone has a place, the place where we put it - here or there. To ask whether place has a place shows just how careful scientific questioning then was in questioning verities we would normally simply take for granted.&lt;/p&gt;&#xA;&lt;p&gt;What was Aristotles answer to the question raised? He wasn't able to come to a definitive answer. He simply said it was a difficult question. And in truth it is. It took the genius of Einstein to show that place actually did have a place. Or rather, spacetime, and which amounts to the same thing here&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-08T05:26:04.420" LastActivityDate="2021-03-08T05:26:04.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7382" PostTypeId="2" ParentId="7379" CreationDate="2018-05-20T22:21:48.437" Score="2" Body="&lt;p&gt;Einstein's objections to QM are best understood by his famous EPR paper written in 1936.  Einstein proposed a thought experiment to determine whether QM was local and real.  The locality part is that no signal of information can travel faster than the speed of light.  The &quot;Reality&quot; part is that nature (let's say an assemblage of particles) is real even when it is not being observed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's proposition is that the current QM theories could not be both real and local unless other hidden variables were added to the theory.  So, this was then known as the issue of QM Completeness -- that is, QM is not complete.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reality requirement of Einstein was directed primarily against the Copenhagen Interpretation where Bohr claims that momentum (for example) does not exist as a measurable quantity if it is not being observed.  Or observation causes momentum of a particle to exist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit that in the above I am glossing over the more subtle arguments in the issue.  However, an excellent book on this very topic that covers all of the sides of Einstein's angst and arguments is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0465096050&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is Real?  The Unfinished Quest for the Meaning of Quantum Physics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2018-05-20T22:21:48.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7383" PostTypeId="1" CreationDate="2018-05-21T00:26:20.980" Score="7" ViewCount="750" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Euclidean algorithm&lt;/a&gt; is an algorithm that produces the greatest common divisor of two integers. It was described by Euclid as early as in 300 BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm&quot; rel=&quot;noreferrer&quot;&gt;extended Euclidean algorithm&lt;/a&gt; extends his algorithm to express the greatest common divisor as an integer-linear combination of the two inputs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who came up with the extended Euclidean algorithm?&lt;/p&gt;&#xA;" OwnerUserId="7439" LastEditorUserId="55" LastEditDate="2018-05-24T08:24:07.993" LastActivityDate="2021-01-15T14:04:38.373" Title="Who extended the Euclidean algorithm to derive the Bezout identity?" Tags="|mathematics|number-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7384" PostTypeId="2" ParentId="7379" CreationDate="2018-05-21T05:40:07.370" Score="2" Body="&lt;p&gt;I don't  think    Einstein  completely  rejected   quantum  Mechanics,  but  he did  disagree  with  other  people's   interpretations     and  opinions.  He   said  &quot;God  does  not  play   dice  with  the universe.&quot; At  the  subatomic  and  quark  scales  of  size,  God  DOES  play    dice   with  the  Universe.  Let's  not  get  diverted  into  whether  God  exists  or  not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd  have  to  look  up   if  he  said  anything  about   about  or  Heisenberg  about  the  Heisenberg  uncertainty   principle,  basically  that  just  by  observing    and  trying  to  measure   something   you  disturb  t  and   change   the  system   assuming  the   system  is  a closed  system,.  LOTS  of  people  disagree   about  whether  the  Universe  is  completely &quot;closed&quot;  system  or  no,   Often   that  is  based  in their  own ignorance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein   was  just  one   of many  people  who   shattered  the  classical  Newtonian    &quot;model&quot;   of a clockwork   Universe.  Disagreement  and  peer  review   is  part  of   of  the  scientific  method,   which   basically  is     a logical  objective   guessing   game   with   flashes  of  serendipity   and  inspiration.   it  took  me  about  15  to  20  minutes   for my  master's  thesis   to  admit   that   and  to   agree   that   usually  most  scientists  will NOT  admit  to  that   because   peer  pressure      and  the  risk  of  damaging  their  won  credibility   with  the  general  public,   and  he  may  have  been   thinking    their  students.   i  was  his  last  graduate   student.    and   retirement   and  becoming   an  Emeritus  professor was less   than   3  years   in  the  future  for  him.  of  course  this   conversation   was  late  Friday   afternoon   conversation   as   well  that  went  on  past  5  pm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Objective, logical, analytical, critical and unbiased thinking is NOT  &quot;normal&quot;    Way  to think   for   anyone,  Including  people  who  think  deductively   rather    inductively.    We   each have  our  own  unique  perspective  of  the  universe.  We  each live  in  out  own  subjective  universes.  &quot;Truth&quot;  is   subjective  is  too.  What  may   be   'true'   for  you,  may   be   fatal   for  me..  Survival  of  the  fittest   is   what   it  is  all  about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't  think  Einstein   rejected  all  of  quantum  mechanics. He  wasn't  that  closed  minded    completely    stuck  in   the  paradigm  that  he  had    shattered   along  with other  people.   As  someone  else  has  already  said,  entire  books   have  been  written  about  this.&lt;/p&gt;&#xA;" OwnerUserId="7440" LastActivityDate="2018-05-21T05:40:07.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7385" PostTypeId="2" ParentId="7383" CreationDate="2018-05-21T12:09:32.993" Score="3" Body="&lt;p&gt;The path seems a bit deep, but here's a start, from   &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclidean_algorithm#Historical_development&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The extended Euclidean algorithm was published by the English&#xA;  mathematician Nicholas Saunderson,[38] who attributed it to Roger&#xA;  Cotes as a method for computing continued fractions efficiently.[39]  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ref 38 is  Saunderson, Nicholas (1740). The Elements of Algebra in Ten Books. &#xA;PDF at &lt;a href=&quot;https://www.e-rara.ch/zut/content/structure/2440626&quot; rel=&quot;nofollow noreferrer&quot;&gt;this archive&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref 39 is    Tattersall, J. J. (2005). Elementary Number Theory in Nine Chapters. Cambridge: Cambridge University Press. ISBN 978-0-521-85014-8.,   &lt;a href=&quot;https://books.google.it/books?id=QGgLbf2oFUYC&amp;amp;pg=PA66&amp;amp;lpg=PA66&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 72–76&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="131" LastEditDate="2018-05-21T15:53:45.330" LastActivityDate="2018-05-21T15:53:45.330" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7386" PostTypeId="1" AcceptedAnswerId="7387" CreationDate="2018-05-22T02:26:51.293" Score="6" ViewCount="426" Body="&lt;p&gt;Grothendieck developed the theory of the fundamental group of a scheme in SGA 1. In order to do so he used the fact that the fundamental group of a topological space is isomorphic to the group of deck transformations  --- fiber permuting isomorphisms of the universal cover --- and as a consequence is isomorphic to the automorphism group a particular functor. This reframing seems necessary because who knows what a &quot;loop in a scheme&quot; is.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Did Grothendieck reframe the theory of the fundamental group of topological spaces as the group of deck transformations? Or, was this already known? Perhaps it was already known, but no one had phrased it in terms of automorphisms of a fiber functor? &lt;/p&gt;&#xA;" OwnerUserId="2958" LastEditorUserId="55" LastEditDate="2018-05-22T03:32:51.683" LastActivityDate="2018-05-25T10:39:36.903" Title="Who first described the fundamental group as the group of deck transformations?" Tags="|topology|algebraic-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7387" PostTypeId="2" ParentId="7386" CreationDate="2018-05-22T03:27:44.047" Score="5" Body="&lt;p&gt;The idea of a relation between fundamental groups and permutations of the universal cover long predates Grothendieck and SGA. It appears implicitly already in Riemann's work on complex surfaces in 19th century. In &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-60039-0_7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy and Puiseux: Two Precursors of Riemann&lt;/a&gt; Papadopoulos even mentions earlier authors, especially Puiseux:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Riemann defined these surfaces as ramified coverings of the plane (more precisely, of the Riemann sphere). The work of Puiseux on algebraic functions, interpreted from a topological point of view, contains in essence the combinatorics of such a surface, giving a description of how its sheets are permuted above a ramification point, and establishing the precise relation between this sheet permutation and the nature of the singularities of the algebraic equation. At the same time, Puiseux’s work makes the relation with group theory.  At the expense of being anachronical, let us mention that the theory of Puiseux expresses the so-called monodromy  homomorphism from the fundamental group of the Riemann sphere with a finite set deleted (the singular set of the algebraic equation) into the permutation group on $d$ symbols.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The key technical step, the path lifting theorem, was made by Weyl in 1913, and full development is due to Seifert and Threlfall's classic Lehrbuch der Topologie. Here is Dieudonné's description from &lt;a href=&quot;https://www.springer.com/it/book/9780817649067&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Algebraic and Differential Topology 1900-1960, p.296-7&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It is quite likely that Poincare and his immediate successors, such as Tietze&#xA;  and Dehn, were aware of that theorem, even if they did not formulate it explicitly (still less try to prove it!). It was stated and proved for surfaces by H. Weyl in 1913 [483]. In 1928 Reidemeister published a short paper on fundamental groups and covering spaces [386] in which he only considered combinatorial manifolds. He deplored the fact that, except for dimension 2,&#xA;  no previous treatment of these questions was available. His own treatment is&#xA;  in fact limited to dimension 3, although he claims it might be generalized to&#xA;  any dimension; it is written in the Poincare style, without genuine proofs, and&#xA;  the path lifting theorem is not mentioned.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;It was only in 1934 that, in their book [421], Seifert and Threlfall gave an &#xA;  admirable and thorough elaboration of the relations between fundamental&#xA;  groups and covering spaces based on the path lifting theorem: although&#xA;  limited to locally finite simplicial complexes (of any dimension), it is essentially definitive, and can be extended to more general spaces with only minor modifications.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to Theo Buehler's answer in  &lt;a href=&quot;https://math.stackexchange.com/a/38334/152568&quot;&gt;Etymology of the name “deck transformation”&lt;/a&gt; thread on Math SE, the term &quot;deck&quot; is also due to Seifert and Threlfall. They speak of Deckbewegungsgruppe, literally, group of covering movements (from German decken, to cover). Reidemeister called them transitive permutations by the fundamental group. The &quot;automorphisms of a fiber functor&quot; language is of course specific to Grothendieck. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-24T03:44:08.980" LastActivityDate="2018-05-24T03:44:08.980" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7388" PostTypeId="2" ParentId="7386" CreationDate="2018-05-22T10:44:08.037" Score="5" Body="&lt;p&gt;The answer to the title question is &lt;strong&gt;Poincaré&lt;/strong&gt;, in the very note &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3071t/f633&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sur l’Analysis situs&lt;/em&gt; (1892)&lt;/a&gt; where he first introduced the fundamental group. Cf. the description by &lt;a href=&quot;http://analysis-situs.math.cnrs.fr/La-genese-du-groupe-fondamental-chez-Poincare.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Saint-Gervais”&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Now Poincaré gives his definition of the fundamental group. To this end he starts by almost copying his 1883 article which we have just discussed. &lt;em&gt;But now he introduces a group!&lt;/em&gt; In other words, he defines the universal covering as before, but &lt;strong&gt;the fundamental group makes its appearance as a permutation group of the fiber above a point&lt;/strong&gt;: &lt;em&gt;the automorphism group of the universal covering&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also Sarkaria (&lt;a href=&quot;//mathscinet.ams.org/mathscinet-getitem?mr=1674912&quot; rel=&quot;nofollow noreferrer&quot;&gt;1999&lt;/a&gt;, &lt;a href=&quot;http://kssarkaria.org/docs/The%20Topological%20Work%20of%20Henri%20Poincare.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 144&lt;/a&gt;), note 20 referring to §12 of &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=pst.000053123732&amp;amp;num=60&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Analysis Situs&lt;/em&gt; (1895)&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Poincaré  gives four approaches to his groups $g$ and $G$. Firstly, as all &lt;em&gt;deck transformations&lt;/em&gt; of a &lt;strong&gt;covering space&lt;/strong&gt; over $M$ (...) Secondly, his differential equations definition (...) Thirdly, his definition using “loops”, “equivalences” and “lacets” (...) Lastly, in §13, for any $M$ obtained from a polytope by facet conjugations, Poincaré defines $\pi_1(M)$ via some simple and elegant (yet intriguing) cyclic relations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note added:&lt;/strong&gt; Of course, on general spaces Poincaré’s four definitions cease to be equivalent. Grothendieck’s approach of sticking with the &lt;em&gt;first&lt;/em&gt; one — thus eschewing loops — likely originates with &lt;a href=&quot;//ncatlab.org/nlab/show/Chevalley+fundamental+group&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chevalley&lt;/a&gt; (&lt;a href=&quot;//zbmath.org/?q=an:0946.22001&quot; rel=&quot;nofollow noreferrer&quot;&gt;1946&lt;/a&gt;), who used it “following a suggestion of H. Cartan” (Dieudonné &lt;a href=&quot;//zbmath.org/?q=an:1180.55001&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989&lt;/a&gt;, p. 300). It is also adopted in e.g. Ganea (&lt;a href=&quot;//zbmath.org/?q=an:0044.37903&quot; rel=&quot;nofollow noreferrer&quot;&gt;1951&lt;/a&gt;), Souriau (&lt;a href=&quot;//zbmath.org/?q=an:0117.22204&quot; rel=&quot;nofollow noreferrer&quot;&gt;1964&lt;/a&gt;, §10 + Note II), Kennison (&lt;a href=&quot;//zbmath.org/?q=an:0674.57002&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989&lt;/a&gt;), etc.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-05-25T10:39:36.903" LastActivityDate="2018-05-25T10:39:36.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7389" PostTypeId="2" ParentId="7383" CreationDate="2018-05-22T19:15:24.977" Score="4" Body="&lt;p&gt;The earliest forms of the extended Euclidean algorithm are ancient, dating back to 5th-6th century A.D. work of Aryabhata - who described the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ku%E1%B9%AD%E1%B9%ADaka&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kuttaka (&amp;quot;pulverizer&amp;quot;) algorithm&lt;/a&gt; for the more general problem of solving linear Diophantine equations &lt;span class=&quot;math-container&quot;&gt;$ ax + by = c$&lt;/span&gt;.  It was independently rediscovered numerous times since, e.g. by Bachet in 1621, and Fermat and Wallis, and by Euler circa 1731.&lt;/p&gt;&#xA;&lt;p&gt;Weil discusses this briefly in his book &lt;a href=&quot;https://www.springer.com/la/book/9780817645656&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Theory: An Approach through History from Hammurapi to Legendre&lt;/a&gt;, excerpted below (from pp. 6-7)&#xA;&lt;a href=&quot;https://i.stack.imgur.com/gE7gW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gE7gW.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/MMJDZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MMJDZ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Euler's rediscovery is mentioned on pp. 176-77:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YkbQ5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YkbQ5.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6iBZ1.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6iBZ1.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It deserves to be more widely known that the algorithm is simpler to execute (and remember) if you are already familiar row-operations such as in Gaussian elimination / triangularization in linear algebra. See &lt;a href=&quot;https://math.stackexchange.com/a/85841/242&quot;&gt;this MSE answer&lt;/a&gt; for a presentation from that viewpoint. This method eliminates the notoriously error-prone back-substitution in the more common presentation of the algorithm. Below is a worked example done this way, computing &lt;span class=&quot;math-container&quot;&gt;$\, \gcd(141,19),\, $&lt;/span&gt; shown firstly  in  full equational form, and secondly in more concise tabular form.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\rm\begin{eqnarray}[\![1]\!]\!\quad \color{#C00}{141}\!\ &amp;amp;=&amp;amp;\,\ \ \ 1&amp;amp;\cdot&amp;amp; 141\, +\ 0&amp;amp;\cdot&amp;amp; 19 \\&#xA;[\![2]\!]\quad\ \color{#C00}{19}\ &amp;amp;=&amp;amp;\,\ \ \ 0&amp;amp;\cdot&amp;amp; 141\, +\ 1&amp;amp;\cdot&amp;amp; 19 \\&#xA;\color{#940}{[\![1]\!]-7\,[\![2]\!]}\, \rightarrow\, [\![3]\!]\quad\ \ \ \color{#C00}{ 8}\ &amp;amp;=&amp;amp;\,\ \ \ 1&amp;amp;\cdot&amp;amp; 141\, -\ 7&amp;amp;\cdot&amp;amp; 19 \\&#xA;\color{#940}{[\![2]\!]-2\,[\![3]\!]}\,\rightarrow\,[\![4]\!]\quad\ \ \ \color{#C00}{3}\ &amp;amp;=&amp;amp;\, {-}2&amp;amp;\cdot&amp;amp; 141\, + 15&amp;amp;\cdot&amp;amp; 19 \\&#xA;\color{#940}{[\![3]\!]-3\,[\![4]\!]}\,\rightarrow\,[\![5]\!]\quad \color{#C00}{{-}1}\ &amp;amp;=&amp;amp;\,\ \ \ 7&amp;amp;\cdot&amp;amp; 141\, -\color{}{ 52}&amp;amp;\cdot&amp;amp; \color{}{19} \end{eqnarray}\qquad\qquad\qquad$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\rm\begin{eqnarray}                  &amp;amp;&amp;amp;[\![1]\!]\quad \color{#C00}{141} &amp;amp;\ \ \  1 &amp;amp;\quad\ \  0 \\&#xA;                                       &amp;amp;&amp;amp;[\![2]\!]\quad\ \color{#C00}{19} &amp;amp;\ \ \  0 &amp;amp;\quad \ \ 1 \\&#xA;\color{#940}{[\![1]\!]-7\,[\![2]\!]}\,\rightarrow\,&amp;amp;&amp;amp;[\![3]\!]\quad\ \ \ \color{#C00}{ 8} &amp;amp;\ \ \ 1 &amp;amp;\  -7\\&#xA;\color{#940}{[\![2]\!]-2\,[\![3]\!]}\,\rightarrow\,&amp;amp;&amp;amp;[\![4]\!]\quad\ \ \ \color{#C00}{3} &amp;amp;    -2 &amp;amp;\ \ \ \, 15 \\&#xA;\color{#940}{[\![3]\!]-3\,[\![4]\!]}\,\rightarrow\,&amp;amp;&amp;amp;[\![5]\!]\quad \color{#C00}{{-}1} &amp;amp;\ \ \  7 &amp;amp; \, \color{}{{-}52} \end{eqnarray}\qquad\qquad\qquad\qquad\qquad\quad\ $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One can optimize even further (e.g. see the fractional form below). It would be interesting to know who first presented the algorithm from this viewpoint. Likely it as at least a few centuries old.&lt;/p&gt;&#xA;&lt;p&gt;It would take immense effort to write a good history of the extended Euclidean algorithm and related ideas, since it occurs in many different guises throughout mathematics, e.g. search on the following keywords: Hermite / Smith normal form,&#xA;invariant factors, lattice basis reduction, continued fractions,&#xA;Farey fractions / mediants, Stern-Brocot tree / diatomic sequence.&lt;/p&gt;&#xA;&lt;p&gt;In fact even in recent times there are useful twists on the algorithm that are discovered, e.g. we can &lt;a href=&quot;https://math.stackexchange.com/a/2053174/242&quot;&gt;present the algorithm efficiently in fractional form&lt;/a&gt; using modular arithmetic, e.g. below I show how we can compute &lt;span class=&quot;math-container&quot;&gt;$\,1/117\equiv\color{#c00}{-72} \pmod{337}\ $&lt;/span&gt; this way.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${\rm mod}\ 337\!:\,\ \dfrac{0}{337} \overset{\large\frown}\equiv \dfrac{1}{117} \overset{\large\frown}\equiv \dfrac{-3}{\color{#0a0}{-14}} \overset{\large\frown}\equiv \dfrac{-23}5 \overset{\large\frown}\equiv\color{#c00}{\dfrac{-72} {1}}.\,$&lt;/span&gt; Equivalently, without fractions&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\qquad\quad \begin{array}{rrl} &#xA;[\![1]\!]\!:\!\!\!&amp;amp; 337\,x\!\!\!&amp;amp;\equiv\ 0\\&#xA;[\![2]\!]\!:\!\!\!&amp;amp; 117\,x\!\!\!&amp;amp;\equiv\ 1\\&#xA;[\![1]\!]-3[\![2]\!]=:[\![3]\!]\!:\!\!\!&amp;amp; \color{#0a0}{{-}14}\,x\!\!\!&amp;amp;\equiv -3\\&#xA;[\![2]\!]+8[\![3]\!]=:[\![4]\!]\!:\!\!\!&amp;amp; 5\,x\!\!\! &amp;amp;\equiv -23\\&#xA;[\![3]\!]+3[\![4]\!]=:[\![5]\!]\!:\!\!\!&amp;amp;  \color{#c00}1\,   x\!\!\! &amp;amp;\equiv \color{#c00}{-72}&#xA;\end{array}$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="1872" LastEditorUserId="1872" LastEditDate="2021-01-15T14:04:38.373" LastActivityDate="2021-01-15T14:04:38.373" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7390" PostTypeId="1" CreationDate="2018-05-23T02:04:54.133" Score="5" ViewCount="325" Body="&lt;p&gt;As I understand it after winning two (un-shared) Nobel prizes Linus Pauling began work which eventually had him recommending large amounts of vitamin C to all and sundry.&lt;/p&gt;&#xA;&lt;p&gt;When another scientist was unable to repeat the results a scandal began. Linus Pauling and/or his people made savage professional attacks on the poor researcher who just wanted to know how to get the same results in an experiment. Once the smoke settled it became clear that Pauling's conclusions were wrong after all and he should be ashamed of himself for treating another scientist so shabbily. Much was made about how far the mighty have fallen, a lesson to us all etc.&lt;/p&gt;&#xA;&lt;p&gt;What I never hear about was Pauling's original claim in this regard and how he reached it. By what error or series of errors did he make such mistakes? I can imagine based on the severity of his defense he might have been long passed the rational portion of his life. But what was the primary claim, and for that matter, why do we seem to still act as if it were all true, that vitamin C is the answer; scurvy notwithstanding.&lt;/p&gt;&#xA;" OwnerUserId="7448" LastEditorUserId="4251" LastEditDate="2021-09-10T06:42:20.527" LastActivityDate="2022-09-18T14:08:42.040" Title="What was Pauling's claim about vitamin C?" Tags="|chemistry|debunking|depiction-of-scientists|chemists|pauling|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7391" PostTypeId="2" ParentId="7390" CreationDate="2018-05-23T12:19:00.337" Score="4" Body="&lt;p&gt;Among other things (I was present at one of Pauling's lectures), Pauling used examples of Vitamin C levels in various mammals as evidence that humans should use more.&lt;br&gt;&#xA;FWIW, it was the opinion of most of his peers and colleagues that he had simply lost the power of rational investigation, using slim evidence of vitamin C's ability to strengthen cell walls and/or antioxidant capability as proof of the need for massive doses.  Obviously he never investigated the excretion levels of vitamin C in overdosed test subjects. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you were asking about what diseases Pauling claimed it helped fight, no surprise that cancer was on the list. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-05-23T12:19:00.337" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7392" PostTypeId="1" AcceptedAnswerId="7396" CreationDate="2018-05-23T14:04:08.083" Score="6" ViewCount="720" Body="&lt;p&gt;I understood all the gears and cogwheels in the Antikythera Machine, but I am not sure what make all these stuffs works inside the box.&lt;/p&gt;&#xA;&lt;p&gt;Is it mechanical movement done in hand-wound style?&lt;/p&gt;&#xA;" OwnerUserId="7443" LastEditorUserId="12724" LastEditDate="2021-03-13T07:31:38.803" LastActivityDate="2021-03-13T07:31:38.803" Title="How was the Antikythera Mechanism moved?" Tags="|ancient-greece|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7393" PostTypeId="1" AcceptedAnswerId="7398" CreationDate="2018-05-23T14:13:12.380" Score="13" ViewCount="756" Body="&lt;p&gt;In Wikipedia on the page dedicated to Walter Pitts (accesses today), it is written that,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He is widely remembered to have spent three days in a library, at the age of 12, reading &lt;em&gt;Principia Mathematica&lt;/em&gt; and sent a letter to Bertrand Russell pointing out what he considered serious problems with the first half of the first volume. Russell was appreciative and invited him to study in the United Kingdom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My questions are,&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What were his objections specifically? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any copy of his letter that is freely downloadable? &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What was Russell's reply to him? Is there any copy of his letter that is freely downloadable?  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="user459" LastEditorUserId="55" LastEditDate="2018-05-23T23:20:15.540" LastActivityDate="2021-04-19T07:09:42.507" Title="What were 12 year old Pitts' objections to Principia Mathematica?" Tags="|biographical-details|mathematical-logic|debunking|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7394" PostTypeId="2" ParentId="7392" CreationDate="2018-05-23T14:30:29.743" Score="3" Body="&lt;p&gt;There is no evidence of any spring-driven motion. Antikythera mechanism was not&#xA;a &quot;clock&quot; but a mechanical &quot;calculator&quot;. One probably had to set some wheels on the date and it showed the position of the Sun and Moon on this date.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, I don't know of any evidence of metal springs from antiquity. The artillery was driven by elastic things of biological origin like sinew of hair, and at the later stage they experimented with compressed air. But not with metal springs. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-05-23T14:30:29.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7395" PostTypeId="2" ParentId="3131" CreationDate="2018-05-23T22:08:10.173" Score="11" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gustav_Roch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gustav Roch&lt;/a&gt;, well-known for his Riemann-Roch theorem, died at the age of 26 of tuberculosis, in 1866, just a few months after Riemann's death. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Wolfgang_Doeblin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfgang Döblin&lt;/a&gt;, interested in probability theory, and had Maurice Fréchet as doctoral advisor, committed suicide at the age of 25.&lt;/p&gt;&#xA;" OwnerUserId="3454" LastEditorUserId="3454" LastEditDate="2019-07-10T08:23:04.693" LastActivityDate="2019-07-10T08:23:04.693" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7396" PostTypeId="2" ParentId="7392" CreationDate="2018-05-23T22:09:34.820" Score="9" Body="&lt;p&gt;&lt;strong&gt;UPDATE (March 2021):&lt;/strong&gt; A comprehensive new reconstruction of the Antikythera mechanism has been published in Nature, &lt;a href=&quot;https://www.nature.com/articles/s41598-021-84310-w&quot; rel=&quot;nofollow noreferrer&quot;&gt;Freeth et al., A Model of the Cosmos in the ancient Greek Antikythera Mechanism&lt;/a&gt;. It confirms an early use of epicycles, previously it was conjectured that the mechanism relied on Babylonian tables rather than geometric models, and suggests that the method for deriving periods was the one described in Plato's &lt;em&gt;Parmenides&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Inscriptions specifying complex planetary periods forced new thinking on the mechanization of this Cosmos, but no previous reconstruction has come close to matching the data. Our discoveries lead to a new model, satisfying and explaining the evidence... Features on the Main Drive Wheel indicate that it calculated planetary motions with a complex epicyclic system (gears mounted on other gears), but its design remained a mystery... Such theories were certainly employed in the Antikythera Mechanism, given that the Moon was mechanized using a similar epicyclic theory... The newly-discovered periods for Venus and Saturn are unknown from studies of Babylonian astronomy... We have developed a new theory about how the Venus and Saturn periods were discovered and apply this to restore the missing planetary periods. A dialogue of Plato19 (fifth-fourth century BC) was named after the philosopher Parmenides of Elea (sixth-fifth century BC).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But the principle of the motion transfer from the driving crank through a gear system remains the same as in the earlier reconstructions discussed in the old answer below.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;May 2018:&lt;/strong&gt; Nothing is known for sure about the Antikythera (c.100 BC) due to its bad preservation and the lack of accompanying commentary. Perhaps the closest thing whose operation principle we do know is &lt;a href=&quot;http://kotsanas.com/gb/exh.php?exhibit=0101002&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heron's puppet theater (c. 50 AD)&lt;/a&gt; that used a complicated series of gears, rods, drums, and ropes to move small wooden figures. The contraption is powered by a lead weight that falls at a steady pace in a clepsydra (water clock). With the counter-clockwise turns threads are wound around axles and reels, and with the clockwise turns they are released. Heron also describes the famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Aeolipile&quot; rel=&quot;nofollow noreferrer&quot;&gt;aeolipile&lt;/a&gt;, a bladeless steam turbine that transforms steam power into rotary motion. None of that would work for the Antikythera. &lt;a href=&quot;https://en.wikipedia.org/wiki/Spring_(device)#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coiled springs and wind-up mechanisms&lt;/a&gt; do not appear until 15th century AD.&lt;/p&gt;&#xA;&lt;p&gt;We do have &lt;a href=&quot;http://hist.science.online.fr/antikythera/DOCS/THE%20PLANETARIUM%20OF%20ARCHIMEDES.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plutarch's description of Archimedes's planetarium&lt;/a&gt;, often seen as Antikythera's precursor. It was driven through a single rotary motion redistributed to multiple ones at different rates via a device that Archimedes invented:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Archimedes deserved special admiration because he had thought out a way to represent accurately by a single device for turning the globe those various and divergent movements with their different rates of speed. And when Gallus moved the globe, it was actually true that the moon was always as many revolutions behind the sun on the bronze contrivance as would agree with the number of days it was behind in the sky.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What it is that Archimedes deserves special admiration for we do not actually know, the obvious conjecture is that he used some sort of gear mechanism, but  there is no surviving documentary evidence of that. In the Antikythera we do see gears and a differential turntable that would do the trick. In &lt;a href=&quot;https://academic.oup.com/astrogeo/article/41/6/6.10/225422&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Antikythera Mechanism: Still a mystery of Greek astronomy?&lt;/a&gt; (Astronomy &amp;amp; Geophysics, 41 (2000) no. 6, pp.10-17) Edmunds describes De Solla Price's conjectural reconstruction of how the motion might have been transferred from the driving crank:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There are 30 known preserved gears in the mechanism. Each gear has been hand cut from a single sheet of bronze. The teeth have all been cut at an angle of 60° and are roughly the same size on all gears. De Solla Price speculates that the mechanism is ‘cranked’ by a now lost handle that turned a contrate (as yet unidentified) gear that in turn rotated a Sun position dial and the main drive wheel. The mechanism then followed three known and one undetermined gear trains. One gear train ended through an inner shaft in the main drive wheel axis and turned a Moon position dial. Another train passes, via the differential turntable (see figure 5), to end up at a synodic month dial and then a lunar year dial. De Solla Price speculates that the third known train drives a four-year dial.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The differential turntable is what, apart from the Mechanism's sheer complexity, sets it apart from all other gearing up to the 16th century. The modern differential is more familiar as the device that allows the inner wheel of a car to travel slower than the outer when turning a corner, but with power maintained to both wheels. The ‘drive’ turns two wheels at different speeds. This process can be reversed so that two ‘drives’ turning at different speeds can then give one output, the sum of the two inputs. The two ‘drives’ in this case come from the Sun and Moon gear trains. One train rotates anticlockwise with the speed of the Moon while the other turns clockwise with the speed of the Sun.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lCFqQ.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lCFqQ.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-03-12T19:05:54.570" LastActivityDate="2021-03-12T19:05:54.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7397" PostTypeId="1" CreationDate="2018-05-23T22:38:15.357" Score="1" ViewCount="99" Body="&lt;p&gt;Server Side Include (SSI) is a specific concept and related feature of web servers. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Server_Side_Includes&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Server_Side_Includes&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was this feature first introduced?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Wikipedia article references a Wayback Machine entry from September 1995, but I know I used this feature sometime between 1992 and February 1995, and that it was likely before Common Gateway Interface came along in 1993. The feature was likely introduced in NCSA HTTPd.&lt;/p&gt;&#xA;" OwnerUserId="7456" LastEditorUserId="7456" LastEditDate="2018-05-23T22:55:22.703" LastActivityDate="2018-05-23T22:55:22.703" Title="When was the Server Side Include feature of web servers first made available?" Tags="|computer-science|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7398" PostTypeId="2" ParentId="7393" CreationDate="2018-05-23T23:16:49.137" Score="20" Body="&lt;p&gt;This story bears characteristic signs of a tall tale, although in this case one can identify the origin. It appears to be an amalgamation of two anecdotes, neither of which is itself very credible. Both are traceable to &lt;a href=&quot;https://en.wikipedia.org/wiki/Warren_Sturgis_McCulloch&quot; rel=&quot;noreferrer&quot;&gt;Warren McCulloch&lt;/a&gt;, Pitts's co-author on &quot;A Logical Calculus of Ideas Immanent in Nervous Activity&quot; (1943), which proposed the first mathematical model of a neural network. There is some vague connection to actual events, which might have prompted the story's retelling.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;He is widely remembered to have spent three days&quot; reading Principia in the library, says Wikipedia, without naming any sources of this wide rememberance. Upon closer examination the story is interpolated from &lt;a href=&quot;https://books.google.com/books?id=-l-yim2lNRUC&amp;amp;printsec=frontcover&amp;amp;dq=Talking+Nets:+An+Oral+History+of+Neural+Networks&amp;amp;hl=en&amp;amp;src=bmrr&amp;amp;ei=cQi3TZT2PIrj0QH61_zFCg&amp;amp;sa=X&amp;amp;oi=book_result&amp;amp;ct=book-thumbnail&amp;amp;resnum=1&amp;amp;ved=0CDEQ6wEwAA#v=onepage&amp;amp;q=walter%20pitts&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Talking Nets by Anderson-Rosenfeld&lt;/a&gt;, not exactly a historical work, and the more reputable &lt;a href=&quot;https://muse.jhu.edu/article/46504&quot; rel=&quot;noreferrer&quot;&gt;Smalheiser's short biography of Pitts&lt;/a&gt;. Both name McCulloch as the anecdote's source, Smalheiser even references his Collected Works:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The story is told that, at age 12, Pitts ran into the public library to hide from some bullies, found a copy of Principia Mathematica by the 20th-century philosophers Bertrand Russell and Alfred North Whitehead, and proceeded to read it cover to cover in the next few weeks [1]. Pitts experienced a metaphysical insight that logic rules the universe,  and  as  a  corollary  he  felt  that  ego — and  his  ego  in  particular — needed  to  be  erased  in  order  to  achieve  an  understanding  of  the  world.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is no mention of the letter here, and &quot;the story is told&quot; is an odd way to introduce a story one believes. He then quotes the other story, about Pitts meeting &quot;Bert&quot; in a park in Chicago when he ran away from home at 15 (Anderson-Rosenfeld make him 14 for that). And then &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Now,  this  Bert  talked  with  Walter  for  some time of philosophy and mathematics, and came to realize that this was no ordinary youngster.  Bert  was  impressed.  He  told  the  boy  that  Carnap,  then  Professor  of Philosophy  at  the  University  of  Chicago,  had  written  a  book  that  would  interest&#xA;  him, and urged him to go and speak to the grand old man. So, Walter got himself&#xA;  a  copy  of  Carnap’s  book  and  read  it.  Later,  Carnap  was  to  recount  the  meeting thus: “This young boy came in to see me and said he had read my book and that a certain  paragraph  on  a  certain  page  was  not  clear  to  him...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Guess who told this story. It was Blum with reference to... McCulloch. This time Smalheiser is even more explicit about its credibility:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Though the story of meeting “Bert” in this manner may be apocryphal, it&#xA;  is true that Pitts sat in on Bertrand Russell’s course when the latter visited&#xA;  Chicago  in  1938,  and  that  he  walked  into  Rudolf  Carnap’s  office  with  a marked  copy  of  his  book  filled  with  corrections  and  suggested  improvements. After his initial contact with Carnap, he disappeared, and almost a year went by before Carnap was able to contact him again, supervise some&#xA;  of his studies, and help him secure a student job.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1938 Pitts was indeed 15, but there is no mention that the marked copy of Carnap's book survived, however. And even if we are to believe McCulloch's recollections, the alleged letter, let alone Russell's reply to it, are additions by later &quot;contributors&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-05-24T00:12:52.337" LastActivityDate="2018-05-24T00:12:52.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7399" PostTypeId="2" ParentId="7392" CreationDate="2018-05-24T15:57:34.867" Score="3" Body="&lt;p&gt;Deepening my researches I found a exploded image who shows all of that structure brilliantly described by Conifold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://en.protothema.gr/wp-content/uploads/2014/11/antikythera-mechanism-scientific-American.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Original file hosted here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2gDnD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2gDnD.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7443" LastActivityDate="2018-05-24T15:57:34.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7400" PostTypeId="2" ParentId="5680" CreationDate="2018-05-24T16:57:05.220" Score="15" Body="&lt;p&gt;I am his son and have some of his manuscripts from his days at Cal Tech when he worked on Boolean algebra. He taught at Georgia Tech (1938-1942), Texas Teach (1942-1945), and then at the University of Arizona (1946-1970) where he became a professor. He died June 2, 1970.&lt;/p&gt;&#xA;" OwnerUserId="7458" LastActivityDate="2018-05-24T16:57:05.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7401" PostTypeId="2" ParentId="7337" CreationDate="2018-05-24T20:20:40.457" Score="0" Body="&lt;p&gt;As others have said, it's likely we'll never know the absolute first because this was before written history was common enough to take note of every discovery. But the first I know of is the pre-socratic philosopher &lt;a href=&quot;https://en.wikipedia.org/wiki/Anaximander&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anaximander&lt;/a&gt; (c. 600 BC). He thought the sun and moon were rings of fire that were enclosed by solid wheels with holes in them. You can see a video model of his cosmos &lt;a href=&quot;https://www.thehistoryofphysics.org/2018/03/02/s1e2-ionian-science/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Granted, that's somewhat similar to points of light on the earth's dome. The next candidate would be &lt;a href=&quot;https://en.wikipedia.org/wiki/Anaxagoras&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anaxagoras&lt;/a&gt; (c. 500 BC). He imagined that the sun and stars are fiery stones and that the moon was just an earthy rock that reflected the sun's light. Incidentally, he was tried for impiety for these unorthodox ideas.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you want something a little more concrete than philosophical debate, &lt;a href=&quot;https://en.wikipedia.org/wiki/Aristarchus_of_Samos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristarchus of Samos&lt;/a&gt; (c. 300 BC) was the first to measure the sizes and distances of the sun and moon by using a little geometry and a poor man's early trigonometry.&lt;/p&gt;&#xA;" OwnerUserId="7317" LastEditorUserId="7317" LastEditDate="2018-05-24T20:44:53.717" LastActivityDate="2018-05-24T20:44:53.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7402" PostTypeId="1" CreationDate="2018-05-25T01:32:58.947" Score="1" ViewCount="82" Body="&lt;p&gt;Does anyone know behind stories? As it seems that $p=U/c$ was known before the advent of special theory of relativity, (Halliday, Resnick, Walker says that Maxwell derived this.) when $E=pc$ was first derived by plugging in $m=0$ for $E^2=m^2c^4+p^2c^2$, were physicists happy that the two formulas agreed? Who first noticed this agreement? What was the reaction? And, who derived $E=pc$ from the special theory of relativity point of view? I know that $p=γmv$ was first derived by Planck in 1906, but I do not know much anything else such as who derived $E^2=m^2c^4+p^2c^2$ and who first plugged in $m=0$.&lt;/p&gt;&#xA;" OwnerUserId="7461" LastEditorUserId="7461" LastEditDate="2018-05-27T06:20:05.960" LastActivityDate="2018-05-27T06:20:05.960" Title="Who first noticed that the radiation pressure formula $p=U/c$ agreed with $E=pc$ from special relativity?" Tags="|relativity-theory|electromagnetism|optics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7403" PostTypeId="2" ParentId="654" CreationDate="2018-05-25T11:00:00.973" Score="0" Body="&lt;p&gt;For the distinction of potential and actual infinity see &lt;a href=&quot;https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Transfinity&lt;/a&gt;, in particular chapter I, but also a lot of sources in chapter V, and for some statements without references but easily to be found in the internet and pointing to further sources: the lecture slides &lt;a href=&quot;https://www.hs-augsburg.de/homes/mueckenh/HI/HI11.PPT&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/homes/mueckenh/HI/HI11.PPT&lt;/a&gt; and &lt;a href=&quot;https://www.hs-augsburg.de/homes/mueckenh/HI/HI12.PPT&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hs-augsburg.de/homes/mueckenh/HI/HI12.PPT&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-05-25T11:00:00.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7405" PostTypeId="1" AcceptedAnswerId="7406" CreationDate="2018-05-27T18:12:19.943" Score="4" ViewCount="166" Body="&lt;p&gt;What are the best works summarizing, discussing or criticizing the work of Imre Lakatos? What are the pros and cons of said works? Which would you recommend picking up first if one has read some but not all of his books?&lt;/p&gt;&#xA;" OwnerUserId="7468" LastActivityDate="2018-05-28T09:43:55.363" Title="Reference Request: Books on the work of Imre Lakatos" Tags="|mathematics|philosophy-of-science|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7406" PostTypeId="2" ParentId="7405" CreationDate="2018-05-27T19:33:51.857" Score="5" Body="&lt;p&gt;I cannot do pros/cons because I am not a specialist in this field, but I am aware of the basic literature, as follows:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/9027706549&quot; rel=&quot;noreferrer&quot;&gt;Essays in Memory of Imre Lakatos&lt;/a&gt;&lt;/em&gt; (1976)&lt;/li&gt;&#xA;&lt;li&gt;A  notable critique of Lakatos: Ian Hacking, &quot;&lt;a href=&quot;https://fenix.ciencias.ulisboa.pt/downloadFile/1688987299217455/I.%20Hacking,%20Lakatos.pdf&quot; rel=&quot;noreferrer&quot;&gt;Imre Lakatos’s Philosophy of Science&lt;/a&gt;&quot; (1979)&lt;/li&gt;&#xA;&lt;li&gt;Partial review integrating and responding to Hacking's critiques: Matteo Motterlini, &quot;&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0039368102000249&quot; rel=&quot;noreferrer&quot;&gt;Reconstructing Lakatos: a reassessment of Lakatos’ epistemological project in the light of the Lakatos Archive&lt;/a&gt;&quot; (2002)&lt;/li&gt;&#xA;&lt;li&gt;Motterlini also wrote &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/B004MKMWLE&quot; rel=&quot;noreferrer&quot;&gt;a 1999 book&lt;/a&gt; introducing Lakatos' letters to Feyerabend (although these letters are not particularly deep in their philosophy).&lt;/li&gt;&#xA;&lt;li&gt;A larger review of Lakatos' work: &lt;em&gt;&lt;a href=&quot;https://www.springer.com/us/book/9781402002267&quot; rel=&quot;noreferrer&quot;&gt;Appraising Lakatos: Mathematics, Methodology, and the Man&lt;/a&gt;&lt;/em&gt; (2002)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7469" LastActivityDate="2018-05-27T19:33:51.857" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7408" PostTypeId="2" ParentId="7405" CreationDate="2018-05-28T09:35:24.663" Score="4" Body="&lt;p&gt;The famous &lt;em&gt;Proofs and Refutations&lt;/em&gt; by Lakatos contains an appendix on Cauchy's infinitesimals.  Later Lakatos revisited the subject, offering a rather different viewpoint in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lakatos, Imre.&#xA;  Cauchy and the continuum: the significance of nonstandard analysis for the history and philosophy of mathematics. &#xA;  Math. Intelligencer 1 (1978), no. 3, 151–161.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lakatos' perspective on Cauchy got a rather shoddy treatment in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Schubring, Gert. Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th Century France and Germany. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2005. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This was in turn analyzed in the 2013 publication&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Błaszczyk, Piotr; Katz, Mikhail G.; Sherry, David. Ten misconceptions from the history of analysis and their debunking. Found. Sci. 18 (2013), no. 1, 43–74&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;as well as the 2017 publication&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Błaszczyk, Piotr; Kanovei, Vladimir; Katz, Mikhail G.; Sherry, David Controversies in the foundations of analysis: comments on Schubring's &lt;em&gt;Conflicts&lt;/em&gt;. Found. Sci. 22 (2017), no. 1, 125–140. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-05-28T09:43:55.363" LastActivityDate="2018-05-28T09:43:55.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7409" PostTypeId="2" ParentId="6206" CreationDate="2018-05-28T21:50:23.163" Score="10" Body="&lt;p&gt;&lt;strong&gt;I agree&lt;/strong&gt;, as there is further evidence that Lagrange got his primes from Euler:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1.&lt;/strong&gt; Everyone since Cajori (&lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=1502648&quot; rel=&quot;noreferrer&quot;&gt;1923&lt;/a&gt;, p. 6; &lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=3363427&quot; rel=&quot;noreferrer&quot;&gt;1929&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/b29980343_0002#page/207&quot; rel=&quot;noreferrer&quot;&gt;p. 207&lt;/a&gt;) quotes &lt;em&gt;Théorie des fonctions analytiques&lt;/em&gt; (&lt;a href=&quot;https://archive.org/stream/theoriedesfoncti00lagr#page/14&quot; rel=&quot;noreferrer&quot;&gt;1797&lt;/a&gt;) for the (sic) “new” notation $\ f'x,$ $\ f''x,$ $\ f'''x,$ $\ f^{\scriptsize{\mathrm{IV}}}x$. But Euler’s much earlier &lt;em&gt;Institutionum calculi integralis vol. 3&lt;/em&gt; (&lt;a href=&quot;http://doi.org/10.3931/e-rara-29421&quot; rel=&quot;noreferrer&quot;&gt;1770&lt;/a&gt;) had already systematically deployed his own $\ f'\!\!:\!x,$ $\ f''\!\!:\!x,$ $\ f'''\!\!:\!x,$ $\ f^{\scriptsize{\mathrm{IV}}}\!\!:\!x$ (see e.g. pp. &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/9549700&quot; rel=&quot;noreferrer&quot;&gt;37-41&lt;/a&gt;, 45-65, 75-88, 93-103, &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/9549777&quot; rel=&quot;noreferrer&quot;&gt;108-149&lt;/a&gt;, 234-237, 242-250, &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/9549930&quot; rel=&quot;noreferrer&quot;&gt;262-268&lt;/a&gt;, etc). Notationally, Lagrange just dropped the evaluation colon. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; Cajori (ibid.) adds that Lagrange wrote $\psi'x$ already in (&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1768-pub1770#page/n256&quot; rel=&quot;noreferrer&quot;&gt;1770&lt;/a&gt;, p. 275). But he had $\varphi't,$ $\varphi''t,$ $\varphi'''t,$ $\varphi^{\scriptsize{\mathrm{IV}}}t$ still earlier in (&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo03soci#page/n414&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;, pp. 206, 212, 275) and (&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo02soci#page/n246&quot; rel=&quot;noreferrer&quot;&gt;1762&lt;/a&gt;, pp. 43, 63), and these two papers are both &lt;em&gt;immediately preceded in the same journal (edited by Lagrange) by contributions of Euler, featuring&lt;/em&gt; $\Gamma'\!\!:\!u,$ $\ \Gamma''\!\!:\!u,$ etc. in (&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo03soci#page/n236&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;, pp. 13, 32, 62, 69, etc.) and $\varphi'z,$ $\ \varphi''z$, etc. in (&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo02soci#page/n236&quot; rel=&quot;noreferrer&quot;&gt;1762&lt;/a&gt;, p. 9).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;3.&lt;/strong&gt; Cajori also quotes Jourdain (&lt;a href=&quot;https://zbmath.org/?q=an:44.0048.06&quot; rel=&quot;noreferrer&quot;&gt;1913&lt;/a&gt;, &lt;a href=&quot;https://web.archive.org/web/20171125040350/http://www.mathunion.org/ICM/ICM1912.2/&quot; rel=&quot;noreferrer&quot;&gt;p. 540&lt;/a&gt;) to the effect that Lagrange likely wrote the section of de Foncenex (&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo02soci#page/299&quot; rel=&quot;noreferrer&quot;&gt;1762&lt;/a&gt;, p. 321) where $\xi'x,$ $\xi''x,$ $\xi'''x,$ $\xi^{\scriptsize{\mathrm{IV}}}x\ $ “denote, as is the usual custom, the successive differences of $\xi x$ divided by $dx$”. But &lt;em&gt;this follows the above-mentioned papers of Euler and Lagrange (same volume)&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;4.&lt;/strong&gt; Anyway, as you found, Euler had used his own primes before, in (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=ucm.532509502x&amp;amp;num=512&quot; rel=&quot;noreferrer&quot;&gt;1749&lt;/a&gt;, p. 521) $=$ (&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1748-pub1750#page/n70&quot; rel=&quot;noreferrer&quot;&gt;1750&lt;/a&gt;, p. 79). He did again in (&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1753-pub1755#page/n198&quot; rel=&quot;noreferrer&quot;&gt;1755&lt;/a&gt;, pp. 212, 215), (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092615350&amp;amp;num=246&quot; rel=&quot;noreferrer&quot;&gt;1764&lt;/a&gt;, p. 256), (&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1759-pub1766#page/n202&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;, pp. 195, 220, 250) — written in 1759 and featuring, p. 258, the remarkable sentence &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;les caracteres $\Phi$ &amp;amp; $\Psi$ marquent des fonctions quelconques régulieres ou irrégulieres, d’où par la différentiation on aura les &lt;strong&gt;&lt;em&gt;fonctions dérivées&lt;/em&gt;&lt;/strong&gt; $\Phi'$ &amp;amp; $\Psi'$,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1765-pub1767#page/n318&quot; rel=&quot;noreferrer&quot;&gt;1767&lt;/a&gt;, pp. 316, 328, 332, 344), (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092611839&amp;amp;num=205&quot; rel=&quot;noreferrer&quot;&gt;1773&lt;/a&gt;, pp. 215, 224, 388, 395-399, 429, 436) which mostly drops the colon, etc. Moreover, published correspondence shows Lagrange aware of Euler’s papers (1750, 1762) and book (1770) “qui roule entièrement sur le calcul des fonctions”, by respectively &lt;a href=&quot;https://archive.org/stream/uvresdelagrange14lagr#page/162&quot; rel=&quot;noreferrer&quot;&gt;October 1759&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/uvresdelagrange14lagr#page/325&quot; rel=&quot;noreferrer&quot;&gt;January&lt;/a&gt; or &lt;a href=&quot;https://archive.org/stream/uvresdelagrange14lagr#page/178&quot; rel=&quot;noreferrer&quot;&gt;June 1760&lt;/a&gt;, and &lt;a href=&quot;https://archive.org/stream/uvresdelagrange13lagr#page/195&quot; rel=&quot;noreferrer&quot;&gt;April 1771&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;5.&lt;/strong&gt; Nor were Euler and Lagrange alone in using prime notation: e.g. Monge (&lt;a href=&quot;https://archive.org/stream/mlangesdephilo05soci#page/n312&quot; rel=&quot;noreferrer&quot;&gt;1773&lt;/a&gt;, p. 85), (&lt;a href=&quot;http://www.archive.org/stream/mmoiresdemath07acad#page/267&quot; rel=&quot;noreferrer&quot;&gt;1776&lt;/a&gt;, pp. 268, 288, 298), (&lt;a href=&quot;http://www.archive.org/stream/mmoiresdelacadm01turigoog#page/n84&quot; rel=&quot;noreferrer&quot;&gt;1786&lt;/a&gt;, p. 22), (&lt;a href=&quot;http://www.archive.org/stream/histoiredelacad84hist#page/85&quot; rel=&quot;noreferrer&quot;&gt;1787&lt;/a&gt;, pp. 87, 89, . . . , 172-185), (&lt;a href=&quot;http://doi.org/10.3931/e-rara-26758&quot; rel=&quot;noreferrer&quot;&gt;1801&lt;/a&gt;, Nº &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/8614336&quot; rel=&quot;noreferrer&quot;&gt;8&lt;/a&gt;, 10, 14, &lt;a href=&quot;http://numix.sabix.org/ice/ice_page_detail.php?type=img&amp;amp;bdd=numix&amp;amp;table=numix&amp;amp;bookId=22&amp;amp;typeofbookDes=Cours-X&amp;amp;pageOrder=73&quot; rel=&quot;noreferrer&quot;&gt;16&lt;/a&gt;, &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/8614433&quot; rel=&quot;noreferrer&quot;&gt;33&lt;/a&gt;), Cousin (&lt;a href=&quot;http://resolver.sub.uni-goettingen.de/purl?PPN597733910&quot; rel=&quot;noreferrer&quot;&gt;1777&lt;/a&gt;, p. 283), Marie (&lt;a href=&quot;https://archive.org/stream/lezionielementa00fontgoog#page/n18&quot; rel=&quot;noreferrer&quot;&gt;1793&lt;/a&gt;, pp. 8, 18), Bossut (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433069082703&amp;amp;num=427&quot; rel=&quot;noreferrer&quot;&gt;1798&lt;/a&gt;, p. 427), . . .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the “earliest uses” site, Cajori, &lt;a href=&quot;https://en.wikipedia.org/wiki/Notation_for_differentiation#Lagrange&amp;#39;s_notation&quot; rel=&quot;noreferrer&quot;&gt;wikipedia&lt;/a&gt;, et al., seem far from telling the whole story.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Euler titles&lt;/strong&gt; (see &lt;em&gt;Opera Omnia&lt;/em&gt; (2) vol. 10, 11 (&lt;a href=&quot;https://d-nb.info/365758337/04&quot; rel=&quot;noreferrer&quot;&gt;1947&lt;/a&gt;, &lt;a href=&quot;https://d-nb.info/365758264/04&quot; rel=&quot;noreferrer&quot;&gt;1957&lt;/a&gt;) and (3) vol. 1 (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k70068/f29&quot; rel=&quot;noreferrer&quot;&gt;1926&lt;/a&gt;)):&lt;br&gt;&#xA;(&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=ucm.532509502x&amp;amp;num=512&quot; rel=&quot;noreferrer&quot;&gt;1749&lt;/a&gt;) E119: De vibratione chordarum exercitatio.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1748-pub1750#page/n70&quot; rel=&quot;noreferrer&quot;&gt;1750&lt;/a&gt;) E140: Sur la vibration des cordes.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1753-pub1755#page/n198&quot; rel=&quot;noreferrer&quot;&gt;1755&lt;/a&gt;) E213: Remarques sur les mémoires précedens de M. Bernoulli.&lt;br&gt;&#xA;(&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo02soci#page/n236&quot; rel=&quot;noreferrer&quot;&gt;1762&lt;/a&gt;) E268: Lettre de M. Euler à M. de la Grange.&lt;br&gt;&#xA;(&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092615350&amp;amp;num=246&quot; rel=&quot;noreferrer&quot;&gt;1764&lt;/a&gt;) E287: De motu vibratorio cordarum inaequaliter crassarum.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1759-pub1766#page/n202&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) E305: De la propagation du son.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1759-pub1766#page/n228&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) E306: Supplément aux recherches sur la propagation du son.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1759-pub1766#page/n261&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) E307: Continuation des recherches sur la propagation du son.&lt;br&gt;&#xA;(&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo03soci#page/n236&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) E317: Éclaircissemens sur le mouvement des cordes vibrantes.&lt;br&gt;&#xA;(&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo03soci#page/n262&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) E318: Recherches sur le mouvement des cordes inégalement grosses.&lt;br&gt;&#xA;(&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo03soci#page/n295&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) E319: Recherches sur l’intégration de l’équation&#xA;$\smash{\bigl(\frac{ddz}{dt^2}\bigr)=aa\bigl(\frac{ddz}{dx^2}\bigr)+\frac bx\bigl(\frac{dz}{dx}\bigr)+\frac c{xx}z}$.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1765-pub1767#page/n318&quot; rel=&quot;noreferrer&quot;&gt;1767&lt;/a&gt;) E339: Sur le mouvement d’une corde qui au commencement n’a été ébranlée que dans une partie.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1765-pub1767#page/n349&quot; rel=&quot;noreferrer&quot;&gt;1767&lt;/a&gt;) E340: Éclaircissemens plus détaillés sur la génération et la propagation du son, et sur la formation de l’echo.&lt;br&gt;&#xA;(&lt;a href=&quot;http://doi.org/10.3931/e-rara-29421&quot; rel=&quot;noreferrer&quot;&gt;1770&lt;/a&gt;) E385: &lt;em&gt;Institutionum calculi integralis volumen tertium&lt;/em&gt;.&lt;br&gt;&#xA;(&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092611839&amp;amp;num=205&quot; rel=&quot;noreferrer&quot;&gt;1773&lt;/a&gt;) E433: Digressio de traiectoriis tam orthogonalibus quam obliquangulis.&lt;br&gt;&#xA;(&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092611839&amp;amp;num=381&quot; rel=&quot;noreferrer&quot;&gt;1773&lt;/a&gt;) E439: De chordis vibrantibus disquisitio ulterior.&lt;br&gt;&#xA;(&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092611839&amp;amp;num=422&quot; rel=&quot;noreferrer&quot;&gt;1773&lt;/a&gt;) E441: De motu vibratorio chordarum ex partibus quotcunque diversae&#xA;crassitiei compositarum.&lt;br&gt;&#xA;(&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=hvd.32044092611839&amp;amp;num=432&quot; rel=&quot;noreferrer&quot;&gt;1773&lt;/a&gt;) E442: De motu vibratorio chordarum crassitie utcunque variabili praeditarum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Lagrange titles&lt;/strong&gt; (see Taton (&lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=505097&quot; rel=&quot;noreferrer&quot;&gt;1974&lt;/a&gt;)):&lt;br&gt;&#xA;(&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo02soci#page/n246&quot; rel=&quot;noreferrer&quot;&gt;1762&lt;/a&gt;) T6: Nouvelles recherches sur la nature et la propagation du son.&lt;br&gt;&#xA;(&lt;a href=&quot;http://www.archive.org/stream/mlangesdephilo03soci#page/n414&quot; rel=&quot;noreferrer&quot;&gt;1766&lt;/a&gt;) T11: Solution de différens problèmes de calcul intégral.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/berlin-histoire-1768-pub1770#page/n256&quot; rel=&quot;noreferrer&quot;&gt;1770&lt;/a&gt;) T18: Nouvelle méthode pour résoudre les équations littérales par le moyen des séries.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/nouv-berlin-1772#page/n270&quot; rel=&quot;noreferrer&quot;&gt;1774&lt;/a&gt;) T33: Sur une nouvelle espece de calcul, rélatif à la différentiation et à l'intégration des quantités variables.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/theoriedesfoncti00lagr#page/14&quot; rel=&quot;noreferrer&quot;&gt;1797&lt;/a&gt;) T102: &lt;em&gt;Théorie des fonctions analytiques&lt;/em&gt;.&lt;br&gt;&#xA;(&lt;a href=&quot;https://archive.org/stream/sancesdescolesn07fragoog#page/n28&quot; rel=&quot;noreferrer&quot;&gt;1801&lt;/a&gt;) T114: &lt;em&gt;Leçons sur le calcul des fonctions&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-06-06T09:14:21.777" LastActivityDate="2018-06-06T09:14:21.777" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7411" PostTypeId="1" CreationDate="2018-05-30T13:45:41.557" Score="0" ViewCount="121" Body="&lt;p&gt;I had a course on History of Mathematics this semester and it seemed that each civilization became known for developing a particular area of mathematics. For example, Arithmetic and Ancient Egypt/Babylonia, Geometry and Ancient Greece, Algebra and Islam, Analysis and northern Europe, etc.&#xA; Is this the case at all, were those areas really more researched, or do we associate one with the other because it was the first time it was studied? If the former is true, what area/areas of mathematics do you think we'll be remembered for developing?&lt;/p&gt;&#xA;" OwnerUserId="7481" LastActivityDate="2018-05-30T16:06:22.433" Title="What field will we be remembered for developing?" Tags="|mathematics|ancient-greece|" AnswerCount="1" CommentCount="2" ClosedDate="2018-05-31T22:26:33.053" ContentLicense="CC BY-SA 4.0" />
  <row Id="7412" PostTypeId="1" CreationDate="2018-05-30T14:34:40.650" Score="0" ViewCount="506" Body="&lt;p&gt;Is there a list of all the people who attempted to prove the parallel postulate (also known as the fifth postulate or the Euclid axiom) in Euclidean geometry? Wikipedia has a page on the subject but the list given there is far too short. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what I have collected so far, any addition is welcome.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Archimedes (~ -287- -212)&#xA;Posidonios (~-135 ~-51)&#xA;Geminus (~-10 ~60)&#xA;Claude Ptoleme (~90 ~168)&#xA;Proclus (412 485)&#xA;Simplicius (~490 ~560)&#xA;Thabit ibn Qurra (826 901)&#xA;Al-Abbās ibn Said al-Jawharī (~800 ~860)&#xA;Abu'l Abbas al-Fadl ibn Hatim Al-Nayrizi (~865 ~922)&#xA;Ibn al-Haytham (965 1039)&#xA;Avicenna (980-1037)&#xA;Omar Khayyam (1048-1131)&#xA;Nasir ad-Din at-Tusi (1201-1274)&#xA;Erazmus Ciołek Witelo (~1230-~1300)&#xA;Rabbi Levi ben Gershom (1288-1344)&#xA;Pietro Antonio Cataldi (1548-1626)&#xA;Giovanni Alfonso Borelli (1608-1679)&#xA;John Wallis (1616-1703)&#xA;Giovanni Girolamo Saccheri (1667-1733)&#xA;Johann Heinrich Lambert (1728-1777)&#xA;Joseph-Louis Lagrange (1736-1813)&#xA;Adrien-Marie Legendre (1752-1833)&#xA;Farkas Bolyai (1775-1856)&#xA;Ferdinand Carl Schweikart (1780-1859)&#xA;Friedrich Ludwig Wachter (1792-1818)&#xA;Franz Adolph Taurinus (1794-1874)&#xA;Viktor Yakovlevich Bunyakovsky (1804-1889)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;EDIT: added Farkas Bolyai, Joseph-Louis Lagrange, as pointed out in the comments.&lt;/p&gt;&#xA;" OwnerUserId="6959" LastEditorUserId="6959" LastEditDate="2018-05-31T22:52:45.257" LastActivityDate="2018-05-31T22:52:45.257" Title="mathematicians attempts at proving Euclid postulate" Tags="|reference-request|euclidean-geometry|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7413" PostTypeId="2" ParentId="7412" CreationDate="2018-05-30T15:42:27.463" Score="1" Body="&lt;p&gt;There are several nearly insurmountable problems with carrying out this project, such as where to draw the line (pun intended) in determining what suffices as &quot;attempted to prove&quot; and the huge number of mathematical papers, books, manuscripts, letters, etc. in over a dozen languages that would have to be analyzed. Also, simply coming up with such a list of names is certainly of little interest, as one would want to know at least something about what each person did in trying to resolve the parallel postulate issue, since such a list of names would probably include a large minority, if not an actual majority, of the mathematicians who lived before the 1800s. For example, I notice Gauss is missing from your list, but surely Gauss must have made some attempts in his early years (perhaps in the 1790s or the first decade of the 1800s) --- see &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.bams/1183486559&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://link.springer.com/chapter/10.1007/0-387-29555-0_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That said, if you really want to dive into this (which will likely be a several decades project), one place to begin is&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/bibliographyofno00sommuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Bibliography of Non-Euclidean Geometry, Including the Theory of Parallels, the Foundations of Geometry, and Space of&lt;/strong&gt; $n$ &lt;strong&gt;Dimensions&lt;/strong&gt;&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Duncan_Sommerville&quot; rel=&quot;nofollow noreferrer&quot;&gt;D. M'L. Y. Sommerville&lt;/a&gt; (1911, xii + 403 pages)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since this book was published in 1911, there is a huge amount of subsequently published historical literature that you'll also need to go through, as I am sure there are many things that have come to light since the time Sommerville published his bibliography.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2018-05-30T15:42:27.463" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7414" PostTypeId="2" ParentId="7411" CreationDate="2018-05-30T16:06:22.433" Score="3" Body="&lt;p&gt;Once I've heard a lecture of David Kazhdan (a very prominent mathematician), who said the following:&#xA;&quot;18th century was the century of Analysis, 19th century was a century of group theory, and 20th century was the century of cohomology&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I disagree with this, but just wanted to communicate his opinion.&#xA;(In general I think that such statements are meaningless. There was SOME geometry&#xA;in Egypt and Babylon, and very much arithmetic in the Ancient Greece, and it is somewhat questionable which civilization invented algebra, except the name. And so on.&#xA;Of course Kazhdan's statement is even more open to crriticism).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-05-30T16:06:22.433" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7415" PostTypeId="2" ParentId="6547" CreationDate="2018-05-30T18:03:22.973" Score="1" Body="&lt;p&gt;As far as I can tell, &quot;coordinatizing the plane&quot; is a method invented by Rene Descartes and is given as an explicit example in his famous book &lt;a href=&quot;https://en.wikipedia.org/wiki/Discourse_on_the_Method&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discourse on the method&lt;/a&gt; of what the method produces when applied to geometry. So I guess that $R^2$ as the plane spread in Europe as fast as the discourse on the method, starting from its publication in 1637. Note that &quot;geometry with coordinates&quot; is often called cartesian geometry, in honor of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rene Descartes&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let me spell out the four precepts of the method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&gt; we are accepting here the standard rules concerning numbers and their relationship to the line. We see the ruler so clearly, as clearly as $2+2=4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&gt; We divide the plane into two lines and decompose all our geometric objects along these lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&gt; How lengthy the resulting computations are, we proceed step by step with the uttermost rigor, without omission and without errors. Well, we proceed with method, but I think that we really own that expression to Descartes!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-&gt; We synthetize everything back to the plane. And it works!&lt;/p&gt;&#xA;" OwnerUserId="6959" LastActivityDate="2018-05-30T18:03:22.973" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7416" PostTypeId="1" CreationDate="2018-05-30T19:41:09.887" Score="5" ViewCount="469" Body="&lt;p&gt;In John Henry Newman's &lt;em&gt;The Idea of a University&lt;/em&gt; (c. 1850), he writes that, in mathematical science, we are told of&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...the existence of an infinite number of curves, which are able to&#xA;  divide a space, into which no straight line, though it be length&#xA;  without breadth, can even enter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He uses this as an example of an idea which, though it might run counter to intuition, need not be immediately rejected.  But, in modern terms, what mathematical concept is Newman referring to?&lt;/p&gt;&#xA;" OwnerUserId="3811" LastActivityDate="2019-05-07T13:03:25.673" Title="What is Newman's &quot;infinite number of curves&quot;?" Tags="|mathematics|infinity|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7417" PostTypeId="1" CreationDate="2018-05-31T12:13:02.223" Score="0" ViewCount="74" Body="&lt;p&gt;Would anyone know a way to figure out who wrote the (rather dithyrambic) review of D'Alembert’s &lt;em&gt;Opuscules mathématiques&lt;/em&gt;, vol. 6 (1773), found in &lt;em&gt;Mercure de France&lt;/em&gt;, April 1773, &lt;a href=&quot;https://books.google.com/books?id=DgOeFJg3484C&amp;amp;pg=RA1-PA128&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 127-132&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems remarkable for using the then-novel(?) phrase &lt;a href=&quot;https://archive.org/stream/sancesdescolesn07fragoog#page/n12&quot; rel=&quot;nofollow noreferrer&quot;&gt;“calcul des fonctions”&lt;/a&gt;, thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;M. d'Alembert (...) emploie, pour démontrer rigoureusement cette loi, une théorie nouvelle, celle du calcul des fonctions dont il est le premier inventeur. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-05-31T18:43:45.623" Title="Author of a review in *Mercure de France*" Tags="|mathematics|calculus|publishing|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7419" PostTypeId="2" ParentId="7417" CreationDate="2018-05-31T18:43:45.623" Score="1" Body="&lt;p&gt;It might be very difficult to get beyond conjecture here, but I would suggest as a possibility that this review of the sixth volume of d'Alembert's independently-published mathematical works could be an early piece by Laplace. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Laplace after arrival in Paris in 1769 had obtained his first post by the help of d'Alembert, and in March 1773, shortly before this review appeared, Laplace had been elected as a &lt;a href=&quot;https://fr.wikipedia.org/wiki/Pierre-Simon_de_Laplace#L%E2%80%99ascension&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;membre adjoint&lt;/em&gt; of the &lt;em&gt;Académie royale des sciences&lt;/em&gt;&lt;/a&gt;. So it seems that the young Laplace would have been both eligible to write and publish such a review of d'Alembert's book, and possibly interested to do so. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know whether there are sources of evidence that could help to test such possibilities: Laplace was a successor who eclipsed his early patron, and in later life was also known to be reticent in acknowledging some of his 'roots' and early activities. As Michel Paty has noted in &lt;a href=&quot;https://halshs.archives-ouvertes.fr/halshs-00004320/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Les recherches actuelles sur d’Alembert (2002)&lt;/a&gt;, the difficult task of historians of science is to bring out hidden or forgotten contributions by taking away those later 'sediments' that have dimmed and blurred them. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-05-31T18:43:45.623" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7420" PostTypeId="1" CreationDate="2018-06-01T01:02:58.127" Score="6" ViewCount="1147" Body="&lt;p&gt;Did it influence the work of Newton or Leibniz, i have often heard that Europeans &quot;stole&quot; calculus from the Kerala school, these are views often parroted by Indian nationalists, but how accurate is it?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastEditorUserId="658" LastEditDate="2018-07-23T06:47:19.663" LastActivityDate="2018-08-04T12:24:15.467" Title="How influential was the Kerala school to European development in Calculus?" Tags="|mathematics|terminology|calculus|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7421" PostTypeId="1" CreationDate="2018-06-01T11:01:23.897" Score="1" ViewCount="136" Body="&lt;p&gt;Aristotle was chatting about forces we later identified as gravity at about 300BC. He made a few mistakes regarding heavy objects falling faster etc, but that is by the by. Was he the first ever to talk about.... &quot;gravity&quot;? or were there others before him?&lt;/p&gt;&#xA;" OwnerUserId="7488" LastEditorUserId="426" LastEditDate="2018-06-03T08:56:18.400" LastActivityDate="2018-06-03T08:56:18.400" Title="Was Aristotle first to talk about gravity?" Tags="|ancient-greece|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="7423" PostTypeId="1" CreationDate="2018-06-01T16:10:58.053" Score="5" ViewCount="129" Body="&lt;p&gt;I'm trying to find a book similiar to Stephen Brush &quot;Kinetic Theory of Gases: An Anthology of Classic Papers With Historical Commentary&quot;.&lt;br&gt;&#xA;The electricity version, let's say!&lt;br&gt;&#xA;I like to see how subjects and theories developed and nothing is more interesting to read classic papers on the subjects.&lt;br&gt;&#xA;If you do know anything, please share!&lt;/p&gt;&#xA;" OwnerUserId="7250" LastActivityDate="2018-06-01T20:14:10.337" Title="Is there an anthology of classic papers on electricity?" Tags="|physics|electromagnetism|resource-recommendation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7424" PostTypeId="2" ParentId="7423" CreationDate="2018-06-01T17:28:09.397" Score="0" Body="&lt;p&gt;This doesn't contain the actual papers, unfortunately, but perhaps you would still find it useful: &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History of the Theories of Aether and Electricity&lt;/em&gt;&lt;/a&gt;. It's fairly detailed and is a second-best alternative if you can't find an anthology of classic papers.&lt;/p&gt;&#xA;" OwnerUserId="7317" LastActivityDate="2018-06-01T17:28:09.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7425" PostTypeId="2" ParentId="7423" CreationDate="2018-06-01T18:20:12.710" Score="1" Body="&lt;p&gt;If French works for you, O. Darrigol's &lt;a href=&quot;https://www.amazon.fr/dp/2701130751&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Les équations de Maxwell: de MacCullagh à Lorentz&lt;/em&gt; (2005)&lt;/a&gt; is such an anthology, with comments drawn from his earlier &lt;a href=&quot;https://zbmath.org/?q=an:0954.78001&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt;. Such collections are likely a dying breed, as all papers (in this case, nine by MacCullagh, Maxwell, Lorenz, Heaviside, Hertz, Lorentz) become freely available online.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-06-01T18:25:16.797" LastActivityDate="2018-06-01T18:25:16.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7426" PostTypeId="2" ParentId="7423" CreationDate="2018-06-01T20:14:10.337" Score="1" Body="&lt;p&gt;See the hefty § &quot;Magnetism and Electricity&quot;, pp. 387-616 of &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5699&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Source Book in Physics&lt;/em&gt;&lt;/a&gt; by William Francis Magie. It contains selections, prefaced with a biography, of 37 scientists including Gilbert, Franklin, Coulomb, Galvani, Volta, Ørsted, Biot and Savart, Arago, Ampère, Ohm, Faraday, Lenz, Gauss, Joule, Maxwell, Hall, Hertz, et al.&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-06-01T20:14:10.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7427" PostTypeId="1" CreationDate="2018-06-02T14:35:57.187" Score="6" ViewCount="95" Body="&lt;p&gt;In 1878, C. S. Peirce performed a calculation that (I think) would be better done using chi-squared testing — but Pearson hasn’t introduced that yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What exactly is Peirce doing here in the last sentence? Is it valid? almost valid?  (When I did a chi-squared test, I got a different answer than he did.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/imQWv.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/imQWv.png&quot; alt=&quot;Peirce calculates the probability that a difference can be due to chance.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first paragraph is straightforward enough. Where we would now say the probable error in the distribution of a sample is within +/- .6745 standard deviations, he has .477 * sqrt(2), and so on down his table.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the next paragraph, he approximates &lt;em&gt;p&lt;/em&gt; = 0.5 and then correctly calculates probable error on the two distributions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then what does he do in the last sentence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Once I figure that out, I’ll need to figure out if this was a common pre-Pearson approach.)&lt;/p&gt;&#xA;" OwnerUserId="7494" LastActivityDate="2018-06-16T16:41:09.797" Title="What is Peirce doing in this pre-Chi-squared example?" Tags="|scientific-method|statistics|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7428" PostTypeId="1" CreationDate="2018-06-03T10:49:40.207" Score="4" ViewCount="76" Body="&lt;p&gt;For the purposes of this question, a technique is considered &lt;em&gt;adversarial&lt;/em&gt; if it develops two systems (or two kinds of system), the first trying to process certain kinds of data well, and the second trying to provide the first with hard-to-process data. Examples include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Adversarial_machine_learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adversarial machine learning&lt;/a&gt;, in which the processor tries to identify security risks, while the agent trying to fool it seeks to bypass security checks;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Generative_adversarial_network&quot; rel=&quot;nofollow noreferrer&quot;&gt;Generative adversarial networks&lt;/a&gt;, in which the processor is discriminative (i.e. it models a conditional probability distribution) and the agent trying to fool it is generative (i.e. it models a joint probability distribution).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Judging by the dates discussed in the above links, AML dates to the early 2000s while GANs date to 2014. These both seem very recent, and I suspect the general meaning of &quot;adversarial&quot; I specify above has been used for much longer. Clearly, the idea of requiring continual adaptation by both agents to shifting goalposts need not be tied to such specific examples as security or Bayesian models. Any simulation of two competing species evolving, e.g. a predator and its prey, will be &quot;adversarial&quot; in the sense of this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is: when was some kind of adversarial technique first used, in the hope it would lead to machines learning to solve problems more effectively than would happen without adversarial techniques? (A simulation that illustrates how biological evolution works, without wanting to solve a specific data-driven problem of human interest, would not qualify.)&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-06-03T20:33:19.830" Title="When were adversarial techniques first used in machine learning?" Tags="|machine-learning|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7429" PostTypeId="2" ParentId="7428" CreationDate="2018-06-03T20:33:19.830" Score="3" Body="&lt;p&gt;I am unsure as to the parameters of what is being asked, so this may not answer it. First, &quot;data-driven problem&quot; is a term linked to the recent rise of data science and projecting it too far into the past risks modernizing people's intentions. Second, biological evolution need not work by &quot;competition&quot; between two or more species, natural selection applies even to a single species in its &quot;competition&quot; with the environment. And in artificial selection (real or simulated) the &lt;em&gt;environment&lt;/em&gt; can be intentionally manipulated to make it an &quot;adversary&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lotka and Volterra developed &lt;a href=&quot;https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;predator-prey models&lt;/a&gt; in 1920-s, but there were no &quot;data machines&quot; then and this probably would not be &quot;wanting to solve a specific data-driven problem of human interest&quot;. Closer is perhaps Turing's celebrated 1950 paper &lt;a href=&quot;https://www.csee.umbc.edu/courses/471/papers/turing.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computing Machinery and Intelligence&lt;/a&gt;, where he proposed his &quot;imitation game&quot; where a machine tries to fool a human interrogator, the origin of the &quot;Turing test&quot; of machine intelligence, and even suggested &quot;evolutionary learning&quot; as a means of achieving it: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We  have  thus  divided  our  problem  into  two  parts.   The child-programme  and the  education  process.   These two remain very  closely connected. We cannot  expect to find a  good child-machine at the  first  attempt. One  must experiment with teaching one such  machine and  see how well it  learns. One can then try another  and  see  if  it  is  better  or  worse.   There  is  an  obvious connection  between  this  process  and  evolution... Equally important is the  fact  that  he  is not  restricted  to  random  mutations.    If  he can trace a cause for  some weakness he can probably think  of the kind  of mutation which will improve it... It  is probably  wise to  include a  random  element  in a  learning machine... Since  there  is  probably  a  very  large number   of   satisfactory   solutions  the   random   method   seems to  be better than the systematic.   It  should  be noticed that  it  is&#xA;  used   in  the  analogous  process   of   evolution.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is not clear to me that Turing proposes &lt;em&gt;automating&lt;/em&gt; both &quot;adversaries&quot; here, but this paper is often cited as seminal on machine learning and &lt;a href=&quot;https://en.wikipedia.org/wiki/Genetic_algorithm#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;genetic algorithms&lt;/a&gt;, where candidate solutions to optimization or search problems are treated as an evolving population of &quot;individuals&quot; with &quot;traits&quot;, and generational selection is implemented according to &quot;fitness&quot;. It is believed that Fraser's 1957 simulations of artificial selection incorporated all the requisite elements of such algorithms, but again, I am not sure if this was &quot;data-driven&quot;. Bremermann in the 1960s used &quot;evolving programs&quot; specifically to solve optimization problems, see e.g. &lt;a href=&quot;https://ieeexplore.ieee.org/document/870787&quot; rel=&quot;nofollow noreferrer&quot;&gt;Revisiting Bremermann’s Genetic Algorithm by Fogel and Anderson&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the more narrowly understood agent vs agent machine learning that is indeed recent, although &lt;a href=&quot;https://en.wikipedia.org/wiki/Computer_Go#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;computer programs playing against each other&lt;/a&gt; dates back to 1980-s.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-06-03T20:33:19.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7430" PostTypeId="1" CreationDate="2018-06-03T21:16:53.400" Score="8" ViewCount="251" Body="&lt;p&gt;What are some of the original examples of uses of differential equations for modeling and analyzing chemical reactions, particularly those relevant to biochemistry, involving proteins and enzymes? Michaelis and Menten's work in 1910s is one example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the earlier examples or antecedents to this work? (references to good historical accounts of this modeling endeavor would also be helpful).&lt;/p&gt;&#xA;" OwnerUserId="7496" LastEditorUserId="55" LastEditDate="2018-06-05T00:35:41.743" LastActivityDate="2018-10-03T14:31:53.400" Title="What were the early uses of differential equations for modeling chemical reactions?" Tags="|mathematics|chemistry|biology|differential-equations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7431" PostTypeId="2" ParentId="7420" CreationDate="2018-06-04T09:21:35.620" Score="10" Body="&lt;p&gt;In my work with primary sources in such authors as &lt;a href=&quot;http://dx.doi.org/10.1007/s10699-017-9542-y&quot; rel=&quot;noreferrer&quot;&gt;Fermat&lt;/a&gt; and &lt;a href=&quot;http://dx.doi.org/10.1007/s10838-016-9334-z&quot; rel=&quot;noreferrer&quot;&gt;Leibniz&lt;/a&gt;, I have occasionally come across references to earlier works by Arab mathematicians, but have never seen references to work by the Kerala school.  Speculations about Kerala influences are based on the existence of trade routes connecting Europe with India.  Apart from this being a highly speculative matter, I have always wondered why the existence of trade routes should be interpreted as evidence in favor of influence of Kerala on Europe rather than influence of Europe of Kerala.  After all, the traffic presumably was not a one-way street.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-06-04T09:21:35.620" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7432" PostTypeId="1" AcceptedAnswerId="7433" CreationDate="2018-06-04T17:55:24.210" Score="0" ViewCount="153" Body="&lt;p&gt;Ginga is a highly creative and elegant form of playing soccer or football. It was the playing style of legendary footballers like Pele and Garrincha.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I want to know is whether there is a similar form or problem solving methodology in mathematics which can be rightly referred to as the &quot;Ginga style&quot; of doing mathematical sciences?&lt;/p&gt;&#xA;" OwnerUserId="6418" LastEditorUserId="55" LastEditDate="2018-06-05T00:06:27.567" LastActivityDate="2018-06-05T00:06:27.567" Title="Is there an analog of Ginga style in mathematics?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ClosedDate="2018-06-14T01:24:21.327" ContentLicense="CC BY-SA 4.0" />
  <row Id="7433" PostTypeId="2" ParentId="7432" CreationDate="2018-06-05T00:05:14.123" Score="5" Body="&lt;p&gt;&quot;Ginga&quot; refers to the game style full of surprise moves or tricks, it is not really a methodology. I suppose solving &lt;a href=&quot;http://www.mathematicalgemstones.com/misc/olympiad-vs-higher-math&quot; rel=&quot;noreferrer&quot;&gt;Olympiad style math problems&lt;/a&gt; would be a loose analog. However, it is not uncontroversially admired, Terence Tao warns, for example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does. (While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Still, it worked out for Tao, an International Math Olympiad gold winner at 13 he went on to become a Fields medalist (mathematical &quot;Nobel&quot; winner) at 31. The famous older examples at a higher level include the style of &quot;trickster&quot; problem-solvers like &lt;a href=&quot;http://blog.stephenwolfram.com/2016/04/who-was-ramanujan&quot; rel=&quot;noreferrer&quot;&gt;Ramanujan&lt;/a&gt; with his neat and unexpected identities, or versatile &lt;a href=&quot;https://theconversation.com/the-man-who-turned-coffee-into-theorems-16008&quot; rel=&quot;noreferrer&quot;&gt;Paul Erdős&lt;/a&gt;. Grothendieck was famously of the diametrically opposite stylistic persuasion, promoting the systematic &quot;rising sea&quot; approach instead:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The unknown thing to be known appeared to me as some stretch of earth or&#xA;  hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance&lt;/em&gt;&quot;. [Quoted from &lt;a href=&quot;http://www.landsburg.com/grothendieck/mclarty1.pdf&quot; rel=&quot;noreferrer&quot;&gt;MacLarty's The Rising Sea: Grothendieck on simplicity and generality&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Grothendieck even complained that Deligne, his student, proved the famous Weil conjectures with a &quot;trick&quot; instead of raising the sea around them. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-06-05T00:05:14.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7434" PostTypeId="2" ParentId="7430" CreationDate="2018-06-05T00:34:35.647" Score="3" Body="&lt;p&gt;An early result in chemical kinetics was Waage and Guldberg's law of mass action, that relates the speed of a chemical reaction to the quantity of the reacting substances. The field came of age in 1884 with van'tHoff's &quot;Etudes de dynamique chimique&quot;, which is filled with differential equations (see the &lt;a href=&quot;https://archive.org/details/studiesinchemica00hoffrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;English 1895 translation&lt;/a&gt;), he was awarded the first Nobel Prize in Chemistry in 1901. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Chemical_kinetics#cite_note-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's History of Chemical Kinetics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The historical context of the Michaelis-Menten 1913 paper Die Kinetik der Invertinwirkung (Biochem. Z. 49, 333–369) is described in &lt;a href=&quot;https://pubs.acs.org/doi/abs/10.1021/bi201284u&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Original Michaelis Constant Johnson and Goody &lt;/a&gt;, who translated it into English. According to them, their chief precursor was Victor Henri in his Lois Générales de l’Action des Diastases (1903):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Perhaps the unsung hero of the early history of enzymology is Victor Henri, who first derived an equation predicting the relationship between rate and substrate concentration based upon a rational model involving the formation of a catalytic enzyme−substrate complex. However, as Michaelis and Menten point out, Henri made two crucial mistakes, which prevented him from confirming the predicted  relationship between rate and substrate concentration. He failed to account for the slow mutarotation of the products of the reaction (equilibration of the α and β anomers of glucose), and he neglected to control pH. Thus, errors in his data precluded an accurate test of the theory. Otherwise, we would probably be writing about the Henri equation.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A more distant inspiration was the Henderson-Hasselbalch equation for pH dependence on acidity derived by Henderson in 1908 and turned into the logarithmic form that Michaelis and Menten imitated by Hasselbalch.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-06-05T00:34:35.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7435" PostTypeId="2" ParentId="7430" CreationDate="2018-06-06T11:40:11.250" Score="2" Body="&lt;p&gt;A celebrated textbook on enzyme kinetics is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fundamentals of Enzyme Kinetics by Athel Cornish-Bowden, Fourth Edition, 2012, Wiley-VCH, Weinheim.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://www.wiley-vch.de/home/fundenzykinet&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wiley-vch.de/home/fundenzykinet&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As a sample, one can download the &lt;a href=&quot;https://application.wiley-vch.de/books/sample/3527330747_c01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;first chapter&lt;/a&gt; for free, which intersperses historical references with a discussion of the basic principles of enzyme kinetics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From that chapter, I think this paragraph is most relevant to the question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chemical kinetics as a science began in the middle of the 19th century, when Wilhelmy was apparently the first to recognize that the rate at which a chemical reaction proceeds follows definite laws, but although his work paved the way for the law of mass action of Waage and Guldberg, it attracted little&#xA;  attention until it was taken up by Ostwald towards the end of the century, as discussed by Laidler. Wilhelmy realized that chemical rates depended on the concentrations of the reactants, but before considering some examples we need to examine how chemical reactions can be classified.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;L. F. Wilhelmy (1850) “Über das Gesetz, nach welchem die Einwirkung der Säuren auf Rohrzucker stattfindet” Poggendorff’s Annalen der Physik und Chemie 81, 413–433, 499–526, &lt;a href=&quot;https://doi.org/10.1002/andp.18501571203&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1002/andp.18501571203&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;P. Waage and C. M. Guldberg (1864) “Studier over Affiniteten” Forhandlinger: Videnskabs-Selskabet i Christiana, 35–40, 111-120. There is an English translation by H. I. Abrash at &lt;a href=&quot;http://chimie.scola.ac-paris.fr/sitedechimie/hist_chi/text_origin/guldberg_waage/Concerning-Affinity.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://chimie.scola.ac-paris.fr/sitedechimie/hist_chi/text_origin/guldberg_waage/Concerning-Affinity.htm&lt;/a&gt; [The link is dead when I'm posting this, this one works: &lt;a href=&quot;http://www.chemteam.info/Chem-History/Concerning-Affinity.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.chemteam.info/Chem-History/Concerning-Affinity.html&lt;/a&gt; ]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;K. J. Laidler (1993) The World of Physical Chemistry, pages 232–289, Oxford University Press, Oxford&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7364" LastActivityDate="2018-06-06T11:40:11.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7436" PostTypeId="2" ParentId="1948" CreationDate="2018-06-06T12:50:38.850" Score="4" Body="&lt;p&gt;Kepler's 3rd law could have been deduced based on the orbital parameters mentioned in Ptolemy's &lt;a href=&quot;https://en.wikipedia.org/wiki/Almagest&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Almagest&lt;/em&gt;&lt;/a&gt; (of the 2nd century CE).  It was not important for this that Kepler had more modern observations of higher accuracy available to him, nor that he had discovered that the orbits were ellipses instead of circles.  The following chart shows the orbital sizes (a) and periods (P) deduced from the &lt;em&gt;Almagest&lt;/em&gt; as little squares, and shows Kepler's 3rd law as a straight line.  The correspondence is good enough to deduce Kepler's 3rd law from the data points.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OnFcb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OnFcb.png&quot; alt=&quot;Kepler&amp;#39;s 3rd law from Ptolemy&amp;#39;s data&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The crucial insight, first obtained by Copernicus (see his &lt;a href=&quot;https://en.wikipedia.org/wiki/Commentariolus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Commentariolus&lt;/a&gt;, written some time before 1514), was that the heliocentric model of the Solar System allowed for the sizes of the orbits (relative to that of the Earth) to be derived from the available observations.  Without this insight, Kepler would not have known the sizes of the planets' orbits, and would not have had the data from which to deduce his 3rd law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an inferior planet, closer to the Sun than Earth is, the size of their orbit relative to that of the Earth determines their greatest angular distance from the Sun (their maximum elongation) as seen from Earth.  In terms of the geocentric model used in the &lt;em&gt;Almagest&lt;/em&gt;, the size of the orbit corresponds to the size of the epicycle of the planet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a superior planet, further from the Sun than the Earth is, the size of the Earth's orbit relative to that of the planet determines the size of the planet's opposition loop where the planet shows apparent retrograde motion near its opposition to the Sun.  In terms of the geocentric model, the size of the epicycle of the planet corresponds to one over the size of the orbit, because the epicycle of the planet is equivalent to the orbit of the Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy gives the sizes of the epicycles as&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mercury: 22;30 = 22.5 out of 60 (&lt;em&gt;Almagest&lt;/em&gt;, IX 9, Toomer Edition)&lt;/li&gt;&#xA;&lt;li&gt;Venus: 43 + 1/6 out of 60 (X 2)&lt;/li&gt;&#xA;&lt;li&gt;Mars: 39;30 = 39.5 out of 60 (X 8)&lt;/li&gt;&#xA;&lt;li&gt;Jupiter: 11;30 = 11.5 out of 60 (XI 2)&lt;/li&gt;&#xA;&lt;li&gt;Saturn: 6;30 = 6.5 out of 60 (XI 6)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The deduced orbital sizes are, relative to that of the Earth:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mercury: 22.5/60 = 0.375&lt;/li&gt;&#xA;&lt;li&gt;Venus: 43.17/60 = 0.7194&lt;/li&gt;&#xA;&lt;li&gt;Earth: 1&lt;/li&gt;&#xA;&lt;li&gt;Mars: 60/39.5 = 1.5190&lt;/li&gt;&#xA;&lt;li&gt;Jupiter: 60/11.5 = 5.2174&lt;/li&gt;&#xA;&lt;li&gt;Saturn: 60/6.5 = 9.2308&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The planet's orbital period follows from their mean daily motion in longitude (for superior planets) or anomaly (for inferior planets).  Ptolemy gives the following mean daily motions in anomaly (degrees per day), i.e., motion along the epicycle:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mercury: 3;6,24,6,59,35,50 = 3.106699 (IX 3)&lt;/li&gt;&#xA;&lt;li&gt;Venus: 0;36,59,25,53,11,28 = 0.6165087 (IX 3)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and the following mean daily motions in longitude (degrees per day), i.e., motion along the deferent:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Sun: 0;59,8,17,13,12,31 = 0.9856353 (III 1)&lt;/li&gt;&#xA;&lt;li&gt;Mars: 0;31,26,36,53,51,33 = 0.5240597 (IX 3)&lt;/li&gt;&#xA;&lt;li&gt;Jupiter: 0;4,59,14,26,46,31 = 0.08312244 (IX 3)&lt;/li&gt;&#xA;&lt;li&gt;Saturn: 0;2,0,33,31,28,51 = 0.03348854 (IX 3)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For the inferior planets, we have to add the Sun's motion in longitude to the planet's motion in anomaly to get their motion relative to the celestial sphere.  The deduced orbital periods in days and years are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Mercury: 360/(3.106699 + 0.9856353) = 87.96935 → 0.2408492&lt;/li&gt;&#xA;&lt;li&gt;Venus: 360/(0.6165087 + 0.9856353) = 224.6989 → 0.6151977&lt;/li&gt;&#xA;&lt;li&gt;Earth: 360/0.9856353 = 365.2467 → 1&lt;/li&gt;&#xA;&lt;li&gt;Mars: 360/0.5240597 = 686.9446 → 1.880769&lt;/li&gt;&#xA;&lt;li&gt;Jupiter: 3606/0.08312244 = 4330.961 → 11.85763&lt;/li&gt;&#xA;&lt;li&gt;Saturn: 360/0.03348854 = 10749.95 → 29.43202&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These orbital sizes and periods are displayed in the chart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It might be that the mean daily motions still need to be corrected for precession (said by Ptolemy to work at a rate of 1 degree per 100 years), but that makes very little difference to the chart.&lt;/p&gt;&#xA;" OwnerUserId="7502" LastActivityDate="2018-06-06T12:50:38.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7437" PostTypeId="1" CreationDate="2018-06-06T16:01:54.110" Score="3" ViewCount="354" Body="&lt;p&gt;I am puzzled as to where or who produced the U235 for the first nuclear pile.  I read that graphite was used as a neutron moderator but only .7 % of natural uranium is U235. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He must have gotten it from Oak Ridge?.  But I always thought that Oak Ride's uranium production came after the feasibility study that a pile was even possible.   So where did the original batch come from ?  Maybe from mass spectroscope or cyclotron but that wouldn't make enough uranium would it?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2021-12-19T11:58:52.857" LastActivityDate="2021-12-30T14:06:55.600" Title="Where did Fermi get the U235 for the first nuclear pile?" Tags="|physics|biographical-details|physicists|nuclear-physics|fermi|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7438" PostTypeId="1" CreationDate="2018-06-06T22:02:41.610" Score="0" ViewCount="44" Body="&lt;p&gt;I asked a question in similar vain for the &quot;Fermi pile&quot;, however this reactor is different.  It's purpose was to produce plutonium...now I understand the advantage of that since plutonium can be separated from the &quot;atomic waste mess&quot; much easier than separating U235 from U238.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But now it was mostly U238 but didn't there have to be some U235 in concentrated form to start the reaction or was it just the case that it was U238 and the carbon moderators.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Atomic Heritage Museum has video on the B reactor construction and I saw both video's but does not say if those were the only two components.....if so then the mystery is solved because apparently all that is needed for plutonium production  is U238 and carbon. The rest comes out in the wash so to speak.  Interesting  video's by the way for the history buff. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-06-06T22:02:41.610" Title="Did the B reactor at Hanford make use of U235 without having U238 enriched ?" Tags="|nuclear-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7439" PostTypeId="2" ParentId="136" CreationDate="2018-06-07T18:03:05.210" Score="2" Body="&lt;p&gt;Practical steam power predates both Watt and Newcomen. Neither one invented it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Taqi_ad-Din_Muhammad_ibn_Ma%27ruf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Taqi al-Din&lt;/a&gt;, an Ottoman, described a rotary steam thing that rotated a barbeque spit. This was in the year 1551. Apparently the device had vanes on a wheel, and a jet of steam hit the vanes. So this would be the first impulse-type steam turbine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would call that the first practical use of a steam engine, but not enough for an industrial revolution. It &lt;strong&gt;could have been&lt;/strong&gt;... It could have been the beginning if people saw it and tried to improve it, like make it bigger, and apply it to something like a paddlewheel boat. But apparently no one did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, it is not the first practical use of steam itself. Steam cooking goes way, way back. Pretty sure dried tea leaves were steamed in China to make all sorts of tea. Steam cooking is different from just boiling something in water. Steam cooking involves the steam itself heating and moistening the foodstuff in a different way than in boiling water.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jer%C3%B3nimo_de_Ayanz_y_Beaumont&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeronimo Beaumont&lt;/a&gt;, a Spaniard, got a patent for a steam-powered water pump that was used to drain mines. This was in 1606. The wikipedia citation says &quot;Garcia, Nicolas (2007). Mas alla de la Leyenda Negra. Valencia: Universidad de Valencia. pp. 443–454. ISBN 9788437067919.&quot; I don't have the book so I wish I could tell you how that steam engine worked. This is pretty amazing to me, as it predates Savery and Necommen by about 100 years...so amazing in fact that I might open a question about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ferdinand_Verbiest#Verbiest&amp;#39;s_&amp;#39;car&amp;#39;&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferdinand Verbiest&lt;/a&gt; designed a steam-powered toy car, 65 cm long, for the Chinese Emperor around 1672. You can see from the drawing that a jet of steam hits paddlewheels. This makes it an impulse-type turbine. (Hero's aeliopile was a reaction-type turbine.) Wikipedia says that no one knows if the toy was ever actually built, with no citation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Savery&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Savery&lt;/a&gt; built a steam engine in 1698 called &quot;The Miner's Friend&quot; and got a patent for it. It was very similar to the one that Newcomen would soon build. In fact, &lt;strong&gt;Newcomen went into partnership with Savery&lt;/strong&gt; due to the patent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I got most of this knowledge from the wiki article on &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_steam_engine&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of the Steam Engine&lt;/a&gt;. It mentions a lot more stuff too, like a Frenchman who heated small sealed boilers until they exploded, demonstrating that steam pressure can do stuff.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So you see, the inventor of steam engines is not Newcomen or Watt, but at the latest, Beaumont. I don't know if Savery knew of Beaumont's steam engine. If not, then he can be credited as an independent inventor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So no offense, but I think the other answer by winwaed is wrong. Even if an improvement can be called an invention in its own right, you still have to credit Savery's and Beaumont's engines as predating them. To the best of my knowledge, they are originals in the sense that they were both independently invented without drawing on each other, or drawing on any other steam &lt;strong&gt;engine&lt;/strong&gt; (not necessarily steam-use thing) invented beforehand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW, it is incorrect to refer to Newcomen's engine as having a piston. Neither Savery's nor Newcomen's engine had a piston. Steam is &lt;strong&gt;not&lt;/strong&gt; pushing against that disc inside the cylinder. It's the cooling of steam that creates less pressure in the cylinder, which lets the outside atmospheric pressure push the disc down. This is why it's sometimes known as the &quot;Atmospheric Engine&quot;. It's atmospheric pressure doing the pushing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe it was Watt who first applied steam directly onto pistons. Maybe this is where the confusion comes from and why Watt is credited as an inventor &quot;steam power&quot;, by which they mean a modern configuration of steam power.&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="2546" LastEditDate="2018-06-07T21:42:48.060" LastActivityDate="2018-06-07T21:42:48.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7440" PostTypeId="1" CreationDate="2018-06-07T18:18:48.617" Score="5" ViewCount="168" Body="&lt;p&gt;Via &lt;a href=&quot;https://hsm.stackexchange.com/questions/136/who-invented-the-steam-engine-newcomen-or-watt/7439#7439&quot;&gt;a recent answer of mine&lt;/a&gt;, an ultimately wikipedia, I've learned that the first Steam Engine used to drain a mine was invented by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jer%C3%B3nimo_de_Ayanz_y_Beaumont&quot; rel=&quot;noreferrer&quot;&gt;Jeronimo Beamont&lt;/a&gt;, in 1606. He got a patent for it too. This was in Spain. &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_steam_engine&quot; rel=&quot;noreferrer&quot;&gt;Another source&lt;/a&gt; from wikipedia, with a different book cited.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This predates Thomas Savery's &quot;Miner's Friend&quot; by about 100 years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why is this? Why did the steam technology not propagate across Europe from Spain? Did Spain not have any printing presses at this time? Is there something in the Spanish Economy hindering this? Did Britain and France just hate Spain?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2018-06-07T18:18:48.617" Title="Why was Beaumont's steam engine not replicated for about 100 years?" Tags="|technology|steam-engines|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7442" PostTypeId="1" CreationDate="2018-06-09T05:47:28.090" Score="1" ViewCount="326" Body="&lt;p&gt;I first saw this quote at the top of p. 186, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1118505085&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Phonetics For Dummies&lt;/em&gt; (2013 1 ed)&lt;/a&gt;.&lt;br&gt;&#xA;A secondary source: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0813365511&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Many Faces Of Science&lt;/em&gt; (2000)&lt;/a&gt;. But I don't know &lt;a href=&quot;https://books.google.com/books?id=b1JPDwAAQBAJ&amp;amp;lpg=PT41&amp;amp;ots=wO9S9vtxGI&amp;amp;dq=%22whoever%20in%20the%20pursuit%20of%20science%2C%20seeks%22&amp;amp;pg=PT42#v=onepage&amp;amp;q=%22whoever%20in%20the%20pursuit%20of%20science,%20seeks%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;the page number&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;  2: According to the classic ideal, the only value recognized by the scientist as such, &#xA;  the only value actually involved in doing science, is that of knowledge for its own &#xA;  sake. She may welcome the possibility of useful applications of her research, but as a &#xA;  scientist she is devoted purely and simply to the extension of human knowledge as &#xA;  an end in itself. In this respect, she is just like her university colleagues the Sanskrit &#xA;  philologist, the medieval historian, and the pure mathematician, who delight in the &#xA;  achievement of new knowledge and understanding even if there is no prospect of its &#xA;  being useful in any practical way As an eloquent nineteenth-century expression of &#xA;  this view, consider this passage from the physicist Hermann von Helmboltz: &lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&lt;strong&gt;Whoever, in the pursuit of science, seeks after immediate practical utility, may generally rest &#xA;    assured that he will seek in vain.&lt;/strong&gt; All that science can achieve is a perfect knowledge and a &#xA;    perfect understanding of the action of natural and moral forces. Each individual student must &#xA;    be content to find his reward in rejoicing over new discoveries, as over new victories of mind &#xA;    over reluctant matter, or in enjoying the aesthetic beauty of a well-ordered field of &#xA;    knowledge, where the connection and filiation of every detail is clear to the mind, he must &#xA;    rest satisfied with the consciousness that he too has contributed something to the increasing &#xA;    fund of knowledge on which the dominion of man over all the forces hostile to intelligence &#xA;    reposes. &#xA;    (Helmholtz 1893, quoted in Ravetz 1971, 39) &lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The emboldened feels false for applied sciences, like biology, &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_neuroscience&quot; rel=&quot;nofollow noreferrer&quot;&gt;neuroscience&lt;/a&gt; or medicine? These unsolved problems do seek answers of &quot;immediate practical utility&quot;? &lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2018-06-09T18:32:41.400" Title="How's Helmholtz correct that &quot;Whoever in the pursuit of science, seeks after immediate practical utility may rest assured that he seeks in vain&quot;?" Tags="|quote|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7443" PostTypeId="2" ParentId="7442" CreationDate="2018-06-09T18:32:41.400" Score="1" Body="&lt;p&gt;That's history: the current (neoliberal) Ideology insists that everything is done for  profit and of course it claims that this has always been  true. Within its limits  one would start  wondering how could people think that a rational action would aim at anything that does not bring immediate benefit. The &lt;a href=&quot;https://en.wikipedia.org/wiki/One-Dimensional_Man&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;One dimensional Man&quot;&lt;/a&gt; was recognized  already in the sixties.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2018-06-09T18:32:41.400" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7444" PostTypeId="1" CreationDate="2018-05-29T12:36:58.727" Score="3" ViewCount="94" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt;&lt;br&gt;&#xA;PID (proportional-integral-derivative) control has been around for a long time.     &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was in textbooks in the early 1900's.  The Routh-Hurwitz stability criterion is from 1895. &lt;a href=&quot;https://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stability_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;  Nyquist stability is from the 1930's. &lt;a href=&quot;https://en.wikipedia.org/wiki/Nyquist_stability_criterion&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;&lt;br&gt;&#xA;What is the oldest example of intentional PID control containing all the components (proportional, integral, and derivative)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would not be surprised to find something between 500 and 2500 years old.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;More details:&lt;/strong&gt;&lt;br&gt;&#xA;PID control, also called PID, is defined (briefly) as the use of a weighted sum of the measured value (aka proportional), a sum (aka integral), and a difference (aka derivative) of some measured quantity, typically a distance from target, to control a system.  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(P) A tax of 10% of the period-over-period increase could be considered&#xA;derivative control, because if the period-over-period value acquired&#xA;is constant then the increase is 0%.&lt;/li&gt;&#xA;&lt;li&gt;(PD) If the tax were 1% of current value, and 10% of year-over-year&#xA;increase, then it would be PD control, because it is taxing the&#xA;measurement (proportional) and the change (derivative).&lt;/li&gt;&#xA;&lt;li&gt;(PID) A tax of 10% on increase, 1% of total, and 1% times the sum of the&#xA;last 2 years less a target, would be PID, because it would have all&#xA;three components.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;These sorts of things have been used (with varying degrees of success) for automobile cruise-control, guidance of heat-seeking missiles, electric coffee-pots, and control of investment portfolios.  Because it is control-systems-engineering it applies to any &quot;system&quot; that meets the applicability criteria, whether electrical, mechanical, thermal, or economic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Ongoing:&lt;/strong&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Antikythera_mechanism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Antikythera mechanism&lt;/a&gt; was an extremely elaborate astronomical clock.  I'm looking for PID indication there, perhaps in anti-backlash gearing.&lt;br&gt;&#xA;&lt;a href=&quot;https://commons.erau.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&amp;amp;httpsredir=1&amp;amp;article=1000&amp;amp;context=pr-honors-coe&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Embry-Riddle paper on the Ancient Control Systems&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7479" OwnerDisplayName="EngrStudent" LastEditorUserId="7479" LastEditDate="2018-06-11T12:05:51.600" LastActivityDate="2018-06-11T12:05:51.600" Title="Oldest example of (intentional) PID control" Tags="|engineering|ancient|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7446" PostTypeId="2" ParentId="7416" CreationDate="2018-06-10T15:02:00.397" Score="1" Body="&lt;p&gt;As others have pointed it out, it's difficult to know what he meant. Remember that he was reporting on something he had been told which he probably hadn't fully comprehended at the time, or of which he had only an inaccurate recollection at the time of writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer is the configuration that came to my mind as having some chance of being the one he had had in mind. (If he didn't have it &lt;em&gt;in mind&lt;/em&gt; but had only been told of at some point, then I expect the example to be more complicated than mine.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When Newman writes 'which are able to divide a space', I take it that the space can be the Euclidean plane; he would have written &lt;em&gt;the space&lt;/em&gt; if he had meant the three-dimensional space. My guess then is a &lt;a href=&quot;https://en.wikipedia.org/wiki/Apollonian_circles#Pencils_of_circles&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperbolic pencil of circles&lt;/a&gt; (the set of blue circles in the drawing is a finite sample of this):&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rFARJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rFARJ.png&quot; alt=&quot;Apollonian_circles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The set of blue circles, including the vertical straight line which is missing from the middle of the drawing, contains infinitely many curves. If you pick countably infinitely many &lt;strike&gt;(or finitely many)&lt;/strike&gt; of them including the straight line such that the sizes of the circles on both sides are unbounded (which also means that there is one arbitrarily near the straight line on either side), then those lines partition the plane ('divide a space') into shapes such that none of those shapes contains a straight line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum.&lt;/strong&gt; A simpler (but equivalent, as we'll see) example is an unbounded set of concentric circles and the concentric annuli defined by them, e.g. with equal spacing:&#xA;$$\Big\{\{x\in\mathbb{R}^2\,|\, n&amp;lt;|x|&amp;lt;n+1\}\ \Big|\ n\in\mathbb{N}\Big\}.$$&#xA;This is &lt;a href=&quot;https://en.wikipedia.org/wiki/Homography&quot; rel=&quot;nofollow noreferrer&quot;&gt;projectively isomorphic&lt;/a&gt; to my first example because there is a &lt;a href=&quot;https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Specifying_a_transformation_by_three_points&quot; rel=&quot;nofollow noreferrer&quot;&gt;Möbius transformation&lt;/a&gt; which maps one configuration to the other. You can map&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the two focuses (&lt;a href=&quot;https://en.wikipedia.org/wiki/Limiting_point_(geometry)&quot; rel=&quot;nofollow noreferrer&quot;&gt;limiting points&lt;/a&gt;) of the pencil of circles, and the point where the vertical line intersects the segment between the two focuses&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;to  &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the centre of the concentric circles, to $\infty$, and to an arbitrary point of any circle of the set of concentric circles, respectively.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Compare with this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A family of concentric circles centered at a single focus C forms a special case of a hyperbolic pencil, in which the other focus is the point at infinity of the complex projective line. The corresponding elliptic pencil consists of the family of straight lines through C; these should be interpreted as circles that all pass through the point at infinity. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Apollonian_circles#Pencils_of_circles&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pencils of circles, Wikipedia&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7364" LastEditorUserId="7364" LastEditDate="2018-06-11T09:04:31.440" LastActivityDate="2018-06-11T09:04:31.440" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7447" PostTypeId="2" ParentId="136" CreationDate="2018-06-11T12:04:24.620" Score="1" Body="&lt;p&gt;There is a Hellenistic Egypt &quot;aeolipile&quot; described by Heron of Alexandria from  ~300BC to ~100AD, which beats Taqi al-Din (born 1526 AD) by no less than ~1400 years.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.britannica.com/technology/aeolipile&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.britannica.com/technology/aeolipile&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/1991/was-the-steam-engine-really-known-in-ancient-egypt&quot;&gt;Was the steam engine really known in ancient Egypt?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.smith.edu/hsc/museum/ancient_inventions/steamengine2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.smith.edu/hsc/museum/ancient_inventions/steamengine2.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://books.google.com/books?id=WioDAAAAMBAJ&amp;amp;lpg=PA141&amp;amp;ots=3dLhNJFEW8&amp;amp;dq=steam%20actuated%20egyptian%20temple%20ancient&amp;amp;pg=PA141#v=onepage&amp;amp;q=steam%20actuated%20egyptian%20temple%20ancient&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Popular Science Article&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://commons.erau.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&amp;amp;httpsredir=1&amp;amp;article=1000&amp;amp;context=pr-honors-coe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Embry-Riddle paper on Ancience Control Systems&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7479" LastActivityDate="2018-06-11T12:04:24.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7451" PostTypeId="1" CreationDate="2018-06-14T04:27:10.667" Score="11" ViewCount="894" Body="&lt;p&gt;I am interested in when the concept of temperature first arose. This seems surprisingly hard to pin down from the web-based research I've done so far. I am mostly interested in how people thought about the concept (or its precursors) prior to the invention of thermodynamics and thermometers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the one hand, the idea that one thing is hotter or colder than another thing seems like such an obvious concept that one might expect it to have existed since prehistoric times. At the very least, people must have known since the invention of ceramics that a bright red glow is hotter than a dull red one, and that this will systematically affect the properties of the resulting material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But on the other hand, it seems that &lt;a href=&quot;https://en.wikipedia.org/wiki/Thermoscope&quot; rel=&quot;noreferrer&quot;&gt;thermoscopes&lt;/a&gt; (including those built by Galileo) were only able to measure the relative hotness or coldness of the thermoscope itself, since they weren't equipped with a scale that would allow comparisons with other objects. Adding a scale to a thermoscope doesn't seem like a technologically difficult challenge, which suggests that the concept of temperature &lt;em&gt;as a numerical value possessed by all objects&lt;/em&gt; did not exist until the 1600s or so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that things seem to become a bit clearer, with the development of thermodynamics leading to a better understanding of what heat and temperature are, how they behave and how to measure them. My interest is in how people thought about the concepts before that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given this, I guess my questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Prior to the 1600s, how did people think about hotness and coldness?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When did the notion arise of temperature (or hotness/coldness) as a one-dimensional quantity? (That is, the idea that for any two objects, either one is hotter than the other or they are the same temperature.) Did that only appear during the development of thermodynamics or was it known long prior to that?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Similarly, when did the notion arise of temperature &lt;em&gt;as distinct from heat&lt;/em&gt;? Did that only appear with the development of thermodynamics, or was there a less formal understanding of the distinction before that?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When Galileo and his contemporaries were building thermoscopes, did they have the concept of temperature, or were they only intending to measure heat? (That is, is the reason that they didn't add scales because they couldn't, or because they didn't know it was a good idea?)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="595" LastEditorUserId="595" LastEditDate="2018-06-15T02:28:36.090" LastActivityDate="2018-06-15T18:05:05.487" Title="When did the concept of temperature first arise?" Tags="|thermodynamics|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7452" PostTypeId="2" ParentId="7451" CreationDate="2018-06-14T18:34:51.440" Score="7" Body="&lt;p&gt;Your link quickly leads to Middleton,  &lt;em&gt;History of the Thermometer and its Use in Meteorology&lt;/em&gt; (&lt;a href=&quot;http://www.worldcat.org/oclc/413443&quot; rel=&quot;noreferrer&quot;&gt;1966&lt;/a&gt;). Pre-1600 (p. 3):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The opposition of “hot” and “cold,” like that of “dry” and&#xA;  “moist,” [was] used by Aristotle in the formation of his doctrine of opposites [with] no attempt to assign numbers to these qualities. The&#xA;  great physician &lt;a href=&quot;https://en.wikipedia.org/wiki/Galen&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Galen&lt;/strong&gt;&lt;/a&gt; [c.129-216] seems to have introduced the idea of&#xA;  “degrees of heat and cold,” four in number each way from a&#xA;  neutral point in the middle. (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Strange as it may seem, the idea of a scale of temperature was&#xA;  familiar to physicians before they had any instrument to measure&#xA;  it with. This is illustrated by the &lt;em&gt;De logistica medica&lt;/em&gt; [&lt;a href=&quot;https://books.google.com/books?id=mxBSAAAAcAAJ&amp;amp;pg=PP31&quot; rel=&quot;noreferrer&quot;&gt;1578&lt;/a&gt;] of &lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Hasler&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Johann&#xA;  Hasler&lt;/strong&gt;&lt;/a&gt; of Berne. Hasler’s very first “Problem” is entitled “To&#xA;  find the natural degree of temperature [“&lt;strong&gt;temperie&lt;/strong&gt;”] of each man, as determined by [latitude] and other influences.” (...) Hasler showed this supposed relationship by an &lt;a href=&quot;https://books.google.com/books?id=mxBSAAAAcAAJ&amp;amp;pg=PP33&quot; rel=&quot;noreferrer&quot;&gt;elaborate table&lt;/a&gt;, in which the 9 degrees of heat in the first column and the Galenic degrees of heat and cold (...) are set opposite the latitude.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(As a &lt;a href=&quot;https://www.persee.fr/doc/rhs_0048-7996_1968_num_21_3_2565_t1_0281_0000_1&quot; rel=&quot;noreferrer&quot;&gt;review&lt;/a&gt; hints, J.-M. Dureau-Lapeyssonnie in &lt;em&gt;Médecine humaine et véterinaire à la fin du Moyen Âge&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=Px0_PRATGaYC&amp;amp;ots=u4DKBL4mOM&amp;amp;pg=PA175&amp;amp;dq=chaud&quot; rel=&quot;noreferrer&quot;&gt;1966&lt;/a&gt;, pp. 209-211) traces scales further to physicians Ricart (c.1360-1422) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Al-Kindi&quot; rel=&quot;noreferrer&quot;&gt;Al-Kindi&lt;/a&gt; (c.801-873). However, this still involved four “elementary qualities” (hot, cold, dry, moist) evolving independently (heat can increase without cold decreasing), so wasn’t one-dimensional. Physicians like &lt;a href=&quot;https://en.wikipedia.org/wiki/Johannes_Baptista_Montanus&quot; rel=&quot;noreferrer&quot;&gt;da Monte&lt;/a&gt; (1498-1551) did write &lt;a href=&quot;https://books.google.com/books?id=30pBAAAAcAAJ&amp;amp;pg=PA253&amp;amp;dq=temperatura&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;temperatura&lt;/em&gt;&lt;/a&gt;, but apparently in the &lt;a href=&quot;https://en.oxforddictionaries.com/definition/temperature&quot; rel=&quot;noreferrer&quot;&gt;earlier sense&lt;/a&gt; “state of being tempered or mixed, later synonymous with temperament”.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Post-1600, Middleton essentially argues that modern temperature is born the day it gets &lt;em&gt;measured&lt;/em&gt; by attaching a scale to a thermoscope, i.e. at the same time as the thermometer (pp. 4-5): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a distinction must be made between the terms &lt;em&gt;thermoscope&lt;/em&gt; and &lt;em&gt;thermometer&lt;/em&gt;, in which a thermometer is simply a thermoscope provided with a scale. (...) I propose to regard it as axiomatic that a&#xA;  “meter” must have a scale or something equivalent. (...) The serious candidates for the honor of having “invented the&#xA;  thermometer” are usually considered to be four in number:&#xA;  Galileo, Santorio (or Sanctorius), Drebbel, and Fludd.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So his earliest quotes involving the new concept are (p. 9) by &lt;a href=&quot;https://en.wikipedia.org/wiki/Santorio_Santorio&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Santorio&lt;/strong&gt;&lt;/a&gt; (1612, reprint 1632):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(&lt;a href=&quot;https://books.google.com/books?id=YcNaAAAAcAAJ&amp;amp;pg=PA612&amp;amp;dq=dimetiri&quot; rel=&quot;noreferrer&quot;&gt;LXXXV.X&lt;/a&gt;): I wish to tell you about a marvellous way in which I am accustomed&#xA;  to measure, with a certain glass instrument, the cold and hot temperature [“&lt;strong&gt;temperatura&lt;/strong&gt;”] of the air of all regions and places, and of all parts of the body;&#xA;  and so exactly, that we can measure with the compass the degrees&#xA;  and ultimate limits of heat and cold at any time of day.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(&lt;a href=&quot;https://books.google.com/books?id=YcNaAAAAcAAJ&amp;amp;pg=PA644&amp;amp;dq=metimur&quot; rel=&quot;noreferrer&quot;&gt;LXXXVI.III&lt;/a&gt;): the temperature [“&lt;strong&gt;temperatura&lt;/strong&gt;”] of the air can be observed not only in so far as it&#xA;  belongs to the body, but also as a thing in itself; so that the range&#xA;  between very hot and cold temperatures [“&lt;strong&gt;temperatura&lt;/strong&gt;”] of the air can be exactly&#xA;  perceived. For we have an instrument with which not only the heat&#xA;  and cold of the air is measured, but all the degrees of heat and cold&#xA;  of all the parts of the body, as we show to our students at Padua,&#xA;  teaching them its uses; and they have heard about this novelty with no little astonishment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and (p. 7) by &lt;a href=&quot;https://en.wikipedia.org/wiki/Giovanni_Francesco_Sagredo&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Sagredo&lt;/strong&gt;&lt;/a&gt; in letters to Galileo (&lt;a href=&quot;https://books.google.com/books?id=V8i_ddk5Xo4C&amp;amp;pg=RA1-PA270&amp;amp;dq=temperie&quot; rel=&quot;noreferrer&quot;&gt;1613&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=V8i_ddk5Xo4C&amp;amp;pg=RA1-PA363&amp;amp;dq=temperamenti&quot; rel=&quot;noreferrer&quot;&gt;1615&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The instrument for measuring heat, invented by your excellent self,&#xA;  has been reduced by me to various very elegant and convenient forms, so that the difference in temperature [“&lt;strong&gt;temperie&lt;/strong&gt;”] between one room and&#xA;  another is seen to be as much as 100 degrees. With these I have found&#xA;  various marvellous things, as, for example, that in winter the air may&#xA;  be colder than ice (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;I have been making additions and changes every day to the instrument for measuring temperatures [“&lt;strong&gt;temperamenti&lt;/strong&gt;”] (...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Galileo&lt;/strong&gt; himself also used the word, e.g. in &lt;em&gt;Pensieri varj&lt;/em&gt; (c.1619, published &lt;a href=&quot;https://archive.org/stream/bibliografiagal00boffgoog#page/n131&quot; rel=&quot;noreferrer&quot;&gt;1744&lt;/a&gt;, reprint &lt;a href=&quot;https://books.google.com/books?id=Edf7UOFLWF0C&amp;amp;pg=PA333&amp;amp;dq=temperie&quot; rel=&quot;noreferrer&quot;&gt;1855&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;granted that air contained in the instrument is at the same temperature [“&lt;strong&gt;temperie&lt;/strong&gt;”] as other air in the ambient room (...)  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-06-14T21:40:12.157" LastActivityDate="2018-06-14T21:40:12.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7453" PostTypeId="2" ParentId="7451" CreationDate="2018-06-14T20:41:44.087" Score="6" Body="&lt;p&gt;As Francois Ziegler pointed out, Galen introduced the idea of four degrees of heat and four degrees of cold on either side of a standard neutral temperature around 150 A.D.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the 1300s, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Oxford_Calculators&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford Calculators&lt;/a&gt;, associated with Merton College, Oxford, talked about temperature as if it were a continuous one-dimensional quantity, akin to quantities such as position and velocity. The important points are described in Lindberg's &lt;a href=&quot;https://books.google.com/books?id=dPUBAkIm2lUC&amp;amp;printsec=frontcover&amp;amp;dq=beginnings%20of%20western%20science&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwi1o9aYk9bbAhVmqlkKHR7dBSQQ6AEIJzAA#v=onepage&amp;amp;q=beginnings%20of%20western%20science&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Beginnings of Western Science&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The fundamental idea was that qualities or forms can exist in various&#xA;  degrees or intensities: there is not just a single degree of warmth or&#xA;  cold, but a range of intensities or degrees running from very cold to&#xA;  very hot...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Reflection about qualities, their intensity, and  their&#xA;  intensification thus led the Mertonians to a new distinction: between&#xA;  the intensity of a quality (defined above) and its quantity (how much&#xA;  of it there is). An example will help us to understand this&#xA;  distinction: it is obvious enough in the case of heat that one hot&#xA;  object can be hotter than another; this is a reference to the&#xA;  intensity of the quality, what we call &quot;temperature.&quot; But we also have&#xA;  a conception of the quantity of heat--how much of it there is. If we&#xA;  have two objects at the same temperature, one of them twice as big,&#xA;  that larger object clearly has twice the &quot;quantity&quot; of this quality of&#xA;  heat.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nicole Oresme then went so far as to represent temperature (as well as things such as pain and grace) &lt;a href=&quot;http://www.dam.brown.edu/people/mumford/beyond/coursenotes/2006PartIb.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;graphically&lt;/a&gt; in his &lt;em&gt;Treatise on the Configuration of Qualities and Motions&lt;/em&gt;. Unfortunately, I can't get my hands on an English translation, but Lindberg describes it like so:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Take a rod AE... heated differentially, so that the heat increases&#xA;  uniformly from one end to the other. At point A and at whatever&#xA;  intervals you choose, erect a vertical line representing the intensity&#xA;  of heat at that point. If (as we have postulated) the temperature&#xA;  increases uniformly from A to E, then the figure will reveal a uniform&#xA;  lengthening of the vertical lines.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Granted, what these guys seem to have been doing was to fill up a rod with a bunch of heat almost like filling a bucket with a liquid, so that there's more heat on one end than the other, then looking at the density of heat (or &quot;intensity of heat&quot;) at a given point, which is not really the modern definition of temperature. But then, of course they wouldn't use the modern definition of temperature, would they? Regardless, I'd say this more or less reflects their thinking about warmth and coldness, or temperature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for why someone wouldn't add scales to a thermoscope, I think it's a little subtle, and you have to put yourself into the shoes of someone from the 1600s or 1700s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem with measuring temperature is a little different than the problem of measuring speed. Everyone knows that speed is a measure of how much distance something travels in a given time. But there are philosophical issues with measuring temperature. What exactly are you measuring? As you said, everyone has an intuitive sense of one body being warmer or colder than another. But what exactly does that mean? The modern definition,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$&#xA;T = \partial{U}/\partial{S}&#xA;$$&#xA;is obviously of no help to someone pre-1600s.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199696253.001.0001/oxfordhb-9780199696253-e-17&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Oxford Handbook of the History of Physics&lt;/a&gt; gives a good description of some of the problems. First, you have to agree on certain &quot;fixed points&quot;. It's obvious today that something like the freezing and boiling point of water works. But is it really that obvious? Before reliable thermometers, or even a definition of temperature, how do you know water always freezes at the same &quot;temperature&quot;? People tried a number of different fixed points before finding agreement. (Blood temperature, temperature of deep caves, melting point of butter, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, once you've come up with your fixed points, it's easy enough to just divide your scale up into 100 equal-spaced degrees, but everyone's scale was different in non-linear ways! There's a reason we use mercury in our thermometers. Once you make a thermometer that is based on a material expanding, like mercury or alcohol, you've essentially defined temperature to be tied to that material's rate of expansion, but materials don't all expand at a uniform rate! In other words, you can make two &quot;thermometers&quot; out of different materials that agree on your fixed points, but disagree in between! How do you know which one is the &quot;correct&quot; one if you don't already have a reliable thermometer (or a quantitative definition of temperature)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These details were all worked out mostly in the 1700s. I think it's safe to say that before then, people's concept of temperature was similar to Oresme's, in that it was clear that some things are hotter than others, and one thing might feel the same temperature as another, but this was all a qualitative thing, similar to the sensation of pain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Edited to add more details and address the valid point made in the comments that Oresme might have been talking more about heat than temperature.)&lt;/p&gt;&#xA;" OwnerUserId="7317" LastEditorUserId="7317" LastEditDate="2018-06-15T18:05:05.487" LastActivityDate="2018-06-15T18:05:05.487" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7454" PostTypeId="1" CreationDate="2018-06-15T08:09:00.107" Score="3" ViewCount="262" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Lemniscate_of_Gerono&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lemniscate of Gerono&lt;/a&gt; is a special case of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lissajous_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lissajous curves&lt;/a&gt;.  The dates for the two mathematicians are fairly close: &lt;a href=&quot;https://en.wikipedia.org/wiki/Camille-Christophe_Gerono&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerono&lt;/a&gt; (1799-1891) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Jules_Antoine_Lissajous&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lissajous&lt;/a&gt; (1822-1880). Historically who has priority for the lemniscate?  Was Lissajous aware of Gerono's work when he introduced his curves?&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2018-06-17T11:24:57.363" LastActivityDate="2018-06-17T11:24:57.363" Title="Priority on lemniscate of Gerono?" Tags="|mathematics|mathematicians|geometry|differential-geometry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7456" PostTypeId="1" CreationDate="2018-06-15T20:01:54.450" Score="2" ViewCount="290" Body="&lt;p&gt;More specifically, what is the oldest evidence of human awareness of what we now call the Pythagorean Theorem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phrase, &quot;evidence of human awareness&quot; was used to exclude a different question of when was the first proof made.  It also constrains answers to those based on artifacts.  (Artifacts form the basis of our knowledge of ancient history.)  Each artifact can have different interpretations.  The dates of the artifacts have errors/uncertainties.  So, the question boils down to, of all the artifacts ever in existence, which probably has the earliest date that also is good evidence of a human being aware of the Pythagorean Theorem?&lt;/p&gt;&#xA;" OwnerUserId="7527" LastEditorUserId="7527" LastEditDate="2018-07-04T00:18:32.333" LastActivityDate="2018-07-04T00:18:32.333" Title="How old is the Pythagorean Theorem?" Tags="|ancient-greece|geometry|ancient-egypt|pythagorean-proposition|" AnswerCount="0" CommentCount="8" ClosedDate="2018-07-02T17:32:40.883" ContentLicense="CC BY-SA 4.0" />
  <row Id="7458" PostTypeId="2" ParentId="7427" CreationDate="2018-06-16T16:41:09.797" Score="1" Body="&lt;p&gt;It looks like he is saying the probable error of the difference between the two proportions is no more than the sum of the two probable errors, so&#xA;$$d_1+d_2=0.0008+0.0003=0.0011$$&#xA;and that&#xA;$$10(d_1+d_2)=0.011\approx 0.0105=\text{the actual discrepancy.}$$&lt;/p&gt;&#xA;" OwnerUserId="144" LastActivityDate="2018-06-16T16:41:09.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7459" PostTypeId="1" AcceptedAnswerId="7460" CreationDate="2018-06-17T17:18:35.820" Score="2" ViewCount="407" Body="&lt;p&gt;Out of curiosity, I was searching for an English translation of the Disquisitiones Arithmeticae, and I found out that there is indeed one. It was translated in English in 1965 by a certain Arthur A. Clarke. Initially, I was amazed since I thought he was the same science fiction author who wrote the Sentinel, the short story that inspired 2001: A Space Odyssey, but then I noticed that the writer is called Arthur C. Clarke, and I could not find a reference on his Wikipedia page that he did this translation. Then I discovered that C. Clarke did graduate in maths and physics, so my doubt remains. Are the two Clarks the same person? If not, could someone give me some information (field of research, university, etc.) about A. Clarke?&lt;/p&gt;&#xA;" OwnerUserId="7531" LastEditorUserId="10106" LastEditDate="2020-06-07T18:32:13.297" LastActivityDate="2020-06-08T06:02:17.820" Title="Are Arthur C. Clarke and Arthur A. Clarke the same?" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7460" PostTypeId="2" ParentId="7459" CreationDate="2018-06-17T17:26:19.420" Score="8" Body="&lt;p&gt;MathSciNet has &#xA;Arthur Albert Clarke, Ph.D. 1964 at Yeshiva University  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that rules out the Sci Fi writer Arthur Charles Clarke&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arthur A. Clarke's translation of &lt;em&gt;Disquisitiones arithmeticae&lt;/em&gt; was published in 1966.  A revision by William C. Waterhouse, Cornelius Greither and A. W. Grootendorst was published in 1986&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="5989" LastEditDate="2020-06-08T06:02:17.820" LastActivityDate="2020-06-08T06:02:17.820" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7461" PostTypeId="1" AcceptedAnswerId="7462" CreationDate="2018-06-17T20:29:21.017" Score="2" ViewCount="166" Body="&lt;p&gt;Ordinal multiplication has always seemed backwards to me. $\alpha \times \beta$ is defined as the concatenation of $\beta$ copies of $\alpha$, not the other way 'round as one might expect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this go back to Cantor? Or did someone else create this convention?&lt;/p&gt;&#xA;" OwnerUserId="340" LastEditorUserId="1932" LastEditDate="2019-01-27T03:03:52.417" LastActivityDate="2019-01-27T03:03:52.417" Title="Who decided on the convention for ordinal multiplication?" Tags="|set-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7462" PostTypeId="2" ParentId="7461" CreationDate="2018-06-17T22:14:21.983" Score="3" Body="&lt;p&gt;Cantor at first defined multiplication of ordinals one way, then later (for what he thought were good reasons) switched to the opposite way.  So YES it is due to Cantor.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-06-17T22:14:21.983" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7463" PostTypeId="1" AcceptedAnswerId="7464" CreationDate="2018-06-17T22:51:43.170" Score="1" ViewCount="174" Body="&lt;p&gt;Of all mathematical creations, the one of using graphs to describe planes and 3D space seem to me the most strange ones. My head almost can't admit that space can be represent by three numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I'm not mistaken it all start with Descartes. Cartesian axes and analytical geometry were both created by Descartes, although I've never found some derivation or short explanation about how this started.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any book on maths history with nice explanations about?&lt;/p&gt;&#xA;" OwnerDisplayName="user7532" LastEditorUserId="131" LastEditDate="2018-06-18T05:53:46.050" LastActivityDate="2018-06-18T14:11:45.080" Title="On the creation of analytical geometry" Tags="|mathematics|reference-request|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7464" PostTypeId="2" ParentId="7463" CreationDate="2018-06-18T05:53:33.080" Score="5" Body="&lt;p&gt;See : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Carl Boyer, &lt;a href=&quot;https://books.google.it/books?id=2T4i5fXZbOYC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;History of Analytic Geometry&lt;/a&gt;, 1st edition : 1956.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For the origins of &quot;cartesian&quot; axes, see also the post : &lt;a href=&quot;https://hsm.stackexchange.com/questions/3312/cartesian-coordinate-system-in-newtons-work&quot;&gt;Cartesian coordinate system in Newton's work&lt;/a&gt; as well as the post : &lt;a href=&quot;https://hsm.stackexchange.com/questions/2378/when-do-we-see-for-the-first-time-the-use-of-the-cartesian-plane&quot;&gt;When do we see for the first time the use of the Cartesian plane ?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-06-18T05:57:47.227" LastActivityDate="2018-06-18T05:57:47.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7465" PostTypeId="1" CreationDate="2018-06-18T06:14:57.873" Score="4" ViewCount="182" Body="&lt;p&gt;In history of algebra, who was the first to add one equation to another equation? Someone gave me the name of an Italian mathematician of Renaissance period, but I lost the email. I wish to make it a bit more precise: adding two equations (hence involving unknowns), not just equalities: the latter was apparently dealt with by Euclid.&lt;/p&gt;&#xA;" OwnerUserId="7534" LastActivityDate="2018-06-19T08:22:59.660" Title="In history of algebra, who was the first to add one equation to another equation?" Tags="|mathematics|algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7466" PostTypeId="2" ParentId="7465" CreationDate="2018-06-18T12:00:02.123" Score="6" Body="&lt;p&gt;For sure, the technique was used by Leonhard Euler (15 April 1707 – 18 September 1783).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See : &lt;a href=&quot;https://books.google.it/books?id=Xf5eAAAAcAAJ&amp;amp;pg=PA208&quot; rel=&quot;noreferrer&quot;&gt;Elements of Algebra (English transl., 1822)&lt;/a&gt;, page 208:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Since the two equations are,&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;$x+y=a$, and&lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;$x-y=b$;&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;if we add the one to the other, we have $2x=a+b$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first German edition of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Elements_of_Algebra&quot; rel=&quot;noreferrer&quot;&gt;Elements&lt;/a&gt; was in 1770.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;An earlier example seems to be in Gerolamo Cardano (24 September 1501 – 21 September 1576)'s &lt;a href=&quot;https://books.google.it/books?id=ZMZY79bvOPgC&amp;amp;pg=PA74&quot; rel=&quot;noreferrer&quot;&gt;Ars Magna (1545)&lt;/a&gt;, but we have to consider that we are reading a modern English translation: the original Latin text can be quite different (see &lt;a href=&quot;http://www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4_s_4.pdf&quot; rel=&quot;noreferrer&quot;&gt;Caput IX (page 21)&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-06-19T08:22:59.660" LastActivityDate="2018-06-19T08:22:59.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7467" PostTypeId="2" ParentId="7463" CreationDate="2018-06-18T14:11:45.080" Score="1" Body="&lt;p&gt;The &lt;a href=&quot;https://missmeyersmathblog.wordpress.com/2012/04/10/descartes-and-the-cartesian-plane/&quot; rel=&quot;nofollow noreferrer&quot;&gt;anecdote&lt;/a&gt; is like this, but how accurate it is seems &lt;a href=&quot;https://skeptics.stackexchange.com/questions/40022/did-ren%C3%A9-descartes-develop-the-cartesian-coordinate-system-by-watching-a-fly-on&quot;&gt;questionable&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While at La Flèche, Descartes suffered health problems and because of this, his teachers allowed Descartes to stay in bed for most of the morning. ...  The story goes that one day when Descartes was in bed, he noticed a fly crawling around on the ceiling. He tried to think of ways to describe where the fly was located and realized that he could do so by describing the fly’s position by its distance from each wall.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/O5h4x.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/O5h4x.jpg&quot; alt=&quot;book&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the story, for beginning readers.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-06-18T14:11:45.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7468" PostTypeId="1" AcceptedAnswerId="7471" CreationDate="2018-06-19T02:51:36.940" Score="13" ViewCount="4146" Body="&lt;p&gt;I was just curious what Richard Feynman thought of Stephen Hawking's Hawking Radiation. Feynman was one of the developers of quantum field theory and Hawking's work would have been cutting edge on the unification of gravity with quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Feynman passed away I believe in 1988 but there would have been plenty of time for them to meet and discuss Hawking radiation since that concept was published around 1974.   I'm not coming up with much on my research,maybe somebody else has better resources.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-06-06T00:39:51.690" LastActivityDate="2021-04-22T08:49:02.383" Title="Did Richard Feynman ever meet Stephen Hawking or comment on Hawking radiation?" Tags="|physics|biographical-details|theoretical-physics|physicists|feynman|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7469" PostTypeId="1" CreationDate="2018-06-19T03:01:03.417" Score="1" ViewCount="56" Body="&lt;p&gt;Very often in the literature of research papers and other articles, and maybe text books, on topics of quantum field theory, a theory may be described as a 3+1 or 0+1, or maybe even 1+1 theory.  I have never read any definition of this notation but I believe that the general form is (d-1,1) where d is the overall spacetime dimension.  Thus, a 4-dimension space time theory might be described as 3+1; or, sometimes (3,1).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few examples I know of are that the motion of a classical particle that is parameterized by time would simply be a 0+1 theory.  Or, I have read that a particular String Theory is described as a 1+1 theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have just exhausted any knowledge I might have gleaned on this question though.  I have never seen any formal definition or motivation for its use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question, who and when was this notation developed and what was the motivation behind it.  In the very simple examples I have seen, it does not seem to add much than what a few words could say more clearly (at least to a philistine such as I).&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2018-06-19T03:01:03.417" Title="What is the history and motivation for the (d-1,1) notation used to describe a field theory?" Tags="|notation|quantum-mechanics|quantum-field-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7470" PostTypeId="2" ParentId="7468" CreationDate="2018-06-19T12:17:27.633" Score="6" Body="&lt;p&gt;Dunno if they ever met in person.  There's an interesting &lt;a href=&quot;http://nautil.us/blog/why-hawking-radiation-was-almost-feynman-radiation&quot; rel=&quot;noreferrer&quot;&gt;video&lt;/a&gt; (one of several sites available) which states that Feynman came up with the &quot;Hawking Radiation&quot; theory well before Hawking did, but didn't publish. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A partial transcript (I cannot speak to its authenticity) from a &lt;a href=&quot;https://motls.blogspot.com/2005/03/kitp-gross-as-actor-feynma_111013427500223724.html&quot; rel=&quot;noreferrer&quot;&gt;blog&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1972 before Hawking came out with the Hawking radiation formula.&#xA;  Feynman was meeting with Kip Thorne's grad students, Bill Press, Saul&#xA;  Teukolsky &amp;amp; Lightman. They discussed a recent calculation of shining&#xA;  light on a rotating black hole and getting more energy out then in at&#xA;  expense of decreasing rotational energy of the hole. They all went&#xA;  back to Lightman's office. Feynman said: &quot;Hey this is like stimulated&#xA;  emission. So he went to black board and did a A &amp;amp; B coefficient model&#xA;  and then when angular momentum J of the black hole J -&gt; 0 there was&#xA;  still &quot;A&quot; spontaneous emission and it was the later Hawking formula!&#xA;  Well, let me admit that I don't quite understand how the formulae from&#xA;  stimulated emission are useful for deriving the &quot;Hawking formula&quot;,&#xA;  whatever it is. It could be interesting to try to reconstruct&#xA;  Feynman's blackboard. A maid erased the board that night before&#xA;  Lightman and the others realized they should have written down what&#xA;  Feynman wrote. Not even Feynman thought it was important enough to&#xA;  write a paper about apparently.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-06-19T12:17:27.633" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7471" PostTypeId="2" ParentId="7468" CreationDate="2018-06-19T20:29:21.763" Score="18" Body="&lt;p&gt;I take that your primary goal is to know what Feynman thought of Hawking's work. While it is possible that they have met I would consider it unlikely given that Feynman mentioned several times how his attendance of the Chapel Hill conference (on general relativity) made such a bad impression on him that he never attended another conference on that theme, so they largely went around different social circles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Yet, regarding Hawking radiation it is clear that he never understood it completely. If you search for &quot;Feynman's last blackboard&quot; you'll find the photograph of the blackboard in Feynman's Caltech Office at the time of his death. In the right upper corner there is a list of things &quot;to learn&quot;, they are: Bethe Ansatz Probl., Kondo (dashed, so presumably he got that one), 2D Hall, accel. temp. and non-linear classical hydro.[If you have trouble finding it using your-favorite-search-engine look for the Reviews of Modern Physics  about the Unruh Effect, authors are Matsas, Crispino and Higuchi]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Accel. Temp. is a clear reference to Acceleration Temperature, that is the Unruh Effect. While Hawking Radiation and Unruh Effect are different physical phenomena (that is, they have different causes), they are very closely related, both physically and in their mathematical derivation, therefore it seems unreasonable to think that he understood Hawking's work, and other related to quantum gravitation, since he did not understand the Unruh Effect, arguably the simplest example of related phenomena. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other answer mentions stimulated emission by rotating black holes (sometimes people call it Zel'dovich Effect in the literature). By Hawkings own recollections, he first thought of Hawking radiation while visiting the USSR in the 1970s, and there Zel'dovich showed him that a electrically charged black hole would amplify electromagnetic waves. He (Hawking) then made the connection with stimulated and spontaneous emission, and back in the UK finished the calculations. [We'll, it's embarrassing but I cannot remember exactly where he wrote that, I'm not in my office for some time now. This version of the story is either in the book &quot;300 years of gravitation&quot; (Hawking was one of it's editors) or in Kip Thorne's &quot;Black Holes and Time Warps&quot;. Sorry for giving no concrete reference, I would bet on Thorne's book right now, also because if Feynman showed the math to Kip Thorne's students then surely it will be mentioned in Thorne's recollection on the history of GR.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope it helps to dig further references, but this is the extend of which I'm aware.&lt;/p&gt;&#xA;" OwnerUserId="5904" LastActivityDate="2018-06-19T20:29:21.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7474" PostTypeId="1" CreationDate="2018-06-24T21:49:47.580" Score="2" ViewCount="446" Body="&lt;p&gt;Who introduced the term sinus cardinalis? I do not mean the abbreviation sinc, which was introduced 1952 by Woodward.&lt;/p&gt;&#xA;" OwnerUserId="7554" LastEditorUserId="131" LastEditDate="2018-06-25T19:12:23.727" LastActivityDate="2018-06-25T19:12:23.727" Title="First appearance of the term sinus cardinalis" Tags="|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7475" PostTypeId="1" AcceptedAnswerId="7482" CreationDate="2018-06-25T08:28:23.480" Score="2" ViewCount="120" Body="&lt;p&gt;Perusing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Radioisotope_thermoelectric_generator&quot; rel=&quot;nofollow noreferrer&quot;&gt;Radioactive Thermoelectric Generator&lt;/a&gt; or &lt;a href=&quot;https://space.stackexchange.com/questions/tagged/rtg&quot;&gt;&lt;code&gt;rtg&lt;/code&gt; tag&lt;/a&gt; in Space SE will show how important these items are for Space exploration both as a source of heat to keep spacecraft instrumentation warm, and to power the thermoelectric generators for electricity. The Curiosity rover on Mars uses it's RTG for both and includes a fluid circulation system to heat (or cool) various sections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;RTG's most often use an alpha decay source like Plutonium-238 or perhaps &lt;a href=&quot;https://space.stackexchange.com/q/27433/12102&quot;&gt;in the future&lt;/a&gt; Americium-241) rather than a fission source, to minimize longer range radiation that can damage the spacecraft.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; I'm not sure if the self-heating of a radioactive sample was first noticed then explained, or predicted and then demonstrated. But in either case, when was a &lt;em&gt;measured&lt;/em&gt; or even &lt;em&gt;noticed&lt;/em&gt; temperature rise first documented or reported?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2018-06-25T08:35:49.720" LastActivityDate="2018-06-29T15:52:20.130" Title="When was it first noticed, or demonstrated, that radioactive material became warm?" Tags="|experimental-physics|nuclear-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7476" PostTypeId="2" ParentId="7474" CreationDate="2018-06-25T12:02:32.390" Score="2" Body="&lt;p&gt;According to sources quoted in Wikipedia's &lt;code&gt;sinc&lt;/code&gt; &quot;talk&quot; page, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The name sinus cardinalis dates back by Edmund T. Whittaker in 1915,&#xA;  where he named the bandlimited or most simple function of a family of&#xA;  cotabular functions (sharing a function table, i.e., values at equally&#xA;  spaced sample points) the cardinal function of this family. One would&#xA;  have to check if he already named the basis functions or if that came&#xA;  later. See A History of interpolation.--LutzL (talk)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Following that &quot;talk&quot; link leads to &lt;a href=&quot;https://imagescience.org/meijering/research/chronology/&quot; rel=&quot;nofollow noreferrer&quot;&gt;a history page&lt;/a&gt;  which includes an image from that paper, From E. T. Whittaker, &quot;On the Functions which are Represented by the Expansions of Interpolation-Theory&quot;, Proceedings of the Royal Society of Edinburgh, vol. 35, 1915, pp. 181-194  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Fg9t7.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Fg9t7.gif&quot; alt=&quot;whittaker excerpt&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="609" LastEditDate="2018-06-25T15:38:15.193" LastActivityDate="2018-06-25T15:38:15.193" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7477" PostTypeId="1" AcceptedAnswerId="15807" CreationDate="2018-06-25T23:48:03.920" Score="9" ViewCount="918" Body="&lt;p&gt;The YouTube video &lt;a href=&quot;https://www.youtube.com/watch?v=cMxbSsRntv4&quot; rel=&quot;noreferrer&quot;&gt;Alan Turing's lost radio broadcast rerecorded&lt;/a&gt; contains a re-enactment of Alan Turing's lecture broadcast by the BBC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the introduction, the narrator (James Grimes, also of the &lt;a href=&quot;http://numberphile.com/team/index.html&quot; rel=&quot;noreferrer&quot;&gt;Numberphile&lt;/a&gt; series of videos) states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;His lecture was titled “Can Digital Computers Think?” and was a part of a series of lectures which featured other leading figures in computing at the time. The other speakers being Douglas Hartree,  Max Newman, Freddie Williams and Maurice Wilkes. Together they represented major new projects in computing  at the Universities of Cambridge and Manchester. Unfortunately, these recordings no longer exist, along with all other recordings of Alan Turing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Was there an intentional purge of all audio recordings of Alan Turing, or is this loss more likely to be accidental/unintended? If intentional, what might have been the reason, and was it restricted to him alone, or included recordings of others relating to mathematics and the future of computing science?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="4251" LastEditDate="2020-07-13T00:32:33.603" LastActivityDate="2023-12-14T01:50:08.740" Title="Was there an intentional purge of all audio recordings of Alan Turing?" Tags="|mathematics|computer-science|computers|turing|computer-scientists|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="7478" PostTypeId="1" CreationDate="2018-06-26T22:36:52.617" Score="7" ViewCount="528" Body="&lt;p&gt;The English-language &lt;a href=&quot;https://en.wikipedia.org/wiki/Bruno_Pontecorvo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; about physicist Bruno Pontecorvo is illustrated with this photo:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0mgKY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0mgKY.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, this man seems distinctly different from most portraits of Pontecorvo around:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/n09B6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/n09B6.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;a href=&quot;https://i.stack.imgur.com/zpML2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zpML2.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;a href=&quot;https://i.stack.imgur.com/5q43P.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5q43P.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;a href=&quot;https://i.stack.imgur.com/sT4TU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sT4TU.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first two come, respectively, from Italian-language and Russian-language Wikipedias (corresponding to two of his nationalities), the other two from &lt;a href=&quot;http://centropontecorvo.df.unipi.it/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bruno Pontecorvo Centre of the University of Pisa&lt;/a&gt;. But just check most of the photos that can be seen in any book or &lt;a href=&quot;https://www.google.it/search?q=%22bruno+pontecorvo%22&amp;amp;newwindow=1&amp;amp;source=lnms&amp;amp;tbm=isch&amp;amp;sa=X&amp;amp;ved=0ahUKEwjj4dyRrPLbAhUDjCwKHcGrDJgQ_AUICigB&amp;amp;biw=1280&amp;amp;bih=703&quot; rel=&quot;nofollow noreferrer&quot;&gt;image search&lt;/a&gt;. The hair seems different, and so do the ears, the eyebrows and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actually this alleged appearance of Pontecorvo seems to derive from a &lt;a href=&quot;https://www.gettyimages.co.uk/photos/bruno-pontecorvo-mondadori?family=editorial&amp;amp;phrase=bruno%20pontecorvo%20mondadori&amp;amp;sort=best&quot; rel=&quot;nofollow noreferrer&quot;&gt;single set of photos&lt;/a&gt; depicting this man during a meeting with Enrico (and Laura?) Fermi, from one of which the first photo of this question is cropped. In particular, in one of those photos, the man wears glasses, which doesn't happen in any other portrait of Pontecorvo that I know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, in Wikimedia Commons these photos are dated “1950s” (see here &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Bruno_Pontecorvo_and_Enrico_Fermi_1950s3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;one of them&lt;/a&gt;, for instance), while in the 1950s Pontecorvo was as far away as possible from Fermi, the former in Soviet Union (out of which he could travel only in 1978) and the latter in the USA (and for half the 1950s Fermi was dead, of course).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: is that man actually Bruno Pontecorvo and for some reason he doesn't look like his other photos? If not, who is he? And – bonus question – in any case, which was the occasion for that meeting between this man and Fermi?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;: I have found further photos that show Enrico and Laura Fermi in the same outfit as the one in the photos with the mysterious man: &lt;a href=&quot;https://www.gettyimages.it/detail/fotografie-di-cronaca/italian-born-american-physicist-enrico-fermi-is-fotografie-di-cronaca/471542472#/italianborn-american-physicist-enrico-fermi-is-wife-laura-capon-and-picture-id471542472&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://www.agefotostock.com/age/en/Stock-Images/Rights-Managed/MDO-AA318818&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;, &lt;a href=&quot;https://www.gettyimages.it/license/471542662&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;. As it might have been surmised, they were all taken at the 1954 conference in Villa Monastero, Varenna, Lake Como (Italy), Fermi's last journey in Europe, and one of his last public appearances. So, the other man is definitely not Pontecorvo (who at the time was in the Soviet Union, with a very limited freedom of internal movement, let alone to go abroad). The question now is: who was this man?&lt;/p&gt;&#xA;" OwnerUserId="634" LastEditorUserId="634" LastEditDate="2018-06-28T20:09:37.017" LastActivityDate="2019-07-11T03:42:08.100" Title="Who was this man (who is not Bruno Pontecorvo)?" Tags="|physics|depiction-of-scientists|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7479" PostTypeId="1" CreationDate="2018-06-28T19:43:45.573" Score="7" ViewCount="5679" Body="&lt;p&gt;I have run across several references to Aristotle's arguments for a spherical earth which &lt;a href=&quot;http://www.discovery.com/dscovrd/nature/no-earth-isnt-flat-heres-how-ancients-proved-it/&quot; rel=&quot;noreferrer&quot;&gt;claim&lt;/a&gt; that he noted that ships sink over horizon hull-first. For instance, Isaac Asimov writes in his essay &lt;a href=&quot;https://chem.tufts.edu/answersinscience/relativityofwrong.htm&quot; rel=&quot;noreferrer&quot;&gt;The Relativity of Wrong&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There were reasons, to be sure, to find the flat-earth theory&#xA;  unsatisfactory and, about 350 B.C., the Greek philosopher Aristotle&#xA;  summarized them. First, certain stars disappeared beyond the Southern&#xA;  Hemisphere as one traveled north, and beyond the Northern Hemisphere&#xA;  as one traveled south. Second, the earth's shadow on the moon during a&#xA;  lunar eclipse was always the arc of a circle. Third, here on the earth&#xA;  itself, ships disappeared beyond the horizon hull-first in whatever&#xA;  direction they were traveling.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, Aristotle's arguments for the sphericity of the earth appear in &lt;a href=&quot;http://classics.mit.edu/Aristotle/heavens.2.ii.html&quot; rel=&quot;noreferrer&quot;&gt;On the Heavens, Book II, Part 14&lt;/a&gt;, and I see no reference to ships sinking below the horizon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Aristotle in fact make this argument somewhere else, and if not, who was the first person to take this as evidence of a spherical earth?&lt;/p&gt;&#xA;" OwnerUserId="7317" LastEditorUserId="4251" LastEditDate="2021-09-14T20:04:20.840" LastActivityDate="2021-09-14T20:04:20.840" Title="Did Aristotle note that ships disappear over the horizon hull-first?" Tags="|ancient-greece|aristotle|antiquity|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7480" PostTypeId="2" ParentId="7479" CreationDate="2018-06-29T12:56:35.147" Score="4" Body="&lt;p&gt;I dug up some text from a translation of &quot;on the heavens.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are similar disputes about the shape of the earth. Some think it&#xA;  is spherical, others that it is flat and drum-shaped. For evidence&#xA;  they bring the fact that, as the sun rises and sets, the&#xA;  part concealed by the earth shows a straight and not a curved edge,&#xA;  whereas if the earth were spherical the line of section would have to&#xA;  be circular. In this they leave out of account the great distance of&#xA;  the sun from the earth and the great size of the circumference, which,&#xA;  seen from a distance on these apparently small circles  appears&#xA;  straight. Such an appearance ought not to make them doubt the circular&#xA;  shape of the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He goes on to discuss why the &quot;whirling&quot; does not require wind or loss of atmosphere.  There does not appear to be any mention of observing ships. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, he invokes eclipses of the moon:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How else would eclipses of the moon show segments ments shaped as we&#xA;  see them? As it is, the shapes which the moon itself each month shows&#xA;  are of every kind straight, gibbous, and concave-but in eclipses the&#xA;  outline is always curved: and, since it is the interposition of the&#xA;  earth that makes the eclipse, the form of this line will be caused by&#xA;  the form of the earth's surface, which is therefore spherical&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Followed by a discussion of the change in star positions, roughly as a function of latitude, to show the earth cannot be flat. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm far from a knowledgeable scholar of ancient history, so cannot state that there's no other document written by Aristotle on this topic.  &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-06-29T12:56:35.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7482" PostTypeId="2" ParentId="7475" CreationDate="2018-06-29T15:52:20.130" Score="4" Body="&lt;p&gt;I found the following at &lt;a href=&quot;https://history.aip.org/history/exhibits/curie/article.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;an AIP site on Marie Curie&lt;/a&gt;, apparently a copy of an article of hers in Century Magazine, pp 461-466 (January 1904). It appears to be a non-technical journal: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Radium possesses the remarkable property of liberating heat spontaneously and continuously. A solid salt of radium develops a quantity of heat such that for each gram of radium contained in the salt there is an emission of one hundred calories per hour. Expressed differently, radium can melt in an hour its weight in ice. When we reflect that radium acts in this manner continuously, we are amazed at the amount of heat produced, for it can be explained by no known chemical reaction.The radium remains apparently unchanged. If, then, we assume that it undergoes a transformation, we must therefore conclude that the change is extremely slow; in an hour it is impossible to detect a change by any known methods.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One possible technical article as the first report, although I can't access it on-line to check, is:&#xA;'On heat spontaneously extricated by salts of radium', P. Curie and A. Laborde, Comptes Rendus 136, 673-675 (1903). Note that the author is Pierre Curie, not Marie Curie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Previous articles, similarly hard to get, may have a comment on the warmth of the various entities that the Curies were isolating, but that would take finding a library with Comptes Rendus from the early 1900's.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2018-06-29T15:52:20.130" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7483" PostTypeId="1" AcceptedAnswerId="7485" CreationDate="2018-06-29T19:04:23.337" Score="7" ViewCount="373" Body="&lt;p&gt;In number theory, we may encounter the notion of &lt;a href=&quot;https://en.wikipedia.org/wiki/S-unit&quot; rel=&quot;noreferrer&quot;&gt;$S$-unit&lt;/a&gt;, $S$-integer, etc. where $S$ is a finite set of prime numbers (for simplicity). For instance, if $S = \{2,3\}$ then the $S$-integers are the elements of $\Bbb Z[1/6]$. I was wondering if there is a particular reason as to why the letter $S$ was chosen to systematically denote a finite set of primes. I wouldn't be surprised if it came from a German word (as &lt;a href=&quot;https://mathoverflow.net/questions/62218/&quot;&gt;many&lt;/a&gt; number-theoretic words).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My first thought that the $S$-integers are related to a &lt;a href=&quot;https://en.wikipedia.org/wiki/Localization_(algebra)&quot; rel=&quot;noreferrer&quot;&gt;localization&lt;/a&gt; of $\Bbb Z$ — as the example above shows. Usually, when $R$ is a commutative ring and $S \subset R$ is multiplicatively closed with $1 \in S$ (e.g. $S$ is the set of powers of $6$), we can construct a new ring denoted by $S^{-1}R$. My second question is: why did we choose $S$ as a letter? Is it just for &quot;&lt;strong&gt;s&lt;/strong&gt;ubset&quot;, or because $S$ is the next letter after $R$?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know where/when/who first introduced these two notions, the one of $S$-integer, and the one of localization. I'm not sure that these two are historically related, at least in the choice of the letter $S$. I would be grateful to any piece of information about this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you very much!&lt;/p&gt;&#xA;" OwnerUserId="3454" LastActivityDate="2018-06-30T02:03:28.677" Title="Where does the letter S in &quot;$S$-units&quot; and in localization $S^{-1} R$ come from?" Tags="|terminology|notation|number-theory|abstract-algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7484" PostTypeId="1" CreationDate="2018-06-29T21:27:41.267" Score="5" ViewCount="199" Body="&lt;p&gt;Gauss's contributions (&lt;strong&gt;Dioptriche Untersuchungen, 1840&lt;/strong&gt;) to the &quot;classical&quot; optical theory (Gaussian optics, i.e calculations made under the paraxial approximation) are &lt;a href=&quot;https://hsm.stackexchange.com/questions/6150/what-results-did-c-f-gauss-add-to-euler-s-dioptrics&quot;&gt;well known&lt;/a&gt;, but long before he published his known treatise he published a small article (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22:[509],%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1817&lt;/a&gt;) with much more impressive results. This article dealt with a qualitative and quantitive description of a new objective for telescopes that is designed to reduce chromatic aberration (the phenomena in which the dispersion of light results in confused colours of the image). This objective consisted of two lenses, and Gauss calculated the shape, refractive powers (refractive indexes) and spatial disposition of the two lenses that is required for maximal elimination of chromatic aberration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What made me ask this question is the article &lt;strong&gt;&quot;On the new form of the achromatic object-glass introduced by Steinheil&quot;&lt;/strong&gt; (&lt;a href=&quot;https://archive.org/details/onnewformofachro00bond&quot; rel=&quot;nofollow noreferrer&quot;&gt;1863&lt;/a&gt;, by  Bond, George Phillips) which claims that the famous optician Steinheil constructed a new form of achromatic objective &quot;according to the system of curves proposed by Gauss&quot;. In addition, I found in Gauss's Nachlass much material in optics (see Gauss's Werke, &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN236020595&quot; rel=&quot;nofollow noreferrer&quot;&gt;volume 11, part 1&lt;/a&gt;, pp. 117-171) which seems to deal with the problem of removal of chromatic aberration, but not enough comments about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I didn't find enough material about this aspect of his optical work in the literature, so I ask if anyone can give a more detailed explanation of what Gauss actually did.  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-21T18:41:52.203" LastActivityDate="2019-10-21T18:41:52.203" Title="What was Gauss's theoretical work related to his invention of the &quot;Double-Gauss lens&quot;?" Tags="|biographical-details|optics|gauss|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7485" PostTypeId="2" ParentId="7483" CreationDate="2018-06-29T23:40:34.337" Score="6" Body="&lt;p&gt;As Francois Ziegler suggests in his comment, the notation $S$ and term $S$-unit might go back to Artin and Whaples in their paper about the product formula:  &quot;Axiomatic Characterization of Fields by the Product Formula for Valuations&quot; (Bull. AMS 51 (1945), 469-492). Here they write $S$ for a finite nonempty set of primes that includes all the archimedean primes and they define $S$-units on p. 487.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The $S$-unit theorem is in a weak form as Theorem 5 (p. 489) and in its full form as Theorem 6 (p. 491), with a footnote attributing the statement of the theorem to Hasse and the proof to Chevalley in Chevalley's paper on class field theory in Annals of Math. 41 (1940), 394-418. There Chevalley writes $E$, not $S$, for a finite set of &quot;prime divisors containing all the infinite prime divisors&quot; and the $S$-unit theorem is stated two paragraphs after Theorem 3. Since Chevalley's paper is the original appearance of the $S$-unit theorem, using notation no longer around today, it looks like the notation $S$ and the term $S$-unit are due to Artin and Whaples.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though the $S$-units are the unit group of the ring of $S$-integers, the concept of an $S$-integer came later since Artin and Whaples don't mention any such type of generalized integers in their paper.  It's sort of historically fitting that $S$-units were created before the ring of which they are units, since Chevalley did the same thing for ideles: he defined that group (1936) and worked with it before anyone had defined the ring of adeles of which the ideles are the unit group. Strange but true. The Wikipedia page for adelic algebraic groups points out that Chevellay started using the term &quot;idele&quot; in 1940, while the adeles were called repartitions and (Tate's thesis) valuation vectors before the term &quot;adele&quot; became standard later in the 1950s.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2018-06-30T02:03:28.677" LastActivityDate="2018-06-30T02:03:28.677" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7486" PostTypeId="2" ParentId="68" CreationDate="2018-06-30T09:33:53.940" Score="0" Body="&lt;p&gt;Danu's answer is true for European cultures. But according to &lt;a href=&quot;https://www.gns.cri.nz/Home/Our-Science/Earth-Science/Earth-s-Magnetic-Field/Discovery-of-the-Earth-s-magnetic-field&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gns.cri.nz/Home/Our-Science/Earth-Science/Earth-s-Magnetic-Field/Discovery-of-the-Earth-s-magnetic-field&lt;/a&gt; but the Chinese were 1200 years ahead of them.&lt;/p&gt;&#xA;" OwnerUserId="7570" LastActivityDate="2018-06-30T09:33:53.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7487" PostTypeId="1" AcceptedAnswerId="16254" CreationDate="2018-06-30T16:27:34.073" Score="1" ViewCount="544" Body="&lt;p&gt;My question refers to a fragment on electrodynamics written by Gauss at around 1835. In this short note (see Gauss' &lt;em&gt;Werke&lt;/em&gt;, volume V, p.617) Gauss wrote down a fundamental equation which describes the general electromagnetic interaction between moving charges with respect to each other, i.e. his formula is a generalization of Coulomb's law for moving charges. I find this formula to be  the most insightful part of his physical work - Gauss wrote it in the context of his meditations on the electromagnetic force after an idea came to his mind that that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;two elements of electricity attract or repel each other in a different way than if they were in a relative state of rest.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;one needs to replace the &amp;quot;action at a distance&amp;quot; theory of charge-charge interaction with a finite speed of propagation of forces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gauss' equation is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F = \frac {{q_1q_2}}{{r^2}}(1 + \frac {{1}}{{c^2}}(u^2 - \frac {{3}}{{2}}\dot r^2))$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the speed of light, &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; the relative velocity of the two charges and &lt;span class=&quot;math-container&quot;&gt;$\dot r$&lt;/span&gt; the radial component of the relative movement of the charges. The Gauss' &lt;em&gt;Nachlass&lt;/em&gt; contains an additional such basic law (see Schaefer's treatise on Gauss physical works):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F = -\frac{u}{r^3} - \frac{\alpha}{u r}(\frac{du}{dt})^2 - \frac{\beta}{r}\frac{d^2 u}{dt^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is a kind of &amp;quot;distance vector&amp;quot; (Gauss terms it &amp;quot;tricomplex number&amp;quot;), &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the scalar distance, &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; time, and &lt;span class=&quot;math-container&quot;&gt;$\alpha, \beta$&lt;/span&gt; are two constants which are related by: &lt;span class=&quot;math-container&quot;&gt;$\beta = 2\alpha$&lt;/span&gt;. Gauss defines  &lt;span class=&quot;math-container&quot;&gt;$\alpha, \beta$&lt;/span&gt; in terms of a third constant &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; which has units of speed: &lt;span class=&quot;math-container&quot;&gt;$\alpha = \frac{1}{8\epsilon^2}, \beta = \frac{1}{4\epsilon^2}$&lt;/span&gt;. Gauss' first electrodynamics law is therefore very similar to Weber's general law, by that the first term is the Coulomb's Law, the second depends on the squared relative speed, and the third is an acceleration term.&lt;/p&gt;&#xA;&lt;p&gt;In the article &amp;quot;The Suppressed Electrodynamics of Ampère-Gauss-Weber&amp;quot;, note 8, the author says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;More than a decade before the publication of Weber’s 1846 paper, one can find an 1835 entry in Gauss’s Notebooks, showing a hypothesized form of the electrodynamic force law, dependent on relative velocity and acceleration, that is essentially equivalent to that which Weber used in the 1846 publication. Interestingly, the Gauss formulation appears on the same page as an alternative formulation, which was the one James Clerk Maxwell chose to use in his text Treatise on Electricity and Magnetism to falsely imply a difference in electrodynamic views among the three collaborators, Gauss, Weber, and Riemann.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I guess that the author refers to Gauss' first electrodynamics law, which appears on p. 616 of the same volume.&lt;/p&gt;&#xA;&lt;p&gt;Now, I found very confusing statements about Gauss' equation in the literature - all the sources agree that this equation doesn't settle completely with Maxwell's description of electromagnetism, but some state that it's consistent with Maxwell in many aspects (though not completely).&lt;/p&gt;&#xA;&lt;p&gt;I checked the modern relativistic formula for the electric field produced by a uniformly moving charge:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$E(r,\theta) = \frac{kq}{r^2}\frac{1-\beta^2}{(1-\beta^2\mathbb{sin}^2\theta)^{3/2}}\hat{r}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which in general is obviously not of the same mathematical form of Gauss's equation for non-accelerating charges. Here &lt;span class=&quot;math-container&quot;&gt;$\beta=u/c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle between the line of sight to the charge and the direction of relative motion. But I still tried to check if the two expressions are equivalent for low speeds (&lt;span class=&quot;math-container&quot;&gt;$\beta&amp;lt;&amp;lt;1$&lt;/span&gt;) by using first-order approximation of the relativistic formula:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$cos\theta = \frac{\dot{r}}{u} \implies \frac{1-\beta^2}{(1-\beta^2\mathbb{sin}^2\theta)^{3/2}}= \frac{1-(u/c)^2}{(1-(u/c)^2\cdot (\frac{u^2-\dot{r}^2}{u^2}))^{3/2}}\approx (1-(u/c)^2)\cdot(1+\frac{3}{2}(\frac{u^2-\dot{r}^2}{c^2}))\approx 1+\frac{1}{c^2}(\frac{1}{2}u^2-\frac{3}{2}\dot{r}^2)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;so there is no agreement of the two expressions even in the limit of low speeds. This is interesting, because apparently (according to articles I found) Gauss's electrodynamics law settles completely with Ampere's force law for current-current interaction, with the last law being an established scientific fact (Ampere derived it from detailed experiments).&lt;/p&gt;&#xA;&lt;p&gt;I'm not to trying to be a promoter of lost and obscure physical theories, (obviously if Maxwell's theory eventually superseded Gauss-Weber approach, then there must be a rational historical reason for that), but I really think i miss a piece of the puzzle in my historical  understanding of pre-maxwellian electrodynamics.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2022-05-06T20:42:46.427" LastActivityDate="2024-02-24T12:18:09.243" Title="Are Gauss' electrodynamics laws for charge-charge interaction correct in the limit of low-speeds?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7488" PostTypeId="2" ParentId="7341" CreationDate="2018-07-01T02:34:05.973" Score="10" Body="&lt;p&gt;I wonder why the insistence on the (English) word &lt;em&gt;&lt;strong&gt;rule&lt;/strong&gt;&lt;/em&gt;, especially as German wikipedia translates / redirects it to &lt;a href=&quot;https://de.wikipedia.org/wiki/Bornsche_Regel&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;interpretation&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;. Isn’t it enough for your purposes to see it stated, named and credited as Born’s &lt;em&gt;Deutung&lt;/em&gt; (Jordan &lt;a href=&quot;https://zbmath.org/?q=an:53.0849.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;https://doi.org/10.1007/BF01390903&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 811&lt;/a&gt;), &lt;em&gt;assumption&lt;/em&gt; (Dirac &lt;a href=&quot;https://zbmath.org/?q=an:53.0847.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;https://doi.org/10.1098/rspa.1927.0039&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 257&lt;/a&gt;), &lt;em&gt;Interpretation&lt;/em&gt; (Hilbert et al. &lt;a href=&quot;https://zbmath.org/?q=an:53.0849.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1928&lt;/a&gt;, &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0098?tify=%7B%22pages%22:%5B33%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 29&lt;/a&gt;), or &lt;em&gt;Auffassung&lt;/em&gt; (Schrödinger &lt;a href=&quot;https://zbmath.org/?q=an:53.0832.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k15385d/f991&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 967&lt;/a&gt;; Handbuch der Physik &lt;a href=&quot;http://zbmath.org/?q=an:54.0959.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1928&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=933yBgAAQBAJ&amp;amp;pg=PA589&amp;amp;dq=Bornsche+Auffassung&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 589&lt;/a&gt;)?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt;&lt;br&gt;To your edit asking “who first linked Born’s name to an explicit statement about measuring &lt;em&gt;arbitrary&lt;/em&gt; observables” (though nothing prevents a hamiltonian from being “arbitrary”...): if not von Neumann (&lt;a href=&quot;https://zbmath.org/?q=an:58.0929.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;1932&lt;/a&gt;, footnotes &lt;a href=&quot;https://google.com/books?id=JLyCo3RO4qUC&amp;amp;ots=4CgJA0RT8z&amp;amp;pg=PA5&amp;amp;dq=statistical&quot; rel=&quot;nofollow noreferrer&quot;&gt;8&lt;/a&gt;, &lt;a href=&quot;https://google.com/books?id=JLyCo3RO4qUC&amp;amp;ots=4CgJA0RT8z&amp;amp;pg=PA198&amp;amp;dq=%22statistical+character%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;118&lt;/a&gt;, &lt;a href=&quot;https://google.com/books?id=JLyCo3RO4qUC&amp;amp;ots=4CgJA0RT8z&amp;amp;pg=PA210&amp;amp;dq=%22statistical+interpretation%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;122&lt;/a&gt;), then I’d guess &lt;strong&gt;E. Bauer&lt;/strong&gt; (&lt;a href=&quot;https://zbmath.org/?q=an:59.1521.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1933&lt;/a&gt;, p. 42; translation &lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=155568&quot; rel=&quot;nofollow noreferrer&quot;&gt;1962&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=oKLq9qmJoL0C&amp;amp;ots=aMq5egGJX8&amp;amp;pg=PA39&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 39&lt;/a&gt;) who writes, for a superposition &lt;span class=&quot;math-container&quot;&gt;$\psi=\smash{\sum_k\beta_k\psi_k}$&lt;/span&gt; of eigenstates belonging to eigenvalues &lt;span class=&quot;math-container&quot;&gt;$\alpha_k$&lt;/span&gt; of a “physical quantity” &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Nous admettrons qu’une expérience faite sur un système à l’état &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; peut nous donner l’une quelconque de ces valeurs &lt;span class=&quot;math-container&quot;&gt;$\alpha_k$&lt;/span&gt; avec une probabilité égale à &lt;span class=&quot;math-container&quot;&gt;$\overline\beta{}_k\beta{}_k$&lt;/span&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ce dernier postulat entrevu par Einstein, &lt;em&gt;&lt;strong&gt;énoncé en toute précision par Born&lt;/strong&gt;&lt;/em&gt; et développé par Dirac est comme la clef de voûte de l’édifice quantique.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(The mention of Einstein is also in Born (&lt;a href=&quot;https://doi.org/10.1007/BF01397184&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;, p. 804; &lt;a href=&quot;https://books.google.com/books?id=mpTvAAAAMAAJ&amp;amp;q=%22guided+by+a+remark+of+Einstein%27s%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1978&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=mpTvAAAAMAAJ&amp;amp;q=%22must+be+regarded+as+the+probability+density%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 232&lt;/a&gt;), Heisenberg (&lt;a href=&quot;https://zbmath.org/?q=an:53.0853.05&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=KF2oBgAAQBAJ&amp;amp;ots=qBkWy5LJMz&amp;amp;pg=PA482&amp;amp;dq=statistische+Bedeutung&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 176&lt;/a&gt;), Pais (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=684484&quot; rel=&quot;nofollow noreferrer&quot;&gt;1982&lt;/a&gt;, pp. 1196-1197).) Bauer may have the first occurrence of “Born’s rule”, too, in Théorie quantique de la valence. Les liaisons homéopolaires. &lt;em&gt;Bull. Soc. Chim. France. Mém.&lt;/em&gt; (5) &lt;strong&gt;1&lt;/strong&gt; (1934) 293-347, &lt;a href=&quot;https://books.google.com/books?id=EXm3AAAAIAAJ&amp;amp;q=%22d%27apr%C3%A8s+la+r%C3%A8gle+de+Born%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 302&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Multiplions d'autre part &lt;span class=&quot;math-container&quot;&gt;$|\psi|^2$&lt;/span&gt; par le volume &lt;span class=&quot;math-container&quot;&gt;$4\pi r^2dr$&lt;/span&gt; compris entre les sphères de rayons &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$r + dr$&lt;/span&gt;, nous obtenons, d’après la &lt;em&gt;&lt;strong&gt;règle de Born&lt;/strong&gt;&lt;/em&gt;, la probabilité de rencontrer l’électron dans la couche sphérique ainsi délimitée, c’est-à-dire à une distance du noyau comprise entre &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$r + dr$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally and perhaps most influentially, I would say that Born’s book &lt;em&gt;Atomic Physics&lt;/em&gt; (1935) has everything you asked for (and essentially insisted he never wrote...):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/in.ernet.dli.2015.176739#page/n153&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;§V.6. The Statistical Interpretation of Wave Mechanics.&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We have already mentioned the interpretation of the wave function &lt;em&gt;&lt;strong&gt;given by the author&lt;/strong&gt;&lt;/em&gt; (p. 83). Let the proper function corresponding to any state be &lt;span class=&quot;math-container&quot;&gt;$\psi_E$&lt;/span&gt;; then &lt;span class=&quot;math-container&quot;&gt;$\smash{|\psi_E|{}^2dv}$&lt;/span&gt; is the probability that the electron (regarded as a corpuscle) is in the volume element &lt;span class=&quot;math-container&quot;&gt;$dv$&lt;/span&gt;. (...)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/in.ernet.dli.2015.176739#page/n343&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Appendix XXII. The Formalism of Quantum Mechanics, and the Uncertainty Relation.&lt;/strong&gt;&lt;/a&gt; (...)&lt;/p&gt;&#xA;&lt;p&gt;The statistical interpretation of quantum mechanics consists in the following assumptions: To each physical quantity or “observable” belongs a real operator &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. The proper functions &lt;span class=&quot;math-container&quot;&gt;$\psi_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\psi_2$&lt;/span&gt;, ... correspond to the quantised states, for which the operator takes the value &lt;span class=&quot;math-container&quot;&gt;$a_1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a_2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$a_3$&lt;/span&gt; ... ; any function &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; is a state, which is composed of these states (...) The coefficients &lt;span class=&quot;math-container&quot;&gt;$c_n$&lt;/span&gt; of the expansion determine the strength with which the quantum state &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; occurs in the general state &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;. The probability of then finding the proper value &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; in a measurement is given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$\smash{w_n=|c_n|^2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In fact, Born already wrote the same things, slightly less polished, in proceedings of the Fall 1927 conferences in &lt;a href=&quot;http://www.worldcat.org/oclc/9537862&quot; rel=&quot;nofollow noreferrer&quot;&gt;Como&lt;/a&gt; (translation &lt;a href=&quot;https://archive.org/stream/physucsinmygener006567mbp#page/n26&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 15&lt;/a&gt;; reprint &lt;a href=&quot;https://doi.org/10.1007/978-3-322-88794-8_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 16&lt;/a&gt;) and &lt;a href=&quot;https://zbmath.org/?q=an:54.0960.06&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brussels&lt;/a&gt; (&lt;a href=&quot;http://www.solvayinstitutes.be/html/solvayconf_physics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 166–170&lt;/a&gt;). There he credits himself (&lt;a href=&quot;https://doi.org/10.1007/BF01397184&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;, &lt;a href=&quot;https://doi.org/10.1007/BF01400360&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;) as well as Dirac (&lt;a href=&quot;https://doi.org/10.1098/rspa.1927.0012&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;), Jordan (&lt;a href=&quot;https://doi.org/10.1007/BF01390903&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;https://doi.org/10.1007/BF01391714&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;) and von Neumann (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN252457811_1927?tify=%7B%22pages%22:%5B5%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, &lt;a href=&quot;http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002271788&quot; rel=&quot;nofollow noreferrer&quot;&gt;1928&lt;/a&gt;). This is likely where Bauer got his information.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-04T16:29:31.297" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7489" PostTypeId="1" AcceptedAnswerId="7497" CreationDate="2018-06-30T17:45:26.797" Score="7" ViewCount="294" Body="&lt;p&gt;In &lt;a href=&quot;https://www.goodreads.com/book/show/37037001-the-gene&quot; rel=&quot;noreferrer&quot;&gt;The Gene: An Intimate History&lt;/a&gt;, author concludes the Mendel's pea experiments publication by these words:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mendel himself requested forty reprints, which he mailed, heavily annotated, to many scientists. It is likely that he sent one to Darwin, but there is no record of Darwin’s having actually read it.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What followed, as one geneticist wrote, was “&lt;strong&gt;&lt;em&gt;one of the strangest&#xA;  silences in the history of biology&lt;/em&gt;&lt;/strong&gt;.” The paper was cited &lt;strong&gt;only four&#xA;  times between 1866 and 1900&lt;/strong&gt;—virtually disappearing from scientific&#xA;  literature. Between 1890 and 1900, even as questions and concerns&#xA;  about human heredity and its manipulation became central to policy&#xA;  makers in America and Europe, Mendel’s name and his work were lost to&#xA;  the world. The study that founded modern biology was buried in the&#xA;  pages of an obscure journal of an obscure scientific society, read&#xA;  mostly by plant breeders in a declining Central European town.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wonder what could be the reason for that. If we attribute only to Mendel being a low-key monk from Brno, then still just 4 citations don't do justice given the Genetics and Heredity was arguably dawning right then. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does it have something to do with his radical ideas - but if that is the case, then Darwin's ideas were much more challenging to the Church in my opinion. I am interested to know the possible reason(s) behind this &quot;&lt;strong&gt;&lt;em&gt;one of the strangest silences in the history of Biology&lt;/em&gt;&lt;/strong&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="7572" OwnerDisplayName="Talha Irfan" LastEditorUserId="4251" LastEditDate="2020-07-06T15:39:40.557" LastActivityDate="2020-07-06T15:39:40.557" Title="Why didn't Mendel's experiments draw attention until the 20th century?" Tags="|biology|genetics|mendel|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7490" PostTypeId="1" AcceptedAnswerId="7491" CreationDate="2018-07-01T07:53:32.663" Score="9" ViewCount="734" Body="&lt;p&gt;I would like to find an accurate translation to the following &lt;a href=&quot;https://archive.org/stream/raumzeitmateriev00weyl#page/48&quot; rel=&quot;nofollow noreferrer&quot;&gt;quote&lt;/a&gt; from Space Time Matter:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Man muß gegen diese Orgien des Formalismus, mit dem man heute sogar die Techniker zu belästigen beginnt, nachdrücklich protestieren.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Google translate gives:&#xA;It is necessary to vigorously protest against the orgies of formalism with which even today the technicians begin to bother.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Henry L. Brose's 1922 English translation &lt;a href=&quot;https://archive.org/stream/spacetimematter00weyluoft#page/54&quot; rel=&quot;nofollow noreferrer&quot;&gt;reads&lt;/a&gt;:&#xA;An emphatic protest must be entered against these orgies of formalism which are threatening the peace of even the technical scientist.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are:&#xA;What is a &quot;technical scientist&quot;? Does he mean &quot;professional&quot;?  Or someone who has a strong technique?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does &quot;even&quot; apply to &quot;today&quot; (i.e. mean &quot;already&quot;) or &quot;technical scientist&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does it really say &quot;threatening the peace&quot;?&lt;/p&gt;&#xA;" OwnerUserId="7571" LastEditorUserId="10106" LastEditDate="2020-06-07T18:46:14.817" LastActivityDate="2020-06-07T18:46:14.817" Title="Help translate from German a quote by Hermann Weyl in Space Time Matter" Tags="|mathematicians|differential-geometry|riemannian-geometry|translations|" AnswerCount="4" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7491" PostTypeId="2" ParentId="7490" CreationDate="2018-07-01T10:24:06.587" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;One must firmly protest these orgies of a formalism with which even technicians are getting harassed today.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Literally: of the formalism, with which one is beginning to harass even technicians today.)&lt;br&gt;To your questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;em&gt;Techniker&lt;/em&gt; is for technicians, engineers, graduates of the &lt;a href=&quot;https://archive.org/stream/spacetimematter00weyluoft#page/n8&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Technische Hochschule&lt;/em&gt;&lt;/a&gt; where Weyl gave these lectures, as opposed to scientists or graduates of the University. Applied, not pure.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;em&gt;sogar&lt;/em&gt; definitely applies to &lt;em&gt;Techniker&lt;/em&gt;, not to &lt;em&gt;heute&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;em&gt;belästigen&lt;/em&gt; &lt;a href=&quot;https://translate.google.com/#de/en/bel%C3%A4stigen&quot; rel=&quot;nofollow noreferrer&quot;&gt;=&lt;/a&gt;&#xA;harass,&#xA;molest,&#xA;bother,&#xA;annoy,&#xA;irritate,&#xA;badger,&#xA;pester,&#xA;trouble,&#xA;importune,&#xA;persecute.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note added:&lt;/strong&gt;&lt;br&gt;&#xA;Weyl’s words sound like a veiled attack on F. Klein’s preface to Schouten’s &lt;em&gt;Affinoranalysis&lt;/em&gt; (&lt;a href=&quot;https://zbmath.org/?q=an:45.0190.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;) and its &lt;a href=&quot;https://archive.org/stream/grundlagenderve00schogoog#page/n108&quot; rel=&quot;nofollow noreferrer&quot;&gt;19+ operations on affinors, deviators, septors, nonors, etc.&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dr. J. A. Schouten was active so far in Rotterdam as an electrical engineer (Elektrotechniker), and got on his own from electrotechnical problems to the theories he outlines in what follows.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The point is to investigate the geometrical quantities that arise in vector analysis and the Gibbs dyads, triads, etc., on the basis of a group theoretical principle I established long ago: that all geometry is invariant theory under a group, which however one has much latitude in choosing.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Mr. Schouten’s investigations are all the more welcome, that it is the first time the developments in question, which alone seem to lead to a rational division of geometrical structures, are taken up by a practitioner. Mr. Schouten’s main achievement is that he consistently implements the principle even in higher cases. Of course, some of the resulting higher-order structures already appeared now and then in mechanics and physics, but they had not yet been enumerated in such systematic completeness as is the case here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Jan_Arnoldus_Schouten#Grundlagen_der_Vektor-_und_Affinoranalysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;even claims&lt;/a&gt; that Weyl’s quote targets this book explicitly. However, this seems to rely on overinterpretation by Reich (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=1353442&quot; rel=&quot;nofollow noreferrer&quot;&gt;1994&lt;/a&gt;, p. 157) of an ill-captioned picture in Rowe (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=979020&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989a&lt;/a&gt;, p. 17; &lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=1007032&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989b&lt;/a&gt;): Struik (&lt;a href=&quot;https://ir.cwi.nl/pub/20360&quot; rel=&quot;nofollow noreferrer&quot;&gt;1971&lt;/a&gt;, p. 2) merely writes that “&lt;strong&gt;Schouten&lt;/strong&gt; later realized that Weyl’s critique of “orgies of formalism” was &lt;strong&gt;also&lt;/strong&gt; applicable to this book” — and Klein also promoted &lt;em&gt;Theory of Screws&lt;/em&gt; (&lt;a href=&quot;https://zbmath.org/?q=an:31.0679.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900&lt;/a&gt;) or &lt;em&gt;Geometrie der Dynamen&lt;/em&gt; (&lt;a href=&quot;https://zbmath.org/?q=an:33.0691.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1903&lt;/a&gt;), among others.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-07-14T06:20:42.073" LastActivityDate="2018-07-14T06:20:42.073" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7492" PostTypeId="2" ParentId="7490" CreationDate="2018-07-01T16:35:15.300" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;One has to object thoroughly to the orgies of formalism, with which recently even technicians have been bothered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Man muß ... protestieren -&gt; One has to object to (in the context of legislative elements like mandatory paperwork)&lt;/li&gt;&#xA;&lt;li&gt;nachdrücklich -&gt; thoroughly (or intensely, not just a little)&lt;/li&gt;&#xA;&lt;li&gt;man heute ... zu belästigen beginnt -&gt; with which recently ... have been bothered (this translation lacks the continuation element that '...en beginnt' has, but I think it matches the intend better, also note that the original uses a different grammatical form that I omitted for better readability)&lt;/li&gt;&#xA;&lt;li&gt;sogar die Techniker -&gt; even the technicians (most likely not scientists but applied craftsmen, even if they have studied)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I found it interesting that there is a 'dem' not a 'denen', so the objection is not against the amount of formalism (so not against the orgies of) but against the formalism at all.&lt;/p&gt;&#xA;" OwnerUserId="7575" LastActivityDate="2018-07-01T16:35:15.300" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7493" PostTypeId="2" ParentId="7490" CreationDate="2018-07-01T17:44:44.773" Score="1" Body="&lt;p&gt;&quot;Man muß gegen die Orgien des Formalismus, mit dem man heute sogar die Techniker zu belästigen beginnt, nachdrücklich protestieren.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both of the translations you got are 'decent'. Technician is indeed the literal translation of the German &quot;Techniker&quot;, and engineer (Ingeneur) following. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Technical scientists are beyond my knowledge, but I would expect graduates of technical fields of academic study&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own, spontaneous, translation of your sentence:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;One has to protest vehemently against the orgies of formalism, which nowadays hamstring (or impair or sabotage) our technicians (or the work of our technicians).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My focus was the information, not literature excellence though.  &lt;/p&gt;&#xA;" OwnerUserId="7576" LastActivityDate="2018-07-01T17:44:44.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7494" PostTypeId="2" ParentId="7489" CreationDate="2018-07-01T18:50:35.727" Score="7" Body="&lt;p&gt;Elizabeth B Gasking summarised the reasons in 1959, &lt;a href=&quot;https://www.jstor.org/stable/pdf/2707967.pdf&quot; rel=&quot;noreferrer&quot;&gt;but you're not reading it without a Jstor account&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes a paper is overlooked because we think about its relevance from the wrong perspective. It seems even Mendel didn't realise what he'd uncovered. The laws of inheritance that now bear his name, insofar as he formulated them in something close to their modern form, were imagined - by him, not just his readers - to have a very limited scope, in terms of which species and traits they applied to. This is partly because, as was typical of biologists of the time, he usually thought about inheritance in terms of blending. He called the paper &lt;em&gt;Experiments on Plant Hybridization&lt;/em&gt;, and the general perception was that it was a minor fact about how certain plants hybridise, not a general realisation that heredity is digital. (Before the theory of computers, that concept may have been a bit beyond us.)&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-07-01T18:50:35.727" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7496" PostTypeId="2" ParentId="6306" CreationDate="2018-07-03T00:18:19.453" Score="7" Body="&lt;p&gt;Bargmann (&lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=58601&quot; rel=&quot;noreferrer&quot;&gt;1954&lt;/a&gt;, p. 1) calls this &lt;strong&gt;“Weyl’s terminology&lt;/strong&gt;, cf. (&lt;a href=&quot;https://archive.org/stream/quantumtheoryofa029235mbp#page/n29&quot; rel=&quot;noreferrer&quot;&gt;1931&lt;/a&gt;, p. 4 and p. 20)&lt;strong&gt;”&lt;/strong&gt;. Ray, ray space and ray representation (p. 181) stand for Weyl’s original &lt;em&gt;Strahl&lt;/em&gt;, &lt;em&gt;Strahlenkörper&lt;/em&gt;, &lt;em&gt;Strahldarstellung&lt;/em&gt;, see (&lt;a href=&quot;https://zbmath.org/?q=an:54.0954.03&quot; rel=&quot;noreferrer&quot;&gt;1928&lt;/a&gt;, pp. 7, 19, 161) or (&lt;a href=&quot;https://zbmath.org/?q=an:53.0848.02&quot; rel=&quot;noreferrer&quot;&gt;1927&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;(p. 5): Zwei von 0 verschiedene Vektoren gehören demselben &lt;strong&gt;Strahl&lt;/strong&gt; an, wenn der eine aus dem anderen dutch Multiplikation mit einer (komplexen, von 0 verschiedenen) Zahl hervorgeht. (...) Fasse ich eine unitäre Abbildung (4) auf nicht als Abbildung des Vektor-, sondern des Strahlenkörpers (homogener Standpunkt), so soll sie kurz eine &lt;strong&gt;Drehung&lt;/strong&gt; heißen.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(p.18): Denn in der Quantenmechanik haben nicht die Vektoren eine Bedeutung, sondern lediglich die Strahlen; sie kennzeichnen die verschiedenen reinen Fälle. (...) So soll das Wort Darstellung in Zukunft verstanden werden: als getreue &lt;strong&gt;Darstellung durch Drehungen des Strahlenkörpers*&lt;/strong&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;* Tiefgehende Untersuchungen über das Darstellungsproblem in diesem Sinne hat &lt;strong&gt;I. Schur&lt;/strong&gt; angestellt: Crelles Journ. &lt;strong&gt;127&lt;/strong&gt;, 20, 1904 und &lt;strong&gt;132&lt;/strong&gt;, 85, 1907.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The word (apparently not used by Schur) likely originates in line geometry (a.k.a. optics) where rays were affine lines (W. R. Hamilton, &lt;em&gt;Theory of Systems of Rays&lt;/em&gt;, &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Rays/&quot; rel=&quot;noreferrer&quot;&gt;1828&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-07-03T00:18:19.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7497" PostTypeId="2" ParentId="7489" CreationDate="2018-07-03T09:13:17.903" Score="4" Body="&lt;p&gt;This is a supplement to J.G.'s useful answer: there is a good deal of illuminating source material in William Bateson's (1909) &quot;Mendel's principles of heredity&quot;, available at (&lt;a href=&quot;https://archive.org/details/mendelsprinciple00bate&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/mendelsprinciple00bate&lt;/a&gt;), which was cited by the Gasking 1959 paper. Bateson gave translations of both of Mendel's major papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some points from Gasking's (1959) paper, cited by J.G.: her explanation for Mendel's neglect argued several points. The prevailing perspective on breeding experiments was that of a farmer/horticulturalist, different from a scientific perspective, and linked with reluctance to seek general conclusions from observational facts gained from breeding experiments. Gasking found that some elements of Mendel's observations had been seen [long] before, e.g. by &quot;workers, such as Goss and Seton, [who] were also breeding peas. Goss (John Goss, &quot;On Variations in Colour of Peas Occasioned by Cross-Impregnation,&quot; Trans. Hort. Soc., V (1824), 235) actually noted that, if a white-seeded variety were crossed with a blue- [= green-] seeded type, the hybrid produced only white seeds. He also remarked, first, that amongst the next generation blue seeds appeared once more, and that, if plants grown from these seeds were planted and allowed to self-fertilize, all the subsequent generations were blue-seeded: the white seeds meanwhile continued to produce both white and blue-seeded plants.  It was perfectly consistent with the aims of these workers that Goss, having made this discovery, was content to record it as a fact, and made no attempt to follow it up, in the hope of establishing a more general law.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gasking also pointed out that the development of thought of the 19th-century had been hindered by &quot;extreme vagueness: experimenters would report simply that a hybrid 'resembled' the maternal or paternal species more or less closely, or that it was intermediate -- they did not even trouble to say in what respects (flower-color, height, etc.) the resemblances appeared.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus there was vagueness in the 'species' concept itself, but &quot;In tackling the theory of inheritance as a problem about unit-characters rather than unit-species, Mendel's work was thus completely novel.&quot; [and at some distance from contemporary ways of thinking]. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-07-03T09:13:17.903" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7498" PostTypeId="2" ParentId="7479" CreationDate="2018-07-03T14:59:58.747" Score="4" Body="&lt;p&gt;From &lt;a href=&quot;http://journals.uchicago.edu/doi/abs/10.1086/349149?journalCode=isis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Queries and Answers,&quot; Isis 41, no. 2&lt;/a&gt;, linked to by sand1 in the comments above, we see that the proof of the sphericity of the earth based on the fact that the masts of a distant ship are visible above the horizon when the body of the ship is no longer visible&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...has been attributed to Aristotle in &lt;a href=&quot;http://classics.mit.edu/Aristotle/heavens.2.ii.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Caelo&lt;/em&gt;&lt;/a&gt;, but it does not&#xA;  appear there. It does, however appear in &lt;em&gt;De Revolutionibus&lt;/em&gt; of&#xA;  Copernicus in reference to a light on the tip of a mast. A Similar&#xA;  statement may be found in Ptolemy's &lt;em&gt;Almagest&lt;/em&gt; where it is stated:&#xA;  &quot;Whenever we sail towards mountains or any high places from whatever&#xA;  angle, in whatever direction, we see their bulk little by little&#xA;  increasing as if they were arising from the sea, whereas before they&#xA;  seemed submerged because of the curvature of the water's surface.&quot;&#xA;  [Ptolemy, C., Mathematical Composition, transl. by R. C. Taliaferro,&#xA;  mimeographed for St John's College, Baltimore, 1938, Book I, Section&#xA;  4.].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A century earlier than Ptolemy, Pliny the Elder, in his &lt;em&gt;Natural&#xA;  History&lt;/em&gt;, Book 2, LXIV. 160, ff. (Loeb Library) included in his&#xA;  discussion of the earth's sphericity, &quot;The same cause explains why the&#xA;  land is not visible from the deck of a ship when in sight from the&#xA;  mast-head; and why as a vessel passes far into the distance, if some&#xA;  shining object is tied to the top of the mast it appears slowly to&#xA;  sink and finally it is hidden from sight.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it appears that while Aristotle had several arguments for the sphericity of earth, he did not in fact argue it based on ships sinking below the horizon. However, it is often falsely attributed to him. The earliest I can find of anyone making this specific argument is then Pliny the Elder, around 77 A.D.&lt;/p&gt;&#xA;" OwnerUserId="7317" LastActivityDate="2018-07-03T14:59:58.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7499" PostTypeId="1" AcceptedAnswerId="7528" CreationDate="2018-07-06T02:57:03.493" Score="2" ViewCount="202" Body="&lt;p&gt;The birthdate of Gotthold Eisenstein is Apr. 16, 1823 as is stated in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gotthold_Eisenstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;. But a letter(whose recipient is Gauss) of Encke on &lt;a href=&quot;https://gauss.adw-goe.de/handle/gauss/2728&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oct. 11, 1852&lt;/a&gt; clearly states that the birthdate of Eisenstein is Apr. 10(see the last page of the quoted letter). The date is also implied by the calculation of Gauss on the same page of the letter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is the exact birthdate of Eisenstein?&lt;/p&gt;&#xA;" OwnerUserId="6450" LastEditorUserId="1932" LastEditDate="2018-07-21T12:34:45.780" LastActivityDate="2018-07-21T12:34:45.780" Title="On the birthdate of Gotthold Eisenstein" Tags="|mathematicians|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7501" PostTypeId="1" AcceptedAnswerId="7502" CreationDate="2018-07-08T02:24:56.573" Score="2" ViewCount="287" Body="&lt;p&gt;I first saw this quote on p. 26: Philip Johnson-Laird &lt;a href=&quot;https://dof.princeton.edu/about/clerk-faculty/emeritus/philip-nicholas-johnson-laird&quot; rel=&quot;nofollow noreferrer&quot;&gt;BA PhD Psychology (UCL)&lt;/a&gt;, Stuart Professor of Psychology Emeritus at Princeton.    (Author isn't  a logician.)    &lt;a href=&quot;https://www.amazon.ca/How-We-Reason-Philip-Johnson-Laird/dp/0199551332&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How We Reason  (1st edn 2008)&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.pitt.edu/%7Ejdnorton/Goodies/Einstein_think/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's the context.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;(A) The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The &lt;strong&gt;psychical entities [I emboldened.]&lt;/strong&gt; which seem to serve as elements in thought are certain signs and more or less clear images which can be &amp;quot;voluntarily&amp;quot; reproduced and combined. There is, of course, a certain connection between those elements and relevant logical concepts. It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of this rather vague play with the above-mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought--before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.&lt;/p&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;p&gt;From &amp;quot;A Mathematician's Mind, Testimonial for &lt;em&gt;An Essay on the Psychology of Invention in the Mathematical Field&lt;/em&gt; by Jacques S. Hadamard, Princeton University Press, 1945.&amp;quot; in &lt;em&gt;Ideas and Opinions&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-07-08T10:33:32.197" Title="What did Albert Einstein mean by &quot;psychical entities&quot;?" Tags="|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7502" PostTypeId="2" ParentId="7501" CreationDate="2018-07-08T10:33:32.197" Score="4" Body="&lt;p&gt;Einstein means : &lt;a href=&quot;https://en.wikipedia.org/wiki/Mental_image&quot; rel=&quot;noreferrer&quot;&gt;mental images&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some background.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A.Einstein answered to a questionnaire issued by the French mathematician  &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacques_Hadamard&quot; rel=&quot;noreferrer&quot;&gt;Jacques Hadamard&lt;/a&gt; regarding &quot;THE WORKING METHODS OF MATHEMATICIANS&quot;, originally issued in &lt;em&gt;L'Enseignement Mathematique&lt;/em&gt; (1902-1904).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein letter is reproduced into : Jacques Hadamard, &lt;a href=&quot;http://worrydream.com/refs/Hadamard%20-%20The%20psychology%20of%20invention%20in%20the%20mathematical%20field.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Psychology of Invention in the Mathematical Field&lt;/a&gt;, original edition: 1949, page 142.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote answer to Hadamard's question n°.30 [see footnote 1, page 142] :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[page 140] 30. It would be very helpful for the purpose of psychological &#xA;  investigation to know what internal or &lt;em&gt;mental images&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;], what kind of &quot;internal word&quot; mathematicians make use of; whether they are motor, auditory, visual, or mixed, depending on the subject which they are studying. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-07-08T10:33:32.197" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7503" PostTypeId="1" CreationDate="2018-07-08T12:12:38.123" Score="3" ViewCount="336" Body="&lt;p&gt;First of all, i appologize if i'm asking too many questions about Gauss (and some will also say &quot;not very interesting questions&quot;); i know i might appear ridiculus - but i really think it's an important goal to map the entire Gaussian contributions to math and science. Now, to my questions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm very interested to learn about Gauss's unpublished fragments on electromagnetism; especially i'm interested in Gauss's &quot;Zur Electrodynamik&quot; (Gauss's werke, volume 5, p. 601 - 630). &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Electric circuits&lt;/strong&gt; - The first titles in this manuscript record, to my opinion, his discovery of Kirchoff's laws for branched electrical circuits. But in addition, it seems that he studies some interesting configurations for electrical circuits (see p.601-604), which i didn't find comments about in the literature. So if anyone knows something about these schematic figures, please explain. By the way, it seems also that title 2 in this manuscript records his discovery of the &quot;principle of minimum heat&quot; (established by Kirchoff in 1848; it's related to minimum entropy principles), which Dunnington mentions in p.161 of his biography of Gauss. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;electromagnetic telegraph&lt;/strong&gt; - was Gauss-Weber's telegraph just a combination of already known principles (binary code,...) and components (electrical wires,commutator)? or did they make essential improvements over these components?&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Sympathetic vibrations&lt;/strong&gt; - Dunnington tells in his biography: &quot;In 1834, he constructed an induction coil, and was enable to recognize the damping of a magnet vibrating in a coil as a result of induced currents; this led him to the construction of a copper damper for his magnetic apparatus, and to the discovery of &quot;sympathetic vibrations&quot;&quot;. I didn't find any comment about &quot;Gauss's discovery of sympathetic vibrations&quot; in the whole literature, and i'm curious to learn more of this.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Mathematical descriptions for induction and electrodynamics&lt;/strong&gt; - there is much more material in &quot;Zur electrodynamics&quot; according to Dunnington: &quot;Grassmann's law&quot;, &quot;Newmann's law of potential&quot;. If anybody knows something about these, i'll be glad to hear. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Thanks sincerely for all the efforts.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-21T18:47:12.597" LastActivityDate="2019-10-21T18:47:12.597" Title="Several questions about Gauss's contributions to electromagnetism" Tags="|biographical-details|electromagnetism|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7504" PostTypeId="1" CreationDate="2018-07-08T16:17:56.127" Score="0" ViewCount="185" Body="&lt;p&gt;In the preface to &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0471504475&quot; rel=&quot;nofollow noreferrer&quot;&gt;Methods of Mathematical Physics&lt;/a&gt; Richard Courant, the author, wrote that the book was suppressed by the National Socialist rulers(Nazi) of Germany. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence, my question. Thanks. &lt;/p&gt;&#xA;" OwnerDisplayName="user7599" LastActivityDate="2018-07-08T16:44:47.467" Title="Why was Courant's &quot;Methods of Mathematical Physics&quot; suppressed, by the Germans, during WW2?" Tags="|mathematics|physics|mathematical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7505" PostTypeId="2" ParentId="7504" CreationDate="2018-07-08T16:44:47.467" Score="6" Body="&lt;p&gt;I think because &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Courant&quot; rel=&quot;noreferrer&quot;&gt;Richard Courant&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;left Germany in 1933, earlier than many Jewish escapees. He did not lose his position due to being Jewish, as his previous service as a front-line soldier exempted him; however, his public membership in the social-democratic left was reason enough (for the Nazis) for dismissal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See the dates of the editions of &lt;a href=&quot;https://en.wikipedia.org/wiki/Methoden_der_mathematischen_Physik&quot; rel=&quot;noreferrer&quot;&gt;Methoden der mathematischen Physik&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-07-08T16:44:47.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7506" PostTypeId="1" AcceptedAnswerId="7507" CreationDate="2018-07-09T06:18:08.583" Score="1" ViewCount="257" Body="&lt;p&gt;I counldn't find precise info on who/ when successfully observed relativistic mass increase. The only thing I found by a Google search is that it was first confirmed in 1908 in experiments with electrons in cathode tube.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I know that in 1905 Einstein already produced is formula E=mc^2, and wiki says that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Mass–energy equivalence arose originally from special relativity as a&#xA;  paradox described by Henri Poincaré.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Can you give a detailed chronology of the development of the notion of relativistic mass and its relation to mass-energy equivalence?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="609" LastEditDate="2018-07-15T14:37:26.593" LastActivityDate="2018-07-15T14:37:26.593" Title="When was relativistic mass first observed?" Tags="|energy|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7507" PostTypeId="2" ParentId="7506" CreationDate="2018-07-09T08:01:04.707" Score="3" Body="&lt;p&gt;In theory, relativistic mass was preceded by the &quot;electromagnetic mass&quot; introduced by J.J. Thomson in 1881, and further developed by Heaviside (1888), Searle (1897), Poincaré (1900), Abraham (1902), Lorentz (1904). &lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Electromagnetic_mass&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Electromagnetic_mass&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, the first experiments precise enough to measure such effects were conducted by Kaufmann in 1901-1905, followed by experiments by Bucherer in 1908 and others, even though it took some time until their meaning was clarified. &lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kaufmann%E2%80%93Bucherer%E2%80%93Neumann_experiments&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Kaufmann%E2%80%93Bucherer%E2%80%93Neumann_experiments&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the advent of special relativity by Einstein in 1905, &quot;electromagnetic mass&quot; was superseded by the concept of &quot;relativistic mass&quot; first formulated by Lewis and Tolman in 1909. However, modern authors try to avoid the concept of relativistic mass, instead they use the concepts of &quot;relativistic momentum&quot; and &quot;relativistic energy&quot; in order to describe the experiments in particle accelerators, while &quot;mass&quot; is now simply denoted as &quot;rest mass&quot; or &quot;invariant mass&quot;.&lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_mass_concept&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_mass_concept&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding the evolution of the &lt;strong&gt;mass energy equivalence&lt;/strong&gt;: In terms of electromagnetism some elements of that concept were known to Poincaré (1900) and Hasenöhrl (1904), while the complete equivalence in terms of all forms of energy was first pointed out by Einstein in 1905.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Electromagnetic_mass#Inertia_of_energy_and_radiation_paradoxes&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Electromagnetic_mass#Inertia_of_energy_and_radiation_paradoxes&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Electromagnetic_mass#Modern_view&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Electromagnetic_mass#Modern_view&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7601" LastEditorUserId="7601" LastEditDate="2018-07-09T08:12:21.407" LastActivityDate="2018-07-09T08:12:21.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7508" PostTypeId="2" ParentId="6511" CreationDate="2018-07-09T14:18:13.307" Score="1" Body="&lt;p&gt;The Schröder-Bernstein Theorem has a remarkable history. Repeatedly stated (and claimed as proven) between 1882 [G. Cantor, letter to R. Dedekind (5 Nov 1882)] and 1895 [G. Cantor, collected works, p. 285] but never really proved by Cantor, this theorem is called after Ernst Schröder and Felix Bernstein, because both proved it. Alwin Korselt however discovered a flaw in Schröder's proof in 1902. Alas the Mathematische Annalen did not publish the correction before 1911. [A. Korselt: &quot;Über einen Beweis des Äquivalenzsatzes&quot;, Math. Ann. 70 (1911) 294] Nevertheless it took some time until this correction received public attention. Ernst Zermelo noted in his edition of Cantor's collected works as late as in 1932: &quot;The theorem, here mentioned without proof [...] has been proved only in 1896 by E. Schröder and 1897 by F. Bernstein. Since then this 'equivalence-theorem' is considered of the highest importance in set theory.&quot; [G. Cantor, collected works, p. 209] We learn from this that wrong proofs can survive in mathematics over many decades.&lt;/p&gt;&#xA;" OwnerUserId="12699" LastActivityDate="2018-07-09T14:18:13.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7509" PostTypeId="2" ParentId="7490" CreationDate="2018-07-09T23:19:13.373" Score="1" Body="&lt;p&gt;Heney L. Beose's translation (last sentence of &lt;a href=&quot;https://books.google.com/books?id=KCgZAQAAIAAJ&amp;amp;pg=PA54#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;ch. 1, § 6, p. 54&lt;/a&gt;) says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An emphatic protest must be entered against these orgies of formalism which are threatening the peace of even the technical scientist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-07-09T23:19:13.373" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7510" PostTypeId="1" AcceptedAnswerId="7511" CreationDate="2018-07-10T23:43:11.587" Score="1" ViewCount="212" Body="&lt;p&gt;Obviously like they say hindsight is 20/20, but it seems to me that all the ingredients for Noether's theorem were in place more than a hundred years before its publication, and to be honest it is not a very hard proof. Let me elaborate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrangian mechanics was formulated in the late 18th century. Just from the Euler-Lagrange equations&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ \frac{d}{dt} \frac{\partial L}{\partial \dot{q_i}} - \frac{\partial L}{\partial q_i} = 0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;one can trivially see a basic version of the theorem, which is that if the Lagrangian $L$ is independent of one coordiante $q_i$, the corresponding momentum $p_i = \partial L / \partial \dot{q_i}$ is constant. The more general version is a one-line proof: If the Lagrangian is invariant under a variation $\delta q_i$ of the coordinates and the equations of motion are satisfied, then&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\delta L = \frac{\partial L}{\partial q_i} \delta q_i + \frac{\partial L}{\partial \dot{q_i}} \frac{d}{dt} \delta q_i = \underbrace{ \left(\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q_i}}\right) }_{=0} \delta q_i + \frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q_i}} \delta q_i \right) = 0,$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so that $Q = (\partial L / \partial \dot{q_i}) \delta q_i$ is constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that this proof uses only theory which was already developed by 1800, and that it is a generalization of a very simple observation, why did it take another hundred years for the theorem to appear?&lt;/p&gt;&#xA;" OwnerUserId="7607" LastActivityDate="2018-07-11T10:42:21.817" Title="Why did Noether's theorem take so long to show up?" Tags="|theoretical-physics|mathematical-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7511" PostTypeId="2" ParentId="7510" CreationDate="2018-07-11T10:42:21.817" Score="6" Body="&lt;p&gt;I suggest you have a look at the book on &lt;a href=&quot;https://www.springer.com/la/book/9780387878676&quot; rel=&quot;noreferrer&quot;&gt;Noether Theorems&lt;/a&gt; by Yvette Kosmann-Schwarzbach where this history is described in much detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There it's clarified how the results you alluded at, were known already to Jacobi (and in the easiest case to Lagrange), so at the beginning of XIXth century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As Dan mentioned, of course, to formalize such statements in the way Noether did the machinery of groups and group actions was needed. Let me mention the fact that also much work on algebraic invariants (like invariants of quadratic forms) developed between the end of XIXth and the beginning of the XXth were fundamental in inspiring Noether's work.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2018-07-11T10:42:21.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7513" PostTypeId="1" CreationDate="2018-07-12T02:13:05.923" Score="5" ViewCount="616" Body="&lt;p&gt;In &lt;a href=&quot;https://www.elsevier.com/books/foundations-of-set-theory/fraenkel/978-0-7204-2270-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Foundations of Set Theory&lt;/em&gt; by Fraenkel, Bar-Hillel, and Levy (1973)&lt;/a&gt;, the authors argue that there have been three distinct periods of crisis in the foundations of mathematics. The first was undergone by the Ancient Greeks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] two discoveries were made that were extremely paradoxical: the first was that not all geometrical entities of the same kind were commensurable with each other, so that, for instance, the diagonal of a given square could not be measured by an aliquot part of its side (in modern terms, that the square root of 2 is not a rational number); the other were the paradoxes of the Eleatic school (Zenon and his circle) developing with many variations the theme of the non-constructibility of finite magnitudes out of infinitely small parts. (pp. 13)&lt;/p&gt;&#xA;&lt;p&gt;This crisis shocked the Greek mathematicians into obtaining two more brilliant achievements: the theory of proportions, as contained in books 5 and 10 of Euclid’s Elements, and the method of exhaustion, as invented by Archimedes, that was nothing less than a strict, though not sufficiently general, forerunner of modern theories of integration. (pp. 13)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second involved the foundations of analysis:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] in the 17th and 18th centuries, the great power and fruitfulness of the newly invented calculus led most mathematicians of those times into feverish applications of the new ideas without caring much for the solidity of the basis upon which the calculus was founded. However, the shakiness of this basis became clear at the beginning of the 19th century, constituting the second crisis in the foundations of mathematics. (pp. 13)&lt;/p&gt;&#xA;&lt;p&gt;In order to overcome this crisis, Cauchy, in the eighteen thirties, showed how to replace the irresponsible use of infinitesimals by a careful use of limits, whereas Weierstrass and others, in the sixties and seventies, demonstrated how all of analysis and function theory could be “arithmetized”. (pp. 13-14)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And the third of course was sparked by investigation into set theory and the discovery of the antinomies:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;More than the mere appearance of antinomies in the basis of set theory, and thereby of analysis, it is the fact that the various attempts to overcome these antinomies, to be dealt with in the subsequent chapters, revealed a far-going and surprising divergence of opinions and conceptions on the most fundamental mathematical notions, such as set and number themselves, which induces us to speak of the third foundational crisis that mathematics is still undergoing. (pp. 14)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The three crises were motivated by either explicit paradoxes or severe philosophical skepticism about the subject matter and/or validity of the mathematical tools and concepts which were under scrutiny. They also were started by people who are part of Western civilizations (mainly European) and can be seen as part of Western mathematical history (at least in origin, the third crises surely has had global contributions made to it).&lt;/p&gt;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&lt;p&gt;Are there any examples in non-Western historical cultures, such as classical India or China for example, which had foundational crises in their work on mathematics? Were there any paradoxes or was there severe philosophical skepticism which motivated and lead directly to new tools or conceptions of mathematics?&lt;/p&gt;&#xA;" OwnerUserId="5664" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2023-09-28T12:16:07.857" Title="Foundational crises in non-Western historical mathematical communities" Tags="|mathematics|discoveries|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7514" PostTypeId="2" ParentId="7513" CreationDate="2018-07-12T14:06:57.873" Score="2" Body="&lt;p&gt;Here is a sort of answer.  It is merely a memory I have, so no reference.  And I cannot say whether it is even true.  Maybe someone knows a reference.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In ancient Egypt, if you go back far enough, you find that the taxes for a plot of farmland were calculated based on the perimeter of the plot.  Over many years, experience showed that different plots (say, a square and a long thin rectangle) with the same perimeter would yield different amounts of grain.  So they had to invent a new notion of &quot;area&quot; to use in computing the taxes.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-07-12T14:06:57.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7515" PostTypeId="1" CreationDate="2018-07-12T13:56:40.847" Score="7" ViewCount="468" Body="&lt;p&gt;This question is from one historical perspective. The question is: how physicists historically found out that one needs quantum fields to describe matter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Being more detailed. Let us consider the electromagnetic field for a while. Classically this was already a field. Now, if I understood the history correctly, in the days of old quantum theory, when Planck proposed the solution to the blackbody radiation problem in terms of quantized energy levels, and when Einstein did the same to solve the photoelectric effect problem, they were essentialy proposing that light (and hence the electromagnetic waves) could be described in terms of photons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since light classicaly was a field it seems to be expected that when this field was properly quantized we would get these particles somehow as proposed by Planck/Einstein. I believe this was done by Heisenberg as soon as he proposed his matrix mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I recall, he applied his methods to the electromagnetic field and found a collection of harmonic oscilators, which would be the particles (photons).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it turns out that today we use fields to describe all matter. Some physicist even say that fields are more fundamental than particles altogether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But other matter (like electrons) classicaly isn't a field like the photon. And in truth, by the historical development of quantum mechanics for the other particles one would expect wave functions instead of fields.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Actualy, it seems Dirac himself proposed his equation as a &quot;wave function equation&quot; rather than a field equation, only later this approach being taken.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: while the photon is classicaly a field and the historical development pointed towards a particle upon quantization, the quantum field point of view seems quite natural. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But for other matter (like electrons, and all other fundamental particles), how physicists found out, historically, that the field viewpoint was needed? What led physicists to realize one needed to describe all matter with fields, and not just the electromagnetic field?&lt;/p&gt;&#xA;" OwnerUserId="4072" OwnerDisplayName="user1620696" LastActivityDate="2018-08-10T03:40:33.990" Title="Historically how it was discovered that we need fields to describe matter?" Tags="|quantum-mechanics|electromagnetism|quantum-field-theory|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7516" PostTypeId="2" ParentId="7515" CreationDate="2018-07-12T16:56:44.730" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_field_theory#Early_development&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia coverage of the history&lt;/a&gt; is pretty spot on, and there is hardly a point in exactly dating the incremental &lt;em&gt;formal&lt;/em&gt; developments of the second quantization picture of Dirac, Jordan, Wigner, Pauli, and Heisenberg, etc. QED has served since as a prototype of arbitrary creation and annihilation of matter and antimatter, made possible by relativity, that I suspect you are already aware of.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here, however, I would wish to emphasize  &lt;a href=&quot;https://doi.org/10.1007/BF01351864&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermi's 1933&lt;/a&gt;  crucial utilization of QFT to do &lt;strong&gt;&lt;em&gt;&quot;real physics&quot;&lt;/em&gt;&lt;/strong&gt; (Wigner);  particle physics as we understand it even today: he established that the creation, annihilation and transmutation of particles in the weak interaction beta decay could best be described in QFT, specifically through his eponymous quartic fermion interaction. &lt;a href=&quot;http://physics.princeton.edu/~mcdonald/examples/EP/yang_ijmp_a27_1230005_12.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;This C N Yang paper is essential reading on the subject&lt;/a&gt;. Note &lt;a href=&quot;https://doi.org/10.1142/S0217751X12300050&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A neutron disappears, a proton, electron and neutrino appear, in a calculable framework.The essence of QFT is collective accounting of &lt;em&gt;indefinite numbers&lt;/em&gt; of creation and destruction of particles of diverse species, namely second quantization in Fock space: rampant emergence and disappearance of particles/excitations subject to the conservation principles of the particular QFT, here conservation of baryon and lepton numbers, charge, etc. With the above reference, Fermi really opened the conceptual floodgates.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2018-07-16T14:20:13.050" LastActivityDate="2018-07-16T14:20:13.050" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7517" PostTypeId="1" AcceptedAnswerId="7518" CreationDate="2018-07-12T22:22:41.073" Score="6" ViewCount="442" Body="&lt;p&gt;The &quot;&lt;strong&gt;Law of the Unconscious Statistician&lt;/strong&gt;&quot; states that, for a random variable $X$ with density $f_X(x)$ and a function of it $h(X)$ we have that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$E[h(X)] = \int_{-\infty}^{\infty} f_X(x)h(x) dx$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In words, we do not need to obtain the distribution and density of $Z=h(X)$ in order to calculate its expected value.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How this strange name for this law came about? (it does sound a little dismissive).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician&quot; rel=&quot;noreferrer&quot;&gt;wikipedia page&lt;/a&gt; does not discuss the issue. The Quora forum has a &lt;a href=&quot;https://www.quora.com/How-did-the-Law-of-the-Unconscious-Statistician-get-its-name&quot; rel=&quot;noreferrer&quot;&gt;related thread&lt;/a&gt;, but no real historical answer regarding the name is given.&lt;/p&gt;&#xA;" OwnerUserId="244" LastActivityDate="2018-07-12T23:47:02.470" Title="Law of the Unconscious Statistician - history of the term?" Tags="|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7518" PostTypeId="2" ParentId="7517" CreationDate="2018-07-12T23:47:02.470" Score="7" Body="&lt;p&gt;This was &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=6786678&quot; rel=&quot;noreferrer&quot;&gt;discussed on Math Forum&lt;/a&gt;. According to Elliot, Halmos called it Fundamental Theorem of the Unconscious Statistician as early as 1946, and according to Bernier, Introduction to the Techniques of Operations Research by Hillier (1965) calls it the &quot;Law of the Unconscious Statistician&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But apparently the nickname only took off after Ross's &quot;Introduction to Probability Models&quot; (1980). Proposition 4.1 of Chapter 2 has a footnote that reads:&quot;&lt;em&gt;This law got its name from &quot;unconscious&quot; statisticians who ave used it as if it were the definition of $E[g(X)]$&lt;/em&gt;&quot;. It did not go over too well, for example Berger and Casella write in Statistical Inference:&quot;&lt;em&gt;Ross 1988 refers to this as the 'law of the unconscious statistician'. We do not find this amusing&lt;/em&gt;&quot;. Apparently, they were not alone, Ross dropped the nickname from subsequent editions. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-07-12T23:47:02.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7519" PostTypeId="2" ParentId="7513" CreationDate="2018-07-13T15:58:26.753" Score="2" Body="&lt;p&gt;I'd suggest that there are evidence-based reasons why the present question, though interesting, may be regarded as very difficult or impossible to answer, at least as it stands. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As &lt;a href=&quot;https://hsm.stackexchange.com/users/131/mauro-allegranza&quot;&gt;Mauro Allegranza's&lt;/a&gt; comment already pointed out, the question presupposes a search for foundations, clearly a precondition before there can be such a thing as a foundational crisis.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In turn, one may question what the conditions are for a search for foundations. When one looks at historical evidence, one sees that at various periods mathematical knowledge may earn acceptance in (at least) three ways: on the basis of authority (because it was taught by a respected teacher); as a praxis (because it works); or on the basis of justification by an intellectually respectable theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The evidence is of several kinds.  Acceptance on the basis of authority is with little doubt a phenomenon known worldwide. In Europe it is known at least from the example of acceptance during many centuries of the works of Aristotle and Ptolemy -- and the many mistaken things that they taught (along with other matters of persisting value or validity).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historical acceptance by practical justification may sometimes be impossible to identify or exclude because of a shortage of evidence. The ancient evidence is often scanty, preferentially showing what was done or used, but not why: it sometimes lacks direct evidence (or even any evidence at all) about reasons for the acceptance and use of the praxis, and whether it was associated with any theoretical justification -- or with any felt need for any such thing. Examples can be seen in parts of the early history of trigonometry, e.g. in &lt;a href=&quot;https://archive.org/details/historyofmathema031897mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;D E Smith's History of Mathematics, vol.2&lt;/a&gt; especially chapter 8. This example shows how motivation for acceptance of a mathematical praxis can for lack of evidence remain a doubtful and conjectural matter, much less secure than the historical evidence for the praxis itself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;When there is acceptance in either of these two ways, there is arguably little room for a search for theoretical foundations or for a foundational crisis. (The western example of the long-accepted authority of Aristotle and Ptolemy also arguably casts doubt on the validity of the present framing of the question in terms of possible disjunction between western and non-western traditions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the question perhaps might be usefully be reframed to include a preliminary enquiry for the preconditions of a foundational crisis.  It may also be asked whether the example of 'three distinct periods of crisis' that underlies the question is heavily overdrawn, even magnifying by rhetoric alone the difference between 'crisis' and non-crisis.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the 19th-century work of Cauchy and then the arithmetists 'overcame' the calculus 'crisis' by clarifying contemporary scruples in a directly applicable way. It is arguable that this 'crisis' had involved a wider canvas anyway than problems of the new calculus and its infinitesimals or limits. There were contributory 17th-century scruples about the mathematical legitimacy of other innovative mathematics that had in common that they were ultimately of a character involving transcendentals and/or the infinite, e.g. the so-called 'tentative' solutions of Kepler's problem to evaluate in effect angle from area, and the impossibility of a direct numerical evaluation without something equivalent to infinite series solution or its approximation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to the first period, it seems difficult to claim that the problem or 'crisis' of the Pythagorean discovery of irrational numbers was in any analogous way 'overcome' by the theory of proportion or by the method of exhaustion.  It arguably raised a particular problem involving the infinite that took a very long time to overcome or answer.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 'periods of crisis' identified in the book cited in the question are arguably similar only by stretched analogy. If the analogy is allowed to be stretched like that, then the different crises also begin to look more like particular instances of a common and deeper and longer-lasting problem about how mathematically to handle aspects of the infinite. The 'periods' of 'crisis' may then be accountable more simply as periods when the problems gained more conspicuous attention, while at other times attentions were turned elsewhere.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What might be (respectfully) suggested here is a reframing to help towards an answer to an at least related question.  It might first be asked whether,  when and where there were historical periods and places where the first two mentioned forms of mathematical acceptance were missing, that is, acceptance as praxis or from a teaching authority.  For those periods and places, one may perhaps then hope with greater likelihood to find signs of deeper mathematical introspection about foundations -- if indeed, as historians may hope, the relevant evidence was ever recorded and has been preserved. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-07-13T15:58:26.753" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7520" PostTypeId="1" CreationDate="2018-07-13T23:35:55.287" Score="1" ViewCount="56" Body="&lt;p&gt;I'm interested in when and how Isaac Newton's mechanics first came to be applied practically in engine design. Did for example &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Newcomen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Newcomen&lt;/a&gt; take advantage on Newtons discoveries when he invented the so called atmospheric engine in 1712?&lt;/p&gt;&#xA;" OwnerUserId="3663" LastEditorUserId="3663" LastEditDate="2018-07-14T06:06:36.993" LastActivityDate="2018-07-14T06:06:36.993" Title="What were the first practical applications of Newton's mechanics on engine design?" Tags="|newton|steam-engines|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7521" PostTypeId="2" ParentId="7513" CreationDate="2018-07-14T00:30:33.627" Score="8" Body="&lt;p&gt;Let me point out that the Fraenkel-Bar-Hillel &quot;march to rigorization&quot; quote reflects the early 20-th century &quot;rational reconstructions&quot; of history that were worked out by mathematician-historians like Klein, van der Warden and Dieudonné and got enshrined in historical sections of many old and still current textbooks, partly due to their didactic convenience. Nonetheless, they lift patterns and concerns from recent history (formalization, axiomatization, foundations, etc.) and project (impose) them onto prior times. As a result, many of their surmises are largely discredited by contemporary scholarship, including the one about &quot;foundational crises&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On early Greek mathematics a good source is &lt;a href=&quot;https://www.maa.org/press/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction&quot; rel=&quot;noreferrer&quot;&gt;Fowler's Mathematics of Plato's Academy&lt;/a&gt;. Concerning the traditional story about the &quot;discovery of irrationals&quot; and the subsequent &quot;crisis&quot;, the reviewer writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It may be somewhat surprising, then, to discover that many, if not most, historians of ancient mathematics disbelieve some or all of this story. In The Mathematics of Plato's Academy, David Fowler gives a convincing account of the reasons for rejecting the standard story, and offers a very interesting alternative reconstruction of the history of early Greek mathematics... One of the important themes in Fowler's book is that one must carefully assess the available sources on early Greek mathematics. He demonstrates that most of the elements of the &quot;standard story&quot; either come from very late Greek sources (500 years or more after the time of Plato) or are reconstructions by modern historians who are really reflecting nineteenth and twentieth century concerns about foundational issues... The resulting picture is quite different from the traditional one. In particular, incommensurability is seen not as a foundational crisis but rather as an interesting discovery that led to significant mathematics.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There was no &quot;shock&quot; or &quot;crisis&quot;, and Eudoxus was not the savior. Much work on incommensurables was done by Pythagoreans like Archytas, Theodorus and Theaetetus before him, some of it survives in books II and X of Elements. Their work was based on anthyphairesis, a geometric version of what is now called Euclidean algorithm producing what is now called continued fractions, the same process that presumably led to the discovery of incommensurables. There is no evidence of the supposed phase before the discovery, when Pythagoreans thought that all magnitudes were commensurable. Eudoxus's motivations, if we are to speculate about them, likely were extra-mathematical and not internal, they might have had more to do with Platonic push to eliminate motion and process from geometry &quot;corrupted&quot; by them. The same tendency is transparent throughout Elements, and we know that Eudoxus's ingenious mathematical astronomy of nested spheres was a response to Plato's call to &quot;save the phenomena&quot; from their sensible disorder too. Let me also mention in passing that another background assumption of textbook histories, &quot;Euclid the axiomatician&quot;, has also been largely discredited, see references in &lt;a href=&quot;https://hsm.stackexchange.com/a/7104/55&quot;&gt;What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;feverish applications&quot; without &quot;solidity of the basis&quot; story about calculus conveniently omits its kinematic (late Newtonian) conception, within which Cauchy worked, and most of algebraic analysis of 18th century, that had much more &quot;solidity&quot;, to manufacture another &quot;crisis&quot; and leap straight to Weierstrass and Cantor. The boost at the beginning of 19-th century had more to do with institutional changes and Fourier inspired broadening of the repertory of functions than with &quot;the basis&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the motivation behind the question is to compare Western mathematics to other cultures to see if it displays unique historical patterns, and this is a valid topic. Unfortunately, the proposed &quot;pattern&quot; in this case is largely fabricated even for the West, so there is no point looking for it elsewhere. An additional difficulty is that our sources for Egyptian, Babylonian, early Chinese and Indian mathematics are too sporadic to form a reliable picture of their development. It would take some work to figure out what a similarly motivated but answerable question might be, perhaps one should look for apparent &quot;big turns&quot;. Of course, it will depend on what counts as &quot;big&quot; or &quot;turn&quot;, but some promising places to look might be medieval Islamic mathematics and the &quot;Chinese Renaissance&quot;. One could perhaps make a case that some traits of &quot;Western&quot; (including Middle East) mathematics were unique, see related &lt;a href=&quot;https://hsm.stackexchange.com/a/2344/55&quot;&gt;Is the Scientific Method uniquely Western?&lt;/a&gt;, but we should be mindful that our &quot;sample&quot; for detecting historical patterns just might be too thin to generate anything but artifacts.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-07-14T00:42:38.423" LastActivityDate="2018-07-14T00:42:38.423" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7522" PostTypeId="1" CreationDate="2018-07-14T12:20:39.223" Score="2" ViewCount="100" Body="&lt;p&gt;When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of $O_K$. We have the famous relation $n = \sum_{i=1}^r e_i f_i$, where $f_i = [O_K / P_i : \Bbb F_p]$.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where do these notations $e, f, r$ come from?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Remarks :&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The letter $r$ is also denoted for the number $r_1$ (resp $r_2$) of real (resp. pairs of complex) embeddings of $K$. It gives the number $n=r_1 + 2r_2$ (with $2=e(w/v)$ for a complex place $w$, and $f(w/v):=1$) of infinite primes above the unique infinite place of $\Bbb Q$. But why would the number of primes above a given (finite/infinite) prime be denoted by $r$? Maybe the &quot;&lt;strong&gt;r&lt;/strong&gt;ank&quot; of some fiber?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It looks like the ramification (Verzweigung) index $e$ denotes &quot;exponent&quot;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The inertia index (inertia degree, Trägheitsgrad) is denoted by the next letter after $e$. Or is it because of &quot;residual &lt;strong&gt;f&lt;/strong&gt;ield&quot; ?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="3454" LastActivityDate="2018-07-14T12:20:39.223" Title="Notation $n=efr$ in algebraic number theory" Tags="|notation|number-theory|abstract-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7523" PostTypeId="2" ParentId="5141" CreationDate="2018-07-14T17:06:07.303" Score="0" Body="&lt;p&gt;Number 22 is at the observatory the Paris. &#xA;Saw it there on 12-07-2018.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Erik &lt;/p&gt;&#xA;" OwnerUserId="7625" LastActivityDate="2018-07-14T17:06:07.303" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7524" PostTypeId="1" CreationDate="2018-07-15T02:21:52.073" Score="5" ViewCount="290" Body="&lt;p&gt;I'm new to this stack community, please bear with me as I try to explain my question properly. Recently I came across with these trigonometric identities (where $ \omega + \phi + \psi = 180^\circ $):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\tan \omega + \tan \phi + \tan \psi = \tan \omega \tan \phi \tan \psi$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\sin (2\psi ) + \sin (2 \phi) + \sin (2 \omega) = 4 \sin \omega \sin \phi \sin \psi$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and a user &lt;a href=&quot;https://math.stackexchange.com/a/1429470&quot;&gt;who answered the first one&lt;/a&gt; mentioned that such identity belonged to the 18th century scientist Antonio Cagnoli. I tried to look for sources regarding this but I only could find the &lt;a href=&quot;https://en.wikipedia.org/wiki/Antonio_Cagnoli&quot; rel=&quot;nofollow noreferrer&quot;&gt;wiki entry&lt;/a&gt; and &lt;a href=&quot;http://worldcat.org/identities/lccn-n86870249/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this other entry&lt;/a&gt; in World cat. The OP of the answer had referenced his proof by &lt;a href=&quot;https://play.google.com/books/reader?id=Lj51QYK1fIcC&amp;amp;printsec=frontcover&amp;amp;output=reader&amp;amp;hl=fr&amp;amp;pg=GBS.PA25&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt; and it is consistent with what it appears on World cat under:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Trigonométrie rectiligne et sphérique by Antonio Cagnoli( Book )&#xA;18 editions published between 1808 and 2017 in French &#xA;and Undetermined and held by 117 WorldCat member libraries worldwide&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Although it does mention in the book about the identity with the tangent function it doesn't specifically say about the one with the double angle sine function, which is the one that has triggered my attention the most.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore my question arises that other than this source (the book mentioned) are any others out there which can reassure or tell if these two belong to Cagnoli?.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Initially I wanted to post this question in the Mathstack site but I felt this place would be more proper. I'm looking forward for your answers.&lt;/p&gt;&#xA;" OwnerUserId="7627" LastEditorUserId="5788" LastEditDate="2018-07-15T14:37:29.410" LastActivityDate="2018-10-26T04:46:38.287" Title="Do these trigonometric identities belong to Antonio Cagnoli?" Tags="|mathematicians|biographical-details|reference-request|elementary-algebra|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7525" PostTypeId="2" ParentId="2633" CreationDate="2018-07-15T10:15:23.263" Score="5" Body="&lt;p&gt;The &lt;em&gt;New York Times&lt;/em&gt; actually did carry your putative headline, on &lt;a href=&quot;https://www.nytimes.com/1928/09/12/archives/science-reported-near-secret-of-life-british-savants-told-oxygen-is.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;September 12, 1928&lt;/a&gt; (p. 1):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\hspace{8em}$&lt;img src=&quot;https://i.stack.imgur.com/Xb3Ei.png&quot; width=&quot;310&quot; height=&quot;300&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Earlier, Heisenberg had been featured on &lt;a href=&quot;https://www.nytimes.com/1927/09/02/archives/details-concepts-of-quantum-theory-heisenberg-of-germany-gives.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;September 2, 1927&lt;/a&gt; and Born on &lt;a href=&quot;https://www.nytimes.com/1926/03/19/archives/his-dynamic-theory-rivals-einsteins-prof-born-expounds-principles.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;March 19, 1926&lt;/a&gt;. If we allow for “old quantum theory”, then Pais (&lt;a href=&quot;https://books.google.com/books?id=b-VBDgAAQBAJ&amp;amp;ots=G10ceIgpkS&amp;amp;pg=PA253&amp;amp;dq=Silliman&quot; rel=&quot;nofollow noreferrer&quot;&gt;1991, p. 253&lt;/a&gt;) says that Bohr’s 1923 tour mentioned by Nicola Ciccoli gave rise to the first extensive coverage (his emphasis):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Far more&#xA;  important (...) are the lengthy and detailed reports of Bohr’s talks,&#xA;  which, every single day following one of his lectures, appeared in the &lt;em&gt;New&#xA;  York Times&lt;/em&gt; [November &lt;a href=&quot;https://www.nytimes.com/1923/11/07/archives/dr-bohr-expounds-theory-of-atoms-begins-a-series-of-lectures-at.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;7&lt;/a&gt;, &lt;a href=&quot;https://www.nytimes.com/1923/11/08/archives/elements-of-atom-described-by-bohr-it-comprises-electric-units-the.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;8&lt;/a&gt;, &lt;a href=&quot;https://www.nytimes.com/1923/11/09/archives/dr-bohr-discusses-bombarding-atoms-tells-his-yale-audience-how.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;9&lt;/a&gt;, &lt;a href=&quot;https://www.nytimes.com/1923/11/14/archives/discusses-atom-from-new-point-dr-bohr-in-yale-lecture-takes-up-the.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;14&lt;/a&gt;, &lt;a href=&quot;https://www.nytimes.com/1923/11/15/archives/shows-vast-spaces-within-the-atom-prof-bohr-pictures-electron-and.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;15&lt;/a&gt;, &lt;a href=&quot;https://www.nytimes.com/1923/11/16/archives/bohr-demonstrates-action-of-the-atom-shows-how-electrons-become.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;16&lt;/a&gt;]. These newspaper articles are of a far higher quality than most&#xA;  of those which today pass for science reporting. They played a central role in bringing twentieth century conceptions about the structure of matter &lt;strong&gt;&lt;em&gt;for&#xA;  the first time to the attention of a wide audience&lt;/em&gt;&lt;/strong&gt;. On &lt;a href=&quot;https://www.nytimes.com/1924/01/06/archives/pictures-the-atom-as-a-small-solar-system-dr-bohr-advances-evidence.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;6&lt;/a&gt; and &lt;a href=&quot;https://www.nytimes.com/1924/01/20/archives/science-promises-an-amazing-future-leaders-of-research-in-twelve.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;20&lt;/a&gt; January 1924 the&#xA;  Sunday editions of the &lt;em&gt;New York Times&lt;/em&gt; carried long articles on Bohr and&#xA;  the atom in their special features section.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Even that wasn’t the first appearance of quanta or wave-particle duality: as far as I can tell from the &lt;em&gt;Times&lt;/em&gt; archive, that distinction goes to similar conferences of Sommerfeld on &lt;a href=&quot;https://www.nytimes.com/1922/09/18/archives/einstein-absorbed-by-quanta-theory-partly-atones-for-upsetting.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;September 18, 1922&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$\hspace{2em}$&lt;img src=&quot;https://i.stack.imgur.com/J6D5r.png&quot; width=&quot;285&quot; height=&quot;536&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/128uJ.png&quot; width=&quot;272&quot; height=&quot;200&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and of Einstein on &lt;a href=&quot;https://www.nytimes.com/1921/04/22/archives/einstein-explains-fallacy-of-ether-space-without-electromagnetic.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;April 22, 1921&lt;/a&gt; (detailing his 1917 proof of Planck’s radiation law). Planck’s Nobel had been announced in &lt;a href=&quot;https://www.nytimes.com/1919/11/15/archives/nobel-prizes-to-germans-three-awards-for-science-go-to-university.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1919&lt;/a&gt; and mistakenly &lt;a href=&quot;https://www.nytimes.com/1908/11/23/archives/nobel-prize-award-marconi-not-in-an-unofficial-list-given-out.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;, but with no mention of quanta.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I’d say media awareness started rather earlier than previous answers suggest, and likely drove the reaction in cultural circles.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-08-04T11:52:04.270" LastActivityDate="2018-08-04T11:52:04.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7526" PostTypeId="1" AcceptedAnswerId="7527" CreationDate="2018-07-17T03:50:07.670" Score="4" ViewCount="398" Body="&lt;p&gt;This question is about &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Benacerraf&quot; rel=&quot;noreferrer&quot;&gt;Paul Benacerraf&lt;/a&gt;, who worked on the philosophy of mathematics, and wrote the 1965 essay &lt;em&gt;What numbers could not be&lt;/em&gt; (see: &lt;a href=&quot;https://en.wikipedia.org/wiki/Benacerraf%27s_identification_problem&quot; rel=&quot;noreferrer&quot;&gt;Benacerraf's identification problem&lt;/a&gt;). He was at Princeton in the 1960s up to the end of the '00s, and presumably appears in at least one of the following two photos from 1968 and 1979 respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hyCzh.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hyCzh.jpg&quot; alt=&quot;Princeton Philosophy department 1968&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://philosophy.princeton.edu/about/sixties-snapshot&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PlGdZ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PlGdZ.jpg&quot; alt=&quot;Princeton Philosophy department 1979&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://philosophy.princeton.edu/about/seventies-snapshot&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, can we identify Benacceraf in either of these photos? My purpose is to use a picture of him roughly contemporaneous with the essay in a seminar for undergraduate students.&lt;/p&gt;&#xA;" OwnerUserId="2531" LastActivityDate="2018-07-18T18:25:57.377" Title="Can we identify Paul Benacerraf in these photos" Tags="|mathematicians|philosophy-of-science|set-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7527" PostTypeId="2" ParentId="7526" CreationDate="2018-07-17T17:37:10.493" Score="5" Body="&lt;p&gt;The site has &lt;a href=&quot;https://philosophy.princeton.edu/gallery/1344&quot; rel=&quot;noreferrer&quot;&gt;legends&lt;/a&gt; that identify him: 4th of back row in 1968, 3rd of front row in 1979.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-07-17T17:37:10.493" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7528" PostTypeId="2" ParentId="7499" CreationDate="2018-07-17T18:08:48.140" Score="3" Body="&lt;p&gt;(Moving my comment here, to take this off the unanswered list:) Wikipedia links a pdf &lt;em&gt;The life of Gotthold Ferdinand Eisenstein&lt;/em&gt; which translates, from his &lt;em&gt;Mathematische Werke&lt;/em&gt; (1975), a &lt;a href=&quot;https://books.google.com/books?id=bXacGQjSYKwC&amp;amp;pg=PA884&amp;amp;dq=Jahres&quot; rel=&quot;nofollow noreferrer&quot;&gt;1843 CV&lt;/a&gt; where he wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I first saw the light of day on April sixteenth, in the year 1823 in Berlin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-07-17T18:08:48.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7529" PostTypeId="2" ParentId="2679" CreationDate="2018-07-18T14:30:44.020" Score="1" Body="&lt;p&gt;There is a very rich (and turbulent) history in trying to understand mass loss due to winds of red giants and supergiants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this article answers your question quite thoroughly.&#xA;&lt;a href=&quot;https://arxiv.org/abs/astro-ph/0507598&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/astro-ph/0507598&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reimer's relation itself is of course physically motivated, as explained here in section 2.1 &lt;a href=&quot;https://arxiv.org/abs/astro-ph/0512326&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/astro-ph/0512326&lt;/a&gt; (it used to be thought that bulk motion through the chromosphere could spill over and cause mass loss). But as the first article explains, there are several deficiencies in Reimer's and subsequent formulations, from which better formulations have been developed since. &lt;/p&gt;&#xA;" OwnerUserId="7638" LastActivityDate="2018-07-18T14:30:44.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7530" PostTypeId="2" ParentId="6695" CreationDate="2018-07-18T15:40:53.317" Score="1" Body="&lt;p&gt;This question seems simple, but the history of mathematics is a turbulent, tangled, mess. As Mauro said, it's hard to find sources that predate Euclid that provide &quot;formal&quot; definitions like we expect to have in modern times. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a physicist, so my impulse answer is that, &quot;humans have always known what a number is as long as they have known that they had more than one finger.&quot; The fundamentals of mathematics are steeped in our experience as pattern seeking creatures. As a child, I always viewed numbers as a physical attribute of a physical object(s), rather than viewing them as abstract objects themselves. I think this is the prototype for humanity - only recently have we begun to view number in a &quot;formal&quot; manner. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that said, it depends on what you mean by &quot;number.&quot; This is tricky, you see, since the concept of &quot;formality&quot; in mathematics is debatable (&lt;a href=&quot;https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, in order to answer this, you have to be willing to be lenient about how you view formalism. A tallying system has no concept of place value, which limits its representation of large numbers, but a tallying system is still immensely useful and technically defines numbers in a relational manner. (Tallying systems are considered the first kind of abstract numbering system.) Prehistoric humans used tallying systems for thousands of years (&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1559210419&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Roots-Civilization-Cognitive-Beginnings-Notation/dp/1559210419&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can find, the first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt. These systems are well before Euclid's time. In terms of &quot;formalism,&quot; for one example, the ancient Egyptians had a formalism for rational numbers (&lt;a href=&quot;https://en.wikipedia.org/wiki/Egyptian_fraction#Early_history&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Egyptian_fraction#Early_history&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without generalizing too much, in the history of mathematics, geometric definitions usually precede algebraic (abstract) definitions. For example, Caspar Wessel and J.R. Argand provided geometric definitions of the complex plane before (I think) Cauchy provided the first algebraic (abstract) definition using functional analysis. (&lt;a href=&quot;http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, regardless of whether you want a geometrical or algebraic definition, Euclid was not the first, since his work was predated by other geometric definitions, and was not algebraically abstract. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(edit) I must also add that Bertrand Russell argued that Frege provided the first truely abstract definition of the concept of number, (&lt;a href=&quot;https://people.umass.edu/klement/imp/imp.html#chapter2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://people.umass.edu/klement/imp/imp.html#chapter2&lt;/a&gt;) see chapter 2. As always, Russell presents a compelling argument. &lt;/p&gt;&#xA;" OwnerUserId="7638" LastActivityDate="2018-07-18T15:40:53.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7533" PostTypeId="1" CreationDate="2018-07-19T02:54:02.687" Score="0" ViewCount="267" Body="&lt;p&gt;An answer on &lt;a href=&quot;https://stats.stackexchange.com/questions/66767/power-of-a-hypothesis-test/66918#66918&quot;&gt;this question on the cross validated stack exchange&lt;/a&gt; compared statistical power to a microscope, such that &quot;in order to see small things you need a powerful microscope&quot; is analogous to &quot;in order to detect small effects you need high statistical power.&quot; I have heard this before from a colleague and I have also used it elsewhere as a pedagogical device to much benefit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a primary source for this metaphor? Who came up with it first? Or is it just a meme whose creator is now a meme (pending answers to this question)?&lt;/p&gt;&#xA;" OwnerUserId="7643" LastActivityDate="2018-07-19T02:54:02.687" Title="Statistical Power as a Microscope Metaphor" Tags="|statistics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7534" PostTypeId="2" ParentId="7524" CreationDate="2018-07-19T05:09:44.800" Score="4" Body="&lt;p&gt;Your first identity is indeed attributed to Cagnoli by e.g. Franchini (&lt;a href=&quot;https://books.google.com/books?id=m-fmAAAAMAAJ&amp;amp;pg=PA158&quot; rel=&quot;nofollow noreferrer&quot;&gt;1805&lt;/a&gt;), &lt;em&gt;Encyclopédie du dix-neuvième siècle&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=g4J9_7WX114C&amp;amp;pg=PA377&quot; rel=&quot;nofollow noreferrer&quot;&gt;1847&lt;/a&gt;), or Le Cointe (&lt;a href=&quot;https://archive.org/stream/leonssurlathori00coingoog#page/n79&quot; rel=&quot;nofollow noreferrer&quot;&gt;1858&lt;/a&gt;, p. 59). It is in the second edition of his &lt;em&gt;Trigonometria&lt;/em&gt; (&lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/15902655&quot; rel=&quot;nofollow noreferrer&quot;&gt;1804&lt;/a&gt;, Chap. IV, nº 173) but apparently not in the first  (&lt;a href=&quot;https://archive.org/details/bub_gb_odSSYA3fokwC&quot; rel=&quot;nofollow noreferrer&quot;&gt;1786&lt;/a&gt;) where Chap. IV ends at nº 128. Woodhouse (&lt;a href=&quot;https://archive.org/stream/atreatiseonplan01woodgoog#page/n58&quot; rel=&quot;nofollow noreferrer&quot;&gt;1819&lt;/a&gt;) writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\tan.(A+B+C)=\frac{t+t'+t''-tt't''}{1-(tt'+tt''+t't'')}.$$&lt;/span&gt;&#xA;  If &lt;span class=&quot;math-container&quot;&gt;$A+B+C=\pi$&lt;/span&gt;, (which is the case when &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, are the three angles of a triangle), since &lt;span class=&quot;math-container&quot;&gt;$\tan.\pi=0$&lt;/span&gt;,&#xA;  &lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;t+t'+t''&amp;amp;-tt't''=0, \text{ or}\\&#xA;t+t'+t''&amp;amp;=tt't'',&#xA;\end{align}&lt;/span&gt;&#xA;  which is the theorem given in the &lt;a href=&quot;https://doi.org/10.1098/rstl.1808.0010&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Phil. Trans.&lt;/em&gt; 1808, p. 122&lt;/a&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;But the theorem has an origin much more remote; for, the above formula for &lt;span class=&quot;math-container&quot;&gt;$\tan.(A+B+C)$&lt;/span&gt; and similar formulas for the tangents of &lt;span class=&quot;math-container&quot;&gt;$A+B+C+D$&lt;/span&gt;, &amp;amp;c. were given as far back as the year 1722, by John Bernoulli, and are inserted in the &lt;em&gt;Leipsic Acts&lt;/em&gt; for that year, p. 361, and in the second volume of his works, at p. 526 [see &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/1052686&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lemma II&lt;/a&gt;].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Your second identity appears among many similar ones in e.g. Crelle (&lt;a href=&quot;https://archive.org/stream/lehrbuchderelem00crelgoog#page/n434&quot; rel=&quot;nofollow noreferrer&quot;&gt;1826&lt;/a&gt;), Grunert (&lt;a href=&quot;https://archive.org/details/bub_gb_mnZXAAAAcAAJ/page/n635&quot; rel=&quot;nofollow noreferrer&quot;&gt;1836&lt;/a&gt;), Serret (&lt;a href=&quot;https://archive.org/stream/traitdetrigonom02serrgoog#page/n82&quot; rel=&quot;nofollow noreferrer&quot;&gt;1850&lt;/a&gt;; &lt;a href=&quot;https://books.google.com/books?id=7eVlBvMYwYcC&amp;amp;pg=PA45&quot; rel=&quot;nofollow noreferrer&quot;&gt;1862&lt;/a&gt;), Colenso (&lt;a href=&quot;https://archive.org/stream/planetrigonomet01colegoog#page/n80&quot; rel=&quot;nofollow noreferrer&quot;&gt;1851&lt;/a&gt;), Müller (&lt;a href=&quot;https://books.google.com/books?id=U-DXBTolGrUC&amp;amp;pg=RA1-PA92&quot; rel=&quot;nofollow noreferrer&quot;&gt;1852&lt;/a&gt;), Le Cointe (&lt;em&gt;loc. cit.&lt;/em&gt;, p. 58), Christensen (&lt;a href=&quot;https://books.google.com/books?id=b_g_AQAAIAAJ&amp;amp;pg=RA2-PA79&quot; rel=&quot;nofollow noreferrer&quot;&gt;1884&lt;/a&gt;), &lt;em&gt;Mathesis&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=iaAKAAAAYAAJ&amp;amp;pg=PA24&quot; rel=&quot;nofollow noreferrer&quot;&gt;1885&lt;/a&gt;), Gelin (&lt;a href=&quot;https://books.google.com/books?id=1Qg5iGbPfLsC&amp;amp;pg=PA89&quot; rel=&quot;nofollow noreferrer&quot;&gt;1888&lt;/a&gt;, pp. 89, 104-106), etc. None of them attribute it to Cagnoli.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-26T04:46:38.287" LastActivityDate="2018-10-26T04:46:38.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7536" PostTypeId="2" ParentId="5614" CreationDate="2018-07-19T23:10:23.780" Score="2" Body="&lt;p&gt;As the saying goes - yesterday's magic is tomorrow's science. But really, humans progress this way on long timescales relative to a human's lifespan. Myth and superstition generally precede rational understanding, but sometimes the transition is messy. This seems to be a consequence of how humans learn by trial and error of their environment to seek patterns - the better we are at finding and modeling patterns the more we can learn (&lt;a href=&quot;https://onlinelibrary.wiley.com/doi/pdf/10.1348/000712606X101808&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://onlinelibrary.wiley.com/doi/pdf/10.1348/000712606X101808&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is not unfair to classify numerology as a superstition. Sometimes the transition of beliefs from pseudoscience/superstitious to scientific/rational is clear, i.e. alchemy mostly died out by the late 1800's (never mind a few occultist revivalist movements) right when modern chemistry started taking off along with statistical physics (&lt;a href=&quot;https://en.wikipedia.org/wiki/Alchemy#Late_modern_period&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Alchemy#Late_modern_period&lt;/a&gt;). And hardly anyone knows about alchemy these days.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of an unclear transition could be from astrology to astronomy. &#xA;Astrology was made up thousands of years ago and astronomy is relatively modern, and even though astrology is immensely outdated and debunked, still its concepts are more widespread through our population than are the objective facts of astronomy (look in a farmer's almanac and it has more about astrology than astronomy).&#xA;(&lt;a href=&quot;https://www.washingtontimes.com/news/2003/aug/17/20030817-105449-9384r/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.washingtontimes.com/news/2003/aug/17/20030817-105449-9384r/&lt;/a&gt;) (&lt;a href=&quot;https://io9.gizmodo.com/more-than-1-3rd-of-americans-think-that-astrology-is-a-1520598014&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://io9.gizmodo.com/more-than-1-3rd-of-americans-think-that-astrology-is-a-1520598014&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this is the case for the numerology to number theory transition, since the origins of numerology and of number theory emerged at around the same time - twas an unclear transition.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My feeling is that arithmetic is very old indeed, and numerology cannot be said to precede arithmetic because it requires arithmetic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think this is instead due to the fact that numerology and arithmetic were developed nearly simultaneously. For instance, as far as I understand these things, the Arabic Abjad system did not require the use of arithmetic. Chinese numerology also does not require the use of arithmetic, since it is more linguistic. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_numerology&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chinese_numerology&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;AS far as we can tell (what records we have), the origin of number theory precede the origin of numerology (assuming it began with Pythagoras), so numerology did not precede number theory. Who knows, maybe a long lost document will be found that shows otherwise.&lt;/p&gt;&#xA;" OwnerUserId="7638" LastActivityDate="2018-07-19T23:10:23.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7538" PostTypeId="1" CreationDate="2018-07-20T00:42:18.727" Score="3" ViewCount="142" Body="&lt;p&gt;I found the following interesting function and its analysis at &lt;a href=&quot;https://en.wikipedia.org/wiki/Non-analytic_smooth_function#A_smooth_function_which_is_nowhere_real_analytic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Non-analytic smooth function&lt;/a&gt; article in Wikipedia. I include a screen capture below for those who don't wish to navigate away:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LQ8wI.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LQ8wI.jpg&quot; alt=&quot;This is from Wikipedia.&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could not find a clear reference in the article as to the source of this example. Hence my question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; who found the example above ?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I found the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fabius_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fabius function article&lt;/a&gt; gives another example of a smooth but nowhere analytic function after reading &lt;a href=&quot;https://math.stackexchange.com/q/1782736/36530&quot;&gt;this great MSE question&lt;/a&gt; but I need a reference for the nameless example I post above. Thanks in advance for your insight on this matter.&lt;/p&gt;&#xA;" OwnerUserId="327" LastActivityDate="2018-07-20T13:05:53.810" Title="Origin of the smooth but nowhere real analytic function built with dyadic rationals" Tags="|reference-request|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7539" PostTypeId="1" CreationDate="2018-07-20T07:53:24.783" Score="2" ViewCount="266" Body="&lt;p&gt;Query 31 in Newton's opticks contains the following words: &quot;&lt;strong&gt;But by reason of the Tenacity of Fluids, and Attrition of their Parts, and the Weakness of Elasticity in Solids, Motion is much more apt to be lost than got, and is always upon the Decay&lt;/strong&gt;&quot;.  One can cautiously interpret this sentence as containing all the elements for a formulation of the second law of thermodynamics. Because for Newton, &quot;motion&quot; is momentum, or organized motion, and not random motion like heat. In other words, Newton states the universe tends to randomality and disorder.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, by &quot;the Weakness of Elasticity in Solids&quot; Newton refers to inelastic collisions, and by &quot;But by reason of the Tenacity of Fluids, and Attrition of their Parts&quot; he reffers to viscocity in fluids.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This interpretation was taken from &lt;a href=&quot;http://www.mathpages.com/rr/s8-02/8-02.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.mathpages.com/rr/s8-02/8-02.htm&lt;/a&gt;.&#xA;Now to my questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;This far-reaching interpreation of the sentence fron Newton's opticks is based on very slight evidence (One can give a simpler interpretation of it). So, was the notion of &quot;loss of motion&quot; a reoccuring topic in Newton's writings, or is this sentence the only reference to such issues?&lt;/li&gt;&#xA;&lt;li&gt;Did Newton's statement have any influence on the first researchers of heat and thermodynamics?   &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2018-07-20T07:53:24.783" Title="Query 31 in Newton's opticks and the second law of thermodynamics" Tags="|biographical-details|thermodynamics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7540" PostTypeId="2" ParentId="7538" CreationDate="2018-07-20T12:45:51.160" Score="3" Body="&lt;p&gt;Mandelbrojt (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=6354&quot; rel=&quot;nofollow noreferrer&quot;&gt;1942&lt;/a&gt;, &lt;a href=&quot;https://core.ac.uk/display/4465385&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 2&lt;/a&gt;) has $\displaystyle f(x)=\sum\limits_{n=1}^\infty e^{-\sqrt{n!}}\cos(n!x)$, with a very similar proof.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-07-20T13:05:53.810" LastActivityDate="2018-07-20T13:05:53.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7541" PostTypeId="1" CreationDate="2018-07-20T13:50:53.977" Score="0" ViewCount="46" Body="&lt;p&gt;I'm interesting of how scientific research covers a specific topic in a time. Scientific field: biology, cellulosic ethanol production&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;topic A&#xA;1940 - 2 publications (maybe with details like: 1 patent, one publication with title &quot;X&quot; in magazin &quot;Y&quot;)&#xA;1941 - 5 publications&#xA;...&#xA;2018 - n publications&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are the sources for such research?&lt;/p&gt;&#xA;" OwnerUserId="7652" LastActivityDate="2018-07-20T13:50:53.977" Title="Coverage of a topic by scientific research through time" Tags="|biology|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7542" PostTypeId="2" ParentId="5579" CreationDate="2018-07-20T15:52:04.333" Score="2" Body="&lt;p&gt;In 1862, the physicist  Kelvin published calculations that fixed the age of Earth at between 20 million and 400 million years.[18][19]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He assumed that Earth had formed as a completely molten object, and determined &lt;em&gt;the amount of time it would take for the near-surface to cool to its present temperature.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This assumed initial condition was the linchpin for Kelvin's entire method .&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bits of material at the surface would sink before solidifying, creating convection currents that kept the Earth at a uniform temperature until solidification began at the core.&#xA;   (Hallam, 110; Knopf, 445).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kelvin needed to know:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) the temperature at Earth's core,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) the temperature gradient with regard to depth below the surface, and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(3) the thermal conductivity of rocks. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The gradient was established to be around one degree Fahrenheit for every fifty feet. Kelvin made his own measurements of conductivity. The problem was determining the temperature at the core. This is where Kelvin's theory of solidification enters the picture. Because the core was thought to be solid rock, its temperature could not exceed the melting point of rocks (Hallam, 110). He constructed the following equation:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;dθ / dx = S / h √(π t)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The temperature gradient is expressed by &lt;strong&gt;dθ / dx&lt;/strong&gt;; &lt;strong&gt;h&lt;/strong&gt; is the thermal conductivity; &lt;strong&gt;x&lt;/strong&gt; is the distance below the surface, and &lt;strong&gt;θ&lt;/strong&gt; is the temperature (Holmes, 445).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1862, Kelvin arrived at a likely age of 100 million years.&#xA; Because of uncertainties in the data, the lower and upper limits were 20 million and 400 million years (Dalyrmple, 26; Hallam, 111). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(According to modern biology, the total evolutionary history from the beginning of life to today has taken place since  3.5 billion  years.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a lecture in 1869, Darwin's great advocate, Thomas H. Huxley, attacked Thomson's calculations, suggesting they appeared precise in themselves but were based on faulty assumptions.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Other scientists backed up Thomson's figures. Charles Darwin's son, the astronomer George H. Darwin, proposed that Earth and Moon had broken apart in their early days when they were both molten. He calculated the amount of time it would have taken for tidal friction to give Earth its current 24-hour day. His value of 56 million years added additional evidence that Thomson was on the right track.[23]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The last estimate Thomson(Lord Kelvin) gave, in 1897, was: &quot;that it was more than 20 and less than 40 million year old, and probably much nearer 20 than 40&quot;.[25] In 1899 and 1900, John Joly calculated the rate at which the oceans should have accumulated salt from erosion processes, and determined that the oceans were about 80 to 100 million years old.[23]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;however modern estimates are-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first geologic time scale that included absolute dates was published in 1913 by the British geologist Arthur Holmes.[15] He greatly furthered the newly created discipline of geochronology and published the world renowned book The Age of the Earth in which he estimated Earth's age to be at least 1.6 billion years.[16] &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;References-&lt;a href=&quot;https://en.wikipedia.org/wiki/Age_of_the_Earth#Early_calculations&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Age_of_the_Earth#Early_calculations&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&lt;a href=&quot;http://apps.usd.edu/esci/creation/age/content/failed_scientific_clocks/kelvin_cooling.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://apps.usd.edu/esci/creation/age/content/failed_scientific_clocks/kelvin_cooling.html&lt;/a&gt;&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastEditorUserId="7635" LastEditDate="2018-07-20T16:15:08.810" LastActivityDate="2018-07-20T16:15:08.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7543" PostTypeId="1" CreationDate="2018-07-22T10:50:17.313" Score="4" ViewCount="131" Body="&lt;p&gt;It is a common belief that we use a base 10 representation of integers because we have 10 fingers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does there exist historical evidence which supports the claim that this is true and that the number of fingers indeed was the major reason that we use 10 over similar candidates like 8 or 12?&lt;/p&gt;&#xA;" OwnerUserId="5906" LastActivityDate="2018-07-22T10:50:17.313" Title="Historical evidence for claim that we use base 10 because of the number of fingers" Tags="|reference-request|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7544" PostTypeId="1" AcceptedAnswerId="7547" CreationDate="2018-07-22T20:35:39.910" Score="6" ViewCount="1071" Body="&lt;p&gt;In older algebra textbooks for high school (mainly 19th century) four properties of ratio and proportions were used. The properties were Invertendo, Alternendo, Componendo, and Dividendo.  This terminology has vanished in most textbooks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could never locate these words in any &lt;em&gt;unabridged&lt;/em&gt; English dictionaries (Oxford's multivolume dictionary, as well as Webster's). Most likely these are Latin words. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know the literal meanings and English equivalents of these four words? Intuitively, they make sense but what is the word form (verb, adjective, noun?).&#xA;&lt;a href=&quot;https://brilliant.org/wiki/componendo-and-dividendo/&quot; rel=&quot;noreferrer&quot;&gt;https://brilliant.org/wiki/componendo-and-dividendo/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-07-24T04:12:38.173" Title="Etymology of Some Terms Used in Ratio and Proportion in Old Algebra Textbooks" Tags="|mathematics|terminology|elementary-algebra|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7545" PostTypeId="1" AcceptedAnswerId="7546" CreationDate="2018-07-23T03:53:16.537" Score="3" ViewCount="775" Body="&lt;p&gt;I was actually curious how the Solar Coronal event of 1859 known as the Carrington Event affected the telegraph system.  Wiki does not seem to have much detail of this but there is mention of it in a number of historical documents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was the telegraph powered by large batteries and I am curious if the effect of the storm was one of temporary disruption or was there physical damage to the electrical connections or other components that would have caused them to be replaced or repaired ?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-07-23T13:30:35.307" Title="Was the telegraph system of 1859 powered by AC or DC and how extensive was it?" Tags="|electromagnetism|electricity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7546" PostTypeId="2" ParentId="7545" CreationDate="2018-07-23T13:30:35.307" Score="7" Body="&lt;p&gt;Google really is your friend.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.history.com/news/a-perfect-solar-superstorm-the-1859-carrington-event&quot; rel=&quot;noreferrer&quot;&gt;history.com&lt;/a&gt; says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;E.W. Culgan, a telegraph manager in Pittsburgh, reported that the&#xA;  resulting currents flowing through the wires were so powerful that&#xA;  platinum contacts were in danger of melting and “streams of fire” were&#xA;  pouring forth from the circuits. In Washington, D.C., telegraph&#xA;  operator Frederick W. Royce was severely shocked as his forehead&#xA;  grazed a ground wire. According to a witness, an arc of fire jumped&#xA;  from Royce’s head to the telegraphic equipment. Some telegraph&#xA;  stations that used chemicals to mark sheets reported that powerful&#xA;  surges caused telegraph paper to combust.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;On the morning of September 2, the magnetic mayhem resulting from the&#xA;  second storm created even more chaos for telegraph operators. When&#xA;  American Telegraph Company employees arrived at their Boston office at&#xA;  8 a.m., they discovered it was impossible to transmit or receive&#xA;  dispatches. The atmosphere was so charged, however, that operators&#xA;  made an incredible discovery: They could unplug their batteries and&#xA;  still transmit messages to Portland, Maine, at 30- to 90-second&#xA;  intervals using only the auroral current.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wired.com/2011/09/0902magnetic-storm-disrupts-telegraphy-carrington-event/&quot; rel=&quot;noreferrer&quot;&gt;Wired&lt;/a&gt; wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On Sept. 2, 1859, at the telegraph office at No. 31 State Street in&#xA;  Boston at 9:30 a.m., the operators' lines were overflowing with&#xA;  current, so they unplugged the batteries connected to their machines,&#xA;  and kept working using just the electricity coursing through the air.When a geomagnetic storm hits the Earth, it shakes the Earth's magnetosphere.[...] As the magnetized plasma pushes the Earth's magnetic field lines around, currents flow. Those currents have their own magnetic fields and soon, down at the ground, strong electromagnetic forces are in play. In other words, your telegraph can run on &quot;auroral current.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-07-23T13:30:35.307" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7547" PostTypeId="2" ParentId="7544" CreationDate="2018-07-23T18:28:30.883" Score="8" Body="&lt;p&gt;These terms derive from the four Latin verbs &lt;em&gt;invertere&lt;/em&gt; &quot;to turn upside down&quot;, &lt;em&gt;alternare&lt;/em&gt; &quot;to alternate, to arrange in alternating order&quot;, &lt;em&gt;componere&lt;/em&gt; &quot;to add together&quot;, and &lt;em&gt;dividere&lt;/em&gt; &quot;to divide up, to separate into parts&quot;. The specific grammatical form is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gerund&quot; rel=&quot;noreferrer&quot;&gt;gerund&lt;/a&gt;, indicated by &lt;em&gt;-nd-&lt;/em&gt;, and the grammatical case used is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ablative_case&quot; rel=&quot;noreferrer&quot;&gt;ablative&lt;/a&gt;, indicated by the suffix &lt;em&gt;-o&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore their literal meaning is: &lt;em&gt;invertendo&lt;/em&gt; &quot;by turning upside down&quot; or &quot;by inverting&quot;, &lt;em&gt;alternando&lt;/em&gt; &quot;by alternating&quot; or &quot;by arranging in alternate order&quot;, &lt;em&gt;componendo&lt;/em&gt; &quot;by adding together&quot;, and &lt;em&gt;dividendo&lt;/em&gt; &quot;by separating&quot;. Except for the final term, they clearly describe the underlying mathematical relationship:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;invertendo&lt;/em&gt;: if $a : b = c : d$, then (by inverting) $b : a = d : c$&lt;br&gt;&#xA;&lt;em&gt;alternando&lt;/em&gt;: if $a : b = c : d$, then (by arranging in alternate order) $ a  :c=b:d$&lt;br&gt;&#xA;&lt;em&gt;componendo&lt;/em&gt;: if $a : b = c : d$, then (by adding together) $(a+b) : b = (c+d):d$&lt;br&gt;&#xA;&lt;em&gt;dividendo&lt;/em&gt;: if $a : b = c : d$, then (by separating) $(a - b ) :b = (c-d):d$&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2018-07-24T04:12:38.173" LastActivityDate="2018-07-24T04:12:38.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7550" PostTypeId="2" ParentId="5066" CreationDate="2018-07-27T11:30:40.160" Score="0" Body="&lt;p&gt;Turns out the answer seems to be the Roman doctor-philosopher Galen, who wrote extensively on food, diet, and the humors. I found an excellent translation of his work on this specific topic on Amazon here: Galen on Food and Diet &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0415232333&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/dp/0415232333/ref=cm_sw_r_cp_api_RhWwBbF900F84&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4537" LastActivityDate="2018-07-27T11:30:40.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7551" PostTypeId="2" ParentId="7478" CreationDate="2018-07-27T20:13:50.150" Score="7" Body="&lt;p&gt;I confirm that it is definitely not Bruno Pontecorvo on the first picture (and on the Wikipedia English article at the moment). I knew Bruno Pontecorvo well for a long time beginning in early 60's, I have seen most of his earlier pictures, it is definitely not him. I noticed that earlier with absolute certainty, when I saw this Wikipedia article. Also pictures from archive, from which this picture is coming, are definitely not showing Bruno Pontecorvo, contrary to what is stated there.   &lt;/p&gt;&#xA;" OwnerUserId="7680" LastActivityDate="2018-07-27T20:13:50.150" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7552" PostTypeId="1" CreationDate="2018-07-28T00:05:47.027" Score="4" ViewCount="9692" Body="&lt;p&gt;Aristotle is often credited with proving the sphericity of Earth from the fact that the shadow of the Earth on the moon during lunar eclipses is always an arc of a round circle (as opposed to arcs of an ellipse, which would be the case e. g. if the Earth were a disc). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what Aristotle actually  wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;As it is, the shapes which the &#xA;  moon itself each month shows are of every kind straight, &#xA;  gibbous, and concave but in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will be caused by the form of the earth's surface, which is therefore spherical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A common interpretation of this (e. g., one in the quote by Asimov featuring in &lt;a href=&quot;https://hsm.stackexchange.com/q/7479/7610&quot;&gt;this question&lt;/a&gt;) is that by &quot;curved&quot;, he meant &quot;part of circular arc&quot;, and then implied that a body whose all shadows are circles must be a sphere, as proven in &lt;a href=&quot;https://mathoverflow.net/q/39127/56624&quot;&gt;this MO entry&lt;/a&gt;. However, there are two concerns with this interpretation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;curved&quot; is a vague word that does not necessarily mean &quot;arc of a circle&quot;;&lt;/li&gt;&#xA;&lt;li&gt;during lunar eclipses, one only sees a small arc of the Earth (at most about 1/12 of the full circle). Is it really possible to tell apart a circular arc from an elliptic one with a naked eye given such a short sample?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Is it believed that Aristotle had sufficiently precise data to conclude that the arcs were round circular, and implied a geometric argument as above? Could he? Or should we take the word &quot;curved&quot; at its face value and rather view the whole argument as one of philosophical flavour rather then mathematical one?&lt;/p&gt;&#xA;" OwnerUserId="7610" LastEditorUserId="7610" LastEditDate="2018-07-30T13:25:49.693" LastActivityDate="2018-09-07T13:02:10.450" Title="Sphericity of Earth from lunar eclipses - is Aristotle's argument valid?" Tags="|astronomy|ancient-greece|geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7553" PostTypeId="1" CreationDate="2018-07-28T06:03:29.887" Score="3" ViewCount="232" Body="&lt;p&gt;Is there any serious doubt of whether the first edition of Russell and Whitehead's &lt;em&gt;Principia Mathematica&lt;/em&gt; used the ramified theory of types?  I am travelling and cannot easily check sources but I do easily verify (in the original and in the more explicit secondary sources that I find) that PM edition one explains and (to all appearances) adopts the ramified theory and the Axiom of Reducibility.&lt;/p&gt;&#xA;&lt;p&gt;Yet Gödel says the simple theory of types &amp;quot;is the system of the first &lt;em&gt;Principia&lt;/em&gt; in an appropriate interpretation&amp;quot; (in his 1944 essay &amp;quot;Russell's mathematical logic&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;I wonder if Gödel was referring loosely to Russell's 1903 &lt;em&gt;Principles of Mathematics&lt;/em&gt; as &amp;quot;the first &lt;em&gt;Principia&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Or did Gödel have some way to read  the first edition of Russell and Whitehead's &lt;em&gt;Principia Mathematica&lt;/em&gt; as using simple type theory?&lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="2324" LastEditDate="2022-09-23T08:50:07.500" LastActivityDate="2022-09-23T08:50:07.500" Title="Checking a Gödel quote on Principia Mathematica" Tags="|mathematical-logic|quote|type-theory|goedel|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7554" PostTypeId="1" CreationDate="2018-07-28T08:52:30.263" Score="5" ViewCount="4266" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Srinivasa_Ramanujan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiki&lt;/a&gt; says 'After his death, his brother Tirunarayanan chronicled Ramanujan's remaining handwritten notes consisting of formulae on singular moduli, hypergeometric series and continued fractions and compiled them' (quoted from Indian express (I would trust not much outside The Hindu but still) and 'When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later'. On &lt;a href=&quot;https://www.thefamouspeople.com/profiles/srinivasa-ramanujan-503.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt; 'His mother gave birth to several children after Ramanujan, but none of them survived infancy' is written. As far as I know the local folklore is Ramanujan or his immediate family had no direct descendants. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Did Srinivasa Ramanujan have a surviving sibling?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="Freeman." LastEditorUserId="7679" LastEditDate="2018-09-03T13:51:18.047" LastActivityDate="2018-11-13T21:53:27.103" Title="Did Srinivasa Ramanujan have a surviving sibling?" Tags="|reference-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7555" PostTypeId="2" ParentId="7554" CreationDate="2018-07-28T11:15:26.637" Score="11" Body="&lt;p&gt;There were actually &lt;em&gt;two&lt;/em&gt; surviving brothers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;IMG SRC=&quot;https://ilorentz.org/beenakker/MO/Ramanujan.png&quot;/&gt;&#xA;&lt;A HREF=&quot;https://www.imsc.res.in/~rao/ramanujan/newnow/familytree.htm&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/A&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also page 12 of &lt;A HREF=&quot;https://archive.org/details/TheManWhoKnewInfinityALifeOfTheGeniusRamanujan&quot; rel=&quot;noreferrer&quot;&gt;Robert Kanigel's biography of Ramanujan.&lt;/A&gt;&lt;/p&gt;&#xA;" OwnerUserId="1697" OwnerDisplayName="Carlo Beenakker" LastActivityDate="2018-07-28T11:44:57.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7556" PostTypeId="1" AcceptedAnswerId="7558" CreationDate="2018-07-28T22:32:33.373" Score="5" ViewCount="342" Body="&lt;p&gt;The title is drawn from a sentence in a Jim Holt article, &#xA;&quot;The Dangerous Idea of the Infinitesimal,&quot;&#xA;now a chapter in his book collection.&lt;sup&gt;1&lt;/sup&gt;&#xA;I found this a striking claim, and perhaps true, as &#xA;the ancient Greeks did not&#xA;have the algebraic means to formulate &quot;scientific laws&quot;&#xA;as we know them today.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. To what extent is it in some sense true that &#xA;  the ancient Greeks did not discover a single scientific law?&#xA;  Are there counterexamples, cases where they did discover &quot;laws,&quot;&#xA;  even if not phrased in equations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The full sentence is: &quot;The Greeks might have amassed much particular&#xA;knowledge of nature, but their love of rigor held them back&#xA;from discovering a single scientific law.&quot; Of course it is unfair&#xA;to Holt to extract this from the surrounding context, but I&#xA;do think the &lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt; above can be discussed outside of Holt's context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;hr /&gt;&#xA;&lt;sup&gt;1&lt;/sup&gt;&#xA;Jim Holt.&#xA;&lt;em&gt;When Einstein Walked with Gödel: Excursions to the Edge of Thought&lt;/em&gt;.&#xA;Farrar, Straus and Giroux, 2018. Chp.13, pp.146-156. Quote p.153.&#xA;(&lt;a href=&quot;https://www.nytimes.com/2018/05/15/books/review/review-when-einstein-walked-with-godel-jim-holt.html&quot; rel=&quot;noreferrer&quot;&gt;NYTimes Review&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="606" LastActivityDate="2018-07-29T19:28:39.953" Title="The Greeks did not discover &quot;a single scientific law&quot;" Tags="|philosophy-of-science|ancient-greece|mathematical-physics|scientific-method|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7557" PostTypeId="1" CreationDate="2018-07-29T02:58:04.267" Score="0" ViewCount="39" Body="&lt;p&gt;My thinking about this topic is vague, and I'm looking to clarify it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not sure what &quot;intellectual progress&quot; is or if that's even a useful abstraction, but it seems like it should include things like...&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;New inventions  &lt;/li&gt;&#xA;&lt;li&gt;Formalizations of new mathematics (the invention/discovery of probability theory, for instance)&lt;/li&gt;&#xA;&lt;li&gt;Scientific breakthroughs&lt;/li&gt;&#xA;&lt;li&gt;The founding of new intellectual domains (originating economics, for instance)&lt;/li&gt;&#xA;&lt;li&gt;Improvements in methodological standards (the spread of the scientific method being the obvious example)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm looking to do a deep dive into rates of intellectual progress over time.&lt;/p&gt;&#xA;" OwnerUserId="7689" LastActivityDate="2018-07-29T02:58:04.267" Title="What are some good metrics for intellectual progress (of all sorts)?" Tags="|discoveries|philosophy-of-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7558" PostTypeId="2" ParentId="7556" CreationDate="2018-07-29T06:13:57.337" Score="12" Body="&lt;p&gt;It is a strange idea that scientific laws can be only expressed with algebraic means. The Greek did discover several scientific laws. The oldest one is attributed to Pythagoras himself: it relates the length of the string to its pitch. This seems to be the oldest scientific law ever discovered. More laws were discovered in Hellenistic times: the law of reflection, for example. Everyone knows the Archimedes laws about floating bodies and the laws of lever.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these are genuine scientific laws, and we still learn them in our schools and universities in the same form as they were discovered. It does not matter how you&#xA;express them: with equations or with words or with pictures.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-07-29T06:22:22.187" LastActivityDate="2018-07-29T06:22:22.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7559" PostTypeId="2" ParentId="7556" CreationDate="2018-07-29T13:19:18.053" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Snell%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snell's law&lt;/a&gt; was discovered in the 16th century or even earlier, whereas the algebraic notation is a post-Viete thing and a product of the 17th century.  Holt seems to be influenced by the ideology that &quot;mathematics is the language of the sciences&quot; which is an idea apparently first popularized by Galileo. While useful as a heuristic principle, this may not be exactly correct.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2018-07-29T13:19:18.053" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7560" PostTypeId="2" ParentId="7556" CreationDate="2018-07-29T19:11:43.607" Score="8" Body="&lt;p&gt;Euclid wrote an &lt;em&gt;Optica&lt;/em&gt; (&lt;a href=&quot;https://doi.org/10.1364/JOSA.35.000357&quot; rel=&quot;noreferrer&quot;&gt;300 BC&lt;/a&gt;) — surely “Visual rays proceed in a straight line indefinitely” ranks with the best physical laws. So did Ptolemy (&lt;a href=&quot;https://archive.org/details/lotticadiclaudi00eugegoog/&quot; rel=&quot;noreferrer&quot;&gt;160 AD&lt;/a&gt;), and Hero wrote  a &lt;em&gt;Catoptrica&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=fNc2AAAAMAAJ&amp;amp;pg=PA334&quot; rel=&quot;noreferrer&quot;&gt;50 AD&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle knew the &lt;a href=&quot;https://en.wikipedia.org/wiki/Virtual_work#History&quot; rel=&quot;noreferrer&quot;&gt;principle of virtual work&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jim Holt’s physics don’t seem to fare much better than his &lt;a href=&quot;https://mathoverflow.net/questions/306851/long-standing-conjectures-in-analysis-often-turn-out-to-be-false&quot;&gt;math&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-07-29T19:28:39.953" LastActivityDate="2018-07-29T19:28:39.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7561" PostTypeId="2" ParentId="7553" CreationDate="2018-07-30T00:00:47.587" Score="8" Body="&lt;p&gt;In responding to your question, I hope that I can convince you of the following two claims:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Gödel held that the formal system of the actual &lt;em&gt;Principia&lt;/em&gt; was that of ramified type thoery (simple type theory with orders built into the syntax).&lt;/li&gt;&#xA;&lt;li&gt;Gödel held that the formal system of &lt;em&gt;Principia&lt;/em&gt; without orders built into the syntax would have been motivated by a kind of realist position.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;But first, I want to address your leading question about the formal system of &lt;em&gt;Principia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there serious doubt of whether the first edition of Russell and Whitehead's Principia Mathematica used the ramified theory of types?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes. The appearance that &lt;em&gt;Principia&lt;/em&gt; has a ramified theory of types (in which orders are part of the syntax) is not accepted by all Russell scholars, and there have been serious challenges to that reading. The most detailed source for the interpretation on which &lt;em&gt;Principia&lt;/em&gt; embraces a simple theory of types (in which orders are only part of the nominalistic intended interpretation) is&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Gregory Landini (1998) &lt;em&gt;Russell's Hidden Substitutional Theory&lt;/em&gt;: &lt;a href=&quot;https://global.oup.com/academic/product/russells-hidden-substitutional-theory-9780195116830?cc=us&amp;amp;lang=en&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;Oxford University Press&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Landini is the leading proponent of the view that &lt;em&gt;Principia&lt;/em&gt; embraces a simple type theory. There is a nice centenary anthology in which the controversy over the formal system of &lt;em&gt;Principia&lt;/em&gt; is treated:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nicholas Griffin and Bernard Linsky (2013) &lt;em&gt;The Palgrave Centenary Companion to Principia Mathematica&lt;/em&gt;: &lt;a href=&quot;https://www.palgrave.com/us/book/9781137344625&quot; rel=&quot;noreferrer&quot;&gt;Palgrave Macmillan&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yet Gödel says the simple theory of types &amp;quot;is the system of the first Principia in an appropriate interpretation&amp;quot; (in his 1944 essay &amp;quot;Russell's mathematical logic&amp;quot;). I wonder if Gödel was referring loosely to Russell's 1903 Principles of Mathematics as &amp;quot;the first Principia.&amp;quot; Or did Gödel have some way to read the first edition of Russell and Whitehead's Principia Mathematica as using simple type theory?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think that the answer to this is decidedly no. Gödel is fairly consistent about calling the formal system of &lt;em&gt;Principia&lt;/em&gt; one on which there is a simple type theory with orders. For there is no textual evidence to suggest that Gödel was referring obliquely to the 1903 &lt;em&gt;Principles&lt;/em&gt; or to the formal system of the actual &lt;em&gt;Principia&lt;/em&gt; as Gödel understood it. In the remark that you cite, Gödel really is suggesting that on a &amp;quot;realist&amp;quot; interpretation of &lt;em&gt;Principia&lt;/em&gt;, the syntax would be just that of a simple type theory. In Gödel's telling, the &amp;quot;constructive&amp;quot; standpoint motivated by a Vicious Circle Principle is wholly unmotivated on a &amp;quot;realist&amp;quot; picture. This reading seems to be supported by Gödel's holding that (1) the theory of simple types and the theory of ramified types are logically independent and (2) the theory of simple types is (a) inconsistent with the &amp;quot;constructive&amp;quot; position and (b) is well-motivated by a &amp;quot;realist&amp;quot; stance with an additional assumption. I quote from the Schilpp volume printing of Gödel's 1944:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) I now come in somewhat more detail to the theory of simple types which appears in &lt;em&gt;Principia&lt;/em&gt; as combined with the theory of orders; the former is, however, (as remarked above) quite independent of the latter... (p. 147; see also pp. 134-135, especially note 17)&lt;/p&gt;&#xA;&lt;p&gt;(2)(a) But, it should be noted that the theory of simple types (in contradistinction to the vicious circle principle) cannot in a strict sense follow from the constructive standpoint... (p. 148)&lt;/p&gt;&#xA;&lt;p&gt;(2)(b) The theory of simple types (in its realistic interpretation) can be considered as a carrying through of this scheme, based, however, on the following additional assumption concerning meaningfulness: ... (p. 149)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So when Gödel calls simple type theory &amp;quot;the system of the first edition of &lt;em&gt;Principia&lt;/em&gt; in an appropriate interpretation&amp;quot; (p. 152), he is referring to &lt;em&gt;Principia&lt;/em&gt; &amp;quot;in its realistic interpretation&amp;quot; - which is not one Gödel himself endorses, nor the one that Gödel identifies with the actual formal system of &lt;em&gt;Principia&lt;/em&gt; but one on which the theory of simple types would be motivated by a kind of &amp;quot;realist&amp;quot; position.&lt;/p&gt;&#xA;&lt;p&gt;It bears mentioning that, Landini and company are right about the formal system of &lt;em&gt;Principia&lt;/em&gt; being just simple type theory (with orders built into the intended interpretation and not built into the syntax) if and only if Gödel is wrong about the formal system of &lt;em&gt;Principia&lt;/em&gt; being ramified type theory (simple type theory with orders built into the syntax). It is a fun debate in Russell scholarship!&lt;/p&gt;&#xA;" OwnerUserId="7691" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-07-30T00:00:47.587" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7562" PostTypeId="2" ParentId="7420" CreationDate="2018-07-30T04:00:42.943" Score="6" Body="&lt;p&gt;So I’ll quote &lt;a href=&quot;https://en.wikipedia.org/wiki/David_Pingree&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pingree&lt;/a&gt; (&lt;a href=&quot;https://www.jstor.org/stable/234257&quot; rel=&quot;nofollow noreferrer&quot;&gt;1992&lt;/a&gt;, p. 562) (to which both comment threads led me via &lt;a href=&quot;https://en.wikipedia.org/wiki/M._Ram_Murty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Murty&lt;/a&gt; (&lt;a href=&quot;https://hrj.episciences.org/volume/view/id/46&quot; rel=&quot;nofollow noreferrer&quot;&gt;2013&lt;/a&gt;)), as it struck me as elucidative in this ugly matter.&lt;/p&gt;&#xA;&lt;p&gt;Briefly, he says that those making the claim are belittling both Indian and Western math because “Hellenophilia” (~ idea that math cannot exist other than in Greek style) rendered them &lt;em&gt;&lt;strong&gt;unable to imagine&lt;/strong&gt;&lt;/em&gt; (p. 563) anything other than plagiarism:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;    (When) the Indian &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mādhava&lt;/a&gt;’s demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions (...) was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians’ discovery of the calculus. (This) resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series &lt;em&gt;&lt;strong&gt;without&lt;/strong&gt;&lt;/em&gt; the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava’s mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.&lt;/p&gt;&#xA;&lt;p&gt;    Other examples of this dangerous tendency abound. For instance, since the 1850s historians ignorant of Mādhava’s work have argued about whether Indian astronomers had the concept of the infinitesimal calculus (...) in a formula for finding the instantaneous velocity of the moon (...) I cannot tell you how that formula was derived, since its author, &lt;a href=&quot;https://en.wikipedia.org/wiki/Var%C4%81hamihira&quot; rel=&quot;nofollow noreferrer&quot;&gt;Varāhamihira&lt;/a&gt;, has not told me; but I find it totally implausible that some Indian discovered the calculus — a discovery for which previous developments in Indian mathematics would not at all have prepared him — applied his discovery only to the problem of the instantaneous velocity of the moon, and then threw it away. The idea that he might have discovered the calculus arises only from the Hellenophilic attitude that what is valuable in the past is what we have in the present.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(For exposition and detailed bibliography of the papers on Mādhava see e.g. Roy (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=2807493&quot; rel=&quot;nofollow noreferrer&quot;&gt;2011&lt;/a&gt;, Chap. 1, &lt;a href=&quot;https://books.google.com/books?id=q8H3CwAAQBAJ&amp;amp;ots=iw09i8RdSU&amp;amp;pg=PT37&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 14-15&lt;/a&gt;).)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-08-04T12:24:15.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7563" PostTypeId="1" AcceptedAnswerId="7567" CreationDate="2018-07-30T20:01:16.827" Score="8" ViewCount="624" Body="&lt;p&gt;I am almost sure I read once that a famous physicist said that anyone who discover a new particle should be fined instead of receiving a prize. The context was that at the time there was more and more particles being found and little theory to give account of that. It was necessary some model based on few elementary particles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately I am not finding such quote anywhere. Is that quote true? If yes, who said that? Any reference would be great.&lt;/p&gt;&#xA;" OwnerUserId="5436" LastActivityDate="2018-07-31T19:14:25.993" Title="Who did say that anyone who discover a new particle should be fined instead of receiving a prize?" Tags="|physics|theoretical-physics|quote|particle-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7564" PostTypeId="1" CreationDate="2018-07-30T22:01:41.970" Score="4" ViewCount="74" Body="&lt;p&gt;I know it sounds like a silly question but didn't both Pauli and Fermi hint at Beta decay missing a particle to carry away a small amount of it's energy or mass but why is this an anti neutrino ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The neutrino was not actually discovered some years latter so how was the Beta decay dubbed with the anti neutrino? &lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-07-31T05:03:27.297" Title="From a historical perspective why is beta decay associated with an anti neutrino instead of a neutrino?" Tags="|nuclear-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7565" PostTypeId="1" CreationDate="2018-07-31T00:29:04.003" Score="2" ViewCount="184" Body="&lt;p&gt;Who discovered the following &lt;a href=&quot;https://math.stackexchange.com/questions/591350/expressing-a-factorial-as-difference-of-powers-sum-r-0n-binomnr-1&quot;&gt;theorem&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;$$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2692" LastEditorUserId="2692" LastEditDate="2019-02-13T15:15:06.577" LastActivityDate="2019-02-15T16:27:50.577" Title="Who discovered the expansion for factorial as a successive difference of integers?" Tags="|mathematics|discoveries|" AnswerCount="1" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7566" PostTypeId="2" ParentId="7564" CreationDate="2018-07-31T05:03:27.297" Score="6" Body="&lt;p&gt;Only after the neutrino's discovery did we come up with the concept of lepton number (based on the unexpected multiple lepton families), which warranted reclassifying &quot;that other thing emitted in beta emission&quot;, which was how we originally &lt;em&gt;defined&lt;/em&gt; the word neutrino, as an antineutrino. Ironically, there's still an outside chance the neutrino is its own antiparticle anyway.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-07-31T05:03:27.297" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7567" PostTypeId="2" ParentId="7563" CreationDate="2018-07-31T11:57:17.440" Score="10" Body="&lt;p&gt;According to a &lt;a href=&quot;http://hst-archive.web.cern.ch/archiv/HST2004/standard_model/introductory_elemetary_particle.ppt&quot; rel=&quot;noreferrer&quot;&gt;slide deck&lt;/a&gt; I found,  it was Willis Lamb. Quote from said deck:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1955, Willis Lamb started his Nobel Prize acceptance speech by&#xA;  saying that “maybe physicists discovering a new particle ought to be&#xA;  fined 10 000$”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastEditorUserId="609" LastEditDate="2018-07-31T12:30:42.070" LastActivityDate="2018-07-31T12:30:42.070" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7568" PostTypeId="1" AcceptedAnswerId="7571" CreationDate="2018-07-31T13:11:25.227" Score="2" ViewCount="587" Body="&lt;p&gt;Does anyone know who started using the symbols A, B, E, T  (First column, left) for showing irreducible representations of symmetry groups? In older maths books I see capital gamma. Herein A= totally symmetric, B= anti-symmetric with respect to rotation, E= doubly degenerate, T is triply degenerate irreducible representation. The symbols in the top row are due to Schonflies (D4h, C, S, Sigma). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many books/ webpages claim that Mulliken started this symbols, and they are indeed called Mulliken symbols &lt;a href=&quot;http://www.cup.uni-muenchen.de/ch/compchem/geom/point3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cup.uni-muenchen.de/ch/compchem/geom/point3.html&lt;/a&gt;, however, many other physicists were using A, B,  E and T symbolism at the same time (1930s). The letter C was not used. It appears that mathematicians must be using these symbols before that but I could not find any example. Is anyone aware of the history of character tables such as the one shown here? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2018-11-04T17:01:20.367" LastActivityDate="2018-11-04T17:01:20.367" Title="History of group theory character tables (as used in physics and chemistry)" Tags="|mathematics|terminology|theoretical-physics|abstract-algebra|group-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7571" PostTypeId="2" ParentId="7568" CreationDate="2018-07-31T21:43:37.067" Score="2" Body="&lt;p&gt;Mulliken credits &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Placzek&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Georg Placzek&lt;/strong&gt;&lt;/a&gt; in his autobiography (&lt;a href=&quot;https://books.google.com/books?id=2eDrCAAAQBAJ&amp;amp;ots=klOUqVqu5v&amp;amp;pg=PA90&amp;amp;dq=Placzek&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989&lt;/a&gt;, p. 90). According to T. Oka (&lt;a href=&quot;https://doi.org/10.1002/9780470749593.hrs074&quot; rel=&quot;nofollow noreferrer&quot;&gt;2011&lt;/a&gt;)(&lt;a href=&quot;http://fermi.uchicago.edu/publications/PDF/oka270.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Placzek (1934) introduced the currently used symbols of irreducible representations such as A (symmetric), B (antisymmetric), E (“entartet”, doubly degenerate), F (triply degenerate), subscript 1 and 2 (to specify symmetry with respect to C&lt;sub&gt;2&lt;/sub&gt;) and superscript ′ and ′′ (to specify symmetry with respect to σ), etc. While a little unsystematic, the much-needed nomenclature was immediately adopted by theorists like Teller, Tisza, Mulliken, Wilson, and others, and has become the fixture in molecular spectroscopy.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;Placzek, G. (1934) Rayleigh-Streuung und Raman-effekt. &lt;em&gt;Marx’s Handbuch der Radiologie&lt;/em&gt;, &lt;strong&gt;VI&lt;/strong&gt;, (Part II), 209–374.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-07-31T22:08:15.423" LastActivityDate="2018-07-31T22:08:15.423" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7572" PostTypeId="1" AcceptedAnswerId="7584" CreationDate="2018-08-01T17:16:05.153" Score="6" ViewCount="174" Body="&lt;p&gt;The metric was originally defined to be valid only from the surface of a black hole outward but somewhere along the line it was extended inward to include the region under the event horizon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This extension of the metric was, of course, what gave us the definition of Schwarzschild black hole and I've been curious for some time about how the extension first came about and who did it.&lt;/p&gt;&#xA;" OwnerUserId="7699" LastEditorUserId="658" LastEditDate="2018-08-08T15:00:49.757" LastActivityDate="2018-08-08T15:00:49.757" Title="When, and by whom was the Schwarzschild metric first taken to be valid for all radii greater than zero?" Tags="|theoretical-physics|relativity-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7574" PostTypeId="1" AcceptedAnswerId="7575" CreationDate="2018-08-02T15:41:16.040" Score="2" ViewCount="157" Body="&lt;p&gt;Hearing to episode 3 of Stephen Fry's podcast &lt;em&gt;Great Leap Years&lt;/em&gt;, he describes this very prestigious French prize for scientist of 19th century called the Volta prize (&lt;em&gt;le prix Volta&lt;/em&gt; in French). The recipients include Humphry Davy and Graham Bell, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Volta_Prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; is very short, is an incomplete list and has some broken links. I would like to find a full list of recipients, do you know where I may find something like that? I tried to make search in French but I didn't find more than the Wiki article.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2018-08-02T16:31:24.140" LastActivityDate="2018-08-03T00:02:25.453" Title="Where to find a list of laurates of the Volta prize?" Tags="|academic-awards|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7575" PostTypeId="2" ParentId="7574" CreationDate="2018-08-02T23:56:28.980" Score="3" Body="&lt;p&gt;Maindron (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k97893365/f145&quot; rel=&quot;nofollow noreferrer&quot;&gt;1881, pp. 131-133&lt;/a&gt;) has the 1852-1880 story, slightly different from what Wikipedia says. Not the same as Napoléon Bonaparte’s &lt;a href=&quot;https://de.wikipedia.org/wiki/Galvanischer_Preis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Prix du galvanisme&lt;/em&gt;&lt;/a&gt; (pp. 69-70) which is what Davy got in 1807.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-08-03T00:02:25.453" LastActivityDate="2018-08-03T00:02:25.453" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7576" PostTypeId="1" CreationDate="2018-08-04T11:22:15.047" Score="9" ViewCount="1830" Body="&lt;p&gt;I would like to have help in producing examples of mathematicians that, in some sense I'll explain below,turned their career into failure. I am mainly interested in examples from XIX and XXth century.&lt;/p&gt;&#xA;&lt;p&gt;I'd like to hear of mathematicians that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;started their career in a very promising manner and then turned to pseudosciences, or devoted all their energies to &amp;quot;lost causes&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;lost years and years in trying to prove statements that turned out to be wrong and/or for which there were not suitablly developed techniques.&lt;/li&gt;&#xA;&lt;li&gt;lost their life in trying to support theories that turned out to be of little use/ of limited interest/ not well granted.&lt;/li&gt;&#xA;&lt;li&gt;developed complete theories that rapidly went &amp;quot;out of fashion&amp;quot; and did never become fashionable again.&lt;/li&gt;&#xA;&lt;li&gt;missed by little big opportunities of great results.&lt;/li&gt;&#xA;&lt;li&gt;were very famous during their lifetime and are completely forgotten nowadays.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am of course thinking of reasonably well-known mathematicians, not of an obscure Molvanian guy that got lost in his studies.&lt;/p&gt;&#xA;&lt;p&gt;I list just one example which falls into some of this categories:&lt;/p&gt;&#xA;&lt;p&gt;Luigi Fantappié that became quite famous in the 30ies for his theory of analytic functionals. This theory was somewhat halted, since it needed a more abstract approach that was possible only after sheaf theory was developed in the 50ies. He then developed an alternative &amp;quot;final relativity theory&amp;quot; that was completely ignored by physicist (missing the opportunity to anticipate of some years the theory of Lie algebra deformations). He also devoted enormous amount of time to the so-called theory of sintropy which turned out being a bizarre mixture of pseudoscience, theological considerations, doubtful philosophical statements.&lt;/p&gt;&#xA;&lt;p&gt;No living mathematicians please.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="1689" LastEditDate="2020-09-26T18:23:17.857" LastActivityDate="2021-07-19T04:25:21.863" Title="Failures in math" Tags="|mathematics|mathematicians|examples|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7577" PostTypeId="2" ParentId="7576" CreationDate="2018-08-04T11:44:46.193" Score="9" Body="&lt;p&gt;One possibility is Ted Kaczynski, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ted_Kaczynski&quot; rel=&quot;nofollow noreferrer&quot;&gt;Unabomber&lt;/a&gt;. He has been described as a mathematical prodigy and &lt;a href=&quot;https://en.wikipedia.org/wiki/Allen_Shields&quot; rel=&quot;nofollow noreferrer&quot;&gt;Allen Shields&lt;/a&gt;, his doctoral adviser, once told that Kaczynski was the best doctoral student he ever directed.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="4251" LastEditDate="2020-06-29T06:25:41.677" LastActivityDate="2020-06-29T06:25:41.677" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7578" PostTypeId="1" CreationDate="2018-08-04T14:17:13.097" Score="6" ViewCount="1162" Body="&lt;p&gt;According to the Jeff Miller's Earliest Known Uses of the Words of Mathematics &quot;Moment was taken into Statistics from Mechanics by Karl Pearson when he treated the frequency-curve (or observation curve) as the sheet enclosed by the curve and the horizontal axis. See his &quot;Asymmetrical Frequency Curves,&quot; Nature October 26th, 1893: &quot;Now the center of gravity of the observation curve is found at once, also its area and its first four moments by easy calculation.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This implies that the term moment was taken from mechanics. On the other hand, a Wikipedia reference Robertson, D.G.E.; Caldwell, G.E.; Hamill, J.; Kamen, G.; and Whittlesey, S.N. (2004) Research Methods in Biomechanics. Champaign, IL: Human Kinetics Publ., p. 285 says that the concept of moments was taken from mathematics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering about the analogy of mechanical moment (force into the distance) with the statistical moment. What would be a mechanical analog of the zeroth moment (area), first moment (center of gravity), and second moment (variance) or there is none? Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-08-10T00:26:45.240" Title="Why statistical moments are called moments?" Tags="|mathematics|terminology|philosophy-of-science|statistics|" AnswerCount="4" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7579" PostTypeId="1" CreationDate="2018-08-04T15:24:45.980" Score="5" ViewCount="453" Body="&lt;p&gt;This question is a continuation of my previously-posted question: &lt;a href=&quot;https://hsm.stackexchange.com/questions/7503/several-questions-about-gausss-contributions-to-electromagnetism?noredirect=1#comment13789_7503&quot;&gt;Several questions about Gauss&amp;#39;s contributions to electromagnetism&lt;/a&gt;.  I wrote it after user vonbrand asked me to split my original question into several.&#xA;I'm very interested to learn about Gauss's unpublished fragments on electromagnetism; especially I'm interested in Gauss's &amp;quot;Zur Electrodynamik&amp;quot; (Gauss's werke, volume 5, p. 601 - 630).&lt;/p&gt;&#xA;&lt;p&gt;The first titles in this manuscript relate directly to electric circuits, and record, to my opinion, his discovery of Kirchoff's laws for branched electrical circuits. But in addition, it seems that he studies some interesting configurations for electrical circuits (&lt;strong&gt;see p.601-604 (titles 1-2) and p. 621-622 (titles 16-17)&lt;/strong&gt;), which I didn't find comments about in the literature. The formulas related to these figures seem to be of a very interesting form (they involve uncommon functions in the basic theory of electric circuits). For these reasons, I  refuse to believe that nobody knows what these figures describe, so if anyone knows something about these schematic figures, please explain.&lt;/p&gt;&#xA;&lt;p&gt;It seems also that title 2 in this manuscript records his discovery of the &amp;quot;&lt;strong&gt;principle of minimum heat&lt;/strong&gt;&amp;quot; (established by Kirchoff in 1848; it's related to minimum entropy principles), which Dunnington mentions in p.161 of his biography of Gauss - Gauss writes: &amp;quot;The basic principle is that &lt;span class=&quot;math-container&quot;&gt;$\sum ri^2$&lt;/span&gt; is in minimum for the actual current distribution&amp;quot;. Here, &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the resistance of a resistor and &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; the electric current (for explanation see this article: &lt;a href=&quot;https://bayes.wustl.edu/etj/articles/min.ent.prod.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Minimum Entropy Production Principle&lt;/a&gt;). So, did Gauss refer to this important &lt;strong&gt;variational principle&lt;/strong&gt; in several of his letters (to Weber, perhaps?)?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2023-07-26T22:29:51.183" LastActivityDate="2023-07-26T22:29:51.183" Title="Question about Gauss's contributions to the theory of electric circuits" Tags="|biographical-details|electricity|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7581" PostTypeId="2" ParentId="7578" CreationDate="2018-08-04T21:39:39.197" Score="4" Body="&lt;p&gt;This seems to depend on who you call a statistician, mathematician, or mechanician. Certainly Pearson sounds like he’s using, as a matter of course, a term also found in e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Stieltjes_moment_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stieltjes&lt;/a&gt; (&lt;a href=&quot;https://zbmath.org/?q=an:25.0326.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1894&lt;/a&gt;, p. 48; &lt;a href=&quot;https://zbmath.org/?q=an:16.0243.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1885&lt;/a&gt;), Wittenbauer (&lt;a href=&quot;https://zbmath.org/?q=an:13.0683.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1881&lt;/a&gt;), Reye (&lt;a href=&quot;https://zbmath.org/?q=an:02.0671.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1870&lt;/a&gt;), Poinsot (&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k433669p/f183&quot; rel=&quot;nofollow noreferrer&quot;&gt;1806&lt;/a&gt;), Euler (&lt;a href=&quot;http://books.google.com/books?id=m5UDAAAAMAAJ&amp;amp;pg=PA185&quot; rel=&quot;nofollow noreferrer&quot;&gt;1752&lt;/a&gt;, p. 192), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hadn’t heard of &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment_(physics)#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Commandino&lt;/a&gt; (&lt;a href=&quot;https://books.google.com/books?id=2jgPAAAAQAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1565&lt;/a&gt;): it would be interesting to see what Greek word he translated into &lt;em&gt;moment[um]&lt;/em&gt; — and whether it was from Archimedes or Pappus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Mechanical versions of 0th and 2nd moment are total mass and &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment_of_inertia&quot; rel=&quot;nofollow noreferrer&quot;&gt;moment of inertia&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-08-09T18:03:55.953" LastActivityDate="2018-08-09T18:03:55.953" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7582" PostTypeId="2" ParentId="7578" CreationDate="2018-08-05T13:26:52.050" Score="4" Body="&lt;p&gt;The &lt;em&gt;Oxford English Dictionary&lt;/em&gt; shows &lt;em&gt;moment of a force&lt;/em&gt; appearing in 1830 in &lt;em&gt;A Treatise on Mechanics&lt;/em&gt; by Henry Kater and Dionysius Lardner.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So perhaps it is reasonable to guess that Stieltjes and/or Pearson took the term from mechanics.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-08-05T13:26:52.050" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7584" PostTypeId="2" ParentId="7572" CreationDate="2018-08-06T02:47:47.300" Score="1" Body="&lt;p&gt;The story is told in Eisenstaedt’s book (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=2264671&quot; rel=&quot;nofollow noreferrer&quot;&gt;2006&lt;/a&gt;, &lt;a href=&quot;https://google.com/books?id=dINaDwAAQBAJ&amp;amp;ots=tgFPNDYCGH&amp;amp;pg=PA255&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chap. 12&lt;/a&gt;) or papers (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=677686&quot; rel=&quot;nofollow noreferrer&quot;&gt;1982&lt;/a&gt;, &lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=920462&quot; rel=&quot;nofollow noreferrer&quot;&gt;1987&lt;/a&gt;, &lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=1200724&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989&lt;/a&gt;) (&lt;a href=&quot;https://www.jstor.org/stable/41133669&quot; rel=&quot;nofollow noreferrer&quot;&gt;also&lt;/a&gt; &lt;a href=&quot;https://www.jstor.org/stable/41133818&quot; rel=&quot;nofollow noreferrer&quot;&gt;jstor&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Briefly he says that, while Schwarzschild and Droste’s original works (&lt;a href=&quot;http://www.archive.org/stream/sitzungsberichte1916deutsch#page/189&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/verslagvandegewo2511916#page/n174&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916&lt;/a&gt;, &lt;a href=&quot;http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publDetail&amp;amp;pId=PU00012346&quot; rel=&quot;nofollow noreferrer&quot;&gt;1917&lt;/a&gt;)  ignored the region $r\leqslant 2m$ (Schwarzschild actually used $\smash{\tilde r=(r^3-8m^3)^{1/3}}$ &gt; 0 as his radial coordinate), other early papers like Hilbert (&lt;a href=&quot;https://emis.de/cgi-bin/JFM-item?46.1298.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1917&lt;/a&gt;, &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN252457811_1917?tify=%7B&amp;quot;pages&amp;quot;:[76]%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 70&lt;/a&gt;), Eddington (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?48.1063.12&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/spacetimegravi00eddi#page/97&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 97&lt;/a&gt;) or Lodge (&lt;a href=&quot;https://archive.org/stream/londonedinburg6411921lond#page/549&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921&lt;/a&gt;, p. 555) &lt;em&gt;did&lt;/em&gt; use the metric for all $r$ &gt; 0. Putative crossing under the $r=2m$ “singularity” was also discussed at a 1922 Paris seminar of Einstein, as reported by Nordmann (&lt;a href=&quot;https://archive.org/stream/p09revuedesdeux1922pariuoft#page/129&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;, pp. 154-156) and Brillouin (&lt;a href=&quot;https://hal.archives-ouvertes.fr/AJP/jpa-00204376v1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1923&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The possibility was dismissed then, and apparently never &lt;a href=&quot;https://en.wikipedia.org/wiki/Schwarzschild_metric#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;seriously&lt;/a&gt; considered again until the early 1960s, when people discussed extended metrics.&#xA;(Eisenstaedt also notes that Weyl had pioneered such an extension in (&lt;a href=&quot;https://emis.de/cgi-bin/JFM-item?46.1303.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1917&lt;/a&gt;, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=nyp.33433090770839&amp;amp;num=131&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 131&lt;/a&gt;; &lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?46.1277.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1918&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=bPhYAAAAYAAJ&amp;amp;pg=PA205&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 205&lt;/a&gt;), but removed it from later editions of his book.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-08-06T02:52:54.410" LastActivityDate="2018-08-06T02:52:54.410" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7585" PostTypeId="1" CreationDate="2018-08-06T12:18:50.660" Score="8" ViewCount="933" Body="&lt;p&gt;Recently i read the article &quot;&lt;strong&gt;Hamilton, Rodrigues, Gauss, Quaternions and Rotations: A Historical Reassessment&lt;/strong&gt;&quot;, which can be found freely on the internet. This article is by far the most comprehensive paper i found on the early history of quaternions: it discusses Euler's glimpses of a quaternions algebra, and Gauss's anticipations of it in his posthomously published 1819 fragment &quot;mutations des raumes&quot; (english: &quot;&lt;strong&gt;rotations of space&lt;/strong&gt;&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article gives a quite complete assessement of this fragment of Gauss: it lists (on p. 11-12) several aspects of Gauss's &quot;algebra of rotations&quot;, which correspond almost exactly to the eight subsections of part I of Gauss's fragment. However, it doesn't comment about subsection 5; in it Gauss writes down, as far as i understand it, several congruences involving the elements (i.e, the quaternion coefficients $a,b,c,d$) of a composition of two spatial rotations (two quaternions) modulo the norm of one quaternion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More specifically, Gauss defines the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$ A = a\alpha - b\beta - c\gamma - d\delta$$&#xA;$$B = a\beta + b\alpha - c\delta + d\gamma $$&#xA;$$C = a\gamma + b\delta + c\alpha - d\beta$$&#xA;$$D = a\delta - b\gamma + c\beta + d\alpha$$ and the norms: $m = a^2 + b^2 + c^2 + d^2, \mu = \alpha^2 + \beta^2 + \gamma^2 + \delta^2$, and then observes that, for example:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$\frac {{C+Di}}{{A + Bi}} \equiv \frac {{c + di}}{{a + bi}} \pmod m$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that immediately came to my mind is what is &quot;$i$&quot;? he didn't define it previously, and my second question is: what's the meaning of this congruence? in fact Gauss writes six such congruences (anyone interested can find it in Gauss's werke, volume 8, p.359)  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2022-09-08T06:19:23.247" LastActivityDate="2022-09-08T06:19:23.247" Title="Gauss's anticipation of quaternions and their relation to congruences" Tags="|biographical-details|abstract-algebra|gauss|quaternions|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7586" PostTypeId="2" ParentId="7578" CreationDate="2018-08-06T12:58:25.897" Score="4" Body="&lt;p&gt;Moments in mechanics and statistics are defined by the same formula:&#xA;$$\int x \rho(x)dx,$$&#xA;for the first moment. In mechanics, $x$ is distance, and $\rho$ is the mass density. In statistics, $x$ is anything (whatever your random variable represents) and $\rho$ is the probability density. So it is not surprising that the name is the same. Moments in mechanics were of course considered much earlier (since Archimedes at least). &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-08-06T12:58:25.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7587" PostTypeId="2" ParentId="7576" CreationDate="2018-08-07T07:23:50.387" Score="-2" Body="&lt;p&gt;Two examples come to my mind but they are not to be taken too &lt;em&gt;seriously&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our set theory professor in his lectures introduced &lt;a href=&quot;https://www.youtube.com/watch?v=1bZv3pSaLtY&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bertrand Russell&lt;/a&gt; jokingly by saying that he had started off as a mathematician, then became a philosopher, then became a peace activist. The implication was that this was clearly a downhill career.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My other example is &lt;a href=&quot;https://en.wikipedia.org/wiki/Oded_Schramm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oded Schramm&lt;/a&gt;. He was a probability theorist, the inventor of Schramm–Loewner evolutions (SLE). His `failure' was that he died too early in a mountaineering accident (at the age of 46). I mean, he could have avoided it by not going into the mountains. I hope I didn't offend anybody's sensitivities. (But then one could list &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89variste_Galois&quot; rel=&quot;nofollow noreferrer&quot;&gt;Évariste Galois&lt;/a&gt; as well, and possibly others.)&lt;/p&gt;&#xA;" OwnerUserId="7364" LastActivityDate="2018-08-07T07:23:50.387" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7588" PostTypeId="1" CreationDate="2018-08-07T15:34:47.180" Score="1" ViewCount="744" Body="&lt;p&gt;Galileo used a refractor i.e. no mirror: but what was the aperture?&lt;/p&gt;&#xA;" OwnerUserId="5924" LastActivityDate="2018-08-07T16:20:01.900" Title="What was the aperture of the first Galileo's telescope?" Tags="|galileo|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7589" PostTypeId="2" ParentId="7588" CreationDate="2018-08-07T16:20:01.900" Score="2" Body="&lt;p&gt;Galileo's most famous invention was the telescope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galileo made his first telescope in 1609, modeled after telescopes produced in other parts of Europe that could magnify objects three times and its aperture was 1.5 cm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He made/assembled two telescopes later in 1612/1620 that could finally magnify objects twenty times.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;With this telescope, he was able to look at the moon, discover the four satellites of Jupiter, observe a supernova, verify the phases of Venus, and discover sunspots. His discoveries proved the Copernican system which states that the earth and other planets revolve around the sun. Prior to the Copernican system, it was held that the universe was geocentric, meaning the sun revolved around the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;                             Aperture          Type       Inventor   (year)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Galileo's 1609 telescope [4]   1.5 cm  .62″    Refractor   Galileo Galilei Italy   (1609)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Galileo's 1612 telescope [4]   2.6 cm   1″     Refractor   Galileo Galilei Italy   (1612)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Galileo's 1620 telescope [4]   3.8 cm   1.5″   Refractor   Galileo Galilei Italy   (1620)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;see also -&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_largest_optical_telescopes_historically&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_largest_optical_telescopes_historically&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://galileo.rice.edu/bio/narrative_6.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://galileo.rice.edu/bio/narrative_6.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastActivityDate="2018-08-07T16:20:01.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7590" PostTypeId="2" ParentId="7576" CreationDate="2018-08-07T16:59:41.453" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Abian&quot; rel=&quot;noreferrer&quot;&gt;Alexander Abian&lt;/a&gt; may fit the bill here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His most notable contribution to mathematics was proving consistency and independence of four of the Zermelo-Fraenkel axioms. &lt;a href=&quot;http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html&quot; rel=&quot;noreferrer&quot;&gt;source: Wolfram Mathworld&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They are : &lt;em&gt;Extensionality&lt;/em&gt;, &lt;em&gt;Replacement&lt;/em&gt;, &lt;em&gt;Power set&lt;/em&gt;, and &lt;em&gt;&quot;Sum-set&quot;&lt;/em&gt; (i.e., &lt;em&gt;Union&lt;/em&gt;). See, for example, Abian and LaMacchia's &lt;a href=&quot;https://pdfs.semanticscholar.org/9dce/c1b889ba513c42ea08dc564b9bdb3665c6b3.pdf&quot; rel=&quot;noreferrer&quot;&gt;Original Paper&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, so good.  However, Abian spent the last 10 years of his life promoting his &quot;Moonless Earth&quot; theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the wikipedia article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Abian gained a degree of international notoriety for his claim that blowing up the Moon would solve virtually every problem of human existence. He made this claim in 1991 in a campus newspaper. Stating that a Moonless Earth wouldn't wobble, eliminating both the seasons and its associated events like heat waves, snowstorms and hurricanes. Refutations were given toward that idea by NASA saying that part of the exploded Moon would come back as a meteorite impacting the Earth and causing sufficient damage to extinguish all life, while restoring the seasons in the process.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Abian said that &quot;Those critics who say 'Dismiss Abian's ideas' are very close to those who dismissed Galileo.&quot; This claim and others, made in thousands of Usenet posts during the last portion of his life, gained Abian mention (not entirely favorable) and even interviews in such diverse publications as Omni, People, Weekly World News, and The Wall Street Journal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2018-08-07T16:59:41.453" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7591" PostTypeId="2" ParentId="7042" CreationDate="2018-08-07T17:23:30.970" Score="2" Body="&lt;p&gt;Robert Millikan’s oil drop experiment presented the first measurement of the electron’s charge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The first results came out in 1910, but the seminal work was a 1913 paper in the Physical Review.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Millikan reported a value for the fundamental electric charge that was within &lt;strong&gt;half a percent of today’s accepted value.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Backdrop:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. J. Thomson discovered the electron in 1897 when he measured the charge-to-mass ratio for electrons in a beam. &#xA;But the value of the charge and whether it was fundamental remained open questions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomson and others &lt;strong&gt;tried to measure&lt;/strong&gt; an irreducible electric charge by looking at clouds of water droplets. Using various techniques, they estimated the smallest charge that a droplet could hold, but &lt;strong&gt;the results were not entirely&lt;/strong&gt; convincing because they &lt;strong&gt;relied on averages over many&lt;/strong&gt; particles of various sizes. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“The evidence for a unitary charge was at the time very ambiguous,” says &lt;strong&gt;science historian Gerald Holton&lt;/strong&gt; of Harvard University.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At the University of Chicago in the 1900s, Millikan and his graduate students realized that ramping up the electric field would disperse a water cloud, so that only &lt;strong&gt;a few droplets&lt;/strong&gt; remained.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;He decided to try isolating single droplets, but it soon became clear that single water droplets evaporated too quickly to make reliable measurements.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;One of his students, Harvey Fletcher, found that long-lasting droplets could be made with a light oil that was used for lubricating clocks.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The oil drop experiment that Millikan and Fletcher designed had two chambers. In the upper chamber, an atomizer  dispersed a fine mist of micron-sized oil droplets. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Individual droplets would fall through a pinhole into the lower chamber, which consisted of two horizontal plates, with &lt;strong&gt;one held 16 millimeters&lt;/strong&gt; above the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The air in this chamber was &lt;strong&gt;ionized with x rays&lt;/strong&gt;, so that ions or free electrons could be captured on the falling droplets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A small window on the side allowed the experimenter to observe the droplets through a telescope. &lt;strong&gt;The droplets fell slowly enough—due to atmospheric drag—that the researchers could measure their downward speed by eye,&lt;/strong&gt; using horizontal lines in the telescope. From this speed, they could estimate the size and mass of each droplet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They then &lt;strong&gt;applied a high voltage across&lt;/strong&gt; the plates and measured the &lt;strong&gt;upward speed&lt;/strong&gt; of the droplet, to determine the electric force and ultimately the charge. &lt;strong&gt;Multiple measurements on a single droplet could be performed by repeatedly turning the electric field on and off.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The droplets had various amounts of charge on them (and they would often gain or lose charge during an observation), but &lt;strong&gt;the data showed that the charge was indeed quantized into integer multiples of a unit charge.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1910 Millikan published the first results of these experiments [1] (Fletcher was not included as an author, based on a deal the two struck [2]). &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Millikan then made several improvements, including an empirical estimate of the drag forces. The culmination of this effort, reported in 1913, was a value of the fundamental charge with an error bar of just 0.2 percent. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The precision acquired was so great that “other experiments did not improve on his result until a decade later,” Holton says.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;But Felix Ehrenhaft of the University of Vienna repeatedly challenged Millikan’s results, based on his own measurements of “sub-electron” charges on small metal particles. The dispute lasted for many years—known as the “Battle over the Electron”—but eventually most physicists sided with Millikan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In more recent years, historians who have examined Millikan’s lab notes have said that he discarded some of the measurements to boost the evidence of a fundamental charge. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;David Goodstein of the California Institute of Technology in Pasadena believes these accusations of fraud are unwarranted. He has analyzed the notes and says that Millikan excluded droplets because their observations were incomplete, not because their implied charge didn’t match his expectations [3]. “Millikan’s oil drop experiment is a classic example of outstanding physics done by one of the giants of his era,” Goodstein says.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Those scientists and other scholars who have carefully reviewed this case have failed  to agree on whether Millikan was guilty of unethical behavior or &quot;bad science&quot; in the treatment and presentation of his data. One of the expressed opinions condemns Millikan on the simple basis of the fact that his published statement is at odds with what can be concluded from an uncritical examination of his laboratory notebooks.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Others exonerate Millikan on the basis of a careful analysis and interpretation of comments on the data that appear in the notebooks. In the opinion of these Millikan defenders, the assertion that all drops were presented in the paper refers to all of the data taken under those conditions when the apparatus was working properly. Some of the scientists who have commented on this case appear to permit Millikan much discretion in the use of his &quot;scientific intuition&quot; to decide which data to include or exclude. This latter view seems to be guided by the principle that any scientist who consistently gets what turns out to be the correct answer is doing &quot;good&quot; science.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;For biographical information about Robert A Millikan and the history of the oil-drop experiment that will provide the context for this case we suggest that one can  read:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Robert A. Millikan,&quot; by Daniel J. Kevles, Scientific American, 240, pp 142-151&#xA;  (January 1979).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;My Work With Millikan On the Oil-drop Experiment,&quot; by Harvey Fletcher, Physics&#xA;  Today, pp 43-47 (June 1982).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;For a detailed analysis of the Millikan's work on the oil-drop experiment, including what he wrote in his laboratory notebooks see:&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Subelectrons, Presuppositions and the Millikan-Ehrenhaft Dispute&quot; in The Scientific Imagination, by Gerald Holton (Cambridge, Mass: Cambridge Univ. Press, 1978).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.onlineethics.org/cms/9726.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.onlineethics.org/cms/9726.aspx&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oil_drop_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Oil_drop_experiment&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastActivityDate="2018-08-07T17:23:30.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7592" PostTypeId="1" AcceptedAnswerId="7593" CreationDate="2018-08-08T02:08:23.547" Score="5" ViewCount="255" Body="&lt;p&gt;It seems a bit of a stretch to go from a battery to electrolysis but this fellow appears to have thought of it only a few weeks after Volta invented the &quot;pile&quot; battery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering if he knew water's elemental components before he assembles his electrolysis device .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What gave him the idea to run electricity through it from one pole to the other I wonder.  Was he expecting hydrogen and oxygen or did he have to run many tests to see what the gas elements were when the water disappeared from the tubes? It seems an amazing insight he must have had. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="55" LastEditDate="2018-08-08T05:01:40.520" LastActivityDate="2022-09-20T23:56:10.567" Title="What inspired Nicholson to break water into hydrogen and oxygen with electricity?" Tags="|chemistry|electricity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7593" PostTypeId="2" ParentId="7592" CreationDate="2018-08-08T03:10:26.230" Score="10" Body="&lt;p&gt;Mander's Carnocycle blog has a post &lt;a href=&quot;https://carnotcycle.wordpress.com/2017/02/01/carlisle-nicholson-and-the-discovery-of-electrolysis/&quot; rel=&quot;noreferrer&quot;&gt;Carlisle, Nicholson and the discovery of electrolysis&lt;/a&gt; with a detailed account of the story. I will only give the highlights. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On March 20th, 1800 Volta wrote to Sir Banks, the President of the Royal Society in London, describing the construction of his pile. It was an alternating assemblage of zinc and silver disc pairs separated by pieces of cloth soaked in a conducting liquid. Banks disclosed the contents of the letter to select few acquaintances, among whom was the surgeon Anthony Carlisle, who in turn shared it with William Nicholson. By 30th April 1800, Carlisle and Nicholson completed a pile of their own.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unlike Volta, who was more preoccupied with describing how his pile electroshocked humans, Carlisle and Nicholson focused on the pile itself. As Nicholson later commented in his paper:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“&lt;em&gt;I cannot here look back without some surprise and observe that … the rapid oxidation of the zinc should constitute no part of his [Volta’s] numerous observations&lt;/em&gt;”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first clue came on May 1st. Steel wires were attached to the pile, and to assist contact a drop of river water was placed on the uppermost disc. Nicholson recorded:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“&lt;em&gt;Mr. Carlisle observed a disengagement of gas round the touching wire. This gas, though very minute in quantity, evidently seemed to me to have the smell afforded by hydrogen. This, with some other facts, led me to propose to break the circuit by the substitution of a tube of water between two wires.&lt;/em&gt;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The next day, after placing the wires into a water filled tube, Carlisle and Nicholson noticed a stream of hydrogen bubbles rising from the wire attached to the zinc disc, while the wire attached to the silver disc blackened from oxidation. It was intriguing that oxygen, presumably produced together with hydrogen, found its way through the water to the other wire. Nicholson wrote that it “&lt;em&gt;seems to point at some general law of the agency of electricity in chemical operations&lt;/em&gt;”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To investigate the &quot;agency&quot; they switched to electrodes made of platinum, which was known to resist oxidation. This time there were two streams of bubbles, large one coming from the zinc side and small one from the silver side, with no oxidation of the wires. &lt;em&gt;“It was natural to conjecture, that the larger stream was hydrogen, and the smaller oxygen”&lt;/em&gt;, Nicholson concluded.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-08-08T18:46:24.183" LastActivityDate="2018-08-08T18:46:24.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7594" PostTypeId="2" ParentId="7552" CreationDate="2018-08-08T12:03:26.790" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it believed that Aristotle had sufficiently precise data to conclude that the arcs were round circular, and implied a geometric argument as above? Could he? Or should we take the word &quot;curved&quot; at its face value and rather view the whole argument as one of philosophical flavour rather than a mathematical one?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I think the conclusive remark of Aristotle about the shape of the earth was not only from the curved line of the shadow but from all the other observations he made or people made during those times.&#xA;As the general environs of interpretations and conjectures contribute to a final outcome or belief. see the notes-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By around 500 B.C., most ancient Greeks believed that Earth was round, not flat. But they had no idea how big the planet is until about &lt;strong&gt;240 B.C.(**much before the Aristotle observation) when **Eratosthenes devised a clever method of estimating its circumference.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was around 500 B.C. that Pythagoras first proposed a spherical Earth, mainly on aesthetic grounds rather than on any physical evidence. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like many Greeks, he believed the sphere was the most perfect shape. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Possibly the first to propose a spherical Earth based on actual physical evidence was Aristotle (384-322 B.C.), who listed several arguments for a spherical Earth: &lt;em&gt;ships disappear hull first when they sail over the horizon, Earth casts a round shadow on the moon during a lunar eclipse, and different constellations are visible at different latitudes.&lt;/em&gt;(italics mine)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Greek philosopher and scientist Aristotle (384-322 BC) proved that the Earth is round by carefully observing lunar eclipses. The shadow that the Earth casts on the Moon during a lunar eclipse, Aristotle noticed, was always curved, something only a sphere will do. If the Earth were flat, it would not consistently cast a curved shadow; after observing a number of lunar eclipses with the same result, Aristotle became convinced – correctly – that the Earth is round.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In about 350 BC, Aristotle wrote, “the earth is spherical…in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will be caused by the form of the earth’s surface, which is therefore spherical.” one  can try Aristotle’s revolutionary experiment for yourself – from any place on Earth a lunar eclipse will always cast a rounded shadow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Around 350 BC, the great Aristotle declared that the Earth was a sphere (based on observations he made &lt;strong&gt;about which constellations you could see in the sky as you travelled further and further away from the equator)&lt;/strong&gt; and during the next hundred years or so, Aristarchus and Eratosthenes actually measured the size of the Earth!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Around this time Greek philosophers had begun to believe the world could be explained by natural processes rather than invoking the gods, and early astronomers began making physical measurements, in part to better predict the seasons. The first person to determine the size of Earth was Eratosthenes of Cyrene, &lt;em&gt;who produced a surprisingly good measurement using a simple scheme that combined geometrical calculations with physical observations.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It has actually been known that the Earth was round since the time of the ancient Greeks.  It was Pythagoras who first proposed that the Earth was round sometime around 500 B.C. As I recall, he based his idea on the fact that he showed the Moon must be round by observing the shape of the terminator (the line between the part of the Moon in light and the part of the Moon in the dark) as it moved through its orbital cycle. Pythagoras reasoned that if the Moon was round, then the Earth must be round as well. &lt;em&gt;After that, sometime between 500 B.C. and 430 B.C., a fellow called Anaxagoras determined the true cause of solar and lunar eclipses - and then the shape of the Earth's shadow on the Moon during a lunar eclipse was also used as evidence that the Earth was round.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;during lunar eclipses, one only sees a small arc of the Earth (at most about 1/12 of the full circle). Is it really possible to tell apart a circular arc from an elliptic one with a naked eye given such a short sample?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding the above as earth is spinning as well as rotating about the Sun, during different observations of the eclipses the same edge of the earth is not necessarily repeated, therefore if there had been major changes in the curvature the lines may have been disturbed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the whole, taking the stages of measurements of those days one can say that His observation was correct(based on mathematical findings rather than philosophical), though we now know that Earth is not a perfect sphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref.-&lt;a href=&quot;http://classics.mit.edu/Aristotle/heavens.2.ii.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://classics.mit.edu/Aristotle/heavens.2.ii.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://www.aps.org/publications/apsnews/200606/history.cfm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aps.org/publications/apsnews/200606/history.cfm&lt;/a&gt;&#xA;  &lt;a href=&quot;https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question54.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question54.html&lt;/a&gt;&#xA;  &lt;a href=&quot;https://www.slooh.com/community/post/4598&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.slooh.com/community/post/4598&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastEditorUserId="7635" LastEditDate="2018-08-08T12:09:24.533" LastActivityDate="2018-08-08T12:09:24.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7595" PostTypeId="2" ParentId="7515" CreationDate="2018-08-09T06:51:31.243" Score="1" Body="&lt;p&gt;The development of concept of &amp;quot;Field' to describe the events happening around us in connection with  various interactions  which operated at &amp;quot;action at a distance'&#xA;has very early  beginning.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Faraday and Maxwell created one of history's most telling changes in our&#xA;physical worldview: the change from particles to fields.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As Albert Einstein put it,“Before Maxwell, Physical Reality …was thought of as consisting in material particles…. Since Maxwell's time, Physical Reality has been thought of as represented by continuous fields, ...and not capable of any mechanical interpretation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;This change in the conception of Reality is the most profound and&#xA;the most fruitful that physics has experienced since the time of Newton.&lt;/strong&gt;”&lt;/p&gt;&#xA;&lt;p&gt;As the preceding quotation shows, Einstein supported a &amp;quot;fields are all there&#xA;is&amp;quot; view of classical (but not necessarily quantum) physics. He put the final logical touch on classical fields &lt;strong&gt;in his 1905 paper&lt;/strong&gt; proposing the special theory of relativity, where he wrote &amp;quot;The introduction of a 'luminiferous' ether will prove to be superfluous.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For Einstein, there was no material ether to support light waves.&#xA;Instead, the &amp;quot;medium&amp;quot; for light was space itself. That is, for Einstein, fields are states or conditions of space. This is the modern view. The implication of special relativity (SR) that energy has inertia further reinforces both Einstein's rejection of the ether and the significance of fields. Since fields have energy, they have inertia and should be considered &amp;quot;substance like&amp;quot; themselves rather than simply states of some substance such as ether.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The general theory of relativity (1916) resolves Newton's dilemma concerning the &amp;quot;absurdity&amp;quot; of gravitational action-at-a-distance. According to general relativity, the universe is full of gravitational fields, and physical processes&#xA;associated with this field occur even in space that is free from matter and EM&#xA;fields.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Thus by 1915 classical physics described all known forces in terms of fields-&#xA;-conditions of space--and Einstein expressed dissatisfaction that matter couldn't be described in the same way.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;However the real inroads into particle concept and an alternative field  description or visualization came in with the advent of Quantum Field  Theory.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1950s, and led to the introduction of quantum electrodynamics (QED). QED was so successful and accurately predictive that efforts were made to apply the same basic concepts for the other forces of nature. By the late 1970s, these efforts successfully utilized gauge theory in the strong nuclear force and weak nuclear force, producing the modern standard model of particle physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let us try to visualise electrons.&lt;/p&gt;&#xA;&lt;p&gt;Everywhere in space  there is a field called the electron field.&#xA;A physical electron isn’t the field, but rather a localized vibration in the field.&lt;/p&gt;&#xA;&lt;p&gt;Electrons aren’t the only particles to consist of localized vibrations of a field; all particles do. There is a field for every known particle say a  photon, a quark, a gluon field and so on.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Even the recently discovered Higgs boson is like this. The Higgs field interacts with particles and gives them their mass, but it is hard to observe this field directly. Instead, we supply energy to the field in particle collisions and cause it to vibrate. When one says “we’ve discovered the Higgs boson,” you should think “we’ve caused the Higgs field to vibrate and observed the vibrations.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This idea gives an entirely different view of how the subatomic world works. Spanning all of space are a great variety of different fields that exist everywhere. What we think of as a particle is simply a vibration of its associated field.&lt;/p&gt;&#xA;&lt;p&gt;This has important consequences on the interaction of particles. For instance, consider a process in which two electrons are fired at one another and get scattered.&lt;/p&gt;&#xA;&lt;p&gt;In the quasi-classical view of scattering, one electron emits a photon and then recoils. The photon travels to the other electron, which also recoils.&lt;/p&gt;&#xA;&lt;p&gt;When the photon makes a quark and antiquark pair, the quark field is vibrating while the other two fields have no excitation.&#xA;Finally, when the quark and antiquark combine to make a gluon, only the gluon field has a vibration.&lt;/p&gt;&#xA;&lt;p&gt;In the QFT approach, a vibration in the electron field induces a vibration in the photon field. The photon field vibration transports energy and momentum to another electron vibration and is absorbed.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the well-known process where a photon converts into an electron and an anti-electron, the photon field vibrations are transferred to the electron field and two sets of vibrations are set up—one consistent with an electron vibration and the other consistent with the anti-electron.&lt;/p&gt;&#xA;&lt;p&gt;This idea of fields and vibrations explains how the universe works at a deep and fundamental level. These fields span all of space. Some fields can “see” other fields while being blind to others.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The photon field can interact with the fields of charged particles but cannot see gluon or neutrino fields. On the other hand, a photon can interact indirectly with the gluon field, first by making quark vibrations which then make gluon vibrations.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quantum fields are really a mind-bending way of thinking. Everything—and I mean everything—is just a consequence of many infinitely-large fields vibrating.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The entire universe is made of fields playing a vast, subatomic symphony. Physicists are trying to understand the melody.&lt;/p&gt;&#xA;&lt;p&gt;Reference-&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1204/1204.4616.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/ftp/arxiv/papers/1204/1204.4616.pdf&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_quantum_field_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/History_of_quantum_field_theory&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.pbs.org/wgbh/nova/blogs/physics/2013/08/the-good-vibrations-of-quantum-field-theories/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.pbs.org/wgbh/nova/blogs/physics/2013/08/the-good-vibrations-of-quantum-field-theories/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-08-10T03:40:33.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7596" PostTypeId="2" ParentId="7578" CreationDate="2018-08-10T00:26:45.240" Score="1" Body="&lt;p&gt;@ Francois, I can see the usage in Stieltjes book but I can't read French. One can see the word moment but the context is missing. I checked Oxford English Dictionary for earliest usage, it turns that moment was first used in calculus as &quot;Mathematics. In Newtonian calculus: the increment in the value of a time-varying quantity that occurs in an infinitesimal period of time; = differential n. 1. Cf. momentane n.   Now hist.&#xA;1706   Phillips's New World of Words (new ed.) (at cited word)   In Mathematicks, Moments are such indeterminate and uncertain Parts of Quantity, as are supposed to be in a perpetual Flux, i.e. either continually encreasing or decreasing.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then another meaning is given &quot;Mathematics. Any of various functions describing torsional effects, generally having the form of the product of a force and a distance; spec. the turning effect produced by a force; the magnitude of this, equal to the product of the force and the perpendicular distance from its line of action to the point about which rotation may occur.&#xA;  moment of a couple n.  [compare French moment d'un couple (1869)] the product of either of the two equal forces comprising the couple and the perpendicular distance between their lines of action.  moment of inertia n.  [compare French moment d'inertie (1786)] the product of the mass of a particle and the square of its distance from a given axis; the sum of such products for all the particles of a body.  moment of momentum n. the vector product of the momentum of a particle and its radius vector from a given point; the sum of such products for all the particles of a body; also called angular momentum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;bending, pitching, rolling moment: see the first element.&#xA;1830   H. Kater &amp;amp; D. Lardner Treat. Mechanics x. 135   The moment of a force is therefore found by multiplying the force by its leverage.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;OED credits Pearson for statistical moments.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-08-10T00:26:45.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7598" PostTypeId="2" ParentId="7576" CreationDate="2018-08-11T02:04:51.943" Score="8" Body="&lt;p&gt;Lucjan Emil Böttcher (1872–1937) might fit the bill. He got his PhD in Leipzig in 1898 under Sophus Lie and lectured at Lwow Polytechnics from 1901 until his retrement in 1935. His publications, starting with his PhD thesis,  are full of ideas anticipating holomorphic dynamics (developed about 20 years later by Pierre Fatou and Gaston Julia), and his theorem on local behavior of a holomorphic function around its superattracting fixed point is well known (and referred to as Böttcher's theorem). It is amazing how much he accomplished without the notion of a normal family, introduced by Paul Montel, which  gave solid foundations to holomorphic dynamics in one variable and accelerated its development.   However, Böttcher's goal was  to put iteration of holomorphic maps into the framework of Lie groups, which was a hopeless task. Besides, his writings were lacking in rigor, which is probably why his applications to extend  his lecturing licence from Lwow Polytechnics to Lwow  University met with failure (he tried 4 times). There were mathematicians at Lwow University familiar with complex variables and special functions (mainly Jozef Puzyna, a student of Weierstrass and an author of the first monograph in Polish on analytic functions, incorporating also elements of set theory and group theory), but they did not appreciate Böottcher's work.  Böttcher continued teaching and popularizing activities (e.g., making Russell's paradox familiar to Lwow scholarly community), but did not publish any mathematical research after 1914. Around that time he became actively involved with spiritism and later published 2 brochures, on turning tables and on afterlife (this interest might have been influenced by his attending Wilhelm Wundt's lectures in Leipzig, possibly also by his earlier contact with Julian Ochorowicz, an inventor, publicist and spiritist, and Wundt's student). Judging from Lwow's newspapers, at the moment of his death he was more recognized as a “spiritual” activist than as a mathematician.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="5989" LastEditDate="2020-09-26T07:34:28.240" LastActivityDate="2020-09-26T07:34:28.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7599" PostTypeId="2" ParentId="6778" CreationDate="2018-08-12T09:35:55.877" Score="4" Body="&lt;p&gt;The idea of a &quot;Chinese Flat Earth&quot; is an ideological old one that comes back from time to time. The last version is &lt;a href=&quot;https://www.academia.edu/22174185/AN_ANCIENT_CHINESE_FLAT_EARTH_COSMOLOGY&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.academia.edu/22174185/AN_ANCIENT_CHINESE_FLAT_EARTH_COSMOLOGY&lt;/a&gt; .&#xA;In about 2,000 BCE the Chinese astronomers had already discovered the cause of eclipses (shadow of the Earth/Moon), in the IV century BCE they had constructed their first (found) Armillary Sphere (&lt;a href=&quot;http://viewofchina.com/ancient-chinese-cosmology/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://viewofchina.com/ancient-chinese-cosmology/&lt;/a&gt;). But what most &quot;scholars&quot; forget to mention is that the Chinese had up to 8 schools of astronomy (variations of 3 basic ones). The two main schools were: the OLDEST one was called &quot;the wise school&quot; and adopted the spherical Sky model, &quot;the unwise school&quot; was nearer to the &quot;flat Earth&quot; model. Many astronomers at the time did not know how to make calculations (inherited positions from fathers, very influential in the court) and repeated old manuals. Maps were more schematics (graphs) than real maps (as we study them today) and most of the maps were square/rectangular with a &quot;square&quot; China and eventually some neighbor or People outside the limits of China. The most cited text for the Chinese Flat Earth is a mathematical manual (without figures) with instructions on how to calculate the dates (aiming the solstice) based on a gnomon and its shadow that takes the sky and the Earth as flat parallel planes (&quot;as above, as bellow&quot;), and based in this imprecise model extrapolates by similarity of triangles some calculations about the distance of the Sun, etc. The format of the Earth was not important for them (they already knew it because of the eclipses and did not care much). The calendrical calculations, the calculation of the eclipses and positions of the Sun/Moon (precise) in the sky (without a geometrical model) and the 5 planets (not so precise) was their focus. &quot;Europeans&quot; (because of the colonization ideology) usually have a hard time to accept that China had almost all what China needed inside China (center of the civilized world for them) and the rest was seen as barbarians after Chinese products. To leave China was a burden. In this site: &lt;a href=&quot;https://www.theepochtimes.com/new-evidence-ancient-chinese-explorers-landed-in-america-excites-experts_1348894.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.theepochtimes.com/new-evidence-ancient-chinese-explorers-landed-in-america-excites-experts_1348894.html&lt;/a&gt; ) and in the famous &quot;1421: The year China discovered the World&quot; there are evidences of the Chinese presence (for trade and discovery and not &quot;colonization&quot;/exploitation) in Central America. Another practical point is map projections. If you need a &quot;precise&quot; Map/Chart a spherical Earth is important, if you are OK with a schematic/relative positions of places draw, them it is not even a problem to be discussed. If you use a geometrical model for astronomical calculations a spherical Earth is important, if you have an algebraic model, it is not. If you torture enough the historical data, your ideological interpretation will make them confess anything. Most People and ancient cultures did not care about most problems our contemporary westernized world care.        &lt;/p&gt;&#xA;" OwnerUserId="7729" LastEditorUserId="7729" LastEditDate="2018-08-12T09:45:43.793" LastActivityDate="2018-08-12T09:45:43.793" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7600" PostTypeId="1" AcceptedAnswerId="8105" CreationDate="2018-08-12T12:08:49.113" Score="9" ViewCount="730" Body="&lt;p&gt;In the last years, I read a lot about a mathematical object that was discovered by John Napier in 1620 and explored much more deeply by Gauss, who called this &amp;quot;&lt;strong&gt;Pentagramma Mirificum&lt;/strong&gt;&amp;quot; (latin for &amp;quot;&lt;strong&gt;the miraculous pentagram&lt;/strong&gt;&amp;quot;). Unfortunately, there is no wikipedia article about this, and I didn't  find lower-level and accesible discussions of it, but only a few articles about it by H.S.M Coxeter and others. This object is related to spherical pentagrams (planar pentagrams are star like figures with 5 sides), but since I didn't find enough good sources on it, I wasn't able to extrapolate if it's a phenomenon of algebraic nature (some references say it was a special case of &amp;quot;cluster algebras&amp;quot;), geometric nature, or perhaps both. I've read that the geometer Coxeter invented something called &amp;quot;frieze paterns&amp;quot;, which gave a modern interpretation of Gauss's calculations about this phenomenon.&lt;/p&gt;&#xA;&lt;p&gt;So, first of all I want to understand what the &amp;quot;Pentagramma Mirificum&amp;quot; is about. Since I know almost nothing about it, I simply listed a few articles that might serve as an aid in an exposition of the subject. I think it will be good not only for me, but for stack exchange, if there will be at least one post about this - I didn't find a single post about it in stack exchange. I'll be glad if someone will also be able to explain a little bit of Gauss's investigations about it (which go quite deeply; for example, in his later fragments he applies Mobius's barycentric calculus to a chain of pentagons infinite in both directions). So here is a list of articles:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://arxiv.org/pdf/1106.3633.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PENTAGRAMMA MIRIFICUM AND ELLIPTIC FUNCTIONS&lt;/a&gt; - this is an exposition article for some of Gauss's fragments. It also gives some background, so I think it's the most suitable article to use in order to answer my question.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18132.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frieze patterns&lt;/a&gt; - an article by H.S.M Coxeter.&lt;/li&gt;&#xA;&lt;li&gt;There is also the reference &amp;quot;Draft Translation of Gauss’ Fragmentary Notes on the Pentagramma Mirificum&amp;quot; (just google these words and you will find it), which is an English translation of Gauss's original fragments.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="12330" LastEditDate="2023-08-25T15:54:32.877" LastActivityDate="2023-08-27T14:51:42.220" Title="What are the modern connections of the Pentagramma Mirificum studied by Gauss?" Tags="|biographical-details|geometry|abstract-algebra|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7602" PostTypeId="2" ParentId="7600" CreationDate="2018-08-13T21:42:47.550" Score="8" Body="&lt;p&gt;The Pentagramma Mirificum is a spherical figure formed by a series of five great circle arcs, each orthogonal to the next, and it probably does deserve a Wikipedia entry. However some elementary information about it is available online. A good start is &lt;a href=&quot;http://mathcentral.uregina.ca/mp/previous2006/nov06sol.php&quot; rel=&quot;noreferrer&quot;&gt;Math Central&lt;/a&gt; where we read that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The story begins in 1602, when Nathaniel Torporley (1564-1632) began to investigate the five &quot;parts&quot; a, A, b, B, c of a right-angled spherical triangle (right-angled at C).  According to De Morgan, Torporley anticipated by a dozen years the famous rules of Napier which Gauss embodied in his pentagramma mirificum... The &quot;core&quot; of the pentagram is a pentagon whose vertices are (obviously) poles of these five arcs; it is thus a self-polar pentagon.  [This means that if a vertex of the inside pentagon is placed at the north pole of the sphere, the opposite side lies along the equator.] The whole figure can be derived from the right-angled triangle ABC (appearing at the top) by extending the sides and drawing also the polar great circles of the vertices A and B.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wikipedia does have an article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Spherical_trigonometry#Napier%27s_rules_for_right_spherical_triangles&quot; rel=&quot;noreferrer&quot;&gt;Napier's rules&lt;/a&gt;, which are a mnemonic for ten identities relating every triple chosen from the set a, b, c, A, B. The identities are algebraic when written in terms of their square tangents. If we call the arc lengths $L_i$ instead, and set $a_i=\tan^2 (L_i)$ then $a_ia_{i+1}=1+a_{i+3}$, adding the indices mod $5$. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss noticed that the first three of these identities imply the other two, incidentally this proves the horizontal $5$-periodicity of the Coxeter's frieze pattern of width $2$. Coxeter's frieze patterns are arrengements of numbers into horizontal rows so that $2\times2$ determinants formed by neighboring four numbers are $1$. For an elementary exposition see the first lecture of &lt;a href=&quot;https://www.math.psu.edu/tabachni/talks/Friezes_lectures.pdf&quot; rel=&quot;noreferrer&quot;&gt;Tabachnikov's Three Lectures on Frieze  Patterns&lt;/a&gt;. For a more extensive survey see &lt;a href=&quot;https://arxiv.org/abs/1503.05049&quot; rel=&quot;noreferrer&quot;&gt;Coxeter's Frieze patterns at the Crossroads of Algebra, Geometry and Combinatorics by Morier-Genoud&lt;/a&gt;, where the pentagramma mirificum is also discussed. Frieze patterns lead into less elementary precincts. For those interested, the space of classical Coxeter’s frieze patterns can be presented as a discrete version of a coadjoint orbit of the Virasoro algebra, and the canonical cluster pre-symplectic form on it is a discretization of the Kirillov symplectic form, see &lt;a href=&quot;http://ftp.math.psu.edu/tabachni/prints/ContFriezeFinal.pdf&quot; rel=&quot;noreferrer&quot;&gt;Coxeter's Frieze Patterns and Discretization of the Virasoro Orbit by Ovsienko and Tabachnikov&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another modern relation is to &lt;a href=&quot;https://en.wikipedia.org/wiki/Cluster_algebra&quot; rel=&quot;noreferrer&quot;&gt;cluster algebras&lt;/a&gt; of Fomin and Zelevinsky, already covered by Wikipedia. The algebras are generated by subsets of fixed size called clusters, whose elements satisfy relations similar to $a_ia_{i+1}=1+a_{i+3}$. Fomin is a great enthusiast of their connection to the pentagramma, see his &lt;a href=&quot;https://www.math.uni-bielefeld.de/~sek/cluster/pentagramma/&quot; rel=&quot;noreferrer&quot;&gt;Bielefeld page&lt;/a&gt;, where we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It would be a stretch to say that Gauss discovered cluster algebras: the pentagon equation, in its algebraic form, was found by W. Spence in 1809. In the context of Pentagramma Mirificum, the realization that the 5-cycle represents a nontrivial algebraic identity seems to appear first in print by Arthur Cayley; see [&lt;a href=&quot;https://archive.org/stream/collectedmathema07cayluoft#page/36&quot; rel=&quot;noreferrer&quot;&gt;A. Cayley, On Gauss's Pentagramma Mirificum&lt;/a&gt;, Philosophical Magazine, vol. XLIL (1871), 311-312; The collected mathematical papers of Arthur Cayley. Vol. 7].&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;nontrivial algebraic identity&quot;, the pentagon equation, expresses the fact that the map $F:(a,b)\mapsto(b,\frac{b+1}a)$ satisfies $F^5=$id. This can be depicted as a pentagon of Laurent polynomials, and leads to quivers and a different view of cluster algebras, see &lt;a href=&quot;http://www.maths.dur.ac.uk/users/philipp.b.lampe/LectureNotes/cluster.pdf&quot; rel=&quot;noreferrer&quot;&gt;Lampe's lecture notes&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-08-16T16:46:31.350" LastActivityDate="2018-08-16T16:46:31.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7603" PostTypeId="1" CreationDate="2018-08-14T12:58:53.167" Score="4" ViewCount="221" Body="&lt;p&gt;Where can I find a copy (online) of Schwarz's paper that proved Clairaut's theorem for mixed partial derivatives?&lt;/p&gt;&#xA;&lt;p&gt;His paper is: Schwarz, H. A., &amp;quot;Communication&amp;quot;, Archives des Sciences Physiques et Naturelles, (1873), v. 48, pp. 38-45.&lt;/p&gt;&#xA;" OwnerUserId="6574" LastEditorUserId="136" LastEditDate="2022-04-04T16:57:21.560" LastActivityDate="2022-04-04T16:57:21.560" Title="Reference - Schwarz's Proof of Clairaut's Theorem" Tags="|mathematics|reference-request|real-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7604" PostTypeId="2" ParentId="7603" CreationDate="2018-08-14T16:30:41.670" Score="4" Body="&lt;p&gt;You can download  the journal here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/5919#/summary&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.biodiversitylibrary.org/bibliography/5919#/summary&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also Collected works of Schwarz:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/gesammeltemathem02schwuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/gesammeltemathem02schwuoft&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-08-14T16:30:41.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7605" PostTypeId="1" AcceptedAnswerId="7610" CreationDate="2018-08-16T07:15:56.353" Score="5" ViewCount="444" Body="&lt;p&gt;In Sec. 2.4 of &lt;em&gt;Inside Interesting Integrals&lt;/em&gt; (2015), Paul J Nahin says of &lt;span class=&quot;math-container&quot;&gt;$$I:=\int_0^{\pi/2}\ln (a\sin x)dx=\int_0^{\pi/2}\ln (a\cos x)dx$$&lt;/span&gt;that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For many years it was commonly claimed in textbooks that these are quite&#xA;difficult integrals to do, best tackled with the powerful techniques of contour&#xA;integration. As you’ll see with the following analysis, however, that is simply not the case.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The real method is&lt;span class=&quot;math-container&quot;&gt;$$2I=\int_0^{\pi/2}\ln\left(\frac{a^2}{2}\sin 2x\right)dx=\frac{\pi}{2}\ln\frac{a}{2}+\frac{1}{2}\int_0^\pi\ln(a\sin y) dy=\frac{\pi}{2}\ln\frac{a}{2}+I,$$&lt;/span&gt;so &lt;span class=&quot;math-container&quot;&gt;$I=\frac{\pi}{2}\ln\frac{a}{2}$&lt;/span&gt;. Presumably Euler's calculation, which Nahin says he did for &lt;span class=&quot;math-container&quot;&gt;$a=1$&lt;/span&gt; in 1769, didn't use the above method, or else textbooks would never have claimed the need for contour integration. So when did the above proof first surface? And when did textbooks first claim the need for complex methods?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2021-01-18T18:25:34.027" LastActivityDate="2021-01-18T18:25:34.027" Title="When was Euler's log-sine integral first computed by real methods?" Tags="|discoveries|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7606" PostTypeId="1" CreationDate="2018-08-16T09:46:04.263" Score="9" ViewCount="874" Body="&lt;h3&gt;Background&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;Mathematics and some areas of physics and computer science have the peculiar appeal that some problems and results are easy to understand and it is conceivable that somebody armed with nothing but the right idea can come up with something groundbreaking. This makes these fields particularly prone to amateurs becoming obsessed with solving famous problems or debunking something established. Some of these erroneously think they succeeded and then proceed to pester professional scientists (and are typically called cranks or crackpots).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, there are often good arguments that it is unlikely that an amateur is actually onto something, but here I am wondering about the empirical side:&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;Question&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;What are the most relevant results (if any) produced by amateurs in these fields? Some specifications:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Eligible results must be pen-and-paper theory, possibly aided by a computer.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finding a simpler or completely different proof for a solved problem is eligible.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Results that can in principle be found by brute-force computing are not eligible (even if they require some search strategy to avoid combinatorial explosion). It’s not that such results are without value, but they do not fit what I am interested in for several reasons (results are easy to check; it’s more plausible to get lucky; obsession may be a virtue; …).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For the purpose of this question, an amateur is somebody who never did any of the following:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;acquire an academic degree in a field that features proofs (mathematics, physics, computer science, …),&lt;/li&gt;&#xA;&lt;li&gt;publish a paper in one of these fields,&lt;/li&gt;&#xA;&lt;li&gt;make a living of performing research in such a field.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The result must have been found after 1960. This is not a hard deadline; I mainly want to ensure that the basics of mathematics and physics had been thoroughly explored and to somewhat ensure that there were no amateurs who would not be amateurs nowadays.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13" LastActivityDate="2018-12-23T21:59:51.450" Title="Did amateurs ever produce important proofs or similar?" Tags="|mathematics|discoveries|theoretical-physics|computer-science|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7607" PostTypeId="2" ParentId="674" CreationDate="2018-08-16T12:54:49.677" Score="2" Body="&lt;p&gt;Perhaps not the record-holder, but a contender:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prof. Noam Elkies is a mathematician at Harvard University and has written or co-written some chess papers:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noam Elkies. On numbers and endgames: Combinatorial game theory in chess endgames. arXiv:math/9905198 [math.CO] &quot;Games of No Choice&quot; (Proceedings of July 1994 MSRI conference on combinatorial games), MSRI Publ. #29 (1996) via CUP, pp.135-150.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noam Elkies. Higher Nimbers in pawn endgames on large chessboards. More Games of No Chance (R.J.Nowakowski, ed.; MSRI Publ. #42, 2002 via CUP, pp.61-78.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Noam Elkies and Richard Stanley. The mathematical knight. Math. Intelligencer 25 #1 (2003), pp.22-34.&lt;/p&gt;&#xA;" OwnerUserId="5020" LastActivityDate="2018-08-16T12:54:49.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7608" PostTypeId="2" ParentId="7606" CreationDate="2018-08-16T14:13:22.367" Score="9" Body="&lt;p&gt;A well-known example is the work of &lt;a href=&quot;https://en.wikipedia.org/wiki/Marjorie_Rice&quot; rel=&quot;noreferrer&quot;&gt;Marjorie Rice&lt;/a&gt; on pentagonal tilings of the plane.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipeidia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In December 1975, Rice came across a Scientific American article on tessellations. Despite having only a high-school education, she began devoting her free time to discovering new pentagonal tilings, ways to tile the plane using pentagons. She developed her own system of notation to represent the constraints on and relationships between the sides and angles of the polygons and used it to discover four new types of tessellating pentagons and over sixty distinct tessellations by pentagons by 1977. Rice's work was eventually examined by mathematics professor Doris Schattschneider, who deciphered the unusual notation and formally announced her discoveries to the mathematics community. Schattschneider has lauded Rice's work as an exciting discovery by an amateur mathematician.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That Wikipedia page links to a &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_amateur_mathematicians&quot; rel=&quot;noreferrer&quot;&gt;List of amateur mathematicians&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-08-16T14:13:22.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7609" PostTypeId="2" ParentId="5059" CreationDate="2018-08-16T17:02:20.407" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/heaviside-oliver&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oliver Heaviside&lt;/a&gt;'s 1893 &lt;a href=&quot;https://archive.org/details/electromagnetict01heavrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Electromagnetic Theory&lt;/em&gt; (vol. 1)&lt;/a&gt; mentions &quot;Ampere's Rule [or 'formula' or 'law'] for deriving the magnetic force from the current&quot; in a handful of places (cf. &lt;a href=&quot;https://archive.org/stream/electromagnetict01heavrich#page/64&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 64&lt;/a&gt;). He calls it &quot;Ampère's 'dodge'&quot; in his 1892 &lt;a href=&quot;https://archive.org/stream/electricalpapers01heavuoft#page/260&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Electrical Papers&lt;/em&gt; (vol. 1) p. 261&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the most curious statement by Heaviside on Ampère is in his paper &quot;&lt;strong&gt;The Mutual Action of a Pair of Rational Current-Elements&lt;/strong&gt;&quot; (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=njp.32101050973542;view=1up;seq=256&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;The Electrician&lt;/em&gt;, Dec. 28, 1888&lt;/strong&gt; (written: 25 Nov. 1888), &lt;strong&gt;p. 230&lt;/strong&gt;&lt;/a&gt; = &lt;a href=&quot;https://archive.org/stream/electricalpapers02heavuoft#page/500&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Electrical Papers&lt;/em&gt; (vol. 2), p. 501&lt;/a&gt;); Heaviside ends the short paper with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It has been stated, on no less authority than that of the great Maxwell&lt;sup&gt;[&lt;a href=&quot;https://archive.org/stream/electricandmag02maxwrich#page/n193&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Treatise&lt;/em&gt; §528&lt;/a&gt;]&lt;/sup&gt;, that Ampère's law of force between a pair of current-elements is the cardinal formula of electrodynamics. If so, should we not be always using it? Do we &lt;em&gt;ever&lt;/em&gt; use it? Did Maxwell, in his treatise? Surely there is some mistake. I do not in the least mean to rob Ampère of the credit of being the father of electrodynamics; I would only transfer the name of cardinal formula to another due to him, expressing the mechanical force on an element of a conductor supporting current in any magnetic field; the vector product of current and induction. There is something real about it; it is not like his force between a pair of unclosed elements; it is fundamental; and, as everybody knows, it is in continual use, either actually or virtually (through electromotive force) both by theorists and practicians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-08-16T17:12:07.377" LastActivityDate="2018-08-16T17:12:07.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7610" PostTypeId="2" ParentId="7605" CreationDate="2018-08-16T21:46:49.653" Score="6" Body="&lt;p&gt;To the title question: In E393 (&lt;a href=&quot;https://archive.org/stream/novicommentariia141impe#page/129&quot; rel=&quot;nofollow noreferrer&quot;&gt;1770&lt;/a&gt;, p. 167) and again E499 (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6966w/f37&quot; rel=&quot;nofollow noreferrer&quot;&gt;1780&lt;/a&gt;, §§6, 7, 10) — reprinted in vol. 4 of his &lt;em&gt;Integral Calculus&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=aoM_AAAAcAAJ&amp;amp;pg=PA154&quot; rel=&quot;nofollow noreferrer&quot;&gt;1794&lt;/a&gt;) — Euler integrated the &lt;a href=&quot;https://books.google.com/books?id=ufOYJWfhyeMC&amp;amp;ots=Dp5toWA_RS&amp;amp;pg=PA195&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier&lt;/a&gt; &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k433675h/f411&quot; rel=&quot;nofollow noreferrer&quot;&gt;series&lt;/a&gt;&#xA;$$&#xA;\cot\varphi =2\bigl(\sin2\varphi+\sin4\varphi+\sin6\varphi+\sin8\varphi+\text{etc.}\bigr)&#xA;$$&#xA;term by term twice to obtain&#xA;$$&#xA;\log(\sin\varphi)=-\log2 - \cos2\varphi - \tfrac12\cos4\varphi - \tfrac13\cos6\varphi - \tfrac14\cos8\varphi - \text{etc.}&#xA;$$&#xA;and&#xA;\begin{align}&#xA;\int_0^{\pi/2}\log(\sin\varphi)\,d\varphi &amp;amp;= \Bigl[-\varphi\log2 - \tfrac12\sin2\varphi - \tfrac18\sin4\varphi - \tfrac1{18}\sin6\varphi - \tfrac1{32}\sin8\varphi-\text{etc.}\Bigr]_0^{\pi/2}\\&#xA;&amp;amp;=\tfrac\pi2\log\tfrac12.&#xA;\end{align}&#xA;Koyama and Kurokawa (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=2111930&quot; rel=&quot;nofollow noreferrer&quot;&gt;2005&lt;/a&gt;) call this “not tricky contrary to the usual explanation.” E393 has a second proof that I haven’t deciphered. Later proofs by real methods:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Poisson (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k433675h/f406&quot; rel=&quot;nofollow noreferrer&quot;&gt;1823&lt;/a&gt;, p. 489): derive $\int_0^{\pi/2}\cos^a x\cos ax\,dx=\frac{\pi}{2^{a+1}}$ with respect to $a$ at $a=0$.&lt;/li&gt;&#xA;&lt;li&gt;Ellis (&lt;a href=&quot;https://archive.org/stream/mathematicalothe00ellirich#page/124&quot; rel=&quot;nofollow noreferrer&quot;&gt;1841&lt;/a&gt;): recognize the log of $\prod_{k=1}^n\sin^2\bigl(\frac{k}{n}\frac{\pi}2\bigr)=\frac{n}{4^{n-1}}$ in a Riemann sum for the integral.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.archive.org/stream/catalogueofscien02royarich#page/936&quot; rel=&quot;nofollow noreferrer&quot;&gt;Goodwin&lt;/a&gt; (&lt;a href=&quot;https://archive.org/stream/cambridgeanddub08unkngoog#page/n180&quot; rel=&quot;nofollow noreferrer&quot;&gt;1843&lt;/a&gt;): your proof, also found in Grunert (&lt;a href=&quot;https://archive.org/stream/archivdermathem12unkngoog#page/n131&quot; rel=&quot;nofollow noreferrer&quot;&gt;1844&lt;/a&gt;), Todhunter (&lt;a href=&quot;https://archive.org/stream/atreatiseoninte02todhgoog#page/n67&quot; rel=&quot;nofollow noreferrer&quot;&gt;1857&lt;/a&gt;), Bierens de Haan (&lt;a href=&quot;https://archive.org/stream/tablesdintegral00haan#page/431&quot; rel=&quot;nofollow noreferrer&quot;&gt;1858&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/verhandelingende08koni#page/256&quot; rel=&quot;nofollow noreferrer&quot;&gt;1862&lt;/a&gt;), etc. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Nahin’s assertion that any book “claimed the need for complex methods” sounds like a straw man. E.g. I see no such claim near the contour integration proofs of Cauchy (&lt;a href=&quot;https://archive.org/stream/mmoiresurlesin00cauc#page/45&quot; rel=&quot;nofollow noreferrer&quot;&gt;1825&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/bub_gb_BmOo00FGCuAC#page/n233&quot; rel=&quot;nofollow noreferrer&quot;&gt;1826&lt;/a&gt;) (he attributes the formula to Euler), Lindelöf (&lt;a href=&quot;https://archive.org/stream/lecalculdesrsid00lindgoog#page/n61&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;), Ahlfors (&lt;a href=&quot;https://archive.org/stream/AhlforsL.V.ComplexAnalysis3rdEd.McGrawHill1979/AhlforsL.V.ComplexAnalysis3rdEd.Mcgraw-hill1979#page/n170&quot; rel=&quot;nofollow noreferrer&quot;&gt;1979&lt;/a&gt;),...&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-08-20T16:09:48.117" LastActivityDate="2018-08-20T16:09:48.117" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7611" PostTypeId="2" ParentId="7606" CreationDate="2018-08-17T07:52:24.730" Score="16" Body="&lt;p&gt;A case from this year is that of Aubrey de Grey.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Aubrey de Grey, a biologist known for his claims that &lt;a href=&quot;https://www.ted.com/talks/aubrey_de_grey_says_we_can_avoid_aging&quot; rel=&quot;noreferrer&quot;&gt;people alive today will live to the age of 1,000&lt;/a&gt;, posted a paper to the scientific preprint site arxiv.org with the title “&lt;a href=&quot;https://arxiv.org/abs/1804.02385&quot; rel=&quot;noreferrer&quot;&gt;The Chromatic Number of the Plane Is at Least 5.&lt;/a&gt;” In it, he describes the construction of a unit-distance graph that can’t be colored with only four colors. The finding represents the first major advance in solving the problem since shortly after it was introduced. “I got extraordinarily lucky,” de Grey said. “It’s not every day that somebody comes up with the solution to a 60-year-old problem.”&lt;/p&gt;&#xA;&lt;p&gt;De Grey appears to be an unlikely mathematical trailblazer. He is the co-founder and chief science officer of an organization that aims to develop technologies for “reversing the negative effects of aging.”&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/&quot; rel=&quot;noreferrer&quot;&gt;https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The excellent &lt;a href=&quot;https://www.quantamagazine.org/mathematics/&quot; rel=&quot;noreferrer&quot;&gt;Quanta Magazine&lt;/a&gt; likes to report about such exciting happenings in the world of science. They have covered those mentioned so far except Kurt Heegner:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/&quot; rel=&quot;noreferrer&quot;&gt;Marjorie Rice&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.quantamagazine.org/yitang-zhang-proves-landmark-theorem-in-distribution-of-prime-numbers-20130519/&quot; rel=&quot;noreferrer&quot;&gt;Yitang Zhang&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Then there other cases of professionals who, like Yitang Zhang, count as outsiders:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As he was brushing his teeth on the morning of July 17, 2014, &lt;strong&gt;Thomas Royen&lt;/strong&gt;, a little-known retired German statistician, suddenly lit upon the proof of a famous conjecture at the intersection of geometry, probability theory and statistics that had eluded top experts for decades.&lt;/p&gt;&#xA;&lt;p&gt;Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 1950s, was posed in its most elegant form in 1972 and has held mathematicians in its thrall ever since.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quantamagazine.org/statistician-proves-gaussian-correlation-inequality-20170328/&quot; rel=&quot;noreferrer&quot;&gt;https://www.quantamagazine.org/statistician-proves-gaussian-correlation-inequality-20170328/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7364" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-08-17T07:52:24.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7614" PostTypeId="1" CreationDate="2018-08-20T16:14:14.517" Score="2" ViewCount="694" Body="&lt;p&gt;When we have a bilinear symmetric/ bilinear anti-symmetric/hermitian form $b$ on a real/complex vector space $V$, one can consider the group of invertible matrices $A \in GL(V)$ which respect $b$, that is&#xA;$$b(Av, Aw) = b(v,w).$$&#xA;We get respectively the orthogonal, symplectic and unitary groups, denoted $O(V,b), Sp(V,b), U(V,b)$ respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If we consider instead the corresponding &lt;em&gt;similitudes&lt;/em&gt;, i.e. invertible matrices $A \in GL(V)$ satisfying&#xA;$$\exists \lambda \in k^{\times},\; b(Av, Aw) = \lambda \cdot b(v,w)$$ (with $k=\Bbb R$ or $\Bbb C$), we obtain the subgroups usually denoted&#xA;$$GO(V,b), GSp(V,b), GU(V,b)$$ respectively.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My question is to know what the letter $G$ stands for, in the notation of these three subgroups.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While the $G$ in $GL(V)$ denotes the &quot;general&quot; linear group, I don't think this has the same origin for the similitude group. Neither the French nor the German translations of &lt;em&gt;similitude&lt;/em&gt; begin with &quot;g&quot;. Notice that writing $SO(V,b)$ would be confusing, because we already have this notation for &quot;special orthogonal&quot; group.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="3454" LastActivityDate="2018-08-20T16:14:14.517" Title="What does the &quot;G&quot; for the similitude groups stand for?" Tags="|terminology|notation|group-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7615" PostTypeId="1" CreationDate="2018-08-21T11:23:57.497" Score="-2" ViewCount="70" Body="&lt;p&gt;Can you explain if einstein's M=C refers to the rest mass of the electron which iss turned into energy by annihilation  or to the relativistic/ electromagnetic mass or both?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since at the time positron, antimatter and annihilation was not known I would say that it referred only to the electromagnetic one,   what is the truth?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2018-08-21T20:40:22.890" Title="Was the equivalence of mass-energy referred to the rest or kinetic mass?" Tags="|energy|" AnswerCount="1" CommentCount="3" ClosedDate="2018-08-22T08:41:21.157" ContentLicense="CC BY-SA 4.0" />
  <row Id="7616" PostTypeId="2" ParentId="7615" CreationDate="2018-08-21T20:40:22.890" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Albert Einstein was only 26 (the year 1905)when he published the brief, 3-page article that announced the equivalence between mass and energy, known today as E=mc2 .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the original paper, Einstein uses V instead of c for the speed of light, and L instead of E for energy. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Today's world famous formula is simply explained in words. Anyway, the message is fascinating: &lt;em&gt;as the speed of light is constant, the energy inherent to a body is proportional to its mass, with a huge constant of proportionality (c2 = 90 billion joules in each kilogram of mass).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Remarkably, Einstein proposed an experiment to test his daring theory - and this is where good science can be instantly recognised.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery opened vast new horizons for physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an example, Arthur Eddington - a forefront supporter of Einstein's theories - realised in 1920 that the mass difference between four hydrogen atoms and one helium atom would provide enough energy to power our Sun, thus solving one of the major physics puzzles of that time.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although Einstein achieved a definite conceptual advance in equating the mass of an object with its total energy content—whether or not it is moving, whether or not it has an electromagnetic field—&lt;strong&gt;we can also credit Hasenöhrl&lt;/strong&gt; for unambiguously recognizing that heat itself possess an equivalent mass, and physicists before him for providing a chain of shoulders on which to stand. E = mc^2 is the short punch line to a long and winding scientific story. for details see-&lt;a href=&quot;https://www.scientificamerican.com/article/was-einstein-the-first-to-invent-e-mc2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scientificamerican.com/article/was-einstein-the-first-to-invent-e-mc2/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;see-&lt;a href=&quot;https://www.euro-fusion.org/news/detail/detail/News/discovery-of-emc2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.euro-fusion.org/news/detail/detail/News/discovery-of-emc2/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7635" LastActivityDate="2018-08-21T20:40:22.890" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7617" PostTypeId="1" CreationDate="2018-08-22T01:14:06.963" Score="0" ViewCount="237" Body="&lt;p&gt;It is not at all obvious that atoms exist and there's no reason why there should be an indivisible unit of matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there an experiment that was conducted which gave strong evidence for the existence of atoms?&lt;/p&gt;&#xA;" OwnerUserId="2656" LastActivityDate="2018-08-22T08:35:21.297" Title="The Discovery of Atoms" Tags="|atomic-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7618" PostTypeId="1" CreationDate="2018-08-21T22:48:38.433" Score="2" ViewCount="234" Body="&lt;p&gt;I am starting a PhD on childbirth in the 18th century and was scheduled to meet a doctor of medicine who specialised in 18th medicine and history. However, this has fallen through and I really wanted to talk to someone who has more knowledge in the field to fill in the gaps in my own ideas around the subject.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone on here know anything about 18th century childbirth? Namely, about the medicinal side of things? Or, maybe, point me in the direction of someone who could help?&lt;/p&gt;&#xA;" OwnerDisplayName="Sarah Parkins" LastEditorUserId="4251" LastEditDate="2021-09-09T06:08:12.223" LastActivityDate="2021-10-09T07:00:40.773" Title="18th century childbirth — does anyone know anything about the tools or process?" Tags="|medicine|18th-century|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7619" PostTypeId="2" ParentId="7617" CreationDate="2018-08-22T07:17:46.913" Score="2" Body="&lt;p&gt;Consult the &lt;a href=&quot;https://en.wikipedia.org/wiki/Atom#First_evidence-based_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page for &lt;em&gt;atom&lt;/em&gt;&lt;/a&gt;. In its historical introduction, it presents a ladder of theories and discoveries establishing our current knowledge:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First evidence-based theory&lt;/li&gt;&#xA;&lt;li&gt;Brownian motion&lt;/li&gt;&#xA;&lt;li&gt;Discovery of the electron&lt;/li&gt;&#xA;&lt;li&gt;Discovery of the nucleus&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;French physicist Jean Perrin used Einstein's work to experimentally determine the mass and dimensions of atoms, thereby conclusively verifying Dalton's atomic theory.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Atom#Brownian_motion&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Atom#Brownian_motion&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;in his studies of the Brownian motion of minute particles suspended in liquids, [Jean Perrin] verified Albert Einstein’s explanation of this phenomenon and thereby confirmed the atomic nature of matter (sedimentation equilibrium).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The discovery of the nucleus by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geiger-Marsden experiment&lt;/a&gt; was another very important step.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1909, Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, bombarded a metal foil with alpha particles to observe how they scattered. They expected all the alpha particles to pass straight through with little deflection, because Thomson's model said that the charges in the atom are so diffuse that their electric fields could not affect the alpha particles much. However, Geiger and Marsden spotted alpha particles being deflected by angles greater than 90°, which was supposed to be impossible according to Thomson's model. To explain this, Rutherford proposed that the positive charge of the atom is concentrated in a tiny nucleus at the center of the atom.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Atom#Discovery_of_the_nucleus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Atom#Discovery_of_the_nucleus&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7364" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-08-22T08:35:21.297" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7620" PostTypeId="1" AcceptedAnswerId="7621" CreationDate="2018-08-22T09:59:02.503" Score="2" ViewCount="249" Body="&lt;p&gt;As Hans Lipperperhey had made a decent telescope albeit worse telescope before Galileo, &lt;a href=&quot;http://galileo.rice.edu/sci/instruments/telescope.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;, would he have been able to theoretically view the moons of Jupiter before Galileo if he had looked?&lt;/p&gt;&#xA;" OwnerUserId="7762" LastEditorUserId="4251" LastEditDate="2020-05-15T00:24:59.560" LastActivityDate="2020-05-15T10:38:42.370" Title="Could someone have discovered the moons of Jupiter before Galileo?" Tags="|astronomy|galileo|telescope|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7621" PostTypeId="2" ParentId="7620" CreationDate="2018-08-22T12:33:47.147" Score="1" Body="&lt;p&gt;Given that it's reasonably easy to see the 4 major moons with a 6 to 8- power binoculars, (you can find hundreds of discussions of planet-gazing with binocs online) it is quite reasonable to suppose that a patient, skilled observer with a 3 or 4 X telescope could have observed these moons.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Keep in mind that Lipperperhey, or users of his 'scope, would have had not only to see the moons, but make enough observations to recognize that they are orbiting Jupiter rather than being other (perhaps even stellar) sky objects.  &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-08-22T12:33:47.147" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7622" PostTypeId="1" AcceptedAnswerId="7796" CreationDate="2018-08-22T19:58:46.977" Score="1" ViewCount="1384" Body="&lt;p&gt;Where can I find Dirichlet's proof of the convergence of Fourier series?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastEditorUserId="4251" LastEditDate="2021-01-25T21:12:17.687" LastActivityDate="2021-01-25T21:12:17.687" Title="Dirichlet's Proof of the Convergence of Fourier Series" Tags="|mathematics|real-analysis|harmonic-analysis|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7623" PostTypeId="2" ParentId="7618" CreationDate="2018-08-22T22:25:00.637" Score="1" Body="&lt;p&gt;There are very knowledgable people on the c18-L and MEDMED-L listserves. You have to sign up on the web pages Google sends you to.  The manager of MEDMED-L is Monica Green, monica.green@asu.edu, a professor of history at Arizona State; she might be able to answer your questions off the top of her head. Also try H-WOMEN H-NET, the same way.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2018-08-22T22:25:00.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7624" PostTypeId="2" ParentId="7622" CreationDate="2018-08-22T22:28:18.803" Score="3" Body="&lt;p&gt;The first line and footnote of &lt;a href=&quot;https://hsm.stackexchange.com/a/6827&quot;&gt;this answer&lt;/a&gt; link Dirichlet’s paper (in French) as well as a translation and expositions in English and German.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-08-22T22:28:18.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7625" PostTypeId="2" ParentId="7618" CreationDate="2018-08-23T00:10:53.217" Score="1" Body="&lt;p&gt;Here I am quoting some interesting  episodes of earlier  researchers:&#xA;An example-&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When a birth began to go horribly wrong, a cesarean section was the last option.  There is one story of an American woman who begged for a C-section when she decided her death was imminent. He put her to sleep with a large dose of laudanum, opened her up and took out a baby girl, as well as his wife's ovaries in the hope that she would, &quot;not be subjected to such an ordeal again.&quot; He then stitched her up and she survived the whole ordeal by becoming the first person to survive a C-section in the United States.&#xA;  &lt;a href=&quot;http://georgianaduchessofdevonshire.blogspot.com/2010/03/joys-of-prenancy.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://georgianaduchessofdevonshire.blogspot.com/2010/03/joys-of-prenancy.html&lt;/a&gt;&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://18thcenturydiary.org.uk/childbirth-2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://18thcenturydiary.org.uk/childbirth-2/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To fulfil the study I found primary and secondary sources that helped me to better understand what it was to be pregnant in 18th century Britain. Knowing that women did not have a secured role in the British society I found it interesting to try and show how the elementary and natural act of pregnancy in a way granted them a part to play in society. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the article “Explaining the Rise in Marital Fertility in England in the ‘Long’ Eighteenth Century” by E. A. Wrigley and the book Or, A Treatise on the Prevention and Cure of Diseases by Regimen and Simple Medicines, William Buchan , 1785 I learned that medicine wondered and evolved around pregnant women and their state of pregnancy to secure the life of both the mother and the unborn child. In the articles “Maternal Health in the English Aristocracy Myths and Realities 1790-1840” by Judith Schneid Lewis and “Childbearing and Female Bonding in Early Modern England” by Linda A. Pollock I learned how the state of pregnancy permitted women to create a female bond and a social group that pressed with all its weigh in the British society. And finally in the article “Illegitimacy in Eighteenth-Century Westminster” by Nicholas Rogers I learned how society organized around illegitimate pregnancies and how the rise of illegitimacy can be a sign of a sexual “revolution” for women.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://18centurybodies.wordpress.com/2013/06/05/the-state-of-pregnancy-and-the-role-of-women-in-18th-century-british-society/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://18centurybodies.wordpress.com/2013/06/05/the-state-of-pregnancy-and-the-role-of-women-in-18th-century-british-society/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastActivityDate="2018-08-23T00:10:53.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7626" PostTypeId="1" CreationDate="2018-08-23T02:01:58.927" Score="9" ViewCount="341" Body="&lt;p&gt;I'm interested in the birth / death / life cycle of scientific fields over time, and looking for quantitative metrics that suggest whether a particular scientific field is in decline. A simple and common metric (from the field of bibliometrics) is the number of publications in a given time period, but I'd like to delve deeper: e.g., consider to what degree researchers in a subfield are building on each others' work, versus pursuing independent and idiosyncratic lines of inquiry that don't materially engage with others. (I'm focusing on fields in social science, in case the context matters.) I have two specific questions:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Suppose I have a database of publications and citations for a particular subfield over a given time period. What network measure would allow me to show how cohesive the field is, as measured by the citation network? (My best guess is the distribution of the cardinality of &lt;a href=&quot;https://en.wikipedia.org/wiki/Connected_component_(graph_theory)&quot; rel=&quot;noreferrer&quot;&gt;connected components&lt;/a&gt;; are there other metrics to consider?) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What other papers should I read, that have measured the decline and death of scientific fields? I'm not the first person to look for metrics on the rise and decline of scientific fields – but all the papers I've found (e.g., &lt;a href=&quot;https://www.researchgate.net/publication/262987677_Growth_rates_of_modern_science_A_bibliometric_analysis_based_on_the_number_of_publications_and_cited_references_Growth_Rates_of_Modern_Science_A_Bibliometric_Analysis_Based_on_the_Number_of_Publicatio&quot; rel=&quot;noreferrer&quot;&gt;this one&lt;/a&gt;) focus mainly on the number of publications in a time period. This seems well suited for establishing that a subfield is well on the way to death, but less good for measuring 'warning signs', e.g., in a field that may be in declining health even as the number of papers remains roughly constant. Are there papers that have used network or bibliometric measures to look for signs of decline, using metrics other than sheer number of publications?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;If there is another forum/community that matches the question better, please let me know. &lt;/p&gt;&#xA;" OwnerUserId="7766" LastActivityDate="2023-01-24T10:47:45.113" Title="Quantitative measures of rise and decline of scientific fields over time (beyond number of publications)" Tags="|academia|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7627" PostTypeId="1" AcceptedAnswerId="7631" CreationDate="2018-08-23T06:14:15.047" Score="1" ViewCount="131" Body="&lt;p&gt;The positron was discovered in 1932, did physicist have any knowledge or  assumed that matter could be annihilated into energy at the start of XX century?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2018-09-24T04:22:17.283" Title="Was annihilation considered in 1905?" Tags="|energy|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7628" PostTypeId="1" AcceptedAnswerId="7629" CreationDate="2018-08-23T11:01:17.183" Score="2" ViewCount="282" Body="&lt;p&gt;In Buhler's biography of Gauss (Gauss: A Biographical Study), at the chapter on modular forms and hypergeometric series, he mentions a function that Gauss called &quot;&lt;strong&gt;Summatorische Function&lt;/strong&gt;&quot;, which he used implicitly without giving an explicit definition to it. According to Buhler, &quot;it is in fact the absolute invariant of the modular group of all linear substitutions $t' = \frac {{\alpha t - i\beta}}{{\delta + i\gamma t}}$ where $\alpha\delta - \beta\gamma = 1$ and $\alpha,\beta,\gamma, \delta$ are integers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, my first question is how is this function interpreted from a modern point of view?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My other questions are also relevant to this post. In Wolfram MathWorld article on &quot;j-Function&quot; appears the fact that &quot;Gauss was apparently aware of the j-function before 1800&quot;. Also, the other name of the j-function (&quot;Klein's absolute invariant&quot;) suggests there might be a connection with the Summatorische function that Buhler mentions. So, where in Gauss's nachlass is the relevant work on the j function? and is it connected with the Summatorische function? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since modular forms are a very advanced mathematical topic (i think mathematicians begin to learn about it only in their second degree), obviously i wasn't able to understand the answer to my question by myself.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-22T07:24:11.473" LastActivityDate="2019-10-22T07:24:11.473" Title="What is the modern interpretation of Gauss's &quot;Summatorische Function&quot;?" Tags="|biographical-details|complex-analysis|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7629" PostTypeId="2" ParentId="7628" CreationDate="2018-08-23T12:22:46.813" Score="2" Body="&lt;p&gt;According to Klein (Lectures on history of mathematics in 19 century),&#xA;he absolute invariant $J$ was introduced by Gauss in his manuscript &quot;On summatory function&quot;, which is reproduced  on p. 386 of volume III of Gauss collected works.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-08-23T12:22:46.813" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7630" PostTypeId="2" ParentId="7626" CreationDate="2018-08-24T11:08:36.207" Score="1" Body="&lt;ol&gt;&#xA;&lt;li&gt;I would look for measures of robusteness of (connected components of) the network. The number of connected components itself is probably a very rough measure. What happens if you randomly delete a small number of knots from your network? How much properties of the whole network are shared with large subnetworks?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2018-08-24T11:08:36.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7631" PostTypeId="2" ParentId="7627" CreationDate="2018-08-24T16:46:25.787" Score="2" Body="&lt;p&gt;&lt;strong&gt;Let us consider the following info on developments prior to 1905.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Before 1905 (The forthcoming of &lt;span class=&quot;math-container&quot;&gt;$E = mc^2$&lt;/span&gt;)&#xA;&lt;strong&gt;1881&lt;/strong&gt; J. J. Thompson proposed that a charged conductor in motion increases its mass by &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{15} e^2/a$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;1904&lt;/strong&gt; H. A. Lorentz proposed that &lt;span class=&quot;math-container&quot;&gt;$m_L = m_0 (1 – v^2/c^2)^{-3/2}$&lt;/span&gt; based on deformable spherical charge.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;1904&lt;/strong&gt; Hasenöhrl derived an apparent mass increase of a moving cavity containing electromagnetic energy E, obtaining  &lt;span class=&quot;math-container&quot;&gt;$m_e = \frac{8}{3} E/c^2$&lt;/span&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;1905 On Abraham’s suggestion, Hasenöhrl corrected this to  &lt;span class=&quot;math-container&quot;&gt;$m_e = \frac{4}{3} E/c^2$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Based on the study of the historical development of concepts, one may deduce that the equation &lt;span class=&quot;math-container&quot;&gt;$E = mc^2$&lt;/span&gt; would have been discovered by other physicists sooner or later. However, it is likely that Einstein merely accelerated the development.&#xA;ref.- &gt;&lt;a href=&quot;https://files.eric.ed.gov/fulltext/EJ848449.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://files.eric.ed.gov/fulltext/EJ848449.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;our contention is that though annihilation of particles was not an acceptable concept prior to 1905 but other ways of energy conversion of mass was being attempted.&lt;/p&gt;&#xA;" OwnerUserId="7635" LastEditorUserId="3869" LastEditDate="2018-09-24T04:22:17.283" LastActivityDate="2018-09-24T04:22:17.283" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7633" PostTypeId="1" CreationDate="2018-08-22T13:15:08.193" Score="4" ViewCount="524" Body="&lt;p&gt;Reading an article I have stumbled across the concept of &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_excluded_middle&quot; rel=&quot;noreferrer&quot;&gt;law of excluded middle&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia mentions that original expression is &lt;code&gt;principium tertii exclusi&lt;/code&gt; which literally translates to &lt;code&gt;principle of the excluded third&lt;/code&gt;. My native language (Romanian) also uses the literally translation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering where does middle come from.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is the origin of &quot;law of excluded middle&quot;? I specifically refer to &lt;code&gt;middle&lt;/code&gt; instead of &lt;code&gt;third&lt;/code&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="Alexei" LastActivityDate="2018-08-24T22:47:31.747" Title="What is the origin of &quot;law of excluded middle&quot;?" Tags="|terminology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7634" PostTypeId="2" ParentId="7633" CreationDate="2018-08-22T19:57:57.877" Score="2" Body="&lt;p&gt;My simple intuition is that if you assume a bivalent logic system, otherwise called two-valued logic system, it means that every proposition is either true or not true, and there is no &quot;middle&quot; ground, so to speak, or a third option, what you called &quot;the excluded third.&quot;&lt;br&gt; &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Principle_of_bivalence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principle of bivalence&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as I can tell Aristotle (3rd century BC) first wrote about this. In a slightly different way he writes about &quot;law of non-contradiction&quot;, basically meaning that something cannot be and not be at the same time.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And it will not be possible to be and not to be the same thing, except&#xA;  in virtue of an ambiguity, just as if one whom we call &quot;man&quot;, and&#xA;  others were to call &quot;not-man&quot;; but the point in question is not this,&#xA;  whether the same thing can at the same time be and not be a man in&#xA;  name, but whether it can be in fact.&lt;br&gt;&#xA;  &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_excluded_middle#Aristotle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle and excluded middle&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This idea, that something cannot both be and not be is expressed in logical notation as ¬(P ∧ ¬P), said &quot;not (P AND NOT P)&quot;. Basically something cannot be and not be at the same time. If this is true, and we a bivalent logic where every proposition is either true or false, then ¬(P ∧ ¬P) leads to (P ∨ ¬P), meaning something has to either be or not be.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle specifically refers to the &quot;intermediate&quot;. He says 'intermediate among two contradictories.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He then proposes that &quot;there cannot be an intermediate between&#xA;  contradictories, but of one subject we must either affirm or deny any&#xA;  one predicate&quot; (Book IV, CH 7, p. 531). In the context of Aristotle's&#xA;  traditional logic, this is a remarkably precise statement of the law&#xA;  of excluded middle, P ∨ ¬P.&lt;br&gt;&#xA;  &lt;a href=&quot;http://He%20then%20proposes%20that%20%22there%20cannot%20be%20an%20intermediate%20between%20contradictories,%20but%20of%20one%20subject%20we%20must%20either%20affirm%20or%20deny%20any%20one%20predicate%22%20(Book%20IV,%20CH%207,%20p.%20531).%20In%20the%20context%20of%20Aristotle&amp;#39;s%20traditional%20logic,%20this%20is%20a%20remarkably%20precise%20statement%20of%20the%20law%20of%20excluded%20middle,%20P%20%E2%88%A8%20%C2%ACP.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristotle and law of excluded middle&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So as to why it's called the excluded middle, it may be because Aristotle called it the disallowed &quot;intermediate between contradictories&quot;, I'm not sure about this. The term makes intuitive sense to me, if &quot;true&quot; is on one side, and &quot;false&quot; is on the other, neither true nor false would fall in between, the middle, and it's excluded, presuming you accept a two-value logic system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article goes on to Leibniz:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Every judgment is either true or false&quot;&lt;br&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And Bertrand Russell: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;Law of excluded middle: &quot;Everything must either be or not be.&quot;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, if I may make a correction, Wikipedia doesn't mention that the original expression is principium tertii exclusi, it says it's &quot;also known as&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The law is also known as the law (or principle) of the excluded third,&#xA;  in Latin principium tertii exclusi.&lt;br&gt;&#xA;  &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_excluded_middle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Law of excluded middle Wikipedia article&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It was originally written, or translated, I should say, from Aristotle, being &quot;intermediate between contradictories&quot;.&lt;/p&gt;&#xA;" OwnerDisplayName="Zebrafish" LastActivityDate="2018-08-23T08:01:46.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7635" PostTypeId="2" ParentId="7633" CreationDate="2018-08-23T19:24:12.210" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://oed.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;OED&lt;/em&gt;&lt;/a&gt; attests 'excluded middle, third' (in the entry for &lt;strong&gt;excluded&lt;/strong&gt;, &lt;em&gt;adj.&lt;/em&gt;, .... &lt;strong&gt;b.&lt;/strong&gt;) from &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=hvd.hnuirt;view=1up;seq=317&quot; rel=&quot;nofollow noreferrer&quot;&gt;1849, in William Thomson's &lt;em&gt;An Outline of the Necessary Laws of Thought&lt;/em&gt;&lt;/a&gt;. Although &lt;em&gt;OED&lt;/em&gt; does not give a quote, merely the citation, the relevant text is this: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;3d Criterion. &lt;em&gt;The principle of the middle being excluded&lt;/em&gt;, (lex exclusi medii.) &quot;Either a given judgement must be true of any subject, or its contradictory; there is no middle course.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I did not find any earlier work containing the phrase &lt;em&gt;lex exclusi medii&lt;/em&gt;. Some earlier works contain the phrase &quot;excluded third&quot;. Among them, the &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015068388480;view=1up;seq=21&quot; rel=&quot;nofollow noreferrer&quot;&gt;1815 &lt;em&gt;Encyclopaedia Londinensis&lt;/em&gt;&lt;/a&gt; mentions the &quot;excluded third&quot; (top of second column): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;&lt;em&gt;The Principle of the Excluded Third, (principium exclusi medii inter duo contradictoria,)&lt;/em&gt; upon which the (logical) &lt;em&gt;necessity&lt;/em&gt; of a Knowledge is grounded; namely, that we must judge so, and not otherwise; i.e. that the opposite is false; for &lt;em&gt;Apodictical Judgements&lt;/em&gt;.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Together, the context of these uses suggests that early appearances in English of both &quot;the principle of the excluded third&quot; and &quot;the principle of the excluded middle&quot; were associated with the Latin &lt;em&gt;exclusi medii&lt;/em&gt;, rather than the Latin &lt;em&gt;exclusi tertii&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The account represented in the 1815 &lt;em&gt;Encyclopaedia Londinensis&lt;/em&gt; is drawn from an 1813 translation of Immanuel Kant's &lt;em&gt;Logic&lt;/em&gt; by Thomas Wirgman (see &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015068388480;view=1up;seq=15;size=175&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 3, op. cit.&lt;/a&gt;). The Latin rendering of the fuller phrase, &lt;em&gt;principium exclusi medii inter duo contradictoria&lt;/em&gt;, may have been drawn from Kant directly, or from a Latin translation of Kant.&lt;/p&gt;&#xA;" OwnerUserId="7775" OwnerDisplayName="JEL" LastActivityDate="2018-08-23T19:42:58.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7636" PostTypeId="1" CreationDate="2018-08-26T13:46:02.110" Score="1" ViewCount="105" Body="&lt;p&gt;I am looking for a book on the history of mathematics that would also serve as a book on the history of physics. In the sense that the history of math is developed along with the developments in physics during the time and explaining some influences both ways. Or the other way, a book that develops the history of physics also with the developments in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many thanks in advance for all your suggestions!&lt;/p&gt;&#xA;" OwnerUserId="7780" LastActivityDate="2018-08-26T13:46:02.110" Title="Reference for Math-Physics history book" Tags="|mathematics|physics|mathematical-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7637" PostTypeId="1" AcceptedAnswerId="7638" CreationDate="2018-08-26T20:09:01.393" Score="3" ViewCount="99" Body="&lt;p&gt;Regarding &lt;em&gt;Conics&lt;/em&gt;, it is widely written, e.g. &lt;a href=&quot;http://sites.math.rutgers.edu/~cherlin/History/Papers1999/schmarge.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rutger's site&lt;/a&gt;,  that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Try as i might, I am unable to locate any information on the origin and whereabouts of the oldest extant source for &lt;em&gt;Conics&lt;/em&gt;; books 1-4 in original Greek. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Bodleian Library, Oxford has an &lt;a href=&quot;https://genius.bodleian.ox.ac.uk/exhibits/browse/conics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arabic translation of books 1 - 7&lt;/a&gt;, giving a date of 1070.  This appears to be the manuscript used by Halley to produce his own edition.  It is not clear to me currently if this is accepted as the oldest extant Arabic translation. Some sites talk of a 9th century translation, but this dating has been challenged.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, despite the many references to the older Greek copies of Books I - IV, I can find no information on their origin and whereabouts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: What is the origin and whereabouts of the oldest, extant source for Apollonius’ &lt;em&gt;Conics&lt;/em&gt;, Books I-IV?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2018-08-26T21:19:20.743" Title="Whereabouts of oldest extant source for Apollonius’ *Conics*, Books I - IV" Tags="|reference-request|ancient-greece|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7638" PostTypeId="2" ParentId="7637" CreationDate="2018-08-26T21:19:20.743" Score="6" Body="&lt;p&gt;Apollonius work have  been reedited recently at W. de Gruyter's (2008-9) &#xA;: Apollonius de Perge, &lt;em&gt;Coniques&lt;/em&gt;,&#xA;Texte grec et arabe etabli, traduit et commenté&#xA;Ed. by &lt;a href=&quot;https://en.wikipedia.org/wiki/Roshdi_Rashed&quot; rel=&quot;noreferrer&quot;&gt;Roshdi Rashed&lt;/a&gt;, M. Decorps-Foulquier, M. Federspiel, (Livre I. Commentaire historique et mathématique, édition et traduction du texte arabe. 1.2: Livre I: Édition et traduction du texte grec Ed. by R. Roshdi, M. Decorps-Foulquier, M. Federspiel)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roshdi in the preface (p.27) and also in a paper online about &lt;a href=&quot;https://core.ac.uk/download/pdf/39213165.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;reediting apollonius&quot;&lt;/a&gt; states that the&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Eutocius’ edition of the first four books, as it has come down to us&#xA;  in a manuscript from the 12th century, the Vaticanus. graec. 206&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2018-08-26T21:19:20.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7639" PostTypeId="1" AcceptedAnswerId="7640" CreationDate="2018-08-28T06:02:36.983" Score="7" ViewCount="221" Body="&lt;p&gt;I'm aware of work done by the Army Operational Research Group from the United Kingdom's Ministry of Supply. I know Stigler's Diet Problem came out in 1939, if I'm recalling that correctly. However, I wasn't able to find out if it had been in use during the war. I know there were some statistics groups in American universities, which supported the war effort, but I have not been able to determine, if they did any operations research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize if this belongs on the History site versus this site.&lt;/p&gt;&#xA;" OwnerUserId="7669" LastActivityDate="2020-05-01T16:48:28.650" Title="Was the United Kingdom the only faction in the Second World War that used Operations Research?" Tags="|mathematics|world-war-2|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7640" PostTypeId="2" ParentId="7639" CreationDate="2018-08-28T06:52:57.863" Score="7" Body="&lt;p&gt;Operations Research is a somewhat eclectic term. The British coined and promoted it, so in the early stages they were the only ones using it even if others were doing similar things under different names, before and after. On the US military uses there is a comprehensive monograph, &lt;a href=&quot;https://history.army.mil/html/books/hist_op_research/CMH_70-102-1.pdf&quot; rel=&quot;noreferrer&quot;&gt;History of Operations Research in the United States Army by Schrader&lt;/a&gt; (freely available), in three volumes. Schrader uses a substantive description rather than the name to trace pre-history of Operations Research all the way back to the Napoleon wars. Here are some excerpts: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The starting date for this study, 1942, was determined by the first efforts to create an OR capability in the  U.S.  Army;  the  ending  date,  1962,  was  determined  by  the  beginning  of  the  major  changes  in  Army  organization  and  procedures  instituted  by  Secretary  of  Defense  Robert  S. McNamara, notably the initiation of efforts to reorganize the  Army  along  functional  lines  and  to  consolidate  related  activities  under  major  functional  commands,  such  as  the  U.S. Army Combat Developments Command and the U.S. &#xA;  Army Materiel Command... Between 1942 and 1962, the main thrust of OR work in the U.S. Army was in fact the improvement of weapons and equipment, organiza-&#xA;  tion, tactical doctrine, and, to a lesser degree, the formulation of higher-level strategy and policy pertaining to the political, economic, and social issues facing the Army.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;The  United  States  Navy  owns  the  distinction  of  having  the  fi  rst  active  OR  group  in  the  U.S.  armed  forces:  the  Mine  Warfare  Operations  Research  Group  (MWORG),  established informally in January 1942.&#xA;  The Navy’s   early   adoption of OR was partly the result of prewar contacts with British OR groups at RAF Coastal Command and the Admiralty working on problems of naval mining, antisubmarine warfare,  and  convoy  organization.  The  work  being  done  in  Britain was directly applicable to the problems faced by the U.S. Navy immediately after the Pearl Harbor attack... MWORG was officially established as part of the U.S. Navy  Bureau  of  Ordnance’s  Re&#xA;  search  Division  on  24  June  1942,  but  it  had  existed  informally  since  late  January  1942  and  thus  merits  distinction  as  the  first  OR  organization  in  the  U.S.  armed  forces.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;MWORG  grew  out  of  work  being  done on countermeasures for German magnetic mines from December 1939 at the Naval Ordnance Laboratory (NOL) in Washington, D.C. The NOL Mine Research Unit, led from early 1940 by Ellis A. Johnson, a Carnegie Institution physicist,  focused  on  techniques  for  sweeping  for  magnetic  mines and degaussing (demagnetizing) U.S. naval and merchant vessels. In July 1940, Ralph D. Bennett, a professor of electrical engineering at MIT, was called to active duty as a  lieutenant  commander,  U.S.  Naval  Reserve  (USNR)  and  was assigned the task of expanding NOL staff, which he did by  bringing  in  accomplished  scientists  of  his  acquaintance, increasing the scientific staff at NOL from a dozen to nearly  one  thousand.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Starting in 1942 Army air, ground and service forces joined the Navy.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-08-28T06:52:57.863" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7642" PostTypeId="1" CreationDate="2018-08-28T15:52:26.847" Score="13" ViewCount="3821" Body="&lt;p&gt;What are some historical examples of theories/ideas that were initially labeled &quot;pseudoscience&quot; and later considered legitimate &quot;science&quot;? I don't mean theories or ideas that were initially not accepted or ignored and later accepted, but things that were derided as pseudoscientific (and labeled as such) but then either evidence or new theories eventually made them considered &quot;science.&quot; Something akin to the hypothetical where &quot;dowsing&quot; was suddenly given a mechanistic and experimental basis that convinced the scientific community to accept it as &quot;real science&quot; (or similarly for parapsychic powers).&lt;/p&gt;&#xA;" OwnerUserId="7496" LastEditorUserId="6609" LastEditDate="2023-01-16T16:22:27.213" LastActivityDate="2023-09-15T13:11:01.530" Title="Historical examples of &quot;pseudoscience&quot; becoming &quot;science&quot;" Tags="|philosophy-of-science|scientific-method|" AnswerCount="8" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="7643" PostTypeId="1" CreationDate="2018-08-28T17:02:57.513" Score="0" ViewCount="640" Body="&lt;p&gt;It is believed by many that Galois penned down his ideas in a letter to his friend the day before his death. He even asked his friend to take that letter to the great Gauss, not to verify its contents but to justify its relevance in that era. What did Gauss reply?&lt;/p&gt;&#xA;" OwnerUserId="7788" LastEditorUserId="4251" LastEditDate="2021-06-22T09:00:52.060" LastActivityDate="2022-10-07T20:05:45.733" Title="Did Gauss reply to Galois?" Tags="|mathematicians|gauss|19th-century|galois|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7644" PostTypeId="1" AcceptedAnswerId="7659" CreationDate="2018-08-28T17:45:59.090" Score="1" ViewCount="362" Body="&lt;p&gt;Who first solved the &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Two-body_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;two-body problem&lt;/a&gt; in 3-dimensions? Was it Laplace?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-09-01T12:04:26.803" Title="Who first solved the two-body problem in 3D?" Tags="|physics|astronomy|mathematical-physics|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7645" PostTypeId="2" ParentId="7639" CreationDate="2018-08-29T02:58:33.137" Score="6" Body="&lt;p&gt;In World War II, Andrey Kolmogorov did what you might call operations research, in Russia, for air defense  problems.  His &lt;a href=&quot;https://en.wikipedia.org/wiki/Andrey_Kolmogorov#Adulthood&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia bio article&lt;/a&gt; says &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;During World War II Kolmogorov contributed to the Russian war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of barrage balloons intended to help protect Moscow from German bombers.[22]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The cited reference is to a modern secondary pop-history work, not to an authoritative primary account, but you might be able to trace back from there.  This 1942 or 1943 &lt;a href=&quot;http://www.dtic.mil/dtic/tr/fulltext/u2/a800403.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;report by Norbert Wiener&lt;/a&gt; references this or similar work at p.11 of the pdf file.  Rodrigo  de Azevedo comments that K's air defense work is also discussed in a  History &#xA; SE &lt;a href=&quot;https://history.stackexchange.com/q/39634/17887&quot;&gt;question&lt;/a&gt;, with further references.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-05-01T16:48:28.650" LastActivityDate="2020-05-01T16:48:28.650" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7646" PostTypeId="1" CreationDate="2018-08-29T12:24:31.587" Score="4" ViewCount="456" Body="&lt;p&gt;This question is a continuation of my previously posted question: &lt;a href=&quot;https://hsm.stackexchange.com/questions/6362/was-gauss-aware-of-the-non-euclidean-implications-of-his-work-on-moduler-forms?rq=1&quot;&gt;Was Gauss aware of the non-euclidean implications of his work on moduler forms?&lt;/a&gt;, and is based on the information given in John Stillwell's book &quot;Mathematics and it's history&quot;, where it's mentioned that using his theory of modular forms Gauss discovered a tessellation of the unit disk by &quot;equilateral triangles&quot; (but curved) of angles: $\pi/4,\pi/4,\pi/4$.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My previous question was about &lt;strong&gt;how Gauss interpreted&lt;/strong&gt; his own computations with modular forms. In this question i don't wish to know if Gauss was aware of the Poincare disk model of hyperbolic geometry, but it's more about placing his calculations in the right context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From looking in his writings i gained the impression that he was quite aware of the action of the modular group in the upper-half plane (although he didn't view it as a model of the hyperbolic plane) - his nachlass contains drawings of the fundamental domain of the modular group, and he, for example, proved a remarkable result on the mutual connection of all values of the arithmetic-geometric mean of two complex numbers (see David Cox's paper: &quot;&lt;a href=&quot;https://www.researchgate.net/publication/248675540_The_Arithmetic-Geometric_Mean_of_Gauss&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;AGM of Gauss&lt;/em&gt;&lt;/a&gt;&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But i didn't find things that look like computations of tessellations of the unit disk (by curved triangle). In his commentary on p. 99-102 of the 8th volume of Gauss's werke (&quot;Three Fragments About Elliptic Modulfunctions&quot;), Fricke gives some related sources (such as one of Gauss's articles on the orbit of pallas, where he is supposed to use some of his analytic work in implicit way), and mentions that the drawing of tessellation of the unit disk was found among his writings. I guess one of the problems is that Google translate doesn't make very good transations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Since i'm not sure that p. 99-102 in volume 8 is really the place where he is supposed to make calculations about tessellations, i'm asking for reference  - if anyone who is famililar enough with these themes can help, i'll be grateful.&#xA;If someone sees a formula that resembles a formula he knows from the theory of tessellations of the unit disk in one of Gauss's writings,i'll be glad if he will mention it here.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-09-18T19:16:42.833" LastActivityDate="2020-09-18T19:16:42.833" Title="What mathematical techniques Gauss used in order to tessellate the unit disk?" Tags="|biographical-details|reference-request|geometry|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7648" PostTypeId="2" ParentId="6008" CreationDate="2018-08-29T15:27:58.160" Score="1" Body="&lt;p&gt;When &lt;a href=&quot;https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chandrasekhar&lt;/a&gt; explained that sufficiently massive white dwarf stars collapse under self-attraction and become neutron stars or black holes, &lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Eddington&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir Arthur Eddington&lt;/a&gt; rejected this possibility and his infuence was such that this idea was, for all practical purposes, suppressed for a long time. This history is very well decribed in Arthur I. Miller's &lt;em&gt;Empire of the stars&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2019-11-17T14:42:10.193" LastActivityDate="2019-11-17T14:42:10.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7649" PostTypeId="2" ParentId="7646" CreationDate="2018-08-29T17:14:59.570" Score="2" Body="&lt;p&gt;It is impossible to answer precisely what Gauss was thinking. All we have is his collected papers (this includes unpublished pieces). A good online source is&#xA;Fortunately they are all&#xA;easily available online:&#xA;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348&lt;/a&gt;.&#xA; The same edition includes extended commentaries. The topic you are asking about was commented on by Klein. If you do not read German, there is a book of Klein, Lectures on history of mathematics in 19th century, which has&#xA;a large chapter on Gauss, and this book is available in English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I understand (from Klein's lectures and other sources), Gauss knew about the action of the modular group on a half-plane,&#xA;(pictures in his manuscripts suggest this), but there is no evidence that he was aware of the interpretation of this action in terms of non-Euclidean geometry). &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-08-29T17:14:59.570" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7650" PostTypeId="2" ParentId="5299" CreationDate="2018-08-30T09:35:28.183" Score="3" Body="&lt;p&gt;Following the suggestion by @Guido Jorg I have found &lt;a href=&quot;https://www.nybooks.com/articles/2012/05/10/crisis-big-science/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; by Steven Weinberg published in The New York Review of Books on the May 10, 2012 issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's called &lt;strong&gt;The Crisis of Big Science&lt;/strong&gt; and it contains the following quotes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Then in 1992 the House of Representatives canceled funding for the SSC. Funding was restored by a House–Senate conference committee, but the next year the same happened again, and this time the House would not go along with the recommendation of the conference committee. After the expenditure of almost two billion dollars and thousands of man-years, the SSC was dead.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;One thing that killed the SSC was an undeserved reputation for over-spending. There was even nonsense in the press about spending on potted plants for the corridors of the administration building. Projected costs did increase, but the main reason was that, year by year, Congress never supplied sufficient funds to keep to the planned rate of spending. This stretched out the time and hence the cost to complete the project. Even so, the SSC met all technical challenges, and could have been completed for about what has been spent on the LHC, and completed a decade earlier.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Spending for the SSC had become a target for a new class of congressmen elected in 1992. They were eager to show that they could cut what they saw as Texas pork, and they didn’t feel that much was at stake. The cold war was over, and discoveries at the SSC were not going to produce anything of immediate practical importance. Physicists can point to technological spin-offs from high-energy physics, ranging from synchotron radiation to the World Wide Web. For promoting invention, big science in this sense is the technological equivalent of war, and it doesn’t kill anyone. But spin-offs can’t be promised in advance.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What really motivates elementary particle physicists is a sense of how the world is ordered—it is, they believe, a world governed by simple universal principles that we are capable of discovering. But not everyone feels the importance of this. During the debate over the SSC, I was on the Larry King radio show with a congressman who opposed it. He said that he wasn’t against spending on science, but that we had to set priorities. I explained that the SSC was going to help us learn the laws of nature, and I asked if that didn’t deserve a high priority. I remember every word of his answer. It was “No.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The article makes for a very interesting, if sad, read. If a personal comment is allowed, it seems to me that in the years which have elapsed since then, the picture has not improved.&lt;/p&gt;&#xA;" OwnerUserId="93" LastActivityDate="2018-08-30T09:35:28.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7651" PostTypeId="1" AcceptedAnswerId="7654" CreationDate="2018-08-30T17:35:02.160" Score="6" ViewCount="1908" Body="&lt;p&gt;Previously what is called now &quot;natural sciences&quot; was called &quot;natural philosophy&quot;. I'm interested in details, what was so wrong with the name &quot;philosophy&quot; so the name &quot;science&quot; became preferred?&lt;/p&gt;&#xA;" OwnerUserId="7339" LastActivityDate="2018-09-01T03:26:57.013" Title="When and why did people stopped using &quot;natural philosophy&quot; term and started using &quot;science&quot;?" Tags="|terminology|naming-conventions|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7652" PostTypeId="2" ParentId="7651" CreationDate="2018-08-30T20:21:54.420" Score="3" Body="&lt;p&gt;It is a gradual process. All sciences were  born from &quot;philosophy&quot;. Philosophy is just a verbal speculation about the thing we do not know exactly. When speculation on a subject reaches some degree of maturity, some truth can be discovered and proved. Then a science is born. Ancients started to speculate about nature. Then gradually, mathematics and astronomy became sciences (when they obtained correct, proved results). Astronomers and mathematicians were still called philosophers for long time.&#xA;The rest remained philosophy. At the time of Newton, mathematics was already completely separate but physics and chemistry were called &quot;natural philosophy&quot;. Then physics was born, then chemistry, etc.&#xA;So the area of phisolophy always shrinks while the area of science gradually expands.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this is a very slow process which started about 6 hundred years BC and still continues. The language lags behind the process. In 19 century one mathematician still could call another mathematician a &quot;philosopher&quot;, but then this was already somewhat unusual.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-08-30T20:21:54.420" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7654" PostTypeId="2" ParentId="7651" CreationDate="2018-08-30T21:02:55.367" Score="9" Body="&lt;p&gt;What was called &quot;natural philosophy&quot; is not what is now called &quot;science&quot;. The latter, with its specific institutions and standards, only started forming in 17th century and was not fully formed until mid 19th. While Newton still called his &lt;em&gt;opus magnum&lt;/em&gt; Mathematical Principles of Natural Philosophy in 1687, after much professionalization and institutional growth in 18th century &quot;men of science&quot; became self-conscious, and sought to distinguish their more disciplined endeavor from contemporary natural philosophy, such as Schelling's, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Naturphilosophie&quot; rel=&quot;noreferrer&quot;&gt;Naturphilosophie&lt;/a&gt;. At the time, German idealists and romantics had their own ideas as to what &quot;science&quot; should mean (recall Hegel's Science of Logic), and &quot;Newtonian science&quot; was not exactly a term of praise to them. It was to be surpassed by some kind of &quot;higher Science&quot; of a more speculative and holistic nature. This did not take.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Whewell#History_and_development_of_science&quot; rel=&quot;noreferrer&quot;&gt;Whewell's promotion of &quot;inductive sciences&quot;&lt;/a&gt; in History of the Inductive Sciences (1837) and Philosophy of the Inductive Sciences (1840) contributed to cementing the change and emphasizing the &quot;inductive method&quot;, which of course goes back to Bacon. &lt;a href=&quot;https://en.wikipedia.org/wiki/Auguste_Comte#Comte&amp;#39;s_positivism&quot; rel=&quot;noreferrer&quot;&gt;Comte distinguished &quot;positive sciences&quot;&lt;/a&gt; in The Course in Positive Philosophy around the same time (1830-1842). Whewell also coined the term &quot;scientist&quot; (and &quot;physicist&quot;), by analogy to &quot;artists&quot;, in an 1834 review of Mary Somerville’s On the Connexion of the Physical Sciences, but it did not prevail over &quot;men of science&quot; until 1870s in the US, and the Royal Society of London, the British Association for the Advancement of Science, the Royal Institution and the Cambridge University Press still rejected “scientist” as late as 1924. The British surrendered only after World War II, see &lt;a href=&quot;http://theconversation.com/how-man-of-science-was-dumped-in-favour-of-scientist-30132&quot; rel=&quot;noreferrer&quot;&gt;How ‘man of science’ was dumped in favour of ‘scientist’&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle's episteme (&quot;theoretical&quot; knowledge), translated into Latin as &lt;em&gt;scientia&lt;/em&gt;, much predates modern science of course, but one can tell that the meaning was different from the observation that what medieval scholasts were doing was called &lt;em&gt;scientia&lt;/em&gt;. Geometry was episteme, but mechanics was techne, &quot;practical&quot; knowledge. Aquinas took the scholastic formula &quot;&lt;em&gt;science means the certitude of knowledge which is gained by demonstration&lt;/em&gt;&quot; almost word for word from Averroes, an Arabic Aristotelian. It referred to intellectual study of any kind, from natural philosophy to theology, all of them mostly speculative, and &quot;demonstrations&quot; included not just Euclid's ones but also proofs of God's existence like Anselm's. Some elements of science in the modern sense, as opposed to natural philosophy, can be attributed to Hellenistic science, see e.g. &lt;a href=&quot;https://www.maa.org/press/maa-reviews/the-forgotten-revolution-how-science-was-born-in-300-bc-and-why-it-had-to-be-reborn&quot; rel=&quot;noreferrer&quot;&gt;Russo's Forgotten Revolution&lt;/a&gt;, and medieval Islamic science, but they were neither self-consciously &quot;positive&quot; nor &quot;inductive&quot; nor institutionalized. It is an attribution in retrospect.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-09-01T03:26:57.013" LastActivityDate="2018-09-01T03:26:57.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7657" PostTypeId="2" ParentId="5299" CreationDate="2018-08-31T04:25:10.140" Score="6" Body="&lt;p&gt;Fermilab director &lt;a href=&quot;http://history.fnal.gov/testimony.html&quot; rel=&quot;noreferrer&quot;&gt;R. R. Wilson’s Congressional Testimony&lt;/a&gt; (April 17, 1969, &lt;a href=&quot;https://purl.stanford.edu/yv824kn8030&quot; rel=&quot;noreferrer&quot;&gt;p. 113&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;SENATOR PASTORE.&lt;/strong&gt; Is there anything connected in the hopes of this accelerator that in any way involves the security of the country?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DR. WILSON.&lt;/strong&gt; No, sir; I do not believe so. (...) It only has to do with the respect with which we regard one another, the dignity of men, our love of culture. It has to do with those things. It has nothing to do with the military. I am sorry.&lt;/p&gt;&#xA;&lt;p&gt;(...)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;SENATOR PASTORE.&lt;/strong&gt; Is there anything here that projects us in a position of being competitive with the Russians, with regard to this race?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;DR. WILSON.&lt;/strong&gt; Only from a long-range point of view, of a developing technology. Otherwise, it has to do with: Are we good painters, good sculptors, great poets? I mean all the things that we really venerate and honor in our country and are patriotic about. In that sense, this new knowledge has all to do with honor and country but it has nothing to do directly with defending our country except to help make it worth defending.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-08-31T04:25:10.140" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7658" PostTypeId="1" AcceptedAnswerId="7661" CreationDate="2018-09-01T10:28:48.877" Score="5" ViewCount="545" Body="&lt;p&gt;It is known that Gauss studied hypergeometric equation&lt;/p&gt;&#xA;&#xA;&lt;p&gt;$$x(1-x) \dfrac {d^2y}{dx^2}+(c-(a+b+1)x)\dfrac {dy}{dx}-aby=0$$&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know something about history of this equation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) If $a=b=c=0$ then what are the solutions of corresponding equation and did they been studied before Gauss?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Why it is called &quot;hypergeometric&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) What are contributions of Gauss regarding this equation?&lt;/p&gt;&#xA;" OwnerUserId="7805" LastActivityDate="2018-09-01T13:45:29.550" Title="History of hypergeometric equation" Tags="|mathematics|differential-equations|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7659" PostTypeId="2" ParentId="7644" CreationDate="2018-09-01T11:30:42.763" Score="4" Body="&lt;p&gt;(1) Newton's consideration of the two-body problem includes right from the start his statement and proof of the planar character of the motion in space, in his first two propositions of the &lt;em&gt;Principia&lt;/em&gt;, which contain no intrinsic or &lt;em&gt;a-priori&lt;/em&gt; restriction to two dimensions: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prop.1  :  &quot;The areas, which revolving bodies describe by radii drawn to an immovable centre of force, do lie in the same immovable planes, and are proportional to the times in which they are described.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prop.2  :  &quot;Every body, that moves in any curve line described in a plane, and by a radius, drawn to a point either immoveable, or moving forward with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(These statements can be seen at pages 57 and 60 in the contemporary (1729) English translation of Newton's Latin (based on the &lt;em&gt;Principia&lt;/em&gt;'s 3rd edition, of 1726), available online at &lt;a href=&quot;https://books.google.com/books?id=Tm0FAAAAQAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?id=Tm0FAAAAQAAJ&lt;/a&gt; .  Each statement is followed by its associated proof, and &lt;em&gt;Conifold&lt;/em&gt;'s comment has already effectively given the gist of the proof of planarity found in Proposition 1.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) The enlargement of the question in a comment (&quot;Does [Newton] just assume central forces?&quot;) appears to presuppose that Newton gave his mathematical demonstrations and physical inferences by running them together. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This presupposition is not justified: Newton's writing for the most part shows particular care to separate mathematical reasoning from physical inferences.  He also emphasised in effect the different (lesser) degree of certainty associated with physical inferences as compared with mathematical demonstration (see e.g. &lt;em&gt;Principia&lt;/em&gt; Book 3, 'Rules of Reasoning in Philosophy', esp. Rule IV ). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of the separation is seen in Book 1 Proposition 4 and its corollaries (with mathematical demonstrations about some consequences of different conceivable laws of centripetal force, including the inverse-square among others) and its scholium (with comments of a more tentative character about how observations by named observers match an inverse-square law). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, in Proposition 1, what Newton shows is theoretical and mathematical, i.e. that &lt;em&gt;if&lt;/em&gt; the force on an orbiting body is directed to a centre which is either at rest or moving uniformly in a straight line &lt;em&gt;then&lt;/em&gt; (among other things) the motion of the orbiting body relative to that centre is confined to a plane. The question whether such a situation exists, exactly or approximately, in the real world, is treated elsewhere as a separate matter.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's identifications and demonstrations of several different features of two-body motion are spread over a number of different propositions and corollaries in the &lt;em&gt;Principia&lt;/em&gt;, and questions of how closely the actually-observed motions of celestial bodies might match this essentially mathematical model are treated separately again.   &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2018-09-01T12:04:26.803" LastActivityDate="2018-09-01T12:04:26.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7660" PostTypeId="2" ParentId="7658" CreationDate="2018-09-01T12:42:29.993" Score="6" Body="&lt;p&gt;1) For $a=b=c=0$ we have equation&#xA;$$&#xA;x(1-x) \dfrac {d^2y}{dx^2}-x\dfrac {dy}{dx}=0&#xA;$$&#xA;with solution $C_1 + C_2\log(1-x)$.  This would probably not be considered hypergeometric, except in a degenerate sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Answer (needs reference):  In a &lt;strong&gt;geometric&lt;/strong&gt; series $\sum a_n$, the ratio $\frac{a_{n+1}}{a_n}$ is constant, does not depend on $n$.  We generailze that to &lt;strong&gt;hypegeomeric&lt;/strong&gt; series where $\frac{a_{n+1}}{a_n}$ is a rational function of $n$.  These series are the $_p F_q$ functions, not only the $_2 F_1$ of Gauss.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Gauss did a lot on this.  But it is beyond the scope of this answer...  Perhaps read the bit in &lt;a href=&quot;https://en.wikipedia.org/wiki/Hypergeometric_function&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; for some idea &lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-09-01T12:42:29.993" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7661" PostTypeId="2" ParentId="7658" CreationDate="2018-09-01T13:25:16.450" Score="8" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;If $a=b=c=0$ we get a first order separable equation for $y'$, viz. $\ x[(1-x)y''-y']=0$ whose solution $y'= c(1-x)^{-1}$ would have been obvious since at least Bernoulli.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Dutka (&lt;a href=&quot;https://zbmath.org/?q=an:0554.01013&quot; rel=&quot;noreferrer&quot;&gt;1984&lt;/a&gt;, &lt;a href=&quot;https://www.jstor.org/stable/41133728&quot; rel=&quot;noreferrer&quot;&gt;p. 16&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In contrast to the geometric progression $a, ar, ar^2,\dots$ in which each term,&#xA;  after the first, is obtained by multiplying the preceding term by a constant ratio, &lt;strong&gt;Wallis&lt;/strong&gt; (&lt;em&gt;Arithmetica infinitorum&lt;/em&gt;, 1656, Scholium to Prop. 190; &lt;em&gt;A Treatise of Algebra&lt;/em&gt;, 1685, pp. 315–316) introduced another type of progression, to which he later gave the name “hypergeometric,” in which the successive multipliers are unequal. (&lt;em&gt;E.g.&lt;/em&gt;, the factorial sequence, 1, 2, 6, 24,... is a hypergeometrical progression.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Andrews, Askey and Roy (&lt;a href=&quot;https://zbmath.org/?q=an:0920.33001&quot; rel=&quot;noreferrer&quot;&gt;1999&lt;/a&gt;, p. 61):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A hypergeometric series is a series $\sum c_n$ such that $c_{n+1}/c_n$ is a rational function of $n$. On factorizing the polynomials in $n$, we obtain&#xA;  $$&#xA;\frac{c_{n+1}}{c_n}=\frac{(n+a_1)\cdots(n+a_p)x}{(n+b_1)\cdots(n+b_q)(n+1)}.&#xA;\tag{2.1.1}&#xA;$$&#xA;  The $x$ occurs because the polynomial may not be monic. The factor $(n+1)$ may result from the factorization, or it may not. If not, add it along with the compensating factor $(n+1)$ in the numerator (...) From (2.1.1) we have&#xA;  $$&#xA;\sum_{n=0}^\infty c_n&#xA;=c_0\sum_{n=0}^\infty\frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n}\frac{x^n}{n!}&#xA;=:c_0\,{}_pF_q\left(\begin{matrix}a_1,\dots,a_p\\b_1,\dots,b_q\end{matrix};x\right).&#xA;\tag{2.1.2}&#xA;$$&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Birkhoff (&lt;a href=&quot;https://zbmath.org/?q=an:0275.01009&quot; rel=&quot;noreferrer&quot;&gt;1973&lt;/a&gt;, p. 61):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Kummer&lt;/strong&gt; (&lt;a href=&quot;http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00214056X&quot; rel=&quot;noreferrer&quot;&gt;1836&lt;/a&gt;) gave the name “hypergeometric function” to $F$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;See Dutka (&lt;em&gt;ibid.&lt;/em&gt;, pp. 29–33). Necessary context is the prior work of Wallis (pp. 15–17), Newton (17), Stirling (17–20), Euler (20–26), Legendre (26), Pfaff (26–29).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-09-01T13:45:29.550" LastActivityDate="2018-09-01T13:45:29.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7664" PostTypeId="1" CreationDate="2018-09-02T23:17:08.650" Score="2" ViewCount="668" Body="&lt;p&gt;So he says that heavier objects fall faster than light ones and size is irrelevant?  (at least as far as I understand)&lt;br&gt;&#xA;This brings 2 easy ways to see the error of this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) throw a big rock and a small one&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) thought experiment 2 weights connected by a thin piece of material  the moment that thin piece of material snaps the 2 objects begin falling at half the rate they used to&lt;/p&gt;&#xA;" OwnerUserId="7812" LastEditorUserId="131" LastEditDate="2018-09-03T09:31:34.417" LastActivityDate="2018-09-03T12:56:40.257" Title="How did people believe Aristotle's law of gravity for so long?" Tags="|physics|galileo|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7665" PostTypeId="2" ParentId="7664" CreationDate="2018-09-03T12:56:40.257" Score="6" Body="&lt;p&gt;The assumption that people believed Aristotle’s law for so long is highly questionable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle’s law occurs in a philosophical context. He introduces it in order to argue that there can be no such thing as an object of infinite weight. It was not intended as a starting point for quantitative science, nor did many readers take it as such. Insofar as people “believed” Aristotle’s law, they did so largely as a by-product of accepting Aristotle’s system as a whole on philosophical grounds that had next to nothing to do with quantitative science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no evidence that any mathematical scientist in antiquity accepted Aristotle’s law, and quite some evidence to the contrary. Both Strato and Hipparchus wrote treatises on falling bodies that are now lost. By all indications they did not follow Aristotle’s law. Philoponus (in his commentary on Aristotle) used the experiment (1) you mention to reject Aristotle’s law of fall more than a thousand years before Galileo. Still earlier, Lucretius clearly stated that, in the absence of air resistance, all objects fall at the same speed regardless of weight (De rerum natura, II:225–239).&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2018-09-03T12:56:40.257" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7666" PostTypeId="1" CreationDate="2018-09-03T13:59:49.350" Score="4" ViewCount="191" Body="&lt;p&gt;I have not found bibliographic data to show what age Noether was in this picture:&lt;a href=&quot;https://i.stack.imgur.com/woMtg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/woMtg.jpg&quot; alt=&quot;Emmy Noether relatively young&quot;&gt;&lt;/a&gt;&#xA;And I cannot estimate well either by her clothes or her face. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone here help me?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the past I have known people to date photographs by the clothes, in ways that often checked out very well.  And this style looks fairly distinctive to me. But I am not now in touch with any of those people. &lt;/p&gt;&#xA;" OwnerUserId="38" LastEditorUserId="55" LastEditDate="2018-09-06T21:07:06.773" LastActivityDate="2018-09-06T21:07:06.773" Title="How old might Emmy Noether be in this picture?" Tags="|mathematicians|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7667" PostTypeId="1" CreationDate="2018-09-05T08:02:47.663" Score="5" ViewCount="466" Body="&lt;p&gt;I've encountered the following claim: If we, as Popper apparantly did, view Marx and Freud's body of work as non-scientific because core tenets are not falsifiable and we apply the same rubic to Darwins theory of evolution, the latter would also appear to have non-falsifiable (core?)-tenets. According to the claim, Popper himself was well aware of that dilemma.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't follow the last part. Here's where I think I follow the argument:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Within Marx' work there's certainly things - the whole theleological attempt to predict history - that are either plain false or so wishy-washy that they're not falsifiable. Marx core axiom, the labor theory of value, is likely untestable. So I kinda follow that for Popper, Marxism is not a science.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Freud posits the unconscious - that we can't observe - as well as a few other concepts that can't be tested, or at least not in isolation. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;What I don't understand is the claim that applying that rubic to Darwins work, we should discard it too. &lt;a href=&quot;https://rationalwiki.org/wiki/Falsifiability_of_evolution&quot; rel=&quot;noreferrer&quot;&gt;Darwin himself said:&lt;/a&gt; &lt;em&gt;&quot;If it could be demonstrated that any complex organ existed which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down. But I can find out no such case.&quot;&lt;/em&gt;   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To me, it appears that there is no dilemma. But maybe I'm wrong. &lt;strong&gt;Is my very brief summary of likely Popperian attacks on Freud and Marx grossly wrong? Did Popper ever comment on the dilemma stated at the beginning of the question, or does it simply not exist?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7816" LastActivityDate="2023-05-08T20:55:38.513" Title="Popper on Marx, Freud and Darwin" Tags="|philosophy-of-science|evolution|psychology|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7668" PostTypeId="2" ParentId="7667" CreationDate="2018-09-05T11:56:09.163" Score="6" Body="&lt;p&gt;Popper's main idea about the &lt;a href=&quot;https://plato.stanford.edu/entries/popper/#ProbDema&quot; rel=&quot;nofollow noreferrer&quot;&gt;demarcation problem&lt;/a&gt; : pseudo-science is untestable, is a marvel of simplicity :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On this criterion of demarcation physics, chemistry, and (non-introspective) psychology, amongst others, are sciences, psychoanalysis is a pre-science (i.e., it undoubtedly contains useful and informative truths, but until such time as psychoanalytical theories can be formulated in such a manner as to be falsifiable, they will not attain the status of scientific theories), and astrology and phrenology are pseudo-sciences.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, is a philosophical (methodological) view, and its application to historical facts can be in same case difficult.&lt;/p&gt;&#xA;&lt;p&gt;According to Popper, scientific doctrines do not emerge inductively from&#xA;facts (contra Bacon and Mill); they are first imagined and then verified empirically.&lt;/p&gt;&#xA;&lt;p&gt;Popper demarcation criteria for scientific theories demands testability.&lt;/p&gt;&#xA;&lt;p&gt;According to Popper, theories which are testable but have been obviously refuted are scientific, for he only demands testability.&lt;/p&gt;&#xA;&lt;p&gt;In Popper's view, metaphysics has a role : metaphysics (again contra Bacon) is sometimes important as a source of scientific inspiration.&lt;/p&gt;&#xA;&lt;p&gt;Marxism (and, more fgenerally &lt;a href=&quot;https://plato.stanford.edu/entries/popper/#ScieKnowHistPred&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historicism&lt;/a&gt;) is rejected by Popper as purely metaphysics (untestable) due to his claim about &amp;quot;unconditional&amp;quot; historical predictions.&lt;/p&gt;&#xA;&lt;p&gt;According to Popper :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;every event in human history is discrete, novel, quite unique, and ntologically distinct from every other historical event. For this reason, it is impossible in principle that unconditional scientific prophecies could be made in relation to human history — the idea that the successful unconditional prediction of eclipses provides us with reasonable grounds for the hope of successful unconditional prediction regarding the evolution of human history turns out to be based upon a gross misconception, and is quite false. As Popper himself concludes,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“The fact that we predict eclipses does not, therefore, provide a valid reason for expecting that we can predict revolutions.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Regarding Darwin's theory, Popper's attitude changed.&lt;/p&gt;&#xA;&lt;p&gt;See : Mehmet Elgin and Elliott Sober, &lt;a href=&quot;https://www.journals.uchicago.edu/doi/10.1086/691119&quot; rel=&quot;nofollow noreferrer&quot;&gt;Popper’s Shifting Appraisal of Evolutionary Theory&lt;/a&gt; (2017) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Karl Popper argued in 1974 that evolutionary theory contains no testable laws and is therefore a metaphysical research program. Four years later, he said that he had changed his mind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We may imagine that Popper's attitude changed due to his personal shift towards &lt;a href=&quot;https://books.google.it/books?id=hcCyXqwTv_cC&amp;amp;dq=popper+objective+knowledge&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=0ahUKEwipvqqv5qPdAhUCyIUKHRl3CU0Q6AEIJzAA&quot; rel=&quot;nofollow noreferrer&quot;&gt;evolutionary approach&lt;/a&gt; regarding knowledge as well as due to the pressure from biology scientific community.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-05T11:56:09.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7669" PostTypeId="1" AcceptedAnswerId="7670" CreationDate="2018-09-06T07:44:05.257" Score="1" ViewCount="51" Body="&lt;p&gt;Do you know who introduced the distinction between active and passive mass?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In what circumstances the necessity for such distinction arose and when it was discarded? Wasn't it quite obvious that a massive body produces gravitational pull and is pulled at the same time?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2018-09-06T09:45:17.150" Title="Who introduced passive/active gravitational mass?" Tags="|gravity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7670" PostTypeId="2" ParentId="7669" CreationDate="2018-09-06T09:45:17.150" Score="1" Body="&lt;p&gt;See Max Jammer, &lt;a href=&quot;https://books.google.it/books?id=lYvz0_8aGsMC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Concepts of Mass in Classical and Modern Physics&lt;/a&gt;, Princeton UP (2000), page 91 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is, of course, difficult, if not impossible, to identify the first individual to use the notion or the term “gravitational mass.” However,&#xA;  records show that in discussions held in 1907 at a convention of the Italian Physical Society, attended by E. Alessandri, G. Castelnuovo, and G. Vailati, among others, the term “massa gravitazionale” was used. It&#xA;  thus seems certain that there was an explicit distinction between $m_i$ and&#xA;  $m_g$ not later than 1907.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It is also difficult to name with certainty the first individual to distinguish&#xA;  between $m_a$ and $m_p$. What is certain, however, is the fact that this distinction played an important role in physical discussions from the time Hermann Bondi publicized it in his often quoted essay on negative mass in general relativity [H. Bondi, “Negative Mass in General Relativity,” Reviews of Modern Physics 29, 423–428 (1957).].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-09-06T09:45:17.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7671" PostTypeId="1" AcceptedAnswerId="7680" CreationDate="2018-09-06T18:18:55.040" Score="3" ViewCount="1527" Body="&lt;p&gt;I know that English is the most popular language to write scientific/mathematical papers after World War 2. I also know that in the second half of 19th century and first half of 20th century, German and French were the main languages to write mathematical (scientific) papers. Before 19th century however, it seems that Latin was the main languages in which most scientific/mathematical papers were written. My question is, what was the main language in which scientific/mathematical papers were written before 1850? I know Newton's Principia Mathematica was in Latin. What were languages used by most 18th to 19th centuries mathematicians like Euler, Gauss, Lagrange, Laplace and etc?&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2018-09-17T17:22:10.303" LastActivityDate="2019-05-06T01:34:11.643" Title="What was the main language in science/mathematics before 1850" Tags="|mathematics|reference-request|publishing|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7672" PostTypeId="2" ParentId="7671" CreationDate="2018-09-06T23:30:50.783" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://www.pri.org/stories/2014-10-06/how-did-english-become-language-science&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; article is interesting, because the argument regarding how German faded out in way of English becoming the current &quot;language of science&quot; is also applicable to how Latin was replaced by German as the &quot;language of science.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;French, German, Italian, and English are all latin based languages. So, as you say, Latin was the main language used prior to the 19th century, and the scientists/mathematicians of the 18th and 19th centuries began using their own local languages (which all branched from Latin) since this time period was a transition away from Latin. Naturally, this means that Laplace wrote in French, Gauss in German, Lagrange in Italian and French, etc... But keep in mind that such intellectuals were very likely to be literate in multiple languages, i.e. Euler wrote in French, German and Latin.&lt;/p&gt;&#xA;" OwnerUserId="7638" LastActivityDate="2018-09-06T23:30:50.783" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7673" PostTypeId="2" ParentId="7671" CreationDate="2018-09-07T04:35:41.847" Score="1" Body="&lt;p&gt;Since the end of 18 century, perhaps even earlier, most papers were probably published in German. And it kept this leading position until 1930s. I am not so sure about other sciences, but this was so in mathematics.&#xA;Once I counted the PhD listed in the Math Genealogy project, by countries. Since the 18th century, the largest number (by far) came from German-speaking countries. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it was &quot;main&quot; in terms of statistics. But certainly, English mathematicians wrote in English, French wrote in French, Italians wrote in Italian, and many (in all countries) wrote in Latin. So German&#xA;was not &quot;dominating&quot; in the same sense as English is now. But in terms of numbers, one can safely state that the largest number of mathematical publications were in German, and the largest number of  mathematicians lived in German-speaking countries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. For example, here is the 18th century distribution in Math Genealogy: Germany 317, Netherlands 122, England 24, France 14, Switzerland 8, Austria 7, Poland 1. Looks strange, of course. And I am not sure how to interpret this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-09-13T12:56:45.207" LastActivityDate="2018-09-13T12:56:45.207" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7674" PostTypeId="2" ParentId="7671" CreationDate="2018-09-07T13:05:05.247" Score="2" Body="&lt;p&gt;Habits and fashions among authors and publishers for choosing languages in which to communicate science and mathematics evolved in a complex as well as an interesting way. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I could not pretend to give an exhaustive answer to the question, but I offer examples below of different linguistic tendencies at work in about the period 1700 to 1850.  The examples given by the memoirs of the academies of science in Berlin, and also in St Petersburg, show that during the 18th century the French language attained an international usage that seems to have been little noticed in previous related questions and the answers to them, which have had focus more on the use of German. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This early prominence of French also seems confirmed by the interesting chart that &lt;em&gt;Francois Ziegler&lt;/em&gt;'s answer has added to the discussion, with his chart showing the language makeup of his database of math/physics papers from about 1690.  In summary, for that database of papers, the 18th-century growth of the literature appears to have been largely in French, then the 19th-century growth more in German and English.      &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Turning now to the more detailed examples, The present question suggested that before the 19th century it seemed &quot;that Latin was the main language in which most scientific/mathematical papers were written&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are signs that the well-known traditional dominance of Latin was beginning to break down in certain respects already in the 17th century.  For example, in England the 'Philosophical Transactions' appeared mainly in English, though they used Latin for many purely mathematical contributions and also for some communications received or copied from authors outside England.  English authors who wrote both on mathematics and on other scientific subjects, e.g. Edmond Halley, can be read in Latin for their mathematical demonstrations, and in English for their other scientific contributions in that journal.  In France, the 'Histoire de l'Académie royale des sciences ... avec les mémoires' from mid-17th-century onwards appeared entirely or almost entirely in French.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scientific and mathematical books still appeared mainly in Latin, sometimes at considerable effort of translation, e.g. in the case of John Wallis' 1672-3 edition of works and letters of the astronomer Jeremiah Horrocks: this was translated by Wallis into Latin expressly for the sake of the international audience, where (some of) the original papers had been in English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far, the influences seen through these examples include the wish of authors and editors to be internationally understood, and the habits and preferences of journal-publishers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the early 18th century onwards it seems there was an increasing tendency towards use of the vernacular at least in in England and France. Books originally in Latin were translated into English and French, and some appeared first in their respective vernacular, for example Newton's 'Opticks' appeared first in English in 1704, though a translated and revised Latin edition followed soon after. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One remarkable apparent exception, for a time, to the tendency to use the vernacular was the publications of the Berlin royal academy of sciences, which appeared mainly or wholly in French and not German, from 1745 to 1804, first as &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige?band=02-hist/1745&quot; rel=&quot;nofollow noreferrer&quot;&gt;Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin&lt;/a&gt;, then as &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige?band=03-nouv/1770&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres&lt;/a&gt;, and finally as &lt;a href=&quot;http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige?band=05-mem/1804&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mémoires de l'Académie Royale des Sciences et Belles-Lettres&lt;/a&gt;.  On the other hand this  use of French outside of France for the Berlin memoirs was perhaps not such an exception, one can also see a mixed use of French alongside the traditional Latin in the memoirs of the 18th-century Petersburg Imperial academy of sciences.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Very possibly the Napoleonic/European wars reinforced a tendency to use the vernacular, but it now seems remarkable that there were some signs of slowness to include German among the internationally-current languages of science. Thus for example Gauss' work of great importance 'Theoria motus corporum coelestium...' (1809) had been written first in German, according to the biography of Gauss by G W Dunnington et al. (2004) &lt;a href=&quot;https://books.google.com/books?id=4mwSrfxBSzkC&amp;amp;pg=PA90&quot; rel=&quot;nofollow noreferrer&quot;&gt;p.90&lt;/a&gt;, but on request from the publisher, F C Perthes, it was translated and published in Latin. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the authors specifically named in the question:-- there are many papers of Euler in the Berlin memoirs in French, while elsewhere Euler wrote in Latin or (seemingly less often) German. Euler's work dated from long before the Napoleonic wars and their related conflicts.  Lagrange and Laplace wrote in French, and I'm not aware of anything from them that first appeared in any other language. (Lagrange was born and educated in Turin before moving to Paris, and published early works there: what I have seen appeared in French but perhaps there were also items in Italian.) Gauss, who was personally much affected by Napoleonic-war conditions, is said to have refused to publish in French; his works appeared in Latin, otherwise in German.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other answers, to this and other questions, have described the growth of German-language publications in science and mathematics once German became established -- and the examples of widespread use of French, and now also &lt;em&gt;Francois Ziegler&lt;/em&gt;'s chart, suggest this hardly happened before the 19th century. The later growth in international use of English is outside the scope of this particular question. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2018-09-08T23:49:22.283" LastActivityDate="2018-09-08T23:49:22.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7675" PostTypeId="1" AcceptedAnswerId="7677" CreationDate="2018-09-07T14:18:40.453" Score="3" ViewCount="1341" Body="&lt;p&gt;I saw that the word &quot;mode&quot; means &quot;popular&quot; in French, and I was wondering if this might be the etymology of the &quot;mode&quot; of a population in stat? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was wondering if anyone had sources for early use of the term?&lt;/p&gt;&#xA;" OwnerUserId="3727" LastActivityDate="2018-09-07T18:51:09.193" Title="What is the etymology of the term &quot;mode&quot; in statistics?" Tags="|terminology|statistics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7676" PostTypeId="1" AcceptedAnswerId="7678" CreationDate="2018-09-07T15:46:41.737" Score="3" ViewCount="191" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;NOTE:&lt;/strong&gt; I am talking about the period before electricity and magnetism were unified. So I am not seeking for answers based on Ampere atomic current model of magnets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Who first figured out that one of the &lt;strong&gt;surfaces&lt;/strong&gt; (pole) of a magnet attract or repel other &lt;strong&gt;surface&lt;/strong&gt; (pole) of another magnet?&lt;/p&gt;&#xA;&lt;p&gt;How did people at that time explain the following possibility:&lt;/p&gt;&#xA;&lt;p&gt;One half of the magnet's &lt;strong&gt;volume&lt;/strong&gt; is north pole and the other half is south pole (as shown in figure below). Do these &amp;quot;volumes&amp;quot; attract or repel each other, or do the &amp;quot;surfaces&amp;quot; attract or repel each other?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bafAs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bafAs.png&quot; alt=&quot;North and south magnetic poles' volumes&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&lt;/em&gt; &lt;strong&gt;I found the following highlighted statement from &lt;a href=&quot;https://archive.org/stream/atreatiseonelec00thomgoog#page/n33&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxwell's treatise Vol II, Article 373&lt;/a&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rNUUU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rNUUU.png&quot; alt=&quot;Maxwell's treatise Vol II, Article 373&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does it suggest that &amp;quot;it has been seen experimentally that &lt;em&gt;&lt;strong&gt;surface end of a long thin magnet attract or repel surface end of another long thin magnet&lt;/strong&gt;&lt;/em&gt; instead of &lt;em&gt;&lt;strong&gt;half of magnet's volume attract or repel half of other magnet's volume&amp;quot;&lt;/strong&gt;&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="5313" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-09T03:46:14.647" Title="Who determined whether surfaces between magnetic poles attract/repel or whether the poles' volumes do?" Tags="|physics|magnetism|" AnswerCount="2" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7677" PostTypeId="2" ParentId="7675" CreationDate="2018-09-07T18:25:11.387" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.ilpost.it/mauriziocodogno/2013/03/12/parole-matematiche-moda/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article (in Italian) by Maurizio Codogno, the origin is in an article by Pearson dated 1895 (&lt;a href=&quot;https://archive.org/details/philtrans06840806&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, in the article by Pearson you can find the following note on page 2:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I have found it convenient to use the term &lt;strong&gt;mode&lt;/strong&gt; for the abscissa corresponding to the ordinate of maximum frequency. Thus the “mean,” the “mode,” and the “median” have all distinct characters important to the statistician.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Codogno further speculates that Pearson might have known the French expression &quot;à la mode&quot; (i.e. fashionable), so that its usage for representing the most &quot;fashionable&quot; value in a distribution is particularly apt. Also, it might have helped (both for Pearson to select it and for the term to gain widespread usage) the fact that the word started with &quot;m&quot; (like mean and median).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is for sure a very nice term (from which also &lt;em&gt;unimodal, bimodal, ...&lt;/em&gt; come from)!&lt;/p&gt;&#xA;" OwnerUserId="7818" LastActivityDate="2018-09-07T18:25:11.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7678" PostTypeId="2" ParentId="7676" CreationDate="2018-09-07T18:41:26.617" Score="4" Body="&lt;p&gt;in 1785, the French physicist &lt;em&gt;Charles-Augustin de Coulomb&lt;/em&gt; published his first three reports of electricity and magnetism where he stated his law.&lt;/p&gt;&#xA;&lt;p&gt;The conclusion that there were two types of magnetism,  was a natural consequence of the discovery that the poles of a magnet either attract or repel one another. The north (i.e. north-seeking) pole was said to be positive, whereas the south pole was negative.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Attempts were made to separate the positive and negative poles of a magnet by breaking the magnet into two pieces,&lt;/em&gt; but it was always found that negative and positive poles, respectively, appeared on the two sides of the break.&lt;/p&gt;&#xA;&lt;p&gt;It was, therefore, reluctantly accepted that the privilege of independent experience, enjoyed by electric charges, is not a prerogative of magnetic poles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Whenever a positive pole is discovered in nature, it is inevitably associated with a negative pole of equal strength.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;The problem of obtaining &amp;quot;free&amp;quot; poles, in order to determine the force between them, was overcome by using long, thin magnets.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The positive pole was &lt;strong&gt;sufficiently far removed&lt;/strong&gt; from its associated negative pole as to be practically isolated from it.&lt;/p&gt;&#xA;&lt;p&gt;The law of force between poles was investigated by Charles Coulomb, using the same torsion balance with which he established the law of force between electric charges, and was found to be similar in form to that for charges.&lt;/p&gt;&#xA;&lt;p&gt;The torsion balance consists of a &lt;strong&gt;bar suspended from its middle by a thin fiber.&lt;/strong&gt; The fiber acts as a very weak torsion spring.&lt;/p&gt;&#xA;&lt;p&gt;As regards the suggestion that one half of  the bar magnet may be taken as one  type of pole of the magnetic field- let us see the plot of lines of forces&lt;/p&gt;&#xA;&lt;p&gt;The density of lines of force is much smaller in the region from the poles to the center of the magnet. (see diagrams in hyperphysics ref.given below))&#xA;In fact, the magnetic dipoles in the material of the magnet form a circuital line and are well connected inside- as this array reaches the ends the lines of force emerge and get connected to any 'magnetic material' outside the magnet- if one traces back these lines- a &lt;strong&gt;pole&lt;/strong&gt;  gets defined.&lt;/p&gt;&#xA;&lt;p&gt;Therefore if ane takes a thin long magnet the effect of the force from poles will be &lt;strong&gt;much larger&lt;/strong&gt; than from surfaces along the length- In the torsion&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;balance experiment Coulomb actually minimized the effects of lateral lines of forces and could measure the deflection due to polar  action.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;ref.-&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.daviddarling.info/encyclopedia/C/Coulombs_law_for_magnets.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.daviddarling.info/encyclopedia/C/Coulombs_law_for_magnets.html&lt;/a&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Coulomb%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Coulomb%27s_law&lt;/a&gt;&#xA;&lt;a href=&quot;http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-08T08:03:48.250" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7679" PostTypeId="2" ParentId="7675" CreationDate="2018-09-07T18:51:09.193" Score="2" Body="&lt;p&gt;The &lt;em&gt;Oxford English Dictionary&lt;/em&gt; divides its definitions of the noun &lt;em&gt;mode&lt;/em&gt; into: &quot;senses derived directly from the Latin&quot; and &quot;senses derived from French&quot;.  The statistical meaning is listed in &quot;senses derived from French&quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-09-07T18:51:09.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7680" PostTypeId="2" ParentId="7671" CreationDate="2018-09-07T19:27:13.470" Score="11" Body="&lt;p&gt;For what it’s worth, here are the languages of the 1645 math/phys paper and book &lt;em&gt;titles&lt;/em&gt; from the years 1690–1919 in a bibtex file I have. Of course unscientific with all kinds of biases, but I imagine the catalogues of &lt;a href=&quot;https://archive.org/details/repertoriumcomm12reusgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reuss&lt;/a&gt; (1808), the &lt;a href=&quot;http://www.archive.org/details/catalogueofscien01royarich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Royal Society&lt;/a&gt; (1800–1883) and &lt;a href=&quot;http://emis.de/MATH/JFM/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jahrbuch&lt;/a&gt; (1868–1942) might give a similar plot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bFblM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bFblM.png&quot; alt=&quot;paper and book titles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Added (from the same file; graph is smoother as a journal appears in every decade it was active, regardless of whether it is represented in that decade):&lt;/p&gt;&#xA;&#xA;&lt;h3&gt;&lt;em&gt;Titles&lt;/em&gt; of math / physics / celestial mechanics journals represented before 1850:&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Hk5YC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Hk5YC.png&quot; alt=&quot;journal titles&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;code&gt;                                                                                    until:&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb343488023/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1665&lt;/a&gt; Le Journal des sçavans (Paris)                                                    1947&#xA;&lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/by/year&quot; rel=&quot;nofollow noreferrer&quot;&gt;1665&lt;/a&gt; Philosophical Transactions of the Royal Society (London)                             &gt;&#xA;     &lt;a href=&quot;https://catalog.hathitrust.org/Record/008383158&quot; rel=&quot;nofollow noreferrer&quot;&gt;1822&lt;/a&gt; Memoirs of the (Royal) Astronomical Society                                  1978&#xA;     &lt;a href=&quot;https://catalog.hathitrust.org/Record/000522950&quot; rel=&quot;nofollow noreferrer&quot;&gt;1827&lt;/a&gt; Monthly notices of the (Royal) Astronomical Society                             &gt;&#xA;     &lt;a href=&quot;http://rspl.royalsocietypublishing.org/content/by/year&quot; rel=&quot;nofollow noreferrer&quot;&gt;1832&lt;/a&gt; Proceedings of the Royal Society                                                &gt;&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb327469896/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1679&lt;/a&gt; Connaissance des temps (Paris)                                                       &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb34354613f/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Acta.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1682&lt;/a&gt; (Nova) Acta Eruditorum (Leipzig)                                                  &lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Acta.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1782&lt;/a&gt;&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/009285328&quot; rel=&quot;nofollow noreferrer&quot;&gt;1692&lt;/a&gt; Memoires de mathematique et de physique de l’Académie des Sciences (Paris)        1693&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/4366&quot; rel=&quot;nofollow noreferrer&quot;&gt;1698&lt;/a&gt; Regiae Scientiarum Academiae Historia                                        1698&#xA;     &lt;a href=&quot;https://catalog.hathitrust.org/Record/100002872&quot; rel=&quot;nofollow noreferrer&quot;&gt;1705&lt;/a&gt; Histoire de l’Academie des Sciences, avec les Memoires                       1788&#xA;     &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb328496507/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1721&lt;/a&gt; Recueil des pièces qui ont remporté le prix de l’Académie des Sciences       1777&#xA;     &lt;a href=&quot;https://catalog.hathitrust.org/Record/009335749&quot; rel=&quot;nofollow noreferrer&quot;&gt;1729&lt;/a&gt; Memoires de l’Académie des Sciences depuis 1666 jusqu’à 1699                 1734&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/4360&quot; rel=&quot;nofollow noreferrer&quot;&gt;1750&lt;/a&gt; Mémoires présentés à l’Académie des Sciences par divers Sçavans              1786&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/16282&quot; rel=&quot;nofollow noreferrer&quot;&gt;1798&lt;/a&gt; Mémoires de l’Institut                                                       &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/4002&quot; rel=&quot;nofollow noreferrer&quot;&gt;1864&lt;/a&gt;&#xA;     &lt;a href=&quot;https://books.google.com/books?id=GAjW7bRdNZQC&quot; rel=&quot;nofollow noreferrer&quot;&gt;1805&lt;/a&gt; Mémoires présentés à l’Institut par divers Savans                            &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb328135052/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1914&lt;/a&gt;&#xA;     &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb343481087/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1835&lt;/a&gt; Comptes rendus de l’Académie des Sciences                                       &gt;&#xA;&lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Berlin.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1710&lt;/a&gt; Miscellanea Berolinensia (Berlin)                                                 1744&#xA;     &lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Berlin.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1746&lt;/a&gt; Histoire de l’Académie des Sciences et des Belles-Lettres                    1771&#xA;     &lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Berlin.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1772&lt;/a&gt; (Nouveaux) Mémoires de l’Académie des Sciences et Belles-Lettres             1807&#xA;     &lt;a href=&quot;https://catalog.hathitrust.org/Record/008614898&quot; rel=&quot;nofollow noreferrer&quot;&gt;1774&lt;/a&gt; Astronomisches Jahrbuch                                                      1959&#xA;     &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/#A7&quot; rel=&quot;nofollow noreferrer&quot;&gt;1815&lt;/a&gt; Abhandlungen der Preußischen Akademie der Wissenschaften                     1919&#xA;     &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/#A8&quot; rel=&quot;nofollow noreferrer&quot;&gt;1836&lt;/a&gt; Berichte der Preußischen Akademie der Wissenschaften                         1938&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/000058853&quot; rel=&quot;nofollow noreferrer&quot;&gt;1710&lt;/a&gt; Giornale de’ letterati d'Italia (Venezia)                                         1740&#xA;&lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Petersburg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1728&lt;/a&gt; (Novi) Commentarii Academiae Scientiarum (St Petersburg)                          1776&#xA;     &lt;a href=&quot;http://www.izwtalt.uni-wuppertal.de/Petersburg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1778&lt;/a&gt; (Nova) Acta Academiae Scientiarum                                            1806&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/9528&quot; rel=&quot;nofollow noreferrer&quot;&gt;1809&lt;/a&gt; Mémoires de l'Académie des Sciences                                          1897&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/53549&quot; rel=&quot;nofollow noreferrer&quot;&gt;1831&lt;/a&gt; Mémoires présentés à l'Académie des Sciences par divers Savans               1859&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/11493&quot; rel=&quot;nofollow noreferrer&quot;&gt;1835&lt;/a&gt; Bulletin de l'Académie des Sciences                                          1894&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/G%C3%B6ttingische_Gelehrte_Anzeigen&quot; rel=&quot;nofollow noreferrer&quot;&gt;1739&lt;/a&gt; Göttingische Gelehrte Anzeigen (Göttingen)                                           &lt;a href=&quot;https://adw-goe.de/ueber-uns/schriftenreihen/regelmaessig-erscheinende-veroeffentlichungen/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/Commentationes_Societatis_Regiae_Scientiarum_Gottingensis_recentiores&quot; rel=&quot;nofollow noreferrer&quot;&gt;1752&lt;/a&gt; Commentarii Societatis Scientiarum (Göttingen)                                    1841&#xA;     &lt;a href=&quot;https://de.wikisource.org/wiki/Abhandlungen_der_K%F6niglichen_Gesellschaft_der_Wissenschaften_zu_G%F6ttingen&quot; rel=&quot;nofollow noreferrer&quot;&gt;1843&lt;/a&gt; Abhandlungen der Gesellschaft der Wissenschaften                                &lt;a href=&quot;https://adw-goe.de/ueber-uns/schriftenreihen/regelmaessig-erscheinende-veroeffentlichungen/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;     &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN668664029&quot; rel=&quot;nofollow noreferrer&quot;&gt;1845&lt;/a&gt; Nachrichten der Gesellschaft der Wissenschaften                                 &lt;a href=&quot;https://rep.adw-goe.de/handle/11858/72&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/7921&quot; rel=&quot;nofollow noreferrer&quot;&gt;1759&lt;/a&gt; Miscellanea Philosophico-Mathematica Societatis Privatae (Torino)                 1759&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/7920&quot; rel=&quot;nofollow noreferrer&quot;&gt;1762&lt;/a&gt; Mélanges de Philosophie et de Mathématique de la Société Royale              1776&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/7823&quot; rel=&quot;nofollow noreferrer&quot;&gt;1786&lt;/a&gt; Mémoires de l’Académie des Sciences                                          1801&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/6366&quot; rel=&quot;nofollow noreferrer&quot;&gt;1818&lt;/a&gt; Memorie della Accademia delle scienze                                           &lt;a href=&quot;https://www.accademiadellescienze.it/attivita/editoria/periodici-e-collane/memorie&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/volumes/id/PPN129323659&quot; rel=&quot;nofollow noreferrer&quot;&gt;1777&lt;/a&gt; (Nouveaux) Mémoires de l’Académie des Sciences et Belles-Lettres (Bruxelles)         &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/volumes/id/PPN129323640&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/8091&quot; rel=&quot;nofollow noreferrer&quot;&gt;1782&lt;/a&gt; Memorie di matematica e di fisica della Società italiana (Modena)                    &lt;a href=&quot;http://www.accademiaxl.it/en/pubblicazioni-2/rendiconti/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/Leipziger_Magazin_f%C3%BCr_reine_und_angewandte_Mathematik&quot; rel=&quot;nofollow noreferrer&quot;&gt;1786&lt;/a&gt; Magazin für reine und angewandte Mathematik (Leipzig)                             1788&#xA;     &lt;a href=&quot;https://de.wikisource.org/wiki/Archiv_der_reinen_und_angewandten_Mathematik&quot; rel=&quot;nofollow noreferrer&quot;&gt;1794&lt;/a&gt; Archiv der reinen und angewandten Mathematik                                 1800&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/973&quot; rel=&quot;nofollow noreferrer&quot;&gt;1787&lt;/a&gt; Transactions of the Royal Irish Academy (Dublin)                                  &lt;a href=&quot;https://www.ria.ie/irish-and-international-journal-collections&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907&lt;/a&gt;&#xA;     &lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/50818&quot; rel=&quot;nofollow noreferrer&quot;&gt;1841&lt;/a&gt; Proceedings of the Royal Irish Academy                                          &lt;a href=&quot;https://www.ria.ie/irish-and-international-journal-collections&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb343481563/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1789&lt;/a&gt; Annales de chimie (Paris)                                                         1815&#xA;     &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb343780820/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1816&lt;/a&gt; Annales de chimie et de physique                                             &lt;a href=&quot;https://www.annphys.org/component/issues/&quot; rel=&quot;nofollow noreferrer&quot;&gt;2009&lt;/a&gt;&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/Annalen_der_Physik&quot; rel=&quot;nofollow noreferrer&quot;&gt;1790&lt;/a&gt; (Neues) Journal der Physik (Leipzig)                                              1797&#xA;     &lt;a href=&quot;https://de.wikisource.org/wiki/Annalen_der_Physik&quot; rel=&quot;nofollow noreferrer&quot;&gt;1799&lt;/a&gt; Annalen der Physik                                                              &gt;&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/44719&quot; rel=&quot;nofollow noreferrer&quot;&gt;1791&lt;/a&gt; (Nouveau) Bulletin des sciences, par la société philomathique (Paris)             1833&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb34378280v/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1794&lt;/a&gt; Journal de l’École polytechnique (Paris)                                          1938&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/58328&quot; rel=&quot;nofollow noreferrer&quot;&gt;1798&lt;/a&gt; Philosophical magazine (London)                                                      &lt;a href=&quot;https://www.tandfonline.com/loi/tphm19&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb32813422c/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1807&lt;/a&gt; Mémoires de physique et de chimie, de la Société d'Arcueil (Paris)                1817&#xA;&lt;a href=&quot;http://www.numdam.org/journals/AMPA&quot; rel=&quot;nofollow noreferrer&quot;&gt;1810&lt;/a&gt; Annales de mathématiques pures et appliquées (Nîmes)                              1832&#xA;     &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb343487840/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1836&lt;/a&gt; Journal de mathématiques pures et appliquées (Paris)                            &gt;&#xA;&lt;a href=&quot;https://archive.org/search.php?query=Effemeridi+astronomiche+di+Milano&amp;amp;sort=date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1814&lt;/a&gt; Effemeridi astronomiche (Milano)                                                  &lt;a href=&quot;https://books.google.com/books?id=uT5RAAAAYAAJ&amp;amp;&amp;amp;pg=RA2-PR3&quot; rel=&quot;nofollow noreferrer&quot;&gt;1873&lt;/a&gt;&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/011437445&quot; rel=&quot;nofollow noreferrer&quot;&gt;1818&lt;/a&gt; Correspondance astronomique, géographique, hydrographique et statistique (Genova) 1826&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/Astronomische_Nachrichten&quot; rel=&quot;nofollow noreferrer&quot;&gt;1821&lt;/a&gt; Astronomische Nachrichten (Altona)                                                   &lt;a href=&quot;http://www.aip.de/AN/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/000526741&quot; rel=&quot;nofollow noreferrer&quot;&gt;1822&lt;/a&gt; Transactions of the Cambridge Philosophical Society (Cambridge)                   1928&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/cb32726560z/date&quot; rel=&quot;nofollow noreferrer&quot;&gt;1824&lt;/a&gt; Bulletin des sciences mathématiques, astronomiques, physiques et chimiques (Paris)1831&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/012306283&quot; rel=&quot;nofollow noreferrer&quot;&gt;1825&lt;/a&gt; Correspondance mathématique et physique (Gand)                                    1839&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik/Inhalt&quot; rel=&quot;nofollow noreferrer&quot;&gt;1826&lt;/a&gt; Journal für die reine und angewandte Mathematik (Berlin)                             &gt;&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/2276&quot; rel=&quot;nofollow noreferrer&quot;&gt;1833&lt;/a&gt; Reports of Meetings of the British Association (London)                           1938&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/000059583&quot; rel=&quot;nofollow noreferrer&quot;&gt;1839&lt;/a&gt; The Cambridge (and Dublin) mathematical journal (Cambridge)                          &lt;a href=&quot;https://en.wikipedia.org/wiki/Quarterly_Journal_of_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://catalog.hathitrust.org/Record/000523170&quot; rel=&quot;nofollow noreferrer&quot;&gt;1841&lt;/a&gt; Archiv der Mathematik und Physik (Leipzig)                                        1920&#xA;&lt;a href=&quot;http://www.numdam.org/journals/NAM&quot; rel=&quot;nofollow noreferrer&quot;&gt;1842&lt;/a&gt; Nouvelles annales de mathématiques (Paris)                                        1927&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/13498&quot; rel=&quot;nofollow noreferrer&quot;&gt;1842&lt;/a&gt; Acta Societatis Scientiarum Fennicae (Helsinki)                                   &lt;a href=&quot;https://catalog.hathitrust.org/Record/000496749&quot; rel=&quot;nofollow noreferrer&quot;&gt;1939&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.biodiversitylibrary.org/bibliography/7832&quot; rel=&quot;nofollow noreferrer&quot;&gt;1847&lt;/a&gt; Atti della Accademia dei Lincei (Roma)                                               &lt;a href=&quot;https://www.ems-ph.org/journals/journal.php?jrn=rlm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://de.wikisource.org/wiki/Sitzungsberichte_der_Kaiserlichen_Akademie_der_Wissenschaften_in_Wien_%E2%80%93_mathematisch-naturwissenschaftliche_Classe/Inhalt&quot; rel=&quot;nofollow noreferrer&quot;&gt;1848&lt;/a&gt; Sitzungsberichte der Akademie der Wissenschaften (Wien)                              &gt;&lt;/code&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While I hesitate to draw conclusions from these graphs, they match other answers’ idea that there was never &lt;em&gt;one main&lt;/em&gt; language before WWII — more like &lt;em&gt;four&lt;/em&gt; by 1850. Publication by foreigners has a longer tradition in French (Huygens, Leibniz, Bernoulli, Euler, Bessel, Jacobi, Dirichlet, etc.) than in other languages (more like &lt;a href=&quot;https://mathoverflow.net/questions/310523/early-examples-of-mathematicians-publishing-from-home-in-a-foreign-language&quot;&gt;late 19th century&lt;/a&gt;), but for a long time international dissemination remained rather via translations: e.g. in &lt;em&gt;Annalen der Physik&lt;/em&gt; (&lt;a href=&quot;//archive.org/stream/annalenderphysi173unkngoog#page/n163&quot; rel=&quot;nofollow noreferrer&quot;&gt;Young&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/annalenderphysi117unkngoog#page/n254&quot; rel=&quot;nofollow noreferrer&quot;&gt;Biot&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/annalenderphysi96unkngoog#page/n10&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère, Fresnel&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/annalenderphysi86pogggoog#page/n132&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sturm&lt;/a&gt;), &lt;em&gt;Liouville’s&lt;/em&gt; and other journals in French (&lt;a href=&quot;//google.com/books?id=a2sEAAAAYAAJ&amp;amp;pg=PA69&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hamilton&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/s1journaldemat03liou#page/60&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacobi&lt;/a&gt;, &lt;a href=&quot;https://archive.org/stream/s1journaldemat07liou#page/217&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacCullagh&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/s1journaldemat07liou#page/369&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neumann&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/s1journaldemat07liou#page/273&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss&lt;/a&gt;, &lt;a href=&quot;http://www.numdam.org/item?id=NAM_1860_1_19__362_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kummer&lt;/a&gt;, &lt;a href=&quot;//google.com/books?id=dKEwAAAAYAAJ&amp;amp;pg=PA370&quot; rel=&quot;nofollow noreferrer&quot;&gt;Christoffel&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/s2journaldemat11liou#page/337&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plücker&lt;/a&gt;, &lt;a href=&quot;http://www.numdam.org/item?id=ASENS_1869_1_6__251_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beltrami&lt;/a&gt;, &lt;a href=&quot;http://www.numdam.org/item?id=ASENS_1886_3_3__303_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kirchhoff&lt;/a&gt;, &lt;a href=&quot;http://www.numdam.org/item?id=ASENS_1891_3_8__87_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Klein&lt;/a&gt;), &lt;em&gt;Philosophical Magazine&lt;/em&gt; and others in English (&lt;a href=&quot;//biodiversitylibrary.org/page/2528925&quot; rel=&quot;nofollow noreferrer&quot;&gt;Foucault&lt;/a&gt;, &lt;a href=&quot;//biodiversitylibrary.org/page/2529203&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhamel&lt;/a&gt;, &lt;a href=&quot;//biodiversitylibrary.org/page/2510001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plücker&lt;/a&gt;, &lt;a href=&quot;//biodiversitylibrary.org/page/2493316&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clausius&lt;/a&gt;, &lt;a href=&quot;//emis.de/cgi-bin/JFM-item?06.0227.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lipschitz&lt;/a&gt;, &lt;a href=&quot;//google.com/books?id=xl9bAAAAcAAJ&amp;amp;pg=PA69&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermite&lt;/a&gt;, &lt;a href=&quot;//biodiversitylibrary.org/page/39587436&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helmholtz&lt;/a&gt;, &lt;a href=&quot;//archive.org/stream/bulletinnewyork00socigoog#page/n253&quot; rel=&quot;nofollow noreferrer&quot;&gt;Klein&lt;/a&gt;), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt;&lt;br&gt;Sommerville’s &lt;em&gt;Bibliography of non-Euclidean geometry&lt;/em&gt; also has some statistics (&lt;a href=&quot;https://archive.org/stream/bibliographyofno00sommuoft#page/n11&quot; rel=&quot;nofollow noreferrer&quot;&gt;1911, p. viii&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-11-11T05:00:38.257" LastActivityDate="2018-11-11T05:00:38.257" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7681" PostTypeId="1" CreationDate="2018-09-08T13:11:42.000" Score="0" ViewCount="82" Body="&lt;p&gt;At least Newton realized that the motion of the Moon on the heaven and the motion of an apple on the earth are governed by the same law. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But who first proposed that the same laws should hold everywhere, regardless of the distance from us? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2018-09-13T03:25:32.647" Title="Who first came up with the idea that heavenly laws and earthly laws should be the same?" Tags="|physics|philosophy-of-science|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7682" PostTypeId="1" CreationDate="2018-09-08T15:42:04.210" Score="0" ViewCount="283" Body="&lt;p&gt;I am aware that approximately two years ago a question was posted on the existence of Pythagoras. After two years, I want to gain more incite on the thought of those on this site. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was drawn to the idea of Pythagoras ever even existing from Simon Critchley, in his book The Book of Dead Philosophers, where he writes, &quot;Sadly, it is now almost universally assumed by classical scholars that Pythagoras never existed. It seems that there was a group of people in southern Italy called Pythagoreans who invented a &quot;Founder&quot; for their beliefs who, accordingly, lived and died in a manner consistent with those beliefs.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since reading this I have dove deeply into the history of Pythagoras. I have read through Wikipedia for sources and have read and reread their quote, &quot;No primary sources about Pythagoras have survived. This article describes the classical interpretation of Pythagoras, which is based on a small set of texts written between 150 AD and 450 AD. As these texts were written 600 to 1000 years after Pythagoras is said to have lived, their accuracy is uncertain. It is postulated that the classical Pythagoras did not exist prior to these biographies: many of the discoveries and life details they attributed to Pythagoras may have been those of other Pythagoreans, if not fiction.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read all of the sources listed on the other Pythagoras question on this site. Are there any other resources out there that can help prove (or disprove) the existence of Pythagoras? I feel like we keep looking at the same stories that were written years after Pythagoras' death and with no written records, how can anything be accurately assumed about him, his life, and his findings? Did those authors have anything concrete to write from? Anything more than a conversation that was had with Pythagoras? OR a story of something he did told by someone who may have been told by someone else?&lt;/p&gt;&#xA;" OwnerUserId="7822" LastActivityDate="2018-09-08T19:34:19.810" Title="Existence of Pythagoras Resources" Tags="|mathematics|mathematicians|resource-recommendation|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7683" PostTypeId="2" ParentId="7682" CreationDate="2018-09-08T19:34:19.810" Score="3" Body="&lt;p&gt;You can see the specific literature:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Walter Burkert, &lt;a href=&quot;https://books.google.it/books?id=0qqp4Vk1zG0C&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lore and Science in Ancient Pythagoreanism&lt;/a&gt;, Harvard UP (1972, German ed.1962);&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Charles Kahn, &lt;a href=&quot;https://books.google.it/books?id=GKUtAwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pythagoras and the Pythagoreans : A Brief History&lt;/a&gt;, Hackett (2001).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-09-08T19:34:19.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7684" PostTypeId="2" ParentId="7676" CreationDate="2018-09-09T03:46:14.647" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://www.newadvent.org/cathen/12079e.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre de Maricourt (Peter the Pilgrim / &lt;em&gt;Petrus Peregrinus&lt;/em&gt;)&lt;/a&gt; showed that one cannot isolate a magnetic pole because when a magnetic is divided, the two halves each have a north and south pole:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.gutenberg.org/files/33810/33810-h/33810-h.htm#BI.5&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qtbj0.jpg&quot; alt=&quot;Petrus Peregrinus&amp;#39;s broken magnet experiment&quot;&gt;&lt;/a&gt;&#xA;&lt;sup&gt;See also: Nicolas Wipf, “&lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=5207&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre Duhem (1861 - 1916) et la théorie du magnétisme fondée sur la thermodynamique&lt;/a&gt;” (Université Lille 1 Sciences et Technologies, 2011) &lt;strong&gt;p. 77&lt;/strong&gt;.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-09-09T03:46:14.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7686" PostTypeId="1" AcceptedAnswerId="7687" CreationDate="2018-09-10T05:38:28.867" Score="10" ViewCount="531" Body="&lt;p&gt;Usually the word &lt;em&gt;center&lt;/em&gt; means the center of a circle. I have encountered the word &lt;em&gt;center&lt;/em&gt; in group theory, but do not see any connection with the &lt;em&gt;center&lt;/em&gt; of a circle. I think the history of group theory probably has something to do with it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone tell me how the terms &lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Center_(group_theory)&quot; rel=&quot;noreferrer&quot;&gt;center&lt;/a&gt;&lt;/strong&gt; and &lt;strong&gt;centralizer&lt;/strong&gt; came up in group theory? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have searched for &quot;center in group theory&quot; here, but it was not previously asked, and did not get anything from the web either.&lt;/p&gt;&#xA;" OwnerUserId="7827" LastEditorUserId="10106" LastEditDate="2022-02-23T16:50:30.737" LastActivityDate="2022-02-23T16:50:30.737" Title="How did the terms &quot;center&quot; and &quot;centralizer&quot; come up in group theory?" Tags="|mathematics|terminology|abstract-algebra|group-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7687" PostTypeId="2" ParentId="7686" CreationDate="2018-09-10T14:44:03.607" Score="6" Body="&lt;p&gt;The center (originally the &lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?40.0188.01&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;central&lt;/em&gt;&lt;/a&gt;) seems to have appeared between the first and second edition of Burnside’s book (&lt;a href=&quot;https://archive.org/stream/117768378#page/62&quot; rel=&quot;noreferrer&quot;&gt;1897, §53&lt;/a&gt; vs. &lt;a href=&quot;https://books.google.com/books?id=rGMGAQAAIAAJ&amp;amp;pg=PA120&quot; rel=&quot;noreferrer&quot;&gt;1911, §93&lt;/a&gt;) and more precisely in &lt;strong&gt;de Séguier&lt;/strong&gt; (&lt;a href=&quot;https://archive.org/stream/thoriedesgroup00sguoft#page/57&quot; rel=&quot;noreferrer&quot;&gt;1904, §51&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ainsi l’ensemble des éléments normaux de $\mathrm G$ est un diviseur normal qui sera dit &lt;em&gt;central&lt;/em&gt; de $\mathrm G$.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?36.0187.02&quot; rel=&quot;noreferrer&quot;&gt;Jahrbuch&lt;/a&gt;: “Die Gesamtheit invarianter Elemente einer Gruppe nennt Verf. le central der Gruppe.”)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Neither gives an explanation. Guess: conjugacy classes are the orbits of the group’s conjugation action on itself, and a (Kepler) orbit is a point iff it lies at the center. (Then again, maybe they meant the kind of &lt;a href=&quot;https://pt.wikipedia.org/wiki/Central_telef%C3%B4nica&quot; rel=&quot;noreferrer&quot;&gt;central&lt;/a&gt; where operators are busy &lt;a href=&quot;https://mathoverflow.net/questions/309857/why-are-commutative-diagrams-called-commutative#comment772034_309857&quot;&gt;commuting&lt;/a&gt;?) &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-09-11T01:06:38.690" LastActivityDate="2018-09-11T01:06:38.690" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7689" PostTypeId="1" CreationDate="2018-09-11T04:23:18.307" Score="5" ViewCount="402" Body="&lt;p&gt;Yukawa received a Nobel Prize in 1949 for predicting the pi meson but while in Japan he published  his theory which explained the interaction between protons and neutrons in 1935. What work did he do during the war years? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume his work was not needed by the United States to complete the A-Bomb but did Japan keep it secret since he worked in Japan  ? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it was published I  am wondering what  the U.S. reaction was considering the U.S. had banned any mention of atomic physics in any form be stripped from any publications in the United States and this also included the press as well as scientists in the U.S. &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="5040" LastEditDate="2018-09-11T04:34:42.040" LastActivityDate="2018-09-11T04:34:42.040" Title="What work did Hideki Yukawa do during the Second World War?" Tags="|atomic-theory|nobel-prize|world-war-2|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="7692" PostTypeId="1" AcceptedAnswerId="7693" CreationDate="2018-09-11T14:24:59.807" Score="6" ViewCount="413" Body="&lt;p&gt;According to Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Pavel_Urysohn&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Pavel_Urysohn&lt;/a&gt;, Urysohn and Alexandrov first formulated the modern definition of compactness. In which paper did they do this? Is there an English translation? If so, where can I find it?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2018-09-11T15:17:34.650" Title="Origin of Compactness" Tags="|mathematics|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7693" PostTypeId="2" ParentId="7692" CreationDate="2018-09-11T15:17:34.650" Score="7" Body="&lt;p&gt;See Wiki's entry on &lt;a href=&quot;https://en.wikipedia.org/wiki/Compact_space#References&quot; rel=&quot;noreferrer&quot;&gt;Compact space&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Alexandrov, Pavel and Urysohn, Pavel (1929), &quot;Mémoire sur les espaces topologiques compacts&quot;, &lt;em&gt;Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam&lt;/em&gt;, Proceedings of the section of mathematical sciences, 14.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-09-11T15:17:34.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7694" PostTypeId="1" CreationDate="2018-09-12T13:43:23.440" Score="0" ViewCount="65" Body="&lt;p&gt;How did &lt;em&gt;people before Ampere (eg-Coulomb)&lt;/em&gt; who used &lt;em&gt;magnetic-pole model&lt;/em&gt;, know that only two surfaces (north and south poles) of a bar magnet exerts and experiences force while the rest of the surfaces is free from magnetic force? Is there a deduction for this or is it just a guess made from experiment with two bar magnets?&lt;/p&gt;&#xA;" OwnerUserId="7837" LastActivityDate="2018-09-12T13:43:23.440" Title="How did people in pre-Ampere period deduce this?" Tags="|physics|electromagnetism|magnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7695" PostTypeId="2" ParentId="7681" CreationDate="2018-09-13T03:25:32.647" Score="3" Body="&lt;h1&gt;Philoponus&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/John_Philoponus.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Philoponus&lt;/a&gt;, &quot;The Grammarian,&quot; who lived in the late 5&lt;sup&gt;th&lt;/sup&gt; – 2nd ½ of 6&lt;sup&gt;th&lt;/sup&gt; century A.D., argued that the sun is fire and of terrestrial-like, corruptible matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;http://www.encyclopedia.com/topic/John_Philoponus.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;his &lt;em&gt;Dictionary of Scientific Biography&lt;/em&gt; entry&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Philoponus’ main significance for the history of science lies in his being, at the close of antiquity, the first thinker to undertake a comprehensive and massive attack on the principal tenets of Aristotle’s physics and cosmology, an attack unequaled in thoroughness until Galileo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;hr&gt;&lt;sup&gt;Philoponus is certainly one of the &quot;&lt;em&gt;grands génies de l'Antiquité&lt;/em&gt;&quot; (&quot;great geniuses of Antiquity&quot;) and &quot;&lt;em&gt;principaux précurseurs de la Science moderne&lt;/em&gt;&quot; (&quot;principle precursers to modern Science&quot;), &lt;a href=&quot;https://archive.org/stream/lesystmedumond01duhe#page/398/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;as Pierre Duhem wrote&lt;/a&gt; in his magisterial, 10 volume work in the history of medieval physics:&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;sup&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/#panel=book_list&amp;amp;search=title:Le%20Syst%C3%A8me%20du%20Monde&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le système du monde: Histoire des doctrines cosmologiques de Platon à Copernic&lt;/a&gt;&lt;/em&gt;. 10 vols. Paris: A. Hermann, 1913-1959,&lt;/sup&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;partially translated in:&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;sup&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem, Pierre Maurice Marie&lt;/a&gt;. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4757&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval Cosmology: Theories of Infinity, Place, Time, Void, and the Plurality of Worlds&lt;/a&gt;&lt;/em&gt;. Edited and translated by Roger Ariew. Chicago: University of Chicago Press, 1985.&lt;/sup&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;cf. also:&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;sup&gt;Hannam, James. &lt;em&gt;&lt;a href=&quot;https://isidore.co/calibre/browse/book/4182&quot; rel=&quot;nofollow noreferrer&quot;&gt;God’s Philosophers: How the Medieval World Laid the Foundations of Modern Science&lt;/a&gt;&lt;/em&gt;. London: Icon Books, 2009.&lt;/sup&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastActivityDate="2018-09-13T03:25:32.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7696" PostTypeId="1" CreationDate="2018-09-13T05:42:52.993" Score="3" ViewCount="141" Body="&lt;p&gt;It seems that when you play around with the &lt;a href=&quot;https://genealogy.math.ndsu.nodak.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Genealogy Project&lt;/a&gt;, starting at some contemporary mathematician and going backwards through their advisor, advisor's advisor, etc., you tend to arrive at a famous mathematician once you get back to around the late 1700s. Does this indicate there were relatively few active mathematicians back then, and if so how many were there? Or is it perhaps an artifact of the tendency to keep better track of famous mathematicians' students? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a more difficult follow-up question: What percentage of mathematicians from the late 1700s have some of their work preserved until the present day? One difficulty in answering this is that a mathematician who has none of their work preserved is much more likely to have their name be forgotten altogether, in which case we might not know of their existence in the first place. Still, I think it should be possible to provide a reasonable estimate.&lt;/p&gt;&#xA;" OwnerUserId="7842" LastActivityDate="2018-09-13T05:42:52.993" Title="How many active mathematicians were there in Euler's time?" Tags="|mathematicians|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7697" PostTypeId="1" AcceptedAnswerId="7699" CreationDate="2018-09-13T07:19:13.943" Score="4" ViewCount="328" Body="&lt;p&gt;Over the years I read different versions of how the point in geometry (and subsequently in maths) came to be defined as an abstract, dimensionless entity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read that it was Architas who influenced Euclid, but also that it was a mistake made by translators of Plato, rendering 'the point &lt;strong&gt;hasn't any parts&lt;/strong&gt;' with the wrong &quot;the point has &lt;strong&gt;no part&lt;/strong&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you give a detailed account of how the debate evolved and who was actually responsible for the definitive decision?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="609" LastEditDate="2018-09-24T02:34:35.413" LastActivityDate="2018-09-24T02:34:35.413" Title="Who originated the concept of making the point dimensionless?" Tags="|euclidean-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7699" PostTypeId="2" ParentId="7697" CreationDate="2018-09-13T08:35:46.323" Score="4" Body="&lt;p&gt;See Sir Thomas Heath's edition of &lt;a href=&quot;https://books.google.it/books?id=HxhEBgAAQBAJ&amp;amp;pg=PA155&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Thirteen Books of Euclid's Elements, Volume 1&lt;/a&gt;, &lt;strong&gt;Book I, Definitions&lt;/strong&gt;, page 155-on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;A point is that which has no part&lt;/a&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Pre-Euclidean definitions&lt;/strong&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It would appear that this was not the definition given in earlier textbooks;&#xA;  for Aristotle (&lt;em&gt;Topics&lt;/em&gt; VI. 4,141 b 20), in speaking of &quot;the definitions&quot;&#xA;  of point, line, and surface, says that they all define the prior by means of the&#xA;  posterior, a point as an extremity of a line, a line of a surface, and a surface&#xA;  of a solid. The first definition of a point of which we hear is that given by the Pythagoreans (cf. Proclus, p. 95, 21); who defined it as a &quot;monad having position&quot; or &quot;with position added&quot;. It is frequently used by Aristotle, either in this exact form (&lt;em&gt;De anima&lt;/em&gt; I. 4, 409 a 6) or its equivalent: e.g. in &lt;em&gt;Metaph&lt;/em&gt;. 1016 b 24 he says that that which is indivisible every way in respect of magnitude and qua magnitude but has not position is a &lt;em&gt;monad&lt;/em&gt;, while that which is similarly indivisible and has position is a point.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Aristotle's conception of a point as that which is indivisible and has position is further illustrated by such observations as that a point is not a body (&lt;em&gt;De caelo&lt;/em&gt; II. 13, 296 a 17) and has no weight (ibid. III. 1, 299 a 30); again, we can make no distinction between'a point and the place where&#xA;  it is (&lt;em&gt;Physics&lt;/em&gt; IV. I, 209 a II). &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Euclid's definition itself is of course practically the same as that which&#xA;  Aristotle's frequent allusions show to have been then current, except that it&#xA;  omits to say that the point must have position. [...] The definition has been over and over again criticised because it is purely negative.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Simplicius (quoted by an-Nairizi) says that &quot;a point is the beginning of&#xA;  magnitudes and that from which they grow; it is also the only thing which,&#xA;  having position, is not divisible.&quot; He, like Aristotle, adds that it is by its&#xA;  motion that a point can generate a magnitude: the particular magnitude can&#xA;  only be &quot;of one dimension,&quot; viz. a line, since the point does not &quot;spread&#xA;  itself&quot;. Simplicius further observes that Euclid defined a point&#xA;  negatively because it was arrived at by detaching surface from body, line from&#xA;  surface, and finally point from line. &quot;Since then body has three dimensions&#xA;  it follows that a point [arrived at after successively eliminating all three&#xA;  dimensions] has none if the dimensions, and has no part.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;On &lt;a href=&quot;https://en.wikipedia.org/wiki/Archytas&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archytas of Tarentum&lt;/a&gt;, see : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Carl Huffman, &lt;a href=&quot;https://books.google.it/books?id=2ietQ7tX8TEC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archytas of Tarentum : Pythagorean, Philosopher and Mathematician King&lt;/a&gt;, Cambridge UP (2005), page 499-on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is one important piece of external evidence that provides at least&#xA;  partial support for assigning the definitions of the point as “the starting&#xA;  point of the line” and the unit as “the starting point of number” to Archytas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We must take into account that :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;No list of &lt;a href=&quot;https://plato.stanford.edu/entries/archytas/#Sour&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archytas’ works&lt;/a&gt; has come down to us from antiquity, so that we don’t know how many books he wrote. In the face of the large mass of spurious works, it is disappointing that only a few fragments of genuine works have survived.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We can see also &lt;a href=&quot;https://plato.stanford.edu/entries/archytas/#Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archytas as Mathematician&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;We can see also Lucio Russo, &lt;a href=&quot;https://www.jstor.org/stable/41134046?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid's Elements&lt;/a&gt;, &lt;em&gt;AHES&lt;/em&gt; (1998), for the interesting thesis that Euclid's definitions has been interpolated in late antiquity into Euclid’s treatise and their source is the &lt;em&gt;Definitiones&lt;/em&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Hero_of_Alexandria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heron of Alexandria (c.10 AD – c.70 AD)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-09-13T17:28:36.420" LastActivityDate="2018-09-13T17:28:36.420" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7701" PostTypeId="2" ParentId="3605" CreationDate="2018-09-13T17:58:25.573" Score="2" Body="&lt;p&gt;A good book to survey is The Chemical Philosophy by Allen G. Debus. It is less about physics specifically but will give you a good picture of what a scientist thought he was supposed to be doing way back when. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see along the way how each expert had their own versions of Claim, Counterclaim, Evidence and Argument or Proof. They can hardly be blamed for their imprecision since nothing then was standardized. Each had their own motivations and suspicions of others. &lt;/p&gt;&#xA;" OwnerUserId="7845" LastActivityDate="2018-09-13T17:58:25.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7702" PostTypeId="1" CreationDate="2018-09-13T19:07:27.010" Score="2" ViewCount="340" Body="&lt;p&gt;Math Genealogy, &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/search.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.genealogy.math.ndsu.nodak.edu/search.php&lt;/a&gt; is a funny site which aims at listing all PhD's in mathematics, with years, place, titles and advisers. Of course it cannot be complete in any sense. But here is a strange thing that I discovered: numbers of PhD by country in each century (the countries are assigned there according to the modern political map).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;17 century: Germany 250, Netherlands 85, Switerland 23, Italy 16, England 15, France 13.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;18 century: Germany 317, Netherlands 122, England 24, France 14, Switzerland 8, Italy 7, Austria 7.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;19 century: Germany 1563, Netherlands 214, USA 214, France 71, Austria 66, Switzerland 54, England 37, Italy 35. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What conclusions can one make from this statistics? That most of mathematics (by far!) in 17-19 century was done in Germany? That Germans (and Dutch) are more likely to report the data then French or British? That Germans and Dutch are better in keeping historical records? Or what?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. The list of universities founded before 1800 on Wikipedia gives a more&#xA;balanced picture: France 40, Italy 37, Germany 36, Netherlands 6, etc. This indicates that Math Genealogy is probably indeed strongly biased towards Germany. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-09-27T14:52:25.333" LastActivityDate="2018-09-27T14:52:25.333" Title="Strange pattern in Math Genealogy" Tags="|mathematics|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="7703" PostTypeId="1" CreationDate="2018-09-16T23:19:00.953" Score="0" ViewCount="49" Body="&lt;p&gt;It is known that in the sun the main reaction is the fusion process which consumes hydrogen and generates helium. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While this is conceivable by the mass-energy equation, did people have any experimental evidence for it in the lab? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2018-09-16T23:19:00.953" Title="experimental evidence of fusion" Tags="|physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7704" PostTypeId="1" CreationDate="2018-09-17T06:52:19.593" Score="12" ViewCount="1863" Body="&lt;p&gt;Below are several quotes suggesting that Newton's notation had the effect of retarding English mathematics by 50 years, 100 years, or even &lt;em&gt;centuries&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is my simplistic two-sentence historical account of what happened (based on my reading of the various writers below):&lt;/p&gt;&#xA;&lt;p&gt;After Newton, English mathematics was in a dark hole years behind the Continent. Then circa 1820, Englishmen adopted Leibnizian notation and by 1830, had mostly caught up and could once again contribute to mathematics.&lt;/p&gt;&#xA;&lt;p&gt;As a layperson who isn't too familiar with this history, it seems rather incredible that notation alone cost 50+ years of progress.&lt;/p&gt;&#xA;&lt;p&gt;My questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Was English mathematics ever significantly behind — by say 50 years, 100 years, or even &lt;em&gt;centuries&lt;/em&gt;?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;If so, to what extent can this be attributed to Newton's poor notation?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Arturo Magidin on Math.SE (&lt;a href=&quot;https://math.stackexchange.com/a/21209/46234&quot;&gt;2011&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In fact, Leibniz's notation is so good, so superior to the prime notation and to Newton's notation, that England fell behind all of Europe for &lt;em&gt;centuries&lt;/em&gt; in mathematics and science because, due to the fight between Newton's and Leibniz's camp over who had invented Calculus and who stole it from whom (consensus is that they each discovered it independently), England's scientific establishment decided to ignore what was being done in Europe with Leibniz notation and stuck to Newton's... and got stuck in the mud in large part because of it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Noah Kennedy, &lt;em&gt;The Industrialization of Intelligence: Mind and Machine in the Modern Age&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=J79aDwAAQBAJ&amp;amp;pg=PT79&amp;amp;dq=%22Disastrously%20for%20English%20mathematics%22&quot; rel=&quot;noreferrer&quot;&gt;1989&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Disastrously for English mathematics, there was an easy way for an English mathematician to affirm his national allegiance to Newton's claim, a way that seemed innocuous enough but was actually destined virtually to cripple mathematical inquiry in England for more than a century. The two men had quite naturally arrived at two entirely different systems of notation for denoting the central concept of differentiation, and quite naturally the English adopted Newton's and the Germans favoured Leibniz's. The problem for the English was that Leibniz's notation was a far more elegant and evocative expression of the concept and lent itself much more easily to various innovations that rippled through mathematics in the wake of the discovery of the calculus. Primarily because of its utility, Leibniz's notation was in general use throughout Europe and precipitated significant innovation, particularly by the French, while in England, mathematical progress was tortuously slow, in part because of the burden of Newton's notation and in part because English mathematicians had effectively isolated themselves from the common language of continental mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Morris Kline, &lt;em&gt;Mathematical Thought From Ancient to Modern Times&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=VOcUBvbUXlEC&amp;amp;pg=PA622&amp;amp;lpg=PA622&amp;amp;dq=%22england%20too%20languished%22&quot; rel=&quot;noreferrer&quot;&gt;1972&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;England too languished. Brook Taylor, Matthew Stewart (1717–85), and Colin Maclaurin were the only prominent mathematicians.  England's poor performance in view of its great activity in the seventeenth century may be surprising, but the explanation is readily found. The English mathematicians had not only isolated themselves personally from the Continentals as a consequence of the controversy between Newton and Leibniz, but also suffered by following the geometrical methods of Newton. The English settled down to study Newton instead of nature. Even in their analytical work they used Newton's notation for fluxions and fluents and refused to read anything written in the notation of Leibniz.&lt;/p&gt;&#xA;&lt;p&gt;In the first quarter of the nineteenth century the British mathematicians began to take interest in the work on the calculus and its extensions, which had proceeded apace on the Continent. The Analytical Society was formed at Cambridge in 1813 to study this work. George Peacock (1791–1858), John Herschel (1792–1871), Charles Babbage and others undertook to study the principles of &amp;quot;d-ism&amp;quot;—that is, the Leibnizian notation in the calculus, as against those of &amp;quot;dot-age,&amp;quot; or the Newtonian notation. Soon the quotient &lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt; replaced &lt;span class=&quot;math-container&quot;&gt;$\dot{y}$&lt;/span&gt;, and the Continental texts and papers became accessible to English students. Babbage, Peacock, and Herschel translated a one-volume edition of Lacroix's &lt;em&gt;Traité&lt;/em&gt; and published it in 1816. By 1830 the English were able to join in the work of the Continentals. Analysis in England did prove to be largely mathematical physics, though some entirely new directions of work, algebraic invariant theory and symbolic logic, were also initiated in that country.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Jason Bardi, &lt;em&gt;The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=6GMQBwAAQBAJ&amp;amp;pg=PT143&amp;amp;dq=%22Nor%20was%20Newton%27s%20notation%20as%20useful%20as%20the%20superior%20notation%20that%20Leibniz%20had%20invented%22&quot; rel=&quot;noreferrer&quot;&gt;2006&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nor was Newton's notation as useful as the superior notation that Leibniz had invented and the advanced calculus that Johann Bernoulli and the other European mathematicians developed throughout the century. Leibniz had correctly surmised that his symbols would make for the easy development of calculus, and these symbols, which he first penned in his notebooks in Paris in 1675, can still be found to this day in every calculus textbook.&lt;/p&gt;&#xA;&lt;p&gt;In this sense, the high esteem in which Newton was held in Britain was not always a good thing, because, many of the mathematicians and scientists living there in the eighteenth century were behind the iron curtain of Newton's fame and glory. Ironically, as much as Leibniz's reputation suffered in Great Britain, the whole country may have suffered a self-inflicted wound by so underappreciating him. After the calculus wars, British mathematicians were prevented from learning calculus using Leibniz's notations, which were largely in use elsewhere, and they were not finally accepted in that country until the early nineteenth century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Christopher D. Green, &amp;quot;Charles Babbage, the Analytical Engine, and the Possibility of a 19th-Century Cognitive Science&amp;quot; (&lt;a href=&quot;http://www.yorku.ca/christo/papers/Babbage-CogSci.htm&quot; rel=&quot;noreferrer&quot;&gt;2001&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The upholding of the Newtonian notation was a matter of some national pride for the British, for the nasty priority dispute between Newton's and Leibniz's partisans over the discovery of calculus still echoed in the very traditional halls of Cambridge. Unfortunately for the British, Newton's notation was difficult to manipulate algebraically, and they were now some 50 years behind the mathematical developments of their Continental colleagues, who had of course used Leibniz's notation from the first. Babbage, Herschel, and Peacock aimed to put a stop to what they called the &amp;quot;dot-age&amp;quot; of Cambridge (a satirical reference to the dots used to indicate derivatives in the Newtonian notation) and replace it with the &amp;quot;pure d-ism&amp;quot; of Leibniz (who used the letter &amp;quot;d&amp;quot; to indicate the same).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gerald L. Alexanderson, &amp;quot;About the Cover - Voltaire, du Châtelet, and Newton&amp;quot; (&lt;a href=&quot;https://www.ams.org/bull/2015-52-01/S0273-0979-2014-01482-X/S0273-0979-2014-01482-X.pdf&quot; rel=&quot;noreferrer&quot;&gt;2014&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton’s clumsy notation may have hampered progress in England.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user67" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-12-08T16:29:03.350" Title="Was English mathematics behind Europe by many years because of Newton's notation?" Tags="|notation|calculus|newton|leibniz|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7706" PostTypeId="1" AcceptedAnswerId="7720" CreationDate="2018-09-17T17:37:10.260" Score="3" ViewCount="390" Body="&lt;p&gt;The second half of 19th century and first half of 20th century are golden age of modern mathematics and science, as many important ideas and theories were proposed and developed within that period of time. I think that German and French were the main languages for the mathematical papers published in that period. After World War II,  it is well known that English has gradually become the predominant language for mathematical papers until this date. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to have more statistics such as articles published in major journals in German/French/English/Others between 1850 and 1950 and beyond. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="16128" LastEditDate="2022-03-25T16:55:15.293" LastActivityDate="2022-03-25T16:55:15.293" Title="What was the main language in science/mathematics between 1850 and 1950 and beyond" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="7707" PostTypeId="1" CreationDate="2018-09-17T19:13:16.083" Score="1" ViewCount="85" Body="&lt;p&gt;The old days being roughly 1600 - 1850, the time when chemistry and the nature of the elements was being investigated empirically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, hydrogen gas was discovered by Robert Boyle in 1671 after pouring acids over iron filings. It was recognized by Henry Cavendish in 1766.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did these people trap the gas? What kind of equipment and setup allows this? And crucially, how pure could it be without mixing with normal ambient air?&lt;/p&gt;&#xA;" OwnerUserId="2546" LastActivityDate="2018-09-17T19:13:16.083" Title="In the old days, how did chemists trap and measure gases?" Tags="|physics|chemistry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7708" PostTypeId="1" CreationDate="2018-09-18T07:35:43.500" Score="0" ViewCount="128" Body="&lt;p&gt;I'm interested in the history of calculus &amp;amp; analysis and  looking for books that examine in some detail the history of writings on these subjects, mainly the history of the 17th-century &quot;Treatise of Fluxions&quot; to the 19th-c. &quot;Cours d' Analyse&quot;.&lt;br&gt;&lt;br&gt;&#xA;Which works were significant and why? Which works, now forgotten, can still be useful in the understanding of the revolutionary rise of infinitesimal analysis? &lt;br&gt;&lt;br&gt; Personal favorites would be welcome too. Any suggestions?&lt;/p&gt;&#xA;" OwnerUserId="5909" LastActivityDate="2018-09-18T07:35:43.500" Title="Books on the history of influential Treatises on Calculus and Analysis" Tags="|calculus|book-recommendation|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7709" PostTypeId="1" CreationDate="2018-09-18T09:56:18.063" Score="1" ViewCount="114" Body="&lt;p&gt;Today we can predict lunar/solar eclipses down to the precision of seconds.&lt;a href=&quot;https://www.timeanddate.com/eclipse/accuracy.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what was the precision like in medieval times? I imagine it must have improved over centuries. To keep the question very specific I'm looking for the precision in prediction of lunar eclipse around the time of Columbus. But if a broad overview of historical development can be given, it'd be very helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Background: Christopher Columbus used the knowledge of an upcoming lunar eclipse to threaten the local tribes against his God's fury. I was wondering if they all knew exactly how long to wait for the event or they just knew the date on which it'd happen but not the precise time.&lt;/p&gt;&#xA;" OwnerUserId="6220" LastActivityDate="2018-09-18T12:21:32.183" Title="Precision of Ancient/Medieval Astronomy (wrt Eclipses)" Tags="|astronomy|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7710" PostTypeId="2" ParentId="7704" CreationDate="2018-09-18T10:15:34.450" Score="10" Body="&lt;p&gt;Several factors come together to suggest that the idea that &quot;English mathematics [was] ever significantly behind -- by say 50 years, 100 years, or even centuries&quot; (i.e. in the post-Newtonian 18th or early 19th centuries) is at best a sweeping over-generalization, although something very like it has clearly become a received view. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two recent valuable studies in particular shed some light on the question: Judith V Grabiner (&lt;a href=&quot;https://www.jstor.org/stable/2974733&quot; rel=&quot;noreferrer&quot;&gt;'&lt;em&gt;...The Continental Influence of Maclaurin's Treatise...&lt;/em&gt;' &lt;em&gt;American Mathematical Monthly&lt;/em&gt;, 104 (1997), 393-410&lt;/a&gt;), and by Niccolo Guicciardini (&lt;a href=&quot;http://www.jstor.org/stable/4130350&quot; rel=&quot;noreferrer&quot;&gt;'&lt;em&gt;...Newton's Mathematical Legacy...&lt;/em&gt;', in '&lt;em&gt;Early Science and Medicine&lt;/em&gt;' 9 (2004), 218-256&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They show for example: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) that Colin Maclaurin's mathematical work (in fluxions) up to the 1740s was well-received by continental mathematicians (Grabiner, 1997); Maclaurin was also awarded two prizes by the Academie royale des sciences in Paris. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;{Edit begins:} In particular, Maclaurin's work gained a special contemporary continental appreciation: he was credited for his important contribution, in his work of 1742, of placing the infinitesimal calculus on a rigorous mathematical foundation, which the methods of Leibnitz in themselves did not provide. Thus he definitely and satisfactorily answered the attacks on the foundations of the calculus that had arisen repeatedly during the 18th century (see also &lt;a href=&quot;https://hsm.stackexchange.com/questions/7988&quot;&gt;Did Michel Rolle say that the calculus is &amp;quot;a collection of ingenious fallacies&amp;quot;?&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jean-%C3%89tienne_Montucla&quot; rel=&quot;noreferrer&quot;&gt;Jean-Étienne Montucla's&lt;/a&gt; &lt;a href=&quot;https://books.google.com/books?id=3_YKVLHHEEQC&quot; rel=&quot;noreferrer&quot;&gt;Histoire des Mathématiques&lt;/a&gt;, 2nd ed. vol.3, was mostly completed by the end of Montucla's life and published 1802 shortly after his death by &lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%A9r%C3%B4me_Lalande&quot; rel=&quot;noreferrer&quot;&gt;Jérôme de Lalande&lt;/a&gt;. It contains appreciations of the defence of the calculus by a number of British mathematicians, but gives this special acknowledgement to Maclaurin (in passages from pp.116 and 118, here in my translation):-- &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Nobody is unaware these days that the infinitesimal calculus is in&#xA;  its fundamentals absolutely the same as what Newton has called the&#xA;  calculus of fluxions.  Now, this latter has nothing that is not in&#xA;  conformity with the most rigorous principles of Geometry, as has been&#xA;  shown at full length. Accordingly, both the one and the other must&#xA;  enjoy the same degree of certainty.&quot; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &quot;It was seemingly in answer to the attacks of Berkeley that Mr&#xA;  Maclaurin undertook his 'Treatise of Fluxions' that appeared in 1742.&#xA;  There the method of Newton is fully demonstrated without any&#xA;  assumption of infinitesimals or anything else capable of lending&#xA;  itself to controversy ...  Mr Maclaurin's demonstrations are of&#xA;  prodigious length ... he could have limited himself to some examples&#xA;  ... . [But] however that may be, one can say that if any doubts could&#xA;  remain about the solidity of Newton's method, they are entirely&#xA;  dissipated by this work of Maclaurin ... .&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also shown by the studies cited above is that {edit ends:}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) Maclaurin's was not an isolated example. Guicciardini (2004) discusses the work and influence of a number of 18th-century mathematicians writing in English, including Brook Taylor, James Stirling, Abraham De Moivre, Thomas Simpson, William Emerson, and others, as well as Maclaurin himself. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another work by Guicciardini offers strong arguments that the differing notational preferences between 18th-century Newtonians and Leibnizians were not as big of an issue as has often since been made out: they were interconvertible and actually interconverted: ('&lt;em&gt;Reading the Principia&lt;/em&gt;', Cambridge, 1999; e.g. points referred to in chapter 9, p.250 et seq.).  Guicciardini recommended that &quot;it is more fruitful, and more adherent to historical evidence, to focus on the amount of shared knowledge between the two [Newtonian and Leibnizian] schools.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Guicciardini (2004, at 220) also notes that the received view &quot;can easily be traced back to the irreverent writings of reformers such as John Playfair, John Toplis, and Robert Woodhouse, but even more so to the fellows of the Cambridge Analytical Society who, at the beginning of the nineteenth century, tried to introduce the algebraic methods of Joseph Louis Lagrange and F. A. Arbogast into Great Britain. Like all reformers, they offered a pessimistic view of the past. Since then, this received view of eighteenth-century Newtonian mathematics has prevailed in the histories of mathematics.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it may be suspected that there is often 'no smoke without fire', and the 'received view' of English mathematics may have some source in institutional rivalries, in which 18th-century British mathematicians lost out to representatives of competing branches of science. After Newton's death in 1727, as Guicciardini (2004 at 250) noticed, there was a contest in the Royal Society between the 'philomaths', who sympathised with Newton's view of the primacy of mathematics, and those who saw themselves as 'naturalists'. The election of Hans Sloane in 1727 as President to succeed Newton, &quot;marked a defeat of the philomaths&quot;.  The primacy of the 'naturalists' and relative discouragement of mathematicians in the Royal society seems to have continued for a long time. Thus, Marie Boas Hall's '&lt;em&gt;All Scientists Now&lt;/em&gt;' (Cambridge, 1984, especially at chapter 1, '&lt;em&gt;The eighteenth-century legacy&lt;/em&gt;') recorded that under the 42-year presidency of botanist Joseph Banks, there was only one attempt to challenge Banks' dominance, and that came in the 1780s from two men of mathematical sympathy, Charles Hutton and Samuel Horsley. Hutton had been foreign secretary of the Society but was dismissed by Banks for reasons not altogether clear.  Horsley &quot;tried to make it a revolt of the mathematical scientists against ... the biological scientists&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What may perhaps be truer than the 'received view' is that mathematics in Britain had a period of some relative institutional discouragement after the death of Newton: and this may have had a repressive effect on the numbers of practicing mathematicians and the opportunities open to them, even while those who were active were in touch and interactive with mathematical trends and mathematicians outside their own country, and not 'behind' by whatever time-units have been suggested. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="-1" LastEditDate="2018-12-07T22:32:17.560" LastActivityDate="2018-12-07T22:32:17.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7711" PostTypeId="2" ParentId="7709" CreationDate="2018-09-18T12:13:00.980" Score="2" Body="&lt;p&gt;First of all there is a big difference between solar and Lunar eclipses. (Lunar eclipses are easy to predict, while solar are difficult). Second, there was no much improvement in the medieval period. Development in astronomy was very slow in the countries of Islam, and non-existent in the West between the time of Ptolemy&#xA;(2nd century) and 16 century. (In the period 2-13 centuries there was even a regress). Third, even the tools of measuring time precisely did not exist. During all this period, Lunar eclipses could be predicted within few hours precision (as Columbus did), but Solar ones could not be reliably predicted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The difference is that Lunar eclipses occur at a prescribed time everywhere, while&#xA;Solar ones only in a small area on the Earth, and the time depends on where you are located. So the best one could say is that a solar eclipse &quot;may occur&quot; on certain date. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both predictions depend mainly on the theory of Moon motion, and the first real improvement of this theory, which had practical implications,  after Ptolemy, was due to Tycho Brahe.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-09-18T12:21:32.183" LastActivityDate="2018-09-18T12:21:32.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7712" PostTypeId="2" ParentId="7704" CreationDate="2018-09-18T21:56:02.293" Score="1" Body="&lt;p&gt;The question reads like an answer (Conifold’s remark). So, answer in the form of a question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Open a modern textbook on calculus or differential equations. To whom are the theorems and methods due?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(The result is not wholly uniform, but there is a trend. &lt;em&gt;Need&lt;/em&gt; it be attributed to notation, “crisis,” or anything? Does people &lt;em&gt;not&lt;/em&gt; discovering things require an explanation?)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-09-18T22:03:32.023" LastActivityDate="2018-09-18T22:03:32.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7713" PostTypeId="1" CreationDate="2018-09-20T06:07:14.387" Score="2" ViewCount="111" Body="&lt;p&gt;Can you give a few details? where/how was it measured, was the value .511MeV found right away, etc..&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Readers who'd like a proper answer to the question may find &lt;a href=&quot;https://www.google.com/url?sa=t&amp;amp;rct=j&amp;amp;q=&amp;amp;esrc=s&amp;amp;source=web&amp;amp;cd=13&amp;amp;cad=rja&amp;amp;uact=8&amp;amp;ved=2ahUKEwiZ2ZeZvs7dAhXLxYUKHafoAkk4ChAWMAJ6BAgIEAI&amp;amp;url=http%3A%2F%2Fscipp.ucsc.edu%2Foutreach%2Finternships%2FIntern_desktop%2FElectron-positron%2520Annihilation-Veronica-David-Mark.ppt&amp;amp;usg=AOvVaw1ut6XaQQ6hbewQcWbl-JlX&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt; useful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The value found by this experiment is .537 MeV&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do hope that someone can answer the question from an historical point of view, specifying when this experiment was made first, and if Einstein's conjecture (we all know that his 1905 paper was not even intended as a proof by himself) was verified by any different experiment.&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="426" LastEditDate="2018-12-21T17:29:25.720" LastActivityDate="2018-12-21T17:29:25.720" Title="When was the energy of the electron first measured?" Tags="|energy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7714" PostTypeId="2" ParentId="7713" CreationDate="2018-09-20T13:33:19.483" Score="5" Body="&lt;p&gt;I suspect what you really are after is the electron's mass, since we can then &quot;easily&quot; get its energy via an obscure :-) equation developed by some fellow named Einstein. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The path thru measuring the charge-to-mass ratio, followed by some famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Millikan_Experiment&quot; rel=&quot;noreferrer&quot;&gt;oil-drop experiments&lt;/a&gt;, is discussed pretty well in a &lt;a href=&quot;https://physics.stackexchange.com/questions/95801/how-to-measure-the-mass-of-the-electron&quot;&gt;physics.SE&lt;/a&gt; page. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-09-20T13:33:19.483" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7715" PostTypeId="2" ParentId="7671" CreationDate="2018-09-20T22:33:36.180" Score="1" Body="&lt;p&gt;As mentioned by others, lots was written in german during the 1700-1800s. All the german mathematicians wrote to each other in german. And lots of people in East-Prussia (big parts of now Poland), Habsburg and later Austro-Hungarian empire was super huge and many of people living there wrote in german even though it was not their mother tongue. It is quite impressive how they managed to learn it so well given how different germanic and slavonic languages are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lots of mathematics was in fact written in communication between mathematicians by letter during this time. Probably much more than was ever published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another question is which language had the most journals and/or number of articles published in such journals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I guess the answer you seek could depend on if you mean total amount written or in which form it was published, as letters or in journals or standalone books or some other form.&lt;/p&gt;&#xA;" OwnerUserId="7268" LastActivityDate="2018-09-20T22:33:36.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7716" PostTypeId="1" CreationDate="2018-09-20T22:56:08.913" Score="5" ViewCount="185" Body="&lt;p&gt;I wonder when was discovered the nine point conic. English Wikipedia article about it  &lt;a href=&quot;https://en.wikipedia.org/wiki/Nine-point_conic&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nine-point_conic&lt;/a&gt; is misleading. The nine point conic wasn't discovered in &lt;strong&gt;1892&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the book Trilinear Coordinates, by William Allen Whitworth,&#xA; published in &lt;strong&gt;1866&lt;/strong&gt;(&lt;a href=&quot;https://ia801405.us.archive.org/21/items/trilinearcoordin029731mbp/trilinearcoordin029731mbp.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://ia801405.us.archive.org/21/items/trilinearcoordin029731mbp/trilinearcoordin029731mbp.pdf&lt;/a&gt;) one can see that the proof of the existence of the nine point conic is proposed as an exercise (page 473, exercises 438 and 439).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My rough guess is that the nine point conic was discovered in the very beginning of the nineteen century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="7866" LastActivityDate="2021-07-19T03:12:49.080" Title="When was the nine point conic discovered?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7720" PostTypeId="2" ParentId="7706" CreationDate="2018-09-23T23:01:11.090" Score="3" Body="&lt;p&gt;Here are plots using &lt;a href=&quot;https://www.emis.de/MATH/JFM/full.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jahrbuch&lt;/a&gt; for 1870–1940 and &lt;a href=&quot;https://zbmath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zentralblatt&lt;/a&gt; after that. A glance at their detailed output suggests some unreliability (e.g. Zentralblatt counts a paper twice when both reviewed it), but hopefully not too much overall. Papers and books in other languages (Danish, Dutch, Hungarian, Japanese, Latin, Norwegian, Polish, Portuguese, Spanish, Swedish,...) are left out as only after 1980 does their &lt;em&gt;sum total&lt;/em&gt; (slightly) surpass Italian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/BTSEW.png&quot; alt=&quot;1870-2009&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Zoom on the requested pre-1950 period:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/QJhQT.png&quot; alt=&quot;1870-1949&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; Similar picture from Ammon (&lt;a href=&quot;//doi.org/10.1515/applirev-2012-0016&quot; rel=&quot;nofollow noreferrer&quot;&gt;2012, p. 338&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ll58w.png&quot; alt=&quot;1880-2005&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-09-24T03:20:52.327" LastActivityDate="2018-09-24T03:20:52.327" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7722" PostTypeId="2" ParentId="162" CreationDate="2018-09-24T23:56:33.317" Score="2" Body="&lt;p&gt;A brief nubbin of an answer, but I think neglected by other answers: Bismarck's big centralized push to subsidize science and industry, in order to overtake Britain in such, in particular (and perhaps also France, et al).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is similar to the U.S.'s NSF-and-other strong federal subsidization of &quot;hard science&quot; after WWII, both because &quot;the bomb won the war&quot; (not to mention crypto...), and to keep ahead of the main competitor, the Russians (who did acquire &quot;the bomb&quot;, and, on top of everything, orbited Sputnick first, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, at the very least, there was another &quot;golden period&quot; of U.S. mathematics, namely, post-WWII until perhaps the collapse of the former Soviet Union c. 1990. Well-funded, high-esteem. This tended to promote English as &quot;the language of science&quot; in and after that period...&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2018-09-24T23:56:33.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7726" PostTypeId="1" CreationDate="2018-09-26T18:25:39.180" Score="2" ViewCount="230" Body="&lt;p&gt;Where does the prefix &quot;super&quot; from &quot;supersymmetry&quot; come from?&lt;/p&gt;&#xA;" OwnerUserId="7893" OwnerDisplayName="AWanderingMind" LastEditorUserId="131" LastEditDate="2018-09-26T18:59:02.460" LastActivityDate="2018-09-28T17:22:34.743" Title="Where does the prefix &quot;super&quot; from &quot;supersymmetry&quot; come from?" Tags="|physics|terminology|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7727" PostTypeId="2" ParentId="7726" CreationDate="2018-09-27T09:31:35.317" Score="0" Body="&lt;p&gt;The question does not seem to be sharply formulated, so the answer will be vague: &quot;super&quot; comes from the way semantic evolves. Words have antonyms, usually conceived as negations, but negation can have different modalities, so instead of a pair, language usually has more terms. This idea has been developed in &lt;a href=&quot;https://en.wikipedia.org/wiki/Semiotic_square&quot; rel=&quot;nofollow noreferrer&quot;&gt; the semiotic square&lt;/a&gt; proposed by Greimas. So we have e.g. 'natural' and 'unnatural' but also 'artificial' and more over 'supernatural'. In science: symmetry and asymmentry and anti-symmetry. I doubt that  'supersymmetric' has any mental relationship with 'supernatural', but one has to know who first started using the word (and here we might recall the details behind the famous 'god particle'). &lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2018-09-27T09:40:15.833" LastActivityDate="2018-09-27T09:40:15.833" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7728" PostTypeId="2" ParentId="7726" CreationDate="2018-09-28T17:22:34.743" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://www.oed.com/view/Entry/194186&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Super-&lt;/em&gt;&lt;/a&gt; means &amp;quot;situated over, above, higher than, or (less commonly) upon another&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The earliest usages the &lt;em&gt;OED&lt;/em&gt; gives for &lt;a href=&quot;http://www.oed.com/view/Entry/290530&quot; rel=&quot;nofollow noreferrer&quot;&gt;supersymmetry&lt;/a&gt;—&amp;quot;Symmetry of a more general type or of a higher order; (&lt;em&gt;Physics&lt;/em&gt;) a very general type of mathematical symmetry proposed as relating fermions and bosons&amp;quot;—in the physics sense are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1958   &lt;em&gt;Progr. Theoret. Physics&lt;/em&gt; &lt;strong&gt;19&lt;/strong&gt; 639   If there is the super symmetry between &lt;span class=&quot;math-container&quot;&gt;$Λ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Σ$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$gK^2/ℏc = e^2/ℏc = 0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Λ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Σ$&lt;/span&gt; must have the same intrinsic parity.&lt;/p&gt;&#xA;&lt;p&gt;1961   S. L. Glashow &amp;amp; M. Gell-Mann in &lt;em&gt;Ann. Physics&lt;/em&gt; &lt;strong&gt;15&lt;/strong&gt; 297   One of them might play a role in the physics of the strong interactions if there is an underlying super-symmetry, transcending charge independence, that is badly broken.&lt;/p&gt;&#xA;&lt;p&gt;1966   &lt;em&gt;Ann. Physics&lt;/em&gt; &lt;strong&gt;38&lt;/strong&gt; 554   The spectrum happens to coincide with that contained in the (6*, 6)–(6, 6*) representation of the recently proposed U6 × U6 supersymmetry.&lt;/p&gt;&#xA;&lt;p&gt;1974   A. Salam &amp;amp; J. Strathdee in &lt;em&gt;Physics Lett.&lt;/em&gt; B. &lt;strong&gt;51&lt;/strong&gt; 353/1   These authors [&lt;em&gt;sc.&lt;/em&gt; J. Weiss and B. Zumino, 1974]..designate this Fermi-Bose symmetry by the expression ‘super-gauge’. Since the word ‘gauge’ has come to be more commonly associated with ‘gauges of the second kind’ or local symmetries, it is confusing to use super-gauge to describe what is indeed a global symmetry of fermions and bosons. We suggest therefore that the expression ‘super-symmetry’ might be more appropriate.&lt;/p&gt;&#xA;&lt;p&gt;1975   S. Ferrara &amp;amp; B. Zumino in &lt;em&gt;Nucl. Physics&lt;/em&gt; B. &lt;strong&gt;87&lt;/strong&gt; 207   Supersymmetry transformations form an extended Lie algebra.&lt;/p&gt;&#xA;&lt;p&gt;1977   &lt;em&gt;Physics Today&lt;/em&gt; Apr. 49/3   As far as I know, the only natural way to keep a scalar boson massless is to have a ‘supersymmetry’,..which puts scalar fields in the same multiplet as massless fermion fields.&lt;/p&gt;&#xA;&lt;p&gt;2001   &lt;em&gt;N.Y. Mag.&lt;/em&gt; 14 May 93/1   Paintings whose images draw on the ‘supersymmetry’ physics theory that every particle of matter possesses a ‘shadow’ particle.&lt;/p&gt;&#xA;&lt;p&gt;2005   L. Randall &lt;em&gt;Warped Passages&lt;/em&gt; xvii. 347   The idea was to communicate supersymmetry breaking not through the graviton, but instead through gauginos, the supersymmetric partners of the gauge bosons.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-28T17:22:34.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7729" PostTypeId="1" AcceptedAnswerId="7730" CreationDate="2018-09-28T18:35:55.133" Score="2" ViewCount="127" Body="&lt;p&gt;This keeps confusing me. I keep imagining &quot;permittivity&quot; as being &quot;The ability for a substance to permit electricity&quot;, when it is the opposite. High permittivity means it is hard (requires a lot of charge) for electric fields to be produced, while low permittivity means it is easy. I can't find the etymology on the internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although it's not a traditional English word, I keep hearing it like it's just a normal English word and deducing &quot;ability to permit&quot; just like  &quot;morality&quot; or &quot;similarity&quot;. High &quot;A-ity&quot; meaning more A/easier to do A.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The inverse doesn't make much sense, such as more permittivity =&gt; more charge, as the field doesn't create the charge - the charge creates the field.&lt;/p&gt;&#xA;" OwnerUserId="7898" LastEditorUserId="7898" LastEditDate="2018-09-28T18:41:38.040" LastActivityDate="2018-09-28T22:07:32.260" Title="Why is the amount of charge needed to generate 1 unit of electric flux, called permittivity?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7730" PostTypeId="2" ParentId="7729" CreationDate="2018-09-28T21:21:47.520" Score="4" Body="&lt;p&gt;Heaviside coined the term &amp;quot;permittivity&amp;quot;. He explains it and related terms in his &lt;a href=&quot;https://archive.org/details/electricalpapers02heavrich/page/124&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Electrical Papers&lt;/em&gt; (vol. 2) pp. 124-5&lt;/a&gt;, § &amp;quot;Nomenclature Scheme&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To explain the word &amp;quot;permittance&amp;quot; that I used in the last Section, I may remark that in stating my views in 1885 in several communications to this journal on the subject of a systematic and convenient electrical nomenclature based upon the explicit recognition of the three fluxes, conduction-current, magnetic induction, and electric displacement, proposing several new words, some of which have found partial acceptance, I remarked upon the unadaptable character of the word &amp;quot;capacity.&amp;quot; It must be the capacity of something or other, as of permitting displacement. I did not then go further in connection with the flux displacement than to use &amp;quot;elastance,&amp;quot; for the reciprocal of electrostatic capacity. The following shows the scheme so far as it is at present developed:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/electricalpapers02heavrich/page/124&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/45sGD.png&quot; alt=&quot;Nomenclature Scheme&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-09-28T22:07:32.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7731" PostTypeId="1" AcceptedAnswerId="7732" CreationDate="2018-09-29T01:37:11.957" Score="2" ViewCount="562" Body="&lt;p&gt;I read below paragraph from the book &quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/0321816196&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Friendly Introduction to Number Theory&lt;/a&gt;&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The use of &quot;&lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;&quot; to denote the square root of negative &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; dates back to&#xA;  the days when people viewed such numbers with great suspicion and,&#xA;  indeed, felt that they were so far fom being real numbers that they&#xA;  deserved to be called imaginar. In these more enlightened times we&#xA;  recognize that all1 numbers are, to some extent, abstractions that can&#xA;  be used to solve certain sorts of problems. For example, negative&#xA;  numbers (which were not used by European mathematicians even in the&#xA;  fourteenth century, although they were in use in India as early as AD&#xA;  600) are not needed for counting cattle, but they are useful in&#xA;  keeping track of who owes how many cattle to whom. Fractions arise&#xA;  naturally when people start dealing with objects that can be&#xA;  subdivided, such as bushels of wheat or felds of corn. Irational&#xA;  numbers-that is, numbers that are not factions-appear in even the&#xA;  simplest sorts of measurements, as the Pythagoreans discovered when&#xA;  they found that the diagonal of certain geometric fgures may be&#xA;  incommensurable with their sides.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...but even if we introduce more general irational numbers, we still&#xA;  won't be able to solve the very simple equation &lt;span class=&quot;math-container&quot;&gt;$x^2 + 1 = 0$&lt;/span&gt;. Since this&#xA;  equation doesn't have any solutions in &quot;real numbers,&quot; there's nothing&#xA;  to stop us fom creating a new sort of number to be a solution and&#xA;  giving that new number the name i. This is no diﬀerent fom observing&#xA;  that since the equation &lt;span class=&quot;math-container&quot;&gt;$x^2 - 5 = 0$&lt;/span&gt; has no solution in factions, we are&#xA;  fee to create a solution and call it &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;. In fact, we're even doing the&#xA;  same thing when we observe that &lt;span class=&quot;math-container&quot;&gt;$3x - 7 = 0$&lt;/span&gt; has no solutions in whole&#xA;  numbers, so we create a solution and call it &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As I recall from other readings, when people &lt;strong&gt;invented&lt;/strong&gt; the &lt;code&gt;complex number&lt;/code&gt;, it &lt;strong&gt;&lt;em&gt;merely&lt;/em&gt;&lt;/strong&gt; served as a &lt;strong&gt;bizarre&lt;/strong&gt; way to solve some hard equations. There was no practical/massive application of such &lt;strong&gt;invention&lt;/strong&gt; until years later in the electrical engineering, communication and other physics areas. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it gives me an impression that mathematics development can sometimes &lt;strong&gt;exceed&lt;/strong&gt; the practical needs. Some currently seemingly &lt;strong&gt;bizarre&lt;/strong&gt; solutions/operations look that way only because &lt;strong&gt;a proper context&lt;/strong&gt; just haven't revealed itself yet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is my understanding sound?&lt;/p&gt;&#xA;" OwnerUserId="592" LastEditorUserId="131" LastEditDate="2018-09-29T16:30:19.100" LastActivityDate="2018-10-04T19:24:39.660" Title="Mathematics development can sometimes **exceed** the practical needs, right?" Tags="|mathematics|complex-analysis|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7732" PostTypeId="2" ParentId="7731" CreationDate="2018-09-29T14:51:19.833" Score="10" Body="&lt;p&gt;Yes, this happens many times.  Mathematics developed purely for internal mathematical reasons, later turns out to have applications elsewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See the &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences&quot; rel=&quot;noreferrer&quot;&gt;essay&lt;/a&gt; by Wigner, &quot;The unreasonable effectiveness of mathematics in the natural sciences&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I heard many years ago a lecture by Heisenberg.  He told how, in finding the first things about quantum mechanics, he came up with a strange sort of multiplication, which was not commutative.  When he told a mathematician, he got the reply, &quot;Oh, yes.  That is called matrix multiplication.  It was intruduced by Cayley a hundred years ago.&quot;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-09-29T14:51:19.833" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7733" PostTypeId="1" CreationDate="2018-09-29T17:26:19.523" Score="3" ViewCount="303" Body="&lt;p&gt;The Wikipedia article on Gottfried Wilhelm Leibniz mentions, in the chapter on symbolic thought, that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Leibniz saw that the uniqueness of prime factorization suggests a central role for prime numbers in the universal characteristic, a striking anticipation of Gödel numbering. Granted, there is no intuitive or mnemonic way to number any set of elementary concepts using the prime numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;First of all, as far as I understand, Gödel numbering is an operation that &lt;strong&gt;does assign a prime number to every elementary symbol (or the elementary concept it represents) or logical operation (such as logical conjunction)&lt;/strong&gt; , and therefore the second sentence is unclear to me.&lt;/p&gt;&#xA;&lt;p&gt;Secondly, I'd like to get a reference about the manuscripts in which Leibniz anticipated Gödel numbering (of which the wikipedia claim is based). I guess it's in of his writings on &amp;quot;characteristic numbers&amp;quot; and the &amp;quot;calculus of propositions&amp;quot;, but the Leibniz's nachlass is too vast and i don't know where to find the relevant material.&lt;/p&gt;&#xA;&lt;p&gt;Everyone with useful information is welcome to contribute to the answer of my question.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2022-09-23T05:59:27.287" LastActivityDate="2023-10-18T12:04:35.587" Title="Question about Leibniz's &quot;characteristic numbers&quot; and propositional logic" Tags="|mathematics|mathematicians|reference-request|mathematical-logic|leibniz|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7734" PostTypeId="1" CreationDate="2018-09-29T20:41:58.497" Score="3" ViewCount="151" Body="&lt;p&gt;I used Wikipedia to try to figure this out but I am not clear on the explanation.  As for Newton's Law of Gravitation does not give the value for the Gravitational constant but it was measured by Cavendish.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering how the derivation was put together to arrive at the fine structure constant given that the units all cancel out or perhaps Sommerfeld calculated it's value.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="658" LastEditDate="2018-10-07T03:49:56.513" LastActivityDate="2018-10-07T03:49:56.513" Title="Did Sommerfeld derive or measure the fine structure constant the first time he introduced it?" Tags="|quantum-mechanics|atomic-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7735" PostTypeId="2" ParentId="7734" CreationDate="2018-09-29T22:11:32.430" Score="4" Body="&lt;p&gt;Computed (as &lt;span class=&quot;math-container&quot;&gt;$\alpha=e^2/\hbar c$&lt;/span&gt;) in equation &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=umn.31951d003144537&amp;amp;num=51&quot; rel=&quot;nofollow noreferrer&quot;&gt;(12a), p. 51&lt;/a&gt; of: Zur Quantentheorie der Spektrallinien, &lt;em&gt;Ann. Physik&lt;/em&gt; (4) &lt;strong&gt;51&lt;/strong&gt; (1916) 1–94.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Earlier he had used &lt;span class=&quot;math-container&quot;&gt;$\alpha = (e^2/2\hbar c)^2$&lt;/span&gt; in equation &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3976519&amp;amp;num=469&quot; rel=&quot;nofollow noreferrer&quot;&gt;(16), p. 469&lt;/a&gt; of Die Feinstruktur der Wasserstoff- und der Wasserstoff-ähnlichen Linien, &lt;em&gt;Münch. Ber.&lt;/em&gt; &lt;strong&gt;1915&lt;/strong&gt; (1916) 459–500. For discussion see e.g. M. Eckert (ed.) &lt;em&gt;Die Bohr-Sommerfeldsche Atomtheorie&lt;/em&gt;, Springer (&lt;a href=&quot;//doi.org/10.1007/978-3-642-35115-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;2013&lt;/a&gt;).)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-09-29T22:22:25.973" LastActivityDate="2018-09-29T22:22:25.973" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7736" PostTypeId="1" AcceptedAnswerId="7737" CreationDate="2018-09-30T06:43:05.097" Score="2" ViewCount="164" Body="&lt;p&gt;In the Wikipedia &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Baroque#Origin_of_word&quot; rel=&quot;nofollow noreferrer&quot;&gt;Baroque&lt;/a&gt;&amp;quot; article I found this quote from &amp;quot;&lt;a href=&quot;https://www.rousseauonline.ch/Text/volume-9-dictionnaire-de-musique.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dictionnaire&#xA;de Musique&lt;/a&gt;&amp;quot; by Jean-Jacques Rousseau:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;BAROQUE. Une Musique Baroque est celle dont l’Harmonie est confuse, chargée de Modulations &amp;amp; Dissonances, le Chant dur &amp;amp; peu naturel, l’Intonation difficile, le Mouvement contraint.&lt;/p&gt;&#xA;&lt;p&gt;Il y a bien de l’apparence que ce terme vient du Baroco des Logiciens.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Being a mathematician, I became intrigued - where to read about this &amp;quot;Baroco des Logiciens&amp;quot;? What is it?&lt;/p&gt;&#xA;" OwnerUserId="3439" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-02T10:50:47.590" Title="What does Rousseau mean by &quot;Baroco des Logiciens&quot;?" Tags="|terminology|mathematical-logic|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7737" PostTypeId="2" ParentId="7736" CreationDate="2018-09-30T12:40:23.467" Score="2" Body="&lt;p&gt;As noted in the comments, &quot;baroco&quot; is a mnemonic for a certain type of syllogism in classical Aristotlean logic.  See &lt;a href=&quot;https://en.wikipedia.org/wiki/Syllogism#Baroco_(AOO-2)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it is no longer believed that &quot;Baroque&quot; comes from it etymologically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;added&lt;/strong&gt; &#xA;In the mnemonic, the three vowels code what type proposition is used for each of the three terms of the syllogism.  Here is the key:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A = all S are P,   E = no S is P,   I = some S are P,   O = some S are not P  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Baroco&lt;/strong&gt; in particular is this syllogism:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All X are Y.&lt;br&gt;&#xA;  Some Z are not Y.&lt;br&gt;&#xA;  Therefore, some Z are not X.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An example from &lt;a href=&quot;https://en.wikipedia.org/wiki/Syllogism#Baroco_(AOO-2)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;All informative things are useful.&lt;br&gt;&#xA;     Some websites are not useful.&lt;br&gt;&#xA;  ∴ Some websites are not informative.   &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2018-10-02T10:50:47.590" LastActivityDate="2018-10-02T10:50:47.590" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7738" PostTypeId="2" ParentId="7731" CreationDate="2018-09-30T13:47:46.250" Score="4" Body="&lt;p&gt;It is well known when and why complex numbers were introduced. When you solve a cubic equation which has 3 real roots, using Cardano's formula, you obtain square roots of negative numbers in your formula, so to obtain the correct result you have to do arithmetic with complex numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, when Cardano's formula was discovered, it had no practical application&#xA;(and perhaps has no practical application to this day). But complex numbers are important in many modern applications. Mathematicians invent things very much ahead of time, if we are speaking of applications. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-09-30T13:47:46.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7739" PostTypeId="1" CreationDate="2018-10-01T13:06:58.530" Score="1" ViewCount="111" Body="&lt;p&gt;In his &amp;quot;Rational Thermodynamics&amp;quot; 1983, 2nd ed., on page 55, Truesdell writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The axiomatization of Caratheodory has left a baneful legacy: Born's endorsement, which for a long time promoted in physical circles widespread complacent belief that thermodynamics had been standing on a general and secure foundation since 1909. Until recently physicists used Caratheodory's name as a mall to smash subsequent fundamental inquiry by the mathematically critical. To this I can speak from an experience of my own a third of a century ago, when I submitted an essay of mine to a nabob who enjoyed positive measure on a Nobel basis in theoretical physics. The Great Man replied on July 8, 1948:&lt;/p&gt;&#xA;&lt;p&gt;[T]he value of such general investigations ... is certainly rather limited.&lt;br /&gt;&#xA;However, I find an axiomatic treatment often very clarifying, and I certainly have no objection against it. I do not think that one can go in this way beyond the well established theories. I am therefore very doubtful whether Murnaghan&#xA;and your generalizations of the classical mechanics of continua, correspond in&#xA;any way to the behaviour of real solids or fluids. ...&#xA;Second, your exposition of thermodynamics certainly goes against the grain. ... I do not understand, why you dismiss the work of Caratheodory (which is a serious and helpful attempt to axiomatise thermodynamics) in one sentence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At first reading I thought Truesdell was talking about Born, but now I am not sure at all. On page 33 in the footnote he quite effusively praises him for&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[In view of my words about Born's baneful influence on thermodynamics perhaps it is not out of place to remark here that his autobiography as well as the reputation he left behind him reflect a fine character as well as charm, learning, letters, and eminent humanity, along with a very human and ridiculous yearning for the approbation of a committee in Sweden. His encounter with the foundations of thermodynamics seems to be the one instance in his life when his physical sense failed him.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So I do not think Born is the &amp;quot;nabob&amp;quot;. My question is, who was this &amp;quot;Great Man&amp;quot; and &amp;quot;nabob who enjoyed positive measure on a Nobel basis in theoretical physics&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-01T13:06:58.530" Title="who was the &quot;nabob&quot; Truesdell thinking of?" Tags="|thermodynamics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7740" PostTypeId="1" AcceptedAnswerId="7742" CreationDate="2018-10-01T17:59:52.957" Score="4" ViewCount="398" Body="&lt;p&gt;Who coined the term &quot;immune system&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;em&gt;OED&lt;/em&gt; lists the following as its earliest example of the term &quot;&lt;a href=&quot;http://www.oed.com/view/Entry/91999#eid117080817&quot; rel=&quot;nofollow noreferrer&quot;&gt;immune system&lt;/a&gt;&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;http://science.sciencemag.org/content/sci/97/2522/405.full.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1943   &lt;em&gt;Science&lt;/em&gt; 30 Apr. 406/1&lt;/a&gt;&lt;br&gt;Complement..is removed by the addition of an antigen and the corresponding rabbit antibody unrelated to the immune system to be analysed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-10-01T21:40:58.017" LastActivityDate="2018-10-02T02:55:15.077" Title="Who coined the term &quot;immune system&quot;?" Tags="|terminology|biology|medicine|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7741" PostTypeId="1" CreationDate="2018-10-01T23:52:29.540" Score="1" ViewCount="84" Body="&lt;p&gt;I wish to know about whether such example exists, could someone please help and tell me some stories about this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;p.s: Also I wonder why such thing can happen. I wish the two main characters in the story to be researchers of the same level. The only thing I can think of that makes this happen is that they are using different methods, but what else can make this happen?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="7911" LastActivityDate="2018-10-01T23:52:29.540" Title="Example of: Two researchers working on the same question but get opposite conclusions" Tags="|mathematicians|philosophy-of-science|scientific-method|" AnswerCount="0" CommentCount="4" ClosedDate="2018-10-06T23:44:58.850" ContentLicense="CC BY-SA 4.0" />
  <row Id="7742" PostTypeId="2" ParentId="7740" CreationDate="2018-10-02T02:55:15.077" Score="4" Body="&lt;p&gt;Translated from G. Toepfer, &lt;em&gt;Historisches Wörterbuch der Biologie&lt;/em&gt; (2016), vol. 3, &lt;a href=&quot;https://books.google.com/books?id=nvrZDAAAQBAJ&amp;amp;ots=IyL4bECHSI&amp;amp;pg=PA227&amp;amp;dq=Immunsystem&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 226–227&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Immune system&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(...) The discovery of a specific &lt;strong&gt;&lt;em&gt;immune system&lt;/em&gt;&lt;/strong&gt; in vertebrates took place in the 20th century.&lt;sup&gt;89&lt;/sup&gt; The immune system concept was already used sporadically in the first decades of the 20th century. Thus in 1909 E. Meyer and E. Emmerich reported their investigation of a “bacteriolytic immune system”&lt;sup&gt;90&lt;/sup&gt; and in 1920 H. Schade spoke of the “hemolytic immune system, consisting of blood cells and associated hemolysin-amboceptor”.&lt;sup&gt;91&lt;/sup&gt; But until the 1950s, the expression covered primarily laboratory investigated defense reactions, viz. the reaction system of antigens and antibodies&lt;sup&gt;92&lt;/sup&gt; — the expression only solidified into the concept of a body’s defense system in the mid-60s, first under the name &lt;em&gt;lymphoid system&lt;/em&gt;,&lt;sup&gt;93&lt;/sup&gt; but soon thereafter as “immune system”.&lt;sup&gt;94&lt;/sup&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;&lt;strong&gt;89&lt;/strong&gt; Cf. Moulin, A. M. (&lt;a href=&quot;http://www.jstor.org/stable/23330349&quot; rel=&quot;nofollow noreferrer&quot;&gt;1989&lt;/a&gt;). The immune system: a key concept for the history of immunology. Hist. Philos. Life Sci. 11, 221–236.&lt;br&gt;&#xA;  &lt;strong&gt;90&lt;/strong&gt; Meyer, E. &amp;amp; Emmerich, E. (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3745906&amp;amp;num=305&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;). Über paroxysmale Hämoglobinurie. Deutsches Archiv für klinische Medizin 96, 287–327: 305.&lt;br&gt;&#xA;  &lt;strong&gt;91&lt;/strong&gt; Schade, H. (1920). Die physikalische Chemie in der inneren Medizin: 135; cf. 2nd ed. (1923): 142.&lt;br&gt;&#xA;  &lt;strong&gt;92&lt;/strong&gt; Bier, O. G., Siqueira, M. &amp;amp; Osler, A. G. (1955). Studies on the mechanism of hypersensitivity phenomena 1. Internat. Arch. Allergy Appl. Immunol. 7, 1–9: 5.&lt;br&gt;&#xA;  &lt;strong&gt;93&lt;/strong&gt; Cooper, M. D., Peterson, R. D. &amp;amp; Good, R. A. (1965). Delineation of the thymic and lymphoid system in the chicken. Nature 205, 143; (...)&lt;br&gt;&#xA;  &lt;strong&gt;94&lt;/strong&gt; New York Times, 24 Oct. 1965: IV. 8/3 (after OED).&#xA;  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-02T02:55:15.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7744" PostTypeId="2" ParentId="7731" CreationDate="2018-10-02T18:27:09.407" Score="2" Body="&lt;p&gt;A great example is binary arithmetic. When Boole first presented his work on the subject he introduced it saying, &quot;I give you a fascinating system that has absolutely no application.&quot;* &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventually the Navy used it to replace base 10 arithmetic in computers they were building, getting us to the world we know today.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Source: Dr. Goldbeck's Calc 1 class.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7914" LastActivityDate="2018-10-02T18:27:09.407" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7745" PostTypeId="1" CreationDate="2018-10-03T09:22:04.850" Score="-4" ViewCount="299" Body="&lt;p&gt;The purpose of this question is trying to know originally how did counting in integer degrees angles from (one degree to &lt;span class=&quot;math-container&quot;&gt;$360$&lt;/span&gt; degrees) being adopted basically in geometry, despite the impossibility or the great difficulty of constructing exactly (without any approximation methods) those many integer degrees angles that aren't divisible by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I thought that there may be a very good unknown historical reasons (mainly by the ancient Babilions or Egyptians) to adopt such systems instead of using the exactly constructible angles defined essentially by the full angle of twice &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanking your opinions&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Editing: October 4, 2018&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No, the point of my question is quite different from the answers provided that well-explain the historical details about the issue, for the basic reason that many angels are impossible construction, where this strictly includes a subset of majority (or exactly two thirds) of integer degree angles of the form (&lt;span class=&quot;math-container&quot;&gt;$3n +/- 1$&lt;/span&gt;), where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a natural number&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other more important point about the true reason of the impossibility of constructing exactly certain angels, which was well proven by Wantzel in 1837, regarding the angle &lt;span class=&quot;math-container&quot;&gt;$\pi/9 = 20$&lt;/span&gt; degrees, where despite the fact of the proof, people (including the proofers) still think that such angle exists but impossible to construct &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally the most important point that might seem shocking and not at all acceptable by the mainstream academic mathematicians (for the first while), which states that such angles as &lt;span class=&quot;math-container&quot;&gt;$\pi/9 = 20$&lt;/span&gt; degrees, doesn’t exist at all, same like all those integer degrees angles of the above-described form &lt;span class=&quot;math-container&quot;&gt;$3n +/- 1$&lt;/span&gt;,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, to say why the angle of &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; degrees is impossible construction is simply because it doesn’t exist at all&#xA;Otherwise, how can an angle or a real number exist but can’t be exactly constructed? Wonder!&#xA;To explain this simple fact, it is impossible existence for a triangle with exactly three known sides to have exactly an angle of &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; degrees, or &lt;span class=&quot;math-container&quot;&gt;$19$&lt;/span&gt; degrees or &lt;span class=&quot;math-container&quot;&gt;$22$&lt;/span&gt; degrees, or generally any integer degrees angle which isn’t divisible by &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, of course one shouldn’t use any approximation methods and concludes or claims the exact construction by means of limits or convergence or cuts or else, since all those methods are basically and strictly with endless number of steps (theoretically unfinished operation)&#xA;So, the hope lies upon the mathematicians to give themselves a little chance not to miss this very important and new point of view&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the whole topic is deeply related to real number concept that had been extended wrongly beyond the real positive constructible numbers (which are the only true real existing numbers), where this had been well-proved and were also published publically in other forums and also some here on SE, but were deleted from SE sections by moderators since they so, unfortunately, contradict strictly many common beliefs among mathematicians    &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;EDIT: NOV. 25th, 2018&#xA;Adding reference: &lt;a href=&quot;https://www.quora.com/Do-most-of-the-named-angles-in-mathematics-truly-exist&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/Do-most-of-the-named-angles-in-mathematics-truly-exist&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought of not adding my content here about my earlier claims regarding the absolute non-existence of majority of well-known angles in current and old mathematics as well since truly most likely it would be immediately deleted as were done to many of my other deleted claims here and generally at SE&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the fact which is more important than all of us, for sure&lt;/p&gt;&#xA;" OwnerUserId="2855" LastEditorUserId="2855" LastEditDate="2018-11-25T13:55:36.827" LastActivityDate="2018-11-25T13:55:36.827" Title="How did the integer degrees angles counting being first adopted in geometry and mathematics?" Tags="|mathematics|mathematicians|geometry|euclidean-geometry|measurement|" AnswerCount="2" CommentCount="7" ClosedDate="2018-10-07T10:39:14.837" ContentLicense="CC BY-SA 4.0" />
  <row Id="7746" PostTypeId="2" ParentId="7745" CreationDate="2018-10-03T13:05:46.673" Score="0" Body="&lt;p&gt;Counting angles in degrees does not really belong to &quot;geometry and mathematics&quot;.&#xA;It was invented in astronomy by the Babylonians. Later, the degrees were used in geography and geodesy. And not only whole degrees were used,&#xA;but when needed, fractions of a degree. Either simple fractions of a degree, or minutes.&#xA;In mathematics, there is a more natural unit of angles: the radian.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-03T13:05:46.673" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7747" PostTypeId="2" ParentId="7745" CreationDate="2018-10-03T13:37:17.237" Score="2" Body="&lt;p&gt;See : Bartel van Der Waerden, &lt;a href=&quot;https://books.google.it/books?id=HK3vCAAAQBAJ&amp;amp;pg=PA37&quot; rel=&quot;nofollow noreferrer&quot;&gt;Science Awakening&lt;/a&gt;, Oxford UP (original ed, 1950), page 37 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The &lt;em&gt;sexagesimal system&lt;/em&gt; was taken over by the Semitic Babylonians from their predecessors, the Sumerians. This remarkable cultural group, which had also invented the cuneiform script, dominated southern Mesopotamia during the third millenium B.C. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Regarding &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, see the &lt;a href=&quot;https://en.wikipedia.org/wiki/Approximations_of_%CF%80#cite_ref-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&lt;/a&gt; for some references to ancient (Egyptian and Babylonian) approximation of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also : Bartel van der Waerden, &lt;a href=&quot;https://books.google.it/books?id=-pTzCAAAQBAJ&amp;amp;pg=PA170&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry and Algebra in Ancient Civilizations&lt;/a&gt;, Springer (1983), page 170 for &lt;em&gt;An Ancient Egyptian Rule for Squaring the Circle&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method used in Problem 50 of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rhind mathematical papyrus (around 1550 BCE)&lt;/a&gt; amounts to assuming : &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pi = (\dfrac {16}{9})^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also page 173 :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Babylonians too used to compute area and circumference of the&#xA;  circle by means of &lt;span class=&quot;math-container&quot;&gt;$C=3d$&lt;/span&gt;. We can express the same facts by saying that&#xA;  the Babylonians and the Egyptian author of Problem 32 [of the &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_ancient_Egyptian_papyri&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cairo papyrus&lt;/a&gt; (first century BCE)] assumed &lt;span class=&quot;math-container&quot;&gt;$\pi=3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-10-04T13:00:26.740" LastActivityDate="2018-10-04T13:00:26.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7748" PostTypeId="2" ParentId="7430" CreationDate="2018-10-03T14:31:53.400" Score="1" Body="&lt;p&gt;Adding to Bence Mélykúti's response, you can find a discussion of the relative contributions of Michaelis and Menten on the one hand and Henri on the other in a paper by Deichmann &lt;em&gt;et al.&lt;/em&gt; &quot;Commemorating the 1913 Michaelis-Menten paper &lt;em&gt;Die Kinetik der Invertinwirkung&lt;/em&gt;: three perspectives&quot; &lt;em&gt;FEBS J.&lt;/em&gt; &lt;strong&gt;281&lt;/strong&gt; (2014) 435-463. The &quot;three perspectives&quot; have different authors because there wasn't full agreement between the four authors of the whole paper about the amount of credit to be assigned, but the editor of the journal wanted one paper, not three.&lt;/p&gt;&#xA;" OwnerUserId="7917" LastActivityDate="2018-10-03T14:31:53.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7749" PostTypeId="1" CreationDate="2018-10-03T16:27:58.490" Score="4" ViewCount="357" Body="&lt;p&gt;I'm thinking specifically of the Jantar Mantar sundial in Jaipur, India, the Samrat Yantra.  It is apparently accurate to within 2 seconds, and was built around 1740.  What would such an accurate clock have been used for at that time in India?  Was the instrument accurate simply because it was built so big, or was the accuracy important for something else?&lt;/p&gt;&#xA;" OwnerUserId="7919" LastActivityDate="2018-10-09T17:53:38.470" Title="Why were extremely accurate sundials necessary in India?" Tags="|time|india|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7750" PostTypeId="1" CreationDate="2018-10-03T17:27:28.480" Score="5" ViewCount="62" Body="&lt;p&gt;When was the first publication of the abbreviation RSA (Rivest, Sharmir, Adleman) because it does not appear in Martin Gardner’s article of 1977 which is at the following url:&#xA;&lt;a href=&quot;https://simson.net/ref/1977/Gardner_RSA.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://simson.net/ref/1977/Gardner_RSA.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7920" LastActivityDate="2018-10-04T00:15:48.650" Title="When was the first appearance of the abbreviation RSA?" Tags="|mathematics|number-theory|computer-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7751" PostTypeId="1" CreationDate="2018-10-03T18:19:59.960" Score="10" ViewCount="40255" Body="&lt;p&gt;According to various sources on the Web, Albert Einstein is likely to have said or written one of the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Probleme kann man niemals mit derselben Denkweise lösen, durch die sie entstanden sind.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(German)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;2&quot;&gt;&#xA;  &lt;li&gt;We cannot solve our problems with the same thinking we used to create them.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(English)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;Problems can never be solved by the way of thinking that first created them.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(English)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest quote I found is &quot;We have to learn to think in a new way.&quot; (&lt;a href=&quot;https://pugwash.org/1955/07/09/statement-manifesto/&quot; rel=&quot;noreferrer&quot;&gt;Russel-Einstein Manifesto, July 9, 1955, London&lt;/a&gt;), which is pretty different.&#xA;Since &lt;a href=&quot;https://www.sec.in.tum.de/~malkis/howToReviewPapers.htm&quot; rel=&quot;noreferrer&quot;&gt;it is well known&lt;/a&gt; that the Web is a huge garbage-bin, this information can be well unreliable, perhaps even invented. Does anyone know the original language and source that one could look up, say, in a library (or a scanned book at worst)?&lt;/p&gt;&#xA;" OwnerDisplayName="user7921" LastEditorDisplayName="user7921" LastEditDate="2018-10-03T19:15:10.660" LastActivityDate="2020-12-31T13:55:22.680" Title="Did Einstein say &quot;We cannot solve our problems with the same thinking we used to create them&quot;?" Tags="|reference-request|debunking|quote|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7752" PostTypeId="1" AcceptedAnswerId="7754" CreationDate="2018-10-03T18:37:09.040" Score="5" ViewCount="358" Body="&lt;p&gt;Is there a proof of the Lebesgue Differentiation Theorem that does not involve the Hardy-Littlewood Maximal Function? For example, did Lebesgue prove it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If there is such a proof, where can I find it?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastEditorUserId="55" LastEditDate="2018-10-03T21:02:42.297" LastActivityDate="2021-11-30T14:18:00.147" Title="Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?" Tags="|mathematics|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7753" PostTypeId="2" ParentId="7751" CreationDate="2018-10-03T18:49:32.013" Score="13" Body="&lt;p&gt;I am afraid there is no original source. Wikipedia has talk pages where sourcing is discussed, and its editors did extensive searches on this one and its variants. It is listed under the heading &lt;a href=&quot;https://en.wikiquote.org/wiki/Talk:Albert_Einstein#Unsourced_and_dubious/overly_modern_sources&quot; rel=&quot;noreferrer&quot;&gt;Unsourced and dubious/overly modern sources&lt;/a&gt;, and the &amp;quot;original&amp;quot; appears to have been made up by Ram Dass around 1970. Dass (born Richard Alpert) is an American academic turned spiritual teacher after an epiphany in 1967, known for associations with Timothy Leary and the Hindu guru Neem Karoli Baba. Dass is also the author of another &amp;quot;Einstein&amp;quot; quote dating to about the same time, &amp;quot;&lt;em&gt;I didn't arrive at my understanding of the fundamental laws of the universe through my rational mind&lt;/em&gt;&amp;quot;. No earlier occurrences of the &amp;quot;quote&amp;quot; have been found so far, Einstein passed away in 1955. Here is the surmise:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Searching for &amp;quot;Einstein&amp;quot; and &amp;quot;level of thinking&amp;quot; rather than &amp;quot;same level of thinking&amp;quot; turns up a much earlier example from The Journal of Transpersonal Psychology, Volumes 1-4, which is dated 1969 by google books though these snippets show it contains pieces from 1969 and 1970. The quote, on p. 124, is &amp;quot;The world that we have made as a result of the level of thinking we have done thus far creates problems that we cannot solve at the same level as the level we created them at.&amp;quot; It's prefaced by &amp;quot;Einstein said an interesting thing&amp;quot;, and the same phrase and quote appears in a 1974 book by Ram Dass (who needs his own wikiquote page!), The Only Dance There Is... so presumably the one in The Journal of Transpersonal Psychology is the same piece by Ram Dass.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Also, the next two earliest versions I find on google books (searching for &amp;quot;Einstein&amp;quot; and &amp;quot;level of thinking&amp;quot; on an advanced search with date range 1900-1979) use wording nearly identical to the version given by Ram Dass... The supposed Einstein quote from Skeptic reads &amp;quot;the world we have made as a result of the level of thinking we have done thus far creates problems that we cannot solve at the level of thinking at which we created them&amp;quot;, and the one from New Age reads &amp;quot;the world that we have made, as a result of the level of thinking we have done thus far, creates problems that we cannot solve at the same level we created them&amp;quot;, both identical to Ram Dass' quote in the first part and very close in the second part.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Since Ram Dass generally spoke extemporaneously without using notes, and The Only Dance There Is says the section with the Einstein quote is from talks he gave to &amp;quot;spiritual seekers&amp;quot; at &amp;quot;the Menninger Foundation in 1970&amp;quot;, it's likely he would have been paraphrasing a bit even if his source was some other unknown one that also misquoted Einstein (as opposed to him being the original source of the misquote as I am speculating). So if the only other sources found in the 1970s repeat Ram Dass' exact wording, it seems pretty likely that Ram Dass (or some secondary source that got the quote from Ram Dass) was the source for them. And the later variants that emerged in the 1980s and 1990s might well be all from authors that got their quotes in a chain of influences that goes back to Ram Dass' version.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-03T18:57:52.117" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7754" PostTypeId="2" ParentId="7752" CreationDate="2018-10-03T19:48:50.227" Score="7" Body="&lt;p&gt;The original proof appears in &lt;a href=&quot;https://archive.org/details/leconsdiscontinues00bairrich/page/n9/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lebesgue's Lecons sur l'integration et la recherche des fonctions primitives, Paris, 1904&lt;/a&gt;, freely available if you read French. He only considered continuous monotone functions. Young gave a proof without assuming continuity in 1911, and in 1932 Riesz gave an elementary proof for the continuous case using his &amp;quot;Rising Sun Lemma&amp;quot;. In 1965 Austin gave a non-elementary geometric proof that requires intricate measure theory including the measurability of the Dini derivatives. Botsko gives a general elementary proof in &lt;a href=&quot;https://www.jstor.org/stable/3647803&quot; rel=&quot;nofollow noreferrer&quot;&gt;American Mathematical Monthly&#xA;Vol. 110, No. 9 (Nov., 2003), pp. 834-838&lt;/a&gt;, and according to him all other proofs use the Vitali Covering Theorem (in proving the estimate for Hardy-Littlewood Maximal Function, for instance), or the Lebesgue Density Theorem.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-30T14:18:00.147" LastActivityDate="2021-11-30T14:18:00.147" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7755" PostTypeId="2" ParentId="7750" CreationDate="2018-10-03T20:08:35.943" Score="5" Body="&lt;p&gt;Nor does it appear in &lt;a href=&quot;https://dl.acm.org/citation.cfm?id=359342&quot; rel=&quot;nofollow noreferrer&quot;&gt;A method for obtaining digital signatures and public-key cryptosystems&lt;/a&gt; (Communications of the ACM Volume 21 Issue 2, Feb. 1978 Pages 120-126), which was Rivest-Shamir-Adleman's original publication. It is hard to pinpoint the watershed with such obvious abbreviations, but MathSciNet shows &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0161-117891853162&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blakley-Blakley  Security of number-theoretic public key cryptosystems against random attack I&lt;/a&gt; (Cryptologia 2 (1978), no. 4, 306–321) as the earliest occurrence. The trio founding RSA Data Security, Inc. in 1982 (now &lt;a href=&quot;https://en.wikipedia.org/wiki/RSA_Security&quot; rel=&quot;nofollow noreferrer&quot;&gt;RSA Security LLC&lt;/a&gt;) probably solidified the usage.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-10-04T00:15:48.650" LastActivityDate="2018-10-04T00:15:48.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7756" PostTypeId="2" ParentId="7749" CreationDate="2018-10-04T00:29:50.730" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Jantar_Mantar,_Jaipur#Purpose&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jantar Mantar monument in Jaipur&lt;/a&gt; was built on the orders of the Rajput king Sawai Jai Singh II, and completed in 1734. Aside from the sundial it has an observatory with various other instruments for astronomical observations with the naked eye. Apparently the zīj at the time, Indian/Islamic version of ephemerides, were not very accurate, and the king noticed. Jantar Mantar was one of several observatories he ordered built to improve on them. The resulting ephemerides, called Zij-i Muhammad Shahi, were used in India for about a century, but in Europe astronomers made far more accurate ephemerides by using telescopes.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-10-04T00:29:50.730" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7757" PostTypeId="1" CreationDate="2018-10-04T09:36:30.990" Score="1" ViewCount="174" Body="&lt;p&gt;I found in some old Latin texts and their translations that polynomials were once called &quot;species&quot; (if I understand correctly that they meant the same thing, but it looks like it). And their constituent parts are called &quot;terms&quot; to this day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&#xA;Why they called them &quot;species&quot; and &quot;terms&quot;? What was the reason behind choosing these particular names for those notions?&lt;/p&gt;&#xA;" OwnerUserId="7926" LastEditorUserId="7926" LastEditDate="2018-10-04T10:21:11.710" LastActivityDate="2018-10-04T19:42:18.253" Title="&quot;Species&quot; and &quot;terms&quot; meaning polynomials and monomials" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7758" PostTypeId="2" ParentId="7757" CreationDate="2018-10-04T15:09:50.700" Score="3" Body="&lt;p&gt;The source of &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te#The_logic_of_species&quot; rel=&quot;nofollow noreferrer&quot;&gt;species&lt;/a&gt; (in mathematics) seems to be &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;François Viète&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;em&gt;Logistice speciosa&lt;/em&gt; (algebra) in contrast to &lt;em&gt;Logistice numerosa&lt;/em&gt; (arithmetic), into his &lt;a href=&quot;https://books.google.it/books?id=omHqrYBzMpAC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;In artem analyticem isagoge&lt;/em&gt; (1591)&lt;/a&gt;, page 19 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Logistice numerosa est quae per numeros, Speciosa quae per species seu formas exhibitur [Numerical logistic is (a logistic) that employs numbers, symbolic logistic one that employs symbols or signs for things].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See Introduction (page 13) for comments :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Samuel Jeake in 1696, has it that this &amp;quot;name ... with the Latins serveth for the Figure, Form or shape of any thing&amp;quot; and that, accordingly, &amp;quot;Species are Quantities or Magnitudes, denoted by Letters, signifying Numbers, Lines, Lineats, Figures Geometrical, &amp;amp;c.&amp;quot; Alexandre Saverein's &lt;em&gt;Dictionnaire Universel de Mathematique et de Physique&lt;/em&gt; (Paris, 1753), vol. I, p. 17, says that the expression &amp;quot;algebre specieuse&amp;quot; derives from that fact that quantities are represented by letters which designate &amp;quot;leur forme et leur espece,&amp;quot; adding &amp;quot;d'ou vient le mot spécieuse.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Smith thinks &lt;a href=&quot;https://en.wikipedia.org/wiki/Diophantus#Arithmetica&quot; rel=&quot;nofollow noreferrer&quot;&gt;Diophantus&lt;/a&gt; &amp;quot;the most likely source for Vieta's use of the word 'species' &amp;quot; and that it is, in effect, his substitute for  Diophantus' &lt;a href=&quot;https://en.wiktionary.org/wiki/%CE%B5%E1%BC%B6%CE%B4%CE%BF%CF%82&quot; rel=&quot;nofollow noreferrer&quot;&gt;εἶδος&lt;/a&gt; : form, image, shape.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-04T19:42:18.253" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7759" PostTypeId="2" ParentId="7731" CreationDate="2018-10-04T19:24:39.660" Score="1" Body="&lt;p&gt;For a skeptical balance I recommend &quot;&lt;a href=&quot;http://www.sfu.ca/~rpyke/cafe/reasonable.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solving Wigners Mystery: The Reasonable (Though Perhaps Limited) Effectiveness of Mathematics in the Natural Sciences&lt;/a&gt;&quot; which argues that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...the connections between mathematics and the natural sciences are, and always have been, rationally although fallibly forged links, not a collection of mysterious parallelisms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7348" LastActivityDate="2018-10-04T19:24:39.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7760" PostTypeId="1" CreationDate="2018-10-04T20:06:40.010" Score="4" ViewCount="336" Body="&lt;p&gt;Luzin was accused of plagiarism and other misconduct by Kolmogorov and other students during the Luzin Affair of 1936. Were these allegations actually true?&lt;/p&gt;&#xA;" OwnerUserId="7927" LastActivityDate="2018-10-05T00:10:07.693" Title="Did Nikolai Luzin plagiarize?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="1" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7761" PostTypeId="2" ParentId="7760" CreationDate="2018-10-04T20:42:49.353" Score="3" Body="&lt;p&gt;It was part of Stalin's political witch hunt, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Great_Purge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Great Purge&lt;/a&gt;. Luzin was denounced in Pravda as the author of &quot;&lt;em&gt;would-be scientific papers&lt;/em&gt;&quot; who &quot;&lt;em&gt;felt no shame in declaring the discoveries of his students to be his own achievements&lt;/em&gt;&quot;, anonymously. Then the corroborating &quot;testimony&quot;, from Aleksandrov and Kolmogorov among others, was obtained by the usual methods of &lt;a href=&quot;https://en.wikipedia.org/wiki/NKVD&quot; rel=&quot;nofollow noreferrer&quot;&gt;NKVD&lt;/a&gt; (later renamed into KGB): the &quot;witnesses&quot; were threatened and blackmailed, if not &quot;persuaded&quot; physically. Then Luzin was declared &quot;&lt;em&gt;enemy under the mask of a Soviet citizen&lt;/em&gt;&quot;, and he was lucky not to get imprisoned or executed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The methods of political insinuations and slander had been used against the old Muscovite professorship already several years before the article in Pravda&lt;/em&gt;&quot;. Luzin's advisor, Egorov, was named among &quot;&lt;em&gt;active counter-revolutionaries among mathematicians&lt;/em&gt;&quot; back in 1930, arrested, and died in prison after a hunger strike in 1931. He had wrong religious beliefs. Luzin's fault, aside from the usual guilt by association, was likely that he published major results in foreign journals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia has a detailed account of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Nikolai_Luzin#Luzin_affair_of_1936&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luzin 1936 affair&lt;/a&gt;, another source is &lt;a href=&quot;https://arxiv.org/abs/math/0702632&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kutateladze's The Tragedy of Mathematics in Russia&lt;/a&gt;, which quotes many contemporary documents:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It is customary to emphasize that Luzin was not so great a mathematician as his students that had persecuted him. Some moral fault is persistently incriminated to Luzin in the untimely death of M. Ya. Suslin (1894–1919) from typhus fever. Luzin is often blamed for all his disasters at least partly. He is ascribed such traits of character as theatricality,&#xA;  envy of others’ success, hypocrisy, plagiarism, and inclination to intriguing. He is said to deserve all punishments and if not all then it is not his student&#xA;  s’ fault but stalinism and the curse of the epoch.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;No person with these defects of personality could ever become the founder of “Lusitania” — the most successful mathematical school in the history of world science. Therefore, there exists as rather likely theory of “two Luzins”: one of the epoch of Lusitania and the other of the times of the Luzin case... If Luzin were guilty then his fault would belong to the sphere of the personal mathematical relations between a teacher and a student. No convincing evidence of Luzin’s plagiarism was ever submitted.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In particular it quotes Lebesgue's preface to Luzin's book on analytic sets:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Anyone will be astonished to find out from Luzin’s book that I had incidentally invented the sieve method and was the first to construct an analytic set. But nobody could be more amazed than me.  Mr.  Luzin feels himself happy only when he has managed to ascribe his own discoveries to someone else.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Curiously enough, even this is sometimes turned against Luzin, he was supposedly trying to &quot;flatter&quot; Lebesgue.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-10-05T00:10:07.693" LastActivityDate="2018-10-05T00:10:07.693" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7762" PostTypeId="1" CreationDate="2018-10-04T21:32:22.777" Score="4" ViewCount="6747" Body="&lt;p&gt;While Schrödinger's cat thought experiment serves a greater purpose than animal cruelty enjoyment, one of the possible results is a dead cat. Gruesome  or not is one's appreciation, but you will always find someone joking about &quot;oh boy this guy must have hated cats&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which leads to a simple question, why did Schrödinger choose a cat in the first place, instead of, say, a lamb or a goldfish?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only thing I was able to find online so far is some idle-guessing-looking &quot;Pictures of Cats&quot; website which doesn't seem really trustworthy (&lt;a href=&quot;https://pictures-of-cats.org/Why-did-Erwin-Schrodinger-choose-a-cat.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Is there a more reliable source?&lt;/p&gt;&#xA;" OwnerUserId="7929" LastActivityDate="2023-08-29T12:44:40.567" Title="Why did Schrödinger choose a cat for his thought experiment?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7763" PostTypeId="2" ParentId="7762" CreationDate="2018-10-04T22:29:25.167" Score="8" Body="&lt;p&gt;Erwin Schrödinger doesn't appear to have personally owned a cat. He did however own a dog.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But even noises have their timbre, from which we may infer what is going on; and &lt;strong&gt;even my dog is familiar with the peculiar noise of the opening of a certain tin box&lt;/strong&gt;, out of which he occasionally receives a biscuit.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;sup&gt;&lt;a href=&quot;https://books.google.co.uk/books?id=dg2bYMwdaBwC&amp;amp;pg=PA176&amp;amp;lpg=PA176&amp;amp;dq=schrodinger%20%22cats%22%20thomas%20becket&amp;amp;source=bl&amp;amp;ots=ekP43gicwi&amp;amp;sig=F4_8u7d-7qIpcTIAFNBICd5LUN4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjVg_vp6O3dAhVJF8AKHdfGAEYQ6AEwCHoECAUQAQ#v=onepage&amp;amp;q=thomas&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;What is Life?: With Mind and Matter and Autobiographical Sketches By Roger Schrodinger, Erwin Schrödinger&lt;/a&gt;&lt;/sub&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His great-aunt owned several cats. He speaks disapprovingly of them on several occasions, making multiple references to &amp;quot;&lt;em&gt;yowling&lt;/em&gt;&amp;quot; cats and making specific reference to a tomcat named Thomas Becket that seems to have made a particular impression on him.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An aunt of my mother's also lived there with her husband, Alfred Kirk,&#xA;and six Angora cats. (In later years there were said to be twenty.) &lt;strong&gt;In&#xA;addition she had an ordinary tomcat who would very often come home&#xA;from his nocturnal adventures in a sad state, so he was given the name&#xA;Thomas Becket&lt;/strong&gt; (referring to the Archbishop of Canterbury who was&#xA;killed in office by order of King Henry II) not that this meant a&#xA;great deal to me then, nor was it very appropriate.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;sup&gt;&lt;a href=&quot;https://books.google.co.uk/books?id=dg2bYMwdaBwC&amp;amp;pg=PA176&amp;amp;lpg=PA176&amp;amp;dq=schrodinger%20%22cats%22%20thomas%20becket&amp;amp;source=bl&amp;amp;ots=ekP43gicwi&amp;amp;sig=F4_8u7d-7qIpcTIAFNBICd5LUN4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjVg_vp6O3dAhVJF8AKHdfGAEYQ6AEwCHoECAUQAQ#v=onepage&amp;amp;q=thomas&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;What is Life?: With Mind and Matter and Autobiographical Sketches By Roger Schrodinger, Erwin Schrödinger&lt;/a&gt;&lt;/sub&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7931" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-04T22:29:25.167" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7764" PostTypeId="1" AcceptedAnswerId="7772" CreationDate="2018-10-05T03:47:32.653" Score="6" ViewCount="466" Body="&lt;p&gt;I am trying to understand why vibrational modes of polyatomic molecules are called &quot;normal&quot; mode of vibrations and with corresponding normal coordinates. What is the origin of the term normal here? I am sure this term must have been borrowed from mechanics. What other mechanical oscillations are possible which may not be considered normal?&#xA;The unabridged Oxford English Dictionary attributes early usage of word normal to W. Thomson &amp;amp; P. G. Tait Treat. Nat. Philos. 1867 I. ii. 274   &lt;em&gt;There are in general..i distinct determinate displacements, which we shall call the normal displacements, fulfilling the condition, that if any one of them be produced alone, and the system then left to itself for an instant at rest, this displacement will diminish and increase periodically according to a simple harmonic function of the time, and consequently every particle of the system will execute a simple harmonic movement in the same period.&lt;/em&gt;&#xA;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="55" LastEditDate="2018-10-05T23:13:53.053" LastActivityDate="2018-10-08T10:41:53.097" Title="What is the origin of &quot;normal&quot; in normal coordinates and normal modes?" Tags="|terminology|philosophy-of-science|experimental-physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7765" PostTypeId="1" CreationDate="2018-10-05T12:44:14.267" Score="2" ViewCount="178" Body="&lt;p&gt;The integral of the secant function was first correctly conjectured by Henry Bond in the 1640s, and Isaac Newton was aware of his conjecture in 1665, although no proof was published until 1668. Of course, once you've guessed the right integral you can prove it works by differentiation, using the Fundamental Theorem of Calculus; was it unknown to Newton at that time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But my main question concerns James Gregory's 1668 proof in Execitationes Geometricae, which &lt;a href=&quot;https://en.wikipedia.org/wiki/Integral_of_the_secant_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; says &quot;was presented in a form that renders it nearly impossible for modern readers to comprehend&quot;. By contrast, Isaac Barrow's 167 proof is a fairly simple one, the &lt;span class=&quot;math-container&quot;&gt;$u=\sin\theta $&lt;/span&gt; substitution I discuss &lt;a href=&quot;https://matheducators.stackexchange.com/questions/14631/why-do-we-state-the-antiderivative-of-sec-x-as-ln-sec-x-tan-x&quot;&gt;here&lt;/a&gt;. But what was Gregory's proof? I can only find his book &lt;a href=&quot;https://books.google.co.uk/books?id=ZtRYqgyD5YsC&amp;amp;printsec=frontcover&amp;amp;dq=Exercitationes+Geometricae&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiBz6fqqO_dAhUOKVAKHfACCecQ6AEIKTAA#v=onepage&amp;amp;q=Exercitationes%20Geometricae&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;in Latin&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-10-05T13:09:47.677" Title="Gregory's integration of $\sec\theta$" Tags="|mathematics|mathematicians|calculus|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7766" PostTypeId="2" ParentId="7765" CreationDate="2018-10-05T13:09:47.677" Score="1" Body="&lt;p&gt;According to Rickey and Turchinsky (&lt;a href=&quot;//ams.org/mathscinet-getitem?mr=600072&quot; rel=&quot;nofollow noreferrer&quot;&gt;1980&lt;/a&gt;, p. 165; my emphasis):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first to prove the conjecture was, to quote Edmund Halley, “the excellent Mr. &lt;em&gt;James&#xA;  Gregory&lt;/em&gt; in his &lt;em&gt;Exercitationes Geometricae&lt;/em&gt;, published &lt;em&gt;Anno&lt;/em&gt; 1668, which he did, not without a long&#xA;  train of Consequences and Complication of Proportions, whereby the evidence of the Demonstration is in a great measure lost, and the Reader wearied before he attain it” ([&lt;a href=&quot;//doi.org/10.1098/rstl.1695.0028&quot; rel=&quot;nofollow noreferrer&quot;&gt;7&lt;/a&gt;], p. 203).&#xA;  Judging by &lt;strong&gt;&lt;em&gt;Turnbull’s modern elucidation&lt;/em&gt;&lt;/strong&gt; [&lt;a href=&quot;//ams.org/mathscinet-getitem?mr=792&quot; rel=&quot;nofollow noreferrer&quot;&gt;19&lt;/a&gt;] &lt;strong&gt;&lt;em&gt;of Gregory’s proof&lt;/em&gt;&lt;/strong&gt;, one would have to agree with&#xA;  Halley. At any rate, Gregory’s proof could not be presented to today’s calculus students, and so&#xA;  we omit it here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So I’d seek a copy of Turnbull’s rare book. (I have before for other reasons, but never successfully.) &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-05T13:09:47.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7767" PostTypeId="2" ParentId="7646" CreationDate="2018-10-05T13:33:33.300" Score="1" Body="&lt;p&gt;Google translation to Ludwig Schlesinger's essay on Gauss's contribution to analysis reveals the answer - on p.105-106 of his essay, appears his explanation of Gauss's result - according to which Gauss was aware of the importance of reflection in circles (inversion with respect to circles) for the composition of certain fundamental substitutions. He than explains the drawing of tessellation of the unit disk from his nachlass as a network of circular arc triangles  completed by their orthogonal circle.&lt;/p&gt;&#xA;&lt;p&gt;Although it appears first that the drawing from p.104 of volume VIII of Gauss's nachlass was made by Fricke (what makes this historical thesis on Gauss's understanding of modular forms weaker), Schlesinger mentions that this drawing was actually found among Gauss's posthumous papers, and published with only few modifications by Fricke.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ga4bP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ga4bP.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Related directly to the questions sorrounding Gauss's drawing is my Mathoverflow post and partial answer: &lt;a href=&quot;https://mathoverflow.net/questions/370190/what-is-the-representation-of-the-generators-of-the-triangle-group-for-the-unifo&quot;&gt;https://mathoverflow.net/questions/370190/what-is-the-representation-of-the-generators-of-the-triangle-group-for-the-unifo&lt;/a&gt;. The main conclusions made in this post are that Gauss's handwritten notes confirm that he recognized the key notion of &amp;quot;&lt;em&gt;&lt;strong&gt;inversion with respect to a circle&lt;/strong&gt;&lt;/em&gt;&amp;quot; (which can be understood as a generalization of the concept of reflection), as well as the principle of generation of curved triangles network by successive reflection of curved triangles with respect to their sides, a process which tiles the whole hyperbolic plane with copies of a fundamental triangle.&lt;/p&gt;&#xA;&lt;p&gt;In addition, the values he gives for the radiuses of curvature of the sides of the first and secondary circles (which give the scale of the fundamental triangle, when it viewed euclidly) show that the border curve for his network is exactly the unit circle, a fact which i'm unable to explain without assuming that Gauss was using the Cayley-Klein metric for the hyperbolic plane (this fact shows that Gauss knew how to calculate the scale of the border circle in terms of the scale of the fundamental triangle).&lt;/p&gt;&#xA;&lt;p&gt;However, it seems that Gauss was led to these consideration solely by his analytic work and it's number theoretic counterparts (i.e the intimate relation to quadratic forms). Gauss attempted to appreciate and illustrate the different aspects of the modular action by casting it in geometric form. Even though he might eventually view these considerations as connected with his meditations on non-euclidean geometry, this connection almost certainly didn't come to him so early in his career (Schlesinger and Fricke date Gauss's drawing to before 1805).&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-09-18T11:57:25.527" LastActivityDate="2020-09-18T11:57:25.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7769" PostTypeId="2" ParentId="7764" CreationDate="2018-10-05T23:13:10.513" Score="3" Body="&lt;p&gt;This term was borrowed from mechanics, but it is not specific to mechanics. In the original use &lt;em&gt;norma&lt;/em&gt; was &quot;carpenter's square&quot;, by 1500 extended generally to &quot;rule, pattern&quot; (c. 1500), with normal meaning &quot;typical, common&quot;. By 1640-s a more specific use split off, &quot;made according to a carpenter's square&quot;, and from that the modern use as &quot;orthogonal, perpendicular&quot; derives. These are the broad and the narrow meanings prevailing today. The broad one comes in two shades, as &quot;generic&quot; and as &quot;patterned&quot;, &quot;rule-governed&quot;. &lt;a href=&quot;https://www.etymonline.com/word/normal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Online Etymology Dictionary&lt;/a&gt; continues:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Meaning &quot;conforming to common standards, usual&quot; is from 1828, but probably older than the record [Barnhart]. As a noun meaning &quot;usual state or condition,&quot; from 1890. Sense of &quot;normal person or thing&quot; is from 1894.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The normal distribution got its name at about the same time, based on the broad meaning, as both something standard and well regimented. In &lt;a href=&quot;http://jeff560.tripod.com/n.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt; we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;According to Kruskal &amp;amp; Stigler, the term normal was used, apparently independently, by Charles S. Peirce (1873) in an appendix to a report of the US Coast Survey (reprinted in Stigler (1980, vol. 2), Wilhelm Lexis Theorie der Massenerscheinungen in der menschlichen Gesellschaft (1877) and Francis Galton 'Typical laws of heredity' (1877).&#xA;  Of the three, Galton had most influence on the development of Statistics in Britain and, through his ‘descendants’ Karl Pearson and R. A. Fisher, on Statistics worldwide. In the 1877 article Galton used the phrase &quot;deviated normally&quot; only once (p. 513)--his name for the distribution was &quot;the law of deviation.&quot; However in the 1880s he began using the term &quot;normal&quot; systematically.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Later Pearson seemed to imply that he had introduced the term: &quot;Many years ago I called the Laplace-Gaussian curve the normal curve ...&quot; (Biometrika, 13, (1920), p. 25). While Pearson did not introduce the term, it is fair to say that his &quot;consistent and exclusive use of this term in his epoch-making publications led to its adoption throughout the statistical community.&quot; (DSB) Curiously a main theme in Pearson’s scientific work was that data did not ‘normally’ follow this distribution and that alternative distributions had to be devised.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To Tait it mattered, apparently, that &quot;normal displacements&quot; when left alone evolved &quot;according to a simple harmonic function&quot;. But it could not have hurt in the eventual spread of the term that the things are commonly occurring, and in many mechanical problems &quot;normal modes&quot; are indeed orthogonal to each other. So all three shades blended well in this context.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-10-05T23:13:10.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7772" PostTypeId="2" ParentId="7764" CreationDate="2018-10-07T07:26:30.620" Score="3" Body="&lt;p&gt;I believe this refers to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Normal_form_(mathematics)#Linear_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;normal form&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt; (sum of squares) a quadratic force function (a.k.a. potential) takes in such coordinates. E.g. Netto (&lt;a href=&quot;https://zbmath.org/?q=an:25.0211.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1893&lt;/a&gt;, pp. 266, 271) implies as much. Stäckel (&lt;a href=&quot;https://zbmath.org/?q=an:39.0733.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=oMLVAAAAMAAJ&amp;amp;pg=PA482&amp;amp;dq=%22normale%20Koordinaten%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 482&lt;/a&gt;) spells out the relation of general to normal (a.k.a. principal) oscillations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By a proposition in the theory of quadratic forms ... one can find &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; linear functions &lt;span class=&quot;math-container&quot;&gt;$s_1,\dots,s_r$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$q_1,\dots,q_r$&lt;/span&gt; such that simultaneously&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sum A_{\alpha\beta}q_\alpha q_\beta=\sum s_\alpha^2,\qquad\sum B_{\alpha\beta}q_\alpha q_\beta=\sum \rho_\alpha^2s_\alpha^2\quad\dots&#xA;$$&lt;/span&gt;&#xA;  On introducing these &lt;em&gt;principal coordinates&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_\alpha$&lt;/span&gt; &lt;sup&gt;95)&lt;/sup&gt; the equations (1) get replaced by the simple equations:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\ddot s_\alpha+\rho_\alpha^2\,s_\alpha=0,&#xA;\tag2&#xA;$$&lt;/span&gt;&#xA;  satisfied by&#xA;  &lt;span class=&quot;math-container&quot;&gt;$&#xA;s_\alpha=\mu_\alpha\sin(\rho_\alpha t+\sigma_\alpha)&#xA;$&lt;/span&gt;... A general oscillation of the system is then the result of superposing &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; principal oscillations (harmonic oscillations) with periods &lt;span class=&quot;math-container&quot;&gt;$\frac{2\pi}{\rho_1},\dots,\frac{2\pi}{\rho_r}$&lt;/span&gt;.&lt;sup&gt;96)&lt;/sup&gt; ... The resulting motion is not in general periodic; for it to be, the ratios of &lt;span class=&quot;math-container&quot;&gt;$\rho_1,\dots,\rho_r$&lt;/span&gt; must be integers.&lt;/p&gt;&#xA;  &#xA;  &lt;hr&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;95) &lt;em&gt;Lagrange&lt;/em&gt; essentially has the principal coordinates already; they are also called harmonic, simple, normal coordinates.&lt;br&gt;&#xA;  96) The principle of superposition of oscillations is due to &lt;em&gt;Daniel Bernoulli&lt;/em&gt;, Berlin Mém. année 1753, p. 173: in any system the alternating motions of the bodies are always a mixture of simple, regular and permanent oscillations of different kinds; see also &lt;em&gt;D. Bernoulli&lt;/em&gt;, Petersburg Nov. Comment. 19 ad annum 1775, p. 239.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Attribution of the terminology to Thomson and Tait is confirmed by Lamb (&lt;a href=&quot;https://zbmath.org/?q=an:38.0799.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1907&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=8Nx7WU0HSAEC&amp;amp;pg=PA223&amp;amp;dq=Normalkoordinaten&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 223&lt;/a&gt;), Pockels (&lt;a href=&quot;https://zbmath.org/?q=an:23.0403.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1891&lt;/a&gt;, p. 44) and Ball (&lt;a href=&quot;https://zbmath.org/?q=an:02.0708.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1869&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=lEY9AQAAMAAJ&amp;amp;pg=PA597&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 597&lt;/a&gt;). It could have been inspired also by the “orthogonality” of the principal axes (e.g. in a different but related context, Thomson (&lt;a href=&quot;https://doi.org/10.1098/rstl.1856.0022&quot; rel=&quot;nofollow noreferrer&quot;&gt;1856, p. 487&lt;/a&gt;) had already written: “&lt;em&gt;Def.&lt;/em&gt; A normal system ... is one in which the strains or stresses of the different types are ... mutually orthogonal”), but I share the impression that perpendicularity relative to general quadratic forms in higher dimensions didn’t become “intuitive” until later:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\hspace{4.5cm}$&lt;/span&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/c/c5/1D_normal_modes.gif&quot; alt=&quot;normal modes&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks:&lt;/strong&gt; 1) &lt;em&gt;Normal form&lt;/em&gt; has an earlier tradition going back to Jacobi (&lt;a href=&quot;https://books.google.com/books?id=KpxGAAAAcAAJ&amp;amp;pg=PA224&amp;amp;dq=formam+normalem&quot; rel=&quot;nofollow noreferrer&quot;&gt;1845&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=egVCAAAAcAAJ&amp;amp;pg=PA85&amp;amp;dq=Normalformen&quot; rel=&quot;nofollow noreferrer&quot;&gt;1848&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=uSC3DBtjqnwC&amp;amp;q=Normalform&quot; rel=&quot;nofollow noreferrer&quot;&gt;1876&lt;/a&gt;), Hesse (&lt;a href=&quot;https://books.google.com/books?id=DwcHAAAAcAAJ&amp;amp;q=Normalform&quot; rel=&quot;nofollow noreferrer&quot;&gt;1865&lt;/a&gt;), Kronecker (&lt;a href=&quot;https://books.google.com/books?id=K-0aAAAAYAAJ&amp;amp;pg=PA600&amp;amp;dq=Normalform&quot; rel=&quot;nofollow noreferrer&quot;&gt;1866&lt;/a&gt;). 2) Thomson and Tait themselves don’t say &lt;em&gt;normal coordinates&lt;/em&gt;. The first to do so may well be Lipschitz (&lt;a href=&quot;https://books.google.com/books?id=1j9bAAAAcAAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1870&lt;/a&gt;) who uses &lt;em&gt;Normalvariabelen&lt;/em&gt; (and &lt;em&gt;Normalform&lt;/em&gt;, &lt;em&gt;Normaltypus&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-08T10:41:53.097" LastActivityDate="2018-10-08T10:41:53.097" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7773" PostTypeId="1" CreationDate="2018-10-07T16:11:57.483" Score="3" ViewCount="97" Body="&lt;p&gt;Is there any known portrait of &lt;a href=&quot;https://en.wikipedia.org/wiki/Rafael_Bombelli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rafael Bombelli&lt;/a&gt;? I don't think so, but if you visit his &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bombelli.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor biography&lt;/a&gt;, you will see there this picture:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MzFk4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MzFk4.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is clear to me that this cannot possibly be a picture of a XVI&lt;sup&gt;th&lt;/sup&gt; century Italian. So:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Does anyone here know who is this person?&lt;/li&gt;&#xA;&lt;li&gt;Is there any known portrait of Bombelli?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-10-07T16:11:57.483" Title="A portrait of Bombelli" Tags="|mathematicians|biographical-details|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7774" PostTypeId="2" ParentId="7360" CreationDate="2018-10-08T03:00:32.170" Score="1" Body="&lt;p&gt;The foregoing are inferred estimates of pi by the ancient Egyptians. The only explicit mention of a circumference-to-diameter ratio by the ancient Egyptians is 3 in the &lt;em&gt;Demotic Mathematical Papyri&lt;/em&gt;.   &lt;/p&gt;&#xA;" OwnerUserId="7941" LastActivityDate="2018-10-08T03:00:32.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7775" PostTypeId="1" AcceptedAnswerId="7779" CreationDate="2018-10-08T12:49:30.527" Score="2" ViewCount="140" Body="&lt;p&gt;In &lt;a href=&quot;https://space.stackexchange.com/a/29980/12102&quot;&gt;this answer&lt;/a&gt; I cite and quote some surprising information; that the real Cyrano de Bergerac (not the one in the play) had written of using a rocket ship to go to the Moon circa 1650.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As strange as it sounds, I'm asking if Cyrano de Bergerac's conceptualization of a space ship include retro-rockets, or at least &lt;em&gt;some propulsive method to decelerate&lt;/em&gt; (slow down)?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2018-10-08T12:54:47.470" LastActivityDate="2018-10-08T13:36:36.973" Title="Did Cyrano de Bergerac's space ship have retro-rockets?" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7776" PostTypeId="1" AcceptedAnswerId="7777" CreationDate="2018-10-08T12:50:13.300" Score="1" ViewCount="650" Body="&lt;p&gt;I'm wondering what is the purpose of Gauss in his letter to János Bolyai when he mentioned &quot;howling Boeotians&quot;? Is it related to some scientists or philosophers that opposed non-Euclidean geometry? I appreciate any information or suggestion.&lt;/p&gt;&#xA;" OwnerUserId="7942" LastEditorUserId="4251" LastEditDate="2020-06-29T18:05:57.570" LastActivityDate="2020-06-29T18:05:57.570" Title="What is the purpose of Gauss in his letter to János Bolyai when he mentioned &quot;howling Boeotians&quot;?" Tags="|mathematics|mathematicians|geometry|gauss|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7777" PostTypeId="2" ParentId="7776" CreationDate="2018-10-08T13:08:35.330" Score="6" Body="&lt;p&gt;Gauss was probably afraid of attacks of some mathematicians and philosophers: the idea looked too radical for that time. The history shows that Gauss was right.&#xA;When non-Euclidean geometry was published (by Lobachevski), it had a poor reception in the beginning. Even among some mathematicians (for example Ostrogradskii). Later when Riemann published his ideas on what is called today Riemannian geometry he was also attacked by many people (including philosophers in his own university, like During).&lt;/p&gt;&#xA;&lt;p&gt;The reason of these attacks is that at that time geometry was supposed to describe the so-called &amp;quot;real world&amp;quot;, and that the geometry of the real world had to satisfy Euclidean axioms was considered &amp;quot;self-evident&amp;quot;. An influential philosopher (Kant) even insisted that our brain is &amp;quot;hard-wired&amp;quot; for Euclidean geometry. Kant's ideas were dominating philosophy in the time of Gauss, at least in Germany. Even in the early 20th century some outstanding mathematicians (Steklov, for example) criticized Riemann's ideas as &amp;quot;unrelated to real world&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-29T00:40:07.213" LastActivityDate="2020-06-29T00:40:07.213" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7778" PostTypeId="2" ParentId="7776" CreationDate="2018-10-08T13:30:47.060" Score="1" Body="&lt;p&gt;Would have been easier if spelled right... sigh.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From  an &lt;a href=&quot;https://www.dictionary.com/browse/boeotian&quot; rel=&quot;nofollow noreferrer&quot;&gt;online dictionary&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;dull; obtuse; without cultural refinement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this case, a cultural slur on the residents of Boeotia, and presumably anyone else that Gauss disapproved of. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-10-08T13:30:47.060" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7779" PostTypeId="2" ParentId="7775" CreationDate="2018-10-08T13:36:36.973" Score="2" Body="&lt;p&gt;Hard to say, seeing as his little &lt;a href=&quot;http://www.photonics.umbc.edu/Menyuk/Schwartz_VOYAGE-TO-THE-MOON.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;essay&lt;/a&gt; about going to the moon was almost certainly a combination of fantasy and satire.  Consider his landing:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After falling a long time, or so I presume, the violence of my fall&#xA;  having kept me from noticing it, the next thing I can recall is&#xA;  finding myself under a tree, tangled in three or four rather thick&#xA;  branches that I had shattered by my fall, with my face smeared with an&#xA;  apple that had smashed into it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-10-08T13:36:36.973" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7780" PostTypeId="2" ParentId="7749" CreationDate="2018-10-09T17:53:38.470" Score="2" Body="&lt;p&gt;Indian and Chinese calendars are astronomical, in that for instance dates are based on the exact moment of astronomical events. So wheter for instance a new Moon falls one second before or after midnight (in the Chinese case) or one seconde before or after sunrise (in the Indian case), can change the date of the start of a lunar month. That's why astronomical precission has been of paramount importance in both countries .&lt;/p&gt;&#xA;" OwnerUserId="661" LastActivityDate="2018-10-09T17:53:38.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7781" PostTypeId="1" AcceptedAnswerId="7782" CreationDate="2018-10-10T06:25:05.060" Score="2" ViewCount="311" Body="&lt;p&gt;Wiki says that his transfinite numbers met opposition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Henri Poincaré referred to his ideas as a &amp;quot;grave disease&amp;quot; infecting&#xA;the discipline of mathematics, and Leopold Kronecker's public&#xA;opposition and personal attacks included describing Cantor as a&#xA;&amp;quot;scientific charlatan&amp;quot;, a &amp;quot;renegade&amp;quot; and a &amp;quot;corrupter of youth&amp;quot;.&#xA;Kronecker objected to Cantor's proofs that the algebraic numbers are&#xA;countable, and that the transcendental numbers are uncountable,&#xA;results now included in a standard mathematics curriculum. Writing&#xA;decades after Cantor's death, Wittgenstein lamented that mathematics&#xA;is &amp;quot;ridden through and through with the pernicious idioms of set&#xA;theory&amp;quot;, which he dismissed as &amp;quot;utter nonsense&amp;quot; that is &amp;quot;laughable&amp;quot;&#xA;and &amp;quot;wrong&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some Christian theologians (particularly neo-Scholastics) saw Cantor's&#xA;work as a challenge to the uniqueness of the absolute infinity in the&#xA;nature of God – on one occasion equating the theory of transfinite&#xA;numbers with pantheism&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Can you briefly explain when the commotion started, if it regarded also his work on number or set theory , show what was the most grounded or shared criticism and when the opposition ended and who turned the tide? Was it Hilbert ? Could you please specify if any doubt is still surviving on any aspect of his theories?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is definitely not a duplicate of the linked question nor the links in the comments provide any help:&lt;/p&gt;&#xA;&lt;p&gt;That question tackles a rather marginal problem: whether Poincaré said it or not? I asked what other ideas met opposition and when / by the help of whom criticism was overcome and, indeed, if it has definitely or if even today there are some detractors?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="4251" LastEditDate="2020-07-25T20:33:23.747" LastActivityDate="2020-07-25T20:33:23.747" Title="Cantor's fortune" Tags="|mathematics|mathematicians|set-theory|cantor|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7782" PostTypeId="2" ParentId="7781" CreationDate="2018-10-10T15:03:46.837" Score="3" Body="&lt;p&gt;This question is too broad, and it is impossible to give a short answer.&lt;/p&gt;&#xA;&lt;p&gt;In the beginning of 20th century, Cantor's discoveries led to what is called a &amp;quot;crisis of foundations&amp;quot; in mathematics. You are right. Hilbert was one of the principal defenders of Cantor's theory, but paradoxes were soon discovered in set theory and the heated discussion continued.&lt;/p&gt;&#xA;&lt;p&gt;There were several programs on how to get rid of these paradoxes, the most famous of them being &amp;quot;Hilbert's program&amp;quot;. Two other alternatives were Intuitionism and Type Theory. Later, Gödel showed that Hilbert program would not work. For the full story, I recommend an excellent book:&lt;/p&gt;&#xA;&lt;p&gt;Fraenkel, Abraham A.; Bar-Hillel, Yehoshua Foundations of set theory. Studies in Logic and the Foundations of Mathematics North-Holland Publishing Co., Amsterdam 1958.&lt;/p&gt;&#xA;&lt;p&gt;One can say that the discussion on the foundations of set theory still continues, but most mathematicians are satisfied with ZFC axioms.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2020-07-05T08:11:16.610" LastActivityDate="2020-07-05T08:11:16.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7783" PostTypeId="1" CreationDate="2018-10-11T02:05:28.563" Score="0" ViewCount="73" Body="&lt;p&gt;What examples are there of scientific effects or correlations that we now know to be real, and were known about for a long time but thought to be coincidences?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give an example of the kind of thing that I have in mind (but I don't necessarily know if these are real or the best examples):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;People used to know that there was a relationship between the moon and the tides, but expert opinion was that this was just a superstition. Now we know the moon really does influence tides.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &quot;hot hand&quot; effect in basketball. Originally this was claimed to be the result of random chance and a psychological bias to see patterns, later some papers claimed that this was a real effect. I'm not sure what the current state of the debate is.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7951" LastActivityDate="2018-10-11T12:46:13.777" Title="Real effects long thought to be coincidences" Tags="|philosophy-of-science|scientific-method|statistics|big-list|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7784" PostTypeId="1" CreationDate="2018-10-11T05:03:56.473" Score="0" ViewCount="147" Body="&lt;p&gt;How to understand Freeman Dyson's Saying:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After quantum mechanics, nature itself suddenly became linear.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QIJFu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QIJFu.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7944" LastActivityDate="2018-11-11T00:02:40.500" Title="How to understand `After quantum mechanics, nature itself suddenly became linear`?" Tags="|quantum-mechanics|linear-algebra|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7785" PostTypeId="1" AcceptedAnswerId="7787" CreationDate="2018-10-11T05:32:41.017" Score="8" ViewCount="619" Body="&lt;p&gt;I got curious who invented the &lt;a href=&quot;https://en.wikipedia.org/wiki/Particle_in_a_box&quot; rel=&quot;noreferrer&quot;&gt;particle-in-a-box&lt;/a&gt; model first. It is really simple and intuitive. I was googling to find the original author who suggested it but I only get textbook or webpages as results. &lt;/p&gt;&#xA;" OwnerUserId="7956" OwnerDisplayName="Suzanka" LastActivityDate="2018-10-11T13:00:11.927" Title="Who solved the particle-in-a-box model first?" Tags="|quantum-mechanics|potential|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7786" PostTypeId="2" ParentId="7783" CreationDate="2018-10-11T12:46:13.777" Score="1" Body="&lt;p&gt;Dunno, how about&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- equivalence of gravitational and inertial mass&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- transmission of disease via contact. Well-known, but not proven until well after microscopes came along. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You could contrast with effects long-thought to be real but which were &lt;strong&gt;disproven&lt;/strong&gt;, such as  night air carrying disease&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-10-11T12:46:13.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7787" PostTypeId="2" ParentId="7785" CreationDate="2018-10-11T13:00:11.927" Score="5" Body="&lt;p&gt;The eigenvalue problem of a free particle constrained by “vessel walls” or an “infinite potential well” is mentioned without details by Schrödinger (&lt;a href=&quot;https://zbmath.org/?q=an:52.0966.01&quot; rel=&quot;noreferrer&quot;&gt;1926&lt;/a&gt;, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k153811/f543&quot; rel=&quot;noreferrer&quot;&gt;p. 523&lt;/a&gt;; &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.211600/page/n48&quot; rel=&quot;noreferrer&quot;&gt;translation&lt;/a&gt;) and Hund (&lt;a href=&quot;https://doi.org/10.1007/BF01400234&quot; rel=&quot;noreferrer&quot;&gt;1927&lt;/a&gt;, p. 747). An early reference giving the energy levels and eigenfunctions seems to be Condon and Morse’s book (&lt;a href=&quot;https://zbmath.org/?q=an:55.1172.09&quot; rel=&quot;noreferrer&quot;&gt;1929&lt;/a&gt;, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.$b568098&amp;amp;num=44&quot; rel=&quot;noreferrer&quot;&gt;p. 44&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-11T13:00:11.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7789" PostTypeId="1" CreationDate="2018-10-11T15:11:34.253" Score="9" ViewCount="273" Body="&lt;p&gt;How did Faraday, Ampere and Hippolyte Pixii know that electricity 'traveled' at all?  Let alone in a certain direction?  And that it was reversing direction with certain early generators?&lt;/p&gt;&#xA;" OwnerUserId="15273" OwnerDisplayName="Kurt Hikes" LastActivityDate="2018-10-12T03:24:23.997" Title="How did early scientists know if a current was changing direction? (AC vs. DC)" Tags="|electromagnetism|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7790" PostTypeId="2" ParentId="7789" CreationDate="2018-10-11T16:44:58.897" Score="10" Body="&lt;p&gt;Daniel Sank is correct; with a magnetized needle on a pivot and a coil of wire you can make a device called a galvanometer with which you can watch current change direction, at least for low frequencies of reversal. The early experimenters could connect galvanometers to their electrochemical experiments and correlate the direction of needle deflection to the direction of charge transport. &lt;/p&gt;&#xA;" OwnerUserId="14713" OwnerDisplayName="niels nielsen" LastActivityDate="2018-10-11T16:44:58.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7791" PostTypeId="2" ParentId="7789" CreationDate="2018-10-11T17:14:57.943" Score="0" Body="&lt;p&gt;Well physicists also conducted experiments on Electrochemistry and stuff like Electroplating and where one metal atom 'travels' to another may have made them think about electricity 'travelling'.&lt;/p&gt;&#xA;" OwnerUserId="7960" OwnerDisplayName="Harshit Joshi" LastActivityDate="2018-10-11T17:14:57.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7792" PostTypeId="2" ParentId="7789" CreationDate="2018-10-11T19:56:27.443" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://www.encyclopedia.com/people/history/us-history-biographies/benjamin-franklin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Benjamin Franklin&lt;/a&gt;, with his lightning kite experiment, was one of the first to realize that electricity travels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère&lt;/a&gt; knew that parallel wires with current going the same (opposite) direction attract (repel). Thus, if there an AC, you will see the wires vibrating like a plucked string.&lt;/p&gt;&#xA;" OwnerUserId="232" OwnerDisplayName="Geremia" LastEditorUserId="232" LastEditDate="2018-10-11T21:21:57.867" LastActivityDate="2018-10-11T21:21:57.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7793" PostTypeId="2" ParentId="5452" CreationDate="2018-10-11T21:54:21.487" Score="1" Body="&lt;p&gt;Other answers made it clear that “commissioned Max Planck” is likely a stretch although Siemens, owner of the oldest German light bulb factory (&lt;a href=&quot;https://www.osram.com/media/resource/HIRES/334233/2591693/history---100-years-of-osram.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Siemens &amp;amp; Halske&lt;/a&gt;), did heavily fund Berlin physics (e.g. co-founded the &lt;a href=&quot;https://books.google.com/books?id=X4lvDO6kcdsC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reichsanstalt&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Physikalisch-Technische_Bundesanstalt#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;gave it land&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if the story could be rooted in conflation of Planck with his engineer brother &lt;a href=&quot;https://books.google.com/books?id=Ph4LAAAAYAAJ&amp;amp;pg=PA256&amp;amp;dq=Adalbert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adalbert&lt;/a&gt; (&lt;a href=&quot;https://books.google.com/books?id=7ck6AQAAMAAJ&amp;amp;pg=PA445&amp;amp;dq=Adelbert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adelbert&lt;/a&gt;?) who &lt;a href=&quot;https://www.siemens.com/history/en/news/1297_siemens-in-munich.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;did&lt;/a&gt; work for Siemens. (This might be &lt;a href=&quot;https://books.google.com/books?id=Qihs7RIvlOYC&amp;amp;pg=PA243&amp;amp;dq=Adalbert+Planck&quot; rel=&quot;nofollow noreferrer&quot;&gt;checkable&lt;/a&gt; in the &lt;a href=&quot;https://www.siemens.com/global/en/home/company/about/history/siemens-historical-institute/archives.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Siemens-Akten-Archiv&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-11T21:54:21.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7794" PostTypeId="2" ParentId="7784" CreationDate="2018-10-11T23:48:54.547" Score="2" Body="&lt;p&gt;Until quantum mechanics, linear differential equations arising in physics were the result of approximation (linearization) of non-linear ones. So it was believed that the fundamental laws of nature are described by non-linear differential equations,&#xA;while linear ones are only used to approximate them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the invention of quantum mechanics we know that the most fundamental equation &#xA;of physics (Schrodinger equation) is a linear differential equation. And that the&#xA;space of states of a physical system is a vector space (in classical mechanics it is&#xA;a manifold, again a non-linear object). As QM is the most fundamental theory that we currently possess about the real world, the statement of Dyson is well justified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;QM is indeed the most fundamental theory: most physical theories are thought to be approximate. This is the only one that seems to be exact, at the present state of our knowledge.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-11T23:48:54.547" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7795" PostTypeId="2" ParentId="7789" CreationDate="2018-10-12T03:24:23.997" Score="0" Body="&lt;p&gt;In the 1830s many people were interested in knowing the &quot;speed&quot; of electricity in wires. Wheatsone designed a visual experiment to determine the speed of electricity the help of rotating mirrors. The original reference is &#xA;Wheatstone, Charles (1834). &quot;An Account of Some Experiments to Measure the Velocity of Electricity and the Duration of Electric Light&quot;. Philosophical Transactions of the Royal Society of London. 124: 583–591.&#xA;Here you might get better and original references.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-10-12T03:24:23.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7796" PostTypeId="2" ParentId="7622" CreationDate="2018-10-12T04:28:56.303" Score="4" Body="&lt;p&gt;Dirichlet's proof of the convergence of the Fourier series was nearly identical to a proof that Fourier offered in his original treatise on Heat Conduction. The reason this seems to have gone unnoticed is that Fourier's original manuscript was banned from publication for over a decade. Dirichlet was Fourier's student, and probably had access to the unpublished manuscript. H.S. Carslaw notes this in the &lt;em&gt;Historical Introduction&lt;/em&gt; to his 1906 book &lt;em&gt;Introduction to the Theory of Fourier’s series and integrals&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;However, it is a mistake to suppose that Fourier did not establish in a rigorous and conclusive manner that a quite arbitrary function (meaning by this any function capable of being represented by an arc of a continuous curve or by successive portions of diﬀerent continuous curves), could be represented by the series with his name, and it is equally wrong to attribute the ﬁrst rigorous demonstration of this theorem to Dirichlet, whose earliest memoir was published in 1829. A closer examination of Fourier’s work will show that the importance of his investigations merits the fullest recognition, and Darboux, in the latest complete edition of Fourier’s mathematical works points out that the method he employed in the ﬁnal discussion of the general case is perfectly sound and practically identical with that used later by Dirichlet in his classical memoir. In this discussion Fourier followed the line of argument which is now customary in dealing with inﬁnite series. He proved that when the values &lt;span class=&quot;math-container&quot;&gt;$a_n,b_n$&lt;/span&gt; are inserted in the terms of the series&quot;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a0 + (a1 \cos x + b1 \sin x) + (a2 \cos 2x + b2 \sin 2x) + …,$$&lt;/span&gt;&#xA;the sum of the terms up to &lt;span class=&quot;math-container&quot;&gt;$\cos(nx)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin(nx)$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{\pi}\int_{-\pi}^{\pi} f(x')\frac{ \sin \frac{1}{2}(2n + 1)(x'−x) }{\sin \frac{1}{2}(x'−x)} dx'$$&lt;/span&gt;&#xA;He then discussed the limiting value of this sum as n becomes inﬁnite, and thus obtained the sum of the series now called Fourier’s Series.&quot;&lt;/p&gt;&#xA;" OwnerUserId="7961" LastActivityDate="2018-10-12T04:28:56.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7799" PostTypeId="2" ParentId="3364" CreationDate="2018-10-13T07:33:06.203" Score="2" Body="&lt;p&gt;Faced with an interruption in the Euler–Lagrange correspondence (late 1762 to early 1765), its editors comment very cautiously (&lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=595168&quot; rel=&quot;nofollow noreferrer&quot;&gt;1980&lt;/a&gt;, p. 49): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=qf8cLkaJ7pgC&amp;amp;ots=UZBu3ob7wZ&amp;amp;pg=PA49&amp;amp;dq=refroidissement&quot; rel=&quot;nofollow noreferrer&quot;&gt;one can attempt to see here a hint of cooling&lt;/a&gt; in the relation between Lagrange and Euler.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As possible reasons, they suggest 1) Lagrange’s frustration with the delay and eventual scrapping of the publication of a treatise on variational calculus he had sent to Euler in 1756, 2) Euler’s hasty generalization and use of Lagrange results on sound propagation, 3) Lagrange visiting Paris and sending greetings only &lt;em&gt;via&lt;/em&gt; Euler’s foe D’Alembert, 4) Lagrange beating Euler to the 1764 Paris prize on Moon libration.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This, and the general downplaying of Euler in history sections of &lt;em&gt;Mécanique Analytique&lt;/em&gt; — famously lamented by Truesdell (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=70568&quot; rel=&quot;nofollow noreferrer&quot;&gt;1954&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=-1ryAAAAMAAJ&amp;amp;q=anteriorities&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. xliii&lt;/a&gt;) (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=131341&quot; rel=&quot;nofollow noreferrer&quot;&gt;1960&lt;/a&gt;, pp. &lt;a href=&quot;https://books.google.com/books?id=pywPAAAAIAAJ&amp;amp;q=%22Lagrange&amp;#39;s+reluctance+to+cite+Euler%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;252&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=pywPAAAAIAAJ&amp;amp;q=%22worse+than+none%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;411&lt;/a&gt;) (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=242616&quot; rel=&quot;nofollow noreferrer&quot;&gt;1968&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=_crsCAAAQBAJ&amp;amp;ots=PXTwDdVeCj&amp;amp;pg=PA134&amp;amp;dq=misjudgment&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 134&lt;/a&gt;) — is further discussed in:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Galletto, Dionigi&lt;/em&gt;, Lagrange e le origini della &lt;em&gt;Mécanique Analytique&lt;/em&gt;, Giornale di Fisica 32, 84-126 (1991).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Galletto, Dionigi&lt;/em&gt;, The genesis of &lt;em&gt;Mécanique analytique&lt;/em&gt;, Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 126, Suppl., 277-370 (1992). &lt;a href=&quot;https://zbmath.org/?q=an:1180.70001&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL1180.70001&lt;/a&gt;. (See &lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=1264671&quot; rel=&quot;nofollow noreferrer&quot;&gt;MR&lt;/a&gt; items (d), (e).)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Capecchi, Danilo; Drago, Antonino&lt;/em&gt;, &lt;a href=&quot;http://dx.doi.org/10.1007/s11012-004-2198-z&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;On Lagrange’s history of mechanics&lt;/strong&gt;&lt;/a&gt;, Meccanica 40, No. 1, 19-33 (2005). &lt;a href=&quot;https://zbmath.org/?q=an:1098.01005&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL1098.01005&lt;/a&gt;. (See end of §2.3.1.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Caparrini, Sandro&lt;/em&gt;, &lt;a href=&quot;http://dx.doi.org/10.1007/s40329-014-0043-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The history of the &lt;em&gt;Méchanique analitique&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;, Lettera Matematica, Int. Ed. 2, No. 1-2, 47-54 (2014). &lt;a href=&quot;https://zbmath.org/?q=an:1332.70005&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL1332.70005&lt;/a&gt;. (Alleges unspecified “sarcastic remarks on Euler” in the Lagrange–D'Alembert correspondence, which is &lt;a href=&quot;http://www.archive.org/details/uvresdelagrange13lagr&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;.) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-13T19:07:47.567" LastActivityDate="2018-10-13T19:07:47.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7800" PostTypeId="1" AcceptedAnswerId="7802" CreationDate="2018-10-13T12:17:31.943" Score="3" ViewCount="657" Body="&lt;p&gt;I am quite interested in the origins of our modern way of understanding analysis. I know that Augustin-Louis Cauchy was one of the pioneers regarding a rigorous foundation towards real and complex analysis, and further he is one of my favorite mathematicians. Therefore I would like to study his original works. The book I am especially interested in is called &quot;Cours d'analyse de l'École royale polytechnique&quot;. It is not hard to find these notes in French (they are linked on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy#Output&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page&lt;/a&gt; of Cauchy for example) but hence I am not capable of reading and understanding French this does not help me at all. Honestly speaking, I have no clue how to search for a translation, which source I can trust and so on.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Therefore I would ask first of all if you are aware of a complete English translation? I am rather sure that something similar to this exists hence the importance of Cauchy and his work within the history of analysis is crucial. Furthermore, could you provide a source where I could - in the best case - directly download the whole papers for example in the form of a PDF or - in the worst case - where I could at least buy a PDF or printed version?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="7971" LastEditorUserId="10106" LastEditDate="2020-06-06T13:48:41.533" LastActivityDate="2020-06-06T13:48:41.533" Title="An English translation of Cauchy's &quot;Cours d'Analyse&quot;" Tags="|mathematics|reference-request|translations|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7801" PostTypeId="1" CreationDate="2018-10-13T16:15:32.830" Score="3" ViewCount="1224" Body="&lt;p&gt;I was looking for some properties of the prime numbers, and I found some curiosities of the number 17 in that &lt;a href=&quot;https://primes.utm.edu/curios/page.php/17.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;site&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of them is this &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;17 was called by the Pythagoreans - opposition, obstruction, and evil,&#xA;  and the day the Devil triumphed over God.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But, unlike the other curiosities, it has no reference. Have you read anything like that? Can you tell me more about this?&lt;/p&gt;&#xA;" OwnerUserId="7972" LastActivityDate="2022-06-15T15:01:46.567" Title="Pythagorean and the number 17" Tags="|number-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7802" PostTypeId="2" ParentId="7800" CreationDate="2018-10-14T00:47:19.770" Score="3" Body="&lt;p&gt;As others have pointed out, &#xA;there seems to be a 2009 &lt;a href=&quot;https://www.springer.com/us/book/9781441905482&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer translation&lt;/a&gt; by Bradley and Sandifer, linked by a different Wikipedia article  &lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2018-10-14T13:14:51.420" LastActivityDate="2018-10-14T13:14:51.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7804" PostTypeId="1" CreationDate="2018-10-14T17:48:54.273" Score="0" ViewCount="682" Body="&lt;p&gt;Just curious about the history of electric flux&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Flux-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of the fact that flux, on its own is a mathematical concept but how did it find its way into physics. Was  it just introduced to replace the &lt;strong&gt;E.A&lt;/strong&gt; in gauss law or maybe in an attempt to explain inverse square law. Or was it because physicists were experimenting by using all possible operations on vector fields? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-Field lines-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a similar doubt regarding regarding field lines. I believe that they were probably introduced to explain the inverse square law by saying that the lines of &lt;strong&gt;force&lt;/strong&gt; get distributed over a surface of a sphere and thus vary inversely with the square of distance.&#xA;Am i correct?&#xA;Thx for any help&lt;/p&gt;&#xA;" OwnerUserId="7967" LastActivityDate="2018-10-15T13:34:45.473" Title="Concept of electric flux" Tags="|electricity|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7805" PostTypeId="1" AcceptedAnswerId="7808" CreationDate="2018-10-14T19:48:47.780" Score="11" ViewCount="520" Body="&lt;p&gt;It's obvious for us, that &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt; can also be written as &lt;span class=&quot;math-container&quot;&gt;$\frac{d}{dx}y$&lt;/span&gt;, but skimming through Leibniz or Eulers writings I couldn't see them write the latter. I speculate that this change happened out of laziness, when the expression for &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; became very long, but I wonder if it was also related with a shift of perspective from treating &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; as the main operator (Leibniz and Euler) to treating &lt;span class=&quot;math-container&quot;&gt;$d/dx$&lt;/span&gt; as main one (maybe starting with Lagrange?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(If this question seems to be about a triviality, consider writing &lt;span class=&quot;math-container&quot;&gt;$\frac{\log}{\log 2}x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(\log/\log 2) x$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\frac{\log x}{\log 2}$&lt;/span&gt; in an article or in front of students.)&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2020-02-13T07:25:16.810" Title="When and why did $\frac{dy}{dx}$ become $\frac{d}{dx}y$?" Tags="|notation|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7806" PostTypeId="2" ParentId="7804" CreationDate="2018-10-14T19:52:59.210" Score="-1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The line of force, in physics, is  the path followed by an electric charge free to move in an electric field  or generally any appropriate test particle in a given force field.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;More abstractly, lines of force &lt;strong&gt;are lines in any such force field the tangent of which at any point gives the field direction at that point&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and &lt;strong&gt;the density of these field lines gives the magnitude of the field.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To investigate the nature of electric force between charges -** Coulomb had performed  an  experiment using  Torsion balance and the inverse square law could be verified for electric field between charges .**&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The concept of lines of force was introduced into physics in the 1830s by the English scientist &lt;strong&gt;Michael Faraday&lt;/strong&gt;, who considered magnetic and electric effects in the region around a magnet or electric charge as a property of the region rather than an effect taking place at a distance from a cause.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The electric lines of force that represent the field of a positive electric charge in space &lt;strong&gt;consist of a family of straight lines radiating uniformly in all directions from the charge where they originate.&lt;/strong&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;A second positive charge placed in the field would travel radially away from the first charge.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;By relating the strength of the Electric field to the density of lines of force crossing unit area of a surface perpendicular to the flux lines led to inverse square dependence on radial distance.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The concept of  Maxwell's tubes of forces were also defined and as area crossing through the field lines or tubes varies as the inverse square of the distance(thereby the density of lines of force will decrease as inverse square) it was understood that the nature of field strength will vary as 1/ r^2.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ref.-&#xA;  &lt;a href=&quot;https://www.britannica.com/science/line-of-force&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.britannica.com/science/line-of-force&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7635" LastEditorUserId="7635" LastEditDate="2018-10-15T13:34:45.473" LastActivityDate="2018-10-15T13:34:45.473" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7807" PostTypeId="1" AcceptedAnswerId="7816" CreationDate="2018-10-14T20:59:11.100" Score="8" ViewCount="637" Body="&lt;p&gt;Recently I have encountered the so-called Umbral calculus. The main idea of this field is to treat indices as exponents, applying simpler techniques available to exponents and switching everything back when the work is done. The Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Umbral_calculus&quot; rel=&quot;noreferrer&quot;&gt;Umbral calculus&lt;/a&gt; states that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These techniques were introduced by John Blissard (1861) and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Further there is a brief historical outline of the development of this technique and even a look on the modern usage of Umbral calculus. Hence I am quite fascinated by this topic I would like to learn more about it; especially its applications for proofs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First of all the Wikipedia side offers a proof of the derivatives of the Bernoulli polynomials by using Umbral calculus. Furthermore I am aware of &lt;a href=&quot;https://math.stackexchange.com/questions/191752/whats-umbral-calculus-about/1085165&quot;&gt;this question on MSE&lt;/a&gt; which was answered by Tom Copeland with some interesting examples. Furthermore within &lt;a href=&quot;https://math.stackexchange.com/q/1823862&quot;&gt;this answer on MSE&lt;/a&gt; the user Count Iblis provided an elegant proof of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ramanujan%27s_master_theorem&quot; rel=&quot;noreferrer&quot;&gt;Ramanujan's Master Theorem&lt;/a&gt; also using umbral calculus. Anyway I have not found that much further one MSE or somewhere else at first sight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Refering to the quote from Wikipedia I was not able to find some proofs by neither É. Lucas nor by J. J. Sylvester actually  using this field of mathematics. By saying &quot;&lt;em&gt;I was not able to find some proofs&lt;/em&gt;&quot; I mean I have found nothing detached from a book of something similiar. It seems like Umbral calculus does not gain that much attention overall.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What I would like to know: &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; Are you aware of proofs using Umbral calculus beside the few I mentioned? I would be interested in some historical examples - for instance the first proof using this technique, some proofs by the named mathematician and other, etc. - as well as mondern examples. &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; Is there a reason why it seems like there are not that much applications of this techniques? Yes, of course the possibility that a proof can be done with Umbral calculus is not that high but however I would say there are enough problems cover something so simple as indices.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="7971" LastEditorUserId="55" LastEditDate="2018-10-15T21:01:21.843" LastActivityDate="2021-07-11T18:29:18.570" Title="Why is umbral calculus not used more widely?" Tags="|mathematics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7808" PostTypeId="2" ParentId="7805" CreationDate="2018-10-15T02:16:18.387" Score="8" Body="&lt;p&gt;If you mean conceptually, I remember reading that Arbogast “&lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Fran%C3%A7ois_Antoine_Arbogast&quot; rel=&quot;nofollow noreferrer&quot;&gt;was the first writer to separate the symbols of operation from those of quantity&lt;/a&gt;.” Among those he influenced were &lt;a href=&quot;https://en.wikipedia.org/wiki/Analytical_Society&quot; rel=&quot;nofollow noreferrer&quot;&gt;Babbage, Herschel, Peacock&lt;/a&gt;, in whose edited translation of Lacroix that phrase is used to justify writing Taylor’s formula for &lt;span class=&quot;math-container&quot;&gt;$u=f(x)$&lt;/span&gt; in the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Delta u =\bigl(e^{h\frac d{dx}}-1\bigr)u&#xA;\tag{*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://archive.org/details/anelementarytre00babbgoog/page/n497&quot; rel=&quot;nofollow noreferrer&quot;&gt;1816&lt;/a&gt;, see pp. 486–489) — an improvement over the rather clumsy &lt;span class=&quot;math-container&quot;&gt;$\Delta u =e^{\frac{du}{dx}h}-1$&lt;/span&gt; of Lagrange (&lt;a href=&quot;https://archive.org/details/nouv-berlin-1772/page/n267&quot; rel=&quot;nofollow noreferrer&quot;&gt;1774&lt;/a&gt;, p. 195), Lacroix (&lt;a href=&quot;https://archive.org/details/traitducalculdi00lacrgoog/page/n28&quot; rel=&quot;nofollow noreferrer&quot;&gt;1800&lt;/a&gt;, p. 13), Brinkley (&lt;a href=&quot;//doi.org/10.1098/rstl.1807.0007&quot; rel=&quot;nofollow noreferrer&quot;&gt;1807&lt;/a&gt;), or Laplace (&lt;a href=&quot;https://archive.org/details/theorieanaldepro00laplrich/page/n230&quot; rel=&quot;nofollow noreferrer&quot;&gt;1820&lt;/a&gt;, p. 41). (Arbogast wrote it &lt;span class=&quot;math-container&quot;&gt;$\Delta u=(e^{h\partial.}-1)\times u$&lt;/span&gt; (&lt;a href=&quot;https://archive.org/details/ducalculdesdri00arbouoft/page/350&quot; rel=&quot;nofollow noreferrer&quot;&gt;1800&lt;/a&gt;, p. 350), and Lacroix adopted (*) in his second edition (&lt;a href=&quot;https://archive.org/details/traitducalculdi04lacrgoog/page/n89&quot; rel=&quot;nofollow noreferrer&quot;&gt;1819&lt;/a&gt;, p. 61).) The ability to meaningfully write such formulas may well be the “why” you are after.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you mean just typographically, I don’t know if earlier examples exist!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; As noted today on &lt;a href=&quot;//mathoverflow.net/questions/352294/earliest-reference-on-the-calculation-of-derivatives-by-fourier-transform&quot;&gt;MO&lt;/a&gt;, Fourier (maybe first?) uses a “bare” &lt;span class=&quot;math-container&quot;&gt;$\smash{\frac d{dx}}$&lt;/span&gt; in &lt;em&gt;Théorie analytique de la chaleur&lt;/em&gt; (&lt;a href=&quot;//archive.org/details/thorieanalytiq00four/page/540&quot; rel=&quot;nofollow noreferrer&quot;&gt;1822&lt;/a&gt;, Art. 399-405, 410-414, 419-422,...). &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-02-13T07:25:16.810" LastActivityDate="2020-02-13T07:25:16.810" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7809" PostTypeId="1" CreationDate="2018-10-15T07:04:12.237" Score="7" ViewCount="391" Body="&lt;p&gt;I find numerous claims that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Basel_problem&quot; rel=&quot;noreferrer&quot;&gt;Basel Problem&lt;/a&gt; was first posed by &lt;a href=&quot;https://en.wikipedia.org/wiki/Pietro_Mengoli&quot; rel=&quot;noreferrer&quot;&gt;Pietro Mengoli&lt;/a&gt; in 1644. However, I am unable to find even the name of the publication (or book or letter) in which this was supposedly done.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am hoping someone will help me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(My suspicion is that this might be an academic urban legend. Mengoli would have been 18 years old in 1644 and the earliest publication I can find of his is his 1650 &lt;em&gt;&lt;a href=&quot;https://archive.org/details/bub_gb_PrKgVx1LcUUC&quot; rel=&quot;noreferrer&quot;&gt;Novae quadraturae arithmeticae&lt;/a&gt;&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Related question: Where (if at all) in &lt;em&gt;Novae quadraturae arithmeticae&lt;/em&gt; (1650) does Mengoli pose the Basel Problem? (Unfortunately I don't know any Latin and there doesn't seem to be any translation of this work.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Below is a list of references claiming that Mengoli first posed the Basel Problem in 1644.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All fail to cite the name of the avenue in which he did so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And only two even bother citing anything at all for their claim. One (Mathworld) does so to another book on this list, while the other's (Benko, 2012) citations are incorrect.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Papers/articles&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;The Riemann Zeta Function With Even Arguments as Sums Over Integer Partitions&quot; (2017, p. &lt;a href=&quot;https://www.jstor.org/stable/10.4169/amer.math.monthly.124.6.554&quot; rel=&quot;noreferrer&quot;&gt;554&lt;/a&gt;) by Mircea Merca.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Explorations in the theory of partition zeta functions&quot; (2016, p. &lt;a href=&quot;https://arxiv.org/pdf/1605.05536&quot; rel=&quot;noreferrer&quot;&gt;2&lt;/a&gt;) by Ono, Reilen, and Schneider.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Some integrals related to the Basel problem&quot; (2016, p. &lt;a href=&quot;https://arxiv.org/pdf/1611.03571&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt;) by Khristo Boyadzhiev.&lt;/li&gt;&#xA;&lt;li&gt;&quot;The Connection between the Basel Problem and a Special Integral&quot; (2014, p. &lt;a href=&quot;https://file.scirp.org/pdf/AM_2014090416224975.pdf&quot; rel=&quot;noreferrer&quot;&gt;2570&lt;/a&gt;) by Zhou and Xu.&lt;/li&gt;&#xA;&lt;li&gt;&quot;The Basel Problem as a Rearrangement of Series&quot; (2013, p. &lt;a href=&quot;https://www.jstor.org/stable/10.4169/college.math.j.44.3.171&quot; rel=&quot;noreferrer&quot;&gt;171&lt;/a&gt;) by David Benko and John Molokach.&lt;/li&gt;&#xA;&lt;li&gt;&quot;The Basel Problem as a Telescoping Series&quot;, (2012, p. &lt;a href=&quot;https://www.jstor.org/stable/10.4169/college.math.j.43.3.244&quot; rel=&quot;noreferrer&quot;&gt;244&lt;/a&gt;) by David Benko.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Benko (2012) incorrectly cites (i) William Dunham who does not give any specific year (&lt;em&gt;Euler: The Master of Us All&lt;/em&gt;, 1999, p. &lt;a href=&quot;https://books.google.com/books?id=x7p4tCPPuXoC&amp;amp;pg=PA42&amp;amp;lpg=PA42&amp;amp;dq=%22Decades%20earlier,%20Pietro%20Mengoli%20had%20raised%20the%20question%22&quot; rel=&quot;noreferrer&quot;&gt;42&lt;/a&gt;); and (ii) Julian Havil who gives the year of 1650 and does not mention 1644 (2003, &lt;em&gt;Gamma: Exploring Euler's Constant&lt;/em&gt;, p. &lt;a href=&quot;https://books.google.com/books?id=7-sDtIy8MNIC&amp;amp;pg=PA38&amp;amp;lpg=PA38&amp;amp;dq=%22following%20its%20appearance%20in%201650,%20when%20Pietro%20Mengoli%22&quot; rel=&quot;noreferrer&quot;&gt;38&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&quot;A simple solution to Basel problem&quot; (2008, p. &lt;a href=&quot;https://www.emis.de/journals/GM/vol16nr4/ivan/ivan.pdf&quot; rel=&quot;noreferrer&quot;&gt;111&lt;/a&gt;) by Mircea Ivan.&lt;/li&gt;&#xA;&lt;li&gt;&quot;Euler’s Solution of the Basel Problem&quot; (2003, p. &lt;a href=&quot;https://faculty.math.illinois.edu/~reznick/sandifer.pdf&quot; rel=&quot;noreferrer&quot;&gt;2&lt;/a&gt;) by Ed Sandifer&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Books&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Anthony Sofo in &lt;em&gt;Mathematical Analysis and Applications: Selected Topics&lt;/em&gt; (&lt;a href=&quot;https://books.google.com.sg/books?id=lMJVDwAAQBAJ&amp;amp;pg=PT903&amp;amp;lpg=PT903&amp;amp;dq=%22an%20italian%20mathematician,%20posed%20the%20following%20problem%20in%201644,%20find%20the%20numerical%20value%20of%22&amp;amp;source=bl&amp;amp;ots=G1dR1C8GsH&amp;amp;sig=N5fJJKi5UugfksiFO1G0bYFRzkE&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=%22an%20italian%20mathematician%2C%20posed%20the%20following%20problem%20in%201644%2C%20find%20the%20numerical%20value%20of%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;2018&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Exploring the Riemann Zeta Function: 190 years from Riemann's Birth&lt;/em&gt; (2017, p. &lt;a href=&quot;https://books.google.com/books?id=iz01DwAAQBAJ&amp;amp;pg=PA224&amp;amp;lpg=PA224&amp;amp;dq=mengoli%201644&quot; rel=&quot;noreferrer&quot;&gt;224&lt;/a&gt;), edited by Montgomery, Nikeghbali, and Rassias.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Phi, Pi, e and i&lt;/em&gt; (2017, p. &lt;a href=&quot;https://books.google.com/books?id=Ot7EDgAAQBAJ&amp;amp;pg=PA47&amp;amp;lpg=PA47&amp;amp;dq=basel%20problem%201644&quot; rel=&quot;noreferrer&quot;&gt;47&lt;/a&gt;), by David Perkins.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads&lt;/em&gt; (2010, p. &lt;a href=&quot;https://books.google.com/books?id=kI7fv-yNGoIC&amp;amp;pg=PA91&amp;amp;lpg=PA91&amp;amp;dq=basel%20problem%201644&quot; rel=&quot;noreferrer&quot;&gt;91&lt;/a&gt;) by Michael Rassias.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;How Euler Did It&lt;/em&gt; (2007, p. &lt;a href=&quot;https://books.google.com/books?id=sohHs7ExOsYC&amp;amp;pg=PA205&amp;amp;lpg=PA205&amp;amp;dq=basel%20problem%201644&quot; rel=&quot;noreferrer&quot;&gt;205&lt;/a&gt;) by Edward Sandifer&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Prime Obsession&lt;/em&gt; (2003, p. &lt;a href=&quot;https://books.google.com/books?id=kk93AgAAQBAJ&amp;amp;pg=PA370&amp;amp;lpg=PA370&amp;amp;dq=%22prime%20obsession%22%20mengoli&quot; rel=&quot;noreferrer&quot;&gt;370&lt;/a&gt;), by John Derbyshire.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Others&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Wikipedia entry on &lt;a href=&quot;https://en.wikipedia.org/wiki/Pietro_Mengoli&quot; rel=&quot;noreferrer&quot;&gt;Pietro Mengoli&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;ProofWiki entry on &lt;a href=&quot;https://proofwiki.org/wiki/Mathematician:Pietro_Mengoli&quot; rel=&quot;noreferrer&quot;&gt;Pietro Mengoli&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Mathworld entry on &quot;&lt;a href=&quot;http://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html&quot; rel=&quot;noreferrer&quot;&gt;Riemann Zeta Function zeta(2)&lt;/a&gt;&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Mathworld cites Derbyshire (2003, p. 370, already given above).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Brendan Sullivan's &lt;a href=&quot;http://math.cmu.edu/~bwsulliv/basel-problem.pdf&quot; rel=&quot;noreferrer&quot;&gt;2013 Beamer presentation&lt;/a&gt; on the Basel Problem (when googling &quot;Basel problem&quot;, this is the second result after Wikipedia).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7862" LastEditorUserId="7862" LastEditDate="2018-10-16T03:56:52.507" LastActivityDate="2018-10-16T03:56:52.507" Title="In which 1644 publication did Pietro Mengoli first pose the Basel Problem?" Tags="|mathematics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7810" PostTypeId="2" ParentId="7809" CreationDate="2018-10-15T07:57:59.077" Score="6" Body="&lt;p&gt;André Weil, in his article &lt;em&gt;Prehistory of the Zeta-Function&lt;/em&gt;, only mentions Mengoli's &lt;em&gt;Novae quadraturae arithmeticae&lt;/em&gt; as the source of the problem. However, the year 1644 also appears there. In that year, Torricelli published his &lt;em&gt;De dimensione parabolae&lt;/em&gt;, in which he states that, given a sequence &lt;span class=&quot;math-container&quot;&gt;$(a_n)_{n\in\mathbb{Z}^+}$&lt;/span&gt;, we always have&lt;span class=&quot;math-container&quot;&gt;$$a_0=\sum_{i=0}^{n-1}(a_i-a_{i+i})+a_n$$&lt;/span&gt;and&lt;span class=&quot;math-container&quot;&gt;$$\sum_{i=0}^\infty(a_i-a_{i+i})=a_0-\lim_{n\to\infty}a_n.\tag1$$&lt;/span&gt;Mengoli, in his book, used &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; to prove that the series&lt;span class=&quot;math-container&quot;&gt;$$\sum_{i=0}^\infty\frac2{(i+1)(i+2)}$$&lt;/span&gt;converges. But this is the series of the reciprocals of the triangular numbers. Then, Mengoli expressed &amp;ldquo;his wonderment at the fact that the reciprocals of the &amp;lsquo;triangular numbers&amp;rsquo; can be summed, but not those of the &amp;lsquo;square numbers&amp;rsquo;.&amp;rdquo; Perhaps that it's from this that the year 1644 came to be mentioned in this context.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2018-10-15T14:57:37.253" LastActivityDate="2018-10-15T14:57:37.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7811" PostTypeId="1" CreationDate="2018-10-15T10:04:46.783" Score="1" ViewCount="91" Body="&lt;p&gt;While today text to speech technology gets so advanced that synthetic voices come quite close to human ones, during my childhood I got quite inspired by &amp;quot;robotic&amp;quot; mechanical voices in Hollywood sci fi movies of that time.&lt;/p&gt;&#xA;&lt;p&gt;Now, my question is - was there an original TTS (text to speech) technology which gave input to those movies, or not?&lt;/p&gt;&#xA;" OwnerUserId="5924" LastEditorUserId="5924" LastEditDate="2021-02-01T18:11:39.080" LastActivityDate="2021-02-01T18:11:39.080" Title="Are there historical origins of &quot;robotic&quot; voices seen in earlier sci fi movies?" Tags="|technology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7812" PostTypeId="2" ParentId="7811" CreationDate="2018-10-15T12:57:32.043" Score="2" Body="&lt;p&gt;There's a stop, or set of stops, on many cathedral organs called &quot;vox humana.&quot;  Granted these can only produce human-like vowel sounds, but that's the first &quot;synth&quot; I know of. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, prior to modern &quot;text to speech&quot; digital software, there was a period in the 1970s and early 80s when a limited, hardwired, voice synth was available. See for example &quot;Firepower&quot; pinball machine.  THese chips, IIRC, stored a collection of waveforms (not digitized voice clips) and used some weak algos to try to &quot;sound out&quot; a limited number of character strings (words). &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-10-15T12:57:32.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7813" PostTypeId="1" AcceptedAnswerId="7814" CreationDate="2018-10-15T15:08:52.050" Score="2" ViewCount="94" Body="&lt;p&gt;In which paper, did Jordan introduce/prove the Jordan canonical form?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2018-10-15T15:56:44.123" Title="Jordan's Paper on the Jordan Canonical Form" Tags="|mathematics|reference-request|linear-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7814" PostTypeId="2" ParentId="7813" CreationDate="2018-10-15T15:33:05.227" Score="3" Body="&lt;p&gt;See Hawkins (&lt;a href=&quot;https://ams.org/mathscinet-getitem?mr=528299&quot; rel=&quot;nofollow noreferrer&quot;&gt;1977&lt;/a&gt;, &lt;a href=&quot;https://www.jstor.org/stable/41133484&quot; rel=&quot;nofollow noreferrer&quot;&gt;§5&lt;/a&gt;): 2 &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; 2 case in (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?01.0040.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1868&lt;/a&gt;, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k16412q/f119&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 111–113&lt;/a&gt;), &lt;em&gt;n&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; &lt;em&gt;n&lt;/em&gt; in &lt;em&gt;Traité des substitutions et des équations algébriques&lt;/em&gt; (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?03.0042.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1870&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/traitdessubsti00jorduoft/page/114&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 114–126&lt;/a&gt;). Also &lt;em&gt;Cours d’analyse&lt;/em&gt;, vol. 3 (&lt;a href=&quot;http://emis.de/cgi-bin/JFM-item?19.0252.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1887&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/coursdanalyspoly03jorduoft/page/173&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 173–175&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-15T15:56:44.123" LastActivityDate="2018-10-15T15:56:44.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7816" PostTypeId="2" ParentId="7807" CreationDate="2018-10-15T20:58:57.483" Score="8" Body="&lt;p&gt;This is odd, since Rota revived it there is a fairly large community of umbral enthusiasts &lt;a href=&quot;http://www.combinatorics.org/files/Surveys/ds3/ds3v2-2000.pdf&quot; rel=&quot;noreferrer&quot;&gt;Di Bucchianico's 1995 survey&lt;/a&gt; has a bibliography 506 entries long. American Mathematical Monthly has an accessible &lt;a href=&quot;https://www.jstor.org/stable/2304144&quot; rel=&quot;noreferrer&quot;&gt;History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life by E.T. Bell&lt;/a&gt;. Normally I would take anything written by Bell with a grain of salt, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2400/55&quot;&gt;What resources are available for lives of recent mathematicians besides E.T. Bell's Men of Mathematics?&lt;/a&gt; But he is a good expositor and in this case he is citing original sources.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It sometimes happens in the history of mathematics that the credit for a particular method is commonly ascribed to another than its originator. In the interests of historical justice, such oversights should be corrected, where the&#xA;  facts are known. A conspicuous instance is the extremely useful symbolic method expounded and widely applied by Edouard Lucas [3, 5],t and usually attributed to him. The kernel of this symbolic method is described below. This representative notation [1], to use the designation of its inventor, or symbolic method, as Lucas [3, 5] called it, or umbral calculus [8, vol. 31], was fully developed in 1861 by John Blissard [1], in a mathematical journal with a wide circulation among mathematicians, fifteen years before Lucas published his first papers [3, 4] on the subject.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[1] John Blissard, Theory of Generic Equations, Quarterly Journal of Pure and Applied Mathematics, vol. 4, 1861, pp. 279-305; vol. 5, 1862, pp. 58-75, 185-208. In nine further papers, ibid., vols. 5-9, Blissard gave applications of his method. He frequently contributed problems to the Educational Times.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[3] Edouard Lucas, Theorie nouvelle des nombres de Bernoulli et d'Euler, Comptes rendus de l'Academie des Sciences (Paris), vol. 83, 1876, pp. 539-541; Annali di Matematica pura ed applicata, Serie 2, vol. 8, 1877, pp. 56-79.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[4] E. Lucas et E. Catalan, Sur le calcul symbolique des nombres de Bernoulli, Nouvelle Correspondance Math6matique, vol. 2, 1876, pp. 328-336.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[5] Edouard Lucas, Theorie des Nombres, 1891, Chap. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-10-16T17:28:12.290" LastActivityDate="2018-10-16T17:28:12.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7817" PostTypeId="2" ParentId="7811" CreationDate="2018-10-15T22:22:21.150" Score="3" Body="&lt;p&gt;The technology is that of the &quot;Voder&quot; and &quot;Vocoder&quot; exhibited at the 1939 World's Fair in New York City.  See the Wikipedia articles on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Vocoder&quot; rel=&quot;nofollow noreferrer&quot;&gt;vocoder&lt;/a&gt;  and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Voder&quot; rel=&quot;nofollow noreferrer&quot;&gt;voder&lt;/a&gt; for some explanations and references to Sci Fi movies using it, and to &lt;a href=&quot;https://www.youtube.com/watch?v=5hyI_dM5cGo#t=3m00s&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 1939 film clip&lt;/a&gt;.  There is a recent pop technology history book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/1612190928&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How to Wreck a Nice Beach&lt;/em&gt;&lt;/a&gt; on the subject.  The first non phone company and non military users of the technology seem to have been musicians.  It lives on in all cell phones and VOIP phones nowadays.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2018-10-15T22:22:21.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7819" PostTypeId="1" AcceptedAnswerId="7820" CreationDate="2018-10-16T08:22:56.630" Score="2" ViewCount="1190" Body="&lt;p&gt;In continuation of this question, &lt;a href=&quot;https://hsm.stackexchange.com/questions/3360/who-introduced-the-creation-and-annihilation-operators-for-the-harmonic-oscillat&quot;&gt;Who introduced the creation and annihilation operators for the harmonic oscillator?&lt;/a&gt;, I got curious who came up with the quantum harmonic oscillator originally. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Harmonic_oscillator#Simple_harmonic_oscillator&quot; rel=&quot;nofollow noreferrer&quot;&gt;simple classical harmonic potential&lt;/a&gt; arises from the definition of &lt;span class=&quot;math-container&quot;&gt;$F = kx$&lt;/span&gt; as a central force, thus &lt;span class=&quot;math-container&quot;&gt;$F = \nabla V$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ V = \frac{1}{2} kx^2 $&lt;/span&gt;. But what was the purpose of plugging such a potential into the Schrödinger equation? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nowadays the harmonic oscillator is the simplest way to model chemical bonds but I assume that it had a different purpose in the dawn of quantum mechanics. &lt;/p&gt;&#xA;" OwnerUserId="7956" LastEditorUserId="55" LastEditDate="2020-02-18T00:28:49.133" LastActivityDate="2020-02-18T00:28:49.133" Title="Who solved the quantum harmonic oscillator?" Tags="|quantum-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7820" PostTypeId="2" ParentId="7819" CreationDate="2018-10-16T13:29:01.687" Score="2" Body="&lt;p&gt;In matrix mechanics: &lt;em&gt;Born, Max; Jordan, Pascual&lt;/em&gt;, &lt;a href=&quot;http://dx.doi.org/10.1007/BF01328531&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Zur Quantenmechanik&lt;/strong&gt;&lt;/a&gt;, Z. f. Physik 34, 858-888 (1925). &lt;a href=&quot;https://zbmath.org/?q=an:51.0728.08&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL51.0728.08&lt;/a&gt;. (27 September 1925)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In wave mechanics: &lt;em&gt;Schrödinger, Erwin&lt;/em&gt;, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k153811/f534&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Quantisierung als Eigenwertproblem. II&lt;/strong&gt;&lt;/a&gt;, Annalen d. Physik (4) 79, 489-527 (1926). &lt;a href=&quot;https://zbmath.org/?q=an:52.0966.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL52.0966.01&lt;/a&gt;. (23 February 1926)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(“Old quantum theory” also discussed oscillators and even &lt;a href=&quot;https://en.wikipedia.org/wiki/Zero-point_energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;zero-point energy&lt;/a&gt;, but I guess that’s outside the scope of this question.) &lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-16T13:29:01.687" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7821" PostTypeId="2" ParentId="7819" CreationDate="2018-10-16T13:50:01.107" Score="1" Body="&lt;p&gt;Harmonic oscillator in one dimension was solved in the very first paper of&#xA;Heisenberg where he proposed quantum mechanics. But it gives the same result&#xA;as the one obtained &quot;old quantum mechanics&quot; of Bohr. To obtain something new, Heisenberg also considers &quot;anharmonic oscillators&quot; in the same paper. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Uber quantumtheoretische Umdeutung kinematischer und mechanischer Beziehungen,&#xA;Z. Phys, 33 (1925).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an English translation in the book: B. L. van der Waerden, Sources of quantum mechanics,&#xA;North Holland Amsterdam 1967.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A translation of Bohr's paper is also included there.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-10-16T18:43:06.777" LastActivityDate="2018-10-16T18:43:06.777" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7822" PostTypeId="1" AcceptedAnswerId="7824" CreationDate="2018-10-17T03:40:53.013" Score="6" ViewCount="473" Body="&lt;p&gt;I have been revisiting math from my highschool through undergrad. I picked Courant’s excellent &lt;em&gt;What is Mathematics?&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The flow is well so far. However, in one of the chapters he introduces inversion - mostly w.r.t. the circle. This seemed a bit abrupt especially since I was never introduced to this concept. I’m aware of mapping planes to spheres for topological purposes etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I’m try to understand is why was this investigated and who started it? Some reference (Googling) point to Apollinius of Perga and then jump straight to someone in 1911. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The formula for circle inversion:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$OP \cdot OP’ = r^2$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; is the center and &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is a point inside the circle and &lt;span class=&quot;math-container&quot;&gt;$P’$&lt;/span&gt; the &lt;em&gt;inverse&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This formula seems rather contrived and doesn’t make intuitive sense to me. Why is this formula used and who investigated inversion and for what purpose? Any interesting references?&lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-10-17T06:48:39.173" Title="When and why was inversive geometry created/studied?" Tags="|mathematics|reference-request|discoveries|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7823" PostTypeId="1" AcceptedAnswerId="7825" CreationDate="2018-10-17T06:34:34.290" Score="2" ViewCount="4405" Body="&lt;p&gt;Does anyone remember who proposed molecular diagrams for simple molecules as taught today in most general chemistry texts? I cannot access Hund's original article, however, Mulliken's early articles on molecular orbitals do not show any diagram like this. For example, The Assignment of Quantum Numbers for Electrons in Molecules (MULLIKEN, Phys. Reviews 1928) does not have a single figure.&#xA;Thanks&#xA;&lt;a href=&quot;https://i.stack.imgur.com/uUAsa.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uUAsa.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-10-19T02:05:42.927" Title="Origins of molecular orbital diagrams?" Tags="|mathematics|physics|terminology|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7824" PostTypeId="2" ParentId="7822" CreationDate="2018-10-17T06:48:39.173" Score="7" Body="&lt;p&gt;The wiki page and links seem good except maybe on history. For that see e.g. Coolidge (&lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=2769&quot; rel=&quot;noreferrer&quot;&gt;1940&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=tmHDAgAAQBAJ&amp;amp;ots=-ET3DDLFDe&amp;amp;pg=PA279&quot; rel=&quot;noreferrer&quot;&gt;pp. 278-282&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;§2. Circular transformations.&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The point transformations of the plane which carry points on a line&#xA;  or circle into points on a line or circle, are surely the simplest and most&#xA;  interesting plane transformations, after the linear ones. The simplest&#xA;  of these is circular inversion (...)&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;OP.OP'=r^2,\qquad&#xA;x'=\frac{r^2x}{x^2+y^2},\qquad&#xA;y'=\frac{r^2y}{x^2+y^2}.&#xA;$$&lt;/span&gt;&#xA;  We saw (&lt;a href=&quot;https://books.google.com/books?id=tmHDAgAAQBAJ&amp;amp;ots=-ET3DDLFDe&amp;amp;pg=PA64&quot; rel=&quot;noreferrer&quot;&gt;pp. 64-65&lt;/a&gt;) that this transformation was first&#xA;  mentioned by &lt;strong&gt;Pappus&lt;/strong&gt;, who knew that it carried a line or circle into a&#xA;  line or circle. It has been discovered subsequently by several writers,&#xA;  the first being perhaps &lt;strong&gt;Steiner&lt;/strong&gt; (in) an unpublished manuscript (...)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It is truly astonishing what can be accomplished by the use of this&#xA;  transformation. If two circles intersect they can be inverted into two&#xA;  straight lines. If they touch they can be inverted into two parallel&#xA;  lines. If they have no common point we can find two intersecting circles&#xA;  cutting both of them orthogonally. Hence we may invert the first two&#xA;  into two circles that cut two intersecting lines orthogonally, i.e. into&#xA;  two concentric circles. The properties of systems of circles tangent to&#xA;  two given circles are most easily found by means of these &#xA;  transformations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Thus the relation with &lt;a href=&quot;https://en.wikipedia.org/wiki/Problem_of_Apollonius&quot; rel=&quot;noreferrer&quot;&gt;Apollonius’s problem&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Circular inversion is but the simplest of all the transformations that&#xA;  carry a line or circle into a line or circle. The first writer to undertake&#xA;  a systematic study of transformations so defined, and by purely &#xA;  geometric means, was &lt;strong&gt;Möbius&lt;/strong&gt; in his &lt;em&gt;Theorie der Kreisverwandschaft&lt;/em&gt; which&#xA;  appeared in 1855 (:)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Möbius' Theorem 4]&lt;/strong&gt;  &lt;em&gt;Every one-to-one transformation of the Euclidean&#xA;  plane that carries points of a line or circle into points of a line or circle&#xA;  can be factored into reflections and inversions and is conformal.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(...) That enables us to write the general analytic form for the transformation&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;z'=\frac{\alpha z+\beta}{\gamma z+\delta}\qquad\text{(2a)},\qquad&#xA;z'=\frac{\alpha \bar z+\beta}{\gamma \bar z+\delta}\qquad\text{(2b)},\qquad&#xA;(\alpha\delta-\beta\gamma\ne0).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(i.e. one of two types, where &lt;span class=&quot;math-container&quot;&gt;$z=x+iy$&lt;/span&gt;; etc.) Earlier Coolidge had written in (&lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=389515&quot; rel=&quot;noreferrer&quot;&gt;1916&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/circleandsphere00coolrich/page/n25&quot; rel=&quot;noreferrer&quot;&gt;p. 22&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This transformation is usually credited to Plücker. See his &lt;em&gt;Analytisch-geometrische Aphorismen&lt;/em&gt;, Crelle, vol. xi, 1836. It was rediscovered a decade later by Sir William Thompson, &lt;em&gt;Principe des images électriques&lt;/em&gt;, Liouville, vol. x, 1845. The most recent view, however, seems to be that the method was found some time previous by Steiner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-17T06:48:39.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7825" PostTypeId="2" ParentId="7823" CreationDate="2018-10-17T07:32:14.917" Score="3" Body="&lt;p&gt;You mean Hund (&lt;a href=&quot;https://hsm.stackexchange.com/a/7787&quot;&gt;1927&lt;/a&gt;)? I don’t know if he was first, but p. 756:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/hs6qu.png&quot; alt=&quot;Hund&quot;&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Similar depictions for Zeeman splitting occur in Sommerfeld (&lt;a href=&quot;https://emis.de/cgi-bin/JFM-item?48.0026.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=g3EdBgAAQBAJ&amp;amp;pg=PA553&amp;amp;dq=Aufspaltung&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 553&lt;/a&gt;) and probably earlier.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; Hall (&lt;a href=&quot;https://en.wikipedia.org/wiki/Molecular_orbital_theory#cite_note-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;1991&lt;/a&gt;) points to &lt;strong&gt;Lennard-Jones&lt;/strong&gt; (&lt;a href=&quot;https://emis.de/cgi-bin/JFM-item?55.1180.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1929&lt;/a&gt;) who “introduced the Linear Combination of Atomic Orbitals approximation for the molecular orbitals” and has (&lt;a href=&quot;//doi.org/10.1039/TF9292500668&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 679&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The possible molecular states are shown diagrammatically in Fig. 4, each being represented by a cell which can contain two electrons of opposite spins. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\hspace{5.5em}$&lt;/span&gt;&lt;img src=&quot;https://i.stack.imgur.com/x7WL9.png&quot; width=&quot;434&quot; height=&quot;514&quot;&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later in (&lt;a href=&quot;https://emis.de/cgi-bin/JFM-item?57.1213.02&quot; rel=&quot;nofollow noreferrer&quot;&gt;1931&lt;/a&gt;, &lt;a href=&quot;//doi.org/10.1112/jlms/s1-6.4.290&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 314–318&lt;/a&gt;) Lennard-Jones also has:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; be two atoms, which, when infinitely distant, are in &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; states of multiplicity &lt;span class=&quot;math-container&quot;&gt;$2s_a+1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2s_b+1$&lt;/span&gt;. (...) The energy levels of the system &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; depend on the distance between the atoms, and the lower levels usually have a minimum, which is the equilibrium distance of the molecule &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt;. An example of the relation between the energy levels of separated atoms and the same atoms when interacting with each other is shown in Fig. 8.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/9b8sK.png&quot; width=&quot;735&quot; height=&quot;370&quot;&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-19T02:05:42.927" LastActivityDate="2018-10-19T02:05:42.927" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7826" PostTypeId="1" AcceptedAnswerId="7829" CreationDate="2018-10-17T17:55:06.023" Score="4" ViewCount="1452" Body="&lt;p&gt;I've read that John Wilkins was Newton's room-mate and they lived together for 20 years. Is there anything about Newton written by Wilkins?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, there is nothing easily found on google.&lt;/p&gt;&#xA;" OwnerDisplayName="user7532" LastEditorUserId="55" LastEditDate="2018-10-17T21:35:33.097" LastActivityDate="2020-11-27T11:34:39.810" Title="Is there anything written by Newton's roommate Wilkins about him?" Tags="|biographical-details|reference-request|newton|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7828" PostTypeId="1" AcceptedAnswerId="7855" CreationDate="2018-10-17T20:16:46.113" Score="5" ViewCount="200" Body="&lt;p&gt;A Foucault pendulum in Mainz is &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Foucault_pendulums#Germany&quot; rel=&quot;noreferrer&quot;&gt;listed on Wikipedia&lt;/a&gt;. The article says that it is in&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;School for Business and Technique, Mainz&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, I didn't find any information about this pendulum on the Internet. I know that this question is not fully in the profile of this site, but maybe someone know that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;where exactly is this Foucault pendulum and when is it visitable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="100" LastActivityDate="2018-10-26T05:29:31.480" Title="Where is the Foucault pendulum in Mainz?" Tags="|physics|reference-request|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7829" PostTypeId="2" ParentId="7826" CreationDate="2018-10-17T21:34:13.900" Score="7" Body="&lt;p&gt;Part of the problem may be that Newton's roommate's name is spelled as Wickins or Wickens, not to be confused with bishop &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Wilkins&quot; rel=&quot;noreferrer&quot;&gt;John Wilkins&lt;/a&gt;, Newton's older contemporary. Since they did live in the same room for 20 years Wickens's testimony would be precious indeed, but alas, as far as I know, the only things Wickens wrote concerning Newton were not about him but to him, letters. Some of the letters might be in the multi-volume &lt;a href=&quot;https://www.cambridge.org/ec/academic/subjects/physics/history-philosophy-and-foundations-physics/correspondence-isaac-newton-volume-2?format=PB&amp;amp;isbn=9780521085991&quot; rel=&quot;noreferrer&quot;&gt;Correspondence of Isaac Newton by Cambridge University Press&lt;/a&gt;, but not it seems in the online &lt;a href=&quot;http://www.newtonproject.ox.ac.uk/texts/correspondence/all&quot; rel=&quot;noreferrer&quot;&gt;Newton’s Correspondence&lt;/a&gt;. It would not help with the most common speculation concerning Wickens, however. According to &lt;a href=&quot;https://mnheritagesongbook.files.wordpress.com/2014/04/alicesevidenceapril2014revision.pdf&quot; rel=&quot;noreferrer&quot;&gt;Waltz's history of autism&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Others — e.g. Gribbin, p. 178&lt;/em&gt; [Science: A History 1543-2001, BCA, 2002] &lt;em&gt;have suggested a homosexual relationship with his long-time roommate John Wickens, but there is no hint of this in their surviving correspondence.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://rsnr.royalsocietypublishing.org/content/34/1/11&quot; rel=&quot;noreferrer&quot;&gt;Spargo and Pounds in Newton's 'Derangement of the Intellect'&lt;/a&gt; quote a remark about Newton by Wickens's son Nicholas, but even that is just a reply to an inquiry forwarded to him by his father rather than some kind of memoir:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He was turning Grey, I think, at Thirty, and when my Father observed y&lt;span class=&quot;math-container&quot;&gt;$^{\textrm{t}}$&lt;/span&gt; to him as y&lt;span class=&quot;math-container&quot;&gt;$^{\textrm{e}}$&lt;/span&gt; Effect of his deep attention of Mind, He would jest w&lt;span class=&quot;math-container&quot;&gt;$^{\textrm{th}}$&lt;/span&gt; y&lt;span class=&quot;math-container&quot;&gt;$^{\textrm{e}}$&lt;/span&gt; Experimy&lt;span class=&quot;math-container&quot;&gt;$^{\textrm{ts}}$&lt;/span&gt; he made so often wth QuickSilver, as if from thence he took so soon that Colour.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The source is a manuscript labeled King's College Library, Cambridge, Keynes MS I37, not something that would be online either.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-10-17T21:34:13.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7830" PostTypeId="1" CreationDate="2018-10-17T16:43:30.720" Score="0" ViewCount="109" Body="&lt;p&gt;Did Planck have an intuition behind the idea of energy quantization of atomic oscillators and radiation, or was it just a mathematical trick to derive his distribution law?&lt;/p&gt;&#xA;" OwnerUserId="7987" OwnerDisplayName="Karim mohie" LastEditorUserId="4251" LastEditDate="2021-09-07T21:08:27.737" LastActivityDate="2021-09-07T21:08:27.737" Title="Max Planck and energy quantization idea" Tags="|physics|quantum-mechanics|physicists|energy|planck|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7832" PostTypeId="1" AcceptedAnswerId="7838" CreationDate="2018-10-19T12:47:31.167" Score="1" ViewCount="104" Body="&lt;p&gt;Historically,how,and by what experiments,the concept of identical and distinguishable particles was discovered?Is it a tagline set by scientists,or nature indeed works in this way?&lt;/p&gt;&#xA;" OwnerUserId="6277" LastActivityDate="2018-10-20T13:51:04.907" Title="Distinguishability of particles in statistical mechanics" Tags="|quantum-mechanics|statistical-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7834" PostTypeId="1" AcceptedAnswerId="7837" CreationDate="2018-10-19T14:04:00.303" Score="1" ViewCount="226" Body="&lt;p&gt;&lt;a href=&quot;http://mathworld.wolfram.com/GausssCircleProblem.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Gauss Circle Problem&lt;/a&gt;: find the number of integer lattice points inside a circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: why was Gauss studying this problem? Was it just math for math's sake, or was this a part of a larger problem he wanted to solve?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm guessing there must be a connection to Gaussian integers, but I tried to find the origin on math history books and couldn't find anything.&lt;/p&gt;&#xA;" OwnerUserId="7995" LastEditorUserId="4251" LastEditDate="2021-01-25T21:12:29.720" LastActivityDate="2021-01-25T21:12:29.720" Title="History of the Gauss Circle Problem" Tags="|mathematics|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7835" PostTypeId="1" AcceptedAnswerId="7836" CreationDate="2018-10-19T22:50:05.660" Score="0" ViewCount="92" Body="&lt;p&gt;Of course, only heliocentric model can explain this. But, was this Copernicus himself or Galilei?&lt;/p&gt;&#xA;" OwnerUserId="7461" LastActivityDate="2018-10-20T13:27:27.933" Title="Who correctly first explained the reason why you can see certain constellations only in certain seasons?" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7836" PostTypeId="2" ParentId="7835" CreationDate="2018-10-20T13:22:14.407" Score="2" Body="&lt;p&gt;Your statement about heliocentric model is wrong. Everything we see in the sky with a naked eye is perfectly explained by geocentric model as well. The reason why we see different constellations in different seasons, is that the Sun does not occupy a fixed position with respect to the stars, and moves along the ecliptic making one turn per year. All the earliest writers on astronomy that we know already knew this. Without this knowledge one cannot make a reasonable calendar. As the calendar is indispensable to those who do agriculture, there is no doubt that his was discovered at the time when people started agriculture, that is in pre-historic period. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there can be different meanings of the word &quot;explain&quot;. One can ask &quot;why the Sun moves as it does&quot;? This was &quot;explained&quot; by the Law of Universal gravity (proved by Newton). But why to stop here, one can also ask &quot;why the Law of gravity&quot; is as it is? And so on. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-10-20T13:27:27.933" LastActivityDate="2018-10-20T13:27:27.933" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7837" PostTypeId="2" ParentId="7834" CreationDate="2018-10-20T13:34:27.080" Score="2" Body="&lt;p&gt;Gauss was motivated by arithmetic. He wanted to study the asymptotic behavior of the number of representations of integers as a sum of two squares (of integers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See, for example, K. Chandrasekharan, Introduction to analytic number theory, Springer, 1968.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-20T13:34:27.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7838" PostTypeId="2" ParentId="7832" CreationDate="2018-10-20T13:51:04.907" Score="1" Body="&lt;p&gt;The discovery was made in trying to explain the spectrum of black body radiation, using statistical mechanics. See Wikipedia articles on &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Bose-Einstein statistics&lt;/a&gt; and&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ultraviolet_catastrophe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ultraviolet catastrophe&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-20T13:51:04.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7839" PostTypeId="1" CreationDate="2018-10-20T16:25:56.257" Score="-4" ViewCount="160" Body="&lt;p&gt;Could I argue that the quantum mechanical relation &quot;discovered&quot; by Heisenberg&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;qp-pq= \mathrm{i}\hbar I&#xA;\tag{01}\label{01}     &#xA;\end{equation}&lt;/span&gt;&#xA;is the greatest scientific discovery of the 20th century and may be of all centuries till now?&lt;/p&gt;&#xA;" OwnerUserId="4376" LastActivityDate="2018-10-20T17:50:24.797" Title="About the quantum mechanical relation $\:qp-pq= \mathrm{i}\hbar I\:$" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="1" ClosedDate="2018-10-21T21:44:00.493" ContentLicense="CC BY-SA 4.0" />
  <row Id="7840" PostTypeId="2" ParentId="7839" CreationDate="2018-10-20T17:50:24.797" Score="2" Body="&lt;p&gt;You could argue it, but who would agree with you?  Maybe limit yourself to &quot;greatest discovery in physics&quot; or something.  After all, penicillin and plate tectonics are also 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here: &lt;a href=&quot;https://historyplex.com/scientific-achievements-top-discoveries-of-20th-century&quot; rel=&quot;nofollow noreferrer&quot;&gt;top ten discoveries of the 20th century&lt;/a&gt; or here: &lt;a href=&quot;https://list25.com/25-biggest-scientific-discoveries-in-history-of-mankind/&quot; rel=&quot;nofollow noreferrer&quot;&gt;25 biggest scientific discoveries in history&lt;/a&gt; it is not even on the list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is number 8 at &lt;a href=&quot;https://owlcation.com/stem/Top-Ten-Beautiful-Physics-Equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;top ten most beautiful equations in physics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-10-20T17:50:24.797" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7841" PostTypeId="2" ParentId="6772" CreationDate="2018-10-20T21:21:09.637" Score="0" Body="&lt;p&gt;Gauss's procedure &lt;strong&gt;does imply&lt;/strong&gt; Bolyai's result on the volume of orthoscheme tetrahedron, as i'll show here. However, Gauss's result is a little bit more limited than Bolyai, since in Gauss's tetrahedron 4 of the 12 face angles of the tetrahedron are right, while Bolyai refers to a slightly more general tetrahedron whose only 3 face angles ar right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UN0Kg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UN0Kg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the sake of consistency, we denote the angles 431, 234, and 214 as &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;, respectively. Now lets look at the link of vertex 3 of the tetrahedron: it's a spherical triangle whose two edges lengths are &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; and one angle is &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; (it is the dihedral angle of edge 31 and it's also equal to &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;). In addition the sides &lt;span class=&quot;math-container&quot;&gt;$\alpha$$, \beta$&lt;/span&gt; of this spherical triangle are orthogonal to each other. Therefore, by a combination of the spherical sine theorem and the spherical pythagoras theorem, we get:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{{sin(arccos(cos\alpha\cdot cos\beta))}}{{sin 90}} = \frac {{sin\beta}}{{sin\gamma}}$$&lt;/span&gt;, or:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$sin\gamma = \frac {{sin\beta}}{{\sqrt{{1 - (cos\alpha \cdot cos\beta)^2}}}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, Gauss's procedure for the calculation of the volume leads to the following integral:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\Delta = \frac {{tan\beta}}{{2 tan \gamma}}\int_{0}^{c}\frac {{x sinh(x) dx}}{{(cosh^2(x) - 1 + \frac {{cosh^2(x)}}{{sin^2\alpha cot^2\beta}})\sqrt{{\frac{{cosh^2x}}{{cos^2\beta}} - 1}} }}$$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the left factor of the denominator &lt;span class=&quot;math-container&quot;&gt;$cosh^2(x)(1 + \frac{{1}}{{sin^2\alpha \ cot^2\beta}})-1$&lt;/span&gt;, is exactly equal to &lt;span class=&quot;math-container&quot;&gt;$cosh^2(x)\cdot \frac{{1}}{{cos^2\gamma}}-1$&lt;/span&gt;, because subtitution of &lt;span class=&quot;math-container&quot;&gt;$cos\gamma = \sqrt {1 - \frac {{sin^2\beta}}{{1-(cos\alpha\cdot cos\beta)^2}}}$&lt;/span&gt; in this expression gives the previous one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Concluding remarks&lt;/strong&gt;: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;As can be seen from this presentation - &lt;a href=&quot;http://www.csu.ru/faculties/Documents/%D0%9F%D1%80%D0%B5%D0%B7%D0%B5%D0%BD%D1%82%D0%B0%D1%86%D0%B8%D1%8F_%D0%90%D0%B1%D1%80%D0%BE%D1%81%D0%B8%D0%BC%D0%BE%D0%B2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyperbolic Volumes and Symmetry&lt;/a&gt;, the Bolyai's volume integral (see Theorem 5, p. 12, at this presentation):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$Vol(T) = \frac {{tan\beta}}{{2 tan \gamma}}\int_{0}^{c}\frac {{x sinh(x) dx}}{{(\frac {{cosh^2(x)}}{{cos^2\gamma}} - 1)\sqrt{{\frac{{cosh^2x}}{{cos^2\beta}} - 1}} }}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;can be seen as treating a slightly more general case then the one treated by Gauss (&lt;strong&gt;note&lt;/strong&gt;: the differences in notation between the Bolyai integral in the presentation and Gauss's integral are just due the different symbols of the angles 431, 234, and 214).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, for the case treated by Gauss, his formulas are absolutely correct. He should also be given credit for the identification of the calculation of the orthoscheme tetrahedron as the basis for volume formulas of general tetrahedrons (without right angles). In one of his letters, he refered to those calculations of volumes as &quot;&lt;strong&gt;die jungle&lt;/strong&gt;&quot; - i guess he refered to the extremely complicated integrals that arise in the attempts to the decompose the general tetrahedron into orthoscemes (this problem was only solved very recently).    &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It's still necessary to understand how Gauss arrived at the formula &lt;span class=&quot;math-container&quot;&gt;$\partial \Delta = -\frac{{1}}{{2}}(24)d(341)$&lt;/span&gt; (he missed the factor &lt;span class=&quot;math-container&quot;&gt;$\frac {{1}}{{2}}$&lt;/span&gt; at the first attempt); the second formula from his note can be derived with relative ease. I think dechipering Gauss's second fragment might serve as a clue for understanding Gauss's reasoning. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-06-10T11:04:54.740" LastActivityDate="2019-06-10T11:04:54.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7842" PostTypeId="1" CreationDate="2018-10-22T09:19:03.003" Score="1" ViewCount="601" Body="&lt;p&gt;I found a &lt;a href=&quot;https://www.school-for-champions.com/science/x-ray_generation.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;web page&lt;/a&gt; that says that it was discovered by Tesla in 1890, is that true or generally acknowledged?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Isn't 1890 too early a date?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="4251" LastEditDate="2021-12-25T09:50:49.643" LastActivityDate="2021-12-25T09:50:49.643" Title="Who discovered Bremsstrahlung?" Tags="|physics|discoveries|electromagnetism|radiation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7843" PostTypeId="2" ParentId="7842" CreationDate="2018-10-22T13:42:33.553" Score="3" Body="&lt;p&gt;See listing here:&#xA;&lt;a href=&quot;http://shadetreephysics.com/bremindx.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Index of Early Bremsstrahlung Articles&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest items are: patents filed by Tesla in 1891 and 1892.  And papers (by others, not Tesla) in &lt;em&gt;Philosophical Magazine&lt;/em&gt; in 1891.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-10-22T13:42:33.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7844" PostTypeId="2" ParentId="7842" CreationDate="2018-10-23T05:45:30.627" Score="0" Body="&lt;p&gt;The term Bremsstrahlung is said to be coined by Arnold Sommerfeld in 1909. Cannot access this chapter but the abstract is available. It must have the original reference.&#xA;&lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-540-70626-7_23&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/chapter/10.1007/978-3-540-70626-7_23&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-10-23T05:45:30.627" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7845" PostTypeId="1" CreationDate="2018-10-23T11:16:24.123" Score="1" ViewCount="286" Body="&lt;p&gt;I was reading &amp;quot;&lt;a href=&quot;https://archive.org/details/dialoguesconcern00galiuoft/page/n9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Two New Sciences&lt;/a&gt;&amp;quot; by Galileo Galilei and therein was a chapter named &lt;a href=&quot;https://archive.org/details/dialoguesconcern00galiuoft/page/n29&quot; rel=&quot;nofollow noreferrer&quot;&gt;third day&lt;/a&gt; which deals with motion. When I started reading that chapter I eventually came across the &lt;a href=&quot;https://archive.org/details/dialoguesconcern00galiuoft/page/154&quot; rel=&quot;nofollow noreferrer&quot;&gt;theorem 1&lt;/a&gt; wherein I could not comprehend two things.&lt;/p&gt;&#xA;&lt;p&gt;ONE: Why do we need to prove such a thing since the statement of the theorem is the direct consequence of this relation speed = distance/time?&lt;/p&gt;&#xA;&lt;p&gt;TWO: Why are the speed and time represented as geometric lines, what does that mean?&lt;/p&gt;&#xA;&lt;p&gt;and as I proceeded to the theorem 2 3, and 4 the same two things were bewildering me a lot and I was no more able to digest the further text in the book.&lt;/p&gt;&#xA;&lt;p&gt;NOTE: I started reading the book from the chapter third day and haven't read the chapters prior to it.&lt;/p&gt;&#xA;" OwnerUserId="7928" LastEditorUserId="12330" LastEditDate="2022-06-25T06:22:14.773" LastActivityDate="2022-06-25T06:22:14.773" Title="Questions regarding &quot;Two New Sciences&quot; by Galileo Galilei" Tags="|galileo|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7846" PostTypeId="2" ParentId="7845" CreationDate="2018-10-23T13:45:26.707" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Why do we need to prove such a thing since the statement of the theorem is the direct consequence of this relation speed = distance/time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is exactly what he tries to prove here. The difference between you and Galileo is that you were taught some concepts in your childhood, which Galileo was not. For example,&#xA;the concept of a (real) number. The ancients did not have this concept. They only had rational numbers. When they discovered that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is not a rational number, they had to abandon the whole concept of number in measuring and substitute for it geometric concepts (length). They developed a highly sophisticated &amp;quot;theory of proportions&amp;quot; to deal with these lengths. This was the mathematics available at the time of Galileo. The modern concept of real number was slowly developing (with some major developments at the time of Galileo), but reached sufficient clarity and was widely accepted only in 19th century. And nowadays we are indoctrinated (I cannot say &amp;quot;taught&amp;quot;, the subject is too subtle) with this concept in elementary school.&lt;/p&gt;&#xA;&lt;p&gt;Even at the time of Newton, a century after Galileo, they phrased the theorems in terms of proportions, rather than real numbers, functions etc.&lt;/p&gt;&#xA;&lt;p&gt;In general, to read and appreciate the old authors like Galileo, one has to study some of the previous development to have an idea of the background these people had.&lt;/p&gt;&#xA;&lt;p&gt;Remark. There is a funny paper,&#xA;N. J. Wildberger, Real fish, real numbers, real jobs, The Mathematical Intelligencer,&#xA;March 1999, Volume 21, Issue 2, pp 4–7,&#xA;which tries to explain to modern students how deep, abstract and counter-intuitive the concept of real number really is. When I was a student, a rigorous theory of real numbers was taught in the mathematics departments of universities. Nowadays it is not, as too complicated for first-year students taking too much time etc. So students are just forced to believe that there is some rigorous theory, and trained to use the results.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="12724" LastEditDate="2022-06-12T07:23:02.840" LastActivityDate="2022-06-12T07:23:02.840" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7847" PostTypeId="1" CreationDate="2018-10-23T23:14:40.393" Score="0" ViewCount="147" Body="&lt;p&gt;First of All, I doubt there has ever been any new idea that did not involve intuition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, most textbooks suggest that the quantization idea was just a mathematical trick, that Planck introduced to be able to drive his distribution law. &#xA;Is this the only right way to think about it? If not, did Planck really have intuition behind quantization idea? &lt;/p&gt;&#xA;" OwnerUserId="7987" LastActivityDate="2019-06-28T01:01:55.277" Title="Planck's quantization idea" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7848" PostTypeId="2" ParentId="2918" CreationDate="2018-10-24T00:47:21.910" Score="-1" Body="&lt;p&gt;Old Kingdom Egyptian architecture featured engaged columns--i.e., columns embedded into walls--that were fluted around their exposed circumferences. Analysis of the columns showed that if they were freestanding columns, they would have 22 flutings of equal width around their circumferences. The diameter of the column from tip of fluting to tip of fluting would be very nearly equal to 7 units of fluting width. It would appear that ancient Egyptian architects were aware of a practical circumference-to-diameter constant ratio of 22/7.            &lt;/p&gt;&#xA;" OwnerUserId="7941" LastActivityDate="2018-10-24T00:47:21.910" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7849" PostTypeId="1" CreationDate="2018-10-24T01:59:09.593" Score="0" ViewCount="101" Body="&lt;p&gt;Are there any problems in Babylonian mathematics that deal with finding the centroid of some plane figure? &lt;/p&gt;&#xA;" OwnerUserId="7941" LastActivityDate="2018-10-24T13:29:41.543" Title="Centroid in Babylonian Mathematics" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7851" PostTypeId="2" ParentId="7849" CreationDate="2018-10-24T13:29:41.543" Score="1" Body="&lt;p&gt;The earliest listed identification on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Centroid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; page is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first explicit statement of this proposition is due to Heron of&#xA;  Alexandria (perhaps the first century CE) and occurs in his Mechanics.&#xA;  It may be added, in passing, that the proposition did not become&#xA;  common in the textbooks on plane geometry until the nineteenth&#xA;  century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;http://users.metu.edu.tr/serge/courses/303-2015/Book-essentials-A4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;different&lt;/a&gt; history page says&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pappus of Alexandria (Πάππος) 290–350 the last great mathematicians of Antiquity&#xA;His work Collection (Synagoge) in 8 volumes mainly survived: it including&#xA;geometry, recreational mathematics, doubling the cube, polygons and&#xA;polyhedra. Famous for:&lt;br&gt;&#xA;1) Pappus's Theorem in projective geometry,&lt;br&gt;&#xA;2) Pappus Hexagon Theorem&lt;br&gt;&#xA;3) Pappus chain of circles (centers are on an&#xA;ellipse)&lt;br&gt;&#xA;&lt;strong&gt;4) Pappus centroid (two) theorems&lt;/strong&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My quick skim of the interwebz did not find any direct mention of Babylonian discovery of centroids.  That doesn't mean they didn't do so, just that it will be obscure. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-10-24T13:29:41.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7853" PostTypeId="1" AcceptedAnswerId="7896" CreationDate="2018-10-25T12:18:36.287" Score="3" ViewCount="309" Body="&lt;p&gt;In his book on quantum mechanics in the chapter on perturbation theory Dirac says in a footnote: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A system with only one stationary state belonging to each energy-level is often called &lt;em&gt;non-degenerate&lt;/em&gt; and one with two or more stationary states belonging to an energy-level is called &lt;em&gt;degenerate&lt;/em&gt;, although these words are &lt;strong&gt;not very appropriate from the modern point of view&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have two questions about this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) Why did Dirac deem the terms &lt;em&gt;(non-)degenerate&lt;/em&gt; inappropriate?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Why do we, with our even more modern point of view, still use them?&lt;/p&gt;&#xA;" OwnerUserId="6179" LastActivityDate="2018-11-06T13:46:28.130" Title="Degenerate States in Quantum Mechanics" Tags="|terminology|quantum-mechanics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7854" PostTypeId="1" CreationDate="2018-10-25T21:39:00.327" Score="0" ViewCount="313" Body="&lt;p&gt;I have always heard that the first and most prominent definition of the metre was to use the length of the seconds pendulum - pendulum with the period of exactly 2 seconds. However, in the end it was abolished because of the fact that gravity is different on every point on Earth, affecting the length of the pendulum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I am confused about one other thing: the period of any pendulum depends not only on its length, but also noticeably on the &lt;em&gt;initial angle&lt;/em&gt; (amplitude) the pendulum makes with the vertical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Why were these scientists so eager to adopt the seconds pendulum definition of the metre in light of the fact that the length would be different for different amplitudes?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="8021" LastActivityDate="2018-10-26T11:42:29.030" Title="Metre definition using a seconds pendulum" Tags="|physics|units|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7855" PostTypeId="2" ParentId="7828" CreationDate="2018-10-26T05:29:31.480" Score="2" Body="&lt;p&gt;Summary of comments, per OP’s request:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As observed by @njuffa, there is no apparent trace of a Foucault pendulum at (closest match) &lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Applied_Sciences,_Mainz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hochschule Mainz&lt;/a&gt;’s School of Business (Wirtschaft) nor School of Engineering (Technik).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Could it, then, be a mixup with — or relocation to — nearby &lt;a href=&quot;https://de.wikipedia.org/wiki/Experiminta&quot; rel=&quot;nofollow noreferrer&quot;&gt;Experiminta Museum&lt;/a&gt; in Frankfurt am &lt;strong&gt;Main&lt;/strong&gt;, which  &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Foucault_Pendel.JPG&quot; rel=&quot;nofollow noreferrer&quot;&gt;does&lt;/a&gt; have one, half an hour away and missing from the wiki list?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-10-26T05:29:31.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7856" PostTypeId="2" ParentId="7854" CreationDate="2018-10-26T11:42:29.030" Score="1" Body="&lt;p&gt;First of all, the amplitude of the pendulum swing has a very small effect until you get to rather large swings.  That's the beauty of the sine function.  :-) .&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, it took rather a while for science to develop the concept of repeatability, let alone repeating this precise pendulum measurement in several locations -- and observing a repeatable difference. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the meantime, it was only 10 years after Wren's original pendulum proposal that  Gabriel Mouton suggested setting the meter relative to the Earth's meridian arc. At that time (late 17th century) this method was undoubtedly more difficult.  &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-10-26T11:42:29.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7857" PostTypeId="2" ParentId="3323" CreationDate="2018-10-26T16:39:13.023" Score="7" Body="&lt;p&gt;The accepted answer leaves no doubt that Leibniz was the first to write &lt;span class=&quot;math-container&quot;&gt;$d^2y/(dx)^2$&lt;/span&gt; for the second derivative. But since I've &lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz%27s_notation#Leibniz_notation_for_higher_derivatives&quot; rel=&quot;noreferrer&quot;&gt;found&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Notation_for_differentiation#Leibniz&amp;#39;s_notation_for_antidifferentiation&quot; rel=&quot;noreferrer&quot;&gt;so&lt;/a&gt; &lt;a href=&quot;https://math.stackexchange.com/questions/25102/why-is-the-2nd-derivative-written-as-frac-mathrm-d2y-mathrm-dx2/25106#25106&quot;&gt;many&lt;/a&gt; &lt;a href=&quot;https://math.stackexchange.com/questions/1612437/notation-of-the-second-derivative-where-does-the-d-go&quot;&gt;misleading&lt;/a&gt; &lt;a href=&quot;https://math.stackexchange.com/questions/475016/leibniz-notation-for-high-order-derivatives?noredirect=1&amp;amp;lq=1&quot;&gt;justifications&lt;/a&gt; for this &lt;a href=&quot;https://math.stackexchange.com/questions/477507/is-d2y-dx-a-valid-mathematical-notation/477523&quot;&gt;notation&lt;/a&gt; &lt;a href=&quot;https://math.stackexchange.com/questions/2156821/multiple-derivative-notation&quot;&gt;online&lt;/a&gt;, I feel that something additional needs to be said about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most justifications in the links above are along the lines of: &quot;by formal manipulation&quot; or &quot;too obviously&quot; &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d}{dx} \frac{d}{dx} =\frac{d^2}{dx^2}.&#xA;\tag1&#xA;$$&lt;/span&gt;&#xA;But Leibniz, the Bernoullis or Euler would &lt;em&gt;not&lt;/em&gt; have approved of this without reservation. Not even if the equation was written in the form &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d \left(\frac{dy}{dx} \right)}{dx} = \frac{d dy}{(dx)^2},&#xA;\tag2&#xA;$$&lt;/span&gt;&#xA;which is closer to the standard of the time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To explain let me make a simple analogy first. No one today would claim that the following is correct&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\log \frac{\log y}{\log x}}{\log x} = \frac{\log \log y}{(\log x)^2},&#xA;\tag3&#xA;$$&lt;/span&gt;&#xA;and everyone can spot the error.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Analogously, for Leibniz, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; was an operator (he might not have called it that way, but he knew it acted on variables just like &lt;span class=&quot;math-container&quot;&gt;$\log$&lt;/span&gt;) and he knew the quotient rule for &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;. So he might have approved of the following general equation &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d \left(\frac{dy}{dx} \right)}{dx} = \frac{d^2y}{(dx)^2} - \frac{dy\cdot d^2x}{(dx)^3}.&#xA;\tag4&#xA;$$&lt;/span&gt;&#xA;The reason the second term on the right disappeared, was because an &lt;em&gt;additional assumption&lt;/em&gt; was often made: it was assumed that the differential of the differential of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is zero (i.e. &lt;span class=&quot;math-container&quot;&gt;$d^2x=0$&lt;/span&gt;), or put differently: &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; was assumed constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This can be seen in the 1693 &lt;a href=&quot;https://books.google.nl/books?id=jz7vFXu8ODUC&amp;amp;dq=%22ddx%20dy2%22&amp;amp;pg=PA459#v=onepage&amp;amp;q=%22ddx%20dy2%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; of Leibniz quoted by @ViktorBlasjo, a line above &lt;span class=&quot;math-container&quot;&gt;$ddx:\overline{dy}^2$&lt;/span&gt;, where he writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;posita &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; constante&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It can also be found in Eulers &lt;em&gt;Institutiones Calculi Differentialis&lt;/em&gt; (&lt;a href=&quot;https://books.google.ch/books?id=AQEMCAAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=euler%20foundations%20of%20differential%20calculus&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwibxaaYwaTeAhVQ-aQKHZb2Dl4Q6AEIKjAA#v=snippet&amp;amp;q=%22increases%20uniformly%2C%20so%20that%20the%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;1743&lt;/a&gt;) § 131.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Now we will proceed under the assumption that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; increases uniformly,&#xA;  so that the first differentials &lt;span class=&quot;math-container&quot;&gt;$dx, dx^I , dx^{II},\ldots$&lt;/span&gt; are equal to each&#xA;  other, so that the second and higher differentials are equal to zero.&#xA;  We can state this condition by saying that the differential of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, that&#xA;  is &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;, is assumed to be constant. Let &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; be any function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And it can be found in Lacroix's &lt;em&gt;Traité du calcul différentiel et du calcul intégral&lt;/em&gt; (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k92729z/f131.image&quot; rel=&quot;noreferrer&quot;&gt;1797&lt;/a&gt;) p.96 &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pour la simplifier nous observons que l'accroissement &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; étant regardé invariable, &lt;span class=&quot;math-container&quot;&gt;$f'(x)dx$&lt;/span&gt; se change en &lt;span class=&quot;math-container&quot;&gt;$f'(x+dx)dx$&lt;/span&gt; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Summarizing: for Leibniz, Euler and others the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d \left(\frac{dy}{dx} \right)}{dx} = \frac{d^2y}{dx^2}&#xA;\tag5&#xA;$$&lt;/span&gt; &#xA;was only true under the additional assumption that &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leaves a question for me, which hopefully someone else can answer: when and why did mathematicians forget this additional assumption and simply adopt the notation &lt;span class=&quot;math-container&quot;&gt;$\frac{d^2}{dx^2}$&lt;/span&gt; for what should actually be written as &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{d}{dx}\right)^2$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="658" LastEditDate="2018-10-26T18:55:38.490" LastActivityDate="2018-10-26T18:55:38.490" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7858" PostTypeId="1" CreationDate="2018-10-26T17:13:42.180" Score="3" ViewCount="67" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Muon-catalyzed_fusion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Muon-catalysed fusion&lt;/a&gt; is obtained at low temperatures, although as of 2018 its energy yield is less than the muon production requirements. The term &amp;quot;cold fusion&amp;quot; was first used in the 1950s, to refer to μCF. The 1989 &lt;a href=&quot;https://en.wikipedia.org/wiki/Cold_fusion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fleischmann-Pons&lt;/a&gt; claim of empirical low-temperature fusion, in heavy water electrolysis with a palladium electrode, was soon debunked. Why did this hypothetical example of fusion take the existing name, &amp;quot;cold fusion&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2023-02-28T10:44:09.993" LastActivityDate="2023-02-28T10:44:09.993" Title="Why did &quot;cold fusion&quot; come to mean Fleischmann-Pons fusion instead of μCF?" Tags="|terminology|nuclear-physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7859" PostTypeId="1" CreationDate="2018-10-27T11:09:56.280" Score="0" ViewCount="82" Body="&lt;p&gt;What did people in the 16th and 17th centuries mean by the term &lt;em&gt;speed&lt;/em&gt;? Did they have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\text{speed} = \frac{ \text{distance} }{ \text{time} }$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;back then? Or did they have some other notion of speed and this relation came into being somewhat later ?&lt;/p&gt;&#xA;" OwnerUserId="7928" LastEditorUserId="4251" LastEditDate="2020-07-11T05:14:39.550" LastActivityDate="2020-07-11T05:14:39.550" Title="How was the term speed treated in the 16th and 17th centuries?" Tags="|physics|terminology|classical-mechanics|16th-century|17th-century|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7860" PostTypeId="2" ParentId="2215" CreationDate="2018-10-27T18:09:28.200" Score="1" Body="&lt;p&gt;SI units are interrelated in such a way that one unit is derived from other units without conversion factors. SI is used in most places around the world, so our use of it allows scientists from disparate regions to use a single standard in communicating scientific data without vocabulary confusion.&#xA;SI units are important because these are common to the people of the entire world, so that people from different countries can communicate with each other conveniently regarding business and science. It makes systematic use of prefixes, making it easy to express very large or very small numbers.&lt;/p&gt;&#xA;" OwnerUserId="8026" LastActivityDate="2018-10-27T18:09:28.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7861" PostTypeId="1" AcceptedAnswerId="7930" CreationDate="2018-10-27T22:08:18.050" Score="2" ViewCount="210" Body="&lt;p&gt;In German there is phrase concerning the complexity of finding the integral of a given function in contrast to the simplicity of finding its derivative. It has various slightly different formulations but can be stated as the following&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Das Differenzieren ist ein Handwerk, das Integrieren eine Kunst.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which can be roughly translated to&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While differentiation is a craft, integration is an art.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nevertheless this would be my &lt;em&gt;own&lt;/em&gt; translation as a native German speaker. Hence it is kind of an idiom in Germany it is hard to find its origins but I would be especially interested in these.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Therefore I am asking for two things: &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; First of all is there an, I would call it &quot;official&quot;, English equivalent which you have encountered somewhere before (e.g. within your mathematical education, in media, etc.)? &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; Are you aware of the origins of this phrase either in German or in English? For example do you know a rather prominent person who has stated something similiar?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am not sure whether my question is suitable here or if it would be better placed on a different site of the Stackexchange community (for example one MSE, or at the specific pages for the English and the German language). I guess especially the historical context - which I am interested in - in connection with the close relation to mathematics makes it fit in here perfectly. Howsoever let me know if it is the case that I should post this question on another site or when the question has been asked and answered before somewhere.&lt;/p&gt;&#xA;" OwnerUserId="7971" LastEditorUserId="658" LastEditDate="2018-11-06T13:42:52.033" LastActivityDate="2018-11-06T14:16:25.270" Title="English equivalent for a German idiom concerning integration" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7862" PostTypeId="1" AcceptedAnswerId="7874" CreationDate="2018-10-27T22:18:37.657" Score="4" ViewCount="192" Body="&lt;p&gt;&lt;sub&gt;[I adapted the question to reflect what I've learned from Alexandre's answer: that Euclid never talks of lengths and areas but only of line segments and figures (like squares). The question itself remains unchanged.]&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;For Euclid, &lt;strong&gt;magnitudes&lt;/strong&gt; were things that can be compared, be equal, smaller or greater than each other, that can be added and subtracted, and that can have proportions. (Next to these, there were &lt;strong&gt;numbers&lt;/strong&gt;, another type of thing with the same properties.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are four magnitudes Euclid explicitly and mainly deals with:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strike&gt;lengths of&lt;/strike&gt; straight line segments&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;angles between straight line segments&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strike&gt;areas of&lt;/strike&gt; plane figures (polygons and circles)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strike&gt;volumes of&lt;/strike&gt; solid figures (polyhedra and spheres)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Only &lt;strike&gt;lengths of&lt;/strike&gt; straight line segments can be multiplied (giving &lt;strike&gt;areas and volumes&lt;/strike&gt; figures), as well as numbers (giving numbers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He also deals with &lt;strike&gt;lengths of&lt;/strike&gt; circle segments, at least with the &lt;strike&gt;length of&lt;/strike&gt; the whole circle (circumference).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But does he anywhere talk explicitly about &lt;strike&gt;the length of&lt;/strike&gt; arbitrary circle segments and compares, adds, or substracts them?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Furthermore: Was he aware that an angle is &quot;proportional&quot; to &lt;strike&gt;the length of&lt;/strike&gt; a corresponding circle segment and did make use of this (which would not have meant for him, that the two were the same)? &lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2018-10-29T12:04:08.413" LastActivityDate="2018-10-29T12:04:08.413" Title="Did Euclid consider circle segments as another magnitude?" Tags="|number-theory|euclidean-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7863" PostTypeId="1" AcceptedAnswerId="7868" CreationDate="2018-10-28T00:27:09.370" Score="4" ViewCount="516" Body="&lt;p&gt;&lt;em&gt;Preliminary note: With &quot;Euclid&quot; I don't mean a person but the mathematicians of the Euclidean period of which Euclid (if he had been one person) was a representative.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I imagine that Euclid could have thought about &lt;a href=&quot;https://math.stackexchange.com/questions/2973915/why-did-euclid-treat-magnitudes-and-numbers-twice&quot;&gt;relating lengths and numbers in a bijective way&lt;/a&gt; even though they were considered completely different things. For rational lengths, it would have worked because for each rational length there is a number (which for Euclid was necessarily rational) and vice versa.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But unfortunately there do exist constructible lengths which provably are not rational (e.g. the length of the diagonal of the unit square) and for which Euclid hadn't got a (rational) number to assign to. So this may have been the main reason that he abondonded the plan (if he had it) to relate lengths and numbers in a bijective way: there were not enough numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But there may have been another (admittedly speculative) reason: For the assignment to work, one has to pick an arbitrary line segment and assign the number 1 (the &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/elements/bookVII/defVII1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;unit&lt;/a&gt;) to its length. Did possibly Euclid dislike the arbitrariness of assigning the &quot;primordial&quot; unit (from the realm of platonic entities, from which all numbers are built) to a &quot;random&quot; line segment (from the realm of &quot;earthly&quot; entities)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But note, that he doesn't define (in &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/elements/bookVII/bookVII.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Definition VII.1&lt;/a&gt;) &lt;strong&gt;the&lt;/strong&gt; unit, but &lt;strong&gt;a&lt;/strong&gt; unit. So this might &lt;strong&gt;not&lt;/strong&gt; have been the reason. But the question arises why he didn't define &lt;strong&gt;the&lt;/strong&gt; unit, which seems much more intuitive. (How would he have distinguished between different units?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;Alternatively, he could have started with two &quot;primordial&quot; points &amp;ndash; named 0 and 1 &amp;ndash; from which &lt;a href=&quot;https://math.stackexchange.com/questions/2911971/covering-the-euclidean-plane-with-constructible-lines-and-circles&quot;&gt;all other points and lengths can be built with straightedge and compass&lt;/a&gt;, and assign the number 1 to the length of the distinguished &quot;primordial&quot; line segment &lt;span class=&quot;math-container&quot;&gt;$\overline{01}$&lt;/span&gt; &amp;ndash; but that's even more speculative.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit that I might completely misunderstand Euclid's way of thinking. Any hint in which respect I do so would be welcome.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To my defense: Euclid's Definition VII.1 is really rather obscure:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lUZTw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lUZTw.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;sup&gt;&lt;a href=&quot;http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf#page=194&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;A unit is that by virtue of which each of the things that exist is called one.&lt;/em&gt;&lt;br&gt;&lt;sup&gt;&lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/elements/bookVII/defVII1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="4251" LastEditDate="2020-07-20T12:25:58.360" LastActivityDate="2020-07-20T12:25:58.360" Title="Why didn't Euclid try to assign numbers to lengths?" Tags="|mathematics|number-theory|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7865" PostTypeId="1" CreationDate="2018-10-28T10:08:07.783" Score="6" ViewCount="718" Body="&lt;p&gt;I'm interested in the mathematical problems proposed for the grand-prix&lt;br&gt;&#xA;of the French Academy Of Sciences, from its beginnigs in 1666 to the present.&#xA;Are there any books or articles with the precise problem statements &amp;amp; the winners?&lt;/p&gt;&#xA;" OwnerUserId="5909" LastActivityDate="2024-02-20T07:41:25.017" Title="Grand Prix in Mathematics of the French Academy of Sciences" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7866" PostTypeId="2" ParentId="7865" CreationDate="2018-10-28T12:24:40.640" Score="7" Body="&lt;p&gt;Ernest Maindron, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k97893365&quot; rel=&quot;nofollow noreferrer&quot;&gt;Les fondations de prix à l’Académie des sciences: Les lauréats de l’Académie, 1714–1880&lt;/a&gt;. Gauthier-Villars, Paris, 1881.&lt;/p&gt;&#xA;&lt;p&gt;It is striking and not well-enough known (e.g. to Wikipedia) how the huge &lt;strong&gt;125,000 pounds&lt;/strong&gt; endowed in 1714 by &lt;a href=&quot;https://fr.wikipedia.org/wiki/Jean-Baptiste_Rouill%C3%A9_de_Meslay&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rouillé de Meslay&lt;/a&gt; (compare £20,000 for the famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Longitude_rewards&quot; rel=&quot;nofollow noreferrer&quot;&gt;Longitude Act&lt;/a&gt; drafted by &lt;a href=&quot;http://books.google.com/books?id=LQxDAAAAcAAJ&amp;amp;pg=PA677&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton&lt;/a&gt;) drove fame and focus to celestial mechanics (for navigation, mostly even years) and shipbuilding (odd years) in 18th century science and technology, until the French Revolution:&lt;/p&gt;&#xA;&lt;p&gt;&lt;code&gt;Year:                                                                    Winners:&#xA;1720  Nature of motion                                                     Crousaz&#xA;1720  Clock stabilization at sea                                             Massy&#xA;1724  Collision laws                                      Mac Laurin, J. Bernoulli&#xA;1725  Hourglass and water clock stabilization at sea                  D. Bernoulli&#xA;1726  Collision laws                                                       Mazière&#xA;1727  Ship masting                                               Bouguer, Le Camus&#xA;1728  Causes of gravity                                                 Bulffinger&#xA;1729  Height measurement at sea                                            Bouguer&#xA;1730  Motion of planetary aphelia                                     J. Bernoulli&#xA;1731  Compass use at sea                                                   Bouguer&#xA;1732  Tilting of planetary orbits                                                ∅&#xA;1733  Ship path tracing                                                     Poleni&#xA;1734  Tilting of planetary orbits                       D. Bernoulli, J. Bernoulli&#xA;1736  Propagation of light                                            J. Bernoulli&#xA;1737  Anchor design          Créqui, Poleni, Trésaguet, D. Bernoulli, J. Bernoulli&#xA;1738  Nature and propagation of fire            Créqui, Lozeran de Fiesc, L. Euler&#xA;1740  Tides                          Cavalleri, Mac Laurin, L. Euler, D. Bernoulli&#xA;1741  Capstan building      de Pontis, Fenel, Delorme, Poleni, Ludot, J. Bernoulli&#xA;1743  Compass building                                      L. Euler, D. Bernoulli&#xA;1746  Compass magnetism               Dutour, L. Euler, D. Bernoulli, J. Bernoulli&#xA;1747  Finding time at sea                                             D. Bernoulli&#xA;1748  Inequalities of Saturn and Jupiter                                  L. Euler&#xA;1750  Inequalities of Saturn and Jupiter                                         ∅&#xA;1751  Estimation of currents at sea                                   D. Bernoulli&#xA;1752  Inequalities of Saturn and Jupiter                        Bošković, L. Euler&#xA;1753  Ship propulsion           Mathon de la Cour, Pereyre, L. Euler, D. Bernoulli&#xA;1754  Planetary perturbations of Earth                                           ∅&#xA;1755  Ship pitch and roll                                                 Chauchot&#xA;1756  Planetary perturbations of Earth                                    L. Euler&#xA;1757  Ship pitch and roll                                             D. Bernoulli&#xA;1758  Celestial body atmospheres                                             Frisi&#xA;1759  Ship pitch and roll                                      Groignard, L. Euler&#xA;1760  Mean motion of planets                                       Frisi, C. Euler&#xA;1761  Ship stowage and ballasting                              Bossut, J. A. Euler&#xA;1762  Lunar acceleration                                                    Bossut&#xA;1764  Lunar libration                                                     Lagrange&#xA;1765  Ship stowage and ballasting  Bourdé de Villehuet, Groignard, Gautier, Bossut&#xA;1766  Satellites of Jupiter                                               Lagrange&#xA;1768  Secular equation of the Moon                                               ∅&#xA;1769  Timekeeping at sea                                                    Le Roy&#xA;1770  Secular equation of the Moon                           J. A. Euler, L. Euler&#xA;1772  Secular equation of the Moon                              Lagrange, L. Euler&#xA;1773  Timekeeping at sea                                        Arsandeaux, Le Roy&#xA;1774  Secular equation of the Moon                                        Lagrange&#xA;1776  Planetary perturbations of comets                                          ∅&#xA;1777  Compass building                                 Magny, Van Swinden, Coulomb&#xA;1778  Planetary perturbations of comets                                       Fuss&#xA;1780  Planetary perturbations of comets                                   Lagrange&#xA;1781  Rope stiffness and friction in winches Delanges, Ximenes, L. Carnot, Coulomb&#xA;1782  Comets of 1532 and 1661                                              Méchain&#xA;1787  Maritime insurance                                     Bicquilley, Delacroix&#xA;1791  Resistance of fluids                                          Gerlach, Romme&#xA;1791  Herschel’s planet                                                   Delambre&#xA;1792  Satellites of Jupiter                                               Delambre&#xA;&lt;/code&gt;&lt;/p&gt;&#xA;&lt;p&gt;More details in:&lt;br&gt;Frédéric Marguet, &lt;a href=&quot;https://web.archive.org/web/20130328041503/http://hydro.marseille.free.fr:80/histoire_nav_marguet/histoire_nav_marguet.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Histoire générale de la navigation du XV&lt;sup&gt;e&lt;/sup&gt; au XX&lt;sup&gt;e&lt;/sup&gt; siècle&lt;/a&gt;. Société d’Éditions Géographiques, Maritimes et Coloniales, Paris, 1931.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="10608" LastEditDate="2024-02-20T07:41:25.017" LastActivityDate="2024-02-20T07:41:25.017" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7867" PostTypeId="1" AcceptedAnswerId="7872" CreationDate="2018-10-28T13:29:21.543" Score="6" ViewCount="287" Body="&lt;p&gt;In September 1962 a nice young man ordered the United States &lt;a href=&quot;https://www.youtube.com/watch?v=ouRbkBAOGEw&quot; rel=&quot;noreferrer&quot;&gt;to commit to landing on the moon in eight years&lt;/a&gt;. Even though none of the technologies, methods, or materials for doing so existed at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On what basis did Kennedy think that this was possible? Did he have any knowledge of the challenge involved, or was he naive to the issues? How off were his estimates of budget for the program at the time of declaration of the goal? Who were his consultants on the matter, and what were their positions?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question is not intended to spark discussion of the budget overruns or opposition that the program would encounter during the 1960's. Rather, this question only addresses the information known to Kennedy and other people involved with the decision to commit to the moon landing, in the timeframe leading up to the &lt;a href=&quot;https://en.wikipedia.org/wiki/We_choose_to_go_to_the_Moon&quot; rel=&quot;noreferrer&quot;&gt;speech at Rice university&lt;/a&gt;. The Wikipedia article for that speech mentions with whom the president had spoken, without providing information on the information (supporting or opposing the landing) that each provided.&lt;/p&gt;&#xA;" OwnerUserId="261" LastActivityDate="2018-10-28T20:11:13.183" Title="On what basis did Kennedy think that humans could land on the moon so quickly?" Tags="|engineering|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7868" PostTypeId="2" ParentId="7863" CreationDate="2018-10-28T15:05:51.673" Score="4" Body="&lt;p&gt;You state the main reason correctly. At the time of Euclid, the only known numbers were rational numbers, and it was discovered that you cannot measure segments constructed in geometry with rational numbers. Therefore, numbers were abandoned in geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead Euclid (or his predecessors) developed a highly sophisticated theory of proportions, which can be shown to be equivalent to our theory of real numbers.&#xA;So they could talk about length and area of a circle, for example. It took two thusand years after Euclid to develop a satisfactory theory of real numbers. So Euclid just could not &quot;assign numbers to lengths&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recommend a very nice book which discusses these things:&#xA;R. Hartshorne, Companion to Euclid, AMS, 1970.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-10-28T15:12:04.503" LastActivityDate="2018-10-28T15:12:04.503" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7869" PostTypeId="2" ParentId="7862" CreationDate="2018-10-28T15:20:43.890" Score="1" Body="&lt;p&gt;Contrary to what you say, Euclid never assigns any numbers to geometric figures.&#xA;He never speaks of &quot;length&quot;, or &quot;angle measure&quot; (he does not use degrees).&#xA;It is more difficult with areas and volumes, which Euclid really compares but without assigning any numbers to them.&#xA;From Euclid's point of view, two polygons have the same area if one can be dissected into parts and then these parts be rearranged to form the other polygon. This does now work with a circle or 3-d polytops, but the area&#xA;of the circle and the volume of a polytop is never really defined in Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All these things are very well explained in the book of R. Hartshorne, Companion to Euclid, which on my opinion is the best exposition of Euclid from the modern point of view.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-10-28T17:25:25.263" LastActivityDate="2018-10-28T17:25:25.263" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7870" PostTypeId="1" CreationDate="2018-10-28T18:14:37.653" Score="6" ViewCount="317" Body="&lt;p&gt;So I am writing a research paper on the properties of the Dirichlet function (the function with 1 if x is rational and 0 if x is irrational), and I wanted to include some historical background on how Dirichlet came up with this function. However, I have found next to nothing about how he came up with this function. All I have found are what the function is and how you prove it, and other applications. But I want to include something on the historical developments of the Dirichlet function for my research paper.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do anybody know of the historical background to the Dirichlet function? And if so, what resource are you getting this information from?&lt;/p&gt;&#xA;" OwnerUserId="8029" LastEditorUserId="4251" LastEditDate="2020-10-02T06:18:21.637" LastActivityDate="2020-10-02T06:18:21.637" Title="Does anybody know the history of how Peter Gustav Lejeune Dirichlet came up with the “nowhere continuous” Dirichlet function?" Tags="|mathematics|mathematicians|real-analysis|19th-century|dirichlet|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7871" PostTypeId="2" ParentId="7870" CreationDate="2018-10-28T19:14:35.917" Score="6" Body="&lt;p&gt;Page 169 of his &lt;a href=&quot;https://hsm.stackexchange.com/questions/6826/dirichlet-integrals-history/6827#6827&quot;&gt;1829 paper&lt;/a&gt;. It arises as the simplest example of a function for which his proof of Fourier’s theorem fails (because it’s &lt;a href=&quot;https://www.google.com/search?q=%22Dirichlet+function%22+%22not+Riemann+integrable%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;not integrable&lt;/a&gt;): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It would remain for us to consider the case where the suppositions we have made upon the number of breaks of continuity and upon that of the maxima and minima values cease to hold. (...) One would have an example of a function which does not satisfy (them), if we supposed &lt;span class=&quot;math-container&quot;&gt;$\phi(x)$&lt;/span&gt; equal to a fixed constant &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; when the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; assumes a rational value, and equal to another constant &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; when this variable is irrational. The function thus defined has finite and determinate values for any value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and yet one cannot substitute it in the series, as the different integrals which occur in this series lose any significance in this case.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-29T13:15:15.353" LastActivityDate="2018-10-29T13:15:15.353" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7872" PostTypeId="2" ParentId="7867" CreationDate="2018-10-28T19:39:16.787" Score="10" Body="&lt;p&gt;The decision to go to the Moon was made much sooner than September 1962. In his &lt;a href=&quot;https://www.presidency.ucsb.edu/documents/special-message-the-congress-urgent-national-needs&quot; rel=&quot;noreferrer&quot;&gt;Special Message to the Congress on Urgent National Needs&lt;/a&gt; made on 25 May 1961, President Kennedy announced that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the earth. No single space project in this period will be more impressive to mankind, or more important for the long-range exploration of space; and none will be so difficult or expensive to accomplish.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This decision was not made in a vacuum. It was made in light of Yuri Gagarin's orbital flight around the Earth on 12 April 1961 and in light of analysis by the National Aeronautics and Space Council, headed by then Vice President Lyndon Johnson. President Kennedy had seen how much Gagarin's flight electrified the world, much more so than did Sputnik 1 four years earlier. On the day after Gagarin's flight, Kennedy said (emphasis mine)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We are, I hope, going to go in other areas where we can be first and which will bring perhaps more long-range benefits to mankind. &lt;strong&gt;But here we are behind.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kennedy's election claim of Soviet dominance in nuclear missiles was false. The Bay of Pigs invasion that he inherited from the previous administration failed less than a week after Gagarin's flight. Kennedy wanted something, anything, first done in space by the US that would electrify the world even more than did Gagarin's flight. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To this end, Kennedy directed Vice President Johnson as head of the National Aeronautics and Space Council to investigate various options via which the US could best the Soviet Union. The options to be investigated and NASA's evaluation included&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Building a manned&lt;sup&gt;1&lt;/sup&gt; laboratory in space. NASA deemed this option as unwinnable.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Landing an unmanned probe on the Moon. NASA deemed this as not particularly exceptional and likely not winnable.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Sending a man around the Moon and back. NASA deemed this as not particularly exceptional and possibly not winnable.&lt;/li&gt;&#xA;&lt;li&gt;Landing a man on the Moon and returning him to Earth. NASA deemed this as the one endeavor in which they had a good chance of beating the Soviet Union and that the world would view as exceptional.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The reply by NASA came on 8 May 1961, three days after Alan Shepard became the first American to travel into space. It was a bit more than two weeks later that Kennedy made his address to Congress.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; I wrote &quot;manned&quot; intentionally. The same applies to the concept of sending a man to the Moon. 1961 was a man's world.&lt;/p&gt;&#xA;" OwnerUserId="1843" LastEditorUserId="1843" LastEditDate="2018-10-28T20:11:13.183" LastActivityDate="2018-10-28T20:11:13.183" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7873" PostTypeId="1" AcceptedAnswerId="7888" CreationDate="2018-10-28T20:28:41.473" Score="0" ViewCount="117" Body="&lt;p&gt;From Wikipedia on &lt;a href=&quot;https://en.wikipedia.org/wiki/Cardinal_number#Formal_definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;cardinal numbers&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The oldest definition of the &lt;strong&gt;cardinality&lt;/strong&gt; of a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; (implicit in&#xA;  Cantor and explicit in Frege and Principia Mathematica) is as the&#xA;  class &lt;span class=&quot;math-container&quot;&gt;$[X]$&lt;/span&gt; of all sets that are equinumerous with &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question is simple:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Who was the first to state explicitly that the &lt;strong&gt;length&lt;/strong&gt; of a line&#xA;  segment &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; is the class &lt;span class=&quot;math-container&quot;&gt;$[l]$&lt;/span&gt; of all line&#xA;  segments that are equal to &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; (in the sense of Euclid)? And who can be assumed to have known or considered this implicitly? Euclid himself?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Probably not the first but a very important author did state it like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vflNx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vflNx.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Hartshorne, &lt;a href=&quot;http://euclid.colorado.edu/~wtaylor/agi.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometry: Euclid and beyond&lt;/a&gt; (1997), p. 3)&lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2018-11-03T11:39:58.567" LastActivityDate="2018-11-03T11:39:58.567" Title="Lengths as equivalence classes" Tags="|euclidean-geometry|set-theory|" AnswerCount="1" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="7874" PostTypeId="2" ParentId="7862" CreationDate="2018-10-28T23:18:22.127" Score="2" Body="&lt;p&gt;Magnitudes could be compared and have proportions, added or subtracted, but &lt;em&gt;only&lt;/em&gt; to like magnitudes. For instances, adding segments meant roughly concatenating them (same with areas), &lt;em&gt;not&lt;/em&gt; relating them to some other type of thing, like number, and then adding those. Lengths could not be compared or related to areas, for example. As for the lengths of circle segments, to Euclid &quot;circumference&quot; applies not just to the whole circle but to circle arcs as well, see e.g. &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/elements/bookVI/bookVI.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book VI, Proposition 33&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Whether circumferences could be related to line segments, the problem of rectification, is of a kind with the problem of quadrature, relating curved areas to rectiliniar ones, and can only be done by using what is now called &quot;method of exhaustion&quot;. Euclid does not take this up in the Elements, that circumferences are as their diameters is enough for his purposes.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-10-28T23:18:22.127" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7875" PostTypeId="2" ParentId="5499" CreationDate="2018-10-29T00:33:20.817" Score="3" Body="&lt;p&gt;As Mauro states, the term &lt;em&gt;&lt;a href=&quot;https://en.wiktionary.org/wiki/logistic&quot; rel=&quot;nofollow noreferrer&quot;&gt;logistic&lt;/a&gt;&lt;/em&gt; is due to the Belgian mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Fran%C3%A7ois_Verhulst&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre François Verhulst&lt;/a&gt;, who invented the &lt;a href=&quot;https://en.wikipedia.org/wiki/Logistic_growth&quot; rel=&quot;nofollow noreferrer&quot;&gt;logistic growth&lt;/a&gt; model, and named it &lt;em&gt;logistic&lt;/em&gt; (French: &lt;em&gt;&lt;a href=&quot;https://en.wiktionary.org/wiki/logistique&quot; rel=&quot;nofollow noreferrer&quot;&gt;logistique&lt;/a&gt;&lt;/em&gt;) in his 1845 &quot;Recherches mathématiques sur la loi d'accroissement de la population&quot;, &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN129323640_0018?tify=%7B%22pages%22:%5B21%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 8&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nous donnerons le nom de &lt;em&gt;logistique&lt;/em&gt; à la courbe&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;We will give the name &lt;em&gt;logistic&lt;/em&gt; to the curve&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He does not explain why he uses this term, but it is presumably by analogy with &lt;em&gt;arithmetic&lt;/em&gt;, &lt;em&gt;geometric&lt;/em&gt;, and in contrast to &lt;em&gt;logarithmic&lt;/em&gt;, as in &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN129323640_0018?tify=%7B%22pages%22:%5B54%5D,%22panX%22:0.487,%22panY%22:0.616,%22view%22:%22info%22,%22zoom%22:0.592%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;figure&lt;/a&gt; below (from the original paper).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The French term &lt;em&gt;logistique&lt;/em&gt; is from Ancient Greek λογιστικός (logistikós, “practiced in arithmetic; rational”), from λογίζομαι (logízomai, “I reason, I calculate”), from λόγος (lógos, “reason, computation”), whence English &lt;em&gt;logos&lt;/em&gt;, &lt;em&gt;logic&lt;/em&gt;, &lt;em&gt;logarithm&lt;/em&gt;, etc. In Ancient Greek mathematics, &lt;em&gt;logistikós&lt;/em&gt; was a division of mathematics: practical computation and accounting, in contrast to ἀριθμητική (arithmētikḗ), the theoretical or philosophical study of numbers. Confusingly, today we call practical computation &lt;em&gt;arithmetic&lt;/em&gt;, and don't use &lt;em&gt;logistic&lt;/em&gt; to refer to computation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Verhulst first discusses the &lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmetic_growth&quot; rel=&quot;nofollow noreferrer&quot;&gt;arithmetic growth&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_growth&quot; rel=&quot;nofollow noreferrer&quot;&gt;geometric growth&lt;/a&gt; models, referring to arithmetic &lt;em&gt;progression&lt;/em&gt; and geometric &lt;em&gt;progression&lt;/em&gt;, and calling the geometric growth curve a &lt;em&gt;logarithmic&lt;/em&gt; curve (confusingly, the modern term is instead &lt;em&gt;exponential&lt;/em&gt; curve, which is the inverse), then follows with his new model of &quot;logistic&quot; growth, which is presumably named by analogy, after a traditional division of mathematics, and in contrast to the logarithmic curve. The term &lt;em&gt;logarithm&lt;/em&gt; is itself derived as &lt;em&gt;log-arithm&lt;/em&gt;, from Ancient Greek λόγος (&lt;em&gt;lógos&lt;/em&gt;) and ἀριθμός (&lt;em&gt;arithmós&lt;/em&gt;), the sources respectively of &lt;em&gt;logistic&lt;/em&gt; and &lt;em&gt;arithmetic&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is no connection with &lt;em&gt;logis&lt;/em&gt; (lodging), though that is the source of the term &lt;em&gt;&lt;a href=&quot;https://en.wiktionary.org/wiki/logistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;logistics&lt;/a&gt;&lt;/em&gt; (1830).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cvJh3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cvJh3.png&quot; alt=&quot;Graph of logistic curve, contrasted with logarithmic curve, Verhulst 1845&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cross-posted at &lt;em&gt;Cross Validated:&lt;/em&gt; &lt;a href=&quot;https://stats.stackexchange.com/a/374825/83275&quot;&gt;Why is the logistic distribution called “logistic”?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Cross-posted at &lt;em&gt;Mathematics:&lt;/em&gt; &lt;a href=&quot;https://math.stackexchange.com/a/2975512/256893&quot;&gt;Why is logistic equation called “logistic”?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="8002" LastEditorUserId="8002" LastEditDate="2018-11-22T03:40:09.170" LastActivityDate="2018-11-22T03:40:09.170" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7876" PostTypeId="1" AcceptedAnswerId="7878" CreationDate="2018-10-29T03:35:11.127" Score="4" ViewCount="362" Body="&lt;p&gt;I'm a little confused when I read that mathematical logic is actually a very recent field (1800's - 1900's), regarding the foundations of mathematics and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By this I refer to writing proofs in classical propositional logic systems (Hilbert-style, natural deduction, sequent calculus, etc), extending up to first-order logic, using that to define theories like ZFC which (to my knowledge) is largely seen as a modern-day foundation for mathematics. Likewise for Peano axioms, Peano arithmetic, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I get confused on this because haven't we been doing mathematics for thousands of years? How were we doing proofs before? How were we doing mathematics with no &quot;foundation&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were we just blindly using arithmetic and real numbers and so on without really defining what they were or how they worked? Did Euler and Gauss do all their advanced number theory stuff on these informal foundations? What about Newton and Leibniz inventing calculus? All of this years before we start talking about logic and model theory? How did people convince each other that this stuff actually worked especially once we start getting into concepts like infinity?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't really understand the timeline of it all or why logic was such a late subject. Did we choose to formalize it so late for some specific reason? What were we trying to solve or achieve? How was it being done before? Why did it take so long before we started asking questions about mathematical foundations?&lt;/p&gt;&#xA;" OwnerUserId="8035" LastActivityDate="2018-10-30T01:00:09.643" Title="Really confused about the history of logic / proofs / etc" Tags="|mathematical-logic|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7877" PostTypeId="2" ParentId="7876" CreationDate="2018-10-29T04:19:27.977" Score="0" Body="&lt;p&gt;I'm not sure it is a choice, more likely a historical accident.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It took from Thale to Euclid to establish what we now call Euclidean geometry as the foundation of logic (and indeed Elements was a textbook of logic in the middle ages), and it pretty much survived intact up till late 18th early 19th century (e.g. Newton actually used Euclidean geometry not calculus to come up with his inverse square law, despite his later lies/half-truths in his quarrel with Leibniz) because it has work well enough.  It wasn't until non-Euclidean geometry come into being that this was challenged -- it took 2000 years to finally have someone (Bolyai and Lobachevsky) brave enough to stand up publicly to challenge the parallel postulate, and sparked the crisis that ultimately change the foundation of modern mathematics away from Euclidean geometry to logic (and later set theory etc.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not to say logic wasn't there in Euclidean geometry.  For example, Euclid's Elements has a number of &lt;em&gt;reductio ad absurdum&lt;/em&gt; proofs.  But ultimately its foundation is geometry, e.g., a number is represented by line segments in Elements.&lt;/p&gt;&#xA;" OwnerUserId="8036" LastActivityDate="2018-10-29T04:19:27.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7878" PostTypeId="2" ParentId="7876" CreationDate="2018-10-29T12:32:24.870" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_logic#History&quot; rel=&quot;noreferrer&quot;&gt;Mathematical logic&lt;/a&gt; is a quite modern discipline : it emerged in the mid-19th century with Boole, Peirce and Frege.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Logic instead, is quite ancient : we can date it at least from &lt;a href=&quot;https://plato.stanford.edu/entries/aristotle-logic/&quot; rel=&quot;noreferrer&quot;&gt;Aristotle (384–322 BCE)&lt;/a&gt; and &lt;a href=&quot;https://plato.stanford.edu/entries/logic-ancient/#Sto&quot; rel=&quot;noreferrer&quot;&gt;the Stoics&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In ancient Greek mathematics we have poofs and axioms : see &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_Elements&quot; rel=&quot;noreferrer&quot;&gt;Euclid's Elements&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Equilibrium_of_Planes&quot; rel=&quot;noreferrer&quot;&gt;Archimedes' treatises&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, the &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Axiomatic_method&quot; rel=&quot;noreferrer&quot;&gt;axiomatic method&lt;/a&gt; is at the core of mathematics since the beginning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a recent study on the Greek origins of &lt;em&gt;proofs&lt;/em&gt;, see Reviel Netz, &lt;a href=&quot;https://books.google.it/books?id=VwggGX0ORLkC&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;The Shaping of Deduction in Greek Mathematics (2003)&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the ancient world (Greece compared to e.g. China) the works of &lt;a href=&quot;https://en.wikipedia.org/wiki/G._E._R._Lloyd&quot; rel=&quot;noreferrer&quot;&gt;G.E.R. Lloyd&lt;/a&gt; are releveant; see e.g. &lt;a href=&quot;https://books.google.it/books?id=Qck-DwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;The Ambivalences of Rationality (2017)&lt;/a&gt; and &lt;a href=&quot;https://books.google.it/books?id=OFzCAwAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;The Ideals of Inquiry (2014)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&quot;Foundational&quot; aspects are present since the origins of science, mathematics and philosophy; see &lt;a href=&quot;https://plato.stanford.edu/entries/aristotle-logic/#DemDemSci&quot; rel=&quot;noreferrer&quot;&gt;Aristotle's &lt;em&gt;Posterior Analytics&lt;/em&gt;&lt;/a&gt; as well as  Descartes' &lt;a href=&quot;https://plato.stanford.edu/entries/descartes-mathematics/&quot; rel=&quot;noreferrer&quot;&gt;Mathematics&lt;/a&gt; and &lt;a href=&quot;https://plato.stanford.edu/entries/descartes-physics/&quot; rel=&quot;noreferrer&quot;&gt;Physics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the concepts of &lt;em&gt;method&lt;/em&gt;, &lt;em&gt;rationality&lt;/em&gt;, &lt;em&gt;certainty&lt;/em&gt; and so on evolved over time, as well as &quot;tools&quot; and &quot;languages&quot; : thus, it makes little sense to ask &quot;how it is possible to do math prior to &lt;span class=&quot;math-container&quot;&gt;$\mathsf {ZFC}$&lt;/span&gt;&quot;. &lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-10-29T19:51:24.443" LastActivityDate="2018-10-29T19:51:24.443" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7879" PostTypeId="1" CreationDate="2018-10-29T12:44:00.823" Score="2" ViewCount="408" Body="&lt;p&gt;I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article &amp;quot;&lt;a href=&quot;https://www.maths.ed.ac.uk/~v1ranick/papers/epple8.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;orbits of asteroids, a braid, and the first link invariant&lt;/a&gt;&amp;quot;. Among the findings of this article is a new historical thesis of the role which the linking integral served in Gauss's ideas on topology, and an interpretation of it as a computational tool.&lt;/p&gt;&#xA;&lt;p&gt;This interpretation of Gauss linking integral, as a computation rather then a definition, seems to be the missing block in the reconstruction of Gauss's thoughts on topological issues, as the many indirect reports (by Mobius, Listing, etc) document. I always had a strong intuitive feeling that Gauss's introduction of the linking number cannot be just a definition, since one doesn't need an  integral in order to define the linking number (one can do it simply by projecting the link into a plane and then counting negative and positive crossings).&lt;/p&gt;&#xA;&lt;p&gt;Since viewing braids as mathematical objects seem to inspire Gauss's &amp;quot;meditations&amp;quot; on topology, i'd like to understand more about the issues discussed in Epple's article.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some basic questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Epple begins his exposition of Gauss's fragment with the observation that &amp;quot;Gauss thought of the braid as being devided into six segments, extending from one crossing to the next&amp;quot;. Even this basic fact is not clear to me, since the crossings devide the four strands to more than six segments.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What are the &amp;quot;half-twists&amp;quot; mentioned in the following paragraphs?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;More difficult questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Why did Gauss used coordinates that are based on complex integers in order to encode a braid? Epple says &amp;quot;That Gauss used complex integers to encode the composition of the braid might be motivated by his well known fascination with these numbers; &lt;strong&gt;it might, however, also have a more serious reason&lt;/strong&gt; (see below)&amp;quot;.&#xA;Unfortunately i coudn't infer from the rest of the text the explanation for why complex numbers are important here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Epple suggests to connect the drawing of the braid found in Gauss's notebook with his later definition of the linking integral; according to his interpretation, &lt;strong&gt;the linking integral represents a computation rather then a definition&lt;/strong&gt; - in the case of two strand braid, this analytic approach (the integral) gives rise to a linking number which equals to an integer multiple of &lt;span class=&quot;math-container&quot;&gt;$\frac {{1}}{{2}}$&lt;/span&gt;. If Gauss did apply a kind of linking integral for braids (how one can define linking number for a braid with more then two strands?), a  question arises - are there any known definitions of a &amp;quot;linking number&amp;quot; for general braids?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It's tempting to connect the 1847 letter from Mobius to Gauss, which mentions a treatise on all possible configurations of a &amp;quot;thread&amp;quot; (which seems to mean a &amp;quot;braid&amp;quot;) as an introduction for a paper on electricity and magnetism, with this fragment, and imagine it as a detailed presentation of his thoughts on braids. However, it seems to me completely unclear how such classical physics issues are related to braids - and therefore i'd like to know if anyone is aware of connections between electromagnetism and braids and can tell something about it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As is being said in my posted answer (which is just intended to give additional information), the indirect reference in Betti's letter states that Gauss was occupied with a certain braid problem that he didn't succeed in solving except in special cases. Is there any term from modern braid theory for the topological problem described in Betti's letter?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;General question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;One thing that i noticed, which seems very interesting to me, is that several results or drawings which some authors refer to, don't appear in any of the 12 volumes of the published version of Gauss's werke. Just as one example, in several articles on the early history of knot theory, the authors mention a collection of 13 knots made by Gauss. So my question is: is there any website which gives access to those obscure writings of Gauss?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2021-01-31T00:16:17.430" LastActivityDate="2021-01-31T00:16:17.430" Title="Several questions about Gauss's mathematical conception of braids" Tags="|biographical-details|reference-request|gauss|topology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7880" PostTypeId="2" ParentId="5499" CreationDate="2018-10-29T15:57:27.527" Score="2" Body="&lt;p&gt;Side remark: Huygens had previously used that name for the graph of logarithm, in &lt;em&gt;Discours de la cause de la pesanteur&lt;/em&gt; (&lt;a href=&quot;//www.dbnl.org/tekst/huyg003oeuv21_01/huyg003oeuv21_01_0084.php#963T&quot; rel=&quot;nofollow noreferrer&quot;&gt;1690&lt;/a&gt;, p. 169):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... there was a curved line, which I had examined &lt;a href=&quot;//dbnl.nl/tekst/huyg003oeuv14_01/huyg003oeuv14_01_0049.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;much before&lt;/a&gt;, which was of great use in this research. One can call it &lt;strong&gt;&lt;em&gt;Logarithmic&lt;/em&gt;&lt;/strong&gt; or &lt;strong&gt;&lt;em&gt;Logistic&lt;/em&gt;&lt;/strong&gt;, for I don’t see that anyone has yet named it, although others also considered it before.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Latin version with elaboration by G. Grandi in &lt;em&gt;Opera reliqua&lt;/em&gt; (&lt;a href=&quot;//doi.org/10.3931/e-rara-4462&quot; rel=&quot;nofollow noreferrer&quot;&gt;1728&lt;/a&gt;, pp. 130, 135, 149–288). Same in the &lt;em&gt;Cyclopaedia&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/Cyclopediachambers-Volume2/page/n114&quot; rel=&quot;nofollow noreferrer&quot;&gt;1728&lt;/a&gt;: Logistic Line, Logistic Spiral, Logistical Arithmetic) and in Roger Boscovich, &lt;em&gt;De cycloide et logistica&lt;/em&gt; (&lt;a href=&quot;//doi.org/10.3931/e-rara-3504&quot; rel=&quot;nofollow noreferrer&quot;&gt;1745&lt;/a&gt;, p. 80) and also &lt;em&gt;Theoria philosophiae naturalis&lt;/em&gt; (&lt;a href=&quot;//books.google.com/books?id=RH5jw_7UOT4C&amp;amp;pg=PA261&amp;amp;dq=logistic+curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;1763&lt;/a&gt;, pp. 260–261). Lalande, already quoted by Mauro Allegranza, confirms in his Vol. 3 (&lt;a href=&quot;//archive.org/details/astronomie03lala/page/750&quot; rel=&quot;nofollow noreferrer&quot;&gt;1771&lt;/a&gt;, nº 3915):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Logistics is the name one used to call algebra; later one gave it to the logarithmic curve; today it is devoted to this little kind of logarithms [“&lt;a href=&quot;https://archive.org/details/lelaroussepourto02laro/page/58&quot; rel=&quot;nofollow noreferrer&quot;&gt;logistic logarithms&lt;/a&gt;”]; perhaps this word comes from &lt;em&gt;λογὶζομαι Colligo&lt;/em&gt;, because algebra packs many things in few characters.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As to “algebra”, Cajori says (&lt;a href=&quot;https://archive.org/details/mathehistory00cajorich/page/26&quot; rel=&quot;nofollow noreferrer&quot;&gt;1896&lt;/a&gt;, p. 26): “Greek mathematicians were in the habit of discriminating between the science of numbers and the art of computation. The former they called &lt;em&gt;arithmetica&lt;/em&gt;, the latter &lt;em&gt;logistica&lt;/em&gt;.” Later Vieta used logistica &lt;em&gt;numerosa&lt;/em&gt; and &lt;em&gt;speciosa&lt;/em&gt;, and maybe not coincidentally, a Belgian biography of Stevin had recapped, shortly before Verhulst’s paper (Goethals &lt;a href=&quot;https://books.google.com/books?id=OAhiAAAAcAAJ&amp;amp;pg=PA13&amp;amp;dq=logistique&quot; rel=&quot;nofollow noreferrer&quot;&gt;1841&lt;/a&gt;, p. 13):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Logistica numerosa&lt;/em&gt;&lt;/strong&gt; was properly arithmetic; &lt;strong&gt;&lt;em&gt;logistica speciosa&lt;/em&gt;&lt;/strong&gt; was algebra, which since Vieta, contemporary of Stevin, split into two branches (...). The &lt;strong&gt;&lt;em&gt;logistic&lt;/em&gt;&lt;/strong&gt; due to Monk Barlaam, who flourished in 1350, was originally the arithmetic of sexagesimal fractions, more curious than useful. This word was long kept; and by the end of Stevin’s career, it was still used by the mathematician Beyer who published, in 1619, a treatise on &lt;strong&gt;&lt;em&gt;decimal and sexagesimal logistic&lt;/em&gt;&lt;/strong&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the word had been through quite a few different meanings when Verhulst (&lt;a href=&quot;https://books.google.com/books?id=b0JUAAAAcAAJ&amp;amp;pg=PA8&amp;amp;dq=logistique&quot; rel=&quot;nofollow noreferrer&quot;&gt;1845&lt;/a&gt;) co-opted it out of obsolescence again — with no apparent explanation. Whence understandable confusions, such as Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Roger_Joseph_Boscovich#Middle_years&quot; rel=&quot;nofollow noreferrer&quot;&gt;linking Boscovich to the wrong graph&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-31T06:37:22.970" LastActivityDate="2018-10-31T06:37:22.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7881" PostTypeId="1" AcceptedAnswerId="7890" CreationDate="2018-10-29T18:41:39.617" Score="3" ViewCount="543" Body="&lt;p&gt;I have looked at several sources, and Newton was right about the fact that the Earth is not a perfect sphere, but an ellipsoid caused the precession of equinoxes, as the Moon's gravitational attraction of the Earth causes torque, if the Earth is not a perfect sphere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I looked at the English translation of Principia&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop2&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[408]&#xA;&quot;And since the mean semi-diameter of the earth, according to Picart's mensuration, is 19615800 Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth will be higher at the equator than at the poles by 85472 feet, or 17 1⁄10 miles. And its height at the equator will be about 19658600 feet, and at the poles 19573000 feet.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also looked at the Latin edition, even though I don't know Latin: &lt;a href=&quot;http://www.gutenberg.org/files/28233/28233-h/28233-h.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.gutenberg.org/files/28233/28233-h/28233-h.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The value was slightly different, but not that much&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Ideoque cùm Terræ semidiameter mediocris, juxta nuperam Gallorum mensuram, sit pedum Parisiensium 19615800 seu milliarium 3923 (posito quod milliare sit mensura pedum 5000;) Terra altior erit ad æquatorem quàm ad polos, excessu pedum 85200 seu milliarium 17.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I searched the Internet and found out that the actual difference is neither 17.1 miles nor 17 miles but 13.3 miles. So, Newton got the radius difference quite wrong. I thought about the possibility that the miles then and now were different, but this was not the case. Internet says that the radius of the earth is 3950~3963 miles, which almostly agrees with Newton's value 3923.16 miles or 3923 miles. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, if Newton got wrong about the radius difference, how did he end up getting the right value? (50 arcseconds per a year)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Axial_precession&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Axial_precession&lt;/a&gt;&#xA;Wikipedia says&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Over a century later precession was explained in Isaac Newton's Philosophiae Naturalis Principia Mathematica (1687), to be a consequence of gravitation (Evans 1998, p. 246). Newton's original precession equations did not work, however, and were revised considerably by Jean le Rond d'Alembert and subsequent scientists.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know what actually happened?&lt;/p&gt;&#xA;" OwnerUserId="7461" LastActivityDate="2018-10-30T14:47:39.547" Title="Was Newton's successful calculation of precession of equinoxes a fluke?" Tags="|physics|astronomy|classical-mechanics|newton|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7882" PostTypeId="1" AcceptedAnswerId="7883" CreationDate="2018-10-29T19:32:37.303" Score="5" ViewCount="749" Body="&lt;p&gt;I've been intrigued by the similarities between what Eudoxus' Theory of Proportions and Dedekind cuts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I wish to question this &quot;perceived similarity&quot; and would like to where the flaws are, in my understanding.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Pythagoreans discovered irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt(2)$&lt;/span&gt; - now the notion of a &quot;number&quot; was threatened and the (Greek) world wondered how could the laws/proofs of &quot;similarity of triangles&quot; be applied correctly (poetic exaggeration)&lt;/li&gt;&#xA;&lt;li&gt;A &lt;em&gt;length&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$\sqrt(2)$&lt;/span&gt; clearly &lt;em&gt;exists&lt;/em&gt; geometrically, but not number theoretically as a diagonal of a unit square. (The latter implying that &lt;span class=&quot;math-container&quot;&gt;$\sqrt(2)$&lt;/span&gt; is only attainable as a limit)&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Eudoxus could see that similarity continues to exists &lt;em&gt;irrespective&lt;/em&gt; of the &quot;numerical magnitude&quot; of the side. So side-stepping number theory, just casting this problem geometrically it's relatively straightforward to  come up with the idea of a &lt;em&gt;proportion&lt;/em&gt; IMHO:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Represent any pair of magnitudes as two straight lines &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Take another pair of magnitudes represented by two straight lines &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$na &amp;lt;=&amp;gt; mb \space\forall (m,n) $&lt;/span&gt; and the lines &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; move similarly (i.e.,  &lt;span class=&quot;math-container&quot;&gt;$nc &amp;lt;=&amp;gt; md \space\forall (m,n)$&lt;/span&gt;) then there is definitely a &quot;relationship&quot; between the two magnitudes. If they move the other way (i.e., increasing one decreases the other or vice versa) then we have an inverse relationship&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;li&gt;Given the above, we can continue with similarity as if there wasn't any problems and voila! we're back, crisis averted, number theory stalled for 2000 years!&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Now, Eudoxus tried to avert infinity by having definitions (per Euclid) like so: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, the definition of equality of proportions seems to apply an infinite process (i.e., Dedekind cut) since &lt;span class=&quot;math-container&quot;&gt;$m, n$&lt;/span&gt; are expected to be integers/rationals to the best of my knowledge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;It doesn't seem that Eudoxus was really partitioning anything - he was looking at the problem to subvert the irrational crisis and just put forth the notion of proportionality geometrically vs. that of considering individual numbers. Is this correct?&lt;/li&gt;&#xA;&lt;li&gt;Was he (or his contemporaries) aware of the &quot;hidden&quot; infinite process in computing &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; since &quot;number theory&quot; was absent - all they were doing was extending &quot;lines&quot; and seeing if another related &lt;em&gt;pair&lt;/em&gt; moved &quot;similarly&quot;. Did they really encounter this infinite process &quot;explicitly&quot; or is this a view from our modern understanding of real numbers, infinity and limits?&lt;/li&gt;&#xA;&lt;li&gt;What may I be missing?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7035" LastEditorUserId="55" LastEditDate="2020-08-27T10:31:42.270" LastActivityDate="2020-08-29T23:44:18.257" Title="Did Eudoxus really set out to present irrationals as Dedekind cuts?" Tags="|mathematics|ancient-greece|real-analysis|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7883" PostTypeId="2" ParentId="7882" CreationDate="2018-10-29T21:34:10.093" Score="4" Body="&lt;p&gt;I'm not sure that my answer will add anything to your understanding of the problems as expressed in your question, but it may be useful to remind us of the conventional reading of Eudoxus.&lt;/p&gt;&#xA;&lt;p&gt;John Stillwell, in his text &lt;em&gt;The Real Numbers&lt;/em&gt;, summarises the situation as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The longest book of the &lt;em&gt;Elements&lt;/em&gt;, Book X, is devoted to the classification of irrational quantities arising in geometry.  And the most subtle book of the &lt;em&gt;Elements&lt;/em&gt;, book V, is devoted to Eudoxus' &amp;quot;theory of proportions,&amp;quot; which seeks to deal with irrational quantities by comparing them with the rationals.&lt;/p&gt;&#xA;&lt;p&gt;In the next chapter we will see how the idea of comparing an irrational quantity with rationals was revived by Dedekind in the nineteenth century to provide a concept of &lt;em&gt;real numbers&lt;/em&gt; making up the &lt;em&gt;number line&lt;/em&gt;, thus providing an arithmetical foundation for geometry.  The novel part of Dedekind's idea is its acceptance of infinite sets - and idea that the Greeks rejected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Stillwell then goes on to introduce Dedekind cuts, closing the chapter by saying:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A slogan to sum up this chapter might be: the basic theory of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; equals Greek mathematics + Infinity (or even Euclid + Infinity).  The ancient Greeks gave us integral and rational numbers and the principle of induction by which their properties may be proved.  They also gave us infinite processes for approaching irrational numbers, though they did not dare to complete them or &amp;quot;take them to the limit&amp;quot;.  Thus Euclid, in his Elements, Book X, proposition 2, gave nontermination of the Euclidean algorithm as a criterion for irrationality.  He gave no example at this point, but his proposition 5 of Book XIII immediately implies periodicity, and hence nontermination, of the Euclidean algorithm on the pair &lt;span class=&quot;math-container&quot;&gt;$\frac{1+\sqrt 5}{2}, 1$&lt;/span&gt;.  This leads us to the continued fraction representation [&lt;span class=&quot;math-container&quot;&gt;$1; 1, 1, $&lt;/span&gt;...] but it would not have been accepted by the Greeks, since it implied &amp;quot;completing&amp;quot; a process that does not end.  The Greeks accepted the &amp;quot;potential&amp;quot; infinity of a process, but not the &amp;quot;actual&amp;quot; infinity of its completion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So while all of the necessary ingredients were available to the Greeks, Stillwell appears to be emphatic in his view that they never took the necessary final step.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-29T21:34:10.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7884" PostTypeId="2" ParentId="7882" CreationDate="2018-10-30T00:50:03.543" Score="0" Body="&lt;p&gt;All work of Eudoxus on the theory of proportions is lost. So we cannot really say what he did. Our knowledge of the theory of proportions is based on Euclid (which is easily available online, but difficult to read. There are several excellent commentaries). Euclid himself does not credit Eudoxus (or anyone else). That the theory of&#xA;proportions really belongs to Eudoxus is known only from the later ancient historians who lived hundreds years after Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Imagine for comparison that all work of Newton is lost, and a modern historian tries to describe what he did based on secondary and tertiary sources). So speculations on&#xA;what Eudoxus REALLY did are groundless. We can only discuss what is written in Euclid on the subject.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-30T00:50:03.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7885" PostTypeId="2" ParentId="7882" CreationDate="2018-10-30T00:51:13.687" Score="5" Body="&lt;p&gt;A reliable source on early Greek mathematics is &lt;a href=&quot;https://archive.org/details/MathematicsOfPlatosAcademyFowler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fowler's Mathematics Of Plato's Academy&lt;/a&gt;, freely available. Unfortunately, there are too many inaccuracies propagated by textbooks and popular sources to address here. So let me start by brief statements.&lt;/p&gt;&#xA;&lt;p&gt;Pythagoreans did not discover irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;, they had no such notion. There is no such thing as &amp;quot;length&amp;quot; of a segment in Euclid. There was no &amp;quot;irrationality crisis&amp;quot;, and there existed a theory of incommensurables, perfected by Theatetus, before Eudoxus got on the scene. The &amp;quot;infinite process&amp;quot; it exploited is called &lt;em&gt;anthyphairesis&lt;/em&gt;. It is a geometric version of the Euclidean algorithm, and the modern arithmetic analog of it is expanding irrationals into continued fractions. Unfortunately, it becomes intractable beyond quadratic irrationals.&lt;/p&gt;&#xA;&lt;p&gt;Eudoxus's invention of the ratio theory, and the double &lt;em&gt;reductio&lt;/em&gt; technique that came with it (currently misnamed &lt;a href=&quot;https://en.wikipedia.org/wiki/Method_of_exhaustion&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;method of exhaustion&amp;quot;&lt;/a&gt;), was indeed a breakthrough that advanced geometry by a mile. It is still underappreciated due to lack of direct sources, we only have Euclid's retelling and scattered secondary references. But it had little to do with the &amp;quot;crisis&amp;quot; of Pythagoreans discovering incommensurability.&lt;/p&gt;&#xA;&lt;p&gt;So why so much inaccuracy? Because it would take too long to convey the context of Greek mathematics, with its strict separation of numbers, magnitudes and ratios, and no analog of Cartesian coordinates or even Vieta's assignment of numbers to geometric objects. So what is done instead is a &amp;quot;translation&amp;quot; into the closest things available today. This translation must collapse the Greek distinctions and inject quick moves they could never make or think of. The moves they did make then become inexplicable and drop out of the story. Fables about the &amp;quot;crisis&amp;quot; and Hippasus being thrown to the sharks are made up in their stead. Incommensurable ratios become &amp;quot;irrationals&amp;quot;, and Eudoxus becomes Dedekind's thesis advisor.&lt;/p&gt;&#xA;&lt;p&gt;There are indeed some similarities in proof moves and techniques in Eudoxus and Dedekind. But consider the differences too. Dedekind &lt;em&gt;constructs&lt;/em&gt; real numbers as cuts, this kind of hypostatic abstraction is unheard of not only in antiquity but even in the early 19th century. Frege still struggled with its analog (the Caesar's problem), as do many modern students when they encounter quotient groups. Eudoxus, on the other hand, defines equality for ratios of &lt;em&gt;pre-constructed&lt;/em&gt; magnitudes. With the standard tools (straightedge and compass) the magnitudes produced correspond to only a small fragment of algebraic numbers, let alone of real numbers.&lt;/p&gt;&#xA;&lt;p&gt;Dedekind then proceeds to define arithmetical &lt;em&gt;operations on cuts&lt;/em&gt;. Eudoxus, on the other hand, can only add/subtract &lt;em&gt;magnitudes&lt;/em&gt;. There are already problems with multiplying them because it changes the type of magnitude involved: the product of lines is an area, but the product of areas already has no geometric interpretation. And even addition and subtraction are hard to transfer to ratios because it requires the Cartesian move of choosing a unit magnitude arbitrarily, something that runs counter to Greeks' invariance intuitions (not unlike those of modern differential geometers). So Eudoxus mainly uses only &lt;em&gt;comparison of ratios&lt;/em&gt; by a process described in &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookV/defV5.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's definition 5 of Book V&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Dedekind &lt;em&gt;proves&lt;/em&gt; continuity for his cuts. Eudoxus, on the other hand, &lt;em&gt;assumes&lt;/em&gt; it (even admitting the &amp;quot;translation&amp;quot;) by assuming that something like line segments, angles and areas &lt;em&gt;are&lt;/em&gt;, in fact, magnitudes. One can see it in the Elements where Euclid wields the &amp;quot;definition&amp;quot; 5 of Book V as a postulate. And so on. This is not to diminish the Eudoxus's breakthrough, which was the root of a tall tree that eventually bloomed with Dedekind and Cantor. But it needed to grow a lot of other branches to get there.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-29T23:44:18.257" LastActivityDate="2020-08-29T23:44:18.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7886" PostTypeId="2" ParentId="7876" CreationDate="2018-10-30T01:00:09.643" Score="1" Body="&lt;p&gt;Here is a simple summary of Mauro Allegranza's excellent answer. Mathematics was done since Euclid by &quot;usual logic&quot;. What we call &quot;mathematical logic&quot; is a later invention, a part of the 19th and 20th century program to formalize mathematics (and logic). And of course, this 19-20 century program is only a continuation of the attempts to formalize logic, starting at least from Aristotle. But the technical term &quot;mathematical logic&quot; is reserved for this modern formalization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics was done at the time of Euclid (and most of it nowadays) using &quot;ordinary&quot; (non-formalized) logic.  &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-30T01:00:09.643" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7887" PostTypeId="2" ParentId="7870" CreationDate="2018-10-30T01:10:16.890" Score="2" Body="&lt;p&gt;History of the modern definition of function is long and very complicated. It begins&#xA;with a discussion between Euler, Bernoulli and d'Alembert related to what was later called Fourier series. The modern definition evolved only by the middle 19 century and&#xA;is usually credited to Dirichlet. He gave this example to illustrate his definition.&#xA;For the whole discussion there is an excellent paper&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR1613935 Luzin, N. Function. I. Amer. Math. Monthly 105 (1998), no. 1, 59–67.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR1615544 Luzin, N. Function. II. Amer. Math. Monthly 105 (1998), no. 3, 263–270.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-30T01:10:16.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7888" PostTypeId="2" ParentId="7873" CreationDate="2018-10-30T01:24:16.710" Score="1" Body="&lt;p&gt;Intuitively, Euclid understood this. Probably. (It is very difficult to speculate&#xA;what a person who lived 2300 years ago, and about whom we really know nothing, understood). Euclid's Elements was rigorously formalized by Hilbert&#xA;(Foundations of Geometry, end of 19th century, there are translations to many languages, including English). He essentially corrected what Euclid &quot;probably understood but could not express well&quot;, and filled many gaps in Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This includes the notion of length. In the modern approach length is a function on the set of intervals which takes values in real numbers. This is equivalent to what you say. Equivalence relation a set is mathematically the same as a function on the set (which sends an element of the set to its equivalence class).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a very accessible discussion of this and other questions about Euclid's Elements I recommend the book by R. Hartshorne, Geometry. Euclid and beyond.&#xA;Hilbert's book is also very readable. But on my opinion, Hartshorne is the best. One can also read Euclid, but this is difficult, and it is recommended to read him with modern comments (by Heath or Hartshorne).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-10-30T01:24:16.710" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7889" PostTypeId="2" ParentId="7881" CreationDate="2018-10-30T02:03:55.997" Score="2" Body="&lt;p&gt;It could have been a fluke, or he could have &amp;quot;adjusted&amp;quot; the calculation to get the &amp;quot;right&amp;quot; value. He did a not very convincing move like that to &amp;quot;explain&amp;quot; why his calculation predicted only half of the observed value for the &lt;a href=&quot;http://sites.apam.columbia.edu/courses/ap1601y/Lunar_Motion_PhysToday_2010.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;motion of lunar apsides&lt;/a&gt;. Here is &lt;a href=&quot;https://books.google.com/books?id=3wIzvqzfUXkC&amp;amp;pg=PA213&amp;amp;lpg=PA213&amp;amp;dq=Newton%27s+precession+of+equinoxes+Jean+le+Rond+d%27Alembert+history&amp;amp;source=bl&amp;amp;ots=ZKQ0HgbNV1&amp;amp;sig=dR1XkXHAPmAc3YQd0sGPFBLY9FI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjk8p_8gq3eAhVEF6wKHeDZALI4ChDoATAHegQIABAB#v=onepage&amp;amp;q=Newton%27s%20precession%20of%20equinoxes%20Jean%20le%20Rond%20d%27Alembert%20history&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilson in Cambridge Companion to Newton&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The nutation, which Newton had not predicted, required an explanation in terms of inverse-square gravitation, and in mid-1748 Jean le Rond d’Alembert (1717–83) set about deriving it. Nutation is a refinement of the precession of the equinoxes, and d’Alembert soon found that Newton’s explanation of the precession (Proposition 66, Corollary 22, Book 1, and Proposition 39, Book 3 with the preceding lemmas) was deeply flawed.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Newton’s basic error arose from his lack of an appropriate dynamics for the rotational motions of solid bodies, and his attempt to treat problems involving such motions in terms of linear momentum rather than angular momentum. D’Alembert now furnished the elements of the appropriate dynamics, and Leonhard Euler systematized it.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Adams and Leverrier apparently got even more lucky with Neptune. Their Bode's law estimate of its orbital period was off by over half a century, and of its mass by 100-200%. As Kelley writes, &amp;quot;&lt;em&gt;the only actual value they were close to, if one looks at the table, is the location in the sky it would be found&lt;/em&gt;&amp;quot;, see &lt;a href=&quot;https://physics.stackexchange.com/q/202342/65263&quot;&gt;Was Leverrier-Adams prediction of Neptune a lucky coincidence?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-10-30T02:03:55.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7890" PostTypeId="2" ParentId="7881" CreationDate="2018-10-30T04:59:43.973" Score="5" Body="&lt;p&gt;Numerical agreement was (at best) a fluke; quoth e.g. A. Berry, &lt;em&gt;A short history of astronomy&lt;/em&gt; (&lt;a href=&quot;//archive.org/details/shorthistoryofas00unse/page/235&quot; rel=&quot;nofollow noreferrer&quot;&gt;1898&lt;/a&gt;, p. 235):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The amount of the precession as&#xA;  calculated by Newton did as a matter of fact agree pretty&#xA;  closely with the observed amount, but this was due to the&#xA;  accidental compensation of two errors, arising from his&#xA;  imperfect knowledge of the form and construction of the&#xA;  earth, as well as from erroneous estimates of the distance&#xA;  of the sun and of the mass of the moon&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first correct computation is attributed to D’Alembert (&lt;a href=&quot;//archive.org/details/bub_gb_82xlL0ilO7UC&quot; rel=&quot;nofollow noreferrer&quot;&gt;1749&lt;/a&gt;, “disorderly”) and then Euler (&lt;a href=&quot;//archive.org/details/berlin-histoire-1749-pub1751/page/n294&quot; rel=&quot;nofollow noreferrer&quot;&gt;1751&lt;/a&gt;, “a model of clarity”) — appreciations taken from Curtis Wilson’s good summary, &lt;em&gt;The precession of the equinoxes from Newton to d’Alembert and Euler&lt;/em&gt; (&lt;a href=&quot;//zbmath.org/?q=an:0859.01001&quot; rel=&quot;nofollow noreferrer&quot;&gt;1995&lt;/a&gt;, pp. 47–54).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For detailed analyses of Newton’s “errors”: D’Alembert (&lt;em&gt;ibid.&lt;/em&gt;) or Wilson’s exposition of it (&lt;a href=&quot;//ams.org/mathscinet-getitem?mr=904328&quot; rel=&quot;nofollow noreferrer&quot;&gt;1987&lt;/a&gt;, §3); Laplace (&lt;a href=&quot;//archive.org/details/oeuvrescomplte05lapluoft/page/273&quot; rel=&quot;nofollow noreferrer&quot;&gt;1825&lt;/a&gt;, pp. 275–278); and Westfall (&lt;a href=&quot;//doi.org/10.1126/science.179.4075.751&quot; rel=&quot;nofollow noreferrer&quot;&gt;1973&lt;/a&gt;; &lt;a href=&quot;//ams.org/mathscinet-getitem?mr=741027&quot; rel=&quot;nofollow noreferrer&quot;&gt;1980&lt;/a&gt;, &lt;a href=&quot;//books.google.com/books?id=3ngEugMMa9YC&amp;amp;ots=rE5mVdsOqQ&amp;amp;pg=PA736&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 736–739&lt;/a&gt;) who convincingly argues that their “accidental compensation” happened quite by design. For instance,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the correction of a faulty lemma in edition one imposed the necessity of an adjustment of more than 50 percent in the remaining numbers. Without even pretending that he had new data, Newton brazenly manipulated the old figures on precession so that he not only covered the apparent discrepancy but carried the demonstration to a higher plane of accuracy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;... whence the changes you noted between the first (Latin) and third (translated) edition. (The correspondence with Cotes, in which they &lt;em&gt;double&lt;/em&gt; the alleged accuracy to reach 1:3000, is a riot.) &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-10-30T14:47:39.547" LastActivityDate="2018-10-30T14:47:39.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7891" PostTypeId="1" CreationDate="2018-10-30T16:21:18.207" Score="5" ViewCount="778" Body="&lt;p&gt;I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing symbols to represent numbers and then manipulating them using rules of arithmetic, but I don't understand where these rules were formalized for so many years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only in recent years did we get the Peano axioms for defining what a natural number is, or a real number in terms of Cauchy sequences, or Peano arithmetic, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We even had calculus, power series, etc, before a lot of these things took off.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We didn't seem to have a &quot;proof theory&quot; where we all agreed what constituted a proof or what was considered a correct / incorrect proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't understand how people defined and used all these terms and techniques before we defined what these things were. Every answer I see on this feels circular to me. Did we just &quot;do the math&quot; without caring too deeply about possible edge cases and issues that arise in certain situations? Were proofs simply an appeal to our senses and whether we found it convincing, as opposed to showing 100% that these things always held? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How exactly did we do mathematics for so long and what necessitated the need to redefine the foundations?&lt;/p&gt;&#xA;" OwnerUserId="8035" LastEditorUserId="4251" LastEditDate="2020-07-25T20:33:44.780" LastActivityDate="2020-07-25T20:33:44.780" Title="What brought about the need for real analysis and formal logic in recent years?" Tags="|mathematics|calculus|mathematical-logic|real-analysis|set-theory|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7892" PostTypeId="2" ParentId="7891" CreationDate="2018-10-30T17:25:24.863" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;We didn't seem to have a &quot;proof theory&quot; where we all agreed what constituted a proof or what was considered a correct / incorrect proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Yes we did. (The &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_logic#Aristotle&quot; rel=&quot;noreferrer&quot;&gt;Greeks&lt;/a&gt; theorized proof by &lt;a href=&quot;https://en.wikipedia.org/wiki/Reductio_ad_absurdum&quot; rel=&quot;noreferrer&quot;&gt;contradiction&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Tertium_non_datur&quot; rel=&quot;noreferrer&quot;&gt;excluded middle&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Modus_tollens&quot; rel=&quot;noreferrer&quot;&gt;contraposition&lt;/a&gt;, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A short answer to the title question (with its emphasis on analysis) is that new “needs” &lt;a href=&quot;https://www.maa.org/press/books/a-radical-approach-to-real-analysis&quot; rel=&quot;noreferrer&quot;&gt;arose&lt;/a&gt; from very concrete problems in exactly when and where Fourier series converge. This forced &lt;a href=&quot;https://books.google.com/books?id=oZpGAAAAcAAJ&amp;amp;pg=PA157&quot; rel=&quot;noreferrer&quot;&gt;Dirichlet&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/mobot31753002059225/page/n193&quot; rel=&quot;noreferrer&quot;&gt;Riemann&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=pzxeAAAAcAAJ&amp;amp;pg=PA355&amp;amp;dq=%22trigonometrischen+Reihen%22&quot; rel=&quot;noreferrer&quot;&gt;Weierstrass&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/mathematischean32behngoog/page/n133&quot; rel=&quot;noreferrer&quot;&gt;Cantor&lt;/a&gt;, their &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Dedekind.html&quot; rel=&quot;noreferrer&quot;&gt;cohort&lt;/a&gt; &lt;a href=&quot;https://archive.org/details/essaysintheoryof00dedeuoft/page/n13&quot; rel=&quot;noreferrer&quot;&gt;Dedekind&lt;/a&gt;, and others into considering more general “functions” and “sets” (not to mention “numbers”) than anyone had &lt;a href=&quot;https://mathoverflow.net/questions/117051/the-origin-of-sets&quot;&gt;before&lt;/a&gt;; and axiomatizing those turned out to be slightly subtler, as more obvious &lt;a href=&quot;https://en.wikipedia.org/wiki/Russell%27s_paradox&quot; rel=&quot;noreferrer&quot;&gt;nonsense&lt;/a&gt; occurs if we’re not careful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In my opinion, the best antidote to the sort of &lt;strong&gt;&lt;em&gt;vertigo&lt;/em&gt;&lt;/strong&gt; you express — and to the general notion that the purpose of mathematical logic is to give mathematics “foundations” or “define everything” (it isn’t: that would indeed be circular) — is to read and understand &lt;a href=&quot;https://www.press.uchicago.edu/ucp/books/book/chicago/W/bo3616369.html&quot; rel=&quot;noreferrer&quot;&gt;Wittgenstein&lt;/a&gt;. It’s therapeutic.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-11-02T20:59:30.443" LastActivityDate="2018-11-02T20:59:30.443" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7893" PostTypeId="2" ParentId="7891" CreationDate="2018-10-30T19:01:47.857" Score="8" Body="&lt;p&gt;As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle.&#xA;(This is called &quot;formal logic&quot;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the later time, the idea arose to use mathematics to formalize logic.&#xA;One of the early proponents of this idea was Leibniz, and it achieved further development in the work of 19 century mathematicians, like George Boole and A. de Morgan. This new approach to logic was called &quot;mathematical logic&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So logic which mathematicians always used in their research is something &quot;outside mathematics&quot;.&#xA;But &quot;mathematical logic&quot; is a part of mathematics. There is no vicious circle here: see &#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Metamathematics&quot; rel=&quot;noreferrer&quot;&gt;Metamathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did people try to formalize logic? There are at least two reasons:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;a) to better understand what the rules of logic really are, and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b) the idea that logical arguments can be performed by a machine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second idea is much older than it may seem to modern students (it goes back to &lt;a href=&quot;https://en.wikipedia.org/wiki/Ramon_Llull&quot; rel=&quot;noreferrer&quot;&gt;Raymond Lully&lt;/a&gt; (1232-1315),&#xA;and apparently was an important motivation for Leibniz and Boole.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-10-30T19:07:10.133" LastActivityDate="2018-10-30T19:07:10.133" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7895" PostTypeId="1" CreationDate="2018-10-31T11:23:26.133" Score="0" ViewCount="239" Body="&lt;p&gt;The number &lt;span class=&quot;math-container&quot;&gt;$360$&lt;/span&gt; as the number of units into which the circle is divided has &lt;a href=&quot;https://hsm.stackexchange.com/questions/1884/origin-of-360-degrees&quot;&gt;some nice properties&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;it has as many divisors as a number of its size can have&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;it's nearly the number of days per year&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But none of these properties is really &quot;magic&quot;, so the number &lt;span class=&quot;math-container&quot;&gt;$360$&lt;/span&gt; &lt;a href=&quot;https://math.stackexchange.com/questions/142735/why-is-a-full-turn-of-the-circle-360-why-not-any-other-number&quot;&gt;isn't really distinguished mathematically&lt;/a&gt; and may have been chosen mainly for practical and logistic reasons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if there is evidence that the number &lt;span class=&quot;math-container&quot;&gt;$360$&lt;/span&gt; has been seen as a somehow &quot;magic&quot; number by ancient mathematicians. For example, it is the number that combines the first three natural numbers &lt;span class=&quot;math-container&quot;&gt;$1,2,3$&lt;/span&gt; with the first three prime numbers &lt;span class=&quot;math-container&quot;&gt;$2,3,5$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$360 = 2^3 \cdot 3^2 \cdot 5^1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that the number &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; (which happens to be the number of months per year) is the number that combines the first two natural numbers &lt;span class=&quot;math-container&quot;&gt;$1,2$&lt;/span&gt; with the first two prime numbers &lt;span class=&quot;math-container&quot;&gt;$2,3$&lt;/span&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$12 = 2^2 \cdot 3^1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; (the number of halves of the year) is just&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2 = 2^1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note, that OEIS knows the sequence &lt;span class=&quot;math-container&quot;&gt;$2,12,360$&lt;/span&gt; under the name of &lt;a href=&quot;https://oeis.org/A006939&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chernoff sequence&lt;/a&gt; but hasn't got a lot to tell about it.&lt;/p&gt;&#xA;" OwnerUserId="6422" LastActivityDate="2018-10-31T11:23:26.133" Title="Was 360 considered a magic number, possibly?" Tags="|ancient-greece|number-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7896" PostTypeId="2" ParentId="7853" CreationDate="2018-10-31T12:17:56.140" Score="3" Body="&lt;p&gt;Here is my interpretation of what Conifold’s unchronological comment may have been trying to say. The term “degeneracy” was first introduced by Schwarzschild (&lt;a href=&quot;http://www.archive.org/stream/sitzungsberichte1916deutsch#page/548&quot; rel=&quot;nofollow noreferrer&quot;&gt;1916&lt;/a&gt;, p. 550) in the context of “old quantum theory”. (Credit by Sommerfeld (&lt;a href=&quot;https://archive.org/details/atombauundspekt00sommgoog/page/n521&quot; rel=&quot;nofollow noreferrer&quot;&gt;1921&lt;/a&gt;, p. 500).)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN252457811_1927?tify=%7B%22pages%22:%5B5%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;1927&lt;/strong&gt;&lt;/a&gt;, p. 4) von Neumann “embedded quantum mechanics into Hilbert spaces” after participating in Hilbert’s &lt;a href=&quot;https://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002271788&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926/27 seminar&lt;/a&gt; (and reading Dirac’s papers). Thereafter, degeneracy took the meaning of operator eigenvalue multiplicity, which is what Dirac must be referring to as “the modern point of view” in his book (1935 or 1947).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Dirac may be objecting to the old QM word having been imported in this way — as opposed to using a “more appropriate” new one to mean “multiplicity &gt; 1”. But no handy alternative emerged, so we still use degeneracy. (It would be interesting to see if similar uses in linear algebra, or algebraic geometry, pre- or post-date this episode.)&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-11-06T13:46:28.130" LastActivityDate="2018-11-06T13:46:28.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7897" PostTypeId="2" ParentId="7891" CreationDate="2018-11-01T02:11:37.627" Score="2" Body="&lt;p&gt;A different thread (in addition to one of my favorites, Fourier's extravagant claim that &quot;every&quot; [sic] function was &quot;expressible&quot; [sic] in trigonometric series (by the way, the pointwise convergence under various hypotheses is not to be purely attributed to Dirichlet, since he had seen a manuscript of Fourier's in which that proof occurred... but for some reason, political?, had its publication obstructed for several years...) ...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;... is Kronecker's description of finite (hence, algebraic) field extensions of a field &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$k[x]/\langle p\rangle$&lt;/span&gt; for irreducible polynomials &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;. This in contrast to the prior style of &quot;believing&quot; that there was/is a larger realm in which all kinds of mystical numbers exist, which we can &quot;adjoin&quot; to the rationals, etc. Of course, that is the spirit. And a first approximation. But... wait a mo', is that legit? :) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, similarly, the number-theoretic notion (Dedekind) that not every field extensions of &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q$&lt;/span&gt; is canonically imbedded in &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;. Rather, the collection of such imbeddings is the object of interest...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In topology, the idea of computing simplicial homology independent of triangulations required some notions of &quot;abstract&quot; groups and such, which would not have made sense decades earlier.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2018-11-01T02:11:37.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7898" PostTypeId="1" CreationDate="2018-11-01T23:17:24.833" Score="0" ViewCount="355" Body="&lt;p&gt;The reason I ask is because he talks about the fine structure constant in one of his undergraduate  lectures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reference to its mystery is analogous to all the special constants of nature such as the cosmological constant and so on and so forth that are attempted to be explained by either the anthropic principle or invoking multiverses &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to work in today's physics and I am curious why Feynman didn't use this argument.  Not that I am saying it is an explanation but I am sure he would have had something to say about it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not really sure what tags I should even use for this question.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-07-06T07:41:11.940" LastActivityDate="2020-07-06T07:41:11.940" Title="Is there any historical mention of Richard Feynman being aware of multiverse theories of cosmology?" Tags="|biographical-details|physicists|cosmology|feynman|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7899" PostTypeId="1" CreationDate="2018-11-02T06:53:25.823" Score="1" ViewCount="191" Body="&lt;p&gt;Ever since learning about projective geometry and its birth in the world of art, I’ve been intrigued to learn more about their union and how they influenced each other. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m specifically looking for references that show how the needs of art led to new math and how math fed back into art, from the renaissance onward. Projections, spherical drawings, Escher’s twisted perspectives, prominent contributors to both fields etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any good books/references that sit at the intersection of the two fields? &lt;/p&gt;&#xA;" OwnerUserId="7035" LastActivityDate="2018-11-03T06:25:03.490" Title="What are some good books that interweave the history of math and art from renaissance onward?" Tags="|mathematics|reference-request|discoveries|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7900" PostTypeId="2" ParentId="7899" CreationDate="2018-11-02T09:00:41.553" Score="2" Body="&lt;p&gt;An exceptional book about that subject is Kirsti Andersen's &lt;em&gt;The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge&lt;/em&gt;, published by Springer.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-11-02T09:00:41.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7901" PostTypeId="1" CreationDate="2018-11-02T10:15:51.563" Score="5" ViewCount="496" Body="&lt;p&gt;The Jewish Encyclopedia (&lt;a href=&quot;https://en.wikipedia.org/wiki/Jewish_Encyclopedia&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Jewish_Encyclopedia&lt;/a&gt;) contains a sizeable entry on one 'Adolph Klein', allegedly born on 1829-05-29 in ther German city of Halle (Saale). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Puzzlingly, hardly anything seems to have survived of his writings (or at least: to have found its way to the internet). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a historical research project, I need to know:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Are there indications (over and above the few that I'll list below) that this 'Adolph Klein' actually lived? If so, was this 'Adolph Klein' Jewish in any meaningful way?&lt;/li&gt;&#xA;&lt;li&gt;Has anything survived of his writings (or of the journals he is said to have edited) and if so, &lt;em&gt;where&lt;/em&gt; can those be read? (There seems to be nothing on the internet.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The above-mentioned encyclopedia-entry lists the following titles of texts Klein allegedly wrote (&lt;strong&gt;none&lt;/strong&gt; of them returned &lt;em&gt;any&lt;/em&gt; result in the currently largest internet search engine, except for one hit only: above-mentioned encyclopedia-entry, which is why one may get the idea that this character might be entirely fictional)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;i. &quot;Anthropologie und Makrobiotik&quot; (i.e. 'anthropology and macrobiotics')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ii. &quot;Der Chemisch-Mechanische Theil des Menschlichen Organismus und Seine Lebenskraft&quot; (i.e. 'the chemico-mechanical part of the human organism and its &lt;em&gt;vis vitalis&lt;/em&gt;')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;iii. &quot;Die Geistbewusste Persönlichkeit und Ihre Unvergänglichkeit als Weltprinzip&quot; (i.e. 'the mind-conscious personality and its imperishability as a world-principle')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;iv. &quot;Die Magnetischen Funktionen des Eisengehaltes im Blute&quot; (i.e. the magnetic functions of the iron-content in the bloodstream')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;v. &quot;Geheimnissvolle Funktionen der Sympathischen Ganglien&quot; (i.e. 'mysterious functions of the sympathetic ganglia'; cf. &lt;a href=&quot;https://en.wikipedia.org/wiki/Sympathetic_nervous_system&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Sympathetic_nervous_system&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vi. &quot;Philosophie und Anthropologie&quot; (i.e. 'philosophy and anthropology')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vii. &quot;Religions-Einigung&quot; (i.e. 'unification of religion(s)')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;viii. &quot;Sozialpolitische Beiträge zur Judenfrage.&quot; (i.e. 'socio-political contributions to the Jewish question')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ix. &quot;Ueberwindung des Wissenschaftlichen Materialismus&quot; (i.e. 'the overcoming of scientific materialism')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;x. &quot;Weshalb Theologie, Materialismus und Spiritualismus in Bezug auf das Menschenräthsel zu Falschen Schlüssen Gelangen?&quot; (i.e. 'why theology, materialism and spiritualism arrive at wrong conclusions in regard to the puzzles of humanities')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;xi. &quot;Wissenschaftliche Beiträge zum Hypnotismus&quot; (i.e. 'scientific contributions to hypnotism')&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also said that this Adolph Klein worked as an editor of the following journals:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Berliner Eisenbahn-Zeitung&quot; (i.e. 'Berlin railroad newspaper) &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Das Rothe Kreuz&quot; (i.e. 'the red cross')&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Deutsche Frauenblätter&quot;  (i.e. 'German women letters')&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;Gross-Lichterfelder Wegweiser&quot; (i.e. 'guidepost for gross-lichterfeld')&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I could not find any scans of these journals (or even of libraries where they might have been preserved).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Adolph Klein is not listed in www.deutsche-biographie.de. (I searched for `Adolf Klein', too, to allow for variant spellings, but---while there are several such entries---none came even close to agreeing with the entry in the Jewish Encyclopedia).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;One 'Friedrich Wilhelm Adolph Klein' is mentioned in an enrollment-book of the University of Königsberger (&quot;Akademisches Erinnerungs-Buch für die, welche in den Jahren 1817 bis 1844 die Königsberger Universität bezogen haben&quot;; &lt;a href=&quot;https://books.google.de/books?id=A8JLAAAAcAAJ&amp;amp;pg=PP5&amp;amp;lpg=PP5&amp;amp;dq=%22Erinnerungsbuch+fur+die+welche+in+den+Jahren+1817+bis+1844+die+Konigsberger&amp;amp;source=bl&amp;amp;ots=8uiawZqQt3&amp;amp;sig=eeawzmw-hVjZdEWhXg62OF2qEoI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjZhPGlu7XeAhXBDuwKHUdkDX4Q6AEwC3oECAEQAQ#v=onepage&amp;amp;q=%22Erinnerungsbuch%20fur%20die%20welche%20in%20den%20Jahren%201817%20bis%201844%20die%20Konigsberger&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.de/books?id=A8JLAAAAcAAJ&amp;amp;pg=PP5&amp;amp;lpg=PP5&amp;amp;dq=%22Erinnerungsbuch+fur+die+welche+in+den+Jahren+1817+bis+1844+die+Konigsberger&amp;amp;source=bl&amp;amp;ots=8uiawZqQt3&amp;amp;sig=eeawzmw-hVjZdEWhXg62OF2qEoI&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjZhPGlu7XeAhXBDuwKHUdkDX4Q6AEwC3oECAEQAQ#v=onepage&amp;amp;q=%22Erinnerungsbuch%20fur%20die%20welche%20in%20den%20Jahren%201817%20bis%201844%20die%20Konigsberger&amp;amp;f=false&lt;/a&gt;). However, while the location of the university of Königsberger is consistent with the information given in the Jewish Encyclopedia according to which Klein worked as a medical doctor in Königsberger, this 'Friedrich Wilhelm Adolph Klein' &lt;strong&gt;cannot&lt;/strong&gt; be the 'Adolph Klein' I am looking for, for the simple reason that he is said to have been born in 1829 while `Friedrich Wilhelm Adolph Klein' is said to have enrolled at University of Königsberg in 1830; no 1-year old has ever seriously enrolled at a university.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The pseudonym `Nielk Floda', given in the entry in the Jewish Encyclopedia is listed as part of a long list of pseudonyms in 'Meyers Konersationslexikon', and is resolved as 'Dr. Adolf Klein', but nothing more than that is said (also, this information is already in the entry in the Jewish Encyclopedia).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6060" LastActivityDate="2018-11-10T00:12:20.567" Title="Has anything survived of the work of alleged science journalist and medical doctor 'Adolph Klein', allegedly born 1829 in Merseburg?" Tags="|biographical-details|medicine|magnetism|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7903" PostTypeId="1" AcceptedAnswerId="7916" CreationDate="2018-11-02T20:37:05.747" Score="4" ViewCount="497" Body="&lt;p&gt;Geometry as described in &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid&lt;/a&gt;'s &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_Elements&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements&lt;/a&gt; originated roughly at the same same time as &lt;a href=&quot;https://en.wikipedia.org/wiki/Democritus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Democritus&lt;/a&gt; described his &lt;a href=&quot;https://en.wikipedia.org/wiki/Atomism#Greek_atomism&quot; rel=&quot;nofollow noreferrer&quot;&gt;atomic theory&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder how close these two points of view were related at those times: mathematical geometry and physical atomism? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there evidence that mathematicians like Euclid considered physical theories like that of Democritus and/or vice versa? &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there mentions in Euclid's Elements that possibly suggest that he&#xA;  was aware of and had thought about Democritus' atomic theory?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Might a mathematician like Euclid have said: &quot;Consider the world surrounding us and neglect all the atoms in it (which master Democritus has shown to exist and describes): What remains is 'pure space' and these are the things existing in it and the laws governing them: ...&quot;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Later atoms could come back into space:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;At each point in time an atom occupies/defines a point in space.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The distance between two atoms is the distance between the points in space they occupy.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Atoms move along straight lines and circles.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;(The same could have been said about celestial bodies &amp;ndash; which from the distance look like atoms.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even later:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Atoms and/or celestial bodies move on epicycles.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Atoms and/or celestial bodies move on conic sections.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="55" LastEditDate="2018-11-05T00:53:12.657" LastActivityDate="2018-11-05T21:34:58.877" Title="What was the relation between Euclid's points and Democritus' atoms?" Tags="|ancient-greece|euclidean-geometry|atomic-theory|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7904" PostTypeId="2" ParentId="7899" CreationDate="2018-11-02T22:57:08.017" Score="1" Body="&lt;p&gt;A fascinating and recent development has been the discovery of &lt;a href=&quot;https://en.wikipedia.org/wiki/Quasicrystal&quot; rel=&quot;nofollow noreferrer&quot;&gt;quasicrystals&lt;/a&gt; and an early account is to be found in&#xA;István Hargittai  (ed) &lt;em&gt;&lt;a href=&quot;https://www.worldscientific.com/worldscibooks/10.1142/1359&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fivefold Symmetry&lt;/a&gt;&lt;/em&gt;, World scientific, 1992. Emil Makovicky, &lt;em&gt;Symmetry: Through the Eyes of Old Masters&lt;/em&gt; (2016) is a newer treatment  (see also some other writings by Makovicky and Hargittai or just get a peek at the &lt;a href=&quot;https://tilings.math.uni-bielefeld.de/&quot; rel=&quot;nofollow noreferrer&quot;&gt;(online) Tilings encyclopedia&lt;/a&gt;). The roots of the topic are in writings about the Fibonacci sequence, the Golden mean, the platonic solids etc and Martha Boles &amp;amp; Rochelle Newman, &lt;em&gt;The Golden Relationship : Art, Math &amp;amp; Nature&lt;/em&gt;, vols 1-2 (1992) might be of interest. &#xA;Other recent topics are fractals and chaos theory with a classic &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Beauty_of_Fractals&quot; rel=&quot;nofollow noreferrer&quot;&gt;The beauty of fractals&lt;/a&gt; by Heinz-Otto Peitgen and Peter Richter (1986).&#xA;A century ago the 4th dimension was a buzz topic and it became entangled in the devlopement of &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubism&quot; rel=&quot;nofollow noreferrer&quot;&gt;cubism&lt;/a&gt; see e.g.&#xA;Tony Robbin, &lt;em&gt;Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought&lt;/em&gt;, 2006&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2018-11-03T06:25:03.490" LastActivityDate="2018-11-03T06:25:03.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7905" PostTypeId="2" ParentId="4876" CreationDate="2018-11-03T02:26:15.160" Score="2" Body="&lt;p&gt;I can think of two major examples offhand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Computer Science, John McCarthy's &lt;a href=&quot;http://www-formal.stanford.edu/jmc/recursive.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;original Lisp paper&lt;/a&gt; was meant to be a foray into using computers to represent knowledge for artificial intelligence. Instead, he ended up accidentally inventing both functional programming and garbage collection.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In mathematics, Eilenberg and Mac Lane were studying algebraic topology, and ended up discovering category theory, which had huge effects, from simplifying Cayley's theorem as the Yoneda lemma, to modelling programming languages, to quantum physics.&lt;/p&gt;&#xA;" OwnerUserId="3044" LastActivityDate="2018-11-03T02:26:15.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7906" PostTypeId="2" ParentId="7901" CreationDate="2018-11-03T10:00:14.217" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;http://www.jewishencyclopedia.com/articles/9371-klein-adolph&quot; rel=&quot;nofollow noreferrer&quot;&gt;entry&lt;/a&gt; is wholly drawn from an ostensibly autobiographical portrait inserted in the “&lt;a href=&quot;https://books.google.com/books?id=uwe1DQAAQBAJ&amp;amp;pg=PA360&amp;amp;dq=%22Das+geistige+Berlin%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;humorvoll&lt;/a&gt;” who’s who &lt;strong&gt;&lt;em&gt;Das geistige Berlin&lt;/em&gt;&lt;/strong&gt; (&lt;a href=&quot;http://www.worldcat.org/oclc/162453784&quot; rel=&quot;nofollow noreferrer&quot;&gt;1897&lt;/a&gt;, p. 245; &lt;a href=&quot;https://books.google.com/books?id=p71u-OCuOFIC&quot; rel=&quot;nofollow noreferrer&quot;&gt;reprint&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Klein,&lt;/strong&gt; Ad., Dr. med. und phil., geboren zu Merseburg an der Saale am 18. Mai 1829, studirte in Leipzig 1848–51, praktizirte in Königsberg in Ostpreussen 1859–70. Vermählte sich 1860 mit Fräulein Marie Josephine geborene Binder aus Thorn an der Weser. Widmete sich seit 1870 der Schriftstellerei und redigirte eine Anzahl von Blättern: Das Rothe Kreuz, &lt;a href=&quot;https://books.google.com/books?id=lVAzAQAAMAAJ&amp;amp;pg=PA253&amp;amp;dq=Frauenblattes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Deutsche Frauenblätter&lt;/a&gt;, Berliner Eisenbahn-Zeitung, Gross-Lichterfelder Wegweiser und Andere. Hauptwerke sind: (&lt;strong&gt;i,  ix, vi, ii, v, iv, iii, xi, &lt;a href=&quot;https://books.google.com/books?id=vDQOAQAAMAAJ&amp;amp;q=%22Religi%C3%B6se+Einigung+und+wirtschaftlich-sittliche+Weltreform%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;vii&lt;/a&gt;, x, &lt;a href=&quot;https://books.google.com/books?id=ejcYAAAAYAAJ&amp;amp;pg=PA209&amp;amp;dq=Adolf+Klein&quot; rel=&quot;nofollow noreferrer&quot;&gt;viii&lt;/a&gt;&lt;/strong&gt;). Ausserdem noch eine Anzahl Schriften humanitären, sozialökonomischen und sozialhygienischen Inhalts. Studien über Uranographie und Allwohlsschriften auch philosophischen Inhalts. Mehrere meiner Schriften sind ihrer Freimüthigkeit halber in New York verlegt. Pseudonym: Dr. Nielk Floda. Ohne Preise, Titel, Orden und Ehrenzeichen (die ich übrigens ablehnen würde). Vorstandsmitglied verschiedener Vereine, insbesondere des „Allwohlsbundes“. Jetziger Wohnort: Gross-Lichterfelde, Villa Klein.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;iv&lt;/strong&gt; has &lt;em&gt;Funktion&lt;/em&gt; and &lt;strong&gt;xi&lt;/strong&gt; reads in full „Wissenschaftliche Beiträge zum Hypnotismus, thierischen Magnetismus, Freiheit des Willens und die &lt;a href=&quot;https://hsm.stackexchange.com/q/7853&quot;&gt;Degeneration&lt;/a&gt; in der Kulturwelt“.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shorter notes appear in &lt;em&gt;Kürschner’s &lt;strong&gt;Deutscher Litteratur-Kalender&lt;/em&gt;&lt;/strong&gt;, years &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=iau.31858002687220&amp;amp;num=156&quot; rel=&quot;nofollow noreferrer&quot;&gt;1887&lt;/a&gt; to &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uiug.30112109581329&amp;amp;num=693&quot; rel=&quot;nofollow noreferrer&quot;&gt;1903&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More crumbs (and one surviving work):&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=EoLMo5A5QQYC&amp;amp;pg=PA123&quot; rel=&quot;nofollow noreferrer&quot;&gt;Antwort von Dr. Adolf Klein&lt;/a&gt;. In: Ferdinand Maack (ed.), Okkultismus: Was ist er? Was will er? Wie erreicht er sein Ziel? Verlag von Paul Zillmann, Zehlendorf b. Berlin, 1898, pp. 123–131.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bericht über die Verhandlungen auf dem zweiten Kongress des „Verbandes Deutscher Okkultisten“ am 7. u. 8. Juni (Pfingsten) 1897 in Dresden. (Berlin–Gross-Lichterfelde, Berlinerstr. 142, Adolf Klein’s Verlagsbuchhandlung, &lt;a href=&quot;https://books.google.com/books?id=Lrn-iMXa41MC&amp;amp;pg=PA656&amp;amp;dq=Adolf+Klein%E2%80%99s&quot; rel=&quot;nofollow noreferrer&quot;&gt;1897&lt;/a&gt;.)&#xA;(Publishing house listed as established in &lt;a href=&quot;https://books.google.com/books?id=fMkMImGF-4sC&amp;amp;pg=PA224&amp;amp;dq=%22Klein,+Adolf%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1880&lt;/a&gt;, disbanded by &lt;a href=&quot;https://books.google.com/books?id=fMkMImGF-4sC&amp;amp;pg=RA4-PA8&amp;amp;dq=%22Klein%2C%20Adolf%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt;. This confirms the address found by @njuffa.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The above 1897 proceedings also appear in &lt;a href=&quot;http://www.worldcat.org/oclc/252399116&quot; rel=&quot;nofollow noreferrer&quot;&gt;oclc notes&lt;/a&gt; on proceedings of a 1896 congress. The latter mention Klein’s election as second chairman of the Bureau (&lt;a href=&quot;http://www.iapsop.com/ssoc/1896__anonymous___kongress_deutscher_okkultisten_partial.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;partial pdf&lt;/a&gt;, p. 6).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-11-06T05:43:32.067" LastActivityDate="2018-11-06T05:43:32.067" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7907" PostTypeId="1" CreationDate="2018-11-03T10:56:57.210" Score="2" ViewCount="386" Body="&lt;p&gt;Why I asked this question : &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k90195m/f54.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gallica.bnf.fr/ark:/12148/bpt6k90195m/f54.image&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;p 50-51, in course of Cauchy, a proof of the intermediate value theorem. Now, that's not a proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And I learned that Cauchy had made a lot of mistakes, so this is not a proof, of a change of standard between our era and that of Cauchy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the question stay : has the standard of mathematical proofs changed over time ?&lt;/p&gt;&#xA;" OwnerUserId="8057" LastEditorUserId="8058" LastEditDate="2018-11-03T22:06:05.507" LastActivityDate="2018-11-03T23:17:18.220" Title="Has the standard of mathematical proofs changed over time?" Tags="|mathematics|mathematical-logic|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7908" PostTypeId="2" ParentId="7907" CreationDate="2018-11-03T14:52:25.590" Score="8" Body="&lt;p&gt;Yes and no. It is better to say that there were always several different standards. Most proofs of Euclides, Archimedes and Apollonius are on the level of modern standards, though gaps in those proofs can be found, and some were found in the ancients time. At the time where calculus was invented, many mathematicians understood that the proofs using new methods are not to the same standard of rigor as the proofs of Archimedes.&#xA;It took more than two centuries to clean all this and to establish calculus on the same degree of rigor as in the work of Archimedes.  Archimedes himself also used sometimes non-rigorous arguments, and he perfectly understood this, and said this explicitly. And this is what happens nowadays too. So at any time, there are several different standards of rigor.   &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-11-03T14:52:25.590" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="7909" PostTypeId="2" ParentId="7903" CreationDate="2018-11-03T15:00:49.517" Score="2" Body="&lt;p&gt;The difference between a point (as in Euclid) and atom in Democritus is that an interval (or a plane or solid figure) in Euclid is infinitely divisible, every interval, no matter how small can be divided further. While in the atomic theory, there is a smallest part of matter which is not further divisible. If applied to the real world, these are two opposite points of view (infinite divisibility of matter against atomic theory) which were both discussed since antiquity. Another important difference is that Euclid did not deal with &quot;real world&quot;, but with idealized mathematical objects, and he probably perfectly understood this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-11-03T15:00:49.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7911" PostTypeId="2" ParentId="7907" CreationDate="2018-11-03T18:25:31.703" Score="2" Body="&lt;p&gt;Not an answer, as Eremenko’s can hardly be improved upon. But, courtesy of M. Audin (&lt;a href=&quot;https://hsm.stackexchange.com/a/5647&quot;&gt;2016&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;— It's still irksome, Cartan said one day. You read a theorem, its proof. Then there is a remark that says “the theorem we have just proved is not always true”.&lt;/p&gt;&#xA;&lt;p&gt;— You read that in Goursat?&lt;/p&gt;&#xA;&lt;p&gt;— No, that one is in Bertrand.&lt;/p&gt;&#xA;&lt;p&gt;— What is the theorem?&lt;/p&gt;&#xA;&lt;p&gt;— Cauchy’s theorem. And, he shows his colors before proving it: “this theorem holds in general, but it often fails”.&lt;/p&gt;&#xA;&lt;p&gt;— And he proves it?&lt;/p&gt;&#xA;&lt;p&gt;— Yes. Then he gives counterexamples.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-11-03T23:17:18.220" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7912" PostTypeId="1" AcceptedAnswerId="7954" CreationDate="2018-11-03T23:33:29.707" Score="0" ViewCount="282" Body="&lt;p&gt;I am looking for the origin of the symbol &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; used by chemists to denote the symmetry operation consisting of a &lt;span class=&quot;math-container&quot;&gt;$\smash{\frac{2\pi}n}$&lt;/span&gt; rotation (&lt;span class=&quot;math-container&quot;&gt;$C_n$&lt;/span&gt;) about an axis &lt;em&gt;and&lt;/em&gt; a reflection in a plane perpendicular to that axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google Scholar traces earliest use of the operation’s name “Drehspiegelung” to A. Schoenflies’, &lt;em&gt;Krystallsysteme und Krystallstructur&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/krystallsysteme00schogoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;1891&lt;/a&gt;). However, that book does not seem to denote it &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Physics books on molecular structure had &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; by the 1930s: an early use can be seen in L. Tisza, &lt;em&gt;Zur Deutung der Spektren mehratomiger Moleküle&lt;/em&gt; (&lt;a href=&quot;//doi.org/10.1007/BF01342114&quot; rel=&quot;nofollow noreferrer&quot;&gt;1933&lt;/a&gt;, p. 59), so it must have started earlier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;strong&gt;N.B.:&lt;/strong&gt; Symbols like &lt;span class=&quot;math-container&quot;&gt;$\mathrm C_n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm S_n$&lt;/span&gt; can also denote symmetry &lt;em&gt;groups&lt;/em&gt;. This should not be confused with the notation for symmetry operations or “elements” — see e.g. &lt;a href=&quot;https://sites.cns.utexas.edu/jones_ch431/symmetry-elements-and-operations&quot; rel=&quot;nofollow noreferrer&quot;&gt;this table&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="658" LastEditDate="2018-11-10T07:01:18.907" LastActivityDate="2018-11-13T07:41:41.627" Title="Who used the symbol $S_n$ for &quot;rotation reflection&quot; as a symmetry operation?" Tags="|mathematics|terminology|mathematical-physics|" AnswerCount="2" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7913" PostTypeId="2" ParentId="7912" CreationDate="2018-11-04T06:14:18.887" Score="1" Body="&lt;p&gt;In the 1906 Encyklopädie  (see comments above), Schonflies gives reference to his book Krystallsysteme und Krystallstructur (1891) in the footnote. His book uses the same symbols, namely &lt;span class=&quot;math-container&quot;&gt;$\mathfrak S$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathfrak A$&lt;/span&gt;(α), &lt;span class=&quot;math-container&quot;&gt;$\overline{\mathfrak A}(\alpha)$&lt;/span&gt; with a bar for Spiegelung, Drehung, Drehspiegelung respectively. However, in the modern literature, the &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;, where n indicated the 360/n rotations is used. It seems Schonflies was switching back and forth in his symbolism. An early use of modern  symbol can be seen in Tisza's 1930s article: &lt;em&gt;Zur Deutung der Spektren mehratomiger Moleküle&lt;/em&gt;&#xA;&lt;a href=&quot;https://link.springer.com/article/10.1007/BF01342114&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/article/10.1007/BF01342114&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2018-11-08T00:38:22.557" LastActivityDate="2018-11-08T00:38:22.557" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7914" PostTypeId="1" CreationDate="2018-11-04T09:31:05.137" Score="4" ViewCount="338" Body="&lt;p&gt;The Gaussian integral&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\intop_{-\infty}^{\infty} dx \exp(-x^2) = \sqrt{\pi} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is done in a very smart way. But where is the original document? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastEditorUserId="609" LastEditDate="2018-11-05T14:05:13.227" LastActivityDate="2018-11-06T14:01:26.280" Title="Original document of the Gaussian integral" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7915" PostTypeId="2" ParentId="7914" CreationDate="2018-11-04T15:50:59.957" Score="6" Body="&lt;p&gt;At first I voted to close because Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates&quot; rel=&quot;nofollow noreferrer&quot;&gt;clearly attributes&lt;/a&gt; that “smart way” to &lt;strong&gt;Poisson&lt;/strong&gt;. But oddly, the linked document only quotes hearsay by Sturm (&lt;a href=&quot;https://archive.org/details/coursdanalysede04prougoog/page/n32&quot; rel=&quot;nofollow noreferrer&quot;&gt;1859&lt;/a&gt;). The actual source is given by Bierens de Haan (&lt;a href=&quot;https://archive.org/details/tablesdintegral00haan/page/81&quot; rel=&quot;nofollow noreferrer&quot;&gt;1858&lt;/a&gt;, Table 40): Poisson’s &lt;em&gt;Théorie mathématique de la chaleur&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/thoriemathma00pois/page/139&quot; rel=&quot;nofollow noreferrer&quot;&gt;1835&lt;/a&gt;, §74).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The integral was computed earlier by other methods — see &lt;em&gt;loc. cit.&lt;/em&gt;, Tables 36 &amp;amp; 37 and &lt;a href=&quot;https://hsm.stackexchange.com/questions/4880/who-derived-int-0-infty-mathrme-x2-dx-frac-sqrt-pi2/4887#4887&quot;&gt;@KCd&lt;/a&gt;.) &lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-11-06T14:01:26.280" LastActivityDate="2018-11-06T14:01:26.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7916" PostTypeId="2" ParentId="7903" CreationDate="2018-11-05T00:52:31.930" Score="5" Body="&lt;p&gt;They were the opposite of close. Geometers and atomists were bitter ideological enemies since before Euclid. The reason for such high passions was that Greek view of geometry was different from the modern relativistic/formalistic view, to them the nature of real space was at stake, not one idealized fiction among others. The indivisibles (atoms) and the infinite divisibility (geometry) could not go together, only one had to be true. As for points, geometers treated them as marks placed at will rather than anything real, Aristotle explicitly rejects the idea that continuum is assembled from points. For a recent review see &lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&amp;amp;httpsredir=1&amp;amp;article=1316&amp;amp;context=jhm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aristidou Some Thoughts on the Epicurean Critique of Mathematics&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Epicurus postulated the least conceivable length called &lt;em&gt;elahiston&lt;/em&gt;, declared that geometry is based on falsehoods, and banned it from the curriculum at his school, he similarly rejected Eudoxian mathematical astronomy because it was based on geometry. The rivalry between contemporary Epicurean and Eudoxian schools was quite intense, see &lt;a href=&quot;https://www.academia.edu/3051043/Epicurus_and_the_mathematicians_of_Cyzicus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sedley's Epicurus and the Mathematicians of Cyzicus&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;That Epicurus believed in a minimal unit of measure out of which not only atoms but also all larger lengths, areas, and volumes are composed,  is  nowadays  widely  accepted;  and  most  would  also agree that it is not merely a physical minimum, contingent upon the  nature  of  matter,  but  a  theoretical  minimum,  than  which nothing smaller is conceivable.  Others both before and since Epicurus  have  been  seduced  by  similar  theories  without  being  led to reject conventional geometry.  Yet this is precisely the penalty which a theory of minimal parts should carry with it, for one of its consequences is to make all lines integral multiples of a single length and therefore commensurable with each other, whereas the incommensurability of lines in geometrical figures had been recognized by Greek mathematicians since the fifth century.  Moreover the principle of infinite divisibility lay at the heart of the geometrical method commonly called the ‘method of exhaustion’, which was fruitfully developed by Eudoxus in the fourth century.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Euclid largely recorded geometry as presented by Eudoxus and other late Pythagoreans, so atomism and motion were alien to him. Conversely, &quot;&lt;em&gt;while  there  were some Epicureans who were — or previously had been — mathematicians, it seems that acceptance of the Epicurean world-view generally  involved,  for  these  individuals,  a  conversion  from  the practice  of  geometry&lt;/em&gt;&quot; (White, What to Say to a Geometer, 1989). Some Epicureans sought to undermine geometric arguments, e.g. Zeno of Sidon, according to Proclus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Since some persons have raised objections to the construction of the equilateral triangle with the thought that they were refuting the  whole  of  geometry,  we  shall  also  briefly  answer  them.   The Zeno whom we mentioned above asserts that, even if we accept the  principles  of  the  geometers,  the  later  consequences  do  not stand unless we allow that two straight lines cannot have a common segment.  For if this is not granted, the construction of the equilateral triangle is not demonstrated.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Archimedes in his Method seems to be at least inspired by Democritus's computation of the volume of the pyramid by slicing it into the indivisibles, &#xA;and writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In the case of the theorems the proof of which Eudoxus was first to discover, that the cone is a third part of the cylinder, and the pyramid of the prism, we should give no small share of the credit to Democritus who was the first to make the assertion with regard to the said figure, though he did not prove it.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But he then explicitly distinguishes &quot;discovering by mechanical methods&quot; from &quot;demonstrating by geometric methods&quot;, and gives clear priority to the latter. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-11-05T21:34:58.877" LastActivityDate="2018-11-05T21:34:58.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7917" PostTypeId="1" CreationDate="2018-11-05T08:47:41.773" Score="4" ViewCount="161" Body="&lt;p&gt;&lt;em&gt;[Even though this question may seem as a duplicate of this question about the &lt;a href=&quot;https://hsm.stackexchange.com/questions/3414/history-of-sine-function&quot;&gt;History of sine function&lt;/a&gt;, I'd like to ask it again - with a more specific title and a more specific focus (on specific references) which hopefully will re-attract some users.]&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse and this is the mental picture one must have in mind:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bxjro.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bxjro.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This or similar pictures can be found in many ancient texts on trigonometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But today, when hearing &quot;sine&quot; many people inevitably have in mind a mental picture like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/I24Bh.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I24Bh.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and it's not so hard to understand how it arises from the first picture (by rotating the radius with constant speed and plotting the sine):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PAN6W.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PAN6W.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was not always so: neither has there been this mental picture of the &#xA;sine function nor have there been physical drawings.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What are the earliest (known) physical drawings of the plot of the sine or&#xA;  another trigonometric function? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Are they Greek, Indian, Chinese or from the Middle Ages? A specific reference would be welcome.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;This may not be the earliest reference, but it's a rather early one, as I suppose:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/awKX6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/awKX6.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Albrecht Dürer: &lt;a href=&quot;https://de.wikisource.org/wiki/Seite%3ADuerer_Underweysung_der_Messung_021.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen&lt;/a&gt;, 1525)&lt;/p&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2018-11-05T11:33:45.173" LastActivityDate="2018-11-05T11:33:45.173" Title="Earliest drawings of the plots of trigonometric functions" Tags="|mathematics|reference-request|notation|geometry|education|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7918" PostTypeId="1" CreationDate="2018-11-05T14:00:10.267" Score="1" ViewCount="169" Body="&lt;p&gt;Related to a more &lt;a href=&quot;https://math.stackexchange.com/questions/2985781/can-one-state-and-proof-that-euclidean-space-has-genus-0-in-hilberts-geometry&quot;&gt;mathematically inclined question&lt;/a&gt;, I'd like to ask the following question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ancient Greeks made use of infinite arguments and processes (limits), e.g. in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Method_of_exhaustion&quot; rel=&quot;nofollow noreferrer&quot;&gt;method of exhaustion&lt;/a&gt; and in &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_infinite_descent&quot; rel=&quot;nofollow noreferrer&quot;&gt;proofs by infinite descent&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if they may have used it in a pre-topological context:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They knew/presupposed that for each length &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; (arbitrary large or small) and each point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, a circle with center &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; can be drawn and is &quot;contained in space&quot;. This can be thought of as &quot;every circle is collapsable to a point which itself is contained in space&quot;. (In modern terms every circle is &lt;em&gt;continuously&lt;/em&gt; collapsable to a point.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any evidence that they were aware of this characteristic feature of (Euclidean) space? And did they actually believe to have a proof of this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or was it too abstract for them to think about, or too &quot;natural&quot; to even mention it? (They probably didn't think about toroidal spaces.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To sum it up:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Were the ancient Greeks aware of the topology of (Euclidean) space?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6422" LastEditorUserId="6422" LastEditDate="2018-11-06T19:15:18.777" LastActivityDate="2018-11-06T19:15:18.777" Title="Were the ancient Greeks aware of the &quot;topology&quot; of (Euclidean) space?" Tags="|euclidean-geometry|topology|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="7919" PostTypeId="1" CreationDate="2018-11-05T15:39:46.640" Score="3" ViewCount="248" Body="&lt;p&gt;Does any extant Greek text prove that the area of a regular polygon inscribed in a fixed circle increases with the number of sides in the polygon?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't find such a proposition in &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/elements.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid&lt;/a&gt;, but the Greeks must surely have known of it, and had a proof, especially as a proof needn't even use Eudoxus's theory of proportion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I think &lt;a href=&quot;https://math.stackexchange.com/a/2984710/213690&quot;&gt;this&lt;/a&gt; is valid, for example. It's something the Greeks could easily have come up with. I may even have overcomplicated it.)&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note on the edited title, &lt;em&gt;Did Greeks know that the area of inscribed regular polygons increases with the number of sides?&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;As stated in the body of the question, I personally don't doubt that&#xA;the Greeks &quot;knew&quot; the result, both in the weak sense of finding it&#xA;intuitively obvious, and in the strong sense of having at least one&#xA;proof (which, in the absence of evidence, can only be guessed at).&#xA;Of course, this is only a personal opinion, not even an educated&#xA;one.  I can't prove it; but on the other hand, I don't want to ask a&#xA;question about it!  Also, the edited title raises an epistemological&#xA;question (what does it mean to &quot;know&quot; a result?), which I would&#xA;rather avoid.  Not only that, but it could lead to a complex&#xA;historical debate, with no clear conclusion.  I prefer my original&#xA;concrete and narrowly-focussed wording.  If it is still necessary to&#xA;reword the question, please say why.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;As a similar proposition about perimeters is being discussed in an answer and its comments, it seems worth reproducing here a comment in which I derive that result as a corollary:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; be the area and &lt;span class=&quot;math-container&quot;&gt;$p_n$&lt;/span&gt; the perimeter of a regular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-gon inscribed&#xA;in a circle of radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;. If the vertices of the &lt;span class=&quot;math-container&quot;&gt;$(2n)$&lt;/span&gt;-gon are &#xA;&lt;span class=&quot;math-container&quot;&gt;$P_0P_1P_2\cdots$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$P_0P_2 \perp OP_1$&lt;/span&gt;, therefore the area of&#xA;&lt;span class=&quot;math-container&quot;&gt;$\triangle OP_1P_2$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{r}{2}\cdot\frac{p_n}{2n}$&lt;/span&gt;; but the same area is&#xA;also equal to &lt;span class=&quot;math-container&quot;&gt;$\frac{a_{2n}}{2n}$&lt;/span&gt;; therefore &lt;span class=&quot;math-container&quot;&gt;$p_n = \frac{2a_{2n}}{r}$&lt;/span&gt;; and&#xA;&lt;span class=&quot;math-container&quot;&gt;$a_{2n}$&lt;/span&gt; increases with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, therefore so does &lt;span class=&quot;math-container&quot;&gt;$p_n$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8068" LastEditorUserId="8068" LastEditDate="2018-11-08T20:26:21.027" LastActivityDate="2018-11-08T20:26:21.027" Title="Does any extant Greek text prove that the area of an inscribed regular polygon increases with the number of sides?" Tags="|mathematics|reference-request|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7920" PostTypeId="1" CreationDate="2018-11-05T18:28:31.643" Score="6" ViewCount="751" Body="&lt;p&gt;Yesterday I took my time to look again into Schlesinger's essay on Gauss's contributions to analysis, and I found something new I didn't know about (so it caught my eye) in the last subsection of the chapter &quot;The posthumous treatise on the agM [apparently 'arithmetico-geometric Mean']. Extension to the theory of module function&quot; (p. 84-117). In subsection (e) of this chapter, named &quot;The last diary notes from 1800. Asymptotic. Midpoint equation. Class number&quot;, Schlesinger comments on the equation of the center, includes some intriguing results of Gauss, and mentions that his formulas agree with that of Jacobi (1849). The relevant pages in Gauss's nachlass are p. 420-428 of volume 10-1 of his works, and they are entitled &quot;&lt;strong&gt;on the convergence of expansions [in series]  of the equation of the center&lt;/strong&gt;&quot; (German: 'Über die Konvergenz&#xA;der Entwicklung der Mittelpunktsgleichung'). It seems that these pages are actually part [VII] of a planned treatise on the convergence of infinite series.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just to be clear, I know nothing about this so called &quot;Mittelpunktsgleichung&quot;, and I'm not even sure if this work was done by Gauss in astronomical contexts or analytic contexts, or perhaps both (I mention astronomical context since it seems that these formulas were connected with his work on &quot;Kepler's equation&quot; in astronomy).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I'm not looking for a very sophisticated answer, just an explanation of what Gauss actually did and the context it was done. I think that if Schlesinger commented on it then it's important.    &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10106" LastEditDate="2020-08-22T05:05:14.270" LastActivityDate="2021-03-20T13:27:32.557" Title="Meaning of passages by Gauss on the &quot;convergence of expansions (in infinite series) of the (elliptical) equation of the center&quot;?" Tags="|mathematics|biographical-details|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7921" PostTypeId="1" CreationDate="2018-11-05T18:50:14.433" Score="4" ViewCount="133" Body="&lt;p&gt;I have heard about a &quot;famous&quot; mistake made on a physics paper by the editor of a scientific journal but I can't seem to find the paper or even to recall what it was about. Has anyone hear of this and would know the paper in question?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The part of the story I remember is the following : It was at a time when personal computers weren't a thing and editorial work was a manual job. The editor erased a point from an experimental plot thinking it was a smudge while it was in fact the central discovery of the article.&lt;/p&gt;&#xA;" OwnerUserId="8069" LastEditorUserId="8069" LastEditDate="2018-11-06T11:04:13.223" LastActivityDate="2018-11-06T11:04:13.223" Title="Name of the paper that suffered a famous editor mistake" Tags="|experimental-physics|publishing|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7922" PostTypeId="1" CreationDate="2018-11-05T20:11:26.240" Score="4" ViewCount="81" Body="&lt;p&gt;[I asked this originally at the Math Stack Exchange, and they suggested I also ask about it here.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I heard about this from a college professor but haven't ever been able to find any other mention of it. I remember it being called &quot;La Guerre&quot; geometry, but I could be wrong there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story was that a French(?) mathematician was taken as a POW in some war (no idea which one, but pre-WWII anyway and possibly hundreds of years before that). He was bored and passed the time doing geometry in the dirt on the floor of his cell. He noticed that if you have circle problems in the Euclidean plane you can sometimes solve them more easily by turning the problem into a 3-space problem with cones instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The general idea is that each circle is the base of a right circular cone (h=r). Then the vertex of each cone (r units above the center of the circle) acts as a stand-in for the whole initial circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something like that anyway.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is, confusingly, also a mathematician named Laguerre, but what I can find on him doesn't seem to be quite what I'm talking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, I'd love to know more about it all. I remember we did some very cool proofs using this method and I just haven't been able to coax Google into finding what I need. Hopefully I'll have more luck here.&lt;/p&gt;&#xA;" OwnerUserId="8070" LastEditorUserId="55" LastEditDate="2018-11-05T21:51:42.653" LastActivityDate="2018-11-05T21:51:42.653" Title="What was the old system of using right circular cones to solve problems about circles in the plane?" Tags="|geometry|euclidean-geometry|projective-geometry|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7923" PostTypeId="2" ParentId="7901" CreationDate="2018-11-05T21:02:10.020" Score="4" Body="&lt;p&gt;Following the &lt;a href=&quot;https://books.google.com/books?id=lVAzAQAAMAAJ&amp;amp;pg=PA253&amp;amp;dq=Frauenblattes&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; given in Francois Ziegler's answer I found a Dr. Klein among the signers of a parliamentary petition calling for the modification of &lt;a href=&quot;https://en.wikipedia.org/wiki/Paragraph_175&quot; rel=&quot;nofollow noreferrer&quot;&gt;section 175&lt;/a&gt; of the German Criminal Code:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Petition an die gesetzgebenden Körperschaften des deutschen Reiches behufs Abänderung des § 175 des R.-Str.-G.-B. und die sich daran anschliessenden Reichstags-Verhandlungen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The main purpose of this section, which was carried forward in various versions of the German Criminal Code until it was eliminated in 1994, was the criminalization of homosexual acts between men. The petition calls for changing this section such that it applies equally to homosexual and heterosexual acts and applies only to cases involving physical violence, persons under 16 years of age, or indecent exposure. The petition's author is listed as a Dr. med. Hirschfeld. This is the German Jewish physician and pioneering sexologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Magnus_Hirschfeld&quot; rel=&quot;nofollow noreferrer&quot;&gt;Magnus Hirschfeld&lt;/a&gt;, as I confirmed via &lt;a href=&quot;https://books.google.com/books?id=dw8jeQvTSaAC&amp;amp;pg=PA262&amp;amp;dq=Petition+an+die+gesetzgebenden+K%C3%B6rperschaften+des+deutschen+Reiches+behufs+Ab%C3%A4nderung+des+%C2%A7+175+des+R.-Str.-G.-B.+und+die+sich+daran+anschliessenden+Reichstags-Verhandlungen.&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwib46zq6L7eAhUhJDQIHVP_DZQQ6AEIKzAA#v=onepage&amp;amp;q=Petition%20an%20die%20gesetzgebenden%20K%C3%B6rperschaften%20des%20deutschen%20Reiches%20behufs%20Ab%C3%A4nderung%20des%20%C2%A7%20175%20des%20R.-Str.-G.-B.%20und%20die%20sich%20daran%20anschliessenden%20Reichstags-Verhandlungen.&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt;, which also states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When Hirschfeld and his allies went looking for signatures, lists of which were regularly published in the yearbook, they did not seek volume. Instead, they took pains to solicit exclusively well-respected, impeccable figures of Wilhelmine society.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Among the petition's signers, Dr. Klein is listed as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dr. Adolf Klein, Chefredakteur des deutschen Frauenblattes, Gross-Lichterfelde&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Berlin Central and Regional Library makes various historic address directories available online, among them the &quot;Adreßbuch für Berlin und seine Vororte&quot; for 1897. In the section for the suburb of Groß-Lichterfelde, one finds the following &lt;a href=&quot;https://digital.zlb.de/viewer/readingmode/34115316_1897/2839/&quot; rel=&quot;nofollow noreferrer&quot;&gt;entry&lt;/a&gt; under the heading &quot;Buchhandlungen&quot; (book sellers):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dr. Klein, Ad., Berlinerstr. 142&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Around 1900, Groß-Lichterfelde was an &lt;a href=&quot;https://de.wikipedia.org/wiki/Gro%C3%9F-Lichterfelde&quot; rel=&quot;nofollow noreferrer&quot;&gt;upscale neighborhood&lt;/a&gt; consisting mostly of &quot;Villen&quot; (i.e. small mansions). This jibes with Dr. Klein's residence being listed as &quot;Villa Klein&quot; in the autobiographical notes. The street has since been renamed to Ostpreußendamm, according to the &lt;a href=&quot;https://de.wikipedia.org/wiki/Liste_der_Stra%C3%9Fen_und_Pl%C3%A4tze_in_Berlin-Lichterfelde#Ostpreu%C3%9Fendamm&quot; rel=&quot;nofollow noreferrer&quot;&gt;German Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ostpreußendamm. Der vorherige Name der vor 1878 angelegten Straße war Berliner Straße.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Google Maps's &lt;a href=&quot;https://www.google.com/maps/place/Ostpreu%C3%9Fendamm+142,+12207+Berlin,+Germany/@52.4238471,13.31417,3a,75y,146.97h,90t/data=!3m7!1e1!3m5!1sACrSHkMTxqlMUJ9_dxvzUg!2e0!6s%2F%2Fgeo0.ggpht.com%2Fcbk%3Fpanoid%3DACrSHkMTxqlMUJ9_dxvzUg%26output%3Dthumbnail%26cb_client%3Dsearch.TACTILE.gps%26thumb%3D2%26w%3D86%26h%3D86%26yaw%3D146.96924%26pitch%3D0%26thumbfov%3D100!7i13312!8i6656!4m5!3m4!1s0x47a85ae2a3039f2b:0x2d00e0a3dc0f15a9!8m2!3d52.42373!4d13.31431&quot; rel=&quot;nofollow noreferrer&quot;&gt;street view&lt;/a&gt; shows modern buildings at 142 Ostpreußendamm. The address directory also lists the inhabitants of Groß-Lichterfelde by street, and there we find Dr. Klein &lt;a href=&quot;https://digital.zlb.de/viewer/readingmode/34115316_1897/2830/&quot; rel=&quot;nofollow noreferrer&quot;&gt;again&lt;/a&gt; under the heading &quot;Berlinerstr.&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;142 &lt;strong&gt;E.&lt;/strong&gt; Klein, A., Dr., Verlagsbuchhdl.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The word &quot;Verlagsbuchhändler&quot; (here abbreviated) indicates that Dr. Klein was a publisher in addition to being a bookseller. I am reasonably sure the &quot;E.&quot; indicates that Dr. Klein was the &quot;Eigentümer&quot; of this house, that is, he owned this property. The &lt;a href=&quot;https://digital.zlb.de/viewer/readingmode/34115316_1902/3506/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1902&lt;/a&gt; and &lt;a href=&quot;https://digital.zlb.de/viewer/readingmode/34115495_1904/3910/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt; editions of the address directory show Dr. Klein still residing at this address. The 1905 &lt;a href=&quot;https://books.google.com/books?id=CAw3AQAAMAAJ&amp;amp;pg=PA1456-IA1&amp;amp;lpg=PA1456-IA1&amp;amp;dq=%22Berliner+Str.+142%22+lichterfelde&amp;amp;source=bl&amp;amp;ots=HVnxLL5FvX&amp;amp;sig=WIhXUD06rLGJSAA3ZIgTfJYAVBU&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiGsJy5xMbeAhXBIDQIHTvkDM4Q6AEwBXoECAQQAQ#v=onepage&amp;amp;q=%22Berliner%20Str.%20142%22%20lichterfelde&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;member directory&lt;/a&gt; of the Association of German Engineers (VDI = &quot;Verein Deutscher Ingenieure&quot;) shows someone else residing at this address:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Otto Bernhardt, Ingenieur bei Bechem &amp;amp; Post, Gr.-Lichterfelde, Berliner Str. 142&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Considering the lead time for the creation of printed directories, one might reasonably &lt;em&gt;assume&lt;/em&gt; that Dr. Klein died between late 1903 and late 1904. Berlin death records (&quot;Sterberegister&quot;) are accessible via &lt;code&gt;ancestry.com&lt;/code&gt;, which requires membership for access to records. A search indicates that they have a death record for an Adolf Klein from 1903. This seems like a promising lead for establishing a definite date of death, as well as confirmation of the day of birth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The autobiographical notes quoted in Francois Ziegler's answer seem to be partially in error or possibly fictitious. Dr. Klein's wife is said to hail from &quot;Thorn an der Weser&quot;, but best I can determine, there has never been such a town. There was however a &quot;Thorn an der Weichsel&quot; (modern-day &lt;a href=&quot;https://en.wikipedia.org/wiki/Toru%C5%84&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toruń&lt;/a&gt; in Poland) in the relevant time frame. A publication &quot;Deutsches Frauenblatt&quot; seems to have existed, based on a single relevant reference I was able to locate &lt;a href=&quot;https://books.google.com/books?id=nsc0AQAAMAAJ&amp;amp;pg=PA552&amp;amp;lpg=PA552&amp;amp;dq=Engelken,+H.,+Deutsche+Lungenheilst%C3%A4tten.+Deutsches+Frauenblatt.&amp;amp;source=bl&amp;amp;ots=gmOJIPkujD&amp;amp;sig=W33tKiIdXZtkaRUotd-6Wjkl4x4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjBopD2kb7eAhWIA3wKHZnOAX8Q6AEwAHoECAkQAQ#v=onepage&amp;amp;q=Engelken%2C%20H.%2C%20Deutsche%20Lungenheilst%C3%A4tten.%20Deutsches%20Frauenblatt.&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Engelken, H., Deutsche Lungenheilstätten. Deutsches Frauenblatt. 1898. No. 1 ff&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If there is an archive of this publication somewhere, it should be straightforward to establish whether Dr. Klein was indeed its editor-in-chief, as the information in the petition claims. I have not been able to establish anything else about &quot;Deutsches Frauenblatt&quot; other than a presumably unrelated publication of the same name was published from 1926 on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the &lt;a href=&quot;http://www.jewishencyclopedia.com/articles/9371-klein-adolph&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jewish Encyclopedia&lt;/a&gt; Dr. Klein received his medical degree from the University of Leipzig in 1851. The university's archive provides a &lt;a href=&quot;https://www.archiv.uni-leipzig.de/geschichte/promotionen-aller-fakultaten-von-1810-bis-1991/&quot; rel=&quot;nofollow noreferrer&quot;&gt;list&lt;/a&gt; of all doctorates (across all faculties) it has awarded since 1810. I cannot find an Adolf Klein or Adolph Klein on that list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the encyclopedia article, Dr. Klein practiced medicine in Königsberg between 1859 to 1870. Checking an &lt;a href=&quot;http://martin-opitz-bibliothek.de/de/elektronischer-lesesaal?action=book&amp;amp;bookId=z01880-1865#lg=1&amp;amp;slide=143&quot; rel=&quot;nofollow noreferrer&quot;&gt;1865 address directory&lt;/a&gt; of Königsberg, I see several people called A. Klein listed, none of whom works in a medical profession.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From what I have been able to piece together, the &quot;Allwohlsbund&quot; mentioned in the autobiographical portrait quoted in Francois Ziegler's answer was an organization advocating for land reform. Founded by &lt;a href=&quot;https://books.google.com/books?id=i7q8c22XlPQC&amp;amp;pg=PA531&amp;amp;lpg=PA531&amp;amp;dq=%22Allwohlsbund%22+%22Theodor+Stamm%22&amp;amp;source=bl&amp;amp;ots=lJOi-HGmqz&amp;amp;sig=YbvRfbdfd4QcQ51H_8me4YzgDWo&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiL89HOxcjeAhVEM3wKHR1rBm8Q6AEwAHoECAEQAQ#v=onepage&amp;amp;q=%22Allwohlsbund%22%20%22Theodor%20Stamm%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theodor Stamm&lt;/a&gt;, it existed between 1888 and 1893. For a while it was led by the progressive Berlin industrialist Heinrich Freese. I have been unable to find any information about other board members or directors. I am therefore unable to verify Dr. Klein's claimed involvement with this organization at this time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found a reference to one of Dr. Klein's books in &lt;a href=&quot;https://books.google.com/books?id=DU8KAQAAIAAJ&amp;amp;pg=PA86&amp;amp;lpg=PA86&amp;amp;dq=%22Adolf+klein%22&amp;amp;source=bl&amp;amp;ots=dsIE76xsm9&amp;amp;sig=mIGbGDDz2kDIOAZNyXVdgafe5XQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjb-Ob3gL_eAhUSHDQIHaKJAywQ6AEwEXoECAcQAQ#v=onepage&amp;amp;q=%22Adolf%20klein%22%20%22juedische%20gemeinde&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;a publication&lt;/a&gt; of The New York Public Library, &quot;List of Works Relating to the History and Condition of the Jews in Various Countries,&quot; (New York 1914), p. 86:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Socialpolitische Beiträge zur Judenfrage in Deutschland. Aus dem praktischen Leben der Provinz und Grossstadt geschildert, Berlin O.: Adolf Klein [188-?]. 44 p. 8°&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A PDF file containing a scan of the entire book can be downloaded from the &lt;a href=&quot;https://digital.slub-dresden.de/werkansicht/dlf/101651/2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saxon State and University Library Dresden&lt;/a&gt;. Reading the foreword, I found that the author expresses strong antisemitic sentiments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found a summary of a 1897 talk given by a Dr. med. Ad. Klein in a &lt;a href=&quot;http://www.iapsop.com/archive/materials/ubersinnliche_welt/ubersinnliche_welt_v5_n4_apr_1897.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;publication&lt;/a&gt; dealing with the occult. &quot;Die Übersinnliche Welt.&quot; Vol. 5, No. 4, April 1897: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Am 10. d. M. hielt in der Wissenschaftlichen Vereinigung &quot;Sphinx&quot; zu Berlin, Unter den Linden 27 (&quot;Hopfenblüthe&quot;), Dr. med. Ad. Klein einen Vortrag über das Thema: &quot;Mystik und Naturwissenschaft&quot; als Einführung in das Gebiet der Physiologie des Magischen. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I cannot tell whether this would be the same Adolf Klein the asker is researching. The same goes for a &lt;a href=&quot;https://www.antiquariatlange.de/kataloge/katalog-5/5-spiritismus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;reference&lt;/a&gt; to an article published in a spiritistic publication:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Adolf Klein: &quot;Über Lebensentstehung&quot;. &lt;em&gt;Zeitschrift für Spiritismus und verwandte Gebiete&lt;/em&gt;. Vol. 3, Leipzig 1899.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2018-11-10T00:12:20.567" LastActivityDate="2018-11-10T00:12:20.567" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7924" PostTypeId="2" ParentId="7922" CreationDate="2018-11-05T21:50:36.053" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Laguerre_plane&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laguerre geometry&lt;/a&gt; is the geometry of Laguerre plane, and it was introduced by Laguerre, who defined it as the geometry of the oriented lines and circles in the real Euclidean plane (modern definition is as an incidence geometry of parabolas and lines). It was studied classically using the so-called cyclographic mapping, which can indeed be defined via projections involving intersections of right angled cones with a plane. &lt;a href=&quot;https://books.google.com/books?id=6ZrqcYKgtE0C&amp;amp;pg=PA366&amp;amp;lpg=PA366&amp;amp;dq=%22La+Guerre%22+geometry,&amp;amp;source=bl&amp;amp;ots=N_jK9Kgojb&amp;amp;sig=TD7tIsUyU1Ard05gWXKHBPDCQpc&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiW2I3Tmr7eAhVLWq0KHeRvCwUQ6AEwD3oECAkQAQ#v=onepage&amp;amp;q=%22La%20Guerre%22%20geometry%2C&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computational Line Geometry by Pottmann and Wallner, p.366ff&lt;/a&gt; gives a description of the method, and references to classical books of Coolidge and Muller-Krames, and to modern applications in computer assisted geometric design.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-11-05T21:50:36.053" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7925" PostTypeId="1" CreationDate="2018-11-04T02:41:51.207" Score="4" ViewCount="1302" Body="&lt;p&gt;What is the historical origin of the size of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Coulomb&quot; rel=&quot;nofollow noreferrer&quot;&gt;coulomb&lt;/a&gt; (and in turn, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ampere&quot; rel=&quot;nofollow noreferrer&quot;&gt;ampere&lt;/a&gt;)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Currently (pardon the pun), the coulomb is defined &lt;a href=&quot;https://physics.stackexchange.com/questions/70651/why-is-the-ampere-a-base-unit-and-not-the-coulomb&quot;&gt;in terms of the ampere&lt;/a&gt;. The ampere is in turn effectively defined by the vacuum permeability, which is exactly defined as &lt;span class=&quot;math-container&quot;&gt;$\mu_0 = 4 \pi \times10^{-7} H / m$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Giancoli, this exact value of &lt;span class=&quot;math-container&quot;&gt;$\mu_0$&lt;/span&gt; was chosen because it very closely matched the value of &lt;span class=&quot;math-container&quot;&gt;$\mu_0$&lt;/span&gt; when measured with an independently defined ampere (which, I assume, was defined as &lt;span class=&quot;math-container&quot;&gt;$1 C/s$&lt;/span&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;None of this, however, answers my question: why is the coulomb the size it is today, an outrageous size that makes it impractical for direct use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I'm more interested in the historical reasoning that gave the ampere and coulomb the sizes they have today, not the current definitions.&lt;/p&gt;&#xA;" OwnerUserId="16004" OwnerDisplayName="theunamedguy" LastEditorUserId="3643" LastEditDate="2018-11-06T17:03:45.073" LastActivityDate="2018-11-06T17:03:45.073" Title="Origin of the coulomb and ampere" Tags="|physics|electricity|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7926" PostTypeId="2" ParentId="7925" CreationDate="2018-11-04T05:12:32.150" Score="0" Body="&lt;p&gt;Wikipedia’s “Ampere” article has a History section which explains how the ampere was originally defined:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“The ampere was originally defined as one tenth of the unit of electric current in the centimetre–gram–second system of units. That unit, now known as the abampere, was defined as the amount of current that generates a force of two dynes per centimetre of length between two wires one centimetre apart. The size of the unit was chosen so that the units derived from it in the MKSA system would be conveniently sized.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the numbers in this definition, other than 1, are 1/10 and 2, so the definition was reasonably natural. The coulomb was simply an ampere-second.&lt;/p&gt;&#xA;" OwnerDisplayName="G. Smith" LastActivityDate="2018-11-04T05:12:32.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7927" PostTypeId="2" ParentId="7925" CreationDate="2018-11-04T05:15:11.367" Score="0" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Ampere&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampere&lt;/a&gt; of wikipedia, SI defines ampere as follows: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to &lt;span class=&quot;math-container&quot;&gt;$2\times10^{-7}$&lt;/span&gt; newtons per metre of length.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Generally, according to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère's force law&lt;/a&gt;, for two straight parallel conductors of infinite length with current &lt;span class=&quot;math-container&quot;&gt;$I_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$I_2$&lt;/span&gt; and separation &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;, the force felt by a part of conductor with length &lt;span class=&quot;math-container&quot;&gt;$\Delta L$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$F = \frac{\mu_0 I_1 I_2 \Delta L}{2\pi d}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say the value of &lt;span class=&quot;math-container&quot;&gt;$\mu_0$&lt;/span&gt; is defined by the definition of Ampere.&lt;/p&gt;&#xA;" OwnerDisplayName="Eric Yang" LastActivityDate="2018-11-04T05:15:11.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7928" PostTypeId="2" ParentId="7925" CreationDate="2018-11-04T10:04:20.877" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;my question: why is the coulomb (and ampere) the size it is today, which an outrageous size that makes it impractical for direct use?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The coulomb has that value because in the mid 19th century electrical engineers needed practical units for submarine cables and telegraphy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1861 a committee of the British Association for the Advancement of Science was appointed to propose a system of units that included electrical and mechanical units. The committee defined two coherent systems for scientists, called electromagnetic and electrostatic units, and added several &quot;practical&quot; units that were decimally compatible with the electromagnetic units, for the electrical engineers involved in telegraphy and submarine cables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1861 the electrical engineer Latimer Clark suggested to the committee that practical voltages were in the range of 1 to 10^6 volt, resistances of conductors and insulators were in the range of 1 to 10^8 ohm, and the smallest current was about 10^-3 ampere (translated to modern units). He proposed names like volt and ohm for these practical units, and the prefix mega for 10^6. The unit of charge was unimportant for submarine cables and telegraphy, so nobody cared about its practical value, it merely had to be coherent with the unit of current.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1881 the International Metre Convention adopted the practical units and their names: volt, ohm, and ampere. These practical units are the ones that scientists and engineers are using today. &lt;/p&gt;&#xA;" OwnerUserId="2667" OwnerDisplayName="jkien" LastActivityDate="2018-11-04T10:53:48.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7929" PostTypeId="2" ParentId="7919" CreationDate="2018-11-06T01:27:39.973" Score="3" Body="&lt;p&gt;It is obvious geometrically when one duplicates the number of sides, but not in general. Archimedes considered up to 96-gons for his approximation of the ratio of the circumference to diameter, but he gets to 96 by successively duplicating sides of the regular triangle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar (and somewhat stronger) proposition, &quot;&lt;em&gt;for regular polygons with the same perimeter, more sides imply greater area&lt;/em&gt;&quot;, is (presumably) proved by &lt;a href=&quot;https://en.wikipedia.org/wiki/Zenodorus_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zenodorus&lt;/a&gt; in On Isoperimetric Figures, the earliest known treatise on the isoperimetric problem. Unfortunately, it is not extant, and we only know about it from Theon’s commentary on Ptolemy’s Almagest, and Book V of Pappus’s Collection where some ideas of the proof are given. See &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/blasjo526.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blasjo's Isoperimetric Problem&lt;/a&gt; for a reconstruction.&#xA;&lt;a href=&quot;https://books.google.com/books?id=_poUuMFvA3oC&amp;amp;pg=PA273&amp;amp;lpg=PA273&amp;amp;dq=isoperimetric+problem+archimedes+simplicius+aristotle+zenodorus&amp;amp;source=bl&amp;amp;ots=xH-Nu2AayC&amp;amp;sig=IS71vG74wKdmqDkkts7JiAUsdK0&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjfrfjzscDeAhUGRK0KHSPdB98Q6AEwAXoECAMQAQ#v=onepage&amp;amp;q=isoperimetric%20problem%20archimedes%20simplicius%20aristotle%20zenodorus&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knorr in The Ancient Tradition of Geometric Problems, p.273&lt;/a&gt; speculates that the proposition was already known to Archimedes, ingredients of a possible proof do occur in the Sand Reckoner. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To get the OP proposition about the inscribed regular polygons one would need to show that their perimeters increase with the number of sides in addition to the Zenodorus's lemma. This can also be proved, in principle, using the inequalities alluded to in the Sand Reckoner.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-11-08T00:57:42.223" LastActivityDate="2018-11-08T00:57:42.223" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7930" PostTypeId="2" ParentId="7861" CreationDate="2018-11-06T14:16:25.270" Score="3" Body="&lt;p&gt;The 1905 textbook &lt;a href=&quot;https://books.google.com/books?id=z5POAAAAMAAJ&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Lehrbuch der Mathematik: für Studierende der Naturwissenschaften und der Technik, Einführung in die Differential-und Integralrechnung und die Analytische Geometrie&lt;/em&gt;&lt;/a&gt; by Georg Scheffers has this zinger on p.550:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Schon hieraus erhellt, daß das Integrieren einen ganz anderen Charakter als das Differenzieren hat. Dies letztere ließe sich als eine Art Handwerk bezeichnen, das Integreieren demgegenüber als eine Kunst...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Which means something like: From this it is clear that integration is unlike  diffentiation in kind.  The latter can be considered a kind of craft, but in contrast integration is an art...)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scheffer's book looks like all calculus books, and it is hard to believe he originated the trope. &lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2018-11-06T14:16:25.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7931" PostTypeId="1" AcceptedAnswerId="7966" CreationDate="2018-11-07T15:45:34.130" Score="1" ViewCount="266" Body="&lt;p&gt;My question refers to some not very well known fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the interior of the ellipse to the interior of the unit circle. These fragments date from 1839, much later then his better-known article from 1822 on conformal mappings. I guess the mathematics involved here is connected somehow with the later Riemann's mapping theorem, which states that every simply-connected subset of the complex plane is &quot;conformally equivalent&quot; to the interior of unit circle. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relevant pages from Gauss's nachlass are in volume 10-1, p. 311-320. Gauss's formulas in this fragment are interesting since they seem to be very complex - this stands in contrast to the reltively simple solution to this problem using the &quot;Joukowsky transform&quot; (as can be seen from this post: &lt;a href=&quot;https://math.stackexchange.com/questions/1582608/conformally-mapping-an-ellipse-into-the-unit-circle&quot;&gt;https://math.stackexchange.com/questions/1582608/conformally-mapping-an-ellipse-into-the-unit-circle&lt;/a&gt;). I even found (in the last pages) some formulas that look like theta functions. It's noteworthy that in his essay, Schlesinger comments on this fragment of Gauss at p. 192, so i think it might be of importance (i didn't find comments about it the literature). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So just to be clear, i ask this question simply because i don't understand the mathematical framework of this problem. I think that unlike other questions i asked before (like the question on Gauss's &quot;conception of braids&quot;), this question can be answered by a mathematician who is not an historian, but has enough knowledge in the theory of conformal mappings. So i hope for a good answer!   &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-21T18:40:52.590" LastActivityDate="2019-10-21T18:40:52.590" Title="Explanation of Gauss's late fragments dealing with &quot;the conformal image of the ellipse&quot;" Tags="|biographical-details|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7933" PostTypeId="1" CreationDate="2018-11-08T09:46:51.677" Score="6" ViewCount="574" Body="&lt;p&gt;I know Euclid's Definition VII.1 of a unit only from English and German translations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A unit is (that) according to which each existing (thing) is said (to&#xA;  be) one. &lt;br&gt;&lt;sub&gt;[translation by &lt;a href=&quot;http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf#page=194&quot; rel=&quot;noreferrer&quot;&gt;Fitzpatrick]&lt;/a&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A unit is that by virtue of which each of the things that exist is&#xA;  called one.&lt;br&gt;&lt;sub&gt;[translation by &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0086%3Abook%3D7%3Atype%3DDef%3Anumber%3D1&quot; rel=&quot;noreferrer&quot;&gt;Heath]&lt;/a&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Eine Einheit ist etwas, das zu einer solchen bestimmt&#xA;  ist.&lt;br&gt;&lt;sub&gt;[translated by &lt;a href=&quot;http://www.opera-platonis.de/euklid/Euklid_Stoicheia.pdf#page=167&quot; rel=&quot;noreferrer&quot;&gt;Haller&lt;/a&gt;: &lt;em&gt;A unit is something that is destined as such.&lt;/em&gt; ]&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But it seems so that what Euclid defines is &lt;strong&gt;the&lt;/strong&gt; unit instead of &lt;strong&gt;a&lt;/strong&gt; unit. At least Joyce in his &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/elements/bookVII/defVII1.html&quot; rel=&quot;noreferrer&quot;&gt;commentary on Definition VII.1&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Definition 1 is meant to define the unit as 1.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From his commentary on Definition VII.2:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Definition 2 says that a number is a multitude of units.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;one could conclude that a number was thought of as a multitude of &quot;&lt;em&gt;copies&lt;/em&gt;&quot; of the unit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can one reconstruct how Euclid actually did think about the concept of &quot;unit&quot;: Did he think, there is one distinguished unit, and numbers were multitudes of copies of the unit? If so: Why did he define &quot;&lt;strong&gt;a&lt;/strong&gt; unit&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If he not thought of one distinguished unit: How did he conceive other units, what did distinguish different units for him?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that it's definitely not a matter of translation, because &lt;span class=&quot;math-container&quot;&gt;$\mu o\nu\acute{\alpha}\varsigma$&lt;/span&gt; (monas) has no article which means is deliberately &lt;a href=&quot;https://en.wikipedia.org/wiki/Ancient_Greek_grammar#Definite_article&quot; rel=&quot;noreferrer&quot;&gt;indefinite&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UUnTk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UUnTk.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Note, that in modern terms, &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Unit_(ring_theory)&quot; rel=&quot;noreferrer&quot;&gt;unit&lt;/a&gt;&quot; is defined as an invertible element of a (unital) ring, as opposed to &quot;unity&quot; (as in &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Root_of_unity&quot; rel=&quot;noreferrer&quot;&gt;root of unity&lt;/a&gt;&quot;) or just &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. Of course, the unity is a unit, but in general not every unit is the unity (only in some rings).&lt;/p&gt;&#xA;" OwnerUserId="6422" LastActivityDate="2018-11-08T20:12:29.557" Title="Why did Euclid define &quot;a unit&quot; instead of &quot;the unit&quot;?" Tags="|terminology|ancient-greece|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7935" PostTypeId="2" ParentId="7933" CreationDate="2018-11-08T20:12:29.557" Score="5" Body="&lt;p&gt;The problem again is that the modern framing of mathematical concepts was completely different from ancient Greek one. Today mathematical objects and entities are seen as abstract, to Greeks they were &lt;em&gt;much&lt;/em&gt; more concrete. The abstract idea of a &lt;em&gt;single&lt;/em&gt; unit did not even arise in their context, and hence there was nothing to make &quot;copies&quot; of. There is a unit for each sort of thing, &quot;&lt;em&gt;How many pairs of socks?' has a different answer from 'how many socks?'&lt;/em&gt;&quot;, as Stein puts it, but they are abstracted from what is already there, not copied from a single template. In modern terms, it is better to think of Greek units as set elements of a pre-given concrete set than universal abstractions that &quot;count&quot; them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even today's notion of &quot;abstraction&quot; itself is far more abstract (pun intended) than it was to Greeks. To Aristotle, Euclid, et al., it meant selective attention to concrete things, and even Plato's forms were forms of concrete things with causal powers. Not disembodied causally inert ghosts, like modern numbers or equivalence classes, at the service of mathematicians by the stroke of a definition. One could say that Greek units &lt;em&gt;are&lt;/em&gt; individuals on which the mind did some work of selective attention (disregarding texture and color, for example). It is for the same reason that the idea of assigning numbers (multitudes of units) to lines or figures did not make sense either. Here are some excerpts from &lt;a href=&quot;http://strangebeautiful.com/other-texts/stein-eudoxos-dedekind.pdf&quot; rel=&quot;noreferrer&quot;&gt;Stein's Eudoxos and Dedekind&lt;/a&gt; that discusses the differences at length:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;According to Aristotle, the objects studied by mathematics have no independent existence, but are separated in thought from the substrate in which they exist, and treated as separable - i.e., are &quot;abstracted&quot; by the mathematician. In particular, numerical attributives or predicates (which answer the question 'how many?') have for &quot;substrate&quot; multitudes with a designated unit. 'How many pairs of socks?' has a different answer from 'how many socks?'. (Cf. Metaph. XIV i 1088a5ff.: &quot;One&#xA;  signifies that it is a measure of a multitude, and number that it is a&#xA;  measured multitude and a multitude of measures&quot;.)... There is perhaps some ambiguity in the quoted passage: the statement, &quot;Number signifies that it is a measured multitude&quot;, might be taken either to identify numbers with finite sets, or to imply that the subjects numbers are predicated of are finite sets. Euclid's definition - &quot;a number is a multitude composed of units&quot; - points to the former reading (which implies, for example, that there are many two's - a particular knife and fork being one of them).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;And indeed one finds, in Euclid's arithmetic and geometry, that &quot;sameness&quot; is never predicated of numbers, lengths, areas, volumes, or angles: ratios, for example, of two areas on the one hand, two lengths on the other, are (in appropriate circumstances) said to be &quot;the same&quot; - but never &quot;equal&quot;; on the other hand, the areas of two figures are said to be &quot;equal&quot;, but never &quot;the same&quot; (indeed, most often it is simply said that &quot;the two figures are equal&quot; - that area is the appropriate magnitude-kind is taken to be understood). This difference is quite alien to our present way of thinking about such matters: for us, to say that two distinct triangles are equal in area is to say that they have &quot;the same area&quot;. But on the suggested reading of the Greek terminology, it would be incorrect to speak of &quot;the area of this triangle&quot;: a triangle does not have an area, it &lt;strong&gt;is&lt;/strong&gt; an area - that is, a finite surface; &lt;strong&gt;this area&lt;/strong&gt; means &lt;strong&gt;this figure&lt;/strong&gt;, and the two distinct triangles are two different, but equal, areas.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Thus, we may say that each species of quantity (whether discrete or continuous) is distinguished in Greek mathematics by its own proper equivalence-relation, called in each case just &quot;equality&quot;; and that where our own practice is to proceed to the corresponding equivalence classes, regarding these as particulars (numbers, lengths, etc.), the Greeks did not, in principle, make this abstraction.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-11-08T20:12:29.557" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7936" PostTypeId="1" AcceptedAnswerId="7939" CreationDate="2018-11-09T03:23:57.150" Score="3" ViewCount="328" Body="&lt;p&gt;Apparently there was some negative reception of Boltzmann's idea of an &quot;atom&quot;. I assume the mathematics used by Einstein in his publication did not use any of Boltzmann's statistical mathematics otherwise Boltzmann would have know about it.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein calculated the size of the atom as well as it's existence in his publication of Brownian motion.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boltzmann apparently passed away a year after he had been vindicated by Albert Einstein.   Was Boltzmann aware of this at the time that he passed away? &lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-06-07T18:44:56.547" LastActivityDate="2020-06-07T18:44:56.547" Title="Did Ludwig Boltzmann read Albert Einstein's publication published on Brownian motion one year before Boltzmann passed away?" Tags="|biographical-details|thermodynamics|atomic-theory|physicists|boltzmann|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7937" PostTypeId="2" ParentId="5829" CreationDate="2018-11-09T06:36:49.667" Score="1" Body="&lt;p&gt;If we write &lt;span class=&quot;math-container&quot;&gt;$0_x$&lt;/span&gt; then it looks like a special kind of 0, since the 0 is above the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Baseline_(typography)&quot; rel=&quot;nofollow noreferrer&quot;&gt;baseline&lt;/a&gt; for typography. So we would expect &lt;span class=&quot;math-container&quot;&gt;$0_x+1_x=1_x$&lt;/span&gt; etc. With the &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; notation it is clear that &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; is a special kind of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, or at any rate the same kind of thing as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="144" LastActivityDate="2018-11-09T06:36:49.667" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7938" PostTypeId="1" CreationDate="2018-11-09T10:59:47.263" Score="11" ViewCount="1260" Body="&lt;p&gt;The Romans did not have Indian numerals. Worse still, they did not have the decimal system.  Yet, they produced amazing works of engineering  and architecture. How was that possible? It's troublesome to make simple sums, but how could they make products and complex calculations?&lt;/p&gt;&#xA;&lt;p&gt;Is there any math textbook that survived to tell us how they made complex calculations?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="4251" LastEditDate="2021-09-19T07:12:44.797" LastActivityDate="2021-09-19T16:48:30.577" Title="How did Romans do multiplications?" Tags="|mathematics|reference-request|roman-empire|antiquity|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7939" PostTypeId="2" ParentId="7936" CreationDate="2018-11-09T13:15:28.280" Score="5" Body="&lt;p&gt;I’m afraid he no longer cared. According to Boltzmann biographer E. Broda (&lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=651515&quot; rel=&quot;noreferrer&quot;&gt;1981&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;//doi.org/10.5170/CERN-1981-010&quot; rel=&quot;noreferrer&quot;&gt;p. 9&lt;/a&gt;): One might have thought Boltzmann would, after 1900, in lectures and writings refer to Planck’s work on radiation, made possible by adoption of his own, Boltzmann’s, statistical methods. This was not the case, however. While Boltzmann continued to lecture and publish till his tragic death in 1906, we find no word about Planck or about his work. Nor did Boltzmann refer to Planck in his classes, as told by his student Lise Meitner&lt;sup&gt;49)&lt;/sup&gt;. By the way, Boltzmann did not refer to Einstein either&lt;sup&gt;49)&lt;/sup&gt;, although Einstein’s explanation, in 1905, of Brownian movement as a fluctuation phenomenon, to which we shall return later, had been another triumph of Boltzmann’s ideas. It looks as if Boltzmann had quite systematically shut himself up in his own world (...)&lt;/p&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;//doi.org/10.5170/CERN-1981-010&quot; rel=&quot;noreferrer&quot;&gt;p. 15&lt;/a&gt;): Independently of Einstein, Smoluchowski developed the quantitative theory of Brownian movement on the basis of fluctuations. A curious fact that does not seem to have been discussed so far: Smoluchowski, who had done his thesis under Stefan, but called himself a student of Boltzmann, had been interested in Brownian movement, to be explained through fluctuations, since 1900. That was 6 years before Boltzmann’s death. Is it possible that Boltzmann was not aware of Smoluchowski’s promising investigations?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;49) L. Meitner, Looking back, Bull. Atom. Scient. &lt;strong&gt;20&lt;/strong&gt;, No. 9, 2 (&lt;a href=&quot;//doi.org/10.1080/00963402.1964.11454713&quot; rel=&quot;noreferrer&quot;&gt;1964&lt;/a&gt;).&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="658" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-11-09T13:15:28.280" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7940" PostTypeId="2" ParentId="7938" CreationDate="2018-11-09T14:35:05.597" Score="9" Body="&lt;p&gt;They used abacus. The techniques used for operations with abacus were understood and were basically the same used also until quite recent time also in China and Japan, as far as I know. This does not require indoarabic numerals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact you shouldn't think &quot;Latins&quot;. Latin numerals were in use in Europe till the XIIIth century at least.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Liber Abaci&lt;/em&gt; by Leonardo Fibonacci is credited as being one of the main sources of introduction of indo-arabic numerals in the Western countries and dates back to 1228.&#xA;One of the reason of its success is exactly that it explained how using such numerals could improve computations. It is, in fact, a Book of Abacus, as the name says, i.e. a book centered around techniques for algebraic computations.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2018-11-09T14:35:05.597" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7941" PostTypeId="2" ParentId="7938" CreationDate="2018-11-09T14:55:39.453" Score="8" Body="&lt;p&gt;You should be more specific when you say &quot;Romans&quot;. If you mean ancient Romans, almost no mathematical text survived in Latin from the times before 2nd century AD.&#xA;From the Roman empire we mostly have Greek texts. &#xA;(See also &lt;a href=&quot;https://hsm.stackexchange.com/questions/2043/roman-engineers&quot;&gt;Roman engineers&lt;/a&gt;). Almost all technical literature which we have from the Roman empire is written in Greek. Greek was also the spoken language of large portion of the empire.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Greek system is described in detail in the book of van der Waerden, Science awakening.&#xA;The digits of the decimal system were denoted by Greek letters. One had to memorize&#xA;(as we do) the multiplication table for digits. That's all one need to multiply numbers. For example,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$265\times265=200\times200+200\times60+200\times5+60\times200+60\times60$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$+60\times5+5\times200+5\times60+5\times5=70225.$$&lt;/span&gt;&#xA;To simplify the task, a counting board was used with counting stones.&#xA;Such a counting board in mentioned in Polybius, for example. &#xA;For computations with simple fractions a more complicated algorithm was used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For astronomical computations, Babylonian positional system with base 60,&#xA;including fractions based on 1/60th was used (but the numbers from 1 to 60 were denoted by pairs of Greek letters). Multiplication tables were used (as Babylonians did before). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-11-09T15:06:31.397" LastActivityDate="2018-11-09T15:06:31.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7942" PostTypeId="1" CreationDate="2018-11-10T09:33:49.207" Score="1" ViewCount="277" Body="&lt;p&gt;Pythagoras  introduced the multiplication table in Southern Italy about 500 BC, do we know how it looked like?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not mean the so called pytagorean/multiplication/times table but the &lt;strong&gt;actual table 'invented' by Pythagoras&lt;/strong&gt; about 500 BC&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="7300" LastEditDate="2018-11-11T09:46:12.000" LastActivityDate="2018-11-11T09:46:12.000" Title="What pythagorean table looked like?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7943" PostTypeId="1" AcceptedAnswerId="7952" CreationDate="2018-11-10T09:55:05.430" Score="16" ViewCount="800" Body="&lt;p&gt;The history of physics is full of examples of phenomena that used to be described independently, until additional insight proved they were the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some famous instances are&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;motion of bullets and motion of stars and planets, until Newton unified Galileo and Kepler visions&lt;/li&gt;&#xA;&lt;li&gt;mechanical work and heat each had their own unit (Joules and calories) until it was realized heat is just another form of energy&lt;/li&gt;&#xA;&lt;li&gt;electricity and magnetism, until Maxwell unified Gauss and Ampere's laws&lt;/li&gt;&#xA;&lt;li&gt;electromagnetism and light propagation&lt;/li&gt;&#xA;&lt;li&gt;spin and special relativity until Dirac's equation&lt;/li&gt;&#xA;&lt;li&gt;etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Throughout history, these unifying concepts have shaped the way we see the world, and it certainly seems to me as if the general trends is towards more unification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is the reverse process also an observed trend, though less famous? In other words, &lt;strong&gt;what are significant advances that have been made in the natural sciences from realizing that concepts that used to be thought the same were actually distinct?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="8086" LastActivityDate="2019-03-09T12:22:16.583" Title="What are natural science concepts that were once thought the same, but grew to be distinguished?" Tags="|discoveries|examples|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7944" PostTypeId="1" AcceptedAnswerId="7947" CreationDate="2018-11-10T15:15:21.037" Score="5" ViewCount="330" Body="&lt;p&gt;Why do we call Chinese monoid &quot;Chinese&quot;? Why not &quot;American&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find the definition of Chinese monoid from Wikipedia. &lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_monoid&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chinese_monoid&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8087" LastActivityDate="2018-11-11T15:47:02.120" Title="Why do we call Chinese monoid &quot;Chinese&quot;? Why not &quot;American&quot;?" Tags="|mathematics|abstract-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7947" PostTypeId="2" ParentId="7944" CreationDate="2018-11-11T12:47:19.153" Score="8" Body="&lt;p&gt;I do not have the paper &quot;Plactic-growth-like monoids&quot; at hand at the moment but the history is correctly rendered in &#xA;[J. Cassaigne, M. Espie, F. Hivert, D. Krob, J.C. Novelli, The chinese monoid, International Journal of Algebra and Computation, 11, (3), 301-334, 2001]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was a question of M.-P. Schützenberger (MPS) to Krob and myself to prove (maybe at the cost of some additional conditions to be found), that only the Plactic Monoid had the same (multivariate) Poincaré-Hilbert series (PH-series) as itself. The first task was then to study the monoids (with multihomogeneous relators) having the same &#xA;PH-series as the plactic. We found three series of ternary monoids. Now, if you arrange their relators along the hexagon formed by the permutations of &quot;abc&quot;, one and only one of these series has the form of a triangle evoking the Chinese hat (here a triangle with angles of 30, 30 and 120 degrees). MPS used to call it &quot;The Chinese hat monoid&quot; which was eventually abbreviated as &quot;The Chinese monoid&quot;.     &lt;/p&gt;&#xA;" OwnerUserId="2793" LastEditorUserId="2793" LastEditDate="2018-11-11T15:47:02.120" LastActivityDate="2018-11-11T15:47:02.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7948" PostTypeId="1" CreationDate="2018-11-11T19:15:37.727" Score="0" ViewCount="130" Body="&lt;p&gt;Who chose the term &quot;model&quot; in model theory?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was their reason for choosing the word &quot;model&quot; to mean what it means now in model theory?&#xA;The current meaning: &quot;[interpretation] I is a &lt;em&gt;model&lt;/em&gt; of [sentence] S&quot; means &quot;interpretation I happens to make [sentence] S state something true&quot; (Wilfrid Hodges on SEP). &lt;a href=&quot;https://plato.stanford.edu/entries/model-theory/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://plato.stanford.edu/entries/model-theory/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In everyday English, &quot;X models Y&quot; means &quot;X is a simplification of Y&quot;, but the &quot;model&quot; in model theory doesn't seem to come from that.&lt;/p&gt;&#xA;" OwnerUserId="8091" LastActivityDate="2018-11-12T12:56:33.100" Title="What motivated the choice of the word &quot;model&quot; in model theory?" Tags="|terminology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7949" PostTypeId="2" ParentId="7948" CreationDate="2018-11-11T19:57:46.143" Score="2" Body="&lt;p&gt;The first appearance of the term “model” as name for an interpretation that satisfies a set of formulas (or theory) is &lt;a href=&quot;https://en.wikipedia.org/wiki/John_von_Neumann&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Neumann&lt;/a&gt;'s &lt;a href=&quot;https://books.google.it/books?id=v4tBTBlU05sC&amp;amp;pg=PA346&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eine Axiomatisierung der Mengenlehre (1925)&lt;/a&gt;, §4. MODELS OF SET THEORY.&lt;/p&gt;&#xA;&lt;p&gt;It seems that the new term owes its success to &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Weyl&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Weyl&lt;/a&gt;’s &lt;a href=&quot;http://krishikosh.egranth.ac.in/bitstream/1/2028666/1/G5751.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philosophy of Mathematics and Natural Science (1st German ed., 1927)&lt;/a&gt;, page 20 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An axiom system must under all circumstances be free from contradictions, in which case it is called &lt;em&gt;consistent&lt;/em&gt; [...]. Not indispensable but desirable is the &lt;em&gt;independence&lt;/em&gt; of the individual axioms of an axiom system.&lt;/p&gt;&#xA;&lt;p&gt;There are three methods at one's disposal of reaching this goal [to establish the independence of a proposition &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; on other propositions &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; (an axiom system)].&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) &lt;em&gt;The construction of a model&lt;/em&gt;. Objects and relations are exhibited which, upon suitable naming, satisfy all of the propositions &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, and yet fail to satisfy &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. This method has been the most successful so far invented. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Klein was the first to offer a Euclidean model for non-Euclidean geometry; the objects of Euclidean geometry itself, upon an assignment of names differing from the customary one, satisfy the non-Euclidean axioms.&lt;/p&gt;&#xA;&lt;p&gt;More than anybody else has Hilbert, through the ingenious construction of suitable arithmetical models, contributed to the clarification of the logical relations that connect the various parts of the geometrical system of axioms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We can find the term used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfred_Tarski&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alfred Tarski&lt;/a&gt; in an address on the foundations of the geometry of solids given to the First Polish Mathematical Congress, held in Lwow in 1927. A summary of the address appeared in French under the title ‘Les fondements de la géométrie des corps’, 1927.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Thus the connection with &amp;quot;representation&amp;quot; is due to the technique of interpreting a geometrical system into a different one.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-11-11T20:33:02.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7951" PostTypeId="1" CreationDate="2018-11-11T21:28:35.567" Score="2" ViewCount="148" Body="&lt;p&gt;Already in antiquity, Greek mathematicians realized that the Earth is round, and the idea was operative in Europe ever since. But how widespread was this belief in the centuries until the Renaissance?&lt;/p&gt;&#xA;&lt;p&gt;It was suggested that my question may be a duplicate of &lt;a href=&quot;https://hsm.stackexchange.com/q/5627&quot;&gt;this question&lt;/a&gt;. That is not so.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastEditorUserId="4251" LastEditDate="2023-06-26T09:50:36.020" LastActivityDate="2023-06-26T09:50:36.020" Title="How widespread was the belief in Europe that the Earth is round until the Renaissance?" Tags="|geography|middle-ages|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="7952" PostTypeId="2" ParentId="7943" CreationDate="2018-11-11T21:37:58.193" Score="13" Body="&lt;p&gt;The two most famous paradigmatic examples of de-unification are phlogiston and ether. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kaluza-Klein theory&lt;/a&gt; of gravity and electromagnetism did not get to spread as far and wide before faltering. The splitting of jade into two distinct minerals, &lt;a href=&quot;https://en.wikipedia.org/wiki/Jade#Nephrite_and_jadeite&quot; rel=&quot;nofollow noreferrer&quot;&gt;nephrite and jadeite&lt;/a&gt;, is a small scale example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phlogiston/caloric theory was able to unify chemical and thermal phenomena in a way that was eliminated by the mechanical theory of heat. Specifically, the processes of heating/cooling were assimilated to chemical reactions with phlogiston (later caloric), and this was one of the key arguments in 18-19th century for preferring this theory. Stahl explained burning and decalcination (formation of metals) by separation of phlogiston from matter in 1703. Black's work in 1757-1764 led to the explanation of latent heat by phlogiston's ability to combine chemically with matter, and of gaseous states as solutions of liquids in phlogiston. Even Lavoisier, who introduced the oxygenation theory of burning in 1780-s, needed a heat fluid (renamed into caloric) to make his theory work. For more details and references see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3470/what-are-the-major-flaws-of-the-caloric-theory-of-heat&quot;&gt;What are the major flaws of the “caloric” theory of heat?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ether offered a unification of wave phenomena with hopes of extending it to an intuitively appealing unified theory of matter. Kelvin's theory of vortex atoms and Lorentz's theory of electrons were steps in that direction. Suffice it to quote &lt;a href=&quot;http://press.princeton.edu/chapters/s6683.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michelson from 1902&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The day seems not far distant when the converging lines from many apparently remote regions of thought will meet... Then the nature of the atoms, and the forces called into play in their chemical union... the explanation of cohesion, elasticity, and gravitation — all these will be marshaled into a single compact and consistent body of scientific knowledge... one of the grandest generalizations of modern science ... that all the phenomena of the physical universe are only different manifestations of the various modes of motion of one all-pervading substance — the ether.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The ether loss was taken particularly hard, it still has its adherents, and even Einstein (1920) mused about reconceptualizing it:&quot;&lt;em&gt;We may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an Aether. According to the general theory of relativity space without Aether is unthinkable...&lt;/em&gt;&quot;. Dirac (1951) suggested identifying it with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Vacuum_state&quot; rel=&quot;nofollow noreferrer&quot;&gt;QED vacuum&lt;/a&gt;:&quot;&lt;em&gt;The Aether is no longer ruled out by relativity, and good reasons can now be advanced for postulating an Aether... We have now the velocity at all points of space-time, playing a fundamental part in electrodynamics. It is natural to regard it as the velocity of some real physical thing.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, de-unifications are examples of what is termed Kuhn loss, benefits of a prior theory that do not carry over to its successors, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/187/examples-of-kuhn-loss&quot;&gt;Examples of Kuhn loss?&lt;/a&gt; The topic is controversial. In recent decades the &lt;a href=&quot;https://plato.stanford.edu/entries/scientific-unity/#Disu&quot; rel=&quot;nofollow noreferrer&quot;&gt;disunity of science&lt;/a&gt; thesis has gained popularity. It is argued that there are insurmountable obstacles to unifying physical and biological concepts, for example, or biological and psychological ones. In &lt;a href=&quot;https://www.jstor.org/stable/20117830&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Conceptual Foundations of Renormalization Theory (1993)&lt;/a&gt; Cao and Schweber describe the effective field theory approach that opposes the &quot;theory of everything&quot; even in physics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This position rejects uncompromisingly the idea successively advanced during the last fifteen years by grand unified theorists, supergravity theorists, and superstring theorists that the development of fundamental physics will end with the discovery of an ultimate, definitive, and conclusive mathematical formalism. Rather, the development is taken as a process of successive extrapolations that is assumed not to have an end, with every step of the extrapolation being justified by a collective reinterpretation of theory and observation before and after the extrapolation&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Presumably, the reinterpretations should be expected to involve de-unifications with previously unified phenomena coming apart at higher resolutions. This effect should be expected even under the most traditional views of the history of science: many early unifications are bound to be premature/superficial, and hence come to be broken in the course of further research.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-01-31T08:54:10.797" LastActivityDate="2019-01-31T08:54:10.797" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7953" PostTypeId="2" ParentId="7948" CreationDate="2018-11-12T12:56:33.100" Score="1" Body="&lt;p&gt;I expect &quot;model&quot; is used in model theory imitating previous uses in mathematics.  Such as for &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane&quot; rel=&quot;nofollow noreferrer&quot;&gt;non-euclidean geometry.&lt;/a&gt;  (the Klein model, the Poincaré disk model, the Poincaré half-plane model, the Lorentz model.)  In the 19th century, the fact that hyperbolic geometry is consistent was shown by exhibiting a model for it inside Euclidean geometry, which was considered to be clearly consistent.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-11-12T12:56:33.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7954" PostTypeId="2" ParentId="7912" CreationDate="2018-11-12T19:53:48.780" Score="2" Body="&lt;p&gt;Some data points from the convoluted history of rotatory-reflection (RR) and its notation:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Auguste_Bravais&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bravais&lt;/a&gt; (a classmate of Galois) was among the first to consider orthogonal symmetry of crystals. His papers (&lt;a href=&quot;//gallica.bnf.fr/ark:/12148/bpt6k9707964q/f153&quot; rel=&quot;nofollow noreferrer&quot;&gt;1849&lt;/a&gt;, p. 144; &lt;a href=&quot;//gallica.bnf.fr/ark:/12148/bpt6k4336880/f4&quot; rel=&quot;nofollow noreferrer&quot;&gt;1850&lt;/a&gt;, p. 93) introduce notation for the rotations (&lt;span class=&quot;math-container&quot;&gt;$\Lambda^n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathrm L^n$&lt;/span&gt; according as the axis is “principal” or not) and also for plane and central reflections (&lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm P$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm C$&lt;/span&gt;). But none for RR, which he deliberately disregards on grounds that the extra crystal class it allows “doesn’t seem to occur in Nature” (&lt;a href=&quot;//gallica.bnf.fr/ark:/12148/bpt6k4336922/f104&quot; rel=&quot;nofollow noreferrer&quot;&gt;1851&lt;/a&gt;, pp. 229–230, 275), though he does mention (p. 237) a shape with such symmetry, viz. &lt;a href=&quot;//archive.org/details/lehrbuchderrein1naum/page/258&quot; rel=&quot;nofollow noreferrer&quot;&gt;Naumann’s sphenoid&lt;/a&gt; (&lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/19067715&quot; rel=&quot;nofollow noreferrer&quot;&gt;a.k.a&lt;/a&gt;. Breithaupt’s &lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/18923909&quot; rel=&quot;nofollow noreferrer&quot;&gt;tetragonal sphenohedron&lt;/a&gt;, from &lt;em&gt;σφήν&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; &lt;a href=&quot;//en.wikipedia.org/wiki/Tetragonal_disphenoid&quot; rel=&quot;nofollow noreferrer&quot;&gt;wedge&lt;/a&gt;):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\hspace{9em}$&lt;/span&gt;&lt;a href=&quot;//www.e-rara.ch/zut/content/pageview/5295242&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iGsuB.png&quot; width=&quot;353&quot; height=&quot;208&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//de.wikipedia.org/wiki/Axel_Wilhelmowitsch_Gadolin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gadolin&lt;/a&gt; (often credited with first deducing the 32 crystal classes) does consider RR under the name &lt;strong&gt;&lt;em&gt;sphenoidal symmetry&lt;/em&gt;&lt;/strong&gt; (and thus gets the “&lt;a href=&quot;https://en.wikipedia.org/wiki/Crystal_system#Crystal_classes&quot; rel=&quot;nofollow noreferrer&quot;&gt;tetragonal-(di)sphenoidal class&lt;/a&gt;” Bravais had missed) (&lt;a href=&quot;//archive.org/details/actasocietatissc91871suom/page/n20&quot; rel=&quot;nofollow noreferrer&quot;&gt;1871&lt;/a&gt;, pp. 18, 28). He apparently has no symbol for the operation.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Pierre_Curie&quot; rel=&quot;nofollow noreferrer&quot;&gt;Curie&lt;/a&gt; (also credited with deducing the 32 classes) completes Bravais’ notation by writing &lt;span class=&quot;math-container&quot;&gt;$\Pi^n$&lt;/span&gt; for RR, which he calls &lt;em&gt;plane of &lt;strong&gt;alternating symmetry&lt;/strong&gt; of order &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&lt;/em&gt; (&lt;a href=&quot;//doi.org/10.3406/bulmi.1884.1888&quot; rel=&quot;nofollow noreferrer&quot;&gt;1884&lt;/a&gt;, pp. 430, 442, 453; &lt;a href=&quot;//gallica.bnf.fr/ark:/12148/bpt6k3056t/f1393&quot; rel=&quot;nofollow noreferrer&quot;&gt;1885&lt;/a&gt;):&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\hspace{4.5em}$&lt;/span&gt;&lt;a href=&quot;//archive.org/details/oeuvresdepierrec00curi/page/89&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9jWwQ.png&quot; width=&quot;304&quot; height=&quot;235&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;//archive.org/details/oeuvresdepierrec00curi/page/111&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LghSE.png&quot; width=&quot;243&quot; height=&quot;235&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://d-nb.info/gnd/117048410&quot; rel=&quot;nofollow noreferrer&quot;&gt;Minnigerode&lt;/a&gt; (also credited) calls RR a &lt;strong&gt;&lt;em&gt;symmetry of the second kind&lt;/em&gt;&lt;/strong&gt;. He has no symbol for it, as he prefers to use rotatory-inversion (&lt;a href=&quot;//zbmath.org/?q=an:16.0871.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1884&lt;/a&gt;, pp. 203, 213; &lt;a href=&quot;//archive.org/details/neuesjahrbuchfr25unkngoog/page/n157&quot; rel=&quot;nofollow noreferrer&quot;&gt;1887&lt;/a&gt;, pp. 151, 162). &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Arthur_Moritz_Schoenflies&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schoenflies&lt;/a&gt; briefly calls RR an &lt;strong&gt;&lt;em&gt;operation &lt;span class=&quot;math-container&quot;&gt;$\smash{\mathfrak S(\frac{2\pi}n)}$&lt;/span&gt;&lt;/em&gt;&lt;/strong&gt; in (&lt;a href=&quot;//zbmath.org/?q=an:21.0529.04&quot; rel=&quot;nofollow noreferrer&quot;&gt;1889&lt;/a&gt;, p. 175), then switches to &lt;span class=&quot;math-container&quot;&gt;$\smash{\overline{\mathfrak A}(\frac{2\pi}n)}$&lt;/span&gt; and the name &lt;strong&gt;&lt;em&gt;Drehspiegelung&lt;/em&gt;&lt;/strong&gt; in his influential book (&lt;a href=&quot;//archive.org/details/krystallsysteme00schogoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;1891&lt;/a&gt;, pp. 29, 49) and &lt;em&gt;Encyklopädie&lt;/em&gt; article (&lt;a href=&quot;//gdz.sub.uni-goettingen.de/id/PPN360709532?tify=%7B&amp;quot;pages&amp;quot;:[468]%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;, p. 444). His later papers keep that notation, e.g. (&lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.31822017778291&amp;amp;seq=831&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;, p. 219).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Evgraf_Fedorov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fedorov&lt;/a&gt; calls RR &lt;strong&gt;&lt;em&gt;combined symmetry&lt;/em&gt;&lt;/strong&gt;, without symbol (&lt;a href=&quot;//books.google.com/books?id=7ZNXAAAAYAAJ&amp;amp;pg=PA25&quot; rel=&quot;nofollow noreferrer&quot;&gt;1892&lt;/a&gt;, pp. 27–28). This terminology is adopted by e.g. &lt;a href=&quot;//de.wikipedia.org/wiki/Charles_Kephart_Swartz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Swartz&lt;/a&gt; (&lt;a href=&quot;//archive.org/details/bulletinofgeolog20190geol/page/369&quot; rel=&quot;nofollow noreferrer&quot;&gt;1909&lt;/a&gt;, pp. 374–377), &lt;a href=&quot;//de.wikipedia.org/wiki/Karl_Jellinek&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jellinek&lt;/a&gt; (&lt;a href=&quot;//archive.org/details/lehrbuchderphys00jellgoog/page/n380&quot; rel=&quot;nofollow noreferrer&quot;&gt;1915&lt;/a&gt;, pp. 355–361).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//de.wikipedia.org/wiki/Carlo_Maria_Viola&quot; rel=&quot;nofollow noreferrer&quot;&gt;Viola&lt;/a&gt; calls an RR axis &lt;strong&gt;&lt;em&gt;Spiegeldrehachse&lt;/em&gt;&lt;/strong&gt;, without symbol (&lt;a href=&quot;//archive.org/details/grundzgederkris00violgoog/page/n273&quot; rel=&quot;nofollow noreferrer&quot;&gt;1904&lt;/a&gt;, p. 246).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Felix_Auerbach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Auerbach&lt;/a&gt; in &lt;em&gt;Handbuch der Physik&lt;/em&gt; calls an RR axis &lt;strong&gt;&lt;em&gt;Spiegeldrehachse&lt;/em&gt;&lt;/strong&gt;, symbol &lt;span class=&quot;math-container&quot;&gt;$\color{red}{S_n}$&lt;/span&gt; (&lt;a href=&quot;//books.google.com/books?id=0KgLAAAAYAAJ&amp;amp;pg=PA58&amp;amp;dq=Spiegeldrehachse&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;, p. 58).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Woldemar_Voigt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voigt&lt;/a&gt; calls an RR axis in direction &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; &lt;strong&gt;&lt;em&gt;Spiegelachse&lt;/em&gt;&lt;/strong&gt;, symbol &lt;span class=&quot;math-container&quot;&gt;$S_r$&lt;/span&gt;; or &lt;strong&gt;&lt;em&gt;Drehspiegelachse&lt;/em&gt;&lt;/strong&gt;, symbol &lt;span class=&quot;math-container&quot;&gt;$\mathfrak S$&lt;/span&gt; (&lt;a href=&quot;//archive.org/details/bub_gb_SvPPAAAAMAAJ/page/n92&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910&lt;/a&gt;, pp. 66–77; &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015056706941&amp;amp;num=237&quot; rel=&quot;nofollow noreferrer&quot;&gt;1918&lt;/a&gt;, pp. 245, 333–336).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Friedrich_Rinne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rinne&lt;/a&gt; calls rotations &lt;em&gt;Gyralen&lt;/em&gt; and RR &lt;strong&gt;&lt;em&gt;Gyroiden&lt;/em&gt;&lt;/strong&gt;, without symbol; in a later edition he denotes them &lt;span class=&quot;math-container&quot;&gt;$\mathrm G_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\underline G{}_n$&lt;/span&gt;  (&lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=mdp.39015069826975&amp;amp;num=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;, p. 1; &lt;a href=&quot;//babel.hathitrust.org/cgi/pt?u=1&amp;amp;id=uc1.b3482138&amp;amp;num=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1922&lt;/a&gt;, p. 1).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.minsocam.org/msa/collectors_corner/arc/rogers.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rogers&lt;/a&gt; calls RR &lt;strong&gt;&lt;em&gt;rotatory-reflection&lt;/em&gt;&lt;/strong&gt;, symbol &lt;span class=&quot;math-container&quot;&gt;$ap_{2\pi/n}$&lt;/span&gt; (&lt;a href=&quot;//archive.org/details/introductiontos00rogegoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;1912&lt;/a&gt;, pp. 5, 7; &lt;a href=&quot;//doi.org/10.2307/20026154&quot; rel=&quot;nofollow noreferrer&quot;&gt;1926&lt;/a&gt;, pp. 167–171; &lt;a href=&quot;//archive.org/details/journalofwashin251935wash/page/476&quot; rel=&quot;nofollow noreferrer&quot;&gt;1935&lt;/a&gt;, p. 478).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Paul_Peter_Ewald&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ewald&lt;/a&gt; in &lt;em&gt;Handbuch der Physik&lt;/em&gt; calls an RR axis &lt;strong&gt;&lt;em&gt;Drehspiegelachse&lt;/em&gt;&lt;/strong&gt;, symbol &lt;span class=&quot;math-container&quot;&gt;$\color{red}{S_n}$&lt;/span&gt;. In a footnote about the nomenclature (of both operations and groups), he adds: “As experience showed that misunderstandings can hardly result, a simplified spelling of Latin letters and subscripted symmetry indices established itself, which has also been applied here” (&lt;a href=&quot;//doi.org/10.1007/978-3-7091-5424-3_4&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927&lt;/a&gt;, pp. 199, 203, 206). He edited &lt;a href=&quot;//catalog.hathitrust.org/Record/009798822&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zeitschrift für Kristallographie&lt;/a&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;//en.wikipedia.org/wiki/Eugene_Wigner&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wigner&lt;/a&gt; adopts Ewald’s &lt;em&gt;Handbuch&lt;/em&gt; notation (&lt;a href=&quot;//zbmath.org/?q=an:56.0689.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;1930&lt;/a&gt;, p. 139).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The linked papers of Swartz (1909) and Rogers (1926) have nice historical accounts in English. In decreasing order of likelihood, I’d say that the &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; seems to come from 1) Spiegeldrehachse, 2) sphenoidal, or 3) Schoenflies’ ephemeral &lt;span class=&quot;math-container&quot;&gt;$\mathfrak S$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="658" LastEditorUserId="658" LastEditDate="2018-11-13T07:41:41.627" LastActivityDate="2018-11-13T07:41:41.627" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7957" PostTypeId="1" AcceptedAnswerId="7958" CreationDate="2018-11-14T01:12:33.557" Score="8" ViewCount="828" Body="&lt;p&gt;I've always grappled with anything related to Fourier since my undergrad days. Recently, when revisiting why I learned what I did, I discovered how Fourier's desire to understand the flow of heat through a solid body led to the creation/discovery of Fourier Series and correspondingly the Fourier Transform.&lt;/p&gt;&#xA;&lt;p&gt;However, I've never been able to make the mental leap from the heat equation to the creation of Fourier Series.&lt;/p&gt;&#xA;&lt;p&gt;Are there any good sources (references, books, videos (most preferable) etc.) that provide a walkthrough to help &amp;quot;discover/create&amp;quot; the findings of Fourier by yourself?&lt;/p&gt;&#xA;&lt;p&gt;This is purely an intellectual exercise out of curiosity - it may not be worth it. Still, I'm curious to be able to understand it &amp;quot;naturally&amp;quot; vs being told it is so.&lt;/p&gt;&#xA;&lt;p&gt;My expected &amp;quot;thought experiment setup&amp;quot; is a &lt;em&gt;thin rod&lt;/em&gt; with thermometers at regular intervals and using that to derive what Fourier did - not sure if this is even possible, but the idea is to go from the simplest abstraction to the actual concept at hand. It's okay if things are mixed with modern calculus to help understand it better since at the time it was still in flux.&lt;/p&gt;&#xA;" OwnerUserId="7035" LastEditorUserId="4251" LastEditDate="2021-03-02T16:17:40.347" LastActivityDate="2021-05-20T17:51:07.147" Title="What are some good references elucidating the discovery/creation of Fourier Series?" Tags="|mathematics|physics|reference-request|harmonic-analysis|fourier|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7958" PostTypeId="2" ParentId="7957" CreationDate="2018-11-14T02:55:38.043" Score="6" Body="&lt;p&gt;§1.1 (+ &lt;a href=&quot;https://www.macalester.edu/aratra/edition2/chapt1.html&quot; rel=&quot;noreferrer&quot;&gt;supplement&lt;/a&gt;) of Bressoud’s &lt;a href=&quot;http://ams.org/mathscinet-getitem?mr=2284828&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A radical approach to real analysis&lt;/em&gt;&lt;/a&gt;, recommended here just &lt;a href=&quot;https://hsm.stackexchange.com/questions/7891/what-brought-about-the-need-for-real-analysis-and-formal-logic-in-recent-years/7892#comment14784_7892&quot;&gt;recently&lt;/a&gt;, does pretty much exactly what you want.&lt;/p&gt;&#xA;" OwnerUserId="658" LastActivityDate="2018-11-14T02:55:38.043" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7959" PostTypeId="2" ParentId="7957" CreationDate="2018-11-14T14:12:10.813" Score="9" Body="&lt;p&gt;Contrary to what the name suggests, Fourier series were not invented/discovered by Fourier. They were considered by Euler and Bernoullis, in relation to the one dimensional wave equation, not the heat equation. This early story is described&#xA;for example in the papers by Luzin in Amer. Math. Monthly:&lt;/p&gt;&#xA;&lt;p&gt;Luzin, N. Function. I. Amer. Math. Monthly 105 (1998), no. 1, 59–67.&lt;/p&gt;&#xA;&lt;p&gt;Luzin, N. Function. II. Amer. Math. Monthly 105 (1998), no. 3, 263–270.&lt;/p&gt;&#xA;&lt;p&gt;(Hovewer, Fourier integral and theta-functions are Fourier's inventions). The book of Fourier is translated into English, and it is still a very interesting reading.&#xA;In it Fourier gave a systematic theory of solving PDE's by the method of separation of the variables, and after its publication, Fourier series became a general tool in mathematics and physics. So the names Fourier series and Fourier analysis are well justified.&lt;/p&gt;&#xA;&lt;p&gt;Remark on comments. Fourier did not establish with a rigorous proof any general criteria of representation of functions by Fourier series.&#xA;His book was criticized as non-rigorous, and this substantially delayed its publication. However he made very convincing arguments. He also checked some of his results by actual experiment with heating various bodies.&lt;/p&gt;&#xA;&lt;p&gt;Fourier was a scientist, not only a pure mathematician.He said that the main goal of mathematics is a profound study of nature. One of the principal motivating questions for his study of heat was determination of the age of the Earth. This was later done by Thomson (Lord Kelvin) using Fourier theory.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="10106" LastEditDate="2020-07-03T11:53:42.043" LastActivityDate="2020-07-03T11:53:42.043" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7960" PostTypeId="1" CreationDate="2018-11-15T07:16:18.457" Score="1" ViewCount="340" Body="&lt;p&gt;Fubini's theorem relates the double integral of a function &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt; to an iterated integral with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;.  The basic idea of this theorem for Riemann integrals of continuous functions was known for centuries, but Lebesgue proved it for bounded Lebesgue measurable functions, then Fubini proved it for Lebesgue integrable functions, then Tonelli fixed a flaw in Fubini's proof and proved it in a bit more generality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The results of Lebesgue, Fubini, and Tonelli all pertain to Lebesgue measure.  Yet today we commonly see in textbooks a Fubini-Tonelli theorem for product measures on products of abstract measure spaces.  My question is, who is the first person who proved this result for abstract measure spaces?  And what technique(a) did they use to prove it?&lt;/p&gt;&#xA;" OwnerUserId="7582" LastEditorUserId="7582" LastEditDate="2019-06-07T15:56:22.367" LastActivityDate="2020-10-04T08:09:45.810" Title="Who first proved Fubini's theorem for abstract measure spaces?" Tags="|mathematics|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7961" PostTypeId="2" ParentId="7960" CreationDate="2018-11-15T08:26:47.720" Score="4" Body="&lt;p&gt;According to Saks (&lt;a href=&quot;https://archive.org/details/theoryoftheinteg032192mbp/page/n99&quot; rel=&quot;nofollow noreferrer&quot;&gt;1937&lt;/a&gt;, p. 88):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Hahn, Hans&lt;/em&gt;, &lt;a href=&quot;http://www.numdam.org/item?id=ASNSP_1933_2_2_4_429_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Über die Multiplikation total-additiver Mengenfunktionen&lt;/strong&gt;&lt;/a&gt;, Ann. Sc. Norm. Super. Pisa, II. Ser. 2, 429-452 (1933). &lt;a href=&quot;https://zbmath.org/?q=an:0007.40203&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL0007.40203&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Ulam, S.&lt;/em&gt;, &lt;a href=&quot;https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1932.2/ICM1932.2.ocr.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Zum Maß begriffe in Produkträumen&lt;/strong&gt;&lt;/a&gt;, Verhandlungen Kongreß Zürich 1932, 2, 118-119 (1932). &lt;a href=&quot;https://zbmath.org/?q=an:58.0267.18&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL58.0267.18&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Łomnicki, Z.; Ulam, S.&lt;/em&gt;, &lt;a href=&quot;http://dx.doi.org/10.4064/fm-23-1-237-278&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités. I. Variables independantes&lt;/strong&gt;&lt;/a&gt;, Fundam. Math. 23, 237-278 (1934). &lt;a href=&quot;https://zbmath.org/?q=an:0009.40601&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL0009.40601&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Feller, W.&lt;/em&gt;, Bemerkungen zur Maß theorie in abstrakten Räumen, (Auszug aus “Rad” 249, 204-224.) Bulletin Acad. Yougoslave Zagreb 28, 30-45 (1934). &lt;a href=&quot;https://zbmath.org/?q=an:60.0977.03&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZBL60.0977.03&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="658" LastEditorUserId="4490" LastEditDate="2020-10-04T08:09:45.810" LastActivityDate="2020-10-04T08:09:45.810" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="7962" PostTypeId="2" ParentId="5829" CreationDate="2018-11-15T09:08:12.337" Score="1" Body="&lt;p&gt;Like Ben Crowell in the comments, I'm not sure I fully understand the question. But interpreting it broadly about the origin of the index notation, I quote Spalt who in &lt;em&gt;Die Analysis im Wandel und im Widerstreit&lt;/em&gt; &lt;a href=&quot;https://books.google.ch/books/about/Die_Analysis_im_Wandel_und_im_Widerstrei.html?id=0fzWrQEACAAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;(2015)&lt;/a&gt;, p. 106 while referring to a figure in Leibniz &lt;em&gt;De quadratura arithmetica circuli ellipseos et hyperbolae&lt;/em&gt; (p. &lt;a href=&quot;http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/VII6.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;528&lt;/a&gt;) claims (my translation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Leibniz is the inventor of the index notation. Today we would write &lt;span class=&quot;math-container&quot;&gt;$D_1, D_2$&lt;/span&gt; etc. for the indexed points &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;. Leibniz does this differently: he writes the numbers &lt;em&gt;in front&lt;/em&gt; of the letter and he does &lt;em&gt;not set them lower&lt;/em&gt;. (In the case of this figure he does not write the numbers smaller [so he wrote &lt;span class=&quot;math-container&quot;&gt;$1D, 2D$&lt;/span&gt; etc.]. But he does in other cases. Sometimes he also sets the numbers lower, but he never writes the numbers smaller and sets them lower at the same time. ...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't know when the passage to lower and smaller and after the letter happened and if one can find explicit reasons for this change.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2018-11-15T09:08:12.337" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7963" PostTypeId="1" CreationDate="2018-11-15T10:08:44.730" Score="2" ViewCount="70" Body="&lt;p&gt;A well-known (but usually only cursorily discussed) feature of General Relativity is the so-called radial (or volume) excess due to the curvature of spacetime in the presence of mass/energy (or more properly the effect of mass/energy on the Ricci curvature). See for instance Chapter 42 of the second volume of the Feynman Lectures, where the radial excess of the earth is estimate to be about 1.5mm: &lt;a href=&quot;http://www.feynmanlectures.caltech.edu/II_42.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.feynmanlectures.caltech.edu/II_42.html&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When was this first explicitly recognised, and by whom? I am specifically interested in the occurrence of the relevant equations in the literature, and numerical estimates of the magnitude of the effect for e.g. the earth. References welcome.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To be clear, my question is specifically about General Relativity, and answers should therefore refer to material after 1915.&lt;/p&gt;&#xA;" OwnerUserId="6496" LastActivityDate="2018-11-16T00:13:48.307" Title="First discussion of radial/volume excess in General Relativity" Tags="|relativity-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7964" PostTypeId="1" AcceptedAnswerId="7973" CreationDate="2018-11-15T18:07:29.360" Score="4" ViewCount="276" Body="&lt;p&gt;The intention behind my question is to understand what are the kind of general problems of which the ideas of Riemann's dissertation (1851) lie at the heart of it's solution methods. In his dissertation, Riemann introduced his mapping theorem (and gave a flawed proof of it based on Dirichlet's principle) and &quot;Riemann surfaces&quot;. According to the historical introduction to Carathéodory's &quot;Conformal representation&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1822 Gauss (1777-1855) stated and completely solved the general problem of finding all conformal transformations which transform a sufficiently small neighbourhood of a point on an arbitrary analytic surface into a plane area. This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the surface can be represented on a portion of the plane. This was first pointed out by Riemann (1826-1866), whose Dissertation (1851) marks a turning-point in the history of the problem which has been decisive for its whole later development; Riemann not only introduced all the ideas which have been at the basis of all subsequent investigation of the problem of conformal representation, but also showed that the problem itself is of fundamental importance for the theory of functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What i'm trying to understand is &lt;strong&gt;the meaning&lt;/strong&gt; of &quot;&lt;strong&gt;conformally representing a finite (not infinitisimal) portion of a surface on a portion of the plane&lt;/strong&gt;&quot;. As far as i understand, a conformal mapping can be visualized as acting (everywhere on the surface) like a composition of rotation (so angles are preserved) and rescaling (magnification of linear element). It should be mentioned that in general, the rotation angle might vary from place to place, as well as the scaling factor. So, conformal mappings already act on finite portions of a surfaces. So what is the meaning of all the references i find to much more difficult problems?     &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2018-11-15T18:55:48.753" LastActivityDate="2018-11-17T14:56:34.193" Title="Why is Riemann's dissertation (from 1851) considered a turning point in the history of the theory of conformal mappings?" Tags="|complex-analysis|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="7966" PostTypeId="2" ParentId="7931" CreationDate="2018-11-16T16:31:21.327" Score="0" Body="&lt;p&gt;According to p.192-194 of Schlesinger's essay, Gauss was led to the problem of conformal mapping of the interior of the ellipse into the interior of the unit circle around the year 1834. His first attempt, from 1834, was futile, since he was unable to find the mathematical expression of the mapping.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His second attempt, dating from 1839, was successful, as he was ocuppied at this period with the (related) theory of the potential in physics and mathematics and composed his treatise &quot;General propositions on inverse-squared forces&quot; (published in 1840). He succeeded in constructing the so-called Green's function of ellipse with the centre at the pole. Gauss then expressed the resulting conformal mapping using his theory of theta series and theorems from the theory of potentials (Schlesinger mentions article 29 from &quot;general propositions&quot; as connected with this problem).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schlesinger also mentions that Gauss's result matches the formula of Hermann amadeus Schwarz from 1870 (although one also has to use a formula of Jacobi in order to make the step from Gauss's to Schwarz's formula).   &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2018-11-17T11:26:38.100" LastActivityDate="2018-11-17T11:26:38.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7967" PostTypeId="1" AcceptedAnswerId="7971" CreationDate="2018-11-16T16:57:15.057" Score="1" ViewCount="269" Body="&lt;p&gt;The Gauss-Newton method can be derived from Newton's method, but I am unable to see how Gauss was linked with this method. It seems unlikely that he himself worked on the method, but I am at a loss.&lt;/p&gt;&#xA;" OwnerUserId="8110" LastEditorUserId="4251" LastEditDate="2020-05-14T20:13:58.177" LastActivityDate="2020-05-14T20:13:58.177" Title="Origin of Gauss-Newton method" Tags="|mathematics|terminology|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7969" PostTypeId="1" AcceptedAnswerId="7970" CreationDate="2018-11-16T19:22:35.893" Score="3" ViewCount="177" Body="&lt;p&gt;From A.P. Yushkevich, &quot;Leonhard Euler, his life and work&quot;, in &quot;Development of Leonhard Euler's ideas and contemporary science&quot;, Nauka, Moscow, 1988, 15--46 (translation from Russian is mine):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;One should note the Euler role in the theory of &lt;strike&gt;movement of the moon&lt;/strike&gt; the moon's motion, which attracted attention of the scientific world because of the discrepancy between the observed orbit and the orbit calculated by Newton on the basis of his law of gravity. The discrepancy between observations and theoretical calculations was so large that for a while D'Alembert, Clairaut and Euler thought to introduce corrections to the law of gravity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were any concrete corrections proposed? Where one can read about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;b&gt;Update&lt;/b&gt;: Many thanks to Mauro ALLEGRANZA for an interesting reference, and to terry-s for a beautifully detailed answer.&lt;/p&gt;&#xA;" OwnerUserId="167" LastEditorUserId="167" LastEditDate="2018-11-20T21:07:55.207" LastActivityDate="2018-11-20T21:07:55.207" Title="Clairaut's proposed correction (reported as &quot;D'Alembert's, Clairaut's and Euler's corrections&quot;) to the Newtonian inverse-square law of gravity" Tags="|reference-request|gravity|euler|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7970" PostTypeId="2" ParentId="7969" CreationDate="2018-11-17T03:31:15.047" Score="6" Body="&lt;p&gt;This answers (with some explanation and references) the two questions, (a) Were any concrete corrections proposed (in the 1740s, to the law of gravitation)? and (b) Where can one read about it?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) Clairaut (but not Euler or d'Alembert) proposed in 1747 a correction to the inverse-square law, that is, an additive term depending on a higher power of the distance than the square, which he discussed in one place (p.337 of his 1747 paper cited below) as possibly an inverse fourth power, and in another (p.362) as possibly an inverse cube. The proposal was semi-quantitative and not fully 'concrete', as it still needed at least a coefficient (and a definite conclusion about the exponent), which Clairaut did not give.  (The source of the quotation given in the question is mistaken as a matter of history: only Clairaut, not the other two, made any proposal for 'correction' of the gravitation law.) Clairaut later (1749) reconsidered and retracted his correction proposal. D'Alembert stated his attitude (retrospectively) in his &lt;a href=&quot;https://books.google.com/books?id=Bw8OAAAAQAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Recherches sur differens points importans du systeme du monde' (part 1, 1754)&lt;/a&gt; (in the &lt;em&gt;Discours preliminaire&lt;/em&gt;): that he had thought some extra force such as magnetism might be involved, but would not propose changing the the law of gravity on the basis of a single phenomenon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) One can find original sources (below) for Clairaut's 1747 proposal, for part of the controversy that it evoked, and his 1749 retraction (the primary sources are only in French, see below for comment on English secondary sources). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.1 -- for Clairaut's paper with his proposal to correct the law of gravitation (proposal itself starts at p.337):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k3543t/f427&quot; rel=&quot;nofollow noreferrer&quot;&gt;15 Nov. 1747: &quot;Du système du monde dans les principes de la gravitation universelle&quot;, Mém.Acad.Roy.Sci. 1745 (1749) pp. 329-364&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The date-marks of the paper are strange: it was read in 1747, but judged sufficiently important to be inserted into the memoirs for the year 1745 -- which were then still in the press -- and the whole eventually appeared in 1749.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.2 -- for some of the controversy that resulted: Clairaut's reply to a critique from Buffon. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k3543t.f629&quot; rel=&quot;nofollow noreferrer&quot;&gt;17 Feb.1748: &quot;Réponse aux réflexions de M. de Buffon sur la loi de l'attraction, et sur le mouvement des apsides&quot;, Mém.Acad.Roy.Sci. 1745 (1749) pp. 529-548&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.3 -- for Clairaut's statement of retraction of his proposal to correct the law of gravitation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k3543t/f678&quot; rel=&quot;nofollow noreferrer&quot;&gt;17 May 1749: &quot;Avertissement de M. Clairaut au sujet des mémoires qu'il a donnez en 1747 et 1748, sur le système du Monde dans les principes de l'attraction&quot;, Mém.Acad.Roy.Sci. 1745 (1749), pp. 577-578&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;b.4 -- for Clairaut's theory of the moon's motion in which he explained that he had reached the revised conclusion (p.83) that &quot;it would not be useful to seek any other cause of the inequalities of the movement of the Moon than solely the attraction that is inversely proportional to the squares of the distances&quot;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k110454w.texteImage&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Théorie de la Lune déduite du seul principe de l'attraction réciproquement proportionelle aux quarrés des distances, par M. Clairaut'&lt;/a&gt; &#xA;('Theory of the moon deduced solely from the principle of attraction inversely proportional to the squares of the distances'). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is notable that many mis-statements have been in circulation for a long time in secondary accounts and commentaries, about what Clairaut and the others actually did and what they claimed.  I have not yet seen any account in English that avoids all misrepresentation of the original sources. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Amongst the better brief accounts is a summary of the 1740s-1752 controversy about the lunar apogee on pages 11-12 of C A Wilson's book of 2010 about the &lt;a href=&quot;https://books.google.com/books?id=JWCIFYLgRWkC&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Hill-Brown theory of the Moon's motion'&lt;/a&gt;. &#xA;Many of the points made in this summary are historically good, but it does appear to misrepresent Clairaut's own account of what he actually had done to convince not only himself, but others later on, that the lunar problem did not after all justify any correction to Newton's gravitational law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;C A Wilson, like some other commentators, described Clairaut's improved calculation as a 'second-order' calculation. But from &#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k110454w/f28.image.texteImage&quot; rel=&quot;nofollow noreferrer&quot;&gt;pages 29-31&lt;/a&gt; of his paper on the theory of the moon, it can be seen that Clairaut expressed his results numerically, and not as a power series of terms of increasing order (which was really a contribution of d'Alembert). Clairaut's improved calculation, as indicated on the same pages, did not concern the 'order' of approximation or adding terms of higher order/power, rather it proceeded by using first a naive approximation for the moon's radius vector, based only an expression for an ellipse with a rotating axis, and then a better approximation that also took into account the effect of the main perturbations of the radius vector. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may be of interest to add, that the reason why such an approximation had to be made at all, lay in the form of differential equation that Clairaut set out to solve. It was a form that can be given schematically as &lt;span class=&quot;math-container&quot;&gt;$ d^2 u / dv^2 + k u = f(u, \Omega, \Pi) $&lt;/span&gt;, where u is the reciprocal radius vector, v is the moon's (angle of) anomaly, k is a constant, and &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt; are perturbing forces, expressed in terms of u and other quantitities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The methods of the time for solving such equations required that references to variables except in the derivatives had to be known and expressed in terms of independent variable v and no other (see, for early-19th-c. methods of solution, Airy's 'Mathematical Tracts', editions from 1826 onwards). But the relation between u and v was the very relation (unknown) that was intended to be found by solving the equation. So the overall solution could only be approached iteratively by starting from a rough account or  guess for &lt;em&gt;u&lt;/em&gt; in terms of &lt;em&gt;v&lt;/em&gt; , and then incorporating an improved solution into the next iteration. Clairaut's paper (e.g. pp.29-31) shows two such iterations, giving an improvement in the coefficient of the apogee motion. But it did not give the apogee motion to full accuracy, and Clairaut at that point announced that he would dispense with the work of further repetition of the 'same calculations' (i.e. further iterations) by adopting straight away for the apogee motion the 'true value given by the observations'.  This can be read at &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k110454w/f30.image.texteImage&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 31&lt;/a&gt; of his paper on the theory of the moon.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2018-11-18T10:57:16.893" LastActivityDate="2018-11-18T10:57:16.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7971" PostTypeId="2" ParentId="7967" CreationDate="2018-11-17T04:20:30.397" Score="3" Body="&lt;p&gt;Undoubtedly, Gauss' name is attached to the name of the Gauss-Newton method due to his pioneering role in the development and application of the least-squares method to non-linear least-squared problems. As to whether he actually conceived of and employed the Gauss-Newton method to solve such problems, the paper &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01585997&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;&lt;em&gt;A Gauss-Newton method for convex composite optimization&lt;/em&gt;”&lt;/a&gt; makes that claim in the introduction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Addendum: After some more searching, I found where Gauss published his variant of Newton's method:  sections 179-180 of his &lt;em&gt;Theoria motus corporum coelestium in sectionibus conicis solem ambientum&lt;/em&gt; (1809).&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2018-11-18T20:19:37.027" LastActivityDate="2018-11-18T20:19:37.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7973" PostTypeId="2" ParentId="7964" CreationDate="2018-11-17T14:49:58.243" Score="2" Body="&lt;p&gt;A homeomorphism between two surfaces is called conformal if it preserves angles.&#xA;Examples of conformal maps were known since antiquity: stereographic projection maps&#xA;conformally the sphere without one point onto the plane (this map was known to the ancient Greeks, but conformality was stated and proved only in 17th century).&#xA;Another example is Mercator's projection which maps the plane onto the sphere minus 2 points (It is not one-to-one, but rather a covering, using the modern terminology. To make it one-to-one, one has to restrict it to a strip in the plane and the image will be the sphere minus a meridian.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss as the first who stated a LOCAL existence theorem: given any surface and a point on it, there exists a region containing this point which can be mapped conformally onto a region in the plane. (From the modern point of view, Gauss&#xA;neither proved this, nor even stated accurately).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Riemann discovered the first general GLOBAL result: for every simply connected region in the plane (other than the plane itself), there exists a conformal map&#xA;of this region onto the unit disk in the plane. Riemann's proof of this contained gaps which were immediately found. As the theorem is really of fundamental importance, this triggered an intensive research, and he first complete proof for&#xA;arbitrary regions was obtained by William Osgood in 1900.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since that time, the theory of conformal mapping is a central part of the complex function theory. The main question of this theory is how the geometric properties&#xA;of the region reflect in the properties of the mapping function. This is an important subject of research still at the present time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An important generalization of Riemann's theorem is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Uniformization_theorem#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Uniformization theorem&lt;/a&gt; proved independently by Poincare and Kobe in 1907. A simple example of&#xA;uniformization is the Merfcator map from the plane to the sphere minus 2 points.&#xA;The theorem of Poincare and Kobe says that a uniformization exists for every surface.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-11-17T14:56:34.193" LastActivityDate="2018-11-17T14:56:34.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7974" PostTypeId="1" AcceptedAnswerId="8031" CreationDate="2018-11-18T03:09:15.740" Score="14" ViewCount="1209" Body="&lt;p&gt;In Christoffel's 1869  &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0070?tify=%7B%22pages%22:%5B50%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt;  in which he introduced the Christoffel symbols on the 3rd and 4th pages, they are written as &lt;span class=&quot;math-container&quot;&gt;$\left[\substack{ij \\ k}\right]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{\substack{ij \\ k}\}$&lt;/span&gt;. The notation &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{kij}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Gamma_{ij}^k$&lt;/span&gt; that is used now is not there.  I would like to know who first introduced this &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;-indexed notation for Christoffel symbols.  Specifically, was it Einstein?&lt;/p&gt;&#xA;&lt;p&gt;Here is what I have been able to find.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The paper of&#xA;Ricci and Levi-Civita (Crelle 1900, pp. 125--201) uses Christoffel's braces notation (see p. 138 &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0054?tify=%7B%22pages%22:%5B144%5D,%22panX%22:0.402,%22panY%22:0.668,%22view%22:%22info%22,%22zoom%22:0.583%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Einstein's 1916 &lt;a href=&quot;http://myweb.rz.uni-augsburg.de/%7Eeckern/adp/history/einstein-papers/1916_49_769-822.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper on general relativity&lt;/a&gt;&#xA;uses &lt;span class=&quot;math-container&quot;&gt;$[\substack{\mu\nu\\ \sigma}]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{\substack{\mu\nu\\ \sigma}\}$&lt;/span&gt; initially (see pp. 791 and 792) but in the middle of p. 802 he writes &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\Gamma_{\mu \nu}^\tau = -\{\substack{\mu\nu\\ \tau}\}$&lt;/span&gt;&amp;quot; and doesn't go back to using braces or brackets except on pp. 817 and 818.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Weyl's &amp;quot;Space, Time, Matter&amp;quot; (1918) uses &lt;span class=&quot;math-container&quot;&gt;$\left[\substack{ij \\ k}\right]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{\substack{ij \\ k}\}$&lt;/span&gt;. See p. 119 of the German edition &lt;a href=&quot;https://archive.org/details/raumzeitmateriev00weyl/page/118&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and p. 196 of the English translation &lt;a href=&quot;http://www.gutenberg.org/files/43006/43006-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Levi-Civita's book &amp;quot;The Absolute Differential Calculus&amp;quot; (1923), which had an English translation in 1928, uses &lt;span class=&quot;math-container&quot;&gt;$[ij,k]$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\{ij,k\}$&lt;/span&gt; (see p.111&#xA;&lt;a href=&quot;https://archive.org/details/TheAbsoluteDifferentialCalculus/page/n131&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So up to a sign factor (the issue of the physical relevance of that sign is discussed on MO &lt;a href=&quot;https://mathoverflow.net/questions/386787/the-einstein-minus-convention-lost&quot;&gt;here&lt;/a&gt;), the &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;-indexed notation was introduced no later than Einstein's work (the books by Weyl and Ricci came after that, and I mention them in order to point out that the original notation of Christoffel was still being used after Einstein). I am not familiar with the history of differential geometry, so I would like either a confirmation that the notation is due to Einstein (his wording just before using it suggests the notation is new) or a pointer to earlier use of it.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="3024" LastEditDate="2021-03-21T18:17:16.750" LastActivityDate="2021-03-21T18:17:16.750" Title="Notation for Christoffel symbols" Tags="|biographical-details|mathematical-physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7975" PostTypeId="2" ParentId="26" CreationDate="2018-11-18T16:11:09.593" Score="5" Body="&lt;p&gt;&lt;em&gt;This is too near a miss to my criteria not to mention it for completeness.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the early years of molecular genetics, it was only known that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Genetic_code&quot; rel=&quot;nofollow noreferrer&quot;&gt;genetic code&lt;/a&gt; used an alphabet of four different bases and encoded twenty amino acids.&#xA;This sprouted several hypotheses on the design of the code, which were all able to explain what was experimentally known at that time.&#xA;However, some of these yielded the correct number of amino acids without further ado, i.e., they did not require this number as a parameter and thus were slightly favourable as per Occam’s razor.&lt;/p&gt;&#xA;&lt;p&gt;For example, &lt;a href=&quot;https://doi.org/10.1073/pnas.43.5.416&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crick et al.&lt;/a&gt; considered comma-less codes that featured an immunity against frameshift errors.&#xA;They showed that, given a codon length of three, there exist codes that can encode twenty amino acids and that it was impossible to have a code encoding more amino acids.&lt;/p&gt;&#xA;&lt;p&gt;Code designs that automatically yielded the correct number of amino acids met particular interest at that time, however, when the real genetic code was discovered it turned out to be of a different type: you could encode up to 63 amino acids with this general design.&lt;/p&gt;&#xA;&lt;p&gt;Now, I can find no contemporary invocation of Occam’s razor.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/What_Mad_Pursuit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crick&lt;/a&gt; even cautioned against it (due to natural selection not being bound to yield the most effective mechanism):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While Ockham's razor is a useful tool in the physical sciences, it can be a very dangerous implement in biology. It is thus very rash to use simplicity and elegance as a guide in biological research.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Still, the razor was mentioned in retrospect, e.g., by &lt;a href=&quot;https://doi.org/10.1017/S1355838201010615&quot; rel=&quot;nofollow noreferrer&quot;&gt;Woese&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The details of Gamow’s coding theories (there was more than one) are no longer of interest, for in their specifics his models were wrong.&#xA;However, his Occam’s razor approach and the impact his thinking had on his contemporaries was a major factor in moulding how gene expression was perceived.&lt;/p&gt;&#xA;&lt;p&gt;[…]&lt;/p&gt;&#xA;&lt;p&gt;But without doubt, the most memorable and influential theory to emerge from this new chapter in the code’s history (in that it retained a biological semblance and theoretical panache) was Crick’s famous “comma-free code”—one of those wonderful, but ephemeral triumphs of intellect over reality (to which theoreticians are predisposed).&#xA;The comma-free code remained based upon the hopeful presumption that the code might be inferred from first principles of some kind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-11-18T16:11:09.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7976" PostTypeId="1" CreationDate="2018-11-19T07:40:41.833" Score="3" ViewCount="82" Body="&lt;p&gt;I just wondered if &lt;a href=&quot;https://en.wikipedia.org/wiki/Hendrik_Lorentz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorentz&lt;/a&gt; was aware of &lt;a href=&quot;https://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le Sage theory&lt;/a&gt;. Seems like he uses similar math. Is there any direct quote from Lorentz regarding Le Sage gravity?&lt;/p&gt;&#xA;" OwnerDisplayName="Anti Fa" LastActivityDate="2018-11-19T08:10:28.613" Title="Was Lorentz aware of Le Sage theory?" Tags="|gravity|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7977" PostTypeId="1" CreationDate="2018-11-19T19:50:11.417" Score="0" ViewCount="129" Body="&lt;p&gt;So, the SI system of units has the basic philosophy of defining all 7 of the base units in terms of universal constants such as the plank constant, the speed of light, etc. These are all measured, then defined to be that number. But what if we we to develop a more accurate measurement apparatus, and obtain higher accuracy. Would this effect the definition?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the speed of light was defined to be exactly 299,792,458 m/s. Now imagine we develop some apparatus that measures it to be 299,792,458.6 m/s, with all experimental error accounted for, etc. How would this effect the definition of the meter, which is based upon the speed of light?&lt;/p&gt;&#xA;" OwnerUserId="8121" LastActivityDate="2018-11-19T21:40:34.893" Title="What if we measure a physical constant to higher accuracy?" Tags="|units|" AnswerCount="1" CommentCount="3" ClosedDate="2018-11-28T08:48:30.447" ContentLicense="CC BY-SA 4.0" />
  <row Id="7978" PostTypeId="2" ParentId="7977" CreationDate="2018-11-19T21:40:34.893" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;For example, the speed of light was defined to be exactly 299,792,458 m/s. Now imagine we develop some apparatus that measures it to be 299,792,458.6 m/s, with all experimental error accounted for, etc. How would this effect the definition of the meter, which is based upon the speed of light?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See the definition of the metre on &lt;a href=&quot;https://en.wikipedia.org/wiki/Metre#Speed_of_light_definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 second.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we'd change our definition of the metre to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458.6 seconds.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the important thing to note here is that the speed of light is constant by definition. It's defined to be constant. See what John Moffat and João Magueijo said in their 2007 &lt;a href=&quot;https://arxiv.org/abs/0705.4507&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comments on “Note on varying speed of light theories”&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;“Can c vary? Could such a variation be measured? As correctly pointed out by Ellis, within the current protocol for measuring time and space the answer is no. The unit of time is defined by an oscillating system or the frequency of an atomic transition, and the unit of space is defined in terms of the distance travelled by light in the unit of time. We therefore have a situation akin to saying that the speed of light is “one light-year per year”, i.e. its constancy has become a tautology or a definition”.&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it isn't constant. It varies in the room you're in. That's why optical clocks go slower when they're lower. Einstein said this in &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;“Second, this consequence shows that the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See my &quot;physics detective&quot; essay &lt;a href=&quot;http://physicsdetective.com/the-speed-of-light/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the speed of light is not constant&lt;/a&gt; for details and references.  &lt;/p&gt;&#xA;" OwnerUserId="3264" LastActivityDate="2018-11-19T21:40:34.893" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="7980" PostTypeId="2" ParentId="464" CreationDate="2018-11-20T18:21:11.877" Score="4" Body="&lt;p&gt;I apologize for the limited information in this answer, but as the question has gone unanswered for 4 years, I plead that something is better than nothing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your fourth version appears in &lt;em&gt;The Mathematical Theory of Relativity&lt;/em&gt; by Eddington, in a section titled &quot;The mathematical notion of a vector&quot;. Eddington writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We have a set of four numbers &lt;span class=&quot;math-container&quot;&gt;$(A_1,A_2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$A_3,A_4)$&lt;/span&gt; which we associate with some  point &lt;span class=&quot;math-container&quot;&gt;$(x_1,x_2,x_3,x_4)$&lt;/span&gt; and with a certain system of coordinates. We make a change of the coordinate system, and we ask, What will these numbers become in the new coordinates? The question is meaningless; they do not automatically &quot;become&quot;    anything. Unless we interfere with them they stay as they were. But the mathematician may say, &quot;When I am using the coordinates &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,x_3,x_4$&lt;/span&gt;, I want to talk about    the numbers &lt;span class=&quot;math-container&quot;&gt;$A_1,A_2,A_3,A_4$&lt;/span&gt;; and when I am using &lt;span class=&quot;math-container&quot;&gt;$x'_1,x'_2,x'_3,x'_4$&lt;/span&gt; I find that ... I shall want to talk about four different numbers &lt;span class=&quot;math-container&quot;&gt;$A'_1,A'_2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$A'_3,A'_4$&lt;/span&gt;. So for brevity I propose to call both sets of numbers by the same symbol &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{A}$&lt;/span&gt;.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The mathematician soon realizes he cannot write down the coordinates for &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{A}$&lt;/span&gt; in every one of an infinite variety of coordinate systems, and continues:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;I see that I must alter my plan. I will give you a general rule to find the new values of &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{A}$&lt;/span&gt; when you pass from one coordinate system to another ...&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In mentioning a &lt;em&gt;rule&lt;/em&gt; the mathematician gives up his arbitrary&#xA;  power... He binds himself down to some kind of regularity. Indeed we&#xA;  might have suspected that our orderly-minded friend would have some&#xA;  principle in his assignment of different meanings to &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{A}$&lt;/span&gt;.&#xA;  Now Eddington asks, can we possibly know what kind of rule the mathematician would adopt, without knowing what problem he is working on? He answers his own question:&#xA;  I think we can; it is not necessary to know anything about the nature of his problem... it is sufficient that we know a little about the nature of a mathematician.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Using various assumptions, Eddington narrows the options until only the transformation laws for contravariant and covariant vectors remain.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eddington then goes on to talk about the physicist's notion of a vector, but that's not relevant to your question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now some speculation. Older treatments that I've seen (i.e., early 20th C.) use the transformation definition. There's a famous quote by Weyl, usually given as &quot;The introduction of numbers as coordinates is an act of violence&quot; (see &lt;a href=&quot;https://math.stackexchange.com/questions/1493341/what-exactly-did-hermann-weyl-mean&quot;&gt;this question&lt;/a&gt; for the actual quote). It would not surprise me if Weyl looked for a coordinate-free definition of tangent vector.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I vaguely remember reading a passage by Weyl, in which he talked about the origin of the concept of a topological space. It went something like this. Differential manifolds are defined via patches with differentiable transition functions; Riemann surfaces via patches with conformal transition functions. What about continuous transition functions? And then the epiphany: in this case you can just axiomatize the patches,  e.g., the intersection of any two patches is a patch. Replace the word &quot;patch&quot; with &quot;open set&quot; and you're good to go. (Of course we've widened the scope from topological manifolds to topological spaces in general.) Weyl then went on to lament that for differential manifolds and Riemann surfaces, no such &quot;purely intrinsic&quot; definition has ever been found. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not been able to locate this passage again. Also this version says nothing about Hausdorff's role, so I fear I must be misremembering some of the details.&lt;/p&gt;&#xA;" OwnerUserId="48" LastActivityDate="2018-11-20T18:21:11.877" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7981" PostTypeId="1" CreationDate="2018-11-20T22:22:08.923" Score="0" ViewCount="65" Body="&lt;p&gt;Just wonder if there was ever any response to Mikhail Lomonsov's theory that Gravity is pressure differences in the aether. This overcomes self interaction and other problems in liberal gravity.&lt;/p&gt;&#xA;" OwnerUserId="8131" LastActivityDate="2018-11-20T22:22:08.923" Title="Lomonsov gravity vs liberal-relativistic gravity" Tags="|theoretical-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7982" PostTypeId="1" AcceptedAnswerId="7990" CreationDate="2018-11-22T09:39:25.223" Score="3" ViewCount="170" Body="&lt;p&gt;I am searching for books and articles about applications of Geometry, in particular to the problem of computing distances and lengths which are apparently beyond human reach. As an example, consider the computations of the size of the Earth by Eratosthenes and by Al-Biruni. I am particularly interested in those cases in which (as in the examples that I mentioned) it is known that those methods were actually used in practice.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="55" LastEditDate="2018-11-24T21:34:42.030" LastActivityDate="2018-11-26T10:54:37.833" Title="What are historical applications of geometry to measuring distances beyond human reach?" Tags="|reference-request|euclidean-geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7983" PostTypeId="1" CreationDate="2018-11-22T13:03:03.610" Score="3" ViewCount="130" Body="&lt;p&gt;Ado's Theorem is a very reelvant result in Lie theory (every finite-dimensional Lie algebra is isomorphic to a matrix Lie algebra).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been, however, unable to find anything more than very basics information on who is the mathematician that proved it. Igor Ado (1910-1983), student of Chebotaryov, graduated in Kazan University in 1935. And basically that's all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know of some more related informations?&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="5989" LastEditDate="2018-11-22T13:34:30.673" LastActivityDate="2019-08-26T18:10:47.693" Title="Biographical informations on Igor Ado" Tags="|biographical-details|reference-request|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7987" PostTypeId="2" ParentId="7983" CreationDate="2018-11-23T09:56:59.683" Score="6" Body="&lt;p&gt;In: D.V. Anosov, M.I. Monastyrskii, M.A. Soloviev, Nas ostalos' tak malo... (So few of us remained...), Istoriko-Matematicheskie Issledovaniya 7 (2002), 166-189 (in Russian) (the periodical is available in many places, for example &lt;a href=&quot;http://old.mathedu.ru/journals-collections/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), Ado is mentioned in passing. It is claimed that Chebotarev failed to create an algebraic school in Kazan' due to bureaucratic/political obstacles; Ado was unable to find a suitable employment in academia, has left pure mathematics (what is apparent from his list of publications), and has switched to numerical mathematics.&lt;/p&gt;&#xA;" OwnerUserId="167" LastActivityDate="2018-11-23T09:56:59.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7988" PostTypeId="1" AcceptedAnswerId="7989" CreationDate="2018-11-24T09:38:28.347" Score="5" ViewCount="793" Body="&lt;p&gt;It is probable that this quote was popularized by the writings of Morris Kline, e.g. in &lt;a href=&quot;https://books.google.com/books?id=lPcyAQAAIAAJ&amp;amp;q=%22the%20calculus%20was%20a%20collection%20of%20ingenious%20fallacies.%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematics for Liberal Arts&lt;/em&gt; (1967)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Michel Rolle] taught that the calculus was a collection of ingenious fallacies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The earliest instance I can find of this quote is in Rouse Ball, &lt;a href=&quot;https://books.google.com/books?redir_esc=y&amp;amp;id=U9zuAAAAMAAJ&amp;amp;dq=%22nothing%20but%20a%20collection%20of%20ingenious%20fallacies%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Short Account of the History of Mathematics&lt;/em&gt; (1888)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He taught that the differential calculus was nothing but a collection of ingenious fallacies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This quote has since been repeated many times elsewhere. For example, Stewart, &lt;a href=&quot;https://books.google.com/books?id=Zv1HDQAAQBAJ&amp;amp;pg=PA284&amp;amp;dq=rolle%20%22ingenious%20fallacies%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calculus&lt;/em&gt; (2016)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He was a vocal critic of the methods of his day and attacked calculus as being a &quot;collection of ingenious fallacies.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bomans &amp;amp; Rogers, &lt;a href=&quot;https://books.google.com/books?id=oYj3BgAAQBAJ&amp;amp;pg=PA123&amp;amp;dq=rolle%20%22ingenious%20fallacies%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;How We Got From There to Here: A Story of Real Analysis&lt;/em&gt; (2014)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In fact, Rolle was disdainful of both Newton and Leibniz's versions of calculus, once deriding them as a collection of &quot;ingenious fallacies.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bradley &amp;amp; Sandifer, &lt;a href=&quot;https://books.google.com/books?id=M0or-HGe7D0C&amp;amp;pg=PR7&amp;amp;dq=rolle%20%22ingenious%20fallacies%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Cauchy’s Cours d’analyse: An Annotated Translation&lt;/em&gt; (2010)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Rolle, for example, said that calculus was &quot;a collection of ingenious fallacies&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Did Rolle ever say/write any such thing?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user67" LastEditorUserId="104" LastEditDate="2018-11-27T08:40:57.880" LastActivityDate="2018-12-05T20:53:41.553" Title="Did Michel Rolle say that the calculus is &quot;a collection of ingenious fallacies&quot;?" Tags="|calculus|quote|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7989" PostTypeId="2" ParentId="7988" CreationDate="2018-11-24T14:33:09.250" Score="12" Body="&lt;p&gt;&quot;Did Rolle ever say/write any such thing (as that the calculus was 'a collection of ingenious fallacies')?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Michel Rolle (France, 1652-1719) certainly did attack the mathematical basis of the infinitesimal calculus.  I haven't found the exact phrase attributed to him by the authors quoted in the question, but there are plenty of broadly similar attacking phrases from Rolle about the calculus in Rolle's tract of 1703 (&lt;a href=&quot;https://books.google.com/books?id=QZA_AAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Remarques De M. Rolle De L'Académie Royale Des Sciences Touchant Le Problesme General Des Tangentes&lt;/em&gt;&lt;/a&gt;) -- such as 'absurd' {'absurde' - p.36}, 'entirely inconceivable' {'tout à fait inconcevable[s]' - p.41}, 'impossibilities and contradictions arise in crowds' {les impossibilitez &amp;amp; les contradictions se presenteroient en foule - p.41}, &amp;amp;c. This tract formed part of a disputatious exchange about the foundations of the new calculus, between Rolle and Joseph Saurin (1659-1737). [Also see edit/postscript below for a second and very relevant paper by Rolle found just now.] (It has been reported that later on Rolle changed his views: Michel Blay, 'La Naissance de la mécanique analytique', Paris, 1992, p.49.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rolle's criticisms had much to do with the seemingly ambiguous character (zero or non-zero?) of an infinitesimal, and he was not alone as a critic of the subject: he lived at a period when the logical basis of newer mathematical methods related to calculus was for a long time in widespread doubt and dispute.  Rolle expressed respect for the work of Descartes, Fermat and Hudde, earlier contributors to what might be called 'pre-calculus', but in 1703 he aimed his violent expressions of criticism especially at the well-known 1696 exposition of Leibniz's infinitesimal calculus, Guillaume (Marquis) de L'Hôpital's 'Analyse des infiniment petits' (see &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-l-hospital-s-analyse-des-infiniment-petits&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-l-hospital-s-analyse-des-infiniment-petits&lt;/a&gt;). Rolle's criticisms had much in common with other and better-known critiques. An earlier stage in these mathematical developments was the 'method of indivisibles' of Bonaventura Cavalieri (c.1598-1647): this had been given poor mathematical justification at best, and it soon attracted fundamental and destructive criticisms, notably from Paul Guldin (Switzerland, 1577-1643).  A much later and still-better-known critique of similar kind, aimed especially at the idea of infinitesimals, was given by George Berkeley in 'The Analyst' (1734) (see &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Analyst&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/The_Analyst&lt;/a&gt;).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Edit/postscript&lt;/em&gt;:  There is another paper by Rolle stating his attacks, which perhaps comes closer (than the paper linked above) to the content of the phrase quoted in the current question -- because its final sentence, following all of the attacks, acknowledges that the infinitesimal analysis is 'very ingenious'. It also gives another set of Rolle's condemnatory conclusions about the calculus: 'fort défectueux' (highly defective), 'ces Infinis fourmilleroient de contradictions' (these infinite[simal]s teem with contradictions), 'insoûtenable' (unsustainable), and the calculus involves 'petition de principe' (a &lt;em&gt;petitio principii&lt;/em&gt; -- one of the well-known forms of fallacy), &amp;amp;c: see Rolle's &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k3483p/f496.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Du nouveau systeme de l'infini&lt;/em&gt;, &lt;em&gt;Mém. acad. roy. des sci.&lt;/em&gt;, année 1703, 312-336&lt;/a&gt;. This combination of contents arguably suggests that the short phrase quoted in the question could have arisen as a brief paraphrase of this paper by a later commentator or possibly by Rolle himself. Blay (1986), cited below, shows that not all of Rolle's attacking papers were printed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among mathematical historians, Carl B Boyer has given a historical account of the long-lasting disputes about the foundations of calculus, in chapter 6 of his 'History of the Calculus and its Conceptual Development' (also published as 'Concepts of the Calculus') (editions of 1949 and 1959). This says rather little about the history of Rolle's  attacks, though, and more detail on these is given by Michel Blay, in &lt;a href=&quot;https://www.persee.fr/doc/rhs_0151-4105_1986_num_39_3_4477&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Deux moments de la critique du calcul infinitésimal : Michel Rolle et George Berkeley&lt;/em&gt;; &lt;em&gt;Revue d'histoire des sciences&lt;/em&gt;, 39 (1986) 223-253&lt;/a&gt;). Blay reports that Rolle's earlier attacks on the Leibnitzian calculus were defended by Pierre Varignon, who made several justificatory appeals to Newton's first section from Book 1 of the &lt;em&gt;Principia&lt;/em&gt;. (The material thus used by Varignon offers justification of Newton's geometrical form of calculus in terms of 'first and last ratios', i.e. limits.) In respect of the arguments relied on by Varignon against Rolle, a recent study argues that Boyer's and some other modern commentaries give unjustly poor assessments of the cogency of these earlier foundations offered for Newton's work in the calculus (Bruce Pourciau (2001), &lt;a href=&quot;https://ac.els-cdn.com/S0315086000923012/1-s2.0-S0315086000923012-main.pdf?_tid=d12d0cd9-7925-4bc5-9e91-8c4d5603e813&amp;amp;acdnat=1543630753_c264b4685ac9bebddd7a2095e9ed345a&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Newton and the notion of limit&lt;/em&gt;, &lt;em&gt;Historia Mathematica&lt;/em&gt; 28, 18-30&lt;/a&gt;).  Also, Jean-Étienne Montucla's well-known &lt;a href=&quot;https://books.google.com/books?id=3_YKVLHHEEQC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Histoire des mathématiques&lt;/em&gt;, vol.3 (1802)&lt;/a&gt;, especially at pp. 110-119, gives more background on the history and merits/demerits of Rolle's attacks, and indicates that in the late 18th-c., it was considered that such attacks had by then been rigorously rebutted. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2018-12-05T20:53:41.553" LastActivityDate="2018-12-05T20:53:41.553" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="7990" PostTypeId="2" ParentId="7982" CreationDate="2018-11-24T20:20:48.883" Score="4" Body="&lt;p&gt;For the down to Earth geometry see &lt;a href=&quot;https://books.google.com/books?id=1Izau5_ihmsC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Surveying Instruments of Greece and Rome by Lewis&lt;/a&gt;. For the astronomical measurements, which aside from the names mentioned would include Aristarchus, Ptolemy, see &lt;a href=&quot;https://books.google.com/books?id=L-yb7GX9mQIC&amp;amp;dq=Aristarchus+Eratosthenes+al-Biruni+geometry+measuring+distances&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley by van Helden&lt;/a&gt;. &lt;a href=&quot;https://books.google.com/books?id=bHD8IBaYN-oC&amp;amp;dq=Aristarchus+Eratosthenes+al-Biruni+geometry+measuring+distances&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mathematics of the Heavens and the Earth by van Brummelen&lt;/a&gt; focuses specifically on applications of trigonometry. The &lt;a href=&quot;https://www.ast.cam.ac.uk/~mjp/history_parallax.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Stellar Measurement website&lt;/a&gt; lists some more recent applications to the parallax measurements. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-11-24T20:20:48.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="7992" PostTypeId="1" CreationDate="2018-11-25T16:48:52.173" Score="4" ViewCount="474" Body="&lt;p&gt;What developments led to the definition of Work as the dot product of Force and Displacement vectors?&lt;/p&gt;&#xA;&lt;p&gt;I did some searching but couldn't find a satisfying answer. According to what I have read, the initial concept of work was associated with measuring work by horses. How did it get associated with energy? Another common answer I got was that &amp;quot;work indicates a transfer of a &lt;em&gt;quantity&lt;/em&gt; that &lt;em&gt;happens&lt;/em&gt; to be conserved in closed systems just like momentum&amp;quot;. I see what that means but how can anyone, possibly, come up with a formula as &lt;span class=&quot;math-container&quot;&gt;$W=F\cdot S$&lt;/span&gt; from that (or anything else?).&lt;/p&gt;&#xA;&lt;p&gt;Also, did conservation of &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; in elastic collisions led to development of work formula and modern concept of energy.&lt;/p&gt;&#xA;" OwnerUserId="8145" LastEditorUserId="4251" LastEditDate="2021-10-18T04:15:33.850" LastActivityDate="2021-10-18T04:15:33.850" Title="How did people figure out the formula for mechanical work, and related it to energy?" Tags="|physics|classical-mechanics|energy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7994" PostTypeId="1" AcceptedAnswerId="7996" CreationDate="2018-11-26T03:39:13.217" Score="4" ViewCount="1934" Body="&lt;p&gt;Gravity is of course something that we can all observe. Stuff falls towards the ground. But not everything: some things like steam or smoke defy this force and instead float up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During Ancient Greek and Roman times (when the two shared a lot of culture and scientific knowledge), how was this natural force explained? What was believed to be the cause of gravity? And why was it that some things obeyed it while steam and smoke defied it?&lt;/p&gt;&#xA;" OwnerUserId="8149" OwnerDisplayName="Thunderforge" LastEditorUserId="62" LastEditDate="2018-11-26T14:48:08.697" LastActivityDate="2018-11-27T20:06:40.803" Title="How was gravity explained in Ancient Greek and Roman times?" Tags="|physics|ancient-greece|gravity|roman-empire|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7995" PostTypeId="2" ParentId="7994" CreationDate="2018-11-26T07:47:36.927" Score="4" Body="&lt;p&gt;The aristotelian theory affirmed that there was a natural 'affinity' between substances that were alike. Thus, a stone fell towards the ground because of a natural, telluric affinity, and fire/smoke rose to the sky because of a natural affinity, too...&lt;/p&gt;&#xA;" OwnerUserId="6222" OwnerDisplayName="xxavier" LastActivityDate="2018-11-26T07:47:36.927" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="7996" PostTypeId="2" ParentId="7994" CreationDate="2018-11-26T16:07:04.743" Score="7" Body="&lt;p&gt;See &lt;a href=&quot;https://plato.stanford.edu/entries/aristotle-natphil/&quot; rel=&quot;noreferrer&quot;&gt;Aristotle's Natural Philosophy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to Aristotle, &lt;em&gt;change&lt;/em&gt; in the natural world can be :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[either] in accordance with the nature of the object — in which case the change is &lt;em&gt;natural&lt;/em&gt; (&lt;em&gt;phusei&lt;/em&gt;) or according to nature, or can happen in the face of a contrary disposition on the part of the nature of the entity — in which case the change is &lt;em&gt;forced&lt;/em&gt; or contrary to nature.&lt;/p&gt;&#xA;&lt;p&gt;A mover can effect a motion which is contrary to its own nature. Aristotle’s example of such an unnatural mover is the lever, an object heavy by nature, with which loads can be lifted.&lt;/p&gt;&#xA;&lt;p&gt;[The] large class of natural motions [is formed by] the natural motions of the &lt;em&gt;elements&lt;/em&gt;. [...] the elements move to their natural places — the light bodies up and the heavy ones down — by an appeal to their respective natures as causes (“that it is simply their nature to move somewhere, and this is what it is to be light and to be heavy,” &lt;em&gt;Physics&lt;/em&gt; Book VIII, 255b13–17).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See Aristotle's text :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[&lt;a href=&quot;http://classics.mit.edu/Aristotle/physics.8.viii.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Phys&lt;/em&gt;, 255b&lt;/a&gt;] The activity of lightness consists in the light thing being in a certain place, namely high up: when it is in the contrary place, it is being prevented. The case is similar also in regard to quantity and quality. But, be it noted, this is the question we are trying to answer — how can we account for the motion of light things and heavy things to their proper places? The reason for it is that they have a natural tendency towards a certain position; and this is what it is to be light or heavy, the former being determined by an upward, the latter by a downward, tendency.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-11-26T16:07:04.743" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="7997" PostTypeId="2" ParentId="7994" CreationDate="2018-11-27T20:06:40.803" Score="-1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How was gravity explained in Ancient Greek and Roman times?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It wasn't. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As xxavier and Mauro say, people like Aristotle claimed that things had a natural affinity or tendency towards a certain location. That's no explanation at all. &lt;em&gt;That's&lt;/em&gt; what's missing from the &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_gravitational_theory#&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; sempaiscuba was referring to. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Wikipedia article does include &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_gravitational_theory#General_relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;a section on general relativity&lt;/a&gt; which purports to explain gravity. However it doesn't. Note this: &lt;em&gt;&quot;Einstein proposed that spacetime is curved by matter, and that free-falling objects are moving along locally straight paths in curved spacetime&quot;.&lt;/em&gt; He didn't, because: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Spacetime curvature is associated with the &lt;em&gt;tidal&lt;/em&gt; force, not the force of gravity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Spacetime models space at all times, so there's no motion through spacetime.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A concentration of energy causes spacetime curvature not just matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Einstein did say is &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;: &lt;em&gt;“As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”.&lt;/em&gt; Unfortunately you tend not to hear about this, which means gravity is generally not explained in modern times either. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gravity is of course something that we can all observe. Stuff falls towards the ground. But not everything: some things like steam or smoke defy this force and instead float up.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However they don't do this in a vacuum. Note that the Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Vacuum&quot; rel=&quot;nofollow noreferrer&quot;&gt;vacuum&lt;/a&gt; article says this: &lt;em&gt;&quot;Vacuum has been a frequent topic of philosophical debate since ancient Greek times, but was not studied empirically until the 17th century&quot;.&lt;/em&gt; The ancient Greeks weren't big on experiment. &lt;/p&gt;&#xA;" OwnerUserId="3264" LastActivityDate="2018-11-27T20:06:40.803" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="7998" PostTypeId="1" AcceptedAnswerId="8008" CreationDate="2018-11-27T22:06:34.947" Score="2" ViewCount="277" Body="&lt;p&gt;In Dirac's 1938 paper on classically radiating electrons, he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Secondly we have the idea of the positron... in which positive and negative values for the mass of an electron play symmetrical roles. This cannot be fitted in with the electromagnetic idea of mass, which insists on all mass being positive, even in the abstract theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What motivated physicists to abandon the idea of anti-particles having a negative mass?&lt;/p&gt;&#xA;" OwnerUserId="2461" LastActivityDate="2018-11-29T22:34:24.943" Title="Why was the idea of anti-particles having negative mass abandoned?" Tags="|theoretical-physics|particle-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="7999" PostTypeId="1" CreationDate="2018-11-28T18:33:12.193" Score="3" ViewCount="2023" Body="&lt;p&gt;Every one of my lecturers have always called it this, as do I, despite the fact that I know its properly called 'tilde'. Does anyone have any clue where this convention comes from and why it might have became (somewhat) standard?&lt;/p&gt;&#xA;" OwnerUserId="8160" LastEditorUserId="3643" LastEditDate="2018-11-28T22:35:37.077" LastActivityDate="2018-11-29T13:05:39.797" Title="Why do mathematicians call ~ 'twiddle'?" Tags="|mathematicians|terminology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8000" PostTypeId="2" ParentId="7999" CreationDate="2018-11-28T21:33:20.383" Score="0" Body="&lt;p&gt;The OED gives a citation for the noun &quot;twiddle&quot; possibly meaning  the tilde sign dating to 1894: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;‘e’ for ‘æ’ is just as much a contraction as ‘r’ with a twiddle for ‘rum’.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This might or might not be a description of an &quot;r&quot; with a tilde,  analogous to  &quot;u&quot; with a tilde meaning &quot;um&quot; which I have seen in 16th century printed books.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2018-11-28T23:48:34.803" LastActivityDate="2018-11-28T23:48:34.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8001" PostTypeId="2" ParentId="7999" CreationDate="2018-11-29T13:05:39.797" Score="2" Body="&lt;p&gt;I'm not convinced they are &lt;em&gt;exactly&lt;/em&gt; the same, as a tilde is an accent mark over a letter, while a twiddle is a relationship operator sitting between letters (or numbers, or whatever).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But you might as well ask why an exclamation point is referred to as &quot;bang&quot; and as &quot;splat&quot; by various software (and math) people.  And don't get me started on the ridiculous renaming of &quot;poundsign&quot; to &quot;hashtag&quot;  . (Insert rant about misusing pound sign vs. number sign vs. musical sharp sign -- yes, they are not the same symbol) &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-11-29T13:05:39.797" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8002" PostTypeId="1" AcceptedAnswerId="8013" CreationDate="2018-11-29T13:46:58.760" Score="6" ViewCount="382" Body="&lt;p&gt;&lt;sup&gt;(I hope this is on-topic on this site)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;I am wondering what are some of the best examples of popular-science books that had large influence in public, but was based on weak science?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;large influence in public&quot; I mean that the book's claims became widely considered factual (at least by the general public, not necessarily by experts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;weak science&quot; I mean any of the following indicators: that the book's &quot;facts&quot; were already known to be wrong or became proven very wrong later, was based on cherry picked studies, its claims are largely contradicted by the evidence, or something similar.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am open to any branch of science whether it be biology or psychology, or whatever.&lt;/p&gt;&#xA;" OwnerUserId="8166" LastEditorUserId="8166" LastEditDate="2018-11-29T22:16:43.263" LastActivityDate="2018-12-01T13:35:27.467" Title="Pop-sci books that were publicly influential but based on weak science" Tags="|reference-request|book-recommendation|big-list|" AnswerCount="7" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8003" PostTypeId="2" ParentId="8002" CreationDate="2018-11-29T15:18:20.123" Score="2" Body="&lt;p&gt;There were a number of pop-sci books written by non-mathematicians (physiciscs, chemists, etc.) where the notion was spread that &lt;span class=&quot;math-container&quot;&gt;$\aleph_1$&lt;/span&gt; denotes the cardinal of the continuum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, George Gamow, &lt;em&gt;One, Two, Three... Infinity&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-11-29T15:18:20.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8004" PostTypeId="1" CreationDate="2018-11-29T17:27:45.537" Score="3" ViewCount="1187" Body="&lt;p&gt;Around the time that Coulomb gave Coulomb's law, how did people measure electric charge?&lt;/p&gt;&#xA;" OwnerUserId="8168" LastEditorUserId="55" LastEditDate="2018-11-29T20:59:40.080" LastActivityDate="2018-11-29T22:10:22.850" Title="How did people measure electric charge at the time of Coulomb?" Tags="|physics|electromagnetism|electricity|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8005" PostTypeId="2" ParentId="8004" CreationDate="2018-11-29T20:58:20.903" Score="3" Body="&lt;p&gt;People did not. They generated and stored static electricity, but measuring it, other than by how strong the electic shock felt, was not yet done. Coulomb was the one who invented a way to do it in 1785. For this he used a device called torsion balance. John Michell used it earlier to measure gravity, but Coulomb re-invented the idea, and made it famous. &lt;a href=&quot;https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/museum/torsion-balance-1785&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maglab.org&lt;/a&gt; has a picture and a description of how it worked:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;A narrow glass tube extends through the top of the glass case. At the top of the tube is a metal sphere, from which the needle hangs by the thread. A small metal sphere is at one end of the needle, which can swing freely due to its suspended state. Protruding through the top of the glass case is also a metal rod with metal spheres at both ends (one inside the case, one outside).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;To use the torsion balance, Coulomb would hold an object near the metal sphere at the upper end of the metal rod. Any charge held by the object being studied would transfer to the metal sphere, then travel along the rod to the sphere at the other end. There the charge could affect the needle suspended in the case, which in its resting state touched the rod's lower sphere. So any charge in that sphere passed to the needle’s sphere. Once the rod’s sphere and the needle’s sphere both became similarly charged, they repelled one another. The repulsion caused the needle to move and the thread holding it to twist.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The twisting action is called torsion, hence the name of the instrument. To determine how much torsion occurred, Coulomb consulted a small scale marked in degrees near the upper end of the narrow glass tube. A second scale encircling the glass case itself allowed him to determine how far the needle moved. As Coulomb realized, the greater the charge, the greater the torsion and displacement he observed.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An additional question from the comments is how Coulomb established proportionality of the electric force to the charges. One &lt;em&gt;could&lt;/em&gt; do it by doubling the charges and measuring what happens to the force, but the answer is... Coulomb did not. Falconer gives a good historical account in &lt;a href=&quot;https://pdfs.semanticscholar.org/b0a6/defd222f58d1c65c78950107757aa2fc05e7.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Augustin Coulomb and the Fundamental Law of Electrostatics&lt;/a&gt;, where she writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The second memoir (1787), on the attraction between dissimilarly charged bodies,  shows  Coulomb’s  newtonian  programme.   It  is  the first to state explicitly &lt;strong&gt;that the force was proportional to the product of the ‘electric masses’ of the two bodies, a proposition that  Coulomb  frequently  asserted  but  never  attempted  to demonstrate&lt;/strong&gt;. Nor did he ever define a unit electric charge,  relying  on  relative  electric  ‘masses’  or  ‘densities’. Throughout, his treatment depends on the unstated assumption that electric mass is localized and analogous to gravitational mass...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JeURo.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JeURo.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-11-29T22:10:22.850" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8007" PostTypeId="2" ParentId="8002" CreationDate="2018-11-29T22:08:43.983" Score="-6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I am wondering what are some of the best examples of popular-science books that had large influence in public, but was based on weak science? By &quot;large influence in public&quot; I mean that the book's claims became widely considered factual.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd say the most obvious example was Stephen Hawking’s 1989 book &lt;a href=&quot;http://www.fisica.net/relatividade/stephen_hawking_a_brief_history_of_time.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Brief History of Time&lt;/a&gt;. There's a lot wrong with this book, and I can't go into it all. But for an example, take a look at the chapter on black holes. Hawking referred to John Michell, talking about &lt;em&gt;&quot;such a strong gravitational field that light could not escape: any light emitted from the surface of the star would be dragged back by the star’s gravitational attraction before it could get very far&quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, you might think Hawking is merely talking about Michell's understanding here, But he isn't. One of Hawking’s “seminal” papers was &lt;a href=&quot;https://www.google.co.uk/search?q=singularities+and+the+geometry+of+spacetime&amp;amp;spf=277&amp;amp;cad=h&quot; rel=&quot;nofollow noreferrer&quot;&gt;singularities and the geometry of spacetime&lt;/a&gt; dating from 1966. On page  page 76 Hawking talked of &lt;em&gt;“such a strong gravitational field that even the ‘outgoing’ light rays from it are dragged back”&lt;/em&gt;. This flatly contradicts Einstein, and is wrong. See Einstein’s &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;fundamental ideas and methods of the theory of relativity&lt;/a&gt;. That’s where Einstein explained why light curves, saying a gravitational field is a place where &lt;em&gt;“the speed of light is spatially variable”&lt;/em&gt;. That's why optical clocks go slower when they're lower. There's something important that you can work out from that. Something rather counter-intuitive. See how PhysicsFAQ editor Don Koks &lt;a href=&quot;http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;describes it&lt;/a&gt;: &lt;em&gt;“light speeds up as it ascends from floor to ceiling, and it slows down as it descends from ceiling to floor; it’s not like a ball that slows on the way up and goes faster on the way down”&lt;/em&gt;. Hawking didn't know this. He didn’t know that in a strong gravitational field, outgoing light rays aren’t dragged back. They speed up even more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The bottom line is that Hawking didn’t understand the first thing about gravity, so he didn’t understand the first thing about black holes either. But people believe what he said, because it's been repeated ad infinitum for decades. There are other examples. See for example my &quot;physics detective&quot; essay on &lt;a href=&quot;http://physicsdetective.com/hawking-radiation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hawking radiation&lt;/a&gt;.   &lt;/p&gt;&#xA;" OwnerUserId="3264" LastActivityDate="2018-11-29T22:08:43.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8008" PostTypeId="2" ParentId="7998" CreationDate="2018-11-29T22:34:24.943" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;What motivated physicists to abandon the idea of anti-particles having a negative mass?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Nothing. They never took the idea seriously. Don't forget that in his 1905 &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol2-trans/186&quot; rel=&quot;nofollow noreferrer&quot;&gt;E=mc² paper&lt;/a&gt; Einstein said &lt;em&gt;“the mass of a body is a measure of its energy-content”&lt;/em&gt;. You can remove energy from a thing just as you can shorten a pencil. However when you remove all the energy from a thing it just isn’t there any more, just as a pencil shortened to 0cm isn’t there any more. You cannot then remove more energy to create a negative-energy particle, just as you cannot shorten a pencil to less than 0cm long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, what we're really talking about here is Dirac’s 1930 negative-energy “hole theory” paper &lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/126/801/360&quot; rel=&quot;nofollow noreferrer&quot;&gt;a theory of electrons and protons&lt;/a&gt;. Oppenheimer shot it down straight away in his 1930 paper &lt;a href=&quot;https://authors.library.caltech.edu/1547/&quot; rel=&quot;nofollow noreferrer&quot;&gt;on the theory of electrons and protons&lt;/a&gt;. He said Dirac’s derivation of the Thomson formula was invalid, and there are several grave difficulties which arise when &lt;em&gt;“one tries to maintain the suggestion that protons are the gaps of negative energy, and that there are no distinctive particles of positive charge”.&lt;/em&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dirac then wrote &lt;a href=&quot;http://rspa.royalsocietypublishing.org/content/133/821/60&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantised singularities in the electromagnetic field&lt;/a&gt;. This is the paper where he’s said to have predicted the positron. But what he was really doing was moving the goalposts is response to Oppenheimer’s paper. Graham Farmelo talks about it in his 2010 article &lt;a href=&quot;http://sci-hub.tw/10.1080/00107510903217214&quot; rel=&quot;nofollow noreferrer&quot;&gt;did Dirac predict the positron?&lt;/a&gt; He says Dirac’s &lt;em&gt;“close friend Patrick Blackett, one of the leading players in the story’s denouement, denied it”&lt;/em&gt;. And that &lt;em&gt;“very few physicists took Dirac’s hole theory seriously”&lt;/em&gt;. He also says &lt;em&gt;“Victor Weisskopf later recalled the idea ‘seemed incredible and unnatural to everybody’”&lt;/em&gt;. I would say it's been glamourised since then, without enough attention to the historical facts.  &lt;/p&gt;&#xA;" OwnerUserId="3264" LastActivityDate="2018-11-29T22:34:24.943" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8009" PostTypeId="2" ParentId="8002" CreationDate="2018-11-29T22:39:38.060" Score="2" Body="&lt;p&gt;The prime example is psychoanalysis. Thousands of books were written, popular and not. But the general consensus today if that this is a very weak science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other examples are homeopathy, astrology, and UFOlogy. Unlike psychoanalysis (which is a weak science) these are not sciences at all. Nevertheless an enormous amount of &#xA;literature was written on each of these subjects.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-11-29T22:39:38.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8010" PostTypeId="1" CreationDate="2018-11-30T03:27:05.430" Score="1" ViewCount="78" Body="&lt;p&gt;In chromatography, the signal is shaped like a Gaussian peak, and it is plotted against time vs. instrument's signal. &lt;a href=&quot;https://en.wikipedia.org/wiki/Chromatography#/media/File:Rt_5_12.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chromatography#/media/File:Rt_5_12.png&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(a) One of the ways to measure chromatographic efficiency is to calculate &quot;number of theoretical plates&quot; which is just (time of peak maximum)^2/ (variance of the peak in time units). It is a dimensionless number which essentially shows how narrow the peak is. I have searched the work of Martin &amp;amp; Synge (Nobel Prize) for they came up with the theoretical plate concept, however, nowhere they used this equation. However, every modern textbook, says the number of theoretical plates is defined as (peak maximum)^2/ (variance of the peak). One can use widths of the peak instead of variance, assuming the peak is Gaussian. No logic is provided as to why this measure was invented.&#xA;&lt;a href=&quot;https://www.shimadzu.com/an/hplc/support/lib/lctalk/theoretical_plate.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.shimadzu.com/an/hplc/support/lib/lctalk/theoretical_plate.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(b) Are there any mathematical analogs as to why one would divide the (mean)^2 by its variance or more rigorously, is there any physical interpretation of first-moment squared divided by the variance of the distribution. &lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-11-30T03:27:05.430" Title="Who came up with a number of the theoretical plates equation?" Tags="|mathematics|terminology|statistics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8011" PostTypeId="2" ParentId="8002" CreationDate="2018-11-30T09:12:35.013" Score="1" Body="&lt;p&gt;A good example are the books written by &lt;a href=&quot;https://en.wikipedia.org/wiki/Percival_Lowell&quot; rel=&quot;nofollow noreferrer&quot;&gt;Percival Lowell&lt;/a&gt; about the canals on Mars: &lt;em&gt;Mars&lt;/em&gt;, &lt;em&gt;Mars and its Canals&lt;/em&gt; and &lt;em&gt;Mars as the Abode of Life&lt;/em&gt;. Lots of people believed that this was a proof of the existence of an advanced Martian civilization.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-11-30T09:12:35.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8012" PostTypeId="1" CreationDate="2018-11-30T11:06:54.737" Score="3" ViewCount="454" Body="&lt;p&gt;It seems to me that ancients (greeks at least) knew that in their geocentric model Venus and Mercury were closer to them than the sun, and correctly differenciated inner (Mercury and Venus) from outer planets (Mars, Jupiter...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm trying to understand how they arrived at that conclusion, but I'm unable to.&lt;/p&gt;&#xA;" OwnerUserId="8170" LastActivityDate="2019-07-23T22:40:38.017" Title="How did ancients differenciate between inner and outer planets?" Tags="|archaeoastronomy|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8013" PostTypeId="2" ParentId="8002" CreationDate="2018-11-30T12:24:45.617" Score="3" Body="&lt;p&gt;All of the self-help books out there belong into the dubious category.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether it is about psychology, interpersonal skills, psychology, medicine or especially nutrition; and the biggest loser has to be economics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many psychology research findings are in a &lt;a href=&quot;https://en.wikipedia.org/wiki/Replication_crisis&quot; rel=&quot;nofollow noreferrer&quot;&gt;replication crisis&lt;/a&gt;, not the least because they are based on findings in &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/20550733&quot; rel=&quot;nofollow noreferrer&quot;&gt;weird people&lt;/a&gt;. Yet, the 'results' are gobbled up by popular books and remakers of ideology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Medicine is as much an art form today as it was in the time of Imhotep. Sure, there is very hard science going around and many actors are performing in the best of faiths. But inherent limitations of smaller studies or big systematic reviews are seldom acknowledged by the popular media. Medical professionals should know that: &lt;a href=&quot;https://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why Most Published Research Findings Are False&lt;/a&gt;. But on the bookshelf of a department store you will just find the opposite. I'd nominate any last one of those books that relates to &quot;cure for cancer&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After proteins were discovered, people were told to eat meat, as that was supposedly good for them. Then vitamins were the cure-all for everyone and everything. But cholesterol was a new nemesis, scaring people away from eggs and meat. &quot;Eat more carbs, eat no-carbs, saturated fat is good for you, cholesterol is harmless.&quot;  We know just not enough about the complexities of nutrition and in developed countries people are well fed and mostly approaching an area of diminishing returns. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Looking at the evidence based reviews like at &lt;a href=&quot;https://www.cochrane.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cochrane&lt;/a&gt; we have to face the fact that indeed most dietary advice from the last 100 years was very dubious if it came from medical professionals. And advice that came from dietary zealots that screamed this time we've got it&quot; fared way worse. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given these broad categories to choose a wealth of nominees from, I'd suggest to consider the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Life_Extension:_A_Practical_Scientific_Approach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Durk Pearson and Sandy Shaw: &quot;Life Extension: A Practical Scientific Approach&quot;, 1982&lt;/a&gt;: &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The book discussed free radicals and the idea that they cause aging, and how antioxidants were said to partially prevent the damage they do.&lt;a href=&quot;https://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt; The book suggests causes of aging and ways to slow them, with material on improving health and various aspects of the quality of life.&#xA;  One notable feature of the book was several full-page pictures of its male and female authors, Durk Pearson and Sandy Shaw, striking bodybuilding poses and showing off some impressive muscles for &quot;sedentary research scientists,&quot; which they claimed was due to the &quot;growth hormone releasers&quot; they took daily.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Medical health experts have dismissed Pearson and Shaw's life-extension formula as dangerous. The American Council on Science and Health reported that the risks of their formula included headaches, intestinal disorders and kidney damage.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Put anything &quot;quantum physics&quot; into a popular book, and see the disaster unfold:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Secret_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rhonda Byrne: &quot;The Secret&quot; 2006.&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The book has been translated into 50 languages and has sold over 30 million copies. Due partly to an appearance on The Oprah Winfrey Show, the book and film had grossed $300 million in sales by 2009. Byrne has subsequently released Secret merchandise and several related books.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In 2009, Barbara Ehrenreich published Bright-Sided: How the Relentless Promotion of Positive Thinking Has Undermined America as a reaction to self-help books such as The Secret, claiming that they promote political complacency and a failure to engage with reality.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Byrne's scientific claims, in particular concerning quantum physics, have been rejected by a range of authors including Christopher Chabris and Daniel Simons at The New York Times and Harvard physicist Lisa Randall. Mary Carmichael and Ben Radford, writing for the Center for Inquiry, have also pointed out that The Secret has no scientific foundation, stating that Byrne's book represents: &quot;a time-worn trick of mixing banal truisms with magical thinking and presenting it as some sort of hidden knowledge: basically, it’s the new New Thought.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A syncope opens up between professional psychologists and kitchen psychology with this 'masterpiece'.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Power_of_Positive_Thinking&quot; rel=&quot;nofollow noreferrer&quot;&gt;Norman Vincent Peale: &quot;The Power of Positive Thinking&quot;, 1952&lt;/a&gt;:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Peale's work came under criticism from various mental health experts, theologians, and academics. One general criticism against Peale's book was the lack of verified sources. The Power of Positive Thinking includes many personal anecdotes that the reader has no way of validating. The book includes stories about “a business executive”, “a man, an alcoholic”, “a famous trapeze artist”, “a friend of [Peale’s], a midwestern businessman”, and other unnamed individuals which cannot be verified from the information Peale presents with each anecdote.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[…] Murphy explains that such repeated hypnosis defeats an individual's self-motivation, sense of reality, and ability to think critically. He describes Peale's understanding of the mind as inaccurate and his description of the workings of the mind as deceptively simplistic and false. […]  Ellis stated that eventually Peale's teachings “lead to failure and disillusionment, and not only boomerang back against people, but often prejudice them against effective therapy.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;While contemporary theologians and mental health experts criticized Peale's teachings in The Power of Positive Thinking, the general public praised the self-help book. The Los Angeles Times estimates that “legions of followers testified that Peale’s message changed their lives for the better and represented the best combination of faith and pragmatism.”&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;This is evidenced by the popularity of Peale's book, which sold more than 5 million copies worldwide and was eventually translated into over 40 languages. In addition, Peale was close friends with American presidents Eisenhower and Nixon, both of which highly regarded his positive thinking teachings. Countless others accredited The Power of Positive Thinking for their success in overcoming obstacles including George Foster, of the Cincinnati Reds, Rev. Robert Schuller, founder and pastor of the Crystal Cathedral in Garden Grove, and Billy Graham, a prominent Southern Baptist Minister.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;President Donald Trump has called Peale “his pastor” and “one of the greatest speakers” he had ever seen. Fred and Mary Trump, President Trump's parents, traveled to the Marble Collegiate Church in Manhattan with their children to hear Peale's sermons. Trump grew up hearing Peale's teachings from his parents, and Peale officiated his first wedding. Trump credits his survival in 1990 after bankruptcy to Peale's positive thinking teachings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And some will argue that the following doesn't even really count, but it claims to be based on science, in its title, and has had a huge influence, with a publication run of more than 83 million copies sold:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Dianetics:_The_Modern_Science_of_Mental_Health&quot; rel=&quot;nofollow noreferrer&quot;&gt;L Ron Hubbard: &quot;Dianetics: The Modern Science of Mental Health&quot;, 1950&lt;/a&gt;:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Although it received an initial positive public response,&lt;a href=&quot;https://en.wikipedia.org/wiki/Dianetics:_The_Modern_Science_of_Mental_Health&quot; rel=&quot;nofollow noreferrer&quot;&gt;8&lt;/a&gt; Dianetics was strongly criticized by scientists and medical professionals for its scientific deficiencies. The American Psychological Association passed a resolution in 1950 stating of Dianetics &quot;the fact that these claims are not supported by empirical evidence of the sort required for the establishment of scientific generalizations.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6224" LastEditorUserId="6224" LastEditDate="2018-12-01T13:35:27.467" LastActivityDate="2018-12-01T13:35:27.467" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8014" PostTypeId="2" ParentId="8002" CreationDate="2018-11-30T12:38:06.743" Score="1" Body="&lt;p&gt;There's always the vonDaniken &quot;Chariots of the Gods&quot; collection of nonsense about alien landing strips in the Andes and other UFO-related fakery. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-11-30T12:38:06.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8015" PostTypeId="2" ParentId="8012" CreationDate="2018-11-30T18:10:31.097" Score="1" Body="&lt;p&gt;This is an immediate conclusion from direct observation of the sky. Mercury and Venus accompany the Sun: are never far away from the Sun. Outer planets behave very differently: they can be at any angular distance from the Sun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, the choice between geocentric and heliocentric system&#xA;has nothing to do with the phenomenon: what we see in the sky can be described within any coordinate system with any precision. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of Ptolemy model, the distinction means that the radius of epicycle&#xA;is smaller than the radius of the deferent for Mercury and Venus,&#xA;while for the outer planets it is larger. The order of the planets corresponds to their periods of revolutions on their epicycles.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-01T15:34:48.463" LastActivityDate="2018-12-01T15:34:48.463" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8016" PostTypeId="2" ParentId="8012" CreationDate="2018-11-30T21:57:25.467" Score="1" Body="&lt;p&gt;Of course everything orbits around the earth.&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/aHmWM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aHmWM.png&quot; alt=&quot;diagram&quot;&gt;&lt;/a&gt;&lt;a href=&quot;https://i.stack.imgur.com/xTHLM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xTHLM.png&quot; alt=&quot;diagram&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://courses.lumenlearning.com/sanjac-earthscience/chapter/introduction-to-the-solar-system/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lumenlearning&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some planets (the inner planets) sometimes pass between the sun and the earth, meaning their orbints are inside the orbit of the sun.  Other planets (the outer planets) never pass between the sun and the earth.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2018-11-30T22:03:39.010" LastActivityDate="2018-11-30T22:03:39.010" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8017" PostTypeId="2" ParentId="8012" CreationDate="2018-12-01T00:04:58.717" Score="2" Body="&lt;p&gt;A useful source on this subject is Olaf Pedersen's &lt;a href=&quot;http://books.google.co.uk/books?id=z7M8AAAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Early Physics and Astronomy: A Historical Introduction&lt;/em&gt;&lt;/a&gt;, (first published 1974, CUP reprint 1993). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much is shrouded in the mists of pre-history, but the question appears to suppose that any model would take it that planets are actual bodies at different distances.  The sources that do exist show that this supposition corresponds to what would have been already a somewhat advanced state of knowledge. Consider, for example, Pedersen's mention (p.40) of &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;the theory of Empedocles of Agrigentum (c.493-433 BC), &#xA;for whom the Sun did not exist at all as a material entity. &#xA;He explained day and night by the assumption that a bright &#xA;and a dark hemisphere revolve around the Earth, the light &#xA;from the bright hemisphere being reflected from the Earth &#xA;back onto the heavens as a strongly illuminated spot which &#xA;we call the Sun.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And then, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In the thinking of Anaximander, for example, a fairly &#xA;complete theory of the solar system had already emerged. &#xA;His conception of the planets as wheels filled with fire &#xA;rotating about the Earth is perhaps the first Greek attempt &#xA;to explain astronomical phenomena by means of a mechanical &#xA;model.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pedersen went on to mention (p.60) what may perhaps have been the first Greek model to incorporate the more modern suppositions made in the question: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Aristotle (who never mentions Pythagoras by name and &#xA;possibly did not believe in his existence) says that the &#xA;so-called Pythagoreans assumed the existence of a central &#xA;fire in the middle of the spherical universe. This 'fire &#xA;of Hestia' ... the Pythagoreans believed, is surrounded by &#xA;ten concentric spherical shells or spheres. The inmost but &#xA;one of these spheres takes the Earth round the central &#xA;fire in the course of one day. The inmost sphere moves &#xA;a globe invisible to us, the 'anti-Earth', which also &#xA;performs a revolution about the centre in the course of &#xA;one day, always opposite to the Earth. Outside the Earth &#xA;are the planets in the following order: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Each of these spheres turns from west to east, completing &#xA;a revolution in a period characteristic of the planet. &#xA;Furthest out is the tenth sphere containing the fixed &#xA;stars, which also moves, but so slowly that it is &#xA;imperceptible to the eye.  In this form the theory is &#xA;usually connected with the rather obscure Pythagorean &#xA;Philolaos, who may have lived at the end of the fifth &#xA;century b.c. &quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nothing is said about how or why Philolaos and the Pythagoreans arrived at this view of the correct order of the bodies. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What seems possible (and I haven't seen any citations of evidence either for or against it) is that (as already suggested elsewhere) the ancients may have made inferences about relative distances based on visible phenomena such as the apparent speed of motion of the visible celestial objects relative to the fixed pattern of the starry background.  In any case, it is clear that these motions of the planets relative to the fixed pattern of the other stars were actually noticed -- because they underlie the very name and idea of 'planet' i.e. 'wanderer'. As others have mentioned, perhaps the inference may have been, that the slower the planet moved, the farther away it was.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mars, Jupiter and Saturn share a visible characteristic, that at some time in each year or a little longer, each is visible in the south at midnight, moving slowly west (what we now call retrograde) relative to the fixed stars. The motion of these three relative to the stars is always slower than that of the Sun, Saturn being the slowest, then Jupiter and Mars.  This order of appearances might plausibly be at the origin of the idea that Saturn is farthest away, &amp;amp;c.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand Mercury and Venus are never seen as far from the Sun as the others, they are never visible at midnight, and at their greatest apparent speed, unlike Mars, Jupiter and Saturn, they move faster than the Sun (Mercury faster than Venus).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many ancient theories accordingly differentiated between Mars, Jupiter and Saturn on the one hand, and Mercury and Venus on the other, but they showed a diversity of view about which is closer, Mercury or Venus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy's 'Almagest' is of little help in answering the present question.  He treats the answer as either a settled consensus (without explaining how it came to be) or else (in respect of Mercury and Venus), a matter of long uncertainty or controversy.  Thus, from Toomer's translation (1984) of Book IX, section 1 &quot;on the order of the spheres of sun, moon and the 5 planets&quot;, Ptolemy writes: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;... we see that almost all the foremost astronomers agree that all &#xA;the spheres are closer to the earth than that of the fixed stars, and &#xA;farther from the earth than that of the moon, and that those of the &#xA;three [outer planets] are farther from the earth than those of the &#xA;other [two] and the sun, Saturn’s being greatest, Jupiter's the next &#xA;in order towards the earth, and Mars' below that. But concerning the &#xA;spheres of Venus and Mercury, we see that they are placed below the &#xA;sun’s by the more ancient astronomers, but by some of their successors &#xA;these too are placed above [the sun’s], for the reason that the sun &#xA;has never been obscured by them [Venus and Mercury] either. To us, &#xA;however, such a criterion seems to have an element of uncertainty ... .&quot; &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-12-01T00:04:58.717" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8018" PostTypeId="2" ParentId="8002" CreationDate="2018-12-01T00:39:59.677" Score="-2" Body="&lt;p&gt;There's a long history of &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Scientific_racism&quot; rel=&quot;nofollow noreferrer&quot;&gt;scientific racism&lt;/a&gt;&quot; that &lt;a href=&quot;https://www.theguardian.com/news/2018/mar/02/the-unwelcome-revival-of-race-science&quot; rel=&quot;nofollow noreferrer&quot;&gt;more recently&lt;/a&gt; has included the chicanery by Charles Murray, Steven Pinker, David Reich, Nicholas Wade, et al.&lt;/p&gt;&#xA;" OwnerUserId="7348" LastEditorUserId="7348" LastEditDate="2018-12-01T02:03:22.813" LastActivityDate="2018-12-01T02:03:22.813" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="8020" PostTypeId="2" ParentId="8012" CreationDate="2018-12-01T10:58:26.577" Score="0" Body="&lt;p&gt;I think the question is historically vague so it could be answered by extending the time frame.  The conviction that the Moon is closer than the Sun could be taken for granted, so what remained to be settled  is to order the 5 wandering stars.  The most simple hypothesis is that planets move at the same speed, so the longer orbit needs a longer period. Identifying Vesper as Phosphor  and establishing that its return to the same place takes longer than a year is sophisticated achievement. What was generally known is that it either precedes or follows the sun, so the opinions where to situate it, above or below, seem to have been divided and the same goes for Mercury. The difference with the 3 outer planets that move much slower is emphasized by the fact that they can be observed at any distance from the Sun. &#xA;(Without looking for a reference I have a dim memory that the Maya listed the planets just as the Greeks)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2018-12-01T10:58:26.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8021" PostTypeId="1" CreationDate="2018-12-01T14:47:22.140" Score="0" ViewCount="137" Body="&lt;p&gt;The meter was initially defined as &lt;span class=&quot;math-container&quot;&gt;$10^{-7}$&lt;/span&gt; times the distance of the north pole to the equator. How exactly was this definition used to fabricate the actual meter sticks from which the standard meter bar -which subsequently actually determined how long one meter was- was selected?&lt;/p&gt;&#xA;" OwnerUserId="7241" LastActivityDate="2018-12-31T16:01:50.327" Title="How was the historical definition of the meter used in practice?" Tags="|physics|units|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8022" PostTypeId="2" ParentId="8021" CreationDate="2018-12-01T15:26:55.700" Score="2" Body="&lt;p&gt;Once the definition in terms of the length of the meridian was accepted, the French sent several expeditions to measure the meridian. On the basis of these expeditions, the reference meter (a stick of a special alloy, which has very little thermal expansion) was made and kept&#xA;under controlled conditions. Later, when more precise measurements of Earth were made, it was decided to change the definition, and use the standard reference stick&#xA;instead of the length of the meridian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See, for example &lt;a href=&quot;https://writingwithrobin.com/2014/04/26/measure-for-measure-delambre-mechain-and-the-meter-the-things-that-endure-part-1-of-2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-12-01T15:26:55.700" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8024" PostTypeId="1" CreationDate="2018-12-01T21:30:28.910" Score="2" ViewCount="201" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Michell&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Michell&lt;/a&gt; proposed black holes in the 18th century, hundreds of years before Schwarzschild and Einstein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His ideas were said to to be away head of his time, that he died in obscurity. I assume racism might play a role in this as he was a black man.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The APS &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Michell&quot; rel=&quot;nofollow noreferrer&quot;&gt;states&lt;/a&gt; that while&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he was one of the most brilliant and original scientists of his time, Michell remains virtually unknown today, in part because he did little to develop and promote his own path-breaking ideas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6459" LastEditorUserId="144" LastEditDate="2018-12-02T10:03:16.623" LastActivityDate="2018-12-02T16:33:31.360" Title="If John Michell was more well known, would he rank above Isaac Newton in the history of science?" Tags="|mathematics|physics|philosophy-of-science|" AnswerCount="2" CommentCount="7" ClosedDate="2018-12-03T22:53:45.363" ContentLicense="CC BY-SA 4.0" />
  <row Id="8025" PostTypeId="2" ParentId="8024" CreationDate="2018-12-02T10:50:00.067" Score="4" Body="&lt;p&gt;I'd argue Michell is more like Schwarzschild (albeit perhaps more prolific than him, but then he did live longer).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Michell's &lt;a href=&quot;https://en.wikipedia.org/wiki/Dark_star_(Newtonian_mechanics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;dark stars&lt;/a&gt; aren't really analogous to Schwarzschild's black holes, because in Newtonian mechanics there's no limit to how fast something can go, and the speed of light is relative like everything else, albeit so fast you couldn't test that in Michell's lifetime. All Michell did was calculate the size of an object needed to get an escape velocity equal to the measured speed of light, and even then he would have made an errant density assumption, which is why he thought it would take 500 solar masses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously Michell had other contributions to physics, but dark stars illustrate a general principle. Newton and Einstein added new fundamental principles to physics that challenged what came before: the former explained motion with second-order differential equations, and said gravity was inverse-square; Einstein challenged Galilean invariance and instant transmission of gravity. Neither Michell nor Schwarzschild did such things; they merely calculated the implications of ideas they took for granted.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-12-02T10:50:00.067" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8026" PostTypeId="2" ParentId="8024" CreationDate="2018-12-02T16:27:30.973" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;If John Michell was more well known, would he rank above Isaac Newton in the history of science?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't think so. Yes, Michell was a smart guy all right. He's the man who devised the torsion balance used by Henry Cavendish to determine the mass of the Earth. And he was &lt;em&gt;way&lt;/em&gt; ahead of his time when it came to dark stars. See his paper &lt;a href=&quot;http://rstl.royalsocietypublishing.org/content/74/35&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Dear Sir, The method, which I mentioned to you when I was last in London, by which it might perhaps be possible to ind the distance, magnitude, and weight of some of the fixed stars, by means of the diminution of the velocity of their light, occurred to me soon after i wrote what is mentioned by Dr. Priestley in his History of optics, concerning the diminution of the velocity of light in consequence of the attraction of the sun; but the extreme difficulty, and perhaps impossibility, of procuring the other data necessary for this purpose appeared to be to be such objections against the scheme, when I first thought of it, that I gave it them no farther consideration. As some late observations, however, begin to give us a little more chance of procuring some at least of these date, I thought it would not be amiss, that astronomers should be apprized of the method, I propose (which, as far as I know, has not been suggested by any one else) left, for want of being aware of the use, which may be made of them, they should neglect to make the proper observations, when in their power; I shall therefore beg the favour of you to present the following paper on this subject to the Royal Society...&quot;&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just love &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=John_Michell&amp;amp;oldid=761228985#Black_holes&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; bit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;“if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability”.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That’s exactly what we’ve done to establish the existence of a supermassive black hole &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Sagittarius_A*&amp;amp;oldid=769794306&quot; rel=&quot;nofollow noreferrer&quot;&gt;in the centre of our galaxy&lt;/a&gt;. But I think Newton was a true genius. I've done a lot of research into the history of physics. and I'm something of an Einstein fan. But I have to say I think Newton is number one. That's why I wrote a little physicsworld article called &lt;a href=&quot;https://www.google.co.uk/search?ei=6QYEXIHmE6XhkgWpgoOoDw&amp;amp;q=%22mr+newton%27s+classroom%22&amp;amp;oq=%22mr+newton%27s+classroom%22&amp;amp;gs_l=psy-ab.3..35i302i39.4023.6001..6213...0.0..0.118.199.1j1......0....1..gws-wiz.......35i39.ZgpT4rqkZeE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mr Newton's classroom&lt;/a&gt; back in 2010. I've done a lot more reading since then, and I still think Newton is number one.  &lt;/p&gt;&#xA;" OwnerUserId="3264" LastEditorUserId="3264" LastEditDate="2018-12-02T16:33:31.360" LastActivityDate="2018-12-02T16:33:31.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8027" PostTypeId="1" AcceptedAnswerId="8034" CreationDate="2018-12-02T17:47:41.987" Score="3" ViewCount="403" Body="&lt;p&gt;First of all, you should be at least a little familiar with &lt;a href=&quot;https://en.wikipedia.org/wiki/Combinatorics&quot; rel=&quot;nofollow noreferrer&quot;&gt;combinatorics&lt;/a&gt; to understand that question.&#xA;Some often used calculator keys in stochastic are the &lt;kbd&gt;nCr&lt;/kbd&gt; and &lt;kbd&gt;nPr&lt;/kbd&gt; ones.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I've first asked this question &lt;a href=&quot;https://german.stackexchange.com/questions/48196/stochastic-why-are-permutations-npr-called-variations-in-german&quot;&gt;for the German&lt;/a&gt;, &lt;a href=&quot;https://english.stackexchange.com/questions/475344/stochastic-why-are-permutations-npr-called-variations-in-non-english-language&quot;&gt;English Stackexchange version&lt;/a&gt; (both places where &quot;mathematics&quot; tags exist) and usual &lt;a href=&quot;https://math.stackexchange.com/questions/3022884/why-are-permutations-npr-called-variations-in-non-english-languages&quot;&gt;Math Stackoverflow&lt;/a&gt;, but as it turned out this is likely also the case for all non-English languages and the community here may be better suited to answer the question, I've also posted it here. Despite that, below the German language is referred to as an example, where it is called differently.&lt;br&gt;&#xA;&lt;strong&gt;Edit2:&lt;/strong&gt; Also posted at &lt;a href=&quot;https://hsm.stackexchange.com/questions/8027/why-are-permutations-npr-called-differently-in-non-english-languages-german&quot;&gt;History of Science and Mathematics&lt;/a&gt; and &lt;a href=&quot;https://linguistics.stackexchange.com/questions/29822/stochastic-why-are-permutations-npr-called-variations-in-non-english-language&quot;&gt;linguistics&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit3:&lt;/strong&gt; In French this &lt;a href=&quot;https://math.stackexchange.com/questions/3022884/why-are-permutations-npr-called-variations-in-non-english-languages#comment6233252_3022884&quot;&gt;seems to be called&lt;/a&gt; &lt;a href=&quot;https://fr.wikipedia.org/wiki/Arrangement&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;arrangement&quot;&lt;/a&gt;. And &lt;a href=&quot;https://math.stackexchange.com/questions/3022884/why-are-permutations-npr-called-variations-in-non-english-languages?noredirect=1#comment6233374_3022884&quot;&gt;in Russian&lt;/a&gt; &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BC%D0%B5%D1%89%D0%B5%D0%BD%D0%B8%D0%B5&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;razmeshhenie&quot;&lt;/a&gt;, &quot;which also means arrangement&quot;.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;&lt;span class=&quot;math-container&quot;&gt;$_nC_r$&lt;/span&gt; = combinations&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;&lt;kbd&gt;nCr&lt;/kbd&gt; is quite obvious. The &quot;C&quot; stands for &lt;a href=&quot;https://en.wikipedia.org/wiki/Combination&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;combinations&quot;&lt;/a&gt; (actually those &lt;a href=&quot;https://de.wikipedia.org/wiki/Kombination_(Kombinatorik)#Kombination_ohne_Wiederholung&quot; rel=&quot;nofollow noreferrer&quot;&gt;without repetition&lt;/a&gt;) and this is how they are called in German and English. That is just the &lt;a href=&quot;https://en.wikipedia.org/wiki/Binomial_coefficient&quot; rel=&quot;nofollow noreferrer&quot;&gt;binomial coefficient&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\binom{n}{k}=\frac{n!}{k!(n-k)!}=_n\!\!C_k$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;&lt;span class=&quot;math-container&quot;&gt;$_nP_r$&lt;/span&gt; = permutations (English)/variations (German)&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Keeping that knowledge in mind, as a German, you would assume &lt;kbd&gt;nPr&lt;/kbd&gt; is for calculating the &lt;a href=&quot;https://de.wikipedia.org/wiki/Permutation#Permutation_ohne_Wiederholung&quot; rel=&quot;nofollow noreferrer&quot;&gt;permutations (without repetition, again)&lt;/a&gt;, i.e. just:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n!$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that's not the case, actually it calculates the &lt;a href=&quot;https://de.wikipedia.org/wiki/Variation_(Kombinatorik)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;variation&quot;&lt;/a&gt;, as it's called in German:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{n!}{(n-k)!} =_n\!\!C_r$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And it is true: &lt;a href=&quot;http://www.rither.de/a/mathematik/stochastik/kombinatorik/variation-und-permutation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Actually the &quot;P&quot; does stand for &quot;permutation&quot;&lt;/a&gt; in English. So the last formula is what they call &lt;a href=&quot;https://en.wikipedia.org/wiki/Permutation#k-permutations_of_n&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;permutation&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Just different names?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;So we could say, these are just different names, but no, it gets more complicated, because – using the German terms here again – &lt;a href=&quot;http://www.rither.de/a/mathematik/stochastik/kombinatorik/variation-und-permutation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;permutations are just a special kind of variations&lt;/strong&gt;&lt;/a&gt;. Essentially, it's the last formula, where &lt;code&gt;k=n&lt;/code&gt;, i.e. you choose all items and do not select a subset when arranging them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Obviously the English mathematics do not use the term &quot;permutations&quot; for the specific version we name it in German, but for the general version.&#xA;Essentially this leads to another problem, however, when we look at nPr &lt;strong&gt;with repetition&lt;/strong&gt;. All examples before where without repetition, but you have formulas for the ones with repetition, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the &quot;permutation with repetition&quot;/&quot;Variation mit Wiederholung&quot; and is easy to calculate, you just:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n^k$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Permutation#Permutations_with_repetition&quot; rel=&quot;nofollow noreferrer&quot;&gt;does not seem to want to acknowledge the English term for that&lt;/a&gt; saying they have &quot;sometimes been referred to&quot; in this way… (Or is this actually something different as the formula is k^n?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, if we assume the term is used like that, we've got another way to have German &quot;Permutationen&quot; &quot;with repetition&quot;. This time, however, as in the German definition of permutations we do not select items, we just have multiple of the same items. So e.g. you have r, s, …, t same elements in n elements you get a formula like that:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{k!}{r!\cdot s! \cdots t!}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And this is what we call &lt;a href=&quot;https://de.wikipedia.org/wiki/Permutation#Permutation_mit_Wiederholung&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Permutation mit Wiederholung&quot;&lt;/a&gt; in German. But what term is then used in the English for this kind of &quot;repetition&quot;?&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Questions&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;So how did this inconsistent naming across languages happen? Is there any &quot;correct&quot; term or has one term been invented before another one, so someone adapted it wrong? &#xA;Do other languages possibly also name it differently, i.e. is the German naming the exception or the English one?&#xA;And what term is used for &quot;Permutationen mit Wiederholung&quot;/same elements in a set in English then?&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;See also&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;If you need some more understanding:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/a/429495&quot;&gt;Here are the formulas for the calculator keys&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.studyhelp.de/online-lernen/mathe/kombinatorik/#MathJax-Element-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is another overview about the subject as Germans see it&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I found something: &lt;a href=&quot;https://en.wikipedia.org/wiki/Variation&quot; rel=&quot;nofollow noreferrer&quot;&gt;The English Wikipedia describes the term &quot;variations&quot; as&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;em&gt;Variations without repetition&lt;/em&gt;, an archaic term in combinatorics still commonly used by non-English authors for k-permutations of n&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;Variations with repetition&lt;/em&gt;, an archaic term in combinatorics still commonly used by non-English authors for n-tuples&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Despite that sounding a little pejorative to me as a German speaker, it raises the question of whether this is really (internationally?) deprecated/outdated? Or what term is supposed to be used?&#xA;Also the relation to tuples, which are – I thought – just a different concept of a list of numbers, is not clear to me. After all, I could not found any of the formulas I've just mentioned &lt;a href=&quot;https://en.wikipedia.org/wiki/Tuple&quot; rel=&quot;nofollow noreferrer&quot;&gt;in the linked article&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="8178" LastEditorUserId="55" LastEditDate="2018-12-05T04:09:10.627" LastActivityDate="2018-12-05T19:17:16.280" Title="Why are permutations ($_nP_r$) called differently in non-English languages (&quot;variations&quot; in German)?" Tags="|mathematics|terminology|language|combinatorics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8029" PostTypeId="2" ParentId="8027" CreationDate="2018-12-02T21:14:14.717" Score="2" Body="&lt;p&gt;&quot;...what term is used for &quot;Permutationen mit Wiederholung&quot;/same elements in a set in English?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The combinatorics expressions you're looking for in English are probably 'permutations/combinations with/without replacement' &amp;amp;c, rather than 'with/without repetition'. See this answer on the math stack exchange:  &lt;a href=&quot;https://math.stackexchange.com/questions/474741/formula-for-combinations-with-replacement&quot;&gt;https://math.stackexchange.com/questions/474741/formula-for-combinations-with-replacement&lt;/a&gt; . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also you might see &lt;a href=&quot;https://web.ma.utexas.edu/users/parker/sampling/repl.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sampling With Replacement and Sampling Without Replacement&lt;/em&gt;&lt;/a&gt; for other usage examples.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, as in many other cases of technical translation, it's not guaranteed that the (nearly) equivalent phrases have exactly the same scope. I hope that helps.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2018-12-02T21:14:14.717" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8030" PostTypeId="2" ParentId="7943" CreationDate="2018-12-02T23:45:46.603" Score="6" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;number versus numeral (a distinction everyone learns in the third grade – except for those who played hooky that day)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;loss of unique-factorization as you move from the reals to the complex numbers – e.g., 26 = (2)(3) and 26 = (1 – 5i)(1 + 5i)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;for two distinct numbers, in the set of positive numbers “is to the left of” and “is farther from 0” are synonymous, but are not synonymous for the set of real numbers.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For two distinct lines, in the plane “are everywhere equidistant from each other” and “do not intersect” are synonymous, but are not synonymous for 3D space.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;rotation versus revolution&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;apparent astronomical movement versus actual astronomical movement&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;dependence of final result on initial conditions, versus, independence&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;rational versus irrational numbers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a polynomial versus the polynomial function associated with it&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In foraging theory, time minimization versus energy maximization (pp. 8-9 of the book ‘Foraging Theory’ by Stephens and Krebs)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;explanation versus prediction&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;tracking versus detection&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;detection versus recognition&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;ordinal numbers versus cardinal numbers&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In foraging, path depletion not equal to negative acceleration of the energy gain function&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;complete information versus perfect information&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a function being analytic versus being infinitely differentiable&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;two types of paraboloid (elliptic and hyperbolic)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;wave/particle duality of light&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;MAD (median absolute deviation) has more than one meaning&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;inertial mass versus rest mass&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;domain of a partial function is ambiguous, depending on the discipline (logic or mathematics)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;multiple, and only partially satisfactory, definitions of tortuosity&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;general life situation versus general life situation (terminology of Kurt Lewin)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;singularities of solutions not necessarily occurring only at singularities of the equation&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;inequality of the types of cardinality for surface area and volume (e.g.: Gabriel’s horn)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;sometimes homeomorphism type is not determined by homotopy type&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;“There are several definitions of R2 that are only sometimes equivalent.”&#xA;(Wikipedia’s article on coefficient of determination)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;coverage probability splits into ‘actual’ and ‘nominal’&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;utility versus exactness – e. g., Agresti and Coull's 1998 paper “Approximate is Better than ‘Exact’ for Interval Estimation of Binomial Proportions.” (cited in the Wikipedia article on binomial proportion confidence intervals)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;having to choose between a statistical estimator that is unbiased or which has better mean squared error&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;There are two types of Hermite polynomials: the ‘probabilists’ Hermite polynomials and the ‘physicists’ Hermite polynomials.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;good for exploratory data analysis versus good for classification applications – e. g., Sammon mapping&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;canonical form vs normal form (see the Wikipedia article on computer algebra)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;A subgroup of a finitely generated group need not be finitely generated.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;exploiting prey versus exploiting patches&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the zero-one law in foraging theory versus Kolmogorov’s zero-one law – the former being prescriptive, and the latter being descriptive&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;double-entry bookkeeping versus single-entry bookkeeping&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;agent-designer’s goals versus agent’s goals&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;how the product topology is defined for finitely many spaces versus how it is defined for infinitely many spaces&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;‘heavy-tailed’ distribution has several meanings&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;non-unique generalization of the single-variable derivative&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a series converging versus getting arbitrarily many digits correct&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;convexity of a set versus convexity of the region bounded by the set&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;dice equivalence versus dice winning against each other with equal probability&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;connectedness versus i-connectedness&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;connectedness versus path-connectedness&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;leaves versus structures that look like leaves (such as that possessed by mosses and leafy liverworts)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the polyphyletic nature of algae versus the situation of living in water and performing photosynthesis&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;whether energy is present versus whether it is available&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;sidereal time versus solar time&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;rolling friction versus static friction&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;planet versus star&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;elastic versus inelastic collision&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;heat versus temperature&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a removable versus a non-removable discontinuity&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;blood versus type-distinguished blood&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;vapor versus gas&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;air versus oxygen&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;whale versus fish&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;gold versus fool’s gold&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;rocket propulsion versus friction-based propulsion&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;physical change versus chemical change&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;chemical combustion versus stellar dynamics&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;warm-blooded versus cold-blooded creatures&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;robustness versus anti-fragility&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;linear response versus nonlinear response&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;chaotic versus non-chaotic phenomena&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;continuity versus differentiability&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;speed of sound in air versus speed of sound in water&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;how others hear us versus how we hear ourselves&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;compound versus element&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bronze-Age creation myth versus Evolution&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;jealousy versus envy&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;perception controlling behavior versus behavior controlling perception&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;conscious versus unconscious mind&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;momentum versus energy&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;potential energy versus kinetic energy&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;radiant energy versus heat&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;24 hour period versus calendar day (as in ‘Around the World in 80 Days’)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;mass versus weight&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;currency versus money&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;sub-sonic versus super-sonic explosions&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;cycloid versus circular arc&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;coma versus death&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;medical intervention versus palliative care&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;data versus information&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;macro versus micro economics&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;weather versus climate&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;strategy versus tactics&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;longitudinal versus transversal waves&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;traditional versus public-key cryptography&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the definition of uniform integrability in measure theory versus probability theory&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nash equilibrium for a game repeated finitely many times versus infinitely many times&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;looking only at truth values versus looking at content (material implication)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Spheroidal coordinates are of two types: oblate and prolate.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;how symmetric groups behave on finite versus on infinite sets&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;optimal behavior in the Prisoners’ Dilemma in the short run (betrayal)&#xA;  versus in the long run (cooperation)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If W is a generalized complex subspace of a generalized complex vector&#xA;space V, then V/W is not necessarily a generalized complex quotient of V.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;topological definition of an object versus geometrical definition&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;stable, versus merely long-lived&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;defining fields by polynomials giving different results in the finite and infinite cases&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;temperature versus conductivity&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;intensive versus extensive properties&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;addition of heat versus a rise in temperature (i.e., the phenomenon of ‘latent heat’)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Subsystems can be connected in series or parallel.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;distinct types of rock (igneous, sedimentary, metamorphic)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;musical time versus metronome time&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;amortized update time of an algorithm versus worst-case update time&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;absence of evidence versus evidence of absence&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;simple versus small (e.g., uncompressed data versus compressed;&#xA; cf: Pascal’s remark about not having time to make his letter shorter)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;‘no students’ (but staff present) versus ‘no school’ (nobody present)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a commonly accepted proxy for a hard-to-pin-down concept (e.g., variance as a&#xA;    proxy for risk); also: measure of an entity versus a (more convenient) proxy&#xA;    measure for the entity (e.g., weight as a proxy measure for mass)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;‘good’ cholesterol versus ‘bad’ cholesterol&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;cis versus trans configurations of unsaturated fatty acids&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;direct current versus alternating current&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;animal classification by its diet versus by the nature of its digestive system&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;frequency of a burden versus its intensity&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;specification versus implementation&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;testing versus debugging&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;impossibility versus probability of 0&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;de jure versus de facto&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;topological convergence versus convergence in measure&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;frontier versus wilderness&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;revenue versus profit&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;voltage versus current&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;dictionary versus encyclopedia&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;conservation as wilderness preservation versus as resource management&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;background versus foreground&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;short term versus long term&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;duality in terms of polar reciprocation versus topological duality&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;a candid photo versus a posed photo&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;mixture versus solution&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;billable time versus elapsed time&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;independent generalizations (such as how Newton’s work stands in&#xA;  relation to Kepler’s work) versus dependent generalizations (such as&#xA;  how the Law of Cosines stands in relation to the Pythagorean Theorem,&#xA;  or how Taylor’s Theorem stands in relation to Maclaurin’s Theorem)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;ability as a conversationalist versus ability as a lecturer&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;archival storage versus working storage&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;complex versus complicated&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;powdered chocolate mix for a cold drink versus for a hot drink&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;form determined by gravity versus by capillarity&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;vector as an arrow (‘something with magnitude and direction’) versus as an element&#xA;    of a vector space&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;see also the website ‘DifferenceBetween.net’:&#xA;&lt;a href=&quot;http://www.differencebetween.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.differencebetween.net/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8073" LastEditorUserId="8073" LastEditDate="2019-03-02T01:06:25.560" LastActivityDate="2019-03-02T01:06:25.560" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8031" PostTypeId="2" ParentId="7974" CreationDate="2018-12-04T00:34:27.030" Score="9" Body="&lt;p&gt;Well, the history is not too simple, even though you have traced some correct events. It was Albert Einstein on November 4th, 1915 in his paper &quot;Zur allgemeinen Relativitätstheorie&quot; (Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte (1915)) in which he officially put &quot;the gravitational field&quot; (English translation) in connection (pun intended!) with the Christoffel symbols as: &lt;span class=&quot;math-container&quot;&gt;$\Gamma^{\tau}_{\mu\nu} = -\left\{^{\mu\nu}_{\tau}\right\}$&lt;/span&gt; (&lt;a href=&quot;http://www.archive.org/stream/sitzungsberichte1915deut#page/782/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;eqn. 15 on page 783&lt;/a&gt;). The Gamma symbol had been previously introduced by Einstein in 1914 in his article &quot;Die formale Grundlage der allgemeinen Relativitätstheorie&quot; on &lt;a href=&quot;https://www.biodiversitylibrary.org/item/92711#page/1140/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;page 1058&lt;/a&gt;. That equation 15 had previously been written as equation (45) of the famous &quot;Zurich Notbook&quot;, a collection of notes written by Einstein from 1912 to 1913 which served him with the development of GR up to 1915. It was this connection between the geometrical quantities of Christoffel and the then-sought theory of gravitation that was made during the year 1912-1913. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, it was Weyl in 1918 in his article in the &quot;Mathematische Zeitschrift&quot; called &quot;Reine Infinitesimalgeometrie&quot; (article starting on &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN266833020_0002?tify=%7B%22pages%22:[394],%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;page 384&lt;/a&gt;) who used the Gamma symbol of 3 indices to denote today's &quot;Weyl affine connection&quot; (unnumbered equation on &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN266833020_0002?tify=%7B%22pages%22:[404],%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;page 394&lt;/a&gt;). He showed later in the article how this new object is a generalization of the Christoffel symbols. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps a more suitable mentioning of the Gamma used as a generalization of the Christoffel symbols is made by the article THE RIEMANN GEOMETRY AND ITS GENERALIZATION  published by Veblen &amp;amp; Eisenhart in 1922, in the PNAS from which I quote: &lt;em&gt;&quot;This generalized geometry has been studied by H. Weyl in his book,&#xA;&quot;Raum, Zeit, Materie,&quot; Berlin, 1919, and in Vol. 1 of the &quot;Mathematische&#xA;Zeitschrift.&quot; It has also been considered by A. S. Eddington in Proc. Roy.&#xA;Soc. London, 99A (1921). Both these authors define it in terms of a&#xA;generalization of Levi-Civita's concept of infinitesimal parallelism rather&#xA;than by the more natural idea of a system of paths.&quot;&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The complete dissolution of the curly-straight backet notation for the Christoffel symbols is hard to track: In 1940 Eisenhart published his Diff. Geom. book, he still used the original notation of Christoffel. In 1949 in his &quot;Riemannian Geometry&quot; it was still there. &lt;/p&gt;&#xA;" OwnerUserId="7069" LastEditorUserId="643" LastEditDate="2018-12-04T02:55:12.730" LastActivityDate="2018-12-04T02:55:12.730" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8032" PostTypeId="1" CreationDate="2018-12-04T03:58:09.383" Score="0" ViewCount="53" Body="&lt;p&gt;I can somewhat understand how the spark-gap was devised to transmit waves and how the waves were first measured in frequency but what lead to the idea  of &quot;modulation&quot; where another electrical signal generated from a voice would be carried on the wave produced by the spark-gap.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fessenden was I believe the first to demonstrate this by constructing large alternating current spark generators but he would still need to know how to &quot;encode&quot; a wave onto another wave and rectify it in such a way at to both transmit and receive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was even wondering if perhaps he was proficient in Fourier analysis or perhaps some other mathematics to figure out how to encode or modulate and demodulate the signal.  I am still researching the topic and the more I think about this the more amazing of a feat it seems to be.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-12-04T03:58:09.383" Title="How did Reginald Fessenden realize that spark-gaps could generate waves and the modulation of those waves with voice?" Tags="|electromagnetism|electricity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8033" PostTypeId="2" ParentId="8027" CreationDate="2018-12-04T19:44:09.943" Score="2" Body="&lt;p&gt;This isn't an answer, but I think it's too long for a comment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes the most literal translation of a technical term doesn't sound quite right in the language you're translating to. For example, Schrödinger used the term &lt;a href=&quot;https://www.collinsdictionary.com/dictionary/german-english/verschranken&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;verschränken&lt;/em&gt;&lt;/a&gt; in German to describe particles related by Einstein's spooky-action-as-a distance, and then he used the word &lt;em&gt;entangled&lt;/em&gt; in English, because a more literal translation of &lt;em&gt;verschränken&lt;/em&gt; didn't have the right feel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to answer this properly, you might have to go back to the history of the words&#xA;&lt;em&gt;arrangement&lt;/em&gt; in French, &lt;em&gt;k-permutation&lt;/em&gt; in English, &lt;em&gt;Variation&lt;/em&gt; in German, find out when they were first used and who first used them, see whether they cited any papers in other languages, and try to decide whether a literal translation would have been perceived as incorrect for some reason. For example, I don't think &lt;em&gt;variation&lt;/em&gt; would sound right in English, although &lt;em&gt;permutation&lt;/em&gt; and &lt;em&gt;arrangement&lt;/em&gt; would both work.&lt;/p&gt;&#xA;" OwnerUserId="3844" LastEditorUserId="3844" LastEditDate="2018-12-04T22:54:53.760" LastActivityDate="2018-12-04T22:54:53.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8034" PostTypeId="2" ParentId="8027" CreationDate="2018-12-05T03:52:14.653" Score="3" Body="&lt;p&gt;A place to start for questions of this sort is Jeff Miller's website &lt;a href=&quot;http://jeff560.tripod.com/p.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;. On permutations and combinations one of his sources is &lt;a href=&quot;https://archive.org/details/historyofmathema031897mbp/page/n3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smith, History Of Mathematics Vol II, p.28&lt;/a&gt; (freely available), where we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first work of any extent that is devoted to the subject was Jacques Bernoulli's Ars Conjectandi. This work contains the essential part of the theory of combinations as known today. In it appears in print for the first time, with the present meaning, the word &amp;quot;permutation&amp;quot;. For this concept Leibniz had used variationes and Wallis had adopted alternationes. The word &amp;quot;combination&amp;quot; was used in the present sense by both Pascal and Wallis. Leibniz used complexiones for the general term, reserving combinationes for groups of two elements and conternationes for groups of three, words which he generalized by writing con2natio, con3natio, and so on.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Miller marks the attribution to Ars Conjectandi (1713) as incorrect and gives two earlier sources for &amp;quot;permutation&amp;quot;, based on OED:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1678 Thomas Strode, A Short Treatise of the Combinations, Elections, Permutations &amp;amp; Composition of Quantities, has: “By Variations, permutation or changes of the Places of Quantities, I mean, how many several ways any given Number of Quantities may be changed.”&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Lexicon Technicum, or an universal English dictionary of arts and sciences (1710) has: “Variation, or Permutation of Quantities, is the changing any number of given Quantities, with respect to their Places.”&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the &amp;quot;inconsistency&amp;quot; was there from the start, as it usually is with ideas from multiple origins, rather than coming from a breakthrough by a single person. In such cases it is the eventual consistency, if it happens, that requires explanation, not the inconsistency. Mathematicians often stick to the use encountered in the native language textbooks, and those tend to emphasize what was introduced by their &amp;quot;own&amp;quot;. The calculus notation story on the continent vs Britain is well known, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/7704/was-english-maths-at-any-point-behind-europe-by-50-100-or-more-years-because-o&quot;&gt;Was English mathematics behind Europe by many years because of Newton&amp;#39;s notation?&lt;/a&gt; When people encounter several different words for the same term they tend to introduce some distinctions, and adapt each word to a subclass of the original meaning. And so on.&lt;/p&gt;&#xA;&lt;p&gt;These issues are not settled by good reasons. There are no &amp;quot;correct&amp;quot; terms, and therefore, no wrong adaptations. This is how language is supposed to evolve. By the way, permutation (&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B5%D1%81%D1%82%D0%B0%D0%BD%D0%BE%D0%B2%D0%BA%D0%B0&quot; rel=&quot;nofollow noreferrer&quot;&gt;perestanovka&lt;/a&gt;) is also used in Russian, although more typically in algebraic rather than combinatorial contexts.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-12-05T19:17:16.280" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8035" PostTypeId="1" AcceptedAnswerId="8048" CreationDate="2018-12-05T14:41:57.883" Score="2" ViewCount="287" Body="&lt;p&gt;In France, they refer to the H-theorem of Boltzmann (&lt;em&gt;Théorème H&lt;/em&gt;) as 'eta'-theorem (&lt;em&gt;théorème 'eta'&lt;/em&gt;). The connection obviously comes from the uppercase version of the Greek letter &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt;, which looks like a capital H. Why would they associate H to &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt; (?) and not to &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; as Hamiltonian?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update: I have added my answer. If anybody has a follow up on the Brush article or another article about this curiosity feel free to comment or answer.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2018-12-13T15:26:12.870" LastActivityDate="2018-12-13T15:26:12.870" Title="Why is the H theorem called the big Eta theorem?" Tags="|terminology|statistical-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8036" PostTypeId="2" ParentId="7957" CreationDate="2018-12-06T06:12:22.667" Score="6" Body="&lt;p&gt;The book &lt;em&gt;Introduction to the Theory of Fourier's Series and Integrals&lt;/em&gt; by H. S. Carslaw answers your questions in the first chapter on the History of this subject. Many commonly held false beliefs are debunked in his first chapter, including the idea that Fourier failed to give a rigorous proof of convergence. Another common false belief is that Fourier discovered the Fourier coefficients. The &quot;orthogonality&quot; conditions were discovered by Clairaut and Euler. This is a fascinating bit of History that is often incorrectly cited.&lt;/p&gt;&#xA;" OwnerUserId="7962" LastActivityDate="2018-12-06T06:12:22.667" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8037" PostTypeId="1" CreationDate="2018-12-06T08:19:56.277" Score="1" ViewCount="182" Body="&lt;p&gt;Did engineers like I.K. Brunel and his contemporaries employ calculus in their constructions? Or did they work just with 'rules of the thumb' and useful 'laws' like the square-cube...? What was the level of maths (beyond simple algebra and trigonometry) used by engineers in those times? &lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2018-12-06T13:19:07.130" Title="Level of maths of engineers in the Industrial Revolution" Tags="|mathematics|calculus|engineering|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8038" PostTypeId="2" ParentId="8037" CreationDate="2018-12-06T10:17:38.570" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Isambard_Kingdom_Brunel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brunel (1806-1859)&lt;/a&gt; in 1822 was a candidate to the &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89cole_Polytechnique&quot; rel=&quot;nofollow noreferrer&quot;&gt;École Polytechnique&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Augustin-Louis Cauchy (1789-1857)&lt;/a&gt; graduated in civil engineering and in 1816 was appointed professor at École Polytechnique.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The seminal &lt;a href=&quot;https://en.wikipedia.org/wiki/Cours_d%27Analyse&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cours d'Analyse&lt;/a&gt; was published by Cauchy in 1821 as &lt;em&gt;Cours d'Analyse de l’École Royale Polytechnique&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On this ground, we have to presume that the calculus was part of Brunel's professional knowledge.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-12-06T13:19:07.130" LastActivityDate="2018-12-06T13:19:07.130" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8039" PostTypeId="1" AcceptedAnswerId="8041" CreationDate="2018-12-07T19:39:12.060" Score="2" ViewCount="259" Body="&lt;p&gt;I've been reading Galois' First Memoir, where he introduces Galois Theory by giving a sufficient and necessary condition for a polynomial to be solvable by radicals. The proofs are a bit sketchy and the terminology is strange, which I'm guessing is normal. Up to this point I have seen no mention of the concept of a basis, which I deem rather interesting since bases are an essential, even necessary pillar of Galois Theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I've read that Galois was able to prove some of his theorems by use of different strategies as the ones taught today, for instance, Peter M. Neumann explains that Galois did never prove that the alternating groups &lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt;&#xA;are simple for &lt;span class=&quot;math-container&quot;&gt;$n&amp;lt;4$&lt;/span&gt;, supposedly, he did not need to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which leads me to wonder: Did Galois make use of bases to prove his Fundamental Theorem? If so, could someone please point me to the source. And if not, how was he able to prove there was a bijection between the intermediate fields of an extension and the subgroups of its Galois group?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I appreciate any thoughts/help!&lt;/p&gt;&#xA;" OwnerUserId="8199" LastEditorUserId="4251" LastEditDate="2021-06-22T09:01:38.870" LastActivityDate="2021-06-22T09:01:38.870" Title="Did Galois make use of the concept of a basis?" Tags="|mathematics|abstract-algebra|group-theory|galois-theory|galois|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8041" PostTypeId="2" ParentId="8039" CreationDate="2018-12-08T09:43:53.293" Score="8" Body="&lt;p&gt;No, he did not. You are used to see Galois theory from the modern point of view, developed by Emmy Noether and two of her students: van der Waerden and Emil Artin. I suggest that you read B. Melvin Kiernan's &lt;em&gt;The Development of Galois Theory from Lagrange to Artin&lt;/em&gt; (Archive for History of Exact Sciences Vol. 8, No. 1/2, pp. 40&amp;ndash;154). &lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-12-08T09:43:53.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8042" PostTypeId="2" ParentId="7704" CreationDate="2018-12-08T16:29:03.350" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;As a layperson who isn't too familiar with this history, it seems rather incredible that notation alone cost 50+ years of progress.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is incredible. Essentially it was the lack of dialogue between British and Continental mathematicians and physicists due to the priority fight on the invention of the calculus. That this fight seemed even neccessary at the time seems perverse since both men already had many achievements to their name. For example, Newton discovered the universal law of gravitation which Liebniez had not; and Liebniz had understood the neccessity of an 'analysis situs' which led to Poincares development of topology (Poincare published a paper with this very name in 1895), something which Newton had not thought about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One might say it was fight fermented by the many lesser followers of both Newton and Liebniz.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2018-12-08T16:29:03.350" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8043" PostTypeId="1" CreationDate="2018-12-09T14:09:16.113" Score="1" ViewCount="140" Body="&lt;p&gt;I have two questions regarding the development of mathematics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;1)&lt;/strong&gt; Is there an example where in mathematics, a collaboration has led to the discovery of another result? I already know something like the Polymath project, or the Hilbert program and the Hamilton's program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;2)&lt;/strong&gt; Is there an example of how individual secrecy and lone effort in mathematics has led to a breakthrough discovery without much contact with the math community? The ones I am aware of are like Andrew Wiles, and Ramanujan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I just want some more examples cause I was quite curious with regards to certain approaches to mathematical development. &lt;/p&gt;&#xA;" OwnerUserId="8205" LastActivityDate="2019-04-22T04:13:26.293" Title="mathematical development" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8044" PostTypeId="2" ParentId="8043" CreationDate="2018-12-09T15:39:01.507" Score="1" Body="&lt;p&gt;According to the Mathscinet database (which covers most publications since approximately 1940),&#xA;37.6% of all mathematical papers have more than one author, so they are results of&#xA;collaboration. This number was much less in the previous centuries. &#xA;This means that examples of a) and b) are too numerous to try to make a list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This site contains a lot of statistics on collaboration in mathematics:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.oakland.edu/enp/trivia/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.oakland.edu/enp/trivia/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-12-09T15:39:01.507" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8045" PostTypeId="1" AcceptedAnswerId="8046" CreationDate="2018-12-09T20:13:33.890" Score="0" ViewCount="105" Body="&lt;p&gt;I was just looking at a YouTube Video interview with Leonard Susskind and did not realize he was the first to discover stings in string theory! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=CQAcLW6qdQY&amp;amp;pbjreload=&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=CQAcLW6qdQY&amp;amp;pbjreload=&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;@12:30&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is an interesting video but I also had read a few articles of other participants in this endeavor such as Veneziano while working at  CERN  and others who were also working on the same idea so I did not think anyone person was really the originator but I believe I am wrong now. I believe he was the first to develop this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am wondering if someone is familiar enough with the topic to perhaps comment on the history of it's original founding? &lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-12-10T04:36:59.470" Title="Who was the first scientist to develop the concept and mathematical formula of string theory?" Tags="|string-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8046" PostTypeId="2" ParentId="8045" CreationDate="2018-12-10T04:36:59.470" Score="1" Body="&lt;p&gt;It was a collective effort, and certainly it is not one formula as the title of your question seems to imply:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the principal &quot;founding fathers&quot; was Gabriele Veneziano.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-12-10T04:36:59.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8047" PostTypeId="1" AcceptedAnswerId="8049" CreationDate="2018-12-10T14:46:36.370" Score="8" ViewCount="596" Body="&lt;p&gt;In his magnificent book &lt;em&gt;Number Theory: An approach through history, from Hammurapi to Legendre&lt;/em&gt;, André Weil quotes the article &lt;em&gt;Solution de quelques questions d'analyse indéterminée&lt;/em&gt;, by L. Aubry (Sphinx-&amp;OElig;dipe, 7&lt;sup&gt;e&lt;/sup&gt; année (1912), pp.&amp;nbsp;81&amp;ndash;84). Can someone tell me something about this author? Or where to find a copy of that article? From what I was able to find online, it seems that all other (few) authors who quote this article are only aware of what André Weil wrote about it.&lt;hr/&gt;&lt;strong&gt;Edit:&lt;/strong&gt; It turns out that Aubry's article had already been quoted by J. W. S. Cassels, in his 1978 book &lt;em&gt;Rational quadratic forms&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="4251" LastEditDate="2021-10-11T10:25:50.823" LastActivityDate="2021-10-11T10:25:50.823" Title="Who was L. Aubry?" Tags="|mathematics|mathematicians|reference-request|number-theory|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8048" PostTypeId="2" ParentId="8035" CreationDate="2018-12-10T15:06:27.143" Score="4" Body="&lt;p&gt;@kimchi lover, gave the answer. The article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Brush, Stephen G. &quot;Boltzmann's “Eta Theorem”: Where's the Evidence?.&quot; American Journal of Physics 35.9 (1967): 892-892.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;talks about the problem of the naming of the theorem. So initially Boltzmann used &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; for entropy, but according to Sidney Chapman, S.H. Burbury changed the variable to &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; (about 1890) and it stayed that way, even for Boltzmann. But up to 1963, nobody has come with a proof that the &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; theorem is an capital Greek letter &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know now if in general the French physicists, or just some, call it 'eta' theorem, but I realize this seems just a generalized misnaming.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Update: I have found the following reference:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hjalmars, Stig. &quot;Evidence for Boltzmann’s H as a capital eta.&quot; American Journal of Physics 45.2 (1977): 214-215.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That hints that by the typographical styles of Greek and Latin letters in Boltzmann papers, and by the generalization of Gibbs from &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; for negative entropy and free energy, the interpretation of &quot;eta theorem&quot; should be favored. &lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2018-12-10T16:37:13.330" LastActivityDate="2018-12-10T16:37:13.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8049" PostTypeId="2" ParentId="8047" CreationDate="2018-12-10T15:08:16.610" Score="7" Body="&lt;p&gt;L. Aubry was a French mathematician (most likely a high school teacher) who published 56 papers in mathematical journals&#xA;and 4 books in the period 1894-1933, (according to Zentralblatt database). The paper you mention is not in this database. Most of his papers are in elementary mathematics journals for school children and amateurs. Such journals are rarely included in mathematics databases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also found references on this particular journal on Internet:&#xA; &lt;a href=&quot;https://www.amazon.fr/Sphinx-Oedipe-journal-mensuel-curiosit%C3%A9-concours/dp/B001C92HGI&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.fr/Sphinx-Oedipe-journal-mensuel-curiosit%C3%A9-concours/dp/B001C92HGI&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that it was published only for 7 years. If I really needed it, I would search in big French libraries. It is highly unlikely that this kind of journal has been digitalized. For example, this library seems to have it,&#xA;and I would try Interlibrary loan (ILL):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://data.bnf.fr/32871768/sphinx-oedipe/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://data.bnf.fr/32871768/sphinx-oedipe/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a Word Catalog entry:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.worldcat.org/title/sphinx-oedipe/oclc/9034835&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.worldcat.org/title/sphinx-oedipe/oclc/9034835&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you have access to a scientific library, these data are sufficient for ordering it through the ILL. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-11T16:56:06.693" LastActivityDate="2018-12-11T16:56:06.693" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="8050" PostTypeId="1" CreationDate="2018-12-11T02:42:24.427" Score="0" ViewCount="177" Body="&lt;p&gt;Pythagoras and his followers believed that all magnitudes are commensurable; that is, the ratio of two magnitudes of the same kind, like two lengths or two areas, is equal to the ratio of natural numbers.  In modern language, this means that all real numbers are rational numbers.  This allowed them to find out whether any two ratios of magnitude are equal: you just find out what ratio of natural numbers each one is equal to, and then compare those ratios to each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Hippassus of Metapontum discovered that the ratio of a diagonal of a square to its side is not equal to a ratio of natural numbers.  So a new definition was needed for when two ratios of magnitudes are equal.  This was done by Eudoxus of Cnidus.  In modern language, his definition says that two real numbers are equal if the same rational numbers are less than, equal to, and greater than them (basically the idea behind Dedekind cuts).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, what results of geometry did the Pythagoreans prove using the incorrect assumption that all real numbers were rational?  Long ago, I read a book, whose name I don't recall, which listed some results, and described how they had to be reproved with Eudoxus' new definition. &lt;/p&gt;&#xA;" OwnerUserId="7582" LastActivityDate="2024-03-14T13:15:17.343" Title="What geometric results were first proven by assuming all real numbers are rational?" Tags="|mathematics|ancient-greece|geometry|euclidean-geometry|real-analysis|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8051" PostTypeId="2" ParentId="8050" CreationDate="2018-12-11T09:23:23.170" Score="4" Body="&lt;p&gt;There are errors in the assumptions of your question:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;That history about Hippasus of Metapontum is highly doubtful. There is the version that you mentioned. There is also the version according to which what he discovered (or revealed) was how to construct a dodecahedron.&lt;/li&gt;&#xA;&lt;li&gt;You claim that the Pythagoreans were able to find out whether any two ratios of magnitude are equal: they just had to &amp;ldquo;find out what ratio of natural numbers each one is equal to, and then compare those ratios to each other.&amp;rdquo; Concerning this idea, &lt;a href=&quot;https://en.wikipedia.org/wiki/David_Fowler_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Fowler&lt;/a&gt; wrote: &amp;ldquo;I know of no explicit evidence for this, early or late.&amp;rdquo; (&lt;em&gt;The Mathematics of Plato's Academy: A new reconstruction&lt;/em&gt;, 2&lt;sup&gt;nd&lt;/sup&gt; edition, Clarendon Press, 1999, §&amp;nbsp;10.1)&lt;/li&gt;&#xA;&lt;li&gt;Finally, I doubt that the Pythagoreans proved any geometrical statement using the idea that all magnitudes are commensurable. And, if they did, no such proof survived.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-12-11T09:23:23.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8052" PostTypeId="2" ParentId="8047" CreationDate="2018-12-11T21:14:11.613" Score="5" Body="&lt;p&gt;&lt;em&gt;Sphinx-Œdipe&lt;/em&gt; was a publication edited by the French mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9_G%C3%A9rardin&quot; rel=&quot;noreferrer&quot;&gt;André Gérardin&lt;/a&gt;, as noted in this recent &lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-01383821&quot; rel=&quot;noreferrer&quot;&gt;report&lt;/a&gt;: Maarten Bullynck, &quot;From exploration to theory-driven tables (and back again). A History of Tables in Number Theory.&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Between 1906 and 1928 Gérardin published a special journal, &lt;em&gt;Sphinx-Oedipe, journal mensuel de la curiosité et de concours&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 2017, a lecturer at the University of Nantes in France gave a &lt;a href=&quot;https://cfv.univ-nantes.fr/publications-cfv-2260708.kjsp&quot; rel=&quot;noreferrer&quot;&gt;talk&lt;/a&gt; on this journal and its editor:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jenny Boucard. &quot;André Gérardin (1879-1953) et la revue Sphinx-Oedipe (1906-19??)&quot;. 2017. Intervention dans le cadre d'une table ronde sur le thème &quot;Les Journaux mathématiques au XXe siècle&quot; (ANR Cirmath) (Paris, France)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The slides from the talk are available &lt;a href=&quot;https://f-origin.hypotheses.org/wp-content/blogs.dir/2187/files/2017/01/2017-02-06-S%C3%A9minaire-Cirmath-Jenny-Boucard-1.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://docplayer.fr/64054745-Les-journaux-mathematiques-au-xx-andre-gerardin-et-la-revue-sphinx-oedipe.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boucard notes that library holdings of the publication are incomplete and tracked down the following volumes of &lt;em&gt;Sphinx-Œdipe&lt;/em&gt;: 1906-1914, 1916-1917, 1920-1926, 1928. It is unclear whether the listed volumes are all available from the &lt;a href=&quot;http://www.bnf.fr/en/tools/a.welcome_to_the_bnf.html&quot; rel=&quot;noreferrer&quot;&gt;BNF&lt;/a&gt; (Bibliothèque nationale de France).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Boucard notes that the publication used contributions from a mix of both regional and international authors, and gives the following examples of regional contributors:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;L. Chanzy, professeur de lycée à Nancy ou L. Aubry, viticulteur à Jouy-les-Reims&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In English: &quot;L. Chanzy, high school teacher in Nancy or L. Aubry, winegrower in Jouy-les-Reims&quot;. The description of L. Aubry as a &quot;viticulteur&quot;, i.e. a winegrower, seems plausible, as there is a champagne producer &lt;a href=&quot;https://www.skurnik.com/producer/l-aubry-fils/&quot; rel=&quot;noreferrer&quot;&gt;L. Aubry Fils&lt;/a&gt; (English: L. Aubry Son) in Jouy-les-Reims today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://pdfs.semanticscholar.org/bcc1/c9acd452032fc16d025d92589dca81a35ee2.pdf&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt; by Pete L. Clark, &quot;Quadratic Forms Over Global Fields&quot;, describes L. Aubry as an amateur mathematician:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] the result goes back to a 1912 paper of the amateur mathematician L. Aubry&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;L. Aubry's first name was Léon, as can be seen from these 1911 &lt;a href=&quot;https://www.e-periodica.ch/cntmng?pid=ens-001:1911:13::185&quot; rel=&quot;noreferrer&quot;&gt;proceedings&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Les travaux de la Section de Mathématiques et d'Astronomie de l'Association Française pour l'Avancement des Sciences. Congrès de Dijon, 31 juillet-5 août 1911.  [...]&lt;br&gt;&#xA;    M. Léon Aubry, de Jouy-les-Reims, adresse deux mémoires intitulés &lt;em&gt;Sur les diviseurs des formes quadratiques&lt;/em&gt; et &lt;em&gt;Démonstration du théorème de Bachet&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also, in this 1924 &lt;a href=&quot;https://www.e-periodica.ch/cntmng?pid=ens-001:1924:24::143&quot; rel=&quot;noreferrer&quot;&gt;conference program&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Les Mathématiques à l'Association française pour l'Avancement des Sciences, Congrès des Grenoble. Juillet 1925 [...]&lt;br&gt;&#xA;  Léon AUBRY, Jouy-les-Reims: Sur la congruence &lt;span class=&quot;math-container&quot;&gt;$$\frac{(x + \sqrt{k})^{n} - (x -\sqrt{k})^n}{2\sqrt{k}}\equiv 0\;\;\;\;   (\mathrm{mod.}\;p)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;L. Aubry also made note of the result referenced in the question in &lt;em&gt;L'Intermédiaire des mathématiciens&lt;/em&gt;. Tome XIX, Paris 1912, &lt;a href=&quot;https://books.google.com/books?id=ysM_AQAAIAAJ&amp;amp;q=%22L.+Aubry%22+%22Sphinx%22&amp;amp;dq=%22L.+Aubry%22+%22Sphinx%22&amp;amp;hl=en&quot; rel=&quot;noreferrer&quot;&gt;p. 177&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Tout nombre décomposable en trois carrés fractionnaires est décomposable en trois carrés entiers. J'ai donné une démonstration directe et très élémentaire de cette proposition, dans Sphinx-Œdipe (1912, p. 81). L. Aubry&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In English: &quot;Any number decomposable into three fractional squares is decomposable into three whole squares. I gave a direct and very basic demonstration of this proposition, in Sphinx-Oedipe (1912, p. 81)&quot;&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2018-12-11T21:44:51.437" LastActivityDate="2018-12-11T21:44:51.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8053" PostTypeId="2" ParentId="7247" CreationDate="2018-12-12T01:12:07.917" Score="4" Body="&lt;p&gt;I first heard this story from Dr. Troy Hicks in the 1980's. I posted the story in a few different places online over the years, probably dating back to the late 90s or early 2000's. Professor Hicks was a rather conservative Christian who was very careful about his words, and was not inclined to exaggerate. When Dr. Hicks told this story, he said that he had heard it at a conference from Mary Ellen Rudin, who was a well-known Topologist, and the wife of Walter Rudin. Dr. Hicks said that, &quot;she told it on her husband Walter.&quot; Walter came back right as the class was ready to end and declared that it really was obvious! Mary Ellen was a Professor at Wisconsin as well; apparently she first heard it from some of Walter's students, soon after it happened. Mrs. Rudin confirmed that it really did happen.&lt;/p&gt;&#xA;" OwnerUserId="7962" LastActivityDate="2018-12-12T01:12:07.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8054" PostTypeId="2" ParentId="7247" CreationDate="2018-12-12T09:15:15.473" Score="7" Body="&lt;p&gt;This story is often told about G. H. Hardy. You will find it in &lt;em&gt;Lion hunting &amp;amp; other mathematical pursuits&lt;/em&gt;, by Ralph P. Boas Jr. It says: “The story is told of G. H. Hardy (and of other people) that during a lecture he said ‘It is obvious… &lt;em&gt;Is&lt;/em&gt; it obvious?’ left the room, and returned fifteen minutes later, saying ‘Yes, it's obvious.’ I was present once when Rogosinski asked Hardy whether the story were true. Hardy would admit only that he might have said ‘It is obvious… &lt;em&gt;Is&lt;/em&gt; it obvious?’ (brief pause) ‘Yes, it's obvious.’”&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2022-01-11T21:33:17.070" LastActivityDate="2022-01-11T21:33:17.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8055" PostTypeId="1" AcceptedAnswerId="8056" CreationDate="2018-12-12T14:26:46.543" Score="0" ViewCount="175" Body="&lt;p&gt;I think it should be an ancient Greek mathematician who was interested in proving all triangle has inner angles summing to 180 degree? Was it Thales?&lt;/p&gt;&#xA;" OwnerUserId="8090" LastActivityDate="2018-12-12T15:22:00.893" Title="Who was the first person who emphasis the importance of proof?" Tags="|mathematicians|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8056" PostTypeId="2" ParentId="8055" CreationDate="2018-12-12T15:12:57.917" Score="2" Body="&lt;p&gt;According to the Greek (Hellenistic) tradition, this was Thales. He is credited with proofs of such things as that &quot;vertical angles formed by two lines are equal&quot;.&#xA;But this was written in the Hellenistic histories of mathematics centuries after Thales, and we have no idea what sort of proofs he really gave to such things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We really do not have primary sources for pre-Euclidean mathematics earlier than Euclid himself, and Euclid does not give credits.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One has to be very cautious about Hellenistic sources. For example they claim almost unanimously that Thales and Pythagoras studied mathematics in Egypt but according to modern historians mathematics in Egypt was on very primitive level, and it is not clear what they could possibly learn in Egypt. It seems that the Greeks invented the proofs themselves, at some time between Thales and Plato, but who and when exactly, we do not know.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-12T15:22:00.893" LastActivityDate="2018-12-12T15:22:00.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8057" PostTypeId="1" AcceptedAnswerId="8058" CreationDate="2018-12-13T00:37:29.853" Score="7" ViewCount="348" Body="&lt;p&gt;Every serious source I consulted, be it Cajori, Struik, Edwards,... discusses the method of exhaustion as the means used by ancient Greeks to avoid “taking limits”, because they “disliked infinity”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as my sources go, exhaustion is defined as proving that a sequence of values gets arbitrarily close to some value L by “doing a double reductio ad absurdum to guarantee the sequence does not get arbitrarily close to something less than L, nor something more than L”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then they all go on to say that this “avoids the step of taking a limit”. HOWEVER, the definition sounds exactly like what the modern definition  of limit is! The only practical difference I see is the consistent use of reductio to reach the result.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What subtlety am I missing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1861" LastActivityDate="2018-12-13T01:00:12.833" Title="The exhausting Greek fear of infinity" Tags="|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8058" PostTypeId="2" ParentId="8057" CreationDate="2018-12-13T00:53:41.380" Score="7" Body="&lt;p&gt;You are right, their arguments are equivalent to our proofs with limits. An important difference however is the assumption that the limit EXISTS which they took for granted, and which they could not justify because they did not have any definition of real number.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Archimedes says that perimeter of any inscribed polygon in a circle is less than perimeter of any circumscribed polygon. Then he proves that the difference of perimeters&#xA;of inscribed and circumscribed polygon can be arbitrary small.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One is tempted to conclude that there is a real number which is greater&#xA;than perimeter of any inscribed polygon and less than perimeter of any circumscribed polygon, and call this number the length of circle.&#xA;But what is a (real) number, and why such a real number exists?&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-13T01:00:12.833" LastActivityDate="2018-12-13T01:00:12.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8059" PostTypeId="1" AcceptedAnswerId="8060" CreationDate="2018-12-13T00:44:20.780" Score="0" ViewCount="55" Body="&lt;p&gt;What came first, Fourier's Law, or the definition of heat capacity?&lt;/p&gt;&#xA;" OwnerUserId="8219" OwnerDisplayName="Christina Daniel" LastActivityDate="2018-12-13T13:13:27.163" Title="Heat Capacity versus Fourier's Law" Tags="|thermodynamics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8060" PostTypeId="2" ParentId="8059" CreationDate="2018-12-13T13:13:27.163" Score="0" Body="&lt;p&gt;Warning -- assumptions made in the following.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fourier's Wiki page mentions that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1822 Fourier published his work on heat flow in Théorie analytique&#xA;  de la chaleur (The Analytical Theory of Heat)&#xA;  Which one could interpret as the first publication of what later became named &quot;Fourier's Law.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Whereas the Wiki page on &lt;em&gt;heat capacity&lt;/em&gt;  states  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a previous theory of heat common in the early modern period, heat&#xA;  was thought to be a measurement of an invisible fluid, known as the&#xA;  caloric. Bodies were capable of holding a certain amount of this&#xA;  fluid, hence the term heat capacity, named and first investigated by&#xA;  Scottish chemist Joseph Black in the 1750s.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus  one can assume (making an ass of you and me) or &lt;a href=&quot;https://xkcd.com/1339/&quot; rel=&quot;nofollow noreferrer&quot;&gt;assert&lt;/a&gt; (see the title-text)  that heat capacity was defined first. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2018-12-13T13:13:27.163" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8061" PostTypeId="1" AcceptedAnswerId="8063" CreationDate="2018-12-13T23:49:36.097" Score="3" ViewCount="151" Body="&lt;p&gt;The fact (!) that the Laplace-Beltrami operator on compact (connected) Riemannian manifolds has compact resolvent, and therefore has purely discrete spectrum, is well known. So far as I can tell, the most common textbook or monograph discussions of things in this vein are treatments of Hodge theory, where most of the foundational analytical details and facts seem to be taken for granted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems also to be well known that the idea of a proof is to smoothly cut up the given manifold into small-enough pieces that they can be continuously imbedded into Euclidean cubes, and, then, into products of circles. On the latter, with the usual flat Laplace-Beltrami operator, the theory of Fourier series easily gives a Rellich compactness lemma... which then gets transported back to the manifold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not that the proof details are necessarily interesting... but... who did such a thing first? I'm having great trouble following bibliographic links back to anything...&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2018-12-15T02:21:42.270" Title="who proved discrete spectrum of Laplace-Beltrami on compact Riemannian manifolds?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8062" PostTypeId="1" AcceptedAnswerId="8069" CreationDate="2018-12-14T16:52:33.073" Score="3" ViewCount="355" Body="&lt;p&gt;In his book “A History of Mathematics”, Carl Boyer mentions that both AlKhwarizmi and Abd ElHamid Ibn Turk wrote their books on Algebra (“Aljabr w Almuqabla” and “Logical Necessities” respectively) at nearly the same time. He also states that they use the same methodology and even an example used for demonstration is exactly the same in both books. He then concludes that this indicates that these books weren't the first on the subject as it's widely believed and that Algebra was already mature enough at that time for them to share the same methodology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Quoting from the book:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Algebra of al-Khwarizmi is usually regarded as the first work&#xA;  on the subject, but a publication in Turkey raises some question about&#xA;  this. A manuscript of a work by ‘Abd-al-Hamid ibn-Turk, titled&#xA;  “Logical Necessities in Mixed Equations,” was part of a book on&#xA;  Al-jabr wa’l muqabalah, which was evidently very much the same as&#xA;  that by al-Khwarizmi and was published at about the same time—&#xA;  possibly even earlier. The surviving chapters on “Logical Necessities”&#xA;  give precisely the same type of geometric demonstration as al-&#xA;  Khwarizmi’s Algebra and in one case the same illustrative example,&#xA;  &lt;span class=&quot;math-container&quot;&gt;$x^2+21=10x$&lt;/span&gt;. In one respect, ‘Abd al-Hamid’s exposition is more&#xA;  thorough than that of al-Khwarizmi for he gives geometric figures to&#xA;  prove that if the discriminant is negative, a quadratic equation has no&#xA;  solution. Similarities in the work of the two men and the systematic&#xA;  organization found in them seem to indicate that algebra in their day&#xA;  was not so recent a development as has usually been assumed. When&#xA;  textbooks with a conventional and well-ordered exposition appear&#xA;  simultaneously, a subject is likely to be considerably beyond the formative stage. Successors of al-Khwarizmi were able to say, once a problem had been reduced to the form of an equation, “Operate according to the rules of algebra and almucabala.” In any case, the survival of al-Khwarizmi’s Algebra&#xA;  can be taken to indicate that it was one of the better textbooks typical of Arabic algebra of the time. It was to algebra what Euclid’s Elements&#xA;  was to geometry—the best elementary exposition available until modern times—&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Did any further investigation go into this and try to trace the steps that led to this claimed maturity? &lt;/p&gt;&#xA;" OwnerUserId="8202" LastEditorUserId="5989" LastEditDate="2018-12-16T08:06:49.617" LastActivityDate="2020-07-13T01:02:41.263" Title="Was further research done about the invention of Algebra?" Tags="|mathematics|algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8063" PostTypeId="2" ParentId="8061" CreationDate="2018-12-14T21:55:30.527" Score="2" Body="&lt;p&gt;I believe that this was proved in full generality for the first time by Hilbert,&#xA;in the series of papers &#xA;Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen,&#xA;1904-1910, later published as a book &#xA;(Leipzig und Berlin: B. G. Teubner. XXVI u. 282 S.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many special cases were known earlier, in particular by Poincare.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. However, it should be mentioned that the formal definition of a smooth manifold was only given by Veblen and Whitehead in 1931, see&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remmert, Reinhold,&#xA;From Riemann surfaces to complex spaces. Matériaux pour l'histoire des mathématiques au XXe siècle (Nice, 1996), 203–241, &#xA;Sémin. Congr., 3, Soc. Math. France, Paris, 1998. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So this is a case when the result was established long before the proper definitions were given, the thing which happens frequently.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-15T02:21:42.270" LastActivityDate="2018-12-15T02:21:42.270" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8064" PostTypeId="1" CreationDate="2018-12-13T20:47:31.793" Score="1" ViewCount="327" Body="&lt;p&gt;AFAIK, partial derivatives made it to the forefront as a result of coming up with an equation for a vibrating string i.e., 1D wave equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From purely a physical phenomena to math mapping POV: &lt;em&gt;Why&lt;/em&gt; does a partial derivative show up? That is, how do we justify that we need something like a &lt;em&gt;partial derivative&lt;/em&gt; to address something that ordinary derivatives are deficient at?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let's assume that the vibrating string was the genesis of partial differentials (AFAIK Bernoulli (one of them) had invented this concept sooner but let's assume otherwise for now). The question then is what aspect of capturing the vibrating string problem's as math equations leads one to &quot;invent&quot; partial derivatives?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been pondering over this for a while ever since I learned about the history and derivation of the 1D wave equation. I don't think I understand the &quot;need for partial derivates&quot; well enough to answer this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: Decided to post in the Physics.SE vs. Math.SE since I'd like to understand PDEs from the POV of physical phenomena and build intuition about their need.&lt;/p&gt;&#xA;" OwnerUserId="7035" OwnerDisplayName="PhD" LastActivityDate="2022-07-11T17:03:50.357" Title="Why are partial derivatives necessary when deriving the equation for a vibrating string?" Tags="|differential-equations|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8065" PostTypeId="2" ParentId="8064" CreationDate="2018-12-14T00:56:30.803" Score="1" Body="&lt;p&gt;I suppose deflection is a function of time and position, and hence the partials are needed to make the notation unambiguous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A small change in displacement is &lt;em&gt;defined&lt;/em&gt; as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ {\rm d}y = \frac{\partial y}{\partial x} {\rm d}x + \frac{\partial y}{\partial t} {\rm d}t $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;without the partials, the above would be confusing.&lt;/p&gt;&#xA;" OwnerUserId="3913" OwnerDisplayName="ja72" LastActivityDate="2018-12-14T00:56:30.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8066" PostTypeId="2" ParentId="3642" CreationDate="2018-12-16T18:30:09.183" Score="0" Body="&lt;p&gt;A good case could be made for Kepler to be the first to pose and &lt;em&gt;solve&lt;/em&gt; a differential equation, although it involved a large amount of trial and error.  In particular, his &lt;em&gt;Astronomia Nova&lt;/em&gt; (1609) is in its essence a treatise on how he figured out how to go from the physical hypothesis that the Sun attracts planets to a set of (angular) equations that embodies this hypothesis:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\dfrac{dE}{dt} \propto \frac{1}{r}\\&#xA;\dfrac{dV}{dt} \propto \frac{1}{r^2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is time, &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the distance to the Sun, &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is the angle about the Sun (called the &lt;strong&gt;true anomaly&lt;/strong&gt;; &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; for latin &lt;em&gt;verus&lt;/em&gt;), and &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is the angle about the center of the orbit (called the &lt;strong&gt;eccentric anomaly&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ultimately, he shows that elliptical motion satisfies this set of equations, a result we now call &lt;strong&gt;Kepler's First and Second Laws of Planetary Motion&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h2&gt;Aside&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;There is plenty of other mathematical and history of science gold in the book, such as:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Proving isomorphism of Ptolemaic, Copernican, and Tychonic systems.&lt;/li&gt;&#xA;&lt;li&gt;Use of first and second order finite differences in analyzing motion.&lt;/li&gt;&#xA;&lt;li&gt;Posing an inverse problem of extreme practical importance for astronomers, but with no closed-form solution. Gauss' attempt to simplify calculations of it led to his first application of hypergeometric series, yielding a highly efficient numerical solution (see his 1809 &lt;em&gt;Theoria motus corporum coelestium...&lt;/em&gt;).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="7233" LastActivityDate="2018-12-16T18:30:09.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8068" PostTypeId="1" AcceptedAnswerId="8070" CreationDate="2018-12-17T01:41:14.453" Score="6" ViewCount="411" Body="&lt;p&gt;The question pretty much says it all, though I have a specific example in mind.  In the mid-1500s while working on his &lt;em&gt;Ars Magna&lt;/em&gt; Cardano asked Tartaglia to solve the cubic &lt;span class=&quot;math-container&quot;&gt;$x^3=9x+10$&lt;/span&gt;.  Using Tartaglia's formula results in having to take the square root of a negative number, which was undefined at the time; neither Tartaglia nor Cardano could solve the equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, with a little work the above equation can be factored into &lt;span class=&quot;math-container&quot;&gt;$(x+2)(x^2-2x-5)$&lt;/span&gt;, and the three roots readily found using nothing more complicated than the quadratic formula which was known at the time.  Clearly neither Tartaglia nor Cardano did this, as they had no solution to the original cubic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All that aside, when was it realized that such higher-order equations could be factored?&lt;/p&gt;&#xA;" OwnerUserId="7074" LastActivityDate="2018-12-17T12:38:13.527" Title="When were polynomial equations first factored?" Tags="|mathematics|discoveries|elementary-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8069" PostTypeId="2" ParentId="8062" CreationDate="2018-12-17T08:43:37.913" Score="3" Body="&lt;p&gt;Nothing new was discovered after Boyer published his textbook. In 2014, &lt;a href=&quot;https://en.wikipedia.org/wiki/Victor_J._Katz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Victor J.Katz&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Karen_Parshall&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karen Parshall&lt;/a&gt; published their book &lt;em&gt;Taming the Unknown: A history of algebra from antiquity to the early twentieth century&lt;/em&gt;, in which they write (chapter 7):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While influencial, al-Khwārizmī's was evidently not the ony algebra text of the time. A fragment of a contemporaneous work remains, namely, the section &lt;em&gt;Logical Necessities in Mixed Equations&lt;/em&gt; from a longer work &lt;em&gt;Kitāb al-jabr wa'l muqabalah&lt;/em&gt; [&amp;hellip;] by ʿAbd al-Hamīd ibn Wase ibn Turk Jili.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, no other algebra textbook from that time was discovered meanwhile.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-12-17T08:43:37.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8070" PostTypeId="2" ParentId="8068" CreationDate="2018-12-17T12:38:13.527" Score="7" Body="&lt;p&gt;I found no trace of factoring polynomials being done before Descartes, in his &lt;em&gt;La Géométrie&lt;/em&gt;, written in 1637. In it, he wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is evident from the above that the sum of an equation having several roots [that is, the polynomial itself] is always divisible by a binomial consisting of the unknown quantity diminished by the false roots. In this way, the degree of an equation can be lowered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is how Descartes stated what we call today the &lt;a href=&quot;https://en.wikipedia.org/wiki/Factor_theorem&quot; rel=&quot;noreferrer&quot;&gt;factor theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-12-17T12:38:13.527" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8071" PostTypeId="1" AcceptedAnswerId="8072" CreationDate="2018-12-17T16:48:04.240" Score="0" ViewCount="353" Body="&lt;p&gt;Until yesterday i thought that nothing direct is known about the &quot;Archimedes's planetarium&quot; - an elaborate ancient planetarium based on complex gear mechanism that could represent the geocentric motions of the seven classical planets (the sun, the moon and the first 5 planets in the solar system). I thought the only information known about this instrument was that Archimedes wrote a (now-lost) manuscript, entitled &quot;on spheres-making&quot;, on the construction of such mechanisms, and the possibility of it related to the &quot;Antikythera Planetarium&quot; (discovered in 1902).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But yesterday i read the abstract of a recently published book &quot;&lt;strong&gt;THE RECOVERED ARCHIMEDES PLANETARIUM&lt;/strong&gt;&quot; by the author GIOVANNI PASTORE, according to which an 2006 archeological excavation found a fragment of gear that might be a part, or relate to the planetarium of Archimedes. The book's abstract makes several claims that seem to me very interesting:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;That in some aspects Archimedes's planetarium was even more advanced than the Antikythera Planetarium sinced it incorporated a special teeth's profile based on mathematical principles rediscovered only in the 17th and 18th century (in a youtube video the speaker mentions the teeth profile was based on the cycloid curve).&lt;/li&gt;&#xA;&lt;li&gt;That Archimedes's device was actually a three dimensional planetrium while the Antikythera Planetarium was an analog computer with two dimensional display.&lt;/li&gt;&#xA;&lt;li&gt;That a certain Roman general (i don't remember the name) deliberately attacked the Mecadonians in a dark night of lunar eclipse (in order to use the element of surprise). It suggests a conjecture that the Roman general knew earlier that there is going to be a lunar eclipse beacuse he saw the date calculated by Archimedes's device.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So, i'd like to know if anyone can tell something about the &quot;gear of Olbia&quot;. Did someone here read this book? is it worth buying? &lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2023-12-15T13:23:48.807" Title="What is the story behind the recently discovered gear fragment of Olbia?" Tags="|ancient-greece|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8072" PostTypeId="2" ParentId="8071" CreationDate="2018-12-17T17:14:53.450" Score="3" Body="&lt;p&gt;Let me cite an authority on Antikythera mechanism (Alexander Jones, &quot;Portable cosmos&quot;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A fragment of a gear, found in 2006 at Olbia,&#xA;  Sardinia, and accompanied by no related mechanical remains but in an&#xA;  archeological context reportedly dating to about 200 BC, has been claimed as&#xA;  part of the original planetarium of Archimedes; see Pastore 2006 and Pastore&#xA;  2013. Its material is brass, and its teeth have rounded, roughly cycloidal shape.&#xA;  Contamination of the archeological context by an artifact of much more modern&#xA;  date is the most probable explanation of the fragment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems that Giovanni Pastore makes too many far-reaching conclusions from a single piece of a wheel of  doubtful origin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Babylonian and Greek astronomers could predict lunar eclipses. And they needed no mechanisms for this. Clockwork toys demonstrating various astronomical phenomena were popular in 18 century, and their purpose was not science but entertaining rich sponsors. I suppose that the purpose of Antikythera mechanism and Archimedes planetarium was similar. Still it is highly interesting to know what kind of planet theory may have been used in Archimedes planetarium. Archimedes lived before Hipparchus, and we know from Ptolemy that Hipparchus had no planet theory.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-17T20:09:00.923" LastActivityDate="2018-12-17T20:09:00.923" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8073" PostTypeId="1" CreationDate="2018-12-17T18:50:54.397" Score="0" ViewCount="174" Body="&lt;p&gt;The rule says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The product (or quotient) of two debts is a fortune&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What I’m struggling with is what exactly &lt;em&gt;is&lt;/em&gt; the product of two debts? What accounting need forces one to multiply debts? How do you interpret something like this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here’s what &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Negative_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; has to say but it doesn’t sound right:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thus (−2) × 3  =  −6 and (−2) × (−3)  =  6. The reason behind the&#xA;  first example is simple: adding three −2's together yields −6: (−2) ×&#xA;  3  =  (−2) + (−2) + (−2)  =  −6. The reasoning behind the second&#xA;  example is more complicated. The idea again is that losing a debt is&#xA;  the same thing as gaining a credit. In this case, losing two debts of&#xA;  three each is the same as gaining a credit of six: (−2 debts ) × (−3&#xA;  each)  =  +6 credit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The trippy thing is &lt;span class=&quot;math-container&quot;&gt;$(-3$&lt;/span&gt; each&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt; - that makes no sense IMHO. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m okay even coming up with a contrived scenario but I can’t swallow the interpretation above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how should one interpret “&lt;em&gt;debt times debt is fortune&lt;/em&gt; from an accounting POV?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/2631/historically-how-did-people-define-multiplication-for-negative-numbers&quot;&gt;This question&lt;/a&gt; seems to have similar intent but the answers there are inconclusive or not &quot;intuitively correct&quot; (multiplication by &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; people for example). Hence the focus of this question is to purely understand it from an accounting POV vs. multiplying negative numbers in general.&lt;/p&gt;&#xA;" OwnerUserId="7035" LastEditorUserId="7035" LastEditDate="2018-12-17T22:03:19.143" LastActivityDate="2018-12-19T07:52:47.787" Title="What is the intuition behind Brahmagupta’s rule for multiplying negative numbers?" Tags="|mathematics|number-theory|ancient-india|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8074" PostTypeId="1" AcceptedAnswerId="8077" CreationDate="2018-12-17T21:23:32.070" Score="0" ViewCount="93" Body="&lt;p&gt;One meaning of the word kernel is the set of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$T(u)=0$&lt;/span&gt;. Another meaning of the word kernel is the &quot;kernel&quot; of an integral transform. Is there any relationship between these two?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, the word &quot;kernel&quot; is used to refer to the kernels of group homomorphisms. &lt;a href=&quot;https://hsm.stackexchange.com/questions/2149&quot;&gt;Here&lt;/a&gt;, a discussion has concluded that group kernels came before and are distinct from linear map kernels. This leaves integral transform kernels, which might be the same as one or the other in some way that is not clear to me, or could also be a third distinct thing with the same name.&lt;/p&gt;&#xA;" OwnerUserId="9230" LastEditorUserId="9230" LastEditDate="2018-12-17T21:34:05.250" LastActivityDate="2018-12-18T20:26:26.203" Title="What is the relationship between the word &quot;kernel&quot; that means nullspace and the &quot;kernel&quot; of an integral transform?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8075" PostTypeId="2" ParentId="8073" CreationDate="2018-12-18T14:16:46.740" Score="3" Body="&lt;p&gt;The interpretation is derived from &quot;stretching&quot; the natural interpretation of &lt;em&gt;product&lt;/em&gt; as repeated &lt;em&gt;sum&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$3 \times 2 = 6$&lt;/span&gt; because &lt;span class=&quot;math-container&quot;&gt;$3 \times 2 = 3+3$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The same for &lt;span class=&quot;math-container&quot;&gt;$(−3) \times 2 = (−3) + (−3) = −6$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Starting from this, we may interpret &lt;span class=&quot;math-container&quot;&gt;$a \times (−2)$&lt;/span&gt; as &quot;repeated subtraction&quot; : we have to &quot;subtract&quot; twice the quantity &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a negative quantitiy, i.e. a debt, to subtract a debt is to earn money : if I have a debt of &lt;span class=&quot;math-container&quot;&gt;$-6$&lt;/span&gt; with you and I give it to you, my account changes from &lt;span class=&quot;math-container&quot;&gt;$−6$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; while your account changes from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$-6$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2018-12-18T14:16:46.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8076" PostTypeId="1" CreationDate="2018-12-18T15:58:05.863" Score="0" ViewCount="50" Body="&lt;p&gt;i'm writing an assignment on firework rockets and their trajectory. Now of course im doing this with a lot of limitation as a realistic rocket calculation would be impossible to execute, at least for me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I derived the classic rocket equation that assumes theres no gravity or aerodynamics through a simple differential equation. I then through simple calculations determined its trajectory if it was to fly straight up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now what i want to do is derive a new equation that includes the drag formula.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a bit lost as to how I'm going to do this.&#xA;The purpose it to find the difference between the normal rocket equation and one with drag integrated into it. I've searched all over for something like this but i haven't found anything.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas?&lt;/p&gt;&#xA;" OwnerUserId="9231" LastActivityDate="2018-12-18T15:58:05.863" Title="Rocket &amp; drag equation?" Tags="|mathematics|physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8077" PostTypeId="2" ParentId="8074" CreationDate="2018-12-18T20:26:26.203" Score="1" Body="&lt;p&gt;There is no relation, except that both come from the German word &quot;Kern&quot; which is usually translated as &quot;nucleus&quot; but has very many meanings in German (see Google translate).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-12-18T20:26:26.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8078" PostTypeId="2" ParentId="8073" CreationDate="2018-12-19T07:52:47.787" Score="0" Body="&lt;p&gt;The intuition is that since debts cancel credits, &lt;span class=&quot;math-container&quot;&gt;$k0=0$&lt;/span&gt; determines &lt;span class=&quot;math-container&quot;&gt;$k(-l)$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$kl$&lt;/span&gt;, even for &lt;span class=&quot;math-container&quot;&gt;$k&amp;lt;0$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$a=(-2)(-3)$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$a-6=(-2)(-3)+(-2)3=(-2)0=0$&lt;/span&gt;. Therefore, &lt;span class=&quot;math-container&quot;&gt;$a=6$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2018-12-19T07:52:47.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8080" PostTypeId="1" AcceptedAnswerId="8081" CreationDate="2018-12-19T14:00:42.513" Score="0" ViewCount="242" Body="&lt;p&gt;Did Newton believe in infinitely small particle theory of matter? Because when he talks about axis of rotation, which is locus of the centers of the circles of the rotating body and particle on the axis does not rotate for that they must have no size. For that, I think he had to considered matter made of infinitely small particles with no size at all.&lt;/p&gt;&#xA;" OwnerUserId="9235" LastEditorUserId="131" LastEditDate="2018-12-19T14:35:43.987" LastActivityDate="2018-12-21T15:35:05.040" Title="What was Newton's view of theory of matter?" Tags="|physics|newton|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8081" PostTypeId="2" ParentId="8080" CreationDate="2018-12-19T14:38:46.820" Score="1" Body="&lt;p&gt;About Newton's theory of matter, see &lt;a href=&quot;https://plato.stanford.edu/entries/atomism-modern/#NewAto&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's Atomism&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The key sources of Newton's stance on atomism in his published work are &lt;a href=&quot;https://books.google.it/books?id=3T1wDwAAQBAJ&amp;amp;pg=PA220&quot; rel=&quot;nofollow noreferrer&quot;&gt;Querie 31&lt;/a&gt; of his &lt;a href=&quot;https://en.wikipedia.org/wiki/Opticks&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Opticks&lt;/em&gt; (1704)&lt;/a&gt;, and a short piece on acids. Atomistic views also make their appearance in the &lt;em&gt;Principia&lt;/em&gt;, where Newton claimed “the least parts of bodies to be—all extended, and hard and impenetrable, and moveable, and endowed with their proper inertia”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ernan McMullin, &lt;a href=&quot;https://books.google.it/books?id=WVsFDgAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton on Matter and Activity&lt;/a&gt;, Notre Dame UP (1978)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A.Rupert Hall and Marie Boas Hall, &lt;a href=&quot;https://www.journals.uchicago.edu/doi/abs/10.1086/348867?journalCode=isis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's Theory of Matter; &lt;em&gt;Isis&lt;/em&gt; (1960)&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;We have to take into account that Newton's &lt;em&gt;Principia&lt;/em&gt; is mainly a book of mathematical physics (the first one).&lt;/p&gt;&#xA;&lt;p&gt;See &lt;strong&gt;Def.VIII&lt;/strong&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For the quantity of motion arises from the celerity multiplied by the quantity of matter; and the motive force arises from the accelerative force multiplied by the same quantity of matter. For the sum of the actions of the accelerative force, upon the several &lt;em&gt;particles&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;] of the body, is the motive force of the whole.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, bodies are made of &amp;quot;particles&amp;quot; (or &amp;quot;corpuscles&amp;quot;) and nothing more than this is postulated in the treatise to manage the laws of motion and mutual attraction of bodies.&lt;/p&gt;&#xA;&lt;p&gt;But see &lt;strong&gt;Scholium&lt;/strong&gt; to &lt;strong&gt;Lemma XI&lt;/strong&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Those things which have been demonstrated of curved lines, and the surfaces which they comprehend, may be easily applied to the curved surfaces and contents of solids. These Lemmas are premised to avoid the tediousness of deducing involved demonstrations ad absurdum, according to the method of the ancient geometers. For demonstrations are shorter by the method of indivisibles; but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the following Propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios, and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles; and now those principles being demonstrated, we may use them with greater safety.&lt;/p&gt;&#xA;&lt;p&gt;Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curved lines for right ones, &lt;em&gt;I would not be understood to mean indivisibles&lt;/em&gt; [&lt;em&gt;emphasis added&lt;/em&gt;], but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-12-21T15:35:05.040" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8082" PostTypeId="2" ParentId="2940" CreationDate="2018-12-19T15:08:21.093" Score="4" Body="&lt;p&gt;To continue on jmite's excellent answer, &lt;a href=&quot;https://en.wikipedia.org/wiki/Nancy_Lynch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nancy Lynch&lt;/a&gt; is a pioneer in the theory of &quot;Distributed Systems&quot; in Computer Science.&#xA;For example, her work with Michael J.Fischer and Mike Paterson showed that &quot;In an asynchronous distributed system, consensus is impossible if there is one processor that crashes&quot; which is a fundamental result in the field.  &lt;/p&gt;&#xA;" OwnerUserId="8202" LastActivityDate="2018-12-19T15:08:21.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8083" PostTypeId="1" AcceptedAnswerId="8085" CreationDate="2018-12-20T00:05:42.450" Score="0" ViewCount="1098" Body="&lt;p&gt;I understand the SI meaning of a Coulomb but how were the individual electrons counted in the measurement to know how many it would take to constitute a Coulomb? &lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2018-12-20T07:32:00.473" Title="How was it discovered how many electrons are in one coulomb?" Tags="|electricity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8084" PostTypeId="2" ParentId="8083" CreationDate="2018-12-20T02:18:40.240" Score="1" Body="&lt;p&gt;This was discovered in the experiments with &lt;a href=&quot;https://en.wikipedia.org/wiki/Electrolysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;electrolysis&lt;/a&gt;. The idea is that when electric current passes through a solution, positive ions accumulate on one side,&#xA;and negative on another. We know the charge in terms of the number of electrons of one ion (+n if n electrons are missing,&#xA;-n if there are n extra electrons). The number of ions that accumulated can be found when you know their atomic weight and the weight of the accumulated material. On the other hand, by measuring time and current&#xA;passing through the solution we can find out how many Coulombs correspond to this charge.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2018-12-20T02:18:40.240" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8085" PostTypeId="2" ParentId="8083" CreationDate="2018-12-20T07:32:00.473" Score="1" Body="&lt;p&gt;Please check the following references at the bottom (from the earliest known usages of Coulomb in unabridged Oxford English Dictionary). Also, check the early definition of ampere which was also defined on the basis of silver nitrate electrolysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;from the wt of Ag+ deposited, by Faraday's law, we have one to one correspondence:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ag+ (solution) + electron ---&gt; Ag (metal), thus from the weight of Ag deposited during electrolysis, you can &lt;em&gt;count&lt;/em&gt; electrons involved in the process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can independently measure charge passed through a solution via time (t) and current (I). Q= Ixt&lt;/p&gt;&#xA;&#xA;&lt;p&gt;from OED:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1881   Nature 29 Sept. 512/2   The name Coulomb to be given to the quantity of electricity defined by the condition that an ampère gives one coulomb per second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1892   Lightning 3 Mar. (Spec. Suppl.) Gloss. Electrical Terms s.v.   The Coulomb is the quantity of electricity, which, when passed through a solution of nitrate of silver, deposits ·001118 of a gramme of silver on the plate by which it leaves the liquid.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2018-12-20T07:32:00.473" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8086" PostTypeId="1" AcceptedAnswerId="8107" CreationDate="2018-12-21T07:48:51.720" Score="4" ViewCount="304" Body="&lt;p&gt;I've had a book for many years called &lt;em&gt;Puzzles, Mazes, and Numbers&lt;/em&gt; which describes a method for performing multiplication as follows called &quot;Russian peasant multiplication&quot;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are two columns, on the left hand side the number is halved, rounding down and on the right hand side, each successive number is doubled. In the algorithm described in the book, rows corresponding to even numbers in the left column are struck out. In this example, rows with even numbers in the left column simply have a blank entry in the right. &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;    45  x   89    89&#xA;    22     178&#xA;    11     356   356&#xA;     5     712   712&#xA;     2    1424&#xA;     1    2848  2848&#xA; -----------------------&#xA;                4005&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;My understanding is that this is similar to the Ancient Egyptian method of multiplication, but that the Ancient Egyptians decomposed the left argument into powers of two by repeatedly subtracting the largest power of two possible, rather than by repeatedly dividing by two and checking the parity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm curious whether a doubling and mediation algorithm with the parity check as described here was ever the/an ordinary of doing multiplication anywhere in the world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One &lt;a href=&quot;http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt; cited by the Wikipedia article claims that the supposed Russian origin of the algorithm is a bit murky and hard to definitively show.&lt;/p&gt;&#xA;" OwnerUserId="9243" LastEditorUserId="9243" LastEditDate="2018-12-21T08:07:29.217" LastActivityDate="2018-12-28T11:28:32.700" Title="Was &quot;peasant multiplication&quot; ever used as the predominant method of multiplication?" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8087" PostTypeId="1" CreationDate="2018-12-21T09:16:00.117" Score="1" ViewCount="63" Body="&lt;p&gt;I write a chapter for my book in logic concerning Gentzen systems and natural deduction. I read the paper of Gentzen and I realized that his motivation was to create a system that works as close as to human reasoning .... In my current book version, I present first Gentzen systems, and then the natural deduction, but after all this info I wonder if I have to present natural deduction first. What is your opinion about this? If there are some classical references I don't know, please let me know. Thanks a lot !!&lt;/p&gt;&#xA;" OwnerUserId="9241" LastActivityDate="2018-12-21T09:16:00.117" Title="Gentzen systems and natural deduction" Tags="|mathematical-logic|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8090" PostTypeId="1" CreationDate="2018-12-21T13:54:07.290" Score="1" ViewCount="451" Body="&lt;p&gt;For example, &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is not an invention, it is a discovery which was natural, that is &lt;strong&gt;ratio of the circumference of a circle to its diameter&lt;/strong&gt;. But when we define a meter it is not a natural value it is an invented unit of measure for length.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is what I want to know: was one mole, which is approximately &lt;span class=&quot;math-container&quot;&gt;$6.022*10^{23}$&lt;/span&gt;, a natural number like &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, which was discovered, or an invention, like,  for example, the meter.&lt;/p&gt;&#xA;" OwnerUserId="9235" LastEditorUserId="55" LastEditDate="2018-12-22T01:46:56.717" LastActivityDate="2018-12-22T20:00:39.360" Title="Was the value of the mole invented or discovered in chemistry?" Tags="|terminology|chemistry|units|measurement|education|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8091" PostTypeId="1" CreationDate="2018-12-21T14:28:26.113" Score="3" ViewCount="165" Body="&lt;p&gt;In 1934, Fermi and his team irradiated uranium (Z=92) with neutrons in order to synthesize element 93. They (and others) did not recognize that they had performed a nuclear fission of uranium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The common explanation for this mistake:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the periodic table at that time (see below, shaded elements are missing, no actinide), the transuranium elements (Z&gt;92) are expected to be located in the subgroups IV, V and VI. Hence the chemical experiments to prove the formation of the new element 93 aimed at the chemical similarity to Mn, Ma(Tc) and Re. With the 'discovery' of the actinides this turned out to be wrong.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ZwOTA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZwOTA.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The lanthanides were known (except for Pm). Their number and position in the periodic table was correctly described by the electric charge of the nucleus (Moseley, 1913) as well as their electronic configuration (Pauli's Aufbau principle, 1920s).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the arrangement of Th, Pa and U based on?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did they not expect a second row in the &quot;f-block&quot;? &lt;/p&gt;&#xA;" OwnerUserId="19387" OwnerDisplayName="faber" LastActivityDate="2018-12-21T18:11:53.027" Title="Historic misidentification of transuranium elements?" Tags="|nuclear-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8092" PostTypeId="2" ParentId="8091" CreationDate="2018-12-21T17:48:11.103" Score="1" Body="&lt;p&gt;The initial placement of &lt;span class=&quot;math-container&quot;&gt;$Th$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Pa$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; on the tail of &lt;span class=&quot;math-container&quot;&gt;$Ac$&lt;/span&gt; but as transition metals (&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;-block) was wrong but not without merit. Coincidence has it that the main valence of &lt;span class=&quot;math-container&quot;&gt;$Th$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$+4$&lt;/span&gt;, as do the elements of Group &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;. Similarly the valence of &lt;span class=&quot;math-container&quot;&gt;$Pa$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$+5$&lt;/span&gt;, as does Group &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;. Even &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt;'s main valence is &lt;span class=&quot;math-container&quot;&gt;$+6$&lt;/span&gt;, as expected of a Group &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; element. Chemically these elements do behave more as transition metals than as Actinides.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later the elements &lt;span class=&quot;math-container&quot;&gt;$Am$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$Cm$&lt;/span&gt; and beyond started showing &lt;span class=&quot;math-container&quot;&gt;$+3$&lt;/span&gt; as an important valence and it became clear they belonged to a second &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;-block. The pieces of the puzzle were then reshuffled to the modern and correct version of the Periodic Table.&lt;/p&gt;&#xA;" OwnerDisplayName="Gert" LastActivityDate="2018-12-21T17:55:08.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8093" PostTypeId="2" ParentId="8090" CreationDate="2018-12-22T01:29:43.827" Score="1" Body="&lt;p&gt;Perrin originally defined the &lt;a href=&quot;https://en.wikipedia.org/wiki/Avogadro_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Avogadro number&lt;/a&gt; as the number of atoms in one gram of hydrogen (later it was redefined as the number of atoms in 12 grams of carbon-12). As such, it was certainly discoverable from measurements. However, picking grams, or hydrogen was not &quot;natural&quot;. Neither was picking circles and diameters in the definition of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. Once they were picked the value became discoverable. An additional difference, of course, is that the value of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; can be established without empirical observations, while the Avogadro number requires them, but that does not seem to matter for invention/discovery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In November 2018 the 26th General Conference on Weights and Measures &lt;a href=&quot;https://en.wikipedia.org/wiki/2019_redefinition_of_SI_base_units&quot; rel=&quot;nofollow noreferrer&quot;&gt;voted to change the definition of the mole&lt;/a&gt; to be the amount of atoms/molecules given by the Avogadro constant, whose numerical value they fixed by decree (of course, it came from the measurements of the Avogadro number). Since the meter is still the length of the path traveled by light in vacuum during 1/299792458 seconds (originally, one ten-millionth of the distance from the North Pole to the equator) it is currently more &quot;discoverable&quot; than the mole, see &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_metre&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of the Meter&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mole displaced the concept of &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalent_weight&quot; rel=&quot;nofollow noreferrer&quot;&gt;equivalent weight&lt;/a&gt; (the weight of a given substance which will combine with or displace a given weight of another substance), and the weight ratio is not up to convention. Mole allowed the standardization of these weights by making an arbitrary choice of unit. It is the same with the meter, the ratios of segments are objective, the &quot;unit&quot; segment is not.&#xA;In all three cases we have elements of convention intertwined with objective elements, so the choice of word (invention/discovery) is a matter of taste.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-12-22T20:00:39.360" LastActivityDate="2018-12-22T20:00:39.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8094" PostTypeId="1" AcceptedAnswerId="8096" CreationDate="2018-12-22T09:02:21.397" Score="3" ViewCount="202" Body="&lt;p&gt;Was Hooke close to developing a physical system of gravity on the same level as Newton. looking at Luka Trkanjec's post on Quora, he insists that Newton was quite overrated in this regard, and that Hooke was close to developing a system of gravity with the inverse square law&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quora.com/Does-Isaac-Newton-deserve-the-praise-and-respect-hes-been-given-throughout-history&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/Does-Isaac-Newton-deserve-the-praise-and-respect-hes-been-given-throughout-history&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2018-12-22T17:19:50.740" Title="How close was Hooke to developing a comprehensive system of gravity?" Tags="|physics|philosophy-of-science|scientific-method|newton|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8095" PostTypeId="1" AcceptedAnswerId="8100" CreationDate="2018-12-22T13:07:58.647" Score="4" ViewCount="224" Body="&lt;p&gt;[ Question copied from &lt;a href=&quot;https://math.stackexchange.com/questions/2541170/euclid-s-proposition-i-3-overused&quot;&gt;https://math.stackexchange.com/questions/2541170/euclid-s-proposition-i-3-overused&lt;/a&gt; ]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although the references to postulates, axioms, and previous propositions are not part of the original text of Euclid's &lt;em&gt;Elements&lt;/em&gt;, all the editions I have seen contain them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In many propositions, it is needed to make &lt;strong&gt;adjacent&lt;/strong&gt; segments, say &lt;span class=&quot;math-container&quot;&gt;$PR$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$PQ$&lt;/span&gt; equal.  Examples include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/propI5.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proposition I.5&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/propI9.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proposition I.9&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/propI11.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proposition I.11&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/propI16.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proposition I.16&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;All the editions linked from &lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/web.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt; and more (I do not have enough reputation to post the links to them): John Casey’s English translation, Richard Fitzpatrick’s edition, Johan Ludvig Heiberg's edition, and Józef Czech's Polish translation refer in those cases to &lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/propI3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proposition I.3&lt;/a&gt;, which is about making &lt;strong&gt;arbitrary&lt;/strong&gt; segments equal (“To cut off from the greater of two given unequal straight lines a straight line equal to the less”).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I understand it, for making &lt;strong&gt;adjacent&lt;/strong&gt; segments equal, &lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/post3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Postulate 3&lt;/a&gt; (“Let it be postulated to draw a circle with any center and radius”) is sufficient.  Why do all those editions of &lt;em&gt;Elements&lt;/em&gt; refer to Proposition I.3 instead of Postulate 3 in the propositions I listed above? The only reason I can think of is editors' inertia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[I am not asking about uncontroversial uses of Proposition I.3, like &lt;a href=&quot;https://mathcs.clarku.edu/~DJoyce/java/elements/bookI/propI6.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proposition I.6&lt;/a&gt;, where &lt;span class=&quot;math-container&quot;&gt;$DB$&lt;/span&gt; is made equal to &lt;span class=&quot;math-container&quot;&gt;$AC$&lt;/span&gt;.]&lt;/p&gt;&#xA;" OwnerUserId="9246" LastActivityDate="2018-12-25T01:12:21.100" Title="Euclid’s Proposition I.3 overused?" Tags="|geometry|euclidean-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8096" PostTypeId="2" ParentId="8094" CreationDate="2018-12-22T16:41:13.847" Score="7" Body="&lt;p&gt;Hooke was not close (as far as we can judge from his surviving work) to what Newton accomplished. Yes, he conjectured the inverse square law. He understood correctly some simple qualitative features of the motion under this law. He probably performed some simple experiments suggesting these features. And he proposed to Newton to prove that the inverse square law would imply Kepler laws. This is known from his correspondence with Newton.&#xA;It is clear from Hooke's letter that he could not prove this himself, and he recognizes this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hooke was not a mathematician, and it was clearly far beyond his power to prove mathematically that the inverse square law implies Kepler laws. The conjecture itself was widely discussed at that time, not only by Hooke. Sir Christopher Wren even proposed a prize for proving it. But nobody could do this except Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you glance at the contents of Newton's Principia, you will see that it contains VERY MUCH more than a proof of Kepler laws from the law of attraction. It contains a comprehensive theory, which essentially lays the ground for mathematical physics. Nothing even close to this can be attributed to any author among Newton's contemporaries. And Newton's contemporaries understood this very well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is another, different thing, that Newton was not fair to many of his contemporaries, and did not give them credit which they deserved. But for the theory of gravitation he did much more than they all combined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Newton's own words (from a letter to Halley):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mathematicians that find out, settle &amp;amp; do all the business&#xA;  must content themselves with being nothing but dry calculators&#xA;  &amp;amp; drudges &amp;amp; another that does nothing but pretend &amp;amp; grasp at all things&#xA;  must carry away all the invention as well of those that were to follow him&#xA;  as of those that went before.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is harsh with respect to Hooke, but correct.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2018-12-22T17:19:50.740" LastActivityDate="2018-12-22T17:19:50.740" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8097" PostTypeId="2" ParentId="8095" CreationDate="2018-12-23T12:38:27.347" Score="2" Body="&lt;p&gt;I suppose that you are right. Besides, what you suggested (using postulate&amp;nbsp;3 when the segments are adjacent) is exactly what Euclid does in his proof of proposition&amp;nbsp;2. At this point, he could not possibly use proposition&amp;nbsp;3, of course. Besides, the proof of proposition&amp;nbsp;3 uses proposition&amp;nbsp;2.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2018-12-23T12:38:27.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8098" PostTypeId="2" ParentId="7606" CreationDate="2018-12-23T21:59:51.450" Score="5" Body="&lt;p&gt;There was a peculiar case when an &lt;em&gt;anonymous&lt;/em&gt; forum commenter made significant progress on a problem. They could well have had substantial formal training (reading their &lt;a href=&quot;https://warosu.org/sci/thread/S3751105#p3751197&quot; rel=&quot;nofollow noreferrer&quot;&gt;posts&lt;/a&gt; I think this is likely).&lt;/p&gt;&#xA;&lt;p&gt;The problem is establishing the formula for sequence &lt;a href=&quot;http://oeis.org/A180632&quot; rel=&quot;nofollow noreferrer&quot;&gt;A180632 of the On-Line Encyclopedia of Integer Sequences (OEIS)&lt;/a&gt;: the &lt;em&gt;Minimum length of a string of letters that contains every permutation of n letters as sub-strings, also known as length of the minimal super-permutation.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[In September 2011] one anonymous 4chan poster set out to solve this puzzle. The person claimed to have worked out a lower bound for the answer...&lt;/p&gt;&#xA;&lt;p&gt;Crucially, they offered a proof – and, in doing so, they provided the world with an answer to a problem that had eluded mathematicians for a quarter of a century.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.iflscience.com/editors-blog/an-anonymous-online-anime-fan-just-solved-a-problem-thats-been-eluding-mathematicians-for-decades/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.iflscience.com/editors-blog/an-anonymous-online-anime-fan-just-solved-a-problem-thats-been-eluding-mathematicians-for-decades/&lt;/a&gt; (30 Oct 2018)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...because in 1993 there had been a publication on the problem that later turned out to be wrong. In October 2018, three professionals checked the proof and published a manuscript with &lt;em&gt;Anonymous 4chan Poster&lt;/em&gt; as the first author:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The above mentioned lower bound was essentially shown in 2011 by an anonymous poster on the internet and, filling in some minor details, brought into a formal form by Houston, Pantone and Vatter (&lt;a href=&quot;https://oeis.org/A180632/a180632.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;see reference&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://oeis.org/A180632&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://oeis.org/A180632&lt;/a&gt; (27 Oct 2018)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7364" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-12-23T21:59:51.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8099" PostTypeId="1" AcceptedAnswerId="8103" CreationDate="2018-12-24T17:49:11.573" Score="6" ViewCount="995" Body="&lt;p&gt;The &quot;modern&quot; understanding of the Galois group of a polynomial is as automorphisms of the splitting field of the polynomial which keep the base field fixed. These concepts were unknown to Galois, who thought about permutations of roots and &quot;substitutions.&quot; These formulations are &lt;a href=&quot;https://math.stackexchange.com/questions/3050732/how-did-galois-understand-the-galois-group/&quot;&gt;entirely equivalent&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is: where did the modern formulation come from? It seems likely that it was already in use in the course given by Noether and Artin in the 1920's, on which van der Waerden's Moderne Algebra was based. (Certainly this is the point of view in the 7th German edition). And it was presumably already well in place by the time Artin's 1948 lectures on Galois theory appeared. But maybe its origin was earlier, with Kronecker, or Dedekind, or Kummer?&lt;/p&gt;&#xA;" OwnerUserId="9253" LastEditorUserId="9253" LastEditDate="2019-01-11T16:23:48.133" LastActivityDate="2019-01-21T19:58:14.210" Title="How did the modern understanding of Galois theory come about?" Tags="|mathematics|abstract-algebra|galois-theory|algebra|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8100" PostTypeId="2" ParentId="8095" CreationDate="2018-12-24T19:02:01.930" Score="4" Body="&lt;p&gt;I think Euclid does refer to previous propositions by echoing the words.  Compare the alleged use of I.3 in I.5, given with a word-for-word translation of the Greek words:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0085%3Abook%3D1%3Atype%3DProp%3Anumber%3D3&quot; rel=&quot;nofollow noreferrer&quot;&gt;I.3 statement&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;δύο δοθεισῶν εὐθειῶν ἀνίσων         % Of two given straights unequal,&#xA;  ἀπὸ τῆς μείζονος                  % from the greater,&#xA;  τῇ ἐλάσσονι ἴσην                  % to the lesser equal&#xA;  εὐθεῖαν                           % (a) straight&#xA;  ἀφελεῖν.                          % to take&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0085%3Abook%3D1%3Atype%3DProp%3Anumber%3D5&quot; rel=&quot;nofollow noreferrer&quot;&gt;I.5, 11-12&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;ἀφῃρήσθω                            % let there have been taken&#xA;  ἀπὸ τῆς μείζονος τῆς ΑΕ           % from the greater AE&#xA;  τῇ ἐλάσσονι τῇ ΑΖ ἴση             % to the lesser AF (ΑΖ) equal&#xA;  ἡ ΑΗ                              % the (straight) AG (ΑΗ)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;To be more pedantic one could interpret the references to the figure as appositives: for example, &quot;ἀπὸ τῆς μείζονος τῆς ΑΕ...&quot; could be read &quot;from the greater, the (straight) AE,....&quot;  The main difference in I.5 is the omission of an explicit statement of what is given and the position of the verb.  The rest is word-for-word the same except for the appositives, which clarify the application to the proposition.  In this way, Euclid is referring to I.3 and no other proposition or postulate.  Throughout the Elements, Euclid refers to the relevant definition, postulate, etc. by echoing the words.  At first such references are word-for-word as above. Later he sometimes shortens it. &lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2018-12-25T01:12:21.100" LastActivityDate="2018-12-25T01:12:21.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8101" PostTypeId="1" CreationDate="2018-12-24T19:07:18.147" Score="3" ViewCount="289" Body="&lt;p&gt;In The Feynman Lectures on Physics volume 2, &lt;a href=&quot;https://www.feynmanlectures.caltech.edu/II_13.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Magnetostatics&lt;/a&gt;, section 13-4 Mr. Feynman tells us that there were &quot;three critical experiments and a brilliant theoretical argument given by Ampère&quot; that helped in determining the the magnetic field by a steady current. However he says he is going to skip those and come directly upon the results.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you please tell me what were those 3 experiments and that argument by Ampere. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="9254" LastEditorUserId="5989" LastEditDate="2018-12-25T01:30:21.407" LastActivityDate="2018-12-25T01:30:21.407" Title="What were the 3 critical experiments and the argument given by Ampère which helped in determining the magnetic field made by electric current?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8102" PostTypeId="2" ParentId="8101" CreationDate="2018-12-24T22:34:35.233" Score="3" Body="&lt;p&gt;Ampère devised more than 3 null experiments in the 1820s, but perhaps Feynman is referring to these:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Galvanic spirals and helices produce the same effects as magnets&quot;:&lt;br&gt;&lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/electrodynamique/index-en.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RtBT8.jpg&quot; alt=&quot;&amp;quot;Galvanic spirals and helices produce the same effects as magnets&amp;quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Experiment showing equivalence between helical current and bar magnet:&lt;br&gt;&lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/electrodynamique/index-en.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Rx9hH.jpg&quot; alt=&quot;helical current = bar magnet&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Experiment of two parallel rectilinear currents:&lt;br&gt;&lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/electrodynamique/index-en.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rIm5H.jpg&quot; alt=&quot;∥ rectilinear currents&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/electrodynamique/index-en.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;images source&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's the English translation of Ampère's work where he explains his experiments and derives &lt;a href=&quot;https://hsm.stackexchange.com/a/6865/232&quot;&gt;his force law&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Assis, André Koch Torres&lt;/a&gt;; Chaib, J. P. M. C; Ampère, André-Marie (2015). &lt;em&gt;&lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience&lt;/em&gt; (PDF)&lt;/a&gt;. Montreal: Apeiron. ISBN 978-1-987980-03-5.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Also, Ampère did not use the field concept, which is due to Faraday and which Ampère would have opposed for being &lt;em&gt;ad hoc&lt;/em&gt;; cf. &lt;a href=&quot;http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;ibid.&lt;/em&gt;&lt;/a&gt; ch. 15 pp. 221ff.&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;See also:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/electrodynamique/index-en.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampère lays the foundations of electrodynamics (September 1820-January 1821)&lt;/a&gt; by Christine Blondel &amp;amp; Bertrand Wolff (tr. Andrew Butrica)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;and &lt;a href=&quot;http://www.ampere.cnrs.fr/parcourspedagogique/index-en.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;the other excellent articles translated into English on &lt;em&gt;Ampère et l'histoire de l'électricité&lt;/em&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, Feynman seems to think the Maxwell equation named after Ampère (&lt;a href=&quot;https://hsm.stackexchange.com/q/5059/232&quot;&gt;&quot;Ampère's&quot; circuital law&lt;/a&gt;) was actually due to Ampère. It was not; &lt;a href=&quot;https://hsm.stackexchange.com/a/7609/232&quot;&gt;it was due to Heaviside&lt;/a&gt;. &lt;a href=&quot;https://hsm.stackexchange.com/a/6865/232&quot;&gt;Ampère's force law&lt;/a&gt; is something completely different.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2018-12-24T22:44:37.520" LastActivityDate="2018-12-24T22:44:37.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8103" PostTypeId="2" ParentId="8099" CreationDate="2018-12-25T01:28:43.280" Score="9" Body="&lt;p&gt;The systematic modern terminology and presentation of the Galois theory is due to Artin, a part of his joint project with Emmy Noether to reformulate the &quot;concrete&quot; older algebra in abstract terms, inspired by Dedekind. It was Artin who finally detached the Galois theory from the problem of solving equations algebraically, and gave a presentation that freely moves between considering the same object as a field, as a group, or as a vector space, as needed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, a book length presentation of the prior history in &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF00327219&quot; rel=&quot;noreferrer&quot;&gt;Kiernan's The development of Galois theory from Lagrange to Artin&lt;/a&gt; traces the ideas behind it to Kronecker, Dedekind and Weber. Kronecker addressed many points of Galois theory in a new light using his domain of rationality concept (not quite the modern field, but close). His treatment was developed in a number of papers since 1853, culminating in &lt;em&gt;Grundzuge einer arithmetischen Theorie der algebraischen Gröβen&lt;/em&gt; (1882, Jen. Werke, II, 237-387). Here is Kiernan, p.127:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Kronecker makes reference in this article to many points of Galois' work. He is primarily concerned with the concept of adjunction alone; so this is not a development of Galois Theory. Nevertheless he does provide a concept of a system, the domain of rationality of the coefficients, to which the roots of the successive auxiliary equations may be adjoined. There can also be recognized, in very primitive fashion, the connection between the group of the equation and the automorphisms on the extension fields of the field of coefficients. Of course, Kronecker could not express the result in this way, since he did not see a domain of rationality as a completed entity, but merely as a region or place which contained the result of any finite number of algebraic operations on its elements, or rather where these operations took place.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Dedekind lectured on the Galois theory at Göttingen in 1856-57, the same time he corresponded with Kronecker about it, and introduced the concept of a field  (&lt;em&gt;rational Gebiet&lt;/em&gt; at the time, &lt;em&gt;Körper&lt;/em&gt; referred only to complex number fields then) in 1857-58. His treatment became standard in German universities for a while, but he reworked it anew in the 1890-s.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In the 1894 edition of the Zahlentheorie, Dedekind developed a theory of field extensions which is in some ways similar to that of Kronecker. In his development, Dedekind specifically refers to Galois and his work on the concept of adjoining elements to the set of coefficients of an equation… Dedekind then develops the idea of a field isomorphism, which he calls a &quot;permutation of the field&quot;... Further, he recognizes that any &quot;permutation&quot; &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; of a field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is an identity mapping on &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, that is, &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; maps each element of the rational field &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; onto itself, no matter what it does to the other elements of &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;. Also, if &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; is a permutation, that is, an isomorphism, of the field K onto K', then, says Dedekind, the set of elements in K left fixed by &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; forms a field containing &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;,&#xA;  called the field belonging to &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;... Many results developed here by Dedekind on the interpretation of an extension field as a vector space over the ground field were later used by Artin in his formulation of Galois Theory… In retrospect it is surprising how little immediate use was made of the work either of Kronecker or of Dedekind in field theory. The modern development of field theory, so brilliantly begun by Dedekind, lay largely dormant for another 30 years.&lt;/em&gt;&quot; [Kiernan, p.128ff]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To the extent that the field concept was used for the Galois theory at all in the late 19th century, it was Kronecker's rather than Dedekind's. One exception to this trend was Weber in his &lt;em&gt;Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie&lt;/em&gt; (1893), and &lt;em&gt;Lehrbuch der Algebra&lt;/em&gt; (1895):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Weber's development of Galois Theory is certainly the first modern treatment of the topic. The development is not restricted to the rational field, but to an arbitrary field. The theory itself is clearly acknowledged as concerned only with the extensions of the ground field and the groups of automorphisms of these extensions. Even the consideration of the process of solution is made&#xA;  secondary to the study of the nature of the solution.&#xA;  But for a paper to be regarded as modern today, it must necessarily have been years ahead of its time in the 1890's. As will be seen in the next section, Weber's presentation stands almost alone in its own generation. It may be viewed as a direct predecessor of Artin's work, forty years later.&lt;/em&gt;&quot; [Kiernan, p.141]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-12-26T09:16:51.477" LastActivityDate="2018-12-26T09:16:51.477" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8104" PostTypeId="1" AcceptedAnswerId="8106" CreationDate="2018-12-26T04:54:38.080" Score="5" ViewCount="573" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Positional_notation#Sexagesimal_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; section of a Wikipedia article says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;[Modern time and angle notation] contrasts with the numbers used by Hellenistic and Renaissance astronomers, who used thirds, fourths, etc. for finer increments. Where we might write 10°25'59.392&quot;, they would have written 10°25′59″23‴31⁗12′′′′′ or 10&lt;sup&gt;0&lt;/sup&gt;25&lt;sup&gt;I&lt;/sup&gt;59&lt;sup&gt;II&lt;/sup&gt;23&lt;sup&gt;III&lt;/sup&gt;31&lt;sup&gt;IV&lt;/sup&gt;12&lt;sup&gt;V&lt;/sup&gt;.&lt;/blockquote&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Are these claims correct? and if so, according to what secondary source?&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Were Roman numerals used in superscript with this meaning?&lt;/li&gt;&#xA;&lt;li&gt;Were divisions beyond the arcsecond used?&lt;/li&gt;&#xA;&lt;/ol&gt;&lt;/li&gt;&#xA;&lt;li&gt;Is the origin of the prime (typographic symbol), and double and triple prime, in the Roman numerals I, II, and III?&lt;/li&gt;&#xA;&lt;li&gt;Is the origin of the degree symbol, similarly, in a superscript zero?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9258" LastEditorUserId="9258" LastEditDate="2018-12-27T02:13:16.533" LastActivityDate="2018-12-27T09:15:14.970" Title="Origin of arcminutes, arcseconds, &quot;arcthirds,&quot; &quot;arcfourths,&quot; etc" Tags="|mathematics|notation|geometry|language|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8105" PostTypeId="2" ParentId="7600" CreationDate="2018-12-26T09:55:34.377" Score="5" Body="&lt;p&gt;[I do not have enough reputation to add comments so I have to make this an answer]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;FWIW, I have written &lt;a href=&quot;https://pl.wikipedia.org/wiki/Pentagramma_mirificum&quot; rel=&quot;nofollow noreferrer&quot;&gt;a Polish Wikipedia article&lt;/a&gt; on &lt;em&gt;pentagramma mirificum&lt;/em&gt;. For the time being, you can try to make sense of it with Google Translate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit (December 28, 2018): Lo and behold, &lt;a href=&quot;https://en.wikipedia.org/wiki/Pentagramma_mirificum&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article in English Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit (January 6, 2019): Also, &lt;a href=&quot;https://youtu.be/wQKMAx6jxGw&quot; rel=&quot;nofollow noreferrer&quot;&gt;a short video about the construction of &lt;em&gt;pentagramma mirificum&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/uLRgp.gif&quot; alt=&quot;Pentagramma mirificum&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="9246" LastEditorUserId="9246" LastEditDate="2019-01-06T07:35:23.293" LastActivityDate="2019-01-06T07:35:23.293" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8106" PostTypeId="2" ParentId="8104" CreationDate="2018-12-26T11:27:05.697" Score="5" Body="&lt;p&gt;See F.Cajori, &lt;a href=&quot;https://books.google.it/books?id=bT5suOONXlgC&amp;amp;pg=PA142&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Mathematical Notations: Vol. II&lt;/a&gt;, para 511 : &lt;em&gt;Origin of the modern symbols for degrees, minutes, and seconds&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Signs resembling those now in use are found in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Almagest&quot; rel=&quot;nofollow noreferrer&quot;&gt;Syntaxis (Almagest)&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ptolemy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy (ca. AD 100 – ca. 170)&lt;/a&gt;, where the Babylonian sexagesimal fractions are used in astronomical calculations. The units were sometimes called &lt;em&gt;μοῖραι&lt;/em&gt; and frequently denoted by the abbreviation &lt;span class=&quot;math-container&quot;&gt;$\mu^°$&lt;/span&gt;. The &lt;em&gt;first sixtieths&lt;/em&gt; or minutes were marked with one accent, the &lt;em&gt;second sixtieths&lt;/em&gt; with two accents. [...] From these facts it would seem that our signs &lt;span class=&quot;math-container&quot;&gt;$^°, ', ''$&lt;/span&gt; for degrees, minutes, and seconds were of Greek origin. But it is difficult to uphold this view, especially for the sign &lt;span class=&quot;math-container&quot;&gt;$^°$&lt;/span&gt; for &quot;degrees.&quot; Such a line of descent has not been established.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See an example of &lt;a href=&quot;https://books.google.it/books?id=4ppKAAAAcAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;medieval astronomical table&lt;/a&gt; : the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfonsine_tables&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alfonsine tables&lt;/a&gt;, compiled in Toledo, Spain, and containing astronomical data starting on January 1, 1252, the date of the coronation of the King &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfonso_X_of_Castile&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alfonso X of Castile&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;During Middle Ages and Renaissance [see &lt;a href=&quot;https://books.google.it/books?id=bT5suOONXlgC&amp;amp;pg=PA143&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori, para 512&lt;/a&gt;]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the names &lt;em&gt;signa, gradus, minutae, secundae&lt;/em&gt;, etc., were used with several abbreviations; the more common ones are &lt;em&gt;Sig., Gr., Min., Sec.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Ulricus Regius in his &lt;a href=&quot;https://books.google.it/books?id=hs85AAAAcAAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Epitome&lt;/em&gt; of 1536&lt;/a&gt; refers to the Alfonsine tables and gives the signs &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$T, s, g, \overline m, \overline s, \overline t, \overline {qr}$&lt;/span&gt;. [&lt;em&gt;Denominator minutorum est unitas. Secundorum binarius, Tertiorum ternarius, Etc.&lt;/em&gt;]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In 1540 &lt;a href=&quot;https://en.wikipedia.org/wiki/Gemma_Frisius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gemma Frisius&lt;/a&gt; wrote :&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$\text{Integr. Mi. 2. 3. 4.}$&lt;/span&gt; &lt;/p&gt;&#xA;    &#xA;    &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\ \ \ \ \ \ \text{36.   30.  24  50  15}$&lt;/span&gt;&quot;&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;for our modern &lt;span class=&quot;math-container&quot;&gt;$36^°30'24''50'''15^{iv}$&lt;/span&gt;. This is the first modern appearance that I have found of &lt;span class=&quot;math-container&quot;&gt;$°$&lt;/span&gt; for &lt;em&gt;integra&lt;/em&gt; or &quot;degrees.&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In 1571 &lt;a href=&quot;https://en.wikipedia.org/wiki/Erasmus_Reinhold&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erasmus Reinhold&lt;/a&gt; gave an elaborate explanation of sexagesimal fractions as applied to angular measure and wrote &quot;&lt;span class=&quot;math-container&quot;&gt;$62° 54' 18''$&lt;/span&gt;.&quot; This notation was adopted by &lt;a href=&quot;https://en.wikipedia.org/wiki/Tycho_Brahe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tycho Brahe&lt;/a&gt; who in his comments of 1573 on his &lt;em&gt;Nova Stella&lt;/em&gt; writes &lt;span class=&quot;math-container&quot;&gt;$75° 5'$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In conclusion : yes, roman numerals were adopted as superscript for the fractions of degree.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;But the &quot;scientific notation&quot; for fractions used in Astronomy was not used in &quot;ordinary&quot; mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See e.g. &lt;a href=&quot;http://jeff560.tripod.com/fractions.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient roman fractions&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Romans did not use numerals to indicate fractions, but instead used words to indicate parts of a whole. A unit of weight was the &lt;em&gt;as&lt;/em&gt; and the &lt;em&gt;uncia&lt;/em&gt; (from which we have the word &quot;ounce&quot;) was a twelfth part of the &lt;em&gt;as&lt;/em&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See e.g. the term &lt;em&gt;deunx&lt;/em&gt; for the fraction &lt;span class=&quot;math-container&quot;&gt;$\dfrac {11} {12}$&lt;/span&gt; (from &lt;em&gt;de uncia&lt;/em&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$\dfrac {1}{12}$&lt;/span&gt; taken away).&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2018-12-27T09:15:14.970" LastActivityDate="2018-12-27T09:15:14.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8107" PostTypeId="2" ParentId="8086" CreationDate="2018-12-28T02:25:49.950" Score="2" Body="&lt;p&gt;That may be considered a cheating answer but indeed there was an era when this algorithm was used intensely. I'm talking about early decades of electronic computers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This algorithm is very easy to implement in binary because multiplication and division by 2 becomes simply a 1 bit shift of the number in the left or the right direction correspondingly. The lowest bit gives you the parity. Thus all you need from you processor is to have a shifter and adder. Before appearance of new algorithms in 60s and possibility to perform several additions simultaneously at a hardware level this was how multiplication was done on computers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The positional system provides you with several techniques relying on the knowledge of the multiplication table for all numbers up to the base. The Ancient Egyptian method is attractive if you don't have any of that (or restricted to the base 2 like computers). I couldn't find (not that I searched very long...) much info on the multiplication methods from Ancient Greek sources. However I find it likely that by the time such sources could exist this Ancient Egyptian multiplication method was not the main one for the people educated in arithmetics (who, again would write such sources). The Babylonians had a positional numerical system and multiplication tables by 2000 bc. While true positional numerical systems were not that popular in the ancient world, the abacus was.&lt;/p&gt;&#xA;" OwnerUserId="9264" LastActivityDate="2018-12-28T02:25:49.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8108" PostTypeId="2" ParentId="8086" CreationDate="2018-12-28T11:28:32.700" Score="2" Body="&lt;p&gt;The analogue for exponentiation, usually referred to as &quot;square-and-multiply&quot; is in common use in applied cryptography, e.g. in the RSA system for public key encryption.  See for example Section 14.6 of Menezes et al's Handbook of Applied Cryptography. &lt;/p&gt;&#xA;" OwnerUserId="9267" LastActivityDate="2018-12-28T11:28:32.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8109" PostTypeId="1" AcceptedAnswerId="8111" CreationDate="2018-12-27T19:03:04.740" Score="58" ViewCount="14278" Body="&lt;p&gt;I realize that &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; are relatively popular terms when wanting to use a placeholder for an unknown English or math term. What is the origin of this term, and why was it &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;; why not the other letters?&lt;/p&gt;&#xA;" OwnerUserId="9268" OwnerDisplayName="Sweet_Cherry" LastEditorUserId="4251" LastEditDate="2022-08-17T09:11:03.807" LastActivityDate="2022-12-13T10:12:19.783" Title="Why are $X$ and $Y$ commonly used as mathematical placeholders?" Tags="|mathematics|notation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8110" PostTypeId="2" ParentId="8109" CreationDate="2018-12-27T19:31:28.697" Score="30" Body="&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/variables.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Earliest uses of mathematical symbols&lt;/em&gt;&lt;/a&gt;, which quotes&#xA;F. Cajori, &lt;em&gt;A History of Mathematical Notations&lt;/em&gt;, 2 volumes (1928-29)  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The use of &lt;em&gt;z, y, x ...&lt;/em&gt; to represent unknowns is due to René Descartes, in his &lt;em&gt;La géometrie&lt;/em&gt; (1637). Without comment, he introduces the use of the first letters of the alphabet to signify known quantities and the use of the last letters to signify unknown quantities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" OwnerDisplayName="Gerald Edgar" LastActivityDate="2018-12-27T19:31:28.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8111" PostTypeId="2" ParentId="8109" CreationDate="2018-12-27T19:26:24.807" Score="71" Body="&lt;p&gt;They became popular because of René Descartes’ usage in his &lt;a href=&quot;https://en.m.wikipedia.org/wiki/La_G%C3%A9om%C3%A9trie&quot; rel=&quot;noreferrer&quot;&gt;La Géométrie&lt;/a&gt;. The letters at the end of the alphabet are chosen as the variables, while those at the beginning are constants. There is speculation about why this might have been done. It is likely to allow the largest number of sequential letters without overlap between the two sets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why &lt;em&gt;x&lt;/em&gt; became the most common is unknown. Some sources attempt to draw a line from the Arabic word for &lt;em&gt;unknown&lt;/em&gt; through the Greek letter &lt;em&gt;chi&lt;/em&gt; (which resembles a capital X), but the claims are unsubstantiated (Arabic being the source of our numerals and Greek being the common letter set for variables).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The link from mathematics to common speech is likely just a simple repurposing of known concepts.&lt;/p&gt;&#xA;" OwnerDisplayName="Ian MacDonald" LastActivityDate="2018-12-28T14:18:58.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8112" PostTypeId="2" ParentId="8109" CreationDate="2018-12-29T10:19:10.397" Score="9" Body="&lt;p&gt;Here is the original source : René Descartes, &lt;em&gt;La Géométrie&lt;/em&gt; (1637), &lt;a href=&quot;https://fr.wikisource.org/wiki/Page%3ADescartes_-_Discours_de_la_m%C3%A9thode%2C_%C3%A9d._1637.djvu/383&quot; rel=&quot;nofollow noreferrer&quot;&gt;I, page 299&lt;/a&gt;, for &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; used to denote &lt;em&gt;parameters&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;And see &lt;a href=&quot;https://fr.wikisource.org/wiki/Page%3ADescartes_-_Discours_de_la_m%C3%A9thode%2C_%C3%A9d._1637.djvu/385&quot; rel=&quot;nofollow noreferrer&quot;&gt;I, page 301&lt;/a&gt; for &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; and &lt;a href=&quot;https://fr.wikisource.org/wiki/Page%3ADescartes_-_Discours_de_la_m%C3%A9thode%2C_%C3%A9d._1637.djvu/387&quot; rel=&quot;nofollow noreferrer&quot;&gt;I, page 303&lt;/a&gt; for &lt;span class=&quot;math-container&quot;&gt;$x,y$&lt;/span&gt; respectively, used to refer to an &lt;em&gt;unknown quantity&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Letters was already used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;François Viète&lt;/a&gt; (but the use of alphabetical variables to represent &lt;em&gt;magnitudes&lt;/em&gt; is due to euclidean geometry).&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://archive.org/details/bub_gb_BWTyywN39KEC/page/n15&quot; rel=&quot;nofollow noreferrer&quot;&gt;In artem analyticem isagoge (1591)&lt;/a&gt;, &lt;strong&gt;Rule III&lt;/strong&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sunto duae magnitudines &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;. [Let there be two magnitudes, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And also :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Oportet &lt;span class=&quot;math-container&quot;&gt;$A \dfrac {\text { plano }}{B}$&lt;/span&gt; addere &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; [Suppose &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; is to be added to &lt;span class=&quot;math-container&quot;&gt;$A^p / B$&lt;/span&gt;].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But obviously the success of Descartes' &amp;quot;new geometry&amp;quot; explains the success of the new algebraic notation.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2022-12-13T10:12:19.783" LastActivityDate="2022-12-13T10:12:19.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8113" PostTypeId="1" CreationDate="2018-12-29T13:13:26.060" Score="2" ViewCount="119" Body="&lt;p&gt;I tried to find Gauss's original paper on his method of quadrature, but in vain. Is it translated into English? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;How about Legendre's paper? &lt;/p&gt;&#xA;" OwnerUserId="4079" LastActivityDate="2018-12-30T16:49:39.623" Title="Original paper of Gauss on his method of quadrature" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8114" PostTypeId="1" AcceptedAnswerId="8120" CreationDate="2018-12-29T15:54:22.327" Score="1" ViewCount="939" Body="&lt;p&gt;I never did find an answer from professors, or even see an acknowledgement in textbooks, on why capital-letter-r is invariably used to represent the constant 0.08206 L-atm/mol-K seen in chemistry everywhere. The only variation I've seen is to use subscripts or overbars to denote incorporation of a particular circumstances.&lt;/p&gt;&#xA;" OwnerUserId="9274" LastEditorDisplayName="user3224" LastEditDate="2019-07-10T01:23:19.070" LastActivityDate="2019-07-10T01:23:19.070" Title="Who came up with R for the universal gas constant?" Tags="|terminology|naming-conventions|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8115" PostTypeId="1" CreationDate="2018-12-29T19:06:17.677" Score="4" ViewCount="212" Body="&lt;p&gt;Einstein realized that uniform acceleration is indistinguishable from a homogeneous gravitational field (the principle of equivalence) through the Einstein Elevator thought experiment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the thought experiment, the physicist will not be able to tell the difference between whether he is homogenous gravitational field or if he is in gravitation free space in uniform acceleration. How did Einstein generalise this one case to conclude as a whole that uniform acceleration creates a uniform gravitational field? This is a fundamental part of general relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it linked with the law of conservation of mass and energy?&lt;/p&gt;&#xA;" OwnerUserId="9276" LastActivityDate="2019-02-02T21:02:40.583" Title="General Relativity, General Covariance and Equivalence Principle" Tags="|philosophy-of-science|relativity-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8116" PostTypeId="1" CreationDate="2018-12-29T20:39:50.760" Score="3" ViewCount="605" Body="&lt;p&gt;According to one historian Mustafa Sarikaya's article in Foundations of Chemistry DOI 10.1007/s10698-011-9128-7, the mole concept was introduced to chemistry earlier than Avogadro’s number. The mole concept was introduced by the German chemist August Horstmann. How did the early chemists determine that if we weigh an amount of a pure compound in grams (solid/liquid/gas) equal to its gram formula weight, it will contain an Avogadro's number of &quot;particles.&quot; &#xA;The second question is related to the arbitrary definition of atomic mass scale of C-12 as 12 (as an exact number). This was a scaling choice made to make physicists and chemistry agree without changing much of their old tabulated values of atomic weights (c.f. chemical scale and physical scale of atomic masses). Does 12.00000000 g exactly contain Avogadro's number of C-12 atoms or the value of Avogadro's number was also adjusted when C-12 was chosen as a standard by the IUPAC? Thanks. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: The key query here is how the gram formulae of any solid, liquid or gas were connected with Avogadro's number in the early 19th century? What &lt;em&gt;experimental evidence&lt;/em&gt; led them to conclude that the gram formula weight of any pure substance has Avogadro's number of particles?&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="8202" LastEditDate="2019-01-03T17:31:54.910" LastActivityDate="2019-02-03T02:09:52.290" Title="How did the early chemists make a connection between gram formula weight with 1 mole and Avogadro's number?" Tags="|terminology|philosophy-of-science|experimental-physics|chemistry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8117" PostTypeId="2" ParentId="8113" CreationDate="2018-12-29T21:26:01.383" Score="4" Body="&lt;p&gt;You can find a link to Gauss' 1815 paper on numerical quadrature, ``&lt;em&gt;Methodus nova integralium valores per approximationem inveniendi&lt;/em&gt;'' &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k2412190.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  I'm not aware of any translation of it into English.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2018-12-30T16:49:39.623" LastActivityDate="2018-12-30T16:49:39.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8118" PostTypeId="1" AcceptedAnswerId="8119" CreationDate="2018-12-30T03:25:51.087" Score="2" ViewCount="413" Body="&lt;p&gt;The Body Mass Index (BMI) compares body masses on the assumption they scale with height squared, not cubed, an example of allometry. BMI is due to Lambert Quetelet. Why did he settle on this power law? Was it a theoretical case such as conserving pressure, or an empirical exponent?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="55" LastEditDate="2018-12-30T09:35:53.087" LastActivityDate="2018-12-30T09:35:53.087" Title="How did Quetelet discover that the body mass is proportional to the squared height?" Tags="|discoveries|biology|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8119" PostTypeId="2" ParentId="8118" CreationDate="2018-12-30T09:32:03.090" Score="4" Body="&lt;p&gt;BMI is now widely used for detecting obesity, but Quetelet's motivation was in defining the characteristics of an ‘average man’. Quetelet was one of the early enthusiasts of what we now call statistical studies and aspired to extend  statistical analysis beyond demographic and anthropometric characteristics to other aptitudes, associated with behaviour, and even mind. A good scientific biography is &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0021968172900276&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eknoyan's Adolphe Quetelet (1796–1874) — the average man and indices of obesity&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1831–1832, at the start of his career, Quetelet conducted the first cross-sectional study of newborns and children based on height and weight, and extended it to the study of adults. The results were published in the Proceedings of the Academy of Sciences as ‘&lt;em&gt;Recherches sur le poids de l’homme aux différent âges&lt;/em&gt;’ (Research on the weight of man at different ages, 1832), where the index is first proposed (he did not call it BMI). The empirical weight to height distribution did not really fit into a Gaussian curve. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on his studies, in 1835, Quetelet published a book, A Treatise on Man subtitled Essays on Social Physics , in three volumes (renamed into Social Physics or an Assay on the Development of the Faculties of Man in 1869). Chapter 2 of the second volume explains his motivation for the index, which is clearly empirical:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If man increased equally in all dimensions, his weight at different ages would be as the cube of his height. Now, this is not what we really observe. The increase of weight is slower, except during the first year after birth; then the proportion we have just pointed out is pretty regularly observed. But after this period, and until near the age of puberty, weight increases nearly as the square of the height. The development of weight again becomes very rapid at puberty, and almost stops after the twenty-fifth year. In general, we do not err much when we assume that during development the squares of the weight at different ages are as the fifth powers of the height; which naturally leads to this conclusion, in supporting the specific gravity constant, that the transverse growth of man is less than the vertical&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The index, known to the experts as the Quetelet Index, remained obscure for over a century, until the relation between body weight and mortality, cardiac disease and diabetes, became a medical concern after World War II. The new studies in the 1960s also suggested that the normal body weight in kilograms was proportional to the square of the height in meters, as Quetelet observed. Keys introduced the name Body Mass Index in a 1972 comparative study of available indices, that confirmed the validity of Quetelet's observation. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2018-12-30T09:32:03.090" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8120" PostTypeId="2" ParentId="8114" CreationDate="2018-12-30T13:31:54.190" Score="5" Body="&lt;p&gt;I quote from &lt;a href=&quot;http://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/100.%20Gas%20Constant.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Ask the Historian&lt;/em&gt;&lt;/a&gt; &quot;The Universal Gas Constant&quot; by William B. Jensen, Department of Chemistry, University of Cincinnati, published in &lt;em&gt;J. Chem. Educ.&lt;/em&gt;, &lt;strong&gt;2003&lt;/strong&gt;, &lt;em&gt;80, 731-732&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;br&gt;&#xA;  Why is the universal gas constant in &lt;span class=&quot;math-container&quot;&gt;$PV = nRT$&lt;/span&gt; represented by the letter &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;?&lt;br&gt;&#xA;  &lt;strong&gt;Answer&lt;/strong&gt;&lt;br&gt;&#xA;  This is best answered by tracing the origins of the ideal gas law itself. One of the first persons to combine Boyle’s law (1662) relating volume and pressure and Gay-Lussac’s law (1802) relating volume and temperature in a single equation appears to have been the French engineer, Benoit-Paul Emile Clapeyron (1799- 1864). In his famous memoir of 1834 on the Carnot cycle, he wrote the combined equation as &#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;pv = R(267 + t) \tag{1}$$&lt;/span&gt;&#xA;  where &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is the temperature in degrees centigrade.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later versions of the law kept the letter &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later in the same reference, Jensen guesses why &quot;&lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;&quot; was chosen by Clapeyron.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So why did Clapeyron choose the letter &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; for the constant in his gas law? The fact is that he doesn’t explicitly tell us why and we are left with two speculative answers: (a) it was arbitrary or (b) it stood for &lt;em&gt;ratio&lt;/em&gt; or one of its French equivalents: &lt;em&gt;raison&lt;/em&gt; or &lt;em&gt;rapport&lt;/em&gt;, since Clapeyron noted that the value of R for each gas was obtained by evaluating the constancy of the ratio &lt;span class=&quot;math-container&quot;&gt;$pv/(267 + t)$&lt;/span&gt; over a range of pressures and temperatures, a point also emphasized by Clausius using the revised ratio &lt;span class=&quot;math-container&quot;&gt;$pv/(273 + t)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2018-12-30T13:31:54.190" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8121" PostTypeId="1" AcceptedAnswerId="8122" CreationDate="2018-12-30T16:11:47.917" Score="2" ViewCount="2301" Body="&lt;p&gt;How did Ruffini discover his method of polynomial division? At that time was it known that polynomial division works similar to integer division?&lt;/p&gt;&#xA;" OwnerDisplayName="user9281" LastEditorUserId="55" LastEditDate="2018-12-30T19:02:56.950" LastActivityDate="2018-12-31T09:51:32.103" Title="How did Ruffini discover his method of polynomial division?" Tags="|mathematics|discoveries|elementary-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8122" PostTypeId="2" ParentId="8121" CreationDate="2018-12-30T19:00:29.717" Score="3" Body="&lt;p&gt;Polynomial division algorithms were known long before Ruffini's &lt;em&gt;Sopra la determinazione delle radici&lt;/em&gt; (1804). For a history of even older numerical division algorithms see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2093/who-invented-short-and-long-division&quot;&gt;Who invented short and long division?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://new.math.uiuc.edu/im2008/rogers/algebra.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some authors&lt;/a&gt;, see also &lt;a href=&quot;http://books.google.com/books/about/The_History_of_Mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Victor Katz's History of Mathematics (7.2.3)&lt;/a&gt;, credit medieval Arabic mathematician al-Samaw'al (1130-1180) for inventing long division, of both polynomials and integers. Al-Samaw'al was first to use tables of coefficients to write and perform calculations with polynomials, he even allowed negative powers. He described a division method underlying the polynomial long division, although his record keeping is more akin to what is now called synthetic division. He also noted the analogy between his way of writing polynomials and the decimal positional notation, and transferred his algorithms to decimal numbers by simply replacing the variable with &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, so he may well be the first to provide a mathematical justification for a positional division algorithm. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method for dividing by a linear factor is presented implicitly in Descartes's &lt;em&gt;La Geometrie&lt;/em&gt; (1637), where it is used as a part of his algebraic algorithm for drawing tangents to algebraic curves (described in &lt;a href=&quot;https://hsm.stackexchange.com/questions/1963/is-there-a-lost-calculus/1968#1968&quot;&gt;Is there a &amp;#39;lost calculus&amp;#39;?&lt;/a&gt;). A more efficient algorithm of polynomial division, an optimized version of synthetic division, was produced by Jan Hudde in a letter included into 1659 edition of Descartes's &lt;em&gt;La Geometrie&lt;/em&gt;, see a description in &lt;a href=&quot;https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/suzuki339.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Suzuki's The Lost Calculus (1637–1670)&lt;/a&gt;. Hudde was a talented Dutch mathematician who had to abandon mathematics for politics to save Netherlands from Spanish invasion. Ruffini's rule is an adaptation of Hudde's, see &lt;a href=&quot;https://boycottholland.wordpress.com/2016/12/22/ruffinis-rule-and-meaning-of-division/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ruffini’s Rule and meaning of division&lt;/a&gt; post. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Ruffini describes generalizes to the modern synthetic division algorithm for polynomials, see &lt;a href=&quot;https://files.eric.ed.gov/fulltext/EJ1121410.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fitzherbert's Ghosts of mathematicians past: Paolo Ruffini&lt;/a&gt;. For subsequent generalizations by Budan and Horner see &lt;a href=&quot;https://books.google.com/books?id=XcDqCAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chabert's  History of Algorithms, 7.9&lt;/a&gt;. As a method of finding roots, historians trace the Ruffini–Horner scheme to the Chinese mathematician Jia Xian (ca. 1010–ca. 1070), who lived even earlier than al-Samaw'al, see &lt;a href=&quot;https://www.cambridge.org/core/journals/arabic-sciences-and-philosophy/article/similarities-between-chinese-and-arabic-mathematical-writings-i-root-extraction1/E6F7BE7877E4C1D4265B7819DD5A2702&quot; rel=&quot;nofollow noreferrer&quot;&gt;Similarities between Chinese and Arabic Mathematical Writings: (I) Root extraction by Chemla&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2018-12-31T09:51:32.103" LastActivityDate="2018-12-31T09:51:32.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8123" PostTypeId="1" AcceptedAnswerId="8127" CreationDate="2018-12-31T04:54:48.870" Score="2" ViewCount="199" Body="&lt;p&gt;The weak equivalence principle (WEP) is often attributed to Galileo. Did Galileo ever state the WEP precisely? For instance as given &lt;a href=&quot;https://www.mathpages.com/home/kmath629/kmath629.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The original principle, usually attributed to Galileo, asserts that all material objects in free fall undergo the same acceleration in a gravitational field, regardless of their mass and composition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7638" LastEditorUserId="55" LastEditDate="2018-12-31T11:22:54.387" LastActivityDate="2018-12-31T12:10:11.110" Title="Did Galileo state the principle of equivalence in full generality?" Tags="|relativity-theory|gravity|galileo|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8124" PostTypeId="1" AcceptedAnswerId="8144" CreationDate="2018-12-31T07:09:03.410" Score="1" ViewCount="251" Body="&lt;p&gt;How did ancient Egyptians know that they have to choose the symbols for multiple of 10 in their Egyptian number system, since at that time hindu-arabic system was not there and no one knows what is 1,2,3,...or 10?&lt;/p&gt;&#xA;" OwnerDisplayName="user9281" LastEditorDisplayName="user9281" LastEditDate="2018-12-31T09:30:03.350" LastActivityDate="2019-01-06T16:16:13.727" Title="Egyptian number system?" Tags="|mathematics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8125" PostTypeId="2" ParentId="8116" CreationDate="2018-12-31T09:24:15.103" Score="1" Body="&lt;p&gt;In 1806, John Dalton published a table of the relative weights of elements. Since elements are very light, he decided to publish the weights of &quot;one pack of X elements, with X chosen such as the weight of the pack for Hydrogen is one gram&quot;. This is called the atomic mass unit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since writing &quot;one pack of X elements, with X chosen such as the weight of the pack for Hydrogen is one&quot; is a bit long, in 1893, Wilhelm Ostwald invented the word &quot;mole&quot; to simplify things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make experiments easier, the definition of the mole has been changed to &quot;one pack of X elements, with X chosen such as the weight of the pack for Carbon-12 is twelve grams&quot; around 1960. It turns out that it basically represents the same thing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this definition is only as precise as the measurement of twelve grams of Carbon-12. To overcome this difficulty, the mole has been defined in 2019 as exactly &lt;span class=&quot;math-container&quot;&gt;$6.02214076×10^{23}$&lt;/span&gt;, and not related to an actual element in any way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TL,DR:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A man named Dalton &lt;strong&gt;decided&lt;/strong&gt; that &lt;span class=&quot;math-container&quot;&gt;$N_a$&lt;/span&gt; (Avogadro constant) elements of Hydrogen would weight one gram.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It stuck.&lt;/p&gt;&#xA;" OwnerUserId="9292" LastActivityDate="2018-12-31T09:24:15.103" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8127" PostTypeId="2" ParentId="8123" CreationDate="2018-12-31T11:22:01.907" Score="3" Body="&lt;p&gt;Not quite. This is what Galileo stated in the Two New Sciences (1638) through his character Sagredo (honest inquirer open to arguments from both sides):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;But I, Simplicio, who have made the test can assure you that a cannon ball weighing one or two hundred pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a pound, provided both are dropped from a height of 200 cubits.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Viviani, in his 1654 biography, reported that Galileo had made the test himself &amp;quot;&lt;em&gt;in front of all the faculty and students assembled&lt;/em&gt;&amp;quot;, but this is unconfirmed. The common anecdote about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo%27s_Leaning_Tower_of_Pisa_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;balls dropped from the leaning tower of Pisa&lt;/a&gt; grew out of his account. Galileo did study the falling bodies since 1589, and did perform less theatrical experiments. Moreover, already in 1586  Stevin simultaneously released two pieces of lead (1 pound and 10 pounds) from a height of 30 feet,  and observed that the sounds of their impacts &amp;quot;&lt;em&gt;could not be separated&lt;/em&gt;&amp;quot;, see &lt;a href=&quot;http://www.numericana.com/answer/record.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michon's Final Answers&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Right after the above quoted line, Salviati (Galileo's mouthpiece) gives a famous thought experiment to the same effect. He does stipulate the &amp;quot;same material&amp;quot; though:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;But, even without further experiment, it is possible to prove clearly, by means of a short and conclusive argument, that a heavier body does not move more rapidly than a lighter one, provided both bodies are of the same material, and in short are such as those mentioned by Aristotle.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;if a large stone moves with a speed of, say, eight, while a smaller stone moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the light one, I infer that the heavier body moves more slowly... We infer therefore that large and small bodies move with the same speed, provided they are of the same specific gravity.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The specific gravity is proportional to the density. See a discussion of the validity of the argument in &lt;a href=&quot;http://thep.housing.rug.nl/sites/default/files/users/user12/127_Galileo_and_Prior_Philosophy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo and Prior Philosophy by Atkinson and Peijnenburg&lt;/a&gt;. The ship mast experiment, mentioned in the comments, is from the earlier Dialogue Concerning the Two Chief World Systems (1632). It is related, but was meant to argue for the idea of rotating Earth.&lt;/p&gt;&#xA;&lt;p&gt;Newton stated the equivalence principle more broadly and precisely in 1687 (&lt;a href=&quot;https://en.wikisource.org/wiki/Page:Newton%27s_Principia_(1846).djvu/400&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principia, Book III, Proposition VI, Theorem VI&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://einstein.stanford.edu/STEP/information/data/gravityhist2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cultural History of Gravity and the Equivalence Principle&lt;/a&gt; for an outline of further developments.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2018-12-31T12:10:11.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8128" PostTypeId="2" ParentId="5839" CreationDate="2019-01-01T19:41:10.177" Score="1" Body="&lt;p&gt;It has been claimed that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacquard_loom&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Jaquard Loom&lt;/strong&gt;&lt;/a&gt; (1804) was a digital computer.  (But not electronic, of course.)  We could even say that it was &lt;strong&gt;binary&lt;/strong&gt;, since on the punched cards that control it, in each locaton on the card there was either a hole or not.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then there were &lt;a href=&quot;https://en.wikipedia.org/wiki/Music_box&quot; rel=&quot;nofollow noreferrer&quot;&gt;music boxes&lt;/a&gt; driven by clockwork, where the notes where controlled by holes punched in a metal disk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lnoy4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lnoy4.jpg&quot; alt=&quot;music box&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2019-01-02T01:47:01.023" LastActivityDate="2019-01-02T01:47:01.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8129" PostTypeId="1" CreationDate="2019-01-02T23:33:48.077" Score="1" ViewCount="94" Body="&lt;p&gt;I read &lt;a href=&quot;http://www.math.hawaii.edu/~tom/history/functions.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that Karl Weierstrass used &quot;| |&quot; to indicate absolute value in 1841. The same sources indicate that Leopold Kronecker wrote of the sign function in 1878. &lt;a href=&quot;https://en.wikipedia.org/wiki/Sign_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; it is indicated that the sign function may be expressed via the absolute value function, &lt;span class=&quot;math-container&quot;&gt;$\textrm{sgn}(x)=\frac{x}{|x|}$&lt;/span&gt;, but no history of the practice is indicated. Any leads on who put them together if it wasn't Kronecker in 1878?&lt;/p&gt;&#xA;" OwnerUserId="9312" LastEditorUserId="55" LastEditDate="2019-01-07T22:40:22.070" LastActivityDate="2019-01-07T22:40:22.070" Title="Who came up with a formula expressing the sign function in terms of the absolute value?" Tags="|mathematicians|notation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8130" PostTypeId="1" CreationDate="2019-01-03T12:37:00.797" Score="1" ViewCount="66" Body="&lt;p&gt;I have encountered various abstract algebra resources that prove the impossibility of number systems with plural additive inverses for a given element, generally through the substitution property of equality. Historically, did these proofs arise in response to ill-conceived number systems with multiple additive inverses or are they more so used to illustrate limitations on the properties of certain algebraic structures?&lt;/p&gt;&#xA;" OwnerUserId="9312" LastEditorUserId="55" LastEditDate="2019-01-07T22:44:17.650" LastActivityDate="2019-01-07T22:44:17.650" Title="How did the proofs of uniqueness of additive inverses originate historically?" Tags="|notation|abstract-algebra|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8131" PostTypeId="2" ParentId="8115" CreationDate="2019-01-03T20:49:47.323" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein realized that uniform acceleration is indistinguishable from a homogeneous gravitational field (the principle of equivalence) through the Einstein Elevator thought experiment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Like Conifold said, Einstein's original insight didn't involve an elevator, it involved a man falling from a roof. This was his &quot;&lt;a href=&quot;https://erenow.net/biographies/einsteinhislifeanduniverse/8.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;happiest thought&lt;/a&gt;&quot;, in 1907, wherein the falling observer doesn’t feel his own weight. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments#Falling_painters_and_accelerating_elevators&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; Conifold linked to talks about it. Einstein didn't start talking about elevators until much later. You can search the Einstein digital papers on &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/~searchResults?searchMode=app&amp;amp;searchParam-start=0&amp;amp;searchParam-page=20&amp;amp;searchParam-terms=elevator&amp;amp;searchParam-titles=full&amp;amp;searchParam-versions=all&amp;amp;searchParam-years=%7B%22operator%22%3A%22%22%7D&amp;amp;searchParam-sort=date&amp;amp;searchParam-context=-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;elevator&lt;/a&gt;. Note the dates. You can also search on &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/~searchResults?searchMode=app&amp;amp;searchParam-start=0&amp;amp;searchParam-page=20&amp;amp;searchParam-terms=chest&amp;amp;searchParam-titles=full&amp;amp;searchParam-versions=all&amp;amp;searchParam-years=%7B%22operator%22%3A%22%22%7D&amp;amp;searchParam-sort=date&amp;amp;searchParam-context=-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;chest&lt;/a&gt; and find &lt;a href=&quot;http://The%20Equality%20of%20Inertial%20and%20Gravitational%20Mass%20as%20an%20Argument%20for%20the%20General%20Postulate%20of%20Relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;chapter 20&lt;/a&gt; of &lt;em&gt;Relativity: The Special and General Theory&lt;/em&gt;. That's where he talks about a spacious chest resembling a room with an observer inside. He also said &lt;em&gt;&quot;to the middle of the lid of the chest is fixed externally a hook with rope attached, and now a &quot;being&quot; (what kind of a being is immaterial to us) begins pulling at this with a constant force&quot;.&lt;/em&gt; Einstein wrote this book in 1916, and it was translated into English in 1920. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From the thought experiment, the physicist will not be able to tell the difference between whether he is homogenous gravitational field or if he is in gravitation free space in uniform acceleration. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's right, but do note that the homogeneous gravitational field is an idealization. In &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol6-trans/333?highlightText=%22entirety%29%20vanishes%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;chapter 20&lt;/a&gt; of &lt;em&gt;Relativity: The Special and General Theory&lt;/em&gt; Einstein said it’s &lt;em&gt;“impossible to choose a body of reference such that, as judged from it, the gravitational field of the Earth (in its entirety) vanishes”.&lt;/em&gt; Also see &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22nowhere%20precisely%20realized%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; where Einstein says special relativity is nowhere precisely realized in the real world. In the real world, there are no homogeneous gravitational fields, and a gravitational field is in itself space which is &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol7-trans/192?highlightText=%22neither%20homogeneous%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;neither homogeneous nor isotropic&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How did Einstein generalise this one case to conclude as a whole that uniform acceleration creates a uniform gravitational field? This is a fundamental part of general relativity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He didn't, and it isn't. See Kevin Brown‘s mathspages article &lt;a href=&quot;https://www.mathpages.com/home/kmath629/kmath629.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;on the many principles of equivalence&lt;/a&gt; where you can read that the equivalence principle has undergone several changes over the years. Brown says this: &lt;em&gt;“the modern statement of the strong equivalence principle, of the assertion that the laws of physics are the same for all frames of reference (i.e. independent of velocity) is also conceptually quite distinct from the original meaning of Einstein’s equivalence principle”&lt;/em&gt;. He's saying the equivalence principle isn't Einstein's equivalence principle. See the &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalence_principle#The_Einstein_equivalence_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein equivalence principle&lt;/a&gt; section of the Wikipedia equivalence principle article. It says &lt;em&gt;“the outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime”&lt;/em&gt;. That isn’t Einstein’s equivalence principle either. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;John D Norton talked about this in his 1985 paper &lt;a href=&quot;https://pdfs.semanticscholar.org/1f1b/65d257caf28f7b3122ce30dacb5707777e62.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;what was Einstein’s principle of equivalence?&lt;/a&gt; He said it was a special relativity principle that dealt only with fields that could be transformed away. He talked of an old view and a new view, and said &lt;em&gt;“the equivalence of all frames embodied in this new view goes well beyond the result that Einstein himself claimed in 1916”&lt;/em&gt;. This is why on pages ix and x in his 1960 preface to &lt;a href=&quot;http://strangebeautiful.com/other-texts/synge-relativity-general.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;relativity: the general theory&lt;/a&gt;, John Synge said the equivalence principle performed the essential office of midwife at the birth of general relativity, but should &lt;em&gt;“be buried with appropriate honours”.&lt;/em&gt; In a nutshell, a lot of what you hear about the equivalence principle isn't historically accurate. See my own article on &lt;a href=&quot;http://physicsdetective.com/the-principle-of-equivalence-and-other-myths/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the  principle of equivalence and other myths&lt;/a&gt; for further details and references. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is it linked with the law of conservation of mass and energy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Not really. People don't agree about the conservation of energy in general relativity. I think it applies full stop, but not everybody shares that sentiment. And there is no law of conservation of mass. If you drop a brick some of its gravitational potential energy, which is mass energy, is converted into kinetic energy. When the brick hits the ground this kinetic energy gets dissipated, leaving you with a mass deficit. See the Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Binding_energy#Mass-energy_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;binding energy&lt;/a&gt; article for more on that.&lt;/p&gt;&#xA;" OwnerUserId="3264" LastActivityDate="2019-01-03T20:49:47.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8132" PostTypeId="1" AcceptedAnswerId="8138" CreationDate="2019-01-04T01:49:17.743" Score="11" ViewCount="1500" Body="&lt;p&gt;The question &lt;a href=&quot;https://hsm.stackexchange.com/questions/6219/why-is-differentiation-under-the-integral-sign-named-feynmans-trick&quot;&gt;here&lt;/a&gt; asked why differentiation under the integral sign is named &quot;Feynman's trick&quot;. That is a comparatively recent name for the method.  Aside from the name &quot;differentiation under the integral sign&quot; for this technique, it is also called Leibniz's rule or, more precisely, the Leibniz integral rule, in many places.  My question is: why is Leibniz's name attached to this result on differentiation of parametric integrals? If there was a particular result of his (something more than the Fundamental Theorem of Calculus, I presume) that is a special case of differentiation under the integral sign, I'd like to be pointed to a place where that appeared. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have looked in several books on the history of calculus or analysis and none of them explain the reason for using Leibniz's name for differentiation under the integral sign. Cauchy, in his &lt;em&gt;Résumé&lt;/em&gt; (1823), discussed differentiation under the integral sign in the 35th lesson &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k90196z/f211.image&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; but he didn't name it after anyone.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="4251" LastEditDate="2020-06-05T21:34:00.650" LastActivityDate="2020-06-05T21:34:00.650" Title="Why is differentiation under the integral sign named the Leibniz rule?" Tags="|mathematics|terminology|calculus|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8133" PostTypeId="2" ParentId="3131" CreationDate="2019-01-04T02:37:00.190" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://www.realclearscience.com/blog/2015/02/mathematicians_die_in_horrible_ways.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.realclearscience.com/blog/2015/02/mathematicians_die_in_horrible_ways.html&lt;/a&gt; refers to &lt;a href=&quot;http://www.kellenmyers.org/deaths.html&quot; rel=&quot;noreferrer&quot;&gt;Kellen Myers's list&lt;/a&gt;, and exemplifies with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Or what about Austrian American mathematician Kurt Gödel? Considered by some to be just as influential a logician and philosopher as Aristotle, he sadly succumbed to crippling paranoia later in life. In his sixties, he became convinced that his food was being poisoned, and would only trust the cooking of his wife Adele. When she was hospitalized for six months in 1977, Gödel refused to eat, and subsequently died of starvation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.usatoday.com/story/tech/2015/03/08/mathematician-deaths/24605901/&quot; rel=&quot;noreferrer&quot;&gt;https://www.usatoday.com/story/tech/2015/03/08/mathematician-deaths/24605901/&lt;/a&gt; exemplifies with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There's Alan Turing and Dénes Konig, both of whom killed themselves.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wikipedia details &lt;a href=&quot;https://en.wikipedia.org/wiki/D%C3%A9nes_K%C5%91nig#Biography&quot; rel=&quot;noreferrer&quot;&gt;Kőnig's rational suicide&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After the &lt;a href=&quot;https://en.wikipedia.org/wiki/Operation_Margarethe&quot; rel=&quot;noreferrer&quot; title=&quot;Operation Margarethe&quot;&gt;occupation of Hungary&lt;/a&gt; by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Nazi&quot; rel=&quot;noreferrer&quot; title=&quot;Nazi&quot;&gt;Nazis&lt;/a&gt;, he worked to help persecuted mathematicians. On October 15, 1944 the National Socialist &lt;a href=&quot;https://en.wikipedia.org/wiki/Arrow_Cross_Party&quot; rel=&quot;noreferrer&quot; title=&quot;Arrow Cross Party&quot;&gt;Arrow Cross Party&lt;/a&gt; took over the country.  Days later on October 19, 1944 he committed suicide to evade persecution from the Nazis being a Hungarian Jew.&lt;a href=&quot;https://en.wikipedia.org/wiki/D%C3%A9nes_K%C5%91nig#cite_note-Richard_McCoart-1&quot; rel=&quot;noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2019-01-04T02:37:00.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8136" PostTypeId="1" AcceptedAnswerId="8137" CreationDate="2019-01-05T09:22:19.000" Score="3" ViewCount="151" Body="&lt;p&gt;The story goes that in the 1880s Newcomb noticed that logarithm tables were more worn down towards the beginning of the book (where the leading digit of the logs were 1). This led him to formulate an equation that describes Benford’s law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this story true? How do we know? If it is true, where was this story first recorded? (e.g. a certain publication)&lt;/p&gt;&#xA;" OwnerUserId="9326" LastActivityDate="2019-01-05T10:51:48.973" Title="Where did the story about Newcomb observing Benford’s Law come from?" Tags="|mathematics|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8137" PostTypeId="2" ParentId="8136" CreationDate="2019-01-05T10:51:48.973" Score="3" Body="&lt;p&gt;It is not a story, Newcomb published his observation in a two page &lt;a href=&quot;https://www.jstor.org/stable/2369148&quot; rel=&quot;nofollow noreferrer&quot;&gt;Note on the Frequency of Use of the Different Digits in Natural Numbers&lt;/a&gt; (American Journal of Mathematics Vol. 4, No. 1 (1881), pp. 39-40), which you can read under the link. Here is the opening paragraph:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;That the ten digits do not occur with equal frequency must be evident to any one making much use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones. The first significant figure is oftener I than any other digit, and the frequency diminishes up to 9. The question naturally arises whether the reverse would be true of logarithms. That is, in a&#xA;  table of anti-logarithms, would the last part be more used than the first, or would every part be used equally? Tne law of frequency in the one case may be deduced from. that in the other. The question we have to consider is, what  is the probability that if a natural number be taken at random its first significant digit will be &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, its second &lt;span class=&quot;math-container&quot;&gt;$n'$&lt;/span&gt;, etc.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He proceeds to select two numbers &quot;at random&quot;, and looks for the probability of the first significant digit of their ratio to be &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. It turns out that the fractional parts of the ratio's logarithms are equally probable, so &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; in about &lt;span class=&quot;math-container&quot;&gt;$30\%$&lt;/span&gt; of cases, and any other one digit in less than &lt;span class=&quot;math-container&quot;&gt;$20\%$&lt;/span&gt; of cases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Benford in  &lt;a href=&quot;https://www.jstor.org/stable/984802&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Law of Anomalous Numbers&lt;/a&gt; (Proceedings of the American Philosophical Society, Vol. 78, No. 4 (Mar. 31, 1938), pp. 551-572) starts with the same observation about logarithmic tables, but proceeds to test the law  on 20 different datasets, and gives an explicit formula for the frequency as &lt;span class=&quot;math-container&quot;&gt;$\log(\frac{n+1}n)$&lt;/span&gt;, as well as its refinements. He does not seem to be aware of Newcomb's note.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-01-05T10:51:48.973" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8138" PostTypeId="2" ParentId="8132" CreationDate="2019-01-05T12:18:23.227" Score="10" Body="&lt;p&gt;This rule is, indeed, due to Leibniz, although it was Johann Bernoulli who realized its broader implications, and there is an interesting story to its discovery. It is told in &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0304020808735667&quot; rel=&quot;noreferrer&quot;&gt;Chapter 3 of Families of Curves and the Origins of Partial Differentiation by Engelsman&lt;/a&gt;. The rule appears in a Leibniz's 1697 letter to Bernoulli, as a side result in their long correspondence on the problem of orthogonal trajectories. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As originally posed in 1694, it was &quot;&lt;em&gt;Given infinitely many curves by position; find the curve that intersects them all at right angles&lt;/em&gt;&quot;, with the motivation that the light rays are orthogonal to the wave fronts in Huygens's wave optics. Leibniz solved the problem the same year as follows: if &lt;span class=&quot;math-container&quot;&gt;$V(x,y,a)=0$&lt;/span&gt; give the family then the trajectories can be found by solving &lt;span class=&quot;math-container&quot;&gt;$V_x(x,y,a)dy-V_y(x,y,a)dx=0$&lt;/span&gt;. At the time, Bernoulli had only algebraic &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; in mind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In June 1696 Bernoulli posed to the readers of Acta Eruditorum his now famous brahistochrone problem. He was able to find orthogonal trajectories to their family, given by &lt;span class=&quot;math-container&quot;&gt;$y=\int_{0}^x\sqrt{\frac{x}{a-x}}\,dx$&lt;/span&gt;, using his optico-mechanical analogy, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3486/did-wave-optics-anticipate-quantum-mechanics/3493&quot;&gt;Did wave optics anticipate quantum mechanics?&lt;/a&gt; What he pointed out in a letter to Leibniz was that his general method did not seem to work for this family, or, more generally, for  families of transcendental curves given by &lt;span class=&quot;math-container&quot;&gt;$y=\int_{x_0}^xp(x,a)\,dx$&lt;/span&gt;. And then came the Leibniz's integral rule.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Johann Bernoulli's great break-through for transcendental curves came in August 1697, and was an immediate consequence of Leibniz's discovery earlier that month of the interchangeability theorem for differentiation and integration. When he received Leibniz's letter containing this theorem, Bernoulli at once recognised that it opened a way to differentiation with respect to the parameter for any type of expression. There had been no difficulty interpreting &lt;span class=&quot;math-container&quot;&gt;$V_a(x,y,a)$&lt;/span&gt; as far as algebraic expressions &lt;span class=&quot;math-container&quot;&gt;$V(x,y,a)$&lt;/span&gt; were concerned, and now the problem of interpreting &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial}{\partial a}\int_{x_0}^xp(x,a)\,dx$&lt;/span&gt; had been solved as well.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I could not get hold of Leibniz's letter, but &lt;a href=&quot;https://books.google.com/books?id=KDSqLsOHc9UC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Cambridge History of Science: Volume 4, Eighteenth-Century Science, p.316&lt;/a&gt; says that he used that the differential of a sum of infinitesimals is equal to the sum of their differentials. While also studying orthogonal trajectories, Euler gave a  different proof in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E044.html&quot; rel=&quot;noreferrer&quot;&gt;De Infinitis Curvis Eiusdem (c. 1734, published 1740)&lt;/a&gt;, by applying antiderivatives to the equality of mixed partials.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-01-30T08:00:13.703" LastActivityDate="2019-01-30T08:00:13.703" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8139" PostTypeId="1" CreationDate="2019-01-05T12:43:47.513" Score="5" ViewCount="94" Body="&lt;p&gt;Prograde burn and retrograde burn are quite basic maneuvers in orbital mechanics. But how did the scientists (or researchers) discover that (the mathematical interpretation)? Did they try any practical experiments or was it all a theoretical achievement? How did they discover orbits in the first place? &lt;/p&gt;&#xA;" OwnerUserId="9303" LastEditorUserId="55" LastEditDate="2019-01-05T23:17:15.000" LastActivityDate="2019-01-05T23:24:34.263" Title="How did scientists (or researchers ) discover prograde and retrograde burns?" Tags="|theoretical-physics|engineering|celestial-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8140" PostTypeId="2" ParentId="8139" CreationDate="2019-01-05T23:14:22.433" Score="5" Body="&lt;p&gt;It was a theoretical invention. A special case, optimal elliptic transfer between two circular orbits, a.k.a. the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hohmann_transfer_orbit#cite_note-5&quot; rel=&quot;noreferrer&quot;&gt;Hohmann transfer&lt;/a&gt;, was described by Hohmann in The Attainability of Heavenly Bodies (1925), see &lt;a href=&quot;https://archive.org/details/nasa_techdoc_19980230631&quot; rel=&quot;noreferrer&quot;&gt;Washington: NASA Technical Translation F-44, 1960 on Internet Archive&lt;/a&gt;. It is accomplished by two burns, one prograde one retrograde. Hohmann, apparently, drew inspiration from science fiction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The above concept, to impart to a body an acceleration opposing gravity by expelling parts of its mass, is not new in itself. It is contained unwittingly in Jule Verne's &amp;quot;Journey Around the Moon,&amp;quot; where mention is made of rockets, which are taken along to decelerate the vehicle, and it is continuously used in Kurd Lasswitz's &amp;quot;On Two Planets.&amp;quot; Here however under the very favorable assumption, that the velocity of expulsion is that of light, resulting in only a minor decrease of the vehicle's mass.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The more recent works of Goddard, Oberth and Valier have already been mentioned. Also the pioneer of aviation, Hermann Ganswiudt, has proclaimed the idea of a rocket vehicle in public talks around I890, at the same time also, the Russian Cielkowsky. Finally, Newton mentioned the possibility of travel in empty space on the occasion of a lecture concerning impulse reaction.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A similar maneuver between two elliptic trajectories, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bi-elliptic_transfer&quot; rel=&quot;noreferrer&quot;&gt;bi-elliptic transfer&lt;/a&gt;, was published by Sternfeld in  &lt;em&gt;Sur les trajectoires permettant d'approcher d'un corps attractif central à partir d'une orbite keplérienne donnée&lt;/em&gt; (1934, On the allowed trajectories for approaching a central attractive body from a given Keplerian orbit), see &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k31506/f711.image.langEN&quot; rel=&quot;noreferrer&quot;&gt;Comptes rendus de l'Académie des sciences, Paris, 198 (1): 711–713 on Gallica&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-05T23:24:34.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8142" PostTypeId="2" ParentId="2135" CreationDate="2019-01-06T02:53:59.907" Score="4" Body="&lt;p&gt;Clearly, Buridan, rector of the university of Paris, made all the basic breakthroughs in dynamics: he viewed force, vectorial force, as what changed impetus, and only force could modify impetus (be it a force from air resistance, or from gravity). Impetus was written by Buridan as equal to (quantity of matter)(velocity).  This is the modern definition of momentum. Remember, that was before equations... &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without force applied, impetus would go on and on, either straight, or circular from gravity: thus the planets kept on turning around (Buridan observed that the heliocentric theory was as compatible with observations as the geocentric theory; his student Oresme added that it was more likely that the Earth moved and rotated than all the stars doing so, because the latter were obviously much more distant and massive; impetus theory insured the air won't be left behind). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Buridan also exposed the essence of the experimental method, decided by observing phenomena, he said. What was not compatible with phenomena, had to be ruled out (that meant Aristotle’s physics was ruled out). Buridan's student Oresme demonstrated (geometrically) the first non trivial theorems of calculus applied to motion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, as rector of the university of Paris, Buridan has had to generalize, and formally enable, in 1340 CE, a notion of &quot;object of knowledge&quot; as potentially unobservable... (Impetus, heliocentrism, etc. were not directly observable…) That was theologically dangerous. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All this was all the more remarkable as Buridan, Oresme (who became dean of the cathedral of Rouen, then a bishop!) had to stay clear from the murderous Inquisition (which considered Aristotle's physics as god-given). Thus some apparent contradictions in Buridan and Oresme, to assuage the Inquisition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, more than a century after his death, all of Buridan's works were outlawed on the pain of death by the Catholic church (all Buridan;s works were put in the Index Librorum Prohibitorum, the Index of Prohibited Books of the Roman Office of the Inquisition)... except in Eastern Europe, which was furious from the burning alive of Jan Hus, rector of the university of Prague in 1415 CE. In Krakow Buridan was mandatory reading at the university, thus teaching Buridan’s heliocentric theory to the young Copernicus...  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ignoring Buridan while just focusing on Newton, should be viewed as a sneaky way to glorify the Anglosphere. That is unfair not just to Buridan and another dozen intervening major physicists, over those three centuries in between. It's also unfair to the process of serious scientific revolution, and the enormous boldness necessary to put it in motion. &lt;/p&gt;&#xA;" OwnerUserId="9331" LastEditorUserId="9331" LastEditDate="2019-01-06T03:29:29.123" LastActivityDate="2019-01-06T03:29:29.123" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8144" PostTypeId="2" ParentId="8124" CreationDate="2019-01-06T16:16:13.727" Score="2" Body="&lt;p&gt;The Egyptian language, like most languages of the world, has a counting system based implicitly on base 10. This is much older than any system of mathematical notation. When the Egyptians began writing numbers they reflected the counting system which existed in their language.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2019-01-06T16:16:13.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8145" PostTypeId="1" CreationDate="2019-01-07T16:50:18.960" Score="3" ViewCount="459" Body="&lt;p&gt;How was the power series form of the exponential function disovered? Was it just observed? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the exponential function, I mean the solution to the differential equation &lt;span class=&quot;math-container&quot;&gt;$\frac{df}{dx} = f$&lt;/span&gt; with the intial condition &lt;span class=&quot;math-container&quot;&gt;$f(0) = 1$&lt;/span&gt;. &lt;/p&gt;&#xA;" OwnerUserId="8168" LastActivityDate="2019-01-07T16:50:18.960" Title="Discovery of the Power Series Form of the Exponential Function" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8146" PostTypeId="1" CreationDate="2019-01-08T19:07:58.373" Score="5" ViewCount="963" Body="&lt;p&gt;In the book &lt;a href=&quot;https://isidore.co/calibre/get/pdf/5513&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Energy - The Subtle Concept' by Energy, the Subtle Concept by Jennifer Coopersmith&lt;/a&gt; the author says that Huygens was the first to use the term &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; in physics. He was considering elastic collisions of bodies in the horizontal plane. The book states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So Huygens, who wanted to keep all considerations of forces out of &#xA;  the picture, had to allow gravity as the force that, in collisions, gave the &#xA;  bodies their initial speeds and soaked up their recoil speeds. (Even so, &#xA;  Huygens was never able to accept Newton’s Law of Universal Gravitation because of the occult requirement for action-at-a-distance.) In this way, height came into a scenario that was totally on the horizontal. This seems like reasoning a long way round, but it had a very interesting by-product. For the height of the centre of gravity to remain constant, the mathematics showed that another quantity had to stay constant: the total &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt;  before and after the collision. This was the first time that the formula &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; made an appearance in physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Can somebody explain &quot;For the height of the centre of gravity... &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; made an appearance in physics&quot;. My question is what was the mathematics that showed that &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; is a constant? Or was it shown experimentally ?&lt;/p&gt;&#xA;" OwnerUserId="8145" LastEditorUserId="55" LastEditDate="2019-01-09T08:57:15.607" LastActivityDate="2019-01-10T16:24:37.780" Title="How did Huygens derive the conservation law for of kinetic energy?" Tags="|physics|energy|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8147" PostTypeId="2" ParentId="8146" CreationDate="2019-01-09T07:32:58.747" Score="5" Body="&lt;p&gt;The &amp;quot;mathematics&amp;quot; was a combination of experiments with falling bodies, imaginative thought experiments, common sense, and geometric reasoning. Part of it is explained in the book. Galileo found that &lt;span class=&quot;math-container&quot;&gt;$v^2$&lt;/span&gt; at the time of impact is proportional to &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;. Torricelli argued that when two bodies are linked together and freely move, but only in a vertical plane, then their center of gravity maintains the same height. Otherwise levitation would be possible (and, as in Raspe's later satire, Baron Munchausen could pull himself, and his horse, out of the swamp by his pigtail).&lt;/p&gt;&#xA;&lt;p&gt;Huygens knew of Galileo's contraptions from Discourses on Two New Sciences that deflected falling bodies into horizontal motion. Conversely, one can deflect horizontal motion into the vertical and see how high the body would rise. If we allow &amp;quot;&lt;em&gt;gravity as the force that, in collisions, gave the bodies their initial speeds and soaked up their recoil speeds&lt;/em&gt;&amp;quot;, then &lt;span class=&quot;math-container&quot;&gt;$v^2$&lt;/span&gt;, or the sum of &lt;span class=&quot;math-container&quot;&gt;$v^2$&lt;/span&gt; times their magnitudes (Huygens means weights) in case of multiple bodies, must stay the same. Otherwise, as Huygens realized, we can get perpetual motion by sequencing deflections upward and downward. &amp;quot;&lt;em&gt;It is a most certain axiom in mechanics that, when bodies move because of their gravity, their common center of gravity cannot rise.&lt;/em&gt;&amp;quot; This was his generalization of Torricelli's observation. On the further fate of &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; see &lt;a href=&quot;https://hsm.stackexchange.com/questions/7992/how-did-people-figure-out-the-formula-for-mechanical-work-and-related-it-to-ene&quot;&gt;How did people figure out the formula for mechanical work, and related it to energy?&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Blackwell's English translation of Huygens's unpublished &lt;a href=&quot;https://www.jstor.org/stable/230011?seq=1#metadata_info_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;De motu corporum ex percussione (1656, published posthumously in 1702, The Motion of Colliding Bodies)&lt;/a&gt; is available on Jstor. One can see that he does not use formulas and carries all arguments geometrically, in the venerable style of Euclid and Archimedes, the same style that Newton will later adopt in Principia. The conservation law is Proposition 11, formulated thus:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;strong&gt;Proposition 11:&lt;/strong&gt; &lt;em&gt;If two bodies collide with each other, and if the ratio of both their magnitudes and their speeds is given either in numbers or in lines, then the sum of their magnitudes multiplied by the squares of their respective velocities is equal before and after the collision.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Blackwell adds that Huygens formulated this law for perfectly elastic collisions as early as 1652, without explanation: &lt;em&gt;Sed necesse est quadrata velocitatum ducta in magnitudinem corporum semper eundem numerum producere&lt;/em&gt; (But it is necessary that the solids on the squares on velocities and the magnitudes of the bodies always produce the same number). The proof uses the techniques of Book II of the Elements, formerly known as &amp;quot;geometric algebra&amp;quot;, particularly II.8:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If a straight line be cut at random, four times the rectangle contained by the whole and one of the segments together with the square of the remaining segment is equal to the square described on the whole and the aforesaid segment as on one straight line&lt;/em&gt;&amp;quot; (algebraically, &lt;span class=&quot;math-container&quot;&gt;$4(AB\times AC) + (CB)^2 = (AB + AC)^2$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To give a taste of the following two-page proof, split into seven different cases, here are two excerpts from the beginning and the end of it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;What must be proven is that the solid figure composed of the line CB multiplied by the square of AD plus the solid of CA times the square of BD is equal to the solid of CB times the square of EA plus the solid of CA times the square of EB.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;But when the square of AD is larger or smaller than the square of AE, then the square of BE will always be respectively larger or smaller than the square of BD. Therefore it is clear that the solid composed of BC times the square of AD always is larger or smaller than the solid composed of BC times the square of AE by the same amount that the solid composed of AC times the square of BE is respectively larger or smaller than the solid composed of AC times the square of BD. Q.E.D.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to Blackwell, one can find algebraic conversions of the proofs in De motu in the editor's notes to Huygens's Oeuvres, Vol. XVI, pp. 29-91.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-10T06:48:21.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8148" PostTypeId="2" ParentId="7992" CreationDate="2019-01-09T08:47:08.950" Score="2" Body="&lt;p&gt;It was a side effect of the vis viva controversy described in &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects/598#598&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt; about what to call the &amp;quot;quantity of motion&amp;quot;, momentum (vis mortua, dead force) or kinetic energy (vis viva, living force). Since the conservation of &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; in elastic collisions led to singling out vis viva as a useful concept one could say that it played a part in relating it to mechanical work, indirectly. Other concepts and interrelations in mechanics were also clarified in the course of this &amp;quot;debate about words&amp;quot; resulting in the classical presentations of mechanics by Euler and D'Alambert that we are used to.&lt;/p&gt;&#xA;&lt;p&gt;The insight on work came from Boscovich in &lt;em&gt;De Viribus Vivis&lt;/em&gt; (1745), it is reflected  in the second edition of D'Alambert's &lt;em&gt;Traite de Dynamique&lt;/em&gt; (1758), which became a standard text. Based on the work of Costabel, in &lt;a href=&quot;https://nature.berkeley.edu/departments/espm/env-hist/articles/1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;D'Alambert and the Vis Viva Controversy Carolyn Iltis&lt;/a&gt; explains Boscovich's thought process, which is metaphysical, and a bit murky:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Employing both the ancient scholastic categories and the new mathematical methods of his time, Boscovich discussed the graphical representation of a pressure applied through a time and a force applied over a distance... The pressure, an active presence (puissance), passes into action not by multiplication of effects but by generating a two-dimensional image adequately rendered only by geometry.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Without taking a position on the definition of force, Boscovich measured&#xA;the velocity acquired as a ratio composed of the pressure and its duration. A geometrical image is generated by the line representing the pressure with time as the second dimension of the diagram. The pressure is thus a function of time. Interpreting this in modern terminology, the momentum &lt;span class=&quot;math-container&quot;&gt;$mv$&lt;/span&gt; would be represented as the integral of these instantaneous pressures (or impulses) over a time, or &lt;span class=&quot;math-container&quot;&gt;$\int mdv = \int pdt$&lt;/span&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Boscovich suggested that, if the time coordinate is replaced by the space traversed and the pressure coordinate by the force which at any instant produces the velocity proportional to it, a second aspect of the phenomenon is represented. Boscovich, however, explained neither this substitution nor the introduction of the concept of force. The new term 'force' must be interpreted as an entity proportional to the velocity engendered at any instant. If the pressure coordinate is changed to the&#xA;force and the time coordinate to the space then the new geometrical image&#xA;producing the velocity would be represented in modern notation as &lt;span class=&quot;math-container&quot;&gt;$\int Fds$&lt;/span&gt;. We would then interpret vis viva as &lt;span class=&quot;math-container&quot;&gt;$\int mvdv = \int Fds$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$ds=vdt$&lt;/span&gt;). Boscovich does not bring the mass into this analysis.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-10-17T20:50:37.563" LastActivityDate="2021-10-17T20:50:37.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8149" PostTypeId="2" ParentId="8146" CreationDate="2019-01-09T16:23:30.407" Score="5" Body="&lt;p&gt;(&quot;How did Huygens derive the conservation law for of kinetic energy?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Huygens in 'The Motion of Colliding Bodies' (&lt;a href=&quot;https://www.jstor.org/stable/230011&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt;) contributed ingenious reasoning, mathematics and thought-experiments based on physical assumptions. &#xA;But this work was all about the collisions of a particular kind of body -- supposed and idealized at some distance away from any realizable physical object, albeit that the result looks rather like an elastic version of a billiard-ball.  There is no concept in there anything like as broad as &quot;the conservation law for kinetic energy&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Not surprisingly, Huygens' work was not taken to have established any such concept or law, as is evident from the later &lt;a href=&quot;https://hsm.stackexchange.com/questions/596/what-was-the-vis-viva-controversy-including-its-philosophical-aspects&quot;&gt;vis-viva controversy&lt;/a&gt;.  The work required to settle such matters was long-drawn-out and involved experimental (physical) evidence as well as mathematical creativity and controversy.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An illustration showing that the matter was still in doubt 25 years after Huygens' death, and one of the stages in the long and painstaking process of gathering necessary physical evidence, is given by Willem 's Gravesande's book &lt;em&gt;Physices elementa mathematica, experimentis confirmata&lt;/em&gt; (1720-21), (English translation published 1747 (&lt;a href=&quot;https://archive.org/details/mathematicalelem01grav/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mathematical Elements of Natural Philosophy Confirm'd by Experiments&lt;/em&gt;&lt;/a&gt;)). Especially notable is Chapter III, pages 183-202, where 's Gravesande describes his comparisons of the 'forces' (we would now say - kinetic energies) needed for a moving heavy object to make equal impressions in an impressionable clay, losing all of its motion while doing so. 's Gravesande found, among other things, that for &#xA;equal impressions, the constant measure of this 'force' was proportional to the mass and to the square of the velocity (and not the first power of velocity). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it took much more than that, of course, to establish the concept of kinetic energy and a law of its conservation. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-01-09T16:34:40.850" LastActivityDate="2019-01-09T16:34:40.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8150" PostTypeId="1" CreationDate="2019-01-09T21:00:02.260" Score="11" ViewCount="1218" Body="&lt;p&gt;I originally asked this &lt;a href=&quot;https://physics.stackexchange.com/questions/453131/what-new-physics-was-discovered-or-needed-as-a-result-of-the-manhattan-project?noredirect=1#comment1017367_453131&quot;&gt;question&lt;/a&gt; on the Physics StackExchange and was told to migrate it here. I've tightened up the question a bit. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recently got into a discussion with colleagues regarding quantum mechanics and the Manhattan Project.  My colleague (not a physicist) conjectured that it was mostly an engineering feat with little need for quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I disagreed, arguing that it would be silly to hire some of the world's top physicists (i.e &lt;a href=&quot;https://en.wikipedia.org/wiki/Enrico_Fermi&quot; rel=&quot;noreferrer&quot;&gt;Enrico Fermi&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hans_Bethe&quot; rel=&quot;noreferrer&quot;&gt;Hans Bethe&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/J._Robert_Oppenheimer&quot; rel=&quot;noreferrer&quot;&gt;Robert Oppenheimer&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernest_Lawrence&quot; rel=&quot;noreferrer&quot;&gt;Ernest Lawrence&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Feynman&quot; rel=&quot;noreferrer&quot;&gt;Richard Feynman&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Eugene_Wigner&quot; rel=&quot;noreferrer&quot;&gt;Eugene Wigner&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Leo_Szilard&quot; rel=&quot;noreferrer&quot;&gt;Leo Szilard&lt;/a&gt;) and enlisting the assistance of others (i.e. &lt;a href=&quot;https://en.wikipedia.org/wiki/Niels_Bohr#Manhattan_Project&quot; rel=&quot;noreferrer&quot;&gt;Niels Borh&lt;/a&gt;) if their physics skills, especially regarding quantum mechanics, weren't needed.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There seems to be a plethora of resources mentioning experimental achievements such as the measurement of &lt;a href=&quot;https://hsm.stackexchange.com/a/6839/9343&quot;&gt;cross-sections&lt;/a&gt;, &lt;a href=&quot;https://hsm.stackexchange.com/a/6837/9343&quot;&gt;isotope separation&lt;/a&gt;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;QUESTION : &#xA;Did the Manhattan project lead to (or require) major theoretical discoveries / contributions to physics? &lt;/p&gt;&#xA;" OwnerUserId="9343" LastEditorUserId="4251" LastEditDate="2021-09-21T07:01:16.283" LastActivityDate="2021-09-21T07:01:16.283" Title="What new physics was discovered or needed as a result of the Manhattan Project?" Tags="|physics|quantum-mechanics|theoretical-physics|nuclear-physics|world-war-2|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8151" PostTypeId="2" ParentId="8146" CreationDate="2019-01-09T22:14:24.107" Score="5" Body="&lt;p&gt;There are several themes in Huygens' unpublished paper &lt;em&gt;De motu corporum ex percussione&lt;/em&gt; (&quot;On the motion of bodies out of collisions&quot;), but maybe the most significant is that he frequently investigates a specific or extreme case (where some factor is zero, one, or infinity) first, and then guesses about a general case which might have those at its boundary. Let's call this his &quot;specific-first&quot; mindset and go into his argument.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Prelude: elastic collisions between equal balls&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Huygens starts from a basic Galilean perspective where bodies in motion continue in uniform motion in a straight line unless impeded. His problem is, he doesn't necessarily know the best way to phrase what makes a collision elastic. So he starts specific-first: if two balls are identical, then by symmetry, if they both come in towards a center of collision with speed &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;, then in a rigid collision they should both leave with speed &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To generalize this, he adds what we now call today the &quot;freedom of choice of reference frame&quot; or the &quot;principle of relativity,&quot; it says that the way physics works on a smooth train/boat ride is the same as it works on the ground, no matter how fast the train/boat is going. He goes specific-first: assume one of the identical balls &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; is stationary and the other &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; comes in from the left at velocity &lt;span class=&quot;math-container&quot;&gt;$v_X = u$&lt;/span&gt;, then I can use relativity (add &lt;span class=&quot;math-container&quot;&gt;$-u/2$&lt;/span&gt; to all velocities) to prove that after the collision &lt;span class=&quot;math-container&quot;&gt;$v_X = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v_Y=u.$&lt;/span&gt; He then generalizes even more: the proper approach is to consider a boat which sails smoothly from the halfway point between &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; to the collision point between them: such a boat has speed &lt;span class=&quot;math-container&quot;&gt;$v_B = (v_X + v_Y)/2$&lt;/span&gt; and the two velocities are &lt;span class=&quot;math-container&quot;&gt;$v_{X,Y} = v_B \pm \delta$&lt;/span&gt;, and the collision trades &lt;span class=&quot;math-container&quot;&gt;$\pm \leftrightarrow \mp$&lt;/span&gt; so he comes to his conclusion that after any collision of identical balls, they swap speeds (and, of course, the velocities reverse direction).  &lt;/p&gt;&#xA;&#xA;&lt;h1&gt;What about non-identical balls?&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;So he has handled a very specific case first, but Huygens now wants to generalize his definition of &quot;rigid collision.&quot; It's no good to him if that definition &lt;em&gt;only&lt;/em&gt; handles identical bodies. But he actually generalizes in a specific-first way, too! He starts with the case of a big unequal ball hitting a small ball at rest, and he says &quot;I won't say &lt;em&gt;exactly&lt;/em&gt; what happens, but surely the smaller ball starts moving forward, and the bigger ball moves forward less.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's important to understand that this is related to what has just been derived. We just saw that if the &quot;bigger&quot; ball were only &lt;em&gt;infinitesimally bigger&lt;/em&gt;, it would approximately &lt;em&gt;stop&lt;/em&gt;. So by making it larger, it presumably must &lt;em&gt;keep going forward&lt;/em&gt;. The limiting case on the other side, that if the smaller ball were infinitesimally small, the bigger ball should continue in uniform motion, also means that the bigger ball never goes &lt;em&gt;faster&lt;/em&gt; than it started from this sort of collision. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Huygens uses the principle of relativity to derive the opposite limiting case: If one takes this entire picture &lt;span class=&quot;math-container&quot;&gt;$X \gg Y, ~(V_X = u,~V_Y=0)\mapsto (u_-, u_+), u_- &amp;lt; u &amp;lt; u_+$&lt;/span&gt; and shifts it by &lt;span class=&quot;math-container&quot;&gt;$-u$&lt;/span&gt; then one finds a picture where &lt;span class=&quot;math-container&quot;&gt;$V_X$&lt;/span&gt; starts at &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and then becomes some negative number &lt;span class=&quot;math-container&quot;&gt;$u_- - u$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$V_Y$&lt;/span&gt; starts at &lt;span class=&quot;math-container&quot;&gt;$-u$&lt;/span&gt; and becomes some positive number &lt;span class=&quot;math-container&quot;&gt;$u_+ - u$&lt;/span&gt;, ricocheting forwards. Huygens doesn't actually talk about the ricochet all that much, but I think that's because the ricochet was already the best previous prediction, due to Descartes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fourth, if the body C were entirely at rest,…and if C were slightly larger than B; the latter could never have the force to move C, no matter how great the speed at which B might approach C. Rather, B would be driven back by C in the opposite direction: because…a body which is at rest puts up more resistance to high speed than to low speed; and this increases in proportion to the differences in the speeds. Consequently, there would always be more force in C to resist than in B to drive, …. (Descartes, &lt;a href=&quot;https://plato.stanford.edu/entries/descartes-physics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;as translated in SEP&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Huygens goes on to clearly call out that he is refuting Descartes: if a big object always moves slower when it impacts a small stationary object, these Galileian principles &lt;em&gt;demand&lt;/em&gt; that the big object is always moved by any collision from a small impacting object. Huygens is saying &quot;no, the big object emphatically does &lt;em&gt;not&lt;/em&gt; stand still!&quot; and that is much more important to him and the physics of his day, than saying that the small object must travel backwards, which was already obvious.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;The broader generalization to unequal balls&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;So Huygens has attacked the problem specific-first and has found these two speeds &lt;span class=&quot;math-container&quot;&gt;$u_- &amp;lt; u &amp;lt; u_+$&lt;/span&gt;, and he knows at least one limiting case, where the mass ratio is &lt;span class=&quot;math-container&quot;&gt;$m_Y/m_X = 1$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$u_-=0$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$u_+ = u.$&lt;/span&gt; He presumably already has an idea of the other limiting case, where the &quot;ball&quot; &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; becomes more of a a &quot;wall&quot; as this mass ratio goes to zero: there are many reasons both experimental and theoretical to imagine that against an immovable wall (&lt;span class=&quot;math-container&quot;&gt;$v_X$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;) a rigid head-on collision should cause a ball to ricochet with the same speed that it impacted (&lt;span class=&quot;math-container&quot;&gt;$v_Y$&lt;/span&gt; goes from &lt;span class=&quot;math-container&quot;&gt;$-u$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;). And he postulates that the general law is therefore as follows: if you can find a frame of reference (lets call it a &quot;central&quot; frame) where one object enters with velocity &lt;span class=&quot;math-container&quot;&gt;$+u$&lt;/span&gt; and then leaves with velocity &lt;span class=&quot;math-container&quot;&gt;$-u$&lt;/span&gt;, the other object must also see its speed unchanged. This is his original formulation of the conservation of energy, that a central frame for &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is, in a rigid collision, also a central frame for &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Huygens proceeds to come up with an alternative formulation of that condition: the relative velocity of the two bodies are the same before and after the collision. In this central frame this is easy, they go from relative velocity &lt;span class=&quot;math-container&quot;&gt;$-|v_X| - |v_Y|$&lt;/span&gt; to relative velocity &lt;span class=&quot;math-container&quot;&gt;$|v_X| + |v_Y|$&lt;/span&gt; and those are only different by a sign difference; furthermore any transformation to any other frame preserves relative velocities. He also derives the other limiting case where the mass ratio is zero: if a massive &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; hits a near-massless &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; then to preserve the relative velocity when &lt;span class=&quot;math-container&quot;&gt;$v_X$&lt;/span&gt; hardly changes, &lt;span class=&quot;math-container&quot;&gt;$v_Y$&lt;/span&gt; must take on the limiting value &lt;span class=&quot;math-container&quot;&gt;$2v_X.$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Adding masses into the picture&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Up until now, Huygens has been reluctant to place mass directly into his reasoning and that is quite understandable: he wants his arguments to hold independently of Galileo's observations on gravity and how things fall and how scales work. But he insists on connecting his mathematics to Galileo's work eventually and it comes to this argument: that this center-of-mass frame amounts to taking the weight of the objects and multiplying it times their speeds and setting them to be equal. He phrases it as &quot;if two bodies whose speeds are inversely proportional to their magnitudes collide with each other, then each rebounds with the same speed which it had before the collision.&quot; So he knows what a &quot;center of gravity&quot; is and he is stating precisely that this central frame for the collision is one where the center of gravity is stationary at the point of collision. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To make the connection, he observes that Galileo showed that falling things fall a distance proportional to time-squared, meaning that their final velocity increases proportional to time. Huygens says that this means that the distance is proportional to velocity squared, and then he says &quot;oh, and I can use this in reverse, too -- because everybody knows that the velocity that something gets as it comes down is exactly what it needs to go back up.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His argument is geometrical so it is a bit clumsy to state in his language, I will rephrase it algebraically. So he says &quot;let's say at time &lt;span class=&quot;math-container&quot;&gt;$t=-T$&lt;/span&gt; you release these two masses from rest from different heights, you put them into some sort of curve like Galileo did so that they change directions to horizontal without changing speed. They collide at &lt;span class=&quot;math-container&quot;&gt;$t=0$&lt;/span&gt; and bounce back, and they return to different heights. We choose the original velocities according to this balance principle, so let me define the mass ratios &lt;span class=&quot;math-container&quot;&gt;$\mu_i = m_i/(m_1 + m_2)$&lt;/span&gt; and we have &lt;span class=&quot;math-container&quot;&gt;$v_1 = u/\mu_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v_2 = -u/\mu_2$&lt;/span&gt; for some number (with dimensions of speed) &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;. Finally, freeze each one when it attains its maximum. Then if they rebound with speeds &lt;span class=&quot;math-container&quot;&gt;$s_{1,2}$&lt;/span&gt; they attain heights &lt;span class=&quot;math-container&quot;&gt;$s_1^2/(2a)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s_2^2/(2a)$&lt;/span&gt; when we stop them in midair, and their center of gravity thus attains a height proportional to &lt;span class=&quot;math-container&quot;&gt;$\mu_1 s_1^2 + \mu_2 s_2^2$&lt;/span&gt; subject to the constraint that &lt;span class=&quot;math-container&quot;&gt;$s_1 + s_2 = u\cdot(1/\mu_1 + 1/\mu_2).$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From here what he observes is essentially that one can derive that this &lt;em&gt;maximum center-of-gravity height&lt;/em&gt; is in turn &lt;em&gt;minimized&lt;/em&gt; by this choice &lt;span class=&quot;math-container&quot;&gt;$s_1 = u / \mu_1, s_2 = u/\mu_2:$&lt;/span&gt; any other choice of &lt;span class=&quot;math-container&quot;&gt;$s_{1,2}$&lt;/span&gt; leads to a higher center of gravity. Why would this matter? Because in the argument Huygens gives, he states that it is &lt;em&gt;impossible&lt;/em&gt; for this collision apparatus to return the center-of-gravity to a greater height than it started. So if we &lt;em&gt;start&lt;/em&gt; from the minimum, and cannot return higher, we must return back to the height that we came, and the velocities must be the same out as in. (We would today say that if there are two configurations with no kinetic energy and there is no external energy input, then gravity is a conservative force and thus the height of the center of mass must be strictly lower as the only possibilities are for frictional loss terms, if that helps.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So the assertion Huygens is making is that, by his day, it is known that when the only external force that is involved is gravity (presumably combined with the forces to fix an object at its maximum height and to change its direction without changing its speed), there is no way for gravity to increase the height of a center of gravity that starts at rest. Since any other allocation of &lt;span class=&quot;math-container&quot;&gt;$s_{1,2}$&lt;/span&gt; would increase it, it follows that if we start with the minimum, only the minimizing allocation &lt;span class=&quot;math-container&quot;&gt;$s_{1,2} = u/\mu_{1,2}$&lt;/span&gt; works.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Finally deriving conservation of kinetic energy&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;So Huygens has now identified the &quot;same speeds&quot; frame with the center-of-mass frame, and he gives a rather complicated geometric argument which I think I can replace (like, I'm not even going to try to match his argument) as follows: we first look at a collision in the center-of-mass frame where we have determined that the initial velocities are &lt;span class=&quot;math-container&quot;&gt;$v_1 = u/\mu_1, v_2 = -u/\mu_2$&lt;/span&gt;, and the final velocities are &lt;span class=&quot;math-container&quot;&gt;$V_1 = -u/\mu_1, V_2 = +u/\mu_2.$&lt;/span&gt; We can see lots of things that are invariant in this frame, for example &lt;span class=&quot;math-container&quot;&gt;$a_1 v_1^2 + a_2 v_2^2 = a_1 V_1^2 + a_2 V_2^2$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$a_{1,2}$&lt;/span&gt; because it's true if &lt;span class=&quot;math-container&quot;&gt;$a_1 = 0, a_2\ne 0$&lt;/span&gt; and vice versa. The squaring is important for getting the negative to disappear but otherwise has no deeper purpose here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But remember that Huygens is interested in the principle of relativity, so he is interested in adding a constant velocity &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; to all of these numbers. When we do that in the case of these square-sums we'll find &lt;span class=&quot;math-container&quot;&gt;$$a_1 (v_1 + c)^2 + a_2 (v_2 + c)^2 = a_1 v_1^2 + a_2 v_2^2 + 2 (a_1 v_1 + a_2 v_2) c + (a_1 + a_2) c^2.$$&lt;/span&gt; Now if we equate this to &lt;span class=&quot;math-container&quot;&gt;$a_1 (V_1 + c)^2 + a_2 (V_2 + c)^2$&lt;/span&gt; to find a quantity that's the same before and after, no matter what &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is, we find that the leading and trailing terms vanish and the middle term we can divide out by &lt;span class=&quot;math-container&quot;&gt;$2c$&lt;/span&gt; leaving just &lt;span class=&quot;math-container&quot;&gt;$$a_1 v_1 + a_2 v_2 = a_1 V_1 + a_2 V_2.$$&lt;/span&gt;There is a problem here though, the right hand side is the negative of the left hand side. Thus they can only equal if they both equal zero, and &lt;span class=&quot;math-container&quot;&gt;$a_1 (u/\mu_1) - a_2 (u/\mu_2) = 0$&lt;/span&gt; leads naturally to choosing &lt;span class=&quot;math-container&quot;&gt;$a_{1,2} \propto m_{1,2}.$&lt;/span&gt; So our conclusion is that the sum of &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; must be the same before and after.&lt;/p&gt;&#xA;" OwnerUserId="3012" LastEditorUserId="3012" LastEditDate="2019-01-10T16:24:37.780" LastActivityDate="2019-01-10T16:24:37.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8152" PostTypeId="2" ParentId="1816" CreationDate="2019-01-10T06:25:23.873" Score="2" Body="&lt;p&gt;Actually, if we are talking about Earth, there is another sense in which Aristotle was (accidentally) right, Earth has an atmosphere, and an atmosphere exerts a drag force on falling objects. It is not difficult to calculate terminal velocity and if we assume drag is proportional to velocity (R = kv), at terminal velocity acceleration and hence resultant force is zero, mg - kv = 0, kv=mg, v=mg/k so if the objects are roughly the same size and shape so k is constant, terminal velocity is proportional to mass! Interesting.&lt;/p&gt;&#xA;" OwnerUserId="9344" LastActivityDate="2019-01-10T06:25:23.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8154" PostTypeId="2" ParentId="8150" CreationDate="2019-01-11T14:59:54.903" Score="7" Body="&lt;p&gt;A side note first - when dealing with non-physicists, they will generally regard quantum mechanics as the end-all-be-all of physics, the coolest weirdest stuff. So, it is not surprising that your colleague focused on quantum mechanics and the rest was just engineering. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, lets look at your list first, then go on to some others of note that worked in the Manhattan Project:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Enrico Fermi: His &lt;a href=&quot;https://www.nobelprize.org/prizes/lists/all-nobel-prizes-in-physics/&quot; rel=&quot;noreferrer&quot;&gt;Nobel Prize&lt;/a&gt;, awarded in 1938, was “for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons”. Well, that seems pretty core to the Project given the need to know deeply about nuclear reactions, neutron-driven reactions, and, of course, the production of large amounts of Plutonium through neutron irradiation needed for Fat Man. Note that Plutonium was only discovered in 1940. (Your list does not have Glenn Seaborg on it, who produced and isolated the first samples of Plutonium in February, 1941. This lead to his Nobel Prize in Chemistry in 1951, as well as having an element named after him, an honor with a higher distinction than a Nobel Prize). Fermi led the design of the Chicago Pile, which you may consider engineering but required deep knowledge of nuclear physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hans Bethe - While his Nobel Prize, “for his contributions to the theory of nuclear reactions, especially his discoveries concerning the energy production in stars”, did not come until 1967, it traces back to his 1939 paper &quot;Energy Production in Stars&quot;, Physical Review 55, 434-456 (1939). Of note is the first sentence in the introduction: 'The progress of nuclear physics in the last few years makes it possible to decide rather definitely which processes can and which cannot occur in the interior of stars.&quot; It is all about analyzing nuclear reaction chains, which is a skill quite useful in the Project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Robert Oppenheimer - perhaps the closest related to quantum mechanics per se, given he is part of the Born-Oppenheimer approach for molecular wave functions. However he is also known for the Oppenheimer-Phillips process that allows adding protons to nuclei using deuterium, lowering the Coulomb barrier. Pre-war he was deeply involved in the nuclear physics activities at Berkeley (more below). His role on the project was as the head of it, so his contributions were more managerial, not technical. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ernest Lawrence: His 1939 Nobel Prize was “for the invention and development of the cyclotron and for results obtained with it, especially with regard to artificial radioactive elements”. The cyclotron which he invented and made practical was a key experimental apparatus in making and understanding nuclear reactions, leading to understanding of nuclear structure and early nuclear physics. His experience with accelerators also led to the use of the Calutrons at Oak Ridge for uranium isotope separation for Little Boy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Richard Feynman - Note that he was a freshly minted PhD who ended up in Los Alamos when basically the entire group of Wheeler was moved there. He worked in the computational group, overseeing the IBM punch card machines. Certainly, being there with all the other names probably helped him come up to speed as a professional. His Nobel Prize in 1965 was “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles”. Given the nuclear physics going on at Los Alamos, one is tempted to draw connections between the Nobel Prize and the Project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eugene Wigner: Another physicist with ties to both quantum and nuclear physics. However, his 1963 Nobel Prize was “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”. So, while I &quot;know&quot; him far better for solid state physics, he was recognized more for his nuclear physics contributions. Wigner led the design of the nuclear reactors to produce plutonium, the first reactors constructed. Doctoral students include Victor Weisskopf (nuclear physics) and John Bardeen (only two-time Nobel Prize winner in Physics). Wigner was also present at the meeting between Leo Szilard and Albert Einstein that lead to the letter to Roosevelt that started the Project.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leo Szilard - a quite wide ranging polymath, he conceived the idea of a nuclear chain reaction in 1933, shortly after the discover of the neutron. He worked with Fermi early in the Project on the Chicago Pile, then moved into reactor design. Post-war he moved into biology, including nuclear medicine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lets look at some other Nobel Prize winners, pre-war. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Werner Heisenberg, 1932, “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen” - he was a leader of the German program.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Paul Dirac, 1933, “for the discovery of new productive forms of atomic theory” - during the war he focused on theory and experiments of gas centrifuges for isotope separation. (his co-winnder, Schrodinger, was in Dublin through the war but remained an Austrian citizen until 1948. Thus Schrodinger did not participate in the Project).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;James Chadwick, 1935, “for the discovery of the neutron”. He was deeply involved with nuclear physics, first under Rutherford and then in his own right at Liverpool. Interned in Germany during the First World War. He was closely associated with Britain's Tube Alloys project, and followed it to Los Alamos. Initial work in Britain included measuring the cross section of U-235, critical to understanding the chain reaction. Chadwick was the head of the British Mission, that oversaw all joint technical work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Carl Anderson, 1936, “for his discovery of the positron”. During the war he did research in rocketry, was not associated with the Project. Anderson's co-winner, Victor Hess, was not involved in the Project either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clinton Davisson, 1937, “for their experimental discovery of the diffraction of electrons by crystals”, did not take part in the project. In contrast, the co-winner George Thomson, had gone into nuclear physiscs. Thomson was the chair of the MAUD committee that determined that an atmoic bomb was feasible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the last 'quantum mechanics' Nobel Prize was awarded in 1933. After that, nuclear physics was coming into being. The confluence of World War 2 and the realization that an atomic weapon was conceivable resulted in a huge physics and engineering effort. It took a huge, concentrated effort to develop theory, mature it, and then implement the theory in engineering. It took a similar effort on the experimental side.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a final note, you casually toss out cross-section measurements as if they are not very important. They are the core of understanding nuclear physics since they are the only way to probe nuclear energy levels and states. In this context it is important to note that John Cockroft and Ernest Walton (with Mark Oliphant) performed the first human-controlled nuclear reaction in 1932, measuring the energy dependence of the Li7(p,&lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;)&lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; reaction. Without these measurements, nuclear physics wasn't going much of anywhere (and their measurement of the energy released is within 1% of the currently accepted Q value - not bad for the first ever measurement). Cockroft was also part of the MAUD Committee, but went into radar work for the war. Cockroft and Walton won the 1951 Nobel Prize “for their pioneer work on the transmutation of atomic nuclei by artificially accelerated atomic particles”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I would agree that 'quantum mechanics' such as an undergraduate takes was not necessary at all for the Manhattan Project. However, in the 1930s, quantum mechanics generally well established, and was rapidly evolving into what we now call nuclear and particle physics. Most of the names you mention are better known for these later efforts. The fact that 1932 saw the discovery of the neutron, the first human-controlled splitting of an atom, and by 1933 the realization that a chain reaction was possible shows how quickly people were taking 'classical' quantum mechanics and applying those concepts to understanding nuclear and particle physics.  During the Manhattan Project huge strides were made, experimentally and theoretically, in understanding nuclear physics and realizing nuclear physics as a weapon.    &lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2019-01-11T14:59:54.903" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8155" PostTypeId="1" CreationDate="2019-01-11T19:20:47.377" Score="0" ViewCount="72" Body="&lt;p&gt;How did Hahn and Kolmogorov prove the Hahn-Kolmogorov Extension Theorem? &lt;/p&gt;&#xA;" OwnerUserId="8168" LastActivityDate="2019-01-11T19:20:47.377" Title="The Hahn-Kolmogorov Extension Theorem" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8156" PostTypeId="1" AcceptedAnswerId="8157" CreationDate="2019-01-12T15:24:53.460" Score="7" ViewCount="328" Body="&lt;p&gt;A &lt;a href=&quot;https://www.merriam-webster.com/dictionary/corollary&quot; rel=&quot;noreferrer&quot;&gt;dictionary&lt;/a&gt; search of the word &quot;corollary&quot; immediately yields the usual definition that all people involved with mathematics are used to dealing with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this surely comes from the Latin &quot;corollarium&quot;, which has arguably a completely unrelated meaning. &lt;a href=&quot;http://www.latin-dictionary.net/definition/14347/corollarium-corollarii&quot; rel=&quot;noreferrer&quot;&gt;Many&lt;/a&gt; Latin dictionaries define the word as &lt;a href=&quot;https://en.wiktionary.org/wiki/corollarium&quot; rel=&quot;noreferrer&quot;&gt;follows&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;Money paid for a garland of flowers. &lt;/li&gt;&#xA;  &lt;li&gt;A gift, present, gratuity.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How did a Latin word referring substantially to a tip become a word used for &quot;secondary&quot; theorems? When did this practice become common?  &lt;/p&gt;&#xA;" OwnerUserId="3853" LastActivityDate="2019-01-13T18:47:56.540" Title="When was the term &quot;corollary&quot; first used in proofs?" Tags="|mathematics|terminology|linguistics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8157" PostTypeId="2" ParentId="8156" CreationDate="2019-01-12T17:49:34.660" Score="11" Body="&lt;p&gt;&amp;quot;Corollary&amp;quot; is similar to the word &amp;quot;bonus&amp;quot;: a little extra (i.e. an extra proposition coming from a demonstration).&lt;/p&gt;&#xA;&lt;p&gt;The term Euclid uses is πόρισμα &amp;quot;porism,&amp;quot; which Liddell-Scott-Jones cite as akin to πορίζω in the sense of &amp;quot;to find (money).&amp;quot;   For instance, after &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0085%3Abook%3D1%3Atype%3DProp%3Anumber%3D15&quot; rel=&quot;noreferrer&quot;&gt;I.15&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Πόρισμα&lt;/p&gt;&#xA;&lt;p&gt;ἐκ δὴ τούτου φανερὸν ὅτι, ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς πρὸς τῇ τομῇ γωνίας τέτρασιν ὀρθαῖς ἴσας ποιήσουσιν.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The earliest usage in English cited by the OED is &lt;a href=&quot;https://books.google.com/books?id=aFgJAAAAQAAJ&amp;amp;pg=PA71&amp;amp;lpg=PA71&amp;amp;dq=%22porysmes%22&amp;amp;source=bl&amp;amp;ots=CM9aDo891g&amp;amp;sig=yZoemHvPwoRLuI4zpJbjqfQPSN4&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjT4_-f1-jfAhUxn-AKHSQPABQQ6AEwD3oECAMQAQ#v=onepage&amp;amp;q=%22porysmes%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Chaucer's translation&lt;/a&gt; of &lt;a href=&quot;https://books.google.com/books?id=Wk7d81B7vOwC&amp;amp;pg=PA27&amp;amp;dq=boethius+De+consolatione+philosophiae+book+III+latin&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwicgLLM4OjfAhUHKqwKHf8mAuoQ6AEIUTAH#v=onepage&amp;amp;q=boethius%20De%20consolatione%20philosophiae%20book%20III%20latin&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Boethius, &lt;em&gt;De consolatione philosophiae&lt;/em&gt;, Bk. III&lt;/a&gt;, who proposes to call a porism a corollary.  Boethius is making a theological argument, not a mathematical one, so he is using corollary as analogous to the mathematical term porism.  Chaucer adds &amp;quot;or a mede of coroune&amp;quot;, that is, &amp;quot;a reward of crown,&amp;quot; as an explanation† of corollary:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;ryȝt as þise geometriens whan þei han shewed her proposiciouns ben wont to bryngen in þinges þat þei clepen porismes or declaraciouns of forseide þinges. ryȝt so wil I ȝeue þe here as a corolarie or a mede of coroune....&lt;/p&gt;&#xA;&lt;p&gt;Super haec, iniquit, veluti geometrae solent demonstratis propositis aliquid inferre, quae πορίσματα ispi vocant, ita ego quoque tibi veluti corollarium dabo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;LSJ also cites Pappus as using porism in the sense &amp;quot;a kind of proposition intermediate between a theorem and a problem.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;†&lt;em&gt;Perhaps &amp;quot;meed of crown&amp;quot; was a phrase in the Middle Ages.  I'm not familiar with it.  Meed can mean reward or bribe or share (of money, honor, etc.).  Possibly &amp;quot;crown&amp;quot; refers to king, lord, and by extension God in this case.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-12T17:49:34.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8158" PostTypeId="2" ParentId="8099" CreationDate="2019-01-12T19:57:40.493" Score="3" Body="&lt;p&gt;My impression is that Artin's role in the development of Galois theory is usually greatly overrated. Kiernan's article jumps from 1900 to Artin in the late 1930s. Certainly a &quot;modern&quot; Galois theory existed during the intervening years --- see for example, Albert's Algebra books. Artin's main contribution was to give a proof of the main theorem of Galois theory without using the existence of a primitive element. Certainly choosing a primitive element is unnatural, but I don't think Artin's proofs are any easier for students than the older proofs (as in Albert). &lt;/p&gt;&#xA;" OwnerUserId="9349" LastEditorUserId="-1" LastEditDate="2019-01-21T19:58:14.210" LastActivityDate="2019-01-21T19:58:14.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8160" PostTypeId="2" ParentId="7585" CreationDate="2019-01-13T12:08:51.350" Score="6" Body="&lt;p&gt;I asked a mathematician who is an expert to abstract algebra and he showed me that Gauss's congruence was actually correct. To prove Gauss congruence let's introduce the following notation: &lt;span class=&quot;math-container&quot;&gt;$$x = a+bi,y = c+ di, u = \alpha + \beta i, v = \gamma + \delta i, X = A + Bi, Y = C + Di$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;First of all, one has to understand that Gauss's notation is different from modern convention in two aspects:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;When Gauss designates a quaternion by a collection of four coefficients &lt;span class=&quot;math-container&quot;&gt;$(a,b,c,d)$&lt;/span&gt;, he means &lt;span class=&quot;math-container&quot;&gt;$a+ib+jc+kd$&lt;/span&gt; (not &lt;span class=&quot;math-container&quot;&gt;$a+bi+cj+dk$&lt;/span&gt;).&lt;/li&gt;&#xA;&lt;li&gt;Secondly, Gauss defines quaternions multiplication in such way that the product of two fundamental quaternions gives the third one with positive sign &lt;strong&gt;if&lt;/strong&gt; the multiplication operation is done &lt;strong&gt;counterclockwise&lt;/strong&gt; (not clockwise like in modern convention); that is, &lt;span class=&quot;math-container&quot;&gt;$ij=-k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$jk = -i$&lt;/span&gt;. That he defines quaternions multiplication in this way is evident from the bilinear expressions he give for the four coefficients &lt;span class=&quot;math-container&quot;&gt;$(A,B,C,D)$&lt;/span&gt; of the product.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Therefore each of the quaternions &lt;span class=&quot;math-container&quot;&gt;$q_1= a + ib + jc +kd $&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$q_2 = \alpha + i\beta  +j\gamma  + k\delta $&lt;/span&gt; can be written as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$q_1 = x + jy$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$q_2 = u + jv$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$q_3 = q_1q_2 = X+jY = (x + jy)(u+jv) = (xu - \bar y v) + j(yu + \bar x v)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The complex conjugate symbols appear because of the non-commutitivity of quaternions algebra; that is, &lt;span class=&quot;math-container&quot;&gt;$jy = \bar yj$&lt;/span&gt;. Finally, to prove Gauss's congruence, let's make the following step (multiplying both sides of the congruence by &lt;span class=&quot;math-container&quot;&gt;$(A+iB)(a+ib)$&lt;/span&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(C+Di)(a + bi)-(A+Bi)(c+di) = Yx - Xy = (yu + \bar x v)x - (xu - \bar y v)y = (x\bar x + y\bar y)v + (xy - xy)u =\rVert x + jy \rVert ^2 v$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$\rVert x + yj \rVert ^2 = m$&lt;/span&gt; one gets that that the result equals the product of m and an integral complex number.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Important note:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The proof of Gauss's congruence as presented above includes one problematic step - which is the multiplication of both sides of the congruence by &lt;span class=&quot;math-container&quot;&gt;$Xx = (A+iB)(a+ib)$&lt;/span&gt; and the conclusion that if the resulting congruences is correct than the original congruence is also correct. It should be more appropriate to prove, indeed, that m divides &lt;span class=&quot;math-container&quot;&gt;$\frac{||x+jy||^2v}{(A+iB)(a+ib)} = m\frac{v}{(A+iB)(a+ib)} = m(\epsilon+i\pi)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\epsilon,\pi$&lt;/span&gt; are rational numbers. However, since we are dealing with modular arithmetic, for prime &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; every rational number has an equivalent integer (for example, &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3} \equiv 5 \pmod 7$&lt;/span&gt;), so after multiplication of the integer equivalent of &lt;span class=&quot;math-container&quot;&gt;$\epsilon + i\pi$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; the resulting Gaussian integer is really congruent to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; modulo &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; (Gauss does not say &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is a prime number, so I guess these congruences are just intended to illustrate a general principle).&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2022-07-13T09:21:06.473" LastActivityDate="2022-07-13T09:21:06.473" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8161" PostTypeId="2" ParentId="8156" CreationDate="2019-01-13T12:54:33.730" Score="5" Body="&lt;p&gt;This is a complement to Michael's excellent answer. The source of Boethius and subsequent use is most likely Proclus's commentary on Euclid's Elements, where he organizes and names various concepts concerning demonstrations (given, lemma, case, reduction, enunciation, exposition, construction, proof, reductio, etc.), apparently extracting some from the Euclidean tradition and/or inventing his own. In particular, he explicitly distinguishes the two different senses of &quot;porism&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The term porism is used also of certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some other theorem is revealed at the same time without our propounding it, which theorem has on this very account been called a porism (corollary) as being a sort of incidental&#xA;  gain arising from the scientific demonstrations.&lt;/em&gt;&quot; (quoted by Heath in The Thirteen Books of the Elements, p.134)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Heath adds that, according to Tannery,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Far from distinguishing his corollaries from the conclusions of his propositions, Euclid inserts them before the closing words &quot;(being) what it was required to do&quot; or &quot;to prove.&quot; In fact the porism-corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Euclid's Porisms are not extant. Heath traces the &quot;intermediate between problem and theorem&quot; sense to Pappus's account of them (p.429).&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-01-13T18:47:56.540" LastActivityDate="2019-01-13T18:47:56.540" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8162" PostTypeId="1" AcceptedAnswerId="8165" CreationDate="2019-01-14T07:57:37.553" Score="5" ViewCount="1248" Body="&lt;p&gt;Joel Spolsky &lt;a href=&quot;https://www.joelonsoftware.com/2006/10/25/the-guerrilla-guide-to-interviewing-version-30/&quot; rel=&quot;nofollow noreferrer&quot;&gt;tells this story&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Serge Lang, a math professor at Yale, used to give his Calculus students a fairly simple algebra problem on the first day of classes, one which almost everyone could solve, but some of them solved it as quickly as they could write while others took a while, and Professor Lang claimed that all of the students who solved the problem as quickly as they could write would get an A in the Calculus course, and all the others wouldn’t. The speed with which they solved a simple algebra problem was as good a predictor of the final grade in Calculus as a whole semester of homework, tests, midterms, and a final.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What could this algebra problem have been? (I considered this question for matheducators.se, but I'm not sure they consider such historical questions on-topic.)&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2019-01-14T17:55:44.333" LastActivityDate="2023-06-11T21:27:55.020" Title="What algebra problem did Serge Lang give to calculus students?" Tags="|calculus|education|algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8163" PostTypeId="1" AcceptedAnswerId="8164" CreationDate="2019-01-14T14:49:07.270" Score="6" ViewCount="440" Body="&lt;p&gt;In &lt;a href=&quot;https://arxiv.org/abs/1405.0323&quot; rel=&quot;noreferrer&quot;&gt;this recent article&lt;/a&gt;, the authors write&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consequently, it is a desideratum to teach general relativity in a way&#xA;  that is based on elementary mathematics only. This objective, &lt;strong&gt;already&#xA;  stated by Einstein in 1916&lt;/strong&gt; (Einstein 1916), has been pursued in many&#xA;  different ways both in the development of teaching materials and in&#xA;  popular science publications&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Where exactly, in his 1916 paper, does Einstein state the goal of simplifying the mathematics of general relativity?  An English translation would be great, but the German would suffice.&lt;/p&gt;&#xA;" OwnerUserId="3811" LastActivityDate="2019-01-16T23:49:50.160" Title="Did Einstein want to reduce general relativity to elementary mathematics?" Tags="|relativity-theory|einstein|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8164" PostTypeId="2" ParentId="8163" CreationDate="2019-01-14T17:55:42.067" Score="8" Body="&lt;p&gt;Not sure about &quot;elementary&quot; but, at least, he tried to reduce it to &quot;&lt;em&gt;the simplest and most intelligible form&lt;/em&gt;&quot;. He assumes the level of a graduating secondary school student, and warns of the &quot;&lt;em&gt;patience and force of will&lt;/em&gt;&quot; required to get through. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked article refers to the German 1916 publication of &lt;em&gt;Über die spezielle und allgemeine Relativitätstheorie&lt;/em&gt;, 1920 reprint is available on the &lt;a href=&quot;https://archive.org/details/berdiespezielle00unkngoog/page/n4&quot; rel=&quot;noreferrer&quot;&gt;Internet Archive&lt;/a&gt;. The first English translation, by Lawson, Relativity: the Special and the General Theory, first appeared also in 1920, and is available on &lt;a href=&quot;https://www.ibiblio.org/ebooks/Einstein/Einstein_Relativity.pdf&quot; rel=&quot;noreferrer&quot;&gt;ibiblio.org&lt;/a&gt;. Here is Einstein's preface in full:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of&#xA;  clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist, L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a “step-motherly” fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for trees. May the book bring someone a few happy hours of suggestive thought!&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-01-15T19:53:31.390" LastActivityDate="2019-01-15T19:53:31.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8165" PostTypeId="2" ParentId="8162" CreationDate="2019-01-14T18:51:52.160" Score="10" Body="&lt;p&gt;In 1969 Lang wrote an article for the Columbia Daily Spectator, &lt;a href=&quot;http://spectatorarchive.library.columbia.edu/cgi-bin/columbia?a=d&amp;amp;d=cs19691208-02.2.8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Don't&#xA;Blame Us if You Flunk Math&lt;/a&gt; (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times have changed:&amp;quot;&lt;em&gt;A fifteen minute quiz raises questions what kind of people should be taught what kinds of math at Columbia&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;He provides a basic introductory test for calculus courses and argues for its adequacy to decide who should or should not take them. The amazing predictive success is not mentioned, though. It seems it was used as a placement test for sections with three different levels of difficulty, but the prediction story is more fun, I guess.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;During the last few years there have been some complaints concerning the&#xA;calculus courses. The situation was particularly tense last year when I&#xA;was away, and the math department got a lot of flak. Since I am departmental&#xA;representative this year I have tried to pin point some of the causes for the&#xA;difficulties. We separate our first and second year calculus students into three groups, A, B, C, of which A is supposed to be the more computational and easiest,&#xA;B more theoretical and harder, and C for the very interested and the talented&#xA;students.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Following complaints which Mr. Wyer regarded as comjng from unqualified students, the course was watered down and became less rewarding to the more responsible students. Mr. Wyer vouches for the opinions of at least ten others who agree with him. Mr. Wyer questions the insight of students criticizing math courses as&#xA;being too &amp;quot;theoretical&amp;quot;. As he says: &amp;quot;It is not an infrequent occurrence in&#xA;the fields of physics and engineering for a student to do very well in his&#xA;introductory mathematics courses and subsequently to run into difficulties&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;After receiving Mr. Wyer's letter, I decided to give a short test to check&#xA;Mr. Wyer's opinions. The test consisted of five problems, and was given to the&#xA;1A sections. One can draw some conclusions: a) The test is very easy and students&#xA;unable to do reasonably well on such a test should not be taking a calculus course.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;BASIC TEST&lt;/p&gt;&#xA;&lt;p&gt;You have fifteen minutes to do the following problems:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve the system of equations: &lt;span class=&quot;math-container&quot;&gt;$3x - 2y = 1;$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$4x + 7y = 15$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve the system of equations: &lt;span class=&quot;math-container&quot;&gt;$3x - y + 2z = 1;$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$x - y - z = 0;$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$2x - 2y + 3z = 3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Solve the following equations:&#xA;a) &lt;span class=&quot;math-container&quot;&gt;$2x^2 - 4x + 5 = 0;$&lt;/span&gt;&#xA;b) &lt;span class=&quot;math-container&quot;&gt;$3x^2 + 2x - 8 = 0$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the sine of an angle of:&#xA;a) 45 degrees  b) 30 degrees  c) 90 degrees&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Show that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$1/(x+y)  - 1/(x-y) = -2y/(x^2 - y^2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-06-11T21:27:55.020" LastActivityDate="2023-06-11T21:27:55.020" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8166" PostTypeId="1" CreationDate="2019-01-15T11:25:54.453" Score="5" ViewCount="178" Body="&lt;p&gt;Who was the first person to use the expression “Fundamental theorem of Algebra”? It is well-known that Gauss called it “Fundamental theorem of algebraic equations”. Grattan-Guiness, in his &lt;em&gt;The Rainbow of Mathematics: A History of the Mathematical Sciences&lt;/em&gt; claims that “The modern name, &amp;lsquo;the fundamental theorem of algebra&amp;rsquo;, seems to have come in late in the 19&lt;sup&gt;th&lt;/sup&gt; century”. But who used it first?&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2019-01-15T15:46:26.250" LastActivityDate="2019-04-09T02:28:41.553" Title="Origin of the expression “Fundamental theorem of Algebra”" Tags="|terminology|algebra|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8167" PostTypeId="1" CreationDate="2019-01-15T17:25:02.703" Score="6" ViewCount="627" Body="&lt;p&gt;How did Newton and Leibniz think about the integral? Did they only see it as an anti-derivative or did they also think of it as the area under a curve?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastEditorUserId="55" LastEditDate="2019-01-15T19:38:29.533" LastActivityDate="2019-01-18T02:49:04.707" Title="How did Newton and Leibniz interpret the integral?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8168" PostTypeId="2" ParentId="8167" CreationDate="2019-01-15T19:38:01.670" Score="6" Body="&lt;p&gt;It was known even before Newton and Leibniz that areas under curves can be found by inverting the &quot;computation of derivatives&quot; (drawing tangents). In explicit geometric form this &quot;fundamental theorem of calculus&quot; was derived by Newton's teacher Barrow, see &lt;a href=&quot;https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/barrows-fundamental-theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barrow's Fundamental Theorem by Wagner&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton and Leibniz developed explicit symbolic methods for computing derivatives (isolated from the general process of drawing tangents), and anti-derivatives. To the extent that they conceptualized integral as such, it was identified with the anti-derivatives (what we call the indefinite integral), but, of course, it was used as a tool for computing areas, among other things. Euler still thought of it this way in 18th century. The idea of the &lt;em&gt;definite&lt;/em&gt; integral does not appear until Cauchy in the 19th, see &lt;a href=&quot;https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0080597&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kallio's History of the Definite Integral&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-01-17T22:14:41.320" LastActivityDate="2019-01-17T22:14:41.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8169" PostTypeId="1" CreationDate="2019-01-15T18:43:00.353" Score="1" ViewCount="88" Body="&lt;p&gt;How did the concept of positive and negative charge originate before the discovery of an electron? My question is related to the historical aspects.&lt;/p&gt;&#xA;" OwnerDisplayName="Iqrar Ullah" LastEditorUserId="6085" LastEditDate="2019-01-17T22:00:08.323" LastActivityDate="2019-01-17T22:00:08.323" Title="Charge concept before discovery of electron" Tags="|electromagnetism|electricity|" AnswerCount="1" CommentCount="0" ClosedDate="2019-01-19T03:47:29.520" ContentLicense="CC BY-SA 4.0" />
  <row Id="8170" PostTypeId="2" ParentId="8169" CreationDate="2019-01-15T19:30:14.377" Score="1" Body="&lt;p&gt;In the 1700s it was established that materials (especially those that we now call 'electrical insulators) when rubbed with other suitable materials, could be divided into two groups. All the rubbed materials repelled materials in the same group, but attracted materials in the other group. Benjamin Franklin called the group containing glass rubbed with silk &quot;positively charged&quot; and the group containing amber (fossilised resin) rubbed with fur &quot;negatively charged&quot;. [He didn't think of positive charge and negative charge as of equal status as we now do; rather he thought of negative charge as an absence of the real stuff: positive charge!] &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason that we now say that an electron has a negative charge and the proton has a positive charge goes back to Franklin's arbitrary decision to say that glass rubbed with silk is positive and amber rubbed with silk is negative. If you hold a rubbed lump of glass near an electron beam it attracts the beam; if you hold a rubbed lump of amber near the beam it repels it!&lt;/p&gt;&#xA;" OwnerUserId="15941" OwnerDisplayName="Philip Wood" LastActivityDate="2019-01-15T19:30:14.377" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8171" PostTypeId="1" CreationDate="2019-01-16T05:29:00.290" Score="5" ViewCount="584" Body="&lt;p&gt;Historically, why has it been important to have a clear distinction between science and other forms of knowledge? The only thing I have come across so far is the debate between evolution and creation, with Popper's idea of scientific theories being falsifiable being used to classify creation as non-scientific.&lt;/p&gt;&#xA;" OwnerUserId="8141" LastActivityDate="2021-04-16T16:55:40.463" Title="Why is the problem of demarcation important in the history of science?" Tags="|philosophy-of-science|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8172" PostTypeId="2" ParentId="8171" CreationDate="2019-01-16T06:51:27.450" Score="3" Body="&lt;p&gt;Warning: the following answer simplifies the history of the philosophy for brevity.&lt;/p&gt;&#xA;&lt;p&gt;In 2019, when &amp;quot;science says X&amp;quot; has all but uncontroversially meant X for a long time, it's easy to misunderstand the historical interest in the demarcation problem. It wasn't about saying, &amp;quot;this is science so is right, while that isn't so it's wrong&amp;quot;. It was about understanding &lt;em&gt;why&lt;/em&gt; science is trustable. (Do you think you know why it is? Maybe you do, but knowledge always has a history.) If you believe science is inductive, as David Hume did, you can't trust science unless you trust induction (which is harder to do, once you've read him). Meanwhile, if before 1900 you think unscientific claims are less reliable - in other words, if you want to make metaphysics (especially theology) the dirty word it gradually became for many - you need a rough idea of what makes it different, or you can't complain about certain ideas not being up to par.&lt;/p&gt;&#xA;&lt;p&gt;Most post-Humean philosophers who commented on science at least somewhat trusted it, if only because of its transformational effect on the era they were living in. But how did they justify it? Mostly, by trusting/defending inductivism. Popper is of historical significance because he took a different approach of saying Hume was &lt;em&gt;right&lt;/em&gt; about induction being unworkable, but &lt;em&gt;wrong&lt;/em&gt; about science being inductive in the first place! But of course, if you say that, you have to say what you think characterises science instead.&lt;/p&gt;&#xA;&lt;p&gt;So what is science? Popper gave his opinion; Kuhn gave his; Lakatos gave his; Feyerabend said it was a mug's game to attempt demarcation; Thagard nonetheless gave yet another view. From Hume 1748 to Popper 1934, science was thought of as inductive. In the decades following that, multiple philosophers prominently gave yet another take on demarcation. Not only did the pace of new proposals accelerate; each new idea led to important new discussions in philosophy about what you can trust in science. For example, if you agree with Kuhn, you'll think modern scientific explanations are incommensurable with older ones, which doesn't bode well for the concept of scientific progress. Similarly, Feyerabend's attack on method is also an attack on scientific exceptionalism, whether you think it deletes scientific knowledge or puts dancing angels back on pinheads. So obviously, whom you agree with matters a lot.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2021-04-16T16:27:22.187" LastActivityDate="2021-04-16T16:27:22.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8173" PostTypeId="1" AcceptedAnswerId="8177" CreationDate="2019-01-16T23:06:40.030" Score="9" ViewCount="5825" Body="&lt;p&gt;I read somewhere, some time ago that Maxwell originally wrote his eponymous equations using the formalism of quaternions and it was only the later intervention of Gibbs and Heaviside that put them into the modern form, that is via vector analysis (I say modern, but its more modern to write them via differential forms or Clifford analysis).&lt;/p&gt;&#xA;&lt;p&gt;Is this true? And if true, how many equations were there in that formalism?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2023-08-04T22:47:10.847" LastActivityDate="2023-08-11T11:56:31.403" Title="Did Maxwell originally write his equations using quaternions?" Tags="|physics|electromagnetism|maxwell|quaternions|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8174" PostTypeId="2" ParentId="8163" CreationDate="2019-01-16T23:49:50.160" Score="4" Body="&lt;p&gt;Only elementary mathematics is required to formulate and understand Special Relativity as a glance of Einsteins 1905 paper &lt;em&gt;On the Electrodyamics of moving bodies&lt;/em&gt; shows, where he put forward the main ideas involved in SR.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is not the case for GR, as Einstein himself was forced to admit. He learned tensorial methods for working with general manifolds from his longtime colleague, Marcel Grossmann and this was the main mathematical technology he used to formulate GR; methods that were different from his rival, Hilbert, who used a variational approach - aka the Einstein-Hilbert action. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it is possible to develop some of the ideas of GR in an elementary way as Robert Geroch did in his small but delightful book, &lt;em&gt;Relativity from A to B&lt;/em&gt;, the book that I first learnt about GR during my last year of high school; basically he uses the fact the Gausian curvature is a product of principal curvatures and this allows him to get to the main ideas of GR without developing the apparatus of tensor fields over manifolds, an apparatus that I might add, whose pedagogical development still has much room for improvement.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-01-16T23:49:50.160" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8175" PostTypeId="2" ParentId="8173" CreationDate="2019-01-17T00:06:59.397" Score="4" Body="&lt;p&gt;Maxwell's equations as they are expressed today, with modern vector notation and not Maxwell's quaternion notation, were first written by &lt;a href=&quot;https://hsm.stackexchange.com/a/7609/232&quot;&gt;Oliver Heaviside&lt;/a&gt; in a few papers in the mid- to late 1880s, but it wasn't until 1893 that the first volume of his &lt;em&gt;Electromagnetic Theory&lt;/em&gt; appeared (and vol. 2 in 1899 and vol. 3 in 1912).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Crowe, Michael J. 1967. &lt;a href=&quot;https://isidore.co/calibre/browse/book/5497&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A history of vector analysis: the evolution of the idea of a vectorial system&lt;/em&gt;&lt;/a&gt;. New York: Dover. &lt;strong&gt;pp. 162-176&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Also, &lt;a href=&quot;https://physics.stackexchange.com/a/444987/10024&quot;&gt;see this modernization of Maxwell's version of the Lorentz force law&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-01-17T21:45:28.447" LastActivityDate="2019-01-17T21:45:28.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8176" PostTypeId="1" AcceptedAnswerId="8180" CreationDate="2019-01-17T07:43:20.763" Score="2" ViewCount="2720" Body="&lt;p&gt;In early history, the relative atomic weight of hydrogen was assigned as 1 (exactly) and all other elements were compared against hydrogen. What is difficult to find who determined the absolute atomic weight of hydrogen as 1.007? The earliest reference I can find is that of Aston by mass spectrograph (1938). What chemical methods were also used to find out the true chemical atomic weight* of H in absolute terms?&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In early literature &lt;em&gt;chemical atomic weights&lt;/em&gt; referred to atomic weights determined by chemical methods and measurements.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.39.164&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://journals.aps.org/pr/abstract/10.1103/PhysRev.39.164&lt;/a&gt; (pdf is open access)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-08-19T18:46:15.023" Title="How did the early chemists determine the atomic weight of hydrogen?" Tags="|physics|terminology|reference-request|philosophy-of-science|chemistry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8177" PostTypeId="2" ParentId="8173" CreationDate="2019-01-17T08:40:36.133" Score="11" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I read somewhere, some time ago that Maxwell originally wrote his eponymous equations using the formalism of quaternions ... Is this true?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems that the answer is &quot;Not quite&quot;. Maxwell originally wrote his equations in components, and later simplified them by using quaternions and some vector calculus. It is true that Heaviside and Gibbs put them into their modern form.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And if true, how many equations were there in that formalism?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My source says 20. I count 12, but it says that &quot;cont. eq. missing here&quot;, so maybe there were 8 equations relating to continuity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: slides from the presentation given at University College London in 2010 &lt;a href=&quot;http://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/MaxwellUCL2.pdf&quot; rel=&quot;noreferrer&quot;&gt;On the changing form of Maxwell’s equations&#xA;during the last 150 years — spotlights on the history of classical electrodynamics&lt;/a&gt; by Friedrich W. Hehl.&lt;/p&gt;&#xA;" OwnerUserId="104" LastEditorUserId="11115" LastEditDate="2020-02-22T04:21:23.200" LastActivityDate="2020-02-22T04:21:23.200" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8179" PostTypeId="1" CreationDate="2019-01-17T20:09:49.793" Score="6" ViewCount="257" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Elasticity_of_a_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elasticity&lt;/a&gt; of a function is a mathematical concept that is widely used in economics. In particular, price elasticity of demand or supply. But generally &lt;a href=&quot;https://en.wikipedia.org/wiki/Elasticity_(economics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;elasticity in economics&lt;/a&gt; is the measurement of how an economic variable responds to a change in another.&lt;/p&gt;&#xA;&lt;p&gt;Yet I found the term ambiguous and I don't understand why we didn't choose &amp;quot;sensitivity&amp;quot; or &amp;quot;reactivity&amp;quot; instead. To make matters worse, economists often call &amp;quot;Price elasticity of the demand&amp;quot; only, &amp;quot;Elasticity of the demand&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;What is the history of the term &amp;quot;elasticity&amp;quot;? Why do we use this term?&lt;/p&gt;&#xA;" OwnerUserId="9332" LastEditorUserId="12330" LastEditDate="2023-11-01T14:49:59.233" LastActivityDate="2023-11-02T17:12:28.343" Title="Where does the term elasticity (of a function) come from?" Tags="|mathematics|economics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8180" PostTypeId="2" ParentId="8176" CreationDate="2019-01-17T21:51:49.177" Score="4" Body="&lt;p&gt;Atomic weights, like much else in chemical science, were the result of a long sequence of developments and discoveries; but a convenient starting-point for the discussion is &lt;a href=&quot;https://archive.org/details/newsystemofchemi01daltuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;A new system of chemical philosophy&quot; (1808, 1810)&lt;/a&gt; by John Dalton (1766-1844)). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In brief summary of the detail offered below, the H = 1 scale was originally an arbitrary (and rather rough) relative scale of atomic weights.  With increased refinements in measurement, the atomic weights on this or any other scale ceased in general to be whole numbers (see also (&lt;a href=&quot;https://hsm.stackexchange.com/questions/6589/before-the-discovery-of-the-neutron-how-did-scientists-explain-standard-atomic/&quot;&gt;Before the discovery of the neutron, how did scientists explain standard atomic weights?&lt;/a&gt;)). The current assignment of H as about 1.007 comes from a re-scaling that by convention put oxygen O = 16 , although as a result of slight adjustments, O itself also now varies slightly from exactly 16, even if only in the fourth decimal place, (see &lt;a href=&quot;https://www.qmul.ac.uk/sbcs/iupac/AtWt/&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Atomic weights of the elements (2017)'&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Returning to John Dalton in 1808, one of his starting points was that there were &quot;ultimate particles&quot; of matter, and also that in &quot;a given volume of any gas, we seem persuaded that, let the divisions be ever so minute, the number of particles must be finite&quot;.  Dalton claimed no originality in the old idea of ultimate particles, but he also saw clearly that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Chemical analysis and synthesis go no farther than to the separation&#xA;  of particles one from another, and to their reunion. No new creation&#xA;  or destruction of matter is within the reach of chemical agency. ...&#xA;  All the changes we can produce, consist in separating particles that&#xA;  are in a state of cohesion or combination, and joining those that were&#xA;  previously at a distance.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Dalton's concept of chemical substances was that for each distinct substance there was a characteristic 'atom', either simple or compoound. His book proposed a (rather awkward) set of symbolic diagrams (see extract below) for several of these 'atoms', &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yuF1E.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yuF1E.gif&quot; alt=&quot;Dalton&amp;#39;s (1808) atomic symbols&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;along with a table (below) that gave estimates of how the compound ones were made up of the simple ones in terms of their relative contribution by weight to the compound. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/r8333.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r8333.gif&quot; alt=&quot;Dalton&amp;#39;s (1808) table of &amp;#39;relative weights&amp;#39; and compositions&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Everybody familiar with basic chemistry will immediately see that many of these early estimates of proportions and weights were quite some way off.  But some ideas persisted through later improvements in precision of measurement and fresh discovery, especially for example &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- that the 'atoms' combined in whole numbers to make a compound atom (what is now called a molecule), &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- that the whole-number presence of (simple) atoms in (compound) atoms reflected that the combining proportions of chemical substances were definite and fixed, and could be expressed in terms of relative or 'equivalent' weights that would combine with nothing left over, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- and that the lightest of the combining equivalent weights was that of hydrogen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This made it natural to assign to hydrogen the weight or mass of '1' on what was originally an arbitrary (and rather rough) relative scale. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The errors and uncertainties in these early assignment of atomic proportions and equivalent weights were given considerable clarification and correction almost simultaneously with Dalton's book, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Louis_Gay-Lussac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph-Louis Gay-Lussac&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His important (1809) law of combining volumes proposed that when gases react together (and the conditions are such that nothing of the original reactants is left over), then the ratio between the volumes of the reactant gases can be expressed in simple whole numbers (when all volumes are measured at the same temperature and pressure). If the reaction product was gaseous, then its volume too could be expressed as a whole-number relative to the reactant volumes. (This gave another way to determine equivalent combining weights, based on the densities and volumes of the combining gases.) See his &lt;a href=&quot;https://books.google.com/books?id=hnJKAAAAYAAJ&amp;amp;pg=PA207&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Mémoire sur la combinaison des substances gazeuses, les unes avec les autres&quot;&lt;/a&gt; &lt;a href=&quot;http://web.lemoyne.edu/~giunta/gaylussac.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation: (Memoir on the combination of gaseous substances with each other)&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the later developments that soon complicated this picture -- and established, through increased refinements in measurement, that atomic weights on any scale ceased in general to be whole numbers-- are described in this answer (&lt;a href=&quot;https://hsm.stackexchange.com/questions/6589/before-the-discovery-of-the-neutron-how-did-scientists-explain-standard-atomic/&quot;&gt;Before the discovery of the neutron, how did scientists explain standard atomic weights?&lt;/a&gt;), which also features the isotope discoveries of F W Aston from about 1919 onwards.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During the two centuries since the works of Dalton and Gay-Lussac, the atomic-weight scale has several times been refined, sometimes in interests of maintaining a close relationship to standard measurement tools. For example, in the 20th-century, atomic weights have been re-scaled to define the atomic weight of the isotope carbon-12 to be exactly 12, &#xA;or oxygen-16 to be exactly 16.  But these alterations have always left the weight assigned to H to be pretty close to 1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.qmul.ac.uk/sbcs/iupac/AtWt/&quot; rel=&quot;nofollow noreferrer&quot;&gt;current table&lt;/a&gt; is derived from a re-scaling based on the convention that oxygen O = 16. This is what leads to H = about 1.007 instead of 1 exactly, but the current scale has undergone very slight adjustments from time to time, e.g. as a result of improved measurements, and O itself also now varies slightly from exactly 16, even if only in the fourth decimal place. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The calibration of the atomic weight scale in terms of a terrestrial standard unit of weight came much later than Dalton and Gay-Lussac, of course, and is really another subject. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-01-17T23:34:05.833" LastActivityDate="2019-01-17T23:34:05.833" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8181" PostTypeId="1" AcceptedAnswerId="8186" CreationDate="2019-01-17T22:04:53.273" Score="5" ViewCount="808" Body="&lt;p&gt;What did Riemann do for the theory of integration?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking because I hear his name a lot in relation to integration and it is often implied that he made large contributions, but I do not know what they are.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To clarify, Cauchy was the first to make the integral rigourous. Then, Riemann then gave a slightly different definition of the integral. Then, Darboux gave a defintion that simplified the proofs to the common theorems of integration. It should also be noted that what is often taught as the Riemann integral is normally the Darboux integral. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Riemann do more for integration than this slight different defintion of the integral, whose main ideas were preceeded by Cauchy and whose related proofs were simplified by Darboux?&lt;/p&gt;&#xA;" OwnerUserId="6574" LastActivityDate="2020-09-30T08:37:11.023" Title="Riemann's Contribution to Integration" Tags="|mathematics|calculus|real-analysis|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8182" PostTypeId="2" ParentId="8181" CreationDate="2019-01-18T00:03:02.563" Score="6" Body="&lt;p&gt;Cauchy made integral rigorous, and proved that integral (in the sense of Cauchy) exists for continuous functions. Riemann proposed a more general definition,&#xA;(integral in the sense of Riemann) and introduced the new class of functions, which are called now Riemann-integrable. This class is strictly larger&#xA;than continuous (or piecewise-continuous) functions. Later, after Riemann, necessary and sufficient conditions were proved for a function to be Riemann-integrable. (A bounded function is Riemann-integrable if and only if &#xA;it is continuous almost everywhere).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sourcs: I. N. Pesin, Classical and modern integration theories, Acad. Press, 1970.&#xA;(Translated from the Russian original, 1966), &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_integral#Integrability&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-01-18T00:03:02.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8183" PostTypeId="2" ParentId="8167" CreationDate="2019-01-18T02:49:04.707" Score="1" Body="&lt;p&gt;Newton proves the upper and lower sums converge when the function (actually curve) is positive and monotonic (decreasing) in &lt;a href=&quot;https://books.google.com/books?id=N-hHAQAAMAAJ&amp;amp;pg=PA95#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lemmas II, III in the &lt;em&gt;Principia.&lt;/em&gt;&lt;/a&gt;  Newton does not formulate the process in terms of integration per se, nor in terms of functions.  It is a solely geometrical process.  The proof is equally valid if the function is negative or increasing; and it is easily extend to the case where the function is piecewise monotonic (i.e., a finite number of monotonic pieces).  Newton does not make these trivial extensions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0080597&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kallio's description&lt;/a&gt;, which @Conifold references, Fermat's work resembles Newton's, although Fermat does not seem to bound the area with inscribed and circumscribed rectangles but just uses (something like a limit of) one sum of rectangles.  Fermat also works algebraically, not geometrically in the &lt;em&gt;Principia&lt;/em&gt;.  Also it seems Newton did not develop further the limit of sums as integration; at least, according to Boyer, as cited by Kallio, future work was in terms of anti-derivatives as @Conifold says.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2019-01-18T02:49:04.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8184" PostTypeId="1" CreationDate="2019-01-18T05:37:47.187" Score="84" ViewCount="26650" Body="&lt;p&gt;What are the most glaring examples — if any — of when the professional scientists or mathematicians were wrong, but the nonprofessionals were right?&lt;/p&gt;&#xA;" OwnerUserId="9370" LastEditorUserId="4251" LastEditDate="2020-07-13T00:33:49.920" LastActivityDate="2024-02-04T12:46:58.030" Title="Examples of when the professional scientists or mathematicians were wrong, but the nonprofessionals were right" Tags="|mathematicians|examples|big-list|scientists|" AnswerCount="18" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="8185" PostTypeId="2" ParentId="8184" CreationDate="2019-01-18T07:25:53.230" Score="27" Body="&lt;p&gt;Take meteorites, for instance. By the end of the XVIII&lt;sup&gt;th&lt;/sup&gt; century, educated people &amp;ldquo;knew&amp;rdquo; that no rock found on Earth could possibly have fallen from the sky, in spite of the evidence (eyewitnesses included) for their existence. As science journalist Kat Eshner &lt;a href=&quot;https://www.smithsonianmag.com/smart-news/1803-rain-rocks-helped-establish-existence-meteorites-180963017/&quot; rel=&quot;noreferrer&quot;&gt;wrote&lt;/a&gt;, &amp;ldquo;eighteenth-century rationalists [&amp;hellip;] thought the stories of rains of iron rocks weren’t real&amp;rdquo;. This state of affairs lasted until 1803, when Jean-Baptiste Biot &lt;a href=&quot;http://www.q-mag.org/whodunit-the-fall-of-a-meteorite-at-laigle-on-the-6th-of-floreal-of-the-year-11.html&quot; rel=&quot;noreferrer&quot;&gt;established the reality of meteorites&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2019-01-24T09:24:07.627" LastActivityDate="2019-01-24T09:24:07.627" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8186" PostTypeId="2" ParentId="8181" CreationDate="2019-01-18T13:25:18.403" Score="15" Body="&lt;p&gt;The following is a slightly edited version of my &lt;a href=&quot;https://groups.google.com/g/sci.math/c/V-p72TzeFS4/m/JVYfY5IKEVQJ&quot; rel=&quot;noreferrer&quot;&gt;31 January 2003 sci.math post&lt;/a&gt; archived at google groups.&lt;/p&gt;&#xA;&lt;p&gt;Riemann &lt;strong&gt;[6]&lt;/strong&gt; introduced his integral in his December 1853 Habilitationsschrift thesis. In his thesis he also gave an example, correctly verified, of a Riemann integrable function whose discontinuities form a dense set. Riemann's thesis wasn't widely known until it was published (by Dedekind in 1867) shortly after his death in 1866. Two years later in 1870, Hankel &lt;strong&gt;[3]&lt;/strong&gt; proved that every Riemann integrable function has a dense set of continuity points. (For more about Hankel's result, see Renfro &lt;strong&gt;[5]&lt;/strong&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Thus, within TWO years after the publication of Riemann's definition of the integral, it was known that Riemann integrable functions are very well behaved in one sense and it was known that they can be very badly behaved in another sense. They are well behaved in the sense that they have a dense set of continuity points, and thus one might think they are so close to being continuous that they can't be any worse behaved than functions such as piecewise continuous functions, the characteristic function of &lt;span class=&quot;math-container&quot;&gt;$\{\frac{1}{2}, \; \frac{1}{3}, \; \frac{1}{4}, \; \ldots \},$&lt;/span&gt; etc. On the other hand, they can be so badly behaved that they can have a dense set of discontinuity points, and thus it would not be intuitively evident to a novice that such a function could be continuous anywhere, except maybe at isolated points or on sets such as &lt;span class=&quot;math-container&quot;&gt;$\{\frac{1}{2}, \; \frac{1}{3}, \; \frac{1}{4}, \; \ldots \}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, the definition that Riemann gave for his integral is a minor modification of Cauchy's definition from 1823. Cauchy used partitions whose associated subintervals have varying lengths, but Cauchy used only left (or only right) endpoints of these subintervals at which to evaluate the function. In fact, Cauchy's seemingly more restrictive definition actually gives rise to the same collection of integrable functions that Riemann's definition does --- see Gillespie &lt;strong&gt;[2]&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Riemann's nontrivial contributions to this topic were:&lt;/strong&gt;&lt;/em&gt; &lt;strong&gt;(A)&lt;/strong&gt; giving a necessary and sufficient condition for integrability based on the behavior of a function; &lt;strong&gt;(B)&lt;/strong&gt; using this condition to prove the integrability of a certain function having a dense set of discontinuities; &lt;strong&gt;(C)&lt;/strong&gt; putting the focus on the collection of functions that are integrable according to some notion of integrability, rather than defining a notion of integrability only for the purpose of being able to prove certain desired integrability properties. Regarding (B), I believe this was the first time a function that was continuous on a dense set and discontinuous on another dense set had been defined (or even contemplated, for that matter). A well known example of such a function is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomae%27s_function&quot; rel=&quot;noreferrer&quot;&gt;ruler function&lt;/a&gt;, which is also called the Thomae function because it first appeared in an 1875 booklet by Thomae.&lt;/p&gt;&#xA;&lt;p&gt;Cauchy, I believe, only proved that functions having at most finitely many discontinuities in every interval are integrable. Moreover, these were the only functions that Cauchy was interested in, and so Cauchy never sought to generalize his results to more pathological functions nor did he consider his definition of the integral as defining a collection of functions that might be worthy of study.&lt;/p&gt;&#xA;&lt;p&gt;The measure zero version for Riemann integrability was obtained independently by Lebesgue &lt;strong&gt;[4]&lt;/strong&gt; (p. 24; p. 254 of published version) and Vitali &lt;strong&gt;[8]&lt;/strong&gt; &lt;strong&gt;[9]&lt;/strong&gt; and Young &lt;strong&gt;[11]&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The Lebesgue/Vitali/Young integrability condition is actually a small step from results already known by 1875. The integrability condition that Riemann gave, what I called contribution (A) above, involved the oscillation of a function &lt;em&gt;in an interval&lt;/em&gt;. In 1870 Hankel &lt;strong&gt;[3]&lt;/strong&gt; reformulated Riemann's condition in terms of the oscillation of a function &lt;em&gt;at a point&lt;/em&gt;, a notion that was also first introduced in this paper. Hankel's reformulated condition is a bit closer to the notion of continuity at a point, since (as Hankel was well aware) it is straightforward to show that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; if and only if the oscillation of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; is zero.&lt;/p&gt;&#xA;&lt;p&gt;Specifically, Hankel said that a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is Riemann integrable if and only if for each &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt; 0$&lt;/span&gt; the set of points at which &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has an oscillation greater than &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; can be covered by a finite collection of intervals whose lengths have an arbitrarily small sum. [This differs from the way Lebesgue measure zero sets are defined only in that finitely many intervals, rather than countably many intervals, are allowed for the coverings.] Hankel's claim is true. However, Hankel's proof that a Riemann integrable function has this property was incorrect. [The other half of Hankle's &amp;quot;iff&amp;quot; claim, namely that this property implies Riemann integrability, was correctly proved by Hankel.] Hankel's error was that he tried to prove this by trying to prove a stronger result --- he tried to prove that every function whose points of continuity form a dense set will be Riemann integrable. Unfortunately, this last result is too strong, and not just because the methods of proof Hankel used were too weak, but in fact the result itself is not even true! The characteristic function of a Cantor set with positive measure is a counterexample. It was precisely to show this error in Hankel's paper that Smith &lt;strong&gt;[7]&lt;/strong&gt; used a Cantor set of positive measure in his 1875 paper, the paper that contained the first ever construction of a Cantor set (whether of measure zero or of positive measure, and Smith gave both kinds).&lt;/p&gt;&#xA;&lt;p&gt;Hankel's proof was repaired by Ascoli &lt;strong&gt;[1]&lt;/strong&gt; in 1875. An even more precise version was given by Volterra &lt;strong&gt;[10]&lt;/strong&gt; in 1881. We say that a set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is &amp;quot;Jordan null&amp;quot; (or has Jordan content zero) if, given any &lt;span class=&quot;math-container&quot;&gt;$\delta &amp;gt; 0,$&lt;/span&gt; there exists a finite collection of intervals covering &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; whose lengths have a sum less than &lt;span class=&quot;math-container&quot;&gt;$\delta.$&lt;/span&gt; Then, using this notion, which I might add is equivalent (when &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is bounded) to the closure of &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; having Lebesgue measure zero), Hankel's integrability condition says that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is Riemann integrable if and only if for each &lt;span class=&quot;math-container&quot;&gt;$\epsilon &amp;gt; 0$&lt;/span&gt; the points at which &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has an oscillation greater than &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; form a Jordan null set. Volterra showed that the result continues to hold if we use the &amp;quot;left oscillation of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;&amp;quot; rather than the oscillation of &lt;span class=&quot;math-container&quot;&gt;$f,$&lt;/span&gt; or equivalently (consider &lt;span class=&quot;math-container&quot;&gt;$-f),$&lt;/span&gt; the right oscillation of &lt;span class=&quot;math-container&quot;&gt;$f.$&lt;/span&gt; [Left and right oscillation at a point are defined in the obvious way.]&lt;/p&gt;&#xA;&lt;p&gt;The jump from these results to Lebesgue's result is small once the notion of a set of measure zero had been considered worthy of consideration, since it's essentially a matter of a countable union of measure zero sets having measure zero. To be explicit, we can obtain a covering of the discontinuities of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; with intervals whose lengths have a sum less than &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; by taking the union over &lt;span class=&quot;math-container&quot;&gt;$n = 1,\; 2, \;3, \; \ldots$&lt;/span&gt; of the following finite collections &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}_n$&lt;/span&gt; of intervals that we know exist by Hankel's condition -- Given &lt;span class=&quot;math-container&quot;&gt;$n,$&lt;/span&gt; let &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}_n$&lt;/span&gt; be a finite collection of intervals covering the points where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has an oscillation greater than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n}$&lt;/span&gt; and such that the sum of the lengths of the intervals in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{C}_n$&lt;/span&gt; is less than &lt;span class=&quot;math-container&quot;&gt;$\frac{\delta}{2^n}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Giulio Ascoli, &lt;a href=&quot;https://books.google.com/books?id=s9tBAQAAMAAJ&amp;amp;pg=PA862&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sul concetto di integrale definito&lt;/em&gt;&lt;/a&gt; [On the notion of definite integral], &lt;strong&gt;Atti della Accademia Reale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali&lt;/strong&gt; (2) 2 (1875), 862-872.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; David Clinton Gillespie, &lt;a href=&quot;https://www.jstor.org/stable/2007121&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Cauchy definition of a definite integral&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Annals of Mathematics&lt;/strong&gt; (2) 17 #2 (December 1915), 61-63.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Hermann Hankel, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k995758&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Untersuchungen über die unendlich oft oszillierenden und unstetigen funktionen&lt;/em&gt;&lt;/a&gt; [Investigations of functions with infinitely many oscillations and discontinuous functions], Memoir presented at the University of Tübingen on 6 March 1870, 51 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=YQkPAAAAIAAJ&amp;amp;pg=PA63&quot; rel=&quot;noreferrer&quot;&gt;Reprinted&lt;/a&gt; in &lt;strong&gt;Mathematische Annalen&lt;/strong&gt; 20 #1 (1882), 63-112 (&lt;a href=&quot;https://books.google.com/books?id=YQkPAAAAIAAJ&amp;amp;pg=PA63&quot; rel=&quot;noreferrer&quot;&gt;google books&lt;/a&gt; version; &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0020?tify=%7B%22pages%22:%5B69%5D,%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;Göttinger Digitalisierungszentrum&lt;/a&gt; version). &lt;a href=&quot;http://www.numdam.org/item/CSHM_1988__9__139_0/&quot; rel=&quot;noreferrer&quot;&gt;A French translation&lt;/a&gt; by Jeanne Peiffer, with some additional remarks by Pierre Dugac and René Taton, appears in &lt;strong&gt;Cahiers du Séminaire d'Histoire des Mathématiques&lt;/strong&gt; 9 (1988), 139-209.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Henri Léon Lebesgue, &lt;a href=&quot;https://archive.org/details/integralelongueu00lebe&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Intégrale, Longueur, Aire&lt;/strong&gt;&lt;/a&gt; [Integral, Length, Area], Ph.D. dissertation (under Félix Edouard Justin Émile Borel), Université Henri Poincaré Nancy 1, 1902, iv + 129 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The published version (&lt;a href=&quot;https://link.springer.com/article/10.1007/BF02420592&quot; rel=&quot;noreferrer&quot;&gt;Springer pay wall version&lt;/a&gt;; &lt;a href=&quot;http://people.unica.it/antoniogreco/files/2018/06/Lebesgue_tesi.pdf&quot; rel=&quot;noreferrer&quot;&gt;another copy&lt;/a&gt;) is in &lt;strong&gt;Annali di Mathematica Pura ed Applicata&lt;/strong&gt; (3) 7 (1902), 231-359.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Dave L. Renfro, &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=1184837&quot; rel=&quot;noreferrer&quot;&gt;28 January 2003 Historia-Matematica post&lt;/a&gt; archived at Math Forum.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Georg Friedrich Bernhard Riemann, &lt;strong&gt;Ueber die Darstellbarkeit einer Function durch eine Trigonometrische Reihe&lt;/strong&gt; [On the Representability of a Function by a Trigonometric Series], Habilitationsschrift thesis (under Johann Carl Friedrich Gauss), Göttingen University, December 1853.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The original version of Riemann’s thesis is a handwritten manuscript that is currently in Riemann’s &lt;em&gt;Nachlass&lt;/em&gt; at the Göttingen University Library Archives. Before 1868 it is likely that only a few people had seen Riemann’s thesis (mainly Berlin mathematicians, and possibly some others while visiting Berlin), with perhaps a few others having heard about its contents by word of mouth. &lt;a href=&quot;https://www.biodiversitylibrary.org/item/108308#page/194/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;The first printed and circulated version&lt;/a&gt; was published in &lt;strong&gt;Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen&lt;/strong&gt; 13 (1867), 87-131. This first publication was due to the efforts of Julius Wilhelm Richard Dedekind. Although Dedekind’s introductory footnote on the first page is dated July 1867 and the date given on the title page is 1867, I believe that printed copies did not actually reach the mathematical public until 1868. A French translation (by Jean Gaston Darboux and Guillaume Jules Hoüel), &lt;em&gt;Sur la possibilité de représenter une fonction par une série trigonométrique&lt;/em&gt;, was published in &lt;strong&gt;Bulletin des Sciences Mathématiques et Astronomiques&lt;/strong&gt; (1) (1873), &lt;a href=&quot;http://www.numdam.org/article/BSMA_1873__5__20_1.pdf&quot; rel=&quot;noreferrer&quot;&gt;20-48&lt;/a&gt; &amp;amp; &lt;a href=&quot;http://www.numdam.org/article/BSMA_1873__5__79_1.pdf&quot; rel=&quot;noreferrer&quot;&gt;79-96&lt;/a&gt;. English translations can be found in &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0974042730&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Bernhard Riemann. Collected Papers&lt;/strong&gt;&lt;/a&gt; (2004, pp. 219-256) and &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0762430044&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;God Created the Integers&lt;/strong&gt;&lt;/a&gt; (2005, pp. 826-865; 2007, pp. 992-1031).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Henry John Stephen Smith, &lt;a href=&quot;http://fa.its.tudelft.nl/%7Ehart/onderwijs/verzamelingenleer/materiaal/smith-1875-discontinuous-functions.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;On the integration of discontinuous functions&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Proceedings of the London Mathematical Society&lt;/strong&gt; (1) 6 (1875), 140-153.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Giuseppe Vitali, &lt;a href=&quot;https://books.google.com/books?id=92sUAAAAYAAJ&amp;amp;pg=PA27&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sulla condizione di integrabilità delle funzioni&lt;/em&gt;&lt;/a&gt; [On the integrability condition of functions], &lt;strong&gt;Bullettino Mensile della Accademia Gioenia di Scienze Naturali in Catania&lt;/strong&gt; (N.S.) 79 (1903), 27-30.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; Giuseppe Vitali, &lt;a href=&quot;https://books.google.com/books?id=qlZQAQAAMAAJ&amp;amp;pg=PA69&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sulla integrabilità delle funzioni&lt;/em&gt;&lt;/a&gt; [On the integrability of functions], &lt;strong&gt;Rendiconti del Reale Istituto Lombardo di Scienze e Lettere&lt;/strong&gt; (2) 37 (1904), 69-73.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; Vito Volterra, &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN599415282_0019?tify=%7B%22pages%22:%5B82%5D,%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Alcune osservazioni sulle funzioni punteggiate discontinue&lt;/em&gt;&lt;/a&gt; [Some observations on pointwise discontinuous functions], &lt;strong&gt;Giornale di Matematiche di Battaglini&lt;/strong&gt; 19 (1881), 76-86.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[11]&lt;/strong&gt; William Henry Young, &lt;a href=&quot;https://books.google.com/books?id=mFGdAAAAMAAJ&amp;amp;pg=PA189&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;A note on the condition of integrability of a function of a real variable&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Quarterly Journal of Pure and Applied Mathematics&lt;/strong&gt; 35 (1904), 189-192.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2020-09-30T08:37:11.023" LastActivityDate="2020-09-30T08:37:11.023" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8187" PostTypeId="2" ParentId="8184" CreationDate="2019-01-18T13:29:21.863" Score="23" Body="&lt;p&gt;Just warning not to include pre-1920s medicine (and a lot of medical mantra thru the 20th century), as there was little to no science involved amongst physicians.  Just look at how difficult it was for Lister et. al. to convince hospitals, midwives, etc. to wash their hands and sterilize operating theatres.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are dozens of incorrect anecdotes purporting to show scientists were wrong.  The &quot;bumblebee can't fly&quot; is one such.  The truth behind the science/engineering theories is rather different. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-01-18T13:29:21.863" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8188" PostTypeId="2" ParentId="8184" CreationDate="2019-01-18T13:33:43.040" Score="64" Body="&lt;p&gt;It seems &lt;strong&gt;ball lightning&lt;/strong&gt; was disbelieved by scientists until around 1960.  See &lt;a href=&quot;https://en.wikipedia.org/wiki/Ball_lightning&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I knew a geologist who told us how his eye-witness account of ball lightning had been ridiculed.  He had learned not to mention it when he interviewed for jobs as a professor of geology.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2019-01-24T14:07:48.580" LastActivityDate="2019-01-24T14:07:48.580" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="8189" PostTypeId="2" ParentId="8184" CreationDate="2019-01-18T14:41:31.330" Score="19" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Green_flash#Observing&quot; rel=&quot;noreferrer&quot;&gt;Green Flash&lt;/a&gt; was&#xA;described for the first time (at least in the Western literature) by Jules Verne,&#xA;a science fiction writer.&#xA;Many scientists did not believe until photographs were taken and published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Herbert Wells in 1914 described the use of nuclear energy for both bombs and peaceful applications. (His novel The world Set Free). At approximately the same time many scientists thought this was impossible.&#xA;(Rutherford is on record for saying this publicly, that nuclear energy will be never used).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-01-18T14:41:31.330" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8190" PostTypeId="2" ParentId="8184" CreationDate="2019-01-18T16:09:10.750" Score="30" Body="&lt;p&gt;I think a famous example is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Monty_Hall_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Monty Hall problem&lt;/a&gt; about switching doors. The problem was answered correctly by &lt;a href=&quot;https://en.wikipedia.org/wiki/Marilyn_vos_Savant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marilyn vos Savant&lt;/a&gt;, but she got baskets of letters from experts that she is wrong.&lt;/p&gt;&#xA;" OwnerUserId="9374" LastEditorUserId="4251" LastEditDate="2023-11-04T17:49:37.163" LastActivityDate="2023-11-04T17:49:37.163" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="8192" PostTypeId="1" CreationDate="2019-01-18T17:25:27.683" Score="3" ViewCount="352" Body="&lt;p&gt;Who first realized that it is possible to define the integral of a function as the limit of the integrals of a sequence of step functions that converge uniformly to the given function?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is normally how the Lebesgue integral is taught now (although the uniform convergence is avoided by Egorov's theorem), but Lebesgue defined the integral in an analogous way to how Cauchy, Riemann and Darboux did (although he sort of flipped everyting horizontally). &lt;/p&gt;&#xA;" OwnerUserId="6574" LastEditorUserId="5989" LastEditDate="2019-01-20T08:21:36.387" LastActivityDate="2019-01-20T08:21:36.387" Title="The Integral as a Uniform Limit of Step Functions" Tags="|mathematics|calculus|real-analysis|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8195" PostTypeId="2" ParentId="8184" CreationDate="2019-01-19T03:19:40.077" Score="56" Body="&lt;p&gt;In 1726's &lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1402743394&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gulliver's Travels&lt;/a&gt;&lt;/em&gt;, Jonathan Swift mocked the learned scientists of Britain for not having solved the &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/080271529X&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Longitude&lt;/a&gt; problem:  Figure out a way to keep track of one's east-west location to within a mile after making a round-trip across the Atlantic.  This was one of the most important scientific challenges of the 18th century.  The British Parliament had an outstanding offer of £ 20,000 for a solution, and had trusted the astronomers at the Royal Observatory with awarding the prize.  The prize was worth several times the modern-day Nobel Prize, and was famous at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1731, a watchmaker named &lt;a href=&quot;https://www.infogalactic.com/info/John_Harrison&quot; rel=&quot;noreferrer&quot;&gt;John Harrison&lt;/a&gt; solved the problem.  The astronomers at the Observatory refused to believe him.  Over the next 40 years, Harrison steadily refined his solution, but the astronomers never did award him the prize.  It took an act of Parliament in 1773 before Harrison was finally given his prize, and east-west navigation could be made safe.&lt;/p&gt;&#xA;" OwnerUserId="9391" LastEditorUserId="9391" LastEditDate="2020-03-23T23:55:11.993" LastActivityDate="2020-03-23T23:55:11.993" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="8196" PostTypeId="2" ParentId="8184" CreationDate="2019-01-19T18:12:29.690" Score="21" Body="&lt;p&gt;Michael Ventris, an amateur philologist, (he was an architect) managed to decipher the Mycenean script known as Linear B, a problem that professional specialists had been trying to solve for decades.&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2019-01-19T18:12:29.690" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8197" PostTypeId="2" ParentId="8184" CreationDate="2019-01-19T23:22:21.407" Score="38" Body="&lt;p&gt;This isn't a topic I'm familiar with, just something I've read on Quanta, but according to &lt;a href=&quot;https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;, Richard Kershner of Johns Hopkins &lt;a href=&quot;https://www.jhuapl.edu/techdigest/views/pdfs/V08_N6_1969/V8_N6_1969_Kershner.pdf&quot; rel=&quot;noreferrer&quot;&gt;claimed to have a complete classification of convex pentagon tilings&lt;/a&gt; in 1968, though he notably said that &quot;The proof that the list in Theorems 1 and 2 is complete is extremely laborious and will be given elsewhere&quot; and that &quot;a complete proof would require a rather large book&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, after Martin Gardner talked about this claim in his column in &lt;em&gt;Scientific American&lt;/em&gt; in 1975, it got to Marjorie Rice, a California housewife with a high school math education, who found four additional families, and Richard James, a computer programmer, who found another. Eventually, Michael Rao proved that there were exactly 15. You can read more about Rice in &lt;a href=&quot;https://www.quantamagazine.org/marjorie-rices-secret-pentagons-20170711/&quot; rel=&quot;noreferrer&quot;&gt;this article by the same author&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Admittedly, this is an instance of a single professional mathematician making a false claim without giving a proof, which mathematicians consider poor form, and a nonprofessional correcting him, rather than the general mathematics community being wrong.&lt;/p&gt;&#xA;" OwnerUserId="7416" LastEditorUserId="7416" LastEditDate="2019-01-22T15:50:20.457" LastActivityDate="2019-01-22T15:50:20.457" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8198" PostTypeId="1" CreationDate="2019-01-20T05:01:38.857" Score="0" ViewCount="55" Body="&lt;p&gt;Somewhere (I wish I remember where) I read that &lt;a href=&quot;https://plato.stanford.edu/entries/frege/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gottlob Frege&lt;/a&gt;, although he didn't invent a logic algorithm, alluded to one (the &lt;a href=&quot;https://en.wikipedia.org/wiki/Quine%E2%80%93McCluskey_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quine–McCluskey algorithm&lt;/a&gt;? something else? converting truth tables to Boolean expressions in a normal form?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did Frege have an early idea regarding a logic algorithm, even though he did not invent an algorithm?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-01-20T05:14:32.250" LastActivityDate="2019-01-20T05:14:32.250" Title="Frege alluded to a logic algorithm?" Tags="|mathematical-logic|computer-science|computation|algorithm|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8199" PostTypeId="2" ParentId="8184" CreationDate="2019-01-20T16:31:48.723" Score="23" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Wright_brothers&quot; rel=&quot;noreferrer&quot;&gt;Wright Brothers&lt;/a&gt;, both bicycle mechanics, solved both the control and power problems of &lt;a href=&quot;https://en.wikipedia.org/wiki/Early_flying_machines#Powered,_controlled_flight&quot; rel=&quot;noreferrer&quot;&gt;heavier-than-air manned flight&lt;/a&gt; in 1902, beating the well funded aeronautical academic scientists of the Smithsonian Institute, notably &lt;a href=&quot;https://en.wikipedia.org/wiki/Samuel_Pierpont_Langley&quot; rel=&quot;noreferrer&quot;&gt;Samuel Pierpont Langley&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9408" LastActivityDate="2019-01-20T16:31:48.723" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8201" PostTypeId="2" ParentId="8184" CreationDate="2019-01-21T00:09:11.607" Score="12" Body="&lt;p&gt;The &lt;a href=&quot;https://medium.com/the-physics-arxiv-blog/why-hot-water-freezes-faster-than-cold-physicists-solve-the-mpemba-effect-d8a2f611e853&quot; rel=&quot;noreferrer&quot;&gt;Mpemba effect&lt;/a&gt;, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;named after a Tanzanian student who discovered that a hot ice cream&#xA;  mix freezes faster than a cold mix in cookery classes in the early&#xA;  1960s&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;was initially ridiculed.  Quoting &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Mpemba_effect&quot; rel=&quot;noreferrer&quot;&gt;the wiki page on this topic&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After [a lecture by Dr. Denis G. Osborne], Erasto Mpemba asked him the question, &quot;If you take two similar containers with equal volumes of water, one at 35 °C (95 °F) and the other at 100 °C (212 °F), and put them into a freezer, the one that started at 100 °C (212 °F) freezes first. Why?&quot;, only to be ridiculed by his classmates and teacher. After initial consternation, Osborne experimented on the issue back at his workplace and confirmed Mpemba's finding. They published the results together in 1969.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7093" LastEditorUserId="-1" LastEditDate="2019-01-25T13:38:24.700" LastActivityDate="2019-01-25T13:38:24.700" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8202" PostTypeId="2" ParentId="8184" CreationDate="2019-01-21T06:24:51.637" Score="11" Body="&lt;p&gt;Rogue/Freak waves. It seems that reports of these were considered myths by science for a long time until they were finally recorded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(However this is not an exact answer to the question - no non-professional had a theory about these waves, it was more of &quot;ignoring observed facts which don't fit the accepted theory)  &lt;/p&gt;&#xA;" OwnerUserId="9423" LastActivityDate="2019-01-21T06:24:51.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8203" PostTypeId="2" ParentId="8184" CreationDate="2019-01-21T07:23:09.313" Score="22" Body="&lt;p&gt;I'd be tempted to add &lt;a href=&quot;https://en.wikipedia.org/wiki/Gregor_Mendel&quot; rel=&quot;noreferrer&quot;&gt;Gregor Mendel&lt;/a&gt;, whose experiments on plants and his analysis demonstrated how genes work, to that list.&lt;/p&gt;&#xA;&lt;p&gt;It wasn't so much that the 'professional' scientists of the time considered that he was wrong — rather that they didn't even know of his results.  In particular, Darwin puzzled over what the mechanism for transfer of traits was and was searching for a reason why traits wouldn't be continually diluted — which was an answer that Mendel had already answered very neatly.  (Darwin argued for pangenesis as he was sure from his observations that blending inheritance would have diluted distinct traits.)&lt;/p&gt;&#xA;&lt;p&gt;I'd argue that Mendel would meet the criteria as being a 'non-professional' as while he did teach physics — he repeatedly failed the teaching exams so he wasn't qualified to teach high school or adults - only young children.  A pedant may point out that as presented his experiments in a couple of meetings and published an ignored paper he should be regarded as a 'professional scientist', but since his own boss in the church banned him from studying mice as it was considered wrong to study animal reproduction — I'd safely argue that it was a very non-scientific profession he was in.&lt;/p&gt;&#xA;" OwnerUserId="9424" LastEditorUserId="4251" LastEditDate="2020-07-12T12:32:52.050" LastActivityDate="2020-07-12T12:32:52.050" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8205" PostTypeId="1" AcceptedAnswerId="8215" CreationDate="2019-01-21T13:51:07.867" Score="0" ViewCount="122" Body="&lt;p&gt;Page 50 of the book&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gustav Karpeles: &lt;em&gt;A Sketch of Jewish History&lt;/em&gt;. Translated from the German [by an anonymous translator]. The Jewish Publication Society of America. 1897.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;contains the following interesting though unsourced boast: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;A Jew was the first to study the refraction of light ; a Jew introduced to Europe the famous work of Dioscorides, the foundation of the whole science of botany ; a Jew wrote the first textbook of geometry in Europe [...]&quot; &lt;span class=&quot;math-container&quot;&gt;${}\hspace{24.76em}$&lt;/span&gt;&lt;strong&gt;(B)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For a historical research project, I need to know:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Which three Jews did Karpeles refer to in &lt;strong&gt;(B)&lt;/strong&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I am mostly interested in the first and the last of the three contributions alleged, whence my headline for this question.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Remarks.&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I made considerable efforts of looking through reference works and online sources, but in none of the three separate domains (1) light refration, (2) Dioscorides' writings, (3) geometry textbooks, did I find references to significant Jewish involvement in the &lt;em&gt;early&lt;/em&gt; stages. (Except perhaps in the case of Karpeler's &lt;em&gt;least interesting&lt;/em&gt; claim, the one about Dioscorides: it is well known that there were many Arab and Jewish &lt;em&gt;translators&lt;/em&gt; of Greek scientific texts around the Mediterranean, from islamized Spain all the way to Greece, and one of those translators probably was on Karpeler's mind.) What I am interested in for the most part are (1) and (3). In the case of (1), Alhazen, of course, is being mentioned a lot as an early contributor to the study of light, but as far as I know, Alhazen was not Jewish, not even secretly so. In the case of (1) and (3) I still have no inkling of whom Karpeler is referring to.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As already mentioned, Karpeler's book does not contain any sources or references, not even for any other statement in the short book; this has a &lt;em&gt;systematic&lt;/em&gt; reason: in the foreword, the book is said to be a translation of stenographic notes of a &lt;em&gt;lecture&lt;/em&gt; given by Karpeler in the winter of 1895/1896 in a B'nai B'rith lodge in Berlin; as such, it is not meant to be a carefully referenced work, but rather has some degree of pep-talk-flavor mixed to it; still, Karpeler's three claims stand and call for an explanation. (If Karpeler's claims turn out to be untenable, there still remains the minor sub-question of whether Karpeler's statements were made from whole cloth, or whether he mixed up some names and e.g. thought of Alhazen as a Jew.)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6060" LastActivityDate="2019-03-04T14:02:05.330" Title="Reference request concerning an alleged Jewish contribution to the early theory of light refraction, and to the first geometry textbook in Europe" Tags="|geometry|light|islamic-science|refraction|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8207" PostTypeId="2" ParentId="8171" CreationDate="2019-01-21T16:17:33.243" Score="-1" Body="&lt;p&gt;In the US, we might say that &quot;science is the kind of knowledge that we should all agree upon.&quot;  That's why it's taught in schools, while we don't teach religion: public schools must refrain from contradicting individual freedom of religion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tend to think that Popper's criterion doesn't classify just creation as non-scientific, but all explanations of &lt;em&gt;origin&lt;/em&gt;, which no experiment could ever be devised to falsify.&lt;/p&gt;&#xA;" OwnerUserId="9436" LastActivityDate="2019-01-21T16:17:33.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8210" PostTypeId="1" CreationDate="2019-01-21T20:22:38.877" Score="1" ViewCount="198" Body="&lt;p&gt;Nowadays, it's nearly impossible to obtain a university degree in a scientific discipline without completing at least some basic coursework in Calculus, and often times advanced courses are required. Calculus itself was more or less discovered by Newton and Leibniz in the 17th and 18th centuries. Scientists functioned for centuries without it, yet today, Calculus forms a foundation for modern quantitative research methods and seems nearly indispensable for any sort of serious empirical research.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At what point did training in Calculus become an essential part of a scientific education? That is, at what point did Calculus switch from being an interesting tool that some people were playing around with (but was little more than a footnote for everyone else) to an essential tool that more or less &lt;em&gt;everyone&lt;/em&gt; in science was expected to use, or at least be able to use? Was this a &lt;em&gt;gradual&lt;/em&gt; process, with Calculus becoming more and more important as &quot;easy&quot; problems that could be solved without it were solved, or was there a watershed event that ushered it in, such as a war or a specific discovery? I'm &lt;em&gt;imagining&lt;/em&gt; this as somewhat similar to the transition from slide rules to pocket calculators, where there was a ten-year (ish) transition period during which both &quot;camps&quot; had at least somewhat strong support (and people who refused to switch could still function), but is this anything close to reality?&lt;/p&gt;&#xA;" OwnerUserId="6976" LastEditorUserId="6976" LastEditDate="2019-01-21T21:10:53.957" LastActivityDate="2019-01-21T21:10:53.957" Title="At what point did Calculus become a required field of study for aspiring scientists?" Tags="|calculus|academia|education|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8211" PostTypeId="1" AcceptedAnswerId="8217" CreationDate="2019-01-21T23:16:47.463" Score="6" ViewCount="311" Body="&lt;p&gt;Charles Lutwidge Dodgson, better known by his pen name of Lewis Carroll, was a mathematics lecturer at Oxford University and today is primarily famous for his fanciful stories laced with mathematical and logical jokes. He also seems to have been a somewhat prolific writer of math &lt;em&gt;textbooks&lt;/em&gt; in his day, but it's not clear whether he made any significant original &lt;em&gt;discoveries&lt;/em&gt; in math.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did he?&lt;/p&gt;&#xA;" OwnerUserId="6976" LastEditorUserId="6976" LastEditDate="2019-01-22T14:27:29.013" LastActivityDate="2019-01-22T17:52:22.990" Title="Did the author of Alice in Wonderland make any substantial original discoveries in mathematics?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8212" PostTypeId="2" ParentId="3268" CreationDate="2019-01-22T05:11:09.473" Score="1" Body="&lt;p&gt;The US debt markets are a relatively recent addition to the financial markets - the US government did not issue debt until the early twentieth century.&lt;/p&gt;&#xA;&lt;p&gt;When the US debt markets started trading, they took their cue from the already established US stock markets where prices were quoted in eighths.&lt;/p&gt;&#xA;&lt;p&gt;The site &lt;a href=&quot;https://money.howstuffworks.com/personal-finance/financial-planning/stock-market-use-fractions.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Stuff Works&lt;/a&gt; give the following description of this tradition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;These early stockbrokers looked to Europe for a model to build their system on and decided to base it on the system of Spain. This was largely due to the fact that the U.S. dollar's value had been based on the value of the Spanish real.&lt;/p&gt;&#xA;&lt;p&gt;The real was the Spanish silver dollar and was divided into eight parts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This strongly implies the following route to thirty-seconds, which I give as an &amp;quot;educated guess&amp;quot; without reference:&lt;/p&gt;&#xA;&lt;p&gt;As market liquidity increased, pressure on the bid/offer spreads prompted brokers (buyers and sellers) to &amp;quot;meet half way&amp;quot;, thus introducing sixteenths pricing.  Continued increasing liquidity and a &amp;quot;meet half way&amp;quot; attitude finally introduced thirty-seconds pricing. (Each of these pricing refinements would have cut into the brokers profit margins, so their loss would have to have been made up by increased liquidity.  Otherwise, they would have resisted the change.)&lt;/p&gt;&#xA;&lt;p&gt;This would then be the pricing practise inherited by the debt markets when they started trading in the early 20th century. The later introduction of options trading and the &amp;quot;hyper-liquidity&amp;quot; of modern debt markets introduced the final refinement of sixty-fourths.&lt;/p&gt;&#xA;&lt;p&gt;As late as 1997 some of the older stocks in the stock market were still quoting in 32nds.  This ended with the introduction of the &amp;quot;Common Cents Stock Pricing Act&amp;quot;.  The debt markets however appear to have decided to stick with the historic pricing practice, perhaps because the nominal price of a treasury bond is $1000 making further fractional price increments cutting into the brokers spreads undesirable to the market makers.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-22T05:11:09.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8213" PostTypeId="2" ParentId="2683" CreationDate="2019-01-22T12:12:30.227" Score="1" Body="&lt;p&gt;&lt;em&gt;Comment (in the answer window for the sole reason of being able to include the orignial quote from Descartes' book):&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This thread still seems to lack the basic remark that probably Newton's reasons were much less a &lt;em&gt;negative&lt;/em&gt; attitude towards building a line from parts than rather a &lt;em&gt;positive attachment&lt;/em&gt; to the long &lt;em&gt;tradition&lt;/em&gt; of favoring &lt;em&gt;contructibility&lt;/em&gt; in geometry (not only in the Greek authors, but also in Descartes, whose work influenced Newton strongly). In 'Livre premier' of La Géométrie, Descartes (1) declares that in his opinion, it is required to find a method of &lt;em&gt;drawing&lt;/em&gt; a curve which one already knows as a set of points, (2) reproaches Pappus of not having sufficiently 'described' curves which he had indirect determined. I submit that normative statements like Descartes' &amp;quot;il est aussy requis de [...] tracer la ligne&amp;quot; have had an effect on Newton. &lt;a href=&quot;https://i.stack.imgur.com/hB6Y9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hB6Y9.png&quot; alt=&quot;Descartes: La Géométrie, first edition, facsimile in a 1925 edition by 'The Open Court Publishing Company'&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Source: Descartes: La Géométrie, first edition, facsimile in a 1925 edition by 'The Open Court Publishing Company')&lt;/p&gt;&#xA;&lt;p&gt;As translations are available, I don't translate the above page. (Except for the archaic orthography, this is easy French, still understandable if one knows contemporary French.)&lt;/p&gt;&#xA;" OwnerUserId="6060" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-22T12:12:30.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8214" PostTypeId="1" CreationDate="2019-01-22T13:39:57.127" Score="3" ViewCount="101" Body="&lt;p&gt;Have you ever seen a scientific paper/document that uses the decimal time introduced during the French Revolution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not looking for any paper that uses decimal time like in astronomy, but instead I look for a scientific paper made during the French Revolution when the metric time was official. Possibly, I'm trying to find a paper where some physical phenomena is measured in the metric seconds/minutes/hours.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2019-01-24T11:38:06.237" LastActivityDate="2019-01-24T11:38:06.237" Title="Did the French ever use the decimal time in science?" Tags="|units|time|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8215" PostTypeId="2" ParentId="8205" CreationDate="2019-01-22T15:18:40.940" Score="2" Body="&lt;p&gt;Possible sources :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Dioscorides, we have &lt;a href=&quot;https://en.wikipedia.org/wiki/Moses_Hamon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moses Hamon&lt;/a&gt; that owned a famous Ms of &lt;em&gt;De Materia Medica&lt;/em&gt; (later known as : &lt;a href=&quot;https://en.wikipedia.org/wiki/Vienna_Dioscurides&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vienna Dioscurides&lt;/a&gt;) and sold it, through &lt;a href=&quot;https://en.wikipedia.org/wiki/Ogier_Ghiselin_de_Busbecq&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ogier Ghiselin de Busbecq&lt;/a&gt; (a Flemish writer, herbalist and diplomat that served as ambassador to the Ottoman Empire in Constantinople) to the Holy Roman Emperor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Regarding geometry, the first Latin translation of Euclid's Elements is attributed to &lt;a href=&quot;https://en.wikipedia.org/wiki/Adelard_of_Bath&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adelard of Bath (ca.1150)&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have the &lt;em&gt;Practica Geometriae&lt;/em&gt; (1220) of &lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fibonacci (Leonardo Pisano)&lt;/a&gt; that &quot;incorporated&quot; some algebra and geometry from the &lt;em&gt;Liber embadorum&lt;/em&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Plato_Tiburtinus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Plato of Tivoli&lt;/a&gt;’s translation, dated 1145, of the &lt;em&gt;Ḥibbur ha-Meshiḥah ve-ha-Tishboret&lt;/em&gt;) of the spanish jew &lt;a href=&quot;https://en.wikipedia.org/wiki/Abraham_bar_Hiyya&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abraham bar Hiyya&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2019-02-02T13:55:48.707" LastActivityDate="2019-02-02T13:55:48.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8216" PostTypeId="1" AcceptedAnswerId="8220" CreationDate="2019-01-22T16:42:13.930" Score="5" ViewCount="264" Body="&lt;p&gt;In modern proofs of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Peano_existence_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peano Existence Theorem&lt;/a&gt; for ordinary differential equations, Ascoli's theorem is used. Ascoli's theorem came after Peano's proof. Did Peano prove a form of Ascoli's theorem in his proof or did he do something else?&lt;/p&gt;&#xA;" OwnerUserId="8168" LastEditorUserId="55" LastEditDate="2019-01-23T00:45:06.597" LastActivityDate="2019-01-23T00:45:06.597" Title="How did Peano prove his existence theorem without Ascoli's theorem?" Tags="|mathematics|real-analysis|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8217" PostTypeId="2" ParentId="8211" CreationDate="2019-01-22T17:52:22.990" Score="5" Body="&lt;p&gt;I used C. L. DODGSQN, Condensation of determinants. Proc. Royal Sot. London 15 (1866). 150-155 in one of my papers. But for some reason I cited a secondary reference instead, D. P. Robbins and H. Rumsey, Jr., Determinants and alternatig sign matrices, Adv. Math. 62 (1986), 169-184.&lt;/p&gt;&#xA;" OwnerUserId="661" LastActivityDate="2019-01-22T17:52:22.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8218" PostTypeId="2" ParentId="8184" CreationDate="2019-01-22T20:37:55.607" Score="14" Body="&lt;p&gt;Mathematicians have been looking for &lt;a href=&quot;https://en.wikipedia.org/wiki/Amicable_numbers&quot; rel=&quot;noreferrer&quot;&gt;amicable numbers&lt;/a&gt; for millenia. The smallest pair &lt;span class=&quot;math-container&quot;&gt;$(220, 284)$&lt;/span&gt; was known to the Pythagoreans, and several larger pairs and a formula for generating them were found by Hindu and Arab mathematicians during the Middle Ages.  Fermat, Descartes, and Euler rediscovered some of these and found some more. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But in 1866, a 16-year old schoolboy, Nicolo I. Paganini (no relation to the composer) found the previously unknown pair &lt;span class=&quot;math-container&quot;&gt;$(1184, 1210)$&lt;/span&gt;, which is actually the second smallest. &lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2019-01-22T20:46:00.260" LastActivityDate="2019-01-22T20:46:00.260" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8219" PostTypeId="1" AcceptedAnswerId="8221" CreationDate="2019-01-22T21:20:08.427" Score="4" ViewCount="547" Body="&lt;p&gt;In Wikipedia and many other sources copying it, it is claimed that Lev Landau was an atheist (I have no problem to believe the claim), however the (two) sources are not satisfying at all, to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know whether Landau wrote something about God and/or atheism, or to read sources that mention his atheism other than &quot;he was an atheist&quot;. I have found none thus far.&lt;/p&gt;&#xA;" OwnerUserId="9469" LastEditorUserId="4251" LastEditDate="2021-09-06T18:59:54.683" LastActivityDate="2021-09-06T18:59:54.683" Title="What are some references that Lev Landau was atheist?" Tags="|biographical-details|physicists|landau|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8220" PostTypeId="2" ParentId="8216" CreationDate="2019-01-23T00:38:57.153" Score="7" Body="&lt;p&gt;Peano did do something else, and not quite right, apparently. In 1969 Kennedy published a note in the American Mathematical Monthly titled &lt;a href=&quot;https://www.jstor.org/stable/2317137&quot; rel=&quot;noreferrer&quot;&gt;Is There an Elementary Proof of Peano's Existence Theorem&lt;/a&gt;, where he wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1886 Giuseppe Peano stated [8] that the initial value problem: &lt;span class=&quot;math-container&quot;&gt;$y' = f(x,y),\,&#xA;y(a) = b$&lt;/span&gt;, has a solution on the sole condition that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous, and he gave an&#xA;  elementary proof of this. His theorem is correct, but a historical investigation&#xA;  into the work of Peano has led to the conclusion that his proof is invalid.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Kennedy's note stimulated a flurry of activity, some of which is reviewed in Pouso's &lt;a href=&quot;https://arxiv.org/abs/1202.1152&quot; rel=&quot;noreferrer&quot;&gt;Peano's Existence Theorem revisited&lt;/a&gt;. He divides proposed proofs into two groups: (A) constructing a sequence of approximate solutions, and (B) using fixed point arguments. There are also combinations of the two, like the one relying on the Arzela–Ascoli theorem:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;For instance, a well–known kind of mix of types (A) and (B) consists in approximating &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; by polinomials &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; (by virtue of the Stone–Weierstrass Theorem) so that the problems (1.1) with &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; replaced by &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; have unique solutions &lt;span class=&quot;math-container&quot;&gt;$y_n$&lt;/span&gt; (by virtue of the Banach contraction principle) which form an approximating sequence.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Peano's attempted proof, on the other hand, is in (A). For those who read Italian, it can be found in his &lt;a href=&quot;https://archive.org/stream/attidellaraccade21real#page/436/mode/2up/search/peanoseen&quot; rel=&quot;noreferrer&quot;&gt;Sull'integrabilità delle equazioni differenziali del primo ordine, Atti Accad. Sci. Torino (1886) 21: 437–445&lt;/a&gt; (1D) or &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF01200235&quot; rel=&quot;noreferrer&quot;&gt;Démonstration de l'intégrabilité des équations différentielles ordinaires, Mathematische Annalen, June (1890) 37, Issue 2, pp. 182–228&lt;/a&gt; (multi D). There are also more accessible presentations using what came to be called Perron's method of superfunctions, apparently anticipated by Peano (Perron's proof appeared in 1915). Here is more from Pouso:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;More elementary proofs were devised in the seventies of the 20th century, partly as a reaction to a question posed in [10]. In [4, 6, 18, 20] we find proofs of type (A) in dimension &lt;span class=&quot;math-container&quot;&gt;$n = 1$&lt;/span&gt; which avoid the Arzela–Ascoli Theorem. As already stated&#xA;  in those references, similar ideas do not work in higher dimension, at least&#xA;  without further assumptions. We also find in [4] a proof which uses Perron’s&#xA;  method, see [16], a refined version of Peano’s own proof in [14]. It is fair&#xA;  to acknowledge that probably the best application of Perron’s method to (1.1) in dimension &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; is due to Goodman [7], who even allowed &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to be discontinuous with respect to the independent variable and whose approach has proven efficient in more general settings, see [1, 2, 8, 12].&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The [10] is Kennedy's note, and the [4] is &#xA;&lt;a href=&quot;https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/elementary-proofs-of-peanos-existence-theorem/4209A0E6DDF1A103ACC01236A4FCCAD2&quot; rel=&quot;noreferrer&quot;&gt;Elementary proofs of Peano’s Existence Theorem by Dow and Vyborny in J. Aust. Math. Soc. 15 (1973), 366–372&lt;/a&gt; (freely available), who characterize Perron's proof as non-elementary, and give an elementary version &quot;&lt;em&gt;in the spirit of Perron's original proof&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-01-23T00:38:57.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8221" PostTypeId="2" ParentId="8219" CreationDate="2019-01-23T22:10:09.130" Score="5" Body="&lt;p&gt;Short answer, yes. The lengthiest quote of Landau's on religion that I know of comes from his article The Bourgeoisie and Modern Physics, which I doubt has been translated into English. Here is the &lt;a href=&quot;http://old.ihst.ru/projects/sohist/material/press/landau35.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russian source, originaly published in Izvestia VCIK in 1935&lt;/a&gt;. My edited Google translation of the relevant passages is below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;However, one should not think that idealism is found only among pseudo-scholars. The physicists themselves are not consistent materialists. Their materialism is limited to a narrow area of ​​directly scientific work, and already their statements about physics contain idealistic nonsense and are almost indistinguishable from the usual arguments of bourgeois intellectuals. The only effect of the aggressively materialistic content of modern physics is the steady spreading of atheism among physicists. There is practically no major physicist who would not be an atheist. Of course, their atheism is not militant, but calmly gets on with the most benevolent attitude towards religion. Most of them lack the courage to openly admit that religion is against science.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Among physicists of the average hand, religious elements are more common, and it is not surprising that their fame exceeds their scientific value. This is the reason for the extraordinary popularity among non-physicists of the names of Eddington, Jeans and Abbot Lemaitre. Eddington and Jeans owe their fame not to insignificant (and far from always correct) scientific works, and not even to really nice popular books written by them, but to the fact that in these books they connect God to modern physics in every possible way.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Preaching physicists are an exception, but atheists do their best to confuse the situation also. It is difficult to describe all the various types of idealistic utterances that can be heard from them about physics.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;However, in their science, real physicists are constrained by the need to solve real problems and cannot afford the luxury of too much incomprehension. In relation to other sciences, they behave much freer. If we do not count history, the ideas about ​​which are, understandably, childishly naive (of course, in the bourgeois style), then biology is the one that gets it the most.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;One of the most brilliant physicists of the world, Niels Bohr, recently published the book &amp;quot;Light and life&amp;quot;, where he tries to convince the reader that modern physics is consistent with the unknowability of life. Needless to say, Bohr’s arguments do not stand up to either physical or logical criticism. It is surprising that a person who understands physics so deeply in scientific work could compromise it so much when it comes to biology.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The publication date, and the writing style, may sow doubt as to the sincerity of the views expressed. Almost everybody had to be an atheist and a Marxist in the Soviet Union of 1935. However, Landau was not one to just carry the Communist Party's tune. In April 1938 he was arrested for comparing Stalinism to Nazism, and was only released after Kapitsa, in a letter to Stalin, threatened to quit the Soviet atomic bomb project otherwise.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, Landau's atheist views are confirmed by his wife of many years, Kora, who, in her memoirs, relates a private conversation from &lt;em&gt;after&lt;/em&gt; Landau's release in 1939. The Russian text of the relevant chapter is &lt;a href=&quot;https://philologist.livejournal.com/9622142.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, again I doubt it has been translated into English. My edited Google translation follows (Dau is Landau's nickname, Lifshitz is his well-known friend and co-author):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Zhenka Lifshitz rushed after me. His first words to Dau: “Now do you understand what an ass you were to come back from your last foreign trip? What magnificent conditions the British offered you, along with the Americans, but you returned to your free country and got prison! Tell me honestly: do you regret that you returned to the Soviet Union?&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Daunka looked at Zhenka in surprise. “Have you fallen off the moon?” No! I do not regret and I will never regret! I look at my imprisonment simply as a nationwide natural calamity. In the Soviet Union, I met Kora. I divide my life into two epochs: before meeting Kora, the first, and the second - after meeting Kora. And then, despite the various distortions in the government system of our state, our socialist system is the fairest on the planet. Understand the main thing: Marxism denies all religions, and capitalism encourages the too multi-faced religion. You are a scientist. Try to combine science with religions. Science and religion are incompatible to the highest degree&lt;/em&gt; [literally, &lt;em&gt;on the international scale&lt;/em&gt;]! &lt;em&gt;Religions are the fooling of workers all over the planet.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-25T01:17:54.063" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8222" PostTypeId="1" AcceptedAnswerId="8226" CreationDate="2019-01-24T06:27:58.387" Score="3" ViewCount="3206" Body="&lt;p&gt;It is well documented that the notation for the electronic configuration (s,p,d,f) of atoms as used today originates from the words sharp, principal, diffuse, fundamental from alkali metal spectra (William B Jensen has written a nice article &lt;a href=&quot;http://www.che.uc.edu/jensen/w.%20b.%20jensen/reprints/137.%20s,%20p,%20d,%20f.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). After searching for a while&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I never got a chance to find the entire alkali metal spectra where we can visually show the students that this set of lines was called sharp, this set of lines is called the principal, and this is the fundamental set. Does anyone recall a photograph of a real spectrum of any alkali metal with the series labeled in any reference or text (not all the lines are in the visible)? Are there any photographic plates which label all the four series?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Who was the first person to associate the descriptors of spectral lines with angular momentum values of l = 0 implies s, l=1 implies p, l=2 is d and so on? If we look at Hund's open preview of  &lt;em&gt;Linienspektren&lt;/em&gt; &quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3642495400&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Linienspektren-Periodisches-Elemente-Eigenschaften-Einzeldarstellungen/dp/3642495400&lt;/a&gt;&quot; pg 3, he uses the s,p,d and calls them Rydberg correction. How come they got associated with angular momentum?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-01-24T15:30:50.997" LastActivityDate="2019-01-25T18:19:15.507" Title="Who associated the sharp, principal, diffuse, fundamental spectral terms with electron's momentum?" Tags="|physics|biographical-details|terminology|philosophy-of-science|atomic-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8223" PostTypeId="1" CreationDate="2019-01-24T10:11:13.427" Score="1" ViewCount="152" Body="&lt;p&gt;From Gallian, Contemporary Abstract Algebra: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...if G is a group and H is a subgroup of G, it is&#xA;  not always true that aH = Ha for all a in G. There are certain situations&#xA;  where this does hold, however, and these cases turn out to be of critical&#xA;  importance in the theory of groups. It was Galois, about 185 years ago,&#xA;  who first recognized that such subgroups were worthy of special&#xA;  attention.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A subgroup H of a group G is called a normal subgroup of G if aH =&#xA;  Ha for all a in G. We denote this by H &lt;span class=&quot;math-container&quot;&gt;$\triangleleft$&lt;/span&gt; G.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Are subgroups with aH = Ha named normal (&quot;usual, typical, or expected&quot;), from they being excess in examples, and not being rare? Or is there any other reason? And is there any reason for choosing the notation &lt;span class=&quot;math-container&quot;&gt;$\triangleleft$&lt;/span&gt;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am asking these questions, as knowing the relation with names and notation, seems to eliminate keeping memory for the mapping of name and to what it corresponds to, via relation.&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2019-01-26T03:20:21.090" Title="Is there any relation of the word &quot;normal&quot; with a subgroup being normal?" Tags="|mathematics|abstract-algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8224" PostTypeId="1" CreationDate="2019-01-24T15:16:30.973" Score="5" ViewCount="123" Body="&lt;p&gt;I'd like to know the earliest reference to the Z-combinator. This could be either where the name was first coined, or even the first discussion of a need for &lt;a href=&quot;https://en.wikipedia.org/wiki/Fixed-point_combinator#Strict_fixed_point_combinator&quot; rel=&quot;noreferrer&quot;&gt;an applicative-order Y combinator&lt;/a&gt;. I didn't find anything in Barendregt, which was my obvious place to look, nor did I see it in Curry's Combinatory Logic books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone point me the right direction?&lt;/p&gt;&#xA;" OwnerUserId="9499" LastActivityDate="2019-01-24T15:16:30.973" Title="Where is the first reference to the &quot;Z combinator&quot;, a call-by-value fix-point combinator?" Tags="|mathematics|terminology|mathematical-logic|computer-science|computation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8225" PostTypeId="1" AcceptedAnswerId="8227" CreationDate="2019-01-24T16:52:56.817" Score="7" ViewCount="1226" Body="&lt;p&gt;How did Cauchy go about Cauchy's integral theorem? What was his motivation?&lt;/p&gt;&#xA;" OwnerUserId="8168" LastEditorUserId="55" LastEditDate="2019-10-05T07:52:08.243" LastActivityDate="2019-10-05T07:52:08.243" Title="What was the motivation for Cauchy's Integral Theorem?" Tags="|mathematics|complex-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8226" PostTypeId="2" ParentId="8222" CreationDate="2019-01-24T20:54:59.143" Score="2" Body="&lt;p&gt;Yes, those terms come from spectroscopy; they refer to the appearance of the spectral lines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It wasn't until quantum mechanics that atomic spectra were understood as an effect of electron energy levels, including electron angular momentum.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Linus Pauling's concise in his &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=4912&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Introduction to Quantum Mechanics&lt;/em&gt;&lt;/a&gt; ch. 5 (&quot;The Hydrogen Atom&quot;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The hydrogen atom has held a prominent place in the development of physical theory. The first spectral series expressed by a simple formula was the Balmer series of hydrogen. Bohr's treatment of the hydrogen atom marked the beginning of the old quantum theory of atomic structure, and wave mechanics had its inception in Schrödinger's first paper, in which he gave the solution of the wave equation for the hydrogen atom. Only in Heisenberg's quantum mechanics was there extensive development of the theory (by Heisenberg, Born, and Jordan) before the treatment of the hydrogen atom, characterized by its difficulty, was finally given by Pauli. In later developments,* beyond the scope of this book, the hydrogen atom retains its important position; Dirac's relativistic quantum theory of the electron is applicable only to one-electron systems, its extension to more complicated systems not yet having been made.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;*Even finer energy levels were later discovered by Willis Lamb of &quot;Lamb shift&quot; fame.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schrödinger was the first who &quot;gave the solution of the wave equation for the hydrogen atom&quot;. In solving the differential equation, he had to introduce integer constants called the&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;total quantum number (&lt;span class=&quot;math-container&quot;&gt;$n=1,2,3,…$&lt;/span&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;azimuthal quantum number (&lt;span class=&quot;math-container&quot;&gt;$\ell=0,1,2,…,n-1$&lt;/span&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;magnetic quantum number (&lt;span class=&quot;math-container&quot;&gt;$m=-\ell,-\ell+1,…,0,\ell-1,\ell$&lt;/span&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;However, these quantum numbers did have analogues in the older quantum theory of Bohr, Sommerfeld, et al.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-01-25T18:19:15.507" LastActivityDate="2019-01-25T18:19:15.507" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8227" PostTypeId="2" ParentId="8225" CreationDate="2019-01-25T00:44:29.000" Score="13" Body="&lt;p&gt;The original motivation, and an inkling of the integral formula, came from Cauchy's attempts to compute improper real integrals. Here is from &lt;a href=&quot;https://radryan1979.wordpress.com/2013/12/10/a-brief-history-of-complex-analysis-in-the-19th-century/&quot; rel=&quot;noreferrer&quot;&gt;A Brief History of Complex Analysis in the 19th Century&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Cauchy’s first work on complex integration appeared in an 1814 paper on definite integrals (improper real integrals) that was presented to the Institute but not published until 1827 (Bottazzini 1986, 132).  In this paper Cauchy describes the method passing from the real to the imaginary realm where one can then calculate an improper integral with ease. This is the first hint of Cauchy’s later famous integral formula and Cauchy-Riemann equations.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first explicit statement of the theorem dates to Cauchy's 1825 memoir, and is not exactly correct:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$f(x + iy)$&lt;/span&gt; is finite and continuous for &lt;span class=&quot;math-container&quot;&gt;$x_0\leq x\leq X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y_0\leq y\leq Y$&lt;/span&gt;, then the value of the integral &lt;span class=&quot;math-container&quot;&gt;$\int_{x_0+iy_0}^{X+iY}f(z) dz$&lt;/span&gt; is independent of the form of the functions &lt;span class=&quot;math-container&quot;&gt;$x=\phi(t)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=\psi(t)$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The proof was not rigorous by modern standards, but it suggests additional motivation, the calculus of variations. After Bernoulli's brachistochrone problem variational problems occupied the best minds of 18th century, including Euler, who also made early contributions to complex analysis. Here is from  &lt;a href=&quot;https://zulfahmed.files.wordpress.com/2018/07/historycauchyintegralformula.pdf&quot; rel=&quot;noreferrer&quot;&gt;Cauchy Integral Theorem: a Histortical Development of its Proof by Scott&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Using the method of calculus of variations, he considered &lt;span class=&quot;math-container&quot;&gt;$\phi(t)+\varepsilon u(t)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\psi(t)+\varepsilon v(t)$&lt;/span&gt; as an alternate path and showed that the first variation of the integral with respect to &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; vanishes. Note that no mention was made of the continuity of the derivative of &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; or even of its existence although Cauchy made use of both in his proof. Kline [10] suggests that a possible explanation for this is that Cauchy believed, as others of his time, that a continuous function was always differentiable and that its derivative was discontinuous only where the function itself was discontinuous.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1846 Cauchy gave the more familiar proof of the theorem based on Green's formula. &#xA;It is unclear whether Cauchy knew of Green's work (it was published in a privately printed booklet in 1828, but in a mathematical journal only in 1850), or rediscovered it independently. According to Scott, &quot;&lt;em&gt;there are indications that he was influenced by Green's work because he extended his integral theorem to areas on curved surfaces.&lt;/em&gt;&quot; In any case, it suggests another motivation, coming from the study of partial differential equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A comprehensive account of Cauchy's thought process is &lt;a href=&quot;https://www.cambridge.org/core/books/cauchy-and-the-creation-of-complex-function-theory/FBE617B0A16336557AC8E02A6E1731F6&quot; rel=&quot;noreferrer&quot;&gt;Cauchy and the Creation of Complex Function Theory by Smithies&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-01-25T00:44:29.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8228" PostTypeId="2" ParentId="8223" CreationDate="2019-01-25T15:23:39.893" Score="1" Body="&lt;p&gt;This is not an answer, but a step towards one.  Perhaps one can find the first user of the term, and find a discussion of the point in question when it is first introduced.&#xA;A stepping stone in this process is to identify early users of the term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heinrich Weber's &lt;em&gt;Lehrbuch der Algebra&lt;/em&gt; (Chelsea reprint of 2nd edition of 1899?),&#xA;volume 2, p.12, defines &quot;Normaltheil&quot; (normal subgroup), and in a footnote gives the synonyms &quot;ausgezeichnete Untergruppe&quot; and &quot;invariente Untergruppe&quot; (distinguished subgroup, invariant subgroup).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of normal subgroups is usually attributed to Galois, who used the term &quot;décomposition propre&quot;, according to the discussion in Hans Wussing's 1969 book, published in translation in 1984 (MIT) and 2007 (Dover) as &lt;em&gt;The genesis of the group concept&lt;/em&gt;.  (See &lt;a href=&quot;https://books.google.com/books?id=Xp3JymnfAq4C&amp;amp;pg=PA115&amp;amp;lpg=PA115&amp;amp;dq=decomposition+propre&amp;amp;source=bl&amp;amp;ots=gmfcfpU4Ei&amp;amp;sig=ACfU3U0cs0USCgqA_dte-e3-5sCpF22prQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiTgIL0u4rgAhUDPN8KHfTXBqQQ6AEwBXoECAgQAQ#v=onepage&amp;amp;q=decomposition%20propre&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; snippet, for example.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So maybe &quot;normal&quot; is a translation of &quot;&lt;em&gt;propre&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2019-01-26T03:20:21.090" LastActivityDate="2019-01-26T03:20:21.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8232" PostTypeId="1" CreationDate="2019-01-27T01:51:44.047" Score="5" ViewCount="300" Body="&lt;p&gt;In modern chemistry textbooks, a DC source (battery symbol) is usually shown for cathode ray tube experiments by Thomson and others. Certainly, no battery can provide very high voltages needed for gas discharge experiments. Similarly, Roentgen needed high voltage sources for X-ray experiments. What type of devices did these early physicists use for conducting their experiments?&#xA;Youtube has several educational videos known as Ruhmkorff coils, but it seems that they are discontinuous sources. Not sure if discontinuous sources were useful for cathode ray experiments or measuring e/m ratio. Is anyone aware of the names of such devices which powered cathode ray tubes used by early physicists?&#xA;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-04-19T15:27:36.893" Title="High voltage / current sources in 19th century cathode ray experiments" Tags="|physics|experimental-physics|electromagnetism|electricity|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8233" PostTypeId="2" ParentId="8232" CreationDate="2019-01-27T11:38:19.000" Score="5" Body="&lt;p&gt;In order to obtain a nonpulsating power source some early investigators used Wimshurst or similar static electricity generators, or batteries of many small storage cells. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(&lt;a href=&quot;https://archive.org/details/TheDiscoveryOfTheElectron/page/n15&quot; rel=&quot;noreferrer&quot;&gt;The discovery of the electron, David L. Anderson&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="2667" LastActivityDate="2019-01-27T11:38:19.000" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8234" PostTypeId="1" CreationDate="2019-01-27T23:05:53.503" Score="3" ViewCount="143" Body="&lt;p&gt;Who discovered fluid viscosity and what is the history behind the concept of fluid viscosity?  Further, who first defined or parametrized viscosity as shown in Equation 1.1-2 below?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Equation 1.1-2 (below) of the textbook &lt;em&gt;Transport Phenomena&lt;/em&gt; by Bird, Stewart, and Lightfoot is given with the following footnote:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\tag{1.1-2} \tau_{yx}=-\mu \frac{dv_x}{dy}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Actually Eq. 1.1-2 does not appear in Sir Isaac Newton's &lt;em&gt;Philosophiae Naturalis Principia Mathematica&lt;/em&gt;, but the germ of the idea is there. For illuminating comments, see D.J. Acheson, &lt;em&gt;Elementary Fluid Dynamics&lt;/em&gt;, Oxford University Press, 1990., Sec. 6.1&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I do not have Acheson's book.&lt;/p&gt;&#xA;" OwnerUserId="3819" LastEditorUserId="3819" LastEditDate="2019-01-28T13:38:04.627" LastActivityDate="2019-01-28T13:38:04.627" Title="Who discovered fluid visocity?" Tags="|discoveries|experimental-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8235" PostTypeId="2" ParentId="8234" CreationDate="2019-01-28T02:12:40.500" Score="2" Body="&lt;p&gt;The following insight appears in the text you referenced (Acheson, Elementary Fluid Dynamics, OUP, 1990). It is basically a quote from &lt;em&gt;Principa&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In Book II of the Principia (1687) Newton writes: &#xA;SECTION IX &#xA;The circular motion of fluids &#xA;HYPOTHESIS &#xA;The resistance arising from the want of lubricity in the parts of a fluid is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another. &#xA;&lt;em&gt;PROPOSITION LI. THEOREM XXXIX&lt;/em&gt; &#xA;If a solid cylinder infinitely long, in an uniform and infinite fluid, revolves with an uniform motion about an axis given in position, and the fluid be forced round by only this impulse of the cylinder, and every part of the fluid continues uniformly in its motion: I say, that the periodic times of the parts of the fluid are as their distances from the axis of the cylinder. This is the essence of what Newton has to say about viscous flow. &#xA;The hypothesis, of course, gets the subject off to a good start, but it is contained and applied wholly within a section on the circular motion of fluids, and it is immediately followed by a proposition which is false&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The unabridged version of Oxford English Dictionary also quotes Robert Boyle from 1669:&#xA;R. Boyle Contin. New Exper. Physico-mech. (1682) ii. 140   &lt;em&gt;That liquor is very thin, and hath no viscosity to resist the pervading body.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-01-28T02:12:40.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8236" PostTypeId="1" AcceptedAnswerId="8240" CreationDate="2019-01-28T14:54:48.720" Score="10" ViewCount="284" Body="&lt;p&gt;I'd like to know what western scientists thought about the causes of earthquakes in the mid to late 18th century (especially pertaining to the one in Lisbon in 1755). I've read that the ancient Greeks thought that they were caused by rocking of the continental place, and that they were later believed to be somehow related to magnetism. Was any of this believed at the time I'm interested in?&lt;/p&gt;&#xA;&lt;p&gt;Are there any sources I could read?&lt;/p&gt;&#xA;" OwnerUserId="6978" LastEditorUserId="4251" LastEditDate="2021-09-18T11:34:50.943" LastActivityDate="2021-09-18T11:34:50.943" Title="What was the scientific explanation of earthquakes in the 18th century?" Tags="|reference-request|earth-sciences|18th-century|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8237" PostTypeId="1" CreationDate="2019-01-28T15:33:33.880" Score="2" ViewCount="77" Body="&lt;p&gt;I am interested in looking at the notebooks, manuscripts, and letters of scientists and mathematicians from history. It is usually possible to search for a particular person, but tracking images and PDFs from many different sources can be time consuming. So, I'm looking for resource recommendations for browsing handwritten or typed informal sentiments and notes from historical scientists and mathematicians. Both online resources, and physical archives in my region (Philadelphia area), would be very useful. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example: Michael Faraday wrote numerous letters, and his notes on chemistry have been digitized and assembled with transcriptions, into a book which is also available online. However, it is difficult to find an indexed list which allows one to search for different topics. Most scientists are much more difficult to track down than Faraday as well, e.g. Laplace, Poisson, Legendre, Riemann, Hamilton, Ampere, Maxwell, Coulomb, Hertz, etc.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2019-01-29T03:48:09.323" Title="Resources for handwritten scientific notes?" Tags="|resource-recommendation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8238" PostTypeId="2" ParentId="8237" CreationDate="2019-01-28T19:07:23.327" Score="2" Body="&lt;p&gt;I don't think there is a systematic way to search for handwritten manuscripts, and they are rarely available on the web. One collection is here:&#xA;&lt;a href=&quot;http://www.claymath.org/node/1370&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.claymath.org/node/1370&lt;/a&gt;&#xA;If you press on the pop-up menu &quot;Publications&quot; in the top, you find&#xA;manuscripts of Klein, Quillen, Riemann and others.&#xA;University of Uppsala has Beurling's archive:&#xA;&lt;a href=&quot;http://www.math.uu.se/beurling/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.uu.se/beurling/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-01-29T03:48:09.323" LastActivityDate="2019-01-29T03:48:09.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8239" PostTypeId="2" ParentId="8237" CreationDate="2019-01-28T22:00:54.203" Score="1" Body="&lt;p&gt;Dirac Shoebox Collection at Florida State University contains various handwritted notes by Paul Dirac. A part of it is available online: &lt;a href=&quot;https://fsu.digital.flvc.org/islandora/object/fsu%3Amarksstaging&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fsu.digital.flvc.org/islandora/object/fsu%3Amarksstaging&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2019-01-28T22:00:54.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8240" PostTypeId="2" ParentId="8236" CreationDate="2019-01-29T00:26:47.897" Score="9" Body="&lt;p&gt;Aristotle's idea of earthquakes caused by &amp;quot;winds within the Earth&amp;quot; had much currency for centuries, if they were ascribed to natural causes at all. Here is from &lt;a href=&quot;https://igppweb.ucsd.edu/%7Eagnew/Pubs/agnew.a66.pdf&quot; rel=&quot;noreferrer&quot;&gt;Agnew's History of Seismology&lt;/a&gt; on the prevailing thought in the 17-18th centuries:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;With the decline of Aristotelian thought in early modern Europe, other ideas were put forward, though many of the writers were from northern Europe and so (unlike the Greeks) had little direct experience of earthquakes. They did, however, know about gunpowder: This new technology of chemical explosives suggested that earthquakes might be explosions in the Earth (or in the air); of the various chemical theories put forward, the&#xA;most popular involved the combustion of pyrites or a reaction of iron with sulfur. Such theories also explained volcanic action; that the most seismic part of Europe, namely Italy, was also volcanic helped to support this association. In the 18th century the development of theories of electricity, and especially their application to lightning, provoked several theories that related earthquakes to electrical discharges.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One of the electrical theories was proposed by Priestley. And as late as 1750, a writer in the Philosophic Transactions of the Royal Society still apologized to &amp;quot;&lt;em&gt;those who are apt to be offended at any attempts to give a natural account of earthquakes&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The Lisbon Earthquake of 1755 was indeed a watershed event. Here is from &lt;a href=&quot;https://projects.eri.ucsb.edu/understanding/history.html&quot; rel=&quot;noreferrer&quot;&gt;A Brief History of Seismology to 1910&lt;/a&gt; based on &lt;a href=&quot;https://books.google.com/books?id=_FoHAwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;The Founders of Seismology by Davison&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Empirical observations of the effects of earthquakes were rare, however, until 1750, when England was uncharacteristically rocked by a series of five strong earthquakes. These earthquakes were followed on Sunday, November 1, 1755, by a cataclysmic shock and tsunami that killed an estimated 70,000 people, leveling the city of Lisbon, Portugal, while many of its residents were in church. This event marks the beginning of the modern era of seismology, prompting numerous studies into the effects, locations, and timing of earthquakes.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Prior to the Lisbon earthquake, scholars had looked almost exclusively to Aristotle, Pliny, and other ancient classical sources for explanations of earthquakes. Following the Lisbon earthquake, this attitude was jettisoned for one that stressed ideas based on modern observations. Cataloging of the times and locations of earthquakes and studying the physical effects of earthquakes began in earnest, led by such people as John Michell in England and Elie Bertrand in Switzerland.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But for almost a century this activity was mostly phenomenological, not explanatory. Michell's 1761 explanation was close to the modern one, and was based on Bevis's History and Philosophy of Earthquakes, where he collected accounts of the Lisbon earthquake from first hand sources. Michell wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Earthquakes were waves set up by the shifting masses of rock miles below the surface.., the motion of the earth in earthquakes is partly tremulous and partly propagated by waves which succeed each another&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He even estimated that the Lisbon earthquake waves had traveled outward at 530 m/sec. Drijhout in 1765 and Young in 1807 also supported the wave theory. However, Michell's work was &amp;quot;&lt;em&gt;very much overlooked&lt;/em&gt;&amp;quot; until a century later, according to Mallet, &amp;quot;&lt;em&gt;the first true seismologist&lt;/em&gt;&amp;quot; who also coined the word &amp;quot;seismology&amp;quot;, see &lt;a href=&quot;https://engineering.purdue.edu/%7Ece597m/Handouts/ConciseHistory_BenMenahem.pdf&quot; rel=&quot;noreferrer&quot;&gt;A concise history of mainstream seismology by Ben-Menahem&lt;/a&gt;. As Agnew writes, the older theories continued to dominate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The type of wave envisaged was like a traveling wrinkle in a carpet; Michell also suggested that the vibrations close to the source were related to waves propagated through the elasticity of the rocks, as sound waves were known to propagate through the elasticity of the air. The elasticity and pressure of gases, more specifically of high-temperature steam, also provided Michell's driving force for the earthquake itself, which he took to be caused by water vaporized by sudden contact with underground fires. (This steam also supported the propagation of waves to great distances.) Michell attempted to locate this &amp;quot;place of origin&amp;quot; by comparing the times of the seiche-inducing wave and the observed sea wave, and also hazarded a guess at the depth. While these ideas were not forgotten, they did not lead to any additional research, and certainly did not&#xA;replace older theories of earthquakes.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-01-29T01:25:07.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8242" PostTypeId="1" CreationDate="2019-01-28T22:51:35.463" Score="0" ViewCount="341" Body="&lt;p&gt;When I asked for proof of &lt;a href=&quot;https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ampere's circuital law&lt;/a&gt; I learned that this law itself is taken as an axiom.&#xA;Okay I just need to know how did he come up with such a law? how to confirm that result experimentally?&#xA;Is it just a total experimental law? &#xA;When I searched a bit more I found a way to derive it from biot and savart law and I'm good with that but what I wanna know how to come up with it in the first place? &lt;/p&gt;&#xA;" OwnerDisplayName="gogo oka" LastActivityDate="2019-01-29T12:31:44.093" Title="How on Earth did Ampere come up with Ampere's circuital law?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8243" PostTypeId="1" CreationDate="2019-01-29T05:35:45.337" Score="6" ViewCount="983" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When first encountering slope fields in calculus or elementary differential equations, students often ask &quot;What is the purpose?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A concise answer is that slope fields provide a way to graphically represent a first-order ODE of the form y' = f(x, y). Graphical representations can provide a range of insights regarding the overall behavior of solutions, at a glance. The same cannot often be said about symbolic representations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This answer would be more compelling if it were possible to point to examples of the use of slope fields throughout history. However, I've had some difficulty charting this history.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The essential idea of a slope field, namely that y' = f(x, y) indicates the slope of the solution passing through any (x, y) that we choose, must have been clear since the beginning of the calculus. Supporting examples can be found in &lt;a href=&quot;https://twitter.com/HigherMathNotes/status/1089091168657264640&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Twitter exchange&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the earliest instance of (or reference to) an actual slope field that I've found so far was around the middle of the 20th century. See, for example, this &lt;a href=&quot;https://maa.tandfonline.com/doi/abs/10.1080/00029890.1944.11999120?needAccess=true&amp;amp;journalCode=uamm20#.XE_jFFVKjZZ&quot; rel=&quot;nofollow noreferrer&quot;&gt;1944 note on the Teaching of Differential Equations&lt;/a&gt; from the American Mathematical Monthly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apart from the Twitter exchange above, I've searched the web, Google Scholar, and my personal library (including many textbooks and a couple of math history books).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: I did see the Stack Exchange question on the &lt;a href=&quot;https://hsm.stackexchange.com/questions/2063/invention-of-the-concept-of-vector-field&quot;&gt;&quot;Invention of the concept of vector field,&quot;&lt;/a&gt; but the concept of vector field is more general, and it's difficult to isolate any reference to slope fields from the answer to that question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the earliest instance of a slope field as a graphical representation of the first-order differential equation y' = f(x, y)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here I refer not to the underlying idea of a slope field, but rather an actual picture, similar to what is now commonly found in introductory textbooks on elementary differential equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I would be happy to see any early examples, not necessarily the very earliest. I would also be happy to see early occurrences of the term &quot;slope field&quot; or &quot;direction field,&quot; since these indicate the existence of actual pictures.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Progress! The question isn't settled yet, but I've discovered some new evidence. Starting from references in the paper by &lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-01484569/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tournès&lt;/a&gt; posted by Blåsjö below, I was able to find the following two sources that add to the narrative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;First&lt;/em&gt;, consider the book &lt;a href=&quot;https://archive.org/details/graphmethods00rungrich/page/n5&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Graphical Methods&lt;/em&gt; by Carl Runge&lt;/a&gt; (based on lectures given in 1909-1910, published in 1912). Starting on p. 134 (numbered as p. 120 in the physical copy) he discusses the graphical solution of first-order ODEs. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In figure 88, he shows the same method as Bernoulli pictured in Blåsjö's answer, with one addition: he draws a &quot;pencil of rays,&quot; which is basically a set of vectors. Since the direction vector indicated by the differential equation is the same for every point on a given isocline, that isocline corresponds to a unique direction vector. The pencil of rays depicts exactly those vectors, and they are labeled to indicate the isoclines to which they correspond.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While Runge evidently has a reason for doing this (he uses the pencil of rays to effect a change of coordinates later on), it's more difficult to quickly sketch multiple solution curves by visual inspection using this graphical representation than it is with a modern slope field, in which the directions are drawn on top of the isoclines or arranged uniformly across a rectangular grid. I did not find any occurrence of a slope field in its modern form, and neither &quot;slope field&quot; nor &quot;direction field&quot; appears in the text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Second&lt;/em&gt;, consider the book &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015064584769;view=1up;seq=128&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guide to &lt;em&gt;Graphical Computing&lt;/em&gt; by Rudolf Mehmke&lt;/a&gt;, published five years later in 1917. Mehmke was aware of Runge's book (he cites a 1914 edition on p. 98). He writes the following (translated from German using Google Translate, with a little editing - I do not read or write German).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;p. 116 (I've substituted equations for equation numbers in the original text):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...differential equations &lt;span class=&quot;math-container&quot;&gt;$F(x, y, \frac{dy}{dx}) = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx} = f(x, y)$&lt;/span&gt; can be inferred from the geometric&#xA;  sense that they (generally speaking)  assign a specific direction to&#xA;  each point of the plane, which must have the tangent of the integral&#xA;  curve going through the point there, or the presented differential&#xA;  equation requires, as one can say, a &quot;directional field&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;p. 121 (footnote omitted here):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If the current direction for a sufficient number of points is&#xA;  indicated by short dashes on... a series of isoclines (see Fig.&#xA;  107), then any number of integral curves can already be drawn in a&#xA;  reasonably correct way according to the eye.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the exactly the modern idea (with the aid of computers, we often space these &quot;dashes&quot; uniformly along a rectangular grid, but I suppose that drawing them along isoclines only can also be considered modern). Moreoever, modern terminology is employed (perhaps even coined).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;The evidence presented above appears to indicate that the modern conception of a slope field emerged between 1912 and 1917.&lt;/em&gt; Of course, there is the (somewhat unlikely) possibility that Runge used the modern form but did not present it in his book on graphical methods. There is also the possibility that Runge didn't use slope fields in their modern form, but others did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Updated Questions&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Can anyone find an instance earlier than 1917, or any other corroborating evidence that modern slope fields did in fact originate close to the year 1917? The evidence is suggestive but not yet convincing. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For the sake of completeness, it would be nice to see early instances of slope fields with segments uniformly distributed across a rectangular grid, as is now common.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thanks for everyone's help so far!&lt;/p&gt;&#xA;" OwnerUserId="9529" LastEditorUserId="9529" LastEditDate="2019-01-31T09:26:14.200" LastActivityDate="2019-01-31T09:26:14.200" Title="Earliest Instances of a Slope/Direction Field for a First-Order ODE" Tags="|mathematics|reference-request|calculus|differential-equations|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8244" PostTypeId="1" AcceptedAnswerId="8252" CreationDate="2019-01-29T10:09:03.770" Score="1" ViewCount="224" Body="&lt;p&gt;In reference [1], Gelman &lt;em&gt;et al.&lt;/em&gt; give an account of one of the first occurrence of Bayesian inference. Laplace used the number of births by gender in Paris from 1745 to 1770 to conclude that the probability that boys have a higher probability to be born than girls is about &lt;span class=&quot;math-container&quot;&gt;$1.15\times 10^{-42}$&lt;/span&gt;, that is, the data he was using showed with extreme confidence that one is more likely to give birth to a girl than a boy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, &lt;span class=&quot;math-container&quot;&gt;$10^{-42}$&lt;/span&gt; is a remarkably small number. Not only for a probability, but also for any physical quantity in SI units (or any unit I know, for that matter). I believe the closest example would be Planck's constant at &lt;span class=&quot;math-container&quot;&gt;$10^{-34} [J\cdot s]$&lt;/span&gt;, and that is still several orders of magnitude off. And so I wondered, was this, at the time, the smallest non-zero number ever encountered?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, my question is: &lt;strong&gt;throughout the history of science and mathematics, what were the smallest &lt;em&gt;finite&lt;/em&gt; non-zero numbers (in absolute value) encountered as the result of a legitimate calculation?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] &lt;em&gt;A. Gelman, J.B. Carlin, H. S. Stern and D. B. Rubin&lt;/em&gt;, &lt;strong&gt;Bayesian Data Analysis&lt;/strong&gt;, 2nd edition (2004) pp 34-36 &lt;/p&gt;&#xA;" OwnerUserId="8086" LastEditorUserId="8086" LastEditDate="2019-01-30T08:19:10.667" LastActivityDate="2019-02-25T01:45:20.857" Title="What were the smallest numbers encountered in history?" Tags="|records|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="8245" PostTypeId="1" CreationDate="2019-01-29T10:09:47.490" Score="6" ViewCount="213" Body="&lt;p&gt;The United States adopted the international mile and foot in 1959, but for some reason they decided to keep alongside the older units as &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mile#U.S._survey_mile&quot; rel=&quot;nofollow noreferrer&quot;&gt;survey mile&lt;/a&gt;&lt;/em&gt; and &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Foot_(unit)#Survey_foot&quot; rel=&quot;nofollow noreferrer&quot;&gt;survey foot&lt;/a&gt;&lt;/em&gt;. These units are &lt;em&gt;almost&lt;/em&gt; equal to the first two, but &quot;almost&quot; means &quot;problem&quot; when it comes to measuring large distances, or areas (when the units are squared), and comparing measurements. I understand that some people did not want to bother with conversion of old measurements, but that was sixty years ago and the National Geodetic Survey wasn't sleeping since then. So why do they keep both units?&lt;/p&gt;&#xA;" OwnerUserId="9530" LastEditorUserId="1930" LastEditDate="2019-02-03T11:54:09.217" LastActivityDate="2021-11-02T20:24:10.873" Title="Why does the US keep survey miles and feet?" Tags="|units|measurement|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8246" PostTypeId="5" CreationDate="2019-01-29T10:12:30.557" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2019-01-29T10:12:30.557" LastActivityDate="2019-01-29T10:12:30.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8247" PostTypeId="4" CreationDate="2019-01-29T10:12:30.557" Score="0" Body="This tag is dedicated to questions about historical world records and milestone achievements in science and mathematics." OwnerUserId="8086" LastEditorUserId="8086" LastEditDate="2019-02-03T11:54:03.780" LastActivityDate="2019-02-03T11:54:03.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8248" PostTypeId="2" ParentId="8245" CreationDate="2019-01-29T12:15:45.947" Score="0" Body="&lt;p&gt;This is more of a political question than scientific, I fear.   The USA is really bad at standardization.  The US Navy, for example, still insists on knots,  fathoms, nautical miles, etc., even though their hardware is mostly full of metric fasteners and dimensions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far as survey-X units go, I wouldn't be surprised to find that they are in use so that all the previous benchmarks (stones hammered into the ground, mostly) don't need to be moved.  ... Peeking around, there's the &quot;acre&quot; and then there's the &quot;acre (Cheshire)&quot; which are radically different.  And hectares, and rods, and furlongs....&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rather as the UK is officially on kg but people still use &quot;stone&quot; , I fear no culture ever gives up its semitraditional units. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-01-29T12:15:45.947" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8249" PostTypeId="2" ParentId="8242" CreationDate="2019-01-29T12:31:44.093" Score="1" Body="&lt;p&gt;Perhaps we should reviewwhat &quot;proof&quot; and &quot;law&quot; mean in Physics.  A 'law' is, as you suggest, similar to a mathematical axiom.  The difference is that physics-laws a selected to be compatible with our observable universe &lt;em&gt;and&lt;/em&gt; that we don't generally find much use for investigating the results of choosing an alternative axiom (compare with nonEuclidean geometry, which was of purely theoretical use for a long time).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And for &quot;proof,&quot; strictly speaking we are trying to show that all observable and testable behavior is consistent with the proposed axiom.   Again, in mathematics, an axiom is just a starting point for constructing a system (think of it as a mathematical universe), but in physics there's not a lot of work &amp;lt;&amp;lt;\cough&gt;&gt;string theory&amp;lt;&amp;lt;\cough&gt;&gt; done with non-real-universe axioms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For Ampere &amp;amp; Maxwell &amp;amp; Gauss, then, the 4 E-M equations were born of experiment, observation, and deep insight, after which a zillion experiments validated their utility.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-01-29T12:31:44.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8250" PostTypeId="1" AcceptedAnswerId="8253" CreationDate="2019-01-29T12:56:33.157" Score="1" ViewCount="98" Body="&lt;p&gt;I'm doing a project on the history of George Boole and wanted some background on how notation has changed through the ages, however do not understand Vives logical notation. Wondering if someone could help me, thank you.&lt;/p&gt;&#xA;" OwnerUserId="9531" LastEditorUserId="131" LastEditDate="2019-01-30T07:44:27.943" LastActivityDate="2019-01-30T15:38:59.180" Title="Could someone explain Vives logical notation?" Tags="|logic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8251" PostTypeId="1" CreationDate="2019-01-30T07:29:18.583" Score="0" ViewCount="192" Body="&lt;p&gt;I'm having a hard time gathering my thoughts about this.&#xA;I'm trying to find a connection or some sort of relation between the first 3 axioms (postulates) of Euclidean geometry (though around Aristotle's time it wasn't called Euclidean geometry yet, but the concepts were there) and Aristotle's theory of elements and motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that Aristotle believed that elements move on a straight line according to their 'weight' (heavy to light), and that objects have 2 natural movements, towards the center of the earth (downwards) or further from the center of the earth (upwards). So that I think kind of covers the first geometrical axiom, of a straight line between 2 points. What about the other two?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware of his adoption of Eudoxus geometrical planetary movements model and his uniform circular motion of the planets, but I can't seem to think of a way to connect them in any way to the geometrical axioms.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any ideas, guys?&lt;/p&gt;&#xA;" OwnerUserId="9534" LastActivityDate="2019-03-01T12:02:35.493" Title="What's the relationship between Aristotle's theory of elements and motion and geometry?" Tags="|physics|astronomy|euclidean-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8252" PostTypeId="2" ParentId="8244" CreationDate="2019-01-30T08:53:44.097" Score="3" Body="&lt;p&gt;There is a problem with using dimensional magnitudes in this context. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck_length&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planck length&lt;/a&gt; in meters, the unit of length that light travels in Planck time, beats the Planck constant, it is &lt;span class=&quot;math-container&quot;&gt;$1.616229(38)×10^{−35}$&lt;/span&gt; meters. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck_time&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planck time&lt;/a&gt; beats even the probability in the OP, it is &lt;span class=&quot;math-container&quot;&gt;$5.39116(13)×10^{−44}$&lt;/span&gt; seconds. And one can beat this easily by choosing milliseconds or nanoseconds as units. A better, and even smaller, example is the dimensionless &lt;a href=&quot;https://en.wikipedia.org/wiki/Gravitational_coupling_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;gravitational coupling constant&lt;/a&gt;, the squared ratio of the electron mass to the Planck mass, &lt;span class=&quot;math-container&quot;&gt;$\alpha_G\approx1.7518×10^{−45}$&lt;/span&gt;. It measures the strength of gravitational attraction between two electrons in Planck scale terms. Its electrostatic analog is the better known, and much bigger, &lt;a href=&quot;https://en.wikipedia.org/wiki/Fine-structure_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;fine structure constant&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\alpha\approx1/137.036$&lt;/span&gt;, hence the idea that &quot;gravity is much weaker than electromagnetism&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probabilities and mathematical constants are also better because they are dimensionless. But &quot;smallest non-zero in absolute value&quot; does not rule out infinitesimals, liberally used by Fermat, Leibniz and Euler in their calculations, and, in recent times, by Robinson and Conway. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Surreal_number#Overview&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conway's scheme&lt;/a&gt; even allows to individuate some of them, like his constant &lt;span class=&quot;math-container&quot;&gt;$ε:=\{ 0 | 1, 1/2, 1/4, 1/8,…\}$&lt;/span&gt;, and perform calculations on them, similar to the ones in positional systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even if we stick to reals or rationals the precision estimates in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Approximations_of_%CF%80&quot; rel=&quot;nofollow noreferrer&quot;&gt;Approximations of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&lt;/a&gt; easily beat all of the above (other than &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;). Wikipedia even makes a point of mentioning that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;the circumference of the largest known object, the observable universe, can be calculated from its diameter (93 billion light-years) to a precision of less than one Planck length... using &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; expressed to just 62 decimal places&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Vega calculated first 140 decimal places, of which the first 126 were correct, already back in 1789. That would be to &lt;span class=&quot;math-container&quot;&gt;$10^{-126}$&lt;/span&gt; precision. As of November 2016, the record precision is &lt;span class=&quot;math-container&quot;&gt;$10^{-22,459,157,718,361 (π^e×10^{12})}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-02-25T01:45:20.857" LastActivityDate="2019-02-25T01:45:20.857" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8253" PostTypeId="2" ParentId="8250" CreationDate="2019-01-30T10:56:31.030" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://plato.stanford.edu/entries/vives/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Juan Luis Vives (1493–1540)&lt;/a&gt; was a Spanish humanist and (like other Humanists : Valla, Ramus) was a harsh critic of medieval formal logic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of his works, relevant for the history of logic are : &lt;em&gt;Adversus pseudodialecticos&lt;/em&gt; (1520) [see R. Guerlac, &lt;a href=&quot;https://books.google.it/books?id=BP70CAAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Juan Luis Vives Against the Pseudodialecticians : A Humanist Attack on Medieval Logic, Springer (1979)&lt;/a&gt; for the English edition] and &lt;a href=&quot;https://books.google.it/books?id=Rxg8AAAAcAAJ&amp;amp;pg=PA6&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De disciplinis libri XX&lt;/em&gt; (1531)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The triangular diagram can be found into : &lt;a href=&quot;https://books.google.co.in/books?id=FJbmaidjCk4C&amp;amp;pg=PA171&quot; rel=&quot;nofollow noreferrer&quot;&gt;De censura veri&lt;/a&gt;, page 171, regarding &lt;em&gt;Comparationes&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;si omne &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; est &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, et omne &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; est &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, omne &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; est &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2019-01-30T15:38:59.180" LastActivityDate="2019-01-30T15:38:59.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8254" PostTypeId="2" ParentId="8251" CreationDate="2019-01-30T11:23:17.047" Score="2" Body="&lt;p&gt;&lt;em&gt;On the Heavens&lt;/em&gt;, bk.1 (chap.2) , is perhaps the place to read carefully. Simple motions are of two types, radial and circular, and combinations of them produce any other motion.  &quot;&quot;Radial&quot; motion is further described as &lt;em&gt;to&lt;/em&gt; the center and &lt;em&gt;from&lt;/em&gt; the center, while the movement around a center is &quot;without an opposite&quot;. Circular motion pertains to the &lt;em&gt;first&lt;/em&gt; element, which later somehow became the &lt;em&gt;fifth&lt;/em&gt; one (quint-essence). It is inexhausible, eternal and does not need a mover, at least in this work.&#xA;Following Eudoxus, Aristotle devised a mechanical model with one self rolling outer sphere whose movement is transmitted through a complicated system of homocentric smaller spheres down to the lunar sphere.&#xA;The world being finite, there are not actually arbitrarily long lines. Physics is the science about Nature (&lt;em&gt;physis&lt;/em&gt;), and it is definitely &lt;em&gt;not Geometry&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2019-01-30T11:29:43.897" LastActivityDate="2019-01-30T11:29:43.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8255" PostTypeId="2" ParentId="8243" CreationDate="2019-01-30T11:39:14.577" Score="5" Body="&lt;p&gt;Johann Bernoulli explains the idea of a direction field quite explicitly (Modus generalis construendi omnes aequationes differentiales primi gradus, Acta Eruditorum, November 1694). He focusses on drawing isoclines rather than slope segments. There is no figure in that work but Bernoulli drew an example in his correspondence:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8lvaA.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8lvaA.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Corresponding to:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eGhLS.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eGhLS.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-01484569&quot; rel=&quot;noreferrer&quot;&gt;Dominique Tournès, Résolution graphique des équations différentielles&lt;/a&gt;, discusses this and many other historical examples.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2019-01-30T11:39:14.577" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8256" PostTypeId="1" AcceptedAnswerId="8264" CreationDate="2019-01-30T23:27:38.860" Score="11" ViewCount="501" Body="&lt;p&gt;Often professors in math and physics academia have inboxes full of people claiming to have solved deep problems such as dark matter, black holes, prime number conjectures and claim that many big names were actually wrong. Most of these &quot;cranks&quot; employ pseudo-scientific/mathematical arguments that often have no actual bearings on the subject.  My question is how long has this been a trait of scientific academia. Did physicists like Einstein and Fermi have to sort through this type of spam?&lt;/p&gt;&#xA;" OwnerUserId="8110" LastActivityDate="2019-02-05T12:54:13.270" Title="What are some of the earliest mentions of scientific &quot;cranks&quot;?" Tags="|social-context|examples|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8258" PostTypeId="2" ParentId="8244" CreationDate="2019-01-31T15:06:44.310" Score="5" Body="&lt;p&gt;Below are some examples of possible interest from my 8 April 2002 sci.math post &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=380011&quot; rel=&quot;noreferrer&quot;&gt;BIG NUMBERS #1&lt;/a&gt;. Many of these are things that I had jotted down in personal notes back in the 1970s and early 1980s, and I believe all of them (except the Besicovitch example and the ASCII google groups example) are in a lengthy handwritten manuscript on cardinal numbers, order types, and ordinal numbers that I wrote in Summer 1985.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Something else of possible interest to estimate is discussed &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=380016&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. Also, &lt;a href=&quot;http://mathforum.org/kb/message.jspa?messageID=380022&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; I wrote: &lt;em&gt;I suppose we could estimate the odds that someone will live long enough to think of every possible thought while waiting to die of statistical asphyxiation&lt;/em&gt; [of the Gamow type]. &lt;em&gt;Seeing as how it's&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; &lt;em&gt;out of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$10^{10^{36}}$&lt;/span&gt; &lt;em&gt;to live at least&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1000$&lt;/span&gt; &lt;em&gt;years, this might get us up to around&lt;/em&gt; 10^^5 &lt;span class=&quot;math-container&quot;&gt;$[ = \sideset{_{}^5}{}{10} = 10^{10^{10^{10^{10}}}}$&lt;/span&gt;] &lt;em&gt;for a number that has some remote significance to the physical world.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5.53 \times 10^{-67}$&lt;/span&gt; -- The gravitational force (in Newtons) between two electrons that are &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; cm apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$6.18 \times 10^{-103}$&lt;/span&gt; -- The gravitational force (in Newtons) between two electrons that are &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; light year apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-200}$&lt;/span&gt; -- This is the probability that an electron in a 1s orbital of a hydrogen atom is at least &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; nanometers from the nucleus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-1080}$&lt;/span&gt; -- This is the probability of flipping a coin once a second for an hour and getting all heads.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-2576}$&lt;/span&gt; --- This is a number that appeared in a paper by a 1928 paper by Besicovitch and that I discussed &lt;a href=&quot;https://math.stackexchange.com/a/98224/13130&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-42,000}$&lt;/span&gt; -- This is the probability of a monkey typing Hamlet by chance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-10^8}$&lt;/span&gt; -- This is the probability of flipping a coin once a second for &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; years and getting all heads.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-2.9 \times 10^{12}}$&lt;/span&gt; -- This is the Yukawa nuclear force between two nucleons that are &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; cm apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-10^{13}}$&lt;/span&gt; -- This is the probability that by randomly typing ASCII characters, a long-lived monkey will type every post, in a pre-selected order, that appears at the Google Groups archive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-5.2 \times 10^{18}}$&lt;/span&gt; -- Using non-relativistic quantum mechanics, this is the probability that an electron in a 1s orbital of a hydrogen atom is at least &lt;span class=&quot;math-container&quot;&gt;$240,000$&lt;/span&gt; miles from the nucleus (distance to the Moon).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-3 \times 10^{26}}$&lt;/span&gt; -- Gamow observes (see Chapter 8.4 of [43]) there is a nonzero probability that all of the air molecules in a room will collect together simultaneously on one side of the room, namely &lt;span class=&quot;math-container&quot;&gt;$(1/2)^n$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$n = 10^{27}$&lt;/span&gt; is the number of air molecules in the room.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-10^{36}}$&lt;/span&gt; -- The probability given by life insurance calculations that someone will live &lt;span class=&quot;math-container&quot;&gt;$1000$&lt;/span&gt; years. [This appears at the beginning of Chapter 1 in Volume I of William Feller's text PROBABILITY THEORY AND ITS APPLICATIONS. Feller writes: &quot;We hesitate to admit that man can grow &lt;span class=&quot;math-container&quot;&gt;$1000$&lt;/span&gt; years old, and yet current actuarial practice admits no such limit to life. According to formulas on which modern mortality tables are based, the proportion of men surviving &lt;span class=&quot;math-container&quot;&gt;$1000$&lt;/span&gt; years is of the order of magnitude of one in &lt;span class=&quot;math-container&quot;&gt;$10^{10^{36}}$&lt;/span&gt; -- a number with &lt;span class=&quot;math-container&quot;&gt;$10^{27}$&lt;/span&gt; billions of digits.&quot;]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-10^{36}}$&lt;/span&gt; -- Using non-relativistic quantum mechanics, this is also the probability that an electron in a 1s orbital of a hydrogen atom is at least &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; billion light years away from the nucleus. (Note: The most distant objects that have been observed are &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_the_most_distant_astronomical_objects&quot; rel=&quot;noreferrer&quot;&gt;a little over &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; billion light years distant&lt;/a&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-2.7 \times 10^{40}}$&lt;/span&gt; -- This is the Yukawa nuclear force between two nucleons that are &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; billion light years apart.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-10^{93}}$&lt;/span&gt; -- Using non-relativistic quantum mechanics, this is the probability that every electron in all the hydrogen atoms in the sun are at least &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; billion light years away.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$10^{-10^{166}}$&lt;/span&gt; --- This is the probability of &quot;randomly typing&quot; the entire space-time universe by selecting the correct particle distribution (out of the &lt;span class=&quot;math-container&quot;&gt;$10^{10^{125}}$&lt;/span&gt; total permutations that all elementary particles in the universe (assumed to be distinct) can be distributed into the particle-sized locations in the universe at any given time), for each of the &lt;span class=&quot;math-container&quot;&gt;$10^{41}$&lt;/span&gt; instants of time since the Big Bang (&quot;instant&quot; equals the time it takes light to travel the diameter of an atomic nucleus).&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2019-01-31T15:06:44.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8259" PostTypeId="2" ParentId="8256" CreationDate="2019-01-31T15:16:27.253" Score="6" Body="&lt;p&gt;In the foreword to G.H.Hardy's &quot;A mathematician's apology&quot; C.P.Snow writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;About his discovery of Ramanujan, he [Hardy] showed &#xA;  no secrecy at all. It was, he wrote, the one romantic incident in his life : anyway, it is an admirable &#xA;  story, and one which showers credit on nearly &#xA;  everyone (with two exceptions) in it. One morning &#xA;  early in 1913, he found, among the letters on his &#xA;  breakfast table, a large untidy envelope decorated &#xA;  with Indian stamps. When he opened it, he found &#xA;  sheets of paper by no means fresh, on which, in a &#xA;  non-English holograph, were line after line of &#xA;  symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, &#xA;  a world famous mathematician : &lt;strong&gt;and world famous &#xA;  mathematicians, he had already discovered, are &#xA;  unusually exposed to cranks&lt;/strong&gt;. He was accustomed &#xA;  to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the &#xA;  revelations of the Elders of Zion, or the cryptograms that Bacon had inserted in the plays of the &#xA;  so-called Shakespeare. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;p.29f in the 1967 CUP edition, my emphasis&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This suggests that this phenomenon occurs as early as 1913. According to Snow, Hardy himself has written this story, but Snow does not provide any references. As far as I can tell, it is not from &quot;A mathematician's apology&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's works were published in the same period, so he has probably encountered the same phenomenon.&lt;/p&gt;&#xA;" OwnerUserId="9456" LastActivityDate="2019-01-31T15:16:27.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8260" PostTypeId="1" CreationDate="2019-01-31T17:32:08.870" Score="0" ViewCount="71" Body="&lt;p&gt;Concerning the role of mathematics in technological inventions: which books would you suggest that examine the historical relation between mathematical analysis &amp;amp; practical wisdom? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if Leonardo Da Vinci had known Newton's laws of motion, would he design much more effective machines, or would he continue to work through trial-and-error guided by his own genius?&lt;/p&gt;&#xA;" OwnerUserId="5909" LastActivityDate="2019-01-31T17:32:08.870" Title="Mathematical analysis vs. Practical genius" Tags="|mathematics|technology|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8262" PostTypeId="1" CreationDate="2019-01-31T18:28:33.877" Score="2" ViewCount="64" Body="&lt;p&gt;Last night I left a bottle of cleanser in my car, and during the 5-degree night the cleanser turned to a sludge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This started me thinking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lots of people living before the development of the thermometer probably noticed that on cold nights some liquids freeze but not others, and that liquids can be put in order by freezing point such that all of those liquids to the left of some point on the spectrum will freeze while all of those to the right remain liquid. I imagine fresh water, sea water, medicines, juices and alcoholic drinks might have been interesting examples, of which many people had plenty of experience.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do we have written record of this train of thought in the ancient world?&lt;/p&gt;&#xA;" OwnerUserId="9545" LastActivityDate="2019-01-31T19:46:17.193" Title="In the world before thermometers did people use freezing points as a sort of proto-thermometer?" Tags="|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8263" PostTypeId="2" ParentId="8262" CreationDate="2019-01-31T19:46:17.193" Score="2" Body="&lt;p&gt;Not really. Using multiple liquids to register a single temperature probably would have been impractical, but crude thermometers appear already in antiquity. Philo of Byzantium c. 240 BC used a tube from a hollow sphere to a jug of water. When the sphere was in the sun bubbles were released as air expanded out of the sphere, when moved to the shade, the water rose in the tube as air in the sphere contracted. Heron c.50 AD described similar devices in Pneumatica, and it is known that such pioneers of modern thermometers as Galileo and Fludd read Heron. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galen c. 170 AD came even closer. He took boiling water as the top and melting ice as the bottom of the scale, with their mixture in equal parts as the neutral point. He then inserted four degrees above and below the neutral point by using the corresponding mixtures, thus making a nine-point scale, see &lt;a href=&quot;http://www.eoht.info/page/Thermometer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thermometer on Hmolpedia&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-01-31T19:46:17.193" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8264" PostTypeId="2" ParentId="8256" CreationDate="2019-01-31T19:50:27.253" Score="13" Body="&lt;p&gt;There are lots of references to cranks in &lt;em&gt;A Budget of Paradoxes&lt;/em&gt;, by Augustus de Morgan (1806–1871), who calls them &amp;ldquo;paradoxers&amp;rdquo;. There, he writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[&amp;hellip;] I say something on my personal knowledge of the class of discoverers that square the circle, upset Newton, etc. I suspect I know more of the English class than any man in Britain. I never kept any reckoning; but I know that one year with another &amp;mdash; and less of late years than in earlier time &amp;mdash; I have talked to more than five in each year, giving more than a hundred and fifty specimens. Of this I am sure, that it is my own fault if they have not been a thousand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-01-31T19:50:27.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8265" PostTypeId="1" CreationDate="2019-01-31T22:45:11.797" Score="11" ViewCount="1380" Body="&lt;p&gt;As discussed &lt;a href=&quot;https://hsm.stackexchange.com/questions/6295/how-did-eratosthenes-knew-the-exact-time-of-the-day&quot;&gt;over here&lt;/a&gt;, Eratosthenes measured the earth’s circumference by comparing shadows cast at apparent noon at two locations separated by a known distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although accounts of the event (like the one cited above, and &lt;a href=&quot;http://www.classichistory.net/archives/eratosthenes-circumference-of-earth&quot; rel=&quot;noreferrer&quot;&gt;this one&lt;/a&gt;, and &lt;a href=&quot;https://www.nationalgeographic.org/thisday/jun19/eratosthenes-measures-circumference-earth/&quot; rel=&quot;noreferrer&quot;&gt;this one&lt;/a&gt;) do not emphasize the issue, it seems that Eratosthenes’ method depended on an assumption that the two measurements were on the same longitude. (When I think through the claim, I don't understand how any inference would be possible if we did not make such an assumption.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How was this known?&lt;/p&gt;&#xA;" OwnerUserId="9545" LastEditorUserId="55" LastEditDate="2019-02-01T09:50:18.150" LastActivityDate="2024-02-12T04:49:59.103" Title="How did Eratosthenes determine that Alexandria and Syene were on the same meridian?" Tags="|astronomy|geometry|geography|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8266" PostTypeId="1" AcceptedAnswerId="8267" CreationDate="2019-02-01T03:10:35.487" Score="7" ViewCount="355" Body="&lt;p&gt;Discussion below the question &lt;a href=&quot;https://earthscience.stackexchange.com/q/16115/6031&quot;&gt;Does the US National Weather Service use Celsius or Fahrenheit?&lt;/a&gt; and correction of my original use of &amp;quot;Centigrade&amp;quot; to the modern Celsius lead me here.&lt;/p&gt;&#xA;&lt;p&gt;For example, in the merriam-webster.com's &lt;a href=&quot;https://www.merriam-webster.com/dictionary/centigrade&quot; rel=&quot;noreferrer&quot;&gt;entry for centigrade&lt;/a&gt; it says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Did You Know?&lt;/p&gt;&#xA;&lt;p&gt;The centigrade scale &lt;strong&gt;is essentially identical&lt;/strong&gt; to the Celsius scale, the standard scale by which temperature is measured in most of the world. Anders Celsius of Sweden first devised the centigrade scale in the early 18th century. But in his version, 100° marked the freezing point of water, and 0° its boiling point. Later users found it less confusing to reverse these two. To convert Fahrenheit degrees to centigrade, subtract 32 and multiply by 5/9. To convert centigrade to Fahrenheit, multiply by 9/5 and add 32.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;seems to equivocate by using the term &amp;quot;essentially&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia's page for &lt;a href=&quot;https://en.wikipedia.org/wiki/Centigrade&quot; rel=&quot;noreferrer&quot;&gt;Centigrade&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Centigrade, a historical forerunner to the Celsius temperature scale, synonymous in modern usage&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;comes closer since synonymous suggests (but doesn't explicitly say) that the two are numerically identical.&lt;/p&gt;&#xA;&lt;p&gt;If they are both linear scales, and have the same two fixed (and non-identical) points, perhaps there's a mathematical proof that they are identical?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; When and how did usage of the term Centigrade give way to Celsius? Are/were they in fact numerically identical?&lt;/p&gt;&#xA;&lt;p&gt;When did &amp;quot;Later users (find) it less confusing to reverse these two.&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-02-01T03:50:28.957" Title="When and how did usage of the term Centigrade give way to Celsius? Are/were they in fact numerically identical?" Tags="|thermodynamics|units|measurement|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8267" PostTypeId="2" ParentId="8266" CreationDate="2019-02-01T03:50:28.957" Score="3" Body="&lt;p&gt;The unabridged Oxford English dictionary clarifies the &quot;when&quot; part of the question. Also consult the paper in Nature (1949) &lt;a href=&quot;https://www.nature.com/articles/163427a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nature.com/articles/163427a0&lt;/a&gt; for the historic discussion (paper is open access). However, the modern reference point is triple point of water, &quot;The Celsius scale is defined using the T.Pt. = 0.01 °C with 1 °C made identical in size to 1 K.&quot; &lt;a href=&quot;http://www1.lsbu.ac.uk/water/water_properties.html#d&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www1.lsbu.ac.uk/water/water_properties.html#d&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;em&gt;OED&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Celsius described a scale in which 0 was the boiling point of water and 100 the freezing point ( Observationer om twänne beständiga Grader på en Thermometer (1742)): this is the inverse of the scale subsequently called the Celsius scale. Linnaeus used a thermometer calibrated with 0 as the freezing point and 100 as the boiling point ( Hortus Upsaliensis (1745) 16 Dec.).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Prior to Celsius, a thermometric scale of 100 degrees had been proposed by P. Elvius ( Dissertatio physico-mechanica de thermometris (1710) vii. 35).&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;In scientific use, the term Celsius was formally adopted in place of centigrade in 1948 (see quot. 1949 at sense B.); see the note at centigrade adj. 1a. The Celsius scale was redefined in 1954 in terms of the Kelvin scale, by the relationship °C = °K − 273.15, thereby effectively replacing the freezing point of water at 0°C as a fixed point by the triple point of water at 0.01°C. (see the note at Kelvin n. 4a).&quot;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-02-01T03:50:28.957" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8269" PostTypeId="2" ParentId="8265" CreationDate="2019-02-01T09:47:39.730" Score="14" Body="&lt;p&gt;We do not know. First, Syene and Alexandria are not on the same meridian, Alexandria is about 3° to the West, and second, Syene is not on the tropic (where the Sun is straight up on the summer solstice) either, which is another assumption Eratosthenes makes, according to Cleomedes.&lt;/p&gt;&#xA;&lt;p&gt;Our primary source on Eratosthenes's Geographica, where these assumptions were, presumably, justified, is, unfortunately, Strabo, whose mastery of geometry leaves much to be desired. Here is what Russo writes in &lt;a href=&quot;https://msp.org/memocs/2013/1-1/memocs-v1-n1-p04-p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy's longitudes and Eratosthenes' measurement of the Earth's Circumference&lt;/a&gt; on his account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Strabo writes in a couple of passages that according to Eratosthenes the Nile flows along the meridian from Syene to Alexandria, but in the same passages the Nile is described as flowing along the same meridian even from Meroë to Syene, while in the book devoted to Egypt, still quoting Eratosthenes, he describes the path of the Nile as far from being a north-south straight line and in more than one instance Strabo appears to confuse distances with their orthogonal projections along a meridian.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If we take Strabo at face value, Eratosthenes could have taken the word of mouth that sailing down the Nile was northward, which could be roughly determined by watching the Sun and the Polaris.  Just as he took the word of mouth that the Sun was straight up in Syene on the summer solstice.&lt;/p&gt;&#xA;&lt;p&gt;But even Shcheglov, who disagrees with Russo on Eratosthenes, is equally unkind to Strabo, see his &lt;a href=&quot;https://grbs.library.duke.edu/article/viewFile/361/441&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hipparchus on the Latitude of Southern India&lt;/a&gt;. And the above does not align well with Eratosthenes's reputation as a meticulous geometer and geographer, who, as the head of Alexandrian library, had a corps of surveyors at his disposal. In the &lt;a href=&quot;https://books.google.com/books/about/The_Forgotten_Revolution.html?id=MOTpnfz7ZuYC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Forgotten Revolution&lt;/a&gt; Russo questions the accuracy of Cleomedes's account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Do we know for sure that Eratosthenes assumed that Alexandria and Syene lay on the same meridian and that Syene was on the tropic? The primary source on the subject, Cleomedes, actually wrote:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Eratosthenes' method, being geometric, seems more difficult [than&#xA;the previously explained method of Posidonius]. What he says will&#xA;become clearer if we allow ourselves to make two assumptions. We&#xA;assume first that Alexandria and Syene are on the same meridian.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Cleomedes does not give a detailed account of the method, that would be pointless, Eratosthenes' work still being available, but a pedagogical précis meant for readers scared of the complex geometric arguments of the original work. He attains his goal of explaining Eratosthenes' method by taking an ideal case obtained by eliminating all technical difficulties; how else could he compress into three pages a work that occupied two books? Cleomedes also rounds off the numbers, evidently so as not to bother the reader with calculations inessential to an understanding of the method... Cleomedes and other authors probably name Syene because it was the Egyptian town closest to the tropic and the most convenient base for an expedition to the tropic. As for the well whose bottom was lit by the sun at the solstice, Pliny says that it was dug out for a demonstration.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Eratosthenes &lt;em&gt;could&lt;/em&gt; have his surveyors do a version of &lt;a href=&quot;https://en.wikipedia.org/wiki/Dead_reckoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;dead reckoning&lt;/a&gt;, i.e. track the angles with the North and the distances traveled in the meandering of the Nile, to determine the longitude more accurately, but his map does draw a meridian through Syene and Alexandria. Then again, this map comes down to us through none other than Strabo, see &lt;a href=&quot;https://www.jstor.org/stable/41133648&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Eratosthenes-Strabo Nile Map by Rawlins&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We do know that a century later, in a treatise characteristically titled &lt;em&gt;Against the ‘Geography’ of Eratosthenes&lt;/em&gt; (see &lt;a href=&quot;https://arxiv.org/pdf/1412.3150.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analysis of the latitudinal data of Eratosthenes and Hipparchus by Marx&lt;/a&gt; on related issues), Hipparchus proposed an astronomical method for measuring longitudes. It relies on comparing absolute and local times, with the former measured by marking when a lunar eclipse began and ended with an accurate clock.&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy &lt;em&gt;implied&lt;/em&gt; that he used this method to determine the longitudes of multiple locations in his &lt;em&gt;Geographike Syntaxis&lt;/em&gt;, but his data suggests that he used Roman itineraries (road distance measurements) instead, along with a widely off value of Earth's circumference. The idea will be vigorously pursued in the early modern times, when what came to be known as &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_longitude#Problem_of_longitude&quot; rel=&quot;nofollow noreferrer&quot;&gt;the problem of longitude&lt;/a&gt; was ultimately solved by designing chronometers, see also &lt;a href=&quot;https://hsm.stackexchange.com/a/2104/55&quot;&gt;When was the issue of time zones at different longitudes first described?&lt;/a&gt; However, those were not available in antiquity.&#xA;Russo's alternative speculation runs as follows&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Fixing longitude has always been much harder, until the advent of chronometers and radio, and we don't know what method Eratosthenes might have employed. Possibly it was the one mentioned by Ptolemy at the beginning of his work, whereby one finds the difference in longitude between two places by determining the difference in latitude and estimating the angle that the line between the two makes with the meridian. In the case of cities linked by a sea to the West lane, this information would have been known approximately to seafarers&#xA;plying the route.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As I mentioned, Russo's speculations are not without detractors, see e.g. &lt;a href=&quot;https://www.journals.uchicago.edu/doi/full/10.1086/689763&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shcheglov's  Accuracy of Ancient Cartography Reassessed&lt;/a&gt; for a much more skeptical view of Eratosthenes's work.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="18903" LastEditDate="2024-02-12T04:49:59.103" LastActivityDate="2024-02-12T04:49:59.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8270" PostTypeId="1" CreationDate="2019-02-01T21:35:49.757" Score="1" ViewCount="927" Body="&lt;p&gt;&quot;I recently came across a text where the electric dipole moment was represented as &quot;&lt;strong&gt;d&lt;/strong&gt;&quot; instead of the more commonly used &quot;&lt;strong&gt;p&lt;/strong&gt;&quot;. The reason behind using &quot;&lt;strong&gt;d&lt;/strong&gt;&quot; here is probably that &quot;&lt;em&gt;d&lt;/em&gt;ipole moment&quot; begins with a &quot;d&quot;. However, it also made me wonder why the most commonly used symbol of electric dipole moment is &quot;&lt;strong&gt;p&lt;/strong&gt;&quot; and not &quot;&lt;strong&gt;d&lt;/strong&gt;&quot;. What is the reason behind using &quot;&lt;strong&gt;p&lt;/strong&gt;&quot; to represent electric dipole moment?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, is there any historical reason for this?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I originally asked the above question at the Physics SE (&lt;a href=&quot;https://physics.stackexchange.com/q/458157&quot;&gt;https://physics.stackexchange.com/q/458157&lt;/a&gt;). There, a user (@Avantgarde) suggested that I should ask this question here too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, another user (@frapadingue) commented that this &quot;&lt;strong&gt;p&lt;/strong&gt;&quot; is from &quot;p&quot; in &lt;em&gt;p&lt;/em&gt;olarization. This led me to wonder about the following questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Historically, what was studied first - electric polarization in materials or the concept of electric dipole moment? If the concept of electric dipole moment was studied first, then, was the representation of this vector quantity changed later when polarization was discovered?&lt;/em&gt;  &lt;/p&gt;&#xA;" OwnerUserId="9551" LastActivityDate="2019-02-04T16:02:16.570" Title="Why is electric dipole moment represented by “p”?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8271" PostTypeId="1" CreationDate="2019-02-02T02:46:20.437" Score="1" ViewCount="108" Body="&lt;p&gt;The concept of relative atomic weight originated from measuring the combining weight of hydrogen with a certain element. In the simplification process H was taken as unity (18th, 19th and 20th century). For example from combining weights of hydrogen and oxygen, one can show that the weight of oxygen is almost 16 times more than H. Chemists did not have any means to measure absolute masses.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now when Aston invented the mass spectrometer in 1940s, he also measured atomic masses. Aston's book is available here &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.205751/page/n14&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/in.ernet.dli.2015.205751/page/n14&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) As per my understanding, a mass spectrometer can measure the &lt;em&gt;absolute mass&lt;/em&gt; based on charge-to-mass ratio. Provided we know the charge on the ion which is usually one, one can determine the absolute mass without any regard to chemical principles, purely on the principles of electricity and magnetism. I am wondering how come the current masses measured by chemical means turn out to be almost the same, &lt;em&gt;in terms of numerical values&lt;/em&gt;, as measured by an independent instrument such as a mass spectrometer. For example, O-16 isotope is still 16 by MS measurements?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Why don't we quote absolute atomic weights given that modern FT-Ion Cyclotron MS can detect extremely delicate mass changes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-02-02T16:13:31.597" LastActivityDate="2019-02-02T16:13:31.597" Title="Is it a historical coincidence that relative atomic weights by chemical methods and mass spectrometry are very close?" Tags="|physics|terminology|discoveries|philosophy-of-science|chemistry|" AnswerCount="0" CommentCount="9" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8273" PostTypeId="2" ParentId="2223" CreationDate="2019-02-02T20:25:47.293" Score="8" Body="&lt;p&gt;Concerning the notation &lt;span class=&quot;math-container&quot;&gt;$\text{Pr}(|\xi|&amp;gt;\varepsilon)$&lt;/span&gt; here's what I've found so far:&lt;/p&gt;&#xA;&lt;p&gt;Cajori's 1929 &lt;em&gt;A History of Mathematical Notations&lt;/em&gt; says nothing on probability theory, which suggest that the subject had not yet developed any special or widely adopted notation around the beginning of the 20th century. This seems to be supported by Jeff Miller, who &lt;a href=&quot;http://jeff560.tripod.com/stat.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;writes&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Symbols for the probability of an event &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; on the pattern of &lt;span class=&quot;math-container&quot;&gt;$P(A)$&lt;/span&gt; or&#xA;&lt;span class=&quot;math-container&quot;&gt;$Pr(A)$&lt;/span&gt; are a relatively recent development given that probability has&#xA;been studied for centuries. A. N. Kolmogorov's &lt;em&gt;Grundbegriffe der&#xA;Wahrscheinlichkeitsrechnung&lt;/em&gt; (1933) used the symbol &lt;span class=&quot;math-container&quot;&gt;$\mathbf{P}(A)$&lt;/span&gt;. The use of&#xA;upper-case letters for events was taken from set theory where they&#xA;referred to sets. H. Cramér's &lt;em&gt;Random Variables and Probability&#xA;Distributions&lt;/em&gt; (1937), &amp;quot;the first modern book on probability in&#xA;English,&amp;quot; used &lt;span class=&quot;math-container&quot;&gt;$P(A)$&lt;/span&gt;. In the same year J. V. Uspensky (&lt;em&gt;Introduction to&#xA;Mathematical Probability&lt;/em&gt;) wrote simply &lt;span class=&quot;math-container&quot;&gt;$(A)$&lt;/span&gt;, following A. A. Markov&#xA;&lt;em&gt;Wahrscheinlichkeitsrechnung&lt;/em&gt; (1912, p. 179). W. Feller's influential &lt;em&gt;An&#xA;Introduction to Probability Theory and its Applications volume 1&lt;/em&gt;&#xA;(1950) uses &lt;span class=&quot;math-container&quot;&gt;$Pr\{A\}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{P}\{A\}$&lt;/span&gt;in later editions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But digging a little deeper one finds a few earlier sources.&lt;/p&gt;&#xA;&lt;p&gt;Before giving some, let me note that I find Miller's claim that upper-case letters for events were taken from set theory quite dubious, given that already Bayes in &lt;em&gt;An Essay towards solving a Problem in the Doctrine of Chances&lt;/em&gt; (&lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1763.0053&quot; rel=&quot;nofollow noreferrer&quot;&gt;1763&lt;/a&gt;) used upper case &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; for an event.&lt;/p&gt;&#xA;&lt;p&gt;Next, let me quote Markov from his &lt;em&gt;Wahrscheinlichkeitsrechnung&lt;/em&gt; (1912) p. &lt;a href=&quot;https://archive.org/details/wahrscheinlich00markrich/page/n27&quot; rel=&quot;nofollow noreferrer&quot;&gt;14-15&lt;/a&gt; (my translation):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We don't find it superfluous to express the theorem of multiplications of probabilities via the formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(AB)=(A)(B,A)=(B)(A,B),&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$(AB)$&lt;/span&gt; denotes the probability of the simultaneous occurrence of both events &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(A)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(B)$&lt;/span&gt; denote respectively the probabilities of events &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(B,A)$&lt;/span&gt; denotes the probability of event &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is known for a fact, and &lt;span class=&quot;math-container&quot;&gt;$(A,B)$&lt;/span&gt; denotes the probability of event &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is known to have occurred.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't have access to his first Russian edition (&lt;em&gt;Исчисление вероятностей&lt;/em&gt; 1900), but Poincaré in his &lt;em&gt;Calcul des probabilités&lt;/em&gt; (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k29064s/f41.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;1896&lt;/a&gt;) p. 37 used almost the same notation as Markov.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;La probabilité pour que &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; se produisent tous deux est égale à la probabilité pour que &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; se produise, multipliée par la probabilité pour que &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; se produise, quand on sait que &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; s'est produit.&lt;/p&gt;&#xA;&lt;p&gt;Ou, inversement, elle est égale à la probabilité pour que &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; se produise, multipliée par la probabilité pour que &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; se produise, quand on suppose que &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; doit se produire.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(A \text{ et } B) =(B) (A \text{ si } B) = (A) (B \text{ si } A).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's interesting that unlike Markov, Poincaré doesn't provided an explicit definition of the notation &lt;span class=&quot;math-container&quot;&gt;$(A)$&lt;/span&gt;. Instead he evolves it from what look like labels to equations some pages earlier, into mathematical objects that can appear in equations.&lt;/p&gt;&#xA;&lt;p&gt;For a notation that's closer to &lt;span class=&quot;math-container&quot;&gt;$\text{Pr}(A)$&lt;/span&gt;, it seems like Hausdorff and his doctor father Bruns played a role. Bruns in his &lt;em&gt;Wahrscheinlichkeitsrechnung und Kollektivmaßlehre&lt;/em&gt; (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k99627p/f6.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;1906&lt;/a&gt;) writes &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{W}(E)$&lt;/span&gt; for the probability of an event &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\mathfrak{W}$&lt;/span&gt; abbreviating &lt;em&gt;Wahrscheinlichkeit&lt;/em&gt;). The book is based on lectures that Bruns gave every two years since the beginning of the 1880's at the University of Leipzig. Hausdorff attended these lectures in 1890 and stenographed them (but I don't have access to Hausdorffs notes).&lt;/p&gt;&#xA;&lt;p&gt;In 1900/01 Hausdorff himself lectured on probability theory and introduced the notation &lt;span class=&quot;math-container&quot;&gt;$p_F(E)$&lt;/span&gt; for conditional probability in his &lt;em&gt;Beiträge zur Wahrscheinlichkeitsrechnung&lt;/em&gt; (1901):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wenn bei dem Versuche, der über das Eintreffen oder Ausbleiben des Ereignisses &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; entscheidet, die Zahl der &lt;em&gt;überhaupt&lt;/em&gt; günstigen gleichberechtigten Fälle durch die Zahl der &lt;em&gt;überhaupt&lt;/em&gt; möglichen gleichberechtigten Fälle dividiert wird, so entsteht die Wahrscheinlichkeit von &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; schlechthin, die &lt;em&gt;absolute Wahrscheinlichkeit&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$p(E)$&lt;/span&gt;. Werden hingegen unter den günstigen und möglichen Fällen nur diejenigen gezählt, die ein bestimmtes anderes&#xA;Ereignis &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; herbeiführen, so entsteht die &lt;em&gt;relative Wahrscheinlichkeit von &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; unter der Voraussetzung, dass &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; verwirklicht sei&lt;/em&gt;, ein&#xA;Begriff, für den sich die Schreibung &lt;span class=&quot;math-container&quot;&gt;$p_F(E)$&lt;/span&gt; und etwa die&#xA;Ausdrucksweise relative Wahrscheinlichkeit von &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, posito &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; empfehlen dürfte.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If in the attempt of deciding the occurrence or absence of an event &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, the number of favourable and equally likely cases is divided by the number of all possible and equally likely cases, the probability of &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; proper emerges, the &lt;em&gt;absolute probability&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$p(E)$&lt;/span&gt;. If on the other hand, among the favourable and possible cases only those are counted, which induce another determined event &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, then the &lt;em&gt;relative probability of &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; under the condition that &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is realised&lt;/em&gt; emerges, a concept for which the notation &lt;span class=&quot;math-container&quot;&gt;$p_F(E)$&lt;/span&gt; and the parlance &lt;em&gt;relative probability of &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, posito &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;&lt;/em&gt; might be recommended.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is a long commentary by W. Purkert appended to this article in &lt;em&gt;Felix Hausdorff, Gesammelte Werke, Band V&lt;/em&gt; (&lt;a href=&quot;https://link.springer.com/chapter/10.1007/3-540-30669-2_8&quot; rel=&quot;nofollow noreferrer&quot;&gt;2005&lt;/a&gt;). There Purkert explains that Hausdorff influenced several authors with it.&lt;/p&gt;&#xA;&lt;p&gt;For instance Bruns, in a footnote of his 1906 book, writes that Bayes' Theorem is a simple corollary from purely arithmetical theorems, if one introduces the notion of ,conditional' probability as Hausdorff did. (But I don't know if Bruns' &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{W}(E)$&lt;/span&gt; was inspired by Hausdorff's &lt;span class=&quot;math-container&quot;&gt;$p(E)$&lt;/span&gt; or if Bruns already used it before 1900.)&lt;/p&gt;&#xA;&lt;p&gt;Czuber, who authored the article on probability theory in Klein's &lt;em&gt;Encyclopedia of mathematics&lt;/em&gt; (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN360504779?tify=%7B%22pages%22:%5B189%5D,%22panX%22:0.495,%22panY%22:0.529,%22view%22:%22info%22,%22zoom%22:0.643%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900&lt;/a&gt;) and the book &lt;em&gt;Wahrscheinlichkeitsrechnung und ihre Anwendung auf Fehlerausgleichung, Statistik und Lehensversicherung&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/wahrscheinlichk01czubgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;1903&lt;/a&gt;) adopted Hausdorffs notation in the second revised edition of his book (&lt;a href=&quot;https://archive.org/details/wahrscheinlichk05czubgoog/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;1908&lt;/a&gt;), writing &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{W}$&lt;/span&gt; like Bruns, instead of Hausdroff's &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;. Czuber acknowledges Hausdorff in a footnote on p. 45 for bringing clarity to the subject with this notation. Also Broggi adopted Hausdorffs notation in &lt;em&gt;Versicherungsmathematik&lt;/em&gt; (1911) (&lt;em&gt;Matematica attuariale&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;In Hausdorff's later lectures, &lt;em&gt;Wahrscheinlichkeitsrechnung&lt;/em&gt; (1923, 1931, Gesammelte Werke Band V p. 595) one also finds a definition of random variable with finitely many values and a (first?) use of an inequality to stand for an event.&lt;/p&gt;&#xA;&lt;p&gt;There he uses the notation &lt;span class=&quot;math-container&quot;&gt;$w(A)$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$p(A)$&lt;/span&gt; and writes (p. 608 Gesammelete Werke Band V, my translation)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$A_1,A_2,\ldots , A_m$&lt;/span&gt; be a complete disjunction, &lt;span class=&quot;math-container&quot;&gt;$w(A_i) = p_i, \sum p_1=1$&lt;/span&gt;. Imagine that to each case &lt;span class=&quot;math-container&quot;&gt;$A_i$&lt;/span&gt; is associated a real number &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; (for instance the number &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;).&#xA;If the &lt;span class=&quot;math-container&quot;&gt;$p_i$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$&amp;gt; 0$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; are pairwise different, and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; denotes one of them, then we call &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; a &lt;em&gt;variable&lt;/em&gt;, which can assume the values &lt;span class=&quot;math-container&quot;&gt;$x_1,\ldots,x_m$&lt;/span&gt;; only that here, compared to the common use of language, the variable is made more precise, in that it can assume each value &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; with a certain probability &lt;span class=&quot;math-container&quot;&gt;$p_i$&lt;/span&gt;. We call the embodiment [Innbegriff] of the &lt;span class=&quot;math-container&quot;&gt;$p_i, x_i$&lt;/span&gt; a distribution of the variabel &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Such distributions play the main role in applications of the calculus of probabilities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Immediately after that he defines the expectation value and the &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-th moment &lt;span class=&quot;math-container&quot;&gt;$\mu_k$&lt;/span&gt; of a variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and on the next page writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is a positive number and the sum &lt;span class=&quot;math-container&quot;&gt;$\overset{*}{\sum}$&lt;/span&gt; only runs over those &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$|x_i|\geq t$&lt;/span&gt;, then for &lt;span class=&quot;math-container&quot;&gt;$k=2,4,\ldots$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mu_k = \sum p_i x^k_i \geq \overset{*}{\sum} p_i x^k_i \geq t^k \cdot \overset{*}{\sum} p_i,&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\overset{*}{\sum} p_i \leq \dfrac{\mu_k}{t^k}&#xA;$$&lt;/span&gt;&#xA;or with the obvious notation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;w(|x| \geq t)\leq \dfrac{\mu_k}{t^k},\quad w(|x| &amp;lt; t) \geq 1 - \dfrac{\mu_k}{t^k}\; (\text{Tschebyscheff}) &#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've searched in Todhunter's &lt;em&gt;A History of the mathematical theory of probability&lt;/em&gt; (&lt;a href=&quot;https://archive.org/details/historyofthemath000979mbp/page/n497&quot; rel=&quot;nofollow noreferrer&quot;&gt;1865&lt;/a&gt;) and Czuber's Encyclopedia article (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN360504779?tify=%7B%22pages%22:%5B189%5D,%22panX%22:0.495,%22panY%22:0.529,%22view%22:%22info%22,%22zoom%22:0.643%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900&lt;/a&gt;) for even earlier uses of notations like &lt;span class=&quot;math-container&quot;&gt;$p(A), P(A), Pr(A)$&lt;/span&gt;, but couldn't find any. Of course, the use of upper/lower-case &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to stand for probability is much older than 1900. Laplace for instance uses it often. But from a modern perspective the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; of Laplace and earlier mathematicians should be understood as a number between 0 and 1, while the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; introduced by Hausdorff or the parenthesis used by Poincare and Markov can be seen as operators, that accept events as inputs and yield numbers between 0 and 1 as outputs.&lt;/p&gt;&#xA;&lt;p&gt;What I haven't looked for is the first use of the notation &lt;span class=&quot;math-container&quot;&gt;$P(A|B)$&lt;/span&gt; for conditional probabilities. A related question is &lt;a href=&quot;https://mathoverflow.net/q/163582/745&quot;&gt;https://mathoverflow.net/q/163582/745&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-02-07T14:24:40.437" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8274" PostTypeId="2" ParentId="8116" CreationDate="2019-02-03T00:47:01.027" Score="3" Body="&lt;p&gt;(Question: &quot;How did the early chemists make a connection between gram formula weight with 1 mole and Avogadro's number?&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article cited in the question has unfortunately led to a misconception, because that cited article starts out from a fundamental mistake about Avogadro's work. Avogadro did not discuss anything like what is now (honorifically) called Avogadro's number, he did not deal in absolute numbers or quantities. His important advance was concerned with &lt;em&gt;relative&lt;/em&gt; combining proportions and not with absolute measures. Even the title of his original work already makes that quite clear, thus: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Essai d'une manière de déterminer les masses &lt;em&gt;relatives&lt;/em&gt; des&#xA;  molécules élémentaires des corps, et les proportions selon lesquelles&#xA;  elles entrent dans ces combinaisons; par A. Avogadro&quot;; Journal de&#xA;  Physique 73, 58-78 (1811): (emphasis added)&#xA;  (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k9608503q/f64.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gallica.bnf.fr/ark:/12148/bpt6k9608503q/f64.image&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This title becomes in English: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Essay on a manner of determining the &lt;em&gt;relative&lt;/em&gt; masses of the&#xA;  elementary molecules of bodies, and the proportions in which they&#xA;  enter into these compounds&quot; (again, emphasis added).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This title comes from a very useful English translation, of which the &lt;a href=&quot;http://web.lemoyne.edu/~giunta/EA/AVOGADROann.HTML&quot; rel=&quot;nofollow noreferrer&quot;&gt;main parts are given here&lt;/a&gt;, of Avogadro's original paper, made in about 1890 by the Alembic Club (Edinburgh).  Their translation is also provided with a large number of helpful notes which should assist in the avoidance of misunderstanding .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notes to the Alembic Club translation also mention the honorific origin of the expression 'Avogadro number', which was named a century after the time of Avogadro himself, after breakthrough experiments by Millikan to measure the charge on single electrons, thus enabling determination of scale factors between macroscopic and atomic-scale measures.  Previous to that experimental achievement, the idea of this number had remained an indeterminate concept that there was a definite number of particles in a standard gram-atomic weight quantity (or a different definite number for any other adopted macroscopic scale-unit), but no way to measure any of these supposed numbers or even to verify that the concept was justified in reality.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is well-expressed in the wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Avogadro_constant#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Avogadro's constant&lt;/a&gt;, thus: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Accurate determinations of the Avogadro constant require the&#xA;  measurement of a single quantity on both the atomic and macroscopic&#xA;  scales using the same unit of measurement. This became possible for&#xA;  the first time when American physicist Robert Millikan measured the&#xA;  charge on an electron in 1910.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words, the Avogadro number is hardly a constant, but rather a calibration factor, and it's also important that the above statement in Wikipedia doesn't specify which quantity is measured nor on which macroscopic scale.  Clearly, different values for the calibration factor &#xA;result from different choices of atomic-weight scale and macroscopic mass-scale, although in practice until recently the uncertainties of measurement practically swamped any differences due to variants in &#xA;the chosen atomic-weight scale. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the answer to the question as put is that the early chemists &lt;em&gt;did not&lt;/em&gt; make a connection between gram formula weight with 1 mole and Avogadro's number, &lt;a href=&quot;https://hsm.stackexchange.com/users/55/conifold&quot;&gt;&lt;em&gt;@conifold&lt;/em&gt;'s&lt;/a&gt; comment is correct, the concepts named in the question were altogether later concepts, and the earlier chemists dealt with relative measures not absolute numbers or quantities. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Avogadro number, thus named much later 'after Avogadro', as an honorific not as something he actually dealt with himself, is thus not a fundamental constant of nature, it provides a scale factor to relate two chosen standardized scales, one on an atomic level and the other on a macroscopic level but both of them essentially arbitrary.  The present-day value of the Avogadro number is effectively the calibration factor between the current relative standard scale of atomic weights, and the current standard scale that includes the gram, i.e. the number of atoms or molecules in the quantity of a pure compound which is its atomic or molecular weight taken to be expressed in grams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It may be helpful to add mention of the main historical 'ingredients', so to speak, of the modern concept, they should really include all of the elements of the history of relative atomic-weight determinations: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- Dalton's (chemical) atomic theory;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- Gay-Lussac's law of combining volumes; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- Avogadro's law (initially hypothesis) about the equality of molecular numbers in the same volumes of different gaseous compounds and elements under standardized temperature and pressure -- to begin with.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then these principles had to be augmented, by further experimental facts that led to the concepts of&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- &lt;a href=&quot;https://en.wikipedia.org/wiki/Valence_(chemistry)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;valency&lt;/em&gt;&lt;/a&gt; and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- alternative &lt;a href=&quot;https://en.wikipedia.org/wiki/Oxidation_state&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;oxidation states&lt;/em&gt;&lt;/a&gt; for many chemical elements, and thus to the need for a distinction between (gram-)atomic weight and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalent_weight&quot; rel=&quot;nofollow noreferrer&quot;&gt;equivalent weight&lt;/a&gt;. (One and the same element, with a single (gram-)atomic weight, would have a different equivalent weight for each of any different oxidation-states or valencies with which it enters into its different compounds: its equivalent weights are related by small-whole-number ratios or fractions with its (gram-)atomic weight.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern value(s) of Avogadro's number thus rests on two kinds of arbitrary decision: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(1) to fix the relative scale of atomic weights, based thus far on ratios, by adopting a convention-based number for the weight of one of the elements in the network of ratios: and then this determines the scale weights of the others &lt;em&gt;upon the adopted convention&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Among the convention-based numbers (scales) that have been historically used are H=1 (Dalton and more general use until near the end of the 19th century); (chemical) O=100 (Berzelius, used for a time in the earlier 19th-century, thanks to &lt;a href=&quot;https://hsm.stackexchange.com/users/6085/m-farooq&quot;&gt;&lt;em&gt;@m-farooq&lt;/em&gt;&lt;/a&gt; for his reminder of this one); and a (chemical) O=16 scale adopted near the end of the nineteenth century. There were slight adjustments made to the atomic weight scale and its basis during the course of the 20th century which have only very slightly altered the numbers, much less than the change from H=1 to O=16, but the value of the Avogadro number depends on these alterations too.)  (For a reference-source, see IUPAC's &lt;a href=&quot;https://www.researchgate.net/publication/44159991_History_of_recommended_atomic_weight_values_from_1882_to_1997_A_comparison_of_differences_from_current_values_to_the_estimated_uncertainties_of_earlier_values&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;History of the recommended atomic-weight values from 1882 to 1997&quot;&lt;/a&gt;, from 'Pure &amp;amp; Applied Chemistry', Vol. 70, No. 1, pp.237-257 (1998), which also discusses the differences between current and older values and the uncertainties of earlier ones).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(2) A second arbitrary element in the definition of the present-day Avogadro number and mole is to assign to each arbitrary unit of the relative atomic weight scale an arbitrary unit of macroscopic weight or mass. For the purposes of chemical science, that unit has been traditionally and arbitrarily the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gram&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard gram&lt;/a&gt;, the definition of which has experienced slight alterations too.  The practical value of such an assignment is to bring the abstracted relative atomic weights back into the concrete world by matching them up with actual measurable weights. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chemical terminology used to include expressions such as 'gram-atomic weight', 'gram-molecular weight', and 'gram-equivalent weight' or even 'gram-formula weight', mostly superseded now by the expression 'mole' which means the same as 'gram-molecular weight' i.e. molecular weight on the arbitrary standard scale, but taken to be expressed in grams.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only after these two steps have been definitely taken and standardised can it make sense to speak of a specific number as the Avogadro number.  The Avogadro number can thus be seen to depend on two standardized arbitraries: the scale-unit assigned for the relative atomic weight scale, and the standard mass-measure to be associated with that scale-unit.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-02-03T02:09:52.290" LastActivityDate="2019-02-03T02:09:52.290" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8275" PostTypeId="2" ParentId="8270" CreationDate="2019-02-04T16:02:16.570" Score="-2" Body="&lt;p&gt;A couple thoughts.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First  the dipole moment is defined in terms of a distance  &lt;span class=&quot;math-container&quot;&gt;$\textbf{d}$&lt;/span&gt;, so that letter is &quot;right out&quot; . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, as the dipole moment in aggregation indicates the degree of polarization of a macro object, so using &lt;span class=&quot;math-container&quot;&gt;$\textbf{p}$&lt;/span&gt; is not unreasonable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was unable to find a &quot;first instance&quot; reference, however. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-02-04T16:02:16.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8276" PostTypeId="1" CreationDate="2019-02-04T19:55:23.993" Score="0" ViewCount="540" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Thales understood similar triangles and right triangles, and what is&#xA;  more, used that knowledge in practical ways. The story is told in DL&#xA;  (loc. cit.) that he measured the height of the pyramids by their&#xA;  shadows at the moment when his own shadow was equal to his height. A&#xA;  right triangle with two equal legs is a 45-degree right triangle, all&#xA;  of which are similar. The length of the pyramid's shadow measured from&#xA;  the center of the pyramid at that moment must have been equal to its&#xA;  height.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;-- &lt;a href=&quot;https://en.wikipedia.org/wiki/Thales_of_Miletus#cite_note-43&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Apparently Thales's measurement had to wait for that special moment when his shadow was equal to his height because was not able to generalize that whatever the ratio of his own height to his own shadow length, the ratio of pyramid height to pyramid shadow length would be equal; because he knew that isosceles right triangles were similar but he did not know of the more general AAA triangle similarity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Right?&lt;/p&gt;&#xA;" OwnerUserId="9545" LastActivityDate="2019-02-04T19:55:23.993" Title="Does the story about Thales and the heights of pyramids illustrate that Thales did not know of AAA triangle similarity?" Tags="|ancient-greece|geometry|ancient-egypt|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8277" PostTypeId="1" CreationDate="2019-02-04T20:49:29.293" Score="3" ViewCount="113" Body="&lt;p&gt;I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal Collocation Revisited” which has a brief section on history of MWR and collocation methods. It’s at: &lt;a href=&quot;https://authors.elsevier.com/a/1YHLy_12dr4lJw&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://authors.elsevier.com/a/1YHLy_12dr4lJw&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have found and read 7 articles on the history of FEM and a few presentations. One paper frequently cited as “a first” is Courant’s 1943 paper (based on 1941 presentation) “Variational Methods for the Solution of Problems of Equilibrium and Vibrations”. It seems the appendix of the paper is responsible for its citation as a first finite element paper. In the appendix he treats a torsion problem, first using a Raleigh-Ritz method with simple one and two term global trial functions. He then checks the results with a finite difference method on grids of triangles. He gives no details of the calculations. He also states:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“…. [the finite difference method] is obviously adaptable to any type of domain. Much more so than the Raleigh-Ritz procedure in which the construction of admissible functions would usually offer decisive obstacles.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since he does not use a variational method on grids of triangles and seems to think this would be difficult, why is the paper considered a first paper on the FEM? &lt;/p&gt;&#xA;" OwnerUserId="9560" LastActivityDate="2019-02-04T20:49:29.293" Title="Courant (1943) and History of Finite Element Method" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8278" PostTypeId="1" CreationDate="2019-02-04T20:59:40.303" Score="2" ViewCount="153" Body="&lt;p&gt;According to an uncited Wikipedia paragraph, Merit-Ptah is the earliest known female scientist.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An ancient Egyptian, Merit-Ptah (c. 2700 BC), described in an inscription as &quot;chief physician&quot;, is the earliest known female scientist named in the history of science.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is this true? Are there figures that could be (even debatable) be described as preceding her as a female scientist? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&#xA;As mentioned in the comments, there were no &quot;scientists&quot; in 2700BC or earlier in the modern sense. What I was effectively asking was whether there were any women who made known interesting contributions to medicine, mathematics, biology or any other field at this time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example of an early(ish) interesting contribution would be Galen's description of the difference between motor and sensory nerves, despite his other mix of partly through &amp;amp; partly incorrect observations &amp;amp; statements.&lt;/p&gt;&#xA;" OwnerUserId="9561" LastEditorUserId="609" LastEditDate="2019-02-05T13:25:10.517" LastActivityDate="2019-02-09T18:23:17.990" Title="Earliest Known reference to a female scientist?" Tags="|reference-request|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="8279" PostTypeId="2" ParentId="8256" CreationDate="2019-02-05T12:54:13.270" Score="4" Body="&lt;p&gt;Alexander von Humboldt describes what we might call scientific cranks in 1799. His activities might also one of the earliest documented accounts of what we now call &quot;public engagement with science&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Describing his visit to Cumana (Venezuela), where he arrived in July 1799, he describes how the local population is very interested in his research and he has to spend much time explaining science, but also meeting some rather complicated characters. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;THE first weeks of our abode at Cumana were employed in verifying our instruments [...] If every thing around us was fitted to inspire us with the most lively interest, our physical and astronomical instruments in their turns excited strongly the curiosity of the inhabitants. We were distracted by frequent visits; and in order not to dissatisfy persons, who appeared so happy to see the spots of the Moon through Dollond`s telescope, the absorption of two gazes in a eudiometrical tube, or the effects of galvanism on the motions of a frog, we were obliged to answer questions often obscure, and repeat for whole hours the same experiments. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And now the cranks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These scenes were renewed for the space of five years, every time that we took up our abode in a place where it was understood, that we were in possession of microscopes, telescopes, and electrical apparatus. They were in general so much the more fatiguing, as the person who visited us had confused notions of astronomy and physics; two sciences, which in the Spanish colonies are designated under the singular name of the new philosophy, nueva filosophia. The half-scientific looked on us with a sort of disdain, when they learnt that we had not brought in our collection of books the Spectacle de la Nature by Abbé Pluche, the Cours de Physique of Sigaud la Fond, or the Dictionary of Valmont de Bomare. These three works, and the Traité d`Economie politique of Baron Bielfeld, are the foreign, works most known and esteemed in Spanish America, from Caraccas and Chili to Guatimala and the north of Mexico. No one is thought learned, who cannot quote their translations; and it is only in the great capitals, at Lima, at Santa Fe de Bogota, and at Mexico, that the names of Haller, Cavendish, and Lavoisier, begin to take the place of those, that have enjoyed popular celebrity for these fifty years past. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But he is impressed by the curiosity of the native inhabitants and their eagerness to learn:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The curiosity excited respecting the phenomena of the heavens, and various objects of the natural sciences, takes a very different character among anciently civilized nations, and among those who have made but little progress in the unfolding of their intellectual faculties. Each of them exhibits in the highest classes of society frequent examples of persons unacquainted with science; but in the colonies, and among new people, curiosity, far from being idle or transient, arises from an ardent desire of instruction, and discovers itself with an ingenuousness and simplicity, which in Europe are the characteristics only of youth. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Quoted from: Alexander von Humboldt: Personal Narrative of Travels to the Equinoctial Regions of the New Continent during the Years 1799-1804, Vol 2, Chapter V, translated into English by Helen Maria Williams. - London 1818.&#xA;&lt;a href=&quot;http://www.avhumboldt.net/humboldt/publications/paragraph/did/3/vid/2/cid/14/tid/518/text/THE-first-weeks-of-our-abode-at-Cumana-were-employed-in-verifying-our-instruments-in-herbalizing-in&quot; rel=&quot;nofollow noreferrer&quot;&gt;Online version&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9563" LastActivityDate="2019-02-05T12:54:13.270" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8281" PostTypeId="2" ParentId="6708" CreationDate="2019-02-06T13:24:18.493" Score="2" Body="&lt;p&gt;Marxism was almost mainstream sociology into the 1980s and that has had a lasting influence on the field despite subsequent marginalization. There is an active &lt;a href=&quot;http://www.asanet.org/asa-communities/sections/marxist-sociology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marxist Sociology section&lt;/a&gt; of the American Sociological Association. While I think this reflects a certain degree of separation of Marxist thought from the mainstream of the discipline, it also shows that there are many sociologists still interested in Marxist theory. Before his recent death, &lt;a href=&quot;https://en.wikipedia.org/wiki/Erik_Olin_Wright&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erik Olin Wright&lt;/a&gt; was probably the most prominent and influential living Marxist sociologist. &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Bellamy_Foster&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Bellamy Foster&lt;/a&gt; is another sociologist well-known for his commitment to Marxism. &lt;/p&gt;&#xA;" OwnerUserId="9568" LastActivityDate="2019-02-06T13:24:18.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8282" PostTypeId="1" CreationDate="2019-02-06T16:06:29.733" Score="3" ViewCount="219" Body="&lt;p&gt;Can it be (are there real examples/anecdotes of) that, say during international&#xA;séminaires or conferences in mathematics, people of different linguistical&#xA;origins could understand each other with the help of pure -or almost- mathematical&#xA;formalization and notation ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="9569" LastActivityDate="2019-02-19T10:51:32.183" Title="Communication using mathematical notation among people of different languages" Tags="|notation|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8283" PostTypeId="1" AcceptedAnswerId="8284" CreationDate="2019-02-07T00:02:22.760" Score="2" ViewCount="1645" Body="&lt;p&gt;According to wikipedia (&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_trigonometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/History_of_trigonometry&lt;/a&gt;) :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the modern word &quot;sine&quot; is derived from the Latin word sinus, which means &quot;bay&quot;, &quot;bosom&quot; or &quot;fold&quot; is indirectly, via Indian, Persian and Arabic transmission, derived from the Greek term khordḗ &quot;bow-string, chord&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I don't see yet why a sine is a bow-string or a chord.&lt;/p&gt;&#xA;" OwnerUserId="9350" LastActivityDate="2019-05-16T00:59:48.620" Title="Where does the word ``sine'' (as in $\sin x$) come from?" Tags="|geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8284" PostTypeId="2" ParentId="8283" CreationDate="2019-02-07T04:48:43.697" Score="2" Body="&lt;p&gt;A detailed answer is already here with an excellent explanation &lt;a href=&quot;https://english.stackexchange.com/questions/40364/what-reasoning-is-behind-the-names-of-the-trigonometric-functions-sine-secan&quot;&gt;https://english.stackexchange.com/questions/40364/what-reasoning-is-behind-the-names-of-the-trigonometric-functions-sine-secan&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A good researcher always double-checks multiple resources. One can never rely on one (history) source especially if it is open to editing by everyone. Wikipedia entry is quite incorrect which has nothing to with Indian, Persian transmission or Greek words. According to the unabridged Oxford Dictionary, the original definition of sine was &quot;the length of a straight line drawn from one end of a circular arc parallel to the tangent at the other end, and terminated by the radius.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is the authentic etymology &quot;&amp;lt; Latin sinus a bend, bay, etc.; also, the hanging fold of the upper part of a toga, the bosom of a garment, and hence used to render the synonymous Arabic &lt;em&gt;jaib&lt;/em&gt; , applied in geometry as in sense 2&quot;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-02-07T04:48:43.697" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8286" PostTypeId="1" CreationDate="2019-02-08T03:08:07.353" Score="3" ViewCount="459" Body="&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=newbYZ0lqzQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;This source&lt;/a&gt; (I believe it's in Ukrainian) claims that Maxwell came very close to creating the relativity theory, however, it was the fact that he was adhering to the traditional scientific frame of thinking of his time that prevented him from doing that. Einstein, on the contrary, was free from any kind of traditions and, therefore, he was the one who became the author of the relativity theory even though he drew a lot from Maxwell's groundwork.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The author of the source mentions that only in passing without elaborating much.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone here provide some details on that? What exactly did Maxwell not do that Einstein did?  &lt;/p&gt;&#xA;" OwnerUserId="9578" LastEditorUserId="4251" LastEditDate="2020-06-06T13:02:29.373" LastActivityDate="2020-06-06T13:02:29.373" Title="How did Maxwell fall short of Einstein?" Tags="|physics|relativity-theory|einstein|maxwell|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8287" PostTypeId="2" ParentId="8282" CreationDate="2019-02-09T09:31:52.550" Score="1" Body="&lt;p&gt;I am not completely sure about the kind of information you're looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that common experience of many scholars coming from non English speaking countries is that their first experience of international congress is something of this kind: you rely 80% on formulas that you see at the blackboard (whiteborad/slides) and 20% ore less form the few words that you're able to understand from the speaker. Many of us passed throught this moment (and, somewhat, survived). Percentages here are to be understood as a half-joke, each case is of course different.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whether this is of help to answer your question I do not know. True is the fact that among scientists mathematicians were always much interested in developing a &lt;em&gt;lingua franca&lt;/em&gt; that allowed to communicate more easily. Leibniz was the first to think that developing a language for logic and at the same time for universal communication was one of the principal concern for science. At the 1900 International Congress of Math there was a session discussing the problem of communication between mathematicians that adressed the problem on whether one should choose a natural language to communicate between different mathematicians or develop an artificial one. It is not by chance that the Italian logician Peano both tried to write an encyclopedi treatise of math relying on set theory (&lt;strong&gt;Formulario Mathematico&lt;/strong&gt;) and developed an artificial international language based on Latin (&lt;em&gt;Latino sine flexione&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find some interesting information of the sort in this paper:&#xA;&lt;a href=&quot;https://jcom.sissa.it/archive/01/01/A010103/jcom0101(2002)A03_it.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://jcom.sissa.it/archive/01/01/A010103/jcom0101(2002)A03_it.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ironically enough, it is in Italian so you'll need help to understand it :D&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2019-02-09T09:31:52.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8288" PostTypeId="1" CreationDate="2019-02-09T13:43:29.450" Score="0" ViewCount="36" Body="&lt;p&gt;Was Einstein the first to propose that observing an ensemble of N particles for time interval of dt is same as observing a single particle of ensemble for time interval of Ndt?&lt;/p&gt;&#xA;" OwnerUserId="9585" LastActivityDate="2019-02-09T13:43:29.450" Title="Duality while studying properties of an ensemble" Tags="|einstein|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8292" PostTypeId="1" AcceptedAnswerId="8293" CreationDate="2019-02-11T06:22:46.107" Score="6" ViewCount="2288" Body="&lt;p&gt;From what I've read in Wikipedia, John von Neumann made a stupendous number of contributions to economics, computer science and mathematics. Why, then, didn't he receive a top award in any of these disciplines?&lt;/p&gt;&#xA;" OwnerUserId="9595" LastEditorUserId="4251" LastEditDate="2021-03-01T00:41:40.517" LastActivityDate="2021-03-01T00:41:40.517" Title="Why didn't John von Neumann win the Turing Award, Fields Medal or Nobel Prize?" Tags="|mathematicians|nobel-prize|academic-awards|von-neumann|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8293" PostTypeId="2" ParentId="8292" CreationDate="2019-02-11T07:40:53.910" Score="12" Body="&lt;p&gt;I don't know about the Fields medal, but he could not have won the Nobel prize since the only appropriate one would have been the one in Economics, which was awarded for the first time in 1969, 12 years after his death. And the Turing award was awarded for the first time in 1966, 9 years after his death.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-02-11T07:40:53.910" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8294" PostTypeId="1" CreationDate="2019-02-11T15:47:32.613" Score="7" ViewCount="151" Body="&lt;p&gt;I am writing a paper and I would like to cite the person(s) who proved that only primes of the form &lt;span class=&quot;math-container&quot;&gt;$4k+1$&lt;/span&gt; can evenly divide odd integers of the form &lt;span class=&quot;math-container&quot;&gt;$n^2+1$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if &lt;span class=&quot;math-container&quot;&gt;$n=8$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$n^2 + 1 = 65$&lt;/span&gt; which is evenly divisible by 5 and 13, both of which are of the form &lt;span class=&quot;math-container&quot;&gt;$4k + 1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9596" LastEditorUserId="55" LastEditDate="2019-02-11T23:21:14.820" LastActivityDate="2019-02-11T23:24:35.320" Title="Who first proved that only primes of the form $4k+1$ divide odd integers of the form $n^2+1$?" Tags="|mathematics|mathematicians|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8295" PostTypeId="2" ParentId="8294" CreationDate="2019-02-11T23:19:33.083" Score="7" Body="&lt;p&gt;It would seem to be Euler. Dickson, in &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick/page/380&quot; rel=&quot;noreferrer&quot;&gt;Ch. XVI of History of the  Theory of Numbers&lt;/a&gt;, writes the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Euler discussed the numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt; is divisible by a prime &lt;span class=&quot;math-container&quot;&gt;$4n+1=r^2+s^2$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; be the convergent preceding &lt;span class=&quot;math-container&quot;&gt;$r/s$&lt;/span&gt; in the continued fraction for &lt;span class=&quot;math-container&quot;&gt;$r/s$&lt;/span&gt;; then &lt;span class=&quot;math-container&quot;&gt;$ps-qr=1$&lt;/span&gt;. Thus every &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is of the form &lt;span class=&quot;math-container&quot;&gt;$(4n+1)m\pm k$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$k=pr+qs$&lt;/span&gt;. Euler gave the &lt;span class=&quot;math-container&quot;&gt;$161$&lt;/span&gt; integers &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt; 1500$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt; is a prime, and the cases &lt;span class=&quot;math-container&quot;&gt;$a= 1, 2, 4, 6, 16, 20, 24, 34$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$a^4+1$&lt;/span&gt; is a prime.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first appearance is in the &lt;a href=&quot;http://eulerarchive.maa.org/correspondence/letters/OO0784.pdf&quot; rel=&quot;noreferrer&quot;&gt;letter to Goldbach from July 9, 1743&lt;/a&gt;. Euler came back to these numbers in 1755 and 1762:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] &lt;em&gt;Euler treated the problem to find all integers for which &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt; is divisible by a given prime &lt;span class=&quot;math-container&quot;&gt;$4n+1=p^2+q^2$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$p^2+q^2$&lt;/span&gt;, there exist integers &lt;span class=&quot;math-container&quot;&gt;$r,s$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$a=pr+qs,b=ps-qr$&lt;/span&gt;. We wish &lt;span class=&quot;math-container&quot;&gt;$b=\pm1$&lt;/span&gt;. Hence we take the convergent &lt;span class=&quot;math-container&quot;&gt;$r/s$&lt;/span&gt; preceding &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; in the continued fraction for &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt;. Thus &lt;span class=&quot;math-container&quot;&gt;$ps-qr=\pm1$&lt;/span&gt;, and our answer is &lt;span class=&quot;math-container&quot;&gt;$a=(pr+qs)$&lt;/span&gt;. He listed all primes &lt;span class=&quot;math-container&quot;&gt;$P=4n+1&amp;lt;2000$&lt;/span&gt; expressed as &lt;span class=&quot;math-container&quot;&gt;$p^2+q^2$&lt;/span&gt;, and listed all the &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;'s for which &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. The table may be used to find all the divisors &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;a$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; given number &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt;. He gave his table and tabulated the values &lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;1500$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$(a2+1)/k$&lt;/span&gt; is a prime, for &lt;span class=&quot;math-container&quot;&gt;$k=2, 5, 10$&lt;/span&gt;. He tabulated all the divisors of &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$a\leq1500$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Fermat is mentioned on the same page, but only in connection with something else, so I doubt that he looked into factorizations of &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt;. &lt;a href=&quot;https://books.google.com/books/about/Guide_to_Tables_in_the_Theory_of_Numbers.html?id=ri4rAAAAYAAJ&quot; rel=&quot;noreferrer&quot;&gt;Lehmer in Guide to Tables in the Theory of Numbers (p.31)&lt;/a&gt; also names Euler as the first to produce a factorization table of &lt;span class=&quot;math-container&quot;&gt;$a^2+1$&lt;/span&gt; numbers in 1762, and nobody else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Fermat, there are almost no &lt;em&gt;proofs&lt;/em&gt; by him in existence, a notable exception is a case of his Last theorem for &lt;span class=&quot;math-container&quot;&gt;$n=4$&lt;/span&gt;. Euler generally made it his task to sort out Fermat's guesses. For example, Fermat &quot;knew&quot; that &lt;span class=&quot;math-container&quot;&gt;$2^{2^n}+1$&lt;/span&gt; were all primes, after checking the first four. Euler computed &lt;span class=&quot;math-container&quot;&gt;$2^{2^5}+1=641\times6700417$&lt;/span&gt;. No other primes of this form turned up since.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-02-11T23:24:35.320" LastActivityDate="2019-02-11T23:24:35.320" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8297" PostTypeId="1" CreationDate="2019-02-14T13:20:05.160" Score="2" ViewCount="623" Body="&lt;p&gt;Before we knew about gases in the atmosphere, what did scientists think was the function of breathing?&lt;/p&gt;&#xA;" OwnerUserId="9608" LastEditorUserId="55" LastEditDate="2019-02-14T21:59:20.493" LastActivityDate="2023-09-07T02:14:58.850" Title="What did scientists think was the function of breathing before the discovery of atmospheric gases?" Tags="|biology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8299" PostTypeId="1" CreationDate="2019-02-14T17:58:22.553" Score="1" ViewCount="66" Body="&lt;p&gt;I have a Felix M Arithmometer (shown below) that appears to follow the same design as an &lt;a href=&quot;https://en.wikipedia.org/wiki/Odhner_Arithmometer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Odhner arithmometer&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing I noticed about the device is that the turn counter (on the left) does not overflow like the addition register does.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means if I want to multiply 37 * 49 I must do the following&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Procedure A:&#xA;1. enter 37&#xA;2. rotate + (9 times)&#xA;3. shift right&#xA;4. rotate + (4 times)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Performing the operation &quot;subtractively&quot; will result in the right answer, but won't display the correct result in the turn counter&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Procedure B&#xA;1. enter 37&#xA;2. rotate - (1 time) (causing underflow)&#xA;3. shift right&#xA;4. rotate + (5 times)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;OR&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;Procedure C&#xA;1. enter 37&#xA;2. shift right&#xA;3. rotate + (5 times)&#xA;4. shift left&#xA;5. rotate - (1 time)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;If I do procedure B or C, then I see &lt;code&gt;51&lt;/code&gt; in the turn counter instead of the expected &lt;code&gt;49&lt;/code&gt; even though the number in the addition register is still &lt;code&gt;1813&lt;/code&gt; as expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's some logic to this. If I multiply 37 * 48 subtractively, then 52 appears in the turn counter, which can be though of as 5 in the tens place and -2 in the units place. So you can check your work by consulting the turn counter if you remember which digits are supposed to be negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that a register with overflow is heavier and more expensive to produce than a register without overflow, so it makes sense that the turn counter wouldn't overflow in exactly the same way as the addition register. However, that still doesn't explain why the turn counter counts up then down instead of always counting up.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were any arithmometers produced anywhere with ordinary overflow behavior in the turn counter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is multiplying &quot;subtractively&quot; and remembering which digits are supposed to be negative when interpreting the turn counter an intended mode of operation for an Odhner arithmometer?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/72IAP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/72IAP.jpg&quot; alt=&quot;A Felix Model M Arithmometer&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9243" LastEditorUserId="9243" LastEditDate="2019-02-14T18:05:00.320" LastActivityDate="2019-02-15T04:19:26.317" Title="Overflow behavior in turn counters of Odhner-style arithmometers" Tags="|computers|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8300" PostTypeId="2" ParentId="8299" CreationDate="2019-02-14T21:29:28.467" Score="1" Body="&lt;p&gt;After a little bit of research, I think the answer is that the turn counter works the way it does to support division and division &lt;em&gt;would not function as intended if the turn counter carried&lt;/em&gt;. &quot;Subtractive&quot; multiplication is not well supported because it would make division less intuitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This video &lt;a href=&quot;https://www.youtube.com/watch?v=aDN4s8ElxqE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=aDN4s8ElxqE&lt;/a&gt; shows an arithmometer of similar design and the intended algorithm for doing division problems starting at about 4:57.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The turn counter shows the quotient in a division problem. Quotients are computed by repeatedly subtracting the divisor multiplied by a power of ten until just before the divisor underflows.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quotient is a positive number and the quotient-so-far (in the turn counter) must &lt;em&gt;increase&lt;/em&gt; each time a multiple of the divisor is subtracted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the turn counter carried, then the result of a division problem would be the nine's complement of the real answer. Alternatively, if the divisor or dividend were complemented, then the quotient would be non-complemented. This is explained in detail here: &lt;a href=&quot;https://www.youtube.com/watch?v=YXMuJco8onQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=YXMuJco8onQ&lt;/a&gt; starting around 5:19.&lt;/p&gt;&#xA;" OwnerUserId="9243" LastEditorUserId="9243" LastEditDate="2019-02-15T04:19:26.317" LastActivityDate="2019-02-15T04:19:26.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8301" PostTypeId="2" ParentId="8297" CreationDate="2019-02-14T21:57:43.507" Score="5" Body="&lt;p&gt;That air is a substance, which is breathed in, was known already in antiquity. Aristotle even suggests that, unlike fire, it has weight in &lt;a href=&quot;http://classics.mit.edu/Aristotle/heavens.4.iv.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Caelo [On the Heavens] Book IV.4&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Earth, then, and bodies in which earth preponderates, must have weight everywhere, while water is heavy anywhere but in earth, and air is heavy when not in water or earth. In its own place each of these bodies has weight excepting fire, even air. Of this we have evidence in the fact that a bladder when inflated weighs more than when empty&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given the weighing methods used, the experiment is considered inconclusive today, but, in any case, this view was common. There is also a work &lt;a href=&quot;https://en.wikipedia.org/wiki/On_Breath&quot; rel=&quot;nofollow noreferrer&quot;&gt;Περὶ Πνεύματος (On Breathing)&lt;/a&gt; in the Aristotelian corpus, although the attribution is controversial. The etymology gives an idea of what role the &lt;em&gt;pneuma&lt;/em&gt;, the &quot;breath of life&quot;, the vital spirit, was thought by &quot;the folk&quot; to play in the body. This certainly influenced the ancient natural philosophy. Aristotle (and/or his immediate successors) talked of the &quot;connate&quot; &lt;em&gt;pneuma&lt;/em&gt; that transmits the capacity to move and feel to the offspring through the sperm. Nonetheless, the pneuma was distinguished from the &lt;em&gt;psyche&lt;/em&gt; (soul, spirit).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ancient Stoics went much further. Their &lt;a href=&quot;https://en.wikipedia.org/wiki/Pneuma_(Stoic)&quot; rel=&quot;nofollow noreferrer&quot;&gt;pneuma&lt;/a&gt;, a mixture of air and fire, was the force of growth (a prototype of the later &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89lan_vital&quot; rel=&quot;nofollow noreferrer&quot;&gt;élan vital&lt;/a&gt;, postulated by Lamarck and others), and in its most refined form, the material of the soul. Indeed, it was elevated to a cosmic principle that both animates and organizes the universe. The Stoics had a curious mixture of materialism with theology, their God/World Soul was made of pneuma. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the more empirical Hellenistic science, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pneuma&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hippocratic tradition&lt;/a&gt; ascribed to the pneuma the preservation of bodily heat and the spread of nourishment. When Praxagoras discovered that arteries in the corpses were empty c. 300 BC, it was taken to mean that they were vessels for transporting pneuma from the lungs to different parts of the body. From &lt;a href=&quot;https://www.springer.com/us/book/9783540200680&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russo's Forgotten Revolution&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Herophilus maintains that pneuma, besides being breathed in, is also present in the arteries. The surviving testimonia unfortunately say nothing about the passage of pneuma from lungs to arteries. It has been said that Herophilus thought that arteries contained only pneuma and&#xA;  no blood, but a closer analysis of the sources reveals that most likely he distinguished between the contents of veins (blood alone) and that of the arteries (blood and pneuma)... According to Galen, Erasistratus believed that the pneuma is communicated from the heart to all parts of the body through the arteries, which are filled with blood. We also know that Herophilus believed that the pulsation of arteries served to spread nourishment.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Pneumatics, the study of machines that used compressed air, such as air pumps, was developed, most notably by &lt;a href=&quot;https://en.wikipedia.org/wiki/Ctesibius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ctesibius (c. 285-222 BC)&lt;/a&gt;. He was likely the first head of the Alexandrian library, and is known as the &quot;father of pneumatics&quot;. The principle of the siphon was known to him, but it is unclear if he linked it to the mechanism of breathing, his works are not extant. &lt;a href=&quot;https://en.wikipedia.org/wiki/Hero_of_Alexandria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heron (c. 10-70 AD)&lt;/a&gt;, in a later treatise, describes pneuma without any vitalistic connotations, as simply air in motion, and links suction to the functioning of lungs. According to Russo:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Heron, most probably drawing from Ctesibius or Strato of Lampsacus, devotes the entire introduction of his Pneumatica to arguing that macroscopic regions of vacuum cannot exist in nature, but can be approximated artificially, as in fact he demonstrates later in the&#xA;  work on several occasions. In particular, Heron explains that the natural&#xA;  distribution of particles and void in air can be changed in both directions&#xA;  under the application of external forces, although the air opposes such&#xA;  changes with an elastic reaction. The elastic properties of air had already been described and used by Ctesibius, as we know from Philo.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Heron also describes a medical device (syringe) based on suction due to expansion (Pneumatica, II, xviii). It does not seem that the idea of suction depending on expansion was clear to Aristotle: he does say that when the lung rises the air comes in, but he also says that when the air comes in the lung rises, so it's not clear whether and in what direction he postulates a causal link. It might be that, following his teleological scheme, he attributed the influx of air to the lung's porousness,&#xA;  which he stresses (De partibus animalium, III, vi, 669a). It may seem that the very early use of bellows should have been enough to suggest the &quot;modern&quot; explanation for suction, but Aristotle's passage shows that this is not so.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-02-17T09:39:16.630" LastActivityDate="2019-02-17T09:39:16.630" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8302" PostTypeId="1" AcceptedAnswerId="8308" CreationDate="2019-02-15T05:13:17.307" Score="2" ViewCount="759" Body="&lt;p&gt;I have tried to find the actual origin of the Heaviside unit step function and could not. I've searched and searched, read one complete biography of Oliver Heaviside, skimmed another, but nowhere can I find what he was doing, why he developed his step function. Guessing, I would say it had something to do with how a continuous wave signal on a telegraph wire is &quot;digitized&quot; into discrete, binary information as we get from this piecewise function, i.e., how dashes and dots are got from the EM signal on the wire. If anyone has the story behind this function's origin, derivation, application, I would greatly appreciate knowing. It's an excellent example of a continuous phenomenon being transformed into a discrete phenomenon.&lt;/p&gt;&#xA;" OwnerUserId="9610" LastEditorUserId="4251" LastEditDate="2020-06-07T18:32:19.713" LastActivityDate="2020-06-07T18:32:19.713" Title="Origin of the Heaviside function?" Tags="|electromagnetism|mathematical-physics|heaviside|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8303" PostTypeId="2" ParentId="7565" CreationDate="2019-02-15T16:27:50.577" Score="3" Body="&lt;p&gt;I firstly encountered this identity while working with related sums. Within this &lt;a href=&quot;https://math.stackexchange.com/questions/2856676/summation-problem-involving-odd-numbers-and-binomial-coefficients-prove-sum&quot;&gt;question on MSE&lt;/a&gt;, asked by myself, the user &lt;em&gt;Robert Z&lt;/em&gt; refered to the particular case you are asking for as &lt;a href=&quot;http://www.ssmrmh.ro/wp-content/uploads/2018/01/TEPPERS-IDENTITY.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tepper's Identity&lt;/a&gt;. In order to be precise Tepper's Identity is a more general result given by&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n!~=~\sum_{k=0}^n(-1)^k\binom nk (a-k)^n,a\in\mathbb R$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The aforementioned proof of Tepper's Identity was given by &lt;em&gt;Ravi Prakash&lt;/em&gt; and was posted on the &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;nd of January &lt;span class=&quot;math-container&quot;&gt;$2018$&lt;/span&gt; on the official website of the Romanian Mathematical Magazine which can be found &lt;a href=&quot;http://www.ssmrmh.ro/2018/01/02/teppers-identity/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Later on it reappeared within &lt;a href=&quot;http://www.ssmrmh.ro/wp-content/uploads/2017/02/RMM-FAMOUS-INEQUALITIES-MARATHON-1-100.compressed.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;RMM - Famous Inequalities Marathon 1 - 100&lt;/a&gt; as problem &lt;span class=&quot;math-container&quot;&gt;$97$&lt;/span&gt;. There are various proofs avaible nowadays, for instance &lt;a href=&quot;https://www.jstor.org/stable/2687866?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Another Proof of Tepper's Identity&lt;/em&gt;&lt;/a&gt; by &lt;em&gt;F. J. Papp&lt;/em&gt; published in the &lt;em&gt;Mathematics Magazine&lt;/em&gt; Vol. &lt;span class=&quot;math-container&quot;&gt;$45$&lt;/span&gt;, No. &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$($&lt;/span&gt;May, &lt;span class=&quot;math-container&quot;&gt;$1972)$&lt;/span&gt;, pp. &lt;span class=&quot;math-container&quot;&gt;$119-121$&lt;/span&gt; and &lt;a href=&quot;https://nttuan.org/2007/10/10/tepper/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Proof of Tepper’s identity&lt;/em&gt;&lt;/a&gt; by &lt;em&gt;Nguyen Trung Tuan&lt;/em&gt; posted on the &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;th of October &lt;span class=&quot;math-container&quot;&gt;$2007$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning the origin it seems like all sources are dating back to an arcticle called &lt;a href=&quot;https://www.jstor.org/stable/2687939?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Factorial Conjecture&lt;/em&gt;&lt;/a&gt; by &lt;em&gt;Myron Tepper&lt;/em&gt; published in the &lt;em&gt;Mathematics Magazine&lt;/em&gt;&#xA;Vol. &lt;span class=&quot;math-container&quot;&gt;$38$&lt;/span&gt;, No. &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$($&lt;/span&gt;Nov., &lt;span class=&quot;math-container&quot;&gt;$1965)$&lt;/span&gt;, pp. &lt;span class=&quot;math-container&quot;&gt;$303-304$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$($&lt;/span&gt;the index to this volume can be found &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0025570X.1965.11975666?journalCode=umma20&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt; proposing the identity, perhaps, for the first time.&lt;/p&gt;&#xA;" OwnerUserId="7971" LastActivityDate="2019-02-15T16:27:50.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8304" PostTypeId="1" AcceptedAnswerId="8311" CreationDate="2019-02-15T23:01:08.750" Score="7" ViewCount="168" Body="&lt;p&gt;From the inaugural lecture of J.J. Sylvester at Oxford, December 12, 1885, &amp;quot;On the method of reciprocants as containing an exhaustive theory of the singularities of curves&amp;quot;, Collected Mathematical Papers, Vol. 4, 278-302 (in Collected Papers it is claimed to be a reprint from Nature in 1886, but I was unable to locate it there), footnote at p.283:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Search has been made in the collected works of Monge and in manuscripts of his own ... I have also made application to the Universal Information Society, who undertake to answer &amp;quot;every conceivable question,&amp;quot; but nothing has so far come out of it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What is this &amp;quot;Universal Information Society&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="167" LastEditorUserId="10106" LastEditDate="2021-05-17T14:49:55.270" LastActivityDate="2021-05-17T20:38:13.500" Title="The &quot;Universal Information Society&quot; of the XIX century?" Tags="|mathematics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8305" PostTypeId="1" AcceptedAnswerId="8312" CreationDate="2019-02-15T23:45:05.243" Score="0" ViewCount="264" Body="&lt;p&gt;The famous Millikan's oil drop experiment in 1908 determined the magnitude of charge of the electron to be 1.59*10^(-19) C. However, it was well known that electron has a negative charge. Although there is no definition of charge &lt;em&gt;per se&lt;/em&gt;, the original experimental identification is still Franklin's experiment i.e., whenever glass is rubbed with silk, the glass becomes negatively charged. I am searching for those original experiments which showed that the charge on the electron behaved the  same way as glass rod rubbed with silk? Thanks&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-02-16T21:12:53.510" Title="What experiments led to the sign of the electron's charge?" Tags="|physics|philosophy-of-science|experimental-physics|electricity|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="8306" PostTypeId="1" AcceptedAnswerId="8314" CreationDate="2019-02-16T06:42:55.583" Score="6" ViewCount="395" Body="&lt;p&gt;In any modern differential geometry textbook (Do Carmo, for example), the fundamental theorem of curves can be found. It states that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;every regular curve in three-dimensional space, with non-zero&#xA;  curvature, has its shape (and size) completely determined by its&#xA;  curvature and torsion &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am wondering who first formulated this theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In D. J Struik's &quot;History of Differential Geometry&quot; he describes papers by Michel Ange Lancret (1774 – 1807). Lancret knew of first curvature and second curvature (torsion).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He wrote two mathematical papers...His first paper is of a more&#xA;  general nature. It contains the two fundamental quantities of the&#xA;  space curve, which he calls &quot; premiere flexion &quot; and &quot; seconde&#xA;  flexion.&quot;...Curvature and torsion appear therefore as differentials,&#xA;  and are not written as finite quantities until CAUCHY.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;LANCRET is therefore the first to take up the systematic theory of&#xA;  space curves after EULER, but it seems in an independent way. The line&#xA;  of progress goes here from CLAIRAUT via EULER and LANCRET to CAUCHY&#xA;  and FRENET.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suspect that the theorem was developed by someone in the above quoted line of progression.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Differential Geometry of Curves and Surfaces&quot;, Manfredo Do Carmo, 1976&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_curves&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Fundamental_theorem_of_curves&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9611" LastEditorUserId="4251" LastEditDate="2021-09-16T05:47:39.027" LastActivityDate="2021-09-16T05:47:39.027" Title="Who developed The Fundamental Theorem of Curves?" Tags="|mathematics|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8307" PostTypeId="1" CreationDate="2019-02-16T12:13:20.133" Score="2" ViewCount="373" Body="&lt;p&gt;Is there a book, which expresses all the questions, or searches for attainment of certain utility/need, or thing, which gave the discovery or invention of all the components of Topology?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I need book to give more math data; I want to know topology from that, than from a non-historical ordered textbook, as abstraction if has come from certain concrete contexts, they may allow knowing on how entire structure of topology is made for those contexts, and later allow making their alternate structures for our other needs. This might allow having memory of topology, from their particular needs which have created them, than having memory from the continuous artificial reknowing?  &lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="4251" LastEditDate="2022-01-01T01:01:15.187" LastActivityDate="2022-01-01T01:01:15.187" Title="Which book covers topology historically?" Tags="|mathematics|topology|book-recommendation|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8308" PostTypeId="2" ParentId="8302" CreationDate="2019-02-16T12:59:32.137" Score="6" Body="&lt;p&gt;&quot;Oliver Heaviside ... what he was doing, why he developed his step function&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A short answer is that Heaviside was interested, as a practical electrical engineer, in transient effects in complex electrical circuits as well as in steady effects. Examples of 'transient' problems: What happens when a switch is flipped (closed), and the circuit is an intricate combination of inductance, capacitance and resistance, partly in distributed form -- as in a long telephone transmission-line? And then what happens, in a more developed and intricate real situation, when a train of impulsive signals is generated, as when a person speaks into a microphone connected to that kind of circuit? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Step functions and their derivatives, unit impulses, specified with variable delay &amp;amp;c, were among the tools used by Heaviside to study these phenomena in their electrical-engineering environment. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;{Edit: }&lt;/em&gt;&lt;/strong&gt; It would be very long on detail to give Heaviside's full motivation for his adoption and use of the step-function in the solution of transient problems such as the 'switch-flip' problem already noted. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But a shorter indication of its logic-flow, as described in &lt;a href=&quot;https://catalog.hathitrust.org/Record/001986461&quot; rel=&quot;noreferrer&quot;&gt;H J Josephs' commentary &lt;em&gt;Heaviside's Electric Circuit Theory&lt;/em&gt;&lt;/a&gt;, may be of interest: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Pages 6-10 in Josephs' monograph build up a set of canonical equations and formulae which] &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... in concentrated form contains the whole theory of the symbolic&#xA;  solution [of the differential equations] of alternating-current&#xA;  problems.  [But classical solution-methods give] only steady-state&#xA;  currents ... and such currents are 'forced' currents. They vary with&#xA;  the time in precisely the same manner as do the electromotive forces.&#xA;  Such currents are, however, only part of the total currents; the&#xA;  complete solution must contain transient as well as steady-states. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If it is found that the steady-state solution of a problem is&#xA;  sufficient, then the need for the use of the Heaviside calculus does&#xA;  not arise. In cases such as this the well-known symbolic method ... is&#xA;  sufficient.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[However, the classical methods usually break down in the attempted&#xA;  solution of the transient problem.]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It was at this point that Heaviside parted with classic analysis and&#xA;  invented his operational calculus ...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In his attack on the general problem Heaviside made use of the&#xA;  principle of superposition. When several electromotive forces are&#xA;  applied simultaneously to a network of fixed parameters, each produces&#xA;  its own effect independently of the others. Thus the effect of each&#xA;  alone may be calculated and the results added to obtain a resultant&#xA;  due to the simultaneous action of the various electromotive forces.&#xA;  This suggested to Heaviside that he could deduce a superposition&#xA;  theorem expressing the independent variables in terms of voltage and&#xA;  admittance functions which have been made to satisfy the necessary&#xA;  terminal conditions of the network.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[In the following description, the step-function mentioned in the question, denoted [ 1 ] below, is defined as identically zero for t&amp;lt;0 and identically 1 for t&gt;=0.]&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qSchw.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qSchw.png&quot; alt=&quot;extract from Josephs (1946)&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What remained was to determine the 'indicial admittance' functions h(t), which Heaviside did by solving 'operational equations' in which differential and integral operators are replaced by symbols &lt;span class=&quot;math-container&quot;&gt;$ p $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ p^(-1) $&lt;/span&gt; , then treated as if algebraic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two parts of Heaviside's operational methods were described in &quot;On operators in physical mathematics, Part 1&quot;, in Proc.Roy.Soc., Vol.52, 504-529, February 1893; then &quot;Part 2&quot; in Proc.Roy.Soc., Vol.54, 105-143, June 1893 (but the third part was rejected for publication). Shorter versions of all three parts were later printed together in volume 2 (1899) of Heaviside's &lt;a href=&quot;https://archive.org/details/electromagnetict02heavrich&quot; rel=&quot;noreferrer&quot;&gt;'Electromagnetic Theory'&lt;/a&gt; -- (see also above) at pages 434-492. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further sources showing the relation between these practical situations and Heaviside's developments in mathematics include:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Oliver Heaviside, &lt;em&gt;Electrical Papers&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vol 1 (1892)&#xA;(&lt;a href=&quot;https://archive.org/details/electricalpapers01heavuoft&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/electricalpapers01heavuoft&lt;/a&gt;)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;vol 2 (1894)&#xA;(&lt;a href=&quot;https://archive.org/details/electricalpaper01heavgoog&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/electricalpaper01heavgoog&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and &#xA;Oliver Heaviside, &lt;em&gt;Electromagnetic theory&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vol 1 (1894)&#xA;(&lt;a href=&quot;https://archive.org/details/electromagnetict01heavrich&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/electromagnetict01heavrich&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vol 2 (1899)&#xA;(&lt;a href=&quot;https://archive.org/details/electromagnetict02heavrich&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/electromagnetict02heavrich&lt;/a&gt;}&lt;/p&gt;&#xA;&#xA;&lt;p&gt;vol 3 (1912)&#xA;(&lt;a href=&quot;https://books.google.co.uk/books?id=yrcMgchWSzEC&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.co.uk/books?id=yrcMgchWSzEC&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides  H J Josephs (1946) &lt;a href=&quot;https://catalog.hathitrust.org/Record/001986461&quot; rel=&quot;noreferrer&quot;&gt;'Heaviside's Electric Circuit Theory'&lt;/a&gt;, &#xA;there are also useful commentaries and developments of Heaviside's work in works by I A Bromwich and by J R Carson. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-02-16T20:35:35.223" LastActivityDate="2019-02-16T20:35:35.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8309" PostTypeId="2" ParentId="8307" CreationDate="2019-02-16T13:28:24.360" Score="7" Body="&lt;p&gt;J. Dieudonne, A History of Algebraic and Differential Topology, 1900 - 1960.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-02-16T13:28:24.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8310" PostTypeId="2" ParentId="8307" CreationDate="2019-02-16T13:42:33.400" Score="6" Body="&lt;p&gt;I. M. James, &lt;em&gt;History of Topology&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a collection of 40 essays by different authors, on topics related mostly to manifolds and algebraic topology.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps the page&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;from the MacTutor History of Mathematics site&#xA;will be of interest for aspects of topology before Poincaré came on the scene. Their general topics history page is &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/HistTopics/&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-01-01T00:17:29.523" LastActivityDate="2022-01-01T00:17:29.523" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8311" PostTypeId="2" ParentId="8304" CreationDate="2019-02-16T14:07:32.853" Score="5" Body="&lt;p&gt;I doubt if this is that, but there was an &lt;a href=&quot;https://books.google.com/books?id=OViw8TibA9YC&amp;amp;pg=PT1&amp;amp;dq=%22universal+information%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjC8bf7rsDgAhVHVd8KHU75DkUQ6AEIVDAI#v=onepage&amp;amp;q=%22universal%20information%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Office of Universal Information, Agency, and Reference&lt;/em&gt;&lt;/a&gt; in 19 Catherine Street in London, in 1848.  (The blurb at the back of the link above has a list of qualifications which seem inconsistent with &amp;quot;every conceivable question&amp;quot;, however.) Its managing director  was &lt;a href=&quot;https://en.wikipedia.org/wiki/Joachim_Hayward_Stocqueler&quot; rel=&quot;nofollow noreferrer&quot;&gt;J.H. Stoqueler&lt;/a&gt;, who seems to have a  lived a more interesting life than mine.&lt;/p&gt;&#xA;&lt;p&gt;Closer in time to Sylvester is a derogatory 1887 reference to a UIS in the &lt;a href=&quot;https://books.google.com/books?id=m1RJAQAAMAAJ&amp;amp;pg=RA1-PA445&amp;amp;dq=%22universal+information+society%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwj19qH4rMDgAhVkmuAKHcGlCw4Q6AEIKDAA#v=onepage&amp;amp;q=%22universal%20information%20society%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saturday Review of Literature,  Politics, Science  and Art&lt;/a&gt;, March 26 1887, p.445, column 1.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mr. Gladstone, who had come down on Tuesday afternoon,&#xA;made a theatrical entry bearing a large volume suggestive of the&#xA;mock-books of explosives carried by the St. Peterburg anachists,&#xA;and feigned to be very busy in taking notes, Mr. Childers&#xA;assisting him somewhat after the fashion of the unfortunate&#xA;bearers of the desks of the Universal Information Society who&#xA;perambulate the streets.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The picture this conjures up, of itinerant know-it-alls from Laputa  carrying desks in the streets,  makes me want more information.&lt;/p&gt;&#xA;&lt;p&gt;According to a column &amp;quot;General Foreign Notes&amp;quot; in the Boston fortnightly review, &lt;a href=&quot;https://www.google.com/books/edition/The_Literary_World/_jQ-AAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22for+universal+information%22&amp;amp;pg=PA297&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Literary World_&lt;/a&gt;, Sept6, 1884, p. 297:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An office for &amp;quot;universal information&amp;quot; has been opened near the British Museum. You pay your shilling, and ask your question, and get your answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The temporal proximity makes me think this might be the factual basis of Sylvester's 1885 reference, and the 1887 parliamentary report. A similar enterprise shows up in the Feb 1 1890 issue of &lt;a href=&quot;https://www.google.com/books/edition/The_Athen%C3%A6um/B21GAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22universal+information%22+%22british+museum%22&amp;amp;pg=PA133&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Athenaeum&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;UNIVERSAL INFORMATION BUREAU -- QUESTIONS ANSWERED on all SUBJECTS -- Fees from 1s -- FRANK HALLER, 56 Chancery Lane, W.C., Letters only.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And the Jan 29 1887 &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=chi.39519050&amp;amp;view=1up&amp;amp;seq=122&amp;amp;q1=universal%20information&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Illustrated London News&lt;/em&gt;&lt;/a&gt; has, in what I suppose is an &amp;quot;agony column&amp;quot; or Victorian analogue of a Stack Exchange post,  (on p. 112) this comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“E. M. J.&amp;quot; who wishes me to tell him the value of a genuine Mulready's cheap postage envelope, is referred to the Universal Information Office, Southampton-street, Bloomsbury. Down with the “dust,” “E. M. J.” (one shilling sterling), and the Southampton-street Bureau will tell you the cause of thunder; the meaning of the proverb, “‘Bate me an ace, quoth Bolton&amp;quot;; and why we should say, “Dancing the Hays,&amp;quot; instead of “Dancing the Hay.” -&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2021-05-17T20:38:13.500" LastActivityDate="2021-05-17T20:38:13.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8312" PostTypeId="2" ParentId="8305" CreationDate="2019-02-16T15:40:00.183" Score="3" Body="&lt;p&gt;I think we have an answer. The original paper is in French by Perrin. The translation is in Nature (1896). NEW EXPERIMENTS ON THE KATHODE RAYS Jean Perrin. It is a two paged article. &lt;a href=&quot;https://www.nature.com/articles/053298a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nature.com/articles/053298a0&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;With the help of magnetic deflection, he directed the cathode rays into a electroscope connected to a Faraday cylinder (see Figure 1). It is a nice experiment. In the end he concludes &quot;In short, the Faraday's cylinder became negatively charged when the kathode rays entered it, and only when they entered it; &lt;em&gt;the kathode rays are then charged with negative electricity&lt;/em&gt; (emphasis is his own). He does not give the details of his electroscope. However, testing the sign of the electroscope was rather an elementary experiment with charged rods.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-02-16T15:40:00.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8313" PostTypeId="2" ParentId="8305" CreationDate="2019-02-16T21:12:53.510" Score="0" Body="&lt;p&gt;I believe this goes back to the Hall effect.  The original experiment, carried by Hall in 1879, allows the &lt;a href=&quot;https://www.nist.gov/pml/engineering-physics-division/popular-links/hall-effect/hall-effect-measurements-introduction&quot; rel=&quot;nofollow noreferrer&quot;&gt;determination of carrier densities&lt;/a&gt;.  The &lt;a href=&quot;https://web.archive.org/web/20110727010116/http://www.stenomuseet.dk/skoletj/elmag/kilde9.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;original paper&lt;/a&gt; by Hall does not, however, mention this application so it is not clear how quickly carrier densities were determined using this method, although one would think this was done before Perrin did his own work approx 17 years later as referenced in another anwser.&lt;/p&gt;&#xA;" OwnerUserId="7093" LastActivityDate="2019-02-16T21:12:53.510" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8314" PostTypeId="2" ParentId="8306" CreationDate="2019-02-17T04:19:04.060" Score="6" Body="&lt;p&gt;Existence claims as theorems became fashionable after Hilbert introduced the axiomatic method. Before that people more often talked about problems and constructions (following Euclid's, or rather Pappus's, distinction). In this case, the construction was of a curve given its curvature and torsion as functions of arclength (the &quot;natural equations&quot;). For the plane curves the construction (naturally, from curvature alone) appears in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E051.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's De constructione aequationum ope motus tractorii aliisque ad methodum tangentium inversam pertinentibus (On the construction of equations using dragged motion, and of other things pertinent to the inverse method of tangents, 1736)&lt;/a&gt;, so McCleary ascribes the &quot;Fundamental Theorem of Plane Curves&quot; to him. Euler first introduces the idea of natural equation (without the name), and does the calculations for the tractrix. Later he used modified versions of natural equations (with angle of inclination instead of curvature) to show that logarithmic spiral was similar to its evolute (1740), and cycloid was congruent to it (1764), see Boyer's &lt;a href=&quot;https://books.google.com/books?id=MJ4oAwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Analytic Geometry, pp.227-8&lt;/a&gt;. According to &lt;a href=&quot;http://webmath2.unito.it/paginepersonali/sergio.console/CurveSuperfici/AG5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gray's Modern Differential Geometry of Curves and Surfaces&lt;/a&gt;, natural equations were subsequently studied by Lacroix and Hill, &lt;a href=&quot;https://epdf.tips/lacroix-and-the-calculus-science-networks-historical-studies.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lacroix and the Calculus by Domigues&lt;/a&gt; gives a lot of background on the early history of differential geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the case of space curves had to wait until Frenet (partly in 1847) and Serret (1851) introduced the system of equations for the moving frame, which are required for the construction. It was then accomplished by &lt;a href=&quot;https://eudml.org/doc/147848&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hoppe in Ueber die Darstellung der Curven durch Krümmung und Torsion (1862)&lt;/a&gt;. Struik does not seem to have been aware of Hoppe's work, as he attributes the Riccati equations method to Lie (1882) and Darboux (1887). &lt;a href=&quot;https://www.jstor.org/stable/2974768&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scofield in Curves of Constant Precession&lt;/a&gt; writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;This activity, called &quot;solving natural equations&quot;, is generally achieved by solving Riccati equations... Although the solution generally exists, it usually cannot be obtained explicitly. Euler [6] found explicit integral formulae for plane curves (where &lt;span class=&quot;math-container&quot;&gt;$\tau=0$&lt;/span&gt;) through direct geometric analysis. Hoppe [9] developed a general method for solving the natural equations for space curves by solving Riccati equations through a complicated sequence of integral transformations. He digressed to obtain formulae for the tangent, normal, and binormal indicatrices for general helices and essentially for curves of constant precession. Enneper [5] obtained explicit closed-form solutions for helices on revolved conic sections through direct geometric analysis.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-28T10:00:34.953" LastActivityDate="2019-06-28T10:00:34.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8315" PostTypeId="1" CreationDate="2019-02-17T05:54:31.067" Score="1" ViewCount="36" Body="&lt;p&gt;I have this vague idea that Astrophysics morphed out of Astronomy as a field of study and research.  I am curious if this is true and when did Astrophysics become separate from Astronomy as a field of study (if it is separate).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was an undergrad physics major in the 1960s and I do not remember the term Astrophysics ever being mentioned.  In fact, I remember when I first heard of it well after school, I thought it was all about rockets and satellites only.  That was likely my own ignorance though.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So when did the name Astrophysics become separate from Astronomy and how much do they overlap.  Would one call Astronomy (today) a specialization of Astrophysics?&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2019-02-17T05:54:31.067" Title="When did the research field of Astrophysics begin?" Tags="|terminology|astronomy|astrophysics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8316" PostTypeId="2" ParentId="8307" CreationDate="2019-02-17T17:10:39.617" Score="7" Body="&lt;p&gt;I don't entirely understand the question, but this book is probably relevant:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. H. Manheim, &lt;em&gt;The Genesis of Point Set Topology&lt;/em&gt; (1964).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I must have borrowed it from a library a long time ago. It is out of print. Used copies go for about a hundred quid. I suppose Dover might look kindly on a request for a reprint.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is very probably also relevant (and it is certainly a very good read):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;José Ferreirós, &lt;em&gt;Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics&lt;/em&gt; (1999; second edition Birkhäuser, Basel 2007).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is expensive, but perhaps it will appear again in one of Springer's annual Yellow Sales.&lt;/p&gt;&#xA;" OwnerUserId="8068" LastActivityDate="2019-02-17T17:10:39.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8317" PostTypeId="1" CreationDate="2019-02-17T19:42:51.557" Score="3" ViewCount="374" Body="&lt;p&gt;&quot;Baba wrote over 60 books, almost everyone on a different topic, writing on issues from astronomy, identified stars that European scientists technology could not discover until the late 1800s, calculus before Newton, identified Saturn’s rings, Jupiter’s moons, the orbit of several star systems including the Sirius star system, plotted the orbit of planets accurately to the year 1990, documented an immense force of density not visible to the naked eye (which many scholars essentially interpret as gravity but that term did not exist in his language at that time), documented the structure of the Milky Way galaxy, documented ovens that cooked to over 2000 degrees and made steel. Baba set up a system of calculation for weights for ships that the Songhai had built which were 100 feet long, 13 feet wide and could carry 80 tons. These ships sailed to South America and China and traded in Ivory, gold and silver.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;if the &lt;strong&gt;&quot;immensive density&quot;&lt;/strong&gt; part is accurate, wouldn't that mean the idea of universal gravitation predates Newton?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quora.com/Who-is-was-the-most-brilliant-scientist-of-African-descent&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/Who-is-was-the-most-brilliant-scientist-of-African-descent&lt;/a&gt;&#xA;&lt;a href=&quot;https://www.google.com/search?q=ahmed+baba+brilliant&amp;amp;rlz=1C1CHBF_enUS794US794&amp;amp;oq=ahmed+baba+brilliant&amp;amp;aqs=chrome..69i57.2207j0j4&amp;amp;sourceid=chrome&amp;amp;ie=UTF-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=ahmed+baba+brilliant&amp;amp;rlz=1C1CHBF_enUS794US794&amp;amp;oq=ahmed+baba+brilliant&amp;amp;aqs=chrome..69i57.2207j0j4&amp;amp;sourceid=chrome&amp;amp;ie=UTF-8&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2019-02-19T10:47:51.157" Title="Did the Idea of Universal Gravitation predate Newton?" Tags="|mathematics|physics|philosophy-of-science|gravity|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="8318" PostTypeId="1" CreationDate="2019-02-17T23:19:24.480" Score="3" ViewCount="358" Body="&lt;p&gt;&amp;quot;Bootstrapping&amp;quot; is a term which in general refers to a self-starting process. It is very heavily used in the field of computer science, but it also has uses elsewhere.&lt;/p&gt;&#xA;&lt;p&gt;For example, in mathematical logic it has been used to refer to the process of taking a set of axioms and then proving useful but tedious lemmas and theorems which will be necessary for future proofs of more interesting theorems. From Samuel Buss' PhD thesis &lt;em&gt;&lt;a href=&quot;http://www.math.ucsd.edu/%7Esbuss/ResearchWeb/BAthesis/Buss_Thesis_OCR.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bounded Arithmetic&lt;/a&gt;&lt;/em&gt;, dated 1985-1986:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term &amp;quot;bootstrapping&amp;quot; is a computer term which describes the process of starting the operations of a computer. It used to be common to power up a computer with only a small amount of software loaded, say about 80 bytes, the amount of data which fits on a Hollerith card. This small amount of software would be responsible for reading from tape or cards the entire operating system, thus making the computer fully operational. This process was called &amp;quot;bootstrapping&amp;quot; from the analogy of &amp;quot;lifting oneself by the bootstraps.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Similarly we need to bootstrap &lt;span class=&quot;math-container&quot;&gt;$\mathsf{S_2^1}$&lt;/span&gt;. That is, we shall have to do a lot of work to define some simple functions and predicates in &lt;span class=&quot;math-container&quot;&gt;$\mathsf{S_2^1}$&lt;/span&gt; (for example, subtraction). Once we have completed the bootstrapping it will be easy to show that &lt;span class=&quot;math-container&quot;&gt;$\mathsf{S_2^1}$&lt;/span&gt; is actually a fairly strong system which can define a variety of functions and predicates.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am wondering if either this term had been used before to describe this process, which is ubiquitous in mathematical logic as in for example defining ordinal arithmetic in ZFC so you can prove arithmetical theorems, or if Buss was just borrowing terminology from computer science given the close relationship between the proof theory of bounded arithmetic and computational complexity theory.&lt;/p&gt;&#xA;&lt;p&gt;For more context which might help, I think that the necessity of bootstrapping axiomatic theories in order to start doing actual work with them and whether or not this is a result of the theories being too generalized is an interesting foundational question. When writing about that question, I would like to be able to say something along the lines of &amp;quot;This process has been referred to as 'bootstrapping', see e.g. [Buss 1986]&amp;quot; so I am interested in the first recorded instances of the term being used.&lt;/p&gt;&#xA;" OwnerUserId="5664" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-02-18T07:10:39.330" Title="Origin of &quot;bootstrapping&quot; in mathematical logic" Tags="|terminology|mathematical-logic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8319" PostTypeId="2" ParentId="8318" CreationDate="2019-02-18T07:01:46.147" Score="3" Body="&lt;p&gt;All early mentions on MathSciNet refer to &lt;a href=&quot;https://en.wikipedia.org/wiki/Bootstrap_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;hadron bootstrap models&lt;/a&gt;, that &quot;that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles&quot;, e.g. &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.132.878&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rockmore Some Equivalent Approaches to the Self-Consistent Bound State, 1963&lt;/a&gt;, or computer science/statistics, e.g. &lt;a href=&quot;https://amstat.tandfonline.com/doi/abs/10.1080/01621459.1967.10500923#.XGpUX6CJh0c&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedman-Rubin On Some Invariant Criteria for Grouping Data, 1967&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We wish to separate the n objects into g groups in some &quot;optimal'' manner. Some definitions of &quot;optimal'' are proposed and some &quot;bootstrap&quot; (iterative) computer methods are proposed, with some numerical examples. One of the proposed methods is invariant with respect to non-singular linear transformations of the space, but the authors are not dogmatically in favour of any one method.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bootstrap for elliptic PDE, where one starts from some weak regularity assumptions about solutions and then proves that they are, in fact, much more regular, seemingly out of nothing (actually, because they are elliptic solutions) appears as early as 1972, e.g. in the review of &lt;a href=&quot;https://www.jstor.org/stable/2038537&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coffman's On the bifurcation theory of semilinear elliptic eigenvalue problems&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The author applies a &quot;bootstrap&quot; method in the bifurcation problem for the semilinear elliptic eigenvalue problem... for which the Dirichlet problem is solvable.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Miller compares the idea that there are no atomic facts to hadron bootstrap in a &lt;a href=&quot;https://philpapers.org/rec/MILTUO-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;commentary on the Tractatus&lt;/a&gt; in 1977 (&quot;&lt;em&gt;The article closes with some perfunctory speculation (put &quot;half seriously, half frivolously'') on the possibility that there are no basic, or atomic, facts. The entertained view is similar to the so-called &quot;nuclear democracy'' or &quot;hadron bootstrap'' view in nuclear physics&lt;/em&gt;&quot;), but I do not think this is relevant. It seems that Buss just took it from computer science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Scare quotes typically indicate novelty, they are absent in Rockmore's case, so hadron model use might have been established by then. As a colloquialism, the expression is much older, although the linkage to Raspe's Baron Munchausen, who &quot;bootstrapped&quot; himself out of the swamp by his own pigtail, appears to be a late addition. According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Bootstrapping#Etymology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's etymology&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The idiom dates at least to 1834, when it appeared in the Workingman's Advocate: &quot;It is conjectured that Mr. Murphee will now be enabled to hand himself over the Cumberland river or a barn yard fence by the straps of his boots.&quot;[4] In 1860 it appeared in a comment on philosophy of mind: &quot;The attempt of the mind to analyze itself [is] an effort analogous to one who would lift himself by his own bootstraps.&quot;&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-02-18T07:10:39.330" LastActivityDate="2019-02-18T07:10:39.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8320" PostTypeId="2" ParentId="8317" CreationDate="2019-02-18T07:29:32.953" Score="10" Body="&lt;p&gt;The &lt;em&gt;idea&lt;/em&gt;, yes, &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_gravitational_theory#India&quot; rel=&quot;noreferrer&quot;&gt;Aryabhata&lt;/a&gt; speculated about something like that as early as c. 500 AD, Brahmagupta called it &lt;em&gt;gurutvākarṣaṇ&lt;/em&gt;. So did Kepler, at about the same time as Ahmad Baba al Massufi (late 1500-s), and much less vaguely.&#xA;Russo even ascribes the idea to Hipparchus (c.150 BC), although this is far fetched. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even the inverse square law for gravity predates Newton, as he acknowledged in Principia, see &lt;a href=&quot;https://hsm.stackexchange.com/a/1975/55&quot;&gt;Who was first to explain intuitively the inverse square law of gravity?&lt;/a&gt; The originator is considered to be Bullialdus, who wrote in 1645: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It turns with the body of the Sun; now, seeing that it is corporeal, it becomes weaker and attenuated at a greater distance or interval, and the ratio of its decrease in strength is the same as in the case of light, namely, the duplicate proportion, but inversely, of the distances.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It was picked up by Huygens, Hooke, Halley, Wren, etc. Newton's fame is not for suggesting the idea but for convincing everyone that it works, universally - by deriving both Galileo's laws of free fall, and Kepler's laws of planetary motion from it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-02-18T18:30:09.607" LastActivityDate="2019-02-18T18:30:09.607" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8321" PostTypeId="2" ParentId="8317" CreationDate="2019-02-18T14:20:24.890" Score="6" Body="&lt;p&gt;'Did the Idea of Universal Gravitation predate Newton?'&#xA;(The question went on to mention the books of Ahmed Baba.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had a look to see whether Baba's work is available in any way online, but found nothing. Could the questioner point to any source, it would help discussion? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Commentators who discuss early origins of gravitational ideas, perhaps as a possible anticipation of Newton, often leave it very unclear just what sort of gravitational ideas they are discussing.  Then, when details do emerge, it has always so far turned out that what is being discussed is quite different from Newtonian gravitation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, the word 'gravity' means just heaviness, and no-one has ever seriously suggested that Newton discovered that, or that it was unknown before him.  Occasionally one reads unconsidered statements that Newton 'discovered gravity', but that is not what he is seriously reputed to have discovered. On the other hand, gravity before Newton was about terrestrial physics -- the properties of bodies at or near the earth's surface. There were older theories about what moved the celestial bodies, but they did not suggest that the same laws of physics applied to celestial bodies as to terrestrial objects. It seems that Newton was the first to suggest, and provide evidence in the form of his 'moon-test', that what moves the celestial bodies is of the same kind as what moves falling terrestrial bodies -- their heaviness, subject to the same laws of motion and inertia as heavy bodies on the earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Newton produced was also very specific. Considered for example in its astronomical application, a Newtonian gravitational force, if I'm not mistaken,  is &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- an accelerative force, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- directed towards a massive body, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- operating as an attraction on another body, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- and its measure is inversely proportional to the square of the distance between them.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides not accounting for terrestrial and celestial motions in the same way, earlier ideas of celestial motion discussed as possible anticipations of Newton have so far turned out either not to involve accelerative force, or not to involve forces directed centrally, etc.  R S Westfall's 1971 book 'Force in Newton's physics' offers detailed discussion of concepts of force before Newton, and shows how some of them were quite vague: the idea was slow to become clarified in the pre-Newtonian world.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Kepler's work is sometimes suggested as an anticipation of Newton, but Kepler's idea of force was not accelerative, he supposed that force maintains velocity, he did not think of it as producing acceleration, and it was not central. Thus, Kepler supposed that the sun's force came from its rotation and somehow impelled the planets along and around their orbital paths, as if their natural state was rest. His ideas about the moon's irregularities of motion were that they are caused by the Sun's illumination of the moon.  Kepler's ideas did not include Newton-like ideas either of force or of inertia, see Bruce Stephenson's &lt;a href=&quot;https://books.google.co.uk/books/about/Kepler_s_Physical_Astronomy.html?id=pxCYAeOqJg8C&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Kepler's Physical Astronomy'&lt;/a&gt; for a good discussion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quotation from Bullialdus introduced by &lt;em&gt;Conifold&lt;/em&gt; expressly states the idea of something that 'turns with the body of the Sun', which appears to show that whatever it is, it is not a central gravitational force: the central forces of Newtonian gravitation do not turn with rotation of the attracting body. (Bullialdus, if he thought that the turning of the Sun's body has anything to do with gravity, may have been expressing a similar idea as the one already mentioned as held by Kepler.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also important to distinguish between the bare idea of an inverse square law and the specifics of Newtonian gravitation. Anything that becomes spread out uniformly through space from a point-source is likely to follow an inverse-square law -- for example, light-intensity, it follows from the geometrical law for the surface area of a sphere, proportional to the square of its radius.  Inverse-square laws of &lt;em&gt;something&lt;/em&gt; are not peculiar to Newton, but whether that &lt;em&gt;something&lt;/em&gt; is anything like Newtonian accelerative gravitation clearly should not be taken for granted and merely assumed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's fame was, among other things, for producing a newly compiled set of laws of motion, which operate together with what was (then) a new law of accelerative gravitational force inversely proportional to the squares of the distances, and together these have enabled celestial motions to be accounted for with previously-unknown mathematical precision and accuracy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There have been many excessive claims for the accuracy of this or that ancient historical account of planetary motions and orbits. What is accurate enough in one age or context might be very rough in another.  It is very difficult to discuss such suggestions without concrete material such as a particular book or description to go on. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-02-19T10:47:51.157" LastActivityDate="2019-02-19T10:47:51.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8322" PostTypeId="2" ParentId="3209" CreationDate="2019-02-19T00:32:53.593" Score="3" Body="&lt;p&gt;You can only answer your question by a bibliometric analysis over the far and recent past. Luckily, there are nowadays quite a &lt;a href=&quot;http://Sabine%20Hossenfelder&quot; rel=&quot;nofollow noreferrer&quot;&gt;lot of tools&lt;/a&gt; to evaluate the data. But this is a research question and some has to do this evaluation or you have to look on google scholar, searching in bibliometrics or philosophy of science journals if someone maybe already has done such analysis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To add to Conifolds answer, there are also physicists with the opinion that there is too much focus on mathematics in cosmology and particle physics nowadays. The recent book &lt;a href=&quot;http://www.math.columbia.edu/~woit/wordpress/?p=10314&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Lost in Math&lt;/em&gt;&lt;/a&gt; by Sabine Hossenfelder shares very interesting interviews with world-leading theoretical physicists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From a more philosophical/meta-physical point of view, if mathematics is the language of nature or just the best mankind/human brain is able to develop is a neverending discussion. But I find it quite interesting to think about if the progress in math/physics is limited by each other! But I would be very much surprised, if the collaboration between both branches has shrinked in the recent 2 decades since internet boosted scientific publications and collaboration. If you could prove this by bibliometric analysis, this would trigger major questions, as it would seem very counter-intuitive.&lt;/p&gt;&#xA;" OwnerUserId="2947" LastActivityDate="2019-02-19T00:32:53.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8323" PostTypeId="2" ParentId="8282" CreationDate="2019-02-19T10:51:32.183" Score="0" Body="&lt;p&gt;I’ve read that certain linguists actually wonder, given the complexity of language and thought, as to how people actually communicate given that they most manifestly do. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don’t really know the research literature on this but my tuppence is that it matters a great deal as to whether people are on the same wavelength, so to speak. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After all, if one is going to a conference then one might suspect that there is already much in common that one already has with the other participants in terms of the major questions at hand. Also, it’s important to recognise that scientific language despite its wide vocabulary - mainly due to its many specialisms - actually uses quite simple language on the whole. This makes it relatively straight-forward to learn scientific English as it only uses a small fragment of the language and its resources. The main problem, is understanding the concepts. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for anecdotes I can’t recall any offhand that is about what you are asking for. I do recall Richard Feynman writing about his teenage years how he by inventing his own symbols for already existing terms wasn’t helping him to communicate and that having a common language to communicate in was important. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-02-19T10:51:32.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8324" PostTypeId="2" ParentId="8286" CreationDate="2019-02-19T11:36:15.757" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Einstein, on the contrary, was free of any kind of traditions ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This I think really says a great deal more about the author, his preconceptions, the Einstein hagiographical industry and the fetishisation of a free-floating and heavily romanticised libertarianism in Western culture than about Einstein himself. And I mean romanticism in the technical sense of the word, that absurd Nietzschean picture of man so proud, so free, so overcoming his mortal nature that he stands by himself in a kind of exponentiated freedom without duties, without responsibilities, without obligations - to others, I mean - but only to himself. The historical record shows that this is just plain wrong. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein was heavily involved with the physical tradition, not only did he read Maxwell closely, he read Hertz, Helmholtz, Kirchhoff, Lorentz and Mach closely. It’s because he was steeped in the physical tradition that he could make the leaps that he did. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It’s also important to recognise that his ideas did not come from nowhere, there were others in pursuit of the same goals, broadly speaking. It’s usually said, for example, that whilst any number of others could have made the correct conceptual leap to Special Relativity - Poincare, for example; that only Einstein could have made the conceptual leap to GR. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This turns out to be wrong. For example, the English mathematician Clifford made the right guess over fifty years before Einstein having learnt of Riemanns theory of geometry, the same theory that Einstein relied upon with GR. Moreover, the Scandinavian physicist, Nordstrom came out with a &lt;em&gt;geometric&lt;/em&gt; theory of gravity that Einstein was aware of and also had the right terms in it - aka the stress-energy tensor and the Riemann curvature - (but he had reduced the terms to scalars rather than kept them as tensors) - and Einstein congratulated him upon his great discovery; even more, he discovered the possibility - which Einstein did not, although Einstein immediately recognised its significance - that a fifth dimension (physically, an additional degree of freedom) helped unify EM &amp;amp; gravity. This was rediscovered twenty years later but didn’t really become part of mainstream physics until the 70s when QFT was reformulated in geometric language. For example, the physicist Yang said that he hadn’t understood the deep geometrical reasoning behind the generalisation of Maxwell’s EM he had worked out with Mills, aka Yang-Mills, until it was explained to him by Chern. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Maxwell did, was to come up with Maxwell’s equations, and this was enough because deep within them lay a symmetry of motion that was different from that of Galileo and Newton - this had to be explained. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-02-19T11:36:15.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8325" PostTypeId="2" ParentId="3209" CreationDate="2019-02-19T15:16:54.780" Score="1" Body="&lt;p&gt;Mathematics and physics have a long standing relationship. In some periods there is more interaction than at others. For example, there was a large amount of work in the 70s when physicists and mathematicians realised that some of the tools and theories they had developed had much in common with each other, and would serve to Illuminate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although I’m not a big fan of string theory, by all accounts it has helped invigorate many areas of mathematics. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-02-19T15:16:54.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8326" PostTypeId="2" ParentId="5834" CreationDate="2019-02-20T05:39:18.253" Score="5" Body="&lt;p&gt;Einstein himself wrote, in a letter of 14 December 1915 to Moritz Schlick, that he was primarily inspired by &lt;a href=&quot;https://en.wikipedia.org/wiki/David_Hume&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Hume&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Mach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ernst Mach&lt;/a&gt;.  According to the paper &lt;a href=&quot;https://www.pitt.edu/%7Ejdnorton/papers/HumeMach.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Hume and Mach Helped Einstein Find Special Relativity&lt;/a&gt;, &amp;quot;It was more Hume than Mach.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Quoting from Einstein's letter to Schlick:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your exposition is also quite right that positivism suggested rel. theory, without&#xA;requiring it. Also you have correctly seen that this line of thought was of great&#xA;influence on my efforts and indeed E. Mach and still much more Hume, whose&#xA;treatise on understanding I studied with eagerness and admiration shortly before&#xA;finding relativity theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Einstein goes on to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is very possible that without these philosophical studies I can not say that the solution would have come.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Einstein's &lt;em&gt;Autobiographical Notes&lt;/em&gt; of 1949 he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Today everyone knows, of course, that all attempts to clarify this paradox [of light&#xA;that leads to special relativity] satisfactorily were condemned to failure as long as the axiom of the absolute character of time, or of simultaneity, was rooted unrecognized&#xA;in the unconscious. To recognize clearly this axiom and its arbitrary character already&#xA;implies the essentials of the solution of the problem. The type of critical reasoning&#xA;required for the discovery of this central point was decisively furthered, in my case,&#xA;especially by the reading of David Hume’s and Ernst Mach’s philosophical writings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT 07, Aug 2022&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In a letter to his old friend Michele Besso, referring to Mach's anti-metaphysical philosophy, Einstein wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mach's way cannot give birth to anything living; it can only exterminate harmful vermin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2022-08-07T18:31:09.360" LastActivityDate="2022-08-07T18:31:09.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8327" PostTypeId="1" CreationDate="2019-02-20T08:30:54.997" Score="10" ViewCount="576" Body="&lt;p&gt;Based on Lorentz factor &lt;span class=&quot;math-container&quot;&gt;$\gamma = \frac{1}{\sqrt {1-\frac{v^2}{c^2}}}$&lt;/span&gt; it is easy to see &lt;span class=&quot;math-container&quot;&gt;$v &amp;lt; c$&lt;/span&gt; since otherwise &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; would be either undefined or a complex number, which is non-physical. Also, as far as I understand this equation was known before Einstein's postulates were published. My question is: why didn't Lorentz himself conclude that no object can go faster than speed of light? Or maybe he did, I do not know. I feel I am missing some context here.&lt;/p&gt;&#xA;" OwnerUserId="9625" LastEditorUserId="9630" LastEditDate="2019-02-20T16:00:09.557" LastActivityDate="2019-02-20T16:00:09.557" Title="Why didn't Lorentz conclude that no object can go faster than light?" Tags="|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8328" PostTypeId="2" ParentId="3131" CreationDate="2019-02-20T09:55:05.293" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oded_Schramm&quot; rel=&quot;noreferrer&quot;&gt;Oded Schramm&lt;/a&gt; died in a climbing accident at the age of 46. He is best known for his work on &lt;a href=&quot;https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution&quot; rel=&quot;noreferrer&quot;&gt;Schramm–Loewner evolution&lt;/a&gt;. He also &lt;a href=&quot;https://en.wikipedia.org/wiki/Special:Contributions/OdedSchramm&quot; rel=&quot;noreferrer&quot;&gt;contributed to Wikipedia&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="9627" LastActivityDate="2019-02-20T09:55:05.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8329" PostTypeId="2" ParentId="8327" CreationDate="2019-02-20T10:03:41.727" Score="14" Body="&lt;p&gt;While Lorentz (before 1905) himself didn't directly address the question whether light speed is a universal limiting speed, there were many physicists before Einstein who argued that the speed of light cannot be reached, at least in the context of electrically charged particles. For instance, since 1881 the concept of electromagnetic mass was used, according to which the inertia of a particle increases at higher velocity due to its electromagnetic self-energy. &lt;a href=&quot;https://en.wikipedia.org/wiki/Electromagnetic_mass#Thomson_and_Searle&quot; rel=&quot;noreferrer&quot;&gt;J.J. Thomson concluded in 1893&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[p. 21] When in the limit v = c, the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain constant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Or &lt;a href=&quot;https://en.wikipedia.org/wiki/Electromagnetic_mass#Thomson_and_Searle&quot; rel=&quot;noreferrer&quot;&gt;G.F.C. Searle in 1897&lt;/a&gt;, whose computation were based on the so-called &quot;Heaviside ellipsoid&quot;, according to which the spherical fields of moving charges become ellipsoids:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;.. when v = c the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1899 and 1904, Lorentz completed his &lt;a href=&quot;https://en.wikipedia.org/wiki/Lorentz_ether_theory&quot; rel=&quot;noreferrer&quot;&gt;electron model&lt;/a&gt; which includes the Lorentz factor as well as alternative formulas for electromagnetic mass. This, together with Searle's concept of the Heaviside ellipsoid, brought &lt;a href=&quot;https://de.wikisource.org/wiki/Erwiderung_auf_die_Kritik_des_Hrn._M._Abraham&quot; rel=&quot;noreferrer&quot;&gt;Wilhelm Wien in 1904&lt;/a&gt; to the conclusion that the Heavisde ellipsoid prohibits faster than light motion, because the Lorentz factor  &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1-(v^{2}/c^{2})}$&lt;/span&gt; becomes imaginary.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Henri Poincaré (1904) came closest to Einsteins postulate of the constancy of the speed of light, &lt;a href=&quot;https://en.wikisource.org/wiki/The_Principles_of_Mathematical_Physics&quot; rel=&quot;noreferrer&quot;&gt;by writing&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From all these results, if they were confirmed, would arise an entirely new mechanics, which would be, above all, characterized by this fact, that no velocity could surpass that of light, [Because bodies would oppose an increasing inertia to the causes which would tend to accelerate their motion; and this inertia would become infinite when one approached the velocity of light.] any more than any temperature can fall below absolute zero. No more for an observer, carried along himself in a translation he does not suspect, could any apparent velocity surpass that of light; and this would be then a contradiction, if we did not recall that this observer would not use the same clocks as a fixed observer, but, indeed, clocks marking 'local time.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There he alluded to the relation between the &quot;apparent&quot; speed of light as a limiting speed, the relativistic mass (which includes the Lorentz factor), the relativity principle, and &quot;local time&quot; based on &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_synchronisation&quot; rel=&quot;noreferrer&quot;&gt;Poincaré-Einstein synchronisation&lt;/a&gt; in contradistinction to &quot;true time&quot; in the aether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Einstein (1905) was the first to note, that all these relations follow from the relativity principle and light speed constancy alone, and are directly related to space and time, without any reference to the aether - which is precisely what we now call &lt;a href=&quot;https://en.wikipedia.org/wiki/Special_relativity&quot; rel=&quot;noreferrer&quot;&gt;special relativity&lt;/a&gt;. This laid the basis of Minkowski's &lt;a href=&quot;https://en.wikipedia.org/wiki/Spacetime&quot; rel=&quot;noreferrer&quot;&gt;spacetime&lt;/a&gt; formulation of relativity.&lt;/p&gt;&#xA;" OwnerUserId="7601" LastEditorUserId="7601" LastEditDate="2019-02-20T10:36:15.480" LastActivityDate="2019-02-20T10:36:15.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8330" PostTypeId="2" ParentId="3131" CreationDate="2019-02-20T13:45:28.750" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes&quot; rel=&quot;noreferrer&quot;&gt;Archimedes of Syracuse&lt;/a&gt; (287 - 212 BC) died in a tragic episode of the Second Punic War between Rome and Carthage. At that time he was already 75 years old.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Syracuse was an important Greek colony on the island of Sicily with good relations to Rome. However, during the Second Punic War Syracuse ended up being under a long Roman siege. One reason for the length of the siege was the ingenuity of Archimedes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, the city fell and Archimedes was killed by an angry Roman soldier as he didn't want to interrupt his geometrical studies. His alleged last words were:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;Noli turbare circulos meos&quot;&lt;/em&gt; (&lt;em&gt;&quot;Do not disturb my circles!&quot;&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9631" LastActivityDate="2019-02-20T13:45:28.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8331" PostTypeId="1" CreationDate="2019-02-21T12:43:12.300" Score="1" ViewCount="189" Body="&lt;p&gt;The question could be answered in a number of ways:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Historically (e.g. Egyptians did for &amp;lt;...&gt; reasons) &lt;/li&gt;&#xA;&lt;li&gt;Mathematically (It is a highly composite number)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;I'm looking for a &lt;em&gt;mathematical&lt;/em&gt; answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've got a basic idea of this, but I want to understand any underlying theories, and I want to be more knowledgeable on this matter so as to be able to see similar patterns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea I've used and been told of is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The ratio of the number of factors of a number to the number itself gives us a good idea of how easily it can be used to count.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The number 12 definitely satisfied this criteria, at least in comparison to many of the numbers that come before it. Hence, there are 12 hours in half a day. It could easily be divided in 4 parts, so that one could do 4 different things, or it could be easily divided into 6 parts... etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please give me further insight.&lt;/p&gt;&#xA;" OwnerUserId="9637" LastEditorUserId="5788" LastEditDate="2019-02-21T23:22:18.087" LastActivityDate="2019-02-21T23:22:18.087" Title="Why are there 24 hours in a day?" Tags="|mathematics|time|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8332" PostTypeId="1" CreationDate="2019-02-22T03:38:39.107" Score="11" ViewCount="904" Body="&lt;p&gt;I hope this question is not inappropriate for this site; I found hsm.stackexchange better suited for it than MathOverflow or math.stackexchange. The motivation for it is just curiosity.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: Which mathematical papers have been co-authored by people who were / are direct blood relatives (i.e. supposedly in most cases: siblings, or parent and child)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Everyone knows the Bernoullis (among the famous ones of which there were also uncle-nephew and cousin relations), but I'm looking for more, and more contemporary, examples.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More generally, I would be interested in family &lt;em&gt;collaboration&lt;/em&gt;, but I ask specifically for co-authored papers because that's the most obvious way to make the notion rigorous.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, I &lt;em&gt;exclude&lt;/em&gt; collaboration between partners, spouses, married couples etc. on purpose, partly because I assume that list would be longer, and partly because such relations have (sometimes vague) start (or end) points in time, allowing discussion about whether a collaboration happened while the relationship existed or not; unless of course there might be an example where e.g. a parenting couple published something together with a child of theirs.&lt;/p&gt;&#xA;" OwnerUserId="6514" LastEditorUserId="604" LastEditDate="2023-06-19T13:40:27.953" LastActivityDate="2023-07-09T17:25:46.467" Title="Examples of papers co-authored by parent/child, or siblings" Tags="|social-context|examples|big-list|weyl|" AnswerCount="17" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8333" PostTypeId="2" ParentId="8332" CreationDate="2019-02-22T07:37:29.103" Score="13" Body="&lt;p&gt;Elie and Henri Cartan (father and son) published a paper together, in 1931 (&lt;em&gt;Les transformations des domaines cerclés bornés&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-02-22T07:37:29.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8334" PostTypeId="2" ParentId="8332" CreationDate="2019-02-22T09:30:32.393" Score="8" Body="&lt;p&gt;Richard K. Guy - well known for work in recreational mathematics,  and son Michael J. T. Guy - computer scientist and mathematician, co-published the paper &quot;On rational Morley triangles&quot; in Acta Arithmetica. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Authors: Bremner, Andrew; Goggins, Joseph R.; Guy, Michael J. T.; Guy, R. K. (2000). &lt;/p&gt;&#xA;" OwnerUserId="6562" LastActivityDate="2019-02-22T09:30:32.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8335" PostTypeId="2" ParentId="8332" CreationDate="2019-02-22T17:28:39.883" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Rubin&quot; rel=&quot;noreferrer&quot;&gt;Arthur Leonard Rubin&lt;/a&gt; co-authored at least two papers with his mother, &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_E._Rubin&quot; rel=&quot;noreferrer&quot;&gt;Jean Estelle Hirsh Rubin&lt;/a&gt;, the first one below when he was 13 years old.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arthur L. Rubin and Jean E. Rubin, &lt;a href=&quot;http://matwbn.icm.edu.pl/ksiazki/fm/fm65/fm65123.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Extended operations and relations on the class of ordinal numbers&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Fundamenta Mathematicae&lt;/strong&gt; 69 #2 (1969), 227-242.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Paul Howard, Arthur L. Rubin, and Jean E. Rubin, &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF01326766&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Kinna-Wagner selection principles, axioms of choice and multiple choice&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Monatshefte für Mathematik&lt;/strong&gt; 123 #4 (December 1997), 309-319.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2019-02-22T17:28:39.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8336" PostTypeId="2" ParentId="8332" CreationDate="2019-02-22T21:24:23.670" Score="18" Body="&lt;p&gt;Check out these mathematicians:  A father, and two sons, all of whom&#xA;co-authored papers in various combinations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/David_Borwein&quot; rel=&quot;noreferrer&quot;&gt;David Borwein&lt;/a&gt;, father&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Peter_Borwein&quot; rel=&quot;noreferrer&quot;&gt;Peter Borwein&lt;/a&gt;, son &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jonathan_Borwein&quot; rel=&quot;noreferrer&quot;&gt;Jonathan Borwein&lt;/a&gt;, son&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peter's &lt;a href=&quot;https://jonborwein.org/2016/08/my-brother-jon-by-peter-borwein/&quot; rel=&quot;noreferrer&quot;&gt;memorial&lt;/a&gt; to his brother Jonahan, in which he speaks of writing&#xA;papers together with his brother.&lt;/p&gt;&#xA;" OwnerUserId="9641" LastActivityDate="2019-02-22T21:24:23.670" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8337" PostTypeId="2" ParentId="8332" CreationDate="2019-02-23T01:35:50.597" Score="12" Body="&lt;p&gt;Brothers Marcel and Frigyes Riesz have a joint paper,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brothers Rolf and Frithiof Nevanlinna have 6 joint papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brothers Alexander and Alexei Zamolodchikov have 8 joint papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brothers David and Gregory Chudnovsky have 80 joint papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This list can be made very long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Father-son collaborations are not so frequent but Herman Weyl has a paper and a book&#xA;written jointly with his son Joachim.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-09T12:54:53.623" LastActivityDate="2019-05-09T12:54:53.623" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8338" PostTypeId="2" ParentId="8332" CreationDate="2019-02-23T01:37:09.877" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ken_Ono&quot; rel=&quot;noreferrer&quot;&gt;Ken Ono&lt;/a&gt; published a paper with his father &lt;a href=&quot;https://en.wikipedia.org/wiki/Takashi_Ono_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Takashi Ono&lt;/a&gt; in 1996: &lt;a href=&quot;https://projecteuclid.org/euclid.pja/1195510211&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;``&lt;/span&gt;Quadratic forms and elliptic curves, III''&lt;/em&gt;&lt;/a&gt; in Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 9, 204–205.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2019-02-23T16:41:08.757" LastActivityDate="2019-02-23T16:41:08.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8339" PostTypeId="1" CreationDate="2019-02-23T03:00:40.240" Score="0" ViewCount="102" Body="&lt;p&gt;&lt;a href=&quot;https://www.brown.edu/academics/applied-mathematics/philip-j-davis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philip J. Davis&lt;/a&gt;' article on the &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Davis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of the gamma function&lt;/a&gt; (PDF) mentions how &lt;a href=&quot;https://www.iep.utm.edu/leib-ove/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz&lt;/a&gt; proposed the &lt;a href=&quot;https://www.mathpages.com/home/kmath616/kmath616.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;iterated differential operator&lt;/a&gt; (p. 851 in the upper right corner, or p. 3 of the PDF, about half-way down the page).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dr. Davis uses a lower-case 'd' in his article.  However, most contemporary literature (er, Web sites), shows an upper-case 'D' for this.  Examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.mathpages.com/home/kmath616/kmath616.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathpages.com/home/kmath616/kmath616.htm&lt;/a&gt; (same link as above)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://chalkdustmagazine.com/features/fractional-calculus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://chalkdustmagazine.com/features/fractional-calculus/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://mathworld.wolfram.com/FractionalDerivative.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://mathworld.wolfram.com/FractionalDerivative.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://medium.com/@notaredpanda/fractional-calculus-48192f4e9c9f&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://medium.com/@notaredpanda/fractional-calculus-48192f4e9c9f&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.hindawi.com/journals/ddns/2011/562494/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hindawi.com/journals/ddns/2011/562494/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.xuru.org/fc/riemannliouville.asp&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.xuru.org/fc/riemannliouville.asp&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm copy-editing Davis' paper (just for fun and to practice LaTeX), and so now I'm curious:&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;What did Leibniz use - 'd' or 'D'?&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Please note that I'm asking specifically about the &lt;em&gt;iterated&lt;/em&gt; differential operator (often called a &lt;a href=&quot;https://www.google.com/search?q=riemann%20liouville%20fractional%20derivative&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann-Liouville fractional derivative&lt;/a&gt;, or RLFD), and not the 'd' in Leibniz's &lt;span class=&quot;math-container&quot;&gt;$\frac{\text{d}y}{\text{d}x}$&lt;/span&gt; notation (although they are related).&lt;/p&gt;&#xA;" OwnerUserId="9642" LastActivityDate="2019-02-23T03:00:40.240" Title="Is it the 'd' or 'D' operator?" Tags="|mathematicians|notation|calculus|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8340" PostTypeId="2" ParentId="8332" CreationDate="2019-02-23T07:30:36.913" Score="10" Body="&lt;p&gt;Luis Alvarez and his son Walter, and two other chemists, co-authored &lt;a href=&quot;http://science.sciencemag.org/content/208/4448/1095&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; 1980 paper on iridium levels at the K/T boundary. The Alvarezes conjectured the now widely accepted impact hypothesis explaining that extinction event.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another example: Katharine Cook Briggs and her daughter Isabel Briggs Myers co-developed &lt;a href=&quot;https://en.wikipedia.org/wiki/Myers%E2%80%93Briggs_Type_Indicator&quot; rel=&quot;nofollow noreferrer&quot;&gt;MBTI&lt;/a&gt; (I'll leave it to the reader to decide whether that is science).&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2019-05-09T08:19:07.593" LastActivityDate="2019-05-09T08:19:07.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8342" PostTypeId="1" AcceptedAnswerId="8384" CreationDate="2019-02-23T16:55:28.050" Score="3" ViewCount="1362" Body="&lt;p&gt;When the &amp;quot;Cartesian coordinate system&amp;quot; was discovered By René Descartes, Algebra and Geometry were connected. How exactly did that affect Newton and Leibniz in the invention of what we know as &amp;quot;Calculus&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Is the concept of graph the way of showing the function geometrically, by using &amp;quot;Cartesian coordinate system&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;If that is true, could we also have connection between &amp;quot;Calculus&amp;quot; and &amp;quot;Geometry&amp;quot; in graphing functions with coordinate system?&lt;/p&gt;&#xA;" OwnerUserId="9644" LastEditorUserId="4251" LastEditDate="2021-02-28T08:09:26.973" LastActivityDate="2021-02-28T08:09:26.973" Title="Historically, how did René Descartes's works affect the invention of calculus?" Tags="|calculus|geometry|newton|descartes|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8343" PostTypeId="1" CreationDate="2019-02-23T07:58:19.567" Score="6" ViewCount="289" Body="&lt;p&gt;In reading &lt;a href=&quot;https://groups.google.com/forum/#!msg/sci.crypt/yLh8TmH-4Bc/5VGTa1HK76QJ&quot; rel=&quot;noreferrer&quot;&gt;a description on Usenet&lt;/a&gt; of a NIST competition for selecting a standard cipher, I read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Consider that the best currently known methods for factoring use&#xA;  randomization:  Construct enough cases at random, and eventually you can&#xA;  paste them together into a set of factors.  &lt;em&gt;This notion of an algorithm&#xA;  is very new - at best, 40-50 years old&lt;/em&gt;, though in terms of actual&#xA;  practice, perhaps no more than 25 years old.  Mathematics in the past&#xA;  has dealt with proofs, which may be constructive or non-constructive.&#xA;  Constructive proofs have historically been given as deterministic&#xA;  algorithms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Emphasis mine. I know for a fact that the concept is much older than that, as there are numerous algorithms named after people from classical antiquity. This makes me wonder how old the concept of the algorithm really is. Wikipedia merely says that it has existed for centuries, and cites the Sieve of Eratosthenes as an early example. Here I define algorithm as an unambiguous series of steps and procedures to solve a specific problem, particularly a mathematical problem. Note that I am not asking about the &lt;em&gt;terminology&lt;/em&gt;, but about the earliest known instance of a well-defined algorithm.&lt;/p&gt;&#xA;" OwnerUserId="9646" OwnerDisplayName="forest" LastEditorUserId="4251" LastEditDate="2020-07-06T15:39:31.887" LastActivityDate="2022-10-31T21:58:11.913" Title="What is the earliest instance of the use of an algorithm to solve problems?" Tags="|mathematics|algorithm|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="8345" PostTypeId="2" ParentId="8343" CreationDate="2019-02-23T22:41:55.647" Score="4" Body="&lt;p&gt;Surely the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rhind Papyrus&lt;/a&gt; (around 1500 BC) contains many algorithms.  At that time, such things were presented this way:  State a specific problem, then show how to solve it.  After several such problems/solutions, the algorithm is supposed to be clear, and the student can solve similar problems in the same way.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-02-23T22:41:55.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8346" PostTypeId="1" CreationDate="2019-02-24T02:01:53.383" Score="7" ViewCount="265" Body="&lt;p&gt;The spectrum of hydrogen was very well studied by the mid-19th century. However, if one were doing experimental spectroscopy for more complex atoms, one would see plenty of spectral lines in the spectra of sodium or potassium. A diagram for lithium by Grotrian &quot;&lt;em&gt;Graphische Darstellung der Spektren von Atomen und Ionen mit ein, zwei und dreivalenzelektronen&lt;/em&gt;&quot; (1927) is attached below. What protocol was used to assign the terms associated with spectral lines?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key question is how did physicists experimentally assign electron transitions say for example, 670 nm emission line is from 2s to 2p levels? How did they calculate energy of 2s and 2p for multi-electron atoms? Is the scale based on the hydrogen atom? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/v7M4e.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/v7M4e.png&quot; alt=&quot;Historical diagram by Grotrian (1927) for atomic spectrum for 1, 2, and 3 valence electrons&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="123" LastEditDate="2019-02-28T15:37:40.260" LastActivityDate="2019-02-28T15:37:40.260" Title="How did early physicists experimentally assign electronic transitions in atoms?" Tags="|terminology|experimental-physics|chemistry|atomic-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8349" PostTypeId="2" ParentId="2129" CreationDate="2019-02-27T15:08:12.467" Score="1" Body="&lt;p&gt;By chance, I was also looking for the original reference for some time. Checked Scientific American reference from 1996 cited above, but it does not seem to exist. The page 10, is nowhere and it has some other topics. I used this quote from Wikipedia a month ago for an editorial, which also appears in several books. I regret that I trusted Wikipedia and other books for this quote  &quot;in this field, almost everything is already discovered, and all that remains is to fill a few unimportant holes&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Only &quot;Max Planck: Wege zur Physikalischen Erkenntnis. Reden und Vorträge, Band 1. Leipzig 1943&quot; has the original version- right from the horse's mouth on page 128. I used DeepL translator to get the idea&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;„als eine hochentwickelte, nahezu voll ausgereifte Wissenschaft die nunmehr, nachdem ihr durch die Entdeckung der Energie gewissermaßen die Krone aufgesetzt sei, wohl bald ihre endgültige stabile Form angenommen haben würde. Wohl gäbe es vielleicht in einem oder dem anderen Winkel noch ein Stäubchen oder ein Bläschen zu prüfen und einzuordnen, aber das System als Ganzes stehe ziemlich gesichert da, und die theoretische Physik nähere sich merklich demjenigen Grade der Vollendung, wie ihn etwa die Geometrie schon seit Jahrhunderten besitzt“.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;as a highly developed, almost fully developed science which, now that it has been crowned, so to speak, by the discovery of energy, would soon have assumed its final stable form. There might still be a little dust or bubble at one or the other angle to check and classify, but the system as a whole is quite secure, and theoretical physics is noticeably closer to that degree of perfection that geometry has had for centuries.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Translated with www.DeepL.com/Translator&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-02-27T16:35:11.520" LastActivityDate="2019-02-27T16:35:11.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8350" PostTypeId="1" AcceptedAnswerId="8354" CreationDate="2019-02-28T14:40:49.373" Score="2" ViewCount="1857" Body="&lt;p&gt;I made a small research about the history before JavaScript was available to use, but I couldn't find much. I only found several plugins that was used, such as Adobe Flash, ActiveX.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, what I want is to talk about why those are replaced with JS. Is that because of slow data-parsing, security issues?&lt;/p&gt;&#xA;" OwnerUserId="9663" LastEditorUserId="217" LastEditDate="2019-03-03T07:46:57.180" LastActivityDate="2022-08-19T09:07:11.323" Title="What technologies were used before JavaScript?" Tags="|computer-science|computers|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8352" PostTypeId="2" ParentId="8332" CreationDate="2019-02-28T22:01:15.753" Score="8" Body="&lt;p&gt;There are four Polish brothers ( Ryszard, Pawel, Michal and Karol Horodecki) all working in quantum information science, and they have several common papers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their most cited paper (with &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;5000$&lt;/span&gt; Google Scholar citations at the time of writing) is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Horodecki, R., Horodecki, P., Horodecki, M., &amp;amp; Horodecki, K. (2009). Quantum entanglement. Reviews of modern physics, 81(2), 865.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This one (without brother Karol)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Horodecki, Michał, Paweł Horodecki, and Ryszard Horodecki. &quot;Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps.&quot; Physics Letters A 283.1-2 (2001): 1-7.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;has collected a little under 4000 Google citations at the time of writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They also all work at Gdańsk University.&lt;/p&gt;&#xA;" OwnerUserId="7093" LastActivityDate="2019-02-28T22:01:15.753" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8353" PostTypeId="1" CreationDate="2019-03-02T21:13:40.033" Score="3" ViewCount="72" Body="&lt;p&gt;Consider a mouse who lives its whole life in a maze in a lab. If you asked the mouse about its understanding of the universe, it might say &quot;it's 3 inches high and 3 inches wide, has two left turns and three right turns, and there's cheese at the end&quot;. Of course the scientist outside the maze has a different--and much broader--conception of the universe. This is because the mouse's powers of observation are limited by its enclosure in the maze. The scientist knows that her model of the universe is more accurate, but the mouse's is consistent with the observable data, and useful for its purposes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or consider pre-Galilean humanity. Their experience of the world was consistent with a flat Earth: that model matches the observable data, and is useful enough for travel on human scales. Once the technology and analytic capacity to observe the stars and planets was invented (and a few centuries later, accepted), the flat-Earth model was discarded in favor of a broader one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a word, phrase, or theory that describes this manner in which (1) a model can be consistent with all local observable data, but later, (2) in the face of more data, broadened, adapted, or replaced?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think there are a few interrelated ideas here, so any pointers to relevant ideas or literature are appreciated!&lt;/p&gt;&#xA;" OwnerUserId="9675" LastActivityDate="2019-03-02T21:13:40.033" Title="Is there a name for the idea that the limitations on the accuracy of a model depend on the point of view of the observer?" Tags="|philosophy-of-science|scientific-method|naming-conventions|resource-recommendation|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8354" PostTypeId="2" ParentId="8350" CreationDate="2019-03-03T01:32:29.557" Score="5" Body="&lt;p&gt;Javascript is older than Flash and ActiveX.&lt;/p&gt;&#xA;&lt;p&gt;Originally, it was developed by the Netscape, as the &amp;quot;small cousine&amp;quot; of the Java.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;World Wide Web&amp;quot; was originally developed only for static content without client-side dynamical behaviors, and with only zero to little server-side dynamical content.&lt;/p&gt;&#xA;&lt;p&gt;Javascript became capable to replace Flash &amp;amp; ActiveX only with HTML5 in the early 2010s. Its main reason was that all client-side web programming technologies were closed or in the hands of a single company, Javascript was the only &amp;quot;least common denominator&amp;quot; between the major browser developers.&lt;/p&gt;&#xA;&lt;p&gt;Thus, initially - roughly at the early nineties - the Javascript didn't replace anything, it was new.&lt;/p&gt;&#xA;&lt;p&gt;There were still some possibilities for dynamical content, mainly by server-side support, but its possibilities were very weak. For example, images could dynamically created and re-used by sending refreshing http headers with images, and then refer them in &lt;code&gt;&amp;lt;img src=...&lt;/code&gt; tags.&lt;/p&gt;&#xA;&lt;p&gt;The most common technology before the WWW was &lt;a href=&quot;https://en.wikipedia.org/wiki/Gopher_(protocol)&quot; rel=&quot;nofollow noreferrer&quot;&gt;gopher&lt;/a&gt;, which supported only static content, too. At this time, non-graphical user environments were more common, they were remotely used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Telnet&quot; rel=&quot;nofollow noreferrer&quot;&gt;telnet&lt;/a&gt;, or more early, by serial terminals.&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="217" LastEditDate="2022-08-19T09:07:11.323" LastActivityDate="2022-08-19T09:07:11.323" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8355" PostTypeId="1" AcceptedAnswerId="8356" CreationDate="2019-03-03T09:09:07.613" Score="4" ViewCount="186" Body="&lt;p&gt;Hilbert's and Cohn-Vossen's wonderful book &quot;Anschauliche Geometrie&quot; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Geometry_and_the_Imagination&quot; rel=&quot;noreferrer&quot;&gt;&quot;Geometry and the Imagination&quot;&lt;/a&gt;) from 1932 contains a lot of great illustrations which, given the time of publication, must have been drawn by hand.  I couldn't find a mention in the book about who did them.  (The photos of the 3D models are mentioned, but not the drawings unless I missed it somehow.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know who created the drawings and how they were made at that time?  Was it some unknown graphical artists or maybe Cohn-Vossen himself?&lt;/p&gt;&#xA;" OwnerUserId="3555" LastEditorUserId="5989" LastEditDate="2019-03-03T09:43:30.957" LastActivityDate="2019-03-03T20:58:01.137" Title="Who did the drawings in Hilbert's and Cohn-Vossen's &quot;Anschauliche Geometrie&quot;?" Tags="|mathematics|geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8356" PostTypeId="2" ParentId="8355" CreationDate="2019-03-03T14:53:57.793" Score="6" Body="&lt;p&gt;I agree with you that the range of demonstrations and illustrations makes it a fascinating book. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Hilbert's preface dated 1932 from Göttingen, in the English translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The line diagrams have all been drawn by K. H. Naumann and  H. Bödeker&#xA;  (Göttingen). The photographic pictures were taken  by W. Jentzsch&#xA;  (Göttingen), and the models he photographed  belong to the collection&#xA;  of the Göttingen Mathematical Institute. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I couldn't find out anything else about the illustrators Naumann &amp;amp; Bödeker.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2019-03-03T14:53:57.793" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8357" PostTypeId="2" ParentId="8355" CreationDate="2019-03-03T20:58:01.137" Score="2" Body="&lt;p&gt;The book is accesible on the internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://w2.gzg-fn.de/mia/mathe/euklid/HilbertCohnVossenAnschaulicheGeometrie.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://w2.gzg-fn.de/mia/mathe/euklid/HilbertCohnVossenAnschaulicheGeometrie.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The prologue, written by Hilbert, mentions the authors of the artwork:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qcVVR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qcVVR.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2019-03-03T20:58:01.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8358" PostTypeId="2" ParentId="3131" CreationDate="2019-03-04T19:10:17.997" Score="10" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lu_Jiaxi_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Lu Jiaxi&lt;/a&gt; (1935 - 1983) was a self-taught Chinese mathematician who solved a major problem in combinatorial design theory. He was born in a poor family and his father died when he was in junior middle school. Working in the factory for several years, he self-studied and got into university majoring in physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fascinated by the Kirkman's schoolgirl problem while in university, he self-studied math and solved a generalized version of the problem in early 1960s, and he wrote a paper on it. However, being a physics teacher at a middle school in a remote city, his paper was rejected by Chinese math journals, and then the Cultural Revolution started. After the Cultural Revolution, he found out that the same problem was later solved and published in 1971 by Ray-Chaudhuri and R. M. Wilson.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He turned his attention to the problem of existence of large sets of disjoint Steiner triple systems, which was very much open at the time. As a school teacher, he could only worked in the evenings on the problem. His health was poor due to heavy work and poverty. He solved the problem and submitted it to Journal of Combinatorial Theory, Series A. The journal published first three of his papers in March 1983, it was then the Chinese mathematicians discovered him. Tragically, after attending the general conference of the Chinese Mathematical Society in October that year, he hurried back home, and in that very evening of his arrival he had a heart attack and died, leaving the seventh paper which was the last part of his proof unfinished. The President of the University of Toronto wrote to the principal of his school asking the principal to allow Lu Jiaxi to be transferred to a university, but it was too late.&lt;/p&gt;&#xA;" OwnerUserId="9682" LastEditorUserId="9682" LastEditDate="2019-03-04T19:24:13.670" LastActivityDate="2019-03-04T19:24:13.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8359" PostTypeId="1" CreationDate="2019-03-07T05:00:08.187" Score="2" ViewCount="87" Body="&lt;p&gt;This is a &lt;a href=&quot;https://stackoverflow.com/questions/54996109/in-left-or-right-rotation-where-did-those-conventions-for-the-directions-come&quot; title=&quot;In left- or right-rotation, where did those conventions for the directions come from?&quot;&gt;repost&lt;/a&gt; from another exchange. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was going to write a paper about code to rotate elements in an array, but I realized I have no idea why rotation towards the start of the array is “left” and “right” goes towards the end. I’ve seen horizontal tape diagrams of arrays, with elements going left-to-right (i.e. Western-culture reading order), but I don’t know if that’s the actual reason. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe finding out who first came up with the convention would help. &lt;/p&gt;&#xA;" OwnerUserId="3289" LastActivityDate="2019-03-07T05:00:08.187" Title="In left- or right-rotation for arrays, where did those conventions for the directions come from?" Tags="|computer-science|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8360" PostTypeId="1" AcceptedAnswerId="12943" CreationDate="2019-03-07T06:47:35.383" Score="5" ViewCount="134" Body="&lt;p&gt;I remember reading a long time ago about some experiments measuring evaporation rate of water in large open containers (tubs) sitting on open ground. It was a simple experiment, each day the water level would be checked and recorded. From what I remember the experiment was in the 1950's or 1960's in Israel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Results generally made sense, the rate of loss was highest on hot sunny days, but something about the analysis lead to the conclusion that in addition to thermal effects (hot water evaporates faster than cold water) there was also a component to the evaporation rate related to the brightness of the light itself; a non-thermal or light-induced evaporation process.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've been searching and so far come up empty-handed. I've even asked a related question in Physics SE (&lt;a href=&quot;https://physics.stackexchange.com/q/443314/83380&quot;&gt;Rate of direct (non-thermal) photon-induced evaporation of water?&#xA;&lt;/a&gt;) several months ago, but so far no bites.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can't remember where I read this, but it may have been as an anecdote in a text about either climate change or long term variability of the solar constant or the power output of the Sun.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-02-26T23:43:33.973" LastActivityDate="2021-03-04T15:02:40.273" Title="Looking for references to an experiment on the evaporation rate of water in sunlight" Tags="|physics|chemistry|earth-sciences|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8361" PostTypeId="1" CreationDate="2019-03-08T20:44:21.920" Score="3" ViewCount="122" Body="&lt;p&gt;Ladyzhenskaya is famous for fluid dynamics and partial differential equations, both of which are beyond my pay grade. And she worked on the Navier-Stokes equations. Does this &lt;a href=&quot;https://www.cnet.com/news/google-doodle-celebrates-mathematician-olga-ladyzhenskaya/&quot; rel=&quot;nofollow noreferrer&quot;&gt;circle with the arrows&lt;/a&gt; represent anything in particular?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I know from differential equations is that they sometimes involve arrows in &quot;direction fields&quot;, which may be a connection.&lt;/p&gt;&#xA;" OwnerUserId="9694" LastEditorUserId="55" LastEditDate="2019-03-09T09:22:04.013" LastActivityDate="2019-03-09T09:22:04.013" Title="Does the “O” in the google doodle for Olga Ladyzhenskaya have anything to do with her work?" Tags="|mathematical-physics|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8362" PostTypeId="2" ParentId="8361" CreationDate="2019-03-09T06:02:09.290" Score="7" Body="&lt;p&gt;The equation in the picture is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations#Compressible_flow&quot; rel=&quot;noreferrer&quot;&gt;Navier–Stokes momentum equation in convective form&lt;/a&gt;. The part inside the O looks  like a cross-section of (the velocity field of) a flow between two rotating cylinders, mentioned in Ladyzhenskaya's &lt;a href=&quot;https://pdfs.semanticscholar.org/23fa/28319d928fa7d67eb72fde26bfa8e7c0611f.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mathematical Analysis of Navier-Stokes Equations for Incompressible Liquids, 6.4&lt;/a&gt;, among other places, the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Taylor%E2%80%93Couette_flow&quot; rel=&quot;noreferrer&quot;&gt;Taylor-Couette flow (spiral vortex)&lt;/a&gt;. But this is not something Ladyzhenskaya worked on specifically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is also similar to a picture of (the direction field of) an unstable spiral surrounded by a &lt;a href=&quot;https://en.wikipedia.org/wiki/Limit_cycle&quot; rel=&quot;noreferrer&quot;&gt;limit cycle&lt;/a&gt;. Those appear in some finite-dimensional reductions of the Navier-Stokes equations (see e.g. &lt;a href=&quot;https://www.scirp.org/journal/PaperInformation.aspx?PaperID=68842&quot; rel=&quot;noreferrer&quot;&gt;Wang Five-Mode System of the Navier-Stokes Equations on a Torus&lt;/a&gt;), but this is even further from what she worked on. So it could just be a cute picture of a &quot;fluid flow&quot; that fits nicely into letter O.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Zcxkb.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Zcxkb.png&quot; alt=&quot;Ladyzhenskaya Google Doodle&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-03-09T06:49:41.920" LastActivityDate="2019-03-09T06:49:41.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8363" PostTypeId="2" ParentId="7943" CreationDate="2019-03-09T12:22:16.583" Score="2" Body="&lt;p&gt;Mass &amp;amp; weight. It’s easy enough to make that mistake since we see objects in a uniform gravitational field. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-03-09T12:22:16.583" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8364" PostTypeId="1" AcceptedAnswerId="8365" CreationDate="2019-03-09T17:39:12.143" Score="4" ViewCount="165" Body="&lt;p&gt;Does anybody knows how this important notion of hydrostatics was discovered?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read that it is about someone walking up and down the mountains of Latin America trying to disprove Cartesian geometry. But I can't remember where I have read that.&lt;/p&gt;&#xA;" OwnerUserId="5421" LastEditorUserId="55" LastEditDate="2019-03-10T08:46:49.210" LastActivityDate="2019-03-10T08:46:49.210" Title="How was the notion of the metacenter of a floating body discovered?" Tags="|mathematics|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8365" PostTypeId="2" ParentId="8364" CreationDate="2019-03-09T20:39:23.277" Score="5" Body="&lt;p&gt;In  1727  the  Parisian  Royal  Academy  of  Sciences  held  a  prize  contest  on  the  optimal placement  of  masts  in  a  sailing  ship. Bouguer and 20-year old Euler, then a student of Johann Bernoulli, submitted solutions. The  optimal  placement  of the sail is directly related to the heel angle of the ship where the heeling  moment  and the  restoring moment are equal. This prompted the investigations of ship's stability that led to the introduction of the metacenter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bouguer  subsequently participated  in an  expedition  to  the Andes  in  Peru,  today  Ecuador,  for  geodesic  measurements  near  the  equator  from  1735  to 1744, where he  worked  on  his  treatise  Traité  du Navire on ship  hydrostatics, published  in  1746 after  his  return  to France. However, according to Nowacki, he likely started working on ship's stability already in 1732, when he tested the 18-gun frigate Gazelle in Le Havre. Euler worked on the same subject in St. Petersburg from 1737 to 1741, and summarized it in  Scientia Navalis published after a delay in 1749.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For infinitesimal angles  of  heel,  Bouguer  introduced  the  metacenter    as  the point of intersection of two infinitesimally neighboring buoyancy directions, that is directions passing through the centers of buoyancy, the centers  of  gravity  of  the  displaced  water (see his diagram below, g is the metacenter).   Generally, the  metacenter  is the center of curvature of the curve traced by the centers of buoyancy as the  ship  heels  continuously. The significance of it is that the ship is stable when the center of its gravity is below the metacenter. Bouguer  constructed  the  metacenter  numerically by using the trapezoidal rule. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler gave an equivalent formulation in terms of moments, without introducing the metacenter explicitly. But Euler's book was written in Latin, lacked worked out examples, and remained almost unknown to shipbuilders, while Bouguer's treatise became a blueprint for standard texts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://www.kayakdesign.org/stabilitet/Paper01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historical Roots of the Theory of Hydrostatic Stability of Ships by Nowicki&lt;/a&gt;, where the following diagram from Bouguer's Traité is reproduced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EEEh5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EEEh5.png&quot; alt=&quot;Bouguer Metacenter&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-03-10T03:53:26.053" LastActivityDate="2019-03-10T03:53:26.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8366" PostTypeId="1" AcceptedAnswerId="8385" CreationDate="2019-03-10T16:53:15.730" Score="22" ViewCount="2020" Body="&lt;p&gt;Most people are familiar with the famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Mad_scientist&quot; rel=&quot;noreferrer&quot;&gt;&quot;mad scientist&quot;&lt;/a&gt; trope from science fiction - the scientist who is typically a loner, often self-funded from family wealth, frequently mentally ill, engaged in secretive research that may be unethical, unusual, highly innovative, or all of the above, having a shaky or nonexistent relationship with &quot;established&quot; research institutions (universities, hospitals, government-funded labs, etc.), often working from a remote castle on a mountaintop or a makeshift lab in an abandoned subway station, and often cast either as a villain or a tragic hero.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Literary examples abound - Shelley's Dr. Frankenstein, Lovecraft's Joseph Curwen and Herbert West, and Doc Brown from the &lt;em&gt;Back to the Future&lt;/em&gt; films.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Have there been any real life &quot;mad scientist&quot; types who have made &lt;em&gt;substantial&lt;/em&gt; scientific discoveries? That is, is the &quot;mad scientist&quot; &lt;em&gt;solely&lt;/em&gt; a literary and theatrical device/trope, or is it inspired by real scientists of the past?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: This question is intended to be a &lt;a href=&quot;https://stackoverflow.blog/2010/09/29/good-subjective-bad-subjective/&quot;&gt;&quot;good subjective&quot;&lt;/a&gt; question that should be at least somewhat answerable based on knowledge of past scientists as well as the literary stereotype.&lt;/p&gt;&#xA;" OwnerUserId="6976" LastActivityDate="2024-03-02T22:18:36.297" Title="Has a stereotypical &quot;mad scientist&quot; ever made a significant discovery?" Tags="|social-context|depiction-of-scientists|" AnswerCount="12" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8367" PostTypeId="1" AcceptedAnswerId="8404" CreationDate="2019-03-10T18:38:23.500" Score="-1" ViewCount="129" Body="&lt;p&gt;This is quite a specific question about radio, but it has been rejected on the &lt;a href=&quot;https://ham.stackexchange.com/questions/12969/can-anybody-identify-an-appropriate-radio-for-this-game&quot;&gt;Amateur Radio&lt;/a&gt; and &lt;a href=&quot;https://history.stackexchange.com/questions/51473/can-anybody-identify-an-appropriate-radio-for-this-game&quot;&gt;History&lt;/a&gt; Stack Exchanges, so I will ask it here...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm a complete radio ignoramous, but I've been playing a game called The Blackout Club, and I'd quite like to make a 3D model of a radio set for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The game is set around 2007 and, as in the picture below, features young teenagers (13-17 years old) monitoring the communications of a sinister conspiracy which uses radio to send commands to their field agents.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/58YlI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/58YlI.png&quot; alt=&quot;One of the tutorial texts from the game&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would be nice if my 3D model fit the tone / period of the game as much as possible - and so I'm turning to you, the specialists! Can anybody suggest a suitable make and model for me? Something that would be suitable for a dispatcher at their home base?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was looking at military radios earlier, but I figure they'd probably be encrypted... this needs to be something that an average person could use, that could be intercepted by a teenager.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you reading my odd request!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;--Rev&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: If you're curious, here's a trailer for the game: &lt;a href=&quot;https://www.youtube.com/watch?v=MKzd6YpBBcY&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=MKzd6YpBBcY&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9699" LastActivityDate="2019-03-22T01:34:13.353" Title="Can anybody identify an appropriate radio for this game?" Tags="|physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8368" PostTypeId="2" ParentId="8366" CreationDate="2019-03-10T19:24:47.327" Score="8" Body="&lt;p&gt;I almost hesitate to offer this answer, because I don't want to suggest in any way that its subject was really a 'mad scientist' at all.&lt;/p&gt;&#xA;&lt;p&gt;But he did show a few of the other characteristics suggested in the question, so his name is arguably not out of place here.&lt;/p&gt;&#xA;&lt;p&gt;For a time in the 1960s something of a loner in his research, even a maverick, he carried out essentially self-funded work in a historic mansion that he had restored from a burnt-out shell -- used partly for his family home, and partly for a private research laboratory. During part of this period he also maintained a lone scientific position against the claims of researchers from three established research centers who thought they had disproved his theories. Some of his results appeared in two little books for which his scientific competitors made fun of him by calling them the 'thoughts of Chairman M'.&lt;/p&gt;&#xA;&lt;p&gt;But his story ended well, he turned out to be essentially in the right, and this was eventually well-recognized. His work and results overcame the initially great reluctance from many in his field to accept them; he was awarded a Nobel Prize, and his theory and results have now become part of the orthodox foundation of knowledge in his field.&lt;/p&gt;&#xA;&lt;p&gt;Readers of this answer who are familiar with biochemistry and cell biology will probably already have recognized that these selected glimpses come from the life and work of Peter Mitchell (1920-1992). His contribution is often not well described in brief accounts (wikipedia for example has &amp;quot;discovery of the mechanism of ATP synthesis&amp;quot; which arguably misses the main point). The issue was not the synthesis of &lt;a href=&quot;https://en.wikipedia.org/wiki/Adenosine_triphosphate&quot; rel=&quot;nofollow noreferrer&quot;&gt;adenosine triphosphate (ATP)&lt;/a&gt; altogether, it was about the harnessing and deployment of the chemical energy needed and used for most of the biochemical building processes especially within the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mitochondrion&quot; rel=&quot;nofollow noreferrer&quot;&gt;mitochondria&lt;/a&gt; of &lt;a href=&quot;https://biologydictionary.net/eukaryotic-cell/&quot; rel=&quot;nofollow noreferrer&quot;&gt;eukaryotic cells&lt;/a&gt;. This involves a kind of recycling, in which the ATP participates in any of a wide range of biosynthetic reactions, 'spending' its chemical energy by giving up its third phosphate group, and leaving behind the corresponding diphosphate ADP (or occasionally the monophosphate AMP), compound(s) of lower chemical activity. The ADP (or AMP) then has to be 'recharged' by reconversion to ATP.  A big question before Mitchell's work was accordingly, how does the cell carry out this 'recharge', to reconvert those lower-energy residual molecules by adding on the extra phosphate groups -- which takes much energy -- so as to replenish the supply of the ATP molecules that then make the chemical energy available to drive biosynthetic processes?  Mitchell's correct answer was that the recharging process does not rely solely on a chain of biochemical reactions (which was the previous orthodox view), it involves subcellular membranous compartments within the mitochondria, into which ionic intermediates are pumped against concentration gradients, and then in turn drive the replenishment of the ATP as they pass out - hence 'chemi-osmotic' theory.  Further details can safely be left to the references:&lt;/p&gt;&#xA;&lt;p&gt;Mitchell's &lt;a href=&quot;http://www1.ci.uc.pt/pessoal/manolo/Mitchell.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1961 statement&lt;/a&gt; of his chemi-osmotic hypothesis (Mitchell, P. (1961), &amp;quot;Coupling of phosphorylation to electron and hydrogen transfer by a chemi-osmotic type of mechanism&amp;quot;. &lt;em&gt;Nature&lt;/em&gt; vol. 191 (No.4784): 144–148);&lt;/p&gt;&#xA;&lt;p&gt;Mitchell's &lt;a href=&quot;https://www.nobelprize.org/uploads/2018/06/mitchell-lecture.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1978 Nobel lecture (.pdf)&lt;/a&gt; and at (&lt;a href=&quot;https://www.nobelprize.org/prizes/chemistry/1978/mitchell/lecture/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nobelprize.org/prizes/chemistry/1978/mitchell/lecture/&lt;/a&gt;);&lt;/p&gt;&#xA;&lt;p&gt;Mitchell's &lt;a href=&quot;http://www.life.illinois.edu/crofts/bioph354/mitchell.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1992 obituary&lt;/a&gt; (by Antony Crofts, June 29th 1992,  originally appeared in &lt;em&gt;Photosynthesis Research&lt;/em&gt;);&lt;/p&gt;&#xA;&lt;p&gt;A &lt;a href=&quot;https://papers.xtremepape.rs/OCR/AS%20and%20A%20Level/Biology/Resources/142690-oxidative-phosphorylation-and-chemiosmotic-theory-june-2013.doc&quot; rel=&quot;nofollow noreferrer&quot;&gt;recent summary (.doc)&lt;/a&gt; of oxidative phosphorylation and chemiosmotic theory (June 2013, from Martin D Brand, Buck Institute, California; Stuart J Ferguson, University of Oxford; J Baz Jackson, University of Birmingham; David G Nicholls, Buck Institute, California; and Peter R Rich, University College, London).&#xA;,&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-04-21T09:57:49.233" LastActivityDate="2022-04-21T09:57:49.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8369" PostTypeId="1" AcceptedAnswerId="8371" CreationDate="2019-03-11T04:02:36.410" Score="6" ViewCount="400" Body="&lt;p&gt;&lt;a href=&quot;https://astronomy.stackexchange.com/a/29817/7982&quot;&gt;This answer&lt;/a&gt; to the question &lt;a href=&quot;https://astronomy.stackexchange.com/q/29741/7982&quot;&gt;First observation that the movement of a planet or asteroid in its orbit was affected by another planet&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1705, with the mathematical assistance of Issac Newton, Edmond Halley published Synopsis of the Astronomy of Comets which made the prediction that the comet observed in 1682 had previously been observed in 1531 and 1607 was the same object returning about every 76 years or so. He included a rough estimation of the gravitational effects of Jupiter and Saturn, and predicted it would return in 1758.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there anything written specifically about mathematics used in a &quot;rough estimation of the gravitational effects of Jupiter and Saturn&quot;? Was the size of the effect on the comet's orbit mentioned, or anything about how the calculation was made?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Just fyi the linked answer also cites Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Halley%27s_Comet&quot; rel=&quot;noreferrer&quot;&gt;Halley's Comet&#xA;&lt;/a&gt; and the passage cites page 86 of P. Lancaster-Brown (1985). Halley &amp;amp; His Comet. Blandford Press. ISBN 0-7137-1447-6.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="5788" LastEditDate="2019-03-11T11:33:59.997" LastActivityDate="2019-03-11T11:33:59.997" Title="What calculation did Halley or anyone else do to estimate the effects of Jupiter and Saturn on Halley's comet's return in 1758/9?" Tags="|mathematics|astronomy|celestial-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8370" PostTypeId="2" ParentId="8366" CreationDate="2019-03-11T07:27:42.407" Score="31" Body="&lt;p&gt;I would say that Henry Cavendish (1731–1810) fits this description. A hugely rich man (at the time of his death he was the largest depositor in the Bank of England) he was also a loner in a huge scale. According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Cavendish&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He could only speak to one person at a time, and only if the person were known to him and male. He conversed little, always dressed in an old-fashioned suit, and developed no known deep personal attachments outside his family. Cavendish was taciturn and solitary and regarded by many as eccentric. He only communicated with his female servants by notes. By one account, Cavendish had a back staircase added to his house to avoid encountering his housekeeper, because he was especially shy of women.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He was also, of course, one of the greatest scientists of all time. For instance, he discovered that water is not an element, but rather the result of a chemical reaction between oxygen and hydrogen (and he was the one who discovered hydrogen). He was also the first person who computed the bending of light while passing near a star.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="4251" LastEditDate="2020-07-14T05:59:07.210" LastActivityDate="2020-07-14T05:59:07.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8371" PostTypeId="2" ParentId="8369" CreationDate="2019-03-11T08:42:19.593" Score="7" Body="&lt;p&gt;Alexis Clairaut designed and led the detailed calculations of planetary perturbations to estimate the return-date in 1759 of what then became known as Halley's comet.  The most detailed account of the work (in which Clairaut was assisted in the calculations by the young Jerome de Lalande and by Mme Lepaute) seems to be that given by Curtis Wilson in the &lt;a href=&quot;http://adsabs.harvard.edu/abs/1993JHA....24....1W&quot; rel=&quot;noreferrer&quot;&gt;Journal for the History of Astronomy v.24 (1993) pp.1-15&lt;/a&gt;: &quot;Clairaut's Calculation of the Eighteenth-century Return of Halley's Comet&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The full text is available online. (I can hardly add anything except to mention that while Wilson cites early sources, he does not attribute to Halley any calculation of the planetary perturbations of the comet by Jupiter or Saturn.  It seems that Halley's contribution here was indeed to identify the probable periodic nature of a number of comets, and to  calculate their orbital paths on an unperturbed parabolic and then elliptical basis.) &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2019-03-11T08:42:19.593" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8372" PostTypeId="2" ParentId="8367" CreationDate="2019-03-11T10:47:40.533" Score="0" Body="&lt;p&gt;Try searching Google images of &quot;amateur radio&quot; to find a lot of these.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TUWL4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TUWL4.jpg&quot; alt=&quot;widipedia&quot;&gt;&lt;/a&gt;  (image from hackaday.com)&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-03-11T10:47:40.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8373" PostTypeId="2" ParentId="8366" CreationDate="2019-03-11T13:00:22.220" Score="6" Body="&lt;p&gt;Kari Mullis, inventor/discoverer of the polymerase chain reaction (PCR) for fast analysis of DNA samples, was into wild LSD rides and espoused near-Scientology-level beliefs in supernatural goings-on. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might suggest Linus Pauling, but all evidence suggests his nuttiness came well after all his discoveries, so he was &quot;of sound mind&quot; while doing the work which led to his Nobel Prize. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-03-11T13:00:22.220" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8374" PostTypeId="2" ParentId="8366" CreationDate="2019-03-12T10:20:17.407" Score="9" Body="&lt;p&gt;Besides the other excellent answers, &lt;em&gt;maybe&lt;/em&gt; &lt;strong&gt;Georg Cantor&lt;/strong&gt; could fit your description, with some serious &lt;em&gt;caveats&lt;/em&gt;. Although the details of his life have been extremely romanticized (e.g. the widespread story that he was driven crazy &quot;after seeing infinity&quot;), he did suffer from chronic depression in the last years of his life, to the point to which he was hospitalized several times. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the main contributors to set theory and the back then shocking concept of transfinite numbers. Saying that his discoveries are significant is kind of an understatement.&lt;/p&gt;&#xA;" OwnerUserId="3853" LastActivityDate="2019-03-12T10:20:17.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8375" PostTypeId="2" ParentId="8366" CreationDate="2019-03-12T14:09:31.570" Score="14" Body="&lt;p&gt;The most commonly mentioned name in this context is Nicola Tesla, he is even featured as such in some fiction (e.g. Tomorrowland). He was brilliant, eccentric, kept to himself, and had some wacky ideas, especially later in life, including persistent search for the ether, see &lt;a href=&quot;https://physics.stackexchange.com/q/176727/65263&quot;&gt;Was Nikola Tesla right about his ether theory?&lt;/a&gt; Conspiracy theories around his late life and work were fueled by FBI seizing his belongings when he died. However, and this is were real life diverges from fiction, if the &amp;quot;mad&amp;quot; scientist is supposed to be successful in his &amp;quot;mad&amp;quot; activities, it is not how it worked out with Tesla. He was most successful, indeed brilliant, in his less &amp;quot;mad&amp;quot; early years. Hartsfield gives an interesting take titled &lt;a href=&quot;https://www.realclearscience.com/blog/2012/10/mad-scientists-of-the-modern-era-nikola-tesla.html&quot; rel=&quot;noreferrer&quot;&gt;Mad Scientists of the Modern Age: Nikola Tesla&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Edison's light bulb is trumpeted as the signal achievement beginning the electrical era. However, it was Tesla who devised most of the alternating current (AC) electrical systems that the world's power grids are modeled upon. He single-handedly patented AC electrical generators, transformers, power lines, and motors. The electrical system formed by these devices proved so much superior to Edison's direct current (DC) electrical system that it was adopted universally across the globe... Tesla was in many ways the opposite of Thomas Edison. Edison was never known as a brilliant man, but as a methodical, determined one. &amp;quot;Genius is 99 percent perspiration and one percent inspiration,&amp;quot; he once said. Tesla was a nearly unparalleled genius, but lacked the business intuition and common sense of Edison. Edison died rich and one of most famous men in the world, while Tesla died penniless and alone, feeding pigeons.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;After years of wildly successful invention, Tesla began to lose his way, making grandiose, eccentric and sometimes completely impossible claims. He fervently spoke of building a &amp;quot;teleforce weapon&amp;quot; capable of melting steel wirelessly from 200 miles away. Upon his death, a box supposedly containing a death ray part was found to contain only a basic piece of old equipment. Tesla believed that he could create a camera to project a brain's thoughts onto a wall. He built a 187 foot-tall tower to try to wirelessly convey electric power across the planet, using the earth itself as an enormous electrical conductor. He wanted to harness the entire sphere of the Earth as a power line! Upon picking up odd signals at one of his labs, Tesla believed that they could have been sent from Mars by using that planet similarly as a conductor.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Celibate his entire life, Tesla became lonely and slightly paranoid in his elder years. He claimed to keep his ideas inside of his brain to keep them from being stolen. He was suspicious of twentieth century physics, as explained by Einstein and others... An entire conspiracy theory movement has formed around Tesla's &amp;quot;hidden&amp;quot; or &amp;quot;suppressed&amp;quot; inventions and ideas. The United States government also unwittingly contributed to Tesla's legacy of mad genius. Shortly after his death in 1943, the FBI seized all of his belongings.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Real Clear Science has a whole series on Mad Scientists of the Modern Age, they name &lt;a href=&quot;https://www.realclearscience.com/blog/2012/10/mad-scientists-of-the-modern-age-vladimir-demikhov.html&quot; rel=&quot;noreferrer&quot;&gt;Vladimir Demikhov&lt;/a&gt; (experimental surgeon known for intrathoracic transplantation), &lt;a href=&quot;https://www.realclearscience.com/blog/2012/10/three-real-life-mad-scientists-of-the-modern-age.html&quot; rel=&quot;noreferrer&quot;&gt;Jack Parsons&lt;/a&gt; (jet and rocket propulsion whiz), &lt;a href=&quot;https://www.realclearscience.com/blog/2013/02/mad-scientists-of-the-modern-age-lynn-margulis.html&quot; rel=&quot;noreferrer&quot;&gt;Lynn Margulis&lt;/a&gt; (the founder of endosymbiotic theory), &lt;a href=&quot;https://www.realclearscience.com/blog/2012/10/mad-scientists-of-the-modern-age-barry-marshall.html&quot; rel=&quot;noreferrer&quot;&gt;Barry Marshall&lt;/a&gt; (discovered the cause of ulcers), and &lt;a href=&quot;https://www.realclearscience.com/blog/2012/10/mad-scientists-of-the-modern-age-josef-mengele.html&quot; rel=&quot;noreferrer&quot;&gt;Josef Mengele&lt;/a&gt; (Nazi death doctor). On the general correlation between brilliance and &amp;quot;madness&amp;quot; see &lt;a href=&quot;https://www.livescience.com/20713-genius-madness-connected.html&quot; rel=&quot;noreferrer&quot;&gt;Why Are Genius and Madness Connected? on Live Science&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Kay Redfield Jamison, a clinical psychologist and professor at Johns Hopkins University School of Medicine, said the findings of some 20 or 30 scientific studies endorse the notion of the &amp;quot;tortured genius.&amp;quot; Of the many varieties of psychosis, creativity appears to be most strongly linked to mood disorders, and especially bipolar disorder, which Jamison suffers from herself. For example, one study tested the intelligence of 700,000 Swedish 16-year-olds and then followed up a decade later to learn which of them had developed mental illnesses. The startling results were published in 2010. &amp;quot;They found that people who excelled when they were 16 years old were four times as likely to go on to develop bipolar disorder... &amp;quot;People with bipolar tend to be creative when they're coming out of deep depression,&amp;quot; Fallon said. When a bipolar patient's mood improves, his brain activity shifts, too: activity dies down in the lower part of a brain region called the frontal lobe, and flares up in a higher part of that lobe. Amazingly, the very same shift happens when people have bouts of creativity. &amp;quot;There [is] this nexus between these circuits that have to do with bipolar and creativity&amp;quot;&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-03-20T17:43:47.990" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8376" PostTypeId="1" AcceptedAnswerId="8377" CreationDate="2019-03-12T17:02:43.217" Score="10" ViewCount="295" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/avwkk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/avwkk.png&quot; alt=&quot;Poincare quotation Theorem of Lie&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this quotation from Henri Poincaré's essay &quot;Non-Euclidean Geometry&quot; published in &lt;em&gt;Nature&lt;/em&gt; in 1892 (No. 1165, Vol 45, p. 406), he refers to a theorem of Sophus Lie. Does anyone know a source for this theorem, or one that discusses it? &lt;/p&gt;&#xA;" OwnerUserId="9709" LastEditorUserId="55" LastEditDate="2019-03-13T09:43:31.103" LastActivityDate="2019-03-15T07:52:24.603" Title="What theorem of Sophus Lie on the number of geometries is H. Poincaré referring to?" Tags="|reference-request|geometry|euclidean-geometry|riemannian-geometry|projective-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8377" PostTypeId="2" ParentId="8376" CreationDate="2019-03-13T09:41:45.387" Score="12" Body="&lt;p&gt;Poincare refers to the Lie's solution of the so-called problem of space, a.k.a. the Helmholtz , or Riemann-Helmholtz, or Helmholtz-Lie problem of space, which amounts to characterizing all manifolds (originally, only 3-dimensional) with free mobility of figures (roughly, homogeneity and isotropy). In modern terms, free mobility amounts to constant Riemannian  curvature. This is what Poincare's second quoted condition amounts to, and the space forms are discussed throughout the paper (see &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Extras/Poincare_non-Euclidean.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor's English translation&lt;/a&gt;). The problem had roots in Helmholtz's objections to Kantian &lt;em&gt;a priori&lt;/em&gt; intuition of space being Euclidean. It seemed to be related to the fundamental questions about the nature of space and our perception of it, and preoccupied some eminent geometers (including also Riemann, Clifford, Klein, Engel and Killing, later Hopf) for a good part of 19th and early 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Helmholtz, all &lt;em&gt;a priori&lt;/em&gt; intuition of space gives us is free mobility, the rest is to be determined empirically. A good modern exposition is in &lt;a href=&quot;https://press.princeton.edu/titles/8960.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weyl's Philosophy of Mathematics and Natural Science&lt;/a&gt;, if you read French see also &lt;a href=&quot;https://arxiv.org/abs/0910.0801&quot; rel=&quot;nofollow noreferrer&quot;&gt;Merker's thesis Sophus Lie, Friedrich Engel and the Riemann-Helmholtz problem&lt;/a&gt;. Poincare more or less agreed with Helmholtz's conception, and himself published a mathematical paper on space forms, &lt;em&gt;&lt;a href=&quot;http://www.numdam.org/item/BSMF_1887__15__203_0/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur les hypothèses fondamentales de la géométrie&lt;/em&gt; (1887)&lt;/a&gt; (the title is borrowed from Riemann's lecture), which uses Lie group theory. Later, in the last of his Analysis Situs papers (fifth complement, 1904), he introduced the &quot;dodecahedron space&quot;, a nontrivial 3-dimensional space form, and formulated the now famous Poincare conjecture.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Helmholtz's formulation of free mobility was axiomatic, according to &lt;a href=&quot;https://books.google.com/books?id=EcLrCAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toretti's Philosophy of Geometry from Riemann to Poincaré&lt;/a&gt;, Lie's group-theoretic reformulation was the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;To determine all the finite continuous groups of &lt;span class=&quot;math-container&quot;&gt;$R_n$&lt;/span&gt;, which have no one-point invariant, exactly one (independent) two-point invariant and no essential &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-point invariant for &lt;span class=&quot;math-container&quot;&gt;$k&amp;gt;2$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Toretti gives a detailed account of the pre-history and the Lie's solution (part of which is that there are finitely many such space forms for each &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The results of Lie’s explorations&#xA;  are contained in the monumental Theory of Transformation&#xA;  Groups, edited with the assistance of Friedrich Engel. Part V of the&#xA;  third volume is devoted to a detailed study of the Helmholtz problem&#xA;  of space. Lie says that this problem was brought to his attention by&#xA;  his friend Klein, who told him that many mathematicians would not&#xA;  accept Helmholtz’s reasoning, and suggested that the problem might&#xA;  be attacked successfully with the resources of Lie’s group theory.&#xA;  Lie reported his results on this matter in 1886, and published them&#xA;  with proofs in 1890. The content of his two papers of 1890 has been&#xA;  inserted in the considerably expanded treatment of Helmholtz’s&#xA;  problem contained in Lie’s big book.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As it turned out, Lie's solution was incomplete because he disregarded global topology, all of his space forms sit in projective spaces. Topological issues came up already in the Klein-Killing correspondence in 1880-s, but the complete solution took a long time to finalize, and is now known as the Hopf-Killing theorem, see &lt;a href=&quot;https://arxiv.org/pdf/math/0305023.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Epple's From Quaternions to Cosmology: Spaces of Constant Curvature,ca. 1873–1925&lt;/a&gt;. A detailed historical account of Killing's contributions is &lt;a href=&quot;https://www.jstor.org/stable/41133645&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilhelm Killing and the Structure of Lie Algebras by Hawkins&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-03-15T07:52:24.603" LastActivityDate="2019-03-15T07:52:24.603" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8378" PostTypeId="1" CreationDate="2019-03-13T10:54:36.337" Score="5" ViewCount="310" Body="&lt;p&gt;This is a question about old physics lab equipment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://space.stackexchange.com/q/27118/12102&quot;&gt;This&lt;/a&gt; and &lt;a href=&quot;https://space.stackexchange.com/q/27124/12102&quot;&gt;this&lt;/a&gt; question led to the question in SciFi SE &lt;a href=&quot;https://scifi.stackexchange.com/q/187145/51174&quot;&gt;Crookes Radiometer at the end of a telescope in old movie scene, something about a comet?&lt;/a&gt; which led to the image below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course one does not use wires to couple a radiometer to a telescope.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, my question is about the (now old, possibly then as well) &lt;a href=&quot;https://en.wikipedia.org/wiki/Crookes_radiometer&quot; rel=&quot;noreferrer&quot;&gt;Crookes Radiometer&lt;/a&gt; shown in the image from 1965. There are two terminals in the wooden base of the radiometer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Did old Crookes Radiometers ever have a pair of terminals? I can imagine a heater used to evaporate trace amounts of some material to adjust the pressure for teaching/demonstration purposes, or even some kind of getter pump to reduce the pressure (as some electron tubes had), but I thought I'd ask here to see if someone familliar with older lab equipment might know.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;below:&lt;/strong&gt; Screen shot from this &lt;a href=&quot;https://www.youtube.com/watch?v=HTxfEWi1AYM&quot; rel=&quot;noreferrer&quot;&gt;video&lt;/a&gt; of Episode 3 (1965) of The Time Tunnel &lt;a href=&quot;https://www.imdb.com/title/tt0723756/&quot; rel=&quot;noreferrer&quot;&gt;The end of the World&lt;/a&gt; after roughly &lt;code&gt;36:15&lt;/code&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0xzCX.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0xzCX.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2022-11-22T19:06:35.860" Title="Did old Crookes Radiometers ever have a pair of terminals?" Tags="|physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8379" PostTypeId="2" ParentId="8366" CreationDate="2019-03-13T12:19:01.767" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Georg_Cantor&quot; rel=&quot;noreferrer&quot;&gt;Georg Cantor&lt;/a&gt; is well known as someone who was admitted to a mental asylum several times in his lifetime, mainly it seems because of the opposition to his ideas on the characterising mathematical infinities; a theory, that today passes without much comment. In fact, Kronecker &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;perceived him as a &quot;corrupter of youth&quot; for teaching his ideas to a younger generation of mathematicians.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One might then view that his sickness was not within, but without; that is a sick society. This is after all, one of the main themes of romanticism, which arose as a reaction to a sick society and fed into science fiction and fantasy. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-03-13T12:19:01.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8380" PostTypeId="2" ParentId="8378" CreationDate="2019-03-13T13:05:53.477" Score="1" Body="&lt;p&gt;Google is your friend here.  I found a forum which contained a reference to &lt;a href=&quot;http://www.ece.drexel.edu/pix/cmf2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this image&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which shows a barely-working generator.  Exerpt from that image:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iL2b9.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iL2b9.jpg&quot; alt=&quot;induction generator&quot;&gt;&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The forums I found generally suggested that it's pretty much impossible to get enough torque to drive any mechanically connected generator system.  So it can be done but I suspect the image you provided is a case of &quot;Let's toss in as much cool-looking lab toys as we can.&quot; &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-03-13T13:05:53.477" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8381" PostTypeId="1" AcceptedAnswerId="8382" CreationDate="2019-03-13T13:27:45.767" Score="4" ViewCount="155" Body="&lt;p&gt;Originally, the gamma function &lt;span class=&quot;math-container&quot;&gt;$\Gamma(x)$&lt;/span&gt; is defined as &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \Gamma(x )=\int_0^\infty e^{-t}  t^{x-1} dt  .$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This definition works for &lt;span class=&quot;math-container&quot;&gt;$Re(x)&amp;gt; 0 $&lt;/span&gt; only. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, who extended into the whole complex plane? This is nontrivial. &lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2019-03-14T14:20:06.427" Title="Who achieved the analytic continuation of the Gamma function?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8382" PostTypeId="2" ParentId="8381" CreationDate="2019-03-13T14:09:01.810" Score="9" Body="&lt;p&gt;The formula for &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt; function which is valid for all complex values of the argument was introduced by Euler,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Gamma(z)=\frac{1}{z}\prod_{n=1}^\infty\left\{\left(1+\frac{1}{n}\right)^z\left(1+\frac{z}{n}\right)^{-1}\right\}.$$&lt;/span&gt;&#xA;This is how he originally defined it, and then he showed that it is equal to the &quot;Euler's integral&quot;. Of course at the time of Euler there was no rigorously defined notion of analytic continuation, but this did not prevent Euler from working with it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A modern definition using the product (which also defines &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt; for all complex &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;) is due to Weierstrass:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac{z}{n}\right)e^{-z/n},$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; is the Euler-Mascheroni constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: Whittaker-Watson, Chapter 12.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-03-14T14:20:06.427" LastActivityDate="2019-03-14T14:20:06.427" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8383" PostTypeId="1" CreationDate="2019-03-13T18:58:51.647" Score="1" ViewCount="199" Body="&lt;p&gt;Western philosophy was founded by Socrates, and Greek science was blossomed by his students and descendants such as Plato, Aristotle, Euclid, Archimedes, and many others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seemed that after the tragic death of Archimedes, Greek science and philosophy started declining into the Middle Ages, a dark period in science and civilization. The trend was snapped only after the Renaissance in Italy from the 15th century. Is this true? More thought is welcome.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2019-03-28T02:41:30.307" LastActivityDate="2019-03-28T02:41:30.307" Title="Did the death of Archimedes mark the turning point of the golden age of Greek science" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="5" ClosedDate="2019-03-14T12:35:58.693" ContentLicense="CC BY-SA 4.0" />
  <row Id="8384" PostTypeId="2" ParentId="8342" CreationDate="2019-03-13T20:15:21.787" Score="1" Body="&lt;p&gt;Calculus is based upon functions that are defined in a coordinate system. So Descartes' work of Cartesian coordinates lay the foundation for Newton and Leibniz to invent the Calculus. For example, in order to calculate the area below a curve, you must have a function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; to express the curve based on Cartesian coordinate before integration and the area is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;A=\int_a^b f(x)\:dx&#xA;$$&lt;/span&gt;&#xA;Furthermore, this fundamental contribution of Descartes goes far beyond geometry because the physical quantities such as force, distance, time, velocity, work and so on, are also based on Cartesian coordinates. This means that many notions in physical sciences like mechanics, electricity, thermodynamics, chemistry and so on, can be based on Cartesian coordinates as well.&lt;/p&gt;&#xA;&lt;p&gt;For example, the acceleration that is proportional to the force is the derivative of the distance function with respect to time&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;F=m\cdot a=m\cdot s''(t)&#xA;$$&lt;/span&gt;&#xA;To calculate work by force, we need first to express force as a function with respect to distance and then integrate it as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;W=\int_C F(s)\:ds&#xA;$$&lt;/span&gt;&#xA;Thus in order to do Calculus, you need functions expressed in the Cartesian coordinate system first.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-02-26T20:31:48.197" LastActivityDate="2021-02-26T20:31:48.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8385" PostTypeId="2" ParentId="8366" CreationDate="2019-03-14T03:16:31.970" Score="9" Body="&lt;p&gt;While this might not be a popular view, I think that Newton would fit this criterion. He was extremely isolated and socially alone. He was surrounded by all sorts of crazy ideas and spent more time on decoding the secret messages from the Bible that he thought could be revealed only to him and on alchemy than he did on actual physics. He did work obsessively, to wit, he invented calculus over a bet. &lt;/p&gt;&#xA;" OwnerUserId="4257" LastActivityDate="2019-03-14T03:16:31.970" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="8386" PostTypeId="2" ParentId="8383" CreationDate="2019-03-14T04:10:36.873" Score="6" Body="&lt;p&gt;After Archimedes, there was Apollonius (a mathematician in the same league as Euclid and Archimedes by consensus of historians of mathematics.) &#xA;Even later (no one knows exactly when) lived Diophantus. Apollonius and Diophantus created algebraic geometry in the same sense as Archimedes&#xA;invented integral calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Whole science of trigonomerty was developed after Archimedes death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are mathematicians. But Greek astronomy even did not even start at the time of Archimedes: the first of the two greatest astronomers of antiquity, Hipparchus, was born after Archimedes death. The second and the most famous, Ptolemy, lived more than 300 years after Archimedes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is true that Greek science experienced the periods of decline.&#xA;But the final decline and the end of it&#xA;came only at the time of the late Roman empire. (Ptolemy or Diophantus would mark&#xA;the turning point). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can certainly try to argue that Archimedes was the greatest of all of them. But continuing this line of thought you may conclude that the turning point of Western European science was Newton's death:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-03-14T04:19:11.867" LastActivityDate="2019-03-14T04:19:11.867" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8387" PostTypeId="1" AcceptedAnswerId="8388" CreationDate="2019-03-14T23:01:53.567" Score="15" ViewCount="6162" Body="&lt;p&gt;I recently came to know of an anecdote about Niels Bohr that &lt;a href=&quot;https://www.youtube.com/watch?v=lTAp3KZwYl0&quot; rel=&quot;noreferrer&quot;&gt;the philosopher Slavoj Zizek claims to have read in a biography of Bohr&lt;/a&gt;. He doesn't specify the author or the biography. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The anecdote goes as follows: Bohr used to keep a horseshoe on the door of his house. In European (and Indian) superstitions, the horseshoe is believed to be an object that guards the house against the evil spirits. A friend, upon seeing the horseshoe on the door of Bohr's house, asked Bohr as to whether he subscribed to the relevant superstitions. Bohr replied that he didn't believe in them but he was told that the horseshoe works whether or not one believes in their power. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to find an authentic historical account of the story but I could only find sparsely sourced articles which only claimed the story to be an anecdote. Is there an authentic historical account of the story being either true or false? Thanks!  &lt;/p&gt;&#xA;" OwnerUserId="4257" LastEditorUserId="55" LastEditDate="2019-03-15T05:44:35.007" LastActivityDate="2019-03-15T05:44:35.007" Title="Is the anecdote about Niels Bohr keeping a horseshoe on his door true?" Tags="|biographical-details|debunking|depiction-of-scientists|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8388" PostTypeId="2" ParentId="8387" CreationDate="2019-03-15T05:42:43.003" Score="12" Body="&lt;p&gt;One can spot a fabricated story by a number of tells: absence of the original citation, shifty dates (in Heisenberg's version, Bohr was telling it in 1927), proliferation of mutually exclusive details (in some versions, the horseshoe was over Bohr's desk). As for the origin, aside from Kenyon's and Droke's popular retellings of 1956, linked in the comments, we read the following in &lt;a href=&quot;https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.25.419&quot; rel=&quot;noreferrer&quot;&gt;Samuel Goudsmit's editorial Bias (Physical Review Letters 25 (1970) 419-20)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There  are still  authors  who believe  that  referees   and editors are biased   against them. We doubt  that  they  can be convinced  that  this  is not so. A  necessary  condition for being  a  successful  research  worker is  a  touch  of paranoia... I  have  known  a few physicists  who lack  this  essential  trait.   When  someone  publishes  their  results,   they  take  it as proof   that  their  work  was  worthwhile  and they are  happy to start  on something  else... I did not feel  it  that  way at all.   Unfortunately  I  could  not blame  any referees  or  editors  for  having been  scooped;  instead  I  just  blamed  my stars... &lt;strong&gt;In my study hangs  a  fine old  horse  shoe,  which  I found  in an abandoned  Western ghost  town.  I don't  believe  in superstitions, but it is  supposed  to work  even for  a nonbeliever&lt;span class=&quot;math-container&quot;&gt;$^2\!\!$&lt;/span&gt;. It hasn't  so far.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;em&gt;For  historians:  This  fact was  conveyed to me in  1941 by I. Bernard Cohen, the historian of science  at Harvard University. I passed it  on to  Niels  Bohr in  1954 when he visited Brookhaven.   It  is  now known as  &amp;quot;Bohr's story&amp;quot;. W. Heisenberg,  in his  book Der  Teil  und  das  Gauze,  incorrectly has  Bohr telling it  already in 1927.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The irony. Those interested in Goudsmit can consult &lt;a href=&quot;http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/goudsmit-samuel.pdf&quot; rel=&quot;noreferrer&quot;&gt;Bederson' scientific biography&lt;/a&gt;, but the story would lose its gloss without Bohr's name in it.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-03-15T05:42:43.003" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8390" PostTypeId="1" AcceptedAnswerId="8392" CreationDate="2019-03-16T21:44:59.460" Score="1" ViewCount="137" Body="&lt;p&gt;From my classes I don't hear about a lot of italian mathematicians, but two of them, Fubini and Tonelli, are both related to multivariable calculus. Is there a reason for this? Just a coincidence? Or was Italy historically stronger in this area?&lt;/p&gt;&#xA;" OwnerUserId="7481" LastEditorUserId="55" LastEditDate="2019-03-17T08:29:32.230" LastActivityDate="2019-03-17T08:36:55.257" Title="Was multivariable calculus particularly prominent in Italy?" Tags="|mathematics|mathematicians|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8391" PostTypeId="1" AcceptedAnswerId="11650" CreationDate="2019-03-17T05:20:58.070" Score="5" ViewCount="361" Body="&lt;p&gt;Dorothy Stein, a biographer of Ada Lovelace, was pretty blunt in her assessment: Lovelace was a mediocre mathematician, for example see &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.184217/page/n43&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder if she's fair to her. The  fact that Lovelace translated a printer's error &quot;cos&quot; (like cosine) literally as &quot;cos&quot; (instead of recognizing the French &quot;cas&quot; was meant, translating it as &quot;case&quot;) could be just some careless mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More serious is her trouble with functional equations, like the following exercise (English and notation slightly modernized):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Show that the equation &lt;span class=&quot;math-container&quot;&gt;$\phi(x+y) + \phi(x-y) = 2\phi(x)\,\phi(y)$&lt;/span&gt;&#xA;   is satisfied by &lt;span class=&quot;math-container&quot;&gt;$\phi(x)=(a^x + a^{-x})/2$&lt;/span&gt; for every value of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So I guess that would go like this: &lt;span class=&quot;math-container&quot;&gt;$$\begin{align}2\phi(x)\,\phi(y) &amp;amp;= (a^x + a^{-x})(a^y + a^{-y})/2 \\&amp;amp;= (a^{x+y}+a^{x-y} + a^{-x+y} + a^{-x-y})/2 \\&amp;amp;= (a^{x+y}+ a^{-(x+y)} + a^{x-y} + a^{-(x-y)} )/2 \\&amp;amp;= \phi(x+y) +\phi(x-y)\,.\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lovelace struggled with the problem and wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I do not know when I have been so tantalized by anything, &amp;amp; should be ashamed to say &lt;em&gt;how&lt;/em&gt; much time I have spent upon it, in vain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now it could be that as a beginner's mistake she misread the question, trying also to prove that this was the &lt;em&gt;only&lt;/em&gt; solution - which would be a much, much harder mathematical problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These are just two examples of the controversy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So I wonder if we &lt;em&gt;can&lt;/em&gt; make a somewhat objective judgment about her talent for mathematics?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wouldn't it be surprising if the evidence is compatible with both extremes:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Ada Lovelace simply wasn't very good at math; other factors like her heritage and social connections allowed her to take the position as Charles Babbage's assistant&lt;/li&gt;&#xA;&lt;li&gt;she was a mathematical genius?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9479" LastEditorUserId="4251" LastEditDate="2021-09-22T06:08:06.787" LastActivityDate="2021-09-22T06:40:41.577" Title="Can we fairly assess Ada Lovelace's talent for mathematics?" Tags="|mathematicians|computer-scientists|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8392" PostTypeId="2" ParentId="8390" CreationDate="2019-03-17T08:36:55.257" Score="5" Body="&lt;p&gt;In a way, but not exactly. From about the same period (late 19th-early 20th century), the names of Peano (differential equations, mathematical logic), Betti (algebraic topology, elasticity theory), Veronese (classical geometry), Castelnuovo, Enriques (algebraic geometry), Beltrami, Bianchi, Ricci-Curbastro, Levi-Civita (Riemannian geometry, tensor calculus), Arzela, Cesaro, Dini, Vitali (mathematical analysis), Volterra (integral equations, mathematical biology) are pretty widely known. Of course, Cavalieri and Torricelli were early calculus pioneers, and both Riemannian geometry and analysis are sophisticated descendants of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://books.google.com/books?id=SVI8ftMVE3gC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Mathematicians in Bologna 1861–1960, p.469&lt;/a&gt;, &quot;&lt;em&gt;the collaboration between Betti and Brioschi was the true driving force behind the initial development of Italian mathematics&lt;/em&gt;&quot;. Betti befriended Riemann, and did a lot to promote his ideas in Italy. He also deserves a lot of credit for fostering top grade mathematicians, both geometers and analysts. Six of the above were his doctoral students (Arzela, Bianchi, Dini, Enriques, Ricci-Curbastro and Volterra), see his &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=26323&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Genealogy page&lt;/a&gt;. Tonelli was Arzela's student, and Fubini was both Dini's and Bianchi's. Cesàro's 1890-s textbooks also promoted both subjects. Brioschi founded Milan's Polytechnic in 1863, geometers Cremona and Beltrami were his students. Beltrami was also active in promoting Riemann's ideas, he translated his works into Italian.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-03-17T08:36:55.257" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8393" PostTypeId="1" AcceptedAnswerId="8394" CreationDate="2019-03-17T11:11:47.177" Score="7" ViewCount="663" Body="&lt;p&gt;I found this &lt;a href=&quot;https://www.quora.com/Was-Ramanujan-a-student-at-Cambridge&quot; rel=&quot;noreferrer&quot;&gt;quote at Quora:&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (This degree was later renamed as Ph.D. from 1920) for his work on Highly composite numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given how important his paper on &lt;a href=&quot;http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf&quot; rel=&quot;noreferrer&quot;&gt;Highly composite numbers&lt;/a&gt; was it &lt;em&gt;appears&lt;/em&gt; that Cambridge upgraded their policy specifically for him;  but I'm unable to find anything more on the internet about this. &lt;/p&gt;&#xA;" OwnerUserId="9723" LastEditorUserId="9723" LastEditDate="2019-03-17T16:26:55.033" LastActivityDate="2019-03-18T03:24:36.803" Title="Did Cambridge change their BSc policy for Ramanujan?" Tags="|mathematicians|biographical-details|reference-request|debunking|publishing|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8394" PostTypeId="2" ParentId="8393" CreationDate="2019-03-18T03:24:36.803" Score="6" Body="&lt;p&gt;It does not appear that Cambridge changed the policy for Ramanujan, although they did use a rather atypical degree they had for him (Bachelor of &lt;em&gt;Arts&lt;/em&gt; by Research, &lt;em&gt;not&lt;/em&gt; Bachelor of Science by Research). PhD in Mathematics was not traditional in Britain, and was only created after the First World War. Ramanujan was a beneficiary of the general conversion in 1920s. Here is from &lt;a href=&quot;http://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm&quot; rel=&quot;noreferrer&quot;&gt;The Mathematics PhD in the United Kingdom: Historical Notes for the Mathematics Genealogy Project&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The UK has an unbroken tradition of 800 years of university mathematics but for nearly all of that time the creation of doctors played no part in the process of preparing the next generation of mathematicians. The PhD is new to the UK, compared to Germany or even to the United States (see NSF US Doctorates). It appeared at the end of the First World War and it was not until after the Second that a PhD became part of the usual preparation of a university mathematician... When looking up British mathematicians in the Mathematics Genealogy Project database:&lt;br/&gt;&#xA;  Do not expect to find many PhDs before 1940.&lt;br/&gt;&#xA;  Do not be surprised to find supervisors (advisors) with only a first degree...&lt;br/&gt;&#xA;  The image of a PhD system has been present since 1850, there has been a PhD degree since 1920 and a functioning PhD system since around 1950.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In the 19th century British university reformers looked to Germany for a model of the modern university and the higher degree was an element in the German system... &lt;strong&gt;In Cambridge there was no feeling that a research degree was necessary for its own graduates but the university created a Bachelor of Arts by Research for graduates from other universities; this could be completed in one year. This was not a popular degree and BritMath lists only one person with it, appropriately the unique Ramanujan...&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;By the beginning of the 20th century the case for a PhD type degree had been made and won — at least outside Cambridge (and Oxford). In 1917 representatives of the universities met and agreed that they would establish a PhD degree... However, deciding to create a degree is not the same as deciding that the degree should matter. There was no resolution to create a system in which the PhD would be an essential part of the preparation of the university academic. The system evolved without anybody planning it... Oxford was the first university to institute such a degree, although its choice of title, DPhil, was idiosyncratic. The first Oxford DPhil in mathematics was awarded in 1921. The first Cambridge PhD in mathematics was awarded in 1924 to an Australian Thomas Cherry.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-03-18T03:24:36.803" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8395" PostTypeId="1" AcceptedAnswerId="8396" CreationDate="2019-03-19T22:43:31.010" Score="3" ViewCount="357" Body="&lt;p&gt;The famous anecdote of the 1903 announcement of the factorization of &lt;span class=&quot;math-container&quot;&gt;$2^{67}-1$&lt;/span&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Frank_Nelson_Cole&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frank Nelson Cole&lt;/a&gt; has recently been discussed, for example in light of the announcement of another &quot;twitter-sized&quot; proof by Andrew Booker that &lt;span class=&quot;math-container&quot;&gt;$x^3+y^3+z^3=33$&lt;/span&gt; has a &lt;a href=&quot;https://www.google.com/search?q=youtube+42+is+the+new+33&amp;amp;rlz=1C1SQJL_enUS776US776&amp;amp;oq=youtube+42+is+the+new+33&amp;amp;aqs=chrome..69i57j69i60l2j69i64.8111j0j4&amp;amp;sourceid=chrome&amp;amp;ie=UTF-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;solution&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although there is already a fantastic &lt;a href=&quot;https://hsm.stackexchange.com/questions/2105/whats-the-famous-story-about-a-mathematician-who-gave-a-talk-without-saying-a-w/2106#2106&quot;&gt;breakdown&lt;/a&gt; of the story on HSM, and an even further review of the &lt;a href=&quot;https://mathoverflow.net/questions/207321/how-did-cole-factor-267-1-in-1903&quot;&gt;mathematics&lt;/a&gt; used by Cole on MathOverflow, I wonder if there's more evidence to be found to shed light on how much, if any, of the Cole story is true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'll take it as granted that Cole actually &lt;em&gt;did&lt;/em&gt; factor &lt;span class=&quot;math-container&quot;&gt;$M_{67}$&lt;/span&gt;, because he has a paper to prove it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest source of the anecdote seems to be E.T. Bell's account from the &quot;Queen of Mathematics&quot; chapter in James Roy Newman's edited &lt;strong&gt;&lt;em&gt;The World of Mathematics, Vol. 1&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bell is silent on how long the multiplication would have taken.  But there are other, secondary sources that expand on this.  For example, &lt;a href=&quot;https://books.google.com/books?id=0Dta7OkNhyoC&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gridgemen&lt;/a&gt; states &quot;Subsequently, in private, Cole said that those &lt;strong&gt;few minutes at the blackboard&lt;/strong&gt; had cost him three years of Sundays.&quot;  However, Wikipedia (March 19, 2019) states &quot;Cole returned to his seat, not having uttered a word during the &lt;strong&gt;hour-long presentation&lt;/strong&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because as far as can be traced the story seems to originate from Bell, many have taken that to establish a &lt;em&gt;prima facie&lt;/em&gt; case that the &lt;em&gt;entirety&lt;/em&gt; of the story is not to be believed.  But, is that harsh to Mr. Bell?  A broken clock is right twice a day.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least one question can be asked and answered:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Granting the necessary parts of the story, how long would it take for Cole to actually to &lt;em&gt;do the multiplication on the blackboard&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Perhaps people would not sit through an hour of silent multiplication as Wikipedia suggests, but would sit through a few minutes, as Gridgemen suggests.  My rough guess is that it's closer to ten minutes than an hour.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will perform the experiment shortly.&lt;/p&gt;&#xA;" OwnerUserId="9482" LastEditorUserId="9482" LastEditDate="2019-03-20T01:19:22.567" LastActivityDate="2020-05-26T11:07:17.930" Title="How long would it have taken Cole to multiply the factors of $M_{67}$ on a blackboard?" Tags="|mathematicians|number-theory|anecdote|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="8396" PostTypeId="2" ParentId="8395" CreationDate="2019-03-20T08:44:00.287" Score="5" Body="&lt;h1&gt;Hand-Computation of &lt;span class=&quot;math-container&quot;&gt;$193707721 \times 761838257287$&lt;/span&gt;&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;This is a &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt;-digit number times a &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;-digit number, and thus would require &lt;span class=&quot;math-container&quot;&gt;$9 \times 12 = 108$&lt;/span&gt; digit-by-digit products, some of which require carrying, and then adding &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; columns each with &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; digits (or adding &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; columns each with &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; digits). Assuming each of the &lt;span class=&quot;math-container&quot;&gt;$108$&lt;/span&gt; multiplications takes &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; seconds (this includes writing things down on the blackboard and sometimes carrying) and that adding the columns takes about &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; minutes, we get roughly &lt;span class=&quot;math-container&quot;&gt;$7.5$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, since there will be three &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;'s in the &quot;bottom number&quot; (no matter which number you use for the &quot;bottom number&quot;), you'll get close to a minute savings because for the second and third times you multiply the &quot;top number&quot; by &lt;span class=&quot;math-container&quot;&gt;$7,$&lt;/span&gt; you simply copy down what you did the first time (with an appropriate number of &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;'s to the right). Also, when multiplying the &quot;top number&quot; by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; you just copy down the &quot;top number&quot;. Finally, Cole had certainly done this several times previously, if only to check his work, so much of the computation would have been familiar to him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I would bet the multiplication was done on the blackboard in roughly &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; minutes &lt;span class=&quot;math-container&quot;&gt;$(2$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; minutes if he was alone, writing on paper), especially given that his skills in carrying out such computations were nowhere near as atrophied as is the case for most everyone today (due to calculators).&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Hand-Computation of &lt;span class=&quot;math-container&quot;&gt;$2^{67} - 1$&lt;/span&gt;&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;My guess is that he began with a power of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; that everyone knew, such as &lt;span class=&quot;math-container&quot;&gt;$2^{10} = 1024,$&lt;/span&gt; and worked up from that. Probably quickest would be to square this, then multiply the result by &lt;span class=&quot;math-container&quot;&gt;$1024$&lt;/span&gt; to obtain &lt;span class=&quot;math-container&quot;&gt;$2^{30},$&lt;/span&gt; then square this last number to get &lt;span class=&quot;math-container&quot;&gt;$2^{60},$&lt;/span&gt; and finally multiply &lt;span class=&quot;math-container&quot;&gt;$2^{60}$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$128.$&lt;/span&gt; My guess is this would take him about &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt; minutes. I used an on-line calculator and getting to &lt;span class=&quot;math-container&quot;&gt;$2^{20}$&lt;/span&gt; took &lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt; seconds (working quickly, but not trying to race the clock), but of course with three of the digits being &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; little work was needed.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Conclusion&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;My guess is that the entire process could probably be done in a relatively leisurely but steady manner within &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; minutes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I used to do long multiplications like this from time to time before anyone in my school had a calculator (roughly when I was age &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$16).$&lt;/span&gt; For example, when checking results I saw in books or articles (such as Martin Gardner's &quot;Mathematical Games&quot; articles I saw in library copies of Scientific American or in a library book of his articles, and similarly for some of Isaac Asimov's well known essays), or working out large powers of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3,$&lt;/span&gt; etc. Anyway, I have a rough idea of how much work this would involve, and while it might appear daunting to someone today, it's not really all that much. (What I did a year ago &lt;a href=&quot;https://math.stackexchange.com/questions/681893&quot;&gt;here&lt;/a&gt;, now that was daunting.) It's just not something one would do in a talk, even back then. Of course, what &quot;cost him three years of Sundays&quot; were the computations needed to discover the factors to begin with.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2019-03-20T10:29:16.747" LastActivityDate="2019-03-20T10:29:16.747" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8397" PostTypeId="2" ParentId="7920" CreationDate="2019-03-20T08:47:35.210" Score="4" Body="&lt;p&gt;The &amp;quot;midpoint equation&amp;quot; is another name for the &amp;quot;equation of the centre&amp;quot;, related to kepler's equation. What Gauss wanted to determine in this investigation is the asymptotic mathematical expression for the coefficients of the trigonometric series (Fourier series) for the difference between true anomaly &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; and mean anomaly &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v - M = \sum_{n=1}^\infty \frac{{1}}{{n}}C_n \sin(nM)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$C_n = f(\epsilon)$&lt;/span&gt;. The problem of determination of this trigonometric series is essentially a modern equivalent of the ancient method of epicycles - this was a model of the celestial positions of the planets which used a series of &amp;quot;wheels&amp;quot; (&amp;quot;epicycles&amp;quot;) with different radiuses and angular speeds to trace the location of the planet. The coefficient &lt;span class=&quot;math-container&quot;&gt;$\frac{C_n}{n}$&lt;/span&gt; is essentially the radius of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th wheel, while it's angular frequency is &lt;span class=&quot;math-container&quot;&gt;$\frac{nM}{t}$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is the time of a given mean anomaly measurement).&lt;/p&gt;&#xA;&lt;p&gt;Gauss's final asymptotic formula for the coefficients is written in the last line of p.423 of volume 10-1 as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$C_n \sim \gamma^n(1+\frac{\sqrt{8}}{3cos^{\frac{3}{2}}\varphi \cdot \sqrt{n\pi}})$$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is the base of natural logs and &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; is the arcsine of the eccentricity &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\varepsilon = \sin\varphi$&lt;/span&gt;), while &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; is defined through a new angle &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\gamma = tan(\frac{\theta}{2})$&lt;/span&gt; which itself is connected to angle &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; by the relation: &lt;span class=&quot;math-container&quot;&gt;$tan(\frac{\theta}{2}) = \tan(\frac{{\varphi}}{{2}})e^{\cos\varphi}$&lt;/span&gt;. Note that this formula descibes the asymptotic behaviour of &lt;span class=&quot;math-container&quot;&gt;$C_n$&lt;/span&gt; - it's not an explicit formula for the coefficients! Gauss's formula for &lt;span class=&quot;math-container&quot;&gt;$C_n$&lt;/span&gt; is therefore identical with Jacobi's formula from his 1849 memoir - &lt;span class=&quot;math-container&quot;&gt;$C_n \sim (\tan(\frac{{\varphi}}{{2}})e^{\cos\varphi})^n(1 + \frac{{4}}{{3}}\frac{{1}}{{\sqrt{{2n\pi \cos^3\varphi}}}})$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It's worth remarking that &lt;span class=&quot;math-container&quot;&gt;$C_n$&lt;/span&gt; tends to zero as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; tends to infinity only when &lt;span class=&quot;math-container&quot;&gt;$\gamma\le1$&lt;/span&gt; (otherwise, it tends to infinity); this means that the trigonometric series converges only when &lt;span class=&quot;math-container&quot;&gt;$\tan(\frac{{\varphi}}{{2}})e^{\cos\varphi}\le 1$&lt;/span&gt;. If we recall the definition of &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; - that &lt;span class=&quot;math-container&quot;&gt;$\varepsilon = \sin\varphi$&lt;/span&gt; - we get after using several trigonometric identities that the critical value of the eccentricity &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; for which the series begins to diverge is the numerical solution of the following equation : &lt;span class=&quot;math-container&quot;&gt;$\frac{\epsilon e^{\sqrt{1-\epsilon^2}}}{1+\sqrt{1-\epsilon^2}} = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is exactly the definition of &amp;quot;Laplace limit&amp;quot; (whose value is approximately &lt;span class=&quot;math-container&quot;&gt;$0.662$&lt;/span&gt;); it's a very significant point since Laplace's discovery of the divergence of this series for sufficiently large eccentricity as well as the calculation of this limit was published only in 1827 (so Gauss preceeded him by more than 20 years). On p. 424 of the same manuscript appears a table of relevant quantities which Gauss calculated for 25 different values of mean anomaly &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; (between 0 and 180 degrees) by cutting the infinite trigonometric series after several terms. The whole table is calculated for a large value of eccentricity - &lt;span class=&quot;math-container&quot;&gt;$0.655$&lt;/span&gt; - a value which is very close to Laplace limit. This table is probably the numerical aspect of the investigation Gauss conducted on &amp;quot;the degree of convergence of trigonometric series&amp;quot;; i.e the rate of convergence of such series.&lt;/p&gt;&#xA;&lt;p&gt;In order to derive the closed form expression for the coefficients of the series Gauss applied a cetain imaginary transformation for the eccentric anomaly &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; (the imaginary transformation is &lt;span class=&quot;math-container&quot;&gt;$E = i\cdot \log\cot(\frac{{\varphi}}{{2}}) + \epsilon$&lt;/span&gt;. Gauss doesn't expain how he arrived at this method, as he is operating entirely with series representations, but later the mathematician Jakob Horn, who was known for his contributions to the theory of asymptotic expansions, showed that one can arrive at this susbtitution by operating with integrals in the complex plane. The following is an excerpt from a letter of Jakob Horn in which he explains the meaning of Gauss's substitution:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For the developement &lt;span class=&quot;math-container&quot;&gt;$$C_n = \frac{1}{\pi}\int_{-\pi}^{+\pi}\frac{dv}{dM}e^{inM}dM = \frac{cos\varphi}{\pi}\int_{-\pi}^{+\pi}e^{inM}\frac{dE}{1-f cosE}$$&lt;/span&gt;&#xA;by introducing &lt;span class=&quot;math-container&quot;&gt;$z = e^{iE}$&lt;/span&gt; we get &lt;span class=&quot;math-container&quot;&gt;$$C_n = -i\frac{cos\varphi}{\pi}\int F(z)(\Phi(z))^ndz$$&lt;/span&gt;&#xA;in which &lt;span class=&quot;math-container&quot;&gt;$$F(z) = \frac{1}{z-\frac{f}{2}z^2-\frac{f}{2}}, \Phi(z) = ze^{-\frac{f}{2}(z-\frac{1}{z})}$$&lt;/span&gt;&#xA;is integrated over the unit circle &lt;span class=&quot;math-container&quot;&gt;$|z| = 1$&lt;/span&gt;. The function &lt;span class=&quot;math-container&quot;&gt;$F(z)$&lt;/span&gt; has the real singular points &lt;span class=&quot;math-container&quot;&gt;$$z_0 = \frac{1-\sqrt{1-f^2}}{f} = \mathbb{tan(\frac{\varphi}{2})} = e^{i\mathfrak R}&amp;lt;1$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$z_1 = \frac{1+\sqrt{1-f^2}}{f} = \mathbb{cot(\frac{\varphi}{2})} = e^{-i\mathfrak R}&amp;gt;1$$&lt;/span&gt;&#xA;which are also zeroes of &lt;span class=&quot;math-container&quot;&gt;$\Phi'(z)$&lt;/span&gt;. The integration path &lt;span class=&quot;math-container&quot;&gt;$$|z| = 1$$&lt;/span&gt; can be replaced by the integration path &lt;span class=&quot;math-container&quot;&gt;$|z| = z_0$&lt;/span&gt;, which only has to avoid the singular point &lt;span class=&quot;math-container&quot;&gt;$z_0$&lt;/span&gt;. One has to set accordingly &lt;span class=&quot;math-container&quot;&gt;$z = z_0e^{i\epsilon}$&lt;/span&gt;, or what is the same &lt;span class=&quot;math-container&quot;&gt;$$E = \mathfrak R +\epsilon = i\mathbb{logcot(\frac{\varphi} {2})}+\epsilon$$&lt;/span&gt;&#xA;which is precisely the substitution used by Gauss. If one takes the circle &lt;span class=&quot;math-container&quot;&gt;$|z| = z_1$&lt;/span&gt; as the integration path, bypassing the singular point &lt;span class=&quot;math-container&quot;&gt;$z_1$&lt;/span&gt;, one has &lt;span class=&quot;math-container&quot;&gt;$$z = z_1e^{i\theta}$$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$$E = \theta - i\mathbb{logcot(\frac{\varphi} {2})}$$&lt;/span&gt;&#xA;a substitution which is used in a paper by Wilhelm Scheibner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Therefore, it seems that the significance of this investigation of Gauss lies in the fact that his method of derivation is based on his ideas on the foundations of complex analysis. In his historic reconstruction of Gauss's line of thought, Schlesinger also mentiones some preliminary research done by Gauss and recorded in several diary entries. These entries are concerned with the divergent series &lt;span class=&quot;math-container&quot;&gt;$0!-1!+2!-3!+...\approx 0.5963$&lt;/span&gt; (Euler's famous summation) and the asymptotic behaviour of Bessel functions and the location of their zeroes (see Gauss's werke, volume 10, p.382-389).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Historical aspects&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Very late in his life (1850 and on) Gauss mentioned in several letters to his friend Schumacher his wish to write a fundamental treatise on the convergence of trigonometric series, which will include the connections to the doctrines of complex quantities (complex functions) and analysis situs (topological ideas about connectivity of surfaces). He claims priority for Jacobi's 1849 researches on kepler's equation, stating that he possesed the solution to the problem of coefficients of the trigonometric series for about &lt;span class=&quot;math-container&quot;&gt;$\pm 50$&lt;/span&gt; years. The exact year at which he made his relevant researches cannot be  dated exactly, but Schlesinger speculates that it was around the year 1805.&lt;/p&gt;&#xA;&lt;p&gt;Actually the first mathematical astronomer who tried to solved this problem was Francesco Carlini, who gave a remarkable but flawed solution in 1817, so Gauss's solution predates even him. For a survey article on Carlini's work, see the article; &amp;quot;&lt;em&gt;&lt;strong&gt;Francesco Carlini: Kepler’s equation and the asymptotic solution to singular differential equations&lt;/strong&gt;&lt;/em&gt;&amp;quot;. Carlini's solution was essentially different from Gauss's, and involved the earliest use of the WKB approximation of quantum mechanics, while Gauss's solution is rather similar to that W. Scheibner has given in 1856, which used Cauchy's theory of residues.&lt;/p&gt;&#xA;&lt;p&gt;The significance of Gauss's way of deriving the solution is that it's incomparably shorter than that of Jacobi. If i'm not mistaken, one of the possible subjects Gauss suggested to Riemann for his doctoral dissertation was trigonometric series, so i suspect that (since it was around the same year of his letters to schumacher) it might be connected somehow with that, but this is entirely a speculation of mine.&lt;/p&gt;&#xA;&lt;p&gt;Finally, it's worth remarking that the famous mathematician Vladimir Arnold, had  the following things to say about Laplace's limit:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In attempting to explain the origin of this constant, Augustin Cauchy created complex analysis. Numerous fundamental mathematical notions and results, such as  Bessel functions, Fourier series, the topological index of vector fields, and the argument principle in the theory of complex functions, also made their first appearence in investigations connected with Kepler's equation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This quotation is taken from Arnold's article &amp;quot;Kepler's second law and the topology of Abelian Integrals (According to Newton)&amp;quot;, which isn't really about  Laplace's constant (it's actually about &amp;quot;Newton's theorem on ovals&amp;quot;), but i mentioned it just to bring the circle of ideas surrounding Gauss's unpublished results to a certain historic closure.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Remaining questions&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I still don't grasp the basic idea behind Gauss's substitution &lt;span class=&quot;math-container&quot;&gt;$E = i\cdot \log\cot(\frac{{\varphi}}{{2}}) + \epsilon$&lt;/span&gt;, and i don't even understand how well defined quantities like &lt;span class=&quot;math-container&quot;&gt;$E,\varphi,\epsilon$&lt;/span&gt; can be related by this equation. Anyway, the meaning of this substitution is the subject of a post on Astronomy stack exchange, and i just keep updating this HSM post as well to increase the chances of getting a response on the meaning of Gauss's substitution.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2021-03-20T13:27:32.557" LastActivityDate="2021-03-20T13:27:32.557" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8398" PostTypeId="1" CreationDate="2019-03-20T15:38:01.707" Score="0" ViewCount="71" Body="&lt;p&gt;This  constant is assignable to charged leptons &lt;span class=&quot;math-container&quot;&gt;$(e, \mu, \tau)$&lt;/span&gt; representing the fraction of their individual magneton &lt;span class=&quot;math-container&quot;&gt;$\hat{m}= e \cdot \hbar/ 2  m$&lt;/span&gt; to their Compton-wavelength &lt;span class=&quot;math-container&quot;&gt;$\lambda_{C} = h / m  c$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{eqnarray*}&#xA;\frac{\hat{m}}{\lambda_{C}} = e \cdot c / 4 \pi   [\frac{A}{m} \cdot m^{2}] &#xA;\end{eqnarray*}&lt;/span&gt;&lt;br&gt;&#xA;Its dimensionality indicates that it determines &#xA;a natural constant of magnetic flux: &lt;span class=&quot;math-container&quot;&gt;$4,8 \times 10^{-18} $&lt;/span&gt; [Wb], however only a fraction &lt;span class=&quot;math-container&quot;&gt;$4 \alpha$&lt;/span&gt; of the magnetic flux-quantum &lt;span class=&quot;math-container&quot;&gt;$\Phi_{0} = h/ 2e$&lt;/span&gt;. The constant may thus define magnetic flux in a leptons dipole-field.&lt;/p&gt;&#xA;" OwnerUserId="4438" LastActivityDate="2019-03-20T15:38:01.707" Title="Which is the physical interpretation of the &quot;strange&quot; constant $e \cdot c/ 4 \pi$?" Tags="|quantum-mechanics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8399" PostTypeId="1" CreationDate="2019-03-20T23:00:28.793" Score="2" ViewCount="172" Body="&lt;p&gt;Today I've come across Michael Stifel's version of Pascal's Triangle, which I've seen referred to as the Figurate Triangle or the Triangle of Figurate Numbers as seen in &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=hRmMp4D12PIC&amp;amp;pg=PA172&amp;amp;lpg=PA172&amp;amp;dq=Stifel%27s%20Triangle&amp;amp;source=bl&amp;amp;ots=nnw61BBxu2&amp;amp;sig=ACfU3U1-yHrj_FRuZRzr_KyS8kPfWRovuQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiiovjb35HhAhWNY98KHcBrDP0Q6AEwBXoECAkQAQ#v=onepage&amp;amp;q=Stifel&amp;#39;s%20Triangle&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Combinatorics: Ancient and Modern&lt;/a&gt;&lt;/em&gt; (page 72) and &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=HUG5LIr49YsC&amp;amp;pg=PA92&amp;amp;lpg=PA92&amp;amp;dq=the%20history%20of%20pascal%27s%20triangle%20stifel&amp;amp;source=bl&amp;amp;ots=7IRaotEpKu&amp;amp;sig=ACfU3U0Ao2qE6NjKUaQpiUwBqjJ9J1RCow&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiAvfGl3JHhAhWvmeAKHXmkBycQ6AEwGXoECAgQAQ#v=onepage&amp;amp;q=the%20history%20of%20pascal&amp;#39;s%20triangle%20stifel&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Between Certainty and Uncertainty&lt;/a&gt;&lt;/em&gt; (page 93)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GBc5X.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GBc5X.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In class we were encouraged to attempt to figure out why his triangle was written the way it was. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through the research I've come up with, it appears that each column appears to be, in order, the natural numbers, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Triangular_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;triangle numbers&lt;/a&gt;, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tetrahedral_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;tetrahedral numbers&lt;/a&gt;, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But he is skipping a few of the initial numbers for each column. For example he is missing the triangle number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, the tetrahedral numbers &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any logical or known reasons why this is the case?&lt;/p&gt;&#xA;" OwnerUserId="9739" LastActivityDate="2019-03-20T23:00:28.793" Title="Why does Michael Stifel's version of Pascal's Triangle look the way it does?" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8400" PostTypeId="1" CreationDate="2019-03-21T12:22:05.137" Score="2" ViewCount="918" Body="&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;A point is that of which there is no part.&lt;/li&gt;&#xA;  &lt;li&gt;And a line is a length without breadth.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If above definition on point, expresses on point as to be indivisible length, as seems to be expressed in the question, &lt;a href=&quot;https://hsm.stackexchange.com/questions/7697/who-originated-the-concept-of-making-the-point-dimensionless/7699#7699&quot;&gt;Who originated the concept of making the point dimensionless?&lt;/a&gt;, or if it expresses point as to be nothing, is there any source which gives data on the reasoning to call point as either of them? If there are any reasons, what are those?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, is any source known on why line is thought to be of no breadth? If there are any reasons here, what are those?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do these definitions allow attaining any intended utility? Were these definitions made to attain utility as quantifying space or measuring space (thus also having origin from culture, from the need for measurement of length?), and were these definitions the only choice? If yes, how?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that, these definitions as to be not those which could be accepted directly (Would one define line as to be of no breadth directly?), and on whether they had their origins in culture or from pre large reasoning, from certain needs which might have made one (Euclid or &lt;a href=&quot;https://hsm.stackexchange.com/questions/8400/what-made-euclid-think-on-line-as-a-length-without-breadth-and-point-as-that-wit#comment16047_8400&quot;&gt;Heron of Alexandria&lt;/a&gt; or the one who defined it) have line as to be breadth less, or point as to be indivisible (this seems a bit acceptable?) or nothing (this might need more reasoning?).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; Euclid’s Elements Of Geometry, The Greek text of J.L. Heiberg (1883–1885)&#xA;from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus&#xA;B.G. Teubneri, 1883–1885&#xA;edited, and provided with a modern English translation, by&#xA;Richard Fitzpatrick&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="55" LastEditDate="2019-12-19T01:30:27.627" LastActivityDate="2019-12-19T01:30:27.627" Title="What made Euclid/Heron define line as a length without breadth and point as that which has no part?" Tags="|geometry|euclidean-geometry|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="8401" PostTypeId="2" ParentId="8400" CreationDate="2019-03-21T18:00:45.923" Score="5" Body="&lt;p&gt;As mentioned in the comments, the definitions were probably inserted by later authors, like Heron, for didactic purposes and/or for compliance with Aristotelian ideas about proper scientific exposition (which starts with &amp;quot;intuitive definitions&amp;quot; that fix the subject matter). There is plenty of textual evidence to support this, e.g. Russo discusses it at length in &lt;a href=&quot;https://link.springer.com/article/10.1007/s004070050016&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid’s Elements&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Many of the ideas here reported by Heron can be found in Aristotle’s writings. Among Aristotle’s passages on this subject we may quote, e.g., Phys., IV, 11, 220a 15 ff., where, among other things, points are said extremities of lines and the analogy between point and instant of time is introduced; De Cael., III, 1, 300a 14, where the above analogy is also considered; Met., V, 6, 1016b 24–30, where that which is indivisible and has position is called point. This definition of a point as a monad having position is in fact an ancient Pythagorean definition, as we know from Proclus (95, 22).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;As another example, we may recall that the definition of lines as extremities of surfaces, which is contained in a passage of Heron’s definition 2 (corresponding to definition I, 6 of the Elements), is the Platonic definition which Aristotle had criticized in Topica, VI, 6, 143 b 11. In other instances Heron may have used later sources than Euclid: for example&#xA;the definition of a straight line, as we shall see in detail in Sect. 10, seems to be drawn from Archimedes.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another, not necessarily incompatible, speculation about the functionality of the Book I definitions is given by Azzouni in &lt;a href=&quot;https://as.tufts.edu/philosophy/sites/all/themes/asbase/assets/documents/azzouniProofAndOntology.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proof and Ontology in Euclidean Mathematics&lt;/a&gt;. It has to do with Euclid's essential use of diagrams, and the dissimilarity of his approach to modern axiomatic method, where such definitions are of no use because nothing is (supposed to be) inferred from the diagrams. This would explain why modern mathematicians are usually perplexed by these definitions, given their, along with the diagrams', &amp;quot;lack&amp;quot; of mathematical utility.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, if one is concerned with which diagrams are admissible for demonstrations then having &amp;quot;fat&amp;quot; points, for example, has to be explicitly proscribed. Indeed, Epicureans objected to Euclid's demonstration of proposition I.1 on the basis that dimensionless points do not exist, the circles intersect over a segment, and the proof fails.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If these uses for diagrams are backtracked to ancient Greek mathematics, an analogous role can perhaps be found for their definitions of primitive terms as well: They are heuristic guides that give the student an intuitive grasp of otherwise difficult mathematical notions. Support for this view can be found in the fact that such definitions play no role in subsequent proofs. Unfortunately, Greek mathematicians took these definitions more seriously than the above interpretation can allow: Their constant tinkering with the definitions for straight lines, points, and so on, suggests they did not see them as heuristic.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Indeed, the same point applies to their diagrams: The Greek focus on whether mechanical methods for constructing curves were admissible suggests to some that (some) Greek mathematicians had constructivist scruples about mathematical objects... It's possible, however, that constructivist scruples (at least as they appear in the tradition leading to book I) were concerned with what sorts of diagrams are admissible in proofs, and not with mathematical objects at all... My conjecture (which only historians can decisively confirm or not) is that we see in the first book of Euclid's Elements a blend of a “pictorial” proof-system (in which diagrams are an essential part of the proofs), and a language-based proof-system (in which diagrams are merely heuristic).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alas, it is doubtful that historians can decisively confirm or infirm such conjectures.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-03-21T19:49:19.427" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8402" PostTypeId="2" ParentId="2064" CreationDate="2019-03-21T18:10:27.750" Score="3" Body="&lt;p&gt;This is a very interesting and complex topic that is far from closed. Here are some of the sources I have found.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Energy the subtle concept&lt;/strong&gt;, by Jennifer Coopersmith, is probably the book you are looking for. However, if you get interested in more critical accounts, continue reading.&#xA;&lt;a href=&quot;http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198716747.001.0001/acprof-9780198716747&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198716747.001.0001/acprof-9780198716747&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Energy conservation as an example of simultaneous discovery&lt;/strong&gt;, a classical (and very critical for its time) article by Thomas Kuhn and a good place to start understanding the subtleties of the history of energy. It has been published in the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;M. Clagett (Ed.), Critical problems in the history of science (pp. 321-356). Madison: University of Wisconsin Press. (Proceedings of the Institute for the History of Science at the University of Wisconsin, September 1-11, 1957); Th. S. Kuhn, The essential tension (pp. 66-104).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;More Heat than Light&lt;/strong&gt;, an extraordinary book by Philip Mirowski. Warning: Mirowski is interested in the link between the conservation of energy in Physics and the conservation of value in neoclassical economics, but its treatment of energy conservation is very useful for your purpose.&#xA;&lt;a href=&quot;https://www.cambridge.org/core/books/more-heat-than-light/4CD2ADE8D5DE8665E43E2922D7E360B3&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cambridge.org/core/books/more-heat-than-light/4CD2ADE8D5DE8665E43E2922D7E360B3&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Science of Energy&lt;/strong&gt;, by Crosbie Smith. While your question is coming from Mechanics, this book is focused on Thermodynamics. However, the first chapters could provide some valuable information and insights.&#xA;&lt;a href=&quot;https://www.press.uchicago.edu/ucp/books/book/chicago/S/bo3628434.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.press.uchicago.edu/ucp/books/book/chicago/S/bo3628434.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As said in comments the work of Mach and Elkana are other important sources but I have not had time to read them.&lt;/p&gt;&#xA;" OwnerUserId="9741" LastEditorUserId="9741" LastEditDate="2019-03-21T18:17:20.883" LastActivityDate="2019-03-21T18:17:20.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8403" PostTypeId="2" ParentId="8400" CreationDate="2019-03-21T19:53:56.923" Score="1" Body="&lt;p&gt;About utility, there is a record of Euclid answering this question himself.&#xA;One student asked him about utility of the whole of this business (axioms, definitions, theorems, etc.) Euclid called a slave and told him &quot;Give this student a small coin and let him go. He is looking for a &quot;utility&quot; everywhere, this is a wrong place for him&quot;. (This is not an exact citation of what Euclid said, but reflects the essence). &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-03-21T19:53:56.923" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8404" PostTypeId="2" ParentId="8367" CreationDate="2019-03-22T01:34:13.353" Score="1" Body="&lt;p&gt;If you are choosing the year 2007, those radios are very much still in operational use.  In fact, there are loads of photos to choose from vendors all over the world.  Consider radios by Yaesu, Kenwood, Icom, Elecraft, Flex-Radio, and so on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The radios I am showing here are all HF (high frequency) radios or HF combined with 6-meter band which is VHF (very high frequency).  HF radios usually cover the 160 to 10 meter bands.  That is 1.8 MHz to 30 MHz.  Technically, 1.8 MHz is actually MF (medium frequency).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is an Icom 756 Pro III that was manufactured from the years 2003 to 2009 when it was discontinued for more modern equipment.  This is a transceiver, capable of transmitting and receiving.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/6V5Hr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6V5Hr.png&quot; alt=&quot;Icom 756 Pro III&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another similar HF radio is from Kenwood and is the TS-590.  The model shown below is the TS-590S from the year 2010+ but earlier models from the year 2007 look just about the same.  This radio is a transceiver.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/VoboM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VoboM.png&quot; alt=&quot;Kenwood TS-590S&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This next radio from Yaesu is smaller, considered for mobile use (such as in a car or truck, sometimes a boat) or portable use (e.g. on a beach in Hawaii using battery power).  Again, it is a HF+6 radio.  This radio is a transceiver.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/iMZlw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iMZlw.png&quot; alt=&quot;Yaesu FT-891&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can find many, many more examples of radios from the 1950s to current times at the web site: &lt;a href=&quot;https://rigpix.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;rigpix.com&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2075" LastActivityDate="2019-03-22T01:34:13.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8405" PostTypeId="1" CreationDate="2019-03-22T12:31:28.813" Score="8" ViewCount="324" Body="&lt;p&gt;I know the story that Cole found the factoring of the big number &lt;span class=&quot;math-container&quot;&gt;$2^{67}-1$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any other remarkable achievement of hand calculation? &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2019-05-04T02:14:06.543" Title="Remarkable numerical calculations before electronic computers" Tags="|mathematics|physics|" AnswerCount="6" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8406" PostTypeId="2" ParentId="6193" CreationDate="2019-03-22T15:17:02.570" Score="3" Body="&lt;p&gt;For the completeness of the discussion in this post i must add a link to another relevant question - &lt;a href=&quot;https://hsm.stackexchange.com/questions/7920/meaning-of-passages-by-gauss-on-the-convergence-of-expansions-in-infinite-seri&quot;&gt;Meaning of passages by Gauss on the &amp;quot;convergence of expansions (in infinite series) of the (elliptical) equation of the center&amp;quot;?&lt;/a&gt; . The answer to this question, if interpreted appropriately, indicates that already in 1805 Gauss had deep insights on the meaning and applicability of integrals in the complex plane; ideas which he used to solve the astronomical problem of finding explicit expression for the coefficients of the fourier series of the &quot;equation of the centre&quot; - the difference between mean and true anomaly of a planet in a keplerian orbit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: another relevant result of Gauss appears as a short note (half page) on p.88 of volume 8 - it's entitled &quot;Beautiful Theorem of Probability Theory&quot;. Gauss left in this note the fourier inversion property of the normal distribution (Gaussian); i.e the fact that the fourier transform of a Gaussian is itself a Gaussian. One can visualize it by saying that the more localized is the Gaussian at the position domain, the wider it is at the frequency domain. The point is that establishing this result requires knowledge of facts from complex integration: one needs it in order to argue that integration of the Gaussian from &lt;span class=&quot;math-container&quot;&gt;$x = -\infty$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x = +\infty$&lt;/span&gt; is the same after tanslation of the real line by an addition of imaginary number to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.     &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-07-29T17:31:54.687" LastActivityDate="2019-07-29T17:31:54.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8407" PostTypeId="2" ParentId="3353" CreationDate="2019-03-22T16:03:16.487" Score="3" Body="&lt;p&gt;A piece of info that is key for energy as a concept is Emmy Noether's brilliant work on symmetry and conservation laws. I have not come accross an explanation of the Noether Theorem that I can understand in detail, but I think this wikipedia article provides a good illustration:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Therefore, when a system behaves symmetrically in time, energy is conserved, something that occurs seldom (if ever) in reality. To me, this result suggests that energy is a mathematical formalism, an abstract concept with very interesting applications in the natural sciences, engineering, etc. but should not be treated as a thing that is actually conserved.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Being a female, Noether was not allowed to become a formal college student and never received the same formal status that her coleagues had, even though the latter respected her work and tried to help her.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are some links that I have found:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/2041/a-book-on-emmy-noethers-life&quot;&gt;A book on Emmy Noether&amp;#39;s life&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nytimes.com/2012/03/27/science/emmy-noether-the-most-significant-mathematician-youve-never-heard-of.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bbc.co.uk/programmes/m00025bw&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.bbc.co.uk/programmes/m00025bw&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9741" LastActivityDate="2019-03-22T16:03:16.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8408" PostTypeId="2" ParentId="8405" CreationDate="2019-03-23T10:03:01.623" Score="4" Body="&lt;p&gt;What about Archimedes computing &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; with such a precision that he was able to prove that&lt;span class=&quot;math-container&quot;&gt;$$3+\frac{10}{71}&amp;lt;\pi&amp;lt;3+\frac17?$$&lt;/span&gt;Or the ancient Babylonians being able to compute&lt;span class=&quot;math-container&quot;&gt;$$\sqrt2\simeq1+\frac{24}{60}+\frac{51}{60^2}+\frac{10}{60^3}?$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2019-03-23T10:09:36.653" LastActivityDate="2019-03-23T10:09:36.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8409" PostTypeId="1" CreationDate="2019-03-23T11:04:25.840" Score="16" ViewCount="1839" Body="&lt;p&gt;I have been reading Point Set Topology for a while, and turns out that there are various possible ways to define a topology. Most popular one is using open set axioms. Another one is using closure axioms, which was introduced by Kuratowski.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested to know name of the researchers who had given the definition of topological spaces in the following ways:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Definition through Open sets (The most popular one)&lt;/li&gt;&#xA;&lt;li&gt;Definition through neighbourhood system&lt;/li&gt;&#xA;&lt;li&gt;Definition through interior points&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="9744" LastActivityDate="2022-09-15T05:53:19.610" Title="History of various definitions of topology" Tags="|mathematics|mathematicians|topology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8410" PostTypeId="2" ParentId="8405" CreationDate="2019-03-23T23:19:47.263" Score="4" Body="&lt;p&gt;One famous hand calculation was that by &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Shanks&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Shanks&lt;/a&gt;, who attempted to calculate the first 707 digits of pi by hand, using Machin's formula and the Maclaurin series for the arctangent function. He did this all by hand. Unfortunately, only the first 527 places were correct. This error was discovered over 70 years later by someone using a desk calculator. The error diminishes the accomplishment, but it is still a &quot;remarkable achievement of hand calculation.&quot; It certainly has been remarked on often.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The linked Wikipedia article also states that &quot;Shanks also calculated e and the Euler–Mascheroni constant γ to many decimal places. He published a table of primes up to 60 000 and found the natural logarithms of 2, 3, 5 and 10 to 137 places.&quot;&lt;/p&gt;&#xA;" OwnerUserId="426" LastActivityDate="2019-03-23T23:19:47.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8411" PostTypeId="1" AcceptedAnswerId="8413" CreationDate="2019-03-24T15:48:27.947" Score="-2" ViewCount="212" Body="&lt;p&gt;As general relativity and quantum mechanics become more accessible, is Newton going to become the new Aristotle, as the example of being wrong and misleading humanity for centuries? (as opposed to the currently popular idea that Newton came up with the right thing and got people out of the delusion created by Aristotle)&lt;/p&gt;&#xA;" OwnerUserId="9748" LastActivityDate="2019-03-24T21:12:10.437" Title="Is Newton going to be the new Aristotle?" Tags="|physics|newton|" AnswerCount="1" CommentCount="4" ClosedDate="2019-03-25T12:38:33.423" ContentLicense="CC BY-SA 4.0" />
  <row Id="8412" PostTypeId="2" ParentId="8405" CreationDate="2019-03-24T20:19:28.757" Score="6" Body="&lt;p&gt;Another example of a &quot;remarkable achievement of hand calculation&quot; was in the field of mathematical astronomy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;During 1758, Alexis Clairaut and his collaborators in Paris worked to refine Edmond Halley's prediction (published in 1705) of a return in about 1758 of the comet that now carries Halley's name.  Halley's original prediction had been for 1758, based on his assessment that the comet of 1682 was periodic, with a mean orbital period of about 75.5 years.  (Halley later somewhat modified the prediction by an informed guess that a close encounter with Jupiter could delay the event by a few months, perhaps to early 1759.)  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clairaut's calculations refined the prediction, forecasting a return with perihelion passage in mid-April 1759 -- give or take a month. Observations of the comet then showed that indeed it was a return of a comet with closely similar orbital elements as the 1682 comet, verifying the return prediction, and showing that it passed perihelion on 13 March, just within the margin of error that Clairaut had allowed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The calculation was remarkable both for what was done and for the public reaction to which it contributed.&#xA;Clairaut, assisted by Jerome Lalande and Mme Lepaute, had calculated by hand a numerical integration of the perturbing effects of Jupiter and Saturn. Lalande wrote that for six months Clairaut and his collaborators in Paris calculated from morning to night, sometimes even continuing at the table at mealtimes. They verified that the perturbing effects acted to delay the return of the comet by some months relative to the original date-estimate of 1758. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Much of the public attention was on the event itself, but the calculations also drew attention and increased Clairaut's fame (not without controversy, spurred partly by rivalries as well as by scientifically-based objections). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- &lt;strong&gt;For the nature of the (hand) calculations&lt;/strong&gt; : &#xA;&lt;a href=&quot;http://adsabs.harvard.edu/abs/1993JHA....24....1W&quot; rel=&quot;noreferrer&quot;&gt;&quot;Clairaut's Calculation of the Eighteenth-century Return of Halley's Comet&quot;&lt;/a&gt; (C A Wilson, Journal for the History of Astronomy, v.24 (1993), pp.1-15).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- &lt;strong&gt;For the public reception of the events and calculations in France&lt;/strong&gt;  : &#xA;&lt;a href=&quot;http://adsabs.harvard.edu/full/1986LAstr.100..397T&quot; rel=&quot;noreferrer&quot;&gt;&quot;Clairaut et le retour de la « comète de Halley » en 1759&quot;&lt;/a&gt; &#xA;(R Taton, &lt;em&gt;L'Astronomie&lt;/em&gt; 100 (1986), 379-408).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- &lt;strong&gt;For a report of other contemporary views of the events&lt;/strong&gt; : &#xA;&lt;a href=&quot;http://adsabs.harvard.edu/full/1986JHA....17....1W&quot; rel=&quot;noreferrer&quot;&gt;&quot;Comet Halley's First Expected Return&quot;&lt;/a&gt; &#xA;(C B Waff, Journal for the History of Astronomy, v.17 (1986), pp.1-37).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;-- For a scientific history of Halley's comet : &#xA;&lt;a href=&quot;https://www.jstor.org/stable/37959&quot; rel=&quot;noreferrer&quot;&gt;&quot;The History of Halley's Comet &quot;&lt;/a&gt; &#xA;(D W Hughes et al., Philosophical Transactions of the Royal Society, Series A, 323 (1987), 349-367). &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2019-03-24T20:19:28.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8413" PostTypeId="2" ParentId="8411" CreationDate="2019-03-24T21:12:10.437" Score="9" Body="&lt;p&gt;I suggest that the baseless suggestion offered in this question can best be answered by Einstein's own words about Newton, written in 1919. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The background was the now-well-known eclipse expedition of 1919, in which the amount of deflection of light from stars close to the sun's limb had been observed during the eclipse. The results gave a probable verification of the predictions of general relativity about the amount of gravitational light-bending -- differing from the amount  predicted by classical physics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 'Times' newspaper carried in November 1919 a report of the event, and the headline to the article included the words &quot;Newton overthrown&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remarkably, Einstein himself, asked to comment, wrote back and disagreed. In his letter he gave some explanation, tailored to his audience, of the nature and basis of relativity theory and how it had had to correct Newtonian physics -- though with retention of Newtonian results in very close approximation, in very many situations. But Einstein also wrote:-  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;No one must think that Newton's great creation  can be overthrown in&#xA;  any real sense by this or  by any other theory.  His clear and wide&#xA;  ideas  will for ever retain their significance as the  foundation on&#xA;  which our modern conceptions of  physics have been built.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Eisntein's letter to 'The Times' (of Nov. 28, 1919) was widely reproduced, and nearly the full text can now most easily be read in &lt;a href=&quot;https://archive.org/details/science511920mich/page/8&quot; rel=&quot;noreferrer&quot;&gt;'Science' (New Series, vol.LI, no.1305, Jan 2, 1920, at pp.8-10)&lt;/a&gt;. (The only omission is of a personal wry joke in a closing paragraph.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The original German was also published in Einstein's 'Mein Weltbild' of 1934, where the paragraph &#xA;quoted above reads: &quot;Niemand aber soll denken, daß durch diese oder irgendeine andere Theorie Newtons große Schöpfung im eigentlichen Sinne verdrängt werden könne. Seine klaren und großen Ideen werden als Fundament unserer ganzen modernen Begriffsbildung auf dem Gebiet der Naturphilosophie ihre eminente Bedeutung in aller Zukunft behalten.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This question repeats one of many myths about Newton, repeated uncritically by those who give little sign of wishing to enquire for the truth.  One of the important reasons for regretting this proliferation of myth, based largely on fiction and gossip, is that it hinders more serious historical enquiry.  Newton was a pioneer over a large field who did make some mistakes and did leave many matters still to be further developed: it could hardly be otherwise, seeing how the lines of investigation for which he set an agenda took centuries after him to be worked out and enriched by many independently ingenious discoverers.  The presence of myth and fiction about the roots of Newtonian science, by creating false preconceptions, tends to spoil the subject for proper historical investigation. But unfortunately there is little hope of an end to such false debunking. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2019-03-24T21:12:10.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8414" PostTypeId="2" ParentId="8405" CreationDate="2019-03-25T15:34:56.780" Score="6" Body="&lt;p&gt;I've always liked logarithms because of their properties, and for some time I wondered who got the idea in the first place and how were the tables computed. It turns out logarithms were developed and computed simultaneously and independently by John Napier and Joost Bürgi. Both of them calculated huge logarithm tables by hand:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Napier computed almost ten million entries from which he selected the appropriate values. Napier himself reckoned that computing this many entries had taken him twenty years, which would put the beginning of his endeavors as far back as 1594.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bürgi computed logarithms for 100000000 to 1000000000. This filled fifty-eight pages of tables for a total of 23,030 entries (23,027, plus an additional 3 entries) to be computed to eight significant digits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-joost-b-rgi-introduces-logarithms&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-joost-b-rgi-introduces-logarithms&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9741" LastActivityDate="2019-03-25T15:34:56.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8415" PostTypeId="2" ParentId="8409" CreationDate="2019-03-25T18:01:45.833" Score="10" Body="&lt;p&gt;By the time these definitions were introduced as definitions there was a body of previous work, where they were convenient side notions for stating theorems in special cases for subsets of real line, plane, and then curve and function spaces (Cantor's accumulation points, derived and closed sets on the line, Weierstrass's theorems, Heine-Borel, Ascoli-Arzelà, Frechet's metric notions, etc.).&lt;/p&gt;&#xA;&lt;p&gt;The first axiomatic definitions are due to Riesz (accumulation points, 1906) and Hausdorff (neighborhoods, 1914). Hausdorff's Grundzüge der Mengenlehre became a model of axiomatic exposition of both set theory and topology. Some details from &lt;a href=&quot;https://books.google.com/books/about/Handbook_of_the_History_of_General_Topol.html?id=m0deV_61e5oC&quot; rel=&quot;noreferrer&quot;&gt;Kreyszig's chapter in the Handbook of History of General Topology, vol.1&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The earliest definition of a topological space was given in 1906 (and somewhat&#xA;obscurely published in 1907) by F. Riesz (1880-1956), with an abstract in&#xA;the Atti of the International Congress of Mathematicians in Rome in 1908 (cf.&#xA;[49], I, 110-154, 155-161). Of the possible concepts to be axiomatized (neighborhood, closure, etc.), Riesz chose accumulation point, as his axioms will show. He followed Hilbert in calling it Verdichtungsstelle (condensation point, which is now used in a different sense).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;...It is not known why Riesz did not make neighborhood his basic concept to&#xA;be axiomatized (as in Hausdorff's definition of topological space). It may be&#xA;that he wished to stay more closely to analysis rather than to geometry. To fully&#xA;understand this, we should recall that the concept of a (general) neighborhood&#xA;of a point developed only slowly, even in the plane, where it was first used as&#xA;late as in 1902, by Hilbert in a paper on the foundations of geometry (Math.&#xA;Ann. 56, 381-422). Neighborhoods ,also occurred later in work by Hedrick and&#xA;by Frechet, mentioned in [63], 1020...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In his classic Grundzüge der Mengenlehre [25], Hausdorff created a useful concept of a topological space by axiomatizing 'neighborhood'. His well-known axioms included his separation axiom &lt;span class=&quot;math-container&quot;&gt;$T_2$&lt;/span&gt; (that any two distinct points have disjoint neighborhoods), so that he obtained a Hausdorff space. Working out his theory of topological and metric spaces abstractly but with applicability in mind, he made his book a landmark that has often been regarded as the beginning of set-theoretic topology as we understand it today.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Subsequently, multiple alternative definitions appeared, which are referenced in &lt;a href=&quot;https://archive.org/details/GeneralTopology&quot; rel=&quot;noreferrer&quot;&gt;Kelley's General Topology (Notes to Chapter 1)&lt;/a&gt;, starting with the volumes of Fundamenta Mathematica following the publication of Hausdorff's book in 1914 and going into 1940-s. I am not sure which one is the &amp;quot;interior point&amp;quot; one.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/t/&quot; rel=&quot;noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses&lt;/a&gt;, the first definition equivalent to the modern one was given by Alexandrov and Hopf in 1935 in terms of &lt;a href=&quot;https://en.wikipedia.org/wiki/Kuratowski_closure_axioms&quot; rel=&quot;noreferrer&quot;&gt;Kuratowski closure axioms&lt;/a&gt; of 1922:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The present usage of the term TOPOLOGICAL SPACE (topologischer Raum) was firmly established in the German language textbook Topologie I (1935) by Alexandroff and Hopf where they use what they call KURATOWSKI'S AXIOMS; they are equivalent to Hausdorff's axioms without axiom &lt;span class=&quot;math-container&quot;&gt;$T_{2}$&lt;/span&gt;​ which is then discussed separately, along with other separation axioms.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The modern open set definition appears in &lt;a href=&quot;https://www.springer.com/us/book/9783540642411&quot; rel=&quot;noreferrer&quot;&gt;Bourbaki's General Topology (1940)&lt;/a&gt;. They do not credit anyone for it, nor does Kelley, but then it is closely related to the neighborhood version and dual to the Kuratowski closure version. Their short surmise of history appears in the Historical Note to Chapter 1:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first attempts to abstract what is common to properties of sets&#xA;of points and sets of functions are due to Frechet [15] and F. Riesz [16].&#xA;The former started from the notion of countable limit and did not succeed&#xA;in constructing a convenient and fruitful system of axioms...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;General topology as it is understood today began with Hausdorff&#xA;([17], Chapters 7, 8, 9), who again took up the concept of neighbourhood&#xA;(by which he meant what in the terminology of this series of volumes is&#xA;called an &amp;quot;open neighbourhood&amp;quot;) and chose from Hilbert's axioms for&#xA;neighbourhoods in the plane those which gave his theory all the precision&#xA;and generality desired. The axioms he took as a starting-point were essentially&#xA;(taking into account the difference between his concept of neighbourhood&#xA;and ours) axioms (V&lt;span class=&quot;math-container&quot;&gt;$_I$&lt;/span&gt;), (V&lt;span class=&quot;math-container&quot;&gt;$_{II}$&lt;/span&gt;), (V&lt;span class=&quot;math-container&quot;&gt;$_{III}$&lt;/span&gt;), (V&lt;span class=&quot;math-container&quot;&gt;$_{IV}$&lt;/span&gt;) of § 1 and (H) of § 8, and the chapter in which he develops the consequences of these axioms&#xA;has remained a model of axiomatic theory, abstract but adapted in advance&#xA;to applications.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;...Finally, the introduction of filters by H. Cartan [20] has brought to topology a valuable instrument, usable in all sorts of applications (in which it replaces to advantage the notion of &amp;quot;Moore-Smith convergence&amp;quot; [18]). Furthermore, the development of the theorem on ultrafilters (Theorem I, § 6), has clarified and simplified the theory.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cartan essentially rediscovered what was done by Vietoris in 1913-19.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-09-15T05:53:19.610" LastActivityDate="2022-09-15T05:53:19.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8416" PostTypeId="1" CreationDate="2019-03-25T20:15:21.423" Score="2" ViewCount="91" Body="&lt;p&gt;Said quickly, solstices are rather perceptible while the equinox is a mental construction. Archeoastronomical evidence shows that neolithic people already had knowledge about the solsticial points on the horizon and eventually they had also some interest in the point that lies in the middle between them.  When the sun rises from this point the duration of the day should be equal to that of the night. Measuring these time lengths with sufficient precision is actually a demanding practical task (and later it was shown that on this date the observable day is slightly longer). But idea is easy to conceive and there should be various mentions in preserved texts. References are what this question is about.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2019-04-03T22:24:02.457" Title="Where to find some early discussions of the Equinox(es)?" Tags="|reference-request|astronomy|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8417" PostTypeId="1" AcceptedAnswerId="8418" CreationDate="2019-03-26T00:42:02.840" Score="13" ViewCount="21955" Body="&lt;p&gt;Browsing Facebook, I often come across posts like this, to test peoples' understanding of order of operations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WMRk7.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WMRk7.png&quot; alt=&quot;Use your brain: 50 + 50 - 25x0 + 2 + 2=?&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This inevitably prompts a deluge of answers that either misunderstand the concept or appear completely unaware of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qoo4s.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qoo4s.png&quot; alt=&quot;A wall of people answering &amp;quot;4&amp;quot; or &amp;quot;79&amp;quot; with a few answering &amp;quot;104&amp;quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Occasionally, people will justify their answer by saying things that they were taught a long time ago and were definitely taught something different in high school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm aware that among mathematicians, the general concept has been accepted since at least the 19th century, but for how long in has it been widely taught in high school mathematics classes?  I assume this will differ for different parts of the world, so please include what countries or regions you can speak for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link to the post in question: &lt;a href=&quot;https://www.facebook.com/postified/posts/1944482075657955&quot; rel=&quot;noreferrer&quot;&gt;https://www.facebook.com/postified/posts/1944482075657955&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9754" LastEditorUserId="5788" LastEditDate="2019-03-26T01:24:32.250" LastActivityDate="2022-10-04T03:18:15.080" Title="How long has the order of priority of arithmetical operations been widely taught in high schools?" Tags="|mathematics|notation|education|algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8418" PostTypeId="2" ParentId="8417" CreationDate="2019-03-26T05:36:24.993" Score="16" Body="&lt;p&gt;This question comes up often, but there is no up to date scholarly study of it that I know of. The most comprehensive recent accounts (and they are rather brief) seem to be &lt;a href=&quot;https://jeff560.tripod.com/operation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Uses of Symbols of Operation&lt;/a&gt; and &lt;a href=&quot;https://web.archive.org/web/20180111045655/http://mathforum.org/library/drmath/view/52582.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peterson's post on Math Forum&lt;/a&gt;, parts of which are aped on various pop-sites, often without attribution. But Peterson admits:&amp;quot;I have not found a definitive answer yet&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Part of the problem is that the most natural source, &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History Of Mathematical Notations&lt;/a&gt;, is inconclusive, possibly because the rules never really became &amp;quot;official&amp;quot;. The &amp;quot;basic rule&amp;quot; that multiplication has precedence over addition is traceable to van Schooten's 1646 edition of Vieta. On the rest Peterson speculates:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I suspect that the concept, and especially the term &amp;quot;order of operations&amp;quot; and the &amp;quot;PEMDAS/BEDMAS&amp;quot; mnemonics, was formalized only in this century, or at least in the late 1800s, with the growth of the textbook industry. I think it has been more important to text authors than to mathematicians, who have just informally agreed without needing to state anything officially.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even if the &amp;quot;basic rule&amp;quot; was enshrined by texts in the early 19th century (the time of the rise of modern academia) this does not apply to the rest. As of Cajori's writing (1920s):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If an arithmetical or algebraical term contains &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; , there is at present no agreement as to which sign shall be used first. &amp;quot;It is best to avoid&#xA;such expressions.&amp;quot; For instance, if in &lt;span class=&quot;math-container&quot;&gt;$24\div4\times2$&lt;/span&gt; the signs are used as&#xA;they occur in the order from left to right, the answer is &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;; if the sign&#xA;&lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; is used first, the answer is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Some authors follow the rule that the multiplications and divisions shall be taken in the order in which they occur. Other textbook writers direct that multiplications in any order be performed first, then divisions as they occur from left to right. The term &lt;span class=&quot;math-container&quot;&gt;$a\div b\times b$&lt;/span&gt; is interpreted by Fisher and Schwatt as &lt;span class=&quot;math-container&quot;&gt;$(a\div b)\times b$&lt;/span&gt;. An English committee recommends the use of brackets to avoid ambiguity in such cases.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In short, it still was not clear &lt;em&gt;what&lt;/em&gt; to teach widely at schools. As of Peterson's writing (2000):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I have not yet found any twentieth-century declarations that resolved these issues, so I do not know how they were resolved... There is still some development in this area, as we frequently hear from students and teachers confused by texts that either teach or imply that implicit multiplication (&lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt;) takes precedence over explicit multiplication and division (&lt;span class=&quot;math-container&quot;&gt;$2\times x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$2/x$&lt;/span&gt;) in expressions such as &lt;span class=&quot;math-container&quot;&gt;$a/2b$&lt;/span&gt;, which they would take as &lt;span class=&quot;math-container&quot;&gt;$a/(2b)$&lt;/span&gt;, contrary to the&#xA;generally accepted rules.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Still, the reference to &amp;quot;generally accepted rules&amp;quot; suggests some &amp;quot;received view&amp;quot; in place. Miller's brief survey of early 20th century texts suggests that it made its way into the standard curriculum around that time:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1892 in Mental Arithmetic, M. A. Bailey advises avoiding expressions containing both ÷ and ×.&lt;br/&gt;&#xA;In 1898 in Text-Book of Algebra by G. E. Fisher and I. J. Schwatt, a÷b×b is interpreted as (a÷b)×b.&lt;br/&gt;&#xA;In 1907 in High School Algebra, Elementary Course by Slaught and Lennes, it is recommended that multiplications in any order be performed first, then divisions as they occur from left to right.&lt;br/&gt;&#xA;In 1910 in First Course of Algebra by Hawkes, Luby, and Touton, the authors write that ÷ and × should be taken in the order in which they occur.&lt;br/&gt;&#xA;In 1912, First Year Algebra by Webster Wells and Walter W. Hart has: &amp;quot;Indicated operations are to be performed in the following order: first, all multiplications and divisions in their order from left to right; then all additions and subtractions from left to right.&amp;quot;&lt;br/&gt;&#xA;In 1913, Second Course in Algebra by Webster Wells and Walter W. Hart has: &amp;quot;Order of operations. In a sequence of the fundamental operations on numbers, it is agreed that operations under radical signs or within symbols of grouping shall be performed before all others; that, otherwise, all multiplications and divisions shall be performed first, proceeding from left to right, and afterwards all additions and subtractions, proceeding again from left to right.&amp;quot;&lt;/em&gt;&amp;quot;&lt;br/&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For my part, I suspect that the rise of computers and formal languages, and the involvement of research mathematicians in school education (see the &lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Math movement&lt;/a&gt;) in the 1960-s put further pressures for the canonization of the Wells-Hart rules. In the Soviet Union, to the threat of which the New Math was a response, this happened even earlier, around 1930s. It was not &lt;em&gt;fully&lt;/em&gt; successful even to this day. Witness not only the aforementioned deviations, but even the questioning of the utility of the whole convention. As of 2007, Wu wrote in &lt;a href=&quot;https://math.berkeley.edu/%7Ewu/order5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Order of operations” and other oddities in school mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;One of the flaws of the school mathematics curriculum is that it wastes time in fruitless exercises in notation, definitions, and conventions, when it should be spending the time on mathematics of substance.  Such flaws manifest themselves in assessment items which assess, not whether students know real mathematics, but  whether  they  could  memorize  arcane  rules  or  senseless  conventions whose raison-d’etre they know nothing about. An example is the convention known as the Rules for the Order of Operations, introduced into the school curriculum in the fifth or sixth grade.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="2277" LastEditDate="2022-10-04T03:18:15.080" LastActivityDate="2022-10-04T03:18:15.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8419" PostTypeId="1" AcceptedAnswerId="8420" CreationDate="2019-03-26T21:50:49.097" Score="4" ViewCount="344" Body="&lt;p&gt;The entropy of a distribution &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; on a discrete set &lt;span class=&quot;math-container&quot;&gt;$\mathcal{X}$&lt;/span&gt; is defined as &lt;span class=&quot;math-container&quot;&gt;$$H(p) = -\sum_{x \in \mathcal{X}} p_x \log p_x.$$&lt;/span&gt; Shannon in his classic paper [1] defines the analogue for continuous distributions with density &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; as (modern notation) &lt;span class=&quot;math-container&quot;&gt;$$h(p) = - \int_{\mathcal{X}} p(x) \log  p(x).$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, Shannon never uses the term differential entropy in his paper. The earliest use of the term I could find was from this paper of Kolmogorov [2] from 1956, and I couldn't go further because the papers he cites are hard to find (which seems to be true for most of the old papers of the Russian school, not that I can read Russian :-/). I also couldn't just start going through the ISIT proceedings from '48-56, because IEEExplore doesn't make them easily accessible. I haven't attempted to check the Trans. Inf. Th. volumes from the early '50s yet. In a different direction, standard texts like Cover and Thomas don't mention the etymology either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have two questions:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;When was the term differential entropy first used in an academic publication (or technical report etc.)?&lt;/li&gt;&#xA;  &lt;li&gt;Why is the term called &lt;em&gt;differential&lt;/em&gt; entropy?&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;2 above is the main reason that I'm asking this question. From a discussion with a colleague today, we concluded that the term could be differential entropy merely because it involves densities (which are derivatives of distributions) and integration and all this calculus-y talk is enough to draw in a term like differential (which seems like an imperfect reason to us), or that it is because the way &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; rigorously arieses is as the difference between the entropy of a quantised version of the random variable in question and a term that can be viewed as a (metric) entropy of the quantising mesh, taken in the limit as the mesh size goes to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; - the key point being it is a difference. However, I'm not sure when this argument arose, although I think this might be from some work of Khintchine in the late '50s, which would be around the right time for the nomenclature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm interested, thus, in if the name arises from either of the above reasons, or from another one altogether.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;[1] &lt;a href=&quot;http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] &lt;a href=&quot;https://pdfs.semanticscholar.org/0e02/b535c582f43140d34c5322d117f89be25799.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pdfs.semanticscholar.org/0e02/b535c582f43140d34c5322d117f89be25799.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. - I'm not too familiar with the tags on this stackexchange, and relevant terms such as etymology and electrical-engineering were no goes, please feel free to edit the tags.&lt;/p&gt;&#xA;" OwnerUserId="9764" LastEditorUserId="4251" LastEditDate="2021-10-02T13:36:47.847" LastActivityDate="2021-10-02T13:36:47.847" Title="Why was the 'differential entropy' from information theory so named?" Tags="|mathematics|terminology|computer-science|information-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8420" PostTypeId="2" ParentId="8419" CreationDate="2019-03-27T11:31:54.800" Score="6" Body="&lt;p&gt;There is a curious discrepancy between the Russian and the English pages of Wikipedia on &amp;quot;differential entropy&amp;quot;. The English page is rather vague on the origin, but the &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%8D%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D0%B8%D1%8F&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russian page&lt;/a&gt; states definitively (in Google's translation):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In information theory, the functional was heuristically introduced by C. Shannon, but he is not the author of the term &amp;quot;differential entropy&amp;quot;. &lt;strong&gt;The term itself was introduced by A. N. Kolmogorov together with I. M. Gelfand and A. M. Yaglom&lt;/strong&gt; and underlines the fact that this concept has a different meaning than the entropy of discrete distributions. They also obtained a rigorous derivation of differential entropy as the first term of the asymptotic expansion of entropy, in which the dependence on the distribution of a random variable is manifested.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Aside from the difficulty of understanding how one can introduce a term &amp;quot;together with&amp;quot;, their source is a Kolmogorov-Gelfand-Yaglom report dated from 1958, while the OP quoted report is from 1956, and by Kolmogorov alone. Unfortunately, the English translation under the link is, shall we say, messed up. In the book &lt;a href=&quot;https://www.twirpx.com/file/82562/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kolmogorov, Theory of Information and Theory of Algorithms&lt;/a&gt; (Google translating the Russian original) we read in a footnote on p.29:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The more detailed exposition of the mathematical side of questions to which was dedicated my report in 1947 is separated into an article [2]. Now the analogous role is played by chapter II of this work, which reproduces, with small changes, the report read by B.V. Gnedenko on my behalf at the symposium on the information theory of the American institute of radioengineers (MIT, September 1956) and based in many parts on a report, read by me together with I.M. Gelfand and A.M. Yaglom at the All-Union mathematical congress in June 1956&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is the congress that the Kolmogorov-Gelfand-Yaglom report is from, the USSR Academy of Sciences only got around to publishing it in 1958. Nonetheless, Yaglom in his book &lt;a href=&quot;https://www.springer.com/us/book/9789027715227&quot; rel=&quot;nofollow noreferrer&quot;&gt;Probability and Information (p.230)&lt;/a&gt; does not use &amp;quot;differential entropy&amp;quot; at all, he calls it &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-entropy&amp;quot;. The text on p. 40 of the Kolmogorov's book gives us the answer: it is to Kolmogorov that the naming credits go, and Kolmogorov alone (compare to the &amp;quot;translation&amp;quot; on p.102 of the linked 1956 report):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the proper sense of the word the entropy of an object &lt;span class=&quot;math-container&quot;&gt;$\xi$&lt;/span&gt; with continuous distribution is always infinite. If the continuous signals can not nevertheless serve to transmit infinitely large information, it is only because they are always observed with limited accuracy. It is natural therefore, by specifying the accuracy of observation &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;, to define the corresponding to it “&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-entropy”  &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(\xi)$&lt;/span&gt; of the object &lt;span class=&quot;math-container&quot;&gt;$\xi$&lt;/span&gt;. This is what was done by Shannon under the name of &amp;quot;speed of creating messages&amp;quot;. Although the choice of a new name does not change the essence of the matter, &lt;strong&gt;I decided to offer such a renaming, which underlines the more broad interest of the notion and its deep analogy with the ordinary exact entropy&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, Kolmogorov is not terribly specific on the exact reason for choosing &amp;quot;differential&amp;quot;. A more explicit explanation, to the effect that it reflects the &amp;quot;differential&amp;quot; nature of probability densities, is given, for example, in &lt;a href=&quot;https://www.twirpx.com/file/1587124/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tarasenko's Russian textbook Introduction to the Course on the Theory of Information (1963, p.77)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In this notation &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; does not appear explicitly, however, that does not mean that &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(x)$&lt;/span&gt; has ceased to be a relative quantity. The number &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(x)$&lt;/span&gt; is usually called the entropy of a continuous random variable. The difference with the discrete entropy values is underlined in the title: &lt;strong&gt;due to the connection of &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(x)$&lt;/span&gt; to the differential law of probability distribution it is often referred to as differential entropy&lt;/strong&gt;; the sometimes used term relative entropy indicates the conventionality, relativity of this characteristic.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Let us also point out that the &amp;quot;non-absoluteness&amp;quot; of the quantity &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(x)$&lt;/span&gt; is compounded by the specificity of its calculation: generally speaking,&#xA;one can set such a law of quantization and such a way of shrinking the quantization intervals to zero, that the limit of the corresponding sums will be different from &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(x)$&lt;/span&gt;. However, by setting once a way of calculating some characteristic and finding out&#xA;its meaning, in the future one can use it for&#xA;comparisons of various distributions. Therefore, the quantity &lt;span class=&quot;math-container&quot;&gt;$H_{\epsilon}(x)$&lt;/span&gt;, despite its relativity, is very important in the theory of information.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-03-29T09:45:57.160" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8421" PostTypeId="1" CreationDate="2019-03-27T14:54:49.883" Score="3" ViewCount="73" Body="&lt;p&gt;Recently, I had an interdisciplinary discussion with some friends about Industry 4.0, internet of things,...which brought up the question, which kind of academic revolutions apart from industrial revolutions have happened so far. Unfortunately, we are all working in STEM and not humanities and are no experts in philsophy of science or sociology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google scholar reveals that the term &quot;academic revolution&quot; exists and is broadly used in scientific literature. But, even on wikipedia, I could not find a good overview of the timeline and kind of academic revolutions that happened so far or are thought of by scientists in humanities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Intuitively and discussed in our group were:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Gutenberg printing press invention&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;International postal service&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Internet&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Artificial intelligence/&quot;singularity&quot; (by Ray Kurzweil)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;...&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I hope there are also some non-STEM users and scientists here who can possibly shed some light on this or link a good review article or website.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally I'm more interested in the methodological and scientific indication of an academic revolution (e.g. strong increase in exchange of scientists between countries and disciplinaries, interdisciplinary referencing, number of research articles, open science, peer review changes etc...). Some of those phenomenological academic discruptions can probably be quite well recognized by bibliometric analysis as in a recent question here about &lt;a href=&quot;https://academia.stackexchange.com/q/126980/41661&quot;&gt;&quot;global number of publications over time&quot;&lt;/a&gt;. But my short googling left it unclear to me, if all of above bullet points are actually agreed on as single academic revolutions by the research community. Also the changing interdependency of universities - industry - government seems to play a bigger role in identifying and arguing for an academic revolution. But apart from technological disruptions, of course also financial disruptions of the funding of academia is an important factor as well as the historical cultural disruptions like the french revolution or the cold war. In the end I would like to be able to extrapolate from a good answer on this question and the past revolutions, which future academic revolutions are to come or if artifical intelligence is the next and last one thought of.&lt;/p&gt;&#xA;" OwnerUserId="2947" LastActivityDate="2019-03-27T14:54:49.883" Title="Which kinds of academic revolutions has philosophy of science indentified so far and when?" Tags="|academia|social-sciences|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8422" PostTypeId="2" ParentId="8409" CreationDate="2019-03-28T02:33:34.233" Score="2" Body="&lt;p&gt;Definition through interior points can be obtained by its dual version first proposed by Kuratowski in 1922, which is known as Kuratowski Closure Algebra. Historically, definition through closure points (Closure Algebra) came first and the one by interior points followed after it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The following three versions for the definition of topological spaces are equivalent:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The neighbourhood system by Hausdorff in his famous work &quot;Grundzüge der Mengenlehre&quot; in 1914.&lt;/li&gt;&#xA;&lt;li&gt;The Closure Algebra by Kuratowski from 1922.&lt;/li&gt;&#xA;&lt;li&gt;The current used system through Open sets by Bourbaki in &quot;Topologie&quot; in 1940.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2019-04-04T18:00:41.323" LastActivityDate="2019-04-04T18:00:41.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8423" PostTypeId="1" AcceptedAnswerId="8426" CreationDate="2019-03-28T19:19:11.613" Score="5" ViewCount="146" Body="&lt;p&gt;G. Waldo Dunnington writes in pages 189-190 of his biography of Gauss:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Among the axioms of geometry which do not depend on the parallel postulate are  those which secure the free mobility of a figure in space. This means that space is the same in nature throuoght and that any figure can be carried along it without tearing or crumpling or stretching ... ... In a note dated about 1827 Gauss called the curved surface of constant negative measure of curvature generated by rotation of the tractrix (pseudo sphere) the &quot;counterpart of the sphere&quot;. The formulas set up by him lead to the theorem that in the pseudo sphere (and only in it) surfaces of rotation are congruent to each other and that, preserving this property, one can move a geodetic triangle on the pseudo sphere just as one can move a spherical triangle on a sphere.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have also found mentions of such sentences in Paul Stackel's treatise on Gauss's contributions to geometry. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have three questions on this quoted passage:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;First of all, what axioms does Dunnington refers to when he says &quot;those which secure the free mobility of figures in space&quot;? are there really &quot;free mobility axioms&quot;?&lt;/li&gt;&#xA;&lt;li&gt;What's the meaning of &quot;in the pseudosphere surfaces of rotation are congruent&quot;? as far as i know the pseudosphere is a surface (not a space!) so how can one talk about surfaces contained in it? this sentence seems completely unrelated and obscure.&lt;/li&gt;&#xA;&lt;li&gt;What's the meaning of &quot;moving a geodetic triangle on the pseudosphere&quot;?&#xA;In euclidean geometry the rigidity of motion is taken as granted so one can speak about motion of figures without distortion, but how can one speak the same about hyperbolic geometry? i really want to understand this point. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Every usefull comment will be blessed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think the article &quot;What did Gauss read in the Appendix?&quot; might be very useful in answering this question. In pages 22-24 of this article the authors analyze the relevant (to our question) fragment from Gauss's nachlass (vol 8, p.255-257; Stackel refers to these pages when he comments on Gauss's formulas about the &quot;mobility of figures&quot;) - in this fragment Gauss assumed a hyperbolic triangle which changes with time (so one can view it as a kind of &quot;motion&quot;) and analyzed the metric and angular relations of it by the process of functional equations.  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-03-28T22:03:53.350" LastActivityDate="2019-03-29T16:18:53.243" Title="Meaning of a cryptic sentence by Gauss on &quot;the mobility of figures in the hyperbolic plane&quot;" Tags="|biographical-details|geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8424" PostTypeId="1" CreationDate="2019-03-29T00:39:55.137" Score="7" ViewCount="1310" Body="&lt;p&gt;A perfect example would be &lt;a href=&quot;https://en.wikipedia.org/wiki/Srinivasa_Ramanujan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Srinivasa Ramanujan&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is known that the conventional community throughout history have been close-minded towards great men of science and mathematics (e.g., Galileo).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Srinivasa Ramanujan being one of them.  Einstein is another one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A lot of history's math and science problems were solved in what would have been considered heresy.&lt;/p&gt;&#xA;" OwnerUserId="9773" LastEditorUserId="72" LastEditDate="2019-12-30T14:58:31.330" LastActivityDate="2021-02-26T20:20:56.163" Title="Who are the top mathematicians who were ignored due to their unconventional approach?" Tags="|mathematics|biographical-details|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="8425" PostTypeId="2" ParentId="8424" CreationDate="2019-03-29T01:54:21.443" Score="10" Body="&lt;p&gt;Hermann Cäsar Hannibal Schubert invented what is called &quot;Schubert calculus&quot; (publ. in 1879). His highly original method was not sufficiently justified. One of the Hilbert problems (1900) was to justify Schubert Calculus. This was achieved in principle in&#xA;the 1920s and the interest to the Schubert Calculus declined, and it was never very strong before that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It experienced a strong revival in the 1970s when mathematicians started to check and re-compute Schubert's results sometimes using computers. Nowadays it is a vigorously developing area of mathematics with many applications inside and outside mathematics. Schubert's book has been recently reprinted. Schubert himself was not sufficiently recognized in his lifetime and worked as a high school (gymnasium)  teacher.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-12T03:15:24.620" LastActivityDate="2019-04-12T03:15:24.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8426" PostTypeId="2" ParentId="8423" CreationDate="2019-03-29T02:11:48.470" Score="5" Body="&lt;p&gt;The answers to your first and third questions are the following. &quot;Free mobility&quot; of a space means that there is a (pseudo-) group of (local) isometries (1-1 maps preserving distances) which acts properly and&#xA;transitively on pairs (point,orthonormal basis in the tangent space). For surfaces, this means that whenever you have&#xA;&lt;span class=&quot;math-container&quot;&gt;$(p_1,v_1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(p_2,v_2)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$p_j$&lt;/span&gt; are points of the space and &lt;span class=&quot;math-container&quot;&gt;$v_j$&lt;/span&gt; are directions (unit tangent vectors) at these points, there is a unique isometry of the space&#xA;which sends &lt;span class=&quot;math-container&quot;&gt;$p_1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$p_2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v_1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$v_2$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular on each surface of constant curvature there is such a group. If you have a triangle on a surface, you can move it to other place so that angles and lengths of sides are preserved. In Euclid this axiom is not stated explicitly but it is used all the time in the form that &quot;triangles&#xA;with two sides and angle between the equal are equal.&quot; The word &quot;equal&quot;&#xA;in Euclid means &quot;can be moved to coincide&quot;, that is there is a unique element of the group mentioned above which sends one object to another.&#xA;It is fundamental for geometry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem of classifying all possible (local) geometries was stated by Riemann&#xA;and Helmholtz and solved by Killing. It is called the Riemann-Helmholtz problem.&#xA;In particular, there are three types of geometries in dimension 2 (spherical,&#xA;hyperbolic and usual, Euclidean).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To your 2-nd question I cannot answer, it is indeed unclear what the author says.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;General remark. If you want to learn mathematics, I do not recommend the books written by historians. Read books written by mathematicians. Historians use imprecise language,&#xA;and frequently do not really understand themselves what they are writing about. Also a Gauss biography is usually a poor source for his mathematics, though it can be a good source about his life. Most biography writers do not really understand mathematics, and rely other sources to describe it, with inevitable distortions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-03-29T16:18:53.243" LastActivityDate="2019-03-29T16:18:53.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8427" PostTypeId="2" ParentId="8405" CreationDate="2019-03-29T02:17:49.967" Score="3" Body="&lt;p&gt;Also in the field of Astronomy: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Urbain_Le_Verrier&quot; rel=&quot;nofollow noreferrer&quot;&gt;Urbain Le Verrier&lt;/a&gt;'s most famous achievement is his prediction of the&#xA;  existence of the then unknown planet Neptune, using only mathematics&#xA;  and astronomical observations of the known planet Uranus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Quoting this time from the Wiki page on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Discovery_of_Neptune&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discovery of Neptune&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was a sensational moment of 19th-century science, and dramatic confirmation of Newtonian gravitational theory. In François Arago's apt phrase, Le Verrier had discovered a planet &quot;with the point of his pen&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Having taught the basics of perturbation theory in classical mechanics, I can attest that this kind of work can only be described a monumental, both from a calculation perspective and from mental ability required to understand and accurately perform such calculations.&lt;/p&gt;&#xA;" OwnerUserId="7093" LastActivityDate="2019-03-29T02:17:49.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8428" PostTypeId="2" ParentId="8424" CreationDate="2019-03-29T17:57:40.840" Score="13" Body="&lt;p&gt;Georg Cantor was a German mathematician who created set theory that has become a fundamental theory in mathematics. However, his original work on infinity and ordinal number was considered unconventional or even unorthodox, which was under heavy attacks from some of the famous contemporary mathematicians for a long time.&lt;/p&gt;&#xA;&lt;p&gt;Another example is Charles Sanders Pierce, an American philosopher and logician who made important contributions to logic, relation theory, pragmatism, semiotics and so on. But his work was largely ignored in his lifetime and he could not even find an academic position. Only after the 1920s, his original work was rediscovered that he is recognized as one of the most original and versatile of American philosophers and America's greatest logicians.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-02-26T20:20:56.163" LastActivityDate="2021-02-26T20:20:56.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8429" PostTypeId="2" ParentId="8417" CreationDate="2019-03-30T15:39:44.743" Score="7" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;but for how long has PEMDAS been widely taught in high school mathematics classes? I assume this will differ for different parts of the world, so please include what countries or regions you can speak for. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The first appearance of an explicit PEMDAS rule in Dutch is in an appendix of a textbook on algebra for the military academy (1838), aimed at military engineers, not elementary school students, nor academic mathematicians. The author explained he had put it in an appendix because average students had to learn the order of operations just by practicing examples; the formal PEMDAS rule merely served to satisfy the curiosity of the happy few. This was repeated in an appendix of a popular textbook for school teachers (1875). Later, teachers apparently could not resist the temptation to teach the explicit PEMDAS rule to all students, with a mnemonic. It is unclear when exactly.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The old Dutch PEMDAS version should actually be written as EMDRAS. The P was omitted because nobody would confuse parentheses with operations, and the R, representing the root symbol, √, was inserted after MD because √2a, without vinculum, was to be interpreted as √(2a) instead of (√2)a.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2019-03-30T20:19:40.930" LastActivityDate="2019-03-30T20:19:40.930" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8430" PostTypeId="1" CreationDate="2019-03-30T15:41:58.433" Score="1" ViewCount="43" Body="&lt;p&gt;In physics, many problems were known at their time to be very challenging, for example the notion of heat, or how to understand the ideal gaz law, or the phase transition criticial behaviors, etc. And they were then solved.&#xA;I'm wondering if people have studied if there are common trends in these problems : what makes some scientific questions difficult but interesting, and what are the various strategies which were used to solve them, especially in the fields of physics.&#xA;Any advice for bibliographic references about this ? &#xA;Thanks !&lt;/p&gt;&#xA;" OwnerUserId="9777" LastActivityDate="2019-03-30T15:41:58.433" Title="Does anyone know articles or books about what the notion of difficulty in science, especially physics?" Tags="|physics|discoveries|philosophy-of-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8431" PostTypeId="1" AcceptedAnswerId="8433" CreationDate="2019-03-31T02:26:54.460" Score="11" ViewCount="2731" Body="&lt;p&gt;This may be borderline off-topic but this is the only place that I can think of ask this particular question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've always seen images of &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt; with a &quot;hat&quot; or head covering that is unfamiliar to me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a name for the head covering shown in Emanuel Handmann 1753 painting of Euler? Does it carry any specific significance of mathematical scholarship?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6GyiQ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6GyiQ.jpg&quot; alt=&quot;Emanuel Handmann 1753 painting of Euler&quot;&gt;&lt;/a&gt; &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Leonhard_Euler.jpg&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2019-03-31T03:38:22.183" LastActivityDate="2019-03-31T12:20:28.037" Title="What's that on Euler's head? Does the head covering shown in Emanuel Handmann 1753 painting signify scholarship?" Tags="|mathematicians|euler|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8432" PostTypeId="1" CreationDate="2019-03-31T09:17:33.970" Score="4" ViewCount="374" Body="&lt;p&gt;Bohm published his interpretation of quantum mechanics in 1952. Comments on Bohm's work from Einstein, Heisenberg und Pauli are cited in the corresponding wikipedia article (&lt;a href=&quot;https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it known what Bohr himself thought? Did he comment on it?&lt;/p&gt;&#xA;" OwnerUserId="7241" LastEditorUserId="55" LastEditDate="2019-04-01T08:22:06.280" LastActivityDate="2019-04-01T08:22:06.280" Title="Did Bohr comment on Bohm's interpretation of quantum mechanics?" Tags="|philosophy-of-science|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8433" PostTypeId="2" ParentId="8431" CreationDate="2019-03-31T11:44:32.747" Score="12" Body="&lt;p&gt;He wears the head wrap in combination with a silk &lt;a href=&quot;https://en.wikipedia.org/wiki/Banyan_(clothing)&quot; rel=&quot;noreferrer&quot;&gt;banyan&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the 18th century...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was fashionable for men of an intellectual or philosophical bent to have their portraits painted while wearing banyans. &lt;a href=&quot;https://en.wikipedia.org/wiki/Benjamin_Rush&quot; rel=&quot;noreferrer&quot;&gt;Benjamin Rush&lt;/a&gt; wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Loose dresses contribute to the easy and vigorous exercise of the faculties of the mind. This remark is so obvious, and so generally known, that &lt;strong&gt;we find studious men are always painted in gowns, when they are seated in their libraries&lt;/strong&gt;&lt;a href=&quot;https://npg.si.edu/exh/franklin/rush.htm&quot; rel=&quot;noreferrer&quot;&gt;[2]&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;… [The banyan] was usually paired with a soft, turban-like cap worn in place of the formal periwig.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The fashion was inspired on a style from (exotic) East Asia.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-03-31T12:20:28.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8434" PostTypeId="1" CreationDate="2019-03-31T12:28:57.313" Score="7" ViewCount="2167" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Avogadro_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia says&lt;/a&gt; the Avogadro number was determined by Perrin in 1908. But Einstein used the value&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$N=6\times 10^{23}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;as early as 1905 in order to estimate the size of the fluctuations of an molecule. How did he know it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Addendum:&lt;/strong&gt; Of course, the concept of a mole existed already. Proponents of the atomic theory of matter knew that a gram of an element contained a certain number of particles. According to &lt;a href=&quot;https://hsm.stackexchange.com/questions/8116/how-did-the-early-chemists-make-a-connection-between-gram-formula-weight-with-1&quot;&gt;this post&lt;/a&gt;, even the relative weights of atoms/molecules were known. However, I don't see there where the absolute scale for the mass of the atoms (and therefore the Avogadro number) was determined for the first time.&lt;/p&gt;&#xA;" OwnerUserId="9779" LastEditorUserId="9779" LastEditDate="2019-04-01T10:18:58.690" LastActivityDate="2019-04-02T10:35:06.627" Title="How did Einstein know the Avogadro Number?" Tags="|thermodynamics|einstein|atomic-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8435" PostTypeId="2" ParentId="8434" CreationDate="2019-03-31T14:29:02.987" Score="3" Body="&lt;p&gt;Although the concept of the existence of the atoms originates from the ancient Greek, for them it was yet more a philosophical concept, they were very far from any experimental proof.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later, mainly the chemistry provided the first suggestion that it might be an experimental fact.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, none of these results was strong enough at the time. In the late XIX. century, it was still a common concept of mainstream science to accept the &quot;atomic theory&quot;, but just as a practical approximation to calculate the possible mass ratios of the elements in their compounds, but without a real meaning. It was so mainly because the lack of experimental proofs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The acceptance of the &quot;atom theory&quot; happened surprisingly late, at the late XIX, early XX. century, which was also the start of Einstein's work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The important thing is: &lt;em&gt;Perrin's result alone was not enough for it, it was accepted because many, essentially different, experiments determined the same Avogadro number. Probably Perrin was the first, but also Einstein provided one, using Brownian motion.&lt;sup&gt;2&lt;/sup&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's method worked by the statistical analysis of the Brownian motion of oil particles suspended in water. He measured the average distance of the drops due to their random Brownian motion in a microscope. It depends on the size of the atoms: if the atoms would be smaller, then the Brownian motion would be slower due to the better statistical leveling of the water molecules around the oil drops.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The experimental challenge was big: he had to find oil drops whose size is big enough to become visible in the microscope, but they are not big enough to make their Brownian motion invisible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;sub&gt;Actually, they didn't really have experimental results. They were good at axiomatizing their theories, but practice and experiments weren't their strong side. Furthermore, science, math and philosophy didn't differ conceptually in their world view: this separation happened millennia later.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt;&lt;sub&gt;As per &lt;a href=&quot;https://www.physik.uni-augsburg.de/theo1/hanggi/History/Renn.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;, he doesn't mention Brownian motion in his paper, although he had probably heard about it. His work is a &quot;new method to calculate the mass of the molecules&quot; (my translation).&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="104" LastEditDate="2019-04-02T10:35:06.627" LastActivityDate="2019-04-02T10:35:06.627" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8436" PostTypeId="1" AcceptedAnswerId="9552" CreationDate="2019-03-31T20:28:41.953" Score="3" ViewCount="3841" Body="&lt;p&gt;I was taught that the numerals {0,1,2,...,9} are from the Arabic alphabet. But they look completely different from today's Arabic letters. I wonder what is the origin of &lt;a href=&quot;https://en.wikipedia.org/wiki/Arabic_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arabic numerals&lt;/a&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: The web page of the above link claims that Arabic numerals were developed in ancient India first and then spread to Arabia, and later to Europe and the rest of the world. This only means numerals themselves as symbols, not the arithmetical system for numerical computations.&lt;/p&gt;&#xA;&lt;p&gt;Note that there are two distinct concepts: arithmetic and numerals. Arithmetic is a number system (often in decimals) that allows easy operations for numbers. Numerals mean symbols for the representation of numbers.&lt;/p&gt;&#xA;&lt;p&gt;Numerals without arithmetic are very inefficient for numerical computations compared to arithmetic. One example is Roman numerals. The widely used numerals with arithmetic today are Arabic numerals.&lt;/p&gt;&#xA;&lt;p&gt;Arithmetic was discovered originally in several countries as early as the 1st century AD. For example, Chinese discovered arithmetic with the decimal system of numbers and rules of numerical operations (Chinese numerals) in &amp;quot;The Nine Chapters on the Mathematical Art&amp;quot; in the 1st century AD. Chinese numerals are in Chinese characters, e.g. 〇, 一, 二, 三, 四, 五, 六, 七, 八, 九 (for &lt;span class=&quot;math-container&quot;&gt;$0,\cdots,9$&lt;/span&gt; respectively), and 十, 百, 千, 万, 亿 (for ten, hundred, thousand, ten thousand, a hundred millions). Thus they are not widely known to the rest of the world.&lt;/p&gt;&#xA;&lt;p&gt;As in the answer by @AChem below, it seems that recently there are many web pages that have highly inaccurate and even wrong information to give India unfair credit for many works in mathematics and science (ancient and modern). Be aware of this situation and do not readily believe this claim from some webpages.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2022-11-08T22:35:34.530" LastActivityDate="2022-11-08T22:35:34.530" Title="What is the origin of Arabic numerals" Tags="|mathematics|reference-request|numbers|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8437" PostTypeId="2" ParentId="8432" CreationDate="2019-04-01T07:18:50.793" Score="4" Body="&lt;p&gt;Bohr did not comment in print. However, in his &lt;a href=&quot;https://archive.org/details/isbn_9780201406351&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bohm's biography Infinite Potential, Peat&lt;/a&gt; describes Bohr's 1953 reaction, as reported by Feyerabend, who visited him in Copenhagen:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first Feyerabend heard of Bohm’s new theory was during a seminar given by Niels Bohr. Following the lecture he asked Bohr to clarify certain points. The Danish physicist’s reaction was, “Have you read Bohm”?  As Feyerabend put it, &amp;quot;It seemed that, for him, the sky was falling in... Bohr was neither dismissive, nor shaken. He was &lt;strong&gt;amazed&lt;/strong&gt;&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In the midst of explaining to Feyerabend why Bohm’s paper so disturbed him, Bohr was called away. The discussion continued without him for two more hours. Some of those present argued that the objections to Bohm’s theory were not at all conclusive. As Feyerabend put it, the orthodox Copenhagen supporters tried to reply “in the Bohrian fashion”. When this attempt was not successful, they said, “But von Neumann has proved...” which ended the discussion. Feyerabend noted, however, that Bohr himself did not use von Neumann’s supposed proof as a crutch in that fashion.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The von Neumann's supposed impossibility of hidden variables proof was comprehensively analyzed in 1965 by &lt;a href=&quot;https://people.isy.liu.se/jalar/kurser/QF/references/Bell1966.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bell in On the Problem of Hidden Variables in Quantum Mechanics (1965)&lt;/a&gt;. He concluded that von Neumann's &amp;quot;&lt;em&gt;essential axioms are unreasonable&lt;/em&gt;&amp;quot; (von Neumann assumed, among other things, that quantum probabilities reflect the distribution of values of physical quantities before the measurement). According to Peat, Bohm realized this much earlier.&lt;/p&gt;&#xA;&lt;p&gt;It should be noted, however, that the overall reaction to Bohm's papers in 1950-s was that of dismissal, although many realized that Bohm's theory was not mathematically flawed. This was compounded by the fact that Bohm was a leftie suspected of communist ties, in the spirit of the times, a traitor. In 1950 he was arrested and suspended from Princeton, and after the acquittal his US passport was confiscated upon arrival to Brazil in 1951.&lt;/p&gt;&#xA;&lt;p&gt;Aside from Oppenheimer's &amp;quot;ignore him&amp;quot; directive, followed by the Princeton group, and the references to von Neumann's theorem, we have Pauli's &amp;quot;&lt;em&gt;your ‘extra wave-mechanical predictions’ are still a check, which cannot be cashed&lt;/em&gt;&amp;quot;, and similar opinions. Feynman, and few others, were an exception. Like Pauli, Rosenfeld, professing to speak on Bohr's behalf, addressed Bohm directly in a condescending letter:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I shall certainly not enter into any controversy with you or anybody else on the subject of complementarity, for the simple reason that there is not the slightest controversial point about it. It is just because we have undergone this process of purification through error that we feel so sure of our results... there is no truth in your suspicion that we may just be talking ourselves into complementarity by a kind of magical incantation. I am inclined to return that it is just among your Parisian admirers that I notice some disquieting signs of primitive mentality.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.tcm.phy.cam.ac.uk/%7Emdt26/pilot_waves.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Towler in Lecture 7 on De Broglie-Bohm Pilot-Wave Theory&lt;/a&gt; gives a survey of contemporary responses, from which the quotes are taken.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-01T07:43:47.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8438" PostTypeId="1" AcceptedAnswerId="8440" CreationDate="2019-04-01T09:21:18.507" Score="7" ViewCount="266" Body="&lt;p&gt;Books I and III seem to receive the bulk of attention by Newton scholars, and historians more generally. However, I'm sure an in-depth study of Book II would shed light on Newton as a thinker -- for that reason it must be of import, no?&lt;/p&gt;&#xA;" OwnerUserId="6867" LastEditorUserId="6867" LastEditDate="2019-04-01T22:48:26.477" LastActivityDate="2019-04-02T13:37:58.207" Title="Why is there little scholarship devoted to Book II of Newton's Principia?" Tags="|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8439" PostTypeId="1" CreationDate="2019-04-01T14:50:38.907" Score="6" ViewCount="266" Body="&lt;p&gt;Minimax theorems are beautiful saddle-point results regarding conditions on a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; under which &lt;span class=&quot;math-container&quot;&gt;$\max_x \min_y f(x,y) = \min_y \max_x f(x,y)$&lt;/span&gt;. In the common &quot;normal form&quot; game case, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are probability distributions corresponding to player strategies and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bilinear. However, the result is true for more general quasiconcave-quasiconvex functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and that distinction is what this question is all about!&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My question is: &lt;strong&gt;What did von Neumann (1928) prove and did Sion (1956) actually generalize it?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Background:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;von Neumann first published a minimax theorem in “Zur Theorie der Gesellschaftsspiele” (1928), &lt;a href=&quot;https://www.ethz.ch/content/dam/ethz/special-interest/gess/chair-of-sociology-dam/documents/articles/Neumann_Zur_Theorie_der_Gesellschaftsspiele_Mathematische_Annalen_1928_Vol_100__1_295-320.pdf&quot; rel=&quot;noreferrer&quot;&gt;possibly available here&lt;/a&gt;. If I could read German then I would not need to ask this question, however &lt;a href=&quot;https://math.stackexchange.com/questions/1387845/english-translation-of-von-neumanns-zur-theorie-der-gesellschaftsspiele-1928&quot;&gt;it seems an English translation does not exist&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1956, Sion wrote &quot;On General Minimax Theorems&quot;, &lt;a href=&quot;https://projecteuclid.org/euclid.pjm/1103040253&quot; rel=&quot;noreferrer&quot;&gt;available here&lt;/a&gt;, in which he claims that von Neumann's minimax theorem was stated and proved for &lt;strong&gt;bilinear forms&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$f(x,y)$&lt;/span&gt;. Sion claims that several others have generalized the theorem since, including Shiffman who &quot;seems to have been the first to have considered &lt;strong&gt;concave-convex&lt;/strong&gt; functions.&quot; Sion proves the result for &lt;strong&gt;quasiconcave-quasiconvex&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. His result has &lt;a href=&quot;https://en.wikipedia.org/wiki/Sion%27s_minimax_theorem&quot; rel=&quot;noreferrer&quot;&gt;a wikipedia page&lt;/a&gt; claiming it generalizes von Neumann's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems final. But if I am correctly reading the very nice historical paper &quot;John von Neumann's conception of the minimax theorem&quot; (Kieldsen, 2001), &lt;a href=&quot;http://web.math.ucsb.edu/~crandall/math201b/vnminimax.pdf&quot; rel=&quot;noreferrer&quot;&gt;available here&lt;/a&gt;, Section 2.3, then von Neumann's original 1928 paper originally already proved the generalization to the quasiconvex-quasiconcave case! The survey directly quotes the German passage from von Neumann's paper and gives a summary (though not complete translation).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was Sion mistaken or unaware of what von Neumann proved? If so, were the references he cited mistaken as well?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One possible reason for confusion is the presentation of the theorem in von Neumann and Morgenstern's famous 1944 English-language book, &quot;Theory of Games and Economic Behavior&quot; (which I find very unpleasant to read, but never mind). Although saddle points are discussed in 13.4 and 13.5, I &lt;em&gt;think&lt;/em&gt; that the only minimax theorem presented in the book is for bilinear forms, in Section 17.6. So if you only read the book, you might think the minimax theorem was only proven for bilinear &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. By the way, that section has a footnote almost a page long concerning the history of the minimax theorem. It is mentioned that a generalization was proved in &quot;tlber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwer'schen Fixpunktsatzes,&quot; von Neumann 1937. This uses fixed-point theorems, as investigated by Kieldsen 2001, but I don't see from that survey anything about whether &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; was only bilinear, or more general, in the 1937 paper. So I would love to know&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Exactly what are the translated statements of the minimax theorems proven in von Neumann (1928) and von Neumann (1937)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1786" LastEditorUserId="4251" LastEditDate="2021-09-21T18:52:58.853" LastActivityDate="2021-09-21T18:52:58.853" Title="The minimax theorem from 1928 to 1956" Tags="|mathematics|mathematicians|discoveries|von-neumann|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8440" PostTypeId="2" ParentId="8438" CreationDate="2019-04-01T17:11:49.463" Score="11" Body="&lt;p&gt;It can certainly be agreed that Book 2 of the &lt;em&gt;Principia&lt;/em&gt; has received less attention than Books 1 and 3. According to I B Cohen (1999):--&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Book 2 of the Principia differs from books 1 and 3 in a variety of&#xA;  ways. One  of the most striking of these is the fact that its major&#xA;  contents, the theoretical and experimental study of the forces of&#xA;  resistance to motion in various types of fluids, is usually not&#xA;  discussed and often not even mentioned by historians of science or&#xA;  even by some historians of mechanics, ... &quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Cohen went on to offer a contributory reason, that:--&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Newton himself was in part responsible for this rather cavalier&#xA;  treatment of book 2 since he suggested to readers that they confine&#xA;  their study to the beginning of book 1 and then skip over to book 3,&#xA;  on the system of the world. For the intellectual historian and&#xA;  philosopher of science, the system of the world has always been a more&#xA;  glamorous field of study than an investigation of the properties of&#xA;  fluids.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Quoted from p.161 of I B Cohen (1999), &quot;&lt;em&gt;A Guide to Newton's Principia&lt;/em&gt;&quot;, pp.1-370 accompanying &quot;&lt;em&gt;The Principia: A New Translation&lt;/em&gt;&quot;, I B Cohen and Anne Whitman (1999, Berkeley, CA, Univ. Cal. Press); for Book 2 see especially pages 161-194.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also another and short possible answer: that Book 2 of the &lt;em&gt;Principia&lt;/em&gt; was less successful than Books 1 and 3, it has not stood the test of time to anything like the same extent, and it interests fewer people also on that account. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Part of the reason for relative lack of success is that the subject of Book 2 really needed for its study experimental observations and data not available until long after Newton's lifetime. The motion of bodies in resisting (fluid) media is affected by phenomena such as viscosity and turbulence and temperature effects. These were hardly accessible to systematic study and observation in the state of experimental and observational science in Newton's time. The effects they contributed therefore remained largely or entirely beyond Newton's observational reach, and their absence from his reasoning limited the correctness and validity of some of his conclusions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To give just one example, Newton in Book 2 calculated or estimated the speed of sound in air; he obtained a poor result, and later, after experimental estimations showed something closer to the true value, he put forward essentially wrong conjectures to account for the difference. The real cause of the errors lay in the temperature component of the gas laws and in temperature effects unknown in his time, thus:--&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;The error in the speed of sound ... reflects nothing more than the &#xA;  impossibility of Newton recognizing the distinction between &#xA;  &lt;a href=&quot;https://en.wikipedia.org/wiki/Adiabatic_process&quot; rel=&quot;noreferrer&quot;&gt;adiabatic&lt;/a&gt; and &#xA;  &lt;a href=&quot;https://en.wikipedia.org/wiki/Isothermal_process&quot; rel=&quot;noreferrer&quot;&gt;isothermal&lt;/a&gt;&#xA;  expansion &#xA;  &lt;strong&gt;in an era before thermometry&lt;/strong&gt;.  Newton’s derivation of  the speed of sound from first principles is correct save for the need to &#xA;  replace Boyle’s law&quot; [&lt;a href=&quot;https://en.wikipedia.org/wiki/Boyle&amp;#39;s_law&quot; rel=&quot;noreferrer&quot;&gt;which&#xA;  considers&lt;/a&gt;  gas pressure&#xA;  and volume but not the effects of temperature] &quot;with the &#xA;  corresponding adiabatic relationship&quot; [in which the effect&#xA;  of temperature is also taken into account] &quot;for expansions and&#xA;  contractions in  sound waves.&quot; (emphasis, comment [ ] and links added):&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;quoted from p.128 in George E Smith's chapter:  &quot;&lt;em&gt;Was Wrong Newton Bad Newton?&lt;/em&gt;&quot;, pp.127-160 in &lt;a href=&quot;https://books.google.com/books?id=xCMhVraUAnMC&quot; rel=&quot;noreferrer&quot;&gt;&quot;&lt;em&gt;Wrong for the Right Reasons&lt;/em&gt;&quot;&lt;/a&gt;, ed. J Z Buchwald and A Franklin (2005).) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(G E Smith in the same chapter also draws attention to errors by Newton in Book 2 arising from incomplete development of some concepts in Book 1: showing, for example, how Newton did not see how to reformulate his second law of motion for the case of angular motion.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Besides the references given above, other sources showing scholarly study of &lt;em&gt;Principia&lt;/em&gt; Book 2 include: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;George E Smith (2001), &lt;a href=&quot;https://books.google.com/books?id=Ppv5h4sgAYwC&quot; rel=&quot;noreferrer&quot;&gt;&quot;&lt;em&gt;The Newtonian Style in Book II of the Principia&lt;/em&gt;&quot;&lt;/a&gt;, ch.9 (pp.249-298) in &#xA;&quot;&lt;em&gt;Isaac Newton's Natural Philosophy&lt;/em&gt;&quot;, eds. J Z Buchwald, I B Cohen, (2001, MIT Press, Cambridge, MA); and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clifford Truesdell (1968), &quot;&lt;em&gt;Essays in the history of mechanics&lt;/em&gt;&quot;, &#xA;pp.138-183 (chap. III): &lt;a href=&quot;https://books.google.de/books?id=_crsCAAAQBAJ&amp;amp;pg=PA138&quot; rel=&quot;noreferrer&quot;&gt;&quot;&lt;em&gt;Reactions of Late Baroque Mechanics to &#xA;Success, Conjecture, Error, and Failure in Newton's &#xA;Principia&lt;/em&gt;&quot;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Altogether it seems hard to tell how much more would be revealed of Newton's thinking by further in-depth study of Book 2. The examples already mentioned show how such study could be specially burdened by the need to distinguish between (a) limitations arising because Newton only partly developed some of the concepts for which foundations are laid in Book 1, and (b) limitations arising from the impossibility of Newton's foreseeing experimental data and phenomena still unknown and inaccessible in his lifetime.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-04-02T13:37:58.207" LastActivityDate="2019-04-02T13:37:58.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8442" PostTypeId="2" ParentId="8439" CreationDate="2019-04-01T20:53:36.217" Score="2" Body="&lt;p&gt;This is too long for a comment, but I was not able to access the translations, so I can not provide details. Kieldsen (p.48ff) does say &quot;&lt;em&gt;Actually von Neumann proved a generalized version of the minimax theorem. Heconsidered a broader class of functions than the bilinear formsh... Today a function with the property (K) is called quasiconcave in ξ and quasiconvex in η&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 1928 paper was translated by Bargman as On the Theory of Games of Strategy, published in &lt;a href=&quot;https://press.princeton.edu/titles/2903.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contributions to the Theory of Games, volume 4&lt;/a&gt;, edited by A. W. Tucker &amp;amp; R. D. Luce, 1959.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The note announcing it was translated just recently by &lt;a href=&quot;https://www.worldscientific.com/doi/10.1142/S0219198910002556&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ben-EL-Mechaiekh&#xA;and Dimand, International Game Theory Review, Vol. 12, No. 2, pp. 115-137 (2010)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The 1937 paper was translated by Morgenstern as &lt;a href=&quot;https://academic.oup.com/restud/article-abstract/13/1/1/1543455&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Model of General Economic Equilibrium, Review of Economic Studies 13 (1945–46), 1–9&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-04-01T20:53:36.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8443" PostTypeId="1" CreationDate="2019-04-02T00:33:29.810" Score="4" ViewCount="128" Body="&lt;p&gt;The Phys.org article &lt;a href=&quot;https://phys.org/news/2018-12-physicists-moon-rotational-dynamics-accounting.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physicists provide first model of moon's rotational dynamics, accounting for the solid inner core&lt;/a&gt; describes the September 2018 paper by Christopher Stys and Mathieu Dumberry &lt;a href=&quot;https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2018JE005607&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cassini State of the Moon's Inner Core&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The plain language summary of the paper is as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It is well known that the reason we see only one face of the Moon is because the period of its revolution around Earth is equal to the period of Moon's rotation around itself. Yet the full description of the Moon's rotation is a bit more complex. The Lunar spin axis is inclined by 1.54° in space, and its direction is precessing, like a spinning top, at a period of 18.6 years. &lt;strong&gt;This precession is caused by the precession of the plane of the Lunar orbit at the same period, an arrangement known as a Cassini state, named after the seventeenth century Italian astronomer.&lt;/strong&gt; Like the Earth, the Moon is suspected to have a metallic iron core, its outer region being fluid but the central part being solid. In this work, we calculate the tilt of the spin axis of this solid inner core so that it obeys a Cassini state. We show that this angle can be large, 17° with respect to the mantle for one specific model but that it is also very sensitive to the specific size and mass of the solid and fluid parts of the Lunar core which are not well known.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Further searching leads to Wikipedia's article &lt;a href=&quot;https://en.wikipedia.org/wiki/Cassini%27s_laws&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cassini's laws&lt;/a&gt;, &quot;...a compact description of the motion of the Moon. They were established in 1693 by Giovanni Domenico Cassini, a prominent scientist of his time.&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;The Moon has a 1:1 spin–orbit resonance. This means that the rotation–orbit ratio of the Moon is such that the same side of it always faces the Earth.&lt;/li&gt;&#xA;  &lt;li&gt;The Moon's rotational axis maintains a constant angle of inclination from the ecliptic plane. The Moon's rotational axis precesses so as to trace out a cone that intersects the ecliptic plane as a circle.&lt;/li&gt;&#xA;  &lt;li&gt;A plane formed from a normal to the ecliptic plane and a normal to the Moon's orbital plane will contain the Moon's rotational axis.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BqViT.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BqViT.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now that I have found out that the Moon's rotational axis is inclined by about 1.54 degrees with respect to its orbital plane, I can finally understand all of the motion in the Moon's disk during its orbit around the earth as described in the question &lt;a href=&quot;https://space.stackexchange.com/q/19936/12102&quot;&gt;Who does these mesmerizing simulations of the phases of the Moon? And how?&lt;/a&gt; and shown below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How did Cassini measure the &quot;Cassini state&quot; of the Moon? What did the data look like? Did he make detailed drawings, or measurements of certain features on the moon, perhaps near the terminator or limb?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A sample of the raw data or an image of his notes would also be wonderful to see if something is available.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Animation used in &lt;a href=&quot;http://earthsky.org/moon-phases/new-moon&quot; rel=&quot;nofollow noreferrer&quot;&gt;EarthSky.org&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; As &lt;a href=&quot;https://space.stackexchange.com/a/20564/12102&quot;&gt;pointed out here&lt;/a&gt; by the GIF's creator, the original image is actually in &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Lunar_libration_with_phase_Oct_2007.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wstX2.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wstX2.gif&quot; alt=&quot;Lunar libration with phase Oct 2007 (GIF)&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Animation from &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Lunar_libration_with_phase2.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/y6oHW.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/y6oHW.gif&quot; alt=&quot;Lunar libration with_phase 2 (GIF)&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2020-02-04T00:50:03.383" LastActivityDate="2020-02-04T00:50:03.383" Title="How did Cassini measure the &quot;Cassini state&quot; of the Moon? What measurements were made; what did the data look like?" Tags="|astronomy|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8444" PostTypeId="2" ParentId="3738" CreationDate="2019-04-02T15:30:45.523" Score="3" Body="&lt;p&gt;The names Pythagoras and Plato are associated with methods of generating Pythagorean triples due to their mention in Proclus's commentary on Euclid.  However, as far as I have been able to determine, there is no mention of such a method in the extant works of Plato, and, as has been mentioned in the comments, attribution of &lt;em&gt;any&lt;/em&gt; mathematical result to Pythagoras or to the early members of his school is problematic.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his commentary on the proof of the Pythagorean theorem in Book I of &lt;em&gt;Elements&lt;/em&gt;, Proclus describes two methods for generating Pythagorean triples, stating that one is attributed to Pythagoras, the other to Plato.  Using an anachronistic algebraic formulation, the &quot;method of Pythagoras&quot; yields the triple&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(n,\frac{n^2-1}{2},\frac{n^2+1}{2}\right)&#xA;$$&lt;/span&gt;&#xA;for any &lt;em&gt;odd&lt;/em&gt; integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.  Examples are &lt;span class=&quot;math-container&quot;&gt;$(3,4,5)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(5,12,13)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(7,24,25)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(9,40,41)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(11,60,61)$&lt;/span&gt;. As two elements of each triple differ by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, these are clearly primitive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;method of Plato&quot; yields&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(n,\left(\frac{n}{2}\right)^2-1,\left(\frac{n}{2}\right)^2+1\right)&#xA;$$&lt;/span&gt;&#xA;for any &lt;em&gt;even&lt;/em&gt; integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.  This is equivalent to the formula you ask about.  Examples are &lt;span class=&quot;math-container&quot;&gt;$(4,3,5)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(6,8,10)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(8,15,17)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(10,24,26)$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(12,35,37)$&lt;/span&gt;.  If &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is twice an odd number, the method yields the same triples as the &quot;method of Pythagoras&quot;, but doubled.  If &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is twice an even integer, the triples are primitive as two of their elements are consecutive odd numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The passage in the question starts on page 340 (location 428.6) in Glenn R. Morrow's translation of &lt;a href=&quot;https://www.scribd.com/document/256343608/Proclus-on-Euclid-s-Elements&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Commentary of the First Book of Euclid's&lt;/em&gt; Elements&lt;/a&gt;.  Here it is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Certain methods have been handed down for finding such triangles, one of them attributed to Plato, the other to Pythagoras. The method of Pythagoras begins with odd numbers,&#xA;  positing a given odd number as being the lesser of the two sides containing the angle, taking its square, subtracting one from it, and positing half of the remainder as the greater of the sides about the right angle; then adding one to this, it gets the remaining side, the one subtending the angle. For example, it takes three, squares it, subtracts one from nine, takes the half of eight, namely, four, then adds one to this and gets five; and thus is found the right-angled triangle with sides of three, four, and five, The Platonic method proceeds from even numbers. It takes a given even number as one of the sides about the right angle, divides it into two and squares the half, then by adding one to the square gets the subtending side, and by subtracting one from the square gets the other side about the right angle. For example, it takes four, halves it and squares the half,&#xA;  namely, two, getting four; then subtracting one it gets three and adding one gets five, and thus it has constructed the same triangle that was reached by the other method. For the square of this number is equal to the square of three and the square of four taken together.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Morrow refers the reader to pages 356&amp;ndash;360 of Volume I of Heath's translation of &lt;em&gt;Elements&lt;/em&gt;, where Heath presents some speculations on how such methods might have been discovered.  Starting on page 360 Heath also discusses at length the early knowledge of the Pythagorean theorem and Pythagorean triples in India that is exhibited in the &lt;em&gt;&amp;Sacute;ulvasūtras&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, in the paragraph preceding the one quoted above, Proclus discusses isosceles and scalene right triangles, echoing Plato's classification in &lt;em&gt;Timaeus&lt;/em&gt;.  But among scalene right triangles, Plato singles out the &lt;span class=&quot;math-container&quot;&gt;$30^\circ$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; triangle, for which, as Proclus notes in the case of the isosceles right triangle, &quot;you cannot find numbers that fit the sides&quot;&amp;mdash;so nothing there about Pythagorean triples.  Later in the paragraph, Proclus says that there are scalene triangles for which &quot;it is possible to find such numbers&quot;, giving the example of &quot;the triangle in the &lt;em&gt;Republic&lt;/em&gt;, in which sides of three and four contain the right angle and five subtends it&quot;.  In a footnote Morrow says the allusion is likely to &lt;em&gt;Rep.&lt;/em&gt; 546c.  In my reading that passage engages in a complicated numerology involving the numbers 3, 4, and 5, but does not say anything about right triangles.  If somewhere in Plato there is a mention Pythagorean triples, that would be interesting, but I have not heard of such a passage.&lt;/p&gt;&#xA;" OwnerUserId="139" LastEditorUserId="139" LastEditDate="2019-04-03T15:40:48.240" LastActivityDate="2019-04-03T15:40:48.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8445" PostTypeId="5" CreationDate="2019-04-02T18:52:57.840" Score="0" Body="&lt;p&gt;Originally it meaned Earth measurement in the Ancient Greek: γεωμετρία; geo- &quot;earth&quot;, -metron &quot;measurement&quot;. On its practical usability was it the first axiomatized branch of the Mathematics.&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="217" LastEditDate="2019-04-10T18:34:27.213" LastActivityDate="2019-04-10T18:34:27.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8446" PostTypeId="4" CreationDate="2019-04-02T18:52:57.840" Score="0" Body="Branch of Mathematics about the properties of the shapes, their similarities and transformations in the space." OwnerUserId="217" LastEditorUserId="217" LastEditDate="2020-01-01T19:17:03.907" LastActivityDate="2020-01-01T19:17:03.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8448" PostTypeId="1" CreationDate="2019-04-03T18:59:42.950" Score="3" ViewCount="95" Body="&lt;p&gt;I was reading Ramsey's 1927 paper &quot;A Contribution to the Theory of Taxation&quot; and came across the following paragraph: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;We have &lt;span class=&quot;math-container&quot;&gt;$\lambda_1 = \mu_1,\ldots,\lambda_m = \mu_m$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; hyperplanes (&lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; folds) whose intersection is a plane &lt;span class=&quot;math-container&quot;&gt;$n-m$&lt;/span&gt; fold which we will call &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.  &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; will cut the hyper-ellipsoids &lt;span class=&quot;math-container&quot;&gt;$u = constant$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$R = constant$&lt;/span&gt; in hyper-ellipsoids which are similar and similarly situated and whose centres are the points &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$Q'$&lt;/span&gt; in which &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is met by the &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;-folds through &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; conjugate to &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$u = c$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$R = c$&lt;/span&gt;.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is a quadratic function such that &lt;span class=&quot;math-container&quot;&gt;$d^2u$&lt;/span&gt; is a negative definite form.  &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the centre of the ellipsoid &lt;span class=&quot;math-container&quot;&gt;$u = constant$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does the term &quot;&lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; folds&quot; here just mean an &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt;-dimensional subspace?  And does &quot;conjugate&quot; here just mean &quot;orthogonal&quot;?  These are the only interpretations of the terms that make sense to me; I would like to just confirm whether these interpretations are correct...Thank you! &lt;/p&gt;&#xA;" OwnerUserId="9788" LastActivityDate="2019-04-03T18:59:42.950" Title="Old geometry terminology" Tags="|terminology|geometry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8449" PostTypeId="2" ParentId="8416" CreationDate="2019-04-03T22:24:02.457" Score="3" Body="&lt;p&gt;This is a partial answer, offering references relevant to ancient &#xA;observation methods for equinoxes and solstices. &#xA;References below cover only the near and middle East (recognizing &#xA;that the question would also cover early knowledge from any &#xA;period or civilization). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;'Preserved [ancient] texts' with descriptions of how to detect and measure &#xA;the times of equinox are rare. Fragments with indirect and &#xA;partial references to this topic are perhaps a little more numerous. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that the best that can be done in answer to the request &#xA;for references, beyond some primary texts, is to give also some major studies &#xA;about them, first because these often gather in some of the content of other &#xA;and fragmentary older works, which will not otherwise appear, and then also &#xA;because the primary texts are sometimes in need of explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Primary texts for present purposes must practically include &#xA;Ptolemy's &lt;em&gt;Almagest&lt;/em&gt;, which has a good 1984 (annotated) translation by &#xA;G J Toomer. There are online &lt;a href=&quot;https://books.google.com/books?id=43XvAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;google reprint metadata&lt;/a&gt; &#xA;about it, but as far as I know no online text. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the absence of online text it may be of interest to add that &#xA;measuring devices and methods relevant to equinoxes &amp;amp;c are described in text &#xA;and accompanying notes at Toomer's p.61...  (Book I.12. 'On the arc between the &#xA;solstices') and p.133... (in part of Book III.1. 'On the length of the year'). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy describes two such measuring devices suitable for taking meridian &#xA;altitudes or zenith distances of the sun around the times of solstice or &#xA;equinox, and later a third device to indicate when there is an equinox.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy first describes an armillary device with two rings, to be mounted &#xA;vertically and in the N-S meridian plane, checking for alignment with a plumb- &#xA;line for the vertical and a N-S meridian line on the ground. (He assumes his &#xA;reader already knows how to locate and mark a meridian line.) The device has &#xA;an outer ring, with graduation marks to indicate degrees from the zenith. An &#xA;inner ring, sliding within the outer, is fitted with two exactly-opposite &#xA;little marker-plates or -blocks, each with a pointer that grazes over the &#xA;nearest graduation marks on the outer ring. At noon, the inner ring is &#xA;adjusted by rotation, sliding within the outer, until the shadow of the &#xA;forward marker cast by the sun falls exactly on the backward marker. Then the &#xA;angular positions of the markers against the graduations can indicate the &#xA;sun's meridian zenith-distance.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A second device is a 'plinthis', a plinth or block that stands on a &#xA;horizontal pavement, and has a flat vertical face in the N-S meridian plane. &#xA;A quadrant sector is marked on this vertical face, with its arc marked in &#xA;degrees. The corner of the quadrant sector (center of the circle of which &#xA;the arc is a part) is at top south. One radius extends horizontally back &#xA;from the corner, towards north, the other radius passes vertically down &#xA;from the corner.  The quadrant is fitted with two pegs or markers: one is &#xA;at the corner of the quadrant, the other vertically below it, on the &#xA;circumference of the arc.  The noonday sun throws a shadow from both &#xA;markers. Ptolemy suggests locating more exactly where the shadow of the &#xA;marker-peg at the corner falls on the graduated arc, by moving some object &#xA;along the arc until it catches the mid-point of the shadow. The purpose &#xA;of the lower marker peg is not explained in Ptolemy's text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy doesn't say much specifically about the equinox measurement, but does &#xA;note that the moment of equinox is somewhat indeterminate. He mentions &#xA;how the extremes of meridian zenith-distance of summer and winter define the &#xA;latitude and the obliquity of the ecliptic (not using that exact expression).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later he mentions a ring of rectangular cross-section mounted in the plane of &#xA;the equator to indicate an equinox when the sun (by change of declination) &#xA;passes though its plane. On such an occasion the inner surface of the ring &#xA;begins to get its light from above instead of below, or vice versa. &#xA;Such a ring had been mounted in Alexandria at the 'square Stoa', and its &#xA;somewhat irregular indications there had been referred to by Hipparchus. &#xA;Ptolemy noticed the practical shortcomings of two such rings in the &#xA;Alexandria of his own time, sometimes 'showing' two 'equinoxes' in a day.&lt;br&gt;&#xA;He put the irregularities down to misalignment rather &#xA;than to any irregular motion of the sun. It is now known that atmospheric &#xA;refraction can also contribute to such errors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also mentioned by Ptolemy is the vertical gnomon. He says that the tip of &#xA;the shadow is hard to discern/measure in winter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are instructive ancient commentaries on Ptolemy by Pappus and &#xA;Theon, which likewise gather in some material from still older sources.&#xA;They have been given modern translations (but so far as I have seen, only &#xA;into French):-- &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Rome (ed.), &lt;em&gt;Commentaires de Pappus et de Theon d'Alexandrie sur l'Almageste&lt;/em&gt;.&#xA;Tome I. Pappus d'Alexandrie, Commentaire sur les livres 5 et 6 de l'Almageste. (Studi e Testi 54). Roma. (1931).&#xA;Tome II. Theon d'Alexandrie, Commentaire sur les livres 1 et 2 de l'Almageste. (Studi e Testi 72). Citta del Vaticano, (1936).&#xA;Tome III. Theon d'Alexandrie. Commentaire sur les livres 3 et 4 de l'Almageste. (Studi e Testi 106). Citta del Vaticano. (1943).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Almagest texts are usefully read in conjunction with three further modern studies: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;O Neugebauer's (1975)&#xA;&lt;a href=&quot;https://books.google.com/books?id=6tkqBAAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;History of Ancient Mathematical Astronomy&lt;/em&gt;&lt;/a&gt;;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=8eaHxE9jUrwC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Survey of the Almagest&lt;/em&gt;&lt;/a&gt;&#xA;With Annotation and New Commentary by Alexander Jones (originally O Pedersen, 1974; new &#xA;edition 2011);&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and for the observational data and methods in the Almagest, also some important &#xA;background and discussion in:&#xA;J P Britton (1992) &lt;a href=&quot;https://archive.org/details/modelsprecisionq00brit/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Models and precision : the quality of Ptolemy's observations &#xA;and parameters&lt;/em&gt;&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further material on Theon's commentary is at&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A. Rome, 'Les observations d'equinoxes et de solstices dans le&#xA;chapitre I du livre 3 du Commentaire sur l'Almageste par Theon&#xA;d'Alexandrie'; [I] Annales de la Societe Scientifique de Bruxelles 57, ser.I,&#xA;(1937), 213-236; [II] ibid. 58, ser.I, (1938), 6-26. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a current research project for producing new editions of works &#xA;not only by Ptolemy but related material by other authors, in &#xA;Arabic and in Latin: the website is growing and already has much bibliographic &#xA;material for searching related ancient sources: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;'Ptolemaeus arabus et Latinus'&lt;/em&gt;, see (&lt;a href=&quot;https://ptolemaeus.badw.de&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ptolemaeus.badw.de&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2019-04-03T22:24:02.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8450" PostTypeId="1" CreationDate="2019-04-03T23:15:57.840" Score="4" ViewCount="133" Body="&lt;p&gt;The value of the electron's spin is &lt;span class=&quot;math-container&quot;&gt;$\pm \frac{\hbar}{2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the paper where Pauli introduces his Pauli matrices he already knows the value of spin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd imagine it was through the Ster-Gerlach experiment or through spectral line-splitting, but those would depend on the electron's intrinsic magnetic moment instead of spin, and the magnetic moment to spin ratio is twice the magnetic moment to angular momentum value that would be classically expected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If that's how it is, then how was it discovered?&lt;/p&gt;&#xA;" OwnerUserId="7065" LastActivityDate="2019-04-04T12:57:53.920" Title="How was the value of the electron's spin ($\pm \frac{\hbar}{2}$) first determined?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8451" PostTypeId="2" ParentId="8450" CreationDate="2019-04-04T12:57:53.920" Score="2" Body="&lt;p&gt;Possibly &lt;a href=&quot;http://cosmology.princeton.edu/~mcdonald/examples/EP/commins_arnps_62_133_12.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; is what you're looking for.  Partial quote follows; the article goes on to discuss the match between spin 1/2 and Zeeman effect, and so on. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;2&quot;&gt;&#xA;  &lt;li&gt;THE DISCOVERY OF ELECTRON SPIN In 1921, Compton (9) suggested that electron spin would be an essential ingredient in any reasonable&#xA;  explanation of bulk paramagnetism and ferromagnetism. Unfortunately,&#xA;  Compton’s proposal had almost no impact on his contemporaries; thus&#xA;  our story really begins in January 1925. At that time, quantum&#xA;  mechanics did not yet exist, and the generally accepted view of the&#xA;  atomic world was still classical, although it was supplemented by the&#xA;  partly successful but confusing quantization rules of the&#xA;  Bohr–Sommerfeld model. The anomalous Zeeman effect was a persistent&#xA;  puzzle that absorbed the attention of several physicists. Alfred&#xA;  Lande, who was in Tubingen, ¨ Germany, in 1925, had devised a useful&#xA;  and well-known semiempirical vector model to describe atomic angular&#xA;  momenta and the Zeeman effect (10), but how and why that model worked&#xA;  were a mystery. Wolfgang Pauli also thought very carefully about the&#xA;  anomalous Zeeman effect, and his deep insight and wide knowledge of&#xA;  atomic spectra led him to formulate the exclusion principle at the end&#xA;  of 1924 (11). Having already attracted attention in 1921 at age 21 for&#xA;  his brilliant monograph on special and general relativity (12),&#xA;  confident of his own extraordinary abilities, and highly critical of&#xA;  others’ work when it appeared superficial or na¨ıve, Pauli would play&#xA;  an important role in the history of electron spin. On January 7, 1925,&#xA;  the 20-year-old Kronig arrived in Tubingen on a traveling fellowship&#xA;  from ¨ Columbia University in New York to spend several weeks working&#xA;  with Lande, W. Gerlach, and E. Back. Lande greeted Kronig on his&#xA;  arrival, told him that Pauli would be visiting Tubingen the ¨ next&#xA;  day, and showed Kronig a letter from Pauli that he had just received.&#xA;  The letter emphasized that to understand the anomalous Zeeman effect,&#xA;  it would be necessary to endow the electron with a fourth quantum&#xA;  number in addition to the three it already possessed in the&#xA;  Bohr–Sommerfeld model, and that the fourth quantum number could take&#xA;  only two discrete values. On reading this letter, Kronig was&#xA;  immediately struck with inspiration, and that very afternoon he&#xA;  invented a concept of electron spin.&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-04-04T12:57:53.920" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8452" PostTypeId="1" AcceptedAnswerId="8453" CreationDate="2019-04-05T01:25:35.670" Score="4" ViewCount="400" Body="&lt;p&gt;I'd like to who are credited for discovering angle transformation formulae &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sin(A\pm B)=\sin(A)\cos( B)\pm\cos(A)\sin(B)&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\cos(A\pm B)=\cos(A)\cos( B)\mp\sin(A)\sin(B)&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tan(A\pm B)=\frac{\tan(A)\pm\tan(B)}{1\mp\tan(A)\tan(B)}&#xA;$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\cot(A\pm B)=\frac{\cot(A)\cot(B)\mp1}{\cot(A)\pm\cot(B)}&#xA;$$&lt;/span&gt;&#xA;and so on, as well as and law of sines&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}&#xA;$$&lt;/span&gt;&#xA;in trigonometry.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2019-04-05T18:33:55.817" LastActivityDate="2019-04-06T06:37:50.437" Title="Who are credited for angle transformation formulae and law of sines in trignometry" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8453" PostTypeId="2" ParentId="8452" CreationDate="2019-04-05T10:54:42.057" Score="4" Body="&lt;p&gt;Your first three formulas (addition laws) were essentially known since the beginning of&#xA;trigonometry, though they were not written in the modern form. These laws actually predate the definition of &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt;.&#xA;The only trigonometric function in the ancient Greece was the chord &lt;span class=&quot;math-container&quot;&gt;${\mathrm{chd}}$&lt;/span&gt;&#xA;which is related to our modern &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; as follows &lt;span class=&quot;math-container&quot;&gt;$\mathrm{chd}\, x=2\sin(x/2)$&lt;/span&gt;. Ptolemy,&#xA;who computed the first trigonometric table makes use of the addition formula for the chord, which is equivalent to Menelaus geometric theorem. Later, when other trigonometric functions were introduced (for convenience of computation) the addition laws were of course written for them as well.&#xA;It is a routine matter to rewrite the addition formula for any trigonometric function, once you know it for one of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your third formula, the sine law for a flat triangle, is the limit case of&#xA;the similar formula for the spherical triangle,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}.$$&lt;/span&gt;&#xA;When the triangle is small, the sines of the sides are approximately equal to the sides themselves. Spherical trigonometry actually predates the flat one. Wikipedia article &quot;Law of sines&quot; credits it to various medieval  astronomers in the Middle East, and in Europe - to Johannes Müller (Regiomontanus).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference. Glen Van Brummelen, in his book Heavenly Mathematics, Princeton 2013&#xA;credits the spherical law of sines to Abu Nasr and Abu 'l-Wafa, and their priority dispute is discussed by Al-Biruni. (The special case of the flat Law of sines was stated much later).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="643" LastEditDate="2019-04-06T06:37:50.437" LastActivityDate="2019-04-06T06:37:50.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8454" PostTypeId="2" ParentId="8452" CreationDate="2019-04-05T14:07:03.543" Score="1" Body="&lt;p&gt;The answer to the question seems to depend in part on how far the questioner is &#xA;prepared to accept translations of the modern formulae in terms of &#xA;earlier mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a part-answer relating to the sine theorem, showing its effective &#xA;equivalence to a result of Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern sine theorem appears only to restate in limited terms a core &#xA;fact proved by Euclid (c. 300 BC) -- but only translated across different &#xA;languages and cultures. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, Euclid's &lt;em&gt;Elements&lt;/em&gt; III.20-21, showed that the angle subtended &#xA;at the center of a circle by any chord is twice the angle subtended by the &#xA;same chord at any point of the circumference.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where the chord is a, and the angle subtended at the cirumference is A, &#xA;then the angle at the center is 2A.  By only translating into modern &#xA;terms, and without proving any further fact,   &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;    a =  chord(2A) ,&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;(by the result of Euclid and the definition or construction of the chord).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the modern sine of A, and the ancient chord of angle 2A inscribed &#xA;in a circle of radius R, are related by definition or by construction as      &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;    sin(A) = 1/2R chord(2A) ,&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;so it only appears to say the same thing as Euclid III.20-21, but in a &#xA;translated and rearranged way, to put &lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;    a / sin(A)  = 2R (a constant), &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;with the same relation clearly applying also where the side and opposite &#xA;angle are b and B, or c and C (or even any x and X, where x is any chord whatever of the same circle with radius R, and X is any of the angles it subtends on the circumference).  The modern sine theorem is clearly included within the generality of that reformulation.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-04-05T14:31:18.330" LastActivityDate="2019-04-05T14:31:18.330" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8455" PostTypeId="1" AcceptedAnswerId="8457" CreationDate="2019-04-05T16:03:13.757" Score="13" ViewCount="824" Body="&lt;p&gt;Scientific theories are rarely accepted immediately. Even if the evidence for a theory is impeccable, there will be some stragglers who resist it for lack of understanding. More interesting is when a theory that ultimately turns out to be true is legitimately resisted at first due to lack of evidence. (For example, for a very long time atoms were rightly not considered a known fact, even though there were many who vigorously argued in support of them.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's laws of motion now seem &quot;pretty obvious,&quot; but that's only because most of us were taught them so long ago. What kind of resistance did they meet at first, and what were the most legitimate counter-arguments?&lt;/p&gt;&#xA;" OwnerUserId="9230" LastEditorUserId="55" LastEditDate="2019-04-05T22:49:13.400" LastActivityDate="2023-07-04T05:28:59.260" Title="Was there early opposition to Newton's mechanics?" Tags="|physics|classical-mechanics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8456" PostTypeId="2" ParentId="8455" CreationDate="2019-04-05T22:38:02.493" Score="6" Body="&lt;p&gt;Newtonian mechanics was resisted throughout its history, all the way until it was replaced by relativity and quantum mechanics. But the criticism did not so much concern the specifics of his laws of motion, there were few Aristotelians around, and, after all, they were pretty well confirmed by experiments, as their interpretations and &amp;quot;metaphysical&amp;quot; implications.&lt;/p&gt;&#xA;&lt;p&gt;When Newton published Principia the prevailing view of physics was Cartesian, and the two of them clashed over the course of 19th century. Cartesian physics, was more qualitative, descriptive and &amp;quot;intuitive&amp;quot;, in other words, it was more like the natural philosophy of old (e.g. Aristotelian or atomistic). Descartes had the luminiferous aether, and the swirling vortices in it moving things around, in place of Newton's faceless mathematical forces, including the action at a distance for gravity, which even Newton himself disliked. In the early 1700-s Cartesian physics was refurbished by Malebranche and Rohault into an alternative to Newton's. Here is from &lt;a href=&quot;https://www.jstor.org/stable/229723&quot; rel=&quot;nofollow noreferrer&quot;&gt;Iltis's The Decline of Cartesianism in Mechanics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Several aspects of Descartes' philosophy of nature were used by Cartesians in the 1720s in attacking mechanical problems. His theory of matter stated that the world was composed of three elements formed from primitive matter. The first element, which formed the material of the sun and stars, consisted of very subtle minute matter capable of moving at enormous speeds and of filling in the small spaces surrounding the other two denser forms of matter. The second element consisted of spherical particles formed from rotation of the original primitive matter. These spheres, the main constituent of celestial matter, moved in large vortices at high speeds and could transmit pressure instantaneously. The third element was coarser and slower and composed the earth and planets.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The properties and action of the spherical aether particles were used by Cartesians in explaining the action of mechanical bodies. As the aether swirled in a centrifugal vortical motion around the earth it caused continual impulses on terrestrial objects. Since the subtle matter moved faster than terrestrial matter and exerted a force toward the center, bodies above the earth would fall toward the earth's center.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By the end of 18th century the Cartesian conception was mostly abandoned, but not the spirit of natural philosophy. The torch was taken up by German romantics and their &lt;a href=&quot;https://en.wikipedia.org/wiki/Naturphilosophie&quot; rel=&quot;nofollow noreferrer&quot;&gt;Naturphilosophie&lt;/a&gt;. Göthe, better known as the author of Faust, also penned &lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_Colours&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theory of Colours&lt;/a&gt;, that critiqued Newton's optics and replaced it with a more &amp;quot;poetic&amp;quot; theory, emphasizing the boundary between &amp;quot;light and darkness&amp;quot;. Although his theory was rightly rejected by physicists, he made a number of astute observations about the human color perception that are now incorporated into chromatics. Schelling and Hegel rejected mechanistic Newtonian science as such, and planned to replace it with an imagined &amp;quot;better&amp;quot; science, see &lt;a href=&quot;https://philosophy.stackexchange.com/a/51101/9148&quot;&gt;Why did Hegel hate Newton?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A seemingly technical issue that consumed a lot of attention of many famous people was the &lt;em&gt;vis viva&lt;/em&gt; controversy, as to what constitutes the &amp;quot;true&amp;quot; quantity of motion, vis viva (now, kinetic energy) or &lt;em&gt;vis mortua&lt;/em&gt; (now, momentum, which Descartes mistakenly believed to be mass times the &lt;em&gt;magnitude&lt;/em&gt; of velocity), see &lt;a href=&quot;https://hsm.stackexchange.com/a/598/55&quot;&gt;What was the vis viva controversy?&lt;/a&gt; A more substantive part of it was a dispute between force-based description of physics, that Newton favored, and energy-based one, favored by Leibniz. Leibniz saw both  Cartesian and Newtonian physics as too &amp;quot;mechanistic&amp;quot;, unable to explain the the &amp;quot;source of the vitality&amp;quot; of matter, which his &lt;em&gt;vis viva&lt;/em&gt;, elevated to a metaphysical status, was meant to do.&lt;/p&gt;&#xA;&lt;p&gt;Newton's absolute space also got its share of detractors. First, Leibniz, who saw space as relational rather than a self-standing metaphysical entity, Berkeley shared similar views and offered an influential criticism of Newton's mechanics, albeit less known now than his criticism of Newton's calculus. Then Kant, who sought to produce a physical construction for the &amp;quot;absolute frame of reference&amp;quot; in almost modern spirit. He associates it with the ideal limit of taking first the frame attached to the Earth, then to the center of mass of the Solar system, etc., see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4020-9510-8_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton and Kant on Absolute Space by Friedman&lt;/a&gt;. These ideas were later be used by Mach to dispense with the absolute space even in the mathematical formalism, and usher in the relativistic conception, adopted and transformed by Einstein.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-12T05:08:57.073" LastActivityDate="2020-08-12T05:08:57.073" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="8457" PostTypeId="2" ParentId="8455" CreationDate="2019-04-06T04:28:35.670" Score="15" Body="&lt;p&gt;There was an opposition. The reasons were mainly philosophical. The main thing which was hard to accept was &amp;quot;action at a distance&amp;quot; through the void space. For example Huygens did not accept this. It contradicted Descartes theory which was prevalent at that time. So the question had too be decided by observations and experiments, as it always happens in science.&lt;/p&gt;&#xA;&lt;p&gt;And it was decided in the most spectacular way. The crucial successes were:&#xA;prediction of the correct shape of the Earth, explanation of inequalities of Moon motion which led to very exact predictions of the Moon position in the sky.&#xA;And perhaps the greatest confirmation: the discovery of new planets, big (Neptune) and small&#xA;(asteroids). Discovery of new planets on the basis of Newton's mechanics continues to this day, by the way. I mean in other star systems, and discovery of smaller objects in the Solar system.&lt;/p&gt;&#xA;&lt;p&gt;After the discovery of Neptune, NO SERIOUS SCIENTIST could have any doubt in the correctness of the theory. In fact we obtain new confirmations of this theory all the time.&lt;/p&gt;&#xA;&lt;p&gt;Here I am in a strong disagreement with Conifold. Newton's mechanics is correct.&#xA;Because it explains and predicts observations and experiments correctly, and&#xA;with very high accuracy.&#xA;It is the cornerstone of the whole physics, and one of the most successful scientific theories. It is taught nowadays at all level from&#xA;elementary school to the universities, and it is still an important research topic. And will always be.&lt;/p&gt;&#xA;&lt;p&gt;And general relativity or quantum mechanics did not diminish its value. One cannot say that&#xA;&amp;quot;it was replaced&amp;quot;. It was complemented with new theories.&lt;/p&gt;&#xA;&lt;p&gt;EDIT. All theories have their limitations. Quantum mechanics does not describe gravity. General relativity has nothing to do with the structure of the atom. Newton's theory is highly successful in a very wide range of natural phenomena, and it is unlikely to be ever &amp;quot;replaced&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-07-04T05:28:59.260" LastActivityDate="2023-07-04T05:28:59.260" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="8458" PostTypeId="1" AcceptedAnswerId="8459" CreationDate="2019-04-06T09:30:54.503" Score="2" ViewCount="126" Body="&lt;p&gt;The concept of an isomorphism is very interesting: a rigorous, formal way of expressing similarity between two objects. When and how did this idea of similarity become formalised as a bijective mapping between two spaces?&lt;/p&gt;&#xA;" OwnerUserId="6867" LastActivityDate="2019-04-06T12:45:34.463" Title="Who is credited for formalising the theory of isomorphisms?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8459" PostTypeId="2" ParentId="8458" CreationDate="2019-04-06T12:45:34.463" Score="6" Body="&lt;p&gt;I found this in &lt;a href=&quot;http://jeff560.tripod.com/h.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;ISOMORPHIC, ISOMORPHISM. The term isomorphism was used early in crystallography. Some books on geology before 1864 referred to “geometrical isomorphism” or “mathematical isomorphism” between crystals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-04-06T12:45:34.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8460" PostTypeId="1" CreationDate="2019-04-06T16:29:24.603" Score="5" ViewCount="153" Body="&lt;p&gt;In a German interview some physicists were asked, what they would ask Einstein, if he were alive today. One of them wanted to know how Einstein came up with the idea of the equivalence principle, that inertial mass is the same as gravitational mass. Correct me if I'm wrong, but I always believed that was known way before Einstein. And I have an experiment in mind where one famous guy standing on a leaning tower and letting two things fall with different mass to test if gravity acts on every object the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, does somebody know the name of this famous person, who conducted the experiment? I can't remember it.&lt;/p&gt;&#xA;" OwnerUserId="9798" OwnerDisplayName="Maxim" LastEditorUserId="3643" LastEditDate="2019-04-07T15:54:32.420" LastActivityDate="2019-04-07T15:54:32.420" Title="Equivalence principle before Einstein" Tags="|physics|" AnswerCount="1" CommentCount="4" ClosedDate="2019-04-07T08:56:57.927" ContentLicense="CC BY-SA 4.0" />
  <row Id="8461" PostTypeId="2" ParentId="8460" CreationDate="2019-04-06T16:36:34.707" Score="11" Body="&lt;p&gt;There are lots of different ways of stating the equivalence principle, and they're not all logically ... er ... equivalent to each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Names that come up in this connection, from before Einstein, are Galileo and Eotvos. Experiments that test the equality of gravitational and inertial mass are called Eotvos experiments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's version of the equivalence principle is more radical than anything that Galileo or Newton envisioned. It entails throwing away the whole concept of a universal reference frame and instead defining inertial frames as local, free-falling frames. Previously, people had imagined that there was a frame of the &quot;fixed stars,&quot; which was as good an approximation to an inertial frame as we would ever need. But according to relativity, observers in New York and Beijing can't even have the same frame of reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Taking the equivalence principle seriously also leads to the prediction of gravitational time dilation, which is something that nobody before Einstein had imagined.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2019-04-06T16:36:34.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8462" PostTypeId="2" ParentId="8366" CreationDate="2019-04-07T13:11:23.347" Score="8" Body="&lt;p&gt;I think &lt;strong&gt;Paul Dirac&lt;/strong&gt; would have certain characteristics which fit this category. Often called as &quot;theorist's theorist&quot;, Dirac was science's archetypal loners,taciturn and devoid of empathy. He has an unhappy childhood, but doesn't mention it for 50 years; learns to speak French, German and Russian, but becomes famous for his long silences; embarks on the wrong career; gets interested in mathematics and ends up at Cambridge, where he becomes famous for his even longer silences; hears about Einstein and gets into advanced physics; and then goes to Copenhagen to meet Niels Bohr, who grumbles to Ernest Rutherford, &quot;This Dirac, he seems to know a lot of physics, but he never says anything&quot;. A lonely boy who becomes a lonely man driven by the concept of mathematical beauty&lt;/p&gt;&#xA;" OwnerUserId="9801" LastActivityDate="2019-04-07T13:11:23.347" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8463" PostTypeId="2" ParentId="3131" CreationDate="2019-04-07T17:52:42.467" Score="5" Body="&lt;p&gt;&lt;strong&gt;C P Ramanujam&lt;/strong&gt;, an Indian Mathematician, not to be confused with Srinivasa Ramanujan, one who worked with GH Hardy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ramanujam's achievements at High School had been outstanding and he had shown that he was extraordinarily gifted, so he entered Loyola College with great expectations. He continued his interest in chemistry but it was mathematics that he specialised in, taking Mathematics Honours after obtaining his Intermediate qualification. He was awarded a B.A. with Honours in Mathematics in 1957 but, strangely for such an outstanding student, he only obtained a second class degree. This may have been a result of starting his university education at so young an age before he was really ready, for the second class degree no way reflected his remarkable mathematical abilities. On the other hand it may have resulted from a lack of belief in himself which haunted Ramanujam throughout his life. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ramanujam felt that he did not have what it takes to solve the big problems of mathematics, and he had no wish to solve small routine problems. Again, as in his undergraduate course, it would appear to be a psychological problem rather than a mathematical one but for Ramanujam, it was a very real problem and he became more and more frustrated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mumford writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was a stimulating experience to know and collaborate with C P Ramanujam. He loved mathematics and he was always ready to take up a new thread or pursue an old one with infectious enthusiasm. He was equally ready to discuss a problem with a first year student or a colleague, to work through an elementary point or puzzle over a deep problem. On the other hand he had high standards. He felt the spirit of mathematics demanded of him not merely routine developments but the right theorem an any given topic. He was sometimes tormented by these high standards, but, in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuine original stamp.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Back in India after his year at the University of Warwick, Ramanujam asked for a Professorship at the Tata Institute ,but be based in Bangalore where a new branch dealing with applications of mathematics was being set up. This was agreed and he taught analysis in Bangalore but, again in the depths of depression caused by his illness, he tried again to leave the Institute and obtain a university teaching post. While waiting for an offer of such a post from the Indian Institute at Simla, he took his life with an overdose of barbiturates.&lt;/p&gt;&#xA;" OwnerUserId="9801" LastEditorUserId="3643" LastEditDate="2019-04-08T00:04:04.197" LastActivityDate="2019-04-08T00:04:04.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8464" PostTypeId="2" ParentId="3131" CreationDate="2019-04-08T04:25:43.203" Score="6" Body="&lt;p&gt;There are a few notable ones from Computer Science and Logic:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alan_Turing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alan Turing&lt;/a&gt; (famously) died at the age of 41, most likely due to suicide from mental issues cause by estrogen he was being forced to take&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gerhard_Gentzen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerhard Gentzen&lt;/a&gt; died at the age of 45, during WWII. His death isn't always viewed as a tragedy, since he was fighting for the Nazis. But he invented both Natural Deduction and Sequent Calculus, and was hugely fundamental in the study of logic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_G%C3%B6del&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Gödel&lt;/a&gt; wasn't young (71), but he suffered from extreme mental illness, so his death was tragic. He died when his wife was hospitalized and he refused to eat any food, for fear that others would poison him.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3044" LastEditorUserId="4251" LastEditDate="2023-12-03T09:04:24.940" LastActivityDate="2023-12-03T09:04:24.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8465" PostTypeId="2" ParentId="3131" CreationDate="2019-04-08T13:24:54.287" Score="7" Body="&lt;p&gt;&lt;strong&gt;Felix Hausdorff&lt;/strong&gt; &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Hausdorff.html&quot; rel=&quot;noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;br&gt;&#xA;This is not &quot;very young&quot; but it is still tragic.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hausdorff was a German Jewish mathematician.  He is also known for his work in astronomy, philosohy, and even literature (under the pseudonym &quot;Paul Mongré&quot;).  When the Nazis came to power in Germany, Hausdorff was retired, and he was allowed to remain in his house in Bonn.  For a long time, the Nazi government ignored him, instead concentrating on purging active Jewish academics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But finally in 1942 Hausdorff heard that they were coming to deport him to a camp.  Felix, his wife, and his wife's sister committed suicide by taking barbiturates.  He was 73.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-04-08T13:24:54.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8467" PostTypeId="1" AcceptedAnswerId="8469" CreationDate="2019-04-09T12:30:30.383" Score="3" ViewCount="733" Body="&lt;p&gt;Anybody with elementary mathematical education will have seen &lt;em&gt;improper&lt;/em&gt; fractions to refer to fractions where the numerator is greater than or equal to the denominator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At a certain point in calculus studies, students are confronted with &lt;em&gt;improper&lt;/em&gt; integrals, that is the limit of a definite integral as the bound(s) tend to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Improper&lt;/em&gt; prior distributions, in Bayesian statistics, are prior distributions that don't integrate to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Personally, as I learned these concepts, I found the use of the word &lt;em&gt;improper&lt;/em&gt; to hint to a negative connotation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A double question here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$1)$&lt;/span&gt; Are there any other objects in mathematical studies where this adjective is used?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$2)$&lt;/span&gt;&#xA;Were these mathematical objects named like this simply because they were extensions (one might also say &quot;dares&quot;, such as taking the limit of a definite integral) of previously consolidated concepts or was this nomenclature used because of possible discord on the existence of such objects while they were being discovered? &lt;/p&gt;&#xA;" OwnerUserId="3853" LastActivityDate="2019-04-09T13:13:29.197" Title="Origin and use of the adjective &quot;improper&quot; in mathematics" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8468" PostTypeId="2" ParentId="8467" CreationDate="2019-04-09T13:11:26.253" Score="3" Body="&lt;p&gt;The usage of &quot;improper&quot; is too old to be usefully traced. It goes back to 1500s. Just checked the Unabridged Oxford English Dictionary. In such cases of relatively odd choices of words, I look up the same terminology in other languages. For instance, what do Germans use for improper fraction? A quick search shows it is &lt;em&gt;unechter&lt;/em&gt; bruch and unechter means fake, false (fraction). It makes perfect sense from an purely arithmetical perspective. How can we have have more than 100 parts out of 100? It must have sounded fake and improper thing to do.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what OED shows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Not properly so called; improper fraction: a fraction whose numerator is greater than (or equal to) its denominator, and whose value is therefore greater than (or equal to) unity. (Formerly applied to analogous fractions in Algebra.) improper diphthong: see quot. 1795.&#xA;1552   R. Record Ground of Artes (rev. ed.) ii. sig. Q.vi   An impropre fraction..that is to say, a fraction in form, which in deed is greater then an vnite.&#xA;1629   J. Fletcher Faithfull Shepheardesse (ed. 2) To Rdr.   They [shepherds and shepherdesses in a Pastoral] are not to be adorn'd with any art, but such improper ones as nature is said to bestow, as singing and Poetry.&#xA;a1690   S. Jeake Λογιστικη Λογια (1696) 44   Improper Fractions have alwayes the Numerator greater than the Denominator.&#xA;1795   L. Murray Eng. Gram. 4   An improper diphthong has but one of the vowels sounded; as, ea in eagle, oa in boat.&#xA;1806   C. Hutton Course Math. (ed. 5) I. 187   To Reduce an Improper Fraction to a Whole or Mixed Quantity.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-04-09T13:11:26.253" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8469" PostTypeId="2" ParentId="8467" CreationDate="2019-04-09T13:13:29.197" Score="4" Body="&lt;p&gt;It is interesting to look at &lt;a href=&quot;http://jeff560.tripod.com/i.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathwords&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;IMPROPER FRACTION was used in English in 1542 by Robert Recorde in The ground of artes, teachyng the worke and practise of arithmetike: &amp;quot;An Improper Fraction...that is to saye, a fraction in forme, which in dede is greater than a Unit.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;IMPROPER INTEGRAL occurs in &amp;quot;Concerning Harnack’s Theory of Improper Definite Integrals&amp;quot; by Eliakim Hastings Moore, Trans. Amer. Math. Soc., July 1901. The term may be much older.&lt;/p&gt;&#xA;&lt;p&gt;PROPER FRACTION appears in English in 1674 in Samuel Jeake Arithmetic (1701): &amp;quot;Proper Fractions always have the Numerator less than the Denominator, for then the parts signified are less than a Unit or Integer&amp;quot; (OED2).&lt;/p&gt;&#xA;&lt;p&gt;PROPER VALUE and VECTOR. See Eigenvalue.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-09T13:13:29.197" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8470" PostTypeId="2" ParentId="8366" CreationDate="2019-04-09T20:13:55.293" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=iQbtCAAAQBAJ&amp;amp;pg=PA351&amp;amp;lpg=PA351&amp;amp;dq=Sophus%20Lie%20mental&amp;amp;source=bl&amp;amp;ots=sPeADKE-YQ&amp;amp;sig=ACfU3U3CwUWb8YL2SlePFxcaqgsShQITog&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiezc646MPhAhVL0KwKHdx_CZw4ChDoATAAegQICRAB#v=onepage&amp;amp;q=Sophus%20Lie%20mental&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sophus Lie&lt;/a&gt; probably qualifies, being both a great mathematician and a patient of mental institutions.&lt;/p&gt;&#xA;&lt;p&gt;According his &lt;a href=&quot;https://en.wikipedia.org/wiki/Sophus_Lie&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiki&lt;/a&gt;, he largely created the theory of &lt;a href=&quot;https://en.wikipedia.org/wiki/Continuous_symmetry&quot; rel=&quot;nofollow noreferrer&quot;&gt;continuous symmetry&lt;/a&gt; and applied it to the study of geometry and differential equations.&lt;/p&gt;&#xA;&lt;p&gt;EDIT (April 2, 2022): Boltzmann probably also qualifies as a greatest physicist and a patient of a sanatorium. Sadly, he committed suicide.&lt;/p&gt;&#xA;&lt;p&gt;EDIT (March 2, 2024): &lt;a href=&quot;https://en.wikipedia.org/wiki/Julius_von_Mayer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julius Mayer&lt;/a&gt; formulated the law of conservation of energy and was a patient of a mental institution.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastEditorUserId="9808" LastEditDate="2024-03-02T22:18:36.297" LastActivityDate="2024-03-02T22:18:36.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8471" PostTypeId="2" ParentId="8366" CreationDate="2019-04-10T05:19:43.543" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_G%C3%B6del&quot; rel=&quot;noreferrer&quot;&gt;Kurt Gödel&lt;/a&gt; achieved some of the most important breakthroughs in logic in the 20th century. As far as mental instability is concerned, quote from Wikipedia: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Later in his life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death. He weighed 29 kilograms (65 lb) when he died.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2019-04-10T05:19:43.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8472" PostTypeId="2" ParentId="8366" CreationDate="2019-04-10T05:29:44.050" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9_Bloch_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;André Bloch&lt;/a&gt; is an extreme case: He murdered three of his family members. Being institutionalised for the rest of his life, he wrote influential mathematical papers (on complex analysis). Quote from Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;On 17 November 1917, while on convalescent leave from service in World War I, Bloch killed his brother Georges, and his aunt and uncle. Several conjectures about the motives for Bloch's crime exist among mathematicians. [...] Bloch told [the medical head of his asylum] that the murders were a eugenic act, in order to eliminate branches of his family affected by mental illness.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;After the murders, Bloch was committed to the asylum at Charenton in Saint Maurice, a suburb of Paris. Bloch continued his mathematical career while confined. All of his publications, including those relating to Bloch's constant, were written while he was committed. Bloch corresponded with several mathematicians [...] giving his return address as only &quot;57 Grande rue, Saint-Maurice&quot;, never mentioning that this was a psychiatric hospital. Several of his correspondents were thus unaware of his situation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2019-04-10T05:29:44.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8473" PostTypeId="1" CreationDate="2019-04-10T18:37:20.890" Score="4" ViewCount="6929" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Why are there four? I would guess it was fixed after the egocentric/&quot;relative&quot; directions (left, right, front, back). Or was it because human used to consider the ground as flat and 2-dimensional?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How are they defined? Wikipedia says their English names have origins in Proto-Germanic words related to sunrise, so probably East is defined as the direction where sun rises. But how exact is such directions? As we know, in the northern hemisphere the sun rises precisely due east only on the equinox. Why was East not picked as a direction somewhat eastnorth or eastsouth? Or did cardinal directions origined in a civilization living near the equator? Did the Inuit have a more southern East?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are they &quot;natural concepts&quot;? I remember I was taught the directions at school when I was a little kid. Could a person develop the concept of cardinal directions without influence from teacher/parents? How important are the directions for hunting/gathering/agriculture in a prehistoric environment? Do animals have concepts of directions?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Did the early civilizations have different definitions for the directions? How do they influence each other? If civ A traded with civ B and noticed they used a &quot;somewhat rotated map&quot;, would any of the civs change their direction system?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9813" LastEditorUserId="1930" LastEditDate="2019-04-22T21:53:19.810" LastActivityDate="2019-04-22T21:53:19.810" Title="What is the origin of the four cardinal directions (North, East, South, and West)?" Tags="|geography|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8474" PostTypeId="1" CreationDate="2019-04-11T08:08:24.263" Score="4" ViewCount="659" Body="&lt;p&gt;Ive never seen a source for this, but I had a professor a few years back that a low tech way of calculating the area under a curve (definite integral) was to use a piece of paper with known thickness/density, plot the function, and carefully cut off paper above the line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Measuring the mass, and then using the density formula, you could calculate the area under the curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was this method used or did I mis-remember the story?&lt;/p&gt;&#xA;" OwnerUserId="9816" LastEditorUserId="609" LastEditDate="2019-04-11T12:39:05.747" LastActivityDate="2019-04-11T20:31:51.167" Title="Using paper of known density to calculate area under a curve" Tags="|mathematics|computation|" AnswerCount="1" CommentCount="7" ClosedDate="2019-04-12T06:25:27.993" ContentLicense="CC BY-SA 4.0" />
  <row Id="8475" PostTypeId="2" ParentId="6957" CreationDate="2019-04-11T19:36:54.043" Score="4" Body="&lt;p&gt;We find some uses of the word &amp;quot;conjecture&amp;quot; in &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathwords&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;CONJECTURE in the sense of &amp;quot;an opinion or supposition based on evidence which is admittedly insufficient&amp;quot; had been in English for a more than a century when Isaac Newton used the term in 1672: &amp;quot;I shall refer him to my former Letter, by which that conjecture will appear to be ungrounded.&amp;quot; Mr. Isaac Newtons Answer to Some Considerations upon His Doctrine of Light and Colors; Which Doctrine Was Printed in Numb. 80. of These Tracts, Phil. Trans. VII. p. 5084. The conjecture in question seemed to concern optics, not mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Jacob Steiner (1796-1863) referred to a result of Poncelet as a conjecture. Poncelet showed in 1822 that in the presence of a given circle with given center, all the Euclidean constructions can be carried out with ruler alone (DSB, article: &amp;quot;Mascheroni&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;In Récréations Mathématiques, tome II, Note II, Sur les nombres de Fermat et de Mersenne (1883), É. Lucas referred to &amp;quot;la conjecture de Fermat.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In his article &amp;quot;Conjecture&amp;quot; (Synthese 111, pp. 197-210, 1997), Barry Mazur writes (bottom of page 207):&lt;/p&gt;&#xA;&lt;p&gt;Since I am not a historian of Mathematics I dare not make any serious pronouncements about the historical use of the term, but I have not come across any appearance of the word Conjecture or its equivalent in other languages with the above meaning [i.e., an opinion or supposition based on evidence which is admittedly insufficient] in mathematical literature except in the twentieth century. The earliest use of the noun conjecture in mathematical writing that I have encountered is in Hilbert’s 1900 address, where it is used exactly once, in reference to Kronecker’s Jugendtraum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-11T19:36:54.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8476" PostTypeId="2" ParentId="8474" CreationDate="2019-04-11T20:31:51.167" Score="0" Body="&lt;p&gt;In fact you do not need to know the thickness of the paper or its density.&#xA;You only have to be sure that the paper is reasonably uniform (has some constant density). Because you can draw a rectangle next to your curve, and weight it too,&#xA;and compare with the weight of the area under the curve.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not sure who invented this (one can never be sure who did some simple thing first) but Galileo used the method for determining the area under a cycloid, which was not known theoretically at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not remember exactly where I read this, but most likely in the book of Drake, Galileo at work. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-04-11T20:31:51.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8477" PostTypeId="2" ParentId="8473" CreationDate="2019-04-11T21:03:50.463" Score="6" Body="&lt;p&gt;The origin is astronomical. North is the direction to the North Pole of the celestial sphere. (To the point about which the sky performs its daily rotation as seen from the Northern hemisphere).&#xA;The opposite direction to this is called South. South is also the direction to&#xA;the Sun in the middle of its daily way over the sky. These directions are of fundamental importance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;East-West are perpendicular directions (in the horizontal plane). East to the side where the Sun rises, and West where it sets. But the Sun rarely rises or sets exactly in the East/West.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why 4? Because the horizontal plane is 2-dimensional. So you need two axes to describe the points and directions in this plane. But it is convenient to have&#xA;different names for the &quot;positive&quot; and &quot;negative&quot; directions of these axes, therefore we have 4. (We are talking of the times very long before negative numbers were introduced, possibly before ANY numbers were introduced. So the subject does not belong to history in the technical sense, but rather to pre-history).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are natural concepts, even long before civilizations. The period in time when they were really needed was the introduction of agriculture. (Long before &quot;civilization&quot; which is usually associated with writing, states, etc.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To do agriculture you need the notions of the year and seasons of the year, when they start and end etc. All these notions are of astronomical nature, and you can &#xA;clearly establish them (and most importantly PREDICT them)&#xA;ONLY by observing the sky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you just begin observing the sky, the first thing that you notice is that the sky as a whole rotates about certain point with period of revolution which we call day-and-night (or simply &quot;day&quot; in English, which is highly confusing, see the below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This point in the sky is seen in the North direction.&#xA;The South direction is even easier to observe: it is where the Sun is&#xA;in the middle of the day.&#xA;These directions never change.&#xA;The directions to the points where the Sun raises and sets do change during the year, but to describe them one needs some fixed points of reference, and these are East and West, so that EW-axis is perpendicular to the NS axis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference. The oldest surviving treatise on agriculture,&#xA;Hesiodes, Days and Works. (Translated from the ancient Greek to all modern languages, I suppose, and free of copyright:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Some languages have a single word for the day-and-night period.&#xA;The technical name in English is nychthemeron (how many English-speaking people have ever heard this?) If your native language does not have this word, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Nychthemeron&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; about it. This word is clearly Greek,&#xA;not native to English or German languages. &#xA;Wikipedia gives a very interesting list of languages which have such a word as a native word. I always wondered what this list says about history or prehistory of various languages and peoples and astronomy etc.:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="1930" LastEditDate="2019-04-13T22:53:53.003" LastActivityDate="2019-04-13T22:53:53.003" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8478" PostTypeId="1" AcceptedAnswerId="8484" CreationDate="2019-04-13T09:25:58.947" Score="3" ViewCount="274" Body="&lt;p&gt;The 18th century saw a rise in the use of mathematical formalisms to account for natural phenomena. Works of Lagrange, Euler, d'Alembert, etc., were groundbreaking in the history of mechanics and mathematics. I know there is an expansive literature on mathematical explanation today; but this 'mathematicization' in the 18th century must have also been met with supporters and dissenters. Was there a 'debate' about the role of mathematics in physics in the 18th century? If so, who were the key figures, and what did they claim?&lt;/p&gt;&#xA;" OwnerUserId="6867" LastActivityDate="2019-04-14T22:24:58.757" Title="How did philosophers and scientists in the 18th century view mathematical explanation?" Tags="|mathematics|physics|mathematicians|mathematical-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8479" PostTypeId="1" AcceptedAnswerId="8482" CreationDate="2019-04-13T14:50:35.887" Score="7" ViewCount="257" Body="&lt;p&gt;A folklore result I have seen used in evaluations of infinite sums is the following clever use of the residue theorem:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}\sum_{1}^\infty f(k)&amp;amp;=\frac1{2\pi i}\oint f(z)\pi\cot\pi z\,\mathrm dz\\\sum_{1}^\infty (-1)^k f(k)&amp;amp;=\frac1{2\pi i}\oint f(z)\pi\csc\pi z\,\mathrm dz\end{align*}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; is analytic and the contour surrounds the appropriate subset of poles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have seen some references ascribe this to Mittag-Leffler, but as far as I aware, his only &lt;a href=&quot;https://en.wikipedia.org/wiki/Mittag-Leffler%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;contribution&lt;/a&gt; was showing that the functions used in the contour integrals above can be decomposed into their poles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that these results do not have a name attached to them makes it hard to search for more information; my limited searching shows me that these results are often quoted in handbooks or textbooks as a mere consequence of the residue theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I must ask: is there any history behind these summation formulae (who came up with it, first appearance in the literature, etc.)? References where I can read more on this and other contour integral summation techniques (e.g. I have seen the variants that use tangent, secant, or even the gamma function itself) are welcome.&lt;/p&gt;&#xA;" OwnerUserId="9821" LastActivityDate="2019-04-13T23:13:29.870" Title="History of a contour integral method for summing series" Tags="|mathematics|complex-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8480" PostTypeId="1" AcceptedAnswerId="8483" CreationDate="2019-04-13T17:08:58.430" Score="9" ViewCount="985" Body="&lt;p&gt;It is commonly known that Feynman's path integral was inspired by Dirac's observation that the kernel is proportional to &lt;span class=&quot;math-container&quot;&gt;$\exp{iS/\hbar}$&lt;/span&gt;. It was Feynman, however, who had the idea of expressing the kernel as the integral of such an expression over all possible paths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it known what led Feynman to try out this idea of an integral over all paths? He provides motivation from a probabilistic understanding of quantum mechanics in &quot;Non-relativistic Quantum Mechanics&quot; and Quantum Mechanics and Path Integrals, but is this motivation what actually led him to try out the idea of the path integral and discover that up to suitable normalization, it recovered the correct expression for the kernel and the Schrodinger equation?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, why was Dirac's observation of the proportionality of the final answer to &lt;span class=&quot;math-container&quot;&gt;$\exp{iS/\hbar}$&lt;/span&gt; insufficient as the basis for an alternative formulation of quantum mechanics.&lt;/p&gt;&#xA;" OwnerUserId="9822" LastEditorUserId="5659" LastEditDate="2020-06-06T12:59:38.110" LastActivityDate="2020-06-06T12:59:38.110" Title="What actually led Feynman to the path integral formulation of quantum mechanics?" Tags="|physics|discoveries|quantum-mechanics|physicists|feynman|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8481" PostTypeId="1" CreationDate="2019-04-13T18:27:09.440" Score="2" ViewCount="77" Body="&lt;p&gt;We know of two ways of representing voltage-charge relations of an assembly of conductors:&#xA;&lt;span class=&quot;math-container&quot;&gt;$V_j=\sum P_iq_i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q_j=\sum C_iv_i$&lt;/span&gt;&#xA;Who introduced these equations for the first time and though one can be modified to other,  yet which one of them was first to be introduced (in Maxwell's treatise, he first mention the former equation and then the latter) ?&lt;/p&gt;&#xA;" OwnerUserId="9585" LastEditorUserId="9585" LastEditDate="2019-04-13T20:14:55.097" LastActivityDate="2019-04-13T20:14:55.097" Title="Who introduced concept of capacitance matrix" Tags="|electromagnetism|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8482" PostTypeId="2" ParentId="8479" CreationDate="2019-04-13T23:13:29.870" Score="6" Body="&lt;p&gt;Whittaker and Watson credit this method to Mittag-Leffler, Acta Soc. Sci. Fenn., XI 1880 273-293, and Acta Math., IV 1884 1-79.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-04-13T23:13:29.870" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8483" PostTypeId="2" ParentId="8480" CreationDate="2019-04-14T10:42:22.180" Score="10" Body="&lt;p&gt;It was Dirac's paper &lt;a href=&quot;https://www.ifi.unicamp.br/~cabrera/teaching/aula%2015%202010s1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Lagrangian in Quantum Mechanics&lt;/em&gt;&lt;/a&gt;. He gave more than just a remark about &lt;span class=&quot;math-container&quot;&gt;$\exp{iS/\hbar}$&lt;/span&gt;, he described the general structure of the path integral expression for the transition amplitude, see &lt;a href=&quot;https://physics.stackexchange.com/q/134209/65263&quot;&gt;Dirac's remark that inspired Feynman when formulating path integral&lt;/a&gt; on Physics SE. But he did not come up with the suggestive intuition of summing over paths, or developed a computational procedure that could be followed, or related it to the semi-classical approximation. Fortunately, Dirac was pretty explicit as to what motivated him:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Quantum mechanics was built up on a foundation of analogy with the Hamiltonian theory of classical mechanics... it would seem desirable to take up the&#xA;  question of what corresponds in the quantum theory to the&#xA;  Lagrangian method of the classical theory. A little consideration shows, however, that one cannot expect to be able to take over the classical Lagrangian equations in any very direct way. These equations involve partial derivatives of the Lagrangian with respect to the coordinates and velocities and no meaning can be given to such derivatives in quantum mechanics... We must therefore seek our quantum Lagrangian theory in an indirect way. We must try to take over the ideas of the classical  Lagrangian theory, not the equations of the classical Lagrangian theory.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He then relates traditional amplitudes to the integral from the least action principle. Feynman uses similar motivation in &lt;a href=&quot;https://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter&quot; rel=&quot;nofollow noreferrer&quot;&gt;QED: The Strange Theory of Light and Matter&lt;/a&gt;, for example, where he introduces the path integrals starting from the various paths a photon could take, and using the Fermat's variational principle.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="5659" LastEditDate="2020-06-06T06:40:09.663" LastActivityDate="2020-06-06T06:40:09.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8484" PostTypeId="2" ParentId="8478" CreationDate="2019-04-14T22:18:04.467" Score="5" Body="&lt;p&gt;This quite difficult question seems to place the mathematization of nature in the 18th century, &#xA;but a substantial body of scholarly opinion writes of it as developing earlier than that, in &#xA;the 17th -- although there can of course be no doubt that its further development in the 18th &#xA;century was very extensive indeed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources on the 17th century earlier developments in this mathematization include &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=ED76ljJ6CD0C&quot; rel=&quot;noreferrer&quot;&gt;R S Westfall (1971) 'The Construction of Modern Science', see esp. pp.120-138&lt;/a&gt;, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;also &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/GeoffreyGorhamBenjaminHillEdwardSlowikEds.TheLanguageOfNatureReassessingTheMathe/page/n3&quot; rel=&quot;noreferrer&quot;&gt;The Language Of Nature : Reassessing The Mathematization Of Natural Philosophy In The 17th Century, G Gorham et al (eds)(2016)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the 18th-century developments, they can be seen as quite heterogeneous in nature and it is possible that a satisfactory history of them has yet to be written. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's work played as might be expected a considerable part in the 18th-century developments and controversies, but it can often be rather difficult to characterize some of the participants in the debates as either Newtonians or opposed, since they would typically accept certain aspects of Newton's work while adopting approaches otherwise opposed to his. There are more than traces of Cartesian abstract thinking especially in subjects hard to reduce to experiments and reliable mathematics: on a Cartesian pattern of thought, acceptability of ideas could depend at least in part on how far they were 'clearly conceivable' or on the other hand 'obscure'.  This is arguably an opposite to Newton's severe requirement to &quot;argue from Phænomena&quot; and to subject everything to experiment or observation even where the results or implications were indeed difficult to conceive. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example may be seen in d'Alembert: although he carried out substantial investigations that relied on Newton as starting-point, he also wrote a &lt;a href=&quot;https://books.google.com/books?id=5R4OAAAAQAAJ&quot; rel=&quot;noreferrer&quot;&gt;Traité de Dynamique (1743)&lt;/a&gt; that tried to account for dynamics without the conceptual difficulty of a concept of force.  There is a not dissimilar tension in some of Euler's early work in mathematical physics. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-04-14T22:24:58.757" LastActivityDate="2019-04-14T22:24:58.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8485" PostTypeId="1" CreationDate="2019-04-15T22:37:50.867" Score="1" ViewCount="31" Body="&lt;p&gt;I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the common set of parameters in use to commonly describe probability distributions and if there is any clear historical path that brought to their definition and use.&lt;/p&gt;&#xA;" OwnerUserId="9828" LastActivityDate="2019-04-15T22:37:50.867" Title="Introduction of shape parameters in the formulation of probability distribution" Tags="|mathematics|mathematical-physics|statistics|probability|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8486" PostTypeId="1" CreationDate="2019-04-16T04:22:19.680" Score="1" ViewCount="123" Body="&lt;p&gt;Answer(s) to the question &lt;a href=&quot;https://physics.stackexchange.com/q/472415/83380&quot;&gt;What is a 3rd-order Fresnel lens?&lt;/a&gt; are disappointing to me, in that the term &lt;em&gt;3rd order&lt;/em&gt; does not refer to anything like a third-order series expansion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But this leads me to a new question: When did the use of an expression like &quot;third order approximation&quot; come into use in naming such a thing as a third-order series expansion?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And if it can be narrowed down sufficiently, is there any idea of who it was who first used it this way?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2019-05-10T09:17:47.947" LastActivityDate="2019-05-10T09:17:47.947" Title="When did the term &quot;order&quot; come into use as the highest exponent in an expansion?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="8487" PostTypeId="1" CreationDate="2019-04-16T05:25:49.477" Score="2" ViewCount="111" Body="&lt;p&gt;Were famous or popular physicists like Galileo, Newton, Einstein, Feynman predominantly mathematicians or scientists (computing, experimenting, engineering, etc.)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am curious if people like the ones mentioned above spent much time on things that are more &quot;purely&quot; mathematical like proofs, or did they use math more as a means to an end and mostly compute number values based on experimentally-gathered inputs?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, there are stories about Feynman being good at computing and integrating or Newton &quot;inventing&quot; calculus, but I haven't heard anything about them proving anything in the way the Greeks did, or more recently Fermat, Euler, Gauss. Also, the university I attended only requires up through calculus and linear algebra for a BS in physics.&lt;/p&gt;&#xA;" OwnerUserId="9831" LastActivityDate="2019-04-16T05:25:49.477" Title="Were notable physicists great at math or computing?" Tags="|mathematics|physics|biographical-details|" AnswerCount="0" CommentCount="6" ClosedDate="2019-04-17T06:49:50.140" ContentLicense="CC BY-SA 4.0" />
  <row Id="8488" PostTypeId="1" AcceptedAnswerId="8489" CreationDate="2019-04-16T23:23:49.827" Score="13" ViewCount="6296" Body="&lt;p&gt;&lt;a href=&quot;https://www.britannica.com/science/vector-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Encyclopedia Britannica&lt;/a&gt; says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In their modern form, vectors appeared late in the 19th century when Josiah Willard Gibbs and Oliver Heaviside (...) independently developed vector analysis to express the new laws of electromagnetism discovered by ... James Clerk Maxwell. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The phrase &quot;in their modern form&quot; suggests vectors were perhaps&#xA;used earlier in other forms. There seems to be no deep mathematical&#xA;idea needed for vectors to be &quot;invented&quot; and used much earlier.&#xA;Can anyone suggest some earlier uses of what we now call vectors?&#xA;E.g., did Newton use vectors? &lt;/p&gt;&#xA;" OwnerUserId="606" LastEditorUserId="606" LastEditDate="2019-04-18T10:34:39.950" LastActivityDate="2019-04-18T10:34:39.950" Title="When were vectors invented?" Tags="|mathematics|mathematical-physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8489" PostTypeId="2" ParentId="8488" CreationDate="2019-04-17T00:01:43.647" Score="22" Body="&lt;p&gt;This question was actually discussed on this site several times, for example here:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/773/when-was-the-vector-notation-in-physics-and-other-sciences-first-introduced&quot;&gt;When was the vector notation in physics and other sciences first introduced?&lt;/a&gt;&#xA;It indeed looks strange to modern people that this simple idea came so late.&#xA;Maxwell never uses vectors in his Treatise on electricity and magnetism,&#xA;which makes his notation somewhat clumsy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact the idea had a predecessor: quaternions. Yes, quaternions were invented before vectors:-) And Newton did not use vectors in the explicit form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead people thought in very &quot;roundabout&quot;, complicated ways about subjects which we routinely treat with vectors nowadays. A striking example is the famous theorem of Apollonius, about &quot;equivalence of excentric and epicycle&quot;. Motion on epicycle means that a point moves on a circle around the center (E), while the planet moves on another circle (of smaller radius) around this point. Excentric means that the planet moves on a circle of large radius whose center is different from E, and this center rotates about E (on a circle of small radius). In complex notation, Apollonius theorem says that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Re^{it(\alpha+\alpha_0)}+re^{it(\beta+\beta_0)}=re^{it(\beta+\beta_0)}+Re^{it(\alpha+\alpha_0)}.$$&lt;/span&gt;&#xA;To us it seems completely evident that these two things are the same. If you look into Apollonius' proof of this, you see that he really proves commutativity of vector addition in the plane. And Ptolemy praises Apollonius for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This frequently happens with simple ideas: after they spread, people start thinking of them as &quot;evident&quot; and wonder how could it happen that their predecessors did not see such simple things. Same thing happened with introduction of matrices to write systems of linear differential equations. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-17T11:41:52.380" LastActivityDate="2019-04-17T11:41:52.380" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="8490" PostTypeId="2" ParentId="8424" CreationDate="2019-04-17T02:31:08.643" Score="6" Body="&lt;p&gt;Not necessarily ignored, but A. Robinson's &lt;a href=&quot;https://en.wikipedia.org/wiki/Non-standard_analysis&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;non standard analysis&lt;/em&gt;&lt;/a&gt; wasn't much well-received either, most probably because he relied on metamathematical machinery to set it up&lt;/p&gt;&#xA;" OwnerDisplayName="user9833" LastActivityDate="2019-04-17T02:31:08.643" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8491" PostTypeId="1" CreationDate="2019-04-17T05:18:27.667" Score="4" ViewCount="158" Body="&lt;p&gt;I found an interesting &lt;a href=&quot;https://math.stackexchange.com/q/1022281/279515&quot;&gt;question&lt;/a&gt; on Math SE asked by &lt;a href=&quot;https://stackexchange.com/users/510255/kconrad&quot;&gt;@KCd&lt;/a&gt;, but it is over four years old without a clear answer. Since it seems to be more on topic here than on Math SE, I thought to post it here in case it receives good answers.&lt;/p&gt;&#xA;&lt;p&gt;(I commented on the post a few months ago asking the OP if they would be interested in posting the question here, but since I did not receive any reply I'm making my own post; I hope there is no objection to this, but if so please let me know.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This question is being asked on behalf of a graduate student in my department. When and where did the tradition start of a seminar or colloquium speaker using just the first initial of the speaker's last name (or initials for the speaker's first and last names) when stating a theorem due to the speaker? Attributions of results due to anyone else, including joint theorems with the speaker, are usually indicated with the other people's full last name.&lt;/p&gt;&#xA;&lt;p&gt;Anything like a photograph of Hilbert giving a talk with &amp;quot;Satz (H.)&amp;quot; at the start of a theorem would be great to see if it exists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3224" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-25T20:51:38.400" Title="The convention for speakers to refer to themselves at the board with a single initial" Tags="|mathematics|mathematicians|notation|naming-conventions|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="8492" PostTypeId="1" CreationDate="2019-04-17T10:19:41.470" Score="2" ViewCount="63" Body="&lt;p&gt;I read that Max Weber supported the anti-positivist approach to sociology in Ludwig Maxmilian University of Munich. So I want to know the academic structure of the first course he appealed for. I doubt if it is possible to answer this but I believe some experienced men will give an answer  as much it is possible to give. I want to know this because I have to start myself in Sociology and I don't want to do it in wrong way.&lt;/p&gt;&#xA;" OwnerUserId="9837" LastActivityDate="2019-04-17T10:19:41.470" Title="What was the structure of first course in sociology?" Tags="|academia|social-sciences|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8493" PostTypeId="2" ParentId="8488" CreationDate="2019-04-17T12:53:40.343" Score="7" Body="&lt;p&gt;On the &lt;strong&gt;non&lt;/strong&gt; modern form of &amp;quot;vector&amp;quot;...  from &lt;a href=&quot;http://jeff560.tripod.com/v.html&quot; rel=&quot;noreferrer&quot;&gt;mathword&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The word VECTOR (which, like the word vehicle, derives ultimately from the Latin vehĕre to carry) was first a technical term in astronomical geometry. The OED’s earliest entry is from a technical dictionary of 1704: J. Harris Lexicon Technicum I. s.v., &amp;quot;A Line supposed to be drawn from any Planet moving round a Center, or the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the New Astronomy, called the Vector; because 'tis that Line by which the Planet seems to be carried round its Center.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Vector usually appeared in the phrase radius vector. The French term was rayon vecteur and can be found in e.g. Laplace’s Traité de mécanique céleste (1799-1825).&lt;/p&gt;&#xA;&lt;p&gt;The term rayon vecteur is used in a non-astronomical context by Monge &amp;quot;Application de l'Analyse à la Géométrie&amp;quot; (1807). On p. 24 he writes, &amp;quot;on nomme la droite r le RAYON VECTEUR du point, et l'origine des  coordonnées devient un pôle, d'où partent les rayons vecteurs des différens points de l'espace.&amp;quot; Monge does not seem to use the terminology later in the text, however. Cauchy in his Leçons sur les Applications du Calcul Infinitésimal à la Géométrie of 1826 uses the term freely after introducing it in the initial &amp;quot;Preliminaries&amp;quot; chapter:  &amp;quot;Une droite AB, menée d'un point A supposé fixe à un point B suppose mobile, sera généralement désignée sous le nom de rayon vecteur.&amp;quot; (p.14) (Information from François Ziegler.) Radius vector appears n English in 1831 in Elements of the Integral Calculus (1839) by J. R. Young: &amp;quot;...when the angle Ω between the radius vector and fixed axis is taken for the independent variable, the formula is....&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Hamilton would create a new meaning for vector (see VECTOR &amp;amp; SCALAR) but he used radius vector in the conventional way in On a General Method in Dynamics Philosophical Transactions Royal Society (part II for 1834, pp. 247-308); see article 14.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-17T12:53:40.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8496" PostTypeId="2" ParentId="6358" CreationDate="2019-04-17T17:11:55.547" Score="23" Body="&lt;p&gt;While it certainly does not confirm (nor refutes) Grothendieck's episode, the fact that Hermann Weyl did commit this mistake might add some 'substance' to it&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The notion of prime number is of course as old and as primitive as that&#xA;  of the multiplication of natural numbers. Hence it is most surprising to find the&#xA;  distribution of primes among all natural numbers is of such a highly irregular&#xA;  and almost mysterious character. While on the whole the prime numbers&#xA;  thin out the further one gets in the sequence of numbers, wide gaps are always followed again by clusters. An old conjecture of Goldbach's maintains&#xA;  that there even come along again and again pairs of primes of the smallest&#xA;  possible difference 2, &lt;strong&gt;like 57&lt;/strong&gt; and 59.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in Weyl, H. (1951). A half-century of mathematics. The American mathematical monthly, 58(8), 523-553.&lt;/p&gt;&#xA;" OwnerDisplayName="user9833" LastActivityDate="2019-04-17T17:11:55.547" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="8497" PostTypeId="1" CreationDate="2019-04-18T07:07:52.790" Score="1" ViewCount="83" Body="&lt;p&gt;And if possible books that could be downloaded for free&lt;/p&gt;&#xA;" OwnerUserId="9840" LastActivityDate="2019-04-18T07:07:52.790" Title="What are some good books on history of mathematical thought?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ClosedDate="2019-04-19T07:36:40.750" ContentLicense="CC BY-SA 4.0" />
  <row Id="8498" PostTypeId="2" ParentId="5836" CreationDate="2019-04-18T14:16:11.330" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;CHEVALLEY depicts HERBRAND as an adventurous, passionate, and often perfectionistic personality who was not only interested in mathematics, but also in poetry and sports. In particular, he seemed to have liked extreme sportive challenges: mountaineering, hiking, and long distance swimming. His interest in philosophical issues and foundational problems of science was developed well beyond his age.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Wirth, C. P., Siekmann, J., Benzmüller, C., &amp;amp; Autexier, S. (2009). Lectures on Jacques Herbrand as a logician. arXiv preprint &lt;a href=&quot;https://arxiv.org/pdf/0902.4682.pdf&quot; rel=&quot;noreferrer&quot;&gt;arXiv:0902.4682.&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerDisplayName="user9833" LastActivityDate="2019-04-18T14:16:11.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8499" PostTypeId="2" ParentId="6362" CreationDate="2019-04-18T22:12:18.377" Score="2" Body="&lt;p&gt;I finally found a reliable source on Gauss's unpublished fragments that actually confirms that this drawing of the hyperbolic tessellation of the unit disk is indeed by Gauss himself: this link - &lt;a href=&quot;https://kalliope-verbund.info/de/search.html?q=Cereri+Palladi+Junoni+sacrum%2C+Febr.+1805&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://kalliope-verbund.info/de/search.html?q=Cereri+Palladi+Junoni+sacrum%2C+Febr.+1805&lt;/a&gt; - gives the notebook in which Gauss drawed this figure. This is a notebook entitled  &amp;quot;&lt;em&gt;&lt;strong&gt;Cereri Palladi Junoni Sacrum&lt;/strong&gt;&lt;/em&gt;&amp;quot;, which Gauss started writing at February 1805, and according to the list of contents given there, Gauss's notes on modular functions are on p. 4-8 of it and include the same figures that are referenced in p.103-105 of volume 8 of the digital version of Gauss's Werke.&lt;/p&gt;&#xA;&lt;p&gt;This website actually gives a list of all of Gauss's notebooks, including much material that is not published in any place of the 12 volumes of Gauss's Werke (maybe I'd get the answer to many of my questions if only these notebooks would be published...). For example, the famous note in which Gauss started the study of braids is on p.283-284 of &amp;quot;Handbuch 7&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As for the non-euclidean implications of Gauss's drawing - my mathoverflow post &lt;a href=&quot;https://mathoverflow.net/questions/370190/what-is-the-representation-of-the-generators-of-the-triangle-group-for-the-unifo&quot;&gt;https://mathoverflow.net/questions/370190/what-is-the-representation-of-the-generators-of-the-triangle-group-for-the-unifo&lt;/a&gt; shows that the formulas stated by Gauss on this tesselation can be derived, apart from one step (which involves the use of Cayley-Klein metric for the Poincaré disk model), by looking at the figure through euclidean eyes (using the concept of inversion in circle). Therefore, perhaps the use of hyperbolic geometry is not necessary here.&lt;/p&gt;&#xA;&lt;p&gt;It's quite possible (even likely) that Gauss saw the connection between his figure and hyperbolic geometry, but since this work was done in analytic context (this fragment contains &amp;quot;notes on modular functions&amp;quot;), one cannot know for sure.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10608" LastEditDate="2024-03-11T10:19:14.577" LastActivityDate="2024-03-11T10:19:14.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8500" PostTypeId="1" CreationDate="2019-04-19T03:12:46.237" Score="1" ViewCount="96" Body="&lt;p&gt;There seems to be a contradiction in the assumptions that Copernicus makes when attempting to explain the motion of the planets:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://dbanach.com/copernicus-commentarilous.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://dbanach.com/copernicus-commentarilous.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Assumptions 1 and 3 both refer to 'celestial spheres', but 1 seems to say that these spheres are not concentric, while 3 says that they are (with their centre at the Sun).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is he referring to the same spheres in each assumption?&lt;/p&gt;&#xA;" OwnerUserId="8141" LastActivityDate="2019-04-27T10:26:18.917" Title="Apparent contradiction in Copernicus' Commentariolus" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8501" PostTypeId="2" ParentId="8500" CreationDate="2019-04-19T06:57:04.310" Score="1" Body="&lt;p&gt;&lt;em&gt;Assumption 1&lt;/em&gt; seems nothing more than a vague distinction of the centers of the lunar and planetary orbits, clarified in the two following postulates.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Assumption 2&lt;/em&gt; : the center of the lunar sphere (and of gravity) is the center of the earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Assumption 3&lt;/em&gt; : all the (planetary ones, included that of the earth) spheres revolve around the sun.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The core doctrine expressed by the above postulates is that the &quot;centre of the universe&quot; (i.e. of the celestial bodies) does not coincide with the &quot;centre of gravity&quot; of terrestial bodies.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2019-04-27T10:26:18.917" LastActivityDate="2019-04-27T10:26:18.917" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8502" PostTypeId="2" ParentId="5836" CreationDate="2019-04-19T09:45:55.903" Score="3" Body="&lt;p&gt;Alan Turing was a very good &lt;a href=&quot;https://therunnereclectic.com/2014/12/01/alan-turing-the-runner/&quot; rel=&quot;nofollow noreferrer&quot;&gt;long&lt;/a&gt; &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Extras/Turing_running.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;distance runner&lt;/a&gt;. No one, even someone with the already considerable natural talent required to run that fast, runs a 2:45 marathon without training a lot. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;While an example like this does nothing to support or debunk a theory such as &quot;physical exercise is in theory linked to better brain function&quot; it could be the case, as is suggested in the links, that Turing thought something along those lines.&lt;/p&gt;&#xA;" OwnerUserId="633" LastActivityDate="2019-04-19T09:45:55.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8503" PostTypeId="1" CreationDate="2019-04-19T23:55:38.153" Score="2" ViewCount="195" Body="&lt;p&gt;First and foremost, I am aware that a similar question has been asked &lt;a href=&quot;https://hsm.stackexchange.com/questions/1881/discovery-and-development-of-coordinate-systems&quot;&gt;here&lt;/a&gt; and has been touched upon elsewhere. I have found these discussions very compelling but a bit light on external reference, and so wanted to approach the topic on different terms and in one place.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;As such, I was wondering if anyone might have recommendations for material on the history of constructions of 'mathematical spaces.'&lt;/strong&gt; By this, I do not mean a 'mathematical space' in perhaps the most restrictive formal sense, i.e., a set with additional structure. I have a more general interest in the history which pertains to the placement of objects—real numbers, shapes, etc.—in relation to one another. I realize, to some of you this may sound the same; I simply mean to say that I wish to approach this topic rather permissively. In this vein, if anyone sees it fit to bring up material on the history of coordinate systems, geometry, rings, groups, lattices or whatever else, I would more than welcome it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've seen this question approached as a topological one—which I suppose it must be to some extent. However, to be clear, if the question were approached as such, I would not be so much interested in the &lt;a href=&quot;https://hsm.stackexchange.com/questions/3136/history-of-covering-spaces&quot;&gt;history of topology&lt;/a&gt; as the history of the various spaces which topology has studied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, I'm very interested to learn of alternative spaces which may have developed in antiquity, so material which deals with the more distant past is very much appreciated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an aside: I am, at least in part, inspired by the Euler Spiral, Ulam Spiral and other kinds of 'spiral space' which have received some, shall I say, more recent attention. That said, I am curious to learn of even more 'out there' ways of constructing mathematical space.&lt;/p&gt;&#xA;" OwnerUserId="9849" LastEditorUserId="4251" LastEditDate="2020-07-12T09:43:04.307" LastActivityDate="2020-07-12T09:43:04.307" Title="Material on the History of Mathematical Spaces" Tags="|mathematics|topology|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="8504" PostTypeId="1" CreationDate="2019-04-20T00:22:13.557" Score="1" ViewCount="226" Body="&lt;p&gt;Kenneth Bainbridge was an early pioneer of mass spectroscopy. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Kenneth_Bainbridge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article about him&lt;/a&gt; says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He used this instrument to verify Albert Einstein's mass-energy equivalence, E = mc2&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with a footnote to &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.44.123.2&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paywalled paper&lt;/a&gt; from 1933. (Thanks, Physical Review, for making sure that I can't read a paper from almost a century ago.) The paper does not seem to be available from SciHub. WP also gives a quote from a 1933 book by Aston, which says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By establishing accurate comparisons of the masses of the light particles concerned in nuclear disintegrations, particularly that of 7Li, discovered by Cockcroft and Walton, he achieved a noteworthy triumph in the experimental proof of the fundamental theory of Einstein of the equivalence of mass and energy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A 1997 obituary in Physics Today says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] he continued to use his mass spectrographs to measure isotopic masses and to compare nuclear mass differences with the energies measured in transitions as a check on Einstein's mass-energy equivalence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Can anyone explain what measurement he actually did? 7Li is a stable element. One quote refers to &quot;transitions,&quot; the other to &quot;nuclear disintegrations.&quot; Would this be a beta decay from 7Be (half-life of 50 days) to 7Li? Or maybe disintegration of 7Li by neutrons or something?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2019-04-20T03:58:55.347" Title="Bainbridge's test of mass-energy equivalence" Tags="|experimental-physics|relativity-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="8505" PostTypeId="2" ParentId="8504" CreationDate="2019-04-20T01:53:56.810" Score="5" Body="&lt;p&gt;I actually managed to get it from SH&lt;a href=&quot;https://i.stack.imgur.com/EG2yh.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EG2yh.png&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user9833" LastActivityDate="2019-04-20T01:53:56.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8506" PostTypeId="2" ParentId="8504" CreationDate="2019-04-20T03:28:35.890" Score="3" Body="&lt;p&gt;Here is a summary of what I was able to figure out from various sources, including the copy of Bainbridge's paper that a.c. bruno uploaded.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the Cavendish laboratory, Cockroft and Walton built a kind of electrostatic generator called a Cockroft-Walton generator, which was capable of creating DC voltages of up to 280 kV. They used this to accelerate beams of protons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;They read a paper by Gamow predicting quantum-mechanical tunneling, and realized that their proton beams ought to be able to fuse with lithium and beryllium, even though the energies were below the Coulomb barrier. Natural lithium is 92% 7Li.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1932, they observed a reaction which they interpreted as &lt;span class=&quot;math-container&quot;&gt;$\text{p}+{}^7\text{Li}\rightarrow2\alpha+17.0\ \text{MeV}$&lt;/span&gt;, for the following reasons. They had seen neutrons from their reactions, and had also hoped to see gammas but didn't. But in this experiment, they knew these were alphas because they stopped in air. Because the beam is much lighter than the target, the lab frame approximately coincides with the center of mass frame, in which the two alphas have equal energy due to conservation of energy and momentum. The range of the alphas in air was 8.4 cm, which determined their energies to be 8.6 MeV each, so the energy released in the reaction was roughly double this amount (but not quite the same due to the beam's kinetic energy and maybe also the fact that the c.m. frame doesn't quite coincide with the lab frame?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was a big deal because they had discovered induced nuclear disintegration, or &quot;splitting the atom.&quot; (Apparently it was not as much of a conceptual exploit to use a proton to knock off a neutron.) They got the Nobel Prize in 1951 for this work.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a bonus, Bainbridge was able to use this as a test of mass-energy equivalence. The energy of 17.0 MeV was equivalent to 0.0182 amu. Aston and Bainbridge had measured the atomic masses involved using sector mass spectrometers, so they could compute that the change in mass was 0.0181(6) amu. This verified &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2019-04-20T03:58:55.347" LastActivityDate="2019-04-20T03:58:55.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8507" PostTypeId="2" ParentId="7751" CreationDate="2019-04-20T14:35:55.340" Score="2" Body="&lt;p&gt;Here is a source from psu.edu in which Marc P quoted Einstein&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problems that exist in the world today cannot be solved by the level of thinking that created them. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.186.4598&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.186.4598&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9855" LastActivityDate="2019-04-20T14:35:55.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8508" PostTypeId="2" ParentId="5836" CreationDate="2019-04-20T18:44:09.490" Score="6" Body="&lt;p&gt;Sophus Lie (1842-1899) often went on  long,  physically demanding hikes. Sometimes he got himself into trouble because of this. In August 1870, after the outbreak of the Franco-Prussian war, he attempted to travel on foot from Paris all the way to Milan, to visit Luigi Cremona and enjoy the scenery of French and  Swiss Alps along the way. Unfortunately, he travelled with an expired Norwegian passport, so in Fontainbleau (about 50 kilometers south of Paris) he was taken for a German spy and imprisoned. Of course being in possession of strange-looking notes with some German names here and there did not help him.  At the intervention of Gaston Darboux he was released after a month and continued his itinerary on a train. But this incidents did not deter him from hiking. Two years  later, when courting his future wife in Norway, he visited her a few times on foot, going about 35-40 kilometers each way. In a letter to her he wrote:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;``I have used my Legs in a decent manner during these Days (...). People think that I have very strange Tastes; but this strengthens the Muscles and Nerves, and besides, I find myself with such good feelings, as I roam the Countryside as a Hobo. By habit, I am always deep in my own Thoughts, and building Castles in the Air for the Future: and You quite understand who it is that plays the Main Role...&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(citation after Arild Stubhaug, &#xA;The Mathematician Sophus Lie.&#xA;It was the Audacity of My Thinking. Springer 2002&#xA;&lt;a href=&quot;https://www.springer.com/us/book/9783540421375&quot; rel=&quot;noreferrer&quot;&gt;https://www.springer.com/us/book/9783540421375&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was trying to pass on his passion for outdoors to his children. His daughter recalled that the family went on on a long tour every Sunday. The children were taught skating and had skis imported from Norway, which were still a novelty (and created a sensation) in the German city of Leipzig, where Lie was appointed a professor of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2019-04-20T18:44:09.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8509" PostTypeId="1" AcceptedAnswerId="8511" CreationDate="2019-04-21T17:38:01.587" Score="6" ViewCount="778" Body="&lt;p&gt;I read the following in C. S. Lewis, &lt;em&gt;Miracles&lt;/em&gt; (page 77-8)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The immensity of the universe is not a recent discovery. More than seventeen hundred years ago Ptolemy taught that in relation to the distance of the fixed stars the whole Earth must be regarded as a point with no magnitude.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There was no reference given, but I assume this would be in the &lt;em&gt;&lt;a href=&quot;https://archive.org/details/PtolemysAlmagestPtolemyClaudiusToomerG.5114_201810/page/n669&quot; rel=&quot;noreferrer&quot;&gt;Almagest&lt;/a&gt;&lt;/em&gt;. However, I don't know where in that work (or elsewhere) Ptolemy discussed this. I want to read his explanation for that view.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hence the question: &lt;em&gt;Where did Ptolemy compare the Earth to the distance of fixed stars?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="9677" LastEditorUserId="4251" LastEditDate="2021-09-10T06:40:30.110" LastActivityDate="2021-09-10T06:40:30.110" Title="Where did Ptolemy compare the Earth to the distance of fixed stars?" Tags="|physics|astronomy|roman-empire|ptolemy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8510" PostTypeId="2" ParentId="8503" CreationDate="2019-04-21T17:46:23.213" Score="0" Body="&lt;p&gt;&lt;strong&gt;I am going to keep a running list of suggested material here:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/1881/discovery-and-development-of-coordinate-systems&quot;&gt;&quot;Discovery and Development of Coordinate Systems&quot;&lt;/a&gt;, Stack Exchange&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://books.google.it/books?id=2T4i5fXZbOYC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;History of Analytic Geometry&lt;/em&gt;&lt;/a&gt;, Carl Boyer&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/2378/when-do-we-see-for-the-first-time-the-use-of-the-cartesian-coordinates/2382#2382&quot;&gt;&quot;When do we see for the first time the use of the Cartesian&#xA;  coordinates?&quot;&lt;/a&gt;, Stack Exchange&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?referer=&amp;amp;httpsredir=1&amp;amp;article=1226&amp;amp;context=jhm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Abscissas and Ordinates&quot;&lt;/a&gt;, David Pierce&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;http://www.dam.brown.edu/people/mumford/beyond/coursenotes/2006PartIb.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Oresme and the Invention of Graphing&quot;&lt;/a&gt;, Mumford&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/225188?seq=1#page_scan_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Outline of a History of Differential Geometry: I&quot;&lt;/a&gt;, D.J. Struik&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1859893/origins-of-differential-geometry-and-the-notion-of-manifold&quot;&gt;&quot;Origins of Differential Geometry and the Notion of Manifold&quot;&lt;/a&gt;,&#xA;  Stack Exchange&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=R83CAgAAQBAJ&amp;amp;printsec=frontcover&amp;amp;dq=concepts%20of%20space&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwip4dqo3uHhAhXVop4KHaKNDKUQ6AEIKjAA#v=onepage&amp;amp;q=concepts%20of%20space&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Concepts of Space&lt;/em&gt;&lt;/a&gt;, Max Jammer&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9849" LastActivityDate="2019-04-21T17:46:23.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8511" PostTypeId="2" ParentId="8509" CreationDate="2019-04-21T19:17:24.477" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k64767c/f100.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;Almagest, Book 1, chap. 5 &lt;/a&gt; contains what you are looking for.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="26" LastEditDate="2019-04-22T21:52:41.347" LastActivityDate="2019-04-22T21:52:41.347" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8512" PostTypeId="2" ParentId="8509" CreationDate="2019-04-21T21:39:07.547" Score="8" Body="&lt;p&gt;Ptolemy's argument can be simply explained in plain English. If the distance to stars&#xA;was comparable to the size of the Earth, the stars would experience a diurnal parallax, that is the visible relative positions of the stars in the sky would change during one night. But this is not observed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, a similar argument will imply that if the Earth is rotating about the Sun&#xA;(or moves in some other way with respect to stars), the distance to the stars must be much larger than the size of this motion. In particular the diameter of the orbit of the Earth around the Sun is negligible in comparison to the distance to the stars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the most serious argument against the Heliocentric system, and it is also&#xA;due to Ptolemy. Until the acceptance of Copernicus theory, people just could not believe that the distance to the stars can be so large. Why would God create such an enormous Universe only to populate a tiny piece of it with humans?&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-22T12:41:27.447" LastActivityDate="2019-04-22T12:41:27.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="8513" PostTypeId="2" ParentId="8503" CreationDate="2019-04-22T03:21:37.580" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Suppes, P. (1973). Some open problems in the philosophy of space and time. In Space, Time and Geometry (pp. 383-401). Springer, Dordrecht.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The development of the concept of space was an evolutionary process in early Greek history. Four major periods have been identified (historically and philosophically) by shifts in the way that the Greek Philosophers as a whole looked at the role of space in physical reality. Within each of these periods, the Mythopoeic, the Material, the Non-Definite and the Definite, those philosophers who made contributions to the overall development of the concept of space are discussed. The major themes discussed throughout this essay deal with defining and refining the concept of space and differentiating space from the concepts of time and matter, while keeping track of the influence of the concept of mathematical as opposed to physical spaces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Beichler, J. E. The Ancient Greek Concept of Space.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although the concept of space is of fundamental importance in both physics and philosophy, until the publication of this book, the idea of space had never been treated in terms of its historical development. It remained for Dr. Jammer, noted scholar and historian of science, to trace the evolution of the idea of space in this comprehensive, thought-provoking study. The focus of the book is on physical, rather than metaphysical, ideas of space; however, philosophical or theological speculations are discussed when relevant. The author has also given special attention to the cultural settings in which the theories developed. Following a Foreword by Albert Einstein and an introductory chapter on the concept of space in antiquity, subsequent chapters consider Judaeo-Christian ideas about space, the emancipation of the space concept from Aristotelianism, Newton's concept of absolute space and the concept of space from the eighteenth century to the present. [...] An abundance of meticulously documented quotations from original sources and numerous bibliographic references make this an exceptionally well-documented book.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Jammer, M. (1954). Concepts of Space: The History of Theories of Space in Physics. Courier Corporation.&lt;/p&gt;&#xA;" OwnerDisplayName="user9833" LastActivityDate="2019-04-22T03:21:37.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8514" PostTypeId="2" ParentId="8043" CreationDate="2019-04-22T04:13:26.293" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Perelman's early work was impressive enough to garner several job offers from US universities. But he turned them down, returning to Russia and a research position with the Steklov Institute of Mathematics in St Petersburg. At that point, Perelman effectively disappeared — he stopped publishing papers or discussing his research with colleagues. “We would occasionally ask where he was,” says a friend. “No one seemed to know what he was doing.” Even people at the Steklov Institute didn't know what he was working on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nature.com/articles/427388a&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nature.com/articles/427388a&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the early 2000s he [Mochizuki] stopped venturing to international meetings, and colleagues say that he rarely leaves the Kyoto prefecture any more. “It requires a special kind of devotion to be able to focus over a period of many years without having collaborators,” says number theorist Brian Conrad of Stanford University in California.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nature.com/news/monumental-proof-to-torment-mathematicians-for-years-to-come-1.20342&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nature.com/news/monumental-proof-to-torment-mathematicians-for-years-to-come-1.20342&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user9833" LastActivityDate="2019-04-22T04:13:26.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8515" PostTypeId="1" AcceptedAnswerId="8516" CreationDate="2019-04-22T06:11:25.930" Score="3" ViewCount="173" Body="&lt;p&gt;I have question regarding the history of the idea of founding mathematics (specially arithmetic) on a logical basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'm interested in knowing is, at what point historically people started to make claims about the properties of the natural numbers let's say, like: commutativity of addition and multiplication (not necessarily general claims) ? And at what point people started to feel that those properties needed&#xA;justification ?&lt;/p&gt;&#xA;" OwnerUserId="9863" LastEditorUserId="55" LastEditDate="2019-04-22T09:01:46.767" LastActivityDate="2019-04-22T09:07:55.190" Title="When did people start to state and justify properties of arithmetical operations?" Tags="|ancient-greece|mathematical-logic|" AnswerCount="1" CommentCount="13" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8516" PostTypeId="2" ParentId="8515" CreationDate="2019-04-22T08:59:40.257" Score="1" Body="&lt;p&gt;At the operational level, the use of properties of addition and multiplication can be traced to Euclid's Elements, and more algebraically, to Diophantus's Arithmetica. They are used by al-Khwarizmi, various medieval treatises on arithmetic, and, eventually symbolized by Vieta and Descartes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the conceptualizing of operations as such, and the study of their properties, only begins in the early 19-th century, when alternative &quot;arithmetics&quot; gain prominence. According to &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;, Servois's  memoir published in Annales de Gergonne (volume V, no. IV, October 1, 1814) mentions commutative and distributive laws. In England, &quot;commutative law&quot; appears in Gregory's 1841 textbook. Hamilton in &quot;On a new Species of Imaginary Quantities connected with a theory of Quaternions&quot; Royal Irish Academy, Proceedings, Nov. 13, 1843, vol. 2, 424-434 writes &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called the associative character of the operation...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In The Mathematical Analysis of Logic (1847) Boole formalized symbolic logic, with inspiration for properties of logical operations explicitly taken from those of arithmetic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for justification, 18th century algebraists were content with Leibniz's vague &quot;generality of algebra&quot; used to transplant idenities from known to new (say, complex) numbers, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2038/55&quot;&gt;Was 18th century algebra more symbolic/formal than the modern conception?&lt;/a&gt; Peacock in &lt;a href=&quot;https://archive.org/details/bub_gb_Ap54gR4l-y8C/page/n4&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Treatise on Algebra (1830)&lt;/a&gt; renamed it into the &quot;principle of permanence&quot;, which he attempted to use, as we would say, axiomatically:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Peacock attempted to put the theory of negative and complex numbers on a firm logical basis by dividing the field of algebra into arithmetical algebra and symbolic algebra. In the former the symbols represented positive integers; in the latter the domain of the symbols was extended by his principle of the permanence of equivalent forms. This principle asserts that rules in arithmetical algebra, which hold only when the values of the variables are restricted, remain valid when the restriction is removed.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;His approach was continued by Hankel in &lt;em&gt;Vorlesungen über die complexen Zahlen und ihre Functionen&lt;/em&gt; (1867), who also brought Grassman's 1844 work on vector algebras out of obscurity. With the proliferation of new algebraic systems in the 19th century (Boole's, Hamilton's, Grassman's etc.), and the new focus on rigor, the attention turned to rigorous construction of arithmetic, and arithmetization of analysis. Dedekind came up with the arithmetization program as early as 1858, and published Stetigkeit und Irrationale Zahlen in 1872. As far as reducing arithmetic to logic, that was a rather bold idea put forward and developed by Frege in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Die Grundlagen der Arithmetik (1884)&lt;/a&gt;. The modern axiomatization of arithmetic appears in Peano's Arithmetices Principia (1889), and is indebted to Grassman's ideas about deriving properties of arithmetic operations from induction on the successor operation, and to Peirce's (1881) and Dedekind's (1888) prior axiomatizations.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-04-22T09:07:55.190" LastActivityDate="2019-04-22T09:07:55.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8517" PostTypeId="1" CreationDate="2019-04-22T23:02:37.103" Score="3" ViewCount="93" Body="&lt;p&gt;There is no link on wikipedia to his work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is really a nice transform. There is coherent theory behind. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am curious what motivated him to invent this transform. &lt;/p&gt;&#xA;" OwnerUserId="1890" LastActivityDate="2019-04-23T02:18:52.383" Title="Mellin's original paper on his transform" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8518" PostTypeId="1" AcceptedAnswerId="8519" CreationDate="2019-04-23T00:11:15.137" Score="26" ViewCount="4280" Body="&lt;p&gt;It is well-known that in 1755 Lagrange was appointed Professor of Mathematics at the Royal Artillery School in Turin. He was 19. His work up until then involves correspondence with Euler. Was he appointed professor by Euler's recommendation? If not, how was his talent recognised so early? &lt;/p&gt;&#xA;" OwnerUserId="6867" LastEditorUserId="4251" LastEditDate="2023-03-31T16:33:14.103" LastActivityDate="2023-03-31T16:33:14.103" Title="How was Lagrange appointed professor of mathematics so early?" Tags="|mathematicians|18th-century|lagrange|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="8519" PostTypeId="2" ParentId="8518" CreationDate="2019-04-23T01:29:07.863" Score="37" Body="&lt;p&gt;Lagrange enrolled to the university at the age of 14, to study law. But he quickly switched to mathematics. According to his biography, his mother was surprised when the French ambassador presented himself to congratulate her and give her the prize of the Paris Academy which Lagrange won in a public competition (by correspondence).&#xA;(His mother was surprised because she assumed that her son was studying law).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In few years he probably made his name known in scientific circles. At that time&#xA;mathematical results were spread mostly by correspondence, there was no regular, established journals yet. Young Lagrange corresponded not only with Euler but also with Count Fagnano, (probably the most famous Italian mathematician of that time) and several other mathematicians. His first published paper appeared only in 1759, in a journal which he and his friends, who were actually his students, published themselves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His correspondence with Euler started in 1754, and he was appointed Professor in 1755.&#xA;The biography does not mention any interference or even recommendation of Euler, but mentions that Euler tried to arrange for him a position in Berlin in 1756. (Euler only succeeded in making him a corresponding member.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lagrange was not unique as a prodigy. For comparison, Alexis Clairaut wrote his first research paper at the age of 12. At the age of 13 he was invited to address the Paris Academy, and became a member of this academy at 18.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;William Rowan Hamilton was appointed Professor of Astronomy shortly before his graduation from college, at the age of 22. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We may conclude that in 18th and early 19th century, the procedures were much less formal than they are nowadays: the number of mathematicians and the number of positions for them were very small, so appointments were decided on individual merits, rather than following rigid rules, as is the case nowadays. As I understand it, the Artillery Academy simply needed a good mathematician, and there were not many around. So they asked other mathematicians, and Lagrange was recommended.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref. I. A. Tiulina, Joseph-Louis Lagrange, &quot;Nauka&quot; Moscow, 1977 (in Russian). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-12T12:46:16.213" LastActivityDate="2020-05-12T12:46:16.213" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="8520" PostTypeId="2" ParentId="8517" CreationDate="2019-04-23T02:12:07.153" Score="2" Body="&lt;p&gt;According to &lt;a href=&quot;http://www-history.mcs.st-andrews.ac.uk/Biographies/Mellin.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He studied the transform which now bears his name and established its reciprocal properties. He applied this technique systematically in a long series of papers to the study of the gamma function, hypergeometric functions,... &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and it gives this &quot;long list of papers&quot;. I can't say much more because most of these papers are in Swedish. His first paper on Gamma function was published in 1883.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-23T02:18:52.383" LastActivityDate="2019-04-23T02:18:52.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9523" PostTypeId="2" ParentId="8491" CreationDate="2019-04-25T20:43:30.093" Score="3" Body="&lt;p&gt;The question probably cannot be precisely answered as stated, but there is an early example close to what you expect to see, at least in terms of time frame and the importance of the mathematician involved. It has to do with printed matter, though, not with talks, and refers to a notion, not a theorem. The notion is that of a Banach space. According to Jeff Miller, &lt;a href=&quot;http://jeff560.tripod.com/b.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://jeff560.tripod.com/b.html&lt;/a&gt;, the name originated with Maurice Fréchet (1878-1973), &#xA;who in  1928 in his Les Espaces Abstraits  wrote about &quot;les espaces de M. Banach.&quot; Yet in his own book Teoria operacji liniowych,  1931, and its French version Théorie des opérations linéaires, 1932, Banach calls such spaces &quot;the spaces of type (B)&quot;. Unfortunately the French text is not digitally available, but the Polish version can be seen here, on p. 96: &lt;a href=&quot;http://kielich.amu.edu.pl/Stefan_Banach/pdf/teoria-operacji-pol/05.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://kielich.amu.edu.pl/Stefan_Banach/pdf/teoria-operacji-pol/05.pdf&lt;/a&gt; &#xA; It is hard to guess if Banach is following the naming convention in question, because in the initial chapters of the book he also refers to (B)-measurability, but he links this notion with the name of Baire. There is also a chapter about &quot;spaces of type (F)&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="3434" LastEditDate="2019-04-25T20:51:38.400" LastActivityDate="2019-04-25T20:51:38.400" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9524" PostTypeId="1" AcceptedAnswerId="9528" CreationDate="2019-04-26T02:37:57.133" Score="1" ViewCount="579" Body="&lt;p&gt;In electrochemistry, all electrode potentials are quoted with reference to the standard hydrogen electrode. Its value is assigned to be 0 volts. I have been searching for the origin of this convention (it is nearly impossible to determine its absolute potential). I searched for a while using Google Scholar/ Google books from the 19th century, but it is quite elusive. Cant't find the inventor of the hydrogen electrode nor the origin of the convention of assigning zero volts to its half cell. The context is given below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Standard hydrogen electrode (abbreviated SHE), is a redox electrode which forms the basis of the thermodynamic scale of oxidation-reduction potentials. Its absolute electrode potential is estimated to be 4.44 ± 0.02 V at 25 °C, but to form a basis for comparison with all other electrode reactions, hydrogen's standard electrode potential (E0) is declared to be zero volts only at 298K. Potentials of any other electrodes are compared with that of the standard hydrogen electrode at the same temperature.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;https://en.wikipedia.org/wiki/Standard_hydrogen_electrode&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="16966" LastEditDate="2022-12-29T10:43:33.857" LastActivityDate="2022-12-29T10:43:33.857" Title="Hydrogen electrode and its electrode potential" Tags="|discoveries|experimental-physics|chemistry|electricity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9525" PostTypeId="1" AcceptedAnswerId="9527" CreationDate="2019-04-26T08:54:45.267" Score="4" ViewCount="86" Body="&lt;p&gt;Poincaré was one of the major precursors of the modern theory of dynamical systems, notably through his famous memoir on the 3 body problem, and subsequent discovery of homoclinic intersections and how they lead to chaos. That was in the late 19th - early 20th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, this part of his work was apparently more or less forgotten in the european and americal mathematical community, until Smale's discovery of horseshoes in the 60's (it seems the russian school of mathematics was more aware of his work at the time). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Parallel to this, Fatou and Julia developpped the basis of holomorphic dynamics, which was itself more or less forgotten until Mandelbrot and the appearance of computers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is : did Poincaré's work on the 3 body problem influence Fatou and Julia ? Surely they were aware of it, since they were also french mathematicians and more or less contemporaries of Poincaré. Were there any parallel noted by Fatou and Julia between the chaos of the 3 body problem and the chaos in a Julia set at the time? &lt;/p&gt;&#xA;" OwnerUserId="9887" LastActivityDate="2019-04-26T12:37:00.520" Title="Influence of Poincaré on Julia and Fatou" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9527" PostTypeId="2" ParentId="9525" CreationDate="2019-04-26T12:13:29.597" Score="7" Body="&lt;p&gt;Poincare's influence is acknowledged by Fatou in his dynamics papers, but not&#xA;the work on 3 body problem. Fatou was inspired by another work of Poincare, namely on Kleinian groups. The analogy between iteration of rational functions (Fatou) and Kleinian groups (Poincare) also plays an important role in the modern development. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Fatou wrote the first book on Kleinian groups which is almost forgotten today, because of his extreme modesty (his name is mentioned on the title page with a small print only). The book is Appele and Goursat, Theorie des fonctions algebriques et de leurs integrales, Tome II (2-eme ed., 1930) In fact Fatou wrote&#xA;the whole second volume (Fonctions automorphes), which is of the same size as vol. I,  but his contribution is only mentioned as &quot;an editor of the second edition&quot;. The first edition was in one volume. The second volume is published a year after Fatou's death.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, there is no doubt that Fatou was very well familiar with Poincare's work on 3 body problem, and the indirect influence of Poincare's ideas&#xA;in dynamics is felt in Fatou's work on holomorphic dynamics. (Julia was  influenced by Poincare to a much lesser extent if at all).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fatou was an astronomer by profession (he worked in Paris observatory for all his career, where his duty was to observe the stars at night). He also made an important contribution to Celestial mechanics, which is almost completely forgotten nowadays. (In particular he obtained the first rigorous results on the&#xA;&quot;Averaging method&quot; which is now universally credited to M. M. Bogolyubov).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I have to add that Fatou's and Juia's main motivation in developing what we call &quot;holomorphic dynamics&quot; was not &quot;dynamics&quot;, but solution of functional equations. This can be even seen from the titles of many of their papers. (Fatou's main work on dynamics is called Sur les equations fonctionnelles.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a comprehensive historical discussion of context of the work of Fatou and Julia I recommend the excellent book of Michele Audin &quot;Fatou, Julia, Montel. The great prize of mathematical science of 1918 and beyond&quot;. &#xA;See also &lt;a href=&quot;http://www.math.purdue.edu/~eremenko/dvi/fatou-b.pdf&quot; rel=&quot;noreferrer&quot;&gt;Fatou's own survey of all his work&lt;/a&gt; written shortly before his death.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-04-26T12:37:00.520" LastActivityDate="2019-04-26T12:37:00.520" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9528" PostTypeId="2" ParentId="9524" CreationDate="2019-04-26T16:07:05.400" Score="3" Body="&lt;p&gt;According to the &lt;a href=&quot;https://www.chemistryworld.com/opinion/heyrovsk%C3%BDs-drops/3008929.article&quot; rel=&quot;nofollow noreferrer&quot;&gt;Royal Society of Chemistry&lt;/a&gt; site, the hydrogen electrode was discovered by &lt;a href=&quot;https://www.chemistryworld.com/opinion/le-blancs-electrode/8154.article&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max Le Blanc&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Chemists like Max Le Blanc to begin to plot ‘polarisation’ curves to investigate the electrochemical processes – his invention of the hydrogen electrode would anchor all measurements to a common standard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/le-blanc-max-julius-louis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia.com&lt;/a&gt; provides further details.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2019-04-26T16:16:14.533" LastActivityDate="2019-04-26T16:16:14.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9529" PostTypeId="1" AcceptedAnswerId="9530" CreationDate="2019-04-27T11:53:16.210" Score="10" ViewCount="4480" Body="&lt;p&gt;The obelus &lt;code&gt;÷&lt;/code&gt; represents division on calculator keyboards, and sometimes in elementary education.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ypKeK.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ypKeK.png&quot; alt=&quot;÷ division button on a calculator&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has a long non-mathematical history starting &lt;a href=&quot;https://episystechpubs.com/2013/07/26/editors-corner-the-obelus/&quot; rel=&quot;noreferrer&quot;&gt;before 200 BC&lt;/a&gt;. Its earliest use as a symbol for division is in &lt;a href=&quot;https://en.wikipedia.org/wiki/Obelus#In_mathematics&quot; rel=&quot;noreferrer&quot;&gt;Johann Rahn's 1659 &lt;em&gt;Teutsche Algebra&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it known why Rahn chose this particular symbol?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people, including me, assume that it came from the fraction bar: &lt;code&gt;÷&lt;/code&gt; looks like a fraction with dots in place of the numerator and denominator. The timeline works out: the fraction bar is attested &lt;a href=&quot;https://en.wikipedia.org/wiki/Abu_Bakr_al-Hassar&quot; rel=&quot;noreferrer&quot;&gt;since the 13th century&lt;/a&gt;, so it must have been known to Rahn. Is there evidence for or against this origin?&lt;/p&gt;&#xA;" OwnerUserId="9895" LastActivityDate="2023-10-20T11:56:52.693" Title="How did the obelus ÷ come to stand for division?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9530" PostTypeId="2" ParentId="9529" CreationDate="2019-04-28T00:55:03.517" Score="9" Body="&lt;p&gt;I doubt that this is known. The formula with the obelus is merely &lt;a href=&quot;http://jeff560.tripod.com/rahn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;tabulated by Rahn&lt;/a&gt; with no text, and if the motivation was known, Cajori would have mentioned it in &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;History Of Mathematical Notations&lt;/a&gt;, which is a classical treatise. It is likely that Rahn (or Pell) simply repurposed the minus sign symbol, it was used from Riese's algebra textbook (1525) onward. Rahn does give a fraction in the familiar form as the translation, and his use of &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; violates the modern convention for the order of operations, in parallel to how fractions are read.&lt;/p&gt;&#xA;&lt;p&gt;It is possible that the adoption of &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; was stimulated by a later pseudo-etymology based on its similarity with fractions. This sort of phenomenon is not uncommon. For example, Wallis's symbol &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; is commonly attributed to &amp;quot;snake biting its own tail&amp;quot;, but this does not come from the author. The shape of the radical symbol coming from &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is similarly a fable, it is Euler's invention 2.5 centuries after, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2265/55&quot;&gt;Why is the radical symbol &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\ \ }$&lt;/span&gt; called “radical”?&lt;/a&gt; But it might have contributed to its continued adoption. &lt;a href=&quot;https://en.wikipedia.org/wiki/False_etymology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pseudo-etymologies&lt;/a&gt; of various words are commonly encountered in lexicography as well.&lt;/p&gt;&#xA;&lt;p&gt;This is what Cajori writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;ln 1659 the Swiss Johann Heinrich Rahn published an algebra in which he introduced &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; as a sign of division (§ 194). Many writers before him had used &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; as a minus sign(§§ 164,208). Rahn's book was translated into English by Thomas Brancker (a graduate of Exeter College, Oxford) and published, with additions from the pen of Joh. Pell, at London in 1668. Rahn's Teutsche Algebra was praised by Leibniz as an &amp;quot;elegant algebra,&amp;quot; nevertheless it did not enjoy popularity in Switzerland and the symbol &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; for division was not adopted by his countrymen.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In England, the course of events was different. The translation met with a favorable reception; Rahn's &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; and some other symbols were adopted by later English writers, and came to be attributed, not to Rahn, but to John Pell. It so happened that Rahn had met Pell in Switzerland, and had received from him (as Rahn informs us) the device in the solution of equations of dividing the page into three columns and registering the successive steps in the solution. Pell and Brancker never claimed for themselves the introduction of the &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; and the other symbols occurring in Rahn's book of 1659. But John Collins got the impression that not only the three-column arrangement of the page, but all the new algebraic symbols were due to Pell. In his extensive correspondence with John Wallis, Isaac Barrow, and others, Collins repeatedly spoke of &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; as &amp;quot;Pell's symbol.&amp;quot; There is no evidence to support this claim(§ 194).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The sign &lt;span class=&quot;math-container&quot;&gt;$\div$&lt;/span&gt; as a symbol for division was adopted by John Wallis and other English writers. It came to be adopted regularly in Great Britain and the United States, but not on the European Continent. The only text not in the English language, known to us as using it, is one&#xA;published in Buenos Aires&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since the time of Cajori's writing (1928) the obelus became more common even in European, especially Russian, textbooks. Cajori also has a paper &lt;a href=&quot;https://www.jstor.org/stable/2300111&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rahn's Algebraic Symbols (American Mathematical Monthly, 31 (1924), no. 2, pp. 65-71)&lt;/a&gt; that gives more details.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-28T22:53:18.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9531" PostTypeId="1" AcceptedAnswerId="9532" CreationDate="2019-04-28T02:19:42.703" Score="2" ViewCount="127" Body="&lt;p&gt;In modern textbooks, students are greeted with plenty of exercises. Usually they are also organized in such a way that you have examples and pointers to what concepts are most important. The elements on the other hand (and for that matter a lot of older books of this sort) greet us instead with a series of propositions followed by proofs. Is there any idea on how these were studied?&lt;/p&gt;&#xA;" OwnerUserId="9898" LastActivityDate="2019-04-28T13:23:11.190" Title="What were the typical ways students were taught the elements when it remained the prime textbook of mathematics?" Tags="|mathematics|ancient-greece|geometry|social-context|education|" AnswerCount="1" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9532" PostTypeId="2" ParentId="9531" CreationDate="2019-04-28T13:23:11.190" Score="7" Body="&lt;p&gt;Students learned theorems, propositions and their proofs. A teacher would call a student to the blackboard and ask to reproduce a proof. A shorter test would be just to state the theorem. Many students memorized theorems and proofs by heart. A good (but not popular) teacher would make his own picture on the blackboard, with his own notation, different from the book, and ask a student to reproduce the proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Exercises are given by the teacher, for homework or in class. Later, separate books of exercises were used. (Including exercises in the main text of the textbook is a British-American custom. In other countries they had separate books of exercises).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was taught by this system in secondary school in Soviet Union in 1960s. They did not use Euclid, but a book very close to it (Kiselev), with slightly modernized language. And a separate exercise book (Rybkin). Euclid was used everywhere until 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-04-28T13:23:11.190" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9533" PostTypeId="1" AcceptedAnswerId="9534" CreationDate="2019-04-28T14:54:27.227" Score="6" ViewCount="1935" Body="&lt;p&gt;Transfer of mathematical knowledge from India to Europe (such as a positional number system with zero) allowed Europeans to develop arithmetic. But was there also a reverse direction (probably via Arab mathematicians) in which knowledge was transferred from Europe to India?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Especially I'm curious when Euclid's Elements (probably the most known ancient mathematical book) was introduced to India.&lt;/p&gt;&#xA;" OwnerUserId="6955" LastEditorUserId="140" LastEditDate="2019-04-29T04:36:40.500" LastActivityDate="2019-05-09T19:19:01.107" Title="When Indian mathematicians learn of Euclid's Elements?" Tags="|geometry|euclidean-geometry|ancient-india|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9534" PostTypeId="2" ParentId="9533" CreationDate="2019-04-28T16:16:48.110" Score="6" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.thehindubusinessline.com/blink/explore/The-peacock-and-the-maharaja/article20849410.ece&quot; rel=&quot;noreferrer&quot;&gt;The Hindu Business Line&lt;/a&gt;, quoting the scholar TA Sarasvati Amma:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was only in the 18th century, nearly 2,000 years after active contact of Indians with the Greeks, that Euclid’s Elements were translated into Sanskrit and even then perhaps the example of the Arabs provided the inspiration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;TL/DR;&lt;/p&gt;&#xA;&lt;p&gt;Concerning Indian mathematics at about the time Euclid, according to &lt;a href=&quot;https://en.wikipedia.org/wiki/Frits_Staal&quot; rel=&quot;noreferrer&quot;&gt;Frits Staal&lt;/a&gt;, professor of Philosophy and South/Southeast Asian Studies at UC Berkley:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The ancient Greeks developed logic and a notion of rationality as deduction best exhibited by Euclid’s geometry. These discoveries contributed substantially to the development of Western science. Ancient Indian civilisation was an oral tradition and the oral transmission of the tradition became the first object of scientific inquiry.&lt;/p&gt;&#xA;&lt;p&gt;Thus arose two human sciences, closely related to each other in their formal structure: the sciences of ritual and language.To begin with, while a number of key contributions were made by Indian mathematicians, they somehow remained in complete darkness about conic sections. These are simply the various dissections of an hourglass (or, a double cone) which are the ellipse, the parabola and the hyperbola. The importance of these curves in the history of science up to the time of Isaac Newton is unparalleled in geometry. Planets were found to move in elliptical orbits, cannonballs and projectiles fell in a parabolic arch under the influence of gravity, and shadows on sundials moved in a hyperbolic path.&lt;/p&gt;&#xA;&lt;p&gt;The other omissions concern solid geometry, and the existence of only five Platonic solids, namely — tetrahedron, cube, octahedron, icosahedron and the dodecahedron. These five elemental solids were used since the time of Plato in pondering the structure of atoms, crystals and matter in general.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, according to Staal, ancient Indian mathematicians were primarily concerned with a scientific exploration of ritual and mysticism, applying their geometry to things like the construction of elaborate altars.&lt;/p&gt;&#xA;&lt;p&gt;Further reading: &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_geometry&quot; rel=&quot;noreferrer&quot;&gt;History of Geometry - wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-04-28T16:16:48.110" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9536" PostTypeId="1" AcceptedAnswerId="9537" CreationDate="2019-04-28T22:57:38.957" Score="4" ViewCount="1242" Body="&lt;p&gt;Stoichiometry, &quot;the quantitative relationship between two or more substances especially in processes involving physical or chemical change&quot; (Merriam-Webster), is currently a major part of the U.S. high school curriculum. Given a balanced chemical equation, a student must be able to convert the quantity (in moles, atoms/molecules/formula units, mass, or volume) of one substance into the quantity of another substance and/or another kind of quantity in the equation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I was not taught this in my own high school chemistry class, about 47 years ago. Others near my age also were not taught this. So when did this subject begin to be a common part of high school chemistry?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that I am not asking when the concept or the term was invented (by Jeremias Benjamin Richter: see &lt;a href=&quot;http://chemed.chem.purdue.edu/genchem/history/stoichiometry.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Stoichiometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;), just when it was commonly taught in American high schools. My web searches can find many pages on doing or teaching stoichiometry or on its origins, but nothing on the history of its teaching.&lt;/p&gt;&#xA;" OwnerUserId="426" LastEditorUserId="426" LastEditDate="2019-04-29T00:00:40.383" LastActivityDate="2019-04-29T01:33:48.000" Title="When did stoichiometry begin to be taught in U.S. high schools?" Tags="|chemistry|teaching|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9537" PostTypeId="2" ParentId="9536" CreationDate="2019-04-28T23:49:54.950" Score="4" Body="&lt;p&gt;It is very hard to separate stoichiometry from chemistry just like separating the concept of addition and subtraction from the teaching of mathematics. As long as atomic weights calculations were taught, knowledge of stoichiometry was essential. I think it was a part and parcel of all chemistry curricula since its beginning in Germany. If you see an essay on &quot;The Outlook for a Better Correlation of Secondary School and College Instruction in Chemistry&quot; written by H. P. Talbot in 1906 and published in &lt;em&gt;Science&lt;/em&gt;, one can see that a mastery of stoichiometry was essential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/1634694?seq=1#metadata_info_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/1634694?seq=1#metadata_info_tab_contents&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/j2jev.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/j2jev.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V0NkS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V0NkS.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-04-29T01:33:48.000" LastActivityDate="2019-04-29T01:33:48.000" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9539" PostTypeId="1" CreationDate="2019-04-29T04:41:16.287" Score="9" ViewCount="197" Body="&lt;p&gt;I am trying to rationalize the meaning of the term kernel, especially when it is associated with distributions. The English and German etymology all show that the literal meaning is corn (English) and grain (in German). For example the term Gaussian kernel is often used in digital filtering to remove noise from signal. According to the Earliest Known Uses of Math Words website kernel &lt;em&gt;entered statistics with the use of Fourier theory in describing estimates for spectral density and probability density functions.&lt;/em&gt; However it does not explain the choice.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the rationalization behind the choice of this word, I mean what is common in the original meaning of kernel and window functions as shown on Wikipedia? The term window function makes much more sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/e3Zsf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/e3Zsf.png&quot; alt=&quot;https://en.wikipedia.org/wiki/Kernel_(statistics)&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-04-29T14:23:39.890" LastActivityDate="2019-04-29T17:54:48.040" Title="How is the word kernel associated with distributions?" Tags="|mathematics|terminology|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9540" PostTypeId="2" ParentId="276" CreationDate="2019-04-29T05:14:12.570" Score="3" Body="&lt;p&gt;Concept of Shunya (Zero)&#xA;The concept of Shunya, or zero void, was originally conceived as the symbol of Brahman, expressing the sum of all distinct forms. The symbol of zero and the decimal system of notation is described in the Atharvaveda[15]. it describes how the number increases by 10 by writing zero in front of it. While there is no explicit mention of zero, it must have been common knowledge based on how it is used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In fact, the concept of shunya was not just mathematical or scientific, but is deeply rooted in all branches of thought - especially metaphysics and cosmology. Shunya is the transition point between oposites, it symboliss the real balance between divergent tendencies. Most ancient mathematicians defined zero as the sum of two equal and opposite quantities. Zero produces all figures, but is itself not limited to a certain value. Zero is the primary or final reservoir of all single numbers. The symbol of zero and the decimal system of notation is described in the Atharvaveda[16]. It describes how the number increases by 10 by writing zero in front of it.&#xA;It is said to be described in Atharva Veda. &lt;a href=&quot;http://www.hindupedia.com/en/Mathematics_of_the_Vedas#Atharva_Veda&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.hindupedia.com/en/Mathematics_of_the_Vedas#Atharva_Veda&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-04-29T05:14:12.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9541" PostTypeId="2" ParentId="9539" CreationDate="2019-04-29T05:57:44.280" Score="5" Body="&lt;p&gt;Rationalizations that &quot;make sense&quot; are often urban legends after the fact, people who introduce terms rarely make a point of it or report their reasons. The process of spreading is largely by accident and loose association, sometimes aided by invented backstories.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fredholm used the French version (&quot;noyau&quot;) for integral equations, for whatever reasons, perhaps because the kernel is the &quot;core&quot; of the integral. It stays while the unknown function varies. Hilbert and Bochner took after him. It is similar to the integral &quot;core&quot; for Fourier and other sums, as well as to integral representations of solutions to boundary value problems. So Zygmund's monograph extended the use. With big names backing it, it spread. When statisticians started using Fourier series, they picked it up from Bochner, Zygmund and others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Gaussian kernel&quot; refers to taking convolutions with the Gaussian distribution, which are similar (identical in the simplest case) to convolutions with the heat kernel to solve the heat equation, or convolutions with the Dirichlet/Fejér kernels to approximate Fourier series, or, indeed, to forming an integral equation. So this association makes more sense than many others.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word is also used in algebra, coming from different people and for unrelated reasons.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-04-29T17:54:48.040" LastActivityDate="2019-04-29T17:54:48.040" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9542" PostTypeId="2" ParentId="3342" CreationDate="2019-04-29T16:11:37.857" Score="1" Body="&lt;p&gt;In India mathematics was considered to be a utilitarian science and arithmetic was widely spread in the population. No abacus was needed and account keeping was very easy. Simple vegetable vendors use arithmetic fluently. It was not considered to be an esoteric discipline.&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-04-29T16:11:37.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9545" PostTypeId="1" CreationDate="2019-04-30T00:20:28.493" Score="4" ViewCount="1273" Body="&lt;p&gt;So I apologize if this question seems a bit nit-picky, but it has bothered me for a while. Usually when a coordinate system is represented in two-dimensions, the x-axis is pointing towards what might be considered &quot;3 o' clock&quot;, while the y-axis is at midnight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nAATK.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nAATK.jpg&quot; alt=&quot;two dimensional coordinate system&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, most representations of 3-dimensional space place the x-axis in the &quot;6 o' clock&quot; direction, and the Y axis in the &quot;3 o'clock&quot; direction.(if you were to lay the clock on the x-y plane).&lt;a href=&quot;https://i.stack.imgur.com/BaZkY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BaZkY.png&quot; alt=&quot;three dimensional coordinate system&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It does not seem like the intuitive way of representing things. Of course a person has the liberty to arrange the axes in whatever fashion they see fit, but the standard seems to be the way shown above. I wonder who thought it was a good idea to arrange the axes in this fashion, and why they chose to do so. &lt;/p&gt;&#xA;" OwnerUserId="9905" LastActivityDate="2020-04-13T17:01:30.293" Title="Why is the representation of the direction of the x and y axes in two dimensions different than in three dimensions?" Tags="|mathematics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9551" PostTypeId="2" ParentId="8405" CreationDate="2019-05-04T02:14:06.543" Score="-1" Body="&lt;p&gt;A few more.&#xA;&quot;Remarkable approximations for pi are given in Indian texts including 3.1416 of Aryabhata (499 AD), 3.14159265359 of Madhava (14th century AD) and 355/113 of Nilakanta (1500 AD). An anonymous work Karanapaddhati (believed to have been written by Putumana Somayajin in the 15th century AD) gives the value 3.14159265358979324 which is correct up to seventeen decimal places.&quot;&#xA;From VoL7,No.4,pp.ll-13;No.IO,p.6,2002. &#xA;Mathematics in Ancient India by &#xA;Amartya Kumar Dutta. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ias.ac.in/article/fulltext/reso/007/04/0004-0019&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ias.ac.in/article/fulltext/reso/007/04/0004-0019&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-05-04T02:14:06.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9552" PostTypeId="2" ParentId="8436" CreationDate="2019-05-05T02:45:09.160" Score="4" Body="&lt;p&gt;The problem with very old historical stuff including the origins of a certain concepts and giving due credit where it is due is one of the most difficult questions in history. The truth is that we may never know the correct picture behind the question who &lt;em&gt;invented&lt;/em&gt; the modern number system? No matter what Wikipedia says. Anyone who is serious scholar of scientific history, Wikipedia should be their &lt;em&gt;last&lt;/em&gt; resort to provide a proof. It must have been a very slow progress based on cross-cultural exchange of knowledge. About a thousand years ago, Middle east and central Asia were the center of knowledge just like today's Europe and North America. Some scientific words from Arabic origin still linger in modern mathematics and chemistry, such as algebra (named after a treatise with this title), algorithm (named after an Arabic mathematician), alcohol, boron, chemistry, alkali, natron etc. This shows that translation of Arabic into Latin was an active process in Europe. Translation is a sign of a progressive society.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For instance, OED entry for Arabic numerals traces the word usage history:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Designating the system of numerals written 0, 1, 2, 3, 4, etc.; of or&#xA;  relating to this system. Chiefly in Arabic numeral. Contrasted with&#xA;  &lt;em&gt;Roman&lt;/em&gt;. Arabic numerals reached western Europe through Arabia by c1200, but probably originated in India&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The earliest usage goes back to 1750s:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1756   Gentleman's Mag. May 239/1   The learned are not quite agreed&#xA;  when the Arabic numerals were first brought into use in this nation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which clearly shows that nobody was even sure when these numbers were brought to Europe. Also note the word &lt;em&gt;probably&lt;/em&gt;. Some falsely imply that these medieval Arab scholars were &lt;em&gt;mere translators&lt;/em&gt; of Greek and Indian works. This is a fabricated story as well because as stated above, translation of scholarly work is a part and parcel of science.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Civilization near the Indus valley (now India, Pakistan) have 3000-4000 year old culture and they had their system of weights for trading. Certainly, they knew how to count as well. China had its own set of philosophers. Since the languages of these cultures was so different, it is extremely difficult to trace the ancient history of science. There is very little primary source of information. Today we can look up 18th century articles and say that this and that person is behind this concept but this does not apply to older works. If the Egyptians could build such huge pyramids thousands of years ago (predating all) which still startle modern civil engineers, can we say that they knew trigonometry very well? Nobody knows, because they did not leave anything which we can understand yet. Who knows how much knowledge was lost with time and then re-invented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently, there are series of rather ridiculous claims in History of Science and Mathematics, and even Physics that everything originated in ancient India. Claims range from invention of alegbraic formulae, sine, cosine, numbers, astronomy, distance between Earth and Sun, ancient aircrafts, stem cell research and above all close relationship of German and Sanskrit. Usually Wikipedia and other dubious sources are quoted. BBC even made a report on it, sadly this is being propagated by some obscure university professors. One can only imagine how many students are misled. Perhaps these awed  students are posting a series of these questions all over the web.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bbc.com/news/world-asia-india-46778879&quot; rel=&quot;nofollow noreferrer&quot;&gt;India: scientists dismiss Einstein theories&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The moral of the story is that paper never refused ink. Web is worse, because everyone can propagate false ideas. Our job is to filter junk vs. good work. &lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-12-22T04:24:25.203" LastActivityDate="2019-12-22T04:24:25.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9554" PostTypeId="2" ParentId="1805" CreationDate="2019-05-05T21:14:26.417" Score="-1" Body="&lt;p&gt;Certainly much before Tyco Brahe. By Hindu Rishis obviously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Indian and Greek Models: &#xA;We first note that the idea that the sun is roughly 500 or so earth diameters away from us is much more ancient that Ptolemy. So Neugebauer was wrong on two counts: first, he did not know of any Indian connections although he admitted that the “study of Hindu astronomy is still at its beginning;” second, he did not recognize that the tradition regarding the distance of the sun might be much older in Greece itself. This greater antiquity is in accordance with the ideas of van der Waerden,4 who ascribes a primitive epicycle theory to the Pythagoreans. But it is more likely that the epicycle theory is itself much older than the Pythagoreans and it is from this earlier source that the later Greek and Indian modifications to this theory emerged which explains why Greek and Indian models differ in crucial details. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It should also be noted that Ptolemy’s Almagest is a 12th cen- tury Arabic book that is likely to include much information of the first millennium astronomy and its original 2nd century form can- not be established with certainty. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did the idea that Rs ≈ 500de originate at about the time of Pancavimsa Brahmana (PB), that is from the second millennium BCE, or is it older? Since this notion is in conflict with the data on the periods of the outer planets, it should predate that knowledge. If it is accepted that the planet periods were known by the end of the third millennium BCE, then this knowledge must be assigned an even earlier epoch. Its appearance in Pancavimsa Brahmana (PB) , a book dealing primarily with ritual, must be explained as a remembrance of an old idea. We do know that PB repeats, almost verbatim, the Rgvedic account of a total solar eclipse. PB is in second millennium BCE. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once the conflict between the planet period information and the supposition that the heavens were 1000 earth diameters away became clear, this supposition was dropped. Presumably, the theory that Rs ≈ 500 de was too entrenched by this time and it became the basis from which different Greek and later Indian models emerged. As mentioned before, Ptolemy considers an Rs equal to 600 de, whereas Ayabhata assumes it to be about 438 de. Thus the Greek and the later Indian modifications to the basic idea proceeded somewhat differently. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The ideas regarding the distance of the sun hardly changed until modern times. The contradictions in the assumption that the luminaries move with uniform mean speed and the requirements imposed by the assumed size of the solar system led to a gradual enlargement of the models of the universe from about twice that of the distance of the sun in PB to one 4.32 × 10^6 times the distance of the sun by the time of Ayabhata. This inflationary model of the universe in AAr makes a distinction between the distance of the sky (edge of the universe) and that of the stars which is taken to be a much smaller sixty times the distance of the sun. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Beyond the visible universe illuminated by the sun and limited by the sky is the infinite invisible universe” this is stated in a commentary onAryabhabaıya (AAr) by BhaskaraI writing in 629CE. The Puranic literature from India, part of which is contemporaneous with Ayabhata, reconciles the finite estimates of the visible universe with the old R. gvedic notion of an infinite universe by postu- lating the existence of an infinite number of universes. &#xA;It is possible that the original notion that the heavens are 1000de away from the earth arose as a metaphor for the large extent of the universe, given that a thousand represents a very great size in Indo-European languages. But it is more likely that some measurements and a theory were at the basis of this supposition. “&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“The Surya Siddhanta 12.84 says: “Any orbit, multiplied by the earth’s diameter and divided by the earth’s circumference, gives the diameter of that orbit; and this, being diminished by the earth’s diameter and halved, gives the distance of the planet.”  The next verses give the distances as follows: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Table 8.1: Distances to planets and stars in yojanas (= 7.5 miles approx )&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moon  324,000&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mercury (conjunction)  1,043,209 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Venus (conjunction)   2,664,637 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sun   4,331,500 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mars   8,146,909 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jupiter  51,375,764 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Saturn  127,668,255 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asterisms 259,890,012 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Brahmanda  (Edge of the Universe)    18,712,080,864,000,000 “&lt;/p&gt;&#xA;&#xA;&lt;p&gt;from  &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.695.536&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.695.536&amp;amp;rep=rep1&amp;amp;type=pdf&lt;/a&gt;&#xA;by   Subash Kak The Astronomical Code of the Ṛgveda (Third Edition) &lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-05-05T23:22:30.433" LastActivityDate="2019-05-05T23:22:30.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9556" PostTypeId="2" ParentId="7224" CreationDate="2019-05-06T01:23:22.367" Score="0" Body="&lt;p&gt;Sushruta was a surgeon who developed cataract surgery and must have seen the human lens.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.aao.org/senior-ophthalmologists/scope/article/sushrutaSushruta&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aao.org/senior-ophthalmologists/scope/article/sushrutaSushruta&lt;/a&gt; in 600 B.C. introduced extraocular expulsion of lens material.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Grzybowski A1, Ascaso FJ.&#xA;Author information&#xA;Abstract&#xA;It is generally accepted that Jacques Daviel introduced in the 18th century the extracapsular technique of extraction of the lens while the couching method of cataract operation had already been practiced since ancient times. Present study analyses the first known cataract surgery description in three translations into English from the original Sanskrit Sushruta textbook and all the available literature on the subject. We found evidences that some sort of extraocular expulsion of lens material through a limbal puncture (paracentesis) was described by the Indian surgeon. Nevertheless, this incision cannot be considered as a classic extracapsular procedure because it was not large enough to allow the extraction of the entire lens.&#xA;© 2013 The Authors. Acta Ophthalmologica © 2012 Acta Ophthalmologica Scandinavica Foundation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/23464869&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ncbi.nlm.nih.gov/pubmed/23464869&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-05-06T01:23:22.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9557" PostTypeId="2" ParentId="19" CreationDate="2019-05-06T03:07:56.487" Score="1" Body="&lt;p&gt;It seems to be true. I dreamed of one myself. How to linearize hot wire measurements of shear stress (and only shear stress) in a turbulent flow? It is actually by squaring hot-wire signals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is in &quot;Impossibility of Linearizing a hot-wire anemometer for measurements in turbulent flows&quot;by S.P. Parthasarthy and D.J. Tritton, AIAA Journal, 1963, 1-5, p1210-1211.Indian Institute of science, Bangalore.&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-05-06T03:07:56.487" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9561" PostTypeId="1" AcceptedAnswerId="9562" CreationDate="2019-05-06T15:21:27.947" Score="5" ViewCount="408" Body="&lt;p&gt;Marc Lange writes (page 29)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was known by about the 19th century that any action at a distance involving electric or magnetic forces would be &lt;em&gt;retarded&lt;/em&gt; and so undermine not only spatial locality, but also temporal locality.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How was it discovered in the 19th century that electric or magnetic forces were not instantaneous? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be interested in survey articles, but I am mainly interested in reading how those who discovered the result described what they found.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Lange, M. (2002). An Introduction to the Philosophy of Physics Locality, Fields, Energy, and Mass. Blackwell Publishing.&lt;/p&gt;&#xA;" OwnerUserId="9677" LastActivityDate="2019-05-06T21:16:17.230" Title="How was it discovered in the 19th century that electric or magnetic forces were not instantaneous?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9562" PostTypeId="2" ParentId="9561" CreationDate="2019-05-06T16:18:35.040" Score="4" Body="&lt;p&gt;This was noticed when observations of eclipses of Jupiter satellites deviated from&#xA;prediction. Before that there could be only speculations (and these speculations existed from antiquity). Jupiter satellites&#xA;gave the first hard evidence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the very beginning, Jupiter satellites were proposed by Galileo as a natural clock for determination of longitude. For that reason, very careful observations were made, and finally the disagreement with Kepler laws was found.&#xA;This happened in 17th century. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Romer and Huygens conjectured that the reason is the finite speed of light and measured it (assuming that Jupiter satellites do obey Kepler laws). They obtained&#xA;a number which was not very precise but of the right order of magnitude. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since then observations were made more and more precise.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By the way, Jupiter satellites gave the most precise way of synchronizing clocks&#xA;at different locations (=measuring of longitude) until the invention of telegraph.&#xA;But the method could only be applied on land, not on a ship. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://gizmodo.com/how-the-speed-of-light-was-first-measured-1138348467&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gizmodo.com/how-the-speed-of-light-was-first-measured-1138348467&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That other electromagnetic oscillations are spread with the same speed as light was discovered by Maxwell.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-06T21:16:17.230" LastActivityDate="2019-05-06T21:16:17.230" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="9564" PostTypeId="1" CreationDate="2019-05-06T16:53:10.653" Score="1" ViewCount="299" Body="&lt;p&gt;I think that The Zodiacal signs are of Vedic origin. Hindus, Greeks and Sumerians all have Zodiacal signs. Which is the most likely origin?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The middle ring is the zodiac in the same between India and Greece.The outer ring is the same as the Greek one for Sumer.  The inner ring in the sumerian one have different images and the Vedic Zodiac there are star names and their represtbtations(the Vedic kalachaka).Later on the Hindus added another star ring displaced with respect to the third ring to make a fourth ring.&#xA;The middle ring is the same for both India and Greece. In the Kalchakra, the stars are arranged in a specific way. 27 stars cover 360 degrees, starting at Meena (also Fish in the Zodiac). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the Sumerian case the inner 7 segments match at Scorpio. &#xA;from&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://shrifreedom.org/freedoms_vedic_zodiac/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://shrifreedom.org/freedoms_vedic_zodiac/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-05-07T23:45:43.393" LastActivityDate="2022-08-21T02:04:20.327" Title="What is the origin of Zodiac (Kaalachakra) signs?" Tags="|astronomy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9565" PostTypeId="2" ParentId="9564" CreationDate="2019-05-06T16:53:10.653" Score="-1" Body="&lt;p&gt;&quot;We present now the basis to our identification of the names of the nakshtras from the Rgveda in a form easier to read: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Vaishaka = Swathi to Anuradha = Vayu, Indragni, Mitra= Vrsa, &lt;em&gt;Bull for Indra,&lt;/em&gt; e.g. RV 8.33; also Vayu is some- times identified with Indra and the two together called Indravayu, and Vayu is also associated with cow (RV 1.134) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Iyasta Anuradha to Mula = Mitra, Varuna, Pitarah. = Mithuna,  Gemini, from the &lt;em&gt;cosmic embrace of Mitra and Varuna&lt;/em&gt; &#xA;Ashada = OurvaAsada to Srona = Apaha., Visvedevaha., Vishnu &#xA;= Karka, circle or Cancer, the sign of Visnu”s chakra (e.g. RV 1.155.6) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shravana Srona to Shatabhisaj = Vishnu, Vasavah, Indra&#xA;= Simha, &lt;em&gt;Lion, after Indra&lt;/em&gt; as in RV 4.16.14 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bhadrapada = Shatabhisaj to UProshthapada = Indra, Aja Ekapada,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ahirbudhnya= Kanya, Virgin, apparently from Aryaman in the opposite side of the zodiac who is the wooer of maidens, kanya (RV 5.3.2) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asvina = UProsthapada to Asvayujau = Ahirbudhnya, Pusan, Asvayujau&#xA;= Tula, Libra, from the &lt;em&gt;Asvins who denote balance&lt;/em&gt; of pairs (e.g. RV 2.39, 5.78, 8.35) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;K ̄artika = Apabharanı to Rohini = Yama, Agni, Prajapati = Ali (Vrshchika), &lt;em&gt;Scorpion, from Krttika, to cut&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Margasırsa = Rohini to Ardra = Prajapati, Soma, Rudra = Dhanus., Archer, from the &lt;em&gt;cosmic archer Rudra&lt;/em&gt; (RV 2.33, 5.42, 10.125) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pausha = Ardra to Pushya = Rudra, Aditi, Brhaspati = &lt;em&gt;Makara, Goat&lt;/em&gt;, Rudra placing goat-head on Prajapati, and goat is the main animal sacrificed at the ritual of which Bruhaspati is the priest &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Magha = Pushya to Magha = Bruhaspati,  Sarpah. , Pitarah. = Kumbha, Water-bearer, from the &lt;em&gt;water-pot offerings&lt;/em&gt; to the pitrus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Phalguna = Phalgunıs to Hasta = Aryaman, Bhaga, Savitar = &lt;em&gt;Mina, Fish,&lt;/em&gt; representing Bhaga (alluded to in RV 10.68) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Caitra = Hasta to Svati = Savitar, Indra, Vayu = Mesha, &lt;em&gt;Ram, from Indra&lt;/em&gt;, see, e.g., RV 1.51 &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We observe that for most solar zodiac segments a plausible name emerges from the name of the deity. The choice of the symbols was also governed by another constraint. The Brahmana texts call the year as the sacrifice and associate different animals with it.21  the short sequence, &lt;em&gt;these animals are goat, sheep, bull, horse, and man.&lt;/em&gt; Beginning with the goat-dragon at number 9 in the sequence starting with Vaishakha, we have sheep at 12, bull at 1, horse (also another name for the sun in India) as the sun-disk at 3, and man as archer at 8. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The kalachkra (Zodiacal signs) are most likely of Hindu origin because they conform to Vedic gods and myths.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;from &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.695.536&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.695.536&amp;amp;rep=rep1&amp;amp;type=pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;by Subash Kak The Astronomical Code of the Ṛgveda (Third Edition)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;archaeoastronomy ancient&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-05-07T07:05:57.220" LastActivityDate="2019-05-07T07:05:57.220" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9566" PostTypeId="2" ParentId="19" CreationDate="2019-05-06T17:57:44.707" Score="2" Body="&lt;p&gt;I read just yesterday that the inventor or the sewing machine had worked on the problem of the needle for a long time. Then in a nightmare, screaming, sweating and all that that he got the answer. The savages that were stabbing him in the dream had the most unusual spears. One's that had a hole here and a hook there and...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article, from one of the &lt;em&gt;Pragmatic&lt;/em&gt; series, was about how the right half of the brain, the non verbal side, waits until pictures can be formed with the answer. Coaxing these out is where dreams and daydreams seem to do the job, if you let them.&lt;/p&gt;&#xA;" OwnerUserId="9940" LastActivityDate="2019-05-06T17:57:44.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9567" PostTypeId="1" AcceptedAnswerId="9576" CreationDate="2019-05-07T00:05:06.110" Score="3" ViewCount="197" Body="&lt;p&gt;In &lt;em&gt;Measurement&lt;/em&gt; by Paul Lockhart&#xA;(&lt;a href=&quot;http://www.hup.harvard.edu/catalog.php?isbn=9780674284388&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harvard Press&lt;/a&gt;),&#xA;he says (p.351):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the classical geometers (as far as I know) never even &lt;em&gt;conceived&lt;/em&gt;&#xA;  of four-dimensional space, whereas adding another variable is&#xA;  an obvious and natural analytic extension.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Q&lt;/em&gt;&lt;/strong&gt;. When was 4D space &quot;conceived of&quot;?&lt;/p&gt;&#xA;" OwnerUserId="606" LastActivityDate="2019-05-08T19:41:56.623" Title="When was 4D space &quot;conceived of&quot;?" Tags="|mathematics|geometry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9568" PostTypeId="2" ParentId="9567" CreationDate="2019-05-07T00:57:14.487" Score="7" Body="&lt;p&gt;According to Klein, the first mathematician who considered space-time as a 4-dimensional manifold was Lagrange, but these ideas were not immediately developed by others. Then he mentions Cauchy and Cayley (Cayley published in 1844 &quot;Chapters on the analytic geometry in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; dimensions&quot;), and credits Grassmann (1844) with the first systematic exposition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While geometers discussed &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; dimensions, people doing statistical&#xA;mechanics (Maxwell, Bolzmann and others) were inevitably lead to consideration&#xA;of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;-dimensional manifolds where &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is of the order of Avogadro number. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref. F. Klein, Vorlesungen uber die Entwicklung der Mathematik im 19 Jahrhundert&#xA;(there is an English transl.), Ch. IV, 3-d part: &quot;Spaces of n dimensions&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-07T11:53:19.837" LastActivityDate="2019-05-07T11:53:19.837" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9569" PostTypeId="2" ParentId="9567" CreationDate="2019-05-07T12:11:05.100" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes (including the six that occur in four dimensions). &lt;a href=&quot;https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;SOURCE&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See the mini-biography of Schläfli at &lt;a href=&quot;https://www-history.mcs.st-andrews.ac.uk/Biographies/Schlafli.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;St Andrews&lt;/a&gt;.  His major treatise was written in 1850-1852, parts were published, but the whole thing was not published until 1901 after his death.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-05-07T12:11:05.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9570" PostTypeId="1" CreationDate="2019-05-07T12:57:28.943" Score="5" ViewCount="1412" Body="&lt;p&gt;Newton's theory of light stated that a light travelled in a straight line as small particles. When these particles travelled in a medium, they experienced an attractive force with the particles of the surrounding medium. This attractive force was equal in all directions, and hence the particle travelled in a straight line. He extended this to explain refraction by saying that when light entered a denser medium, the particles would experience differing levels of attractive forces, as the denser medium would have more particles. This would lead it to bend towards/away from the normal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in this case, how did he explain why different colours of light refracted at different angles?&lt;/p&gt;&#xA;" OwnerUserId="9941" LastActivityDate="2019-05-07T13:09:39.647" Title="Refraction in Newton's Corpuscular Theory of Light" Tags="|physics|newton|optics|refraction|" AnswerCount="1" CommentCount="4" ClosedDate="2019-05-08T18:22:36.410" ContentLicense="CC BY-SA 4.0" />
  <row Id="9571" PostTypeId="2" ParentId="9570" CreationDate="2019-05-07T13:09:39.647" Score="1" Body="&lt;p&gt;Consider one &lt;a href=&quot;http://www.thestargarden.co.uk/Newtons-theory-of-light.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;description&lt;/a&gt; of Newton's theories:&#xA;(reference: Newton, I., Philosophical Transactions of the Royal Society 1672, 80, 3075–3087.) &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Newton introduced the term ‘colour spectrum’ and although the spectrum&#xA;  appears continuous, with no distinct boundaries between the colours,&#xA;  he chose to divide it into seven: red, orange, yellow, green, blue,&#xA;  indigo, and violet. Newton chose the number seven because of the&#xA;  Ancient Greek belief that seven is a mystical number.[5]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Newton showed that every colour has a unique angle of refraction that&#xA;  can be calculated using a suitable prism. He saw that all objects&#xA;  appear to be the same colour as the beam of coloured light that&#xA;  illuminates them, and that a beam of coloured light will stay the same&#xA;  colour no matter how many times it is reflected or refracted. This led&#xA;  him to conclude that colour is a property of the light that reflects&#xA;  from objects, not a property of the objects themselves&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;BTW, you can find excellent transcriptions of the original Opticks at many places online, including gutenberg.org&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-05-07T13:09:39.647" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9573" PostTypeId="2" ParentId="5060" CreationDate="2019-05-08T04:23:18.183" Score="0" Body="&lt;p&gt;I kept looking information about this, and after some time (I'm posting this answer almost 3 years after the question) I found this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Antipositivism is a theoretical stance of social sciences. In the other hand, the first use of the term &quot;social science&quot;, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;first appeared in the 1824 book An Inquiry into the Principles of the&#xA;  Distribution of Wealth Most Conducive to Human Happiness; applied to&#xA;  the Newly Proposed System of Voluntary Equality of Wealth by William&#xA;  Thompson (1775–1833)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_social_sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Social Sciences&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I guess we could say no, interpretivism wasnt always considered to be &quot;scientific&quot;, since it was part of social disciplines, which werent considered social sciences until early XIX century.&lt;/p&gt;&#xA;" OwnerUserId="4512" LastActivityDate="2019-05-08T04:23:18.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9574" PostTypeId="1" AcceptedAnswerId="9578" CreationDate="2019-05-08T13:16:27.320" Score="5" ViewCount="798" Body="&lt;p&gt;This famous photo depicts Margaret Hamilton, leader of the software engineering team for the Apollo Project, next a print out of the code she and her team wrote for the mission. &lt;a href=&quot;https://i.stack.imgur.com/NDqP0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NDqP0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This information is correct according to &lt;a href=&quot;https://skeptics.stackexchange.com/questions/31602/is-this-a-photo-of-margaret-hamilton-standing-next-to-apollo-project-code-that-s&quot;&gt;this&lt;/a&gt; Skeptics SE post and links therein. The question I haven't found an answer to is: &lt;em&gt;why&lt;/em&gt; would they print the code on paper?&lt;/p&gt;&#xA;" OwnerUserId="6485" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-05-08T23:32:43.160" Title="Why would Margaret Hamilton and her team at NASA print the code on paper?" Tags="|computer-science|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9575" PostTypeId="1" AcceptedAnswerId="9577" CreationDate="2019-05-08T19:03:42.653" Score="8" ViewCount="5245" Body="&lt;p&gt;I have seen a claim in some literature that there are three crises in the history of mathematics. The first is the discovery of &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; being irrational in Greek time which shook the belief that every point on a line can be measured by a rational number at that time. The second is the infinitesimal used in the 17th century in Calculus by Newton and Leibniz. The third one is Russell's paradox and other paradoxes discovered from the turn of the 20th century which has shaken the foundation of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, after I googled it online (in English), I can hardly find any discussion or support of such a claim. So I wonder if this is widely accepted in the study of the history of mathematics, or simply a statement without much merit.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2019-05-18T17:33:15.943" LastActivityDate="2019-05-18T17:33:15.943" Title="What is the status of the three crises in the history of mathematics?" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9576" PostTypeId="2" ParentId="9567" CreationDate="2019-05-08T19:35:48.460" Score="5" Body="&lt;p&gt;In his &lt;em&gt;A First Course in Calculus&lt;/em&gt; Serge Lang mentions on pages 525-526 that d'Alembert wrote in an article on dimension in Diderot's &lt;em&gt;Encyclopédie&lt;/em&gt; in 1754 that &quot;duration&quot; could be considered a fourth dimension so the product of time by space has four dimensions, and that  whether or not this is useful it at least has the merit of being new.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The page &lt;a href=&quot;https://www.chstm.org/content/house-where-spacetime-began&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; also mentions d'Alembert in 1754 and gives the year of Lagrange's work as 1797. If credit needs to be given to one person, it should be d'Alembert before Lagrange. &lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2019-05-08T19:41:56.623" LastActivityDate="2019-05-08T19:41:56.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9577" PostTypeId="2" ParentId="9575" CreationDate="2019-05-08T21:59:24.793" Score="16" Body="&lt;p&gt;No, it is not widely accepted. The language of &quot;crises&quot; is rather obsolete and mostly reflects the attitudes of the early 20th century projected backwards. At that time, the contemporaries did indeed characterize the situation in mathematics (and physics) as a crisis. For example,  Weyl's 1920 address was titled “The new foundational crisis in mathematics”, to which Hilbert saw fit to respond with what is now called the Hilbert program (see &lt;a href=&quot;https://pdfs.semanticscholar.org/71ae/8b3bec8e292f56ef71fa4080400c8d5dd1e2.pdf&quot; rel=&quot;noreferrer&quot;&gt;van Dalen's paper&lt;/a&gt;). But the language fits well with the language of (Kuhnian) revolutions, which are precipitated by &quot;crises&quot; (accumulations of anomalies), and provides a convenient shorthand for undergraduate texts to get through the history expediently. After all, the retellings of history in textbooks largely serve the practical purpose of bringing students up to speed on the current mathematics, and this is easier to do by translating all of mathematics into modern terminology, and marking the places where major curriculum pieces were introduced as &quot;crises&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first &quot;crisis&quot; is almost completely made up based on discredited accounts of the early Pythagorean history. The rational numbers or the real line  are sheer modernizations with little relation to the ancient Greece, and even the story of ratios and incommensurables is a dramatized fable. On early Greek mathematics a good source is &lt;a href=&quot;https://www.maa.org/press/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction&quot; rel=&quot;noreferrer&quot;&gt;Fowler's Mathematics of Plato's Academy&lt;/a&gt;. Concerning the traditional story about the &quot;discovery of irrationals&quot; and the subsequent &quot;crisis&quot;, a reviewer writes: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It may be somewhat surprising, then, to discover that many, if not most, historians of ancient mathematics disbelieve some or all of this story. In The Mathematics of Plato's Academy, David Fowler gives a convincing account of the reasons for rejecting the standard story, and offers a very interesting alternative reconstruction of the history of early Greek mathematics... One of the important themes in Fowler's book is that one must carefully assess the available sources on early Greek mathematics. He demonstrates that most of the elements of the &quot;standard story&quot; either come from very late Greek sources (500 years or more after the time of Plato) or are reconstructions by modern historians who are really reflecting nineteenth and twentieth century concerns about foundational issues... The resulting picture is quite different from the traditional one. In particular, incommensurability is seen not as a foundational crisis but rather as an interesting discovery that led to significant mathematics.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The second &quot;crisis&quot; is usually associated not with the creation of calculus but with its subsequent problems, such as the inconsistencies with infinitesimals. It was a pretty slow moving &quot;crisis&quot; though, and the kinematic (late Newtonian) conception of calculus, within which Cauchy worked, and most of algebraic analysis of 18th century, were not seen as having some cardinal problems at the time. Only injecting later foundational attitudes and concerns about rigor lets one detect them. Another potential crisis, with a better claim to the name (in the Kuhnian sense at least), were the struggles with the notion of function that followed the discovery of Fourier series, see the introductory chapters to &lt;a href=&quot;https://books.google.com/books/about/A_Radical_Approach_to_Real_Analysis.html?id=YSe4hUBM7uEC&quot; rel=&quot;noreferrer&quot;&gt;Bressoud's Radical Approach to Real Analysis&lt;/a&gt;, who names his opening chapter Crisis in Mathematics: Fourier's Series.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-09T00:13:07.997" LastActivityDate="2019-05-09T00:13:07.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9578" PostTypeId="2" ParentId="9574" CreationDate="2019-05-08T23:26:48.117" Score="11" Body="&lt;p&gt;This is how computer programming was done in those days. There was no such thing as a modern&#xA;terminal with a screen, and no keyboards. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A programmer would write her code first on paper (handwritten), then a specialist typist will transfer it to punch cards, using some machine, a kind of typewriter. After several rounds of proofreading, the punch cards will be fed to a computer. The output will be printed on&#xA;a wide paper strip as rows of text and numbers. What you see in this picture is probably the output. This output was called &quot;wide print&quot; (I don't know the exact English term, just translating from the Russian). I performed all this process as a student in the early 1970-s. Except the punch cards: they were produced by a specialist typist. So proofreading process was painful.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-08T23:32:43.160" LastActivityDate="2019-05-08T23:32:43.160" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9579" PostTypeId="2" ParentId="9575" CreationDate="2019-05-08T23:44:06.743" Score="6" Body="&lt;p&gt;These notions were popular in the middle of 20th century, in the aftermath of the third crisis. More precisely they were called the &quot;crises of foundations of mathematics&quot;. Of course the first two were essentially the reconstructions of historians of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics was conceived as a rigorous method of obtaining true statements by proofs. The starting point was a set of premises which were considered self-evident without any doubt. This basic believe was seriously shaken several times in the course of history,&#xA;and the principal such episodes are called &quot;crises of foundations&quot;. You can read about this in the middle 20th century histories of mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;About the &quot;first crisis&quot; we can only speculate. The earliest complete mathematical writings which we have originate at the time after the crisis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second one is well documented. It is related to invention of Calculus which it was difficult to justify to the same level of rigor which was considered standard since the Greeks. It took about 200 years to obtain full clarity about foundations of calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third crisis was triggered by Cantor's discoveries, then the discovery of paradoxes, attempts to create new rigorous foundations, and spectacular failure of these attempts. In some sense this third crisis continues, since it was never resolved. But my impression is that most mathematicians just lost interest to these questions, so it is not perceived as a crisis anymore.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-09T01:59:56.333" LastActivityDate="2019-05-09T01:59:56.333" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9580" PostTypeId="1" AcceptedAnswerId="9590" CreationDate="2019-05-09T18:10:18.917" Score="12" ViewCount="3121" Body="&lt;p&gt;I have read that the notion of limit became rigorous two centuries after the discover of calculus &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Newton had in his mind regarding the notion of limit?&lt;/p&gt;&#xA;" OwnerUserId="5421" LastEditorUserId="643" LastEditDate="2019-05-09T21:21:19.637" LastActivityDate="2021-02-26T20:29:00.563" Title="What was the notion of limit that Newton used?" Tags="|mathematics|" AnswerCount="3" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="9581" PostTypeId="2" ParentId="9580" CreationDate="2019-05-09T18:35:40.893" Score="3" Body="&lt;p&gt;Newton did not have the rigorous concept of limit as in &lt;span class=&quot;math-container&quot;&gt;$\epsilon/N$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\epsilon/\delta$&lt;/span&gt; formulas. Instead, he had a vague idea of limit in terms of motion and used the notion of infinitesimal to calculate derivatives and integrals. For example, calculating the derivative of &lt;span class=&quot;math-container&quot;&gt;$y=x^2$&lt;/span&gt; is like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\dot{y}=\frac{\Delta y}{\Delta x}=\frac{(x+\Delta x)^2-x^2}{\Delta x}=2x+\Delta x=2x&#xA;$$&lt;/span&gt;&#xA;(Newton used &lt;span class=&quot;math-container&quot;&gt;$\dot{x}$&lt;/span&gt; for derivative and later Leibniz improved to &lt;span class=&quot;math-container&quot;&gt;$\frac{dy}{dx}$&lt;/span&gt;). In the last step, &lt;span class=&quot;math-container&quot;&gt;$\Delta x=0$&lt;/span&gt;, but in &lt;span class=&quot;math-container&quot;&gt;$\frac{\Delta y}{\Delta x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; can not be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\frac0{0}$&lt;/span&gt; makes no sense. This means that &lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt; (infinitesimal) sometimes is zero and sometimes is not, a fact that Newton could not explain. Nor did Leibniz know the solution. However, this defect of infinitesimal has been largely ignored because the powerful method of Calculus has solved so many and important problems that mankind has even never dreamed of before.&lt;/p&gt;&#xA;&lt;p&gt;The rigorous explanation of infinitesimal through the notion of limit (in forms of &lt;span class=&quot;math-container&quot;&gt;$\epsilon/N$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\epsilon/\delta$&lt;/span&gt; formulas), however,  was not completed until two hundred years after Newton, through the work of  Cauchy and Weierstrass in the 19th century. So it is an overstatement to say that Newton knew the exact notion of limit and the rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is again an overstatement to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-02-26T20:29:00.563" LastActivityDate="2021-02-26T20:29:00.563" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9582" PostTypeId="1" CreationDate="2019-05-09T18:50:10.707" Score="6" ViewCount="166" Body="&lt;p&gt;In &lt;em&gt;Two New Sciences&lt;/em&gt;, Galileo writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I take a perfectly round brass ball about the size of a walnut and project it along the surface of a metallic mirror held in a nearly upright position, so that the ball in its motion will press slightly upon the mirror and trace out a fine sharp parabolic line...The above experiment furnishes clear and tangible evidence that the path of a projectile is a parabola... In the execution of this method, it is advisable to slightly heat and moisten the ball by rolling in the hand in order that its trace upon the mirror may be more distinct.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am confused about three points:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What exactly does this experiment describe?&lt;/li&gt;&#xA;&lt;li&gt;Does the experiment as described work?&lt;/li&gt;&#xA;&lt;li&gt;Did Galileo actually perform this experiment?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;My best understanding is as follows: Galileo took some flat, polished, rectangular metal surface, and placed it at an incline on the ground. He then takes a brass ball, slightly warm and moist, and rolls it up the ramp. He rolls it not &lt;em&gt;straight&lt;/em&gt; up the ramp, but slightly to the side. This causes the ball to follow a parabolic path against the mirror. As it does so, it leaves some kind of mark of its path on the metal surface.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Will this experiment actually work? I don't know enough about mirrors in Galileo's time to be sure what kind of mirror he would use. Also, I'm not sure &lt;em&gt;why&lt;/em&gt; the brass ball would leave a mark on the mirror - would something like &lt;a href=&quot;https://en.wikipedia.org/wiki/Galvanic_corrosion&quot; rel=&quot;noreferrer&quot;&gt;galvanic corrosion&lt;/a&gt; apply in this case? Would it depend on the type of metal used in the mirror?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most accounts I find describe Galileo using a different method to discover that the path of a projectile is a parabola; for example, see &lt;a href=&quot;https://www.journals.uchicago.edu/doi/pdfplus/10.1086/351124&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; by Stillman Drake. I have done a small amount of research on Google/Google Scholar to see if any researcher has commented on the &quot;mirror method&quot;; thus far, I have not found anything (though my searches are limited to English).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, did Galileo even perform this experiment? Even if he didn't, would the experiment still work?&lt;/p&gt;&#xA;" OwnerUserId="9953" LastActivityDate="2019-05-10T13:36:58.153" Title="Does Galileo's method of tracing a parabola actually work?" Tags="|experimental-physics|galileo|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9583" PostTypeId="2" ParentId="9580" CreationDate="2019-05-09T19:01:45.573" Score="10" Body="&lt;p&gt;Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the &lt;em&gt;Principia&lt;/em&gt; immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the &lt;em&gt;Principia&lt;/em&gt;, with a focus on parts of Book 1, section 1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(When I return to my sources, I'm away from base right now, I will put in online references to the &lt;em&gt;Principia&lt;/em&gt; in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in &lt;a href=&quot;https://books.google.com/books?id=Tm0FAAAAQAAJ&amp;amp;pg=PA41&quot; rel=&quot;noreferrer&quot;&gt;The Mathematical Principles of Natural Philosophy, vol.1 of 2&lt;/a&gt;, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'.  When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri.  Newton clearly considered such methods as not well justified, hence his reliance on limits. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is further material that contributes to an answer to the current question in &lt;a href=&quot;https://hsm.stackexchange.com/questions/2362/why-is-calculus-missing-from-newtons-principia/6535#6535&quot;&gt;Why is calculus missing from Newton&amp;#39;s Principia?&lt;/a&gt; , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in &lt;a href=&quot;https://hsm.stackexchange.com/questions/7988/did-michel-rolle-say-that-the-calculus-is-a-collection-of-ingenious-fallacies/7989#7989&quot;&gt;Did Michel Rolle say that the calculus is &amp;quot;a collection of ingenious fallacies&amp;quot;?&lt;/a&gt; . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's &lt;em&gt;Principia&lt;/em&gt; to provide the justification that did not appear to be available elsewhere.  Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.  &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2019-05-09T19:21:49.793" LastActivityDate="2019-05-09T19:21:49.793" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9585" PostTypeId="1" AcceptedAnswerId="9586" CreationDate="2019-05-10T01:38:41.967" Score="10" ViewCount="923" Body="&lt;p&gt;I've heard anecdotally that Bernhard Riemann was interested in applying his discovery/invention of Riemannian geometry to model gravitation about 50 years before Einstein managed to do so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this &lt;a href=&quot;https://www.youtube.com/watch?v=AmUI2qf9uyo&amp;amp;t=3m44s&quot; rel=&quot;noreferrer&quot;&gt;video&lt;/a&gt;, we hear Edward Witten say,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Riemann almost did it. Riemann had the conception that gravity was based on curved [...] space. He didn't conceive of curved spacetime, because special relativity hadn't happen yet, so time and space haven't been unified. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In this &lt;a href=&quot;https://en.wikipedia.org/wiki/Mechanical_explanations_of_gravitation&quot; rel=&quot;noreferrer&quot;&gt;wikipedia article&lt;/a&gt;, we see written,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Similar to Newton, but mathematically in greater detail, Bernhard Riemann assumed in 1853 that the gravitational aether is an incompressible fluid and normal matter represents sinks in this aether. So if the aether is destroyed or absorbed proportionally to the masses within the bodies, a stream arises and carries all surrounding bodies into the direction of the central mass. Riemann speculated that the absorbed aether is transferred into another world or dimension.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These quotes sparked my interest in finding all the sources (preferably primary sources) where Riemann writes about gravity. This is for both where he considers curved space and the aether-based models. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any sources to back up either one of the quotes? &lt;/p&gt;&#xA;" OwnerUserId="6965" LastEditorUserId="55" LastEditDate="2019-05-10T05:33:49.623" LastActivityDate="2019-05-10T08:55:11.327" Title="What are the references for Riemann's discussion of gravity?" Tags="|gravity|riemannian-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9586" PostTypeId="2" ParentId="9585" CreationDate="2019-05-10T05:30:48.903" Score="10" Body="&lt;p&gt;Riemann discussed a &quot;unified field theory&quot;, including light, electromagnetism and gravity, in the unpublished paper Neue Mathematische Principien der Naturphilosophie (New Mathematical Principles of Natural Philosophy, 1853, the title obviously alludes to Newton's), and in Gravitation und Licht (Gravity and Light), the last section of his Fragmente on Naturphilosophie (both published posthumously in 1876). For English translation see e.g. &lt;a href=&quot;https://openlibrary.org/books/OL11801405M/Bernhard_Riemann_Collected_Papers&quot; rel=&quot;noreferrer&quot;&gt;Collected Papers, translated by Baker, Christenson, and Orde, 2004&lt;/a&gt;. Useful summaries are given by &lt;a href=&quot;http://www.numdam.org/article/RHM_1995__1_1_3_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;Bottazzini-Tazzioli in Naturphilosophie and its role in Riemann's mathematics&lt;/a&gt; and &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-60039-0_2&quot; rel=&quot;noreferrer&quot;&gt;Gray in Riemann on Geometry, Physics,&#xA;and Philosophy&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Witten is largely off, what he says is more applicable to Clifford, Wikipedia is closer, although Riemann's ether is not exclusively &quot;gravitational&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Most of Riemann's natural philosophy was motivated by early Herbart's interpretation of Kant (1802), including the idea that the worldspace (Weltraum) is filled with ether which flows through the atoms and then dematerializes (Herbart is also the only person, besides Gauss, credited by name in Riemann's famous geometry lecture). Some quotes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;My main work concerns a new conception of the known laws of nature — an expression of them by means of other fundamental concepts — whereby the use of experimental data concerning the interaction between heat, light, magnetism, and electricity would make possible an investigation of their interrelationship. I was led to this primarily through the study of the works of Newton, Euler and, on the other side, Herbart.&lt;/em&gt; [undated, 1851?]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Both classes of phenomena may be explained, if we suppose that the whole of infinite space is filled with a uniform substance, and each particle of substance acts only on its immediate neighbourhood. The mathematical law according to which this occurs can be considered as divided into &lt;br&gt;&#xA;  1) the resistance of a particle of substance to alteration in volume; &lt;br&gt;&#xA;  2) the resistance of a physical line element to alteration in length. &lt;br&gt;&#xA;  Gravitation and electrostatic attraction and repulsion are founded on the first part; propagation of light and heat, and electrodynamic or magnetic attraction and repulsion on the second.&lt;/em&gt;&quot; [1853]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mathematically, according to Bottazzini-Tazzioli, Riemann’s model amounts to supposing that ether is an elastic, homogeneous, isotropic medium, a popular theory at the time, to which Cauchy, Lame and others contributed earlier. It was vaguely discussed &lt;a href=&quot;https://en.wikipedia.org/wiki/Luminiferous_aether#Particles_vs._waves&quot; rel=&quot;noreferrer&quot;&gt;already by Newton&lt;/a&gt;. Euler expressed even closer ideas in his Letters to German Princess (1760s), but they were largely forgotten. In 1858 Riemann wrote a paper deriving equations for a theory of electrodynamics, but withdrew it, probably due to a mistake in exchanging the order of integration. In his 1861 lectures he tried to relate it to the propagation of light. In Gravity and Light Riemann even writes &quot;&lt;em&gt;this substance can therefore be conceived as a physical space whose points move in geometrical space&lt;/em&gt;&quot;, but he never related gravity to curvature or curved space, and even wrote that &quot;&lt;em&gt;the basis for the metric relations must be sought outside it, in binding forces acting upon it&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea that dynamics of matter is determined by the curving of space itself was first mentioned by Sylvester in 1869, with credit to Clifford, who delivered his own lecture on the subject to the Cambridge Philosophical Society in 1870. Only a summary of it survives, published in 1876 &lt;a href=&quot;https://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter&quot; rel=&quot;noreferrer&quot;&gt;On the Space-Theory of Matter&lt;/a&gt;, very vague and telegraphic, with explicit credit to Riemann for inspiration. He does not mention gravity specifically, only application to the &quot;double refraction&quot; (presumably, a reference to the then unexplained polarization of skylight, see commentary by &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1807/1807.09230.pdf&quot; rel=&quot;noreferrer&quot;&gt;Galindo and Cervantes-Cota&lt;/a&gt;). In the unfinished book &lt;a href=&quot;https://archive.org/details/commexscien00clifuoft/page/n14&quot; rel=&quot;noreferrer&quot;&gt;The Common Sense of the Exact Sciences (1885)&lt;/a&gt;, completed and posthumously published by Pearson, the latter wondered more pointedly &quot;&lt;em&gt;whether physicists might not find it simpler to assume&#xA;that space is capable of a varying curvature, and of a resistance to that&#xA;variation, than to suppose the existence of a subtle medium pervading&#xA;an invariable homaloidal space&lt;/em&gt;&quot;. But even he does not mention gravity by name. Clifford did see himself as following in Riemann's footsteps, but, as Gray writes, &lt;em&gt;there is no evidence that Riemann took that step&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-10T08:55:11.327" LastActivityDate="2019-05-10T08:55:11.327" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9588" PostTypeId="1" AcceptedAnswerId="9589" CreationDate="2019-05-10T16:44:07.407" Score="3" ViewCount="1225" Body="&lt;p&gt;What was Isaac Newton's contribution to optics? I know that he believed in the corpuscular theory of light. Did he know about the laws of reflection ad refraction?&lt;/p&gt;&#xA;" OwnerUserId="9960" LastActivityDate="2019-05-10T17:45:57.513" Title="What was Isaac Newton's contribution to optics?" Tags="|physics|newton|optics|light|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9589" PostTypeId="2" ParentId="9588" CreationDate="2019-05-10T17:38:21.617" Score="5" Body="&lt;p&gt;Contribution of Newton to optics is enormous. He is considered a founding father of physical optics.&#xA;I can only give some examples. His main discovery was that the sunlight can be dissolved into colors (spectrum). The discovery which lead to spectroscopy, and eventually to quantum mechanics. He also analysed what is called &quot;Newton rings&quot; (discovered by Hooke short time before). On the practical side he made one of the first reflecting telescopes, this kind of telescope is still called &quot;Newtonian&quot;. His book &quot;Opticks&quot; lays the foundation of the subject of &quot;physical optics&quot;, and can be compared in its influence only to Principia of the same Newton.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-10T17:45:57.513" LastActivityDate="2019-05-10T17:45:57.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9590" PostTypeId="2" ParentId="9580" CreationDate="2019-05-11T09:54:17.387" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookI-I&quot; rel=&quot;nofollow noreferrer&quot;&gt;Section 1 of book 1 of Principia&lt;/a&gt; opens with a lemma that can strike us as sounding almost modern:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But a second thought raises some doubts. First, this is a &lt;em&gt;lemma&lt;/em&gt;, not a definition of limit. The meaning of &quot;converge continually&quot; and &quot;approach&quot; are assumed to be already understood, the lemma is meant to derive a &lt;em&gt;property&lt;/em&gt; from that. Second, it talks of &quot;quantities&quot;. We also talk of quantities and their limits. But Newton surely can not refer to our notion of functions which assign values to arguments that first appears in Dirichlet's work from 19th century. Or even to &quot;analytic expressions&quot; featured in Euler's 18th century textbooks. Finally, Newton does not even have our idea of a real line assembled from points serving as arguments and values, the arithmetical continuum of Weierstrass, Dedekind and Cantor. The 17th century line is still Euclidean/Aristotelian, with points as merely external marks on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It becomes clearer that Newton's limits, whatever they are, can not be modern, his primitives are different, he works in a different system of mathematical concepts. There might be a sense in which Porciau's opinion that Newton &quot;&lt;em&gt;was the first to present an epsilon argument&lt;/em&gt;&quot; is justified, but it would be similar to the sense in which Eudoxus was first to work with Dedekind cuts. It only means that some of their manipulations can be closely mimicked by modern ones, and we agree to ignore the meaning of what is being manipulated. And that there is an evolutionary chain connecting one to the other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are Newton's &quot;quantities&quot; then? We find an explicit description in his &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Newton/Quadratura/HarrisIQ.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quadrature of Curves (1692)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I don’t here consider Mathematical Quantities as composed of Parts extreamly small, but as generated by a continual motion.  Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux, and so in the rest. These Geneses are founded upon Nature, and are every Day seen in the motion of Bodies&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now it becomes clear where the pre-assumed understanding of &quot;converge continually&quot; and &quot;approach&quot; comes from. Newton takes the idea of motion as &lt;em&gt;intuitively given&lt;/em&gt;, limits with their properties are then founded on it. In the Scholium to Lemma XI of the same section Newton explicitly appeals to the intuitive idea of instantaneous velocity to justify existence of limits, for example. And in &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookII-II&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lemma 2 of Book 2&lt;/a&gt; he talks of &quot;genita&quot;, &quot;&lt;em&gt;quantities I here consider as variable and indetermined, and increasing or decreasing, as it were, by a perpetual motion or flux&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This conception of limits, and calculus generally, relying on the given intuition of motion and its observed properties, came to be called &lt;em&gt;kinematic&lt;/em&gt;. It has roots in some works of Archimedes, such as On Spirals, where he seems to rely on something like the parallelogram of velocities to draw tangents. Newton's teacher Barrow lectured on Archimedes, and the kinematic conception of curves is explicit in his &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasures-isaac-barrows-geometrical-lectures&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometric Lectures&lt;/a&gt;, which Newton helped prepare for publication, see &lt;a href=&quot;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyer's History of Calculus, p.189&lt;/a&gt;. But in the early years Newton also used manipulations with infinitesimals, inherited through Barrow from Fermat, which he later found objectionable. So I would have to agree with Ferraro's assessment in &lt;a href=&quot;https://halshs.archives-ouvertes.fr/halshs-00655601/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some Mathematical Aspects of Newton’s Principia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him... Indeed, I think that Newton’s concept of  first and ultimate  ratio can  be  reduced  to  the  modern  concept  of limits: it is true that Newton has a clear idea of what meaning “approaching  a limit” [is], but this is only an intuitive and non-mathematical idea  that  is  entirely  different  from  the  modern,  mathematical  concept of limit.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Indeed, the &quot;mechanical&quot; aspect of Newton's calculus was explicitly criticized in the 18th century, as &quot;foreign&quot; to pure mathematics, by D'Alambert and l'Huillier, among others. A comprehensive study of 17th century mathematical conceptions is &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF00327940.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whiteside's Patterns of Mathematical Thought in the later Seventeenth Century&lt;/a&gt; (p.374ff on Newton specifically), on Principia see also his &lt;a href=&quot;http://adsabs.harvard.edu/full/1970JHA.....1..116W&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Principles Underlying Newton's Principia Mathematica&lt;/a&gt;. Arthur in &lt;a href=&quot;https://www.humanities.mcmaster.ca/~rarthur/papers/Leery.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leery Bedfellows: Newton and Leibniz&#xA;on the Status of Infinitesimals&lt;/a&gt; contrasts Newton's kinematic conception of calculus to the Leibniz's one, including a detailed discussion of &quot;quantities&quot; and Lemma 1, and the changes from early to late works. On the later fates of the kinematic conception, developed by McLaurin and still more than visible in Cauchy (despite his common assimilation to Weierstrass, his &quot;variables&quot; are not unlike Newton's &quot;quantities&quot;) see &lt;a href=&quot;http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/GrabinerOriginsofCauchysRigorousCalculus.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grabiner's book Origins of Cauchy's Rigorous Calculus&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-14T05:11:05.157" LastActivityDate="2019-05-14T05:11:05.157" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="9591" PostTypeId="2" ParentId="6240" CreationDate="2019-05-11T16:04:49.893" Score="3" Body="&lt;p&gt;The reference to the book on wave propagation (Leçons sur la propagation des ondes et les équations de l'hydrodynamique. Cours du collège de France, Hermann, 1903) &#xA;available at &lt;a href=&quot;https://archive.org/details/leonssurlaprop00hada/page/n12&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/leonssurlaprop00hada/page/n12&lt;/a&gt; &#xA;indeed contains this lemma in the Note I, on p. 352, under the form:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;if &lt;span class=&quot;math-container&quot;&gt;$ p \in \mathbb{N}^* $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ F \in \mathcal{C}^p (\mathbb{R}^d) $&lt;/span&gt;, then there exist &lt;span class=&quot;math-container&quot;&gt;$ \varphi_1 , \dotsc , \varphi_d \in \mathcal{C}^{p-1} (\mathbb{R}^d \times \mathbb{R}^d ) $&lt;/span&gt; such that for every &lt;span class=&quot;math-container&quot;&gt;$(x_1, \dotsc , x_d )$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$( y_1 , \dotsc , y_d ) \in \mathbb{R}^d $&lt;/span&gt;, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; F( x_1 , \dotsc , x_d ) - F( y_1 , \dotsc , y_d ) &#xA; = \sum_{j=1}^d (x_j - y_j ) \varphi_j ( x_1 , \dotsc , x_d , y_1 , \dotsc , y_d ) .&#xA;$$&lt;/span&gt;&#xA;For the proof, J. Hadamard claims that for &lt;span class=&quot;math-container&quot;&gt;$ d =1 $&lt;/span&gt;, it is &quot;easy to verify&quot; that &#xA;&lt;span class=&quot;math-container&quot;&gt;$ (x,y) \mapsto \frac{F(x)-F(y)}{x-y} $&lt;/span&gt; is of class &lt;span class=&quot;math-container&quot;&gt;$ \mathcal{C}^{p-1} (\mathbb{R}) $&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$ F \in \mathcal{C}^p (\mathbb{R}) $&lt;/span&gt;, and deduce the general case by writing&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA; F( x_1 , \dotsc , x_d ) - F( y_1 , \dotsc , y_d ) &#xA; &amp;amp; = [ F( x_1 , x_2 , \dotsc , x_d ) - F( y_1 , x_2 , \dotsc , x_d ) ] &#xA; \\&#xA; &amp;amp; \quad + [ F( y_1, x_2 , \dotsc , x_d ) - F( y_1 , y_2 , x_3 , \dotsc , x_d ) ] &#xA; \\&#xA; &amp;amp; \quad + \dotsc + [ F( y_1, y_2 , \dotsc , y_{d-1} , x_d ) - F( y_1 , y_2 , \dotsc , y_{d-1} , y_d ) ] .&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="9963" LastActivityDate="2019-05-11T16:04:49.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9593" PostTypeId="1" AcceptedAnswerId="9595" CreationDate="2019-05-12T01:26:25.527" Score="1" ViewCount="41" Body="&lt;p&gt;Alfred Tarski writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The whole of the old traditional logic...can almost entirely be reduced to the theory of the fundamental relations among classes, that is, to a small fragment of the entire theory of classes. (page 39)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a footnote he remarks that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;These relations were first investigated in an exhaustive manner by J. D. GERGONNE (1771-1859)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm aware that &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Diez_Gergonne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gergonne&lt;/a&gt; started his own journal &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Annales_de_Gergonne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Annales de Mathématiques Pures et Appliquées&lt;/a&gt;&lt;/em&gt;, but I don't know where in his journal (or elsewhere) he published his results on the fundamental relations among classes. I have found some of these &lt;a href=&quot;https://archive.org/details/gergonne-annales&quot; rel=&quot;nofollow noreferrer&quot;&gt;Annales on Internet Archive&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where can I find sources for Gergonne's published work on the fundamental relations among classes?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Tarski, A. (1996). Introduction to Logic and to the Methodology of the Deductive Sciences. Translated by Olaf Helmer. Dover Publications.&lt;/p&gt;&#xA;" OwnerUserId="9677" LastActivityDate="2019-05-12T06:15:55.173" Title="Where can I find sources for Gergonne's research on the fundamental relations among classes?" Tags="|mathematics|logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9594" PostTypeId="1" CreationDate="2019-05-12T04:58:53.650" Score="1" ViewCount="139" Body="&lt;p&gt;Had this &lt;strong&gt;non&lt;/strong&gt;-analytic theorem ever been known before Euler discovered his analytic formula: &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Cos}(x)+i\operatorname{Sin}(x)= e^{ix}$&lt;/span&gt; ?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please notice the distinction between, the theorem indicated in the title, and De Moivre's theorem - which is &lt;em&gt;not&lt;/em&gt; about a &lt;em&gt;sum&lt;/em&gt; of angles.&lt;/p&gt;&#xA;" OwnerUserId="9966" LastEditorUserId="9966" LastEditDate="2019-05-13T13:04:37.197" LastActivityDate="2019-05-13T13:04:37.197" Title="$[\operatorname{Cos}(x)+i\operatorname{Sin}(x)]\cdot[\operatorname{Cos}(y)+i\operatorname{Sin}(y)]=\operatorname{Cos}(x+y)+i\operatorname{Sin}(x+y)$" Tags="|complex-analysis|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9595" PostTypeId="2" ParentId="9593" CreationDate="2019-05-12T06:15:55.173" Score="1" Body="&lt;p&gt;See &lt;a href=&quot;https://eudml.org/doc/79642&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essai de dialectique rationnelle (1816-17)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A brief description can be found in Leila Haaparanta (editor), &lt;a href=&quot;https://books.google.it/books?id=0jXavKsArnIC&amp;amp;pg=PA133&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Development of Modern Logic (2009)&lt;/a&gt;, &lt;strong&gt;Ch.3 : Logic from Humanism to Kant&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2019-05-12T06:15:55.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9596" PostTypeId="1" CreationDate="2019-05-12T12:20:00.613" Score="6" ViewCount="318" Body="&lt;p&gt;Cylindrical coordinates&lt;span class=&quot;math-container&quot;&gt;$ x=r\cos θ, y=r\sin θ, z=w$&lt;/span&gt; seem to be a simple generalization of polar coordinates. When did they appear first? Also, who came up with the name?&lt;/p&gt;&#xA;" OwnerUserId="9964" LastEditorUserId="9722" LastEditDate="2019-05-23T16:59:04.313" LastActivityDate="2019-05-23T16:59:04.313" Title="Who introduced cylindrical coordinates?" Tags="|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9598" PostTypeId="2" ParentId="8332" CreationDate="2019-05-13T10:28:38.150" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://www.math.ucdavis.edu/~greg/&quot; rel=&quot;noreferrer&quot;&gt;Greg Kuperberg&lt;/a&gt;, mathematician and son of the couple of mathematicians Wlodymierz and Krystina Kuperberg, published, separately, a paper with his father on geometric combinatorics, at the very start of his career (1990) and later on a relevant paper (Annals of Math - 1996) on counterxamples to the Seifert conjecture with his mother. Later on he published yet another paper with his father and a third coauthor.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2019-05-13T10:28:38.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9599" PostTypeId="2" ParentId="9596" CreationDate="2019-05-13T22:45:56.870" Score="6" Body="&lt;p&gt;It is interesting that Coolidge, who has an entire chapter on various coordinate systems in his &lt;a href=&quot;https://archive.org/details/historyofgeometr00cool&quot; rel=&quot;noreferrer&quot;&gt;History of Geometrical Methods&lt;/a&gt; does not even mention cylindrical coordinates. This is probably because they amount to just adding an extra variable to polar coordinates, not exactly a notable move. The earliest explicit example I recall (without the name, of course) is by Euler in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E302.html&quot; rel=&quot;noreferrer&quot;&gt;De motu vibratorio tympanorum (On the motion of vibrations in drums)&lt;/a&gt;, &lt;em&gt;Novi Commentarii academiae scientiarum Petropolitanae&lt;/em&gt; 10, 1766. Dutka describes it as follows in &lt;a href=&quot;https://www.jstor.org/stable/41134001&quot; rel=&quot;noreferrer&quot;&gt;On the Early History of Bessel Functions&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Euler sets up the equations of motion for a homogeneous membrane in which the tensions in the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; directions are equal with no external forces acting on it... He obtains an equation of the form&lt;/em&gt;&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\partial^2z}{\partial t^2}=a^2\left(\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}\right)\hspace{1cm} (5)&#xA;$$&lt;/span&gt;&#xA;  &lt;em&gt;where the right member of the equation is proportional to the sum of the forces in the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; directions, and &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; depends on the tension and density of the membrane.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Euler's solution for the case of a rectangular membrane is proportional to a product of sinusoidal functions of &lt;span class=&quot;math-container&quot;&gt;$x, y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; respectively. He then proceeds to consider the case of a circular membrane in which the tension along the boundary is uniform. He introduces what are essentially cylindrical coordinates &lt;span class=&quot;math-container&quot;&gt;$(r, \theta, z)$&lt;/span&gt; in place of the rectangular coordinates and substitutes into (5).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What Euler gets is the now standard equation for the Bessel functions, also known as the cylindrical functions. This is why. He develops the, also standard, power series formula for these functions. Even before Euler (and Bessel) Daniel Bernoulli studied them already, although not nearly as thoroughly as Bessel. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the name, Google Scholar does not have non-spurious hits before 1870-s, &lt;a href=&quot;https://books.google.com/books?id=XMsGAAAAYAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Cambridge's Solutions of the Senate-House Problems and Riders (1875)&lt;/a&gt; is the earliest. &lt;a href=&quot;https://books.google.com/books?id=ym1LAAAAMAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Frost's Solid Geometry textbook (1886)&lt;/a&gt; already has a section named after cylindrical coordinates.  &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-14T05:06:39.773" LastActivityDate="2019-05-14T05:06:39.773" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9600" PostTypeId="1" CreationDate="2019-05-14T04:09:42.213" Score="1" ViewCount="171" Body="&lt;p&gt;Early civilizations of india, Greece, Babylon, China and Egypt have progressed in astronomy. Is there any shared knowledge among them and if any how can we find out?&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="55" LastEditDate="2019-05-14T05:03:36.617" LastActivityDate="2019-05-15T00:30:58.777" Title="Did knowledge of astronomy spread among Indo-European civilizations in antiquity?" Tags="|astronomy|archaeoastronomy|ancient|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9601" PostTypeId="2" ParentId="9600" CreationDate="2019-05-14T04:56:57.633" Score="2" Body="&lt;p&gt;Much of the ancient transmission went through Greece. Astronomical (and other) knowledge was transmitted from Egypt to Greece at least by the time of early Pythagoreans (c. 500 BC), likely earlier. According to some sources, Pythagoras and Democritus visited Egypt personally to study with the priests. There might have been some early transmission from Babylon, such as of the Metonic and Saros cycles. After the Alexander's conquests (c. 320 BC), Babylonian astronomical observations and computational methods (including sexagesimal arithmetic) came to be widely used by Hellenistic astronomers, starting with Hipparchus (c.250 BC), see &lt;a href=&quot;https://en.wikipedia.org/wiki/Hipparchus#Babylonian_sources&quot; rel=&quot;nofollow noreferrer&quot;&gt;Babylonian sources&lt;/a&gt;. Recently, evidence of Babylon to Hellenistic Egypt transmission came to light, see &lt;a href=&quot;http://www.nino-leiden.nl/message/ancient-knowledge-transfer-egyptian-astronomical-instructions-based-on-babylonian-methods&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient Knowledge Transfer: Egyptian Astronomical Instructions based on Babylonian Methods&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Egyptian astronomers computed the position of the planet Mercury with methods originating from Babylonia. This is proven by a study of two unpublished Egyptian instructional texts from the Ashmolean Museum (Oxford)... The texts date to 1-50 AD and are written in the Demotic language, a late stage of ancient Egyptian, on two ostraca (potsherds). They are the only known texts from Greco-Roman Egypt with instructions for computing astronomical phenomena with Babylonian methods. The instructions correspond exactly to methods invented in Babylonia several centuries earlier (400-300 BC). Surprisingly, the ostraca employ a mathematical formulation not found in Babylonian texts but whose existence has long been suspected by historians of astronomy.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ptolemy employed Babylonian methods, inherited from Hipparchus, extensively in the Almagest (c.150 AD), the &lt;em&gt;opus magnum&lt;/em&gt; of geocentric astronomy. At least parts of the Almagest were known in India by the time of Aryabhata (c.500 AD), see &lt;a href=&quot;https://people.sc.fsu.edu/~dduke/india8.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duke's The Equant in India&lt;/a&gt;. China remained isolated until much later, flat Earth beliefs were still current by the time of Jesuit arrival in 1600-s, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/6778/why-was-china-slow-to-recognise-the-sphericity-of-earth&quot;&gt;Why was China slow to recognise the sphericity of Earth?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=xgAKDAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Circulation of Astronomical Knowledge in the Ancient World&lt;/a&gt; is a comprehensive scholarly collection, highlights are given in &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's History of Astronomy&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-14T05:05:27.450" LastActivityDate="2019-05-14T05:05:27.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9605" PostTypeId="1" CreationDate="2019-05-15T14:44:16.740" Score="3" ViewCount="386" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; the visually similar &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; are often used to represent biological fitness in population genetics models.  (Sometimes &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is used for absolute fitness, and &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; for relative fitness.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These symbols appear in numerous textbooks and journal articles.  I could fill up the page with examples.  Here are two well-known books that use &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;, an introductory textbook, and an advanced reference, respectively: John Gillespie's &lt;em&gt;Population Genetics: A Concise Guide&lt;/em&gt;, 1st edition, 1998; Warren J. Ewens, &lt;em&gt;Mathematical Population Genetics I: Theoretical Introduction&lt;/em&gt;, 2004.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both &quot;w&quot; and &quot;&lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;&quot; seem like odd choices, though, especially given that the people who first developed population genetics (Fisher, Wright, Dobzhansky, et al.) mainly published in English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was &quot;w&quot; (or &quot;&lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;&quot;) intended to symbolize, or was it an arbitrary choice?  Who introduced it, and when?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(There are other symbols in use, such as &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; for &quot;selection cofficient&quot;, which generally has a different meaning, but is closely related to most uses of &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="277" LastEditorUserId="4251" LastEditDate="2021-09-08T06:12:06.803" LastActivityDate="2021-09-08T06:12:06.803" Title="When and why was the symbol $w$ introduced for biological fitness?" Tags="|notation|biology|evolution|genetics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9606" PostTypeId="1" CreationDate="2019-05-15T07:08:06.597" Score="2" ViewCount="75" Body="&lt;p&gt;How was the concept of &lt;span class=&quot;math-container&quot;&gt;$\ln(x)$&lt;/span&gt; found before the man knows that it is the area under hyperbola or it is related to the power of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; (base of logarithm). How did Napier compute the value of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\ln(x)$&lt;/span&gt; before the concept of power of numbers was used in math?&lt;/p&gt;&#xA;" OwnerDisplayName="user359791" LastActivityDate="2019-05-15T15:18:41.697" Title="How was logarithm discovered?" Tags="|calculus|" AnswerCount="0" CommentCount="6" ClosedDate="2019-05-17T06:41:42.573" ContentLicense="CC BY-SA 4.0" />
  <row Id="9608" PostTypeId="2" ParentId="6638" CreationDate="2019-05-15T20:09:00.730" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://assets.press.princeton.edu/chapters/s9308.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://assets.press.princeton.edu/chapters/s9308.pdf&lt;/a&gt;&#xA;is about history of maths. It has 3 diagrams.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 1.1: Thee “classical” Eurocentric trajectory &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 1.2: A modicied Eurocentric trajectory &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Figure 1.3: An alternative trajectory for the “Dark Ages”  (the real thing)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main message of figure 1.3 is that it is dangerous to characterize the history of mathematics solely in terms of European developments. The darkness that was supposed to have descended over Europe for a thousand years before the illumination that came with the Renaissance did not interrupt mathematical activity elsewhere. Indeed, as we shall see in later chapters, the period saw not only a mathematical renaissance in the Islamic world but also high points of Indian and Chinese mathematics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;￼This figure includes history of knowledge transmissions through the ages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The link is &lt;a href=&quot;http://assets.press.princeton.edu/chapters/s9308.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://assets.press.princeton.edu/chapters/s9308.pdf&lt;/a&gt;&#xA;figure 1.4 &#xA;The author is CK Raju.&#xA;Sorry. I couldn't figure out how to insert the figures. But you can see them in the pdf file.&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-05-15T20:09:00.730" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9609" PostTypeId="2" ParentId="8283" CreationDate="2019-05-16T00:59:48.620" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jy%C4%81,_koti-jy%C4%81_and_utkrama-jy%C4%81&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Jyā,_koti-jyā_and_utkrama-jyā&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;defines sine and cosine, mis-translations of jya and kojya (or koti-jya) as jiba by the Arabs again mis translated to Latin &quot;sinus&quot; and co-sine.&#xA;The sequence is jiva (Sanskrit) to jab (Arabic) to sine (bosom in Latin). &#xA;The figure is here&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/File:Jya_and_kotijya.JPG&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/File:Jya_and_kotijya.JPG&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-05-16T00:59:48.620" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9611" PostTypeId="2" ParentId="9605" CreationDate="2019-05-16T19:33:08.483" Score="2" Body="&lt;p&gt;It seems to be a game of broken telephone with Boltzmann's formula for the entropy at the beginning, and &lt;a href=&quot;https://archive.org/details/geneticaltheoryo031631mbp/page/n12&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fisher's Genetical Theory of Natural Selection (1930)&lt;/a&gt; in the middle. Fisher's statistical measures of fitness are the reproductive value denoted &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, &quot;&lt;em&gt;the present value of the future offspring&lt;/em&gt;&quot;, by analogy to compound interest in economics, and what he calls the Malthusian parameter denoted &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, &quot;&lt;em&gt;the relative rate of increase or decrease of a population when in the steady state&lt;/em&gt;&quot;. He then relates the latter to the gene frequencies &lt;span class=&quot;math-container&quot;&gt;$p,q$&lt;/span&gt; (pp.34-35):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The two groups of individuals bearing alternative genes,&#xA;  and consequently the genes themselves, will necessarily either have&#xA;  equal or unequal rates of increase, and the difference between the&#xA;  appropriate values of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; will be represented by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, similarly the&#xA;  average effect upon &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; of the gene substitution will be represented&#xA;  by &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; measures fitness to survive by the objective fact of&#xA;  representation in future generations, the quantity &lt;span class=&quot;math-container&quot;&gt;$pqa\alpha$&lt;/span&gt; will represent&#xA;  the contribution of each factor to the genetic variance in fitness;&#xA;  the total genetic variance in fitness being the sum of these contributions... whence it follows that, &lt;span class=&quot;math-container&quot;&gt;$\alpha dp =pqa\alpha dt$&lt;/span&gt; and, taking all factors into consideration, the total increase in fitness,&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\Sigma (\alpha dp) = \Sigma (pqa\alpha)dt = Wdt.$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;If therefore the time element &lt;span class=&quot;math-container&quot;&gt;$dt$&lt;/span&gt; is positive, the total change of fitness&#xA;  &lt;span class=&quot;math-container&quot;&gt;$Wdt$&lt;/span&gt; is also positive, and indeed the rate of increase in fitness due&#xA;  to all changes in gene ratio is exactly equal to the genetic variance of&#xA;  fitness &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; which the population exhibits. We may consequently state&#xA;  the fundamental theorem of Natural Selection in the form&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;The rate of increase in fitness of any organism at any time is equal&#xA;  to its genetic variance in fitness at that time.&lt;/em&gt;&lt;/strong&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So for Fisher &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is not the fitness itself, but the genetic variance of fitness. Why he chose &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; Fisher does not say, but he later discusses the analogy to the second law of thermodynamics, as derived from the statistical mechanics. This, of course, reminds us of the Boltzmann formula for the entropy &lt;span class=&quot;math-container&quot;&gt;$S=k\ln W$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$p(t)=p_0e^{mt}$&lt;/span&gt;, as in the Malthusian model, then &lt;span class=&quot;math-container&quot;&gt;$p(t+1)=e^{m}p(t)$&lt;/span&gt;, and with &lt;a href=&quot;https://en.wikipedia.org/wiki/Fitness_(biology)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; as defined today&lt;/a&gt; we have &lt;span class=&quot;math-container&quot;&gt;$m=\ln W$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; in the Boltzmann formula is already a case of the broken telephone. It was intended to stand for the &lt;em&gt;Wahrscheinlichkeit&lt;/em&gt; (probability in German) of a macrostate for some distribution of microstates, but was misinterpreted early to stand for the number of microstates, as it does now. If this is what he had in mind, Fisher's use is more true to the original since &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is often described as the probability &quot;&lt;em&gt;that the individual will be included among the group selected as parents of the next generation&lt;/em&gt;&quot; (Maynard Smith). Fisher writes (p.36):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It will be noticed that the fundamental theorem proved above&#xA;  bears some remarkable resemblances to the second law of thermodynamics.&#xA;  Both are properties of populations, or aggregates, true&#xA;  irrespective of the nature of the units which compose them ; both are&#xA;  statistical laws ; each requires the constant increase of a measurable&#xA;  quantity, in the one case the entropy of a physical system and in the&#xA;  other the fitness, measured by &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, of a biological population.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-16T19:47:31.030" LastActivityDate="2019-05-16T19:47:31.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9612" PostTypeId="1" AcceptedAnswerId="9614" CreationDate="2019-05-16T20:25:31.270" Score="4" ViewCount="864" Body="&lt;p&gt;This crossed my mind today...&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is Schrödinger's &lt;strong&gt;cat&lt;/strong&gt; and Newton's &lt;strong&gt;apple&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any other famous animals/plants featured in physics in a similar way?&lt;/p&gt;&#xA;" OwnerUserId="9983" LastEditorUserId="55" LastEditDate="2019-05-17T00:57:56.213" LastActivityDate="2020-04-09T07:47:02.157" Title="What animals or plants were used to illustrate ideas of physics?" Tags="|physics|terminology|" AnswerCount="6" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9613" PostTypeId="2" ParentId="9612" CreationDate="2019-05-16T21:22:51.070" Score="5" Body="&lt;p&gt;There's &quot;&lt;a href=&quot;https://plato.stanford.edu/entries/buridan/#ref-56&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Buridan's ass&lt;/strong&gt;&lt;/a&gt;&quot; in logic, which says that a &quot;hungry donkey&quot; will not be able to decide &quot;between two completely alike bales of hay&quot; (&lt;a href=&quot;https://www.researchgate.net/publication/325430181_Galileo&amp;#39;s_Precursors_Translation_of_Studies_on_Leonardo_da_Vinci_vol_3_by_Pierre_Duhem#pdf-reader-page-31&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duhem 2018 p. 13&lt;/a&gt;) and thus will starve. It's attributed to medieval physicist &lt;a href=&quot;https://www.encyclopedia.com/people/philosophy-and-religion/philosophy-biographies/jean-buridan#proper-ad-encyclopedia_content_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Buridan&lt;/a&gt; (1295-1360), but a physics (not logic) version of it can be found in Aristotle's &lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm#2-25&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Caelo&lt;/em&gt; 295&lt;sup&gt;b&lt;/sup&gt;32 [375.]&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the man who, though exceedingly hungry and thirsty, and both equally, yet being equidistant from food and drink, is therefore bound to stay where he is&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A good treatment of Buridan's ass can be found in Nicholas Rescher's &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5625&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Scholastic Meditations&lt;/em&gt;&lt;/a&gt; (ch. 1) or &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5623&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Studies in the History of Logic&lt;/em&gt;&lt;/a&gt; (ch. 7), &quot;Choice without Preference: The Problem of Buridan's Ass&quot;.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-05-16T21:36:44.080" LastActivityDate="2019-05-16T21:36:44.080" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9614" PostTypeId="2" ParentId="9612" CreationDate="2019-05-16T23:57:21.837" Score="4" Body="&lt;p&gt;Arhytas made the first known steam powered toy in the shape of a pigeon, see &lt;a href=&quot;https://www.ancient-origins.net/ancient-technology/steam-powered-pigeon-archytas-flying-machine-antiquity-002179&quot; rel=&quot;nofollow noreferrer&quot;&gt;The steam-powered pigeon of Archytas&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bbc.com/bitesize/guides/zq8s2nb/revision/4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Priestley put a mint plant&lt;/a&gt; in a closed container with a burning candle. The candle flame went out after using up the oxygen, but after 27 days Priestley re-lit the candle, demonstrating that mint produces oxygen of its own. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/August_Kekul%C3%A9#Kekul%C3%A9%27s_dream&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kekulé claimed&lt;/a&gt;, 25 years later, that he discovered the shape of the benzene molecule after having a dream of a snake biting its own tail. As with the apple, there are doubts that this actually happened. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Snake biting its own tail is an alchemic symbol called the ouroboros. It is sometimes also associated with Wallis's &lt;a href=&quot;https://en.wikipedia.org/wiki/Infinity_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;symbol for mathematical infinity&lt;/a&gt; (he did not make this association).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_conditioning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pavlov's dog&lt;/a&gt; became a common metaphor for psychological conditioning after Pavlov's experiments with causing salivation in dogs by associating food with turning on a light bulb.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Asimov (who is a biochemist, in addition to the science fiction author) wrote a story about a &lt;a href=&quot;https://en.wikipedia.org/wiki/P%C3%A2t%C3%A9_de_Foie_Gras_(short_story)&quot; rel=&quot;nofollow noreferrer&quot;&gt;goose that lays golden eggs&lt;/a&gt;, by transmuting oxygen-18 to gold-197 via an enzyme-catalyzed nuclear process. The goose can not reproduce due to heavy metal poisoning of the eggs, and a biopsy of the liver needed to uncover its secret would kill it. Asimov's riddle is to figure out how to make it reproduce (there is a scientific solution).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://thoughtexperiments.net/people-who-divide-like-an-amoeba/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Parfit's “people who divide like an amoeba”&lt;/a&gt; illustrate a conundrum associated with brain transplants:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;My brain is divided, and each half is housed in a new body. Both resulting people have my character and apparent memories of my life. What happens to me?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Jackson's &lt;a href=&quot;https://en.wikipedia.org/wiki/Qualia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mary, the color scientist&lt;/a&gt;, who knows everything about the physics of colors, but grew up in a colorless room, sees a red tomato for the first time, and learns something new nonetheless. This illustrates the problem with explaining the so-called qualia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A related earlier illustration is due to Wittgenstein, who imagined that everyone has a &lt;a href=&quot;https://www.philosophyonline.co.uk/oldsite/pom/pom_behaviourism_wittgenstein.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;box where they keep a &quot;beetle&quot;&lt;/a&gt;. That is everybody calls it &quot;beetle&quot;, but &quot;&lt;em&gt;no one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle&lt;/em&gt;&quot;. The point is to dispel the idea of a special non-physical entity, &quot;mind&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many of these sorts of things, featuring creatures or not, are what Mach called &lt;a href=&quot;https://plato.stanford.edu/entries/thought-experiment/&quot; rel=&quot;nofollow noreferrer&quot;&gt;thought experiments&lt;/a&gt;, and there is extensive literature on them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-17T00:55:59.780" LastActivityDate="2019-05-17T00:55:59.780" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9615" PostTypeId="1" AcceptedAnswerId="9621" CreationDate="2019-05-17T02:04:02.753" Score="5" ViewCount="201" Body="&lt;p&gt;I have a circular slide rule, but I can't figure out what the markings on the back are used for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The front face has two logarithmic scales, the outer one runs from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt; (exclusive), and the inner scale runs from &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, so it's easy to take square roots or square numbers using the front face.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's a demonstration of multiplying &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; . The slide rule doesn't have instructions, so I'm just guessing on what the proper procedure is for multiplying.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, rotate the multiplicand so it's right under the marker under the top button, and rotate the needle so it points to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, the initial multiplier. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8lV8g.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8lV8g.jpg&quot; alt=&quot;before multiplication&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Next, rotate so that the needle points to the desired multiplier (in this case &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;). The top marker now points to &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; as desired.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3ku2G.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3ku2G.jpg&quot; alt=&quot;after multiplication&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, that all makes sense, but the &lt;em&gt;back&lt;/em&gt; of the slide rule is completely inscrutable to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to have multiple scales, with the innermost scale arranged in some kind of spiral. The scales appear to logarithmically spaced, similar to the scales on the front. However, I can't figure out what computation the scale on the back is supposed to be used for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WhTbR.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WhTbR.jpg&quot; alt=&quot;back&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9243" LastActivityDate="2019-05-17T22:03:09.983" Title="Usage of reverse side of circular slide rule" Tags="|mathematics|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9616" PostTypeId="2" ParentId="9615" CreationDate="2019-05-17T07:15:28.503" Score="4" Body="&lt;p&gt;What an entirely admirable object!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The scales on the back seem to me to be easier to understand than on the front, but without having it in my hand that may be just an illusion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The spiral scale is, like the circular one, one decade per revolution. So stretching it out into a line, you have the ex-circular scale being one decade long, at so many inches per decade, while the spiral scale is &lt;em&gt;two&lt;/em&gt; decades long at the same number of inches per decade. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could be useful (for example) if you want to get 3 x 4 = 12 rather than 3 x 4 = 1.2 “and then to position the decimal point, count on your fingers how many times you have got past the 1 on the scale”. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps, but it is all a total guess. &lt;/p&gt;&#xA;&#xA;&lt;h2&gt;jumping in here...(CWitthoft)&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;this is exactly correct. See this &lt;a href=&quot;http://nsg.upor.net/slide/cratlas.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;explanation page&lt;/a&gt; that has a representative device you can play with.&lt;/p&gt;&#xA;" OwnerUserId="4209" LastEditorUserId="609" LastEditDate="2019-05-17T11:37:52.557" LastActivityDate="2019-05-17T11:37:52.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9617" PostTypeId="1" CreationDate="2019-05-17T07:48:42.660" Score="1" ViewCount="379" Body="&lt;p&gt;I was searching for all the names who discovered electricity. As I was  seeing the names, the first name came of William Gilbert and I searched how had he discovered I got to know. Then the second name came was of Otto von guericke and I searched how had he discovered and I got to know. Next name was of Robert Boyle and when I searched I couldn't find what he did in field of electricity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: How Robert Boyle discovered electricity after Otto von Guericke?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="217" LastEditDate="2019-05-21T00:14:21.510" LastActivityDate="2019-05-21T00:14:21.510" Title="Robert Boyle discovery of electricity" Tags="|biographical-details|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9618" PostTypeId="1" AcceptedAnswerId="9624" CreationDate="2019-05-17T09:16:49.247" Score="4" ViewCount="542" Body="&lt;p&gt;In my life¹, I have never seen a symbol other than ≈ used in handwriting to express “approximately equal”.&#xA;The symbol ~ was only used for more mathematical purposes such as equivalence, proportionality, or “is distributed as”.&#xA;Yet, I often see the symbol ~ used for “approximately equal” in the scientific literature, in particular in fields where TeX use is scarce.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not want to debate what is right™ here, but I am just curious whether this usage is a result of the limitations of typewriters and predominant keyboard layouts and encodings from the early days of computing, which make ≈ difficult to produce or impossible to encode, while ~ is readily available.&#xA;By contrast, I would expect that in moveable type (and similar systems that do not allow superimposing characters), you either had both or neither glyphs available, as I am not aware of any relevant application for the ~ glyph besides equations in such systems².&#xA;Also, in TeX, both symbols are comparably difficult to use (&lt;code&gt;\approx&lt;/code&gt; and &lt;code&gt;\sim&lt;/code&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any harder evidence that the usage of ~ for “approximately equal” is a consequence of it being a makeshift solution in light of technical restrictions?&#xA;For example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Examples of ~ being used for this purpose in systems that do not technically favour this symbol.&lt;/li&gt;&#xA;&lt;li&gt;Works of the same author using ~ only when technical restrictions applied.&lt;/li&gt;&#xA;&lt;li&gt;Historical guidelines on symbol usage.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&#xA;¹ I went to school and studied physics and math in Germany.&lt;br&gt;&#xA;² On a typewriter, the ~ character could be superimposed with, e.g., n to obtain ñ.&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="13" LastActivityDate="2019-05-18T18:36:35.447" Title="Is using ~ for “approximately equal” a relic of the typewriter and ASCII era?" Tags="|notation|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9619" PostTypeId="2" ParentId="9618" CreationDate="2019-05-17T11:34:25.063" Score="-1" Body="&lt;p&gt;In my experience, the tilde , &quot;~&quot;  ,  is used in mathematics to indicate a loose relationship. In geometry, it specifically means &quot;similar,&quot; i.e. same angles, different length sizes.&lt;br&gt;&#xA; &lt;span class=&quot;math-container&quot;&gt;$\approx$&lt;/span&gt;  is used only for items &quot;close to equality&quot; . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far as I know, there was never a case when either math or physics chose a symbol based on &quot;ease of typing.&quot;   (The only case I've seen where 'ease of typing' was involved is a couple computer languages that used &quot; &amp;lt;-&quot; as the replacement operator, because once upon a time some computer keyboards had a single key for that symbol pair!) &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-05-17T11:34:25.063" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9620" PostTypeId="1" CreationDate="2019-05-17T11:36:32.253" Score="-1" ViewCount="251" Body="&lt;p&gt;How he get to know which one is positive and which one is negative as stated by him that there are two types of fluid?&#xA;Also tell how he thought of two fluids?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="9984" LastEditDate="2019-05-18T03:47:05.423" LastActivityDate="2019-05-23T03:53:22.543" Title="Charles du fay experiment" Tags="|biographical-details|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9621" PostTypeId="2" ParentId="9615" CreationDate="2019-05-17T19:40:42.307" Score="3" Body="&lt;p&gt;The S scale maps degrees to sines on the outer scale. The T scale maps degrees to tangents. &lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2019-05-17T19:40:42.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9622" PostTypeId="2" ParentId="9615" CreationDate="2019-05-17T21:53:46.200" Score="4" Body="&lt;p&gt;The outermost scale for the sine, the scale adjacent to it is degrees, it is marked by S and has angles from 0 to 90. For example, sin 30=0.5. However sin 8=0.139.&#xA;The part of the outer scale from 1 to 2 has a finer division.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The inner spiral scale marked T is also degrees&#xA;from 0 to 45, and it is used to read the tangent on the outermost scale.&#xA;You put the arrow against degree on the inner (spiral) scale and read tangent&#xA;on the outer scale. For example sin 45=0.707, tan 45=1 tan 12=0.21.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-17T22:03:09.983" LastActivityDate="2019-05-17T22:03:09.983" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9624" PostTypeId="2" ParentId="9618" CreationDate="2019-05-18T06:16:41.500" Score="5" Body="&lt;p&gt;There is evidence that the story is more complicated than adoption of pre-existing paper and pencil practice being obstructed by keyboard limitations. Cajori makes some interesting remarks on the symbols for “nearly equal” in &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp/page/n327&quot; rel=&quot;nofollow noreferrer&quot;&gt;History Of Mathematical Notations&lt;/a&gt; (1928):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Among the many uses made in recent years of the sign &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; is that of “nearly equal to,” as in “&lt;span class=&quot;math-container&quot;&gt;$e\sim\frac14$&lt;/span&gt;” [sic!]; similarly, &lt;span class=&quot;math-container&quot;&gt;$e\cong\frac14$&lt;/span&gt; is allowed to stand for “equal or nearly equal to.” [reference to Kratzer] A. Eucken lets &lt;span class=&quot;math-container&quot;&gt;$\simeq$&lt;/span&gt; stand for the lower limit, as in “&lt;span class=&quot;math-container&quot;&gt;$J\simeq45\cdot10^{-40}$&lt;/span&gt; (untere Grenze),” where J means a mean moment of inertia. Greenhill denotes approximate equality by &lt;a href=&quot;https://i.stack.imgur.com/78WI6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/78WI6.png&quot; alt=&quot;two parallel sawtooth lines&quot;&gt;&lt;/a&gt;. An early suggestion due to Fischer was the sign &lt;span class=&quot;math-container&quot;&gt;$\asymp$&lt;/span&gt; for “approximately equal to.” This and three other symbols were proposed by Boon who designed also four symbols for “greater than but approximately equal to” and four symbols for “less than but approximately equal to.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Some comments: Cajori published in 1928, which means that symbols for “nearly equal” were recent then, with the exception of Fisher’s (1829) and Greenhill’s (1892). Second, the now dominant &lt;span class=&quot;math-container&quot;&gt;$\approx$&lt;/span&gt; is not even among them (Greenhill’s &lt;a href=&quot;https://i.stack.imgur.com/78WI6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/78WI6.png&quot; alt=&quot;two parallel sawtooth lines&quot;&gt;&lt;/a&gt; would be the closest). Third, it seems that all these symbols were proposed/designed rather than adopted from pre-existing paper and pencil usage. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said, we do generally see the rise of new symbology towards the end of 19th century (think of Boole, Frege and Peano, for example), which does correlate with the commercial adoption of &lt;a href=&quot;https://en.wikipedia.org/wiki/Typewriter#Early_innovations&quot; rel=&quot;nofollow noreferrer&quot;&gt;typewriters&lt;/a&gt;, and their standardization in 1893–1910. The trend is not exclusive to mathematicians, Kratzer’s &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; appears in a journal for physics, and Eucken’s &lt;span class=&quot;math-container&quot;&gt;$\simeq$&lt;/span&gt; in a journal for physical chemistry. And it would have been hard for paper and pencil usage to spread without typographic support. So it seems fair to infer that the increase in typographic flexibility spurred development and proliferation of new symbols, rather than distortion of existing ones to fit what was available. Of course, this does not mean that at a later time, after the use of &lt;span class=&quot;math-container&quot;&gt;$\approx$&lt;/span&gt; was established, keyboard or ASCII did not motivate authors to make pragmatic substitutions.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="13" LastEditDate="2019-05-18T18:36:35.447" LastActivityDate="2019-05-18T18:36:35.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9625" PostTypeId="2" ParentId="9620" CreationDate="2019-05-18T08:51:49.870" Score="0" Body="&lt;p&gt;He firstly rubbed a glass rod with silk and moved the glass towards the light objects (paper, foil) - they got attracted. Then he rubbed amber with silk and the same thing happened.&lt;br /&gt;&#xA;From these phenomena he was clear that if a body is charged then it will attract light objects but not the heavy object (meaning a body heavier than the charged body).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then he thought: if a heavy body is charged what will happen? So he brought amber near the glass (both charged). As we know amber is little bit heavier than glass. Here he also saw attraction hence he thought if a body is heavy but charged then another charged body can show attraction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then he went to prove this using a different material and the next material was two charged glass objects. He brought those close to each other and he found that they repel.&lt;br /&gt;&#xA;He was astonished as it had been known that electricity phenomena only show attraction not repulsion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So these experiment with various materials sometimes show attraction and sometimes repulsion. From this he was clear that there are two types of fluid, one vitreous (glass like or positive) and the other resinous (amber like or negative). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He thought: when a body is not charged there are equal amounts of fluid in both, but when a body is charged it has two kinds of charges or fluids. He concluded there are two types of fluid because of attraction (happening with unequal things) and repulsion (happening with equal things).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He defined &lt;em&gt;positive&lt;/em&gt; meaning excess of fluid and &lt;em&gt;negative&lt;/em&gt; meaning shortage of fluid. He took as convention that amber is negative and glass is positive.&lt;/p&gt;&#xA;" OwnerUserId="9993" LastEditorDisplayName="user8032" LastEditDate="2019-05-22T21:04:43.823" LastActivityDate="2019-05-22T21:04:43.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9626" PostTypeId="2" ParentId="9617" CreationDate="2019-05-18T08:59:19.480" Score="2" Body="&lt;p&gt;Robert Boyle does the same experiment on electricity that is taking a body and rubbing it to make it a charged body and seeing attraction of light object.He concludes that electric phenomena only shows attraction and not repulsion like the magnet which shows both attraction and repulsion&lt;/p&gt;&#xA;" OwnerUserId="9993" LastActivityDate="2019-05-18T08:59:19.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9627" PostTypeId="1" CreationDate="2019-05-18T15:00:30.203" Score="3" ViewCount="329" Body="&lt;p&gt;I see Peano arithmetic so often abbreviated as &quot;P&quot; or &quot;PA&quot;. Why is Robinson Arithmetic &quot;Q&quot;? Following the obvious pattern, I would have expected R&quot; or &quot;RA&quot;. &lt;/p&gt;&#xA;" OwnerUserId="9499" LastEditorUserId="131" LastEditDate="2019-05-18T19:40:40.147" LastActivityDate="2023-09-10T18:41:50.690" Title="Why is Robinson arithmetic &quot;Q&quot;?" Tags="|mathematics|mathematical-logic|naming-conventions|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9629" PostTypeId="2" ParentId="9627" CreationDate="2019-05-18T19:40:22.673" Score="2" Body="&lt;p&gt;I think that there is no special reason here.&lt;/p&gt;&#xA;&lt;p&gt;See Alfred Tarski &amp;amp; Andrzej Mostowski &amp;amp; Raphael Robinson, &lt;a href=&quot;https://books.google.it/books?id=XtLbjZjB1B8C&amp;amp;pg=PA37&quot; rel=&quot;nofollow noreferrer&quot;&gt;Undecidable Theories (North Holland, 1971)&lt;/a&gt;; &lt;strong&gt;Ch.2 : Undecidability in Arithmetic&lt;/strong&gt; (page 51) deals with the :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;formalized arithmetic of natural numbers [...] referred to as Theory &lt;span class=&quot;math-container&quot;&gt;$\mathsf N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We shall be interested in some axiomatic subtheories of &lt;span class=&quot;math-container&quot;&gt;$\mathsf N$&lt;/span&gt; referred to as Theories &lt;span class=&quot;math-container&quot;&gt;$\mathsf P, \mathsf Q, \mathsf R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And see also the footnote on page 39:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A finitely axiomatizable and essentially undecidable subtheory &lt;span class=&quot;math-container&quot;&gt;$\overline {\mathsf Q}$&lt;/span&gt; of the arithmetic of natural numbers was first constructed by Mostowski and Tarski in 1939. [...] R.M.Robinson has shown that Theory &lt;span class=&quot;math-container&quot;&gt;$\overline {\mathsf Q}$&lt;/span&gt; can be replaced by a weaker theory based upon a simpler axiom system, in fact, by Theory &lt;span class=&quot;math-container&quot;&gt;$\mathsf Q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-05-21T07:35:41.103" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9630" PostTypeId="2" ParentId="8436" CreationDate="2019-05-19T14:32:38.947" Score="1" Body="&lt;p&gt;You can see the origin of modern numerals and zero from Sanskrit, carved in granite, also at&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://youtu.be/pElvQdcaGXE?t=101&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtu.be/pElvQdcaGXE?t=101&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in a Temple for zero in a talk by Marcus du Sautoy who looks at the contribution of Ancient Indians like Aryabhata, Brahmagupta, Bhaskara, Madhava, etc &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Programme: BBC - The Story of Maths&#xA;Presenter: Marcus du Sautoy&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-05-20T20:21:27.680" LastActivityDate="2019-05-20T20:21:27.680" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9631" PostTypeId="1" AcceptedAnswerId="9645" CreationDate="2019-05-19T16:18:03.880" Score="9" ViewCount="2035" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;An elegant experiment in 1772 by Henry Cavendish. Cavendish charged a spherical conducting shell that contained within it, and temporarily connected to it, a smaller sphere. The outer shell was then separated into two halves and carefully removed, the inner sphere having been first disconnected. This sphere was tested for charge, the absence of which would confirm the inverse-square law. Assuming that a deviation from the inverse-square law could be expressed as a difference in the exponent, 2 + δ, say, instead of 2, Cavendish concluded that δ must be less than 0.03. This experiment of Cavendish remained largely unknown until Maxwell discovered and published Cavendish’s notes a century later (1876). At that time also, Maxwell repeated the experiment with improved apparatus, pushing the limit down to δ &amp;lt; 10^(-6). The present limit on δ is a fantastically small number – about one part in 10^(16);&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this experiment he deduced the inverse square law in electrostatics even about 14 years before Coulomb. But I want to know how exactly did he deduce it? The experiment isn't proving the law directly. So how did he do it exactly?&lt;/p&gt;&#xA;" OwnerUserId="9999" LastActivityDate="2019-05-23T14:45:12.763" Title="How did Henry Cavendish deduce the inverse square law in electrostatics from his experiment in 1772?" Tags="|physics|experimental-physics|electromagnetism|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9632" PostTypeId="2" ParentId="9631" CreationDate="2019-05-19T17:47:57.700" Score="7" Body="&lt;p&gt;Newton proved that if the attraction obeys the inverse square law, then the force inside a uniformly charged sphere is zero. It follows from the description that you give that Cavendish used the converse statement. In fact this converse statement is true though I doubt that Cavendish had a proof of it in full generality. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is very common for physicists (and other non-mathematicians) to confuse a direct theorem with the converse one, necessary conditions with sufficient etc. (Even Newton did this). Of course, one can make various a priori assumptions with which this statement becomes easy: for example that the law of attraction is a power. Then the power must be -2. I suppose this is what Cavendish had in mind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref. for a discussion of the converse of Newton's theorem: MR2125274 &#xA;S. Stein,&#xA;Remarks on the gravity equation. &#xA;Amer. Math. Monthly 112 (2005), no. 4, 322–333. (He discusses another similar theorem of Newton, but this one can be treated in the same way).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Coulomb did more than that. He was able to measure the force, to determine that it is proportional to the charge, not only inverse to the square distance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 2. Physics book from which I studied electricity as a child credits this argument to B. Franklin. But gives no reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark 3. &lt;a href=&quot;http://www.math.purdue.edu/~eremenko/newtpr.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is&lt;/a&gt; an exposition of Newton's proof of his theorem. (The statement and an application are &lt;a href=&quot;http://www.math.purdue.edu/~eremenko/train.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-05-20T12:41:13.197" LastActivityDate="2019-05-20T12:41:13.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9634" PostTypeId="1" CreationDate="2019-05-20T08:43:24.433" Score="-1" ViewCount="347" Body="&lt;p&gt;I am searching it on Google but I am not finding,who had  discovered static electric spark for the first time,that later Benjamin Franklin proved that this spark(already discovered spark) is same as that of the outside lighting? &#xA;As I know scientists have seen attraction (charged amber and charged glass rod attraction) and repulsion(charged glass rod and charged glass rod repulsion) but no electric spark they have seen then from where did the scientist know about electric spark before Benjamin Franklin?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="9984" LastEditDate="2019-05-21T04:05:43.727" LastActivityDate="2019-05-21T08:47:36.167" Title="Electric spark discovery" Tags="|biographical-details|experimental-physics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9635" PostTypeId="2" ParentId="9634" CreationDate="2019-05-20T20:44:33.620" Score="2" Body="&lt;p&gt;As comments suggest, the static spark is (like fire) found in nature - it was not discovered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In terms of who first harnessed the static spark or created one purposefully, that is also likely to date to time immemorial, since commonly used textiles (like wool, especially) tend to create static easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first experiments by &quot;scientists&quot; (a role that in the past was not a specific professional occupation, and would have been less clearly defined) with electric sparks probably began around the time of the invention of the chemical battery.&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2019-05-20T20:44:33.620" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9637" PostTypeId="2" ParentId="9634" CreationDate="2019-05-21T08:47:36.167" Score="-1" Body="&lt;p&gt;From my prospect von kliest, the discoverer of leyden jar,is the first scientist to know about spark. &#xA;As he might have seen when he took the charged glass rod near the&lt;br&gt;&#xA;nail of the jar then the glass rod lost its charge as it can't attract any more and taking his finger towards the nail he could see the spark and feel the shock. &lt;/p&gt;&#xA;" OwnerUserId="9993" LastActivityDate="2019-05-21T08:47:36.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9638" PostTypeId="1" AcceptedAnswerId="9640" CreationDate="2019-05-21T14:26:44.407" Score="0" ViewCount="92" Body="&lt;p&gt;In Benjamin Franklin's kite given in the link below &lt;a href=&quot;http://www.benjamin-franklin-history.org/wp-content/uploads/2016/12/Franklins-kite.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.benjamin-franklin-history.org/wp-content/uploads/2016/12/Franklins-kite.jpg&lt;/a&gt; &#xA;I want to know how electric charge pass through strings as well as sticks both being an insulator after passing through iron? &lt;/p&gt;&#xA;" OwnerUserId="9993" LastActivityDate="2019-05-22T04:31:11.810" Title="Kite experiment" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9640" PostTypeId="2" ParentId="9638" CreationDate="2019-05-22T04:31:11.810" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;I want to know how electric charge pass through strings as well as sticks both being an insulator after passing through iron&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Good point, remember that water is a conductor of electricity when it has traces of salts and ions. Rain water has enough ions to make it conductive.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you check Wikipedia, you will see the same question answered.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to the 1767 Priestley account, Franklin realized the&#xA;  dangers of using conductive rods and instead used the conductivity of&#xA;  a wet hemp string attached to a kite. This allowed him to stay on the&#xA;  ground while his son assisted him to fly the kite from the shelter of&#xA;  a nearby shed. This enabled Franklin and his son to keep the silk&#xA;  string of the kite dry to insulate them while the hemp string to the&#xA;  kite was allowed to get wet in the rain to provide conductivity. A&#xA;  house key belonging to Benjamin Loxley was attached to the hemp string&#xA;  and connected to a Leyden jar; a silk string was attached to this. &quot;At&#xA;  this key he charged phials, and from the electric fire thus obtained,&#xA;  he kindled spirits, and performed all other electrical experiments&#xA;  which are usually exhibited by an excited globe or tube.&quot; The kite was&#xA;  not struck by visible lightning; had it been, Franklin would almost&#xA;  certainly have been killed. However, Franklin did notice that&#xA;  loose threads of the kite string were repelling each other and deduced&#xA;  that the Leyden jar was being charged. He moved his hand near the key&#xA;  and observed an electric spark, proving the electric nature of&#xA;  lightning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-05-22T04:31:11.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9641" PostTypeId="1" CreationDate="2019-05-22T05:47:36.607" Score="0" ViewCount="67" Body="&lt;p&gt;I read how Benjamin Franklin does the kite experiment in rain but keeping himself and the bottom end dry.I want to know how he know that water can conduct electricity.Please refer the link below and please see the Franklin's kite experiment section which is marked in green colour&#xA;&lt;a href=&quot;http://lnr.li/0rvd6&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://lnr.li/0rvd6&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9984" LastActivityDate="2019-05-22T05:47:36.607" Title="Water conductivity discovery" Tags="|biographical-details|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9642" PostTypeId="1" CreationDate="2019-05-23T00:47:15.230" Score="3" ViewCount="232" Body="&lt;p&gt;In his &lt;em&gt;Ars Magna&lt;/em&gt; Cardano specifies procedures to &quot;depress&quot; a cubic - a means to convert an equation such as &lt;span class=&quot;math-container&quot;&gt;$x^3+6x^2=100$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$y^3=84+12y$&lt;/span&gt;, eliminating the &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was he the one who discovered how to do this?  (Or was he just the first to publish it?  (Or neither?))&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his &lt;em&gt;Quesiti&lt;/em&gt; Tartaglia claimed to have known how to solve cubics of the form &lt;span class=&quot;math-container&quot;&gt;$x^3+ax^2=c$&lt;/span&gt;, which made me wonder if he knew how to eliminate the &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; term. So far I haven't found anything definitive that indicates who first discovered the appropriate substitution.&lt;/p&gt;&#xA;" OwnerUserId="7074" LastActivityDate="2019-05-23T00:47:15.230" Title="Who first &quot;depressed&quot; the cubic equation?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9643" PostTypeId="2" ParentId="9620" CreationDate="2019-05-23T02:04:14.563" Score="1" Body="&lt;p&gt;I may add to the answer above. The electroscope was the main instrument for assessing and checking the electrical sign of charged bodies. Again the reference was amber and glass. Anything electrified like glass, was assigned the same sign as glass i.e. positive and the other one was negative. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is said he could have chosen violet and red to differentiate charges but he chose the words resinous and vitreous to convey two different &quot;types&quot; of charges.&#xA;Let us read the original work: &lt;a href=&quot;http://hipstwiki.wikifoundry.com/page/Charles+d%C3%BA+Fay+%E2%80%93+Explorative+Experiments%3A+Describing+and+Explaining+Electrical+Phenomena&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://hipstwiki.wikifoundry.com/page/Charles+d%C3%BA+Fay+%E2%80%93+Explorative+Experiments%3A+Describing+and+Explaining+Electrical+Phenomena&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iRJWC.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iRJWC.png&quot; alt=&quot;Original words&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9dExg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9dExg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-05-23T03:53:22.543" LastActivityDate="2019-05-23T03:53:22.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9645" PostTypeId="2" ParentId="9631" CreationDate="2019-05-23T14:45:12.763" Score="5" Body="&lt;p&gt;Alexandre Eremenko's answer is great, but I figure the page could benefit from an explanation of the method in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Cavendish's Experiment&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question Cavendish was facing was this: Given that we know that charged bodies exhibit attractive and repulsive forces, what is the likelihood that, like Newton's law of gravitation, the electric force obeys an inverse square law with distance? (Not probabilistic likelihood, I mean just inquisitively). Newton had shown that &lt;em&gt;if&lt;/em&gt; the force law is an inverse square law, &lt;em&gt;then&lt;/em&gt; a hollow uniformly massive sphere will not produce a net gravitational force at any point within the sphere. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cavendish took the converse of this to investigate the behavior of the electric force. That is, he assumed &lt;em&gt;if&lt;/em&gt; a hollow uniformly charged sphere does not produce a net electric force at any point within the sphere, &lt;em&gt;then&lt;/em&gt; the electric force obeys an inverse square law. Note that, as mentioned by Alexandre, this is logically invalid, because a statement does not imply its converse. Cavendish may not have been aware of this, but nonetheless, his experiment was designed to prove that if the electric force obeys a &lt;em&gt;power&lt;/em&gt; law with distance, then the power must be &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The experiment assumes that (as was known at the time) conductors allow electric &quot;fluid&quot; to move freely. Then a hollow conducting sphere, when charged, will have a uniform charge distribution. If we have a conducting sphere on the inside of this shell, but not in contact, then we expect that if the net electric force at any given point is not zero, then the electric &quot;fluid&quot; (charge) will be distributed about the inner sphere unevenly. If you then connect a small wire from the charged shell to the inner sphere, we would expect a net charge flow because of the net force on points in the sphere. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Net charge flow means that, when we remove the charged shell connection and remove the charged shell, the inner sphere will be &lt;em&gt;charged&lt;/em&gt; if there was net force, so if the inner sphere is &lt;em&gt;not&lt;/em&gt; charged, then there was &lt;em&gt;no net force&lt;/em&gt;. The absence of net force is what Cavendish took as evidence of the inverse square law, so after finding that the inner sphere was not charged, he accepted that the electric force was nearly an inverse square law. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By estimating the sensitivity of his electroscope, Cavendish was able to approximate the uncertainty in his null-charge measurement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Maxwell's Proof&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell, in his Treatise on Electricity and Magnetism (vol. 1), provides a fairly rigorous (but still flawed) proof of the fact that if the power is an inverse power, then it must be &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, which he credits partly to Laplace. One problem with the proof is that it depends not on simply testing the null-charge result for one sphere radius, but for &lt;em&gt;every&lt;/em&gt; radius, to truly be confirmed. Another issue, is that while the electric force is clearly a monotonically decreasing function, it was not necessarily an inverse power law. This comes back to the issue of &quot;a statement does not imply its converse.&quot; However, the evidence for an inverse-square electric force was building, and the assumption seemed increasingly valid. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References: Maxwell's &lt;a href=&quot;https://ia802302.us.archive.org/25/items/ATreatiseOnElectricityMagnetism-Volume1/Maxwell-ATreatiseOnElectricityMagnetismVolume1_text.pdf&quot; rel=&quot;noreferrer&quot;&gt;Treatise on Electricity and Magnetism&lt;/a&gt;, art. 74, (p. 80).&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2019-05-23T14:45:12.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9646" PostTypeId="1" AcceptedAnswerId="9702" CreationDate="2019-05-24T02:11:43.617" Score="3" ViewCount="336" Body="&lt;p&gt;There's a story told about Feynman in a non-scientific book (whether written by him or another, I don't recall) that I'm trying to track down.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If I recall correctly: he's part of a group (perhaps while still a student) that's trying to think of an explanation for some phenomenon. He offers one - perhaps based in physics - and is praised for it. He then offers a completely different one, and claims that the exercise is nonsensical: he can always fit a theory to such limited data.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have a reference for (or correction to!) this?&lt;/p&gt;&#xA;" OwnerUserId="10014" LastEditorUserId="4251" LastEditDate="2020-07-06T07:12:12.157" LastActivityDate="2020-07-06T07:12:12.157" Title="Feynman story: providing two different explanations for the same phenomenon" Tags="|biographical-details|scientific-method|feynman|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9648" PostTypeId="2" ParentId="3230" CreationDate="2019-05-24T21:31:42.010" Score="7" Body="&lt;p&gt;Complementing Conifold's answer, it might perhaps be instructive to quote from Minkowski's presentation &lt;a href=&quot;https://de.wikisource.org/wiki/Raum_und_Zeit_(Minkowski)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Raum und Zeit&lt;/em&gt;&lt;/a&gt; at the 80th Congress of German Natural Scientists in Cologne on the 21st of September 1908, a short few months before his own death.  In this speech he makes it clear the physical origin of his eponymous spacetime:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mein Herren!  Die Anschauungen über Raum und Zeit, die ich Ihnen entwickeln möchte, sind auf experimentell-physikalischem Boden erwachsen. Darin liegt ihre Stärke. Ihre Tendenz ist eine radikale. Von Stund′ an sollen Raum für sich und Zeit für sich völlig zu Schatten herabsinken und nur noch eine Art Union der beiden soll Selbständigkeit bewahren.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;or in translation&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gentlemen!  The views of space and time that I wish to lay before you have sprung from the soil of experimental physics and therein lies their strength.  They are radical.  Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="653" LastActivityDate="2019-05-24T21:31:42.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9649" PostTypeId="1" AcceptedAnswerId="9653" CreationDate="2019-05-26T11:19:28.677" Score="1" ViewCount="236" Body="&lt;p&gt;I read that he made a modified version of Otto von Guericke's electrostatic generator but I want to know how he came to know that mercury can glow light in his generator at that time?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="55" LastEditDate="2019-05-27T09:06:26.870" LastActivityDate="2019-05-27T09:06:26.870" Title="How did Francis Hauksbee know that mercury would glow in his electrostatic generator?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9650" PostTypeId="1" AcceptedAnswerId="9652" CreationDate="2019-05-26T15:42:09.210" Score="5" ViewCount="153" Body="&lt;p&gt;I am looking for an English translation of Euler's derivation of Euler's method for ODEs, namely the update&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;y_{n+1} = y_n + h f(y_n, t_n)&#xA;$$&lt;/span&gt;&#xA;What motivated Euler to consider this problem, and how does he develop the idea?&lt;/p&gt;&#xA;" OwnerUserId="10024" LastActivityDate="2019-05-27T09:42:07.763" Title="Euler's Derivation of Euler's Method for ODEs" Tags="|mathematics|reference-request|euler|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9651" PostTypeId="2" ParentId="9646" CreationDate="2019-05-27T03:12:06.883" Score="1" Body="&lt;p&gt;I suspect you're referring to the Feynman anecdote  about a submerged S-shaped sprinkler, which is mentioned in Part 2 of the book &quot;&lt;em&gt;Surely You're Joking, Mr. Feynman&lt;/em&gt;&quot;  (pages 63-65).  The question he was discussing with other students was if the sprinkler is completely submerged in water and sucks in water rather than expelling it, which direction would it rotate?  He described some of the back and forth of the conflicting arguments, and then described his attempted experiment in the Princeton cyclotron lab.  Although the problem was not original with Feynman, the device is now sometimes referred to as a &lt;a href=&quot;https://en.wikipedia.org/wiki/Feynman_sprinkler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feynman Sprinkler&lt;/a&gt; because of this story.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2019-05-27T03:12:06.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9652" PostTypeId="2" ParentId="9650" CreationDate="2019-05-27T07:30:49.530" Score="6" Body="&lt;p&gt;What motivated Euler was not any problem in particular, but rather the general need to solve differential equations approximately when an analytic solution could not be found. He explains the method in a general form in Section 2, chapter VII of volume I of &lt;a href=&quot;http://eulerarchive.maa.org/&quot; rel=&quot;noreferrer&quot;&gt;Institutionum Calculi Integralis (Foundations of Integral Calculus, 1768)&lt;/a&gt;, his textbook on integral calculus. It closely follows the method of step by step numerical integration described in Section 1, chapter VII. The example he considers is illustrative, in modern notation &lt;span class=&quot;math-container&quot;&gt;$y'=\frac{a^2}{x^2-y^2}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The chapter is titled De integratione aequationum differentialium per approximationem (On approximate integration of differential equations), English translation of the relevant portion and commentary are available in &lt;a href=&quot;http://www.math.miami.edu/%7Ehk/csc210/week5/euler.pdf&quot; rel=&quot;noreferrer&quot;&gt;Jardine's Euler's Method in Euler's Words&lt;/a&gt; (note that Euler uses &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt; where we would use &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The pair of variables &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; appear in an integral of the form &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial y}{\partial x}=V$&lt;/span&gt;, the function &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; itself a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. We desire the complete integral, which is interpreted that as long as &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is assigned a certain value &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;, the other variable &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; takes on a given value &lt;span class=&quot;math-container&quot;&gt;$y = b$&lt;/span&gt;. Therefore our primary goal is to find the value of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; so that when &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; takes on a value that differs little from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, or we assume &lt;span class=&quot;math-container&quot;&gt;$x = a + \omega$&lt;/span&gt;, then we can find &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; is a very small quantity, then the value of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; itself differs minimally from &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;; so while &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; varies a little from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$a+\omega$&lt;/span&gt;, one may consider the quantity &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; as a constant.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;When we specify &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = b$&lt;/span&gt; then by virtue of the small change we have &lt;span class=&quot;math-container&quot;&gt;$V=A$&lt;/span&gt;, for that reason when integrating &lt;span class=&quot;math-container&quot;&gt;$y= b+X(x-a)$&lt;/span&gt;, a constant being added of course so that when &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt; we have &lt;span class=&quot;math-container&quot;&gt;$y = b$&lt;/span&gt;. Therefore given the initial values  &lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = b$&lt;/span&gt;, we obtain the approximate next values &lt;span class=&quot;math-container&quot;&gt;$x = a + \omega$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y = b + A\omega$&lt;/span&gt;, so that proceeding further in a similar way over the small intervals, in the end arriving at values as distant as we would like from the earlier values.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Euler was aware of the problems with the accumulation of errors in this method, which he explicitly mentions in the Corollary 2:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Where smaller intervals are taken, through which the values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; progress iteratively, so much the more accurate values are obtained one at a time. However the errors committed one at a time, even if they may be very small, accumulate because of the multitude.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-05-27T09:42:07.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9653" PostTypeId="2" ParentId="9649" CreationDate="2019-05-27T07:48:35.423" Score="4" Body="&lt;p&gt;The glow of mercury in a vacuum is known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Barometric_light&quot; rel=&quot;nofollow noreferrer&quot;&gt;barometric light&lt;/a&gt;. It was discovered at the end of 1675 by Picard, a French astronomer (he was using barometric measurements to correct for refraction in his observations of stars): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Towards the year 1676, Monsieur Picard was transporting his barometer from the Observatory to Port Saint Michel during the night, [when] he noticed a light in a part of the tube where the mercury was moving; this phenomenon having surprised him, he immediately reported it to the sçavans...&lt;/em&gt;&quot; [&quot;Sur la lumière du baromètre&quot; [On the light of the barometer], Histoire de l'Académie Royale des sciences de Paris,vol. 2, pages 202-203 (1694)]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Johann Bernoulli demonstrated the phenomenon to the French Academy in 1700, where Hauksbee learned of it, and experimented with it extensively. He was also the first to use a glass ball in an electrostatic generator, so the idea of combining the effects was natural. In 1705 he discovered that if he created vacuum inside, rubbed the ball, and put his hand on it, the mercury glowed quite brightly. In 1706 he invented an &quot;influence machine&quot; that demonstrated the effect. Later it was used in gas-discharge lamps, neon lights, and mercury vapor lamps, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Francis_Hauksbee#Scientific_contributions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hauksbee's scientific contributions&lt;/a&gt; that references &lt;a href=&quot;https://en.wikipedia.org/wiki/Connections_(TV_series)#Connections_(1978)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Burke's Connections book, p.75&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-05-27T08:07:51.130" LastActivityDate="2019-05-27T08:07:51.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9654" PostTypeId="1" AcceptedAnswerId="9655" CreationDate="2019-05-27T09:45:26.970" Score="4" ViewCount="175" Body="&lt;p&gt;In electronics, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; is used for a square root of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, because &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; is current. Who introduced this and when? And was it really necessary, given that (at least now) current's symbol is capitalised? Perhaps it wasn't at the time, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Electric_current#Symbol&quot; rel=&quot;noreferrer&quot;&gt;as best I can tell&lt;/a&gt; Ampère didn't use lower case.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2019-05-27T16:54:52.913" Title="Who changed $i$ to $j$ in electronics?" Tags="|notation|electricity|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9655" PostTypeId="2" ParentId="9654" CreationDate="2019-05-27T16:41:51.950" Score="4" Body="&lt;p&gt;The symbol &lt;em&gt;i&lt;/em&gt; is still used in mathematics; &lt;em&gt;j&lt;/em&gt; is an invention of an electrical engineers, usually attributed to Charles P. Steinmetz. See for example&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/5082/introduction-of-imath-and-jmath-notations-for-the-imaginary-unit&quot;&gt;Introduction of $\imath$ and $\jmath$ notations for the imaginary unit&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As to why &lt;em&gt;j&lt;/em&gt; was chosen, the following is quoted from a 1921 book, &quot;Tables of Complex Hyperbolic and Circular Functions&quot; by Arthur Edwin Kennelly. I am sure older ones will also explain the same.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/B0EV0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/B0EV0.png&quot; alt=&quot;Book image&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My feeling is that since &lt;em&gt;j&lt;/em&gt; was chosen by a German-born American (Steinmetz), it may be possible to attribute the similarity of &lt;em&gt;i&lt;/em&gt; and &lt;em&gt;j&lt;/em&gt; in German e.g. Iodine (symbol I) is Jod (symbol J) in German. However the letter &lt;em&gt;i&lt;/em&gt; was also introduced by Gauss, who happened to be German, but at that time, Latin was the language of science. This is just my pure speculation.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-05-27T16:54:52.913" LastActivityDate="2019-05-27T16:54:52.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9656" PostTypeId="1" AcceptedAnswerId="9780" CreationDate="2019-05-27T22:04:16.627" Score="0" ViewCount="336" Body="&lt;p&gt;I remember reading that in a couple of places that ancient trigonometry, particularly Ancient Greek trigonometry, used the &lt;em&gt;chord&lt;/em&gt; as the primitive concept instead of sine and cosine. I can't tell whether the &lt;em&gt;chord&lt;/em&gt; or &lt;em&gt;half-chord&lt;/em&gt; was treated as primitive in the Indian mathematical tradition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are two couple equivalent definitions of the chord function &lt;em&gt;with an argument anachronistically written in terms of radians&lt;/em&gt;, written &lt;span class=&quot;math-container&quot;&gt;$k(t)$&lt;/span&gt; . This function also technically assigns &lt;em&gt;negative&lt;/em&gt; lengths to chords when the angle &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; goes around the circle an &lt;em&gt;odd number of complete revolutions&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ k(t) = 2\sin\left(\frac{t}{2}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ k'' = -\frac{k}{4} \;\;\;\text{where}\;\;\; k(0) = 0 \;\;\text{and}\;\; k'(0) = 1 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The chord is also the length of the line segment between two points that are &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; radians apart on the unit circle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you are &lt;em&gt;primarily working with circles&lt;/em&gt; and don't have a notion of Cartesian coordinates, then it makes sense that &lt;em&gt;chord&lt;/em&gt; would be the primitive trigonometric function. It seems to me that &lt;em&gt;chord&lt;/em&gt; is easier to explain than &lt;em&gt;sine&lt;/em&gt; if you're accustomed to thinking of geometric drawings &lt;em&gt;up to a rigid motion&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So my question is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Which mathematical traditions used the &lt;em&gt;chord&lt;/em&gt; as one of their primitive trigonometric functions?&lt;/li&gt;&#xA;&lt;li&gt;When did those traditions, or derivatives thereof, move away from using the &lt;em&gt;chord&lt;/em&gt; function? Did this shift correspond to a key discovery?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9243" LastActivityDate="2019-07-05T14:07:41.950" Title="When did trigonometry move away from treating chord as primitive?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9657" PostTypeId="1" CreationDate="2019-05-28T02:52:43.320" Score="5" ViewCount="99" Body="&lt;p&gt;On October 7th 1959, the soviet Luna 3 probe took the first picture of the far side of the moon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It has been known since ancient times that the moon was a sphere. It would have doubtless been observed that the moon always seemed to present the same face to earth. The back face would have been a complete mystery, and I imagine &lt;strong&gt;suspect to much speculation.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The moon's &lt;a href=&quot;https://en.wikipedia.org/wiki/Libration&quot; rel=&quot;noreferrer&quot;&gt;libration&lt;/a&gt; allows attentive astronomers to see up to 57% of the lunar surface. Nevertheless, the thought of the unknown on the moon's far side must have been very tantalising to many astronomers, in the same way the appearance of Ceres and Pluto was to us prior to 2015, if not more strongly so.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are there any interesting or notable historical theories on what the far side of the moon looked like or contained?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EAlAz.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EAlAz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/zwrco.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zwrco.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;Top image: Luna 3 first image. Bottom image: Lunar Reconnaisance Orbiter composite image.&#xA;Source: wikipedia&lt;/p&gt;&#xA;" OwnerUserId="10028" LastActivityDate="2019-05-29T17:08:25.823" Title="Notable theories on the far side of the moon pre 1959" Tags="|astronomy|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9658" PostTypeId="1" CreationDate="2019-05-28T10:59:29.593" Score="-4" ViewCount="488" Body="&lt;p&gt;I want to know who had used the word pressure for the first time and what was the reason for him to use the word pressure.&#xA;As I searched on google but it is only showing about barometer discovery&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="9984" LastEditDate="2019-05-29T05:16:19.193" LastActivityDate="2019-06-01T13:42:10.057" Title="History behind the word pressure" Tags="|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9659" PostTypeId="1" CreationDate="2019-05-28T12:05:05.053" Score="1" ViewCount="154" Body="&lt;p&gt;I want to know how he thought that when his siphon could not raise water more than 10m it may be because of vacuum?&#xA;Please refer the link below &lt;a href=&quot;https://wikivisually.com/wiki/Gasparo_Berti&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://wikivisually.com/wiki/Gasparo_Berti&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="9984" LastEditDate="2019-05-29T04:45:05.190" LastActivityDate="2019-05-29T04:45:05.190" Title="Galileo discovery of vacuum" Tags="|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9660" PostTypeId="2" ParentId="7847" CreationDate="2019-05-29T00:29:14.357" Score="-1" Body="&lt;p&gt;Quantization first appears in Balmer's equation for the line spectrum of hydrogen, although he didn't assign any physical significance to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1888, Rydberg refined the equation as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\lambda} = 1.097*10^7( \frac{1}{2^2} - \frac{1}{n^2})$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where n is an integer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My feeling is that idea of quantization was floating around and formalized by Planck.&lt;/p&gt;&#xA;" OwnerUserId="10031" LastActivityDate="2019-05-29T00:29:14.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9662" PostTypeId="1" CreationDate="2019-05-29T11:19:10.833" Score="4" ViewCount="221" Body="&lt;p&gt;On p.98 of the book &quot;Mathematics of the 19th Century: Geometry, Analytic Function Theory&quot;, the authors mention a note written by Gauss in 1840:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1840 Gauss wrote a note in which he introduced the concept of one- and two-dimensional manifolds, which he called a &quot;line&quot; (Zug) and a &quot;layer&quot; (Schicht), and posed the question of the decomposition of a &quot;layer&quot; bounded by several &quot;lines&quot; into several &quot;layers&quot; bounded by one &quot;line&quot;. What is actually involved here is the problem of connectedness of a surface, later solved by Riemann and subsequently became one of the most important problems of combinatorial topology.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My question is mainly about getting references. I tried to search for it in volumes 8, 10 and 12 (i think it's most likely to be found in these volumes). Can anyone help locate it in his writings?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-07-09T20:18:45.383" LastActivityDate="2020-08-08T21:01:10.107" Title="Where in Gauss's nachlass did he pose the problem of connectedness of a surface?" Tags="|mathematics|reference-request|geometry|gauss|19th-century|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9663" PostTypeId="1" AcceptedAnswerId="9665" CreationDate="2019-05-29T14:01:18.370" Score="16" ViewCount="3551" Body="&lt;p&gt;I am reading &lt;a href=&quot;https://arxiv.org/pdf/math/9205211.pdf&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; by Donald E. Knuth and get stuck by this sentence: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Our mathematical language continues to improve, just as “the d-ism of&#xA;  Leibniz overtook the dotage of Newton” in past centuries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I know I can get some hints from the reference 4 but I don't think I can fathom the history clearly. Could anyone please help explain what that means in simple English? Thanks in advance. &lt;/p&gt;&#xA;" OwnerUserId="10033" LastActivityDate="2020-05-07T05:48:53.877" Title="What does it mean by &quot;d-ism of Leibniz&quot; and &quot;dotage of Newton&quot; in simple English?" Tags="|mathematicians|newton|leibniz|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9665" PostTypeId="2" ParentId="9663" CreationDate="2019-05-29T14:30:51.960" Score="29" Body="&lt;p&gt;It is a play of words by Charles Babbage. Deism was a religious belief or rather a movement promoting the idea that God exists but it does not interfere with whatever happens in this world. This old philosophy according to the Wikipedia &amp;quot;...asserts God's existence as the cause of all things, and admits its perfection (and usually the existence of natural law and Providence) but rejects divine revelation.&amp;quot; Now, Babbage writes d-ism which sounds the same, but now the reference is Leibniz who was one of the founders of calculus (there was a major controversy whether Newton invented calculus or Leibniz). We still use his d's in differential calculus.  Newton on the other hand was using dots in his calculations for symbolic purposes. Dotage also means someone who is becoming senile (see comments for other meanings). Basically, the writer is making fun of Newton, who is an Englishman himself, and asking his peers to adopt the symbolism of Leibniz.&lt;/p&gt;&#xA;&lt;p&gt;P.S. In some European textbooks, one might see dots over a variable to show a differential operator usually with respect to time. No &amp;quot;d&amp;quot; is used then.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V8GF1.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V8GF1.png&quot; alt=&quot;Google Books on The Methodist Review&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Google Books, The Methodist Review&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-05-30T21:01:00.050" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9667" PostTypeId="1" CreationDate="2019-05-29T21:04:04.173" Score="3" ViewCount="314" Body="&lt;p&gt;Heard he wrote a bunch of rambling thoughts near the end of his life and doubted his own theory of relativity...it was in Einstein: His Life and Universe by Walter Isaacson&lt;/p&gt;&#xA;" OwnerUserId="10034" LastActivityDate="2019-05-29T21:36:06.477" Title="Is it true Einstein doubted relativity near the end of his life?" Tags="|relativity-theory|einstein|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9668" PostTypeId="2" ParentId="9667" CreationDate="2019-05-29T21:36:06.477" Score="8" Body="&lt;p&gt;No.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He opposed Quantum Mechanics from the beginning until death. He did it because its inherent non-determinism (&quot;God does not gamble&quot;). In his opinion, the cause of the non-determinism of the QM is that &quot;it is not ready&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, he worked on the further development of the GR until death. A possible direction was to describe also the electromagnetism as spacetime curvature. This makes highly unlikely that he had doubted it.&lt;/p&gt;&#xA;" OwnerUserId="217" LastActivityDate="2019-05-29T21:36:06.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9669" PostTypeId="2" ParentId="9663" CreationDate="2019-05-30T06:56:12.463" Score="8" Body="&lt;p&gt;The first answer is excellent but just for context on the actual math:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton notation for derivative of f(x):&#xA;&lt;span class=&quot;math-container&quot;&gt;$ \dot f(x) $&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Leibniz notation for derivative of f(x):&#xA;&lt;span class=&quot;math-container&quot;&gt;$ \frac {df}{dx} $&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's notation is fine for very basic single variable derivation and generally the derivative for the described cases but the Leibniz notation is general purpose and demonstrates the process of finding complex derivatives/partial derivatives very clearly. So the newton notation was abandoned (in most cases other than a single variable) in favor of the more consistent and concise notation. Thus our programmer friend Knuth insinuates that we should judge something's value with logical evaluation and not emotional attachment (England's mathematicians went on to use Newton's notation for nationalistic reasons since Newton was an English lad but Leibniz was German/Prussian/Subject of the holy roman empire /Not English so their calculus development stagnated compared to the rest of Europe according to an old math professor of mine, don't know the validity of this statement though).&lt;/p&gt;&#xA;" OwnerUserId="10037" LastEditorUserId="5989" LastEditDate="2020-05-07T05:48:53.877" LastActivityDate="2020-05-07T05:48:53.877" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9670" PostTypeId="2" ParentId="9663" CreationDate="2019-05-30T11:55:30.560" Score="4" Body="&lt;p&gt;I think the two other answers here when combined offer a complete explanation of Knuth's word game.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He calls Leibniz notation &quot;d-ism&quot; to combine the English word &quot;deism&quot; with  Leibniz's use of the prefix &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; to suggest an infinitesimal change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. So Leibniz would write &lt;span class=&quot;math-container&quot;&gt;$dx/dt$&lt;/span&gt; for the rate of change of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with respect to time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. Believing in infinitesimals is akin to believing in a deity - but writing with infinitesimals is good notation because it is extraordinarily productive mathematically  - it helps you understand what is really happening and suggests new theorems. Then if you wish you can provide formal proofs that replace the theological infinitesimals with limits. (It is also possible to make the reasoning with infinitesimals rigorous.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That notation prevailed over Newton's use of &lt;span class=&quot;math-container&quot;&gt;$\dot{x}$&lt;/span&gt; for the rate of change of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with respect to time (in part) because it was so suggestive. The pun for Newton is the connection between the dot over the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &quot;dotage&quot;, suggesting senility or old age. Newton's notation (mostly) faded away.&lt;/p&gt;&#xA;" OwnerUserId="9852" LastEditorUserId="9852" LastEditDate="2019-05-30T12:02:02.103" LastActivityDate="2019-05-30T12:02:02.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9671" PostTypeId="1" AcceptedAnswerId="9673" CreationDate="2019-05-30T12:26:18.000" Score="-5" ViewCount="208" Body="&lt;p&gt;What are the prerequisites to read his book? Why Richard Phillips Feynman is so famous? What are great works of Richard Phillips Feynman?&lt;/p&gt;&#xA;" OwnerUserId="10040" LastEditorUserId="6085" LastEditDate="2019-05-31T17:38:59.850" LastActivityDate="2019-05-31T17:55:01.310" Title="What are the great works of Richard Phillips Feynman?" Tags="|mathematics|physics|discoveries|mathematical-physics|" AnswerCount="2" CommentCount="3" ClosedDate="2019-06-02T00:26:48.273" ContentLicense="CC BY-SA 4.0" />
  <row Id="9672" PostTypeId="1" CreationDate="2019-05-30T17:25:47.327" Score="5" ViewCount="525" Body="&lt;p&gt;I would like to know is there a book that is both a history of mathematics and a collection of open problems?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that there exist many books that cover either larger or smaller periods of the history of mathematics and also a number of books that are collections of open problems, but, a symbiotic approach I have not seen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, a book that I would like to read would inform us about development of mathematics at least from a period of Greek mathematics to the twentieth century, and it would be a story also about some specific problems that led to the development of open problems that are not resolved till the present day, and that book could end with Hilbert problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that development of open problems is also an important aspect of the history of mathematics, and such a book could be filled with some &quot;high&quot; mathematics, but not with too much of it, so that it does not diverge from the historic and informational character and it does not turn into a blending of &quot;high&quot; mathematics with some  historical remarks and content.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I prefer that it does cover open problems from &quot;higher&quot; mathematics but written from an elementary standpoint, so that it more involves a description of open problems than their advanced formulation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is there some book that closely resembles this description?&lt;/p&gt;&#xA;" OwnerDisplayName="user10044" LastEditorDisplayName="user10044" LastEditDate="2019-05-30T17:37:26.477" LastActivityDate="2022-11-19T00:08:46.137" Title="I am searching for a book of this form and content, is there any?" Tags="|mathematics|book-recommendation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9673" PostTypeId="2" ParentId="9671" CreationDate="2019-05-31T03:23:01.193" Score="2" Body="&lt;p&gt;It generally happens that some excellent researchers don't teach very well, especially to undergraduates. Sometimes universities prefer that celebrated scientists just focus on research and teach one or two PhD level courses (and bring in lots of money). The reverse is also sometimes true, brilliant and popular teachers don't make a big name in hardcore sciences. Feynman was an exceptional teacher as well as a brilliant physicist. He loved to teach to younger people i.e., undergraduates. As a result, he was not only popular among the physicists (got a Nobel), he was extremely popular among students. Imagine how many young minds he must have influenced in his life. A typical auditorium in the US has 200-400 students. He must have had a big fan base. If you read his &lt;em&gt;Lectures in Physics&lt;/em&gt;, you will notice that the style is colloquial and informal-as if he is talking (those books originated from his actual lectures). I bought his books as a reference in my university days, even though physics was not my major.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-05-31T05:36:30.633" LastActivityDate="2019-05-31T05:36:30.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9674" PostTypeId="1" CreationDate="2019-05-31T10:52:31.687" Score="-3" ViewCount="136" Body="&lt;p&gt;Why did &lt;a href=&quot;https://en.wikipedia.org/wiki/Evangelista_Torricelli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Torricelli&lt;/a&gt; use a denser liquid like mercury? Why didn't he think of using lighter liquids like water?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="10106" LastEditDate="2021-09-06T01:48:45.557" LastActivityDate="2021-09-06T01:48:45.557" Title="Why didn't Torricelli use other liquids?" Tags="|physics|biographical-details|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9675" PostTypeId="2" ParentId="9674" CreationDate="2019-05-31T11:49:38.740" Score="1" Body="&lt;p&gt;It's not clear whether you understand the mechanism of a barometer, so I apologize if this is redundant.&lt;br&gt;&#xA;  The denser the fluid, the smaller the height of fluid required to balance 1 atmophere pressure.  Simplify your engineering at all times.&lt;br&gt;&#xA;In addition, selecting a fluid with very low vapor pressure means you have the least amount of error caused by the not-quite-vacuum region above the fluid in the closed tower. In fact, precision barometers have a thin layer of specialized oil placed on top of the Hg because this oil's vapor pressure is even lower than that of Hg.  (and it's trivial to calculate the mass offset of this layer). &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-05-31T11:49:38.740" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9676" PostTypeId="2" ParentId="7579" CreationDate="2019-05-31T12:48:09.183" Score="1" Body="&lt;p&gt;I'm quite not an electricity engineer, but it seems that the figure on p. 603 (title 2) - which actually represents an interesting electric circuit with a current source &quot;a&quot; (to use Gauss's designation) connected with a configuration of resistors that includes a &quot;Y&quot; shaped collection of 3 resistors (c,e,f in Gauss's notation) positioned in the centre of a circle of three other resistors (b,d,a in Gauss's notation) in a series (i.e, the  resistors system is neither serial nor parallel) - is related to the &lt;span class=&quot;math-container&quot;&gt;$Y-\Delta$&lt;/span&gt; transform invented in 1899 by Arthur Edwin Kennelly. Gauss gives in this note several formulas which describe the effective resistence of this configuration, as well as the intensity of current in each of the six resistors:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss defines:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(b+c)(d+e)+f(b+d+c+e) = p$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$bcde(\frac{{1}}{{b}}+\frac{{1}}{{c}}+\frac{{1}}{{d}}+\frac{{1}}{{e}}) + f(b+d)(c+e) = q$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and then notes that the total resistence of the configuration without the piece &quot;a&quot; is: &lt;span class=&quot;math-container&quot;&gt;$q/p$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These formulas seem to be very complicated so one can take it as an indication that he used later techniques from network analysis to derive them. Here is an image of the figure from Gauss's fragment.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/ek8GR.png&quot; alt=&quot;Gauss&amp;#39;s electrical circuit&quot;&gt;&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-06-07T18:52:51.670" LastActivityDate="2019-06-07T18:52:51.670" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9677" PostTypeId="2" ParentId="9671" CreationDate="2019-05-31T17:55:01.310" Score="0" Body="&lt;p&gt;It is worth noting Feynman's book; &lt;em&gt;QED: The Strange Theory of Light and Matter&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was written specifically to explain the theory of Quantum Electrodynamics for which he shared his Nobel prize to a non-physicist audience. It assumes no initial understanding of physics, just a willingness to follow the discussion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;All along the way he keeps reiterating &quot;No one knows why this is. But it is precisely so.&quot; A very satisfying read.&lt;/p&gt;&#xA;" OwnerUserId="10058" LastActivityDate="2019-05-31T17:55:01.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9678" PostTypeId="2" ParentId="2043" CreationDate="2019-05-31T18:06:58.807" Score="0" Body="&lt;p&gt;Lord Byron once stated, “While stands the Colosseum, Rome shall stand;/ When falls the Colosseum, Rome shall fall;/ And when Rome falls- the World.” The Colosseum not only depicted the incredible architectural skill of the Romans but also their superiority to others across the globe. Influencing most of the culture and traditions that have been integrated into our modern society, the Roman Empire is a stimulating model of how a single cultural group could shift the architectural world forever. Although inspired by the Greeks, the Roman style is entirely independent and distinct from all others; the Colosseum is only one of many of their historical masterpieces.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Romans constructed the first ostentatious stadia. These colossal structures were architecturally unique due to the Romans’ inimitable engineering technique. The Romans used a skill different than that of the Greeks; instead of carving large masses of stone, the Romans had successfully implemented pouring concrete for their designs. One of the most world-renowned structures and most visited attractions in Italy is the Colosseum, attracting an astounding four and a half million tourists a year. Standing at a height of approximately 45 meters, with respected dimensions of 189 meters by 156 meters, the name is only fitting for this enormous structure (Claridge 1998). Vespasian; the Roman emperor ruling at the time, had commenced its construction in approximately 69 AD near the lake of Nero’s Domus Aurea which took approximately ten years to complete (Claridge 1998). This was a remarkably rapid construction, considering the limited technology that the Romans had access to.&lt;/p&gt;&#xA;" OwnerUserId="10040" LastActivityDate="2019-05-31T18:06:58.807" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9679" PostTypeId="1" CreationDate="2019-05-31T22:06:12.190" Score="4" ViewCount="560" Body="&lt;p&gt;Current estimates for the number of stars in the universe are about 10^22. However, that number has changed several times as new observations have come forth. How has the estimate of the number of stars in the universe changed over time? What was the estimate after Hubble's findings? What was the estimate in the 70s, 90s, etc., and what new discoveries forced those estimates to change?&lt;/p&gt;&#xA;" OwnerUserId="10061" LastActivityDate="2019-06-18T04:40:49.083" Title="How has the estimate of the number of stars in the universe changed over time?" Tags="|astronomy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9681" PostTypeId="1" CreationDate="2019-06-01T00:22:08.557" Score="4" ViewCount="164" Body="&lt;p&gt;I am wondering which is the earliest record of the fact that the plane can be tiled by regular hexagons (in addition to triangles and squares, which may be slightly more obvious).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Had a look in the Loeb volume &lt;em&gt;Greek Mathematics from Thales to Euclid&lt;/em&gt;, but could not find it (they discuss hexagons but not tiling the plane).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kepler, &lt;em&gt;Harmony of the World&lt;/em&gt; (1619) is the earliest I have found so far.&lt;/p&gt;&#xA;" OwnerUserId="10063" LastEditorUserId="10063" LastEditDate="2019-06-01T11:49:39.680" LastActivityDate="2019-06-01T11:49:39.680" Title="Which is the earliest written record of hexagonal tesselation of the plane?" Tags="|geometry|euclidean-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9682" PostTypeId="2" ParentId="9681" CreationDate="2019-06-01T00:48:03.503" Score="3" Body="&lt;p&gt;Ornaments corresponding to all possible discrete groups of motions acting on the plane can be found in the wall decoration of  Alhambra palace in Granada (Spain).&#xA;This was built in 13-15 centuries (it was rebuilt several times, and it is not exactly known who and when made these decorations). It is amazing that they knew all these groups.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This gives an upper estimate for the date you ask. I am sure that the hexagonal tiling was known long before, but this knowledge is hard to confirm with documents and artifacts.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-06-01T00:48:03.503" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="9683" PostTypeId="2" ParentId="9679" CreationDate="2019-06-01T01:00:28.450" Score="2" Body="&lt;p&gt;Ptolemy's star catalog, which probably goes back to Hypparchus contains 1022 stars. &#xA;It contains most of the stars visible by the naked eye in the Northern hemisphere.&#xA;I do not think this number was substantially increased until the invention of the telescope. Flamsteed star catalog (made when the telescopes were already widely used)&#xA;contains 2554 stars, but it was already known at that time that Milky way consists of stars. Nobody even tried to count them. That there are other galaxies besides the Milky way was discovered only in 20th century, and their stars cannot be counted but their number can be only roughly estimated, from average density of stars in a galaxy and its side. Similarly we estimate the total numbers of stars, using the average density of galaxies, number of stars in a galaxy and the radius of the universe (which comes from the Hubble law).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, there was a period when people speculated that there are infinitely many stars, roughly speaking, this period lasted from Copernicus and Bruno to the end of 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-06-01T01:00:28.450" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9684" PostTypeId="2" ParentId="2043" CreationDate="2019-06-01T07:07:56.963" Score="3" Body="&lt;p&gt;One of the most important contributions of Latins to engineering was &lt;strong&gt;road-building&lt;/strong&gt;. This certainly pre-dates contacts with the Greek civilization:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;so called &quot;strade consolari&quot; departing from Rome (thus the Italian way of saying &quot;all roads lead to Rome&quot;) and connecting the principal area of Italy were built already starting from the early Latin period (when Rome was still a kingdom) and mostly developed during III Century B.C. The beginning of road (and thus bridges) construction, in fact, was probably developed in connection with Etruscan civilization.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the reason why we do not have famous &quot;names&quot; from this period is that road engineers where soldiers. Roads were basically built by soldiers (the word &quot;mile&quot; comes from &quot;miles=soldier&quot; in Latin). What we do have is that nothing similar to the net of roads developed in Italy was present in Greece: 150.000km of &quot;strade consolari&quot;. Many many names of Italian cities descend from numbers which were signalling their position along the roads (Treviso, Quartu Sant'Elena, Tor di Quinto, Sesto Fiorentino, Settimo Milanese, Nonantola, Decimomannu - are city names where you can find the words meaning 3rd, 4th, ... 10th).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The second important contribution to engineering were &lt;strong&gt;acqueducts&lt;/strong&gt;: they were usually the construction that immediately followed roads. Also the first acqueducts predates connection with the Greeks. &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The construction of &lt;strong&gt;Cloaca Maxima&lt;/strong&gt;, Rome sewer system, goes back to shortly after the foundation of the city. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;strong&gt;Appian aqueduct&lt;/strong&gt;, realized by &lt;strong&gt;Appio Claudio&lt;/strong&gt;, here you have the name of the consul , goes back to 310B.c.- parallel the the construction of Via Appia, 16km of underground aqueduct. Again I do not think anything analogous was present in Greece at around the same time. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The aqueduct &lt;strong&gt;Anio Vetus&lt;/strong&gt; (65km) was realized in 270 b.c. under &lt;strong&gt;Manio Curio Dentato&lt;/strong&gt; e and &lt;strong&gt;Flavio Flacco&lt;/strong&gt; that were nominated exactly for this purpose. But I guess that their role was mainly political and organizational and we do not have the names of engineers that practically did the computations and projects. Also &lt;strong&gt;Aqua Marcia&lt;/strong&gt; (144 b.c., 92km, under &lt;strong&gt;Quinto Marcio Re&lt;/strong&gt;) is at best contemporary to the annexion of Greece.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;in any case we do not have a culture of thermal public baths in the Greek civilization even vaguely comparable with the one developed by Latins and that brought them to constructin &lt;em&gt;thermae&lt;/em&gt; throughout the Roma empire, from Cartago to the English city of Bath.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here I bet that, as I already said, engineers themselves were basically considered just technicians whose names were note deserving memory;  we rather have the name of politicians that directed the cosntruction of this public buildings. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many historical documents attribute to Latin civilization this engineering ability: Strabo, Dionisus of Alycarnassus, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also for what concerns &lt;strong&gt;military engineering&lt;/strong&gt; you have to consider that the Roman legions were, already from the Republican era, travelling together with a supporting &lt;em&gt;praefectus fabrum&lt;/em&gt; that was basically an engineer concerned with road-building, camp fortifications and artillery constructions. The sieges of Veio (450 b.c) and Lilibeo (1st punic war 250 b.c.) saw the usage of &quot;war-machines&quot;. So I guess there was a culture predating contacts with Greeks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When you move to the Imperial era there are some factors to be considered. For Latins the Empire was a unique structure and famous emperors, writers, artists, scientists came from all over it and considered just Latin despite middleeastern-spanish-greek-african origins. Still it is true that in the so-called Hellenistic period everything that came from Greece was regarded as superior when it came to arts. It is therefore no surprise that for what concerned &lt;em&gt;artistic decoration&lt;/em&gt; of buildings many of them were done by artists coming from the Greek area (in the large) and it was mainly name of artists rather than name of engineers that remained in history. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To summarize a bit: names of engineers usually did not remain in history (we do not know any of the people responsible for the project of Colosseum - to comment on Thoriya). We mainly know names of politicians under whose supervision many operas were built. (so, for example, &lt;em&gt;Sesto Giulio Frontino&lt;/em&gt;, writing an opera &lt;em&gt;De acqueductus urbis Romae&lt;/em&gt;, was more a politician than a technician, &lt;em&gt;magister acquae&lt;/em&gt; in the I century b.C, born in Gallia Nerborense, now South of France). Certainly even Greek writers confirms the idea that road and bridges buildings and hydraulic constructions were mainly developed by Romans.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2019-06-01T07:07:56.963" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="9685" PostTypeId="2" ParentId="9681" CreationDate="2019-06-01T09:26:55.733" Score="4" Body="&lt;p&gt;It depends somewhat on what qualifies as a &amp;quot;record&amp;quot;. People observed honeycombs in bee-hives since the prehistoric times, so any evidence of honey consumption would be a record of sorts. Aristotle mentions honeycombs explicitly in his &lt;a href=&quot;http://classics.mit.edu/Aristotle/history_anim.5.v.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Animals&lt;/a&gt;, for example.&lt;/p&gt;&#xA;&lt;p&gt;As for a theoretical geometric study of tesselations, it occurs surprisingly late. The earliest surviving one is &lt;a href=&quot;https://en.wikipedia.org/wiki/Tessellation#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;indeed Kepler's of 1619&lt;/a&gt;, unless we count Platonic and Archimedean solids as tesselations of the sphere. It is interesting that (presumably) &lt;a href=&quot;https://en.wikipedia.org/wiki/Platonic_solid#Geometric_proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theaetetus's angle counting argument in the Elements&lt;/a&gt; for the existence of five Platonic solids is very similar to the standard modern argument for the existence of three plane tesselations. But we can only speculate whether he took any interest in those. Some Roman mosaics display hexagonal tiling patterns, especially those themed on Orpheus, see e.g. &lt;a href=&quot;http://www.getty.edu/publications/romanmosaics/assets/downloads/RomanMosaics_Belis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roman Mosaics in Getty Museum&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is more indirect evidence from medieval Islamic art. Lu and Broug even claim in &lt;a href=&quot;https://archive.bridgesmathart.org/2016/bridges2016-583.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Classifying Hexagonal Tilings in Islamic Architecture&lt;/a&gt; that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The drafting of a hexagonal honeycomb grid using a compass and straightedge was well understood both by the ancient Greeks and by medieval Islamic architects and designers. The specific sequence of steps using the compass and straightedge to create the honeycomb tessellation of hexagons has been well illustrated in a number of publications.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, this seems to be an indirect inference. They give no examples from antiquity, and the earliest Islamic examples are the Dado panel in Nishapur, Iran from the 10th century AD (currently kept in the Metropolitan Museum of Art in New York), and the Great Mosque of Cordoba, Spain (987 AD). Abdullahi and Bin Embi confirm the timeline in &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S2095263513000216&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evolution of Islamic Geometric Patterns&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;These developments began with simple geometrical shapes constructed from a circle and a set of tangential circles with the same radius (Critchlow, 1976), as seen in the Great Mosque of Kairouan during the early 9th century. By the late 9th century, grids of circles were introduced in the Ibn-Tulun Mosque. The grids were used as constructive bases for the simplest regular and semi-regular tiling with equilateral triangles, squares, hexagons, and octagons.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Then, contemporary to rise of Persian philosophers and cosmologists from Abu Sahl Al-Tustari to Sohravadi who had debates and important contribution to nature of numbers and their relation with that of nature (Critchlow, 1989), mystical Tetractys motifs and symbol merged to traditional geometric patterns. The result was the invention of abstract 6-point geometrical patterns based on the Tetractys symbol and 12-point star patterns that are associated with 12 zodiacal sectors. These decorative elements adorn the facades of the Tomb of Kharaqan (1067) in Iran.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/25597485&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abu al-Wafa' Buzjani's (959-998 AD) Geometry Needed by Craftsmen&lt;/a&gt; also dates to this period. But it is unclear that the early girih designs were conceptualized as tilings, the dominant early method was &amp;quot;direct strapwork&amp;quot; of zigzagging lines using straightedge and compass. &lt;a href=&quot;http://physics.princeton.edu/%7Esteinh/LuSteinhardt2007.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lu and Steinhardt&lt;/a&gt; argue that the shift occurred later:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;On the basis of our examination of a large&#xA;number of girih patterns decorating medieval&#xA;Islamic buildings, architectural scrolls, and other&#xA;forms of medieval Islamic art, we suggest that&#xA;by 1200 C.E. there was an important breakthrough in Islamic mathematics and design: the&#xA;discovery of an entirely new way to conceptualize and construct girih line patterns as decorated tessellations using a set of five tile types, which&#xA;we call “girih tiles”... We further show how the girih-tile&#xA;approach opened the path to creating new types&#xA;of extraordinarily complex patterns, including a&#xA;nearly perfect quasi-crystalline Penrose pattern&#xA;on the Darb-i Imam shrine (Isfahan, Iran, 1453&#xA;C.E.), whose underlying mathematics were not&#xA;understood for another five centuries in the West.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-06-01T09:26:55.733" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9686" PostTypeId="1" AcceptedAnswerId="9690" CreationDate="2019-06-01T07:07:26.630" Score="2" ViewCount="338" Body="&lt;p&gt;I’m reading “What is Real” by Adam Becker and this is mentioned without any more details. &lt;a href=&quot;https://books.google.com/books?id=X4QqDwAAQBAJ&amp;amp;pg=PT78&amp;amp;lpg=PT78&amp;amp;dq=schrodinger+apologized+to+einstein+for+his+duplicity&amp;amp;source=bl&amp;amp;ots=3c30bxaoz_&amp;amp;sig=ACfU3U2rwFRi-lk6bNb0wtSr4r57_mpNtw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjC-J2c2sfiAhUYCTQIHWHDALEQ6AEwDHoECAkQAQ#v=onepage&amp;amp;q=schrodinger%20apologized%20to%20einstein%20for%20his%20duplicity&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is the excerpt&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It’s very possible that it refers to something that was mentioned earlier, which I missed - but I couldn’t find anything.)&lt;/p&gt;&#xA;" OwnerUserId="10211" OwnerDisplayName="Gal" LastActivityDate="2019-06-01T19:47:35.110" Title="Why did Schrödinger write a letter to Einstein “apologizing profusely for his duplicity”?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9689" PostTypeId="2" ParentId="9658" CreationDate="2019-06-01T13:42:10.057" Score="-1" Body="&lt;p&gt;From my prospect the word pressure has been used since Aristotle's time but they were not calling it pressure but instead of that they were calling it weight of something been applied on something like water pressure as water is applying its weight on glass tube &lt;/p&gt;&#xA;" OwnerUserId="9993" LastActivityDate="2019-06-01T13:42:10.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9690" PostTypeId="2" ParentId="9686" CreationDate="2019-06-01T19:47:35.110" Score="4" Body="&lt;p&gt;Walter J. Moore's &lt;a href=&quot;https://books.google.com/books?id=m-YF1glKWLoC&amp;amp;pg=PA338#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Schrödinger: Life and Thought&lt;/em&gt; p. 338&lt;/a&gt; quotes the 19 July 1939 letter, which explains why:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I naturally knew there was a certain danger when I went back to Austria. But that the fortress would be surrendered without striking a blow, that I never reckoned until the end. Just a few days before, I was with a section chief in the Ministry and said to him: if you put a rifle in my hand I will be glad to defend myself, but don't let me remain as a hostage in nazified Graz. You can imagine with what feeling just a few weeks after the overthrow [&lt;em&gt;Umschmiss&lt;/em&gt;] I read the signature of the same gentleman under the orders of the new Minister! I hope you have not seriously taken amiss my certainly quite cowardly statement afterwards. I &lt;em&gt;wanted&lt;/em&gt; to remain free – and could not do so without great duplicity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2019-06-01T19:47:35.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9691" PostTypeId="1" AcceptedAnswerId="9693" CreationDate="2019-06-03T18:39:11.200" Score="1" ViewCount="139" Body="&lt;p&gt;Wikipedia notes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Early European astronomers such as Tycho Brahe identified new stars in the night sky (later termed novae), suggesting that the heavens were not immutable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Who were the other European astronomers who identified new stars around the time of Tycho Brahe?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Wikipedia contributors. (2019, May 23). Star. In Wikipedia, The Free Encyclopedia. Retrieved 18:38, June 3, 2019, from &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Star&amp;amp;oldid=898392700&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/w/index.php?title=Star&amp;amp;oldid=898392700&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9677" LastEditorUserId="4251" LastEditDate="2021-10-09T09:31:37.647" LastActivityDate="2021-10-09T09:31:37.647" Title="Who were the other European astronomers who identified new stars around the time of Tycho Brahe?" Tags="|astronomy|astronomers|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9692" PostTypeId="2" ParentId="9691" CreationDate="2019-06-04T12:13:42.017" Score="2" Body="&lt;p&gt;This sentence in Wikipedia does not say &quot;around the time of Brahe&quot;. One early astronomer who identified a new star was Hipparchus. According to the information about him that reached us, this was the reason for him to make the first star catalog. In 1054 Chinese astronomers observed a supernova in Crab Nebula, whose remains can be seen now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Astronomers who observed the  nova of 1604 were Kepler and Galileo (who lived approximately at the time of Brahe. There is no doubt that many other, less famous astronomers observed it as well. Brahe's nova was in 1572. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6085" LastEditDate="2019-06-05T05:32:40.580" LastActivityDate="2019-06-05T05:32:40.580" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9693" PostTypeId="2" ParentId="9691" CreationDate="2019-06-04T21:45:33.437" Score="1" Body="&lt;p&gt;For novae there is a listing at &lt;a href=&quot;http://www.cbat.eps.harvard.edu/nova_list.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;CBAT&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here are the first few listed.  The discoverer is in the last column.&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;DESIG.         DATE (UT)        R.A. (2000.0) Decl.         Mag. Range  REF.    DISCOVERER(S)  &#xA;04550 Leo N?? N1612 03 30       09 37.3     +15 15           4                  Scheiner                                            &#xA;CK    Vul N:  N1670 06 20       19 47.6     +27 19      1    2.6   20.7         Anthelm                                             &#xA;03846 Pup N?   1673 01 12       07 59.0     -43 49           3     20           Richer                                              &#xA;V529  Ori N?  N1678 03 28       05 58.4     +20 15           6     20.5         Hevelius                                            &#xA;WY    Sge N   N1783             19 32.7     +17 45      1    6     19.5         d'Agelet &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2019-06-05T00:04:56.513" LastActivityDate="2019-06-05T00:04:56.513" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9694" PostTypeId="1" AcceptedAnswerId="9697" CreationDate="2019-06-05T00:41:31.463" Score="7" ViewCount="2392" Body="&lt;p&gt;There is a quote from a 19-20th century scientist that goes (and I am paraphrasing):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;New scientific theories are never accepted until old scientist die. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Who is this cynical quote attributed to, and what is the exact phrasing of the quote? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on my memory of the quote I would put the following odds:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;90% it was a physicist from the late 19 or early 20th century&lt;/li&gt;&#xA;&lt;li&gt;70% it was Niels Bohr or Erwin Schrodinger — I did of course google for a list of known quotes but didn’t see the above one anywhere. &lt;/li&gt;&#xA;&lt;li&gt;20% it was in reference to quantum mechanics. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10073" LastActivityDate="2019-06-05T19:26:32.573" Title="Which physicist is this quote attributed to?" Tags="|physics|quantum-mechanics|quote|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9695" PostTypeId="2" ParentId="9694" CreationDate="2019-06-05T02:14:01.837" Score="2" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;90% it was a physicist from the late 19 or early 20th century&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Excellent memory! It is actually from Thomas Kuhn who was a physicist, and who later turned into a philosopher. His book &lt;em&gt;The Structure of Scientific Revolution&lt;/em&gt; mentions something like this on page 152. This is the third edition 1962 reprinted in 1996.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edits:&lt;/strong&gt; On page 151, he also quotes Planck.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DroDy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DroDy.png&quot; alt=&quot;Book image&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-06-05T13:56:29.340" LastActivityDate="2019-06-05T13:56:29.340" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="9696" PostTypeId="2" ParentId="9646" CreationDate="2019-06-05T03:58:31.283" Score="0" Body="&lt;p&gt;I suspect you're thinking of Feynman's story &quot;Is Electricity Fire?&quot;. See&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.e-reading.club/chapter.php/71262/46/Feynman_-_Surely_Youre_Joking,_Mr._Feynman__Adventures_of_a_Curious_Character.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.e-reading.club/chapter.php/71262/46/Feynman_-_Surely_Youre_Joking,_Mr._Feynman__Adventures_of_a_Curious_Character.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;where he has frustrating discussions with people from the humanities at a conference on the &quot;the ethics of equality&quot; and also discussions about science with rabbinical students at the residence where he stayed during that conference.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2019-06-05T03:58:31.283" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9697" PostTypeId="2" ParentId="9694" CreationDate="2019-06-05T05:00:13.233" Score="15" Body="&lt;p&gt;Max Planck, &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=7213&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Scientific Autobiography and Other Papers&lt;/em&gt;&lt;/a&gt; (Westport, CT: Greenwood, 1949), &lt;a href=&quot;https://books.google.com/books?id=F3iMBAAAQBAJ&amp;amp;pg=PT24&quot; rel=&quot;noreferrer&quot;&gt;pp. 33-34&lt;/a&gt;:&lt;blockquote&gt;A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.&lt;/blockquote&gt;&lt;sup&gt;as quoted in:&lt;br&gt;• M. López Corredoira and C. Castro Perelman, &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5032&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Against the Tide: A Critical Review by Scientists of How Physics and Astronomy Get Done&lt;/em&gt;&lt;/a&gt; (Boca Raton, Fla.: Universal Publishers, 2008), p. 12&lt;br&gt;or&lt;br&gt;• Peter Pesic, &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5685&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Music and the Making of Modern Science&lt;/em&gt;&lt;/a&gt; (MIT Press, 2014), p. 265.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-06-05T19:26:32.573" LastActivityDate="2019-06-05T19:26:32.573" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9698" PostTypeId="1" CreationDate="2019-06-05T07:23:17.070" Score="7" ViewCount="2420" Body="&lt;p&gt;I wanted to know about the experiments that Newton performed which led to the formulation of laws of motion. In particular I am interested in the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did he measure the forces in order to discover that the net force was directly proportional to the acceleration produced? (My teacher told me that he probably used springs but doesn't that require assuming Hooke's Law to be true. If this was the case how did Hooke measure forces before formulating his law).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this translation of Newton's work&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;... and the same is know by the weight of each body; for it (mass) is proportional to the weight as I have found by &lt;strong&gt;experiments on pendulums&lt;/strong&gt; very accurately made .....&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could somebody please clarify?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="8145" LastEditorUserId="55" LastEditDate="2019-06-06T05:46:12.233" LastActivityDate="2022-02-26T13:50:19.720" Title="How did Newton measure forces in his experiments to establish the laws of motion?" Tags="|classical-mechanics|newton|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9699" PostTypeId="1" CreationDate="2019-06-05T09:51:18.340" Score="0" ViewCount="210" Body="&lt;p&gt;I am researching the presence of important scientists in Nordic countries — Iceland, Norway, Denmark, Sweden and Finland. I was hoping that someone could help me since I've not been able to find relevant information.&lt;/p&gt;&#xA;&lt;p&gt;It is well known that Eugene Paul Wigner had to travel to Stockholm, Sweden, to receive his Nobel Prize. I thought that he also travelled to Copenhagen, mainly to discuss quantum models with Niels Bohr, like many scientists at the time did, e.g., Paul Dirac, Albert Einstein or John Wheeler. I am pretty convinced that he travelled at least once to Denmark, but did not find anything. So, maybe someone here knows more about this.&lt;/p&gt;&#xA;&lt;p&gt;Also, I have not found any information that says that Wigner travelled at least once to Norway or Finland. I honestly don't think that any scientist at that time travelled to Iceland, so I'm going to ignore this country.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I have not found any information that says that John von Neumann travelled at least once to any Nordic country. But, again, this is weird for me. Didn't he go at least once to Copenhagen?&lt;/p&gt;&#xA;&lt;p&gt;So, did they travelled to any Nordic countries? Didn't they give conferences, for example, at these countries at least once in their lifes?&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your help in advance!&lt;/p&gt;&#xA;" OwnerUserId="10076" LastEditorUserId="4251" LastEditDate="2021-09-20T13:36:20.170" LastActivityDate="2021-09-20T13:36:20.170" Title="Did John von Neumann ever go to any Nordic Country? Did Eugene Paul Wigner ever go to any other Nordic country apart from Sweden?" Tags="|mathematicians|biographical-details|physicists|von-neumann|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9700" PostTypeId="2" ParentId="9698" CreationDate="2019-06-05T12:03:34.057" Score="2" Body="&lt;p&gt;Newton performed no experiments to discover the laws of motion. All necessary experiments were performed by Galileo and other predecessors. Newton's contribution to the laws of motion was a clear mathematical formulation of what was essentially known at that time. To this he added the law of gravity which also required no experiments but just contemplation of the (essentially known) motion of the Moon. In this area (laws of motion and gravity) Newton's contribution was theoretical. (Unlike his contribution to other areas of physics like optics). This can be compared with Maxwell's contribution to electrodynamics where almost all laws (except one) have been previously established by experiments, and the last one was figured out by Maxwell, on purely theoretic grounds. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-06-05T12:03:34.057" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9701" PostTypeId="2" ParentId="5673" CreationDate="2019-06-05T15:45:44.040" Score="3" Body="&lt;p&gt;From the reading I have done, it is mainly Gibbs. Boltzmann introduced the idea of a factor having the form of &lt;span class=&quot;math-container&quot;&gt;$\exp(-\alpha E)$&lt;/span&gt; in his re-derivation of Maxwell's distribution for the speeds of gases in 1868. Later the constant alpha was determined to be &lt;span class=&quot;math-container&quot;&gt;$1/kT$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; Boltzmann's constant -- I believe this determination was also carried out by Boltzmann. But Boltzmann's definition of entropy was contained in the &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;-theorem (1872) and later the formula &lt;span class=&quot;math-container&quot;&gt;$S = k \log \Omega$&lt;/span&gt; (1878), which deal with an average over single-particle trajectories in phase space, and a constant-energy average, respectively. It was Gibbs who came up with the idea of the canonical and grand canonical ensembles, where the partition function is the essential quantity. Because of this, I've seen the Boltzmann factor referred to as the Gibbs factor in a couple of places. So while I can't point to one website or source that attributes the partition function to Gibbs, I think all this evidence points very strongly to Gibbs having developed it. A good source on Gibbs' work in this area is Lynde Wheeler's biography, Chapter 10.&lt;/p&gt;&#xA;" OwnerUserId="10078" LastEditorUserId="6609" LastEditDate="2023-04-28T13:05:33.347" LastActivityDate="2023-04-28T13:05:33.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9702" PostTypeId="2" ParentId="9646" CreationDate="2019-06-05T18:00:33.827" Score="1" Body="&lt;p&gt;See the story &quot;Meeeeeeeeeee!&quot; at &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.e-reading.club/chapter.php/71262/17/Feynman_-_Surely_Youre_Joking%2C_Mr._Feynman__Adventures_of_a_Curious_Character.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.e-reading.club/chapter.php/71262/17/Feynman_-_Surely_Youre_Joking%2C_Mr._Feynman__Adventures_of_a_Curious_Character.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;in which he gives comparisons between poetry and physics but considers such analogies not to be meaningful since he thinks he could do the same between poetry and any other subject.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2019-06-05T18:00:33.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9703" PostTypeId="1" AcceptedAnswerId="9708" CreationDate="2019-06-05T22:39:49.917" Score="5" ViewCount="245" Body="&lt;p&gt;Is the map &lt;span class=&quot;math-container&quot;&gt;$y=2x+3$&lt;/span&gt; linear?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Of course it is.&quot; -- a high school teacher will answer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Nope; it's affine, but not linear.&quot; -- a college student will contradict.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This difference terminology that basic ought to have historical roots. Could you point toward them?&lt;/p&gt;&#xA;" OwnerUserId="2393" LastActivityDate="2019-06-07T02:27:49.710" Title="Why is the meaning of &quot;linear&quot; different in school and college use?" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9704" PostTypeId="2" ParentId="7292" CreationDate="2019-06-06T01:49:47.753" Score="8" Body="&lt;p&gt;Paul Krugman's research was caused by Isaac Azimov's &lt;em&gt;Foundation&lt;/em&gt; novels.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(For this answer, you have to accept economics as a science and you have to accept &quot;caused scientists to do real research&quot; in the sense of &lt;em&gt;motivating&lt;/em&gt; them to do it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From his interview, December 2008, on the Nobel website.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Yes, yes, I know, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nobelprize.org/prizes/economic-sciences/2008/krugman/interview/&quot; rel=&quot;noreferrer&quot;&gt;https://www.nobelprize.org/prizes/economic-sciences/2008/krugman/interview/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Q: &quot;What turned you on to being an economist in the first place?&quot; ...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;A: &quot;I was an avid science fiction reader when I was a teenager and&#xA;  there's the classic set of novels by Isaac Azimov: the Foundation&#xA;  novels, which are about how a group of social scientists who save&#xA;  galactic civilization through their understanding of the laws that&#xA;  determine the behavior of societies, and I wanted to be one of those&#xA;  guys, and the closest you can get at this point, I'm afraid, is being&#xA;  an economist&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10089" LastActivityDate="2019-06-06T01:49:47.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9705" PostTypeId="2" ParentId="9698" CreationDate="2019-06-06T05:46:16.373" Score="3" Body="&lt;p&gt;Newton did not measure forces for this purpose, nor could he, the second law itself is needed to make sense of measuring forces. What is actually measured are velocities and accelerations, or rather times and distances traveled, so any &quot;measuring&quot; of force has to &lt;em&gt;presuppose&lt;/em&gt; at some point that the law holds (establishing the Hooke's law would have to presuppose it too, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/2687/did-hookes-law-come-from-experiments-or-was-it-mathematically-derived-from-new&quot;&gt;Did Hooke&amp;#39;s law come from experiments, or was it mathematically derived from Newtonian mechanics?&lt;/a&gt;). What Newton measured were the effects it predicted (for collisions, pendulums, circular motion, etc.), not the forces themselves:&quot;&lt;em&gt;On the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notion of force in circulation before Newton was that of an &quot;internal&quot; force that imparts a body with a quantity of motion (our momentum) and keeps it moving, and can roughly be identified with &lt;span class=&quot;math-container&quot;&gt;$Fdt$&lt;/span&gt; or its integral. For impulsive forces in collisions its action was instantaneous, and so imparted finite momentum in zero time. This is the notion that appears in Newton's early works. Eventually, he realized that this Aristotelian idea was incompatible with the Galileo's principle of inertia, and switched to calling the force that which affected a &lt;em&gt;change&lt;/em&gt; in the state of uniform motion, &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;. &quot;&lt;em&gt;Tis known by the light of nature, that equal forces shall effect an equal change in equal bodies&lt;/em&gt;&quot;, he wrote, meaning &lt;span class=&quot;math-container&quot;&gt;$Fdt\sim dv$&lt;/span&gt;. If we denote the proportionality constant &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; we get the second law &lt;span class=&quot;math-container&quot;&gt;$Fdt=mdv$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;The light of nature&quot; is Descartes's term for the innate knowledge of our reason. But the reason was helped, in this case, by the prior work of Galileo on the falling bodies and inertia, Descartes and Huygens on collisions, and Huygens on circular motion, which discredited the Aristotelian &quot;internal forces&quot;. The OP quote is taken from &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Definitions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Definition I of Principia&lt;/a&gt;, and refers not to establishing the proportionality of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\frac{dv}{dt}$&lt;/span&gt;, but to determining that &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, mass, is proportional to weight, what we call the equivalence of inertial and gravitational mass:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The quantity of matter is the measure of the same, arising from its density and bulk conjunctly... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The accurately made experiments are described in &lt;a href=&quot;https://en.wikisource.org/wiki/Page:Newton%27s_Principia_(1846).djvu/400&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principia, Book III, Proposition VI, Theorem VI&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat... The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As expected, the conclusion &lt;em&gt;presupposes&lt;/em&gt; the second law, and the third one as well. To verify the third law, Newton collided two pendulums with &lt;em&gt;unequal&lt;/em&gt; masses, as described in the Scholium to &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Axioms,_or_Laws_of_Motion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Axioms, or Laws of Motion in Principia&lt;/a&gt;. The impacts/forces were &quot;measured&quot; by how far the pendulums rebounded. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fowler gives a brief account of Newton's evolution on the notion of force, and the second law, in &lt;a href=&quot;http://galileoandeinstein.physics.virginia.edu/lectures/Newtons2ndLaw1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton Clarifies the Concept of Force lecture&lt;/a&gt;, for a more detailed account see &lt;a href=&quot;https://books.google.com/books/about/Never_at_Rest.html?id=3ngEugMMa9YC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Westfall's biography of Newton, Never at Rest&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-07T03:45:11.760" LastActivityDate="2019-06-07T03:45:11.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9707" PostTypeId="2" ParentId="9699" CreationDate="2019-06-06T22:41:17.463" Score="7" Body="&lt;p&gt;“In Copenhagen again! The brothers Bohr fetched me at the pier, and now I’m established in Niels Bohr’s private palace.  I had numerous conversations with the Bohrs and Mrs. Bohr, of course mostly political—but we even managed to talk an hour and a half on ‘the interpretation of quantum mechanics.’  I’m sure we were showing off, the both of us: giving an exhibition, that we can worry about physics in September 1938.  It’s all like a dream, a dream of a peculiarly mad quality . . . the Bohrs quarreling, whether Tcheckoslovakia ought to give in—and whether there is any hope for causality in quantum theory.” &lt;/p&gt;&#xA;&#xA;&lt;p&gt;￼John von Neumann to Klára von Neumann, 18 September 1938. &lt;/p&gt;&#xA;" OwnerUserId="10092" LastActivityDate="2019-06-06T22:41:17.463" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9708" PostTypeId="2" ParentId="9703" CreationDate="2019-06-07T02:27:49.710" Score="8" Body="&lt;p&gt;Yes, it has historic roots. The term &quot;linear&quot; is much older than &quot;affine&quot; and the function &lt;span class=&quot;math-container&quot;&gt;$ax+b$&lt;/span&gt; is &quot;linear&quot; because its graph is a straight line. With the invention of linear algebra, the meaning of the world linear changed. But educators, especially on the lower level are very conservative, and also reluctant to introduce extra Greek terms (which on their opinion intimidate students). So the terminology in lower levels of mathematical education lags behind the development of mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similar situation we have with the words &quot;equal&quot; and &quot;congruent&quot; in geometry. Euclid and later educators called two figures &quot;equal&quot; if one can be moved to coincide with another. With the spread of set theory, the word &quot;equal&quot; was reserved for equality in the set-theoretic sense. But school education (in most countries) continues to use the word &quot;equal&quot; in Euclid's sense.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-06-07T02:27:49.710" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9709" PostTypeId="2" ParentId="7292" CreationDate="2019-06-07T15:56:06.817" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Alcubierre_drive&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alcubierre drive&lt;/a&gt; concept. Miguel Alcubierre published his first and only paper on the concept while still a PhD student, stating that he drew inspiration from the Star Trek television series. (See the associated tab on the Wiki link.)&lt;/p&gt;&#xA;" OwnerUserId="9704" LastActivityDate="2019-06-07T15:56:06.817" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9710" PostTypeId="1" CreationDate="2019-06-07T21:43:53.680" Score="8" ViewCount="456" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1.&lt;/strong&gt; When did the modern definition of a &lt;strong&gt;polynomial&lt;/strong&gt; (as a sequence of coefficients, with multiplication defined by &lt;span class=&quot;math-container&quot;&gt;$\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$&lt;/span&gt;) emerge?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Let me clarify: People have been working with some notion of polynomials for centuries, but mostly without a rigorous concept of what a polynomial is. From what I understand, &amp;quot;equations&amp;quot; rather than polynomials were the focus, and demarcating the precise extent of what an &amp;quot;equation&amp;quot; is was secondary to actually solving equations of given, fairly specific types. When the need for a rigorous definition became clear, it seems that the go-to solution was to define a polynomial as a polynomial &lt;strong&gt;function&lt;/strong&gt; (e.g., §1 of &lt;a href=&quot;https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV0733&quot; rel=&quot;noreferrer&quot;&gt;Roland Weitzenböck, &lt;em&gt;Invariantentheorie&lt;/em&gt;, 1923&lt;/a&gt;). This worked well for polynomials over &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, but not so much for polynomials over finite fields (which may be why E. H. Moore had to define &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_{p^n}$&lt;/span&gt; as a quotient ring of a polynomial ring over &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt; rather than of a polynomial ring over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_p$&lt;/span&gt; &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.bams/1183407788&quot; rel=&quot;noreferrer&quot;&gt;in his 1893 talk&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;What I call the &amp;quot;modern definition&amp;quot; of a polynomial is one of the following two definitions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;either as an infinite sequence &lt;span class=&quot;math-container&quot;&gt;$\left(a_0, a_1, a_2, \ldots\right)$&lt;/span&gt; of coefficients (with all but finitely many &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$a_n = 0$&lt;/span&gt;), with addition defined entrywise and multiplication defined by the &lt;span class=&quot;math-container&quot;&gt;$\left(ab\right)_n = \sum\limits_{k=0}^n a_k b_{n-k}$&lt;/span&gt; formula (and the indeterminate defined as the sequence &lt;span class=&quot;math-container&quot;&gt;$\left(0,1,0,0,0,\ldots\right)$&lt;/span&gt; rather than being some mythic &amp;quot;symbol&amp;quot; or &amp;quot;indeterminate value&amp;quot; or &amp;quot;variable&amp;quot;),&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;or as the monoid ring of the additive monoid &lt;span class=&quot;math-container&quot;&gt;$\left\{0,1,2,\ldots\right\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The second of these two definitions appears in §15 of B. L. van der Waerden, &lt;em&gt;Moderne Algebra I&lt;/em&gt;, 2te Auflage 1937. (I can't check the 1st edition, as I don't have it.) This gives a lower bound on the age of the modern definition. As for an upper bound, I am perplexed. Nicolas Bourbaki's &lt;em&gt;Elements of the History of Mathematics&lt;/em&gt; grazes the question on its page 82:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In all this research, the fields that arise consist of &amp;quot;concrete&amp;quot; elements&#xA;in the sense of classical mathematics - (complex) numbers or functions of&#xA;complex variables. 32 But already Kronecker, in 1882, takes full account of&#xA;the fact (obscurely sensed by GaUBB and Galois) that &amp;quot;indeterminates&amp;quot; only&#xA;play the role in his theory of the basis elements of an algebra, and not that&#xA;of variables in the sense of Analysis ([186 a], v. II, pp. 339-340); and, in 1887,&#xA;he develops this idea, linked to a vast programme that aims at nothing less&#xA;than recasting all mathematics by rejecting all that cannot be reduced to&#xA;algebraic operations on the integers. It is on this occasion that, taking up an&#xA;idea of Cauchy ([56 a], (1), v. X, pp. 312 and 351) who had defined the field&#xA;&lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt; of complex numbers as the field of remainders &lt;span class=&quot;math-container&quot;&gt;$\mathbb R\left[X\right]/\left(X^2+1\right)$&lt;/span&gt;, Kronecker&#xA;shows how the theory of algebraic numbers is completely independent of the&#xA;&amp;quot;fundamental theorem of algebra&amp;quot; and even of the theory of real numbers,&#xA;all fields of algebraic numbers (of finite degree) being isomorphic to a field of&#xA;remainders &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q\left(X\right)/\left(f\right)$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; an irreducible polynomial over &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q$&lt;/span&gt;)([186 a], v. 1111,&#xA;pp. 211-240).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Following the references here, I see some signs of the modern definition slowly dawning in the 19th Century: [Note: &amp;quot;Polynomials&amp;quot; are often called &amp;quot;forms&amp;quot; before ca. the 1930s, particularly when they are homogeneous.]&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I don't know what exactly Cauchy had done; the [56 a] reference goes to his Oeuvres, which I don't have.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The first Kronecker reference goes to &lt;a href=&quot;https://archive.org/details/grundzgeeinerar01krongoog/page/n129&quot; rel=&quot;noreferrer&quot;&gt;his &lt;em&gt;Grundzüge einer arithmetischen theorie der algebraischen Grössen&lt;/em&gt; (1882)&lt;/a&gt;, where he talks of &amp;quot;veränderliche oder unbestimmte Grössen&amp;quot; but never constructs any.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://projecteuclid.org/euclid.acta/1485888021&quot; rel=&quot;noreferrer&quot;&gt;J. Molk, &lt;em&gt;Sur une notion qui comprend celle de la divisibilité et sur la théorie générale de l'élimination&lt;/em&gt;, Acta Math. &lt;strong&gt;6&lt;/strong&gt; (1885), pp. 1--165&lt;/a&gt;, spends its section 2 (and maybe the whole Chapitre I?) arguing about the difference between variables and indeterminates. Alas, the argument appears to veer into philosophical terrain rather quickly, which puts it beyond my French comprehension skills; thus I have no idea what conclusion he actually comes to.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://eudml.org/doc/157689&quot; rel=&quot;noreferrer&quot;&gt;H. Weber, &lt;em&gt;Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie&lt;/em&gt;, Mathematische Annalen &lt;strong&gt;43&lt;/strong&gt; (1893), pp. 521--549&lt;/a&gt; tries to define polynomials in its §3 (after defining the notions of groups and fields in ways that are equivalent to their modern definitions). The definition pays obvious tribute to the idea that polynomials should not be functions; in particular it explicitly says that two polynomials are understood to be equal if and only if their respective coefficients are equal (rather than their values being equal). However, a vague notion of &amp;quot;expressions of the form &lt;span class=&quot;math-container&quot;&gt;$\Phi\left(x,y,z,\ldots\right) = \sum a x^r y^s z^t \cdots$&lt;/span&gt;&amp;quot; underlies this definition, and (e.g.) the multiplication is never explicitly defined (nor its associativity proven). (To this day, some algebra textbooks geared towards non-math majors work on this level.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So my question is: Who completed this definition? Weber (later)? Kronecker? Dedekind? Noether? van der Waerden? And while at that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2.&lt;/strong&gt; Who defined formal power series?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 3.&lt;/strong&gt; Who defined polynomials in noncommutative indeterminates?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These are litmus tests for Question 1; a rigorous definition of polynomials needs only a few trivial modifications to define formal power series, and a definition of noncommutative polynomials is not far away either.&lt;/p&gt;&#xA;&lt;p&gt;The concept of a polynomial is, in a way, similar to the number &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;: It's an idea that, on its own, appears pedantic and meatless. But once it is established, it becomes a building block of a whole discipline that no one would want to be missing. (19th Century algebraists like Sylvester tend to use the unique factorization property of a polynomial ring without ever explaining what it is that they are factoring; from a modern perspective, this is a gaping hole in their proofs.) My hope is that, as this specific matter is much younger than the number &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, we may have better clues to its authors.&lt;/p&gt;&#xA;" OwnerUserId="10096" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-06-07T21:49:22.420" Title="Who first defined polynomials as sequences?" Tags="|mathematics|abstract-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9711" PostTypeId="1" AcceptedAnswerId="9712" CreationDate="2019-06-08T07:01:15.213" Score="4" ViewCount="920" Body="&lt;p&gt;Before the advent of calculators they had useful ready made tables for the main functions:sines,cosines logs etc..., do you know if tables of square roots were ever produced or in use?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I never heard of it, yet, it would be very easy to produce, since it is enough to find the roots of numbers from 1 to 100.&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2019-06-16T15:49:02.443" Title="Were tables of square roots ever in use?" Tags="|mathematics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9712" PostTypeId="2" ParentId="9711" CreationDate="2019-06-08T08:52:13.017" Score="15" Body="&lt;p&gt;Pretty much &lt;strong&gt;every mathematics textbook&lt;/strong&gt; (school or college) before the early 1980s (and many even up to the late 1980s), at the algebra level or above, as well as many (most?) chemistry and physics and engineering textbooks, had such a table at the back of the book (as an appendix or something, where &quot;selected answers&quot; and &quot;index&quot; and &quot;glossary&quot; would appear). Often the entries would be for both &lt;span class=&quot;math-container&quot;&gt;$\sqrt{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{10n},$&lt;/span&gt; which was enough to allow you to easily get approximations for any magnitude-size numbers. For example, to find an approximation for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{3880},$&lt;/span&gt; use &lt;span class=&quot;math-container&quot;&gt;$n=3.88$&lt;/span&gt; and look in the &lt;span class=&quot;math-container&quot;&gt;$\sqrt{10n}$&lt;/span&gt; entries, since&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sqrt{3880} \; =\;  \sqrt{1000\times 3.88} \; = \; 10\sqrt{10 \times 3.88}.  $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were stand-alone (i.e. as separate books) tables also, such as the following, where square roots from &lt;span class=&quot;math-container&quot;&gt;$1.00$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$9.99$&lt;/span&gt; by increments of &lt;span class=&quot;math-container&quot;&gt;$0.01$&lt;/span&gt; are on pp. 16-17 and square roots from &lt;span class=&quot;math-container&quot;&gt;$10.0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$99.9$&lt;/span&gt; by increments of &lt;span class=&quot;math-container&quot;&gt;$0.1$&lt;/span&gt; are on pp. 18-19:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Mathematical Tables for Class-Room Use&lt;/strong&gt; by Mansfield Merriman (1915)&#xA;&lt;a href=&quot;https://archive.org/details/mathematicaltabl00merrrich&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/mathematicaltabl00merrrich&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I was in high school I owned (purchased in 1974) and used the 20th edition (1973) of the &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00ACAR1X2&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;CRC Standard Mathematical Tables&lt;/strong&gt;&lt;/a&gt;. On pp. 71-90 you'll find a table having column entries for &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{10n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n^{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{10n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{100n}$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$n=1000$&lt;/span&gt; by increments of &lt;span class=&quot;math-container&quot;&gt;$1.$&lt;/span&gt; In high school I also owned (I no longer seem to have it, however) &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00121UH3Q&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Logarithmic and Trigonometric Tables to Five Places&lt;/strong&gt;&lt;/a&gt; by Kaj L. Nelson (in the well known Barnes and Noble &lt;em&gt;College Outline Series&lt;/em&gt; of books), and other people I knew (in college) had the Schaum's Outline Series of &lt;a href=&quot;https://archive.org/details/MathematicalHandbookOfFormulasAndTables&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Mathematical Handbook of Formulas and Tables&lt;/strong&gt;&lt;/a&gt; by Murray R. Spiegel (1968), which I didn't have a copy of back then but a few years ago I saw and purchased a copy of a later printing (the 1990 printing) at a local used bookstore (square roots are on pp. 238-239). However, in looking at the Schaum's book now, it's more of a handbook of formulas (algebraic, trigonometric, calculus, series, special functions, etc.) than a table of numerical values for computational use.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For other such books, try &lt;a href=&quot;https://www.google.com/search?q=%22mathematical+tables%22+%22square+roots%22+archive.org&quot; rel=&quot;noreferrer&quot;&gt;this search&lt;/a&gt; and similar searches. For tables at the back of textbooks, simple archive.org and google-books searches will give you hundreds (if not thousands) of examples where you'll find square root tables at the back of the book.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2019-06-08T08:52:13.017" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9713" PostTypeId="2" ParentId="9711" CreationDate="2019-06-08T14:13:22.273" Score="4" Body="&lt;p&gt;The Babylonian clay tablet from around 1700 BC known as YBC7289, since it's one of many in the Yale Babylonian Collection has a diagram of a square with one side marked as having length 1/2. They took this length, multiplied it by the square root of 2, and got the length of the diagonal. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given that square roots were in use then, it would have made sense to create a table of such, rather than repeating the labour in calculating the result. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2019-06-08T14:13:22.273" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9716" PostTypeId="1" CreationDate="2019-06-10T06:51:28.620" Score="7" ViewCount="497" Body="&lt;p&gt;Who is the first person defined the concept of a &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Random_variable&quot; rel=&quot;noreferrer&quot;&gt;random variable&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastActivityDate="2024-01-09T06:56:06.580" Title="Who coined the term random variable?" Tags="|mathematics|mathematicians|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9717" PostTypeId="1" AcceptedAnswerId="9719" CreationDate="2019-06-10T11:53:39.260" Score="5" ViewCount="674" Body="&lt;p&gt;The use of negative numbers in most of today's calculations is natural. But how did the use of negative numbers began in physics? What physical quantity required the introduction of negative numbers and why?&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2019-07-12T18:09:08.133" LastActivityDate="2019-07-12T18:09:08.133" Title="How were negative numbers first used in physics?" Tags="|physics|mathematical-physics|measurement|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9718" PostTypeId="2" ParentId="9716" CreationDate="2019-06-10T21:38:27.220" Score="10" Body="&lt;p&gt;The interesting material from &lt;a href=&quot;http://jeff560.tripod.com/r.html&quot; rel=&quot;noreferrer&quot;&gt;mathwords&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Random variable&lt;/strong&gt; is found in 1914 in &lt;em&gt;Biometrika&lt;/em&gt;: “nDx and nDy are now random variables independent of time.” [OED]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Variabile casuale&lt;/strong&gt; is found in 1916 in F. P. Cantelli, “La Tendenza ad un limite nel senso del calcolo delle probabilità,” &lt;em&gt;Rendiconti del Circolo Matematico di Palermo&lt;/em&gt;, 41, 191-201 (David, 1998). A. N. Kolmogorov used the term &lt;strong&gt;zufällige Gröβe&lt;/strong&gt; in the &lt;em&gt;Grundbegriffe der Wahrscheinlichkeitsrechnung&lt;/em&gt; (1933).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Random variable&lt;/strong&gt; is found in 1934 in A. Winter, “On Analytic Convolutions of Bernoulli Distributions,” &lt;em&gt;American Journal of Mathematics&lt;/em&gt;, 56, 659-663 and more visibly in H. Cramér’s &lt;em&gt;Random Variables and Probability Distributions&lt;/em&gt; (1937) (David, 1998). Other, perhaps better, terms, including &lt;strong&gt;chance variable&lt;/strong&gt; in Doob &lt;em&gt;Annals of Mathematical Statistics&lt;/em&gt;, 6, (1935), p. 160 and &lt;strong&gt;stochastic variable&lt;/strong&gt; in Wald &amp;amp; Wolfowitz, &lt;em&gt;Annals of Mathematical Statistics&lt;/em&gt;, 10, (1939), p. 106 did not survive. J. L. Doob recalled the time when he was writing &lt;em&gt;Stochastic Processes&lt;/em&gt; and W. Feller was writing his &lt;em&gt;Introduction to Probability Theory and its Applications&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;I had an argument with Feller. He asserted that everyone said “random variable” and I asserted that everyone said “chance variable.” We obviously had to use the same name in our books, so we decided the issue by a stochastic procedure. That is, we tossed for it and he won.&#xA;From “A Conversation with Joe Doob,” &lt;em&gt;Statistical Science&lt;/em&gt; 1997 (p. 307) Project Euclid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-06-10T21:38:27.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9719" PostTypeId="2" ParentId="9717" CreationDate="2019-06-10T23:11:11.963" Score="9" Body="&lt;p&gt;Ancient Greeks painstakingly avoided negative numbers, although they could have come handy in astronomical calculations and number theory, among other places. Brahmagupta in Correctly Established Doctrine of Brahma (c. 630 AD) uses the language of &quot;fortunes&quot; and &quot;debts&quot;, which suggests the merchant origin of the negative number concept, but that remains a speculation.  Bhaskara II casually mentions negative sides when solving triangles in &lt;a href=&quot;https://en.wikipedia.org/wiki/L%C4%ABl%C4%81vat%C4%AB&quot; rel=&quot;noreferrer&quot;&gt;Lîlâvatî (c. 1150)&lt;/a&gt;:&quot;[The result] &lt;em&gt;is negative, that is to say, in the contrary direction.&lt;/em&gt;&quot; Even earlier Liu Hui (c. 260 AD) and later medieval Islamic mathematicians accepted the negatives simply to make the algebra work more generally, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2635/55&quot;&gt;Historically, how did people define multiplication for negative numbers?&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for physics, it is not that any  problem &quot;required&quot; them. Unlike many other concepts, where physics was an engine driving mathematical advances, when negative numbers finally made their way into physics it was as an afterthought. Medieval natural philosophers, and even Galileo, did not follow Bhaskara. They show considerable ingenuity in &lt;em&gt;avoiding&lt;/em&gt; negatives, in the fashion of ancient Greeks, even where they would have been natural, such as in velocities or temperatures. Oresme (c. 1360 AD), the pioneer of coordinates and (bar) graphs, always adds a constant to his velocity and temperature graphs to avoid negative values. Even Fermat and Descartes still followed suit. Galileo in the &lt;a href=&quot;https://archive.org/details/dialoguesconcern00galiuoft/page/n10&quot; rel=&quot;noreferrer&quot;&gt;Dialogues concerning two new sciences (1638)&lt;/a&gt; specifically splits projectile parabolas into two semi-parabolas to avoid having the velocity change sign. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first physical negatives were, apparently, distances and their derivatives. According to &lt;a href=&quot;http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mumford's What’s so Baffling About Negative&#xA;Numbers?&lt;/a&gt;, we should attribute the first application to Wallis and Newton, who worked independently. &lt;a href=&quot;https://archive.org/details/atreatisealgebr03maclgoog/page/n10&quot; rel=&quot;noreferrer&quot;&gt;Wallis in his Treatise on Algebra (1685)&lt;/a&gt; speaks of negative distances traveled:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As for instance: Supposing a man to have advanced or moved forward (from A to B) 5 yards; and then to retreat (from B to C) 2 yards; If it be asked, how much had he Advanced (upon the whole march) when at C? I find... he has Advanced 3 Yards. But if, having Advanced 5 Yards to B, he thence retreat 8 Yards to D; and it then be asked, How much is he Advanced when at D, or how much Forwarder than when he was at A: I say –3 Yards.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Newton allowed negative numbers for velocities, accelerations, etc., in earlier notes, but did not spell it out in print until after Wallis. In the &lt;a href=&quot;https://archive.org/details/universalarithm00wildgoog/page/n14&quot; rel=&quot;noreferrer&quot;&gt;Universal Arithmetick (1707)&lt;/a&gt; he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Quantities are either Affirmative, or greater than nothing; or Negative, or less than nothing. Thus in human affairs, possessions or stock may be called affirmative goods, and debts negative ones. And so in local motion, progression may be called affirmative motion, and regression negative motion; because the first augments, and the other diminishes the length of the way made.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-11T06:02:16.943" LastActivityDate="2019-06-11T06:02:16.943" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9721" PostTypeId="1" AcceptedAnswerId="9722" CreationDate="2019-06-12T09:06:11.133" Score="10" ViewCount="335" Body="&lt;p&gt;Someone drew my attention to &lt;a href=&quot;http://ilib.mccme.ru/djvu/bib-mat-kr/num-fig.htm&quot; rel=&quot;noreferrer&quot;&gt;the Russian translation&lt;/a&gt; of Rademacher and Toeplitz's &lt;em&gt;Von Zahlen und Figuren&lt;/em&gt; (&lt;em&gt;The Enjoyment of Mathematics&lt;/em&gt; in the English translation). In the chapter on set theory the Russian editor, Isaak Yaglom, added a footnote saying &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Research in mathematical logic (especially by K. Goedel) led to a conviction that (at least in most axiomatic set theories) the problem of continuum is undecidable, meaning that it cannot be deduced from the basic assumptions of set theory.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the Russian translation was published in 1962, and Cohen's results appeared in 1963 (nor is Cohen named in the footnote). The footnote evidently references Goedel's work on CH, which is not mentioned in the text of the chapter because the original is from 1930. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Goedel showed that the axioms of set theory cannot refute the continuum hypothesis, but Yaglom seems to be indicating that before Cohen's result people were already convinced that the CH was not provable either. Was that really the case, and what was the evidence thought to be for or against the independence of CH before Cohen?&lt;/p&gt;&#xA;" OwnerUserId="377" LastActivityDate="2019-06-13T19:06:02.493" Title="Was the continuum hypothesis commonly believed to be independent before Cohen's proof?" Tags="|set-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9722" PostTypeId="2" ParentId="9721" CreationDate="2019-06-12T19:19:44.897" Score="15" Body="&lt;p&gt;According to Cohen himself, no. Most people, apparently, did not  bother to believe one way or the other, and those who did (perhaps with notable exceptions of Lusin and Sierpinski) relied on Gödel's semi-philosophical remarks in a popular paper, and word of mouth that he already partially solved the problem (originating from him, presumably). This is probably what Yaglom refers to by &amp;quot;research in mathematical logic (especially by K. Gödel)&amp;quot;. The paper was &lt;a href=&quot;https://www.jstor.org/stable/2304666&quot; rel=&quot;noreferrer&quot;&gt;What is Cantor's Continuum Problem? (1947)&lt;/a&gt; written for the American Mathematical Monthly, where he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There are (assuming the consistency of the axioms) a priori three possibilities for Cantor's conjecture: It may be either demonstrable or disprovable or undecidable. The third alternative (which is only a precise formulation of the conjecture stated above that the difficulties of the problem are perhaps not purely mathematical) is the most likely, and to seek a proof for it is at present one of the most promising ways of attacking the problem.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He proceeds to give two arguments in favor of this opinion. One is that the standard axioms do not seem to settle whether the sets talked about should be taken platonistically, or as &amp;quot;&lt;em&gt;extensions of definable properties&lt;/em&gt;&amp;quot; (a reference to Gödel's own constructible universe). And investigations of cardinal arithmetic since Cantor suggested that settling questions about it would require additional axioms based on a choice between the two.&lt;/p&gt;&#xA;&lt;p&gt;The second argument (for which Gödel credits Lusin and Sierpinski) is the existence of &amp;quot;&lt;em&gt;certain facts (not known or not existing at Cantor's time) which seem to indicate that Cantor's conjecture will turn out to be wrong&lt;/em&gt;&amp;quot;, while proving it wrong, as Gödel already showed, was impossible from the accepted axioms. The facts cited are &amp;quot;meager&amp;quot; sets (first category on every perfect set, coverable by infinitely many intervals of any given length, etc.) that are provably uncountable, but with no hope of showing that they have the cardinality continuum; and some &amp;quot;paradoxical&amp;quot; consequences (e.g. a set of reals of the cardinality continuum such that any dense open set covers all but countably many of its points).&lt;/p&gt;&#xA;&lt;p&gt;He also makes a famous remark about accepting new axioms based on the &amp;quot;&lt;em&gt;fruitfulness in consequences and in particular in &amp;quot;verifiable&amp;quot; consequences, i.e., consequences demonstrable without the new axiom, whose proofs by means of the new axiom, however, are considerably simpler and easier to discover&lt;/em&gt;&amp;quot;, and sketches what came to be known as the large cardinal program. Much later (1970s) Gödel claimed to have &amp;quot;proved&amp;quot; that the continuum is no more than aleph two, and speculated that it is exactly that, see &lt;a href=&quot;https://books.google.com/books?id=83Attf6BsJ4C&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Kurt Gödel: Essays for his Centennial, p.175&lt;/a&gt;. Gödel's reputation was such that his name attached to a conjecture might have been enough evidence for a &amp;quot;conviction&amp;quot;. But, in hindsight, this &amp;quot;evidence&amp;quot; turned out to be pretty thin. From &lt;a href=&quot;https://projecteuclid.org/euclid.rmjm/1181070010&quot; rel=&quot;noreferrer&quot;&gt;Cohen's The Discovery of Forcing&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;After the publication of Gödel’s result in 1939 and the appearance about a year later of a detailed exposition as lecture notes, there were a total of four papers to my knowledge which in any way dealt with his construction, until my own work in 1963. One of these, actually a series of three papers, was used by Shepherdson who showed that the method of inner models used by Godel and von Neumann could never show the consistency of the negation of AC or CH. This result evidently received insufficient attention because, when I rediscovered them in 1962, I was urged to publish them despite some reservations I had.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;From general impressions I had of the proof, there was a finality to it, an impression that somehow Gödel had mathematicized a philosophical concept, i.e., constructibility, and there seemed no possibility of doing this again, especially because the negation of CH and AC were regarded as pathological. I repeat that these hazy and even self-contradictory impressions I had were strictly my own, but, nevertheless, I think that it is very possible that others had similar impressions. For example, as a graduate student I had looked at Kleene’s large book, and there was very little emphasis or even discussion of the entire matter.... Furthermore, there was little mention of the problem of showing the consistency of the negation of CH. One reference was an expository article of Gödel, in which he refers to it as a likely outcome, but hardly seems to refer to it as a pressing problem for research.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;A rumor had circulated, very well known in all circles of logicians, that Gödel had actually partially solved the problem, specifically as I heard it, for AC and only for the theory of types (years later, after my own proof of the independence of CH, AC, etc., I asked Godel directly about this and he confirmed that he had found such a method, specifically contradicted the idea that type theory was involved, but would tell me absolutely nothing of what he had done)... It seems that from 1941 to 1946 he devoted himself to attempts to&#xA;prove the independence. In 1967 in a letter he wrote that he had indeed&#xA;obtained some results in 1942 but could only reconstruct the proof of&#xA;the independence of the axiom of constructibility, not that of AC, and&#xA;in type theory (contradicting what he had told me in 1966). After 1946 he seems to have devoted himself entirely to philosophy.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-06-13T19:06:02.493" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9723" PostTypeId="2" ParentId="608" CreationDate="2019-06-13T13:34:02.203" Score="-2" Body="&lt;p&gt;The first mention of 4 dimensions is in the Bible at &lt;a href=&quot;https://www.biblegateway.com/passage/?search=Ephesians%203%3A19-20&amp;amp;version=NIV&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ephesians 3:19&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;19 and to know this love that surpasses knowledge—that you may be&#xA;  filled to the measure of all the fullness of God.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;20 Now to him who is able to do immeasurably more than all we ask or&#xA;  imagine, according to his power that is at work within us,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10113" LastEditorUserId="217" LastEditDate="2019-06-15T06:36:56.077" LastActivityDate="2019-06-15T06:36:56.077" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9724" PostTypeId="2" ParentId="2940" CreationDate="2019-06-13T17:04:01.187" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Karen_Uhlenbeck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karen Uhlenbeck&lt;/a&gt; made huge contribution to gauge theory and more. &lt;/p&gt;&#xA;" OwnerUserId="2393" LastActivityDate="2019-06-13T17:04:01.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9725" PostTypeId="2" ParentId="2940" CreationDate="2019-06-13T18:05:04.180" Score="6" Body="&lt;ol&gt;&#xA;&lt;li&gt;Hypatia of Alexandria (AD 350 or 370-AD 417)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;She worked on several researches most significant of which included her commentaries on the Greek text-book, Arithmetica and On the Conics of Apollonius. She is remembered especially for her detailed description of the early hydrometer.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Émilie du Châtelet (1706-1749)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;A French physicist, mathematician and writer during the Enlightenment era in Europe. In 1740, Châtelet published a book on philosophy and science called Institutions de Physique and later translated and commented on Newton’s Principia Mathematica which is its best known translation.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Maria Agnesi (1718-1799)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;She wrote a book on math that still survives, that is: Analytical Institutions for the Use of Italian Youth in English. Another pioneering contribution was the Witch of Agnesi- a curve that she wrote the equation for.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Sophie Germain (1776-1831)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Sophie Germain’s paper on elasticity theory made her the first woman to be awarded from the Paris Academy of Sciences in 1816. She was also a major contributor in proving Fermat’s Last Theorem.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Ada Lovelace (1815-1852)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;When asked to translate the memoir of Charles Babbage, the Analytical Engine, Lovelace went ahead and added her own comments and notes about a method of calculating a sequence of Bernoulli numbers: what is today known as the world’s first ever computer program subsequently making Lovelace renowned as the world’s first computer programmer.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;Sofia Kovalevskaya (1850-1891)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;She gave the Cauchy-Kovalevskaya Theorem its end result in 1875, worked on a paper in which she invented the Kovalevskaya Top and published ten papers based on mathematics and mathematical physics.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;7&quot;&gt;&#xA;&lt;li&gt;Emmy Noether (1882-1935)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Emmy Noether is famous for coining the Noether’s Theorem that clarifies the relationship between conservation laws and symmetry, as well as Noether’s Ring that changed the basics of abstract algebra. Noether is also famous for other theories based on non-commutative algebras, hyper-complex numbers and commutative rings.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;8&quot;&gt;&#xA;&lt;li&gt;Mary Cartwright (1900-1998)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;She authored over a 100 papers which include her work on level curves, functions in the unit disk, topology and ordinary differential equations among others.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;9&quot;&gt;&#xA;&lt;li&gt;Julia Robinson (1919-1985)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;She is well regarded for her work on Hilbert’s Tenth problem and decision problems.&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;10&quot;&gt;&#xA;&lt;li&gt;Shafi Goldwasser (1958-Date)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Her research emphasizes on zero-knowledge proof, complexity theory, computation number theory and cryptography.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2019-06-13T18:56:29.040" LastActivityDate="2019-06-13T18:56:29.040" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9726" PostTypeId="2" ParentId="59" CreationDate="2019-06-13T20:34:34.063" Score="0" Body="&lt;p&gt;Frankly, I don't understand how anybody in 19th century could have thought that intermediate value property implies continuity. Take &lt;span class=&quot;math-container&quot;&gt;$y(x)=0$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=sin(1/x)$&lt;/span&gt; otherwise and you have your counterexample.&lt;/p&gt;&#xA;" OwnerUserId="2393" LastActivityDate="2019-06-13T20:34:34.063" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9727" PostTypeId="1" AcceptedAnswerId="9731" CreationDate="2019-06-14T05:07:39.803" Score="2" ViewCount="378" Body="&lt;p&gt;I recall reading &lt;a href=&quot;https://philosophynow.org/issues/94/I_Re-Read_Therefore_I_Understand&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; that was written to explain how Descartes read philosophy effectively. I am wondering if such analogous tips have been made by past mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="9898" LastActivityDate="2021-08-04T08:23:37.567" Title="Are there any sources of mathematicians talking about their research methods?" Tags="|mathematics|mathematicians|education|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9728" PostTypeId="1" CreationDate="2019-06-14T05:39:09.060" Score="-1" ViewCount="271" Body="&lt;p&gt;As I know that before Galileo,all the scientists believe that vacuum cannot exists. But how suction pump(which itself remove air)was discovered before Galileo. What was the steps involved in it to make suction pump?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastActivityDate="2019-07-14T18:02:55.393" Title="Discovery of Suction pump" Tags="|biographical-details|depiction-of-scientists|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9729" PostTypeId="2" ParentId="9728" CreationDate="2019-06-14T06:47:02.177" Score="1" Body="&lt;p&gt;Aristotle believed that no void(vacuum) could occur naturally, because the denser surrounding material continuum would immediately fill any incipient rarity that might give rise to a void. Even in his Physics, book IV, Aristotle offered numerous arguments against the void: for example, that motion through a medium which offered no impediment could continue ad infinitum, there being no reason that something would come to rest anywhere in particular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But much before Galileo, in the 13th and 14th century, considerable attention was given on these issues by European scholars such as Roger Bacon, Blasius of Parma and Walter Burley . Eventually following Stoic physics in this instance, scholars from the 14th century onward increasingly departed from the Aristotelian perspective in favor of a supernatural void beyond the confines of the cosmos itself, a conclusion widely acknowledged by the 17th century, which helped to segregate natural and theological concerns.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using geometry, Ibn al-Haytham(965–1039) mathematically demonstrated that place is the imagined three-dimensional void between the inner surfaces of a containing body. According to Ahmad Dallal, Abū Rayhān al-Bīrūnī also states that &quot;there is no observable evidence that rules out the possibility of vacuum&quot;. These steps led to the making of suction pump which appeared in Europe from the 15th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, it is not true that scientists before Galileo did not believe about the existence of vacuum or void but were aware of it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Vacuum&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2019-06-14T17:43:28.063" LastActivityDate="2019-06-14T17:43:28.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9731" PostTypeId="2" ParentId="9727" CreationDate="2019-06-14T13:46:03.633" Score="6" Body="&lt;p&gt;Mathematicians rarely describe the process which led them to their discoveries.&#xA;One notable exception was Euler.&#xA;Some books on the subject written by great mathematicians are:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J. Hadamard, The Mathematician's Mind.&#xA;The Psychology of Invention in the Mathematical Field (English translation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;G. Polya, a) Mathematics and plausible reasoning. b) Mathematical discovery.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-06-14T13:46:03.633" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9732" PostTypeId="1" CreationDate="2019-06-15T14:59:04.663" Score="-1" ViewCount="47" Body="&lt;p&gt;I want to know where has he placed the tube and how he adjust the height ?&#xA;Please refer the link below marked in violet &lt;a href=&quot;http://lnr.li/m19fP&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://lnr.li/m19fP&lt;/a&gt; &#xA;Please explain using simple diagram for better understanding&lt;/p&gt;&#xA;" OwnerUserId="9993" LastActivityDate="2019-06-17T14:23:11.490" Title="Ctesibius discovery of compressed air" Tags="|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9733" PostTypeId="2" ParentId="9727" CreationDate="2019-06-15T20:21:57.013" Score="7" Body="&lt;p&gt;Cédric Villani's &lt;em&gt;Birth of a Theorem&lt;/em&gt; does exactly that. A great read for scientists, even non-mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;Edit: The English title of the book is &lt;em&gt;Birth of a Theorem&lt;/em&gt; and not &lt;em&gt;Living Theorem&lt;/em&gt; (the French title is &lt;em&gt;Théorème Vivant&lt;/em&gt;). Thanks to Torsten for pointing it out.&lt;/p&gt;&#xA;" OwnerUserId="10122" LastEditorUserId="10122" LastEditDate="2021-08-04T08:23:37.567" LastActivityDate="2021-08-04T08:23:37.567" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9734" PostTypeId="2" ParentId="9727" CreationDate="2019-06-16T09:31:03.077" Score="4" Body="&lt;p&gt;The first mathematician who did that was Archimedes, in his &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Method of Mechanical Theorems&lt;/em&gt;&lt;/a&gt; (usually known as &lt;em&gt;The Method&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-06-16T09:31:03.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9735" PostTypeId="2" ParentId="9711" CreationDate="2019-06-16T15:49:02.443" Score="2" Body="&lt;p&gt;In addition to the existence of such tables, (in the later 1960's, for example) high school math classes included some lessons on (linear) &lt;em&gt;interpolation&lt;/em&gt; from the values given in tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And some of us wondered about higher-order interpolation, and so on, which does nicely lead to many topics in calculus ... and splines and other stuff.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Slide rules were a very good competitor to &quot;tables&quot;, in many situations, since they could give quicker answers (tho' somewhat lower precision) and were more portable.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2019-06-16T15:49:02.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9736" PostTypeId="1" AcceptedAnswerId="9741" CreationDate="2019-06-16T18:01:14.467" Score="7" ViewCount="218" Body="&lt;p&gt;Galois obtained necessary and sufficient conditions for an algebraic equation (in one variable) to be solvable by extraction of a chain of square roots.  A beautiful application of this is to the problem of solvability of geometric problems by ruler and compass. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, when was the equivalence of ruler and compass solvability and solution by radicals originally identified?  Also, who first applied Galois’ results to prove that a regular polygon with a prime number &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; of sides is constructible by ruler and compass if and only if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is of the form &lt;span class=&quot;math-container&quot;&gt;$2^{2n}+1$&lt;/span&gt;.  (We know that Gauss proved the “if” part, and that the same methods can be used to prove the impossibility of trisecting an angle and doubling a cube.) &lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2019-06-16T18:29:08.753" LastActivityDate="2019-06-17T17:06:47.047" Title="The relationship between solvability of ruler and compass problems and solvability of algebraic equations by radicals" Tags="|mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9737" PostTypeId="2" ParentId="6611" CreationDate="2019-06-16T20:00:08.660" Score="5" Body="&lt;p&gt;Yes, Gauss wrote in several places about his proofs of the reciprocity theorem. Here's one really interesting passage from 1808, worth quoting at length:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Arithmetical truths] are frequently of such a nature that they may be arrived at by many distinct paths and that the first paths to be discovered are not always the shortest. It is therefore a great pleasure after one has fruitlessly pondered over a truth and has later been able to prove it in a round-about way to find at last the simplest and most natural way to its proof. ...For a whole year [the reciprocity theorem] tormented me and absorbed my greatest efforts until at last I obtained a proof given in the fourth section of [the &lt;em&gt;Disquisitiones&lt;/em&gt;]. Later I ran across three other proofs which were built on entirely different principles. One of these I have already given in the fifth section; the others, which do not compare with it in elegance, I have reserved for future publication. Although these proofs leave nothing to be desired as regards rigor, they are derived from sources much too remote, except perhaps the first, which however proceeds with laborious arguments and is overloaded with extended operations. I do not hesitate to say that till now a &lt;em&gt;natural&lt;/em&gt; proof has not been produced. I leave it to the authorities to judge whether the following proof which I have recently been fortunate enough to discover deserves this description. (Translated in David Eugene Smith, &lt;em&gt;A Source Book in Mathematics, Volume One&lt;/em&gt; (Dover, 1959), 113.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;When Gauss wrote this, he'd previously published two proofs of quadratic reciprocity (the ones appearing in DA -- the first by induction, and the second via his new theory of quadratic forms). He'd also found two more proofs which wouldn't be published until after his death (involving quadratic periods). Here, he's about to introduce what's usually known as his third proof. This is the one using Gauss's Lemma that Eisenstein would later improve and geometrize, turning it into the &quot;lattice point counting&quot; proof that you often see in textbooks today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss's third proof was apparently his favorite, as the passage shows. He found the earlier proofs unsatisfactory because they were either too complicated or &quot;derived from sources much too remote&quot;. The third proof was the only one he considered natural.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly, although Gauss is often (and rightly) praised for his deep mathematics, his ideal proof was evidently a simple and transparent one rather than one of great profundity and abstractness.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The other proof about which he had some approving things to say was the &quot;Gauss sums&quot; proof (usually called the sixth proof due to publication order, although it was the eight and last to be discovered). This is the other one that crops up a lot today, in part because it's relatively easy to generalize to get proofs of cubic and quartic reciprocity. Gauss was very interested in finding a proof like this, which is one reason he kept looking for new approaches to QR:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From 1805 onwards I have investigated the theory of cubic and biquadratic residues... Theorems were found by induction... which had a wonderful analogy with the theorems for quadratic residues. On the other hand, for a long time all attempts at complete proofs have been futile. This was the motive for endeavoring to add yet more proofs to those already known for quadratic residues, in the hope that of the many different methods given, one or the other would contribute to the illumination of the related arguments [for higher reciprocity]. (Translated in David Cox, &lt;em&gt;Primes of the Form &lt;span class=&quot;math-container&quot;&gt;$x^2+ny^2$&lt;/span&gt;&lt;/em&gt;, 2nd edition (Wiley, 2013), 78).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Here's what Gauss has to say about the sixth proof, which finally gave him what he wanted:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Although the arithmetical theorem dealt with here [i.e., QR] has, by earlier efforts, been provided with four entirely different proofs, and can seem to be completely finished, I return here to some new observations and present two new proofs, which shed a new light on these facts... [T]he sixth proof calls upon a completely different and most subtle principle, and gives a new example of the wonderful connection between arithmetic truths that at first glance seem to lie very far from one another... Still another reason was provided, which led me to publish a new proof now that I had already spoken of 9 years earlier. In fact, as I was occupied in 1805 with the theory of cubic and biquadratic residues, where a very difficult circumstance that I had begun to work through, happened to draw on almost the same skill as originally in the theory of quadratic residues. Without further notice, indeed, were those theorems -- which had handled those questions completely, and had presented a wonderful analogy with the corresponding theorems about quadratic residues -- found by induction as soon as they were only sought for in a proper way, and all remaining researches led on all sides to a complete proof that for a long time I had sought in vain. This was indeed the impulse that I so much sought to add more and more proofs of the already-known theorems on quadratic residues, in the hope that from these many different methods, one or another could illuminate something in the related circumstances. This hope was in no way idle, and untiring efforts were finally crowned with success. Shortly, I will be in a position to publish the fruits of my studies, but before I undertake that difficult work I have decided to return once more to the theory of quadratic residues, and to say what is still to be said, and so in a certain sense to bid farewell to this part of the higher arithmetic. (Translated in Jeremy Gray, &lt;em&gt;A History of Abstract Algebra: From Algebraic Equations to Modern Algebra&lt;/em&gt; (Springer, 2018), 333).)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Actually, Gauss never got around to publishing proofs of cubic or quartic reciprocity, and it seems to be unclear to what extent he had them fully worked out. But he at least realized that the sixth proof was the right place to start.&lt;/p&gt;&#xA;" OwnerUserId="10123" LastEditorUserId="10123" LastEditDate="2019-06-17T16:10:15.103" LastActivityDate="2019-06-17T16:10:15.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9738" PostTypeId="2" ParentId="608" CreationDate="2019-06-16T21:17:28.223" Score="-2" Body="&lt;p&gt;From &lt;a href=&quot;https://patriceayme.wordpress.com/2017/01/25/we-are-mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://patriceayme.wordpress.com/2017/01/25/we-are-mathematics/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We have, from Patrice Ayme, &quot; And not just mathematics, but even Infinitesimal Calculus! It is indeed clear that animals such as dogs have a mastery of calculus: experiences have shown this, and anybody with a dog throwing a stick sideways in water will see the dog running along the shore a bit, and then jump in the water, so as to minimize the time to reach the stick, a typical calculus problem. Dogs can do calculus, because they can make algebraic geometry in their brains, having a reference frame made of these grid cells! (If they had no grid cells, they would not be able to do calculus.)&quot;&#xA;This means all animals have this concept of 3 dimensions built-in, thanks to evolution!&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-06-16T23:58:35.440" LastActivityDate="2019-06-16T23:58:35.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9740" PostTypeId="2" ParentId="9736" CreationDate="2019-06-17T05:28:18.097" Score="4" Body="&lt;p&gt;There is no &quot;equivalence of ruler and compass solvability and solution by radicals&quot;.&#xA;For example, cubic root of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is evidently a radical but cannot be constructed with&#xA;a compass and ruler. The relation between construction and solvability in radicals is the following: only those segments can be constructed whose length is expressed by&#xA;arithmetic operations and extraction of SQUARE roots. The result is due to Gauss,&#xA;and it pre-dates Galois theory. It was Gauss who determined completely which regular &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-gons can be constructed with a compass and a ruler. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref.  Robin Hartshorne, Geometry: Euclid and beyond, Springer-Verlag, New York, 2000.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-06-17T05:28:18.097" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9741" PostTypeId="2" ParentId="9736" CreationDate="2019-06-17T07:24:02.723" Score="7" Body="&lt;p&gt;The idea of coding ruler and compass constructions algebraically was more or less known to Vieta, as described in &lt;em&gt;Supplementum Geometriae&lt;/em&gt; (1593), see &lt;a href=&quot;https://hsm.stackexchange.com/a/2408/55&quot;&gt;Viète's Relevance and his Connection to Euler&lt;/a&gt;. It was developed further by Descartes in &lt;em&gt;La Geometrie&lt;/em&gt; (1637), where the idea of constructing segment arithmetic (including square roots) geometrically also appears. Gauss was first to investigate the algebraic/geometric correspondence (what we would call) rigorously, and proved that certain regular polygons can be inscribed into the circle by converting the geometric problem into one about the cyclotomic fields (1796). He &lt;em&gt;claimed&lt;/em&gt; that only those he described are constructible, but there is no proof in his Nachlass. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The proof was only given &lt;a href=&quot;https://en.wikipedia.org/wiki/Constructible_polygon#Conditions_for_constructibility&quot; rel=&quot;noreferrer&quot;&gt;by Wantzel in 1837&lt;/a&gt; (along with a proof of the impossibility of trisection), 5 years after Galois' death, although he did not rely on his work, which remained little known for a while. Explicitly, in modern terms, the correspondence between straightedge and compass constructions and quadratic extensions is proved and utilized extensively in Hilbert's Grundlagen der Geometrie (1899). It is based on developing Descartes's segment arithmetic.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-17T17:06:47.047" LastActivityDate="2019-06-17T17:06:47.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9742" PostTypeId="1" AcceptedAnswerId="9744" CreationDate="2019-06-17T10:46:38.520" Score="9" ViewCount="2344" Body="&lt;p&gt;... since the very idea of infinitesimal was foreshadowed by Cavalieri ( &quot;limit&quot;) in 1635, then put forward in an indirect way  by John Wallis (&lt;span class=&quot;math-container&quot;&gt;$1/\infty$&lt;/span&gt;) in 1655, and then formalized by Newton ( &quot;&lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;&quot; ) in 1666, and by Leibniz still a few years later?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="7300" LastEditDate="2019-06-19T13:53:40.197" LastActivityDate="2019-06-19T13:53:40.197" Title="What triggered jesuits' ban on infinitesimals in 1632?" Tags="|mathematics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9743" PostTypeId="2" ParentId="9742" CreationDate="2019-06-17T12:15:54.900" Score="3" Body="&lt;p&gt;The issue regards more &lt;em&gt;indivisibles&lt;/em&gt; than &lt;em&gt;infinitesimals&lt;/em&gt; and must be located in the context of the Early Modern European debate about the &quot;revamping&quot; of &lt;em&gt;atomism&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See : Vincent Jullien (editor), &lt;a href=&quot;https://books.google.it/books?id=8Vt1CQAAQBAJ&amp;amp;pg=PA87&quot; rel=&quot;nofollow noreferrer&quot;&gt;Seventeenth-Century Indivisibles Revisited (2015, Birkhauser)&lt;/a&gt; for details about the works of Kepler (1609), Cavalieri (1635) and Guldin (1640).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cavalieri developed his theory of geometry during the years 1620–1622.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Vincent Jullien's chapter dedicated to &lt;em&gt;Indivisibles in the Work of Galileo&lt;/em&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;on 7 May 1610, Galileo wrote, in a letter to the secretary of the Grand Duke of Tuscany, that he was planning a piece of work on the &lt;em&gt;De Compositione continui&lt;/em&gt;. In February and March 1626, Cavalieri reminded him of the project: “do you remember the work on indivisibles that you had decided to write?”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Indivisibles are implicitly mentioned in part of the second day of the &lt;a href=&quot;https://it.wikisource.org/wiki/Dialogo_sopra_i_due_massimi_sistemi_del_mondo_tolemaico_e_copernicano_(Favaro)/Giornata_seconda&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Dialogo&lt;/em&gt; (1632)&lt;/a&gt;, at&#xA;the beginning of the demonstration of the law of falling bodies.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And see : Galileo's &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Assayer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saggiatore (1623)&lt;/a&gt; and the reply by the jesuit &lt;a href=&quot;https://en.wikipedia.org/wiki/Orazio_Grassi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Orazio Grassi&lt;/a&gt; (&lt;em&gt;Libra&lt;/em&gt;, published under the name : Lotario Grassi) &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;asserting that Galileo's book advanced an atomic theory of matter, and that this conflicted with the Catholic doctrine of the Eucharist, because atomism would make transubstantiation impossible. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Grassi's second response to &lt;em&gt;Il Saggiatore&lt;/em&gt;, the &lt;em&gt;Ratio ponderum librae et simbellae&lt;/em&gt; (1626), focused mainly on doctrinal issues.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;unlike The Assayer, which had recourse to the lethal polemical weapons of satire and the new philosophy, the &lt;em&gt;Ratio&lt;/em&gt; used those no-less-lethal weapons of doctrinal and dialectical retort based on religious and philosophical orthodoxy.[Pietro Redondi, &lt;em&gt;Galileo Heretic&lt;/em&gt;, Princeton University Press, 1987. p.191]&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The jesuit mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Guldin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Guldin&lt;/a&gt; was an harsh critic of Cavalieri's method of &lt;em&gt;indivisibles&lt;/em&gt; into his &lt;em&gt;De centro gravitatis&lt;/em&gt; (or &lt;em&gt;Centrobaryca&lt;/em&gt;, three volumes, 1635-41), on mathematical grounds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See also Mordechai Feingold (editor), &lt;a href=&quot;https://books.google.it/books?id=oSSEpTB15VEC&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jesuit Science and the Republic of Letters (MIT Press, 2002)&lt;/a&gt;, page 28-29, for details about jesuit &lt;a href=&quot;https://en.wikipedia.org/wiki/Rodrigo_de_Arriaga&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rodrogo de Arriaga&lt;/a&gt;'s  &lt;em&gt;Cursus philosophicus&lt;/em&gt; (Anversa, 1632) condemnation of 1632, concerning &quot;mathematical atomism&quot; and &quot;the opinion on quantity made up of indivisibles&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that the modern source is Egidio Festa, &lt;a href=&quot;http://hal.in2p3.fr/in2p3-00004757&quot; rel=&quot;nofollow noreferrer&quot;&gt;La querelle de l'atomisme: Galilee, Cavalieri et les jesuites (1990)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The easily available sources (Wiki,etc.) about the condemnation of &lt;em&gt;indivisibles&lt;/em&gt; of Galileo and Cavalieri (dated August 10, 1632), led the Revisors General of the Jesuits &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacob_Bidermann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacob Bidermann&lt;/a&gt; all ref to Amir Alexander's book.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2019-06-17T18:34:47.860" LastActivityDate="2019-06-17T18:34:47.860" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9744" PostTypeId="2" ParentId="9742" CreationDate="2019-06-17T12:19:55.757" Score="8" Body="&lt;p&gt;See &lt;a href=&quot;https://math.stackexchange.com/q/661999/442&quot;&gt;this question&lt;/a&gt; at math.se &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;infinitesimals&quot; we are talking about (actually more correctly called &quot;indivisibles&quot;) is the assertion that a plane area is made up of parallel line segments;  or that a solid region is made up of parallel plane sections.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Amir book is not a novel, it is historical research. (And it is very interesting!) The short answer (see &lt;a href=&quot;https://math.stackexchange.com/a/831221/442&quot;&gt;my answer there&lt;/a&gt;): Jesuits prohibited indivisibles for use in their education system because indivisibles were viewed as contrary to Aristotle.  &lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2019-06-19T12:28:10.620" LastActivityDate="2019-06-19T12:28:10.620" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9745" PostTypeId="2" ParentId="9732" CreationDate="2019-06-17T14:23:11.490" Score="1" Body="&lt;p&gt;I think the setup is too trivial for most history pages to bother with, but there is a picture in the book &quot;Young Explorers’ Guide : The Search for Ingenious Treasures&quot; . Unfortunately, that picture does not display in the Google Books search.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is a simple case of a lever arm: the mirror remains at a fixed distance from the pivot, while the lead weight can move.  According to other descriptions, the mirror is positioned by means of a pulley&amp;amp; rope which presumably is tied down.  The movement of the lead weight is almost irrelevant other than to provide stability for the entire lever arm at any mirror height.    &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-06-17T14:23:11.490" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9746" PostTypeId="1" AcceptedAnswerId="9747" CreationDate="2019-06-17T17:32:31.760" Score="4" ViewCount="148" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Vitalism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vitalism&lt;/a&gt; is the notion now discredited as pseudoscientific that there is some fundamental cleavage between living and non-living matter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.lesswrong.com/posts/6i3zToomS86oj9bS6/mysterious-answers-to-mysterious-questions#fn1x38&quot; rel=&quot;nofollow noreferrer&quot;&gt;This blog post&lt;/a&gt; contains some quotes that suggest Lord Kelvin was either a proponent of vitalism, or at least seriously entertained it at some point.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I reproduce two of the relevant quotes here: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It seemed to me then, and it still seems to me, most probable that the&#xA;  animal body does not act as a thermodynamic engine . . . The influence&#xA;  of animal or vegetable life on matter is infinitely beyond the range&#xA;  of any scientific inquiry hitherto entered on. Its power of directing&#xA;  the motions of moving particles, in the demonstrated daily miracle of&#xA;  our human free-will, and in the growth of generation after generation&#xA;  of plants from a single seed, are infinitely different from any&#xA;  possible result of the fortuitous concourse of atoms[.]&#xA;  Lord Kelvin, “On the Dissipation of Energy: Geology and General Physics,” in Popular Lectures and Addresses, vol. ii (London: Macmillan, 1894).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Modern biologists are coming once more to a firm acceptance of&#xA;  something beyond mere gravitational, chemical, and physical forces;&#xA;  and that unknown thing is a vital principle.&#xA;  Silvanus Phillips Thompson, The Life of Lord Kelvin (American Mathematical Society, 2005).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that I am not interested particularly in the contents of the blog I posted a link to, beyond the suggestion that Lord Kelvin in some sense believed vitalism warranted serious scientific consideration. Was this the case, or were the quotes somehow cherry-picked to support this conclusion? Further references are welcome.&lt;/p&gt;&#xA;" OwnerUserId="6496" LastActivityDate="2019-06-18T08:28:29.427" Title="Was Lord Kelvin at any point a proponent of vitalism?" Tags="|biographical-details|biology|thermodynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9747" PostTypeId="2" ParentId="9746" CreationDate="2019-06-17T19:33:32.420" Score="5" Body="&lt;p&gt;Kelvin did not merely believe that vitalism warranted serious scientific consideration, he thought it to be &amp;quot;&lt;em&gt;absolutely forced by science&lt;/em&gt;&amp;quot;. In less strong terms, it was a popular idea at the time, given the state of both physics and biology. There was also interest in reconciling physics with free will, etc. Helmholtz, Maxwell, and Boltzmann, among others, were sympathetic. The idea remained respectable even in the early 20th century, Bergson's Creative Evolution (1907) was quite influential.&lt;/p&gt;&#xA;&lt;p&gt;Kelvin was a devout Christian, an elder in the Church of Scotland and the Chairman of the Christian Evidence Society in London, see &lt;a href=&quot;https://www.asa3.org/ASA/PSCF/1985/JASA6-85Seeger.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kelvin, Humble Christian&lt;/a&gt;. He advocated divine &amp;quot;creative power&amp;quot; in nature as attested to by science in various addresses for over 30 years. E.g. in the &lt;a href=&quot;https://www.christianheritageedinburgh.org.uk/2016/08/23/sir-william-thomson-lord-kelvin-1824-1907/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chairman's address from 1889&lt;/a&gt; we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I have long felt that there was a general impression in the non-scientific world [that] believes Science has discovered ways of explaining all the facts of nature without adopting any definite belief in a Creator. I have never doubted that impression was utterly groundless. It seems to me that when a scientific man says – as it has been said from time to time – that there is no God, he does not express his own ideas clearly.  He is, perhaps, struggling with difficulties; but when he says that he does not believe in a creative power I am convinced he does not faithfully express what is in his mind. He is out of this depth…&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While he admitted evolution, it was to be reconciled with God's creative power. &amp;quot;&lt;em&gt;We must pause, face to face with the mystery and miracle of the creation of living creatures&lt;/em&gt;&amp;quot;, he said in 1897. After a course of lectures on Christian Apologetics, given by Henslow to the Christian Association of the University College, Kelvin made public remarks that linked his creationism to the vital principle by name. Kelvin's arguments, I am afraid,  were neither original nor very developed. The version of the report &lt;a href=&quot;https://zapatopi.net/kelvin/papers/science_affirms_creative_power.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;in The Times on May 2, 1903&lt;/a&gt; edited by Kelvin himself, reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I am in thorough sympathy with Professor Henslow in the fundamentals of his lecture. I do not say that, with regard to the origin of life, science neither affirms nor denies creative power. Science positively affirms creative power. Science makes every one feel a miracle in himself. It is not in dead matter that we live and move and have our being, but in the creating and directive Power which science compels us to accept as an article of belief. We cannot escape from that conclusion when we study the physics and dynamics of living and dead matter all around. &lt;strong&gt;Modern biologists are coming once more to a firm acceptance of something beyond mere gravitational, chemical, and physical forces; and that unknown thing is a vital principle.&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;We have an unknown object put before us in science. In thinking of that object we are all agnostics. We only know God in His works, but we are absolutely forced by science to admit and to believe with absolute confidence in a Directive Power in an influence other than physical, or dynamical, or electrical forces. Cicero, editor of Lucretius, denied that men and plants and animals could have come into existence by a fortuitous concourse of atoms. There is nothing between absolute scientific belief in Creative Power and the acceptance of the theory of a fortuitous concourse of atoms. Just think of a number of atoms falling together of their own accord and making a crystal, a sprig of moss, a microbe, a living animal.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-06-18T08:28:29.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9748" PostTypeId="2" ParentId="1848" CreationDate="2019-06-17T19:49:41.513" Score="1" Body="&lt;p&gt;This might not be quite what you mean, but I would suggest NP-complete problems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There have been many, many problems, including Sudoku, scheduling, graph colouring, SAT, and thousands of other, which have all been proved to be NP-complete, so turn out to be &quot;equivalent&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3096" LastActivityDate="2019-06-17T19:49:41.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9749" PostTypeId="2" ParentId="1848" CreationDate="2019-06-18T03:21:59.490" Score="1" Body="&lt;p&gt;Matrix mechanics and the Schrödinger wave formulation of quantum mechanics.  &lt;a href=&quot;https://dialnet.unirioja.es/descarga/articulo/2735593.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A brief history of the mathematical equivalence between the two quantum mechanics&lt;/a&gt;, Carlos M. Madrid Casado, Lat. Am. J. Phys. Educ. Vol. 2, No. 2, May 2008.&lt;/p&gt;&#xA;" OwnerUserId="6501" LastEditorUserId="6501" LastEditDate="2019-06-19T00:55:01.940" LastActivityDate="2019-06-19T00:55:01.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9750" PostTypeId="2" ParentId="9679" CreationDate="2019-06-18T04:40:49.083" Score="3" Body="&lt;p&gt;Do you mean the observable universe? We can only see stars whose light has reached us since the big bang, but the universe is much larger. We don't even know whether it is infinite. &lt;a href=&quot;https://www.space.com/26078-how-many-stars-are-there.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;This article&lt;/a&gt; discusses estimates since 1995, giving a current estimate of &lt;span class=&quot;math-container&quot;&gt;$10^{24}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6501" LastActivityDate="2019-06-18T04:40:49.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9754" PostTypeId="2" ParentId="9727" CreationDate="2019-06-19T17:54:32.067" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Grothendieck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexander Grothendieck&lt;/a&gt;, without a doubt one of the most creative mathematicians in the last 100 years, wrote a voluminous manuscript &lt;em&gt;Récoltes et Semailles&lt;/em&gt; about (among other things) his approach to mathematics. Since he later requested that it be withdrawn from the public, it is hard (but probably not impossible) to find the full text on the internet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One little snippet of this which has gained some popularity among mathematicians is the &lt;a href=&quot;https://ncatlab.org/nlab/show/The+Rising+Sea&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Rising Sea&quot; metaphor&lt;/a&gt;. There is an &lt;a href=&quot;http://www.landsburg.com/grothendieck/mclarty1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; by &lt;a href=&quot;https://hsm.stackexchange.com/users/38/colin-mclarty&quot;&gt;Colin McLarty&lt;/a&gt; which discusses this, quoting at length translated passages from &lt;em&gt;Récoltes et Semailles&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6514" LastEditorUserId="6514" LastEditDate="2019-06-19T19:42:09.783" LastActivityDate="2019-06-19T19:42:09.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9755" PostTypeId="1" CreationDate="2019-06-19T19:41:25.363" Score="5" ViewCount="188" Body="&lt;p&gt;Is there a consensus amongst historians as to the principles of the historiography of science or as to how to write/create/preserve/record history? If there is, then what are those principles? If there isn't, what are the major positions regarding the historiography of the sciences? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;References that center on such questions would be very much appreciated!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;BTW tried to associate a tag 'historiography' or otherwise a tag 'history of history of science' with my question but could not find either such...&lt;/p&gt;&#xA;" OwnerUserId="10142" LastEditorUserId="10142" LastEditDate="2019-06-19T21:16:31.107" LastActivityDate="2019-06-19T21:16:31.107" Title="What are the principles of historiography of science?" Tags="|philosophy-of-science|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9756" PostTypeId="1" CreationDate="2019-06-20T11:16:38.747" Score="4" ViewCount="251" Body="&lt;p&gt;Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his mathematical diary (Mathematisches Tagebuch, full scan available via &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3382323&quot; rel=&quot;noreferrer&quot;&gt;https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3382323&lt;/a&gt;) on page 3 regarding the divergent series&#xA;&lt;span class=&quot;math-container&quot;&gt;$$1-2+8-64...$$&lt;/span&gt;&#xA;in relation to the continued fraction &#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1+\frac{2}{1+\frac{2}{1+\frac{8}{1+\frac{12}{1+\frac{32}{1+\frac{56}{1+128}}}}}}}$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He states also - if I read it correctly - &lt;em&gt;Transformatio seriei&lt;/em&gt; which could mean &lt;em&gt;series transformation&lt;/em&gt;, but I don't see how he transforms from the series to the continued fraction resp. which transformation or rule he applied.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The OEIS has an entry (&lt;a href=&quot;https://oeis.org/A014236&quot; rel=&quot;noreferrer&quot;&gt;https://oeis.org/A014236&lt;/a&gt;) for the sequence &lt;span class=&quot;math-container&quot;&gt;$2,2,8,12,32,56,128$&lt;/span&gt;, but I don't see the connection either.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone help or clarify?&lt;/p&gt;&#xA;" OwnerUserId="6392" LastActivityDate="2019-06-20T11:16:38.747" Title="Entry in Gauss' Mathematisches Tagebuch (Mathematical Diary)" Tags="|mathematics|gauss|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9758" PostTypeId="1" AcceptedAnswerId="9759" CreationDate="2019-06-22T15:18:35.560" Score="7" ViewCount="392" Body="&lt;p&gt;Physical constants (e.g. &lt;em&gt;c, h, G, alpha&lt;/em&gt; and so on) play a central role in our scientific theories and they have yet drawn much of controversial flavor into questions concerning the foundational status of those theories. From a historical point of view - who was the first to suggest a notion/idea of a physical constant? When and how did the notion/idea emerge?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any references on this issue and on closely related issues would be greatly appreciated!&lt;/p&gt;&#xA;" OwnerUserId="10142" LastEditorUserId="10142" LastEditDate="2019-06-30T23:54:08.753" LastActivityDate="2019-06-30T23:54:08.753" Title="When and how did the notion/idea of physical constant emerge?" Tags="|physics|reference-request|philosophy-of-science|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9759" PostTypeId="2" ParentId="9758" CreationDate="2019-06-23T08:48:11.083" Score="8" Body="&lt;p&gt;Going by Wikipedia's definition, a &lt;a href=&quot;https://en.wikipedia.org/wiki/Physical_constant&quot; rel=&quot;noreferrer&quot;&gt;physical constant&lt;/a&gt; is a number &amp;quot;generally believed to be both universal in nature and have constant value in time&amp;quot;. The significance of these constants began to be recognized in the late 19th century, partly as a result of the standardization of the measurement system. But their modern prominence is due to Eddington's semi-philosophical project from 1920-30s.&lt;/p&gt;&#xA;&lt;p&gt;I will start with pre-history. The first quantity acknowledged as universal was the gravitational constant. Of course, at the time the emphasis was not on the universality of the number but rather on the universality of the law it features in. But it was the unification of the Galileo's laws of free fall and Kepler's laws of planetary motion, Heavens and Earth, as it were, that allowed Newton to make a point of &amp;quot;universal gravity&amp;quot; in Principia (1687). As for the constant itself, he only suggested a way of measuring it, and indirectly gave a rough estimate for the order of magnitude. In a way, &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, known since ancient times, is a physical constant too. It measures the angle sum of physical triangles, and hence the curvature of physical space. The speed of light was also measured by Rømer somewhat earlier (1676), but its universality did not really come into view until Maxwell's unification of optics with electromagnetism in 19th century.&lt;/p&gt;&#xA;&lt;p&gt;The efforts to standardize and unify units were undertaken first by the committee on weights and measures after the French revolution (1799), and then by an international effort in 1860-80s that implemented Maxwell's proposal of taking  units of length, mass and time as base units and reducing the rest to them see &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_metric_system&quot; rel=&quot;noreferrer&quot;&gt;History of the metric system&lt;/a&gt;. Many constants (not necessarily universal) naturally appeared as conversion coefficients. Avogadro constant is an example, see &lt;a href=&quot;https://arxiv.org/abs/physics/0506173&quot; rel=&quot;noreferrer&quot;&gt;Fundamental Physical Constants: Looking from Different Angles by Karshenboim&lt;/a&gt; for more details. By the end of 19th century the notion of a physical constant was part of the folklore. For example, Planck was aware that he was proposing a new fundamental constant in his black body radiation formula.&lt;/p&gt;&#xA;&lt;p&gt;However, it was not until 1920s that Eddington undertook a systematic study of &amp;quot;constants of nature&amp;quot; and their relationships in his efforts to effect a grand unification of all physical theories. They start appearing as a theme in his  Mathematical Theory of Relativity (1923), and come into prominence in&#xA;The Charge of an Electron (1929) and New Pathways in Science (1935). He promoted the special role of &lt;em&gt;dimensionless&lt;/em&gt; constants, the only ones that are &amp;quot;truly&amp;quot; universal, in particular with his numerological speculations about the fine structure constant. From &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1510/1510.04046.pdf&quot; rel=&quot;noreferrer&quot;&gt;Kragh's&#xA;On Arthur Eddington’s Theory of Everything&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The fine-structure constant was not the only constant of nature that attracted the attention of Eddington. On the contrary, he was obsessed by the fundamental constants of nature, which he conceived as the building blocks of the universe and compared to the notes making up a musical scale: “We may … look on the universe as a symphony played on seven primitive constants as music played on the seven notes of a scale” (Eddington 1935, p. 227).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The recognition of the importance of constants of nature is of relatively recent origin, going back to the 1880s, and Eddington was instrumental in raising them to the significance they have in modern physics (Barrow 2002; Kragh 2011, pp. 93-99). Whereas the fundamental constants, such as the mass of the electron and Planck’s quantum constant are generally conceived to be irreducible and essentially contingent quantities, according to Eddington this was not the case. Not only did he believe that their numerical values could be calculated, he also believed that he had succeeded in actually calculating them from a purely theoretical basis.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While Eddington's numerology and grand unification did not survive the test of time, the prominence of universal constants in physical theories, and efforts to derive them from some sort of grand unified theory continue to this day.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-06-23T08:53:25.420" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9760" PostTypeId="1" CreationDate="2019-06-23T13:00:35.267" Score="2" ViewCount="249" Body="&lt;p&gt;Could I ask you for the relation between Wheeler's ideas and the multiverse? Do you know if these are related?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I ask you this because I found this very interesting article written by Kip Thorne with Wojciech Zurek and Charles W Misner (&lt;a href=&quot;https://www.its.caltech.edu/~kip/PubScans/VI-50.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.its.caltech.edu/~kip/PubScans/VI-50.pdf&lt;/a&gt;) where they seem to indicate that Wheeler was a multiverse proponent&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, at some part of the article they wrote:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Wheeler described to his two former students his idea that the laws of physics are mutable:&#xA;Those laws must have come into being in our universe’s Big Bang birth, and surely there are other universes, each with its own set of laws. “What principles determine which laws emerge in our universe and which in another?” he asked&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and also, at the end of it, they wrote &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Wheeler speculated that universes governed by sensible laws could come to be inhabited by observer–participators.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But then I contacted a biographer and analyzer of Wheeler's works and he told me that Wheeler's ideas are completely different to any kind of multiverse and that Wheeler was not happy with the idea of a multiverse. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;He said:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;There are a few hundred essays that Wheeler wrote from the 70s to the 90s where he built up his ideas. If you can find something in his essays about how his approach to quantum theory is the same as the multiverse approach, I will stand corrected. But I have looked over these for years, and have not seen anything other than his complicated relationship with Hugh Everett's Many Worlds. I have seen many physicists just guess what Wheeler was talking about, without citing anything that he actually wrote When you ask them their source it is usually &quot;Well that is what my colleagues think&quot; and when you ask their colleagues, they were just giving an interpretation of the phrase &quot;it from bit&quot; without reading any of the essays.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also he cited Wheeler &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Its [MWI/Everett] infinitely many unobservable worlds makes a heavy load of metaphysical baggage.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(He said that His approach (It from Bit) was meant to alleviate this philosophical issue, i.e to eliminate the multiverse in Everett's Interpretation)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, are Wheeler's ideas and the multiverse compatible or not? Is the biographer that I contacted right? Is Wheeler's &quot;It from Bit&quot;/&quot;Participatory Universe&quot;/&quot;Pregeometry and Law without Law&quot; and the Multiverse related? Apart from this, do you know of any quote of Wheeler that indicates that his ideas are compatible with the idea of a multiverse?&lt;/p&gt;&#xA;" OwnerUserId="10154" LastEditorUserId="55" LastEditDate="2019-06-23T23:32:08.453" LastActivityDate="2019-06-23T23:32:08.453" Title="Are Wheeler's It from Bit/Participatory Universe and the Multiverse related?" Tags="|physics|quantum-mechanics|cosmology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9761" PostTypeId="2" ParentId="9760" CreationDate="2019-06-23T23:20:12.140" Score="2" Body="&lt;p&gt;The dissonance comes from using the term &quot;multiverse&quot; too loosely. As far as Everett's many worlds and its derivatives, Wheeler supported Everett as his student, but distanced himself from his interpretations. He was committed to Bohr's Copenhagen interpretation with its role of the observers in &quot;shaping&quot; reality. In a letter to Stern he even coopted Everett to the Copenhagen interpretation against his own explicit words, see &lt;a href=&quot;https://www.scientificamerican.com/article/hugh-everett-biography/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Many Worlds of Hugh Everett in Scientific American&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, Wheeler drew a stark contrast between his &quot;participatory universe&quot; proposal and popular multiverse (&quot;ensemble of universes&quot;) conceptions in &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-94-009-3061-2_7&quot; rel=&quot;nofollow noreferrer&quot;&gt;World as system self-synthesized by quantum networking (1988)&lt;/a&gt;. The subsection is characteristically titled: No ensemble, no factory for making universes. Observer participancy the whole source of the &quot;out there&quot; plus life, mind, communication:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Those constants must have nearly the values they do, Robert H. Dicke, Brandon Carter, and others point out, if life is ever to be possible—not merely life as we know it, but life of almost any conceivable form. This observation has led some investigators to the idea of an ensemble of universes, one differing from another in the values of the dimensionless constants... There operates on such an ensemble of universes, Charles Pantin argued in 1951, something &quot;analogous to the principle of Natural Selection, that only in certain Universes, which happen to include ours, are the conditions suitable for the existence of life, and unless that condition is fulfilled there will be no observers to note the fact&quot;...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The contrast between the two views could hardly be&#xA;  greater: selection-from-an-ensemble and observer participancy.&#xA;  The one not only adopts the concept of&#xA;  universe, and this universe as machine, it also has to&#xA;  postulate, explicitly or implicitly, a supermachine, a scheme,&#xA;  a device, a miracle, which will turn out universes in infinite&#xA;  variety and infinite number. The other takes as foundation&#xA;  notion a higgledy-piggledy multitude of existences, each&#xA;  characterized, directly or indirectly, by the soliciting and&#xA;  receiving of answers to yes-no questions, and linked by&#xA;  exchange of information.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does this mean that Wheeler's self-synthesized worlds can not be subsumed under some sort of &quot;multiverse&quot; in a vague sense? He does not use the language but emphasizes and repeats constantly that the synthesis depends on multiple choices made by the &quot;multitude of existences&quot;, the observers. These choices determine not just the present but even the past, since it only exists to the extent that there is information about it in the present. And that information depends on the choice of questions to be asked. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, Wheeler's &quot;law without law&quot; slogan means that not just the past (initial conditions) but even the physical laws themselves are ultimately the result of those choices. Different choices, different answers, different worlds. This idea was later termed &quot;It from Bit&quot; (1989). The first OP quote refers to this picture, not to (what Wheeler called) &quot;ensemble of universes&quot;. But one is free to call it the &quot;multiverse&quot; of observer created worlds if one so wishes. In &lt;a href=&quot;https://link.springer.com/chapter/10.1007%2F978-1-4613-2443-0_25&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bohr’s “Phenomenon” and “Law Without Law” (1985)&lt;/a&gt; we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;So far as we can see today, the laws of physics cannot have&#xA;  existed from everlasting to everlasting. They must have come into&#xA;  being at the big bang. There were no gears and pinions,no Swiss watchmakers to put things together, not even a pre-existing&#xA;  plan. If this assessment is correct, every law of physics must be&#xA;  at bottom like the second law of thermodynamics, higgledy-piggledy&#xA;  in character, based on blind chance, insofar as it is not mere&#xA;  mathematical identity.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;The elementary quantum phenomenon has much of the character of&#xA;  an elementary act of creation. The phenomenon that we call &quot;a&#xA;  photon from a distant quasar&quot; reaches into the present from billions&#xA;  of years in the past. It is wrong to think of that past as &quot;already&#xA;  existing&quot; in all detail. The &quot;past&quot; is theory. The past has no&#xA;  existence except as it is recorded in the present. By deciding&#xA;  what questions our quantum registering equipment shall put in the&#xA;  present, we have an undeniable choice in what we have the right to&#xA;  say about the past.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-06-23T23:31:13.067" LastActivityDate="2019-06-23T23:31:13.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9762" PostTypeId="1" CreationDate="2019-06-24T01:03:16.597" Score="5" ViewCount="2083" Body="&lt;p&gt;I am looking for the earliest reference which proposed the convolution theorem which is often utilized in signal processing (i.e., convolution becomes multiplication in the Fourier domain).&lt;/p&gt;&#xA;&lt;p&gt;The Earliest Known Uses of the Word of Mathematics websites give a lot of details on the word convolution. Still, I want to know who was the first person to specifically show the above-mentioned property- the connection of Fourier transforms with convolution?&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="10106" LastEditDate="2020-06-22T12:20:54.097" LastActivityDate="2020-06-22T12:20:54.097" Title="Who came up with the convolution theorem?" Tags="|mathematics|reference-request|harmonic-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9763" PostTypeId="2" ParentId="9762" CreationDate="2019-06-24T04:09:28.507" Score="4" Body="&lt;p&gt;Dominguez in &lt;a href=&quot;https://pulse.embs.org/january-2015/history-convolution-operation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of the Convolution Operation&lt;/a&gt; poured through the original sources, and found many of Miller's and Gardner-Barnes's claims and citations to be inaccurate or erroneous. He devotes a separate section to main theorems associated with the convolution, where we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;On the other hand, another important theorem related to the CCO is the so-called convolution theorem. In general, the ­theorem establishes that the Laplace transform of the CCO (5) is the product of the Laplace transform of each input function. A similar result holds for (6) when the integral transform is the Fourier transform. In both cases, it is difficult to determine when both theorems appeared for the very first time. However, it can be said that the convolution theorem for (5) appeared in 1899 in Borel’s memoir about divergent series. For the case of (6), the convolution theorem appeared in the 1920 conference by Daniell about Stieltjes–Volterra products. In it, Daniell defined the convolution of any two measures over the real line, and then he applied the two-sided Laplace transform obtaining the corresponding convolution theorem. From it, he implied the result for the Fourier transform&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the other hand, the rule of multiplying polynomials equivalent to convolving their coefficients (without the terminology or modern notation) appears already in the work of 12th century Islamic mathematician al-Samawal, see &lt;a href=&quot;https://books.google.com/books/about/A_History_of_Mathematics.html?id=h50fAQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Katz, History of mathematics, 9.3.3&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-06-24T04:09:28.507" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9765" PostTypeId="1" AcceptedAnswerId="9771" CreationDate="2019-06-28T22:43:15.620" Score="5" ViewCount="189" Body="&lt;p&gt;My understanding is that in Cartesian geometry, all coordinate axes of an n-dimensional space may intersect at one point. I would like to know how that point--whether (0, 0), (0,0,0), ... -- came to be called &quot;the origin.&quot; What else has it been called? How did the use of the word &quot;origin&quot; become commonplace in geometry?&lt;/p&gt;&#xA;" OwnerUserId="9312" LastActivityDate="2019-07-02T06:16:34.390" Title="How did the use of the word &quot;origin&quot; become commonplace in geometry?" Tags="|terminology|geometry|euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9766" PostTypeId="1" CreationDate="2019-06-29T20:05:56.650" Score="6" ViewCount="1605" Body="&lt;p&gt;Authors writing about history of physics describe that before the writing of the Principia several scholars were aware that if the orbits of the planets would be circular (which they knew wasn't the case) then Kepler's third law was consistent with an inverse square law for gravity. These scholars included figures such as Christopher Wren, John Hooke, and Edmond Halley.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems to me that this implies that during that time the relation F=ma was already generally accepted as valid. Since without F=ma you cannot even evaluate the case of circular motion. However, I cannot find descriptions of how that came to be.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder: was F=ma blindly assumed? It seems so.&#xA;Of course, since F=ma is in fact correct assuming it was justified, but I find it odd that I cannot find any history of how F=ma came to be established.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To my knowledge the earliest attempts at experimental investigation of the relation between acceleration and force was by Galilei. To my recollection historians describe that Galilei had small balls rolling down inclines. A small vessel with a smaal hole near the bottom released a slow stream of water. Galilei (according to the description) would use several different lengths of inclines. He would allow the water to start flowing on release, and stop the flowing when the ball reached the end of the incline. the weight difference between start and end would then be a proportional measure of the time it took the ball to roll down the incline.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This experiment may have actually been performed by Galilei, or possibly it was only a description of something that one &lt;em&gt;could&lt;/em&gt; do (similar to the description of simultaneous dropping of two different weights from a high tower that was described as possible demonstration, with later authors erroneously describing that as something that Galilei had actually done.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried looking up information about the views of Pierre Gassendi, who, as I understand it, is recognized as the first to formulate the modern notion of what today is referred to as 'Newton's first law'. But I didn't find description of Gassendi proposing F=ma&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking in general:&lt;br&gt;&#xA;If you do &lt;em&gt;not&lt;/em&gt; assume F=ma then it seems to me you cannot make any progress when it comes to formulating a theory of mechanics. And of course when you want to do science you must believe that you are in a position where progress is &lt;em&gt;possible&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Once newtonian mechanics became established the law F=ma became knows as Newton's second law, and I suppose the vast majority of authors simply assume that Newton was to first to formulate it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I argue that cannot be, as earlier established work is dependent on the presupposition of F=ma.  &lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2020-05-11T22:12:52.627" Title="The origin of F=ma" Tags="|classical-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9767" PostTypeId="1" CreationDate="2019-06-29T20:16:45.537" Score="1" ViewCount="123" Body="&lt;p&gt;Newton's gravity equations are in an F = ma configuration. Multiply the acceleration formula by m, you get the force formula. So did he have F = ma (or some equivalent - I think the concept of acceleration may not have been a direct quantity in those days), before he got those two gravity equations? Did he get to one of them from the other, via F = ma?&lt;/p&gt;&#xA;" OwnerUserId="10175" LastActivityDate="2019-06-29T20:56:03.293" Title="Did Newton already have F= ma (or equivalent) before finding his gravity equations?" Tags="|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9768" PostTypeId="2" ParentId="9767" CreationDate="2019-06-29T20:56:03.293" Score="2" Body="&lt;p&gt;It is abundantly clear that in Newton's time F=ma was already firmly established.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to illustrate that is to llook at the way the three laws of motion are described in the Principia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is how F=ma is dealt with:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The alteration of motion is ever proportional to the motive force&#xA;  impress'd; and is made in the direction of the right line in which&#xA;  that force is impress'd&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If any force generates a motion, a double force will generate double&#xA;  the motion, a triple force triple the motion, whether that force be&#xA;  impress'd altogether and at once, or gradually and successively. And&#xA;  this motion (being always directed the same way with the generating&#xA;  force) if the body moved before, is added to or subducted from the&#xA;  former motion, according as they directly conspire with or are&#xA;  directly contrary to each other; or obliquely joyned, when they are&#xA;  oblique, so as to produce a new motion compounded from the&#xA;  determination of both.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Clearly, Newton felt no need to convince the reader of F=ma. That in itself shows that in Newton's time F=ma was firmly established.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The purpose of writing down the first second and third law right at the beginning of the Principia was not to introduce them. In a sense it was a formality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like other scientific books of the time the structure of the Principia was modeled after the 'Elements' (The 'Elements' written by the greek mathematician Euclid). Thoughout centuries of science Euclid's Elements have been tremendously influential.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid had demonstrated his mastery of his subject by identifying the smallest set of statements that as a set of statements is sufficient to serve as the logical basis of an entire body of geometrical mathematics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton presented the three laws as a statement of logic: according to Newton those three axioms were sufficient to serve as the logical basis of all of mechanics&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: by the looks of it: in Newton's time F=ma was firmly established. &lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2019-06-29T20:56:03.293" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9769" PostTypeId="1" CreationDate="2019-06-30T22:17:39.817" Score="0" ViewCount="106" Body="&lt;p&gt;My question is about Newton's bucket experiment. If a sphere filled (say) one-third with water is rotated very very fast, will the water eventually spread out across and coat the entire interior surface of the sphere? If so, then does this mean that the sphere's inside is coated with water whose inertial force is everywhere at right angles to the surface? And if this is so, then does this phenomenon offer a primitive physical model for the topological &quot;gluing&quot; together of two 3-dimensional spheres to yield a 4-dimensional sphere (a 3-sphere), especially when the gluing is imagined according to the method of progressive longitudinal &quot;slices&quot;? (Unless it is better to imagine the two 3-d spheres &quot;put through&quot; each other point for point, accordingly with the mechanical, force-oriented nature of Newton's experiment?) I'm trying to visualize William Blake's Mundane Shell, which Bronowski first half-recognized as a 3-sphere in 1942.&lt;/p&gt;&#xA;" OwnerUserId="10178" LastEditorUserId="10178" LastEditDate="2019-06-30T22:23:31.237" LastActivityDate="2019-06-30T22:23:31.237" Title="What if Newton's bucket had been a sphere?" Tags="|topology|newton|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9770" PostTypeId="1" CreationDate="2019-07-01T13:30:52.077" Score="4" ViewCount="327" Body="&lt;p&gt;I chose year 1900 because of:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know about some of them, to mention:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) &lt;a href=&quot;https://en.wikipedia.org/wiki/Legendre%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Legendre´s conjecture&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) &lt;a href=&quot;https://en.wikipedia.org/wiki/Polignac%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Polignac´s conjecture&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) &lt;a href=&quot;https://en.wikipedia.org/wiki/Bunyakovsky_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Bunyakovsky conjecture&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;4) &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_hypothesis&quot; rel=&quot;noreferrer&quot;&gt;Riemann hypothesis&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;5) Are there any odd perfect numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;6) &lt;a href=&quot;https://en.wikipedia.org/wiki/Goldbach%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Goldbach's conjecture&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;7) Are there infinitely many Mersenne primes?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;8) &lt;a href=&quot;https://en.wikipedia.org/wiki/Brocard%27s_problem&quot; rel=&quot;noreferrer&quot;&gt;Brocard's problem&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But, I am not so sure is there a huge number of mathematical unsolved problems posed prior to the year 1900 or just a &quot;few&quot; of them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you list some of them in your answer?&lt;/p&gt;&#xA;" OwnerUserId="10177" LastActivityDate="2019-07-02T12:11:26.703" Title="What are some of the unsolved mathematical problems posed and stated clearly prior to the year 1900?" Tags="|mathematics|big-list|conjecture|" AnswerCount="1" CommentCount="5" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9771" PostTypeId="2" ParentId="9765" CreationDate="2019-07-01T18:01:40.613" Score="7" Body="&lt;p&gt;A good place to look for such things is &lt;a href=&quot;http://jeff560.tripod.com/o.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;, where we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Boyer (page 404) seems to attribute the term origin to Philippe de Lahire (1640-1718). The term presumably appears in Sections Coniques by Marquis de l'Hopital, since the OED shows a use of the term in English in a 1723 translation of this work&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Boyer writes the following in his &lt;a href=&quot;https://archive.org/details/AHistoryOfMathematics/page/n419&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;About the only mathematician of stature in France at the time was Philippe de Lahire (1640-1718), a disciple of Desargues and, like his master, an&#xA;  architect. Pure geometry obviously appealed to him, and his first work on conics in 1673 was synthetic, but he did not break with the analytic wave of&#xA;  the future. Lahire kept an eye out for a patron; hence, in his Nouveaux Elemens des Sections Coniques of 1679, dedicated to Jean Baptiste Colbert,&#xA;  the methods of Descartes came to the fore... But Lahire carried over into analytic geometry some of Desargues’ language. The axis of abscissas was the &#xA;  &quot;trunk,&quot; points on it were &quot;knots,&quot; and ordinates were &quot;branches&quot;.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;Of his analytic language, only the term “origin” has survived.&lt;/strong&gt; Perhaps it was because of his terminology that contemporaries did not give proper weight to a significant point in his Nouveaux Elemens. Lahire provided one of the first examples of a surface given analytically through an equation in three unknowns, which was the first real step toward solid analytic geometry. He, like Fermat and Descartes, had only a single reference point or origin on a single line of reference or axis OB, to which he now added the reference or coordinate plane OBA.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;l'Hopital's Traité Anlytique des Sections Coniques appeared in 1707. Presumably, he, or his 1723 English translators, borrowed the term &quot;origin&quot; from Lahire.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-02T06:16:34.390" LastActivityDate="2019-07-02T06:16:34.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9772" PostTypeId="1" CreationDate="2019-07-01T18:23:49.493" Score="2" ViewCount="185" Body="&lt;p&gt;In Havil's 2003 book &lt;em&gt;&lt;a href=&quot;https://books.google.com/books/about/Gamma.html?id=lQX6Oy_SuOgC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gamma&lt;/a&gt;&lt;/em&gt; he states that Hardy offered up his chair in Oxford to whoever could prove that the Euler-Mascheroni constant &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; is irrational.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm almost positive I had heard a source of this prior to Havil.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what is the earliest source of this?  Did Hardy really make the offer, and if so was it serious or just as a statement meant to attest to the difficulty of such a question?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nick R points to &lt;a href=&quot;http://mathworld.wolfram.com/Euler-MascheroniConstant.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathworld&lt;/a&gt; as indicating that there is no written reference for this quote.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So then I'll change my question - what was Havil's source?  His book was quite solid otherwise.&lt;/p&gt;&#xA;" OwnerUserId="9482" LastEditorUserId="2905" LastEditDate="2020-02-20T02:11:18.660" LastActivityDate="2020-02-20T02:11:18.660" Title="What was Havil's source for the statement that G.H. Hardy would offer his Savillian chair to whoever could prove $\gamma$ irrational?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9774" PostTypeId="2" ParentId="9770" CreationDate="2019-07-02T12:11:26.703" Score="4" Body="&lt;p&gt;Turns out this is discussed in detail over at &lt;a href=&quot;https://mathoverflow.net/questions/27075/what-is-the-oldest-open-problem-in-mathematics&quot;&gt;MathOverflow&lt;/a&gt;.   I'm in no way educated enough in this field to comment further, other than to mention that the top answer there states it's &quot;Existence or nonexistence of odd perfect numbers,&quot; dating from Nicomachus of Gerasa around 100 AD.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-07-02T12:11:26.703" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9775" PostTypeId="2" ParentId="9766" CreationDate="2019-07-03T20:10:43.453" Score="6" Body="&lt;p&gt;The following is not a self-written answer; it's a comment. For obvious reasons here on stackexchange comment space is limited. This comment is large.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Descartes struggled at formulating a set of laws of collision, he didn't get things right, that illustrates how opaque the problem is. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I mentioned in an a comment, the kind of experiment that was readily doable at the time was to construct dual pendulums, with the bobs colliding with each other at the lowest point. Newton describes that kind of setup in the 'Scholium' to the 'Axioms, or laws of motion'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;About collisions:&#xA;As long as you formulate all problems in dynamics in terms of collisions taking place you can get by without a committed position on how to think about force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Novel in Newton's exposition in the Principia, it seems, is that Newton went all in on putting static force and dynamic force on &lt;em&gt;equal footing&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One way to illustrate this is to consider how Newton is motivating the third law. Newton is switching between cases of static force (tension in a rope tied to something immovable) and cases of dynamic force (objects exerting a force upon each other, accelerating each other)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what I didn't appreciate when I submitted the question:&lt;br&gt;&#xA;In Newton's time, when you consider the possibility that the planets move according to a inverse square law, it isn't &lt;em&gt;necessary&lt;/em&gt; to think of the planets being pulled by a &lt;em&gt;force&lt;/em&gt;. You can think of the planets as subject to some accelerating influence, not otherwise specified.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Using the inverse square rule for the magnitude of the acceleration, combined with knowledge of the required centripetal force to sustain circular motion you can see the tantalizing consistency of an inverse square law with Keplers third law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's novel approach:&lt;br&gt;&#xA;Newton is committed to thinking of gravity as a &lt;em&gt;force&lt;/em&gt;. To my understanding: Newton regards the Sun's gravity as literally &lt;em&gt;tugging&lt;/em&gt; at the planets, just like in daily life rope is used for tugging things.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To treat gravity as literally tugging you do need a law that relates force and acceleration. &#xA;F=ma may have been proposed before, but Newton needs it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Still, it is odd to me that in the Principia Newton doesn't work particularly hard to convince the reader that F=ma holds good. One rather gets the impression that Newton is confident that the reader will readily accept F=ma.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Comparison: the case for gravitational mass being equal to inertial mass.&lt;br&gt;&#xA;If gravitational mass would &lt;em&gt;not&lt;/em&gt; be equal to inertial mass then in order to calculate the orbit of a planet you have to know its mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Conversely, if gravitational mass &lt;em&gt;is&lt;/em&gt; equal to inertial mass then every object, &lt;em&gt;independent of its mass&lt;/em&gt;, will follow the same orbit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So it was necessary for Newton to present a strong case that gravitational mass is equal to inertial mass. One part of that is to assure the readers that that pendulums of unequal mass but with the same string length have the same period of oscillation. In the Principia Newton also considers that the other planets may have a composition that is different from the Earth. Therefore Newton describes experiments (pendulum-type setup) he conducted to address that issue. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So: in the case of whether gravitational mass is equal to inertial mass we do see that Newton puts in effort to convince the reader.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why no effort to convince the reader of F=ma?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible hypothesis (but I think it hardly plausible):&lt;br&gt;&#xA;Newton suggests to the readers that the laws of motion he presents arise from prior work by other scholars: Christopher Wren, Christiaan Huygens, John Wallis. However, that prior work was not as advanced as Newton suggests it was. (Conifold describes this as 'Newton assigning credit retroactively') Possibly Newton sougth to smuggle some ideas past his readers. I don't know. &lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2019-07-03T20:10:43.453" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9776" PostTypeId="1" CreationDate="2019-07-04T11:38:43.930" Score="3" ViewCount="207" Body="&lt;p&gt;I just want to check if anyone has additional information about a private astronomical research begun by Gauss in 1819, which dealt with the &quot;&lt;strong&gt;movements of the solar system&lt;/strong&gt;&quot; in the galaxy (motion of the solar system relative to the fixed stars). In addition to the material in his nachlass (volumes 6 and 11:1), I found several interesting sources which show that through his correspondences and unpublished fragments Gauss was involved in attempts to try and define the movement of the sun by the statistical motion of the fixed stars.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since i'm completely unfamiliar with those scientific matters, my question is therefore intended to be a framework for learning in a general way on this aspect of observational astronomy (i.e, mathematial tools, methodology) by achieving better understanding of Gauss's work on this. To be more specific, these are my questions:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;One such correspondence is his 1838 letter to  Argelander (just Google &quot;&lt;strong&gt;On a Letter from Gauss to Argelander&lt;/strong&gt;&quot; and you will find it). My first question is therefore a request for explanation of what Gauss actually did in this letter.&lt;/li&gt;&#xA;&lt;li&gt;In Gauss's werke, volume 11-1, p.390, appears a short note &quot;&lt;strong&gt;About the frequency of optical double stars&lt;/strong&gt;&quot;. Despite it's neglible length, Martin Brendel (the editor of Gauss's astronomical works) comments on it lengthfully, and the interesting title of the note just makes my curiousity greater. So my second question is just for explanation.&lt;/li&gt;&#xA;&lt;li&gt;Any additional useful information will be blessed.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;As i said several times before, one of my main objectives in using this site is to gain a complete picture of all of Gauss's contributions to math and science, so once again, i appologize for being so annoying and tedious with my obssesive questions-asking about Gauss.     &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-07-04T19:04:19.340" LastActivityDate="2020-03-27T11:09:38.403" Title="Several questions about Gauss's contributions to observational astronomy" Tags="|biographical-details|astronomy|gauss|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9777" PostTypeId="1" AcceptedAnswerId="9778" CreationDate="2019-07-05T00:45:29.987" Score="9" ViewCount="1039" Body="&lt;p&gt;In the context of partially ordered sets, why are the words for supremum and infimum &quot;join&quot; and &quot;meet&quot;? I find the nomenclature puzzling, especially since the English words &quot;join&quot; and &quot;meet&quot; are synonyms, but denote opposite concepts when talking about posets.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know how these concepts got these names?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/3283516/why-are-join-and-meet-named-as-they-are&quot;&gt;(Cross-posted on MSE)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10192" LastActivityDate="2022-03-19T04:08:46.020" Title="Why are &quot;join&quot; and &quot;meet&quot; named as they are?" Tags="|mathematics|terminology|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9778" PostTypeId="2" ParentId="9777" CreationDate="2019-07-05T12:23:43.430" Score="7" Body="&lt;p&gt;The &quot;Elements&quot; of modern lattice theory, from which the terminology spread is &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.166886&quot; rel=&quot;nofollow noreferrer&quot;&gt;Birkhoff's Lattice Theory (1940)&lt;/a&gt;. After defining lattices and introducing cups and caps (rather than wedges and vees) for inf and sup he says the following on the subject of meets and joins (p.17):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A large fraction of the most important partly ordered systems &#xA;  considered in mathematics are lattices. Moreover in these systems, the opera &#xA;  tions &lt;span class=&quot;math-container&quot;&gt;$\frown$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\smile$&lt;/span&gt; usually correspond to familiar and significant constructions.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Example 1. Let &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; consist of all the subsets of any aggregate &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;, and let inclusion mean set-inclusion. Then &quot;joins&quot; are sums of sets and &quot;meets&quot; are &#xA;  set-products...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Example 3. Let &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; consist of the subgroups of any group, and let inclusion &#xA;  mean set-inclusion. Then the terms &quot;join&quot; and &quot;meet&quot; have their usual meaning (also called union and intersection).&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In other words, &quot;join&quot; joins (unites) subgroups together, and &quot;meet&quot; is where they meet (intersect). It is curious that Birkhoff considers the &quot;usual meaning&quot; to be for subgroups, not for sets. He does not credit anybody for the names, although he does mention that the definition is due to Peirce, and that Whitney suggested &quot;cap&quot; and &quot;cup&quot; names (Whitney himself used them for chains and cochains in homology). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest appearance of meets and joins I found is in Birkhoff's first paper on lattices, &lt;a href=&quot;https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-the-combination-of-subalgebras/848B24E326A568DCECF2294F626AC977&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the combination of subalgebras (1933)&lt;/a&gt;, where they are applied to subalgebras, subgroups and subrings, and again not credited to anybody. Birkhoff wrote this paper before he became aware of Dedekind's work. Von Neumann picked them up from Birkhoff in his later 1930-s papers. They do not appear in &lt;a href=&quot;https://www.jstor.org/stable/2369451&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peirce's On the Algebra of Logic (1885)&lt;/a&gt;, or in &lt;a href=&quot;https://de.wikisource.org/wiki/Jahresbericht_der_Deutschen_Mathematiker-Vereinigung/Inhalt#Band_38:_1929&quot; rel=&quot;nofollow noreferrer&quot;&gt;Klein's expository paper Einige distributive Systeme in Mathematik und Logik (1929)&lt;/a&gt; that mentions many examples and names like conjuction/disjunction or union/intersection. Menger uses &lt;em&gt;Summe&lt;/em&gt; and &lt;em&gt;Durchschnitt&lt;/em&gt; (average) in &lt;a href=&quot;https://de.wikisource.org/wiki/Jahresbericht_der_Deutschen_Mathematiker-Vereinigung/Inhalt#Band_37:_1928&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bemerkungen zu Grundlagenfragen (1928)&lt;/a&gt;, but his 1936 &lt;a href=&quot;https://www.jstor.org/stable/1968458&quot; rel=&quot;nofollow noreferrer&quot;&gt;New Foundations of Projective and Affine Geometry&lt;/a&gt; already has joins. I am going to speculate that Birkhoff was the terms' originator.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The beginnings of lattice theory go back to Boole, Dedekind and Peirce, but they did not have much of a follow-up until Emmy Noether revived Dedekind's work in her reshaping of algebra. &lt;a href=&quot;https://dml.cz/bitstream/handle/10338.dmlcz/401261/DejinyMat_17-2001-1_31.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bilova in Lattice Theory – Its Birth and Life&lt;/a&gt; says that Birkhoff himself came up with the term &quot;lattice&quot; (under the influence of Hasse's drawings):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Lattice structures started to be studied again at the end of 1920’s –&#xA;  this time in still another area of mathematics. Karl Menger presented&#xA;  the set of axioms characterizing projective geometries which are&#xA;  in fact complemented modular lattices. His investigations did not attract&#xA;  immediate attention, however, lattice structures soon appeared&#xA;  also in the field of formal logic (Fritz Klein who gave lattices its German&#xA;  name: “Verband”) and mainly algebra (Robert Remak, Oystein&#xA;  Ore). The biggest merits in the early developments of lattice theory&#xA;  belong to Garrett Birkhoff who also approached it from the side of&#xA;  algebra and united its various applications. In his first article about&#xA;  lattice structures [4] he rediscovered, apart from others, Dedekind’s&#xA;  results, and only after its publishing it was revealed that the studies&#xA;  of dual groups are identical with Birkhoff’s approach. G. Birkhoff&#xA;  also introduced the English word “lattice”, which is not the translation&#xA;  of its German equivalent, but was inspired by the image of some Hasse&#xA;  diagrams presenting lattices.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The [4] is the aforementioned 1933 paper. I should also mention that aside from lattice theory meet and join are now commonly used in modern geometric algebra/exterior calculus (not to be confused with van der Waerden's term for Euclid's book II), where they are applied to subspaces and Grassmann's multivectors. This was Rota's contribution, see his with co-authors &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0021869385900432&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the exterior calculus of invariant theory&lt;/a&gt;, and &lt;a href=&quot;https://pdfs.semanticscholar.org/4331/196e34ee35988cf16920f631f03e2ff03834.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Many Lives of Lattice Theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-06T11:38:01.000" LastActivityDate="2019-07-06T11:38:01.000" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9779" PostTypeId="2" ParentId="9777" CreationDate="2019-07-05T12:26:11.877" Score="1" Body="&lt;p&gt;I did a quick survey on the origins of join and meet on Google books. Join and meet do not have German origins. See some early references.&#xA;Ref: Information and Control 31, 312-340 (1976), Some Properties of Fuzzy Sets of Type 2&#xA;Masaharu Mizumoto &amp;amp; Kokichi Tanaka&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/W68XX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/W68XX.png&quot; alt=&quot;join and meet&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Another work rationalizes this terminology by Jerry M. Mendel and Robert I. Bob John (2002)&#xA;&lt;a href=&quot;https://i.stack.imgur.com/GfZQQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GfZQQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/88QZ5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/88QZ5.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; Confifold has provided early usages of meet and join.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-07-05T12:26:11.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9780" PostTypeId="2" ParentId="9656" CreationDate="2019-07-05T14:07:41.950" Score="2" Body="&lt;p&gt;The chord function appears in Greek astronomical works. The most accessible is Ptolemy’s &lt;em&gt;Almagest&lt;/em&gt;; the first chapter explains the theory and how the table is computed. The switch to half-chords happened in India, again in astronomical works. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The details can be found in Glen van Brummelen’s history of early trigonometry, &lt;em&gt;The Mathematics of the Heavens and the Earth&lt;/em&gt;. Already in Ptolemy one can find passages in which he connects chords to the angle at the circumference instead of the central angle. That is half the central angle, of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One can also see in the &lt;em&gt;Almagest&lt;/em&gt; contexts in which the key function is half the chord (on pp. 95-96 van Brummelen gives an example from the solar theory). So the move from chords to sines is motivated by computational convenience: it saves factors of two in various formulas.&lt;/p&gt;&#xA;" OwnerUserId="4289" LastActivityDate="2019-07-05T14:07:41.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9782" PostTypeId="1" CreationDate="2019-07-05T21:53:44.133" Score="0" ViewCount="132" Body="&lt;p&gt;When John Napier and Joost Burgi developed logarithms in the 16th century, they succeeded in replacing long, tedious, error-prone multiplications with table-look-up and addition, giving other mathematicians the benefit of their stored work in the form of their extensive tables.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So what makes this better than developing extensive multiplication tables, which would replace multiplication with table-look-up only, instead of table-look-up and addition, as with logarithms?&lt;/p&gt;&#xA;" OwnerUserId="10194" LastActivityDate="2019-07-06T10:20:50.750" Title="Logarithm tables vs multiplication tables" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9783" PostTypeId="2" ParentId="9782" CreationDate="2019-07-06T10:20:50.750" Score="6" Body="&lt;p&gt;I am explaining the comment of Conifold. The main point is that log tables tabulate a function of ONE variable. For example, Napier's original table is organized in 1230 lines and occupies 41 pages (30 lines per page). And it permits interpolation.&#xA;Imagine a similar table for a function of TWO variables. Calculation of such a table by hand is impossible, and its use is impractical. The sheer volume will be like 1230&#xA;of Napier's volumes. Not even mentioning interpolation.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-07-06T10:20:50.750" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9785" PostTypeId="1" CreationDate="2019-07-07T00:11:40.823" Score="4" ViewCount="1028" Body="&lt;p&gt;In the arena of art, it is not uncommon to find great writers, composers or painters who suffer from chaotic personal lives (e.g., lifelong alcoholism, addiction to prostitutes, stormy marriages and affairs), but it seems that most of the biographical sketches of great mathematicians I read portray their lives as peaceful in general. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A counterexample I can think of is &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Grothendieck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexander Grothendieck&lt;/a&gt;, who estranged his wife and children, seeing other women while married. I understand that the criteria I'm setting is quite vague, but I suppose the example serves an illustrative purpose. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What are some other examples of mathematicians or scientists who led tumultuous private lives?&lt;/p&gt;&#xA;" OwnerUserId="2503" LastEditorUserId="4251" LastEditDate="2020-07-12T09:42:50.683" LastActivityDate="2024-03-09T18:54:13.960" Title="Great scientists with chaotic private lives" Tags="|mathematicians|biographical-details|big-list|scientists|" AnswerCount="7" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="9786" PostTypeId="2" ParentId="9785" CreationDate="2019-07-07T05:14:26.853" Score="7" Body="&lt;p&gt;There is nothing special about great scientists, their personal life is in general like the personal life of other people. If you mean by &quot;chaotic personal life&quot; things like&#xA;&quot;estranged his wife and children, seeing other women while married,&quot; read biographies of Einstein or Schrodinger, for example. Several great mathematicians&#xA;suffered from alcoholism (T. Carleman, and probably S. Banach). Another great mathematician, Andre Bloch, murdered his family and spent all all his math career in a mental asylum.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-07-07T05:14:26.853" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9787" PostTypeId="1" AcceptedAnswerId="9788" CreationDate="2019-07-09T07:15:34.617" Score="3" ViewCount="547" Body="&lt;p&gt;&lt;strong&gt;Claims I'm evaluating&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've read in multiple sources that Leibniz formulated separation of variables for ODEs in 1691. A couple example sources are below.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/MathematicalThoughtFromAncientToModernTimes/page/n491&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Thought from Ancient to Modern Times, by Morris Kline&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://reference.wolfram.com/language/tutorial/DSolveIntroductionToODEs.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfram Language Documentation&lt;/a&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My Question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know the original source for this claim? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 1: If it's a handwritten document (e.g. a letter), then I'd be delighted to see a digital copy of the original.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 2: If there are English (or Spanish) translations available, that would be helpful.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 3: If the original source is not available online, then any information about the specific equation (or equations) that Leibniz solved, or the context in which the equation arose, would be interesting.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note 4: If anyone happens to know where I could find John (Johann) Bernoulli's Acta Eruditorum article of 1694, in which he apparently elaborates on the method of separation of variables, then that would be helpful as well.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What I've found so far&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest source I've found is &lt;a href=&quot;https://archive.org/details/derbriefwechselv00leibuoft/page/682&quot; rel=&quot;nofollow noreferrer&quot;&gt;a letter from Leibniz to Huygens&lt;/a&gt;, written on the 29th of December, 1691. The letter is in French, and I read French only a little (just what I can pick up from knowing another romance language). Nonetheless, Leibniz clearly applies separation of variables in this letter to the equation dy = y/a dx.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Kline (see link above), Leibniz's introduction of separation of variables is said to occur in a letter to Huygens, but I don't see any references to a specific letter. Since Leibniz and Huygens corresponded quite a bit in 1691, it's possible that another letter is being referenced.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thank you for any assistance you can provide!&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="9529" LastActivityDate="2019-07-09T18:58:57.093" Title="Source of claim that Leibniz discovered separation of variables for ODEs in 1691?" Tags="|reference-request|discoveries|calculus|differential-equations|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9788" PostTypeId="2" ParentId="9787" CreationDate="2019-07-09T10:17:17.203" Score="7" Body="&lt;p&gt;The reference is probably to a treatise sent to Huygens on 5 October 1691, where Leibniz says (and illustrates with several examples) that &quot;Whenever the subtangent [&lt;span class=&quot;math-container&quot;&gt;$=y/y'$&lt;/span&gt;, but it would also work for just the tangent &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt;] is a product of two quantities or formulas, of which one is given purely in terms of the abscissa &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, and the other in terms of the ordinate &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, then the problem reduces to quadratures [i.e., to integration].&quot; (page 186 of &lt;a href=&quot;http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/III5A.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.gwlb.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/III5A.pdf&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that was hardly a new discovery at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Leibniz's first paper on the calculus, in 1684, he determines a curve from its tangent property, which arguably amounts to a separation of variables. See page 8 of &lt;a href=&quot;http://www.17centurymaths.com/contents/Leibniz/nova1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.17centurymaths.com/contents/Leibniz/nova1.pdf&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his first published paper on the integral calculus, in 1686, Leibniz again effectively uses separation of variables. See page 297 and the note below the image at &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1687, Leibniz separates variables again to solve the differential equation of a descent problem. See page 4 of &lt;a href=&quot;http://www.17centurymaths.com/contents/Leibniz/ae3a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.17centurymaths.com/contents/Leibniz/ae3a.pdf&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his lectures on the calculus of 1691-92, Johann Bernoulli uses separation of variables throughout and treats it as one of the most basic ideas of the calculus. See &lt;a href=&quot;https://archive.org/details/dieersteintegra00kowagoog/page/n19&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/dieersteintegra00kowagoog/page/n19&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Because of examples such as these I do not think it makes sense to view separation of variables as a discovery separate from the Leibnizian calculus, made only in 1691. Separation of variables seems to have been taken for granted as a natural and self-evident part of calculus itself. Separation of variables does not stand out as a distinctive technique in these works because Leibniz et al. worked with relations between differentials in a flexible way rather than expressing all such relations in terms of the derivative &lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt; as we are inclined to do today.&lt;/p&gt;&#xA;" OwnerUserId="370" LastEditorUserId="370" LastEditDate="2019-07-09T18:58:57.093" LastActivityDate="2019-07-09T18:58:57.093" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9789" PostTypeId="1" AcceptedAnswerId="9793" CreationDate="2019-07-09T14:06:43.853" Score="5" ViewCount="356" Body="&lt;p&gt;Hubbard and Hubbard (in &quot;Vector Calculus...&quot;) attribute the following quote to Felix Klein: &quot;Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know where this comes from?  I'd be interested in the original German text.&lt;/p&gt;&#xA;" OwnerUserId="3555" LastActivityDate="2021-01-10T12:54:01.680" Title="Source for Felix Klein quote about curves" Tags="|reference-request|quote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9790" PostTypeId="1" CreationDate="2019-07-09T15:17:32.917" Score="0" ViewCount="62" Body="&lt;p&gt;The development of the fast Fourier transform (FFT) is attributed to Cooley &amp;amp; Tukey, both of whom have written a lot about its historical development. However, I am searching for early publications that showed how people made the discrete frequency axis after FFT.&lt;/p&gt;&#xA;&lt;p&gt;For example, if we have &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; data points (where &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is an even or odd number), the FFT output also consists of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; data points. The specific question is:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Who came up with the equation or the proof that the frequency spacing in discrete Fourier transform is &lt;span class=&quot;math-container&quot;&gt;$\frac k L$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$k = 0,1,2,3,\dots, N-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is the total duration of time for which the signal was acquired.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I am sure the concept of folding or aliasing must be in the same work.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Here is one earlier example by Tukey himself from 1950s: &lt;a href=&quot;https://archive.org/details/TheMeasurementOfPowerSpectra/page/n24&quot; rel=&quot;nofollow noreferrer&quot;&gt;The measurement of power spectra&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="4251" LastEditDate="2020-06-22T11:25:07.210" LastActivityDate="2020-06-22T11:25:07.210" Title="On the origin of the concept of aliasing &amp; the Discrete Fourier Transform frequency axis" Tags="|mathematics|harmonic-analysis|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9791" PostTypeId="1" AcceptedAnswerId="11819" CreationDate="2019-07-09T17:07:53.760" Score="7" ViewCount="1460" Body="&lt;p&gt;It is often said that Romans (see below) had a terrible number system, which made computations a mess. I do believe this, but I'm suspicious of the claim that nobody had better ways to do computations until Fibonacci brought Eastern techniques to Europe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is even more suspicious when you think how big the Roman empire was and that it featured a lot of fiscal checks (such as census), infrastructural works (hence engineering and logistics) and, above all, economic activity (can you believe greedy merchants didn't think of better ways to make calculations?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So, were arithmetical algorithms so inefficient before Arabic numerals were adopted in Western Europe?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 1: I use Romans as a proxy for &quot;ancient Western Europe civilization&quot;. I suppose anything known to Europeans (e.g. Greeks) in the last centuries BC and first centuries AD was known to Romans as well. Also, I assume anything known before was inherited, i.e. if Egyptians/Babylonians/Etruscan had wonderful arithmetical techniques, I doubt they got lost in history, since these are the kind of things that tend to be spread (see merchants, see what happened with Arab-Indian numerals).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2: I'm not asking about numerals, but about techniques of computation. From the comments, I see that the two things had disjoint developments indeed.&lt;/p&gt;&#xA;" OwnerUserId="6884" LastEditorUserId="4251" LastEditDate="2021-09-24T07:56:43.357" LastActivityDate="2021-09-24T07:56:43.357" Title="Were ancient Romans so bad at computations before Arab numerals?" Tags="|mathematics|elementary-algebra|algorithm|roman-empire|antiquity|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9792" PostTypeId="1" CreationDate="2019-07-09T19:29:21.737" Score="4" ViewCount="262" Body="&lt;p&gt;I remember that I read, some time ago, that Erathostenes calculated the circumfernce of the Earth using the distance between Syene and Alexandria, calculated in turn by others starting from the circumference of the Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To prove It, the author used arguments that convinced me at the time, yet I don't remember the name of the book.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So I guess the question Is: Is someone aware of this hypothesis? Is there a book you could point me toward?&lt;/p&gt;&#xA;" OwnerUserId="6996" LastEditorUserId="55" LastEditDate="2019-07-11T01:17:41.060" LastActivityDate="2019-07-11T01:17:41.060" Title="Circular reasoning in Eratosthenes's measurement of the Earth's circumference" Tags="|geography|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9793" PostTypeId="2" ParentId="9789" CreationDate="2019-07-09T20:06:38.487" Score="8" Body="&lt;p&gt;&lt;em&gt;Elementarmathematik vom höheren Standpunkte aus&lt;/em&gt;, Bd.2 &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Was eine Kurve ist, glaubt jeder Mensch zu wissen, bis er so viel&#xA;  Mathematik gelernt hat, daß ihn die unzähligen möglichen Abnormitäten&#xA;  verwirrt gemacht haben.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;see also &lt;a href=&quot;https://www-history.mcs.st-andrews.ac.uk/Quotations/Klein.html&quot; rel=&quot;noreferrer&quot;&gt;Quotations by Felix Klein&lt;/a&gt; for the English version.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3555" LastEditDate="2019-07-10T06:31:49.017" LastActivityDate="2019-07-10T06:31:49.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9794" PostTypeId="2" ParentId="9785" CreationDate="2019-07-09T22:28:36.187" Score="8" Body="&lt;p&gt;Kurt Gödel was certainly known as an eccentric in his life. In fact it is said that he starved to death when his wife died as he refused to take meals from anyone but his wife. &#xA;In the language of today, he would probably be diagnosed as suffering from mental health issues.&lt;/p&gt;&#xA;" OwnerUserId="10210" LastActivityDate="2019-07-09T22:28:36.187" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9795" PostTypeId="2" ParentId="9792" CreationDate="2019-07-10T03:41:21.937" Score="5" Body="&lt;p&gt;This might refer to the wrong circumference hypothesis, but it concerns not Eratosthenes but Ptolemy. There were many issues with Erathostenes's measurement, but circularity was not one of them, see &lt;a href=&quot;https://hsm.stackexchange.com/a/8269/55&quot;&gt;How did Eratosthenes determine that Alexandria and Syene were on the same meridian?&lt;/a&gt; Ptolemy, on the other hand, as might have been his custom (he is alleged to have done similar things with Hipparchus's data), used Roman itineraries (road distance measurements), and a (widely off) value of Earth's circumference taken from Posidonius, to determine longitudes of various locations on his maps. Then he implied that he determined the longitudes from astronomical measurements (proposed by Hipparchus), and used them to &quot;derive&quot; the distances. At least, so the hypothesis goes. Here is Scheglov's description from The &lt;a href=&quot;https://www.journals.uchicago.edu/doi/full/10.1086/689763&quot; rel=&quot;noreferrer&quot;&gt;Accuracy of Ancient Cartography Reassessed: The Longitude Error in Ptolemy’s Map&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In recent years a number of researchers—Dennis Rawlins (Baltimore), Lucio Russo (Rome), Irina Tupikova (Dresden), and Klaus Geus (Berlin)—have independently proposed essentially the same hypothesis. They all suggest that the longitudinal stretching of Ptolemy’s map can be explained completely by a single factor: using the wrong value for the circumference of the Earth... On the basis of these observations, it has been argued that the east–west stretching of Ptolemy’s map was caused by a change in the proposed value of the Earth’s circumference. This means that all the linear distances (in stades, miles, etc.) underlying his map had initially been supposed to be measured on a sphere with Eratosthenes’ circumference, but eventually were converted to angular units (degrees) on a much smaller sphere. As a result, all the distances when expressed in angular units increased by a factor of 1.4... Ptolemy committed the same “crime” that was imputed to his astronomical works by Robert Newton: he not only appropriated the achievements of his predecessors but also perverted them completely.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The issue is controversial, and Shcheglov argues for an alternative explanation that has to do with differing lengths of the stade used in various reported values of the circumference. In that case, Eratosthenes's values were as off as Ptolemy's, but in the opposite direction, and likely because he relied on inaccurate distance measurements reported by travelers.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-07-10T03:41:21.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9796" PostTypeId="1" AcceptedAnswerId="9798" CreationDate="2019-07-10T14:09:12.067" Score="6" ViewCount="276" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Bloch_sphere&quot; rel=&quot;noreferrer&quot;&gt;Bloch sphere&lt;/a&gt; is a geometric representation of a single qubit. I am having trouble figuring out when it came into common usage in the scientific literature. The wikipedia article, as at the time of writing this question, references a journal article by Bloch (Bloch, Felix (Oct 1946). &quot;Nuclear induction&quot;. Phys. Rev. 70 (7–8): 460–474) but the paper contains neither an image of a Bloch sphere, nor (as far as a cursory read suggests) a description of one.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I realise that &lt;a href=&quot;https://en.wikipedia.org/wiki/Polarization_(waves)#Poincar%C3%A9_sphere&quot; rel=&quot;noreferrer&quot;&gt;the Poincaré sphere&lt;/a&gt; is mathematically the same object, but I am interested in when it was first explicitly used to represent a two state quantum system, and specifically when a picture of it first occurred in a scientific publication to that end.&lt;/p&gt;&#xA;" OwnerUserId="6496" LastActivityDate="2019-07-11T01:16:41.313" Title="First occurrence of the Bloch sphere in the scientific literature" Tags="|reference-request|theoretical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9798" PostTypeId="2" ParentId="9796" CreationDate="2019-07-10T19:16:13.627" Score="5" Body="&lt;p&gt;&quot;Qubits&quot; were only named by Schumacher in 1995, and even early ideas about &quot;quantum computing&quot; do not appear until 1960-s. &quot;Bloch sphere&quot; refers to representing the pure states of a 2D quantum system (2 refers to complex dimension). What comes to be called &quot;the Bloch sphere&quot; is spelled out by Feynman, Vernon and Hellwarth in &lt;a href=&quot;https://aip.scitation.org/doi/10.1063/1.1722572&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometrical Representation of the Schrödinger Equation for Solving Maser Problems (1957)&lt;/a&gt;, and they do have a picture on p. 52 of &quot;oscillator diagram in rotating coordinates&quot;, at least: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A simple, rigorous geometrical representation for the Schrödinger equation is developed to describe the behavior of an ensemble of two quantum‐level, noninteracting systems which are under the influence of a perturbation. In this case the Schrödinger equation may be written, after a suitable transformation, in the form of the real three‐dimensional vector equation &lt;span class=&quot;math-container&quot;&gt;$dr/dt=ω×r$&lt;/span&gt;, where the components of the vector r uniquely determine ψ of a given system and the components of ω represent the perturbation. When magnetic interaction with a spin ½ system is under consideration, &quot;r&quot; space reduces to physical space. By analogy the techniques developed for analyzing the magnetic resonance precession model can be adapted for use in any two‐level problems. The quantum‐mechanical behavior of the state of a system under various different conditions is easily visualized by simply observing how r varies under the action of different types of ω.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the subject of the origins they only say:&quot;&lt;em&gt;The extensive and explicit use of rotating coordinate procedures, as was introduced by Bloch, Ramsey, Rabi, and Schwinger for special kinds of magnetic transitions, is generally applicable in dealing with the &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; space&lt;/em&gt;&quot;. The name &quot;Bloch sphere&quot;, with a picture, appears, for example in &lt;a href=&quot;https://journals.aps.org/pra/abstract/10.1103/PhysRevA.5.1094&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stroud et al., Superradiant Effects in Systems of Two-Level Atoms (1972), p. 1103&lt;/a&gt;, and by then they call it &quot;the familiar Bloch-vector form&quot;. &quot;Rotating coordinate procedures&quot; (inspired by Majorana) appear e.g. in Bloch's paper with Rabi &lt;a href=&quot;https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.17.237&quot; rel=&quot;nofollow noreferrer&quot;&gt;Atoms in Variable Magnetic Fields (1945)&lt;/a&gt;. There are no pictures: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As an essential feature in the derivation of his formulae, Majorana has shown that the problem of a system with arbitrary angular momentum can be reduced to the consideration of &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; representative points on the unit sphere, each representing the direction of an angular momentum with value &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt;... Majorana's method, while remarkable in its&#xA;  elegance, has the disadvantage of somewhat obscuring the physical significance of the representative systems with spin &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt;. It is clear that a simple intuitive understanding of the procedure and of the essential formulae will be very useful&#xA;  to many. In this paper we shall arrive at such an understanding by the application of the familiar vector model where the total spin operator &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is treated as a sum of &lt;span class=&quot;math-container&quot;&gt;$2j$&lt;/span&gt; spin operators each representing a system with angular momentum &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; and with the same &quot;&lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt;&quot; value as the total system. Mathematically this will be expressed by a different representation of the system in which the variables of the Schroedinger wave function &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; are the spin variables of the constituent systems with spin &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt; and which we shall call the &quot;composite&quot; representation in contrast to the usual representation which uses only a single spin variable, referring to the total system of spin &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But the mathematics of &quot;rotating coordinate procedures&quot; appears already in Bloch's PhD thesis &lt;a href=&quot;https://courses.physics.ucsd.edu/2018/Fall/physics211a/topic/bloch.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Mechanics of the Electrons in Crystal Lattices (1929)&lt;/a&gt; written under Heisenberg, or, for that matter, Poincare's optical paper from 1892, but outside the quantum mechanical context.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-11T01:16:41.313" LastActivityDate="2019-07-11T01:16:41.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9799" PostTypeId="1" CreationDate="2019-07-10T21:35:42.220" Score="2" ViewCount="440" Body="&lt;p&gt;For example, in proposition 1, Euclid assumes that the instersection of the two circles exist, when he shouldn't have. This, among many other things, was corrected quite recently (by Hilbert and others, I believe). Were these mistakes only found out then? Did no one mention them for over 2000 years? &lt;/p&gt;&#xA;" OwnerUserId="10219" LastActivityDate="2019-07-17T07:06:03.850" Title="Were people aware of the &quot;mistakes&quot; in Euclid's Elements before the start of the formalization of Mathematics?" Tags="|mathematics|geometry|euclidean-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9800" PostTypeId="2" ParentId="9799" CreationDate="2019-07-10T23:29:49.670" Score="2" Body="&lt;p&gt;Yes.  There were numerous commentaries on Euclid in the ancient world, and many of them pointed out flaws in the various propositions.  Zeno of Sidon (~100BC) for example, pointed out the Proposition I.1 requires the assumption that unique straight lines can meet in at most one point, which isn't explicitly stated in the postulates or common notions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, finding issues with Euclid (and proposing fixes) was something of a cottage industry amongst the neo-platonists of late antiquity.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(as to the specific issue with I.1 you point out, I'm not sure if it was pointed out so early.  Heath oddly mentions that it's a &quot;common place objection&quot;, but rather uncharacteristically doesn't actually cite any ancient sources that discuss it.)&lt;/p&gt;&#xA;" OwnerUserId="1731" LastActivityDate="2019-07-10T23:29:49.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9802" PostTypeId="1" AcceptedAnswerId="9803" CreationDate="2019-07-11T04:22:18.403" Score="3" ViewCount="274" Body="&lt;p&gt;Newton had realised that the acceleration due to gravitational force had to be inversely dependent on the square of the distance between two bodies. But how did he arrive at the conclusion that the force is also directly proportional to the product of the masses of the two bodies? And then how could have the terms &quot;inertial&quot; and &quot;gravitational&quot; mass and the confusion of them being equal started?&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2019-07-17T09:34:05.607" Title="How did Newton come up with mass terms in his gravitation formula and then what led to a confusion between inertial and gravitational mass?" Tags="|gravity|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9803" PostTypeId="2" ParentId="9802" CreationDate="2019-07-11T19:25:20.240" Score="5" Body="&lt;p&gt;The terminology of inertial and gravitational masses does not appear until Einstein, and had to do with his  transitioning from classical mechanics to general relativity, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalence_principle#Development_of_gravitational_theory&quot; rel=&quot;noreferrer&quot;&gt;Development of gravitational theory&lt;/a&gt;. In classical mechanics it is more natural to think of the same &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; entering both &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F=G\frac{Mm}{r^2}$&lt;/span&gt; than about two separate masses which are then made &quot;equivalent&quot;. It was not really a &quot;confusion&quot;. Here is how  Newton phrases it in &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Definitions&quot; rel=&quot;noreferrer&quot;&gt;Definition I of Principia&lt;/a&gt; he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The quantity of matter is the measure of the same, arising from its density and bulk conjunctly... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As to the proportionality of the gravitational force to masses, it manifests (along with the &quot;equivalence&quot;) in the acceleration of free fall being the same for all bodies of equal weight. To get a universal &lt;span class=&quot;math-container&quot;&gt;$a=G\frac{M}{r^2}$&lt;/span&gt; from the second law the gravitational force has to be proportional to &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. The universality itself was, of course, established already by Galileo, but Newton confirmed it with his own experiments described in &lt;a href=&quot;https://en.wikisource.org/wiki/Page:Newton%27s_Principia_(1846).djvu/400&quot; rel=&quot;noreferrer&quot;&gt;Principia, Book III, Proposition VI, Theorem VI&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat... The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-07-11T19:25:20.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9804" PostTypeId="2" ParentId="9785" CreationDate="2019-07-11T19:44:57.910" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Georg_Cantor#Life_of_Georg_Cantor&quot; rel=&quot;noreferrer&quot;&gt;Georg Cantor&lt;/a&gt; faced many difficulties and died in a sanatorium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alan_Turing#Personal_life&quot; rel=&quot;noreferrer&quot;&gt;Alan Turing&lt;/a&gt; faced persecution and ultimately committed suicide.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both are titans of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="7348" LastActivityDate="2019-07-11T19:44:57.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9808" PostTypeId="1" CreationDate="2019-07-14T00:57:50.067" Score="2" ViewCount="388" Body="&lt;p&gt;I have been trying to find out who discovered Maxwell-Faraday equation, &lt;span class=&quot;math-container&quot;&gt;$\nabla\times E=-\frac{\partial B}{\partial t}$&lt;/span&gt;. Was it Maxwell himself, or was it Franz Ernst Neumann who derived it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The equation means that a time varying magnetic field induces a spatially varying electric field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While searching for the answer I stumbled upon an important piece which is given below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Maxwell's unique contribution was his &quot;Ampère's&quot; circuital law (both forms: with and without the displacement current term). Faraday's Law was derived by Franz Neumann in &lt;span class=&quot;math-container&quot;&gt;$1845$&lt;/span&gt;. Gauss (and Lagrange before him) knew the compatibility between Coulomb's law and &quot;Gauss&quot;'s law. I'm not sure who was the first to derive &lt;span class=&quot;math-container&quot;&gt;$\nabla\cdot B=0$&lt;/span&gt;&lt;/em&gt;&quot;.– Geremia&#xA;Source: &lt;a href=&quot;https://hsm.stackexchange.com/questions/5055/why-is-maxwell-and-not-amp%C3%A8re-credited-for-unifying-electricity-and-magnetism#comment8029_5076&quot;&gt;Why is Maxwell and not Amp&amp;#232;re credited for unifying electricity and magnetism?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Faraday's law in itself was more about voltage generated as a result of magnetic field. To confuse things more, it is said that it was Neumann who first put the result of Faraday's experiment into a mathematical form somewhere around &lt;span class=&quot;math-container&quot;&gt;$1845$&lt;/span&gt;. Why did it take so long to mathematize the result of an experiment? Why didn't Faraday himself converted the result of his experiment(s) into a mathematical form?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Related discussion: &lt;a href=&quot;https://hsm.stackexchange.com/questions/5078/who-discovered-the-magnetic-vector-potential-veca/5080#5080&quot;&gt;Who discovered the magnetic vector potential, $\vec{A}$?&lt;/a&gt; &#xA;&quot;&lt;em&gt;The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845.&lt;/em&gt;&quot; -&lt;a href=&quot;https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you please help me to find the answers to these queries? Thanks for your help!&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="4251" LastEditDate="2020-07-12T11:32:12.690" LastActivityDate="2024-02-08T16:09:16.137" Title="Who discovered Maxwell-Faraday equation, $\nabla\times E=-\frac{\partial B}{\partial t}$? Was it Maxwell or Neumann?" Tags="|physics|electromagnetism|physicists|maxwell|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9809" PostTypeId="1" CreationDate="2019-07-14T21:08:48.463" Score="1" ViewCount="387" Body="&lt;p&gt;There was a specific account of Stephen Hawking by one of his contemporaries in which Hawking was in his late undergrad or early postgrad.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As I remember, there was a small class of physics students (each one was very capable in their own right) who had been given (I think) ten difficult questions to have a go at. At the next tutorial, the person who was giving the account was very proud that he managed to solve one of them and make significant progress on another. Another person had managed to solve two questions and the tutor was very impressed by this. Stephen showed up late, as he tended to do, and apologised that he only managed to solve eight of them in time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember reading the above tale a long time ago, and thought it was a nice story, but I recently wanted to check if it was true and haven't been able to find any reference to it.&lt;/p&gt;&#xA;" OwnerUserId="10028" LastEditorUserId="10106" LastEditDate="2020-05-28T22:20:42.000" LastActivityDate="2020-05-28T22:20:42.000" Title="Looking for a specific story about young Stephen Hawking" Tags="|biographical-details|debunking|depiction-of-scientists|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9811" PostTypeId="2" ParentId="9802" CreationDate="2019-07-16T11:07:31.673" Score="1" Body="&lt;p&gt;I'd like to point out that the proportionality to &lt;span class=&quot;math-container&quot;&gt;$m_1m_2$&lt;/span&gt; comes from the &lt;em&gt;universality&lt;/em&gt; of Newton's proposed gravitational law. For, suppose two identical unit particles exert equal and opposite forces &lt;span class=&quot;math-container&quot;&gt;$f=G/r^2$&lt;/span&gt;, and let two masses &lt;span class=&quot;math-container&quot;&gt;$m_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt; consist of &lt;span class=&quot;math-container&quot;&gt;$N_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$N_2$&lt;/span&gt; small particles. Then each small particle in &lt;span class=&quot;math-container&quot;&gt;$m_1$&lt;/span&gt; will feel a total force of &lt;span class=&quot;math-container&quot;&gt;$N_2f$&lt;/span&gt; by universality, and since there are &lt;span class=&quot;math-container&quot;&gt;$N_1$&lt;/span&gt; of them, the total force is of &lt;span class=&quot;math-container&quot;&gt;$$F=N_1(N_2f)=\frac{GN_1N_2}{r^2}=\frac{Gm_1m_2}{r^2}$$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$m=N$&lt;/span&gt; units of mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton never writes &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt; in Principia; instead he uses proportionalities (see &lt;a href=&quot;https://plato.stanford.edu/entries/newton-principia/#NewLawMot&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://plato.stanford.edu/entries/newton-principia/#NewLawMot&lt;/a&gt;). So the distinction between inertial and gravitational mass did not arise.&lt;/p&gt;&#xA;" OwnerUserId="9714" LastEditorUserId="9714" LastEditDate="2019-07-17T07:26:37.623" LastActivityDate="2019-07-17T07:26:37.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9812" PostTypeId="1" CreationDate="2019-07-16T15:34:34.930" Score="1" ViewCount="179" Body="&lt;p&gt;As I was reading on the topic plane angle then I came across a term radian.Hence I want to know how radians were discovered as well ?&lt;/p&gt;&#xA;" OwnerUserId="9984" LastEditorUserId="9984" LastEditDate="2019-07-18T02:57:57.423" LastActivityDate="2019-07-18T02:57:57.423" Title="Invention of the SI unit of plane angle:Radian" Tags="|discoveries|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9814" PostTypeId="1" AcceptedAnswerId="9815" CreationDate="2019-07-17T03:09:35.630" Score="2" ViewCount="692" Body="&lt;p&gt;Stephen Hawking is a famous scientist who despite being disabled made a big contribution to the scientific world. I want to know other such scientists who have a big influence on us.&lt;/p&gt;&#xA;" OwnerDisplayName="user10238" LastEditorUserId="4251" LastEditDate="2020-07-13T00:33:38.397" LastActivityDate="2020-07-13T00:33:38.397" Title="Scientists who made a mark in the world despite their disabilities" Tags="|biographical-details|depiction-of-scientists|scientists|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9815" PostTypeId="2" ParentId="9814" CreationDate="2019-07-17T03:30:36.633" Score="3" Body="&lt;p&gt;These are some of the scientists whom I know about:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Thomas_Edison&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Edison&lt;/a&gt;&lt;br /&gt;&#xA;Thomas Alva Edison, an American inventor and businessman, has been described as &lt;strong&gt;America's greatest inventor&lt;/strong&gt;. A bout of scarlet fever as a youth had left him &lt;strong&gt;almost entirely deaf in both ears&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Temple_Grandin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Temple Grandin&lt;/a&gt;&lt;br /&gt;&#xA;Animal behaviourist and autism activist Grandin was diagnosed with &lt;strong&gt;autism&lt;/strong&gt; as a child. Grandin’s work on animal behaviour has been cited as an inspiration for multiple changes to the livestock industry, including how animals are cared for at meat processing plants.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Gustav_Kirchhoff&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gustav Kirchoff&lt;/a&gt;&lt;br /&gt;&#xA;Physicist Kirchhoff’s work in the 1800s is still relevant to our understanding of electricity today. He made headway into the still-young field despite an &lt;strong&gt;unknown disability that restricted his movement to a wheelchair&lt;/strong&gt; or crutches for most of his life.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Edwin_G._Krebs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edwin Krebs&lt;/a&gt;&lt;br /&gt;&#xA;Krebs was &lt;strong&gt;hearing-impaired&lt;/strong&gt; but made a sensational discovery in the 1950s about cellular activity in the human body that led to greater understanding about hormones, cell life spans, and even how the body can reject transplanted organs. He won the &lt;strong&gt;1992 Nobel Prize for medicine&lt;/strong&gt; for this commendable feat.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-07-12T02:55:54.107" LastActivityDate="2020-07-12T02:55:54.107" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9820" PostTypeId="2" ParentId="9802" CreationDate="2019-07-17T09:34:05.607" Score="1" Body="&lt;p&gt;As for the mass terms in the gravitational formula, they manifest after the study of bodies in free fall as said by Conifold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the confusion of the inertial and gravitational mass, there was no reason why they should be the same. The two masses appear in phenomena that seemingly had nothing to do with one another. I apply a force to an object, measure the acceleration, and divide. A principle of dynamics. Why should it have any relationship at all to the nature-provided force that holds the Moon to the Earth?&lt;/p&gt;&#xA;" OwnerDisplayName="user10238" LastActivityDate="2019-07-17T09:34:05.607" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9821" PostTypeId="2" ParentId="9814" CreationDate="2019-07-17T13:18:42.213" Score="4" Body="&lt;p&gt;I am not sure if you count pure mathematics as having a big influence on you, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Solomon_Lefschetz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solomon Lefschetz&lt;/a&gt; did a lot of important work after he lost both his hands in an accident. See also &lt;a href=&quot;https://mathoverflow.net/questions/334726/how-did-lefschetz-do-mathematics-without-hands&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2785" LastActivityDate="2019-07-17T13:18:42.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9822" PostTypeId="2" ParentId="9814" CreationDate="2019-07-17T13:22:28.247" Score="9" Body="&lt;p&gt;Leonard Euler, whom many consider the greatest mathematician of all times, and also the most prolific one, was partially or completely blind during most of his career. His sight began to deteriorate at the age of 31 and he lost his sight completely at 59. He died at 76 and continued to work hard until his death.&#xA;(At the age of 68 he published a paper every week in the average!)&lt;/p&gt;&#xA;&lt;p&gt;Some other famous mathematicians who were blind were:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Pontryagin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lev Pontryagin&lt;/a&gt; (from his childhood, as a result of an accident).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Vitushkin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anatoli Vitushkin&lt;/a&gt; (from his youth, as a result of an accident).&lt;/p&gt;&#xA;&lt;p&gt;Mathematician &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Bloch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Andre Bloch&lt;/a&gt; can be probably also considered as handicapped: he spent all his mathematical career in a mental asylum (after having murdered several people). His mental disorder was apparently the result of severe wound during WWI. (This was the reason of confining him rather than executing).&lt;/p&gt;&#xA;&lt;p&gt;Another mathematician who was severely wounded in WWI was &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Julia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaston Julia&lt;/a&gt; (he lost his nose).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2020-07-12T11:28:31.997" LastActivityDate="2020-07-12T11:28:31.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9823" PostTypeId="1" CreationDate="2019-07-18T00:10:18.820" Score="4" ViewCount="2203" Body="&lt;p&gt;As far as I remember, Calculus was invented/discovered/founded by Newton.&lt;/p&gt;&#xA;&lt;p&gt;What was he trying to achieve that made him find the limit of differences approaching zero?&lt;/p&gt;&#xA;&lt;p&gt;How far did he get into Calculus? Did he also find integration? Differential equations?&lt;/p&gt;&#xA;" OwnerUserId="5860" LastEditorUserId="4251" LastEditDate="2020-06-24T20:28:41.990" LastActivityDate="2020-06-24T20:28:41.990" Title="What problem led to the discovery of Calculus?" Tags="|mathematics|mathematicians|calculus|newton|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9824" PostTypeId="1" CreationDate="2019-07-18T02:03:49.617" Score="6" ViewCount="1240" Body="&lt;p&gt;I have occasionally run into &lt;a href=&quot;https://www.newscientist.com/letter/mg16922837-800-einsteins-uncle/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the following story&lt;/a&gt; (or a variation of it) about Albert Einstein:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Young Albert did not like algebra, and his uncle is supposed to have aroused his curiosity by telling him to think of it as a detective story, where x was the criminal who had to be identified by following the “clues” in the equations. Once the boy had grasped this idea he never looked back.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am wondering if anybody can provide a reliable source for this story.  Did Einstein ever refer to it in any of his writings, or is it purely apocryphal?&lt;/p&gt;&#xA;" OwnerUserId="294" LastEditorUserId="5989" LastEditDate="2020-03-10T06:51:24.667" LastActivityDate="2020-03-10T06:51:24.667" Title="Source for this story about Einstein learning algebra from his uncle" Tags="|einstein|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9825" PostTypeId="2" ParentId="9823" CreationDate="2019-07-18T04:23:42.087" Score="7" Body="&lt;p&gt;You remember incorrectly. Calculus was found by Archimedes, Gregory of Saint-Vincent, Galileo, Kepler, Descartes, Pascal, Cavalieri, Fermat, Barrow,&#xA;Wallis, Brounker, Huygens, Leibniz, J. Gregory, N. Mercator, Newton, Cotes, Taylor, Torricelli, Bernoulli brothers, to name only the most famous ones.&#xA;As every big enterprise, this was a collective enterprise.&lt;/p&gt;&#xA;&lt;p&gt;The problems which led to its development are: finding areas and volumes (integration), finding tangents to curves (differentiation), finding maxima and minima of functions and functionals (calculus of variations) and expansion of functions into power series which was used for solving differential equations arising in geometry and physics.&lt;/p&gt;&#xA;&lt;p&gt;But if by &amp;quot;calculus&amp;quot; you only mean the differentiation rules and Newton-Leibniz formula, these were found by Newton and Leibniz, independently.&#xA;But this is just one theorem of calculus.&lt;/p&gt;&#xA;&lt;p&gt;To answer your second question, yes, Newton (and Leibniz and Bernoulli) also knew integration and differential equations. Integration was developed by Eudoxus and Archimedes, and this is the oldest part of calculus. Differentiation as a tool of finding extrema was also used by Archimedes&#xA;(and by Fermat, and by others).&lt;/p&gt;&#xA;&lt;p&gt;Ref. N. Bourbaki, Elements of the history of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Remark. Since my mentioning of Archimedes triggered so many comments, let me cite Nicolas Bourbaki, the essay on History of Calculus (my own translation):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The greatest mathematical discovery of the Greeks was their method of treatment of problems which we call integral calculus. Eudoxus gave the first examples of application of this method when he determined the volumes of a cone and a pyramid; this reached us in more or less adequate description by Euclid (VII, Prop. 7, 10). But most importantly, almost all works of Archimedes are devoted to these problems, because of an exceptional luck we can read them in the originals, in his beautiful Doric dialect..&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He also mentions that Archimedes was by far the most cited mathematician in&#xA;17th century.&lt;/p&gt;&#xA;&lt;p&gt;Let me add that all surviving works of Archimedes are easily available in English translation to which I send all those who have any doubts about who invented integration. And many commentaries to them are also available. But for a short and non-technical history of calculus in 17th century (and the role of the Greek heritage in it) I recommend the article of Bourbaki cited above.&lt;/p&gt;&#xA;&lt;p&gt;BTW, Newton himself described his main contribution to calculus as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One can solve any differential equation by plugging a power series with indetermined coefficients to it, and find the coefficients one-by one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I slightly modernized his language). This is not taught in modern elementary courses.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-24T00:41:01.000" LastActivityDate="2020-06-24T00:41:01.000" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9828" PostTypeId="1" CreationDate="2019-07-19T08:57:41.210" Score="7" ViewCount="1084" Body="&lt;p&gt;In its article about how the electron was discovered, &lt;a href=&quot;https://en.wikipedia.org/wiki/J._J._Thomson#Discovery_of_the_electron&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; says that Thomson and his students performed experiments which suggested that cathode rays were negatively charged &quot;particles&quot;. But even after reading about his experiments I am unable to understand how he came to the conclusion that the cathode rays are made up &lt;em&gt;particles&lt;/em&gt;. I can see how he came to the conclusion that they are negatively charged, but how did he find out about their particle nature? &lt;/p&gt;&#xA;" OwnerUserId="10245" LastEditorUserId="5989" LastEditDate="2019-07-20T07:12:07.943" LastActivityDate="2019-07-20T07:12:07.943" Title="How did J. J. Thomson establish the particle nature of the electron?" Tags="|discoveries|experimental-physics|particle-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9829" PostTypeId="2" ParentId="9824" CreationDate="2019-07-19T09:08:36.393" Score="11" Body="&lt;p&gt;The story is genuine. Isaacson retells it in his scientific biography &lt;a href=&quot;https://books.google.com/books/about/Einstein.html?id=cdxWNE7NY6QC&quot; rel=&quot;noreferrer&quot;&gt;Einstein: His Life and Universe&lt;/a&gt; based on recollections of his younger sister Maja, along with other evidence that dispels the early &quot;Einstein was a slow starter and flanked school&quot; story exaggerated by his parents:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;His uncle Jakob Einstein, the engineer, introduced him to the joys of algebra. &quot;It's a merry science,&quot; he explained. &quot;When the animal that we are hunting cannot be caught, we call it X temporarily and continue to hunt until it is bagged.&quot; He went on to give the boy even more difficult challenges, Maja recalled, &quot;with good-natured doubts &#xA;  about his ability to solve them.&quot; When Einstein triumphed, as he &#xA;  invariably did, he &quot;was overcome with great happiness and was already &#xA;  then aware of the direction in which his talents were leading him.&quot;&#xA;  Among the concepts that Uncle Jakob threw at him was the Pythagorean theorem (the square of the lengths of the legs of a right triangle add &#xA;  up to the square of the length of the hypotenuse). &quot;After much effort I &#xA;  succeeded in 'proving' this theorem on the basis of the similarity of &#xA;  triangles,&quot; Einstein recalled. Once again he was thinking in pictures. &#xA;  &quot;It seemed to me 'evident' that the relations of the sides of the &#xA;  right-angled triangles would have to be completely determined by one of&#xA;  the acute angles.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Galina Weinstein compiled a useful collection of sources on Einstein's early life in &lt;a href=&quot;https://arxiv.org/abs/1205.4509&quot; rel=&quot;noreferrer&quot;&gt;Albert Einstein: Rebellious Wunderkind&lt;/a&gt;, she cites Winteler-Einstein, 1924, CPAE, Vol. 1, p. xx, pp. 1xi-1xii as the source for Maja's recollections. They are (partly) confirmed by Einstein's own Autobiographical Notes,  &lt;a href=&quot;https://books.google.com/books/about/Albert_Einstein.html?id=qjHPAAAAMAAJ&quot; rel=&quot;noreferrer&quot;&gt;Reiser's 1930 biography&lt;/a&gt;, and a medical student Max Talmoud (later changed to Talmey), who dined with the family on Sabbaths and published his own Personal Recollections of Einstein's Boyhood and Youth (Scripta Mathematica, New York, 1932, Vol. 1, pp. 68-71). According to Talmey:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Contrary to popular belief he [Albert] had an unusual predilection for mathematics, and because of this fact I gave him, after his promotion to the fourth grade, Spieker's textbook on geometry... I used to visit his home every week and whenever I came he delighted in showing me his solutions of new problems which he had found in the book [Spieker's geometry book …] and soon he had mastered the whole textbook. He then turned to higher mathematics [...]. His progress in mathematics was so rapid that very soon I was no longer a match for him in the subject.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Talmey also gave Einstein two books on physics, including Buchner's Force and Matter and Bernstein's illustrated series People's Books on Natural Science, &quot;&lt;em&gt;a work which I read with breathless attention&lt;/em&gt;&quot;, according to Einstein himself. It  also came to light that the electrical engineering enterprise of uncle Jakob and his father Hermann played a part in fostering Einstein's later fascination with fundamental physics. &lt;a href=&quot;https://www.nytimes.com/1984/02/14/science/einstein-revealed-as-brilliant-in-youth.html&quot; rel=&quot;noreferrer&quot;&gt;NYT Sullivan's Einstein Revealed as Brilliant in Youth (1984)&lt;/a&gt; references Stachel, the Princeton University Press editor of Einstein's papers, and records acquired from Swiss archives:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The more recent acquisitions also document, as never before, the scope of the electrical manufacturing activity of Einstein's father Hermann and of his more scientifically minded Uncle Jakob. It now appears that the Einstein enterprise was internationally recognized and considerably more innovative than previously suspected. Technical journals in Britain, France, Italy and Germany reveal that one of its devices, an electric meter, was patented in the United States, and that the Einstein company built the central power station for a Munich suburb... This has led Dr. Stachel to suspect that Einstein's upbringing in a home where manipulations of electricity and magnetism were a daily preoccupation helped set him on a road that led to his first relativity theory. The effect on the young Einstein, in his view, ''was much more significant than usually appreciated&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-07-19T09:08:36.393" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9830" PostTypeId="2" ParentId="9828" CreationDate="2019-07-19T10:18:07.763" Score="14" Body="&lt;p&gt;The idea that matter was made up of &quot;primordial&quot; particles, and currents in metals consisted of them was well established by then. Stoney suggested the name &quot;electron&quot; in 1891, and Lorentz's theory of electrons dates back to 1892, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Electron#Discovery_of_free_electrons_outside_matter&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia's timeline&lt;/a&gt;. Thomson himself mentioned Prout as the source for the view, and saw his task at hand more in establishing the &quot;nature&quot; of the particles, not that they are particles. And he is very clear on the hypothetico-deductive approach to it: he does not attempt to &quot;derive&quot; the particle hypothesis from experiments, he only confirms that the experimental results fit well with what is hypothesized. And not just his, but previous ones as well. Here is Thomson's own explanation from &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786449708621070&quot; rel=&quot;noreferrer&quot;&gt;Cathode Rays (1897)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As the cathode rays carry a charge of negative electricity, are deflected by an electrostatic force as if they were negatively electrified and are acted on by a magnetic force in just the way in which this force would act on a negatively electrified body moving along the path of these rays I can see no escape from the conclusion that they are charges of negative electricity carried by particles of matter. The question next arises, What are these particles? are they atoms or molecules, or matter in a still finer state of subdivision? To throw some light on this point, I have made a series of measurements of the ratio of the mass of these particles to the charge carried by it.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;His measurements established that &lt;span class=&quot;math-container&quot;&gt;$m/e$&lt;/span&gt; for different gases and pressures had the same value, and the paths of particles depended on the density of the medium. He then reasoned further:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The explanation which seems to me to account in the most simple and straightforward manner for the facts is founded on a view of the constitution of the chemical elements which has been favourably entertained by many chemists: this view is that the atoms of the different chemical elements are different aggregations of atoms of the same kind. In the form in which this hypothesis was enunciated by Prout, the atoms of the&#xA;  different elements were hydrogen atoms; in this precise form the hypothesis is not tenable, but if we substitute for hydrogen some unknown primordial substance X, there is nothing known which is inconsistent with this hypothesis, which is one that&#xA;  has been recently supported by Sir Norman Lockyer for reasons derived from the study of the stellar spectra.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;If, in the very intense electric field in the neighbourhood of the cathode, the molecules of the gas are dissociated and are split up, not into the ordinary chemical atoms, but into these primordial atoms, which we shall for brevity call corpuscles;&#xA;  and if these corpuscles are charged with electricity and projected from the cathode by the electric field, they would behave exactly like the cathode rays. They would evidently give a value of &lt;span class=&quot;math-container&quot;&gt;$m/e$&lt;/span&gt; which is independent of the nature of the gas and its pressure, for the carriers are the same whatever the gas may be; again, the mean free paths of these corpuscles would depend solely upon the density of the medium through&#xA;  which they pass.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-19T21:57:49.780" LastActivityDate="2019-07-19T21:57:49.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9831" PostTypeId="1" AcceptedAnswerId="9832" CreationDate="2019-07-19T16:15:27.803" Score="0" ViewCount="305" Body="&lt;p&gt;I am interested in the early proofs of the theorem. It is often called Cauchy mean value theorem, so perhaps Cauchy proved it first. In all the proofs that I have seen we construct a contrived function, applying Rolle's theorem to which works out just right. It seems like a miracle that one would come up with such a function. Perhaps the older proofs were more intuitive.&lt;/p&gt;&#xA;" OwnerUserId="9676" LastEditorUserId="55" LastEditDate="2019-07-20T05:34:51.593" LastActivityDate="2019-07-20T05:40:31.010" Title="Was there a more intuitive early proof of the generalized mean value theorem?" Tags="|mathematics|discoveries|calculus|real-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9832" PostTypeId="2" ParentId="9831" CreationDate="2019-07-20T05:28:07.270" Score="2" Body="&lt;p&gt;Cauchy was indeed the first, although his version was weaker than the modern one. An intuitive geometric interpretation of the theorem, along with a proof motivated by it, is due to Bonnet. A very well sourced and illustrated account of history of the mean value theorem is &lt;a href=&quot;https://abesenyei.web.elte.hu/publications/meanvalue.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A brief history of the mean value theorem by Besenyei&lt;/a&gt;, which describes many historically given proofs of it and its generalization, along with controversies surrounding some of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cauchy's proof appeared in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k62404287/f94.item.zoom&quot; rel=&quot;nofollow noreferrer&quot;&gt;Resume des Lecons sur le Calcul Infinitesimal (1823)&lt;/a&gt;. He does not use Rolle's theorem, but he also does not prove the modern form of it involving derivatives at an intermediate point. In Besenyei's reconstruction, we assume &lt;span class=&quot;math-container&quot;&gt;$g′ &amp;gt; 0$&lt;/span&gt;, and denote &lt;span class=&quot;math-container&quot;&gt;$A := \min_{[a,b]} f ′/g′$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$B := \max_{[a,b]} f ′/g′$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$f' − Ag'\geq0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Bg' − f'\geq0$&lt;/span&gt;, and hence &lt;span class=&quot;math-container&quot;&gt;$f − Ag$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Bg − f$&lt;/span&gt; are monotone increasing. Therefore,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f (b)−f (a)−A(g(b)−g(a)) ≥ 0\ \textrm{and}\ B(g(b)−g(a))−(f (b)−f (a)) ≥ 0.&#xA;$$&lt;/span&gt;&#xA;These are easily manipulated into the double inequality &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;A ≤ \frac{f (b) − f (a)}{g(b) − g(a)} ≤ B, &#xA;$$&lt;/span&gt;&#xA;which is Cauchy's version of the theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The modern version appeared in Serret's Cours de Calcul Infinitesimal (1868). The auxiliary function Serret introduced is directly inspired by Cauchy's proof, it is &lt;span class=&quot;math-container&quot;&gt;$\varphi(x) = (f(x) − Ag(x)) − (f(a) − Ag(a))$&lt;/span&gt;, but Rolle's theorem is not used. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bonnet gave an intuitive geometric interpretation of the  theorem, by treating &lt;span class=&quot;math-container&quot;&gt;$g(t),f(t)$&lt;/span&gt; as parametric equations of a plane curve. Then the theorem amounts to saying that the tangent to the curve segment at some point is parallel to the chord connecting its endpoints, &lt;span class=&quot;math-container&quot;&gt;$\big(g(a),f(a)\big)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\big(g(b),f(b)\big)$&lt;/span&gt;. The slope of the chord is &lt;span class=&quot;math-container&quot;&gt;$\frac{f(b)−f(a)}{g(b)−g(a)}$&lt;/span&gt;, which suggests introducing an auxiliary function&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;h(t) = f (t) − f (a) − \frac{f(b)−f(a)}{g(b)−g(a)}(g(t) − g(a)).&#xA;$$&lt;/span&gt;&#xA;Geometrically, this is the vertical distance between &lt;span class=&quot;math-container&quot;&gt;$(g(t),f(t))$&lt;/span&gt; and the point on the chord right above or below it. The generalized mean value theorem follows by applying Rolle's theorem to &lt;span class=&quot;math-container&quot;&gt;$h(t)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-20T05:40:31.010" LastActivityDate="2019-07-20T05:40:31.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9833" PostTypeId="1" AcceptedAnswerId="9841" CreationDate="2019-07-20T11:00:05.310" Score="3" ViewCount="177" Body="&lt;p&gt;I once heard a cautionary tale about the dangers of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Observer-expectancy_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;observer-expectancy effect&lt;/a&gt;. It was at least presented as a true example from the history of science, but I'm having trouble identifying the incident now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A team of experimental physicists announced they had discovered a previously unknown effect of some sort. The experiment involved some solid sample (a metal or crystal?) of something reasonably available. The setup somehow produced an image related to the sample or its surroundings (maybe a diffraction pattern?). When some additional stimulus or condition was added, the effect was said to cause a specific sort of pattern in the image.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other groups were interested, but attempts to replicate the experiment were inconsistent at best. The original group explained the experiment is delicate, and invited others to come to their lab to observe it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The demo went pretty normally: the hosts first showed and explained the apparatus. One researcher set things up for a few &quot;control&quot; runs, and another reported each time the pattern was not visible in the image. Then a few runs were done with the extra condition applied, and the viewer reported each time the pattern was visible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the demo concluded, one visiting skeptic revealed they had removed the sample before the measurements started! (My vague memory suggests possibly this skeptic is also known for other things in science or philosophy?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does this describe some actual incident, perhaps embellished a bit? If so, what was it?&lt;/p&gt;&#xA;" OwnerUserId="10253" LastActivityDate="2019-07-22T14:18:01.967" Title="What was this experiment tainted by the observer effect?" Tags="|physics|experimental-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9834" PostTypeId="1" CreationDate="2019-07-20T15:36:42.823" Score="5" ViewCount="142" Body="&lt;p&gt;In J. J. Thomson's paper (Phil. Mag. S. 5. Vol. 44. No. 269. Oct. 1897) concerned with cathode rays, Thomson writes, that the experiment by Perrin that supposedly proves that cathode rays are made of negatively charged particles is open to objection. I don't see why, here is his description of Perrin's experiment:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This has been proved to be the case&#xA;  by Perrln, who placed in front of a plane cathode two coaxial&#xA;  metallic cylinders which were insulated from each other: the&#xA;  outer of these cylinders was connected with the earth, the&#xA;  inner with a gold-leaf electroscope. These cylinders were&#xA;  closed except for two small holes, one in each cylinder, placed&#xA;  so that the cathode rays could pass through them into the&#xA;  inside of the inner cylinder. Perrin found that when the rays&#xA;  passed into the inner cylinder the electroscope received a&#xA;  charge of negative electricity, while no charge went to the&#xA;  electroscope when the rays were deflected by a magnet so&#xA;  as no longer to pass through the hole.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The way Thomson describes Perrin's experiment it seems like the cathode rays by default pass through the holes in the coaxial cylinders and fall onto the inner one, charging it. When the cathode ray is deflected by a magnet no charge is collected.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomson then describes his own experiment. To me the difference to Perrin's experiment is, that the cathode rays by default hit the glassbulb and the electroscope shows a charge only when Thomson directs the cathode ray onto the cylinder connected to the electroscope with a magnet.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I feel like both experiments are suited to show that cathode rays are made of negatively charged particles. Why is Perrin's experiment open to objection but Thomson's is not?&lt;/p&gt;&#xA;" OwnerUserId="10255" LastActivityDate="2023-08-03T04:46:01.343" Title="Difference between Perrin's and J. J. Thomson's experiment" Tags="|discoveries|electromagnetism|particle-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9835" PostTypeId="2" ParentId="7579" CreationDate="2019-07-20T19:06:09.133" Score="1" Body="&lt;p&gt;As for the figures on p.621, I think (but I'm not sure) that the problem which Gauss tries to solve in it is (in modern terms): &amp;quot;&lt;strong&gt;to find the self-inductance of a single circular loop&lt;/strong&gt;&amp;quot;. This is a very difficult problem in mathematical physics that is often omitted in textbooks on electromagnetism - who usually adhere to the simple problem of calculating the magnetic field and self-inductance of an infinite coil (that is made of an infinite number of nearly circular loops).&lt;/p&gt;&#xA;&lt;p&gt;What made me make this conclusion is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Of course, what Gauss himself writes in his verbal description of fragments 16-17.&lt;/li&gt;&#xA;&lt;li&gt;The formula which he arrives involves the term &lt;span class=&quot;math-container&quot;&gt;$\log(\frac{8r}{ae^2}) = \log(\frac{8r}{a} )-2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the radius of the circular coil and &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the radius of the wire which the loop is made of. This is exactly the log term in the modern result for self-inductance of a single circular coil: see this link &lt;a href=&quot;http://users.telenet.be/minnaert/blog/The_self_inductance_of_a_circular_loop.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://users.telenet.be/minnaert/blog/The_self_inductance_of_a_circular_loop.htm&lt;/a&gt;. What makes this thesis stronger is Gauss's words and figure in fragment 17, (although he switches there the notation for the radius of the wire from &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt;). In addition, the factor ''&lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;'' (from the modern result) is absent from Gauss's formula because he refers to the induced electric field, not to the induced voltage (so Gauss's formula is derived from the modern result from dividing by &lt;span class=&quot;math-container&quot;&gt;$2\pi R$&lt;/span&gt; ).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The only thing that is lacking for complete agreement is the difference between the numerical factors of the two formulas; a &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is missing from Gauss's. That might be the outcome of a different system of unit that Gauss used (maybe cgs?), but i'm not sure.&lt;/p&gt;&#xA;&lt;p&gt;i've added here a figure of p.621 from Gauss's nachlass:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/1XLai.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1XLai.png&quot; alt=&quot;Gauss's calculation of self-inductance of a single circular loop &quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Notice&lt;/strong&gt;: to prevent misunderstandings, the fact that the self-inductance of a single circular loop can be infinite (in the limit &lt;span class=&quot;math-container&quot;&gt;$r&amp;gt;&amp;gt;a$&lt;/span&gt;) doesn't contradict the finite result for the self-inductance of an infinite coil (&lt;span class=&quot;math-container&quot;&gt;$L = n^2\mu_0\pi r^2 $&lt;/span&gt;), because the derivation of magnetic field inside an infinite solenoid is valid onle when the distances between consecutive loops are comparable to the radius of the wire from which the coil is made of. Otherwise the assumption that the magnetic field inside the solenoid is uniform is simply incorrect. In the case where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; tends to zero, &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (the density of the loops) goes to infinity, so one gets consistent results.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some history&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;According to this article - &lt;a href=&quot;https://nvlpubs.nist.gov/nistpubs/bulletin/04/nbsbulletinv4n1p149_A2b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://nvlpubs.nist.gov/nistpubs/bulletin/04/nbsbulletinv4n1p149_A2b.pdf&lt;/a&gt; (&amp;quot;on the self-inductance of circles&amp;quot;), the first to publish a formula for the self-inductance of circle is Kirchhoff in 1864; his formula was the first in a series of more accurate results by Maxwell, Max Wien and others. So Gauss's (unpublished) formula can be interpreted as a first result in this direction. Gauss's fragment 17 contains several additional formulas, one of which is identical to Kirchhoff's.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update (26.07.2023)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;An accidental search in Maxwell's &amp;quot;treatise on electricity and magnetism&amp;quot; led to a surprising find. In p.345-346 of volume 2 of his treatise, Maxwell refers to this calculation of Gauss in the context of the following problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To find the form of a coil for which the coefficient of self-induction is in a maximum, when the the total length and thickness of the wire are given.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;, and states that the relevant result was first found by Gauss. Gauss's result states that for a coil with circular cross section, the maximal self-induction is obtained when the mean radius &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; of the coil is &lt;span class=&quot;math-container&quot;&gt;$3.22$&lt;/span&gt; times the radius &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; of the cross section of the coil (the condition for maximality is &lt;span class=&quot;math-container&quot;&gt;$a=3.22c$&lt;/span&gt;). I guess this result was particulary beneficial for electrical circuits designers, since it allowed the introduction of coil inductors with optimised self-induction.&lt;/p&gt;&#xA;&lt;p&gt;(note that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is not the arithmetic mean of the radiuses of the toroidal coil, but actually a kind of geometric mean of the radiuses; I am not exactly sure of the details right now. Anyway, I believe the use of the geometric mean was one of Gauss's main innovations in this calculation.)&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2023-07-26T22:11:55.380" LastActivityDate="2023-07-26T22:11:55.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9836" PostTypeId="1" CreationDate="2019-07-20T20:46:25.043" Score="2" ViewCount="253" Body="&lt;p&gt;I have read and I think that I agree with the idea that if we have to choose probability distribution for an unknown system then it is a good idea to choose a distribution that has the least bias.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I understand why Boltzmann chose to assume equal probability distribution because for me, intuitively, it means the least bias possible for the guess of the distribution.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But apparently, when we do statistical mechanics we usually maximize Gibbs entropy subject to some constraints on average values of macroscopic parameters. I am interested in the intuition behind how one would develop this approach and maybe how one would go from Boltzmann's hypothesis to Gibbs hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When I read similar questions online I get explanations using information theory and Shannon's entropy which are extremely interesting and give a lot of insight. But I am also wondering about how did people came up with this definition and how did they justify it before Shannon's papers.&lt;/p&gt;&#xA;" OwnerUserId="10256" LastEditorUserId="217" LastEditDate="2019-07-21T06:49:32.563" LastActivityDate="2022-12-10T19:23:37.103" Title="How did Gibbs discover Gibbs entropy?" Tags="|physics|entropy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9837" PostTypeId="2" ParentId="4" CreationDate="2019-07-20T22:12:58.757" Score="0" Body="&lt;p&gt;0 and the place value notion is required for good arithmetic. Without it modern Science, Engineering and Astronomy and an accurate Calendar and Navigational Charts would not have developed. An example is shown below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The importance of 0 is for accurate practical utility and not merely notional advantages of easy descriptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Bakhsh1ali Formula (from 4th century manuscript)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Suppose we wish to find sqrt(N), where N &gt; 0. We express &#xA;N as N = A^2 + b, where |b| is small in comparison &#xA;with A (as small as possible within the limits of easy guesswork&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By taking A=3 and b=1,the fraction 3 and 1/6 = 3.167; &#xA;computation, using only rational operations, i.e., +, -, &#xA;* , /). Using the linear approximation &#xA;sqrt(1+x)= 1+x/2 ,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for x ~ 0 (this is the tangent approximation) we get: &#xA;sqrt(A^2+b)= A+ b/2A.&#xA;This formula, which was known to the Babylonians, &#xA;yields fairly good approximations; e.g., for SQRT(10) it yields, &#xA;A=3 and b=1,the fraction 3 1⁄6 = 3.167 compared to  3:162. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the Bakhsh1ali formula goes further, by inserting an extra term. It states, in effect: &#xA;In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction. This is subtracted and will give the corrected root. &#xA;After one has deciphered this recipe, it turns to be equivalent to the following formula: &#xA; &#xA;(A^2 +b)^1/2 = A +b/2A - (b/2A)^2 / 2(A +b/2A).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, if N = 11 we may take A = 3 and b = 2. The formula then gives: &#xA; 3 + 1/3 - 1/9 /2(3+1/3) =3+1/3-1/60 = 3.31667&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Compare this with the true value: (11)^1/2 =  3.31662. We see that though b is far from being `small' in comparison with A, we have still got four decimal place accuracy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Choosing A=3.3 and b = 11- 3.3^2 the accuracy jumps up to &#xA;79201/23880 which is 3.316624790, accurate to 9 decimals.&#xA;from&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ias.ac.in/article/fulltext/reso/017/09/0884-0894&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ias.ac.in/article/fulltext/reso/017/09/0884-0894&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-07-21T17:22:01.407" LastActivityDate="2019-07-21T17:22:01.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9838" PostTypeId="1" CreationDate="2019-07-21T04:43:13.843" Score="0" ViewCount="254" Body="&lt;p&gt;I have seen this mentioned on the interwebs a few times. people have mentioned that some Greek thinkers and Islamic astronomers came up with heliocentrism before Copernicus and that Copernicus copied Al-Haytham's model.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it Eurocentrism to give credit solely to Copernicus?&lt;/p&gt;&#xA;" OwnerUserId="6459" LastActivityDate="2019-07-22T11:58:10.997" Title="Does Heliocentrism predate Copernicus?" Tags="|mathematics|physics|astronomy|philosophy-of-science|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9839" PostTypeId="2" ParentId="9838" CreationDate="2019-07-21T06:07:44.430" Score="0" Body="&lt;p&gt;Aristarchus of Samos is credited with the first heliocentric model. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aristarchus_of_Samos&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Aristarchus_of_Samos&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The scientists of that day were capable of sophisticated reasoning. They pointed out that if the earth moves around the sun, then we should see parallax in the stars six months apart. But we see no parallax. Therefore either the stars must be unimaginably far away, which is unlikely; or else heliocentrism must be wrong. Of course today we know that the stars actually are so far away as to not exhibit parallax.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a fantastic website that recounts the history of our ideas of the solar system. Well worth reading. It's called The Great Ptolemaic Smackdown. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://tofspot.blogspot.com/2013/10/the-great-ptolemaic-smackdown-table-of.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://tofspot.blogspot.com/2013/10/the-great-ptolemaic-smackdown-table-of.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="340" LastActivityDate="2019-07-21T06:07:44.430" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9840" PostTypeId="2" ParentId="9838" CreationDate="2019-07-21T06:17:56.157" Score="2" Body="&lt;p&gt;There is an evidence that heliocentrism was proposed by some ancient Greek philosophers, in particular Aristarchus of Samos. His work did not survive, and all we know about it is a brief mentioning in one work of Archimedes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning Ibn al-Haytham, Wikipedia mentions his works where he criticized Ptolemy,&#xA;but does not mention that he ever advocated heliocentrism. If you have a source which says this, please give a reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is highly unlikely that Copernicus could read Ibn al-Haytham. (I doubt that he could read Arabic and that al-Haytham was translated into Latin). But Copernicus mentions his ancient Greek predecessors. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Concerning &quot;giving credit&quot; this is a question of opinion.&#xA;On my opinion, crediting Copernicus has nothing to do with &quot;eurocentrism&quot;. &#xA;(Aristarchus was also a European, and he is given some credit, though his work did not survive and we know almost nothing about it. About al-Haytham, I have never heard that he developed a heliocentric system. Please give a source if you know one.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-07-21T16:52:24.577" LastActivityDate="2019-07-21T16:52:24.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9841" PostTypeId="2" ParentId="9833" CreationDate="2019-07-21T15:54:40.703" Score="5" Body="&lt;p&gt;This is the story of the &quot;discovery&quot; of &quot;N Rays&quot;, as described in a &lt;a href=&quot;https://en.wikipedia.org/wiki/N_ray&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;. The gist of the story told there is as follows:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1903 the physicist Prosper-René Blondlot (1849 –  1930) thought he found something similar to the then-new X rays but not the same.  Since the phenomenon  could not be   duplicated in others' laboratories the community smelled a rat.  Robert W. Wood (1868-1955) attended a demonstration in Blodot's lab. After the effect was once again exhibited, Wood revealed that he had surreptitiously removed an important part of  Blondot's apparatus, so what was exhibited was not the result of the experiment as described.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The effect was the result of a kind of self-delusion.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2019-07-22T14:18:01.967" LastActivityDate="2019-07-22T14:18:01.967" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9843" PostTypeId="2" ParentId="1884" CreationDate="2019-07-21T18:36:30.067" Score="0" Body="&lt;p&gt;The base 60 system of the Babylonians requires 360 degrees for a full circle as a matter of ease of handling. Anything else would be inconvenient.In the base 10 system 1,2,5,10...sequence is most useful. Similarly, in the sexagesimal system, 1,2,6,36,60..will be most useful.&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-07-22T00:34:29.627" LastActivityDate="2019-07-22T00:34:29.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9844" PostTypeId="2" ParentId="2588" CreationDate="2019-07-21T21:16:46.713" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa_numeration&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Āryabhaṭa_numeration&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aryabhata used the opposite way as shown in the above article.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example:  299,792,458&lt;/p&gt;&#xA;&#xA;&lt;p&gt;10^0 10^1   10^2 10^3   10^4 10^5   10^6 10&amp;amp;7   10^8&lt;/p&gt;&#xA;&#xA;&lt;p&gt;85,     42,     97,     99,    2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;जल  घिनि    झुशु    झृसृ    खॢ&lt;/p&gt;&#xA;&#xA;&lt;p&gt;ja-la   ghi-ni  jhu-śu  jhṛ-sṛ  khḷ&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He used alphabets to represent numbers, most useful for brevity and poetry. Most math rules are given as SUTRAS!&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-07-21T21:16:46.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9845" PostTypeId="1" CreationDate="2019-07-21T22:25:26.733" Score="0" ViewCount="809" Body="&lt;p&gt;I am recently learning about teaching the history of mathematics. Gradually, impressions about mathematicians in different countries develops in my mind. I just want to ask, whether it is helpful to think in this way:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;English mathematicians are very likely to be good physicists as well. We have Newton. We have the &quot;Merton Calculators&quot;. Although many other mathematicians (such as Bernoulli) are good physicists as well, I just feel that British math are more prone to be physics. Even today, staff like quantum mechanics is important for many math students.&lt;/li&gt;&#xA;&lt;li&gt;French mathematicians are good at pure and rigorous things. For example, we have Cauchy and Lagrange who made analysis more rigorous. The name &quot;Lagrange&quot; also appears in abstract algebra.&lt;/li&gt;&#xA;&lt;li&gt;I don't know much about German mathematicians, but people like Hilbert and Cantor are the ones who introduces crazy modern ideas in math. (Also Poincare, Gödel, etc!) Germans seem to play a huge role in more modern math. Of course, we also have great analysist like Karl Weierstrass.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Are there any other &quot;general trends&quot; that we can find for mathematicians in different countries (doesn't have to be limited to the three above)? &lt;strong&gt;Are those trends just appearing in history books, or are they still existing in mathematicians in different countries nowadays?&lt;/strong&gt; &lt;/p&gt;&#xA;" OwnerUserId="7088" LastActivityDate="2019-07-21T22:25:26.733" Title="Is there any general difference between British, French and German mathematicians?" Tags="|mathematicians|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9847" PostTypeId="2" ParentId="9838" CreationDate="2019-07-22T11:58:10.997" Score="0" Body="&lt;p&gt;As the other answers have stated, the heliocentric model goes back to Aristarchus, but it had no followers in the Middle Ages, whether in Christendom or in Islam. Some Indian (e.g. Aryabhatta) and Muslim (e.g. al-Biruni) astronomers did consider the possibility that the Earth rotates daily on its axis, but they did not deny that the Earth stands in the centre of the cosmos.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2019-07-22T11:58:10.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9849" PostTypeId="1" CreationDate="2019-07-23T12:43:01.390" Score="3" ViewCount="3098" Body="&lt;p&gt;I graduated high school in 1980, and as a second career, am helping HS students with their math. I just finished my 6th year of this, and have just one question about the change noted in the title. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I recall Septagon, Octagon, Nonagon, Decagon, being figures with sides numbering 7,8,9,10. As a child, I noted that these prefixes reflected those of the last 4 months on the calendar. From this a teacher explained that the calendar months got shifted around, and that at one time September thru December were, in, fact, months 7-10. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few years ago, I ran into a seven sided figure and a student telling me it was a Heptagon. I do understand that Sept is Latin and Hept is Greek, but would like to know when and more important, why, this change was made. &lt;/p&gt;&#xA;" OwnerUserId="2390" LastActivityDate="2019-07-23T13:13:18.090" Title="When/Why did Septagon change to Heptagon?" Tags="|reference-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9850" PostTypeId="2" ParentId="9849" CreationDate="2019-07-23T13:01:41.627" Score="5" Body="&lt;p&gt;It is only a recent mistake.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See : Anthony Lo Bello, &lt;a href=&quot;https://books.google.it/books?id=LfEEAQAAQBAJ&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Origins of mathematical words (John Hopkins UP, 2013)&lt;/a&gt;,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Septagon&lt;/em&gt; This vox nullius is a learned mistake for heptagon. People of some education make a certain type of error not committed by the multitude, and this word is an example of one such mistake, viz., the confusion of languages. Knowing that they need a foreign word for seven, they take the familiar Latin word instead of the required but unfamiliar Greek word and concoct the hybrid septagon on the analogy of the common term pentagon. Related absurdities are automobile, homosexual, neuroscience, sociopath, television, etc. Any such word may be immediately identified as modern. The word septagon appears in&#xA;  Herstein’s &lt;em&gt;Topics in Algebra&lt;/em&gt; (first edition, 1964) on pages 190 and 341, an example of a mathematical Homer nodding. It was corrected to heptagon on page 242 of &lt;em&gt;Abstract Algebra&lt;/em&gt;, Macmillan, New York, 1986.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://jeff560.tripod.com/h.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tripod : Heptagon&lt;/a&gt; : In 1551 in &lt;em&gt;Pathway to Knowledge&lt;/em&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_Recorde&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Recorde&lt;/a&gt; used &lt;em&gt;septangle&lt;/em&gt;. &lt;em&gt;Heptagon&lt;/em&gt; appears in English in 1570 in &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Billingsley&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir Henry Billingsley&lt;/a&gt;’s translation of Euclid’s &lt;em&gt;Elements&lt;/em&gt;. &lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2019-07-23T13:13:18.090" LastActivityDate="2019-07-23T13:13:18.090" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9851" PostTypeId="1" AcceptedAnswerId="9854" CreationDate="2019-07-23T18:22:37.743" Score="7" ViewCount="893" Body="&lt;p&gt;Haar measure is a well-known concept in measure theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am looking for a good reference on the history of Haar measure.&lt;/p&gt;&#xA;" OwnerUserId="10265" LastActivityDate="2019-07-23T20:41:38.573" Title="On the history of Haar measure" Tags="|mathematics|topology|group-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9852" PostTypeId="2" ParentId="9851" CreationDate="2019-07-23T18:31:15.040" Score="5" Body="&lt;p&gt;Try these references:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Section 7.5 of &lt;em&gt;History of Topology&lt;/em&gt;, edited by I. M. James.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Section 2.2 of the chapter &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-94-017-0470-0_5&quot; rel=&quot;noreferrer&quot;&gt;&quot;Topological Features of Topological Groups&quot;&lt;/a&gt; in &lt;em&gt;Handbook of the History of General Topology&lt;/em&gt;, volume 3, edited by C.E. All and R. Lowen&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="186" LastActivityDate="2019-07-23T18:31:15.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9853" PostTypeId="1" AcceptedAnswerId="9857" CreationDate="2019-07-23T19:22:02.117" Score="3" ViewCount="743" Body="&lt;p&gt;I'd like to get an explanation why Maxwell chose the name &quot;displacement current&quot; for the term &lt;span class=&quot;math-container&quot;&gt;$\mu_0\epsilon_0\frac{\partial E}{\partial t}$&lt;/span&gt; he added to Ampere's law (Maxwell's correction). I'm familiar with the basic inconsistencies in electromagnetic theory that led Maxwell to indroduce it (like the charging capacitor example), but i'm note sure i truly understand the naming of the term. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've a kind of an intuitive feeling that Maxwell meant in his name that the &quot;displacement current&quot; is a kind of continuation of the conduction current (like a displacement of motion of actual charges into the rest of space), somewhat in a similar vein to &quot;analytic continuation&quot; of functions. But these are just conjectures of mine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, can anyone explain why Mawxell chose this name? and where in his writings he explains it? &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-06-04T18:55:25.980" LastActivityDate="2020-06-04T18:55:25.980" Title="What is the idea behind Maxwell's &quot;displacement current&quot; in electromagnetism?" Tags="|physics|electromagnetism|maxwell|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9854" PostTypeId="2" ParentId="9851" CreationDate="2019-07-23T20:41:38.573" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4899-0399-0_9&quot; rel=&quot;noreferrer&quot;&gt;Cohn himself&lt;/a&gt; recommends historical notes at the end of sections 15, 16 of &lt;a href=&quot;https://books.google.com/books/about/Abstract_Harmonic_Analysis.html?id=innmCAAAQBAJ&quot; rel=&quot;noreferrer&quot;&gt;Abstract Harmonic Analysis by Hewitt and Ross, volume 1&lt;/a&gt;. Here is an excerpt:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Invariant integration on one or another special class of groups has&#xA;  long been known and used. A detailed computation of the invariant&#xA;  integral on &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{SD}(n)$&lt;/span&gt; was given in 1897 by HURWITZ [1]. SCHUR and&#xA;  FROBEKius in the years 1900-1920 made frequent use of averages over finite groups; for references, see the notes in WEYL [3]. SCHUR in [ 1]&#xA;  computed and applied intensively the invariant integrals for &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{SD}(n)$&lt;/span&gt;&#xA;  and &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{D}(n)$&lt;/span&gt;. WEYL in [1] computed the invariant integrals for &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{U}(n)$&lt;/span&gt;,&#xA;  &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{SD}(n)$&lt;/span&gt;, the unitary subgroup of the symplectic group, and [more or less&#xA;  explicitly] for certain other compact Lie groups. WEYL and PETER in [1]&#xA;  showed the existence of an invariant integral for any compact Lie group.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The decisive step in founding modern harmonic analysis was taken by&#xA;  A. HAAR [3] in 1933. He proved directly the existence [but not the&#xA;  uniqueness] of left Haar measure on a locally compact group with a&#xA;  countable open basis. His construction was reformulated in t erms of&#xA;  linear functionals and extended to arbitrary locally compact groups by&#xA;  A. WEIL [1], [2], and [4], pp. 33 -38. KAKUTANI [2] pointed out also&#xA;  that HAAR's construction can be extended to all locally compact groups.&#xA;  Theorem ( 15. S) as stated is thus due to WEIL. The proof we present&#xA;  is due to H. CARTAN [1].&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;For an arbitrary compact group G, VON NEUMANN [5] proved the&#xA;  existence and uniqueness of the Haar integral, as well as its two-sided&#xA;  and inversion invariance. In [6], VON NEUMANN proved the uniqueness&#xA;  of left Haar measure for locally compact G with a countable open basis;&#xA;  a special case was also established by Sz.-NAGY [1 ]. WEIL [ 4], pp. 37-38,&#xA;  proved the uniqueness of the left Haar integral for all locally compact&#xA;  groups.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-07-23T20:41:38.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9855" PostTypeId="2" ParentId="8012" CreationDate="2019-07-23T22:26:36.803" Score="0" Body="&lt;p&gt;Inner planets in their wanderings cover only a small region of the Zodiac (or Kalachakra) but the outer planets move all around the Zodiac. They can't be observed all the time but only before Sunrise or after sunset. They move with the Sun plus or minus an angle smallest for Mercury and larger for Venus (called evening star for obvious reason). The Sun itself travels a full orbit per year along the Zodiac.&lt;/p&gt;&#xA;" OwnerUserId="9899" LastEditorUserId="9899" LastEditDate="2019-07-23T22:40:38.017" LastActivityDate="2019-07-23T22:40:38.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9857" PostTypeId="2" ParentId="9853" CreationDate="2019-07-24T04:19:20.030" Score="6" Body="&lt;p&gt;Maxwell did not think of the displacement current as a continuation of the conduction current, and his motivations are generally entwined with his mechanical models of ether. But the naming itself comes from the analogy between the added term and the polarization field induced by electric field in a dielectric, caused by atomic charges slightly separating and creating electric dipoles. Here is Maxwell in his own words:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Electromotive force acting on a dielectric produces a state of polarization of its parts similar in distribution to the polarity of the particles iron under the influence of a magnet, and, like the magnetic polarization, capable of being described as a state in which every particle has poles in opposite conditions... The effect of this action on the whole dielectric mass is to produce general displacement of the electricity in a certain direction.&lt;/em&gt;&amp;quot; [Maxwell, SP, 1, 496-498; quoted from Whittaker, A History of the Theories of Aether and Electricity.]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Siegel gives a detailed account of Maxwell's thought process and modern controversies surrounding it in &lt;a href=&quot;https://www.jstor.org/stable/27757576&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origin of the Displacement Current&lt;/a&gt;. He explains the analogy as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Going back to the charging capacitor circuit, consider the case where there is a block of dielectric material in the space between the capacitor plates, as in figure le. In this situation, the electric field &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; attributable to the accumulating charge on the plates would tend to polarize the molecules of the dielectric as indicated, so that the current attributable to the growing polarization would be in the proper direction to close the loop... This physical argument is only suggestive, rather than determinative, as the value of the polarization current would in all cases be less than is required to completely close the loop, and there would be no polarization in vacuo.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The mathematical paradigm for the treatment of polarized media was Simeon Denis Poisson's treatment of the magnetic case; this analysis had been taken over for the electrical case by Ottaviano Mossotti, whom Maxwell cited explicitly; and standing behind Maxwell's physical interpretation of all of this was Faraday's work on dielectrics, also explicitly cited by Maxwell. The equation that Maxwell wrote for the displacement represented an amalgam of these influences, as well as the requirements of his own specific situation in &amp;quot;Physical lines&amp;quot;.&lt;/em&gt; [Reference to On Physical Lines of Force (1861-62).]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It should be added that the motivation often cited in modern texts for adding the displacement current term, namely to reconcile the divergence-free nature of &lt;span class=&quot;math-container&quot;&gt;$J=\textrm{curl}\,H$&lt;/span&gt; (because &lt;span class=&quot;math-container&quot;&gt;$\textrm{curl}$&lt;/span&gt; is divergence-free) with open circuits, where electric charge is accumulating, is not supported by the historical record. Maxwell himself discusses the solenoidal current only in later works. Originally, he talked in terms of molecular vortices in the pervading magneto-electric medium of ether, and introduced a pattern of elastic strain in the neighborhood of an electric charge, a mechanical interpretation of the electrostatic field. In other words, he was thinking in terms of elastic theories of ether, popular in the 19th century, and sought a mechanical unification of electromagnetic and optical phenomena. In this context the terminology of &amp;quot;displacement&amp;quot; comes naturally. But it would be hard to bring up that context in modern etherless expositions, and invoking a purely mathematical inconsistency is far more expedient.&lt;/p&gt;&#xA;&lt;p&gt;The magneto-electric medium was pictured as consisting of near spherical (perhaps, dodecahedral) rotating cells (molecular vortices) filled with a fluid substance, later replaced by a solid elastic material. The cell walls consisted of a layer of small, spherical particles. When the magnetic field was nonuniform, adjacent vortices rotated with slightly different angular velocities, and the small particles were carried along with the more swiftly moving vortex. Maxwell was faithful to Faraday's ideas about the &amp;quot;primacy of the field&amp;quot;, and considered charges and currents to be objectionable (because unobservable) and derivative. The causality for him is reversed compared to the modern picture (initiated by Lorentz): the equations enable us to &amp;quot;&lt;em&gt;deduce the distribution of the currents of electricity whenever we know the values of &lt;span class=&quot;math-container&quot;&gt;$\alpha,\beta,\gamma$&lt;/span&gt;, the magnetic intensities&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Maxwell calculated that the motions of the small particles are related to the spatial derivatives of the magnetic field. Identifying the small particles as electricity, and their density as the electric current, he interpreted the original Ampere's law as a relation between the (rigid) rotations of the vortices and the motions of the small particles. The role of the added term was to make the vortices deformable, to &amp;quot;&lt;em&gt;correct the equation... of electric currents for the effect due to the elasticity of the medium&lt;/em&gt;&amp;quot;, as Maxwell explained in Physical Lines.&lt;/p&gt;&#xA;&lt;p&gt;That the substance of the vortices was at the same time assumed to be a fluid (to represent magnetic forces) and an elastic solid (to represent their rotations) was one of those contradictory properties assigned to ether that led to Lorentz's ghostly version of it, and eventually its elimination. Siegel gives detailed illustrated explanations of Maxwell's mechanical models, and many references to the secondary literature.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-07-25T01:24:22.177" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9859" PostTypeId="2" ParentId="375" CreationDate="2019-07-24T10:22:47.990" Score="0" Body="&lt;p&gt;Another answer is in&#xA;&lt;a href=&quot;https://astronomy.stackexchange.com/questions/10213/naming-of-the-planets-of-the-solar-system&quot;&gt;https://astronomy.stackexchange.com/questions/10213/naming-of-the-planets-of-the-solar-system&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Except for Venus the order is with increasing distance from the Sun. Why Venus is the odd man out is not clear.The names are in Sanskrit and mean something in Sanskrit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Chandra is the Moon&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Surya is the Sun&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mangala is Mars&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bhuda is Mercury&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Jupiter is Guru (weighty in Sanskrit as its size was known from Surya Siddhanta)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Shukra is Venus&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Saturn is Shanischara (slow-moving in Sanskrit)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The week days are ordered in that order of increasing distances except for Venus (I do not know why.) All planets are considered gods!&lt;/p&gt;&#xA;" OwnerUserId="9899" LastActivityDate="2019-07-24T10:22:47.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9860" PostTypeId="1" AcceptedAnswerId="9863" CreationDate="2019-07-24T17:32:19.313" Score="9" ViewCount="1125" Body="&lt;p&gt;In connection to the question &quot;&lt;a href=&quot;https://hsm.stackexchange.com/questions/5287/who-invented-the-gradient&quot;&gt;Who invented the gradient?&lt;/a&gt;&quot;, I want to know who invented the gradient descent algorithm?&lt;/p&gt;&#xA;" OwnerUserId="10272" LastEditorUserId="10106" LastEditDate="2019-07-24T18:06:34.353" LastActivityDate="2019-07-25T18:20:47.643" Title="Who invented the gradient descent algorithm?" Tags="|mathematics|calculus|linear-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9863" PostTypeId="2" ParentId="9860" CreationDate="2019-07-25T06:51:46.810" Score="9" Body="&lt;p&gt;The &quot;gradient descent&quot; algorithm was invented before the gradient. It is described in equivalent form by Cauchy in a 3-page paper in Comptes Rendus, &lt;a href=&quot;http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_CAUCHY_1_10_399_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Méthode générale pour la résolution des systèmes d'équations simultanées (1847)&lt;/a&gt;. Here is an &lt;a href=&quot;https://cs.uwaterloo.ca/~y328yu/classics/cauchy-en.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation&lt;/a&gt;. A good secondary source is &lt;a href=&quot;https://www.math.uni-bielefeld.de/documenta/vol-ismp/40_lemarechal-claude.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy and the Gradient Method by Lemaréchal&lt;/a&gt;, who comments:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Cauchy is motivated by astronomic calculations which, as everybody knows,&#xA;  are normally very voluminous. To compute the orbit of a heavenly body, he&#xA;  wants to solve &quot;not the differential equations, but the [algebraic] equations rep-&#xA;  resenting the motion of this body, taking as unknowns the elements of the orbit&#xA;  themselves&quot;. To solve a system of equations in those days, &quot;one ordinarily starts by reducing them to a single one by successive eliminations, to eventually solve for good the resulting equation, if possible. But it is important to observe that 1◦ in many&#xA;  cases, the elimination cannot be performed in any way; 2◦ the resulting equation&#xA;  is usually very complicated, even though the given equations are rather simple&quot;.&#xA;  Something else is wanted.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Instead of the gradient vector Cauchy simply works with the partial derivatives &lt;span class=&quot;math-container&quot;&gt;$X=f'_x,Y=f'_y,Z=f'_z,...$&lt;/span&gt;. He takes small &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;, sets up the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Theta=f(X-\theta x, Y-\theta y, Z-\theta z,...),$$&lt;/span&gt;&#xA;and, in one of the variants, recommends finding &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$\Theta'_\theta=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] &lt;em&gt;One iteration of the gradient method is thus stated, with two variants: (2) (Armijo-type line-search) or (3) (steepest descent)... Convergence is just sloppily mentioned... Cauchy does not seem to believe that the method always finds a solution; yet, he also seems to hope it: see the excerpt of foot&#xA;  note 4. Anyway a simple picture reveals that the least-squares function in (4) may display positive local minima, playing the role of “parasitic” solutions.&#xA;  On the other hand, he seems convinced that, being decreasing, the sequence of&#xA;  u-values has to converge to a (local) minimum, or at least a stationary point.&#xA;  Thus, the above excerpt is fairly interesting, coming from a mathematician&#xA;  among the most rigorous of his century. Admittedly, Cauchy has not given&#xA;  deep thought to the problem: &quot;I’ll restrict myself here to outlining the principles&#xA;  underlying [my method], with the intention to come again over the same subject,&#xA;  in a paper to follow&quot;. However, the “paper to follow” does not seem to exist.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also related post &lt;a href=&quot;https://stats.stackexchange.com/q/313681/250912&quot;&gt;Who invented stochastic gradient descent? on Cross Validated&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-25T18:20:47.643" LastActivityDate="2019-07-25T18:20:47.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9864" PostTypeId="1" CreationDate="2019-07-25T08:40:45.000" Score="2" ViewCount="158" Body="&lt;p&gt;In ODE, in Frobenius method, there's an equation called &quot;Indicial Equation.&quot; Is there any particular contextual/historical reason that it is named so?&lt;/p&gt;&#xA;" OwnerUserId="10274" LastEditorUserId="10106" LastEditDate="2019-07-30T05:35:12.840" LastActivityDate="2019-07-30T05:35:12.840" Title="Why was Indicial equations named so?" Tags="|mathematics|terminology|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9865" PostTypeId="2" ParentId="9864" CreationDate="2019-07-25T09:36:02.380" Score="2" Body="&lt;p&gt;&quot;Indicial&quot; is derived from &quot;indices&quot;, in this case indices of the coefficients. The Latin word &lt;em&gt;indicium&lt;/em&gt; originally meant a sign, indicator. &quot;Indicial equation&quot; is most often used in the context of solving equations of the form &lt;span class=&quot;math-container&quot;&gt;$a_nx^n\frac{d^n}{dx^n}+\dots+a_1x\frac{dy}{dx}+a_0y=0$&lt;/span&gt;. A substitution, known already to Euler, reduces it to a linear equation with constant coefficients, and the characteristic equation of the latter is called the &quot;indicial equation&quot; of the original. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euler, who gave a general method for solving these types of equations, used neither the term nor the notation. Most notably, his coefficients did not have the subscripts, like &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt;, making the arguments somewhat cumbersome. In  modern notation, the indicial equation is formed directly from the indexed coefficients. In &lt;a href=&quot;https://www.jstor.org/stable/10.4169/college.math.j.47.3.191&quot; rel=&quot;nofollow noreferrer&quot;&gt;What′s in a Name: Why Cauchy and Euler Share the Cauchy–Euler Equation&lt;/a&gt; Parker quotes Sandifer's view of the significance of that (from his Some Facets of Euler’s work on Series, in the volume &lt;a href=&quot;https://books.google.com/books/about/Leonhard_Euler.html?id=YxdEDwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonhard Euler: Life, Work and Legacy&lt;/a&gt; volume):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Indeed, C. E. Sandifer noticed [19, p. 281] &quot;...the many circumlocutions of notation Euler used to do what modern mathematicians&#xA;  would do with indicial notations like subscripts and superscripts. This&#xA;  will demonstrate how the lack of a convenient and powerful notation distorted certain kinds of mathematical developments of the era...&quot;.&#xA;  The argument above&lt;/em&gt; [i.e. Euler's solution] &lt;em&gt;could be simplified by using subscripts in the coefficients of the differential equation. Sandifer claims that Euler’s lack of indices hindered the development of other areas such as Bernoulli numbers, Gamma functions, and approximations of integrals.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-07-25T18:18:14.220" LastActivityDate="2019-07-25T18:18:14.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9866" PostTypeId="1" CreationDate="2019-07-25T18:36:12.990" Score="4" ViewCount="216" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Risch_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Risch algorithm&lt;/a&gt; for computing symbolic integrals was developed by Robert Risch in the 1968-70 time frame.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the &lt;a href=&quot;https://de.wikipedia.org/wiki/Robert_Risch&quot; rel=&quot;nofollow noreferrer&quot;&gt;German Wikipedia article&lt;/a&gt;, I know that Risch was awarded a Ph.D. by U.C. Berkeley in 1968, worked for IBM at their T.J. Watson research center at about that time, and was a member of the Institute for Advanced Studies from 1970-1972.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, nothing.  &lt;a href=&quot;https://www.researchgate.net/scientific-contributions/2015301959_Robert_H_Risch&quot; rel=&quot;nofollow noreferrer&quot;&gt;researchgate.net&lt;/a&gt; lists only two papers that he authored, the most recent in 1970.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anybody have any further information about Robert Risch?&lt;/p&gt;&#xA;" OwnerUserId="10276" LastEditorUserId="55" LastEditDate="2019-07-26T07:03:04.310" LastActivityDate="2019-07-26T07:03:04.310" Title="Is there a biography of Robert Risch?" Tags="|mathematicians|biographical-details|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9867" PostTypeId="2" ParentId="2143" CreationDate="2019-07-25T23:29:47.950" Score="3" Body="&lt;p&gt;Aristotle provided a number of arguments for the sphericity of the world in his writings. For example, (and this one has already been mentioned but I don't believe with any attribution to Aristotle) in &lt;em&gt;On the Heavens&lt;/em&gt;, he noted that during a lunar eclipse the shadow of the earth is circular in shape.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also in &lt;em&gt;On the Heavens&lt;/em&gt;, Aristotle observed that as one moves to the south, new constellations become visible. Furthermore, a relatively short trip to the north also reveals new stars. So, in addition to concluding that the earth is spherical, he also ascertained that it cannot be of great size.   &lt;/p&gt;&#xA;" OwnerUserId="18820" LastActivityDate="2019-07-25T23:29:47.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9871" PostTypeId="1" CreationDate="2019-07-27T03:33:53.773" Score="4" ViewCount="320" Body="&lt;p&gt;Have we figured out how this series was derived by Ramanjuan?&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{\pi}=\frac{\sqrt{8}}{9801}\sum_{n=0}^{\infty}\frac{(4n)!}{(n!)^4}\frac{26390n+1103}{396^{4n}}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="10239" LastEditorUserId="62" LastEditDate="2019-08-08T21:31:12.797" LastActivityDate="2019-08-08T21:31:12.797" Title="How was this equation for pi derived by Ramanujan?" Tags="|mathematics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9872" PostTypeId="2" ParentId="2875" CreationDate="2019-07-27T13:12:03.467" Score="1" Body="&lt;p&gt;Ramanujan was mainly self-taught, he took good use of this book whose name is &lt;a href=&quot;https://archive.org/details/asynopsiselemen00carrgoog/page/n7&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;A Synopsis of Elementary Results in Pure and Applied Mathematics&quot;&lt;/a&gt; by George Shoobridge Carr, and was well inspired by it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Read the book from the link below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/asynopsiselemen00carrgoog/page/n7&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/asynopsiselemen00carrgoog/page/n7&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10283" LastActivityDate="2019-07-27T13:12:03.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9873" PostTypeId="2" ParentId="3227" CreationDate="2019-07-28T04:18:17.347" Score="-1" Body="&lt;p&gt;Well the original post seems to be assuming Einstein is lying. I see no reason why he should lie about his original proof. Strogatz/Schroeder had numerous mutual associates to Einstein through Erst Straus; such communication would not have occurred without confirming with Einstein first. Not all of Einstein's letters after 1930 have been vetted yet. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's use of differential equations on curved manifolds should give him the benefit of the doubt. We likely have more evidence that he completed such a proof than we do of the true origins of the Pythagorean Theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He taught himself differential and integral calculus at 14. He would've made a brilliant geometer if he was interested in it. It's very unlikely that such a proof would have survived into adulthood, but the proof cited above has the hallmarks of Einstein's cognitive elegance. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The story referenced in his autobiographical notes is significant because he consciously (or subconsciously) left out many of his seminal contributions to quantum theory in his autobiographical notes (he doesn't even mention his prescient work on spontaneous and stimulated emission). I have a hard time believing Einstein would just make something like this up - and again, those differential equations I have to struggle with in GR suggest his command of geometry was exceptional. GR is hard sledging, his geometry chops were top notch. &lt;/p&gt;&#xA;" OwnerUserId="4315" LastActivityDate="2019-07-28T04:18:17.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9874" PostTypeId="2" ParentId="7503" CreationDate="2019-07-28T11:26:07.633" Score="1" Body="&lt;p&gt;As for the third question (on &quot;sympathetic vibrations&quot;), p. 127-136 of Schaefer's treatise are the best source on this. According to these pages, much of Gauss's experimental investigations on magnetism was concerned with the determination of the magnetic moment of needles by measuring their period of oscillation; his main objective was to use such precise measurements to determine the constants of the different versions of his magnetometer (unifiliar and bifilar) and thus help measure the components of the earth's magnetic field.  In the course of these investigations Gauss attempted to solve two &quot;core&quot; problems:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The damping of oscillation of magnetic needle in the presence of a coil around it; the movement of the needle creates changing nagnetic field and thus induces current in the coil so it losses energy as a result (see Gauss's werke, volume 11, p.67-68 - &quot;calculation of damped movements of the magnetic needle&quot; ). &lt;/li&gt;&#xA;&lt;li&gt;The problem of mixed oscillations - Gauss attempted to describe the &quot;collective oscillation&quot; of several needles placed next to each other (see Gauss's werke, volume 11, p.63-65) . This is the context at which he discovered &quot;sympathetic vibrations&quot;. There are additional details on this in his letters. Schaefer comments on this:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gauss hat in diesen Versuchen die Theorie der erzwungenen Schwingungen eines Massenpunktes auf die induzierten Schwingungen des Magnetometers gewendet; anscheinend ist er der erste oder jedenfalls einer der ersten, der überhaupt das mechanische Problem der erzwungenen Schwingungen behandelt hat. Auffällig ist es, dass Gauss den allgemeineren Fall, den er im Briefe an Olbers andeutet, nämlich den, dass die »kleineren Nadeln« eine Rück- wirkung auf die grosse Nadel ausüben, nicht behandelt hat; er hat sich offenbar damit begnügt, durch Rechnung festzustellen, dass in seinem Falle diese Rückwirkung unmerkbar klein war; er hätte sonst das Phänomen der Schwebungen und das Hin- und Herpendeln der Energie finden müssen. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and in english this passage translates to: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gauss, in these experiments, turned the theory of the forced vibrations of a mass point to the induced oscillations of the magnetometer; he seems to be the first, or at least one of the first, to deal with the mechanical problem of forced vibrations. It is remarkable that Gauss did not deal with the more general case he suggests in the letter to Olbers, namely, that the &quot;smaller needles&quot; have an effect on the big needle; he was evidently content to ascertain by calculation that in his case this repercussion was imperceptibly small; otherwise he would have had to find the phenomenon of beating and the oscillation of energy. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The whole picture of this aspect of Gauss's work is still not very clear to me (as there is so few information on it), so i posted this &quot;answer&quot; just to make some advance.  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="6514" LastEditDate="2019-08-31T17:38:22.220" LastActivityDate="2019-08-31T17:38:22.220" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9875" PostTypeId="1" CreationDate="2019-07-29T23:33:12.027" Score="1" ViewCount="307" Body="&lt;p&gt;Did physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/David_Bohm&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Bohm&lt;/a&gt; ever propose the existence of any kind of &lt;a href=&quot;https://en.wikipedia.org/wiki/Multiverse&quot; rel=&quot;nofollow noreferrer&quot;&gt;multiverse&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="10288" LastEditorUserId="10106" LastEditDate="2019-07-30T07:08:13.507" LastActivityDate="2020-08-30T04:40:44.997" Title="Did physicist David Bohm ever propose the existence of a multiverse?" Tags="|physics|theoretical-physics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9877" PostTypeId="1" CreationDate="2019-07-30T15:20:05.780" Score="6" ViewCount="246" Body="&lt;p&gt;I was wondering if anyone knows if Foucault actually gave a mathematical description of his observations related to the apparent motion of the pendulum. And if he didn't prove it, who described the motion of the pendulum first? &lt;/p&gt;&#xA;" OwnerUserId="10292" LastEditorUserId="609" LastEditDate="2019-07-31T13:24:14.690" LastActivityDate="2019-08-02T14:15:33.477" Title="Foucault pendulum historical question" Tags="|classical-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9878" PostTypeId="2" ParentId="9877" CreationDate="2019-07-30T16:20:17.000" Score="9" Body="&lt;p&gt;Yes. Léon Foucault in 1851 published in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Comptes_rendus_de_l%27Acad%C3%A9mie_des_Sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comptes rendus&lt;/a&gt; a paper &lt;a href=&quot;https://fr.wikisource.org/wiki/D%C3%A9monstration_physique_du_mouvement_de_rotation_de_la_Terre_au_moyen_du_pendule&quot; rel=&quot;nofollow noreferrer&quot;&gt;Démonstration physique du mouvement de rotation de la Terre au moyen du pendule&lt;/a&gt; detailing his experiment and the mathematical justification for it. (“Physical demonstration of the rotational movement of the Earth by means of the pendulum” is the English translation of that title.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He did not write any equations to describe it directly but instead took it to a limiting case (if the pendulum is at the North Pole) and described that correctly without equations. Then he described how at the equator the pendulum must not rotate, and so he reasoned that therefore up in France it’s got to be somewhere in the middle. He did not derive the sine-of-the-latitude dependence in this short note which connects the two, but he did end on a quick note connecting his effect to Poisson's 1837 &lt;a href=&quot;https://archive.org/details/bub_gb_A1B9m3uYC8IC/page/n3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Recherches sur le mouvement des projectiles dans l’air&lt;/a&gt;, “Research on the movement of projectiles through the air,” which concerns the deviation of the movement of bullets and cannon balls by the Coriolis force, and which has real equations: if you know that this is the only cause of the precession of Foucault’s pendulum, then the sine-of-the-latitude dependence should follow from that directly. He also indicates that this not-quite-a-full-rotation-per-day effect is visible in his experiment, too.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly he also resolves a question in this letter which I myself had long had, whether the rope can torque the pendulum and thus somehow alter the plane of rotation that way. Foucault says,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;L’indépendance du plan d’oscillation et du point de suspension peut être rendue évidente par une expérience qui m’a mis sur la voie et qui est très-facile à répéter. Après avoir fixé, sur l’arbre d’un tour et dans la direction de l’axe, une verge d’acier ronde et flexible, on la met en vibration en l’écartant de sa position d’équilibre et en l’abandonnant à elle-même. Ainsi l’on détermine un plan d’oscillation qui, par la persistance des impressions visuelles, se trouve nettement dessiné dans l’espace ; or on remarque qu’en faisant tourner à la main l’arbre qui sert de support à cette verge vibrante, on n’entraîne pas le plan de vibration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;translated,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The independence of the plane of oscillation and the point of suspension can be rendered evident by an experiment which put me on this path, and which is very simple to replicate. After having fixed, on the shaft of a lathe and in the direction of the axis, a rod of steel round and flexible, it is put into vibration by the displacement of its position from equilibrium and then abandoning it to move on its own. Thus it determines a plane of oscillation that, by the persistence of visual impressions, is located clearly-drawn through space; but we remark that making turns, by hand, of the shaft that supports this vibrating rod, we do not entrain the plane of vibration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So that’s really cool.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think this all raises an interesting point which is how much “mathematics” you consider to be “mathematics.” There are no equations in this paper but there is a beautiful piece of abstract reasoning; ‘take the situation to the North Pole, then this law of inertia must force the pendulum to swing back and forth in a fixed plane, while the Earth if it is truly rotating must rotate out-from-under it and thus someone standing on the Earth will see the pendulum rotate once per day—this effect has vanished completely at the equator but must be partially visible in the middle.’&lt;/p&gt;&#xA;" OwnerUserId="3012" LastEditorUserId="3012" LastEditDate="2019-08-02T14:15:33.477" LastActivityDate="2019-08-02T14:15:33.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9879" PostTypeId="2" ParentId="9875" CreationDate="2019-07-30T19:58:41.787" Score="2" Body="&lt;p&gt;This probably should be a comment, because it addresses the nature of your question rather than makes an attempt to answer it, but I didn't want to splice the following through several connected comments, where the formatting would be less easy to follow.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think you need to be a lot more precise as to what you mean by &quot;multiverse&quot;, as there have been speculations ad nauseam for centuries (and probably for millennia) about our universe being one reality among many. Such speculations became especially prevalent, and a little more founded on mathematical/physical principles, beginning in the late 1800s when fourth dimensional space possibilities became generally known to the educated public (Abbott's 1884 book &lt;a href=&quot;https://en.wikipedia.org/wiki/Flatland&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Flatland&lt;/strong&gt;&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Howard_Hinton&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Hinton's&lt;/a&gt; extensive writings in the 1880s, etc.). Then there was Einstein's relativity (1900s and 1910s, and extensively popularized in the 1920s), the rise of pulp science fiction magazines (1920s onward), etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below are excerpts from a couple of things I know about off the top of my head (the H. P. Manning book and the Hasse SF story are among the top 10-15 memorable items I read during my early teen years) and a couple of things I &lt;a href=&quot;https://www.google.com/search?q=%22fourth+dimension%22+%22universes%22&amp;amp;tbs=cdr:1,cd_min:1/1/1800,cd_max:12/31/1940&amp;amp;tbm=bks&quot; rel=&quot;nofollow noreferrer&quot;&gt;found almost immediately by googling&lt;/a&gt; (e.g. the Newcomb articles), probably none of which fit what you're looking for, but I strongly suspect Bohm would have been aware of these kinds of speculations. Thus, you should probably edit your question in a way that excludes these types of speculations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.ams.org/journals/bull/1894-03-04/S0002-9904-1894-00177-3/S0002-9904-1894-00177-3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Modern mathematical thought&lt;/em&gt;&lt;/a&gt; by Simon Newcomb (1894)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from p. 105)&lt;/strong&gt; Add a fourth dimension to space, and there is room for an indefinite number of universes, all alongside of each other, as there is for an indefinite number of sheets of paper when we pile them upon each other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=z41PAQAAMAAJ&amp;amp;pg=PA4&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The philosophy of hyper-space&lt;/em&gt;&lt;/a&gt; by Simon Newcomb (1898)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from p. 4, right column)&lt;/strong&gt; Our conclusion is that space of four dimensions, with its resulting possibility of an infinite number of universes alongside of our own, is a perfectly legitimate mathematical hypothesis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The Fourth Dimension Simply Explained&lt;/strong&gt; edited by Henry Parker Manning (&lt;a href=&quot;https://books.google.com/books?id=Y7cEAAAAMAAJ&amp;amp;pg=PA136&quot; rel=&quot;nofollow noreferrer&quot;&gt;1910 original printing&lt;/a&gt; and &lt;a href=&quot;https://archive.org/details/TheFourthDimensionSimplyExplained/page/n135&quot; rel=&quot;nofollow noreferrer&quot;&gt;1960 Dover Publications reprint&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from p. 136)&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$[\ldots ]$&lt;/span&gt; similarly, a universe of four dimensions would of necessity contain an infinite number of universes such as ours.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/Amazing_Stories_v10n11_1936-08/page/n13&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;He who shrank&lt;/em&gt;&lt;/a&gt; (see also &lt;a href=&quot;https://en.wikipedia.org/wiki/He_Who_Shrank&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) by Henry Louis Hasse (1936)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from p. 44, right column)&lt;/strong&gt; HOW many of the infinitely smaller atomic cycles I have passed into, I do not know. I tried to keep count of them at first, but somewhere between twenty and thirty I gave it up; and that was long ago. Each time I would think: “This cannot go on forever—it &lt;em&gt;cannot;&lt;/em&gt; surely this next time I must reach the end.” But I have not reached the end.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastActivityDate="2019-07-30T19:58:41.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9880" PostTypeId="1" CreationDate="2019-07-30T20:35:22.823" Score="2" ViewCount="94" Body="&lt;p&gt;Approximately 2300 years ago, Aristarchus proposed a method for determining the relative distances of the sun and the moon in relation to the earth. Specifically, he asserted that when the moon is in its half-phase, a right triangle is formed between the &lt;strong&gt;Sun&lt;/strong&gt;, the &lt;strong&gt;Moon&lt;/strong&gt;, and the &lt;strong&gt;Earth&lt;/strong&gt;. (This is quite true.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From there, he determined that &lt;span class=&quot;math-container&quot;&gt;$\angle ESM = 3^{\circ}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\angle SEM = 87^{\circ}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\angle SME = 90^{\circ}$&lt;/span&gt; and proceeded to calculate that the Sun is approximately 19 times farther away from the Earth than the Moon is. (Actually, a more precise value for &lt;span class=&quot;math-container&quot;&gt;$\angle SEM$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$89^{\circ}50'$&lt;/span&gt; which would put the Sun approximately 400 times farther away from the Earth than the Moon is.)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But my question is, what forms of trigonometry were available to Aristarchus to solve this problem, given that it was about this time that trigonometric methods were being developed for geometrical and astronomical applications? Also, does anyone know how Aristarchus came to discover that a right triangle exists between the Sun, Moon, and Earth when the moon is in half-phase?  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Truly, today, a problem like this might appear in a pre-Calculus book as a homework exercise and is a straightforward computation with a calculator. But I would like to know how an ancient astronomer would have arrived at the answer. Thank you. &lt;/p&gt;&#xA;" OwnerUserId="18820" LastEditorUserId="18820" LastEditDate="2019-07-31T17:06:41.040" LastActivityDate="2019-07-31T17:06:41.040" Title="On Trigonometric Methods Available to Aristarchus" Tags="|mathematics|astronomy|ancient-greece|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9881" PostTypeId="1" CreationDate="2019-07-31T02:07:00.983" Score="2" ViewCount="96" Body="&lt;p&gt;It is known that the physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Archibald_Wheeler&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Archibald Wheeler&lt;/a&gt; proposed the existence of a multiverse in the form of a set of cycles of Big Bangs and Big Crunches (&lt;a href=&quot;https://en.wikipedia.org/wiki/Cyclic_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cyclic/Oscillatory Multiverse Model&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I have found some evidence that indicates that, like the philosopher Robert Nozick, he was also a proponent of a multiverse where multiple universes with fundamentally different laws of nature would exist simultaneously.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In science historian Helge Kragh's book  &quot;&lt;em&gt;Higher Speculations: Grand Theories and Failed Revolutions in Physics and Cosmology&lt;/em&gt;&quot;, it is said at the end of chapter 10, in the notes, that Wheeler's ideas and Nozick's Principle of Plenitude are related, and both of them are also related to the multiverse hypothesis&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Nozick 1981, p. 129. The fecundity assumption is related to the principle of plenitude, but the latter refers only to the realization of possibilities in the actual world we live in. Whereas Nozick did not mention the anthropic principle, he did refer to Wheeler’s speculations of many universes with different laws of physics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This caught my attention, since I was told some years ago by a biographer of Wheeler's work and life, that he did not like the idea of having multiple universes existing in a hypothetical multiverse (he did not even like his student's thesis at all (Everett's many-worlds interpretation) because of this)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, as indicated in a &lt;a href=&quot;https://www.nature.com/articles/278605a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nature article by Rees and Carr&lt;/a&gt; in 1979, and in this &lt;a href=&quot;https://books.google.es/books?id=S7avUXmRTKkC&amp;amp;pg=PA450&amp;amp;lpg=PA450&amp;amp;dq=%22rees%22%20%22carr%22%20%22nature%22%20%221979%22%20%22infinite%20ensemble%22%20%22wheeler%22&amp;amp;source=bl&amp;amp;ots=Rr7RbNS1s4&amp;amp;sig=ACfU3U04-G3CcV-8Y1rc-EArDePaAd7K5A&amp;amp;hl=es&amp;amp;sa=X&amp;amp;ved=2ahUKEwie1syxjt7jAhULkRQKHVMcCCcQ6AEwC3oECAkQAQ#v=onepage&amp;amp;q=%22rees%22%20%22carr%22%20%22nature%22%20%221979%22%20%22infinite%20ensemble%22%20%22wheeler%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt;.:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wheeler envisages an infinite ensemble if universes, all with different coupling constants. Most of these universes are completely uninteresting; only those which start off with the right coupling constants become &quot;aware of themselves&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I have contacted with Helge Kragh and he told me that Nozick referred to &quot;Everett-Wheeler's Interpretation of Quantum Mechanics&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But how can this be true? Why did Nozick refer to Everett and Wheeler's many worlds interpretation if their universes would not have different fundamental laws?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mean according to this interpretation all universes would have the same fundamental laws, so all of this does not make much sense to me.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, how can all of this be compatible? Did Wheeler propose any kind or alternative version to the &quot;traditional&quot; MWI where universes would have fundamental different laws and rules?&lt;/p&gt;&#xA;" OwnerUserId="10288" LastEditorUserId="10288" LastEditDate="2019-08-09T02:26:12.373" LastActivityDate="2019-08-09T02:26:12.373" Title="Did Wheeler ever propose the existence of a multiverse containing parallel simultaneous universes?" Tags="|physics|cosmology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9882" PostTypeId="1" AcceptedAnswerId="9889" CreationDate="2019-07-31T23:28:16.683" Score="1" ViewCount="156" Body="&lt;p&gt;In an answer to a programming question, I included the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The default behavior of [library function in question that displays an image] is to put the origin of the coordinate system in the &lt;em&gt;upper&lt;/em&gt; left corner. This is different from plotting scientific data, such as two entities &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; against each other, where the origin, i.e. the point corresponding to the coordinate &lt;span class=&quot;math-container&quot;&gt;$(0,0)$&lt;/span&gt;, is in the &lt;em&gt;lower&lt;/em&gt; left corner, with the (positive) &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis extending to the right and the (positive) &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis extending towards the top.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The latter is just scientific convention, though one that goes back centuries. Arguably (albeit near impossible to back up with historical evidence), the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis is traditionally aligned left-to-right because that's how text is written in many languages, while the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis is oriented towards the top as that's how people intuit an &lt;em&gt;increase&lt;/em&gt; — much like the elevation of terrain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It goes on to explain how that is different when plotting images. To be clear, my question here is not relevant to my original answer. The point of the above was to make the technical aspects more relatable. It was solely added for context.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am curious, however, if &quot;near impossible to back up with historical evidence&quot; is in fact true. Or if there's any historical evidence that would contradict my statement.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cartesian_coordinate_system#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;history section in the Wikipedia article on the Cartesian coordinate system&lt;/a&gt; (though I do not currently have access to the references cited therein) as well as the question and answers regarding the &lt;a href=&quot;https://hsm.stackexchange.com/q/3312/10139&quot;&gt;Cartesian coordinate system in Newton's work&lt;/a&gt; here on this site.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Historically, what is the reason for the modern scientific convention that the horizontal plot axis is oriented left-to-right and the vertical plot axis bottom-to-top?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="10139" LastEditorUserId="10139" LastEditDate="2019-08-01T01:04:32.740" LastActivityDate="2019-08-04T02:16:04.720" Title="Conventional orientation of axes in scientific plots" Tags="|terminology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9883" PostTypeId="1" CreationDate="2019-08-01T07:06:39.590" Score="3" ViewCount="155" Body="&lt;p&gt;Oliver Heaviside achieved a very high level in mathematics and physics by self-study, starting from a modest school-level math, and working alone in his room, without a tutor. Is it known what textbooks he used?&lt;/p&gt;&#xA;" OwnerUserId="6222" LastEditorUserId="4251" LastEditDate="2020-06-06T18:45:24.967" LastActivityDate="2021-02-18T20:55:06.917" Title="Textbooks used by Oliver Heaviside" Tags="|mathematicians|biographical-details|heaviside|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9884" PostTypeId="1" CreationDate="2019-08-01T17:25:36.040" Score="3" ViewCount="249" Body="&lt;p&gt;According to p.200-202 of the book &quot;&lt;strong&gt;Reciprocity Laws: From Euler to Eisenstein&lt;/strong&gt;&quot; by Franz Lemmermeyer, there are two proofs for the biquadratic reciprocity law in Gauss's nachlass: one is a cyclotomic proof using Gauss sums which is very similar to Eisenstein's proof from 1844, and the other is an original contribution of Gauss - a sketch of a geometric proof of this theorem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Gauss's cyclotomic proof - according to Bachmann (the editor of Gauss's number theoretical works) it cannot be dated, so it's unclear if it's original with Gauss or based on Eisenstein's proof. Since Lemmermeyer states that many of the proofs published after 1850 are just variations on Eisenstein's proof , i'm more interested in Gauss's geometric proof . &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2019-10-21T18:46:01.000" LastActivityDate="2020-06-07T21:05:32.967" Title="Where in Gauss's nachlass apears his sketch of a geometric proof for the biquadratic reciprocity law?" Tags="|reference-request|number-theory|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9885" PostTypeId="1" CreationDate="2019-08-03T13:22:22.547" Score="4" ViewCount="154" Body="&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/658d1.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found &quot;The Arabian scholar Abu'l Raihan al'biruni&quot; has attributed to Archimedes the theorem of the broken chord in web. But I couldn't find a book or papers that this Arabian scholar wrote.&lt;/p&gt;&#xA;" OwnerUserId="10303" LastEditorUserId="4251" LastEditDate="2020-06-04T07:08:34.103" LastActivityDate="2020-06-04T07:08:34.103" Title="I want to know the source of why Archimedes' broken chord theorem is Archimedes good" Tags="|geometry|archimedes|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9886" PostTypeId="1" CreationDate="2019-08-03T17:44:15.773" Score="2" ViewCount="1506" Body="&lt;p&gt;The &lt;strong&gt;J&lt;/strong&gt; symbol can represent the &lt;strong&gt;unit&lt;/strong&gt; of energy but it's also the symbol for the &lt;strong&gt;dimension&lt;/strong&gt; of the candela (or &lt;a href=&quot;https://en.wikipedia.org/wiki/Luminous_intensity&quot; rel=&quot;nofollow noreferrer&quot;&gt;luminous intensity&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the energy &lt;strong&gt;unit&lt;/strong&gt;, it clearly comes from the family name of the English physicist James Prescott &lt;strong&gt;Joule&lt;/strong&gt; (1818–1889), but for the luminous intensity &lt;strong&gt;dimension&lt;/strong&gt;, what's the origin? What does it stand for?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm also asking because the energy &lt;strong&gt;unit&lt;/strong&gt; has the &lt;strong&gt;dimension&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathsf{L}^2\cdot\mathsf{M}\cdot\mathsf{T}^{-2}$&lt;/span&gt; which doesn't contain the &lt;strong&gt;dimension&lt;/strong&gt; of the luminous intensity.&lt;/p&gt;&#xA;" OwnerUserId="10306" OwnerDisplayName="maxagaz" LastEditorDisplayName="user2223" LastEditDate="2019-08-06T07:16:34.510" LastActivityDate="2019-08-06T07:16:34.510" Title="Dimension of the candela unit: What does J stand for?" Tags="|terminology|notation|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="9887" PostTypeId="2" ParentId="9886" CreationDate="2019-08-03T20:15:39.573" Score="-2" Body="&lt;p&gt;Candela contains the power emitted and the power is energy per sec or Joules/sec. So the J in candela is still Joules.&lt;/p&gt;&#xA;" OwnerDisplayName="jmh" LastActivityDate="2019-08-03T20:15:39.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9888" PostTypeId="2" ParentId="9886" CreationDate="2019-08-03T23:27:44.113" Score="3" Body="&lt;p&gt;The symbol for candela is cd according to the NIST's website.&lt;/p&gt;&#xA;&lt;p&gt;I gather you are mixing the concept of dimensions symbols (M, L, T) with dimensional analysis and energy units. The symbol for the mass dimension is &lt;strong&gt;M&lt;/strong&gt;, it can be in any kg, mg, g etc, Length is &lt;strong&gt;L&lt;/strong&gt; which can be millimeters, yards, kilometers etc. The dimension symbol for temperature is &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$\Theta$&lt;/span&gt;&lt;/strong&gt;. So what should be the dimensional symbol for candela? It is &lt;strong&gt;J&lt;/strong&gt;. &lt;em&gt;Does it have to anything with Joule? No&lt;/em&gt;. It is by chance that the symbol matches with Joules. The problem is that English has only 26 letters. Have you heard of the pigeonhole principle?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematics, the pigeonhole principle states that if &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; items are put into &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; containers, with &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;m$&lt;/span&gt;, then at least one container must contain more than one item. This theorem is exemplified in real life by truisms like &amp;quot;in any group of three gloves there must be at least two left gloves or at least two right gloves&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;─ &lt;a href=&quot;https://en.wikipedia.org/wiki/Pigeonhole_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The same applies on symbols and quantities. From the official website of SI Units, this is the list of symbols to be used for these dimensions:&#xA;&lt;a href=&quot;https://www.bipm.org/en/publications/si-brochure/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.bipm.org/en/publications/si-brochure/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IjtVw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IjtVw.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/International_System_of_Units#Base_units&quot; rel=&quot;nofollow noreferrer&quot;&gt;the section on base units on Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-08-05T14:12:06.047" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9889" PostTypeId="2" ParentId="9882" CreationDate="2019-08-04T01:51:55.260" Score="2" Body="&lt;p&gt;From the description in &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicole_Oresme#Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; it seems that graphs in the modern scientific sense were used  by Nichole Oreseme (1323-1382), with the modern conventions about axis orientation.  The &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Oresme.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mactutor article&lt;/a&gt; on him asserts he invented graphing in the modern sense.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2019-08-04T02:16:04.720" LastActivityDate="2019-08-04T02:16:04.720" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9890" PostTypeId="1" CreationDate="2019-08-04T09:05:03.717" Score="1" ViewCount="47" Body="&lt;p&gt;I am interested to know when and why was the concept of mechanical work (W=F.s) introduced? What was the motivation behind it/ why was it considered meaningful before the work energy theorem?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thankyou&lt;/p&gt;&#xA;" OwnerUserId="8145" LastActivityDate="2019-08-04T09:05:03.717" Title="Origin of mechanical work before work energy theorem" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9891" PostTypeId="2" ParentId="9884" CreationDate="2019-08-04T13:59:41.617" Score="4" Body="&lt;p&gt;Fortunately (and surprisingly) i found the answer very quickly - at the website of Springer they allow the readers to see the first 2 pages of each chapter, and to see the complete list of references for this book. Since the desired reference is reference 279, the relevant pages of Gauss's Nachlass are p.56-57 of volume X,1, which are entitled &quot;&lt;strong&gt;Hauptmomente des Beweises fur die biquadratischen Reste&lt;/strong&gt;&quot;. It depicts two interesting geometric figures whose meaning and connection to biquadratic reciprocity is unclear to me (by the way, there is a similar geometric fragment by Gauss which deals with &quot;cubic residues&quot;). Bachmann comments on it and says it's connected with the second chapter of the unpublished treatise &quot;&lt;strong&gt;preconceptions to the general theory of biquadratic residues&lt;/strong&gt;&quot; (Gauss's werke, volume II,p.326-331).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Lemmermeyer, Gauss's geometric proof was reconstructed by the mathematician B. Spies in his dissertation &quot;Der Gauss'sche Beweis des biquadratischen Reziprozitiitsgesetzes&quot; (T.U. Braunschweig, 1983). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So i found the reference, but if anyone still wants to add usefull comments or information, i'll be glad to hear it!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Updates&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;I found a paper by Richard Hill, entitled &quot;A GEOMETRIC PROOF OF A RECIPROCITY LAW&quot; which states that geometric methods were applied by Gauss to reciprocity laws for the cases &lt;span class=&quot;math-container&quot;&gt;$K = Q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K = Q(\sqrt{-1})$&lt;/span&gt; (i.e, quadratic reciprocity and biquadratic reciprocity), and in addition this paper references the same long treatise by Gauss which i mentioned before (&quot;preconceptions to the general theory of biquadratic residues&quot;,Gauss's werke, volume II, p.313-385).&#xA;This shows that there is really an interesting point here on Gauss's geometric treatment of biquadratic residues, a point which was somehow neglected by mainstream historians of mathematics.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;After a long and thorough searching on the internet, i found a most valuable source on these matters - it's the article &quot;&lt;strong&gt;A foundation of class field theory applying properties of spatial figures&lt;/strong&gt;&quot; by Tomio Kubota, a famous researcher of number theory. On p. 14-15 he writes:&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... Gauss didn't publish any proof of biquadratic reciprocity law, but one of his posthumous manuscripts did include a proof that was based on his own idea, which is rather similar to that given in that article and is actually geometric but is not sufficiently generalizable because it requires exact counts of lattice points in spatial figures. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-06-07T21:05:32.967" LastActivityDate="2020-06-07T21:05:32.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9892" PostTypeId="1" CreationDate="2019-08-05T07:57:57.027" Score="1" ViewCount="106" Body="&lt;p&gt;Initially I used to believe that before Newton gave his Laws of Motion, people thought that Force was something that caused motion. It was Newton who first said that Force is something that causes acceleration and gave a mathematical equation for it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But recently I came across this &lt;a href=&quot;https://www.grc.nasa.gov/www/k-12/airplane/newton.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; which says that Newton had developed his Law of Gravitation by 1666 and published his Laws of Motion in 1686. How could Newton have given an equation for Gravitation when there was no mathematical definition of force existing at that time?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On further study I came to know that &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Hooke&amp;#39;s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hooke's Law&lt;/a&gt; was given by Hooke in 1676, again a date before Newton gave Laws of Motion. Doesn't that imply that before Newton gave his Laws of Motion, people had some mathematical intuition about Force? Or may be people meant something else by the term Force.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2019-08-05T07:57:57.027" Title="What did people mean by the term Force in earlier times?" Tags="|physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9893" PostTypeId="1" CreationDate="2019-08-05T13:03:22.117" Score="1" ViewCount="132" Body="&lt;p&gt;The current &lt;a href=&quot;https://www.bipm.org/en/publications/si-brochure/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SI Brochure&lt;/a&gt; defines the &lt;em&gt;dimension&lt;/em&gt; of a certain physical quantity as something separate from its units, and treated in detail in its §2.3.3:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pxxC4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pxxC4.png&quot; width=&quot;500&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is, by now, pretty bread-and-butter stuff in the handling of dimensional analysis, and the usage is extremely widespread (cf. e.g. &lt;a href=&quot;https://physics.stackexchange.com/questions/77690/square-bracket-notation-for-dimensions-and-units-usage-and-conventions&quot;&gt;this thread&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;However&lt;/strong&gt;, when looking things up for &lt;a href=&quot;https://hsm.stackexchange.com/questions/9886/dimension-of-the-candela-unit-what-does-j-stand-for&quot;&gt;this question&lt;/a&gt;, which asks why the letter &lt;span class=&quot;math-container&quot;&gt;$\mathsf J$&lt;/span&gt; was chosen as the dimension symbol for luminous intensity, I was extremely surprised to learn that this usage is &lt;em&gt;extremely&lt;/em&gt; recent as far as the SI brochure is concerned: Over its almost fifty-year history, and nine editions, any talk of quantity dimension as something separate from the units is absent, all the way up to the 7th edition (1998), and it is only present in the 8th edition (2006) and the present 9th edition (2019).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which then leads me to my question: &lt;strong&gt;how did this happen?&lt;/strong&gt; Which is to say, what basis in the literature, both formal (say, CGPM resolutions?) and informal, was used to bring this in? What was the status of this distinction prior to 2006? How well-codified were these conventions at the time, and where were they documented? How and why did they make the leap to the standardized SI Brochure, and where is that process documented?&lt;/p&gt;&#xA;" OwnerUserId="272" LastActivityDate="2019-08-05T13:03:22.117" Title="Where did the notation for dimensions of a quantity come from?" Tags="|units|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9894" PostTypeId="1" CreationDate="2019-08-05T18:52:52.920" Score="2" ViewCount="115" Body="&lt;p&gt;In the answer to &lt;a href=&quot;https://scifi.stackexchange.com/questions/216975/earliest-story-to-mention-the-different-surface-gravity-on-the-moon/217035?noredirect=1#comment593527_217035&quot;&gt;this question&lt;/a&gt;, a user provided a quote from the 1638 novel &lt;em&gt;&lt;a href=&quot;http://www.isfdb.org/cgi-bin/title.cgi?21079&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Man in the Moone&lt;/a&gt;&lt;/em&gt; by Bishop Francis Godwin:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The manner of our Travel to the Palace of Pylonas was more strange and&#xA;  incredible than any thing we have related, for at our first setting&#xA;  forth there were delivered to each of us two Feather Fans, like those&#xA;  our Ladies in Spain cool themselves with in Summer: You must&#xA;  understand, that the Globe of the Moon has likewise an attractive&#xA;  Power, yet so much weaker than the Earth, that if a Man do but spring&#xA;  upward with all his Strength, as Dancers do in shewing their Tricks,&#xA;  he will be able to mount fifty or sixty Foot high; and being then&#xA;  above all Attraction from the Moon's Earth, he falls down no more, but&#xA;  by the Help of these Fans, as with Wings, they convey themselves in&#xA;  the Air in a short Space, (though not quite so swift as Birds) whither&#xA;  they please. In two hours Time (as I could guess) by the Help of these&#xA;  Fans, we were carried through the Air those five Leagues, in all about&#xA;  sixty Persons.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;How is it possible that Godwin was aware that the force of gravity would be reduced on the Moon?  As far as I am aware, the idea that mass and gravity were related was Newton's invention, and at the time of this novel's writing Newton had not yet been born.&lt;/p&gt;&#xA;" OwnerUserId="10312" LastActivityDate="2019-08-05T18:52:52.920" Title="How did Bishop Francis Godwin Know About Mass and Gravity Pre-Newton?" Tags="|gravity|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9897" PostTypeId="1" AcceptedAnswerId="9901" CreationDate="2019-08-06T05:18:43.690" Score="10" ViewCount="352" Body="&lt;p&gt;In &lt;a href=&quot;http://www.math.caltech.edu/~jimlb/abel.pdf&quot; rel=&quot;noreferrer&quot;&gt;this document&lt;/a&gt; by Jim Brown it is claimed (on Section 3, pg 5) that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Descartes] believed that all polynomials of degree &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;4$&lt;/span&gt; could be solved with the same methods as had been applied to the quadratic, the cubic, and the quartic. In fact, he left the solution of higher degree equations as an exercise to the reader.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is a well known fact that the general quintic, and all higher degree polynomials, can not be completely resolved in terms of radicals and elementary operations. So, such an exercise would be impossible to complete, making this an interesting historical example of mathematical hubris.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it's not clear where the author is getting this information from, and I couldn't find any first-hand sources for this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it true that Descartes left solving the quintic, and higher degree polynomials as an exercise to his readers? If so, could I get a specific reference for this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I contacted the author of the above document, and he replied saying he doesn't recall where he got this information.&lt;/p&gt;&#xA;" OwnerUserId="10314" LastEditorUserId="4251" LastEditDate="2020-06-22T11:51:34.350" LastActivityDate="2020-06-22T11:51:34.350" Title="Did Descartes leave solving the quintic as an exercise to his readers?" Tags="|mathematics|reference-request|algebra|galois-theory|descartes|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9899" PostTypeId="1" AcceptedAnswerId="9903" CreationDate="2019-08-06T20:38:21.597" Score="5" ViewCount="148" Body="&lt;p&gt;Much of the early theoretical work in electricity and magnetism appears to have been the result of applying celestial mechanical principles to electrostatics. Examples include Cavendish's inverse-square law experiment (which had an analogous gravitational statement due to Newton), and Poisson's application of potential theory to electrostatics (in fact, generally, Poisson's work in electrostatics involved using analytical tools developed by himself and e.g. Laplace, Legendre, Lagrange, and Biot, mostly in relation to mechanics, to Coulomb's experiments). Not to mention the relationship between magnetism and the Earth, which almost blends the two together. Considering the existence of a large body of work on gravity and celestial mechanics, it seems like the analytical approach to electricity would have borrowed heavily from it. How much does E&amp;amp;M owe to celestial mechanics and the mathematicians who developed it? Were there many essential developments which were due mostly or entirely to celestial mechanics?&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="609" LastEditDate="2019-08-07T13:16:25.380" LastActivityDate="2021-01-26T21:19:13.537" Title="Did the principles of celestial mechanics affect the development of electromagnetism?" Tags="|electromagnetism|celestial-mechanics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9901" PostTypeId="2" ParentId="9897" CreationDate="2019-08-07T16:33:47.643" Score="9" Body="&lt;p&gt;It is in the third book of &lt;em&gt;La Géometrie&lt;/em&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I could also add rules for equations of the fifth, sixth, and higher &#xA;  degrees, but I prefer to consider them all together and to state the &#xA;  following general rule : &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...and, consequently, if it is of the &#xA;  third or fourth degree, the problem depending upon it is solid; if of &#xA;  the fifth or sixth, the problem is one degree more complex, and so &#xA;  on. I have also omitted here the demonstration of most of my state- &#xA;  ments, because they seem to me so easy that if you take the trouble &#xA;  to examine them systematically the demonstrations will present themselves to you and it will be of much more value to you to learn them &#xA;  in that way than by reading them. -- Dover Publ., p. 192 = &lt;a href=&quot;https://archive.org/stream/TheGeometry/Descartes-TheGeometry_djvu.txt&quot; rel=&quot;noreferrer&quot;&gt;Archive.org&lt;/a&gt;, p. 192.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;One question that arises for me is what a &quot;solution&quot; is for Descartes. I'm just not familiar enough with Descartes to say authoritatively. He is at some pains to argue that one ought to &quot;admit&quot; into geometry curves beyond the circle and conic sections in solving problems. In the section quoted above, he is discussing roots of equations, which may be interpreted as points of intersection.  However, the solutions tend to be given in algebraic form, although many of the examples are geometric problems and the solutions are effected by the construction of curves.  He shows, as an example, how to solve a degree 6 problem (intersection of a circle and a cubic), and he gives a method to construct the cubic (&quot;In this way we can find as many &#xA;points of the curve as may be desired&quot;, p. 228).  But it is clear from the discussion that when you actually draw the curves, you can run into difficulties:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It should be remarked, &#xA;  however, that in many of these problems it may happen that the circle &#xA;  cuts the parabola of the second class so obliquely that it is hard to &#xA;  determine the exact point of intersection. In such cases this construc- &#xA;  tion is not of practical value. (p. 239)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He sums up in the conclusion by asserting the generality of his method for solving all problems, the problems being, I take it, geometrical:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;...furthermore, having &#xA;  constructed all plane problems by the cutting of a circle by a straight &#xA;  line, and all solid problems by the cutting of a circle by a parabola ; and, &#xA;  finally, all that are but one degree more complex by cutting a circle by &#xA;  a curve but one degree higher than the parabola, it is only necessary to &#xA;  follow the same general method to construct all problems, more and &#xA;  more complex, ad infinitum... (p. 240)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My reading is that Descartes is not approaching the solution of problems with the same restrictions in mind that &quot;solution by radicals&quot; entails.  But my reading might be too cursory.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2019-08-07T16:33:47.643" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9902" PostTypeId="1" AcceptedAnswerId="9905" CreationDate="2019-08-07T21:57:43.493" Score="5" ViewCount="5736" Body="&lt;p&gt;A while back I came across an interesting method to do multiplication. I don't know what it's called and am interested in when (and who) developed this method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I don't know if it's a mathematical curiosity or if it was developed in a time and place before hindu / place numbers became commonplace.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Say one wants to multiply 21 x 31&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Draw 2 lines, leave some space and then draw 1 line.&#xA;You've just written &quot;21&quot; (See: Figure 1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For &quot;31&quot; draw 3 lines perpendicular to the original set of lines; then leave a space and draw one line.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WcXHg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WcXHg.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To multiply one counts the intersection points at the top left to ascertain how many 100s.(In this case it's 6)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, to determine the tens, one counts the intersection points at the top right and bottom left (In this case it's 2+3).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To determine the ones place count the intersections at the bottom right. (In this case 1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6pR4F.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6pR4F.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've played with numerous variations. It gets a little tedious drawing all the lines for 52x78 but it works. (There is a carrying action when numbers pass 10).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;TL/DR: What is this method called? When was it created?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="4191" LastEditorUserId="4191" LastEditDate="2019-08-08T17:50:53.060" LastActivityDate="2019-08-09T22:07:52.803" Title="What is the origin of the Chinese Stick Multiplication method?" Tags="|mathematics|debunking|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9903" PostTypeId="2" ParentId="9899" CreationDate="2019-08-07T22:38:46.290" Score="4" Body="&lt;p&gt;General references on the subject are &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whittaker's History of the Theories of Aether and Electricity&lt;/a&gt; and &lt;a href=&quot;https://www.electricityforum.com/a-timeline-of-history-of-electricity.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Timeline Of History Of Electricity&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Newton's persuasive evidence for the inverse square law of gravity was a defining achievent of new science, so of course it invited imitation wherever possible. Newton himself already approaches magnetism with gravity as a template, writing in Principia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The power of magnetism... in receding from the magnet, decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1750 Michell clarified that this was due to the dipole effect:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Each pole attracts or repels exactly equally... attraction and repulsion of magnets increases, as the squares of the distances from the respective poles increase&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After Franklin observed that a charge is not attracted to the walls of an electrified can when inside it, Priestley recalled that Newton proved in Principia that a mass inside a spherical shell is not attracted to its walls either. He conjectured in 1766 that the electrostatic force has to fall as the inverse square of the distance for this to happen as well. Cavendish came to the same conclusion in 1771, but &amp;quot;&lt;em&gt;indifferent to fame he neglected to communicate this and other work of importance&lt;/em&gt;&amp;quot;, as Whittaker put it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://etc.usf.edu/lit2go/218/a-short-account-of-the-history-of-mathematics/5536/joseph-louis-lagrange/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange's 1773 idea of the source potential&lt;/a&gt; came up in the context of gravity, and led Laplace to the now famous equation for it in 1782. Poisson made their potential theory a centerpiece of his electrostatic (&lt;a href=&quot;https://archive.org/details/atreatisemechan00poisgoog/page/n3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Treatise of Mechanics&lt;/a&gt;(1811)) and magnetostatic (1824) theories, which indirectly influenced Maxwell, see &lt;a href=&quot;https://www.jstor.org/stable/27757576&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origin of the Displacement Current by Siegel&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &amp;quot;&lt;em&gt;The mathematical paradigm for the treatment of polarized media was Simeon Denis Poisson's treatment of the magnetic case; this analysis had been taken over for the electrical case by Ottaviano Mossotti, whom Maxwell cited explicitly; and standing behind Maxwell's physical interpretation of all of this was Faraday's work on dielectrics, also explicitly cited by Maxwell. The equation that Maxwell wrote for the displacement represented an amalgam of these influences, as well as the requirements of his own specific situation in &amp;quot;Physical lines&amp;quot;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, we do not see field treatments of gravity until much later because Laplace demonstrated in &lt;a href=&quot;http://en.wikipedia.org/wiki/Speed_of_gravity#Laplace&quot; rel=&quot;nofollow noreferrer&quot;&gt;Celestial Mechanics (1799)&lt;/a&gt; that non-zero latency (forces directed at retarded positions of the gravitating bodies rather than their current ones) would lead to planets flying off of their current orbits almost instantly. Unless, gravity propagation was millions of times faster than light.&lt;/p&gt;&#xA;&lt;p&gt;If anything, the influence of celestial mechanics on electromagnetism in 19th century was counterproductive. For example, it led Gauss and Weber to constructing alternative electromagnetic theories based on particle interaction laws rather than field concepts, which did not work out, see Hecht's (somewhat cranky) &lt;a href=&quot;http://21sci-tech.com/articles/spring01/Electrodynamics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Suppressed Electrodynamics of Ampère-Gauss-Weber&lt;/a&gt;. Fluid mechanics and elasticity theory (in the popular after Fresnel elastic theories of ether) supplied more of the field theoretic mathematical machinery.&lt;/p&gt;&#xA;&lt;p&gt;After Poisson, it is the influence in the other direction, stimulated by Faraday's and Maxwell's field intuitions, that proved to be more fruitful, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2395/55&quot;&gt;What 19th century developments contributed to the General theory of Relativity?&lt;/a&gt; For example, Mossotti's 1830 idea that the electric attraction and repulsion do not balance each other exactly, &lt;a href=&quot;http://en.wikisource.org/wiki/Scientific_Memoirs/1/On_the_Forces_which_regulate_the_Internal_Constitution_of_Bodies&quot; rel=&quot;nofollow noreferrer&quot;&gt;and the difference is gravity&lt;/a&gt;, was fateful. In 1900 &lt;a href=&quot;http://en.wikipedia.org/wiki/Lorentz_ether_theory#Gravitation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorentz&lt;/a&gt; showed that such Maxwellian gravity did not suffer from Laplace's latency problem, because the correction to the force's direction was of the order &lt;span class=&quot;math-container&quot;&gt;$v^2/c^2$&lt;/span&gt; rather than of &lt;span class=&quot;math-container&quot;&gt;$v/c$&lt;/span&gt;, as in Laplace's setting. That was the consequence of the Lorentz invariance compensating for the effects of retardation in the first order, as Poincare pointed out in 1905, and it paved the way to field treatments of gravity, including Einstein's relativistic theory.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-26T21:19:13.537" LastActivityDate="2021-01-26T21:19:13.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9905" PostTypeId="2" ParentId="9902" CreationDate="2019-08-08T05:42:11.343" Score="10" Body="&lt;p&gt;It is a fun method but it appears to be very recent. It is characterized as Chinese, Japanese, Korean, Indian, or even Mayan method in various internet posts, all of them recent, and without attribution, naturally. The &quot;ancient origin&quot; story is most likely made up, it is reminiscent of the Lucas's towers of Hanoi hoax, still credited on the internet to the &quot;ancient Indian priests from the Temple of Brahma&quot;, see &lt;a href=&quot;https://hsm.stackexchange.com/a/766/55&quot;&gt;How certain is it that Lucas invented the Towers of Hanoi puzzle?&lt;/a&gt; All current mentions seem to be traceable to a &lt;a href=&quot;http://www.metacafe.com/watch/296904/easy_graphical_multiplication_trick/&quot; rel=&quot;nofollow noreferrer&quot;&gt;2006 MetaCafe video by Akahad&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, as some posters pointed out, the mathematical idea behind counting intersections is a truly old idea of adding partial products, see e.g. detailed explanation in &lt;a href=&quot;https://tapintoteenminds.com/japanese-multiplication/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Japanese Multiplication: The Real Reason Why It Works&lt;/a&gt;. &lt;a href=&quot;https://history-computer.com/CalculatingTools/NapiersBones.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Napier's bones (rabdology)&lt;/a&gt; is another implementation of the same idea. It was derived from the earlier &lt;em&gt;gelosia&lt;/em&gt; method, described by Luca Paccioli in Summa de Arithmetica, Geometrica, Proportioni et Proportionalita (Everything about Arithmetic, Geometry, and Proportion, 1494), and transmitted to Italy from Indians or the Chinese through Arabs. The ancient Chinese did use sticks to implement partial products, but it was not the same, more table-like and suitable for abacus, see &lt;a href=&quot;https://books.google.com/books?id=3kVdDwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numbers Through the Ages by Flegg, pp.190-2&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Since the era before the birth of Christ the Chinese used little bamboo or wooden sticks as calculating pieces (chou) on their counting board... There is a lot of evidence that from the Warring States period (fourth and third centuries BC) calculations were made with stick numerals on a counting board, using a place-value system. Stick numerals are depicted on the coins of that period; they are also described in literary works. Perhaps the most famous instance of the latter is in the Tao Te Ching, where Lao Tzu says 'Good mathematicians do not use counting-rods'&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The question about the origin of the method of the Akahad's video was asked repeatedly on Math SE, see &lt;a href=&quot;https://math.stackexchange.com/q/886177/152568&quot;&gt;Where does the “Visual Multiplication” technique originate from?&lt;/a&gt; and links in the comment thread. @WillOrrick found a &lt;a href=&quot;https://webcache.googleusercontent.com/search?q=cache:BOFRb2igtDUJ:https://artistoon.org/how-to-multiply-numbers-and-algebra-equations-by-drawing-lines/+&amp;amp;cd=13&amp;amp;hl=en&amp;amp;ct=clnk&amp;amp;gl=us&amp;amp;client=safari&quot; rel=&quot;nofollow noreferrer&quot;&gt;comment by Bill Hart&lt;/a&gt; to a &lt;a href=&quot;https://youtu.be/a-e8fzqv3CE&quot; rel=&quot;nofollow noreferrer&quot;&gt;video by Vi Hart&lt;/a&gt; that sheds some light on it. Apparently, Akahad learned the method from his Chinese girlfriend, who, in turn, learned it from her Chinese schoolteacher. She believed him to be the inventor. Here is the comment in full:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Vi, you might be interested to know this method seems to have originated with a school teacher in China. It was first taught to a school girl in China. She taught it to her boyfriend, Akahad, who made a video on MetaCafe on Nov 16th 2006. Akahad was criticised for the fact that it is inefficient for numbers with large digits. However he claimed it was not intended to be an actually efficient method, but only &quot;meant to be a little trick to show to friends and kids who hate maths&quot;. The video was so popular it made $2000 in 4 days. The school teacher who introduced it apparently did so to get kids interested in maths and the criss-cross pattern was used because it reminded the school children of the stools they sat on. It is commonly referred to as the Vedic or Mayan or Japanese method. But perhaps we should be calling it a Chinese method (though there are other Chinese methods perhaps more worthy of the appellation)!?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 2014 Presh Talwalkar made another (YouTube) video about the stick multiplication, which received over a million views in a month. He then wrote a book about it, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00N85DCTU&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Multiply Numbers By Drawing Lines&lt;/a&gt;, and did some research on its origins. Bill Hart's comment above is taken from his post. Here is more:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;From my extensive research, line multiplication dates to Nov 16th 2006 from a Chinese teacher (see links below). If you know of an earlier source, please let us all know with proper proof.&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;To people who say this method is well-known, so why isn't line multiplication mentioned on Wikipedia? It's really, really important that research meet certain standards to be part of academic literature. I would love to see the history of the method, and its uses, as part of the multiplication algorithm – like lattice multiplication is listed as a method. I think line multiplication merits an entry and mention, but the active community of Wikipedia can let be the experts. These are the closest examples of pages where I think the method could be mentioned (or have its own page too):&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Multiplication_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Multiplication_algorithm&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lattice_multiplication&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Lattice_multiplication&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ol start=&quot;3&quot;&gt;&#xA;  &lt;li&gt;To my knowledge, I am the only person's who has published this method in a book Multiply Numbers By Drawing Lines&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00N85DCTU&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/gp/product/B00N85DCTU&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Of course I'm interested in other sources. I've been searching for 5 years and no one has yet sent me anything.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Sources for 1&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;What is the origin of line multiplication? Math StackExchange post authored by me July 2014&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/877120/what-is-the-origin-of-how-the-japanese-multiply-line-multiplication&quot;&gt;https://math.stackexchange.com/questions/877120/what-is-the-origin-of-how-the-japanese-multiply-line-multiplication&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Alas, the Chinese schoolteacher's name is not mentioned by anyone. Perhaps, he was inspired by the ancient stick numerals, but we may never know.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-09T22:07:52.803" LastActivityDate="2019-08-09T22:07:52.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9907" PostTypeId="2" ParentId="9897" CreationDate="2019-08-08T10:36:04.053" Score="6" Body="&lt;p&gt;It is easy to find fault with Descartes and Leibniz spent his life doing it (see Belaval Y., &lt;em&gt;Lz critique de Desc.&lt;/em&gt;, P.1960). Descartes knew that some problems of higher degrees are reductible and erroneously believed that it is the general case. The question here however concerns a paraphrase without reference and asks for a good match. such as e.g. &lt;em&gt;La Geometrie&lt;/em&gt; p.192. An other one could be &lt;a href=&quot;https://archive.org/stream/TheGeometry/Descartes-TheGeometry_djvu.txt&quot; rel=&quot;noreferrer&quot;&gt;p.43.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mais parce que i’espere que d’orenavant ceux qui auront l’adresse de&#xA;  se servir du calcul Geometrique icy proposé, ne trouueront pas assés&#xA;  de quoy s’arester touchant les problesmes plans, ou solides ; je croy&#xA;  qu’il est à propos que je les invite à d’autres recherches, où ils ne&#xA;  manqueront iamais d’exercice.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;= I hope that hereafter those who are clever enough to make use of the  geometric methods herein suggested will find no great difficulty in  applying them to plane or solid problems. I therefore think it proper  to suggest to such a more extended line of investigation which will  furnish abundant opportunities for practice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The text here suggests that more simple case have been rendered trivial and invites the readers to &lt;em&gt;exercice&lt;/em&gt; themselves with more complicated ones, promising that they will never lack &quot;opportunity for practice&quot;. This might also be a reference for the distorted paraphrase. &lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2019-08-08T10:36:04.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9908" PostTypeId="1" CreationDate="2019-08-08T11:00:57.127" Score="3" ViewCount="153" Body="&lt;p&gt;In the last page of the abstract of Gauss's paper on capillarity &lt;em&gt;&amp;quot;Principia generalia theoriae figurae fluidorum in statu aequilibrii&amp;quot;&lt;/em&gt; (1829), the author (who is he?) mentions (Gauss's werke, volume V, p.292) that Gauss's equation for the free surface of a liquid agrees fully with Laplace's equation (Laplace discovered that a free liquid surface in a cylindrical container assumes the shape of a surface of rotation of &amp;quot;elastica&amp;quot;). But in addition, he makes the following remark:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...secondly, for the boundaries of this surface, in the equation &lt;span class=&quot;math-container&quot;&gt;$sin \frac{1}{2}i = \frac{\delta}{\alpha} $&lt;/span&gt;, if &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; denotes the angle of inclination between the planes touching the free surface of the liquid and the surface of the vessel, measured outside the liquid just the second main sentence used by Laplace without justification.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This passage doesn't have correct grammar (I used Google Translate), but I copied it as it was to prevent misunderstandings. This formula isn't clear to me; the verbal description by the author seems to indicate it's connected with the mathematical condition for &amp;quot;contact angle&amp;quot; (wetting angle), but the formula for contact angle is different than this. So my question is about the meaning of this formula; the author describes it as &amp;quot;Laplace's second main sentence&amp;quot;, so perhaps someone familiar with Laplace's capillary work can answer.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My second question is less important, and deals with a very small contribution of Gauss to the calculus of variations and elastica theory. In a fragment from volume 12 (p.49-52), Gauss deals with three problems of the calculus of variations; one of them concerns the derivation of the shape of &amp;quot;elastic curve&amp;quot;, a problem that Euler worked about very intensively.&lt;/p&gt;&#xA;&lt;p&gt;According to Oscar Bolza, Gauss offered an alternative solution to the same problem of variation &amp;quot;which is unlikely to be surpassed in elegance&amp;quot;. The basic variational principle of the theory (the so called Euler-Bernoulli theory) is that the shape of the elastica minimizes the integral of the squared curvature, and Gauss solves the problem of shape of elastica that is tangent to two given straight lines at its end points.&lt;/p&gt;&#xA;&lt;p&gt;Gauss's derivation seems to be altogether very short and uncomplicated, so it doesn't look like a significant aspect of his work, but Bolza states the same method was applied by Max Born in his 1906 work on elastic curves in the plane and space. So this question is about explanation of what Gauss has actually done in this note.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="7205" LastEditDate="2022-10-25T16:25:12.987" LastActivityDate="2022-10-25T16:25:12.987" Title="Two questions about Gauss's contributions to capillarity and the calculus of variations" Tags="|biographical-details|calculus|gauss|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9910" PostTypeId="1" CreationDate="2019-08-10T20:51:04.837" Score="8" ViewCount="241" Body="&lt;p&gt;In &lt;em&gt;Science and Hypothesis&lt;/em&gt;, Poincare cryptically describes a &quot;Fourth Geometry.&quot; Can anyone clarify what he is talking about? Is there a standard name for this geometry?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;The Fourth Geometry&lt;/em&gt;.—Among these explicit axioms there is one which&#xA;  seems to me to deserve some attention, because when we abandon it we&#xA;  can construct a fourth geometry as coherent as those of Euclid,&#xA;  Lobatschewsky, and Riemann. To prove that we can always draw a per-&#xA;  pendicular at a point A to a straight line AB, we consider a straight&#xA;  line AC movable about the point A, and initially identical with the&#xA;  fixed straight line AB. We then can make it turn about the point A&#xA;  until it lies in AB produced. Thus we assume two propositions—first,&#xA;  that such a rotation is possible, and then that it may continue until&#xA;  the two lines lie the one in the other produced. If the first point is&#xA;  conceded and the second rejected, we are led to a series of theorems&#xA;  even stranger than those of Lobatschewsky and Riemann, but equally&#xA;  free from contradiction. I shall give only one of these theorems, and&#xA;  I shall not choose the least remarkable of them. &lt;em&gt;A real straight line&#xA;  may be perpendicular to itself&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10330" LastActivityDate="2019-08-11T11:50:46.407" Title="What is Poincare's &quot;Fourth Geometry&quot;?" Tags="|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9911" PostTypeId="2" ParentId="9910" CreationDate="2019-08-10T22:21:11.580" Score="11" Body="&lt;p&gt;It is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Minkowski_plane&quot; rel=&quot;noreferrer&quot;&gt;Minkowski plane&lt;/a&gt;, the lightlike lines are &amp;ldquo;perpendicular to themselves&amp;rdquo;, see e.g. &lt;a href=&quot;https://www.journals.uchicago.edu/doi/full/10.1086/687847?mobileUi=0&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;Stachel, Poincaré and the Origins of Special Relativity&lt;/a&gt;. To get this geometry, one needs to expand what &amp;ldquo;geometry&amp;rdquo; means. The classical geometries of Euclid, Lobachevsky, and Riemann have what is now called sign-definite metrics, which define real distances. In contrast, the Minkowski plane has an indefinite metric with signature &lt;span class=&quot;math-container&quot;&gt;$1,1$&lt;/span&gt;, so the &amp;ldquo;distances&amp;rdquo; can be imaginary (along the timelike lines). It does have constant curvature, like the classical geometries, but is only homogeneous, not isotropic, spacelike and timelike directions can not be interchanged by Lorentz transformations (which preserve the metric). It can also be realized as the Lorentz geometry on the hyperboloid of one sheet, and is sometimes denoted &lt;span class=&quot;math-container&quot;&gt;$H^{1,1}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Minkowski plane first appeared on a long list of geometries from &lt;a href=&quot;https://en.wikipedia.org/wiki/Erlangen_program&quot; rel=&quot;noreferrer&quot;&gt;Klein's Erlangen program (1872)&lt;/a&gt;, where he classified them based on a choice of &amp;ldquo;absolute&amp;rdquo;, a conic in the projective plane preserved by a subgroup of projective transformations. But it remained largely unexplored until Minkowski connected a four-dimensional version of it to the spacetime of special relativity in 1907. Poincare originally singled it out in &lt;em&gt;Les Géométries non Euclidiennes&lt;/em&gt; (1891) as deserving more attention, and, ironically, used it in &lt;a href=&quot;https://en.wikipedia.org/wiki/Science_and_Hypothesis&quot; rel=&quot;noreferrer&quot;&gt;Science and Hypothesis (1902)&lt;/a&gt; to illustrate his views on the conventional nature of choosing a physical geometry, see &lt;a href=&quot;https://books.google.com/books?id=FWIHCgAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Einstein's Pathway to the Special Theory of Relativity by Weinstein, pp.517-18&lt;/a&gt;. Only the pair physics + geometry is empirically determined, according to him, not each piece separately, and we are free to keep Euclidean geometry, as the simplest, come what may, by modifying the physics. This was the reason he favored Lorentz's interpretation of special relativity, with its Euclidean background space, over Einstein's to the end of his life.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-11T11:50:46.407" LastActivityDate="2019-08-11T11:50:46.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9912" PostTypeId="1" CreationDate="2019-08-11T10:52:22.053" Score="0" ViewCount="112" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H}_k = (h_1,h_2,\cdots,h_k)$&lt;/span&gt; be an admissible k-tuple.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The k-tuple conjecture&lt;/strong&gt; predicts that the number of primes &lt;span class=&quot;math-container&quot;&gt;$(b+h_1,b+h_2,\cdots,b+h_k)\in \mathbb{P}^k$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$b+h_k \leq x$&lt;/span&gt; is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\pi_{\mathcal{H}_k}(x) \sim \left(\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}} \right)\dfrac{x}{\log(x)^k}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$w(\mathcal{H}_k, p)$&lt;/span&gt; is the number of distinct residues &lt;span class=&quot;math-container&quot;&gt;$\pmod p$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathcal{H}_k$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can deduce this formula from &lt;strong&gt;Cramer's random model&lt;/strong&gt;, but Hardy and Littlewood get the same result using the circle method.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did Hardy and Littlewood formulate this conjecture using the circle method?&lt;/p&gt;&#xA;" OwnerUserId="10332" LastEditorUserId="3853" LastEditDate="2019-08-12T11:27:06.297" LastActivityDate="2019-08-12T11:27:06.297" Title="How did Hardy and Littlewood formulate the k-tuple conjecture?" Tags="|number-theory|complex-analysis|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9913" PostTypeId="1" CreationDate="2019-08-11T16:51:02.490" Score="0" ViewCount="270" Body="&lt;p&gt;Was there someone who got a medical education, took an interest in physics and later became a known theoretical physicist?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a medical student studying graduate mathematics and consider becoming a theoretical physicist myself.&lt;/p&gt;&#xA;" OwnerUserId="10334" LastEditorUserId="4251" LastEditDate="2020-06-06T12:58:11.737" LastActivityDate="2020-06-06T12:58:11.737" Title="Did anyone with formal medical education become a known theoretical physicist?" Tags="|biographical-details|medicine|physicists|" AnswerCount="2" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9914" PostTypeId="1" CreationDate="2019-08-11T18:43:26.793" Score="3" ViewCount="53" Body="&lt;p&gt;Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials? I heard it was Pierre Louis Lions, but I found no records.&lt;/p&gt;&#xA;" OwnerUserId="10335" LastActivityDate="2019-08-11T18:43:26.793" Title="Asymptotically Periodic Potentials" Tags="|mathematicians|differential-geometry|differential-equations|functional-analysis|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9915" PostTypeId="2" ParentId="9913" CreationDate="2019-08-11T22:31:15.660" Score="4" Body="&lt;p&gt;One name that comes to mind is &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfred_Kleiner&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alfred Kleiner&lt;/a&gt;, Einstein's doctoral advisor, although he was more distinguished as an experimentalist than a theoretician. He graduated as a medical student from the University of Zurich in 1872, and obtained bis Dr. Med in 1874. During his Berlin studies in 1873-1879 he became attracted to physics, and worked as an assistant at the physical institute of &lt;em&gt;Polytechnikum&lt;/em&gt; until 1885. Here is from &lt;a href=&quot;https://books.google.com/books?id=AwmQTGB8LwMC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Historical Development of Quantum Theory, Volume 5, Part 1, p.280&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;In 1885 he was promoted to full professor and director of the new physics institute in Ramistraße. He established his own experimental research programme in physics, systematically investigating the following topics: temperature dependence of specific heats and heats of solutions; dielectric behaviour of non-ideal insulators; frequency dependence of the resistance and inductance of coils; and the influence of intermediate matter on gravity. Kleiner was an experienced experimentalist, who constructed, for example, a quadrant electrometer of very high sensitivity; he also trained a number of talented&#xA;  students, such as Friedrich Adler (1879-1960), son of the Austrian Socialist Leader Victor Adler.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Einstein turned to Kleiner after his falling out with Weber in 1901, and changed his topic from thermoelectricity to molecular kinetics. A 1901 letter to Mileva Marić mentions that he had &quot;&lt;em&gt;spent the whole afternoon with Kleiner in Zurich and explained my ideas on the electrodynamics of moving bodies to him. ...He advised me to publish my ideas about the electromagnetic theory of light for moving bodies together with the experimental method.&lt;/em&gt;&quot; &lt;a href=&quot;https://en.wikipedia.org/wiki/Annus_Mirabilis_papers#Special_relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Electrodynamics of Moving Bodies&lt;/a&gt; was the paper where Einstein developed special relativity in 1905. However, when in 1908 Kleiner proposed an extraordinary professorship for theoretical physics, he thought that Adler was the best candidate. Adler declined, saying that Einstein's theoretical work was superior, and Einstein was then elected to the position. Einstein's overall characterization of Kleiner (in a letter to Laub) was ambivalent:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] &quot;&lt;em&gt;Professor Kleiner, the chief of our institute is perhaps not a magnificent physicist, but he is a wonderful man, whom I enjoy. It seems to me that scientific distinction and the worth of a personality do not always go together.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-11T23:33:45.467" LastActivityDate="2019-08-11T23:33:45.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9916" PostTypeId="1" CreationDate="2019-08-12T18:36:50.220" Score="0" ViewCount="105" Body="&lt;p&gt;Barrow surely discovered that the tangent to the area curve of a function at a point equals the value of the function at that point. Also, I’ve seen geometric proofs of this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But did he also discover that the area function of the tangent function (or derivative, in modern terms) of a function at a point also evaluated to the function value at that point plus some constant for all the points in the domain?&lt;/p&gt;&#xA;" OwnerUserId="10338" LastActivityDate="2019-08-12T21:49:32.610" Title="Did Isaac Barrow also discover the other thing about the inverse relation between area and tangent?" Tags="|mathematics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9917" PostTypeId="2" ParentId="9913" CreationDate="2019-08-12T21:07:35.440" Score="5" Body="&lt;p&gt;Hermann von Helmholtz not only had a formal education in medicine but even made his living of it for some time. Further in the past this was quite common among naturalists, philosophers and mathematicians, but designation of profession as&#xA;&quot;theoretical physicist&quot; narrows the time frame. There is no doubt however that Helmgoltz was a theoretical physicist, making seminal discoveries in physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He was a military doctor. When he published his paper &quot;On preservation of living force&quot; (vis viva in Latin, &quot;energy&quot; in modern language) he was praised by his military superiors for&#xA;finally doing some useful research:-)  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-08-26T12:38:59.283" LastActivityDate="2019-08-26T12:38:59.283" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9918" PostTypeId="2" ParentId="9916" CreationDate="2019-08-12T21:49:32.610" Score="5" Body="&lt;p&gt;Insofar as Barrow realised the first theorem, he also realised the second. Barrow's theorem X.11 can be interpreted as FTC1 &lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dt}\int_{a}^t y(x)dx = y(t)$$&lt;/span&gt; while his theorem XI.19 can be interpreted as FTC2 &lt;span class=&quot;math-container&quot;&gt;$$\int_{a}^b y'(x) dx = y(b)-y(a)$$&lt;/span&gt; But according to most historians it is more accurate to say that he did not realise either, and that this anachronistic interpretation of what he said does not reflect his own way of thinking. As Katz says in his history of mathematics, &quot;Barrow presented all of his work in a classic geometric form. It does not appear that he was aware of the fundamental nature of the two theorems [equivalent to FTC1 and FTC2] presented in the text. Barrow did not mention that they are particularly important; he just presented them as two among many geometrical results dealing with tangents and areas.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The work by Barrow in question has twice been translated into English:&#xA;&lt;a href=&quot;https://archive.org/details/geometricallect00newtgoog&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/geometricallect00newtgoog&lt;/a&gt;&#xA;&lt;a href=&quot;https://archive.org/details/geometricallectu00barruoft&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/geometricallectu00barruoft&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2019-08-12T21:49:32.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9919" PostTypeId="1" AcceptedAnswerId="9924" CreationDate="2019-08-13T04:00:27.877" Score="0" ViewCount="112" Body="&lt;p&gt;I am interested in some reference text, for example, textbook that would describe going from old times (say, Greece) till modern times:&#xA;1) definitions of units and measurements &#xA;2) describing experiments to motivate models like Newton's first law, Faraday's law&#xA;3) describing predictions made by models and what things it predicts right and which experiments show that, and what things it predicts wrong and which experiments show that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it might be possible that such text does not exist, but I would be happy to see some kind of subset of the things I described.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found many great historical books but most of them skip on the mathematical part by which I mean - let us take this specific idea (let's say equation), and find a prediction, and see that it works/fails in these experiments. Instead, usually, either the equation is just stated, or some results are just being told without specifically saying how that corresponds/agrees/disagrees with specific models of that time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, it might be possible that such text does not exist, but I would be happy to see some kind of subset of the things I described.&lt;/p&gt;&#xA;" OwnerUserId="10341" LastEditorUserId="426" LastEditDate="2019-08-14T07:51:26.237" LastActivityDate="2019-08-14T07:51:26.237" Title="Reference for history of physics with description of mathematical models" Tags="|physics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9920" PostTypeId="1" AcceptedAnswerId="9922" CreationDate="2019-08-13T04:46:45.050" Score="5" ViewCount="349" Body="&lt;p&gt;Today, &quot;reciprocity&quot; is the standard mathematical word used for &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratic_reciprocity&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadratic reciprocity&lt;/a&gt; and its generalizations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found that the name dates back to no later than 1832, when a paper of Dirichlet (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0009?tify=%7B%22pages%22:[385],%22view%22:%22scan%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crelle's Journal vol. 9 p. 379&lt;/a&gt;) made reference to the &quot;theorem so famous under the name of the law of reciprocity&quot; &lt;em&gt;(&quot;théorème si célèbre sous le nom de loi de réciprocité&quot;)&lt;/em&gt;. I don't actually know French, but it seems Dirichlet is saying that the name pre-dates his work and was already well-known by then.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, early writings on the subject do not use the term &quot;reciprocity&quot;. Gauss in &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt;, published in 1801, generally referred to quadratic reciprocity as the &quot;fundamental theorem&quot; &lt;em&gt;(theorema fundamentale)&lt;/em&gt;. He also called it the &quot;golden theorem&quot; &lt;em&gt;(theorema aureum)&lt;/em&gt; in his diary in 1796. From this, I would guess that the name &quot;reciprocity&quot; had not yet been coined then, or at least not yet popularized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who, then, introduced the term &quot;reciprocity&quot;? Did Gauss, or his predecessors such as Euler, use the word?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would be grateful for references attesting to the earliest uses of &quot;reciprocity&quot; in any language.&lt;/p&gt;&#xA;" OwnerUserId="2772" LastEditorUserId="55" LastEditDate="2019-08-13T09:09:45.073" LastActivityDate="2019-08-14T04:17:08.390" Title="Origin of the &quot;law of quadratic reciprocity&quot;" Tags="|mathematics|terminology|discoveries|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9921" PostTypeId="1" CreationDate="2019-08-13T08:36:31.433" Score="0" ViewCount="112" Body="&lt;ol&gt;&#xA;&lt;li&gt;Even if Babbitt taught merely elementary calculus, don't universities hire someone with at least an undergraduate degree in math to teach it? &#xA;&lt;a href=&quot;http://schoenblog.com/?p=943&quot; rel=&quot;nofollow noreferrer&quot;&gt;I quote Schoenblog.com&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Babbitt came to teach at Princeton in 1938—before there was even a music department. We did not have offices. We lived under very Spartan conditions.” When the U.S. entered World War II in 1941, Babbitt was working toward one of the first graduate degrees offered in music at Princeton. During the second semester of that year, he taught music and math (mostly &lt;strong&gt;elementary calculus [I bolded.]&lt;/strong&gt;). “Everybody suddenly began taking math courses that semester after Pearl Harbor. I was teaching on Saturdays.” The music section disappeared as everyone was either drafted or preparing for technical jobs in the army. Babbitt was sent by the Army to Washington, but soon was sent back to teach, because mathematicians were considered top priority. Babbitt stayed with the math department until the end of the war, when he made the not-so-difficult decision to return to music.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;In avouching his 'mathematical research in Washington, D.C', &lt;a href=&quot;https://en.wikipedia.org/wiki/Milton_Babbitt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; hints that Babbitt was skilled at more than elementary calculus. &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Babbitt's father was a mathematician, and it was &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Mathematics&quot;&gt;mathematics&lt;/a&gt; that Babbitt intended to study when he entered the &lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_Pennsylvania&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;University of Pennsylvania&quot;&gt;University of Pennsylvania&lt;/a&gt; in 1931. However, he soon left and went to &lt;a href=&quot;https://en.wikipedia.org/wiki/New_York_University&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;New York University&quot;&gt;New York University&lt;/a&gt; instead, where he studied music with &lt;a href=&quot;https://en.wikipedia.org/wiki/Philip_James&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Philip James&quot;&gt;Philip James&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Marion_Bauer&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Marion Bauer&quot;&gt;Marion Bauer&lt;/a&gt;. There he became interested in the music of the composers of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Second_Viennese_School&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Second Viennese School&quot;&gt;Second Viennese School&lt;/a&gt; and went on to write a number of articles on &lt;a href=&quot;https://en.wikipedia.org/wiki/Twelve_tone_music&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Twelve tone music&quot;&gt;twelve tone music&lt;/a&gt;, including the first description of &lt;a href=&quot;https://en.wikipedia.org/wiki/Combinatoriality&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Combinatoriality&quot;&gt;combinatoriality&lt;/a&gt; and a serial &quot;time-point&quot; technique. After receiving his bachelor of arts degree from &lt;a href=&quot;https://en.wikipedia.org/wiki/New_York_University_College_of_Arts_%26_Science&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;New York University College of Arts &amp;amp; Science&quot;&gt;New York University College of Arts &amp;amp; Science&lt;/a&gt; in 1935 with &lt;a href=&quot;https://en.wikipedia.org/wiki/Phi_Beta_Kappa&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Phi Beta Kappa&quot;&gt;Phi Beta Kappa&lt;/a&gt; honors, he studied under &lt;a href=&quot;https://en.wikipedia.org/wiki/Roger_Sessions&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Roger Sessions&quot;&gt;Roger Sessions&lt;/a&gt;, first privately and then later at &lt;a href=&quot;https://en.wikipedia.org/wiki/Princeton_University&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Princeton University&quot;&gt;Princeton University&lt;/a&gt;. At the university, he joined the music faculty in 1938 and received one of Princeton's first Master of Fine Arts degrees in 1942 (&lt;a href=&quot;https://en.wikipedia.org/wiki/Milton_Babbitt#CITEREFBarkin_&amp;amp;_Brody2001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barkin &amp;amp; Brody 2001&lt;/a&gt;). During the &lt;a href=&quot;https://en.wikipedia.org/wiki/Second_World_War&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Second World War&quot;&gt;Second World War&lt;/a&gt;, Babbitt divided his time &lt;strong&gt;between mathematical research in Washington, D.C.&lt;/strong&gt;, and Princeton, where he became a member of the mathematics faculty from 1943 to 1945 (&lt;a href=&quot;https://en.wikipedia.org/wiki/Milton_Babbitt#CITEREFBarkin_&amp;amp;_Brody2001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barkin &amp;amp; Brody 2001&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2019-08-13T22:52:14.700" Title="How did Milton Babitt teach math at Princeton and research math, without an undergraduate degree in math?" Tags="|teaching|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="9922" PostTypeId="2" ParentId="9920" CreationDate="2019-08-13T08:43:57.627" Score="7" Body="&lt;p&gt;The term (&lt;em&gt;loi de réciprocité&lt;/em&gt;) was introduced by Legendre in his book &lt;a href=&quot;https://archive.org/details/essaisurlathor00lege/page/214&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essai sur la Théorie des Nombres (1797)&lt;/a&gt;. It has (p.214) a section titled &lt;em&gt;Théoréme contenant une loi de réciprocité qui existe entre deux nombres premiers quelconques&lt;/em&gt; (Theorem containing a law of reciprocity that exists between&#xA;two arbitrary prime numbers), and he writes in the Preface (p. viii, my translation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A memoir that I published in the volume of the Academy&#xA;  Sciences for the year 1785, offers the means of demonstrating&#xA;  the principle in question, and contains, moreover,&#xA;  propositions that seem to advance the science of numbers.&#xA;  I gave &#xA;  1. A proof of a theorem to determine the possibility or impossibility of solving an indeterminate equation of the second degree reduced to the form &lt;span class=&quot;math-container&quot;&gt;$ax^2+by^2=cz^2$&lt;/span&gt;;&#xA;  2. A demonstration of a general law that exists between two&#xA;  arbitrary prime numbers, and which may be called the law of&#xA;  reciprocity...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is also the book where Legendre introduced what is now called the Legendre symbol, and formulated the law in terms of it. Neither the term nor the symbol appear in the mentioned &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k35847/f649.item.zoom&quot; rel=&quot;nofollow noreferrer&quot;&gt;Recherches d'analyse indéterminée (1785)&lt;/a&gt;, but instead we find an equivalent formulation split into eight separate cases. A good historical commentary is &lt;a href=&quot;https://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.210&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Legendre’s Work on the Law of Quadratic Reciprocity by Weintraub&lt;/a&gt;, who argues that the changes from 1785 to 1797 were more than cosmetic:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The first change is the introduction of what we now call the Legendre symbol &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{m}{n}\right)$&lt;/span&gt;. Here &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a prime and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is an arbitrary integer not divisible by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. From Theorem A he knows that &lt;span class=&quot;math-container&quot;&gt;$m^{(n-1)/2}$&lt;/span&gt; leaves a remainder of &lt;span class=&quot;math-container&quot;&gt;$\pm1$&lt;/span&gt; when divided by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, and then he sets &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{m}{n}\right)=1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; as this remainder is &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;. Legendre occasionally uses this symbol when &lt;span class=&quot;math-container&quot;&gt;$n=1$&lt;/span&gt;, in which case he sets &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{m}{n}\right)=1$&lt;/span&gt; for every nonzero integer &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. In the opinion of the author, this is more than a notational convenience. In introducing this notion, Legendre reifies this concept, and makes it into an object of independent study. This line of thought later led to the Jacobi symbol and the Hilbert symbol.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The second change is the introduction of the term “reciprocity”... Again,in the opinion of the author, this general heading reflects not only an appreciation of the importance of this overall result, but a conception of it as a single result (rather than a collection of eight results), and a conception of it as reflecting a&#xA;  relationship between distinct odd primes.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Euler conjectured an equivalent result in &lt;a href=&quot;http://eulerarchive.maa.org/pages/E552.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Observationes circa divisionem quadratorum per numeros primos (1772)&lt;/a&gt;, but did not call it anything in particular. He did introduce &quot;residue&quot;, and &quot;quadratic residue&quot;. Gauss became aware of Legendre's work late into his thinking on the subject, and was politely unimpressed by it, hence did not adopt the term. Weintraub writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We have mentioned that Euler stated a conjecture equivalent&#xA;  to the law of quadratic reciprocity (though it takes a bit of work to see that), but Euler’s&#xA;  statement seems not to have had any influence on either Legendre or Gauss. Euler&#xA;  coined the terms quadratic residue and nonresidue, which were not used by Legendre&#xA;  but were used by Gauss. Legendre coined the term quadratic reciprocity, but this was&#xA;  never used by Gauss, who always referred to this result as the “fundamental theorem&#xA;  (in the theory of quadratic residues),” nor did Gauss ever use the Legendre symbol in&#xA;  any of his works on the subject.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In the introduction to [3] Gauss writes that his work there had been done without&#xA;  knowledge of prior results in the subject. He also writes there that in the meanwhile,&#xA;  the “excellent” work [11] of the “highly deserving” Legendre appeared, but that he&#xA;  did not rewrite [3] to take it into account, only adding a few additional remarks in the&#xA;  Appendix.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As for Legendre's &quot;demonstration&quot;, he repeatedly uses the theorem on primes in arithmetic progressions, which is harder and was only proved by Dirichlet in 1837, and falls into what Cox called &quot;&lt;em&gt;a mess of circular reasoning. [...] At the 3rd edition (1830) of his Essay, there had been enough criticism of Legendre's proof for him to add Gauss's 3rd proof of reciprocity, as well as a statement by Jacobi (all while maintaining that his first proof was valid)&lt;/em&gt;&quot; (&lt;a href=&quot;https://books.google.com/books/about/Primes_of_the_form_x2_+_ny2.html?id=pSMlAQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cox, Primes of the Form &lt;span class=&quot;math-container&quot;&gt;$x^2 + Ny^2$&lt;/span&gt;&lt;/a&gt;). &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-14T04:17:08.390" LastActivityDate="2019-08-14T04:17:08.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9923" PostTypeId="2" ParentId="9921" CreationDate="2019-08-13T15:46:26.297" Score="1" Body="&lt;p&gt;It is important to realize that classes and operations at US universities during World War 2 had little in common with today, or even with their operations pre-war. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As mentioned in &lt;a href=&quot;https://blogs.princeton.edu/mudd/2010/09/soldiers-on-campus-circa-1943/&quot; rel=&quot;nofollow noreferrer&quot;&gt;a blog from the Princeton Archives&lt;/a&gt;, before Pearl Harbor Princeton had an undergraduate enrollment of about 2400. In November 1943, there were about 3100 military personnel and some 650 non-military students in residence. The military personnel were a variety of programs, mostly officer training and technical schools. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There would have been a huge demand for fairly basic math instruction, and the easiest way to accommodate that would be to find instructors from the current faculty. So, tapping a music faculty with a math professor father would be entirely reasonable. This was not about providing graduate level math, it was about providing simple instruction. I highly doubt anything above calculus was required, and there would be significant need for lower levels than that. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for math research in DC, there would be plenty of things to be sorted out in ballistics, operations research, etc. that also do not require advanced math beyond calculus. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;To add further, in an &lt;a href=&quot;http://musicmavericks.publicradio.org/features/interview_babbitt.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Interview&lt;/a&gt; he is quoted as saying:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I was teaching there for four years before the war came, and then I was sent to Washington to spend a few years doing what, I cannot reveal. Then, by one of these incredible accidents of the military or government or we not know what, I was sent back to Princeton to teach mathematics. There was a moment in the war when people were being called back from the Battle of the Bulge to teach mathematics; it was the highest priority undertaking because there was radar, there was sonar, there were all these things and there were these kids who didn't have enough math to learn how to do it. Even MIT graduates had to be retrained, because they didn't know how to use a slide rule. So I did that at Princeton—it was by accident that it was Princeton, I'll never know how it became Princeton, again, because it had nothing to do with my music department or anything else&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2019-08-13T22:52:14.700" LastActivityDate="2019-08-13T22:52:14.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9924" PostTypeId="2" ParentId="9919" CreationDate="2019-08-13T15:55:15.693" Score="1" Body="&lt;p&gt;I would recommend&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;René Dugas, &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=4429&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;A History of Mechanics&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;, trans. John Royden Maddox.&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.goodreads.com/book/isbn/0486656322&quot; rel=&quot;nofollow noreferrer&quot;&gt;GoodReads reviews&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;blockquote&gt;Monumental study traces the history of mechanical principles chronologically from their earliest roots in antiquity through the Middle Ages to the revolutions in relativistic mechanics and wave and quantum mechanics of the early 20th century. Contributions of ancient Greeks, Leonardo, Galileo, Kepler, Lagrange, many other important figures. 116 black-and-white illustrations.&lt;/blockquote&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;It starts with Hellenistic mechanics and ends with quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For electromagnetism:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;E. T. Whittaker, &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=3705&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History of the Theories of Aether &amp;amp; Electricity&lt;/em&gt;&lt;/a&gt;, 1910.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For thermodynamics:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Clifford Ambrose Truesdell, &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=4548&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Tragicomical History of Thermodynamics, 1822-1854&lt;/em&gt;&lt;/a&gt; (New York, NY: Springer New York, 1980).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastActivityDate="2019-08-13T15:55:15.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9925" PostTypeId="2" ParentId="3131" CreationDate="2019-08-14T18:37:41.990" Score="6" Body="&lt;p&gt;Roger Cotes, a contemporary of Newton and best known for the Newton-Cotes quadrature rules in numerical integration, died of a fever aged only 33.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton supposedly said of his death 'If he had lived we would have known something': this is extremely high praise given that Newton was generally not inclined to praise the work or intellect of others. &lt;/p&gt;&#xA;" OwnerUserId="10345" LastActivityDate="2019-08-14T18:37:41.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9927" PostTypeId="1" CreationDate="2019-08-15T16:21:09.617" Score="3" ViewCount="477" Body="&lt;p&gt;I've read a lot about the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Magnus_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;Magnus effect&lt;/a&gt; altering the trajectories of cannonballs and musketballs. Robins noticed it with Musket balls and Magnus with canonballs, but presumably they weren't the first to notice. Does anyone know of any examples of where the Magnus effect was noticed in the theatre of war itself, and if so, any effects it had on the battle?&lt;/p&gt;&#xA;" OwnerUserId="10347" LastEditorUserId="9523" LastEditDate="2019-08-28T10:24:18.853" LastActivityDate="2019-08-28T10:24:18.853" Title="Does anyone know of any examples of the Magnus effect in a real battle?" Tags="|experimental-physics|mathematical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9928" PostTypeId="2" ParentId="9927" CreationDate="2019-08-16T07:56:48.687" Score="8" Body="&lt;p&gt;I am afraid nobody noticed it, because nobody could have noticed it. A deviation is only a deviation when one has something that it is a deviation from. To &quot;notice&quot; the Magnus effect one has to operate from a theory that predicts a parabolic trajectory, and be able to separate the subtle deviations caused by spin from other deviations. Hall even wrote that the smoothbore &quot;&lt;em&gt;gun itself was so inconsistent in its behavior that great accuracy in preliminary work, even in the lay of the gun itself, was labour in vain&lt;/em&gt;&quot;. So it is not something that can be simply &quot;observed&quot; without theoretical analysis, and the prerequisites for that were not in place prior to Newton. If then, since it requires fluid mechanics. Even after that, separating the Magnus effect from everything else required careful experimental setups impossible in real battles. Robins (1747), Magnus (1852), Lafay (1912) all staged controlled experiments to &quot;observe&quot; the effect, or rather to confirm theoretical predictions that took it into account.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tartaglia, for example, depicted trajectories of cannonballs as straight line segments connected by a circular turn to vertical descent. Even Galileo in Two New Sciences (1638) treated &quot;these violent shots&quot; as a special case, mentioned that the “beginning of the parabola” may be flatter, listed a number of complicating factors, and admitted his inability to calculate their effects. As Brugh summarizes in &lt;a href=&quot;http://springer.iq-technikum.de/referenceworkentry/10.1007/978-3-319-02848-4_902-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ballistics in Renaissance Science&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The Magnus effect (1852) and Bernoulli’s principle (1738) have helped to explain the inaccuracy of smoothbore ballistics in retrospect, but the operators of early modern firearms were more interested in developing battlefield tactics for overcoming these shortfalls than explaining them. Since smoothbore weapons fired rounded shot from unrifled gun barrels, slight imperfections in the shot or the barrel could send the shot spinning on an axis, giving it unpredictable angular momentum... Thanks to the inaccuracy and unreliability of early modern firearms, tacticians and commanders of the sixteenth and seventeenth (even eighteenth) centuries took a stochastic approach to gunpowder weapons for both the infantry and artillery. Infantry and heavy cavalry units, especially in the late 1500s, developed tactics and drills like the countermarch, the caraçol, and weapons drill that sought to sustain high rates of fire on the battlefield in order to increase the chances of hitting targets.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Newton mentioned the effect in connection with tennis balls rather than warfare in a 1671 letter to Oldenburg, but he more speculated than observed it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I remembered that I had often seen a tennis ball struck with an oblique racket describe such a curved line. For a circular as well as progressive motion being communicated to it by that stroke, its parts on that side where the motions conspire must press and beat the contiguous air more violently, and there excite a reluctancy and reaction of the air proportionately greater.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it is not surprising that no real &quot;observations&quot; are known before Robins &quot;&lt;em&gt;who was in gunnery what the immortal Newton was in philosophy, the founder of a new system&#xA;deduced from experiment and nature, the service of artillery was a mere matter of chance, founded on no principles, or at best, but erroneous ones&lt;/em&gt;&quot;, as an artillery officer wrote in 1789. In particular, Robins was first to account for various effects of the air-resistance on high-speed projectiles, and to separate the effects of the spin. His best known work is New Principles of Gunnery (1742), but the &quot;Magnus&quot; effect is predicted and illustrated experimentally in Of the Nature and Advantage of Rifled Barrel Pieces presented to the Royal Society in 1747. Here is from &lt;a href=&quot;https://www.researchgate.net/publication/274799077_Muskets_and_Pendulums_Benjamin_Robins_Leonhard_Euler_and_the_Ballistics_Revolution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Steele's Muskets and Pendulums: Benjamin Robins, Leonhard Euler, and the Ballistics Revolution&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In one of his experiments, he bent a fixed musket barrel a few&#xA;  degrees to the left. The bullet he shot had a trajectory that completely&#xA;  coincided with his theory. The bullet holes in the tissue-paper curtains&#xA;  showed that the bullet first began moving toward the left, in the&#xA;  direction of the musket's deflection. Eventually the bullet reversed its&#xA;  lateral direction of motion and crossed to the right side of the musket.&#xA;  Robins explained this apparent paradox by noting that the deflected&#xA;  musket forced the bullet to rotate from the left to the right. This would&#xA;  cause the air flow to be greater on the right-hand side; hence, the&#xA;  friction or &quot;pressure&quot; force pushed the projectile to the right, which its&#xA;  trajectory demonstrated. This phenomenon is now called the Magnus&#xA;  effect, named after a 19th-century German physicist who investigated it&#xA;  with full knowledge of Robins's work.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;While Robins's explanation is very similar to Newton's, he was first to back it up by theoretical calculations and a convincing experiment.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-18T02:08:56.540" LastActivityDate="2019-08-18T02:08:56.540" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9929" PostTypeId="1" AcceptedAnswerId="9930" CreationDate="2019-08-17T06:18:25.733" Score="4" ViewCount="375" Body="&lt;p&gt;I recently read &lt;a href=&quot;https://www.japantimes.co.jp/news/2019/08/15/business/hagoromo-japans-rolls-royce-chalk-continues-writing-legacy-south-korea/&quot; rel=&quot;noreferrer&quot;&gt;Hagoromo, the 'Rolls Royce of chalk,' continues writing its legacy in South Korea&lt;/a&gt; article recently, and was fascinated by the huge amount of attention this specific chalk is getting. However, it was my first time hearing about it, even though I am quite an avid follower of such news. When did this brand of chalk start to gain popularity in the West?&lt;/p&gt;&#xA;" OwnerDisplayName="user10205" LastEditorUserId="55" LastEditDate="2019-08-17T07:20:44.217" LastActivityDate="2019-08-18T08:17:18.683" Title="How did Hagoromo Fulltouch chalk gain so much popularity among mathematicians in the West?" Tags="|mathematics|mathematicians|social-context|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9930" PostTypeId="2" ParentId="9929" CreationDate="2019-08-17T07:16:10.417" Score="4" Body="&lt;p&gt;The hype is recent, although the company, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hagoromo_Bungu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hagoromo Bungu&lt;/a&gt;, operated in Japan since 1965, and the brand was popular in Japanese classrooms. Apparently, it started with David Eisenbud, then a math professor at UC Berkeley and the director of MSRI, a prestigious research institute, and Kyung Lee, a Japanese filmmaker of Korean descent. He came (to the University of Tokyo in the early 2000s), he saw, he was swept off his feet. She became his supplier. Although the story has an air of an anecdote, Eisenbud himself tells an abbreviated version of it in the video &lt;a href=&quot;https://www.youtube.com/watch?v=PhNUjg9X4g8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why the World’s Best Mathematicians Are Hoarding Chalk&lt;/a&gt;, where he and other well-known mathematicians sing praises to the Hageromo Fulltouch. Here is the full story in &lt;a href=&quot;https://alumni.berkeley.edu/california-magazine/just-in/2019-06-14/chalk-market-where-mathematicians-go-get-good-stuff&quot; rel=&quot;nofollow noreferrer&quot;&gt;McDonald's retell&lt;/a&gt;, published in Berkeley's alumni magazine:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It all started a year prior when UC Berkeley math professor David Eisenbud visited the University of Tokyo. His host said to him, “You know, we have better chalk than you.” Eisenbud replied, “No you don’t. Chalk is chalk.” When he returned to his office, there was a box of Hagoromo brand chalk waiting for him. The moment he placed a piece of that chalk against a blackboard, felt the smooth action, and saw the crisp, thick line it produced, Eisenbud realized he’d been wrong: There’s chalk, and then there’s Hagoromo. Back in his role as director of the Mathematics Sciences Research Institute (MSRI)—the prestigious “Top Gun” of math perched high in Berkeley hills—he spoke effusively about the chalk to anyone who would listen, lamenting that it couldn’t be obtained in the U.S. One of those who overheard his ravings was Lee, who at the time was editing a film sponsored by MSRI about prime numbers.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;On her next trip to Japan, she paid a visit to Hagoromo Bungu, the small factory that had produced the chalk since 1965 (an earlier incarnation of the business was destroyed in WWII). There she met with the company president, Takayasu Watanabe, who showed her how the chalk was made. “It’s complete craftsmanship,” she says. “They change the portion of ingredients constantly, like they were experimenting with it. It is a good product.” Watanabe told her that Hagoromo Bungu had never extended its market beyond Japan and Korea, in part because he was uncomfortable doing business in English. Lee left with more than sixty cases of Hagoromo chalk, poised to become the only source for the precious stuff in the United States.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Her first customer was Eisenbud. The rest were mathematicians he sent her way. In America, chalk is cheap. Lee’s went for around $18 a box. Nonetheless, when she kicked off her small operation in 2012, grad students, post docs, and professors from all over the country flocked to her because they had nowhere else to turn to get their fix. Hagoromo chalk is a bit thicker than standard American chalk. And it’s coated, giving it a slick feel and protecting the fingers from dust. It glides smoothly across the blackboard, producing a consistent line with sharply defined edges. And it’s durable, difficult to snap between the fingers&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-08-18T08:17:18.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9931" PostTypeId="1" CreationDate="2019-08-18T05:39:56.607" Score="6" ViewCount="5497" Body="&lt;p&gt;It's supposedly a &quot;widely circulated&quot; false claim that wings generate lift because of their asymmetric shape, forcing air above to travel faster so that they meet up on the trailing edge at the same time as air below, and by Bernoulli's equation, the pressure below the wing must therefore be higher.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/24VM8.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/24VM8.gif&quot; alt=&quot;Explanation (with diagram) of equal transit-time theory&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Source:  &lt;a href=&quot;https://www.grc.nasa.gov/WWW/k-12/airplane/wrong1.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.grc.nasa.gov/WWW/k-12/airplane/wrong1.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I've only ever encountered this &quot;equal transit-time&quot; theory when it's being debunked, so it seems like its more common to see it pointed out as wrong rather than alleged to be true.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was this incorrect theory ever a common misconception? And did this misconception originate from a particular source?&lt;/p&gt;&#xA;" OwnerUserId="8004" LastEditorUserId="55" LastEditDate="2019-08-18T09:32:58.187" LastActivityDate="2021-11-03T06:21:23.123" Title="Where did the false &quot;equal transit-time&quot; explanation of lift originate from?" Tags="|physics|debunking|education|aerodynamics|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9932" PostTypeId="2" ParentId="9931" CreationDate="2019-08-18T09:32:02.120" Score="5" Body="&lt;p&gt;The popularity of the equal transit-time fallacy is a bit more complicated than a mistake spreading from a single source. It is simple, intuitively appealing (blowing over an airfoil is often invoked, along with an erroneous picture of flow lines around an asymmetric wing), and gets things done quickly. Just like the &quot;explanation&quot; of seasons by the Earth's changing distance to the Sun, or textbook pseudo-explanations of history, like the &quot;crisis&quot; over the discovery of irrationals, or Maxwell's &quot;mathematical&quot; reason for adding an extra term to Ampere's equation. &quot;The truth is what works&quot;. Eastwell in &lt;a href=&quot;https://eric.ed.gov/?id=EJ1050910&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernoulli? Perhaps, but What about Viscosity?&lt;/a&gt; lists the following motivations:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;• Simplified Bernoulli explanations are quick, sound logical, and make correct predictions. As Brusca (1986b) said with a sense of satisfaction, the prediction is “in complete agreement with what happens in practice” (p. 15). This would be fabulous if it wasn’t for&#xA;  the fact that these explanations are also wrong!&lt;br&gt;&#xA;  • Statements like “as the speed of a moving fluid increases, the pressure within the fluid decreases” facilitate a misunderstanding of Bernoulli’s principle and, when used in a sweeping sense and therefore out of context, are wrong.&lt;br&gt;&#xA;  • Viscosity, entrainment, and the Coanda effect are not to be found in lower-level literature, despite the fact that such literature deals with phenomena that rely on these concepts.&lt;br&gt;&#xA;  • There appears to be a desire to have a single, best explanation for an observed behaviour when in fact a combination of factors may be “at play”&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The fallacy was likely independently reinvented by multiple textbooks authors and physics teachers. Norman Smith, who was one of the first to sound the alarm back in 1972, says the following in &lt;a href=&quot;https://aapt.scitation.org/doi/10.1119/1.2352317&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernoulli and Newton in Fluid Mechanics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Millions of children in science classes are being... told&#xA;  that Bernoulli's theorem is responsible for lift on the airplane wing and for the force that makes a spinning baseball travel in a curved path. Unfortunately, the &quot;dynamic lift&quot; involved in each of these items is not properly explained by Bernoulli's theorem.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;First, an airfoil need not have more curvature&#xA;  on its top than on its bottom. Airplanes can and do fly&#xA;  with perfectly symmetrical airfoils; that is, with airfoils&#xA;  that have the same curvature top and bottom. Second,&#xA;  even if a humped-up (cambered) shape is used, the claim&#xA;  that the air must traverse the curved top surface in the&#xA;  same time as it does the flat bottom surface (or that the&#xA;  molecules must meet again) is fictional. We can quote no&#xA;  physical law that tells us this. Third - and this is the&#xA;  most serious - the common textbook explanation, and&#xA;  the diagrams that accompany it, describe a force on the&#xA;  wing with no net disturbance to the airstream. This&#xA;  constitutes a violation of Newton's third law.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;The use of Bernoulli apparently began in this country some 30 years or more ago and has spread throughout school science books and popular literature to exclude virtually any mention of Newton and momentum.&#xA;  College-level aerodynamics textbooks generally do not use this approach, though it would not be surprising to find a few exceptions... It might be well to point out that the Bernoulli explanation is used mostly in the United States British, for example, seem to use Newton's third law in textbooks and popular literature&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Writing in 2007, Eastwell notes that things did not change much, if not gotten worse:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Such a pressure difference is commonly justified, on the basis of Bernoulli's Principle, by statements such as “when air sweeps across a surface at high speed the pressure on that surface is lowered” (“Bernoulli Station,” 1989, p. 308) or “as the speed of a moving fluid increases, the pressure within the fluid decreases” (Mitchell, n.d., n. 1), or it is implied on this basis (e.g., Brusca, 1986b). This reasoning for the pressure difference, found not only in popular writings but also in specialist, peer-reviewed journals (e.g., see also Bauman &amp;amp; Schwaneberg, 1994; Holmes, 1996), is wrong.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now that even Wikipedia has a subheading &lt;a href=&quot;https://en.wikipedia.org/wiki/Lift_(force)#False_explanation_based_on_equal_transit-time&quot; rel=&quot;nofollow noreferrer&quot;&gt;False explanation based on equal transit-time&lt;/a&gt;, perhaps the tide has finally turned. It is not that the use of Bernoulli's equation is entirely wrong, it is rather that  it comes with a number of erroneous add-ons. As Beatty pointed out in &lt;a href=&quot;http://amasci.com/wing/airfoil.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Airfoil Lifting Force Misconception&#xA;Widespread in K-6 Textbooks&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Air-deflection and Newton's Laws explain 100% of the lifting force. Air velocity and Bernoulli's equation also explains 100% of the lift. There is no 60% of one and 40% of the other. One of them looks at pressure forces, the other looks at &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt; accelerated mass. For the most part they're just two different ways of simplifying a single complicated subject. Much of the controversy arises because one side or the other insists that only &lt;strong&gt;their&lt;/strong&gt; view is correct.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Social psychology aside, there are also several serious mistakes usually associated with the &quot;popular&quot; explanation described above. Those who believe the &quot;popular&quot; explanation are wrongly insisting that any parcels of air divided by the wing's leading edge must meet again at the trailing edge. This is incorrect. Actually it doesn't even occur: experiments easily show that the air above a wing far outraces the air below, and parcels never meet again. (In fact, if a wing is adjusted so the parcels really do merge, this is always the zero-lift configuration!) The same people also believe that wings fly only because of pressure, and that wings don't need to deflect the oncoming air downwards. Also incorrect. These and several other mistakes commonly appear in elementary science texts, as well as in popular articles about aircraft physics. These mistakes change the popular &quot;airfoil-shape&quot; explanation into a system of misconceptions.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-19T03:28:02.750" LastActivityDate="2019-08-19T03:28:02.750" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="9934" PostTypeId="2" ParentId="9931" CreationDate="2019-08-18T18:54:19.657" Score="3" Body="&lt;p&gt;The asymmetric shape (not its explanation with &quot;equal transit time&quot;!)  comes from the first mathematical theory that explained the lifting force (Chaplygin's formula, known in the West as &quot;Blasius theorem&quot;).&#xA;I suppose that the theory of &quot;equal times&quot; was developed in attempts to explain the Chaplygin-Joukowski theory to non-mathematicians:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Under certain circumstances this theory gives a good agreement with experiment. Since a complete mathematical theory does not exist to this day,&#xA;the subject is open to all sorts of speculations on &quot;what causes the lifting force&quot;.&#xA;A sound argument against the &quot;asymmetric theory&quot; is that an airplane can fly upside down:-) &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-08-18T18:54:19.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9936" PostTypeId="2" ParentId="9931" CreationDate="2019-08-19T08:41:08.123" Score="3" Body="&lt;p&gt;This explanation has probably been independently 'rediscovered' many times, and it is unlikely to be able to point to a particular origin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ackroyd says in &lt;a href=&quot;https://www.aerosociety.com/media/4841/babinsky-s-demonstration-the-theory-of-flight-and-its-historical.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Babinsky's Demonstration: The Theory of Flight and Its Historical Background' (Journal of Aeronautical History, 2015)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Imagine that two adjacent air elements ... A and B ... are about to&#xA;  reach the aerofoil’s nose. Element A passes above the aerofoil whilst&#xA;  element B passes beneath. Eventually element A reaches the aerofoil’s&#xA;  tail or trailing edge and element B, according to this argument, must&#xA;  reach the tail at the same time so as to regain its position adjacent&#xA;  to element A ... this flawed explanation of flight has survived and&#xA;  still enjoys wide belief, probably because it has the appeal of&#xA;  simplicity.  Its origin now seems to be lost in the mists of time&#xA;  although it may be that its use in aircrew training manuals gained for&#xA;  it a popularity which extended to school textbooks and even the&#xA;  occasional British A Level examination paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h1&gt;Some sources say this explanation is sometimes used in flight training manuals.&lt;/h1&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The above explanation is extremely wide-spread. It can be found&#xA;  in many textbooks and, to my knowledge, it is also used in the &#xA;  RAF's instruction manuals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: Holger Babinsky (2003). How do wings work?. Phys. Educ. Vol. 38. p.498. &lt;br&gt;&#xA;Accessed: &lt;a href=&quot;http://www3.eng.cam.ac.uk/outreach/Project-resources/Wind-turbine/howwingswork.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www3.eng.cam.ac.uk/outreach/Project-resources/Wind-turbine/howwingswork.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Ballow speaking about his experience when learning from Air Force training manuals while a member of the Nevada Air National Guard:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The training covered the theory of flight as outlined and explained in&#xA;  the Air Force Training Manual [...] Lift is generated by the air&#xA;  passing over the wing. since the upper surface of the wing is curved&#xA;  and the lower surface is straight, the air passing over the upper&#xA;  surface of the wing has farther to travel, and so it is thinned out.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: Tom Ballow (2005). The New Science of Flight and Movement. p.13. &lt;br&gt;&#xA;Accessed: &lt;a href=&quot;https://books.google.com.au/books?id=Wba1JfqHmDYC&amp;amp;pg=PA13&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com.au/books?id=Wba1JfqHmDYC&amp;amp;pg=PA13&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Discussion on the r/flying subreddit:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.reddit.com/r/flying/comments/70klbo/dpe_used_the_equal_transit_time_theory_of_lift_as/&quot; rel=&quot;nofollow noreferrer&quot;&gt;DPE used the 'equal transit time theory' of lift as the correct way to explain lift. What do?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Posts on Aviation Stack Exchange:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://aviation.stackexchange.com/q/1157&quot;&gt;Why is the wrong explanation of “air travels a longer distance and creates a lift” so popular?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://aviation.stackexchange.com/q/8281&quot;&gt;Principle of aerodynamic lift: are misconceptions also taught in flight schools?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h1&gt;Here are some specific instances of where the &quot;equal transit-time&quot; argument is used.&lt;/h1&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The distance over the top of the blade with the angle of attack is&#xA;  greater than the distance along the bottom surface of the rotor blade.&#xA;  Air molecules that pass over the top must move faster than those&#xA;  passing under the bottom to meet at the same time along the trailing&#xA;  edge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: Federal Aviation Administration (2012). Helicopter Instructor's Handbook, Chapter 3 Aerodynamics of Flight. p.2. &lt;br&gt;&#xA;Accessed: &lt;a href=&quot;https://www.faa.gov/regulations_policies/handbooks_manuals/aviation/media/FAA-H-8083-4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.faa.gov/regulations_policies/handbooks_manuals/aviation/media/FAA-H-8083-4.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The shape of a wing looks like an elongated water drop laying on its&#xA;  side ... Usually the top is curved more than the bottom making the&#xA;  upper surface slightly longer than the bottom. Since air passing over&#xA;  the top and bottom must reach the rear of the wing at the same time,&#xA;  the air passing over the top must not only travel faster, but also changes direction&#xA;  and is deflected downward.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: DoD 101 - An Introduction to the Military, Aircraft for Amateurs &lt;br&gt;&#xA;Accessed: &lt;a href=&quot;https://fas.org/man/dod-101/sys/ac/intro.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://fas.org/man/dod-101/sys/ac/intro.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The next examples are from older journal papers.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;This acceleration of the particles over the top side of &quot;the shape&quot;&#xA;  will hasten the upper air-particles toward the trailing edge, so as to&#xA;  make them &quot;join up&quot; with their immediate bottomside neighbors, from&#xA;  which they were separated at the leading edge. Particle pairs thus hit&#xA;  the leading edge together, are separated, and yet arrive at the&#xA;  trailing edge together—although the topside particle had to travel&#xA;  much faster to do so.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: Norwood Russell Hanson (1965). Aristotle (And Others) on Motion through Air. The Review of Metaphysics, Vol. 19, No. 1. &lt;br&gt;&#xA;Link: &lt;a href=&quot;https://www.jstor.org/stable/20124101&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/20124101&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If we consider a wing passing through stationary air, the principal&#xA;  effect on the air is the downward thrust. To the approximation that&#xA;  this is the only net effect, the time required for passage over and&#xA;  under the wing is the same. The air above must therefore be traveling&#xA;  faster (relative to the wing) to keep up with the lower streamlines.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source: Robert P. Bauman and Rolf Schwaneberg (1994). Interpretation of Bernoulli’s equation. The Physics Teacher Vol. 32. p.485. &lt;br&gt;&#xA;Link: &lt;a href=&quot;https://aapt.scitation.org/doi/pdf/10.1119/1.2344087&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://aapt.scitation.org/doi/pdf/10.1119/1.2344087&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;h1&gt;As for why &quot;equal transit-time&quot; ever had to be invoked.&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;I could not find any physical justifications for it, perhaps its surface level intuitiveness stems from being conflated with some kind of continuity equation/condition, almost as if air parcels split at the leading edge, must rejoin at the trailing edge otherwise there would be a &quot;build up of air&quot; on the side with the longer path (the &quot;equal transit-time&quot; requirement is of course not necessary to avoid this). This Quora question seems to suggest there is some confusion related to this: &lt;a href=&quot;https://qr.ae/TWrahV&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why continuity equation must be followed by aerofoil?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8004" LastEditorUserId="8004" LastEditDate="2019-08-23T05:21:59.487" LastActivityDate="2019-08-23T05:21:59.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9938" PostTypeId="1" AcceptedAnswerId="9939" CreationDate="2019-08-19T13:32:53.470" Score="4" ViewCount="170" Body="&lt;p&gt;I read in a book of popularization of Mathematics that in 1830 &lt;a href=&quot;https://en.wikipedia.org/wiki/Mikhail_Ostrogradsky&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mikhail Ostrogradsky&lt;/a&gt; wrote, about non-Euclidean Geometry, that he did not see why anyone would care about that, since the world is obviously Euclidean. Has anyone a good source for this quotation (assuming that it is real)? The closest thing that I was able to find was this sentence, from Jeremy Gray's &lt;em&gt;Worlds out of Nothing&lt;/em&gt;: &amp;ldquo;Russian authorities like Ostrogradsky denigrated Lobachevsky's work, which they simply did not understand.&amp;rdquo;&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="55" LastEditDate="2019-08-20T00:11:45.240" LastActivityDate="2019-08-20T21:03:10.847" Title="Did Ostrogradsky dismiss Lobachevsky's book on non-Euclidean geometry &quot;because the world is obviously Euclidean&quot;?" Tags="|euclidean-geometry|quote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9939" PostTypeId="2" ParentId="9938" CreationDate="2019-08-20T00:10:37.363" Score="5" Body="&lt;p&gt;Ostrogradsky is well-known for his negative reaction to Lobachevsky's work, but in his signed reviews, at least, his complaints were different. Lobachevsky, in contrast to Ostrogradsky, was not a good expositor, and the paper he submitted to the Academy was obscurely written. Apparently, Ostrogradsky was only able to make sense of two integrals computed using hyperbolic geometry, in one of which he saw an error (the manuscript was generally full of typos, and some authors say that Ostrogradsky himself made a mistake). The text of Ostrogradsky's review does not seem to be available online, but &lt;a href=&quot;https://arxiv.org/abs/1307.4902&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prasolov-Skopenkov&lt;/a&gt; and &lt;a href=&quot;http://www.mathnet.ru/links/f75e961602683bf903b7ae3e8891a3b4/mp331.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Papadopoulos&lt;/a&gt; quote Rosenfeld's summary of it. Here is edited Google translation: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Pointing out that of the two definite integrals, on the calculation of which, using his new method,  Mr. Lobachevsky laid claim, one is already known and easily derived using integral calculus and the other is wrong, Mr. Ostrogradsky remarks, moreover, that the work is done with such little diligence that most of it is incomprehensible. Therefore, he believes that this work of Mr. Lobachevsky did not merit the attention of the Academy.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What comes closest to &quot;since the world is obviously Euclidean&quot; (and I am not sure it is that close) is not  Ostrogradsky's 1832 review, but a review published two years later in Bulgarin's journal The Son of the Fatherland, which many call a smear, and some associate with Ostrogradsky.  It does touch on the same themes as his review (obscurity, error in the integral), but is written in an exquisitely condescending and mocking tone aimed at public humiliation. The lengthiest quotes are given in &lt;a href=&quot;https://books.google.com/books/about/In_The_Search_For_Beauty_Unravelling_Non.html?id=CdiEDwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smilga's In the Search for Beauty: Unravelling Non-Euclidean Geometry&lt;/a&gt;, along with detailed analysis of its rhetorical style:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Many of our first-rate mathematicians have read it, thought about it and still do not see the point.&#xA;  After that I hardly need say that I, having&#xA;  thought over this book for some time, could&#xA;  think of nothing to say; in other words, I hardIy&#xA;  understood a single idea... It is oven difficult to understand how Mr. Lobachecvsky was able&#xA;  to concoct out of the simplest and clearest chapter   of mathematics that we know geometry to&#xA;  be - how he could build such an abstruse, murky&#xA;  and impenetrable theory, if it were not that&#xA;  he himself helped us by saying that his Geometry&#xA;  differs from the common kind that we all&#xA;  studied and which, most likely, we cannot unlearn,&#xA;  and is only an imaginary geometry. Yes,&#xA;  that makes things clear indeed.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Just try to picture what a lively, yet monstrous,&#xA;  imagination can conjure up! Why, for instance, not try to imagine black to be white, round to be quadrangular, the sum of all side angles in a right triangle to be less than two right angles and one and the same definite integral to be equal first to &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt;, then to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;. Very, very possible, yet to normal reason it is meaningless.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;It would seem that after a few definitions,&#xA;  composed with the same art and the same&#xA;  precision as the preceding ones, the author says&#xA;  something about triangles, about the dependence&#xA;  of the angles in them upon two sides- therein&#xA;  lies the difference between his geometry&#xA;  and ours - he then proposes a new theory of&#xA;  parallels, which - and he admits as much - nobody&#xA;  is capable of proving whether it exists&#xA;  in nature or not; finally, this is followed by a&#xA;  consideration of how, in this imaginary geometry,&#xA;  one determines the magnitude of curved&#xA;  lines, of areas, of curved surfaces and volumes&#xA;  of solids. And all of this, I must repeat once&#xA;  again, is written so that nothing at all can be&#xA;  understood... I cannot&#xA;  refrain from blaming him slightly for not&#xA;  giving his book a proper title and compelling&#xA;  us to cogitate so wastelully for such a long&#xA;  time. Why, instead of the title The Principles&#xA;  of Geometry, did he not have named it,&#xA;  say, A Satire On Geometry, A Caricature On&#xA;  Geometry, or something of that nature?&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As Smilga says, &quot;&lt;em&gt;after reviews of that kind, people take to their beds, give up work altogether, or even&#xA;commit suicide&lt;/em&gt;&quot;. Some Russian sources associate the &quot;review&quot; (signed SS) with Zeleny and Burachek, who taught at the same Navy college as Ostrogradsky,  Burachek worked for The Son of the Fatherland. But the evidence is circumstantial, and there are alternative speculations involving the Academy's secretary Fuss, for example. Even if Ostrogradsky was the inspiration, apparently the matter was complicated by Lobachevsky's terse style, and another Lobachevsky (distant relative) submitting a cranky paper on the circle quadrature that Ostrogradsky also happened to review.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-20T08:09:50.253" LastActivityDate="2019-08-20T08:09:50.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9940" PostTypeId="1" CreationDate="2019-08-20T07:16:24.223" Score="1" ViewCount="313" Body="&lt;p&gt;The &lt;a href=&quot;http://www.oed.com/view/Entry/132362&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;OED&lt;/em&gt;&lt;/a&gt; doesn't expound what semantic notions underlie  y-coordinate and the Latin etymon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/StaDp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/StaDp.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Etymology:&lt;/strong&gt; &amp;lt; classical Latin &lt;em&gt;ōrdinātus&lt;/em&gt; orderly, regular, regulated, (in geometry) in alignment, parallel, use as adjective of past participle of &lt;em&gt;ōrdināre&lt;/em&gt; &lt;a href=&quot;https://www-oed-com.ezproxy.torontopubliclibrary.ca/view/Entry/132325#eid33278308&quot; rel=&quot;nofollow noreferrer&quot;&gt;ordain v.&lt;/a&gt;  Compare slightly earlier &lt;a href=&quot;https://www-oed-com.ezproxy.torontopubliclibrary.ca/view/Entry/132364#eid33306637&quot; rel=&quot;nofollow noreferrer&quot;&gt;ordinately adv.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2019-08-20T17:39:12.107" Title="Why was 'ordinate' adopted to signify y-coordinate?" Tags="|terminology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9941" PostTypeId="2" ParentId="9940" CreationDate="2019-08-20T08:09:02.277" Score="6" Body="&lt;p&gt;Descartes was motivated by proto-coordinate constructions in Apollonius's Conica, who regularly drew parallel segments (&quot;ordinates&quot;) to a diameter of a conic in his demonstrations. As for the word itself, it might go back to the terminology of Roman surveyors (&lt;em&gt;lineae ordinatae&lt;/em&gt;), the early medieval occurrences are in &lt;em&gt;rectae ordinatim applicatae&lt;/em&gt; (straight lines designated in order), but as a mathematical term it was coined by Leibniz some time between 1677 and 1692. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an early edition of his &lt;a href=&quot;https://www.gutenberg.org/ebooks/31061&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Mathematics (1909), p. 216&lt;/a&gt; Cajori boldly wrote &quot;&lt;em&gt;The Latin term for &quot;ordinate&quot; used by Descartes comes from the expression lineae ordinatae, employed by Roman surveyors for parallel lines&lt;/em&gt;&quot;. But in the later editions, e.g. the &lt;a href=&quot;https://archive.org/details/ahistorymathema03cajogoog/page/n188&quot; rel=&quot;nofollow noreferrer&quot;&gt;1929 edition, p.175&lt;/a&gt;, he changed his mind:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Here Descartes follows Apollonius who related the points of &#xA;  a conic to the points of a diameter, by distances (ordinates) which make a constant angle with the diameter and are determined in length by the position of the point on the diameter. This constant angle is with Descartes usually a right angle... It is also noteworthy that Descartes does not formally introduce a second axis, our &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis. Such formal introduction is found in G. Cramer's Introduction d l'andyse des lignes courbes algibrique, 1750; earlier publications by de Gua, L. Euler, W. Murdoch and others contain only occasional references to a &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis. The words &quot;abscissa&quot;, &quot;ordinate&quot; were not used by Descartes.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In the strictly technical sense of analytics as one of the coordinates of a point, the word &quot;ordinate&quot; was used by Leibniz in 1694, but in a less restricted sense such expressions as &quot;ordinatim applicatae&quot; occur much earlier in F. Commandinus and others. The technical use of &quot;abscissa&quot; is observed in the eighteenth century by C. Wolf and others. In the more general sense of a &quot;distance&quot; it was used earlier by B. Cavalieri in his Indivisibles, by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome, and by others. Leibniz introduced the word &quot;coordinatae&quot; in 1692.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://jeff560.tripod.com/o.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt; gives the following earlier citations for Leibniz's uses:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Leibniz used the phrase &quot;per differentias ordinatarum&quot; in a letter to Newton on June 21, 1677 (Scott, page 155).&lt;br&gt;&#xA;  Leibniz used the term ordinata in 1692 in Acta Eruditorum 11 (Struik, page 272).&lt;br&gt;&#xA;  For the word in English the OED has a passage from 1706: H. Ditton An Institution of Fluxions p. 31 “'Tis required to find the relation of the Fluxion of the Ordinate to the Fluxion of the Abscisse.”&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/B-001-001-112&quot; rel=&quot;nofollow noreferrer&quot;&gt;Struik's Source Book in Mathematics, 1200-1800&lt;/a&gt; in a footnote to the cited page has the following, confirming some of Cajori's claims:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Note the Latin term abscissa. This term, which was not new in Leibniz's day, was made by him into a standard term, as were so many other technical terms. In the article &quot;De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata... ,&quot; Acta Eruditorum 11 (1692), 168-171 (Leibniz, Mathematische Schriften, Abth. 2, Band I (1858),&#xA;  266-269), in which Leibniz discusses evolutes, he presents a collection of technical terms.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Here we find ordinata, evolutio, differentiare, parameter, differentiabilis, functio, and ordinata&#xA;  and abscissa together designated as coordinatae. Here he also points out that ordinates may be given not only along straight but also along curved lines. The term ordinate is derived&#xA;  from rectae ordinatim applicatae, &quot;straight lines designated in order,&quot; such as parallel lines. The term functio appears in the sentence: &quot;the tangent and some other functions depending&#xA;  on it, such as perpendiculars from the axis conducted to the tangent.&quot;&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Scott in History of Mathematics quotes directly the cited Leibniz's letter to Newton, explaining the differential triangle:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;From this it is clear that to find the tangents is nothing more than to find the differences of the ordinates corresponding to given (equal) differences of the abscissae&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-20T17:39:12.107" LastActivityDate="2019-08-20T17:39:12.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9942" PostTypeId="1" AcceptedAnswerId="9947" CreationDate="2019-08-20T08:16:09.933" Score="5" ViewCount="199" Body="&lt;p&gt;I'm not sure if it's a good question but I was reading about the decomposition of any function, f(x), as a sum of even and odd functions; f(x) is not an even or odd function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is it possible to know when this fact (or, theorem?) was discovered and who found it? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Helpful link:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Even_and_odd_functions#Even%E2%80%93odd_decomposition&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Even_and_odd_functions#Even%E2%80%93odd_decomposition&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9341" LastActivityDate="2019-08-20T23:58:12.317" Title="When was it found that a function could be decomposed as a sum of even and odd functions?" Tags="|discoveries|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9943" PostTypeId="1" CreationDate="2019-08-20T17:24:56.483" Score="1" ViewCount="184" Body="&lt;p&gt;I was watching a video (&lt;a href=&quot;https://www.youtube.com/watch?v=muJYTeQlvC4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=muJYTeQlvC4&lt;/a&gt;) where the director of a videogame company speaks about one of its most successful games. This game is set in a floating city which would have &quot;windows&quot; to other parallel or alternative universes. Around minute 2:35, he refers to these multiple universes and mentions that, in reality, Werner Heisenberg was one of the proponents of these multiple universes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But is this right? I have not been able to find any reference or quote by Heisenberg that indicates that he was happy with the idea of having multiple universes. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, is this right? Did Werner Heisenberg ever agree or propose the existence of some kind of multiverse?&lt;/p&gt;&#xA;" OwnerUserId="10288" LastActivityDate="2019-08-20T17:24:56.483" Title="Did Werner Heisenberg ever agree or propose the existence of some kind of multiverse?" Tags="|physics|quantum-mechanics|theoretical-physics|cosmology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9944" PostTypeId="1" CreationDate="2019-08-20T18:56:45.647" Score="0" ViewCount="622" Body="&lt;p&gt;In Ancient Greek, most people like Pythagoras thought 1 (monad, unity) is no number, but it is ruler and beginning of all other numbers. And Pythagoras thought everything is number. But they found irrational numbers which can not be measured by 1, and their theory broke up because of it. But even though ancient Greek found irrational numbers, for example, Euclid who was born after Pythagoras and also knew about irrational number, said every number is made by 1 (monad). Not only Ancient Greek, but also Medival arthimeticians say 1 is source of all other numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q1. Why did they say 1 is source of numbers even though they found existence of irrational numbers? Is it because they only treated natural numbers as numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q2. If so, (only natural numbers can be numbers) what was irrational number for them?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Q3. According to ancient Greeks, If 'one' can not be divided, how can fractions exist? (1/2 or 1/3 etc..)&lt;/p&gt;&#xA;" OwnerUserId="10363" LastEditorUserId="2905" LastEditDate="2020-02-08T11:52:52.110" LastActivityDate="2020-03-09T12:01:35.167" Title="Why 1 was source of numbers even though ancient Greeks knew about irrational number?" Tags="|mathematics|ancient-greece|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9945" PostTypeId="2" ParentId="9938" CreationDate="2019-08-20T20:59:30.903" Score="2" Body="&lt;p&gt;No, Ostrogradski did not say this in his report on Lobachevski's work. His main argument was that this is a &quot;futile phantasy&quot;, with no applications. He did not&#xA;dispute the logical consistency of the theory, but at that time the common opinion was that mathematics has to describe the so-called &quot;real world&quot;. He criticized the absence of applications, or more precisely that there are too few of them and that they are insignificant. (I remember he mentioned that Lobachevski just &quot;computed one new integral&quot;, and this does not justify the development of such a theory).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="5989" LastEditDate="2019-08-20T21:03:10.847" LastActivityDate="2019-08-20T21:03:10.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9946" PostTypeId="2" ParentId="9931" CreationDate="2019-08-20T23:06:16.903" Score="4" Body="&lt;p&gt;Klaus Weltner and Martin Ingelman-Sundberg (from the Department of Physics at the University Frankfurt) in an ultimately unpublished paper, &lt;a href=&quot;http://www.angelfire.com/dc/nova/flight/PHYSIC4.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Physics of Flight — reviewed”&lt;/a&gt;, submitted to the European Journal of Physics in 2003, argue that the original source for the misconception of “equal transit time” was, somewhat unwittingly, aerodynamics pioneer &lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Prandtl&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ludwig Prandtl&lt;/a&gt; circa 1922.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Their article leads with a summary of &lt;a href=&quot;https://books.google.ch/books?id=WzlLAAAACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Darstellung der Aerodynamik in Schulphysikbüchern”&lt;/a&gt;, a book published in 1989 by Klaus Niermann:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Niermann analyzed all German language textbooks published within the last one hundred years investigating the explanations of aerodynamic lift. The main result is that explanations based on Bernoulli’s law are dominating since the 1920s. He found the same to be true for a survey of contemporary American and English textbooks.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The authors eventually go on to conclude:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thus the origin of the path length reasoning may be found in a diagram given by Prandtl in 1922.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Wd7mP.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Wd7mP.gif&quot; alt=&quot;diagram by Prandtl&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;With this diagram Prandtl tried to show that air at the inner layers stick to the surface. In respect to one point this diagram is not correct. It indicates that the volumes remain at the same vertical position and it indicates that originally adjacent air meets again at the end of the airfoil. […]&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Diagrams of this type might have misled scholars to the hypothesis that adjacent air has to meet again after passing the airfoil.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The text says 1922, which may very well refer to the original German publication, though the actual reference the authors cite is the English translation, produced by the National Advisory Committee for Aeronautics, titled &lt;a href=&quot;https://ntrs.nasa.gov/search.jsp?R=19930091180&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Applications of Modern Hydrodynamics to Aeronautics”&lt;/a&gt; and published in 1923.&lt;/p&gt;&#xA;" OwnerUserId="10139" LastEditorUserId="10139" LastEditDate="2019-08-23T08:07:15.917" LastActivityDate="2019-08-23T08:07:15.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9947" PostTypeId="2" ParentId="9942" CreationDate="2019-08-20T23:58:12.317" Score="5" Body="&lt;p&gt;It depends partly on what &quot;function&quot; means. According to a definition prevalent before Dirichlet it meant something expressed by a formula and expandable into a power series. The decomposition claim then is obvious, as long as one introduced the notions of even and odd and related them to even and odd powers. Euler does both in &#xA;&lt;a href=&quot;http://eulerarchive.maa.org/docs/originals/E005&quot; rel=&quot;noreferrer&quot;&gt;Traiectoriarum Reciprocarum Solutio (1727), XIX-XXI&lt;/a&gt; (he uses Latin words &lt;em&gt;pares&lt;/em&gt; and &lt;em&gt;impares&lt;/em&gt;). Luckily, this is one of the papers &lt;a href=&quot;http://www.17centurymaths.com/contents/eulercontents.html&quot; rel=&quot;noreferrer&quot;&gt;translated by Ian Bruce&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;§17. In the first place functions are to be noted that I call even, of which this is the&#xA;  property: that they remain unchanged if &lt;span class=&quot;math-container&quot;&gt;$- x$&lt;/span&gt; is put in place of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. All the powers of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with&#xA;  even exponents are of this kind, and of fractions with even numerators and odd denominators. Then, any such functions of powers of this kind composed either by addition or subtraction, or by multiplication or division, or hence any such construction raised to sum are likewise even powers, such as &lt;span class=&quot;math-container&quot;&gt;$x^{\frac45}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(ax^2+bx^{\frac23})^n$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;§18. In the second place, I take heed of odd functions, which in short produce the&#xA;  negative of these, if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is changed to &lt;span class=&quot;math-container&quot;&gt;$-x$&lt;/span&gt;. All the powers of x with odd exponents, such as&#xA;  &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; itself, &lt;span class=&quot;math-container&quot;&gt;$x^3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^5$&lt;/span&gt;, etc. are of this kind; or fractions of which the numerators and&#xA;  denominators are odd; also functions composed of these powers either added or&#xA;  subtracted, and which are also dignified to be raised to odd exponents, such as &lt;span class=&quot;math-container&quot;&gt;$x^{\frac45}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$(ax^3+bx^{\frac57})^3$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Euler did not have much use for them, or decomposition into them, although, as &lt;a href=&quot;https://www.maa.org/sites/default/files/321922717729.pdf.bannered.pdf&quot; rel=&quot;noreferrer&quot;&gt;Barnett notes&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In fact, the expressions &lt;span class=&quot;math-container&quot;&gt;$(e^x + e^{-x})/2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(e^x - e^{-x})/2$&lt;/span&gt; do make an appearance in Volume I of Euler's Introductio in analysin infinitorum (1745, 1748). Euler's interest in these expressions seems natural in view of the equations &lt;span class=&quot;math-container&quot;&gt;$\cos x=(e^{\sqrt{-1}x} + e^{-\sqrt{-1}x})/2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}\sin x=(e^{\sqrt{-1}x} + e^{-\sqrt{-1}x})/2$&lt;/span&gt; that he derived in this text. However, Euler's interest in what we call hyperbolic functions appears to have been limited to their role in deriving infinite product representations for the sine and cosine functions. Euler did not use the word hyperbolic in reference to the expressions... nor did he provide any special notation or name for them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In a Hacker News discussion &lt;a href=&quot;https://news.ycombinator.com/item?id=12193469&quot; rel=&quot;noreferrer&quot;&gt;Decomposing a function into its even and odd parts&lt;/a&gt; David Goldblatt suggested that the idea might go back to earlier times:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I don't have any references for you right now, but I would be surprised if the usage of the terms doesn't pre-date your citation of Euler by at least several decades. The basic technique for deriving Taylor series is known to date to at least 1671, to the correspondence between James Gregory and John Collins; it also shows up in letters from de Moivre to Johann Bernoulli in 1694 and from Leibniz to Bernoulli in 1708. Newton had a geometric means of deriving the coefficients of power series when he was writing the Principa (published in 1687), as demonstrated by the proof of his tenth proposition, and he included a description of the algebraic technique in an early draft of his Quadrature of Curves (but removed it before publication in 1706)... Perhaps nobody bothered to give the concept a name until Euler in 1727, and perhaps nobody even considered the concept with regard to a more general class of functions until even later&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-08-20T23:58:12.317" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9948" PostTypeId="1" CreationDate="2019-08-21T12:59:13.423" Score="4" ViewCount="125" Body="&lt;p&gt;I read in a PDF document where the author made a comment that it is “dangerous” to use indirect proof method/contradiction proof method (as far as I can remember, and of course &lt;strong&gt;&lt;em&gt;I am paraphrasing&lt;/em&gt;&lt;/strong&gt;) as it may lead to false result. He gave an example of John G. Thompson’s famous proof of finite group theory saying that the first proof Thompson submitted to journal, had to be retracted  immediately due to the gap caused by the indirect proof method. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;I have forgotten where I read that, I can’t remember the name of author, and can’t find the PDF file either.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone confirm what I stated above? If possible please provide reference. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Also if anyone read the PDF file I am referring to please comment. Also if you have any suggestion how can I find that document please share.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The document I am referring to was about proof methods.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="5926" LastEditorUserId="5926" LastEditDate="2019-08-25T10:15:40.123" LastActivityDate="2019-08-25T10:15:40.123" Title="Reference Request: Comment about Contradictions Proof Method Related to John G. Thompson" Tags="|mathematics|reference-request|scientific-method|quote|anecdote|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="9949" PostTypeId="1" CreationDate="2019-08-21T16:09:49.377" Score="2" ViewCount="113" Body="&lt;p&gt;I remember reading in the line that Feynman saying if the best technicians were tasked to build a radio they would make longer and thicker cables and transfer message but it took a physicist to realize that one can manipulate the motion of electrons to generate radio waves which then could be used for communication. I am trying to find reference to it but don't seem to find it anywhere. Did Feynman said something similar and if so where can I find the statement?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="10367" LastActivityDate="2019-08-21T16:09:49.377" Title="Feynman's statement about the importance of basic science" Tags="|physics|electromagnetism|technology|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9950" PostTypeId="1" CreationDate="2019-08-21T18:33:26.260" Score="-1" ViewCount="450" Body="&lt;p&gt;I'm trying to understand the development of the calculus. Does this sound right as one of the stages?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton knows the binomial theorem, which gives&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x+y)^n={n\choose0}x^ny^0+...\;\;\;\;\;\;\;\;\;\;(1)$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Letting &lt;span class=&quot;math-container&quot;&gt;$y=δ_x$&lt;/span&gt; (I realise this is Leizbniz's notation but I find it easier to follow than Newton's &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;), we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x+δ_x)^n={n\choose0}x^nδ_x^0+...\;\;\;\;\;\;\;\;\;\;(2)$$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Considering &lt;span class=&quot;math-container&quot;&gt;$δ_x$&lt;/span&gt; as the base of a differential triangle under a curve, the vertical of the triangle is given by &lt;span class=&quot;math-container&quot;&gt;$(x+δ_x)^n-x^n$&lt;/span&gt;, which gives us&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x+δ_x)^n-x^n={n\choose0}x^nδ_x^0+...-x^n\;\;\;\;\;\;\;\;\;\;(3)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But &lt;span class=&quot;math-container&quot;&gt;${n\choose0}x^nδ_x^0=x^n$&lt;/span&gt;, so the first part of the expansion disappears and everything else moves up one place to the left and we get&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(x+δ_x)^n-x^n={n\choose1}x^{n-1}δ_x^1+...\;\;\;\;\;\;\;\;\;\;(4)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, the vertical &lt;span class=&quot;math-container&quot;&gt;$(x+δ_x)^n-x^n$&lt;/span&gt; can be called &lt;span class=&quot;math-container&quot;&gt;$δ_y$&lt;/span&gt;, so now we can write&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{δ_y}{δ_x}=\frac{(x+δ_x)^n-x^n}{δ_x}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ = \frac{1}{δ_x}{n\choose1}x^{n-1}δ_x^1+...\;\;\;\;\;\;\;\;\;\;(5)$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And now we can use the biniomial theorem to make a differential triangle under any polynomial; and as &lt;span class=&quot;math-container&quot;&gt;$δ_x$&lt;/span&gt; gets small, the gradient approaches instantaneity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, that would also make &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{δ_x}$&lt;/span&gt; get large, so I guess I've made a mistake somewhere.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: mrtaurho points out that the &lt;span class=&quot;math-container&quot;&gt;$δ_x^1$&lt;/span&gt; term cancels the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{δ_x}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10194" LastEditorUserId="10194" LastEditDate="2019-08-21T19:04:51.870" LastActivityDate="2019-08-21T21:09:21.047" Title="Newton's calculus and the binomial theorem" Tags="|calculus|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="9951" PostTypeId="2" ParentId="9950" CreationDate="2019-08-21T21:09:21.047" Score="1" Body="&lt;p&gt;What you stated is not Newton's binomial theorem. Binomial theorem for integer exponent was known long before Newton. Newton discovered the binomial theorem for&#xA;non-integer exponent (an infinite series which is called the binomial series nowadays).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you wish to understand what is the relation to Calculus, I advise reading Newton's Mathematical papers, or at least his&#xA;two letters to Leibniz where he described the essence of his discovery.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main contribution of Newton to Calculus is his realization of importance of power series, and systematic use of them for evaluation of integrals and solving differential equations,&#xA;of which the binomial series is an example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reading Newton is difficult, but there are good modetn expositions of the topic, for example, Arnold's little but very informative book &quot;Huygens and Barrow, Newton and Hooke&quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-08-21T21:09:21.047" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9953" PostTypeId="2" ParentId="7776" CreationDate="2019-08-22T13:58:43.040" Score="0" Body="&lt;p&gt;Gauss was referring to the followers of Kant. The German philosopher was very influential and had just published on the theme of seeing is believing and to dismiss the unverifiable and unobservable. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This could never be compatible with Non_euclidean geometry&lt;/p&gt;&#xA;" OwnerUserId="10371" LastActivityDate="2019-08-22T13:58:43.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9954" PostTypeId="1" CreationDate="2019-08-24T04:01:51.110" Score="1" ViewCount="202" Body="&lt;p&gt;I am Korean, and I thought I can understand majority of english sentences, but this is really hard to translate literally for me. Even though I asked it to my English teacher, he did not know either. Although it is English question, I think it is way better to ask in math forum, because it is about math. Here is my sentence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid Element VII def 1 (Heath edition)&#xA;An unit is that by virtue of which each of the things that exist is called one.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Why he used 'an unit' rather than 'a unit'? I thought 'a unit' is more grammatically right, but this is not so important question anyway.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How should I interpret this? Of course, I know what that sentence means because there are also Korean version of Euclid. But it was liberal translation. We use 'by virtue of~' simillar as 'because of'. Then, there should be consequence in front of 'by virtue of'. For example, 'I won first prize by virtue of my father'. Like this sentence, 'won first prize' this is consequence. But that Euclid text, there is no consequence in front of 'by virtue of'. Please let me know how should I interpret it. (For literal translation. As I told, I know what it means as liberal translation)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10364" LastActivityDate="2019-08-25T21:30:52.453" Title="I can't comprehend the sentence in Euclid Elements" Tags="|mathematics|ancient-greece|euclidean-geometry|" AnswerCount="1" CommentCount="6" ClosedDate="2019-08-28T10:17:34.907" ContentLicense="CC BY-SA 4.0" />
  <row Id="9957" PostTypeId="2" ParentId="9954" CreationDate="2019-08-25T21:30:52.453" Score="2" Body="&lt;p&gt;The &quot;an unit&quot; is an antiquated spelling based on an old pronunciation of unit which is no longer used. The translation should read &quot;A unit is ...&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Your confusion over the definition is nothing new. Even in ancient times, philosophers and mathematicians used to argue about how to interpret this definition.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heath's translation is somewhat obscure. A more modern interpretive translation might be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A unit is that by which each thing exists in and of itself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3497" LastActivityDate="2019-08-25T21:30:52.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9959" PostTypeId="2" ParentId="7983" CreationDate="2019-08-26T18:10:47.693" Score="5" Body="&lt;p&gt;There is the one-page obituary mentioned by Keith Conrad. V. Gubarev and me have uploaded an English translation to the arXiv, see &lt;a href=&quot;https://arxiv.org/abs/1908.08361&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. Furthermore I have tried to provide a more detailed English wikipedia page for Igor Dmitrievich Ado, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Igor_Dmitrievich_Ado&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. I am grateful to Yurii Neretin for providing me with information.&lt;/p&gt;&#xA;" OwnerUserId="10388" LastActivityDate="2019-08-26T18:10:47.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9960" PostTypeId="2" ParentId="9944" CreationDate="2019-08-26T22:20:15.560" Score="-1" Body="&lt;p&gt;In the time of Pythagoras, it was widely believed among Greek mathematicians and philosophers that any line (or object) is made up of basic elements of same length (also known as &lt;em&gt;atom&lt;/em&gt;).  Thus if the length of natural number &lt;em&gt;1&lt;/em&gt; consists of &lt;em&gt;n&lt;/em&gt; atoms, then each atom would have the length of &lt;span class=&quot;math-container&quot;&gt;$1/n$&lt;/span&gt;, and a line of any length consisting of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; atoms would have the length of a rational number &lt;span class=&quot;math-container&quot;&gt;$m/n$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The discovery of the fact that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is not a rational number completely shattered this belief because it means that the line can not be made up of atoms, for otherwise there would be many holes left on the line that can not be measured by rational numbers. Since no one could explain the nature of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;, it was known as an irrational number, which means number of no sense. The name of &lt;em&gt;irrational number&lt;/em&gt; has become widely in use until today.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2019-08-30T01:32:27.763" LastActivityDate="2019-08-30T01:32:27.763" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="9961" PostTypeId="1" CreationDate="2019-08-27T19:32:28.437" Score="7" ViewCount="345" Body="&lt;p&gt;This question is a request for general explanation of an astronomical phenomenon known as &amp;quot;Pallas libration&amp;quot;, and isn't much about the details of Gauss's  mathematical model of this libration. As is well known (see this post: &lt;a href=&quot;https://hsm.stackexchange.com/questions/2323/calculation-of-gauss-leading-to-187-resonance-between-orbits-of-jupiter-and-pal?noredirect=1&amp;amp;lq=1&quot;&gt;Calculation of Gauss leading to 18:7 resonance between orbits of Jupiter and Pallas&lt;/a&gt;), Gauss discovered a &lt;span class=&quot;math-container&quot;&gt;$18:7$&lt;/span&gt; resonance of Pallas with Jupiter, and according to the source given in &lt;a href=&quot;https://aas.org/archives/BAAS/v31n5/aas195/431.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://aas.org/archives/BAAS/v31n5/aas195/431.htm&lt;/a&gt; (&amp;quot;&lt;strong&gt;On the Commensurate Motion of Pallas&lt;/strong&gt;&amp;quot;) he developed a theory of the periodic libration of Pallas with a period of &lt;span class=&quot;math-container&quot;&gt;$737$&lt;/span&gt; Jupiter sidereal periods (Werke v. 7, p. 421, p. 559). Martin Brendel, in his essay on Gauss's astronomical works, also commented about Gauss's theory of libration (he actually devoted an entire section to it in his essay).&lt;/p&gt;&#xA;&lt;p&gt;The basic point i don't understand is the use of the word &amp;quot;libration&amp;quot; within the context of the motion of asteroids - as far as i know, libration is a phenomenon in which an observer on a planet (Earth, for example) is able to see more than a half of an orbiting moon which is in a &amp;quot;tidal lock&amp;quot; with the planet (the effect in which tidal forces and moments synchronize the rotational angular motion of the moon with his orbital angular motion). This is caused by visual effects such as the parallax derived from the change in positions of the observer. But Pallas isn't even a moon of Jupiter, let alone in &amp;quot;tidal lock&amp;quot; with Jupiter. So I'd really like to get an explanation on this.&lt;/p&gt;&#xA;&lt;p&gt;Although this question is more about the physics involved then about the historical circumstances, I chose to mention Gauss and ask it on this HSM Stack Exchange, simply because he was one of the few astronomers that attempted to formulate such a theory (which is quite a &amp;quot;niche&amp;quot; in astronomical sciences), so one can't really separate the science from its history and discoverers in this case.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="7205" LastEditDate="2022-09-28T10:32:09.957" LastActivityDate="2022-09-28T10:32:09.957" Title="What was the &quot;libration of Pallas&quot; that Gauss studied, when Pallas is not a moon of Jupiter?" Tags="|biographical-details|astronomy|gauss|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9962" PostTypeId="2" ParentId="9961" CreationDate="2019-08-27T23:24:46.487" Score="8" Body="&lt;p&gt;&quot;Libration&quot; loosely refers to longitudinal oscillations due to orbital resonances. The tidal lock and the trojans, the only situations that &lt;a href=&quot;https://en.wikipedia.org/wiki/Libration&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia mentions&lt;/a&gt;, are just special cases. The existence of resonances does not require one body to revolve around the other, and Jupiter exerts a major pull on the asteroid belt just as a planet would on its moons.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, according to &lt;a href=&quot;http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1922BAN.....1..161W&amp;amp;data_type=PDF_HIGH&amp;amp;whole_paper=YES&amp;amp;type=PRINTER&amp;amp;filetype=.pdf&quot; rel=&quot;noreferrer&quot;&gt;Woltjer's On the supposed libration in the motion of Pallas&lt;/a&gt;, Gauss's &quot;libration&quot; of Pallas was likely a numerical artifact:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;One of the results of GAUSS' investigation on&#xA;  the motion of Pallas as disturbed by the attraction&#xA;  of Jupiter consists in the commensurability of the&#xA;  mean motions &#xA;  &lt;span class=&quot;math-container&quot;&gt;$$18n'-7n=0.\ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$&lt;/span&gt; This equation shows a remarkable difference with the corresponding equations for some of the best known cases of libration in the solar system. In all these cases the algebraic sum of the integral numbers from the coefficients of the mean motions in the equations analogous to &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; is zero; here it is &lt;span class=&quot;math-container&quot;&gt;$+11$&lt;/span&gt;...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The origin of this strange result contained in &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; lies in the fact that GAUSS developed the perturbative function for definite numerical values of the elements. If &lt;span class=&quot;math-container&quot;&gt;$\lambda, \varpi, \Omega$&lt;/span&gt; denote the mean longitude, longitude of&#xA;  perihelium, longitude of ascending node of Pallas, and the corresponding quantities for Jupiter are denoted by accented symbols, we have the following arguments in the perturbative function associated with the commensurability &lt;span class=&quot;math-container&quot;&gt;$18:7$&lt;/span&gt;&lt;/em&gt;:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;18\lambda'-7\lambda-11\varpi\\&#xA;18\lambda'-7\lambda-10\varpi-\Omega\ \ \ \ \ \ (2)&#xA;$$&lt;/span&gt;&#xA;  &lt;em&gt;All the corresponding terms have been coalesced&#xA;  into one by GAUSS' numerical development; If however&#xA;  one of these terms, say the first; represents a case&#xA;  of libration, then the other's arguments have mean&#xA;  motions, such as for the second term &lt;span class=&quot;math-container&quot;&gt;$\varpi-\Omega$&lt;/span&gt;, which&#xA;  in general are different from zero.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Thus the coefficient &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; in the differential equation for the argument of the libration u:&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{d^2u}{dt^2}=-7\alpha n^2\sin u&#xA;$$&lt;/span&gt;&#xA;  should correspond to one of the arguments &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; only,&#xA;  and not to the coefficient resulting from the addition of all the terms &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; for definite numerical values of &lt;span class=&quot;math-container&quot;&gt;$\varpi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\varpi'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt;. Consequently it seems to me that we have no reason&#xA;  to assume the existence of libration in the motion&#xA;  of Pallas, until a more detailed investigation of the problem has been made by considering each of the&#xA;  terms &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; separately.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1982MNRAS.199..255T&amp;amp;defaultprint=YES&amp;amp;filetype=.pdf&quot; rel=&quot;noreferrer&quot;&gt;Taylor in The secular motion of Pallas&lt;/a&gt; summarizes this as &quot;&lt;em&gt;Woltjer (1922) showed that the &lt;span class=&quot;math-container&quot;&gt;$18:7$&lt;/span&gt; commensurability was unlikely to lead to a libration in the motion of Pallas&lt;/em&gt;&quot;, and discusses the effect of resonances with Jupiter on the motion of asteroids more generally.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-08-28T05:21:31.740" LastActivityDate="2019-08-28T05:21:31.740" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9963" PostTypeId="1" CreationDate="2019-08-28T02:34:09.487" Score="4" ViewCount="157" Body="&lt;p&gt;Prominent physicist Paul Dirac proposed a &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;hypothesis&lt;/a&gt; that said that constants and laws of physics would evolve with time into different constants and laws of nature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This hypothesis was used by &lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_H._Dicke&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Dicke&lt;/a&gt; to argue in favour of the anthropic principle and the existence of multiple universes each one with its own set of rules and laws. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though Dicke modified Dirac's hypothesis, I have read that Dirac himself once said that 'Dicke's reasoning was sound'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does that mean that Dirac, like Dicke, considered the existence of multiple universes with their own set of rules?&lt;/p&gt;&#xA;" OwnerUserId="10288" LastEditorUserId="10106" LastEditDate="2019-08-28T10:24:33.737" LastActivityDate="2019-08-28T10:24:33.737" Title="Did Paul Dirac believe in multiple universes?" Tags="|physics|biographical-details|theoretical-physics|cosmology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="9964" PostTypeId="1" AcceptedAnswerId="9965" CreationDate="2019-08-29T14:11:42.640" Score="3" ViewCount="110" Body="&lt;p&gt;When and by whom was the word 'polytropic' first used to describe an equation of state or process of the form &lt;span class=&quot;math-container&quot;&gt;$P \propto V^n$&lt;/span&gt;? Why was this particular word chosen?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The earliest use of the word that I could find by searching on Google Books was in &lt;a href=&quot;https://doi.org/10.1080/14786445408647467&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1080/14786445408647467&lt;/a&gt;, which seems to be unrelated to the use I am looking for. The other 'contemporaneous' results I could find seemed to be mostly newer books mislabelled as older ones, or works in which the word was used without any explanation, suggesting it was already well-established.&lt;/p&gt;&#xA;" OwnerUserId="10397" LastActivityDate="2019-08-29T14:56:40.590" Title="Earliest use of the word 'polytropic' to describe an equation of state or process" Tags="|thermodynamics|astrophysics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9965" PostTypeId="2" ParentId="9964" CreationDate="2019-08-29T14:56:40.590" Score="3" Body="&lt;p&gt;The best source of word origins and their earliest usages is the Unabridged Oxford English Dictionary. You need a library subscription.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first meaning could be of mathematical original since n can assume&#xA;  several values  Mathematics. Of a function: taking several different&#xA;  values for each value of the variable; many-valued. Cf. monotropic&#xA;  adj. 1. Obsolete. rare—0.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;1889   Cent. Dict. at Function   Polydromic or polytropic function, one which is not monotropic.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then it defines the usage in thermodynamics and shows a 1907 paper where the authors use this term specifically in thermodynamic sense&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;1907   J. F. Klein tr. G. Zeuner Techn. Thermodynamics I. xxix. 152   If the initial condition is given by p1 and v1, we accordingly have pvn = p1v1n..as the equation of the sought pressure curve, which we will hereafter call the polytropic curve.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Now you should search the 1907 paper by J. F. Klein.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2019-08-29T14:56:40.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9966" PostTypeId="1" AcceptedAnswerId="9967" CreationDate="2019-08-30T20:03:00.170" Score="3" ViewCount="119" Body="&lt;p&gt;In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link &lt;a href=&quot;https://www.iro.umontreal.ca/~mignotte/IFT2425/Documents/Solving_Differential_Equations_In_R_Soetaert_K.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is presented as a system of ODEs, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x' = \cos (\phi) &#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;y' = \sin (\phi) &#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\phi' = \kappa&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\kappa' = F \cos (\phi)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and when I looked for where these ODEs come from I could not find a clear derivation anywhere. There are references to Bernoulli, Euler, but the formulation looked different. Does anyone know a book, reference that shows where these equations come from? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks,&lt;/p&gt;&#xA;" OwnerUserId="10406" LastEditorUserId="217" LastEditDate="2019-08-31T13:38:59.050" LastActivityDate="2019-08-31T13:38:59.050" Title="From where the so-named &quot;elastica problem&quot; is coming from?" Tags="|calculus|classical-mechanics|differential-geometry|differential-equations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9967" PostTypeId="2" ParentId="9966" CreationDate="2019-08-31T01:13:15.187" Score="3" Body="&lt;p&gt;A. G. Greenhill, The applications of elliptic functions, Macmillan, London &amp;amp; NY, 1892,&#xA;pp. 87-88.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you read French, a much clearer and more comprehensive discussion is in&#xA;G.-H. Halphen, Traite de fonctions elliptiques et de leurs applications,&#xA;2-eme partie, Paris Gauthier-Villars et fils, 1888.&#xA;Chapitre V.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-08-31T01:22:54.947" LastActivityDate="2019-08-31T01:22:54.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9968" PostTypeId="2" ParentId="8047" CreationDate="2019-09-03T08:11:59.300" Score="15" Body="&lt;p&gt;I am Camille Aubry, granddaughter of Léon Aubry (1882-1947), and I thank you for your interest in my great-grandfather.&#xA;He was a wine grower, farmer, beekeeper, in Jouy-lès-Reims (51).&#xA;He was also a self-taught mathematician and he was published in the journal Sphinx-Oedipe, in l’Intermédiaire des mathématiciens, l’Enseignement mathématique, by Gauthier-Villars and he participated in AFAS congresses. &#xA;According to my researches and the bibliography that I carried out, he was published between 1911 and 1926, date of the end of the publication of Sphinx-Oedipe but he remained active until his death in 1947. &#xA;I am finishing a manuscript on his mathematical work and I hope to see it published soon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mr Santos, have you finished your article that you mentioned during our email exchange?&#xA;Best regards,&lt;/p&gt;&#xA;" OwnerUserId="10415" LastActivityDate="2019-09-03T08:11:59.300" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="9971" PostTypeId="1" AcceptedAnswerId="9972" CreationDate="2019-09-07T03:06:31.217" Score="6" ViewCount="385" Body="&lt;p&gt;In Max Born's Nobel lecture, he alludes to Einstein's proposed interpretation of EM wave amplitude (squared) as being the probability density of detecting a photon:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In which of his papers did Einstein first propose that interpretation?&lt;/p&gt;&#xA;" OwnerUserId="10435" LastEditorUserId="55" LastEditDate="2019-09-08T09:35:53.033" LastActivityDate="2019-09-14T05:53:23.770" Title="Where did Einstein propose interpreting the square of amplitude as probability density (Born's rule)?" Tags="|quantum-mechanics|einstein|optics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9972" PostTypeId="2" ParentId="9971" CreationDate="2019-09-07T05:30:44.267" Score="7" Body="&lt;p&gt;There are no published papers where such interpretation is explicitly made. Born's Nobel lecture mention first appears in his celebrated &lt;em&gt;Quantenmechanik der Stossvorgange&lt;/em&gt; (1926, &lt;em&gt;Zeitschrift fur Physik&lt;/em&gt; 38, 803-827), also with no reference. According to Pais, well-known Einstein's biographer, the inspiration came from Einstein's never published remarks about the &quot;ghost field&quot; from early 1920-s, which are not extant. Their existence is known from Wigner, who does not give many details either. As Pais writes in &lt;a href=&quot;https://books.google.com/books/about/Inward_Bound.html?id=mREnwpAqz-YC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inward Bound&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Nevertheless, it is true that Born's inspiration came from Einstein: not Einstein's statistical papers bearing on light, but his never published speculations during the early 1920s on the dynamics of light-quanta and wave fields. Born states this explicitly in his second paper: 'I start from a remark by Einstein on the relation between [a] wave field and light-quanta; he said approximately that the waves are only there to show the way to the corpuscular light-quanta,&#xA;  and talked in this sense of a &quot;ghost field&quot; [Gespensterfeld] [which] determines the &lt;strong&gt;probability&lt;/strong&gt; [my italics] for a light-quantum ... to take a definite path'.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Little concrete is known about his ideas of a ghost field or guiding field (Führungsfeld). The best description we have is from Wigner, who knew Einstein personally in the 1920s: '[Einstein's) picture has a great similarity with the present picture of quantum mechanics. Yet Einstein, though in a way he was fond of it, never published it. He realized that it is in conflict with the conservation principles... This Einstein never could accept and hence never took his idea of the guiding field quite seriously ... The problem was solved as we know, by Schroedinger's theory'.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Born was even more explicit about his source of inspiration in a letter to Einstein written in November 1926 (for reasons not clear to me this letter is not found in the published Born–Einstein correspondence): 'About me it can be told that physicswise I am entirely satisfied since my idea to look upon Schroedinger's wave field as a &quot;Gespensterfeld&quot; in your sense proves better all the time. Pauli and Jordan have made beautiful advances in this direction.'...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, another connection that comes to mind is to Einstein's 1905 &quot;photoeffect paper&quot; (historians consider this nickname a misnomer). But it has neither &quot;photoeffect&quot; nor &quot;photons&quot; in it, and certainly no complex-valued waves. It studies radiation in an enclosure, and even the photoelectric equation is derived in passing when treating radiation thermodynamically, see &lt;a href=&quot;http://www.thphys.uni-heidelberg.de/~amendola/otherstuff/einstein-paper-v2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Revolution in Physics: Einstein’s Papers of 1905 Made Simple&lt;/a&gt;. Pais is also skeptical of this connection:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;On the face of it, this appears to be a perfectly natural explanation. Had Einstein not stated that light of low intensity behaves as if it consisted of energy packets &lt;span class=&quot;math-container&quot;&gt;$h\nu$&lt;/span&gt;? And is the intensity of light not a function quadratic in the electromagnetic fields?... Recall that Born initially thought, however briefly, that &lt;span class=&quot;math-container&quot;&gt;$|\psi|$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$|\psi|^2$&lt;/span&gt; was a measure of the probability. I find this impossible to understand if it were true that, at that time, he had been stimulated by Einstein's brilliant discussions of the fluctuations of quadratic quantities (in terms of fields) referring to radiation&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://core.ac.uk/download/pdf/81209121.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Creating Modern Probability, von Plato&lt;/a&gt; makes a somewhat tenuous connection to &lt;a href=&quot;http://www.thphys.uni-heidelberg.de/~amendola/otherstuff/einstein-paper-v2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein's Quantum Theory of a Monoatomic Ideal Gas, Second Communication (1925, p.9)&lt;/a&gt;. This paper, and its 1924 predecessor, are best known for developing a theory based on what is now called the Bose-Einstein statistics. In the second one, Einstein derives a formula for the mean-square fluctuation in the number of molecules, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \overline{\left(\frac{\Delta_\nu}{n_\nu}\right)^2}=\frac1{n_\nu}+\frac1{z_\nu}$&lt;/span&gt;, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$n_\nu$&lt;/span&gt; is the number of molecules, and, as Einstein writes, &quot;&lt;em&gt;...a scalar wave field can be adjoined to a gas, and I have assured myself through computation that &lt;span class=&quot;math-container&quot;&gt;$\frac1{z_\nu}$&lt;/span&gt; is the average square of fluctuation of this wave field&lt;/em&gt;&quot;. Pais's objection would still apply to this connection, but, perhaps, this field is an echo of the aforementioned &quot;ghost field&quot;, as von Plato speculates (p.146):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;By adjoining a scalar wave field to matter, Einstein arrived at a computation of his 'interference fluctuation' &lt;span class=&quot;math-container&quot;&gt;$\frac1{z_\nu}$&lt;/span&gt; , writing presumably with interference in mind that 'here more than a mere analogy is involved' (1925, p. 9). What did he use matter waves for? He used them for determining the mean-square fluctuation of &lt;span class=&quot;math-container&quot;&gt;$n_\nu$&lt;/span&gt;. But this can only be if the wave field determines a statistical&#xA;  law that the number of particles in the volume &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; obeys. As we shall see, in some way this connection between matter wave and probability was used in Born's probabilistic interpretation of Schrodinger's wave function. At the same time it pointed at a characteristic feature of quantum probability: the appearance of interference terms.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Other papers of Einstein's that feature both statistics and quantum mechanics are &lt;em&gt;Zum gegenwartigen Stand des Strahlungsproblems&lt;/em&gt; (1909, &lt;em&gt;Physikalische Zeitschrift&lt;/em&gt;, 10, 185-193) and &lt;em&gt;Zur Quantentheorie der Strahlung&lt;/em&gt; (1917, &lt;em&gt;Physikalische Zeitschrift&lt;/em&gt;, 18, 121-128), but their relevance seems even more remote.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-14T05:53:23.770" LastActivityDate="2019-09-14T05:53:23.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="9973" PostTypeId="1" CreationDate="2019-09-07T22:28:12.220" Score="0" ViewCount="66" Body="&lt;p&gt;I am aware the notion of disease 'having a course' comes from &lt;a href=&quot;https://en.wikipedia.org/wiki/Hippocrates&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hippocrates&lt;/a&gt;, who is known as the father of modern medicine in part due to this reason. From this fact I wonder if someone knows and could pinpoint the exact year when Hippocrates introduce 'the historical conception of disease'.&lt;/p&gt;&#xA;" OwnerUserId="3120" LastActivityDate="2019-09-07T22:28:12.220" Title="In which year did Hippocrates introduce the historical conception of disease?" Tags="|medicine|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="9974" PostTypeId="1" AcceptedAnswerId="10996" CreationDate="2019-09-09T03:57:20.617" Score="2" ViewCount="208" Body="&lt;p&gt;First of all, a comment, before this gets marked as a duplicate:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I &lt;em&gt;have&lt;/em&gt; searched this website for the question I’m asking and I’m aware that this exact question &lt;a href=&quot;https://hsm.stackexchange.com/q/5130/6514&quot;&gt;has been asked before&lt;/a&gt;. However, Eric Temple Bell (1915) was &lt;em&gt;not&lt;/em&gt; the first time the Unique Prime Factorization Theorem was called the “Fundamental Theorem of Arithmetic”.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bell’s 1915 book called it the FTA for the first time &lt;em&gt;in the English language&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another mathematician, probably a German one, called it “Fundamental Theorem of Arithmetic”, and I know it was a few years before Eric Temple Bell.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have spent quite a few hours researching this for something, but can’t seem to find the original “FTA” reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks in advance for any answer, help or suggestion.&lt;/p&gt;&#xA;" OwnerDisplayName="user10440" LastEditorUserId="6514" LastEditDate="2019-09-18T05:53:06.553" LastActivityDate="2020-04-16T04:48:56.463" Title="First time the unique factorization theorem was called FTA" Tags="|mathematics|mathematicians|terminology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="9975" PostTypeId="1" CreationDate="2019-09-10T21:14:48.150" Score="22" ViewCount="6219" Body="&lt;p&gt;I read many Einstein's Biographies, but Minkowski was never mentioned, though his discovery of the union of  space and time created the basis for GR. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Minkowski was Einstein's teacher of mathematics in Zürich. How did Einstein comment on Minkowski's revolutionary theory?&lt;/p&gt;&#xA;" OwnerUserId="4438" LastEditorUserId="55" LastEditDate="2019-09-10T23:57:01.447" LastActivityDate="2023-12-14T19:05:52.703" Title="What was the relationship between Einstein and Minkowski?" Tags="|biographical-details|theoretical-physics|relativity-theory|einstein|riemannian-geometry|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="9977" PostTypeId="2" ParentId="9975" CreationDate="2019-09-10T23:54:42.223" Score="26" Body="&lt;p&gt;A good account is &lt;a href=&quot;https://arxiv.org/abs/1210.6929&quot; rel=&quot;noreferrer&quot;&gt;Weinstein, Max Born, Albert Einstein and Hermann Minkowski's Space-Time Formalism of Special Relativity&lt;/a&gt;. They did no have much of a relationship, what it was is well-summarized by Sommerfeld:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Strangely enough no personal contacts resulted between his teacher of mathematics, Hermann Minkowski, and Einstein. When, later on, Minkowski built up the special theory of relativity into his 'world-geometry', Einstein said on one occasion: 'Since the mathematicians have invaded the theory of relativity, I do not understand it myself any more'. But soon thereafter, at the time of the conception of the general theory of relativity, he readily acknowledged the indispensability of the four-dimensional scheme of Minkowski&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Minkowski was one of Einstein's professors of mathematics at the Zürich Polytechnic (1896-1900), but Einstein was not interested in mathematics at the time, and skipped most of his lectures. As Minkowski himself told Born, &quot;&lt;em&gt;in his student days Einstein had been a real lazybones. He never bothered about mathematics at all&lt;/em&gt;&quot;. Einstein confirms this in his Autobiographical Notes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I had excellent teachers (for example, [Adolf] Hurwitz, Minkowski), so that I should have been able to obtain a mathematical training in depth. I worked most of the time in physical laboratory, however, fascinated by the direct contact with experience. The balance of the time I used, in the main, in order to study at home the works of Kirchhoff, Helmholtz, Hertz, etc.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1902 Minkowski moved to Göttingen. According to Born, he was interested in electrodynamics since 1905, but the interest was based on the works of Hertz, Fitzgerald, Larmor, Lorentz, and Poincaré. The now famous spacetime formalism was developed in 1907-08, with the first glimpse of it described in the lecture The Relativity Principle on November 5, 1907. It treats the principle in terms of ether, and names Lorentz as its discoverer. Einstein is credited with clarifying it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A detailed description of the spacetime formalism first appeared in the Minkowski's December 21, 1907 lecture The Basic Equations for Electromagnetic Processes in Moving Bodies, and was published in April, 1908, in &lt;em&gt;Göttinger Nachrichten&lt;/em&gt;. It was the only publication on the subject before his death on January 12, 1909. But it gained traction already after the September 12, 1908 talk at the Cologne Congress. Einstein was supposed to meet Planck there, but ended up not coming. His attitude towards Minkowski's formalism is politely expressed in a 1908 paper with Laub:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In a recent published study Mr. Minkowski has presented the fundamental equations for the electromagnetic processes in moving bodies. In view of the fact that this study makes rather great demands on the reader in its mathematical aspects, we do not consider it superfluous to derive here these important equations in an elementary way, which, is, by the way, essentially in agreement with that of Minkowski.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to Sommerfeld, Einstein was even more blunt privately. So most of the early work on general relativity owed very little to Minkowski's formalism, if anything, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5195/55&quot;&gt;How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?&lt;/a&gt; Einstein's attitude only changed in 1912, when he wrote to Sommerfeld on October 29:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I am now occupied exclusively with the gravitational problem, and believe that I can overcome all difficulties with the help of a local mathematician friend [Marcel Grossmann]. But one thing is certain, never before in my life have I troubled myself over anything so much, and that I have gained great respect for mathematics, whose more subtle parts I considered until now, in my ignorance, as pure luxury! Compared with this problem, the original theory of relativity is childish&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But by then his main influence was tensor calculus through Grossman and Levi-Civita, with whom he corresponded.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-12T11:54:21.900" LastActivityDate="2019-09-12T11:54:21.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10976" PostTypeId="1" AcceptedAnswerId="10977" CreationDate="2019-09-11T01:35:35.143" Score="3" ViewCount="141" Body="&lt;h2&gt;Background&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/Correspondence_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Bohr's correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large.[5] A. Sommerfeld (1924) referred to the principle as &quot;Bohrs Zauberstab&quot; (Bohr's magic wand).&quot;&lt;/p&gt;&#xA;&#xA;&lt;h2&gt;Question&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;Does A. Sommerfeld make a hidden pee pee joke over here? From the context and translation of this German I feel this is likely; since it's used in the context of taking the limit of a system to &quot;large&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know if this joke would make sense in German? Is there any evidence to further strengthen my case? Or am I reading too much &quot;in between the lines&quot;?    &lt;/p&gt;&#xA;" OwnerUserId="10067" LastEditorUserId="55" LastEditDate="2019-09-11T06:33:53.017" LastActivityDate="2019-09-11T06:57:24.987" Title="What did Sommerfeld mean by Bohr's magic wand?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="10977" PostTypeId="2" ParentId="10976" CreationDate="2019-09-11T06:30:48.890" Score="3" Body="&lt;p&gt;“&lt;em&gt;A cigar is sometimes just a cigar&lt;/em&gt;”, Freud. The original appearance is in 1919's edition of &lt;a href=&quot;https://archive.org/details/atombauundspekt00sommgoog/page/n10&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Atombau und Spektrallinien&lt;/em&gt;&lt;/a&gt;, not Wikipedia's 1924's, long before the &quot;new&quot; quantum mechanics. Bohr did wield his  &lt;em&gt;Zauberstab&lt;/em&gt; to import classical results into quantum physics, despite the inconsistency between the two, and to match the right answers from the spectral measurements. The context is discussed in &lt;a href=&quot;http://www.weatherglass.eu/PDFs/selectionRulesSubmittedPDF.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Borelli's The emergence of selection rules and their encounter with group theory, 1913-1927&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Using classical electromagnetism, Rubinowicz computed the angular momentum of a&#xA;  spherical wave and showed that, when renormalized by &lt;span class=&quot;math-container&quot;&gt;$h/2π$&lt;/span&gt;, it had an absolute value equal or less than one. (Rubinowicz, 1918a, pp. 443-444). This led to a selection principle for the azimuthal&#xA;  quantum number: &lt;span class=&quot;math-container&quot;&gt;$\Delta n = 0, ±1$&lt;/span&gt; (Rubinowicz, 1918a, pp. 444-445)... It was left to Sommerfeld, in the first edition of Atombau und Spektrallinien (1919), to connect Bohr and Rubinowicz's results to his own interpretation of spectral terms as linked to&#xA;  azimuthal quantum numbers. Sommerfeld showed a very clear preference for the theory of his&#xA;  former assistant Rubinowicz, which he discussed at length (Sommerfeld, 1919, pp. 390-411).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Rubinowicz's &quot;selection principle&quot; was for Sommerfeld a means to bridge the gap between classical and quantum physics... He admitted that Bohr's condition &lt;span class=&quot;math-container&quot;&gt;$\Delta n = ±1$&lt;/span&gt; fitted much better&#xA;  the Rydberg-Ritz formula than Rubinowicz's, but made clear the epistemological gap he perceived&#xA;  between a theoretically significant selection principle like Rubinowicz's, on the one side, and Bohr's&#xA;  empirically successful condition, on the other. The latter, he described as a &quot;magic wand&quot;&#xA;  (&quot;Zauberstab&quot;) to make quantum theory useful in practice (Sommerfeld, 1919, pp. 406-411, quote&#xA;  from p. 402)&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-11T06:57:24.987" LastActivityDate="2019-09-11T06:57:24.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10978" PostTypeId="1" AcceptedAnswerId="10982" CreationDate="2019-09-11T15:30:59.190" Score="7" ViewCount="674" Body="&lt;p&gt;I looked through both of &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Boole&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Boole&lt;/a&gt;'s treatises (&lt;a href=&quot;http://www.gutenberg.org/files/36884/36884-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; and &lt;a href=&quot;https://www.gutenberg.org/files/15114/15114-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;), but there is nothing like implication as I have seen it, with&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F \rightarrow F = T$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F \rightarrow T = T$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$T \rightarrow F = F$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$T \rightarrow T = T$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, if George Boole didn't create this construct, where did it come from?&lt;/p&gt;&#xA;&lt;p&gt;The reason I ask is that there are a few cognitive traps from implication, particularly when we try to view it &amp;quot;intuitively&amp;quot;, and I suspect that this is because the intuitive explanations are all somewhat nonsensical.&lt;/p&gt;&#xA;&lt;p&gt;Just as a quick for instance of what I mean: if I ask a the classic question&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When we say &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, what can we say about whether &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are necessary or sufficient for one another?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;there are a series of cognitive traps for students.  First is that we have taught implication as a proposition that could be true or false, but we are now using it as an unfalsifiable predicate, so we would not even consider the entry in the truth table in which &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is false.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;What I think happened&lt;/h3&gt;&#xA;&lt;p&gt;If we create a table of all of the possible truth results from two inputs (using 1 for true and 0 for false), we get:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XWlkx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XWlkx.png&quot; alt=&quot;all possible values&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using conjunction, exclusive and inclusive disjunction, not, and tautology/contradiction, we can label these thus:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jjZaV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jjZaV.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This leaves us with the four slots in which three of the four answers are identical, and one of the two middle answers is the only different answer.&lt;/p&gt;&#xA;&lt;p&gt;What I believe happened is that someone essentially looked at this table, realized that those four could be reduced to one asymmetric operation, and looked back to Aristotle or Philo and chose &lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt; and &amp;quot;implies&amp;quot; to fill in this relationship.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rIE1F.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rIE1F.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In essence, then, implication is a purely axiomatic relationship, and all of the &amp;quot;intuitive&amp;quot; (but not really) ways of thinking about it can be used as a good way to think about it, but not what it truly is.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;However, George Boole did not address this in his treatises as far as I can see.  So, where did we get the modern relationship that we now call &amp;quot;implies&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="10448" LastEditorUserId="4251" LastEditDate="2020-07-25T19:45:39.180" LastActivityDate="2020-07-25T19:45:39.180" Title="Where does the material implication come from, if not from George Boole?" Tags="|mathematical-logic|logic|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10979" PostTypeId="2" ParentId="9975" CreationDate="2019-09-11T16:09:31.480" Score="2" Body="&lt;p&gt;There is a monolithic Einstein myth consolidated by scores of popular books which is a great inconvenience for anybody trying to make some sense of the history. Cornifold's answer and refs already point at the main isuue: at first Einstein did not understand and avoided mentioning Minkowski's approach which nevertheless  turned to be crucial for his later work.&lt;a href=&quot;https://www.jstor.org/stable/41134033&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Hermann Minkowski and the Postulate of Relativity&lt;/em&gt;&lt;/a&gt; by Leo Corry offers many details missing in superficial histories. It also emphasizes the link with Hilbert and the mathematical community in general but somehow omits the important connection to the Erlangen Progarm. Klein wrote in 1910 &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“One could replace ‘theory of invariants relative to a group of transformations’ by the words ‘relativity theory with respect to a group’.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The book &lt;em&gt;Sophus Lie and Felix Klein: The Erlangen Program&#xA;and Its Impact in Mathematics and Physics&lt;/em&gt; (2015) ed. by L. Ji&#xA;and A. Papadopoulos, makes evident how Minkowski's work turned a particular physical fact into a major theoretical issue.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2019-09-11T16:09:31.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10980" PostTypeId="2" ParentId="10978" CreationDate="2019-09-11T17:30:47.317" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Material_conditional&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; we find a quote from Dorothy Edgington, The Stanford Encyclopedia of Philosophy, &amp;quot;Conditionals&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was taken up enthusiastically by Russell (who called it &amp;quot;material implication&amp;quot;), Wittgenstein in the Tractatus, and the logical positivists, and it is now found in every logic text.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The date for Boole's &lt;em&gt;Laws of Thought&lt;/em&gt; is 1854.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-09-11T17:30:47.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10981" PostTypeId="2" ParentId="10978" CreationDate="2019-09-11T17:40:23.177" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Sanders_Peirce&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Sanders Peirce&lt;/a&gt; is credited with the introduction of truth tables in an unpublished manuscript dated 1893.  This includes a truth table for what we now call material implication. A detailed account is provided in the paper &lt;a href=&quot;https://arxiv.org/pdf/1108.2429.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peirce’s Truth-Functional Analysis and the Origin of Truth Tables&lt;/a&gt; by I. Anellis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Peirce used the term &lt;em&gt;illiation&lt;/em&gt; to denote material implication.  In his 1880 paper &lt;em&gt;The Algebra of Logic&lt;/em&gt;, Peirce explicitly defines &lt;em&gt;illiation&lt;/em&gt; as &quot;P implies Q&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A typed manuscript of one of Russell's 1912 lectures features a handwritten truth table for material implication on the verso (in the hand of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Wittgenstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wittgenstein&lt;/a&gt;) along with a truth table for negation (in Russell's hand). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The definition of material implication &lt;span class=&quot;math-container&quot;&gt;$P \rightarrow Q$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\lnot P \lor Q$&lt;/span&gt; is found in &lt;a href=&quot;https://en.wikipedia.org/wiki/Bertrand_Russell&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfred_North_Whitehead&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whitehead&lt;/a&gt;'s &lt;a href=&quot;https://en.wikipedia.org/wiki/Principia_Mathematica&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principia Mathematica&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;implies&quot; as used here expresses nothing else than the connection between &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; also expressed by the disjunction &quot;&lt;span class=&quot;math-container&quot;&gt;$\text {not-}p \text { or } q$&lt;/span&gt;&quot; The symbol employed for &quot;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&quot; i.e. for &quot;&lt;span class=&quot;math-container&quot;&gt;$\lnot p \lor  q$&lt;/span&gt;&quot; is &quot;&lt;span class=&quot;math-container&quot;&gt;$p ⊃ q$&lt;/span&gt;.&quot; This symbol may also be read &quot;if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2019-09-11T17:40:23.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10982" PostTypeId="2" ParentId="10978" CreationDate="2019-09-11T23:34:40.190" Score="10" Body="&lt;p&gt;It may be surprising, but the material implication does not come from truth tables, the truth table definition is a late development. Neither de Morgan, nor Peirce, nor Frege, nor even Russell came up with it or justified it by matching Boolean operations to something in Plato and Aristotle. A detailed story can be found in &lt;a href=&quot;https://archive.org/details/AHistoryOfMathematicalNotationVolII&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History of mathematical Notations, vol. II&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It came from a very common idea of classical logicians of identifying propositions with classes (intensions), and classes with sets (extensions). Accordingly, the early definitions of implication interpret &amp;quot;X implies Y&amp;quot; as &amp;quot;X is contained in Y&amp;quot;. Originally, it was only applied to syllogisms, where it matches the intuition.&lt;/p&gt;&#xA;&lt;p&gt;In the 19th century, its scope was expanded with the algebraization of logic by Boole and de Morgan. The chain of transmission went from de Morgan's &lt;code&gt;)&lt;/code&gt; (1847, same year as Boole's first treatise), to Peirce's claw ―&amp;lt; (1867), to Schröder's &lt;span class=&quot;math-container&quot;&gt;$\supset$&lt;/span&gt; (1890), and Peano's Ↄ. Later Peano, and after him Russell, adopted Schröder's &lt;span class=&quot;math-container&quot;&gt;$\supset$&lt;/span&gt; (note that the meaning is reversed compared to the modern set inclusion). Peirce called the material implication &amp;quot;the copula of inclusion&amp;quot; (also &amp;quot;illation&amp;quot;), and Frege (whose notation for it was clumsy and never reproduced later) even criticized Schröder for &amp;quot;confusing&amp;quot; it with class inclusion.&lt;/p&gt;&#xA;&lt;p&gt;The identification itself predates Boole and even Leibniz, it can be traced back to Aristotle, and was implicit in scholastic logic (for syllogisms). Russell's Principia still has a trace of it, the class/set identification only goes away after Hausdorff's &lt;em&gt;Grundzuge der Mengenlehre&lt;/em&gt; (1914), see &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.9839&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kanamori's The empty set, the singleton, and the ordered pair&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, X is not contained in Y if and only if there is something in X which is not in Y. Frege and Peirce understood this truth functional consequence of the proposition/class identification, and made it definitional when transitioning to a logic with quantifiers. For example, Peirce wrote in 1883 (quoted from &lt;a href=&quot;https://www.jstor.org/stable/20127536&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dipert, Peirce's Propositional Logic&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;To say that an inference is correct is to&#xA;say that if the premisses are true the conclusion is also true; or that every possible state of things would be included among the possible state of things in which the conclusion would be true. We are thus led&#xA;to the copula of inclusion&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Frege's work remained buried until Russell brought it back from obscurity in Principia. The rest (including Peano and, through him, Russell) adopted notation and conventions of Schröder's Algebra of Logic, which followed Peirce, where an equivalent of &lt;span class=&quot;math-container&quot;&gt;$A\supset B=\lnot A \lor B$&lt;/span&gt; already appears, see &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01445348408837062&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dipert Peirce, Frege, the logic of relations, and Church's theorem&lt;/a&gt;. But Peirce used truth tables only sporadically and in unpublished manuscripts (1893 and 1902), so they did not become common until Russell and Wittgenstein reinvented them in 1912.&lt;/p&gt;&#xA;&lt;p&gt;So the material conditional gradually emerged from a cluster of intuitions about propositions, classes and sets. But there are only two cases where it fully applies in its modern form:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Conceptual containment in syllogism (a la Aristotle and Kant).  This form is too narrow to cover our intuitive notion of inference.&lt;/li&gt;&#xA;&lt;li&gt;The model-theoretic definition of extensional entailment in modern mathematics, a.k.a. &lt;a href=&quot;https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence&quot; rel=&quot;nofollow noreferrer&quot;&gt;semantic consequence&lt;/a&gt;, a la Tarski.  This model does not entirely match the intuitive &lt;a href=&quot;https://en.wikipedia.org/wiki/Indicative_conditional&quot; rel=&quot;nofollow noreferrer&quot;&gt;indicative conditional&lt;/a&gt;. Hence the cognitive traps:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The material conditional allows implications to be true even when the antecedent is irrelevant to the consequent. For example, it's commonly accepted that the sun is made of plasma, on one hand, and that 3 is a prime number, on the other. The standard definition of implication allows us to conclude that, if the sun is made of plasma, then 3 is a prime number. This is arguably synonymous to the following: the sun's being made of plasma renders 3 a prime number. Many people intuitively think that this is false, because the sun and the number three simply have nothing to do with one another...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;...Another issue is that the material conditional is not designed to deal with counterfactuals and other cases that people often find in if-then reasoning... A further problem is that the material conditional is such that (P ∧ ¬P) → Q, regardless of what Q is taken to mean. That is, a contradiction implies that absolutely everything is true.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An interesting reconstruction of how truth functional connectives became implicit in the vernacular of mathematical proofs is in &lt;a href=&quot;https://as.tufts.edu/philosophy/sites/all/themes/asbase/assets/documents/azzouniStillASense.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Azzouni's paper, pp. 37-38&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-09-15T06:20:16.890" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="10983" PostTypeId="2" ParentId="10978" CreationDate="2019-09-12T11:40:57.090" Score="4" Body="&lt;p&gt;The &quot;discoverer&quot; of what we call today &lt;em&gt;material conditional&lt;/em&gt;, i.e. the truth-functional definition of &quot;if ... then&quot;, is &lt;a href=&quot;https://en.wikipedia.org/wiki/Philo_the_Dialectician&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philo the Dialectician (ca.300 BCE)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;https://plato.stanford.edu/entries/logic-ancient/#DioCroPhiLog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient Logic&lt;/a&gt; :&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A conditional was considered a non-simple proposition composed of two propositions and the connecting particle ‘if’. Philo, who may be credited with introducing truth-functionality into logic, provided the following criterion for their truth: A conditional is false when and only when its antecedent is true and its consequent is false, and it is true in the three remaining truth-value combinations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also Benson Mates, &lt;a href=&quot;https://books.google.it/books?id=3jmQoAEACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stoic Logic (California UP, 2nd ed.1961)&lt;/a&gt;, &lt;em&gt;Ch.4 Propositional connectives&lt;/em&gt;, page 43.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2019-09-12T12:15:40.933" LastActivityDate="2019-09-12T12:15:40.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10984" PostTypeId="1" AcceptedAnswerId="10994" CreationDate="2019-09-12T11:52:54.993" Score="5" ViewCount="2192" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Position&lt;/strong&gt; is a vector. &lt;strong&gt;Distance/length&lt;/strong&gt; is a name of its magnitude. &lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Velocity&lt;/strong&gt; is a vector. &lt;strong&gt;Speed&lt;/strong&gt; is a name of its magnitude.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Acceleration&lt;/strong&gt; is a name of a vector &lt;em&gt;and&lt;/em&gt; its magnitude.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Force&lt;/strong&gt; is a name of a vector &lt;em&gt;and&lt;/em&gt; its magnitude.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Momentum&lt;/strong&gt; is a name of a vector &lt;em&gt;and&lt;/em&gt; its magnitude.&lt;/li&gt;&#xA;&lt;li&gt;...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Of all the vastly many types of vector quantities we traditionally have defined in physics (and other technical sciences), two leap out at us as oddly different, regarding their terminology: &lt;strong&gt;Position&lt;/strong&gt; and &lt;strong&gt;velocity&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These two vectors have scalar magnitudes that are named &lt;em&gt;differently&lt;/em&gt; than themselves. For all other vectors, the naming convention is to use the same term for both the vector itself as well as for its scalar magnitude. If we are talking about the vector or the scalar then depends on context - at least we don't have to memorise two terms for each defined vector quantity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware of the use in the English language. These many different words exist in English, sure, such as &lt;em&gt;distance&lt;/em&gt;, &lt;em&gt;length&lt;/em&gt;, &lt;em&gt;displacement&lt;/em&gt; in relation to position, and such as &lt;em&gt;speed&lt;/em&gt; in relation to velocity. But just because many words exist in the shared language, this doesn't have to require scientists to include all those words into accurate definitions in physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Are there any historical reasons for why only these two vector quantities have differently named scalar magnitudes&lt;/strong&gt;, a practice which breaks the otherwise consistent terminology pattern, and a practice that confuses and complicates the introductory teaching of physics (these two quantites are after all the very first ones you learn about in your high-school and/or university science class).&lt;/p&gt;&#xA;" OwnerUserId="4885" LastEditorUserId="4885" LastEditDate="2019-09-12T12:14:44.440" LastActivityDate="2023-06-12T10:32:30.613" Title="Why are 'speed' and 'velocity' not given the same name?" Tags="|physics|terminology|" AnswerCount="2" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="10986" PostTypeId="1" CreationDate="2019-09-12T17:12:04.197" Score="0" ViewCount="33" Body="&lt;p&gt;Probability function is of two types in general. They are unconditional probability and conditional probability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the terms prior probability and posterior probability are used in place of unconditional probability and conditional probability respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In which year do the usage of terms prior and posterior started in probability?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastActivityDate="2019-09-12T17:12:04.197" Title="Usage of terms prior and posterior in probability" Tags="|terminology|probability|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="10987" PostTypeId="1" CreationDate="2019-09-12T19:55:15.137" Score="3" ViewCount="106" Body="&lt;p&gt;What are some pre-19&lt;sup&gt;th&lt;/sup&gt; or pre-20&lt;sup&gt;th&lt;/sup&gt; century sources on information theory? Do they exist?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="4251" LastEditDate="2021-10-02T14:03:11.510" LastActivityDate="2021-10-02T14:03:11.510" Title="Pre-20th century sources on information theory?" Tags="|reference-request|information-theory|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="10988" PostTypeId="1" CreationDate="2019-09-13T02:07:17.283" Score="0" ViewCount="168" Body="&lt;p&gt;I read an article published by &lt;em&gt;Scientific American&lt;/em&gt; stating that Einstein did not prove the equation within the context of Special Relativity but I am skeptical. I was hoping someone could point me in the right direction on the history of this famous equation.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="72" LastEditDate="2019-12-29T03:13:16.653" LastActivityDate="2019-12-29T03:13:16.653" Title="Does Einstein's original derivation of $E=mc^2$ exist and what is it?" Tags="|physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="10990" PostTypeId="1" AcceptedAnswerId="10993" CreationDate="2019-09-13T18:13:36.700" Score="12" ViewCount="4091" Body="&lt;p&gt;Equations involving &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck_constant&quot; rel=&quot;noreferrer&quot;&gt;Planck's constant&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$h ,$&lt;/span&gt; are often simplified by instead writing them in terms of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reduced_Planck_constant&quot; rel=&quot;noreferrer&quot;&gt;reduced Planck's constant&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi}.$&lt;/span&gt;  But where did the symbol for the reduced Planck's constant, &lt;span class=&quot;math-container&quot;&gt;$\hbar ,$&lt;/span&gt; come from?&lt;/p&gt;&#xA;" OwnerUserId="6913" LastEditorUserId="609" LastEditDate="2019-09-16T12:05:09.820" LastActivityDate="2023-10-03T02:10:52.693" Title="What is the origin of the $\hbar$ symbol?" Tags="|physics|notation|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="10991" PostTypeId="2" ParentId="10990" CreationDate="2019-09-14T01:47:31.220" Score="6" Body="&lt;p&gt;There is another myth that h is a short form of &lt;em&gt;Hilfsgrösse&lt;/em&gt;, with no proof whatsoever (see the excerpt below). Thus &quot;h-bar&quot; is no different myth, no matter how reliable it sounds. A very valid question is who introduced the h-bar notation. Since h-bar is also called Dirac h, I checked his book, and indeed there it is on page 87, of his famous book &quot;&lt;em&gt;Principles of Quantum Mechanics&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;span class=&quot;math-container&quot;&gt;$uv-vu$&lt;/span&gt;=&lt;span class=&quot;math-container&quot;&gt;$\hbar$$i$&lt;/span&gt;[u,v], where &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; is a new universal constant.&#xA;  It has the dimensions of action. In order that theory may agree with&#xA;  experiment, we must take &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; equal to &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;/2&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is the&#xA;  universal constant that was introduced by Planck, known as Planck's&#xA;  constant.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Have a look at this anecdote &lt;a href=&quot;http://www.raczar.es/webracz/ImageServlet?mod=publicaciones&amp;amp;subMod=revistas&amp;amp;car=revista58&amp;amp;archivo=091.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Thermal Radiation Formula of Planck (1900)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8FkqI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8FkqI.png&quot; alt=&quot;Myth about symbols&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Long time ago we were having a chemistry exam in school. A student said what if there is question, &quot;why a beaker is called a beaker?&quot;. In all seriousness another one quipped that &quot;a beaker is a beaker because it has a &lt;em&gt;beak&lt;/em&gt;.&quot; I was impressed, thinking that indeed the beaker's spout looks like a bird's beak and thought this is the right answer. When I came home and checked the dictionary, this cute story had nothing to do with reality. &lt;strong&gt;Don't trust whatever you find on the web.&lt;/strong&gt; A prime example is fake mentioned anecdote, just like I discovered yesterday that nobody knows who coined &lt;em&gt;sinc function&lt;/em&gt;'s full name. Books, webpages, all say it is &lt;em&gt;sinus cardinalis&lt;/em&gt; or cardinal sine. It may be, but the whoever came up with this &lt;em&gt;full name&lt;/em&gt; is not known and all wrong names are associated with it.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="2277" LastEditDate="2020-05-05T21:29:02.957" LastActivityDate="2020-05-05T21:29:02.957" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="10992" PostTypeId="1" AcceptedAnswerId="10995" CreationDate="2019-09-14T12:04:20.117" Score="5" ViewCount="372" Body="&lt;p&gt;In the 19th and 20th centuries, the student of classical Greek geometry used &quot;straight edge and compass&quot;. &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF01395658&quot; rel=&quot;nofollow noreferrer&quot;&gt;A. Seidenberg&lt;/a&gt; uses the terminology &quot;peg and cord&quot; in proposing that altar construction motivated the development of classical Greek and Indian geometry. So one supposes that a transition occurred in which &quot;peg and cord&quot; were replaced by &quot;straight edge and compass&quot;. Is this correct, and if so do we have evidence as to when this transition occurred? &lt;/p&gt;&#xA;" OwnerUserId="10102" LastEditorUserId="55" LastEditDate="2019-09-16T04:42:40.910" LastActivityDate="2019-09-16T04:42:40.910" Title="When did mathematicians transition from peg and rope to straightedege and compass?" Tags="|mathematics|geometry|engineering|architectural-engineering|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="10993" PostTypeId="2" ParentId="10990" CreationDate="2019-09-15T07:23:25.610" Score="18" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;{\def\Target#1{\rlap{\smash{\label{#1}\phantom{\tag{#1}}}}}}&#xA;{\def\BackUp{\raise{0.25em}{\Tiny{\boxed{\boldsymbol{\Uparrow} \hspace{-2px}}}}}}&#xA;$&lt;/span&gt;&lt;strong&gt;&lt;em&gt;tl;dr&lt;/em&gt;–&lt;/strong&gt; It's unclear.  The symbol &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; itself wasn't anything new.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Dirac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Dirac&lt;/a&gt; defined &lt;a href=&quot;https://en.wikipedia.org/wiki/Reduced_Planck_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi}$&lt;/span&gt;&lt;/a&gt; in &lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Ref4&quot;&gt;a 1930 book&lt;/a&gt;, but didn't explain why it was used when defining it.  It might still be possible for someone to figure out the reason for this unusual symbol if they were to examine Dirac's personal notebooks or correspondence, or perhaps a later retrospective publication, but no explanation is apparently found in the original public appearances of &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi} .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;span class=&quot;math-container&quot;&gt;$\Target{Timeline} \textbf{Timeline}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Early&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\left[\texttt{????} \right] {:} \Target{TimelineEarly}$&lt;/span&gt;&lt;/a&gt;  Symbols proximally &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; appear in multiple alphabets, as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Astronomical_symbols&quot; rel=&quot;nofollow noreferrer&quot;&gt;astronomical symbol&lt;/a&gt; for the planet &lt;a href=&quot;https://en.wikipedia.org/wiki/Saturn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saturn&lt;/a&gt;, and as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Alchemical_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;alchemical symbol&lt;/a&gt; for &lt;a href=&quot;https://en.wikipedia.org/wiki/Lead&quot; rel=&quot;nofollow noreferrer&quot;&gt;lead&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-In1900&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\left[\texttt{1900} \right] {:} \Target{Timeline1900}$&lt;/span&gt;&lt;/a&gt;  The symbol &lt;span class=&quot;math-container&quot;&gt;$`` h &quot;$&lt;/span&gt; is used to refer to &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planck's constant&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-In1913&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\left[\texttt{1913} \right] {:} \Target{Timeline1913}$&lt;/span&gt;&lt;/a&gt;  The value &lt;span class=&quot;math-container&quot;&gt;$\frac{h}{2 \pi}$&lt;/span&gt; becomes significant due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bohr's model of atoms&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-In1926&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\left[\texttt{1926} \right] {:} \Target{Timeline1926}$&lt;/span&gt;&lt;/a&gt;  Both &lt;span class=&quot;math-container&quot;&gt;$K \equiv \frac{h}{2 \pi}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h \equiv \frac{h}{2 \pi}$&lt;/span&gt; appear in the literature, from &lt;a href=&quot;https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erwin Schrödinger&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Dirac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Dirac&lt;/a&gt;, respectively.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-In1930&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\left[\texttt{1930} \right] {:} \Target{Timeline1930}$&lt;/span&gt;&lt;/a&gt;  Paul Dirac publishes &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi}$&lt;/span&gt; in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Principles_of_Quantum_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;&lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt;&amp;quot;&lt;/a&gt;.  Dirac doesn't explain his choice of &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; when defining it.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Conclusion&quot;&gt;In short&lt;/a&gt;, while it seems reasonable to assume Dirac selected &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; in part due to its similarity to &lt;span class=&quot;math-container&quot;&gt;$`` h &quot; ,$&lt;/span&gt; it's still unclear what else may've played into his choice.  More information might be gleaned from Dirac's personal journals or correspondence.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-TimelineEarly&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{Early} \textbf{Early history:}~~`` \mathbf{\hbar} &quot; ~\textbf{appears in various old alphabets.}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;The symbol itself, &lt;span class=&quot;math-container&quot;&gt;$\hbar ,$&lt;/span&gt; is nothing new.  Glancing at Wikipedia real quick, looks like it's earlier referenced as:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Latin_alphabet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Latin alphabet&lt;/a&gt;;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Slavic_languages&quot; rel=&quot;nofollow noreferrer&quot;&gt;Slavic&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Cyrillic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cyrillic&lt;/a&gt; letter, &lt;a href=&quot;https://en.wikipedia.org/wiki/Tshe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tshe&lt;/a&gt;;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Astronomical_symbols&quot; rel=&quot;nofollow noreferrer&quot;&gt;astronomical symbol&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Saturn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saturn&lt;/a&gt;;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Alchemical_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;alchemical symbol&lt;/a&gt; for &lt;a href=&quot;https://en.wikipedia.org/wiki/Lead&quot; rel=&quot;nofollow noreferrer&quot;&gt;lead&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Ħ&lt;/strong&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Lower_case&quot; rel=&quot;nofollow noreferrer&quot;&gt;minuscule&lt;/a&gt;: &lt;strong&gt;ħ&lt;/strong&gt;) is a letter of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Latin_alphabet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Latin alphabet&lt;/a&gt;, derived from &lt;a href=&quot;https://en.wikipedia.org/wiki/H&quot; rel=&quot;nofollow noreferrer&quot;&gt;H&lt;/a&gt; with the addition of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Bar_(diacritic)&quot; rel=&quot;nofollow noreferrer&quot;&gt;bar&lt;/a&gt;. It is used in &lt;a href=&quot;https://en.wikipedia.org/wiki/Maltese_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maltese&lt;/a&gt; and in &lt;a href=&quot;https://en.wikipedia.org/wiki/Tunisian_Arabic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tunisian Arabic&lt;/a&gt; transliteration (based on Maltese with additional letters) for a &lt;a href=&quot;https://en.wikipedia.org/wiki/Voiceless_pharyngeal_fricative&quot; rel=&quot;nofollow noreferrer&quot;&gt;voiceless pharyngeal fricative&lt;/a&gt; consonant (corresponding to the letter &lt;a href=&quot;https://en.wikipedia.org/wiki/Heth_(letter)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;heth&lt;/em&gt;&lt;/a&gt; of Semitic &lt;a href=&quot;https://en.wikipedia.org/wiki/Abjad&quot; rel=&quot;nofollow noreferrer&quot;&gt;abjads&lt;/a&gt;). Lowercase ħ is used in the &lt;a href=&quot;https://en.wikipedia.org/wiki/International_Phonetic_Alphabet&quot; rel=&quot;nofollow noreferrer&quot;&gt;International Phonetic Alphabet&lt;/a&gt; for the same sound.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantum mechanics&lt;/a&gt;, an &lt;a href=&quot;https://en.wikipedia.org/wiki/Italic_type&quot; rel=&quot;nofollow noreferrer&quot;&gt;italic&lt;/a&gt; &lt;em&gt;ℏ&lt;/em&gt; (U+210F) with a line, represents the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reduced_Planck_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;reduced Planck constant&lt;/a&gt;. In this context, it is pronounced &amp;quot;h-bar&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The lowercase resembles the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cyrillic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cyrillic&lt;/a&gt; letter &lt;a href=&quot;https://en.wikipedia.org/wiki/Tshe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tshe&lt;/a&gt; (ћ), or the &lt;a href=&quot;https://en.wikipedia.org/wiki/Astronomical_symbols&quot; rel=&quot;nofollow noreferrer&quot;&gt;astronomical symbol&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Saturn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saturn&lt;/a&gt; (♄).&lt;/p&gt;&#xA;&lt;p&gt;—&lt;a href=&quot;https://en.wikipedia.org/wiki/H_with_stroke&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;&lt;em&gt;H with stroke&lt;/em&gt;&amp;quot;&lt;/a&gt;, Wikipedia&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Due to this history, we can at least say that it doesn't appear to be a new symbol made up for &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi} ,$&lt;/span&gt; but rather a preexisting symbol.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline1900&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{In1900} \textbf{In 1900:} ~~ \textbf{Planck's constant,}~ `` h &quot;, ~\textbf{appears.}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;In 1900, &lt;a href=&quot;https://en.wikipedia.org/wiki/Max_Planck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max Planck&lt;/a&gt; came up with &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planck's law&lt;/a&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;{B}_{\nu} \left( \nu, T \right)&#xA;~=~&#xA;\frac{2 h {\nu}^{3}}{c^2}&#xA;\frac{1}{{e}^{\frac{h \nu}{k_{\text{B}} T}} - 1}&#xA;\,,&#xA;$$&lt;/span&gt;&#xA;where&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${B}_{\nu} \left( \nu, T \right)$&lt;/span&gt; is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Spectral_radiance&quot; rel=&quot;nofollow noreferrer&quot;&gt;spectral radiance&lt;/a&gt; of the black-body radiation;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; is the frequency of emitted black-body radiation;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is the temperature of the black-body emitting the radiation;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$k_{\text{B}}$&lt;/span&gt; is &lt;a href=&quot;https://en.wikipedia.org/wiki/Boltzmann_constant&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Boltzmann constant&lt;/a&gt;;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is the Planck constant;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the speed of light in the medium.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As a heuristically established law, it involved an unspecified value that came to be known as Planck's constant, &lt;span class=&quot;math-container&quot;&gt;$h .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline1913&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{In1913} \textbf{In 1913:}~~\textbf{The value}~{\frac{h}{2 \pi}}~\textbf{becomes notable.}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;In 1913, &lt;a href=&quot;https://en.wikipedia.org/wiki/Niels_Bohr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Niels Bohr&lt;/a&gt; proposed the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bohr model of the atom&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Bohr's model included stationary electron orbitals in which electrons had an angular momentum consistent with&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;m_{\text{electron}}vr&#xA;~=~&#xA;n \frac{h}{2\pi}&#xA;\,,$$&lt;/span&gt;&#xA;where:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$m_{\text{electron}}$&lt;/span&gt; is the mass of an electron;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is the orbital velocity of the electron;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the radius of the electron's orbit;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is Planck's constant;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is the circle-constant;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{N}$&lt;/span&gt; is a non-zero, non-negative integer value.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This can be more concisely written as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;m_{\text{electron}}vr&#xA;~=~&#xA;n \hbar&#xA;\,,$$&lt;/span&gt;&#xA;such that there's now some motivation to have a symbol that's &lt;span class=&quot;math-container&quot;&gt;$\equiv \frac{h}{2 \pi} .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline1926&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{In1926} \textbf{In 1926:}~~\textbf{Papers define both }~{K \equiv \frac{h}{2 \pi}}~\textbf{and}~{h \equiv \frac{h}{2 \pi}\,}\textbf{.}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;In 1926, both &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; are defined as &lt;span class=&quot;math-container&quot;&gt;$\frac{h}{2 \pi} .$&lt;/span&gt;&lt;sup&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Ref1&quot;&gt;(Ref. 1)&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erwin Schrödinger&lt;/a&gt; defined &lt;span class=&quot;math-container&quot;&gt;$K \equiv \frac{h}{2 \pi}$&lt;/span&gt; in&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Schrödinger, &lt;em&gt;Ann. D. Phys.&lt;/em&gt;, &lt;strong&gt;79&lt;/strong&gt;, 361-376 (1926).&lt;sup&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Ref2&quot;&gt;(Ref. 2)&lt;/a&gt;&lt;/sup&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Dirac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Dirac&lt;/a&gt; defined &lt;span class=&quot;math-container&quot;&gt;$h \equiv \frac{h}{2 \pi}$&lt;/span&gt; in:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Dirac, &lt;em&gt;Proc. Roy. Soc.&lt;/em&gt;, A&lt;strong&gt;112&lt;/strong&gt;, 661-677 (1926).&lt;sup&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Ref3&quot;&gt;(Ref. 3)&lt;/a&gt;&lt;/sup&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Dirac's 1926 publication appears to suggest that Dirac might not have been using the symbol &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; yet.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline1930&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{In1930} \textbf{In 1930:}~~\textbf{Dirac publishes}~ {\hbar \equiv \frac{h}{2 \pi}} ~ \textbf{in a book.}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;In 1930, Paul Dirac publishes a book, &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Principles_of_Quantum_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;&lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt;&amp;quot;&lt;/a&gt;, which defines &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi} .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Dirac doesn't explain why the symbol &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; was selected when defining it.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{Conclusion} \textbf{Conclusion:}~~\textbf{It's unclear exactly why}~ \mathbf{`` \hbar &quot;} ~\textbf{was selected.}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;We can reasonably estimate that &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi}$&lt;/span&gt; was selected by Paul Dirac (or someone close to him) at some point between 1913 (at which point the value became notable) and 1930 (at which point the definition was published).  Probably after his publication in 1926, as Dirac didn't use &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; in that publication.&lt;/p&gt;&#xA;&lt;p&gt;I think it's a pretty safe bet that the symbol &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; was selected in part due to its similarity to the symbol for Planck's constant, &lt;span class=&quot;math-container&quot;&gt;$`` h &quot;.$&lt;/span&gt;  This seems like a perk over Schrödinger's contemporaneous &lt;span class=&quot;math-container&quot;&gt;$K \equiv \frac{h}{2 \pi} .$&lt;/span&gt;  &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; probably got a boost over alternatives, e.g. &lt;span class=&quot;math-container&quot;&gt;$`` K &quot; ,$&lt;/span&gt; due to appearing in Dirac's influential book in 1930.&lt;/p&gt;&#xA;&lt;p&gt;However, it's unclear why Paul Dirac may've chosen &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; over some other variant of &lt;span class=&quot;math-container&quot;&gt;$`` h &quot; .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;More information on the topic might come from an examination of Paul Dirac's personal notebooks or correspondence, though at the moment, the exact history seems unclear.&lt;/p&gt;&#xA;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{Errata} \textbf{Errata}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Ref1&quot;&gt;(Ref. 1)&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; was introduced in &lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Ref3&quot;&gt;Dirac's 1926 paper&lt;/a&gt;.  However, the that claims appears to have been incorrect, as that online copies of that paper don't appear to include the symbol &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot; ,$&lt;/span&gt; but rather define &lt;span class=&quot;math-container&quot;&gt;$h = \frac{h}{2 \pi} .$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since this is an early paper with a special symbol, perhaps it's possible that other printings of the same paper used &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$`` h &quot; ,$&lt;/span&gt; as claimed by (Ref. 1)?  However, this could just be a misattribution on their part.&lt;/p&gt;&#xA;&lt;p&gt;If this is just an error, then Dirac's 1930 book, &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Principles_of_Quantum_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;&lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt;&amp;quot;&lt;/a&gt;, would seem to be the next earliest sighting of &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi}$&lt;/span&gt; found so far, assuming it actually appears like this in the first edition of the 1930-book.  So far, I've just checked the third-edition, which wasn't published until 1947.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/what-is-the-origin-of-the-hbar-symbol#mjx-eqn-Timeline&quot;&gt;&lt;span class=&quot;math-container&quot;&gt;$\BackUp$&lt;/span&gt;&lt;/a&gt;  &lt;span class=&quot;math-container&quot;&gt;$\Target{References} \textbf{References}$&lt;/span&gt;&lt;/h3&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\Target{Ref1}$&lt;/span&gt;&lt;strong&gt;&amp;quot;The Planck constant &lt;em&gt;h&lt;/em&gt; and the Dirac constant &lt;em&gt;ħ&lt;/em&gt;.  Their units and their history&amp;quot;&lt;/strong&gt;,&lt;br/&gt;   by Ian Mills and P. R. Bunker.&lt;br/&gt;   &lt;a href=&quot;https://chemphys.ca/pbunker/PlanckWiki.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[PDF]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\Target{Ref2}$&lt;/span&gt;&lt;strong&gt;&amp;quot;Quantisierung als Eigenwertproblem&amp;quot;&lt;/strong&gt; (&amp;quot;Quantization as eigenvalue problem&amp;quot;),&lt;br/&gt;   by Erwin Schrödinger (1926)&lt;br/&gt;   doi:&lt;a href=&quot;https://doi.org/10.1002/andp.19263851302&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1002/andp.19263851302&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\Target{Ref3}$&lt;/span&gt;&lt;strong&gt;&amp;quot;On the theory of quantum mechanics&amp;quot;&lt;/strong&gt;,&lt;br/&gt;   by Paul Adrien Maurice Dirac (1926-10-01).&lt;br/&gt;   doi:&lt;a href=&quot;https://doi.org/10.1098/rspa.1926.0133&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1098/rspa.1926.0133&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\Target{Ref4}$&lt;/span&gt;&lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Principles_of_Quantum_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Principles of Quantum Mechanics&amp;quot;&lt;/a&gt;&lt;/strong&gt;,&lt;br/&gt;   by Paul Adrien Maurice Dirac (1930)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6913" LastEditorUserId="6913" LastEditDate="2023-10-03T02:10:52.693" LastActivityDate="2023-10-03T02:10:52.693" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="10994" PostTypeId="2" ParentId="10984" CreationDate="2019-09-15T09:00:39.027" Score="7" Body="&lt;p&gt;Gibbs, the father of vector analysis in physics, or his student Edwin Bidwell Wilson, seems to have established the tradition of using the word speed for the scalar, and the word velocity for the vector. It seems to me that using these two different names was meant to be helpful when introducing vectors, rather than to confuse and complicate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gibbs's lecture notes on vector analysis and its use in physics circulated in a small circle since 1881. Wilson was asked to expand on these notes and write a textbook that would be more suitable as a first introduction to the subject. His book &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015000962285&amp;amp;view=1up&amp;amp;seq=149&quot; rel=&quot;noreferrer&quot;&gt;Vector Analysis&lt;/a&gt; (1901), had a huge influence, and it helped to standardize the notation and vocabulary. In the book, Wilson recommended using the words speed and velocity as follows: &lt;em&gt;Velocity is a vector quantity. Its direction is the direction of the tangent of the curve described by the particle. The term speed is used frequently to denote merely the scalar value of the velocity. This convention will be followed here.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That recommendation was repeated by a popular physics textbook that was published a few years later: &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015065523790&amp;amp;view=1up&amp;amp;seq=31&quot; rel=&quot;noreferrer&quot;&gt;A Textbook on Physics&lt;/a&gt; by Duff (1909): &lt;em&gt;For clearness such a phrase as 'twenty miles an hour' may be called the statement of a speed, which means the mere magnitude of a velocity without reference to the direction.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Before Gibbs the distinction between the words speed and velocity was less clear. For example, Maxwell discussed speed, velocity, vectors and scalars in his book &lt;a href=&quot;https://books.google.nl/books?id=KfC1BshMZ7QC&amp;amp;pg=PA19&amp;amp;dq=maxwell+%22The+rate+or+speed+of+the+motion+is+called+the+velocity%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjeoOu58dLkAhWQLVAKHRdVBJQQ6AEIKjAA#v=onepage&amp;amp;q=maxwell%20%22The%20rate%20or%20speed%20of%20the%20motion%20is%20called%20the%20velocity%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Matter and Motion&lt;/a&gt; (1877). He said: &lt;em&gt;The rate or speed of the motion is called the velocity of the particle, and its magnitude is expressed by saying that it is such a distance in such a time, as, for instance, ten miles an hour, or one metre per second.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(copied from a post by me in Physics Forums)&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2019-09-15T13:07:07.573" LastActivityDate="2019-09-15T13:07:07.573" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="10995" PostTypeId="2" ParentId="10992" CreationDate="2019-09-16T04:37:06.507" Score="6" Body="&lt;p&gt;The transition that occurred took place outside of mathematics, although it eventually affected mathematical practice and teaching. Seidenberg documents early construction practices that used pegs and ropes, and the traditional story is that Pythagoras learned geometry from Egyptian rope-stretchers. The story is not very credible, but it testifies to the Egyptian practices at the time, and the fact that they were known in Greece. Not much is known about drawing in construction practices during the classical and Hellenistic period, see &lt;a href=&quot;https://www.academia.edu/29833752/Geometry_in_Greece_The_Use_of_Geometry_by_Ancient_Greek_Architects&quot; rel=&quot;noreferrer&quot;&gt;Geometry in Greece The Use of Geometry by Ancient Greek Architects by Leonardis&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The tools required for the actual drawing were simple. Drawings could be made using charcoal sticks made from grapevines, used widely in the Mediterranean even today as they provide a sharp, hard point.  A pair of dividers could be used as a compass. Whitened boards were the likely surface for drawings that needed to be retained for a while... There is no doubt that ancient Greek architects used drawings in planning their work, though the nature of their drawings has sometimes been called into question (Coulton 1977: 51–73; Senseney 2011: 32–34). The discovery (noted earlier in the chapter) of incised drawings on the inner walls of the Temple of Apollo at Didyma finally confirmed what had been mostly speculative. The diagrams there are just that: diagrams that are representative of a long, confident tradition of both accurate drawing and geometry (Haselberger 1983, 1985, 1999). Vitruvius frequently refers to drawings with Greek titles that he and others could use for solving architectural problems.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What we do know is that classical geometers, from Eulid to Pappus, stayed away from either pegs and ropes or rods and dividers, and we know why. &quot;Workman tools&quot; were considered alien to &quot;pure science&quot; of geometry. Plato in the Republic disdained even the more abstract language of construction that we find in Euclid, who only talks about producing lines and describing circles without mentioning any tools:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;They speak, I suppose, very laughably and perforce, for they mention squaring, applying, and adding, and state all their claims as if they are engaged in action and fashioning all their proofs for the sake of action; but the fact is, I presume, that the whole science is pursued for the sake of knowledge&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This caused an early controversy between his heirs and Menaechmus, a prominent geometer of the time, see &lt;a href=&quot;https://hsm.stackexchange.com/a/6902/55&quot;&gt;When were the concepts of pure and applied Mathematics introduced?&lt;/a&gt; Earliest copies of Euclid's Elements we have do not even have diagrams, and in practice geometers drew in the specially prepared sand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclidean constructions spread into the architectural practice, as we can see already in Vitruvius, and during the middle ages this was continued in the largely oral tradition of stonemasons, see &lt;a href=&quot;http://medieval-architectural-geometry.com&quot; rel=&quot;noreferrer&quot;&gt;Medieval Architectural Geometry&lt;/a&gt;. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Square_and_Compasses&quot; rel=&quot;noreferrer&quot;&gt;square and compasses&lt;/a&gt; symbol of freemasons was taken from the medieval stonemason guilds. The transition that eventually brought them &quot;officially&quot; into mathematics happened during the Renaissance. Architecture separated from masonry, and, together with perspective in painting, finally became worthy of science, the new science: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As one has seen from Schmuttermayer’s and Roriczer’s booklets, (and other similar documents), by the late fifteenth century, in south Germany, the situation had changed. Non-qualified persons, unskilled in the traditions of masonry, were entering the field of the professional architect, and were introducing Classical Antique ideas from Italy. In reaction to this development the older architects were attempting to repress those ideas and maintain traditional skills by producing their instructional booklets on the importance and correct use of geometry in any design context... As the Renaissance gained strength and took control of European culture, bringing the printing industry, grammar schools, new drawing instruments, machinery, literacy, universities, and increased trade and travel, the architectural profession became separated from that of the mason.  Books were published, such as Langley (1767) giving scholarly details of the classical Roman styles of architecture.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Early modern mathematicians did not have Platonic preconceptions about the nature of geometry, and were happy not only to take inspiration and tools from applications, but even to improve on them. &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.87.8516&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;The Sector: its History, Scales, and Uses&lt;/a&gt; gives a nice overview of the importation of &quot;workman tools&quot; into mathematics during the late Renaissance. Tartaglia proposed gunner's compass in 1537, the more familiar looking reduction compass was in use by mathematicians since before 1570, Galileo (and independently Thomas Hood), are known for inventing the more advanced sector compass in 1590s. From that time on the terminology of &quot;straightedge and compass&quot; gets attached to Euclid's dry language in geometry textbooks:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Tartagula&lt;/em&gt; [sic!], &lt;em&gt;in investigating&#xA;  the “new science” of artillery, describes a gunner’s quadrant&#xA;  used to set the elevation of cannon. This device (shown in&#xA;  use in Figure 2), which resembled a carpenter’s square with a&#xA;  quadrant attached, was often inscribed with various tables...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The reduction compass (Figure 3) was a pair of dividers with&#xA;  the addition of a number of sharp points that could be slid up and&#xA;  down the arms to provide a device capable of giving set measurements&#xA;  in proportion to how far the legs of the dividers were&#xA;  opened. As important was the fact that the sharp points entering a drawing at right angles provided repeatable, accurate, results. The&#xA;  reduction compass appears to have been invented sometime before&#xA;  1570 as a precise drafting instrument and it soon became&#xA;  used for finding proportions between figures. The device is said&#xA;  to have fascinated Giordano Bruno because it was more accurate&#xA;  that other devices available at the time.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Figure 3 depicting Denorry's reduction compass from 1588 is reproduced below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hi6ha.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hi6ha.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-09-16T04:37:06.507" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="10996" PostTypeId="2" ParentId="9974" CreationDate="2019-09-16T18:21:39.180" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kurt_Hensel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Hensel&lt;/a&gt; gave a eulogy commemorating the 100th birthday of E. Kummer in 1910 (&lt;a href=&quot;https://link.springer.com/chapter/10.1007%2F978-3-7091-9534-5_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Gedächtnisrede auf Ernst Eduard Kummer&quot;&lt;/a&gt;; &lt;a href=&quot;https://quod.lib.umich.edu/u/umhistmath/ACD4271.0007.001?rgn=main;view=fulltext&quot; rel=&quot;nofollow noreferrer&quot;&gt;free transcribed version&lt;/a&gt;). On page 20 he lists the irreducible elements among the Gaussian integers &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z[i]$&lt;/span&gt; and then writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Dies sind nun aber auch alle Primzahlen im Bereiche dieser komplexen Zahlen, und hier, wie in der Theorie der reellen Zahlen besteht der &lt;strong&gt;Fundamentalsatz&lt;/strong&gt; [emphasis added]: Jede komplexe Zahl kann stets und nur auf eine einzige Weise in ein Produkt von komplexen Primzahlen zerlegt werden.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;But these, now, are all the primes among these complex numbers, and here, as in the theory of real numbers [sc. integers], we have the Fundamental Theorem: Each complex number [sc. Gaussian integer] can always be decomposed in one and only one way into a product of complex prime numbers [sc. irreducible Gaussian integers].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He goes on to explain, following Kummer, that (in modern language) if such number rings are UFD's, one can prove cases of Fermat's Last Theorem with that, but notes how in general such integer rings of numbers fields are not. Page 21:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Will man also auf dieses allgemeine FERMATsche Problem die Methoden und Ergebnisse der Zahlenlehre anwenden, so muß man zunächst fragen, ob auch für diese Zahlen, in &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;, wie ich sie nennen will, die &lt;strong&gt;Grundsätze der Arithmetik&lt;/strong&gt; [emphasis added] gelten, zunächst also, ob sich jede solche Zahl stets als Produkt von nicht weiter zerlegbaren Zahlen in &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; darstellen läßt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So if one wants to apply the methods and results of number theory to this general  FERMAT Problem, one has to ask first whether for these numbers, in &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;, as I want to call them, the foundational theorems of arithmetic remain true, first and foremost, whether each such number can always be written as a product of numbers without further decomposition in &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Further, in his book &quot;Zahentheorie&quot; published in 1913, Hensel indents the theorem as &quot;Fundamentalsatz&quot; again (&lt;a href=&quot;http://www.gutenberg.org/files/38986/38986-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.gutenberg.org/files/38986/38986-pdf.pdf&lt;/a&gt;, bottom of p. 41 (56 of the pdf file)) and calls it (with his own emphasis) &quot;foundational for the entire multiplicative theory of numbers&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;So if we are very precise, he does not actually call it the Fundamental Theorem of Arithmetic, but &quot;Fundamentalsatz&quot; (without &quot;der Arithmetik&quot;), and a few lines later he describes it as the first among the &quot;Grundsätze der Arithmetik&quot;. But otherwise this might be what your source had in mind, as Hensel was German, and this was a few years before Bell's 1915 book.&lt;/p&gt;&#xA;" OwnerUserId="6514" LastEditorUserId="6514" LastEditDate="2020-04-16T04:48:56.463" LastActivityDate="2020-04-16T04:48:56.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="10997" PostTypeId="1" AcceptedAnswerId="11006" CreationDate="2019-09-17T12:15:14.080" Score="3" ViewCount="239" Body="&lt;p&gt;In chapter V of volume 2 of Dickson's &quot;History of the theory of numbers&quot; (p.191-195), which collects results on &quot;rational&quot; triangles (triangles with integer side lengths), apear several results on the so called &quot;Curtius's problem&quot; (also called &quot;&lt;strong&gt;the problem of three shooters&lt;/strong&gt;&quot;). This problem is concerned with establishing a rule for finding rational triangles which also have circumscribed circles with integer radius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On p.195, appears the following remark:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;C.F Gauss, whose attention had been called to Curtius's problem by Schumacher, stated that the sides of every triangle such that each side and the radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; of the circumscribed circle are integers are of the form: &lt;span class=&quot;math-container&quot;&gt;$$4abfg(a^2+b^2), \pm4ab(f+g)(a^2f-b^2g), 4ab(a^2f^2+b^2g^2)$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a,b,f,g$&lt;/span&gt; are positive integers, while &lt;span class=&quot;math-container&quot;&gt;$r = (a^2+b^2)(a^2f^2+b^2g^2)$&lt;/span&gt;... Many writers derived Gauss's formula.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I checked in Gauss's werke, and those formulas are given in a letter to Schumacher from 1847, which appears in volume 12 under the title &quot;solution to arithmetical task&quot;. What i want to know is how to put this isolated result in modern mathematical framework? i didn't find much material in the internet on problems related to rational triangles. In particular i'd like to know how is Gauss's solution to this problem related to number theoretical ideas (to which branch of number theory is it related?)?  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="55" LastEditDate="2019-09-20T09:51:01.430" LastActivityDate="2019-09-20T09:51:01.430" Title="What is the modern context of Gauss's work on triangles with integer sides and circumradius?" Tags="|number-theory|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="10998" PostTypeId="1" AcceptedAnswerId="11004" CreationDate="2019-09-17T12:45:45.410" Score="0" ViewCount="131" Body="&lt;p&gt;We read that &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Babbage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Babbage&lt;/a&gt; was &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_engine&quot; rel=&quot;nofollow noreferrer&quot;&gt;interested in building a machine&lt;/a&gt; to tabulate polynomial functions. This came from &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-babbage-s-tables-of-logarithms&quot; rel=&quot;nofollow noreferrer&quot;&gt;writing a book of log tables&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quora.com/How-do-I-use-log-table&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here we see an example&lt;/a&gt; of how to use a log table:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/obztO.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/obztO.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;Did Charles Babbage lay out his log table (from his book) like this?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-17T13:40:30.657" LastActivityDate="2019-09-18T11:09:14.327" Title="Did Charles Babbage lay out his log table like this?" Tags="|logarithm|charles-babbage|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="10999" PostTypeId="5" CreationDate="2019-09-17T12:46:40.437" Score="0" Body="&lt;p&gt;In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the &quot;logarithm to base 10&quot; of 1000 is 3. &lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-18T05:53:09.957" LastActivityDate="2019-09-18T05:53:09.957" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11000" PostTypeId="4" CreationDate="2019-09-17T12:46:40.437" Score="0" Body="In mathematics, the logarithm is the inverse function to exponentiation." OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-18T05:52:48.353" LastActivityDate="2019-09-18T05:52:48.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11001" PostTypeId="5" CreationDate="2019-09-17T12:47:48.573" Score="0" Body="&lt;p&gt;Considered by some to be a &quot;father of the computer&quot;, Babbage is credited with inventing the first mechanical computer that eventually led to more complex electronic designs, though all the essential ideas of modern computers are to be found in Babbage's analytical engine. His varied work in other fields has led him to be described as &quot;pre-eminent&quot; among the many polymaths of his century.&lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-18T05:52:33.553" LastActivityDate="2019-09-18T05:52:33.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11002" PostTypeId="4" CreationDate="2019-09-17T12:47:48.573" Score="0" Body="Charles Babbage KH FRS ( 26 December 1791 – 18 October 1871) was an English polymath. A mathematician, philosopher, inventor and mechanical engineer, Babbage originated the concept of a digital programmable computer." OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-18T05:52:29.180" LastActivityDate="2019-09-18T05:52:29.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11003" PostTypeId="2" ParentId="10990" CreationDate="2019-09-17T17:15:43.193" Score="7" Body="&lt;p&gt;Dirac was not free to create a new symbol, because publishing would be prohibitively expensive due to printing costs. So the choice was limited to existing symbols. Many printers probably had the IPA-symbols, as it was used in dictionaries. Around 1930, h-bar had been added to IPA. (&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_International_Phonetic_Alphabet#1888_alphabet&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2019-09-29T08:06:13.150" LastActivityDate="2019-09-29T08:06:13.150" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11004" PostTypeId="2" ParentId="10998" CreationDate="2019-09-18T11:09:14.327" Score="2" Body="&lt;p&gt;Yes and No. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a wonderful mathematical simplicity to the way that &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Babbage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Babbage&lt;/a&gt; lays out his log tables. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the first couple of pages he lays it out like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/t6sVD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/t6sVD.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then when he gets to the thousands, he lays it out like this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eQqYR.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eQqYR.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastActivityDate="2019-09-18T11:09:14.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11005" PostTypeId="1" CreationDate="2019-09-18T22:05:35.583" Score="3" ViewCount="96" Body="&lt;p&gt;&lt;em&gt;Update: It can be traced back to Fraenkel-Bar-Hillel's &lt;strong&gt;Foundations of Set Theory&lt;/strong&gt;, originally published in 1958. Further discussions can be seen at the linked question.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The notion of &amp;quot;three crises of mathematics&amp;quot; is extremely popular in China. Open any book, textbook or encyclopedia about the history of mathematics, one will find a version or variant of this standard description.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Throughout the history of Western mathematics, there have been three major crises on its foundation due to new discoveries, and led to major advancements in mathematics. Namely,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The first crisis&lt;/strong&gt; is the discovery of &lt;span class=&quot;math-container&quot;&gt;$\sqrt 2$&lt;/span&gt; (by Hippasus according to the legend) and other incommensurables that challenged the foundation of Pythagoreanism, not resolved until Eudoxus of Cnidus and Euclid offered geometric treatments.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The second crisis&lt;/strong&gt; is the inconsistent uses of infinity and infinitesimal by Newton and Leibniz that raised doubts on the theoretical soundness of calculus, with vocal critics such as George Berkeley, eventually fully resolved by Cauchy who came up with rigorous definitions that are widely accepted, e.g. the epsilon–delta method.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The third crisis&lt;/strong&gt; is the discovery of various paradoxes in Cantor's naive set theory, such as Russell's paradox, it threatened the set-theoretic foundation of mathematical logic. The attempts of solving the problem included Russell's type theory and the Zermelo set theory, and eventually led to the creation of ZF and ZFC set theory.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It was persuasive to me. Until I realized the notion of &amp;quot;three crises of mathematics&amp;quot; does not exist in English at all.&lt;/p&gt;&#xA;&lt;p&gt;In English literature, although almost everyone recognizes the treatment of incommensurables by Greeks or infinitesimal by Newton as major theoretical problems in history, some authors even used the word &amp;quot;crises&amp;quot;, but different authors presented these issues from different perspectives, for example, In &lt;em&gt;The third great crisis in mathematics&lt;/em&gt; by C.K. Gordon, published in 1968 in IEEE Spectrum (doi:10.1109/MSPEC.1968.5214636), focused on later technical problems in set theory, such as the well-ordering theorem and the Banach-Tarski paradox. Another paper by Ernst Snapper, &lt;em&gt;&lt;a href=&quot;http://www.personal.psu.edu/faculty/e/c/ecb5/Courses/M475W/WeeklyReadings/Week%2014/Three_Crises_in_Mathematics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Three Crises in Mathematics: Logicism, Intuitionism and Formalism&lt;/a&gt;&lt;/em&gt;, discusses modern issues as the title indicated, not historical ones. In conclusion, &lt;strong&gt;a single standard version of &amp;quot;three crises of mathematics&amp;quot; doesn't exist.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, a critical read can raise even more questions.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;According to &lt;em&gt;&lt;a href=&quot;https://arxiv.org/abs/1202.4153&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ten Misconceptions from the History of Analysis and Their Debunking&lt;/a&gt;&lt;/em&gt;, Berkeley’s logical criticism was not effective (and his metaphysical criticism was largely ideological), &amp;quot;Fermat planted the seeds of the answer to the logical criticism of the infinitesimal, a century before George Berkeley ever lifted up his pen&amp;quot;, and Cauchy did not replace infinitesimals by modern, meaning epsilon-delta, definition of continuity.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Only the so-called &amp;quot;third crisis&amp;quot;, known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis&quot; rel=&quot;nofollow noreferrer&quot;&gt;foundational crisis of mathematics&lt;/a&gt; in the 20th century, is recognized in history, the &amp;quot;first&amp;quot; and the &amp;quot;second&amp;quot; crises were simply the subjective assessments by finding similar examples in history, and well before the establishment of modern mathematical logic.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The description of these crises is incomplete. The debates in the 20th century were important, but this popular account only mentioned Russell's paradox in naive set theory. Many other issues, such as Gödel's incompleteness theorems, which had much more impact on the foundation of mathematics, were not even mentioned at all. The halting problem was ignored as well.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And arguably, the historical examples are incomplete. Personally, I think that, if finding persuasive examples in history regardless of context is the primary goal, I would definitely add the discovery of non-Euclidean geometry into the list, as it raised doubts on the notion of absolute rationality and authority, which triggered some great debates among philosophers. Also, more recent issues, such as Vladimir Voevodsky's claim of his suspicion that an inconsistency in first-order Peano arithmetic might be discovered in the future, summarized in the essay &lt;em&gt;&lt;a href=&quot;http://timothychow.net/consistent.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Consistency of Arithmetic&lt;/a&gt;&lt;/em&gt;, deserves an inclusion.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h3&gt;Question&lt;/h3&gt;&#xA;&lt;p&gt;My question is, where did the standard notion of &amp;quot;three crises of mathematics&amp;quot; come from?&lt;/p&gt;&#xA;&lt;p&gt;I'm not referring to three independent issues, but specifically asking for the origin of the &amp;quot;three issues as a complete package&amp;quot; version in the beginning of my question. I suspect it must come from somewhere authoritative, I suspect it came from either German literature or Soviet Union literature, or possibly less known English-speaking historians, but I'm not sure.&lt;/p&gt;&#xA;&lt;p&gt;Note: I don't think this question is a duplicate of &lt;a href=&quot;https://hsm.stackexchange.com/questions/9575/what-is-the-status-of-the-three-crises-in-the-history-of-mathematics&quot;&gt;What is the status of the three crises in the history of mathematics?&lt;/a&gt; or &lt;a href=&quot;https://hsm.stackexchange.com/questions/7513/foundational-crises-in-non-western-historical-mathematical-communities&quot;&gt;Foundational crises in non-Western historical mathematical communities&lt;/a&gt;. Although the same concept was mentioned, neither were an &lt;strong&gt;explicit requests of sources or origins&lt;/strong&gt; of this concept.&lt;/p&gt;&#xA;" OwnerUserId="10481" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-09-19T13:31:20.610" Title="Where does the notion of &quot;three crises of mathematics&quot; come from?" Tags="|mathematics|terminology|reference-request|mathematical-logic|" AnswerCount="0" CommentCount="5" ClosedDate="2019-09-19T13:29:17.073" ContentLicense="CC BY-SA 4.0" />
  <row Id="11006" PostTypeId="2" ParentId="10997" CreationDate="2019-09-20T05:04:27.393" Score="3" Body="&lt;p&gt;&quot;Curtius's problem&quot; is Schumacher's personal name repeated only by Dickson. By scaling, one can reduce the problem to that about triangles with rational sides and the circumradius, which is equivalent to the more commonly considered problem about triangles whose sides and the area are rationals, since &lt;span class=&quot;math-container&quot;&gt;$r=abc/4A$&lt;/span&gt;. These, or the version where the sides and the area are integers, are now called &lt;a href=&quot;https://en.wikipedia.org/wiki/Heronian_triangle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heron triangles&lt;/a&gt;. The solution, for the integer version, goes back to Brahmagupta, and one can see that Gauss's formulas are related to Brahmagupta's (who stated them in words, of course):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a=n(m^2+k^2);\ \ b=m(n^2+k^2);\ \ c=(m+n)(mn-k^2).$$&lt;/span&gt;&#xA;These formulas, just like those for Pythagorean triples, relate Heron triangles to Diophantine equations. The long known connection between Diophantine equations and elliptic curves was made particularly prominent in the Wiles's proof of the Last Fermat theorem, and elliptic curves provide the most common perspective on Heron triangles in recent research, see e.g. &lt;a href=&quot;https://people.math.ethz.ch/~halorenz/publications/pdf/Heron.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heron Triangles and their Elliptic Curves by Halbeisen-Hungerbuhler&lt;/a&gt;. They are also quite popular in recreational mathematics, and e.g. &lt;a href=&quot;http://math.fau.edu/yiu/Southern080216.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yiu's survey&lt;/a&gt; relates them to Gaussian integers. But I did not see a connection explicitly made to Curtius's version of the problem, or Gauss's contribution to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More remotely, with the additional condition that the triangle be right angled, this leads to the still unsolved &lt;a href=&quot;https://en.wikipedia.org/wiki/Congruent_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;problem of congruent numbers&lt;/a&gt;. The problem goes back to al-Khazin (c. 940 AD), and a number is congruent when it is the area of such a triangle. A simple criterion for a number to be congruent would follow from a very famous conjecture, also about elliptic curves, the Birch and Swinnerton-Dyer conjecture. It is one of the millennium problems.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-20T06:08:15.523" LastActivityDate="2019-09-20T06:08:15.523" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11007" PostTypeId="1" AcceptedAnswerId="11025" CreationDate="2019-09-20T12:04:21.340" Score="0" ViewCount="509" Body="&lt;p&gt;At the London Science Museum, we see a model built in 1999 of Charles Babbage's Difference Engine No.2 based on the plans he left:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/o0Mhk.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/o0Mhk.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We &lt;a href=&quot;https://books.google.com.au/books?id=piNNAAAAMAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;know that&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Babbage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Babbage&lt;/a&gt; wrote a book of &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_table#Tables_of_logarithms&quot; rel=&quot;nofollow noreferrer&quot;&gt;log tables&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wO96p.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wO96p.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hvdD6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hvdD6.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In doing this, Babbage was inspired to create Log Tables mechanically, leading to his design of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_engine#targetText=A%20difference%20engine%2C%20first%20created,small%20set%20of%20polynomial%20coefficients.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Difference Engine&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Difference Engine produced a 'tabulation of polynomial functions'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can see that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_engine#Operation&quot; rel=&quot;nofollow noreferrer&quot;&gt;operation&lt;/a&gt; of the Difference engine could produce a table that &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_engine#Operation&quot; rel=&quot;nofollow noreferrer&quot;&gt;looked&lt;/a&gt; like this. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|c|}&#xA;\hline&#xA;x&amp;amp; p(x) = 2x^2 − 3x + 2 &amp;amp; diff1(x) = ( p(x + 1) − p(x) ) &amp;amp; diff2(x) = ( diff1(x + 1) − diff1(x) ) \\ \hline&#xA;0 &amp;amp; 2&amp;amp; -1&amp;amp;4\\ \hline&#xA;1 &amp;amp; 1 &amp;amp; 3 &amp;amp;4\\ \hline&#xA;2 &amp;amp; 4 &amp;amp; 7&amp;amp;4\\ \hline&#xA;3 &amp;amp; 11 &amp;amp; 11 &amp;amp;\\ \hline&#xA;4 &amp;amp; 22 &amp;amp; &amp;amp;\\ \hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is fine - but how do you turn this into a log table? I feel like I'm missing a step in the process. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;How does Charles Babbage's Difference Engine solve his quest for automating Log Tables?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-28T11:38:08.600" LastActivityDate="2019-09-28T13:31:15.850" Title="How does Charles Babbage's Difference Engine solve his quest for automating Log Tables?" Tags="|logarithm|charles-babbage|difference-engine|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11008" PostTypeId="5" CreationDate="2019-09-20T12:05:23.487" Score="0" Body="&lt;p&gt;Its name is derived from the method of divided differences, a way to interpolate or tabulate functions by using a small set of polynomial coefficients. Most mathematical functions commonly used by engineers, scientists and navigators, including logarithmic and trigonometric functions, can be approximated by polynomials, so a difference engine can compute many useful tables of numbers.&lt;/p&gt;&#xA;" OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-11-17T15:56:31.693" LastActivityDate="2019-11-17T15:56:31.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11009" PostTypeId="4" CreationDate="2019-09-20T12:05:23.487" Score="0" Body="A difference engine, first created by Charles Babbage, is an automatic mechanical calculator designed to tabulate polynomial functions." OwnerUserId="7307" LastEditorUserId="7307" LastEditDate="2019-09-25T16:36:17.803" LastActivityDate="2019-09-25T16:36:17.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11010" PostTypeId="2" ParentId="11007" CreationDate="2019-09-20T12:53:47.227" Score="0" Body="&lt;p&gt;You tabulate a function for a set of consecutive numbers, then compute a column with the differences of the tabulated values, then the second, third, etc. differences, until you get a column with a constant value. Then, by reversing the process, you can construct a table of that function. If the original tabulated function is a polynomial approximating the log function, you'll construct  a table of logarithms... It's a purely mechanical operation...&lt;/p&gt;&#xA;" OwnerUserId="6222" LastEditorUserId="6222" LastEditDate="2019-09-21T07:26:49.097" LastActivityDate="2019-09-21T07:26:49.097" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11011" PostTypeId="2" ParentId="267" CreationDate="2019-09-20T21:52:34.403" Score="1" Body="&lt;p&gt;The 3-volume book by Thomas Muir &lt;em&gt;The Theory of Determinants in the Historical Order of Development&lt;/em&gt; covers a narrower subject, but its first sections are very interesting to understand the early history of linear algebra. The notion of determinant predates other notions of linear algebra, and the book gives an exhaustive listing of all its early occurrences from Leibniz onwards.&lt;/p&gt;&#xA;" OwnerUserId="2575" LastActivityDate="2019-09-20T21:52:34.403" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11012" PostTypeId="1" AcceptedAnswerId="11013" CreationDate="2019-09-20T22:14:11.267" Score="5" ViewCount="251" Body="&lt;p&gt;The story of Bombelli solving the equation &lt;span class=&quot;math-container&quot;&gt;$x^3=15x+4$&lt;/span&gt; in &lt;em&gt;L'Algebra&lt;/em&gt; and introducing &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; is well known. The equation, having obvious root &lt;span class=&quot;math-container&quot;&gt;$x=4$&lt;/span&gt;, is solved using Cardano's formula:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x=\sqrt[3]{\frac{q}{2}+\sqrt{(\frac{q}{2})^2-(\frac{p}{3})^3}}+\sqrt[3]{\frac{q}{2}-\sqrt{(\frac{q}{2})^2-(\frac{p}{3})^3}}&#xA;$$&lt;/span&gt;&#xA;getting &lt;span class=&quot;math-container&quot;&gt;$x=\sqrt[3]{2+11\sqrt{-1}}+\sqrt[3]{2-11\sqrt{-1}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After that, all the accounts I've read so far merely check that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (2+\sqrt{-1})^3=2+11\sqrt{-1} $$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$(2-\sqrt{-1})^3=2-11\sqrt{-1}$$&lt;/span&gt;&#xA;to conclude that the formula gives the correct answer &lt;span class=&quot;math-container&quot;&gt;$x=(2+\sqrt{-1})+(2-\sqrt{-1})=4$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question is: how did Bombelli transform &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{2\pm11\sqrt{-1}}$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$2\pm\sqrt{-1}$&lt;/span&gt;? Is it detailed in the book? Did he know a procedure to simplify &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{m+n\sqrt{k}}$&lt;/span&gt; in the form &lt;span class=&quot;math-container&quot;&gt;$a+b\sqrt{k}$&lt;/span&gt; for positive &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;? Was this procedure exposed in earlier chapters of &lt;em&gt;L'Algebra&lt;/em&gt;, or in other books of the time, in relation to Cardano's formula?&lt;/p&gt;&#xA;" OwnerUserId="2575" LastEditorUserId="2575" LastEditDate="2019-09-21T03:59:49.063" LastActivityDate="2019-09-21T09:00:34.283" Title="How did Bombelli transform $\sqrt[3]{2\pm11\sqrt{-1}}$ into $2\pm\sqrt{-1}$?" Tags="|algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11013" PostTypeId="2" ParentId="11012" CreationDate="2019-09-21T08:51:30.583" Score="8" Body="&lt;p&gt;Bombelli did not have a general method for such transformation, nor does it exist. Doing so is equivalent to solving the &lt;a href=&quot;https://en.wikipedia.org/wiki/Casus_irreducibilis&quot; rel=&quot;noreferrer&quot;&gt;irreducible case of the cubic&lt;/a&gt;, which is impossible in (real) radicals. It is solvable in inverse trigonometric functions, as Vieta showed later, but simplifying his trigonometric expression is itself a chore.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when everything is an integer one can find the solutions by educated guessing. In this case, according to &lt;a href=&quot;https://www.academia.edu/35397288/Rd_Edition_Victor_J._Katz_A_History_of_Mathema&quot; rel=&quot;noreferrer&quot;&gt;Katz's History of Mathematics, 12.3.2&lt;/a&gt;, Bombelli used indeterminates &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; to set up (in modern notation)&#xA;&lt;span class=&quot;math-container&quot;&gt;$(a+\sqrt{-b})^3=2+11\sqrt{-1}$&lt;/span&gt;. Given his rules for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;, this quickly leads to &lt;span class=&quot;math-container&quot;&gt;$(a-\sqrt{-b})^3=2-11\sqrt{-1}$&lt;/span&gt;, and then to &#xA;&lt;span class=&quot;math-container&quot;&gt;$$a^2+b=5;\ \ a(a^2-3b)=2.$$&lt;/span&gt;&#xA;By simple divisibility considerations, there are only four choices for &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and Bombelli shows that &lt;span class=&quot;math-container&quot;&gt;$a=2$&lt;/span&gt; is the only viable one, with &lt;span class=&quot;math-container&quot;&gt;$b=1$&lt;/span&gt;. Note that attempting to solve this &quot;honestly&quot; leads to another cubic &lt;span class=&quot;math-container&quot;&gt;$4a^3=15a+2$&lt;/span&gt;, no better than the original one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/article/10.1007/BF03025306&quot; rel=&quot;noreferrer&quot;&gt;LaNave-Mazur's Reading Bombelli&lt;/a&gt; gives more insight into the way Bombelli handled cubics, and complex numbers that came up in them. As they note, Bombelli's &quot;procedure&quot; was generally not very tidy:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;if you open&#xA;  Bombelli's treatise you discover nothing resembling complex&#xA;  numbers until page 133, at which point certain mathematical&#xA;  objects (that might be regarded by a modern as&#xA;  &quot;complex numbers&quot;) burst onto the scene, in hill battle array,&#xA;  in the middle of an on-going discussion. Here is how&#xA;  Bombelli introduces these mathematical objects. He writes,&#xA;  &quot;I have found another sort of cubic radical which behaves&#xA;  in a very different way from the others&quot;... Complex numbers, when they occur in Gerolamo Cardano's earlier treatise Ars Magna, occur neatly as quantities&#xA;  like &lt;span class=&quot;math-container&quot;&gt;$2 + \sqrt{-15}$&lt;/span&gt;. But they appear initially in Bombelli's&#xA;  treatise as cubic radicals of the type of quantities discussed&#xA;  by Cardano; a somewhat complicated way for them to arise&#xA;  in a treatise that is thought of as an organized exposition&#xA;  of the formal properties of complex numbers!&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-21T09:00:34.283" LastActivityDate="2019-09-21T09:00:34.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11015" PostTypeId="1" AcceptedAnswerId="11017" CreationDate="2019-09-22T14:51:19.900" Score="4" ViewCount="431" Body="&lt;p&gt;Some appear to argue that much of the Elements by Euclid is a compilation of knowledge handed down to Euclid from his predecessors. On the other hand, some credit the proof, of the Pythagorean theorem in particular, that appears in the Elements to Euclid himself. How do we know that the proof is due to Euclid and not to a predecessor? More generally, is there anything in the Elements that we know originates with Euclid? &lt;/p&gt;&#xA;" OwnerUserId="10102" LastEditorUserId="4251" LastEditDate="2020-07-21T04:22:25.840" LastActivityDate="2020-07-21T04:22:25.840" Title="Are Euclid's theorems and proofs due to Euclid?" Tags="|mathematics|ancient-greece|euclid|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11016" PostTypeId="2" ParentId="2611" CreationDate="2019-09-22T23:25:18.453" Score="4" Body="&lt;p&gt;The book &lt;em&gt;Lebesgue's theory of integration: Its origins and development&lt;/em&gt; by T. Hawkins.&lt;/p&gt;&#xA;" OwnerUserId="10493" LastEditorUserId="5989" LastEditDate="2019-09-23T07:59:51.630" LastActivityDate="2019-09-23T07:59:51.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11017" PostTypeId="2" ParentId="11015" CreationDate="2019-09-23T00:17:46.123" Score="6" Body="&lt;p&gt;We do not know for sure, but most of them probably not, the proof of the Pythagorean theorem included. The current consensus on the priority of the results in Euclid's Elements is summarized by Lambek in &lt;a href=&quot;https://books.google.com/books/about/The_Heritage_of_Thales.html?id=mZfXHRgJpmQC&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Heritage of Thales&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;None of the theorems contained in the 13 books can with certainty be&#xA;  ascribed to Euclid himself. It is believed that the Pythagoreans, including&#xA;  Archytas, were responsible for much of what appears in Books I, II, VI,&#xA;  VII, VIII, IX and XI and that Hippocrates was behind Books III and IV.&#xA;  For Books V and XII we are to thank Eudoxus, and Books X and XIII are&#xA;  said to be based on the work of Theaetetus. However, the logical organization of the Elements is undoubtedly Euclid's contribution.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We know from Proclus (who quotes Eudemus's historical survey) that a number of authors wrote Elements before Euclid. Unfortunately, he is not very informative as to their content. Hippocrates of Chios was the first, then Leon &quot;&lt;em&gt;was able to compile a book of elements more carefully designed to take account of the number of propositions and of their utility&lt;/em&gt;&quot;, then Eudoxus of Cnidus &quot;&lt;em&gt;was the first to increase the number of the so-called general theorems&lt;/em&gt;&quot;, then Theudius of Magnesia &quot;&lt;em&gt;produced an admirable arrangement of the elements and made many&#xA;partial theorems more general&lt;/em&gt;&quot;, then Hermotimus of Colophon &quot;&lt;em&gt;discovered&#xA;many propositions in the elements&lt;/em&gt;&quot;, and so on. Finally, &quot;&lt;em&gt;came Euclid, who brought together the elements&lt;/em&gt;&quot;. This confirms the attribution of logical organization to Euclid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speculations about Euclid's priority, or lack thereof, have a venerable history, but there was always little firm basis in evidence for them. For example, it is visible in the text that the author tries hard to avoid using congruence, which leads to proliferation of auxiliary triangles in the proofs. The &lt;a href=&quot;https://www.cut-the-knot.org/ctk/BridesChair.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;bride's chair proof&lt;/a&gt; of the Pythagorean theorem I.47 is a particularly vivid illustration. Presumably, this has something to do with Parmenides's and Plato's qualms about the use of motion in &quot;true science&quot;. But that does not mean that the effort is due to Euclid, several prior Elements' authors were Plato's students and/or associates, including Leon, Eudoxus and Theudius. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid also avoids using the parallel postulate as far as possible in book 1. This is probably our best bet on ascribing a group of results to Euclid himself, but how scant the basis for it is can be seen from &lt;a href=&quot;https://archive.org/details/thirteenbookseu03heibgoog/page/n8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heath's commentary&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Although Aristotle gives a clear idea of what he understood by a postulate,&#xA;  he does not give any instances from geometry; still less has he any allusion&#xA;  recalling the particular postulates found in Euclid. We naturally infer that&#xA;  the formulation of these postulates was Euclid's own work. There is a more&#xA;  positive indication of the originality of Postulate 5, since in the passage (Anal.pnor. II: 16, 65 a 4) quoted above in the note on the definition of parallels he alludes to some petitio principii involved in the theory of parallels current in his time.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Naturally&quot;. The aforementioned Theudius and Hermotimus were Aristotle's contemporaries. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Modern scholars in the subsequent decades studied at length the text and the deductive structure of the Elements. Aside from the above, in books 1-4 Euclid avoids using proportions. Presumably, because Eudoxian treatment of them is very non-elementary, and elementary Pythagorean treatment relied on commensurability and was logically flawed. This block is termed &quot;proportion free Elements&quot;, and it  likely predates Euclid. Pythagorean theorem and application of areas from book 2 are instrumental in this avoidance, just as auxiliary triangles are in the avoidance of congruence. But identifying individual contributors and their contributions remains highly circumstantial and speculative. Artmann in &lt;a href=&quot;https://www.degruyter.com/view/j/apeiron.1991.24.4/apeiron.1991.24.4.1/apeiron.1991.24.4.1.xml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's Elements and its Prehistory&lt;/a&gt; speculates that it is due to Theudius, but admits that his arguments are &quot;not conclusive&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Because Euclid defines proportions and studies their properties for the&#xA;  first time in Book V, he cannot use proportions in the first four books.&#xA;  However, these books include some theorems with curiously involved&#xA;  proofs, which could be proved much more simply using proportion.&#xA;  Closer inspection reveals a standard way of transforming statements&#xA;  about proportions into statements about areas using VI.16... A clever combination of II.5 or 6 with the theorem of Pythagoras gives the desired result without invoking proportionality.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In Artmann (1985) I present a detailed study of the Books I- IV, based&#xA;  on the investigations of Neuenschwander (1972) and Mueller (1981).&#xA;  Logical and stylistic observations confirm the conclusion that, essentially,&#xA;  Euclid's Books I - IV are the 'Proportion Free Elements' mentioned&#xA;  by Proclus. However, these Elements were probably altered in at&#xA;  least two ways before being incorporated in Euclid's work. The first is&#xA;  the addition of the theory of parallels and the consequent modifications&#xA;  of the beginning of Book I. The second is the treatment of parallelograms,&#xA;  also in Book I. In Artmann (1985) I tentatively identified&#xA;  Theudius as the author of the 'Proportion Free Elements', but the&#xA;  arguments for this identification are not conclusive. Whoever the&#xA;  author of the 'Proportion Free Elements' was, his aim was to go as far&#xA;  as possible in geometry without proportions. There is no reason to&#xA;  think that he added new theorems; he merely gave new proofs for&#xA;  known results. That this was no mean achievement can be seen, for&#xA;  instance, in 111.35 or in the construction of the regular pentagon in IV.10&#xA;  and II.27.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF00348585&quot; rel=&quot;nofollow noreferrer&quot;&gt;Artmann's Über voreuklidische „Elemente“, deren Autor Proportionen vermied (1985)&lt;/a&gt; and &lt;a href=&quot;https://www.jstor.org/stable/41133382&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neuenschwander's Die ersten vier Bücher der Elemente Euklids (1972)&lt;/a&gt; are not translated from German. &lt;a href=&quot;https://www.maa.org/press/maa-reviews/philosophy-of-mathematics-and-deductive-structure-in-euclids-elements&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mueller's Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1981)&lt;/a&gt; is also a classic.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-25T12:02:15.860" LastActivityDate="2019-09-25T12:02:15.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11018" PostTypeId="1" CreationDate="2019-09-23T14:22:46.050" Score="0" ViewCount="83" Body="&lt;p&gt;In chapter 39.1 of Gravitation, by Misner, Thorne and Wheeler, it is asserted, quite reasonably, that a theory of gravitation (and, I assume, quite generally) must satisfy the three following criteria : &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Self-consistency (no contradictions)&lt;/li&gt;&#xA;&lt;li&gt;Agreement with past experiments&lt;/li&gt;&#xA;&lt;li&gt;Completeness (every experiment within its domain of applicability should be predictible)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This seems like some rather common criteria, but no source is given for these. Is there an origin to that list of criteria?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2019-09-23T14:22:46.050" Title="Theory criteria from Misner-Thorne-Wheeler" Tags="|physics|philosophy-of-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11023" PostTypeId="1" AcceptedAnswerId="11024" CreationDate="2019-09-27T17:35:42.320" Score="3" ViewCount="260" Body="&lt;p&gt;I've read on &lt;a href=&quot;https://www.keionline.org/charity-nih-funding-related-to-zolgensma&quot; rel=&quot;nofollow noreferrer&quot;&gt;KEI website&lt;/a&gt; that in the early 2000s Dr Martine Barkats, from France, successfuly treated mice with spinal muscular atrophy (SMA) with an AAV9 viral vector; and that Dr Brian Kaspar, an AveXis executive, took it even further by treating pigs and later humans. AveXis licensed the cure for SMA under the name Zolgensma.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What did Dr Kaspar change about Dr Barkats' cure and of what significance is this change compared to Dr Barkats' work?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;NOTE: Please keep the expressions and explanations as simple as possible. I am only a high school student so my biology and chemistry knowledge is limited. &lt;/p&gt;&#xA;" OwnerUserId="10508" OwnerDisplayName="Hinko Pih Pih" LastEditorUserId="55" LastEditDate="2019-09-28T09:06:38.993" LastActivityDate="2019-09-28T19:34:12.903" Title="What did Kaspar contribute to the gene therapy for spinal muscular atrophy?" Tags="|biology|medicine|genetics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11024" PostTypeId="2" ParentId="11023" CreationDate="2019-09-28T09:04:37.253" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://books.google.com/books?id=6_1PCwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spinal Muscular Atrophy volume&lt;/a&gt; gives some historical information on the disease and its treatment in the opening sections. Their timeline and attribution of credit is rather different from that of KEI, especially where Kaspar is concerned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gene therapy for SMA was made possible by identifying the responsible gene in 1990 by Gilliam’s group in New York and soon after by Melki’s group in Paris. The key problem was delivery to the motor neuron. The idea that eventually became successful was suggested by Kevin Foust at the Cure SMA research meeting in Washington, DC in 2003. He was then a graduate student in Flotte’s laboratory, and he suggested testing AAV (adeno-associated virus) gene therapy using AAV2 and AAV5 genes. Later he moved to Kaspar’s laboratory at Nationwide Children’s Hospital. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2008, Barkats, working in Paris, and Foust at NCH discovered independently that self-complementary AAV9 gene could cross the blood–brain barrier and transduce motor neurons when injected in the bloodstream. Barkats presented the research first at the 3rd Myology Meeting. But it was Foust (working with Kaspar's group) that eventually developed the treatment for human patients. The crucial difference, however, was already present in the mice experiments, and had to do with different tagging of survival motor neurons (SMN):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;One crucial difference between the constructs&#xA;  is that the Barkats group in their early constructs used&#xA;  a tag at the 3′ end of SMN. Unfortunately, this rendered&#xA;  SMN nonfunctional. In Columbus, the experiments on&#xA;  Δ7SMA mice continued, and the collaboration among&#xA;  the Burghes, Kaspar, and Mark Rich (Wright State in&#xA;  Dayton) laboratories was born. Indeed, a remarkable&#xA;  effect was observed on Δ7 mice with a major increase&#xA;  of survival from 14 days to over a year. The function of&#xA;  the neuromuscular junction as well as motor unit number&#xA;  estimation (MUNE) and compound muscle action&#xA;  potential (CMAP) measures, which are used in patients,&#xA;  were restored.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In both original publications (below), Foust's name appears first, and Kaspar's last. According to the conventions in the field, this usually means that Foust is credited as the main discoverer, and Kaspar as the head of the lab where the discovery was made. Kaspar, now the chief scientific officer at AveXis, also managed multiple grants from National Institute of Health, National Institute of Neurological Disorders and Stroke, and various charities that funded the pre-clinical research. But he did not &quot;discover that the AAV9 vector was capable of crossing the blood brain barrier&quot;, as KEI says. The patents for Zolgensma are owned by the University of Pennsylvania, Nationwide Children’s Hospital (NCH) and Genethon, a French charity. AveXis entered licensing agreement with Genethon in 2018, and FDA approved the treatment in May 2019. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Foust KD, Nurre E, Montgomery CL, Hernandez A, Chan CM, Kaspar BK. &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/19098898&quot; rel=&quot;nofollow noreferrer&quot;&gt;Intravascular AAV9 preferentially targets neonatal neurons and adult astrocytes&lt;/a&gt;, Nat Biotechnol 27(1):59-65, 2009.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Foust KD, Wang X, McGovern VL, Braun L, Bevan AK, Haidet AM, Le TT, Morales PR, Rich MM, Burghes AH, Kaspar BK. &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/20190738&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rescue of the spinal muscular atrophy phenotype in a mouse model by early postnatal delivery of SMN&lt;/a&gt;, Nat Biotechnol 28(3):271-274, 2010.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-28T19:34:12.903" LastActivityDate="2019-09-28T19:34:12.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11025" PostTypeId="2" ParentId="11007" CreationDate="2019-09-28T13:31:15.850" Score="1" Body="&lt;p&gt;Charles Babbage took took much longer than he thought to design his Difference Engine, only a small part got built in his lifetime. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wis5o.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wis5o.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;This is from the Science Museum in London. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are a number of parts to this answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do you calculate &lt;span class=&quot;math-container&quot;&gt;$\log_{10}60$&lt;/span&gt; without a calculator?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0P1sH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0P1sH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can see the row for &lt;span class=&quot;math-container&quot;&gt;$\log_{10}60$&lt;/span&gt; in the book above. First we will cover off some assumed knowledge in Babbage's book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The logarithm is the inverse function to exponentiation. &#xA;This means:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_{10}1$&lt;/span&gt; = 0&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_{10}10$&lt;/span&gt; = 1&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_{10}100$&lt;/span&gt; = 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So &lt;span class=&quot;math-container&quot;&gt;$\log_{10}60$&lt;/span&gt; must be between 1 and 2. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now as @xxavier has suggested - we can do logarithms using a &lt;a href=&quot;https://en.wikipedia.org/wiki/Taylor_series#Natural_logarithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Taylor series&lt;/a&gt; - ie&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;but this approximation only works for logs between &lt;span class=&quot;math-container&quot;&gt;$\log_{10}n$&lt;/span&gt; where 0 &amp;lt; n &amp;lt; 1. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also note that this is natural log (&lt;span class=&quot;math-container&quot;&gt;$\log_{e}n$&lt;/span&gt;) not base 10 log, so we'll have to convert back later. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So how do we get to &lt;span class=&quot;math-container&quot;&gt;$\log_{10}60$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We look at another Logarithmic identity:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_{b}(xy) = log_{b}x + log_{b}y$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How does that help?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can take what we did before - and simplify things:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_{10}100$&lt;/span&gt; = 2&lt;/p&gt;&#xA;&#xA;&lt;p&gt;so&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\log_{10}60 = log_{10}100 + log_{10}0.6 = 2 + log_{10}0.6 $&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we can go back to our Taylor series approximation:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log_{e}x = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3  + \frac{1}{4}(x-1)^4  + \frac{1}{5}(x-1)^5  + \frac{1}{6}(x-1)^6 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log_{e}0.6 \approx -0.510464 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which is pretty close to a value provided by a calculator (-0.510826).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we convert back to &lt;span class=&quot;math-container&quot;&gt;$$ \log_{10}n$$&lt;/span&gt; using another logarithmic identity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log_{b}x = \frac{log_{d}x }{ log_{d}b} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This means we can do:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log_{10}x = \frac{log_{e}x }{ log_{e}10} $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to convert back to &lt;span class=&quot;math-container&quot;&gt;$$ \log_{10}x $$&lt;/span&gt; we can divide by &lt;span class=&quot;math-container&quot;&gt;$$ \log_{e}10 $$&lt;/span&gt; which is approx. 2.302. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So now&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log_{10}0.6 \approx -0.222 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log_{10}60 \approx -0.222 + log_{10}100 \approx -0.222 + 2 \approx 1.778 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which looks close to what Babbage got:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0P1sH.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0P1sH.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we can confidently calculate a logarithm by hand. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do you calculate a logarithm the &lt;a href=&quot;https://en.wikipedia.org/wiki/Finite_difference_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;Finite Differences Method&lt;/a&gt;?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Babbage &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaspard_de_Prony#Influence_on_Charles_Babbage&quot; rel=&quot;nofollow noreferrer&quot;&gt;used&lt;/a&gt; the approach of &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaspard_de_Prony&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaspard de Prony&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We'll use our Binomial expansion above to the sixth order:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We'll do this for a sixth order polynomial:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3  + \frac{1}{4}(x-1)^4  + \frac{1}{5}(x-1)^5  + \frac{1}{6}(x-1)^6 $$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As an excel function - this looks like this:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;=(C161-1)-(1/2)*POWER((C161-1),2)+(1/3)*POWER((C161-1),3)-(1/4)*POWER((C161-1),4)+(1/5)*POWER((C161-1),5)-(1/6)*POWER((C161-1),6)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Now we build a table for this polynomial of sum of the differences between 0 and 1:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|c|c|c|c|}&#xA;\hline&#xA;index &amp;amp; binomial calculation &amp;amp; 1st  diff &amp;amp; 2nd diff &amp;amp; 3rd diff &amp;amp; 4th diff &amp;amp; 5th diff &amp;amp; 6th diff \\ \hline&#xA;1&amp;amp;0.00000&amp;amp;&amp;amp;&amp;amp;&amp;amp;&amp;amp;&amp;amp;\\ \hline&#xA;0.9&amp;amp;-0.10536&amp;amp;-0.10536&amp;amp;&amp;amp;&amp;amp;&amp;amp;&amp;amp;\\ \hline&#xA;0.8&amp;amp;-0.22314&amp;amp;-0.11778&amp;amp;-0.01242&amp;amp;&amp;amp;&amp;amp;&amp;amp;\\ \hline&#xA;0.7&amp;amp;-0.35663&amp;amp;-0.13349&amp;amp;-0.01571&amp;amp;-0.00329&amp;amp;&amp;amp;&amp;amp;\\ \hline&#xA;0.6&amp;amp;-0.51046&amp;amp;-0.15383&amp;amp;-0.02034&amp;amp;-0.00463&amp;amp;-0.00134&amp;amp;&amp;amp;\\ \hline&#xA;0.5&amp;amp;-0.69115&amp;amp;-0.18068&amp;amp;-0.02685&amp;amp;-0.00651&amp;amp;-0.00188&amp;amp;-0.00054&amp;amp;\\ \hline&#xA;0.4&amp;amp;-0.90773&amp;amp;-0.21658&amp;amp;-0.03590&amp;amp;-0.00905&amp;amp;-0.00254&amp;amp;-0.00066&amp;amp;-0.00012\\ \hline&#xA;0.3&amp;amp;-1.17258&amp;amp;-0.26485&amp;amp;-0.04827&amp;amp;-0.01237&amp;amp;-0.00332&amp;amp;-0.00078&amp;amp;-0.00012\\ \hline&#xA;0.2&amp;amp;-1.50229&amp;amp;-0.32971&amp;amp;-0.06486&amp;amp;-0.01659&amp;amp;-0.00422&amp;amp;-0.00090&amp;amp;-0.00012\\ \hline&#xA;0.1&amp;amp;-1.91870&amp;amp;-0.41640&amp;amp;-0.08669&amp;amp;-0.02183&amp;amp;-0.00524&amp;amp;-0.00102&amp;amp;-0.00012\\ \hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now we shuffle top of columns to top row&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|c|c|c|c|}&#xA;\hline&#xA;index &amp;amp; binomial calculation &amp;amp; 1st  diff &amp;amp; 2nd diff &amp;amp; 3rd diff &amp;amp; 4th diff &amp;amp; 5th diff &amp;amp; 6th diff \\ \hline&#xA;1&amp;amp;0.00000&amp;amp;-0.10536&amp;amp;-0.01242&amp;amp;-0.00329&amp;amp;-0.00134&amp;amp;-0.00054&amp;amp;-0.00012\\ \hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we fill in first row with precomputed values, then populate each cell  by adding the cell above and the cell above right:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|c|c|c|c|}&#xA;\hline&#xA;index &amp;amp; binomial calculation &amp;amp; 1st  diff &amp;amp; 2nd diff &amp;amp; 3rd diff &amp;amp; 4th diff &amp;amp; 5th diff &amp;amp; 6th diff \\ \hline&#xA;1&amp;amp;0.00000&amp;amp;-0.10536&amp;amp;-0.01242&amp;amp;-0.00329&amp;amp;-0.00134&amp;amp;-0.00054&amp;amp;-0.00012\\ \hline&#xA;0.9&amp;amp;-0.10536&amp;amp;-0.11778&amp;amp;-0.01571&amp;amp;-0.00463&amp;amp;-0.00188&amp;amp;-0.00066&amp;amp;-0.00012\\ \hline&#xA;0.8&amp;amp;-0.22314&amp;amp;-0.13349&amp;amp;-0.02034&amp;amp;-0.00651&amp;amp;-0.00254&amp;amp;-0.00078&amp;amp;-0.00012\\ \hline&#xA;0.7&amp;amp;-0.35663&amp;amp;-0.15383&amp;amp;-0.02685&amp;amp;-0.00905&amp;amp;-0.00332&amp;amp;-0.00090&amp;amp;-0.00012\\ \hline&#xA;0.6&amp;amp;-0.51046&amp;amp;-0.18068&amp;amp;-0.03590&amp;amp;-0.01237&amp;amp;-0.00422&amp;amp;-0.00102&amp;amp;-0.00012\\ \hline&#xA;0.5&amp;amp;-0.69115&amp;amp;-0.21658&amp;amp;-0.04827&amp;amp;-0.01659&amp;amp;-0.00524&amp;amp;-0.00114&amp;amp;-0.00012\\ \hline&#xA;0.4&amp;amp;-0.90773&amp;amp;-0.26485&amp;amp;-0.06486&amp;amp;-0.02183&amp;amp;-0.00638&amp;amp;-0.00126&amp;amp;-0.00012\\ \hline&#xA;0.3&amp;amp;-1.17258&amp;amp;-0.32971&amp;amp;-0.08669&amp;amp;-0.02821&amp;amp;-0.00764&amp;amp;-0.00138&amp;amp;-0.00012\\ \hline&#xA;0.2&amp;amp;-1.50229&amp;amp;-0.41640&amp;amp;-0.11490&amp;amp;-0.03585&amp;amp;-0.00902&amp;amp;-0.00150&amp;amp;-0.00012\\ \hline&#xA;0.1&amp;amp;-1.91870&amp;amp;-0.53130&amp;amp;-0.15075&amp;amp;-0.04487&amp;amp;-0.01052&amp;amp;-0.00162&amp;amp;-0.00012\\ \hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So again we get &lt;span class=&quot;math-container&quot;&gt;$$ \log_{e}0.6 \approx -0.51046 $$&lt;/span&gt; which we converted back to base 10 above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see a detailed video of the finite differences method &lt;a href=&quot;https://www.youtube.com/watch?v=PFMBU17eo_4&amp;amp;feature=youtu.be&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How do we use Finite Differences in relation to the Difference Engine?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When initialising the engine - you can set values:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/twHMM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/twHMM.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The values you are setting are the top row of the Finite Differences method from before. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|c|c|c|c|}&#xA;\hline&#xA;index &amp;amp; binomial calculation &amp;amp; 1st  diff &amp;amp; 2nd diff &amp;amp; 3rd diff &amp;amp; 4th diff &amp;amp; 5th diff &amp;amp; 6th diff \\ \hline&#xA;1&amp;amp;0.00000&amp;amp;-0.10536&amp;amp;-0.01242&amp;amp;-0.00329&amp;amp;-0.00134&amp;amp;-0.00054&amp;amp;-0.00012\\ \hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Babbages Difference Engine No.2 &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_engine#Charles_Babbage&amp;#39;s_difference_engines&quot; rel=&quot;nofollow noreferrer&quot;&gt;was designed for 7th order polynomials&lt;/a&gt; (our calcualation was sixth order). This means our Finite Differences Method would have had 7 columns. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the Difference Engine each vertical axle represents a number with 31 digits - with a gear corresponding to each digit, the most significant digit at the top:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MNruL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MNruL.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the Difference Engine &lt;a href=&quot;https://en.wikipedia.org/wiki/Difference_engine#Subtraction&quot; rel=&quot;nofollow noreferrer&quot;&gt;represents negative numbers&lt;/a&gt; using &lt;a href=&quot;https://en.wikipedia.org/wiki/Method_of_complements&quot; rel=&quot;nofollow noreferrer&quot;&gt;tens complement&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So to enter our first value -0.10536 - we have to convert it to tens complement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;10's complement of a decimal number can be found by adding 1 to the 9's complement of that decimal number. It is just like 2s compliment in binary number representation. For example, let us take a decimal number 10536, 9's complement of this number will be 99999-10536 which will be 89463. Now 10s compliment will be 89463+1=89464.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I'm assuming - similar to digital logic - there is a way to flag a 10s complement number - but that wasn't available to me at the time of writing.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So assuming we are entering the value 0.89464 - this means we'll need to turn the 31 wheels to look like:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{|c|c|c|c|c|c|c|}&#xA;\hline&#xA;gear index &amp;amp; value  \\ \hline&#xA;31&amp;amp;0\\ \hline&#xA;30&amp;amp;0\\ \hline&#xA;...&amp;amp;...\\ \hline&#xA;5&amp;amp;0\\ \hline&#xA;4&amp;amp;8\\ \hline&#xA;3&amp;amp;9\\ \hline&#xA;2&amp;amp;4\\ \hline&#xA;1&amp;amp;6\\ \hline&#xA;0&amp;amp;4\\ \hline&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(We have entered a decimal fraction as an integer representing the numbers scaled up - and assume that it will be scaled back down later). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then we repeat this activity for the other 6 columns of the table - against the next 6 vertical axles of gear-values. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gmCZD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gmCZD.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How does the Difference Engine Add Numbers?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In our Finite Differences method - we had to take the cell above, and the cell above right. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here &lt;a href=&quot;https://www.youtube.com/watch?v=vdra5Ms__9s&quot; rel=&quot;nofollow noreferrer&quot;&gt;we have&lt;/a&gt; two wheels representing values in the calculation, and the gear in the middle that adds.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cAatZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cAatZ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;How does the Difference Engine carry values to the next significant digit during addition?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To carry numbers up - &lt;a href=&quot;https://www.youtube.com/watch?v=vdra5Ms__9s&quot; rel=&quot;nofollow noreferrer&quot;&gt;there was&lt;/a&gt; an external device: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eSPVv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eSPVv.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was stacked in an vertical array for each value gear on the axle:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NCrOo.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NCrOo.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can see the addition gear and the carry mechanism here:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wM5be.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wM5be.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7307" LastActivityDate="2019-09-28T13:31:15.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11026" PostTypeId="1" AcceptedAnswerId="11027" CreationDate="2019-09-28T15:55:48.820" Score="6" ViewCount="760" Body="&lt;p&gt;In math stack exchange  I often see notations like &lt;span class=&quot;math-container&quot;&gt;$\{x\in\mathbb Q:x^2&amp;lt;2\}$&lt;/span&gt; being called instances of  &lt;em&gt;set builder notation&lt;/em&gt;.  When I went to school we (that is, I, my fellow students, my teachers, and authors of textbooks such as P. Halmos, N. Bourbaki, &lt;em&gt;et al.&lt;/em&gt;) used such notations all the time but never with that description.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, when was this name introduced?  Are the references given in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Set-builder_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; the original introducers, or just modern users of a recent term for an older idea?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added, after comments and after Conifold's superb answer goaded me to look a tiny bit harder:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A 1948 University of Chicago Press publication &lt;em&gt;Fundamental Mathematics Volume 1 Prepared for the General Course 1 in the College&lt;/em&gt; (&quot;by the College mathematics staff&quot; with a list of 13 names)  uses the term &quot;SET-BUILDER&quot; on p.25:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is a property, we use the expression &quot;&lt;span class=&quot;math-container&quot;&gt;$S_x[W(x)]$&lt;/span&gt;&quot;&#xA;  to stand for &quot;the set of all objects &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; which have the property &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt;&quot;&#xA;  or &quot;the set of all objects &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; for which '&lt;span class=&quot;math-container&quot;&gt;$W(x)$&lt;/span&gt;' is true'' or &quot;the&#xA;  set of all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$W(x)$&lt;/span&gt;&quot;. The sign &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is known as the SET-BUILDER.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is photolithographed from a typescript original.  What I render as &lt;span class=&quot;math-container&quot;&gt;$S_x$&lt;/span&gt; has the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; directly under the &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar Chicago 1954 book, &lt;em&gt;Concepts and structure of mathematics, by the College mathematics staff&lt;/em&gt;, uses the term &quot;set-builder notation&quot;, but Google Books doesn't give enough context to show that it is exactly the same as the 2019 sense of the term.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="4251" LastEditDate="2020-07-17T23:39:39.963" LastActivityDate="2020-07-17T23:39:39.963" Title="Where did the term &quot;set-builder notation&quot; come from?" Tags="|mathematics|terminology|notation|social-context|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11027" PostTypeId="2" ParentId="11026" CreationDate="2019-09-28T23:55:13.227" Score="10" Body="&lt;p&gt;The nickname appears to be a creation of the &lt;a href=&quot;https://en.wikipedia.org/wiki/New_Math&quot; rel=&quot;noreferrer&quot;&gt;New Math movement&lt;/a&gt;, and spread from the math education literature. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notation itself in its modern form can be traced back to Lefschetz's Algebraic Topology (1942), and variants appear already in Principia (1910) and &lt;a href=&quot;http://bbi-math.narod.ru/newmann/newmann.html&quot; rel=&quot;noreferrer&quot;&gt;von Neumann's Zur Einführung der transfiniten Zahlen (1923)&lt;/a&gt;. See &lt;a href=&quot;https://mathoverflow.net/q/327706&quot;&gt;Who first discovered the concept corresponding to the symbol of class comprehension?&lt;/a&gt; for many more details. However, mathematicians did not use the name &quot;set-builder&quot;. Bernays (1958) calls it &quot;class operator&quot; and Suppes (1960) &quot;definition by abstraction&quot;. The name does not appear before 1957, but in 1958 we find it in the lively discussions of the high school curriculum in The Mathematics Teacher. E.g. &lt;a href=&quot;https://www.jstor.org/stable/27955572&quot; rel=&quot;noreferrer&quot;&gt;Rourke's Some implications of twentieth century mathematics for high schools&lt;/a&gt; explains:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We have a convenient notation for de noting solution sets, using the set-builder :&#xA;   &lt;span class=&quot;math-container&quot;&gt;$\{x\mid\ \}$&lt;/span&gt;. The braces &quot;&lt;span class=&quot;math-container&quot;&gt;$\{\ \}$&lt;/span&gt;&quot; are read &quot;set&quot;; the vertical &quot;&lt;span class=&quot;math-container&quot;&gt;$|$&lt;/span&gt;&quot; is read &quot;such that.&quot; We put the variable on the left-hand side of the vertical bar, and the sentence on the  right-hand side.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And &lt;a href=&quot;https://www.jstor.org/stable/27955672&quot; rel=&quot;noreferrer&quot;&gt;Duren's The maneuvers in set thinking&lt;/a&gt; goes deep into the pedagogy:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;We do not have any system of individual names for sets  like the decimal representations of the real numbers. Hence we have no way of  giving the name of a particular set which  is the &quot;answer&quot; to a problem except by such indirect devices as the set-builder: &lt;span class=&quot;math-container&quot;&gt;$\{x e X | A \&amp;amp; B \&amp;amp; C\}$&lt;/span&gt; = &quot;The set of all elements in X having properties A and B and C&quot;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The student journal &lt;a href=&quot;http://www.pme-math.org/journal/issues/PMEJ.Vol.3.No.3.pdf&quot; rel=&quot;noreferrer&quot;&gt;Pi Mu Epsilon&lt;/a&gt; still uses scare quotes around set-builder when reviewing Suppes's textbook in 1960. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What happened in 1957 is that the Soviet Union launched an orbital satellite, and a period of existential anxiety in the Western countries known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sputnik_crisis&quot; rel=&quot;noreferrer&quot;&gt;Sputnik crisis&lt;/a&gt;. One of the responses was to pack high school curriculum with symbolic logic, matrices and sets, among other things, to &quot;catch up&quot; to the Soviet advances (in fairness, some reforms date back to he University of Illinois Committee on School Mathematics from 1951 on). Ironically, while Soviet high school mathematics was reformed in the 1930s, it was not this way. Nor is there a Russian analog of the &quot;set-builder&quot; nickname, according to &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE&quot; rel=&quot;noreferrer&quot;&gt;Russian Wikipedia&lt;/a&gt;. After 1958, the New Math, and &quot;set-builder&quot;, rapidly spread into textbooks. The earliest I found is the teacher's edition of &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED173151.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mathematics for High School, p.16 (1959)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The braces &lt;span class=&quot;math-container&quot;&gt;$\{\ \}$&lt;/span&gt; used to enclose the elements of a set call&#xA;  attention to the fact that we are to think of the collection as&#xA;  a singl entity. The set-builder notation &lt;span class=&quot;math-container&quot;&gt;$\{x:\ x..... \}$&lt;/span&gt; is a&#xA;  useful way to represent a set which is characterized by some rule or property, nothing more. In some treatments of the subject vertical bar is used in place of the colon in the set-builder notation. We prefer the colon for typographical reasons.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;New Math was controversial from the start, and heavy criticism pushed most of it out of high schools by the end of 1960s. But the nickname stuck.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-09-29T00:00:13.307" LastActivityDate="2019-09-29T00:00:13.307" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11029" PostTypeId="2" ParentId="6106" CreationDate="2019-09-30T15:15:44.840" Score="2" Body="&lt;p&gt;A short answer to the question might be that, after the Nobel award to Einstein in 1922, many people were given the impression that it had been awarded for relativity, and that consequently he received few or no further nominations for any second Nobel prize. I recall seeing a biographical statement to this effect (but can't now locate the reference).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What does emerge from citable records, however, and makes any shortage of further nominations understandable, is that much in the record could well give rise to a mistaken idea that Einstein's prize actually had been awarded to him for the theory of relativity.  The following examples would contribute to such a mistaken impression. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, there was confusion in the media about the subject of the award.  When the &lt;em&gt;New York Times&lt;/em&gt;  announced the award to Einstein, it ambiguously mentioned relativity, but not a word about the actual subject of the award, Einstein's study and results about the photoelectric effect: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;NOBEL PRIZE FOR EINSTEIN. The Nobel committee has awarded the physics&#xA;  prize for 1921 to Professor Dr. Albert Einstein of Germany, identified&#xA;  with the theory of relativity, and that for 1922 to Professor Neils&#xA;  {sic} Bohr of Copenhagen.&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-- from &lt;a href=&quot;https://static01.nyt.com/packages/html/books/nobel-Benavente.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;New York Times, Nov. 10, 1922, page 4&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Second, confusion also arises from the circumstances of the award and the lecture that followed it. The usual pattern is that Nobel laureates give a lecture about the subject of their prize.  There is indeed a lecture by Einstein in the Nobel archives, but it is not about the subject of his prize, it is about relativity: &lt;a href=&quot;https://www.nobelprize.org/prizes/physics/1921/einstein/lecture/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Nobel Prize in Physics 1921 -- Einstein Nobel lecture (July 11, 1923)&lt;/a&gt;.  The background explains this to some extent. When Einstein received news of his award, he was already on his way to Japan, and perhaps could not attend the award ceremonies in person, at all events he did not attend (Abraham Pais (1982, Oxford UP), &lt;em&gt;'Subtle is the Lord' : the science and the life of Albert Einstein'&lt;/em&gt; at p.503).  At the award ceremony on December 10, 1922, Rudolf Nadolny, German ambassador to Sweden, accepted the Nobel prize in Einstein's name and made diplomatic acknowledgements on his behalf -- and naturally there was no scientific content in what was said. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;After the Nobel ceremonies, Einstein was invited to Sweden to lecture on a subject 'of his choice'.  Arguably this muddied the waters and made confusion worse about the subject of the prize. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;In March 1923 Svante Arrhenius, one of the Committee members, wrote&#xA;  to Einstein suggesting that the latter not wait until December for his&#xA;  visit to Sweden but that he come in July. He could then attend a&#xA;  meeting of the Scandinavian Society of Science in Goteborg on the&#xA;  occasion of the 300th anniversary of the founding of that city.&#xA;  Arrhenius left to Einstein the choice of topic for a general lecture,&#xA;  'but it is certain that one would be most grateful for a lecture about&#xA;  your relativity theory'.  Einstein replied that he was agreeable to&#xA;  this suggestion, though he would have preferred to speak on unified&#xA;  field theory.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;-- Pais (1982, cited above), p.505.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein, when he finally arrived in Sweden several months after the Nobel award ceremonies, did indeed lecture about relativity. His lecture was not formally linked to the Nobel award, it was given at a scientific meeting in Gothenburg.  But it is preserved in the archives among the Nobel lectures, and it is a little difficult to find indications that make clear the relation between the lecture and the prize. There is (now) a fine-print footnote added to the English translation of Einstein's lecture (but not with the original German version). The material is headed 'Nobel lecture' and the footnote to the English version runs: &quot;The Lecture was not delivered on the occasion of the Nobel Prize award, and did not, therefore, concern the discovery of the photoelectric effect.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This combination of events and records was inherently likely to reinforce any confused idea that actually Einstein had already received a Nobel prize for relativity.  Mistakes and myths have arisen from lesser causes than that. &lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2019-09-30T15:15:44.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11030" PostTypeId="1" CreationDate="2019-10-02T14:16:33.107" Score="0" ViewCount="42" Body="&lt;p&gt;Even with present-day knowledge of celestial mechanics, if you asked me to go outside and set up a sundial without the aid of any other timekeeping devices, I would have a very hard time. The time between sun-up and sun-down vary from day to day, so even though I could draw a mark at noon (the time when the shadows are the shortest), I would be unable to figure out how big to make the hour sections. How did people mark out the hours on the first sundials?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I imagine that I could wait for an equinox, mark sun-up and sun-down, and then divide that in to twelve equal segments, but without a clock I don't see how I could prove that the day and night of the equinox were indeed of equal length.&lt;/p&gt;&#xA;" OwnerUserId="9230" LastActivityDate="2019-10-02T14:16:33.107" Title="How were sundials first divided up into hour-long segments?" Tags="|astronomy|time|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11033" PostTypeId="1" AcceptedAnswerId="11035" CreationDate="2019-10-03T00:12:25.250" Score="-1" ViewCount="87" Body="&lt;p&gt;Who first distinguished number theory and numerology?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2019-10-03T09:19:19.073" Title="Who first distinguished number theory and numerology?" Tags="|mathematics|mathematicians|reference-request|number-theory|" AnswerCount="1" CommentCount="3" ClosedDate="2019-10-03T15:00:21.267" ContentLicense="CC BY-SA 4.0" />
  <row Id="11034" PostTypeId="1" AcceptedAnswerId="11036" CreationDate="2019-10-03T01:28:05.390" Score="0" ViewCount="446" Body="&lt;p&gt;Thinking of properties of materials, who discovered anisotropy and what materials were they studying?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I had read a claim that the first recorded experiments to study anisotropy are those by Duhamel who in 1832 studied anisotropic conductivity by measuring the elliptic shape of the melting front around a small heat source embedded in crystals coated with paraffin. Thus, the relation &lt;span class=&quot;math-container&quot;&gt;$q_i =-k_{ij} \partial_j T$&lt;/span&gt; is known as the Fourier-Duhamel law of heat conduction.&lt;/p&gt;&#xA;" OwnerUserId="3819" LastEditorUserId="3819" LastEditDate="2019-10-03T13:33:09.787" LastActivityDate="2019-10-03T13:33:09.787" Title="Who discovered anisotropic materials, and what were they studying?" Tags="|discoveries|optics|materials-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11035" PostTypeId="2" ParentId="11033" CreationDate="2019-10-03T09:19:19.073" Score="0" Body="&lt;p&gt;Looking for methods is theory, looking for results is... whatever. The interest in pythagorean triples or magic squares seems to have evolved from knowing some items to knowing a method to generate all of them. But that does not preclude a motivation e.g. some kind of power, magical or other, to be derived: knowledge is power.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A similar question aks when astronomy separated from  astrology and there is no good answer. According to Westman Copernic was still looking for a better astrology, but in his approach he was following Ptolemy who had already separated the method, that is astronomy, from the application - astrology.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems obvious that Euclid's proof about the infinity of primes is number theory, so at least this a terminus ante quem. Ever since Burkert's book the view about so called pythagoreans has radically changed and currently Zhmud has worked out details various writings. Also one should remember that little is known about ancient Greek isopsephy which became popular at the same time when neopythagoreanism was in vogue.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2019-10-03T09:19:19.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11036" PostTypeId="2" ParentId="11034" CreationDate="2019-10-03T11:15:18.717" Score="2" Body="&lt;p&gt;It depends on what counts as &quot;discovery&quot;. Interestingly, optical anisotropy was discovered before the elastic one, and the first anisotropic &quot;material&quot; modeled was... the luminiferous ether.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bartholinus discovered double refraction in calc-spar, a type of calcite, back in 1669, and Huygens showed in 1690 that two rays arising from refraction by calcite are extinguished by passing them through the same crystal rotated about the direction of the rays. Thus, calcite was optically anisotropic. Newton explained it by suggesting that light particles have &quot;sides&quot;, and used it as an argument against Hooke's wave optics. At the time, only longitudinal (acoustic) waves were known, and they could not explain transverse polarization. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The phenomenon was not much investigated until Malus's polarization experiments with calcite in 1808, and Arago's with quartz in 1811, see  &lt;a href=&quot;https://arago.elte.hu/sites/default/files/DSc-Thesis-2003-GaborHorvath-01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brief History of the Discovery of Phenomena Concerning Light Polarization&lt;/a&gt;. At that time, Young and Fresnel made wave optics indispensable, and that posed a problem. In isotropic elastic media transverse waves were always accompanied by longitudinal waves, but polarization experiments ruled out their presence. If light propagated in elastic ether, that ether had to be anisotropic. Elastic theories of ether became quite popular, with Cauchy, Lame, Green and others studying them. Green's 1838 equations already took anisotropy into account, but it was Kelvin's Elements of a Mathematical&#xA;Theory of Elasticity (1856) that redirected the theory towards real solid materials rather than the hypothetical ether, see &lt;a href=&quot;https://library.seg.org/doi/10.1190/1.2122407&quot; rel=&quot;nofollow noreferrer&quot;&gt;75-plus years of anisotropy by Helbig and Thomsen&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Thus, the first articles on elasticwave&#xA;  propagation already took anisotropy into account. For&#xA;  example, Green (1838) was the first to use strain energy, and&#xA;  he strongly supported the notion that there could be as many&#xA;  as 21 elastic constants. In 1856, Lord Kelvin published “Elements of a mathematical theory of elasticity”, which exclusively discussed solids.&#xA;  This is not to be taken as an indication that he did not believe&#xA;  in the elastic ether, but only that he was interested in metals at&#xA;  that time and thus needed a solid foundation of the theory of&#xA;  elasticity.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;For this purpose, he invented concepts that became&#xA;  common only much later, such as vectors and vector spaces&#xA;  (in 6D space!), tensors, and eigensystems. With these tools, he&#xA;  could describe the elastic tensor in coordinate-free form. His&#xA;  ideas were so much ahead of his time that his paper — and a&#xA;  re-publication (in the “Elasticity” listing of the 1886 edition of&#xA;  the Encyclopedia Britannica) — were regarded by some of his&#xA;  contemporaries as scientifically unsound (despite his stature)&#xA;  and thus made no impact on the development of the theory of&#xA;  anisotropy... Kelvin was also the first to formulate the elastic-wave equation for anisotropic media. (He solved it for a simple case.)&#xA;  Since this achievement was published as part of his “no impact”&#xA;  papers, it was also overlooked. Hence, today the solution&#xA;  of the wave equation is attributed to Christoffel (1877).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In the end, anisotropic ether did not work out either, but the elasticity theory motivated by it came in handy in material science.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-10-03T11:15:18.717" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11038" PostTypeId="2" ParentId="5614" CreationDate="2019-10-03T15:40:15.757" Score="1" Body="&lt;p&gt;Reviel Netz's &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=5693&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History&lt;/em&gt;&lt;/a&gt; p. 278:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Astrology and number-lore present a different kind of problem. Putting them apart from mathematicians in the ‘Euclidean’ sense would not necessarily reflect ancient views. &lt;em&gt;Mathematicus&lt;/em&gt; came to mean in Latin ‘astrologer’.&lt;sup&gt;35&lt;/sup&gt; As for number-lore, Plato’s lecture had already led directly from the mathematical sciences to the conclusion that the good is unity.&lt;sup&gt;36&lt;/sup&gt; But I will not count astrologers and numerologists as mathematicians. This is not cultural snobbery. Rather, I am concentrating on the issues which are important for my study, which are cognitive. Astrology and numerology, in themselves, do not impose the same cognitive regime as ‘Euclidean’ mathematical sciences do. A person who has devoted himself to astrology alone, with no element of astronomy, while he may have, say, calculated a great deal, has never proved anything, and he therefore cannot be seen as a mathematician from the perspective adopted in this study.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;hr&gt;&lt;sup&gt;35. &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dmathematicus&quot; rel=&quot;nofollow noreferrer&quot;&gt;s.v. Lewis &amp;amp; Short&lt;/a&gt;, 2. ‘post-Aug.’&lt;br&gt;36. Aristoxenus, El. Harm. II.30.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Reviewer &lt;a href=&quot;https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=27535&quot; rel=&quot;nofollow noreferrer&quot;&gt;B. Artmann&lt;/a&gt; argues (&lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=1683176&quot; rel=&quot;nofollow noreferrer&quot;&gt;MR1683176&lt;/a&gt;) that &quot;&lt;a href=&quot;https://archive.org/details/NicomachusIntroToArithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicomachus' arithmetic&lt;/a&gt;&quot; is &quot;admittedly a book on the border of mathematics and numerology.&quot;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2019-10-03T15:40:15.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11040" PostTypeId="1" AcceptedAnswerId="11041" CreationDate="2019-10-03T18:48:48.937" Score="4" ViewCount="141" Body="&lt;p&gt;In &lt;a href=&quot;https://hsm.stackexchange.com/questions/10990/&quot;&gt;another question&lt;/a&gt;, I was asking about the origin of the reduced Planck's constant, &lt;span class=&quot;math-container&quot;&gt;$\hbar \equiv \frac{h}{2 \pi} .$&lt;/span&gt;  Specifically, I wanted to know why the symbol &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; was selected for the value &lt;span class=&quot;math-container&quot;&gt;$\frac{h}{2 \pi} ,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is Planck's constant.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found a presentation that claimed it was introduced in a 1926 article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KiAjJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KiAjJ.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&amp;ndash;Page 17 of &quot;&lt;em&gt;The Planck constant h and the Dirac constant ħ.  Their units and their history&lt;/em&gt;&quot;, by Ian Mills and P. R. Bunker.  &lt;a href=&quot;https://chemphys.ca/pbunker/PlanckWiki.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[PDF]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This seems to make it clear that that article used the symbol, &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot; .$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in checking the cited journal article, it seems to use the symbol &lt;span class=&quot;math-container&quot;&gt;$`` h &quot;$&lt;/span&gt; instead.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zc5kq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zc5kq.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&amp;ndash;Page 1 (printed-page 661) of &quot;&lt;em&gt;On the theory of quantum mechanics&lt;/em&gt;&quot;, Paul Dirac (1926-10-01).  doi:&lt;a href=&quot;https://doi.org/10.1098/rspa.1926.0133&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1098/rspa.1926.0133&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This would seem to suggest that the presentation claiming that &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; appeared in the 1926 article was mistaken, &lt;strong&gt;&lt;em&gt;if&lt;/em&gt;&lt;/strong&gt; the above screenshot is representative of the 1926 article.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as noted in &lt;a href=&quot;https://hsm.stackexchange.com/a/11003/6913&quot;&gt;one answer&lt;/a&gt; to my question, this article was printed around the time it became possible to use &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot; .$&lt;/span&gt;  So, it seems possible that some printers may've actually used &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot; ,$&lt;/span&gt; as claimed in the presentation, while the screenshot above shows a different printing that fell back on &lt;span class=&quot;math-container&quot;&gt;$`` h &quot; .$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Were such printing variations something that tended to happen in the past?  Or, were printings kept more uniform, e.g. done by a single shop in a single go?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Historically, have printings of the same journal article contained minor typographical variations?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there an easy way to check if printings of a particular article had a particular variation between printings?  For example, is there an easy way to check if the 1926 article discussed above used &lt;span class=&quot;math-container&quot;&gt;$`` h &quot;$&lt;/span&gt; in some printings vs. &lt;span class=&quot;math-container&quot;&gt;$`` \hbar &quot;$&lt;/span&gt; in other printings?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6913" LastEditorUserId="6913" LastEditDate="2019-10-03T19:07:01.590" LastActivityDate="2019-10-04T18:21:28.850" Title="Were typographical variations between printings of the same journal article common?" Tags="|notation|publishing|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11041" PostTypeId="2" ParentId="11040" CreationDate="2019-10-04T18:21:28.850" Score="4" Body="&lt;p&gt;Having looked through a number of quantum mechanics papers from the era, it is pretty clear that, as @M.Farooq points out, what Dirac is using in the supposed paper-of-origin is just an italic &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, which he redefines as Plancks' constant (who originally used the &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;) divided by &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;. So, that is for 1926. In 1931, Dirac is still doing the same thing (see, say, Quantised Singularities in the Electromagnetic Field).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, in 1931, Oppenheimer &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.38.725&quot; rel=&quot;nofollow noreferrer&quot;&gt;Note on Light Quanta and the Electromagnetic Field&lt;/a&gt; is explicitly using &lt;span class=&quot;math-container&quot;&gt;$h/2\pi$&lt;/span&gt; and not &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$h/2\pi$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;By February of 1934, &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.45.245&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Theory of the Electron and Positive&lt;/a&gt;, W.H. Furry and J. R. Oppenheimer, Physical Review 45 245 (1934) one gets a real &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TuIKf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TuIKf.png&quot; alt=&quot;1934 hbar&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By 1937, Dirac seems to have relented (or the editors have insisted) and an &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; appears in &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1937.0094&quot; rel=&quot;nofollow noreferrer&quot;&gt;Complex Variables in Quantum Mechanics&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Sy46N.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Sy46N.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Without spending more time perusing the literature, it is clear that Dirac for some time just preferred to use &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; instead of an &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$h/2\pi$&lt;/span&gt;, thereby redefining &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; for his purposes. On the American side of the ocean, Oppenheimer shows a change somewhere between 1931 and 1934, moving from &lt;span class=&quot;math-container&quot;&gt;$h/2\pi$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, all this being said, the answer to the question in the title is a firm No - journal articles were printed and that was that. They would not be reprinted, if at all, for quite some time. Now, Dirac may well have started using &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; in place of &lt;span class=&quot;math-container&quot;&gt;$h/2\pi$&lt;/span&gt;, which sets the stage for somebody inventing &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; to avoid the obvious confusion, but he did not actually use &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; until later.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2019-10-04T18:21:28.850" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11042" PostTypeId="1" AcceptedAnswerId="11044" CreationDate="2019-10-05T02:46:09.140" Score="6" ViewCount="335" Body="&lt;p&gt;Did Cauchy ever deal with double or triple integrals? Did he give rigorous proofs of multivariable integral calculus like what came to be called Stokes's theorem, the divergence theorem, etc.?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="4251" LastEditDate="2021-06-22T18:01:28.427" LastActivityDate="2021-06-22T18:01:28.427" Title="Did Cauchy ever deal with double or triple integrals?" Tags="|mathematics|mathematicians|calculus|real-analysis|cauchy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11044" PostTypeId="2" ParentId="11042" CreationDate="2019-10-05T07:54:30.907" Score="8" Body="&lt;p&gt;Yes, he did, multiple times. &lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Singular double integrals (1814)&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_CAUCHY_1_1_319_0&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mémoire sur les intégrales définies&lt;/em&gt; (1814)&lt;/a&gt; Cauchy studied why switching the order of integration in a double integral can sometimes lead to different results. This led him to introduce the notion of &quot;singular integral&quot;. For &lt;span class=&quot;math-container&quot;&gt;$K = \frac{z}{x^2 + z^2}$&lt;/span&gt; he showed that &lt;span class=&quot;math-container&quot;&gt;$\int_0^1\int_0^1\frac{\partial K}{\partial z}dzdx$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt;, but integrated in the reverse order it is &lt;span class=&quot;math-container&quot;&gt;$-\pi/4$&lt;/span&gt;. Good secondary sources are &lt;a href=&quot;https://link.springer.com/chapter/10.1007%2F978-3-0348-7811-1_10&quot; rel=&quot;noreferrer&quot;&gt;The entry of Cauchy: complex variables and differential equations, 1810–1822 by Grattan-Guinness&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books/about/The_Real_and_the_Complex_A_History_of_An.html?id=zpDDCgAAQBAJ&quot; rel=&quot;noreferrer&quot;&gt;Gray's The Real and the Complex: A History of Analysis in the 19th Century&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Mathematicians of the period, with their somewhat formal attitude to the processes&#xA;  of the calculus, not only tended to regard a double integral as a nested pair of single integrals but to be indifferent to the order of integration. This formal approach precluded interpreting double integrals in terms of line integrals along the boundaries of a rectangle in the plane of complex numbers; they were strictly about integrals on a rectangle in the real plane. However, it was known that in some cases this attitude was too naive and led to conflicting results, and it was this problem that Cauchy addressed in the second part of the Mémoire&lt;/em&gt;. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Cauchy showed varying the order of integration of a double integral can yield two distinct definite values when the function becomes infinite or indeterminate for certain values of the variables within the domain of integration. To investigate the matter further, he introduced what he called singular integrals (“intégrales singulières”), which he defined (Cauchy 1814, 334, see Sect. 5.3) as limits as the end points of the integral tend to the same fixed value “without the integrals being zero.” Singular integrals are a profound idea. They were to crop up repeatedly in real analysis and, in an intriguing different way, in Cauchy’s later work on complex integrals.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h1&gt;Connection to repeated integration (1823)&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Basic theory of double and repeated integrals, is laid out in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k62404287.texteImage&quot; rel=&quot;noreferrer&quot;&gt;Cauchy's &lt;em&gt;Résumé des leçons données à l'École royale polytechnique sur le calcul&lt;/em&gt; (1823)&lt;/a&gt;, a lecture course for students of &lt;em&gt;École Polytechnique&lt;/em&gt;, lectures 34-35, see &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/B9780444508713501066&quot; rel=&quot;noreferrer&quot;&gt;Landmark Writings in Western Mathematics by Grattan-Guinness&lt;/a&gt;. In particular, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration&quot; rel=&quot;noreferrer&quot;&gt;Cauchy formula for repeated integration&lt;/a&gt; is derived there and is used to evaluate some elementary integrals.&lt;/p&gt;&#xA;&#xA;&lt;h1&gt;Connection to the potential theory and complex integrals (1846)&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Grattan-Guinness gives a panoramic view of the development of multiple integration in the context of the potential theory in &lt;a href=&quot;https://www.jstor.org/stable/2975029&quot; rel=&quot;noreferrer&quot;&gt;Why did George Green Write His Essay of 1828 on Electricity and Magnetism?&lt;/a&gt; Lagrange (1762) was a pioneer, who formed volume integrals and integrated them by parts to form surface integrals. This was continued in Poisson's work on electrostatics (1812), and led to the divergence theorems such as Green's formula (1828).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k90190w/f77&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur les intégrales qui s'étendent à tous les points d'une courbe fermée&lt;/em&gt; (1846)&lt;/a&gt; Cauchy gives a proof of the Cauchy integral theorem based on the application of the Green's formula (in modern notation):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_C P dx + Q dy = \iint_R \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}dx\,dy.$$&lt;/span&gt;&#xA;On the earlier 1825 proof that did not use double integrals see &lt;a href=&quot;https://hsm.stackexchange.com/a/8227/55&quot;&gt;What was the motivation for Cauchy's Integral Theorem?&lt;/a&gt; It is unclear if Cauchy was influenced by Green's 1828 work, see &lt;a href=&quot;https://zulfahmed.files.wordpress.com/2018/07/historycauchyintegralformula.pdf&quot; rel=&quot;noreferrer&quot;&gt;Cauchy Integral Theorem: a Histortical Development of its Proof by Scott&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Whether Cauchy himself proved this result independently&#xA;  of George Green or actually utilized Green's work is not certain, for although Green's Theorem was published in a privately printed booklet in&#xA;  1828, it was not published in a mathematical journal until 1850. So it&#xA;  is possible that Cauchy conceived of this result independently; however,&#xA;  there are indications that he was influenced by Green's work because he&#xA;  extended his integral theorem to areas on curved surfaces.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;h1&gt;Volume integrals (1848)&lt;/h1&gt;&#xA;&#xA;&lt;p&gt;Although generalities of the &lt;em&gt;Résumé&lt;/em&gt; apply to volume integrals, Cauchy did not study them specifically. They are scarce even where one would expect to find them, in his extensive work on elasticity theory (1822ff) and the optics of elastic ether (1829ff). His best known contribution, &lt;a href=&quot;https://arxiv.org/abs/1706.08518&quot; rel=&quot;noreferrer&quot;&gt;Cauchy's tetrahedron argument&lt;/a&gt;, is typically presented in modern textbooks on continuous mechanics in terms of volume integrals and the divergence theorem, but Cauchy himself did not write any integrals. Everything is typically expressed in differential form, see e.g. &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k90200c/f194&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur les équations qui expriment les conditions d'équilibre, ou les lois du mouvement intérieur d'un corps solide, élastique, ou non élastique&lt;/em&gt; (1828)&lt;/a&gt;. A couple of triple integrals occur in passing in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k90200c/f264.image&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mémoires sur les conditions relatives aux limites des corps et en particulier sur celles qui conduisent aux lois de la réflexion et de la réfraction de la lumiére&lt;/em&gt; (1848, p.265)&lt;/a&gt;, but they are immediately reduced to a single integral using repeated integration in spherical coordinates. On Cauchy's elastic optics generally see &lt;a href=&quot;https://books.google.com/books/about/A_History_of_Optics_from_Greek_Antiquity.html?id=ImM62wvWE_cC&quot; rel=&quot;noreferrer&quot;&gt;Darrigol's History of Optics, ch.6&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-06T09:14:32.503" LastActivityDate="2019-10-06T09:14:32.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11045" PostTypeId="1" CreationDate="2019-10-05T18:32:59.207" Score="4" ViewCount="288" Body="&lt;p&gt;&lt;em&gt;Petit Point: A Candid Portrait on the Aberrations of Science&lt;/em&gt; is a charming book written by the French physicist Pierre-Gilles de Gennes containing short essays on some of the prominent scientists that he encountered in his career. For obvious reasons, their names have been replaced with pseudonyms.&lt;/p&gt;&#xA;&lt;p&gt;I list the chapter names from the book: Mastoc, Vera, Lanterne, Leduc, Emmy, Breton, Smirnoff, Pluvieux, Beziers, Kuba, Vladimir, Aglae, Subtil, Chazot, Anchor, Croesus, Caesar, Guru, Dourakine, Saplir, Manfred, Robert, Polymorph, Revizor, Feston, Philostrate, Elise, Spiros, Akbar&lt;/p&gt;&#xA;&lt;p&gt;The scientists I could identify: Caesar - Benoit Mandelbrot, Manfred - Brian Josephson, Subtil - Bernd Matthias, Chazot - Per Bak.&lt;/p&gt;&#xA;&lt;p&gt;Other potential candidates: Robert - de Gennes himself, Emmy - Susan Coppersmith&lt;/p&gt;&#xA;&lt;p&gt;Could someone identify the remaining names listed ? Thanks&lt;/p&gt;&#xA;" OwnerUserId="10536" LastEditorUserId="10536" LastEditDate="2022-03-04T03:28:32.203" LastActivityDate="2022-03-04T03:28:32.203" Title="Identification of scientists in de Gennes book" Tags="|physics|biographical-details|scientists|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11046" PostTypeId="1" AcceptedAnswerId="11049" CreationDate="2019-10-08T18:25:40.487" Score="8" ViewCount="1407" Body="&lt;p&gt;As I'm advancing my study in Electromagnetism I'm getting introduced to more mathematical operators which are exclusively used in Electromagnetism and Fluid Dynamics only.&lt;/p&gt;&#xA;&lt;p&gt;Let me try to explain myself through an example: Electric flux is defined as &lt;span class=&quot;math-container&quot;&gt;$$ \int\limits_S \mathbf{E} \cdot \hat n~~  dS $$&lt;/span&gt; and now by the &lt;strong&gt;Gauss's Law&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$$ \int\limits_S \mathbf{E} \cdot \hat n~~ dS = \int\limits_V \nabla \cdot \mathbf{E} ~~dV$$&lt;/span&gt;&#xA;and now if we take such a surface such that &lt;span class=&quot;math-container&quot;&gt;$\hat n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{E}$&lt;/span&gt; are parallel (let's say a sphere with a positive charge density &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; and total charge q at center) and then we will have &lt;span class=&quot;math-container&quot;&gt;$$ \frac {1}{4\pi\epsilon_0} \frac{q}{r^2} \times 4\pi r^2=  \int\limits_V \nabla \cdot \mathbf{E} dV$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$ \frac{d}{dV} \left[ \frac{q}{\epsilon_0} \right] = \nabla \cdot \mathbf{E}$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$ \nabla \cdot\mathbf{E} = \frac{\rho}{\epsilon_0}$$&lt;/span&gt; and we call that last equation the first of &lt;strong&gt;Maxwell's Equations&lt;/strong&gt; . I respect those great physicists and mathematicians , I really do and my question is not to doubt them but rather to ask.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why did we care about that divergence notation and started giving intuitive explanation like how much a thing is coming out or going in&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I mean if it was just for notational easiness or something else. There are a lot of videos explaining the intuitive sense of operators of &lt;strong&gt;curl&lt;/strong&gt; and &lt;strong&gt;divergence&lt;/strong&gt; but as you have seen it's really a mathematical consequence of the way we define those two operators. So, my question is&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why there was a need to define divergence and curl&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;p&gt;When I was studying line integrals from &lt;strong&gt;Stewart's Calculus&lt;/strong&gt; there it was written clearly that line integrals were invented to solve the problems of fluid flow. I think that the concept of surface integral was also invented only for defining &lt;em&gt;flux&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9837" LastEditorUserId="10106" LastEditDate="2022-08-31T03:57:29.050" LastActivityDate="2022-08-31T03:57:29.050" Title="What was the real need of divergence and curl operators?" Tags="|physics|calculus|mathematical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11047" PostTypeId="1" CreationDate="2019-10-08T18:31:28.963" Score="3" ViewCount="116" Body="&lt;p&gt;Who first proved a generalized &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Fubini_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fubini theorem&lt;/a&gt; for integrals of order &lt;span class=&quot;math-container&quot;&gt;$≥3$&lt;/span&gt;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&lt;sup&gt;th&lt;/sup&gt; order integral is &lt;span class=&quot;math-container&quot;&gt;$$\underbrace{\underset{x_n}\int\underset{x_{n-1}}\int\ldots\underset{x_1}\int}_{n} f(x_1,x_2,\ldots,x_n)dx_1dx_2\ldots dx_n.$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By &quot;a generalized &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Fubini_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fubini theorem&lt;/a&gt;,&quot; I mean that all permutations of the  &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; produce valid and equivalent integrals.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-10-10T17:58:12.600" LastActivityDate="2019-10-10T17:58:12.600" Title="Who first proved Fubini's theorem $n$th order integrals?" Tags="|mathematics|reference-request|calculus|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11048" PostTypeId="1" AcceptedAnswerId="11056" CreationDate="2019-10-08T21:41:59.880" Score="3" ViewCount="141" Body="&lt;p&gt;We have recently proven in our functional analysis II lecture that the spectrum of an operator is contained in the closure of its numerical range.&#xA;On the German wikipedia &lt;a href=&quot;https://de.wikipedia.org/wiki/Numerischer_Wertebereich#Versionen_f%C3%BCr_Operatoren&quot; rel=&quot;nofollow noreferrer&quot;&gt;page&lt;/a&gt; for the numerical radius they say that Otto Toeplitz already investigated the numerical range in 1918.&#xA;But who has proven this result (first)?&lt;/p&gt;&#xA;" OwnerUserId="10550" LastEditorUserId="10550" LastEditDate="2020-01-06T01:15:46.667" LastActivityDate="2020-01-06T01:15:46.667" Title="Who first proved that the spectrum of an operator is contained in the closure of its numerical range?" Tags="|mathematics|functional-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11049" PostTypeId="2" ParentId="11046" CreationDate="2019-10-08T21:46:31.367" Score="10" Body="&lt;p&gt;These operations arose from the study of quaternions see e.g. Thomson and Tait's &lt;em&gt;Treatise on Natural Philosophy&lt;/em&gt;, that should probably have information on the sort of math. Stokes's theorem originated with Thomson (Lord Kelvin) around 1850 and in there the expression for curl appears. The history of quaternions is certainly interesting, starting with Hamilton in 1843. Evidently, the significance of a sum and difference of partial derivatives was recognized for its utility. &lt;/p&gt;&#xA;&#xA;&lt;h3&gt;PLAUSIBLE INTUITION&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;There is very good reason to use these operators, strictly from an analytical perspective. The question of how to define a derivative in multivariate calculus motivates us to consider first a two-dimensional vector field case. In such a case, the change in direction can be described by taking derivatives of the &lt;span class=&quot;math-container&quot;&gt;$x-y$&lt;/span&gt; components of the field with respect to various directions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\mathbf{A}$&lt;/span&gt; be the vector field with two components, &lt;span class=&quot;math-container&quot;&gt;$A_x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A_y$&lt;/span&gt;, each functions of the position &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt;. Then the total change can first be considered by the change in &lt;span class=&quot;math-container&quot;&gt;$A_x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and the change in &lt;span class=&quot;math-container&quot;&gt;$A_y$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, derivatives in the direction of the components, and second by the change in &lt;span class=&quot;math-container&quot;&gt;$A_x$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A_y$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. It is relatively straightforward to show that we should sum the first two to maintain their useful properties, &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y}$&lt;/span&gt;, and we should take the difference of the cross derivatives for their useful properties, &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}$&lt;/span&gt;. These become our two most useful derivatives, which are the divergence and curl respectively. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Extending this to three dimensions, we can readily adapt the divergence derivative by simply adding another partial derivative term. The cross derivative terms, however, require careful handling, as there are now two cross derivatives for each component, for a total of 9 partial derivatives. To extend the curl derivative, we consider the 2D curl derivatives in a given plane, for the three principle planes (&lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$xz$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$yz$&lt;/span&gt;), and we allow each curl in a plane to be a component in a vector. Thus we lose no information. The vector component we should use would be the component normal to the plane. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the language of vectors, this curl operator would be:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \left (\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right )\mathbf{i} + \left (\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right )\mathbf{j} + \left (\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right )\mathbf{k} $$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;h3&gt;HISTORICAL DEVELOPMENT&lt;/h3&gt;&#xA;&#xA;&lt;p&gt;A modern explanation with vectors is one thing, but historical development is of most concern for us here. One could arguably draw a connection directly from Stokes's theorem to the introduction of curl. From (Katz 1979), there were some hints of Stokes's theorem leading to its appearance in the Smith's Prize Exam, but the curl operation is made quite clear looking at the problem itself. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If X,Y,Z be functions of the rectangular coordinates &lt;span class=&quot;math-container&quot;&gt;$x,y,z$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt; an element of any limited surface, &lt;span class=&quot;math-container&quot;&gt;$l,m,n$&lt;/span&gt; the cosines of the inclinations of the normal at &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt; to the axes, &lt;span class=&quot;math-container&quot;&gt;$ds$&lt;/span&gt; an element of the boundary line, shew that&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$\iint l \left (\frac{\partial Z}{\partial y} - \frac{\partial Y}{\partial z}\right ) + m\left (\frac{\partial X}{\partial z} - \frac{\partial Z}{\partial x}\right ) + n \left (\frac{\partial Y}{\partial x} - \frac{\partial X}{\partial y}\right )  dS = \int \left (X \frac{dx}{ds} + Y \frac{dy}{ds} + Z \frac{dz}{ds} \right )$$&lt;/span&gt;&#xA;  ... the single integral being taken all around the perimeter of the surface. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Apparently, the left hand side had already appeared in a few earlier works of Stokes, although I'm not aware of any earlier history of the expression. As for the introduction of divergence, I'm sure the sum of derivatives often appeared in the calculus of quaternions, so it might take more work to track the origin of this expression. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general, a &lt;em&gt;complete differential&lt;/em&gt; is a differential form which is the differential of some scalar function. Today, we would say the differential is &lt;em&gt;exact&lt;/em&gt;. This idea comes very naturally from potential theory, developed by Green, Hamilton, Gauss, and others. It was known that the condition for e.g. a quaternion differential,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \mathbf{A} = A_x dx + A_y dy + A_z dz $$&lt;/span&gt;&#xA;to be complete is that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \left (\frac{\partial A_y}{\partial z} - \frac{\partial A_z}{\partial y}\right ) = \left (\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right ) = \left (\frac{\partial A_x}{\partial y} - \frac{\partial A_y}{\partial x}\right ) = 0$$&lt;/span&gt;&#xA;And so we are left to wonder about the behavior of quaternion fields for which this is not true, and we ask whether there is sufficient information to uniquely characterize a quaternion field using both the differential form and the cross derivatives above. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This leads us to an idea introduced in 1858 by Helmholtz that quaternion fields can be decomposed into their curl and divergence expressions which we have given above. Helmholtz's work &lt;em&gt;On Integrals of the Hydrodynamical Equations, which Express Vortex-Motion&lt;/em&gt; (1858) presents similar quantities to divergence and curl in the context of fluid dynamics and vortices. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the notations and notions of divergence and curl, the introduction of the symbol &lt;span class=&quot;math-container&quot;&gt;$\nabla$&lt;/span&gt; certainly brought something to the issue. This was, in fact, introduced by Hamilton, and was used greatly by Maxwell and Tait among others. Maxwell in particular was one to attach physical meaning to divergence and curl, though his conventions were slightly different from ours today. He used the &lt;em&gt;convergence&lt;/em&gt; which is the negative divergence, and he called curl the &lt;em&gt;rotation&lt;/em&gt;. These are plainly laid out in sec. 25 (p.30) of Volume 1 of his treatise. &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Selected References&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Katz, Victor J. &lt;em&gt;The History of Stokes' Theorem.&lt;/em&gt; 1979.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Helmholtz, H. &lt;em&gt;On Integrals of the Hydrodynamical Equations, which Express Vortex-Motion.&lt;/em&gt; 1858.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maxwell, James C. &lt;em&gt;Treatise on Electricity and Magnetism, volume 1&lt;/em&gt;. 1873.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Tait. &lt;em&gt;On Green's Theorem&lt;/em&gt;. 1872.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomson and Tait. &lt;em&gt;Treatise on Natural Philosophy.&lt;/em&gt; 1879.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Further Reading&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Crowe, Michael J. &lt;em&gt;A History of Vector Analysis.&lt;/em&gt; 1967.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;P.S. I am not very familiar with quaternions, so some of the stuff above where I say &quot;quaternion field&quot; etc I'm projecting vector field concepts to quaternions. I'm sure this would be outrageous to someone more familiar, so edits in this way (and others) are welcome. &lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2019-10-08T21:46:31.367" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11050" PostTypeId="1" CreationDate="2019-10-09T04:57:24.440" Score="1" ViewCount="141" Body="&lt;p&gt;When a Nobel prize is awarded to multiple people, the prize money is split. But what about the actual medal - how do they determine who will keep and store it? Been reading through the pages on &lt;a href=&quot;https://www.nobelprize.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;nobelprize.org&lt;/a&gt; but there is nothing concrete about this. &lt;/p&gt;&#xA;" OwnerUserId="10551" LastActivityDate="2019-10-09T16:02:45.497" Title="For shared Nobel prizes, where does the medal go?" Tags="|nobel-prize|academic-awards|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11051" PostTypeId="1" AcceptedAnswerId="11052" CreationDate="2019-10-04T23:52:07.947" Score="9" ViewCount="115" Body="&lt;p&gt;I'm not sure if this is the most suitable site for the question. Please feel free to modify or move my post!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have heard that people really walked a lot in Königsberg, trying to solve that &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;noreferrer&quot;&gt;seven bridges puzzle&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm just a bit curious: before the proof of Euler, was there already a common belief that it was impossible, given that so much effort failed?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the proof of Euler is quite important in the history of mathematics. But were people surprised by the negative answer (like, &quot;aha, I wasted so much time on something impossible!&quot;), or did they just say, &quot;ha, we all knew it's impossible, but that guy just cooked up a paper on it!&quot;&lt;/p&gt;&#xA;" OwnerDisplayName="WhatsUp" LastEditorUserId="2277" LastEditDate="2019-10-09T15:48:58.147" LastActivityDate="2019-10-09T15:48:58.147" Title="Seven bridges of Königsberg - did people know that it was impossible?" Tags="|mathematics|euler|" AnswerCount="1" CommentCount="1" ClosedDate="2019-10-09T15:48:46.590" ContentLicense="CC BY-SA 4.0" />
  <row Id="11052" PostTypeId="2" ParentId="11051" CreationDate="2019-10-05T09:06:53.980" Score="3" Body="&lt;p&gt;It looks like there wasn't really a common belief; see the question &lt;a href=&quot;https://hsm.stackexchange.com/q/5848/2277&quot;&gt;What was the origin of the Seven Bridges of Königsberg problem before Euler?&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://hsm.stackexchange.com/a/5849/2277&quot;&gt;answer&lt;/a&gt; by user @Conifold quotes researcher Teo Paloetti:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;According to lore, the citizens of Königsberg used to spend Sunday afternoons walking around their beautiful city. While walking, the people of the city decided to create a game for themselves, their goal being to devise a way in which they could walk around the city, crossing each of the seven bridges only once. &lt;strong&gt;Even though none of the citizens of Königsberg could invent a route that would allow them to cross each of the bridges only once, still they could not prove that it was impossible.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and Euler himself:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The problem, which I am told is widely known, is as follows: in Königsberg in Prussia, there is an island A called the Kneiphof; the river which surrounds it is divided into two branches, as can be seen in Fig. [1.2], and these branches are crossed by seven bridges, a, b, c, d, e, f and g. Concerning these bridges it was asked whether anyone could arrange a route in such a way that he would cross each bridge once and only once. &lt;strong&gt;I was told that some people asserted that this was impossible, while others were in doubt; but nobody would actually assert that it could be done.&lt;/strong&gt; From this, I have formulated the general problem: whatever be the arrangement and division of the river into branches, and however many bridges there be, can one find out whether or not it is possible to cross each bridge exactly once?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(emphasis mine)&lt;/p&gt;&#xA;" OwnerUserId="2277" OwnerDisplayName="Glorfindel" LastEditorUserId="2277" LastEditDate="2019-10-09T15:29:10.543" LastActivityDate="2019-10-09T15:29:10.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11053" PostTypeId="2" ParentId="11050" CreationDate="2019-10-09T16:02:45.497" Score="1" Body="&lt;p&gt;For shared nobel prize.....&#xA;Suppose two persons share a nobel prize then both will get a medal but prize money will be diveded in two parts depending upon contribution say 50% and 50% or 75% and 25% etc.&lt;/p&gt;&#xA;" OwnerUserId="10552" LastActivityDate="2019-10-09T16:02:45.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11054" PostTypeId="1" CreationDate="2019-10-09T16:40:55.777" Score="3" ViewCount="157" Body="&lt;p&gt;Are there any translations of the following into English, German, or Russian?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nicolas Bourbaki, &lt;a href=&quot;https://doi.org/10.1007/BF02036949&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur le théorème de Zorn&lt;/a&gt;, Archiv der Mathematik, Volume 2, pages 434–437, November 1949.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Any help is appreciated.&lt;/p&gt;&#xA;" OwnerDisplayName="user10553" LastEditorUserId="4251" LastEditorDisplayName="user10553" LastEditDate="2020-07-18T06:55:32.183" LastActivityDate="2020-07-18T06:55:32.183" Title="Translations of &quot;Sur le théorème de Zorn&quot;?" Tags="|mathematics|mathematical-logic|set-theory|translations|bourbaki|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11056" PostTypeId="2" ParentId="11048" CreationDate="2019-10-09T19:54:47.380" Score="6" Body="&lt;p&gt;This is called the spectral inclusion theorem. Toeplitz's 1918 paper &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF01212904&quot; rel=&quot;nofollow noreferrer&quot;&gt;Das algebraische Analogon zu einem Satze von Fejér&lt;/a&gt; already has it for finite dimensional operators as Satz 4. This was before the language of operators and Hilbert spaces introduced by von Neumann in 1927-29, see &lt;a href=&quot;https://www.jstor.org/stable/2319079&quot; rel=&quot;nofollow noreferrer&quot;&gt;Highlights in the History of Spectral Theory by Steen&lt;/a&gt;, so it is phrased in terms of eigenvalues and bilinear forms:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Die Eigenwerte einer beliebigen Bilinearform C gehoren alle zu ihrem Wertvorrat&lt;/em&gt;&quot;. [The eigenvalues of any bilinear form belong to its value range.] &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://books.google.com/books?id=ICfoBwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gustafson and Rao's Numerical Range, p.8&lt;/a&gt;, for bounded operators in Hilbert space the theorem was first proved by &lt;a href=&quot;https://en.wikipedia.org/wiki/Aurel_Wintner&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wintner&lt;/a&gt; in &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01187766&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Theorie der beschränkten Bilinearformen (1929)&lt;/a&gt;, who has spectrum and Hilbert spaces, but does not use the term &quot;operator&quot; either. For unbounded operators, the spectral inclusion is false. &lt;a href=&quot;https://books.google.com/books/about/Linear_Transformations_in_Hilbert_Space.html?id=uebNAwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stone in Linear Transformations in Hilbert Space (1932)&lt;/a&gt; mentions that for normal operators the closure of the numerical range is the convex hull of the spectrum. A generalization to non-linear operators is due to &lt;a href=&quot;https://projecteuclid.org/euclid.pjm/1102992107&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zarantonello (1967)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10550" LastEditDate="2019-10-11T13:19:30.703" LastActivityDate="2019-10-11T13:19:30.703" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11057" PostTypeId="1" CreationDate="2019-10-09T22:35:24.240" Score="25" ViewCount="2971" Body="&lt;p&gt;Who discovered this? It is quite nontrivial and very important in quantum mechanics.  &lt;/p&gt;&#xA;" OwnerUserId="3428" LastEditorUserId="55" LastEditDate="2019-10-09T23:46:33.227" LastActivityDate="2020-02-24T04:09:02.550" Title="Who discovered the covering homomorphism between SU(2) and SO(3)?" Tags="|mathematics|physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11058" PostTypeId="2" ParentId="11057" CreationDate="2019-10-09T23:45:53.250" Score="23" Body="&lt;p&gt;Hamilton  and Klein, Klein was more explicit about it. &lt;a href=&quot;https://archive.org/details/lecturesonquater00hami/page/n12&quot; rel=&quot;noreferrer&quot;&gt;Hamilton in Lectures on Quaternions (1853)&lt;/a&gt; realized that his representation of rotations of rigid bodies by the unit quaternions was not &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. &lt;a href=&quot;https://archive.org/details/cu31924059413439/page/n10&quot; rel=&quot;noreferrer&quot;&gt;Klein in Lectures on the Ikosahedron and the Solution of Equations&#xA;of the Fifth Degree (1888)&lt;/a&gt; replaced the unit quaternions by &lt;span class=&quot;math-container&quot;&gt;$2 × 2$&lt;/span&gt; unitary matrices with the determinant &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, now denoted &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt;. He then more or less spelled out that the unit quaternions and &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt; are isomorphic groups, which are &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; epimorphic onto the group of 3D rotations &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pauli proposed the &quot;&lt;em&gt;two-valuedness not describable classically&lt;/em&gt;&quot;, which was later identified with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Spin_(physics)#History&quot; rel=&quot;noreferrer&quot;&gt;electron spin&lt;/a&gt;, in 1924, and formalized it in the matrix form in 1927. In 1932 Heisenberg and Ivanenko guessed that the same effect regulates protons/neutrons as the states of a single particle, later dubbed nucleon, and incorporated it into their &lt;a href=&quot;https://en.wikipedia.org/wiki/Discovery_of_the_neutron#Proton%E2%80%93neutron_model_of_the_nucleus&quot; rel=&quot;noreferrer&quot;&gt;proton–neutron model of the nucleus&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://academic.oup.com/philmat/article-abstract/17/1/1/1433907&quot; rel=&quot;noreferrer&quot;&gt;Steiner&lt;/a&gt; cites this homomorphism as a prime example of &quot;unreasonable effectiveness&quot; of mathematics. Both times the mathematical machinery developed was not aimed, even indirectly, at the application it ended up being useful for. In the case of nucleus, any visible connection to rotations and 3D space is missing altogether.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-10T06:30:02.400" LastActivityDate="2019-10-10T06:30:02.400" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="11059" PostTypeId="1" CreationDate="2019-10-10T12:42:25.640" Score="0" ViewCount="89" Body="&lt;p&gt;The Encyclopaedia Britannica in its history of Science article states that Newton integrals were made of infinitesimals, whereas Leibniz' were made of sticks and that the former's theory prevailed.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But very often I read, even in some question here, that both theories are concerned with infinitesimals, and yet my trust in EB suggests they cannot be wrong.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know that Leibniz was not clear in is published works, but can anyone discuss the issue in one sense or another?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2019-10-10T12:42:25.640" Title="Were fluxions the only infinitesimals?" Tags="|calculus|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11060" PostTypeId="2" ParentId="11057" CreationDate="2019-10-10T22:17:08.597" Score="18" Body="&lt;p&gt;Before Hamilton (1847) one should cite Euler (1771), Gauss (1819), Rodrigues (1840), and Cayley (1845). Detailed references in e.g.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Pujol, J.&lt;/em&gt;, &lt;a href=&quot;http://projecteuclid.org/euclid.cma/1349803591&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Hamilton, Rodrigues, Gauss, quaternions, and rotations: a historical reassessment&lt;/strong&gt;&lt;/a&gt;, Commun. Math. Anal. 13, No. 2, 1-14 (2012). &lt;a href=&quot;https://zbmath.org/?q=an:1268.01010&quot; rel=&quot;noreferrer&quot;&gt;ZBL1268.01010&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, to four numbers &lt;span class=&quot;math-container&quot;&gt;$p,q,r,s$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$pp+qq+rr+ss=u$&lt;/span&gt;, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6956m/f341.image&quot; rel=&quot;noreferrer&quot;&gt;Euler&lt;/a&gt; (1771, §33) attached&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pqrkb.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pqrkb.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which is precisely (the transpose of) the rotation attached to &lt;span class=&quot;math-container&quot;&gt;$(a,b,c,d)=\dfrac{(p,q,r,s)}{\sqrt u}$&lt;/span&gt; in &lt;a href=&quot;https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Conversion_to_and_from_the_matrix_representation&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;R = \begin{pmatrix}&#xA;a^2+b^2-c^2-d^2&amp;amp;2bc-2ad        &amp;amp;2bd+2ac        \\&#xA;2bc+2ad        &amp;amp;a^2-b^2+c^2-d^2&amp;amp;2cd-2ab        \\&#xA;2bd-2ac        &amp;amp;2cd+2ab        &amp;amp;a^2-b^2-c^2+d^2\\&#xA;\end{pmatrix}.&#xA;\tag1&#xA;$$&lt;/span&gt;&#xA;So he had the &lt;em&gt;map&lt;/em&gt;, though maybe not the group law on 4-tuples that makes it a &lt;em&gt;homomorphism&lt;/em&gt;:&lt;sup&gt;1&lt;/sup&gt; that (or less anachronistically, a &quot;formula for the parameters of a composite rotation&quot;) is commonly attributed to &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k163849/f416.image&quot; rel=&quot;noreferrer&quot;&gt;Rodrigues&lt;/a&gt; (1840, p. 408), who put everything in the notation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(a,b,c,d)=\left(\cos\tfrac\theta2,\ \sin\tfrac\theta2\cos g,\ \sin\tfrac\theta2\cos h,\ \sin\tfrac\theta2\cos l\right).&#xA;\tag2&#xA;$$&lt;/span&gt;&#xA;Then &lt;a href=&quot;https://archive.org/details/collmathpapers01caylrich/page/n144&quot; rel=&quot;noreferrer&quot;&gt;Cayley&lt;/a&gt; (1845, pp. 123-124) identified Rodrigues' multiplication of 4-tuples &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; as quaternion multiplication, and the map &lt;span class=&quot;math-container&quot;&gt;$(a,b,c,d)\mapsto R$&lt;/span&gt; as what we would call the adjoint representation of &lt;span class=&quot;math-container&quot;&gt;$\mathit{Sp}(1)$&lt;/span&gt;; and &lt;a href=&quot;https://archive.org/details/proceedingsofroy03proc/page/n30&quot; rel=&quot;noreferrer&quot;&gt;Hamilton&lt;/a&gt; (1847, pp. 13-14) concurred — as did also &lt;a href=&quot;https://archive.org/details/londonedinburghp33lond/page/279&quot; rel=&quot;noreferrer&quot;&gt;Boole&lt;/a&gt; (1848) and &lt;a href=&quot;https://archive.org/details/londonedinburghp01maga/page/189&quot; rel=&quot;noreferrer&quot;&gt;Donkin&lt;/a&gt; (1851).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One may wonder why Euler wasn't cited at the time. As far as I can tell, it's because &lt;a href=&quot;https://archive.org/details/TO00188449_1/page/99&quot; rel=&quot;noreferrer&quot;&gt;Monge&lt;/a&gt; (1786) had (ostensibly independently) published &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; in the alternative notation &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(a,b,c,d)=\left(\sqrt{\tfrac{M\vphantom Q}4},\sqrt{\tfrac{N\vphantom Q}4},\sqrt{\tfrac{P\vphantom Q}4},\sqrt{\tfrac{Q}4}\,\right),&#xA;\tag3&#xA;$$&lt;/span&gt;&#xA;and for many years everyone,&lt;sup&gt;2&lt;/sup&gt; up to and including Rodrigues (p. 405), cited that instead. Only once Euler's paper was reprinted in a &lt;a href=&quot;https://archive.org/details/bub_gb_K2JDAAAAcAAJ/page/n528&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt; (1849, p. 440) did everyone&lt;sup&gt;3&lt;/sup&gt; switch to citing him. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far everything has been in terms of the sphere &lt;span class=&quot;math-container&quot;&gt;$S^3\subset\mathbb R^4$&lt;/span&gt;, or unit quaternions. If you want the homomorphism in terms of &lt;em&gt;literally&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;SU(2) =\left\{\begin{pmatrix}a+bi &amp;amp;-c+di\\c+di&amp;amp;\phantom{-}a-bi\end{pmatrix}: (a,b,c,d)\in S^3\right\},&#xA;\tag4&#xA;$$&lt;/span&gt;&#xA;the question becomes who first represented quaternions this way. The first memoirs on matrices by &lt;a href=&quot;https://archive.org/details/collmathpapers02caylrich/page/n508&quot; rel=&quot;noreferrer&quot;&gt;Cayley&lt;/a&gt; (1858), &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k433698b/f234.image&quot; rel=&quot;noreferrer&quot;&gt;Laguerre&lt;/a&gt; (1867), and &lt;a href=&quot;https://archive.org/details/journalfrdierei55crelgoog/page/n73&quot; rel=&quot;noreferrer&quot;&gt;Frobenius&lt;/a&gt; (1877) all mention the &lt;em&gt;possibility&lt;/em&gt;, but apparently left it to be done explicitly by &lt;a href=&quot;https://archive.org/details/circulars02univgoog/page/n192&quot; rel=&quot;noreferrer&quot;&gt;Peirce&lt;/a&gt; (1882) and &lt;a href=&quot;https://archive.org/details/collectedmathema03jjsy/page/n658&quot; rel=&quot;noreferrer&quot;&gt;four&lt;/a&gt; &lt;a href=&quot;https://archive.org/details/collectedmathema03jjsy/page/n662&quot; rel=&quot;noreferrer&quot;&gt;papers&lt;/a&gt; &lt;a href=&quot;https://archive.org/details/collectedmathem04sylvrich/page/n161&quot; rel=&quot;noreferrer&quot;&gt;by&lt;/a&gt; &lt;a href=&quot;https://archive.org/details/collectedmathem04sylvrich/page/n168&quot; rel=&quot;noreferrer&quot;&gt;Sylvester&lt;/a&gt; (1882-83). On the other hand, one might argue that &lt;a href=&quot;https://archive.org/details/journalfrdierei98crelgoog/page/n312&quot; rel=&quot;noreferrer&quot;&gt;Hermite&lt;/a&gt; (1850, footnote) had it &quot;before matrices&quot;, or ask who first identified rotations &lt;em&gt;of the (Riemann) sphere&lt;/em&gt; as homographies &lt;span class=&quot;math-container&quot;&gt;$\smash{z\mapsto\frac{Az+B}{Cz+D}}$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\smash{\left(\begin{smallmatrix}A&amp;amp;B\\C&amp;amp;D\end{smallmatrix}\right)}$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$(4)$&lt;/span&gt;: for this, &lt;a href=&quot;https://archive.org/details/vorlesungenber00kleiuoft/page/34&quot; rel=&quot;noreferrer&quot;&gt;Klein&lt;/a&gt; clearly credits &lt;a href=&quot;https://archive.org/details/collmathpapers10caylrich/page/n170&quot; rel=&quot;noreferrer&quot;&gt;Cayley&lt;/a&gt; (1879).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, as is often the case, it later emerged that unpublished papers of &lt;a href=&quot;https://archive.org/details/werkearithmetiku08cfga/page/n360&quot; rel=&quot;noreferrer&quot;&gt;Gauss&lt;/a&gt; (c.1819) already had both quaternion multiplication (p. 359) and rotations as homographies (p. 355).&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1. Opinions differ: e.g. Cartan-Study (link below) say that Euler had the composition formula. Maybe they consider he'd have found it obvious, or thought of the &lt;em&gt;bijection&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\smash{\mathbb{RP}^3}\to SO(3)$&lt;/span&gt; rather than the covering &lt;span class=&quot;math-container&quot;&gt;$\smash{S^3}\to SO(3)$&lt;/span&gt;, making the question moot; or they understand, better than I do, why his next § displayed a &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6956m/f343.image&quot; rel=&quot;noreferrer&quot;&gt;4 x 4 array&lt;/a&gt; which is very nearly a quaternion product: it is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(\begin{array}{rr|rr}&#xA;p&amp;amp;-q&amp;amp;-r&amp;amp;-s\\&#xA;q&amp;amp;p&amp;amp;s&amp;amp;-r\\&#xA;\hline&#xA;r&amp;amp;-s&amp;amp;p&amp;amp;q\\&#xA;s&amp;amp;r&amp;amp;-q&amp;amp;p&#xA;\end{array}\right)&#xA;\begin{pmatrix}&#xA;1&amp;amp;\\&#xA;&amp;amp;-1\\&#xA;&amp;amp;&amp;amp;-1\\&#xA;&amp;amp;&amp;amp;&amp;amp;-1&#xA;\end{pmatrix}&#xA;\left(\begin{array}{rr|rr}&#xA;a&amp;amp;-b&amp;amp;-c&amp;amp;-d\\&#xA;b&amp;amp;a&amp;amp;d&amp;amp;-c\\&#xA;\hline&#xA;c&amp;amp;-d&amp;amp;a&amp;amp;b\\&#xA;d&amp;amp;c&amp;amp;-b&amp;amp;a&#xA;\end{array}\right).&#xA;\tag5&#xA;$$&lt;/span&gt;&#xA;(The first column of this array is right out of his &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_four-square_identity&quot; rel=&quot;noreferrer&quot;&gt;famous&lt;/a&gt; &lt;a href=&quot;https://archive.org/details/correspondancem01goldgoog/page/n608&quot; rel=&quot;noreferrer&quot;&gt;letter to Goldbach&lt;/a&gt; (1748), and is also in his papers &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6952c/f409.image&quot; rel=&quot;noreferrer&quot;&gt;E242&lt;/a&gt; (1760) and &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6953p/f266.image&quot; rel=&quot;noreferrer&quot;&gt;E445&lt;/a&gt; (1773), as well as in &lt;a href=&quot;https://archive.org/details/uvresdelagrange03lagr/page/201&quot; rel=&quot;noreferrer&quot;&gt;Lagrange&lt;/a&gt; (1772), &lt;a href=&quot;https://archive.org/details/essaitheorienomb00legerich/page/n229&quot; rel=&quot;noreferrer&quot;&gt;Legendre&lt;/a&gt; (1797) and the English translation of his &lt;a href=&quot;https://www.e-rara.ch/zut/content/pageview/16109634&quot; rel=&quot;noreferrer&quot;&gt;Algebra&lt;/a&gt; (1810).)&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;2. E.g. &lt;a href=&quot;https://archive.org/details/histoiredelacad84hist/page/114&quot; rel=&quot;noreferrer&quot;&gt;Monge&lt;/a&gt; (1787), &lt;a href=&quot;https://archive.org/details/traitducalculdi02lacrgoog/page/n496&quot; rel=&quot;noreferrer&quot;&gt;Lacroix&lt;/a&gt; (1797), &lt;a href=&quot;https://archive.org/details/bub_gb_yJBY1_PahvUC/page/n275&quot; rel=&quot;noreferrer&quot;&gt;Hachette&lt;/a&gt; (1813), &lt;a href=&quot;https://books.google.com/books?id=e-pEAAAAcAAJ&amp;amp;pg=PA308#v=twopage&quot; rel=&quot;noreferrer&quot;&gt;Encke&lt;/a&gt; (1830), &lt;a href=&quot;https://archive.org/details/journalfrdierei12crelgoog/page/n172&quot; rel=&quot;noreferrer&quot;&gt;Grunert&lt;/a&gt; (1832), &lt;a href=&quot;https://archive.org/details/bub_gb_c6pEAAAAcAAJ/page/n491&quot; rel=&quot;noreferrer&quot;&gt;Grunert&lt;/a&gt; (1833), &lt;a href=&quot;https://archive.org/details/collmathpapers04caylrich/page/n578&quot; rel=&quot;noreferrer&quot;&gt;Cayley&lt;/a&gt; (1862).&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;3. E.g. &lt;a href=&quot;https://archive.org/details/lecturesonquater1853hami/page/539&quot; rel=&quot;noreferrer&quot;&gt;Hamilton&lt;/a&gt; (1853), &lt;a href=&quot;https://archive.org/details/collmathpapers02caylrich/page/n231&quot; rel=&quot;noreferrer&quot;&gt;Cayley&lt;/a&gt; (1855), &lt;a href=&quot;http://www.numdam.org/item/NAM_1856_1_15__403_0/&quot; rel=&quot;noreferrer&quot;&gt;Lebesgue&lt;/a&gt; (1856), &lt;a href=&quot;https://archive.org/details/higheralgebra00salmrich/page/36&quot; rel=&quot;noreferrer&quot;&gt;Salmon&lt;/a&gt; (1866), &lt;a href=&quot;https://archive.org/details/bub_gb_bU9rkSdWlFAC/page/n209&quot; rel=&quot;noreferrer&quot;&gt;Hankel&lt;/a&gt; (1867), &lt;a href=&quot;https://archive.org/details/thorielmentaire01hogoog/page/n246&quot; rel=&quot;noreferrer&quot;&gt;Hoüel&lt;/a&gt; (1874), &lt;a href=&quot;https://archive.org/details/gesammeltewerke03cgjj/page/n609&quot; rel=&quot;noreferrer&quot;&gt;Jacobi&lt;/a&gt; (1884), &lt;a href=&quot;https://archive.org/details/leconsurlagendes01darbrich/page/n45&quot; rel=&quot;noreferrer&quot;&gt;Darboux&lt;/a&gt; (1887), &lt;a href=&quot;https://books.google.com/books?id=PCFMAQAAIAAJ&amp;amp;pg=PA350#v=twopage&quot; rel=&quot;noreferrer&quot;&gt;Study&lt;/a&gt; (1890), &lt;a href=&quot;https://books.google.com/books?id=LN84AAAAIAAJ&amp;amp;pg=PA54&amp;amp;dq=Euler&quot; rel=&quot;noreferrer&quot;&gt;Beez&lt;/a&gt; (1896), &lt;a href=&quot;https://archive.org/details/cinematiqueprof00keonrich/page/n363&quot; rel=&quot;noreferrer&quot;&gt;Koenigs&lt;/a&gt; (1897), &lt;a href=&quot;https://archive.org/details/encyklomath104encyrich/page/n227&quot; rel=&quot;noreferrer&quot;&gt;Schoenflies&lt;/a&gt; (1902), &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k2440f/f236.image&quot; rel=&quot;noreferrer&quot;&gt;Cartan-Study&lt;/a&gt; (1908), &lt;a href=&quot;https://archive.org/details/encyklomath1103encyrich/page/n794&quot; rel=&quot;noreferrer&quot;&gt;Müller&lt;/a&gt; (1910), &lt;a href=&quot;https://archive.org/details/theoryofdetermin01muiruoft/page/305&quot; rel=&quot;noreferrer&quot;&gt;Muir&lt;/a&gt; (1911), &lt;a href=&quot;https://archive.org/details/treatisanalytdyn00whitrich/page/n25&quot; rel=&quot;noreferrer&quot;&gt;Whittaker&lt;/a&gt; (1917), &lt;a href=&quot;https://books.google.com/books?id=csgswR8TlQ8C&amp;amp;pg=PA164&quot; rel=&quot;noreferrer&quot;&gt;Bourbaki&lt;/a&gt;.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="10561" LastEditorUserId="10561" LastEditDate="2020-02-24T04:09:02.550" LastActivityDate="2020-02-24T04:09:02.550" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11061" PostTypeId="1" CreationDate="2019-10-11T08:07:44.907" Score="4" ViewCount="145" Body="&lt;p&gt;I'm looking for more details of a 14th century mathematician Ala-El-Din Muhammad El Ferjumedhi.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm translating a popular book on the history of mathematics, and there is one figure that shows a mathematical manuscript in Arabic. I could find it on &lt;a href=&quot;https://www.gettyimages.de/detail/nachrichtenfoto/treatise-on-the-question-of-arithmetic-code-by-the-nachrichtenfoto/142086185&quot; rel=&quot;nofollow noreferrer&quot;&gt;Getty Images&lt;/a&gt;, where it carries the description &quot;SAUDI ARABIA - CIRCA 2002: Treatise on the Question of Arithmetic Code by The Master Ala-El-Din Muhammed El Ferjumedhi, Saudi Arabia, 14th Century.&quot;, but I cannot find this mathematician anywhere, even when I search for different possible spellings of his name. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IYj7a.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IYj7a.jpg&quot; alt=&quot;Treatise on the Question of Arithmetic Code&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any more information about this is greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="9563" LastEditorUserId="72" LastEditDate="2019-12-29T11:04:10.597" LastActivityDate="2019-12-29T11:04:10.597" Title="Ala-El-Din Muhammed El Ferjumedhi" Tags="|algebra|islamic-science|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11064" PostTypeId="1" CreationDate="2019-10-13T13:21:15.600" Score="-1" ViewCount="87" Body="&lt;p&gt;Can you briefly sketch the sequence of math theories that were necessary for Einstein to figure out a convincing background for relativity?  &lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="7300" LastEditDate="2019-10-14T08:34:09.173" LastActivityDate="2019-10-14T08:34:09.173" Title="What was the chain of theories that led to relativity?" Tags="|mathematics|geometry|relativity-theory|" AnswerCount="1" CommentCount="5" ClosedDate="2019-10-14T19:09:08.723" ContentLicense="CC BY-SA 4.0" />
  <row Id="11065" PostTypeId="2" ParentId="11064" CreationDate="2019-10-13T14:17:46.630" Score="1" Body="&lt;p&gt;There are two theories which are called &quot;relativity&quot;.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Special relativity. It required no advanced mathematics at all. Minkovski space, as a proper mathematical background was proposed after the theory itself.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;General relativity. Mathematical background is Riemannian geometry. Riemannian geometry was proposed by Riemann, as a very general outline in his lecture in 1854,&#xA;and the technical development is due to Riemann himself, G. Ricci-Cubastro and his student  T. Levi-Civita.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;All this has nothing to do with Cantor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT. Cantor first paper on set theory was published in 1874. But his ideas were not immediately accepted: the ideas of set theory begin to penetrate the mainstream mathematics only in 20th century. The &quot;revolution&quot; happened much later.  Riemann died in 1866. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-10-13T17:11:57.543" LastActivityDate="2019-10-13T17:11:57.543" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11066" PostTypeId="1" AcceptedAnswerId="11077" CreationDate="2019-10-14T13:24:47.553" Score="4" ViewCount="279" Body="&lt;p&gt;As a layperson, when one learns about Quantum Mechanics, one also learns about the various interpretations of QM-- Many Worlds, Collapse, Pilot Wave theory, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, as far as I can tell (from google searches), other modern domains of science-- condensed matter physics, fluid mechanics, high energy physics-- don't have interpretations, in the same way that there are for QM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this true? Why does QM garner interpretations of its theory, when other branches don't?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt; I feel my initial wording was perhaps too parsimonious. I want to flesh out more what I am asking. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Throughout the history of science, various hypotheses and theories are proposed, and experiments are run, specifically designed to disprove a theory. Over time, generally, one theory wins out over the others, and comes to be the commonly accepted theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A scientific theory has certain properties, such as that it's falsifiable-- that there is, in theory, some experiment that can be designed such that, if a certain result was measured, it would show that the theory was not true. That a theory is not disproved by such experiments is taken as evidence of its accuracy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, I've noticed something peculiar about Quantum Mechanics. In addition to the theory, which has passed many different experiments over the decades, and is generally well-accepted, QM also has &quot;interpretations&quot; or &quot;explanations&quot;. Some of these, such as Pilot Wave theory, are falsifiable-- if no Pilot Wave is measured in an experiment, then the theory can be taken to be false, or, not an explanation of the phenomenon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, there is an interpretation called Many Worlds, which makes no testable predictions, and no observations can be made of the proposed Worlds. But, this shouldn't be a problem, because it's not a theory, but rather an &quot;interpretation&quot; or &quot;explanation&quot;. While the concept of a scientific theory is more strict, in that there are some criteria that a scientific must meet, I have found no similar definition of an interpretation or explanation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Furthermore, after looking, I haven't even found interpretations &lt;em&gt;at all&lt;/em&gt; of other scientific theories! It seems to me that QM is unique in that respect. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wonder, is it true that only QM has garnered interpretations? Why might this be so, when no other theory has interpretations?&lt;/p&gt;&#xA;" OwnerUserId="2708" LastEditorUserId="2708" LastEditDate="2019-10-18T15:46:02.330" LastActivityDate="2019-10-20T22:26:00.327" Title="Do other branches of modern science have interpretations like Quantum Mechanics does?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11067" PostTypeId="1" AcceptedAnswerId="11070" CreationDate="2019-10-14T19:13:09.423" Score="2" ViewCount="664" Body="&lt;p&gt;When did people start using Revolutions per Minute (RPM) to measure motors, engines, other devices and where did the term originate? Why do we continue to use it instead of an SI unit like Hz? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From car engines to disk drives and fans we measure RPM in the thousands instead of HZ, e.g. 3000 rpm or 50 Hz. I realize its a bit of a convention now but it seems odd.&lt;/p&gt;&#xA;" OwnerUserId="10581" LastEditorUserId="55" LastEditDate="2019-10-15T11:24:13.147" LastActivityDate="2019-10-20T22:29:35.413" Title="Why are revolutions per minute (RPM) still used instead of hertz (Hz)?" Tags="|terminology|units|social-context|measurement|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11069" PostTypeId="1" AcceptedAnswerId="11071" CreationDate="2019-10-15T06:59:32.217" Score="0" ViewCount="1550" Body="&lt;p&gt;I have stumbled upon the following alleged Newton's quotes, but I could not find them in any of their works.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;No great discovery was ever made without a bold guess.&lt;/p&gt;&#xA;&lt;p&gt;No sciences are better attested than the religion of the Bible.&lt;/p&gt;&#xA;&lt;p&gt;What we know is a drop, what we don't know is an ocean.&lt;/p&gt;&#xA;&lt;p&gt;I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Where exactly do they come from? Are they apocryphal? Is there anywhere a comprehensive compilation of alleged Newton quotes indicating their authenticity and source?&lt;/p&gt;&#xA;" OwnerUserId="10442" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-10-15T11:00:37.747" Title="Are these Newton's quotes apocryphal?" Tags="|physics|newton|quote|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11070" PostTypeId="2" ParentId="11067" CreationDate="2019-10-15T10:46:45.603" Score="6" Body="&lt;p&gt;We often forget that even the &lt;a href=&quot;https://en.wikipedia.org/wiki/Minute&quot; rel=&quot;nofollow noreferrer&quot;&gt;minute&lt;/a&gt; is not an SI unit, only the second and its decimal multiples and fractions are. It is a leftover of the sexagesimal system (base 60), whose use predates the decimals by many centuries, and goes back to the ancient Babylon. So are the angular degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Heinrich Hertz was born in 1857, and the International Electrotechnical Commission adopted his name as an SI unit in 1930. But back in 1827, for example, we find revolutions per minute (RPM) throughout &lt;a href=&quot;https://archive.org/details/treatiseonsteame01fareuoft/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Farey's Treatise on the Steam Engine&lt;/a&gt;. On p. 620, for example, it shares the page with such still current non-SI units as feet, pounds and horse-powers:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A cast iron neck of 1 inch diameter, and 1 1/5 inches long, is capable of sustaining a force of torsion, amounting to 88 lbs. acting at half a foot distance from the centre of motion, without any risk of injury, for it will require ten times that force, or 880 pounds to twist off the neck. If such a neck made one turn per minute, the mechanical power it would transmit, may be represented by a force of 88 lbs., acting uniformly through the circumference of a circle one foot diameter, or a space of 3.1416 feet per minute; which is equal to 276.46 lbs., acting through a space of one foot per minute. This, divided by 33 000 lbs. for a horse-power, gives .00838 of a horse-power, which may be transmitted by a cast iron neck one inch diameter, and 1 1/5 inc. long, when making one turn per minute. Or conversely, such a neck must make &#xA;  119.37 revolutions per minute, in order to transmit one horse-power.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is followed by detailed instructions on how &quot;&lt;em&gt;to find the dimensions for the cast iron neck of an axis, which is to transmit any given number of horse-power, when it makes a given number of revolutions per minute&lt;/em&gt;&quot;, with a reference to millwrights and Watt's steam engines. Indeed, revolutions per minute are at least as old as horse-powers, see &lt;a href=&quot;http://essexsteamtrain.com/news_letter/the-origin-of-horsepower/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origin of “Horsepower”&lt;/a&gt;. In 1760s, when Watt wanted to impress upon people how powerful his newly minted steam engines were, he calculated how many horses just one of them could replace. It was PLAN/33000, where P is the mean effective pressure in the cylinder, L is the length of stroke in feet, A is the area of the piston in inches, and N is... yes, the number of revolutions per minute. The 33,000, that we meet again, is just 1 horse-power, in foot-pounds per minute. In the SI, it is 745.7 Watts, and the former has the same advantage over the latter as RPM over Hz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So why are degrees, minutes, feet, pounds, horse-powers, etc., still around? People grow up with them, they make their way into various designs and official documents, accumulate a tail of working intuitions and connotations in practical contexts, textbooks and manuals are written around them. Abstract arguments about advantages of unified measurement units, even supplemented by government decrees, are as effective in washing them out of general use as the efforts to prescribe the use of words through dictionaries generally, i.e. not very. And RPM have an extra advantage of intuitive clarity. It is somewhat different with the use of terms and units by the scientific community, which is much more receptive and disciplined, but even scientists are people too.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-20T22:29:35.413" LastActivityDate="2019-10-20T22:29:35.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11071" PostTypeId="2" ParentId="11069" CreationDate="2019-10-15T11:00:37.747" Score="4" Body="&lt;p&gt;The fourth quote is found in the posthumous work &quot;&lt;a href=&quot;https://books.google.com.mt/books?id=JU0JAAAAQAAJ&amp;amp;printsec=frontcover&amp;amp;dq=joseph+spence+anecdotes&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiVl8mpjJ7lAhXFy6QKHXzkALkQ6AEIJzAA#v=onepage&amp;amp;q=joseph%20spence%20anecdotes&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anecdotes, Observations and Characters, of Books and Men&lt;/a&gt;&quot; Vol 1 (page 158), by the historian &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Spence_(author)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Spence&lt;/a&gt; (1820). They are reported to have been uttered by Newton just before his death (1727) to Chevalier Andrew Michael Ramsey (though the latter was recorded to be in France at the time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The second probably refers to this quote: &quot;I find more sure marks of the authenticity of the Bible than in any profane history whatever.&quot; It was published by &lt;a href=&quot;https://en.wikipedia.org/wiki/Richard_Watson_(bishop_of_Llandaff)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bishop Richard Watson&lt;/a&gt; (and professor in Cambridge) in &quot;&lt;a href=&quot;https://books.google.com.mt/books?id=3MAvt_UW4XwC&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Apology for Christianity&lt;/a&gt;&quot; (1776, page 91), as related to him by his former teacher Dr Robert Smith, Master of Trinity College, who heard it directly from Newton. The book was meant to be a response to Gibbon's book which laid the blame for the Roman empire's decline to Christianity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The third quote seems to be a paraphrase of the fourth. The first is probably apocryphal.&lt;/p&gt;&#xA;" OwnerUserId="9714" LastActivityDate="2019-10-15T11:00:37.747" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11072" PostTypeId="2" ParentId="3610" CreationDate="2019-10-16T17:12:48.940" Score="6" Body="&lt;p&gt;This is a supplement to the previous answer to the question, confirming what has been written above.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Illustreret Nyhedsblad&lt;/em&gt;, where prof. Chr. Hansteen in 1862 first published some letters from Abel, is available online from the &lt;a href=&quot;https://www.nb.no/&quot; rel=&quot;noreferrer&quot;&gt;National Library of Norway&lt;/a&gt;. The relevant letter is the one Abel wrote on December 5th, 1825, which is quoted above, and which can be found in issue No. 9, March 2nd, 1862. At the end of the comment on Gauss' obscureness, Hansteen has inserted a footnote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;En Tydsk Student sagde om ham i den Anledning: &quot;er macht es wie der&#xA;  Fuchs, der wischt mit dem Schwanze seine Spuren im Sande aus&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the quote is here attributed to an unnamed German student. When C. A. Bjerknes wrote his biography on Abel, he put together the material on Gauss from Abel's letter with the footnote from Hansteen, and in the French edition the reference to Hansteen was lost. The French text has then been retranslated to German, giving a slightly different wording from the original.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hansteen had an extensive scientific correspondence, mostly on the subject of magnetism, and he had visited Gauss in 1839 to discuss his theories on the subject. It is not surprising that he was well acquainted with anecdotes about Gauss.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2019-10-16T17:12:48.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11073" PostTypeId="1" CreationDate="2019-10-18T09:10:40.777" Score="4" ViewCount="199" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lorentz_force#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;We know&lt;/a&gt; that the magnetic force on a particle moving in a magnetic field was found by J.J. Thomson in 1881, with a slight mistake, and then corrected by Heaviside in 1885 to &lt;span class=&quot;math-container&quot;&gt;$F_M = q\,v\times B$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can you tell me when was discovered that its ratio to the electric force was &lt;span class=&quot;math-container&quot;&gt;$v^2/c^2$&lt;/span&gt;, and how was it determined? I suppose it could have been done only in a cyclotron.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, lastly, do you know when and who decided that &lt;span class=&quot;math-container&quot;&gt;$F_M$&lt;/span&gt; was just &lt;span class=&quot;math-container&quot;&gt;$F_E$&lt;/span&gt; seen in a different system exploiting the resemblance of the ratio with a Lorentz transformation, unifying the two forces ?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="6085" LastEditDate="2019-10-19T14:17:34.930" LastActivityDate="2019-10-19T14:17:34.930" Title="When was the ratio between electric and magnetic forces in an electromagnetic field established?" Tags="|discoveries|electromagnetism|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11074" PostTypeId="1" CreationDate="2019-10-18T11:54:49.410" Score="1" ViewCount="498" Body="&lt;p&gt;I have been told by a computer scientist that most people in that field believe that algebra was invented (along with algorithms) by the 9th century mathematician Muhammad ibn Musa al-Khwarizmi. (I have no idea what most mathematicians believe about this, if anything.)  Yet, the Indian mathematician Brahmagupta gave a clear statement of the quadratic formula over a century before al-Khwarizmi was active. Also, Euclid presents algebra in geometric form in his Elements, and the Babylonians knew how to solve some quadratic equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to the Wikipedia article on Algebra, some have argued that Diophantus is the &quot;father of algebra&quot;.  (Is that the same as &quot;inventing&quot; it?)   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you google &quot;inventor of algebra&quot;, you are told in no uncertain terms that it was al-Khwarizmi. Undoubtedly his book played a crucial role in organizing algebra into a subject (as did Euclid's Elements in geometry), and in bringing these ideas to Europe.  But it seems to me that this is far from &quot;inventing&quot; algebra, especially given the previous work of Brahmagupta and others. It is apparently clear that al-Khwarizmi knew of some of Brahmagupta's work, which seems to have reached Baghdad before he was born, but it is not clear if he knew of Brahmagupta's algebra.   &lt;/p&gt;&#xA;" OwnerUserId="10595" LastActivityDate="2019-10-18T13:26:35.650" Title="Can any one person be credited with &quot;inventing&quot; algebra?" Tags="|algebra|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11075" PostTypeId="2" ParentId="11074" CreationDate="2019-10-18T13:26:35.650" Score="1" Body="&lt;p&gt;There most certainly were multiple 'inventors' -- or 'discoverers,' depending on your phiilosophy, of basic algebra.  Even today, there are youngsters who develop algebra independently.&lt;br&gt;&#xA;As to who should get credit for bringing a given chunk of math to the world,  that's not really an answerable question.  The growth of knowledge is nonmonotonic; civilizations grow and die, and much of their knowledge can be lost until the next civilization comes along. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Further, take a look at Edison and Jobs (or Gates if you prefer). They are credited with major technological revolutions but in fact they were in the right place at the right time and managed to beat out rivals who may or may not have preceded them in a given discovery or invention. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-10-18T13:26:35.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11076" PostTypeId="1" CreationDate="2019-10-19T14:57:00.753" Score="2" ViewCount="265" Body="&lt;p&gt;Wikipedia claims that Euler was the first do derive the wave equation in 3d. In which of his writings can I find this?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2019-10-19T14:57:00.753" Title="Where did Euler derive the wave equation in 3d?" Tags="|mathematical-physics|euler|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11077" PostTypeId="2" ParentId="11066" CreationDate="2019-10-20T01:33:00.870" Score="4" Body="&lt;p&gt;Every theory requires an interpretation to be more than a linguistic or mathematical exercise. At the very minimum, it has to outline procedures connecting its constructs to something empirically observed, so-called operational or minimal interpretation. &lt;a href=&quot;https://www.math.columbia.edu/~woit/wordpress/?p=9649&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Shut up and calculate&quot;&lt;/a&gt; is a parody of such interpretation, as it also needs to provide instructions on matching what is calculated to what is measured. Incorporating prevailing intuitions about reasoning in and applying a theory usually requires more than that. This can get complicated considering that most physical theories have entities that are not observed directly, and make general claims that can only be tested in roundabout ways. They have to rely on some pragmatic extra assumptions about observational and experimental setups (such as lack of unknown confounding factors or of &lt;a href=&quot;https://en.wikipedia.org/wiki/Superdeterminism&quot; rel=&quot;nofollow noreferrer&quot;&gt;superdeterminism&lt;/a&gt; that forces experimenters to measure what they &quot;choose&quot; to measure, for example).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, determinism was traditionally a big part of how classical mechanics was interpreted in the 19th century (and even today), but it is not forced by its mathematical formalism alone. There was a lively discussion of determinism back then, with some prominent figures (like Maxwell) backing alternative, indeterministic, interpretations, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2680/55&quot;&gt;History of the study of indeterminism in classical mechanics&lt;/a&gt;. Modern discussions of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Norton%27s_dome&quot; rel=&quot;nofollow noreferrer&quot;&gt;Norton dome&lt;/a&gt; are a distant echo of those debates. But the incentive for pondering interpretations of classical mechanics is much reduced now. We already know it is a non-fundamental theory that holds only approximately. Whatever interpretation is adopted, its answers to &quot;big questions&quot; about the &quot;nature of reality&quot; have diminished value. The &quot;burning&quot; questions from 18th century interpretations: are live or dead forces (energy or momenta) more fundamental, see &lt;a href=&quot;https://hsm.stackexchange.com/a/598/55&quot;&gt;What was the vis viva controversy&lt;/a&gt;; does nature follow &quot;mechanical&quot; causes or &quot;strives&quot; to minimize action, etc., seem almost comical today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Special relativity also had two competing interpretations in its hey day of the early 20th century. The idea of absolute space, and especially time, is quite ingrained in common perceptions, a new physical theory upsetting it met much resistance. The original version of relativistic mathematical formalism, developed by Lorentz and Poincare, came with an interpretation that did not require such a leap. It postulated an absolute frame of ether, but with dynamical effects (length contractions and time dilations) that forever veiled its presence from our eye. Eventually, most of the public, like Einstein originally, got over parting ways with ether. But not all. Those attuned to the subjective perception of time duration and dissatisfied with &quot;mechanization&quot; of time still invoke &lt;a href=&quot;https://en.wikipedia.org/wiki/Lorentz_ether_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorentz's theory of ether&lt;/a&gt; as an alternative interpretation of special relativity to reconcile their intuitions with physics, see &lt;a href=&quot;https://plato.stanford.edu/entries/presentism/#RelaPhys&quot; rel=&quot;nofollow noreferrer&quot;&gt;presentism&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But again, special relativity is no longer a fundamental theory, general relativity is. It challenges common intuitions still further, suggesting that spacetime itself, even relativistic, is merely a construct. Alternatives with a fixed background, in the spirit of Lorentz's interpretation, popped up, see e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Anatoly_Logunov#Relativistic_theory_of_gravitation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Logunov's relativistic theory of gravitation&lt;/a&gt;. It is not quite an interpretation of GR, as it makes distinguishable in principle predictions, but it could be modified to be mathematically equivalent to GR, if one so wished. There are already both substantivalist (spacetime is real) and non-substantivalist (spacetime is just a bookeeping device) &lt;a href=&quot;https://plato.stanford.edu/entries/spacetime-holearg/#WhaRepSpaManSub&quot; rel=&quot;nofollow noreferrer&quot;&gt;interpretations proper of GR&lt;/a&gt;. On the strength of the much discussed hole argument, Einstein was leaning towards the latter. But the psychological importance of this is undermined by the almost universally accepted fact that GR will have to be replaced by quantum gravity, and it is from there that the &quot;final&quot; answers about spacetime should come.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And quantum gravity, whatever it turns out to be, will inherit the quantum part. Ether and absolute (space)time were one thing, suggesting that the reality itself is observer-dependent, that its &quot;elements&quot; only come to be when looked at, as in the original Copenhagen interpretation, was too much even for Einstein. Strictly speaking, quantum mechanics is not a fundamental theory anymore, it has been supplanted by quantum field theory. But the interpretational issues of QFT (indeterminism, observer dependence, nature of collapse, etc.) are quite similar to those of QM, and the latter are much easier to explain and illustrate to the general public. Hence the proliferation of its interpretations and focus on them in the popular culture.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-20T22:26:00.327" LastActivityDate="2019-10-20T22:26:00.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11079" PostTypeId="1" CreationDate="2019-10-21T16:23:07.933" Score="6" ViewCount="259" Body="&lt;p&gt;In the work of early physicists like Newton, everything is explained in terms of cumbersome (in today's standards) geometry. They don't talk about &quot;cosines&quot; of certain angle, but about proportions between the sides of triangles. But aren't sines and cosines known since antiquity? So why Newton, Copernicus, etc didn't make use of them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When and how did trigonometric functions become mainstream?&lt;/p&gt;&#xA;" OwnerUserId="10602" LastActivityDate="2019-10-21T17:16:31.723" Title="When did the use of Sine and Cosine as functions become mainstream?" Tags="|physics|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11080" PostTypeId="2" ParentId="11079" CreationDate="2019-10-21T17:16:31.723" Score="7" Body="&lt;p&gt;Trigonometric functions became &quot;mainstream&quot; since the publication by Ptolemy (II AD) of trigonometric tables. To be sure he did not use our modern sine and cosine,&#xA;but a single trigonometric function, the chord (&lt;span class=&quot;math-container&quot;&gt;$=2\sin(t/2)$&lt;/span&gt;).&#xA;Modern definitions of sine and cosine were introduced by Indian mathematicians (Surya Siddhanta (V century AD), and reached Europe through Middle Eastern &#xA;mathematicians. After the fall of Constantinople, this knowledge quickly spread in Europe.  The first treatises on trigonometry in Europe were written by Peuerbach and Johann Muller (Regiomontanus) in  1464. Copernicus of course used trigonometry extensively, as well as Kepler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Theory of proportions has nothing to do with this, by the way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may look to Wikipedia: &lt;a href=&quot;https://en.wikipedia.org/wiki/Trigonometry#History&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Trigonometry#History&lt;/a&gt; to begin with.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-10-21T17:16:31.723" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11081" PostTypeId="1" AcceptedAnswerId="11083" CreationDate="2019-10-21T20:01:05.820" Score="4" ViewCount="250" Body="&lt;p&gt;According to Peter Machamer's &quot;A Brief Historical Introduction to the Philosophy of Science&quot; (&lt;em&gt;The Blackwell Guide to the Philosophy of Science&lt;/em&gt;, p. 7)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... classical mechanics was not axiomatized until 1949 ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Machamer does not include a citation here, but nor does this fact seem to belong to &quot;general knowledge&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who first axiomatized classical mechanics (in 1949), and where can I read more about historical attempts to axiomatize classical mechanics?&lt;/p&gt;&#xA;" OwnerUserId="3811" LastActivityDate="2019-10-22T03:53:48.240" Title="Who axiomatized classical mechanics in 1949?" Tags="|reference-request|classical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11082" PostTypeId="1" CreationDate="2019-10-21T20:02:55.793" Score="9" ViewCount="527" Body="&lt;p&gt;I'd like to read something about the history of women in mathematics. I'd love to have reading suggestions of books in English or Italian, of 3 kinds:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1) History of math books, academic style;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) Popular mathematics books, for a broader audience;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;3) Books for young girls (8-11yo), possibly in Italian.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="10603" LastActivityDate="2022-02-20T00:57:03.440" Title="Women in mathematics" Tags="|biographical-details|book-recommendation|" AnswerCount="8" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11083" PostTypeId="2" ParentId="11081" CreationDate="2019-10-21T21:54:16.630" Score="6" Body="&lt;p&gt;Not sure who Machamer and Silberstein had in mind specifically, but, in any case, they are mistaken. Simon published &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786444708561148?journalCode=tphm18&quot; rel=&quot;nofollow noreferrer&quot;&gt;The axioms of Newtonian mechanics&lt;/a&gt; in 1947, Hermes &lt;em&gt;Eine Axiomatisierung der allgemeine Mechanik&lt;/em&gt; in 1938, and Hamel &lt;em&gt;Die Axiome der Mechanik&lt;/em&gt; in 1927. Von Neumann even &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#Postulates_of_quantum_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;axiomatized quantum mechanics&lt;/a&gt; back in 1932. In a &lt;a href=&quot;https://www.jstor.org/stable/185955?seq=1#metadata_info_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;follow-up note (1954)&lt;/a&gt; Simon mentions an alternative axiomatization of Kinsey-Sugar-Suppes from 1953, but nothing from 1949. Kinsey-Sugar-Suppes cite Hamel, Hermes and Simon, but the only citation from 1949 is Hamel's monograph &lt;a href=&quot;https://books.google.com/books/about/Theoretische_Mechanik.html?id=ie_OBgAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Theoretische Mechanik&lt;/em&gt;&lt;/a&gt;. In a later &lt;a href=&quot;https://www.jstor.org/stable/20114347&quot; rel=&quot;nofollow noreferrer&quot;&gt;survey paper&lt;/a&gt;, Suppes also mentions  Khinchin's Statistical Mechanics from 1949, but there is no axiomatization there. So perhaps Machamer and Silberstein refer to Hamel's book or misremembered 7 as 9.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See Tiemeyer's monograph &lt;a href=&quot;https://books.google.com/books?id=z3oYDQAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Axiome der Klassischen Mechanik&lt;/a&gt; for a more comprehensive outlook.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-22T03:53:48.240" LastActivityDate="2019-10-22T03:53:48.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11084" PostTypeId="2" ParentId="11082" CreationDate="2019-10-22T04:30:03.897" Score="2" Body="&lt;p&gt;One such book is &lt;em&gt;She Does Math!&lt;/em&gt;, by Marla Parker, published by the American Mathematical Society.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-10-22T04:30:03.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11085" PostTypeId="1" CreationDate="2019-10-22T09:55:40.327" Score="0" ViewCount="231" Body="&lt;p&gt;Down to the eighteenth century physics was called &lt;strong&gt;philosophia naturalis&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When were the terms Physics, Science and Scientist, introduced?  By whom?  When did they supplant the old ones?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="12724" LastEditDate="2021-03-16T07:53:38.933" LastActivityDate="2021-03-16T07:53:38.933" Title="When did the term 'scientist, physicist, science, physicist' come in use?" Tags="|terminology|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11086" PostTypeId="2" ParentId="11085" CreationDate="2019-10-22T13:55:51.040" Score="1" Body="&lt;p&gt;Some references :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;consider that Lord Kelvin published is well-known textbook in 1867 with the title &lt;a href=&quot;https://en.wikipedia.org/wiki/Treatise_on_Natural_Philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Treatise on Natural Philosophy&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;J.C.Maxwell begins his &lt;a href=&quot;https://books.google.it/books?id=MWoOAAAAIAAJ&amp;amp;pg=PA3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matter and Motion (1876)&lt;/a&gt; with &quot;Physical Science…&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In K.Pearson's &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Grammar_of_Science&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Grammar of Science (1892)&lt;/a&gt;, the terms : &lt;em&gt;physics&lt;/em&gt; and &lt;em&gt;science/scientist&lt;/em&gt;, are definitely there.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2019-10-22T13:55:51.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11087" PostTypeId="2" ParentId="11085" CreationDate="2019-10-24T16:01:37.727" Score="3" Body="&lt;p&gt;I looked in the &lt;a href=&quot;https://www.oed.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Oxford English Dictionary&lt;/em&gt;&lt;/a&gt; (subscription required) for English usages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;physics&lt;/em&gt; used in the medical sense, I do not report on.  &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;physics&lt;/em&gt; used for &quot;natural science&quot; in general:  The oldest instance they have is from 1487.  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;J. Skelton tr. Diodorus Siculus Bibliotheca Historica iii. 174   Among the Grekes, as it is many tymes seen, it is long or they can atteigne vnto thentellective of the &lt;strong&gt;phisikes&lt;/strong&gt; [L. philosophiam, Gk. ϕιλοσοϕίας] by-cause they long perseuere not therin, but geve theym vnto lucrative science.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He uses &quot;phisikes&quot; to translate Latin &quot;philosophiam&quot; and Greek &quot;ϕιλοσοϕίας&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;physics&lt;/em&gt; used in the more modern sense: The oldest instance they have is from 1715.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;tr. D. Gregory Elements Astron. I. Pref. ii   The Celestial &lt;strong&gt;Physics&lt;/strong&gt;, or Physical Astronomy [L. Astronomia Physica], is not only the first in dignity of all inquiries into Nature whatever.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He uses &quot;Celestial Physics&quot; to translate the Latin &quot;Astronomia Physica&quot;.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;science&lt;/em&gt;  Various obsolete, archaic, and rare usages I do not report on.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;science&lt;/em&gt; paired or contrasted with &lt;em&gt;art&lt;/em&gt;:  The oldest instance they have is from 1387.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;J. Trevisa tr. R. Higden Polychron. (St. John's Cambr.) (1879) VII. 69 (MED)   He..fliȝ into..Spayne, forto lerne curious and sotil artes and &lt;strong&gt;sciens&lt;/strong&gt; þere.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;science&lt;/em&gt; in a more modern sense: there is this definition from 1600&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;W. Vaughan Golden-groue i. lxv. sig. Mv   The name of &lt;em&gt;science&lt;/em&gt; is taken more strictly for a habit gotten by demonstration separated from wisedome; in which last signification Naturall philosophy, &amp;amp; the Mathematickes are called Sciences.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So: what had formerly been called &quot;Naturall philosophy&quot; and &quot;Mathematickes&quot; are now called &quot;Sciences&quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-10-24T16:01:37.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11088" PostTypeId="2" ParentId="11082" CreationDate="2019-10-24T21:45:48.630" Score="6" Body="&lt;p&gt;Margaret Murray's &quot;Women Becoming Mathematicians&quot; (MIT Press) is great.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, Prof. Murray has a website with even more info:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://womenbecomingmathematicians.net/&quot; rel=&quot;noreferrer&quot;&gt;https://womenbecomingmathematicians.net/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(All in English.)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2019-10-24T21:45:48.630" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11089" PostTypeId="1" CreationDate="2019-10-25T20:34:11.710" Score="8" ViewCount="2129" Body="&lt;p&gt;The names of many technical and scientific subjects, like mathematics, physics, statistics, etc., etc., end with letters &quot;ics&quot;. What is meant by this, if anything? Was there any logic behind it or is it just random?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have discussed this with some experts on these subjects, but the answer was that it is random. But I want to make sure, is there nothing behind it?&lt;/p&gt;&#xA;" OwnerUserId="10619" LastEditorUserId="55" LastEditDate="2019-10-25T21:50:57.110" LastActivityDate="2019-10-26T06:47:32.223" Title="Why do many names of technical and scientific subjects end with &quot;ics&quot;?" Tags="|mathematics|physics|quantum-mechanics|thermodynamics|statistics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11090" PostTypeId="2" ParentId="11089" CreationDate="2019-10-25T21:46:33.640" Score="19" Body="&lt;p&gt;It is not random. These names are of Greek origin, and -ic or -ics are Anglicizations of the Greek suffix -ikos, which meant &quot;pertaining to&quot;. In other languages it can be rendered as -ika or -ica, Wolfram's &quot;Mathematica&quot; uses such a version. From the &lt;a href=&quot;https://www.etymonline.com/word/-ics&quot; rel=&quot;noreferrer&quot;&gt;Online Etymology Dictionary&lt;/a&gt;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;-ics in the names of sciences or disciplines (acoustics, aerobics, economics, etc.), a 16c. revival of the classical custom of using the neuter plural of adjectives with Greek -ikos &quot;pertaining to&quot; (see -ic) to mean &quot;matters relevant to&quot; and also as the titles of treatises about them. Subject matters that acquired their English names before c. 1500, however, tend to be singular in form (arithmetic, logic, magic, music, rhetoric). The grammatical number of words in -ics (mathematics is/mathematics are) is a confused question&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-26T06:47:32.223" LastActivityDate="2019-10-26T06:47:32.223" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11091" PostTypeId="2" ParentId="11085" CreationDate="2019-10-26T00:43:02.497" Score="2" Body="&lt;p&gt;According to  &quot;The Philosophical Breakfast club&quot;,  &lt;a href=&quot;https://library.si.edu/digital-library/book/philosophicalbr00snyd&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://library.si.edu/digital-library/book/philosophicalbr00snyd&lt;/a&gt;, by Laura J. Snyder, William Whewell first coined the term &quot;scientist&quot; in 1833 at a meeting of the British Association for the Advancement of Science: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;...If &quot;philosophers&quot; was taken to be &quot;too wide and lofty&quot; a term, then &quot;by analogy with artist we may form scientist. This was, as far as anyone knows, the first time the word &quot;scientist was uttered in public.&quot; (page 4).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Snyder goes on to say Whewell also also coined the words &quot;cathode, anode and ion&quot; at Faraday's request.&lt;/p&gt;&#xA;" OwnerUserId="4561" LastActivityDate="2019-10-26T00:43:02.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11092" PostTypeId="1" CreationDate="2019-10-26T05:38:54.033" Score="4" ViewCount="844" Body="&lt;p&gt;It is commonly[1,2] held that Simon Stevin introduced the decimal number system with the decimal point (at least in Europe) in his 1585 book &lt;em&gt;De Thiende&lt;/em&gt;. However, in della Porta's book &lt;em&gt;Magia Naturalis&lt;/em&gt;, Book 17, Chapter 16, there is a clear reference to a decimal point as 360.0.0.0. You can see the 1658 English translation &lt;a href=&quot;https://www.loc.gov/resource/rbc0001.2009pre23451/?sp=380&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and the 1619 Latin &lt;a href=&quot;https://archive.org/details/B-001-003-706/page/n599&quot; rel=&quot;nofollow noreferrer&quot;&gt;version&lt;/a&gt;. I do not have access to the original 1558 version. Are these post factum additions of the 17th century (possibly by della Porta himself) or is it truly an anticipation of the decimal system?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[1] J.T. Derrese, G Varden Berghe, &quot;The Wonderful World of Simon Stevin&quot; Chapter 3; page 58: &quot;It is clear that Stevin regarded De Thiende as his own invention and that he was very proud of it.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[2] Dirk Jan Struik, &quot;A Source Book in Mathematics, 1200-1800&quot;, page 7, &quot;The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet &lt;em&gt;De Thiende&lt;/em&gt;&quot;&lt;/p&gt;&#xA;" OwnerUserId="9714" LastEditorUserId="55" LastEditDate="2019-10-26T21:53:04.213" LastActivityDate="2019-10-26T22:57:22.977" Title="Were decimal fractions known in Europe before Stevin?" Tags="|mathematics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11093" PostTypeId="2" ParentId="11092" CreationDate="2019-10-26T22:42:32.477" Score="1" Body="&lt;p&gt;For a history of decimal fractions see &lt;a href=&quot;https://archive.org/details/historyofmathema031897mbp/page/n255&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smith's History Of Mathematics, vol II, pp. 238ff&lt;/a&gt;. In the Middle East, Smith gives credit to al-Kashi (c.1400), but the relevant algorithms, in a table notation, appear already in al-Samawal (c.1150), see &lt;a href=&quot;http://books.google.com/books/about/The_History_of_Mathematics.html?id=pI8_AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Katz's History of Mathematics, 7.2.3&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notationally, the fractional separator was initially a bar placed over the digit of units, which was later replaced by a point, possibly for ease of printing. Pellos used the decimal point in print already in 1492, but Cardano still used a bar in 1539.  However, neither Pellos nor Cardano appreciated the algorithmic significance of the separator in calculations. According to Smith, the first man to do so in Europe was Rudolff, who performed a sample calculation in &lt;em&gt;Exempel Büchlin&lt;/em&gt; (1530). He also used the bar rather than the decimal point, and his work was not appreciated at the time. Smith does not mention della Porta, but his use was likely purely notational, a la Pellos. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first comprehensive explanation of decimal fraction calculations in Europe is indeed due to Stevin's &lt;em&gt;De Thiende&lt;/em&gt; (1585), and after him they became more common. But their final triumph only came after the invention of logarithms by Napier. However, Stevin did not use either the bar or the point, and his notation was, in fact, quite unwieldy: he wrote the power of &lt;span class=&quot;math-container&quot;&gt;$1/10$&lt;/span&gt; in a circle after each digit (see facsimile in Smith, p.243). The propagation of the modern notation was due to Bürgi, Kepler, Beyer and Napier. Bürgi used a dot or a comma, and his example was followed. Napier did not use the decimal point in his original publication of the logarithmic tables in 1614, but it appears in their English translation by Wright (1616), and Napier adopted it in Rabdologie (1617). Kepler and Beyer used both comma and &lt;span class=&quot;math-container&quot;&gt;$','',''',''''$&lt;/span&gt; placed over digits (as in the ancient sexagesimal notation) in 1616. Here is from Smith:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Another influence leading to the invention of the decimal &#xA;  fraction was the rule for dividing numbers of the form &lt;span class=&quot;math-container&quot;&gt;$a\cdot10^n$&lt;/span&gt;, &#xA;  attributed by Cardan (1539) to Regiomontanus... Borghi (1484) elaborates &#xA;  this rule, but it appears in its most interesting form in the rare &#xA;  arithmetic of Pellos (1492), who unwittingly made use of the &#xA;  decimal point for the first time in a printed work (p. 239). &#xA;  The use of the dot before and after integers had been common &#xA;  in the medieval manuscripts, as in the case of Chuquet's work &#xA;  already mentioned, but its use to separate the integer from &#xA;  what is practically a decimal fraction is first seen here. Later &#xA;  writers commonly used a bar for this purpose, as was the case &#xA;  with Rudolff (1530; see page 241), Cardan (1539), Cataneo &#xA;  (1546), and various other writers... Pellos, however, did not recognize the significance of the decimal point, as is evident from the facsimile on page 239, and no more did Cardan appreciate the significance of the bar that he &#xA;  used for the same purpose.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;The first man who gave evidence of having &#xA;  fully comprehended the significance of all this preliminary &#xA;  work seems to have been Christoff Rudolff, whose Exempel- &#xA;  Büchlin appeared at Augsburg in 1530. In this work he solved &#xA;  an example in compound interest, and used the bar precisely &#xA;  as we should use a decimal point today (see page 241). If any &#xA;  particular individual were to be named as having the best rea- &#xA;  son to be called the inventor of decimal fractions, Rudolff &#xA;  would seem to be the man, because he apparently knew how to &#xA;  operate with these forms as well as merely to write them, as &#xA;  various predecessors had done. His work, however, was not &#xA;  appreciated, and apparently was not understood, and it was not &#xA;  until 1585 that a book upon the subject appeared.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The first to show by a special treatise that he understood the &#xA;  significance of the decimal fraction was Stevin, who published &#xA;  a work upon the subject in Flemish, followed in the same year (1585) by a French translation. This work, entitled in French &#xA;  La Disme, set forth the method by which all business calculations involving fractions can be performed as readily as if they &#xA;  involved only integers. Stevin even went so far as to say that &#xA;  the government should adopt and enforce the use of the decimal system, thus anticipating the modern metric system. He was the first to lay down definite rules for operating with decimal fractions, and his treatment of the subject left little further to be done except to improve the symbolism... The improvement in the symbolism was &#xA;  due largely to Bürgi, Kepler, and Beyer, and to the English &#xA;  followers of Napier... It is unquestionably true that the invention of logarithms had more to do with the use of decimal fractions than any &#xA;  other single influence.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-10-26T22:57:22.977" LastActivityDate="2019-10-26T22:57:22.977" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11094" PostTypeId="1" CreationDate="2019-10-27T04:48:20.983" Score="0" ViewCount="124" Body="&lt;p&gt;Wheatstone Bridge was invented by Mr. Samuel Hunter Christie. So, How come it carries Mr Charles Wheatstone's name?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know what each of them contributed toward the 'Wheatstone bridge' besides Mr Samuel inventing it in the first place.&lt;/p&gt;&#xA;" OwnerUserId="10625" LastActivityDate="2019-10-29T12:48:52.507" Title="Why does 'Wheatstone bridge' carries Wheatstone's name when it was invented by someone else?" Tags="|electricity|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11095" PostTypeId="1" CreationDate="2019-10-27T09:52:08.887" Score="4" ViewCount="161" Body="&lt;p&gt;Note: This question was originally asked &lt;a href=&quot;https://history.stackexchange.com/questions/51681/what-happened-to-the-undergrad-students-who-attended-the-feynman-lecture-series&quot;&gt;here&lt;/a&gt;, but I was wondering if I could get further clarification as this has truly intrigued me. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the academic years of 1962-2 and 1962-3, Richard Feynman gave his celebrated series of lectures to a cohort of undergraduate physics students at Caltech. ... The careers of all those students, now aged somewhere in their mid 70's, have all ended or are winding down. Has anyone done an analysis of performance, relative to expectations, of the careers of these fortunate students?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10630" LastActivityDate="2019-10-27T09:52:08.887" Title="What happened to the undergrad students who attended the Feynman Lecture Series in 1961-63?" Tags="|physics|biographical-details|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11097" PostTypeId="1" AcceptedAnswerId="11099" CreationDate="2019-10-29T02:41:06.827" Score="6" ViewCount="495" Body="&lt;p&gt;In the book &lt;em&gt;What do you care what other people think?&lt;/em&gt;, Feynman talks about how in the 16th century &lt;a href=&quot;https://en.wikipedia.org/wiki/Niccol%C3%B2_Fontana_Tartaglia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Niccolo Tartaglia&lt;/a&gt; discovered a solution to cubic equations.  He says while this was not a major discovery, it was a boost to the confidence of mathematicians of the day, since not even the great Greek mathematicians had been able to solve cubic equations.  In Feynman's words:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;They were very upset when I said that the development of greatest importance to mathematics in Europe was the discovery by Tartaglia that you can solve a cubic equation: although it is of very little use in itself, the discovery must have been a psychologically wonderful because it showed that a modern man could do something no ancient Greek could do.  It therefore helped in the Renaissance, which was the freeing of man from the intimidation of the ancients.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this true?  I.e., did mathematicians in Europe before Tartagalia consider themselves inferior to Greeks? And did Tartaglias's discovery help dispel this inferiority complex?&lt;/p&gt;&#xA;" OwnerUserId="10644" LastEditorUserId="4251" LastEditDate="2023-06-26T09:35:32.863" LastActivityDate="2023-06-26T09:35:32.863" Title="Did Renaissance mathematicians once consider themselves inferior to the great ancient mathematicians?" Tags="|mathematics|ancient-greece|debunking|social-context|renaissance|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11098" PostTypeId="2" ParentId="11097" CreationDate="2019-10-29T05:11:37.810" Score="11" Body="&lt;p&gt;Feynman is being... liberally creative. What he says is his own interpolation that &amp;quot;makes sense&amp;quot; from the perch of today. &amp;quot;Must have been psychologically wonderful&amp;quot;, perhaps, but &amp;quot;freeing of man from the intimidation of the ancients&amp;quot; is not how the men of Renaissance generally felt. The intimidation they sought the freeing from was not of the ancients, but of religious and scholastic orthodoxy, to which &amp;quot;the wisdom of the ancients&amp;quot; was something of a cultural ground and counter. The feelings toward the ancients were more along the lines of awe and admiration. Ancient works were avidly sought after, translated, transmitted, and enthusiastically built upon. The Renaissance does translate as re-birth, it is not liberation from the ancients that it championed, but their revival, and validation by them.&lt;/p&gt;&#xA;&lt;p&gt;As for the specifics. First, the discovery of the solution to the cubic was made 30 years before Tartaglia, by del Ferro, and Tartaglia knew about it. Del Ferro did not make it public, but entrusted the secret to his son-in-law Nave and to his student Fiore. Tartaglia rediscovered it independently, but knowing that a solution exists is a powerful incentive for looking for it, a nerve-wrecking experience as we know from Tartaglia's own description, and not giving up. In fact, he rediscovered it while preparing for a &amp;quot;mathematical duel&amp;quot;, a common practice at the time, he was called to by Fiore, see &lt;a href=&quot;https://hsm.stackexchange.com/a/351/55&quot;&gt;Why is “Cardano's Formula” (wrongly) attributed to him?&lt;/a&gt; and &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/17513472.2014.933552&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saiber's Niccolò Tartaglia's poetic solution to the cubic equation&lt;/a&gt; for more on the story.&lt;/p&gt;&#xA;&lt;p&gt;Second, it is not clear that the men of Renaissance believed that the methods they discovered were unknown to the ancients, I am not aware of Tartaglia, or any of his contemporaries, expressing a Feynman-like sentiment. Instead, there was at the time a popular &amp;quot;conspiracy theory&amp;quot; that the ancients deliberately hid some of their &amp;quot;secrets of the art&amp;quot;. Around the time of Tartaglia, Copernicus, looked for affirmation of his heliocentric system in the writings of ancient Pythagoreans, like Philolaus and Ecphantus, and his contemporaries saw it (mistakenly) as a revival of the Pythagorean system. In the preface to &lt;em&gt;De Revolutionibus&lt;/em&gt;, he also writes of the reluctance of the Pythagoreans to share their discoveries with the unlearned commoners, see &lt;a href=&quot;https://plato.stanford.edu/entries/pythagoreanism/#ren&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, The Renaissance: Ficino, Pico, Reuchlin, Copernicus and Kepler&lt;/a&gt;. Even later, when it came to the calculus methods, Torricelli wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I would not dare to affirm that this new Geometry of indivisibles is a totally new discovery. I could more easily believe that the ancient Geometers have made use of this method of discovery of the most difficult theorems, though they preferred another way in demonstrations, either to conceal the secret of the art, or not to give any occasion of contradiction to their jealous detractors&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Wallis and others expressed similar opinions. After quoting them in Cavalieri and Euclid, see &lt;a href=&quot;https://books.google.com/books?id=IZlUDwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Revolution and Continuity, p. 159&lt;/a&gt;, de Gandt further comments:&amp;quot;&lt;em&gt;By claiming that the ancients possessed analogous instruments, the creators felt justified in adopting new ways. Their invention was a restitution of the ancient skill...&lt;/em&gt;&amp;quot;. Solution to the cubic was, no doubt, inspiring, but not in the way Feynman imagined.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-06-20T05:23:30.697" LastActivityDate="2023-06-20T05:23:30.697" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11099" PostTypeId="2" ParentId="11097" CreationDate="2019-10-29T12:34:59.707" Score="7" Body="&lt;p&gt;The answer to your question in the title is definitely yes. For example, Fermat wrote in a letter: &quot;Perhaps, posterity will thank me for having shown that the ancients did not know everything&quot;. Fermat lived a century after Tartaglia, and the general opinion was still that &quot;the ancients knew everything&quot;. This opinion began to change only after the invention of Calculus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what Feynman says is not quite true. Tartaglia's discovery (as well as some earlier works, like those of Fibonacci) was not so significant to change the general&#xA;opinion of superiority of the ancient mathematics. In fact, the top achievements of the ancients were not fully understood and incorporated to contemporary mathematics until the 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-10-29T12:34:59.707" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11100" PostTypeId="2" ParentId="11094" CreationDate="2019-10-29T12:48:52.507" Score="1" Body="&lt;p&gt;This happens all the time -- inventions,  techniques,  diseases, disease &lt;em&gt;cures&lt;/em&gt;,  all tend to get named for the person who brought them into general use (or simply crushed the competition)   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;See, for examples,  Edison vs. Tesla,  Baird &amp;amp; Farnsworth vs. Zworkin or Sarnoff,  Apple Computing vs. PARC.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-10-29T12:48:52.507" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11101" PostTypeId="1" CreationDate="2019-10-30T04:36:15.700" Score="2" ViewCount="107" Body="&lt;p&gt;In &lt;em&gt;The Copernican Revolution&lt;/em&gt;, Thomas Kuhn states that &quot;...only the Western civilizations which descend from Hellenic Greece have paid much attention to the appearance of the heavens in arriving at [answers to cosmological questions]&quot; (p. 5). If we exclude ancient Babylonian and Egyptian science and philosophy, which Kuhn discusses in the text, are there other prominent examples of cultures whose speculative thought (e.g., about cosmology) evolved similarly? In other words, has Kuhn isolated something unique about the Greek tradition, or was he simply ignorant about the variety of ways that cultures have made made speculative thought answerable to empirical investigation?&lt;/p&gt;&#xA;" OwnerUserId="10648" LastActivityDate="2019-10-30T04:36:15.700" Title="Thomas Kuhn and the Relationship between Astronomy and Cosmology" Tags="|astronomy|cosmology|kuhn|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11103" PostTypeId="2" ParentId="9672" CreationDate="2019-10-30T13:51:32.560" Score="1" Body="&lt;p&gt;I think you would be hard pressed to find one book that covers this whole period for all topics. For example, &lt;em&gt;The Code Book&lt;/em&gt; (by Simon Singh) is a wonderfully written book which covers the history of the subject and ends with the the current problems in the topic which are being researched (i.e. quantum computing). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If it's more the open problems part you are looking for, I would suggest looking at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Millennium_Prize_Problems&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Wikipedia&quot;&gt;Millennium Prize Problems&lt;/a&gt; and researching the history behind them. &lt;/p&gt;&#xA;" OwnerUserId="10652" LastEditorUserId="5989" LastEditDate="2019-10-31T08:23:29.630" LastActivityDate="2019-10-31T08:23:29.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11106" PostTypeId="1" AcceptedAnswerId="11107" CreationDate="2019-10-31T19:04:56.370" Score="11" ViewCount="202" Body="&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Samuel&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; on Pierre Samuel claims (uncitedly):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He was a member of the Bourbaki group, and filmed some of their meetings. A French television documentary on Bourbaki broadcast some of this footage in 2000.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does anyone know the name of this documentary? Better yet, is any of Samuel's footage available online?&lt;/p&gt;&#xA;" OwnerUserId="7842" LastEditorUserId="4251" LastEditDate="2020-07-17T22:30:02.883" LastActivityDate="2020-07-17T22:30:02.883" Title="Where are Pierre Samuel's videos of Bourbaki proceedings available?" Tags="|mathematics|abstract-algebra|bourbaki|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11107" PostTypeId="2" ParentId="11106" CreationDate="2019-11-01T06:44:40.977" Score="7" Body="&lt;p&gt;It appears to be from the following broadcast (&lt;a href=&quot;https://poincare.univ-lorraine.fr/sites/poincare.univ-lorraine.fr/files/users/documents/fichier_page/b-biblio-b-iii2013.pdf&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;ARTE (FRANCE TÉLÉVISION - LA CINQUIÈME) Émission « Archimède » du 14 novembre 2000 consacrée à Bourbaki, réalisée avec la collaboration de Maurice Boulanger et Pierre Samuel. On a présenté de courts extraits de films d’amateur tournés par Pierre Samuel, ancien membre de Bourbaki, durant divers congrès Bourbaki. Mise en&#xA;  ligne d’un résumé de cette émission&#xA;  &lt;a href=&quot;http://archives.arte-tv.com/hebdo/archimed/20001114/ftext/sujet5.html&quot; rel=&quot;noreferrer&quot;&gt;http://archives.arte-tv.com/hebdo/archimed/20001114/ftext/sujet5.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Which (roughly) translates as:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;ARTE (FRANCE TÉLÉVISION - LA CINQUIÈME) &quot;Archimedes&quot; broadcast of November 14, 2000 dedicated to Bourbaki, produced with the collaboration of Maurice Boulanger and Pierre Samuel.  Short films of amateur films made by Pierre Samuel, a former member of Bourbaki, were presented during various Bourbaki congresses. Online summary of this show&#xA;   &lt;a href=&quot;http://archives.arte-tv.com/hebdo/archimed/20001114/ftext/sujet5.html&quot; rel=&quot;noreferrer&quot;&gt;http://archives.arte-tv.com/hebdo/archimed/20001114/ftext/sujet5.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The online summary is no longer available, but has been &lt;a href=&quot;https://web.archive.org/web/20050115230349/http://archives.arte-tv.com/hebdo/archimed/20001114/ftext/sujet5.html&quot; rel=&quot;noreferrer&quot;&gt;archived by the WayBack Machine&lt;/a&gt;. Sadly it seems to date from 5 years after the broadcast and doesn't seem to mention the films at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I couldn't find the broadcast online either, though it may be possible to contact Arte directly to enquire. &lt;/p&gt;&#xA;" OwnerUserId="1912" LastActivityDate="2019-11-01T06:44:40.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11109" PostTypeId="1" AcceptedAnswerId="11110" CreationDate="2019-11-03T11:41:52.533" Score="2" ViewCount="385" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.sciencefocus.com/the-human-body/is-it-coincidental-that-the-human-menstrual-cycle-is-about-the-same-length-as-the-moon-cycle/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; from the BBC's &lt;em&gt;Science Focus&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Charles Darwin thought that the 28-day human menstrual cycle was&#xA;  evidence that our ancestors lived on the seashore and needed to&#xA;  synchronise with the tides.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Did Darwin ever say such a thing?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest I have been able to find from a cursory search is &lt;a href=&quot;http://www.darwinproject.ac.uk/letter/?docId=letters/DCP-LETT-7494.xml&quot; rel=&quot;nofollow noreferrer&quot;&gt;this letter&lt;/a&gt; from T.H. Huxley in the Darwin Correspondence Project, where Huxley writes&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fancy lunacy &amp;amp; menstruation coming out of the primary fact that one’s&#xA;  nth. ancestor lived between tide-marks! I declare it’s the grandest&#xA;  suggestion I have heard of for an age&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there any evidence that Darwin suggested this?&lt;/p&gt;&#xA;" OwnerUserId="9430" LastEditorUserId="4251" LastEditDate="2020-07-12T15:54:20.347" LastActivityDate="2020-07-12T16:35:51.923" Title="Did Darwin say that the human menstrual cycle length was influenced by the tides?" Tags="|biology|quote|evolution|darwin|biologists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11110" PostTypeId="2" ParentId="11109" CreationDate="2019-11-03T16:19:55.617" Score="6" Body="&lt;p&gt;Yes. The menstrual cycle is surely one of the &amp;quot;some of our functions&amp;quot; that Darwin speaks of in this passage from Chapter VI of &lt;a href=&quot;http://www.gutenberg.org/ebooks/2300&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Descent of Man, and Selection in Relation to Sex&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The progenitors of man must have been aquatic in their habits. [...] In the lunar or weekly recurrent periods of some of our functions we apparently still retain traces of our primordial birthplace, a shore washed by the tides.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="370" LastEditorUserId="4251" LastEditDate="2020-07-12T16:35:51.923" LastActivityDate="2020-07-12T16:35:51.923" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11111" PostTypeId="1" AcceptedAnswerId="11112" CreationDate="2019-11-03T19:02:42.950" Score="10" ViewCount="1149" Body="&lt;p&gt;It is said that &lt;a href=&quot;https://en.wikipedia.org/wiki/Antoine_Lavoisier#Pioneer_of_stoichiometry&quot; rel=&quot;noreferrer&quot;&gt;Lavoisier and Lomonosov&lt;/a&gt; both discovered that the cumulated mass of reactants is conserved. My question is simple: who (and when) started weighing chemical products before and after forcing them react? &lt;/p&gt;&#xA;" OwnerUserId="7069" LastEditorUserId="55" LastEditDate="2019-11-03T23:46:24.513" LastActivityDate="2019-11-04T05:18:00.773" Title="Who was the first to weigh chemical reaction products?" Tags="|chemistry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11112" PostTypeId="2" ParentId="11111" CreationDate="2019-11-03T23:36:10.593" Score="14" Body="&lt;p&gt;We shall never know who was first. The term &lt;a href=&quot;http://www.chemistryexplained.com/St-Te/Stoichiometry.html#ixzz64GFX96KG&quot; rel=&quot;noreferrer&quot;&gt;&quot;stoichiometry&quot;&lt;/a&gt; was introduced by Richter in The Art of Measuring the Chemical Elements only in 1792, but the practice long predates modern chemistry. It was done by metal workers and alchemists in the centuries past, although they had few means to control for confounding factors, such as escaping gases or overlooked ingredients. Not to mention the roughness of the balances used. &lt;a href=&quot;https://www.jstor.org/stable/27757160&quot; rel=&quot;noreferrer&quot;&gt;Forbes in On the Origin of Alchemy&lt;/a&gt; mentions &#xA;practices of ancient alchemists like Bolos in this regard:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Therefore, after its incubation period in Mesopotamia, alchemy with other theories came to Alexandria, the great pooling center of the Hellenistic sciences, where it was crystallized into a clear-cut doctrine by the logical-minded Greeks. This codification period starts with the &quot;Physika&quot; of Bolos Democritos of Mendes (200 B.C.)... It seems that Bolos and the alchemists were interested in the transmutation of matter, which they believed to be effected and indicated by the color changes produced in alloying and dyeing operations... The color change, which to these alchemists seems the proof of this transmutation, effected by dyeing, by varnishing and alloying and is quantitatively controlled in some cases by the increase of weight.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It got more precise during middle ages. Van Helmont (1577-1644) staged the following curious experiment, see &lt;a href=&quot;http://www.chymist.com/History%20Alchemy.pdf&quot; rel=&quot;noreferrer&quot;&gt;History of Alchemy and Early Chemistry by Katz&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;He  took  200  pounds  of  earth  dried  in  an  oven,  and  having  put  it  into  an  earth en  vessel  and  moistened  it  with rain water, he planted in it the trunk f a willow tree  of  five  pounds  weight;  this  he  watered,  as  need  required,  with  rain  or  distilled  water;  and  to  keep  the  neighbouring  earth  from  getting  into  the  vessel,  he  employed  a  plate  of  iron  tinned  over  and  perforated  with many holes. Five years having elapsed, he took out the tree and weighed it, and (including the weight of the leaves that fell during the four autumns) he found it to weigh 169 pounds 3 ounces. And having again dried the earth it grew in, he found it only about 2 ounces short of its former weight of 200 pounds; so that 164 pounds of the roots, leaves, wood, and bark, which constituted the tree appeared to have sprung from the water alone.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Rey in 1630 published an essay On an Inquiry Into the Cause&#xA;Wherefore Tin and Lead Increase in Weight on Calcination, indicating that the phenomenon was common knowledge at the time. This is the essay where he defended conservation of mass on speculative atomistic grounds, see &lt;a href=&quot;https://pubs.acs.org/doi/abs/10.1021/ed052p658&quot; rel=&quot;noreferrer&quot;&gt;An historical note on the conservation of mass by Whitaker&lt;/a&gt;.&#xA;When the first scientific theory of chemistry, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Phlogiston_theory&quot; rel=&quot;noreferrer&quot;&gt;phlogiston theory&lt;/a&gt;, was introduced by Stahl in 1667, this observation was used as an objection to it. According to Stahl, decalcination of metals amounts to liberating phlogiston contained in metal ores (calces), contrary to calces weighing more than the resulting metals. Some chemists even ascribed negative weight (buoyant effect) to phlogiston to explain the phenomenon, implicitly relying on the conservation law. Newton, who was obsessed with alchemy in his later life, measured weights meticulously, as recorded in his notebooks, see facsimile in Katz's paper, pp. 60-61.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;By mid-18th century the weighing was standard practice among experimental chemists. For example, Black in Experiments upon Magnesia Alba,&#xA;Quicklime and Some Other Alcaline Substances (1755) noticed that the&#xA;amount of sulfuric acid required to dissolve calcined magnesia&#xA;was slightly less (by 1-9%) than that required to dissolve the&#xA;amount of magnesia originally heated for calcination. He also anticipated the conservation law in trying to explain the difference&#xA;(quoted from Whitaker): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As in the separation of the volatile from the fixed parts of bodies,&#xA;  by means of heat, a small quantity of the latter is generally&#xA;  raised with the former; so the air and water, originally contained&#xA;  in the magnesia, and afterwards dissipated by the fire, seem to&#xA;  have carried off a small part of the fixed earth of this substance.&#xA;  This is probably the reason why calcined magnesia is saturated&#xA;  with a quantity of acid, somewhat less than what is required to&#xA;  dissolve it before calcination: and the same may be assigned as&#xA;  one cause which hinders us from restoring the whole of its original&#xA;  weight, by solution and precipitation.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similar observations can be found in Cavendish's notes. This predates Lomonosov's experiments (1756), let alone his letter to Euler (1774), where the conservation law was communicated, and Lavoisier's formulation of it (1773). &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-04T05:18:00.773" LastActivityDate="2019-11-04T05:18:00.773" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11115" PostTypeId="1" CreationDate="2019-11-04T16:30:32.917" Score="2" ViewCount="136" Body="&lt;p&gt;How was (non-instantaneous) electric current first discovered and what were some of the main first thoughts on it?&lt;/p&gt;&#xA;" OwnerUserId="8168" LastEditorUserId="55" LastEditDate="2019-11-05T00:42:58.680" LastActivityDate="2019-11-08T06:03:54.167" Title="How was (non-instantaneous) electric current first discovered?" Tags="|physics|electromagnetism|electricity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11116" PostTypeId="2" ParentId="11106" CreationDate="2019-11-04T18:10:05.627" Score="3" Body="&lt;p&gt;These videos (some of them, maybe) were on view during an exhibition about Bourbaki and his early collaborators at Bibliothèque de l'École normale supérieure. &#xA;Presumably the Association des collaborateurs de N. Bourbaki still has a copy of some of these videos, I don't know if, nor how, it can share them though.&lt;/p&gt;&#xA;" OwnerUserId="7474" LastActivityDate="2019-11-04T18:10:05.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11117" PostTypeId="2" ParentId="9672" CreationDate="2019-11-04T19:14:27.860" Score="1" Body="&lt;p&gt;I can recommend Colin C. Adams: &quot;the knot book&quot; for a very accessible introduction to knot theory. It covers some history of knot theory, esp. of knot tabulation, in the first few chapters, mentions many open problems that are at least easy to understand (I could even solve one of them, and I was a beginner), and contains many exercises.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Like @Bee, I think it will be difficult to find a broader book with a similar taste. Well, Dominic Olivastro's &quot;ancient puzzles&quot; is very broad and presents one (1!) open problem in graph theory at the end.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In theoretical computer science, &quot;the art of computer programming&quot; by Donald E. Knuth follows the approach you describe, developing each of the treated problems from their historic perspective. But many people (including me) find it hard to read, although both the writing and the coverage of the whole history of ideas is excellent. Well, I admit that I skipped most of the literally thousands of exercises, which are presented along with accessible research problems. If you dare, I'd recommend looking first into volume 4A, which contains more combinatorial puzzles and less low-level bit-fiddling than the other volumes.&lt;/p&gt;&#xA;" OwnerUserId="9523" LastActivityDate="2019-11-04T19:14:27.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11118" PostTypeId="2" ParentId="11115" CreationDate="2019-11-04T21:48:34.080" Score="5" Body="&lt;p&gt;I will assume &quot;non-instantaneous&quot; means something other than electric discharge in the atmosphere, from animals like eels and torpedo fishes, or electrostatic generators like the Leyden jar or the van der Graaf generator. Then the answer is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Voltaic_pile&quot; rel=&quot;nofollow noreferrer&quot;&gt;voltaic pile invented by Volta in 1799&lt;/a&gt;, &quot;&lt;em&gt;the first electrical battery that could continuously provide an electric current to a circuit&lt;/em&gt;&quot;. The invention was inspired by Galvani's experiments with animal electricity, but Galvani himself used electric sparks. Since prior to Volta only static electricity was dealt with there was a controversy at to explaining phenomena with the newly discovered steady current. Despite Volta's own intuitions about the electric fluid propelled into &quot;perpetual impulsion&quot; by an &quot;electromotive contact force&quot;, the original theory of the pile, adopted by an illustrious  committee of Laplace, Coulomb, Fourcroy, Guyton, and Biot, was quasi-static, the current was interpreted as a series of electrostatic discharges. Science getting decided by committee is rarely a good sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The means to measure current did not appear until Oersted's observation of its effect on magnetic needles. Ampere, an outsider to the French scientific establishment, dismissed Biot's quai-static theory in 1820-s, and reintroduced Volta's original intuitions about perpetual circular movement of the two electric fluids in the wire propelled by the pile, &quot;reunions in the wire&quot; as he called them. But even Ampere's theory was intermediary, it sharply separated electrostatic effects in the pile, and electrodynamic ones in the wire, tension electricity and current electricity, as they were called. Only in 1827, when Ohm formulated his law, the modern definition and theory of the electric current emerged. Ohm's law required a reconceptualization of electricity that was seen as radical at the time, and met much resistance. To his contemporaries, his now familiar law looked like a formula linking the intensity of light to its frequency, a nonsensical connection. For more details see &lt;a href=&quot;https://hsm.stackexchange.com/a/5150/55&quot;&gt;What is the history of electric current and resistance?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-08T06:03:54.167" LastActivityDate="2019-11-08T06:03:54.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11120" PostTypeId="1" CreationDate="2019-11-05T13:31:27.857" Score="1" ViewCount="305" Body="&lt;p&gt;Who had direct inside knowledge of the canon through the alleged secret oral tradition? Some possible examples that have been alluded to include Pythagoras, Plato, Euclid, Copernicus, Galileo, Kepler, Leibniz and Newton. Grattan-Guinness mentions the oral tradition in &lt;em&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0393046508&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Norton History of the Mathematical Sciences&lt;/a&gt;&lt;/em&gt; but he gives no sources. Partial introduction to what is recorded in written history by Conifold &lt;a href=&quot;https://hsm.stackexchange.com/questions/1962/did-geometric-patterns-in-nature-suggest-the-early-notions-of-proportion-and-rat&quot;&gt;Did geometric patterns in nature suggest the early notions of proportion and ratios?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2020-08-23T23:18:24.913" Title="What is the ancient cosmic canon of proportion and its role in the history of science?" Tags="|mathematics|physics|cosmology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11121" PostTypeId="1" CreationDate="2019-11-05T15:13:40.157" Score="1" ViewCount="235" Body="&lt;p&gt;In Saunders Mac Lane's autobiography he described how he visited, I think Königsberg, then the centre of mathematics in Germany. He also reported he that he saw Hitler somewhere and that he wished that he had a gun so he could shoot him. Had he done it, it might have saved a great deal of trouble all round. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q. Where did Mac Lane report seeing him? Was it in an academic setting and what was he - I mean Hitler - doing there?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="4490" LastEditDate="2019-11-12T07:15:36.193" LastActivityDate="2019-11-12T07:15:36.193" Title="Where did Mac Lane say he saw Hitler and wished that he had a gun so he could have shot him?" Tags="|mathematicians|mathematics-social-history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11122" PostTypeId="1" AcceptedAnswerId="11123" CreationDate="2019-11-05T16:46:16.010" Score="4" ViewCount="94" Body="&lt;p&gt;Did Newton write down his process of building a reflector telescope? Even with modern knowledge and technology, building one is no easy task so I'm curious of exactly what Newton did. &lt;/p&gt;&#xA;" OwnerUserId="10602" LastEditorUserId="55" LastEditDate="2019-11-07T20:21:53.580" LastActivityDate="2019-11-07T20:21:53.580" Title="Is there a record of how Newton built his telescope?" Tags="|newton|telescope|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11123" PostTypeId="2" ParentId="11122" CreationDate="2019-11-06T01:13:59.617" Score="5" Body="&lt;p&gt;We know relatively little about Newton's original model from 1668, modern versions draw lineage from the improved 1671 version. There is a record with its detailed description, with some comments on construction, although not the entire &quot;process of building&quot;. Here is from &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rsnr.1979.0009&quot; rel=&quot;noreferrer&quot;&gt;Newton's telescope by Mills&#xA;and Turvey&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The prime source of information on this improved instrument must be the&#xA;  formal description drawn up by Oldenburg (19), sent to Newton (20), and&#xA;  carefully checked by him (21) before its dispatch to Huygens (22). The Latin&#xA;  text is accompanied by the illustration shown here as Plate lJ. The most&#xA;  obvious refinement is the use of a substantial wooden ball to provide universal&#xA;  adjustment. This telescope is stated to contain a 2 in. diameter primary speculum (18, 23) with a focal length quoted as 6 1/3 in. in the original correspondence and as about 6 1/4 in. in the Opticks. It seems likely that Newton made a new, thicker, speculum - probably as the best of a pair - rather than attempt to repolish the defective mirror(s) made for his first telescope. A description based on Oldenburg (19) appeared in the Philosophical Transactions for 25 March 1672 (24), but is accompanied by an engraving differing in a number of respects from the drawing with the above manuscript. However, engravings of this period are notoriously inexact. A version obviously redrawn from the Phil. Trans. (24) appears in Hutton's abridgment of 1809 (25).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mills and Turvey have photographs of the model, and facsimile of drawings from the original sources. They also review the description and Newton's remarks on construction in detail. For example, Newton remarked that in the second model &quot;&lt;em&gt;In that metall which I sent to London (i.e. the second telescope) there was noe Arsenick, but a small proportion of silver: as I remember is in 3 ounces of metall&lt;/em&gt;&quot;. Much later, in Opticks, he also revealed his method for polishing the lenses:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I had two round Copper Plates, each six Inches in Diameter, the one&#xA;  convex, the other concave, ground very true to one another. On the convex&#xA;  I ground the Object-Metal or Concave which was to be polish'd, 'till it had&#xA;  taken the Figure of the Convex and was ready for a Polish. Then I pitched&#xA;  over the convex very thinly, by dropping melted Pitch upon it, and warming&#xA;  it to keep the Pitch soft, whilst I ground it with the concave Copper&#xA;  wetted to make it spread eavenly all over the convex. Thus by working it&#xA;  well I made it as thin as a Groat, and after the convex was cold I ground it&#xA;  again to give it as true a Figure as I could. Then I took Putty which I had&#xA;  made very fine by washing it from all its grosser Particles, and laying a little of this upon the Pitch, I ground it upon the Pitch with the concave Copper, till it had done making a Noise; and then upon the Pitch I ground the&#xA;  Object-Metal with a brisk motion, for about two or three Minutes of time,&#xA;  leaning hard upon it. Then I put fresh Putty upon the Pitch, and ground it&#xA;  again till it had done making a noise, and afterwards ground the ObjectMetal&#xA;  upon it as before. And this Work I repeated till the Metal was&#xA;  polished, grinding it the last time with all my strength for a good while&#xA;  together, and frequently breathing upon the Pitch, to keep it moist without&#xA;  laying on any more fresh Putty.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-11-06T01:13:59.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11124" PostTypeId="2" ParentId="11121" CreationDate="2019-11-06T07:37:05.677" Score="4" Body="&lt;p&gt;This incident is recounted in Mac Lane's autobiography, in chapter 6, starting on page 56. It was not Königsberg, but rather Weimar, and he saw Hitler at a performance of a Wagner opera on Wagner's birthday. It's only mentioned in a single paragraph:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Once I actually saw Hitler. While in Göttingen it suddenly occurred to me that I had never visited Weimar, the famous court where Goethe lived and worked. So I took the train and found a hotel, then went to the opera house. The tickets for the Wagnerian concert that evening were all sold out -- it was Wagner's birthday, May 22 -- but someone with an extra ticket came along and sold it to me. I went and enjoyed the opera, and went out into the lobby for the intermission. There, across the lobby, stood Hitler and Göring -- they were easy to recognize from the many pictures I had seen in the papers, and it was well known that Hitler like Wagnerian opera. I had no gun with me. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4490" LastActivityDate="2019-11-06T07:37:05.677" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11125" PostTypeId="2" ParentId="11082" CreationDate="2019-11-07T08:22:11.937" Score="5" Body="&lt;p&gt;Ciao Alberto,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I do not know if you ever came across &quot;&lt;a href=&quot;https://www.springer.com/gp/book/9780857299284&quot; rel=&quot;noreferrer&quot;&gt;Remembering Sophia Kovalevskaja&lt;/a&gt;&quot; which is a quite curious book written by Michele Audin. It is not &quot;popular&quot; in that the math discussions in the book are at a very high level and it is not &quot;academic&quot; in that the way in which it is written is more towards literary style. A very interesting reading in my opinion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also Teri Perl wrote a couple of books popularizing figures of women mathematicians, quite some years ago. A colleague of mine has a very good opinion on this:&#xA;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0201057093&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math Equals &lt;/a&gt;, which I think is out of print now. But this is just a second hand comment.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2019-11-07T08:22:11.937" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11126" PostTypeId="1" CreationDate="2019-11-07T12:01:44.447" Score="4" ViewCount="231" Body="&lt;p&gt;It is natural for physicists to consider the group &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt;. Presumably, &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt; came into physics because of quantum mechanics. How did people realize that when studying rotation of a physical system, sometimes it is the group &lt;span class=&quot;math-container&quot;&gt;$SU(2)$&lt;/span&gt; instead of the group &lt;span class=&quot;math-container&quot;&gt;$SO(3)$&lt;/span&gt; that really matters? &lt;/p&gt;&#xA;" OwnerUserId="3428" LastActivityDate="2019-11-08T14:26:10.260" Title="How did $SU(2)$ came into physics?" Tags="|physics|quantum-mechanics|group-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11127" PostTypeId="2" ParentId="11126" CreationDate="2019-11-07T13:57:46.737" Score="8" Body="&lt;p&gt;It came to physics a bit earlier than quantum mechanics. The homomorphism&#xA;&lt;span class=&quot;math-container&quot;&gt;$SU(2)\to SO(3)$&lt;/span&gt; was discovered by Cayley (1843), Hamilton (1847), and Klein (1875) in their pure mathematical studies, &#xA;and came to the attention of physicists through the theory of rigid body rotation (classical mechanics). It was Klein who brought it to the attention&#xA;of physicists.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In old books on mechanics this is called the &quot;Cayley-Klein's parameters&quot;. Klein explained this correspondence to physicists in his 1895/6 lectures which were later published as a book under the title &quot;Theory of the top&quot; (joint with his student A. Sommerfeld). (A concise exposition in English is in his 1897 Princeton lectures on the same subject. But the 4 volumes of Klein and Sommerfeld has been also translated into English).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his Princeton lectures Klein says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Instead of either of these commonly used systems of parameters, &#xA;  I propose to introduce another, which so far as I know has not yet been&#xA;  employed in dynamics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later he writes in the same lectures:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is nothing essentially new in these considerations.&#xA;  I have merely attempted to throw a method already well known into&#xA;  the most convenient form for applications in mechanics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Among the predecessors of Cayley, Hamilton, and Klein, Euler has to be mentioned: he discovered a rational parametrization of the rotation group which is equivalent to quaternion parametrization, in his work on arithmetic. It was apparently not noticed by physicists, but Klein credits the &quot;quaternion parametrization&quot; to Euler.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-11-08T14:26:10.260" LastActivityDate="2019-11-08T14:26:10.260" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11128" PostTypeId="1" AcceptedAnswerId="11129" CreationDate="2019-11-07T17:42:25.103" Score="6" ViewCount="489" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hungarian_algorithm#cite_note-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; says:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 2006, it was discovered that Carl Gustav Jacobi had solved the assignment problem in the 19th century, and the solution had been published posthumously in 1890 in Latin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The provided reference gives no support to the statement. Do you know the story? Is it true? Who discovered it? &lt;/p&gt;&#xA;" OwnerUserId="3303" LastEditorUserId="4251" LastEditDate="2020-05-12T16:24:16.377" LastActivityDate="2020-05-12T16:24:16.377" Title="Did Jacobi invent the Hungarian algorithm for the assignment problem over a century before Kőnig and Egerváry?" Tags="|discoveries|combinatorics|algorithm|optimization|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11129" PostTypeId="2" ParentId="11128" CreationDate="2019-11-07T23:09:29.943" Score="8" Body="&lt;p&gt;It is true. Wikipedia links to the website of Ollivier, which hosts both the original posthumous publication in Latin, &lt;em&gt;De investigando ordine systematis aequationum differentialum vulgarium cujuscunque&lt;/em&gt;, and its English translation, About the Research of the Order of a System of Arbitrary Ordinary Differential Equations, by Ollivier himself. Unfortunately, navigation within the site is not reflected in the urls, so you have to click on Translations on the top left to see them. Ollivier is the one who &quot;found&quot; it, and he describes the full story of the problem in &lt;a href=&quot;https://pdfs.semanticscholar.org/bf87/4940df54c5c8aefe9383ef786b7a58187adb.pdf?_ga=2.135322033.1079558760.1573168768-645931297.1573168768&quot; rel=&quot;noreferrer&quot;&gt;Jacobi's bound. Transmission and oblivion of a mathematical notion&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Jacobi himself is possibly the first to have forgotten his own work. According to Koenigsberger ([13]), his manuscripts on this subject were written around 1836 and were intended to be a part of a forsaken project of long paper on differential equations. Part of this work was incorporated in his great paper on the last multiplier, but the bound himself was never published in his lifetime.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Jacobi's widow gave the manuscripts he left to Dirichlet who began to work for their publication with his friends Borchardt and Joachimsthal. Very few documents remain from that work and the best source seems to be Koenigsberger ([13]). It seems that the paper were in great disorder... Borchardt entrusted Sigismund Cohn, who worked on the publication of some others manuscripts of Jacobi, with the documents related to the bound... After his death, the work was continued by Borchardt who published the 1st paper [8] in his journal in 1865. The second [9] was published by Clebsch in the volume Vorlesungen uber Dynamik ([FD]) in 1866. This one was quoted by Sofya Kovalevskaya in one of her most famous papers ([16]) in 1875.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For details on the assignment problem itself and the &quot;Hungarian&quot; algortihm for solving it, known already to Jacobi, see &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.725.8898&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;Jeno Egervary: from the origins of the&#xA;Hungarian algorithm to satellite communication&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Jacobi introduces a bound on the order of a&#xA;  system of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; ordinary differential equations in &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; unknowns, and observes that its computation can be reduced to the following problem... in which however the assignment problem is easily recognized:&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&lt;em&gt;Arrange &lt;span class=&quot;math-container&quot;&gt;$nn$&lt;/span&gt; arbitrary numbers &lt;span class=&quot;math-container&quot;&gt;$h^{(i)}_k$&lt;/span&gt; in a square table so that there are&#xA;    n horizontal series and n vertical series, each having n numbers. Among&#xA;    these numbers, we want to select &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; transversals, i.e., all placed in different&#xA;    horizontal and vertical series, which may be done in &lt;span class=&quot;math-container&quot;&gt;$1\cdot2\cdots n$&lt;/span&gt; ways; among&#xA;    all these ways, we want to find one that gives the maximum sum of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&#xA;    selected numbers.)&lt;/em&gt;&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Not only Jacobi defined the AP, but, more importantly, he gave a polynomial-&#xA;  time algorithm to solve it. Although a thorough analysis of this method&#xA;  has not been provided yet, Ollivier and Sadik [29] recently observed that it is&#xA;  essentially identical to the Hungarian algorithm, thus anticipating by many&#xA;  decades the results we have examined in the previous sections. As wittily&#xA;  observed by Kuhn [24], this makes another application of Stigler's law&#xA;  of eponymy&quot;: No scientific discovery is named after its original discoverer&#xA;  (Stigler [34], see also Kuhn [23]).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-08T00:59:37.177" LastActivityDate="2019-11-08T00:59:37.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11130" PostTypeId="2" ParentId="9545" CreationDate="2019-11-09T21:12:13.793" Score="1" Body="&lt;p&gt;So I think I know the answer to this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, note that both coordinate systems relegate the &lt;strong&gt;vertical axis&lt;/strong&gt; to the &lt;strong&gt;last coordinate&lt;/strong&gt; (y in 2D, z in 3D); that this feels 'natural' is perhaps a result of Latin script being written horizontally. Now while this fully defines the 2D coordinate system, there are still two possibilities for the 3D system: a so-called &lt;strong&gt;left-handed&lt;/strong&gt; system and a &lt;strong&gt;right-handed&lt;/strong&gt; system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lM7C2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lM7C2.png&quot; alt=&quot;3D coordinate systems&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason (I believe) that the right-handed system is standard in both mathematics and physics is that it agrees with the definition of the &lt;strong&gt;vector cross product&lt;/strong&gt;, which by convention follows the &lt;strong&gt;right-hand rule&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$\mathbb{x} \times \mathbb{y} = \mathbb{z}$&lt;/span&gt;. This definition is fundamental to vector algebra and much of physics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6WbHS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6WbHS.png&quot; alt=&quot;right-hand rule&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This definition, in turn, follows directly from Hamilton's 1843 formulation or &lt;strong&gt;quaternions&lt;/strong&gt;, defined by &lt;span class=&quot;math-container&quot;&gt;$ij=k$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$jk=i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ki=j$&lt;/span&gt;, by identifying &lt;span class=&quot;math-container&quot;&gt;$i, j, k$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x, y, z$&lt;/span&gt;. Had Hamilton instead defined them by &lt;span class=&quot;math-container&quot;&gt;$ji=k$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$kj=i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ik=j$&lt;/span&gt;, there would likely have been a left-handed rule instead.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now the final question of why representations of right-handed 3D coordinates typically show the y axis at &quot;3 o'clock&quot; and the x axis &quot;somewhere past 6 o'clock&quot; (like the picture below, but unlike the first picture), I expect that's simply to separate the axes on the page as far as possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Y0TRd.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Y0TRd.png&quot; alt=&quot;3D coordinate system&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2019-11-15T16:38:51.733" LastActivityDate="2019-11-15T16:38:51.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11132" PostTypeId="1" AcceptedAnswerId="11133" CreationDate="2019-11-12T21:14:52.343" Score="2" ViewCount="115" Body="&lt;p&gt;I happened to find that one of the most important inventors of the renormalization group, &lt;a href=&quot;https://en.wikipedia.org/wiki/Kenneth_G._Wilson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kenneth G. Wilson&lt;/a&gt;, won the Leroy Randle Grumman Award Medal in 1986, 4 years after his Nobel prize. Details in &lt;a href=&quot;https://natedsanders.com/leroy_randle_grumman_award_medal_presented_to_theo-lot42882.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;this site&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This Leroy Randle Grumman Award is obviously related to &lt;a href=&quot;https://en.wikipedia.org/wiki/Leroy_Grumman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leroy Randle Grumman&lt;/a&gt;, one of the cofounders of the Grumman company, now part of Northrop Grumman, one of the top defense contractors in the world and is famous for, e.g., the stealth aircraft.&lt;/p&gt;&#xA;&lt;p&gt;I am curious about this prize for physicists and who the recipients were. However, I couldn't find any more info of this prize except that another applied physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Petr_Ufimtsev&quot; rel=&quot;nofollow noreferrer&quot;&gt;Petr Ufimtsev&lt;/a&gt;, father of stealth technology, won it for his electromagnetic wave diffraction theory.&lt;/p&gt;&#xA;" OwnerUserId="10697" LastEditorUserId="10697" LastEditDate="2020-07-08T17:05:33.757" LastActivityDate="2020-07-08T17:05:33.757" Title="What is the Leroy Grumman Medal won by the theoretical physicist Kenneth G. Wilson?" Tags="|biographical-details|physicists|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11133" PostTypeId="2" ParentId="11132" CreationDate="2019-11-13T06:40:48.440" Score="4" Body="&lt;p&gt;Curious award, issued by &lt;a href=&quot;https://en.wikipedia.org/wiki/Northrop_Grumman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Northrop Grumman Corp.&lt;/a&gt;, the common abbreviation seems to be &quot;Grumman medal&quot;. Appears to be out of circulation since 1990s. Aside from Wilson (1986) and Ufimtsev (1991), some recipients were &lt;a href=&quot;https://www.aps.anl.gov/files/APS-Uploads/APS-Colloquium/chu2002oct.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Chu&lt;/a&gt; (1987), &lt;a href=&quot;https://www.spiedigitallibrary.org/journals/Journal-of-Nanophotonics/volume-2/issue-01/021999/Special-Section-Editorial-Russell-Messiers-Lifetime-of-Nanomorphology/10.1117/1.3006418.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell Messier&lt;/a&gt; (1987), &lt;a href=&quot;https://www.nationalaviation.org/our-enshrinees/rutan-elbert-burt/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Burt Rutan&lt;/a&gt; (1989), &lt;a href=&quot;https://urresearch.rochester.edu/fileDownloadForInstitutionalItem.action?itemId=4108&amp;amp;itemFileId=6128&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Huizenga (see p.50)&lt;/a&gt; (1991), and &lt;a href=&quot;https://books.google.com/books?id=E-NGFqfq1LsC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward Stone (see p.37)&lt;/a&gt; (?). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, the sources do not specify what exactly the medal was awarded for, other than &quot;Outstanding Scientific Achievement&quot;. Some of the later recipients were involved with aerospace technology. Ufimtsev's acoustic diffraction work was used in Lockheed's stealth fighter F-117. After getting the medal in 1991 he went to work for Grumman Corp. in 1995-2000. Rutan is an aerospace engineer known for designing light, energy-efficient aircraft. Grumman Corp. held a stake in his Scaled Composites (founded in 1982), and &lt;a href=&quot;https://www.space.com/4104-northrop-grumman-buys-builder-spaceshipone.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;acquired it in 2007&lt;/a&gt;. Stone worked on Voyager missions since 1972, and was the director of JPL (Jet Propulsion Lab) in 1991-2001. Others do not seem to have a special connection to aeronautics. Wilson was a theoretical particle physicist, Huizenga was an experimental nuclear scientist, Chu is known for his work on superconductors and dielectrics, and Messier for nanomorphology of thin films.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-11-13T06:40:48.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11134" PostTypeId="1" CreationDate="2019-11-13T07:51:22.600" Score="4" ViewCount="954" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Human_brain#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;early reference&lt;/a&gt; to the brain is found in Edwin Smith Papyrus&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Edwin Smith Papyrus, an ancient Egyptian medical treatise &lt;strong&gt;written&lt;/strong&gt;&#xA;  in the 17th century BC, contains the earliest recorded reference to&#xA;  the brain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Edwin Smith Papyrus is &lt;a href=&quot;https://www.britannica.com/topic/Edwin-Smith-papyrus&quot; rel=&quot;nofollow noreferrer&quot;&gt;believed&lt;/a&gt; to be around 3000 BC&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Edwin Smith papyrus, (c. 1600 BC), ancient Egyptian medical treatise,&#xA;  believed to be a copy of a work dating from c. 3000 BC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although the text contains about brain, the idea of brain's thinking  capability  was not mentioned in it. And fifth century BC is the time of origin for the idea that brain has the &lt;a href=&quot;https://en.wikipedia.org/wiki/Human_brain#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;capability&lt;/a&gt; of thinking.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the &lt;strong&gt;fifth century BC,&lt;/strong&gt; Alcmaeon of Croton in Magna Grecia, first&#xA;  considered the brain to be the seat of the mind. Also in the&#xA;  fifth century BC in Athens, the unknown author of On the Sacred&#xA;  Disease, a medical treatise which is part of the Hippocratic Corpus&#xA;  and traditionally attributed to Hippocrates, believed the brain to be&#xA;  the seat of intelligence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So summarizing all, the idea that &lt;strong&gt;human head&lt;/strong&gt; has the capability of thinking was started from fifth century BC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is my inference correct? Or are there any early mentions that &lt;strong&gt;thoughts arise from human head&lt;/strong&gt; before fifth century BC?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Please note that I am &lt;strong&gt;not&lt;/strong&gt; asking for the idea thoughts arise from human brain. It is enough if there is any idea of thoughts arising from human head before the time mentioned above.&lt;/p&gt;&#xA;" OwnerUserId="2692" LastEditorUserId="55" LastEditDate="2019-11-13T22:15:43.653" LastActivityDate="2024-02-06T21:58:25.393" Title="When did the idea of thoughts occuring in the human head originate?" Tags="|biology|neuroscience|physiology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11135" PostTypeId="2" ParentId="11082" CreationDate="2019-11-13T21:41:35.547" Score="3" Body="&lt;p&gt;Clara Viola, Donne matematiche. (Italian) [Women mathematicians] Matematiche Complementari, 3. Aracne Editrice, Rome, 2015. 83 pp. ISBN: 978-88-548-8037-5&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My Italian is limited, but it seems that this collection of biographies of selected women mathematicians (including Maria Gaetana Agnesi and Sophie Germain, among others) is for readers older than 12 years. Still, it looks pretty accessible.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2019-11-13T21:41:35.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11136" PostTypeId="2" ParentId="11134" CreationDate="2019-11-13T22:07:47.707" Score="5" Body="&lt;p&gt;We should note that the underlying evidence behind the spiced up modern titles like &quot;seat of the mind, &quot;seat of intelligence&quot;, etc., is generally very obscure, and Wikipedia's phrasing is copied almost verbatim from an old and sketchy &lt;a href=&quot;https://web.archive.org/web/20130505044949/http://www.princeton.edu/~cggross/Hist_Neurosci_Ency_neurosci.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper of Gross&lt;/a&gt;. &lt;a href=&quot;https://plato.stanford.edu/entries/alcmaeon&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP's&lt;/a&gt; recent account is more ambiguous as to what it was that Alcmaeon thought the brain did: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;There is no explicit evidence, however, as to what Alcmaeon meant by understanding. The word translated as understanding here is suniêmi, which in its earliest uses means “to bring together,” so that it is possible that Alcmaeon simply meant that humans are able to bring the information provided by the senses together in a way that animals cannot (Solmsen 1961, 151)... It is possible that we should use a passage in Plato’s Phaedo (96a-b = A11) to explicate further Alcmaeon’s epistemology. The passage is part of Socrates’ report of his early infatuation with natural science and with questions such as whether it is the blood, or air, or fire with which we think. He also reports the view that it is the brain that furnishes the sensations of hearing, sight, and smell. This corresponds very well with Alcmaeon’s view of the brain as the central sensory organ&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But even the brain as the sense central might have been a superficial speculation. In any case, it failed to convince most Greeks who continued to designate the heart as &quot;the seat&quot;, following the Egyptians before them:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Alcmaeon’s conclusion that all of the senses are connected to the brain may have been drawn from nothing more that the excision of the eye and the general observation that the sense organs for sight, hearing, smell, and taste are located on the head and appear connected to passages which lead inward towards the brain... His identification of the brain as the seat of human intelligence influenced Philolaus (DK, B13), the Hippocratic Treatise, On the Sacred Disease, and Plato (Timaeus 44d), although a number of thinkers including Empedocles and Aristotle continued to regard the heart as the seat of perception and intelligence.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At this level of detail and confidence, there are earlier mentions that can be interpreted as assigning &quot;thinking&quot; to the brain in the ancient Mesopotamia (c. 2000 BC). For example, after a long interpretive analysis of the sources, Goodnick and Sigrist conclude in &lt;a href=&quot;http://ancientworldonline.blogspot.com/2016/06/newly-open-access-journal-journal-des.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Brain, the Marrow and the Seat of Cognition in Mespotamian Tradition&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In conclusion, according to our argument the Sumerians noted the existence of the brain, which they designated ugu-dig, and believed that understanding passed through the ear to the seat of the intelligence. On the other hand, the Akkadians not only collapsed the lexical distinction between the Sumerian lemmas, ugu «cranium» and ugu-dig &quot;brain&quot; but also equated both with the Semitic muhhu «marrow» (of bones including the cranium). The ancient physicians did not know the function of the brain as an organ and it is quite likely that they considered it a type of skull marrow. Nevertheless, the ancient metaphysical interpretations, both Sumerian and Akkadian, place understanding, reason and wisdom in the ear. Thus, despite the heart being seat of will, the head or brain, the place where temu «reason» is found, must be the seat of cognition, whether or not the medical profession understood the functions of the brain as an organ.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A comprehensive review of the early history, with analysis of the sources, including the Smith and Ebers papyri, is the collection &lt;a href=&quot;https://books.google.com/books?id=uTTYCgAAQBAJ&amp;amp;lr=&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Neurology&lt;/a&gt;. &lt;a href=&quot;https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780195181821.001.0001/acprof-9780195181821-chapter-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Minds Behind the Brain: A history of the pioneers and their discoveries&lt;/a&gt;. According to York and Steinberg's contribution, even with Egypt we have to interpolate:&quot;&lt;em&gt;cavalier treatment of the brain suggests that the ancient Egyptians had no real conception of brain function, or of the primacy of the brain in cognition, though certainty on the subject is not possible&lt;/em&gt;&quot;. The earliest place that offers something more cogent and definitive than speculative interpolations is the aforementioned &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Sacred_Disease&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hippocratic treatise On the Sacred Disease (c. 400 BC)&lt;/a&gt;, discussed by Karenberg:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I hold that the brain is the most powerful organ of the human body, for when it is healthy it is an interpreter to us of the phenomena caused by air, as it is the air that gives it intelligence. Eyes, ears, tongue, hands and feet act in accordance with the discernment of the brain... Wherefore I assert that the brain is the interpreter of consciousness... Men ought to know that from the brain, and from the brain only, arise our pleasures, joys, laughter and jests, as well as our sorrows, pains, griefs and tears. Through it, in particular, we think, see, hear, and distinguish the ugly from the beautiful, the bad from the good, the pleasant from the unpleasant...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A more popular recent history text is &lt;a href=&quot;https://books.google.com/books/about/Minds_Behind_the_Brain.html?id=3OWU1wnOy84C&quot; rel=&quot;nofollow noreferrer&quot;&gt;Finger's Minds Behind the Brain&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-14T02:46:14.593" LastActivityDate="2019-11-14T02:46:14.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11137" PostTypeId="1" CreationDate="2019-11-13T23:16:14.070" Score="-5" ViewCount="232" Body="&lt;p&gt;I read recently in an in-depth book-length study by an investigative journalist on the Rwandan Genocide that the work of the French psychologist, Robert Muchielli, was implicated in organised propaganda preceding the genocide. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is an &lt;a href=&quot;https://fr.wikipedia.org/wiki/Roger_Mucchielli&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; in the French Wikipedia, which I note, calls him a 'psychopedagogue', which in French, probably has a normative meaning, but which if the allegation is correct, has overtones in English which are difficult not to miss in this context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;His work has been compared to the Nazi propagandist Goebbels by Human Rights Watch in their report &lt;em&gt;Propaganda and Practise&lt;/em&gt;. I'm looking for a more in depth study of the influence of his work on the &lt;em&gt;genocidaires&lt;/em&gt; of Rwanda.. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2019-11-21T16:58:40.780" LastActivityDate="2019-11-21T16:58:40.780" Title="What was the work of Robert Muchielli's, a French psychologist, role in the Rwandan Genocide?" Tags="|reference-request|psychology|" AnswerCount="1" CommentCount="9" ClosedDate="2019-11-14T19:17:11.800" ContentLicense="CC BY-SA 4.0" />
  <row Id="11138" PostTypeId="2" ParentId="11137" CreationDate="2019-11-13T23:47:53.790" Score="2" Body="&lt;p&gt;&lt;sub&gt; –– Note: this currently answers the f&lt;a href=&quot;https://hsm.stackexchange.com/revisions/11137/1&quot;&gt;irst revision of the question.&lt;/a&gt; The current edit invalidates this answer.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems to be a bit more indirect than how I read your question. But here are the relevant parts from one such reference:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Propagandists echoed and magnified the hatred and suspicion sown by Habyarimana and officials around him. Under the cover of the newly-established freedom of the press, they blared forth messages disseminated more discreetly by officials, such as many of the conclusions about the “enemy” presented in the military memorandum of September 21, 1992.&lt;/p&gt;&#xA;&lt;p&gt;Propagandists developed the same themes over and over, both before and during the genocide. While some of the similarities in their messages may result simply from sharing the same cultural milieu, other similarities in technique suggest deliberate coordination among propagandists and between them and government officials. In a mimeographed document entitled “Note Relative à la Propagande d’Expansion et de Recrutement,” found in Butare prefecture, one propagandist tells others how to sway the public most effectively. Obviously someone who had studied at university level, the author of the note presents a detailed analysis of a book called &lt;strong&gt;Psychologie de la publicité et de la propagande, by Roger Mucchielli, published in Paris in 1970.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;The author&lt;/em&gt; of the note claims to convey lessons learned from the book and drawn from Lenin and Goebbels. He advocates using lies, exaggeration, ridicule, and innuendo to attack the opponent,&lt;/strong&gt; in both his public and his private life. He suggests that moral considerations are irrelevant, except when they happen to offer another weapon against the other side. He adds that it is important not to underestimate the strength of the adversary nor to overestimate the intelligence of the general public targeted by the campaign. Propagandists must aim both to win over the uncommitted and to cause divisions among supporters of the other point of view. They must persuade the public that the adversary stands for war, death, slavery, repression, injustice, and sadistic cruelty.&lt;/p&gt;&#xA;&lt;p&gt;In addition to these suggestions, the propagandist proposes two techniques that were to become often used in Rwanda. The first is to “create” events to lend credence to propaganda. He remarks that this tactic is not honest, but that it works well, provided the deception is not discovered. The “attack” on Kigali on October 4-5, 1990 was such a “created” event, as were others—the reported discovery of hidden arms, the passage of a stranger with a mysterious bag, the discovery of radio communications equipment—that were exploited later, especially during the genocide.&lt;/p&gt;&#xA;&lt;p&gt;The propagandist calls his second proposal “Accusation in a mirror,” meaning his colleagues should impute to enemies exactly what they and their own party are planning to do. He explains, “In this way, the party which is using terror will accuse the enemy of using terror.” With such a tactic, propagandists can persuade listeners and “honest people” that they are being attacked and are justified in taking whatever measures are necessary “for legitimate [self-] defense.” This tactic worked extremely well, both in specific cases such as the Bugesera massacre of March 1992 described below and in the broader campaign to convince Hutu that Tutsi planned to exterminate them. There is no proof that officials and propagandists who “created” events and made “accusations in a mirror” were familiar with this particular document, but they regularly used the techniques that it described.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;April, 1994&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;April 3: RTLM broadcast a prediction that the RPF would do “a little something” with its bullets and grenades on April 3 to April 5 and again from April 7 to 8. This may have been an “accusation in a mirror”—like that advocated by the disciple of the propaganda expert Mucchielli—with Hutu hard-liners accusing Tutsi of preparing to do just what they themselves were planning.137 The prediction increased fears in an already tense situation. Some people who felt at risk sent their children away from Kigali while others took refuge in places thought to be safe havens.&lt;/p&gt;&#xA;&lt;p&gt;Like the “spontaneous anger” justification, this effort at legitimating violence through “self- defense” was meant both to quiet foreign critics and to incite Hutu to kill more. When the propagandist who disseminated his summary of the work of Mucchielli wrote about “accusations in a mirror,” he recommended that adversaries be accused of terrorism because “honest people” will take action if they believe they are legitimately defending themselves.112 Officials and propagandists alike encouraged Hutu to feel righteous anger at the Tutsi and to give “them the punishment they deserve.”&lt;/p&gt;&#xA;&lt;p&gt;In another incident that afternoon, a Tutsi charged the burgomaster with a machete in an attempt to kill him. According to one survivor:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Just after he [the burgomaster] spoke, one man ran up to try to kill him. He said, “I am going to die, but I will save a lot of people.” He was stopped, of course, but this frightened the burgomaster, so he left right after that. He took his family to safety in Butare and went to get more military men.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another informant who saw the event confirms this version. She reports that the assailant, who was her uncle, was stopped before he even reached the burgomaster and that the burgomaster was not injured in the attack.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Ntaganzwa and his supporters made full use of these incidents to heighten fear of the Tutsi exactly as the disciple of the propaganda expert Mucchielli had directed&lt;/strong&gt;. The burgomaster traveled throughout the commune with his head bandaged warning the population that RPF soldiers were in the church, hiding in the midst of the Tutsi civilians. He insisted that everyone must help defend the commune. A Hutu witness from Rutobwe recounts:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The burgomaster went around doing propaganda meetings, during which he said that the people of Cyahinda had thrown a grenade at him and that he had escaped by a miracle. There were witnesses who said that it was only a stone, but the burgomaster said that it was a grenade. He got in his pickup truck with his head bandaged and went around telling the population: “They tried to kill me!” People saw that his head was bandaged and they believed. I believed itmyself when I saw his bandage. Only later did I find out that he had taken advantage of the stone to arouse anger in the commune: a stone had become a grenade. And the truth? Well, people saw the bandage and believed it was the truth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;–– Alison Liebhafsky Des Forges: &amp;quot;Leave None to Tell the Story: Genocide in Rwanda&amp;quot;, Human Rights Watch, Year: 1999.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6224" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-11-15T23:04:03.820" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11139" PostTypeId="1" AcceptedAnswerId="11140" CreationDate="2019-11-14T08:57:13.623" Score="4" ViewCount="274" Body="&lt;p&gt;In 1932 positron and annihilation were discovered, but in 1928 Dirac had provided a formula allowing for the positron&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did anyone between that date and the discovery of electron ever imagine or suggest of antimatter or a positive electron. How did they interpret the electron?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="64" LastEditDate="2019-11-14T13:47:33.567" LastActivityDate="2019-11-23T13:00:14.923" Title="Did anyone mention the possibility of antimatter before 1928?" Tags="|physics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11140" PostTypeId="2" ParentId="11139" CreationDate="2019-11-14T10:51:37.943" Score="4" Body="&lt;p&gt;Not quite as specific as what you're looking for, but &lt;a href=&quot;https://www.google.com/search?q=%22positive+electron%22&amp;amp;tbs=cdr:1,cd_min:1800,cd_max:1931&amp;amp;tbm=bks&quot; rel=&quot;nofollow noreferrer&quot;&gt;searching google-books using the phase &quot;positive electron&quot; for 1800-1931&lt;/a&gt; gives several pages of hits. Most of these use the phrase in connection with atomic nuclei (or with alpha particles, prior to Rutherford's famous scattering experiment), but some appear to involve speculation about the existence of a positively charged analog of the electron, such the following two. I don't have any more details about the 1928 reference because this journal does not seem to be freely available on the internet. However, the journal can probably be found in most major U.S. universities (e.g. all print volumes from 1861 are at The University of Iowa). I tried adding &quot;Dirac&quot; to the search results, and two or three things looked promising, but none seemed to be freely available, and thus I'll leave it to you or others to pursue further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ernest Rutherford, &lt;a href=&quot;https://books.google.com/books?id=1iMDAAAAMBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Present problems in radioactivity&lt;/em&gt;&lt;/a&gt; (also &lt;a href=&quot;https://archive.org/details/popularsciencemo67newy/page/8&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), &lt;strong&gt;Popular Science Monthly&lt;/strong&gt; 67 [#1] (May 1905), 6-34.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from bottom of p. 9)&lt;/strong&gt; ... no evidence has been obtained of the existence of a positive electron corresponding to the negative electron.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?tbm=bks&amp;amp;q=%22picture+a+true+equivalent+to+the+negative+electron%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Chemical News and Journal of Industrial Science&lt;/strong&gt; 136 (1928)&lt;/a&gt; (also &lt;a href=&quot;https://catalog.hathitrust.org/Record/007085564&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;(from p. 194)&lt;/strong&gt; Let us, for instance, picture a true equivalent to the negative electron---a positive electron of the same infinitesimal atomic weight. Then if ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2019-11-14T10:56:50.200" LastActivityDate="2019-11-14T10:56:50.200" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11142" PostTypeId="2" ParentId="11082" CreationDate="2019-11-14T12:21:28.617" Score="1" Body="&lt;p&gt;You can search a book in the site &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://matematica.unibocconi.it&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://matematica.unibocconi.it&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here you will surely find books on the history of mathematics, and they also hold conferences for secondary school teachers.&#xA;This is a book I recommend:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.it/books?id=nx2J7hqZ43EC&amp;amp;printsec=frontcover&amp;amp;dq=MATEpristem&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=0ahUKEwiNzYyi1-nlAhUHalAKHUZeBFs4ChDoAQgnMAA#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vite Matematiche, protagonisti del '900 da Hilbert a Wiles, Springer&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;written by Bocconi professors&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://redooc.com/it/medie/scienze/donne-matematica&quot; rel=&quot;nofollow noreferrer&quot;&gt;Donne nella matematica&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.editorialescienza.it/it/collana/donne-nella-scienza.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Donne nella scienza&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.it/Donne-scienza-cambiato-mondo-colori/dp/886526912X/ref=sr_1_2?adgrpid=49392159581&amp;amp;gclid=EAIaIQobChMIi5_pirLz5QIVSeN3Ch1WrQ0AEAAYASAAEgKyj_D_BwE&amp;amp;hvadid=255215006158&amp;amp;hvdev=c&amp;amp;hvlocphy=1008463&amp;amp;hvnetw=g&amp;amp;hvpos=1t1&amp;amp;hvqmt=b&amp;amp;hvrand=17941874553284635504&amp;amp;hvtargid=kwd-305742285104&amp;amp;hydadcr=28433_1717417&amp;amp;keywords=donne%20scienza&amp;amp;qid=1574067429&amp;amp;sr=8-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Donne di scienza. 50 donne che hanno cambiato il mondo&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.it/Storie-superdonne-hanno-scienza-colori/dp/8893813483/ref=pd_sim_14_3/257-9491307-8760216?_encoding=UTF8&amp;amp;pd_rd_i=8893813483&amp;amp;pd_rd_r=3fc1852d-ab5c-43cf-bdc6-a816d047d7ab&amp;amp;pd_rd_w=hCWlQ&amp;amp;pd_rd_wg=WqNpv&amp;amp;pf_rd_p=9aa20f03-8d38-4980-9e7f-216203b2f33d&amp;amp;pf_rd_r=PDNK79GKH6CMKNQTP61J&amp;amp;psc=1&amp;amp;refRID=PDNK79GKH6CMKNQTP61J&quot; rel=&quot;nofollow noreferrer&quot;&gt;Superdonne di scienza&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.editorialescienza.it/it/libro/la-trottola-di-sofia.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;La trottola di Sofia&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8171" LastEditorUserId="26" LastEditDate="2019-12-04T22:35:10.617" LastActivityDate="2019-12-04T22:35:10.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11143" PostTypeId="2" ParentId="11082" CreationDate="2019-11-14T14:02:49.310" Score="1" Body="&lt;p&gt;arXiv:1910.01730&#xA;Twenty Female Mathematicians&#xA;by&#xA;Hollis Williams,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;arXiv:1903.02543&#xA;A data analysis of women's trails among ICM speakers&#xA;by Helena Mihaljević, Marie-Françoise Roy&lt;/p&gt;&#xA;&#xA;&lt;p&gt;arXiv:1502.07597&#xA;Women mathematicians in France in the mid-twentieth century&#xA;by Yvette Kosmann-Schwarzbach&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-11-14T14:02:49.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11144" PostTypeId="2" ParentId="11139" CreationDate="2019-11-14T14:22:27.117" Score="-3" Body="&lt;p&gt;Newton had already discussed the possibility of repulsive forces between particles. This was obviously a long time before the discovery of electricity so it wasn't phrased in terms of positive and negatively charged particles. But the kernel of the idea was there.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to add, that Aristotle (which obviously Newton had read), had noted that all ancient philosophers were agreed that principles are contraries, that is they come as opposites. This might have a bearing on the above physically grounded speculation y Newton. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2019-11-14T15:53:23.653" LastActivityDate="2019-11-14T15:53:23.653" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11145" PostTypeId="2" ParentId="11082" CreationDate="2019-11-14T14:58:15.697" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sofya_Kovalevskaya#Novel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sofya Vasilevneya Kovalevskaya&lt;/a&gt; is a famous female Russian mathematician:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;who made noteworthy contributions to analysis, partial differential equations and mechanics. She was a pioneer for women in mathematics around the world – the first woman to obtain a doctorate (in the modern sense) in mathematics, the first woman appointed to a full professorship in Northern Europe and one of the first women to work for a scientific journal as an editor.[1] According to historian of science Ann Hibner Koblitz, Kovalevskaia was &amp;quot;the greatest known woman scientist before the twentieth century&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the 20th Century, the most famous female mathematician is &lt;a href=&quot;https://en.wikipedia.org/wiki/Emmy_Noether&quot; rel=&quot;nofollow noreferrer&quot;&gt;Emmy Noether&lt;/a&gt; and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;she was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The articles above list a number of books describing their achievements and life-story, amongst them are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Kennedy, Don H. (1983). Little Sparrow, a Portrait of Sofia Kovalevsky. Athens: Ohio University Press.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Koblitz, Ann Hibner (1993). A Convergence of Lives: Sofia Kovalevskaia -- Scientist, Writer, Revolutionary.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Tent, M.B.W. (2008), Emmy Noether: The Mother of Modern Algebra&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I can't resist mentioning that Sofia also wrote a memoir later in life, titled &lt;em&gt;A Russian Childhood&lt;/em&gt;, where she described Dostoyevsky as ugly and also how she first discovered a fascination for mathematics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;She used to stare fascinated at one of the walls of one of the rooms in the Palibino estate which had been covered with the lithographed noted of a calculus course ... Kovalevskaya apparently believed that this haphazard introduction to the subject nevertheless gave her a familiarity with the symbols and formulas of the calculus.&lt;/p&gt;&#xA;&lt;p&gt;(quoted from &lt;em&gt;The Mathematics of Sofia Kovalevskaya&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for a book about for young girls and which is about the paradoxes and peculiarities of logic, language and reasoning and written in a vivid, personable style you can't beat &lt;em&gt;Alice in Wonderland&lt;/em&gt; and &lt;em&gt;Alice through the Looking-Glass&lt;/em&gt;. I expect this is available in Italian.&lt;/p&gt;&#xA;&lt;p&gt;Another possibility is &lt;a href=&quot;https://en.wikipedia.org/wiki/Rosalind_Franklin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rosalind Franklin&lt;/a&gt;, who although is not a mathematician, as an X-ray crystallographer would have definitely have used mathematics in her work in obtaining the data for the structure of DNA. Her life is outlined in:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cambridge Women: twelve Portraits (CUP).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-11-15T11:00:13.980" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11146" PostTypeId="2" ParentId="11139" CreationDate="2019-11-14T15:42:59.757" Score="-3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The term antimatter was first used by Arthur Schuster in two rather&#xA;  whimsical letters to Nature in 1898,[13] in which he coined the term.&#xA;  He hypothesized antiatoms, as well as whole antimatter solar systems,&#xA;  and discussed the possibility of matter and antimatter annihilating&#xA;  each other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Antimatter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Antimatter From Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://english.stackexchange.com/questions/15994/when-was-antimatter-first-used&quot;&gt;https://english.stackexchange.com/questions/15994/when-was-antimatter-first-used&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The idea of antimatter seems to have been part of the ancient secret canon that Newton, Kepler and others were privy to: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;Viktor Schauberger;&#xA;The Repulsine, UFO's and Flying Saucers of Nazi Germany&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;It was nearly the end of WWII. At that same time, scientist Viktor Schauberger worked on a secret project. Johannes Kepler, whose ideas&#xA;  Schauberger followed, had knowledge of the secret teachings of&#xA;  Pythagoras that had been adopted and kept secret. It was the knowledge&#xA;  of Implosion (in this case the utilization of the potential of the&#xA;  inner worlds in the outer world). Hitler knew - as did the Thule and&#xA;  Vril people - that the divine principle was always constructive. A&#xA;  technology however that is based on explosion and therefore is&#xA;  destructive runs against the divine principle. Thus they wanted to&#xA;  create a technology based on Implosion. Schauberger's theory of&#xA;  oscillation (principle of the overtone sequence, monochord) takes up&#xA;  the knowledge of Implosion. To put it simply: Implosion instead of&#xA;  Explosion! Following the energy paths of the monochord and the&#xA;  implosion technology one reaches the realm of antimatter and thus the&#xA;  cancellation of gravity.&lt;/em&gt;  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Van Helsing, E. S., &amp;amp; Canaria, G. (1995). &lt;em&gt;Secret Societies and Their Power in the 20th Century&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Schauberger's theory is also claimed by others to have had connections with the ancient Vimanas and Sumerian legends of antimatter and implosion that the Nazis were working on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;German UFO literature very often conforms largely to documented history on the following points: ...&#xA;Nazi Germany conducted research into advanced propulsion technology, ... Viktor Schauberger's engine research, ....&quot;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nazi_UFOs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nazi_UFOs&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;This article deals with a military black project.  Because of the nature of such projects, the most authoritative sources (any involved governments and defense contractors) may not even acknowledge its existence. The most reliable sources may be highly speculative.&quot;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Talk:Nazi_UFOs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Talk:Nazi_UFOs&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=8ZxNCyIFdqE&amp;amp;t=422s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Third Reich Operation UFO Nazi Base In Antarctica Complete Documentary&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Trustworthy, authentic information about Viktor Schauberger directly from primary sources.  ....  The sharing of this information and knowledge is supported by the PKS Organisation (Pythagoras Kepler System according to Viktor and Walter Schauberger).&quot; &lt;a href=&quot;http://pks.or.at/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://pks.or.at/en/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Implosion is no invention in the conventional sense, but rather the renaissance of ancient knowledge, lost over the course of time.&quot;&#xA;&lt;em&gt;Implosion Magazine&lt;/em&gt;, No. 83, p. 16  in Callum Coats: &lt;em&gt;Energy Evolution&lt;/em&gt;, 2000.&#xA;&quot;You must learn to think one octave higher. Only then will you learn how implosion energy works.&quot;&#xA;&lt;em&gt;Implosion Magazine&lt;/em&gt;, No. 83, p. 27 in Callum Coats: &lt;em&gt;Energy Evolution&lt;/em&gt;, 2000.&#xA;&lt;a href=&quot;https://en.wikiquote.org/wiki/Viktor_Schauberger&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikiquote.org/wiki/Viktor_Schauberger&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The implosive forces in Schauberger's turbines were compared to those found in tornadoes and hurricanes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fishman, Gerald J. &quot;Positrons observed to originate from thunderstorms.&quot; &lt;em&gt;Eos, Transactions American Geophysical Union&lt;/em&gt; 92.22 (2011): 185-186.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Castelvecchi, Davide. &quot;Rogue antimatter found in thunderclouds.&quot; &lt;em&gt;Nature News&lt;/em&gt; 521.7551 (2015): 135.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bowers, G. S., et al. &quot;A Terrestrial Gamma‐Ray Flash inside the Eyewall of Hurricane Patricia.&quot; &lt;em&gt;Journal of Geophysical Research: Atmosphere&lt;/em&gt;s 123.10 (2018): 4977-4987. &quot;We report the first observation of gamma‐ray emission from lightning within a hurricane eyewall, consistent with production by a downward beam of positrons.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1950" LastEditorUserId="1950" LastEditDate="2019-11-23T13:00:14.923" LastActivityDate="2019-11-23T13:00:14.923" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11147" PostTypeId="1" AcceptedAnswerId="11148" CreationDate="2019-11-14T21:52:32.777" Score="1" ViewCount="164" Body="&lt;p&gt;How as light intensity measured or determined around 1900? For example, when it was determined that the photoelectric effect was independent of the intensity of the light beam, but rather that it depended on the wavelength, how was the intensity measured?&lt;/p&gt;&#xA;" OwnerUserId="3295" LastEditorUserId="55" LastEditDate="2019-11-15T08:51:09.577" LastActivityDate="2019-11-15T11:38:16.297" Title="How was it determined that the photoelectric effect is independent of the intensity of light?" Tags="|quantum-mechanics|experimental-physics|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11148" PostTypeId="2" ParentId="11147" CreationDate="2019-11-15T08:48:56.873" Score="4" Body="&lt;p&gt;There was no need to measure the intensity directly to determine that the photoeffect is independent of it. One could vary intensity by moving the metal plate closer to or further away from the source. How intensity falls with the distance is well-known. This is what Lenard did in his 1902 experiments that prompted Einstein's 1905 work.&lt;/p&gt;&#xA;&lt;p&gt;If there &lt;em&gt;was&lt;/em&gt; a need to measure intensity, Lenard's preferred method was to use phosphorescent materials to make the ultraviolet (or cathode, now X) rays visible, rather than materials that get heated by the light. However, absolute units of intensity were poorly standardized at the time even for visible light. In Germany, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hefner_lamp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hefnerkerze (Hefner lamp)&lt;/a&gt; developed by Hefner in 1884, was used for the purpose. One &lt;a href=&quot;https://www.sizes.com/units/hefner.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;hefner&lt;/a&gt; was defined as the horizontal intensity of the light from such a lamp, and is approximately 0.903 candela. Candelas were only standardized in 1948.&lt;/p&gt;&#xA;&lt;p&gt;Here is Lenard's original paper &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19023130510&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Ueber die lichtelektrische Wirkung&lt;/em&gt; (About the photoelectric effect, 1902)&lt;/a&gt;. In English, he recounts his methodology in the &lt;a href=&quot;https://www.nobelprize.org/uploads/2018/06/lenard-lecture.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nobel lecture On Cathode Rays (1906)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It must be noted that the rays are not directly visible; it would be useless to put one’s eye to the window, as this organ is not receptive to cathode rays. On the other hand, materials that are capable of becoming luminous without heat, phosphorescent materials as they are called, are suitable for making the rays visible. It is best to use sheets of paper coated with such materials, e.g. a certain ketone, platinum cyanide, or an alkaline-earth phosphor and to hold them as a screen against the rays. If the screen glows, it indicates that it has been hit by the rays. The rays can also be photographed directly.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;These are the same methods that are used to make visible ultraviolet light, at that time the only known example of such demonstrable invisible radiation. When we use the phosphorescent screen, we find it glowing brightly close to the window; as the distance from the window increases, the intensity of the rays progressively diminishes until at a distance of about 8 cm the screen remains quite dark.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;I have also&#xA;found that the velocity is independent of the ultraviolet light intensity (M), and&#xA;thus concluded that the energy at escape does not come from the light at all,&#xA;but from the interior of the particular atom. The light only has an initiating&#xA;action, rather like that of the fuse in firing a loaded gun. I find this conclusion&#xA;important since from it we learn that not only the atoms of radium - the&#xA;properties of which were just beginning to be discerned in more detail at that&#xA;time - contain reserves of energy, but also the atoms of the other elements;&#xA;these too are capable of emitting radiation and in doing so perhaps completely&#xA;break down, corresponding to the disintegration and roughening of the&#xA;substances in ultraviolet light.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The last paragraph sketches Lenard's (classical) explanation of photoeffect, the &amp;quot;triggering hypothesis&amp;quot;, which was largely accepted until 1911, contrary to the now popular opinion, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2306/55&quot;&gt;Were there serious attempts to model the photoelectric effect classically?&lt;/a&gt; It was disprioved by Lenard himself, ionization in gases by ultraviolet light did not&#xA;produce strong ionization unaccompanied by absorption&#xA;of the light. For a historical commentary see &lt;a href=&quot;https://www.jstor.org/stable/27757381&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philipp Lenard and the Photoelectric Effect, 1889-1911, by Wheaton&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is only since 1911 that the photoeffect has been interpreted as a transformation of light energy into electron kinetic energy. In 1902 Lenard concluded that the effect is a resonance phenomenon, but one in which the light contributes no energy to electrons. The light only selects which electrons are to be ejected from an atom. The velocity of a released electron is predetermined within the atom. Consequently a study of the velocity distribution of photoelectrons promised to shed light on atomic constitution.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;For nine years the triggering hypothesis was the accepted explanation of the photoelectric effect. It enabled physicists to avoid the serious difficulties which obstructed formulation of a consistent mechanism for the absorption of classical electromagnetic waves. The light quantum proposed by Einstein to answer more general issues incidentally offered another explanation of the photoeffect. But other physicists preferred the triggering hypothesis.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-11-15T11:38:16.297" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11149" PostTypeId="1" AcceptedAnswerId="11150" CreationDate="2019-11-15T09:45:16.443" Score="5" ViewCount="155" Body="&lt;p&gt;In old physics and engineering publications from the 1950s or so, it's common to see non-SI &quot;double prefixes&quot;, such as a &quot;millimicrosecond pulse&quot;, or a &quot;10 micromicrofarad&quot; capacitor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These units are considered obsolete today. However, I failed to find a historical review on the use of these units.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In which year did units like &quot;nano-&quot; and &quot;pico-&quot; become parts the SI-standard?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;When did the use of standard SI-prefix become common and largely replaced the &quot;double prefixes&quot; in engineering?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10481" LastEditorUserId="55" LastEditDate="2019-11-15T10:45:07.563" LastActivityDate="2019-11-15T10:45:07.563" Title="When did non-SI double prefixes go out of use?" Tags="|terminology|units|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11150" PostTypeId="2" ParentId="11149" CreationDate="2019-11-15T10:43:59.883" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://www.bipm.org/en/CGPM/db/11/12/&quot; rel=&quot;noreferrer&quot;&gt;Resolution 12&lt;/a&gt; of the 11th Conférence Générale des Poids et Mesures (CGPM) adopted 12 SI prefixes in 1960, including pico- and nano-. Google Ngrams show steep decline in the use of &lt;a href=&quot;https://books.google.com/ngrams/graph?content=millimicro&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Cmillimicro%3B%2Cc0#t1%3B%2Cmillimicro%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;millimicro-&lt;/a&gt; after 1964, and &lt;a href=&quot;https://books.google.com/ngrams/graph?content=micromicro&amp;amp;year_start=1800&amp;amp;year_end=2000&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Cmicromicro%3B%2Cc0#t1%3B%2Cmicromicro%3B%2Cc1&quot; rel=&quot;noreferrer&quot;&gt;micromicro-&lt;/a&gt; after 1966. Some other &lt;a href=&quot;https://en.wikipedia.org/wiki/Metric_prefix#Double_prefixes&quot; rel=&quot;noreferrer&quot;&gt;double prefixes&lt;/a&gt;, like kilomega- and hectokilo- were also in use, decimilli- was even standardized in Frace until 1961.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-11-15T10:43:59.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11151" PostTypeId="2" ParentId="9545" CreationDate="2019-11-15T11:23:08.707" Score="0" Body="&lt;p&gt;I'd say that the arrangement of axes in 3d is due to the conventions we have for the cross product, as the cross product &lt;em&gt;U x V&lt;/em&gt; is normal to the plane spanned by the vectors &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;; normal, suggests vertical or perpendicularity and hence we have the z-axis pointing towards 12 O'Clock. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worth noting that the diagrams you've drawn are all for &lt;em&gt;spatial&lt;/em&gt; diagrams, when one of the axes is time then diagrams modelling classical mechanics usually have time along the 3 O'clock direction and height at the 12 O'clock direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However this convention is reversed in special and general relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect, the first convention is simply due to the fact that when we model the trajectory of a thrown stone then this trajectory follows a parabola, and this is the same curve that is drawn on the diagram using this convention. Obviously this is of no relevance to SR/GR.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A final observation is that axes actually define charts and whilst the coordinate frame is special in that the axes are all perpendicular, the notion of a manifold uses charts in a democratic sense - all charts are equally important. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2019-11-15T14:30:37.163" LastActivityDate="2019-11-15T14:30:37.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11152" PostTypeId="1" AcceptedAnswerId="11155" CreationDate="2019-11-15T15:00:39.483" Score="3" ViewCount="154" Body="&lt;p&gt;Fritz Peter (1899–1949) is known mainly as one of the authors of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peter-Weyl theorem&lt;/a&gt;. This theorem appears in a paper (&lt;em&gt;Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe&lt;/em&gt;) published by him and Hermann Weyl in 1926. It was one of the only two scientific papers (and the only mathematical one) published by him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Thomas Hawkins' &lt;em&gt;Emergence of the Theory of Lie Groups&lt;/em&gt; it is written that &amp;ldquo;Peter, whom Weyl referred to as one of his students […] had received his doctorate from Göttingen in 1923 with a  work on the indices of refraction and the absorption constants of diamonds […]. It would seem that this paper and his joint paper with Weyl were his only scientific publications. He  became assistant master (&lt;em&gt;Studienrat&lt;/em&gt;) and then headmaster (&lt;em&gt;Oberstudiendirektor&lt;/em&gt;) of a secondary school (&lt;em&gt;Schule Schlo&amp;beta; Salem&lt;/em&gt;). Exactly what his contribution to the paper was is unknown to me […]&amp;rdquo;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know where to get more information about Fritz Peter?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I remember having read once that Hermann Weyl thought that Peter's choice of leaving mathematical research and going to work in a school in the Bavarian mountains with an idyllic view was a very wise one, and I would like to know where can I possibly have read this.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-11-16T18:10:28.440" Title="Where to get more biographical information about Fritz Peter?" Tags="|mathematicians|biographical-details|reference-request|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11153" PostTypeId="1" CreationDate="2019-11-15T18:25:09.270" Score="2" ViewCount="90" Body="&lt;p&gt;Several years ago, I remember reading about an experiment, and I thought it was in Newton's Principia, so the experiment was conducted in that time frame. The experiment involved timing the fall of a glass ball from a great height indoors (like the inside of a cathedral). Timing was done by watching a pendulum. There were several observers for each experiment, each one reporting their answer independently and then the final answer was obtained by taking the mean of each individual timing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The thing that caught my eye is that the times were reported in seconds and thirds (1 third=1/60 second). That was the first time I had ever heard of a second as being the &lt;em&gt;second&lt;/em&gt; division of an hour, (the first being minutes), and there being further divisions by 60 each time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have skimmed a paper copy of Principia in my school library and didn't find that experiment. I'm pretty sure I didn't imagine the experiment (I wouldn't have come up with thirds) but I can't find a reference to it. I want something to point to when I point out that a &quot;third&quot; is a real unit of time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone help me find that experiment?&lt;/p&gt;&#xA;" OwnerUserId="10704" LastActivityDate="2019-11-18T13:26:27.860" Title="Glass ball drop experiment" Tags="|reference-request|newton|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11154" PostTypeId="1" CreationDate="2019-11-15T20:59:45.773" Score="0" ViewCount="37" Body="&lt;p&gt;I want some references that point the inventor of Path algebras and history/evolution of these algebras from the first idea. If possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried to search in many different places, but all times, the route stops at Gabriel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS: I asked this question in &lt;a href=&quot;http://%20%20I%20want%20some%20references%20that%20point%20the%20inventor%20of%20Path%20algebras%20and%20history/evolution%20of%20these%20algebras%20from%20the%20first%20idea.%20If%20possible.%20%20I%20tried%20to%20search%20in%20many%20different%20places,%20but%20all%20times,%20the%20route%20stops%20at%20Gabriel.%20%20Thanks.&quot; rel=&quot;nofollow noreferrer&quot;&gt;math.stackexchange.com&lt;/a&gt; and they suggested me to ask it here.&lt;/p&gt;&#xA;" OwnerUserId="10705" LastActivityDate="2019-11-15T20:59:45.773" Title="History of Path algebras" Tags="|mathematics|terminology|abstract-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11155" PostTypeId="2" ParentId="11152" CreationDate="2019-11-16T00:48:34.397" Score="6" Body="&lt;p&gt;A significant obstacle to doing an online search for Fritz Peter is that both Fritz and Peter are common as German first and last names. The &lt;a href=&quot;https://de.wikipedia.org/wiki/Fritz_Peter_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;German Wikipedia&lt;/a&gt; provides some basic pointers, such as his 1923 dissertation in Göttingen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Arne Schirrmacher, &quot;Establishing Quantum Physics in Göttingen: David Hilbert, Max Born, and Peter Debye in Context, 1900-1926.&quot; Springer, 2019, &lt;a href=&quot;https://books.google.com/books?id=P7OmDwAAQBAJ&amp;amp;pg=PA87&amp;amp;dq=%22Peter,+fritz%22+%22Born%22&amp;amp;hl=en&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;sa=X&amp;amp;ved=2ahUKEwjJ17a1sO3lAhXUs54KHc9eCXwQ6AEwCHoECAkQAg#v=onepage&amp;amp;q=%22Peter%2C%20fritz%22%20%22Born%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 87&lt;/a&gt;:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Table 5.1 Dissertations concluded under Born's supervision in Göttingen 1921-1925&lt;br&gt;&#xA;  [...]&lt;br&gt;&#xA;  Peter, Fritz  Über Brechungsindizes und Absorptionskonstanten des Diamanten zwischen λ 644 und 266  1923 Z. Phys.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Note that the title of the dissertation differs slightly (266 vs 226) from the published paper based on it, as can be seen &lt;a href=&quot;https://books.google.com/books?id=vH4tAB4LEb0C&amp;amp;q=ber+Brechungsindizes+und+Absorptionskonstanten+des+Diamanten&amp;amp;dq=ber+Brechungsindizes+und+Absorptionskonstanten+des+Diamanten&amp;amp;hl=en&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;sa=X&amp;amp;ved=2ahUKEwjAwO34se3lAhWPsJ4KHa7GA9sQ6wEwAXoECAIQAQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;from this scan of Z. Phys.&lt;/a&gt; It is:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fritz Peter, &quot;Über Brechungsindizes und Absorptionskonstanten des Diamanten zwischen 644 und 226 mμ&quot;, Zeitschrift für Physik 15 (1923), No. 1, pp. 358-368&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I &lt;em&gt;assume&lt;/em&gt; mμ are milli microns as an old-fashioned way to refer to nanometers. The library of the University of Göttingen has Peter's dissertation available in paper and micro-fiche format. Since German dissertations typically list the birth place of the author (&quot;aus ...&quot;) on the title page, it would probably be helpful to get access to this document. The online search mask for the library is at &lt;a href=&quot;https://opac.sub.uni-goettingen.de&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://opac.sub.uni-goettingen.de&lt;/a&gt; . Select search for &lt;code&gt;[PER] person/author&lt;/code&gt; and enter &lt;code&gt;Peter, Fritz&lt;/code&gt; as the search term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The German Wikipedia mentions that Peter was a student of Hermann Weyl, and Weyl's writings confirm this. &lt;a href=&quot;https://books.google.com/books?id=vImIJE0kQNcC&amp;amp;pg=PA165&amp;amp;dq=A+number+of+years+later,+in+Zurich,+a+pupil+of+mine,+F.+Peter,+and+I+applied+integral+equations&amp;amp;hl=en&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;sa=X&amp;amp;ved=2ahUKEwiL_s_3su3lAhWI_J4KHYLLBcUQ6AEwAHoECAQQAg#v=onepage&amp;amp;q=A%20number%20of%20years%20later%2C%20in%20Zurich%2C%20a%20pupil%20of%20mine%2C%20F.%20Peter%2C%20and%20I%20applied%20integral%20equations&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;For example&lt;/a&gt;:&#xA;Hermann Weyl, &quot;Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics&quot; [ed. Peter Pesic], Princeton University Press 2009, p. 165:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A number of years later, in Zurich, a pupil of mine, F. Peter, and I applied integral equations to the construction of a complete set of inequivalent irreducible representations of a compact Lie group.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Peter is also mentioned as one of Weyl's students on &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1927:26#387&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 233&lt;/a&gt; in the following paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;H. Weyl, &quot;Sur la representation des groupes continus&quot;, &lt;em&gt;L'Enseignement Mathématique&lt;/em&gt;, Vol. 26, 1927, pp. 226-239&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Le chemin que nous avons suivi, un de mes élèves, M. F. Peter et moi, met le problème de la représentation en relation avec la théorie des équations intégrales.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From the &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0097?tify=%7B%22pages%22:[743],%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;the scan&lt;/a&gt; of the paper&lt;/p&gt;&#xA;&#xA;&lt;p&gt;F. Peter and H. Weyl, &quot;Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe&quot;, &lt;em&gt;Mathematische Annalen&lt;/em&gt;, Vol. 97, 1927, pp. 737-755&lt;/p&gt;&#xA;&#xA;&lt;p&gt;we can see that Peter lived in &lt;a href=&quot;https://en.wikipedia.org/wiki/Karlsruhe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karlsruhe&lt;/a&gt;, a major city in Southwestern Germany, at the time of publication.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I cannot find anything about Fritz Peter's time as a teacher (of mathematics and / or physics?) at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schule_Schloss_Salem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schule Schloss Salem&lt;/a&gt;, a private boarding school in Southwestern Germany near Lake Constance. The school was founded in 1920 by Kurt Hahn with support of Prince Maximilian of Baden. There is a &lt;a href=&quot;https://www.bodenseekreis.de/fileadmin/06_bildung_kultur/kultur/kreisarchiv/kurt-hahn-archiv/2015_05_KHA_Flyer_engl.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Hahn Archive&lt;/a&gt; which keeps records of the school's founder and apparently also the school itself:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In addition, it holds a considerable amount of documentation about Schule Schloss Salem, including the estates of Salem teachers and information about other organizations which have followed Kurt Hahn’s educational ideas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A &lt;a href=&quot;https://books.google.com/books?id=_cA_AQAAIAAJ&amp;amp;q=%22Peter+Fritz%22+Mathematiker&amp;amp;dq=%22Peter+Fritz%22+Mathematiker&amp;amp;hl=en&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;sa=X&amp;amp;ved=2ahUKEwi6-_uZ_OzlAhVQsp4KHflwCoQQ6AEwAHoECAIQAg&quot; rel=&quot;nofollow noreferrer&quot;&gt;conference record&lt;/a&gt; from a mathematical conference held in Tübingen in Southwestern Germany in 1946 lists Fritz Peter as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Peter, Fritz, Stud.-Dir. Dr., Überlingen (Bodensee), St. Ulrichstr. 36&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given that Peter died only three years after this conference, it would be a reasonable assumption that he died in Überlingen and that you &lt;em&gt;might&lt;/em&gt; be able to access his death record at the local registration office (&quot;&lt;a href=&quot;https://www.ueberlingen.de/startseite/politik+_+verwaltung/standesamt.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Standesamt&lt;/a&gt;&quot;), unless privacy laws prohibit that (e.g. not a family member, not enough time passed).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The small town of Überlingen is located directly on the shores of Lake Constance (&quot;Bodensee&quot; in German), about 10 km from Schule Schloss Salem. In the 19th century Überlingen was part of the Grand Duchy of Baden. From 1945 to 1949 it was in the French-occupied portion of Baden. Since the founding of the state (&quot;Bundesland&quot;) of Baden-Württemberg in 1952 it has been part of that state.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Studiendirektor&quot;, here abbreviated as &quot;Stud.-Dir.&quot;, is a title for senior teachers  at a &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gymnasium_(school)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gymnasium&lt;/a&gt;&quot; (a particular type of German high school) who also have some administrative responsibilities. The principal of such a school normally holds the title of &quot;Oberstudiendirektor&quot; (modern abbreviation: OstD), one level up from the rank of &quot;Studiendirektor&quot;. I am not very knowledgeable how quickly a teacher can rise to the rank of Studiendirektor, but I think it is about 15 to 25 years. This would very &lt;em&gt;speculatively&lt;/em&gt; fix the start of Peter's teaching career as no later than 1930.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;I found a potential further &lt;a href=&quot;https://books.google.com/books?id=utHNAAAAMAAJ&amp;amp;q=%22Fritz+Peter%22+%22Geographische%22&amp;amp;dq=%22Fritz+Peter%22+%22Geographische%22&amp;amp;hl=en&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;sa=X&amp;amp;ved=2ahUKEwiRoqvcw-3lAhUGvZ4KHe0JAJ8Q6AEwAXoECAIQAg&quot; rel=&quot;nofollow noreferrer&quot;&gt;publication&lt;/a&gt; that &lt;em&gt;might&lt;/em&gt; be attributable to the same Fritz Peter the question is concerned about. I base my hypothesis on a match of the author's name, the author's place of living (Freiburg in Southwestern Germany), date of publication, and the fact that it appeared in a publication dedicated to teaching in math and natural sciences. The fact that the author's name is not given as Dr. Fritz Peter raises doubt about this hypothesis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fritz Peter, &quot;Geographische Ortsbestimmung durch Höhenmessungen.&quot;&#xA;&lt;em&gt;Unterrichtsblätter für Mathematik und Naturwissenschaften&lt;/em&gt;, (Otto Salle, Frankfurt a.M.) Vol. 44, 1938, p.23&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, only a &lt;a href=&quot;https://www.google.com/search?tbm=bks&amp;amp;ei=nEbPXdKWC8zQ-gS3rryADw&amp;amp;q=%22Fritz+Peter+in+Freiburg%22&amp;amp;oq=%22Fritz+Peter+in+Freiburg%22&amp;amp;gs_l=psy-ab.3..33i299k1.126719.134827.0.135172.25.25.0.0.0.0.109.1617.23j2.25.0....0...1c.1.64.psy-ab..0.19.1236...0j33i10k1.0.wTm6LOOvTYY&quot; rel=&quot;nofollow noreferrer&quot;&gt;snippet&lt;/a&gt; is available at Google Books:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Von Fritz Peter in Freiburg. / Zum eisernen Bestand der Aufgaben zur sphärischen Trigonometrie gehört die geographische Ortsbestimmung durch Messung der Zeit, der Höhe und des Azimuts eines Sternes oder der Sonne, wenn Deklination ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2019-11-16T18:10:28.440" LastActivityDate="2019-11-16T18:10:28.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11156" PostTypeId="1" CreationDate="2019-11-16T03:40:37.673" Score="3" ViewCount="1215" Body="&lt;p&gt;I am looking for the origin of the concept of reduced mass as used in vibrational spectroscopy e.g. vibration of a diatomic molecule. Most of the texts simply define reduced mass as the sum of the inverses of masses of two bodies. Surely, this concept must have existed in mechanics long time ago. Does anyone know when and who introduced the concept of reduced mass?&#xA;For the definition, see &lt;a href=&quot;http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#rm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reduced Mass&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-11-24T00:14:27.650" Title="What is the origin of the concept of reduced mass?" Tags="|physics|quantum-mechanics|classical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11157" PostTypeId="2" ParentId="11156" CreationDate="2019-11-16T08:38:09.040" Score="6" Body="&lt;p&gt;The use of reduced mass in spectroscopy goes back to Bohr's planetary model of the atom. Nasri explains the context in his &lt;a href=&quot;https://faculty.uaeu.ac.ae/snasri/Notes/My-Intro-QM-Dec-2016.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;notes on quantum mechanics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1912, Alfred Fowler showed that similar lines can be produced in a laboratory mixture of hydrogen and helium gas. Bohr noticed that they have the same spectrum of spectral lines as of hydrogen but with wavelength four times shorter and argued that they could correspond to the spectral lines of ionized helium. In Bohr’s theory this means that the corresponding Rydberg constant for the helium, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{R}_{He}$&lt;/span&gt;, is four times &lt;span class=&quot;math-container&quot;&gt;$\mathcal{R}_{H}$&lt;/span&gt; for the hydrogen atom. However, Fowler was not convinced and sent a letter to Nature where he pointed out that the ratio &lt;span class=&quot;math-container&quot;&gt;$\mathcal{R}_{He}/\mathcal{R}_{H}$&lt;/span&gt; is not simply a factor of &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, but instead is &lt;span class=&quot;math-container&quot;&gt;$4.0016$&lt;/span&gt;. Bohr understood that this small discrepancy is due to the fact that one neglected the effect of the finite mass of the nucleus. So, in the above expressions&#xA;  of &lt;span class=&quot;math-container&quot;&gt;$W_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\nu_{rev}$&lt;/span&gt;, the electron mass must be replaced by the reduced mass of the atom or the ion.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bohr replied to Fowler also in a short note in Nature, &lt;a href=&quot;https://www.nature.com/articles/092231d0&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Spectra of Helium and Hydrogen (1913)&lt;/a&gt;. The predicted value is &lt;span class=&quot;math-container&quot;&gt;$4.00163$&lt;/span&gt;, it is this agreement that allegedly prompted Einstein to say &quot;&lt;em&gt;This is a tremendous result. The theory of Bohr must then be right&lt;/em&gt;&quot; (Pais, Inward  Bound). The reduced mass expression is there, but not the name. However, many spectroscopic works in 1920-s already use &quot;reduced mass&quot; as a matter of course, some with reference to Bohr, e.g. &lt;a href=&quot;http://adsabs.harvard.edu/full/1920ApJ....51..230C&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fine Structure of the Near Infra-Red Absorption Bands of the Halogen Acids by Colby (1920)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Bohr has written the equilibrium conditions for such&#xA;  a system and for this problem it is only necessary to add one term&#xA;  to take care of the molecular rotation. This term is in fact of&#xA;  the same type as the one Bohr has used to indicate the centrifugal&#xA;  force on the electrons in the valence ring. &lt;span class=&quot;math-container&quot;&gt;$2x$&lt;/span&gt; indicates the&#xA;  distance between the nuclei, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; the radius of the valence ring, &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is&#xA;  the reduced mass of the rotating system &lt;span class=&quot;math-container&quot;&gt;$\left[I = M(2x)^2\right]$&lt;/span&gt;, other symbols have their usual significance.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I could not pinpoint who coined the term &quot;reduced mass&quot;, but it appears naturally when expressing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reduced_mass#Moment_of_inertia_of_two_point_masses_in_a_line&quot; rel=&quot;nofollow noreferrer&quot;&gt;moment of inertia &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; of two point masses&lt;/a&gt;. The moment of inertia is implicit in the work of Huygens,  was introduced explicitly by Euler (1758), and later studied by Lagrange, Poinsot, Jacobi and others. Already Lagrange applied it to celestial mechanics (lunar libration, 1764). For a detailed account see &lt;a href=&quot;http://homes.chass.utoronto.ca/~cfraser/Lagrangemech.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics by Fraser&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Although D'Alembert had used results about rigid bodies in his work in theoretical astronomy, the true foundations of the subject were laid by Euler in three important memoirs submitted to the Berlin Academy between 1750 and 1758. In the first of these memoirs&#xA;  Euler lays down the principle of linear momentum as his fundamental&#xA;  dynamical axiom and derives for a general rigid body the 'Euler' equations of motion relative to space fixed axes. In the later memoirs he introduces the concepts and terminology that have since become standard, the notions of principal axis, moment of inertia, Eulerian angle, etc. , and obtains the equations of motion with respect to reference axes fixed in the rigid body.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;In &quot;Recherches sur la connoisance mecanique des corps&quot; (1758) (op. tit. n. 43) Euler introduces the concept of 'moment of inertia' of a rigid body about a given line. He defines a 'principal axis' to be that line through the center of gravity for which the value of the moment of inertia is an extremum. This condition leads to two equations in the angles defining the position of the axis.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-24T00:14:27.650" LastActivityDate="2019-11-24T00:14:27.650" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11158" PostTypeId="1" CreationDate="2019-11-16T11:33:47.467" Score="1" ViewCount="446" Body="&lt;p&gt;I'm writing a physics article with significant historical content, and I'm struggling to find something. Forgetting about the anomalous magnetic dipole moment for a minute, the electron's &lt;a href=&quot;https://en.wikipedia.org/wiki/G-factor_(physics)#Electron_spin_g-factor&quot; rel=&quot;nofollow noreferrer&quot;&gt;g factor&lt;/a&gt; is approximately 2. This is twice the &quot;classical&quot; value. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Who predicted and/or discovered or measured this g factor of 2?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm currently look at H Stanley Allen's 1921 paper &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786442108636201?journalCode=tphm17&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Angular Momentum and Some Related Properties of the Ring Electron&lt;/a&gt;. It dates from 1921. He  gives the magnetic moment as 9.232 x 10ˉ²¹ EMU, ½(e/m)(h/2π)&lt;/p&gt;&#xA;" OwnerUserId="3264" LastEditorUserId="26" LastEditDate="2021-02-20T15:21:49.007" LastActivityDate="2021-02-20T15:21:49.007" Title="Who discovered or predicted an electron g factor of circa 2?" Tags="|electromagnetism|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11159" PostTypeId="2" ParentId="11158" CreationDate="2019-11-16T19:32:12.927" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.nature.com/articles/s41567-018-0341-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Nature Physics article&lt;/a&gt;, it appears to have been Paul Dirac:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From his theory of relativistic quantum mechanics, Paul Dirac predicted that the g-factor of elementary spin-1/2 particles, such as the electron, is exactly 2.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Further corrections to this value come from QED.&lt;/p&gt;&#xA;" OwnerUserId="10710" LastActivityDate="2019-11-16T19:32:12.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11160" PostTypeId="2" ParentId="11158" CreationDate="2019-11-16T20:44:24.917" Score="3" Body="&lt;p&gt;From Crane, H. R. &quot;The g factor of the electron.&quot; &lt;em&gt;Scientific American&lt;/em&gt; 218.1 (1968): 72-85.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Samuel A. Goudsmit and George E. Uhlenbeck proposed  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... in 1925 that the electron itself had an angular momentum and a&#xA;  magnetic moment. ... That the electron should have in addition an&#xA;  intrinsic magnetic moment was part and parcel of the idea that it was&#xA;  spinning. ... Thus at the time of the discovery of electron spin the g&#xA;  factor of the electron could be expressed as the number of &quot;natural&#xA;  units&quot; of magnetic moment (...) divided by the number of &quot;natural&#xA;  units&quot; of angular momentum (...). When defined in this way, the g&#xA;  factor for the orbital motion of the electron in its lowest energy&#xA;  state in hydrogen is 1, whereas the g factor of the free electron is&#xA;  2. (The g factor as a term designating a ratio of magnetic moment to angular momentum in these special units had been introduced a few&#xA;  years earlier for the case of the atom by the German physicist Alfred&#xA;  Lande.) ... In the case of the g factor of the electron, a strong&#xA;  reinforcement for the belief in the exactness of the value 2 came in&#xA;  the late 1920's from the new formulation of quantum mechanics by P. A.&#xA;  M. Dirac.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Also helpful is Tomonaga, S. &lt;em&gt;The story of spin&lt;/em&gt;. University of Chicago Press, 1997&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Enz, Charles P. &lt;em&gt;No time to be brief: A scientific biography of Wolfgang Pauli&lt;/em&gt;. Oxford University Press on Demand, 2010.&#xA;Chp. 5 The Hamburg years: The curious history of spin.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;But who should now be considered the discoverer of the electron's spin, Pauli, Kronig, or Uhlenbech and Goudsmit?&quot; [p.116].&#xA;&quot;That in the history of modern physics the idea of spin has stirred up so much controversy is a strange fact.&quot; [p.119].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Pais, Abraham. &lt;em&gt;Inward bound: of matter and forces in the physical world&lt;/em&gt;. Oxford: Clarendon Press, 1986.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Following a hint by Ehrenfest, George [Uhlenbeck] found in an old article by [M.] Abraham [1903] that an electron considered as a rigid sphere with surface charge only does have g = 2.&quot;  [p.277]. Similar proposals to Abraham were made by Fitzgerald [1900], Compton [1921] and Kennard [1922]. Pauli made a proposal of spin which influenced Goudsmit and Uhlenbeck in 1924. [p.279]. Pais quotes Dirac on the Dirac equation, &quot;It was found that this equation gave the particle a spin of half a quantum. And also gave it a magnetic moment. It gave just the properties that one needed for an electron. That was really an unexpected bonus for me, completely unexpected.&quot; [p.286].&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Pippard, Brian. &quot;Magnetic Moments.&quot; Rev. of &lt;em&gt;Inward Bound: Of Matter and Forces in the Physical World&lt;/em&gt;, by Abraham Pais. &lt;em&gt;London Review of Books&lt;/em&gt; 8.15 (1986): 17-18. 16 Nov. 2019 &lt;a href=&quot;https://www.lrb.co.uk/v08/n15/brian-pippard/magnetic-moments&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.lrb.co.uk/v08/n15/brian-pippard/magnetic-moments&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Why should anyone wish to know the magnetic moment of the electron to&#xA;  ten-figure accuracy? The answer is worth giving in some detail as an&#xA;  exemplary account of the research process. We must return to the early&#xA;  days of quantum mechanics and the task Dirac set himself in 1927 to&#xA;  make it consistent with relativity theory, as it was not when first&#xA;  formulated. In one of the masterpieces of physics he showed that the&#xA;  restrictions imposed by relativity, as well as others which he&#xA;  regarded as essential, allowed only one formulation. Applied to the&#xA;  electron, its solutions described, without any extra assumptions, a&#xA;  number of properties which the electron had been found experimentally&#xA;  to possess, but which had not been explained hitherto. Among them was&#xA;  the magnetic moment whose magnitude is conventionally expressed by a&#xA;  number g; in Dirac’s theory g turns out to be exactly 2, and the&#xA;  spectroscopists at the time were satisfied that this agreed with their&#xA;  extremely precise measurements. There was, however, an exceedingly&#xA;  curious consequence of the theory: that it permitted electrons to have&#xA;  negative as well as positive energy, something that is inconceivable&#xA;  in classical mechanics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/11139/did-anyone-mention-the-possibility-of-antimatter-before-1928&quot;&gt;Did anyone mention the possibility of antimatter before 1928?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2019-11-16T20:44:24.917" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11161" PostTypeId="1" CreationDate="2019-11-17T01:50:21.393" Score="4" ViewCount="1376" Body="&lt;p&gt;I've been re-reading &quot;Surely you're Joking Mr. Feynman&quot;, and at one point he talks about how at Los Alamos they were asked to write down any idea, no matter how obvious involving nuclear technology, and the government would patent it.  Feynman thinks this is ridiculous, and says he'll be talking to them all day with endless ideas of using a nuclear reactor to power things like a submarine, and airplane, a rocket, etc.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He winds up getting the patent for the airplane and the rocket, but someone already got the patent for the nuclear submarine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While reading this, it suddenly occurred to me that patent searches are incredibly easy these days, and available on Google.  But try as I might I couldn't find any patent issued to Richard Feynman at all.  There were certainly other patents issued to the other nuclear scientists, but none to Feynman.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, Feynman also tells a story about how after the war, someone had called him up (Long Distance from all the way from California!) and wanted him to be the director of a laboratory that was going to develop a nuclear powered airplane, because his name was on the patent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can guess at a few different possibilities.  &lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Feynman is mistaken about this, and there weren't any patents, it was something else entirely or he made the whole story up.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The patent(s) was applied for, but never issued because the examiner thought it was too obvious, or there was some other problem with the application.  The story about someone calling him and offering him to be a director arose because they saw the patent application, not the issued patent.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The patent(s) was granted, but either can't be found for some reason under Feynman's name (miss-spelling, etc)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Possibility one seems unlikely, especially because of the LD call story.  I tried investigating possibility two, but patent applications only appear to be searchable online since 2001.&lt;/p&gt;&#xA;" OwnerUserId="10644" LastEditorUserId="10644" LastEditDate="2020-06-05T13:59:46.217" LastActivityDate="2020-12-31T04:50:19.003" Title="Was Richard Feynman really awarded a patent for a nuclear Airplane and Rocket?" Tags="|biographical-details|nuclear-physics|world-war-2|physicists|feynman|" AnswerCount="0" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="11162" PostTypeId="1" AcceptedAnswerId="11163" CreationDate="2019-11-17T08:24:51.467" Score="6" ViewCount="592" Body="&lt;p&gt;How did professor Buchberger discover Gröbner (Groebner) bases for polynomial ideals? What was the problem(s) that lead to such a discovery?&lt;/p&gt;&#xA;" OwnerUserId="6416" LastEditorUserId="55" LastEditDate="2019-11-17T09:45:02.953" LastActivityDate="2019-11-18T00:29:03.007" Title="What was the motive for inventing Gröbner bases?" Tags="|abstract-algebra|algebraic-geometry|algorithm|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11163" PostTypeId="2" ParentId="11162" CreationDate="2019-11-17T09:44:00.980" Score="11" Body="&lt;p&gt;Fortunately, Buchberger himself described the context of his discovery, see &lt;a href=&quot;https://dl.acm.org/citation.cfm?id=1610301&quot; rel=&quot;noreferrer&quot;&gt;Historical background to Gröbner's paper by Abramson&lt;/a&gt;. The method, in general outline, was known to Gröbner long before the &lt;a href=&quot;https://www3.risc.jku.at/research/theorema/Groebner-Bases-Bibliography/gbbib_files/publication_706.pdf&quot; rel=&quot;noreferrer&quot;&gt;Buchberger's thesis (1965)&lt;/a&gt;. In the 1950 paper &lt;em&gt;Über die Eliminationstheorie&lt;/em&gt; (&lt;a href=&quot;https://dl.acm.org/citation.cfm?id=297071&quot; rel=&quot;noreferrer&quot;&gt;Abramson's English translation&lt;/a&gt;) it is applied to finding&#xA;bases of integrals of differential equations, but he alludes to many other applications:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;For about 17 years, I have applied and tested these methods in the most varied and complicated cases,&#xA;  and I believe I can say, on the basis of my experiences, that they in fact represent a useful and valuable&#xA;  tool for the solution of these and similar ideal theoretic tasks in every case. Because I have often been&#xA;  asked how one can most easily find the reduced representation of a polynomial ideal, I have now decided&#xA;  to publish the essential features of these methods, omitting the details.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1964 Gröbner explained the method in examples at a seminar, but it was not clear when the algorithm would terminate. Buchberger took up the problem, already in a general and abstract form, in 1965. In brief, it was to determine effectively whether a generated polynomial ideal is zero dimensional or not, or, when it was, whether only finitely many residue classes remained and whether they were linearly independent. What he later called Gröbner bases were designed to resolve these questions. Here is Buchberger's own account of the details:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The method mentioned here is the method Gröbner presented in his seminar 1964, in which I took&#xA;  part as a student and after which I asked Gröbner whether I could take the study of his method as my&#xA;  PhD thesis subject. In fact, Gröbner described the method only in examples, the method was not really&#xA;  described as a general algorithm. More importantly, it was not at all clear when exactly, for a given input&#xA;  polynomial set, the method could be terminated. This point is subtle for the following reasons:&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;• &lt;em&gt;Even if one knew (in a specific) example that the polynomial ideal is zero dimensional (i.e. the residue&#xA;  class ring has a finite vector space basis) it was by no means clear whether or not by Gröbner’s method&#xA;  a situation will always be reached, in which only finitely many linearly independent residue classes&#xA;  remained (which is the case if each indeterminate appears purely in one of the basis polyomials).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;• &lt;em&gt;Even if a situation was reached in which, by the above criterion, it was clear that only finitely many&#xA;  residue classes remained, it was not at all clear whether they are linearly independent: By considering&#xA;  higher degrees, it might well happen that dependencies could appear that were not known at the&#xA;  moment when the method was stopped. Hence, the question was: How high do we have to go with&#xA;  the degrees of the polynomials in order to be sure that no more linear dependencies will appear?&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;• &lt;em&gt;In general, if an arbitrary polynomial set is given, we do&#xA;  &lt;strong&gt;not know whether or not the polynomial ideal generated is zero dimensional or not. This question just needs the computation of a Gröbner basis for deciding it!&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;These were basically the questions Gröbner gave to me for my PhD thesis. In fact, they were not clearly separated but, in his wording, the question was: When can one terminate computation by this method?&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;On the way towards answering these questions, I then invented the notion of S-polynomials and showed&#xA;  its relevance for answering all three questions. Based on S-polynomials, I then gave a new algorithm that&#xA;  computes polynomials sets (which I later called Gröbner bases) from which as a by-product also a linearly&#xA;  independent vector space basis of the residue class ring of the input polynomial set could be read off. For&#xA;  this algorithm I also gave a complete correctness and termination proof (which guarantees termination&#xA;  also for the non-zero dimensional case).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-18T00:29:03.007" LastActivityDate="2019-11-18T00:29:03.007" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11167" PostTypeId="2" ParentId="11153" CreationDate="2019-11-18T13:26:27.860" Score="2" Body="&lt;p&gt;I think I've found what you are remembering.  There's a free epub available &lt;a href=&quot;https://play.google.com/books/reader?id=9F8TAAAAYAAJ&amp;amp;hl=en&amp;amp;pg=GBS.PA4&quot; rel=&quot;nofollow noreferrer&quot;&gt;at google books&lt;/a&gt;, &quot;Memoirs of the Royal Society&quot; .  Around page 386, I found:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Mx3Fp.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Mx3Fp.jpg&quot; alt=&quot;glass balls filled with stuff&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Or possibly this one, also free epub  &lt;a href=&quot;https://play.google.com/books/reader?id=WU414fkgrcgC&amp;amp;hl=en&amp;amp;pg=GBS.PP12&quot; rel=&quot;nofollow noreferrer&quot;&gt;from google books&lt;/a&gt;. Perhaps they are multiple descriptions of the same thing -- I'll leave it up to you to research these two publications :-) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LfUVz.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LfUVz.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-11-18T13:26:27.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11169" PostTypeId="2" ParentId="11120" CreationDate="2019-11-18T16:25:49.813" Score="0" Body="&lt;p&gt;&lt;em&gt;The Canon: An Exposition of the Pagan Mystery Perpetuated in the Cabala As the Rule of All Arts&lt;/em&gt; by William Stirling  (architect)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Canon is the numerical interpretation of the Cosmic law and, since&#xA;  man is a microcosm of the universe, it is also the law of human&#xA;  nature. The history of all civilizations is the history of their&#xA;  interpretation of the Canon. Codified by ancient philosophers and&#xA;  guarded by priests in the temples, the Canon sanctified the society&#xA;  that possessed it, regulating its institutions in accordance with the&#xA;  laws of nature. Plato, writing in &quot;The Laws of the Egyptian Canon, &quot;&#xA;  claimed that by its use the high civilization of ancient Egypt had&#xA;  been preserved from deterioration for 10,000 years.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The Canon was instituted by the ancient theocracies as a standard in&#xA;  music, architecture and all other arts. Even after the decline of the&#xA;  old world order and of the canonical standard, knowledge of the Canon&#xA;  was preserved in the mystery schools of Greece and Alexandria, and in&#xA;  the tradition of masonic and other craft guilds.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;One of the most revolutionary books of the late 19th century, The&#xA;  Canon owes its high reputation to the august literary style of its&#xA;  author, as well as to its remarkable contents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The old Masonic writers declare that the true system of the universe,&#xA;  and the foundation of all geometrical proportion is to be found in the&#xA;  forty-seventh proposition of Euclid, as expounded by Pythagoras, and&#xA;  communicated in writing by Plato, .... [p.56].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=lDYWKbYDm8cC&amp;amp;dq=The%20Canon:%20An%20Exposition%20of%20the%20Pagan%20Mystery%20Perpetuated%20in%20the%20Cabala%20As%20the%20Rule%20of%20All%20Arts&amp;amp;lr=&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stirling, William. The Canon: An Exposition of the Pagan Mystery Perpetuated in the Cabala as the Rule of All Arts. Weiser Books, 1999.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books/about/Uriel_s_Machine.html?id=p0OTclorcMoC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Knight, Christopher, and Robert Lomas. Uriel's Machine: The Ancient Origins of Science. Random House, 2000.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=-9I2Iek5BlwC&amp;amp;lr=&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guénon, René. The Reign of Quantity &amp;amp; The Signs of the Times. Sophia Perennis, 2001.&lt;/a&gt; &quot;Critique of modern Western civilization from the point of view of traditional metaphysics.&quot;&lt;/p&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2019-11-18T16:25:49.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11170" PostTypeId="1" AcceptedAnswerId="11171" CreationDate="2019-11-19T15:35:32.213" Score="2" ViewCount="162" Body="&lt;p&gt;In modern papers in statistics, it is common to use &lt;a href=&quot;https://en.wikipedia.org/wiki/Expected_value&quot; rel=&quot;nofollow noreferrer&quot;&gt;the symbol &lt;span class=&quot;math-container&quot;&gt;$E[X]$&lt;/span&gt;&lt;/a&gt; to refer to the expectation of a random variable &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While reading (a translated version of) &lt;a href=&quot;http://www.mathnet.ru/links/c76f9e938c6a2afcef461444db55d75c/ppi1003.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Convergence Rate of Nonparametric Estimates of Maximum-Likelihood Type&quot; by A. S. Nemirovskii, B. T. Polyak, A. B. Tsybakov&lt;/a&gt;, I noticed that they seem to use the notation &lt;span class=&quot;math-container&quot;&gt;$M(X)$&lt;/span&gt; to refer to the expectation integral.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I tried Google translating &quot;expectation&quot;, &quot;mean&quot;, &quot;average&quot; and &quot;integral&quot; to Russian, but none of these translations seem to begin with &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd be interested in finding out what the &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; stands for and when this convention was first introduced.&lt;/p&gt;&#xA;" OwnerUserId="10719" LastEditorUserId="10106" LastEditDate="2020-06-08T06:00:44.657" LastActivityDate="2020-06-08T06:00:44.657" Title="Where does $M$ for expected value in Russian papers come from?" Tags="|mathematics|notation|statistics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11171" PostTypeId="2" ParentId="11170" CreationDate="2019-11-20T10:46:59.200" Score="6" Body="&lt;p&gt;One would think that Russian usage stems from Kolmogorov's seminal works on probability. However, in &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01459098&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Über die Summen durch den Zufall bestimmter unabhängiger Größen&lt;/em&gt; (1928)&lt;/a&gt; he uses &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{M}$&lt;/span&gt; to denote probability (presumably from &lt;em&gt;messen&lt;/em&gt;, measure), not mathematical expectation. In the famous &lt;a href=&quot;https://link.springer.com/book/10.1007/978-3-642-49888-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Grundbegriffe der Wahrscheinlichkeitsrechnung&lt;/em&gt; (1933)&lt;/a&gt;, which gave the modern axiomatization of probability theory, mathematical expectation is called &lt;em&gt;die mathematische Erwartungen&lt;/em&gt; and denoted &lt;span class=&quot;math-container&quot;&gt;$\mathsf{E}$&lt;/span&gt;. Unfortunately, its 1936 Russian translation does not seem to be available online, and the &lt;a href=&quot;http://khpi-iip.mipk.kharkiv.edu/library/tv/4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1974 edition&lt;/a&gt;, which is and uses &lt;span class=&quot;math-container&quot;&gt;$\mathsf{M}$&lt;/span&gt;, can not be trusted since it says in the preface that notation has been modernized. Khinchin (Kolmogorov's student) also uses &lt;span class=&quot;math-container&quot;&gt;$\mathsf{E}$&lt;/span&gt; in &lt;a href=&quot;https://techlibrary.ru/b1/3d1j1o1y1j1o_2h.3n._2u1a1t1f1n1a1t1j1y1f1s1l1j1f_1p1s1o1p1c1a1o1j2g_1s1t1a1t1j1s1t1j1y1f1s1l1p1k_1n1f1w1a1o1j1l1j.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Математические основания статистической механики&lt;/em&gt;&lt;/a&gt; (Mathematical foundations of statistical mechanics, written in 1941).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But even the English translation of Gnedenko-Kolmogorov's &lt;a href=&quot;https://archive.org/details/limitdistributio00gned_0/page/n283&quot; rel=&quot;nofollow noreferrer&quot;&gt;Limit distributions for sums of independent random variables (1954, Russian original 1949)&lt;/a&gt; uses &lt;span class=&quot;math-container&quot;&gt;$\mathsf{M}$&lt;/span&gt; and features &lt;em&gt;математическое ожидание&lt;/em&gt; (mathematical expectation) prominently around the definition. &lt;em&gt;Mittlewert&lt;/em&gt; or mean (&lt;em&gt;среднее&lt;/em&gt;) do not start with м in Russian, are not mentioned nearby, and Kolmogorov's previous usage does not make motivation by a foreign spelling likely. Late 1940-s in Russia were characterized by promotion of Russian nationalism and condemnation of &lt;a href=&quot;https://en.wikipedia.org/wiki/Rootless_cosmopolitan&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;rootless cosmopolitans&quot;&lt;/a&gt;, making it even less likely.&#xA;Gnedenko's &lt;em&gt;Курс теории вероятностей&lt;/em&gt; (Course of Probability Theory, 1950), which became the standard text on the subject in Russian universities, introduces &lt;span class=&quot;math-container&quot;&gt;$\mathsf{M}$&lt;/span&gt; in a similar way. So even in the unlikely event that Gnedenko and Kolmogorov had something else in mind, &lt;span class=&quot;math-container&quot;&gt;$\mathsf{M}$&lt;/span&gt; for &lt;strong&gt;&lt;em&gt;M&lt;/strong&gt;атематическое (ожидание)&lt;/em&gt; is what Russian readers would have taken away from the text.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Speaking generally, motivation of the original user is not necessarily the motivation for the subsequent widespread adoption of a notation. In Kolmogorov's case, &lt;span class=&quot;math-container&quot;&gt;$\mathsf{E}$&lt;/span&gt; comes from German spelling. That it happens to match English &quot;&lt;strong&gt;E&lt;/strong&gt;xpected&quot; is just a coincidence, but it may well have played a decisive role in its adoption in the Anglophone literature. His &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{M}$&lt;/span&gt; for probability was not similarly adopted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/pdf/math/0606533.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Sources of Kolmogorov’s Grundbegriffe by Shafer and Vovk&lt;/a&gt; is a good source on Kolmogorov's early work on probability.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-11-21T04:36:24.897" LastActivityDate="2019-11-21T04:36:24.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11172" PostTypeId="1" CreationDate="2019-11-20T11:03:44.960" Score="5" ViewCount="368" Body="&lt;p&gt;The following quote (in English) is fairly widely attributed to Fourier, but I can't substantiate it.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is, I can't verify that he actually said or wrote it (in any language). Can anyone help me out?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, the statement appears in &lt;a href=&quot;http://books.google.com/books?vid=ISBN1441915451&amp;amp;pg=PA19&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gao and Yan (2011)&lt;/a&gt;:  It seems it has been attributed both to &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k33707/f220n7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier's 1807 dissertation&lt;/a&gt; and to his 1822 &lt;a href=&quot;http://books.google.com/books?vid=04X2vlqZx7hydlQUWEq&amp;amp;id=TDQJAAAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;Analytical Theory of Heat&quot;&lt;/a&gt; monograph. The 1822 work has long ago been translated into English by Alexander Freeman and is available on &lt;a href=&quot;http://www.archive.org/details/analyticaltheory00fourrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;archive.org&lt;/a&gt;. However, I can't find any text written by Fourier that is a close match to the claimed quote. Anyone have a solid reference (in any language)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have already asked what is basically &lt;a href=&quot;https://math.stackexchange.com/questions/3442031&quot;&gt;this same question on Math SE&lt;/a&gt;, where it was suggested that I ask it here on HSM SE.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of special interest in the claimed quote is the explicit phrase, &quot;continuous or with discontinuities&quot; (even though this can be inferred implicitly from &quot;fonctions etièrement arbitraires&quot;). The explicit reference to both continuous and discontinuous was, historically, a revolutionary claim. Before the Fourier expansions with coefficients calculated using integrals and bases of sinusoids, there was the Taylor expansions with coefficients calculated using derivatives and bases of polynomials &lt;span class=&quot;math-container&quot;&gt;$(x-a)^n$&lt;/span&gt;. The Taylor expansions were limited to analytic functions...and thus global expansions of discontinuous functions were simply out of the question. Then along came Fourier with a truly revolutionary claim that his expansion worked not only for continuous functions, but discontinuous as well! And to add insult to injury, Fourier wasn't even really a mathematician (but rather an engineer among other things). For more information, see Enders A. Robinson's paper, &lt;a href=&quot;http://www.archive.ece.cmu.edu/~ece792/handouts/Robinson82.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;A Historical Perspective of Spectrum Estimation&quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10723" LastEditorUserId="4251" LastEditDate="2020-11-22T21:01:29.443" LastActivityDate="2020-11-22T21:01:29.443" Title="Substantiating claimed Fourier quote about “an arbitrarily capricious graph”" Tags="|mathematicians|quote|mathematics-social-history|fourier|harmonic-analysis|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11173" PostTypeId="2" ParentId="11172" CreationDate="2019-11-20T12:29:36.747" Score="7" Body="&lt;p&gt;You can find the quote (or something similar) in multiple places in the &lt;em&gt;Théorie analytique de la chaleur&lt;/em&gt;, for example in&lt;/p&gt;&#xA;&lt;p&gt;Chapter I,&lt;/p&gt;&#xA;&lt;p&gt;paragraph (article) 14:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;L'examen de cette condition fait connaître que l'on peut développer en séries convergentes, ou exprimer par des intégrales définies, les fonctions qui &lt;strong&gt;ne sont point assujéties à une loi constant&lt;/strong&gt;e, et qui représentent les ordonnées des &lt;strong&gt;lignes irrégulières ou discontinues&lt;/strong&gt;. Cette propriété jette un nouveau jour sur la Théorie des équations aux différences partielles, et étend l'usage des fonctions arbitraires en les soumettant aux procédés ordinaires de l'analyse.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Chapter IV, Section VI,&lt;/p&gt;&#xA;&lt;p&gt;paragraph 219:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Nous avons supposé jusqu'ici que la fonction dont on demande le développement en séries de sinus d'arcs multiples, &lt;strong&gt;peut être développée en une série ordonnée&lt;/strong&gt;, suivant les puissances de la variable x, et qu'il n'entre dans cette dernière série que des puissances impaires. &lt;strong&gt;On peut étendre les mêmes conséquences à des fonctions quelconques&lt;/strong&gt;, même à celles qui seraient &lt;strong&gt;discontinues et entièrement arbitraires&lt;/strong&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;paragraph 220:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Cette remarque est importante, en ce qu'elle fait connaître comment les fonctions &lt;strong&gt;entièrement arbitraires&lt;/strong&gt; peuvent aussi être développées en séries de sinus d'arcs multiples.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;paragraph 229:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Les suites formées de sinus ou de cosinus d'arcs multiples sont donc propres à représenter &lt;strong&gt;entre des limites déterminées&lt;/strong&gt;, &lt;strong&gt;toutes les fonctions possibles&lt;/strong&gt;, et les ordonnées &lt;strong&gt;des lignes ou des surfaces&lt;/strong&gt; dont la loi est &lt;strong&gt;discontinue&lt;/strong&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;paragraph 278:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;[...] Ainsi ce théorême qui donne, &lt;strong&gt;entre des limites assignées&lt;/strong&gt;, le développement d'une &lt;strong&gt;fonction arbitraire&lt;/strong&gt; en séries de sinus ou de cosinus d'arcs multiples se déduit des règles élémentaires du calcul. [...] Dans la question suivante, on réduit encore la fonction arbitraire en une série de sinus&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;After some search, I think I've finally found the source of the quote (which is not Fourier).&lt;/p&gt;&#xA;&lt;p&gt;The phrase appear almost verbatim at the very beginning of &lt;a href=&quot;https://en.wikipedia.org/wiki/Cornelius_Lanczos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cornelius Lanczos&lt;/a&gt;'s &lt;em&gt;Discourse on Fourier Series&lt;/em&gt; (1966):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a memorable session of the French Academy on the 21st of December 1807, the mathematician and engineer Joseph Fourier announced a thesis which inaugurated a new chapter in the history of mathematics. [...] &lt;strong&gt;Fourier claimed that an arbitrary function, defined in a finite interval by an arbitrary capricious graph, can always be resolved into a sum of pure sine and cosine functions.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now the questions are: is this reconstruction historically correct? Is Fourier's claim reported accurately?&lt;/p&gt;&#xA;&lt;p&gt;First of all, it's really hard to say that the audience considered that session to be truly &lt;em&gt;memorable&lt;/em&gt;.&#xA;The &lt;em&gt;procès-verbal&lt;/em&gt; of the seat is exactly the &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k3296d/f638.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;following&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;M. Fourier lit un Mémoire sur la &lt;em&gt;Propagation de la chaleur sur le solides&lt;/em&gt;.&#xA;MM: Lagrange, Laplace, Monge et Lacroix.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fourier reads a Memoir on &lt;em&gt;Heat Propagation on Solids&lt;/em&gt;.&#xA;Lagrange, Laplace, Monge et Lacroix [were present].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Subsequently, a rather cold account, written by Poisson and simply signed by &amp;quot;P.&amp;quot;, appears in March 1808 in the &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k10535481/f12.item.zoom&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bulletin des Sciences&lt;/a&gt;, where he mentions the heat equation but not the &amp;quot;Fourier&amp;quot; series.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, now we know, following Lanczos, that the claim was made by Fourier either orally during the seat of the French Academy or written in his original 1807 thesis.&#xA;The former hypothesis should be discarded, as there are no detailed records of the session, and the latter must be rejected because in the &lt;em&gt;Théorie de la propagation de la chaleur dans les solides&lt;/em&gt;, published only in 1822 under the name of &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k9646913z/f25.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Théorie analytique de la chaleur&lt;/em&gt;&lt;/a&gt;, there's no occurrence of terms like &amp;quot;capricieux&amp;quot; or &amp;quot;graph(e)&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So, considering also the anachronism and the inaccuracy of a sentence such as &amp;quot;[a] function [...] defined [...] by an arbitrarily capricious graph&amp;quot;, we must conclude that @Daniel J. Greenhoe's answer is correct:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Possibly what happened was that someone wrote a commentary/summary regarding Fourier's work, and then later someone else came along thinking it was an actual quote [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Moreover, when Lanczos want to quote exact words, also in translation, he uses quotation marks, as in the following (p. 8):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The gradual clarification of our ideas concerning the concept of a function has much to do with the discovery of Fourier in 1807. Fourier was well aware of the fact that a function need not be everywhere differentiable in order to be representable by the sum of an infinite sequence of sine and cosine functions. He talk of &amp;quot;arbitrary function&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$y=f(x)$&lt;/span&gt; which could represented by his series.&#xA;The lack of precision in his language was heavily criticised by his contemporaries, although in fact Fourier came nearer to the truth than his critics, who eanted to restrict the validity of the Fourier series to the class of infinitely differentiable functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here quotation marks around &lt;em&gt;arbitrary function&lt;/em&gt; are correct, as Fourier actually uses the expression &lt;em&gt;fonctions entièrement arbitraires&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;We must not blame Lanczos for this, and himself was well aware that his book was not was not a faithful account of the history of Fourier series, as reported in the foreword of the 2016 edition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;His book is not dry; a person inhabits it. &amp;quot;And why is it not allowed,&amp;quot; he says in the video, &amp;quot;to put in a certain emotional emphasis? If you're enthused about something, why not say so?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-06-19T13:11:03.707" LastActivityDate="2020-06-19T13:11:03.707" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11175" PostTypeId="2" ParentId="3131" CreationDate="2019-11-22T02:23:12.830" Score="2" Body="&lt;p&gt;The list is already too long, and can be made much longer, so let me refer to two lists instead of mentioning individuals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The book by A. Goodman, Univalent functions, vol. II (Mariner Publishing Co., Inc., Tampa, FL, 1983, MR0704184) has an Appendix which contains a long list of mathematicians victims of the Nazi regime in Europe. Certainly not all of them died young, but they died in a tragic way, no doubt.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a (incomplete) list on Wikipedia of mathematicians who committed suicide &lt;a href=&quot;https://en.wikipedia.org/wiki/Category:Mathematicians_who_committed_suicide&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Category:Mathematicians_who_committed_suicide&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2019-11-22T02:23:12.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11176" PostTypeId="1" CreationDate="2019-11-22T16:52:29.113" Score="1" ViewCount="103" Body="&lt;p&gt;After reading &lt;a href=&quot;https://hsm.stackexchange.com/q/7037/10738&quot;&gt;this&lt;/a&gt; question, I remembered seeing on our department webpage somewhere that for PhD studies one must show a competency in either German, French, or Russian, but Italian was acceptable depending on specialty.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is this &quot;specialty&quot;? Is there something that that Italian's a specifically known for or is this ambiguous and the answer would differ depending on what department you spoke to?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;My only guess would be Algebraic Geometry since there is a Wiki Page Named: &lt;a href=&quot;https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Italian School of Algebraic Geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But what separates them from the French, for instance, who are well know for their Algebraic Geometry? Is there something that they did and are still doing well that is notable in some regard? &lt;/p&gt;&#xA;" OwnerDisplayName="user10738" LastActivityDate="2019-11-22T16:52:29.113" Title="Italian Mathematics" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11178" PostTypeId="2" ParentId="11172" CreationDate="2019-11-23T12:53:54.400" Score="1" Body="&lt;p&gt;I think the conclusion here is that there is no proof that the claimed Fourier quote is an actual Fourier quote (in any language). &lt;em&gt;However&lt;/em&gt;,&lt;br/&gt;&#xA;  (1) Absence of proof is not proof of absence.&lt;br/&gt;&#xA;  (2) Although Fourier may have not communicated the quote in substance, he did arguably communicate the quote in essence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Possibly what happened was that someone wrote a commentary/summary regarding Fourier's work, and then later someone else came along thinking it was an actual quote, published the commentary with quotes around it... and thus a mathematical urban legend was born(?).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Any author who is looking for a substantiated Fourier quote to replace the claimed quote could consider extracting one from Fourier 1822 Section 219, as referenced by @user6530:&lt;br/&gt;&#xA;&quot;Up to this point we have supposed that the function whose development is required in a series of sines of multiple arcs can be developed in a series arranged according to powers of the variable x,... We can extend the same results to any functions, even to those which are discontinuous and entirely arbitrary.&quot;&lt;br/&gt; &#xA;(&lt;a href=&quot;http://www.archive.org/details/analyticaltheory00fourrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier (1822)&lt;/a&gt;, §219, page 184, translation by Alexander Freeman).&lt;/p&gt;&#xA;" OwnerUserId="10723" LastActivityDate="2019-11-23T12:53:54.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11179" PostTypeId="1" CreationDate="2019-11-23T17:45:39.933" Score="3" ViewCount="150" Body="&lt;p&gt;I think it is often said that one great motivation for the invention of calculus was to have a tool allowing to calculate the slope of a tangent to a curve &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; at a given point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, and even, to find the equation of this tangent. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What made this goal so important for mathematicians (and more broadly for scientists)  at that time? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In which contexts did the tangent-problem reveal so crucial?&lt;/p&gt;&#xA;" OwnerDisplayName="user9853" LastEditorUserId="4251" LastEditDate="2020-06-06T13:00:27.270" LastActivityDate="2020-06-06T13:00:27.270" Title="Before differential calculus was discovered, why were mathematicians interested in tangents?" Tags="|mathematics|calculus|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11181" PostTypeId="1" CreationDate="2019-11-23T22:02:20.933" Score="4" ViewCount="66" Body="&lt;p&gt;In physics, stress essentially captures &lt;em&gt;forces&lt;/em&gt; in a body, where as strain captures &lt;em&gt;displacements&lt;/em&gt;. Two dimensionally very different concepts. If you look it up in a thesaurus, stress and strain are synonyms. How did they come to denote such different things in physics? &lt;/p&gt;&#xA;" OwnerUserId="5975" LastActivityDate="2019-11-23T22:02:20.933" Title="How did the terms stress and strain come to describe two different things?" Tags="|physics|terminology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11182" PostTypeId="1" CreationDate="2019-11-24T08:52:45.153" Score="0" ViewCount="101" Body="&lt;p&gt;On Wikipedia, these two pages are mentioned :&lt;a href=&quot;https://books.google.co.in/books?id=4XkAAAAAMAAJ&amp;amp;pg=PA619&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt; and &lt;a href=&quot;https://books.google.co.in/books?id=0TxeAAAAcAAJ&amp;amp;pg=PA3&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss&lt;/a&gt;&#xA;however I am an English speaker and can not comprehend any of the one, can someone provide the translated pages or shortly their line of reasoning?&lt;/p&gt;&#xA;" OwnerUserId="9585" LastActivityDate="2019-11-24T08:52:45.153" Title="How did Gauss and Lagrange derive Gauss law of Electrostatics" Tags="|electromagnetism|gauss|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11183" PostTypeId="1" CreationDate="2019-11-24T17:11:31.823" Score="5" ViewCount="106" Body="&lt;p&gt;A &lt;a href=&quot;https://arxiv.org/abs/1903.07393&quot; rel=&quot;noreferrer&quot;&gt;recent work&lt;/a&gt; proposed that &lt;a href=&quot;https://commons.wikimedia.org/wiki/Category:Newgrange#/media/File:Flickr_-_bastique_-_Entrance_stone_at_Newgrange_(1).jpg&quot; rel=&quot;noreferrer&quot;&gt;the double spiral motif&lt;/a&gt; found at &lt;a href=&quot;https://en.wikipedia.org/wiki/Newgrange&quot; rel=&quot;noreferrer&quot;&gt;Newgrange&lt;/a&gt; (ca. 3200 BC) is a calendrical representation. Looking south the northern peoples observed that the arc traced by the sun widens steadily from winter solstice to midsummer. A natural  supposition is that what is actually observed is just a part of a spiral, its lower half being hidden below the horizon.  This seems to be strikingly original in contrast with our habit to think about circles. The author mentions just two refs from the 1980's (M.Brennan 1983, N.L. Thomas, 1988), where this interpretation has been made. Besides vague generalisations, such as &quot;solar motif&quot;, are there earlier mentions of the idea as explained along the lines given here?&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-05-16T17:02:15.220" Title="Spirals as calendrical representations" Tags="|archaeoastronomy|calendar|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11185" PostTypeId="1" CreationDate="2019-11-25T06:34:10.717" Score="4" ViewCount="262" Body="&lt;p&gt;The following has been attributed to the German physicist Werner Heisenberg:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;History legitimizes Germany to rule Europe and later the world. Only a nation that rules ruthlessly can maintain itself. Democracy cannot develop sufficient energy to rule Europe. There are, therefore, only two possibilities: Germany and Russia, and perhaps a Europe under German leadership is the lesser evil.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Quoted sources include:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A 1943 letter from Heisenberg to Dutch scientist Hendrik B. G. Casimir (Blood and Water, Dan Kurzman, 1997, p. 35, ISBN 0-8050-3206-1 (from &lt;a href=&quot;http://www.scientificlib.com/en/Physics/Biographies/WernerHeisenberg.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Third-hand report: &quot;&lt;em&gt;it was filed in 1945 by the Dutch-American astronomer Gerard Kuiper, a member of the Alsos mission, to whom it had been relate by Casimir. Casimir alleged that Heisenberg had said this to him during a private walk in Leiden, during which he had also learnt that Heisenberg knew at the time about the German concentration camps. It isn't clear that Casimir's report can be taken at face value.&lt;/em&gt;&quot; (from &lt;a href=&quot;https://books.google.de/books?id=a2HgvgZC1H4C&amp;amp;lpg=PT226&amp;amp;dq=heisenberg%20%22History%20legitimizes%20Germany%20to%20rule%20Europe%22&amp;amp;pg=PT226#v=onepage&amp;amp;q=heisenberg%20%22History%20legitimizes%20Germany%20to%20rule%20Europe%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Is this quote correctly attributed/better documented? Or is there any other evidence Heisenberg was this sort of nationalist?&lt;/p&gt;&#xA;" OwnerUserId="10299" LastEditorUserId="4251" LastEditDate="2020-06-07T11:52:23.103" LastActivityDate="2020-06-07T15:07:10.537" Title="Did Heisenberg defend that Germany should rule Europe?" Tags="|biographical-details|physicists|nazi-germany|heisenberg|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11186" PostTypeId="1" CreationDate="2019-11-25T08:49:13.787" Score="1" ViewCount="118" Body="&lt;p&gt;I read a previous question on Roman engineers and I was surprized that nobody referred to the Pantheon: its dome is an unrivalled wonder of architecture. Roads and aqueducts can be built without theoritical notions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is amazing that people with such a crippled numeral system and little or no math could build such a temple, in comparison to which pyramids and parthenon are child's play.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do you know if there is any record of formulae the architects used to calculate the shape and the load-carryng capacity of materials and structures?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2019-11-25T08:49:13.787" Title="Did Romans know formulae?" Tags="|roman-empire|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11187" PostTypeId="1" AcceptedAnswerId="11188" CreationDate="2019-11-25T16:30:29.740" Score="4" ViewCount="546" Body="&lt;h2&gt;Question&lt;/h2&gt;&#xA;&#xA;&lt;p&gt;I'm searching for the origin of the stereotype (regardless of validity) that physicists hate philosophy? This opinion seems to be more mainstream in the public domain. I do concede they are &lt;a href=&quot;https://www.youtube.com/watch?v=X8aWBcPVPMo&quot; rel=&quot;nofollow noreferrer&quot;&gt;some&lt;/a&gt;(/many) &lt;a href=&quot;https://www.telegraph.co.uk/technology/google/8520033/Stephen-Hawking-tells-Google-philosophy-is-dead.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;physicists&lt;/a&gt; &lt;a href=&quot;https://www.npr.org/sections/13.7/2012/05/01/151752815/blackboard-rumble-why-are-physicists-hating-on-philosophy-and-philosophers&quot; rel=&quot;nofollow noreferrer&quot;&gt;who do&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Historically I know the word scientist has not been around for &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientist&quot; rel=&quot;nofollow noreferrer&quot;&gt;too long&lt;/a&gt;. I'm guessing somewhere between then and now something happened?&lt;/p&gt;&#xA;" OwnerUserId="10067" LastActivityDate="2019-11-25T23:02:42.190" Title="When did it become a mainstream stereotype that physicists hate philosophy?" Tags="|philosophy-of-science|physics-social-history|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11188" PostTypeId="2" ParentId="11187" CreationDate="2019-11-25T22:33:36.213" Score="4" Body="&lt;p&gt;The tipping point most probably was the so called Science Wars,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Science_wars&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Science_wars&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;particularly the Sokal affair&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Sokal_affair&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Sokal_affair&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Alan Sokal submitted for publication an article title &quot;Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity&quot;,  proposing that quantum gravity is a social and linguistic construct.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1996, Sokal submitted an article to Social Text, an academic&#xA;  journal of postmodern cultural studies. The submission was an&#xA;  experiment to test the journal's intellectual rigor and, specifically,&#xA;  to investigate whether &quot;a leading North American journal of cultural&#xA;  studies—whose editorial collective includes such luminaries as Fredric&#xA;  Jameson and Andrew Ross—[would] publish an article liberally salted&#xA;  with nonsense if (a) it sounded good and (b) it flattered the editors'&#xA;  ideological preconceptions&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The origin of these Science Wars can be traced back to the publication of Thomas Kuhn's The Structure of Scientific Revolutions in 1962, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;and the subsequent interpretation by some philosophers that Kuhn's ideas meant that scientific theories were, either wholly or in part, social constructs, &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note: the &quot;hate&quot; is/was against postmodernist philosophy, particularly Post-structuralism (Derrida and such luminaries!).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Post-structuralism&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Post-structuralism&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="8161" LastEditorUserId="8161" LastEditDate="2019-11-25T22:38:42.767" LastActivityDate="2019-11-25T22:38:42.767" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11189" PostTypeId="2" ParentId="11187" CreationDate="2019-11-25T22:56:31.333" Score="5" Body="&lt;p&gt;&lt;em&gt;Gillings, R. J.&lt;/em&gt;, &lt;a href=&quot;http://dx.doi.org/10.2307/2307789&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The so-called Euler-Diderot incident&lt;/strong&gt;&lt;/a&gt;, Am. Math. Mon. 61, 77-80 (1954). &lt;a href=&quot;https://zbmath.org/?q=an:0055.00203&quot; rel=&quot;noreferrer&quot;&gt;ZBL0055.00203&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/messouvenirsdevi03thie/page/142&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VlQMS.png&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10561" LastEditorUserId="10561" LastEditDate="2019-11-25T23:02:42.190" LastActivityDate="2019-11-25T23:02:42.190" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11190" PostTypeId="1" AcceptedAnswerId="11193" CreationDate="2019-11-26T14:57:19.917" Score="4" ViewCount="911" Body="&lt;p&gt;When recently asked for a practical application of parabolas, I responded by talking about objects in free-fall. Afterwards as I was re-thinking this conversation it occurred to me that an object in orbit is also in a sort of free-fall, but its path is not parabolic but elliptical.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And a question began to form.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We can only turn to these contexts to illustrate conic sections because we live in the modern world. And math books often illustrate conic sections with technological examples like satellite dishes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But during the very long interval between the first ancient Greek discussions of conic sections and the work of Kepler and Newton, what did everyone think that conic sections were good for? Were they interested in them purely as ideas, without thought of utility? That would be oddly beautiful and somewhat surprising.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked this question at &lt;a href=&quot;https://matheducators.stackexchange.com/questions/17488/what-were-the-applications-of-conic-sections-before-newton/17494?noredirect=1#comment46274_17494&quot;&gt;another site&lt;/a&gt;, and (just before I was put on hold as off-topic) someone informed me of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Doubling_the_cube&quot; rel=&quot;nofollow noreferrer&quot;&gt;Delian Problem&lt;/a&gt;, in which context a discussion of conic sections arose. I see a distinction which I cannot make perfectly clear, but here goes: that's a purely theoretical setting, of interest to mathematicians, while the applications in physics seem to answer more accessible questions, describing real things that exist outside of the heads of mathematicians. I'm wondering if such associations with conic sections existed before Kepler and his association of conic sections with (what would eventually be called) &quot;inertial movement.&quot;&lt;/p&gt;&#xA;" OwnerUserId="9545" LastEditorUserId="9545" LastEditDate="2019-11-26T17:23:48.193" LastActivityDate="2023-05-03T03:21:44.490" Title="What were the applications of conic sections before Kepler?" Tags="|astronomy|euclidean-geometry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11191" PostTypeId="2" ParentId="11190" CreationDate="2019-11-26T16:56:38.937" Score="5" Body="&lt;p&gt;Apparently the conic sections were invented for solving the equation &lt;span class=&quot;math-container&quot;&gt;$x^3=2$&lt;/span&gt; (Delian problem), which is pure mathematics, of course. (Menaechmus, 4th century bc).&lt;/p&gt;&#xA;&lt;p&gt;Some other applications may include construction of sundials (the end of the gnomon shade moves approximately on a conic section in the plane), some authors claimed that this application could be the primary reason for considering them, but there is little evidence for this point of view.&lt;/p&gt;&#xA;&lt;p&gt;Another is the property of an elliptic mirror to focus the light from a source at one focus at another focus,&#xA;and the corresponding property of parabola that rays become parallel. There is a legend that Archimedes used this&#xA;last property to burn Roman ships during the siege of Syracuse. Of course this is impossible in practice, but the existence of the legend shows that this &amp;quot;real world&amp;quot; application was at&#xA;least known theoretically to the ancient writers (Diocles, 2-3d centuries bc). It could be used to focus the light of a lighthouse, but we do not know whether it was actually used in lighthouses in antiquity or medieval times. I suppose nothing of those antique lighthouses survives, and know no detailed ancient descriptions of their construction.&lt;/p&gt;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653&quot;&gt;https://mathoverflow.net/questions/116627/useless-math-that-became-useful/116653#116653&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-01-15T14:15:49.303" LastActivityDate="2021-01-15T14:15:49.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11192" PostTypeId="1" CreationDate="2019-11-26T18:51:09.563" Score="3" ViewCount="250" Body="&lt;p&gt;In group theory, writing functions on the right is a common, though not universal practice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, given mappings &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; and group element &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, one might write &lt;span class=&quot;math-container&quot;&gt;$\alpha f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha (f \circ g)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$f(\alpha)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(g(\alpha))$&lt;/span&gt; respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can see how this makes sense in group theory, especially in relation to permutations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am unable to locate any information on when this practice was introduced and I suspect that it may be rather difficult to identify its origins.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: When was the practice of writing functions on the right introduced in group theory?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2019-11-26T18:51:09.563" Title="Writing functions on the right" Tags="|mathematics|group-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11193" PostTypeId="2" ParentId="11190" CreationDate="2019-11-27T09:45:09.350" Score="7" Body="&lt;p&gt;There is no doubt that ancient Greeks were primarily interested in conic sections for their intra-mathematical uses. Even aside from the Delic problem, the legendary motivation of the discoverer Menaechmus, Euclid and Apollonius explored their various properties for the sake of the art, and Pappus presents a classification of mathematical problems in the Mathematical Collection into plane (solvable with straightedge and compass), solid (solvable with conic sections) and the rest. But some extra-mathematical applications were known already in antiquity&lt;/p&gt;&#xA;&lt;p&gt;#Sundials&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Besides the quite&#xA;obvious presence of conic sections as the lines of the daily motion of the shadow of the&#xA;tip of a gnomon on a plane surface that are usually represented on planar sundials for&#xA;some special days in the year, such lines appear in a completely different context in&#xA;one other type of ancient Greek sundials: For the largest group of sundials, the shadow&#xA;receiving surface is part of the surface of a right cone whose axis directs towards the&#xA;poles. Since all those sundials have a plane top surface that is parallel to the horizon, the&#xA;intersection of those two surfaces is a non-circular conic section.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Conical sundials make up the largest group (about 35%) among the preserved sun&#xA;dials of the Greek type. The oldest sundials with a design as described above date to the beginning of the 2nd century BCE. Some conical sundials with different corpus forms are even slightly older... Due to the relative positions of cone and block, the edge between the conical surface and the planar top surface is a conic section. On the preserved objects we can observe three different types of curves: ellipses, parabolas, and hyperbolas&lt;/em&gt;&amp;quot; &lt;a href=&quot;https://edition-topoi.org/article/1317-ancient-greek-sundials-and-the-theory-of-conic-sections-reconsidered/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rinner, Ancient Greek Sundials and the Theory of Conic&#xA;Sections Reconsidered&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This does not make sundials good candidates for motivating the discovery of conic sections, Menaechmus lived c. 350 BC, but they were in use by the time of Apollonius.&lt;/p&gt;&#xA;&lt;p&gt;#Burning Mirrors&lt;/p&gt;&#xA;&lt;p&gt;Despite the anecdotal story of Archimedes burning the Roman fleet with a parabolic mirror, our actual source is On Burning Mirrors by Diocles (contemporary of Apollonius), who attributes the discovery of the focal property of the parabola (without proof) to Archimedes's friend Dositheus. Although the study of mirrors was quite popular (Euclid wrote a treatise on the issue), the irony is that the focal property was likely discovered as a by-product of proving a property of parabola's tangents, a purely mathematical pursuit. The optical properties of ellipse and hyperbola also seem to have a similarly intra-mathematical origin, see &lt;a href=&quot;https://hsm.stackexchange.com/a/7103/55&quot;&gt;How was the focus/directrix property of conic sections discovered?&lt;/a&gt; Ironically, it was Kepler who singled out the focal properties of conic sections as optical, and introduced the term &lt;em&gt;focus&lt;/em&gt; (Latin for hearth), see &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/003936818690004X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Field, Two Mathematical Inventions in Kepler's 'Ad Vitellionem Paralipomena'&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Unlike Desargues, Kepler introduces the point at infinity in a context not of pure mathematics but of geometrical optics. Indeed, his reason for discussing conics in Ad Vitellionem Paralipomena is that they give possible shapes for burning mirrors, having the property that light emitted at one ‘focus’ (as Kepler was to call it) would be reflected to the other.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;#Geocentric Astronomy&lt;/p&gt;&#xA;&lt;p&gt;Ellipses did appear in astronomy before Kepler, albeit in a different capacity. According to Eudoxus's model, planets were attached to linked homocentric spheres that rotated at an angle to each other. This allowed to reproduce the retrograde motion of the planets without violating Plato's prescription of allowing uniform circular motions only for the celestial bodies. The projection onto the ecliptic of the latitudinal circle, along which the planet spins in the Eudoxian configuration, is an ellipse, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s004070100039&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yavetz, A New Role for the Hippopede of Eudoxus&lt;/a&gt;, although Eudoxus was not familiar with the concept. Yavetz remarks in passing that &amp;quot;&lt;em&gt;a point on an epicycle that spins at twice the rate of its deferent in the opposite sense, traces the projection of an inclined circle. This gives another definition of the ellipse as a special kind of epicyclic motion&lt;/em&gt;&amp;quot;; and suggests that similar observations might have led Apollonius to simplifying Eudoxian spheres to deferents and epicycles. Menaechmus was a student of Eudoxus, and &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF00776065&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riddell in Eudoxan Mathematics and the Eudoxan Spheres&lt;/a&gt; explains how homocentric configurations could lead to his construction of the solution to the Delic problem via conic sections. Of course, this is highly speculative.&lt;/p&gt;&#xA;&lt;p&gt;#Perspective&lt;/p&gt;&#xA;&lt;p&gt;Oblique view of shields and wheels appears already in ancient Greek pottery. Anaxagoras and Democritus apparently studied some basic geometry of perspective (&lt;em&gt;skenographia&lt;/em&gt;), but their works are not extant. Euclid's Optics explicitly postulates that our field of vision is a cone, and in proofs of Propositions 18-21 he even uses a plane cutting this cone of vision. Nonetheless, conic sections are never explicitly mentioned, see &lt;a href=&quot;https://www.jstor.org/stable/41133617&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's Optics and its Compatibility with Linear Perspective by Brownson&lt;/a&gt;. Ptolemy, when using the stereographic projection for his maps in &lt;em&gt;Geographia&lt;/em&gt;, emphasizes that the tropics, the ecliptic, the equator, the arctic and the antarctic circles have an &amp;quot;egg-shaped&amp;quot; form on the map. Whether he was aware that the &amp;quot;eggs&amp;quot; are ellipses is unclear, but it was within his means to prove. Arabic commentators Ibn Sarjûn and al-Kindî discuss the perspective of a wheel and characterize it as an ellipse explicitly, see &lt;a href=&quot;https://halshs.archives-ouvertes.fr/halshs-00479821/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why Did Geometrical Optics not Lead to Perspective in Medieval Islam? by Raynaud&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Ibn Sarjûn and al-Kindî use the words mu‘awwaj 'distorted' and manhir 'deviated', instead of nuqsân 'ellipse, lack', to refer to the appearance of the circle. So they describe the ellipse by the angle formed by the visual ray falling on the centre of the circle and the line perpendicular to the circle's plane, what&#xA;presupposes an affine transformation &lt;span class=&quot;math-container&quot;&gt;$K = fa / fb = gc / gd$&lt;/span&gt; (Fig. 2). This geometrical property was known from Archimedes' Conoids and Spheroids, prop. 4 (1970: 166-170) and Ibrâhîm b. Sinân expounded on it in his Maqâla fîrasm al-qutû‘al-thalâtha (Epistle on the Drawing of the Three Sections) (Rashed and Bellosta, 2000: 263-89).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, Renaissance scholars and painters were much much more explicit and comprehensive about the circle perspective. Alberti's grid construction of an ellipse as circle's perspective from &lt;a href=&quot;https://en.wikipedia.org/wiki/De_pictura&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Pictura&lt;/em&gt; (1435)&lt;/a&gt; is shown below. It is interesting that Kepler himself introduced the perspectival points at infinity in the aforementioned discussion of conic sections in &lt;em&gt;Ad Vitellionem Paralipomena&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vh0xL.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vh0xL.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-05-03T03:21:44.490" LastActivityDate="2023-05-03T03:21:44.490" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11194" PostTypeId="2" ParentId="2309" CreationDate="2019-11-28T11:26:03.287" Score="4" Body="&lt;p&gt;I beg to differ with Conifold's answer, and say that in this case the truth is more likely, that we do know why, reasonably well. There are at least two strong practical reasons to study conics, besides the mathematical interest per se, both from physics, precisely optics and acoustics. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We do know that the famous Lighthouse of Alexandria had a parabolic projector, which was visible at a distance almost &lt;em&gt;50 km&lt;/em&gt;, according to Iosephus. Knowing the focal properties of the parabola, discovered by the Greek geometers, are necessary in order to build such an advanced technological device, which was so admirable to be named &quot;the seventh  wonder of the world&quot;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;We know that theaters were extremely important in the culture and social life of the Greek world, and we do know that they were carefully projected and built, in particular, in order that the voice of the actors be clearly audible by the (often more than 10,000) spectators. Again, the knowledge of the focal properties of conics, that we do know they did know, are mandatory to this aim. The acoustic effect is still observable in the remaining Greek theaters, where the conic shapes and the location of the focus can be measured.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For a detailed account of the Hellenistic science I recommend Lucio Russo &lt;em&gt;The Forgotten Revolution&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6987" LastActivityDate="2019-11-28T11:26:03.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11195" PostTypeId="1" AcceptedAnswerId="11196" CreationDate="2019-11-29T14:30:31.543" Score="8" ViewCount="746" Body="&lt;p&gt;There are two related historical questions that I'm trying to find answers to:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Who was the first to introduce the concept of magnetic monopoles?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Griffiths in his textbook &lt;em&gt;Introduction to Electrodynamics&lt;/em&gt; refers to a model of magnetic dipole, formed analogously to electric dipole via pair of magnetic monopoles, as a &quot;Gilbert model&quot; (presumably giving a tribute to William Gilbert and his classical work &quot;De Magnete&quot;). Who was the first to consider such model of magnetic dipole?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Several comments:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The earliest reference that is sometimes mentioned in this context is a letter by Petrus Peregrinus de Maricourt from 1269 (translation available at &lt;a href=&quot;http://www.gutenberg.org/files/50524/50524-0.txt&quot; rel=&quot;noreferrer&quot;&gt;Project Gutenberg&lt;/a&gt;), which contains an early (first?) identification of two distinct poles of magnets, but I'm suspicious if this can be interpreted as the introduction of the concept of isolated magnetic poles.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maxwell himself in seminal paper &quot;On Physical Lines of Forces&quot; (1861) (cf. &lt;a href=&quot;https://en.wikisource.org/wiki/On_Physical_Lines_of_Force&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;), on the right-hand side of the equation (6) has quantity &quot;&lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;&quot; which looks like something which in modern notation would correspond to density of magnetic charges, &lt;span class=&quot;math-container&quot;&gt;$\rho_m$&lt;/span&gt; (please, correct me if I'm misinterpreting this!).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;P. Curie in his paper &quot;Sur la possibilité d'existence de la conductibilité magnétique et du magnétisme libre&quot; (1894) (&lt;a href=&quot;https://doi.org/10.1051/jphystap:018940030041501&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;J. Phys. Theor. Appl.&lt;/em&gt; &lt;strong&gt;3&lt;/strong&gt;, 415-417 (1894)&lt;/a&gt;) speculates on possibility of isolated magnetic charges.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;H. Poincaré in his paper &quot;Remarques sur une expérience de M. Birkeland&quot; (1896) (available at &lt;a href=&quot;http://www.bibnum.education.fr/physique/electricite-electromagnetisme/remarques-sur-une-experience-de-m-birkeland&quot; rel=&quot;noreferrer&quot;&gt;bibnum&lt;/a&gt;) considers the problem of motion of point electric charge in a field of magnetic monopole (in his words: &quot;Supposons un seul pôle magnétique...&quot;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;J.J. Thomson in his book &quot;Electricity and Matter&quot; (1904) considers isolated (?) &quot;magnetic poles&quot; (see e.g. p. 25 in &lt;a href=&quot;https://archive.org/details/electricitymatte00thomiala/page/24&quot; rel=&quot;noreferrer&quot;&gt;the copy at the Internet Archive&lt;/a&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10760" LastEditorUserId="26" LastEditDate="2021-02-20T15:21:38.167" LastActivityDate="2021-02-20T15:21:38.167" Title="Historical origin of magnetic monopoles" Tags="|electromagnetism|magnetism|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11196" PostTypeId="2" ParentId="11195" CreationDate="2019-11-30T01:55:11.570" Score="7" Body="&lt;p&gt;The modern concept of magnetic &lt;em&gt;monopole&lt;/em&gt; (as a real isolated charge) is due to Dirac in 1931, although Curie speculated about the possibility earlier. Even electric charges, as in particles, only appeared in 19th century, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Electron#Discovery_of_two_kinds_of_charges&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's Discovery of two kinds of charges&lt;/a&gt;. Before that electricity and magnetism were mostly viewed as produced by fluids, one or two. That was Franklin's (one) and Coulomb's (two) view, for example.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The identification of two magnetic &lt;em&gt;poles&lt;/em&gt; in magnets is suggested by Peregrinus, and more explicitly by Gilbert. The first quantitative model that interprets magnets as dipoles with the poles attracting/repulsing according to the inverse square law appears in &lt;a href=&quot;https://archive.org/details/b30532978/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michell's Treatise of Artificial Magnets (1750)&lt;/a&gt;. Here is from &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich/page/n6&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of the Theories of Aether and Electricity&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In this he states the principles of magnetic theory as follows&lt;/em&gt;:&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&quot;&lt;em&gt;Wherever any Magnetism, is found, whether in the Magnet&#xA;    itself, or any piece of Iron, etc., excited by the Magnet, there are&#xA;    always found two Poles, which are generally called North and&#xA;    South; and the North Pole of one Magnet always attracts the&#xA;    South Pole, and repels the North Pole of another: and vice versa&lt;/em&gt;&quot;&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;This is of course adopted from Gilbert. &quot;Each Pole attracts or repels exactly equally, at equal distances, in every direction.&quot; This, it may be observed, overthrows&#xA;  the theory of vortices, with which it is irreconcilable. &quot;The Magnetical Attraction and Repulsion are exactly equal to each other.&quot; This, obvious though it may seem to us, was really a most important advance, for, as he remarks,&lt;/p&gt;&#xA;  &#xA;  &lt;blockquote&gt;&#xA;    &lt;p&gt;&quot;&lt;em&gt;Most people, who have mention'd any thing relating to this property of the Magnet,&#xA;    have agreed, not only that the Attraction and Repulsion of Magnets are not equal to each other, but that also, they do not observe the same rule of increase and decrease... The Attraction and Repulsion of Magnets decreases, as the&#xA;    Squares of the distances from the respective poles increase.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;  &lt;/blockquote&gt;&#xA;  &#xA;  &lt;p&gt;This great discovery, which is the basis of the mathematical&#xA;  theory of Magnetism, was deduced partly from his own observations,&#xA;  and partly from those of previous investigators (e.g.&#xA;  Dr. Brook Taylor and P. Muschenbroek), who, as he observes,&#xA;  had made accurate experiments, but had failed to take into&#xA;  account all the considerations necessary for a sound theoretical&#xA;  discussion of them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Interestingly, the inverse cube law for dipoles that follows from it was noted experimentally already by Newton in Principia, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2566/55&quot;&gt;Is Coulomb's law the earliest mathematical formula describing electricity?&lt;/a&gt; Aepinus suggested identifying the poles with shortfalls and excesses of the magnetic fluid in 1759, with a mechanistic explanation for the inseparability of the poles, and Brugmans and Wilcke instead introduced two opposing fluids, called &quot;boreal&quot; and &quot;austral&quot;. Michell's dipole model was supported by Mayer and Lambert, and became standard after Coulomb's experiments (1777), whose sources were not particle charges, but boreal and austral fluids jointly locked within molecules. Here is from Whittaker again: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Coulomb rendered great services to magnetic theory. It was&#xA;  he who in 1777, by simple mechanical reasoning, completed&#xA;  the overthrow of the hypothesis of vortices. He also, in the&#xA;  second of the Memoirs already quoted, confirmed Michell's&#xA;  law, according to which the particles of the magnetic fluids&#xA;  attract or repel each other with forces proportional to the&#xA;  inverse square of the distance.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Coulomb, however, went beyond&#xA;  this, and endeavoured to account for the fact that the two&#xA;  magnetic fluids, unlike the two electric fluids, cannot be&#xA;  obtained separately; for when a magnet is broken into&#xA;  two pieces, one containing its north and the other its south&#xA;  pole, it is found that each piece is an independent magnet&#xA;  possessing two poles of its own, so that it is impossible&#xA;  to obtain a north or south pole in a state of isolation.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Coulomb explained this by supposing that the magnetic&#xA;  fluids are permanently imprisoned within the molecules&#xA;  of magnetic bodies, so as to be incapable of crossing from&#xA;  one molecule to the next; each molecule therefore under all&#xA;  circumstances contains as much of the boreal as of the austral fluid, and magnetization consists simply in a separation&#xA;  of the two fluids to opposite ends of each molecule. Such&#xA;  a hypothesis evidently accounts for the impossibility of&#xA;  separating the two fluids to opposite ends of a body of finite&#xA;  size. The same idea, here introduced for the first time, has&#xA;  since been applied with success in other departments of&#xA;  electrical philosophy.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Ampere's &quot;molecular currents&quot; replaced Coulomb's magnetic fluids with circulating electric ones, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5150/55&quot;&gt;What is the history of electric current and resistance?&lt;/a&gt; There was little motivation to ponder magnetic charges before the existence of electric ones was accepted in the second half of 19th century.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-12-04T22:49:26.830" LastActivityDate="2019-12-04T22:49:26.830" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11198" PostTypeId="1" AcceptedAnswerId="11201" CreationDate="2019-12-02T14:42:07.907" Score="7" ViewCount="310" Body="&lt;p&gt;There are a number of photos of bits of Galois's notes floating around online, such as &lt;a href=&quot;http://www.gutenberg.org/files/40213/40213-pdf.pdf?session_id=cacdcb683f105e0d90ea81986953ab5c38da2848&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.storyofmathematics.com/19th_galois.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. But where are they, physically? I'm hoping what remained of them wasn't tossed out once photos were taken.&lt;/p&gt;&#xA;" OwnerUserId="10767" LastActivityDate="2019-12-02T20:27:41.023" Title="Where are Galois's notes?" Tags="|galois-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11199" PostTypeId="1" AcceptedAnswerId="11200" CreationDate="2019-11-29T13:21:47.153" Score="2" ViewCount="295" Body="&lt;p&gt;I'm writing a paper which involves a lot of quantum mechanics.&#xA;As a result of this, I'm finding it necessary to cite the likes of Einstein, Born and Jordan.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Particularly in the case of Einstein, due to the amount of times he has been quoted online and the fact that most of his original works were in private letters or German journals, I'm struggling to find the original source for some of his quotations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Specifically, I want to quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We must use sometimes the one theory and sometimes the other, while at times we may use either.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with reference to wave-particle duality. It's a pretty well established translation of a quotation of Einstein, however, I've been unable to find the original source of this in over an hour of searching.&lt;/p&gt;&#xA;" OwnerDisplayName="Persistence" LastActivityDate="2019-12-02T15:02:19.993" Title="What is the source of this quote by Einstein on wave–particle duality?" Tags="|reference-request|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11200" PostTypeId="2" ParentId="11199" CreationDate="2019-11-29T16:52:09.547" Score="2" Body="&lt;p&gt;You can find it on &lt;a href=&quot;https://scholar.google.com/citations?user=qc6CJjYAAAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein’s Google citations&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can then use Google Translate.&lt;/p&gt;&#xA;" OwnerDisplayName="deags" LastActivityDate="2019-12-02T14:59:57.020" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11201" PostTypeId="2" ParentId="11198" CreationDate="2019-12-02T20:27:41.023" Score="10" Body="&lt;p&gt;At the Library of  l'Institut de France that is &lt;a href=&quot;https://rhpst.huma-num.fr/items/show/122&quot; rel=&quot;noreferrer&quot;&gt;Bibliotheque de L'Institut de France&lt;/a&gt;: &quot;285 feuillets en 27 dossiers. Formats divers. Demi-reliure.&quot; There is a research tool &lt;a href=&quot;http://www.calames.abes.fr/pub/#details?id=IF2B14057&quot; rel=&quot;noreferrer&quot;&gt;online.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2019-12-02T20:27:41.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11202" PostTypeId="1" AcceptedAnswerId="11205" CreationDate="2019-12-03T15:05:01.773" Score="4" ViewCount="131" Body="&lt;p&gt;Do you know anything about it? I couldn't really find something useful on web.&lt;/p&gt;&#xA;" OwnerUserId="10729" LastEditorUserId="10729" LastEditDate="2019-12-03T16:11:27.320" LastActivityDate="2019-12-04T06:33:41.480" Title="When was the inverse quadratic interpolation method first used?" Tags="|mathematics|discoveries|zero|algorithm|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11204" PostTypeId="1" AcceptedAnswerId="11210" CreationDate="2019-12-04T01:26:08.127" Score="6" ViewCount="437" Body="&lt;p&gt;Do North Koreans use Latin (and Greek) letters in their equations?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the one hand, being such an isolationist country, I wouldn't be surprised if they used the Korean alphabet (조선글) in their equations. On the other hand, the similarly isolationist Soviet Union used Latin letters in its equations rather than Cyrillic (Кириллица) throughout its history, even in the space and nuclear programs.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nearby China uses Latin as well (imagine using characters (漢字) as variables! …Actually that would be so cool. But never mind)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another possibility is that they use the Korean alphabet in elementary education and switch to Latin for more advanced subjects (such as their own rocket and nuclear programs).&lt;/p&gt;&#xA;" OwnerUserId="10773" LastActivityDate="2021-02-26T20:27:51.447" Title="Do North Koreans use Latin letters in their equations?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="11205" PostTypeId="2" ParentId="11202" CreationDate="2019-12-04T06:33:41.480" Score="6" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0025570X.1960.11975188&quot; rel=&quot;noreferrer&quot;&gt;Maley's Higher Order Approximations to Solutions of&#xA;Transcendental Systems (1960)&lt;/a&gt;, the earliest occurrence of the inverse quadratic interpolation for finding roots is in &lt;a href=&quot;https://archive.org/details/mmoiresdelacad31826acad/page/n55&quot; rel=&quot;noreferrer&quot;&gt;Dandelin's Recherches sur la resolution des equations numeriques (1826)&lt;/a&gt; published in &lt;em&gt;Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Bruxelles&lt;/em&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Germinal_Pierre_Dandelin&quot; rel=&quot;noreferrer&quot;&gt;Dandelin&lt;/a&gt; was a Belgian mathematician now mostly remembered for a clever construction that proves focal properties of conic sections (Dandelin spheres).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The term &quot;inverse quadratic interpolation&quot; is not used by Dandelin. The earliest mention I found is in the MathSciNet review of &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/10.1002/zamm.19480280404&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Praktische Interpolation höherer Ordnung&lt;/em&gt; by Rubbert (1948)&lt;/a&gt;, and &lt;a href=&quot;https://books.google.com/ngrams/graph?content=inverse+quadratic+interpolation&amp;amp;year_start=1800&amp;amp;year_end=2008&amp;amp;corpus=15&amp;amp;smoothing=3&amp;amp;share=&amp;amp;direct_url=t1%3B%2Cinverse%20quadratic%20interpolation%3B%2Cc0#t1%3B%2Cinverse%20quadratic%20interpolation%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;Google Ngram&lt;/a&gt; shows that its use does not pick up until 1959, perhaps due to the use on computers. Muller adapted the method for one of the first computers, ILLIAC, in 1956, see &lt;a href=&quot;https://www.jstor.org/stable/2001916&quot; rel=&quot;noreferrer&quot;&gt;Method for Solving Algebraic Equations Using an Automatic Computer&lt;/a&gt;.  It gained its modern prominence after Brent combined it with bisection to speed up Dekker's algorithm for finding zeros of a function without taking derivatives, see &lt;a href=&quot;https://maths-people.anu.edu.au/~brent/pd/rpb005.pdf&quot; rel=&quot;noreferrer&quot;&gt;Brent, An algorithm with guaranteed convergence for finding a zero of a function (1971)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-12-04T06:33:41.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11206" PostTypeId="1" CreationDate="2019-12-04T08:22:55.010" Score="2" ViewCount="73" Body="&lt;p&gt;I am wondering as how does the concept of centre of mass came into being. Considering the usefulness of the concept how were the physicist able to define such a quantity. I mean was it just trial and error or something else? A brief history of the concept would do the job! &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thanks!&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user10638" LastActivityDate="2019-12-04T14:01:28.967" Title="How does the concept of centre of mass came to existence?" Tags="|physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11209" PostTypeId="1" CreationDate="2019-12-04T21:24:38.160" Score="1" ViewCount="55" Body="&lt;p&gt;For this doesn't make sense - we see one &lt;em&gt;half&lt;/em&gt; of moon, not one &lt;em&gt;quarter&lt;/em&gt; (at least compared to full moon).&lt;/p&gt;&#xA;" OwnerUserId="9778" LastActivityDate="2019-12-04T21:24:38.160" Title="When was the first quarter first used for moon phase?" Tags="|astronomy|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11210" PostTypeId="2" ParentId="11204" CreationDate="2019-12-04T22:42:13.237" Score="6" Body="&lt;p&gt;North Korea has its own academy of science in which there is a mathematics institute. Its mathematicians are trained (mostly) in Russia (formal Soviet Union) and other Eastern European countries like Hungary, East Germany and Poland. North Korean mathematics is far worse than South Koreans in terms of research but is OK in terms of the application of mathematics as in building its nukes and missiles.&lt;/p&gt;&#xA;&lt;p&gt;So North Korean mathematicians use Latin and Greek alphabets for mathematical symbols rather than Korean's. So are Chinese mathematicians, who were (mostly) trained in the west (Germany, England and France) before World War II, and in the Eastern European countries like the Soviet Union and Poland in the 1950s and 1960s, and have been trained (mostly) in North America after the 1980s.&lt;/p&gt;&#xA;&lt;p&gt;In the ancient time, Chinese mathematics was written in Chinese characters, e.g. Chinese numerals like 〇, 一, 二, 三, 四, 五, 六, 七, 八, 九 (for &lt;span class=&quot;math-container&quot;&gt;$0,\cdots,9$&lt;/span&gt; respectively), and so on. Note that they are simple compared to other Chinese characters for easy writing. In the modern time, Chinese mathematics adopts Latin and Greek alphabets and abandons the Chinese numerals because Chinese learn modern mathematics (entirely) from the west.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-02-26T20:27:51.447" LastActivityDate="2021-02-26T20:27:51.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11212" PostTypeId="1" AcceptedAnswerId="11213" CreationDate="2019-12-05T14:09:49.013" Score="4" ViewCount="3110" Body="&lt;p&gt;Except Navier-Stokes equation, are there any other interesting open problems in partial differential equations?   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to know the collection of problems, which are easy to understand but difficult to solve. Like in number theory, we have &lt;a href=&quot;https://en.wikipedia.org/wiki/Goldbach%27s_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Goldbach conjecture&lt;/a&gt; which is easy to understand, but still unsolved.&lt;/p&gt;&#xA;" OwnerUserId="10768" LastActivityDate="2019-12-05T16:01:10.010" Title="Collection of open problems in Partial differential equations" Tags="|mathematics|mathematical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11213" PostTypeId="2" ParentId="11212" CreationDate="2019-12-05T16:01:10.010" Score="9" Body="&lt;p&gt;This is a collection of open problems concerning various areas in function theory, functional analysis, theory of linear and nonlinear partial differential equations.&#xA;&lt;a href=&quot;https://link.springer.com/article/10.1007/s00020-018-2460-8&quot; rel=&quot;noreferrer&quot;&gt;Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9801" LastActivityDate="2019-12-05T16:01:10.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11214" PostTypeId="2" ParentId="11097" CreationDate="2019-12-05T20:16:57.403" Score="1" Body="&lt;p&gt;Yes, before Tartaglia, Europe had experienced a dark age in culture and science (the Middle Ages) over a thousand years. So it was natural for scholars at that time to think they were inferior to Greeks. Tartaglia was in the period of Renaissance that marked the resurrection of art and scientific activities from the golden ages of Greeks science and knowledge led by Plato, Aristotle and Archimedes.&lt;/p&gt;&#xA;&lt;p&gt;The Renaissance marks the fundamental and comprehensive changes of European culture during the 15th and 16th centuries, which led to the demise of the Middle Ages, and for the first time ever, embodied the values of  the modern world. Achievements from polymath artists led by Da Vinci and scientists such as Copernicus, Galileo, Kepler, and so on, proclaimed that their age had progressed beyond the intellectual decline of the Middle Ages and accomplished resurrection from the ancient Greek and Roman civilizations.&lt;/p&gt;&#xA;&lt;p&gt;The scientific revolution started in the Renaissance continued into the 17 century and led to the discovery of new scientific fields like Analytic Geometry, Calculus, Mechanics, and so on, discovered by Newton, Descartes, Leibniz and others. This revolution marks that the golden ages of Greek science and knowledge had been completely surpassed.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-02-26T20:22:45.923" LastActivityDate="2021-02-26T20:22:45.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11215" PostTypeId="1" AcceptedAnswerId="11229" CreationDate="2019-12-08T03:07:07.167" Score="3" ViewCount="183" Body="&lt;p&gt;p. 153 of&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Raúl M. Falcón Ganfornina and Juan Núñez Valdés, “&lt;a href=&quot;https://isidore.co/calibre/get/PDF/5327&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Foundations of Santilli Isotopies&lt;/a&gt;,” trans. Alan Aversa, &lt;a href=&quot;http://hadronicpress.com/AGG/agg-contents.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Algebras, Groups, and Geometries&lt;/em&gt;&lt;/a&gt; &lt;strong&gt;32&lt;/strong&gt; (2015): 135–308.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;quotes (but does not cite) Einstein saying that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;without the discoveries of Lie, the Theory of Relativity would probably never have been born.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Did Einstein actually say this (or something similar)?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can only find &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/~searchResults?searchMode=app&amp;amp;searchParam-start=0&amp;amp;searchParam-page=20&amp;amp;searchParam-terms=%22Sophus%20Lie%22&amp;amp;searchParam-depth=fulltext&amp;amp;searchParam-versions=all&amp;amp;searchParam-years=%7B%22operator%22%3A%22%22%7D&amp;amp;searchParam-sort=date&amp;amp;searchParam-context=-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;a few references to Sophus Lie in the Einstein Papers&lt;/a&gt;, and they are only in letters addressed to Einstein.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2019-12-11T17:32:50.073" LastActivityDate="2019-12-11T17:53:28.583" Title="Einstein praising Sophus Lie" Tags="|mathematics|physics|reference-request|relativity-theory|einstein|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11216" PostTypeId="1" AcceptedAnswerId="11219" CreationDate="2019-12-08T05:03:03.127" Score="2" ViewCount="124" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The idea of a spin angular momentum was first proposed in a 1925 publication by George Uhlenbeck and Samuel Goudsmit to explain hyperfine splitting in atomic spectra.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At what time was it measured that the subatomic particles have magnetic dipole moments?&lt;/p&gt;&#xA;" OwnerUserId="3223" LastEditorUserId="55" LastEditDate="2019-12-08T10:14:14.720" LastActivityDate="2019-12-08T10:14:14.720" Title="When was spin of subatomic particles other than the electron discovered?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11217" PostTypeId="1" CreationDate="2019-12-08T05:40:36.103" Score="6" ViewCount="253" Body="&lt;p&gt;I'd prefer a year-by-year graph, something like (not necessarily the &quot;demand&quot;, just &quot;supply&quot;)...&lt;br&gt;&#xA;&amp;nbsp; &amp;nbsp; &lt;a href=&quot;https://www.statista.com/statistics/751749/worldwide-data-storage-capacity-and-demand/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.statista.com/statistics/751749/worldwide-data-storage-capacity-and-demand/&lt;/a&gt; &lt;br&gt;&#xA;Google found me that, which goes back to 2009, but I couldn't get it to cough up anything going back as far as 1969 (or even earlier).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp; &amp;nbsp; &lt;strong&gt;E d i t&lt;/strong&gt;&lt;br&gt;--------------&lt;br&gt;&#xA;Re @GeraldEdgar's comment. Tapes, maybe: in 1969 the 6250bpi 9-track&#xA;tapes were just becoming available, replacing 1600bpi (I still recall&#xA;pressing Stop/Rewind/Unload and the subsequent vacuum column &quot;whine&quot;).&#xA;So a 1200-foot tape would contain 11.25MB 8-bit bytes,&#xA;or let's just say 100tapes/GB, or 50tapes/GB using 2400-foot tapes.&#xA;And the typical datacenter might have a GB or so of disk,&#xA;so I'd guess you're likely right that tape storage exceeded disk.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But a box of 2000 80-column cards, even if binary-punched,&#xA;would store 160KB, whereby a GB requires 6250 boxes.&#xA;And, wow, I don't recall ever seeing that many boxes&#xA;of cards in one place (or maybe ever at all, for that matter).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I asked the question with respect to disk because of an&#xA;off-the-cuff remark I made to a co-worker that my home-office&#xA;disk capacity of ~75TB (mostly backups, six 10TB WD EasyStore's,&#xA;a few 4TB 2.5&quot; externals, plus about 10TB of internal disks)&#xA;might well exceed the entire planet's disk capacity back in 1969,&#xA;when I got my first computer job.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That conjecture was met with extreme skepticism. But I figured&#xA;that at ~1GB/datacenter (which may well be an overestimate),&#xA;and maybe ~35,000 datacenters worldwide in 1969,&#xA;I might even have double the entire planet's disk capacity&#xA;back then. (Note: I also couldn't google the number of datacenters&#xA;in 1969, but seem to vaguely recall hearing something like ~35,000&#xA;back at that time).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp; &amp;nbsp; &lt;strong&gt;E d i t # 2&lt;/strong&gt;&lt;br&gt;-------------------&lt;br&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, below's a figure I found at&#xA;&lt;a href=&quot;https://ethw.org/w/images/5/52/EPC_-_fig_6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ethw.org/w/images/5/52/EPC_-_fig_6.jpg&lt;/a&gt;,&#xA;which comes from an interesting historical article at&#xA;&lt;a href=&quot;https://ethw.org/Early_Popular_Computers,_1950_-_1970&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ethw.org/Early_Popular_Computers,&lt;em&gt;1950&lt;/em&gt;-_1970&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The numbers seem roughly in line with my remark above about&#xA;a &quot;vague recollection&quot; of 35,000 datacenters worldwide during&#xA;the illustrated timeframe. And if we then also accept my&#xA;perhaps-generous guesstimate of 1GBdisk/datacenter, then my&#xA;home office disk capacity of 75TB pretty much equals the total&#xA;planet's disk capacity circa 1968.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course, the predictable hockey-stick shape of this curve&#xA;quickly overwhelmed my little boast.&#xA;But it's still kind of interesting to contemplate, &lt;em&gt;i.e.,&lt;/em&gt;&#xA;a mind-boggling way to illustrate how incredibly far and fast&#xA;the technology has advanced. I could instead say that my desktop&#xA;i7 is about a million times faster than the IBM&amp;nbsp;1620&#xA;I first programmed in the fall of 1966. And that's indeed&#xA;impressive, but doesn't quite capture the imagination in the&#xA;same way as saying my home office disk capacity equals the&#xA;entire planet's disk capacity in that same year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3P0Em.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3P0Em.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10787" LastEditorUserId="10787" LastEditDate="2019-12-19T12:19:37.230" LastActivityDate="2019-12-19T12:19:37.230" Title="What was worldwide supply of disk storage in 1969?" Tags="|computers|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11218" PostTypeId="1" AcceptedAnswerId="11220" CreationDate="2019-12-08T08:13:08.557" Score="12" ViewCount="596" Body="&lt;p&gt;On the &lt;a href=&quot;https://grothendieck.umontpellier.fr/&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt; that now displays the part of Grothendieck's archives that had been held at the University of Montpellier, it is mentioned that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dans une lettre adressée à Monsieur Lefranc datée du 13 octobre 1985 (voir cote 183/97), Grothendieck évoque la disparition de sa correspondance allant de S à Z. Il affirme que ces documents se sont volatilisés à l’occasion du « sac » de son bureau la même année.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(&lt;em&gt;Own translation&lt;/em&gt;: In a letter addressed to Monsieur Lefranc dated October 13, 1985 (see reference 183/97), Grothendieck addresses the disappearance of his correspondence from S to Z. He asserts that these documents vanished on the occasion of the &amp;quot;burglary&amp;quot; of his office that same year.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm interested in facts about this.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Is it true that these letters were stolen?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;If so, then who might have been the perpetrators and what might have been their motives? And is it then possible that historically interesting letters, from perhaps Tate or Serre, might have been lost?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6151" LastEditorUserId="4251" LastEditDate="2021-02-28T16:39:46.577" LastActivityDate="2021-02-28T16:39:46.577" Title="Alexander Grothendieck's &quot;stolen&quot; correspondence in 1985?" Tags="|mathematics|mathematicians|biographical-details|grothendieck|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11219" PostTypeId="2" ParentId="11216" CreationDate="2019-12-08T10:05:00.110" Score="4" Body="&lt;p&gt;Subatomic particles, other than electron, photon and proton were not yet discovered in 1925. On the proton (which at the time was only known as hydrogen nucleus &quot;inside&quot; other nuclei), Compton, who conjectured the electron spin in &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S0016003221909177&quot; rel=&quot;nofollow noreferrer&quot;&gt;The magnetic electron (1921)&lt;/a&gt;, wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Besides the molecule and the atom we have the other two fundamental&#xA;  divisions of matter, the atomic nucleus and the electron.&#xA;  The sign of the Richardson-Barnett effect indicates that it is&#xA;  negative electricity which is chiefly responsible for magnetic effects,&#xA;  which makes the view that the positive nucleus is the elementary&#xA;  magnet difficult to defend.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Uhlenbeck and Goudsmit followed Compton, even their follow-up expose in Nature &lt;a href=&quot;http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Uhlenbeck_Goudsmit_1927.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spinning Electrons and the Structure of Spectra&lt;/a&gt; only talks of spinning electrons. As for the photon, according to &lt;a href=&quot;http://dspace.rri.res.in/bitstream/2289/2123/1/1931%20IJP%20V6%20p353.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Experimental proof of the  spin  of the photon by Raman-Bhagavantam&lt;/a&gt;, Bose considered the possibility back in 1924&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In his well-known derivation of the Planck radiation formula from quantum&#xA;  statistics, Prof. Bose obtained an expression for the number of cells in phasespace&#xA;  occupied by the radiation, and found himself obliged to multiply it by a numerical factor &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; in order to derive from it the correct number of possible&#xA;  arrangements of the quantum in unit volume. The paper as published did not&#xA;  contain a detailed discussion of the necessity for the introduction of this factor,&#xA;  but we understand from a personal communication by Prof. Bose that he&#xA;  envisaged the possibility of the quantum possessing besides energy &lt;span class=&quot;math-container&quot;&gt;$h\nu$&lt;/span&gt; and linear&#xA;  momentum &lt;span class=&quot;math-container&quot;&gt;$h\nu/c$&lt;/span&gt; also an intrinsic spin or angular momentum &lt;span class=&quot;math-container&quot;&gt;$\pm h/2\pi$&lt;/span&gt; round an&#xA;  axis parallel to the direction of its motion. The weight factor &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; thus arises from the possibility of the spin of the quantum being either right-handed or left-handed,&#xA;  corresponding to the two alternative signs of the angular momentum.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For further developments see &lt;a href=&quot;https://arxiv.org/pdf/1311.5016.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Milner, A Short History of Spin&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In the 1930s, Rabi and collaborators (inc. N. Ramsey and J. Za&#xA;  charias) using molecular beams&#xA;  in a weak magnetic  field measured  the magnetic  moments and nuclear  spins  of hydrogen,  deuterium, and heavier nuclei [22].&#xA;  By the end of the 1940s, the nuclear shell model had been established [23]. This explained the&#xA;  properties and structure of atomic nuclei and underscored the essential role of proton and neutron&#xA;  spin.  A key aspect was the strong role of spin-orbit coupling, which was suggested to Goeppert-Mayer by a question from Fermi... By the middle of the twentieth century, the intrinsic spin of&#xA;  subatomic particles was a cornerstone of the physicist’s theoretical understanding of the fundamental structure of matter.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-12-08T10:05:00.110" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11220" PostTypeId="2" ParentId="11218" CreationDate="2019-12-08T20:54:22.520" Score="14" Body="&lt;p&gt;The &lt;a href=&quot;https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality/Spirituality13.pdf&quot; rel=&quot;noreferrer&quot;&gt;Grothendieck Circle&lt;/a&gt; site suggests a more innocent explanation for the loss these letters.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having left Montpellier in 1984....&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In May of that year [1985] a&#xA;  secretary informed him that his office on the fourth floor of the institute had&#xA;  been cleared out. Seeing this incident as an egregious example of the general&#xA;  decline of mores, an outraged Grothendieck appealed to M. Lefranc, the&#xA;  director of the institute, to the president of the university, to Mme. Charles,&#xA;  the person directly responsible for issues concerning offices, and, in an open&#xA;  letter, to his colleagues from the mathematical institute. (In another letter&#xA;  written later, he excused himself for having, in his haste, omitted to send the&#xA;  open letter to the staff and the graduate students.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In her answer, Mme. Charles described the situation from her point of&#xA;  view, and pointed out that Grothendieck lived very far away, was difficult to&#xA;  reach, and had not been seen at the institute for a long time. A meeting of&#xA;  lecturers from the institute (UER 5) took place in the presence of&#xA;  Grothendieck, and the director of the institute sent him a&#xA;  collective letter of apology: “Les enseignants de Mathématiques présentent leurs excuses à Monsieur Grothendieck”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2019-12-09T01:14:25.593" LastActivityDate="2019-12-09T01:14:25.593" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11222" PostTypeId="1" CreationDate="2019-12-09T16:54:43.033" Score="0" ViewCount="146" Body="&lt;p&gt;I am not asking the proof of why the Carnot Engine is the most efficient engine.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wish to know the history of the development of idea of the most efficient engine. I wish to know how Carnot came up with the idea of Carnot Engine being the most efficient engine? and how did he know that it was the most efficient engine?&lt;/p&gt;&#xA;" OwnerUserId="10791" LastActivityDate="2019-12-10T17:56:29.103" Title="How did Carnot come up with the idea of Carnot Engine being the most efficient engine?" Tags="|thermodynamics|entropy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11224" PostTypeId="1" CreationDate="2019-12-09T17:26:09.010" Score="6" ViewCount="249" Body="&lt;p&gt;I know that when two bodies of different temperature are kept in contact &quot;heat&quot; flows from hotter body to colder. But how did anyone know that it is the &quot;hotness&quot; that flows, one could have said that the &quot;coldness&quot; flows from colder body to hotter one. Why was heat given a preference as both cases were equally possible.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think that the theory that hotness flows from one body to another came long before the kinetic energy theory of energy transfer as heat. Also the caloric theory was given long before kinetic energy theory. In a way everyone knew that it was heat that is flowing and coldness is absence of heat. But how did they know it?&lt;/p&gt;&#xA;" OwnerUserId="10793" LastEditorUserId="10793" LastEditDate="2019-12-10T14:55:15.787" LastActivityDate="2019-12-13T23:14:20.010" Title="What is the history behind defining temperature as measure of hotness?" Tags="|physics|thermodynamics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11225" PostTypeId="1" CreationDate="2019-12-09T18:10:27.183" Score="4" ViewCount="169" Body="&lt;p&gt;Nowadays, it seems just common sense to write a program in a high-level programming language and let it be compiler (or interpreted) into machine code to run a computer. However, when did the precursors of computer scientists realized that what they needed was a 'programming language' with syntax, grammar, well-defined expressions and symbols,  for example. Couldn't they just thought that any machine that processes information would just get mathematical formula as input? Have 19th century loom 'programmers' already used something akin to a language, in the same way they used punched cards? Was there an eureka moment at all, when people realized that computing machines were not just some really big calculator (to do the tedious math stuff for you), but that could be used for much more (like sorting lists, printing tailored letters, and so on)?&lt;/p&gt;&#xA;" OwnerUserId="10794" LastEditorUserId="10794" LastEditDate="2019-12-09T18:28:31.950" LastActivityDate="2019-12-13T08:48:43.810" Title="When has the notion of &quot;programming language for machines&quot; emerged?" Tags="|computer-science|language|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11227" PostTypeId="2" ParentId="11222" CreationDate="2019-12-10T17:56:29.103" Score="2" Body="&lt;p&gt;Carnot wanted to answer two questions about the operation of heat engines: &quot;Is the work available from a heat source potentially unbounded?&quot; and &quot;Can heat engines in principle be improved by replacing the steam with some other working fluid or gas?&quot; He attempted to answer these in a memoir, published as a popular work in 1824 when he was only 28 years old. It was entitled &lt;em&gt;Réflexions sur la Puissance Motrice du Feu&lt;/em&gt; (&quot;Reflections on the Motive Power of Fire&quot;). The book was plainly intended to cover a rather wide range of topics about heat engines in a rather popular fashion; equations were kept to a minimum and called for little more than simple algebra and arithmetic, except occasionally in the footnotes, where he indulged in a few arguments involving some calculus. He discussed the relative merits of air and steam as working fluids, the merits of various aspects of steam engine design, and even included some ideas of his own regarding possible improvements of the practical nature. The most important part of the book was devoted to an abstract presentation of an idealized engine that could be used to understand and clarify the fundamental principles that are generally applied to all heat engines, independent of their design.&lt;/p&gt;&#xA;" OwnerUserId="9801" LastActivityDate="2019-12-10T17:56:29.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11229" PostTypeId="2" ParentId="11215" CreationDate="2019-12-11T14:50:12.800" Score="8" Body="&lt;p&gt;It is hard to prove that answer is negative, but I suspect that that's the case. That sentence looks familiar. In fact, in his book &lt;em&gt;Relativity: The Special &amp;amp; the General Theory&lt;/em&gt;, Einstein wrote &amp;ldquo;Without it the general theory of relativity [&amp;hellip;] would perhaps have got no farther than its long clothes.&amp;rdquo; But here Einstein is talking about his former teacher Hermann Minkowski, not about Sophus Lie.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only connection I can think of between the origin of the General Theory of Relativity and Lie's theory lies in the contribution of Emmy Noether. She created her &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;theorems&lt;/a&gt; about how to derive conserved quantities from symmetries in the context of General Relativity and she had to use Lie's theory. But I don't think that that was enough to call Einstein's attention to it (although he seems to have been quite impressed by Emmy Noether's contribution).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2019-12-11T17:53:28.583" LastActivityDate="2019-12-11T17:53:28.583" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11230" PostTypeId="2" ParentId="11224" CreationDate="2019-12-12T01:48:59.653" Score="5" Body="&lt;p&gt;It makes no difference for either measuring temperature, or calculating heat flow, what flows there, if anything. So experimental basis for measuring temperature was established long before the nature of what was measured became clear. As Fowler writes in  &lt;a href=&quot;http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/What%20is%20Heat.htm&quot; rel=&quot;noreferrer&quot;&gt;Early Attempts to Understand Heat&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;By the late 1700’s, the experiments of Fahrenheit, Black and others had established a systematic, quantitative way of measuring temperatures, heat flows and heat capacities — but this didn’t really throw any new light on just what was flowing&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The situation was somewhat similar to electricity, which was also analogized to fluid(s) at the time, but there were competing &lt;a href=&quot;https://en.wikipedia.org/wiki/Fluid_theory_of_electricity&quot; rel=&quot;noreferrer&quot;&gt;theories with one and two fluids&lt;/a&gt;. In contrast, there never was a two-fluid theory of heat, or a theory with a cold fluid. Both Stahl's phlogiston (1703) and Lavoisier's caloric (1787) were carrying heat, not cold, see &lt;a href=&quot;https://hsm.stackexchange.com/a/3474/55&quot;&gt;What are the major flaws of the “caloric” theory of heat?&lt;/a&gt; Indeed, when heated materials usually expand, which suggests something being added. Fowler explains more of the intuitive motivation in Lavoiser's case:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;However, in contrast to electricity, which had no noticeable effect on the appearance of a charged object, when heat was added to a solid things changed considerably.  First the material expanded, then it changed to a liquid and finally to a gas, if sufficient heat could be delivered.  Further heating expanded the gas, or increased its pressure if it was held in a fixed container.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;To interpret this sequence of events in terms of a caloric fluid being fed into the material, one could imagine the fluid flowing between the atoms of the solid and lessening their attraction for each other, until the solid melted into a liquid, whereupon the caloric continued to accumulate around the atoms until they were pushed apart into a gas. It was thought that in the gas each atom or molecule was surrounded by a ball of caloric, like a springy ball of wool, and these balls were packed in a container like oranges in a crate, except that the caloric balls could expand indefinitely as heat was poured in.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-12T01:48:59.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11231" PostTypeId="1" AcceptedAnswerId="11253" CreationDate="2019-12-12T15:58:01.727" Score="2" ViewCount="1670" Body="&lt;p&gt;Historically, why was it necessary to define the Zeroth Law of Thermodynamics after defining the other Laws? In our current understanding of science, the way we study it in schools, it seems obvious. So, what is so special about Zeroth Law? Why is it defined as a Law? What knowledge were we lacking without defining it? What was the need to define it?&lt;/p&gt;&#xA;" OwnerUserId="10791" LastEditorUserId="4931" LastEditDate="2019-12-13T20:01:55.663" LastActivityDate="2019-12-22T16:05:43.860" Title="Why was it necessary to define the &quot;Zeroth Law of Thermodynamics&quot;?" Tags="|thermodynamics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11233" PostTypeId="2" ParentId="11225" CreationDate="2019-12-13T08:48:43.810" Score="2" Body="&lt;p&gt;As Conifold says, Turing's proof was a decisive intellectual step, along with Church's Lambda Calculus and Goedel's proof. And as he says, real computers then arrived with assembler language, &quot;bottom up&quot; language design. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fortran and Cobol were the first major steps towards high level languages. McCarthy's LISP was also an important step: he designed it as a way to express algorithms, &quot;top down&quot;, and we reportedly surprised when one of his graduate students decided to implement it. &lt;/p&gt;&#xA;" OwnerUserId="10805" LastActivityDate="2019-12-13T08:48:43.810" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11234" PostTypeId="2" ParentId="11224" CreationDate="2019-12-13T23:14:20.010" Score="7" Body="&lt;p&gt;&lt;strong&gt;In a way everyone knew that it was heat that is flowing and coldness is absence of heat. But how did they know it?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The answer, quite simply, is that &lt;em&gt;they didn't know it.&lt;/em&gt;  Coldness was frequently measured in &lt;em&gt;degrees&lt;/em&gt; just as heat was, and terms like &lt;a href=&quot;https://en.wikipedia.org/wiki/Degree_of_frost&quot; rel=&quot;noreferrer&quot;&gt;degrees of frost&lt;/a&gt; were in common use even into the early 20th century.  Alternative temperature scales like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Delisle_scale&quot; rel=&quot;noreferrer&quot;&gt;Delisle scale&lt;/a&gt; and the original Celsius scale that increased with degrees of cold were widely recognized historically.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the book &lt;em&gt;Inventing Temperature&lt;/em&gt;, Hasok Chang includes an in-depth discussion of historical theories of cold, which the following summary is derived from.  Chang begins (p. 160):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Practical thermometry achieved a good deal of reliability and&#xA;  precision before people could say with any confidence what it was that&#xA;  thermometers measured.  A curious fact in the history of meteorology&#xA;  gives us a glimpse into that situation.  The common attribution of the&#xA;  centigrade thermometer to the Swedish astronomer Anders Celsius&#xA;  (1701-1744) is correct enough, but his scale had the boiling point of&#xA;  water as 0&amp;deg; and the freezing point as 100&amp;deg;.  In fact, Celsius&#xA;  was not alone in adopting such an &quot;upside-down&quot; thermometric scale.&#xA;  [...] These &quot;upside-down&quot; scales were in serious scientific use up to&#xA;  the middle of the eighteenth century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Chang goes on to give plenty of examples of such scales of &quot;coldness,&quot; tracing the history of the concept over time (p. 162).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There have been a number of perfectly capable philosophers and&#xA;  scientists through the ages who regarded cold as real as&#xA;  heat--starting with Aristotle, who took cold and hot as opposite&#xA;  qualities on an equal footing, as two of the four fundamental&#xA;  qualities in the terrestrial world. [...] Although many of [the&#xA;  mechanical philosophers of the seventeenth century] subscribed to&#xA;  theories that understood heat as motion and cold as the lack of it,&#xA;  the mechanical philosophy did not rule out giving equal ontological&#xA;  status to heat and cold.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Francis Bacon viewed cold as a type of contractive motion, the opposite of the expansive motion of heat.  Robert Boyle tried to rule out the reality of positive cold, but admitted he couldn't do it conclusively.  Pierre Gassendi postulated &quot;frigorific atoms&quot; that were the equivalent of the &quot;calorific atoms&quot; that supposedly caused heating.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thomas Thomson (1773-1852), an early historian of chemistry, claimed that the general opinion of philosophers (i.e., scientists) in the early eighteenth century was that cold was a &quot;positive something, of a peculiar body endowed with specific qualities. [...] According to [these philosophers], cold is a substance of a saline nature, very much resembling nitre, constantly floating in the air, and wafted about by the wind in very minute corpuscles, to which they gave the name of frigorific particles.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even in the late 1700s, this question had not been settled, as the &lt;em&gt;Encyclopaedia Britannica&lt;/em&gt; reported in 1778 that there was no consensus on this question, and came down on the side of the existence of cold as a separate force/entity of some sort.  Experiments in the late 1700s and early 1800s seemed to confirm the existence of independent &lt;em&gt;cold&lt;/em&gt; of some sort.  It would be a bit involved to try to describe these here, but suffice it to say that there was considerable evidence marshaled by some scientists even into the 1800s for the concept of &lt;em&gt;positive cold&lt;/em&gt;.  One of the last prominent theories of this type was promoted by Count Rumford (1753-1814), who argued for a type of &quot;frigorific radiation&quot; that was a kind of analogue to radiant heat.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the end, Chang argues that &lt;em&gt;positive cold&lt;/em&gt; didn't go away because it was conclusively debunked at the time, but rather because the caloric theory of heat became so systematized and clearly articulated that it made it difficult to retain a place for &lt;em&gt;positive cold&lt;/em&gt; within it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the reason why &quot;upside-down&quot; thermometers gradually fell into disuse even before this time, there's no clear rationale given in historical treatises.  But one might theorize that the fact that thermometers had scales with a rising expanding liquid might naturally tend toward having a scale marked to indicate the increasing volume by increasing numbers.  Again, note that this trend began likely as a matter of computational and theoretical convenience before any consensus had been reached on whether or not &lt;em&gt;cold&lt;/em&gt; existed as an independent entity.  And, as pointed out in a comment, it is much easier to practically measure the effects of added heat, but much more difficult to create &quot;added cold.&quot;  Scientists certainly tried to do the latter too, but it's likely the bias of experimentation involving heat sources led to the caloric theory focusing on heat rather than cold, which ultimately led to the demise of the notion of independent coldness.&lt;/p&gt;&#xA;" OwnerUserId="2429" LastActivityDate="2019-12-13T23:14:20.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11235" PostTypeId="1" AcceptedAnswerId="11236" CreationDate="2019-12-14T01:55:31.473" Score="13" ViewCount="1487" Body="&lt;p&gt;As I searched for linear algebra, I found it odd that the linear map composition corresponds to the multiplication of matrices. Considering the intuition that the repetition of addition is multiplication, and the definition of a module, I thought it was better to call the scalar multiplication &lt;em&gt;matrix multiplication&lt;/em&gt;. Of course, the scalar multiplication can be defined for any field, and people would have liked to call the operation between matrices &quot;matrix multiplication&quot;. But I feel that &lt;em&gt;matrix composition&lt;/em&gt; is a more logical term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;All I'd like to ask is about history. Was the multiplication of matrices &quot;multiplication&quot; from the beginning?&lt;/p&gt;&#xA;" OwnerUserId="10808" LastEditorUserId="10808" LastEditDate="2019-12-14T07:27:51.747" LastActivityDate="2019-12-14T07:27:51.747" Title="Who started calling the matrix multiplication &quot;multiplication&quot;?" Tags="|mathematics|terminology|linear-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11236" PostTypeId="2" ParentId="11235" CreationDate="2019-12-14T05:03:58.540" Score="15" Body="&lt;p&gt;The same person who introduced it, Cayley. Sylvester first used the term &quot;matrix&quot; (womb in Latin) for an array of numbers in 1848, but did not do much with it. Cayley started developing matrix algebra in 1855 and summarized his theory in &lt;a href=&quot;https://www.jstor.org/stable/108649&quot; rel=&quot;noreferrer&quot;&gt;A Memoir on the Theory of Matrices (1858)&lt;/a&gt;. In the opening paragraphs he writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It will be, seen that matrices (attending only to those&#xA;   of the same order) comport themselves as single quantities; they may be added, multiplied or compounded together, &amp;amp;c.: the law of the addition of matrices is precisely similar to that for the addition of ordinary algebraical quantities; as regards their multiplication (or composition), there is the peculiarity that matrices are not in general&#xA;  convertible; it is nevertheless possible to form the powers (positive or negative, integral or fractional) of a matrix, and thence to arrive at the notion of a rational and integral function, or generally of any algebraical function of a matrix&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Later, he first defines addition and multiplication by scalars, and then, symbolically introduces what is now called &quot;row-column rule&quot;: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...as the rule for the multiplication or composition of two matrices. It is to be observed, that the operation is not a commutative one; the component matrices may be distinguished as the first or further component matrix, and the second or nearer component matrix, and the rule of composition is as follows, viz. any line of the compound matrix is obtained by combining the corresponding line of the first or further&#xA;  component matrix successively with the several columns of the second or nearer compound matrix.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Composition&quot; and &quot;multiplication&quot; are used interchangeably throughout the paper. He goes on to define &quot;inverse or reciprocal&quot;, and prove what is now called the Cayley-Hamilton theorem. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cayley's use of &quot;multiplication&quot; should not be too surprising. Algebraic systems alternative to the usual arithmetic were actively developed at the time, and the words &quot;addition&quot; and &quot;multiplication&quot; came to be used abstractly. This is exactly Cayley's context in defining his algebra of matrices and discussing associative and other laws by analogy to the arithmetical ones. Hamilton did it with quaternions in 1843, although part of his motivation was composing rotations. &lt;a href=&quot;https://www.gutenberg.org/files/15114/15114-pdf.pdf&quot; rel=&quot;noreferrer&quot;&gt;Boole in his Laws of Thought (1854)&lt;/a&gt; did not hesitate to call logical conjunction &quot;multiplication&quot;, explaining &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;it is not affirmed that the process of multiplication in Algebra, of which the fundamental law is expressed by the equation &lt;span class=&quot;math-container&quot;&gt;$xy=yx$&lt;/span&gt;, possesses in itself any analogy with that process of logical combination which &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt; has been made to represent above; but only that if the arithmetical and the logical process are expressed in the same manner, their symbolical expressions will be subject to the same formal law. The evidence of that subjection is in the two cases quite distinct&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-12-14T07:11:29.043" LastActivityDate="2019-12-14T07:11:29.043" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11237" PostTypeId="2" ParentId="7489" CreationDate="2019-12-14T20:00:19.047" Score="3" Body="&lt;p&gt;I can answer this part of the question: &quot;It is likely that he sent one to Darwin, but there is no record of Darwin’s having actually read it.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;No, Darwin never received Mendel's paper. That author's speculation is wrong.  None of his biographers have found evidence of this paper, or that he was aware of its publication. It was in a quite obscure journal.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am intimately acquainted with Darwin research notes from before he formulated his theory until his death. He makes no mention of Mendel. Darwin would have been very interested in the research, since he conducted many breeding experiments with plants, and had a garden just for this at his home.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Darwin accumulated tens of thousands of research notes. I have transcribed, edited, or otherwise worked with thousands of these, and I've never seen a mention of Mendel. About 20,000 of these notes, plus manuscripts, are available on the Darwin Manuscripts Project site at the American Museum of Natural History.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our editor-in-chief, Professor David Kohn, was a Darwin scholar for his entire career. He assures me Darwin had no knowledge of Mendel or his work.&lt;/p&gt;&#xA;" OwnerUserId="10810" LastActivityDate="2019-12-14T20:00:19.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11238" PostTypeId="2" ParentId="5890" CreationDate="2019-12-14T20:19:18.903" Score="11" Body="&lt;p&gt;After quite a bit of searching, I was unable to find any conclusive authoritative source that weighs in specifically on this question.  [Though see &lt;strong&gt;EDIT&lt;/strong&gt; below.]  However, circumstantial evidence suggests that Turing certainly popularized the term &quot;imitation game&quot; (whether or not he coined it himself).  Furthermore, I've found no evidence of a &lt;em&gt;popular&lt;/em&gt; parlor game resembling Turing's description, played in the Victorian era or otherwise.  That doesn't mean it didn't exist, but what seems clear from publications around the time of Turing's original paper is that if Turing's supposed parlor game existed already, no one seemed to reference it or understand what Turing's reference was to.  Hence, if Turing didn't make it up himself for his example, it likely was a rather obscure game, rather than a &quot;popular Victorian game&quot; as is sometimes claimed.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;To provide some further details, let's first address the term &quot;imitation game&quot; itself.  Aside from the Google Ngrams data in the question that suggests the term's popularity increased significantly after Turing's paper, there are several sources in the years immediately following Turing's paper that imply that the term &quot;Imitation Game&quot; was of Turing's invention.  For example, note the wording of Leonard Pinsky in the very first academic reply to Turing's paper &lt;a href=&quot;https://ztcprep.com/images/sitestore/IELTS/Books/The_Turing_Test_Verbal_Behavior.pdf#page=138&quot; rel=&quot;noreferrer&quot;&gt;&quot;Do Machines Think about Machines Thinking&quot;&lt;/a&gt; (1951):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The new formulation [of the question whether machines can think] was&#xA;  based upon &lt;strong&gt;what Mr. Turing called the &quot;imitation game&quot;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Follow-up papers similarly refer to the test as &quot;Turing's 'Imitation Game'&quot; (&lt;a href=&quot;https://ieeexplore.ieee.org/abstract/document/5392646/&quot; rel=&quot;noreferrer&quot;&gt;1958&lt;/a&gt;) or &quot;Turing's well-known 'Imitation Game'&quot; (&lt;a href=&quot;https://dl.acm.org/citation.cfm?id=1460381&quot; rel=&quot;noreferrer&quot;&gt;1959&lt;/a&gt;).  Note the quotation marks in each case, likely indicating a novel term, rather than a previously known one.  From this (and other contemporary examples) it seems likely that Turing popularized the term &quot;imitation game&quot; and this specific use of the phrase.  If the phrase existed earlier, it appears Turing's usage was novel at a minimum, leading to the quotation marks in early citations.  As further evidence, I'd note that early references like this that don't specifically credit Turing will still say something like &quot;Turing describes &lt;strong&gt;an&lt;/strong&gt; imitation game&quot; or &quot;&lt;em&gt;a&lt;/em&gt; game of sorts.&quot;  I haven't come upon a single reference that claims there was a &lt;em&gt;specific&lt;/em&gt; pre-existing known &quot;imitation game&quot; (i.e., &lt;em&gt;the&lt;/em&gt; Imitation Game) that Turing might have been referring to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the possibility of a pre-existing parlor game, I came upon several modern references that are more specific than the BBC link in the question.  Perhaps the most intriguing is a 2014 &lt;a href=&quot;https://www.smithsonianmag.com/innovation/turing-test-measures-something-but-not-intelligence-180951702/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Smithsonian Magazine&lt;/em&gt; article&lt;/a&gt; claiming:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Turing] took the idea from a traditional Victorian parlor game, where you&#xA;  try to figure out if the person hidden behind a curtain is a man or a&#xA;  woman, just by asking questions. (The answers to the questions had to&#xA;  be written down, because the voice would be a giveaway.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That's much more detailed than most descriptions, mentioning the idea of a &quot;curtain&quot; as well as the parameter that &quot;the questions had to be written down&quot; (though from a practical standpoint, it would have been the &lt;em&gt;answers&lt;/em&gt; rather than the questions that would have to be written down).  This article, like all the other recent ones that assume the existence of the Victorian parlor game, provides no citation or clear reference detail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, with these details in hand, I skimmed through descriptions of at least a couple hundred different Victorian parlor games in various collections of them published in the 1800s, and I haven't seen anything mentioned that resembles this game.  One of the only popular games involving a &quot;curtain&quot; was about casting shadows on the curtain (or sheet) and requiring a person behind the curtain to guess who was making the shadows.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other recent sources claim that the people had to be taken to a different room, which actually makes more sense, as visual and auditory cues would likely make it much easier to guess who was who simply behind a curtain or sheet in the same room.  Nevertheless, while there are many Victorian parlor games involving one or two people leaving the room, they inevitably tend to quickly return -- usually having concocted some sort of scheme or idea that the rest of the group has to guess.  After all, the goal of Victorian parlor games was socialization, and a game that banished a few members of the party to another room for a long period of writing responses wouldn't be particularly social.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In any case, having reviewed several popular books that each collected many dozens of common games, I'm ready to say this game -- if it existed in Victorian times -- likely wasn't a well-known or very popular game.  (And there was certainly no game known commonly as the &quot;imitation game.&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There's also a lot of circumstantial evidence that these modern assumptions of a &quot;Victorian game&quot; are referencing a myth.  Careful writers frequently avoid attributing a specific reference to a pre-existing game.  The &lt;em&gt;Smithsonian&lt;/em&gt; quotation above is one of the few I've seen in a reputable source that seems to provide any details at all.  Others often hedge their bets, like the BBC article quoted in the question, which only claims that the competition was &lt;strong&gt;&quot;Victorian-style&quot;&lt;/strong&gt;, not an actual Victorian game.  And the BBC lets open the possible implication that Turing himself came up with this &quot;adaptation&quot; of a game he might have himself invented.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That latter implication is also made by several other careful writers.  Daniel Dennett says Turing &quot;described a parlor game of sorts, the 'imitation game', to be played by a man, a woman, and a judge (of either gender).&quot;  Note the reference to a &quot;parlor game &lt;strong&gt;of sorts&lt;/strong&gt;,&quot; which seems to argue against a pre-existing game.  Similarly, a 2019 &lt;a href=&quot;http://nybooks.com/daily/2019/05/06/what-turing-told-us-about-the-digital-threat-to-a-human-future/&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;New York Review of Books&lt;/em&gt; article&lt;/a&gt; describes Turing's scenario as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Turing’s imitation game, as he set it out in 1950, has &lt;strong&gt;two stages&lt;/strong&gt;.&#xA;  In the first, we measure how well humans can distinguish between a&#xA;  woman and a man who is impersonating a woman. Then we see whether&#xA;  humans are better or worse at telling the difference between a woman&#xA;  and a computer imitating a woman. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;As Turing described it, &lt;strong&gt;three people would take part in the first&#xA;  stage of the game.&lt;/strong&gt; [Description of the man, woman, and judge scenario&#xA;  follows.] Now, asked Turing, “What will happen when a machine takes&#xA;  the part of A in this game?” In the second stage, the man in the&#xA;  second room is replaced by a computer program.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There's no mention of any pre-existing &quot;parlor game&quot; here, Victorian or otherwise.  The article simply assumes there are two &quot;stages&quot; to the game, all apparently attributed to Turing himself.  Given how much discussion there has been about gender issues in this game, it's also rather surprising that I haven't been able to find a single such source that connects this back to an actual parlor game.  (See, for example, recent articles &lt;a href=&quot;https://www.wired.com/2005/07/the-other-turing-test/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.popsci.com/blog-network/zero-moment/lie-lady-profoundly-weird-gender-specific-roots-turing-test/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;: the first merely noting that Turing &quot;suggested an 'imitation game,' which plays like 20 Questions for transsexuals&quot; and the second simply calling this game a &quot;scenario.&quot;  Neither makes any reference to a &quot;parlor&quot; or Victorians.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed, in &lt;a href=&quot;https://ztcprep.com/images/sitestore/IELTS/Books/The_Turing_Test_Verbal_Behavior.pdf&quot; rel=&quot;noreferrer&quot;&gt;this entire 2004 book&lt;/a&gt; on the Turing Test, with numerous reprinted historical essays and references to various controversies around it, there's no mention of the &quot;parlor game&quot; interpretation.  In fact, the only time the term &quot;parlor game&quot; is even used in the entire book is in Daniel Dennett's essay (referenced above), where it's clear that Dennett doesn't think this is a pre-existing game, just a &quot;parlor game &lt;em&gt;of sorts&lt;/em&gt;.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nevertheless, Turing's original game does resemble other actual Victorian games in some ways.  There are plenty of Victorian parlor games involving someone hiding a secret, or requiring someone to guess the identity that another person is trying to portray, or games where people try to deceive others by disguising their identity in some fashion.   (Usually the games involve blindfolding the person guessing.)  Hence, the BBC could still accurately describe Turing's scenario as a &quot;&lt;em&gt;Victorian-style&lt;/em&gt; competition.&quot;  In fact, even contemporary references around the time of Turing's paper make comparisons to Victorian-era games, such as W. Mays's 1952 article &lt;a href=&quot;https://www.jstor.org/stable/pdf/3747787&quot; rel=&quot;noreferrer&quot;&gt;&quot;Can Machines Think?&quot;&lt;/a&gt;, which describes Turing's scenario like this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[Turing] substitutes a definition in terms of the machine's capacity&#xA;  to play  an imitation game which seems to be a variation of the radio&#xA;  game  of Twenty Questions, the part of the witness being taken by a &#xA;  computing machine, the part of the interrogator by a human being.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Twenty_Questions&quot; rel=&quot;noreferrer&quot;&gt;Twenty Questions&lt;/a&gt; was a radio show at the time, but it also had a long history going back to Victorian parlor games.  Clearly, even to Turing's contemporaries, his &quot;imitation game&quot; sounded something like a parlor game to them, just not like any precise game they'd ever had heard of (or else they'd reference that actual game, rather than something like Twenty Questions).  Nevertheless, despite many recent references to a supposed popular Victorian game (maybe involving a curtain and other details), I couldn't find any evidence of its existence prior to Turing's imaginings.  If it previously existed, it clearly wasn't very well-known.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; I did manage to find a pretty authoritative academic source that appears to weigh in rather conclusively.  &lt;a href=&quot;https://www.gold.ac.uk/computing/staff/m-bishop/&quot; rel=&quot;noreferrer&quot;&gt;Mark Bishop&lt;/a&gt;, a professor of cognitive computing at the University of London, began an issue of the journal &lt;em&gt;Kybernetes&lt;/em&gt; devoted to the Turing Test in 2010 with an introduction called &lt;a href=&quot;https://philarchive.org/archive/BISTIG&quot; rel=&quot;noreferrer&quot;&gt;&quot;The Imitation Game&quot;&lt;/a&gt; by noting:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Famously, in his 1950 paper ‘Computing Machinery and Intelligence’ ,&#xA;  the British mathematician Alan Turing suggested replacing this&#xA;  question [&quot;Can a Machine Think?] - which he found “too meaningless to&#xA;  deserve discussion” - with a simple [behaviourial] test &lt;strong&gt;based on an&#xA;  imagined Victorian-esque pastime he entitled the ‘imitation game’.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bishop minces no words here: the &quot;imitation game&quot; was named by Turing, and it was only an &lt;em&gt;imagined&lt;/em&gt; Victorian-esque pastime.  Given this claim in a prominent place in an issue of an academic journal devoted to discussion of the Turing Test, I'd guess that Bishop would have been pretty sure of himself before writing these words.  Nonetheless, he's not the only academic to say something about this topic, and there are contradictory examples, such as Barry Luokkala (from the Department of Physics at Carnegie-Mellon) who writes in &lt;a href=&quot;https://books.google.com/books?id=GJO7DwAAQBAJ&amp;amp;lpg=PA130&amp;amp;ots=clw1GREH7o&amp;amp;dq=parlor%20game%20turing%20luokkala&amp;amp;pg=PA143#v=onepage&amp;amp;q=parlor&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Exploring Science Through Science Fiction&lt;/em&gt;&lt;/a&gt; (2019, p. 130):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The [Turing] test was based on the Imitation Game, a parlor game&#xA;  popular in the pre-television era.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But Luokkala, like other similar sources, provides no citation or evidence supporting this claim.  In the absence of &lt;em&gt;anyone anywhere&lt;/em&gt; providing &lt;em&gt;any&lt;/em&gt; citation of this supposedly &quot;popular&quot; game, and many academics like Bishop assuming it was of Turing's invention, I'm personally willing to declare the pre-existing &quot;popular Victorian game&quot; to be a myth.&lt;/p&gt;&#xA;" OwnerUserId="2429" LastEditorUserId="2429" LastEditDate="2019-12-15T16:07:02.517" LastActivityDate="2019-12-15T16:07:02.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11239" PostTypeId="1" AcceptedAnswerId="11242" CreationDate="2019-12-15T09:55:28.403" Score="2" ViewCount="616" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XKUT9.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XKUT9.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$Lenna$&lt;/span&gt; despite being an playboy model posing nude here and there, is one of the most used images for image processing. How she came into the world of image processing? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Was it done intentionally by some male scientist who were very much fond of the playboy stuff or is it just a coincidence?&lt;/p&gt;&#xA;" OwnerUserId="10815" LastActivityDate="2019-12-16T13:25:54.980" Title="How did Lenna become the most used image in image processing?" Tags="|computer-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11240" PostTypeId="1" CreationDate="2019-12-15T17:10:06.817" Score="8" ViewCount="892" Body="&lt;p&gt;The &lt;a href=&quot;https://www.quora.com/How-does-one-explain-the-pi-4-%E2%80%98proof%E2%80%99-regarding-a-square-reduced-to-a-circle&quot; rel=&quot;noreferrer&quot;&gt;staircase 'paradox'&lt;/a&gt; has been discussed here and elsewhere a few times (search for staircase + paradox). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is whether this puzzle has been discussed in the academic literature or historically in mathematics. I can't find any reference to it except in fora, but I want to reference it in a paper.&lt;/p&gt;&#xA;" OwnerUserId="10819" LastEditorUserId="55" LastEditDate="2019-12-16T04:48:49.613" LastActivityDate="2020-01-18T16:37:26.297" Title="What is the history of staircase or =4 paradox?" Tags="|calculus|geometry|differential-geometry|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11241" PostTypeId="2" ParentId="11240" CreationDate="2019-12-16T02:07:39.380" Score="8" Body="&lt;p&gt;The &quot;staircase paradox&quot; (or &quot;Pythagoras paradox&quot;) name appears to be recent, so it is hard to search for it. Wolfram calls it &lt;a href=&quot;http://mathworld.wolfram.com/DiagonalParadox.html&quot; rel=&quot;noreferrer&quot;&gt;&quot;diagonal paradox&quot;&lt;/a&gt;, but that may be conflating it with a different paradox due to Leibniz, which he used to argue against the actual existence of indivisibles, see &lt;a href=&quot;https://www.jstor.org/stable/41220656&quot; rel=&quot;noreferrer&quot;&gt;The Philosophical Assumptions Underlying Leibniz's Use of the Diagonal Paradox in 1672&lt;/a&gt;. The horizontal zigzag/sawtooth variant appears to be older, Lebesgue reports it from as early as 1890-s. Mathpages calls it &lt;a href=&quot;https://www.mathpages.com/home/kmath063.htm&quot; rel=&quot;noreferrer&quot;&gt;limit paradox&lt;/a&gt; and Lukowski (p.13) &lt;a href=&quot;https://books.google.com/books?id=p0bpyag497oC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;paradox of approximation&lt;/a&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Koch_snowflake&quot; rel=&quot;noreferrer&quot;&gt;Koch's snowflake construction (1904)&lt;/a&gt; uses a related idea, as does Mandelbrot's &lt;a href=&quot;https://en.wikipedia.org/wiki/Coastline_paradox#Mathematical_aspects&quot; rel=&quot;noreferrer&quot;&gt;&quot;coastline paradox&quot;&lt;/a&gt;, which he attributes to Richardson (1961). In 1890 Schwarz gave a construction, which can be seen as a clever 2D generalization of the sawtooth paradox, of polyhedral surfaces inscribed into a cylinder and converging to it (&lt;a href=&quot;https://en.wikipedia.org/wiki/Schwarz_lantern&quot; rel=&quot;noreferrer&quot;&gt;Schwarz lanterns&lt;/a&gt;), with the surface areas growing to infinity, see &lt;a href=&quot;https://www.maa.org/programs/maa-awards/writing-awards/surface-area-and-the-cylinder-area-paradox&quot; rel=&quot;noreferrer&quot;&gt;Surface Area and the Cylinder Area Paradox&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/numbersfunfacts0000frie/page/n231&quot; rel=&quot;noreferrer&quot;&gt;Friend in Numbers: fun &amp;amp; facts (1954), p.72&lt;/a&gt; gives the sawtooth paradox under the title &quot;the field of barley&quot;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From&quot; rel=&quot;noreferrer&quot;&gt;Lakoff and Núñez in Where mathematics comes from (2000)&lt;/a&gt; discuss a variation with semicircles as a &quot;classic paradox of infinity&quot;. It &quot;shows&quot; that &lt;span class=&quot;math-container&quot;&gt;$π=2$&lt;/span&gt;. Their version, along with the sawtooth version &quot;showing&quot; that &lt;span class=&quot;math-container&quot;&gt;$2=1$&lt;/span&gt;, appears in &lt;a href=&quot;https://stemtec.aut.ac.nz/__data/assets/pdf_file/0004/57640/Paradoxes-and-Sophisms-in-Calculus-MAA-e-book.pdf&quot; rel=&quot;noreferrer&quot;&gt;Paradoxes and Sophisms in Calculus, pp. 30-31&lt;/a&gt;. In 1997 there was a lively pedagogical discussion involving it in MAA journals, see &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/07468342.1997.11973887&quot; rel=&quot;noreferrer&quot;&gt;On Arc Length by Barry&lt;/a&gt; and references therein. &lt;a href=&quot;https://www.tandfonline.com/doi/pdf/10.1080/07468342.1997.11973828&quot; rel=&quot;noreferrer&quot;&gt;Barbeau in the Fallacies, Flaws, and Flimflam section of CMJ&lt;/a&gt; mentions the Lebesgue story that Dave Renfro is probably recalling:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A request to the newsgroup math-history-list@maa.org elicited responses from&#xA;  John Conway, Roger Cooke, Mark McKinzie, and Rick Otten, who provided the&#xA;  following references. L. C. Young [7, 8] quotes an anecdote from Lebesgue's book,&#xA;  In the Margin of the Calculus of Variations, in which the paradox was presented as&#xA;  a &quot;joke&quot; at the College de Beauvais.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Young's book is &lt;a href=&quot;https://books.google.com/books?id=F3mZ0Y-FUi4C&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Lectures on the Calculus of Variations and Optimal Control theory (1981), p.152&lt;/a&gt;, and he quotes Lebesgue directly (see image below):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;All my papers [on this subject] are connected with a schoolboy's 'joke.' At the College&#xA;  de Beauvais, we used to show that, in a triangle, one side is equal to the sum of the other two.&#xA;  Let &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt; be a triangle. If &lt;span class=&quot;math-container&quot;&gt;$A_1, B_1, C_1$&lt;/span&gt; are the mid-points of its sides, we have&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;BA+AC = BC_1 + C_1A_1 + A_1B_1 + B_1C.&#xA;$$&lt;/span&gt;&#xA;  On each of the triangles &lt;span class=&quot;math-container&quot;&gt;$BC_1A_1, A_1B_1C$&lt;/span&gt;, proceed as on &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;. We obtain a broken line, formed of&#xA;  eight segments, and equal to &lt;span class=&quot;math-container&quot;&gt;$BA + AC$&lt;/span&gt;. By continuing in this way, we obtain a sequence of&#xA;  broken lines, which stray less and less from the side &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt;, and which still have as length the sum of the two other sides of our original triangle. The pupils at Beauvais concluded from this,&#xA;  that the segment BC, the geometrical limit of our broken lines, had as length the sum of the&#xA;  two other sides &lt;span class=&quot;math-container&quot;&gt;$BA + AC$&lt;/span&gt;. My schoolfellows saw there no more than a good joke. To me,&#xA;  the argument appeared most disturbing, since I could see no difference between it and proofs&#xA;  relating to the areas and surfaces of cylinders, cones, spheres, and to the length of a circumference.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B001BM63YS&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lebesgue's &lt;em&gt;En marge du calcul des variations&lt;/em&gt;&lt;/a&gt; does not seem to be translated into English. It was only found after his death and published in 1963. Four of the six chapters were previously published as papers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the end, Lebesgue probably refers to the classical approximations of arc lengths and surface areas by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Method_of_exhaustion&quot; rel=&quot;noreferrer&quot;&gt;&quot;method of exhaustion&quot;&lt;/a&gt;. In the calculus of variations there was active work at the time on the &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/blasjo526.pdf&quot; rel=&quot;noreferrer&quot;&gt;isoperimetric problem&lt;/a&gt; that raised related analytic issues. It turned out that the classical proofs, from Zenodorus to Steiner, had gaps concerning existence of limit figures. Weierstrass and Edler gave first rigorous proofs for curves in 1879 and 1882, and Schwarz for surfaces in 1890. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/56kSs.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/56kSs.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-12-16T14:40:29.537" LastActivityDate="2019-12-16T14:40:29.537" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11242" PostTypeId="2" ParentId="11239" CreationDate="2019-12-16T13:25:54.980" Score="9" Body="&lt;p&gt;Speaking as someone who worked in imaging tech and related fields from '79 to whenever, I can offer a little more.  This image does contain a wide range of spatial frequencies as well as intensity range, making it useful to &quot;torture test&quot; sensors and image processing algorithms.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In general,  a human face is a preferred subject because us humans are very good at noticing small errors in images of faces.  I'm sure the fact that she was attractive didn't hurt :-) .  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And finally,  once a few labs started using this photo, it became a &quot;standard reference&quot; that everyone could use to compare their hardware and software against.   It's not really a case of &quot;going viral&quot; the way funny cat pictures or stupid memes propogate, but rather that it has functional utility. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2019-12-16T13:25:54.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11244" PostTypeId="1" AcceptedAnswerId="11620" CreationDate="2019-12-18T06:08:41.287" Score="8" ViewCount="353" Body="&lt;p&gt;&lt;em&gt;This is a copy of my question on MSE (&lt;a href=&quot;https://math.stackexchange.com/questions/3372432&quot;&gt;https://math.stackexchange.com/questions/3372432&lt;/a&gt;) because this forum seems better suited for historical questions:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In 1985, Gosper used the not-yet-proven formula by Ramanujan&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{ 1 }{\pi } = \frac{2\sqrt{2}}{99^2}\cdot \sum_{n=0}^\infty \frac{(4n)!}{(n!)^4}\cdot\frac{26390 n+1103}{396^{4n}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;to compute &lt;span class=&quot;math-container&quot;&gt;$17\cdot10^6$&lt;/span&gt; digits of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, at that time a new world record.&lt;/p&gt;&#xA;&lt;p&gt;Here (&lt;a href=&quot;https://www.cs.princeton.edu/courses/archive/fall98/cs126/refs/pi-ref.txt&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cs.princeton.edu/courses/archive/fall98/cs126/refs/pi-ref.txt&lt;/a&gt;) it reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There were a few interesting things about Gosper's computation. First, when he decided to use that particular formula, there was no proof that it actually converged to pi! Ramanujan never gave the math behind his work, and the Borweins had not yet been able to prove it, because there was some very heavy math that needed to be worked through. It appears that Ramanujan simply observed the equations were converging to the 1103 in the formula, and then &lt;em&gt;assumed&lt;/em&gt; it must actually be 1103. (Ramanujan was &lt;em&gt;not&lt;/em&gt; known for rigor in his math, or for providing any proofs or intermediate math in his formulas.) The math of the Borwein's proof was such that after he had computed 10 million digits, and verified them against a known calculation, his computation became part of the proof. Basically it was like, if you have two integers differing by less than one, then they have to be the same integer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Now my historical question&lt;/strong&gt;: Who was the first to prove this formula? Was it Gosper because he added the last piece of the proof, or was it the Borweins, afterwards? And was Gosper aware of this proof when he did his computation?&lt;/p&gt;&#xA;" OwnerUserId="10829" LastEditorUserId="10829" LastEditDate="2021-03-16T11:16:31.810" LastActivityDate="2021-03-16T11:16:31.810" Title="Did Gosper or the Borweins first prove Ramanujans formula?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11245" PostTypeId="1" AcceptedAnswerId="11249" CreationDate="2019-12-18T14:51:46.997" Score="1" ViewCount="400" Body="&lt;p&gt;When does the concept of &quot;normed vector space&quot; emerge? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who is the first mathematician to consider this setting?&lt;/p&gt;&#xA;" OwnerUserId="10471" LastActivityDate="2019-12-19T03:54:18.437" Title="Normed vector space : when and who?" Tags="|topology|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11246" PostTypeId="2" ParentId="11183" CreationDate="2019-12-18T16:53:47.520" Score="1" Body="&lt;p&gt;New is well forgotten old, it seems. In a couple of papers from 1916 Marcel Baudouin discusses as matter of fact the linking of spiral and the Sun's path:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The spiral represents the Sun's path in the sky between a solsticial&#xA;  rising and setting, that is between two solsticial points&#xA;  corresponding to Summer and Winter, as far as what we know about this&#xA;  symbol during the Age of metals.(1)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The same is repeated in a second paper (2).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ref1. Baudouin M., &lt;a href=&quot;https://www.persee.fr/doc/bmsap_0037-8984_1916_num_7_1_8777&quot; rel=&quot;nofollow noreferrer&quot;&gt;''La préhistoire des étoiles: Les Pléiades au Néolithique''&lt;/a&gt;. In: Bulletins et Mémoires de la Société d'anthropologie de Paris, VI° Série. Tome 7 fascicule 1, 1916. pp. 25-103; Ref2. &lt;a href=&quot;https://www.persee.fr/doc/bmsap_0037-8984_1916_num_7_5_8812&quot; rel=&quot;nofollow noreferrer&quot;&gt; ''La préhistoire des étoiles au paléolithique. Les pléiades à l'époque aurignacienne et le culte stello-solaire typique au solutréen''&lt;/a&gt;. In: Bull. et Mémoires de la Soc. d'anthropologie de Paris, VI° Série. Tome 7 fasc. 5-6, 1916. pp. 274-317&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2019-12-18T16:53:47.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11247" PostTypeId="1" CreationDate="2019-12-18T17:42:56.807" Score="1" ViewCount="143" Body="&lt;p&gt;While I was reading translated into Korean version of Thomas Heath Euclid Elements, I found something weird. And I am doubting whether that translation is wrong. I will retranslate it so you guys can rate it. Before, please read the original text of Thomas Heath Elements.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/EuclidsElementsBooksIIIVolume1Heath/page/n164&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/EuclidsElementsBooksIIIVolume1Heath/page/n164&lt;/a&gt;&#xA;(liked above, book page 155, definition 1)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now I will recite Korean version below, which I will retranslate it. Bold sentences are different sentences compare to original text. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Definition 1: A point is that which is indivisible into parts.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The original text says &quot;A point is that which has no part.&quot; But this is incorrect translation. It is better to say &quot;A point is that which is indivisible into parts.&quot;&lt;/strong&gt; An exactly parallel use of (greek word) in the singular is found in Aristotle, literally &quot;There is a part even of the form&quot;; Bonitz translates as if the plural were used, &quot;even the form is divisible(into parts).&quot; So in this case, it is reasonable to say &quot;A point is that which is indivisible into parts.&quot; Capella(5th C.A.D) alone or almost alone translated differently. &quot;a point is that a part of which is nothing.&quot; Simon has adopted this translation, but this does not make sense. &lt;strong&gt;If point does not have part,&lt;/strong&gt; Euclid might as well have said that a point is itself &quot;nothing&quot; , which of course he does not to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, I will write down my understanding below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Thomas Heath&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A point is that which has no part &lt;strong&gt;right&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;A point is that which is indivisible into parts &lt;strong&gt;right&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;A point is that a part of which is nothing &lt;strong&gt;wrong&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Korean translator&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A point is that which has no part &lt;strong&gt;wrong&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;A point is that which is indivisible into parts &lt;strong&gt;right&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;A point is that a part of which is nothing &lt;strong&gt;wrong&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;So what Korean translator think is, &lt;strong&gt;A point is that which has no part&lt;/strong&gt; and &lt;strong&gt;A point is that a part of which is nothing&lt;/strong&gt;, these two sentences are same. But I don't think it is same in view of Thomas Heath's original purpose. Am I wrong? If so, please help me to understand it.&lt;/p&gt;&#xA;" OwnerUserId="10364" LastActivityDate="2019-12-18T17:42:56.807" Title="Question about Euclid Elements book 1, definition 1" Tags="|euclidean-geometry|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="11248" PostTypeId="2" ParentId="11231" CreationDate="2019-12-18T19:56:15.817" Score="1" Body="&lt;p&gt;Without the Zeroth Law it would not be possible to measure the temperature of objects or substances using another object, i.e. a &quot;thermometer&quot;.&lt;/p&gt;&#xA;" OwnerUserId="3540" LastActivityDate="2019-12-18T19:56:15.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11249" PostTypeId="2" ParentId="11245" CreationDate="2019-12-19T01:22:04.530" Score="6" Body="&lt;p&gt;A good source for these types of questions is Miller's site &lt;a href=&quot;http://jeff560.tripod.com/n.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;. On the norms in vector spaces we find the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;On page 57 of his 1908 paper “Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten,” Rendiconti del Circolo Matematico di Palermo, 25 (1908), Erhard Schmidt defines for a function A(x), x = 1, 2, 3 ..., which, he says in a footnote, can be regarded as a vector in an infinite-dimensional space, a positive quantity ||A||, the Euclidean norm, which he calls the length, and calls a vector “normirt,” if its length is equal to 1.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1921 Albert A. Bennett &amp;quot;Normalized Geometric Systems,&amp;quot; Proc. National Acad. Sci. U.S.A. 7 p. 84: &amp;quot;The notion of norm or numerical value of a complex quantity, c = a + b√- 1, namely, |c| = √(a2 + b2), as it arises in algebra, has a more or less immediate generalization to more extensive matric systems.&amp;quot;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1922 S. Banach defined “la norme” for an abstract linear space in “Sur les opérations dans les ensembles abstraits et leur application aux équations integrales”, Fundamenta Mathematicae, 3, pp. 135-6. Among the examples (pp. 167-8) is &lt;span class=&quot;math-container&quot;&gt;$║\varphi║$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$\sqrt[r]{\int_a^l|\varphi|^r\,dx}$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As Dave Renfro mentioned in the comment, the theory of normed vector spaces was developed in the 1920-s by Banach, Hahn and Wiener (concrete functional spaces with a norm, without the name, were studied before that). Their use became a standard tool Banach's 1932 book &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.493033&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Théorie des opérations linéaires&lt;/em&gt; (Theory of Linear Operations, 1932)&lt;/a&gt;, which flashed out the role of completeness and systematically developed the theory of linear operators on complete (Banach) spaces. Additional impulse was provided by &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Neumann's Mathematical Foundations of Quantum Mechanics&lt;/a&gt; published the same year, which systematically utilized Hilbert spaces and operators on them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-19T03:54:18.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11250" PostTypeId="1" AcceptedAnswerId="11251" CreationDate="2019-12-19T02:57:03.130" Score="6" ViewCount="1276" Body="&lt;p&gt;I have a lot of questions that can be summed up by &quot;whats the history of independent and dependent variables?&quot; Here is a list of those questions:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where does our conception of independent and dependent variables come from? When did science get a notion of variable? How was this notion distinct from one quantity changing as another changes? What did they call measured quantities in science? When were these measured quantities first called &quot;variables&quot;. Was there ever a time when mathematical variables were regarded as a distinct notion from measured quantities? Where does this particular terminology come from? When did this terminology become so established as it is today? Was it always central to science and hence a part of all science curricula or is its prevalence due only to some educational reform?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I look into these questions I always get an introduction to independent and dependent variables as if I have never seen them before, and those sources always present the material as if the ideas have been around forever with no historical dimension. The most insight I've gotten into these questions so far has been on this sight here: &lt;a href=&quot;https://hsm.stackexchange.com/questions/2372/is-it-true-that-leibniz-introduced-constant-variable-and-function&quot;&gt;Is it true that Leibniz introduced &amp;quot;constant,&amp;quot; &amp;quot;variable,&amp;quot; and &amp;quot;function&amp;quot;?&lt;/a&gt; . So variables, at least in mathematical contexts, have been around for a while, but nothing else is mentioned about variables in experimental contexts. I've also searched for the 'independent and dependent variables' on the Stanford encyclopedia of philosophy but those articles will at best discuss &quot;causally independent variables&quot; or &quot;probabilistically independent random variables&quot;. There is no discussion even acknowledging the basic ideas learned in school. All of this suggests to me that the notion of independent and dependent variables is a modern invention for teaching science, but I don't have any proof of this. Also, even if this were the case the notion would have taken from some established ideas in science and I'm curious what that history would be.&lt;/p&gt;&#xA;" OwnerUserId="10836" LastActivityDate="2019-12-19T12:55:16.320" Title="History of &quot;independent and dependent variables&quot;" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11251" PostTypeId="2" ParentId="11250" CreationDate="2019-12-19T08:39:47.430" Score="7" Body="&lt;p&gt;It is helpful to separate concepts from words. The sharp separation between mathematical variables and physical quantities they represent is a modern phenomenon. Ancient Greeks had &lt;em&gt;magnitudes&lt;/em&gt;, which included numbers and lines, areas, etc., without taking them as something abstract and separated from reality. They had the idea of &amp;quot;independent and dependent variables&amp;quot; as well, both &amp;quot;mathematical&amp;quot; and &amp;quot;experimental&amp;quot;. Ptolemy (c.150 AD) tabulates values of chords vs angles in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Almagest&quot; rel=&quot;noreferrer&quot;&gt;Almagest&lt;/a&gt;, and incident vs refracted angles in &lt;a href=&quot;https://en.wikipedia.org/wiki/Optics_(Ptolemy)&quot; rel=&quot;noreferrer&quot;&gt;Optics&lt;/a&gt;. He even explains a version of linear interpolation. Diophantus (c. 250 AD) already had an algebraic symbol for a variable, an equivalent of modern &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Let's recall that many of the creators of new mathematics in the 17-18th centuries, including Newton, Leibniz, Euler, Laplace, etc., were also doing physics, and the latter often inspired the former. Leibniz used the Latin word for &amp;quot;variable&amp;quot; in 1692, but not quite in the modern sense, since his notion of the function was not modern either. Bernoulli's 1698 notion of function as analytic dependence was closer. Newton talked of quantities and &lt;em&gt;fluents&lt;/em&gt; in his papers even before &lt;em&gt;Principia&lt;/em&gt; (1687), and the applied and &amp;quot;pure&amp;quot; contexts were intermixed, see &lt;a href=&quot;https://hsm.stackexchange.com/a/9590/55&quot;&gt;What was the notion of limit that Newton used?&lt;/a&gt; Physical forces and masses are quantities, fluents &amp;quot;depend&amp;quot; on them, and velocities are &lt;em&gt;fluxions&lt;/em&gt; of position. By 1710 &amp;quot;variable quantities&amp;quot; made their way into Harris's &lt;em&gt;Lexicon Technicum&lt;/em&gt;, according to &lt;a href=&quot;http://jeff560.tripod.com/v.html&quot; rel=&quot;noreferrer&quot;&gt;Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;, and they bear resemblance to Greek magnitudes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Variable Quantities, in Fluxions, are such as are supposed to be continually increasing or decreasing; and so do by the motion of their said Increase or Decrease Generate Lines, Areas or Solidities&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for the words, in English &amp;quot;variable&amp;quot; on its own and  &amp;quot;independent/dependent variables&amp;quot; crop up only at the beginning of 19th century. Memoirs of the Analytical Society from 1813 contain:&amp;quot;&lt;em&gt;The method of Laplace for reducing an equation of the first order, where the difference of the independent variable is any function of the variable itself, to one wherein that difference is constant, is well known.&lt;/em&gt;&amp;quot; English translation of Lacroix's  Differential and Integral Calculus (1816) had:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;&amp;quot;The limit of the ratio... will be obtained by dividing the differential of the function by that of the variable&lt;/em&gt;&amp;quot;, &amp;quot;&lt;em&gt;Treating the subordinate variables as implicit functions of the indepdndent [sic] ones&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Further from Miller:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Dependent variable appears in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: &amp;quot;On account of this dependence of the value of the function upon that of the variable the former, that is &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, is called the dependent variable, and the latter, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the independent variable&amp;quot;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In Statistics R. A  Fisher used the terms dependent and independent in his presentation of regression analysis: see Section 25 of Statistical Methods for Research Workers (1925) on &amp;quot;Regression coefficients&amp;quot;. For Fisher these terms qualified variate but later writers have generally favoured variable.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-19T12:55:16.320" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11252" PostTypeId="1" AcceptedAnswerId="11254" CreationDate="2019-12-22T12:42:09.460" Score="0" ViewCount="102" Body="&lt;p&gt;Suppose you go to a book store and you see two books titled:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;The Science of Cooking&#xA;The Art of Cooking&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;Before opening the books, how would you expect them to differ? What kinds of things would you expect to be in The Science of Cooking that you would not expect to be in The Art of Cooking, and vice versa?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question applies not only to cooking but to many other things:&lt;/p&gt;&#xA;&#xA;&lt;pre&gt;&lt;code&gt;The Science of Artificial Intelligence&#xA;The Art of Artificial Intelligence&#xA;&#xA;The Science of Home Decorating&#xA;The Art of Home Decorating&#xA;&#xA;The Science of Astronomy&#xA;The Art of Astronomy&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&#xA;&lt;p&gt;In order for a book to legitimately claim that it is &quot;The Science of,&quot; what must it contain? How must it differ from books titled &quot;The Art of&quot; or &quot;The Engineering of&quot; or &quot;The Philosophy of&quot; or &quot;The Discipline of&quot;?&lt;/p&gt;&#xA;" OwnerUserId="10842" LastEditorUserId="131" LastEditDate="2019-12-23T14:40:56.657" LastActivityDate="2019-12-23T14:40:56.657" Title="Differences between &quot;The Science of&quot; versus &quot;The Art of?" Tags="|philosophy-of-science|scientific-method|" AnswerCount="1" CommentCount="7" ClosedDate="2019-12-30T03:09:52.383" ContentLicense="CC BY-SA 4.0" />
  <row Id="11253" PostTypeId="2" ParentId="11231" CreationDate="2019-12-22T16:05:43.860" Score="3" Body="&lt;p&gt;If two unequal thermally insulated cylinders are connected with a pipe with a piston in it, then these two cylinders are said to be in &lt;strong&gt;&quot;Mechanical Equilibrium&quot;&lt;/strong&gt; if the piston does not move and we conclude that the &lt;strong&gt;&quot;Pressure&quot;&lt;/strong&gt; in both of them is same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Similarly, if two bodies are brought in contact with each other and there is no exchange of heat between them then we say that these two bodies are in &lt;strong&gt;&quot;Thermodynamic Equilibrium&quot;&lt;/strong&gt; and hence there must be a property which these bodies must be sharing analogous to &quot;Pressure&quot; by the virtue of which there is no exchange of heat in between them, and this property is defined by Zeroth Law of Thermodynamics as &lt;strong&gt;&quot;Temperature&quot;&lt;/strong&gt;. &#xA;Zeroth Law of Thermodynamics defines the thermodynamic property - &lt;strong&gt;&quot;Temperature&quot;&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The reason it is called Zeroth Law and not the first law because it was defined after having defined the First and Second Laws of Thermodynamics. By the time scientists felt the need for Zeroth Law of thermodynamics, the First and Second Laws had become so popular that it was not possible to change their names or numbering, so this law was defined as Zeroth Law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is worth noting down these different laws of thermodynamics define different thermodynamic properties. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Zeroth Law defines Temperature. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;First Law defines Internal Energy and Energy Conservation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Second Law defines the idea of Entropy. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Third Law defines the idea of Zero Kelvin Temperature and also establishes that it cannot be achieved.&lt;/p&gt;&#xA;" OwnerUserId="10791" LastActivityDate="2019-12-22T16:05:43.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11254" PostTypeId="2" ParentId="11252" CreationDate="2019-12-22T20:05:45.427" Score="2" Body="&lt;p&gt;It is an open ended question. In my understanding, when we use the word &quot;science&quot;, it is often implied that one can explain all the observations from known scientific principles. On the other hand, a book or a paper with a title of &quot;Art of Something&quot; is more like a personal account of tips and tricks used by the author and the author may not explain all his/her actions on a scientific basis.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider food chemists, who may know all the chemical changes that take place during cooking, but certainly they may not have excellent cooking skills, because they may not have personal experience or know good tips/tricks  (=art) of cooking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The dictionary definition of art is &lt;em&gt;&quot;A practical application of knowledge; (hence) something which can be achieved or understood by the employment of skill and knowledge; (in early use also) a body or system of rules serving to facilitate the carrying out of certain principles.&quot;&lt;/em&gt; (Unabridged Oxford English Dictionary)&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2019-12-22T20:29:55.143" LastActivityDate="2019-12-22T20:29:55.143" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11255" PostTypeId="1" AcceptedAnswerId="11256" CreationDate="2019-12-23T20:21:18.913" Score="6" ViewCount="135" Body="&lt;p&gt;Context: One of the key reasons Einstein first suspected space is curved by mass is due to the falling elevator thought experiment. This experiment implies that mass must curve the path of a light ray. From this, he inferred that space must be curved. But why come to that conclusion? (It could rather be for instance that space is flat and the light simply &quot;falls&quot; through flat space like any other object). Apparently the reason is that it was believed that light must follow a geodesic. Thus if space was flat, light would have to travel in a straight line. The thought experiment suggested light actually follows a curved path, and hence by implication, space must curved. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What first lead physicists to believe that light must follow a geodesic, and why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suspect the reason may be just a logical extension of Fermat's theorem for light, but this may be entirely wrong.&lt;/p&gt;&#xA;" OwnerUserId="10845" LastEditorUserId="5989" LastEditDate="2019-12-24T08:12:49.037" LastActivityDate="2019-12-24T08:12:49.037" Title="What was the original reason physicists first believed that light must follow a geodesic?" Tags="|relativity-theory|light|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11256" PostTypeId="2" ParentId="11255" CreationDate="2019-12-23T22:48:20.200" Score="4" Body="&lt;p&gt;It is the so-called Fermat Principle in geometric optics: light chooses the path which takes it least time to travel. Since the speed of light (in the given homogeneous medium) is constant, this must be the shortest path. Later refinements tell us that it is not always shortest but anyway its length must be stationary. Which exactly means a geodesic for some metric. For empty space this is Minkowski's metric, otherwise, some other metric.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This was the simple answer. But one can give a deeper answer, if one looks at the notions of &quot;geodesic&quot; and &quot;straight line&quot; in their historic development.&#xA;Long before the invention of mathematics, people came with the notion of a straight line, the line on which light travels. (Just think of practical method of checking that a line is straight. The most basic of them just compare it with a light ray.) Mathematics is a later invention. The notion of the straight line (distilled from from observations of nature!) was axiomatized and&#xA;became a part of mathematics. The ancients also knew that a straight line minimizes distance. Calculus of variations is a still later invention, as well as the general idea of idea of geodesic. So it should not come as a surprise if in the end we obtain that light travels on geodesics:-)&#xA;All these relevant notions of mathematics developed from the original observations of light rays. This explanation depends on the point of view that our mathematics comes from the observation of nature as its final source. This point of view is not shared by everyone.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-12-23T23:41:08.503" LastActivityDate="2019-12-23T23:41:08.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11257" PostTypeId="1" AcceptedAnswerId="11271" CreationDate="2019-12-24T08:28:20.707" Score="3" ViewCount="527" Body="&lt;p&gt;From what I understand Euclid avoided infinity, and so I'm wondering how Euclid might have dealt with the concept of 0 in the Elements.&lt;/p&gt;&#xA;" OwnerUserId="10846" LastEditorUserId="5989" LastEditDate="2019-12-25T09:38:40.667" LastActivityDate="2019-12-26T07:13:19.470" Title="Does Euclid's Elements acknowledge a concept of 0, either directly or indirectly?" Tags="|mathematics|zero|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11258" PostTypeId="2" ParentId="11257" CreationDate="2019-12-24T10:58:29.550" Score="6" Body="&lt;p&gt;He did not. There is no reference to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; in the &lt;em&gt;Elements&lt;/em&gt;. As a matter of fact, for Euclid even &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; was not a number. For him, the numbers are &lt;span class=&quot;math-container&quot;&gt;$2,3,4,\ldots$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2019-12-24T10:58:29.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11259" PostTypeId="1" CreationDate="2019-12-25T05:44:56.253" Score="3" ViewCount="2041" Body="&lt;p&gt;Who was the first to define &lt;strong&gt;congruent triangles&lt;/strong&gt;? I couldn't find the definition in Euclid's &lt;em&gt;Elements&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9595" LastEditorUserId="55" LastEditDate="2019-12-25T07:22:59.540" LastActivityDate="2019-12-26T06:06:14.890" Title="Who first gave a definition of congruent triangles?" Tags="|terminology|euclidean-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11260" PostTypeId="1" AcceptedAnswerId="11262" CreationDate="2019-12-25T07:16:39.337" Score="1" ViewCount="125" Body="&lt;p&gt;&quot;Positive definite matrix&quot; is a standard term in mathematics, espeically linear algebra. Are there grammatical, linguistic, or historical reasons why it was not called &quot;positively definite matrix&quot;?&lt;/p&gt;&#xA;" OwnerUserId="7937" LastEditorUserId="26" LastEditDate="2021-02-20T15:21:11.237" LastActivityDate="2021-02-20T15:21:11.237" Title="Why positive definite matrix rather than positively definite matrix?" Tags="|mathematics|terminology|linear-algebra|" AnswerCount="1" CommentCount="0" ClosedDate="2019-12-25T22:17:47.003" ContentLicense="CC BY-SA 4.0" />
  <row Id="11261" PostTypeId="2" ParentId="11259" CreationDate="2019-12-25T07:22:17.227" Score="4" Body="&lt;p&gt;It depends on how the question is interpreted. Common notion 4, &amp;quot;&lt;em&gt;things which coincide with one another equal one another&lt;/em&gt;&amp;quot;, is not exactly a definition of congruent triangles or even of congruence generally. Likely due to Platonist strictures, Euclid deliberately avoids using congruence in his demonstrations. The idea of moving and superimposing figures Plato judged as &amp;quot;corrupting the good of geometry&amp;quot;, which is why we do not find &amp;quot;workman tools&amp;quot; like straightedge and compass, or mechanically generated curves, in the Elements either, see &lt;a href=&quot;https://hsm.stackexchange.com/a/6902/55&quot;&gt;When were the concepts of pure and applied Mathematics introduced?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Many Euclidean demonstrations display proliferation of auxiliary triangles where the use of congruence would have simplified the clutter. But at the very beginning Euclid has no choice, as he needs tests of equality for his auxiliary triangles. The first such test is Proposition I.4:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;All sides and angles are equal to each other is a way to define &amp;quot;congruence of triangles&amp;quot; without congruence. Euclid could have just made I.4 another postulate, but as he wanted to give a demonstration he had to resort to superimposing. Even so, the use of imposition is buried inside the demonstration and unacknowledged. Joyce comments on this in a &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/propI26.html#cong&quot; rel=&quot;nofollow noreferrer&quot;&gt;note to I.26 of his translation of the Elements&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Euclid’s congruence theorems are I.4 (side-angle-side), I.8 (side-side-side), and this one, I.26 (side and two angles). Calling them congruence theorems is anachronistic, since Euclid did not explicitly use the concept of congruence. We would say that two triangles ABC and DEF are congruent if the angles A, B, and C equal the angles D, E, and F respectively, and the sides AB, BC, and AC equal the sides DE, EF, and DF respectively, and the triangle ABC equals the triangle DEF (by which is meant that they have the same area).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The word &amp;quot;congruent&amp;quot; for superimposing (&lt;em&gt;superpono&lt;/em&gt;) appears in 16-17th century Latin translations of Euclid, whose authors had no preconceptions about motion. &amp;quot;Congruent&amp;quot;  replaces &amp;quot;coincide&amp;quot; in Common notion 4. &lt;a href=&quot;https://archive.org/details/euclidiselement00barrgoog/page/n8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clavius in &lt;em&gt;Euclidis Elementorum&lt;/em&gt; (1589)&lt;/a&gt; writes, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Hinc enim fit, ut aequalitas angulorum ejusdem generis requirat eandem inclinationem linearum, ita ut lineae unius conveniant omnino lineis alterius, si unus alteri superponatur. Ea enim aequalia sunt, quae sibi mutuo congruunt.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For want of this, equality of the angles of the same kind requires the same inclination of the lines, and lines to lines coincide when placed on top of each other. For those things are equal, that are congruent with each other.&lt;/em&gt;&amp;quot; [translation mine]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miller's Earliest Known Uses&lt;/a&gt; the word is first used systematically in something like the modern sense only by Leibniz in 1679:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;As a more technical term for a relation between figures, congruent seems to have originated with Gottfried Wilhelm Leibniz (1646-1716), writing in Latin and French. His manuscript &amp;quot;Characteristica Geometrica&amp;quot; of August 10, 1679, is in his Gesammelte Werke, dritte Folge: mathematische Schriften, Band 5. On p. 150 he says that if a figure can be applied exactly to another without distortion, they are said to be congruent...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;His Figure 39 shows two radii of a circle, with the center labelled both A and C. Later (p. 154) he points out that &amp;quot;congruent&amp;quot; is the same as &amp;quot;similar and equal.&amp;quot; He used &amp;quot;congruent&amp;quot; in the modern (Hilbert) sense, applied to line segments and various other things as well as triangles.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Shortly afterwards, on September 8, 1679, he included a similar definition in a letter to Hugens (sic) van Zulichem. In his ges. Werke etc. as above, volume 2, p. 22, he illustrates congruence with a pair of triangles, and says that they &amp;quot;peuvent occuper exactement la meme place, et qu'on peut appliquer ou mettre l'un sur l'autre sans rien changer dans ces deux figures que la place.&amp;quot; [Ken Pledger and Julio González Cabillón]&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-26T06:06:14.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11262" PostTypeId="2" ParentId="11260" CreationDate="2019-12-25T08:08:15.123" Score="2" Body="&lt;p&gt;According to &lt;a href=&quot;http://jeff560.tripod.com/p.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miller's Earliest Known Uses&lt;/a&gt; positive definite, as applied to forms, appears first in &lt;a href=&quot;https://archive.org/details/lecturesonthethe032088mbp/page/n1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierpont's Lectures On The Theory Of Functions Of Real Variables (1905)&lt;/a&gt;. Pierpont uses the reversed expression for numbers, e.g. &quot;&lt;em&gt;For, if &lt;span class=&quot;math-container&quot;&gt;$p\neq q$&lt;/span&gt;, say &lt;span class=&quot;math-container&quot;&gt;$p &amp;gt; q$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$p-q$&lt;/span&gt; is a definite positive rational number; call it d&lt;/em&gt;&quot;. Here &quot;definite&quot; and &quot;positive&quot; are thought as two separate characterizations rather than &quot;definite&quot; adverbially modifying &quot;positive&quot;. Later he makes it explicit with forms:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A form... which has always one sign, except at the origin where it necessarily vanishes, is called definite... If the sign of a definite form is positive, it is called a positive definite form; if negative, it is a negative definite form.&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Apparently, he does not have in mind adverbial modifiers as in &quot;positively definite&quot; or &quot;definitely positive&quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2019-12-25T08:08:15.123" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11263" PostTypeId="1" AcceptedAnswerId="11264" CreationDate="2019-12-24T21:35:53.233" Score="7" ViewCount="351" Body="&lt;p&gt;The term &lt;em&gt;positive definite matrix&lt;/em&gt; is a standard one used in mathematics, especially in linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there grammatical, linguistic, or historical reasons why it was not called a &lt;em&gt;positive&lt;strong&gt;ly&lt;/strong&gt; definite matrix&lt;/em&gt; instead?&lt;/p&gt;&#xA;" OwnerUserId="7937" OwnerDisplayName="modnar" LastEditorUserId="26" LastEditDate="2021-02-20T15:21:19.920" LastActivityDate="2021-02-20T15:21:19.920" Title="Why do we call it a &quot;positive definite matrix&quot; rather than a &quot;positively definite matrix&quot;?" Tags="|mathematics|terminology|linear-algebra|linguistics|" AnswerCount="2" CommentCount="12" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11264" PostTypeId="2" ParentId="11263" CreationDate="2019-12-25T17:31:50.850" Score="6" Body="&lt;p&gt;Yes, it seems that there are linguistic reasons&lt;sup&gt;1&lt;/sup&gt; why &lt;em&gt;positive definite&lt;/em&gt; works better than &lt;em&gt;positively definite&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;BTW, for that reason, I think that it was a mistake to migrate this question from the English Language and Usage (EL&amp;amp;U) StackExchange to the History of Science and Mathematics (HSM) StackExchange.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems that when we are picking adjectives as labels for types, in English we prefer the adjective+adjective construction to the adverb+adjective one. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is unlikely we can &lt;em&gt;really&lt;/em&gt; answer why we have that preference any more than we can answer why we prefer e.g. &lt;em&gt;all friends of Kim's&lt;/em&gt; to &lt;em&gt;all Kim's friends&lt;/em&gt;; at the present state of our knowledge, probably the best we can reliably do is state trends and tendencies. Despite that, I will speculate about one possible reason, below.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Discussion&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider this example (I'm of course not saying that the actual claims made in the example are correct):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There are two types of Indic languages, namely&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;(a) &lt;strong&gt;anciently Indic&lt;/strong&gt; languages and &lt;strong&gt;contemporarily Indic&lt;/strong&gt; languages.&lt;br&gt;&#xA;  (b) &lt;strong&gt;ancient-Indic&lt;/strong&gt; languages and &lt;strong&gt;contemporary-Indic&lt;/strong&gt; languages.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In (a), &lt;em&gt;anciently Indic&lt;/em&gt; is a &lt;strong&gt;syntactic construction&lt;/strong&gt;: it is an adjective phrase (AdjP) whose &lt;a href=&quot;https://en.wikipedia.org/wiki/Head_(linguistics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;head&lt;/a&gt; is the adjective &lt;em&gt;Indic&lt;/em&gt; and which has the adverb &lt;em&gt;anciently&lt;/em&gt; as a modifier. The meaning is 'Indic in an ancient manner' (and analogously for &lt;em&gt;contemporarily Indic&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521431468&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;CGEL&lt;/a&gt; (pp. 1657–1658), in (b), &lt;em&gt;ancient-Indic&lt;/em&gt; is a &lt;strong&gt;morphological compound&lt;/strong&gt;: &lt;em&gt;ancient&lt;/em&gt; and &lt;em&gt;Indic&lt;/em&gt; combine to produce a new word (and similarly for &lt;em&gt;contemporary-Indic&lt;/em&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The point is that, to my ear at least, &lt;strong&gt;(b) is clearly preferable to (a).&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I admit that I don't understand why CGEL is so sure that the construction in (b) is morphological and not syntactical (I will probably post a separate question about it). But if CGEL is correct about this, then perhaps this might be the reason why we prefer (b): we would like the type labels to be &lt;strong&gt;syntactically simple,&lt;/strong&gt; to be actual terms, &lt;strong&gt;lexical items,&lt;/strong&gt; as opposed to syntactic phrases.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are numerous other examples that could be constructed: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We manufacture sandals in two types of blue, so we have &lt;strong&gt;light-blue&lt;/strong&gt; sandals and &lt;strong&gt;dark-blue&lt;/strong&gt; sandals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(preferred to &lt;strong&gt;&lt;em&gt;lightly blue&lt;/strong&gt; sandals&lt;/em&gt; and &lt;strong&gt;&lt;em&gt;darkly blue&lt;/strong&gt; sandals&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thus you end up with two types of cream: the &lt;strong&gt;steaming-hot&lt;/strong&gt; cream and &lt;strong&gt;icy-cold&lt;/strong&gt; cream.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(preferred to &lt;strong&gt;&lt;em&gt;steamingly hot&lt;/strong&gt; cream&lt;/em&gt; and &lt;strong&gt;&lt;em&gt;icyly cold&lt;/strong&gt; cream&lt;/em&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Positive definite&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as &lt;em&gt;positive definite&lt;/em&gt;, let's start with some definitions. First of all, when it comes to the notion of &lt;strong&gt;definiteness&lt;/strong&gt;, the basic object is a &lt;strong&gt;quadratic form&lt;/strong&gt;. The next step is to realize that to every matrix, one can associate a quadratic form. A matrix is then called &lt;strong&gt;&lt;em&gt;definite&lt;/em&gt;&lt;/strong&gt; if the associated quadratic form is definite; it's called &lt;strong&gt;&lt;em&gt;positive definite&lt;/em&gt;&lt;/strong&gt; if the associated quadratic form is &lt;em&gt;positive definite&lt;/em&gt;, etc. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;For a diagonalizable matrix, these properties of definiteness are easily relatable to the properties of the eigenvalues of the matrix, and some sources proceed to simply define &lt;em&gt;positive definiteness&lt;/em&gt; of a matrix in terms of the properties of its eigenvalues. I personally dislike that practice, but that's definitely a matter of taste.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So let's talk about quadratic forms now. We have this &lt;a href=&quot;https://books.google.com/books?id=zshyDwAAQBAJ&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;lpg=PA197&amp;amp;dq=%22a%20definite%20quadratic%20form%22&amp;amp;pg=PA197#v=onepage&amp;amp;q=%22a%20definite%20quadratic%20form%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;(source)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Definite Quadratic Forms.&lt;/strong&gt; Since it is homogeneous, every quadratic form is zero at the origin. We call the quadratic form &lt;em&gt;Q&lt;/em&gt; &lt;strong&gt;definite&lt;/strong&gt; if it is nonzero everywhere else: &lt;em&gt;Q(&lt;strong&gt;x&lt;/strong&gt;)&amp;ne;0&lt;/em&gt; for &lt;strong&gt;&lt;em&gt;x&lt;/strong&gt;&amp;ne;0&lt;/em&gt;.&lt;br&gt; &#xA;  …&lt;br&gt;&#xA;  If &lt;em&gt;Q(&lt;strong&gt;x&lt;/strong&gt;)&lt;/em&gt; is a definite quadratic form, then one of the following inequalities holds:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Q(&lt;strong&gt;x&lt;/strong&gt;)&gt;0&lt;/em&gt; for all &lt;strong&gt;&lt;em&gt;x&lt;/strong&gt;&amp;ne;0&lt;/em&gt; (&lt;em&gt;Q&lt;/em&gt; is &lt;strong&gt;positive definite&lt;/strong&gt;), or&lt;br&gt;&#xA;  &lt;em&gt;Q(&lt;strong&gt;x&lt;/strong&gt;)&amp;lt;0&lt;/em&gt; for all &lt;strong&gt;&lt;em&gt;x&lt;/strong&gt;&amp;ne;0&lt;/em&gt; (&lt;em&gt;Q&lt;/em&gt; is &lt;strong&gt;negative definite&lt;/strong&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus we have two types of definite quadratic forms: &lt;strong&gt;positive definite&lt;/strong&gt; and &lt;strong&gt;negative definite&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The analogy with what I said above about &lt;em&gt;ancient-Indic&lt;/em&gt;, &lt;em&gt;dark-blue&lt;/em&gt;, etc. would be complete if &lt;em&gt;positive definite&lt;/em&gt; were hyphenated. It is definitely &lt;em&gt;sometimes&lt;/em&gt; hyphenated, including in two of the examples of usage in the OED (the ones from 1904 and 1957):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;positive definite&lt;/strong&gt; &lt;em&gt;adj.&lt;/em&gt; &lt;em&gt;Mathematics&lt;/em&gt; (of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its image under the operator is greater than zero.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;&#xA;  1904&amp;nbsp;&amp;nbsp;&amp;nbsp;Trans. Amer. Math. Soc. &lt;strong&gt;5&lt;/strong&gt; 464&amp;nbsp;&amp;nbsp;&amp;nbsp;It is well known that there is always one such invariant, a &lt;strong&gt;positive-definite&lt;/strong&gt; Hermitian form.&lt;br&gt;&#xA;  1948&amp;nbsp;&amp;nbsp;&amp;nbsp;W. V. Houston &lt;em&gt;Princ. Math. Physics&lt;/em&gt; (ed. 2) vii. 120&amp;nbsp;&amp;nbsp;&amp;nbsp;The potential energy will be a quadratic expression in the coordinates that, if the equilibrium is stable, will be a &lt;strong&gt;positive definite&lt;/strong&gt; expression.&lt;br&gt;&#xA;  1957&amp;nbsp;&amp;nbsp;&amp;nbsp;L. Fox &lt;em&gt;Numerical Solution Two-point Boundary Probl.&lt;/em&gt; vii. 179&amp;nbsp;&amp;nbsp;&amp;nbsp;If all the &lt;em&gt;λ&lt;sup&gt;τ&lt;/sup&gt;&lt;/em&gt; are positive, which is the case in many physical problems, and corresponds to some structure of the differential system corresponding to a &lt;strong&gt;positive-definite&lt;/strong&gt; matrix &lt;em&gt;A&lt;/em&gt;.., we can also assert [etc.].&lt;br&gt;&#xA;  1990&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;em&gt;IMA Jrnl. Numerical Anal.&lt;/em&gt; 10 546&amp;nbsp;&amp;nbsp;&amp;nbsp;&lt;em&gt;H&lt;sup&gt;k&lt;/sup&gt;&lt;/em&gt; is a positive &lt;strong&gt;definite matrix&lt;/strong&gt; that approximates the inverse reduced Hessian matrix.&#xA;  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It is an interesting question why hyphenation came to be disfavored. But the fact that it came to be disfavored does not, I think, make this case substantially different from   &lt;em&gt;ancient-Indic&lt;/em&gt;, &lt;em&gt;dark-blue&lt;/em&gt;, etc.&lt;/p&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2019-12-26T19:06:52.967" LastActivityDate="2019-12-26T19:06:52.967" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11266" PostTypeId="2" ParentId="11263" CreationDate="2019-12-25T19:50:42.110" Score="4" Body="&lt;p&gt;As far as I know, the first appearance of the concept of positive/negative definiteness (and of indefiniteness) is in the article 271 of Gauss' Disquisitiones Arithmeticae about ternary forms. Of course the Disquisitiones are written in Latin, but maybe the original context can help in clarify the terminology also in English.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Quaedam formae ternariae ita sunt comparatae, ut per ipsas [...] possint repraesentari numeri positivi et negativi [...] &lt;em&gt;formae indefinitae&lt;/em&gt; vocabuntur. [...] Contra per alias numeri negativi repraesentari nequeunt [...] quare &lt;em&gt;formae positivae&lt;/em&gt; dicentur [...] formae positivae et negativae nomine communi &lt;em&gt;formae definitae&lt;/em&gt; dicentur.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;that is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Certain ternary forms are so constructed that positive and negative numbers can be represented by them [...] they will be called &lt;em&gt;indefinite forms&lt;/em&gt;. [...] On the other hand by some other forms, negative numbers cannot be represented [...] and so they will be called &lt;em&gt;positive forms&lt;/em&gt; [...] positive and negative forms will be called by the communal name of &lt;em&gt;definite forms&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So it is clear that the terms &quot;positive&quot; and &quot;negative&quot; are not referred to &quot;definite&quot;, and then it is absolutely correct to say &quot;positive definite form&quot;. Maybe the real question is why we call them &quot;positive definite forms&quot; when &quot;positive forms&quot; would be (at least) equally correct.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed note that Gauss never wrote &quot;positive definite form&quot; but simply &quot;positive form&quot; or &quot;definite form&quot; for a form that can be positive or negative.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here an example in the same article:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;formae definitae semper adiunctam esse definita et &lt;strong&gt;quidem&lt;/strong&gt; (my emphasis) &lt;em&gt;negativam&lt;/em&gt; [...] &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;that is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the adjoint of a definite form is always definite and &lt;strong&gt;more precisely&lt;/strong&gt; &lt;em&gt;negative&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;where it s clear that the first time &quot;definite form&quot; means &quot;positive or negative&quot;, while the Latin term &quot;quidem&quot; is used to express emphasis and to further specify that the adjoint form is not simply a definite form but a negative one.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2019-12-25T19:50:42.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11267" PostTypeId="1" CreationDate="2019-12-26T00:30:03.930" Score="3" ViewCount="1053" Body="&lt;p&gt;Now-a-days it is well known that sun never rise or set but appears to human&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The sun stays in its position at the center of our solar system. &lt;strong&gt;It&#xA;  doesn't rise and set. But it appears to rise and set&lt;/strong&gt; because of the&#xA;  Earth's rotation on its axis. &lt;a href=&quot;https://journeynorth.org/tm/mclass/SunriseSetAns.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://starchild.gsfc.nasa.gov/docs/StarChild/questions/question14.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;An Indian text named &lt;a href=&quot;https://en.wikipedia.org/wiki/Aitareya_Brahmana&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aitareya Brahmana&lt;/a&gt; of Rig veda contains this fact&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;The sun does never rise or set. When people think the sun is setting&lt;/strong&gt; (it is not so). For after having arrived at the end of the&#xA;  day it makes itself produce two opposite effects, making night to what&#xA;  is below and day to what is on the other side. &lt;strong&gt;When they believe it&#xA;  rises in the morning&lt;/strong&gt; (this supposed rising is thus to be explained&#xA;  for). Having reached the end of the night, it makes itself produce two&#xA;  opposite effects, making night to what is below and day to what is on&#xA;  the other side.&quot;  &lt;a href=&quot;https://archive.org/details/aitareyabrahmana04hauguoft/page/162&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/rigvedabrahmana00keitgoog/page/n210&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although there is no unanimous conclusion on dating of the text, it may be as old as &lt;a href=&quot;https://en.wikipedia.org/wiki/Rigveda#Dating_and_historical_context&quot; rel=&quot;nofollow noreferrer&quot;&gt;2000 BC&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Aitareya_Brahmana#Date_of_composition&quot; rel=&quot;nofollow noreferrer&quot;&gt;1st millennium BCE&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any other text that contains this fact before or in the time range of the above mentioned text?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastEditorUserId="5788" LastEditDate="2022-06-28T21:32:46.503" LastActivityDate="2022-06-28T21:32:46.503" Title="Historical mentions (as if fact) that Sun never rises or sets" Tags="|astronomy|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11268" PostTypeId="1" AcceptedAnswerId="11269" CreationDate="2019-12-26T01:39:31.027" Score="10" ViewCount="2107" Body="&lt;p&gt;My sense is that Franklin was a scientist more like Faraday than Maxwell. Given also that Calculus was fairly new when Franklin was in school (and as far as I know, Benjamin Franklin did not get very far in school anyway) how likely was it that he could use Calculus or even really knew what it was?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="7265" LastEditDate="2019-12-27T03:07:40.790" LastActivityDate="2019-12-27T03:07:40.790" Title="Did Benjamin Franklin know calculus?" Tags="|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11269" PostTypeId="2" ParentId="11268" CreationDate="2019-12-26T05:02:07.837" Score="13" Body="&lt;p&gt;Apparently, no. &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/benjamin-franklins-numbers-an-unsung-mathematical-odyssey&quot; rel=&quot;noreferrer&quot;&gt;Boman's biography Benjamin Franklin's Numbers&lt;/a&gt; specifically focuses on Franklin's mathematical activities, and calculus is not among them. He was deep into magic squares, and anticipated some basic ideas of statistics and population dynamics. In a 1772 &lt;a href=&quot;http://www.historycarper.com/1772/09/19/moral-or-prudential-algebra/&quot; rel=&quot;noreferrer&quot;&gt;letter to Joseph Priestley&lt;/a&gt; he described what  he called Moral or Prudential Algebra for decision making, but that was more what is now called cost-benefit analysis.&#xA;In &lt;a href=&quot;https://www.npr.org/templates/story/story.php?storyId=18045610&quot; rel=&quot;noreferrer&quot;&gt;Keith Devlin's description&lt;/a&gt;, the closest he came to calculus topics is predicting exponential growth of populations before Malthus in a two-page essay with no formulas:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Franklin had a lifelong interest in magic squares... Now, to the outsider these look like just doodling with numbers but, in fact, to construct one of those things, you have to get deep into the mathematics and into the patterns. And Franklin did, in his lifetime, spent a lot of time constructing these things and you can only do that if you have a certain deep feeling for numbers.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The other example is what we find in his political and business writings. He is very early, much early than many other people in realizing that you could use what we would now call basic statistics in order to make political and business decisions.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;... he wrote an article... in 1751 called &lt;a href=&quot;https://founders.archives.gov/documents/Franklin/01-04-02-0080&quot; rel=&quot;noreferrer&quot;&gt;Observations Concerning the Increase of Mankind and the Peopling of Countries&lt;/a&gt;. And that was really one of the first ever works in what we now call demographics, using mathematical techniques to look at how populations grow and how people move and how societies develop. In fact, Franklin was the first person who speculated that populations probably increase exponentially. Now, we always associated that with Thomas Malthus, who was the one that demonstrated that. But, in fact, Malthus had already read Franklin's work and cited it when he did his work.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Although Franklin did not do calculus he knew of it at least since 1725 at the age of 19. According to &lt;a href=&quot;https://books.google.com/books?id=KranDwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Bunker's Young Benjamin Franklin&lt;/a&gt;, when in London he met Pemberton, well versed in calculus and Newtonian physics, who was then preparing a new edition of &lt;em&gt;Principia&lt;/em&gt;. Pemberton himself wrote an essay attempting to use Newtonian mechanics to explain the functioning of muscles, which Franklin read and which stimulated his interest in experimental science. A personal meeting with Newton, promised by Pemberton, never came to pass, but Franklin's interests shifted away from metaphysics, and four decades later he ordered a portrait of himself with Newton in the background.&lt;/p&gt;&#xA;&lt;p&gt;In 1754 Franklin became one of the trustees of the Logan library established by the sons of late James Logan &amp;quot;&lt;em&gt;a classicist who in the margins of his books crossed swords with greatest&#xA;European editors, and a scientist who described the fertilization of corn by pollen, understood and used the new inventions of calculus, wrote on optics, and made astronomical observations, the Quaker virtuoso brought books to feed the wide-ranging&#xA;appetite of his mind&lt;/em&gt;&amp;quot;, see &lt;a href=&quot;http://librarycompany.org/about/Instance.pdf&quot; rel=&quot;noreferrer&quot;&gt;At the Instance of Benjamin Franklin&lt;/a&gt;. But they add:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Although&#xA;Franklin in his promotional tract for the establishment of a college in Philadelphia had&#xA;described Logan's library as a valuable book resource available to professors and&#xA;students, use was seldom made of the scholarly works in the collection. In the eighteenth&#xA;century there was furthermore little interest in the classics and advanced mathematical&#xA;sciences on the part of merchants and artisans in Philadelphia.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-26T06:31:48.960" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11270" PostTypeId="2" ParentId="11267" CreationDate="2019-12-26T05:28:06.803" Score="3" Body="&lt;p&gt;The reference [22] from Wikipedia on &quot;Aitareya Brahmana&quot; quotes an English work &lt;a href=&quot;https://zenodo.org/record/1428644#.XgQ-MUdTlPY&quot; rel=&quot;nofollow noreferrer&quot;&gt;A remarkable Vedic Theory&lt;/a&gt;, which in turn is based on a German work. I used DeepL to translate the quoted text in the reference:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;Interessant ist die Angabe des Ai. Bra. (iii, 44), dass die Sonne wirklich weder aufnoch untergeht, sondern dadurch dass sie sich umdreht, inden unteren Regionen, d.h. auf der Erde, abwechselnd Tag und Nacht hervorbringt. Wie die Sonne vom Westen zum Punktedes Anfangs zuruckkehrt, daruber geben die vedischen Texte keinen Aufschluss.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[Translation] Interesting is the indication of the Ai. Bra. (iii,&#xA;  44), that the sun really neither rises nor sets, but by turning&#xA;  around, in the lower regions, i.e. on the earth, alternately produces&#xA;  day and night. The Vedic texts do not explain how the sun returns from&#xA;  the west to the point of the beginning.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The OP provides a reference &quot;Rigveda Brahmanas&quot; from the Internet Archive, which quotes the similar text that the Sun never sets. A quick search of the word Sun shows that it occurs mainly in the context of Sun being treated as a deity as well as a home of deities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One such example is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the sun is the world of the gods, and the fathers are the world of the &#xA;  fathers, verily thus they ascend from the world of the fathers to the&#xA;  world of the gods.&#xA;  or&#xA;  The divine lordly power is the sun ; the sun is the overlord of these&#xA;  beings.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Therefore, it is not surprising to find the main context of Sun never setting or rising statement. Immediately after the Sun never setting or rising, a statement like this appears&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he (sun) never really sets or rises. In that they think of him He is&#xA;  setting ', verily having reached the end of the day, he inverts&#xA;  himself ; thus  he makes evening below, day above. Again in that they&#xA;  think of him ' He  is rising in the morning, verily having reached&#xA;  the end of night he inverts  'himself ; thus he makes day below, night&#xA;  above. He never sets ; indeed he  never sets, union with him and&#xA;  identity of form and world he attains&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The use of capital H in &quot;He&quot; makes it clear that Sun is being treated as a deity, which inverts itself to cause day and night. Thus it is hard to associate a scientific meaning with this statement. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Praying and worshipping the Sun, is apparently prehistoric. One should not extract a scientific meaning from a coincidental mention of Sun never setting in a very old literature, which may have changed over a course of centuries. It is very clear that Sun is being referred to a deity in the context that it will not set. This Sun worship practice dates even before the time of prophet Abraham (the forefather of Judaism, Christianity and Islam), and apparently he also briefly worshipped the Sun before quickly denouncing and devoting himself to a single God. Certainly the dating of this Sun worship practice must be more than 2000 years old.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="10905" LastEditDate="2020-01-01T19:16:57.077" LastActivityDate="2020-01-01T19:16:57.077" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11271" PostTypeId="2" ParentId="11257" CreationDate="2019-12-26T07:13:19.470" Score="2" Body="&lt;p&gt;The arithmetic books of Elements are books VII - IX.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Euclid begins &lt;a href=&quot;https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;book VII&lt;/a&gt; with his definition of number:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A &lt;em&gt;number&lt;/em&gt; is a multitude composed of units.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(This is taken from the Pythagorean notion of &lt;em&gt;number&lt;/em&gt;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since &quot;zero&quot; is not a multitude of units it does not satisfy Euclid's definition of number.  Euclid did not consider &quot;one&quot; to be a number for the same reason.  However, Euclid appears to have been somewhat ambivalent on this point and did treat the unit as a number on those occasions when it helped in stating a general proposition or proving them.  For example, Euclid's famous proof (IX.20) that there are infinitely many prime numbers.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2019-12-26T07:13:19.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11272" PostTypeId="2" ParentId="5908" CreationDate="2019-12-26T14:20:27.287" Score="0" Body="&lt;p&gt;In the article 267 of Disquisitiones Arithmeticae Gauss considers ternary form of second degree like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ax^2+a'x'^2+a''x''^2+2bx'x''+2b'xx''+2b''xx'$$&lt;/span&gt;&#xA;and defines the adjoint to such form as &#xA;&lt;span class=&quot;math-container&quot;&gt;$$Ax^2+A'x'^2+A''x''^2+2Bx'x''+2B'xx''+2B''xx'$$&lt;/span&gt;&#xA;where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \begin{array}{lll}&#xA;    bb-a'a''=A,&amp;amp;b'b'-aa''=A',&amp;amp;b''b''-aa'=A'' \\&#xA;    ab-b'b''=B,&amp;amp;a'b'-bb''=B',&amp;amp;a''b''-bb'=B''&#xA;  \end{array}&#xA;$$&lt;/span&gt;&#xA;Gauss clearly cannot speak about &quot;adjoint to a matrix&quot; (the term matrix in the usual sense was introduced after 1850), and moreover do not use 3x3 (symmetric) matrices to represent ternary forms (while uses something like 3x3 matrices to represent atransformation between forms) but it is clear that the matrix&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \left(\begin{array}{lll}&#xA;    A &amp;amp; B'' &amp;amp; B' \\&#xA;    B'' &amp;amp; A' &amp;amp; B \\&#xA;    B' &amp;amp; B &amp;amp; A'' \\&#xA;  \end{array}\right)&#xA;$$&lt;/span&gt;&#xA;is the &lt;strong&gt;opposite&lt;/strong&gt; (this is due to the fact that Gauss defines the determinant of a form in such a way that it is actually the opposite of the determinant we use nowadays) of the adjoint to the matrix &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;  \left(\begin{array}{lll}&#xA;    a &amp;amp; b'' &amp;amp; b' \\&#xA;    b'' &amp;amp; a' &amp;amp; b \\&#xA;    b' &amp;amp; b &amp;amp; a'' \\&#xA;  \end{array}\right)&#xA;$$&lt;/span&gt;&#xA;which represents the original form.&#xA;So it is possible that this is the first use in linear algebra of the term &quot;adjoint&quot; in a similar sense to the modern one.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2019-12-26T14:20:27.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11273" PostTypeId="1" CreationDate="2019-12-26T15:09:31.103" Score="0" ViewCount="121" Body="&lt;p&gt;I think the &lt;em&gt;Elements&lt;/em&gt; would have been a lot more concise if he did.&lt;/p&gt;&#xA;" OwnerUserId="9595" LastActivityDate="2019-12-26T15:09:31.103" Title="Why didn't Euclid use equations or numerals in his proofs?" Tags="|mathematics|mathematicians|euclidean-geometry|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11274" PostTypeId="1" AcceptedAnswerId="11278" CreationDate="2019-12-26T16:01:21.440" Score="15" ViewCount="4416" Body="&lt;p&gt;Related:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/730/6976&quot;&gt;Are there any theorems that become &amp;quot;lost&amp;quot; and discarded over time?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/1963/6976&quot;&gt;Is there a &amp;#39;lost calculus&amp;#39;?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The questions above use the term 'lost' to refer to theorems that exist in the literature, but may be so obscure and/or not commonly useful nowadays that few people actually bother to learn them. Are there any theorems that were truly &lt;em&gt;lost&lt;/em&gt; - that is, they were reasonably well-received by the mathematical community, were used for some time, and then were &lt;em&gt;truly lost&lt;/em&gt; to history, unavailable in any extant literature, textbook, or other resource. All available copies might have been lost due to fire, war, or an intentional purge, or available copies might remain but only in an undeciphered code. Examples would include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Theorems that have been truly forgotten and literally cannot be learned today from any source&lt;/li&gt;&#xA;&lt;li&gt;Theorems that were truly lost but were rediscovered many years later, after the original author and all of their students were deceased&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am &lt;em&gt;not&lt;/em&gt; talking about theorems that are commonly classified as &lt;em&gt;lost&lt;/em&gt; simply due to obscurity or uselessness, but can be learned by a student determined enough to see out applicable resources, instructors, etc. These are what the two questions at the top are speaking of.&lt;/p&gt;&#xA;&lt;p&gt;To some extent, &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermat's Last Theorem&lt;/a&gt; seems to match this, but I understand that there is quite a bit of controversy whether Fermat ever even &lt;em&gt;had&lt;/em&gt; a proof to begin with.&lt;/p&gt;&#xA;&lt;p&gt;To be clear, I am obviously not asking about theorems that nobody has ever heard of, but theorems that we are fairly certain existed (e.g. there are citations to them), but for which all copies have apparently been lost. For example, something like this could answer the question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We currently have proof that this expression converges when N &amp;gt; 3. A proof that it converges for all N (Smith and Higgins, 1724) has been cited by Jones (1745) and Hooper (1763), but the only known copy of the full proof was reported lost in a fire at the Bodelian Library in 1789. Searches were undertaken in 1812 and 1826 to find a copy in another library, but these searches were unfruitful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6976" LastEditorUserId="6976" LastEditDate="2021-02-18T13:02:59.740" LastActivityDate="2021-07-14T13:24:35.593" Title="Are there theorems that have been truly lost?" Tags="|mathematics|publishing|" AnswerCount="5" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11275" PostTypeId="2" ParentId="11274" CreationDate="2019-12-26T18:16:51.847" Score="8" Body="&lt;p&gt;The Rogers-Ramanujan identities?&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The formulas have a curious history, having been proved by Rogers (1894) in a paper that was completely ignored, then rediscovered (without proof) by Ramanujan sometime before 1913. The formulas were communicated to MacMahon, who published them in his famous text, still without proof. Then, in 1917, Ramanujan accidentally found Roger's 1894 paper while leafing through a journal. In the meantime, Schur (1917) independently rediscovered and published proofs for the identities (Hardy 1999, p. 91).&lt;br&gt;&#xA;  &lt;a href=&quot;http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html&quot; rel=&quot;noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2019-12-26T18:16:51.847" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11276" PostTypeId="1" AcceptedAnswerId="11279" CreationDate="2019-12-26T20:16:48.043" Score="4" ViewCount="306" Body="&lt;p&gt;Was it believed early on that signals sent via wire moved at exactly the speed of light or simply very fast? Who was the first to estimate the speed? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT: Given that they move at less than the speed of light in the vacuum, what I am wondering is, what earlier investigators thought the speed was and how they attempted to measure it. The experiments to measure speed of light using visible light would not have worked.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="55" LastEditDate="2020-02-10T08:29:53.067" LastActivityDate="2020-02-10T08:39:16.113" Title="Was it suspected that the speed of electricity was equal to the speed of light?" Tags="|electromagnetism|electricity|light|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11277" PostTypeId="2" ParentId="11274" CreationDate="2019-12-26T21:23:26.510" Score="6" Body="&lt;p&gt;I did once write an email to Robert Geroch, because an unpublished paper of his was listed as a bibliographical reference &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01646486&quot; rel=&quot;noreferrer&quot;&gt;in a paper of Hajicek&lt;/a&gt;, which contained a few theorems regarding non-Hausdorff spacetimes. He apparently had no recollection of this paper, and it was never published. I assume there are many other such papers, the myriad of unpublished papers, private communications and unwritten conference presentations.&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2019-12-26T21:23:26.510" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11278" PostTypeId="2" ParentId="11274" CreationDate="2019-12-27T02:54:25.320" Score="17" Body="&lt;p&gt;Archimedes' books, &lt;a href=&quot;https://en.wikipedia.org/wiki/Stomachion&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Stomachion&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Method of Mechanical Theorems&lt;/em&gt;&lt;/a&gt; were lost until &lt;a href=&quot;https://news.stanford.edu/news/2006/august9/arch-080906.html&quot; rel=&quot;noreferrer&quot;&gt;rediscovered in 2006&lt;/a&gt;.  The only known copy is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes_Palimpsest#The_Method_of_Mechanical_Theorems&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Archimedes Palimpsest&lt;/em&gt;&lt;/a&gt;.  These two texts comprise many theorems.  The Method describes Archimedes' very early use of Riemann sums to compute areas and a variation of Dedekind cuts (via a pair, one a strictly monotonically increasing sequence of rationals and the other strictly monotonically decreasing, where the difference between sequence member of the same index decreases below any specified bound) to identify real numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A few other books of Archimedes are still &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes_of_Syracuse#Writings&quot; rel=&quot;noreferrer&quot;&gt;lost&lt;/a&gt;, titles known only through indices and reference by other authors: &lt;em&gt;Catoptrica&lt;/em&gt;, a work on optics; Principles, explaining the number system used in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Sand_Reckoner&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Sand Reckoner&lt;/em&gt;&lt;/a&gt;; &lt;em&gt;On Balances and Levers&lt;/em&gt;; &lt;em&gt;On Centers of Gravity&lt;/em&gt;; and &lt;em&gt;On the Calendar&lt;/em&gt;.  We do not know how many of these theorems we do not know.&lt;/p&gt;&#xA;" OwnerUserId="10865" LastEditorUserId="10865" LastEditDate="2019-12-27T21:01:02.483" LastActivityDate="2019-12-27T21:01:02.483" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="11279" PostTypeId="2" ParentId="11276" CreationDate="2019-12-27T05:25:31.583" Score="12" Body="&lt;p&gt;For a long time it was not only believed but even ascertained that electric signals moved not just as fast but &lt;em&gt;faster&lt;/em&gt; than light, even &quot;instantaneously&quot;. The original experiments involving electrostatic discharges of the Leyden jar were made even before wires were introduced. According to &lt;a href=&quot;https://archive.org/details/cu31924031221249/page/n10&quot; rel=&quot;noreferrer&quot;&gt;Fahie's History of Electric Telegraphy&lt;/a&gt;, one of the early experimenters, Winkler &quot;&lt;em&gt;in 1744, ascertained that the rapidity of an electric discharge was exceedingly great and comparable with the speed of lightning&lt;/em&gt;&quot;. He used a battery of three jars connected by an insulated wire, laid along the bank of the river Pleisse, whose waters closed the circuit. Lemonnier made another demonstration:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In April 1746, in the court of the Carthusians, he so laid out two parallel wires of 5700 feet each, that all four ends were close together. Between one pair he placed a jar, and grasped the other extremities himself; then on causing the circuit to be completed, he &#xA;  could not distinguish any interval (so short was it) between the spark at the jar, and the shock through his arms.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1746 Nollet performed experiments on the propagation speed with 200 monks hand in hand, forming a circle of about 1.6 km and concluded that the speed of electricity was very high but finite, see &lt;a href=&quot;https://ieeexplore.ieee.org/document/7386799&quot; rel=&quot;noreferrer&quot;&gt;Guarnieri, The Rise of Light&lt;/a&gt;. But 1747 grand scale systematic experiments conducted by a committee of the Royal Society under Watson including Folkes, Cavendish and Bevis, among others, led to a different conclusion: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;...On the 14th August at Shooter's Hill, an &#xA;  experiment was made &quot;to try whether the electric &#xA;  shock was perceptible at twice the distance to which it &#xA;  had yet been carried, in ground perfectly dry, and &#xA;  where no water was near; and also to distinguish if &#xA;  possible its velocity as compared with that of sound.&quot; &#xA;  The circuit consisted of two miles of wire, and two &#xA;  miles of perfectly dry ground, but one shower of rain &#xA;  having fallen in the previous five weeks. The wire &#xA;  from the inner coating of the jar was 6732 feet long, &#xA;  and was supported all the way upon baked sticks, and &#xA;  that which communicated with the outer coating was &#xA;  similarly insulated, and was 3.68 feet long.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The observers placed at the ends of these wires, two miles &#xA;  apart, were provided with stop watches with which to &#xA;  note the moment that they felt the shock. The result &#xA;  of a series of careful observations was that &quot;as far as &#xA;  could be distinguished the time in which the electric &#xA;  matter performed its circuit might have been instantaneous&quot;&lt;/em&gt;&quot;. [quoted from Fahie]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It should be noted that at the time gravity was also believed to act instantaneously, to Newton's chagrin, and certainly faster than light. In Celestial Mechanics (1799) Laplace introduced velocity dependence into the gravity law, and showed that the planets would quickly fly off of their orbits unless gravity was millions of times faster than light, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2395/55&quot;&gt;What 19th century developments contributed to the General theory of Relativity?&lt;/a&gt; As late as 1837, the instantaneous propagation of signals in the wire was still asserted in Alexander's telegraph proposal, based on Ampere's and Ritchie's ideas, and published in multiple Edinburgh and London journals:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It has been found by experiments made with &#xA;  a view to ascertaining the velocity of electricity, that &#xA;  it is transmitted instantaneously, by means of a &#xA;  common iron wire, a distance of eight miles; and &#xA;  electricians of the first eminence have declared their &#xA;  opinion that, judging from all scientific experience, &#xA;  the electric or galvanic influence would be almost &#xA;  instantaneously transmitted from one end to the &#xA;  other of a metallic conductor, such as ordinary &#xA;  copper wire of moderate thickness, of some hundred &#xA;  miles in length.&lt;/em&gt;&quot; [quoted from Fahie]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to Guarnieri, in 1854 Thomson (later Lord Kelvin), while laying the transatlantic telegraph cables, derived the first version of the &lt;a href=&quot;https://mysite.du.edu/~jcalvert/tech/cable.htm&quot; rel=&quot;noreferrer&quot;&gt;&quot;telegrapher's equation&quot;&lt;/a&gt; (second-order PDE) for the electromagnetic propagation in a telegraph line. But he ignored inductance and&#xA;treated telegraphy as a diffusion rather than wave propagation. Hence the signal speed was inverse proportional to the square length of the cable. In 1857 Kirchhoff (1824–1887) derived  a wave equation for the current and charge in a long line, taking into account self-induction effects, and computed that the speed was equal to the speed of light. The modern derivation of the telegrapher's equation from Kirchhoff’s circuit laws was first given by Heaviside in 1876. In 1886 he introduced the term impedance, and in 1887 developed the modern model of a “distortionless” transmission line, see historical note in &lt;a href=&quot;http://www.hep.princeton.edu/~mcdonald/examples/distortionless.pdf&quot; rel=&quot;noreferrer&quot;&gt;McDonald's Distortionless Transmission Line&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Back in 1855 (published 1856) Weber and Kohlrausch noted that &quot;&lt;em&gt;the ratio of the absolute electromagnetic unit of charge to the absolute electrostatic unit of charge&lt;/em&gt;&quot;, in modern notation &lt;span class=&quot;math-container&quot;&gt;$\frac1{\sqrt{μ_0ε_0}}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$μ_0,ε_0$&lt;/span&gt; are the magnetic permeability and electric permittivity of the vacuum, respectively, had the units of velocity, and determined experimentally that it was remarkably close to the speed of light, see references in &lt;a href=&quot;https://hsm.stackexchange.com/a/5525/55&quot;&gt;Electromagnetic constants and the speed of light&lt;/a&gt;. Towards the end of 1861 Maxwell derived the general formula &lt;span class=&quot;math-container&quot;&gt;$v=\frac1{\sqrt{με}}$&lt;/span&gt; for the electromagnetic propagation speed in a medium in part III of his paper On Physical Lines of Force, and suggested, in particular, that light was a form of electromagnetic radiation, see &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations#Relationships_among_electricity,_magnetism,_and_the_speed_of_light&quot; rel=&quot;noreferrer&quot;&gt;History of Maxwell's equations&lt;/a&gt;. It also resolved the issue of the speed of electromagnetic propagation in a medium. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And, as it turned out, Laplace's estimates were based on a wrong sort of velocity dependence (in modern terms, non-Lorentz invariant). Electromagnetic theories of gravity, implying its propagation at the speed of light, were offered at the end of 19th century, notably by Lorentz himself. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-02-10T08:39:16.113" LastActivityDate="2020-02-10T08:39:16.113" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11280" PostTypeId="1" AcceptedAnswerId="11282" CreationDate="2019-12-27T21:11:31.450" Score="4" ViewCount="175" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Orthogonal_matrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Orthogonal_matrix&lt;/a&gt;,&lt;br&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Unitary_matrix&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Unitary_matrix&lt;/a&gt;, and &#xA;Friedberg et al.'s &lt;em&gt;Linear Algebra&lt;/em&gt; (4th edition), a matrix &lt;span class=&quot;math-container&quot;&gt;$A\in F^{n\times n}$&lt;/span&gt; &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;is &lt;em&gt;unitary&lt;/em&gt; if &lt;span class=&quot;math-container&quot;&gt;$A^*=A^{-1}$&lt;/span&gt; and&lt;/li&gt;&#xA;&lt;li&gt;is &lt;em&gt;orthogonal&lt;/em&gt; if &lt;span class=&quot;math-container&quot;&gt;$A^T=A^{-1}$&lt;/span&gt; &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;no matter whether &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&#xA;Therefore, if &lt;span class=&quot;math-container&quot;&gt;$F=\mathbb{R}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is unitary if and only if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is orthogonal (since &lt;span class=&quot;math-container&quot;&gt;$A^* = A^t$&lt;/span&gt;). However, if &lt;span class=&quot;math-container&quot;&gt;$F=\mathbb{C}$&lt;/span&gt;, then a unitary matrix (e.g., the matrix consisting of a single entry &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;) is not necessarily orthogonal.&#xA;I have seen a lot of unitary (i.e., orthogonal) real matrices and unitary complex matrices. But, I cannot find anything related to orthogonal complex matrices. Have orthogonal complex matrices appeared or been studied in the literature? If they have, can someone please provide a reference?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, if almost all orthogonal matrices in the literature are unitary, then  are there any reasons why the definition of orthogonal matrix stays the current way? Wouldn't things be easier to define orthgonal matrix as unitary real matrix, just like orthogonal linear operator is defined as unitary linear operator on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Orthogonal_transformation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Orthogonal_transformation&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Unitary_transformation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Unitary_transformation&lt;/a&gt;, and Friedberg et al.'s book)?&lt;/p&gt;&#xA;" OwnerUserId="7937" LastEditorUserId="26" LastEditDate="2021-02-20T15:21:05.597" LastActivityDate="2021-02-20T15:21:05.597" Title="Have orthogonal complex matrices appeared in the literature?" Tags="|mathematics|terminology|reference-request|linear-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11282" PostTypeId="2" ParentId="11280" CreationDate="2019-12-28T10:36:17.370" Score="4" Body="&lt;p&gt;They did. A natural way to treat such matrices is to introduce an indefinite inner product on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^n$&lt;/span&gt;, a non-degenerate bilinear form &lt;span class=&quot;math-container&quot;&gt;$(z,w):=z_1w_1+z_2w_2+\dots z_nw_n$&lt;/span&gt;, instead of the usual sesquilinear one. Then &lt;span class=&quot;math-container&quot;&gt;$A^T=A^{-1}$&lt;/span&gt; is equivalent to &lt;span class=&quot;math-container&quot;&gt;$(Az,Aw)=(z,w)$&lt;/span&gt;, i.e. complex orthogonal matrices are isometries of this space. The &quot;orthogonal&quot;/&quot;unitary&quot; split in terminology is useful in signaling whether real or complex spaces are involved, exactly because complex orthogonal matrices are rarely considered.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;General bilinear forms are not as technically handy as sign definite sesquilinear forms, so they are less studied. On complex orthogonal matrices specifically the volume of publications is fairly small. In 1982 Choudhury defended a thesis, The Role of Complex Orthogonal Matrices in Matrix Factorization. Some of the results are published in a paper with Horn, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/03081088708817787&quot; rel=&quot;nofollow noreferrer&quot;&gt;An analog of the singular value decomposition for complex orthogonal equivalence (1987)&lt;/a&gt;, see also &lt;a href=&quot;https://core.ac.uk/download/pdf/82769111.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Horn-Merino, The Jordan Canonical Forms of complex&#xA;orthogonal and skew-symmetric matrices (1999)&lt;/a&gt;. The most recent entry in MathSciNet is &lt;a href=&quot;https://projecteuclid.org/euclid.jgsp/1499306418&quot; rel=&quot;nofollow noreferrer&quot;&gt;Celakoska-Hadzieva, On complex homogeneous space of vectors with constraints (2017)&lt;/a&gt;. Occasionally they are asked about on Math SE, e.g. &lt;a href=&quot;https://math.stackexchange.com/q/1658707/152568&quot;&gt;Is there an orthogonal matrix that is not unitary?&lt;/a&gt; and &lt;a href=&quot;https://math.stackexchange.com/q/2290362/152568&quot;&gt;On eigenvalues of complex-orthogonal matrices&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, some &lt;a href=&quot;https://en.wikipedia.org/wiki/Indefinite_inner_product_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;linear spaces with indefinite inner products&lt;/a&gt; (not of the sort introduced at the top), including infinite-dimensional ones, have been studied in functional analysis since 1940-s. Pontryagin's paper, Hermitian operators in spaces with indefinite metric, is from 1944. After Krein's work in 1950-s a subclass of them (including those considered by Pontryagin) got the name Krein spaces, those admit direct decomposition into two subspaces on which the form is sign definite. Early work of Phillips, Krein and Naimark on operators in such spaces is discussed e.g. in &lt;a href=&quot;http://www.math.ucsd.edu/~helton/BILLSPAPERSscanned/H70a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helton, Unitary &#xA;Operators on a Space with an Indefinite Inner Product&lt;/a&gt;, a book treatment is &lt;a href=&quot;https://books.google.com/books/about/Linear_operators_in_spaces_with_an_indef.html?id=uVmqAAAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Azizov-Iokhvidov, Linear operators in spaces with an indefinite metric&lt;/a&gt;. From 1973 Mathematics Subject Classification includes 47B50: Operators on spaces with an indefinite metric.  &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2019-12-28T11:59:15.410" LastActivityDate="2019-12-28T11:59:15.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11284" PostTypeId="2" ParentId="11280" CreationDate="2019-12-28T16:10:35.117" Score="5" Body="&lt;p&gt;Yes, orthogonal matrices with complex entries appeared at least as early as 1900, in E. Cartan's classification of simple Lie algebras (and Lie groups). In many ways, the complex numbers could be replaced by any algebraically closed field of characteristic &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thinking of complex orthogonal groups as real Lie groups (&quot;forgetting&quot; the complex structure) does produce a family of semi-simple real Lie groups, about which we have many structure theorems (Iwasawa decomposition, Cartan decomposition, Bruhat decomposition). The representation theory of such groups, both finite-dimensional and even more so infinite-dimensional, is of interest both in its own right, and in applications to automorphic forms and number theory (e.g., Langlands programme and such).&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2019-12-28T16:10:35.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11285" PostTypeId="2" ParentId="5763" CreationDate="2019-12-28T19:30:28.850" Score="2" Body="&lt;p&gt;For a discussion of Cauchy's contribution, see pages 92-131 of T. Muir, &lt;em&gt;The Theory of Determinants in the Historical Order of Development&lt;/em&gt;, vol. 1, second edition, MacMillan, London, 1906.&lt;/p&gt;&#xA;" OwnerUserId="10879" LastActivityDate="2019-12-28T19:30:28.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11286" PostTypeId="1" AcceptedAnswerId="11292" CreationDate="2019-12-29T01:22:22.423" Score="4" ViewCount="241" Body="&lt;p&gt;Outstanding French mathematician had also his Brasilian chapter in his life (or two chapters?). I'd like to know more.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The en.wikipedia tells us that Yoccoz was born, raised and educated in France,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;Yoccoz attended the Lycée Louis-le-Grand,[3] during which time he was a silver medalist at the 1973 International Mathematical Olympiad and a gold medalist in 1974.[4][5] He entered the École Normale Supérieure in 1975, and completed an agrégation in mathematics in 1977.&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You'd think that he would immerse himself in mathematics totally. However, suddenly, he had completed &lt;em&gt;military service in Brazil&lt;/em&gt;. How come?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;He joined the French Academy of Sciences and Brazilian Academy of Sciences in 1994, became a chevalier in the French Legion of Honor in 1995, and was awarded the Grand Cross of the Brazilian National Order of Scientific Merit in 1998.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Well, by this time, Yoccoz had so many mathematical and formal achievements that it is no wonder that he was celebrated also by Brasil (in view of his military service there).&lt;/p&gt;&#xA;" OwnerUserId="10855" LastEditorUserId="10855" LastEditDate="2019-12-30T01:38:22.350" LastActivityDate="2019-12-30T16:59:25.630" Title="Jean-Christophe Yoccoz (May 29, 1957 – September 3, 2016)" Tags="|biographical-details|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11287" PostTypeId="2" ParentId="11274" CreationDate="2019-12-29T05:08:49.750" Score="5" Body="&lt;p&gt;&lt;strong&gt;I'd be willing to bet that quite a number of theorems were lost simply due to the Library of Alexandria getting destroyed.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://history.stackexchange.com/questions/677/what-knowledge-may-have-been-lost-at-the-library-of-alexandria&quot;&gt;Take a look at this History Stack Exchange question and some of the answers.&lt;/a&gt;  Particularly where it mentions that all mathematics before Euclid was lost, and that the mathematics of that age wasn't really surpassed until the 19th century.  Given that level of loss and backtracking, it's pretty safe to say that there were some theories that ancient greek mathematic scholars would have received well, but were lost over the years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(The post in question is referencing Lucio Russo's 'Forgotten Revolution', which covers the knowledge of the Hellenistic world that was lost through history; the author is both a historian and a mathematician, which is pretty on-point for the question.)&lt;/p&gt;&#xA;" OwnerUserId="10881" LastActivityDate="2019-12-29T05:08:49.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11288" PostTypeId="1" AcceptedAnswerId="11290" CreationDate="2019-12-29T12:40:12.717" Score="6" ViewCount="3528" Body="&lt;p&gt;We have &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\aleph_0$&lt;/span&gt; borrowed from Greek and Hebrew alphabets. Are there widely accepted math symbols using non-Latin alphabets or characters other than Greek and Hebrew?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A related question on Math.SE: &lt;a href=&quot;https://math.stackexchange.com/questions/489136/cyrillic-alphabet-in-math&quot;&gt;Cyrillic alphabet in math&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Wikipedia's list of letters used in math and science math symbols&#xA;contains &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_letters_used_in_mathematics_and_science#More&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list of non-Latin/Greek letters&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7937" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:57.763" LastActivityDate="2021-02-20T15:20:57.763" Title="Are there widely accepted math symbols using non-Latin alphabets or characters other than Greek and Hebrew?" Tags="|mathematics|terminology|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11289" PostTypeId="2" ParentId="11288" CreationDate="2019-12-29T13:22:06.713" Score="11" Body="&lt;p&gt;There are several non-alphabetic symbols, the best known is the integral sign &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;&#xA;and the Weierstrass &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;-function &lt;span class=&quot;math-container&quot;&gt;$\wp$&lt;/span&gt;. To be sure their origins are letters of Latin alphabet, but they are special stylized symbols, and as far as I know there is no computer code for them in the standard sets of computer characters. Strictly speaking they do not belong to any alphabet. &lt;span class=&quot;math-container&quot;&gt;$\wp$&lt;/span&gt; imitates Weierstrass handwriting. Symbols &lt;span class=&quot;math-container&quot;&gt;$\partial$&lt;/span&gt; for the partial derivative and &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; also belong to this list.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are also mathematicians who use Cyrillic letters, for example John Milnor used Л&#xA;for some standard function is hyperbolic geometry, but this can hardly be called&#xA;&quot;widely accepted&quot;. Л is the first letter in Lobachevski's name (Лобачевский).&#xA;I've seen other Cyrillic letters used, but again this is not &quot;widely accepted&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An interesting case is the &quot;letter&quot; &lt;span class=&quot;math-container&quot;&gt;$\nabla$&lt;/span&gt; usually used for the gradient. This is a apparently not a letter from any alphabet, and it is called &quot;Nabla&quot; which is the Greek word for some Phoenician musical instrument. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Nabla_symbol&quot; rel=&quot;noreferrer&quot;&gt;Nabla symbol&lt;/a&gt;. Unlike &lt;span class=&quot;math-container&quot;&gt;$\wp$&lt;/span&gt; it is included in Unicode.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2019-12-29T13:50:25.010" LastActivityDate="2019-12-29T13:50:25.010" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11290" PostTypeId="2" ParentId="11288" CreationDate="2019-12-29T13:43:29.963" Score="17" Body="&lt;p&gt;The letter Ш (sha) of the Cyrillic alphabet is widely accepted in theoretical computer science as the symbol for the &lt;em&gt;sh&lt;/em&gt;uffle product, which gives the &lt;a href=&quot;https://en.wikipedia.org/wiki/Shuffle_algebra#Shuffle_product&quot; rel=&quot;noreferrer&quot;&gt;shuffle algebra&lt;/a&gt;.&#xA;The same letter is also used to denote the Tate-Shafarevich group, but I'm not sure if it's really a standard (the letter was introduced by Cassels &lt;del&gt;only in 1990&lt;/del&gt; in 1962 instead of TS, see below the KCd's comment).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sometimes, see for example &lt;a href=&quot;https://arxiv.org/pdf/1506.08870.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, the Hiragana symbol よ is used to denote the &lt;a href=&quot;https://en.wikipedia.org/wiki/Yoneda_lemma#The_Yoneda_embedding&quot; rel=&quot;noreferrer&quot;&gt;Yoneda embedding&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2019-12-30T20:43:37.217" LastActivityDate="2019-12-30T20:43:37.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11291" PostTypeId="2" ParentId="11288" CreationDate="2019-12-29T14:33:59.367" Score="11" Body="&lt;p&gt;It is sometimes asserted that &lt;span class=&quot;math-container&quot;&gt;$\varnothing$&lt;/span&gt; for the empty set was introduced by Bourbaki using a Danish and Norwegian letter.&lt;/p&gt;&#xA;&lt;p&gt;EDIT:&lt;/p&gt;&#xA;&lt;p&gt;The source is the Weil autobiography, cited &lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;in Jeff Miller's collection&lt;/a&gt; of the origins of mathematical expressions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;André Weil (1906-1998) says in his autobiography that he was responsible for the symbol:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wisely, we had decided to publish an installment establishing the system of notation for set theory, rather than wait for the detailed treatment that was to follow: it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval. Much later, my own part in these discussions earned me the respect of my daughter Nicolette, when she learned the symbol Ø for the empty set at school and I told her that I had been personally responsible for its adoption. The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The citation above is from page 114 of André Weil's The Apprenticeship of a Mathematician, Birkhaeuser Verlag, Basel-Boston-Berlin, 1992.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2019-12-31T13:55:47.457" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11292" PostTypeId="2" ParentId="11286" CreationDate="2019-12-30T16:59:25.630" Score="5" Body="&lt;p&gt;At that time, France had compulsory military service for all males, but they could replace it by work in a school overseas.The French version of the Wikipedia article on Yoccoz specifies:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Pendant cette période, au début des années 1980, il effectue son&#xA;  service national en coopération à l’Instituto de Matematica Pura e&#xA;  Aplicada de Rio de Janeiro&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;That is: He did his National Service &quot;in cooperation with&quot; the Instituto de Matematica Pura e Aplicada in Rio de Janeiro.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2019-12-30T16:59:25.630" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11293" PostTypeId="2" ParentId="11288" CreationDate="2019-12-31T00:07:17.380" Score="5" Body="&lt;p&gt;The cyrillic letter Ш (sha)is -- for obvious reasons when looking at the graph) also used to denote the &quot;function&quot; (well, it is a distribution if you want to be picky) given by the sum of integral displacements of the Dirac-delta function, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirac_comb&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Dirac_comb&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10893" LastActivityDate="2019-12-31T00:07:17.380" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11294" PostTypeId="1" CreationDate="2019-12-31T11:29:43.373" Score="5" ViewCount="1275" Body="&lt;p&gt;Carl Friedrich Gauss said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;When a philosopher says something that is true then it is trivial. When he says something that is not trivial then it is false.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On one occasion I read that supposedly Gauss has added in parentheses that Aristotle was an exception. Do you know more about it?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If any, I'd be sure that Gauss rather said that &quot;&lt;em&gt;especially Aristotle&lt;/em&gt;&quot;, i.e. that &quot;&lt;em&gt;Aristotle was &lt;strong&gt;except&lt;/strong&gt;ionally bad&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;QUESTION&lt;/em&gt;&lt;/strong&gt; &amp;nbsp; what Gauss really said?&lt;/p&gt;&#xA;" OwnerUserId="10855" LastEditorUserId="10855" LastEditDate="2019-12-31T19:24:12.087" LastActivityDate="2019-12-31T19:24:12.087" Title="Gauss on philosophers" Tags="|biographical-details|philosophy-of-science|quote|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11295" PostTypeId="1" AcceptedAnswerId="11297" CreationDate="2019-12-31T12:16:11.880" Score="40" ViewCount="5770" Body="&lt;p&gt;Why is the volt not identical to the full name Volta, unlike the other electrical units ohm, ampere, coulomb, tesla, weber and henry? Is there a historical explanation, was the volt introduced at a different time?&lt;/p&gt;&#xA;" OwnerUserId="10894" LastEditorUserId="595" LastEditDate="2020-01-04T23:35:12.467" LastActivityDate="2020-01-04T23:35:12.467" Title="Why volt instead of volta?" Tags="|electricity|units|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11296" PostTypeId="2" ParentId="11294" CreationDate="2019-12-31T13:00:00.793" Score="10" Body="&lt;p&gt;The quote is not accurate but Gauss actually wrote something similar to Schumacher in the letter of 1 November 1844 cited &lt;a href=&quot;https://books.google.it/books?id=rykSDAAAQBAJ&amp;amp;pg=PA293&amp;amp;lpg=PA293&amp;amp;dq=gauss%20Schumacher%20letter%201%20November%201844&amp;amp;source=bl&amp;amp;ots=3hjc9HrHYD&amp;amp;sig=ACfU3U1v-V1gIWLTHg-2qwJXKSJ9mcLSZA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiy9pjp99_mAhUI56QKHVPGC8cQ6AEwAnoECAUQAg#v=onepage&amp;amp;q=gauss%20Schumacher%20letter%201%20November%201844&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, where he complains about concepts and definitions given in math books by philosopher that are not mathematicians, namely&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] look around at modern philosophers [...] don't their definitions make your hair stand on end? Read in the history of ancient philosophy what the men of the days, Plato and others (I except Aristotle), gave as explanation.&#xA;  And even in Kant matters are often not much better; his distinction between analytic and synthetic propositions seems to me either a triviality or false. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So Gauss considers Aristotle to be the exception between ancient (and modern) philosophers in the sense that his definitions are free of confusion. &lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2019-12-31T16:00:15.443" LastActivityDate="2019-12-31T16:00:15.443" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11297" PostTypeId="2" ParentId="11295" CreationDate="2019-12-31T14:04:02.130" Score="60" Body="&lt;p&gt;The volt, ohm and farad were introduced by the same person, Latimer Clark, a cable engineer, in a &lt;a href=&quot;https://books.google.nl/books?id=VaEEAAAAQAAJ&amp;amp;pg=PA131&amp;amp;dq=ohma%20galvat&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwiQoeai56TSAhXEVRQKHaZOAvsQ6AEIHzAB#v=onepage&amp;amp;q=ohma%20galvat&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;paper in 1861&lt;/a&gt;. He started the tradition of naming units after scientists. He initially distorted all names: the unit names ohma, volt, galvat (from Galvani), and farad (from Faraday). In his words, he &quot;neglected etymological rules&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that same year, a committee of the British Association for the Advancement of Science began developing a coherent system of electrical units, the cgs system. Varley, a committee member, liked Clark's proposal, but he advocated adding a French name such as Ampere, to facilitate the international acceptance of the new units (&quot;I should like to introduce a French name into the list. We have Germany, England and Italy represented&quot;). For the same reason he may have pleaded for not distorting the French and German names. In addition, Varley objected to the galvat &quot;because Galvani discovered next to nothing&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Latimer Clark adapted the unit names in his 'Elementary treatise on electrical measurement' (1868). He changed ohma to ohm, and dropped the galvat. Maxwell acknowledged Clark's practical units and their names in his 'Treatise on electricity and magnetism' (1873). In 1881, at the International Electrical Congress in Paris), two French names, ampere and coulomb, were added. The &lt;a href=&quot;https://web.archive.org/web/20160909183227/http://seaus.free.fr/spip.php?article964&quot; rel=&quot;noreferrer&quot;&gt;diplomacy&lt;/a&gt; worked, and the new system of electrical units was adopted by all countries. The only units that kept the initial name distortion were the volt and the farad.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastActivityDate="2019-12-31T14:04:02.130" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11298" PostTypeId="1" AcceptedAnswerId="11302" CreationDate="2020-01-01T15:39:02.060" Score="7" ViewCount="708" Body="&lt;p&gt;&lt;a href=&quot;https://astronomy.stackexchange.com/a/34556/7982&quot;&gt;This answer&lt;/a&gt; to &lt;em&gt;Has great eyesight been necessary for astronomers?&lt;/em&gt; mentions Astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Jocelyn_Bell_Burnell&quot; rel=&quot;noreferrer&quot;&gt;Jocelyn Bell Burnell&lt;/a&gt;'s recounting of a likely first &lt;em&gt;visual observation&lt;/em&gt; of a pulsar. This can be found for example in Nature's &lt;a href=&quot;https://www.nature.com/articles/448974a&quot; rel=&quot;noreferrer&quot;&gt;Air force had early warning of pulsars&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The work preceded by several months the observations made by Bell Burnell, then at the University of Cambridge, UK, which led to the first paper on the subject. A Nobel prize for the discovery was subsequently awarded to her supervisor Antony Hewish, but, controversially, not to her. Schisler was not the only one to “pre-discover” a pulsar, though, according to Bell Burnell. &lt;strong&gt;“There are actually a lot of stories,” she says. In the 1950s, a woman visiting the observatory at the University of Chicago, Illinois, pointed out that there was a regularly pulsating source of visible light in the Crab Nebula. Elliot Moore, an astronomer at the university, dismissed the woman's claim, telling her that all stars seem to flicker.&lt;/strong&gt; Another radio astronomer she knows of will, after a drink or two, confess to having dismissed observations of a pulsating source as the result of faulty equipment. “He's a bit embarrassed now,” says Bell Burnell.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is also summarized in Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Crab_Pulsar&quot; rel=&quot;noreferrer&quot;&gt;Crab Pulsar&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jocelyn Bell Burnell, who co-discovered the first pulsar PSR B1919+21 in 1967, relates that in the late 1950s a woman viewed the Crab Nebula source at the University of Chicago's telescope, then open to the public, and noted that it appeared to be flashing. The astronomer she spoke to, Elliot Moore, disregarded the effect as scintillation, despite the woman's protestation that as a qualified pilot she understood scintillation and this was something else. Bell Burnell notes that &lt;strong&gt;the 30 Hz frequency of the Crab Nebula optical pulsar is difficult for many people to see.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there any information on who it might have been who noticed the visibly pulsing light at this observatory in Chicago, or which telescope it was and roughly the aperture size? &lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="55" LastEditDate="2020-01-01T23:54:37.260" LastActivityDate="2021-02-24T03:49:50.847" Title="Who first reported the Crab pulsar's pulsing but was dismissed by an astronomer?" Tags="|discoveries|astronomy|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11299" PostTypeId="1" AcceptedAnswerId="11301" CreationDate="2020-01-01T17:49:07.103" Score="3" ViewCount="485" Body="&lt;p&gt;We have a lot of things around us, that are named after the principle guys behind it. Is the naming done by the respective person or someone else (like IUPAC for physics)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Example 1: We have Ampere's Circuital Law. Did Ampere name it that way or someone else gave that name?&#xA;Example 2: Plank's Constant. Did Plank publish his paper with his name for the constant?&#xA;Example 3: Nicol's prism. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If the naming is because of someone else:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What will happen if a scientist publishes a paper with his name for something?&lt;/li&gt;&#xA;&lt;li&gt;How do scientists refer a work named after them (read the example)?&#xA;Example : Say Maxwell published the equations and now they got the name &quot;Maxwell's equations&quot;. Now Maxwell wants to publish something new based on the equations. Will he write &quot;As per Maxwell's equation...&quot; or &quot;As per my equations...&quot; ?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Note : There are questions already asked here which sound similar, like &lt;a href=&quot;https://hsm.stackexchange.com/questions/3027/when-are-units-that-are-named-after-persons-given-their-names&quot;&gt;When are units that are named after persons given their names?&lt;/a&gt; but they ask &quot;what&quot; and not &quot;how&quot;.&lt;/p&gt;&#xA;" OwnerUserId="10904" LastActivityDate="2020-01-01T20:51:27.710" Title="Do scientists and mathematicians name their discovery or inventions themselves?" Tags="|naming-conventions|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11300" PostTypeId="2" ParentId="753" CreationDate="2020-01-01T18:58:16.310" Score="2" Body="&lt;p&gt;It is not really a coincidence that the different proposals had similar lengths, because the metre (as it is spelled in French, and in British English) was created in a world which already contained many units of measurement. These varied in two ways:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Based on where they were used, and who decreed their correct size; for instance, an &quot;inch&quot; in one country would not be the same as an &quot;inch&quot; in another, as &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Inch_converter.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 19th-century conversion chart&lt;/a&gt; demonstrates.&lt;/li&gt;&#xA;&lt;li&gt;Based on what they were used for; for instance, a length of string might be measured in inches, cloth in yards, but a farm in furlongs. These different measures were originally independent, not related by the exact multiples that have been standardised today.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The creators of the metre were aiming to come up with a measure that had some scientific basis (rather than the decree of a monarch), but which would be useful for the same purposes as existing measures.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest unit in France at the time was apparently the &lt;a href=&quot;https://en.wikipedia.org/wiki/Toise&quot; rel=&quot;nofollow noreferrer&quot;&gt;toise&lt;/a&gt;, which was equivalent to about 1.9 metres, slightly longer than an English &lt;a href=&quot;https://en.wikipedia.org/wiki/Fathom&quot; rel=&quot;nofollow noreferrer&quot;&gt;fathom&lt;/a&gt;; the more common unit in England was the &lt;a href=&quot;https://en.wikipedia.org/wiki/Yard&quot; rel=&quot;nofollow noreferrer&quot;&gt;yard&lt;/a&gt;, roughly half as long. Other historical units of similar length include the &lt;a href=&quot;https://en.wikipedia.org/wiki/Klafter&quot; rel=&quot;nofollow noreferrer&quot;&gt;klafter&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubit&quot; rel=&quot;nofollow noreferrer&quot;&gt;cubit&lt;/a&gt;, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Ell&quot; rel=&quot;nofollow noreferrer&quot;&gt;ell&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;More fundamentally, these units are based on the size of the human body - the span of outstretched arms, a pace forward, and the height of a grown man are all in this range. Those sizes are easy to picture, and useful to proportion things based on.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The two proposed standards were therefore looking for a value somewhere in the range of these existing units, so that it could be used for the same purposes. Two ways to define it were suggested:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The length of a pendulum with a particular period; choosing &lt;a href=&quot;https://en.wikipedia.org/wiki/Seconds_pendulum&quot; rel=&quot;nofollow noreferrer&quot;&gt;a period of two seconds&lt;/a&gt; happens to give a close value to the yard, ell, etc.&lt;/li&gt;&#xA;&lt;li&gt;A fraction of the Earth's circumference; taking one ten-millionth of the distance from the North Pole to the equator gives the right sized unit.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The proposal at the time was to replace all units with decimal systems, &lt;a href=&quot;https://en.wikipedia.org/wiki/Decimal_time&quot; rel=&quot;nofollow noreferrer&quot;&gt;including a new decimal second&lt;/a&gt;, which made the pendulum definition less appealing. The divisor of 10 million fitted better into this decimal system, but required significant research to define.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In practice, the origin of the definition was soon irrelevant, as neither is reproducible with sufficient accuracy - the Earth is hard to measure, and irregularly shaped; and a pendulum swings slightly differently in different locations. So the actual definition of the metre, as with most previous units, was a specimen against which other measuring devices could be checked.&lt;/p&gt;&#xA;" OwnerUserId="10905" LastEditorUserId="10905" LastEditDate="2020-01-02T10:12:53.940" LastActivityDate="2020-01-02T10:12:53.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11301" PostTypeId="2" ParentId="11299" CreationDate="2020-01-01T20:34:57.463" Score="7" Body="&lt;p&gt;Usually not, though there are exceptions. Names in honor of X are usually given by others than X herself. And it cannot be otherwise: the name of an object must be accepted by the research community. It is considered immodest to suggest your own name yourself. Many various naming patterns, with examples, are analysed in my answer and other answers to this similar MathOverflow question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/285627/how-are-constants-functions-named-after-their-discoverer/285644#285644&quot;&gt;https://mathoverflow.net/questions/285627/how-are-constants-functions-named-after-their-discoverer/285644#285644&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Same applies to other sciences. None of the objects you listed was named by the inventor himself. Moreover, most mathematicians and scientists are shy to use their own name as a name of an object, even when it is established.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Examples: Hilbert used to ask &quot;What is a Hilbert space?&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sturm, once said in a lecture: &quot;And now I am going to explain the theorem whose name I have an honor to bear...&quot;  &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-01-01T20:51:27.710" LastActivityDate="2020-01-01T20:51:27.710" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11302" PostTypeId="2" ParentId="11298" CreationDate="2020-01-01T23:52:46.597" Score="5" Body="&lt;p&gt;Bernell recounts the stories herself in &lt;a href=&quot;https://pos.sissa.it/099/014/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reflections on the Discovery of Pulsars&lt;/a&gt;. She gives a bit more detail, in particular that the telescope was 82 in, but does not name the woman. She also mentions Sue Simkin who noticed pulsations of the Minkowski star in 1965, also before Schisler:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The first dates from summer 1957 when Elliott Moore helped out at a public &#xA;  observing night at the 82 inch McDonald Observatory telescope (now known as the &#xA;  Otto Struve &#xA;  telescope). For the final observation of the night the telescope was set on &#xA;  the inner part of the Crab Nebula. One member of the public was a young woman &#xA;  probably in her late twenties; she was familiar with the night sky because her job was &#xA;  flying planes a&#xA;  nd in the dark cockpit there was little else to do except watch the stars. &#xA;  After her turn at the telescope she reported that one of the stars in the Crab nebula was &#xA;  blinking rapidly, and described which one. Elliott Moore looked but could not see the &#xA;  effect. He suggested it was scintillation, but later realised that with such a large aperture &#xA;  telescope it couldn’t be.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The second dates from December 1965. Sue Simkin was taking UV spectra with &#xA;  the Carnegie Image Spectrograph at the Kitt Peak 84 inch telescope.  She was asked by Lo Woltjer to take a UV spectrum of Minkowski’s star (now known to be the pulsar). She said the spectrum was dull, but as she took it she observed flickering, or as if there &#xA;  were waves going out from it. Lo maintained there couldn’t be, but after the discovery &#xA;  of the pulsar agreed that must have been what she had seen. [The Crab pulsar, with a period of 30Hz, is sufficiently fast that many people cannot see it; however some people can see 30Hz and Sue Simkin knows she is one of them – as a child she lived in an area &#xA;  where the mains power supply was at 30Hz.]&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-02T01:04:33.383" LastActivityDate="2020-01-02T01:04:33.383" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11304" PostTypeId="2" ParentId="11267" CreationDate="2020-01-02T14:25:22.060" Score="4" Body="&lt;p&gt;The sun, and its daily and yearly cycles, is one of the most important aspects of life on Earth, so it features prominently in many traditions, &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_solar_deities&quot; rel=&quot;nofollow noreferrer&quot;&gt;often in the form of a &quot;solar deity&quot;&lt;/a&gt;. Within these myths, there are &lt;a href=&quot;http://www.mythencyclopedia.com/Sp-Tl/Sun.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;many different mythological explanations for sunset and sunrise&lt;/a&gt; - the sun might die and be reborn, travel out of sight through another land, or simply rest until the morning.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Given this variety of explanations, there is nothing particularly surprising about the idea that the sun turns to face some other land at the end of the day. There is a contrast in the cited passage with some other explanation, but it's not clear what the original writer would have meant by the phrase &quot;when people think the sun is setting&quot; - sunset is, after all, a real phenomenon. Possibly there was an alternative tradition that the sun deity died and was reborn (as did, for instance, the Ancient Egyptian &lt;a href=&quot;https://en.wikipedia.org/wiki/Ra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ra&lt;/a&gt;), and this writer was asserting that it &quot;lived&quot; continuously (as did, for instance, the Ancient Greek &lt;a href=&quot;https://en.wikipedia.org/wiki/Helios&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helios&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nor does the passage quoted actually match our modern scientific understanding in any detail. There is no singular moment where all observers would see the sun &quot;reaching the end of the day&quot;; rather, at every moment, it is sunset somewhere in the world. Nor does the sun &quot;make itself produce two opposite effects&quot;; it continues its same brightness, just obscured by the shadow of the Earth.&lt;/p&gt;&#xA;" OwnerUserId="10905" LastActivityDate="2020-01-02T14:25:22.060" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11305" PostTypeId="1" CreationDate="2020-01-02T17:14:12.280" Score="1" ViewCount="127" Body="&lt;p&gt;We all know Bohr's quantization rule as &lt;span class=&quot;math-container&quot;&gt;$L = \frac{nh}{2\pi}$&lt;/span&gt;. How did Bohr derive this?&lt;/p&gt;&#xA;&lt;p&gt;If it is a guess, what is the reason behind choosing &lt;span class=&quot;math-container&quot;&gt;$\frac{h}{2\pi}$&lt;/span&gt; and not some other constant? Why did he equate it with angular momentum and not anything else?&lt;/p&gt;&#xA;" OwnerUserId="10904" LastEditorUserId="4251" LastEditDate="2020-08-13T07:42:52.330" LastActivityDate="2020-08-13T07:42:52.330" Title="How did Bohr guess the quantization rule?" Tags="|physics|quantum-mechanics|bohr|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11307" PostTypeId="1" AcceptedAnswerId="11310" CreationDate="2020-01-02T20:34:00.763" Score="6" ViewCount="148" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_third_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;this wiki article&lt;/a&gt;, Dehn solved Hilbert's third problem within a year. &#xA;Did Dehn start to work on the third problem after Hilbert's talk? Since Dehn is Hilbert's student, they were likely to know what problems each other had been working on. &#xA;Can it be the case that Dehn already work on the problem for a while before Hilbert honored it to be among the 23 prestigious open problems perhaps to encourage Dehn's efforts on the problem?&lt;/p&gt;&#xA;" OwnerUserId="7937" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:52.170" LastActivityDate="2021-02-20T15:20:52.170" Title="When did Dehn start to work on Hilbert's third problem?" Tags="|mathematicians|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11308" PostTypeId="1" AcceptedAnswerId="11309" CreationDate="2020-01-03T02:37:59.933" Score="39" ViewCount="6361" Body="&lt;p&gt;Why do we use the fist name in &lt;em&gt;Tychonic system&lt;/em&gt; or &lt;em&gt;Tycho's comet&lt;/em&gt; of 1577, instead of using the last name of Tycho Brahe? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For comparison, we have the &lt;em&gt;Ptolemaic system&lt;/em&gt; and the &lt;em&gt;Copernican system&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am guessing it might have something to do with the Danish naming convention of the time, but Google searches do not seem to reveal anything.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="4251" LastEditDate="2021-10-09T09:30:56.513" LastActivityDate="2021-11-24T12:13:05.973" Title="Why do we call Tycho Brahe by his first name?" Tags="|terminology|astronomy|brahe|astronomers|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11309" PostTypeId="2" ParentId="11308" CreationDate="2020-01-03T05:58:15.237" Score="51" Body="&lt;p&gt;The short answer is that this is how he referred to himself. He was born Tyge Otteson Brahe but at the age of 15 (1561) changed 'Tyge' to Latinized 'Tycho', see &lt;a href=&quot;https://www.space.com/19623-tycho-brahe-biography.html&quot; rel=&quot;noreferrer&quot;&gt;Redd's biography of Tycho&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books/about/The_Lord_of_Uraniborg.html?id=F5a83U4B8XkC&quot; rel=&quot;noreferrer&quot;&gt;Thoren's book Lord of Uraniborg&lt;/a&gt;. Johannes Müller, a well-known German astronomer, wrote under 'Regio Monte', which Melanchton, an educational authority at the Copenhagen university where Tycho studied, Latinized into 'Regiomontanus' in 1534. Mononyms were common among ancient Latin authors (Pliny, Vitruvius), especially poets (Ovid, Horace, Virgil), although those were derived from middle names. Tycho was very much into Latin poetry, and composed himself, see &lt;a href=&quot;http://adsabs.harvard.edu/full/1998HisSc..36..467C&quot; rel=&quot;noreferrer&quot;&gt;Tycho Brahe in Scandinavian Scholarship by Christianson&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Zeeberg began by placing Tycho in the first rank of Danish Latin poets and examined&#xA;  chemical images in one of his major poems (Zeeberg 1987). Next, he identified&#xA;  Ovid as Tycho's model for a poetic persona and style and went on to analyse a&#xA;  poem of 1584, dedicated to Tycho's friend, Erik Lange, which used a conception of&#xA;  love (amor) derived from Marsilio Ficino (Zeeberg 1988, see also Zeeberg 1991b).&#xA;  The poem was replete with chemical imagery and aimed to turn Lange's obsessive&#xA;  love of alchemical goldmaking into a true love focused on the Paracelsian chemistry&#xA;  of Uraniborg.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;In his first Urania poem&#xA;  of 1573, Tycho pictured himself as Urania's humble pupil and described astronomy&#xA;  as an eternal and divine discipline, comparing it to the ignoble, earthbound pursuits&#xA;  of most noblemen. Tycho rejected the customary noble lifestyle of &quot;ambition, moneymaking,&#xA;  ignorance, and luxury&quot;, and this rejection, according to Zeeberg, &quot;had an&#xA;  influence on the colossal self-confidence and superior attitude towards norms and&#xA;  rules which became so characteristic of him&quot;. Later, when Tycho became the noble&#xA;  lord of Uraniborg, he decided that noble rank and intellectual achievement were not&#xA;  irreconcilable and began to ascribe the worldly lifestyle, not to noblemen, but to &quot;the uncomprehending mob&quot; (Zeeberg 1991a, Zeeberg 199lb). Tycho's friendship&#xA;  poems of 1584-85 adapted the Renaissance theme of translatio musarum, depicting&#xA;  Urania as refugee in many lands before she finally found her home on the island&#xA;  of Hven.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Royals and nobles, which Tycho also was, were often known by first names, like his benefactors, Frederik and later Rudolf, and Tycho had many siblings who shared his family name. Many Italian Renaissance figures, mostly poets and artists (Dante, Michelangelo), but some scientists too (Leonardo, Galileo) used first name mononyms, possibly in imitation of revered Greek ancients (Plato, Euclid, Archimedes, Hipparchus, etc.). Any combination of those reasons might have influenced Tycho's choice. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;More is known about the naming practice in Italy, as in the cases of Galileo and Leonardo, see &lt;a href=&quot;https://slate.com/news-and-politics/2009/08/why-we-don-t-use-galileo-s-last-name.html&quot; rel=&quot;noreferrer&quot;&gt;Why Do We Call Galileo Galilei by His First Name?&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Galileo went on to become one of the most recognized names in scientific history. But why do we call him by his first name only?&#xA;  Because that’s how he referred to himself. At the time of Galileo’s birth in 1564, surnames were optional in Italy. In daily interactions, an Italian would use the name his parents gave him at birth—what we’d now call a first name — and, if further clarification were required, add on his father’s name (like di Antonio, or “son of Antonio”), his birthplace (Romano, or “from Rome”), his occupation (Panettiere,  meaning “baker”), or a traditional family surname (if one existed, like Galilei). The Italian astronomer’s name is unusually confusing because both Galileo and Galilei were surnames used by his family for generations. (An equivalent might be “William Williams.”) This was not a particularly common practice at the time. Moreover, the name Galileo itself, although not completely unique, was quite rare. This is part of the reason we continue to use his first name only — it’s unambiguous.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In Renaissance Italy, individuals didn’t even stick with the same second, or identifying, name throughout their lives. Many used their family surnames one day and place of birth the next, depending on the circumstances. Take Leonardo da Vinci. Because Vinci was a very small town, calling himself Leonardo from the town of Vinci left little room for confusion—unless, of course, he was in Vinci at the time. (Leonardo was a common name.) In that case, the artist would probably have called himself Leonardo di ser Piero da Vinci, making reference to his father. Once he became famous, he often signed his name simply “Leonardo.” Galileo referred to himself sometimes by first name only, sometimes as Galileo Galilei, and sometimes as Galileo Galilei Linceo (a nod to his alliance with a progressive group of scientists, which served, in part, as a kind of honorific).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-05T02:06:18.933" LastActivityDate="2020-01-05T02:06:18.933" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11310" PostTypeId="2" ParentId="11307" CreationDate="2020-01-03T07:00:05.367" Score="8" Body="&lt;p&gt;It depends on what counts as working &quot;on it&quot;. His prior work under Hilbert was related to this area of geometry. Bolyai (1832) and Gerwein (1833) proved that polygons of equal area are equidecomposable, and Gauss urged a 3D extension in letters to Gerling mentioned by Hilbert. It was in the works in 1890-s. In 1896 Bricard reproved Gerling's 1844 result that mirror-image polyhedra are equidecomposable, and Hill showed that some specific tetrahedra are equidecomposable with a cube. Bricard claimed a negative result in general, but his proof was incomplete. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Nonetheless, it became the basis of Dehn's 1900 proof, see &lt;a href=&quot;http://sciencecow.mit.edu/me/hilberts_third_problem.pdf&quot; rel=&quot;noreferrer&quot;&gt;Krasilnikova, Hilbert’s Third Problem&lt;/a&gt;. Dehn was in Göttingen from 1899.  &lt;a href=&quot;https://www.maths.ed.ac.uk/~v1ranick/papers/guggenheim.pdf&quot; rel=&quot;noreferrer&quot;&gt;Guggenheimer's Jordan Curve Theorem&lt;/a&gt; discusses an unpublished Dehn's manuscript from 1899, where he proved the Jordan curve theorem for polygons and then extended it to polyhedra, indicating that 3D geometry was already on his mind then. In his doctoral dissertation (1900) he showed that the Archimedean postulate was needed to prove that the sum of the angles of a triangle does not exceed 180°. His solution to the &quot;third problem&quot; meant showing that it is also needed to prove that tetrahedra of equal base and height have equal volumes (this is &lt;em&gt;not&lt;/em&gt; the case for triangles).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem itself was not even stated among the 10 in Hilbert's August 8 Sorbonne lecture at the 1900 International Congress of Mathematicians. It only appeared in the printed version on the expanded list of 23, by which time Dehn already solved it, see &lt;a href=&quot;https://www.ams.org/notices/200209/fea-dawson.pdf&quot; rel=&quot;noreferrer&quot;&gt;Max Dehn, Kurt Gödel, and the Trans-Siberian Escape Route by Dawson&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;After graduating from the Gymnasium in Hamburg, Max went first&#xA;  to Freiburg and later to Göttingen, where he received his doctorate in 1900 under Hilbert’s supervision. In his dissertation he established that the Archimedean postulate is essential in order to&#xA;  prove in neutral geometry that the sum of the angles of a triangle does not exceed 180° (Legendre’s theorem).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Later that same year, soon after Hilbert’s address on “Problems of Mathematics” at the International&#xA;  Congress of Mathematicians in Paris (and before the appearance of its printed version, in which the list of problems was expanded from ten to twenty-three), Dehn established a related result that solved&#xA;  the third of the published problems (one of those left unstated during the lecture [8]): By exhibiting two tetrahedra with the same base and height that are neither equidecomposable into finite, congruent parts nor equicomplementable by such parts to produce two polyhedra that are equidecomposable, he demonstrated that the Archimedean postulate is also needed in order to prove that two tetrahedra of equal base and height have equal volumes. For his solution of Hilbert’s third problem Dehn was awarded his Habilitation at Münster, where he served as a Privatdozent from 1901 until 1911.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-03T08:04:08.063" LastActivityDate="2020-01-03T08:04:08.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11311" PostTypeId="1" CreationDate="2020-01-03T21:25:39.013" Score="2" ViewCount="69" Body="&lt;p&gt;Lagrange defines &quot;similar functions&quot; as functions of the roots of an equation where they change values only at the same kind of permutations of the roots.&#xA;What's a possible predecessor of the idea of similar functions?&#xA;Euler seems to have used some sort of a function permutation concept in his attempt to prove the fundamental theorem of algebra.&lt;/p&gt;&#xA;" OwnerUserId="10927" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:47.400" LastActivityDate="2021-02-20T15:20:47.400" Title="Any idea on how Lagrange came up with similar functions concept in (proto)group theory?" Tags="|mathematics|group-theory|euler|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11312" PostTypeId="1" AcceptedAnswerId="11413" CreationDate="2020-01-04T11:23:03.227" Score="4" ViewCount="8594" Body="&lt;p&gt;On &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_numerals#Zero&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The number zero did not originally have its own Roman numeral, but the word &lt;strong&gt;nulla&lt;/strong&gt; (the Latin word meaning &amp;quot;none&amp;quot;) was used by medieval scholars to represent 0. Dionysius Exiguus was known to use &lt;strong&gt;nulla&lt;/strong&gt; alongside Roman numerals in 525. About 725, Bede or one of his colleagues used the letter &lt;strong&gt;N&lt;/strong&gt;, the initial of &lt;strong&gt;nulla&lt;/strong&gt; or of &lt;strong&gt;nihil&lt;/strong&gt; (the Latin word for &amp;quot;nothing&amp;quot;) for 0, in a table of epacts, all written in Roman numerals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Additionally, on &lt;a href=&quot;https://sites.google.com/site/s10659003/home/chapter2/2-3-non-positional-number-system&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although Romans used the word &lt;strong&gt;nulla&lt;/strong&gt; (nothing) to convey the concept of zero, the Roman numerals lack a zero digit in their system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is interesting. If the Romans were aware in the mathematical concept of zero, or at least, some concept of &amp;quot;nothingness&amp;quot;, then they would have been likely to have a digit for it. However, they didn't, as the two quotes say above.&lt;/p&gt;&#xA;&lt;p&gt;Well, then what's the reason? Why did they only use unofficial symbols such as &lt;strong&gt;nulla&lt;/strong&gt; or the letter &lt;strong&gt;N&lt;/strong&gt; to represent the concept of zero, and not an actual zero digit?&lt;/p&gt;&#xA;" OwnerDisplayName="user10930" LastEditorDisplayName="user10930" LastEditDate="2020-08-04T05:47:35.760" LastActivityDate="2022-12-16T21:48:21.623" Title="Why didn't the Roman numeral system have a zero digit of its own?" Tags="|zero|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11313" PostTypeId="1" AcceptedAnswerId="11315" CreationDate="2020-01-05T00:33:37.557" Score="6" ViewCount="747" Body="&lt;p&gt;I've been reading about the &quot;paradoxa&quot; section of Carl Linnaeus's Systema Naturae, where he debunk some of the more far fetched ideas about animals. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia includes this translation of what Linnaeus wrote about the Phoenix.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Species of bird, of which only one individual exists in the world, and which when decrepit [arises?] from [its] pyre made of aromatic [plants?] is said fabulously to become again young, to undergo happy former periods of life. In reality it is the date palm&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and he cited the work of Engelbert Kaempfer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why does he claim that a mythical bird is actually a real tree? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The closest I see to an explanation is that the English Wiktionary page for φοῖνιξ that the the same word was used for both of these in Greek. It also suggests that they might be etymologically related either due to place of origin or color. It doesn't say anything about one meaning arising from a translation error, or about a tree specimen being claimed to be a bird specimen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit: I want to clarify that my question isn't just about the etymology, but about why Linnaeus said &quot;In reality it is the date palm&quot;, implying that someone confused the two.&lt;/p&gt;&#xA;" OwnerUserId="10935" LastEditorUserId="10935" LastEditDate="2020-01-12T18:26:50.087" LastActivityDate="2020-01-12T18:26:50.087" Title="Why did Linnaeus equate the phoenix, the mythical bird, with Phoenix, a palm genus?" Tags="|terminology|biology|taxonomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11314" PostTypeId="1" CreationDate="2020-01-05T00:58:38.687" Score="3" ViewCount="91" Body="&lt;p&gt;In &lt;a href=&quot;https://books.google.ca/books/about/Eclipse.html?id=BxcxDgAAQBAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eclipse: Journeys to the Dark Side of the Moon&lt;/a&gt; by Frank Close, it is stated that the 1584 lunar eclipse was used to determine the width of the Atlantic, by measuring the times in London and in Virginia. However, I cannot seem to find any other source, except at &lt;a href=&quot;https://www.youtube.com/watch?v=w6JZSzVcKgo&quot; rel=&quot;nofollow noreferrer&quot;&gt;his RI talk&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know if this claim is true, and if yes, some more historical details (such as the names of the people involved) on it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that the question is &lt;em&gt;not&lt;/em&gt; about the methodology, but about the &lt;em&gt;historical fact&lt;/em&gt;, whether or not the claimed measurement happened. It would also be great to know about histories of measurements similar to this.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast, &lt;a href=&quot;https://hsm.stackexchange.com/questions/6616/why-was-in-1584-the-eclipse-of-the-moon-used-to-calculate-the-width-of-the-atlan&quot;&gt;this related question&lt;/a&gt; focuses on the basic methodical aspect of such measurements.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="1800" LastEditDate="2020-01-05T02:59:32.957" LastActivityDate="2020-01-05T02:59:32.957" Title="Was the width of the Atlantic first determined in 1584?" Tags="|astronomy|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11315" PostTypeId="2" ParentId="11313" CreationDate="2020-01-05T03:19:00.403" Score="6" Body="&lt;p&gt;The real etymology is lost, I am afraid, but various speculations have been offered since antiquity and continue to this day. In Egypt the tree was known since at least 2600 BC. Egyptians already used the same name ('Bennu') for the bird and the date fruits. &lt;a href=&quot;https://books.google.com/books/about/The_Date_Palm.html?id=rLK0OgAACAAJ&quot; rel=&quot;noreferrer&quot;&gt;Bircher in The Date Palm A boon For Mankind&lt;/a&gt; speculates as to the reason:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This bird was found on the mural paintings which decorate the tombs of the kings and the nobles, this bird was assimilated to the great 'Sun-God' 'Ra' and the sun itself. The terms 'Bennu', 'bnr', 'bnr.t' were also applied to the date palm fruit and to everything sweet. The date palm was linked in ancient Egypt to the sun-bird and them giving both the same name indicated the importance of this tree to their life (Bircher,1990).&lt;/em&gt;&amp;quot; [quoted from &lt;a href=&quot;https://www.alchemywebsite.com/islam08.html&quot; rel=&quot;noreferrer&quot;&gt;Alchemy in Islamic Times&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Phoenix&amp;quot; was the name the father of botany, Theophrastus, gave&#xA;the palm tree (although he was describing its Cretan species rather than the &lt;em&gt;dactylifera&lt;/em&gt;). Most scholars suggest that it was so named as &amp;quot;the tree of Phoenicians&amp;quot;, who widely used it as a sacred and national symbol. But it is unclear if Theophrastus was adopting a name already in use, or if the bird's name predates him or the tree's, or whether he was thinking of the bird when naming the tree. The Egyptian mythical bird was presumably so named by Greeks for the palm's ability to recover after fire damage. Or so says Pliny (the Elder), see &lt;a href=&quot;http://www.fao.org/3/Y4360E/y4360e05.htm&quot; rel=&quot;noreferrer&quot;&gt;Zaid and de Wet, Botanical and Systematic Description of the Date Palm&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The botanical name of the date palm, Phoenix dactylifera L., is presumably derived from a Phoenician name &amp;quot;phoenix&amp;quot;, which means date palm, and &amp;quot;dactylifera&amp;quot; derived from a Greek word &amp;quot;daktulos&amp;quot; meaning a finger, illustrating the fruit's form (Linné, 1734).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Another source refers this botanical name to the legendary Egyptian bird, &amp;quot;Phoenix&amp;quot;, which lived to be 500 years old, and cast itself into a fire from which it rose with renewed growth (Pliny, 1489; Van Zyl, 1983). This resemblance to the date palm, which can also re-grow after fire damage, makes the bird and the date palm share this name, while &amp;quot;dactylifera&amp;quot; originates from the Hebrew word &amp;quot;dachel&amp;quot; which describes the fruit's shape (Popenoe, 1938).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bircher speculates that the fruit color might have been a factor:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This region&lt;/em&gt; [Phoenicia] &lt;em&gt;was inhabited with a population with famous purple colour from the Murex shellfish, this colour called 'Phoenix' in their language. Theophrastus may have derived the word 'Phoenix' and gave it to date palm fruits which appear purple on ripening (Bircher,1990).&lt;/em&gt;&amp;quot; [quoted from &lt;a href=&quot;https://www.alchemywebsite.com/islam08.html&quot; rel=&quot;noreferrer&quot;&gt;Alchemy in Islamic Times&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On Phoenix the bird there is a detailed history by &lt;a href=&quot;https://books.google.com/books?id=ckUhDQAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Nigg, Phoenix: An Unnatural Biography of a Mythical Beast&lt;/a&gt;, who notes the influence of &lt;a href=&quot;https://archive.org/details/theworksoflactan00lactuoft/page/n225&quot; rel=&quot;noreferrer&quot;&gt;Lactantius's poem The Phoenix&lt;/a&gt;, and his alternative &amp;quot;etymology&amp;quot;. According to him, contra Pliny, it is the tree, and Phoenicia itself, that was named after the bird, who weaved its nest on top of it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He is said to be the first to contend that the bird named Phoenicia&#xA;after itself. Then, à la Pliny, he points out that the bird and the date&#xA;palm share the same name, except that he says the tree was named for&#xA;the bird, the opposite of what Pliny suggested. While Aeolus impedes&#xA;the winds, Lactantius’s Phoenix, like Ovid’s, builds a nest of spices in&#xA;a tall date palm. The nest, similar to Ovid’s “crib” and “tomb,” is “cradle&#xA;or sepulcher — which you will— for she dies to live and yet begets herself.”&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Linnaeus refers to yet another late &amp;quot;etymology&amp;quot;, by Kaempfer, who traveled Persia in 1680-s and left a treatise on the cultivation of date palm there, &lt;em&gt;Phoenix Persicus&lt;/em&gt;. According to Kaempfer (see Schirg's Phoenix Going Bananas in &lt;a href=&quot;https://books.google.com/books?id=gAvEDgAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Apotheosis of the North: The Swedish Appropriation of Classical Antiquity. p.32&lt;/a&gt;), the bird's name comes from a metaphor: the palm is the bird, the leaves are the wings, and the heat of Arabia is the fire, which &amp;quot;&lt;em&gt;brings forth this wonderful plant and its nourishing fruit&lt;/em&gt;&amp;quot;. Hence, &amp;quot;&lt;em&gt;in reality it is the date palm, see Kæmpf&lt;/em&gt;&amp;quot;, as Linnaeus writes of the Phoenix.&lt;/p&gt;&#xA;&lt;p&gt;Babylonians and Assirians revered it as the Tree of Life. Popenoe gives more historical information on the tree and its cult in &lt;a href=&quot;https://www.jstor.org/stable/7328&quot; rel=&quot;noreferrer&quot;&gt;The Date-Palm in Antiquity (1924)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It was emphatically a Tree of Life and is accordingly represented time after time on the monuments of Babylonia and Assyria... The palm cult, probably carried northward from Babylonia, found no lack of adherents among the Phoenicians and Syrians. It is frequently to be detected among the heathen gods to whom the Old Testament prophets animadvert... Phoenician traders had carried the cult to all parts of the Mediterranean as early as the Neolithic period. In graves of this age in Spain and Portugal clear evidence of it has been found.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;That the Greeks obtained their knowledge of the palm from the same source is evident from the name they gave it - Phoenix, the tree of the Phoenicians. As the symbol of that country it is found figured on the Phoenician and, later, the Carthaginian coins struck in Sicily... It is not mentioned in the Iliad, but appears in the Odyssey, particularly in the well-known scene where the far-traveled Ulysses approaches Nausicaa on the strand and flatteringly beseeches her assistance... The palm near the temple of Phoebus in Delos, to which Ulysses refers, was one of the most famous in classical history: at clasping the trunk with her arms, Leta was fabled to have given birth to her son Apollo. The identical tree was shown to the credulous as late as the time of Cicero and Pliny, both of whom speak of it... By the beginning of the empire the date-palm is so well known that the elder Pliny is able to give an extended account of it, from Spain eastward to Persia...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-09T22:11:28.377" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11316" PostTypeId="2" ParentId="2663" CreationDate="2020-01-05T07:47:31.013" Score="1" Body="&lt;p&gt;The infinite series had originated in India by the 14th c. An explicit formula for the sum of an infinite (anantya) geometric series is given by the 15th-16th c. Nilkantha in his Aryabhatyabhasya.(Sastri 1970, commentary on Ganita 17, p. 142.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;एवं यासतुल्यच्छेदपरभागपरम्पराया अनन्ताया अपि संयोग&lt;/p&gt;&#xA;&#xA;&lt;p&gt;तस्यानंतानांपि कल्प्यामान्स्य योगस्याद्यावयविनः &lt;/p&gt;&#xA;&#xA;&lt;p&gt;परंपरांशच्छेदादेकोनच्छेदांशसामयं सर्वत्रापी समानमेव ।&lt;/p&gt;&#xA;&#xA;&lt;p&gt;which may be translated:(cf. Sarma 1972, p. 17)&#xA;The sum of an infinite [anantya] series, whose later terms (after the first) are got by dividing the preceding one by the same divisor everywhere, is equal to the first term multiplied by the common divisor, and divided by one less than the common divisor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That is a+a + a +···= ad/d-1 .(It is assumed that the divisor d&gt;1, so&#xA;that the common ratio is less than 1.)   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Reference: &lt;a href=&quot;http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-1-final.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ckraju.net/papers/Springer/ckr-Springer-encyclopedia-calculus-1-final.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10936" LastActivityDate="2020-01-05T07:47:31.013" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11317" PostTypeId="1" CreationDate="2020-01-05T16:46:58.287" Score="1" ViewCount="167" Body="&lt;p&gt;Does anybody know a freely available overview of the history of  Reciprocity Laws, especially the Cubic and Biquadratic ones? Wikipedia &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know about Franz Lemmermeyer's book &quot;Reciprocity Laws&quot;, but sadly it is not available to me. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far I have only found the following PDF about the history of the Quadratic Reciprocity Law:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://seanelvidge.com/wp-content/uploads/2011/04/HistoryQR.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://seanelvidge.com/wp-content/uploads/2011/04/HistoryQR.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10938" LastActivityDate="2020-01-05T16:46:58.287" Title="History of Reciprocity Laws" Tags="|mathematics|number-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11318" PostTypeId="1" AcceptedAnswerId="11319" CreationDate="2020-01-05T17:22:55.027" Score="6" ViewCount="1769" Body="&lt;p&gt;Try as I may, I cannot find the original source where Einstein described entanglement as &quot;Spooky Action at the Distance.&quot; The work &quot;spooky&quot; is not in the &lt;a href=&quot;https://www.drchinese.com/David/EPR.pdf&quot; rel=&quot;noreferrer&quot;&gt;1935 EPR paper&lt;/a&gt;, which was written in English and thus there is no issue of translation from German. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Spooky Action at a Distance&quot; is widely repeated, but I cannot find from where it is originally cited. So where did Einstein say or write this?&lt;/p&gt;&#xA;" OwnerUserId="6112" LastEditorUserId="55" LastEditDate="2020-01-06T00:33:49.250" LastActivityDate="2020-01-06T03:39:53.647" Title="What is the original source for Einstein's description of entanglement as &quot;Spooky Action at a Distance&quot;?" Tags="|quantum-mechanics|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11319" PostTypeId="2" ParentId="11318" CreationDate="2020-01-06T00:24:06.077" Score="9" Body="&lt;p&gt;Einstein was bothered by &quot;action at a distance&quot; long before the 1935 EPR paper, and it was not specific to entanglement. In his debates with Bohr at the 1927 Solvay congress he used the single slit experiment to illustrate it, see &lt;a href=&quot;https://www3.nd.edu/~dhoward1/Revisiting%20the%20Einstein-Bohr%20Dialogue.pdf&quot; rel=&quot;noreferrer&quot;&gt;Howard, Revisiting the Einstein-Bohr Dialogue&lt;/a&gt;. An objective wave function describing a particle hitting the screen after passing through the slit  &quot;&lt;em&gt;presupposes a very particular mechanism of action at a distance which would prevent the wave continuously distributed in space from acting at two places of the screen&lt;/em&gt;&quot; [English translation of Proceedings of Solvay, quoted from Howard]. Einstein pointed out that the problem does not arise if the wave function is interpreted statistically, as describing an ensemble of particles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, according to &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1806/1806.07925.pdf&quot; rel=&quot;noreferrer&quot;&gt;Boughn's There Is No Action at a Distance in Quantum Mechanics, Spooky or Otherwise&lt;/a&gt;, the word &quot;&lt;em&gt;spukhafte&lt;/em&gt;&quot; was only added much later, in March 3, 1947 letter to Born, where it is not specific to entanglement either. One can argue over the translation of &lt;em&gt;spukhafte&lt;/em&gt; as &quot;spooky&quot;, but it is the &quot;literal&quot; translation, and is used in the English translation of &lt;a href=&quot;https://archive.org/details/TheBornEinsteinLetters/page/n89&quot; rel=&quot;noreferrer&quot;&gt;Einstein-Born correspondence&lt;/a&gt; available on the Internet Archive. The relevant passage is on p. 158 (emphasis mine):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I cannot make a case for my attitude in physics which you &#xA;  would consider at all reasonable. I admit, of course, that there &#xA;  is a considerable amount of validity in the statistical approach &#xA;  which you were the first to recognise clearly as necessary given &#xA;  the framework of the existing formalism. I cannot seriously &#xA;  believe in it because the theory cannot be reconciled with the &#xA;  idea that physics should represent a reality in time and space, &#xA;  free from &lt;strong&gt;spooky actions at a distance&lt;/em&gt;&lt;/strong&gt;. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;I am, however, not yet firmly convinced that it can really be achieved with a continuous field theory, although I have discovered a possible way of doing this which so far seems quite reasonable. The calculation difficulties are so great that I will be biting the dust long before I &#xA;  myself can be fully convinced of it. But I am quite convinced &#xA;  that someone will eventually come up with a theory whose &#xA;  objects, connected by laws, are not probabilities but considered &#xA;  facts, as used to be taken for granted until quite recently. I &#xA;  cannot, however, base this conviction on logical reasons, but can &#xA;  only produce my little finger as witness, that is, I offer no &#xA;  authority which would be able to command any kind of respect &#xA;  outside of my own hand.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-06T03:39:53.647" LastActivityDate="2020-01-06T03:39:53.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11320" PostTypeId="1" AcceptedAnswerId="11322" CreationDate="2020-01-06T01:16:27.017" Score="4" ViewCount="134" Body="&lt;p&gt;After getting such an informative response on my &lt;a href=&quot;https://hsm.stackexchange.com/questions/11048/who-first-proved-that-the-spectrum-of-an-operator-is-contained-in-the-closure-of&quot;&gt;first question&lt;/a&gt;, I have another theorem discussed in our lecture who´s origin I am interested in (sometimes called the &quot;acute angle principle&quot;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Corollary (&lt;a href=&quot;https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brouwer's fixed-point theorem&lt;/a&gt;)&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$h: \mathbb{R}^m \supset \overline{B}(0,R) \to \mathbb{R}^m$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$R &amp;gt; 0$&lt;/span&gt; be continuous and fulfil &lt;span class=&quot;math-container&quot;&gt;$h(z) \cdot z \ge 0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$| z | = R$&lt;/span&gt;.&#xA;  Then there exists an &lt;span class=&quot;math-container&quot;&gt;$z^*$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$h(z^*) = 0$&lt;/span&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10550" LastEditorUserId="55" LastEditDate="2020-01-07T02:11:45.963" LastActivityDate="2020-03-30T20:35:48.850" Title="Who first proved the &quot;acute angle principle&quot; in fixed point theory?" Tags="|mathematics|functional-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11322" PostTypeId="2" ParentId="11320" CreationDate="2020-01-06T17:57:59.513" Score="6" Body="&lt;p&gt;Good sources on the history of fixed point theorems are &lt;a href=&quot;http://www.math.ac.vn/publications/vjm/vjm_27/No.3/187-222_Park.PDF&quot; rel=&quot;nofollow noreferrer&quot;&gt;Park, Ninety Years of the Brouwer Fixed Point Theorem&lt;/a&gt; and &lt;a href=&quot;http://www.utgjiu.ro/math/sma/v08/p08.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kumar, A Short Survey of the Development of Fixed Point Theory&lt;/a&gt;. According to both, early versions of fixed point theorem concerned self-maps, from a ball or some other set to itself. The first version for non-self maps is in &lt;a href=&quot;http://www.numdam.org/item/CM_1938__5__177_0/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rothe's &lt;em&gt;Zur Theorie der topologischen Ordnung und der Vektorfelder in Banachschen Räumen&lt;/em&gt; (1938)&lt;/a&gt;, and is now known as Rothe's fixed point theorem: if &lt;span class=&quot;math-container&quot;&gt;$B\to \mathbb{R}^n$&lt;/span&gt; is continuous and &lt;span class=&quot;math-container&quot;&gt;$f(\partial B)\subseteq B$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has a fixed point. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1955 Altman showed in A Fixed Point Theorem for Completely Continuous Operators in Banach Spaces (Bulletin of the Polish Academy of Sciences, 3 (1955) pp. 409-413) that Rothe's condition is fulfilled if &lt;span class=&quot;math-container&quot;&gt;$|f(x)-x|^2\geq|f(x)|^2-|x|^2$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\partial B$&lt;/span&gt;. The advantage of Altman's condition is that it is not tied to an inner product, and hence can be imposed in Banach spaces as well (when the image of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is compact). However, when the norm does come from an inner product, and we set &lt;span class=&quot;math-container&quot;&gt;$h(x):=x-f(x)$&lt;/span&gt; then it easily simplifies to &lt;span class=&quot;math-container&quot;&gt;$0\geq-2 h(x)\cdot x$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$h(x)\cdot x\geq0$&lt;/span&gt;. A fixed point of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; will be a zero of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;acute angle condition&quot; or &quot;acute angle principle&quot; terminology, referring to the angle between &lt;span class=&quot;math-container&quot;&gt;$h(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, is more recent. MathSciNet only shows it since late 1980-s, the earliest I found is in a Russian paper &lt;a href=&quot;https://iopscience.iop.org/article/10.1070/SM1972v016n03ABEH001428/meta&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Some Noncoercive Nonlinear Equations by Dubinski (1972)&lt;/a&gt;, who refers to  Lions's   &lt;em&gt;Quelques methodes de resolution des problemes aux limites non lineaires&lt;/em&gt; (1969) as the source (but I could not find Lions calling it that).  &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-03-30T20:35:48.850" LastActivityDate="2020-03-30T20:35:48.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11325" PostTypeId="1" CreationDate="2020-01-07T22:24:54.307" Score="1" ViewCount="276" Body="&lt;p&gt;In &lt;em&gt;Hilbert&lt;/em&gt; by Reid, it is noted that Hilbert and Minkowski were great friends and often worked together on problems, seminars, and lectures. This is a similar relationship to the partnership between Hardy and Littlewood. Are there similar examples of strong and lengthy collaborations in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="8110" LastActivityDate="2020-01-13T10:51:28.870" Title="Pairs of Mathematicians" Tags="|mathematicians|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11326" PostTypeId="1" CreationDate="2020-01-08T01:20:30.460" Score="0" ViewCount="299" Body="&lt;p&gt;Nobel laureate Hans Bethe was a friend of mathematician-physicist John von Neumann, and he once said:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;[von Neumann's] brain indicated a new species, an evolution beyond man&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This indicates that Bethe admired von Neumann's work very much. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Did he think the same about von Neumann's quantum logic? Did he like it?&lt;/p&gt;&#xA;" OwnerUserId="10952" LastActivityDate="2020-01-08T01:20:30.460" Title="What did Hans Bethe think of von Neumann's quantum logic?" Tags="|physics|biographical-details|philosophy-of-science|quantum-mechanics|mathematical-logic|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11329" PostTypeId="1" CreationDate="2020-01-08T18:50:55.177" Score="7" ViewCount="97" Body="&lt;p&gt;I just want to check if anybody knows a website where one can find a chronology of proofs of more difficult reciprocity theorems (such as the cubic and biquadratic cases) similar to the (already existing) chronology of 246+ proofs of the law of quadratic reciprocity (here is a link: &lt;a href=&quot;https://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html&lt;/a&gt; ). I borrowed franz lemermeyer's book on the history of reciprocity laws from a library, and this book does include an extensive list of references, but i'm unable to find many of the proofs which the book refers to.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So i'll be grateful for anyone offering useful information!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2020-01-08T18:50:55.177" Title="Chronology of proofs of cubic and biquadratic reciprocity laws" Tags="|reference-request|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11331" PostTypeId="1" CreationDate="2020-01-09T15:46:00.257" Score="1" ViewCount="156" Body="&lt;p&gt;It is understandable that the length of a solar year can be found out using the time of solstices. But how did they find the length of a year with respect to the stars?&lt;/p&gt;&#xA;" OwnerUserId="10959" LastActivityDate="2020-01-10T12:36:16.877" Title="What is the historical basis for the length of a sidereal year?" Tags="|astronomy|time|archaeoastronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11332" PostTypeId="2" ParentId="11331" CreationDate="2020-01-10T04:08:25.267" Score="2" Body="&lt;p&gt;By looking at stars position with respect to the point of intersection of the ecliptic and equator (this is a position of the Sun at the equinox). Or which is the same at the position of this point among the stars. This position varies slowly (this is called precession of the equinox), and it was discovered by Hipparchus. Hipparchus work did not survive and it is known from Ptolemy. Since the precession is rather slow, about 1.4 degree per century, to detect it one has to compare two observations at the great time difference (several centuries). Since the time of Hipparchus this was done by many astronomers arriving to more and more precise values of the sidereal year.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course the practical problem is that Sun and stars are not visible simultaneously. But fortunately for astronomers: a) we have Moon, and b) Sun and Moon move approximatelty on the same circle. This permits to measure the angular distance between the Sun and Moon when they are both visible and then from the Moon to a star when they are both visible. For this reason, Ptolemy has to consider the motion of the Sun first, then the motion of the Moon and only after that he explains how stars positions with respect to the equinox point are measured. There is a continuing discussion lasting more than 100 years on the question &quot;who made the first star catalog, Hipparchus or Ptolemy?&quot;  But Ptolemy's work is the only surviving ancient work where all this is explained in detail.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-01-10T12:36:16.877" LastActivityDate="2020-01-10T12:36:16.877" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11334" PostTypeId="1" AcceptedAnswerId="11335" CreationDate="2020-01-10T11:51:19.397" Score="3" ViewCount="323" Body="&lt;p&gt;I'm reading &lt;a href=&quot;https://b-ok.cc/book/532642/fc9758&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo's Two New Sciences&lt;/a&gt;. But as other Scientific Book it is hard. I am having trouble in First day, where SIMPLICO asks this question(Page 7): &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;But how can one make a rope one hundred cubits long out of hempen&#xA;  fibres which are not more than two or three cubits long, and still&#xA;  give it so much strength?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then SAGR appeals SALV to answer the question. So SALV goes on to answer it in the following way(Page 8):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[....]How are fibres, each not more than two or three cubits in&#xA;  length, so tightly bound together in the case of a rope one hundred&#xA;  cubits long that great force [violenza] is required to break it? Now&#xA;  tell me, Simplicio, can you not hold a hempen fibre so tightly between&#xA;  your fingers that I, pulling by the other end, would break it before&#xA;  drawing it away from you? Certainly you can. And now when the fibres&#xA;  of hemp are held not only at the ends, but are grasped by the&#xA;  surrounding medium through- throughout their entire length is it not&#xA;  manifestly more difficult to tear them loose from what holds them than&#xA;  to break them ? But in the case of the rope the very act of twisting&#xA;  causes the threads to bind one another in such a way that when the&#xA;  rope is stretched with a great force the fibres break rather than&#xA;  separate from each other. At the point where a rope parts the fibres&#xA;  are, as everyone knows, very short, nothing like a cubit long, as they&#xA;  would be if the parting of the rope occurred, not by the breaking of&#xA;  the filaments, but by their slipping one over the other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm not able to understand what SALV means by example of holding fibers. And how does this answer the question of SIMP?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks. &lt;/p&gt;&#xA;" OwnerDisplayName="user10963" LastEditorUserId="55" LastEditDate="2020-01-11T02:35:59.567" LastActivityDate="2020-01-11T02:35:59.567" Title="What is the explanation of rope's strength in Galileo's Two New Sciences?" Tags="|physics|galileo|materials-science|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11335" PostTypeId="2" ParentId="11334" CreationDate="2020-01-11T02:33:48.887" Score="2" Body="&lt;p&gt;In this passage Galileo is explaining how a bundle of multiple short fibers twisted into a rope functions. He is analogizing a fiber in the bundle to a fiber held between fingers, and notes that it would sooner break then slip out because the friction is large when the fingers are pressed together hard. Same with the rope, its fibers would sooner break than be pulled apart, and it is hard to break so many of them at once. This is why a long rope manages to hold together despite its fibers being short, and why they give it so much strength. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-01-11T02:33:48.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11336" PostTypeId="1" AcceptedAnswerId="11337" CreationDate="2020-01-13T03:30:56.220" Score="3" ViewCount="71" Body="&lt;p&gt;I've spent some time with a search engine trying to find out about Wanner, a person whose surname is mentioned in the name of Rosenbrock-Wanner (ROW) methods primarily used for iteratively solving stiff differential equations, but without much success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone know who this person is/was, and the full name of this person?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I apologize if my question does not belong here. In this case please suggest where it can be posted.&lt;/p&gt;&#xA;" OwnerUserId="10111" LastEditorUserId="4251" LastEditDate="2020-07-06T06:58:45.470" LastActivityDate="2020-07-06T06:58:45.470" Title="Who is Wanner from Rosenrock-Wanner (ROW) methods?" Tags="|mathematics|mathematicians|differential-equations|computation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11337" PostTypeId="2" ParentId="11336" CreationDate="2020-01-13T06:43:24.953" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;https://de.wikipedia.org/wiki/Rosenbrock-Wanner-Verfahren&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, it is the Austrian mathematician &lt;a href=&quot;https://de.wikipedia.org/wiki/Gerhard_Wanner_(Mathematiker)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerhard Wanner&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-01-13T06:43:24.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11338" PostTypeId="2" ParentId="11325" CreationDate="2020-01-13T10:51:28.870" Score="4" Body="&lt;p&gt;This is quite common. Some examples of pairs of very strong researchers who published at least &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; papers together:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Guillemin and Sternberg. &lt;/li&gt;&#xA;&lt;li&gt;Harvey and Lawson.  &lt;/li&gt;&#xA;&lt;li&gt;Colding and Minicozzi&lt;/li&gt;&#xA;&lt;li&gt;Ozsvath and Szabo&lt;/li&gt;&#xA;&lt;li&gt;Green and Tao.&lt;/li&gt;&#xA;&lt;li&gt;Schoen and Yau. &lt;/li&gt;&#xA;&lt;li&gt;Mrowka and Kronheimer&lt;/li&gt;&#xA;&lt;li&gt;Atiyah and Bott. &lt;/li&gt;&#xA;&lt;li&gt;Atiyah and Singer.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;There are of course many more examples, as this list mostly reflects my own interests.&lt;/p&gt;&#xA;" OwnerUserId="633" LastActivityDate="2020-01-13T10:51:28.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11339" PostTypeId="2" ParentId="783" CreationDate="2020-01-13T14:33:50.837" Score="2" Body="&lt;p&gt;My vote goes to three out of {Michał Dziewicki, Stanisław Zaremba, Jan Śleszyński, &lt;strong&gt;Witold Wilkosz&lt;/strong&gt;, &lt;strong&gt;Otto Nikodym&lt;/strong&gt;, &lt;strong&gt;Leon Chwistek&lt;/strong&gt;, Stanisław Bilski}; most probably the three whose names I put in bold.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On May 10, 1915, Russell wrote to Ludwig Wittgenstein who was doing military service in Krakow: &quot;If you have time, you should visit in Krakow a lonely old logician — 11, Szczepańska Street. He studied &lt;em&gt;Principia Mathematica&lt;/em&gt; and probably knows nobody who deals with modern logic.&quot; [My retranslation into English from a Polish translation]. The old logician was &lt;strong&gt;Michał Dziewicki&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On September 20, 1919, Dziewicki wrote to Russell that professors of the Jagiellonian University in Krakow, &lt;strong&gt;Stanisław Zaremba&lt;/strong&gt; and &lt;strong&gt;Jan Śleszyński&lt;/strong&gt; were interested in Russell's works, and that Śleszyński had read part I of &lt;em&gt;Principia&lt;/em&gt; from *1 to *43.51 and considered it a work of a genius.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On July 6, 1923, Dziewicki wrote to Russell about professors &lt;strong&gt;Witold Wilkosz&lt;/strong&gt;, &lt;strong&gt;Leon Chwistek&lt;/strong&gt;, and &lt;strong&gt;Otto Nikodym&lt;/strong&gt; who admired the work of Russell and claimed to understand his theory of types. Dziewicki added that a young student, &lt;strong&gt;Stanisław Bilski&lt;/strong&gt; was preparing a dissertation in philosophy, had borrowed Dziewicki's copies of Russell's books, and discussed Russell for hours whenever he met Dziewicki.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;See &lt;a href=&quot;http://journals.pan.pl/Content/100390/PDF/P.Fil.+3-16+13-J.Bremer.pdf?handler=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michał Dziewicki – edytor, logik, lektor, tłumacz, powieściopisarz&lt;/a&gt; (in Polish) for more details.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Side note: &lt;a href=&quot;http://mathematics-in-europe.eu/?p=569&quot; rel=&quot;nofollow noreferrer&quot;&gt;here is a photo of a bench in Krakow&lt;/a&gt; that commemorates Hugo Steinhaus's overhearing the words &quot;Lebesgue integral&quot; from Otto Nikodym and Stefan Banach in 1916.&lt;/p&gt;&#xA;" OwnerUserId="9246" LastActivityDate="2020-01-13T14:33:50.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11340" PostTypeId="1" CreationDate="2020-01-13T20:42:19.940" Score="0" ViewCount="117" Body="&lt;p&gt;Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Positron&quot; rel=&quot;nofollow noreferrer&quot;&gt;says&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Robert Oppenheimer argued strongly against the proton being the negative-energy electron solution to Dirac's equation. He asserted that if it were, the hydrogen atom would rapidly self-destruct.[5] Persuaded by Oppenheimer's argument, Dirac published a paper in 1931 that predicted the existence of an as-yet-unobserved particle that he called an &quot;anti-electron&quot; that would have the same mass and the opposite charge as an electron and that would mutually annihilate upon contact with an electron.[6] &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the other hand, &lt;a href=&quot;https://arxiv.org/abs/1907.04623&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt; by Kragh says:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In his paper of 1931 Dirac also predicted the antielectron as a new particle&#xA;  soon to be known as the positron. The previous year, at a time when he still believed that the antielectron was a proton, he proposed proton-electron annihilation (+  − → 2) with the sole argument, “There appears to be no reason why such  processes should not actually occur somewhere in the world. They would be consistent with all the general laws of Nature” (Dirac 1930).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the surface, these appear to be contradictory. Can anyone explain what's going on? Did Dirac originally imagine that the rate of annihilation would just be very low for some reason?&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2020-01-13T20:42:19.940" Title="Dirac and proton-electron annihilation" Tags="|physics|particle-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11341" PostTypeId="1" AcceptedAnswerId="11342" CreationDate="2020-01-14T09:39:23.333" Score="8" ViewCount="1409" Body="&lt;p&gt;Peano introduced a number of logical symbols still used today:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$∨$&lt;/span&gt; (from Latin &lt;strong&gt;&lt;em&gt;v&lt;/strong&gt;el&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$∧$&lt;/span&gt; (inverted &lt;span class=&quot;math-container&quot;&gt;$∨$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$∃$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This inversion of Latin letters as symbols (and inversion of symbols to signify their 'opposite' operation) was followed by later logicians:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$∀$&lt;/span&gt; (Gentzen, &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN266833020_0039?tify=%7B%22pages%22:%5B184%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;1935&lt;/a&gt;: inverted A from &lt;em&gt;&quot;&lt;strong&gt;A&lt;/strong&gt;ll-Zeichen&quot; / &quot;Für &lt;strong&gt;A&lt;/strong&gt;lle&quot;&lt;/em&gt;, by analogy to &lt;span class=&quot;math-container&quot;&gt;$∃$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$⊥$&lt;/span&gt; (inverted &lt;span class=&quot;math-container&quot;&gt;$⊤$&lt;/span&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I had always assumed that ∃ stood for &quot;E&quot; in &lt;strong&gt;&lt;em&gt;&quot;E&lt;/strong&gt;xistential&quot; / &quot;there &lt;strong&gt;E&lt;/strong&gt;xists&quot;&lt;/em&gt; (or some cognate thereof), but Peano did not appear to use any words beginning 'e' in the paragraph this symbol was introduced:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Mais nous préférons l'indiquer par la nouvelle notation&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$Ǝa$$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;qu'on peut lire « il y a des &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; ».&lt;/p&gt;&#xA;  &#xA;  &lt;ul&gt;&#xA;  &lt;li&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k84142s/f48.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Formulaire de mathématiques&lt;/em&gt;&lt;/a&gt;, Peano (1897)&lt;/li&gt;&#xA;  &lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So why did he choose an inverted &quot;E&quot;?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;a href=&quot;http://jeff560.tripod.com/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/em&gt;&lt;/a&gt;&lt;/sup&gt;&lt;br&gt;&#xA;&lt;sup&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/5817/55&quot;&gt;Is the symbol for set membership $\in$ derived from greek letter $\epsilon$?&lt;/a&gt;&lt;/sup&gt;&lt;br&gt;&#xA;&lt;sup&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/1183278/what-came-first-the-forall-or-the-exists&quot;&gt;Math SE: What came first, the ∀ or the ∃?&lt;/a&gt;&lt;/sup&gt;  &lt;/p&gt;&#xA;" OwnerUserId="10974" LastEditorUserId="4251" LastEditDate="2020-05-13T14:09:04.843" LastActivityDate="2020-05-13T14:09:04.843" Title="Why is the existential quantifier symbol ∃ a backwards &quot;E&quot;?" Tags="|terminology|notation|mathematical-logic|logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11342" PostTypeId="2" ParentId="11341" CreationDate="2020-01-14T15:40:06.777" Score="9" Body="&lt;p&gt;When introducing the older terminology in the previous sentence, Peano describes it thus:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... signifie &quot;il y a des &lt;em&gt;a&lt;/em&gt;&quot;, &quot;les &lt;em&gt;a&lt;/em&gt; &lt;strong&gt;e&lt;/strong&gt;xistent&quot;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems likely this is the source of the inverted &quot;E&quot;.&lt;/p&gt;&#xA;" OwnerUserId="10974" LastActivityDate="2020-01-14T15:40:06.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11343" PostTypeId="1" CreationDate="2020-01-17T00:09:08.563" Score="3" ViewCount="1109" Body="&lt;p&gt;Is there any evidence showing that a unit circle approach was used by early mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="10984" LastActivityDate="2021-08-24T19:44:35.567" Title="When did mathematicians invent the unit circle to extend the trig functions?" Tags="|euclidean-geometry|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11344" PostTypeId="1" CreationDate="2020-01-17T11:05:57.323" Score="3" ViewCount="280" Body="&lt;p&gt;I guess Desargues did not use that term. Anyone could help me know where did it appear for the first time? Thanks&lt;/p&gt;&#xA;" OwnerUserId="10985" LastActivityDate="2023-11-28T17:50:22.523" Title="Who wrote &quot;projective geometry&quot; for the first time?" Tags="|projective-geometry|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11345" PostTypeId="2" ParentId="11344" CreationDate="2020-01-17T14:11:07.873" Score="4" Body="&lt;p&gt;I looked in &lt;a href=&quot;http://jeff560.tripod.com/p.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathWords&lt;/a&gt; and found this&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;PROJECTIVE GEOMETRY. In his account of the origins of projective geometry Katz (1993) mentions Pascal, Desargues and Monge before saying that J.-V.Poncelet composed the first text in synthetic projective geometry: his &lt;em&gt;Traité des Propriétés Projectives&lt;/em&gt; of 1822. See here for an extract from Poncelet’s Introduction.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;em&gt;OED&lt;/em&gt; finds the term &lt;em&gt;projective geometry&lt;/em&gt; used in English in 1885 in Charles Leudesdorf’s &lt;em&gt;Elements of Projective Geometry&lt;/em&gt;, a translation of Cremona’s &lt;em&gt;Elementi di geometria proiettiva&lt;/em&gt; of 1873.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, perhaps the first use in English was 1885, but that was a translation from Italian.  First uses in other languages are (of course) not answered by this.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-17T14:11:07.873" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11346" PostTypeId="1" AcceptedAnswerId="11350" CreationDate="2020-01-17T20:35:46.680" Score="3" ViewCount="758" Body="&lt;p&gt;I'm interested in really understanding what Copernicus was getting at with the quote 'mathematics is written for mathematicians'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He clearly isn't referring to the rigorous nature of mathematical papers since he lived in the year 1500.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What was he trying to describe in saying this? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="10988" LastActivityDate="2020-01-18T17:45:21.737" Title="What was Copernicus trying to mean with 'Mathematics is Written for Mathematicians'?" Tags="|mathematics|quote|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11348" PostTypeId="2" ParentId="11240" CreationDate="2020-01-18T16:37:26.297" Score="2" Body="&lt;p&gt;This is a follow-up to some of my comments to the OP and to the answer that @Conifold gave. A few days ago I purchased a copy of the book that I had mentioned, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0006BOPSG&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Measure and the Integral&lt;/strong&gt;&lt;/a&gt; by Lebesgue (1966). I was correct that this book is where, many years ago (1980s, possibly even late 1970s), I had read about Lebesgue's fascination with this “paradox”. However, contrary to what I thought, Lebesgue's fascination is not discussed in Kenneth O. May's biographical essay, but instead it appears in Section 66 on pp. 97-98. Those wishing to read more about these issues involving arc length and surface area will find Chapter V: &lt;em&gt;Lengths of Curves. Areas of Surfaces&lt;/em&gt; (Sections 62-83; pp. 92-124) especially instructive. Below is all of Section 66, followed by some bibliographic information about this 1966 book.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;66&lt;/strong&gt;&lt;/em&gt; &lt;em&gt;A similar paradox for lengths&lt;/em&gt;. &lt;span class=&quot;math-container&quot;&gt;$\;\;\;\;$&lt;/span&gt; If mathematicians had not been hypnotized by the word “inscribed,” if they had not forgotten that inscribing had been chosen only as one way of approximating, they would have seen that the difficulty encountered for areas existed equally for curves. Now it was just this difference between curves and surfaces that was most shocking. Allow me to refer to my own recollections.&lt;/p&gt;&#xA;&lt;p&gt;When I was a schoolboy, it was agreed in France, as I have said [previously], that one could evaluate lengths, areas, and volumes by passing to the limit. Soon doubts began to appear in the textbooks. The students who had heard Schwarz’ objections in Hermite’s analysis course had now in their turn become teachers. Besides, everything then predisposed us to a critical analysis of concepts: researches on functions of a real variable and on sets, which people were beginning to consider, Tannery’s teaching, which had aroused in many of his students the desire for complete comprehension or at least verbal precision. People began to doubt, sometimes without knowing what they doubted. For example, the determination of the area of a circle by means of the areas of the polygons that it contained or that contained it (see section 42) was confused with an argument about limits.&lt;/p&gt;&#xA;&lt;p&gt;Formerly, when I was a schoolboy, the teachers and pupils had been satisfied with this reasoning by passage to the limit. However, it ceased to satisfy me when some of my schoolmates showed me, along about my fifteenth year, that one side of a triangle is equal to the sum of the other two and that &lt;span class=&quot;math-container&quot;&gt;$\pi = 2.$&lt;/span&gt; Suppose that &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt; is an equilateral triangle and that &lt;span class=&quot;math-container&quot;&gt;$D,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$E,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; are the midpoints of &lt;span class=&quot;math-container&quot;&gt;$BA,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$BC,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$CA.$&lt;/span&gt; The length of the broken line [= polygonal path] &lt;span class=&quot;math-container&quot;&gt;$BDEFC$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$AB + AC.$&lt;/span&gt; If we repeat this procedure with the triangles &lt;span class=&quot;math-container&quot;&gt;$DBE$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$FEC,$&lt;/span&gt; we get a broken line of the same length made up of eight segments, etc. Now these broken lines have &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt; as their limit, and hence the limit of their lengths, that is, their common length &lt;span class=&quot;math-container&quot;&gt;$AB + AC,$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$BC.$&lt;/span&gt; The reasoning with regard to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is analogous.&lt;/p&gt;&#xA;&lt;p&gt;Nothing, absolutely nothing, distinguishes this reasoning from what we used to evaluate the circumference and area of a circle, the surface and volume of a cylinder, a cone, and a sphere. This result has been extremely instructive to me.&lt;/p&gt;&#xA;&lt;p&gt;Besides, every paradox is highly instructive. In my opinion, the critical examination of paradoxes and the correction of erroneous reasoning should be standard exercises, frequently repeated at the secondary level.&lt;/p&gt;&#xA;&lt;p&gt;The preceding example shows that passing to the limit for lengths, areas, or volumes requires justification, and, like Schwarz’ example, it is enough to arouse all one’s suspicions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Lebesgue.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henri Léon Lebesgue (1875-1941)&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0006BOPSG&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Measure and the Integral&lt;/strong&gt;&lt;/a&gt;, edited with a biographical essay by Kenneth Ownsworth May (1915-1977), The Mathesis Series, Holden-Day, 1966, xii + 194 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book is an English translation of two works by Lebesgue. The first work is on pp. 12-175 and the second work is on pp. 178-194.&lt;/p&gt;&#xA;&lt;p&gt;The first work was originally published in &lt;strong&gt;L’Enseignement Mathématique&lt;/strong&gt; with the title &lt;em&gt;Sur la mesure des grandeurs&lt;/em&gt; and consists of an Introduction and 8 sections published in 6 parts: &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1932:31#327&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;(i)&lt;/strong&gt; L’E. M. (1) 31 #2 (1932), pp. 173-206&lt;/a&gt; [&lt;em&gt;Introduction&lt;/em&gt; (pp. 173-174); &lt;em&gt;I. Comparaison des Collections; Nombres Entiers&lt;/em&gt; (pp. 175-181); &lt;em&gt;II. Longueurs; Nombres&lt;/em&gt; (pp. 182-206)]. &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1933:32#302&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;(ii)&lt;/strong&gt; L’E. M. (1) 32 #1 (1933), pp. 23-51&lt;/a&gt; [&lt;em&gt;III. Aires&lt;/em&gt; (pp. 23-51)]. &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1934:33#167&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;(iii)&lt;/strong&gt; L’E. M. (1) 33 #1 (1934), pp. 22-48&lt;/a&gt; [&lt;em&gt;IV. Volumes&lt;/em&gt; (pp. 22-48)]. &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1934:33#324&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;(iv)&lt;/strong&gt; L’E. M. (1) 33 #2 (1934), pp. 177-213&lt;/a&gt; [&lt;em&gt;V. Longueurs des Courbes. Aires des Surfaces&lt;/em&gt; (pp. 177-213)]. &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1934:33#417&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;(v)&lt;/strong&gt; L’E. M. (1) 33 #3 (1934), pp. 270-284&lt;/a&gt; [&lt;em&gt;VI. Grandeurs Mesurables&lt;/em&gt; (pp. 270-284)]. &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001:1935:34#337&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;(vi)&lt;/strong&gt; L’E. M. (1) 34 #2 (1935), pp. 176-219&lt;/a&gt; [&lt;em&gt;VII. Intégration et Dérivation&lt;/em&gt; (pp. 176-212); &lt;em&gt;VIII. Conclusions&lt;/em&gt; (pp. 212-219)]. &lt;strong&gt;[[&lt;/strong&gt; Note: “(1) 33 #3” means “series 1, volume 33, issue 3”. I do not know the precise dates of the issues, or even whether such more precise dates exist, so the years are for the volumes. &lt;strong&gt;]]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The first work was published as a book with the title &lt;a href=&quot;https://repositorio.ufsc.br/handle/123456789/179685&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Sur la Mesure des Grandeurs&lt;/strong&gt;&lt;/a&gt; by Gauthier-Villars (Paris) and L’Enseignement Mathématique (Geneva) in 1956 (iv + 184 pages), which was reprinted with the title &lt;a href=&quot;https://www.worldcat.org/title/mesure-des-grandeurs/oclc/313417897&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;La Mesure des Grandeurs&lt;/strong&gt;&lt;/a&gt; by Albert Blanchard (Paris) in 1975 (iv + 184 pages).&lt;/p&gt;&#xA;&lt;p&gt;The second work is the published version of a conference talk that Lebesgue gave in Copenhagen on 8 May 1926 and it was originally published with the title &lt;a href=&quot;https://www.jstor.org/stable/24530055&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sur le développement de la notion d’intégrale&lt;/em&gt;&lt;/a&gt; in &lt;strong&gt;Matematisk Tidsskrift B&lt;/strong&gt; [&lt;em&gt;after 1952&lt;/em&gt;: &lt;strong&gt;Mathematica Scandinavica&lt;/strong&gt;] 1926 (1926), pp. 54-74, and &lt;a href=&quot;https://www.jstor.org/stable/40896441&quot; rel=&quot;nofollow noreferrer&quot;&gt;reprinted with the same title&lt;/a&gt; in &lt;strong&gt;Revue de Métaphysique et de Morale&lt;/strong&gt; 34 #2 (April-June 1927), pp. 149-167, and translated into Spanish and published in &lt;strong&gt;Revista Matemática Hispano-Americana&lt;/strong&gt; with the title &lt;a href=&quot;http://www.edu.xunta.gal/centros/ieseusebioguarda/system/files/u37/REVISTA_MATEM__TICA_HISPANOAMERICANA.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Evolución de la noción de integral&lt;/em&gt;&lt;/a&gt; and published in 2 parts: &lt;strong&gt;(i)&lt;/strong&gt; R. M. H.-M. (2) 2 #3 (March 1927), pp. 65-74. &lt;strong&gt;(ii)&lt;/strong&gt; R. M. H.-M. (2) 2 #4 (April 1927), pp. 97-106.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Book Reviews I know of:&lt;/em&gt; &lt;a href=&quot;https://www.jstor.org/stable/1720870&quot; rel=&quot;nofollow noreferrer&quot;&gt;Truman Arthur Botts&lt;/a&gt;, &lt;strong&gt;Science&lt;/strong&gt; (N.S.) 155 #3765 (24 February 1967), p. 992; &lt;a href=&quot;https://www.jstor.org/stable/24063307&quot; rel=&quot;nofollow noreferrer&quot;&gt;A. S. G.&lt;/a&gt;, &lt;strong&gt;Current Science&lt;/strong&gt; 36 #7 (5 April 1967), p. 194; &lt;a href=&quot;https://www.jstor.org/stable/2313835&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas William Hawkins&lt;/a&gt;, &lt;strong&gt;American Mathematical Monthly&lt;/strong&gt; 75 #6 (June-July 1968), pp. 696-697; &lt;a href=&quot;https://books.google.com/books?id=98I6D8hLp9oC&amp;amp;pg=PA753&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roger Philip Rigelhof&lt;/a&gt;, &lt;strong&gt;Canadian Mathematical Bulletin&lt;/strong&gt; 11 #5 (December 1968), pp. 753-754; &lt;a href=&quot;https://www.jstor.org/stable/3611895&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mark Edward Noble&lt;/a&gt;, &lt;strong&gt;Mathematical Gazette&lt;/strong&gt; 52 #382 (December 1968), 412-413; &lt;a href=&quot;https://www.jstor.org/stable/41092269&quot; rel=&quot;nofollow noreferrer&quot;&gt;André Reix&lt;/a&gt;, &lt;strong&gt;Revue Philosophique de la France et de l'Étranger&lt;/strong&gt; 166 #4 (October-December 1976), 437-438 (in French).&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-18T16:37:26.297" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11350" PostTypeId="2" ParentId="11346" CreationDate="2020-01-18T17:45:21.737" Score="6" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;“Mathemata mathematicis scribuntur.”&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is the original Latin of Copernic which is easily translated as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Mathematics is written for mathematicians.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but Edward Rosen chose to translate this famous passage as &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;“Astronomy is written for astronomers.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Obviously &quot;astronomy&quot; is not the author's word and also it is generally agreed that there was no mathematicians around if the word denotes what we mean today. The quote is commented by Westmann (&lt;em&gt;The Copernican Question&lt;/em&gt;, 2011, p.37):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Edward Rosen chose to translate this famous passage as &quot;Astronomy is&#xA;  written for astronomers,”  straining to get the text’s language into&#xA;  agreement with his own assumptions about how Copernicus conceived his&#xA;  role. But neither my rendering nor Rosen’s is quite satisfactory&#xA;  without further qualification. For the historian to call Copernicus a&#xA;   mathematician evokes confusing associations with the current domain&#xA;  of  meaning, in which mathematicians may or may not test hypotheses&#xA;  against the physical world; and to call him an astronomer overrides&#xA;  the meaning that mathematicus had in the sixteenth century, that is,&#xA;  someone skilled in any subject that involved mathematics—for example,&#xA;  optics, music, statics, or astrology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is rather early in his book and it is mostly a rhetorical move: obviously &lt;em&gt;De revolutionibus&lt;/em&gt; is not for someone skilled in optics, music or statics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;History of Science tends to get badly distorted when people refuse to acknowledge that from Hellenistic times down to Galileo astrology was a major branch of interest. Attempts to erase it have been made first by devout Christians and later by the Enlightenment. Westman has made a significant contribution to put things back where they belong (but that is an other topic).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Rosen's rendering needs qualification as the Latin words astronomia/astronomer appear in Osiander's preface. Latin translation of Aristotle use the word 'astrologia' for which modern editions put 'astronomy'. Mathematics is referred to as 'geometry' and this is also the meaning of the (fake) motto &quot;Let no one untrained in geometry/ageometretos/ enter here&quot;. The distinction &quot;mathematics&quot; vs. &quot;geometry&quot; is clearly present in Classical Latin as seen it the Codex Justinianeus (9.18.0, quoted above).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, the textual problem is how much astrology was the 16th.c. Church willing to allow? There is no good answer, at least not before Galileo's trial. Liberal minded authorities tended to present it as 'mathematics' while zealots saw it as noxious false prophecies. Copernic composed his dedication well informed for whom it was written.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-01-18T17:45:21.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11351" PostTypeId="1" CreationDate="2020-01-19T00:35:11.893" Score="8" ViewCount="308" Body="&lt;p&gt;A &lt;a href=&quot;https://space.stackexchange.com/questions/39207/what-would-be-different-about-satellite-orbits-if-earth-were-prolate-would-we-h?noredirect=1#comment133181_39207&quot;&gt;comment&lt;/a&gt; below &lt;em&gt;What would be different about satellite orbits if Earth were prolate? Would we have Sun-synchronous and Molniya orbits?&lt;/em&gt; got me reading Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacobi_ellipsoid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacobi ellipsoid&lt;/a&gt; which begins:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Before Jacobi, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Maclaurin_spheroid&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maclaurin spheroid&lt;/a&gt;, which was formulated in 1742, was considered to be the only type of ellipsoid which can be in equilibrium. Lagrange in 1811&lt;a href=&quot;https://en.wikipedia.org/wiki/Jacobi_ellipsoid#cite_note-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt; considered the possibility of a tri-axial ellipsoid being in equilibrium, &lt;strong&gt;but concluded that the two equatorial axes of the ellipsoid must be equal,&lt;/strong&gt; leading back to the solution of Maclaurin spheroid. But &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacobi&lt;/a&gt; realized that &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt;'s demonstration is a sufficiency condition, but not necessary. He remarked, &quot;One would make &lt;strong&gt;a grave mistake&lt;/strong&gt; if one supposed that the spheroids of revolution are the only admissible figures of equilibrium even under the restrictive assumption of second degree surfaces&quot; and further adds that &quot;In fact a simple consideration shows that ellipsoids with three unequal axes can very well be figures of equilibrium; and that one can assume an ellipse of arbitrary shape for the equatorial section and determine the third axis (which is also the least of the three axes) and the angular velocity of rotation such that the ellipsoid is a figure of equilibrium.&lt;a href=&quot;https://en.wikipedia.org/wiki/Jacobi_ellipsoid#cite_note-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;If I understand correctly, Lagrange came to the conclusion that a rotating body under hydrostatic equilibrium (such as a planet) would be of necessity a spheroid and so the cross-section through the equator would be a circle. Jacobi then found that this is not necessarily true, and a rotating triaxial ellipsoid could also be in hydrostatic equilibrium.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is my understanding of the situation correct?&lt;/li&gt;&#xA;&lt;li&gt;Did &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. L. Lagrange&lt;/a&gt; make an error in this case? If so, what was it exactly? Was it intuitive or mathematical in nature?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-08-18T14:05:26.693" LastActivityDate="2021-08-18T14:05:26.693" Title="What exactly was Lagrange's &quot;grave mistake&quot; with respect to rotating bodies under hydrostatic equilibrium?" Tags="|mathematics|physics|classical-mechanics|lagrange|fluid-dynamics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11352" PostTypeId="1" AcceptedAnswerId="11353" CreationDate="2020-01-19T19:47:51.410" Score="10" ViewCount="559" Body="&lt;p&gt;In Weyl's &quot;Algebraic Theory of Numbers&quot;, which was written in 1940, there are many symbols that look handwritten, such as Fraktur (or Sütterlin, whatever you want to call it) letters for ideals. His symbol for the rational numbers looks to me like something else entirely. You can see it a few times at the start of Chapter IV:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/71OOF.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/71OOF.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is that symbol?!?  I suppose it should be a letter, but when I look online at samples of handwritten Fraktur nothing resembles what Weyl is writing for rational numbers. I thought maybe it could be a lower-case q, but the way it curves on the bottom is going the wrong way for it to be a convincing q.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2020-01-19T20:54:06.510" Title="Notation from Weyl's algebraic number theory book" Tags="|notation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11353" PostTypeId="2" ParentId="11352" CreationDate="2020-01-19T20:54:06.510" Score="12" Body="&lt;p&gt;On page 10 of that book the author wrote&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The most important example of a ground field is the field of common rational numbers for which I use the freely invented symbol 9...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;where he uses the symbol in question instead of the 9 that I used above.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2020-01-19T20:54:06.510" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11355" PostTypeId="1" AcceptedAnswerId="11359" CreationDate="2020-01-21T12:59:24.290" Score="4" ViewCount="453" Body="&lt;p&gt;As I'm reading a scientific paper from 1964, I'm wondering how researchers were able to write papers without the use of a computer. Cursive fonts, different font sizes, math, etc. are all present in these (old) papers.&lt;/p&gt;&#xA;" OwnerUserId="10998" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:42.640" LastActivityDate="2021-02-20T15:20:42.640" Title="What technology did researchers use to write scientific papers before the computer?" Tags="|scientific-method|computers|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11356" PostTypeId="1" AcceptedAnswerId="11357" CreationDate="2020-01-21T13:28:31.820" Score="2" ViewCount="756" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Catamorphism#Terminology_and_history&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Banana&amp;quot; brackets&lt;/a&gt; are used to denote catamorphisms:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another notation found in the literature is &lt;a href=&quot;https://i.stack.imgur.com/CYEy8m.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CYEy8m.png&quot; alt=&quot;(!|f|!)&quot; /&gt;&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These symbols are very similar to the composition of a &lt;span class=&quot;math-container&quot;&gt;$($&lt;/span&gt; and a &lt;span class=&quot;math-container&quot;&gt;$|$&lt;/span&gt;, is this similarity more than coincidence? I.e. is there some motivation where they represent the same as &lt;span class=&quot;math-container&quot;&gt;$|...|$&lt;/span&gt; followed by &lt;span class=&quot;math-container&quot;&gt;$(...)$&lt;/span&gt; ?&lt;/p&gt;&#xA;" OwnerUserId="10974" LastEditorUserId="10974" LastEditDate="2021-04-11T09:28:52.977" LastActivityDate="2021-04-11T09:28:52.977" Title="What is the origin of &quot;banana brackets&quot;?" Tags="|mathematics|notation|computer-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11357" PostTypeId="2" ParentId="11356" CreationDate="2020-01-21T13:28:31.820" Score="3" Body="&lt;p&gt;The earliest reference I can find to &amp;quot;banana brackets&amp;quot; is in:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iN1Gj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iN1Gj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;G. Malcolm. &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0167642390900237&quot; rel=&quot;nofollow noreferrer&quot;&gt;Data structures and program transformation&lt;/a&gt;. Science of Computer Programming, 14(2-3):255-280, October 1990.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Where they are clearly crescent-moon/banana shaped symbols: ⦅...⦆.&lt;/p&gt;&#xA;&lt;p&gt;The later style using &lt;span class=&quot;math-container&quot;&gt;$($&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|$&lt;/span&gt; seems to be a typographic practicality, and is used by Malcolm himself in another publication the same year:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GTBhM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GTBhM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;G. Malcolm. &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.221.4281&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Algebraic data types and program transformation&lt;/a&gt;. PhD thesis, Groningen University, 1990.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10974" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-21T14:16:30.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11358" PostTypeId="1" AcceptedAnswerId="11360" CreationDate="2020-01-21T13:52:11.890" Score="1" ViewCount="148" Body="&lt;p&gt;Where in Eulers writings can I find a proof of his &lt;a href=&quot;https://en.wikipedia.org/wiki/Homogeneous_function#Positive_homogeneity&quot; rel=&quot;nofollow noreferrer&quot;&gt;homogeneous function theorem&lt;/a&gt;: &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is a homogeneous function of degree &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$x_1,\ldots,x_n$&lt;/span&gt; iff  &lt;span class=&quot;math-container&quot;&gt;$ky = \sum_{i=1}^n x_i\frac{\partial y}{\partial x_i}$&lt;/span&gt;&lt;/em&gt;? None of the sources I've seen cite him.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm curious because in his &lt;em&gt;Introduction to the analysis of the infinite&lt;/em&gt; he defines a homogeneous function as one &quot;in which each term has the same degree&quot; and goes on to discuss several examples (not just polynomials). But I don't see him writing down the &quot;modern&quot; definition &#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(\lambda{\bf x})=\lambda^k f({\bf x})$$&lt;/span&gt;&#xA; and wonder how he would prove the result without this. (I know he used function notation, but very rarely.)&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="2905" LastEditDate="2020-02-07T07:30:02.310" LastActivityDate="2020-02-07T07:30:02.310" Title="Where did Euler prove 'his' theorem on homogeneous functions?" Tags="|mathematics|notation|calculus|euler|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11359" PostTypeId="2" ParentId="11355" CreationDate="2020-01-21T15:46:28.810" Score="9" Body="&lt;p&gt;There were several layers of craft technology now partly superseded by pdflatex and friends.  There were author's instructions for marking up manuscripts with type font indications, there were technical typists adept at using special math symbol shapes with their modified typewriters, there were Monotype operators who ran typesetting machines from the marked up copy supplied, and so on.  `&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The period 1960-1980 saw a lot of  technology churn as different kinds of phototypesetting machines came into use, of varying typographic quality (or lack of quality).  Stabilizing this situation was one of Knuth's reasons for getting interested in typesetting: he wanted his second edition (phototypeset) of his magnum opus to look like the first (monotype hot metal set).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added 23 Jan:  And, as the &lt;a href=&quot;https://mathoverflow.net/questions/19930/writing-papers-in-pre-latex-era&quot;&gt;thread&lt;/a&gt; mentioned in a comment by Gerald Edgar  makes clear, there was also considerable churn on the software side.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-01-24T01:58:31.943" LastActivityDate="2020-01-24T01:58:31.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11360" PostTypeId="2" ParentId="11358" CreationDate="2020-01-21T16:01:11.593" Score="4" Body="&lt;p&gt;Lucky you! (or me :-) ).  This question was answered a while back on &lt;a href=&quot;https://math.stackexchange.com/questions/3457070/where-did-euler-first-write-about-homogeneous-function&quot;&gt;Math.SE&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I found his original thoughts in the translated version of&#xA;  &quot;Institutiones calculi differentialis cum eius usu in analysi&#xA;  finitorum ac doctrina serierum, volume 1&quot;, chapter 7. The translation&#xA;  is called &quot;Foundations of Differential Calculus&quot; and a link is found&#xA;  here &lt;a href=&quot;https://link.springer.com/book/10.1007%2Fb97699&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/book/10.1007%2Fb97699&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-01-21T16:01:11.593" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11361" PostTypeId="1" AcceptedAnswerId="11366" CreationDate="2020-01-21T18:47:53.900" Score="0" ViewCount="156" Body="&lt;p&gt;In the book &quot;The Road to Reality&quot;, Penrose claims that Joseph Lagrange was aware of Hamiltonian mechanics. Is there any evidence that this statement is true?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10106" LastEditDate="2020-05-16T18:44:57.047" LastActivityDate="2020-05-16T18:44:57.047" Title="Was Lagrange aware of Hamiltonian mechanics?" Tags="|mathematicians|biographical-details|lagrange|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11363" PostTypeId="1" AcceptedAnswerId="11364" CreationDate="2020-01-21T22:09:48.823" Score="2" ViewCount="104" Body="&lt;p&gt;Diophantus’ notation for higher powers of the unknown includes a superscript, usually written as a lowercase or uppercase upsilon - e.g.,  &lt;span class=&quot;math-container&quot;&gt;$\Delta^{\upsilon}$&lt;/span&gt; as the square.  It is not clear to me what the role of this superscript is.  Neither &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; nor &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; (cube) is used in the Greek number system so there is no risk of ambiguity (though curiously &lt;span class=&quot;math-container&quot;&gt;$\upsilon$&lt;/span&gt; is part of the number system as &lt;span class=&quot;math-container&quot;&gt;$400$&lt;/span&gt;). The superscript appears to be superfluous. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: What is the role of the superscript here?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Diophantus includes a special symbol - &lt;span class=&quot;math-container&quot;&gt;$\mathring{M}$&lt;/span&gt; - to identify the constant term in an expression.  On the one hand, we can see why this is necessary formally in order to avoid ambiguity that may arise as a result of his practice of writing coefficients on the right and the absence of a symbol for addition. On the other hand, again from a formal point of view, it suggests that he is treating the constant term as a separate class of object, distinct from number, square, cube, etc., and with a separate designation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: Did Diophantus’ inventory of types treat constants as a separate type?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2020-01-22T07:57:13.437" LastActivityDate="2020-01-22T18:55:18.160" Title="Superscripts and constants in Diophantus’ Arithmetica" Tags="|mathematics|ancient-greece|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11364" PostTypeId="2" ParentId="11363" CreationDate="2020-01-22T01:47:40.807" Score="4" Body="&lt;p&gt;The letter &lt;span class=&quot;math-container&quot;&gt;$\upsilon$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\Upsilon$&lt;/span&gt; is the second letter of &lt;em&gt;δύναμις&lt;/em&gt;, &lt;em&gt;κύβος&lt;/em&gt; and &lt;em&gt;δυναμοδύναμις&lt;/em&gt; (square, cube and bisquare) of the words the notation abbreviates. Being the same letter, it plays an additional role of identifying the power symbols as being of a kind, marking Diophantus's &quot;numerical species&quot; (squares, cubes, etc., multiplied by coefficients). Notably, his power symbols have the same exponents, whereas ours have the same base. In other words, he thinks of them not as we do, a power marker modifying the &quot;unknown&quot;, but in reverse, the implicit &quot;unknown&quot; modifying a generic power, given by the base of the power symbols. As Hettle puts it in The &lt;a href=&quot;https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1190&amp;amp;context=jhm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Symbolic and Mathematical Influence of Diophantus’s&#xA;Arithmetica&lt;/a&gt;: &quot;&lt;em&gt;Diophantus defined powers of unknowns not as a power of the unknown, but as an unknown with the property of being that power of a positive rational number; accordingly he refers to each power as a “species” of number&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In some sense then the letter &lt;span class=&quot;math-container&quot;&gt;$\upsilon$&lt;/span&gt; is the mark of genericity, or of the unknown. However, the unknown by itself is denoted &lt;em&gt;ς&lt;/em&gt;, a contraction of the first two letters of &lt;em&gt;ἀριθμος&lt;/em&gt;, &quot;number&quot;, according to Heath, which Diophantus uses to name it in the text. In contrast to the earlier tradition going back to Pythagoreans, he for the first time admits (positive) rationals as &quot;numbers&quot;, and they are the only possible values of the unknown and legitimate solutions to equations he considers. Although it would seem natural to us to assimilate &lt;em&gt;ς&lt;/em&gt; as marking &quot;linear species&quot;, the difference in notation suggests that he does not think that way. Perhaps, this is a trace of Pythagoreans not viewing &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; as a &quot;number&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Numerical species are also sharply separated by Diophantus, notationally and terminologically, from free terms, to us, &quot;zero power species&quot;, marked by &lt;span class=&quot;math-container&quot;&gt;$\mathring{M}$&lt;/span&gt; for &lt;em&gt;monad&lt;/em&gt;. They are Pythagorean and Euclid's &quot;numbers&quot; (positive integers greater than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, i.e. multitudes of &lt;em&gt;monads&lt;/em&gt;) to which square, cube, etc., can be applied numerically. Apparently, he was concerned with possible confusion of his species  with perfect squares, cubes, etc., even aside from the fact that zero concept, let alone zero power, was not available at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a more detailed commentary from &lt;a href=&quot;https://grbs.library.duke.edu/article/view/15185&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi's Unaccountable Numbers&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;At the beginning of his treatise, Diophantus explains the&#xA;  notation that he will use throughout; he is the first Greek&#xA;  mathematician who consistently adopts a set of signs in order&#xA;  to make his text more concise and, in a sense, conducive to the&#xA;  kind of “algebraic” manipulations forming the technical core of&#xA;  his method for solving numerical problems. In particular, he&#xA;  establishes a terminology to denote what in algebraic language are the powers of the “unknown” &lt;span class=&quot;math-container&quot;&gt;$x^2, x^3, …$&lt;/span&gt;; in Diophantus’&#xA;  theoretical framework, these are abstract numerical &lt;em&gt;εἴδη&lt;/em&gt;&#xA;  “species,” namely generic square, cube, … numbers. The&#xA;  species introduced are assigned a denomination and a conventional&#xA;  sign; the sign is made of the first letter of each component&#xA;  of the denomination, possibly supplemented with the&#xA;  second letter (this always happens to be upsilon): to the generic&#xA;  square number (&lt;em&gt;δύναμις&lt;/em&gt;) corresponds the sign &lt;span class=&quot;math-container&quot;&gt;$Δ^\Upsilon$&lt;/span&gt;, to the &lt;em&gt;κύβος&lt;/em&gt;&#xA;  the sign &lt;span class=&quot;math-container&quot;&gt;$K^\Upsilon$&lt;/span&gt;, to the fourth power (&lt;em&gt;δυναμοδύναμις&lt;/em&gt;) the sign &lt;span class=&quot;math-container&quot;&gt;$Δ^\UpsilonΔ$&lt;/span&gt;, etc... At the &lt;em&gt;end&lt;/em&gt; of the list of species, Diophantus also assigns a&#xA;  denomination and a conventional sign to the most generic abstract&#xA;  number, namely one that neither is a particular number&#xA;  nor can be said to have the features characterizing one of the&#xA;  aforementioned species; I shall call it, with a slight abuse of&#xA;  language, the “1-species”; it corresponds to the “unknown” of present-day algebra.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] Capital &lt;span class=&quot;math-container&quot;&gt;$Δ$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\Upsilon$&lt;/span&gt; are currently printed, but&#xA;  of course no indication to that effect is contained in the text. It is quite&#xA;  obvious that our notation owes very much, both in conception and in the&#xA;  form of the signs, to Diophantus’: note his use of the term &lt;em&gt;δύναμις&lt;/em&gt; “power”&#xA;  and the idea of putting a part of the conventional sign “at the exponent.”&#xA;  One crucial difference is that we conceive of the species as powers of the&#xA;  “unknown,” whereas Diophantus draws a sharp distinction between these&#xA;  notions, as we shall see presently. This difference is made particularly conspicuous&#xA;  by the fact that Diophantus’ conventional signs all have the same&#xA;  exponent (the insignificant letter &lt;em&gt;upsilon&lt;/em&gt;) and a variable “base” indicating the&#xA;  species (letters &lt;span class=&quot;math-container&quot;&gt;$Δ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, possibly doubled), whereas modern algebraic signs&#xA;  all have the same base (the most significant “unknown” &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;) and a variable exponent&#xA;  indicating the power to which the base is to be raised.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] These species must not be confused with particular numbers that happen to be square, cube, fourth powers. On a terminological level, Diophantus settles the problem of separating&#xA;  particular square numbers from the species “square” by&#xA;  means of the opposition &lt;em&gt;τετράγωνος/δύναμις&lt;/em&gt;; a lexical ambiguity&#xA;  (admittedly quite harmless) remains in the case of the&#xA;  &lt;em&gt;κύβος&lt;/em&gt;, which may designate both a particular cube number&#xA;  (such as &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;) and the species “cube.”&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Diophantus highlights this difference when he alludes to the Euclidean&#xA;  definition of number (Elem. 7.def.2) and when he defines a square number:&#xA;  in both cases he adds a &lt;em&gt;τινος&lt;/em&gt;, either to &lt;em&gt;πλήθους&lt;/em&gt; or to &lt;em&gt;ἀριθμοῦ&lt;/em&gt;. This means&#xA;  that the object so qualified is particular, yet generic... The following considerations&#xA;  may help further clarify the point. Diophantine numerical&#xA;  species were invoked by the fourth-century mathematical polymath and&#xA;  commentator Theon of Alexandria (In Alm. 452.21–453.16 Rome) to explain&#xA;  the structure of orders within the sexagesimal system used by the&#xA;  astronomers. The sexagesimal orders are in fact numerical &lt;em&gt;εἴδη&lt;/em&gt;; they correspond&#xA;  to the orders of magnitude in the decimal system... these numerical species do not coincide with particular numbers: indeed, &lt;span class=&quot;math-container&quot;&gt;$300$&lt;/span&gt; is not a square, but &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; items of the square &lt;em&gt;εἶδος&lt;/em&gt; “hundreds”;&#xA;  conversely, the species “hundreds” is not a number (it does not even&#xA;  coincide with number &lt;span class=&quot;math-container&quot;&gt;$100$&lt;/span&gt;).&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-22T18:55:18.160" LastActivityDate="2020-01-22T18:55:18.160" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11366" PostTypeId="2" ParentId="11361" CreationDate="2020-01-22T09:21:01.023" Score="3" Body="&lt;p&gt;Yes, he was, at least in a particular example. Near the end of his life, in 1808-09, Lagrange studied perturbative dynamics of a planet on an elliptic orbit, and derived what came to be called Hamiltonian equations for it in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k229225j/f811&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Second mémoire sur la théorie de la variation des constantes arbitraires dans les problèmes de mécanique, dans lequel on simplifie l'application des formules générales à ces problèmes&lt;/em&gt; (1809)&lt;/a&gt;. A modern discussion with details can be found in &lt;a href=&quot;https://projecteuclid.org/euclid.bams/1183548217&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weinstein's Symplectic Geometry&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The first symplectic manifold&#xA;  was introduced by Lagrange [LAI] in 1808. In studying the motion of the&#xA;  planets under the influence of their mutual gravitational interaction, he took&#xA;  as a starting point the elliptical motion of a single planet around the sun.&#xA;  (Actually, the focus of the ellipse is the center of the mass of the sun-planet&#xA;  system, but we will ignore this point here.) This ellipse was then considered to&#xA;  &quot;drift&quot; under the disturbing influence of the other planets. The drift was&#xA;  described by a system of differential equations, and Lagrange sought to put&#xA;  these equations in the simplest possible form.&lt;/em&gt; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lagrange described the motion by six elements (real functions) &lt;span class=&quot;math-container&quot;&gt;$a_i$&lt;/span&gt;, introduced what is now called Lagrange brackets for them, and was able to write the equations in the form &#xA;&lt;span class=&quot;math-container&quot;&gt;$\frac{da_i}{dt}=\sum_j b_{ij}\frac{\partial\Phi}{\partial a_j}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$b_{ij}$&lt;/span&gt; was antisymmetric. &quot;&lt;em&gt;At the time of Lagrange, this advance was of&#xA;advantage largely because it shortened by a factor of six some rather&#xA;complicated computations&lt;/em&gt;&quot;. The &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; was what Lagrange &quot;&lt;em&gt;called the disturbing&#xA;function, which depends on the perturbing forces but is independent of the&#xA;choice of elements&lt;/em&gt;&quot;. The equations meant that the total time derivative of &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; along the trajectories was &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, i.e. it was conserved. In 1809 he found a change of variables from &lt;span class=&quot;math-container&quot;&gt;$a_i$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$q_i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p_i$&lt;/span&gt;, that satisfied the canonical commutation relations for his brackets. In terms of these elements, the&#xA;equations he derived &quot;&lt;em&gt;are known today as Hamilton's equations&lt;/em&gt;&quot;, with &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; as the Hamiltonian.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-01-22T09:21:01.023" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11369" PostTypeId="2" ParentId="11355" CreationDate="2020-01-22T17:39:52.170" Score="4" Body="&lt;p&gt;My theoretical physics papers and thesis were written by hand, then typed on an IBM &quot;golf ball&quot; typewriter.  The golf ball was so called because the type head was about the same size as a golf ball.  The type heads were interchangeable with different fonts and symbols. Corrections were made using tippex. 40 years later, I'm still amazed at the skill of the typist. &lt;/p&gt;&#xA;" OwnerUserId="6162" LastActivityDate="2020-01-22T17:39:52.170" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11370" PostTypeId="2" ParentId="8184" CreationDate="2020-01-22T19:31:56.390" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wiktionary.org/wiki/abacist&quot; rel=&quot;nofollow noreferrer&quot;&gt;abacists&lt;/a&gt; of 16th century Italy correctly identified the rule of signs from multiplication.  Typical of their arguments (using modern notation) is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$64 = (10 - 2) \times (10 - 2) = \dots =100 + (-40) + (-2\times-2) = 60 + (-2\times-2),$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;so that &quot;subtract 2 times subtract 2&quot; must equal four. (There was still no concept of a negative number.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gerolamo_Cardano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cardano&lt;/a&gt; insisted that the abacists were mistaken, giving a geometric argument that the arithmetic situation here was (using modern algebraic notation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$64 = (10 - 2) \times (10 - 2) =(10\times10) - (2\times20) + (2\times2),$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;where the &quot;&lt;span class=&quot;math-container&quot;&gt;$-(2\times20)$&lt;/span&gt;&quot; corresponds to removing two rectangles from the larger square and the &quot;&lt;span class=&quot;math-container&quot;&gt;$+ (2\times2)&quot;$&lt;/span&gt; adds back the smallest square that has now been removed twice.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-01-22T19:31:56.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11371" PostTypeId="1" AcceptedAnswerId="11373" CreationDate="2020-01-23T20:07:52.673" Score="3" ViewCount="158" Body="&lt;p&gt;I am trying to comprehend an article about primary decomposition of ideals. Zero - dimensional ideals are quite emphasized there. I wonder where zero - dimensional ideals come from, what is the history behind? I found that,in fact &lt;span class=&quot;math-container&quot;&gt;$Krull$&lt;/span&gt; dimension is zero. But how did Krull invent it? What was he searching  for?  Sincere thanks.&lt;/p&gt;&#xA;" OwnerUserId="6416" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:37.260" LastActivityDate="2021-02-20T15:20:37.260" Title="Where did Krull dimension and zero-dimensional ideals come from?" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11373" PostTypeId="2" ParentId="11371" CreationDate="2020-01-24T08:37:39.020" Score="8" Body="&lt;p&gt;He was concerned with providing an abstract analog to the geometric notion of dimension for algebraic varieties given by polynomial rings. Zero-dimensional ideals are abstract algebraic analogs of discrete collections of points. The idea goes back to Hilbert's conversion of algebraic geometry into the language of rings and ideals, and Dedekind inspired Noether's program of abstracting algebra from the more concrete notions for number and polynomial rings. It was embodied in van der Waerden's &lt;em&gt;Moderne Algebra&lt;/em&gt; (1930, based on Artin's and Noether's lectures in 1924-28), and arithmetization of algebraic geometry by Weil and Zariski in 1930-40s. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Krull was influenced by Klein's geometric vision at Erlangen, where he became a professor in 1928, and remarked that &quot;&lt;em&gt;Dedekind's language is better adapted to the needs of arithmetic, but experience shows it seems strange to every beginner&lt;/em&gt;&quot;. According to &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Krull.html&quot; rel=&quot;noreferrer&quot;&gt;McTutor's biography of Krull&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If Emmy Noether had the greatest influence on the topics which Krull would spend his life researching, it can be seen from this inaugural address that it was Klein who had the greatest influence on Krull's large scale view of mathematics... In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held. Perhaps the reason that the idea of the Krull dimension is such a natural concept is that it encapsulates in an abstract setting the analogues of geometric dimensions.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ems-ph.org/journals/newsletter/pdf/1999-06-32.pdf&quot; rel=&quot;noreferrer&quot;&gt;Gray's biographical note on Krull for the EMS Newsletter&lt;/a&gt; explains:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;A significant achievement in ring theory was his introduction of what today is called the Krull   dimension of   a   commutative Noetherian  ring,  and  the  proof  of  the principal  ideal  theorem  in  this  setting.&#xA;  This  result  was  quickly  recognised  as  a decisive advance in Noether’s programme&#xA;  of emancipating abstract ring theory form the  theory  of  polynomial  rings...  The  theorem  asserts that  if  a  local  ring  with  unique  maximal ideal &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;&#xA;  has  (Krull)  dimension  &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the smallest number such that there are &#xA;  &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; distinct  elements  of  &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; which  are  not  contained in any other prime ideal of the ring. Geometrically, if &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the maximal ideal of&#xA;  functions  that  vanish  at  a  point  on  a  variety, this is the claim that at least &#xA;  &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; functions are needed to pick out a point.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;More details can be found in &lt;a href=&quot;https://books.google.com/books?id=8MK_AwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Episodes in the History of Modern Algebra (1800-1950)&lt;/a&gt; and McLarty's entry to &lt;a href=&quot;https://books.google.com/books?id=GywTDAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;The Architecture of Modern Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-24T09:47:13.467" LastActivityDate="2020-01-24T09:47:13.467" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11374" PostTypeId="1" AcceptedAnswerId="11378" CreationDate="2020-01-24T13:08:39.423" Score="8" ViewCount="382" Body="&lt;p&gt;A &lt;a href=&quot;https://books.google.com/books?id=JlR-Ehk35XkC&amp;amp;pg=PA4&amp;amp;lpg=PA4&amp;amp;dq=enderton%20%22the%20reasons%20for%20using%20the%20word%20power%20in%20this%20context%20are%20not%20very%20convincing%22&amp;amp;source=bl&amp;amp;ots=mbz1LgYBjS&amp;amp;sa=X&amp;amp;ved=2ahUKEwjWgoe-pJznAhUcI6YKHXQDDRwQ6AEwAHoECAgQAQ#v=onepage&amp;amp;q=enderton%20%22the%20reasons%20for%20using%20the%20word%20power%20in%20this%20context%20are%20not%20very%20convincing%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;footnote&lt;/a&gt; of Enderton's &lt;em&gt;Elements of Set Theory&lt;/em&gt; (1977, page 4) for the definition of power set states that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the reasons for using the word &quot;power&quot; in this context are not very convincing, but the usage is now well established.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I wonder what the not-so-convincing reasons were. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 1: A question regarding the etymology of power is &lt;a href=&quot;https://hsm.stackexchange.com/questions/3254/etymology-of-power-math&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;EDIT 2: On &lt;a href=&quot;https://books.google.com/books?id=JlR-Ehk35XkC&amp;amp;pg=PA141&amp;amp;dq=%22the%20term%20power%20set%20is%20rooted%20in%20the%20fact%22&amp;amp;sa=X&amp;amp;ved=0ahUKEwj9v6qSlKnnAhUQCpQKHWOfC1sQ6wEILjAA#v=onepage&amp;amp;q=%22the%20term%20power%20set%20is%20rooted%20in%20the%20fact%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 141&lt;/a&gt;, Enderton wrote that the term &quot;power set&quot; is rooted in the fact that card &lt;span class=&quot;math-container&quot;&gt;$\mathscr{P}A$&lt;/span&gt; equals &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; raised to the power card &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7937" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:31.487" LastActivityDate="2021-02-20T15:20:31.487" Title="What were the not-so-convincing reasons for using the word &quot;power&quot; for power sets?" Tags="|terminology|set-theory|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11375" PostTypeId="1" AcceptedAnswerId="11382" CreationDate="2020-01-24T15:02:02.373" Score="6" ViewCount="165" Body="&lt;p&gt;I'm asking this question looking at the unit circle, and thinking that greek mathematicians didn't use negative numbers. Maybe that can give enough insight into what I'm asking?&lt;/p&gt;&#xA;" OwnerUserId="11013" LastEditorUserId="26" LastEditDate="2021-02-20T15:20:22.217" LastActivityDate="2021-02-20T15:20:22.217" Title="When did trigonometry start using negative numbers?" Tags="|euclidean-geometry|trigonometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11378" PostTypeId="2" ParentId="11374" CreationDate="2020-01-25T07:47:05.833" Score="7" Body="&lt;p&gt;As best as I can tell, the term &lt;em&gt;Potenzbegriff&lt;/em&gt; (powerclass, later variation &lt;em&gt;Potenzmenge&lt;/em&gt;, powerset) was introduced by Bernstein in the late 1890s (Cantor did not use it in his papers). In his Habilitation dissertation &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01457734&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Untersuchungen aus der Mengenlehre&lt;/em&gt; (1901, published 1905)&lt;/a&gt; he states in the introduction (my translation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The introduction of the concept of powerclass, and the proof of the equivalence theorem to be mentioned later has now made it possible to arrive at conclusions that were previously only possible through laborious detours by an almost elementary calculation.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The word &lt;em&gt;begriff&lt;/em&gt; literally means &quot;concept&quot;, and intensional classes (falling under a concept) were not fully separated from extensional sets until Hausdorff's &lt;em&gt;Grundzüge der Mengenlehre&lt;/em&gt; (1914), although Zermelo and others operated with the latter earlier. The &quot;equivalence theorem&quot; is now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schröder–Bernstein theorem&lt;/a&gt;, and Bernstein gives a proof in §1 of the dissertation. The original proof was given at Cantor's seminar in Halle in 1897, and does not survive (Dedekind gave a proof in 1887, but did not publish, Schröder's 1896 announcement was of a proof that apparently had a flaw). The set power (&lt;em&gt;Potenz&lt;/em&gt;) is introduced in §2 as follows:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; are two sets, we call that set which - in the sense of a known expression - contains all combinations of elements from &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; to the classes of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;, the power &lt;span class=&quot;math-container&quot;&gt;$M^N$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; raised to &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;). Regarding the application to addition, multiplication and powerclasses of commutative and associative laws, they are the same as for finite numbers&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the apparent motivation for &lt;em&gt;Potenz&lt;/em&gt; (power) seems to be as commonly assumed, by analogy to raising numbers to powers. Bernstein does not use &lt;span class=&quot;math-container&quot;&gt;$2^N$&lt;/span&gt; generally, or calls the class of all subsets the powerclass. But he does write &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_\alpha}$&lt;/span&gt;, meaning set cardinality (&lt;em&gt;Mächtigkeit&lt;/em&gt;, often also translated as &quot;power&quot;) in §9, when discussing the continuum hypothesis.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-03-25T17:56:51.053" LastActivityDate="2020-03-25T17:56:51.053" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11379" PostTypeId="1" AcceptedAnswerId="11392" CreationDate="2020-01-25T17:49:43.607" Score="0" ViewCount="70" Body="&lt;p&gt;Did Lagrange make any noteworthy contribution to the Knights Tour Problem?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10106" LastEditDate="2020-05-16T13:36:09.763" LastActivityDate="2020-05-16T13:36:09.763" Title="What was Lagrange's contribution to the Knights Tour Problem?" Tags="|mathematicians|biographical-details|lagrange|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11380" PostTypeId="2" ParentId="11343" CreationDate="2020-01-26T09:46:57.390" Score="5" Body="&lt;p&gt;The modern convention, like much of modern notation, goes back to &lt;a href=&quot;http://eulerarchive.maa.org/pages/E101.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's Introductio in Analysin Infinitorum (1748), chapter VIII&lt;/a&gt;, there is an &lt;a href=&quot;https://books.google.com/books/about/Introduction_to_Analysis_of_the_Infinite.html?id=se3UBwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation by Blanton&lt;/a&gt;. The unit circle was not invented to extend trigonometric functions, power series already accomplished that even for complex values, it was rather an illustrative device. At the beginning of the chapter Euler writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;After having considered logarithms and exponentials, we must now&#xA;turn to circular arcs with their sines and cosines. This is not only because these&#xA;are further genera of transcendental quantities, but also since they arise from logarithms&#xA;and exponentials when complex values are used. This will become&#xA;clearer in the development to follow.&#xA;We let the radius, or total sine, of a circle be equal to 1, then it is clear&#xA;enough that the circumference of the circle cannot be expressed exactly as a&#xA;rational number.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He then denotes half of the circumference by the familiar &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, and gives the standard suite of trig identities.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-08-24T19:44:35.567" LastActivityDate="2021-08-24T19:44:35.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11381" PostTypeId="1" CreationDate="2020-01-26T10:06:06.320" Score="3" ViewCount="139" Body="&lt;p&gt;I would assume that the early estimates were wildly off and wonder how this measurement was done.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2020-01-27T01:33:49.653" Title="When was the speed of a bullet or cannon ball first measured?" Tags="|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11382" PostTypeId="2" ParentId="11375" CreationDate="2020-01-26T10:13:12.157" Score="3" Body="&lt;p&gt;Seventeenth century mathematicians were still hesitatnt to use negative numbers even when it came to Cartesian coordinates. According to Tannery:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;One frequently attributes wrongly to Descartes the introduction of the convention of reckoning coordinates positively and negatively, in the sense in which we start them from the origin. The truth is that in this respect the Geometrie of 1637 contains only certain remarks touching the interpretation of real or false (positive or negative) roots of equations.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...If then we examine with care the rules given by Descartes in his Geometrie as well as his application of them, we notice that he&#xA;adopts as a principle that an equation of a geometric locus is not&#xA;valid except for the angle of the coordinates (quadrant) in which it&#xA;was established, and all his contemporaries do likewise. The extension&#xA;of an equation to other angles (quadrants) was freely made in particular cases for the interpretation of the negative roots of equations;&#xA;but while it served particular conventions (for example for reckoning&#xA;distances as positive and negative), it was in reality quite long in&#xA;completely establishing itself, and one cannot attribute the honor for&#xA;it to any particular geometer.&lt;/em&gt;&amp;quot;&#xA;[quoted from &lt;a href=&quot;https://archive.org/details/ahistorymathema02cajogoog/page/n186/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori, History of Mathematics, p.175-6&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even Newton and Leibniz, although they knew the power series for sine and cosine, did not treat them as functions of a real variable. The view of them as functions (in the analytic sense of 18th century) derives from &lt;a href=&quot;http://eulerarchive.maa.org/pages/E101.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's Introductio in Analysin Infinitorum (1748), chapter VIII&lt;/a&gt;, where he uses not only negative but even complex values as arguments, adopts the &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; symbol, and introduces the unit circle conventions. There is an &lt;a href=&quot;https://books.google.com/books/about/Introduction_to_Analysis_of_the_Infinite.html?id=se3UBwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation by Blanton&lt;/a&gt;. At the beginning of the chapter Euler writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;After having considered logarithms and exponentials, we must now&#xA;turn to circular arcs with their sines and cosines. This is not only because these&#xA;are further genera of transcendental quantities, but also since they arise from logarithms&#xA;and exponentials when complex values are used.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Later he notes explicitly that formulas like &lt;span class=&quot;math-container&quot;&gt;$\sin\left(\frac{4n+1}2\pi+z\right)=+\cos z$&lt;/span&gt; &amp;quot;&lt;em&gt;hold whether &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive or a negative integer&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-26T10:21:31.187" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11383" PostTypeId="2" ParentId="11381" CreationDate="2020-01-26T12:01:28.587" Score="5" Body="&lt;p&gt;Some prehistory involving Aristotle, Tartaglia, Galileo and Torricelli can be found in &lt;a href=&quot;http://fredrickey.info/talks/08-10-25-Ballistics-ARL/08-10-23-BallisticsARL-pulished.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Impact of Ballistics on Mathematics by McMurran and Rickey&lt;/a&gt;. It was more speculative than experimental, and focused more on the trajectories and range of projectiles rather than speed. Euler wrote a short note &lt;a href=&quot;http://eulerarchive.maa.org/pages/E853.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Meditatio in experimenta explosione tormentorum nuper instituta&lt;/em&gt; (1727-28)&lt;/a&gt; (better known for the employment of letter &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; for the base of natural logarithms) reporting on some early experiments with cannon balls, possibly conducted by Daniel Bernoulli. According to &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/002074039290061K&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johnson's Encounters between Robins, and Euler and the Bernoullis&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The circumstance of its employment there described was to report on seven experiments conducted on cannon balls of different weights projected by different weights of gun&#xA;  powder, for different angles of elevation; range and time of flight (time of ascent and descent&#xA;  were calculated separately) based on some of Euler's own calculations which rest on his&#xA;  assumptions about air resistance. The original title of the 10 page paper written in Latin,&#xA;  when translated, is Meditation upon Experiments made recently in the firing of Cannon. From the use of the word 'recently' it is deduced that the original article was&#xA;  written in 1727 or 1728. However, the manuscript of the paper was first printed at Petrograd&lt;/em&gt; [sic!]&#xA;  &lt;em&gt;only in 1862 in his Postumous Mathematical and Physical Work (pp. 468-477), and it would&#xA;  seem therefore to have been published much too late to be of value for artillerists at large...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;It is likely that these experiments were made by Daniel Bernoulli because all we see&#xA;  presented are various quantities (given) which are substituted into certain equations and&#xA;  then magnitudes such as time of flight, etc., determined. No experimental details are&#xA;  reported, e.g. details about the gun used, charges, shot, calibre and the weather, etc.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But historians of ballistics usually credit Robins with the first reliable measurements, conducted using a clever contraption he invented, known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Ballistic_pendulum&quot; rel=&quot;nofollow noreferrer&quot;&gt;ballistic pendulum&lt;/a&gt;. The momentum of a fired bullet was either measured directly, by firing it into the pendulum, or  by attaching the gun to the pendulum and measuring the recoil. The firing velocity could be found by dividing the momentum by the mass. Robins published his results, on this and many other ballistic topics, in New Principles of Gunnery (1742), which became the bible of modern ballistics. Euler translated it into German in 1745, with a commentary four times the size of the original that mathematized the subject. McMurran and Rickey give a detailed discussion of both. Here is from  &lt;a href=&quot;https://archive.org/details/exteriorballist00unkngoog/page/n24/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Exterior Ballistics in the Plane of Fire by Ingalls (1886)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Benjamin Robins was the &#xA;  first to execute a systematic and intelligent series of experi&#xA;  ments to determine the velocity of projectiles and the effect &#xA;  of the resistance of the air, not only in retarding but in de&#xA;  flecting them from the plane of fire. He was the inventor &#xA;  of the ballistic pendulum, an instrument for measuring the &#xA;  momenta of projectiles and thence their velocities. He also &#xA;  invented the Whirling Machine for determining the resistance &#xA;  of air to bodies of different forms moving with low velo- &#xA;  cities. His &quot; New Principles of Gunnery,&quot; containing the &#xA;  results of his labors, was published in 1742, and immediately &#xA;  attracted the attention of the great Euler, who translated it &#xA;  into French&lt;/em&gt; [sic!].&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The next series of experiments of any value were made &#xA;  toward the close of the last century by Dr. Hutton, of the &#xA;  Royal Military Academy, Woolwich. He improved the &#xA;  apparatus invented by Robins, and used heavier projectiles &#xA;  with higher velocities. His experiments showed that the &#xA;  resistance is approximately proportional to the square of &#xA;  the diameter of the projectile, and that it increases more &#xA;  rapidly than the square of the velocity up to about 1440 f. s., &#xA;  and nearly&lt;/em&gt; as the square of the velocity from 1440 f. s. to &#xA;  1968 f. s.*&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-27T01:33:49.653" LastActivityDate="2020-01-27T01:33:49.653" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="11385" PostTypeId="1" CreationDate="2020-01-26T15:37:53.553" Score="3" ViewCount="123" Body="&lt;p&gt;My question refers to p.307-335 of volume 8 of Gauss's werke, which are part of a section of volume 8 which deals with Gauss's applications of complex numbers for geometry. The pages mentioned are entitled &quot;&lt;strong&gt;Pothenot's task and the square&lt;/strong&gt;&quot;, and deal in a comprehensive way with the &quot;Snellius–Pothenot problem&quot;. First of all, the length of his treatment suggests there is something interesting here; the fact that almost 30 pages of this volume are dedicated to this problem seems to indicate that Gauss considered this problem to be very important.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &quot;&lt;strong&gt;Snellius–Pothenot problem&lt;/strong&gt;&quot; is an early example of a problem of resection - determination of a observer position by observations of angles between lines of sight to known points. Like the triangulation method, this kind of methods is very important, for example, in surveying (geodesy) and astronomy (in satellite navigation, for example), so that partially answers the question why Gauss was occupied with this problem.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, my questions are:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;why the solution of this problem involves such complications? If i recall correctly, triangulation-like problems can be solved very easily by graphical method and also algebrically with a bit more effort. what am i missing here?  &lt;/li&gt;&#xA;&lt;li&gt;Why Gauss dedicated lengthy computations to it? why did he attach importance to it?&lt;/li&gt;&#xA;&lt;li&gt;how exactly did he use complex numbers to solve this problem? what advantages do they offer in this case?&lt;/li&gt;&#xA;&lt;li&gt;From a more modern point of view, do those computations represent a significant idea? &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any usefull comment/good reference/answer, will be blessed!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-01-26T16:12:18.797" LastActivityDate="2020-01-26T16:12:18.797" Title="What are the context and motivation for Gauss's study of the &quot;Pothenot problem&quot;?" Tags="|biographical-details|geometry|gauss|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11386" PostTypeId="2" ParentId="11097" CreationDate="2020-01-26T19:36:46.167" Score="2" Body="&lt;p&gt;The writings of Descartes and Viète do indeed suggest a certain &amp;quot;inferiority complex&amp;quot;, believing that their own work was simply rediscovering Greek analytic methods which had been lost by the &amp;quot;barbarians&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;From Descartes' &lt;a href=&quot;http://dt.pepperdine.edu/descartes-rules-for-direction-of-the-mind.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rules for the Direction of the Mind&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But when I afterwards bethought myself how it could be that the earliest pioneers of Philosophy in bygone ages refused to admit to the study of wisdom any one who was not versed in Mathematics, evidently believing that this was the easiest and most indispensable mental exercise and preparation for laying hold of other more important sciences, I was confirmed in my suspicion that they had knowledge of a species of Mathematics very different from that which passes current in our time. I do not indeed imagine that they had a perfect knowledge of it .... But I am convinced that certain primary germs of truth implanted by nature in human minds ... had a very great vitality in that rude and unsophisticated age of the ancient world. ... Indeed I seem to recognize certain traces of this true Mathematics in Pappus and Diophantus, who though not belonging to the earliest age, yet lived many centuries before our own times.&lt;/p&gt;&#xA;&lt;p&gt;But my opinion is that these writers then with a sort of low cunning, deplorable indeed, suppressed this knowledge. Possibly they acted just as many inventors are known to have done in the case of their discoveries, i.e. they feared that their method being so easy and simple would become cheapened on being divulged, and they preferred to exhibit in its place certain barren truths, deductively demonstrated with show enough of ingenuity, as the results of their art, in order to win from us our admiration for these achievements, rather than to disclose to us that method itself which would have wholly annulled the admiration accorded. Finally there have been certain men of talent who in the present age have tried to revive this same art. For it seems to be precisely that science known by the barbarous name Algebra if only we could extricate it from that vast array of numbers and inexplicable figures by which it is overwhelmed, so that it might display the clearness and simplicity which, we imagine ought to exist in a genuine Mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Leo Correy, in his recent text &lt;em&gt;A Brief History of Numbers&lt;/em&gt;, writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Viète, for one thing, saw his own contribution as part of a more general effort to reconstruct the assumed Greek analytic methods that had been ruined by the &amp;quot;barbarians&amp;quot; (namely, the Arabs). In order to stress this attitude as an essential part of his work, he consistently declined to use the term &amp;quot;algebra&amp;quot; in favour of &amp;quot;analysis&amp;quot;. He also introduced some new Greek-sounding terms of his own invention such as &lt;em&gt;poristic&lt;/em&gt;, &lt;em&gt;exegetic&lt;/em&gt;, and &lt;em&gt;zeteic&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-26T19:36:46.167" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11387" PostTypeId="1" AcceptedAnswerId="11390" CreationDate="2020-01-26T22:13:23.357" Score="3" ViewCount="298" Body="&lt;p&gt;I have been reading one of Lagrange's works where he mentions palindromic sequences in the context of Periodic continued fractions.&#xA;I am wondering if that was the first time that palindromic sequences were defined?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="2277" LastEditDate="2020-01-27T08:55:42.490" LastActivityDate="2020-01-27T08:55:42.490" Title="When did the concept of palindromic sequences first appear in mathematics?" Tags="|mathematics|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11388" PostTypeId="2" ParentId="5212" CreationDate="2020-01-26T22:51:41.340" Score="0" Body="&lt;p&gt;Below are couple of links for a sample of how one may have nonzero number system. The author was inspired by tinkering a Chinese abacus.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;SAMPLE OF COUNTING NUMBERS WITHOUT USE OF ZERO &lt;a href=&quot;https://drive.google.com/file/d/1mbrCvB67fPbG5oXGF0uhdZE1AHyBVApu/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://drive.google.com/file/d/1mbrCvB67fPbG5oXGF0uhdZE1AHyBVApu/view&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;THE MAKING OF LINGO FOR DECIMAL NUMBERS WITH NONZERO DIGITS &lt;a href=&quot;https://drive.google.com/file/d/1QbVsowd4wTny-4RQM-UUr8tXzxPOStXe/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://drive.google.com/file/d/1QbVsowd4wTny-4RQM-UUr8tXzxPOStXe/view&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11002" LastActivityDate="2020-01-26T22:51:41.340" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11389" PostTypeId="1" AcceptedAnswerId="11395" CreationDate="2020-01-26T23:39:45.217" Score="5" ViewCount="473" Body="&lt;p&gt;Have there have been any interesting cases of one person stealing another mathematician's results and publishing them as his own?  In particular, are there any interesting cases of this happening in the late 1800s or early 1900s?&lt;/p&gt;&#xA;" OwnerUserId="11019" LastEditorUserId="55" LastEditDate="2020-01-27T05:42:00.237" LastActivityDate="2020-02-03T09:58:42.203" Title="Plagiarism at the turn of 20th century" Tags="|mathematics-social-history|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11390" PostTypeId="2" ParentId="11387" CreationDate="2020-01-27T04:08:48.020" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.semanticscholar.org/paper/Ratio-in-early-Greek-mathematics-Fowler/800617386f9c80eaf0e2e9d46cc24ce20fcf0c7e&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fowler's Ratio in Early Greek Mathematics&lt;/a&gt;, palindromic patterns in continued fractions of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{p}:\sqrt{q}$&lt;/span&gt; with primes &lt;span class=&quot;math-container&quot;&gt;$p&amp;gt;q$&lt;/span&gt; were likely known already to Pythagoreans. In the modern times the interest in continued fractions was revived largely due to Euler's work (from 1731), although Wallis and Huygens worked on them earlier, see &lt;a href=&quot;http://www.irem.univ-mrs.fr/IMG/pdf/huygens-delft.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Christiaan Huygens’ Planetarium&lt;/a&gt;. The palindromic pattern is noted in Euler's &lt;a href=&quot;http://eulerarchive.maa.org/pages/E323.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De usu novi algorithmi in problemate Pelliano solvendo&lt;/em&gt; (presented 1759)&lt;/a&gt;. Lagrange followed Euler's lead starting in 1768, see &lt;a href=&quot;https://books.google.com/books?id=rxzsCAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brezinski, History of Continued Fractions and Padé Approximants, p. 108&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-27T04:17:38.080" LastActivityDate="2020-01-27T04:17:38.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11391" PostTypeId="2" ParentId="11389" CreationDate="2020-01-27T05:41:27.157" Score="2" Body="&lt;p&gt;A famous case from this time period concerns &lt;a href=&quot;https://en.wikipedia.org/wiki/Relativity_priority_dispute#General_relativity_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;priority dispute between Hilbert and Einstein&lt;/a&gt; over the deriving the field equations of general relativity. Hilbert completed the general theory field equations &quot;at least 5 days before&quot; Einstein submitted his final paper in 1915, Einstein accused Hilbert of &quot;nostrifying&quot; him. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word is interesting, &quot;nostrification&quot; officially means adoption of a foreign diploma, here it is applied to adoption of ideas. Borrowing chords is accepted in music, for example, and is even considered a boon to creativity. Van der Waerden characterized “nostrification” in 1930 as &quot;&lt;em&gt;reformulating other people’s&#xA;best ideas with increased sharpness and generality, and from then on citing the local&#xA;reformulation&lt;/em&gt;&quot;. But the line between that and plagiarism is not bright, Einstein was very unhappy. Corry writes in &lt;a href=&quot;https://www.springer.com/gp/book/9781402027772&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Hilbert and the Axiomatization of Physics&lt;/a&gt; that the practice was widespread in Hilbert's Göttingen:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was widely understood, among German mathematicians at least, that “nostrification” encapsulated the peculiar&#xA;  style of creating and developing scientific ideas in Göttingen, and not least because of the pervasive influence&#xA;  of Hilbert.  Of course, “nostrification” should not be understood as mere plagiarism. It was a central trait of&#xA;  a hectic scientific culture that stimulated an intense and sustained interaction&#xA;  between professors, young docents, and students, across departments and across&#xA;  universities. The Göttingen atmosphere implied a constant discussion and adoption&#xA;  of new ideas, techniques, and problems that had originally been created or suggested&#xA;  by others, either at home or outside. Whenever these ideas appeared to be fruitful&#xA;  and relevant to current concerns of the local community, they were immediately&#xA;  absorbed into the common scientific patrimony.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Weyl, for example, rediscovered the axioms of a vector space&#xA;in 1918 without citing Peano's 1898 version, but, perhaps, he did not know. In&#xA;&lt;em&gt;Grundlagen der Geometrie&lt;/em&gt; &quot;&lt;em&gt;Hilbert “nostrified” much of the contributions of the Italian school, and the&#xA;only work he explicitly referred to was that of Veronese&lt;/em&gt;&quot;. As for the dispute with Einstein, printer's proofs of his paper found by Corry in 1997 seem to back Einstein's opinion, at least in that Hilbert did not beat him to it. The version of the paper submitted &quot;at least 5 days before&quot; did not contain the correct equations. Einstein and Hilbert patched things up quickly, even before the publication. Einstein wrote in a private letter:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;There has been a certain resentment between us,&#xA;  the cause of which I do not want analyze any&#xA;  further. I have fought against the feeling of bitterness&#xA;  associated with it, and with complete&#xA;  success. I again think of you with undiminished&#xA;  kindness and I ask you to attempt the same with&#xA;  me. It is objectively a pity if two guys that have&#xA;  somewhat liberated themselves from this shabby&#xA;  world are not giving pleasure to each other.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hilbert, after some meandering, added to the printed version of his paper a  reference  to  Einstein’s paper and a concession of his priority. But the &quot;Hilbert first&quot; story continued to circulate widely. Here is from the &lt;a href=&quot;https://science.sciencemag.org/content/278/5341/1270.abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;Belated Decision in the Hilbert-Einstein Priority Dispute by Corry, Renn and Stachel (1997)&lt;/a&gt;, where Corry's find was reported and analyzed:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;According to the commonly accepted view, David Hilbert completed the general theory&#xA;  of relativity at least 5 days before Albert Einstein submitted his conclusive paper on this&#xA;  theory  on  25  November  1915.  Hilbert’s  article,  bearing  the  date  of  submission  20&#xA;  November 1915 but published only on 31 March 1916, presents a generally covariant&#xA;  theory of gravitation, including field equations essentially equivalent to those in Einstein’s&#xA;  paper. A close analysis of archival material reveals that Hilbert did not anticipate Einstein.&#xA;  The first set of proofs of Hilbert’s paper shows that the theory he originally submitted&#xA;  is not generally covariant and does not include the explicit form of the field equations&#xA;  of general relativity.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Both the proofs and the final version of&#xA;  Hilbert’s first communication (3) are dated&#xA;  “submitted on 20 November 1915,” presumably&#xA;  referring to the original manuscript. A&#xA;  copy of the proofs, preserved in his archives&#xA;  and marked in his own hand “First proofs of&#xA;  my first note,” bears a printer’s stamp dated&#xA;  6 December 1915 (Fig. 1). However, the&#xA;  cover of the issue in which the heavily&#xA;  revised published version appeared is dated&#xA;  31 March 1916. Its first note cites Einstein’s&#xA;  conclusive paper, in which he reached the&#xA;  final form of his generally covariant theory&#xA;  (11), submitted on 25 November 1915 and&#xA;  published on 2 December 1915. Thus, Hilbert&#xA;  could have revised his paper in response&#xA;  to Einstein’s work.&#xA;  Differences between the proofs and this&#xA;  published version of Hilbert’s paper confirm&#xA;  this view.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;To summarize: Initially, Hilbert did not&#xA;  give the explicit form of the field equations;&#xA;  then, after Einstein had published his field&#xA;  equations, Hilbert claimed that no calculation&#xA;  is necessary; finally, he conceded that&#xA;  one is. Taken together, this sequence suggests&#xA;  that knowledge of Einstein’s result&#xA;  may have been crucial to Hilbert’s introduction&#xA;  of the trace term into his field&#xA;  equations.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-27T13:36:17.413" LastActivityDate="2020-01-27T13:36:17.413" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11392" PostTypeId="2" ParentId="11379" CreationDate="2020-01-27T06:40:43.870" Score="2" Body="&lt;p&gt;It does not seem like there was much of a contribution, if any Lagrange is sometimes mentioned, without reference, e.g. by &lt;a href=&quot;https://pdfs.semanticscholar.org/0783/7a7a4221b6390808e6d52728aedff0434d7a.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Borell&lt;/a&gt;, along with Taylor and de Moivre, as one who &amp;quot;worked&amp;quot; on it. But while Euler's work is called &amp;quot;serious&amp;quot; and &amp;quot;influential&amp;quot; not much else is said about the other three. A detailed history in &lt;a href=&quot;https://www.mayhematics.com/t/1b.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rediscovery of the Knight's Problem by Jelliss&lt;/a&gt; questions Taylor's involvement and does not even mention Lagrange. Neither does Rouse Ball who has a section on it in &lt;a href=&quot;https://www.gutenberg.org/files/26839/26839-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Recreations and Essays&lt;/a&gt;. He does mention Legendre, and I wonder if there was some name conflation. From Jeliss:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The modern study of the knight's problem appears to have begun in the 18th century without knowledge of the mediaeval work, save perhaps for the half-board tour in Guarini's work. The subject first reappeared in Jacques Ozanam's Récréations Mathématiques et Physiques... The first edition of Ozanam's work was published in 1694 but (according to one of the later editors, C. Hutton) Ozanam died in 1717.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The tours were supplied to the editor of that edition of Ozanam in 1722 by de Mairan, who was Director of the Académie Royale des Sciences, Paris. This date is printed in the margin of the book. Rouse Ball (1939) says the tours by de Montmort and de Moivre “were sent by their authors to Brook Taylor who seems to have previously suggested the problem”; unfortunately he gives no reference to where he learnt of the involvement of Taylor (another well known mathematician) and I have been unable to verify this. Similarly Kraitchik attaches the date 1708 to the de Montmort tour, but it does not appear in de Montmort's famous work l'Essai d'analyse sur les jeux de hasard, Paris 1708. A slight variation of the de Moivre tour in which the last three moves are reflected is mentioned in the text and is sometimes diagrammed in later accounts.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The first mathematical paper analysing knight's tours was presented by the most productive mathematician of the eighteenth century, Leonhard Euler (1707–1783), to the Academy of Sciences at Berlin in 1759 (but not printed until 1766). This paper has been of considerable influence, the starting point (but often alas the end also) of many later accounts of the subject.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-27T06:52:20.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11393" PostTypeId="1" AcceptedAnswerId="11394" CreationDate="2020-01-27T09:37:13.610" Score="11" ViewCount="2145" Body="&lt;p&gt;The principle was used as early as late 1760s by Lagrange but are there any earlier uses of it in math?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2021-03-03T05:29:06.783" Title="Who discovered pigeonhole principle in mathematics?" Tags="|mathematics|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11394" PostTypeId="2" ParentId="11393" CreationDate="2020-01-27T10:41:26.543" Score="12" Body="&lt;p&gt;I assume this refers to Lagrange's 1768 proof of the Diophantine approximation theorem. The proof was simplified by Dirichlet in 1842, using the idea twice. He named it &lt;em&gt;Schubfachprinzip&lt;/em&gt; (drawer principle), and it is with Dirichlet that the principle came to be most commonly associated. Many authors date Dirichlet's use back to 1834, but without any reference. The French name was &lt;em&gt;le principe des tiroirs de Dirichlet&lt;/em&gt;, the principle of Dirichlet's drawers. Cabillón traced the first appearance of pigeons in it to &lt;a href=&quot;https://www.semanticscholar.org/paper/On-the-simultaneous-approximation-of-two-real-Robinson/8c2a2727c1d3d3e056f08315a0e3fd9ca4bfe39d&quot; rel=&quot;noreferrer&quot;&gt;Robinson's On the Simultaneous Approximation of Two Real Numbers (1941)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, neither Dirichlet nor Lagrange were the first, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s00283-013-9389-1&quot; rel=&quot;noreferrer&quot;&gt;The pigeonhole principle, two centuries before&#xA;Dirichlet by Rittaud and Heeffer&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In Selectae Propositiones, a book written in latin in 1622 by the French Jesuit&#xA;  Jean Leurechon, the pigeonhole principle is indirectly mentioned in a single short&#xA;  sentence, given without any further elaboration ([13], p. 2): &quot;It is necessary that&#xA;  two men have the same number of hairs, gold, and others.&quot; The famous Marin&#xA;  Mersenne copied several propositions on arithmetic and music from Leurechon,&#xA;  including this mathematical principle in an early work of 1625, acknowledging&#xA;  the &quot;excellent conclusions obtained from arithmetics&quot; by the mathematicians&#xA;  from Pont-a-Mousson.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Jean Leurechon was a factor of stability at Pont-a-Mousson, teaching mathematics&#xA;  on and off between 1614 and 1629. His short Selectae Propositiones&#xA;  from 1622 was a collection of propositions in mixed mathematics that were&#xA;  used for teaching. The booklet is one of the earliest witness accounts of the&#xA;  new mathematics curriculum at Jesuit colleges. Teaching was organised in lectures&#xA;  (lectiones), rehearsals (repetitiones) and discussions (disputationes).&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;It is now established that an immensely popular work published at Pont-&#xA;  a-Mousson in 1624 resulted from these disputationes [12]. Entitled Recreation&#xA;  mathematicque [1], this French work is commonly attributed to Jean Leurechon,&#xA;  but there are good reasons to believe that this attribution is wrong (see&#xA;  [11]). More than seventy editions and translations were published during the&#xA;  seventeenth century. (The First English translation appeared in 1633 [2].) The&#xA;  pigeonhole principle appears at the end of the book (just after some elementary remarks inspired by Archimedes' Sand Reckoner about the number of grains of&#xA;  sand that could fill the universe). Unfortunately, this remarkable part of the&#xA;  book does not appear in the English edition.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;It is quite astonishing that the example of men with the same number of hairs&#xA;  is still a very common illustration of the principle (see for example [4] p. 3&#xA;  where the case of New York City is considered, or [10] with the inhabitants of&#xA;  Madrid, etc.)... Indeed, the problem of men with the same&#xA;  number of hairs appeared in 1737, in a totally different context, in a French&#xA;  book by Charles-Irenee Castel de Saint-Pierre, which is in no way related to&#xA;  mathematics...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In the middle of the nineteenth century, the famous French writer Charles-&#xA;  Augustin Sainte-Beuve quoted extensively this passage&lt;/em&gt; [from de Saint-Pierre] &lt;em&gt;in several publications&#xA;  ([16], [17], [18]), at the very same time Dirichlet made use of the principle in a&#xA;  purely mathematical context... So, do we have to replace the old Dirichlet's pigeonhole principle&quot; by a&#xA;  Leurechon's pigeonhole principle&quot;? We do not know for sure if Leurechon was&#xA;  really the first to publish the result. David Singmaster's comprehensive Sources&#xA;  in Recreational Mathematics [19] and our survey of Italian abbaco manuscripts&#xA;  revealed no earlier instances.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-01-27T10:48:23.707" LastActivityDate="2020-01-27T10:48:23.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11395" PostTypeId="2" ParentId="11389" CreationDate="2020-01-27T13:36:01.210" Score="12" Body="&lt;p&gt;Date: 1930's.&lt;br&gt;&#xA;There is the story of the &quot;ergodic theorem&quot;. Young mathematician John von Neumann proved the so-called &quot;mean ergodic theorem&quot;.  He wrote it up and sent it to the &lt;em&gt;Proceedings of the National Academy of Sciences&lt;/em&gt;.  It was reviewed by one of the editors of the &lt;em&gt;Proceedings&lt;/em&gt;, senior mathematician G. D. Birkhoff.  In due course, Birkhoff recommended publication.  It appeared in 1932.  Meanwhile, inspired by von Neumann's result, Birkhoff worked furiously for a few months and obtained an improved result, the so-called &quot;pointwise ergodic theorem&quot;.  He had it published in the same journal.  It appeared in 1931.  Birkhoff's paper made no mention of von Neumann's paper.  Only many years later, in a paper with Birkhoff as co-author, was it admitted that von Neumann had the priority.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2020-01-27T13:36:01.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11396" PostTypeId="1" AcceptedAnswerId="11398" CreationDate="2020-01-27T18:32:59.870" Score="0" ViewCount="965" Body="&lt;p&gt;In NCERT textbook for class X mathematics, it is given that Baudhayan gave the relation between the hypotenuse square and sum of squares of the other sides of a right angled triangle. Is it true?&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Sj93f.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Sj93f.png&quot; alt=&quot;From: NCERT Class X Mathematics Textbook&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Research efforts:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;From Wikipedia (&lt;a href=&quot;https://en.wikipedia.org/wiki/Baudhayana_sutras&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Baudhayana_sutras&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Baudhāyana Śulbasûtra is noted for containing several early mathematical results, including an approximation of the square root of 2 and the statement of the Pythagorean theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;See also (&lt;a href=&quot;https://en.wikipedia.org/wiki/Shulba_sutras&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Shulba_sutras&lt;/a&gt;).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Facts Legend (&lt;a href=&quot;https://factslegend.org/indian-mathematician-baudhayana-originally-discovered-pythagorean-theorem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://factslegend.org/indian-mathematician-baudhayana-originally-discovered-pythagorean-theorem/&lt;/a&gt;)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;Sanskrit Magazine (&lt;a href=&quot;https://www.sanskritimagazine.com/vedic_science/baudhayana-the-original-mathematician-behind-pythagoras-theorem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sanskritimagazine.com/vedic_science/baudhayana-the-original-mathematician-behind-pythagoras-theorem/&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;PPArihar (&lt;a href=&quot;https://pparihar.com/2014/08/31/baudhayana-pythagoras-theorem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pparihar.com/2014/08/31/baudhayana-pythagoras-theorem/&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Simple web search [by duckduckwent, not googling :)] (&lt;a href=&quot;https://duckduckgo.com/?q=baudhayan+pythagoras+theorem&amp;amp;t=ffab&amp;amp;atb=v192-1&amp;amp;ia=web&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://duckduckgo.com/?q=baudhayan+pythagoras+theorem&amp;amp;t=ffab&amp;amp;atb=v192-1&amp;amp;ia=web&lt;/a&gt;)  &lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;As you can see, a lot of websites which support Baudhayan as original discoverer are either Indian or Hinduism oriented, so they may not be valid sources.  The Wikipedia articles are not in this category, but an answer has asserted that Wikipedia is a major source of biased information about Indian science and mathematics.&lt;/p&gt;&#xA;" OwnerUserId="10904" LastEditorUserId="139" LastEditDate="2020-02-18T15:30:17.700" LastActivityDate="2020-03-02T09:33:52.160" Title="Did Baudhayan publish Pythagoras theorem before Pythagoras?" Tags="|ancient-india|pythagorean-proposition|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11397" PostTypeId="1" CreationDate="2020-01-27T18:39:57.387" Score="1" ViewCount="740" Body="&lt;p&gt;A lot of people have contributed to the development of science (which includes mathematics). I have read that Newton's popularity was powerful enough to dominate the wave-model. Similarly, do we have any scientist whose work was underestimated, was not accepted by the community and was left unnoticed during his/her lifetime, but when a popular scientist published the same thing, got a wide appreciation?&lt;/p&gt;&#xA;" OwnerUserId="10904" LastEditorUserId="4251" LastEditDate="2020-07-13T11:49:19.237" LastActivityDate="2020-07-15T06:20:27.033" Title="Is there any scientist whose work was not recognised during his/her lifetime and was underestimated?" Tags="|biographical-details|depiction-of-scientists|scientists|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11398" PostTypeId="2" ParentId="11396" CreationDate="2020-01-28T04:50:20.670" Score="3" Body="&lt;p&gt;The textbook you quoted is that of high school and probably written by high school/college teachers. Most of these side notes are often traditional hearsay. As a good scientist of future, do not trust whatever appears in books and Wikipedia should be your last resort to verify historical facts. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are they all true? If yes, did they have any effect on other scientists whom we know today? And why Indians are then given less or no credit in books and articles?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Sorry to say but most of those ancient &quot;inventions&quot; are bogus. Wikipedia has become a major source of this non-sense. When someone tries to challenge that discovery, it is reverted. The problem with very old historical stuff is that there is no proof. Suppose a nation A claims today that they invented sines and cosines thousands of years ago. What is the proof? Is there any surviving manuscript? You see the problem with very old historical stuff including the origins of a certain concepts and giving due credit where it is due is one of the most difficult questions in history. Human history is often dependent on the author. You have to be critical when you read a rosy picture of any civilization.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Recently, there are series of rather ridiculous claims in History of Science and Mathematics, and even Physics that everything originated in ancient India. Claims range from invention of alegbraic formulae, sine, cosine, numbers, astronomy, distance between Earth and Sun, ancient air crafts, stem cell research and above all close relationship of German and Sanskrit. Usually Wikipedia and other dubious sources are quoted. BBC even made a report on it, sadly this is being propagated by some obscure university professors.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bbc.com/news/world-asia-india-46778879&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.bbc.com/news/world-asia-india-46778879&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;However the easy question is what is the contribution of modern Indian scientists?&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some Indian scientists during the time of British made really major discoveries while working in India such as Sir Chandrashekhara Venkata Raman (Raman effect), Bose (physicist),  Meghnad Saha (Ionization equation), Ramanujan and many others. Yes they faced difficulties in publishing in major journals and even faced inappropriate comments. However, later on they earned the respect from a majority of the scientists in the world. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;An example is that of the interview of Arthur Compton's wife. She recalled her encounter with Raman as Ref: C. V. Raman and the Discovery of the Raman Effect, Phys. perspect. 4 (2002) 399–420,&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot; In an interview in 1968, although she could not at first remember Raman’s name, she stated: Anyway he [Raman] was very dark, just as black as could be, but he had a beautiful Scotch accent. He would be asking questions from the back of the hall, and it was so disconcerting to have this person, black as coal, with this beautiful Scotch accent. He was the one who said, ‘‘Compton, you answer questions well; you’re a good debater, but the truth isn’t in you.*’’&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today such labels would not be acceptable such as &quot;&lt;em&gt;black as coal&lt;/em&gt;&quot; in any community. &lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2020-01-28T21:13:04.273" LastActivityDate="2020-01-28T21:13:04.273" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="11399" PostTypeId="1" CreationDate="2020-01-28T10:36:21.207" Score="1" ViewCount="118" Body="&lt;p&gt;I have been searching for books on electronics and mathematics of early 20th century.I found a link which was released in 1950 but that was in russian language.Can I find these books in universities like harvard etc?&lt;/p&gt;&#xA;" OwnerUserId="11032" LastActivityDate="2020-02-04T11:00:10.023" Title="Where can I get early 20th century books on electronics and mathematics?" Tags="|mathematics|electricity|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11400" PostTypeId="2" ParentId="11399" CreationDate="2020-01-28T14:30:32.563" Score="1" Body="&lt;p&gt;Internet archive is one of the best resources for historical mathematical books from the 18th/19th century. Another professional level historical resource is Hathi Trust. This is not open access.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2020-01-28T14:30:32.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11401" PostTypeId="1" CreationDate="2020-01-28T17:36:40.593" Score="3" ViewCount="158" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Convex_hull&quot; rel=&quot;nofollow noreferrer&quot;&gt;convex hull&lt;/a&gt; of a set of points appears recognizably in &lt;a href=&quot;http://www.newtonproject.ox.ac.uk/catalogue/record/NATP00196&quot; rel=&quot;nofollow noreferrer&quot;&gt;a 1676 letter from Newton to Henry Oldenburg describing Newton polygons&lt;/a&gt;. Is there an earlier precedent for convex hulls or is this their first known appearance in the mathematical literature?&lt;/p&gt;&#xA;" OwnerUserId="11029" LastEditorUserId="55" LastEditDate="2020-01-29T00:47:07.273" LastActivityDate="2020-01-29T00:47:07.273" Title="Did Newton invent convex hulls?" Tags="|mathematics|geometry|newton|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11402" PostTypeId="1" CreationDate="2020-01-28T16:19:08.030" Score="4" ViewCount="458" Body="&lt;p&gt;As we know, Newton showed that the celestial bodies in the heaven and the objects on the earth obey the same laws of motion, through his laws of motion and gravitation.&lt;br /&gt;&#xA;However, in order to know this, People in Newton's era should understand that the motion of objects on earth are affected by the resistance (or the drag force) including air resistance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is : When and How did people know the existence of resistance including air resistance?&lt;/p&gt;&#xA;" OwnerUserId="11033" OwnerDisplayName="pdh0710" LastEditorUserId="55" LastEditDate="2020-01-29T12:42:49.027" LastActivityDate="2020-01-29T13:38:50.937" Title="When and how did people learn of air resistance?" Tags="|classical-mechanics|aerodynamics|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="11403" PostTypeId="1" CreationDate="2020-01-28T19:57:23.457" Score="8" ViewCount="1086" Body="&lt;p&gt;Ramanujan didn't know modern mathematics. he lacked idea regarding analysis.&#xA;I found in Wikipedia-&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hardy tried his best to fill the gaps in Ramanujan's education and to&#xA;  mentor him in the need for formal proofs to support his results,&#xA;  without hindering his inspiration—a conflict that neither found easy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;What topics exactly Hardy teach Ramanujan during his stay in England?&lt;/p&gt;&#xA;" OwnerUserId="10482" LastEditorUserId="4251" LastEditDate="2020-07-06T12:24:01.270" LastActivityDate="2020-07-06T12:24:01.270" Title="What did G.H. Hardy teach Ramanujan?" Tags="|mathematics|mathematicians|teaching|mathematics-social-history|ramanujan|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11405" PostTypeId="1" CreationDate="2020-01-28T21:24:40.563" Score="0" ViewCount="210" Body="&lt;p&gt;I can imagine that some discoveries are not made for a long time despite availability of all the information and tools that one would in principle need.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there clear examples of that?&lt;/p&gt;&#xA;" OwnerUserId="11034" LastEditorUserId="11034" LastEditDate="2020-01-29T07:33:04.117" LastActivityDate="2020-01-30T05:16:51.460" Title="Are there any important scientific discoveries, or inventions, that could have been made much earlier in history?" Tags="|discoveries|social-context|" AnswerCount="1" CommentCount="7" ClosedDate="2020-02-09T02:08:11.227" ContentLicense="CC BY-SA 4.0" />
  <row Id="11406" PostTypeId="2" ParentId="8184" CreationDate="2020-01-28T22:00:00.733" Score="3" Body="&lt;p&gt;Amateur makes fools of the experts, Marcus Chown September 1995,&#xA;[&lt;a href=&quot;https://www.newscientist.com/article/mg14719952-200-amateur-makes-fools-of-the-experts/]&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.newscientist.com/article/mg14719952-200-amateur-makes-fools-of-the-experts/]&lt;/a&gt; &#xA;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;An experiment  that for eighty years has been cited by physicists as a&#xA;  confirmation of Einstein’s special theory of relativity is nothing of&#xA;  the sort. This remarkable discovery has been made not by a&#xA;  professional physicist but by an enthusiastic amateur.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Prominent physicists are often pestered by people who claim to have&#xA;  found a flaw in our understanding of the Universe. So Arthur Swift of&#xA;  the University of Massachusetts in Amherst at first ignored the&#xA;  letters he received from Gerald Pellegrini of Worcester,&#xA;  Massachusetts. “He has spent twenty years in the wilderness mulling&#xA;  over fundamental problems,” says Swift.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;However, Pellegrini’s insistence that there was something amiss with a&#xA;  classic 1913 experiment eventually persuaded Swift to take a closer&#xA;  look at the theory behind the original experiment. This experiment,&#xA;  reported in the Proceedings of the Royal Society A, was an attempt by&#xA;  the husband-and-wife team Marjorie Wilson and H. A. Wilson of Rice&#xA;  University in Houston to measure the electric field created in a&#xA;  magnetised nonconducting material moving at high speed through a&#xA;  uniform magnetic field. Five years earlier, Einstein had used his&#xA;  special theory of relativity to predict that such an electric field&#xA;  would be stronger than that predicted using the classical theory&#xA;  devised by the Dutch physicist Hendrik Lorentz.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Amateur Scientists Just Proved Einstein Wrong Alfredo Carpineti May 2018&#xA;&lt;a href=&quot;https://www.iflscience.com/physics/one-of-einsteins-major-theories-just-got-disproved-by-a-bunch-of-amateurs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.iflscience.com/physics/one-of-einsteins-major-theories-just-got-disproved-by-a-bunch-of-amateurs/&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Proving Albert Einstein wrong is something that only a small number of&#xA;  scientists can claim to have done but now, more than 100,000 gamers&#xA;  can join that exclusive club and enjoy the smugness that comes with&#xA;  it. In 2016, scientists from around the world led by the Institute of&#xA;  Photonic Sciences in Barcelona (ICFO) asked people to play a simple&#xA;  game online, and the results were used to disprove one of Einstein’s&#xA;  claims about quantum mechanics. The results are published in Nature.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;One of the things Einstein truly disliked about quantum mechanics is&#xA;  how the experimenter plays a role in the results obtained from an&#xA;  experiment. He believed the universe to be independent of our actions&#xA;  and quantum mechanics to be governed by the principle of local&#xA;  realism.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;It Doesn't Take a Rocket Scientist: Great Amateurs of Science&lt;/em&gt; John Malone&#xA;&lt;a href=&quot;https://archive.org/details/ItDoesntTakeARocketScientist-GreatAmateursInScience/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/ItDoesntTakeARocketScientist-GreatAmateursInScience/mode/2up&lt;/a&gt;&#xA;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;It Doesn't Take a Rocket Scientist&lt;/em&gt; examines the lives and work of ten amateur scientists whose investigations yielded insights and&#xA;  discoveries that eluded their highly educated counterparts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&#xA;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Amateur Scientists Making Significant Discoveries While Fighting To Receive Recognition And Respect Bruce Bigelow Jun 10, 1996&#xA;&lt;a href=&quot;https://www.the-scientist.com/news/amateur-scientists-making-significant-discoveries-while-fighting-to-receive-recognition-and-respect-57972&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.the-scientist.com/news/amateur-scientists-making-significant-discoveries-while-fighting-to-receive-recognition-and-respect-57972&lt;/a&gt; &#xA;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Discoveries While Fighting To Receive Recognition And Respect In&#xA;  Tucson, Ariz., David Levy is a freelance writer and educator who&#xA;  spends his nights watching the desert sky for comets. STRUGGLE:&#xA;  Forrest Mims III, shown during a trip to the Mauna Loa Observatory,&#xA;  has encountered resistance to his efforts to work at the site. In&#xA;  Miami, Randy McCranie is a postal worker who has conducted extensive&#xA;  investigations as a field biologist in Honduras. And in Seguin, Texas,&#xA;  Forrest M. Mims III is a pro.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;.&#xA;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The British amateur who debunked the mathematics of happiness Andrew Anthony Jan. 2014&#xA;&lt;a href=&quot;https://www.theguardian.com/science/2014/jan/19/mathematics-of-happiness-debunked-nick-brown&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.theguardian.com/science/2014/jan/19/mathematics-of-happiness-debunked-nick-brown&lt;/a&gt;&#xA;.&#xA;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The astonishing story of Nick Brown, the British man who began a&#xA;  part-time psychology course in his 50s – and ended up taking on&#xA;  America's academic establishment&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2020-01-28T22:00:00.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11407" PostTypeId="2" ParentId="11403" CreationDate="2020-01-29T11:04:34.370" Score="6" Body="&lt;p&gt;For Ramanujan's background see &lt;a href=&quot;https://hsm.stackexchange.com/a/2880/55&quot;&gt;How did Ramanujan learn to do mathematics?&lt;/a&gt; According to Hardy himself, he did not teach him any topics, only the idea and perhaps some methods of proof, see his lecture &lt;a href=&quot;https://www.jstor.org/stable/2301659&quot; rel=&quot;nofollow noreferrer&quot;&gt;Indian Mathematician Ramanujan&lt;/a&gt;. Ramanujan did pick up sporadic bits and pieces of modern mathematics from various sources that Hardy is not too sure about, other than &lt;a href=&quot;https://archive.org/details/asynopsiselemen00carrgoog/page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carr's Synopsis of Elementary Results in Pure and Applied Mathematics&lt;/a&gt;. What he learned, with major gaps, of classical and analytic number theory, elliptic functions, asymptotic analysis, hypergeometric series and continued fractions, was self-learning that Hardy does not credit to himself:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;He had been carrying an impossible handicap, a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe... It was impossible to teach him systematically, but he gradually absorbed new points of view. In particular he learnt what was meant by proof, and his later papers, while in some ways as odd and individual as ever, read like the works of a well-informed mathematician. His methods and his weapons, however, remained essentially the same. One would have thought that such a formalist as Ramanujan would have revelled in Cauchy's Theorem, but he practically never used it, and the most astonishing testimony to his formal genius is that he never seemed to feel the want of it in the least.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;For example, in the analytic theory of numbers he had, in a sense, discovered a great deal, but he was a very long way from understanding the real difficulties of the subject. And there is some of his work, mostly in the theory of elliptic functions, about which some mystery still remains; it is not possible, after all the work of Watson and Mordell, to draw the line between what he may have picked up somehow and what he must have found for himself... Here I must admit that I am to blame, since there is a good deal which we should like to know now and which I could have discovered quite easily... I never even asked him whether (as I think he must have done) he had seen Cayley's or Greenhill's Elliptic Functions.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Ramanujan's theory of primes was vitiated by his ignorance of the theory of functions of a complex variable. It was (so to say) what the theory might be if the Zeta-function had no complex zeros. His method depended upon a wholesale use of divergent series.... That his proofs should have been invalid was only to be expected. But the mistakes went deeper than that, and many of the actual results were false. He had obtained the dominant terms of the classical formulae, although by invalid methods; but none of them are such close approximations as he supposed.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;I do not think that Ramanujan discovered much in the classical theory of numbers, or indeed that he ever knew a great deal. He had no knowledge at all, at any time, of the general theory of arithmetical forms. I doubt whether he knew the law of quadratic reciprocity before he came here. Diophantine equations should have suited him, but he did comparatively little with them, and what he did do was not his best.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;In algebra, Ramanujan's main work was concerned with hypergeometric series and continued fractions (I use the word algebra, of course, in its old-fashioned sense). These subjects suited him exactly, and here he was unquestionably one of the great masters... As regards hypergeometric series one may say, roughly, that he rediscovered the formal theory, set out in Bailey's tract, as it was known up to 1920. There is something about it in Carr, and more in Chrystal's Algebra, and no doubt he got his start from that.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;In analysis proper Ramanujan's work is inevitably less impressive, since he knew no theory of functions, and you cannot do real analysis without it, and since the formal side of the integral calculus, which was all that he could learn from Carr or any other book, has been worked over so repeatedly and so intensively. Still, Ramanujan rediscovered an astonishing number of the most beautiful analytic identities.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The aforementioned Carr's &lt;em&gt;Synopsis of Elementary Results in Pure and Applied Mathematics&lt;/em&gt; had a big impact on Ramanujan early in life:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;It was a book of a very different kind, Carr's Synopsis, which first aroused Ramanujan's full powers... The book is not in any sense a great one, but Ramanujan has made it famous, and there is no doubt that it influenced him profoundly and that his acquaintance with it marked the real starting point of his career... Carr has sections on the obvious subjects, algebra, trigonometry, calculus and analytical geometry, but some sections are developed disproportionally, and particularly the formal side of the integral calculus. This seems to have been Carr's pet subject, and the treatment of it is very full and in its way definitely good. There is no theory of functions... What is more surprising, in view of Carr's own tastes and Ramanujan's later work, is that there is no elliptic functions. However Ramanujan may have acquired his very peculiar knowledge of this theory, it was not from Carr.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-03-23T17:47:51.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11408" PostTypeId="1" CreationDate="2020-01-29T11:18:11.290" Score="5" ViewCount="131" Body="&lt;p&gt;I'm searching for a quote by Littlewood about Hardy not giving proper credit. The story (as I remember it) is that Littlewood claimed uncredited authorship of something Hardy wrote, Hardy claimed it was trivial, and Littlewood made some analogy to the interactions between oppressors and oppressed. The analogy is what Im looking for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ive searched quotes from Littlewood's Miscellany and A Mathematician's Apology to no avail.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe it's connected to (but different from) this quote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I read in the proof sheets of Hardy on Ramanujan: “As someone said, each of the positive integers was one of his personal friends.” My reaction was, “I wonder who said that; I wish I had.” In the next proof-sheets I read (what now stands), “It was Littlewood who said…”. What had happened was that Hardy had received the remark in silence and with poker face, and I wrote it off as a dud. - Littlewood&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11035" LastActivityDate="2020-01-29T11:18:11.290" Title="What was Littlewood's quip about Hardy and plagiarism?" Tags="|mathematics|mathematicians|mathematics-social-history|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11409" PostTypeId="2" ParentId="11402" CreationDate="2020-01-29T12:36:49.893" Score="4" Body="&lt;p&gt;It is hard to answer the question as asked because it presumes that people &lt;em&gt;first&lt;/em&gt; developed the idea of air drag force that alters motion of objects on Earth, and &lt;em&gt;then&lt;/em&gt; realized that laws of motion can be made universal by taking that into account.  This is not, however, how things progressed.&lt;/p&gt;&#xA;&lt;p&gt;People knew of air resistance, in some fashion, since prehistoric times. Aristotle already considered air to be a corporeal fluid that resists motion. It is also true that he sharply separated the laws of motion on Earth and in the Heavens by ascribing &amp;quot;natural&amp;quot; motions towards the Earth's center for the former, and circular ones for the latter. According to his idea of &amp;quot;violent&amp;quot; motion, a &amp;quot;force&amp;quot; had to be constantly applied to produce it, and this force was proportional to the resulting speed and inverse proportional to the medium resitance (at least, in modernized reconstructions of what he says, see &lt;a href=&quot;https://arxiv.org/abs/1312.4057&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rovelli, Aristotle's Physics: a Physicist's Look (And Others)&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Aristotle's explanation of arrow's flight was based on a quaint idea of &lt;a href=&quot;https://en.wikipedia.org/wiki/Antiperistasis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;antiperistasis&lt;/em&gt;&lt;/a&gt;, where an arrow is resisted by air from the front, but pushed by it from behind, as it &amp;quot;&lt;em&gt;fills the void left by the advancing tail&lt;/em&gt;&amp;quot; because &amp;quot;nature abhors vacuum&amp;quot;. This idea was so quaint that he himself expressed doubts about it in &lt;em&gt;Physica VIII&lt;/em&gt;, and it was outright mocked already in antiquity by Philoponus, who sarcastically suggested making an arrow move by waiving hands behind it. With Philoponus (or possibly already Hipparchus) originates the idea of &lt;em&gt;impetus&lt;/em&gt;, stored energy and/or momentum, that maintains motion without application of force, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2136/55&quot;&gt;Why don't we learn Buridan's laws of motion?&lt;/a&gt; This was the prototype of inertia, which turned air into a source of resistance only. But the conception of force, for him and medieval theorists of impetus, remained basically Aristotelian until Leonardo and Galileo, see &lt;a href=&quot;https://www.jstor.org/stable/20124101&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hanson, Aristotle on Motion through Air&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The questioning of the fundamental difference between the Earth and the Heavens became natural after Copernicus, who displaced Earth from its place of pride at the center, and especially Kepler, but air drag played little role in it. The study of air and vacuum progressed on its own course, and we find the idea that air resistance can be disregarded when studying fundamentals of motion clearly spelled out already by Galileo. But, while he rejected the Aristotelian conception of force, he did not yet offer an alternative of his own. Descartes, Huygens and others working on collisions and pendulums contributed to its emergence in Newton's work, see &lt;a href=&quot;https://hsm.stackexchange.com/a/9705/55&quot;&gt;How did Newton measure forces in his experiments to establish the laws of motion?&lt;/a&gt; So the fully modern notion of drag force, accounting for resistance in a non-Aristotelian fashion, and the idea of air being the &lt;em&gt;something&lt;/em&gt; different between the motion on Earth and in the Heavens are not put together until book II of Newton's &lt;em&gt;Principia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is from Hanson:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Regarding an object's motion through an aerial medium, Aristotle correctly noted the latter's resisting and assisting effect on the former. Aristotle's doctrine of projectile motion as founded on the assisting action of the air (itself construed as a fluid medium), was opposed in the 6th century by Philoponus... This Greek grammarian urged that the casting of a projectile imparts to it a certain energy, or impetus, which is transported from the throwing agent and somehow &amp;quot;transfused&amp;quot; into that which is thrown... Philoponus felt that it was this energy, or impetus, which maintained the arrow in motion during its trajectory long after having left the bowstring. The air only resisted its progress forward.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Aristotle's medium theory and Philoponus' first intimations of an impetus theory remained in opposition to each other until the collective insights of Descartes, Galileo and Newton established the Law of Inertia beyond question. By the late 17th century, all serious controversy on the matter was virtually terminated; the air was thenceforth conceived of only as a factor of resistance to the motion of thrown or flown objects. No one continued to argue, a la Aristotle, that the aerial flow could actually assist in some manner the forward and upward flight of a projectile, or of a bird.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Leonardo da Vinci was an interesting interim figure, albeit historically ineffectual. Before 1506 he had assumed that some action of the air did assist motion. After this date, however, he thought of air largely as a resisting medium, which resistance he ascribed to its &amp;quot;compressibility.&amp;quot; It was this compressibility, indeed, which Leonardo identified as the cause of lift in bird flight.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Galileo, whose Mechanics crushed the Aristotelian theory of the &amp;quot;pushing&amp;quot; medium, recognized a proportionality between the air's compressibility and the velocity of objects moving through it. One might hardly have expected anything else: the porpoise would ply through thick oil, or liquid mercury, or a tank of sand, much slower than through water. Galileo's work with pendulum pairs, involving precise calculations for the vacuum condition, led to his conclusion that projectile paths, and pendulums, could be described in a manner unaltered by air resistance.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-01-29T13:38:50.937" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11411" PostTypeId="1" AcceptedAnswerId="11427" CreationDate="2020-01-30T02:13:25.523" Score="5" ViewCount="510" Body="&lt;p&gt;I found lots of background information about the discovery of both imaginary and complex numbers, and enough information about the first two types of hypercomplex numbers; quaternions and octonions (also known as Cayley Numbers). However, I haven't found who were the mathematicians that pioneered the study of the sedenions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is known that this can be achieved through the Cayley-Dickson doubling process. This question is not about the usefulness of the sedenions, or the operations that are allowed on them, but who pioneered their study and when.&lt;/p&gt;&#xA;" OwnerUserId="11038" LastEditorUserId="11038" LastEditDate="2020-02-01T06:52:10.970" LastActivityDate="2020-03-06T16:40:52.397" Title="Who pioneered the study of the sedenions?" Tags="|mathematics|number-theory|mathematical-physics|complex-analysis|algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11412" PostTypeId="2" ParentId="11405" CreationDate="2020-01-30T05:16:51.460" Score="2" Body="&lt;p&gt;To me the most striking example of an invention that could have been invented long before it did is the Antikythera mechanism (&lt;a href=&quot;https://en.wikipedia.org/wiki/Antikythera_mechanism&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Antikythera_mechanism&lt;/a&gt;), a calculating machine that predates those of Pascal and Napier by almost 2000 years, but which was lost and forgotten, until its discovery in 1901. If this invention had become known before it sank at sea, the development of Science might have advanced several centuries ahead.&lt;/p&gt;&#xA;" OwnerUserId="11040" LastActivityDate="2020-01-30T05:16:51.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11413" PostTypeId="2" ParentId="11312" CreationDate="2020-01-30T05:29:27.517" Score="12" Body="&lt;p&gt;The digit 0 was not needed in the Roman numeral system because this is not a &lt;a href=&quot;https://en.wikipedia.org/wiki/Positional_notation&quot; rel=&quot;noreferrer&quot;&gt;positional system.&lt;/a&gt; The only case when it was used was &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_numerals#Zero&quot; rel=&quot;noreferrer&quot;&gt;when the number was actually zero,&lt;/a&gt; which they called &lt;em&gt;nulla&lt;/em&gt;.&lt;br&gt;&#xA;Roman digits have a fixed value independent of where they are in a number. For example, the letter X means 10 &lt;em&gt;per se&lt;/em&gt;. Its value is added to the other digits in a number, with the exception that if it comes before a higher value digit it is subtracted instead of added, e.g. XL means 40 (50-10) and LX means 60 (50+10).&lt;br&gt;&#xA;In the Indian or Arabic system however, each digit has a value which depends on its position, e.g. the digit 2 can have the value 2 if it is in the last position, 20 if it is in the next position, or 200, etc...&#xA;In this positional system you need to fill every position in the number to preserve its meaning. That's why number zero becomes necessary to indicate an empty position.&lt;/p&gt;&#xA;" OwnerUserId="11040" LastEditorDisplayName="user10930" LastEditDate="2020-02-24T09:40:12.377" LastActivityDate="2020-02-24T09:40:12.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11414" PostTypeId="2" ParentId="11411" CreationDate="2020-01-30T05:38:52.867" Score="6" Body="&lt;p&gt;There are (at least) two different types of numbers called &quot;sedenions&quot;.  The first ones were introduced by Muses in 1980, who called them &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt;-ions, and renamed into &quot;sedenions&quot; by Carmody in 1988. However, Sorgsepp and Lohmus beat him to the name in &lt;a href=&quot;https://inis.iaea.org/search/search.aspx?orig_q=RN:14743619&quot; rel=&quot;nofollow noreferrer&quot;&gt;Binary and Ternary Sedenions (1981)&lt;/a&gt;, which they used more in line with &quot;quaternions&quot; and &quot;octonions&quot; (&lt;em&gt;sedecim&lt;/em&gt; is Latin for sixteen), i.e. for the output of the Cayley-Dickson construction:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The possibility of an extension of the 8-dimensional Cayley algebra into a 16-dimensional sedenion algebra is investigated. For retaining some desirable properties, the binary sedenion algebra obtained by the Cayley-Dickson process is modified into an algebra with a ternary product. Both for binary and ternary sedenion algebras, the generalized Cauchy-Riemann and Laplace equations are written out. The finite projective geometries formed by hypercomplex sedenion units are briefly discussed.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As Imaeda and Imaeda, who studied the Cayley-Dickson sedenions further, explain in &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S009630039900140X?via%3Dihub&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sedenions: algebra and analysis (2000)&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Some parts of our work paralells that of other authors [3-5]; but unlike the sedenions discussed in [3,4] our sedenions are non-alternative, non-modular&#xA;  and posess real norm. In [5] alternativity is restored by introducing ternary&#xA;  products. Should Hurwitz theorem be one possible indication of non-conformity&#xA;  of sedenions to current mathematical methods in physics? It is clear from&#xA;  Muses [6] and Gursey [7], and other authors that some new algebraic and&#xA;  analytical structures, such that might be offered by a &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt;-dimensional algebra,&#xA;  may hold a key to better formulation of physical theories. Furthermore, its&#xA;  relation to the Lie group &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt; remains compelling under any circumstances.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The &quot;other authors&quot; are Carmody ([3-4]) and Sorgsepp-Lohmus ([5]). In &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S0096300396000513&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carmody's Circular and hyperbolic quaternions, octonions, and sedenions — Further results (1997)&lt;/a&gt; we find a prefatory note by Muses, written at Carmody's request, which reads in particular:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;From the prevalence of counterquaternions in&#xA;  quantum physics, I feel sure that counteroctonions and sedonions will inevitably be entrained as well on the frontiers of physics. By 1980 I had discovered/invented them and called them 16-ions [3, p. 76] and&#xA;  perhaps that designation is least opaque. In any case, their 16 elementary units (&lt;span class=&quot;math-container&quot;&gt;$i_0$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$i_7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$ε_0$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$ε_7$&lt;/span&gt;) plus the &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt;-arithmetic (&lt;span class=&quot;math-container&quot;&gt;$w_0$&lt;/span&gt; through &lt;span class=&quot;math-container&quot;&gt;$w_7$&lt;/span&gt;) form a&#xA;  &lt;span class=&quot;math-container&quot;&gt;$24$&lt;/span&gt;-dimensional vector space. This is a watershed in hypernumber arithmetic and algebra because the equation &lt;span class=&quot;math-container&quot;&gt;$x^0 - 1 = 0$&lt;/span&gt; has elementary hypernumber solutions only if &lt;span class=&quot;math-container&quot;&gt;$x=\pm i_n$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x=\pm ε_n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$x=\pm w_n$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The reference [3] is to &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0096300380900168?via%3Dihub&quot; rel=&quot;nofollow noreferrer&quot;&gt;Muses's Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries (1980)&lt;/a&gt; that details his previous announcements in 1976 and 1978. In the abstract we read:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It turns out that more than three kinds of &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;-type hypemumbers and more&#xA;  than three hinds of the &lt;span class=&quot;math-container&quot;&gt;$ε$&lt;/span&gt;-type are needed to ensure the necessary nilpotent and&#xA;  noncommutative algebra required in unified field theory. It is also shown that “more&#xA;  than three” here means “at least seven”; and it turns out that a l6-dimensional&#xA;  arithmetic is needed for such computations. The following paper contextualizes, characterizes, and specifies that arithmetic at the apex of a hierarchy susceptible of clear geometric definition. And the hypemumbers needed in quantized unified field theory are specified.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Carmody's paper quoted above is a follow-up to his &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0096300388901336&quot; rel=&quot;nofollow noreferrer&quot;&gt;Circular and hyperbolic quaternions, octonions, and sedenions (1988)&lt;/a&gt; that names them &quot;sedenions&quot; with the following explanation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Muses has exhibited the fact that a 16-dimensional system comprising the elements &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$i_1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$i_7$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$ε_1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$ε_7$&lt;/span&gt;, and a sixteenth element is possible. This system is called the sedenions after the Latin sedecim, sixteen.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10106" LastEditDate="2020-02-01T10:08:01.310" LastActivityDate="2020-02-01T10:08:01.310" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11415" PostTypeId="2" ParentId="11397" CreationDate="2020-01-30T06:44:14.050" Score="4" Body="&lt;p&gt;I can think of&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Alfred Wegener&lt;/strong&gt;, the discoverer of plate tectonics, who died completely ignored before his theory gained acceptance.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ludwig Boltzmann&lt;/strong&gt;, defender of statistical mechanics in Physics.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Giordano Bruno&lt;/strong&gt;, who was burnt at the stake for defending that the Universe is infinite and not centered around the Earth.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Galileo Galilei&lt;/strong&gt;, who was kept under house arrest for the same reason.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Gregor Mendel&lt;/strong&gt;, discoverer of the laws of genetic inheritance, who died completely unknown.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11040" LastEditorUserId="4251" LastEditDate="2020-07-13T11:29:13.863" LastActivityDate="2020-07-13T11:29:13.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11416" PostTypeId="2" ParentId="8436" CreationDate="2020-01-30T07:00:28.400" Score="2" Body="&lt;p&gt;The first 3 digits can be easily explained by the rapid tracing of strokes without lifting the writing stylus from the surface of the paper. If you look at the Chinese characters for 1, 2 and 3: 一, 二, 三, you see these represent obvious numbers of strokes. If you write 二 quickly without lifting the pen, this becomes Z, and from there 2. The same goes for 三 to become 3.&lt;br&gt;&#xA;The digit 4 derives from Brahmi script, where it was represented as a +, possibly a representation of 4 lines or 4 directions. This is reminiscent of the Kharoṣṭhī script, where it was represented as a X. A slanted stroke was added to + to become 4 for aesthetic and stylish reasons.&lt;br&gt;&#xA;Digit 0 is the representation of an empty space.&lt;/p&gt;&#xA;" OwnerUserId="11040" LastEditorUserId="11040" LastEditDate="2020-01-30T09:55:28.880" LastActivityDate="2020-01-30T09:55:28.880" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11417" PostTypeId="1" CreationDate="2020-01-30T10:57:18.343" Score="1" ViewCount="57" Body="&lt;p&gt;The 'square' subset symbols are sometimes used to express &lt;a href=&quot;https://math.stackexchange.com/questions/1569400/does-sqsubset-have-any-special-meaning&quot;&gt;analogous concepts to subsets&lt;/a&gt;, like prefixes or suffixes.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However their use in &lt;a href=&quot;https://en.wikipedia.org/wiki/Communicating_sequential_processes&quot; rel=&quot;nofollow noreferrer&quot;&gt;CSP&lt;/a&gt; seems to be counterintuitive to their shape: &lt;span class=&quot;math-container&quot;&gt;$⊑$&lt;/span&gt; appears analogous to &lt;span class=&quot;math-container&quot;&gt;$⊇$&lt;/span&gt; (as opposed to the more similar looking &lt;span class=&quot;math-container&quot;&gt;$⊆$&lt;/span&gt;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;A process &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; refines &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, denoted as &lt;span class=&quot;math-container&quot;&gt;$Q⊑_TP$&lt;/span&gt;, iff &lt;span class=&quot;math-container&quot;&gt;$traces(P) ⊆ traces(Q)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Does Hoare explain anywhere why he chose this orientation of the symbol?&lt;/p&gt;&#xA;" OwnerUserId="10974" LastActivityDate="2020-01-30T10:57:18.343" Title="Is there a reason $⊑$ in CSP is analogous to $⊇$ (as opposed to $⊆$)?" Tags="|terminology|computer-science|language|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11418" PostTypeId="2" ParentId="11397" CreationDate="2020-01-31T04:33:31.007" Score="5" Body="&lt;p&gt;&lt;strong&gt;Julius Robert Mayer&lt;/strong&gt; is now known for enunciating in 1841 one of the original statements of the conservation of energy (First Law of Thermodynamics). But his achievements were overlooked, and priority for the discovery of the mechanical equivalence of heat was attributed to James Joule. According to &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Julius_von_Mayer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;He attempted suicide on 18 May 1850 and was committed to a mental institution. After he was released, he was a broken man and only timidly re-entered public life in 1860. However, in the meantime, his scientific fame had grown, and &lt;strong&gt;he received a late appreciation of his achievement, although perhaps at a stage where he was no longer able to enjoy it.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-02-28T16:39:02.497" LastActivityDate="2020-02-28T16:39:02.497" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11424" PostTypeId="5" CreationDate="2020-02-01T17:22:15.177" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-02-01T17:22:15.177" LastActivityDate="2020-02-01T17:22:15.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11425" PostTypeId="4" CreationDate="2020-02-01T17:22:15.177" Score="0" Body="For questions asking about how a particular scientist came to being depicted to the general public as he/she is today" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-02-10T03:43:29.457" LastActivityDate="2020-02-10T03:43:29.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11426" PostTypeId="1" CreationDate="2020-02-02T12:52:17.747" Score="10" ViewCount="3476" Body="&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\dagger$&lt;/span&gt; symbol is often used in quantum mechanics,and also often in general mathematics to represent the conjugate transpose operation.For Hermitian matrices we can write &lt;span class=&quot;math-container&quot;&gt;$$A^\dagger=A$$&lt;/span&gt;Who introduced this notation?&lt;/p&gt;&#xA;" OwnerUserId="11051" LastActivityDate="2020-02-11T15:09:20.340" Title="Who introduced the &quot;dagger&quot;symbol as conjugate transpose in quantum mechanics?" Tags="|physics|quantum-mechanics|mathematical-physics|linear-algebra|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11427" PostTypeId="2" ParentId="11411" CreationDate="2020-02-03T05:03:12.813" Score="4" Body="&lt;p&gt;On Bibliography of Quaternions and Allied Mathematics by Alexander Macfarlane I found this:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On page 72; James Byrnie Shaw&#xA;1896 Sedenions (title). American Assoc. Proc., 45, 26.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I couldn't find this reference, but the same author wrote this book:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Synopsis of Linear Associative Algebra: A Report on its Natural Development and Results Reached up to the Present Time. 1907.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From its Table of Contents; Part II: Particular Algebras. Section XVIII: Triquaternions and Quadriquaternions. Page 91.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Early on Sedenions were also known as &quot;quadriquaternions&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Section XIX: Sylvester Algebras. Page 93.&#xA;Covers &quot;Nonions&quot; (9-ions), and &quot;Sedenions&quot; (16-ions).&#xA;Here the Sedenions are attributed to James Joseph Sylvester.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On page 76; James Joseph Sylvester&#xA;1883-4 On quaternions, nonions, sedenions, etc. Johns Hopkins&#xA;Univ. Circ., 3. Nos. 7 and 9. 4, No. 28.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This second reference can be found among The Collected Mathematical Papers of James Joseph Sylvester, [Volume IV(1882—1897)]:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sylvester, James Joseph (1973) [1904], Baker, Henry Frederick (ed.), The collected mathematical papers of James Joseph Sylvester, IV, New York: AMS Chelsea Publishing, ISBN 978-0-8218-4238-6&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, as far as I understood, James Joseph Sylvester appears on the literature as the proponent of two Algebras; the one from the Nonions and the Sedenions. I will leave as an open question for the community to confirm, correct, or debunk that J.J. Sylvester &quot;discovered&quot; or &quot;was the pioneer&quot; of their study in 1883.&lt;/p&gt;&#xA;" OwnerUserId="11038" LastActivityDate="2020-02-03T05:03:12.813" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11428" PostTypeId="2" ParentId="11389" CreationDate="2020-02-03T09:58:42.203" Score="1" Body="&lt;p&gt;No. The question is badly conceived and phrased. You cannot 'steal' a result and proving the theft is unconvincing. There is a sensible difference between a priority dispute and crime. It was not the Facebook Age and people were more interested in valid results and not in the person behind them. Objective validity  is  impersonal and, except for taking out papers from somebody's drawer, stealing is a metaphor. There might have been bad feelings and rivalities but it is obvious that accusations are hard to prove. Freeman Dyson recounts about Feynman saying &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientific_priority#cite_ref-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;(see)&lt;/a&gt;he was&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Always giv[ing] the bastards more credit than they deserve.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(Of course this not how Einstein felt, nor his hagiographers. But that seems to be the spirit that prevailed then in science and mathematics)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-02-03T09:58:42.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11429" PostTypeId="2" ParentId="11399" CreationDate="2020-02-04T11:00:10.023" Score="2" Body="&lt;p&gt;Most of these books will be available at academic libraries. You can search on worldcat to see which academic libraries have the book near you. &lt;a href=&quot;https://www.worldcat.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.worldcat.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11059" LastActivityDate="2020-02-04T11:00:10.023" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11430" PostTypeId="1" CreationDate="2020-02-04T15:27:53.357" Score="0" ViewCount="60" Body="&lt;p&gt;The answer to the question would appear to be &quot;no&quot; looking at the transcription of Riemann's 1854 &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die Hypothesen, welche der Geometrie zu Grunde liegen&lt;/a&gt; lecture. However, Bernhard Riemann (1826-1866) and Elwin Bruno Christoffel (1829–1900) were contemporaries.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there any manuscript in which Riemann adopted the use of Christoffel symbols?&lt;/p&gt;&#xA;" OwnerUserId="2352" LastEditorUserId="2352" LastEditDate="2020-02-04T16:19:27.913" LastActivityDate="2020-02-04T16:19:27.913" Title="Were Christoffel symbols used by Riemann?" Tags="|biographical-details|relativity-theory|differential-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11432" PostTypeId="2" ParentId="11426" CreationDate="2020-02-05T06:00:16.293" Score="10" Body="&lt;p&gt;In a now-deleted comment,  Consigliere ZARF listed a number of papers published in &lt;em&gt;Zeitschrift für Physik&lt;/em&gt; in the late 1920's that used this notation.  The earliest was Pascual Jordan's 1927 &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01390903&quot; rel=&quot;noreferrer&quot;&gt;&quot;Über eine neue Begründung der Quantenmechanik&quot;&lt;/a&gt;, using the notation on pp.816-817; with about 10 other papers published in the following few years, all in the &lt;em&gt;ZfP&lt;/em&gt;, all behind a paywall.&#xA;According to Duncan and Janssen &lt;a href=&quot;https://arxiv.org/pdf/1204.6511.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;(Never) Mind your p’s and q’s:&#xA;Von Neumann versus Jordan on the&#xA;Foundations of Quantum Theory&quot;&lt;/a&gt;, p.57, the Jordan paper was where the notation was introduced.  The &lt;a href=&quot;https://eudml.org/doc/159201&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt; &quot;Über die Grundlagen der Quantenmechanik&quot; by &#xA;D. Hilbert, J. von Neumann, and L. Nordheim, &lt;em&gt;Mathematische Annalen&lt;/em&gt; (1928) uses (on page 19) &lt;span class=&quot;math-container&quot;&gt;$A^+$&lt;/span&gt; for the adjoint of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Up to the point reading Consigliere ZARF's comment I had focused on English-language literature, which seemed to lag about a decade behind the German.  The rest of this post describes what I found there.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;An approximate answer: Google books search for &quot;dagger+Hermitian&quot;  led to these early uses of the notation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Page 59 of W. H. &lt;a href=&quot;https://en.wikipedia.org/wiki/Wendell_H._Furry&quot; rel=&quot;noreferrer&quot;&gt;Furry&lt;/a&gt;'s 1938 &lt;a href=&quot;https://pdfs.semanticscholar.org/6c7f/0814480b494cc30ac24e421441ecfd96f270.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;Note on the Theory of the Neutral Particle&quot;&lt;/a&gt;,&#xA;Phys. Rev. vol 54, p.56  has this:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In dealing with such matrices we shall use&#xA;  an asterisk (*) to indicate the complex conjugate&#xA;  matrix, a prime (') to indicate the transposed&#xA;  matrix, and a dagger (&lt;span class=&quot;math-container&quot;&gt;$\dagger$&lt;/span&gt;) to indicate the Hermitian adjoint matrix.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which seems to me to be an indication that this notation needed explaining.  A few years later, Booth and Wilson, in &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1940.0071&quot; rel=&quot;noreferrer&quot;&gt;&quot;Radiative processes involving fast mesons&quot;&lt;/a&gt; (&lt;em&gt;Proc. Roy Soc A&lt;/em&gt;, v175, July 1940) wrote this in a footnote on p.487:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Note that in order to&#xA;  conform to the usual custom we modify Kemmer’s notation and use a dagger to denote&#xA;  the Hermitian conjugate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;indicating that the notation had by then become standard (although not universal, or else Kemmer's notation from the year before would not have needed adjusting).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For all I know, it is possible that Furry was the true original introducer of the dagger notation.  It is certain that the notation was in use in the late 1930's. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Added: Furry cites an earlier paper of his (Phys Rev v51, p.125 (1937) and &lt;a href=&quot;http://www.numdam.org/article/AIHP_1936__6_2_109_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;another&lt;/a&gt; by Pauli (&quot;Contributions mathématiques à la théorie des matrices de Dirac&quot;, &lt;em&gt;Annales de l'institut Henri Poincaré&lt;/em&gt;, Volume 6 (1936) no. 2, p. 109-136) , that use a superscript plus sign &lt;span class=&quot;math-container&quot;&gt;$A^+$&lt;/span&gt; for the same thing.  Page 119 of Pauli's paper has&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;En général, nous désignerons par &lt;span class=&quot;math-container&quot;&gt;$\gamma^{+\mu}$&lt;/span&gt; les matrices conjuguées hermitiques (transposées et conjuguées) des &lt;span class=&quot;math-container&quot;&gt;$\gamma^\mu$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and so on, with what I think is a typo in the next bit: &quot;(c’est-à-dire telles que &lt;span class=&quot;math-container&quot;&gt;$\gamma^{+\mu}_{\rho\sigma}=\gamma^{*\mu}_{\rho\sigma}$&lt;/span&gt;)&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The leap from superscript &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; to superscript &lt;span class=&quot;math-container&quot;&gt;$\dagger$&lt;/span&gt; is understandable. It was taken, for instance,  by &lt;a href=&quot;https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.15.175&quot; rel=&quot;noreferrer&quot;&gt;Pauli&lt;/a&gt; in his 1943 paper, &quot;On Dirac's New Method of Field Quantization&quot;, if not earlier. (See equation (4) on p.176.)  But not, evidently, by Schwinger in his 1948 paper &lt;a href=&quot;https://journals.aps.org/pr/pdf/10.1103/PhysRev.74.1439&quot; rel=&quot;noreferrer&quot;&gt;&quot;Quantum Electrodynamics. I. A Covariant Formulation&quot;&lt;/a&gt;, who uses &lt;span class=&quot;math-container&quot;&gt;$C^+$&lt;/span&gt; for the Hermitian conjugate of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, on p.1441.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-02-11T15:09:20.340" LastActivityDate="2020-02-11T15:09:20.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11433" PostTypeId="1" AcceptedAnswerId="11438" CreationDate="2020-02-06T11:54:29.987" Score="9" ViewCount="3364" Body="&lt;p&gt;Linear inertia is called &lt;strong&gt;mass&lt;/strong&gt;. Rotational inertia is called &lt;strong&gt;moment of inertia&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Moment of inertia&lt;/em&gt; is an odd choice for the term for this property. It doesn't seem to &quot;fit&quot; with the style or pattern of a simple term like &lt;em&gt;mass&lt;/em&gt;, although it is the rotational equivalent to mass. Other rotational properties keep a pattern, &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;like &lt;em&gt;angular velocity&lt;/em&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;angular acceleration&lt;/em&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;angular momentum&lt;/em&gt; etc.;&lt;/li&gt;&#xA;&lt;li&gt;but there is no &quot;angular mass&quot;. It is instead out-of-the-blue called &lt;em&gt;moment of inertia&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Sure, there is also no &quot;angular force&quot;, but at least that is instead given a simple, clean term: &lt;em&gt;torque&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Moment of inertia&lt;/em&gt; really seems like the odd one in the group, breaking any chance of a pattern.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, Euler &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment_of_inertia&quot; rel=&quot;nofollow noreferrer&quot;&gt;introduced this term&lt;/a&gt; in his work &quot;&lt;em&gt;Theoria motus corporum solidorum seu rigidorum&lt;/em&gt;&quot; from 1765. From excerpts I can find from that book (unfortunately I cannot read the Latin of the book itself), I cannot find any reasoning behind using that particular term.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &lt;em&gt;moment&lt;/em&gt; &lt;a href=&quot;https://www.etymonline.com/search?q=moment&quot; rel=&quot;nofollow noreferrer&quot;&gt;seems to originate from&lt;/a&gt; the Latin word &lt;em&gt;momentum&lt;/em&gt;, meaning &lt;em&gt;movement/change/alteration&lt;/em&gt;, and thus it can make sense to not purely use the word as &quot;a brief duration&quot; but also in relation to physical motion. But that does not explain much about why &lt;em&gt;moment of inertia&lt;/em&gt; suddenly is the choice for the property.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the reasoning for using the term &lt;strong&gt;moment of inertia&lt;/strong&gt; as rotational inertia?&lt;/p&gt;&#xA;" OwnerUserId="4885" LastEditorUserId="26" LastEditDate="2021-02-20T15:19:50.463" LastActivityDate="2021-02-20T15:19:50.463" Title="What is the reasoning behind using &quot;moment&quot; in the &quot;moment of inertia&quot;?" Tags="|terminology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11434" PostTypeId="1" AcceptedAnswerId="11436" CreationDate="2020-02-06T12:34:21.263" Score="6" ViewCount="1822" Body="&lt;p&gt;I saw a letter of Euler to Lagrange congratulating him on his usage of imaginary numbers in the &quot;analysis devoted to rational numbers alone&quot;, was that the first known such usage? What was the likely inspiration of Lagrange(and Euler)?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-02-06T17:32:25.880" Title="When was the first recorded occurence of irrational and imaginary number usage in number theory?" Tags="|biographical-details|number-theory|real-analysis|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11435" PostTypeId="2" ParentId="11433" CreationDate="2020-02-06T13:26:12.257" Score="0" Body="&lt;p&gt;For one thing,  &quot;inertial mass&quot;  could have turned out to be different from &quot;gravitational mass,&quot;  and observing that they are the same was a big &quot;win&quot; for physics theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For another, quoting wikipedia, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Inertia is one of the primary manifestations of mass, which is a&#xA;  quantitative property of physical systems  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So inertia , linear or otherwise, is not the same as mass. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-02-06T13:26:12.257" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11436" PostTypeId="2" ParentId="11434" CreationDate="2020-02-06T15:43:28.160" Score="10" Body="&lt;p&gt;Irrational numbers were used by the ancient Greeks when they were discovered. The earliest texts did not survive but there are plenty of them in Euclid. Though they are not called numbers. The theory of proportions of Eudoxus-Euclid is equivalent to the theory of real numbers. Euclid's book X contains some very complicated theory of some irrational numbers.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ancient theories only dealt with algebraic numbers (like square and cubic root of integers), they did not know that there are transcendental numbers.&#xA;They did not know that &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is transcendental.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first general result on all irrational numbers is due to Nicole Oresme (14th century):&#xA;he proved that the fractional parts of &lt;span class=&quot;math-container&quot;&gt;$na$&lt;/span&gt; are dense on &lt;span class=&quot;math-container&quot;&gt;$[0,1]$&lt;/span&gt; for any irrational &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Imaginary numbers occur for the first time in the theory of cubic and quartic equations (16th century). When the formulas for them were discovered, it was noticed that even in the case when the final answer is real, the intermediate calculation involves square roots of negative numbers. For example try to solve &lt;span class=&quot;math-container&quot;&gt;$x^3-x=0$&lt;/span&gt; using the Cardano formula. So they started to develop arithmetic of complex numbers.&#xA;This was a surprisingly long development achieving a clear and complete theory only in the end of 18th century. Before this was achieved, many looked at their use with a suspicion. This suspicion did not dissolve completely until the later part of the 19th century. (They say that Chebyshev was against the use of complex numbers. If this is true, he was probably the last great mathematician who did not recognize them. His students used them routinely). &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-02-06T15:59:55.157" LastActivityDate="2020-02-06T15:59:55.157" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11437" PostTypeId="2" ParentId="11434" CreationDate="2020-02-06T17:32:25.880" Score="4" Body="&lt;p&gt;Regarding the headline question, Newton, in his text &lt;em&gt;Universal Arithmetick&lt;/em&gt;, gave what Leo Corry states may be the first definition of &lt;em&gt;number&lt;/em&gt; that included both positive and negative integers, fractions, and irrationals.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is Newton's definition:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;By &lt;em&gt;number&lt;/em&gt; we understand, not so much a &lt;em&gt;Multitude of Unities&lt;/em&gt; as an abstracted ratio of any Quantity, to another Quantity of the same kind, which we take for unity. And this is threefold; integer, fracted, and surd: An &lt;em&gt;Integer&lt;/em&gt;, is what is measured by Unity;  a &lt;em&gt;fraction&lt;/em&gt;, that which a submultiple Part of Unity measures; and a &lt;em&gt;Surd&lt;/em&gt;, to which unity is incommencurable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On this definition, Corry writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The unit, the integers, the fractions, and the irrational numbers appear here - perhaps for the first time and certainly in an influential text in such clear-cut terms - all as mathematical entities of one and the same kind, the differences between them being circumscribed to a single feature clearly discernible in terms of a property of ratios. ... Moreover, and very importantly, numbers are &lt;em&gt;abstract entities&lt;/em&gt;: themselves they are not quantities, but they may represent either quantities or a ratio between quantities. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Regarding imaginary number, Newton is more hesitant.  He does not refer to them as &quot;imaginary&quot; or &quot;fictions&quot;, as did Descartes and Wallis.  Newton uses the term &quot;impossible&quot;. Quoting Corry:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;And what he meant by &quot;impossible&quot; he explained by reference to the solution to the equation &lt;span class=&quot;math-container&quot;&gt;$$x^2 + 2ax + b^2 = 0.$$&lt;/span&gt;  Here, we obtain two roots, namely &lt;span class=&quot;math-container&quot;&gt;$$a + \sqrt{a^2 - b^2} \text{  and   } a - \sqrt{a^2 - b^2}.$$&lt;/span&gt;  Now, when &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; is greater than &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt; - Newton wrote - the roots are &quot;real&quot;.  In the opposite case, when &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt; is greater than &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt;, of course, the root is &quot;impossible&quot;.  But, interestingly, Newton nevertheless went on to stress that both expressions are roots of the polynomial, for the simple reason that when they are introduced in the equation in place of the unknowns, the equation is satisfied because &quot;their factors eliminate each other&quot;. In other words, a square root of a negative number is an impossibility and hence does not represent a number in the proper sense of the word, but expressions containing such impossible entities are legitimate roots of an equation and allow for an appealing formulation of the fundamental theorem of algebra, as Newton conceived of it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[Source: Leo Corry, &lt;em&gt;A Brief History of Numbers&lt;/em&gt;.]&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-02-06T17:32:25.880" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11438" PostTypeId="2" ParentId="11433" CreationDate="2020-02-06T21:51:16.643" Score="8" Body="&lt;p&gt;If Euler introduced the term and did not explain his reasoning we can only speculate as to what he had in mind. Euler himself was followed on many notational and terminological choices simply because he put them together in well structured and comprehensive books. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But Euler likely followed the precedent with the &quot;moment of force&quot;. According to &lt;a href=&quot;https://archive.org/details/dynamicsofrotati00wortuoft/page/6/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Worthington's Dynamics of Rotation (1900)&lt;/a&gt;, it came from a curious metamorphosis of the meaning of the word &quot;moment&quot; (originally, short duration), and the older meaning was still live at least at the time of Worthington's writing. &lt;a href=&quot;https://www.etymonline.com/word/moment&quot; rel=&quot;noreferrer&quot;&gt;According to EtymOnline&lt;/a&gt;, &quot;moment&quot; is used in old French as &quot;importance&quot; since the 12th century (in English since 1520s, think of &quot;momentous&quot;), such use in Latin must have come even earlier. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Equilibrium_of_Planes&quot; rel=&quot;noreferrer&quot;&gt;On the Equilibrium of Planes&lt;/a&gt; Archimedes referred to the joint &quot;importance&quot; of force and the place where it was applied to maintaining the equilibrium. Commandino's Latin translation of Archimedes in &lt;a href=&quot;https://archive.org/details/bub_gb_fvtPrJ6BL4wC/page/n8/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Liber De Centro Gravitatis Solidorum (1565)&lt;/a&gt; was:&quot;&lt;em&gt;The center of gravity of each solid figure is that point within it, about which on all sides parts of equal moment stand&lt;/em&gt;&quot;. Now, one can read &quot;moment&quot; here as &quot;importance&quot;, but one can also read it as referring to some specific quantity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;And that is, apparently, how it came to be read. &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment_(physics)&quot; rel=&quot;noreferrer&quot;&gt;&quot;Moment&quot; in physics&lt;/a&gt; now generally refers to quantities that are something times distance, and sums or integrals thereof. Higher moments involve the powers of distance. &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment_(mathematics)&quot; rel=&quot;noreferrer&quot;&gt;&quot;Moment&quot; in mathematics&lt;/a&gt; has a similar meaning, only applied more abstractly. Here is Worthington:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The word moment was first used in Mechanics in its now rather old-fashioned &#xA;  sense of 'importance' or 'consequence' and the moment of a &#xA;  force about an axis meant the importance of the force with &#xA;  respect to its power to generate in matter rotation about the &#xA;  axis; and again, the moment of inertia of a body with respect &#xA;  to an axis is a phrase invented to express the importance of &#xA;  the inertia of the body when we endeavour to turn it about &#xA;  the axis. When we say that the moment of a force about an &#xA;  axis varies as the force, and as the distance of its line of action from the axis, we are not so much defining the phrase &#xA;  'moment of a force' as expressing the result of experiments &#xA;  made with a view to ascertaining the circumstances under &#xA;  which forces are equivalent to each other as regards their &#xA;  turning power. It is important that the student should bear &#xA;  in mind this original meaning of the word, so that such &#xA;  phrases as 'moment of a force' and 'moment of inertia' may &#xA;  at once call up an idea instead of merely a quantity.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;But the word 'moment' has also come to be used by analogy &#xA;  in a purely technical sense, in such expressions as the 'moment of a mass about an axis' or 'the moment of an area with &#xA;  respect to a plane', which require definition in each case. In &#xA;  these instances there is not always any corresponding physical &#xA;  idea, and such phrases stand, both historically and scientifically, on a different footing.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-02-06T21:51:16.643" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11442" PostTypeId="1" CreationDate="2020-02-08T00:26:07.113" Score="1" ViewCount="356" Body="&lt;p&gt;When I did my MSc in astrophysics back in '95-'96, I was told that there had been an attempt around the early 20th century to account for the advance of the perihelion of Mercury by altering the Newtonian gravitational equation by changing the exponent of radial distance r to something like 2.0016.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now obviously this was rapidly debunked by the Einstein theory, but it still strikes me as very odd because in Newtonian theory the exponent has to be exactly 2 and so what could have motivated this complete kludge with no basis in physical or mathematical principle. It is unlike the Nordstrom theory because that, though wrong, is actually mathematically coherent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My problem is that this was an aside from a lecture 25 years ago and I can't find any details on it. I would like to know more on the reasoning behind this idea and when and by whom this was suggested. If possible I'd like to know if it gained any traction whatsoever.&lt;/p&gt;&#xA;" OwnerUserId="11074" LastEditorUserId="26" LastEditDate="2020-02-09T11:25:39.163" LastActivityDate="2020-02-09T11:25:39.163" Title="Advance of the perihelion of Mercury" Tags="|physics|gravity|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11443" PostTypeId="2" ParentId="11397" CreationDate="2020-02-08T00:43:21.400" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://www.nottingham.ac.uk/physics/about/history/george-green.aspx&quot; rel=&quot;noreferrer&quot;&gt;George Green&lt;/a&gt; springs to mind. Whilst he had a very interesting, and not unsuccessful life, it was only after his death and Lord Kelvin looking through his works on mathematical physics that he was appreciated for quite how significant he truly was. &lt;/p&gt;&#xA;" OwnerUserId="11074" LastActivityDate="2020-02-08T00:43:21.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11444" PostTypeId="2" ParentId="11442" CreationDate="2020-02-08T02:06:53.587" Score="4" Body="&lt;p&gt;The idea that the exponent in the law of gravitation is not exactly 2 was around since 18th century when people were trying to work the precise theory of the Moon. At some point, Clairaut thought that deviations in the Moon motion prove that the exponent cannot be 2. A satisfactory theory of the Moon was only developed in the middle of 18th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, it is not clear why you write that &quot;in Newtonian theory the exponent has to be exactly 2&quot;. On what grounds? After all, physical theories have to be checked against experiments and observations, and when a good theory does not account for some small deviations, it must be modified. This is exactly what happened when General relativity was developed. Of course it is a much more elegant modification than just replacing the exponent, but how could one know in advance what kind of modification is required?&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-02-08T02:06:53.587" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11445" PostTypeId="2" ParentId="11442" CreationDate="2020-02-08T07:50:32.090" Score="4" Body="&lt;p&gt;In &lt;a href=&quot;https://www.mathpages.com/rr/s6-02/6-02.htm&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt; K. Brown writes:  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The failure to arrive at a realistic Newtonian explanation for the&#xA;  anomalous precession led some researchers, notably Asaph Hall and&#xA;  Simon Newcomb, to consider the possibility that Newtonian theory&#xA;  itself was at fault, i.e., that perhaps gravity isn't exactly an&#xA;  inverse square law. Hall noted that he could account for Mercury's&#xA;  precession if the law of gravity, instead of falling off as &lt;span class=&quot;math-container&quot;&gt;$1/r$&lt;/span&gt; ,&#xA;  actually falls of as &lt;span class=&quot;math-container&quot;&gt;$1/r^n$&lt;/span&gt; where the exponent &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is 2.00000016. However,&#xA;  most people didn't (and still don't) find that idea to be very&#xA;  appealing, since it conflicts with basic conservation laws, e.g.,&#xA;  Gauss's Law, unless we also postulate a correspondingly modified&#xA;  metric for space (ironically enough). Several researchers attempted to&#xA;  explain Mercury’s precession by imposing a finite speed of gravity on&#xA;  Newton’s theory, but these efforts too gave unsatisfactory results.&#xA;  (See Section 8.10.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It seems that the first one to propose this solution was &lt;strong&gt;Asaph Hall&lt;/strong&gt;:&lt;br&gt;&#xA;&lt;em&gt;A suggestion in the theory of Mercury&lt;/em&gt;, Astronomical Journal, vol. 14, iss. 319, p. 49-51 (1894), which can be found &lt;a href=&quot;http://adsabs.harvard.edu/full/1894AJ.....14...49H&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11040" LastEditorUserId="64" LastEditDate="2020-02-08T13:32:35.393" LastActivityDate="2020-02-08T13:32:35.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11447" PostTypeId="1" CreationDate="2020-02-09T02:12:49.217" Score="4" ViewCount="50" Body="&lt;p&gt;Harold Locke Hazen and Vannevar Bush created the Differential Analyzer at MIT between 1928 and 1931 to solve differential equations, according to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Differential_analyser&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia Article on the topic&lt;/a&gt;. I've read this other places as well. But the only differential equations I've ever read about being solved are those for artillery. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What other kinds of differential equations were solved?&lt;/p&gt;&#xA;" OwnerUserId="6112" LastActivityDate="2020-02-09T02:12:49.217" Title="What kinds of differential equations were solved on the early differential analyzer?" Tags="|differential-equations|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11448" PostTypeId="1" CreationDate="2020-02-09T11:29:17.453" Score="3" ViewCount="89" Body="&lt;p&gt;In &amp;quot;A Panorama of Pure Mathematics&amp;quot; by Dieudonne, he said&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The history of mathematics shows that a theory almost always originates in efforts to solve a specific problem (for example, the duplication of the cube in Greek mathematics). It may happen that these efforts are fruitless, and we have our first category of problems:&lt;/p&gt;&#xA;&lt;p&gt;(I) Stillborn problems (examples: the determination of Fermat primes, or the irrationality of Euler's constant).&lt;/p&gt;&#xA;&lt;p&gt;A second possibility is that the problem is solved but does not lead to progress on any other problem. This gives a second class:&lt;/p&gt;&#xA;&lt;p&gt;(II) Problems without issue (this class includes many problems arising from &amp;quot;combinatorics&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;A more favorable situation is one in which an examination of the techniques used to solve the original problem enables one to apply them (perhaps by making them considerably more complicated) to other similar or more difficult problems, without necessarily feeling that one really understands why they work. We may call these&lt;/p&gt;&#xA;&lt;p&gt;(III) Problems that beget a method (analytic number theory and the theory of finite groups provide many examples).&lt;/p&gt;&#xA;&lt;p&gt;In a few rather rare cases the study of the problem ultimately (and perhaps only after a long time) reveals the existence of unsuspected underlying structures that not only illuminate the original question but also provide powerful general methods for elucidating a host of other problems in other areas; thus we have&lt;/p&gt;&#xA;&lt;p&gt;(IV) Problems that belong to an active and fertile general theory (the theory of Lie groups and algebraic topology are typical examples at the present time).&lt;/p&gt;&#xA;&lt;p&gt;However, as Hilbert emphasized, a mathematical theory cannot flourish without a constant influx of new problems. It has often happened that once the problems that are of the greatest importance for their consequences and their connections with other branches of mathematics have been solved, the theory tends to concentrate more and more on special and isolated questions (possibly very difficult ones). Hence we have yet another category:&lt;/p&gt;&#xA;&lt;p&gt;(V) Theories in decline (at least for the time being: invariant theory, for example, has passed through this phase several times).&lt;/p&gt;&#xA;&lt;p&gt;Finally, if a happy choice of axioms, motivated by specific problems, has led to the development of techniques of great efficacy in many areas of mathematics, it may happen that attempts are made with no apparent motive to modify these axioms somewhat arbitrarily, in the hope of repeating the success of the original theory. This hope is usually in vain, and thus we have, in the phrase of Polya and Szego&lt;/p&gt;&#xA;&lt;p&gt;(VI) &lt;strong&gt;Theories in a state of dilution&lt;/strong&gt; (following the example of these authors, we shall cite no instances of this).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What could such theories be? Dieudonne unfortunately didn't give any examples.&lt;/p&gt;&#xA;" OwnerUserId="11081" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-02-09T11:29:17.453" Title="What did Dieudonne mean by &quot;Theories in a state of dilution&quot;?" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11449" PostTypeId="1" AcceptedAnswerId="11890" CreationDate="2020-02-09T18:18:24.700" Score="3" ViewCount="458" Body="&lt;p&gt;My problem is my lack of knowledge on the historical development of the constant used for the permittivity of free space. I know instead of asking the question I should read Wiki on the history of &quot;permittivity&quot; and I did.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The article in some some sense seems to imply Maxwell's Equations were necessary in order to complete the value of the constant but I am not sure I understand why so this prompted me to ask the above question. I will keep reading the Wiki article again and again till I understand it but thought I would ask the question.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-06-05T17:38:34.997" LastActivityDate="2020-06-09T22:39:32.227" Title="Did James Clerk Maxwell derive the permittivity of free space from his &quot;Maxwell's&quot; equations or was the parameter already known by him?" Tags="|physics|electromagnetism|electricity|maxwell|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11450" PostTypeId="1" CreationDate="2020-02-09T19:32:24.090" Score="3" ViewCount="13381" Body="&lt;p&gt;When and how was the water cycle discovered? Did people, like ancient Greek philosophers, have any correct ideas on the water cycle?  Was its discovery more of a recent thing?&lt;/p&gt;&#xA;" OwnerUserId="11083" LastEditorUserId="4251" LastEditDate="2021-09-17T06:30:19.950" LastActivityDate="2021-09-17T06:30:19.950" Title="When and how was the water cycle discovered?" Tags="|discoveries|earth-sciences|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11451" PostTypeId="1" AcceptedAnswerId="11452" CreationDate="2020-02-09T21:20:15.017" Score="2" ViewCount="154" Body="&lt;p&gt;Please, advise a good review of Isaac Newton's work &quot;The Mathematical Principles of Natural Philosophy&quot; with a detailed analysis of his mathematical ideas. Smth about 20-25 pages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you in advance&lt;/p&gt;&#xA;" OwnerUserId="11084" LastActivityDate="2020-02-10T12:58:24.580" Title="The Mathematical Principles of Natural Philosophy" Tags="|mathematics|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11452" PostTypeId="2" ParentId="11451" CreationDate="2020-02-10T05:36:32.007" Score="5" Body="&lt;p&gt;Your requirements are contradictive: there CANNOT BE a &quot;detailed analysis&quot;&#xA;of mathematical ideas of all of Principia in 25-30 pages!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the best reviews covering all three books of Principia that I know is the Guide to Newton's Principia by I. Bernard Cohen. It is attached to the new translation of Principia itself, U. of California press, 1999. The Guide is 370 pages long.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also a book of S. Chandrasekhar, Newton's Principia for the common reader, &#xA;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/019852675X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.com/Newtons-Principia-Common-Reader-Chandrasekhar/dp/019852675X&lt;/a&gt;&#xA;but it covers only Book I. It is about 600 pages, and does not even completely cover Book I. The author, a Nobel prize winner astronomer and mathematician goes more in depth than Cohen. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;A very short and popular account of SOME principal results of Book I is contained in&#xA;the book of V. I. Arnold, Huygens and Barrow, Newton and Hooke. The length of the part describing Principia is about as you request but of course he analyses only 2 or 3 main results in it. It is written by a famous mathematician but oriented at beginner mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-02-10T12:58:24.580" LastActivityDate="2020-02-10T12:58:24.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11453" PostTypeId="2" ParentId="11397" CreationDate="2020-02-10T13:00:16.490" Score="3" Body="&lt;p&gt;Three people whose work in mathematics was not properly recognized in their lifetime but its importance was  appreciated later are Galois, Grassmann, and Heegner. &lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="10106" LastEditDate="2020-02-11T00:41:22.490" LastActivityDate="2020-02-11T00:41:22.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11454" PostTypeId="2" ParentId="9662" CreationDate="2020-02-10T17:55:53.530" Score="1" Body="&lt;p&gt;P.199-206 of Schlesinger's treatise about Gauss's analytic work give the answer. This section, entitled &quot;Anniversary letter Plan of a treatise on convergence of series&quot;, deals with mathematical themes that occupied Gauss in the last years of his life (the years 1849 - 1855), and includes a discussion of his last proof of the fundamental theorem of the algebra (published in 1849 and topological in it's nature) as well as a summary of the main ideas in an incomplete treatise on convergence of infinite series. The part of Schlesinger's discussion relevant to this question, is on p. 203-206. Schlesinger mentions that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Articles 2-5 describe in terms of &quot;layers&quot; delimited by &quot;lines&quot; in the plane of complex variables the basics of topology, whereby it is said with reference to Article 5 of the jubilee that this geometrical interpretation aim ... To facilitate the field of abstract complex sizes and to convey a language for them. A layer can be delimited by one or even several closed lines, and always determines a certain &quot;sequence&quot; in these lines, &quot;similar to when the entire shore lines of a lake including one or more islands are traversed in such a way, that he ... the lake ... either always on the right or always on the left. Art. 5 then shows that &quot;a layer S, which is delimited by two or more closed lines, can be broken down into as many partial layers, each delimited by only one line&quot;; if it is a question of applications, in which two lines, which are identical in themselves but occur in sequential orders, can be viewed as destructive to one another, the totality of the border lines of 2. Put right now, the totality of the simple border lines all of the, , Decomposing layers. It is not difficult to guess which applications Gauss had in mind here...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This translation was made by Google translate with a few corrections i made. I guess that when Gauss speaks about &quot;a lake including one or more islands&quot; he describes a topolgical picture similar to that of a doughnut, and in the next lines he describes the problem of connectedness of a surface. I think this is the reference i searched for.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Although i found the desired reference ,this passage arises new &quot;nuts to crack&quot;; for example, what does Gauss mean when he speaks about &quot;two lines... occur in sequential orders... can be viewed as destructive to one another...&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also, it seems that since Gauss speaks of connectivity of surfaces with reference to the complex plane, this treatise can be interpreted as a first step in the creation of geometric foundations for complex function theory (i.e Riemann surface). However, Schlesinger says in p.205 that several things Gauss said confirm that he has not fully penetrated the nature of complex integrals (and multivalued functions).  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-02-10T18:11:37.693" LastActivityDate="2020-02-10T18:11:37.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11455" PostTypeId="1" CreationDate="2020-02-10T18:49:54.090" Score="3" ViewCount="218" Body="&lt;p&gt;In Hungarian you spell the surname of &lt;em&gt;Dénes Kőnig&lt;/em&gt; with a double acute accent, whereas &lt;a href=&quot;https://de.wikipedia.org/wiki/Lemma_von_K%C3%B6nig&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Lemma von König&lt;/em&gt;&lt;/a&gt; is written in German with an umlaut.  Why the difference? Is the reason behind it purely typographical or is there something deeper?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In particular, if you write a German text, which variants do you use nowadays: &lt;em&gt;Dénes Kőnig&lt;/em&gt; and &lt;em&gt;Lemma von König&lt;/em&gt; or something else?&lt;/p&gt;&#xA;" OwnerDisplayName="user11087" LastEditorUserId="4251" LastEditorDisplayName="user11087" LastEditDate="2021-12-15T13:38:38.930" LastActivityDate="2021-12-15T13:38:38.930" Title="Dénes Kőnig vs. König's Lemma" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11458" PostTypeId="2" ParentId="7318" CreationDate="2020-02-11T13:01:46.397" Score="2" Body="&lt;p&gt;There has been a similar question here: &lt;a href=&quot;https://www.quora.com/Which-are-the-best-PhD-theses-ever-in-pure-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/Which-are-the-best-PhD-theses-ever-in-pure-mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would suggest Kurt Godel. His doctorate thesis proved the completeness theorem, and a year later he published his incompleteness theorems.&lt;/p&gt;&#xA;" OwnerUserId="11081" LastActivityDate="2020-02-11T13:01:46.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11460" PostTypeId="2" ParentId="11455" CreationDate="2020-02-11T15:13:29.897" Score="5" Body="&lt;p&gt;&quot;ő&quot; exists in Hungarian, it is pronounced as a longer &quot;ö&quot; (and &quot;ö&quot; is pronounced like the first vowel in the English word &quot;dirty&quot;). But in German, only &quot;ö&quot; exists (and it depends on the actual dialect/context, how long is it pronounced).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;König&quot; is originally a German word, and it means &quot;King&quot;. His name has German origin, already the &quot;Kőnig&quot; version was a Hungarianized form (it sounds simply more natural for native speakers).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While in the region and the era was/is it common to use some simplification/translation of the names, mostly to adapt better foreign-speaker environments, he and his father lived their entire life in Hungary and they did not need to do that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The probable reason today to use the &quot;ö&quot;-Form is that &quot;ö&quot; exists in the most common extension of the ascii encoding (so-named &quot;latin1&quot;), but &quot;ő&quot; does not. Another probable reason was that the typewriters and pre-unicode computers of the era could write &quot;ö&quot; far more easily.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In a German text I would use the &quot;ö&quot;-version on decency reason.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Both the English and the German Wikipedia use the Kőnig name.&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="217" LastEditDate="2020-02-11T18:36:31.423" LastActivityDate="2020-02-11T18:36:31.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11462" PostTypeId="1" AcceptedAnswerId="11475" CreationDate="2020-02-11T19:56:16.213" Score="0" ViewCount="149" Body="&lt;p&gt;This &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Extras/Fourier_teachers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; contains excerpts of Fourier's descriptions of some of his math professors  but doesn't really provide any source.&#xA;What's the name of the original document containing these descriptions?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-02-11T21:30:21.493" LastActivityDate="2020-02-18T00:28:29.687" Title="Where can one find the source of Fourier's recollections of his professors?" Tags="|biographical-details|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11463" PostTypeId="2" ParentId="11462" CreationDate="2020-02-12T09:41:58.177" Score="4" Body="&lt;p&gt;On pp. 3-4 of &lt;strong&gt;[1]&lt;/strong&gt; Grattan-Guinness writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The professors were drawn from among the foremost men in the land: for example, [Lagrange, Laplace, Monge are mentioned] &lt;span class=&quot;math-container&quot;&gt;$[\ldots]$&lt;/span&gt; The school opened on 20 January 1795, but lasted only a few months. The failure was partly due to the severity of the winter weather, which caused the already impoverished students to live under conditions of great physical hardship; but there were other difficulties, caused by the poor organization of the teaching. Fourier himself described some of them in a letter &lt;span class=&quot;math-container&quot;&gt;$[\dots]$&lt;/span&gt; Both Lagrange and Laplace were quiet speakers, with Lagrange outstanding for his Italian accent and Laplace for the rapidity of his teaching &lt;span class=&quot;math-container&quot;&gt;$[\dots]$&lt;/span&gt; Monge caught Fourier's imagination, speaking with a loud voice and showing great skill in both theoretical and practical subjects.&lt;span class=&quot;math-container&quot;&gt;$^6$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Footnote &lt;span class=&quot;math-container&quot;&gt;$^6$&lt;/span&gt; (bottom of p. 4) begins with: &quot;Undated letter to Bonard, in A. Challe &lt;em&gt;1858a&lt;/em&gt;, 115−120; also in E. Duché &lt;em&gt;1871a&lt;/em&gt;, 257−261. [&lt;span class=&quot;math-container&quot;&gt;$\ldots]$&lt;/span&gt;&quot; As with pretty much anything before 1900, these two references are freely available on the internet (see &lt;strong&gt;[2]&lt;/strong&gt; and &lt;strong&gt;[3]&lt;/strong&gt;). These two references are in French, so for me it would be difficult to determine whether the precise MacTutor History of Mathematics archive quotes are there, and if so, where, but perhaps someone else who can easily read French can comment further.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Ivor Grattan-Guinness, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0262571781&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Joseph Fourier 1768−1830&lt;/strong&gt;&lt;/a&gt;, in collaboration with Jerome Raymond Ravetz, MIT Press, 1972, xii + 516 pages.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Ambroise Challe, &lt;em&gt;Lettres de Joseph Fourier&lt;/em&gt; [Letters from Joseph Fourier], &lt;strong&gt;Bulletin de la Société des Sciences Historiques et Naturelles de l’Yonne&lt;/strong&gt; (1) 12 (1858), 105-134.&#xA;(&lt;a href=&quot;https://books.google.com/books?id=eVg4AAAAMAAJ&amp;amp;pg=PA105&quot; rel=&quot;nofollow noreferrer&quot;&gt;google books version&lt;/a&gt; and &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k298644h/f123.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gallica version&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Émile Duché, &lt;em&gt;Joseph Fourier, sa vie et ses travaux&lt;/em&gt; [Joseph Fourier, his life and his works], &lt;strong&gt;Bulletin de la Société des Sciences Historiques et Naturelles de l’Yonne&lt;/strong&gt; (2) 5 (1871), 217-262. [see pp. 257-261]&#xA;(&lt;a href=&quot;https://books.google.com/books?id=A8kYAAAAYAAJ&amp;amp;pg=257&quot; rel=&quot;nofollow noreferrer&quot;&gt;google books version&lt;/a&gt; and &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k298667b/f272.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gallica version&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2020-02-12T09:41:58.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11464" PostTypeId="1" AcceptedAnswerId="11466" CreationDate="2020-02-12T11:26:52.957" Score="6" ViewCount="617" Body="&lt;p&gt;I have &lt;a href=&quot;https://www.famousscientists.org/william-thomson/&quot; rel=&quot;noreferrer&quot;&gt;read&lt;/a&gt; that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1847, he became aware of physicist James Prescott Joule’s argument for the mutual convertibility of heat and mechanical work and for their mechanical equivalence.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;We study these things (I mean the above said thing which is quoted) in high schools and we tend to study it a little differently. I have found &lt;a href=&quot;https://en.m.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin&quot; rel=&quot;noreferrer&quot;&gt;something more&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Thomson became intrigued with Fourier's Théorie analytique de la chaleur and committed himself to study the &quot;Continental&quot; mathematics resisted by a British establishment still working in the shadow of Sir Isaac Newton. Unsurprisingly, Fourier's work had been attacked by domestic mathematicians, Philip Kelland authoring a critical book. The book motivated Thomson to write his first published scientific paper[12] under the pseudonym P.Q.R., defending Fourier, and submitted to the Cambridge Mathematical Journal by his father. A second P.Q.R. paper followed almost immediately.[13]  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I mean to say that (if we take my example and I'm from India) we are doing everything systematically, our academic institute makes a syllabus and defines a route to be followed for studying things but these physicists (I have given the example of only one but you all know that they all used to study publications of others for studying something) were reading &lt;em&gt;actual texts&lt;/em&gt; of other physicists not the textbooks that we study.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I want to know how do those 19 th century physicists study? Do they learn from reading the original papers? My question says undergraduate/post graduate because the things that I have mentioned in quotes and other things developed by those 19 th century physicists are generally studied in undergraduat/post graduate courses.  &lt;/p&gt;&#xA;" OwnerUserId="9837" LastEditorUserId="55" LastEditDate="2020-02-12T12:26:03.410" LastActivityDate="2020-02-14T18:42:56.183" Title="How did 19th century physicists do their undergraduate/graduate studies?" Tags="|physics|theoretical-physics|academia|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11465" PostTypeId="1" CreationDate="2020-02-12T20:44:43.867" Score="2" ViewCount="368" Body="&lt;p&gt;I'm trying to find the first paper in which the concept of four-vectors was introduced. I read &lt;em&gt;&quot;Principle of Relativity&quot;&lt;/em&gt; by H. Minkowski but he only presents the notion of metric and invariant space-time interval, but not specifically four-vectors. I'm searching this because I want to find the physical motivation for four-vectors other than &quot;the components must transform by Lorentz transformations&quot; because I think this is a random statement. If this articles exists, please tell me the title, thanks&lt;/p&gt;&#xA;" OwnerUserId="11101" LastEditorUserId="26" LastEditDate="2021-02-20T15:19:44.760" LastActivityDate="2021-02-20T15:19:44.760" Title="First paper introducing the concept of four-vectors" Tags="|relativity-theory|einstein|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="11466" PostTypeId="2" ParentId="11464" CreationDate="2020-02-13T22:09:29.590" Score="11" Body="&lt;p&gt;The status of scientific education in the 19th century is a very complicated mess, especially for the fact that every country worked different than the others. I know of no book or article that tackles this problem in general, so I'll attempt an answer based on various readings, majorly biographies.&lt;/p&gt;&#xA;&lt;h1&gt;1) Basic Education in the 19th century&lt;/h1&gt;&#xA;&lt;p&gt;To understand what happened in the universities one must understand the state of primary and secondary education in the 19th century. Until the 1870s there were few laws regarding education. Most of the population were illiterate. Wealthier people would send their kids to primary education in order to learn how to read, and secondary education would consist chiefly of Latin, Greek, Classic Authors and Religion. While I'm not certain, it is possible that schools would teach the very basics of euclidean geometry. What is certain is that more advanced mathematical topics like analytical geometry were left to higher education. The sciences were not seen as relevant to an education which existed to create an intellectual elite. So it is not that what we see today in schools was not taught, but no science at all. This situation was somewhat maintained up to the turn of the 20th century to various degrees. Heisenberg's autobiography (W. Heisenberg, Physics and Beyond) mentions that he studied the concept of atoms at school where the book illustrated a chemical bond as the atoms having hooks. In the UK, Dirac's biography (G. Farmelo, The Strangest Man: The Hidden Life of Paul Dirac) says he did study science and modern languages at school, but that was unusual because he was lower class and attended technical college, while more wealthy people still went to grammar schools which focused in Classics. Widespread science teaching in secondary schools is a cold war phenomena where countries put a lot of effort in creating a lot of engineers and other science professions, see arms race and space race.&lt;/p&gt;&#xA;&lt;h1&gt;2) The State of Physics in the 19th century&lt;/h1&gt;&#xA;&lt;p&gt;What we now study in undegrad physics were not seen as unified field at that time. Mechanics and Optics (as well as astronomy) were seen as part of mathematics, and the experimental sciences were grouped as natural philosophy. The term &amp;quot;physicist&amp;quot;, meaning someone who used deductive reasoning in experimental sciences was only coined in 1840 by William Whewell, and took some time to get going (which you can see using google ngram). So if you wanted to study mechanics you would get a maths degree, and if you cared for experiments you would study natural sciences (the natural sciences degree at Cambridge started in 1851, although I think it may have precedents in Germany). Getting both together is a late 19th/early 20th century phenomena. Plank's autobiography (M. Planck, Scientific Autobiography and Other Papers) mentions that the concept of theoretical physics was starting to be discussed by the time he took his PhD in the 1880s, majorly because of problems involving thermodynamics, like the blackbody spectrum. By the way, this distinction of physics in two parts is the reason why theoretical physics is still in the mathematics department in the UK and former colonies (and I suspect in the Soviet Union)&lt;/p&gt;&#xA;&lt;h1&gt;3) The State of Physics Education in the 19th Century&lt;/h1&gt;&#xA;&lt;p&gt;Your question is mostly focused on what did people in fact study. Here it gets messier because each country had it's own system. If we take Cambridge as representative of the UK, there the undergrad education was based in the Tripos system. Basically you hanged around the university for 3-4 years taking random classes and studying with a private tutor. At the end of your time you took the Tripos exam, a single test lasting days, and if you passed you got a degree. There was no PhD studies per se, after the Tripos people that showed promise might be offered a fellowship, a stipend by one of the colleges, to teach and tutor. If they did research it was usually to be connected to Royal Institutions of some science of another.&lt;/p&gt;&#xA;&lt;p&gt;Germany is the birthplace of the undergrad/PhD system in the early 1800s. During undergrad you were expected to attend whichever classes you wanted. Matter of fact a lot of professor did not receive pay from the university, but rather they held a license to teach and were paid according to the number of students attending the classes they offered. The PhD was when your worked under a supervisor and had to produce a dissertation to be examined.&lt;/p&gt;&#xA;&lt;p&gt;As far as I understand France was the place where you had rigid curriculum. The system of Grande Ecoles and Ecoles Normales had predetermined classes and was based very strongly on the lecture. France did not have PhDs until the late 19th century, nor the idea of research university (although of course a lot of french scientists did research, mainly in the british style of connection to other institutions).&lt;/p&gt;&#xA;&lt;p&gt;Bottomline, during most of the 19th century the universities did not have rigid curriculum, so to answer what people did study you have to dig the examinations they had to pass. It is noteworthy that in the 1850s the Stokes' Theorem appeared in the examination for the Smith's Prize, an optional part of the math Tripos at Cambridge that evaluated original work instead of previous study, as a question to generalize Green's Theorem to 3 dimension, from what we surmise that it was not explicitly taught. That may give you some idea of at least the math level. In any case most classes were made by the professors from his personal notes, so say taking a ODEs class from professor A might be very different from taking the class with the same name from professor B.&lt;/p&gt;&#xA;&lt;p&gt;[as an addendum, the lack of widespread PhD system is the reason why the Math Genealogy Project has to make some educated guesses when establishing supervisor relations before the 20th century for a lot of cases]&lt;/p&gt;&#xA;&lt;h1&gt;4) The State of Thermodynamics Education in the 19th Century&lt;/h1&gt;&#xA;&lt;p&gt;Since I think the answer was less than satisfactory, I'll try to point out that the idea of thermodynamics as based on the first and second laws, and the understanding that thermodynamical energy was just another form of energy took a long time to develop. Planck's biography (the same as before) mentions that he struggled in his early years to cobble together what was known into a unified frame, and that only later he discovered that Gibbs had done the same in the US (both 1870s), while Silvanus Thompson's Elementary Lessons in Electricity and Magnetism (1890) says it was still discussed if electric energy was an energy like the others, so it is not surprising that Kelvin was learning from original papers about the first law of thermodynamics (Joule did the experiment, Helmholtz the theory).&lt;/p&gt;&#xA;&lt;p&gt;As for Fourier's theory of heat conduction, is good to note that the Heat Equation is a poor empirical description of heat transport in most situations (Joel Lebowitz has a paper on the Heat Equation where he discusses that only in a copule of material, with low temperature differences one can actually model the transport as in Fourier's work). The big news here was Fourier suggesting that all functions could be represented by a trigonometric series in a given interval. By that time issues of convergence were poorly understood, I guess up until Weierstrass in the late 19th century, so the polemic seems to be normal.&lt;/p&gt;&#xA;&lt;p&gt;[I've tried to remember things as best as I could from all the biographies and books on the history of mathematics and physics, but I'm sure there are a couple of errors creeping in, so take everything with lots of grains of salt. Hopefully the comments will be filled with corrections and I'll update accordingly. I do apologize for any mistake that might mislead someone.]&lt;/p&gt;&#xA;" OwnerUserId="5904" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-02-14T18:42:56.183" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11468" PostTypeId="2" ParentId="7276" CreationDate="2020-02-14T23:07:26.907" Score="3" Body="&lt;p&gt;This is a very partial answer, but i had to write down the information i collected up to now.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the best sources on Gauss's article on ellipsoids attraction is Harald Geppert's treatise on Gauss's activities in fields related to classical mechanics. According to this treatise, the structure of Gauss's paper can be described in this way:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Historical background - this takes 2 pages of Gauss's article and lists the contributions made to this problem from the time of Newton up to more recent work made primarily by the french school of analysis ( Lagrange, Legendre, Laplace).&lt;/li&gt;&#xA;&lt;li&gt;The first part of the article - several integral theorems - this apparently includes the known Gauss's divergence theorem. It includes six integral propositions, that fall into two parallel groups - the first three are based on the idea of spatial division of space, while the following three are based on a division of space into elementary cones.&lt;/li&gt;&#xA;&lt;li&gt;The second part of the article contains the special application of the preceeding theorems to the attraction of ellipsoid.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;For the calculation of the attraction of ellipsoids, Gauss first employs the parametric representation of locations on the ellipsoids, and then applies Maclauren's theorem on attraction of confocal ellipsoids in order to carry out several integrations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then Gauss breaks the problem of ellipsoid attraction into two sub-problems, the first of which is done by solving cubic equation with one real root, while the second requires integration of a certain expression that leads to elliptic integrals of the second kind, which in the case of spheroid (&quot;ellipsoid of revolution&quot;) reduce to elementary functions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Geppert remarks that Gauss's solution is of great elegence and avoids convergence issues that arise in previous solutions to this problem.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-02-15T13:47:34.450" LastActivityDate="2020-02-15T13:47:34.450" CommentCount="0" CommunityOwnedDate="2020-02-14T23:07:26.907" ContentLicense="CC BY-SA 4.0" />
  <row Id="11469" PostTypeId="1" AcceptedAnswerId="11472" CreationDate="2020-02-15T21:04:45.960" Score="7" ViewCount="364" Body="&lt;p&gt;Recently i was striked by a quotation of Gauss from a letter to his student Gerling from the date June 23, 1846. This letter states in very concise words that the distinction between right-handed and left-handed 3D cartesian coordinates system is not of aprioric nature; it cannot be determined by the first principles of spatial logic. According to what i read up to now, Gauss remarked this in order to state his disagreement with Kant's philosophy of space and of space perception (Gauss expressed similar disagreement with Kant in his private letters on non-euclidean geometry). Gauss's relevant remarks are those:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;One cannot reduce to concepts the distinction between two systems of three straight lines each (directed lines, of which the one system points forward, upward to the right, the other forward, upward to the left) but one can only demonstrate by holding to actually concrete spatial things. Two minds cannot reach agreement about it unless their views connect up with one and the same system present in the real world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and there is also this quote from his letter to Schumacher (Februar 8 ,1846):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The distinction between right and left cannot be defined, but only shown, so that it is thereby a case similar to sweet and bitter...two such minds, however, cannot make themselves directly understood concerning right and left unless one and the same individual thing forms a bridge between them... i find in it the striking refutation of Kant's imagination that space is merely the form of our external perception.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The reason why i was striked by reading Gauss's quotes is that it anticipates- not in the mathematical details, but rather in the philosophical level - the notion of orientability of surfaces, and perhaps also - though not in the mathematical details - the possibility of non-orientable surfaces and spaces like Mobius's strip and it's higher dimensional analogs. As an illustration of the impossibility of drawing a distinction between right and left at certain spaces, i added here this gif:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9SueG.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9SueG.gif&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A crab walking on a Mobius strip cannot decide between right and left.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What i'd like to know is:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can anyone tell something about the state of &quot;philosophy of right and left&quot; at  Gauss's time? because perhaps i'm misinterpreting Gauss's letter and he didn't really stood against Kant's view.&lt;/li&gt;&#xA;&lt;li&gt;Can anyone tell a little bit about the background to this correspondence of Gauss?  &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="55" LastEditDate="2020-02-17T06:27:12.947" LastActivityDate="2020-02-17T18:59:58.750" Title="Were Gauss's 1846 remarks on the distinction between right and left related to orientability of surfaces?" Tags="|biographical-details|philosophy-of-science|gauss|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11470" PostTypeId="1" AcceptedAnswerId="11471" CreationDate="2020-02-16T16:06:20.250" Score="2" ViewCount="191" Body="&lt;p&gt;Did these scientists know about each other's work and consult from each other or did they discover the electroweak theory each independent of the other?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Below is a quote from nobelprize.org.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to modern physics, four fundamental forces exist in nature. Electromagnetic interaction is one of these. The weak interaction - responsible, for example, for the beta decay of nuclei - is another. Thanks to contributions made by Abdus Salam, Sheldon Glashow,and Steven Weinberg in 1968, these two interactions were unified to one single, called electroweak. The theory predicted, for example, that weak interaction manifests itself in &quot;neutral weak currents&quot; when certain elementary particles interact. This was later confirmed.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2020-03-07T16:54:53.843" Title="Did the recipients of the Nobel Prize in physics for the electroweak theory work independently of each other?" Tags="|theoretical-physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11471" PostTypeId="2" ParentId="11470" CreationDate="2020-02-17T01:20:52.140" Score="5" Body="&lt;p&gt;More or less. Glashow produced an early blueprint in 1961 before the Higgs mechanism was worked out, unaware of it Salam and Ward reproduced it in 1964. Weinberg and Salam presented the final theory independently in 1967. However, they knew each other well, and collaborated on some papers since 1962, so were likely aware of each other's thinking. After a key experimental confirmation in 1973 the Nobel prize was awarded in 1979, Ward was left out.&lt;/p&gt;&#xA;&lt;p&gt;Kibble has helpful notes on &lt;a href=&quot;https://indico.cern.ch/event/328215/contributions/765036/attachments/638254/878389/HistElWkSymBrk_Kibble_Discrete2014.pdf&quot; rel=&quot;noreferrer&quot;&gt;History of Electroweak Symmetry Breaking&lt;/a&gt;, based, in part, on personal recollections. Here are the relevant parts:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Glashow (1961) proposed a model with symmetry group &lt;span class=&quot;math-container&quot;&gt;$SU(2)\times U(1)$&lt;/span&gt; and a fourth gauge boson &lt;span class=&quot;math-container&quot;&gt;$Z^0$&lt;/span&gt;, showing that the parity problem could be solved by a mixing between the two neutral gauge bosons. Salam and Ward (1964), unaware of Glashow's work, proposed a similar model, also based on &lt;span class=&quot;math-container&quot;&gt;$SU(2)\times U(1)$&lt;/span&gt;. Salam was convinced that a unified theory must be a gauge theory... But in all these models symmetry breaking, giving the &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; bosons masses, had to be inserted by hand — spin-&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; bosons with explicit mass were known to be non-renormalizable.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;By 1964 both the&lt;/em&gt; [Higgs] &lt;em&gt;mechanism and Glashow’s (and Salam and Ward’s) &lt;span class=&quot;math-container&quot;&gt;$SU(2)\times U(1)$&lt;/span&gt; model were in place, but it still took three more years to put the two together. Unified model of weak and electromagnetic interactions of leptons proposed by Weinberg (1967) — essentially the same model was presented independently by Salam in lectures at IC in autumn of 1967 and published in a Nobel symposium in 1968 — he called it the electroweak theory.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Salam and Weinberg speculated that their theory was&#xA;renormalizable. This was proved by Gerard ’t Hooft in 1971 — a tour de force&#xA;using methods of his supervisor, Tini Veltman, especially &lt;a href=&quot;https://en.wikipedia.org/wiki/Schoonschip&quot; rel=&quot;noreferrer&quot;&gt;Schoonschip&lt;/a&gt;. In 1973 the key prediction of the theory, the existence of neutral current interactions — those mediated by &lt;span class=&quot;math-container&quot;&gt;$Z^0$&lt;/span&gt; — was confirmed at CERN. This led to the Nobel Prize for Glashow, Salam &amp;amp; Weinberg  in 1979 — but Ward was left out (because of the ‘rule of three’?). ’t Hooft and Veltman gained their Nobel Prizes in 1999. In 1983 the &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; particles were discovered at CERN.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-02-17T04:30:24.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11472" PostTypeId="2" ParentId="11469" CreationDate="2020-02-17T06:23:30.603" Score="5" Body="&lt;p&gt;The short answer to the title question is likely yes. There are two separate issues discussed in the OP, one is more philosophical on Gauss's disagreements with Kant's apriorism about space, and the other, concerning his insights into the orientation of surfaces and non-orientable surfaces. The first question is addressed to some extent in &lt;a href=&quot;https://philosophy.stackexchange.com/a/49465/9148&quot;&gt;Was Kant a factor in forming Gauss's abstract view of mathematical objects?&lt;/a&gt;, so I will focus on the second.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss corresponded with Gerling for a long time, at least since 1819, when he mentioned the angle sum defect (&quot;Gauss-Bonnet theorem&quot;) for hyperbolic triangles in a letter to him. At the time, the main subject was non-Euclidean geometry, and this is discussed at length in &lt;a href=&quot;https://reader.elsevier.com/reader/sd/pii/S0315086012000122&quot; rel=&quot;nofollow noreferrer&quot;&gt;What did Gauss read in the Appendix? by Abardia, Reventós and Rodríguez&lt;/a&gt; (the reference is to Bolyai's Appendix from 1832). Issues related to what Gauss called &lt;em&gt;theorema egregium&lt;/em&gt; (invariance of curvature under isometries) are discussed in 1825 letters to Schumacher. Gauss's celebrated work on geometry of curved surfaces &lt;a href=&quot;https://archive.org/details/disquisitionesg00gausgoog/page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Disquisitiones Generales circa Superficies Curvas&lt;/em&gt;&lt;/a&gt; came out in 1827. His relevant writings on the subject, both published and unpublished, are discussed at length in &lt;a href=&quot;http://www.numdam.org/issue/AST_1979__62__R1_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;150 years after Gauss’ «&lt;em&gt;Disquisitiones generales circa superficies curvas&lt;/em&gt;» by Dombrowski&lt;/a&gt;. Dombrowski quotes an unpublished 1825 draft of &lt;em&gt;Disquisitiones Generales&lt;/em&gt;, which shows that Gauss possessed the concept of surface orientation in 1825 (and possibly as early as 1794). In his modernized &quot;free translation&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The sum of the angles of a (small) geodesic triangle Δ in a curved surface  in  &lt;span class=&quot;math-container&quot;&gt;$\mathbb{E}^3$&lt;/span&gt;  is equal to the sum of π and the oriented surface area of the spherical image of Δ, where the oriented area is taken to be positive or negative according to whether the bounda­ry of the spherical image of Δ winds around the image in the same direction or in the opposite direction as the boundary of Δ winds around Δ.&lt;/em&gt;&quot; (57)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Gauss then generalizes it to geodesic polygons. Dombrowski discusses the context and speculates that Gauss was led to the general statement for curved surfaces by the hyperbolic case combined with his practical experience with geodesic measurements in 1812-1822:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The result (57) was already widely known in the special case of a deve­lopable or a spherical surface. Gauss possessed  the result analogous to &#xA;  (57) for hyperbolic geometry  (and therefore  essentially  for surfaces of constant &#xA;  negative curvature, the author) already in 1794 (see G.W. 8, p. 266), and announced  it to Gerling in a letter in 1819 (see G.W. 8,  p. 182). It is there­fore easy to imagine that Gauss was led to a geometrically intuitive &quot;insight&quot; concerning the validity  of (57) for the general case of a surface of non-constant Gaussian curvature  from the knowledge  of this fundamental case and on the basis of his rich differential-geometric  experience  in geodesy (with geodesics and trigonometry spheroids, but also with questions of mapping and bending) obtained  in the years &#xA;  1812 to 1822.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;Although Gauss had studied intensively the concept of &quot;oriented surface area&quot; as used by him in (57) (see G.W. 8, p. 398, line 2 from below), he did &#xA;  not think that his corresponding studies had &quot;matured&quot; sufficiently... The proof (with its so impressive geometric arguments, which we have just retraced  from the fragment (55)) was never published by Gauss! The reason  for this lies on the one hand certainly  in his  self-criticism of his own above mentioned sketch of a proof for (57), and particularly of his concept  of the &quot;oriented surface area of the spherical image on a curved surface&quot; (involved  in that proof) which he has not defined with that analytic rigour usually applied by him.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This result leads to &lt;em&gt;theorema egregium&lt;/em&gt; discussed in 1825 letters to Schumacher. The Gauss-Bonnet theorem itself is brought up in the 10 October 1846 letter to Gerling (Dombrowski's translation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The theorem which Mr. Schweikart mentioned to vou,that in any eeometrv the sum of all outer angles of a polygon differs from 360 by a quantity,..., which is proportional to the surface area, is the first theorem lying almost on the threshold  of that theory, a theorem whose necessity I already recognized  in 1794&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So the appearance of orientation related issues in the letters to Gerling and Schumacher in 1846 (connected to the older themes of his agreements and disagreements with Kant) is not surprising. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the non-orientable surfaces, the issue is much more murky. The &quot;Möbius band&quot; is described in private papers of Listing and Möbius in 1858, almost simultaneously and in similar terms, Listing's description is earlier by several months. Gauss died in 1855, but both were Gauss's students, and many historians attribute many of Listing's topological ideas to Gauss. Listing himself stated something to this effect, although without specifics. Biggs in his chapter in &lt;a href=&quot;https://archive.org/details/mobiushisbandmat00john/page/n9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Möbius and his band&lt;/a&gt; explicitly speculates that Gauss might have been the common source here, but admits that the issue is undecidable:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Both of them describe the construction in&#xA;  very similar terms. Is this another one of those istances which sometimes&#xA;  happen in scientific discovery, where an idea whose time is ripe&#xA;  appears iI1dependentiy in different places but at the same time? That&#xA;  is certainly a possibility. Or was there a common reason for the fact that both Möbius and Listing described the Möbius band around&#xA;  the same time? If the latter, then the likelihood is that the common&#xA;  reason was connected with the work or Gauss who, as we know, had&#xA;  been very interested in this kind or topic. Gauss had died in 1855, so&#xA;  the idea cannot have been communicated by him directly, but the&#xA;  possibility of some link with his work remains. I doubt that the question will ever be completely resolved.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-02-17T18:59:58.750" LastActivityDate="2020-02-17T18:59:58.750" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11473" PostTypeId="1" AcceptedAnswerId="11474" CreationDate="2020-02-17T07:08:09.050" Score="6" ViewCount="2686" Body="&lt;p&gt;There is a question &lt;a href=&quot;https://hsm.stackexchange.com/q/7819/55&quot;&gt;Who solved the quantum harmonic oscillator?&lt;/a&gt;, but not one for the classical oscillator. Wikipedia's article &lt;a href=&quot;https://en.wikipedia.org/wiki/Harmonic_oscillator&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harmonic Oscillator&lt;/a&gt; does not have historical information either. So who first solved the &lt;strong&gt;classical&lt;/strong&gt; harmonic oscillator equation? &lt;/p&gt;&#xA;" OwnerUserId="11109" LastEditorUserId="55" LastEditDate="2020-02-18T00:26:47.267" LastActivityDate="2020-02-18T06:54:23.203" Title="Who first solved the classical harmonic oscillator?" Tags="|physics|discoveries|classical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11474" PostTypeId="2" ParentId="11473" CreationDate="2020-02-17T12:47:54.437" Score="15" Body="&lt;p&gt;It was &quot;solved&quot; by Huygens in &lt;a href=&quot;https://en.wikipedia.org/wiki/Horologium_Oscillatorium&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Horologium Oscillatorum&lt;/em&gt; (1673)&lt;/a&gt;. The scare quotes are there because he never wrote down the equation, and even Newton's laws were not yet explicitly formulated. Huygens considered the motion of pendula, and for simple cases knew the &quot;&lt;em&gt;law of the conservation of living force&lt;/em&gt;&quot; (mechanical energy), as Bernoullis later called it, see &lt;a href=&quot;https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34251/31151029395534.pdf&quot; rel=&quot;noreferrer&quot;&gt;Mach, History and Root of the Principle of the Conservation of Energy, p. 30&lt;/a&gt;. In modern terms, this would be the first integral, or quadrature, of the corresponding equation of motion. With that, he was able to derive the period formula for pendulum motion with small amplitudes, &lt;span class=&quot;math-container&quot;&gt;$T=2\pi\sqrt{\frac{l}{g}}$&lt;/span&gt; in modern notation, which he also did not use. Here is from &lt;a href=&quot;https://www.jstor.org/stable/41134260&quot; rel=&quot;noreferrer&quot;&gt;Acoustic origins of harmonic analysis by Darrigol, p.351&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The first intimation that harmonic (sine-like) motion plays a basic role in acoustics is found in Christiaan Huygens's theory of musical strings. In his celebrated&#xA;   Horologium Oscillatorium of 1673, Huygens showed that the pendulous motion&#xA;   of a body sliding down on a cycloid was harmonic and isochronous. Around that&#xA;   time, he also understood that the force responsible for this motion was proportional to the distance traveled by the body from the point of equilibrium. Probably noticing that a similar circumstance held in the case of a tense, weightless&#xA;   elastic string loaded with one mass in the middle, he derived the oscillation&#xA;   frequency as a function of tension, length, and mass. The reasoning implied&#xA;   harmonic oscillations for the loaded string. He also sketched a generalization&#xA;   to a string loaded with several masses, in which he assumed all the masses to&#xA;   perform harmonic oscillations of the same frequency and phase.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Taylor, who was studying vibrating strings in 1713, had the benefit of Newton's mechanics. Still, he did not write down the equation, but used pendulum analogies, and realized that the sine shape was a solution for the string. Johann Bernoulli followed Taylor's lead, and represented strings as connected masses. In a 1727 letter to his son Daniel he explicitly wrote the harmonic oscillator equation for each and integrated it analytically. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In print, the first modern treatment of the harmonic oscillator is &lt;a href=&quot;http://eulerarchive.maa.org/pages/E126.html&quot; rel=&quot;noreferrer&quot;&gt;Euler's De Novo Genere Oscillationum (presented 1738-9, published 1750)&lt;/a&gt;. He solved in quadratures not only the equation of the free oscillator, but also of the oscillator driven by harmonic force. This was the first analytic treatment of resonance, see &lt;a href=&quot;https://books.google.com/books?id=VOcUBvbUXlEC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Kline, Mathematical Thought From Ancient to Modern Times, v.2, pp. 479-482&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In his effort to treat the vibrating string, John Bernoulli, in a letter of 1727 to his son Daniel and in a paper, considered the weightless elastic string loaded with &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; equal and equally spaced masses... John&#xA;  recognized that the force on each mass is &lt;span class=&quot;math-container&quot;&gt;$-K$&lt;/span&gt; times its displacement, and solved &lt;span class=&quot;math-container&quot;&gt;$\frac{d^2x}{dt^2} = - Kx$&lt;/span&gt;, thus integrating the equation of simple harmonic motion by analytic methods&lt;/em&gt;.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;In 1728 Euler began to consider second order equations. His interest&#xA;  in these was aroused partly by his work in mechanics. He had worked, for&#xA;  example, on pendulum motion in resisting media, which leads to a second&#xA;  order differential equation... In a paper of 1739 Euler took up the differential equations of the&#xA;  harmonic oscillator &lt;span class=&quot;math-container&quot;&gt;$\ddot{x} + Kx = 0$&lt;/span&gt; and the forced oscillation of the harmonic oscillator &lt;span class=&quot;math-container&quot;&gt;$M\ddot{x} + Kx = F\sin(\omega t)$&lt;/span&gt;. He obtained the solutions by quadratures and discovered (really rediscovered,&#xA;  since others had found it earlier) the phenomenon of resonance&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-02-18T06:54:23.203" LastActivityDate="2020-02-18T06:54:23.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11475" PostTypeId="2" ParentId="11462" CreationDate="2020-02-17T23:45:02.010" Score="3" Body="&lt;p&gt;As GEP as asked for a translation into english of some parts of the Gallica text, here it is :&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;I have noticed Cousin, Lallande, Brisson, librarian Pancouke, some teachers from the Lycée. Some of them are dropped by national carriages or with representatives : no teachers come in another way. Here are some details about the professors : these tiny details will look superfluous, but I write them because you don't find them reported in newspapers. Lagrange, the first among european scientists, looks between 50 and 60, although he is younger : his face testifies dignity and refinement : he looks rather thin and pale ; his voice is very low, unless he gets angry ; he has a very strong italian accent and pronounces &quot;s&quot; like &quot;z&quot; ; he is dressed very modestly in black or brown ; he speaks in a very straightforward manner, somewhat with difficulty; he has in his speech the embarassment and simplicity of a child. Everybody sees he is an extraordinary man, but you must have seen him to recognize him as such. He speaks only in conferences and &#xA;some of his sentences are laughable : &quot;there remains on this subject many important things to say, but I will not say them&quot;. The students, most of them unable to appreciate him, do not especially welcome him, but the professors do it in counterpart. Laplace, who has as well a chair of professor of Analysis, had been appointed in the Melun school, he had accepted it ; the government has repeared this administrative error. Laplace is first rank among scholars, he is known all over Europe as an excellent geometer, physicist and chemist. He looks rather young, has a little voice, though clear, he speaks with precision, albeit with some difficulty ; his exterior appearance is rather nice ; he has a very simple clothing ; his height is in the average. His mathematical teaching, by no way extraordinary, is given at a very fast pace.&#xA;Haüy, who has been formerly a priest, is extraordinarily simple and modest; he isn't old, his clothes are almost that of a clergyman ; moreover he has refused to swear [the civil constitution of the clergy]. His voice is very clear, very audible, and he speaks with much elegance and easiness. It is impossible to speak better. Some say that he memorizes his lessons ; it is said that he reads a part, something that is difficult to check because the professors are far away and always have their lecture notes in front of them. He is so shy that if somebody in the audience asks for some precision, he gets confused, answers in a wrong way or do not answer. He has a wide knowledge, and, though not as brilliant, without the genius of the first two, he is shining by his method and the way he provides the most elegant proofs.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I will probably add his opinion on Monge in the near future.&lt;/p&gt;&#xA;" OwnerUserId="3730" LastEditorUserId="3730" LastEditDate="2020-02-18T00:28:29.687" LastActivityDate="2020-02-18T00:28:29.687" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11476" PostTypeId="2" ParentId="11397" CreationDate="2020-02-19T16:38:27.847" Score="3" Body="&lt;p&gt;Ninety years ago yesterday (Feb. 18), astronomer &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Clyde_Tombaugh&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clyde Tombaugh&lt;/a&gt; gathered the data that proved the existence of what would eventually be dubbed Pluto — but it wouldn't have been possible, astronomers have since realised, without the calculations of a mathematician whom history has forgotten.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;That mathematician was &lt;strong&gt;Elizabeth Williams&lt;/strong&gt;, who worked for astronomer Percival Lowell, who first theorised the existence of a ninth planet. Lowell died before his successor, Tombaugh, finally spotted the elusive Pluto, but both men relied on calculations that Williams made. But the math got lost in the discovery it enabled, and so did Williams. Even her &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Elizabeth_Langdon_Williams&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; page is quite concise and states:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Williams continued to work at Lowell Observatory after Lowell's death, moving from Boston to the observatory itself at Flagstaff in 1919. In 1922, Williams married another astronomer, George Hall Hamilton. She was then dismissed from her position at the observatory by Constance Lowell since it was considered inappropriate to employ a married woman. In 1935, William's husband died. &lt;strong&gt;She moved to New Hampshire and subsequently died in poverty.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-02-28T16:37:23.223" LastActivityDate="2020-02-28T16:37:23.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11477" PostTypeId="1" AcceptedAnswerId="11545" CreationDate="2020-02-20T11:12:27.850" Score="3" ViewCount="509" Body="&lt;p&gt;Recently, I came across this book «&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0465094252&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lost in math&lt;/a&gt;» that aroused my interest. Having read about half of it, I have to admit that I am not a big fan of Mrs. Hossenfelder's informal popular style of writing. Apart from that I have feeling that sometimes she jumps from one subject to another too freely and can't help but feel lost (I am not criticising, of course, merely describing my personal impression; judging by rather high Amazon's rating a lot of people have different opinion).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would prefer some thing more strict, more structured, more academic, if you like (but not too academic for I am not a mathematician or theoretical physicist), about the same subject that is, as I would put it, elements of philosophical analysis of modern mathematics, its touch with reality and its relations with theoretical physics. Can you give me any suggestions please?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you happen to have read classic book «Mathematics and the search for knowledge» by Morris Kline you, probably, understand what I mean. A nice book. It is somewhat outdated, though.&lt;/p&gt;&#xA;" OwnerUserId="11124" LastActivityDate="2020-03-14T22:26:40.910" Title="Alternatives to «Lost in math: how beauty leads physics astray» by Sabine Hossenfelder?" Tags="|mathematics|philosophy-of-science|theoretical-physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11479" PostTypeId="1" CreationDate="2020-02-21T22:01:01.410" Score="4" ViewCount="127" Body="&lt;p&gt;Recently I read the English Wikipedia article about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_calendar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chinese calendar&lt;/a&gt;. Surprisingly, the article stated that the oldest Chinese calendar was a solar calendar (or solar calendars).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The traditional Chinese calendar was developed between 771 and 476 BC, during the &lt;a href=&quot;https://en.wikipedia.org/wiki/Spring_and_Autumn_period&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spring and Autumn period&lt;/a&gt; of the Eastern Zhou dynasty. Before the &lt;a href=&quot;https://en.wikipedia.org/wiki/Zhou_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zhou dynasty&lt;/a&gt;, solar calendars were used.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, as far as I know, the oldest Chinese calendar was a lunisolar calendar, which was recorded in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Oracle_bone&quot; rel=&quot;nofollow noreferrer&quot;&gt;oracle bones&lt;/a&gt; of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Shang_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shang Dynasty&lt;/a&gt;. Since then, the traditional Chinese calendar continued the tradition of the lunisolar calendar. Am I wrong?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;br /&gt;&#xA;* Below is a list of early Chinese dynasties.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Xia_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Xia dynasty&lt;/a&gt;(legendary) → &lt;a href=&quot;https://en.wikipedia.org/wiki/Shang_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shang(Yin) dynasty&lt;/a&gt; → &lt;a href=&quot;https://en.wikipedia.org/wiki/Zhou_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zhou dynasty&lt;/a&gt; → &lt;a href=&quot;https://en.wikipedia.org/wiki/Spring_and_Autumn_period&quot; rel=&quot;nofollow noreferrer&quot;&gt;Spring and Autumn period&lt;/a&gt; → &lt;a href=&quot;https://en.wikipedia.org/wiki/Warring_States_period&quot; rel=&quot;nofollow noreferrer&quot;&gt;Warring States period&lt;/a&gt; → &lt;a href=&quot;https://en.wikipedia.org/wiki/Qin_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Qin dynasty&lt;/a&gt; → &lt;a href=&quot;https://en.wikipedia.org/wiki/Han_dynasty&quot; rel=&quot;nofollow noreferrer&quot;&gt;Han dynasty&lt;/a&gt; → .......&lt;/p&gt;&#xA;" OwnerUserId="11033" LastEditorUserId="11033" LastEditDate="2020-06-18T02:29:59.697" LastActivityDate="2020-06-18T02:29:59.697" Title="Was the oldest Chinese calendar a solar calendar?" Tags="|calendar|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11481" PostTypeId="1" CreationDate="2020-02-22T04:23:36.110" Score="0" ViewCount="42" Body="&lt;p&gt;Was Mach, in his formulation of &quot;Mach's principle,&quot; influenced by pre-Galilean astronomy, such as that of Aristotle in &lt;a href=&quot;https://isidore.co/aquinas/DeCoelo.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On the Heavens&lt;/em&gt;&lt;/a&gt;, where heavenly bodies influence terrestrial ones?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2020-02-22T04:23:36.110" Title="Did pre-Galilean astronomy inspire Mach's principle?" Tags="|physics|astronomy|gravity|cosmology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11482" PostTypeId="1" CreationDate="2020-02-22T07:19:25.057" Score="1" ViewCount="2265" Body="&lt;p&gt;Newton's law of gravitation operative near the earth is the same law causes the earth and the other planets to go around the sun and other heavenly phenomena. Was it a giant leap of faith by Newton or did he religiously studied astronomical data to come to this conclusion? &lt;/p&gt;&#xA;" OwnerUserId="9960" LastActivityDate="2020-02-25T00:00:52.610" Title="What made Newton realize that the law of gravitation was 'universal'?" Tags="|physics|astronomy|newton|gravity|" AnswerCount="3" CommentCount="3" ClosedDate="2020-02-24T23:43:25.450" ContentLicense="CC BY-SA 4.0" />
  <row Id="11483" PostTypeId="1" AcceptedAnswerId="11495" CreationDate="2020-02-22T09:28:21.340" Score="2" ViewCount="258" Body="&lt;p&gt;Heisenberg already in 1927 quotes this equation as a &quot;known equation&quot;.&lt;sup&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ix4iB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ix4iB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ix4iB.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then what is the origin of it? And what is its further history of it, ending in its death?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ix4iB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeits. f. Phys. 43 172 1927. (p.8. in &lt;a href=&quot;https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19840008978.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf in English&lt;/a&gt;), &lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="11131" LastEditorUserId="11131" LastEditDate="2020-02-26T05:18:32.713" LastActivityDate="2020-02-26T05:18:32.713" Title="The history of the equation $\boldsymbol {Et}-\boldsymbol {tE}=\frac{h}{2\pi i}$" Tags="|quantum-mechanics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11484" PostTypeId="1" CreationDate="2020-02-22T04:01:36.403" Score="1" ViewCount="948" Body="&lt;p&gt;Did scientists at some point believe that &lt;em&gt;everything&lt;/em&gt; is made out of atoms? Or were atoms always accompanied by other &quot;elementary particles&quot;?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I myself did not realize that there existed other &quot;elementary particles&quot; besides atoms until quite recently. I assumed that both solid objects, liquids, the wind blowing, fog, and light was all made out of atoms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Apparently, at least light is not made from atoms, but &quot;photons&quot;. While trying to determine what wind and fog is made of, I opened a metaphorical &quot;can of worms&quot; which caused my head to start spinning from all the cryptic and complicated Wikipedia articles, talking about numerous &quot;elementary particles&quot; to the point where I lost all ability to make any sense of the whole situation, as so often is the case when I try to &quot;simply look something up&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Would appreciate if this were straightened out once and for all, both in historical and current understanding.&lt;/p&gt;&#xA;" OwnerDisplayName="Grahm" LastEditorUserId="26" LastEditDate="2021-02-20T15:19:37.340" LastActivityDate="2021-02-20T15:19:37.340" Title="Did scientists at some point believe that *everything* is made out of atoms?" Tags="|particle-physics|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11485" PostTypeId="2" ParentId="11484" CreationDate="2020-02-22T04:25:23.810" Score="6" Body="&lt;p&gt;There was a time in the late 1800's-to-early 1900's when it became fairly clear that all substances in the material world were made of atoms. This &quot;atomic theory&quot; was not adopted by all scientists, however; there were significant holdouts for years after, and all the while the question of what an atom was actually made of was still unsettled. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, in the first 35 years of the 20th century it became fairly clear (again) that atoms were made of three types of subatomic particle: protons, neutrons and electrons. These are the constituents of all ordinary atoms we encounter in our everyday lives. The so-called &quot;zoo&quot; of other subatomic particles discovered since then in laboratory experiments are not found in ordinary matter because their lifetimes are very short, and the conditions needed to create them (extremely high temperatures &amp;amp; energies) don't exist except in exotic places like particle accelerators and the central core of the sun. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Photons are indeed not atoms- they are bundles of energy that have no mass and always travel at the speed of light. The interactions between photons and electrons (which are the outermost constituents of atoms) give rise to most of the ordinary properties of matter that we experience in our daily lives. &lt;/p&gt;&#xA;" OwnerUserId="14713" OwnerDisplayName="niels nielsen" LastActivityDate="2020-02-22T04:25:23.810" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11486" PostTypeId="2" ParentId="11484" CreationDate="2020-02-22T05:06:08.727" Score="6" Body="&lt;p&gt;I’ll supplement Niels’ answer by answering some of your questions that he didn’t.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wind is mostly made of nitrogen and oxygen molecules, which make up most of the atmosphere. Nitrogen molecules have two nitrogen atoms joined together and oxygen molecules have two oxygen atoms similarly joined.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The part of fog that looks white or grey is mainly made of water molecules, which have two hydrogen atoms and one oxygen atom joined together. The water molecules are mixed in with the nitrogen and oxygen molecules that make up most of the colorless air.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The protons and neutrons in the nuclei of atoms are actually made of even more fundamental particles called quarks, held together by gluons. Specifically a proton has two “up” quarks and one “down” quark, while a neutron is the opposite. As far as we can tell today, quarks, gluons, electrons, and photons are not made of anything even more fundamental.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a detailed and very successful mathematical theory of elementary particles called the Standard Model. It actually has seventeen different kinds of particles, by one popular way of counting them which you can see in the diagram with 17 boxes &lt;a href=&quot;https://en.wikipedia.org/wiki/Standard_Model&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some of the particles are fundamental but unstable and quickly decay into stable fundamental particles. Decaying into other particles does not mean that those other particles were “inside” the unstable ones. Instead, the unstable particles are able to mutate into other kinds of particles according to certain rules.&lt;/p&gt;&#xA;" OwnerDisplayName="G. Smith" LastActivityDate="2020-02-22T05:23:00.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11487" PostTypeId="2" ParentId="11484" CreationDate="2020-02-22T07:25:16.787" Score="4" Body="&lt;p&gt;Historically, the concept of &quot;atoms&quot; (from the Greek atomos, indivisible) just denoted the smallest particles of matter. Democritus and followers argued that the only things that existed were atoms and the void. The idea of subatomic particles would not have made sense to them - atoms were simply the smallest indivisible parts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When physics in the 19th century started to discover the elementary parts of matter they often talked about them as &quot;corpuscles&quot; (&quot;small bodies&quot;). The term atom became attached to the smallest constituent of chemical elements, and &lt;em&gt;then&lt;/em&gt; people found the electron and other particles, so the terminology got confused. &lt;/p&gt;&#xA;" OwnerDisplayName="Anders Sandberg" LastActivityDate="2020-02-22T07:25:16.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11488" PostTypeId="2" ParentId="11483" CreationDate="2020-02-22T15:32:23.650" Score="3" Body="&lt;p&gt;Since you do not explain what &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; mean, I suppose that this is the equation&#xA;written by&#xA;M. Born and P. Jordan, Zur Quantenmechanik, Z. Phys. 34 (1925), Eq. 38,&#xA;few weeks after Heisenberg published his first version of Quantum Mechanics.&#xA;Of course, by 1927 it was already well-known. Development was very fast in this area.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-02-22T15:32:23.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11489" PostTypeId="1" CreationDate="2020-02-22T21:29:21.900" Score="2" ViewCount="102" Body="&lt;p&gt;Is there a specific date when Laguerre published his root finding method? I found his 1880 note &lt;a href=&quot;https://books.google.com/books?id=vthMAAAAYAAJ&amp;amp;pg=PA1&amp;amp;lpg=PA1&amp;amp;dq=Laguerre+Notes+sur+la+r%C3%A9solution+des+%C3%A9quations+num%C3%A9riques&amp;amp;source=bl&amp;amp;ots=LhVsXy3SDj&amp;amp;sig=ACfU3U2qErqeVdqFg4O1MMCH59ohQo2jbQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjRq9a_zOfnAhWgknIEHfs0BgUQ6AEwA3oECAkQAQ#v=onepage&amp;amp;q=Laguerre%20Notes%20sur%20la%20r%C3%A9solution%20des%20%C3%A9quations%20num%C3%A9riques&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Résolution des équations numériques&lt;/a&gt;, but I am not sure if this is the source because I can not read French.&lt;/p&gt;&#xA;" OwnerUserId="10729" LastEditorUserId="26" LastEditDate="2021-02-20T15:16:53.130" LastActivityDate="2021-02-20T15:16:53.130" Title="When was the Laguerre's method first used to approximate roots?" Tags="|mathematics|discoveries|algorithm|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11490" PostTypeId="2" ParentId="11482" CreationDate="2020-02-23T02:09:00.040" Score="3" Body="&lt;p&gt;According to his own recollections, Newton arrived to the idea of universal gravitation when he compared the acceleration of the moon in its orbit with the acceleration of falling objects on the Earth surface. This was his first calculation. This was a rough, approximate estimate with many simplifying assumptions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many years later, Newton was able to show that the his attraction law is equivalent to the Kepler Law. This was a major step which made the universality of the law of gravitation much more plausible. After that he started to investigate other phenomena: the three bodies problem (that is deviations of the Moon from the Kepler laws), tides, the shape of the Earth, etc. And was convinced that the gravitation law can indeed explain these phenomena.&#xA;He wrote his famous book, where he gave explanations of many celestial and other phenomena&#xA;from the gravitation law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His results were far from complete. For example, he could not explain all known at that time irregularities of the Moon motion. His prediction of the shape of the Earth was confirmed after many measurements (mostly French).&#xA;His theory of tides was eventually developed to the extent which made tide prediction possible. A reasonably complete theory of Moon motion was developed (with enormous efforts of best mathematicians) by the middle of&#xA;18 century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;These investigations were curried on by his successors, gradually convincing more and more in the exactness and universality of the law. Some major steps were calculations of the orbits of small planets and the discovery of Neptun&#xA;in the first half of 19 century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-02-23T02:14:45.333" LastActivityDate="2020-02-23T02:14:45.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11491" PostTypeId="1" AcceptedAnswerId="11494" CreationDate="2020-02-23T12:48:34.663" Score="1" ViewCount="164" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/50xwZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/50xwZ.png&quot; alt=&quot;[1]: https://i.stack.imgur.com/zAuui.png&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did Weibull or any other mathematician arrive at the above expression? I saw the 1951 paper, but it is not clear to me. In 1939 he had published a book called &quot;A Statistical Theory of the Strength of Materials&quot;, and I am currently searching for this book. In the meantime can anyone explain me.&lt;/p&gt;&#xA;" OwnerUserId="11134" LastEditorUserId="10106" LastEditDate="2020-02-24T07:18:35.453" LastActivityDate="2020-02-24T07:18:35.453" Title="How did Weibull derive the three parameter Weibull distribution?" Tags="|mathematics|reference-request|statistics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11492" PostTypeId="2" ParentId="11489" CreationDate="2020-02-23T15:24:36.100" Score="1" Body="&lt;p&gt;The scanned pages might be more readily available &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=uiug.30112066045045&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Indeed the method is presented as a novel; it is only compared to Newton's method. It is first introduced as a method for finding root enclosures and approximations of the largest and smallest roots of polynomials &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;under the condition that all roots are real.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Under this condition the usual &quot;trick&quot; in the derivation also makes sense, as when &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is the largest root, and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a close approximation, then &lt;span class=&quot;math-container&quot;&gt;$\frac1{x-\alpha}$&lt;/span&gt; is large in&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;G=\frac{f'(x)}{f(x)}=\frac1{x-\alpha}+\frac1{x-\beta}+...&#xA;$$&lt;/span&gt;&#xA;and &#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;H=\frac{f'(x)^2-f(x)f''(x)}{f(x)^2}=\frac1{(x-\alpha)^2}+\frac1{(x-\beta)^2}+...&#xA;$$&lt;/span&gt;&#xA;and the remaining terms are small and of the same sign and can thus be considered as approximately equal. Setting them equal to zero, as Laguerre discusses first, gives in the first equation Newton's method and in the second&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0&amp;lt;\frac1{(x-\alpha)^2}\le H&#xA;\\&#xA;\implies &#xA;\alpha\le x-\frac{|f(x)|}{\sqrt{f'(x)^2-f(x)f''(x)}} ~\text{ or }~ x+\frac{|f(x)|}{\sqrt{f'(x)^2-f(x)f''(x)}}\le \alpha&#xA;$$&lt;/span&gt;&#xA;giving a root exclusion region around &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6894" LastEditorUserId="10106" LastEditDate="2020-02-23T15:49:55.123" LastActivityDate="2020-02-23T15:49:55.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11493" PostTypeId="2" ParentId="11482" CreationDate="2020-02-23T19:46:06.217" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Newton considered himself to be not only rediscovering the mathematical principles of  the ancient philosophers in his &lt;em&gt;Principia&lt;/em&gt;, but also rediscovering the ancient geometrical methods. In an intended preface for the second edition written in 1710, Newton attributed knowledge of universal gravitation to the ancient Chaldaean and claimed that Pythagoras imported it to the Greeks and Romans. [p.23].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/Isaac_Newton_s_Temple_of_Solomon_and_his/XgbeWGGMkSEC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isaac Newton's Temple of Solomon and His Reconstruction of Sacred Architecture Tessa Morrison&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1950" LastActivityDate="2020-02-23T19:46:06.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11494" PostTypeId="2" ParentId="11491" CreationDate="2020-02-23T20:36:17.040" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0266892094900043&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; &quot;Strength of materials and the Weibull distribution&quot; by  Eric S. Lindquist in &lt;em&gt;Probabilistic Engineering Mechanics&lt;/em&gt; &lt;strong&gt;9&lt;/strong&gt; (1994) 191-194 probably has what you want.&#xA;I found several online copies with a &quot;weibull's original paper&quot; google search, but they might be behind institutional paywalls.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Weibull's 1939  paper &quot;A statistical theory of the strength of materials&quot; seems to have appeared in what &lt;a href=&quot;https://www.worldcat.org/title/ingeniorsvetenskapsakademiens-handlingar-proceedings-royal-swedish-academy-of-engineering-sciences-the-royal-swedish-institute-for-scientific-engineering-research-formerly-proceedings-the-royal-swedish-institute-for-engineering-research/oclc/888476918?referer=di&amp;amp;ht=edition&quot; rel=&quot;nofollow noreferrer&quot;&gt;WorldCat&lt;/a&gt; says was this journal:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Ingeniörsvetenskapsakademiens handlingar = Proceedings / Royal Swedish Academy of engineering sciences the Royal Swedish Institute for scientific engineering research = formerly: Proceedings / the Royal Swedish Institute for engineering research&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;in volume 151, pp.1-45.  I have not found any on-line copies of this paper.&#xA;WorldCat lists libraries on 3 continents with this journal; the nearset to me is about 100 km away from where I live, the 6-th nearest in another country.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two of the 3 parameters, &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\eta$&lt;/span&gt; (this is me talking, not Lindquist) are location and scale parameters, more or less inevitable in any technological application. The third, &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; is the interesting one.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-02-23T21:54:10.970" LastActivityDate="2020-02-23T21:54:10.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11495" PostTypeId="2" ParentId="11483" CreationDate="2020-02-24T07:12:56.630" Score="1" Body="&lt;p&gt;As a comment (deleted since) pointed out, the source is mentioneed in in §5.4, p. 140 of Jammer, &lt;a href=&quot;//books.google.com/books?id=uqbvAAAAMAAJ&amp;amp;q=%22earliest+discussion%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Philosophy of Quantum Mechanics&lt;/a&gt;. It is   P. A. M. Dirac, &lt;em&gt;Relativity quantum mechanics with an application to Compton scattering&lt;/em&gt; Proc. R. Soc. Lond. A 1926 111, 405-423. The equation in question is among eqs. (7) (the energy operator is denoted here by &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt;):&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/v5nDw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/v5nDw.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This formula was proved to be senseless by Wolfgang Pauli, in &lt;em&gt;Handbuch der Physik, zweite Auflage, Band XXIV, Ersten Teil - Quantentheorie&lt;/em&gt;, Springer-Verlag Berlin Heidelberg GmbH, 1933, p 140:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HDgRK.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HDgRK.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The English translation of the marked sentence can be found in &lt;em&gt;Fundamental Questions in Quantum Mechanics&lt;/em&gt;, &lt;a href=&quot;https://books.google.hu/books?id=x-X3gdWvX6sC&amp;amp;pg=PA266&amp;amp;dq=%22We%20conclude%20that%20the%20introduction%20of%20an%20operator%20T%20must%20fundamentally%20be%20%22abandoned%22%20and%20that%20the%20time%20t%20in%20quantum%20mechanics%20has%20to%20be%20regarded%20as%20an%20ordinary%20number.%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwit5YPCte7nAhWRs4sKHT02C8kQ6AEIKTAA#v=onepage&amp;amp;q=%22We%20conclude%20that%20the%20introduction%20of%20an%20operator%20T%20must%20fundamentally%20be%20%22abandoned%22%20and%20that%20the%20time%20t%20in%20quantum%20mechanics%20has%20to%20be%20regarded%20as%20an%20ordinary%20number.%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 266&lt;/a&gt; as&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We conclude that the introduction of an operator t must fundamentally&#xA;  be abandoned and that the time t in quantum mechanics has to be&#xA;  regarded as an ordinary number (c-number')&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11131" LastEditorUserId="11131" LastEditDate="2020-02-26T04:53:15.077" LastActivityDate="2020-02-26T04:53:15.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11496" PostTypeId="1" AcceptedAnswerId="11501" CreationDate="2020-02-24T21:36:34.320" Score="4" ViewCount="2537" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;The metre is defined as the length of the path travelled by light in&#xA;  a vacuum in 1/299 792 458 of a second&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Why is this so? Who decided that &lt;code&gt;1/299,792,458&lt;/code&gt; of a second is a suitable measurement? What significance does the number 299,792,458 have? It seems so arbitrary.&lt;/p&gt;&#xA;" OwnerUserId="11144" LastActivityDate="2023-04-21T08:24:28.393" Title="Why is one meter as long as it is?" Tags="|terminology|geometry|units|measurement|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11497" PostTypeId="1" AcceptedAnswerId="11500" CreationDate="2020-02-24T23:00:23.903" Score="1" ViewCount="266" Body="&lt;p&gt;A paper published in 1975 references The Art of Computer Programming (then published in 1969), so it's probably the first edition.  I can't even find this edition for sale anywhere.  Would you be so kind to photograph (or something) pages 7, 8 and 25?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even though I'm able to more or less match what the possible content was (by looking at the newer edition), I'm interested in knowing precisely what was written on these pages because I'm interested in the historical development of some ideas, so I'm trying to infer what influenced the researcher.&lt;/p&gt;&#xA;" OwnerUserId="11145" LastEditorUserId="11145" LastEditDate="2020-02-25T15:57:46.460" LastActivityDate="2020-03-02T03:45:34.217" Title="Can you provide pages 7, 8, 25 of the 1969 edition of the Art of Computer Programming?" Tags="|computer-science|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11498" PostTypeId="2" ParentId="11482" CreationDate="2020-02-24T23:33:35.073" Score="2" Body="&lt;p&gt;The meaning of saying that gravitation is 'universal', as in the question posed in at least the title, is sometimes misunderstood. It refers primarily to the appreciation that all massive bodies attract each other: i.e. that heavy bodies gravitate at the same time towards &lt;em&gt;all&lt;/em&gt; other massive bodies, and not only towards one local source of attraction.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;There isn't one single historical document that explains how Newton arrived at this conception. But the time-period within which he arrived at the idea can be narrowed down, probably, to some time from late 1684 to early 1685, by two of the documents he wrote in that period. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The first of these relevant documents, from late 1684, is &lt;a href=&quot;https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum&quot; rel=&quot;nofollow noreferrer&quot;&gt;'De motu corporum'&lt;/a&gt;, an early precursor of what became the &lt;em&gt;Principia&lt;/em&gt;: it contains no mention of anything that points to the universality of gravitation, so it may provide what can be called a &lt;em&gt;terminus a quo&lt;/em&gt;. Its investigations focus entirely on the relation between the trajectory of a body in motion and a &lt;em&gt;single&lt;/em&gt; source of centripetal force. For this idealised and simplified system, 'De motu corporum' demonstrates among other things the 'equal-area' character of the motion, and the relation between elliptical trajectories and an inverse-square law of centripetal force.  But by its limitation to considering only one center of attraction, it lacks any sign that Newton at the time of writing it appreciated the universality of gravitation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably the first document that does show a sign that Newton considered gravitation to be universal is a manuscript passage from early 1685 often known as the 'Copernican scholium' because it directly asserts among other things that the Copernican system is 'proved a priori' from gravitational considerations. The apparent purpose of this text was to memorialise just how immensely complex are the motions in the solar-system.  The text appeals expressly to the laws of motion, and then also implicitly to ideas of the relative masses of the planets, mentioning in passing &quot;the [attractive] action of all these [planets] on each other&quot;. (Newton's way to measure the relative masses of Sun, earth, Jupiter and Saturn did not appear in print until Book III of the Principia, but the text of the 'Copernican scholium' appears to presuppose and to indicate that when Newton wrote it he already had some of his ideas of solar and planetary masses and mutual actions of the planets, as well as appealing to his demonstrations of motion under centripetal accelerative force). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The message of the scholium (translated from Latin to English by J Herivel (1965) in 'The Background to Newton's Principia') is, in part,  &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;... the planets neither move exactly in ellipses nor revolve twice in&#xA;  the same orbit.  So that there are as many orbits to a planet as it&#xA;  has revolutions, as in the motion of the Moon, and the orbit of any&#xA;  one planet depends on the combined motion of all the planets, not to&#xA;  mention the actions of all these on each other.  But to consider&#xA;  simultaneously all these causes of motion and to define these motions&#xA;  by exact laws allowing of convenient calculation exceeds, unless I am&#xA;  mistaken, the force of the entire human intellect.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus the passage implies or presupposes a generalisation of the attractive actions of the planets, all exerting actions on each other.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It also seems that in early 1685 Newton thought it possible that such motions -- including the actions of all the planets on each other -- were so complex that they might never be solved mathematically. He did later develop initial steps towards solving such many-body motions (e.g. in Proposition 66 of Book I of the &lt;em&gt;Principia&lt;/em&gt;) but the subject continued to provide many thorny problems for celestial mechanicians for centuries after Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far this answer concerns &lt;em&gt;when&lt;/em&gt; rather than &lt;em&gt;how&lt;/em&gt; Newton reached his idea of the universality of gravitation. For the 'how', there is much uncertainty. But if the logic and layout of Book III of the &lt;em&gt;Principia&lt;/em&gt; are any indications, then it seems that Newton considered that the behavior of the satellites of Jupiter had great demonstrative significance for universal gravitation.  The satellites clearly make their revolutions around Jupiter, with something as close to equal-area motion as observations of Newton's time could establish.  Newton may have inferred initially that not only the Sun but also at least Jupiter must be sources of attraction -- and if both of those, then probably also the others, by his 'rules of reasoning in philosophy' (a heading that made its appearance only in later editions of the &lt;em&gt;Principia&lt;/em&gt;, even though core principles were present in the first edition). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's 'moon-test' appears not to concern this question about the &lt;em&gt;universality&lt;/em&gt; of gravitation. The test had the purpose and effect of showing that the force that makes the moon deviate from a straight-line trajectory into its curved orbit around the earth has the same magnitude, when allowance is made for the inverse-square effect of distance, as the force that makes terrestrial heavy bodies tend to fall to earth. Only an attraction by the earth was considered: the moon-test was not designed to consider whether the moon (or terrestrial bodies) is also attracted by anything other than the earth.  Newton's quantitative arguments used and acknowledged Huygens's meaure of the accelerative force of terrestrial gravity from pendulum experiments. The result supports that these forces are of the same kind as what causes the heaviness of ordinary bodies near the earth's surface. That was what Newton believed to justify describing the forces that accelerate the moon and by inference the planets as 'gravitational'.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The contributions of Hooke also do not seem to go to the universality of gravitation: he was a fine observer but had too little mathematics to develop the theoretical potential of his observations.  His telescopic observations included noticing the character of the lunar craters, from which he made the astute inference that they arose by impacts, suggesting that heavy bodies would fall to the moon's surface as they fall to the earth's surface. But Hooke also wrote that the effects of the planets were exerted within their 'sphere of influence' -- a remark that appears to stop short of arriving at any idea of &lt;em&gt;universal&lt;/em&gt; gravitation.  &lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2020-02-25T00:00:52.610" LastActivityDate="2020-02-25T00:00:52.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11499" PostTypeId="1" AcceptedAnswerId="11502" CreationDate="2020-02-25T01:23:23.993" Score="9" ViewCount="302" Body="&lt;p&gt;Vinogradov introduced the notation &#xA;&lt;span class=&quot;math-container&quot;&gt;$$f(x) \ll g(x)$$&lt;/span&gt;&#xA;to denote that for some &lt;span class=&quot;math-container&quot;&gt;$C&amp;gt;0$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$|f(x)|\leqslant C\,g(x)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; under consideration; usually for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; larger than a fixed constant (equivalently &lt;span class=&quot;math-container&quot;&gt;$f(x) = O(g(x))$&lt;/span&gt; is used).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I've also been made aware that this is sometimes used in less formal settings to mean &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is &quot;much less than&quot; &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$A\ll B$&lt;/span&gt;). My question is, where did this notation come from, and what was it used for first? Did Vinogradov invent it, or was it used in this informal sense prior to that?&lt;/p&gt;&#xA;" OwnerUserId="10340" LastActivityDate="2024-03-21T16:58:02.637" Title="Origin of $\ll$ notation" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11500" PostTypeId="2" ParentId="11497" CreationDate="2020-02-25T02:54:36.387" Score="5" Body="&lt;p&gt;If you search for the book &lt;a href=&quot;http://www.worldcat.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, you will be able to find the libraries near you that have this edition. You may be able to arrange an interlibrary loan if it is too far to travel.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I also checked a couple of used book search engines (&lt;a href=&quot;http://used.addall.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; and &lt;a href=&quot;http://www.isbns.net&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;) and found that there are a number of copies of the 1969 edition for sale.&lt;/p&gt;&#xA;" OwnerUserId="11146" LastEditorUserId="10106" LastEditDate="2020-03-02T03:45:34.217" LastActivityDate="2020-03-02T03:45:34.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11501" PostTypeId="2" ParentId="11496" CreationDate="2020-02-25T11:20:24.217" Score="25" Body="&lt;p&gt;This number has no significance. Its origin is historical. Originally the meter was defined as 1/40,000,000 part of the Paris meridian. Based on the measurement of this&#xA;meridian, they made a standard rod in Paris. Since it is inconvenient to base the definition on something which is difficult to measure, the meter was soon redefined simply as the length of this rod. Then when they decided to use another, more convenient and more accurate standard, defined in terms of light, they just measured the rod and obtained this number.&lt;/p&gt;&#xA;&lt;p&gt;The reason why 1/40,000,000 was originally chosen is interesting. At the end of 18-century, the French revolutionary government decided to introduce the decimal system everywhere. This included angle and time measurement. So the angle of 90 ordinary degrees became 100 decimal degrees. Therefore, the whole circle had 400 degrees.&#xA;Each decimal degree was divided into 100 decimal minutes. So kilometer is one decimal&#xA;minute of the Paris meridian. (Similarly to the nautical mile which is one ordinary minute of a meridian). According to the project, also &lt;a href=&quot;https://en.wikipedia.org/wiki/Nychthemeron&quot; rel=&quot;noreferrer&quot;&gt;nychthemeron&lt;/a&gt; (day+night)&#xA;were divided into 20 decimal hours, and decimal hours into 100 decimal time-minutes.&lt;/p&gt;&#xA;&lt;p&gt;Unlike other decimal units, these did not survive, and were abolished by Napoleon.&#xA;One can still find on e-bay French decimal watches and angle measuring instruments made at that epoch. Unlike decimal hours, decimal degrees are &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Grad_protractor.png&quot; rel=&quot;noreferrer&quot;&gt;still in use&lt;/a&gt;; they are called grads.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="10106" LastEditDate="2021-06-02T16:14:46.450" LastActivityDate="2021-06-02T16:14:46.450" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11502" PostTypeId="2" ParentId="11499" CreationDate="2020-02-25T14:05:10.993" Score="10" Body="&lt;p&gt;Vinogradov likely adapted &lt;span class=&quot;math-container&quot;&gt;$\ll$&lt;/span&gt; from Poincaré and Borel, who used it for asymptotic series in 1890s (Cajori cites &lt;a href=&quot;https://archive.org/details/leconssurlesseri00boreuoft/page/n8/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Borel, *Leçons sur les séries divergentes&amp;quot;, 1901&lt;/a&gt;). Physicists used it for vague &amp;quot;much less than&amp;quot; as early as 1918 (&lt;a href=&quot;https://lup.lub.lu.se/search/publication/5c45785b-95cd-4176-a709-994190fa85f1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heurlinger's doctoral dissertation &lt;em&gt;Untersuchungen über die Struktur der Bandenspektra&lt;/em&gt;&lt;/a&gt;). Whether they reinterpreted Poincaré's symbol or simply adapted what was typographically available is unclear. Here is the entry on the subject from &lt;a href=&quot;https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n143/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori's History of Mathematical Notations, v.II, §486&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The sign &lt;span class=&quot;math-container&quot;&gt;$\ll$&lt;/span&gt; was introduced by H. Poincaré and by E. Borel in comparing series like:&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$u_0+u_1z+u_2z^2+\dots\ll M\left[1+\frac{z}{R^1}+\left(\frac{z}{R^1}\right)^2+\dots\right],$$&lt;/span&gt; &lt;em&gt;where the second series has positive coefficients; the modulus of each&#xA;coefficient of the first series is less than the corresponding coefficient&#xA;of the second series. The signs &lt;span class=&quot;math-container&quot;&gt;$\ll$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gg$&lt;/span&gt; are also used for &amp;quot;much&#xA;less than&amp;quot; and &amp;quot;much greater than&amp;quot;. Zeitschrift für Physik, Vol. XIII (1923), p. 366 (the sign was used by H. A. Kramers and W. Pauli); Torsten Heurlinger, Untersuchungen über die Struktur der Bandenspectra (Lund, 1918), p. 39&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Kramers-Pauli paper is &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01328226&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Theorie der Bandenspektren&lt;/a&gt;, uses it as a matter of course without comment or explanation. Heurlinger is not cited, but, given the shared topic, they might have inherited the symbol through intermediaries.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10608" LastEditDate="2024-03-21T16:58:02.637" LastActivityDate="2024-03-21T16:58:02.637" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11503" PostTypeId="2" ParentId="11496" CreationDate="2020-02-25T14:49:51.497" Score="8" Body="&lt;p&gt;It's no more arbitrary than any other measurement unit,  including the second.   Nearly all modern values were chosed to try to avoid changing existing units' values while providing a source less subject to variation.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The most well-known example is the kilogram.  There's a standard cylinder platinum&amp;amp;iridium of which served as the original kilogram for quite a while.  But even that is subject to evaporation as well as impurities.  There were/are plans to carve a perfect crystalline silicon sphere, which would be less subject to those problems.  However, as mentioned in Wikipedia, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;2019: The kilogram is redefined in terms of the Planck constant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So ultimately all ISO / NIST units now refer back to fundamental constants.  We can only hope these are as (literally) universal as we believe they are. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-02-25T14:49:51.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11504" PostTypeId="2" ParentId="5673" CreationDate="2020-02-25T14:52:26.820" Score="4" Body="&lt;p&gt;I think the name &amp;quot;partition function&amp;quot; was first used, and therefore introduced, by Charles George Darwin and R.H. Fowler in their article &amp;quot;On the partition of energy&amp;quot; Phil. Mag. 44(1922) 450–479, 823–842. (§9, p. 469).&lt;/p&gt;&#xA;&lt;p&gt;Charles Galton Darwin was a grandson of the well-known Charles Darwin.&lt;/p&gt;&#xA;" OwnerUserId="11148" LastEditorUserId="6609" LastEditDate="2023-05-06T11:17:17.037" LastActivityDate="2023-05-06T11:17:17.037" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11505" PostTypeId="1" CreationDate="2020-02-25T23:45:49.180" Score="1" ViewCount="57" Body="&lt;p&gt;I am writing a short exposition on the central Delannoy numbers and would like to find the year which Henri Delannoy first introduced them in a formal setting.&lt;/p&gt;&#xA;&lt;p&gt;I &lt;em&gt;believe&lt;/em&gt; Delannoy's initial presentation of the Delannoy numbers was through his article &amp;quot;Emploi de l'échiquier pour la résolution de certains problèmes de probabilités,&amp;quot; which can be found on page 70 of &lt;a href=&quot;https://books.google.com/books?id=6CiVGxzVHW8C&amp;amp;pg=PA70&amp;amp;lpg=PA70&amp;amp;dq=%E2%80%99%20%CC%81echiquier%20pour%20lar%20%CC%81esolution%20de%20certains%20problemes%20de%20probabilite%20%CC%81es&amp;amp;source=bl&amp;amp;ots=KcvNXaEcip&amp;amp;sig=ACfU3U03otjL2j7BYyif2gqcPXNKyHkyWQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjHkviJ5e3nAhVGVs0KHWPjAYcQ6AEwAHoECAgQAQ#v=onepage&amp;amp;q=%E2%80%99%20%CC%81echiquier%20pour%20lar%20%CC%81esolution%20de%20certains%20problemes%20de%20probabilite%20%CC%81es&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; work. However, I cannot read French, and am unsure.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question:&lt;/strong&gt; Is this Delannoy's initial presentation of what are now called the Delannoy numbers?&lt;/p&gt;&#xA;" OwnerUserId="10767" LastEditorUserId="10608" LastEditDate="2024-02-28T07:53:21.263" LastActivityDate="2024-02-28T07:53:21.263" Title="Where did the Delannoy numbers make their first appearance?" Tags="|mathematics|mathematicians|combinatorics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11507" PostTypeId="1" AcceptedAnswerId="11511" CreationDate="2020-02-27T14:33:20.933" Score="3" ViewCount="308" Body="&lt;p&gt;I saw this from  Wikipedia.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The name &quot;Newton's method&quot; is derived from Isaac Newton's description of a special case of the method in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it has no citations. When is its first use?&lt;/p&gt;&#xA;" OwnerUserId="10729" LastEditorUserId="26" LastEditDate="2021-02-20T15:16:42.293" LastActivityDate="2021-02-20T15:16:42.293" Title="When is the first use of Newton's method for root finding?" Tags="|mathematics|discoveries|algorithm|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11509" PostTypeId="2" ParentId="11499" CreationDate="2020-02-28T21:50:49.417" Score="4" Body="&lt;p&gt;A Bourbaki &amp;quot;Note Historique&amp;quot; led me to this: Du Bois-Reymond, P. Sur la grandeur relative des infinis des fonctions. &lt;em&gt;Annali di Matematica&lt;/em&gt; 4, 338–353 (1870), available behind a &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF02420041&quot; rel=&quot;nofollow noreferrer&quot;&gt;paywall&lt;/a&gt;, uses the notations &lt;span class=&quot;math-container&quot;&gt;$$f(x)\succ\phi(x),\qquad  f(x)\sim\phi(x), \qquad f(x)\prec\phi(x)$$&lt;/span&gt; to mean&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim\frac{f(x)}{\phi(x)}=\infty, \qquad\lim\frac{f(x)}{\phi(x)}\text{ is finite }, \qquad\lim\frac{f(x)}{\phi(x)}=0.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A brief look at some 1880 papers by Poincare, Borel, and Stieltjes (found mentioned in Edelyi's little &lt;em&gt;Asymptotic Expansions&lt;/em&gt;) did not show any version of the &lt;span class=&quot;math-container&quot;&gt;$\ll$&lt;/span&gt; sign.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-08-06T17:19:44.500" LastActivityDate="2020-08-06T17:19:44.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11510" PostTypeId="2" ParentId="11497" CreationDate="2020-02-28T22:57:29.003" Score="3" Body="&lt;p&gt;I'm not precisely sure, but are these the pages you're looking for? If not, can you describe the content rather than page numbers?&lt;br&gt;&#xA;&lt;strong&gt;------------------------------------------------------------------------------&lt;/strong&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/NLHXu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NLHXu.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;strong&gt;------------------------------------------------------------------------------&lt;/strong&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/8EOCV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8EOCV.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;strong&gt;------------------------------------------------------------------------------&lt;/strong&gt;&lt;br&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/dA4Ia.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dA4Ia.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;strong&gt;------------------------------------------------------------------------------&lt;/strong&gt;&lt;br&gt;&lt;/p&gt;&#xA;" OwnerUserId="10787" LastActivityDate="2020-02-28T22:57:29.003" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11511" PostTypeId="2" ParentId="11507" CreationDate="2020-03-01T09:34:49.470" Score="3" Body="&lt;p&gt;TL; DR. This is one of those cases where &amp;quot;first use&amp;quot; very much depends on what is meant. Depending on that, it can be ascribed to Babylonians (c. 1600 BC), al-Tusi (c. 1250), Briggs (1633), Newton (1669), Raphson (1690) or Simpson (1740).&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;Babylonian rule (1800–1600 BC)&lt;/a&gt; for approximating square roots converted into modern notation gives the same formula as the &amp;quot;Newton's method&amp;quot;, see &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086098922091&quot; rel=&quot;nofollow noreferrer&quot;&gt;Square Root Approximations in Old Babylonian Mathematics by Fowler and Robson&lt;/a&gt;. It does not come with even a vague anticipation of conceptions involved in the modern method. The same rule is described by Heron in Metrica (c. 50 AD), and is called &amp;quot;Heron's method&amp;quot; in older books. Heron also gives a related rule for cube roots, which does not reduce to the &amp;quot;Newton's method&amp;quot;, suggesting that the connection is not too deep, see &lt;a href=&quot;https://www.researchgate.net/publication/313383940_Remarks_on_Heron%27s_cubic_root_iteration_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Remarks on Heron's cubic root iteration formula&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For the &amp;quot;warmer&amp;quot; part of history see &lt;a href=&quot;https://cedar.wwu.edu/math_facpubs/93/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historical Development of the Newton-Raphson Method by Ypma&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;A method algebraically equivalent to Newton's method was known to the 12th century&#xA;algebraist Sharaf al-Din al-Tusi [13], and the 15th century Arabic mathematician Al-Kashi&#xA;used a form of it in solving &lt;span class=&quot;math-container&quot;&gt;$x^p- N = 0$&lt;/span&gt; to find roots of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;. In western Europe a similar method&#xA;was used by Henry Briggs in his Trigonometria Britannica, published in 1633, though Newton&#xA;appears to have been unaware of this.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Instead, he worked from a more cumbersome perturbative method for solving polynomial equations laid out by Vieta in&#xA;&lt;em&gt;De numerosa potestatum&lt;/em&gt; (c.1600), with additional inspiration from a very old method of false position, a.k.a. &lt;a href=&quot;https://en.wikipedia.org/wiki/Regula_falsi&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Regula Falsi&lt;/em&gt;&lt;/a&gt;. It is close to what is now called the secant method. A simplified version of Vieta's method was published by van Schooten in 1646 and reproduced by Oughtred in &lt;em&gt;Clavis Mathematica&lt;/em&gt; (1647), this was Newton's source.&lt;/p&gt;&#xA;&lt;p&gt;Detailed remarks on it are found in Newton's unpublished 1664 notebook, and by 1669 he linearized Vieta's successive polynomials to produce something convertible into the modern version in &lt;em&gt;De analysi per aequationes numero terminorum infinita&lt;/em&gt; (better known for the first formulation of the method of fluxions). But, as with Babylonians, Heron or al-Tusi, there is little evidence that &amp;quot;his&amp;quot; method was the modern one conceptually. Here is Ypma:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Newton's tract... is the first recorded discussion by Newton of what we may recognize as an instance of the&#xA;Newton-Raphson method (1.1), although the formulation differs considerably from the now&#xA;conventional form, the computations are much more tedious than in the current formulation,&#xA;and the method is given only in the context of solving a polynomial equation. No calculus&#xA;is used in the presentation, and references to fluxional derivatives first appear later in that&#xA;tract, suggesting that Newton regarded this as a purely algebraic procedure. In several other&#xA;instances Newton is known to have used more traditional methods and notations in an effort&#xA;to make his ideas more accessible to a wider audience, but there is no clear evidence that at&#xA;that time he perceived this particular technique as an application of the calculus or derived it&#xA;using the techniques of calculus.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The manuscript was not published until 1711, but private copies circulated earlier and the relevant content is reproduced in Wallis's Treatise of Algebra both Historical and Practical (1685). In &lt;em&gt;Principia&lt;/em&gt; (1687) a similar procedure is applied to solving the Kepler's equation &lt;span class=&quot;math-container&quot;&gt;$x-e\sin x=M$&lt;/span&gt;, but again there is &amp;quot;&lt;em&gt;no clear evidence that Newton associated his technique with the use of the calculus. There are numerous ways to derive this process that do not require the use&#xA;of calculus&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Raphson simplified the technique further in 1690 by eliminating successive polynomials completely, and making the scheme iterative. He already felt that the resulting method was different from Newton's. However, the modern conception with derivatives (fluxions) does not appear until &lt;a href=&quot;https://archive.org/details/essaysonseveralc00simp/page/n6/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simpson's Essays on Several Curious and Useful Subjects (1740)&lt;/a&gt;, where he does not credit any predecessors, and explicitly contrasts his calculus based procedure to the previous algebraic ones.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-17T12:05:04.203" LastActivityDate="2020-09-17T12:05:04.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11512" PostTypeId="2" ParentId="6152" CreationDate="2020-03-01T16:58:55.110" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;How or where did the Wright brothers develop the expertise to formulate quadratic equations to design their first propeller?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Wright brothers developed no serious theory for designing propellers.&#xA;Wilbur Wright, in one of his notebooks, wrote the calculations you can see in the attached picture. He applied no calculus, no quadratic equations, but used a combination of empirical formulas already in existence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KqOGb.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KqOGb.jpg&quot; alt=&quot;The propeller theory of the Wright brothers&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;&lt;em&gt;This is the propeller theory of the Wright brothers (dated March 6, 1903).&lt;/em&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The only things the Wright brothers published before being seen flying, on August 8, 1908, in France, were a few articles containing pictures of their gliders and also they filed 5 patents (see the book below). None of these documents contain drawings or photos of powered flying machines with engine and propellers. The pictures, allegedly made in 1903, 1904 and 1905, showing their airplanes flying or on the ground, were published for the first time in September 1908.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gu5gX.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gu5gX.jpg&quot; alt=&quot;The Wright brothers’ patents and their low importance for aviation&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://wright1903dec17.altervista.org/Wright-patents.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&lt;em&gt;The Wright brothers’ patents (full text) and their low importance for aviation&lt;/em&gt;&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11165" LastActivityDate="2020-03-01T16:58:55.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11513" PostTypeId="5" CreationDate="2020-03-02T02:43:11.100" Score="0" Body="&lt;p&gt;This tag should be used along with &lt;a href=&quot;/questions/tagged/physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;physics&lt;/a&gt; tag.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-17T08:02:29.863" LastActivityDate="2022-03-17T08:02:29.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11514" PostTypeId="4" CreationDate="2020-03-02T02:43:11.100" Score="0" Body="For questions about Classical Mechanics - a branch of Physics involving the study of the motion of macroscopic bodies under the influence of Forces" OwnerUserId="217" LastEditorUserId="10106" LastEditDate="2022-03-17T08:02:25.803" LastActivityDate="2022-03-17T08:02:25.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11515" PostTypeId="1" AcceptedAnswerId="11516" CreationDate="2020-03-02T04:29:50.143" Score="8" ViewCount="3912" Body="&lt;p&gt;When I was originally introduced to vectors, I was told to write them with an arrow above the variable, like so: &lt;span class=&quot;math-container&quot;&gt;$$\vec{x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As soon as I began taking vector-heavy classes, I found that those teachers dropped the arrow head: &lt;span class=&quot;math-container&quot;&gt;$$\overline{x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Eventually, I also had a few professors who used underlines to represent vectors: &lt;span class=&quot;math-container&quot;&gt;$$\underline{x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Around the same time as I started seeing the underline notation, I learned that these lines are all a substitute for bold letters, which work well in print, but not in handwriting: &lt;span class=&quot;math-container&quot;&gt;$$\bf{x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While the rationale for dropping the arrow head seems apparent to me (less effort, doesn't crowd the area near the superscript), the rationale for preferring the overbar or underline is not. I use the underline notation myself, and while I have several reasons for preferring it, I seem to be in the minority. Is there a reason why both notation systems are in use at the same time?&lt;/p&gt;&#xA;" OwnerUserId="11168" LastActivityDate="2020-03-02T18:55:05.273" Title="Why do some people represent vectors with overbars while others use underlines?" Tags="|notation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11516" PostTypeId="2" ParentId="11515" CreationDate="2020-03-02T11:14:00.867" Score="9" Body="&lt;p&gt;According to &lt;a href=&quot;https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n153/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Cajori's History of Mathematical Notations, v.2, §502 (1929)&lt;/a&gt; the overbar, overarrow and boldface notations are older, which may be one reason why they are more common. &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=pst.000052363498&amp;amp;view=1up&amp;amp;seq=410&quot; rel=&quot;noreferrer&quot;&gt;Shaw's 1912 survey&lt;/a&gt; of vector notations also mentions boldface, italics, Gothics, overbar, etc., but not underbar. Judging by &lt;a href=&quot;https://academia.stackexchange.com/q/98257&quot;&gt;How to represent a tensor/matrix/vector/array in blackboard?&lt;/a&gt; thread on Academia SE, the underbar is interpreted as a handwritten equivalent of boldface, and preferred by engineers. From the accepted answer by Massimo Ortolano:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I usually employ underbarred symbols for four reasons:&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;em&gt;Of course, I'm an engineer!&lt;/em&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;A bar is faster to draw then an arrow.&lt;/em&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;My handwriting is awful, and I think that simpler symbols improve readability; at the same time, though, I don't want to abandon the categorization of quantities through different symbols.&lt;/em&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;em&gt;The interpretation of underbar is frequently that of bold, and I prefer bold symbols for vectors over symbols with an arrow.&lt;/em&gt;&quot;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Such field specific variations are not uncommon, compare to &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; in mathematics vs &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; in engineering for the imaginary unit. In more advanced mathematical texts and classes vector distinguishing marks are dropped altogether, and whether something is a vector is expected to be inferred from context. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wikipedia describes some &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_notation#History&quot; rel=&quot;noreferrer&quot;&gt;vector notation standardization efforts&lt;/a&gt; at the beginning of 20th century involving Macfarlane, Klein and Sommerfeld. Here is from Cajori:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Argand adopted for a directed line or vector the notation &lt;span class=&quot;math-container&quot;&gt;$\overline{AB}$&lt;/span&gt;. Möbius used the designation &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is the origin and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; the terminal point of the vector. This notation has met with wide adoption. It was used in Italy by G. Bellavitis, in France by J. Tannery, by E. Rouche and Ch. de Comberousse and P. Appell. On the other hand, W. R. Hamilton, P. G. Tait, and J. W. Gibbs designated vectors by small Greek letters... We have already called attention to the designation &lt;span class=&quot;math-container&quot;&gt;$a\tiny{\alpha}$&lt;/span&gt; of Frarncais and Cauchy, which is found in some elementary books of the&#xA;  present century; Frank Castle lets &lt;span class=&quot;math-container&quot;&gt;$A\scriptsize{\alpha}$&lt;/span&gt; designate a vector. Frequently an arrow is used in marking a vector, thus &lt;span class=&quot;math-container&quot;&gt;$\vec{R}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; represents the length of the vector&lt;/em&gt;. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;W. Voigt in 1896 distinguished between &quot;polar&#xA;  vectors&quot; and axial vectors&quot; (representing an axis of rotation). P. Langevin distinguishes between them by writing polar vectors &lt;span class=&quot;math-container&quot;&gt;$\vec{a}$&lt;/span&gt;,&#xA;  axial vectors &lt;span class=&quot;math-container&quot;&gt;$\overset{\scriptsize\circlearrowleft}{a}$&lt;/span&gt;. Some authors, in Europe and America, represent a&#xA;  vector by bold-faced type &lt;span class=&quot;math-container&quot;&gt;$\mathbf{a}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbf{b}$&lt;/span&gt; which is satisfactory for print, but&#xA;  inconvenient for writing. Cauchy in 1853 wrote a radius vector &lt;span class=&quot;math-container&quot;&gt;$\overline{r}$&lt;/span&gt; and its projections upon the co-ordinate axes &lt;span class=&quot;math-container&quot;&gt;$\overline{x}, \overline{y}, \overline{z}$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$\overline{r}=\overline{x}+\overline{y}+\overline{z}$&lt;/span&gt;. Schell in 1879 wrote&#xA;  &lt;span class=&quot;math-container&quot;&gt;$[AB]$&lt;/span&gt;; some others preferred &lt;span class=&quot;math-container&quot;&gt;$(AB)$&lt;/span&gt;.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-03-02T18:55:05.273" LastActivityDate="2020-03-02T18:55:05.273" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11517" PostTypeId="2" ParentId="9809" CreationDate="2020-03-02T13:21:42.110" Score="5" Body="&lt;p&gt;This story has been depicted in &lt;strong&gt;The Theory of Everything&lt;/strong&gt;, a 2014 biographical romantic drama film directed by James Marsh detailing the life of the theoretical physicist Stephen Hawking.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In an early scene, it has been shown that Stephen Hawking, who has just begun his PhD at the University of Cambridge, and other fellow students was given a problem set of 10 questions by their PhD supervisor Dennis Sciama of which &lt;strong&gt;Stephen could do &quot;only&quot; nine&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Though I could find &lt;strong&gt;no historical proof&lt;/strong&gt; for such a story and it only seems (as commented by others) to be a cooked up anecdote for the sake of publicity. Moreover, the film has been criticized for being utterly dishonest on many occasions. Writing for the film blog of UK daily newspaper The Guardian, Michelle Dean noted:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The Theory of Everything's marketing materials will tell you it is based on Jane Hawking's memoir of her marriage, a book published in the UK as Music to Move the Stars, and then re-issued as Travelling to Infinity. But the screenwriters rearranged the facts to suit certain dramatic conventions. And while that always happens in these based-on-a-true-story films, &lt;strong&gt;the scale of the departure in The Theory of Everything is unusually wide&lt;/strong&gt;. The film becomes almost dishonest–in a way that feels unfair to both parties, and oddly, particularly Jane Hawking herself.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-05-28T17:39:19.990" LastActivityDate="2020-05-28T17:39:19.990" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11518" PostTypeId="1" CreationDate="2020-03-02T21:08:14.690" Score="3" ViewCount="145" Body="&lt;p&gt;I have some questions about von Weizsäcker's views in physics, which I find them generally interesting. One of them is that he thought that time was fundamental (he even thought that logic, which is atemporal, was derived from a more fundamental temporal logic). He thought that time was a fundamental characteristic of physics and quantum mechanics in particular.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Do you know if von Weizsäcker changed his mind towards time being fundamental or if he eventually considered quantum models in which time was not fundamental?&lt;/p&gt;&#xA;" OwnerUserId="10956" OwnerDisplayName="vengaq" LastEditorUserId="55" LastEditDate="2020-03-03T00:35:34.363" LastActivityDate="2020-03-03T00:47:57.113" Title="Did Weizsäcker's change his mind about time being a fundamental object?" Tags="|quantum-mechanics|time|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11519" PostTypeId="2" ParentId="11518" CreationDate="2020-03-03T00:34:12.347" Score="2" Body="&lt;p&gt;Weizsäcker did not change his mind and continued to develop &quot;temporal logic&quot; until 1990-s (e.g. in &lt;em&gt;Zeit und Wissen&lt;/em&gt;, 1992). It is central to his general idea of the relationship between logic and physics inspired by Kant's idea that time is a precondition of experience of events/phenomena. Weizsäcker's views were quite distinct from then (and even now) common interpretations of quantum mechanics, as &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pubmed/24974602&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stöckler&lt;/a&gt; pointed out:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;His approach is not any longer present in contemporary debates. One reason is that Weizsäcker is mainly affected by classical philosophy (Plato, Aristotle, Kant). He could not esteem the philosophy of science that was developed in the spirit of logical empiricism. So he lost interest in disputes with Anglo-Saxon philosophy of quantum mechanics.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In some respects, already back in 1950-s Weizsäcker anticipated what is now called quantum information approach. He even had an analog of qubits, called &lt;em&gt;urs&lt;/em&gt;, that served as 'elements of reality', not unlike Wheeler's later &quot;it from bit&quot;, see &lt;a href=&quot;https://hsm.stackexchange.com/a/9761/55&quot;&gt;Are Wheeler's It from Bit/Participatory Universe and the Multiverse related?&lt;/a&gt; Moreover, according to Weizsäcker, the ordinary logic deals with structural, timeless (&quot;time-bridging&quot;) propositions, while physics deals with contingent ones. The temporal logic is supposed to give a more fundamental description that both of them derive from, see &lt;a href=&quot;https://link.springer.com/chapter/10.1007978-3-662-10557-3_17&quot; rel=&quot;nofollow noreferrer&quot;&gt;Görnitz and Ischebeck, Introduction to Carl Friedrich von Weizsäcker's Program for a Reconstruction of Quantum Theory&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In Komplementarität und Logik, a &quot;logic of complementarity&quot; is abstracted from&#xA;  quantum theory whose core is Birckhoff and v. Neumann's quantum logic [1]. It is&#xA;  understood as the physical quantum theory, expressed as a logical formalism... Weizsäcker went on to design&#xA;  a &quot;temporal logic&quot; which would be the common basis of logic, as the theory of&#xA;  time-bridging propositions and schematic proofs, and of physics, as the theory of&#xA;  predictions and empirical decisions... Weizsäcker&#xA;  reports on his work in logic in Aujbau der Physik [19] and Zeit und Wissen [20].&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;The papers on Komplementarität und Logic introduce the ur, the quantized binary&#xA;  alternative and quantum bit of information, as the basic concept in the reconstruction.&#xA;  The ur-hypothesis was formulated that particles and fields, as well as space can be built&#xA;  from urs. This means a construction of matter and fields out of quantum information... Weizsäcker, however, gives&#xA;  binary alternatives a basic physical significance. Physical objects should be built from&#xA;  urs. The ur-hypothesis follows the rule that mathematical objects of physical theory&#xA;  should be representative of physical objects as much as possible&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;For more details on the temporal logic see &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00262078&quot; rel=&quot;nofollow noreferrer&quot;&gt;Drieschner, Is (quantum) logic empirical?&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-03-03T00:47:57.113" LastActivityDate="2020-03-03T00:47:57.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11520" PostTypeId="1" CreationDate="2020-03-03T11:16:19.790" Score="0" ViewCount="260" Body="&lt;p&gt;Tesla, especially later in his life, was constantly monitored by the FBI. When he died, all his notes and other belongings were taken and, for a very long time, nothing was returned to his family. Has anything like this ever happened to another civilian scientist?&lt;/p&gt;&#xA;" OwnerUserId="11170" LastEditorUserId="4251" LastEditDate="2021-12-25T09:45:08.570" LastActivityDate="2021-12-25T09:45:08.570" Title="Was Tesla the only scientist whose possessions were confiscated by security services?" Tags="|biographical-details|scientists|tesla|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11521" PostTypeId="1" AcceptedAnswerId="11523" CreationDate="2020-03-03T19:46:57.997" Score="6" ViewCount="2007" Body="&lt;p&gt;The state space of a dynamical system is often called a &quot;phase space&quot;. What is the etymology of this?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Note that I'm not asking about the history of the &lt;em&gt;concept&lt;/em&gt;, but rather about the history of the &lt;em&gt;term&lt;/em&gt;. How did it come about, in which language, and why &quot;phase&quot;?)&lt;/p&gt;&#xA;" OwnerUserId="11172" LastEditorUserId="10561" LastEditDate="2020-03-05T07:11:13.653" LastActivityDate="2020-05-06T03:45:58.493" Title="What is the etymology of &quot;phase space&quot; of a dynamical system?" Tags="|terminology|classical-mechanics|thermodynamics|differential-equations|statistical-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11523" PostTypeId="2" ParentId="11521" CreationDate="2020-03-04T05:35:43.910" Score="6" Body="&lt;p&gt;Here is a direct link to &lt;a href=&quot;https://physicstoday.scitation.org/doi/10.1063/1.3397041&quot; rel=&quot;noreferrer&quot;&gt;Nolte's Tangled Tale of Phase Space on Physics Today&lt;/a&gt;. Big takeaways: the name did not come from Liouville's oft-cited 1838 paper, and Boltzmann used &amp;quot;phase&amp;quot; without &amp;quot;space&amp;quot; in the right context back in 1872, and he is the one who fully developed the concept, with a big help from Jacobi's 1842-43 work.&lt;/p&gt;&#xA;&lt;p&gt;Nolte also suggests that the combination &amp;quot;phase space&amp;quot; originates in &amp;quot;&lt;em&gt;a now obscure encyclopedia article published in 1911 that had etymological side effects not fully intended by its authors&lt;/em&gt;&amp;quot;, referring to Ehrenfests' article for the Encyclopedia of Mathematical Sciences written at Klein's request.&lt;/p&gt;&#xA;&lt;p&gt;However, as  @ConsigliereZARF pointed out in the comment, German &lt;em&gt;Phasenraum&lt;/em&gt; appears earlier in  two reviews of &lt;a href=&quot;https://archive.org/details/ElementaryPrinciplesInStatisticalMechanics/page/n25/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Gibbs's Elementary Principles in Statistical Mechanics (1902)&lt;/a&gt;. One by &lt;a href=&quot;https://books.google.com/books?id=bKRovgAACAAJ&amp;amp;source=gbs_book_other_versions_r&amp;amp;cad=2&quot; rel=&quot;noreferrer&quot;&gt;Abraham in &lt;em&gt;Physikalische Zeitschrift&lt;/em&gt;, v. 3, p. 582&lt;/a&gt;, and  the other by &lt;a href=&quot;https://books.google.com/books/about/Annalen_der_Physik_und_Chemie.html?id=BhpbAAAAYAAJ&quot; rel=&quot;noreferrer&quot;&gt;Planck in &lt;em&gt;Annalen der Physik und Chemie&lt;/em&gt;, v. 27, p.748&lt;/a&gt;. Although Gibbs himself does not quite form the combination &amp;quot;phase space&amp;quot;, he writes the following in a footnote on p.11:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If we regard a phase as represented by a point in space of &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; dimensions, the changes which take place in the course of time in our ensemble of systems will be represented by a current in such space. This current will be steady so long as the external coordinates are not varied.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He later analogizes the flow in the &amp;quot;space of &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; dimensions&amp;quot; to that of incompressible fluid. From there the shorthand of &amp;quot;phase space&amp;quot; comes naturally. In a different context, &lt;em&gt;Phasenraum&lt;/em&gt; is mentioned in passing in a short note &lt;a href=&quot;http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1895AN....139..171P&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Rauchkeilbeobachtungen von η Aquilae&lt;/em&gt; by Plassman&lt;/a&gt; back in 1895.&lt;/p&gt;&#xA;&lt;p&gt;Here is Nolte's account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In fact, in his paper Liouville makes no mention of phase space, let alone dynamical systems. Liouville’s paper is purely mathematical, on the behavior of a class of solutions to a specific kind of differential equation. Though he lived for another 44 years, he was apparently unaware of his work’s application to statistical mechanics by others  even within his lifetime... Explicitly referencing Liouville’s 1838 paper in the 1866 publication of his lectures of 1842-43, Jacobi was the first to put Liouville’s mathematical theorem into a mechanical context. What Jacobi did not do, and indeed could not do in his time, was to represent mechanical systems within a generalized space.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;In 1872 Boltzmann used the term “phase” for the first time in a paper on the further studies of the equipartition theory of gas molecules... Boltzmann made the distinction between kind of motion (Bewegungsart, such as translational and rotational motion, which contribute to the total energy) and the phase of the motion (Bewegungsphase, such as the changing coordinate and momentum values of the motion). That is the defining moment for the word “phase” in phase space... Boltzmann, for his part, did not use the term “phase” after his 1872 paper until the publication of his Vorlesungen über Gastheorie (Lectures on Gas Theory) in 1896.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Why does Liouville get the credit when it was Jacobi and Boltzmann who invented phase space and discovered conserved volumes in it? The answer is that Boltzmann himself gives Liouville the credit in his Lectures. Although Boltzmann had known, at the time of his early papers, of Jacobi’s reference to Liouville’s theorem, it was only later in his Lectures of 1896 that Boltzmann first placed Liouville’s name on the conservation theorem in a way that stuck. Had it not been for Jacobi’s reference to Liouville in his Lectures, Boltzmann likely never would have known of Liouville’s paper. In turn, had Boltzmann not given the credit to Liouville, then the conservation of phase-space volume could very reasonably have been called Boltzmann’s theorem. Ironically, by naming it “Liouville’s Theorem” Boltzmann obscured his own role in the discovery and use of phase space.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-05-06T03:45:58.493" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11524" PostTypeId="1" AcceptedAnswerId="11526" CreationDate="2020-03-04T10:43:36.430" Score="0" ViewCount="718" Body="&lt;p&gt;In the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Pierre-Simon_Laplace#Religious_opinions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Religious opinions&lt;/a&gt; section of the Wikipedia article on Laplace, there is an excerpt from Antommarchi's &quot;The Last Moments of Napoleon&quot;, in which it is written that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the name of &lt;strong&gt;God&lt;/strong&gt; appeared endlessly in the works of Lagrange.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is this true, and if yes, did he use it poetically? Can you provide some excerpts from Lagrange's work that contain the word &quot;God&quot;?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10106" LastEditDate="2020-05-16T13:59:14.517" LastActivityDate="2020-05-16T13:59:14.517" Title="Does the word &quot;God&quot; appear regularly in the works of Joseph Lagrange?" Tags="|mathematicians|biographical-details|lagrange|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11525" PostTypeId="2" ParentId="11524" CreationDate="2020-03-04T18:56:45.417" Score="3" Body="&lt;p&gt;Contrary to Antommarchi's &quot;The Last Moments of Napoleon&quot;, the text &lt;a href=&quot;https://archive.org/details/talksofnapoleona00gour/page/274/mode/2up/search/lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;Talks of Napoleon at St. Helena&lt;/a&gt; quotes Napoleon (bottom of page 274):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Napolean replies: How comes it, then, that Laplace was an atheist?  At the Institute neither he nor Monge, nor Bertollet, nor &lt;strong&gt;Lagrange&lt;/strong&gt; believe in God.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Similarly, from Morris Kline's &lt;a href=&quot;https://archive.org/details/MathematicsAndTheSearchForKnowledge/page/n113/mode/2up/search/lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics and the Search for Knowledge&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lagrange and Laplace, though of Catholic parentage, were agnostics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-03-04T18:56:45.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11526" PostTypeId="2" ParentId="11524" CreationDate="2020-03-04T21:40:42.793" Score="1" Body="&lt;p&gt;This is an anecdote, probably invented to complement the other anecdote about Laplace. It is hard to say why Lagrange in particular was picked to fill the part, as he is a bad fit. Probably, it is just that he was seen with Napoleon at Mass, and the name is suitably famous. Many Lagrange's works are translated into English, e.g. his &lt;em&gt;opus magnum&lt;/em&gt; &lt;a href=&quot;https://archive.org/details/springer_10.1007-978-94-015-8903-1/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analytical Mechanics&lt;/a&gt;, so one can do a search to see if he mentions God &quot;endlessly&quot;. In fact, he does not mention him there at all. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Instead, we find God mentioned in the translators' biographical remarks on Lagrange's religious views, which do not exactly leave an impression of excessive devoutness. To the contrary, he was so reserved in his expressions of faith that some considered him an atheist, although this is probably an exaggeration. And he definitely did not think that God can explain much in mechanics, as the anecdote suggests (with Lagrange allegedly exclaiming:&quot;&lt;em&gt;But it is a beautiful hypothesis that explains so many things&lt;/em&gt;&quot;):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It appears that he had no interest in Maupertuis’ &#xA;  attempt to harmonize the Principle of Least Action with the existence of a deity. In &#xA;  his view, it was simply a mathematical proposition... The endless discussions over this question, which appeared to lead &#xA;  nowhere, appeared meaningless to Lagrange. Hence, he undertook to describe the &#xA;  mechanical behavior of material systems without any metaphysical commentary &#xA;  at all...  Euler had written a work — Lettres a &#xA;  une Princesse d’Allemagne — in which he attempted to reconcile science and &#xA;  religion. Lagrange held that this effort was inappropriate for a man of science.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[...] &lt;em&gt;The question of what were Lagrange’s religious beliefs is difficult to answer. &#xA;  Authors who have investigated this aspect of Lagrange’s personality have not &#xA;  been able to reach a conclusion... Many of Lagrange’s colleagues were not members of the &#xA;  Christian church and therefore, Lagrange, as prudent as he was in the conduct of &#xA;  his personal affairs, kept his religious beliefs to himself. He may have viewed &#xA;  religious belief as he did metaphysical arguments, that is, with the attitude that they &#xA;  would not produce any tangible results.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Secondly, his private library contained &#xA;  a number of books on religion — not all on the Christian religion but bibles and &#xA;  works on various world religions. These books indicate an interest in religion on &#xA;  the part of Lagrange... On the other hand, in his letters to his family, where it is not necessary to be &#xA;  prudent, there are no references to religion. Even when a death in the near family &#xA;  seemed to require words of consolation such as the death of his mother, brother &#xA;  and father, there are none.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;But there was never any indication from Lagrange that he was an atheist. In fact, he attended Mass regularly and during his sojourn in Paris during &#xA;  the Empire, there is a record of his many chance meetings with Napoleon at Mass. Lagrange’s name is found in the Dictionnaire des Athees Anciens et Modernes by &#xA;  Sylvain Marechal along with the names of Napoleon, Frederick the Great, Laplace, &#xA;  Monge, de Prony, Lalande, Peyrard, Fourcroy, etc. Marechal noted that these men &#xA;  held that it was impossible to prove the existence of God. He referred to them &#xA;  as atheists but it would have been more accurate for Marechal to point out that a &#xA;  declaration of agnosticism rather than atheism was really in question.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-03-04T21:48:59.330" LastActivityDate="2020-03-04T21:48:59.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11531" PostTypeId="1" AcceptedAnswerId="11541" CreationDate="2020-03-06T04:19:26.343" Score="3" ViewCount="128" Body="&lt;p&gt;I know Gauss made it clear that he wanted an efficient solution to the problem in 1801.  In his Disquisitiones Arithmeticae he calls all factoring algorithms &quot;laborious and prolix&quot;, surely a 19th-century notion of &lt;a href=&quot;https://en.wikipedia.org/wiki/Time_complexity#Superpolynomial_time&quot; rel=&quot;nofollow noreferrer&quot;&gt;superpolynomial&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know Euclid's Elements essentially contains the fundamental theorem of arithmetic, so perhaps Euclid and ``his peers'' were already interested in the question in some more or less implicit way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know the first publication that has --- like Gauss --- made it loud and clear that the problem should be given due attention.  So, for example, the first factoring algorithm effectively does that.  The author must have studied enough to find it and, therefore, I could make the case that he saw the importance.  (Even if he found the algorithm sort by chance or indirectly.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Essentially, I'm interested in giving a date of when the problem becomes, say, pressing.&lt;/p&gt;&#xA;" OwnerUserId="11145" LastEditorUserId="55" LastEditDate="2020-03-08T03:27:39.870" LastActivityDate="2020-03-08T03:27:39.870" Title="When was the problem of factoring integers explicitly considered, what was the oldest factoring algorithm?" Tags="|number-theory|algorithm|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11532" PostTypeId="1" CreationDate="2020-03-06T12:28:05.017" Score="1" ViewCount="92" Body="&lt;p&gt;In his &lt;em&gt;Réflexions&lt;/em&gt; paper, Lagrange is using one of Cramer's results on the Elimination Theory for the proof of a &lt;a href=&quot;https://hsm.stackexchange.com/questions/11527/was-it-common-for-lagrange-to-make-his-theorems-so-general-to-the-point-of-ambig&quot;&gt;theorem&lt;/a&gt;. It seems that Cramer did some early work on the permutations of variables in connection to Elimination Theory. Is there a deeper connection between Lagrange's (and Vandermonde's) work on permutations and Cramer?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10106" LastEditDate="2020-03-07T06:16:18.237" LastActivityDate="2020-03-07T06:16:18.237" Title="How did Cramer's work on permutations influence early pre-group theory (particularly the work of Lagrange and Vandermonde)?" Tags="|biographical-details|abstract-algebra|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11533" PostTypeId="2" ParentId="1923" CreationDate="2020-03-06T15:40:44.973" Score="0" Body="&lt;p&gt;Check out the wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo%27s_Leaning_Tower_of_Pisa_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;where&lt;/a&gt;  they explain:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version which predicted that bodies of the same material falling through the same medium would fall at the same speed.&lt;/p&gt;&#xA;&lt;p&gt;This was contrary to what Aristotle had taught: that heavy objects fall faster than lighter ones, in direct proportion to their weight. While this story has been retold in popular accounts, there is no account by Galileo himself of such an experiment, and it is accepted by most historians that it was a thought experiment which did not actually take place.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't actually agree with this - its Aristotelianism that taught this; and not Aristotle; one ought to recall that Aristotle was at the very beginning of science and he had was commenting on the problem of change, and he had elucidated that change had several modes, and the predominant mode of change was motion - this was a huge advance for the time and this is why his name is justly revered; that he got details wrong is not a good reason to knock him, as physicists often do today - after all, by this basis we could say that Newtons theory of gravity was wrong in the light of Einstiens theory - but this is not the proper mode of historical assessment as it is ahistorical.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An exception is Stillman Drake, who argues that it took place, more or less as Viviani described it, as a demonstration for students.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They also remark that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Galileo arrived at his hypothesis by a famous thought experiment outlined in his book &lt;em&gt;On Motion&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This experiment runs as follows: Imagine two objects, one light and one heavier than the other one, are connected to each other by a string. Drop this system of objects from the top of a tower. If we assume heavier objects do indeed fall faster than lighter ones (and conversely, lighter objects fall slower), the string will soon pull taut as the lighter object retards the fall of the heavier object. But the system considered as a whole is heavier than the heavy object alone, and therefore should fall faster. This contradiction leads one to conclude the assumption is false.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In other words, the scholarly consensus is that he came to his hypothesis/conjecture about motion through a thought experiment based on continuity and symmetry rather than through direct experiment; and nor had he directly verified it. Given this, it seems likely that the notion that Galileo actually carried out this experiment at the leaning tower of Pisa is a type of scientific myth akin to urban myths; it really ought to have died a death by now.&lt;/p&gt;&#xA;&lt;p&gt;On a personal note, I came up with more or less the same idea just as I was finishing high school; which suggests to me other people may have had the same idea; take three balls of plasticine of the same size and hold them above the earth at the same height and at equal distances from each other - obviously they will hit the ground at the same time. Then move the middle ball a little towards the left and repeat the experiment; obviously again they will hit the ground at the same time; eventually, the middle ball will be right next to the left ball; repeating the experiment again will show that the balls again hit the surface of the earth at the same time; now squeeze the middle plasticine ball into the left - again repeating the experiment we see there is no difference; eventually the left and middle ball merge and we have a ball twice the size of the right and they both fall at the same rate.&lt;/p&gt;&#xA;&lt;p&gt;Galileo used weights and a string attached between them.&#xA;as one can see, the argument hinges on symmetry and continuity (and the observation that how something falls does not actually vary by location).&lt;/p&gt;&#xA;&lt;p&gt;Thinking about it, I think I used stones originally...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-03-10T09:37:49.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11538" PostTypeId="2" ParentId="2810" CreationDate="2020-03-06T17:01:10.033" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to Russell's paradox, which ultimately lead to the development of Zermelo-Fraenkel set theory, thus putting mathematics back on solid formal foundations.&lt;/p&gt;&#xA;&lt;p&gt;My question is, how much of a revolution did this process cause in the rest of mathematics, outside of the formal foundations?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are two things here that need to be distinguished:&lt;/p&gt;&#xA;&lt;p&gt;a. setting all of mathematics on a formal basis&lt;/p&gt;&#xA;&lt;p&gt;b. setting set theory on a formal basis&lt;/p&gt;&#xA;&lt;p&gt;It was believed that set theory solved the first; thus formalising set theory, would help formalise mathematics; my personal take on this is that this does a dis-service to the nature of mathematics, which is a human endeavour, and mistakenly understands that mathematics is merely a deductive system, when it is not; be that as it may.&lt;/p&gt;&#xA;&lt;p&gt;The discovery of Russells paradox put paid to a naive formalisation of set theory; ZFC succeeds by more or less ignoring Russells paradox; the other option is to embrace it and see that sets come in a hierarchy of types; this is type theory, and works as an alternative foundation for mathematics. In fact a naive type theory works with ZFC; where ordinary sets are what we use there, and the only larger sets are classes; this is not all abstruse, because in category Theory we actually do need larger sets than ZFC is capable of giving to us to use.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-03-06T17:01:10.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11539" PostTypeId="2" ParentId="11484" CreationDate="2020-03-06T17:23:14.117" Score="0" Body="&lt;p&gt;Well, Richard feynman, did write in his first volume in his lectures on physics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He neglected to say that such a theory had been discovered over two millenia ago, and in two places - Buddhist India and Ancient Greece. No cataclysm occured. Its merely that these were understood as the esoteric musing of so called natural philosophers, and perhaps not to be taken too seriously by the ordinary man. Thus they have led a kind of secret and underground life known only to specialists of ancient and medieval philosophy; until the rennaisance where they burst again into new life. Newton notion of a corpuscule would have been unthinkable without this prior discovery. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, these two ancient conceptions, which on the face of it are similar are also dis-similar; for example, the Greek atoms are permanent elements of reality; whereas the buddhist conception, given their notion of Pratītyasamutpāda (dependent origin or arising), is not; &lt;a href=&quot;https://en.wikipedia.org/wiki/Buddhist_atomism#:~:text=Buddhist%20atomism%20is%20a%20school,%2C%20Aristotelian%2Dstyle%20atomic%20theory.&quot; rel=&quot;nofollow noreferrer&quot;&gt;according&lt;/a&gt; to Noa Ronkin, this kind of atomism was developed in the Sarvastivada and Sautrantika schools for whom material reality can be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;reduced to discrete momentary atoms, namely, the four primary elements. These momentary atoms, through their spatial arrangement and by their concatenation with prior and posterior atoms of the same type, create the illusion of persisting things as they appear in our everyday experience. Atomic reality is thus understood first and foremost as change, though not in the sense of a thing x transforming into y. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;That is, change itself is the very nature of atomic reality rather than its being made of enduring substances the qualities of which undergo change. Atoms that appear to endure are, in fact, a series of momentary events that ascend and fall in rapid succession and in accordance with causal relations. Unlike the atoms of the Vaifesika, the atoms of the Sarvastivada-Vaibhasika and the Sautrantika are not permanent: they come into being and cease from one moment to the next going through a process of birth, continuance, decay and destruction. Yet the material compounds that consist of these atoms are real, if only in the minimal, phenomenological sense&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus the buddhist atoms, as described in popular book, the Tao of Physics, is much closer to the modern conception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, it ought to be understood that atoms aren't the whole story. Aristotle certainly didn't think so, and nor did Thales and Parmenides. One might consider this the otherside of the story, the story of continua, and the way that continua carry motion is through waves.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Again, it is only in the last century that they have been understood as two sides of the same question - the question of what constitutes ontological change, that is physical.  &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-03-06T17:23:14.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11540" PostTypeId="2" ParentId="11470" CreationDate="2020-03-07T15:55:35.847" Score="3" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Did these scientists know about each other's work and consult from each other or did they discover the electroweak theory each independent of the other?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's actually quite a common occurrence for researchers working independently to come up with solutions more or less around the same time; the most famous example is of Newton and Liebniz with the calculus, but one could also mention the Higgs mechanism, or the knot polynomial; or as here, the unified electroweak theory. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is simply because in any field the most important questions are well-known, moreover, every serious researcher knows what possible techniques can be used. For example here, Salam and Weinberg would have both seen the work on how spontaneously symmetry breaking gives mass to bosons and and how QED was the first successful quantised field theory. Moreover, they would have known of each others work, as its likely they would be publishing in the same journals and the serious theoretical physics world is actually quite a small one. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="6085" LastEditDate="2020-03-07T16:54:53.843" LastActivityDate="2020-03-07T16:54:53.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11541" PostTypeId="2" ParentId="11531" CreationDate="2020-03-07T22:48:51.960" Score="3" Body="&lt;p&gt;Trial division was used even before Pythagoreans, Euclid's number theory allowed restricting trials to primes, with the sieve of Eratosthenes identifying the primes. Al-Banna explicitly limited the size of tried factors by the square root, as did Fibonacci later. More sophisticated methods before Gauss were given by Fermat and Euler. There is no cutoff date, I am afraid, the &quot;pressure&quot; grew gradually, see &lt;a href=&quot;https://www.jstor.org/stable/3219180&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mollin, Brief History of Factoring and Primality Testing&lt;/a&gt;. &lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-03-07T22:48:51.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11542" PostTypeId="1" AcceptedAnswerId="11543" CreationDate="2020-03-08T13:00:14.577" Score="6" ViewCount="409" Body="&lt;p&gt;I read more than once a story which took place, if memory serves me well, in England, in the XIX&lt;sup&gt;th&lt;/sup&gt; century. A statistician (or a mathematician) was making computations about life expectancy (or mortality rate). Those computations involved the normal distribution and therefore the number &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. Someone asked that person what that symbol meant and the answer was that it is the ratio between the perimeter of a circle and its diameter. The person who asked then replied something like &amp;ldquo;That is surely a delusion, since the ratio between the perimeter and the diameter of a circle cannot possibly have anything to do with [whatever the other fellow was computing]&amp;rdquo;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can anyone find the whole story?&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-03-09T15:44:05.360" Title="Quotation about $\pi$ and the number of deaths" Tags="|statistics|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11543" PostTypeId="2" ParentId="11542" CreationDate="2020-03-09T00:23:48.840" Score="11" Body="&lt;p&gt;Almost. &lt;a href=&quot;http://www.dartmouth.edu/%7Ematc/MathDrama/reading/Wigner.html&quot; rel=&quot;noreferrer&quot;&gt;E P Wigner (1960), &lt;em&gt;Communications on Pure and Applied Mathematics&lt;/em&gt; &lt;strong&gt;13&lt;/strong&gt;   1–14&lt;/a&gt;&#xA;&lt;em&gt;The Unreasonable Effectiveness of Mathematics in the Natural Sciences&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. &amp;quot;How can you know that?&amp;quot; was his query. &amp;quot;And what is this symbol here?&amp;quot; &amp;quot;Oh,&amp;quot; said the statistician, &amp;quot;this is pi.&amp;quot; &amp;quot;What is that?&amp;quot; &amp;quot;The ratio of the circumference of the circle to its diameter.&amp;quot; &amp;quot;Well, now you are pushing your joke too far,&amp;quot; said the classmate, &amp;quot;surely the population has nothing to do with the circumference of the circle.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. &amp;quot;How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?&amp;quot; It has to be admitted that we have no definite evidence that there is no such theory.&lt;/p&gt;&#xA;&lt;p&gt;The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-03-09T15:44:05.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11544" PostTypeId="2" ParentId="11520" CreationDate="2020-03-09T11:08:23.137" Score="1" Body="&lt;p&gt;Andrew Weil, the mathematician was in Finland when WWII broke out and he was arrested on suspicions of spying; in fact, it was during his internment in Rouen he did the work he was most famous for and he &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Extras/Weil_prison.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;wrote&lt;/a&gt; a letter to his wife saying:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;My mathematics work is proceeding beyond my wildest hopes, and I am even a bit worried - if it's only in prison that I work so well, will I have to arrange to spend two or three months locked up every year? In the meantime, I am contemplating writing a report to the proper authorities, as follows: &quot;To the Director of Scientific Research: Having recently been in a position to discover through personal experience the considerable advantages afforded to pure and disinterested research by a stay in the establishments of the Penitentiary System, I take the liberty of, etc. etc.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Lawvere became politically active during the Vietnam war and as he &lt;a href=&quot;http://www.mat.uc.pt/~picado/lawvere/interview.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;explains&lt;/a&gt; in an interview:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Some members of the team, including myself, became active against the Vietnam&#xA;  war and later against the War Measures Act proclaimed by Trudeau. That Act,&#xA;  similar in many ways to the Patriot Act 35 years later in the US, suspended civil&#xA;  liberties under the pretext of a terrorist danger. (The alleged danger at the time&#xA;  was a Quebec group later revealed to be infiltrated by the RCMP, the Canadian&#xA;  secret police.) Twelve communist bookstores in Quebec (unrelated to the terrorists) were burned down by police; several political activists from various groups&#xA;  across Canada were incarcerated in mental hospitals, etc. etc. I publicly opposed&#xA;  the consolidation of this fascist law, both in the university senate and in public&#xA;  demonstrations. The administration of the university declared me guilty of “disruption of academic activities”. Rumors began to be circulated, for example, that&#xA;  my categorical arrow diagrams were actually plans for attacking the administration&#xA;  building. My contract was not renewed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;There is a famous case of Luzin in the former Soviet Union when he was &lt;a href=&quot;http://www.math.nsc.ru/LBRT/g2/english/ssk/case_e.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;accused&lt;/a&gt; by the Soviet scientific establishment of being:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a paltry individual who pretends to champion “pure science” but betrays the interests of science, merchandizing it to appease you former bosses—the present-day masters of faschistoid science. The Soviet community will perceive the story about Academician Luzin as another object-lesson of the fact that the adversary never lays down his arms, that he camouflages himself more skilfully, that the methods of his mimicry becomes more diverse, and that vigilance remains the most demanded trait of every Bolshevik and every Soviet citizen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I imagine that there were plenty of other cases.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-03-09T11:17:26.387" LastActivityDate="2020-03-09T11:17:26.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11545" PostTypeId="2" ParentId="11477" CreationDate="2020-03-09T16:49:09.877" Score="1" Body="&lt;p&gt;If you are looking for a books exploring similar territory to Sabine Hossenfelders book, you can try looking at&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Lee Smolin, &lt;em&gt;The Trouble with Physics&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;David Lindleys, &lt;em&gt;The End of Physics: The Myth of a Unified Theory&lt;/em&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Peter Woit, &lt;em&gt;It's not Even Wrong&lt;/em&gt;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;If you are looking for a more positive spin on the physics, try&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Rickles, &lt;em&gt;A Brief History of String Theory&lt;/em&gt; (although it is quite long)&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Brian Greenes, &lt;em&gt;The Elegant Universe&lt;/em&gt; &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Both of these are quite dense - as far as popular science goes - but they do not use any mathematics; one that is quite mathematical, in that it actually has maths in it - but is still written for the lay mathematician; is quite short, and has plenty of illustrations in it, is Sossinskys &lt;em&gt;Knot Theory&lt;/em&gt;. In many ways, it's my favourite; too bad, more people do not know how to write well and with dash and panache; it's short, but it packs a great deal of punch.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is also another book which fits exactly, I think, what you are looking for; its sober and academic without actually being academic in the 'textbook' sense; moreover, it's not given to bursts of either undue optimism or pessimism (although it leans on the optimistic side) its called &lt;em&gt;On Spacetime&lt;/em&gt; and is a book of essays edited by Shan Majid and features essays by Andrew Taylor on Dark energy and dark matter; Penrose (on causality), Connes (on non-Commutative geometry) and Shann Majid on Quantum groups - moreover it has Polkinghorne and Heller discussing larger issues of metaphysics (which may or may not be to your taste - but they are ineteresting essays nonetheless); I recommend it highly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I found all these books in my local bookshop (depending where I was living at the time); if you have a good one nearby then I would suggest browsing the shelves to see what they have; there is a great deal more good science writing today, than there was when Kline was writing thirty years ago ...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-03-14T22:26:40.910" LastActivityDate="2020-03-14T22:26:40.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11548" PostTypeId="1" CreationDate="2020-03-11T16:09:31.493" Score="2" ViewCount="67" Body="&lt;p&gt;I guess that every International Congress of Mathematics (Mathematicians) brings up some unresolved issues of various types but that also that at least some (if not all) ICM´s present some new unsolved problems (conjectures).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there some comprehensive monograph (or handbook) about almost all the open problems (conjectures) mentioned (and proposed) at all the ICM´s?&lt;/p&gt;&#xA;" OwnerDisplayName="user11203" LastEditorDisplayName="user11203" LastEditDate="2020-03-11T16:15:14.700" LastActivityDate="2020-03-11T16:15:14.700" Title="Conjectures mentioned (and proposed) at International Congresses of Mathematics" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11549" PostTypeId="1" AcceptedAnswerId="11551" CreationDate="2020-03-12T13:23:05.047" Score="1" ViewCount="78" Body="&lt;p&gt;There is a oft-repeated story (&lt;a href=&quot;https://en.wikipedia.org/wiki/Emilio_Segr%C3%A8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Emilio_Segr%C3%A8&lt;/a&gt;) about Lawrence, upon discovering that Segre was legally trapped in the USA, unable to return to Italy due to fascist racial policies, reduced Segre's monthly salary to 116 bucks. Even in that time, I think this was not a lot of money, under 2000 dollars/month today and Segre had already discovered an element and would eventually be a Nobelist in physics. I have some sense from reading what rents and food costs were like and I suspect this would have been a very big hardship for Segre, just as living on that kind of money would be today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is, is there further information about this apparently pretty terrible behavior? Did Lawrence eventually increase this amount? Was he criticized by others in academia and are there other instances of this sort of behavior by Lawrence?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2020-03-12T23:00:24.193" Title="Lawrence and Segre: Further information about relationship" Tags="|physics|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11550" PostTypeId="1" CreationDate="2020-03-12T17:52:08.227" Score="6" ViewCount="376" Body="&lt;p&gt;I was looking at how did von Neumann and Birkhoff formulate their Quantum Logic formalism back in 1936. To solve some questions, I contacted via email a philosopher who studied this topic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I thought that Von Neumann and Birkhoff's original formalism was strongly related to projective geometry (and that basically, Quantum Logic was fundamentally based on projective geometry, as they said in their original paper: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hence, we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I also thought that Von Neumann disliked the Hilbert space formulation of quantum mechanics before publishing his Quantum Logic with Birkhoff, since in a letter to Birkhoff in 1935, Von Neumann said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;— John von Neumann, letter to Garrett Birkhoff, 1935&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, in 1954, almost two decades after his original paper with Birkhoff, he still studied and considered the non-boolean/non-commutative/non-distributive characteristics of quantum logic, since according to Jeffrey Bub: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;They (probabilities) are “uniquely given from the start” as a feature of the non-Boolean structure, to quote von Neumann,[10] related to the angles in Hilbert space, not measures over states as they are in a classical or Boolean theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[10] John von Neumann, “Unsolved problems in mathematics,” an address to the International Mathematical Congress, Amsterdam, September 2, 1954.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;The thing is that the philosopher I contacted with contradicted all of this. He said:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;In the strict sense, quantum logic is a non-distributive consequence relation, &lt;strong&gt;not a geometry or an algebra&lt;/strong&gt;. In this sense of quantum logic, von Neumann abandoned his interest in it (&lt;strong&gt;and in the Hilbert state formulation&lt;/strong&gt; of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have a few questions about this:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is this philosopher right? Or on the contrary, Von Neumann and Birkhoff's quantum logic is fundamentally based in an abstract projective geometry?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Didn't Von Neumann already dislike or abandon the Hilbert space formalism &lt;em&gt;before&lt;/em&gt; 1936 (according to the letter that von Neumann sent to Birkhoff in 1935)?&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10956" LastEditorUserId="10956" LastEditDate="2020-03-13T23:25:09.253" LastActivityDate="2020-03-22T10:36:28.687" Title="Was Von Neumann and Birkhoff's original formulation of Quantum Logic related with projective geometry?" Tags="|quantum-mechanics|theoretical-physics|mathematical-logic|mathematical-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11551" PostTypeId="2" ParentId="11549" CreationDate="2020-03-12T23:00:24.193" Score="4" Body="&lt;p&gt;No, Lawrence did not increase the amount, but in 1940 Segre got a contract directly from University of California that brought his monthly salary back to $300. Below are details. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Segre's memoirs &quot;A Mind Always in Motion: the Autobiography of Emilio Segrè.&quot; Berkeley, California: University of California Press, 1993.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In 1939, \$300 a month was a good salary, and it got me out of the woods for some time, but after six months, in July 1939, Lawrence, who by then must have realized my situation, asked me if I could return to Palermo. I answered by telling him the truth, and he immediately interjected: &quot;But then why should I pay you \$300 per month? From now on I will give you \$116.&quot; I was stunned, and even now, so many years afterward, I marvel at Lawrence's impulsiveness; he did not think for a second of the impression he conveyed. With a minimum of reflection and diplomacy, he could have saved his \$184 a month without cutting a horrible figure in my eyes. However, although I have not forgotten his conduct, I now see it in a different light than I did then. I did not know what the salaries of other members of the Rad Lab were at the time. It was I who, with \$300 a month, was the exception, and salaries around \$116 were not rare. If I had known this and how Lawrence behaved with Americans who were unemployed because of the depression, I might have viewed the episode somewhat differently. I was somewhat older and better established as a scientist than some excellent Americans were, but they too got meager salaries, although they had academic positions and guaranteed careers. In any case, with a wife and a child, a salary of \$116 was scant, but it was not totally impossible to live on it. With \$200 a month I could make ends meet, sparingly, but without deprivation, and I could bring my salary to that level by using my private funds, the existence of which was known only to Elfriede and to my friend Emo, whose discretion was absolute.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the summer of 1940, we were visited at Berkeley by the head of the physics department at Purdue University, Karl Lark-Horovitz. He was of Austrian origin, of a touchy and sullen character that made him many enemies, but at heart a very decent man and a good physicist. At Purdue he had created an excellent physics department and ruled it autocratically, but with good results. He recognized quality in science and helped whoever he thought merited it. When I became acquainted with him, I felt I could openly tell him my personal problems. He invited me to go to Purdue for a limited period. He could not secure a permanent job for me, but thought that even a temporary appointment would greatly improve my position at Berkeley. Thus in the fall of 1940, Elfriede, Claudio, and I took the train for West Lafayette, Indiana. At Purdue, we settled in at Union Hall, which was strongly heated and &#xA;  extremely dry, so much that in walking on its insulating carpets, one got highly electrified, and I could amuse Claudio by pulling sparks from his nose. I had also prepared some experiments I could easily and rapidly carry out, and I did my best to elicit appreciation for my lectures. Lark Horovitz's plan worked to a fault. Things went well, and I was pleased.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;At Berkeley, when they perceived that I was really leaving, Lawrence and the head of the physics department, Raymond T. Birge, found the money to pay me and wired me with the offer of a lecturership. It was not a tenured position, but it was great progress. I had, more or less, returned to the position with which I had started in 1938, before they knew I could not return to Italy. I asked for a contract with the University of California, not with the Rad Lab, of which I had had enough. I knew that Birge kept his word, whereas Lawrence's intentions and capabilities could change at any time. In any case I would have access to the cyclotron, and I hoped that relations with Lawrence would improve, because having good work done by me in the Rad Lab at no expense to himself, he would be happy and sweet, as indeed happened. Thanks to Lark-Horovitz, I had \$300 a month and a much improved position.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2020-03-12T23:00:24.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11552" PostTypeId="1" CreationDate="2020-03-12T23:51:30.930" Score="8" ViewCount="155" Body="&lt;p&gt;In my theory computation class, I was told that early computer pioneers didn't realize that some problems are intrinsically hard—what we now call &lt;strong&gt;NP-hard&lt;/strong&gt; problems. Instead, it took a while to realize that some problems could be solved with better programming, and others... well, they couldn't.&lt;/p&gt;&#xA;&lt;p&gt;For example, early sort programs were relatively slow, because they were limited by both available memory and algorithmic issues. I read the sort algorithm for the Univac and it's a merge-sort algorithm that uses multiple tape drives to sort more data than will fit in the computer's memory. As memories got larger, merge sort and heapsort became dominant. Then Hoare invented Quicksort in 1959, and people were really surprised, both because of its elegance and its efficiency.&lt;/p&gt;&#xA;&lt;p&gt;Other problems didn't get a fantastic speedup—for example, scheduling. We now know that many of these are &lt;strong&gt;NP-complete&lt;/strong&gt;, and possibly even harder than NP-hard. (PSPACE problems might be harder than NP problems, but then again, they might not be.)&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that NP-hard predates NP-complete, and I recently found this lovely article by Johnson, &lt;a href=&quot;https://www.math.uni-bielefeld.de/documenta/vol-ismp/50_johnson-david.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;A Brief History of NP-Completeness 1954-2012&amp;quot;&lt;/a&gt;, which traces the history of the concept.&lt;/p&gt;&#xA;&lt;p&gt;What I would like to know is when people working in the field did realize that some problems were intrinsically hard and that they weren't going to get dramatically faster solutions with simply more clever programming (absent a significant mathematical breakthrough). After all, linear programming was invented in World War II, and so with that, my guess is that many problems that seemed hard also seemed solvable. So, when did people discover that some problems are not?&lt;/p&gt;&#xA;" OwnerUserId="6112" LastEditorUserId="4251" LastEditDate="2020-07-10T01:24:02.400" LastActivityDate="2020-07-10T01:24:02.400" Title="When did computer pioneers realize that some problems are intrinsically hard?" Tags="|computer-science|computers|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11553" PostTypeId="1" CreationDate="2020-03-13T16:40:02.400" Score="4" ViewCount="262" Body="&lt;p&gt;Camille Jordan's famous canonical form for matrices over algebraically closed fields, is considered an important result nowadays, commonly taught to all students of mathematics in undergraduate linear algebra courses. The vast majority of them only get to see it applied in one case: in the study of linear systems of differential equations with constant coefficients, where it is used very effectively to solve all such systems. Other applications that are sometimes seen are its use in systems of linear recurrence relations, and in Markov chains, but that's about it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, originally, Jordan did not discover his form to solve any of these problems - it was, in fact, a problem in Galois theory that motivated his research. Given that I cannot read french, and Jordan's &lt;em&gt;Traité&lt;/em&gt; was never translated to english, I give the only english reference to it that I could find, from Hawkins' &quot;The Mathematics of Frobenius in Context&quot; (2013), pages 137-138:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Jordan... preferred to make the consideration of homogeneous linear substitutions fundamental, and in a paper of 1867, he indicated their important role in the problem of determining all the irreducible equations of a given degree that are solvable by radicals. In connection with this problem he sought, in a paper of 1868, to determine the solvable subgroups of the group of linear substitutions in two variables...  To do it he used the fact that by a linear change of variables, a linear substitution S could be put in one of a limited number of “canonical forms” depending on the nature of the roots of det(S−kI) ≡ 0 (mod p). His method of constructing solvable subgroups was to build them up from their composition series, and this involved determining all linear substitutions that commute with a given substitution S. To this end, he introduced the possible canonical forms for S.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Hawkins then mentions that Jordan further generalized his theorem to linear substitutions in n variables,  and that a few years later he realized the relevance of his theorem to systems of differential equations. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Two points are still unclear to me. First, from this summary it seems that Jordan was interested in matrices (here termed linear substitutions) over finite fields - given that all the work is done mod p. Finite fields, however, are never algebraically complete - so which theorem did Jordan even prove? The usual theorem taught is not applicable here. Second, this summary explains what is the problem that Jordan was interested in, but only sketches very roughly how he solved it. How exactly is this classification of matrices (up to similarity) relevant for determining which irredicuble polynomials are solvable?&lt;/p&gt;&#xA;" OwnerUserId="11209" LastEditorUserId="11209" LastEditDate="2020-03-13T16:58:11.743" LastActivityDate="2021-01-24T12:03:01.970" Title="Why did Jordan introduce his canonical form?" Tags="|mathematics|abstract-algebra|linear-algebra|galois-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11554" PostTypeId="2" ParentId="3768" CreationDate="2020-03-13T22:48:47.460" Score="4" Body="&lt;p&gt;A contibution to a discussion that ended nearly four years may appear quixotic but it is difficult to let stand an answer which, though accurately summing up Boyer's views on the subject, is completely out of date and misleading. The state of the question of the origin and evolution of fractions still today is to be found in the acts of an international conference on the question, published as: &lt;em&gt;Histoire de fractions, fractions d'histoire&lt;/em&gt; (Basel: Birkhäuser) 1992 — unfortunately very little has been done in the field since. The specific question of Ancient Egyptian fractions is treated in the editors' introduction, &quot;The earliest fractions&quot; (pp. 3-15) and in the chapter by James Ritter, &quot;Metrology and the prehistory of fractions&quot; (pp. 19-34). (Note that though the bulk of the book is in French, these two chapters are in English).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Summing up rapidly, the earliest Egyptian fractions date back to the earliest period of Egyptian writing and state formation (1st Dynasty, roughly 3000 BCE). They appear first as elements of particular systems of weights and measures but later are integrated into a system of abstract numbers used in doing mathematics. They are indeed &quot;unit fractions&quot; (except for 2/3) but they did not combine to make up our rational fractions p/q; rather, they were conceptualized as unique values, specific parts of a whole, and were written, for the most part, as inverses of integers (a dot placed above the integer), the exceptions being 1/2, 1/3, (2/3), and 1/4 which had special signs, with no relation to the integers 2, 3 and 4. (Remember that Egyptians did not normally write in hieroglyphics but in a cursive form called hieratic; hieroglyphs were reserved for the most part for monumental inscriptions) By the way, this way of writing fractions was adopted by the Ancient Greeks (see &quot;The earliest fractions&quot;, cited above, pp. 12-15).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The rest of Boyer's presentation of Egyptian calculations is also wrong but since it is irrelevant to the question of fractions, no point would be served by correcting it here.&lt;/p&gt;&#xA;" OwnerUserId="11210" LastActivityDate="2020-03-13T22:48:47.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11555" PostTypeId="2" ParentId="629" CreationDate="2020-03-14T11:55:39.470" Score="6" Body="&lt;p&gt;It is true that after 1914 (and the &quot;Foundations of General Relativity&quot; article), Einstein only published a few articles, exclusively replies to criticism, in the &lt;em&gt;Annalen der Physik&lt;/em&gt;. The major reason why he shifted away from that journal was simply because he was elected in 1914 to membership in the Prussian Academy of Science. There he could publish automatically (without peer review) anything he presented to the Academy; moreover the journal, the &lt;em&gt;Sitzungsberichte der Preussichen Akademie der Wissenschaften&lt;/em&gt;, appeared every week so that the material would appear almost as quickly as it was written. This journal became his main  publishing choice until the arrival of Hitler in 1933 and Einstein's exile and ejection from the Academy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another reason might be also that with his work on unified theories, starting in 1919, his public became more and more mathematical. German mathematicians read their equivalent of &lt;em&gt;Annalen&lt;/em&gt;, which was the &lt;em&gt;Mathematische Annalen&lt;/em&gt; journal. Indeed Einstein became one of the editors of the latter journal just after the First World War. While &lt;em&gt;Annalen der Physik&lt;/em&gt; became the major journal for longish articles in quantum physics, work in general relativity and in unified theories were often published in mathematical journals or in the general &lt;em&gt;Sitzungsberichte&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question has been treated in some detail in Catherine Goldstein and Jim Ritter, &quot;The Varieties of Unity: Soundings in Unified Theories 1920-1930&quot; in Abhay Ashtekar et al. (eds.), &lt;em&gt;Revisiting the Foundations of Relativistic Physics&lt;/em&gt;. (Dordrecht: Kluwer) 2003, pp. 93-149.&lt;/p&gt;&#xA;" OwnerUserId="11210" LastActivityDate="2020-03-14T11:55:39.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11556" PostTypeId="1" AcceptedAnswerId="11557" CreationDate="2020-03-14T20:21:44.733" Score="10" ViewCount="375" Body="&lt;p&gt;I am looking for the first reference of a Nondeterministic Turing machine. There is no mention of them in the 1936 paper &lt;a href=&quot;https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf&quot; rel=&quot;noreferrer&quot;&gt;&quot;On Computable Numbers.&quot;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6112" LastActivityDate="2020-03-14T21:38:54.867" Title="What is the first reference to a nondeterministic Turing machine?" Tags="|computer-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11557" PostTypeId="2" ParentId="11556" CreationDate="2020-03-14T21:38:54.867" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://plato.stanford.edu/entries/turing-machine/#VariTuriMach&quot; rel=&quot;noreferrer&quot;&gt;According to SEP&lt;/a&gt;, non-determinism was introduced by &lt;a href=&quot;http://www.cse.chalmers.se/~coquand/AUTOMATA/rs.pdf&quot; rel=&quot;noreferrer&quot;&gt;Rabin and Scott in Finite Automata and their Decision Problems (IBM Journal of Research and Development, 3(2) 114–125, 1959)&lt;/a&gt;. Their main result was that it does not lead to a stronger notion of computability:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The automata used throughout Chapter I were strictly&#xA;  deterministic in their tape-reading action, which was&#xA;  uniquely determined by the table of moves, since there&#xA;  was one and only one way the machine would change its&#xA;  state in any particular situation. Requiring all machines to&#xA;  be of this form can lead to rather cumbersome details, in&#xA;  view of the large number of internal states needed even&#xA;  for some relatively elementary operations. In this section&#xA;  we introduce the notion of a nondeterministic automaton&#xA;  and show that any set of tapes defined by such a&#xA;  machine could also be defined by an ordinary automaton.&#xA;  The main advantage of these machines is the small&#xA;  number of internal states that they require in many&#xA;  cases and the ease in which specific machines can be described.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-03-14T21:38:54.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11558" PostTypeId="2" ParentId="11550" CreationDate="2020-03-15T18:04:32.387" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;In the strict sense, quantum logic is a non-distributive consequence relation, not a geometry or an algebra. In this sense of quantum logic, von Neumann abandoned his interest in it (and in the Hilbert state formulation of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'd say that your philosopher hasn't really understood what von Neumann &amp;amp; Birkhoff were aiming at; their notion of a Quantum Logic was merely a preliminary attempt to ask questions on the foundations of classical physics; the physics of which - before quantum mechanics - were understood to hold without exception.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its undoubtedly this that gave a strong motivation and impetus to investigate non-classical logics and geometry; that this has been a fruitful line of inquiry one only needs to look at how toposes, which can be considered as a non-classical set theory with an underlying intuitionistic logic and are moreover geometric have shown their usefulness in physics, and moreover in quantum physics; for example, take the reformulation of the Kochen-Specker statement in the foundations of quantum mechanics: the spectral presheaf of the algebra of bounded operators has no global elements when the dimension of the underlying Hilbert space is greater than 2.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;I also thought that Von Neumann disliked the Hilbert space formulation of quantum mechanics before publishing his Quantum Logic with Birkhoff, since in a letter to Birkhoff in 1935, Von Neumann said:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&quot;I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;First, a couple of points; one should realise that von Neumann, was primarily a mathematician rather than a physicist, thus his confession above; secondly, it was von Neumann himself that had promoted the notion of a Hilbert space in his book-length treatment in axiomatising Quantum Mechanics. Why Von Neumann thought to be 'immoral', I don't know; most likely by then, he understood that &lt;em&gt;his&lt;/em&gt; formalism of the Hilbert space was not sufficient for the purposes of QM, never mind QFT? But then it would have been 'immoral' to keep quiet; to make a confession would then be moral, especially if one was the main protagonist in pushing forward this concept.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Paul Dirac of course made no such confession; to him, mathematics was a tool used in order to discover more physics; this is why in QM physicists use his bra-ket notation - which superficially looks like the inner product of Hilbert spaces but is not - because it works - we can't always wait for mathematicians to play catch up (in fact, they did catch up - but much later, after the discovery of distributions).&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;.. von Neumann abandoned his interest in it (and in the Hilbert state formulation of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).and shifted it to type II factor algebras (which we now call von Neumann algebras).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;These are merely particular types of C*-algebras - those that have a predual (and in fact are also called W*-algebras); and since the interest in C*-algebras primarily arose from their use in Quantum Physics, it seems merely that von Neumann was still investigating the foundations of QM, but not as a physicist, but as a mathematician would by looking at their 'prime' factors.      &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-03-22T10:36:28.687" LastActivityDate="2020-03-22T10:36:28.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11559" PostTypeId="2" ParentId="3409" CreationDate="2020-03-16T18:53:31.773" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Conservation of information is a term with a short history. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is more or less true and it's good to see that someone has actually come out and said this given the hype surrounding 'information' these days. Until someone comes out and says what actually constitutes information - physically speaking - its hard to say how and why it is conserved; in fact, one could rephrase the above and say:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Information is a term with a short history&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This does not mean that the notion of 'information' has no meaning at all to it; in physics, for example, it's a concept that is heavily promoted by Susskind and he does actually give a remarkable argument for black hole entropy based on this which is outlined in his popular book, &lt;em&gt;The Black Hole Wars&lt;/em&gt;; given that we are discussing entropy, it is perhaps not surprising the notion of information has turned up here - nevertheless, here, it a measure of all the possible microstates consistent with the macrostate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The notion of 'information' shouldn't be confused with other notions that appear to deal with 'information' in some not quite precisely defined ways; Unitarity, for example, is conservation of probability and &lt;strong&gt;not&lt;/strong&gt; of information.  &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-03-16T18:53:31.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11560" PostTypeId="1" CreationDate="2020-03-17T23:15:13.500" Score="1" ViewCount="274" Body="&lt;p&gt;A very common expression I see in pop science is &quot;the spacetime continuum&quot;. This expression isn't commonly used in modern discussions of general relativity, but looking at &lt;a href=&quot;https://sjcrothers.plasmaresources.com/Weyl-2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;some&lt;/a&gt; &lt;a href=&quot;http://adsabs.harvard.edu/full/1923JRASC..17..387E&quot; rel=&quot;nofollow noreferrer&quot;&gt;older&lt;/a&gt; papers on the topic, it does seem to pop up occasionally back then.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a formal definition of what a continuum is? Is it some terminology for a manifold of the era, or a more general space?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2020-09-03T03:29:43.480" Title="What is a spacetime continuum?" Tags="|terminology|relativity-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11562" PostTypeId="1" CreationDate="2020-03-20T18:36:50.993" Score="2" ViewCount="79" Body="&lt;p&gt;Although, it seems very easy to find the magnetic field produced by a solenoid or a toroid, all we got to do is to make a suitable an &lt;strong&gt;Amperian Loop&lt;/strong&gt; and take the &lt;span class=&quot;math-container&quot;&gt;$\mathbf B$&lt;/span&gt; out of the integral and so on.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the main step is to find the direction of magnetic field. For example see these derivations of Griffiths:  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fgt1b.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toroid 1&lt;/a&gt;.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AjDUI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;Toroid 2&lt;/a&gt;  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to know who actually drived those equations, who actually worked out for the directions of magnetic field (Griffiths has done his)?  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I searched for magnetic field by solenoid and toroid but found nothing historical. I aimed at finding something from Wikipedia’s “Refernces” or “External Links”. I read somewhere that Wilhelm Eduard Weber did work out the field for some geometrical shapes.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I request for some refernces where I can find the actual derivations of magnetic field produced by solenoid and toroid. &lt;/p&gt;&#xA;" OwnerUserId="9837" LastActivityDate="2020-03-20T18:36:50.993" Title="Who originally worked out the magnetic field produced by a solenoid and toroid?" Tags="|physics|electromagnetism|mathematical-physics|magnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11563" PostTypeId="1" CreationDate="2020-03-20T19:24:08.063" Score="3" ViewCount="525" Body="&lt;p&gt;The general quintic equation cannot be solved by radicals and is shown in a landmark and far reaching work of Galois from 1832, which became a template of modern Group theory and Galois theory. However, the general quintic equation can be reduced to a Bring radical form (using the Tschirnhaus transformation), and this form of the quintic can somehow be solved using ideas from elliptic functions theory. I'm really not familiar with these materials, and that is why I am asking this question.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to what I have read, Hermite based his construction on results by Jacobi and remarks made by Galois himself (in Galois' last paper) - these results are concerned with the so-called &quot;transformation problem of elliptic integrals&quot;; an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th order transformation of elliptic integral leads to a modular equation which is actually an &lt;span class=&quot;math-container&quot;&gt;$(n+1)$&lt;/span&gt;th degree polynomial equation in two variables, and these variables are somehow connected to the elliptic integral. Hermite based his proof on the &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt;th order transformation of Jacobi.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For the completeness of historical discussion, according to p.4 of the book &quot;Hessian Polyhedra, Invariant Theory And Appell Hypergeometric Functions&quot;, transformations of order 3,5 and 7 had been known to Gauss since 1808, and according to another source Gauss also made some very significant remarks on the problem of transformation of any odd order.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My questions are therefore both explanatory and historical:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What is the &quot;transformation problem of elliptic integrals&quot; and how is it related to the solution of polynomial equations? I want to understand a bit more about the meaning of Hermite's achievement.&lt;/li&gt;&#xA;&lt;li&gt;I just want to know if anyone can help locate these transformations in Gauss' Nachlass. I think it's somewhere in the section on elliptic functions in volume 3 of his work, but I'm completely unable to recognize familiar patterns in the flood of formulae in those writings.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10106" LastEditDate="2020-03-21T09:12:56.300" LastActivityDate="2020-03-21T23:28:35.647" Title="Question about Hermite's 1858 solution to the quintic equation using elliptic modular functions and it's relation to Gauss' and Jacobi's work" Tags="|biographical-details|gauss|galois-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11564" PostTypeId="2" ParentId="11563" CreationDate="2020-03-20T22:37:18.913" Score="4" Body="&lt;p&gt;In general, transformation of elliptic integrals (or differentials) is&#xA;finding algebraic solutions &lt;span class=&quot;math-container&quot;&gt;$F(x,y)=0$&lt;/span&gt; of a differential equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{dx}{\sqrt{f(x)}}=\frac{dy}{\sqrt{g(y)}},$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are polynomials of degree 3.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First such transformation was discovered by Landen in 1775, and it is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Landen%27s_transformation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Landen's transformation&lt;/a&gt;. Independently it was discovered by Gauss 1790 when he studied&#xA;the arithmetic-geometric mean (previously studied by Lagrange in 1785).&#xA;But as usual, Gauss gets all credit for anything he touched.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The theory of transformations leads to certain algebraic equations called&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_modular_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;classical modular equations&lt;/a&gt; which Hermite used to solve the quintic. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can read the details in Klein's book Lectures on Icosahedron and solution of 5-th degree equation. For a modern exposition, you can look to the book of Jonathan and Peter Borwein, Pi and AGM.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Original papers: Hermite C. R. 46 (1858) 508-515.&#xA;Kronecker (a simplified proof): C. R. 46 (1858) 1150-1152.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Generalization to equations of arbitrary degree: H. Umemura, Solving algebraic equations with theta-constants, Appendix I to the book of D. Mumford, Tata lectures on Theta, 1983. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-03-21T23:28:35.647" LastActivityDate="2020-03-21T23:28:35.647" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11565" PostTypeId="1" CreationDate="2020-03-20T22:50:06.263" Score="4" ViewCount="231" Body="&lt;p&gt;Although I do not doubt in Riemann˙s originality, I would like to know how much complex analysis was developed up to the day when Riemann conjectured what is today called Riemann hypothesis and how much itself was the theory of zeta-functions developed up to that day?&lt;/p&gt;&#xA;" OwnerDisplayName="user11203" LastEditorUserId="10106" LastEditDate="2022-08-17T12:32:46.807" LastActivityDate="2024-03-01T03:12:00.817" Title="How much ground was prepared for Riemann so that he could conjecture Riemann hypothesis?" Tags="|mathematics|mathematicians|complex-analysis|riemann|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11566" PostTypeId="1" CreationDate="2020-03-21T13:56:24.963" Score="5" ViewCount="150" Body="&lt;p&gt;The Arabic numerals I am referring to are “1234567890”. I have read that Thai numerals, “๑๒๓๔๕๖๗๘๙๐”, are actually distantly related. Both descend from the numeral system invented by the Phoenicians, which then developed into separate branches. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is there a flow chart to show how the Thai numerals ended up as such? Why do both systems look so vastly different, when they are supposed to be based on the same “shape” given by the original Phoenician numerals?&lt;/p&gt;&#xA;" OwnerUserId="11236" LastActivityDate="2020-03-21T13:56:24.963" Title="Why do Thai numerals look so different than Arabic numerals?" Tags="|mathematics|discoveries|notation|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11567" PostTypeId="1" CreationDate="2020-03-21T14:45:01.473" Score="3" ViewCount="80" Body="&lt;p&gt;I am looking for any documented, historical reference to an oft-cited humorous reply by German astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Walter_Baade&quot; rel=&quot;nofollow noreferrer&quot;&gt;Walter Baade&lt;/a&gt; (1893 – 1960).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After having asked if other astronomers were familliar with the quote below in  &lt;a href=&quot;https://astronomy.stackexchange.com/q/35516/11806&quot;&gt;astronomy SE&lt;/a&gt;, I managed to contact the person I had heard it from and they sent me the following which I have posted there as an answer:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Baade was asked:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If you had your life to live over, would you be an astronomer again?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;To which he answered:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Only if the ratio of total to selective absorption is everywhere the same.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which was meant to be a humorous reply. Comments there point to the web page &lt;a href=&quot;https://pages.astronomy.ua.edu/keel/galaxies/dust.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dust in Galaxies&lt;/a&gt; that also recounts the quote but with a slightly different wording but alas does also not provide a source. A comment there mentioned this SE community, so I came here to ask a related but different question:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there any historical records or published accounts of the above exchange taking place?&lt;/p&gt;&#xA;" OwnerUserId="11237" LastEditorUserId="3869" LastEditDate="2020-03-21T15:39:47.323" LastActivityDate="2020-03-21T15:39:47.323" Title="Historical accounts of a oft-cited quip by astronomer Walter Baade" Tags="|physics|astronomy|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11568" PostTypeId="1" CreationDate="2020-03-21T22:43:25.720" Score="5" ViewCount="661" Body="&lt;p&gt;According to Subhash C. Kak:  Early Theories on the Distance to the Sun: “Pancavimsa Brahmana states that the heavens are 1000 earth diameters, de,  away from the earth. The sun was also taken to be halfway to the heavens, so this suggests a distance of the sun, Rs, about 500 earth diameters from the earth. The confirmation for this supposition comes from the later theories (c. 500 AD) from the same region and from Greek ideas that speak of roughly the same distance.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In contrast to Hipparchus and Ptolemy’s accounts, I assume that the Veda scriptures do not mention any measurement approach. However, given the errors – se below - we must assume prevailed in the measurements of the e.g. Sun-Moon-Earth right angle triangle at half Moon  makes one wonder if the distance  to the Sun (in Earth units) was perhaps fitted to earlier references such as the one above, assuming there were exchanges of information perhaps via Babylonian astronomers. Is there any account of the actual measurement available?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Nt8Ck.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Nt8Ck.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From Erik Høg:  Astrometric accuracy during the past 2000 years, Niels Bohr Institute, Copenhagen, &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1707/1707.01020.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/ftp/arxiv/papers/1707/1707.01020.pdf&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5602" LastEditorUserId="26" LastEditDate="2021-02-20T15:16:24.320" LastActivityDate="2021-02-20T15:16:24.320" Title="Did Ptolemy and other Greek scientists actually measure the distance to the Sun?" Tags="|astronomy|ancient-greece|experimental-physics|ptolemy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11569" PostTypeId="1" CreationDate="2020-03-21T22:53:24.273" Score="4" ViewCount="121" Body="&lt;p&gt;In the theory of elliptic integrals, one encounters the terms &quot;amplitude&quot; and &quot;modular angle&quot; in relation to incomplete integrals of the first kind, which are two variables that denote the upper limit of the elliptic integral (the amplitude) and a certain parameter of the integral (the modular angle). I guess the naming of the term &quot;amplitude&quot; is remniscent of the origins elliptic integrals have in physical problems- in particular, the problem of arbitrary amplitude pendulums (not under the restriction of small angles). According to this interpretation, the amplitude of an incomplete elliptic integral of the first kind corresponds directly to the amplitude of the corresponding pendulum oscillation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem is that this interpretation doesn't fit the formulas; according to several sources, the amplitude of the elliptic integral that calculates the period of a pendulum is &lt;span class=&quot;math-container&quot;&gt;$\pi/2$&lt;/span&gt; regardless of the pendulum's amplitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In addition, i've no guess about the naming of the term &quot;modular angle&quot;, so i'd also like to know the etymology of that.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-03-22T14:00:03.303" LastActivityDate="2020-03-23T02:14:40.337" Title="Etymology of certain terms in the theory of elliptic integrals" Tags="|terminology|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11570" PostTypeId="2" ParentId="11568" CreationDate="2020-03-21T23:32:29.287" Score="5" Body="&lt;p&gt;No, they did not. Several methods were proposed but they do not give &quot;correct&quot; distances. Of course, all depends on the exact meaning of the word &quot;measure&quot; and &quot;correct&quot;. But their estimates were orders of magnitude away from the true numbers. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some details are given here: &#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/6021/historical-knowledge-of-distance-of-earth-from-sun/6025#6025&quot;&gt;Historical knowledge of Distance of Earth from Sun&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;and also here the related subjects are discussed:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/1796/who-first-measured-the-distance-to-the-moon-how-was-it-done/1798#1798&quot;&gt;Who first measured the distance to the Moon? How was it done?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/8509/where-did-ptolemy-compare-the-earth-to-the-distance-of-fixed-stars/8512#8512&quot;&gt;Where did Ptolemy compare the Earth to the distance of fixed stars?&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/6869/how-did-ptolemy-calculate-the-distance-to-the-moon/6871#6871&quot;&gt;How did Ptolemy calculate the distance to the moon&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. On the source of the number 10000 seconds. This is about 1/4 of a degree, which is the usual precision of numbers given in Ptolemy. With this accuracy Ptolemy knew that &quot;the Sun has no parallax&quot;. Which gives a LOWER estimate of the distance. Speaking of Hipparchus, almost none of his writings survive (the only surviving text is not mathematical), so our principal source on Greek astronomy is Ptolemy, and it is open to wide speculation what exactly in Ptolemy comes from Hipparchus or other earlier astronomers. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-03-22T13:36:27.783" LastActivityDate="2020-03-22T13:36:27.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11571" PostTypeId="1" CreationDate="2020-03-22T02:05:06.367" Score="8" ViewCount="202" Body="&lt;p&gt;In the end of his address to Annual Meeting of the Mathematical Association in 1933 titled &amp;quot;The marquis and the land agent: a tale of the 18th century&amp;quot;, the Association president G. N. Watson says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My final task is to express my gratitude to Prof. Neville for having made the extremelly happy suggestion of an exhibition of books on elliptic functions...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and the editor adds:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The exhibition consisted of 97 volumes, and included all but very few existing books on elliptic functions....Before the collection is dispersed, the most careful bibliographical record will be compiled, and it is hoped that this list will become of permanent value to librarians and scholars.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My source: Math. Gazette 17 (1933) 5-17. The opening page of this issue explains the circumstances under which the exhibition was held:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Annual Meeting of the Mathematical Association was held at the Institute of Education on 5th-6th January, 1933. . . . A Publishers' Exhibition was open during the two days.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question: &lt;strong&gt;Is there any way to find this list (if it was ever published)?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I searched for publications in this journal with title words &amp;quot;books&amp;quot;, &amp;quot;elliptic&amp;quot; but had no success. (Watson's talk is about the founders of the theory, Fagnano and Landen). The following issue (May 1933) of this journal contains reprints of most of the talks that were given during the Annual Meeting, but it does not contain the list of books at the Exhibition.&lt;/p&gt;&#xA;&lt;p&gt;Remark. Zentralblatt finds only 15 books with &amp;quot;Elliptische Funktionen&amp;quot; in the title published in 1850-1933, and 4 books with Elliptische Integrale.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-12-19T06:00:44.890" LastActivityDate="2022-12-19T06:00:44.890" Title="Books on elliptic functions" Tags="|mathematics|reference-request|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11573" PostTypeId="1" CreationDate="2020-03-22T08:46:44.297" Score="5" ViewCount="339" Body="&lt;p&gt;Emmy Noether was the first woman in Germany to obtain  &lt;a href=&quot;https://thonyc.wordpress.com/2010/03/24/emmy-and-the-habilitation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;habilitation in 1919&lt;/a&gt;. But I remember to have heard in the debate concerning the situation of women in academic mathematics that took place on the occasion of the International Congress of Mathematicians 2006 in Madrid (mentioned in this &lt;a href=&quot;https://www.google.com/url?q=https://www.drivehq.com/file/df.aspx/isGallerytrue/shareID8755087/fileID6097463843%3F1%3D1&amp;amp;sa=U&amp;amp;ved=2ahUKEwiInaqi5a3oAhUF_RQKHd-0BMMQFjAJegQIAxAB&amp;amp;usg=AOvVaw3Cl8dZ8Zx7BJeDNaOMiDuG&quot; rel=&quot;nofollow noreferrer&quot;&gt;report from the AWM&lt;/a&gt;),  that, in fact, she has been one of the very few female mathematicians at  Göttingen University to get habilitation for many years. Another one was &lt;a href=&quot;https://en.wikipedia.org/wiki/Hel_Braun&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helene Braun&lt;/a&gt; (who, by the way, doesn't appear in the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/List_of_University_of_G%C3%B6ttingen_people&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia list of University of Göttingen people&lt;/a&gt;): &#xA; according to what she explained in her book &lt;a href=&quot;https://books.google.es/books?id=t-OoBgAAQBAJ&amp;amp;pg=PP1&amp;amp;lpg=PP1&amp;amp;dq=%22Eine+Frau+und+die+Mathematik%22&amp;amp;source=bl&amp;amp;ots=jqmk-XSiXr&amp;amp;sig=ACfU3U1r1RwhtmVFb0HRYR_NGsx0gyodIQ&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwiwlbib_q7oAhUHkRQKHTU9B-YQ6AEwCXoECAQQAQ#v=onepage&amp;amp;q=Habilitation&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Eine Frau und die Mathematik 1933–1940: Der Beginn einer wissenschaftlichen Laufbahn&lt;/em&gt;&lt;/a&gt;, she achieved habilitation in Göttingen at the end of 1940 (or maybe at the beginning of 1941: I can only read an excerpt of the book).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'm not completely sure if they really were the only two famale  mathematicians at Göttingen who were able to obtain habilitation during these years, but I remember to have heard that there were no habilitations of female mathematicians at Göttingen for many years. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe this was mentioned in the film &lt;a href=&quot;https://books.google.es/books?id=PNIJDAAAQBAJ&amp;amp;pg=PA124&amp;amp;lpg=PA124&amp;amp;dq=film+%22women+and+mathematics+across+cultures%22&amp;amp;source=bl&amp;amp;ots=iDsSG1yjmO&amp;amp;sig=ACfU3U3N2gwv7tMfuTkVa-wt8_8XewdfPg&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwjrgpGh8q3oAhWQ3eAKHRumA10Q6AEwAXoECAkQAQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Women and mathematics across cultures&lt;/em&gt;&lt;/a&gt;, which has been projected before the debate, but I'm not completely sure about that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Does anyone have some details about this?&lt;/p&gt;&#xA;" OwnerUserId="10392" LastEditorUserId="4251" LastEditDate="2021-03-09T11:16:48.733" LastActivityDate="2021-03-09T11:16:48.733" Title="For many years, were Emmy Noether and Helene Braun the only female mathematicians to obtain habilitation at Göttingen University?" Tags="|mathematics|mathematicians|academia|gottingen|germany|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11575" PostTypeId="2" ParentId="11569" CreationDate="2020-03-22T15:06:40.180" Score="2" Body="&lt;p&gt;I don't understand French, but it seems that Legendre is the key person who coined these terms. You can use DeepL (www.deepl.com) to translate French. So if one can decipher the definition of &lt;em&gt;le module&lt;/em&gt; then &lt;em&gt;l'angle du module&lt;/em&gt; will begin to make sense. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Module itself is from Latin so there is no difference in French/German/ or English. Can we make sense if we take module as a &quot;measure&quot;. This is how the unabridged Oxford Dict. tells us:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&amp;lt; classical Latin modulus standard unit of measurement, rhythmic&#xA;  measure, pipe for controlling flow of water &amp;lt; modus mode n. + -ulus&#xA;  -ule suffix. Compare Middle French, French module (1547 in sense 3b, 1611 in Cotgrave in sense 4, 1762 in sense 6), Italian modulo (14th&#xA;  cent. in sense 3b, 1806 in sense 6, 1837 in sense ‘unit of measurement&#xA;  of running water’: compare sense 13).&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From his 1825 book, &lt;em&gt;Traité des fonctions elliptiques et des intégrales Eulériennes&lt;/em&gt;, (available from Google books) shows the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kbhYp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kbhYp.png&quot; alt=&quot;Book image&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;From now on we will call transcendental functions or elliptical functions, the integrals included in this formula. The transcendental H will be assumed to vanish or begin when phi=0; its extent will be determined by the variable phi, which we will call the amplitude; the constant c, always smaller than unity, will be called the modulus; we can always represent c by sin-theta, and theta will be what we call the angle of the modulus.&quot; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Translated with www.DeepL.com/Translator (free version)&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2020-03-23T02:14:40.337" LastActivityDate="2020-03-23T02:14:40.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11576" PostTypeId="1" CreationDate="2020-03-22T18:43:15.217" Score="12" ViewCount="1099" Body="&lt;p&gt;I can think of three. Saul Kripke quite famously could only be begrudged to finish his undergraduate degree at Harvard before being hired as a full professor. Donald Martin (the set theorist of Martin's Axiom fame) seems to have been hired into professorship out of a post-baccalaureate fellowship, without pursuing graduate study. Finally, I can't find evidence that Chaitin ever got any graduate degrees.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there other famous examples of great logicians or mathematicians making it into the academy while skipping over the PhD? Examples from before the era of the modern academic degree structure are not interesting for the purpose of this inquiry.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of my professors told me that these days, someone in a similar situation would likely be put into a no-effort PhD program until the residency requirements are met, with some important publication or another of theirs being declared a &quot;dissertation.&quot; None of the professors I asked this to could think of a fourth example. Thus, it seems the no-doctorate logician is a dying breed, one worth examining and cataloguing.&lt;/p&gt;&#xA;" OwnerUserId="11241" LastEditorUserId="4251" LastEditDate="2021-01-25T21:12:36.203" LastActivityDate="2023-06-04T23:44:48.570" Title="How many important logicians did NOT receive doctoral degrees?" Tags="|set-theory|mathematics-social-history|logic|logicians|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11577" PostTypeId="2" ParentId="663" CreationDate="2020-03-22T19:00:20.733" Score="4" Body="&lt;p&gt;At &lt;a href=&quot;https://www.uni-math.gwdg.de/en/burmann.xhtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;, you can find a loose translation of „Die Entwicklung der Mathematik an der Universität Göttingen&quot; by Erwin Neuenschwander and Hans-Wilhelm Burmann, an interesting article published in the book &lt;a href=&quot;https://books.google.es/books?id=SZPuC7VGVbIC&amp;amp;pg=PA8&amp;amp;lpg=PA8&amp;amp;dq=%E2%80%9EDie+Entwicklung+der+Mathematik+an+der+Universit%C3%A4t+G%C3%B6ttingen%22&amp;amp;source=bl&amp;amp;ots=W1ftbwy_Vv&amp;amp;sig=ACfU3U0z9ZLygE6if5IwgydKlj0I5GB4OQ&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwjr9dj7v67oAhUQCxoKHbsCAjIQ6AEwAnoECAkQAQ#v=onepage&amp;amp;q=%E2%80%9EDie%20Entwicklung%20der%20Mathematik%20an%20der%20Universit%C3%A4t%20G%C3%B6ttingen%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Die Geschichte der Verfassung und der Fachbereiche der Georg-August-Universität zu Göttingen&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the early times, Segner, Penther and Tobias Mayer played an important role in the development of Göttingen as a relevant center of mathematics and science in Germany:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;At the founding of the university, only five chairs were planned in the Philosophical Faculty. Its first mathematician, appointed in 1735, was Johann Andreas von Segner, who had studied philosophy, mathematics and medicine in Jena and had practiced as a physician before starting his academic career. Apart from mathematics, Segner was responsible for the lectures in physics, wrote numerous textbooks on both subjects, and occasionally lectured on chemistry. His prolific activity soon brought Segner a reputation as one of the best living mathematicians in Germany. Friedrich Penther, originally hired as overseer of the academic buildings in Göttingen, was appointed professor for mathematical topics in areas such as economics, architecture, and mapmaking. After Penther’s death his place was occupied by the well-known astronomer and mapmaker Tobias Mayer. Already before Mayer’s arrival in Göttingen, Segner had campaigned for the establishment of an astronomical observatory, which he and Mayer then shared for some time; but as competition over the observatory grew and the authorities in Hanover generally supported Mayer, Segner left Göttingen in 1755 to take over the chair vacated by Christian Wolff at the University of Halle.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Once Segner left Göttingen, Kästner, his successor for almost 50 years, was defined by the German physicist Georg Christoph Lichtenberg in one of his aphorisms as &quot;an encyclopaedic dictionary&quot;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Segner's successor was Abraham Gotthelf Kästner, who worked in Göttingen for almost 50 years. When not busy with mathematics, he worked especially in physics and astronomy, as well as in chemistry, botany, anatomy, philosophy, literature, and many other areas, and served as director of the observatory after Mayer’s death. In one of his now classic aphorisms, Georg Christoph Lichtenberg called Kästner &quot;an encyclopaedic dictionary.&quot; One of his many influential books was the four-volume &lt;em&gt;Anfangsgruende der Mathematik&lt;/em&gt;, or &quot;starting points of mathematics,&quot; arranged by field and treating mechanics and hydrodynamics in addition to arithmetic, geometry, analysis, algebra, and applied mathematics. In Kästner's time Albrecht Ludwig Friedrich Meister and Mayer's brother-in-law Georg Moritz Lowitz taught applied mathematics, Lichtenberg and Johann Christian Polykarp Erxleben taught physics, and Karl Felix Seyffer taught astronomy. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But, after Kästner's death, there was a period of crisis until the arrival of Gauss:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;[...] no suitable replacement could be found, and for a time there was no towering personality of the kind the University of Göttingen would find some years later in Gauss. At the same time, Hanover experienced a political crisis, submitting to Prussian and then French occupation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then, of course, as mentioned in other answers, there was Gauss, the most important mathematian of his time and one of the most relevant astronomers and physicists in Europe:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Carl Friedrich Gauss came from Braunschweig, or Brunswick, where he spent his youth. Thanks to a scholarship from the duke of Braunschweig, who recognized and supported his talent, he went to Göttingen in 1795 to study mathematics there. He came into particularly close contact with Seyffer, the professor of astronomy at the time, and thought highly of Lichtenberg, but was less impressed by Kästner and the other mathematics lecturers. Other German mathematicians, however, may not have been able to meet his standards either. In 1798 Gauss returned to Braunschweig, where he worked on his magnum opus &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; and received a doctorate in 1799 in Helmstedt. With the publication of the &lt;em&gt;Disquisitiones&lt;/em&gt; and the sighting of the dwarf planet Ceres based on his computations in 1801, Gauss became one of the most prominent astronomers and mathematicians in Europe. He received a salary increase from the Duke of Braunschweig and an offer to direct the observatory in St. Petersburg, upon which Göttingen Professor Wilhelm Olbers stepped into negotiations with the University of Göttingen to do as much as possible to keep Gauss in Germany. Among Gauss’s main reasons for accepting the resulting appointment in 1807 were the Hanover government’s plan to establish a new observatory in Göttingen and the relatively light obligations which he had to take over within the university. After the completion of the new observatory in 1816, Gauss moved in as its director and lived there until his death in 1855. &#xA;  &lt;br&gt;&lt;br&gt;&#xA;  Gauss was the leading mathematician of his time, and many regard him simply as the greatest mathematician. At 18 he discovered that the regular seventeen-sided polygon can be constructed with a ruler and compass, the first major discovery in this area in 2000 years. His &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; provided the foundations for modern number theory. His methods for the calculation of planetary motion have still not been essentially improved upon even today. Just how far Gauss was ahead of his contemporaries in complex analysis, and especially in his insights about non-Euclidean geometry, became clear only with the publication of his posthumous works, letters, and scientific diary. His interest in geodesy —he started a project to survey the lands of the Kingdom of Hanover in 1818 and carried its main workload for some years— led him to the investigation of curved surfaces, culminating in the Theorema Egregium and other results developed further by Riemann. Together with William Weber he investigated electricity and magnetism, leading to the invention of the electrical telegraph as a byproduct of their research. These are only a few of the highlights of Gauss's enormous output. He made crucial discoveries in almost all areas of mathematics, and shaped the entire field with his demands for mathematical rigor. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;At Gauss's time, also Thibaut played an important role as a prominent lecturer:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Gauss seems to have had an aversion to lecturing at least in his earlier years, and appeared content when his lectures were cancelled due to a lack of students. In Gauss's time, basic mathematics lectures were conducted largely by Thibaut, mentioned above, and later by George Karl Justus Ulrich and Moritz Abraham Stern. Thibaut usually covered pure mathematics, differential equations and integrals, and finite analysis, while Ulrich lectured on solid geometry and trigonometry, applied geometry, mechanics and civil architecture. Thibaut was known as the best lecturer in Göttingen, and because of his perfected rhetorical style was even compared with Goethe. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;After Gauss's death, there was again a period of crisis, which seemed to have arrived to its end with the arrival of Dirichlet. But, his premature death in 1859 prevented him for working in Göttingen for a long time. Riemann succeed Dirichlet, but he also died from tuberculosis in 1866, so, once again, Göttingen remained without a leading figure:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The death of Gauss left the University of Göttingen with a gap that could not easily be filled. The directorship of the observatory remained vacant for several years and was administered intermittently by Weber until 1868, when it was finally divided between William Klinkerfues and Ernst Schering. In the mathematical areas a worthy successor was found in Peter Gustav Lejeune Dirichlet, who at that time enjoyed the highest reputation among German mathematicians. Dirichlet's work united two currents, number theory in the tradition of Gauss and applied mathematics from the French school, as for example in his theorem on prime numbers in arithmetic progressions together with its beautiful proof, and in the extraordinarily fruitful Dirichlet principle. Dirichlet was an inspiring teacher, and his lectures on number theory, published after his early death by Richard Dedekind, remained the standard work for decades.&lt;br&gt;&#xA;  &lt;br&gt;&lt;br&gt;&#xA;  Dirichlet's successor was Bernhard Riemann, who had studied first in Göttingen and then under Dirichlet in Berlin and obtained his doctorate and Habilitation under Gauss in 1851 and 1853. It was a great moment in mathematics when the young Riemann delivered his Habilitation lecture &quot;On the Hypotheses that Lie at the Foundations of Geometry&quot; in front of Gauss; Gauss was deeply impressed by it and, according to Weber, had the highest expectations about him, and Riemannian geometry would later supply the mathematical framework for Einstein's relativity theory. Riemann's innovative ideas were understood in their full depth only gradually, but their effects were lasting and provide mathematics with essential stimulation even today. He set the theory of analytic functions on a firm foundation and gave it a new dimension with the Riemann mapping theorem, which he proved with the help of the Dirichlet principle. The notion of Riemann surface shed light on the preceding half century's investigations of algebraic curves, unified and standardized them and opened the way for many future developments in algebraic geometry and topology. In an eight-page work on number theory, Riemann provided the key to problems of prime number distribution. Hilbert remarked in a lecture (W.S. 1896/97, p. 264) that &quot;only very rarely has an essay of such shortness, sharpness and genius flowed from the pen of a human being, as did this masterwork of one of the greatest spirits of our science.&quot; The Riemann hypothesis, whose demonstration would greatly extend our knowledge of the distribution of prime numbers, has defied all attempts at proof and refutation until today.&#xA;  &lt;br&gt;&lt;br&gt;&#xA;  Unfortunately, Riemann, like Dirichlet, was prevented from working for a very long time as a full professor in Göttingen, since he contracted tuberculosis just three years after his appointment and then spent much of his remaining time at health resorts in Italy. After Riemann came Alfred Clebsch, who also died a few years after his appointment, and then Lazarus Fuchs, who one year later followed a call to Heidelberg. For a brief period Göttingen mathematics once again lacked a long-term leading figure, and ceded supremacy to Berlin, just as the nearby capital Hanover ceded much of its political power to Berlin around this time.  &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Then, there was the time of Schwarz:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After the departure of Fuchs in 1875, Hermann Amandus Schwarz was appointed from Zurich. Schwarz organized his lectures according to a well worked out study plan and arranged them into two cycles. The first, introductory cycle covered differential and integral calculus, analytic geometry, surfaces of second degree, curved surfaces and double curvature, and synthetic geometry. The second, simultaneously running cycle covered analytic functions, elliptical functions, minimal surfaces, the hypergeometric series, and other areas of function theory. Schwarz, supported by Stern, spearheaded the creation of a circulating library for the Mathematics-Physics Seminar in 1878, and it remained under his administration until his departure for Berlin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And, as explained in Mikhail Katz's answer, Klein played a relevant role as a dynamizer and a great organizer of mathematics and physics in Göttingen:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Felix Klein, the great organizer of Göttingen mathematics and physics, studied in Bonn, Göttingen, Berlin and Paris, particularly with Plücker and Clebsch, and received his &lt;em&gt;Habilitation&lt;/em&gt; in Göttingen in 1871. In 1872 he was appointed full professor at Erlangen, from where he went to Munich in 1875, to Leipzig in 1880, and finally back to Göttingen upon Stern's retirement in 1886. In his 1872 &lt;em&gt;Erlangen Program&lt;/em&gt; Klein used the notion of group to formulate a classification principle for geometry that had a lasting effect on geometrical thought. He saw himself as a developer of the brilliant ideas of Riemann, whose geometrical core he worked out further and brought into the investigation of model functions and automorphic functions.&#xA;  &lt;br&gt;&lt;br&gt;&#xA;  Klein's main objectives apart from pure mathematics were the reinforcement of the connections between mathematics, the natural sciences and technology and the restructuring of education in mathematics and science from the earliest grades to the university. Both of these goals shaped much of his time in Göttingen. Already in his appointment negotiations, Klein advocated the creation of a mathematical reading room with a reference library like the one he had organized in Leipzig. His request was approved by the time he came to Göttingen, and immediately after his arrival the Reading Room of the Mathematics–Physics Seminar was opened. At that time it was in Auditorium No. 20 on the upper floor of the auditorium building at Weender Gate, directly beside the model collection and the lecture rooms of the Mathematics–Physics Seminar. At first its furnishing was relatively modest, with 20 working places and a library of about 500 volumes around 1890.&lt;br&gt;&#xA;  &lt;br&gt;&lt;br&gt;&#xA;  With Schwarz's departure for Berlin as successor to Weierstrass in 1892 and the subsequent departure of Heinrich Weber, Klein had the freedom to reorganize mathematical instruction in Göttingen according to his wishes. The small seminar library developed by Schwarz was combined with the library of the mathematics reading room and put under Klein's direction. Klein introduced term fees for the gradually expanding student body and applied for 3000 Marks for the library's further development, a sum five times the annual budget for mathematics. He also took over the collection of mathematical instruments and models and obtained funding for a special assistant for it, and took part in the establishment of the Mathematical Society and the first edition of study plans, distributed to students free of charge upon matriculation.&lt;br&gt;&#xA;  &lt;br&gt;&lt;br&gt;&#xA;  At the same time, Klein worked on his second main goal: strengthening the ties between mathematics, the natural sciences and technology. Gathering together interested professors and industry leaders, he created the Göttingen Association for the Advancement of Applied Physics and Mathematics in 1898. This association raised over 200,000 Marks over the next ten years to support these sciences in Göttingen, allowing for the establishment of numerous new buildings and institutes. At the 10-year anniversary celebration, Klein could report that since the Association’s inception the Institutes for Applied Electricity, Applied Mathematics and Mechanics and the Geophysics Institute had been founded, and the number of professors in mathematics and physics had doubled. This led, among other things, to the appointment of Hermann Theordor Simon, Carl Runge, Ludwig Prandtl and Emil Wiechert.&lt;br&gt;&#xA;  &lt;br&gt;&lt;br&gt;&#xA;  The number of mathematics students and reading room users was constantly rising, and space shortages and the move of the Physics Institute to Bunsenstrasse left the mathematics and physics facilities scattered across the city. Klein felt a growing need for a new Mathematics Institute. By 1911 he believed this dream almost attained, as the Göttingen Association had allocated 200,000 Marks for the purpose and its chairman had purchased a suitable property on Bunsenstrasse in direct proximity to the physics institutes. But the realization of Klein's plans was delayed by the First World War and later by the struggling economy and spiralling inflation, and one of his great projects did not come to fruition in his time. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Klein attracted Hilbert to Göttingen and both of them brought Minkowski and Landau:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the meantime, however, Klein continued to further mathematics in many other ways. He served for several years as chairman of the International Commission on Mathematical Instruction and oversaw several series of publications associated with the Commission. He served as editor of the leading mathematics journal &lt;em&gt;Mathematische Annalen&lt;/em&gt; and the monumental &lt;em&gt;Encyclopedia of the Mathematical Sciences and their Applications&lt;/em&gt;. Among his greatest victories for Göttingen was his success, thanks to his excellent relations with Prussian Ministry Director Friedrich Althoff, in drawing David Hilbert to Göttingen in 1895 and in keeping him despite repeated calls to other universities. After one such competing offer in 1902, Hilbert and Klein convinced Althoff to create a third mathematics professorship in Göttingen, which was then occupied first by Hermann Minkowski and then by Edmund Landau. Thus Göttingen mathematics rose to first place among German universities even in the number of professors. &#xA;  &lt;br&gt;&lt;br&gt;&#xA;  The mathematical center of Göttingen at that time was undoubtedly David Hilbert, who shaped 20th century mathematics as no one else has. Hermann Weyl wrote in a letter in 1927: &quot;The spirit in which we do mathematics, we received from him.&quot; In his universality —every few years he turned to a completely new sphere of activity— he is comparable only with Gauss. He ended the classical era of invariant theory with his new &quot;transcendental&quot; methods. His unification of algebraic number theory, commonly known as the &lt;em&gt;Zahlbericht&lt;/em&gt; or &quot;report on numbers,&quot; was for at least a half century the classic work for everyone concerned with the field. His &lt;em&gt;Foundations of Geometry&lt;/em&gt; inaugurated the axiomatic method. He introduced a method for the justification of the Dirichlet principle which now belongs to the essential tools of analysis. In his investigations of integrals he recognized the necessity of treating infinite-dimensional areas —one speaks today of Hilbert spaces— and he developed a spectral theory that provided the foundations for quantum theory and gave rise to functional analysis, a powerful branch of modern mathematics. His program for the proof of the consistency of mathematics could not attain its goal, as Kurt Gödel showed in 1933, but it directed attention to mathematical models of calculating machines and to the theory of formal languages, which today form the basis of computer science and computer engineering. In his famous lecture at the Second International Congress of Mathematicians in Paris in 1900, Hilbert put forth a now classic list of 23 unsolved problems in a range of areas of mathematics, setting the agenda for much of 20th century research. The overall course of his lectures is of tremendous span, covering all areas of pure mathematics and extending to physical topics such as mechanics, electromagnetic oscillations, and relativity theory, and to philosophical topics such as the nature of mathematical knowledge. He attracted many students, and among his 69 graduate students were eminent figures such as Otto Blumenthal, Richard Courant, and Hermann Weyl.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And, from these period, Göttingen became an attraction pole for many of the most relevant mathematicians and physicists of the time.&lt;/p&gt;&#xA;" OwnerUserId="10392" LastActivityDate="2020-03-22T19:00:20.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11578" PostTypeId="2" ParentId="11576" CreationDate="2020-03-23T00:21:54.023" Score="8" Body="&lt;p&gt;Thoralf Skolem could perhaps be counted as a quasi example of that. He did not enroll as a PhD candidate before becoming a docent and a member of the Norwegian Academy of Science and Letters, and it seems that he still needed some convincing from his colleagues in order to actually submit a thesis, several years later. When he finally defended it, his formal supervisor, Axel Thue, had already been deceased for no less than four years.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Do note, though, that this could actually be seen as a century-old example of the claim made by your professor.&lt;/p&gt;&#xA;" OwnerUserId="11242" LastActivityDate="2020-03-23T00:21:54.023" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11579" PostTypeId="1" CreationDate="2020-03-23T19:41:29.773" Score="3" ViewCount="674" Body="&lt;p&gt;Does anybody know of a portrait of British physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Stephen_Butterworth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stephen Butterworth&lt;/a&gt; (1885-1958)?&lt;/p&gt;&#xA;" OwnerUserId="11246" LastEditorUserId="4251" LastEditDate="2021-09-22T11:58:29.967" LastActivityDate="2021-09-22T17:45:07.017" Title="Portrait of Stephen Butterworth" Tags="|biographical-details|physicists|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11581" PostTypeId="2" ParentId="8184" CreationDate="2020-03-24T07:00:40.223" Score="3" Body="&lt;p&gt;Einstein of course.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He had been working in a patent office for three years when he formulated the theory of relativity, gave an explanation of the photo-electric effect and an explanation of Brownian motion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One shouldn't discount the fact he had been trained as a physicist, nevertheless, he was a non-professional physicist at the time he published his papers. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-03-24T14:58:43.770" LastActivityDate="2020-03-24T14:58:43.770" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11582" PostTypeId="2" ParentId="198" CreationDate="2020-03-24T07:51:15.760" Score="1" Body="&lt;p&gt;First a little &lt;a href=&quot;https://en.wikipedia.org/wiki/Linear_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;history&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is discussed in Roger Hart, &lt;em&gt;The Chinese Roots of Linear Algebra&lt;/em&gt;; however, in Euope&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Systems of linear equations arose with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. However, the first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In fact, Leibniz considered there to be a theory of 'extension' or 'logical characteristic' but wasn't able to come up with a viable such theory; in 1844 a prize competition was instituted on exactly this problem; this was won by Grassmann who had entered an essay 'Geometrische analsye...' after being persuaded by Mobius to enter; this included foundational new topics of what is today called linear algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was around this time (actually 1843), that Hamilton discovered the quaternions which prompted the discovery of other hyper-complex systems and then five years later, the English mathematician, James Joseph Sylvester introduced the term matrix (which is Latin for womb); it was another English mathematician, William Clifford who combined both Grassmanns theory and the theory of hyper-complex systems into what are now known as Clifford algebras. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the transition from the early quantum mechanics to conventional quantum mechanics, Heisenberg and Jordan rediscovered matrix multiplication in 1925 (although Connes says that this would be better understood through groupoids). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was Emmy Noether and her school who pioneered the study of abstract algebraic structures per se placing them in a systematic foundation; and in 1930, Van der Waerden published his &lt;em&gt;Modern Algebra&lt;/em&gt; which 'forever changed' how algebra was taught in universities.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would consider that it was all of these developments that pushed the undergraduate curriculum to the consideration of abstract algebra per se and of intrinsic structures and not just quantum mechanics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(On a personal note, matrix mathematics wasn't just taught at universities, I distinctly remember being taught matrix mathematics at school). &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-03-24T07:51:15.760" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11583" PostTypeId="2" ParentId="11573" CreationDate="2020-03-24T08:53:39.337" Score="-1" Body="&lt;p&gt;Have you considered &lt;a href=&quot;https://en.wikipedia.org/wiki/Sofya_Kovalevskaya&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sofya Kovalevskaya&lt;/a&gt;?  In 1874 she presented three papers — on partial differential equations, on the dynamics of Saturn's rings, and on elliptic integrals — to the University of Göttingen as her doctoral dissertation and with the support of Weierstrass, this earned her a doctorate in mathematics and thereby became the first woman to have been awarded a doctorate at a European university.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She was born in Moscow in 1850 where her father was a Lieutenant General in the Imperial Russian Army (there may also have been some Romani ancestry on the father's side). When she was 8 or 9 years old, she was intrigued by the symbols of calculus when the wall of her bedroom had been papered with pages from lecture notes by Ostrogradsky, left over from her father's student days. She describes this in her memoir, &lt;em&gt;A Russian Childhood&lt;/em&gt; (I've read this and recommend it highly).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;She first studied under Strannoliubskii, who was also a well-known advocate of higher education for women in St. Petersburg; however,  in order to study abroad, Kovalevskaya needed written permission from her father or husband; and thus, in 1868 she contracted a 'fictitious marriage' with Vladimir Kovalevskij, a young paleontology student, book publisher and radical, who was the first to translate and publish the works of Charles Darwin in Russia. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1869, they first moved to Vienna where she attended classes on physics, and then to Heidelberg, where, after great effort, she was allowed to attend classes given by Helmholtz, Kirchhoff and Bunsen. Then, a year later, she moved to Berlin, where she spent the next three years - apart from a short period in France where she helped out with the Paris commune where her sister, Anyuta was active -  taking  private lessons with Weierstrass who taught her the same material that comprised his lectures at the university.  &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-03-24T08:53:39.337" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11584" PostTypeId="2" ParentId="5972" CreationDate="2020-03-24T09:08:37.703" Score="0" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Why did Tesla disagree with Einstein? ... How can this be? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It wasn't just Tesla; there was Millikan who also disagreed with Einstein; though much later he accepted without ever mentioning his initial disagreement - presumably due to embarassment.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;How can this be? If Einstein's theory is falsifiable yet very reinforced by evidence (the balance for a good scientific theory) why did Tesla disavowed it so strongly?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Einsteins theory at the time wasn't falsifiable; we simply did not have the technological know-how to look for gravitational waves or for precise measurements of deviations from Newtons theory. The evidence, at best, was circumstantial. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But more, isn't this normative with new and innovative theories? Consider today, where we have several competing theories in quantum gravity; this is very speculative area where not much can be done by direct experiment. Thus you will get disagreement even though they are aiming towards the same end - the quantisation of gravity - and even though they are consistent with what is already understood.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-03-24T09:08:37.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11585" PostTypeId="2" ParentId="11403" CreationDate="2020-03-24T09:27:23.293" Score="2" Body="&lt;p&gt;I wouldn't trust Hardys assessment; he after all said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;he had been carrying an impossible handicap, a poor and solitary Hindu pitting his brains against the accumulated wisdom of Europe... It was impossible to teach him systematically, but he gradually absorbed new points of view.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Given that Ramanujan, by his own admission, had learnt mathematics from European textbooks, such as Carr's Synopsis, one can hardly say he was 'pitting his brains against the accumulated wisdom of Europe'; moreover, his decision to move to Europe showed that he was keen to involve himself in European mathematics. I would say, that when he moved to Europe he was already a fully-fledged mathematician with his own ideas about how to do mathematics, and more importantly - what interested him. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What did G.H. Hardy teach Ramanujan?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Very little, what Hardy gave Ramanujan was entry into the mathematical milieu which was the most important thing at that time in Ramanujans mathematical career.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-03-24T09:27:23.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11586" PostTypeId="1" CreationDate="2020-03-25T09:16:27.433" Score="1" ViewCount="201" Body="&lt;p&gt;It's well-known that if &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are two sets, then the set of all functions from &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; can be denoted by &lt;span class=&quot;math-container&quot;&gt;$B^A$&lt;/span&gt;: explanations of this particular notation can be found in many places:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/901735/meaning-of-a-set-in-the-exponent&quot;&gt;https://math.stackexchange.com/questions/901735/meaning-of-a-set-in-the-exponent&lt;/a&gt;&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/63960/what-does-it-mean-when-a-set-is-the-exponent&quot;&gt;https://math.stackexchange.com/questions/63960/what-does-it-mean-when-a-set-is-the-exponent&lt;/a&gt;&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/709184/why-is-the-exponential-of-sets-the-function-set&quot;&gt;https://math.stackexchange.com/questions/709184/why-is-the-exponential-of-sets-the-function-set&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I ask for is: &lt;em&gt;when this notation was first introduced and in which context?&lt;/em&gt;&#xA;(So this question &lt;strong&gt;is not&lt;/strong&gt; about the meaning or the rationale behind it.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The older occurrence I can find is in Bourbaki's &lt;a href=&quot;http://tomlr.free.fr/Math%E9matiques/Bourbaki/Theorie%20Des%20Ensembles.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Théorie des ensembles&lt;/a&gt;, of 1954, E.R.20, but is it the first? &lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2020-03-25T17:54:21.343" Title="History of exponential notation for the set of functions between two sets" Tags="|notation|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11588" PostTypeId="2" ParentId="11586" CreationDate="2020-03-25T17:54:21.343" Score="4" Body="&lt;p&gt;It comes from Bernstein's Habilitation dissertation &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01457734&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Untersuchungen aus der Mengenlehre&lt;/em&gt; (1901, published 1905)&lt;/a&gt;, where he also introduced the now common symbolism for cardinal arithmetic. The exponential notation is introduced in §2 as follows (my translation):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; are two sets, we call that set which - in the sense of a known expression - contains all combinations of elements from &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; to the classes of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;, the power &lt;span class=&quot;math-container&quot;&gt;$M^N$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; raised to &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;). Regarding the application to addition, multiplication and powerclasses of commutative and associative laws, they are the same as for finite numbers&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Bernstein does not use &lt;span class=&quot;math-container&quot;&gt;$2^N$&lt;/span&gt; for the powerset, but he does write &lt;span class=&quot;math-container&quot;&gt;$2^{\aleph_\alpha}$&lt;/span&gt;, meaning set cardinality, in §9, when discussing the continuum hypothesis.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-03-25T17:54:21.343" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11589" PostTypeId="2" ParentId="11579" CreationDate="2020-03-25T18:44:33.027" Score="4" Body="&lt;p&gt;Edit: Apparently, Prof. Silage made a mistake and used a wrong picture for his book. So, the mystery remains. The obituary in &amp;quot;Times,&amp;quot; if someone can access it, might contain a picture.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Old answer:&lt;/p&gt;&#xA;&lt;p&gt;Here is a picture. The page is taken from &lt;a href=&quot;https://astro.temple.edu/%7Esilage/Chapter2MS.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chapter 2, &amp;quot;Frequency Domain Analysis&amp;quot;&lt;/a&gt; in Dennis Silage, &amp;quot;Analog and Digital Communications.&amp;quot; Where did Silage find the photograph I do not know, you can email him and ask.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KGlHe.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KGlHe.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2021-09-22T17:45:07.017" LastActivityDate="2021-09-22T17:45:07.017" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11590" PostTypeId="1" CreationDate="2020-03-25T20:44:30.643" Score="5" ViewCount="728" Body="&lt;p&gt;Both Heisenberg and Schrödinger found each other's way of formulating Quantum Mechanics quite repellent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: Did any of the two change their views towards each other (specially Heisenberg)? Did he eventually accept Schrödinger's formulation or relax his attitude towards it, specially considering that both formulations were shown to be mathematically equivalent?&lt;/p&gt;&#xA;" OwnerUserId="10956" LastEditorUserId="4251" LastEditDate="2020-06-07T07:00:57.843" LastActivityDate="2021-03-27T17:07:30.750" Title="Did Heisenberg ever accept Schrödinger's formulation of quantum mechanics, or at least did he relax his negative views about it?" Tags="|physics|biographical-details|theoretical-physics|depiction-of-scientists|heisenberg|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11592" PostTypeId="2" ParentId="9776" CreationDate="2020-03-26T20:53:55.027" Score="1" Body="&lt;p&gt;Concerning the note &quot;&lt;strong&gt;About the frequency of optical double stars&lt;/strong&gt;&quot;, i think that Gauss attempted to solve in this note a kind of probabilistic problem that is relevant to astronomy. Gauss tries to find a pattern in the most updated maps of stars up to his time - what is the fraction of double stars in a ball (sphere with volume) centred at the earth, among  a total of &lt;span class=&quot;math-container&quot;&gt;$kM$&lt;/span&gt; observed stars, and assuming uniform prior distribution (Gauss writes in his  note:&quot;...&lt;span class=&quot;math-container&quot;&gt;$kM$&lt;/span&gt; stars &lt;strong&gt;scattered without a rule&lt;/strong&gt;&quot;)? A double optical star is actually two stars whose angular distance on the celestial sphere doesn't exceed &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gauss gives the answer in a form of a Poisson distribution with a rate parameter &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; for which he gives a formula in terms of &lt;span class=&quot;math-container&quot;&gt;$\lambda, M, k$&lt;/span&gt;. This note is therefore noteworthy for two reasons:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The first mathematician to apply Poisson distribution for describing statistical behaviour of stars is, according to wikipedia, Simon Newcomb. Gauss's attempt therefore preceeds him.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I guess the main problem is how to evaluate the rate parameter &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;, and that is what Brendel attempts to reconstruct in his commentary on Gauss's note.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;I'm still not sure if Gauss tries to solve this problem on a sphere or on a ball; it's more reasonable that he tries to solve it on a ball because in this way his calculation has direct relevance to astronomy.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-03-27T11:09:38.403" LastActivityDate="2020-03-27T11:09:38.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11593" PostTypeId="1" CreationDate="2020-03-27T13:26:02.830" Score="1" ViewCount="74" Body="&lt;p&gt;I did much searching about the N-95 mask but couldn't find any clue about its main  country, i.e. who launched it. Can anyone let me know which country really launched it?&lt;/p&gt;&#xA;" OwnerDisplayName="user25488" LastEditorUserId="2277" LastEditDate="2020-04-13T21:14:10.490" LastActivityDate="2020-04-13T21:14:10.490" Title="which country launched N-95 mask for the first time?" Tags="|medicine|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11594" PostTypeId="2" ParentId="11593" CreationDate="2020-03-27T15:21:26.797" Score="3" Body="&lt;p&gt;According to the article &lt;a href=&quot;https://www.fastcompany.com/90479846/the-untold-origin-story-of-the-n95-mask&quot; rel=&quot;nofollow noreferrer&quot;&gt;The untold origin story of the N95 mask&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The first single-use N95 “dust” respirator as we know it was developed by 3M, according to the company, and approved on May 25, 1972. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The article details the use and development of the precursors to the N95, noting that initial use of the N95 was for industrial applications.  It was the appearance of drug resistant tuberculosis and HIV in the 1990s which brought the mask into clinical use.  &lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-03-27T15:21:26.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11595" PostTypeId="1" CreationDate="2020-03-27T16:16:56.797" Score="9" ViewCount="218" Body="&lt;p&gt;A fairly detailed (14 page) account of  Apéry’s original proof of the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt; is given in Julian Havil’s book &lt;em&gt;The Irrationals&lt;/em&gt; which states that Apéry’s starting point is the mysterious recurrence relation &lt;span class=&quot;math-container&quot;&gt;$$n^3u_n + (n-1)^3u_{n-2} = (34n^3 - 51n^2 + 27n - 5)u_{n-1}.$$&lt;/span&gt;&#xA;From this, he defines two sequences using two pairs of boundary conditions.  He then gives, without proof, closed-form expressions for these sequences on which Havil  comments “... and it becomes easy to appreciate the incredulity of those who attended that conference.”  Finally, he proves that the term-wise ratio of these two sequences converges to &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt; and then uses Dirichlet’s irrationality criterion to prove irrationality.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Q&lt;/strong&gt;: Did Apéry ever explain how he obtained this recurrence? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have been able to locate two different derivations of this recurrence relation, however both appear to be more recent than Apéry’s work and indeed appear to be motivated by it.  For example,  in the article &lt;a href=&quot;https://video.ias.edu/sites/video/files/IASColloquium3.pdf&quot; rel=&quot;noreferrer&quot;&gt;Irrationality Proofs for Zeta Values and Dinner Parties&lt;/a&gt; (page 15/32), regarding the Apéry recurrence relation it states “There are numerous interpretations of this recurrence relation as a Picard-Fuchs equation of a family of varieties.”  The other derivation is via the relatively new concept of Cellular Integrals and Beuker Integrals - see, for example &lt;a href=&quot;https://books.google.ca/books?id=CPeFDwAAQBAJ&amp;amp;pg=PA349&amp;amp;lpg=PA349&amp;amp;dq=cellular%20integral%20apery%20recurrence&amp;amp;source=bl&amp;amp;ots=1MUD3Wz8AO&amp;amp;sig=ACfU3U0LCl4o3CrXxM9-tC8JM7yeRWHZoQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjco4G7grvoAhVFsZ4KHTmGDAEQ6AEwAnoECAoQAQ#v=onepage&amp;amp;q=cellular%20integral%20apery%20recurrence&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Elliptical Integerals, Elliptic Functions and Modular Forms in Quantum Field Theory&lt;/a&gt;, starting on page 342.  (Beuker presented a simplified version of Apéry’s proof using &quot;Beuker integrals&quot;.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(nb: The Wikipedia page on &lt;a href=&quot;https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;Apéry’s Theorem&lt;/a&gt; makes no mention of this recurrence relation in its summary of the proof.  However if you read Apéry’s summary of his proof (linked on wiki), then we see the two sequences he uses do indeed come from the recurrence.)&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-03-27T16:16:56.797" Title="Apéry’s mysterious recurrence relation" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11596" PostTypeId="1" AcceptedAnswerId="11597" CreationDate="2020-03-27T20:07:00.110" Score="11" ViewCount="320" Body="&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$R\subseteq A\times A$&lt;/span&gt; is an equivalence relation (i.e., a relation that is reflexive on &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, symmetric, and transitive), then for each element &lt;span class=&quot;math-container&quot;&gt;$x\in A$&lt;/span&gt;, the subset &lt;span class=&quot;math-container&quot;&gt;$[x]_R=\{y\in A: \langle x,y\rangle\in R\}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is called an &lt;em&gt;equivalence class&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. In set theory, the word class usually refers to a collection of elements which is not necessarily a set. I wonder why &lt;span class=&quot;math-container&quot;&gt;$[x]_R$&lt;/span&gt; was not called an equivalence set or an equivalence subset.&lt;/p&gt;&#xA;" OwnerUserId="11260" LastEditorUserId="55" LastEditDate="2020-03-27T21:01:13.373" LastActivityDate="2020-03-28T20:15:03.450" Title="Why were equivalence classes named classes rather than sets?" Tags="|terminology|set-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11597" PostTypeId="2" ParentId="11596" CreationDate="2020-03-27T20:58:47.590" Score="7" Body="&lt;p&gt;One detailed account of the history of equivalence relations and associated terminology is &lt;a href=&quot;http://philsci-archive.pitt.edu/14261/1/Equivalence%20An%20Attempt%20at%20a%20History%20of%20the%20Idea.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Equivalence: An Attempt at a History of the Idea by Ashgari&lt;/a&gt; largely based on Fowler's posts on the &lt;em&gt;Historia Mathematica&lt;/em&gt; forum. The terminology was a long time in the making, with &amp;quot;equivalence relation&amp;quot; appearing much earlier than &amp;quot;equivalence class&amp;quot;. Although neither the sources nor Ashgari directly answer why the choice of &amp;quot;class&amp;quot; was preferred there is one clue as to the possible reason.&lt;/p&gt;&#xA;&lt;p&gt;Even before &amp;quot;equivalence relation&amp;quot; was so named abstractly by Jourdain, a particular use of &amp;quot;equivalence&amp;quot; and its classes was prominent, and remained so for a while after. It is the equivalence of sets by equipotence or cardinality, the relation being existence of a bijective correspondence between them, introduced by Hume and popularized by Cantor. In this case, the equivalence classes (of all sets with the same cardinality) are &lt;em&gt;not&lt;/em&gt; themselves sets. As this seemed to be the paradigmatic example, by the time abstract terminology was being established the authors might have favored a combination of words they were already used to hearing. And this way there was no need to explain that in the case of cardinality the &amp;quot;equivalence sets&amp;quot; are not sets.&lt;/p&gt;&#xA;&lt;p&gt;Von Neumann introduced the set/class distinction in set theory explicitly in 1925 (it became the basis of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;set theory axiomatization NBG&lt;/a&gt;, alternative to classless ZFC), but in 1926 and 1929 he used the expression &amp;quot;equivalence class&amp;quot; in some algebraic contexts nonetheless. Aside from cardinality, there was some pre-existing terminology with &amp;quot;classes&amp;quot; in algebra, e.g. Weber introduced &amp;quot;class field&amp;quot; in 1891, see &lt;a href=&quot;http://jeff560.tripod.com/c.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miller's Earliest Known Uses&lt;/a&gt;, &lt;a href=&quot;https://www.jstor.org/stable/2370015&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jordan in a paper on group characters&lt;/a&gt; talked of &amp;quot;classes of conjugate substitutions&amp;quot; (to become conjugacy classes) in 1907. So it seems that the terminological transition from classes to sets only happened narrowly, where it mattered, and older terminology was generally preferred where it was already established.&lt;/p&gt;&#xA;&lt;p&gt;Here are some relevant passages from Ashgari:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I call a relation which is reflexive, symmetrical, and transitive an isoid relation. Philip Jourdain (1912) presented his article On isoid relations and theories of irrational number to the Fifth International Congress of Mathematics, starting with the sentence above. At the time, the process of “definition by abstraction” (Russell 1903, pp. 219-220) was quite well established but the term “equivalence” was mainly attached to the context of cardinal numbers... By 1919, neither the combination “equivalence relation” nor “equivalence class” was in use.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Under the section Äquivalenzrelationen und Klasseneinteilungen of the first chapter of the first edition of Höhere Algebra, Hasse (1926) wrote:&amp;quot;We call such a decomposition a partition of &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, and the subsets thereby determined its classes&amp;quot;. (Hasse 1954, p. 22; Higher Algebra is the English translation of the third edition of Höhere Algebra published in 1933)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In the third edition of Mengenlehre (Hausdorff 1914; the third edition was published in 1937 and then as Set Theory in English in 1957), the term equivalence is still attached only to the context of cardinality. Even in 1942 and in a paper with the title Theory of Equivalence Relations, Oystein Ore called them “the blocks of the partition &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;&amp;quot; (Ore 1942,p.574)&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Von Neumann (1926, 1929), Hopf (1930), and Seifert and Threlfall (1934) used the term “Äquivalenzklasse”, von Neumann (1936) “equivalence-class” (with a hyphen) and Solomon Lefschetz (1938, 1942) “equivalence class” (without a hyphen). The terminology was for denoting the classes constructed by certain contextualized equivalences, none of which was referred to as an “equivalence relation”. Moreover, none of these authors felt obliged to define “equivalence classes” in general terms and outside the contexts that they were working.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;A foundational treatment is found in the first chapter of Tukey’s Convergence and Uniformity in Topology (Tukey 1940, p. 4), where the combinations “equivalence relation” and “equivalence class” appear in the same place... Generally, the combination “equivalence class” became common much later than the “equivalence relation” while closer to 1950 we can observe clear uses of both combinations alongside each other.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-03-28T20:15:03.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11598" PostTypeId="1" CreationDate="2020-03-28T10:26:05.087" Score="9" ViewCount="1405" Body="&lt;p&gt;In his well-known measurement of the Earth, and according to Cleomedes, Eratosthenes estimated in 5000 stades the distance between Aswan and Alexandria. Modern accounts state that he calculated the distance from the time taken by caravans (improbable, as the Nile was navigable all the way, downstream from Aswan, so caravans weren't needed) or that he hired walking surveyors for that task. For example, in&#xA;&lt;a href=&quot;https://www.aps.org/publications/apsnews/200606/history.cfm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.aps.org/publications/apsnews/200606/history.cfm&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He realized that if he knew the distance from Alexandria to Syene, he could easily calculate the circumference of Earth. But in those days it was extremely difficult to determine distance with any accuracy. Some distances between cities were measured by the time it took a camel caravan to travel from one city to the other. But camels have a tendency to wander and to walk at varying speeds. So Eratosthenes hired bematists, &lt;strong&gt;professional surveyors&lt;/strong&gt; trained to walk with equal length steps. They found that Syene lies about 5000 stadia from Alexandria.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've been unable to confirm this in old, Greek or Latin accounts.&lt;/p&gt;&#xA;&lt;p&gt;Any comments?&lt;/p&gt;&#xA;" OwnerUserId="6222" LastEditorUserId="4931" LastEditDate="2021-07-22T17:09:36.553" LastActivityDate="2024-02-24T06:57:29.593" Title="How did Eratosthenes know the distance between Aswan and Alexandria?" Tags="|reference-request|geometry|ancient-greece|geography|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11599" PostTypeId="2" ParentId="6423" CreationDate="2020-03-28T17:56:26.400" Score="5" Body="&lt;p&gt;The earliest reference appears to be:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;... it is easy to see how &lt;strong&gt;a machine could be programmed so that it appeared&#xA;  to learn&lt;/strong&gt;... whether it would in principle be possible to construct a generalized learning programme which would enable an operator, if he had sufficient patience,&#xA;  to 'teach' the machine any subject he chose...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can Machines Think?, M. V. Wilkes (1951)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;C.f:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786441208520256&quot; rel=&quot;nofollow noreferrer&quot;&gt;Programming a digital computer to learn&lt;/a&gt;, A. G. Oettinger (1952)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://dl.acm.org/doi/10.1147/rd.21.0002&quot; rel=&quot;nofollow noreferrer&quot;&gt;A learning machine&lt;/a&gt;, R. M. Friedberg (1958)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Before the term &quot;machine-learning&quot; itself gained currency:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Some experiments in machine learning, H. Campaigne (Mar 1959)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ieeexplore.ieee.org/abstract/document/5392560&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some Studies in Machine Learning Using the Game of Checkers&lt;/a&gt;, A. L. Samuel (Jul 1959)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://dl.acm.org/doi/abs/10.1145/612201.612227&quot; rel=&quot;nofollow noreferrer&quot;&gt;Generalization of learning in a machine&lt;/a&gt;, R. J. Lee (Sep 1959)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0019995859800140&quot; rel=&quot;nofollow noreferrer&quot;&gt;Two notes on machine “Learning”&lt;/a&gt;, H. H. Martens (Dec 1959)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10974" LastEditorUserId="10974" LastEditDate="2020-04-25T18:08:53.743" LastActivityDate="2020-04-25T18:08:53.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11600" PostTypeId="1" CreationDate="2020-03-29T06:51:29.980" Score="5" ViewCount="311" Body="&lt;p&gt;It is common to find accounts of Democritus explaining his thought experiment to demonstrate the existence of atoms by taking a piece of rock/shell/cheese, and breaking it in smaller and smaller bits until no more division is possible. I recall the cheese version, and searching recently, found many instances of the shell version.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What is the main source of the story?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Our main reference on Democritus is Diogenes Laërtius, and I could not find the story there.&lt;/p&gt;&#xA;" OwnerUserId="1861" LastEditorUserId="55" LastEditDate="2020-03-29T10:37:05.300" LastActivityDate="2020-03-29T10:37:05.300" Title="What are the sources for Democritus’ experiment of dividing a shell down to its atoms?" Tags="|reference-request|ancient-greece|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11601" PostTypeId="2" ParentId="11600" CreationDate="2020-03-29T10:28:39.457" Score="5" Body="&lt;p&gt;It would be nice if people stopped ascribing to the source what they think is a neat illustration of the source's idea, especially when it does not really make sense. Not much cogent on Democritus's natural philosophy survives, and what there is is in retellings of Aristotle, Diogenes Laertius and Simplicius. Neither of them tells anything close to the shell story. The obvious flavor borrowed from Zeno's Dichotomy (in some narrations the shell is first split in half, then a half is itself split in half, and so on) suggests how it might have been invented. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more cautious academic accounts forego the shell and simply state that Leucippus and Democritus developed atomism in response to Zeno's (and Parmenides's) arguments. Even that is somewhat controversial, but most modern scholars accept it (see &lt;a href=&quot;https://plato.stanford.edu/entries/atomism-ancient/#AtomBefoLeuc&quot; rel=&quot;noreferrer&quot;&gt;SEP, Atomism before Leucippus?&lt;/a&gt;) based on Aristotle's passages in &lt;a href=&quot;https://archive.org/details/oncomingtobepass00arisuoft/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;De Generatione et Corruptione&lt;/em&gt; I, 8&lt;/a&gt; and in Physics A3, 187, a1-3, see &lt;a href=&quot;http://www.filosofia.unimi.it/zucchi/NuoviFile/Barnes%20%20-%20Physics.pdf&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt;. The rest is an interpolation with a rock/shell/cheese &quot;experiment&quot; added for vividness and &quot;scientificity&quot;. Perhaps, following the lead of Lucretius, who already popularized arguments for atomism for the general audience in &lt;em&gt;De Rerum Natura&lt;/em&gt;, but more competently.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The relevant passages are discussed in detail in &lt;a href=&quot;https://books.google.com/books/about/Two_Studies_in_the_Greek_Atomists.html?id=z07WCgAAQBAJ&quot; rel=&quot;noreferrer&quot;&gt;Furley's Two Studies in the Greek Atomists, chapter 6&lt;/a&gt;. The closest Aristotle comes to linking atomists to Dichotomy is a passage from Physics:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Some gave in to both of these [sc. Eleatic] arguments - to the&#xA;  argument that all is one if &quot;what is&quot; means one, by saying that&#xA;  not-being exists, and to the argument from Dichotomy, by positing&#xA;  atomic magnitudes.&lt;/em&gt;&quot; [Furley's translation]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Alas, ancient commentators of this passage thought that &quot;some&quot; refers to Plato and Xenocrates, so even this is controversial. And if Leucippus or Democritus sought to answer Zeno the &quot;experiment&quot; with &quot;indivisible&quot; shell powder is just silly, Zeno's argument does not depend on how far things can be &lt;em&gt;physically&lt;/em&gt; split, as Furley himself remarks.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;em&gt;De Generatione et Corruptione&lt;/em&gt; Democritus is at least mentioned by name around an argument &quot;&lt;em&gt;which appears to prove that there are indivisible magnitudes&lt;/em&gt;&quot;. The argument is theoretical and involves &lt;em&gt;potentially&lt;/em&gt; (i.e. conceptually, not physically) dividing &quot;&lt;em&gt;a stick or something else... at whatever point&lt;/em&gt;&quot;, which makes more sense, see &lt;a href=&quot;https://www.academia.edu/790551/Aristotles_diagnosis_of_atomism&quot; rel=&quot;noreferrer&quot;&gt;Aristotle's Diagnosis of Atomism by Hasper&lt;/a&gt; for another reconstruction. Although this is the closest we have to &quot;Democritus's exposition&quot;, the prominence of Aristotle's own potentiality in it makes some scholars dispute even this attribution. Here is Aristotle's summary of the conclusion:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Neither, then, may one dividing in successive stages bring about an&#xA;  infinite process of breaking, nor is it possible for the magnitudes to be divided at every point (κατά πάν σημεϊον) at the same time (for it is not possible), but [only] up to a limit (άλλα μέχρι του). It is necessary, therefore, that there are invisible atomic magnitudes in it.&lt;/em&gt;&quot; [Hasper's translation]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although Zeno is not mentioned, Luria observed that the argument Aristotle retells seems to parallel one of his lesser known arguments reported by Simplicius. As for Lucretius, he does talk of physical splitting in &lt;em&gt;De Rerum Natura&lt;/em&gt; I.746-752. But there is still no shell, and he does not suggest that the powder particles we see are the atoms. Even his is not an &quot;experiment&quot; but only an analogy:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Next, because they assume that there is no end to the cutting&#xA;  up of bodies, and that no stop is made in their breaking, and&#xA;  indeed that there is no minimum at all in things; though we see&#xA;  that there exists in everything that last extremity which is seen&#xA;  to be the minimum with respect to our senses, so that you may&#xA;  infer from this that of those things which you cannot see the last&#xA;  extremity is the minimum&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-03-29T10:35:31.930" LastActivityDate="2020-03-29T10:35:31.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11602" PostTypeId="2" ParentId="11560" CreationDate="2020-03-29T15:52:24.473" Score="1" Body="&lt;p&gt;William Clifford didn't define a spacetime continuum, but he did write in &lt;em&gt;On the Spacetime theory of Matter&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.&lt;/p&gt;&#xA;&lt;p&gt;I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact&lt;/p&gt;&#xA;&lt;p&gt;(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.&lt;/p&gt;&#xA;&lt;p&gt;(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.&lt;/p&gt;&#xA;&lt;p&gt;(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.&lt;/p&gt;&#xA;&lt;p&gt;(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This conjecture was formed in 1876, fifty years before Einstein wrote down his equations for GR! Moreover, he went onto write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am endeavouring in a general way to explain the laws of double refraction on this hypothesis ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So although he didn't dream up gravitational lensing, he could see that, given his conjecture, there were implications for how light travelled in space.&lt;/p&gt;&#xA;&lt;p&gt;Another important precursor is Hertz who, thinking that there were two fundamental concepts - inertia and force - felt that this was one concept too many and theorised that all forces could be rethought as constraint forces and to accomplish this introduced the notion of higher spaces, that is higher dimensional spaces. This is an important conceptual advance for both GR (gravity is a constraint force) and Kaluza-Klien unification which was revived in the much later String Theory.&lt;/p&gt;&#xA;&lt;p&gt;It's also worth noting that Schrodinger in his essay &lt;em&gt;Science &amp;amp; Humanism&lt;/em&gt; cautioned identifying the mathematical or geometrical continuum with the physical continuum:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However painful its loss may be, by losing it we probably lose something that is very well worth losing. It seems simple to us, because the idea of the continuum seems simple to us. We have somehow lost sight of the difficulties it implies ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And in fact the same had been said by Einstein, who against the popular view of GR as a geometric theory insisted on its physicality by pointing out that the equation of motion of a free particle in GR, geometrically seen as the geodesic equation, was for him, pre-eminently a unification of inertia and gravity - both physical rather than geometric concepts.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-09-03T03:29:43.480" LastActivityDate="2020-09-03T03:29:43.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11603" PostTypeId="2" ParentId="8332" CreationDate="2020-03-31T02:17:28.323" Score="8" Body="&lt;p&gt;I am aware that this is a physics answer rather than mathematics, but in science probably the most famous example is William Bragg and son Lawrence Bragg who did pioneering experimental work with X-rays.  They are so far the only father-son team to win a Nobel Prize and Lawrence remains the youngest person to ever win a Nobel Prize in Physics, being aged only 25.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was also going to mention the Kuperbergs, but I have seen someone else has already mentioned them.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastActivityDate="2020-03-31T02:17:28.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11604" PostTypeId="1" AcceptedAnswerId="11609" CreationDate="2020-03-31T12:03:53.120" Score="4" ViewCount="862" Body="&lt;p&gt;I was reading the wikipedia article about the normal distribution and it's attribution by some historians of science to De Moivre(althought he lacked the concept of the probability density function) and I was wondering as to when was the probability density function defined.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="55" LastEditDate="2020-04-01T06:18:53.267" LastActivityDate="2020-04-01T06:38:03.223" Title="When did the concept of probability density explicitly appear in mathematics?" Tags="|mathematics|biographical-details|probability|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11605" PostTypeId="1" AcceptedAnswerId="11608" CreationDate="2020-03-31T16:25:48.173" Score="6" ViewCount="374" Body="&lt;p&gt;Bertrand Russel gave an exhaustive treatment of creating mathematics from logic in Principia Mathematica (1910-1913), using the logical notation created by Frege and Peano. As monumental as this is, I find it difficult to read, having learned logic with modern notation.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;⊢:.p.q.⊃.∼r:≡:q.r.⊃.∼p&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;is &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;p &amp;amp; q ⊃ ∼ r ≡ q &amp;amp; r ⊃ ∼p&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;with current expectations of precedence.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is: &lt;strong&gt;when did this newer notation become more popular and who were the ones who first started using it?&lt;/strong&gt; &lt;a href=&quot;https://plato.stanford.edu/entries/goedel/&quot; rel=&quot;noreferrer&quot;&gt;Goedel (1930)&lt;/a&gt; seems not to use it, but that's a twenty-year difference, hardly time for a change to come about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There are all sorts of &lt;a href=&quot;https://plato.stanford.edu/entries/pm-notation/&quot; rel=&quot;noreferrer&quot;&gt;references explaining the old way&lt;/a&gt;, but nothing ever seems to mention how the newer way started to become popular.&lt;/p&gt;&#xA;" OwnerUserId="8179" LastEditorUserId="10106" LastEditDate="2020-04-01T01:09:31.277" LastActivityDate="2020-04-03T00:25:17.047" Title="Who superseded Peano's dot notation in symbolic logic and when?" Tags="|notation|mathematical-logic|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11606" PostTypeId="2" ParentId="2450" CreationDate="2020-03-31T18:42:03.427" Score="1" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;Were there other related definitions defined around this time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In physics textbooks, vectors are introduced first, and then vector fields; given the complexity of the notion of tensors, one would suppose tensors would then be introduced and then tensor fields. Instead, a quite complex and obscure definition of a tensor field (and not a tensor) is presented without enough motivation to the curious - this is the definition you have more or less presented. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That the definition is obscure has been understood, with various physicists, such as  Dirac, in his short book on GR, noticing the explanatory lacuna and who suggests an 'affine' description of a tensor would be worth looking at before moving to the notion of a tensor in general; but he doesn't attempt a description of an 'affine' tensor. Weinberg, in his book &lt;em&gt;Gravitation and Cosmology&lt;/em&gt;, does make an attempt at it but the result is not much more clear than the original definition. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, one would suppose that Einstein himself, given the importance of tensors in his theory of gravity, would make an attempt to explain the notion in his book with Infeld, &lt;em&gt;The Evolution of Physics&lt;/em&gt;; but whilst they pays careful attention to the notion of a vector - they, unfortunately, they do not discuss the notion of a tensor. This is simply, as I've already noted, that the actual definition used by physicists, isn't intuitive (in fact, it is, when thought through but nevertheless, it is long-winded and relies on an operational definition rather than an intrinsic definition).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main alternative presentation of tensors is most often used by mathematicians rather than physicists, however, this tends to be increasingly not the case with more advanced work. It's presented, for example, in MacLanes book, &lt;em&gt;Categories for the Working Mathematician&lt;/em&gt; (first published in 1971) by what is known as universal properties. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Given modules &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; over a commutative ring &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, a tensor product is a universal element of the set &lt;span class=&quot;math-container&quot;&gt;$\text{Bilin}(A,B)$&lt;/span&gt; of bilinear functions &lt;span class=&quot;math-container&quot;&gt;$\beta:A \times B \rightarrow C$&lt;/span&gt; to some third &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-module &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;. This set is (the object function of) a functor of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;. To get a solution set for given &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, it suffices to consider only those bilinear &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; which span &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; (do not factor through a proper submodule of &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;). Then &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; consists of all finite sums &lt;span class=&quot;math-container&quot;&gt;$\sum(\beta(a_i,b_i))$&lt;/span&gt; ... [so] a tensor product &lt;span class=&quot;math-container&quot;&gt;$\otimes : A \times B \rightarrow A \otimes B$&lt;/span&gt; exists. The usual (more explicit) construction is wholly needless, since all the properties of the tensor product follow directly from the universality. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;He notes, that although Bourbaki, in principle, came close to this definition, but given that they didn't have the notion of a universal property to hand - they missed it.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="58" LastEditDate="2020-06-08T21:13:40.300" LastActivityDate="2020-06-08T21:13:40.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11607" PostTypeId="2" ParentId="8332" CreationDate="2020-03-31T20:11:59.733" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Arto_Salomaa&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arto Salomaa&lt;/a&gt; and his son &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Kai_Salomaa&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kai Salomaa&lt;/a&gt; both work in theoretical computer science. They have several joint research papers. An notable example is the following:&lt;/p&gt;&#xA;&lt;p&gt;Tao Jiang, Arto Salomaa, Kai Salomaa, Sheng Yu:&#xA;Decision Problems for Patterns. Journal of Computer and System Sciences 50(1): 53-63 (1995)&lt;/p&gt;&#xA;&lt;p&gt;Marcin and Michał Philipczuk are &lt;a href=&quot;https://www.mimuw.edu.pl/%7Emp248287/links.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;brothers&lt;/a&gt;. They work together very often - and their &lt;a href=&quot;https://dblp.uni-trier.de/pers/hd/p/Pilipczuk:Marcin&quot; rel=&quot;nofollow noreferrer&quot;&gt;dblp records&lt;/a&gt; already show more than 100 joint publications. To give an example:&lt;/p&gt;&#xA;&lt;p&gt;Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh: Minimum Bisection Is Fixed-Parameter Tractable. SIAM Journal on Computíng, 48(2), 417–450.&lt;/p&gt;&#xA;&lt;p&gt;(I assume that theoretical computer science papers qualify as mathematical papers, since they contain mathematical theorems and rigorous proofs.)&lt;/p&gt;&#xA;&lt;p&gt;The computer scientists Dorothea Blostein and Lippold Haken are children of &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Wolfgang_Haken&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wolfgang Haken&lt;/a&gt;. According to Blostein's Wikipedia &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Dorothea_Blostein&quot; rel=&quot;nofollow noreferrer&quot;&gt;entry&lt;/a&gt;, &amp;quot;while she was an undergraduate student she helped check her father's proof of the four color theorem.&amp;quot; Less mathematically, the two siblings Dorothea and Lippold wrote several joint papers on computer music, for example:&lt;/p&gt;&#xA;&lt;p&gt;Dorothea Blostein, Lippold Haken: The Lime Music Editor: a Diagram Editor Involving Complex Transformations. Software - Practice and Experience 24(3): 289-306 (1994)&lt;/p&gt;&#xA;" OwnerUserId="9523" LastEditorUserId="9523" LastEditDate="2021-02-02T19:07:55.813" LastActivityDate="2021-02-02T19:07:55.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11608" PostTypeId="2" ParentId="11605" CreationDate="2020-04-01T00:03:04.093" Score="8" Body="&lt;p&gt;This is not so straightforward, see &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01445348408837062&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peirce, Frege, the Logic of Relations, and Church's Theorem by Dipert&lt;/a&gt; for a sketch of history. The notation Russell used was not created by Peano, and certainly not by Frege. Nobody used Frege's convoluted notation except his own &lt;em&gt;Begriffsschrift&lt;/em&gt; (1879), not even Frege himself afterwards. Peano did contribute, but the basis of &lt;em&gt;Principia's&lt;/em&gt; notation and logical content was actually &lt;a href=&quot;https://archive.org/details/vorlesungenberd03mlgoog/page/n8/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schröder's &lt;em&gt;Vorlesungen über die Algebra der Logik&lt;/em&gt; (1895)&lt;/a&gt;, &lt;em&gt;Principia's&lt;/em&gt; precursor in symbolic treatment of modern logic. In its turn, Schröder's treatment goes back not to Frege but to Peirce, who invented predicate logic with quantifiers in 1881-85 independently of Frege. Here is Dipert's surmise:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Neither Peirce nor&#xA;  Schröder had the services of such an excellent propagandist as Russell. The Peirce-&#xA;  Schröder calculus was portrayed as purely algebraic, without the variable-binding&#xA;  operators Peirce regarded as essential and to which Schröder usually resorted; its&#xA;  weaknesses were rhetorically exploited with the bon mot 'too complicated'; its&#xA;  subtlest achievements were ignored (e.g., clever theorems proven and Peirce's&#xA;  insights into the differences between the monadic and polyadic predicate logic); and,&#xA;  in a final injustice, the development of the theory of relations in Principia Mathematica&#xA;  owes most, especially in notation, to Schröder via the influence of Peano rather&#xA;  than to Frege, but it was presented without substantial acknowledgement.&#xA;  Alfred Tarski is one of the few logicians or historians writing in the 20th century&#xA;  who seems to realize the proportions of this injustice.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The Peirce-Schröder's notation was more mathematician friendly and structurally closer to the modern one than Russell's. They used &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Pi$&lt;/span&gt; for the existential and universal quantifiers, and did not use dotting for punctuation. That was Peano's creation, see &lt;a href=&quot;https://plato.stanford.edu/entries/pm-notation/#UseDotsForPunc&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, Use of Dots for Punctuation&lt;/a&gt;. He also changed &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\exists$&lt;/span&gt; for the existential quantifier, and &lt;span class=&quot;math-container&quot;&gt;$(x)$&lt;/span&gt; for the universal was Russell's preference. &lt;a href=&quot;https://www.ams.org/journals/bull/1930-36-01/S0002-9904-1930-04859-4/S0002-9904-1930-04859-4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert-Ackerman's &lt;em&gt;Grundzüge der Theoretischen Logik&lt;/em&gt; (1928)&lt;/a&gt;, the first textbook on mathematical logic, used &lt;em&gt;Principia's&lt;/em&gt; notation, but the dotting is already gone in &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.358238/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert-Bernays's &lt;em&gt;Grundlagen Der Mathematik&lt;/em&gt; (1934)&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Gentzen proposed &lt;span class=&quot;math-container&quot;&gt;$\forall x$&lt;/span&gt; in 1935, by analogy to &lt;span class=&quot;math-container&quot;&gt;$\exists$&lt;/span&gt;, but &lt;em&gt;Principia&lt;/em&gt;'s version and the dotting persisted well into 1950-s. Church, Gödel, Quine used it, and so did &lt;a href=&quot;https://archive.org/details/logicformathemat00ross/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rosser's Logic for Mathematicians (1953)&lt;/a&gt;. However, Quine's From a Logical Point of View (1953) already uses parentheses instead of dots. The switch is likely associated with the &quot;transfer of ownership&quot; of logic from philosophers to mathematicians in the late 1950s early 1960s. The catalyst might have been Bourbaki's systematization of mathematics in &lt;a href=&quot;http://archives-bourbaki.ahp-numerique.fr/elements-mathematique#elements-math1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Éléments de Mathématique&lt;/em&gt;&lt;/a&gt;. Summary of the first volume, &lt;em&gt;Théorie des Ensembles&lt;/em&gt; (&lt;em&gt;Fascicule de Résultats&lt;/em&gt;), came out in 1939, and the chapters on formal logic and set theory in 1954. Bourbaki used Gentzen's quantifier and no dotting. Mendelsohn's Introduction to Mathematical Logic (1964), which became a standard textbook, already used modern notation.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-03T00:25:17.047" LastActivityDate="2020-04-03T00:25:17.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11609" PostTypeId="2" ParentId="11604" CreationDate="2020-04-01T06:18:30.353" Score="8" Body="&lt;p&gt;The answer depends somewhat on what &quot;explicitly&quot; means. The appearance of continuous distributions, and their densities, is generally attributed to Simpson. Namely, his 1757 response to Bayes's critique of his 1755 letter on de Moivre's theory of errors. As Stigler writes in &lt;a href=&quot;https://www.hup.harvard.edu/catalog.php?isbn=9780674403413&quot; rel=&quot;noreferrer&quot;&gt;The History of Statistics&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Simpson's 1757 republication of the letter incorporated one other&#xA;  change - three and a half pages of additional material that is frequently&#xA;  cited now as the first publication of a continuous error distribution... Simpson&#xA;  (1757, p. 71) proposed to consider the situation ''when the error admits of&#xA;  any value whatever, whole or broken, within the proposed limits, or when&#xA;  the result of each observation is supposed to be accurately known.&quot; His&#xA;  analysis, in the spirit of the time, consisted of viewing the interval of possible errors (AB in Figure 2.3) as being divided into an indefinitely&#xA;  large number of indefinitely small subintervals, and effectively passing to&#xA;  the limit in his earlier expression for the discrete case...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;In modern terminology, Simpson's idea is that we may consider the&#xA;  triangle ABD as a probability density along the axis AB... the curve AFEFB represented the&#xA;  probability density for the mean error. He integrated the resulting density&#xA;  to find that the probability that the mean of t errors did not exceed&#xA;  &lt;span class=&quot;math-container&quot;&gt;$(1 - y/t)\times$&lt;/span&gt;CA in absolute value&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This said, Simpson is certainly very far from abstract notions of probability distribution and density. Bayes's 1764 essay is noted as a step forward, &quot;&lt;em&gt;remarkable for its careful treatment of probability densities,&#xA;treating all probabilities as areas and not relying (in the later manner&#xA;of both French and English mathematicians) on the interpretation of densities&#xA;as attaching infinitesimal probability to points&lt;/em&gt;&quot; (as Maxwell did as late as 1860). But he still deals with a specific density function. Finally, Laplace wrote a number of memoirs on the &quot;error curve&quot; in 1770s, where the said curve (density) gets denoted generally &lt;span class=&quot;math-container&quot;&gt;$\phi(x)$&lt;/span&gt;. He was aiming at the normal distribution density, but did not succeed at identifying it at the time, and had to treat his &lt;span class=&quot;math-container&quot;&gt;$\phi(x)$&lt;/span&gt; more abstractly:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;All&#xA;  early workers, from Simpson on, accepted it as given that the curve should&#xA;  be symmetrical and that the chance of an error should decrease toward&#xA;  zero as the magnitude of the error increased. Laplace himself had repeated&#xA;  these conditions in his 1774 memoir, as I have noted. The problem&#xA;  was that there were too many possibilities, and the choice of just one was&#xA;  critical to the mean to be obtained. As Laplace wrote, &quot;But of an infinite&#xA;  number of possible functions, which choice is to be preferred?&quot;&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Earlier writers had confronted the choice of an error curve by making&#xA;  an arbitrary selection. Simpson, for example, had based his choice of the&#xA;  uniform and triangular distributions upon mathematical expediency. For&#xA;  his limited aim (showing that an arithmetic mean was better than a single&#xA;  observation) this arbitrariness was not a major drawback - his audience&#xA;  only needed to accept the distribution as qualitatively correct in order to&#xA;  believe his qualitative conclusion - that averaging increased accuracy.&#xA;  Laplace, however, was attempting an exact computation, and he needed&#xA;  both a convincing case for any choice of &lt;span class=&quot;math-container&quot;&gt;$\phi(x)$&lt;/span&gt; he might make and a mathematical&#xA;  analysis equal to the task of calculating the best mean for that curve.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As for the terminology, it is quite recent. According to &lt;a href=&quot;https://www.leidenuniv.nl/fsw/verduin/stathist/1stword.htm&quot; rel=&quot;noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Probability &amp;amp; Statistics&lt;/a&gt; Markoff came up with &quot;&lt;em&gt;Wahrscheinlichkeitsdichte&lt;/em&gt;&quot; (probability density) only in 1912, and the &quot;distribution function&quot;, of which it it the derivative, only appears in von Mises's &lt;em&gt;Grundlagen der Wahrscheinlichkeitsrechnung&lt;/em&gt;  (1919).&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-01T06:35:44.780" LastActivityDate="2020-04-01T06:35:44.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11610" PostTypeId="2" ParentId="11604" CreationDate="2020-04-01T06:27:37.053" Score="2" Body="&lt;p&gt;Hans Fischer, in his &lt;em&gt;A History of the Central Limit Theorem&lt;/em&gt; writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The history of the CLT as a universal law begins with Laplace; all relevant studies in the 18th century, starting with [de Moivre 1733], essentially contained&#xA;  only approximations of the binominal distribution and their scope of application remained narrow. Laplace’s finding of 1810, according to which the additive&#xA;  coaction of a large number of independent random variables generally leads to&#xA;  probabilities that can, at least approximately, be calculated according to the normal distribution, substantially expanded the numerical possibilities of probability&#xA;  theory, especially in the discussion of mass phenomena&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In one of his first published papers [in 1776], Laplace had already set out to&#xA;  determine the probability that the sum of the angles of inclination of comet orbits (or&#xA;  the arithmetic mean of these angles respectively) is within given limits. He assumed&#xA;  that all angles, which had to be measured against the ecliptic, were distributed randomly according to a uniform distribution between 0 and 90 degrees (and also tacitly&#xA;  presupposed that all angles were stochastically independent). Laplace succeeded in&#xA;  calculating these probabilities for an arbitrary number of comets via induction (with&#xA;  a minor mistake which was subsequently corrected in his paper of 1781) . In this [latter] paper, Laplace even introduced a general — however very intricate — method, based&#xA;  on convolutions of &lt;em&gt;density functions&lt;/em&gt;, in order to exactly determine the probability&#xA;  that a sum of independent random variables (“quantités variables,” as Laplace put it)&#xA;  was within given limits&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Personally, I feel that the notion of a probability density would have had earlier provenance; since, after the integral calculus had been invented, it wouldn't taken much to realise that the generalisation from finite probability to a continuous probability would require integration over a density (and the term density suggests that the notion migrated from physical theory). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It's probably worth adding that Kolmolgorov introduced the notion of a probability density in measure theoretic terms (the now standard notion) in his book, &lt;em&gt;The Foundations of the Theory of Probability&lt;/em&gt;, first published in 1933.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-04-01T06:38:03.223" LastActivityDate="2020-04-01T06:38:03.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11613" PostTypeId="1" AcceptedAnswerId="11614" CreationDate="2020-04-03T00:40:35.203" Score="6" ViewCount="848" Body="&lt;p&gt;When were the concepts of latitude and longitude first introduced? How did cartographers determine them?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="4251" LastEditDate="2020-07-09T20:43:47.073" LastActivityDate="2020-07-09T20:43:47.073" Title="How did cartographers of past centuries determine latitude and longitude?" Tags="|geography|navigation|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11614" PostTypeId="2" ParentId="11613" CreationDate="2020-04-03T12:54:07.270" Score="10" Body="&lt;p&gt;The concepts of latitude and longitude were introduced in antiquity, and our principal source for that period are the surviving books by Ptolemy (Geography and&#xA;Almagest). These notions were understood long before Ptolemy,&#xA;at the time of Eratosthenes, but we have very little sources on astronomy before Ptolemy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Determining latitude is easy: all one has to measure is&#xA;the altitude of Sun at noon. Or of some star at the moment of culmination. This was&#xA;known (and actually performed) by Hellenistic astronomers in 3-2 centuries BC. One can use the simplest of all astronomical tools: the gnomon. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Determining longitude is much more difficult, and this was done by the ancient cartographers by estimating the distance from the travel time needed to reach one point from another.&#xA;Of course all these determinations could be only very rough. The problem of longitude is equivalent to the problem of synchronizing clocks at a large distance. Before the invention of telegraph only astronomical methods were available.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ptolemy was aware of the principle that longitude can be found from observations of&#xA;Lunar eclipses (unlike Solar eclipses, they appear at the same time everywhere on Earth), but there is no ancient record of any serious attempts to do this in practice. This, and observation of star occultations&#xA;(moments of covering of a star by the Moon)  were the only methods available in antiquity, but they lead to large mistakes because there was no good theory of the Moon motion. For example, Columbus tried to estimate the&#xA;longitude of Americas by observing a Lunar eclipse, but with no much success.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The next step was made by Galileo who discovered Jupiter's satellites, and proposed&#xA;to use their eclipses by Jupiter to determine longitude. This was a very good method for cartography, but despite many attempts one could not make it work at sea (from a ship) because it requires a large telescope, and there is no way to stabilize&#xA;a telescope on a ship. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Since your question is about cartography, I will skip&#xA;the further development of methods suitable for observations from a ship  which were developed in&#xA;18th century (Lunar distances and chronometers). But for long time even after that, the method based on Jupiter's satellites remained the most accurate method. (Transportation of a chronometer keeping good time over land is even more difficult than transportation by sea. For example, Lewis and Clark used Lunar distances when mapping North America. Lunar distances&#xA;are measured by a small  portable instrument, called sextant, which is much handier than a large telescope needed for Jupiter satellites.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method of Lunar distances is based on determining the exact position&#xA;of the Moon in the sky by measuring distances from the Moon to stars.&#xA;If a good, exact theory of Moon motion is available, you can find the time&#xA;from the position of the Moon.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From the perspective of a cartographer, the methods based on Jupiter satellites and Lunar distances were superseded only with the advent of&#xA;long-distance telegraph cables,&#xA;when synchronizing clocks by telegraph became possible.&#xA;This happened in the second half of 19th century. Since then, we know the longitudes of all places (where a nearby telegraph is available) with high accuracy. The first &lt;a href=&quot;https://en.wikipedia.org/wiki/Transatlantic_telegraph_cable#Great_Eastern&quot; rel=&quot;nofollow noreferrer&quot;&gt;transatlantic cable&lt;/a&gt; was laid in 1866, and this permitted for the first time to determine longitude of places in Americas with higher accuracy then it was possible by astronomical methods.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then, in the beginning of 20th century, radio broadcast of time signals gave a simple and convenient solution for all locations. The last step was technologies based on artificial satellites which came in the late 1970s.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-04-10T00:48:39.103" LastActivityDate="2020-04-10T00:48:39.103" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11615" PostTypeId="1" CreationDate="2020-04-03T17:37:44.413" Score="1" ViewCount="86" Body="&lt;p&gt;Who first defined temperature &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; such that  &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{T} = \frac{\delta S}{\delta U}$&lt;/span&gt;, and where and when?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It could not have been Thomson, right? He did not have an entropy concept. Clausius? &lt;/p&gt;&#xA;" OwnerUserId="7494" LastActivityDate="2020-04-06T17:31:41.437" Title="Who introduced 1/T=δS/δU and when?" Tags="|thermodynamics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11616" PostTypeId="1" AcceptedAnswerId="11619" CreationDate="2020-04-04T17:08:58.837" Score="1" ViewCount="133" Body="&lt;p&gt;I was reading a book about computational complexity theory and the author made a claim that the study of time complexity of algorithms started with a result on the upper bound on the number of operations needed by euclid's algorithm on two numbers by Gabriel Lamé in the early/mid 1800s.&#xA;Since analyses of speed of convergence of algorithms can also be thought as sort of proto-time complexity analyses and date back at least to Lagrange I was wondering if they  are even earlier than Lagrange.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-04-08T10:54:16.843" LastActivityDate="2020-04-08T10:54:16.843" Title="When did the study of the rate of convergence of algorithms begin?" Tags="|mathematics|biographical-details|computer-science|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11617" PostTypeId="1" AcceptedAnswerId="11626" CreationDate="2020-04-04T17:34:22.360" Score="2" ViewCount="177" Body="&lt;p&gt;I just took a read of Clausius' original writing from 1867 on the second law. He refers to the &quot;&lt;em&gt;unit of heat&lt;/em&gt;&quot; as &quot;&lt;em&gt;that amount of heat which is required to increase the temperature of unit-weight of water by one degree&lt;/em&gt;&quot;. This makes sense to me as a unit of heat, but I am using my modern understanding of heat, in terms of kinetic energy of particles, to make sense of this. What would Clausius have meant by &quot;heat&quot; itself?  What would he have meant by &quot;&lt;em&gt;heat&lt;/em&gt; required to raise something by a temperature&quot;? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;They apparently understood that &quot;&lt;em&gt;there must be an equivalence between mechanical work and heat, as between cause and effect&lt;/em&gt;&quot; (wikipedia), but this doesn't explain what they actually thought heat was, or what it means to &quot;put heat into something to increase its temperature&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_heat&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page on Theory of heat&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/pdf/10.1080/14786446808640002?casa_token=1kxIVIewloEAAAAA:dtUQSK2-VxUdsFNA91jxay5Wo0VTiJETxZ6rDIlEf76ZMemO8w2KOf40uTbtQ8IVkTzTnX8Wb8hY&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Second Fundamental Theorem of the Mechanical&#xA;Theory of Heat (Clausius, 1867)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4556" LastEditorUserId="55" LastEditDate="2020-04-06T09:03:05.093" LastActivityDate="2020-04-06T17:21:54.667" Title="What did people understand heat and temperature to be in Clausius' time?" Tags="|physics|thermodynamics|statistical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11618" PostTypeId="1" AcceptedAnswerId="11622" CreationDate="2020-04-05T01:40:30.567" Score="9" ViewCount="199" Body="&lt;p&gt;In other words, how old is the practice of submitting mathematical work for peer review to specialized magazines? When/where it started to become the norm?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My question is oriented toward the evolution of specialized publications in mathematics, until they reached the status of becoming the de-facto certificate of quality (also known as the &lt;em&gt;publish or perish&lt;/em&gt; mandate).&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="12271" LastEditDate="2020-04-05T10:45:20.850" LastActivityDate="2021-05-01T02:29:20.780" Title="When and where did scientific publications become the norm in mathematics?" Tags="|mathematics|publishing|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11619" PostTypeId="2" ParentId="11616" CreationDate="2020-04-05T02:24:02.193" Score="4" Body="&lt;p&gt;Rate of convergence estimates are typically not counted as estimating complexity of algorithms because the relation between number of steps in them and computational complexity is not very straightforward. And the algebraic analysis conceptions of 18th century make comparisons to the modern complexity theory even more remote, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2038/55&quot;&gt;Was 18th century algebra more symbolic/formal than the modern conception?&lt;/a&gt; This is likely why its origins are only traced to the work of Reynaud (1811), Finck (1841) and Lame (1844) on the Euclidean algorithm, where both issues are absent.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, both the rate of convergence and its speed up were commonplace in the investigation of series in the early 18th century, albeit not in the way resembling the modern one. Ferraro has two papers &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00033790010028179&quot; rel=&quot;nofollow noreferrer&quot;&gt;Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730&lt;/a&gt; and &lt;a href=&quot;https://www.academia.edu/1486699/Convergence_and_formal_manipulation_in_the_theory_of_series_from_1730_to_1815?auto=download&quot; rel=&quot;nofollow noreferrer&quot;&gt;Convergence and formal manipulation in the theory of series from 1730 to 1815&lt;/a&gt;, on the subject. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Jacob Bernoulli explicitly introduced and estimated remainders for the geometric series and the Mercator method in &lt;em&gt;Ars Conjectandi&lt;/em&gt; (1713). But these sorts of considerations happily coexisted with the formal conception of series and manipulating divergent series, which got him in trouble with the &quot;sum&quot; of inverse square roots. Attempts to improve rates of convergence themselves often involved formal manipulations, and led to the appearance of divergent asymptotic series, as in Stirling's &lt;em&gt;Methodus Differentialis&lt;/em&gt; (1730). Even d'Alembert, who criticized the use of divergent series and made detailed rate of convergence estimates for the binomial series in &lt;em&gt;Réflexions sur les Suites et sur les Racines Imaginaires&lt;/em&gt; (1768), was no exception.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Although N. Bernoulli's and d'Alembert's conception might seem close to modern ideas about series, it was in fact rather problematic... I stress that d'Alembert accepted the principle of formal manipulation upon which the early theory was based: this was precisely the principle that had led to asymptotic series, recurrent series, etc. In&#xA;  particular, d'Alembert did not consider series as autonomous objects but the result of transformations of given closed analytical expressions. Although it is true that d'Alembert used the inequality technique, it was only a tool for the numerical evaluation of a function. In no case was this technique used to prove&#xA;  the existence of a limit. D’Alembert had no knowledge of the ratio test, if by this term we refer to a criterion to establish whether a given series has a finite sum or not. For him the condition &lt;span class=&quot;math-container&quot;&gt;$a_{n+1}/a_n &amp;lt; 1$&lt;/span&gt; served to establish where the series approximated its known sum: the “ratio test” was not used to prove the existence of the sum but was only a procedure to determine the bounds of errors. In conclusion, criticisms of the use of divergent series were weak because all mathematicians used formal procedures even if some of them, such as Euler, Lagrange, and Laplace, used them in a stronger form.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This said, Lagrange did publish two long papers on the numerical solution of equations in 1769/70, and continued to work on the subject with his work eventually summarized in his book &lt;a href=&quot;https://archive.org/details/traitdelarso00lagr/page/n8/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Traite de la Resolution des Equations Numeriques de Tous les Degres&lt;/em&gt; (1798, revised 1808)&lt;/a&gt;. There is also a chapter on them in his &lt;a href=&quot;https://archive.org/details/lecturesoneleme00lagrgoog/page/n23/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lectures on Elementary Mathematics&lt;/a&gt;, translated into English. He paid special attention to the rate of convergence and accelerating it, both for series and continued fractions. In particular, he replaced Newton's method with a continued fractions based one that provided error estimates, and introduced generalized continued fractions to speed up the convergence, see &lt;a href=&quot;http://www.uco.es/~ma1mamaa/GIHEM/documentos/lagrange-and-the-solution.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange and the Solution of Numerical Equations by Laubenbacher and McGrath&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-05T02:31:19.937" LastActivityDate="2020-04-05T02:31:19.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11620" PostTypeId="2" ParentId="11244" CreationDate="2020-04-05T06:38:44.760" Score="7" Body="&lt;p&gt;I asked Bill Gosper, and here is is response:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There was email on this at the time, which I may be able to dig up.  But, as I recall, when I began&#xA;  my computation, the Borwein brothers had proved that if Ramanujan's formula did not equal π,&#xA;  it differed from π by at least 10^-3000000, so that as my computation passed the 3000000 mark&#xA;  in agreement with Kanada's 16000000 digit AGM computation, it served to complete the Borwein&#xA;  proof.  But by the time my computation reached 17000000, the Borwein's resolved their ambiguity&#xA;  without my empirical confirmation.  Their completed proof is almost certainly in their Pi and the AGM&#xA;  book.  Tito Piezas and the Chudnovsky brothers have presumably put the matter to rest with their&#xA;  climactic series based on √163, the final Heegner number.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Just to clarify, I did not crank out π as a decimal string.  I resumably summed Ramanujan's series&#xA;  as an exact rational number on a Symbolics computer with unlimited integers, converting to decimal&#xA;  now and then for comparison with Kanada, but with the ultimate purpose of computing the&#xA;  {3,7,15,1,292,...} continued fraction, which is actually mathematically interesting, as opposed to&#xA;  useless decimal or binary, which is actually an encryption.  Regrettably, (almost) everybody ignored&#xA;  my continued fraction and wasted their time computing (eventually trillions of) useless digits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10829" LastActivityDate="2020-04-05T06:38:44.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11622" PostTypeId="2" ParentId="11618" CreationDate="2020-04-05T13:36:36.923" Score="13" Body="&lt;p&gt;The earliest journals were multidisciplinary, they were published by academies since&#xA;the second half of&#xA;17th century. Before that time, communication was only by books and letters. Letters sometimes were copied and circulated. The first French, English and German journals were:&lt;/p&gt;&#xA;&lt;p&gt;Journal des sçavans, 1665 (published papers in all sciences and humanities)&lt;/p&gt;&#xA;&lt;p&gt;Philosophical transactions of the Royal Society, 1665 (all natural sciences and mathematics)&lt;/p&gt;&#xA;&lt;p&gt;Acta eruditorum, 1682 (Started by Leibniz, published in physics and mathematics).&lt;/p&gt;&#xA;&lt;p&gt;A curious example of multidisciplinary journal of 18th century was&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Ladies%27_Diary&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ladies' Diary&lt;/a&gt; started in 1704, where many mathematical papers were published, for example the&#xA;seminal papers of Landen on &amp;quot;Landen's transform&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Specialized mathematical journals appear in early 19 century.&#xA;The oldest specialized mathematical journal which still continues is the Journal fur die reine und angewandte Mathematik (a. k. a. Crelle's journal), started in 1826. A French analog (with identical title) Journal de Mathematiques pures et appliquee (Liouville's journal) started in 1836.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-05-01T02:29:20.780" LastActivityDate="2021-05-01T02:29:20.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11623" PostTypeId="1" CreationDate="2020-04-05T13:49:10.250" Score="4" ViewCount="482" Body="&lt;p&gt;Who discovered the indeterminate forms and how did they discover them? How did someone come to know that a particular form (fraction, product, sum/difference, exponent) is indeterminate? For example, &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt; is an indeterminate form. How did they come to know that when both numerator and denominator approach zero the fraction can be any number (which depends upon their respective rates)? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’ll be highly grateful if you explain it comprehensively, I’m a high school student. &lt;/p&gt;&#xA;" OwnerUserId="7928" LastEditorUserId="55" LastEditDate="2020-04-07T11:11:54.763" LastActivityDate="2020-04-07T11:23:46.987" Title="Who discovered the indeterminate forms like 0/0?" Tags="|mathematics|calculus|elementary-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11624" PostTypeId="1" AcceptedAnswerId="11625" CreationDate="2020-04-05T16:12:42.860" Score="8" ViewCount="1404" Body="&lt;p&gt;A &lt;strong&gt;non-derogatory matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;&lt;/strong&gt; is one, whose minimal polynomial &lt;span class=&quot;math-container&quot;&gt;$m(z)$&lt;/span&gt; equals its characteristic polynomial &lt;span class=&quot;math-container&quot;&gt;$p(z)$&lt;/span&gt;, where we apply the convention &lt;span class=&quot;math-container&quot;&gt;$p(z) = det(zI-A)$&lt;/span&gt;, while a matrix is &lt;strong&gt;derogatory&lt;/strong&gt;, if they do not coincide. I have certainly never felt particularly offended when dealing with the identity matrix &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;, so seriously &lt;strong&gt;I wonder, where that strange name has its origin&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Some &lt;strong&gt;background&lt;/strong&gt;: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would like to know about it, because I teach some advanced pupils at a german gymnasium school, who will be considering a field of study in the STEM-area, some linear algebra using examples from polynomial geometry. In that context, certain polynomials generate matrices with a double eigenvalue, which are either non-derogatory or diagonalizable, and where the non-derogatory ones are indeed &lt;strong&gt;preferable&lt;/strong&gt; to the diagonalizable ones due to the particular application. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In order to make the rather involved notion more palatable to the audience, it is always good to have a story to tell about it, even if it turns out to be rather dull in the end. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;My native language is german and I know of only one place in the german literature, where non-derogatory matrices get a special name. That is in E.Brieskorn &quot;Lineare Algebra und analytische Geometrie II&quot;, where they are called &quot;regular&quot;, which the author immediately distinguishes from another use of that word as invertible. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the contrary, the english denomination sems to be well-established when refering to Google and I found it already mentioned in the first chapter of J.H. Wilkinson &quot;The Algebraic Eigenvalue Problem&quot; from 1965 without further comment, so it seems to have been established by then. &lt;/p&gt;&#xA;" OwnerUserId="12287" OwnerDisplayName="thomashennecke" LastActivityDate="2020-04-06T07:49:23.307" Title="What is the origin of the term (non-)derogatory matrix?" Tags="|reference-request|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11625" PostTypeId="2" ParentId="11624" CreationDate="2020-04-06T07:49:23.307" Score="6" Body="&lt;p&gt;The original meaning is derived from Latin &lt;em&gt;derogare&lt;/em&gt;, &quot;to take away, detract from, diminish&quot;, see &lt;a href=&quot;https://www.etymonline.com/word/derogatory&quot; rel=&quot;noreferrer&quot;&gt;EtymOnline&lt;/a&gt;. Association with offense is from later, the 16th century. For matrices, the term was introduced by Sylvester in the early 1880s, in his matrix based reworking of the Hamilton's theory of quaternions, see &lt;a href=&quot;https://books.google.com/books/about/The_Emergence_of_the_American_Mathematic.html?id=uMvcfEYr6tsC&quot; rel=&quot;noreferrer&quot;&gt;The Emergence of the American Mathematical Research Community, 1876-1900, by Parshall and Rowe, p. 136&lt;/a&gt;. The &quot;diminishment&quot; probably refers to the degree of the characteristic polynomial being diminished in the minimal polynomial. On the other hand, Sylvester's another suggestion, to rename characteristic into &quot;latent&quot; polynomial, did not catch on.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-04-06T07:49:23.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11626" PostTypeId="2" ParentId="11617" CreationDate="2020-04-06T09:01:29.333" Score="4" Body="&lt;p&gt;Until 1850s the dominant theory was that the heat is caloric fluid, and the phrasing reflected that. The caloric theory was a revision by Lavoisier of the earlier phlogiston theory, and Carnot and others still worked with it. Only after the work of Mayer and Joule in 1840s, promoted by Kelvin and interpreted by Clausius, that the paradigm began to change in 1850s. While Clausius reinterpreted heat as mechanical energy of particle motion, he continued to use the established terminology from Carnot, as we still do. It is the same metaphorical phrasing as in &amp;quot;put energy into work&amp;quot;, or &amp;quot;energy drain&amp;quot; see &lt;a href=&quot;https://hsm.stackexchange.com/a/3474/55&quot;&gt;What are the major flaws of the “caloric” theory of heat?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Clausius was a firm, if at first secretive, believer in the kinetic theory. Since the temperature was a measure of caloric density, to him it was a measure of energy density. The vague idea about temperature as measuring intensity of micromotion, which in turn stems from analogizing spread of commotion in a particle medium to spread of fluid, goes back to Rumford and his 1798 inconclusive experiments on the mechanical equivalent of heat. The work of Joule and Mayer was much more definitive. Fortunately, Clausius did not need to define temperature, unlike entropy or heat it was a directly measurable quantity, and he used it as a given to define entropy instead. This was in line with his general preference for quantities that directly depended on measurements rather than speculative molecular assumptions.&lt;/p&gt;&#xA;&lt;p&gt;While Clausius was a champion of kinetic theory, his attitude towards the use of statistical methods was much more ambivalent. He preferred algebraic approaches that were to become a staple thermodynamics, in particular, because they depended much less on specific hypotheses about parameters of molecules and the nature of intermolecular forces, that could not be verified at the time.&lt;/p&gt;&#xA;&lt;p&gt;Nonetheless, Clausius's 1857 paper inspired Maxwell to apply statistical methods to the theory of heat. Clausius only introduced mean speeds and mean free paths of particles, while Maxwell worked directly with a velocity distribution, and hence could distinguish between squares of mean speeds and means of squared speeds. The flip side was that Maxwell had to introduce very specialized hypotheses to derive his (normal) distribution function, which Clausius criticized him for. Ultimately, it led Clausius to abandon statistical approach altogether, and even Maxwell himself had to reboot, taking inspiration from Boltzmann's work, see &lt;a href=&quot;https://www.jstor.org/stable/27757309&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clausius and Maxwell's Kinetic Theory of Gases by Garber&lt;/a&gt;. A clear statistical interpretation of temperature as mean kinetic energy of molecules is due to Boltzmann. Here is from Grabiner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;By 1857, when he first began to publish papers on gases, Clausius had already established a reputation as a leading theoretical physicist with his clear, incisive papers on thermodynamics. These latter papers reveal his ability to separate the essential physical principles from the special matter-theory assumptions that he had used in his development of the subject (all of his thermodynamics papers published after 1860 were based upon his concept of the structure of matter). In 1857 Clausius was still cautious about revealing these assumptions.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;While Clausius regarded Maxwell's theory as a serious attempt to solve the problems of gaseous behavior, he criticized it on basic physical grounds. He considered Maxwell's use of a spherically symmetric distribution function in analyzing transport properties as incorrect. He developed his own theory, taking into account the additional vis viva associated with motion in the direction of the gradient within the gas. The factor, ignored by Maxwell, was the cause of the phenomenon, and, in the case of thermal conduction, the gas molecules possessed additional momentum in the direction of the temperature gradient; therefore, the distribution of molecular motions could not be spherically symmetric.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The statistical method might be correct in principle, but Maxwell's particular method was inaccurate. Clausius' acceptance of a statistical method was only tentative and he finally rejected it altogether. His hesitancy in using the distribution function reflects the difference between his and Maxwell's concept of matter and gases... Clausius did not use the statistical distribution even after Maxwell derived it independently of the mean free path. He only used the distribution function once, to find the mechanical expression for the heat added to a gas... In 1862 he had revealed the molecular ideas that had guided his thermodynamics, and his later papers were all directed to the problem of expressing disgregation and entropy in mechanical form. Unlike Boltzmann, Clausius never fully appreciated the power of the statistical method, and he continued to work within the traditional, yet sophisticated, field of analytical mechanics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-04-06T17:21:54.667" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11627" PostTypeId="2" ParentId="9612" CreationDate="2020-04-06T11:51:48.703" Score="2" Body="&lt;p&gt;Penguin diagrams are used in quantum field theory to describe CP violation.&#xA;&lt;a href=&quot;https://arxiv.org/pdf/hep-ph/9510397.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; the history about how penguins enter in Feynman diagrams: &lt;a href=&quot;https://en.wikipedia.org/wiki/Mikhail_Shifman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mikhail Shifman&lt;/a&gt; remembers that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the first paper where the graphs that are now called penguins were considered in&#xA;  the weak decays appeared in JETP Letters in 1975, and there they did not look like&#xA;  penguins at all. Later on they were made look line penguins [...] The penguin name came in 1977, as follows. [...] That summer, there was a student at CERN, &lt;a href=&quot;https://en.wikipedia.org/wiki/Melissa_Franklin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Melissa Franklin&lt;/a&gt; who is now an experimentalist at Harvard. One evening, she, I and &lt;a href=&quot;https://en.wikipedia.org/wiki/Serge_Rudaz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Serge [Rudaz]&lt;/a&gt; went to a pub, and she and I started a game of darts. We made a bet that if I lost I had to put the word penguin into my next paper. She actually left the darts game before the end, and was replaced by Serge, who beat me. Nevertheless, I felt obligated to carry out the conditions of the bet. [...] For some time, it was not clear to me how to get the word into this b quark paper [then I] visit some friends living in Meyrin where I smoked some illegal substance. Later, when I got back to my apartment and continued working on our paper, I had a sudden flash that the famous diagrams look like penguins.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So not only animals are involved in this story, but also a plant, I suppose..., maybe this also answers to another question on &lt;a href=&quot;https://hsm.stackexchange.com/questions/3781/claims-of-mathematical-breakthroughs-while-in-an-altered-state-of-consciousness/3782#3782&quot;&gt;hsm&lt;/a&gt; even if we cannot say that this was a &lt;em&gt;&quot;mathematical breakthroughs&quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another Feynman diagram whose name derives from the similarity with an animal is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tadpole_(physics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;tadpole diagram&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Biological similarity are more often used in mathematics and computer science, see for example butterfly diagrams (in fast Fourier transforms), onion diagrams/models and spider diagrams (in Euler-Venn diagrams), ... &lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-04-06T12:01:31.803" LastActivityDate="2020-04-06T12:01:31.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11628" PostTypeId="2" ParentId="3781" CreationDate="2020-04-06T12:57:27.557" Score="1" Body="&lt;p&gt;As Conifold noticed, only occasionally scientists report the circumstances of their creative process. The following story does not describe an altered state of consciousness induced by psychotropic substances, but rather a connection between ideas belonging to different domains made in a reduced level of consciousness due to tiredness, similar to Poincaré's altered state. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/L-system&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lindenmayer systems&lt;/a&gt; are a special type of formal grammar developed by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;L. especially studied yeast and fungi and wanted to develop a general formal model of the structure and development of such simple organisms. He found a perfect basis [...] in the theory of formal languages [...] &#xA;  Before long, L. might have found formal languages as the model for his system by himself, but a nice coincidence made it happen earlier: L. was absorbed in studyng photosynthetic organism called &quot;algae&quot; as he passed the lecture hall where a lecture on formal languages was taking place. The lecturer repeatedly used the term &quot;L(G)&quot;, which cause the biologist to immediately experience an aha moment: &lt;em&gt;&quot;Algae, that's it!&quot;&lt;/em&gt; (cited from &lt;em&gt;Arto Salooma: Mathematician, Computer Scientist, and Teacher&lt;/em&gt;, Jukka Paakki, p. 111.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2020-04-06T12:57:27.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11629" PostTypeId="1" CreationDate="2020-04-06T17:21:57.383" Score="0" ViewCount="94" Body="&lt;p&gt;It is my understanding that Einstein produced the EPR paper to necessarily elucidate the existence of physical properties that could not exist prior to measurement. That was his assumption which was later proven incorrect. However, there is another consequence of quantum mechanics: &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_tunnelling&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantum tunneling&lt;/a&gt;. Did he not know about this? Was it unaware at that time that tunneling exists or was there no empirical evidence to support tunneling?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here's why I think quantum tunneling would've dissuaded Einstein for making his case against quantum mechanics. If position eigenstates are really determined prior to measurement, then it seems to be no reason whatsoever why you would expect a particle to cross a high energy barrier unless you accept the probabilistic wavefunction as the correct description for the position eigenfunctions. &lt;/p&gt;&#xA;" OwnerUserId="3353" LastActivityDate="2020-04-06T17:21:57.383" Title="Did Einstein or the scientific community know about quantum tunneling during the EPR debates?" Tags="|quantum-mechanics|einstein|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11630" PostTypeId="2" ParentId="11615" CreationDate="2020-04-06T17:31:41.437" Score="2" Body="&lt;p&gt;&lt;em&gt;Thermodynamic&lt;/em&gt; temperature is defined as pV = RT, where p is pressure, V is the Volume, R the ideal gas constant and T the temperature.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Clausius in &quot;The Mechanical Theory of Heat: With Its Applications to the Steam-engine and ...&quot; (1864 ?) first introduced the concept of entropy as the integral dQ/T = dS.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The form that you give came much later and is the statistical mechanics definition of temperature.  When exactly it was first defined I can't say.&lt;/p&gt;&#xA;" OwnerUserId="12291" LastActivityDate="2020-04-06T17:31:41.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11631" PostTypeId="1" CreationDate="2020-04-06T20:24:03.027" Score="5" ViewCount="119" Body="&lt;p&gt;I read &lt;em&gt;Men of Mathematics&lt;/em&gt; by E.T. Bell long ago, and this quote he attributed to Jacobi stuck with me:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Certainly I have sometimes endangered my health from overwork, but what of it? Only cabbages have no nerves, no worries. And what do they get out of their perfect wellbeing?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I'm aware that the book is not to be taken as a work of historical scholarship, so I was looking for some other reference.  I googled “Jacobi cabbage” and didn't find anything but a Google Books hit back to &lt;em&gt;Men of Mathematics&lt;/em&gt;, and many recipes.  So is this story totally aprocryphal?&lt;/p&gt;&#xA;" OwnerUserId="12292" LastEditorUserId="26" LastEditDate="2021-02-20T15:16:04.623" LastActivityDate="2021-02-20T15:16:04.623" Title="Is there any historical evidence of this quote E.T. Bell attributed to C.G.J. Jacobi?" Tags="|mathematicians|reference-request|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11632" PostTypeId="2" ParentId="9612" CreationDate="2020-04-07T08:54:20.353" Score="2" Body="&lt;p&gt;In mathematics (with relevance to physics), Émile Borel in 1913 illustrated concepts of probability with monkeys, this lead to what's now called the &quot;Inifinite Monkey theorem&quot;. He wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Concevons qu’on ait dressé un million de singes à frapper au hasard sur les touches d’une machine à écrire et que, sous la surveillance de contremaîtres illettrés, ces singes dactylographes travaillent avec ardeur dix heures par jour avec un million de machines à écrire de types variés.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Translation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Let us imagine that a million monkeys have been trained to strike at random on the keys of a typewriter and that, under the supervision of illiterate foremen, these typing monkeys work eagerly ten hours a day with a million typewriters of various types.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Source:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Borel, Émile. „La mécanique statique et l’irréversibilité“. Journal de Physique Théorique et Appliquée 3, Nr. 1 (1913): 189–96. &lt;a href=&quot;https://doi.org/10.1051/jphystap:019130030018900&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1051/jphystap:019130030018900&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9563" LastActivityDate="2020-04-07T08:54:20.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11633" PostTypeId="2" ParentId="9612" CreationDate="2020-04-07T09:08:23.067" Score="1" Body="&lt;p&gt;Descartes allegedly developed coordinate systems after observing a fly on the ceiling of his bedroom; he realised he could define its position by counting the number of wooden panels from two adjacent walls. This is ascribed to Descartes in many popular maths books, but so far I have not been able to find a definite primary source.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(It's maths, but clearly very relevant to physics...)&lt;/p&gt;&#xA;" OwnerUserId="9563" LastActivityDate="2020-04-07T09:08:23.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11634" PostTypeId="2" ParentId="11623" CreationDate="2020-04-07T11:02:47.137" Score="10" Body="&lt;p&gt;Special cases were handled algebraically even before the &quot;l'Hopital's&quot; rule, which appears in l'Hopital's 1696 transcription of tips on calculus he purchased (literally) from Johann Bernoulli in 1694, see &lt;a href=&quot;https://www.jstor.org/stable/2691133&quot; rel=&quot;noreferrer&quot;&gt;Indeterminate Forms Revisited, by Boas&lt;/a&gt;. For example, Descartes's method of finding tangents involved resolving &quot;indeterminate forms&quot; like &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt;, see &lt;a href=&quot;https://hsm.stackexchange.com/a/1968/55&quot;&gt;Is there a 'lost calculus'?&lt;/a&gt; So the phenomenon was known from examples (without a name or special attention paid to it) by the time it was singled out by Bernoulli, and then comprehensively systematized by Euler. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;That was done in Euler's textbook &lt;em&gt;Institutionum Calculi Differentialis&lt;/em&gt; (1755), chapter 15 of part II. Fortunately, there is an &lt;a href=&quot;http://www.17centurymaths.com/contents/differentialcalculus.htm&quot; rel=&quot;noreferrer&quot;&gt;English translation by Bruce&lt;/a&gt;. At the beginning Euler explains how &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; come up, why they are &quot;indeterminate&quot;, and then gives some tricks for resolving them, including cancellation, &quot;l'Hopital's&quot; rule and logarithmic differentiation. He freely manipulates infinitesimals, and at the end even deduces the famous sum of the Basil series by applying the &quot;l'Hopital's&quot; rule thrice. Here is from the opening, where he shows that arbitrary value is possible with a remarkably simple example:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;If the fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{Q}$&lt;/span&gt; were some function &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, the numerator and the denominator of&#xA;  which likewise may vanish on putting a certain value in place of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, then in that case the&#xA;  fraction &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{Q}$&lt;/span&gt;&#xA;  may arise expressing the value of the function &lt;span class=&quot;math-container&quot;&gt;$y=\frac00$&lt;/span&gt;; which expression thus&#xA;  may be considered indeterminate, since for each quantity either finite or infinite, or&#xA;  infinitely small it may become equal to, from that evidently in this case the value of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&#xA;  cannot be deduced. Yet meanwhile it is easily seen, because in addition in this case the&#xA;  function &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; takes a determined value always, whatever may be substituted for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, also in this&#xA;  case an indeterminate value of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; cannot be possible. This is made clear from this example,&#xA;  if there were &lt;span class=&quot;math-container&quot;&gt;$y=\frac{aa-xx}{a-x}$&lt;/span&gt;, so that on making &lt;span class=&quot;math-container&quot;&gt;$x=a$&lt;/span&gt; certainly there becomes &lt;span class=&quot;math-container&quot;&gt;$y=\frac00$&lt;/span&gt;. But since&#xA;  with the numerator divided by the denominator it may become &lt;span class=&quot;math-container&quot;&gt;$y = a + x$&lt;/span&gt; , it is evident, if&#xA;  there is put &lt;span class=&quot;math-container&quot;&gt;$x= a$&lt;/span&gt; to become &lt;span class=&quot;math-container&quot;&gt;$y=2a$&lt;/span&gt;, thus so that in this case that fraction &lt;span class=&quot;math-container&quot;&gt;$\frac00$&lt;/span&gt; may be&#xA;  equivalent to the quantity &lt;span class=&quot;math-container&quot;&gt;$2a$&lt;/span&gt;.&lt;/em&gt;&quot;     &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Although Euler uses the (Latin version of) &quot;indeterminate&quot; he does not call them &quot;indeterminate forms&quot; or introduces handy notations and classifications encountered in modern textbooks. According to &lt;a href=&quot;http://jeff560.tripod.com/i.html&quot; rel=&quot;noreferrer&quot;&gt;Jeff Miller's Earliest Uses&lt;/a&gt;, this taxonomic process began in 1840s:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;&lt;em&gt;The term INDETERMINATE FORM is used in French in 1840 in Moigno, abbé (François Napoléon Marie), (1804-1884): Leçons de calcul différentiel et de calcul intégral, rédigées d'après les méthodes et les ouvrages publiés ou inédits de M. A.-L. Cauchy, par M. l'abbé Moigno. Indeterminate Forms is found in English as a chapter title in 1841 in An Elementary Treatise on Curves, Functions, and Forces by Benjamin Pierce. Forms such as &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; are called singular values and singular forms in in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson. In Primary Elements of Algebra for Common Schools and Academies (1866) by Joseph Ray, &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; is called &quot;the symbol of indetermination.&lt;/em&gt;&quot;    &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-07T11:23:46.987" LastActivityDate="2020-04-07T11:23:46.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11635" PostTypeId="1" CreationDate="2020-04-08T03:49:35.970" Score="7" ViewCount="129" Body="&lt;p&gt;In Konrad Zuse's Plankalkül &lt;a href=&quot;http://zuse.zib.de/pdfs&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZIA ID 0020&lt;/a&gt; from 1972, in his patent submission Z23624 &quot;Rechenmaschine&quot; &lt;a href=&quot;http://zuse.zib.de/pdfs&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZIA ID 0177&lt;/a&gt; from 1936 and modern &lt;a href=&quot;https://de.wikipedia.org/wiki/Dualsystem&quot; rel=&quot;nofollow noreferrer&quot;&gt;German Wikipedia article on the dyadic system&lt;/a&gt;, 2020-01-17 we see L for one and 0 for zero.  In certain modern lecture materials I cannot cite here I saw a variant with O/L instead of 0/L.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mr. Zuse wrote he chose 0, L as the binary digits (so as to distinguish them from the decimal digits 0 and 1) but did not explain why the binary one is L and not some other letter (say, A, B, C, D, ...).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the origin of the usage of 0/L and O/L as representations of no/yes and false/true?&lt;/p&gt;&#xA;" OwnerDisplayName="user11087" LastEditorUserId="1930" LastEditDate="2020-04-09T23:21:55.193" LastActivityDate="2020-04-10T06:13:20.927" Title="Origin of O/L for false/true in German computer-science texts" Tags="|notation|computer-science|naming-conventions|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11636" PostTypeId="1" AcceptedAnswerId="11638" CreationDate="2020-04-09T03:40:20.460" Score="3" ViewCount="326" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Woldemar_Voigt#The_Voigt_transformation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's Voigt transformation&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Lorentz did not adopt this transformation, as he found in 1904 that&#xA;  only the Lorentz contraction corresponds to the principle of&#xA;  relativity. Since Voigt's transformation preserves the speed of light&#xA;  in all frames, the Michelson–Morley experiment and the&#xA;  Kennedy–Thorndike experiment can not distinguish between the two&#xA;  transformations. The crucial question is the issue of time dilation.&#xA;  The experimental measurement of time dilation by Ives and Stillwell&#xA;  (1938) and others settled the issue in favor of the Lorentz&#xA;  transformation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Einstein derived the Lorentz transformation using the two postulates, see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4899-2674-6_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein’s Postulates and the Lorentz Transformations by Terletskii&lt;/a&gt;. Einstein also derived the equations for transverse and longitudinal mass and in the derivation time dilation equation was used. I'm a beginner and just trying to understand the relativity at basic level.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If both Einstein's and Lorentz's derivations were the same then wouldn't it mean that Voigt's time dilation equation had more chance of being wrong? If it was not decided until the experiment of Ives and Stillwell that which version of time dilation equation was right then it would also mean that though Einstein postulates were correct but still until 1938 there was still some doubt regarding some quantitative results derived by Einstein. In other words, if Voigt's time dilation was found correct, how would it have affected the other quantitative results or formulas?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I'd really appreciate if you could help me with it. I'm posting it here because IMHO it has more to do with the history.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="55" LastEditDate="2020-04-09T07:32:15.967" LastActivityDate="2020-04-09T07:32:15.967" Title="Were there doubts that Voigt's time dilation was correct rather than Einstein's?" Tags="|relativity-theory|einstein|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11637" PostTypeId="2" ParentId="9612" CreationDate="2020-04-09T07:06:18.400" Score="1" Body="&lt;p&gt;The frog was used in early electrical experiments, such as the invention of &lt;a href=&quot;https://en.wikipedia.org/wiki/Frog_galvanoscope&quot; rel=&quot;nofollow noreferrer&quot;&gt;galvanoscopes&lt;/a&gt; and &lt;a href=&quot;https://www.aps.org/publications/apsnews/200603/history.cfm&quot; rel=&quot;nofollow noreferrer&quot;&gt;batteries&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1780, Luigi Galvani experimented with frog legs, mounted on iron or brass hooks, and saw the leg twitch when touched with a probe made of another metal. They would also jump when hanging on a metal fence in a lightning storm.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Galvanized, Alessandro Volta realized that the crucial feature of Galvani’s experiments was the two dissimilar metals – the iron or brass hook - and the probe of some other metal. The metals were generating the current.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To show that the generation of an electric current did not require animal parts, he built a stack of alternating zinc and silver discs, separated by brine-soaked cloth, in imitation of the electric organ of the &lt;em&gt;torpedo fish&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;When a wire was connected to both ends of the pile, a current flowed. He found that different types of metal could change the amount of current produced, and he could increase the current by adding disks to the stack.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The battery was a huge success. In 1800 Nicholson and Carlisle used its current to decompose water into hydrogen and oxygen. In the 1830s Faraday used a battery in his studies of electromagnetism. Other inventors made improvements on Volta’s design, and soon it was powering telegraphs and doorbells.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Monkey_and_hunter&quot; rel=&quot;nofollow noreferrer&quot;&gt;monkey and hunter&lt;/a&gt; thought experiment on the effect of gravity on projectile motion.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If human bodies are allowed there is, for example: &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Archimedes' &lt;a href=&quot;https://en.wikipedia.org/wiki/Eureka_(word)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eureka&lt;/a&gt; realization of a way to measure the volume of irregular objects.&lt;/li&gt;&#xA;&lt;li&gt;Fleming's &lt;a href=&quot;https://en.wikipedia.org/wiki/Fleming%27s_left-hand_rule_for_motors&quot; rel=&quot;nofollow noreferrer&quot;&gt;left-hand&lt;/a&gt; rule for motors and &lt;a href=&quot;https://en.wikipedia.org/wiki/Fleming%27s_right-hand_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;right-hand&lt;/a&gt; rule for generators.&lt;/li&gt;&#xA;&lt;li&gt;Einstein's &quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments#Falling_painters_and_accelerating_elevators&quot; rel=&quot;nofollow noreferrer&quot;&gt;happiest thought of my life&lt;/a&gt;&quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6562" LastEditorUserId="6562" LastEditDate="2020-04-09T07:47:02.157" LastActivityDate="2020-04-09T07:47:02.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11638" PostTypeId="2" ParentId="11636" CreationDate="2020-04-09T07:27:04.933" Score="5" Body="&lt;p&gt;If Einstein's postulates are correct then Voigt's transformation is incorrect. Einstein showed in his 1905 paper that the Lorentz transformations were the only ones consistent with &lt;a href=&quot;https://en.wikipedia.org/wiki/Postulates_of_special_relativity&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;two&lt;/em&gt; postulates&lt;/a&gt;: invariance of the speed of light in all inertial frames &lt;em&gt;and&lt;/em&gt; form-invariance of the laws of physics in all inertial frames (principle of relativity). His mathematics was checked by many prominent mathematicians, including Hilbert and Minkowski, so there were no doubts about his derivations long before 1938. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But invariance of the speed of light &lt;em&gt;by itself&lt;/em&gt; is not enough. &lt;em&gt;If&lt;/em&gt; Voigt's transformation turned out to be correct it would have meant that the principle of relativity is false, and the frame of ether &lt;em&gt;can&lt;/em&gt; be detected after all, albeit not by Michelson–Morley or Kennedy–Thorndike type measurements. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, special relativity had so many indirect confirmations by the end of 1910s (electrodynamic effects) that the Nobel committee felt comfortable enough to consider awarding Nobel prize for it (it was &lt;a href=&quot;https://www.discovermagazine.com/the-sciences/einstein-vs-the-nobel-prize&quot; rel=&quot;noreferrer&quot;&gt;not awarded for political reasons&lt;/a&gt;), and Miller's seeming &lt;a href=&quot;https://en.wikipedia.org/wiki/Dayton_Miller#Aether_research&quot; rel=&quot;noreferrer&quot;&gt;detection of the ether wind in 1920s&lt;/a&gt; was not taken seriously (his measurement errors were not explained until 1950s). Even general relativity, based on a more sweeping relativity principle, was confirmed by &lt;a href=&quot;https://en.wikipedia.org/wiki/Eddington_experiment&quot; rel=&quot;noreferrer&quot;&gt;Eddington's observations of the 1919 eclipse&lt;/a&gt; (gravitational bending of light). So the Ives–Stilwell experiment of 1938 was more about crossing the t's and dotting the i's by a direct test of time dilation than any real doubts. Various high precision tests of special relativity are performed to this day, see &lt;a href=&quot;http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html&quot; rel=&quot;noreferrer&quot;&gt;experimental basis of Special Relativity&lt;/a&gt; bibliography.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-04-09T07:27:04.933" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="11639" PostTypeId="1" AcceptedAnswerId="11640" CreationDate="2020-04-09T12:07:54.857" Score="4" ViewCount="211" Body="&lt;p&gt;I wonder how the Romans were able to control so vast territories, peoples, armies, ammunition, taxes, etc., without the help of a number system that allowed easy computing (e.g. multiplication and division) of (relatively) large quantities.&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="55" LastEditDate="2020-04-09T23:38:24.003" LastActivityDate="2020-04-09T23:46:14.300" Title="How could the Roman Empire manage their accounting without a proper number system?" Tags="|roman-empire|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11640" PostTypeId="2" ParentId="11639" CreationDate="2020-04-09T23:34:17.287" Score="5" Body="&lt;p&gt;Romans used tricks for arithmetic that worked well enough, see &lt;a href=&quot;https://scienceblogs.com/goodmath/2006/08/16/roman-numerals-and-arithmetic&quot; rel=&quot;noreferrer&quot;&gt;Roman Numerals and Arithmetic on ScienceBlogs&lt;/a&gt;. Addition and subtraction was quite simple, as for multiplication:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Multiplication using Roman numerals is not particularly easy or obvious. You can do the trivial thing, which is repeated addition. But it should be pretty obvious that that's not practical for large numbers. The trick that they used was actually pretty nifty. It's basically a strange version of binary multiplication. You need to be able to add and divide by two, but those are both pretty easy things to do. So here goes:&lt;/em&gt;&lt;br&gt;&#xA;  &lt;em&gt;Given A×B, you create two columns, and write A in the left column, and B in the right. Then:&lt;/em&gt;&lt;br&gt;&#xA;  1. &lt;em&gt;Divide the number in the left column by two, discarding the remainder. Write it down in the next row of the left column.&lt;/em&gt;&lt;br&gt;&#xA;  2. &lt;em&gt;Multiply the number in the right column by two. Write it down in the right column next the the result from step 1.&lt;/em&gt;&lt;br&gt;&#xA;  3. &lt;em&gt;Repeat from step 1 until the value in the left column is 1.&lt;/em&gt;&lt;br&gt;&#xA;  4. &lt;em&gt;Go down the table, and cross out every row where the number in the left column is even.&lt;/em&gt;&lt;br&gt;&#xA;  5. &lt;em&gt;Add up the remaining values in the right column.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Division is the biggest problem in roman numerals. There is no good trick that works in general. It really comes down to repeated subtraction. The only thing you can do to simplify is variations on finding a common factor of both numbers that's easy to factor out. For example, if both numbers are even, you can divide each of them by two before starting the repeated subtraction. It's also fairly easy to recognize when both numbers are multiples of 5 or 10, and to do the division by 5 or 10 on both numbers. But beyond that, you take a guess, do the multiplication, subtract, repeat.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;But for the more serious calculations they did have the &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_abacus&quot; rel=&quot;noreferrer&quot;&gt;Roman abacus&lt;/a&gt; and reckoning boards. They effectively implemented a base 10 system. Here is a reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz from the bronze original held by the &lt;em&gt;Bibliothèque nationale de France&lt;/em&gt; in Paris. Many counter beads are missing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Z5ie6.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z5ie6.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://books.google.com/books/about/Number_Words_and_Number_Symbols.html?id=PKGozmluj-IC&quot; rel=&quot;noreferrer&quot;&gt;Menninger's Number Words and Number Symbols, p.315&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, gives us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi. The Latin word calx means 'pebble' or 'gravel stone'; calculi are thus little stones (used as counters).&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-09T23:46:14.300" LastActivityDate="2020-04-09T23:46:14.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11641" PostTypeId="2" ParentId="11635" CreationDate="2020-04-10T02:22:33.160" Score="4" Body="&lt;p&gt;Likely relevant resources are Konrad Zuse, &quot;Der Computer ― Mein Lebenswerk&quot; (Springer, Berlin 1984) and Raúl Rojas (ed.), &quot;Die Rechenmachinen Konrad Zuses&quot; (Springer, Berlin 1998). &lt;em&gt;F.L. Bauer, H. Wössner, The &quot;Plankalkül&quot; of Konrad Zuse: A Forerunner of Today’s Programming Languages. Communications of the ACM, Vol. 15, No. 7, July 1972, pp. 678-685&lt;/em&gt; references an earlier edition of the former book as Z70 in a footnote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;From a remark in [Z70, p. 157], one can infer that Zuse already during his Berlin period, that is before 1944, used L and 0, which he called &lt;em&gt;Sekundalziffern&lt;/em&gt; (see also [Z70, p. 68] in his diary entry of June 20, 1937).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Based on the Google Books preview of the latter book it includes the full text of Zuse's German patent application Z391 from 1941, which contains (&lt;a href=&quot;https://www.google.com/books/edition/Die_Rechenmaschinen_von_Konrad_Zuse/WteiBgAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Sekundalzahlen%22&amp;amp;pg=PA116&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 116 of the book&lt;/a&gt;; &lt;a href=&quot;http://zuse.zib.de/file/trJ1nquciiskKrG2/57/99/48/a4-ec1f-452a-87ee-fabd323fb666/0/original/0496767598866f2641235f18479773b7.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;document 005/011&lt;/a&gt; in the online Zuse archive) the following sentence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Um Verwechslungen zu vermeiden, wird bei Sekundalzahlen die Ziffer 1 als L geschrieben. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Sekundalzahlen&quot; is Zuse's term for binary numbers. The sentence states that 'L' is used instead of '1' to prevent mixups. A plausible &lt;em&gt;hypothesis&lt;/em&gt; as to the nature of the mixup in question is that typewriters in the 1930s and 1940s often lacked a dedicated '1'-key, requiring typists to use the &lt;em&gt;lowercase&lt;/em&gt; letter 'l' instead. A perusal of a site dealing with trade in old typewriters that includes pictures of typewriters available for sale shows that this restriction also applied to at least some models of the German brands Adler, Olympia, and AEG. The use of the &lt;em&gt;uppercase&lt;/em&gt; letter 'L' would therefore have been motivated by a desire to distinguish the binary digit from an ordinary decimal '1' and the lowercase letter 'l'.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am not aware of any sources that suggest an origin of the 0/L notation in works preceding Zuse's, but cannot with certainty eliminate the possibility that he adapted it from some other, pre-existing work.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2020-04-10T06:13:20.927" LastActivityDate="2020-04-10T06:13:20.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11642" PostTypeId="2" ParentId="3781" CreationDate="2020-04-10T11:15:22.487" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kary_Mullis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kary Mullis&lt;/a&gt; was awarded a 1994 Nobel Prize in Chemistry for his invention of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Polymerase_chain_reaction&quot; rel=&quot;nofollow noreferrer&quot;&gt;polymerase chain reaction&lt;/a&gt; (PCR) technique.&lt;/p&gt;&#xA;&lt;p&gt;Mullis reputedly said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Back in the 1960s and early 1970s I took plenty of LSD [...] and I found it to be a mind-opening experience. It was certainly much more important than any courses I ever took.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;During a symposium held for centenarian Albert Hofmann, Hofmann said Mullis had told him that LSD had &amp;quot;helped him develop the polymerase chain reaction that helps amplify specific DNA sequences&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In a Notices of the AMS &lt;a href=&quot;https://www.ams.org/notices/201003/rtx100300391p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;interview with Mikhail Gromov&lt;/a&gt; [PDF], Gromov says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I realized that there had been a huge development in molecular biology in the 1980s, after the discoveries of genetic engineering and of PCR (polymerase chain reaction). It was really mathematical procedures applied to living cells. Mathematicians could invent PCR. It didn’t happen, but mathematicians could have invented PCR.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So perhaps this is a mathematical breakthrough while in an altered state of consciousness.&lt;/p&gt;&#xA;" OwnerUserId="6562" LastEditorUserId="4251" LastEditDate="2020-07-14T06:00:05.650" LastActivityDate="2020-07-14T06:00:05.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11644" PostTypeId="2" ParentId="3781" CreationDate="2020-04-10T17:50:29.637" Score="2" Body="&lt;p&gt;Bob Thomason attributed of co-authorship to his deceased (former??) collagorator Tom Trobaugh for a significant paper, due to Trobaugh's coming to him in a dream with critical ideas. (Thomason has a Wikipedia page. I'm not sure which paper this was, as it all happened 25+ years ago. Perhaps it was the &quot;Higher algebraic K-theory of schemes and of derived categories&quot;, in the Grothendieck Festschrift.)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2020-04-10T17:50:29.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11646" PostTypeId="1" AcceptedAnswerId="11651" CreationDate="2020-04-11T15:08:11.623" Score="6" ViewCount="184" Body="&lt;p&gt;Edward Jenner was a British doctor in the 18th century who in 1798 developed &quot;vaccination,&quot; that is exposing humans to cow pox in order to prevent them from acquiring small pox. Vaccination was quickly taken up and Jenner seen as a hero. Vaccination replaced inoculation (actually exposing people to small pox in an attenuated form) as a way to prevent small pox.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, the germ theory of disease only began to take hold with Pasteur and Koch in the mid-to-late 1800s. Until then there was significant debate between theories of disease as contagious, or whether disease was simply caused by one's environment (the miasmatic theory of disease). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thus, I do not understand how people conceptualized/explained the fact of immunity conferred by exposure to small pox or cow pox. People knew before Pastuer and Koch that being exposed to small pox meant one would not contract small pox later; how did they explain the fact with their understandings of disease at the time? I cannot find any commentary on this matter other than the description that inoculation and vaccination quickly became widespread because observation revealed they work, but I cannot find any discussion of explanations advanced at the time for &lt;em&gt;why&lt;/em&gt; they worked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, Edward Jenner's papers to the royal society on vaccination can be found here: &lt;a href=&quot;https://biotech.law.lsu.edu/cphl/history/articles/jenner.htm&quot; rel=&quot;noreferrer&quot;&gt;https://biotech.law.lsu.edu/cphl/history/articles/jenner.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But they are entirely empirical and offer very little attempt at an explanation for why his vaccination works.&lt;/p&gt;&#xA;" OwnerUserId="9822" LastEditorUserId="26" LastEditDate="2021-02-20T15:15:58.763" LastActivityDate="2021-02-20T15:15:58.763" Title="How did Jenner and his contemporaries understand vaccination before germ theory of disease?" Tags="|biology|medicine|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11648" PostTypeId="1" CreationDate="2020-04-11T19:09:31.187" Score="6" ViewCount="211" Body="&lt;p&gt;My simplified historical understanding is as follows. Euclidean geometry accepted a limited number of geometrical objects (straight-edge and compass constructions, conics). Descartes' &lt;em&gt;Géométrie&lt;/em&gt; expanded the geometrical universe and introduced methods to deal with algebraic curves in general. The next centuries a further relaxation took place, allowing also non-algebraic curves and (analytic) functions in the modern sense.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ancient mathematical astronomy, it is said, makes use of &lt;a href=&quot;https://en.wikipedia.org/wiki/Epicycloid&quot; rel=&quot;noreferrer&quot;&gt;epicycloids&lt;/a&gt;. But these are (with exception of special cases of ratio between the radii) not even algebraic curves. Descartes would not have accepted them and therefore (?) neither would have Greek geometers. My assumption is roughly that &quot;Greek geometrical objects&quot; &lt;span class=&quot;math-container&quot;&gt;$\subsetneq$&lt;/span&gt; &quot;Cartesian geometrical objects&quot; though this probably needs further qualification.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Ancient geometers/astronomers did not write down parametric equations of epicycloids. But were these curves even acceptable objects in the eyes of mathematicians? Was generation of a curve by circles rolling on each other even allowed according to Greek geometrical standards? There was much debate on using complex curves (spiral, chonchoid, etc.) to solve the famous mathematical problems (trisection of angle etc.). So why was the use of epicycloids not highly controversial at best and &quot;forbidden&quot; at worst?&lt;/p&gt;&#xA;" OwnerUserId="12319" LastEditorUserId="55" LastEditDate="2020-04-11T20:54:38.617" LastActivityDate="2020-04-12T03:21:53.287" Title="Were epicycloids from astronomy acceptable curves in Greek geometry?" Tags="|mathematics|astronomy|geometry|euclidean-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11649" PostTypeId="2" ParentId="11648" CreationDate="2020-04-11T20:53:23.150" Score="4" Body="&lt;p&gt;It is not quite accurate to say that ancient astronomy made use of epicycloids. &lt;em&gt;Epicycles&lt;/em&gt;, yes, but curiously enough, hypocycloids, epicycloids and cycloids were not studied as geometric curves (as far as we know), despite the use of epicycles in astronomy. It would have been possible to draw tangents to them by the same technique that Archimedes used for his spiral, for example, but it was not done until modern times. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Epicycles were viewed as merely computational devices, and their geometry was of little relevance even in geometric constructions used by Ptolemy and his Islamic successors, see e.g. &lt;a href=&quot;https://books.google.com/books?id=kZnxBwAAQBAJ&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Episodes From the Early History of Astronomy by Aaboe&lt;/a&gt;. These curves visualize motion of the planets, but for determining measurable parameters of orbits line and circle constructions were enough. Perhaps, geometers wanted to keep their external applications apart, or perhaps they did not prove as useful internally, but their geometry was not explicitly studied until 16th century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said, Greeks were not restricted to straight lines and conics for their curves. Acerbi gives an overview of curve constructions available to them in &lt;a href=&quot;https://www.academia.edu/8016021/Homeomeric_Lines_in_Greek_Mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Homeomeric Lines in Greek Mathematics by Acerbi&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;strong&gt;&lt;em&gt;1)&lt;/strong&gt; Generative constructions. These normally amount to cutting a surface with a plane.&#xA;  In this way the conic (Con. I.11–3) and toric sections (iE, 112.4–8) were defined.&lt;/em&gt; &lt;br&gt;&#xA;  &lt;strong&gt;2)&lt;/strong&gt; &lt;em&gt;Pointwise geometrical constructions. A procedure is given to find isolated points&#xA;  that lie on the curve; the latter is then approximated by joining such points by line&#xA;  segments or arcs of other known lines...&lt;/em&gt;&lt;br&gt;&#xA;  &lt;strong&gt;3)&lt;/strong&gt; &lt;em&gt;Intersections of surfaces, in their turn obtained by rotation of plane figures (conic,&#xA;  cylindrical, toric surfaces). This is the case of the curves implicit in Archytas’&#xA;  method for solving the problem of duplication of the cube (Eutocius, on the&#xA;  authority of Eudemus, in AOO III:84.12–88.2).&lt;/em&gt;&lt;br&gt;&#xA;  &lt;strong&gt;4)&lt;/strong&gt; &lt;em&gt;“Mechanical” constructions, in which some geometrical objects, for instance&#xA;  straight lines, are allowed to move, the curve being generated by the motion&#xA;  of some suitable point on them...&lt;/em&gt;&lt;br&gt;&#xA;  &lt;strong&gt;5)&lt;/strong&gt; &lt;em&gt;Setting forward a property that univocally identifies the curve...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&quot;Mechanical curves&quot;, studied as such, included &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadratrix_of_Hippias&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadratrix&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedean_spiral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedean spiral&lt;/a&gt;, &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Apollonius.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;helix&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Conchoid_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;conchoid&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Cissoid_of_Diocles&quot; rel=&quot;nofollow noreferrer&quot;&gt;cissoid&lt;/a&gt;. Conchoid and cissoid are algebraic curves, the rest are transcendental. Astroid, the evolute of ellipse, and those of parabola and hyperbola, appear implicitly in Apollonius's Conica, in the solution to the problem of the number of normals that can be drawn to a conic from a point, see &lt;a href=&quot;https://www.researchgate.net/publication/228887613_Apollonius%27_ellipse_and_evolute_revisited&quot; rel=&quot;nofollow noreferrer&quot;&gt;Apollonius' ellipse and evolute revisited by Hartmann-Jantzen&lt;/a&gt;. But he does not study them as curves. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for acceptance, there was a stigma attached to mechanical (and even intersectional) curves, because they were &quot;impure&quot; under Plato's strictures, and &quot;corrupted the good of geometry&quot;, as he put it, see &lt;a href=&quot;https://hsm.stackexchange.com/a/6902/55&quot;&gt;When were the concepts of pure and applied Mathematics introduced?&lt;/a&gt; Nonetheless, they were acceptable in applied contexts. Plato himself called on mathematicians to &quot;save the phenomena&quot; in astronomy by resorting to &quot;impure&quot; methods, and Eudoxus, his close associate, cleverly used one of the spiric sections, hippopede, in his models of the retrograde motion of the planets, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s004070100039&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yavetz's A New Role for the Hippopede of Eudoxus&lt;/a&gt; and &lt;a href=&quot;http://link.springer.com/article/10.1007%2FBF00776065&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riddel's Eudoxan Mathematics and the Eudoxan Spheres&lt;/a&gt;. The original construction of the hippopede was also mechanical, by composing motions of spheres revolving at angles to each other, and, according to some reconstructions, it was the simplification of replacing spheres with flat circles that led Apollonius to introducing epicycles.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, when it came to solving mathematical problems proper, mechanical curves were frowned upon. &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Pappus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pappus in his Collection&lt;/a&gt; classifies problems into plane (solvable by straightedge and compass), solid (solvable by conics) and the rest, strangely, to us, dubbed &quot;linear&quot; (to Greeks &quot;line&quot; meant what we call &quot;curve&quot;) Pappus makes it clear that each type of problem ought to be solved by &quot;appropriate&quot; methods, with mechanical curves being the last resort, when all else fails. As Acerbi writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was obvious to ancient geometers that any&#xA;  plane problem could also be solved by solid methods, etc., and quite clear to them that&#xA;  any single problem falls by its essence into only one of these categories if minimality&#xA;  of mathematical tools employed is required – even if no proof is offered of this&#xA;  (cf. Pappus, Coll. III.21 and IV.59; the normative character of Pappus’ prescription&#xA;  is likely to originate with Apollonius, as we shall see in the companion paper).&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The quadratrix,  spiral and helix were used just so, by Dinostratus, Archimedes and Apollonius, respectively, for one such problem, squaring of the circle. Eudoxus, and his teacher Archytas, used the hippopede for duplicating the cube, as Diocles used the cissoid, quadratrix and conchoid were used for the angle trisection. But Menaechmus's solution by intersecting conics was preferred for both. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-12T03:21:53.287" LastActivityDate="2020-04-12T03:21:53.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11650" PostTypeId="2" ParentId="8391" CreationDate="2020-04-11T22:18:08.897" Score="5" Body="&lt;p&gt;As mentioned in the comment by &lt;em&gt;Conifold&lt;/em&gt;, it is very hard and subjective to assess a talent for anything.&lt;/p&gt;&#xA;&lt;p&gt;It may be true that Ada Lovelace did some basics mistakes and she had some typos or misuderstanding in her translations, however, this does not mean that she had no talent for mathematics. Very often people with gift for science and technology are poor in languages and often they do not see solution of things obvious for normal people.&lt;/p&gt;&#xA;&lt;p&gt;However, I do not want to dive much deep into speculation and more or less philosophy. What is fact that Ada Lovelace was visionary and she saw applicaiton of technology yet to be discovered and designed in her time.&lt;/p&gt;&#xA;&lt;p&gt;I would recommed to read her notes to article &lt;a href=&quot;https://www.fourmilab.ch/babbage/sketch.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sketch of analytical engine invented by Charles Babbage&lt;/a&gt; authored by Luis Menabrea. Interestingly her notes are longer than an original article itself. The notes contain what is considered to be the first computer algorithm, in particular a method how Bernoulli numbers can be computed on Babbage's analytical engine. Interestingly, a physical implementation (or programming) of the algorithm on the analytical engine is very similar to programming on early electronic computers in 1950's (especially how variables are stored in registers) but there is more than one century gap in time. Additionaly architecture of analytical engine is very similar to von Neumann scheme - there are I/O devices, memory, controler and &lt;em&gt;mill&lt;/em&gt; (i.e. arithmetical - logical unit in modern terms). Ada also talks about controlling card and number cards - instructions and data in modern terms.&lt;/p&gt;&#xA;&lt;p&gt;Overall, Ada Lovelace and Charles Babbage paved a way to modern computing (Ada to software and Charles to hardware) about one century before electronic computers come to existence. I think that it is not important wheter they were geniuses but their contribution to knowledge of humankind should be taken into account.&lt;/p&gt;&#xA;" OwnerUserId="12316" LastEditorUserId="5989" LastEditDate="2021-09-22T06:40:41.577" LastActivityDate="2021-09-22T06:40:41.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11651" PostTypeId="2" ParentId="11646" CreationDate="2020-04-12T03:19:03.383" Score="7" Body="&lt;p&gt;The germ theory dates to late middle ages, and was favored by Avicenna among others, although it did not gain much currency in Europe until the mid 19th century. But it was not needed to discover the success of inoculations, which long predate Jenner, and which were later developed into more elaborate vaccinations. That much was established empirically, see &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1200696/&quot; rel=&quot;noreferrer&quot;&gt;Edward Jenner and the history of smallpox and vaccination by Riedel&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was common knowledge that survivors of smallpox became immune to the disease. As early as 430 BC, survivors of smallpox were called upon to nurse the afflicted (9)... However, the most successful way of combating smallpox before the discovery of vaccination was inoculation. The word is derived from the Latin inoculare, meaning “to graft.” Inoculation referred to the subcutaneous instillation of smallpox virus into nonimmune individuals... Inoculation, hereafter referred to as variolation, was likely practiced in Africa, India, and China long before the 18th century, when it was introduced to Europe&lt;/em&gt;&quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Jenner himself was inoculated with smallpox in 1757, at 8 years old, developed a mild form of the disease, and recovered, so he needed no theory to be convinced. He became a learned man, but those were Newtonian times, when experimental evidence was valued over theoretical speculations in the spirit of &lt;em&gt;hypotheses non fingo&lt;/em&gt;. Old school &quot;explanations&quot; were mocked, as in Moliere's &quot;&lt;em&gt;opium puts you to sleep because it has sleep-inducing virtue&lt;/em&gt;&quot;, and as Jenner's teacher John Hunter wrote in a letter to him:“&lt;em&gt;I think your solution is just. But why think? Why not try the experiment?&lt;/em&gt;”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So Jenner collected sixteen case histories since the 1770s, and staged twelve inoculation experiments of his own in 1796-97 instead. Even his idea of replacing infected tissue from human hosts by that from infected animals (which weakens the virus) came from empirical observations, and not theory, see &lt;a href=&quot;https://www.healthaffairs.org/doi/full/10.1377/hlthaff.24.3.611&quot; rel=&quot;noreferrer&quot;&gt;The History Of Vaccines And Immunization by Stern and Markel&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This said, pre-Pasteur theories of inoculation's effectiveness did exist, even as Jenner had no interest in them, see &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0008874980902452&quot; rel=&quot;noreferrer&quot;&gt;A History of Theories of Acquired Immunity by Silverstein and Bialasiewicz&lt;/a&gt;. An early one, based on the humoral theory of disease is due to Rhazes (c. 900), and in Drake's 1707 theory smallpox &lt;em&gt;can&lt;/em&gt; recur, but the symptoms do not manifest because the &quot;morbid matter&quot; would escape through the now extended pores and glands as fast as it forms.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the 18th century, the age of inoculations in Europe, depletion theories came to be popular, which envisioned instead of imbalance of humors some external agent entering the body as the cause, and some depletable substrate in the body as enabling it. Fracastro offered an early such theory back in 1546, which relied on an early form of germ theory inspired by Avicenna:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;It was Fracastoro who first gave formal currency to the idea that not only was&#xA;  disease caused by small seeds (seminaria), but that the contagion might spread&#xA;  directly from person to person, indirectly by means of infected clothing, etc., or&#xA;  even at a distance... The seminaria of&#xA;  smallpox, he felt, not only had an affinity for blood as Rhazes had suggested, but&#xA;  more specifically they had an affinity for that trace of menstrual blood with which&#xA;  each of us was supposed to be tainted in utero, and which thenceforth contaminates&#xA;  our own blood. In this, Fracastoro picked up and expanded upon an idea advanced&#xA;  early in the 11th century by Avicenna... As with Rhazes’ theory, that of Fracastoro appeared to explain all of the known&#xA;  phenomena associated with smallpox, with acquired immunity in this case resulting&#xA;  from the expulsion during the first illness of the menstrual blood contaminant&#xA;  without which clinical disease could not recur.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mather, best known for his role in the Salem witch trials, offered another such theory in 1724, but without identifying the external agent as some kind of germ. Kirkpatrick postulated a depletable “pabulum” in the blood, with which contagious “primordia” united (1754), and Gatti (1764) compared the body to dry wood that a single spark can set afire, but which &#xA;becomes “incombustible” thereafter.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-04-12T03:19:03.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11652" PostTypeId="1" CreationDate="2020-04-12T06:51:48.300" Score="3" ViewCount="101" Body="&lt;p&gt;Newton founded the Law of Gravitation based on the observations of Kepler and Galilei. He later invented the mathematical framework which eventually helped him formulate his theory of gravitation. That same framework was used to predict the position of the moon and other heavenly bodies in the night sky.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Many people claim that an apple falling of a tree is what inspired him to pursue the theory of gravitation. That may even perhaps be the case, however, that may not be enough for our generation to understand it. I think it is probably the realization that all heavenly objects are falling into each other. The moon, for e.g is also falling into earth, but it doesn't reach earth surface since there is some heavenly force making it sustain an orbit around the Earth.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Along this line of reasoning it becomes apparent that Newton had these same ideas. He would have been naturally curious as to what set these heavenly bodies in motion. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am just curious to know what was his theory of the universe at that point.&lt;/p&gt;&#xA;" OwnerUserId="6084" LastEditorUserId="217" LastEditDate="2020-04-13T10:04:01.580" LastActivityDate="2020-04-13T10:04:01.580" Title="What was Sir Issac Newton's thoughts on the origin of the Universe?" Tags="|newton|gravity|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11653" PostTypeId="1" AcceptedAnswerId="11654" CreationDate="2020-04-12T07:54:01.190" Score="13" ViewCount="4701" Body="&lt;p&gt;&lt;strong&gt;I would like to know what is considered to be the first electronic digital computer.&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A literature ambiguous on this. I found these claims:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;ENIAC - a computer constructed by Mr. Eckert and Mr. Mauchly at the University of Pennsylvania &lt;/li&gt;&#xA;&lt;li&gt;Colossus - a computer used in Bletchley Park to break the Enigma code, connected with Alan Turing (&lt;em&gt;correction provided in comments: in fact Colossus was used for breaking Lorentz SZ40 enciphered teleprinter&lt;/em&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Harvard Mark I - a computer constructed in collaboration between Howard Aiken and IBM&lt;/li&gt;&#xA;&lt;li&gt;Z1, Z2 and Z3 - a computers constructed by German scientist Konrad Zuse&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Personally, I would say that ENIAC is the right answer, because Colossus was single purpose electromechanical machine, Harvard Mark I was rather an electromechanical calculator than a computer, and Zuse's computers were highly experimental devices (Z1 and Z2 were electromechanical, Z3 was never completely constructed). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;ENIAC was completely electronicised programmable universal computer. But on the other hand it used decimal coding of numbers instead of binary one. If we take this into account, the first computer should be EDVAC or BINAC.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Could anybody please shed more light on this? Some resources would be appreciated as well.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="12316" LastEditorUserId="12316" LastEditDate="2020-05-01T18:56:34.923" LastActivityDate="2020-05-01T18:56:34.923" Title="What was the first electronic computer?" Tags="|computation|computers|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11654" PostTypeId="2" ParentId="11653" CreationDate="2020-04-12T10:16:05.313" Score="25" Body="&lt;p&gt;The literature is ambiguous because the question is ambiguous. The characteristics of these early machines are well known, but one can set the criteria for being &quot;electronic computer&quot; according to their preference. Does it have to be completely non-mechanical? Turing universal? digital? binary? History can tell us what happened, not what choice of words we should prefer. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One early candidate missing from the OP list is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Atanasoff%E2%80%93Berry_computer&quot; rel=&quot;noreferrer&quot;&gt;Atanasoff-Berry computer&lt;/a&gt;, which Wikipedia calls &quot;the first automatic electronic digital computer&quot;. It was conceived by Atanasoff in 1937 and built in Iowa in 1942 with Berry's help to solve systems of linear equations. It had binary arithmetic and electronic switches, but was neither programmable, nor Turing universal. Unlike the ENIAC of 1947, which it inspired and which Wikipedia calls &quot;the first electronic general-purpose digital computer&quot;. In 1973 a US court even invalidated the ENIAC patent, ruling that the idea of &quot;electronic digital computer&quot; was derived from Atanasoff. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly enough, &quot;the world's first electronic analog computer&quot; was &lt;a href=&quot;https://en.wikipedia.org/wiki/Helmut_H%C3%B6lzer&quot; rel=&quot;noreferrer&quot;&gt;built also in 1942 by Hölzer&lt;/a&gt;, an associate of Nazi rocket designer von Braun, to simulate V-2 rocket trajectories.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-04-12T10:16:05.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11655" PostTypeId="1" CreationDate="2020-04-12T20:05:44.107" Score="10" ViewCount="1027" Body="&lt;p&gt;Apparently Dijkstra wrote in an article in Datamation&lt;sup&gt;1&lt;/sup&gt; in 1977:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It's very illuminating to think about the fact that some – at most four&#xA;  hundred – years ago, professors at European universities would tell the&#xA;  brilliant students that if they were very diligent, it was not&#xA;  impossible to learn how to do long division. You see, the poor guys&#xA;  had to do it in Roman numerals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Is there any evidence for Dijkstra's claim? I had always thought that people had been happily doing all the usual arithmetic operations using abacuses (i.e., using decimal notation) or on clay/slate/parchment/paper using decimal or sexagesimal notation since the time of the Babylonians.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; M.W. Cashman, An Interview with Prof. Edsger W. Dijkstra. &lt;em&gt;Datamation&lt;/em&gt;, 23 (5) (1977), pp. 164-166.&lt;/p&gt;&#xA;" OwnerUserId="6443" LastEditorUserId="4201" LastEditDate="2020-04-24T18:24:05.380" LastActivityDate="2020-04-24T18:24:05.380" Title="Arithmetic calculation before the 17th Century" Tags="|mathematics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11656" PostTypeId="2" ParentId="11655" CreationDate="2020-04-12T21:50:00.683" Score="6" Body="&lt;p&gt;There were at least two very different kinds of calculations performed in&#xA;medieval Europe: a) calculation with integers, for example in accounting, and &#xA;b) calculation with long fractional numbers in astronomy. Positional, sexagesimal system was invented in Babylon for astronomy and other theoretical purposes, and it was not used in Europe,&#xA;in ordinary daily life, for example for accounting.  Since Dijkstra speaks of&#xA;&quot;university professors&quot; he probably means astronomy. Accounting was not taught in the universities at those times.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Decimal system was introduced by Fibonacci 1170-1275 for integers, and for accounting, and only&#xA;in the end of 16th century Simon Stevin 1548-1620 popularized decimal fractions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Division using Roman numerals is indeed awkward. But even in a positional (say, decimal or sexagesimal) system,&#xA;division and multiplication of long numbers are tiresome. Have you actually ever tried to multiply two random 6 digit numbers in decimal system?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You do not have to be very brilliant for this, but&#xA;once you try, you will probably agree that this is a time consuming process requiring great attention. This problem was solved in the late 16th century by Napier. The previous solutions of this problem &lt;a href=&quot;https://en.wikipedia.org/wiki/Prosthaphaeresis&quot; rel=&quot;noreferrer&quot;&gt;Prosthapheresis&lt;/a&gt; were much more complicated.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To conclude, one can say that &quot;brilliant&quot; is an exaggeration, one should probably say &quot;assiduous&quot;. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-04-12T21:50:00.683" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11658" PostTypeId="1" AcceptedAnswerId="11662" CreationDate="2020-04-13T00:50:30.067" Score="6" ViewCount="1005" Body="&lt;p&gt;The sentence is Latin for &lt;em&gt;nature doesn't make jumps&lt;/em&gt;. It refers to the fact that, in most physical processes, quantities vary continuously. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The principle was used by Leibniz, Kant and Darwin among others. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The best source I found for this sentence, tracking back from &lt;a href=&quot;https://en.wikipedia.org/wiki/Natura_non_facit_saltus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;,  &lt;a href=&quot;https://www.textlog.de/1850.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;is this website&lt;/a&gt; (in german). There it is stated that the sentence appeared already  in   &lt;em&gt;Discours véritable de la vie... du géant&lt;/em&gt; Theutobocus (1613), where one finds the sentence in its singular (saltum) version:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Natura in operationibus suis non facit saltum&lt;/em&gt;. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Is this the first occurence of this phrase? It's possible that a single author cannot be found, in any case when did the phrase first appear? Should the phrase be attributed to Theutobocus?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12327" LastEditorUserId="4251" LastEditDate="2020-07-12T22:13:46.047" LastActivityDate="2020-07-12T22:13:46.047" Title="Natura non facit saltus (nature does not make jumps), who said that?" Tags="|physics|reference-request|biology|quote|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11659" PostTypeId="2" ParentId="11653" CreationDate="2020-04-13T01:16:32.323" Score="8" Body="&lt;p&gt;Why does being decimal disqualify ENIAC?  Decimal electronic stored-program computers used to be a thing, for example the &lt;a href=&quot;https://en.wikipedia.org/wiki/IBM_1401&quot; rel=&quot;noreferrer&quot;&gt;commercially successful IBM 1401&lt;/a&gt;. It was a matter of &quot;binary for scientific use, decimal for business use&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My own bias as a programmer disqualifies ENIAC for not being a stored-program digital computer, or more accurately, I'm more interested in stored-program systems. The honors for first place then go to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Manchester_Baby&quot; rel=&quot;noreferrer&quot;&gt;Manchester Baby&lt;/a&gt;. The EDVAC was designed before Baby, but for actually running, Baby wins. That's another bias: a working system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As the other answer says, the question is ambiguous.&lt;/p&gt;&#xA;" OwnerUserId="12328" LastActivityDate="2020-04-13T01:16:32.323" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11660" PostTypeId="2" ParentId="11655" CreationDate="2020-04-13T01:55:58.883" Score="11" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; Dijkstra basically got it right.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The primary means for computing with Roman numerals were the &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_abacus&quot; rel=&quot;noreferrer&quot;&gt;abacus&lt;/a&gt; and the reckoning board. The use of small pebbles in this manner of computation is the origin of our word &lt;a href=&quot;https://www.etymonline.com/search?q=calculus&quot; rel=&quot;noreferrer&quot;&gt;calculus&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Roger Cooke, &quot;The History of Mathematics. A Brief Course 2nd. ed.&quot;, Wiley 2005, p. 144, gives a brief but useful overview how and when the switch took place to our current decimal system using Arabic numerals employing pen-and-paper methods of computation:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For computations these cumbersome numerals were supplanted centuries ago by the Hindu-Arabic place-value decimal system. Before that time, computation had been carried out using common fractions, although for geometric and astronomical computations, the sexagesimal system inherited from the Middle East was also used. It was through contact with the Muslim culture that Europeans became familiar with the decimal place-value system, and such mathematicians as Gerbert of Aurillac encouraged the use of the new numbers in connection with the abacus. In the thirteenth century Leonardo of Pisa also helped to introduce this system of calculation into Europe, and in 1478 an arithmetic was published in Treviso, Italy, explaining the use of Hindu-Arabic numerals [...] In the sixteenth century many scholars, including Robert Recorde (1510-1558) in Britain and Adam Ries (1492-1559) in Germany, advocated the use of the Hindu-Arabic system and established it as a universal standard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Leonardo of Pisa (c. 1170 – 1240), called &lt;a href=&quot;https://en.wikipedia.org/wiki/Fibonacci&quot; rel=&quot;noreferrer&quot;&gt;Fibonacci&lt;/a&gt;, popularized the use of Arabic numerals in his book &lt;em&gt;Liber Abaci&lt;/em&gt; published in 1202. It already contained early variants of our modern long-hand methods for performing basic arithmetic as well as extracting square roots. However, a wide-spread switch to these new methods of doing arithmetic did not take place until roughly the first half of the 16&lt;sup&gt;th&lt;/sup&gt; century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There is a &lt;a href=&quot;https://www.drthomasoshea.com/typus-arithmeticae.html&quot; rel=&quot;noreferrer&quot;&gt;famous allegorical woodcut&lt;/a&gt; from a &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-reisch-s-margarita-philosophica&quot; rel=&quot;noreferrer&quot;&gt;work&lt;/a&gt; by &lt;a href=&quot;https://de.wikipedia.org/wiki/Gregor_Reisch&quot; rel=&quot;noreferrer&quot;&gt;Gregor Reisch&lt;/a&gt; from 1503 that shows a competition between Pythagoras, computing in the traditional method with an abacus, and Boethius, computing with pen and paper using Arabic numerals. From their respective facial expressions it is clear that the latter won. We also note that the goddess Arithmetica in the background smiles favorably upon Boethius. This is a clear expression that the superiority of paper-and-pen computations with Arabic numerals had been recognized.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Adam_Ries&quot; rel=&quot;noreferrer&quot;&gt;Adam Ries&lt;/a&gt; published several popular textbooks in German. It is interesting to note that his first book &lt;em&gt;Rechnung auff der linihen&lt;/em&gt; (1518) explains the use of the abacus, while his second book &lt;em&gt;Rechenung auff der linihen vnd federn&lt;/em&gt; (1522) explains the use of both abacus and pen-and-paper computation, indicating a shift toward the latter manner of computation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Google provides a &lt;a href=&quot;https://books.google.com/books?id=7sVZAAAAcAAJ&amp;amp;dq&quot; rel=&quot;noreferrer&quot;&gt;full scan&lt;/a&gt; of the book &quot;Die Coss Christoffs Rudolffs, Mit schönen Exempeln der Cosz. Durch Michael Stifel Gebessert vnd sehr gemehrt&quot; of 1571. This is a book on algebra written originally by &lt;a href=&quot;https://en.wikipedia.org/wiki/Christoph_Rudolff&quot; rel=&quot;noreferrer&quot;&gt;Christoph Rudolff&lt;/a&gt; (1499-1545) that was improved and expanded by Stifel. Beyond basic arithmetic it covers the extraction of square roots and cube roots. The methods demonstrated are basically identical with the long-hand methods in use today.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2020-04-13T07:30:46.530" LastActivityDate="2020-04-13T07:30:46.530" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11661" PostTypeId="2" ParentId="7801" CreationDate="2020-04-13T03:44:15.403" Score="3" Body="&lt;p&gt;See page 97 of &lt;em&gt;The Mathematical Experience&lt;/em&gt; by Davis and Hersh for an explanation of why the Pythagoreans abominated the number 17.  There it says Plutarch wrote that the Egyptians related that the death of Osiris occurred on the seventeenth of the month, when the full moon is most obviously waning.  Also, the number seventeen is between the two numbers (sixteen and eighteen) that can be simultaneously perimeters and areas of rectangles (sides 4 and 4 or 3 and 6).  &lt;/p&gt;&#xA;" OwnerUserId="12330" LastActivityDate="2020-04-13T03:44:15.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11662" PostTypeId="2" ParentId="11658" CreationDate="2020-04-13T05:04:47.933" Score="7" Body="&lt;p&gt;The Latin quip was derived from Aristotle's History of Animals, where it is rather specific to the context of the &quot;scale of life&quot;. Hence Linnaeus's and Darwin's references are more on point than more sweeping post-Aristotelian generalizations about &quot;most physical processes&quot;, some quite remarkable in how obviously false they are. It should also be said that Darwin's use had a very specific target, Georges Cuvier and his paleontological catastrophism (which was already replaced by Lyell's gradualism, Darwin's inspiration) extended to biology. Here is Aristotle:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Nature proceeds little by little from things lifeless to animal life in such a way&#xA;  that it is impossible to determine the exact line of demarcation, nor on which&#xA;  side thereof an intermediate form should lie. Thus, next after lifeless things in&#xA;  the upward scale comes the plant, and of plants one will differ from another&#xA;  as to its amount of apparent vitality; and, in a word, the whole genus of plants,&#xA;  whilst it is devoid of life as compared with an animal, is endowed with life as&#xA;  compared with other corporeal entities. Indeed, as we just remarked, there is&#xA;  observed in plants a continuous scale of ascent towards the animal.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Already Pliny the Elder in &lt;em&gt;Naturalis Historia&lt;/em&gt; extended Aristotle's scale to &lt;em&gt;scala naturae&lt;/em&gt;, and this is the &quot;ladder&quot; Kant refers to. Plotinus and Christian authors took it all the way to the Heaven as a &quot;great chain of being&quot; much earlier, see &lt;a href=&quot;https://books.google.com/books/about/Aristotle_s_Ladder_Darwin_s_Tree.html?id=YVAZAwAAQBAJ&quot; rel=&quot;noreferrer&quot;&gt;Archibald, Aristotle’s Ladder, Darwin’s Tree&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for the quip, &quot;Theutobocus&quot; is not the author name, the reference is to &lt;a href=&quot;https://fr.wikisource.org/wiki/Discours_v%C3%A9ritable_de_la_vie,_mort,_et_des_os_du_g%C3%A9ant_Theutobocus&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Discours véritable de la vie, mort et des os du géant Theutobocus&lt;/em&gt;&lt;/a&gt; by a rather obscure Jacques Tissot (Lyon, 1613). The &quot;bones of giant &lt;a href=&quot;https://fr.wikipedia.org/wiki/Theutobocus&quot; rel=&quot;noreferrer&quot;&gt;Theutobocus&lt;/a&gt;&quot; apparently referred to bones of a mastodon presented as those of a human giant in a hoax. That a French work inserts a Latin quip, calling it an &quot;axiom&quot; no less, suggests that it was already in circulation by then. &lt;a href=&quot;https://books.google.com/books?id=yoUVAAAAMAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Classical and Foreign Quotations, p. 209&lt;/a&gt; confirms it:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Tissot is quoting an old and well-established principle of physics. &quot;Operatur natura&quot;, he says, &quot;quantum et quandim potest, sans neant moins faire aucun sault ab extremis ad extrema. Natura enim in operationibus suis non facit saltum&quot;. His contemporary, Sir E. Coke, applies it to law:&quot;Natura non facit saltus, ita nec lex&quot;. Coke upon Littleton, pp 238b, 239.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And then it goes to Leibniz and Linnaeus. &lt;a href=&quot;https://books.google.com/books?id=LXIrAAAAYAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Wicksteed in  Studies in Theology, p. 30&lt;/a&gt; adds the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;I have not been able to trace it further back in the form of an aphorism, but the principle is taken for granted by Dante as holding good in the physical universe, and is applied by him to the intellectual order.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;A lengthy quote from &lt;a href=&quot;https://digitaldante.columbia.edu/text/library/the-convivio/book-03/#07&quot; rel=&quot;noreferrer&quot;&gt;Dante's &lt;em&gt;Convivio&lt;/em&gt;, III.7&lt;/a&gt; follows, where the &quot;great chain of being&quot; is on full display:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;And since in the intellectual order of the universe the ascent and descent are almost by continuous gradations from the lowest form to the highest and from the highest to the lowest, as we see in the order of beings capable of sensation; and since between the angelic nature, which is intellectual being, and the human nature there is no gradation but rather the one is, as it were, continuous with the other by the order of gradation; and since between the human soul and the most perfect soul of the brute animals there is also no intermediary gradation, so it is that we see many men so vile and in such a state of baseness that they seem to be almost nothing but beasts. Consequently it must be stated and firmly believed that there are some so noble and so lofty in nature that they are almost nothing but angels, for otherwise the human species would not be continuous in both directions, which is impossible.&lt;/em&gt;&quot; &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-13T07:22:57.003" LastActivityDate="2020-04-13T07:22:57.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11663" PostTypeId="1" AcceptedAnswerId="11665" CreationDate="2020-04-13T11:28:58.570" Score="5" ViewCount="139" Body="&lt;p&gt;Fibonacci is &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system&quot; rel=&quot;nofollow noreferrer&quot;&gt;credited&lt;/a&gt; with the introduction of Arabic arithmetic to Europe. Did he tried hard to share this knowledge with other mathematicians? With whom? How did they react?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="12271" LastEditDate="2020-04-13T14:01:26.337" LastActivityDate="2020-04-13T22:05:25.783" Title="What actions and communications did Fibonacci undergo to introduce the Hindu-Arabic Number System to Europe?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11664" PostTypeId="1" CreationDate="2020-04-13T15:57:41.183" Score="3" ViewCount="291" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat_number#Factorization_of_Fermat_numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Wikipedia Page on Fermat numbers&lt;/a&gt; states that &lt;span class=&quot;math-container&quot;&gt;$F_5$&lt;/span&gt; was &quot;fully factored&quot; in 1732.  This appears to be the same time that Euler found that any factor of a Fermat number &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt; was of the form&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2^{n+1}k+1$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;for some &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; (later improved by Lucas) and found the smaller factor of &lt;span class=&quot;math-container&quot;&gt;$F_5$&lt;/span&gt;, the prime &lt;span class=&quot;math-container&quot;&gt;$641$&lt;/span&gt;, by trial and error.  What isn't clear to me is whether Euler also verified that the larger factor &lt;span class=&quot;math-container&quot;&gt;$F_5 / 641 = 6,700,417$&lt;/span&gt; was prime (&lt;a href=&quot;https://www.wolframalpha.com/input/?i=factor+6700417&quot; rel=&quot;nofollow noreferrer&quot;&gt;it is&lt;/a&gt;). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;C. Edward Sandifer's book &lt;em&gt;&lt;a href=&quot;https://books.google.com/books?id=3c6iBQAAQBAJ&amp;amp;pg=PA43&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Euler did Even More&lt;/a&gt;&lt;/em&gt; contains the passage&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Euler did not speculate in print on whether the other factor, 6,700,417, is prime. It is prime, but&#xA;  there is no evidence that Euler ever tried to find out. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;but continues along the lines of how easy it is to prove, using the same tools Euler used to find the smaller factor.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You can't say a number is &quot;fully factored&quot; until you know all of its factors are prime. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was originally going to ask whether Euler proved &lt;span class=&quot;math-container&quot;&gt;$6700417$&lt;/span&gt; was prime, but in light of the above passage, I guess we'll never know. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can we really say that &lt;span class=&quot;math-container&quot;&gt;$F_5$&lt;/span&gt; was fully factored in 1732, or is there some date for the verification of &lt;span class=&quot;math-container&quot;&gt;$6700417$&lt;/span&gt; that's more appropriate?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(user6530 pointed me to a cleaner version of the text than &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2007-03.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Euler Archive preprint&lt;/a&gt; I'd originally linked to.)&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2020-04-13T19:44:42.237" LastActivityDate="2020-09-21T01:00:40.077" Title="What is the basis of the claim that $F_5$ was fully factored in 1732?" Tags="|mathematics|number-theory|euler|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11665" PostTypeId="2" ParentId="11663" CreationDate="2020-04-13T20:32:05.507" Score="8" Body="&lt;p&gt;Sharing with other mathematicians is not how things were done in the 13th century Europe. The only other mathematician of note in Europe whose years of life overlap with Fibonacci's was Nemorarius (1225-1260), and he was not even born when &lt;em&gt;Liber Abaci&lt;/em&gt; came out (1202). Nonetheless, Fibonacci was quite active in promoting mathematics both to the folk and to the learned, those who were around, and he had a knack for popularization of mathematics in applications. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Fibonacci.html&quot; rel=&quot;noreferrer&quot;&gt;MacTutor's biography&lt;/a&gt; is quite informative in this regard. For example, he corresponded with Michael Scotus, the astrologer, Theodorus Physicus, the philosopher, and Dominicus Hispanus since his return to Pisa c. 1200. This got him a favorable reception at the court of Frederick II, who became the Holy Roman emperor in 1220, and spent most of 1220s in Italy solidifying his control there. Frederick was a benefactor of scholars and founded the University of Naples in 1224. Johannes of Palermo, a Frederick's courtier, presented a number of problems to Fibonacci, which he solved and published in &lt;em&gt;Flos&lt;/em&gt; (1225), adding to his fame. &lt;em&gt;Liber Abaci&lt;/em&gt; itself was presciently dedicated to Scotus, who became the court astrologer. Its second section contained problems concerning the price of goods, calculating profits on transactions, and converting between currencies, clearly designed to make the decimal system attractive to merchants. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fibonacci also wrote a separate book on commercial arithmetic &lt;em&gt;Di Minor Guisa&lt;/em&gt;, which is not extant. His &lt;em&gt;Practica geometriae&lt;/em&gt; (1220) had a similarly practical bent. &lt;em&gt;Liber Abaci&lt;/em&gt; became quite popular with the folk as well as the court, and so did Fibonacci. In 1240 his home city of Pisa awarded a salary to &quot;&lt;em&gt;the serious and learned Master Leonardo Bigollo&lt;/em&gt;&quot; (&quot;Fibonacci&quot; is a nickname invented by a historian Libri in 1838) for his troubles with advising and teaching the citizens new ways of accounting and land surveying. Vogel in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dictionary_of_Scientific_Biography&quot; rel=&quot;noreferrer&quot;&gt;Dictionary of Scientific Biography&lt;/a&gt; writes the following on his influence:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Direct influence was exerted only by those portions of the &quot;Liber abaci&quot; and of the &quot;Practica&quot; that served to introduce Indian-Arabic numerals and methods and contributed to the mastering of the problems of daily life. Here Fibonacci became the teacher of the masters of computation and of the surveyors, as one learns from the &quot;Summa&quot; of Luca Pacioli...&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, as Schaller points out in &lt;a href=&quot;https://www.hec.ca/finance/Fichier/Schaller.pdf&quot; rel=&quot;noreferrer&quot;&gt;Fibonacci, Pacioli, and the Slow Diffusion of Knowledge&lt;/a&gt;, comparison to Pacioli also shows limitations of Fibonacci's efforts, which explain why decimal calculations did not spread widely until 16th century:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In the 1460s, while Luca Pacioli, a Franciscan monk, was working in Venice, he&#xA;  discovered a copy of Fibonacci's Liber Abaci. Because he had worked six years as a merchant's assistant, Pacioli was able to recognize the importance of the book for the practical world of business... Pacioli's summary of contemporary mathematical knowledge appeared in 1494 under the&#xA;  title Summa de arithmetica, geometria, proportioni, et proportionalita...&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;There are two key differences between Fibonacci's Liber Abaci and Pacioli's Summa, despite the fact that the intellectual content of Pacioli's chapters on Arabic numerals and the&#xA;  basic principles of arithmetic follow Fibonacci's book very closely. First, the Summa was&#xA;  printed, rather than handwritten. The dramatic decrease in cost made it possible for merchants to&#xA;  buy the book. Second, Pacioli wrote almost all of his book in the vernacular, making it accessible&#xA;  to business people.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-04-13T22:05:25.783" LastActivityDate="2020-04-13T22:05:25.783" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11667" PostTypeId="2" ParentId="11653" CreationDate="2020-04-14T07:19:55.030" Score="4" Body="&lt;p&gt;My personal marker is stored program control, ie, the program is stored in the computer memory. Early machines like the original ENIAC, Colossus, Zuse's Z-machines, were all programmed by changing the wiring in some fashion. The Z-machines used relays, ENIAC was an advance because it used electronics (valves, the transistor hadn't been invented). The first stored program computer was the Manchester Baby in 1948.&lt;/p&gt;&#xA;" OwnerUserId="12344" LastActivityDate="2020-04-14T07:19:55.030" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11669" PostTypeId="1" AcceptedAnswerId="11670" CreationDate="2020-04-15T14:21:12.877" Score="3" ViewCount="128" Body="&lt;p&gt;I've only heard personal anecdotes about perceived stimuli and economic support to research in pure mathematics in the 1960s, presumably tied to the space race of the time. Is this something we can claim to have been planned by the governments engaged in the race?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="10106" LastEditDate="2020-05-12T19:22:03.960" LastActivityDate="2020-05-12T19:22:03.960" Title="Is it fair to say that the Space Race incentivized research in pure mathematics?" Tags="|mathematics|cold-war|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11670" PostTypeId="2" ParentId="11669" CreationDate="2020-04-16T11:49:58.893" Score="9" Body="&lt;p&gt;Indeed, funding of mathematics, together with other science and engineering fields&#xA;experienced a sharp increase between 1957 and 1970. The number of PhD awarded in these fields in USA tripled during this period, and this was the fastest growth for&#xA;any period after WWII.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nsf.gov/statistics/2018/nsf18304/static/report/nsf18304-report.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.nsf.gov/statistics/2018/nsf18304/static/report/nsf18304-report.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the &quot;race to the Moon&quot; was not the primary reason. The primary reason was arms race during the Cold war. The race to the Moon itself, is a byproduct of this arms race. This happened both in Soviet Union and in the US. Soviets demonstrated their success in ICBM development by launching the first artificial satellite in 1957.&#xA;This made an enormous impression on American public, and the &quot;space race&quot; began. Much of mathematical research at that time was funded by the US Navy and Air Force, in the same way as it is funded nowadays by NSF. &#xA;In Soviet Union itself, the sharp increase in S and E research was triggered by the race to make a nuclear bomb, and then it was stimulated by&#xA;development of new weapons, like ICBM and submarines.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am a &quot;pure mathematician&quot;, and my career began in Soviet Union in the 1970s. Everyone around was strongly encouraged to make some contribution to research directly related to submarine warfare at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Edit. In general, on my opinion, the arms race strongly benefited mathematics, especially in Soviet Union. They made the careers of professor and researcher very attractive by relatively high salaries. They developed a system of mathematical olimpiads to select talented children. The competition for students to enroll in the best math departments was high.&#xA;Some top mathematicians were chosen to advise the government, and could influence some decisions. For example, the initiative of the space exploration came from scientists directly involved in defense research. The ideological interference into physics and mathematics was minimal (as compared for example to biology, and related medical and agricultural sciences, not even mentioning humanities which were under complete ideological control). And so on. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-04-16T16:06:32.787" LastActivityDate="2020-04-16T16:06:32.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11671" PostTypeId="1" AcceptedAnswerId="11672" CreationDate="2020-04-16T16:52:44.000" Score="5" ViewCount="1023" Body="&lt;p&gt;I guess it is an experimental law, so what was the experiment?&lt;/p&gt;&#xA;" OwnerUserId="12320" LastEditorUserId="12316" LastEditDate="2020-04-16T19:19:59.517" LastActivityDate="2020-04-27T08:40:10.713" Title="How did Newton prove his third law of motion?" Tags="|physics|experimental-physics|newton|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11672" PostTypeId="2" ParentId="11671" CreationDate="2020-04-16T23:52:38.237" Score="6" Body="&lt;p&gt;It was not &amp;quot;the experiment&amp;quot;. First, Newton considered &amp;quot;his&amp;quot; laws to be &amp;quot;common knowledge&amp;quot; already &amp;quot;abundantly&amp;quot; confirmed and accepted by experts (he names Galileo, Wallis, Wren, and Huygens). In &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Axioms,_or_Laws_of_Motion&quot; rel=&quot;noreferrer&quot;&gt;Axioms, or Laws of Motion section of Principia&lt;/a&gt;, where they are laid down, he writes in particular:&amp;quot;&lt;em&gt;Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments&lt;/em&gt;&amp;quot;. Nonetheless, he describes plenty of experiments of his own used to reconfirm the laws.&lt;/p&gt;&#xA;&lt;p&gt;But second, Newton also understood that it is the theory as a whole that is being confirmed, not each law by itself. He usually needed all of them to derive any one prediction, so no one experiment could prove them one at a time. He writes in the Scholium:&amp;quot;&lt;em&gt;And thus the third Law, so far as it regards percussions and reflexions, is proved by a theory exactly agreeing with experience.&lt;/em&gt;&amp;quot; Indeed, the third law is first introduced with appeal to common experiences:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;proofs&amp;quot; in the Scholium refer to more formally staged experiments, which include colliding two pendula of various masses, sizes and densities, and measuring the rebound, bringing a system of iron and loadstone into equilibrium, and measurements on simple machines, like balances, wedges and pulleys. They are not proofs in the literal sense (as in deriving a law from the experiment), but rather confirmations that all of the above are predicted to behave as observed when the third law (along with the first and second) is used to make the predictions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Thus trying the thing with pendulums of ten feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and, of consequence, that the action and reaction were always equal... I must add, that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to diminish the reflexion in such a certain proportion as the quantity of the elastic force requires.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;In attractions, I briefly demonstrate the thing after this manner... I made the experiment on the loadstone and iron. If these, placed apart in proper vessels, are made to float by one another in standing water, neither of them will propel the other; but, by being equally attracted, they will sustain each other's pressure, and rest at last in an equilibrium.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary... I was only willing to show by those examples the great extent and certainty of the third Law of motion. For if we estimate the action of the agent from its force and velocity conjunctly, and likewise the reaction of the impediment conjunctly from the velocities of its several parts, and from the forces of resistance arising from the attrition, cohesion, weight, and acceleration of those parts, the action and reaction in the use of all sorts of machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon the resisting body, the ultimate determination of the action will be always contrary to the determination of the reaction.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-04-16T23:58:13.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11673" PostTypeId="2" ParentId="3131" CreationDate="2020-04-17T10:57:39.747" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Moj%C5%BCesz_Presburger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mojżesz Presburger&lt;/a&gt; was one of the victims of the Holocaust.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;/questions/tagged/mathematicians&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;mathematicians&amp;#39;&quot; rel=&quot;tag&quot;&gt;mathematicians&lt;/a&gt; &lt;a href=&quot;/questions/tagged/logicians&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;logicians&amp;#39;&quot; rel=&quot;tag&quot;&gt;logicians&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-02-28T11:54:31.473" LastActivityDate="2021-02-28T11:54:31.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11674" PostTypeId="1" AcceptedAnswerId="11680" CreationDate="2020-04-17T14:22:43.797" Score="0" ViewCount="284" Body="&lt;p&gt;At the beginning of the electronic computing era in the 1950s, computers were mutually incompatible both in terms of hardware and software. It was impossible to use peripherals (e.g. punched card or tape readers, printers, etc.) for some computer with another one as there was no common interface. The same was mostly true for software. Programs created for one computer often could not run on another because programming languages (or rather assemblers or microcode) were unique to each computer (although first versions of Fortran, Cobol and Algol were introduced at the end of 1950s). So when a customer bought a new computer, new peripherals also had to be purchased and software completely (or at least partially) reprogrammed. This changed in 1964 when IBM introduced System/360, a scalable computing system with mutually compatible peripherals, common programming language(s) and operating system. It was even possible to emulate IBM older models (e.g. 1401) on System/360. After that, some other computer manufacturers started to produce computers compatible with IBM systems.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can understand that at the beginning, there was a lack of standards, and each manufacturer tried to lock its customers in and avoid switching to competitor's products. However, I do not understand why computers produced by manufacturers were incompatible with each other, and software had to be rewritten after an upgrade. In IBM, there were even two rival teams of engineers, placed in different factories. The first team produced smaller computers such as 600 and 1400 series and second one mainframe computers (series 700 and later 7000). Each product line had its operating system, programming language and peripheries as these teams did not collaborate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;So, why did it take so long (until mid of the 1960s) to introduce common standards even within one company? Was it more about technical reasons or business ones (e.g. development of System/360 cost IBM about 30 billion of today US dollars)?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Note: Information about IBM is based on the book &quot;IBM: The Rise and Fall and Reinvention of Global Icon&quot; by James W. Cortada.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="12316" LastEditorUserId="26" LastEditDate="2021-02-20T15:15:51.010" LastActivityDate="2021-02-20T15:15:51.010" Title="Why were early electronic computers mutually incompatible?" Tags="|computation|computers|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11675" PostTypeId="2" ParentId="11674" CreationDate="2020-04-18T01:33:43.893" Score="5" Body="&lt;p&gt;First machines were one-off affairs, custom built (often for a very narrow purpose). There just weren't enough of those around to make standarization of anything surrounding them worthwhile. Thomas Watson (IBM president at the time) accurately estimated a demand of half a dozen computers worldwide in that timeframe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remember the idea behind the Multics proyect (GE -- AT&amp;amp;T -- others) was to build a machine to essentially serve all computing needs of a larger area. Not &quot;a machine in each home&quot;, an access point. Here in Chile by the early '80s even banks didn't have their own computers, they rented computer time from companies that specialized in doing the processing for a lot of even large clients.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Later on, each company built their own machines, and had their own standards (i.e., IBM machines worked --mostly-- with IBM peripherals), some copycats came up (e.g. Amdahl had a very successful IBM-compatible line of mainframes), Digital specialized on minicomputers -- small machines for e.g. a department, later workstations --powerful machines for personal use with graphics screens-- became common. All very much incompatible, software-wise (porting a program to the brand-own Unix dialect was &lt;em&gt;real&lt;/em&gt; fun) and absolutely hardware-wise. For example, IBM has EBCDIC as character set standard (up to this day, I'm told), they used to use a weird coaxial cable with two  central conductors to connect assorted equipment. But even so, often the new model was very much incompatible with the last one. War stories of machines running an emulator of an earlier one, on which to run an emulator for an even older one, on which to run some legacy software abound.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today's much simpler world, in which &quot;everything is a PC&quot; is a rather recent phenomenon (since around the time Apple migrated to intel), largeish &quot;servers&quot; now also are PC-type hardware, not SPARC or Alpha or some other architecture. Earlier phenomenon was the mistaken idea that &lt;a href=&quot;http://catb.org/~esr/jargon/html/V/vaxocentrism.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&quot;everything's a VAX&quot;&lt;/a&gt; in the Unix world.&lt;/p&gt;&#xA;" OwnerUserId="2905" LastEditorUserId="2905" LastEditDate="2020-04-18T22:21:43.187" LastActivityDate="2020-04-18T22:21:43.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11676" PostTypeId="1" CreationDate="2020-04-18T16:10:46.123" Score="0" ViewCount="62" Body="&lt;p&gt;Did Garrett Birkhoff ever work with Feynman's path integral? Did he ever work in his Many-Histories interpretation? Or at least, did he mention it in any of his articles?&lt;/p&gt;&#xA;" OwnerUserId="10956" LastEditorUserId="10956" LastEditDate="2020-04-27T17:17:56.843" LastActivityDate="2020-04-27T17:17:56.843" Title="Did the mathematician Garrett Birkhoff ever work with or mention Feynman's path integrals?" Tags="|mathematics|physics|mathematicians|quantum-mechanics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11677" PostTypeId="1" CreationDate="2020-04-18T16:37:58.683" Score="8" ViewCount="514" Body="&lt;p&gt;The early voltaic pile has cells consisting of a [&lt;strong&gt;copper&lt;/strong&gt;||electroltye||&lt;strong&gt;zinc&lt;/strong&gt;] unit, and then another, so all of the zinc-&gt;copper junctions are metal-on-metal and all of the copper-&gt;zinc junctions are metal-on-electrolyte-on-metal, as neatly illustrated by this illustration from Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uwNzt.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uwNzt.png&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where did the decision come from to name the Zinc side the negative side and the copper side the positive side? Was this Volta's doing, or someone else? It's confusing for a lot of people, because we speak of &quot;positive charge&quot; where we now know that positive charge is really the absence of electrons. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So where does this confusion originate?&lt;/p&gt;&#xA;" OwnerUserId="6112" LastActivityDate="2020-04-19T07:38:14.670" Title="Where did the names positive and negative for the early voltaic piles come from?" Tags="|electricity|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11678" PostTypeId="1" CreationDate="2020-04-18T16:47:14.570" Score="3" ViewCount="100" Body="&lt;p&gt;I learned about the push-pull construction from &lt;a href=&quot;https://www.mpim-bonn.mpg.de/sites/default/files/videos/download/freedman01.mp4&quot; rel=&quot;nofollow noreferrer&quot;&gt;a video lecture by Freedman&lt;/a&gt; in which it is explained starting around 39:08. It is somewhat long and technical to describe in detail, but the main idea is that it is a technique to control an homeomorphism in one linear direction.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The example application is to show that if &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; are compact metric spaces such that &lt;span class=&quot;math-container&quot;&gt;$X\times\Bbb R$&lt;/span&gt; is homeomorphic to &lt;span class=&quot;math-container&quot;&gt;$Y\times\Bbb R$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$X\times S^1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y\times S^1$&lt;/span&gt; are also homeomorphic. The way this is done is by taking an homeomorphism &lt;span class=&quot;math-container&quot;&gt;$h\colon X\times\Bbb R\to Y\times\Bbb R$&lt;/span&gt; and using the push-pull construction to modify it to another homeomorphism between the two spaces which is periodic in the &lt;span class=&quot;math-container&quot;&gt;$\Bbb R$&lt;/span&gt; coordinate, so that it descends to an homeomorphism &lt;span class=&quot;math-container&quot;&gt;$X\times S^1\to Y\times S^1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Morse used the construction in 1960 to establish the Schoenflies conjecture &quot;Every bicollared embedding &lt;span class=&quot;math-container&quot;&gt;$S^d\to\Bbb R^{d+1}$&lt;/span&gt; extends to an embedding &lt;span class=&quot;math-container&quot;&gt;$D^{d+1}\to\Bbb R^{d+1}$&lt;/span&gt;&quot;. In particular he used the push-pull construction to show that every such bicollared embedding admits a so called flat spot after composing with a self-homeomorphism of &lt;span class=&quot;math-container&quot;&gt;$\Bbb R^{d+1}$&lt;/span&gt;, and the conjecture had already been established by Mazur for bicollared embeddings with a flat spot.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However Freedman mentions that the push-pull construction was not introduced by Morse, and in fact that it was already well known by the time he used to to settle Schoenflies's conjecture, hence my question: who invented the push-pull construction?&lt;/p&gt;&#xA;" OwnerUserId="12363" LastActivityDate="2020-04-18T16:47:14.570" Title="Who invented the push-pull construction?" Tags="|geometry|topology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11680" PostTypeId="2" ParentId="11674" CreationDate="2020-04-19T02:56:16.217" Score="6" Body="&lt;p&gt;There is a heap of answers to this question, and they cover a wide range of aspects of the early computer industry.&#xA;As has been pointed out, the very first machines were hand built lab experiments. It was not at all obvious how to build a digital computer that would work. Remember Babbage's Analytical Engine had foundered on the challenge of actually building it. &#xA;Go back to the 1940s and 50s. Not merely no integrated circuits (invented 1959), but no transistors (invented 1948).&#xA;Until the widespread inclusion of microcode ( invented 1951), instruction sets were intimately tied to the physical architecture. So a different architecture meant a different instruction set. Incidentally, a hard wired architecture will out perform a microcode implementation of same, but at an enormous cost in flexibility. And we all know that every design is perfect when first implemented, right? &#xA;Why would IBM have entirely separate architectures in their product line-up? Part of it was market control - the incompatibilities meant customers were locked in. There was more competition in the midrange (a market pioneered by DEC), but they didn't want high paying mainframe customers jumping down to midrange where profits were lower. They have finished up at the other extreme now, iSeries, pSeries and zSeries all use Power architecture chips (but still have incompatible instructions and OS's).&#xA;RISC architecture promised massive performance/efficiency gains (which is why Sun, Motorola and IBM all jumped in that direction), but seemed to fall by the wayside, until the rise of mobile devices in the last 10 years (most of which use ARM, a RISC architecture). That begot the SBCs such as Raspberry Pi, as well, IoT is another growth area. &#xA;It's sometimes hard for people who grew up with the Wintel stack to see this, but it's by no means the only solution, and it's dominance owes more to history, market power and market forces than technical merit. &lt;/p&gt;&#xA;" OwnerUserId="12344" LastActivityDate="2020-04-19T02:56:16.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11681" PostTypeId="2" ParentId="11677" CreationDate="2020-04-19T03:18:07.867" Score="9" Body="&lt;p&gt;The &amp;quot;decision&amp;quot; came from Franklin c. 1750, and predates voltaic piles. Franklin followed the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fluid_theory_of_electricity#One-fluid_theory&quot; rel=&quot;noreferrer&quot;&gt;one-fluid theory of electricity&lt;/a&gt; proposed by Watson in 1746, and explained  electric discharges in a Leyden jar by the fluid flowing from the side with its excess (+) to the side with its lack (-). Which side had the excess/lack he decided more or less randomly, so when the glass was rubbed the fluid was supposed to go into it while it was coming out of resins, contrary to the actual movement of electrons. In any case, it was argued on Physics SE that Franklin's choice was fortuitous rather than confusing, and that generally identifying conventional current with the flow of electrons in reverse causes the real confusion, see &lt;a href=&quot;https://physics.stackexchange.com/a/17131/65263&quot;&gt;Why is the charge naming convention wrong?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When Volta proposed his pile in 1799 what went on in it was originally interpreted as a sequence of electrostatic discharges, so, naturally, Franklin's convention was followed, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5150/55&quot;&gt;What is the history of electric current and resistance?&lt;/a&gt; By the time electrons were discovered and the nature of current identified the convention has become too entrenched to change, especially considering that the real direction made no difference to electric circuit calculations, and that ions can also move, so the direction of the overall current had to be assigned by convention anyway. Here is from &lt;a href=&quot;https://www-spof.gsfc.nasa.gov/Education/woppos.html&quot; rel=&quot;noreferrer&quot;&gt;Stern's The Direction Assigned to Electric Currents&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The choice of which type of electricity is called &amp;quot;positive&amp;quot; and which &amp;quot;negative&amp;quot; was made around 1750 by Ben Franklin... Franklin knew of two types of electric charge, depending on the material one rubbed. He thought that one kind signified a little excess of the &amp;quot;electric fluid&amp;quot; over the usual amount, and he called that &amp;quot;positive&amp;quot; electricity (marked by +), while the other kind was &amp;quot;negative&amp;quot; (marked -), signifying a slight deficiency.  It is not known whether he tossed a coin before deciding to call the kind produced by rubbing glass &amp;quot;positive&amp;quot; and the other &amp;quot;resinous&amp;quot; type &amp;quot;negative&amp;quot; (rather than the other way around), but he might just as well have.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Later, when electric batteries were discovered, scientists naturally assigned the direction of the flow of current to be from (+) to (-). A century after that electrons were discovered and it was suddenly realized that in metal wires the electrons were the ones that carried the current, moving in exactly the opposite direction. Also, it was an excess of electrons which produced a negative electric charge. However, it was much too late to change Franklin's naming convention.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-04-19T07:38:14.670" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11682" PostTypeId="2" ParentId="11677" CreationDate="2020-04-19T04:06:59.713" Score="4" Body="&lt;p&gt;You are indeed right, these two signs (+) and (-) have confused generations of scientists. I have searched for more than 10 years on Volta's assignment on the signs of battery terminals. I will summarize my findings below. What is disappointing when these chemistry or physics teachers say...well, it was just a convention. Sorry, this is not true, scientists from the previous centuries were as good as any other scientists today. There is a very valid reason why they labelled the batteries that way. What confuses most people is the transition from the signs of pure electrostatics to current flow-like a battery. How did the batteries get their labels?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Forget about electrons for the time being. This is a relatively recent discovery. Let say you are in the 18th century. You will have to follow the train of the thought of these early physicists. First of all, you might already be aware that Benjamin Franklin called one type of charge as positive and the other type of charge as negative. You have already got the details by Conifold by the time I was typing the answer. The &lt;em&gt;functional&lt;/em&gt; definition is positive charge is that it is the charge that is obtained by rubbing glass with silk. Label the glass rod as positive electricity. Rub the resinous bodies, and you get negative electricity. This is the functional definition of the negative electricity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key point is that all electrical signs on batteries can now be traced these functional definitions given above. Keep in mind we still do not know what an electron is. Now there is a device called the condensing electroscope. Volta himself invented it. This can sense very small charges, as found in batteries, and causes the gold leaves to diverge. Let us say you have an unknown charged body, you would connect the object to be tested with the condensing electroscope and charge the electroscope. The leaves will diverge. Note that ordinary electroscope won't show any response. Next bring your charged glass rod in contact with the condensing electroscope. If the leaves diverge further, your test body had the same positive electricity. Your test body will get a (+) sign. In same way you can test it with a rubbed resinous body and if the leaves diverge your test body will get a (-) sign.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So far there is no need to invoke electrons and protons. Now you may guess that how Volta would have tested the sign of the ends of this pile using his condensing electroscopes. Note that electrometers were a fancier version of determining the signs of charged objects. This term will occur in the suggested paper below.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;With that you should this 1930's paper from Nature (Open access) in order to find the answer to your query &lt;a href=&quot;https://www.nature.com/articles/125349a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Original Mode of Constructing a Voltaic Pile&lt;/a&gt;. It is a single page article but it will clarify the junk our modern textbooks have created about the signs of the Voltaic pile.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Food for thought: &#xA;a) Rarely any physics or chemistry textbook talks about about determining the sign of the electron's charge? How was this determined I will let you find out. Hint, it is also traceable to resinous electricity definition.&#xA;b) How did the early scientists find out that the charge carriers in wires are negatively charged particles?&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2020-04-19T04:43:46.077" LastActivityDate="2020-04-19T04:43:46.077" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11686" PostTypeId="1" CreationDate="2020-04-20T15:53:23.197" Score="2" ViewCount="188" Body="&lt;p&gt;I am interested in nineteenth-century astronomy and Laplace's &lt;a href=&quot;https://en.wikipedia.org/wiki/Trait%C3%A9_de_m%C3%A9canique_c%C3%A9leste&quot; rel=&quot;nofollow noreferrer&quot;&gt;Treatise of celestial mechanics&lt;/a&gt; is often mentioned as one of the most significant contributions to science in this period. The more than thousand pages long tomb is impenetrable to me. Most secondary literature just claims that it is a synthesis of the progress of eighteenth-century Newtonian science, but nobody goes into more detail. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is a &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/References/Laplace.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;list&lt;/a&gt; of some literature, but none really provides a detailed outline of Laplace's main work. Alas, there is no secondary literature of his main work as there is for figures like Galileo, Newton, Euler and Einstein.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I can understand why there is no secondary literature: you need to have a very strong mathematical background (problems like shape of earth, and lunar motion are complex) and in addition you also need to have much historical knowledge of empirical sciences on tides, lunar motions, astronomical data. This seems to be an extremely demanding skillset. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was hoping somebody could fill the gap between the popularizing presentions and the technical subjects of the book. You may also refer to relevant secondary literature. Partial answers are also welcome. I don't expect a complete summary of his main work. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The main topics in the work can be read off the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trait%C3%A9_de_m%C3%A9canique_c%C3%A9leste&quot; rel=&quot;nofollow noreferrer&quot;&gt;table of contents&lt;/a&gt;. The kind of questions I have are for example:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;what &lt;em&gt;exactly&lt;/em&gt; did the book accomplish? What did Laplace himself said it accomplishes?&lt;/li&gt;&#xA;&lt;li&gt;what kind of (mathematical) methods and (theoretical) ideas are used to accomplish the main results?&lt;/li&gt;&#xA;&lt;li&gt;how to draw a distinction between his own contributions and his predecessors?&lt;/li&gt;&#xA;&lt;li&gt;was the book constructed according to some larger plan or does he just deal with different domains in each chapter?&lt;/li&gt;&#xA;&lt;li&gt;was some of his theory unsatisfactory? Were there errors in his reasoning or some problematic aspects?&lt;/li&gt;&#xA;&lt;li&gt;did Laplace's theoretical work resolve the needs of empirical science of his day (especially in astronomy)?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;This is a non-exhaustive list of some questions (other contribution are also welcome).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;p.s. the english translation of the four books is &lt;a href=&quot;https://en.wikipedia.org/wiki/Trait%C3%A9_de_m%C3%A9canique_c%C3%A9leste#External_links&quot; rel=&quot;nofollow noreferrer&quot;&gt;freely available online&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12376" LastEditorUserId="12376" LastEditDate="2020-04-20T16:04:50.687" LastActivityDate="2020-04-21T09:14:49.547" Title="Laplace's Treatise of celestial mechanics" Tags="|mathematics|astronomy|celestial-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11688" PostTypeId="2" ParentId="11686" CreationDate="2020-04-21T09:14:49.547" Score="2" Body="&lt;p&gt;In An IX du calenedrier republicain (1801) &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean-Baptiste_Biot&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean-Baptiste Biot&lt;/a&gt; published his &lt;a href=&quot;https://books.google.bg/books?id=ZFBZAAAAcAAJ&amp;amp;pg=PA1&amp;amp;dq=biot+analyse+traite&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjqgYCSkPnoAhUv_SoKHfViDYcQ6AEIVjAF#v=onepage&amp;amp;q=biot%20analyse%20traite&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Analyse du traité de mécanique céleste de P.S. Laplace&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;To develop the relationships existing between movements and forces&#xA;  that produce them, and from here to deduce the nature of the force&#xA;  that would animate celestial bodies in such a way that their movements&#xA;  are the ones presented by observation, and thus ascend to the&#xA;  principle of universal gravitation then descend from this principe to&#xA;  the explanation of all celestial phenomena in their minutest details,&#xA;  that is object of &lt;em&gt;Mecanique celeste&lt;/em&gt;, such is the aim of LaPlace's&#xA;  work. (p.3)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AoyuK.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AoyuK.jpg&quot; alt=&quot;Title page and p.3 from Biot&amp;#39;s booklet&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-04-21T09:14:49.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11689" PostTypeId="1" CreationDate="2020-04-21T12:05:18.733" Score="2" ViewCount="47" Body="&lt;p&gt;I'm looking for the earliest known written register of an improper fraction, that is, a numerical fraction in which the numerator is greater than the denominator (like 3/2). By the way, who invented the term &quot;improper fraction&quot; for such fractions? I've tried to look at the traditional references (Cajori's book of mathematical notation) for these informations, but no luck.&lt;/p&gt;&#xA;" OwnerUserId="12383" LastActivityDate="2020-04-21T12:05:18.733" Title="First historical register of an improper fraction" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="4" ClosedDate="2020-04-22T14:57:22.687" ContentLicense="CC BY-SA 4.0" />
  <row Id="11690" PostTypeId="1" AcceptedAnswerId="11708" CreationDate="2020-04-21T13:01:17.100" Score="2" ViewCount="108" Body="&lt;p&gt;In a statistical hypotheses testing a significance level &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; has to be set. The most often, &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is set to be 5 %, sometimes 1 % and 10 % values are used. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Value of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; tells us what is a maximal tolerated probability of error of first kind (rejecting true hypothesis). There is also error of second kind (not rejecting false hypothesis). The convention is to anchor maximal probability of the error of first kind. But there is no reason to prefer error of first kind to error of second kind.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My questions are these:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Who first devised to use values 1 %, 5 % and 10 % for statistical signifance and why?&lt;/li&gt;&#xA;&lt;li&gt;Why error of first kind is preferred?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12316" LastActivityDate="2020-04-25T22:22:07.240" Title="Significance level $\alpha$ values - who devised to use $\alpha = 5 \%$?" Tags="|mathematics|statistics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11691" PostTypeId="1" CreationDate="2020-04-20T16:06:46.873" Score="2" ViewCount="128" Body="&lt;p&gt;My apologies if this is too naive a question.&#xA;I keep wondering what steered people away from assuming the new moon to be an eclipse and hence go on to figure the angular difference in the orbital planes. I'm trying to understand the hypothesis and the observations regarding the same. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The search on this question just results in the explanation of the angular differences in the orbital planes. I'm curious how through observations, the scientists concluded the new new moon is not the earth's shadow on the moon? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Through simple home-made experiments, are there ways to identify the relative positions of the earth, sun and moon? Like shadow experiments for e.g.&lt;/p&gt;&#xA;" OwnerDisplayName="Kadirvel Vanniarajan" LastEditorUserId="26" LastEditDate="2021-02-20T15:15:44.253" LastActivityDate="2021-02-20T15:15:44.253" Title="How did the initial scientists conclude the new moon phase is not an eclipse?" Tags="|astronomy|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11695" PostTypeId="2" ParentId="3131" CreationDate="2020-04-22T19:00:00.520" Score="3" Body="&lt;p&gt;&lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Lyapunov.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aleksandr Mikhailovich Lyapunov&lt;/a&gt; (Александр Михайлович Ляпунов) committed suicide:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1917 Lyapunov left St Petersburg to take up a post at the university in Odessa, on the Black Sea coast. He taught at the university but in the spring of 1918 his wife's health began to deteriorate rapidly. Natalia Rafailovna suffered from a form of tuberculosis and Lyapunov was greatly disturbed to watch her health fail. On 31 October 1918 Lyapunov's wife died and later that day Lyapunov shot himself. He died three days later in hospital.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;/questions/tagged/mathematicians&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;mathematicians&amp;#39;&quot; rel=&quot;tag&quot;&gt;mathematicians&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-02-28T11:55:21.950" LastActivityDate="2021-02-28T11:55:21.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11696" PostTypeId="1" AcceptedAnswerId="11700" CreationDate="2020-04-22T23:30:42.260" Score="1" ViewCount="108" Body="&lt;p&gt;I was trying to read an &lt;a href=&quot;https://doi.org/10.1007/BF01917434&quot; rel=&quot;nofollow noreferrer&quot;&gt;older paper&lt;/a&gt; about treewidth by Halin (1976). He used different hand-written labels, obviously Sütterlin:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7X9sg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7X9sg.png&quot; alt=&quot;&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I assume the hand-written letters to be (from top to bottom and from left to right) &quot;F*(n)&quot;, &quot;C*(n)&quot;, &quot;D*(n)&quot;. No I am poring over the next letter. Is it an &quot;Y*(n)&quot;? Or a &quot;H*(n)&quot;? Or even a strangely-written &quot;E*(n)&quot;?&lt;/p&gt;&#xA;" OwnerUserId="12392" LastEditorUserId="9523" LastEditDate="2020-05-15T07:12:57.280" LastActivityDate="2020-05-15T07:12:57.280" Title="What letters are used in this paper by Halin (1976)?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11697" PostTypeId="1" AcceptedAnswerId="11698" CreationDate="2020-04-23T00:02:20.293" Score="5" ViewCount="1615" Body="&lt;p&gt;Who first discovered or invented the concept of &quot;muscle memory,&quot; that our habitual bodily movements can be trained?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Aristotle, in his &lt;a href=&quot;https://isidore.co/aquinas/MemoriaReminiscentia.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De memoria et reminiscentia&lt;/em&gt;&lt;/a&gt;, realized that memory depends on our bodies, a fact later proved by modern experimental psychology. Violinists and typists, for example, don't have to be conscious of every &lt;strong&gt;habit&lt;/strong&gt;ual, &quot;quasi-voluntary&quot; muscle movement of their activities. &lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-04-23T00:31:39.843" LastActivityDate="2020-04-23T13:46:50.887" Title="Who first discovered or invented the concept of &quot;muscle memory&quot;?" Tags="|psychology|physiology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11698" PostTypeId="2" ParentId="11697" CreationDate="2020-04-23T00:20:50.663" Score="3" Body="&lt;p&gt;The earliest quotation of &quot;&lt;a href=&quot;https://oed.com/view/Entry/124020#eid35535069&quot; rel=&quot;nofollow noreferrer&quot;&gt;muscular memory&lt;/a&gt;&quot; the &lt;em&gt;OED&lt;/em&gt; gives is:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;1883   F[rancis] Galton [1822-1911] &lt;em&gt;&lt;a href=&quot;https://www.gutenberg.org/files/11562/11562-h/11562-h.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inq[uiries into] Human Faculty&lt;/a&gt;&lt;/em&gt; 106&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;which gives an example from painting:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;There is abundant evidence that the visualising faculty admits&#xA;  of being developed by education. The testimony on which I would&#xA;  lay especial stress is derived from the published experiences of&#xA;  M. Lecoq de Boisbaudran, late director of the École&#xA;  Nationale de Dessein, in Paris, which are related in his &#xA;  &lt;em&gt;Education de la Mémoire Pittoresque&lt;/em&gt;.&lt;sup&gt;7&lt;/sup&gt; He&#xA;  trained his pupils with extraordinary success, beginning with the&#xA;  simplest figures. They were made to study the models thoroughly&#xA;  before they tried to draw them from memory. One favourite&#xA;  expedient was to associate the sight memory with the &lt;strong&gt;muscular&#xA;  memory&lt;/strong&gt;, by making his pupils follow at a distance the outlines of&#xA;  the figures with a pencil held in their hands. After three or&#xA;  four months' practice, their visual memory became greatly&#xA;  strengthened. They had no difficulty in summoning images at will,&#xA;  in holding them steady, and in drawing them. Their copies&#xA;  were executed with marvellous fidelity, as attested by a&#xA;  commission of the Institute, appointed in 1852 to inquire into&#xA;  the matter, of which the eminent painter Horace Vernet was a&#xA;  member. The present Slade Professor of Fine Arts at University&#xA;  College, M. Légros, was a pupil of M. de Boisbaudran. He&#xA;  has expressed to me his indebtedness to the system, and he has&#xA;  assured me of his own success in teaching others in a somewhat&#xA;  similar way.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sup&gt;[Footnote 7: Republished in an 8vo, entitled &lt;a href=&quot;https://archive.org/details/enseignementarti00leco/page/44/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Enseignment&#xA;  Artistique&lt;/em&gt;&lt;/a&gt;. Morel et C&lt;sup&gt;ie&lt;/sup&gt;. Paris, 1879.]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2020-04-23T00:20:50.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11700" PostTypeId="2" ParentId="11696" CreationDate="2020-04-23T01:19:49.663" Score="6" Body="&lt;p&gt;It's clear to me that the letters with * are capital Fraktur forms of F, C, D, and H, respectively, corresponding to the lower case &quot;S-functions&quot; f, c, d, and h, respectively.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I like this &lt;a href=&quot;https://web.library.yale.edu/cataloging/music/fraktur&quot; rel=&quot;noreferrer&quot;&gt;cheat sheet&lt;/a&gt;, which gives the handwritten forms of the letter shapes. &lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2020-04-23T01:19:49.663" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11701" PostTypeId="2" ParentId="11697" CreationDate="2020-04-23T09:09:30.087" Score="7" Body="&lt;p&gt;Galton's 1883 mention of &amp;quot;muscular memory&amp;quot; quoted by OED is not its earliest occurrence. The loose idea had common currency at the time, mostly based on self-introspection and common observations, combined with mixing older mentalistic concepts with freshly prominent &lt;a href=&quot;https://en.wikipedia.org/wiki/Psychophysics#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;psychophysics&amp;quot;&lt;/a&gt; (a prototype of psychophysiology) championed by Fehner since 1860.&lt;/p&gt;&#xA;&lt;p&gt;Hillard, in a pop-sci article &lt;a href=&quot;https://quod.lib.umich.edu/m/moajrnl/acw8433.2-04.003/268:10?page=root;rgn=full+text;size=100;view=image&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reading Aloud, Appleton's Journal, 4 (1878) no.3, p. 258&lt;/a&gt;, already  uses &amp;quot;muscular memory&amp;quot; in connection with one &amp;quot;&lt;em&gt;Dr. Streeter, of Boston, who spent years in experimenting upon his own vocal organs, and has succeeded in formulating ideas which many teachers have dimly perceived, but have failed to embody in words&lt;/em&gt;&amp;quot;, whose &amp;quot;&lt;em&gt;book is, unfortunately, written in so arrogant and unpolished a style as to revolt many critical readers to whom its theories would otherwise appeal&lt;/em&gt;&amp;quot;. The book is &lt;a href=&quot;https://books.google.com/books?id=9H0BAAAAYAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Primary Elements of Music (1873)&lt;/a&gt;, and Streeter bills himself as a progressive teacher of &amp;quot;voice building&amp;quot;. Galton's use is of a similar kind. In 1893 William James mentions &amp;quot;&lt;em&gt;facility of musical execution which, M. Arreat suggests, may be the result of good muscular memory&lt;/em&gt;&amp;quot; in a &lt;a href=&quot;https://www.pdcnet.org/phr/content/phr_1893_0002_0005_0590_0594&quot; rel=&quot;nofollow noreferrer&quot;&gt;review of a book&lt;/a&gt;, which, he says, would have been entitled Anecdotes 30-40 years prior.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, Beaunis published, in the spirit of Fehner's more rigorous experimental psychology, &lt;em&gt;Recherches sur la memoire des sensations musculaires&lt;/em&gt; (1888) in &lt;em&gt;Revue Scientifique&lt;/em&gt; (25:257-263, 361-368, 757-764). According to a &lt;a href=&quot;https://www.jstor.org/stable/1411858&quot; rel=&quot;nofollow noreferrer&quot;&gt;1889 review of memory theories by Burnham&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Beaunis has reported the provisional results of somewhat similar&#xA;experiments upon muscular sensations. He found three phases in&#xA;the vanishing of &lt;strong&gt;muscular memory&lt;/strong&gt; when tested by the power to&#xA;reproduce the extent and direction of movements: 1st, the phase of&#xA;conscious memory; 2d, the phase of unconscious memory, where conscious memory has vanished, but it is still possible to reproduce the&#xA;given movements; 3d, the phase of total obliviscence. See Rev.&#xA;Philos., XXV, pp. 369-574.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The review is interesting in its own right in describing the debates over the new &amp;quot;physical explanation&amp;quot; vs old &amp;quot;mental explanation&amp;quot; of memory. Smith authored a long article &lt;a href=&quot;https://www.jstor.org/stable/1411845&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Muscular Memory&lt;/a&gt; published in The American Journal of Psychology, 7 (1896), no. 4, pp. 453-490, which got considerable attention. Bentley in a historical sketch from &lt;a href=&quot;https://books.google.com/books?id=u5QZAAAAMAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Memory Image and Its Qualitative Fidelity, p.4&lt;/a&gt; puts it into a general context that contrasts the new experimental approach to the old associationist school of thought on memory, starting from Aristotle and going through Hume, Herbart and Spencer, among others.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://pdfs.semanticscholar.org/8518/f3b3bc259cf3ff525e893f4afb7e238c529e.pdf?_ga=2.226508181.905625345.1587625192-1612486891.1575183510&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adams in Historical Review and Appraisal of Research on the Learning, Retention, and Transfer of Human Motor Skills&lt;/a&gt; characterizes this &amp;quot;early period&amp;quot; as &amp;quot;&lt;em&gt;struggling for direction and, with the exception of the first topic—the form of learning curves&lt;/em&gt;&amp;quot;, and names Bryan and Harter (1897,1899) as &amp;quot;&lt;em&gt;fathering research on complex&#xA;skills&lt;/em&gt;&amp;quot;, specifically learning to send and receive Morse code. Thorndike is generally credited with having put research in this field on firmer ground from 1898 on.&lt;/p&gt;&#xA;&lt;p&gt;In Hillard's retelling of Streeter:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The proper control of the tongue, which enables us to take the form of &lt;span class=&quot;math-container&quot;&gt;$\overline{a}$&lt;/span&gt; and hold it as long as we like, can only be acquired by practice before a mirror until the muscles learn their lesson, and respond to our volitions. &amp;quot;Muscles,&amp;quot; says Dr. Streeter, &amp;quot;retain, to an almost indefinite extent, habits acquired,&amp;quot; and it is upon this &lt;strong&gt;muscular memory&lt;/strong&gt; that we all depend, to an extent of which we are quite unconscious. That there can be false intonation in speaking as well as in singing, I think we are all aware. The sudden fall or rise of a voice in speaking will strike a sensitive ear at once.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The power of musical intonation, we are told, depends on the power of accurate adaptation of the muscular parts concerned in the production of the voice to a state known to be capable of producing the required sound. This state is at first recognized by the effect on the ear, and afterward directly through the &amp;quot;muscular sensibility&amp;quot; of the muscles concerned (&amp;quot;muscular sensibility&amp;quot; is, in other words, muscular consciousness, which is the first step toward &lt;strong&gt;muscular memory&lt;/strong&gt;). &amp;quot;Thus we have two memories stored up - the memory of a definite sound and the memory of a definite state of muscles. After a time the process becomes automatic.&amp;quot; That is, we have acquired that &lt;strong&gt;muscular memory&lt;/strong&gt; of which we have just spoken...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-04-23T13:46:50.887" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11702" PostTypeId="1" AcceptedAnswerId="11705" CreationDate="2020-04-23T18:03:10.813" Score="6" ViewCount="194" Body="&lt;p&gt;If you want a crash course in &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras and probability spaces, you should absolutely read this amazing &lt;a href=&quot;https://stats.stackexchange.com/a/199323/137037&quot;&gt;answer&lt;/a&gt; by @Sycorax on Cross Validated. Sycorax says something in particular though that jumps out at me:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The three requirements of a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-field can be considered as consequences of what we would like to do with probability: A &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-field is a set that has three properties:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Closure under countable unions.&lt;/li&gt;&#xA;&lt;li&gt;Closure under countable intersections.&lt;/li&gt;&#xA;&lt;li&gt;Closure under complements.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to my literal interpretation, this suggests that the defining characteristics of a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra (which are usually phrased a bit &lt;a href=&quot;https://en.wikipedia.org/wiki/%CE%A3-algebra#Definition&quot; rel=&quot;noreferrer&quot;&gt;differently&lt;/a&gt; it seems) were formulated with probability spaces in mind.  Then again, the &lt;em&gt;notion&lt;/em&gt; of something and its formal &lt;em&gt;definition&lt;/em&gt; often come at different times.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, I have two closely related questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Was the &lt;strong&gt;notion&lt;/strong&gt; of a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra devised with probability spaces in mind?&lt;/li&gt;&#xA;&lt;li&gt;Were the points in the (widespread) &lt;strong&gt;definition&lt;/strong&gt; of &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras formulated with probability spaces in mind?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I haven’t found any online sources that reference the history of &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras other than &lt;a href=&quot;https://math.stackexchange.com/questions/2171400/what-the-meaning-of-sigma-s-in-sigma-algebra&quot;&gt;this&lt;/a&gt; explanation of the etymology of the &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6806" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-04-24T02:19:37.853" Title="Were $\sigma$-algebras defined for probability?" Tags="|mathematics|set-theory|probability|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11703" PostTypeId="2" ParentId="11691" CreationDate="2020-04-23T18:11:40.847" Score="2" Body="&lt;p&gt;I don't think any scientist seriously thought that moon phases (in particular new moon) is caused be Earth's shadow. One can often see moon crescent and sun at the same time. It is clear that there is no Earth between them. I think people understand this, way before they understand that eclipses caused by Earth's shadow.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/84Hrx.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/84Hrx.jpg&quot; alt=&quot;There is no Earth between Moon and Sun&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-04-23T18:11:40.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11704" PostTypeId="2" ParentId="11664" CreationDate="2020-04-23T23:37:42.643" Score="2" Body="&lt;p&gt;Once you know that factors of &lt;span class=&quot;math-container&quot;&gt;$F_5$&lt;/span&gt; must be of the form &lt;span class=&quot;math-container&quot;&gt;$64k+1$&lt;/span&gt;, then it is very easy to try all of the possible factors of 6700417. You need try &lt;span class=&quot;math-container&quot;&gt;$64k+1$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$k\le 40$&lt;/span&gt;. You can easily use Sieve of Eratosthenes to eliminate at least some non prime numbers. For example eliminating numbers that are divided by 3,5,7,11 you left only with sixteen possible values for &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;: 3,4,7,9,10,12,15,18,22,24,25,30,33,37,40.&#xA;It took for me less than two minutes to try one of the number using only paper and pencil and about half an hour to try all these numbers (hint: it is useful to use base 64 arithmetic). So, even Euler never claimed that 6700417 is prime, we can be almost sure that he knew it. And even if he didn't test it, it would be a simple exercise for his readers.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-04-23T23:37:42.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11705" PostTypeId="2" ParentId="11702" CreationDate="2020-04-24T02:19:37.853" Score="6" Body="&lt;p&gt;No and no, I am afraid. Those things are not specific to probability, we do them with logical connectives, which parallel set operations, and with areas and volumes, just as well as with probabilities. And &lt;em&gt;countable&lt;/em&gt; additivity is not necessarily natural for some classical applications (e.g. to natural number frequencies), it is a technical assumption. Indeed, Kolmogorov's &lt;a href=&quot;https://archive.org/details/foundationsofthe00kolm/page/2/mode/2upThe&quot; rel=&quot;noreferrer&quot;&gt;Foundations of the theory of probability (1933)&lt;/a&gt;, that axiomatized probability in terms of measure theory, does not ask that the collection of sets be a &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-field in the initial five axioms, only a field, and that the probability measure be &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-additive, only finitely additive. Countable fields only appear in chapter II, and he formulates the last axiom in terms of measure continuity, not additivity. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The concept of set field he borrows from Hausdorff's &lt;em&gt;Grundzüge der Mengenlehre&lt;/em&gt; (1914, &lt;a href=&quot;https://archive.org/details/settheory0000haus&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt;), which is not addressing probability at all. And the idea of considering collections of sets (or rather classes back then) closed under standard operations goes back to Boole and de Morgan, who were not even doing measure theory, they were interested in logic. Measure theory was originally developed for the case of real number line, with Peano-Jordan, Borel and Lebesgue introducing set functions on systems of sets, whose properties were eventually packaged into set fields, &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-fields, and &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebras. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Borel was the first to require &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-additivity in &lt;a href=&quot;https://archive.org/details/leconstheoriefon00borerich/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Leçons sur la théorie des fonctions&lt;/em&gt; (1898)&lt;/a&gt;, but his interest in probability only develops  around 1905. Lebesgue took a more abstract view of sets beyond subsets of a line in &#xA;&lt;a href=&quot;https://archive.org/details/leconegrarecher00leberich/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Leçons sur l'intégration et la recherche des fonctions primitives&lt;/em&gt; (1904)&lt;/a&gt;, although he states his conditions in terms of integrals rather than measures. Neither Lebesgue nor Hausdorff were much involved with probability theory. A short review of this gradual measure-theoretic abstraction process is &lt;a href=&quot;http://culturemath.ens.fr/histoire%20des%20maths/htm/villeneuve2009/theories-de-la-mesure.html#passage&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Le processus d'abstraction dans le développement des premières théories de la mesure by Villeneuve&lt;/em&gt;&lt;/a&gt; (it is in French but Google produces a decent translation).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For longer treatments see &lt;a href=&quot;https://books.google.com/books?id=vr7iBQAAQBAJ&amp;amp;dq=pesin+integration&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Classical and Modern Integration Theories by Pesin&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=oV1aLqag6WwC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;Lebesgue's Theory of Integration: Its Origins and Development by Hawkins&lt;/a&gt;, and on connections to probability specifically &lt;a href=&quot;https://arxiv.org/abs/math/0606533&quot; rel=&quot;noreferrer&quot;&gt;Sources of Kolmogorov's &lt;em&gt;Grundbegriffe&lt;/em&gt; by Shafer and Vovk&lt;/a&gt;, and  &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007%2F978-3-319-16859-3_5.pdf&quot; rel=&quot;noreferrer&quot;&gt;Feller’s Early Work on Measure Theory and Mathematical Foundations of Probability by Fisher&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-04-24T02:19:37.853" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11706" PostTypeId="1" CreationDate="2020-04-24T08:32:32.493" Score="1" ViewCount="80" Body="&lt;p&gt;Is there any indication that Viete read Bombelli or Stifel before he discovered his algebraic notation?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-04-24T08:32:32.493" Title="Is there any paper on how Viete may have came up with his algebraic notation?" Tags="|biographical-details|algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11707" PostTypeId="2" ParentId="5645" CreationDate="2020-04-25T21:36:08.493" Score="3" Body="&lt;p&gt;He is not a mathematician per se, but J.M. Coetzee studied mathematics at university and worked as a computer programmer when he was younger.  He is probably the most mathematically minded person to win the Nobel Prize in Literature and has won the Man Booker Prize on multiple occasions.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;James Clerk Maxwell also wrote poetry.  Hamilton was at one point spending quite a significant amount of time on poetry until his friend William Wordsworth advised him to focus on his strengths and not dedicate too much time to literature.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastActivityDate="2020-04-25T21:36:08.493" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11708" PostTypeId="2" ParentId="11690" CreationDate="2020-04-25T22:22:07.240" Score="2" Body="&lt;p&gt;Based on comments, the reasons for using &lt;span class=&quot;math-container&quot;&gt;$\alpha = 5\%$&lt;/span&gt; are these:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The value was used by prominent statisticians (such as R. A. Fisher). In order to have statistical analysis comparable with others, statisticans started to follow their example.&lt;/li&gt;&#xA;&lt;li&gt;Probability 5 % and 95 % is connected with &lt;span class=&quot;math-container&quot;&gt;$2\sigma$&lt;/span&gt; interval under normal distribution&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The reason for using &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; equal to &lt;span class=&quot;math-container&quot;&gt;$1 \%, 5\%$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\,10\%$&lt;/span&gt; is given by the fact that these values have more or less equal distribution on logaritmic scale.&lt;/p&gt;&#xA;" OwnerUserId="12316" LastActivityDate="2020-04-25T22:22:07.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11709" PostTypeId="2" ParentId="3131" CreationDate="2020-04-25T22:44:41.967" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Renato_Caccioppoli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Renato Caccioppoli&lt;/a&gt; committed suicide at 55, mainly due to his disappointments towards politics.&#xA;He is known for his work on functional analysis, his masterpiece is &quot;Measure and integration of dimensionally oriented sets&quot; (Misura e integrazione degli insiemi dimensionalmente orientati).&#xA;He was nephew of the Russian revolutionary Mikhail Bakunin, he was also arrested due to its anti-fascism. His life (and death) are portraited in the Venice Film Festival awarded film &lt;a href=&quot;https://en.wikipedia.org/wiki/Death_of_a_Neapolitan_Mathematician&quot; rel=&quot;nofollow noreferrer&quot;&gt;Death of a Neapolitan Mathematician&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2020-04-25T22:44:41.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11711" PostTypeId="1" AcceptedAnswerId="11768" CreationDate="2020-04-26T08:48:43.393" Score="5" ViewCount="674" Body="&lt;p&gt;Do we have anything  of what Gauss thought of these people from his works or anecdotes ?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="26" LastEditDate="2021-02-20T15:15:04.393" LastActivityDate="2021-02-20T15:15:04.393" Title="What was Gauss' opinion of Lagrange,Laplace and Euler?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11712" PostTypeId="1" AcceptedAnswerId="11713" CreationDate="2020-04-26T21:25:13.770" Score="5" ViewCount="348" Body="&lt;p&gt;I first read the term in an introduction of Fourier transform on locally compact groups. In this article on &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Character_of_a_group&quot; rel=&quot;noreferrer&quot;&gt;Character of a group&lt;/a&gt; from Encyclopedia of Mathematics, a &lt;em&gt;character of a group&lt;/em&gt; is defined as a homomorphism of the given group into some standard Abelian group &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. It is said that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The concept of a character of a group was originally introduced for finite groups &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$A=T:=\{z\in\mathbb{C}:|z|=1\}$&lt;/span&gt; (in this case every character &lt;span class=&quot;math-container&quot;&gt;$G\to \mathbb{C}^*$&lt;/span&gt; takes values in &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Who in history coined the term &quot;character&quot; of a group and why is it called so? &lt;/p&gt;&#xA;" OwnerDisplayName="user4178" LastActivityDate="2020-05-02T01:27:07.117" Title="Who in history coined the term &quot;character&quot; of a group and why is it called so?" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11713" PostTypeId="2" ParentId="11712" CreationDate="2020-04-27T03:35:37.903" Score="4" Body="&lt;p&gt;With credit to @ConsigliereZARF for helpful comments and references. The earliest definition of group character (&quot;&lt;em&gt;Charakter&lt;/em&gt;&quot;) for Abelian groups is likely due to Weber (1881-2), and it was generalized to general groups by Frobenius (1896). According to &lt;a href=&quot;https://www.ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/home.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mackey's survey Harmonic analysis as the exploitation of symmetry&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;In 1881 Weber defined a character of a finite commutative group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; to be a complex valued function &lt;span class=&quot;math-container&quot;&gt;$\chi$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\chi(xy) = \chi(x)\chi(y)$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. This definition was an abstract generalization of one given three years earlier by Dedekind in connection with his work on algebraic number theory, which was inspired in turn by early work of Gauss and Dirichlet (see sections 6 and 12). While Weber's definition makes sense for arbitrary finite groups, it is more or less vacuous except insofar as the group has commutative aspects. Specifically, every character is identically one on the commutator&#xA;  subgroup and consequently the only characters not identically one are&#xA;  derived trivially from characters of commutative quotient groups.&lt;/em&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;em&gt;Group&#xA;  theory acquired a powerful new tool that was soon to become almost indispensable&#xA;  when G. Frobenius (1849-1917) published a paper in 1896 showing&#xA;  that there is a natural generalization of the character notion that involves&#xA;  the whole group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; in a significant and interesting way—even when &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is&#xA;  non-commutative.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Mackey appears to refer to Weber's &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01443599&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist&lt;/em&gt; (1882)&lt;/a&gt;, where he generalizes Dirichlet's prime number theorem. Weber introduces the character using a basis of Abelian group and remarks that it always satisfies the multiplicative property. In a footnote he credits &lt;a href=&quot;https://archive.org/details/vorlesungenberz01dirigoog/page/n604/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dedekind's Supplement XI to the third edition of Dirichlet's &lt;em&gt;Vorlesungen über Zahlentheorie&lt;/em&gt; (1879), p. 581&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dirichlet uses the word &quot;&lt;em&gt;Charaktere&lt;/em&gt;&quot; throughout &lt;a href=&quot;https://archive.org/details/vorlesungenberz02dirigoog/page/n636/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Vorlesungen über Zahlentheorie&lt;/em&gt;&lt;/a&gt;, the earliest on p.77 in connection with &lt;a href=&quot;https://en.wikipedia.org/wiki/Legendre_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;quadratic reciprocity&lt;/a&gt;, and then expands the use of the name (p. 316). &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirichlet_character#Dirichlet_characters&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirichlet's characters&lt;/a&gt; are not quite group characters, and the notion of a group was not in place at the time of Gauss's or Dirichlet's writing. However,  concerning Gauss's &quot;total character&quot; of a quadratic form from &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; (1801), Mackey writes:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;The word character as used today stems directly from Gauss's use of the&#xA;  term in his theory of binary quadratic forms... Dedekind makes the same definition for the special case of the ideal class group&#xA;  of an algebraic number field. As Dedekind himself had pointed out earlier,&#xA;  the ideal class group is a generalization of Gauss's group of equivalence&#xA;  classes of binary quadratic forms. Dedekind's mode of expression is such as&#xA;  to make it clear that he regards his definition as a generalization of that of&#xA;  Gauss.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Before Gauss, quadratic characters, later generalized by Jacobi and Dirichlet, appear in   Legendre's &lt;a href=&quot;https://archive.org/details/essaisurlathor00lege/page/n6/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Essai sur la theorie des nombres&lt;/em&gt; (1798)&lt;/a&gt;. Introducing the quadratic character symbol, he simply wrote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;As the quantities analogous to &lt;span class=&quot;math-container&quot;&gt;$N^{\frac{c-1}2}$&lt;/span&gt; will frequently appear in the course of our research, we will use the abbreviated &lt;strong&gt;character&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{N}{c}\right)$&lt;/span&gt; to express the residue from dividing &lt;span class=&quot;math-container&quot;&gt;$N^{\frac{c-1}2}$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;; the residue which, according to what we have just seen, can only be &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;&lt;/em&gt;.&quot;[ See &lt;a href=&quot;https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n53/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cajori, History of Mathematical Notation, p.30&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So to him &quot;character&quot; meant nothing more than &quot;symbol&quot;. Whether this is where Gauss took the word from is unclear, Mackey says that large parts of &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; were written before Legendre's &lt;em&gt;Essai&lt;/em&gt; was published, but Gauss &quot;&lt;em&gt;acknowledges that he was inspired to study quadratic forms by the work of Lagrange and Legendre&lt;/em&gt;&quot;. Legendre divided classes of forms into cosets that Gauss called genera, and forms were said to be in the same genus if they had the same &quot;total character&quot;, a system of &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt;s and &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;s canonically associated to them. Each &lt;span class=&quot;math-container&quot;&gt;$\pm1$&lt;/span&gt; was produced by a function on forms what we today call a group character.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-05-02T01:27:07.117" LastActivityDate="2020-05-02T01:27:07.117" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11714" PostTypeId="2" ParentId="5645" CreationDate="2020-04-27T04:26:07.163" Score="6" Body="&lt;p&gt;Lewis Carroll, a logician, wrote &lt;em&gt;Alice in Wonderland&lt;/em&gt;.&#xA;Probably the best selling work of fiction by a mathematician ever.&lt;/p&gt;&#xA;&lt;p&gt;Sofya Kovalevskaya wrote a &lt;em&gt;Russian Childhood&lt;/em&gt; where she mentions&#xA;how she was inspired to learn mathematics by the wallpaper in her bedroom. It had&#xA;been papered over by the pages of a textbook. She also mentions how she&#xA;was annoyed by Dostoyevsky.&lt;/p&gt;&#xA;&lt;p&gt;The Irish mathematician and physicist, Hamilton wrote poetry, but was advised by his friend, Wordsworth, to stick to maths and physics.&lt;/p&gt;&#xA;&lt;p&gt;Mathilde Marcolli who has worked with Alain Connes on Non-Commutative Geometry&#xA;has written a series of anarchist/dada inspired texts. She's also working on a book&#xA;investigating the notions of spatiality in mathematics and art.&lt;/p&gt;&#xA;&lt;p&gt;David Brin who trained as an astrophysicist wrote &lt;em&gt;The Postman&lt;/em&gt;, a science-fiction classic&#xA;but which was made into a terrible film.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-10-24T07:06:33.607" LastActivityDate="2021-10-24T07:06:33.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11715" PostTypeId="2" ParentId="11671" CreationDate="2020-04-27T04:49:43.210" Score="1" Body="&lt;p&gt;A good question. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Julian Barbour, the British physicist, asks the same question in one of his books and he didn't answer it. The real answer is that we don't know. Newton didn't write out how he came to his laws. But then again he didn't write the &lt;em&gt;Principia&lt;/em&gt; in the language of the calculus because he knew just how conservative mathematicians were - he wrote it in the language of the day - Euclids geometry (which kind of suggests just how much mathematics/physics today goes under the radar because mathematicians/physicists can't invest the time to learn new mathematics/physics. Of course one can't blame mathematicians/physicists for all this given how much of higher education has been neo-liberalised into a market logic of publish or perish ... but I digress). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However I think we can reconstruct it once we know that he was cogniscent of the notion of an atom, after all, he based his theory of the corpuscular nature of light on the same as well as supposing atoms were formed from 'nothingness' by the mere force of Gods will.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He's also known to have written quite heavily over his copy of Lucretious &lt;em&gt;On Nature&lt;/em&gt; and obviously he would have known about Aristotles notion of force: that something moves if it is capable of moving and if an external force is applied by &lt;em&gt;contact&lt;/em&gt;. This is one half of the first law and one-half of the third law.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now, Aristotle was merely expounding on a general nature of change whereas Newton specifically wanted to consider motion. Suppose one atom collided with another. They are exactly the same. The force on the second must be due to the impact of the first (see Aristotles law above). Then by &lt;em&gt;symmetry&lt;/em&gt;, given both atoms are the same, a force must be applied by the second on the first, and again by &lt;em&gt;symmetry&lt;/em&gt;, we see that they are the same!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Then by induction we generalise to large macroscopic bodies made up of atoms. We then test this physical hypothesis by seeing what it says about motion. Because they work, we argue &lt;em&gt;a posteriori&lt;/em&gt; that the hypothesis is correct and thus the hypothesis is alchemised into a law. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Notice how he used Aristotles law to come up with his own. This is no different to how today mathematcians/physicists work. When Newton said he could only do what he did because &quot;he stood on the shoulders of giants&quot; - he meant it. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, when he said that it was &quot;I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.&quot; He also meant that. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-04-27T08:40:10.713" LastActivityDate="2020-04-27T08:40:10.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11716" PostTypeId="2" ParentId="7134" CreationDate="2020-04-27T05:00:03.300" Score="1" Body="&lt;p&gt;The basic reason is that the straightline and circle are the most basic of shapes. Aristotle theorises about them in his &lt;em&gt;Physics&lt;/em&gt;, for example.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its still the case today. Except of course the idea has ramified into many new and different ways. For example, when we think topologically, the straight-line is the standard example of a non-compact space and the circle, the standard example of a compact space.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-04-27T05:00:03.300" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11717" PostTypeId="1" CreationDate="2020-04-27T12:15:32.777" Score="0" ViewCount="53" Body="&lt;p&gt;Could someone please explain how the concept of electricity was originally conceived and how it became connected with time? Also how did people come about the relationship between time and electricity? &lt;/p&gt;&#xA;" OwnerUserId="12420" LastEditorUserId="26" LastEditDate="2021-02-20T15:14:55.840" LastActivityDate="2021-02-20T15:14:55.840" Title="What is the history of electricity" Tags="|electricity|" AnswerCount="1" CommentCount="2" ClosedDate="2020-04-27T19:18:53.487" ContentLicense="CC BY-SA 4.0" />
  <row Id="11718" PostTypeId="2" ParentId="11717" CreationDate="2020-04-27T12:51:29.200" Score="2" Body="&lt;p&gt;An answer to this question requires a whole book, and such book indeed exists:&#xA;E. T. Whittaker, A history of the theories of aether and electricity from the age of Descartes to the close of nineteenth century. There are many editions. The later editions, in 2 volumes, bring the history to the 1920th,&#xA;including quantum mechanics and general relativity.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Complete (classical) theory of electricity was created by Maxwell (based on earlier research of many people like Volta, Galvani, Ampere, Ohm, Kirkhoff, Faraday and many others). His&#xA;book Treatise on electricity and magnetism also contains a complete history,&#xA;up to the end of 19th century. &lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-04-29T00:44:47.317" LastActivityDate="2020-04-29T00:44:47.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11719" PostTypeId="1" CreationDate="2020-04-27T13:16:32.747" Score="5" ViewCount="151" Body="&lt;p&gt;Joseph Priestley is often said to have discovered Oxygen, or not due to calling it Dephlogisticated air - depending on one's preference. However, regardless of this, it is often said in popular accounts, such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Priestley&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, that due to his continued use of the theory of &lt;a href=&quot;https://en.wikipedia.org/wiki/Phlogiston_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Phlogiston&lt;/a&gt;, he became isolated from the rest of the academic community and the chemical revolution. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have read some of the works of Priestley, and the related work on Cavendish - who used the Phlogiston theory up to the late 1700s. It is reported in &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Cavendish&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; that in 1787 Cavendish was one of the first people to use the Oxygen theory outside of France. The riot that burned Priestley's house was in 1791, and Priestley moved to America in 1794 as a result of his political disagreements. He rapidly became friends with Franklin (perhaps due to his work on electricity, but I have not read up on that). He became ill in 1801 and died a few years later, and before that was simply hampered by lack of contact with the scientific centers in Europe. On the other hand, he was connected to Franklin. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This does not feel to me as though Priestley became scientifically isolated due specifically a unique (others also defended the theory) tendency to support Phlogiston. And indeed no more isolated than would have been likely due to events in his later life. The isolation, such as it was, seems not to have been long, and not really to have related simply to support of the Phlogiston theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Are there more details that can be added to support the suggestion of isolation due to support of the Phlogiston theory?&lt;/p&gt;&#xA;&#xA;  &lt;hr&gt;&#xA;&#xA;&lt;p&gt;Addendum:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I mention Cavendish in respect of 1787 being the &lt;em&gt;first&lt;/em&gt; used of Lavoisier's theory outside of France. Hence, giving a timeline for the acceptance. Further, having read some of Cavendish's papers of the time, I see that he actually supported the Phlogiston theory over Oxygen - for &lt;i&gt;entirely pragmatic&lt;/i&gt; reasons. Cavendish was reclusive, and I do not put him there to counter claims of Priestley isolation - Priestley was well published and otherwise connected. On the other hand Cavendish also was an FRS and did attend dinners regularly, even if he would not look at anyone. So, I would not describe Cavendish as isolated. (Thanks to @Conifold for bringing up this issue).&lt;/p&gt;&#xA;" OwnerUserId="12421" LastEditorUserId="10106" LastEditDate="2020-04-29T04:16:20.270" LastActivityDate="2020-04-30T05:27:13.740" Title="Was Joseph Priestley &quot;isolated&quot;" Tags="|physics|biographical-details|chemistry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11720" PostTypeId="1" AcceptedAnswerId="11722" CreationDate="2020-04-28T04:25:06.633" Score="-1" ViewCount="143" Body="&lt;p&gt;An ancient scholar said a leader cannot be neutral, that he must chose a side or he will be viewed as an enemy by both sides. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am trying to find who said/wrote that. Anyone know? &#xA;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="12422" LastActivityDate="2020-04-28T10:55:28.833" Title="Quote from ancient scholar: that a leader cannot be neutral, he must chose a side" Tags="|quote|" AnswerCount="1" CommentCount="0" FavoriteCount="0" ClosedDate="2020-04-29T06:38:43.570" ContentLicense="CC BY-SA 4.0" />
  <row Id="11722" PostTypeId="2" ParentId="11720" CreationDate="2020-04-28T10:55:28.833" Score="1" Body="&lt;p&gt;The answer appears to be &quot;everyone.&quot;  First hit included:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“The hottest places in hell are reserved for those who, in times of great moral crisis, maintain their neutrality.” &#xA;― Dante Alighieri&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“If you are neutral in situations of injustice, you have chosen the side of the oppressor. If an elephant has its foot on the tail of a mouse, and you say that you are neutral, the mouse will not appreciate your neutrality.” &#xA;― Desmond Tutu (Foreword)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“Neutrality wears upon it the bias dress, whether direct or indirect way since no one shows and proves naturally or habitually, to be a natural. It demonstrates a legal, virtuous and impeccable idiocy.” &#xA;― Ehsan Sehgal&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-04-28T10:55:28.833" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11723" PostTypeId="1" CreationDate="2020-04-28T12:36:39.680" Score="2" ViewCount="72" Body="&lt;p&gt;Temperature measurements started with the developments of thermoscopes and thermometers. They were based on the fact that liquids in general expanded significantly on change in temperature (especially mercury). The initial mercury thermometers were calibrated against common physical phenomenon and divided linearly (for example, the Celcius scale).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But now we know that materials do not expand linearly with temperature. How was this knowledge incorporated to get a material independent temperature scale, because if we continued using a linear scale we would face problems at high temperatures?&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2020-04-29T14:25:51.247" Title="How was a material independent temperature scale created?" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11724" PostTypeId="1" CreationDate="2020-04-28T17:52:30.693" Score="2" ViewCount="106" Body="&lt;p&gt;This is a cross post from &lt;a href=&quot;https://math.stackexchange.com/q/3647129/272127&quot;&gt;MSE&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I want to know &lt;em&gt;why&lt;/em&gt; are faithful actions called faithful and &lt;em&gt;who&lt;/em&gt; first called them  faithful?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definition:&lt;/strong&gt; An action &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is faithful when &lt;span class=&quot;math-container&quot;&gt;${g_1 \neq g_2  \Rightarrow  g_1 x \neq g_2 x}$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;${x \in X}$&lt;/span&gt; (different elements of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; act differently at some point).&lt;/p&gt;&#xA;" OwnerUserId="12424" LastEditorUserId="26" LastEditDate="2021-02-20T15:14:10.943" LastActivityDate="2021-02-20T15:14:10.943" Title="Why are faithful actions called faithful and who first called them faithful?" Tags="|terminology|group-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11725" PostTypeId="1" CreationDate="2020-04-28T19:48:03.063" Score="3" ViewCount="416" Body="&lt;p&gt;I would like to know how Planck went from an expression for the number of ways energy can be distributed in oscillators (denoted &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt;) via the Boltzmann equation (&lt;span class=&quot;math-container&quot;&gt;$ S = k \ln W$&lt;/span&gt;) to the Planck radiation law - the distribution &lt;span class=&quot;math-container&quot;&gt;$B(\nu, T)$&lt;/span&gt; of energy from a blackbody:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;B(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{\exp(\frac{h\nu}{kT}) - 1}.&#xA;$$&lt;/span&gt;&#xA;I know this is achieved by finding the number of states using combinatorics arguments to show that &lt;span class=&quot;math-container&quot;&gt;$W = (P+N-1)!/P!(N-1)!$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the number of energy quanta &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; to be distributed over &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; oscillators.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think it hinges upon the relation between entropy and temperature:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{dS}{dE} = \frac{1}{T}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am reading a history of quantum mechanics and want to see how quantising the oscillators emerged and how that then relates to later developments. As such, I specifically want an argument in the line of how Planck would have done it, rather than deriving the law from physics that didn’t exist in Planck’s time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I’m okay with maths and technical details etc. - just really want to get my head around this.&lt;/p&gt;&#xA;" OwnerUserId="12428" LastEditorUserId="6229" LastEditDate="2020-05-01T15:08:05.877" LastActivityDate="2020-05-05T17:49:23.163" Title="How to get to Planck’s radiation law as Planck did it (warts and all)?" Tags="|quantum-mechanics|theoretical-physics|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11726" PostTypeId="2" ParentId="11723" CreationDate="2020-04-29T14:25:51.247" Score="3" Body="&lt;p&gt;Fundamentally, as the field of Statistical Mechanics (aka Thermodynamics) developed, it became possible to define temperature based on fundamental units and constants.  As you probably know,  temperature in general is defined as something proportional to the logarithm of the number of available energy states in the system.   To keep people's brains from exploding, the numeric values were 'adjusted' so that, e.g.,  zero Celsius could stay where it was originally defined (freezing point at standard pressure).  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, short answer is that the rigorous mathematical definitions of temperature are based on quantum energy levels as well as  phonons, etc., and the final numerical values were chosen to align with the then-current (but slightly inaccurate) values for various phase transitions. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-04-29T14:25:51.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11727" PostTypeId="1" CreationDate="2020-04-30T02:14:00.987" Score="0" ViewCount="185" Body="&lt;p&gt;I vaguely remember reading a quote/listening to a statement by John Conway which I can paraphrase as follows: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;&quot;I do not know what object the Monster group acts on, but it seems to exist because string theory needs/wants/demands it.&quot;&lt;/em&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, I do not know whence, nor do I know the exact phrasing. Does anyone know of the quote to which I am referring, and perhaps has a reference? &lt;/p&gt;&#xA;" OwnerDisplayName="user12433" LastActivityDate="2020-04-30T12:13:18.980" Title="Exact quote (and source of) by John Conway regarding the action/origin of the Monster?" Tags="|quote|group-theory|string-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11728" PostTypeId="2" ParentId="11719" CreationDate="2020-04-30T05:27:13.740" Score="3" Body="&lt;p&gt;In researching this question and discussing with @conifold I came to two conclusions. Firstly, that Priestly was active and interactive in the scientific community almost to the month of his death. Secondly, that there is far less support for the suggestion that it is believed that he was rejected from the scientific community due to the Phlogiston debate than was my original impression based on some popular accounts. There is a great deal of believe that Priestley was wrong, but that is not the question I asked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Having said that, I do believe that a question should be answered in the spirit in which it was asked (this time by me). So with acknowledgement to @conifold for helpful comments (see the chat) I give the result of my research on the matter.&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;It has been stated in several popular sources that in his later career, Joseph &#xA;Priestly became isolated from the orthodox scientific community due to his &#xA;continued and unpopular support for the Phlogiston theory then eschewed by the &#xA;community at large.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Priestley&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Joseph_Priestley&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;However, Priestley's determination to defend phlogiston theory and to reject &#xA; what would become the chemical revolution eventually left him isolated within &#xA; the scientific community&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This seems to be a misrepresentation of the situation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Priestly did not start the theory of phlogiston, he used it.&#xA;Stahl is usually credited with developing the theory in 1703, &#xA;but the word had been used in 1606 in more or less the same way. &#xA;Priestly was born 1733, so by the time he was say, 30, it was&#xA;like being a follower of Einstein in the 1960s. Stahl was generally&#xA;recognised as giving a good rational theory of some chemical reactions.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Priestly was not a recluse. He had engaged in the late 1700s in a variety&#xA;of pamphlet campaigns, some of which were considered rather inflamatory&#xA;and he spoke out against the reformed church. He was described by his&#xA;opponents as a conflict between a priestly politician and a political&#xA;priest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lavoisier wrote on oxygen (which he mistakenly named acid maker) in 1775 &#xA;with his Easter Paper to the French Academy. And then started the anti&#xA;phlogiston movement by responding to Priestly's complaints that Lavoisier &#xA;had not acknoweldged Priestly's work in the matter, by claiming that Priestly &#xA;totally misunderstood the experiments and deserved no credit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cavendish in England is said to be &#xA;the first person to start working with the oxygen theory around 1787, &#xA;however, his papers on this at this time show a mild favoring of the&#xA;Phlogiston theory. He also developed a theory that effectively &#xA;equated Phlogiston with Hydrogen, and did a better job of explaining&#xA;acid reactions than did Lavoisier.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, by 1797, Kant reported that the Phlogiston theory had been&#xA;replaced by the Oxygen theory in a rapid change of opinion within the&#xA;scientific establishment. In 1797, Priestly wrote Observations on &#xA;Phlogiston and the decomposition of water - which was effectively an&#xA;explicit defence of the Phlogiston theory against the Oxygen theory.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1800, Priestly, himself, was explicitly identifying Phlogiston with&#xA;Hydrogen rather than oxygen. This is was not a turn around, as the &#xA;identification of Phlogiston with negative oxygen, a common notion &#xA;today, was not inevitable. The core issue was to explain the chemical &#xA;reactions by combinatorics of elements. The details had been under&#xA;flux for a couple of centuries and would continue to be so for another &#xA;century.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In November of 1801, Priestly responded to a paper by Cruickshank &#xA;regarding the chemistry of carbon monoxide. Priestly supported the&#xA;Phlogiston theory. However, for 1802 and 1803 he seems to have focused&#xA;on other scientific work. In Priestly's career, the Phlogiston theory&#xA;was not everything, he defended it strongly and rationally, and does not &#xA;seem to ever have been converted to the Lavoisier school. But, he defended &#xA;it in interaction with, and even with some support from, the scientific&#xA;community of the time. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Priestly died in early 1804. It is hard to say what would have happened&#xA;next. He might have become a crank ranting about a discarded theory, or&#xA;he might have been converted, or he might have continued to be no longer&#xA;interested enough to continue the debate. But, he died, and there seems&#xA;to be no period of his life in which the image of Priestly rejected by&#xA;the orthodox science for his loudly proclaimed views is justified.&lt;/p&gt;&#xA;" OwnerUserId="12421" LastActivityDate="2020-04-30T05:27:13.740" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11729" PostTypeId="2" ParentId="11727" CreationDate="2020-04-30T12:13:18.980" Score="1" Body="&lt;p&gt;I found this statement in a short &lt;a href=&quot;https://www.ias.edu/ideas/2013/roberts-monster&quot; rel=&quot;nofollow noreferrer&quot;&gt;IAS&lt;/a&gt; article, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In [Conway's] view, conformal field theory is too complicated to&#xA;  understand, and thus too complicated to be the only answer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However,  seeing as Conway's work on the Monster group and 'moonshine' seems to predate its proposed application to string theory by a number of years, I'm a bit skeptical that he would have made a statement like that.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-04-30T12:13:18.980" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11730" PostTypeId="1" AcceptedAnswerId="11734" CreationDate="2020-04-30T16:27:42.713" Score="10" ViewCount="2224" Body="&lt;p&gt;Today the standard interpretation of intuitionistic logic is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Brouwer%E2%80%93Heyting%E2%80%93Kolmogorov_interpretation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brouwer-Heyting-Kolmogorov interpretation&lt;/a&gt; which was presented independently by Arend Heyting and Andrei Nikolajewitsch Kolmogorow. While Heyting was a student of Brouwer and an intuitionist himself, to me it does not look like Kolmogorov was an intuitionist.&lt;/p&gt;&#xA;&lt;p&gt;Is there a text where Kolmogorov describes his own view in the philosophy of mathematics? Or, more general, is it known which views he was holding?&lt;/p&gt;&#xA;" OwnerUserId="4096" LastEditorUserId="4251" LastEditDate="2020-07-06T12:23:25.457" LastActivityDate="2020-07-06T12:23:25.457" Title="What was Kolmogorov’s point of view in the philosophy of mathematics?" Tags="|mathematics|mathematicians|reference-request|philosophy-of-science|kolmogorov|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11733" PostTypeId="2" ParentId="11730" CreationDate="2020-04-30T22:52:20.323" Score="9" Body="&lt;p&gt;Kolmogovov expressed his views in this paper:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR2278817 Kolmogorov, A. N. Modern debates on the nature of mathematics. (Russian). With a commentary by V. A. Uspenskiĭ. Reprinted from Nauchnoe Slovo 1929, no. 6, 41–54. Problemy Peredachi Informatsii 42 (2006), no. 4, 129–141; translation in Probl. Inf. Transm. 42 (2006), no. 4, 379–389.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is a reprint of his paper of 1929 (that is before Goedel theorem) with modern comments.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His views were not intuitionist. Formalization of intuitionist logic for him was a pure&#xA;mathematical exercise. He actually wrote several papers on philosophy and foundation, expressing his general views on mathematics. The most famous is his entry &quot;Mathematics&quot;&#xA;in the Great Soviet Encyclopedia:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;MR2236304 Kolmogorov, Andrei Nikolaievich, Mathematics (Spanish). Translated from the Russian original (Great Soviet Encyclopedia of 1936). Gac. R. Soc. Mat. Esp. 9 (2006), no. 1, 108–141. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I don't know whether an English translation exists. I was surprised that Encyclopedia Britannica (1960) had no entry with this title:-)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark. Soviet authorities did not permit any free discussion of philosophy by unauthorized scientists. In the Stalin period and some time later, the word &quot;intuitionism&quot; was banned in Soviet Union. Not as strictly as genetics, and they did not shoot people for intuitionism, but anyone using this word would have problems with the authorities and would be unable to publish. So soviet intuitionists (A. A. Markov Jr., N. A. Shanin, Esenin-Volpin) used an euphemism: &quot;constructive mathematics&quot;. This remark may be helpful when you search the literature.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-06T22:59:08.113" LastActivityDate="2020-05-06T22:59:08.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11734" PostTypeId="2" ParentId="11730" CreationDate="2020-04-30T23:30:23.837" Score="16" Body="&lt;p&gt;Kolmogorov was not exactly free to express his views considering the situation in the Soviet Union. Philosophical issues, even concerning mathematics, were ideologically sensitive, and everyone had to express allegiance, in one form or another, to the dialectical materialism of Marx and Engels. It went beyond that, as the only grand philosophy available it had genuine impact on the thinking of people immersed in Soviet life.&lt;/p&gt;&#xA;&lt;p&gt;#1) &amp;quot;Extremes&amp;quot; of formalism and intuitionism&lt;/p&gt;&#xA;&lt;p&gt;The closest we have is Kolmogorov's 1929 article &lt;a href=&quot;http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;amp;jrnid=ppi&amp;amp;paperid=67&amp;amp;option_lang=rus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Современные споры о природе математики&lt;/a&gt;, published before the total ideologization of 1930s. (English version &lt;a href=&quot;https://link.springer.com/article/10.1134/S0032946006040107&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contemporary debates on the nature of mathematics&lt;/a&gt;, trans. V.A. Uspensky, 2006.) In it, he gives a very lucid presentation of the state of affairs in Hilbert's formalism vs Brouwer's intuitionism (platonism is not even considered as a serious contender), and criticizes both as &amp;quot;extreme&amp;quot;, although with a clear sympathy to intuitionism. Still, his own position seems to be not intuitionism, but a form of empiricism, with &amp;quot;intuition&amp;quot; distilling ways of dealing with material objects through &amp;quot;idealization&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The appearance of these extreme points of view is explained by the fact that the combination of both&#xA;aspects of set-theoretic mathematics led to great difficulties and even contradictions. A common source of these difficulties is the following. Mathematicians are accustomed to dealing with numbers, functions, sets as&#xA;as if they were things of the real world, in everything similar to material ones.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;From the epistemological side, Hilbert's point of view reduces to a strict&#xA;limitation to the finite; all mathematical sentences in which the infinity enters one way or the other are declared devoid of any meaning. True, with a&#xA;brilliant skill, Hilbert recovers rejected mathematical&#xA;theories in the form of a formal consistent game of symbols. Yet, this way out, giving no explanation of what sustained mathematics to date, of why, while expressing judgments about infinity that have no meaning,&#xA;mathematicians understood each other, is dictated only by inability to find a more satisfactory way out.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This makes us pay special attention to Brouwer, who, without shying away from the problem, promises to find out the nature of the infinite. But it is possible to doubt that intuition and the construction of new images, proceeding from a natural number, will turn out to be reliable guides. In particular, Brouwer studies the continuum in the form of infinite sequences of natural&#xA;numbers, since only in this form it is natural to obtain it by purely logical means. Historically, the idea of ​​a continuum was created through idealization of truly observable continuous media; so far it is hard to imagine&#xA;how this can provide a basis for the development of mathematical theory, but only this would be a direct path to understanding the nature of the mathematical continuum.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;#2) Kolmogorov's &amp;quot;dialectical materialism&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;http://staff.ulsu.ru/baranetz/files/2011/06/baranec_verevkin_koncep_matemat_kolmogorova.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Conception of Mathematics of A.N. Kolmogorov (Russian)&lt;/a&gt; Baranets and Veryovkin try to disentangle Kolmogorov's own views from ideological declarations in his articles written for the Great Soviet Encyclopedia in 1930s and 50s. Ostensibly, he copies Engels's definition of mathematics from &lt;a href=&quot;https://en.wikipedia.org/wiki/Dialectics_of_Nature&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dialectics of Nature&lt;/a&gt;, which is also empiricist in spirit, albeit outdated:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Pure mathematics has as its subject matter spatial forms and quantitative relations of the real world, i.e. a very real content... to study these forms and relations in pure form, one should&#xA;detach them from their content, eliminate it as something irrelevant to the matter&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To this Kolmogorov adds from himself:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We will see that this very definition is fraught with development opportunities, acquiring a new, more&#xA;broad meaning with the growth of science. We will also note narrower definitions, which mathematics has already&#xA;outgrown... Both as a result of M.'s internal needs, as well as of new demands of natural science, the range of quantitative relations and spatial forms studied by&#xA;M. is greatly expanded&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Moreover, Kolmogorov included mathematical logic, which does not exactly fit even into &amp;quot;expanded&amp;quot; Engels's definition, in his list of modern mathematical disciplines.&lt;/p&gt;&#xA;&lt;p&gt;#3) Interpretation of intuitionism&lt;/p&gt;&#xA;&lt;p&gt;On intuitionism specifically we also have Kolmogorov's works of a more technical character. In &lt;a href=&quot;https://books.google.com/books?id=ikN59GkYJKIC&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;On&#xA;the Principle &amp;quot;&lt;em&gt;Tertium non Datur&lt;/em&gt;&amp;quot; (1925)&lt;/a&gt;, where he gives  interpretation (later rediscovered by Gödel) of classical theorems as intuitionistic theorems when their terms are replaced by their double negations, and concludes that &amp;quot;&lt;em&gt;along with the Brouwerian presentation of mathematics without the&lt;/em&gt; tertium non datur &lt;em&gt;principle it is necessary to retain the ordinary presentation, albeit only as a presentation of pseudo-truth mathematics&lt;/em&gt;&amp;quot;, which is apparently his view of the classical mathematics. Later commenting on this paper in a &lt;a href=&quot;https://iopscience.iop.org/article/10.1070/RM1988v043n06ABEH001986/pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;letter to Heyting&lt;/a&gt; he remarks &amp;quot;&lt;em&gt;I believe it is possible to go even further in this direction&#xA;and so prove, from an intuitionistic point of view, the consistency of a large part of classical mathematics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;These and other remarks show that Kolmogorov's attitude was the opposite to that of Gödel with his platonism. While Gödel saw the classical interpretation of intuitionism as a limited validation of the latter, it was the opposite for Kolmogorov. In &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01186549&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Zur Deutung der intuitionistischen Logik&lt;/em&gt; (1932)&lt;/a&gt; Kolmogorov also gives his own, non-Brouwerian, interpretation of intuitionism as a calculus not of statements but of &amp;quot;problems&amp;quot;, expressing &amp;quot;&lt;em&gt;the intention to find a certain construction&lt;/em&gt;&amp;quot;, as he put it in another letter to Heyting. Intuitionism/constructivism emerges not as a logic of statements, let alone as self-sufficient foundation of mathematics, but as a logic of problems.&lt;/p&gt;&#xA;&lt;p&gt;Uspenskii assembled a &lt;a href=&quot;https://math.ru/lib/533&quot; rel=&quot;nofollow noreferrer&quot;&gt;comprehensive bibliography&lt;/a&gt; of Kolmogorov's works with  relation to philosophy (in Russian). &lt;a href=&quot;https://pdfs.semanticscholar.org/e7b2/23d1924a60d1f5092d6cc856370882d3bdda.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kolmogorov and mathematical logic&lt;/a&gt; is his commentary on the more technical works. Translations from Russian above are lightly edited Google translations.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-07-06T06:43:47.283" LastActivityDate="2020-07-06T06:43:47.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11735" PostTypeId="1" CreationDate="2020-05-01T04:16:41.423" Score="2" ViewCount="91" Body="&lt;p&gt;Who discovered Napier's number?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I read Bernoulli calculated it, Napier discovered it but it is e because of Euler.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I know how Bernoulli calculated it but where did Napier see it?&lt;/p&gt;&#xA;" OwnerUserId="12437" LastActivityDate="2020-05-01T04:16:41.423" Title="Who discovered Napier's number?" Tags="|mathematics|euler|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11736" PostTypeId="1" CreationDate="2020-05-01T18:18:05.783" Score="5" ViewCount="401" Body="&lt;p&gt;I am looking for some background information on the principle of explosion. When was it first formulated, how was it justified, and how did it affect interpretation and development of logic? Please include references where I can find more information.&lt;/p&gt;&#xA;" OwnerUserId="12442" LastEditorUserId="26" LastEditDate="2021-02-20T15:14:02.920" LastActivityDate="2021-02-20T15:14:02.920" Title="How did the principle of explosion come up and was developed historically?" Tags="|mathematical-logic|" AnswerCount="1" CommentCount="4" ClosedDate="2020-05-04T13:51:04.100" ContentLicense="CC BY-SA 4.0" />
  <row Id="11737" PostTypeId="1" CreationDate="2020-05-01T20:29:40.867" Score="4" ViewCount="89" Body="&lt;p&gt;In his paper &lt;a href=&quot;https://www.jstor.org/stable/pdf/42968457.pdf?refreqid=excelsior%3A4baf5687c1940da30b8dd8f546e5c7d8&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Two notes on the foundations of set-theory&lt;/em&gt;&lt;/a&gt;, Kreisel begins:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Recall that, naively, sets present themselves in a number of distinct contexts. [...] One may therefore doubt whether any definite general notion (of set) is involved here; it look[s] much more like a mixture of notions. As a matter of historical fact &lt;strong&gt;this was the common feeling&lt;/strong&gt; among Cantor's contemporaries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;(Emphasis mine.)&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kreisel does not give a source for the emphasized claim. My question is whether in fact this is accurate.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;One possible difficulty, at least in looking for sources slightly &lt;em&gt;pre&lt;/em&gt;-Cantor, is likely to be that insofar as that was the common feeling there may not be much positive evidence of that - if nobody sees a particularly unified conception of X, there's no obvious impetus to &lt;em&gt;say&lt;/em&gt; so. However, I'm still hopeful.&lt;/p&gt;&#xA;" OwnerUserId="8037" LastEditorUserId="131" LastEditDate="2020-05-02T10:10:57.513" LastActivityDate="2020-05-02T10:10:57.513" Title="Kreisel on &quot;set&quot; as a coherent idea" Tags="|set-theory|logic|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11738" PostTypeId="5" CreationDate="2020-05-01T22:20:02.710" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-05-01T22:20:02.710" LastActivityDate="2020-05-01T22:20:02.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11739" PostTypeId="4" CreationDate="2020-05-01T22:20:02.710" Score="0" Body="For question related to computers (including both analog and digital and both mechanical and electronic), calculators and any machines used for computation." OwnerUserId="12316" LastEditorUserId="12316" LastEditDate="2020-05-04T21:50:40.970" LastActivityDate="2020-05-04T21:50:40.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11740" PostTypeId="5" CreationDate="2020-05-02T06:16:49.713" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-05-02T06:16:49.713" LastActivityDate="2020-05-02T06:16:49.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11741" PostTypeId="4" CreationDate="2020-05-02T06:16:49.713" Score="0" Body="Cryptology is the science concerned with data communication and storage in secure and usually secret form. It encompasses both cryptography and cryptanalysis." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-05-04T21:50:32.910" LastActivityDate="2020-05-04T21:50:32.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11742" PostTypeId="2" ParentId="11736" CreationDate="2020-05-02T10:09:42.597" Score="4" Body="&lt;p&gt;The principle/law of explosion had a curious history. &lt;a href=&quot;https://projecteuclid.org/euclid.rml/1312203776&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anellis in his review of Handbook of the History of Logic&lt;/a&gt; suggests that it was known already to ancient Stoics. Indeed, it is closely linked to the idea of the material conditional, which is true whenever the premise is false, and which goes back to &lt;a href=&quot;https://en.wikipedia.org/wiki/Philo_the_Dialectician&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philo the Dialectician, c. 300 BC&lt;/a&gt;. It was not accepted in antiquity, however, Aristotle in Prior Analytic II 4 57b3 asserted quite the opposite:&amp;quot;&lt;em&gt;it is impossible that the same thing should be necessitated by the being and by the not-being of the same thing&lt;/em&gt;&amp;quot;. But it reappeared in the 12th century AD in Paris, where &lt;a href=&quot;https://en.wikipedia.org/wiki/William_of_Soissons&quot; rel=&quot;nofollow noreferrer&quot;&gt;William of Soissons&lt;/a&gt; produced a derivation of it reconstructed by C.I. Lewis. Martin describes the path of the medieval theories of logical consequence towards William's derivation, going through Boethius and Abelard among others, in &lt;a href=&quot;https://www.pdcnet.org/jphil/content/jphil_1986_0083_0010_0564_0572&quot; rel=&quot;nofollow noreferrer&quot;&gt;William's Machine&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The law was widely accepted in Europe by 14th century, when it appears in Questions on Aristotle’s Prior Analytics originally ascribed to a prominent scholastic Duns Scotus (authorship and dating are controversial), and in Book I of &lt;em&gt;Tractatus de Consequentiis&lt;/em&gt; (c. 1330) by Buridan, a Parisian, who advanced remarkably modern notions of logical consequence and material conditional, see &lt;a href=&quot;https://philosophy.stackexchange.com/a/59995/9148&quot;&gt;What were the historical interpretations of Aristotle's definition of validity/logical consequence?&lt;/a&gt; Łukasiewicz mentions pseudo-Scotus's example:&amp;quot;&lt;em&gt;If Socrates exists and Socrates does not exist then Plato is an ass&lt;/em&gt;&amp;quot; in a 1925 paper. On Buridan there is a study by D'Ors &lt;em&gt;Ex impossibili quodlibet sequitur&lt;/em&gt; (Jean Buridan), which is hard to find, and a &lt;a href=&quot;https://brill.com/view/journals/viv/53/2-4/article-p353_12.xml&quot; rel=&quot;nofollow noreferrer&quot;&gt;commentary on it by Normore&lt;/a&gt;. Buridan is explicit about the connection of the law with the definition of the material conditional, see &lt;a href=&quot;https://plato.stanford.edu/entries/consequence-medieval/#BurParTra&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, Medieval Theories of Consequence&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But wide does not mean universal. In the 15th century William's derivation was rejected by the school of Cologne, who objected to a step in it now called disjunctive syllogism, see &lt;a href=&quot;https://plato.stanford.edu/entries/logic-paraconsistent/#BrieHistExContQuod&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, Paraconsistent Logic&lt;/a&gt;. And in another paper, &lt;a href=&quot;https://www.jstor.org/stable/20117822&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Necessity in Deduction&lt;/a&gt;, Normore observes that the formal notion of logic that prevailed in the 14th century was replaced by a psychologistic one in the 16th (Leibniz was a rare late exception to the trend):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Scholastic logic disappeared from the curriculum in Paris during the 1520s and disappeared from much of Europe by the middle of the century. The logical climate in which Descartes grew up was one in which the classical conception of validity that we share with the high Middle Ages was being eclipsed in favor of conceptions that emphasized the psychological dimension of reasoning. Within such a framework the Cartesian stress on the role of the natural light makes sense and so does the suggestion that what a deduction is to preserve is not necessity but certain truth.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If this is correct, then the logical revolution of the first half of the sixteenth century, a revolution that has seemed to most recent historians of logic to be a revolution away from logic, can perhaps be better seen as a revolutionary redirection of the conception of correct inference.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Formal logic and the material conditional, made a comeback again in the second half of 19th century, when Boole and De Morgan algebraized propositional logic, then Frege and Peirce invented predicate logic, and Frege and Husserl sharply criticized logical psychologism. And &lt;em&gt;ex falso quodlibet&lt;/em&gt; came back along with them.&lt;/p&gt;&#xA;&lt;p&gt;A detailed study of its early modern history, in connection with the emergence of the paraconsistent logic, is &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/01445340903340033?scroll=top&amp;amp;needAccess=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Origins of the Use of the Argument of Trivialization in the Twentieth Century by Bobenrieth&lt;/a&gt;. Russell-Whitehead's &lt;em&gt;Principia&lt;/em&gt; presents an &amp;quot;immediate consequence&amp;quot; of the principles, the formula &lt;span class=&quot;math-container&quot;&gt;$∼ p · ⊃ · p ⊃ q$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\neg p\to (p\to q)$&lt;/span&gt;), as formalizing that &amp;quot;one false proposition entails any proposition&amp;quot;. Hilbert derived &lt;span class=&quot;math-container&quot;&gt;$(A\&amp;amp;¬A)→B$&lt;/span&gt; from the &amp;quot;principle of contradiction&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$(A → (B \&amp;amp; ¬B)) →¬A$&lt;/span&gt; in his papers in 1920s. The first textbook on mathematical logic, Hilbert-Ackermann's &lt;em&gt;Grundzüge der theoretischen Logik&lt;/em&gt; (1928), put into wide circulation the now familiar argument for consistency as a means of avoiding trivialization:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It might seem as though we were giving a preferred position to one particular logical principle — the principle of contradiction. The fact is, however, that the occurrence of a formal contradiction, i.e. the provability of two formulas &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-A$&lt;/span&gt;, would condemn the entire calculus as meaningless; for we have observed above that if two sentences of form &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-A$&lt;/span&gt; were provable the same would be true of any other sentences whatsoever. Thus consistency of the calculus in the sense of the definition has the same meaning as the stipulation that not every arbitrary formula be provable.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This argument was soon disputed by Wittgenstein in his Lecture XXI on the Foundations of Mathematics:&amp;quot;&lt;em&gt;Well then, don’t draw any conclusions from a contradiction.  Make that a rule&lt;/em&gt;&amp;quot;. He was challenged by Turing, who pointed out that arbitrary formulas can still be derived even without going through a contradiction explicitly. Eventually, this motivated a paraconsistent reworking of classical logic, where such roundabout ways are blocked as well, by dialetheists like Priest. To some extent, both the argument from trivialization and the ways around it were also anticipated by Łukasiewicz.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-05-05T13:07:42.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11743" PostTypeId="1" AcceptedAnswerId="11749" CreationDate="2020-05-02T10:15:38.247" Score="16" ViewCount="3510" Body="&lt;p&gt;Liouville published Galois' work a decade after the death of this singular mathematician. Are there other cases of results being rescued by the mathematical community long after their authors were gone? Please include results whose importance went unnoticed at their time. &lt;em&gt;Rediscoveries&lt;/em&gt; may be interesting too.&lt;/p&gt;&#xA;" OwnerUserId="12271" LastActivityDate="2023-01-24T13:17:34.620" Title="Mathematical results that became known long after their authors passed away" Tags="|mathematicians|mathematics-social-history|" AnswerCount="7" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11744" PostTypeId="1" CreationDate="2020-05-02T16:37:54.307" Score="6" ViewCount="296" Body="&lt;p&gt;I located Joseph Fourier's book, &lt;em&gt;Analytical Theory of Heat&lt;/em&gt; (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?&lt;/p&gt;&#xA;" OwnerUserId="8219" LastEditorUserId="4251" LastEditDate="2020-11-18T01:55:38.917" LastActivityDate="2020-11-18T01:55:38.917" Title="Origin of the Fourier transform (1878)" Tags="|mathematics|physics|19th-century|fourier|harmonic-analysis|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11746" PostTypeId="2" ParentId="11743" CreationDate="2020-05-03T08:17:13.743" Score="11" Body="&lt;p&gt;Is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fast_Fourier_transform&quot; rel=&quot;noreferrer&quot;&gt;Fast Fourier Transform&lt;/a&gt; a mathematical result? The point might be debated but its history has been well researched (e.g.  Heideman et al., (1984). &lt;a href=&quot;http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Gauss and the history of the fast FFT&lt;/em&gt; &lt;/a&gt;. IEEE ASSP Magazine). In 1987 One of the modern (re)discoverers also wrote on the topic.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The method and the general idea of an FFT was popularized by a publication of Cooley and Tukey in 1965, but it was later established that they had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805, and subsequently rediscovered several times in limited forms. Backtracking leads to Gauss's unpublished work from 1805  needed to interpolate the orbit of asteroids . While Gauss's work predated even Joseph Fourier's results in 1822, he did not analyze the computation time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[Links and refs are in the wikipedia article which has been used here]&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-05-03T08:17:13.743" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11747" PostTypeId="2" ParentId="9931" CreationDate="2020-05-03T09:53:15.480" Score="3" Body="&lt;p&gt;&quot;Hat der Körper eine gleichmäßige Geschwindigkeit in ruhender Luft, so wird auch in der den Körper umgebenden Luft eine gleichmäßige Bewegung eintreten, die im wesentlichen darin besteht, daß die Luft vor dem Körper sich auseinander thut und hinter dem Körper wieder zusammengeht.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Otto Lilienthal 1889&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Which translates as&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;If the body has a constant speed in still air, then there will also be a uniform movement in the air surrounding the body, which essentially consists in the air separating from the body and coming together again behind the body.&quot;&lt;/p&gt;&#xA;" OwnerUserId="12454" LastActivityDate="2020-05-03T09:53:15.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11748" PostTypeId="2" ParentId="11743" CreationDate="2020-05-03T10:57:55.780" Score="20" Body="&lt;p&gt;Bolzano.&lt;/p&gt;&#xA;&lt;p&gt;Here is a copy of an answer of mine from &lt;a href=&quot;https://mathoverflow.net/a/176430/454&quot;&gt;MathOverflow&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Bernhard Bolzano .... ( &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Bolzano/&quot; rel=&quot;nofollow noreferrer&quot;&gt;interesting reading&lt;/a&gt; )  Much of his work was unpublished until much later (for reasons see the link), thus remaining largely unknown.  For example, a theorem of Weierstrass is now known as the &amp;quot;Bolzano-Weierstrass theorem&amp;quot;, acknowledging that Bolzano had proved it previously.  Bolzano anticipated Cantor and Dedekind in work on doing calculus without infinitesimals.  His example of a continuous nowhere-differentiable function is in a manuscript from 1830, but only published in 1930.&lt;/p&gt;&#xA;&lt;p&gt;(See also the other answers to that MathOverflow question.)&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="5788" LastEditDate="2023-01-24T13:17:34.620" LastActivityDate="2023-01-24T13:17:34.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11749" PostTypeId="2" ParentId="11743" CreationDate="2020-05-03T11:41:58.373" Score="21" Body="&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Ramanujan%27s_lost_notebook&quot; rel=&quot;noreferrer&quot;&gt;Ramanujan's Lost Notebook&lt;/a&gt; is one such collection of mathematical results. It consists of loose and unordered sheets of paper in which the Indian mathematician &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Srinivasa_Ramanujan&quot; rel=&quot;noreferrer&quot;&gt;Srinivasa Ramanujan&lt;/a&gt; recorded the mathematical discoveries of the last year (1919–1920) of his life.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Its whereabouts were unknown to all but a few mathematicians until it was rediscovered by George Andrews in 1976, at the Wren Library at Trinity College, Cambridge.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to Wikipedia:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Berndt says of the notebook's discovery: &quot;&lt;strong&gt;The discovery of this 'Lost Notebook' caused roughly as much stir in the mathematical world as the discovery of Beethoven’s tenth symphony would cause in the musical world.&lt;/strong&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The majority of the formulas are about q-series and mock theta functions, about a third are about modular equations and singular moduli, and the remaining formulas are mainly about integrals, Dirichlet series, congruences, and asymptotics. &lt;strong&gt;The mock theta functions in the notebook have been found to be useful for calculating the entropy of black holes.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-05-03T11:50:36.760" LastActivityDate="2020-05-03T11:50:36.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11750" PostTypeId="1" AcceptedAnswerId="11756" CreationDate="2020-05-03T12:08:01.353" Score="5" ViewCount="313" Body="&lt;p&gt;A theory of strong interaction is called &lt;em&gt;quantum chromodynamics&lt;/em&gt; (QCD). Particles interacting strongly are supposed to have &lt;em&gt;color charge&lt;/em&gt;, for example quarks appear in three &quot;mutations&quot; - red, green and blue. A hadron composed of quarks has to be &lt;em&gt;white&lt;/em&gt;, for example baryons - three quarks in RGB mutations resulting  in &lt;em&gt;white&lt;/em&gt; or mesons - pair of quark-antiquark (color + &lt;em&gt;anti&lt;/em&gt;-color is considered to be &lt;em&gt;white&lt;/em&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question are:&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Who coined the term &lt;em&gt;chromodynamics&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;li&gt;Why colors are used and not simply numbers or some characters?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12316" LastEditorUserId="4251" LastEditDate="2023-12-09T19:59:43.687" LastActivityDate="2023-12-09T19:59:43.687" Title="On the naming of quantum chromodynamics" Tags="|physics|quantum-mechanics|physicists|quantum-field-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11751" PostTypeId="2" ParentId="11743" CreationDate="2020-05-03T13:41:30.653" Score="8" Body="&lt;p&gt;One of the most famous examples is the Gauss's diary which was discovered in 1897.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-05-03T13:41:30.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11752" PostTypeId="1" CreationDate="2020-05-02T23:44:22.627" Score="3" ViewCount="90" Body="&lt;p&gt;It is well known that the tobacco industry used the tactic of funding research which would come to conclusions that were in their interests, even though it contested other research findings. Specifically, they attempted to have research demonstrate that smoking cigarettes was okay.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Did any of the industry funded research arguing that smoking cigarettes is not bad for you ever end up being published in respectable peer reviewed journals?&lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Update: I have updated the question to be more specific. While I do realize that many industries fund research for their benefit, I wanted to keep the scope on the history of the tobacco industry, specifically their research that attempted to state that smoking cigarettes was safe.&lt;/p&gt;&#xA;" OwnerUserId="12476" OwnerDisplayName="m-y" LastEditorUserId="12476" LastEditDate="2020-05-06T13:18:56.647" LastActivityDate="2020-05-06T13:18:56.647" Title="Were any tobacco industry funded studies published in peer reviewed journals?" Tags="|academia|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11754" PostTypeId="2" ParentId="11725" CreationDate="2020-05-03T16:58:35.450" Score="3" Body="&lt;p&gt;Thanks for all the comments. Using the paper linked by the helpful commenter (to which I found a pdf version, see &lt;a href=&quot;http://strangepaths.com/files/planck1901.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) - I could see how Planck did it (reminding me of all my old statistical mechanics in the process). Thought I'd sketch it out here to answer the OP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;We want to identify the energy distribution of a blackbody. Planck conceptualised the situation as a system of oscillators capable of absorbing and emitting quanta of radiation. In thermal equilibrium, he used some thermodynamic arguments to arrive at the distribution of energy of such a system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;First, define the overall energy of the system of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; oscillators as &lt;span class=&quot;math-container&quot;&gt;$E_N = NE$&lt;/span&gt;, where E is the average energy of one oscillator. Similarly the entropy is extensive: total entropy of the system is &lt;span class=&quot;math-container&quot;&gt;$S_N = NS$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quantisation step is to realise that to arrive at the known right answer (Planck found the energy distribution essentially empirically, before working out how to derive it) you must assume the energy stored by any oscillator to be quantised (not continuous) such that the total energy &lt;span class=&quot;math-container&quot;&gt;$E_N$&lt;/span&gt; can be decomposed into &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; units of a quanta of energy &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$E_N = P\epsilon$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The number of ways of arranging the &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; units across all &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; oscillators is (as written above) given by the weight function:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;W = \frac{(N+P-1)!}{P!(N-1)!} \simeq \frac{(N+P)!}{P!N!} \simeq \frac{(N+P)^{N+P}}{P^PN^N}&#xA;$$&lt;/span&gt;&#xA;where we've assumed number of oscillators and number of energy units are large and then applied Stirling's approximation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What I'd forgotten was the clever steps to utilise this within Boltzmann's equation (which was actually written by Planck for the first time) to get the entropy of the set of N resonators:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;S_N = k \ln W &amp;amp;= k \big[ (N+P)\ln (N+P) - N\ln N - P\ln P \big] \\&#xA; &amp;amp;= kN \big[ \left(1+\frac{P}{N}\right) \left( \ln N + \ln\left(1+\frac{P}{N}\right) \right)  - \ln N - \frac{P}{N}\ln P \big] \\&#xA; &amp;amp;= kN \big[ \left(1+\frac{P}{N}\right) \ln\left(1+\frac{P}{N}\right)  - \frac{P}{N}\ln \frac{P}{N} \big].&#xA;\end{align}&lt;/span&gt;&#xA;Using &lt;span class=&quot;math-container&quot;&gt;$P = E_N/\epsilon = NE/\epsilon$&lt;/span&gt; and dividing out the number of oscillators to get the average entropy per oscillator we get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S = k \left[ \left(1+\frac{E}{\epsilon}\right) \ln\left(1+\frac{E}{\epsilon}\right)  - \frac{E}{\epsilon}\ln \frac{E}{\epsilon}  \right]&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is all framed in terms of the unknown energy quanta Planck introduced to be able to generate the form of the weight function he knew would lead to the correct result. To go beyond this he uses Wien's displacement law, which states that the peak blackbody intensity occurs at a frequency proportional to temperature of the blackbody - i.e. &lt;span class=&quot;math-container&quot;&gt;$\nu_{peak} \propto T$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Planck goes through a series of steps (which I have to admit I haven't looked at in detail) to show that this implies that the entropy of a resonator sitting in a diathermic medium (which I think means basically in thermodynamic equilibrium) must be a function of &lt;span class=&quot;math-container&quot;&gt;$E/\nu$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$S = f(E/\nu)$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Our expression for &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is framed in terms of &lt;span class=&quot;math-container&quot;&gt;$E/\epsilon$&lt;/span&gt; which means that (up to a constant of proportionality) &lt;span class=&quot;math-container&quot;&gt;$\epsilon \propto \nu$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$\epsilon = h\nu$&lt;/span&gt; introducing Planck's constant. This 1901 paper (and a &lt;a href=&quot;http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Planck%20(1900),%20Distribution%20Law.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1900 paper&lt;/a&gt; that seems to cover mostly the same ground) I think are some of the first places where the Planck constant is introduced and quantified - which I thought was quite cool!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Anyway, substituting this into the entropy equation we can find an expression for the energy of an average oscillator related to the temperature &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1}{T} = \frac{dS}{dE} = \frac{k}{h\nu} \ln\left( \frac{1 + E/h\nu}{E/h\nu} \right) = \frac{k}{\epsilon} \ln\left( 1 + \frac{h\nu}{E} \right) \Rightarrow E = \frac{h\nu}{e^{h\nu/kT}-1}.&#xA;$$&lt;/span&gt;&#xA;To go all the way to the energy radiance, it is possible to re-use the Rayleigh-Jeans law, namely that the radiance of a blackbody is &lt;span class=&quot;math-container&quot;&gt;$B_\nu = \frac{2\nu^2}{c^2}kT = \frac{2\nu^2}{c^2} E$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is the average energy of a mode of radiation in the blackbody. This expression is incorrect since it assumes that each mode of radiation has (according to equipartition of energy) average energy &lt;span class=&quot;math-container&quot;&gt;$kT$&lt;/span&gt;. The reason I'm borrowing it here is because it contains the results of doing various integrals to deal with the 3D geometry so we don't have to (Planck refers to a previous paper of his to sidestep doing this in his paper). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As a result we can convert from energy of a mode of radiation to spectral energy radiance - i.e. the result quoted above in the OP - using Planck's expression for the average energy (as derived by considering the distribution of energy quanta across oscillators in the cavity wall), giving (finally) the result:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;B_\nu = \frac{2\nu^2}{c^2} E = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT}-1}.&#xA;$$&lt;/span&gt; &#xA;It is worth noting in the paper linked above that Planck writes out the spectral energy density not the energy radiance as I've done - the two can be switched between according to the relation: &lt;span class=&quot;math-container&quot;&gt;$u_\nu = \frac{4\pi}{c}B_\nu$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12428" LastActivityDate="2020-05-03T16:58:35.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11756" PostTypeId="2" ParentId="11750" CreationDate="2020-05-04T07:29:43.403" Score="6" Body="&lt;p&gt;The use of &lt;em&gt;color&lt;/em&gt; to refer to the &quot;color charge&quot; characteristic of QCD is present in Gell-Mann, Fritzsch and Bardeen's &lt;a href=&quot;http://cds.cern.ch/record/593283/files/0211388.pdf&quot; rel=&quot;noreferrer&quot;&gt;original work from 1972&lt;/a&gt;. This terminology was apparently due to Gell-Mann:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;They had actually begun exploring this possibility in their earlier paper at the Tel Aviv conference. Now, working with Bardeen, they elaborated on the idea. What Nambu called &quot;charm&quot; Murray started calling &quot;color&quot; The three flavors of quarks—up, down, and strange-would also come in three different colors-red, white, and blue. Feeling cosmopolitan there in Geneva, Murray picked the arbitrary labels in honor of the French flag. It was a natural choice of terminology. Gell-Mann and Feynman had used &quot;color&quot; for years at Caltech as a fanciful name for new particle properties-there were red and blue neutrinos, for example. (But it was not an original notion. An Indiana University physicist had written about &quot;color&quot; in a textbook published in 1970. And other physicists had used the term as well.) Like Neapolitan ice cream, every baryon would consist of three differently colored quarks. A delta would be made from a red up, a white up, and a blue up, each would have different quantum numbers, and the Pauli principle would be saved.&lt;/em&gt;&quot; – Strange Beauty by George Johnson [Random House, 2011]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The later switch from red-white-blue to red-green-blue was apparently due to Fritzsch:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Initially Gell-Mann liked to use the French national colors red, white, and blue since he lived in a house not far from the village of Gex, in the Jura mountains near Geneva. But Fritzsch preferred the colors red, green, and blue, because white is not a primary color. When one mixes red, green, and blue, one obtains white, and that turned out to be very useful later.&lt;/em&gt;&quot; – The Fundamental Constants by Harald Fritzsch [World Scientific, 2009]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;According to his New York Times obituary, &quot;&lt;em&gt;it was Dr. Gell-Mann who named the theory quantum chromodynamics&lt;/em&gt;&quot;, a term which follows naturally from his choice of metaphor (chromo = color, dynamics = the study of forces).&lt;/p&gt;&#xA;" OwnerUserId="1912" LastActivityDate="2020-05-04T07:29:43.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11757" PostTypeId="2" ParentId="11743" CreationDate="2020-05-04T11:31:40.700" Score="6" Body="&lt;p&gt;Jean-Robert Argand published his geometrical interpretation of the complex numbers as points of the plane in 1806. It become a standard way of dealing with these numbers and now sometimes the complex plane is called the Argand plane. However, the same idea had been published in 1799 by Caspar Wessel, a norwegian surveyor, and it was forgotten. Wessel's paper was rediscovered in 1895, when Christian Juel draw attention to it. In the same year, Sophus Lie republished the paper.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-05-04T11:31:40.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11758" PostTypeId="2" ParentId="11743" CreationDate="2020-05-04T14:08:51.220" Score="5" Body="&lt;p&gt;&lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Rogers_James.html&quot; rel=&quot;noreferrer&quot;&gt;Leonard James Rogers&lt;/a&gt; (1862 - 1933) obtained degrees in Mathematics, Classics and Music from Oxford. During 1888-1919 he was Professor of Mathematics at Yorkshire College, before returning to his Alma mater. In 1894 he published the paper 'On the expansion of some infinite products'. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This contains the Rogers-Ramanujan identities, so called because they were rediscovered, without proof, by Ramanujan before 1913. In 1917 Ramanujan chanced upon Rogers' paper and expressed great admiration. A correspondence followed, and Rogers was led to a considerable simplification of his original proof.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 1936 Atle Selberg, published a 'generalization' of the Rogers-Ramanujan identities which turned out, in fact, to be another special case of Rogers' original result.&lt;/p&gt;&#xA;" OwnerUserId="6562" LastActivityDate="2020-05-04T14:08:51.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11759" PostTypeId="2" ParentId="11743" CreationDate="2020-05-04T14:46:28.037" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bayes%27_theorem#History&quot; rel=&quot;noreferrer&quot;&gt;Bayes' Theorem&lt;/a&gt;, fundamental in Bayesian statistics, was considered unremarkable by Thomas Bayes and so not published.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After Bayes' death, Richard Price edited Bayes' manuscript for reading at the Royal Society for which he was elected a Fellow.&lt;/p&gt;&#xA;" OwnerUserId="12462" LastActivityDate="2020-05-04T14:46:28.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11760" PostTypeId="1" CreationDate="2020-05-05T05:03:22.403" Score="4" ViewCount="99" Body="&lt;p&gt;I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from Felix Klein, though the lemma is sometimes referred as Schottky's criterion. I would also like to know what the original formulation looks like, if anyone knows it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thank you so much for your help!&lt;/p&gt;&#xA;" OwnerUserId="12468" LastActivityDate="2020-05-05T05:03:22.403" Title="First uses of the Ping-Pong Lemma" Tags="|mathematics|group-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11765" PostTypeId="1" CreationDate="2020-05-06T17:11:11.630" Score="1" ViewCount="78" Body="&lt;p&gt;I'm mostly interested in the degree of mathematisation with which physics was taught. Intuitively, I believe to recall that Europe has had a long tradition of mathematical physics, while physics higher education in the U.S. has generally been seen as less mathematical and more based on intuition (as an illustration, my intuition on the difference may be summed up as the traditions which P. Dirac or F. Hund followed vs. the one of R. Feynman or M. Gell-Mann). How mathematically has physics (esp. theoretical physics) been taught at U.S. universities throughout the 20th century?, i.e has the education been closer to mathematical physics or phenomenological theoretical physics?&lt;/p&gt;&#xA;" OwnerUserId="12364" LastActivityDate="2020-05-06T17:11:11.630" Title="How has the degree of mathematisation in physics university education in the U.S. changed in the 20th century?" Tags="|physics|theoretical-physics|education|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11766" PostTypeId="1" CreationDate="2020-05-07T19:41:53.277" Score="3" ViewCount="95" Body="&lt;p&gt;The index of a subgroup &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; in a group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is the number of distinct cosets of &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Why did someone decide to call this an 'index'?&lt;/p&gt;&#xA;" OwnerUserId="6208" LastActivityDate="2020-05-07T19:41:53.277" Title="Origin of the term 'index of a subgroup'" Tags="|mathematics|terminology|abstract-algebra|group-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11767" PostTypeId="1" CreationDate="2020-05-08T01:27:58.763" Score="5" ViewCount="93" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem#Explanations_attributed_to_Brouwer&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; attributes the remark to Brouwer himself, but I am extremely skeptical. Their &lt;a href=&quot;https://archive.is/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html&quot; rel=&quot;noreferrer&quot;&gt;citation&lt;/a&gt; goes to a webpage of a ? French educational TV show, where the remark appears to be a fictionalized explanation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I suspect the &quot;crumpled paper&quot; explanation was due to someone else trying to give a down-to-earth explanation of the result. It was certainly in widespread circulation by the 1970s.&lt;/p&gt;&#xA;" OwnerUserId="9253" LastActivityDate="2020-05-08T01:27:58.763" Title="Who first called the Brouwer Fixed Point Theorem &quot;the crumpled paper theorem&quot;?" Tags="|mathematics|topology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11768" PostTypeId="2" ParentId="11711" CreationDate="2020-05-08T03:24:44.183" Score="5" Body="&lt;p&gt;There is a &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Quotations/Gauss.html&quot; rel=&quot;noreferrer&quot;&gt;quote&lt;/a&gt; where Gauss mentions Laplace:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;Sin^2 φ is odious to me, even though Laplace made use of it; should it be feared that sin^2 φ might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin(φ^2),well then, let us write (sin φ)^2, but not sin^2 φ, which by analogy should signify sin (sin φ).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This quote suggests that Gauss respected Laplace, even if Laplace used a formula that he (Gauss) disliked.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link: &lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Quotations/Gauss.html&quot; rel=&quot;noreferrer&quot;&gt;http://mathshistory.st-andrews.ac.uk/Quotations/Gauss.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Euler.html&quot; rel=&quot;noreferrer&quot;&gt;Gauss worked&lt;/a&gt; on a few of Euler's integrals and rediscovered other pieces of his work, according to this link: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;http://mathshistory.st-andrews.ac.uk/Biographies/Euler.html&quot; rel=&quot;noreferrer&quot;&gt;http://mathshistory.st-andrews.ac.uk/Biographies/Euler.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This may imply that Gauss admired Euler's work to the point of studying it intensely.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for Lagrange, for whom Euler was a &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=38586&quot; rel=&quot;noreferrer&quot;&gt;mentor&lt;/a&gt;, the sources for any idea of their thoughts about each other remain elusive as of this writing.&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2020-05-08T03:24:44.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11769" PostTypeId="1" CreationDate="2020-05-08T21:18:20.150" Score="3" ViewCount="95" Body="&lt;p&gt;I &lt;a href=&quot;https://math.stackexchange.com/q/3664182/588411&quot;&gt;asked&lt;/a&gt; this question on math.se and they sent me here.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is well known that Mayan people &lt;a href=&quot;https://en.wikipedia.org/wiki/Maya_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;used&lt;/a&gt; vigesimal (base 20) numeral system, and had had an advanced &lt;a href=&quot;https://en.wikipedia.org/wiki/Maya_calendar&quot; rel=&quot;nofollow noreferrer&quot;&gt;calendar system&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Except for these facts, I'm couldn't find any more information about Mayan mathematics. I would like to know a little bit more about Mayan geometry or astronomy. Does someone know a good resource which could be a good starting point for my research? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;P.S. While researching this topic, I have seen that there is a lot of resources that are almost pseudo-scientific. Of course, I'm not interested in these references.&lt;/p&gt;&#xA;" OwnerUserId="12493" LastActivityDate="2020-05-08T21:50:34.460" Title="Literature on Mayan mathematics" Tags="|mathematics|reference-request|astronomy|geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11771" PostTypeId="1" AcceptedAnswerId="11778" CreationDate="2020-05-09T07:44:23.923" Score="10" ViewCount="1875" Body="&lt;p&gt;Heinrich Kornblum was a student of Edmund Landau who is known for his generalization of Dirichlet's theorem on primes in arithmetic progressions to function fields. He was born in Wolow, Poland in 1890, and he died in 1914 as a soldier in WWI at the age of 24 before he had a chance to write up his work, which was to be his doctoral thesis. Landau edited and published his results posthumously as &lt;em&gt;Über die Primfunktionen in einer Arithmetischen Progression&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is, as far as I can tell, the sum total of all knowledge that has been preserved about Heinrich Kornblum. I cannot find any other papers written by him, and every source I have seen that mentions him cites only the narrative I have reproduced above. Does anyone know any more information about him, or is he a tragic mathematician lost to history?&lt;/p&gt;&#xA;" OwnerUserId="12496" LastEditorUserId="4251" LastEditDate="2021-09-06T17:06:00.223" LastActivityDate="2021-09-06T17:06:00.223" Title="Who was Heinrich Kornblum?" Tags="|mathematicians|mathematics-social-history|20th-century|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11772" PostTypeId="1" CreationDate="2020-05-09T12:55:32.803" Score="3" ViewCount="143" Body="&lt;p&gt;&lt;strong&gt;Structure of this post&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I want to better understand how Bohr responded to Einstein's 1930 attempt to demonstrate that quantum mechanics was inherently contradictory under the Copenhagen interpretation.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This post is long because I really wanted to be clear in what I think Einstein and Bohr said in setting up my question. But I've put a brief summary of problem and question at the top with all the nitty-gritty details underneath.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary of problem and question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Issue.&lt;/em&gt; Einstein proposed an experiment in 1930 that contradicted energy-time uncertainty. If you set it up so a box of light opened for a brief period &lt;span class=&quot;math-container&quot;&gt;$ \Delta t $&lt;/span&gt; (meaning you know when the even happened within a fixed time-interval) and a photon escaped, by measuring the mass of the box afterwards, you could find the energy &lt;span class=&quot;math-container&quot;&gt;$ E = m c^2$&lt;/span&gt; exactly, i.e. &lt;span class=&quot;math-container&quot;&gt;$\Delta E \rightarrow 0$&lt;/span&gt; - contravening &lt;span class=&quot;math-container&quot;&gt;$\Delta E \Delta t &amp;gt; h$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Bohr solution.&lt;/em&gt; Bohr rebuts this by envisaging what the experiment would look like. He pictures a box hanging from a spring, with a pointer aligned against a scale. After the photon escapes, the box starts oscillating. By adding weights we can centre the box against the original position of the scale, so finding the mass lost with arbitrary accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The process of measuring the mass by looking at a scale means we are working to an accuracy determined by an uncertainty in position &lt;span class=&quot;math-container&quot;&gt;$\Delta q$&lt;/span&gt;. The better we know this, the worse we know the momentum of the box according to &lt;span class=&quot;math-container&quot;&gt;$\Delta p &amp;gt; h/\Delta q$&lt;/span&gt;. By linking the uncertainty in mass with the uncertainty in momentum, Bohr pupports to show that to minimise the uncertainty in position &lt;span class=&quot;math-container&quot;&gt;$\Delta q$&lt;/span&gt; we must maximise the length of time &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; over which we measure the mass of the box. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In his final move, he notes that the time interval in the box's reference frame is different to that in the laboratory rest frame since the box is moving in a gravitational field. The uncertainty in the time interval - &lt;span class=&quot;math-container&quot;&gt;$\Delta T$&lt;/span&gt; - is a function of &lt;span class=&quot;math-container&quot;&gt;$\Delta q$&lt;/span&gt;. By putting all this together, Bohr reproduces the energy time uncertainty relation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Delta E \Delta T &amp;gt; h.&#xA;$$&lt;/span&gt;&#xA;Phew for Copenhagen.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;My big issue with this.&lt;/em&gt; I have a lot of issues with this derivation, but first and foremost among them is: Einstein was concerned with the length of time the box was open &lt;span class=&quot;math-container&quot;&gt;$\Delta t$&lt;/span&gt; and our ability to measure the energy of the associated photon escaping the box to an arbitrary accuracy.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bohr introduces a totally different time-scale - that of the mass measurement (what I've called &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; to differentiate it) and then derives the energy-time uncertainty for the measurement of that.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What on earth is the relationship between the time interval Bohr introduces and what Einstein was taking about.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This is driving me nuts - as the convoluted post below will attest. Please help!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The above question described out in lots of gory detail&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein fundamentally disliked the prevailing Copenhagen interpretation of QM that emerged in the 20s and 30s. A couple of times, he devised thought experiments to try and illustrate that uncertainty - a fundamental tenet of QM - was violated. The below is his most successful attempt - taking Bohr a fretful night to try and rebuke it (which he did, see below).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What Einstein did was try and show that you could measure the energy of something perfectly within a fixed time period - violating the energy-time uncertainty principle &lt;span class=&quot;math-container&quot;&gt;$\Delta E \Delta t &amp;gt; h$&lt;/span&gt; (I'll ignore factors of &lt;span class=&quot;math-container&quot;&gt;$4\pi$&lt;/span&gt; in the following for convenience). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What Einstein said to Bohr&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Imagine a box with photons bouncing around in it (perfectly reflective walls etc; all that usual thought experiment / idealisation stuff).&lt;/li&gt;&#xA;&lt;li&gt;The box contains a clock to keep an eye on the time.&lt;/li&gt;&#xA;&lt;li&gt;For a length of time &lt;span class=&quot;math-container&quot;&gt;$\Delta t$&lt;/span&gt; - as determined by the clock - a hole in the box is opened and a photon flies out.&lt;/li&gt;&#xA;&lt;li&gt;We know to within a fixed time-scale how long the box is open - &lt;span class=&quot;math-container&quot;&gt;$\Delta t$&lt;/span&gt; - i.e. the time-window in which we are attempting to calculate the energy of the photon.&lt;/li&gt;&#xA;&lt;li&gt;According to the indeterminacy principle, our knowledge of the energy of the photon should be bounded by &lt;span class=&quot;math-container&quot;&gt;$\Delta E &amp;gt; h / \Delta t$&lt;/span&gt; .&lt;/li&gt;&#xA;&lt;li&gt;What Einstein aimed to demonstrate was that it &lt;em&gt;was&lt;/em&gt; possible to calculate the energy as precisely as you like, i.e. &lt;span class=&quot;math-container&quot;&gt;$\Delta E \rightarrow 0$&lt;/span&gt; - in violation of Heisenberg. &lt;/li&gt;&#xA;&lt;li&gt;He 'showed' this by noting that &lt;span class=&quot;math-container&quot;&gt;$\Delta E = \Delta m c^2$&lt;/span&gt;. By simply weighing the box (which we can do at our leisure so do it as accurately as possible) we can infer the energy change as the difference in two exactly known masses, meaning &lt;span class=&quot;math-container&quot;&gt;$\Delta E \rightarrow 0$&lt;/span&gt; while the time interval is fixed.&lt;/li&gt;&#xA;&lt;li&gt;This obviously violates uncertainty.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Bohr's reply&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To rebut this, Bohr went into a lot more detail about the box and how you'd actually go about carrying out the thought experiment.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;He pictured the weighing apparatus as follows: the box is attached to an undamped spring (undamped so energy, i.e. mass, is not lost) and lined up against a scale to measure its mass.&lt;/li&gt;&#xA;&lt;li&gt;After the photon is released, the box will be oscillating; we can add smaller and smaller weights to the box to get it back to the zero on the scale - i.e. where it will be (almost) at rest. The masses we add equals the mass lost (and so gives us the energy change perfectly, according to Einstein).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Now I will attempt to relate Bohr's version of his reply as he later told it in 1949, see &lt;a href=&quot;https://www.marxists.org/reference/subject/philosophy/works/dk/bohr.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I'm not saying I agree with this - and if it's obtuse it's because Bohr is very brief in his discussion.&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The box's mass can be measured with any accuracy &lt;span class=&quot;math-container&quot;&gt;$\Delta m$&lt;/span&gt; by addition of suitable loads. To know we've added the correct masses we must line the box up at the zero of the aligned scale. In other words, to assess the mass we must limit uncertainty in the measurement of position of the box &lt;span class=&quot;math-container&quot;&gt;$\Delta q$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;li&gt;This will require smaller and smaller masses to be added over a long time-interval I'll denote &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;,which Bohr calls the &quot;interval of the balancing procedure&quot; (that this interval is 'long' is shown below). &lt;/li&gt;&#xA;&lt;li&gt;Through the position-momentum uncertainty relation, measuring the position of the box up to some uncertainty &lt;span class=&quot;math-container&quot;&gt;$\Delta q$&lt;/span&gt; places a lower bound on our knowledge of the momentum of the box: &lt;span class=&quot;math-container&quot;&gt;$\Delta p &amp;gt; h/\Delta q$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The residual uncertainty in our knowledge of the mass (&lt;span class=&quot;math-container&quot;&gt;$\Delta m$&lt;/span&gt;) will leave the box moving slightly (since the spring system is out of equilibrium unless &lt;span class=&quot;math-container&quot;&gt;$\Delta m = 0$&lt;/span&gt;, which would require an infinite precision in the experiment which is impossible). &lt;/li&gt;&#xA;&lt;li&gt;Bohr then states that the &quot;latitude&quot; in momentum will &quot;obviously&quot; be smaller than the total impulse given by the force of gravity to a body of mass &lt;span class=&quot;math-container&quot;&gt;$\Delta m$&lt;/span&gt;, leading to:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Delta p &amp;lt; gT \Delta m.&#xA;$$&lt;/span&gt; &lt;/li&gt;&#xA;&lt;li&gt;So we have &lt;span class=&quot;math-container&quot;&gt;$\Delta p &amp;lt; gT \Delta m $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta p &amp;gt; h/\Delta q$&lt;/span&gt;. Taking the left equation as an equality we can put these together to show: &lt;span class=&quot;math-container&quot;&gt;$\Delta m &amp;gt; \frac{h}{gT\Delta q}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;What this shows is that in order to render our reading of the mass very accurate - which necessitates us minimising the uncertainty in the measurement of &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$\Delta q \rightarrow 0$&lt;/span&gt; - then we need to perform the weight measurement over a very long time interval: &lt;span class=&quot;math-container&quot;&gt;$T \rightarrow \infty $&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;The following is Bohr's coup-de-grâce:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;7&quot;&gt;&#xA;&lt;li&gt;A clock displaced by an amount &lt;span class=&quot;math-container&quot;&gt;$\Delta q$&lt;/span&gt; will change its rate in such a way that its reading over a time interval &lt;span class=&quot;math-container&quot;&gt;$\Delta T$&lt;/span&gt; will differ by an amount:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\Delta T}{T} = \frac{g \Delta q}{c^2}.&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;This imposes a limit on the accuracy of the time interval measurement according to: &lt;span class=&quot;math-container&quot;&gt;$\Delta T = Tg\Delta q/c^2$&lt;/span&gt;, which upon combining with the equation derived above relating position and mass uncertainty (point 6) results in:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Delta mc^2 \Delta T &amp;gt; h.&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;In other words, we re-derive time-energy uncertainty and so there is no issue. As Bohr stated in 1949: &quot;Consequently, a use of the apparatus as a means of accurately measuring the energy of the photon will prevent us from controlling the moment of its escape.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;My question (again)&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have so many questions about this I could write probably ten. But I'll keep it to one clear and simple one: assuming no other issues with the above, it seems to me that Bohr has related uncertainty in our knowledge over the time interval that the weighing procedure is carried out &lt;span class=&quot;math-container&quot;&gt;$\Delta T$&lt;/span&gt;, with uncertainty in the mass of the box. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;So he has shown that, for an oscillating box, we cannot measure the energy of the box over a given time &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; to greater than an accuracy of &lt;span class=&quot;math-container&quot;&gt;$\Delta T \Delta E &amp;gt; h$&lt;/span&gt;. This is just a statement of the uncertainty principle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the critical time here is surely the length of time the box is open, not the time over which we measure the energy of the box? How is Bohr happy with this justification; and how did Einstein not pick him up on introducing a totally different time interval? &lt;/p&gt;&#xA;" OwnerUserId="12428" LastActivityDate="2020-05-09T12:55:32.803" Title="Understanding Bohr's response to a 1930 Einstein thought experiment that challenged the uncertainty principle" Tags="|quantum-mechanics|einstein|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11773" PostTypeId="1" AcceptedAnswerId="11775" CreationDate="2020-05-09T15:10:02.280" Score="2" ViewCount="427" Body="&lt;p&gt;Some mathematicians are considered the &quot;greatest of their era&quot;. For instance, Archimedes is generally considered the greatest mathematician of Antiquity. Is there any mathematician who stood out similarly from the rest in that timeline?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2020-06-06T12:20:33.453" LastActivityDate="2020-06-13T19:59:39.250" Title="Who is considered the greatest mathematician between the years 400 AD and 1300 AD?" Tags="|mathematicians|biographical-details|middle-ages|" AnswerCount="2" CommentCount="7" ClosedDate="2020-05-10T03:22:40.090" ContentLicense="CC BY-SA 4.0" />
  <row Id="11774" PostTypeId="2" ParentId="11773" CreationDate="2020-05-09T15:58:23.380" Score="4" Body="&lt;p&gt;There are several prominent mathematicians of the medieval era, most notably Leonardo of Pisa, or more commonly called &quot;Fibonacci&quot;. He developed the Fibonacci sequence and he also introduced the Hindu - Arabic numeral system to Europe in the 13th century. As a result, the Roman numeral system was discarded in favor of the former.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link: &lt;a href=&quot;https://www.storyofmathematics.com/medieval_fibonacci.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.storyofmathematics.com/medieval_fibonacci.html&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another notable mathematician is Aryabhata. Aryabhata was an astronomer and mathematician.&#xA;He wrote the influential &quot;Aryabhatiya&quot;, where he gave rules for square roots, quadratic equations, and predicting eclipses. He also gave an approximation for the irrational constant pi (π).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Link: &lt;a href=&quot;https://www.britannica.com/biography/Aryabhata-I&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.britannica.com/biography/Aryabhata-I&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="12487" LastEditDate="2020-06-13T19:59:39.250" LastActivityDate="2020-06-13T19:59:39.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11775" PostTypeId="2" ParentId="11773" CreationDate="2020-05-09T18:29:40.170" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Al-Khwarizmi&lt;/a&gt; was a ninth-century mathematician who created many of the most basic techniques for how we perform calculations. His greatest contributions were in the realm of developing formal, systematic ways of doing arithmetic and solving equations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His works mark the beginning of what we today understand as &lt;strong&gt;Algebra&lt;/strong&gt;. One of his principal achievements in algebra was his demonstration of how to solve quadratic equations by &lt;strong&gt;completing the square&lt;/strong&gt;, for which he provided geometric justifications. Indeed, the word &quot;algebra&quot; comes from part of the title of his book on solving equations (&lt;a href=&quot;https://en.m.wikipedia.org/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Al-jabr&lt;/a&gt;), and the word &quot;algorithm,&quot; meaning a systematic set of rules used to solve a problem, descends from his name (&lt;strong&gt;in Latin, his name became ‘Algoritmi’ and his systematic calculating procedures were called algorisms — later, algorithms&lt;/strong&gt;).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Al-Khwarizmi's &quot;On the Calculation with Hindu Numerals&quot;, written about 820, &lt;strong&gt;was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-05-18T18:11:17.347" LastActivityDate="2020-05-18T18:11:17.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11778" PostTypeId="2" ParentId="11771" CreationDate="2020-05-09T22:02:19.747" Score="23" Body="&lt;p&gt;The Göttingen Digitalization Center provides a &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN266833020_0005?tify=%7B%22pages%22:[106],%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;digital scan&lt;/a&gt; of the paper H. Kornblum: &quot;Über die Primfunktionen in einer arithmetischen Progression.&quot; &lt;em&gt;Mathematische Zeitschrift&lt;/em&gt; 5 (1919), 100–111 online. The first page features a footnote by Landau that provides a few biographical details: Heinrich Kornblum was born in Wohlau on August 23, 1890. He volunteered at the start of the war and fell in October 1914 near Poël-Capelle.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Wohlau (today: Wołów in Poland) was a small town in the Prussian province of Silesia, which in 1890 had about 2,500 inhabitants according to the German &lt;a href=&quot;https://de.wikipedia.org/wiki/Wo%C5%82%C3%B3w&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;. It is located in the vicinity of Breslau (today: Wrocław in Poland). Poël-Capelle appears to be an older spelling of &lt;a href=&quot;https://en.wikipedia.org/wiki/Langemark-Poelkapelle&quot; rel=&quot;noreferrer&quot;&gt;Poelkapelle&lt;/a&gt;, a municipality in Flanders (in Belgium). In October 1914, the area around Poelkapelle and Langemarck was involved in the &lt;a href=&quot;https://en.wikipedia.org/wiki/First_Battle_of_Ypres&quot; rel=&quot;noreferrer&quot;&gt;First Battle of Ypres&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I managed to locate a &lt;a href=&quot;http://digital.ub.uni-duesseldorf.de/download/pdf/7465810?name=1908-1909&quot; rel=&quot;noreferrer&quot;&gt;digital scan&lt;/a&gt; of the annual report for the school year 1908-1909 of the high school Kornblum attended in his hometown of Wohlau:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Dr. Fritz Sattig, &quot;Königliches Gymnasium zu Wohlau. Bericht über das Schuljahr 1908-1909&quot;. Wohlau: Schulze 1909&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Among the list of graduates on p. 13, we find Heinrich Kornblum, born August 23, 1890. He attended the school for nine years and graduated Easter 1909. His father is listed as a physician (general practitioner) residing in Wohlau, and he intended to study mathematics. The family appears to have been well-off, because on p. 12 it is stated that Kornblum's father donated 25 marks to the school.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the report, there are indications that Kornblum was an excellent student. On p. 11 we read that he did not have to participate in the oral part of final exams, which (to my knowledge) was a privilege only accorded the best students. Also on p. 11 we learn that Kornblum was awarded a prize in the form of a book &quot;Deutschlands Seemacht&quot; (German Naval Power) by Georg Wislicenus on the occasion of the birthday of his majesty the emperor (back then, January 27).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Kornblum's father was likely Dr. Hermann Kornblum, who had been approved as a resident physician in Wohlau in 1883, per an &lt;a href=&quot;https://books.google.com/books?id=efUbAQAAMAAJ&amp;amp;pg=PA260&amp;amp;lpg=PA260&amp;amp;dq=%22Hermann+Kornblum%22+%22Wohlau%22&amp;amp;source=bl&amp;amp;ots=W8ac3zRm0b&amp;amp;sig=ACfU3U06qgDRxkiEXcm41l8m-URXcePuOg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwj5sKbRvqfpAhVEop4KHdKTATcQ6AEwAHoECAIQAQ#v=onepage&amp;amp;q=%22Hermann%20Kornblum%22%20%22Wohlau%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;official notice&lt;/a&gt; published in &lt;em&gt;Berliner klinische Wochenschrift&lt;/em&gt;, vol. 20, no. 17, p. 260. He &lt;em&gt;may&lt;/em&gt; be identical with Dr. Hermann Kornblum whose medical dissertation from 1878, &quot;Zur Aetiologie und Therapie des Cephalämatoms Neugeborener&quot; at the University of Breslau was &lt;a href=&quot;https://books.google.com/books?id=rtJ5vcg6nekC&amp;amp;pg=PA111&amp;amp;lpg=PA111&amp;amp;dq=Zur+Aetiologie+und+Therapie+des+Cephal%C3%A4matoms+Neugeborener;+von+Hermann+Kornblum&amp;amp;source=bl&amp;amp;ots=fndvXbDLRO&amp;amp;sig=ACfU3U26KVrLivneZzThiYuK8snrfx4esQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwi4xuib1afpAhXoJDQIHTg-BKEQ6AEwAHoECAIQAQ#v=onepage&amp;amp;q=Zur%20Aetiologie%20und%20Therapie%20des%20Cephal%C3%A4matoms%20Neugeborener%3B%20von%20Hermann%20Kornblum&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;reviewed&lt;/a&gt; in &lt;em&gt;Schmidt's Jahrbücher der in- und ausländischen gesammten Medicin&lt;/em&gt;, vol. 182 (1879), p. 111. Possible confirmation is an entry in a &lt;a href=&quot;https://books.google.com/books?id=WJ_YaXpKVYoC&amp;amp;pg=RA1-PA127&amp;amp;lpg=RA1-PA127&amp;amp;dq=%22Dr.+Kornblum%22+%22Wohlau%22&amp;amp;source=bl&amp;amp;ots=iFr-nZCiEg&amp;amp;sig=ACfU3U2e1aoeJmERslWHvwx0aQRfOcw25Q&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiOp9yi66fpAhWMr54KHSfaBbAQ6AEwAXoECAQQAQ#v=onepage&amp;amp;q=%22Dr.%20Kornblum%22%20%22Wohlau%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;list of practicing physicians&lt;/a&gt; in &lt;em&gt;Dr. Paul Börner's Reichs-Medicinal-Kalender für Deutschland&lt;/em&gt;, Leipzig: George Thieme 1887, p. 127 which states: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Wohlau (Kr. Wohlau, 3144 E.) Dr. Kornblum, Herm. 78, Kr.-W.-A.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;where &quot;78&quot; appears to refer to (by comparison with other entries) 1878, the year of his dissertation. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Based on the &lt;a href=&quot;https://books.google.com/books?id=LgY9AAAAYAAJ&amp;amp;pg=PA154&amp;amp;lpg=PA154&amp;amp;dq=Breslau+Kornblum+Branitz&amp;amp;source=bl&amp;amp;ots=vmmYocwyEg&amp;amp;sig=ACfU3U0JndOh2a8mjHSsW3cypjks2F5KXQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjFlIiR96jpAhXKrJ4KHZhIB0UQ6AEwA3oECAsQAQ#v=onepage&amp;amp;q=Breslau%20Kornblum%20Branitz&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;dissertation records&lt;/a&gt; of the University of Breslau, Hermann Kornblum was born on March 2, 1854 in Branitz in Silesia (today: &lt;a href=&quot;https://en.wikipedia.org/wiki/Branice&quot; rel=&quot;noreferrer&quot;&gt;Branice&lt;/a&gt; in Poland). I &lt;a href=&quot;https://books.google.com/books?id=aJtaAAAAcAAJ&amp;amp;pg=RA1-PA15&amp;amp;lpg=RA1-PA15&amp;amp;dq=%22Kornblum%22+%22Branitz%22&amp;amp;source=bl&amp;amp;ots=4x5suixSBc&amp;amp;sig=ACfU3U1qG_E3LuY0gCmYS9H3242DOsWrxg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwid8M708qjpAhVLrZ4KHeVyBxcQ6AEwAnoECAcQAQ#v=onepage&amp;amp;q=%22Kornblum%22%20%22Branitz%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;found&lt;/a&gt; that the attended the &quot;Königliches Gymnasium&quot; in the Silesian town of Ratibor (today: [Racibórz] in Poland), from which he graduated Easter 1873 with the stated intention to study medicine in Breslau. His deceased father is listed as &quot;Kaufmann&quot; (merchant, shopkeeper) in Branitz.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is interesting to note that Hermann Kornblum's graduation record shows his religious affiliation as Jewish, while Heinrich Kornblum's graduation record shows his religious affiliation as protestant. A potential sister of Heinrich Kornblum is &lt;a href=&quot;https://geschichte.charite.de/aeik/biografie.php?ID=AEIK00608&quot; rel=&quot;noreferrer&quot;&gt;Fri[e]da Kornblum&lt;/a&gt;, born September 23, 1888 in Wohlau in Silesia whose father is listed as &quot;Sanitätsrat&quot;, an honorary title that in Prussia was often accorded physicians in private practice after twenty years of practicing their profession. She became a physician and was later restricted in her professional practice under Nazi laws on account of her Jewish ancestry.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2020-05-10T09:36:45.000" LastActivityDate="2020-05-10T09:36:45.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11781" PostTypeId="1" CreationDate="2020-05-11T13:28:42.823" Score="2" ViewCount="225" Body="&lt;p&gt;&lt;a href=&quot;https://astronomy.stackexchange.com/a/36143/7982&quot;&gt;This answer&lt;/a&gt; to &lt;em&gt;What kind of triangle is formed by three unequal masses in a circular restricted three body orbit?&lt;/em&gt; explains that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In the Newtonian limit, an equilateral 3-body solution exists for any combination of masses&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and links to &lt;a href=&quot;https://arxiv.org/abs/1212.0754&quot; rel=&quot;nofollow noreferrer&quot;&gt;Triangular solution to general relativistic three-body problem for general masses&lt;/a&gt; which stars with &quot;Lagrange’s equilateral triangular solution for the three-body problem&quot; and looks at relativistic (&quot;post-Newtonian&quot;) effects.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://en.wikipedia.org/wiki/Three-body_problem#Restricted_three-body_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;circular restricted three body problem&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrangian_point&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrangian points&lt;/a&gt; fame one body is a &quot;test mass&quot; with no significant effect on the other two, but apparently three massive bodies can also have a circular orbit solution where they still form an equilateral triangle, as &lt;span class=&quot;math-container&quot;&gt;$L_4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L_5$&lt;/span&gt; do in the CR3BP.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Where did Lagrange first write about the equilateral triangular solution for the three-body problem where all three bodies have non-zero mass?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2020-09-20T04:28:30.423" LastActivityDate="2023-01-22T03:29:53.047" Title="Where can I find Lagrange’s original equilateral triangular solution for arbitrary masses?" Tags="|mathematics|physics|newton|resource-recommendation|lagrange|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11783" PostTypeId="2" ParentId="9766" CreationDate="2020-05-11T22:12:52.627" Score="2" Body="&lt;p&gt;Buridan introduced p = mv, called it “impetus” and stated that it did not change if no force was applied. &#xA;Buridan’s Inertia Law is known as Newton’s First Law.&#xA;If one reads Buridan carefully, one sees he asserts: Force = Deviation from Trajectory, and dp/dt = Force.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton needed this law: dp/dt = F (where  F is the Force, by definition). It’s an axiom. (Weirdly the Second Law implies the First…)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe one should establish historical priority and call it Buridan's law? That was more than three centuries before Newton... And some of Buridan's students would establish some of the first theorems of calculus (through graphical methods they invented).&lt;/p&gt;&#xA;" OwnerUserId="9331" LastActivityDate="2020-05-11T22:12:52.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11784" PostTypeId="2" ParentId="11573" CreationDate="2020-05-12T15:10:01.807" Score="4" Body="&lt;p&gt;In fact, according to the book &lt;a href=&quot;https://books.google.es/books?id=f_WhvNKnJG8C&amp;amp;pg=PA237#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&quot;Des Kennenlernens werth&quot;: bedeutende Frauen Göttingens&lt;/em&gt;&lt;/a&gt;, only one other woman apart from Emy Noether achieved habilitation in Göttingen during the Weimar Republic: the physicist Herta Sponer, who obtained habilitation in 1925. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;At least until 1945, the fact that very few famale mathematicians had achieved habilitation wasn't indeed something specific to Göttingen, but to the whole Germany. In the book &lt;a href=&quot;https://books.google.es/books?id=eDVBDwAAQBAJ&amp;amp;pg=PA57#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America&lt;/em&gt;&lt;/a&gt;, I've found that only six female mathematicians got habilitation in Germany between 1913 and 1945, including Noether and Hel Braun in Göttingen:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Renate Tobies notes that there were only six women who completed a postdoctoral thesis (&lt;em&gt;Habilitationsschrift&lt;/em&gt;) in Germany between 1913 and 1945. As Tobies put it, &quot;[a]lthough each of these women produced outstanding research, they could not obtain a paid professorship unless they went into exile abroad (Noether and Geiringer), or persevered until after the end the war&quot; [60, p. 286]. In the group of female graduates in Germany before 1937, only Noether, Hilda Geiringer, Ruth Moufang, Maria Pia Geppert, and Hel Braun held positions as professor and pursued mathematical research [58, p. 151].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;And according to the book &lt;a href=&quot;https://books.google.es/books?id=gwR2mmlpgoQC&amp;amp;pg=PA181#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&quot;Aller Männerkultur zum Trotz&quot;: Frauen in Mathematik und Naturwissenschaften&lt;/em&gt;&lt;/a&gt;, by Renate Tobies, Ruth Moufang was the third woman to complete an habilitation in mathematics in Germany after Emmy Noether and Hilda Pollazcek-Geiringer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I finally managed to find the full text of the film &lt;em&gt;Women and mathematics across cultures&lt;/em&gt; at pages 20-26 of the &lt;a href=&quot;http://downloads.hindawi.com/books/9789775945006.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proceedings of the eighth general meeting &lt;em&gt;European Women in Mathematics&lt;/em&gt;&lt;/a&gt;. This film includes an interview with Mara D. Neusel held at Madrid in 1995, which contains this information:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Without&#xA;  definite details, I want to remark that at the university where I studied, Göttingen, only&#xA;  two women achieved their Habilitation Thesis, this was Emmy Noether and Helene&#xA;  Braun and the last one 53 years ago.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So, yes, after Noether and Braun habilitations, there were at least 53 years without any other famale mathematician habilitation in Göttingen.&lt;/p&gt;&#xA;" OwnerUserId="10392" LastActivityDate="2020-05-12T15:10:01.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11786" PostTypeId="1" AcceptedAnswerId="11787" CreationDate="2020-05-13T17:44:59.937" Score="5" ViewCount="68" Body="&lt;p&gt;What did Emmy Noether contribute to the theory of &lt;a href=&quot;https://www.encyclopediaofmath.org/index.php/Integral_invariant&quot; rel=&quot;noreferrer&quot;&gt;integral invariant&lt;/a&gt;s that wasn't already done by, e.g., Sophus Lie in his 1902 &lt;a href=&quot;https://www.gutenberg.org/ebooks/25157&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Über Integralinvarianten und Differentialgleichungen&lt;/em&gt;&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2020-05-13T18:53:33.373" Title="What did Noether contribute to the theory of integral invariants?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11787" PostTypeId="2" ParentId="11786" CreationDate="2020-05-13T18:40:05.707" Score="3" Body="&lt;p&gt;Noether was the first to, as she said in the abstract of her 1918 &quot;&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/PCZVNCHS/Noether%20-%201971%20-%20Invariant%20variation%20problems.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Invariant Variational Problems&lt;/a&gt;&quot; paper (&lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=6355&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Noether Theorems&lt;/em&gt;&lt;/a&gt; p. 3), formulate&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;a combination of the methods of the formal calculus of variations and of Lie’s theory of groups.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;She says&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For those differential equations that arise from&#xA;  variational problems, the statements that can be formulated are much more precise&#xA;  than for the arbitrary differential equations that are invariant under a group, which&#xA;  are the subject of Lie’s researches. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;and recognized that&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;For certain groups and variational problems this combination is not new&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-05-13T18:53:33.373" LastActivityDate="2020-05-13T18:53:33.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11796" PostTypeId="1" AcceptedAnswerId="11799" CreationDate="2020-05-14T22:11:54.217" Score="4" ViewCount="199" Body="&lt;p&gt;&lt;strong&gt;&lt;em&gt;When and WHY did mathematicians start turning their attention to imaginary exponents?&lt;/em&gt;&lt;/strong&gt; I read on Wikipedia about Euler's correspondences with Bernouille and such, but it doesn't answer what exactly triggered this. What did they hope to achieve by exploring such things? Why did they not do this earlier? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have recently read the history of fractional exponents and how they were a suggested notation according to the laws of exponents by Nicole Oresme for numbers which were not &quot;nice&quot; (natural number) powers of each other (e.g. 4 and 8, &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt;; and expressions like &lt;span class=&quot;math-container&quot;&gt;$5^\frac{1}{3}$&lt;/span&gt; were taken to mean the number whose cube is 5). I don't know whether Oresme's work had the same philosophy of exponents as Euler's though! &lt;/p&gt;&#xA;" OwnerUserId="9747" LastActivityDate="2020-05-15T11:43:01.823" Title="When and WHY did mathematicians start turning their attention to imaginary exponents?" Tags="|mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11797" PostTypeId="1" AcceptedAnswerId="11906" CreationDate="2020-05-14T22:14:59.063" Score="2" ViewCount="167" Body="&lt;p&gt;At some point in history, theoretical computer science emerged as a sub-discipline of computer science. Later, apparently it segregated into &quot;volume A&quot; work and &quot;volume B&quot; work. This can be seen for example at the Handbook of Theoretical Computer Science (1990), which is divided into Volume A: Algorithms and Complexity, and Volume B: Formal Models and Sematics. Other witnesses of the divide are the journal Theoretical Computer Science and the International Colloquium on Automata, Languages, and Programming, which also use Track A and B (with another Track C later). What were the reasons (in the sense of developments) that led to the divide, what were the cultural consequences (e.g. in the US, theory A is predominant), and when did it take place? Can we observe some point in time when the terms &quot;A&quot; and &quot;B&quot; (or similar) were used for the first time to denote this divide? &lt;/p&gt;&#xA;" OwnerUserId="9523" LastEditorUserId="9523" LastEditDate="2020-05-15T21:57:52.240" LastActivityDate="2020-07-13T23:03:10.260" Title="Origins and reason for Volume A and B divide in Theoretical Computer Science" Tags="|computer-science|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11799" PostTypeId="2" ParentId="11796" CreationDate="2020-05-15T11:35:04.883" Score="6" Body="&lt;p&gt;The original object of study was logarithm, rather then exponent. Logarithm (discovered by Napier) was an important function, both in pure and applied mathematics. So people started asking what the log of a negative number can be.&#xA;Research on this question quickly lead to paradoxes. (Which still cause difficulties to students).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The solution of these paradoxes is mostly due to Euler, who explained that understanding of logarithm was impossible without considering complex values (of argument and of the function). Moreover, he understood that logarithm is a &quot;multivalued function&quot;. To simplify the matter, he&#xA;introduced the inverse which is an honest, single valued function of a complex variable. This is the genesis of the exponential function.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;L. Euler, &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/807/&quot; rel=&quot;noreferrer&quot;&gt;Sur les logatithms des nombres negatifs et imaginaires&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;L. Euler, &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/168/&quot; rel=&quot;noreferrer&quot;&gt;De la controverse entre Mrs. Leibnitz et Bernoulli sur les logarithmes des nombres négatifs et imaginaires&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-15T11:43:01.823" LastActivityDate="2020-05-15T11:43:01.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11800" PostTypeId="2" ParentId="7370" CreationDate="2020-05-15T14:59:52.527" Score="1" Body="&lt;p&gt;To answer your title question, the first &quot;programmable, digital computer&quot; &lt;a href=&quot;https://www.computerhistory.org/babbage/engines/&quot; rel=&quot;nofollow noreferrer&quot;&gt;was designed by Charles Babbage and Ada Lovelace&lt;/a&gt;, known as the Analytical Engine. Turing, von Neumann, and other computer scientists and mathematicians approached the concept of a programmable computer both independently and building on the work of Babbage, Lovelace and their contemporaries. Turing devised the concept of a &lt;a href=&quot;https://blogs.bodleian.ox.ac.uk/adalovelace/2018/07/26/ada-lovelace-and-the-analytical-engine/&quot; rel=&quot;nofollow noreferrer&quot;&gt;general-purpose computer&lt;/a&gt; in the 1930s, which the Analytical Engine was theorized to be. A quote exists from the second link which illustrates how the modern computer was theorized through Babbage's engine:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;em&gt;&quot;Charles Babbage started work on his Analytical Engine in the mid-1830s, with the idea of creating a new calculating machine that could “eat its own tail”, by which he meant that it could modify its calculation while it was running. It would do this by pausing during a calculation, and using the values it had already determined to choose between two possible next steps. Babbage listed the basic operations that such a machine, with a large enough memory, would need if it were to execute “the whole of the developments and operations of analysis”, in other words any calculation that could be conceived of at the time. We now know that the basic operations that he described are what are needed to compute anything that can be calculated by any modern computer. This means that the Analytical Engine would have been, in modern terms, a general-purpose computer, a concept first identified by Alan Turing in the 1930s.&quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Many mathematicians and computer scientists interpreted this concept through their distinct vision, and they also built on it through various manifestations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Hope this helps!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Helpful links:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.computerhistory.org/babbage/engines/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.computerhistory.org/babbage/engines/&lt;/a&gt;&#xA; &lt;a href=&quot;https://blogs.bodleian.ox.ac.uk/adalovelace/2018/07/26/ada-lovelace-and-the-analytical-engine/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blogs.bodleian.ox.ac.uk/adalovelace/2018/07/26/ada-lovelace-and-the-analytical-engine/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2020-05-15T14:59:52.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11801" PostTypeId="1" AcceptedAnswerId="11802" CreationDate="2020-05-15T18:39:14.540" Score="9" ViewCount="929" Body="&lt;p&gt;Who was the first person to notice logarithms of negatives numbers and why? When did they first arise naturally? I thought I saw somewhere that it had to do with integration but I can't find it anymore and it's really bothering me. What was this integral (if indeed that's when mathematicians first stumbled upon logarithms of negatives)? &lt;/p&gt;&#xA;&#xA;&lt;hr&gt;&#xA;&#xA;&lt;p&gt;Also, this is somewhat of a follow-up question to: &lt;a href=&quot;https://hsm.stackexchange.com/questions/11796/when-and-why-did-mathematicians-start-turning-their-attention-to-imaginary-expon/11799#11799&quot;&gt;When and WHY did mathematicians start turning their attention to imaginary exponents?&lt;/a&gt; &lt;/p&gt;&#xA;" OwnerUserId="9747" LastActivityDate="2020-05-16T08:04:25.007" Title="Who was the first person to notice logarithms of negatives numbers and for what reason?" Tags="|complex-analysis|logarithm|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11802" PostTypeId="2" ParentId="11801" CreationDate="2020-05-15T19:07:37.517" Score="10" Body="&lt;p&gt;As far as I known, logarithms of negative numbers first appeared (in the modern sense) in the 1751 paper of Euler &lt;em&gt;De la controverse entre Mrs. Leibniz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires&lt;/em&gt;, you can find &lt;a href=&quot;http://eulerarchive.maa.org/pages/E168.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; the original and the English translation. As you will see, Euler discuss the different positions of Bernoulli and Leibniz about this kind of quantities, so of course they considered logarithms of negative numbers &lt;em&gt;before&lt;/em&gt; Euler. Their correspondence from 1694 to 1716 was published in 1745 as the two-volume collection &lt;em&gt;Virorum celeberr. Got. Gul. Leibnitii et Johan. Bernoullii Commercium philosophicum et mathematicum&lt;/em&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Euler write &lt;span class=&quot;math-container&quot;&gt;$l-a$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\log(-a)$&lt;/span&gt;)&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;POSITION OF M. BERNOULLI&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;M. Bernoulli holds that the logarithms of negative numbers are the same as those of positive numbers,&#xA;  in other words that the logarithm of the negative number &lt;span class=&quot;math-container&quot;&gt;$−a$&lt;/span&gt; is equal to the logarithm of the positive number&#xA;  &lt;span class=&quot;math-container&quot;&gt;$+a$&lt;/span&gt;. Thus, the position of M . Bernoulli implies that &lt;span class=&quot;math-container&quot;&gt;$l−a = l+a$&lt;/span&gt;.&#xA;  M. Leibniz gave occasion to this declaration of M. Bernoulli when he argued, in Letter CXC of the Exchange, that the ratio of &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt;, or of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt;, was imaginary, because the logarithm of the measure of that ratio, that is to say the logarithm of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt;, which is the exponent of that ratio, was imaginary. To this,&#xA;  M. Bernoulli declared, in Letter CXCIII, that he was not of this opinion, and that he believed in fact&#xA;  that the logarithms of negative numbers were not only real, but also equal to the logarithms of the same&#xA;  numbers, taken positively. M. Bernoulli supported his position with the following proofs. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;POSITION OF M. LEIBNIZ&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;M. Leibniz holds that the logarithms of all negative numbers, and even more so those of imaginary&#xA;  numbers, are imaginary; thus, since &lt;span class=&quot;math-container&quot;&gt;$l−a = la + l−1$&lt;/span&gt;, he holds that &lt;span class=&quot;math-container&quot;&gt;$l−1$&lt;/span&gt; is an imaginary quantity.&#xA;  I have already remarked that M. Leibniz held that the ratio of +1 to &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; or of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; is imaginary, since the logarithm of that ratio or &lt;span class=&quot;math-container&quot;&gt;$l−1$&lt;/span&gt; is imaginary. We see, of course, that all the objections which were&#xA;  made against the system of M. Bernoulli serve to strengthen this position, and that the reasons advanced&#xA;  to support M. Bernoulli's position must be contrary to those of M. Leibniz. Nevertheless, one can bring&#xA;  forward particular proofs to confirm M. Leibniz's position, which will be the subject of my examination which follows.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;PROOF 1.&#xA;  Having observed that the logarithm of the number &lt;span class=&quot;math-container&quot;&gt;$1 + x$&lt;/span&gt; is equal to the sum of this series&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$l(1 + x) = x − \frac{1}{2}x^2+ \frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5-\frac{1}{6}x^6+ecc.&#xA;$$&lt;/span&gt;&#xA;  from which we see to begin with that if &lt;span class=&quot;math-container&quot;&gt;$x = 0$&lt;/span&gt;, it follows that &lt;span class=&quot;math-container&quot;&gt;$l1 = 0$&lt;/span&gt;, now to obtain the logarithm of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; we&#xA;  must set &lt;span class=&quot;math-container&quot;&gt;$x = −2$&lt;/span&gt;, whence we obtain&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$l−1  = -2 − \frac{1}{2}\cdot 4+ \frac{1}{3}\cdot 8-\frac{1}{4}\cdot 16+\frac{1}{5}\cdot 32-\frac{1}{6}\cdot 64-ecc.$$&lt;/span&gt;&#xA;  Now, there is no doubt that the sum of this divergent series could not be &lt;span class=&quot;math-container&quot;&gt;$= 0$&lt;/span&gt;; thus, it is certain that &lt;span class=&quot;math-container&quot;&gt;$l−1$&lt;/span&gt;&#xA;  is not &lt;span class=&quot;math-container&quot;&gt;$= 0$&lt;/span&gt;. The logarithm of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; will thus be imaginary, since it is also clear that it could not be real, i.e., positive or negative.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It would thusappear that M. Leibniz’s view is better-founded, since it is not contrary to the discovery&#xA;  of M. Bernoulli, that is&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$l\sqrt{}−1 = \frac{1}{2}\pi\sqrt{}−1,$$&lt;/span&gt;&#xA;  since M. Leibniz holds that the logarithm of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt;, and even more so that of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{}−1$&lt;/span&gt;, is imaginary. But, in&#xA;  adopting M. Leibniz's position, we plunge into the aforementioned difficulties and contradictions. For, if&#xA;  &lt;span class=&quot;math-container&quot;&gt;$l−1$&lt;/span&gt; were imaginary, its double, i.e. the logarithm of &lt;span class=&quot;math-container&quot;&gt;$(−1)^2 = +1$&lt;/span&gt;, would also be imaginary, which doesn't&#xA;  accord with the first principle of the theory of logarithms, in virtue of which we suppose that &lt;span class=&quot;math-container&quot;&gt;$l+1 = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;SOLUTION OF THE PRECEDING DIFFICULTIES&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;It must first be stated that if the idea which Messrs. Leibniz and Bernoulli have attached to the term&#xA;  ‘logarithm’, and which all Mathematicianshave had up to now, were perfectly correct, it would be absolutely&#xA;  impossible to rescue the theory of logarithms from the contradictions which I have been propounding. Now,&#xA;  the idea of logarithms having been derived from an origin of which we have a perfect understanding, how is&#xA;  it possible that it could be defective? When we say that the logarithm of a given number is the exponent of&#xA;  the power of a certain number taken arbitrarily, which becomes equal to the given number, it appears that&#xA;  nothing is lacking to the correctness of that idea. And that is perfectly true; but we generally put that idea&#xA;  together with a condition which does not suit it at all: that is, we ordinarily suppose, almost without noticing&#xA;  it, that to each number there corresponds only one logarithm; now, with only a little consideration, we will&#xA;  find that all the difficulties and contradictions by which the theory of logarithms appears to be embarrassed,&#xA;  persist only to the extent that we suppose that to each number there corresponds only one logarithm.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The problem doesn't seem to be related to some integrals, but rather, as one can see also in the following, to infinite series. Moreover, from Leibniz and Euler consideration, it seems to me that logarithms of negative quantities arise as a natural extension of &quot;real&quot; logarithms (in Leibniz words: &quot;the logarithm of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; [...] is the exponent of the ratio of &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt;, or of &lt;span class=&quot;math-container&quot;&gt;$−1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt;&quot; see above), provided that they are &quot;in accord with the [...] principle[s] of the theory of logarithms&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you can read French I also suggest &lt;a href=&quot;https://webcache.googleusercontent.com/search?q=cache:kuiTy0yGPRIJ:https://www.accademiadellescienze.it/media/309+&amp;amp;cd=17&amp;amp;hl=en&amp;amp;ct=clnk&amp;amp;gl=it&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; modern paper (&lt;em&gt;Euler, d'Alembert et la controverse sur les logarithmes&lt;/em&gt;), ad also &lt;a href=&quot;https://archive.org/details/miscellaneaphilo01acca/page/112/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; 1759 original paper (&lt;em&gt;Réflexions sur les quantités imaginaires&lt;/em&gt;) of Daviet de Foncenex, expecially at page 126.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-05-15T20:01:18.400" LastActivityDate="2020-05-15T20:01:18.400" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11805" PostTypeId="1" AcceptedAnswerId="11806" CreationDate="2020-05-16T15:44:06.037" Score="4" ViewCount="108" Body="&lt;p&gt;(Excuse me in advance if I've hereby launched a duplicate question, but so far I didn't came across any opinions.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In other words, I was wondering: Will scientific progress finally stall, because it will take too long for a researcher to reach the level of previous research? (Improved health care will definitely provide some boost here, although I would judge this would be a minor effect?)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe one would need to distinct between research in a broad sense vs. in a deep sense. In the latter case, research couldn't be continued without understanding the previously discovered concepts. And maybe there's also a limit on how much you can split a scientific area into a subfield. In the end you're working on a phenomenon that is a subsubsub.... field.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore, is the portion of &lt;em&gt;all&lt;/em&gt; knowledge that humankind can ever get a grip on, limited?&#xA;So AI-powered robot scientists like 'Adam' and 'Eve' will take it from there?&lt;/p&gt;&#xA;" OwnerUserId="12536" LastActivityDate="2020-05-16T19:00:47.067" Title="Eventually impossible to contribute to science?" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="0" ClosedDate="2020-05-16T19:50:52.240" ContentLicense="CC BY-SA 4.0" />
  <row Id="11806" PostTypeId="2" ParentId="11805" CreationDate="2020-05-16T19:00:47.067" Score="3" Body="&lt;p&gt;It seems to me that what you're grasping at is the &lt;em&gt;cost of reproduction&lt;/em&gt; of scientific knowledge. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;This must impose a limit in theory, but there are a few things that occur to me offhand.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Firstly, economic technologies that arise from science often reduce the amount of labour we must perform merely to survive on a daily basis, and this creates slack for people to engage in additional education and study.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Secondly, scientific progress is often also measured by the amount of knowledge that we &lt;em&gt;stop&lt;/em&gt; reproducing because it was wrong. In other words, scientific progress does not relentlessly accrete additional knowledge, instead there are often periods in which huge simplifications occur and much thought is jettisoned.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Thirdly, once scientific knowledge is recorded, it is often far easier to reproduce thereafter - it can be remarkably easy to climb up onto the shoulders of giants.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fourthly, very few of the population are actually engaged in producing scientific knowledge, and are only modestly engaged in reproducing it, with much intellectual labour inefficiently deployed, so there is still far more untapped potential remaining than is presently consumed.&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2020-05-16T19:00:47.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11808" PostTypeId="1" CreationDate="2020-05-17T09:39:18.930" Score="0" ViewCount="75" Body="&lt;p&gt;I could get a little intuition about preset, poset, and toset. e.g. A toset is, in effect, a linearly ordered set, and a poset is a set in which no more than one element in the same order exists, although not all elements need to be comparable.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But I thought, if I knew where this definition was initially needed and how it appeared, I could get a better understanding. And it was hard for me to find out enough explanation. I'd appreciate it if you could let me know about this. In addition, if you have any references to the history of other &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_order_theory_topics&quot; rel=&quot;nofollow noreferrer&quot;&gt;ordered sets&lt;/a&gt;, please recommend it.&lt;/p&gt;&#xA;" OwnerUserId="10808" LastActivityDate="2020-05-17T09:39:18.930" Title="How did the definition of each &quot;ordered set&quot; come about?" Tags="|mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11811" PostTypeId="1" AcceptedAnswerId="11812" CreationDate="2020-05-17T18:11:14.300" Score="1" ViewCount="125" Body="&lt;p&gt;My questions deal more properly with pre-Newtonian modern physics in its Cartesian or Hobbesian versions. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word &quot;mechanical&quot; comes from a Greek word meaning &quot;machine&quot;. However, the received definition of mechanical philosophy does not contain the concept of a machine. This school of thought is said to adopt the principle according to which &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;everything in nature can be explained by the size, the shape and the&#xA;  movement of bodies or material objects (one body acting on the other,&#xA;  through collision).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, which concept of a &quot;machine&quot; allows calling &quot;mechanical&quot; such a principle?&lt;/p&gt;&#xA;" OwnerDisplayName="user11223" LastEditorUserId="10106" LastEditDate="2020-05-26T11:26:24.783" LastActivityDate="2020-05-26T11:26:24.783" Title="Which conception of a &quot;machine&quot; allows to call &quot;mechanical&quot; the new physics of the 17th century?" Tags="|physics|philosophy-of-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11812" PostTypeId="2" ParentId="11811" CreationDate="2020-05-17T19:12:58.113" Score="3" Body="&lt;p&gt;The etymology of the word &quot;mechanics&quot; is irrelevant here. In 18th century, when this concept prevailed &quot;mechanics&quot; meant a part of physics as we know it now: mechanics created by Galileo and Newton.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The word itself arises in Hellenistic Greece, when the only part of mechanics known at that time was statics. It had a lot of technical applications (building, war engines, etc.) so mechanics was associated with machines. Essentially the same machines were available in the 18th century,&#xA;with an important addition: clocks and watches, and other clockwork mechanisms. Comparison of the Solar system with a clockwork was common among philosophers at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But the scope of mechanics was already much wider: it included celestial mechanics first of all.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-17T20:54:15.923" LastActivityDate="2020-05-17T20:54:15.923" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11813" PostTypeId="1" AcceptedAnswerId="11823" CreationDate="2020-05-17T21:45:05.700" Score="1" ViewCount="127" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Troughton_scale&quot; rel=&quot;nofollow noreferrer&quot;&gt;Troughton scale&lt;/a&gt; served as the first (de facto) standard of length in the United States, from 1832 until 1856. It is a measurement scale of 82 inches, subdivided to tenths of inches, on a silver inlay in a brass bar about 86 inches long (see &lt;a href=&quot;https://web.archive.org/web/20200517175327/https://www.unav.es/gep/LenzenMetrology1965.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, p. 31). On p. 33 of the same reference, which dates to 1965, it is stated that the Troughton scale is located 'in a&#xA;case on the right' relative to 'the Standards Vault at the National Bureau of Standards, Washington, D. C.' &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The scale was brought to the United States and metrologically characterized by &lt;a href=&quot;https://en.wikipedia.org/wiki/Ferdinand_Rudolph_Hassler&quot; rel=&quot;nofollow noreferrer&quot;&gt;F. R. Hassler&lt;/a&gt;. (Another useful reference on these matters is &lt;a href=&quot;https://web.archive.org/web/20180604174515/https://nvlpubs.nist.gov/nistpubs/bulletin/01/nbsbulletinv1n3p365_A2b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of standard weights and measures of the United States&lt;/a&gt;, published in 1905, esp. pp. 370 and 372.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In 2004, a descendant of F. R. Hassler gave a presentation which is included in the NIST Special Publication 1068 ('Ferdinand Rudolph Hassler (1770-1843): A Twenty Year Retrospective, 1987-2007', available &lt;a href=&quot;https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication1068.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). In the presentation, there is a picture whose caption reads (on p. 106, which is p. 130 of the NIST pdf), &lt;em&gt;My father was Ferdinand Rudolph IV, on the left, with his brother, John, seated next to him. Here, they pose with some FRH's instruments that are now in the NIST Museum.&lt;/em&gt; True, the Troughton scale was not exactly one of F. R. Hassler's instruments, and in any case does not appear to be in that photo. Nevertheless, the NIST Museum would be a logical place for it. However, I was unable to find any record of it &lt;a href=&quot;https://www.nist.gov/nist-museum/museum-collection/&quot; rel=&quot;nofollow noreferrer&quot;&gt;on the website of the museum&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Another logical place where it could be located is the Smithsonian, but I couldn't locate it &lt;a href=&quot;https://naturalhistory.si.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;there,&lt;/a&gt; either. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Finally, I did a lot of googling, including on google books. So far, the 1965 reference I mentioned above is the most recent reference I have been able to find that specifically mentions the location of the Troughton scale at the time of the writing.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would, I think, be quite surprising if the Troughton scale is not in the NIST museum, and it would be &lt;strong&gt;very&lt;/strong&gt; surprising if it has been lost or destroyed. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Can someone confirm the present location of the Troughton scale?&lt;/p&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2020-05-18T16:29:03.267" LastActivityDate="2022-06-10T16:28:31.027" Title="Where is the Troughton scale kept?" Tags="|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11815" PostTypeId="1" CreationDate="2020-05-18T14:49:41.003" Score="2" ViewCount="99" Body="&lt;p&gt;Do you know any handbook where original proofs of  mathematicians' of the past theorems and facts are in modern notation? For example, for the Archimedean spiral etc&lt;/p&gt;&#xA;" OwnerUserId="11084" LastActivityDate="2020-05-18T14:49:41.003" Title="Handbook of proofs" Tags="|mathematics|book-recommendation|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11816" PostTypeId="1" CreationDate="2020-05-18T17:23:10.733" Score="9" ViewCount="854" Body="&lt;p&gt;The peculiar agricultural terminology commonly used in algebraic geometry and category theory, &quot;sheaf&quot;, &quot;stalk&quot;, &quot;germ&quot;, is quite well-known. A sheaf is pictured as something like a bundle of stalks, in which reside germs. Very roughly and intuitively, a germ is a localized datum capable of being developed or extended to a function. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is commonly noted that Jean Leray introduced sheaves (and also spectral sequences) while he was interned as a prisoner of war (in Austria during World War II). In French, the terms &quot;&lt;em&gt;faisceau, fibre, germe&lt;/em&gt;&quot; are used, with &lt;em&gt;faisceau&lt;/em&gt; making its first appearance in Leray. The term &quot;fiber&quot; is another well-known term in mathematical English, as in &quot;fiber bundle&quot;, etc. -- such terms are conceptual neighbors of the sheaf-theoretic terms above, although I suppose the need was felt to translate this sense of the French &lt;em&gt;fibre&lt;/em&gt; into something other than &quot;fiber&quot; (for example, the fiber of a vector bundle over a point of the base space is something different from the stalk (over the same point) of the corresponding sheaf of modules). Hence the English &quot;stalk&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am interested in knowing more about the provenance and etymology of the mathematical terms. Here is the precise question: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Are there any sources that attest to why Leray chose &lt;em&gt;faisceau&lt;/em&gt;? Relatedly, are there earlier sources (predating Leray) which use for example the term &quot;germ&quot; (or any of its cognates in other languages)? &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;(I must confess that I haven't looked at Leray's seminal papers. I'm pretty sure I'm not alone in this, as many mathematicians from that time commented and complained about how obscure Leray's presentations were, and everyone nowadays seems to learn sheaf theory and spectral sequences from other sources.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I might as well share my hunch or private folk etymology which leads me to ask this question: it was &quot;germ/&lt;em&gt;germe&lt;/em&gt;&quot; that came first, with other terms like &quot;&lt;em&gt;faisceau&lt;/em&gt;/sheaf&quot; and &quot;&lt;em&gt;fibre&lt;/em&gt;/stalk&quot; later being built around it, and that &quot;germ&quot; itself might be rooted in the theory of Riemann surfaces, with later carry-over into the modern foundations of differentiable manifolds and algebraic varieties. The idea is that starting with a germ of an analytic function, i.e., its local behavior in arbitrarily small neighborhoods of a point, there &lt;em&gt;germinates&lt;/em&gt; by analytic continuation a development into a maximal connected Riemann surface in which the analytic function naturally &quot;lives&quot; (possibly as a multi-valued meromorphic function). I think I probably first got this idea from Chapter 8 of Complex Analysis by Ahlfors. &lt;/p&gt;&#xA;" OwnerUserId="1987" LastActivityDate="2020-05-22T09:05:52.870" Title="What is the etymology of the mathematical terms &quot;sheaf, stalk, germ&quot;?" Tags="|terminology|differential-geometry|algebraic-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11817" PostTypeId="1" AcceptedAnswerId="11818" CreationDate="2020-05-18T18:04:34.520" Score="2" ViewCount="251" Body="&lt;p&gt;In pure mathematics, a &lt;em&gt;sequence&lt;/em&gt; is a list of terms, for instance &lt;span class=&quot;math-container&quot;&gt;$1, \frac12, \frac14, \dots, \frac{1}{2^k},\dots$&lt;/span&gt;, and a &lt;em&gt;series&lt;/em&gt; is the sum of an infinite sequence, for instance &lt;span class=&quot;math-container&quot;&gt;$\sum_{k=1}^\infty \frac{1}{2^k}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, in many applied contexts, a sequence of data is called a &lt;em&gt;series&lt;/em&gt;, for instance &lt;a href=&quot;https://en.wikipedia.org/wiki/Time_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;time series&lt;/a&gt; in statistics.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How did this difference in usage originate?&lt;/p&gt;&#xA;" OwnerUserId="3529" LastEditorUserId="3529" LastEditDate="2020-05-19T10:31:26.317" LastActivityDate="2020-05-19T14:29:58.557" Title="Why is a time series not called a time sequence?" Tags="|terminology|real-analysis|statistics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11818" PostTypeId="2" ParentId="11817" CreationDate="2020-05-19T11:49:36.177" Score="5" Body="&lt;p&gt;As @Dave L Renfro noticed, the distinction between series and sequence is not old, and it was possible for the same author to use the two terms with different meanings (also in the same article).&#xA;Consider e.g. Gauss's &lt;a href=&quot;http://dx.doi.org/10.3931/e-rara-61066&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theoria Residuorum Biquadraticorum Commentatio Prima&lt;/a&gt; &amp;amp; &lt;a href=&quot;http://dx.doi.org/10.3931/e-rara-61067&quot; rel=&quot;nofollow noreferrer&quot;&gt;Commentatio Secunda&lt;/a&gt;, we have some examples:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;article 5&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;cunctos numeros &lt;span class=&quot;math-container&quot;&gt;$1, 2, 3 \ldots p-1$&lt;/span&gt; inter quatuor &lt;strong&gt;series&lt;/strong&gt; A , B , C , D ita distribui, vt ...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;all the numbers &lt;span class=&quot;math-container&quot;&gt;$1, 2, 3 \ldots p-1$&lt;/span&gt; divide themselves into the four series A,&#xA;  B, C, D, such that ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;here A, B, C and D are (obviously finite) subsets of &lt;span class=&quot;math-container&quot;&gt;$[1,p-1]$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;article 8&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Sit &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; radix primitiva pro modulo &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, i. e. numerus talis, vt in &lt;strong&gt;serie&lt;/strong&gt; potestatum &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$gg$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$g^3$&lt;/span&gt; ... nulla ante hanc &lt;span class=&quot;math-container&quot;&gt;$g^{p-1}$&lt;/span&gt; vnitati secundum modulum &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; congrua evadat.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Let g be a primitive root for the modulus &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, i.e. a number of such a type that in the series of powers &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$gg$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$g^3$&lt;/span&gt; ... no power before &lt;span class=&quot;math-container&quot;&gt;$g^{p-1}$&lt;/span&gt; will be congruent to unity relative to modulus &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;here series means the (finite) sequence of the power of &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$g^{p-1}$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;article 37&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Quum singulae normae sint vel numeri primi reales (e &lt;strong&gt;serie&lt;/strong&gt; 2, 5, 13, 17 etc.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Since the individual norms are either real prime numbers (from the series 2, 5, 13, 17, ...)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;here series is the infinite sequence of prime numbers with a prescribed form&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;article 73&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Disponamus partes ipsius &lt;span class=&quot;math-container&quot;&gt;$\varphi(2b,a-b)$&lt;/span&gt; sequenti modo&#xA;  &lt;span class=&quot;math-container&quot;&gt;\begin{equation*}&#xA;\begin{split}&#xA;&amp;amp;[\frac{a-b}{2b}]+[\frac{3(a-b)}{2b}]+[\frac{5(a-b)}{2b}]+ecc+[\frac{(b-1)(a-b)}{2b}]\\&#xA;+&amp;amp;[\frac{a-b}{b}]+[\frac{2(a-b)}{b}]+[\frac{3(a-b)}{b}]+ecc+[\frac{\frac{1}{2}b(a-b)}{b}].&#xA;\end{split}&#xA;\end{equation*}&lt;/span&gt;&#xA;  &lt;strong&gt;Series&lt;/strong&gt; secunda manifesto fit &lt;span class=&quot;math-container&quot;&gt;$=\varphi(b,a-b)=\varphi(b,a)-1-2-3-ecc-\frac{1}{2}b=\varphi(b,a)-\frac{1}{8}(bb+2b)$&lt;/span&gt; ...&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;If we order the terms &lt;span class=&quot;math-container&quot;&gt;$\varphi(2b,a-b)$&lt;/span&gt; in the following way:&#xA;  &lt;span class=&quot;math-container&quot;&gt;\begin{equation*}&#xA;\begin{split}&#xA;&amp;amp;[\frac{a-b}{2b}]+[\frac{3(a-b)}{2b}]+[\frac{5(a-b)}{2b}]+ecc+[\frac{(b-1)(a-b)}{2b}]\\&#xA;+&amp;amp;[\frac{a-b}{b}]+[\frac{2(a-b)}{b}]+[\frac{3(a-b)}{b}]+ecc+[\frac{\frac{1}{2}b(a-b)}{b}].&#xA;\end{split}&#xA;\end{equation*}&lt;/span&gt;&#xA;  the second series will evidently be equal to &lt;span class=&quot;math-container&quot;&gt;$=\varphi(b,a-b)=\varphi(b,a)-1-2-3-ecc-\frac{1}{2}b=\varphi(b,a)-\frac{1}{8}(bb+2b)$&lt;/span&gt; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;here series is a finite sum (whose result Gauss calls &quot;summatio&quot;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In &lt;a href=&quot;http://resolver.sub.uni-goettingen.de/purl?PPN35283028X_0001_2NS&quot; rel=&quot;nofollow noreferrer&quot;&gt;Summatio quarundam serierum singularium&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;article 6&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Duas iam &lt;strong&gt;progressiones&lt;/strong&gt; considerabimus [...] Progressio prima haec est&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;1-\frac{1-x^{m}}{1-x}+\frac{(1-x^{m})(1-x^{m-1})}{(1-x)(1-xx)}- \frac{(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}+ecc&#xA;$$&lt;/span&gt;&#xA;  sive&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;1-(m,1)+(m,2)-(m,3)+(m,4)-ecc&#xA;$$&lt;/span&gt;&#xA;  [...] obvium est quoties, &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; fit numerus integer positivus hanc &lt;strong&gt;seriem&lt;/strong&gt; post terminum suum &lt;span class=&quot;math-container&quot;&gt;$m+1^{\rm tum}$&lt;/span&gt; abrumpi [...] adeoque [...] summam esse debere functionem finitam integram ipsius &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Now we will consider two &quot;progressions&quot; [...] The first one is&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;1-\frac{1-x^{m}}{1-x}+\frac{(1-x^{m})(1-x^{m-1})}{(1-x)(1-xx)}- \frac{(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}+ecc&#xA;$$&lt;/span&gt;&#xA;  or&#xA;  &lt;span class=&quot;math-container&quot;&gt;$$&#xA;1-(m,1)+(m,2)-(m,3)+(m,4)-ecc&#xA;$$&lt;/span&gt;&#xA;  [...] it is clear that if &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is a positive integer, then this series end after the &lt;span class=&quot;math-container&quot;&gt;$m+1^{\rm th}$&lt;/span&gt; term [...] and so [...] the sum must be a polynomial in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;here series (or progression) is a general sum, finite of infinite&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;article 8&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;sed propter rei singularitatem etiam de casibus iis ubi &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; vel fractus vel negativus est pauca adiecisse haud poenitebit. Manifesto tunc &lt;strong&gt;series&lt;/strong&gt; nostra haud amplius abrumpetur, sed in infinitum excurret, facileque insuper perspicitur, divergentem eam fieri, quoties ipsi &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; valor minor quam 1 tribuatur, quapropter ipsius summatio ad valores ipsius &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; qui sin maiores quam 1 restrinci debebit&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;However, for the particularity of this argument will not be unpleasant if we add a few words also about the case in which &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is fractional or negative. In these cases, evidently, our series is not interrupted, but extends to infinity, and moreover it is easily seen that it is divergent every time we assign to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; a value less than 1, which is why we have to limit the sum to the values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; that are greater than 1.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;finally here Gauss uses the term &lt;em&gt;series&lt;/em&gt; in its &quot;modern&quot; meaning of infinite sum (either convergent or divergent).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also consider that (in latin, of course) &quot;time series&quot; means a sequence of items ordered in time, so it is used in statistics with its original meaning: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Virgilium vidi tantum: nec avara Tibullo / tempus amicitiae fata dedere meae. / successor fuit hic tibi, Galle, Propertius illi; / quartus ab his &lt;strong&gt;serie temporis&lt;/strong&gt; ipse fui. (Ovid, &lt;em&gt;Tristia&lt;/em&gt;, 4, X, 51-54)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Vergil I only saw, and to Tibullus greedy fate gave no time for friendship with me. Tibullus was thy successor, Gallus, and Propertius his; after them came I, fourth &lt;strong&gt;in order of time&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-05-19T14:29:58.557" LastActivityDate="2020-05-19T14:29:58.557" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11819" PostTypeId="2" ParentId="9791" CreationDate="2020-05-19T19:23:27.223" Score="4" Body="&lt;p&gt;No, &lt;strong&gt;Romans were not at all bad at computations before Arab numerals were introduced to them&lt;/strong&gt;. In fact, Romans had a perfectly fine way of doing computations that was every bit as good as Arabic numerals. They used a table and stones to do computations, similar to an &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Roman_abacus&quot; rel=&quot;nofollow noreferrer&quot;&gt;abacus&lt;/a&gt;. Roman numerals were just used to record numbers, not to do long computations. Though shorter computations were, in fact, easier in the Roman system than in the Arab system. This is explained &lt;a href=&quot;https://www.washingtoncitypaper.com/columns/straight-dope/article/20854121/how-did-anyone-do-math-in-roman-numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and is a fascinating read.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Pope Sylvester II (946-1003) introduced Arabic numerals to Europe but &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Pope_Sylvester_II#Abbacus_and_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;without 0&lt;/a&gt;. Without 0, Arabic numerals can't compete with an abacus for computation. Thus, they didn't spread for another 200 years, until Fibonacci successfully introduced them through his book &lt;em&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Liber_Abaci&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liber Abaci&lt;/a&gt;&lt;/em&gt; and merchants and others started learning how to use them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Even so, it took a couple of hundred years more for Europeans to make the conversion. Computations aren't easier with Arabic numerals than with an abacus, and using them for computation requires paper for those computations, so there was a good reason not to use them. One minor advantage is that when you write down the computations, there's a record you can check. It was of this reason that Arab numerals were adopted and not because the Roman system was inefficient.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt; Note: Everywhere here I have used the term &quot;Arab&quot; numerals or system, but ideally it should be the &quot;Hindu-Arabic&quot; numerals or system. I used this terminology just because in the question it was referred to as Arab. &lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2020-05-19T19:23:27.223" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11821" PostTypeId="1" CreationDate="2020-05-20T00:39:33.560" Score="3" ViewCount="81" Body="&lt;p&gt;How did the light theories of Newton and Young/Fresnel explain the transparency of materials such as glass, or opacity of materials such as paper and wood? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton's theory of light was largely based on particles or corpuscles, but he did use the aether to explain phenomena such as diffraction. Newton might have thought that materials such as glass let the corpuscles pass through them freely and opaque materials just absorb the corpuscles. Then, there were materials such as metallic surfaces which were opaque but reflective, and it could have been thought that these materials neither let the corpuscles pass through them nor absorb them, they simply reflect them.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the wave theory of light, the aether was assumed to permeate all material bodies. If it was present in all materials, then why wouldn't all materials be transparent and light could pass through all the materials? Sound waves don't pass through a solid wall because no medium, i.e. air, is present inside the wall.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Could you please help me with it? &lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="5989" LastEditDate="2020-05-20T06:36:11.657" LastActivityDate="2020-05-20T06:36:11.657" Title="How transparency was explained by corpuscular theory and Young/Fresnel's wave theory" Tags="|physics|optics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11822" PostTypeId="2" ParentId="8292" CreationDate="2020-05-20T18:09:39.680" Score="5" Body="&lt;p&gt;At the inception of the Fields medal it was intended to encourage young mathematicians, hence the age restriction. Nash was famously overlooked for the Fields medal because it was thought that ''he didn't need encouragement'' and was already on his way.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Also why did Poincare not win a Nobel Prize in Physics despite being nominated many times? One reason is the sheer breadth and variety of his achievements which made it very hard to zoom in on one particular achievement that could be used as the basis of being awarded the prize, probably a similar phenomenon happened with von Neumann as the range of his achievements was too much for anyone to even keep up with.  &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Polymaths typically don't win the prizes you might expect as no-one can keep up with their work, let alone single out a few things that can be made into a case for an award.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastActivityDate="2020-05-20T18:09:39.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11823" PostTypeId="2" ParentId="11813" CreationDate="2020-05-21T15:31:01.670" Score="2" Body="&lt;p&gt;The NIST museum confirms in an email that the Troughton scale is still in a case located to the right of the Standards Vault, although no longer in Washington, D.C. but rather on the NIST campus in Gaithersburg, MD. There is an archived photo of it &lt;a href=&quot;https://web.archive.org/web/20160322202936/http://museum.nist.gov/object.asp?ObjID=11&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2022-06-10T16:28:31.027" LastActivityDate="2022-06-10T16:28:31.027" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11825" PostTypeId="2" ParentId="11816" CreationDate="2020-05-22T08:29:59.693" Score="2" Body="&lt;p&gt;Todd and Alexandre, I just want to offer a perspective to your discussion from a German language point of view. I wanted to put it as a comment, but it was getting too long so I am putting it as an answer (recognizing it does not exactly answer your question). But just wanted to offer an additional perspective.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If you translate sheaf-stalk-germ into German, you get as a rather general translation Bündel-Stiel-Keim.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;But if you relate it more strictly to an agricultural setting, the words that are used there are actually Garbe-Halm-Korn.&#xA;(by the way in German mathematics texts, sheaf is commonly translated as Garbe, i.e. really taking it from the agricultural setting)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Now here is the interesting thing, and maybe this helps a bit for your question. If you take these agricultural expressions Garbe-Halm-Korn and translate those into English, you get the following:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Garbe – sheaf, burst of fire&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Halm –culm, blade (of grass), stalk&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Korn – grain, seed, corn; also: grit&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maybe you find something there that connects well with the intuition behind mathematical sheaves? I am not an expert in that field, so cannot really judge, just wanted to offer.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;“burst of fire” certainly does not seem to fit the agricultural setting, like grit, and obviously comes from a different context.&lt;/p&gt;&#xA;" OwnerUserId="12555" LastEditorUserId="12555" LastEditDate="2020-05-22T09:05:52.870" LastActivityDate="2020-05-22T09:05:52.870" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11827" PostTypeId="2" ParentId="3131" CreationDate="2020-05-23T13:40:11.650" Score="1" Body="&lt;p&gt;Matvei Petrovich Bronstein was a Soviet era theoretical physicist who was arrested in the night of August 6th during Stalins Great Terror. His crime was to believe in communism but not in Stalinism. He was aged just thirty and was executed in a Leningrad prison several months later, in February 1938. In fact, at the time of his arrest he already knew he was under observation by Stalins security apparatus. His wife, Lydia Chukovskaya, was finally allowed to see her husbands inquisition file in 1990 after the collapse of the Soviet regime. At the time of his arrest she was told he had been sentenced to a labour camp for ten years without the right of correspondence. In his file she found the arrest warrant issued by the Kiev state security department on the 5th August, it said:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;M. P. Bronstein who is trying to escape arrest should be detained for an active involvement of a Leningrad counter-revolutionary organisation. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;They were friends of Andrei Sakharov, a physicist who was involved in the Soviet nuclear weapons programme and was later known for advocating civil liberties and reforms in the Soviet Union. In fact, he won the Nobel peace prize for his efforts and the Sakharov prize is named in his honour and awarded annually by the European Parliament for people and organisations dedicated to human rights and freedoms. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bronstein counts amongst one of the first pioneers of quantum gravity. Landau had claimed that the Heisenberg uncertainty principle called into question the reality of the electromagnetic field in that it could not be probed with arbitrary precision. Bronstein showed that Landaus analysis was incorrect by showing that it could so be probed by an arbitrarily massive measuring apparatus, but he showed that there was a quantum limit when it came to gravity and this limit became ma icrst when particles approached what is now called the Planck mass. This is the energy scale at which quantum gravity effects are expected to become manifest. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Bronstein of course was not the only physicist to suffer at Stalins hands. According to Gennady Gorelik, his biographer, two other talented physicists were arrested at the same time: Alexander Witt from Moscow and Semen Shubin from Sverdlosk. They were sentenced to five and eight years forced labour respectively. They also both died in the winter of 1938 in the Kolyma. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One imagines these examples can be multiplied many times. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-05-23T13:40:11.650" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11828" PostTypeId="2" ParentId="7100" CreationDate="2020-05-23T14:02:52.367" Score="1" Body="&lt;p&gt;Virtually nobody reads Newton's &lt;em&gt;Principia&lt;/em&gt;, yet virtually all physicists start off with Newton's laws. Likewise with geometry, we learn from Euclid even if his book is not right there before us. Even now. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Take for example non-Euclidean geometry. One might think that this is far from Euclid geometry as one might imagine. Yet, it still conceived in exactly the same framework he proposed. Merely one of the pillars has been altered slightly. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, Euclid popularised the axiomatic method which is used whether or one reasons analytically, synthetically or otherwise. And that method is still going strong even now - consider the description of homology given by the Eilenberg-Steenrod axioms in algebraic topology. This is definitely geometry - but is it analytic or synthetic?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Mathematics, like every other human endeavour has a multiplicity of modes and methods and it proceeds first by one, and then the other, and sometimes both; they mix and intermix, and then separate only then to come together again. One might say the term dialectic or sublation is apposite here.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-05-23T14:02:52.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11829" PostTypeId="1" AcceptedAnswerId="11831" CreationDate="2020-05-25T03:04:10.670" Score="3" ViewCount="618" Body="&lt;p&gt;Since Sir Issac Newton invented a lot of modern mathematics, what mathematics did he already know? Since he was standing on the shoulders of giants which giants was he speaking of?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I presume he knew &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pythagoras's theorem. Trigonometry?&lt;/li&gt;&#xA;&lt;li&gt;Maybe how to solve quadratic equations.&lt;/li&gt;&#xA;&lt;li&gt;Euclidean geometry. &lt;/li&gt;&#xA;&lt;li&gt;Perhaps even the infinitessimal calculus of volumes from Archimedes?&lt;/li&gt;&#xA;&lt;li&gt;Archimedes lever principle to do with forces? &lt;/li&gt;&#xA;&lt;li&gt;Galilean relativity principles?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Any other things?&lt;/p&gt;&#xA;" OwnerUserId="7722" LastEditorUserId="12316" LastEditDate="2020-05-26T11:26:13.057" LastActivityDate="2022-06-11T01:59:45.327" Title="What mathematics did Isaac Newton learn at school?" Tags="|mathematics|geometry|newton|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11830" PostTypeId="1" AcceptedAnswerId="11832" CreationDate="2020-05-25T14:34:51.423" Score="-2" ViewCount="218" Body="&lt;p&gt;I heard that mathematics in France is very hard because it didn't follow the decimal positional system (promoted by Leonardo Fibonacci in Europe), and I searched and found from &lt;a href=&quot;https://en.wikipedia.org/wiki/Positional_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; that:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;After the French Revolution (1789–1799), the new French government promoted the extension of the decimal system. Some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I learned that the system includes vigesimal and sexagesimal, and France ranked 2nd in &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_countries_by_number_of_Fields_Medalists&quot; rel=&quot;nofollow noreferrer&quot;&gt;countries by number of Fields Medalists&lt;/a&gt;. I wonder why only France cannot completely adopt the decimal positional system and how and why does its own system benefit its mathematics?&lt;/p&gt;&#xA;" OwnerUserId="10033" LastEditorUserId="26" LastEditDate="2021-02-20T15:13:42.200" LastActivityDate="2021-02-20T15:13:42.200" Title="Why didn't French people take the decimal positional system?" Tags="|france|" AnswerCount="1" CommentCount="1" ClosedDate="2020-05-26T06:36:44.673" ContentLicense="CC BY-SA 4.0" />
  <row Id="11831" PostTypeId="2" ParentId="11829" CreationDate="2020-05-25T20:37:30.460" Score="11" Body="&lt;p&gt;Newton studied at school and at the university, but he mostly taught himself by reading. (At his secondary school he certainly learned Latin, &#xA;Greek, the Bible and some arithmetic. In the universities, they mostly studied Aristotle at that time,&#xA;which has nothing to do with mathematics). Besides textbooks that existed at that time he mastered Euclid, and then Descartes. He also read Archimedes and Apollonius, and certainly the works of contemporary mathematicians which were published is Transactions of the Royal society, of which he was a member and later the president. There is a comprehensive discussion of his reading in various periods of his life in the book&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Westfall, Never at rest.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It is based on a catalog of his personal library, which survives.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Newton certainly created a lot of modern mathematics, but he was aware of the whole enormous body of mathematics which already existed before him, and discoveries of his contemporaries. The list of all the &quot;giants on whose shoulders he stood&quot; would be too long.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-05-30T23:33:20.347" LastActivityDate="2020-05-30T23:33:20.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11832" PostTypeId="2" ParentId="11830" CreationDate="2020-05-26T04:42:56.730" Score="4" Body="&lt;p&gt;What you heard is not true. The French do indeed use the decimal position system, which you can verify in any French textbook on mathematics. Consequently, mathematics is no easier or harder in France than it is anywhere else.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;You may be confused by the names of numbers in French. For example, 70 is &quot;soixante-dix&quot; (literally translated as &quot;sixty-ten&quot;), 71 is &quot;soixante-onze&quot; (&quot;sixty-eleven&quot;), 80 is &quot;quatre-vingt&quot; (&quot;four-twenty&quot;), 90 is &quot;quatre-vingt-dix&quot; (&quot;four-twenty-ten&quot;), and so on. (As a small aside, I've also heard 90 called &quot;nonante&quot; by Swiss French speakers, so there are dialect variations in the names as well.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But these are just the names of numbers, and have nothing to do with using the decimal  position system or not. The names might have been originally derived from vigesimal or sexagesimal, but French speakers do not think about the etymology when doing mathematics any more or less than they or anyone else thinks about the etymology of any word while using it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The quote from Wikipedia that you cite in your question has more to do with the metric system, which is yet another thing entirely. The metric system is certainly inspired by the decimal position system, but it is a system of measurements and not a system of numerals for representing numbers. For example, the Americans use the decimal  position system, but have not adopted the metric system.&lt;/p&gt;&#xA;" OwnerUserId="12570" LastActivityDate="2020-05-26T04:42:56.730" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11833" PostTypeId="1" CreationDate="2020-05-26T06:52:55.250" Score="0" ViewCount="149" Body="&lt;p&gt;In the movie &quot;&lt;em&gt;The Imitation Game&lt;/em&gt;&quot;, Alan Turing along with his team crack the German encryption machine Enigma but advises his superiors to not act on all decrypted intelligence, as that might lead to the Germans getting suspicious about Enigma being cracked, possibly resulting in a change of its design to thwart further decryption efforts.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;His character specifically says that his team &quot;&lt;em&gt;would develop a system to help determine how much intelligence to act on; which attacks to stop, which to let through, using statistical analysis; find the minimal number of actions it would take to win the war, but the maximal number that can be taken before the Germans get suspicious&lt;/em&gt;&quot;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Is this historically accurate? If so, how was it done? What branch of mathematics/statistics deals with this?&lt;/p&gt;&#xA;" OwnerUserId="12571" LastEditorUserId="26" LastEditDate="2021-02-20T15:13:23.120" LastActivityDate="2021-02-20T15:13:23.120" Title="How was mathematics used in World War II to &quot;act on the right amount of intelligence&quot;?" Tags="|mathematics|statistics|world-war-2|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11834" PostTypeId="1" AcceptedAnswerId="11838" CreationDate="2020-05-26T07:07:20.293" Score="2" ViewCount="364" Body="&lt;p&gt;In linear algebra, we care a lot about dimensions. I get why it’s useful but not why it’s such a big deal. So I wondered what problem was solved historically by introducing a rigorous definition of dimension of a vector space?&lt;/p&gt;&#xA;" OwnerUserId="12572" LastEditorUserId="26" LastEditDate="2021-02-20T15:13:15.090" LastActivityDate="2021-02-20T15:13:15.090" Title="What problem was solved by introducing the dimension of a vector space?" Tags="|mathematics|linear-algebra|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11835" PostTypeId="2" ParentId="8395" CreationDate="2020-05-26T11:07:17.930" Score="1" Body="&lt;p&gt;As it happens, the same question was covered in a submission to SIGBOVIK 2019 dated 13 March 2019 and published as pages 225 to 229 of the &lt;a href=&quot;http://sigbovik.org/2019/proceedings.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;proceedings&lt;/a&gt; -- four methods were examined (the author describes long multiplication as lattice multiplication but worse and thus it was not tested) with the following results:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Lattice multiplication: 11 minutes&lt;/li&gt;&#xA;&lt;li&gt;Double and halve: 21 minutes, gave incorrect answer&lt;/li&gt;&#xA;&lt;li&gt;Quarter-square multiplication: Not tested, but given lookup tables up to 200,000 the author estimates 12 lookups would be required, with each lookup taking 30 seconds to a minute&lt;/li&gt;&#xA;&lt;li&gt;Karatsuba multiplication: 17 minutes, acknowledged as anachronistic&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The tests were performed on paper; the overhead of performing multiplications on chalkboard was not examined.&lt;/p&gt;&#xA;" OwnerUserId="12575" LastActivityDate="2020-05-26T11:07:17.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11836" PostTypeId="2" ParentId="11834" CreationDate="2020-05-26T12:33:13.643" Score="0" Body="&lt;p&gt;I'm not convinced there was any specific &quot;problem,&quot; but simply that a full formal definition allowed people to use multidimensional notation as a shorthand way of defining and managing large amounts of variables.  Sort of like working out the rules for exponentiation, for example. &lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-05-26T12:33:13.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11838" PostTypeId="2" ParentId="11834" CreationDate="2020-05-26T15:39:42.703" Score="4" Body="&lt;p&gt;I will begin by 2 examples where the notion of dimension is essential :&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The increasing sequence of subspaces &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\{0\}\subset\ker(A-\lambda I)\subset\ker(A-\lambda I)^2\subset\ker(A-\lambda I)^3\subset\dots$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(with ultimate stabilisation at a certain step) and the connection of their dimensions with the so important Jordan decomposition of a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; relatively to eigenvalue &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; (well described in this answer by Bernard &lt;a href=&quot;https://math.stackexchange.com/q/2736505&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the concept of degree of field extensions (see paragraph &quot;examples&quot; &lt;a href=&quot;https://en.wikipedia.org/wiki/Field_extension&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) where for example &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q} (\sqrt {2},\sqrt {3})$&lt;/span&gt; can be considered in two different ways as a vector space of dimension &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;. &lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;But, on an historical point of view, the concept of dimension finds its roots in the connection with &lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;the concept of rank, which was initially defined as the maximal size of a sub-determinant one could extract from a determinant ,[in modern terms : replace in the previous sentence : &quot;determinant&quot; by &quot;matrices&quot;]. We were far, when this concept has emerged, from &quot;rank = dimension of the range of the corresponding linear mapping&quot;. This concept of &quot;rank&quot; has emerged (not with this name) when Rouché has published an article in Journal de l'Ecole Polytechnique (1880) explaining the kind of solutions one could expect from a linear system of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; equations with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; unknowns and rank &lt;span class=&quot;math-container&quot;&gt;$r \le \min(n,p)$&lt;/span&gt; .&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;[connected to the previous item] the &quot;rank-nullity&quot; theorem. Mathematicians have been accustomed more and more to identify a subspace either as the &lt;strong&gt;kernel&lt;/strong&gt; or the &lt;strong&gt;range&lt;/strong&gt; of a certain linear mapping. This can be traced back to 1884 where Sylvester has defined the &quot;nullity&quot; of a square matrix&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;Here is a quotation of &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/&lt;/a&gt; :&#xA;Sylvester defined the nullity of &lt;span class=&quot;math-container&quot;&gt;$n(A)$&lt;/span&gt; of matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, to be the largest &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; such that every minor of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; of order &lt;span class=&quot;math-container&quot;&gt;$n-i+1$&lt;/span&gt; is zero. He was interested in invariants of matrices, that is properties which are not changed by certain transformations. He proved that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\max \{n(A), n(B)\} \le n(AB) \le n(A) + n(B)$$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Remark : The concept of dimension/rank can be enlarged&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&quot;The rank of a mathematical object is defined whenever that object is free. In general, the rank of a free object is the cardinal number of the free generating subset G&quot; as explained in &lt;a href=&quot;https://mathworld.wolfram.com/Rank.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathworld.wolfram.com/Rank.html&lt;/a&gt;. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;it is also to be connected to the &quot;number of degrees of freedom&quot;, a somewhat looser concept that can be much more complicated when you are not in the linear case, with &lt;strong&gt;manifolds&lt;/strong&gt; (generalized surfaces) instead of vector spaces.  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;it has been discovered in the 1960s in the framework of numerical analysis that there is a notion of &lt;a href=&quot;https://en.wikipedia.org/wiki/Low-rank_approximation&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;low rank approximation&lt;/em&gt;&lt;/a&gt; expressing that for example a set of points in the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional space is &quot;close to belong to &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-dimensional subspace&quot;, using the concept of &quot;singular value decomposition&quot; (a generalization of eigenvalues/vectors decomposition). These concepts are very fundamental in present-day treatments of the so-called &quot;big-data&quot;.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3730" LastEditorUserId="3730" LastEditDate="2020-06-05T08:40:04.207" LastActivityDate="2020-06-05T08:40:04.207" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11839" PostTypeId="1" AcceptedAnswerId="11840" CreationDate="2020-05-26T16:35:40.030" Score="3" ViewCount="4093" Body="&lt;p&gt;Einstein's attribution of 'spooky action at a distance' to the Copenhagen interpretation of quantum mechanics is obviously in wide use as a quote. However, I've been googling the issue for a while and searched a relevant book on the history behind quantum mechanics, but haven't found the source of this quote so far. In some thread, people pointed to the Einstein-Bohr correspondence, but without giving the specific instance.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The source of the supposedly original German quotation &quot;spukhafte Fernwirkung&quot; would actually be appreciated even more.&lt;/p&gt;&#xA;" OwnerUserId="12583" OwnerDisplayName="Markus Gratis" LastEditorUserId="26" LastEditDate="2021-02-20T15:13:06.097" LastActivityDate="2021-02-20T15:13:06.097" Title="What is the source of Einstein's quote &quot;spooky action at a distance&quot;?" Tags="|einstein|" AnswerCount="1" CommentCount="0" ClosedDate="2020-05-27T09:51:48.053" ContentLicense="CC BY-SA 4.0" />
  <row Id="11840" PostTypeId="2" ParentId="11839" CreationDate="2020-05-26T19:25:18.537" Score="4" Body="&lt;p&gt;Paragraph from &lt;a href=&quot;https://www.troubador.co.uk/bookshop/computing-science-education/light-after-dark-ii/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Large and the Small&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;In a letter to Max Born in 1947 Einstein said of the statistical approach to&#xA;  quantum mechanics, which he attributed to Born, &lt;em&gt;“I cannot seriously believe in it&#xA;  because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky action at a distance”&lt;/em&gt; (the actual phrase used by Einstein was German, &lt;em&gt;“spukhafte Fernwirkung”&lt;/em&gt;. Spooky, or ghostly, is a reasonable translation, although &lt;em&gt;“spooky”&lt;/em&gt; was not in common usage in English in 1947).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The correspondence has been published in &lt;em&gt;The Born-Einstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 1916–1955, with commentaries by Max Born&lt;/em&gt;. Macmillan. 1971. p. 158.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein's problem stems from the assumption of &lt;em&gt;a reality in time and space&lt;/em&gt;. Although it is implicit in the assumptions of his paper on special relativity that spacetime consists only of relationships found in measurement, he was never able to fully embrace the implications of his argument, that there is no spacetime independent of measurement. Had he done so, he would have been able to recognise that the statistical approach, formalised by von Neumann, is indeed the correct way to understand quantum mechanics.&lt;/p&gt;&#xA;" OwnerUserId="12581" LastActivityDate="2020-05-26T19:25:18.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11841" PostTypeId="1" CreationDate="2020-05-26T19:26:27.780" Score="1" ViewCount="99" Body="&lt;p&gt;What is the name of this identity? Who discovered this identity? What is the history behind this? I have looked up on Wikipedia with little  documentation of the identity&lt;/p&gt;&#xA;&#xA;&lt;p&gt;see finite product of trigfunctions here:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulae&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulae&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Cross post : &lt;a href=&quot;https://math.stackexchange.com/questions/3692240/history-of-sinnx-2n-1-prod-0n-1-sin-leftx-frac-pi-kn-rig?noredirect=1#comment7587656_3692240&quot;&gt;https://math.stackexchange.com/questions/3692240/history-of-sinnx-2n-1-prod-0n-1-sin-leftx-frac-pi-kn-rig?noredirect=1#comment7587656_3692240&lt;/a&gt;&#xA;tl;dr: looking for the history behind this identity&lt;/p&gt;&#xA;" OwnerDisplayName="user12582" LastActivityDate="2020-05-28T19:05:44.150" Title="History of $\sin(nx) = 2^{n-1} \prod_{0}^{n-1} \sin\left(x + \frac{\pi k}{n}\right)$" Tags="|mathematics|" AnswerCount="0" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11842" PostTypeId="1" CreationDate="2020-05-28T11:29:29.993" Score="4" ViewCount="66" Body="&lt;p&gt;I'm working on an analysis of the use of the term &quot;dogma/dogmatism&quot; in 19th-century scientific periodicals, two of which are &lt;em&gt;Nature&lt;/em&gt; (1869-) and &lt;em&gt;Science&lt;/em&gt; (1880-). Basically, I search every issue for the term to then understand how often and in which context it was used.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I already found several online archives which provide (partially) access to the early issues: HathiTrust, Archive.org and several smaller or more specialised websites - apart from the journals' official archives. &#xA;However, analysing every issue through those websites is nearly impossible. The official archives are only paid-to-use, while my usual databases (f.ex. HathiTrust) only have a very limited number of issues and archive.org or digicoll.library.wisc.edu have very poor search engines which a) are not very reliable and b) take a very long time to browse through one issue.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I wanted to ask &lt;strong&gt;if somebody has experience with those journals and can tell me if there is a better way to finish this task?&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I am aware that I would probably be able to search them by wading through all those fragments of sources (or pay a lot of money to enter the official archives), which I will do if there is no other possibility. But I hope somebody can help me finding a smarter option.&lt;/p&gt;&#xA;" OwnerUserId="12587" LastActivityDate="2020-05-28T11:29:29.993" Title="Is there a way to search through the early journal issues of &quot;Nature&quot; and &quot;Science&quot; and read the issues online?" Tags="|publishing|resource-recommendation|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11843" PostTypeId="1" CreationDate="2020-05-28T15:38:23.710" Score="2" ViewCount="628" Body="&lt;p&gt;I remember my brother telling me an anecdote about Carl F. Gauss three-four years ago, I want to know if anybody here also know about it, or can provide the source of it. The anecdote goes like this&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One day Carl came from school, he was probably a kid of 9 or 10 years, and smilingly handed a letter to his mother and said&lt;/p&gt;&#xA;&lt;p&gt;“Mother, my teacher has given me this letter and told me to hand over it to you without reading it”.&lt;/p&gt;&#xA;&lt;p&gt;His mother opened the letter and read it (in her mind, not loud out), after reading it she remain quiet for few minutes. Gauss asked her many times about what was written in the letter, finally she replied&lt;/p&gt;&#xA;&lt;p&gt;“Your teacher has written that you have far more ability than what the school can offer and develop, so your teacher suggested you to leave the school and work on your own great mind, develop it by your own as no one is able enough to teach you my child”&lt;/p&gt;&#xA;&lt;p&gt;After that, Gauss became an autodidact, motivated by the letter and did many charismatic works in Mathematics. Things went on good, Mathematical societies were surprised by his discoveries. After many years when his mother died, he became sad and tried to go  for her things, suddenly in her old box he found that same letter and read it, something like this was written in it&lt;/p&gt;&#xA;&lt;p&gt;“We are sorry Mrs. Gauss but your son Carl is a very stupid kid and we cannot keep him in our school anymore.”&lt;/p&gt;&#xA;&lt;p&gt;Gauss murmured “It’s my mother who has made me what I am, It’s only because of my mother that I have reached here in my life&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Well, the thing is that wikipedia asserts that Carl was a prodigy but being a prodigy doesn't at all means that one would perform outstandingly in school, he might be disagreeing with his teachers and that could have made them angry to write that letter.&lt;/p&gt;&#xA;&lt;p&gt;Actually, I request you all to tell me if you have every heard this kind of anecdotes about any other mathematicians, as there are chances that my brother could be mistaken, but I'm sure he didn't make up that story, he would have read it somewhere. So, this kind of story is important as it shows up the effect of encouragement in developing someone's life, especially an intellectual life.&lt;/p&gt;&#xA;&lt;p&gt;Please help me in finding the source of this story or at least let me know if you ever came to hear about an anecdote like this.&lt;/p&gt;&#xA;" OwnerUserId="9837" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-06-04T14:22:08.733" Title="Do you know about this anecdote or its source where the mother reads out the letter to Gauss and made him Gauss, the mathematician?" Tags="|mathematicians|biographical-details|debunking|gauss|depiction-of-scientists|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11844" PostTypeId="2" ParentId="11843" CreationDate="2020-05-28T17:24:36.200" Score="5" Body="&lt;p&gt;This story has often been associated with Thomas Edison but is, in fact, a made-up anecdote. The origin of this story is because of the fact that Edison spent only a few months in a formal classroom before being taken out of school and educated by his mother, who was a teacher by training.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;According to a &lt;a href=&quot;https://www.loc.gov/collections/edison-company-motion-pictures-and-sound-recordings/articles-and-essays/biography/life-of-thomas-alva-edison/&quot; rel=&quot;noreferrer&quot;&gt;U.S. Library of Congress biography of Edison&lt;/a&gt;, he was indeed labelled “addled” by a school administrator, an event that led his mother to remove him from school. Still, this information was not conveyed to her in a letter, nor was it hidden by her from young Thomas:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Edison was a poor student. When a schoolmaster called Edison “addled,” his furious mother took him out of the school and proceeded to teach him at home. Edison said many years later, “My mother was the making of me. She was so true, so sure of me, and I felt I had someone to live for, someone I must not disappoint.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2020-05-28T17:24:36.200" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11847" PostTypeId="2" ParentId="11843" CreationDate="2020-05-29T16:25:30.427" Score="6" Body="&lt;p&gt;The following is taken from the book&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Walter K. Buhler, &quot;Gauss: a biographical study.&quot; Springer Verlag, 1981. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Only a very few characteristic and interesting facts are known from the&#xA;  childhood and youth of Gauss. Basically, we have to satisfy ourselves with&#xA;  the bare data of his biography and that kind of information which can be&#xA;  induced from a general familiarity with the social and political situation of&#xA;  the time. Our only major specific source is Gauss himself; the stories of his&#xA;  childhood that he liked to tell as an old man were preserved and &#xA;  transmitted by his students and friends and are now part of our traditional&#xA;  picture of Gauss. Many of these anecdotes cannot be substantiated, nor are&#xA;  they of serious interest.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The story told by your brother appears to be one of these anecdotes and its essence is inconsistent with the known facts (see below). &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Neither of Gauss's parents had much education: his father, as we can&#xA;  gather from his jobs, at least could read and write; he also knew elementary&#xA;  arithmetic. His mother probably could read, but not write.* Gauss does not&#xA;  seem to have been close to his father, and we know from later remarks that&#xA;  he traced his genius back to his mother. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, we do not even know for fact that Gauss mother could read. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The factual information that is available to us starts with the year 1784&#xA;  when the young Carl Friedrich entered elementary school. That he actually&#xA;  went to school was not unusual at the time, for children who grew up in a&#xA;  major city usually had this opportunity. He was unusually lucky in another&#xA;  way, for his teacher, Buttner, seems to have been competent and concerned.&#xA;  He took a personal interest in the boy, trying to help and encourage him.&#xA;  In retrospect, it is easy to see how the bright youngster must have excelled&#xA;  among the other students. &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;...Gauss reportedly knew how to read and write&#xA;  before entering school, skills he appears to have picked up without any help&#xA;  from his parents. When he was barely three years old he could count and&#xA;  perform elementary calculations. Buttner was impressed by the boy; in 1786&#xA;  he obtained from Hamburg a special arithmetic text for his exceptional&#xA;  student—there was nothing in the standard primers which would have been&#xA;  new to him. Biittner's assistant during these years was Martin Bartels (1769—&#xA;  1836), later professor of mathematics at the University of Kazan and only&#xA;  eight years older than Gauss. He soon recognized Gauss's genius and&#xA;  devoted special attention to the young Carl Friedrich. We do not know&#xA;  how the parents encouraged their prodigious son, if they did so at all; the&#xA;  times, against a background of need and poverty, were utilitarian and not&#xA;  conducive to appreciating the advantages of good schooling and academic&#xA;  success. The parents' benevolent but disinterested astonishment at their son's&#xA;  achievements was most certainly not coupled with any expectation of an&#xA;  extraordinary career. In their narrow and limited world, after all, there were&#xA;  more important and promising gifts for a son of a dependent laborer to have&#xA;  than a curious facility in counting and arithmetic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Of course, it is your choice whom to believe, but if you are interested in knowing facts, I suggest you ask your brother for the source of his story. As for the sentiment expressed in your comment that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&quot;Something like that must have happened that’s why we got some story like this, else it is not possible to totally make up a story like that.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;it strikes me a counter-factual. (Some) people make up stories all the time with no basis whatsoever, or misremember or/and misattribute stories that have heard. It reminds me one of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Radio_Yerevan_jokes&quot; rel=&quot;noreferrer&quot;&gt;Radio Yerevan jokes&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Radio Yerevan was asked: &quot;Is it correct that Grigori Grigorievich Grigoriev won a luxury car at the All-Union Championship in Moscow?&quot;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Radio Yerevan answered: &quot;In principle, yes. But first of all it was not Grigori Grigorievich Grigoriev, but Vassili Vassilievich Vassiliev; second, it was not at the All-Union Championship in Moscow, but at a Collective Farm Sports Festival in Smolensk; third, it was not a car, but a bicycle; and fourth he didn't win it, but rather it was stolen from him.&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2020-05-29T16:25:30.427" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11848" PostTypeId="1" AcceptedAnswerId="11849" CreationDate="2020-05-29T16:53:14.593" Score="0" ViewCount="643" Body="&lt;p&gt;According to the laws of nature progression is always downward. One of these laws is the Second Law of Thermodynamics which says that isolated systems cannot increase in complexity. For example, a wearing suit will eventually decompose. Thus Duane Gish said ”This [evolution] is a direct violation of the Second Law of Thermodynamics.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Additionally, some say that Albert Einstein, the greatest physicist who ever lived, felt that evolution is a breach of natural laws (laws of conversation of energy and mass) since progression is always downward.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;How can we reconcile evolution with natural laws and did Einstein reject evolution?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Sources, please&lt;/p&gt;&#xA;" OwnerUserId="12598" LastEditorUserId="12598" LastEditDate="2020-05-29T17:00:40.127" LastActivityDate="2020-05-29T21:58:24.633" Title="Did Einstein oppose evolution?" Tags="|thermodynamics|einstein|evolution|" AnswerCount="1" CommentCount="1" ClosedDate="2020-05-31T23:45:08.533" ContentLicense="CC BY-SA 4.0" />
  <row Id="11849" PostTypeId="2" ParentId="11848" CreationDate="2020-05-29T20:46:47.310" Score="6" Body="&lt;ol&gt;&#xA;&lt;li&gt;First of all, I did not study in any detail what Einstein's views on evolution were; the following is just a trivial observation:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Our sun is constantly pumping energy into the ecosystem of the Earth. Most of the energy that the life forms on Earth are getting, can be traced to the sun (plus, the heat from the Earth mantle). All past, present and (presumably) future life on Earth depends on this energy source which obviously has to be taken into account when applying the 2nd Law of Thermodynamics to evolution. One can, of course, consider a larger system, namely the solar system. It is still not isolated, but let's ignore this for a moment. The increase in complexity on Earth due to evolution has to be balanced against the decrease in complexity of the Sun and the Earth core due to nuclear processes (which are behind the energy that we are getting). Once these two are exhausted (about &lt;a href=&quot;https://en.wikipedia.org/wiki/Future_of_Earth&quot; rel=&quot;noreferrer&quot;&gt;5-7 billions of years from now&lt;/a&gt;), that will be the end of life and, hence, evolution, on Earth. (I am simplifying quite a bit here.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Einstein, of course, understood all this and would not have objected to a/the evolutionary theory on the basis of thermodynamics. &lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Here are two Einstein quotes regarding evolution taken from:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;*What about Darwin? All Species of Opinion from Scientists, Sages, Friends, and Enemies Who Met, Read, and Discussed the Naturalist Who Changed the World&quot;, by Thomas Glick, The Johns Hopkins University Press, 2010. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;In my opinion, a great era of atomic science cannot be assured by organizing science in the way large corporations are organized. One can organize the application of a discovery already made, but one cannot organize the discovery itself. Only a free individual can make a discovery. However, there can be a kind of organization wherein the scientist is assured freedom and proper conditions of work. Professors of science in American universities, for instance, should be relieved of some of their teaching so as to have more time for research. Can you imagine an organization of scientists making the discoveries of Charles Darwin?&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Albert Einstein, “On the Atomic Bomb,” Atlantic Monthly, November 1945, 43–45, reprinted in Einstein on Politics, ed. David E. Rowe and Robert Schulmann (Princeton, NJ: Princeton University Press, 2007), p. 376.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;Darwin’s theory of the struggle for existence and the selectivity connected with it has by many people been cited as authorization of the spirit of competition. Some people also in such a way have tried to prove pseudo-scientifically the necessity of the destructive economic struggle of competition between individuals. But this is wrong, because man owes his strength in the struggle for existence to the fact that he is a socially living animal. As little as a battle between single ants of a hill is essential for survival, just so little is this the case with the individual members of a human community.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;em&gt;Albert Einstein, Out of My Later Years (New York: Citadel, 1995), p. 34.&lt;/em&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Lastly, from &lt;a href=&quot;https://ncse.ngo/albert-einsteins-voice-evolution&quot; rel=&quot;noreferrer&quot;&gt;Albert Einstein’s Voice for Evolution&lt;/a&gt;, by Glenn Branch:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Over the course of his life, Einstein wrote a lot and spoke a lot and moreover was widely consulted for comment on the issues of his day, so I hesitate to offer any definitive pronouncement, but it looks as though he never wrote or spoke extensively about evolution or evolution education. There are a few suggestive snippets here and there, however. For example, in 1939, speaking at Princeton Theological Seminary, Einstein famously decried conflicts arising “when a religious community insists on the absolute truthfulness of all statements recorded in the Bible.” The result of such an insistence, he explained, is “an intervention on the part of religion into the sphere of science; this is where the struggle of the Church against the doctrines of Galileo and Darwin belongs.” It’s hard not to believe that the Scopes trial was in his thoughts then. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2020-05-29T21:58:24.633" LastActivityDate="2020-05-29T21:58:24.633" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11850" PostTypeId="2" ParentId="480" CreationDate="2020-06-01T09:32:04.637" Score="2" Body="&lt;p&gt;We can find the first peer review procedures as early as the mid-19th century when the importance of scientific journals increased. As Aileen Fyfe and Noah Moxham could show in their analyses of the &lt;em&gt;Philosophical Transactions&lt;/em&gt; of the Royal Society (2019) and Alex Ciscar in his book on scientific journals (2018), first referee-reports can be found as early as 1832. &#xA;In 1865, &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;refereeing was a well-established practice at the Royal Society and some similar societies, but it was rarely used by nonsociety journals or any journal where speed of publication was prioritized. [Fyfe, Moxham:2019]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;In 1856, Adam Sedgwick complained to the board of the Geological Society that the board had rejected his paper based on the anonymous (and seemingly established) review of the Council, which he then called &quot;dogmatic&quot;. The president defends the procedure in the editorial of the journal, since they had, in fact, not rejected the paper but asked to alter/leave out a small passage. (Quarterly Journal of the Geological Society of London, Vol. 12 (1856), p. XVLI)&#xA;By then, Sedgwick had already published the paper elsewhere, causing some uproar which shows us, that this practise, too, was known and frowned upon at that time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The anonymous peer-review is also closely related to the tradition of anonymous reviews in general, which should give the reviewer's words more weight becoming the voice of the scientific community or the journal etc. Anonymity should also guarantee security for the reviewer and avoid open discussions doubting the decisions. &#xA;In Britain, the importance of anonymous reviews increased in the 19th century, due to the fact that open scientific debates were restricted (Barton 2016). This fact, closely related to the traditions of learned societies such as the Royal Society, to not pass much information about intern procedures in general, make the anonymous peer-review a logical step.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have not come across open reviews in the long 19th century so far, though, and would be very surprised, if so, by the aforesaid reasons.&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Fyfe, A. &amp;amp; Moxham, N. (2019), &lt;a href=&quot;https://journals.sagepub.com/doi/10.1177/0162243919862868&quot; rel=&quot;nofollow noreferrer&quot;&gt;Managing the Growth of Peer review at the Royal Society Journals&lt;/a&gt;, 1865-1965.&lt;/li&gt;&#xA;&lt;li&gt;Csiszar, A. (2018), The Scientific Journal. Authorship and the Politics of Knowledge in the Nineteenth Century. The University of Chicago Press.&lt;/li&gt;&#xA;&lt;li&gt;Barton, R. (2018), The X Club. Power and Authority in Victorian Science. The University of Chicago Press.&lt;/li&gt;&#xA;&lt;li&gt;Fyfe, A. (2020, forthcoming), Editors, Referees and Committees. Distributing editorial work at the Royal Society journals in the late nineteenth and twentieth centuries. In: Centaurus.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12587" LastActivityDate="2020-06-01T09:32:04.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11851" PostTypeId="1" CreationDate="2020-06-01T18:08:57.960" Score="0" ViewCount="95" Body="&lt;p&gt;Many creationists use the 2nd law of Thermodynamics &quot;Progression is always downward, a law,&quot; to reject evolution. When did this misunderstanding of Thermodynamics start and how to refute it? &lt;/p&gt;&#xA;" OwnerUserId="12598" LastActivityDate="2020-06-01T18:40:33.803" Title="When did they say 'progression is downward?'" Tags="|thermodynamics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11852" PostTypeId="2" ParentId="11851" CreationDate="2020-06-01T18:40:33.803" Score="3" Body="&lt;p&gt;The general idea started probably around the 18th century, but some illustrious names in the 19th century like Sir William Thompson talked of the &quot;heat death&quot; of the universe, so originally it appears physicists did understand thermodynamics as a &quot;downward progression&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the same time, there has always been a competing view that energy is neither created nor destroyed, and that too has a long history of illustrious support.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The problem in the 19th century is that it wasn't particularly obvious how things like burned coal or exhausted stars were created or recycled. Even today these things are known to occur only on scales far greater than human civilisation, but the picture that emerges is less of &quot;downward progression&quot; but of energy merely moving and transforming around the universe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Such cosmological questions are not yet settled in that they are not fully understood or modelled, but no physicist nowadays would assert in their scientific capacity that &quot;downward progression&quot; was a &quot;law&quot; - merely an opinion or a hypothesis held by some.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Therefore the 2nd law is nowadays stated in terms that both schools can agree with, that there is &lt;em&gt;no upward progression&lt;/em&gt; (leaving it unstated whether there is an overall steady state or whether there is &quot;downward progression&quot;).&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2020-06-01T18:40:33.803" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11854" PostTypeId="2" ParentId="11834" CreationDate="2020-06-01T22:07:27.603" Score="1" Body="&lt;p&gt;The idea of dimension was already intuitively understood. For example, the ancient Greeks distinguished between plane and solid geometry. And early modern physicists used the notion of degrees of freedom. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;It was Liebniz who called for an algebra that would treat the figure as directly as it treated quantity which he called &lt;em&gt;analysis situs&lt;/em&gt;. In fact, it was on the basis of his theorising that a competition was instituted which was won by Hermann Grassmann. In fact, if I recall correctly it was Mobius that encouraged Grassmann to enter it. This was on the basis of his book, &lt;em&gt;The Theory of Linear Extension: A New Theory of Mathematics&lt;/em&gt;, and what now is known as linear algebra, and more precisely as the exterior or Grassmann algebra. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In this theory the intuitive understanding of dimension is given a rigorous foundation. This is simply the cardinality of any basis. &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;So I was wondered what problem was solved historically by introducing a rigorous definition of dimension. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;The historical problem was the one posed by Liebniz: to give a rigorous exposition of an algebra that treats extension directly. This is the theory of vectors and tensors. An outcome of this theory was a rigorous definition of dimension which showed that this theory was on the right track. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I believe Grassmann actually credited his father with the crucial notion that helped him theorise about extension synthetically. &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-06-05T04:45:39.827" LastActivityDate="2020-06-05T04:45:39.827" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11856" PostTypeId="2" ParentId="771" CreationDate="2020-06-02T17:41:43.417" Score="1" Body="&lt;p&gt;We don't know how Archimedes compute square root of 3 not because there is no known method, but because there are several methods that Archimedes could use, and we don't know which one he actually used.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-06-02T17:41:43.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11857" PostTypeId="5" CreationDate="2020-06-03T07:21:14.810" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-03T07:21:14.810" LastActivityDate="2020-06-03T07:21:14.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11858" PostTypeId="4" CreationDate="2020-06-03T07:21:14.810" Score="0" Body="Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-04T07:08:10.010" LastActivityDate="2020-06-04T07:08:10.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11860" PostTypeId="1" AcceptedAnswerId="11861" CreationDate="2020-06-04T07:22:29.383" Score="3" ViewCount="126" Body="&lt;p&gt;&lt;em&gt;Coulomb's Law&lt;/em&gt; states that :&#xA;&lt;span class=&quot;math-container&quot;&gt;$$F_e = K_e\dfrac{q_1q_2}{r^2}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$q_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q_2$&lt;/span&gt; are magnitudes of the two point charges, &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the distance between them and &lt;span class=&quot;math-container&quot;&gt;$K_e$&lt;/span&gt; is &#xA;&lt;em&gt;Coulomb's Constant&lt;/em&gt; (aka &lt;em&gt;Dielectric Constant&lt;/em&gt;). In my Physics textbook, it's mentioned that &lt;span class=&quot;math-container&quot;&gt;$K_e$&lt;/span&gt; can also be written as &lt;span class=&quot;math-container&quot;&gt;$\dfrac{1}{4\pi\varepsilon_m}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\varepsilon_m$&lt;/span&gt; is ther permittivity of a medium &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; but no details about its establishing or derivation are given.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So, I'd like to know who established this (I think it was Gauss, though), thanks!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;PS : If it was Gauss, was it established by using Gauss's Law?&lt;/p&gt;&#xA;" OwnerUserId="12601" LastEditorUserId="10106" LastEditDate="2020-06-04T08:25:57.367" LastActivityDate="2020-06-04T13:18:52.597" Title="Who established that $K_e = \frac{1}{4 \pi \varepsilon _m}$?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11861" PostTypeId="2" ParentId="11860" CreationDate="2020-06-04T10:48:08.480" Score="4" Body="&lt;p&gt;It is not that a particular person established this relation. This relation is a result of the historical development of the idea of assigning units or rather dimensionality to charge &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;. According to the Wikipedia article on &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Vacuum_permittivity#Historical_origin_of_the_parameter_%CE%B50&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vacuum permittivity&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The experiments of Coulomb and others showed that the force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; between&#xA;  two equal point-like &quot;amounts&quot; of electricity, situated a distance &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;&#xA;  apart in free space, should be given by a formula that has the form&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$F=k_{\text{e}}{\frac {Q^{2}}{r^{2}}}$&lt;/span&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is a quantity that represents the amount of electricity&#xA;  present at each of the two points, and &lt;span class=&quot;math-container&quot;&gt;$k_e$&lt;/span&gt; is the Coulomb constant. If&#xA;  one is starting with no constraints, then the value of &lt;span class=&quot;math-container&quot;&gt;$k_{\text{e}}$&lt;/span&gt; may be&#xA;  chosen arbitrarily. For each different choice of &lt;span class=&quot;math-container&quot;&gt;$k_e$&lt;/span&gt; there is a&#xA;  different &quot;interpretation&quot; of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;: to avoid confusion, each different&#xA;  &quot;interpretation&quot; has to be allocated a distinctive name and symbol.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In one of the systems of equations and units agreed in the late 19th&#xA;  century, called the &quot;centimetre–gram–second electrostatic system of&#xA;  units&quot; (the cgs esu system), the constant &lt;span class=&quot;math-container&quot;&gt;$k_e$&lt;/span&gt; was taken equal to 1, and&#xA;  a quantity now called &quot;gaussian electric charge&quot; &lt;span class=&quot;math-container&quot;&gt;$q_s$&lt;/span&gt; was defined by the&#xA;  resulting equation&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$F={\frac {{q_{\text{s}}}^{2}}{r^{2}}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Thus, charge was expressed in esu, a derived CGS unit with dimensions as&#xA;&lt;span class=&quot;math-container&quot;&gt;$[M]^{1/2}[L]^{3/2}[T]^{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The idea subsequently developed that it would be better, in situations&#xA;  of spherical geometry, to include a factor &lt;span class=&quot;math-container&quot;&gt;$4π$&lt;/span&gt; in equations like&#xA;  Coulomb's law, and write it in the form:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$F=k'_{\text{e}}{\frac {{q'_{\text{s}}}^{2}}{4\pi r^{2}}}$&lt;/span&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The quantities &lt;span class=&quot;math-container&quot;&gt;$q'_s$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k'_e$&lt;/span&gt; are not the same as those in the older convention. Putting &lt;span class=&quot;math-container&quot;&gt;$k_e = 1$&lt;/span&gt; generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Lorentz%E2%80%93Heaviside_units&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lorentz–Heaviside&lt;/a&gt; system of units. Lorentz–Heaviside units, unlike Gaussian units, are rationalised, meaning that there are no factors of &lt;span class=&quot;math-container&quot;&gt;$4π$&lt;/span&gt; appearing explicitly in Maxwell's equations.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The next step was to treat the quantity representing &quot;amount of&#xA;  electricity&quot; as a fundamental quantity in its own right, denoted by&#xA;  the symbol &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, and to write Coulomb's Law in its modern form:&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$F={\frac {1}{4\pi \varepsilon _{0}}}{\frac{q^{2}}{r^{2}}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;This is the SI system with Ampere as a fundamental unit for electric current. Coulomb is the unit of charge with one Coulomb being equal to one Ampere-second. This makes the dimensionality of charge as &lt;span class=&quot;math-container&quot;&gt;$[I]^{1}[T]^{1}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-06-04T13:18:52.597" LastActivityDate="2020-06-04T13:18:52.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11863" PostTypeId="1" CreationDate="2020-06-04T13:45:01.523" Score="1" ViewCount="110" Body="&lt;p&gt;According to Wikipedia, Sakharov in his early scientific career investigated cosmic rays. It offers no other details. It claims that it was for this work that he was awarded his doctorate. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However according to Spartacus Educational, he joined the Lebedev Institute in 1945 to work on his doctorate under the supervision of Igor Tamm and defended his thesis on nuclear physics two years later. There is no mention of cosmic rays here. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Of course it is likely that Sakharov worked on a number of projects during this period.  What research did he specifically carry out in investigating these rays? &lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="4251" LastEditDate="2020-07-08T17:06:19.957" LastActivityDate="2020-08-07T18:03:25.347" Title="What was the content of Sakharov's research into cosmic rays?" Tags="|biographical-details|experimental-physics|physicists|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11864" PostTypeId="5" CreationDate="2020-06-04T17:06:59.157" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-04T17:06:59.157" LastActivityDate="2020-06-04T17:06:59.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11865" PostTypeId="4" CreationDate="2020-06-04T17:06:59.157" Score="0" Body="James Clerk Maxwell (1831-1879) was a Scottish mathematical physicist whose most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. " OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:15.253" LastActivityDate="2020-06-12T21:58:15.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11866" PostTypeId="2" ParentId="11863" CreationDate="2020-06-05T09:50:54.023" Score="2" Body="&lt;p&gt;According to a biography of Sakharov, his first published paper was in the &lt;em&gt;Journal of Theoretical and Experimental Physics&lt;/em&gt;, and 'estimated the cross-section for the production of mesons as the hard components of cosmic rays according to a model proposed by Tamm' &lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-06-05T09:50:54.023" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11867" PostTypeId="1" AcceptedAnswerId="11868" CreationDate="2020-06-05T21:24:49.923" Score="2" ViewCount="74" Body="&lt;p&gt;I'm referring to the Hans Bauer who is the author of &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01999447&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; from 1915 (H. Bauer, &quot;Der Perronsche Integralbegriff und seine Beziehung auf Lebesguesschen&quot; Monatsh. Math. Phys. , 26 (1915) pp. 153–198). When was this man born and when did he die? I'm asking because I have mentioned his name in the beginning of my thesis and I would like to provide the reader with some additional information. &lt;/p&gt;&#xA;" OwnerUserId="12624" LastEditorUserId="26" LastEditDate="2021-02-20T15:12:53.280" LastActivityDate="2021-02-20T15:12:53.280" Title="Who was Hans Bauer who worked on the Perron integral?" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11868" PostTypeId="2" ParentId="11867" CreationDate="2020-06-05T21:31:59.457" Score="3" Body="&lt;p&gt;Hans (Adolf) Bauer was Pauli's private tutor in mathematics and physics, more datails &lt;a href=&quot;https://books.google.it/books?id=8tUVMSsC9wAC&amp;amp;pg=PA380&amp;amp;lpg=PA380&amp;amp;dq=hans+bauer+mathematician&amp;amp;source=bl&amp;amp;ots=mHT7QSAq2w&amp;amp;sig=ACfU3U0lNI5bZVd5edt3rkdSMk0XQR4yZg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjhleG7z-vpAhVKm6QKHbJqAwsQ6AEwA3oECAoQAQ#v=onepage&amp;amp;q=bauer&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Born in Wien in 1891 became Privatdozent in 1925 and then professor of theoretical physics in 1937 at the Technische Hochscule in Wien. He published paper on relativity heory and wave mechanics; he died in 1953.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;He published two books on relativity and wave mechanics: &lt;/p&gt;&#xA;&#xA;&lt;p&gt;*Mathematic Einfuhrung in De Gravitationstheorie Einsteins (1922)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;*&lt;a href=&quot;https://www.springer.com/de/book/9783662361221&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grundlagen der Atomphysik&lt;/a&gt; (1938)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;From &lt;a href=&quot;https://books.google.it/books?id=5mGZno8CvnQC&amp;amp;pg=PA176&amp;amp;lpg=PA176&amp;amp;dq=hans+bauer+wolfgang+pauli&amp;amp;source=bl&amp;amp;ots=FecEL7ZJzr&amp;amp;sig=ACfU3U26j3O7r0qPXZbSPSEncXzondnTRg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiluKzx0evpAhX9SBUIHb65ASMQ6AEwA3oECAsQAQ#v=onepage&amp;amp;q=hans%20bauer%20wolfgang%20pauli&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Bauer discussed introducing Einstein's principle in of general relativity in high school instruction. He also gave lectures on general relativity at the Verein deutscher Mathematiker uns Physiker at the university and expanded them into a book (1922) which was intended to form a bridge between many popular expositions and the highly technical ones. His research on Einstein's equations resulted in a very lenghtly, regrettably almost totally overlooked, study of spherically symmetric fluid solutions with a linear equation of state, and later in a similar one of the general static spherically symmetric solution.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-06-05T21:49:10.023" LastActivityDate="2020-06-05T21:49:10.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11869" PostTypeId="1" CreationDate="2020-06-06T07:35:15.057" Score="2" ViewCount="100" Body="&lt;p&gt;William Playfair was a Scottish engineer and economist, who invented the pie and bar charts as well as the line graph, which have all played an indubitably ubiquitous role in modern statistics.&#xA;I hadn't actually come across Playfair, before I browsed Colin Stuart's Maths 100 Numbers book. The book put William Playfair as follows:&#xA;&quot;It's fair to say the inventor of the ubiquitous line graph, bar chart and pie chart is not a household name.&quot;&#xA;What is it that deterred him from being a household name, given the contributions he made to statistics? Even in the scientific community, since statistics plays such a fundamental role in modelling numerous phenomena?&lt;/p&gt;&#xA;" OwnerUserId="12626" LastActivityDate="2020-06-06T07:35:15.057" Title="Why is William Playfair seldom heard about in mathematics?" Tags="|statistics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11870" PostTypeId="5" CreationDate="2020-06-06T07:43:34.553" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-06T07:43:34.553" LastActivityDate="2020-06-06T07:43:34.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11871" PostTypeId="4" CreationDate="2020-06-06T07:43:34.553" Score="0" Body="Richard Phillips Feynman (1918-1988) was an American theoretical physicist known for his work on the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, as well as his work in particle physics." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:11.280" LastActivityDate="2020-06-12T21:58:11.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11872" PostTypeId="5" CreationDate="2020-06-06T08:13:53.500" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-06T08:13:53.500" LastActivityDate="2020-06-06T08:13:53.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11873" PostTypeId="4" CreationDate="2020-06-06T08:13:53.500" Score="0" Body="Galileo di Vincenzo Bonaulti de Galilei (1564-1642) was an Italian astronomer, physicist and engineer from Pisa." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:08.343" LastActivityDate="2020-06-12T21:58:08.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11874" PostTypeId="5" CreationDate="2020-06-06T13:25:59.293" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-06T13:25:59.293" LastActivityDate="2020-06-06T13:25:59.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11875" PostTypeId="4" CreationDate="2020-06-06T13:25:59.293" Score="0" Body="Georg Ferdinand Ludwig Philipp Cantor (1845-1918) was a German mathematician who created set theory." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:21.413" LastActivityDate="2020-06-12T21:58:21.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11876" PostTypeId="5" CreationDate="2020-06-06T15:26:30.863" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-06T15:26:30.863" LastActivityDate="2020-06-06T15:26:30.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11877" PostTypeId="4" CreationDate="2020-06-06T15:26:30.863" Score="0" Body="Werner Karl Heisenberg (1901-1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:18.157" LastActivityDate="2020-06-12T21:58:18.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11878" PostTypeId="2" ParentId="11185" CreationDate="2020-06-06T23:16:55.853" Score="4" Body="&lt;p&gt;The following is taken from&lt;/p&gt;&#xA;&lt;p&gt;H.Casimir, &amp;quot;Haphazard Reality: Half a Century of Science,&amp;quot; Amsterdam University Press, 2010, pp. 207-210.&lt;/p&gt;&#xA;&lt;p&gt;I added some paragraph breaks since the text is hard to read otherwise.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Heisenberg came to Holland in 1943. I believe he made use of the same arrangement for cultural exchange to visit his old friends. Although on principle we did not favor such exchanges, we were glad to see Heisenberg. I am convinced that he really cared how physicists in Holland were making out and might have tried to help in special cases (but he considered it was quite beyond his powers to do anything at all for Jewish physicists or their families). Kramers arranged a program for his visit.&lt;/p&gt;&#xA;&lt;p&gt;I remember that I met Heisenberg and that we went for a brisk walk. I cannot retrace the itinerary, but in my mind it is associated with a curving road and a viaduct. I even have a feeling that I might recognize the spot today-if it were still there unchanged, which is improbable considering the extent of roadbuilding that has been going on in my country since the Second World War.&lt;/p&gt;&#xA;&lt;p&gt;It was during that walk that Heisenberg began to lecture on history and world politics. He explained that &lt;strong&gt;it had always been the historic mission of Germany to defend the West and its culture against the onslaught of eastern hordes and that the present conflict was one more example. Neither France nor England would have been sufficiently determined and sufficiently strong to play a leading role in such a defense,&lt;/strong&gt; and his conclusion was-and now I repeat in German the exact words he used-&lt;strong&gt;&amp;quot;da waere vielleicht doch ein Europa unter deutscher Fuehrung das kleinere Uebel&amp;quot; (and so, perhaps, a Europe under German leadership might be the lesser evil).&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course, I objected that the many inequities of the Nazi regime, and especially their mad and cruel anti-Semitism, made this unacceptable. Heisenberg did not attempt to deny, still less to defend, these things; but he said one should expect a change for the better once the war was over. And one had to recognize that they were a consequence of the great power of the leader that was also part of the German tradition.&lt;/p&gt;&#xA;&lt;p&gt;I cannot deny that I was somewhat impressed: I had always admired Heisenberg, not only as a physicist. For me he represented much of what was valuable in German culture. He was a good musician and a good sportsman, and his knowledge of classical languages was far superior to mine. But when later in the evening I thought over what he had said, I realized with a shock that his ideas came uncomfortably close to the usual German propaganda.&lt;/p&gt;&#xA;&lt;p&gt;In later years I have often wondered why Heisenberg spoke the way he did. Did he still believe Germany was going to win the war? Then it would be understandable that he tried to console me by pointing out that this might not be altogether bad. And perhaps he might already have been thinking of the possibility of finding in me, in the long run, someone who could be of some help in rebuilding continental European research. After all, he had more or less offered me a position way back in 1934. The trouble with this explanation is that it no longer did look as if Germany were going to win and that Heisenberg had anyway been convinced from the beginning that Germany was going to lose. About this there can be little doubt: in this respect, his widow's recollections are in complete agreement with what I heard from former students of his, to whom he had said: &amp;quot;Once the United States gets into the war, as it is bound to do, it will be like a game of chess in which you start out with only one rook against your opponent's two.&amp;quot; I was also told that one of the reasons he gave just prior to the war for not wanting to stay in the U.S.A. was that he felt that after the unavoidable defeat of Germany he would be able to do more for its cultural resurrection if he remained on the spot....&lt;/p&gt;&#xA;&lt;p&gt;The third visitor was Hans Kopfermann. We knew each other fairly well. He was an experimental physicist, but he belonged to the Copenhagen clan: he had worked on spectra at Bohr's Institute, and was a specialist on hyperfine structures, a field in which I had done theoretical research. He came to Philips on some business connected with the purchase of scientific apparatus, and very discreetly, without showing that we knew each other, asked me whether he could see me in private. I went to see him at his hotel in the evening and proposed that we should go to my home for a quiet talk. While we were walking along the dark streets of Eindhoven, he said, &amp;quot;Won't you get into difficulties with your colleagues if they should find out that you are receiving a German at home?&amp;quot; I had a ready answer: &amp;quot;Now that you have said that, I don't care.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Becker: Die Rechnung wird schon presentiert werden (The bill will be presented).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Heisenberg: Vielleicht waere ein Europa unter deutscher Fuehrung das kleinere Uebel (Perhaps a Europe under German leadership might be the lesser evil).&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A genius is someone who can create things that are initially beyond his own understanding. In that sense Heisenberg was certainly a genius, and this goes but rarely together with a special gift for understanding the feelings and the ways of thinking of others. Heisenberg did not have that gift. Perhaps his greatest shortcoming was that he was unable to grasp the full measure of depravity of what was then the ruling group in Germany. A Europe under German leadership would have meant a Europe under Nazi leadership, and it would have been a horrible thing. That became increasingly clear thereafter. At the time of our meeting the persecution and annihilation of the Jews had not yet reached its climax.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit.&lt;/strong&gt; I checked the book by Kurzman: There is no mentioning of any letters, he simply repeats what Heisenberg told Casimir.&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/iHuhy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iHuhy.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-06-07T15:07:10.537" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11879" PostTypeId="1" CreationDate="2020-06-07T01:17:38.577" Score="2" ViewCount="108" Body="&lt;p&gt;I am reading several sources and there seems to be a lack of clarity, and some contradiction, about the origins of the most recognised prime counting function approximations:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pi(n) \sim \frac{n}{\ln(n)}$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\pi(n) \sim \int_{0}^{n}\frac{1}{ln(x)}dx$&lt;/span&gt;&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;One very famous mathematician in a popular book has stated that (1) was conjectured by Gauss and (2) by Dirichlet. But others say Gauss published (2) in his 1849 letter to Encke (&lt;a href=&quot;https://gauss.adw-goe.de/handle/gauss/199&quot; rel=&quot;nofollow noreferrer&quot;&gt;ref.&lt;/a&gt;), saying he came up with it when he was about 15.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Other books and papers, which presumably have been edited and reviewed, suggest slightly different versions of the history. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;What is the correct history? How were these approximations originally developed, and by whom?&lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="26" LastEditDate="2021-02-20T15:12:46.970" LastActivityDate="2021-02-20T15:12:46.970" Title="What is the history of these prime counting function approximations?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11880" PostTypeId="1" AcceptedAnswerId="15762" CreationDate="2020-06-07T12:59:33.223" Score="9" ViewCount="277" Body="&lt;p&gt;It's a well-known, &lt;a href=&quot;https://www.discovermagazine.com/technology/why-nuclear-fusion-is-always-30-years-away&quot; rel=&quot;noreferrer&quot;&gt;running joke&lt;/a&gt; (or criticism) in the fusion community that &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Fusion is always 30 years away.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;refering to the considerable difficulties that harnessing &lt;a href=&quot;https://en.wikipedia.org/wiki/Fusion_power#History&quot; rel=&quot;noreferrer&quot;&gt;nuclear fusion&lt;/a&gt; as an energy source has faced since the 1940's.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Who said that first?&lt;/p&gt;&#xA;" OwnerUserId="10299" LastActivityDate="2023-09-19T14:35:55.330" Title="&quot;Nuclear fusion is 30 years away&quot; since when?" Tags="|physics|engineering|energy|nuclear-physics|technology|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11881" PostTypeId="2" ParentId="11879" CreationDate="2020-06-07T13:37:15.283" Score="1" Body="&lt;p&gt;Are you asking about the history of conjectures or the history of results?&#xA;All conjectures were made on the basis of study of the tables of primes.&#xA;The first proved result about this is due to Chebyshev. He proved that&#xA;there exist constants &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ax/\ln x&amp;lt;\pi(x)&amp;lt;Ax/\ln x.$$&lt;/span&gt;&#xA;Then Hadamard and Valle-Poussin independently proved that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\pi(x)\sim\frac{x}{\ln x},$$&lt;/span&gt;&#xA;which is the same as your 1 or 2 (they are equivalent if you are not discussing the more precise error term).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-06-07T13:37:15.283" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11882" PostTypeId="5" CreationDate="2020-06-07T20:34:04.313" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-07T20:34:04.313" LastActivityDate="2020-06-07T20:34:04.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11883" PostTypeId="4" CreationDate="2020-06-07T20:34:04.313" Score="0" Body="Max Karl Ernst Ludwig Planck (1858-1947) was a German theoretical physicist who originated quantum theory." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:27.153" LastActivityDate="2020-06-12T21:58:27.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11884" PostTypeId="1" CreationDate="2020-06-08T08:47:25.340" Score="1" ViewCount="103" Body="&lt;p&gt;A famous problem in electronics or physics course,is the following---&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Consider an infinite 2d grid of resistors having resistance of equal value.Find the resistance between any two nodes in the grid.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;This problem has been discussed in nerd sniping,and in-&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1.&lt;a href=&quot;https://physics.stackexchange.com/questions/2072/&quot;&gt;https://physics.stackexchange.com/questions/2072/&lt;/a&gt;&#xA;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;2.&lt;a href=&quot;https://www.mbeckler.org/resistor_grid/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mbeckler.org/resistor_grid/&lt;/a&gt;&#xA;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;3.&lt;a href=&quot;https://sites.google.com/site/resistorgrid/node1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sites.google.com/site/resistorgrid/node1&lt;/a&gt;&#xA;and &lt;/p&gt;&#xA;&#xA;&lt;p&gt;4.&lt;a href=&quot;https://www.mathpages.com/home/kmath668/kmath668.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathpages.com/home/kmath668/kmath668.htm&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Simpler versions of this often crops up in various college/university/competitive exams,and is now-a-days considered a &quot;famous&quot; problem---&#xA;(see last paragraph of &lt;a href=&quot;https://www.mathpages.com/home/kmath668/kmath668.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathpages.com/home/kmath668/kmath668.htm&lt;/a&gt;)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interestingly,I could not find it's reference in most classic circuit analysis/electronics textbooks! So,How did it rise to popularity and what is the original source of this?Did this come up on the internet or excercise of some book or otherwise?&lt;/p&gt;&#xA;" OwnerUserId="11051" LastEditorUserId="11051" LastEditDate="2020-06-17T04:25:08.177" LastActivityDate="2020-06-17T04:25:08.177" Title="What is the original source of the problem of finding equivalent resistance between two nodes in an infinite grid of resistors?" Tags="|physics|reference-request|electricity|resource-recommendation|book-recommendation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11886" PostTypeId="5" CreationDate="2020-06-08T10:54:35.087" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-08T10:54:35.087" LastActivityDate="2020-06-08T10:54:35.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11887" PostTypeId="4" CreationDate="2020-06-08T10:54:35.087" Score="0" Body="Dmitri Ivanovich Mendeleev (1834-1907) was a Russian chemist and inventor who developed the periodic classification of the elements." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-12T21:58:29.773" LastActivityDate="2020-06-12T21:58:29.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11888" PostTypeId="1" AcceptedAnswerId="11889" CreationDate="2020-06-08T17:03:19.080" Score="8" ViewCount="400" Body="&lt;p&gt;Gauss, in his 1849 letter to Encke, mentions that he noticed the primes have a density approx &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{\ln(n)}$&lt;/span&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;In that letter, he also mentions an integral function for approximating the prime counting function &lt;span class=&quot;math-container&quot;&gt;$\pi(n)$&lt;/span&gt;. His notation slightly unconventional, he wrote &lt;span class=&quot;math-container&quot;&gt;$\int \frac{{\rm d} n}{\log(n)}$&lt;/span&gt;, which today we consider the logarithmic integral approximation &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int^n_2\frac{1}{\ln(x)} {\rm d} x$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;If Gauss came up with the logarithmic integral approximation, who came up with the more discrete &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{\ln(n)}$&lt;/span&gt; ?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Today, it seems an obvious leap from the densities, but this is a question of maths history — who, when and how.&lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="26" LastEditDate="2021-02-20T15:12:26.860" LastActivityDate="2021-02-20T15:12:26.860" Title="Who first identified $\frac{n}{\ln(n)}$ as an approximation of a prime counting function?" Tags="|mathematics|number-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11889" PostTypeId="2" ParentId="11888" CreationDate="2020-06-09T19:50:12.300" Score="9" Body="&lt;p&gt;Gauss also came up with the more discrete &lt;span class=&quot;math-container&quot;&gt;$n/\ln (n)$&lt;/span&gt;- in volume 10 of his collected works appears a short (5-6 pages) fragment entitled &quot;&lt;strong&gt;asymptotic laws of arithmetics&lt;/strong&gt;&quot;, which is dated to the year 1791. In [1] of this fragment Gauss states this approximation of the primes counting function, as well as additional conjecture on the asymptotics of k-prime numbers. By examination of tables of prime numbers up to very large numbers, he stated the prime numbers theorem (that the average density of primes between &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1/\log (n)$&lt;/span&gt;) as an empirical law, and on this basis derived his more general conjecture on k-prime numbers. Only later (it seems that in his 1849 letter to Encke) he refined his conjecture so that primes have a density &lt;span class=&quot;math-container&quot;&gt;$1/\log (n)$&lt;/span&gt; &quot;around&quot; the number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, what leads directly to the (better) logarithmic integral approximation &lt;span class=&quot;math-container&quot;&gt;$\pi(n)\propto \int_{2}^{n}\frac{dx}{\ln x}$&lt;/span&gt;.   &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For an expanded discussion of the relation of Gauss's asymptotics for k-prime numbers to the prime numbers theorem, you might look at  &lt;a href=&quot;https://math.stackexchange.com/questions/2587733/what-was-the-heuristics-behind-gausss-guess-on-the-asymptotic-distribution-of-k&quot;&gt;https://math.stackexchange.com/questions/2587733/what-was-the-heuristics-behind-gausss-guess-on-the-asymptotic-distribution-of-k&lt;/a&gt; . &lt;/p&gt;&#xA;&#xA;&lt;p&gt;As for online sources, Gauss's collected works can be read freely at the digital library of the goettingen university. Here is a link for Gauss's collected works: &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235957348&quot; rel=&quot;noreferrer&quot;&gt;https://gdz.sub.uni-goettingen.de/id/PPN235957348&lt;/a&gt;. Specifically, what you are looking for (the prime numbers theorem) is in the begining of volume 10-1  &lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="55" LastEditDate="2020-06-11T21:14:37.853" LastActivityDate="2020-06-11T21:14:37.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11890" PostTypeId="2" ParentId="11449" CreationDate="2020-06-09T22:39:32.227" Score="3" Body="&lt;p&gt;This answer is a repost of a comment of Conifold on the question, which the asker indicated gave a satisfactory answer to their question:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations#Relationships_among_electricity,_magnetism,_and_the_speed_of_light&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiki's History of Maxwell's equations&lt;/a&gt; implies that the value was measured by Weber and Kohlrausch in 1855, its relation to the speed of light noted by Kirchoff in 1857, and Maxwell was aware of it since 1861. You may want to check the original references linked in &lt;a href=&quot;https://hsm.stackexchange.com/a/5525/55&quot;&gt;Geremia's answer to Electromagnetic constants and the speed of light&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="26" LastActivityDate="2020-06-09T22:39:32.227" CommentCount="0" CommunityOwnedDate="2020-06-09T22:39:32.227" ContentLicense="CC BY-SA 4.0" />
  <row Id="11891" PostTypeId="1" CreationDate="2020-06-10T04:13:22.343" Score="7" ViewCount="235" Body="&lt;p&gt;There are of course many physics dictionaries and &lt;a href=&quot;https://en.wikipedia.org/wiki/Glossary_of_physics&quot; rel=&quot;noreferrer&quot;&gt;glossaries&lt;/a&gt; and some words can be found in &lt;a href=&quot;https://www.etymonline.com/&quot; rel=&quot;noreferrer&quot;&gt;general etymological dictionaries&lt;/a&gt; and even &lt;a href=&quot;https://www.merriam-webster.com/&quot; rel=&quot;noreferrer&quot;&gt;English dictionaries&lt;/a&gt;; but is there a Physics Etymological Dictionary, such as there is &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0883855119&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;for math&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="10299" LastEditorUserId="10106" LastEditDate="2020-06-10T04:21:43.713" LastActivityDate="2020-06-15T17:35:07.983" Title="Is there an etymological dictionary of terms in physics?" Tags="|physics|resource-recommendation|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11892" PostTypeId="2" ParentId="11891" CreationDate="2020-06-10T10:36:45.100" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0395065712&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Asimov&lt;/a&gt; wrote one.&lt;/p&gt;&#xA;&lt;h1&gt;edit&lt;/h1&gt;&#xA;&lt;p&gt;Ooops,   he wrote  &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0395137225&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;two&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-06-15T17:35:07.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11893" PostTypeId="1" AcceptedAnswerId="11898" CreationDate="2020-06-10T18:29:05.113" Score="5" ViewCount="389" Body="&lt;p&gt;Today, when the term &amp;quot;field&amp;quot; is defined in algebra, it is almost always stipulated that all fields are commutative. However, the author of &lt;a href=&quot;http://alpha.math.uga.edu/%7Epete/FieldTheory.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;these lectures&lt;/a&gt; says that this has not always been the case:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In older terminology, a field could be non-commutative, i.e., any ring in which each nonzero element has a two-sided multiplicative inverse. We now call such things “division rings” or “division algebras.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this true? If so, why and when did this shift in the definition of &amp;quot;field&amp;quot; occur? What could I read to learn more?&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance for your help!&lt;/p&gt;&#xA;" OwnerUserId="12646" LastEditorUserId="205" LastEditDate="2022-04-27T02:40:32.290" LastActivityDate="2022-04-27T02:40:32.290" Title="Could a &quot;field&quot; have non-commutative multiplication originally?" Tags="|mathematics|terminology|abstract-algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11894" PostTypeId="2" ParentId="11888" CreationDate="2020-06-10T23:56:39.927" Score="5" Body="&lt;p&gt;With all due credit to Gauß, see the other answer, it seems to have been Adrien-Marie Legendre who first &lt;em&gt;published&lt;/em&gt; this conjecturally. More precisely, on the last page of the introduction (p. 19) of the &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/btv1b8626880r/f55.image&quot; rel=&quot;noreferrer&quot;&gt;first edition of his &lt;em&gt;Essai sur la théorie des nombres&lt;/em&gt;&lt;/a&gt; (1798), he says in a footnote:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Au reste, il est vraisemblable que la formule rigoureouse qui donne la valeur de &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; [primes &lt;span class=&quot;math-container&quot;&gt;$\le a$&lt;/span&gt;] lorsque &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; est tres-grand, est de la forme &lt;span class=&quot;math-container&quot;&gt;$b = \dfrac{a}{A \,log. a +B}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; étant des coefficiens constans, et &lt;span class=&quot;math-container&quot;&gt;$log.a$&lt;/span&gt; désignant un logarithme hyperbolique [i.e. &lt;span class=&quot;math-container&quot;&gt;$ln(a)$&lt;/span&gt;]. La détermination exacte de ces coefficiens seroit un problême curieux et digne d'exercer la sagacité des Analystes.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;[My translation: &quot;Besides, it is likely that the rigorous formula that gives the value of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; [primes &lt;span class=&quot;math-container&quot;&gt;$\le a$&lt;/span&gt;] when &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is very large, is of the form &lt;span class=&quot;math-container&quot;&gt;$b = \dfrac{a}{A \,log. a +B}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are constant coefficients, and &lt;span class=&quot;math-container&quot;&gt;$log.a$&lt;/span&gt; denotes a hyperbolic logarithm [i.e. &lt;span class=&quot;math-container&quot;&gt;$ln(a)$&lt;/span&gt;]. The exact determination of these coefficients is a curious problem, worthy for Analysts to exercise their sagacity.&quot;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;So in modern notation he conjectures that &lt;span class=&quot;math-container&quot;&gt;$\pi(x) \approx \dfrac{x}{A \ln(x) +B}$&lt;/span&gt; with some constants &lt;span class=&quot;math-container&quot;&gt;$A, B$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k62826k/f420.image&quot; rel=&quot;noreferrer&quot;&gt;second edition of the same work&lt;/a&gt; (1808), he turned this into an entire section (§VIII) in the fourth part, starting on p. 394. He now claims that &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{x}{\ln(x)-1.08366}$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;is the best approximation, i.e. some time between 1798 and 1808 he made up his mind that of his constants, &lt;span class=&quot;math-container&quot;&gt;$A=1$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$B\approx -1.08366$&lt;/span&gt;. In volume 2 of &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k42612x/f80.image&quot; rel=&quot;noreferrer&quot;&gt;the extended third edition (1830)&lt;/a&gt;, the last published in Legendre's life time, he sticks with this claim (fourth part, §VIII, p. 65).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The somewhat curious number &lt;span class=&quot;math-container&quot;&gt;$(\pm)1.08366$&lt;/span&gt; has, somewhat unfairly, become known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Legendre%27s_constant&quot; rel=&quot;noreferrer&quot;&gt;Legendre's constant&lt;/a&gt;. According to the Wikipedia article, Chebyshev proved in 1849 that if the formula is valid for any &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; at all, then the value must be &lt;span class=&quot;math-container&quot;&gt;$B=(-)1$&lt;/span&gt;, and according to the prime number theorem, it really is. So Legendre was actually right on his first account, just that one of his constants turned out to be much simpler than he thought.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It seems likely that Legendre came up with this independently from Gauß (if only because it seems Gauß conjectured the right constants rightaway).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;It would actually be interesting to know where Legendre got his confidence from that those five decimals &lt;span class=&quot;math-container&quot;&gt;$.08366$&lt;/span&gt; were right; because even though they are not that far off in the range for which prime number tables were available at that time, neither would have been some other numbers. Cf. &lt;a href=&quot;https://www.mathpages.com/home/kmath032.htm&quot; rel=&quot;noreferrer&quot;&gt;https://www.mathpages.com/home/kmath032.htm&lt;/a&gt;. (E.g. sure, if we look at &lt;a href=&quot;https://en.wikipedia.org/wiki/Legendre%27s_constant#/media/File:Legendre%27s_constant.svg&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Legendre%27s_constant#/media/File:Legendre%27s_constant.svg&lt;/a&gt;, the blue line matches somewhat, but if we moved it up or down by &lt;span class=&quot;math-container&quot;&gt;$\pm 0.01$&lt;/span&gt;, it still would look reasonable; let alone going to &lt;span class=&quot;math-container&quot;&gt;$1.08361$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1.08379$&lt;/span&gt; or ...) Likely there is something about that to be found in that §VIII of the fourth part of the later editions, which to be honest I did not read thoroughly now (and would lack understanding for anyway); maybe someone with better knowledge of number theory could help here.&lt;/p&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2020-06-10T23:56:39.927" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11895" PostTypeId="2" ParentId="2567" CreationDate="2020-06-11T05:01:04.980" Score="3" Body="&lt;p&gt;This is an addendum to the mostly correct &lt;a href=&quot;https://hsm.stackexchange.com/a/2610/10854&quot;&gt;answer by jkien&lt;/a&gt;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;The overall presentation is largely borrowed from &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Kilogram&amp;amp;oldid=961923859#Kilogram_becoming_a_base_unit:_the_role_of_units_for_electromagnetism&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, but the actual facts come mostly from these sources: &lt;a href=&quot;https://ieeexplore.ieee.org/document/8458060&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://www.biodiversitylibrary.org/page/29853513#page/322/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://archive.org/stream/encyclopaediabri27chisrich#page/740/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Introduction&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One has to be careful when saying that the practical units (the volt, the ampere, etc.) were 'coherent with' the meter-kilogram-second (MKS) system. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;If by 'MKS' we mean a &lt;em&gt;three-dimensional&lt;/em&gt; mass-length-time system, then the volt, the ampere, and the ohm were most definitely &lt;strong&gt;not&lt;/strong&gt; coherent units in it. (&lt;strong&gt;However&lt;/strong&gt;—and this will be the key—the product volt&amp;nbsp;&amp;times;&amp;nbsp;ampere, which was named the watt, is a purely mechanical unit—of power—which &lt;em&gt;was&lt;/em&gt; coherent in MKS. It is &lt;em&gt;that&lt;/em&gt; fact—that the watt is coherent in MKS—that truly was a lucky accident.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, if by 'MKS' we really mean a &lt;em&gt;four-dimensional&lt;/em&gt;, MKSX system, where 'X' is the unit of some non-mechanical electric quantity,&lt;sup&gt;1&lt;/sup&gt; then it is incorrect to say that it is an accident that the volt, the ampere, etc. were coherent units in such a system—of course they were, since one of them was chosen to be a base unit!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;Serious consideration was given to proposals where X was either the coulomb, or the ampere, or the ohm, or the volt. Eventually, metrological considerations turned out to favor the ampere.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The key here are the units that straddle both the electric and the mechanical domains—in particular, the watt. It is those units that would have made it impossible to extend the CGS system by e.g. adding the ampere to it as a fourth independent base unit: the watt is not equal to the erg per second, and so would not be a coherent derived unit in such a system. However, the watt does turn out to be kg&amp;nbsp;&amp;times;&amp;nbsp;meter&lt;sup&gt;2&lt;/sup&gt;/second&lt;sup&gt;3&lt;/sup&gt;, and so the practical electric units can be integrated with the MKS system by adding a fourth independent base unit.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Discussion&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the time the 'practical' electric units were adopted (1873–1893), everyone as a matter of course assumed that a scientific system of units should be &lt;a href=&quot;https://www.google.com/books/edition/A_Dictionary_of_Weights_Measures_and_Uni/RZfGWl-RCQcC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22absolute%20system%20electromagnetics%20Metric%20The%20evolution%20of%20telecommunications%20and%20other%20applications%20of%20electricity%20in%20the%2019th%20century%20led%20to%20a%20plethora%20of%20units%20for%20associated%20measurement.%20Concomitant%20realization%20that%20the%20electrical%20and%20magnetic%20effects%20could%20be%20represented%20in%20mechanical%20terms%20led%20to%20the%20expression%20of%20electromotive%20force%20and%20current%2C%20and%20thereby%20resistance%2C%20in%20terms%20of%20the%20metre%2C%20gram%2C%20and%20second.%20Such%20expression%20of%20the%20electrical%20in%20terms%20purely%20of%20the%20established%20mechanical%20units%20was%20called%20&amp;#39;absolute&amp;#39;.%22&amp;amp;pg=PA3&amp;amp;printsec=frontcover&amp;amp;bsq=%22absolute%20system%20electromagnetics%20Metric%20The%20evolution%20of%20telecommunications%20and%20other%20applications%20of%20electricity%20in%20the%2019th%20century%20led%20to%20a%20plethora%20of%20units%20for%20associated%20measurement.%20Concomitant%20realization%20that%20the%20electrical%20and%20magnetic%20effects%20could%20be%20represented%20in%20mechanical%20terms%20led%20to%20the%20expression%20of%20electromotive%20force%20and%20current%2C%20and%20thereby%20resistance%2C%20in%20terms%20of%20the%20metre%2C%20gram%2C%20and%20second.%20Such%20expression%20of%20the%20electrical%20in%20terms%20purely%20of%20the%20established%20mechanical%20units%20was%20called%20&amp;#39;absolute&amp;#39;.%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;absolute&lt;/em&gt;&lt;/a&gt;, meaning that the base dimensions should be just the three mechanical ones: length, mass, and time. The abvolt, for example, is g&lt;sup&gt;1/2&lt;/sup&gt;cm&lt;sup&gt;3/2&lt;/sup&gt;/s&lt;sup&gt;2&lt;/sup&gt; when expressed in the base cgs units (see &lt;a href=&quot;https://www.google.com/books/edition/Encyclopaedia_of_Historical_Metrology_We/XnRVDwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Encyclopaedia%20of%20Historical%20Metrology%2C%20Weights%2C%20and%20Measures%22%20vol%201&amp;amp;pg=PA25&amp;amp;printsec=frontcover&amp;amp;bsq=%22Electric%20potential%20(V)%20abvolt%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Now, there is indeed an absolute (i.e. a three-dimensional, length-mass-time) system is which the practical units are coherent, but it is &lt;em&gt;not&lt;/em&gt; the meter-kilogram-second system. It is, rather, a system in which the base unit of length is 10&lt;sup&gt;7&lt;/sup&gt; meters (called a &lt;a href=&quot;https://www.google.com/books/edition/A_Dictionary_of_Weights_Measures_and_Uni/RZfGWl-RCQcC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22quadrant%20from%20the%20North%20Pole%20to%20the%20Equator%22&amp;amp;pg=PP1&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;quadrant&lt;/em&gt;&lt;/a&gt;, as it is very nearly one half of a meridian of the Earth), and the base unit of mass is 10&lt;sup&gt;-11&lt;/sup&gt; grams (an &lt;a href=&quot;https://www.biodiversitylibrary.org/page/29853513#page/323/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;eleventh-gram&lt;/em&gt;&lt;/a&gt;): the &lt;a href=&quot;https://www.google.com/books/edition/Bulletin_of_the_University_of_Wisconsin/QWFBAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=a%20quadrant-elevth-gramme%20and%20second%20system%20or%20Q.E.S.&amp;amp;pg=PA346&amp;amp;printsec=frontcover&amp;amp;bsq=%22the%20practical%20system%20as%20the%20Quadrant%2C%20Eleventh-gram%2C%20Second%20(Q.E.S.)%20System%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;quadrant-eleventh-gram-second&lt;/em&gt;&lt;/a&gt; (QES) system. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sub&gt;This can be derived from the following facts. The practical units were defined &lt;a href=&quot;https://www.biodiversitylibrary.org/page/29853513#page/323/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;in 1873&lt;/a&gt; as decimal multiples and submultiples of the 'electromagnetic' absolute cgs units, cgs-emu. We will somewhat anachronistically use the following names for the emu units: the 'abvolt' for the potential, the 'abampere' for the current, etc.&lt;sup&gt;1&lt;/sup&gt;  When expressed in the base cgs units, the abvolt is g&lt;sup&gt;1/2&lt;/sup&gt;cm&lt;sup&gt;3/2&lt;/sup&gt;/s&lt;sup&gt;2&lt;/sup&gt;, the abampere is g&lt;sup&gt;1/2&lt;/sup&gt;cm&lt;sup&gt;1/2&lt;/sup&gt;/s, and the abcoulomb is g&lt;sup&gt;1/2&lt;/sup&gt;cm&lt;sup&gt;1/2&lt;/sup&gt; (see &lt;a href=&quot;https://www.google.com/books/edition/Encyclopaedia_of_Historical_Metrology_We/XnRVDwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Encyclopaedia%20of%20Historical%20Metrology%2C%20Weights%2C%20and%20Measures%22%20vol%201&amp;amp;pg=PA25&amp;amp;printsec=frontcover&amp;amp;bsq=%22Electric%20potential%20(V)%20abvolt%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://www.google.com/books/edition/Encyclopaedia_of_Historical_Metrology_We/XnRVDwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22Current%20(I)%20abamp%22&amp;amp;dq=%22Encyclopaedia%20of%20Historical%20Metrology%2C%20Weights%2C%20and%20Measures%22%20vol%201&amp;amp;pg=PA25&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;https://www.google.com/books/edition/Encyclopaedia_of_Historical_Metrology_We/XnRVDwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Charge%20(Q)%20abcoul%22&amp;amp;pg=PA25&amp;amp;printsec=frontcover&amp;amp;bsq=%22Charge%20(Q)%20abcoul%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). On the other hand, the volt was defined as 10&lt;sup&gt;8&lt;/sup&gt; abvolts, the ampere as 0.1 abamperes, and the coulomb as 0.1 abcoulombs (see the same three links).  Now imagine we change the base units of lenghts, mass, and time by factors of M, L, and T, respectively. Then the base unit of potental will become (M g)&lt;sup&gt;1/2&lt;/sup&gt;(L cm)&lt;sup&gt;3/2&lt;/sup&gt;/(T s)&lt;sup&gt;2&lt;/sup&gt; = M&lt;sup&gt;1/2&lt;/sup&gt;L&lt;sup&gt;3/2&lt;/sup&gt;/T&lt;sup&gt;2&lt;/sup&gt;&amp;nbsp;&amp;times;&amp;nbsp;g&lt;sup&gt;1/2&lt;/sup&gt;cm&lt;sup&gt;3/2&lt;/sup&gt;/s&lt;sup&gt;2&lt;/sup&gt; = M&lt;sup&gt;1/2&lt;/sup&gt;L&lt;sup&gt;3/2&lt;/sup&gt;/T&lt;sup&gt;2&lt;/sup&gt; abvolts. We want this new unit to be the volt, so we must have M&lt;sup&gt;1/2&lt;/sup&gt;L&lt;sup&gt;3/2&lt;/sup&gt;/T&lt;sup&gt;2&lt;/sup&gt; = 10&lt;sup&gt;8&lt;/sup&gt;. Similarly, if we want the new unit for current to be the ampere, we obtain that M&lt;sup&gt;1/2&lt;/sup&gt;L&lt;sup&gt;1/2&lt;/sup&gt;/T = 0.1, and if we want the new unit of charge to be the coulomb, we obtain that M&lt;sup&gt;1/2&lt;/sup&gt;L&lt;sup&gt;1/2&lt;/sup&gt; = 0.1. We thus have a system of three equations with three unknwns. The solution is L = 10&lt;sup&gt;9&lt;/sup&gt; (so the base unif of length should be 10&lt;sup&gt;9&lt;/sup&gt; cm = 10&lt;sup&gt;7&lt;/sup&gt; m), M = 10&lt;sup&gt;-11&lt;/sup&gt; (so the base unit of mass should be 10&lt;sup&gt;-11&lt;/sup&gt; g), and T = 1 (so the second remains the base unit of time).&lt;/sub&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;This naming convention, where the name of the emu unit is formed by adding a prefix 'ab-' (short for 'absolute') to the name of the corresponding practical unit, came &lt;a href=&quot;https://www.google.com/books/edition/Transactions/lTFDAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22The%20expedient%20suggests%20itself%20of%20attaching%20the%20prefix%20ab%20or%20abs%20to%20a%20practical%20or%20Q.%20E.%20S.%20unit%2C%20in%20order%20to%20express%20the%20absolute%20or%20corresponding%20C.%20G.%20S.%20magnetic%20unit.%22&amp;amp;pg=PA1412&amp;amp;printsec=frontcover&amp;amp;bsq=%22The%20expedient%20suggests%20itself%20of%20attaching%20the%20prefix%20ab%20or%20abs%20to%20a%20practical%20or%20Q.%20E.%20S.%20unit%2C%20in%20order%20to%20express%20the%20absolute%20or%20corresponding%20C.%20G.%20S.%20magnetic%20unit.%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;only in 1903&lt;/a&gt;, three decades after the practical units were originally defined in terms of the absolute emu units. At that earlier time, the absolute cgs electric units themselves didn't have any special names. One just used 'e.m.u.' or 'C.G.S', as in 'a current of 5 e.m.u.' or '5 C.G.S. units of current' (or perhaps one could also use the base units, e.g.  a current of 5 g&lt;sup&gt;1/2&lt;/sup&gt;cm&lt;sup&gt;1/2&lt;/sup&gt;/s) However, for convenience, we will use 'abvolt', 'abampere', etc. in what follows.&lt;sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The way the meter-kilogram-second system enters the story is this. In addition to the purely electric and magnetic units such as the ohm, the volt, the ampere, etc., the practical system of units also had to include some purely mechanical units. This is because of relations such as voltage&amp;nbsp;&amp;times;&amp;nbsp;current = power. In particular, the volt times the ampere gives a unit of power, which was &lt;a href=&quot;https://www.google.com/books/edition/The_Popular_Science_Monthly/5dMWAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22might+be+appropriately+called+a+Watt%22&amp;amp;pg=PA53&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;in 1882&lt;/a&gt; given a special name: the watt. Then the watt times the second gives a unit of energy, which was named &lt;a href=&quot;https://www.google.com/books/edition/The_Popular_Science_Monthly/5dMWAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22might%20be%20appropriately%20called%20a%20Watt%22&amp;amp;pg=PA48&amp;amp;printsec=frontcover&amp;amp;bsq=%22called%20the%20Joule%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;the joule&lt;/a&gt;. Of course, these purely mechanical practical units were coherent in the QES system. However, they are in fact coherent in a whole family of systems. To see why that is so, recall that the dimensions of power are ML&lt;sup&gt;2&lt;/sup&gt;/T&lt;sup&gt;3&lt;/sup&gt;. It follows that if the watt is coherent in a system, it will also be coherent in any system obtained from the original system by simultaneously changing the base unit of length by a factor of L and the base unit of mass by a factor of M in such a way that ML&lt;sup&gt;2&lt;/sup&gt;=1, i.e. in such a way that M=L&lt;sup&gt;-2&lt;/sup&gt;. We are told that the watt is coherent in the QES system; thus, it will also be coherent in any system in which the base unit of length is L&amp;nbsp;&amp;times;&amp;nbsp;10&lt;sup&gt;7&lt;/sup&gt; meters while the base unit of mass is L&lt;sup&gt;-2&lt;/sup&gt;&amp;nbsp;&amp;times;&amp;nbsp;10&lt;sup&gt;-11&lt;/sup&gt; grams. Picking L = 10&lt;sup&gt;-7&lt;/sup&gt; gives the meter and the kilogram. Moreover, it is easy to check that, if we insist that the new base units should be decimal multiples or submultiples of the meter and the gram, then the choice L = 10&lt;sup&gt;-7&lt;/sup&gt; is the &lt;em&gt;only&lt;/em&gt; choice that produces base units of practical sizes. For example, if we pick L = 10&lt;sup&gt;-8&lt;/sup&gt;, so the base unit of length is the decimeter, then the base unit of mass becomes 10&lt;sup&gt;16&lt;/sup&gt;&amp;nbsp;&amp;times;&amp;nbsp;10&lt;sup&gt;-11&lt;/sup&gt; grams&#xA;=&amp;nbsp;10&lt;sup&gt;5&lt;/sup&gt; grams = 100 kg, which is impractically large.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Probably many people noticed that the watt is coherent in the meter-kilogram-second system, but it was Giovanni Giorgi who really took note of it. He had the further insight—which was sort of iconoclastic at the time—that while the purely electric and magnetic units cannot be made coherent in the three-dimensional meter-kilogram-second system, they &lt;em&gt;could&lt;/em&gt; be made coherent in a four-dimensional extension of that system. Thus he proposed, in 1901, to introduce a fourth base dimension, which would be purely electric or magnetic. In principle, this fourth independent dimension could be any electromagnetic quantity, but only four received serious consideration: electric charge, electric current, electric resistance, and electric potential. Eventually, electric current was chosen because it was most advantageous metrologically. Another selling point of Giorgi's system was that it made it possible to rationalize (i.e. remove the awkward factors of 4π from) Maxwell's equations &lt;em&gt;without&lt;/em&gt; a corresponding redefinition of units by factors of (4π)&lt;sup&gt;1/2&lt;/sup&gt; (which is what happens when the Gaussian system is rationalized, giving the Lorentz-Heaviside system).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The Giorgi proposal (with the ampere as the fourth base unit) was adopted by the International Electrotechnical Commission in 1935 and by the CGPM in 1946; the CGMP later incorporated it into the SI system.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Summary&lt;/strong&gt; &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The fact that the kilogram rather than the gram is the base unit of mass in the SI is all the more remarkable given that, for about a century, the scientific community had been almost universally using the centimeter-gram-second system. Let me summarize the main reason why the CGS was abandoned and the meter-kilogram-second (MKS) was adopted. The main background facts to be aware of are that (a) by the end of the 19th century, the so-called 'practical system' of electric units had become nearly universaly accepted in practical applications of electricity such as telegraphy, and (b) this system of units included the volt and the ampere, and therefore also their product; but this product is a purely mechanical unit (of power), and if one multiplies that by the second, one gets another purely mechanical unit (of energy). In 1882, these two units were named, respectively, the watt and the joule. Now: the MKS is the &lt;em&gt;unique&lt;/em&gt; system which has all three of the following characteristics (and which keeps the second as a unit of time): 1. the watt and the joule are coherent, 2. the base units of length and mass are decimal multiples of the meter and the gram (so that the system is 'properly metric'), and 3. the sizes of the base units of length and mass are convenient (more or less) for practical work. All this assumes that the second remains the base unit of time; but it is definitely true that any proposal to replace the second would have been dismissed out of hand. The non-mechanical units such as the volt, the ampere, etc. are not coherent in a three-dimensional MKS system, which is why a fourth independent dimension was added: the ampere became a new base unit, dimensionally independent from the meter, the kilogram, and the second.&lt;/sup&gt; &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2020-06-16T23:00:13.310" LastActivityDate="2020-06-16T23:00:13.310" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11896" PostTypeId="1" CreationDate="2020-06-11T18:15:25.377" Score="7" ViewCount="561" Body="&lt;p&gt;It is sometimes stated that the early metric units of 'weight' really were meant to be exactly that: units of &lt;em&gt;weight&lt;/em&gt; (i.e. &lt;em&gt;force&lt;/em&gt;), not &lt;em&gt;mass&lt;/em&gt;. However, is that really so? In fact, did the people responsible for the creation of these units in the 1790s&lt;sup&gt;1&lt;/sup&gt; make a decision on this question one way or the other?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;Namely, the Commission of Weights and Measures of the French Academy of Sciences, which was renamed 'temporary commission' in September, 1793.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As best as I can tell, there are five possibilities:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;the Commission really did want to define standards of weight (force) and not mass. True, they completely ignored the fact that &lt;em&gt;g&lt;/em&gt; varies from place to place, and therefore one can also ask, given the state of metrology at the time, how good or poor job they did; but that's a separate question. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission really did want to define standards of mass and not weight. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission was aware that there is a difference between specifying standards for weight and mass; however, they decided to leave it open whether the new standards should be understood as standards of weight or of mass (perhaps because they thought—rightly or wrongly—that the variation in &lt;em&gt;g&lt;/em&gt; wasn't metrologically important for their purposes).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission members didn't clearly distinguish between mass and weight in their own minds (which wouldn't be very surprising in those days).&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission took it as a matter of course that, even though mass and weight are separate quantities, one can use the same unit to measure both. This is similar to how we have the same unit for inertial mass and gravitational mass: we know that these two are, in principle, separate quantities&lt;sup&gt;1&lt;/sup&gt;, but since they are always proportional to each other (with the same constant of proportionality for all objects), we may choose to have the same unit for both—or, alternatively, we may say that one may be quantified by the other. Analogously, mass and weight are always proportional to each other (with the same constant of proportionality for all objects&lt;sup&gt;2&lt;/sup&gt;), and so we may choose to have the same unit for both—or, alternatively, we may say that one may be quantified by the other.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;If you think that General Relativity eliminated this possibility, imagine this was written sometime between 1901 and 1915.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;2&lt;/sup&gt;We know (from their discussions about using a seconds pendulum to define the standard of length) that the Commission knew that the constant of proportionality, &lt;em&gt;g&lt;/em&gt;, does somewhat depend on the latitude. It is not clear what role this fact had on their thinking as far as the unit of 'weight'.&lt;/sup&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Perhaps there are other options, but I can't think of any at the moment. My question is, which one of these four is correct (if any?), and based on what historical evidence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The law of 18 germinal, year 3&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I would say that the phrasing of &lt;a href=&quot;https://web.archive.org/web/20160817122340/http://www.metrodiff.org/cmsms/index.php?page=18_germinal_an_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;this law&lt;/a&gt; is not decisive. The law says that the gram is&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;the absolute weight &lt;em&gt;(le poids absolu)&lt;/em&gt; of a volume of pure water equal to the cube of the hundredth part of the meter, and at the temperature of melting ice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;As I explain in detail in Update 2 below, contemporary texts explain that 'the absolute weight' of an object is the product of the object's volume and the object's specific gravity. According to the same texts, specific gravity is a relative quantity: a ratio of the weight (contemporary texts do clearly mean weight, not mass) of a body of interest to a body of same volume but made of some reference material. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;However, it was well understood (and emphasized in the texts) that the ratio of weights is, if all the weighing is done at one and the same location, equal to the ratio of masses. Therefore, specific gravity is also equal to the ratio of mass densities. It is perfectly possible that when people used the concept of specific gravity in the 1790s, they meant two things by it: experimentally, it was a ratio of weights, but conceptually, its meaning was a ratio of mass densities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;If this is so, the definition of the gram may be interpreted as either a unit of weight or a unit of mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An example of the claim&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;What has become of the kilogram, once the unit of &lt;strong&gt;weight&lt;/strong&gt; in the metric system? It has been transformed from the metric unit of &lt;strong&gt;weight&lt;/strong&gt; (force) into the unit of &lt;strong&gt;mass&lt;/strong&gt; (amount of matter), an even more basic role in the world of measurements. &lt;br&gt;&#xA;  …&lt;br&gt;&#xA;  The earliest kilogram, then designated as a measure of &lt;strong&gt;weight&lt;/strong&gt; rather than &lt;strong&gt;mass,&lt;/strong&gt; was derived, with difficulty, from the first meter of length. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;Herbert A. Klein, &lt;em&gt;The science of measurement : a historical survey&lt;/em&gt;&lt;br&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;(Dover, New York, 1988). &lt;br&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;See &lt;a href=&quot;https://www.google.com/books/edition/The_Science_of_Measurement/CrmuSiCFyikC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22What%20has%20become%20of%20the%20kilogram%2C%20once%20the%20unit%20of%20weight%20in%20the%20metric%20system%3F%20%22&amp;amp;pg=PA198&amp;amp;printsec=frontcover&amp;amp;bsq=%22What%20has%20become%20of%20the%20kilogram%2C%20once%20the%20unit%20of%20weight%20in%20the%20metric%20system%3F%20%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.google.com/books/edition/The_Science_of_Measurement/CrmuSiCFyikC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22The%20earliest%20kilogram%2C%20then%20designated%20as%20a%20measure%20of%20weight%20rather%20than%20mass%22&amp;amp;pg=PA199&amp;amp;printsec=frontcover&amp;amp;bsq=%22The%20earliest%20kilogram%2C%20then%20designated%20as%20a%20measure%20of%20weight%20rather%20than%20mass%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;An example of a contrary claim&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;It was intended that the kilogram should have the same &lt;strong&gt;mass&lt;/strong&gt; as a cubic decimeter of pure water at maximum density, and the experimental determination of that mass was made by finding the difference of &lt;strong&gt;weight&lt;/strong&gt; in air and in water of a hollow brass cylinder whose exterior dimensions at a temperature of 17.6 &amp;deg;C were, height = 2.437672 decimeters, diameter = 2.428368 decimeters, volume = 11.2900054 cubic decimeters.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;W. Harkness, &quot;The progress of science as exemplified in the art of&lt;br&gt; &#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;weighing and measuring.&quot; Presidential address delivered before&lt;br&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;the Philosophical Society of Washington, December 10, 1887&lt;br&gt;&#xA;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;(Bulletin Phil. Soc. vol 10, pp. 39–86). &lt;a href=&quot;https://www.google.com/books/edition/United_States_Congressional_serial_set/qhI2AQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=weight%20rather%20than%20mass%20kilogram&amp;amp;pg=PA614&amp;amp;printsec=frontcover&amp;amp;bsq=%22It%20was%20intended%20that%20the%20kilogram%20should%20have%20the%20same%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;(source)&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Confusion between mass and weight&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, it is well documented that the concepts of weight and mass were embedded in much confusion, so much so that, as late as 1901, the 3rd CGPM felt compelled to pass the following resolution:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&lt;strong&gt;Declaration on the unit of mass and on the definition of weight; conventional&#xA;  value of &lt;em&gt;g&lt;/em&gt;&lt;sub&gt;n&lt;/sub&gt;&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;Taking into account&lt;/strong&gt; the decision of the Comité International des Poids et Mesures of&#xA;  15 October 1887, according to which the kilogram has been defined as unit of mass;&lt;br&gt;&#xA;  &lt;strong&gt;Taking into account&lt;/strong&gt; the decision contained in the sanction of the prototypes of the Metric&#xA;  System, unanimously accepted by the Conférence Générale des Poids et Mesures on&#xA;  26 September 1889;&lt;br&gt;&#xA;  &lt;strong&gt;Considering&lt;/strong&gt; the necessity to put an end to the ambiguity which in current practice still exists on the meaning of the word &lt;em&gt;weight,&lt;/em&gt; used sometimes for &lt;em&gt;mass,&lt;/em&gt; sometimes for &lt;em&gt;mechanical force&lt;/em&gt;;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;The Conference declares&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;ol&gt;&#xA;  &lt;li&gt;&lt;p&gt;The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram;&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;The word “weight” denotes a quantity of the same nature as a “force”: the weight of a body is&#xA;  the product of its mass and the acceleration due to gravity; in particular, the standard weight&#xA;  of a body is the product of its mass and the standard acceleration due to gravity;&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;li&gt;&lt;p&gt;The value adopted in the International Service of Weights and Measures for the standard&#xA;  acceleration due to gravity is 980.665 cm/s&lt;sup&gt;2&lt;/sup&gt;, value already stated in the laws of some countries.&lt;br&gt;&lt;/p&gt;&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; (See &lt;a href=&quot;https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9-EN.pdf#page=159&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, p. 159.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;The question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What evidence is there that can help us understand whether, in the 1790s, the Commission of Weights and Measures of the French Academy of Sciences meant for the grave, the gram, the kilogram, etc. to be the units of weight (force) or the units of mass? &lt;/p&gt;&#xA;&#xA;&lt;p&gt;For what it's worth, I wouldn't be surprised if the Commission never seriously thought about the question and would be fine with either interpretation of these units. It is also possible they did think about it, but left it intentionally ambiguous whether these were units of force or of mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that it will not suffice to point out that e.g. the available texts keep using the word 'weight' rather than 'mass' in connection with these units—&lt;em&gt;unless&lt;/em&gt; it can be shown that the relevant author clearly meant to use those as distinct technical terms whose meaning is more or less the same as it is today.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Evidence from some texts from the 1820s and 1830s&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Fourier&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;In Fourier's &lt;em&gt;Analytic Theory of Heat&lt;/em&gt;, published in 1822, the author keeps using the construction &lt;em&gt;a mass having a certain weight&lt;/em&gt;. For example,&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Suppose a &lt;strong&gt;mass&lt;/strong&gt; of ice having a definite &lt;strong&gt;weight&lt;/strong&gt; (a kilogramme) to be at temperature 0&lt;br&gt;&#xA;  &lt;sub&gt;On suppose qu'une &lt;strong&gt;masse&lt;/strong&gt; de glace d'un &lt;strong&gt;poids&lt;/strong&gt; determine (un kilogramme) soit a la temperature 0&lt;/sub&gt; (§25)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;To raise a metallic &lt;strong&gt;mass&lt;/strong&gt; having a certain &lt;strong&gt;weight,&lt;/strong&gt; a kilogramme of iron for example, from the temperature 0 to the temperature 1, a new quantity of heat must be added to that which is already contained in the mass.&lt;br&gt;&#xA;  &lt;sub&gt;Pour elever une &lt;strong&gt;masse&lt;/strong&gt; metallique d'un certain &lt;strong&gt;poids,&lt;/strong&gt; par exemple, un kilogramme de fer, depuis la temperature o jusqu'a la temperature 1, il est necessaire d'ajouter une nouvelle quantite de chaleur a celle qui etait deja contenue dans cette masse.&lt;/sub&gt;  (§26)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;The choice of this unit would have been preferable in many respects to that of the quantity of heat required to convert a &lt;strong&gt;mass&lt;/strong&gt; of ice of a given &lt;strong&gt;weight,&lt;/strong&gt; into an equal mass of water at 0, without raising its temperature.&lt;br&gt;&#xA;  &lt;sub&gt;Le choix de cette unite serait preferable a plusieurs egards a ceiui de la quantite' de chaleur ne'eessaire pour convertir une &lt;strong&gt;masse&lt;/strong&gt; de glace d'un &lt;strong&gt;poids&lt;/strong&gt; donne, en une masse pareille deau, sans elever la temperature 0.&lt;/sub&gt;  (§157)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Unfortunately, it is never made clear in the text if 'weight' &lt;em&gt;(poids)&lt;/em&gt; is really meant to be understood as a force, or if it is merely a synonym for mass. To my ear, it sounds as if he wants to use &lt;em&gt;weight&lt;/em&gt; as a term for force (and thus the kilogram as a unit of force). However, I don't think we can rule out the possibility that it was meant as a unit of mass after all, or that Fourier really didn't give much thought to the distinction and, for his purposes, didn't care which way it was interpreted.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Gauss&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As another example, consider Gauss's &lt;a href=&quot;http://21stcenturysciencetech.com/translations/gaussMagnetic.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;'The intensity of the Earth's magnetic force reduced to absolute measurement'&lt;/a&gt;, published in 1832. On the one hand, the author clearly says that by 'weight' he means a force, and by 'mass', the inertial mass:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Hence the effect of a given amount of magnetic flux on a given amount of either the same or the opposite flux at a given distance is comparable to a given motive force, i.e. with the effect of a given accelerating force on a given &lt;strong&gt;mass&lt;/strong&gt;, and since the magnetic fluxes themselves can be known only through the effects, which they bring forth, the latter must directly serve to measure the former.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In order, however, that we may be able to reduce this measurement to definite concepts, units must above all be established for three kinds of magnitudes, namely, the unit of distance, the unit of ponderable &lt;strong&gt;mass&lt;/strong&gt;, and the unit of acceleration. For the third, the force of gravity at the locus of observation can be assumed.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sub&gt;Actio itaque quantitatis datae fluidi magnetici in quantitatem datam vel eiusdem fluidi vel alterius in distantia data comparabilis erit cum vi motrice data, i.e. cum actione vis acceleratricis datae in &lt;strong&gt;massam&lt;/strong&gt; datam, et quum fluida magnetica ipsa non nisi per effectus quos producunt cognoscere liceat, hi ipsi illorum mensurae inseruire debent.&lt;/sub&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sub&gt;Quo igitur hanc mensuram ad notiones distinctas reuocare possimus, ante omnia circa tria quantitatum genera vnitates stabilire oportet, puta vnitatem distantiarum, vnitatem &lt;strong&gt;massarum&lt;/strong&gt; ponderabilium, vnitatem virium acceleratricium . Pro tertia accipi potest grauitas in loco obseruationis&lt;/sub&gt; &lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;… we want to choose &lt;em&gt;A'&lt;/em&gt; for the center of gravity, and, denoting by &lt;em&gt;p&lt;/em&gt; the &lt;strong&gt;weight&lt;/strong&gt; of the body, i.e. the motive force of gravity on its &lt;strong&gt;mass&lt;/strong&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sub&gt;…pro lubitu assumere liceat , pro &lt;em&gt;A'&lt;/em&gt; adoptabimus centrum grauitatis , et denotato &lt;strong&gt;pondere&lt;/strong&gt; corporis,  i .e . vi motrice quam grauitas &lt;strong&gt;massae&lt;/strong&gt; corporis inducit , per &lt;em&gt;p&lt;/em&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;However, he uses the gram as a unit for both weight and mass. Here it is as a unit of weight…&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;…  the thread holds a needle with the usual equipment alone, where the total &lt;strong&gt;weight&lt;/strong&gt; was 496.2 g&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sub&gt;dum filum portabat acum cum sola supellectile ordinaria, vbi &lt;strong&gt;pondus&lt;/strong&gt; integrum erat 496,2 grammatum&lt;/sub&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In experiments I-VIII, in fact, different needles were used, although they had the same &lt;strong&gt;weight&lt;/strong&gt; and the same length (the &lt;strong&gt;weight&lt;/strong&gt; was between 400 g and 440 g)…&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sub&gt;In experimentis I—VIII adhibitae sunt acus diuersae quidem, sed eiusdem fere &lt;strong&gt;ponderis&lt;/strong&gt; et longitudinis (&lt;strong&gt;pondus&lt;/strong&gt; erat inter 400 et 440 grammata)&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;… and here (as a milligram) as a unit of mass:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;If we take the second, the millimeter and the milligram for the units of time, distance&#xA;  and &lt;strong&gt;mass&lt;/strong&gt;,…&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;sub&gt;Accipiendo pro vnitatibus temporis, distantiae et &lt;strong&gt;massae&lt;/strong&gt; minutum secundum, millimetrum et milligramma, …&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Based on this admittedly limited evidence, it seems to me that, in the first several decades of the 19th century, it was possible for authors to alternate between using the gram (or the kilogram) as a unit of mass and using it as a unit of weight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;A summary of the question&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Fourier and Gauss are important, but my main question is what the members of the Commission of Weights and Measures thought about the issue: did they think they were coming up with a unit of mass or with a unit of weight? Did they care about the distinction? Indeed, one part of my question is whether they thought about that issue at all. Either way, I would like to know what the story was!&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update 1&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/a/11899/10854&quot;&gt;It has been suggested&lt;/a&gt; that the important point is this: since &lt;em&gt;g&lt;/em&gt; varies from place to place and the Commission made no mention of that, and since the Commission carried out all comparisons between objects side by side at the same location, the conclusion is that the Commission &lt;em&gt;in fact&lt;/em&gt; came up with a standard of mass, not wieght.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;While that is a fair point, it does not address what I am asking, which is not about what it &lt;em&gt;turns out&lt;/em&gt; that the Commission did, but about what the Commission &lt;em&gt;intended&lt;/em&gt; to do, or what it &lt;em&gt;thought&lt;/em&gt; it did once it finished its job. As best as I can tell, there are four possibilities:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;the Commission really did want to define standards of weight (force) and not mass; since they ignored the variation in &lt;em&gt;g&lt;/em&gt;, one can then also ask how good or poor job they did, but that's a separate question. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission really did want to define standards of mass and not weight. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission was aware that there is a difference between specifying standards for weight and mass; however, they decided to leave it open whether the new standards should be understood as standards of weight or of mass (perhaps because they thought—rightly or wrongly—that the variation in &lt;em&gt;g&lt;/em&gt; wasn't metrologically important for their purposes).  &lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Commission members didn't clearly distinguish between mass and weight in their own minds. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Perhaps there are other options, but I can't think of any at the moment. My question is, which one of these four is correct, and based on what historical evidence?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(An update of the update: I did think of another possibility. Here it is:&lt;/p&gt;&#xA;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;The Commission took it as a matter of course that, even though mass and weight are separate quantities, one can use the same unit to measure both—or, one can say that mass 'is quantified by' its weight.)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update 2&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As pointed out in &lt;a href=&quot;https://hsm.stackexchange.com/a/11901/10854&quot;&gt;this answer&lt;/a&gt;, a law from &lt;a href=&quot;https://web.archive.org/web/20160817122340/http://www.metrodiff.org/cmsms/index.php?page=18_germinal_an_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;18 germinal year 3&lt;/a&gt; (7 Apr 1795) uses the following phrasing (in Article 5):&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;ol start=&quot;5&quot;&gt;&#xA;  &lt;li&gt;The new measures will henceforth be distinguished by the nickname of republicans; their nomenclature is definitively adopted as follows:&lt;/li&gt;&#xA;  &lt;/ol&gt;&#xA;  &#xA;  &lt;p&gt;We will call: …&lt;br&gt;&#xA;  Gram, the absolute weight &lt;em&gt;(le poids absolu)&lt;/em&gt; of a volume of pure water equal to the cube of the hundredth part of the meter, and at the temperature of melting ice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I was able to find two French sources from the second half of the 1700s that explain what &lt;em&gt;le poids absolu&lt;/em&gt; means: &lt;a href=&quot;https://books.google.com/books?id=ok8VAAAAQAAJ&amp;amp;dq=%22le+poids+absolu%22&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;an encyclopedia from 1774&lt;/a&gt; and &lt;a href=&quot;https://www.google.com/books/edition/Trait%C3%A9_th%C3%A9orique_et_experimental_d_hyd/1yqxX5ASZ2YC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;a treatise on hydrodynamics&lt;/a&gt; from Year 4 of the Republic (23 Sep 1795–22 Sep 1796).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The encyclopedia says that 'absolute weight' is the product of volume and specific gravity (&lt;em&gt;Le poids absolu d'un corps est égal au produit de sa grandeur par sa pesanteur spécfique&lt;/em&gt;, see p. 210). And it defines specific gravity just like we still do: 'This term is understood to mean the weight of a body of a certain volume, compared to the weight of another body of the same volume' (&lt;em&gt;L'on entend par ce terme, le poids d'un corps d'un certain volume, comparé au poids d'un autre corps de même volume&lt;/em&gt;, p. 205).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The treatise on hydrodynamics has &lt;a href=&quot;https://www.google.com/books/edition/Trait%C3%A9_th%C3%A9orique_et_experimental_d_hyd/1yqxX5ASZ2YC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA7&amp;amp;printsec=frontcover&amp;amp;bsq=%22De%20m%C3%AAme%2C%20la%20pesanteur%20sp%C3%A9cifique%20d&amp;#39;un%20corps%20est%20le%20poids%20de%20ce%20corps%20sous%20un%20volume%20donn%C3%A9%20pris%20pour%20unit%C3%A9%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;this discussion&lt;/a&gt;. I will reproduce a translation (via Google translate, which does an amazing job) of the whole short section:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;We must remember that the density of a body (solid or fluid) is the quantity of matter of this body, included in a given volume that we take for unity; or, which comes to the same thing, the quotient of the mass of the body, divided by the number of cubic feet or cubic inches (depending on whether you take the cubic foot or the cubic inch as a unit of measure of volume), which form its total volume. Thus, by naming &lt;em&gt;M&lt;/em&gt; the mass, &lt;em&gt;G&lt;/em&gt; its volume or its size, &lt;em&gt;D&lt;/em&gt; the density, we have &lt;em&gt;D&lt;/em&gt; = &lt;em&gt;M&lt;/em&gt; / &lt;em&gt;G&lt;/em&gt;; and therefore &lt;em&gt;M&lt;/em&gt; = &lt;em&gt;G&lt;/em&gt; x &lt;em&gt;D&lt;/em&gt;, that is, that mass is equal to the product of volume by density. We see that the density of a body is always relative to that of another. Care must be taken to assess the volumes of the two bodies in units of the same species.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;Likewise, the specific gravity of a body is the weight of this body under a given volume taken, for unity; or, which comes to the same thing, the quotient of the absolute weight of body, divided by the number of measurements of its volume. If we therefore name &lt;em&gt;P&lt;/em&gt; the absolute weight of a body, &lt;em&gt;G&lt;/em&gt; its volume, &lt;em&gt;p&lt;/em&gt; its specific gravity, we have &lt;em&gt;p&lt;/em&gt; = &lt;em&gt;P&lt;/em&gt; / &lt;em&gt;G&lt;/em&gt;; and therefore &lt;em&gt;P&lt;/em&gt; = &lt;em&gt;G&lt;/em&gt; x &lt;em&gt;p&lt;/em&gt;, that is to say that the absolute weight is equal to the product of the volume by the specific gravity. For example, either the proposed body, fresh water; we know that a cubic foot of fresh water weighs 70 pounds, with very little difference; therefore taking the weight of a cubic foot of water, for the specific gravity of this fluid, we will have &lt;em&gt;p&lt;/em&gt; = 70 pounds, and &lt;em&gt;P&lt;/em&gt; = G x 70 pounds; if &lt;em&gt;G&lt;/em&gt; is 100 cubic feet, it will come &lt;em&gt;P&lt;/em&gt; = 7000 pounds; if &lt;em&gt;G&lt;/em&gt; = 25 cubic feet, we will have &lt;em&gt;P&lt;/em&gt; = 1750 pounds.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;In the same place on earth, or at equal latitudes, the masses are proportional to the weights; we must therefore assume that the densities of two bodies are proportional to their specific gravities, since the densities of these two bodies are masses included in the same volume, and that their specific gravities are two weights also included in the same volume.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I am actually not clear where this leaves us. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One thing that seems to be the case (and is consistent with what is often written about the period) is that the scientists of the day considered both mass and weight to be derived quantities, and density and specific gravity as more fundamental. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;But it's less clear whether absolute weight is meant to be a measure of weight or a measure of mass; here is why.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;As far as what's in the encyclopedia, in today's language, we would say that specific gravity is a dimensionless number (a ratio of weights), and so absolute weight has the dimensions of volume. Now, for the definition to make sense, one has to specify a reference material, but I suppose we may assume that the default reference material is water. This would mean that the specific gravity is exactly 1, so the absolute weight is then numerically exactly equal to the volume, here presumably measured in cm&lt;sup&gt;3&lt;/sup&gt;, so that the gram is then equal to the absolute weight of '1 cm&lt;sup&gt;3&lt;/sup&gt;'. As far as whether this means that the gram is a unit of weight or a unit of mass… let's first consider what the hydrodynamics treatise says, and then come back to it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The hydrodynamics treatise uses the word 'weight' in an unqualified fashion, which makes it sound as if it is truly a force; this would be consistent with Herbert A. Klein's claim that the standard was designated as a measure of weight rather than mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note also that the text uses the same units, pounds &lt;em&gt;(livres)&lt;/em&gt;, for both absolute weight and specific gravity. Though I suppose this would tend to favor the interpretation of absolute weight as true weight (force), and so favor Kelin's claim, I confess I don't really know how to make sense of that. What happened to the units of volume??&lt;/p&gt;&#xA;&#xA;&lt;p&gt;On the other hand, note the following sentence in the first paragraph of the treatise: &lt;em&gt;We see that the density of a body is always relative to that of another.&lt;/em&gt; The paragraph on absolute weight begins with 'likewise', so it looks like what is said about specific gravity should be taken as analogous to what was said about density; and what was said about density is that it is a relative quantity. If that is so, then the treatise is consistent with the encyclopedia after all. (The units that are used are inconsistent, as I've already noted, and are thus of limited help.) &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Moreover, in the third paragraph, the treatise says that if all measurements are done at the same location (so that &lt;em&gt;g&lt;/em&gt; is constant), then densities are proportional to specific gravities. This suggests that it is density that one really wants, and that measured relationships between specific gravities are merely a convenient route to get at the relationships between densities.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Interpreted &lt;em&gt;that&lt;/em&gt; way, not only is the treatise is consistent with the encyclopedia, but also both are then at least &lt;em&gt;not inconsistent&lt;/em&gt; with what W. Harkness says: that although the actual measurements compared weights, from those measurements one was &lt;strong&gt;meant&lt;/strong&gt; to deduce relationships between masses; and so the gram was meant as a measure of mass after all.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;After all this, I am still not clear whether the Commission meant the gram to be taken as a unit of mass or of weight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Update 3&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The more I read contemporary sources, the more I think Option 5 is correct: that the Commission took it for granted that although weight and mass are separate quantities, at least locally one is always proportional to the other; and so one may be quantified by the other; and so we may use the same units for both. This is much like we continue to use the same units for inertial mass and gravitational mass because they are always proportional to each other, with the same constant of proportionality for all objects. (If you think that General Relativity has shown that these two kinds of mass are in fact one and the same, imagine it's some year between 1901 and 1915.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;A typical example, consider the following text from a &lt;a href=&quot;https://www.google.com/books/edition/Traite_elementaire_d_hydrodynamique_ouvr/cfPS9Vjgn7wC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%226.%20La%20masse%20d%27un%20corps%2C%20soit%20solide%2C%20%22&amp;amp;pg=PA3&amp;amp;printsec=frontcover&amp;amp;bsq=%226.%20La%20masse%20d%27un%20corps%2C%20soit%20solide%2C%20%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;1771 treatise on hydrodynamics&lt;/a&gt;, which says literally the following:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;The mass of a body …. is the quantity of proper matter of which this body is composed. &lt;strong&gt;It is known by weight&lt;/strong&gt;; therefore if we have, for example, two bodies A &amp;amp; B, the first of which weighs two or three times as much as the second, the first has two or three times as much mass as the second. This proportionality of the weights to the masses is demonstrated by experience which teaches that in the same empty space of air, under the container of the pneumatic machine, all the bodies, whatever difference there is in their figures &amp;amp; in their dimensions, fall with the same speed…&lt;br&gt;&#xA;  …&lt;br&gt;&#xA;  By mass measurements, I mean pounds, marcs, ounces, etc., that mass &lt;strong&gt;weighs&lt;/strong&gt;…. in measurements of &lt;strong&gt;weight or mass&lt;/strong&gt;, we can take the ounce as a fundamental unit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;From this perspective, the quotes from Fourier's book above make even more sense. In 1822, Fourier still wrote in a way that suggests that mass is quantified by weight.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;And recall that ten years later, Gauss alternated between using the gram as a unit of weight and as a unit of mass.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;At the moment, my bets are on Option 5.&lt;/p&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2020-06-15T12:18:27.613" LastActivityDate="2020-06-15T15:27:21.440" Title="Were the kilogram, the grave, etc. meant to be units of weight, units of mass, or was it ambiguous at the time?" Tags="|units|gauss|fourier|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11897" PostTypeId="2" ParentId="11891" CreationDate="2020-06-11T23:44:15.237" Score="5" Body="&lt;p&gt;After going through so many etymological dictionaries of science (mainly chemistry &amp;amp; mathematics), I still feel that the &lt;em&gt;unabridged&lt;/em&gt; Oxford Dictionary is the best. It is not available for public use but one can buy subscription for a year. Most universities have it.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There you can find the etymology for sure, the best part is that you can see the earliest use with original reference.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;For example, you might wish to see who introduced the term &quot;flavor&quot; in particle physics?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Here is what a typical entry looks like:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Particle Physics.  [An arbitrary choice of name.] A quantized property&#xA;  of quarks which differentiates them into (at least) six varieties&#xA;  (called up and down, charmed and strange, top and bottom) and which&#xA;  can be changed by the weak interaction; an analogous property of&#xA;  leptons which differentiates the electron, the muon, the tau, and&#xA;  their respective neutrinos. Also, a quark or lepton of a particular&#xA;  flavour.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1975   Sci. Amer. Oct. 38/1   In the whimsical terminology that has&#xA;  evolved for the discussion of quarks they are said to come in four&#xA;  flavors, and each flavor is said to come in three colors. (‘Flavor’&#xA;  and ‘color’ are, of course, arbitrary labels; they have no relation to&#xA;  the usual meanings of those words.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1975   Sci. Amer. Oct. 38/1   In the whimsical terminology that has&#xA;  evolved for the discussion of quarks they are said to come in four&#xA;  flavors, and each flavor is said to come in three colors. (‘Flavor’&#xA;  and ‘color’ are, of course, arbitrary labels; they have no relation to&#xA;  the usual meanings of those words.)&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1978   Nature 2 Feb. 406/2   The quarks in the proton and neutron are&#xA;  of two varieties (or flavours), ‘up’ (u) and ‘down’ (d).&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1980   Sci. Amer. July 60/1   For almost 20 years it has been well&#xA;  established that there are at least two flavors, or kinds, of&#xA;  neutrino; one flavor can appear only in association with an electron&#xA;  and the other is always created together with a muon.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1981   M. Gell-Mann in J. H. Mulvey Nature of Matter viii. 176   The&#xA;  electromagnetic and weak forces are ‘flavour forces’: the electric&#xA;  charge of a particle depends on its flavour; weak forces are flavour&#xA;  exchange forces.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1981   Sci. Amer. Feb. 65/1   A total of five flavors have definitely&#xA;  been observed (they are called up, down, strange, charm and bottom)&#xA;  and the existence of a sixth flavor (top) is all but certain.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1982   Sci. Amer. Nov. 134/1   Charm, like up, down and strange, is a&#xA;  quark flavor.&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;1985   Sci. Amer. Apr. 66/3   Like leptons, the quarks experience weak&#xA;  interactions that change one species, or flavor, into another.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2020-06-11T23:44:15.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11898" PostTypeId="2" ParentId="11893" CreationDate="2020-06-12T04:38:22.430" Score="3" Body="&lt;p&gt;Not quite. There was some vagueness in Dedekind's early formulations, but the tendency was to use &quot;&lt;em&gt;Körper&lt;/em&gt;&quot; or &quot;field&quot; when the multiplication is commutative from the start. As a curiosity, in Russian general division algebra is called &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BB%D0%BE_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;тело&lt;/em&gt;&lt;/a&gt;, literally &quot;body&quot;, which is apparently the translation of German &lt;em&gt;Körper&lt;/em&gt;, as opposed to commutative &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;поле&lt;/em&gt;&lt;/a&gt;, the translation of English &quot;field&quot;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The original use of German &lt;em&gt;Zahlenkörper&lt;/em&gt; (literally, number body), or &lt;em&gt;Körper&lt;/em&gt; for short, now translated as field, is attributed to Dedekind's &lt;a href=&quot;https://archive.org/details/vorlesungenberz01dirigoog/page/n456/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Supplement XI to Dirichlet’s &lt;em&gt;Vorlesungenueber Zahlentheorie&lt;/em&gt;, §159 (1871)&lt;/a&gt; and his also &lt;em&gt;Stetigkeit und irrationale Zahlen&lt;/em&gt; (1872) (the latter has been translated into English in &lt;a href=&quot;https://www.gutenberg.org/files/21016/21016-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essays on the Theory of Numbers (1901)&lt;/a&gt;). Here is the relevant passage from Supplement XI:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;So far, we have only the considered the whole &#xA;  numbers &lt;span class=&quot;math-container&quot;&gt;$0, ± 1, ± 2, ± 3, ± 4...$&lt;/span&gt;, namely all those numbers that arise from the number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; by repeated addition and subtraction; these numbers are reproduced by addition, subtraction and multiplication, or in other words, the sums, differences and products of two whole numbers are again whole numbers. In contrast, the fourth basic operation, the division, leads to the&#xA;  more comprehensive notion of the rational numbers under which name the quotients of any two integers [with non-zero denominator] will be understood; these rational numbers are evidently reproduced by all four basic operations.&lt;/em&gt;&lt;/p&gt;&#xA;  &#xA;  &lt;p&gt;&lt;strong&gt;&lt;em&gt;In the future, we want to call every system of real or complex numbers that has this fundamental property of reproduction Zahlenkörper or Körper for short&lt;/strong&gt;; the system &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; of all rational numbers is one such Körper, and it is the simplest example of one. This Körper &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; of rational numbers now consists of&#xA;  whole numbers and fractions, i. e. non-integers; the former we want to call rational integers to distinguish them from new integers to be introduced.&lt;/em&gt;&quot; [my translation and emphasis]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;On the one hand, Dedekind does not explicitly require multiplication to be commutative in this &quot;definition&quot;, and this would be in line with Galois' definition of a group, which was not required to be Abelian. On the other hand, he appears to be talking about subfields of &quot;real or complex numbers&quot;, which are commutative, and all of his examples are fields of algebraic numbers (Gaussian rationals are considered right after the definition). In particular, he considers neither non-commutative division algebras nor finite fields, even though again, his &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is not formally required to be infinite.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Systems of hypercomplex numbers were popular around that time, and some of them (quaternions) had non-commutative multiplication and inverses for non-zeros. Whether Dedekind was inclined to call them &lt;em&gt;Körper&lt;/em&gt; as well is hard to say. According to &lt;a href=&quot;http://jeff560.tripod.com/f.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;, Wildstrom argued that he was. Miller also suggests that the term did not catch on until early 1890-s, in particular, Moore translated it into English as &quot;field&quot; in 1893 (but the aforementioned 1901 translation of Dedekind uses &quot;body&quot; instead). Here is Huntington's surmise from a paper presented to AMS in 1904:&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;&quot;&lt;em&gt;Closely connected with the theory of groups is the theory of fields, suggested by GALOIS, and due, in concrete form, to DEDEKIND in 1871. The word field is the English equivalent for DEDEKIND’s term Körper; KRONECKER’s term Rationalitätsbereich, which is often used as a synonym, had originally a somewhat different meaning. The earliest expositions of the theory from the general or abstract point of view were given independently by WEBER and by Moore, in 1893 [...] The earliest sets of independent postulates for abstract fields were given in 1903 by Professor Dickson and myself; all these sets were the natural extensions of the sets of independent postulates that had already been given for groups.&lt;/em&gt;&quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;Again, if we want &quot;fields&quot; to be a generalization of groups, rather than Abelian groups, it would have to cover what is now called division algebras. However, in the aforementioned &lt;a href=&quot;https://archive.org/details/mathematicalpape00inteiala/page/210/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore's paper of 1893&lt;/a&gt;, and in both 1903 papers of &lt;a href=&quot;https://www.jstor.org/stable/1986446&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dickson, Definitions of a Field by Independent Postulates&lt;/a&gt; and &lt;a href=&quot;https://www.jstor.org/stable/1986449&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huntington, Definitions of a Field by Sets of Independent Postulates&lt;/a&gt; the commutativity of multiplication is explicitly assumed. &lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-06-15T20:43:26.570" LastActivityDate="2020-06-15T20:43:26.570" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11899" PostTypeId="2" ParentId="11896" CreationDate="2020-06-12T04:42:14.823" Score="7" Body="&lt;p&gt;It was Jean Richer who demonstrated the fact that, described in a modern conceptual framework, there were variations in the strength of the earth's apparent gravitational field (in the earth-fixed frame) from point to point. His result was published in 1679. However, the interpretation was complicated and led to a controversy between Newton and Cassini as to whether the earth was prolate or oblate. This controversy began in 1720, and the physics was not correctly analyzed for a long time. E.g., John Bernoulli wrote an analysis in 1734 using Descartes' vortices. Further expeditions were organized in 1735 to high and low latitudes in order to do high-precision astronomical observations.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;So we need to keep in mind that science didn't move as fast in those days as it does now, and also that there is a general tendency when looking back at the history of science to imagine that scientific confusions were resolved quickly when in fact they took a long time to clear up. The people developing a system of measures in 1790 had probably had their educations and spent their working lives in an environment where the variation of g was exotic cutting-edge science, not something you would learn about in the first chapter of a freshman physics textbook.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Language is conservative, and therefore it's not surprising that people speaking French and English hadn't yet decided to start using two different words for mass and weight. Even today, English lacks a commonly used verb meaning to measure the mass of something -- we have &quot;to weigh,&quot; but &quot;to mass&quot; is an uncommon usage. We still use phrases like &quot;Cavendish weighed the earth.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;What actually matters is what implicit operational definition the Académie des sciences carried out. Their methods all involved working with objects (distilled water, a hollow brass cylinder, and a &lt;a href=&quot;https://en.wikipedia.org/wiki/Grave_(unit)&quot; rel=&quot;noreferrer&quot;&gt;solid metallic weight&lt;/a&gt;) &lt;em&gt;in Paris&lt;/em&gt; (which is, after all, the center of the universe). The standards produced in this way would be duplicated and transported to various places, but because comparisons were always carried out side by side, the different copies would all have the same &lt;em&gt;mass&lt;/em&gt;, not the same &lt;em&gt;weight&lt;/em&gt;. Therefore the operational definition they implemented was a definition of a unit of mass, regardless of what words were current in scientific usage at the time.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[EDIT] The OP asks in comments whether the state of mind of the academy's committee was #1, #2, #3, or #4. It is clearly not the case that the committee considered this issue, that it came to a consensus, and that this consensus was recorded by history -- for if so, then it would never have been necessary in 1901 for the CGPM to make the clarification quoted by the OP. Most likely the various members of the committee had different states of mind on this issue. It's possible that at least some members of the commission had in mind a possibility #5, which was to forgo conceptual definitions in favor of operational definitions. (Operationalism was explicitly laid out by Bridgman as a broad philosophical stance until 1927, but in the narrower context of defining a system of units it seems pretty obvious to me that it's the only reasonable approach.) In any case I doubt that we will ever know what any member of the committee wanted, thought, or intended as a conceptual definition, unless, e.g., some historical source turns up such as a letter from one member of the committee to another.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The question was very well researched and fleshed out with some interesting quotes from various sources. However, I disagree with the OP's contention that these sources contradict one another. Harkness is just saying that the unit was in operational terms a unit of mass, but that the operationalization of it was carried out by comparing gravitational forces at a given location. Klein is describing the spread of a conceptual shift, not a redefinition.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-06-12T23:09:05.387" LastActivityDate="2020-06-12T23:09:05.387" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11901" PostTypeId="2" ParentId="11896" CreationDate="2020-06-12T21:57:38.840" Score="0" Body="&lt;p&gt;Whatever the scientists on the committee intended, or even the operational reality of using a standard chunk of metal in different parts of the world, the gram was legislated as a measure of &lt;strong&gt;weight&lt;/strong&gt;, not mass.&lt;/p&gt;&#xA;&lt;p&gt;Following the citations in &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_metric_system#The_units_of_weight_and_length&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Wikipedia article about the history of the metric system&lt;/a&gt;, we find the text of the 1795 law that established the metric system, containing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Les nouvelles mesures seront distinguées dorénavant par le surnom de républicaines; leur nomenclature est définitivement adoptée comme il suit:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;Gramme, le &lt;strong&gt;poids&lt;/strong&gt; absolu d'un volume d'eau pure égal au cube de la centième partie du mètre , et à la température de la glace fondante&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;The new measures will henceforth be distinguished by the sobriquet of republicans; their nomenclature is definitively adopted as follows:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;Gram, the absolute &lt;strong&gt;weight&lt;/strong&gt; of a volume of pure water equal to the cube of the hundredth part of the meter, and to the temperature of melting ice.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;-- &lt;em&gt;Décret relatif aux poids et aux mesures du 18 germinal an 3 (7 avril 1795)&lt;/em&gt;, cited by Wikipedia  at&#xA;&lt;a href=&quot;https://web.archive.org/web/20160817122340/http://www.metrodiff.org/cmsms/index.php?page=18_germinal_an_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.archive.org/web/20160817122340/http://www.metrodiff.org/cmsms/index.php?page=18_germinal_an_3&lt;/a&gt; A live version can be found at &lt;a href=&quot;https://www.metrodiff.org/wp/metrologie/histoire-de-la-metrologie/18-germinal-an-3/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.metrodiff.org/wp/metrologie/histoire-de-la-metrologie/18-germinal-an-3/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The original question shows that the French word corresponding to mass is &lt;em&gt;masse&lt;/em&gt; and weight is &lt;em&gt;poids&lt;/em&gt;, so I don't think there is any ambiguity.&lt;/p&gt;&#xA;&lt;p&gt;It's quite ironic that they adopted Lavoisier's unit of weight after they chopped his head off.&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="-1" LastEditDate="2020-06-18T08:32:24.750" LastActivityDate="2020-06-12T22:23:18.170" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11902" PostTypeId="1" CreationDate="2020-06-13T00:21:01.370" Score="3" ViewCount="515" Body="&lt;p&gt;There is no Nobel Prize for mathematics. However, have there been any Nobel Prizes for the use of applied mathematics to model the real world?&lt;/p&gt;&#xA;" OwnerUserId="12655" LastEditorUserId="26" LastEditDate="2021-02-20T15:12:06.657" LastActivityDate="2021-02-26T20:24:52.597" Title="Nobel Prize for Applied Mathematics" Tags="|mathematics|nobel-prize|" AnswerCount="2" CommentCount="11" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11903" PostTypeId="1" CreationDate="2020-06-13T00:54:17.513" Score="4" ViewCount="152" Body="&lt;p&gt;Since Europe has had numerous people in various fields make several contributions over the centuries,&#xA;was the same nature of study and academia reflected in the colonies of the western hemisphere? Specifically, &#xA;the question is about British America, where most of the modern Ivy League (and The College of William and Mary) &#xA;had already been established before the American Revolutionary War. Was there a colonial equivalent of the Royal Society?&#xA;Did academics who studied in the thirteen American colonies travel overseas to work with their contemporaries and colleagues?&#xA;I am aware that Benjamin Franklin established the University of Pennsylvania, but were there any scientists who had been working&#xA;on groundbreaking research by the time the colonies started to rebel against the United Kingdom?&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="26" LastEditDate="2021-02-20T15:11:59.557" LastActivityDate="2022-04-15T17:23:22.417" Title="What did scientific research look like during the colonial era of British America?" Tags="|academia|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11905" PostTypeId="1" CreationDate="2020-06-13T19:46:44.023" Score="3" ViewCount="139" Body="&lt;p&gt;1) Zeno's oxymoronic fleet, stationary arrow: One of the earliest infinity paradoxes, of course, is the flying arrow of Zeno which can't possibly be moving since it takes a finite amount of time to move half the distance AB from one point A to another C and again a finite time for half of that, the length AB/2, ad infinitum, and, since the sum of an infinite number of finite times must be infinite, the arrow could never arrive at C. The resolution comes with the introduction of the concept of the convergerce of an infinite sum, which the paradox points to. (Note &lt;a href=&quot;https://mathoverflow.net/questions/116445/positive-results-coming-from-paradoxes?noredirect=1&quot;&gt;Kalai's answer&lt;/a&gt; here.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;2) The two-equal-one paradox: Take a horizontal line segment L of length one, form an equilateral triangle from that, and then collapse the triangle by pushing its apex down to L, forming two triangles. Iterate on the apex of each of the two triangles, and then the next four triangles, ad infinitum, and we arrive at the apparent merging of a curve of length two at each iteration to a curve of length one. Resolution and lesson? You'd better impose smoothness, or differentialbility on the collapsing curve. (See this &lt;a href=&quot;https://hsm.stackexchange.com/questions/11240/what-is-the-history-of-staircase-or-4-paradox/11348#11348&quot;&gt;HSE question&lt;/a&gt; pointed out to me by David Renfro and &quot;In Search of Infinity&quot; by Vilenkin.)&#xA;. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;What other apparent paradoxes involving going to a geometric limit have been historically instructive and how, i.e., what can they teach us?&lt;/strong&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I was thinking of explicitly excluding Berkeley's ghosts, but that led to omega-epsilon gymnastics and eventually to Robinson's nonstandard analysis, I believe.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(I believe there are some very sophistcated topological paradoxes of a similar nature, but can't recall their names or where I read of them.)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;Note that this excludes primarily set-theoretic or logical paradoxes such as Russell's Barbershop or the Hilbert/Cantor Hotel paradoxes although they are fascinating and important. Bertrand's probability paradox might suit the bill though since it relies on geometrical constructs.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="1830" LastEditDate="2020-06-14T12:01:31.120" LastActivityDate="2020-06-14T12:01:31.120" Title="Lessons from apparent paradoxes in geometric limits" Tags="|mathematics|geometry|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="11906" PostTypeId="2" ParentId="11797" CreationDate="2020-06-13T22:21:41.377" Score="0" Body="&lt;p&gt;I later realized that I posed several questions. I try to give a partial answer the simplest of those questions (when). An article by Ute and Wilfried Brauer for the &lt;a href=&quot;http://eatcs.org/index.php/component/content/article/497&quot; rel=&quot;nofollow noreferrer&quot;&gt;silver jubilee of the European Association for Theoretical Computer Science&lt;/a&gt; mentions that Maurice Nivat (then editor-in-chief) communicated the decision of dividing the journal Theoretical Computer Science into Track A and B in Bulletin of the EATCS no. 45, p.2,3, October 1991. (I could not access that article yet). &lt;/p&gt;&#xA;&#xA;&lt;p&gt;Maurice Nivat wrote an &lt;a href=&quot;https://doi.org/10.1016/0304-3975(51)90001-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Editorial&lt;/a&gt; to Theoretical Computer Science 78(1) as well &amp;ndash; he mentions &quot;the huge increase of the number of subject areas in computer science that are open to formal, mathematical or logic methods and the huge increase of the number of researchers at work&quot;. Consequently, it was decided that the number of volumes per year would be increased by fifty percent, and that they would split the editorial board into Part A and Part B. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The conference ICALP started to have plenary sessions, and also two parallel sessions (with Track A and Track B papers) only in 1997. In the &lt;a href=&quot;https://doi.org/10.1007/3-540-63165-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;foreword&lt;/a&gt;, Pierpaolo Degano, Roberto Gorrieri and Alberto Marchetti-Spaccamela write: &lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;ICALP '97 was organised differently than before and accommodated further&#xA;  events, to react positively to the new challenges that the theoretical science community&#xA;  faces in the information technology society. Indeed, our community has&#xA;  developed and now utilizes several approaches and different methodologies that&#xA;  require increased specialization. As a consequence, there is a growing number&#xA;  of specialized conferences and workshops, and it is difficult for researchers to&#xA;  follow the recent developments on specialized research topics. ICALP '97 was&#xA;  a first step towards having a conference offering a single unifying environment&#xA;  while leaving room for specialization. In such an event, the computer science&#xA;  community interested in the development of formal methods and methodologies&#xA;  can stress the relationships that exist among different branches.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;I suspect that the A/B divide was caused by the growth of the field of theoretical computer science, and the development of specialized methods used in its subfields, and that it gradually took place before the &quot;formal&quot; split we can observe in the handbook of TCS, the journal TCS and ICALP. (This summary sounds much like a platitude to me, and there's certainly more to say about it.)&lt;/p&gt;&#xA;" OwnerUserId="9523" LastActivityDate="2020-06-13T22:21:41.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11907" PostTypeId="2" ParentId="11902" CreationDate="2020-06-14T02:19:58.947" Score="7" Body="&lt;p&gt;Here is a list of &quot;Mathematicians who were awarded Nobel prize&quot; taken from&#xA;&lt;a href=&quot;https://www.maa.org/external_archive/devlin/devlin_04_04.html&quot; rel=&quot;noreferrer&quot;&gt;this paper&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1902 Lorentz (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1904 Rayleigh (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1911 Wien (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1918 Planck (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1921 Einstein (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1922 Bohr (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1929 de Broglie (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1932 Heisenberg (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1933 Schroedinger (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1933 Dirac (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1945 Pauli (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1950 Russell (Literature)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1954 Born (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1962 Landau (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1963 Wigner (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1965 Schwinger (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1965 Feynman (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1969 Tinbergen (Economics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1975 Kantorovich (Economics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1983 Chandrasekhar (Physics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1994 Selten (Economics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;1994 Nash (Economics)&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The work of all of them, except Russell qualifies as &quot;applied mathematics&quot;. &lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-06-14T02:19:58.947" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="11908" PostTypeId="2" ParentId="11896" CreationDate="2020-06-14T21:41:27.973" Score="2" Body="&lt;p&gt;I would choose #2. The commission reported that Lavoisier and Hauy used a beam scale to determine the mass of the unit volume of water. A beam scale measures mass. As it was the standard weighing device in 1790, there was no need to state that it does not measure the force of weight. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;One of the commission members was Laplace. Laplace knew the difference between mass and the force of weight, and he knew that gravity varies on the surface of the earth. He had shown that the measured relation between g and latitude λ (Krafft's formula: g = (1 + 0,0052848 sin&lt;sup&gt;2&lt;/sup&gt;λ) g&lt;sub&gt;0&lt;/sub&gt;) can be reconciled with the measured oblateness of the earth by assuming that the earth is composed of concentric shells of different density. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;The entire commission knew that gravity varies with latitude because in 1790 the initial proposal was to define the metre by the seconds pendulum at 45 degrees latitude. This latititude was specified because gravity varies with latitude.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;[EDIT] in reply to the comments:&lt;/p&gt;&#xA;&#xA;&lt;p&gt;My source for stating that Lavoisier used a balance in 1793 was &lt;a href=&quot;https://www.persee.fr/doc/rhs_0048-7996_1959_num_12_1_3698&quot; rel=&quot;nofollow noreferrer&quot;&gt;Birembaut&lt;/a&gt; (&quot;la grande balance de précision que Fortin avait construite pour Lavoisier&quot;). The conclusion that a balance measures mass, not force, was mine. &lt;/p&gt;&#xA;&#xA;&lt;p&gt;I think &lt;a href=&quot;https://books.google.nl/books?id=lsdJAAAAMAAJ&amp;amp;pg=PA597&amp;amp;lpg=PA597&amp;amp;dq=%22weight+in+commerce%22+hutton&amp;amp;source=bl&amp;amp;ots=Hah5yyvDe6&amp;amp;sig=ACfU3U1CfoEUrlnyd2CyWVlC0qaTV7yh0A&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiM8ZHpj4TqAhUDzaQKHZWMBx4Q6AEwCnoECAcQAQ#v=onepage&amp;amp;q=%22weight%20in%20commerce%22%20hutton&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hutton's dictionary&lt;/a&gt; (1815) is a useful source describing what weight meant at the time. Hutton distinguishes weight in physics, and weight in commerce:&lt;/p&gt;&#xA;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Weight, in physics, .. on different parts of the earth's surface, the weight of the same body is different.  ... the weights of the same body, at different distances above the earth, are inversely as the squares of the distances from the centre. ... A body loses so much of its weight, as is equal to the weight of the .. fluid.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Weight, in commerce, denotes a body of a known weight, appointed to be put into a balance against other bodies, whose weight is required to to be known. &lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&#xA;&lt;p&gt;The &lt;a href=&quot;https://books.google.nl/books?id=FufDNJHvgFEC&amp;amp;pg=RA1-PA278&amp;amp;lpg=RA1-PA278&amp;amp;dq=18841+grains+grave&amp;amp;source=bl&amp;amp;ots=B8Aenskenh&amp;amp;sig=zanQ-d8GXWWN3q6bG2YBbh9bZOM&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjy0-C5rr7MAhWBLMAKHcG2AhwQ6AEIKTAD#v=onepage&amp;amp;q=%22question%20principale%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;report of 19 january 1793&lt;/a&gt; shows that, for the french government, a primary purpose of the new weight standard was commercial: it had to be sufficiently reliable for the specification of the silver content of the new coin, the franc.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="2667" LastEditDate="2020-06-15T15:27:21.440" LastActivityDate="2020-06-15T15:27:21.440" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="11909" PostTypeId="2" ParentId="11674" CreationDate="2020-06-15T02:07:50.903" Score="4" Body="&lt;blockquote&gt;&#xA;  &lt;p&gt;This changed in 1964 when IBM introduced System/360, a scalable&#xA;  computing system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;It's obvious now that we have scalable computing systems that we can have scalable computing systems.  Before then, it wasn't.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;There were in my opinion two insights that made this possible:&lt;/p&gt;&#xA;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Firstly, the separation of architecture from implementation: that the machine as seen by the programmer, and the machine as seen by the hardware designers, were different things.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Secondly, that the cost of upgrading of a computer system was heavily skewed towards the cost of rewriting working programs.&lt;/p&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&#xA;&lt;p&gt;Prior to that, there was more of a design culture oriented to the immediate goals for the machine being designed.  Don't forget, also, the division into &quot;scientific&quot; and &quot;business&quot; computers, optimizing for different targets. Scientific computers had fixed word lengths and fast floating point and relatively poor I/O; business computers tended to decimal-string arithmetic with extensive I/O capability.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;To build a family of computers requires some careful design to ensure that architectural features don't add undue cost to the low end (alternatively stated, that you're willing to accept the extra cost at the low end for the sake of the family), and consideration for the low end doesn't badly constrain performance at the high end.  For that, the state of the art needs to have matured a little from the very early days.&lt;/p&gt;&#xA;" OwnerUserId="12665" LastActivityDate="2020-06-15T02:07:50.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11910" PostTypeId="1" CreationDate="2020-06-16T15:10:32.867" Score="3" ViewCount="206" Body="&lt;p&gt;In the following &lt;a href=&quot;https://hsm.stackexchange.com/questions/2788/was-riemanns-zeta-function-paper-based-on-eulers-unproven-observation#&quot;&gt;answer&lt;/a&gt; it has been claimed that &quot;The reference here is not to Euler's 1737 &quot;factorization&quot; of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity for &lt;span class=&quot;math-container&quot;&gt;$s&amp;gt;1$&lt;/span&gt; appears.&quot; Can anyone please tell me on which page can I find the following fact? I  found a book by Euler, which is translated into English and it is titled as 'Introduction to  Analysis of the  Infinite ' Book 1. I found at page 235 of that book that Euler was aware of the fact that harmonic series, i.e. when &lt;span class=&quot;math-container&quot;&gt;$s=1$&lt;/span&gt;, is divergent. But I can't still find the case where &lt;span class=&quot;math-container&quot;&gt;$s&amp;gt;1$&lt;/span&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;Here is &lt;a href=&quot;http://www.claymath.org/sites/default/files/ezeta.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilkins's translation of Riemann's paper&lt;/a&gt;. Remarks at the&#xA;  start of &lt;a href=&quot;http://www.claymath.org/sites/default/files/ezeta.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;the paper&lt;/a&gt; show that Riemann was well aware of the&#xA;  convergence issues:&quot;&lt;em&gt;For this investigation my point of departure is&#xA;  provided by the observation of Euler that the product&lt;/em&gt;&#xA;  &lt;span class=&quot;math-container&quot;&gt;$\prod\frac{1}{1-\frac{1}{p^{s}}}=\sum\frac{1}{n^{s}}$&lt;/span&gt; &lt;em&gt;if one&#xA;  substitutes for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; all prime numbers, and for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; all whole numbers.&#xA;  The function of the complex variable &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; which is represented by these&#xA;  two expressions, wherever they converge, I denote by &lt;span class=&quot;math-container&quot;&gt;$\zeta(s)$&lt;/span&gt;. Both&#xA;  expressions converge only when the real part of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is greater than&#xA;  &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;; at the same time an expression for the function can easily be&#xA;  found which always remains valid&lt;/em&gt;&quot;. &lt;strong&gt;The reference here is not to&#xA;  Euler's 1737 &quot;factorization&quot; of the harmonic series but to 1748&#xA;  Introductio in Analysin Infinitorum, where the identity for &lt;span class=&quot;math-container&quot;&gt;$s&amp;gt;1$&lt;/span&gt;&#xA;  appears.&lt;/strong&gt; Although Euler did not work with convergence in the modern&#xA;  sense he already knew the difference between &lt;span class=&quot;math-container&quot;&gt;$s=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$s&amp;gt;1$&lt;/span&gt; cases&#xA;  since he summed the Basel series in 1735 (&lt;span class=&quot;math-container&quot;&gt;$s=2$&lt;/span&gt;, later all even &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;),&#xA;  and used unlimited growth of &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2006-03.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;harmonic series to prove infinitude of&#xA;  primes&lt;/a&gt; in that 1737 paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12673" LastEditorUserId="4251" LastEditDate="2020-06-22T07:12:41.330" LastActivityDate="2020-06-22T07:12:41.330" Title="Reference for Euler's Introductio in Analysin Infinitorum" Tags="|mathematics|number-theory|euler|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11911" PostTypeId="2" ParentId="11910" CreationDate="2020-06-16T16:12:59.857" Score="4" Body="&lt;p&gt;Yes, the reference is to &lt;a href=&quot;https://en.wikipedia.org/wiki/Introductio_in_analysin_infinitorum&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Wikipedia - Introductio in analysin infinitorum&quot;&gt;Euler's &lt;em&gt;Introductio in Analysin Infinitorum (Introduction to Analysis of the Infinite)&lt;/em&gt;&lt;/a&gt; in which Euler investigates infinite series. The reference seems to point to &lt;strong&gt;Volume 1&lt;/strong&gt;, &lt;strong&gt;Chapter XV&lt;/strong&gt;, which is titled &quot;Concerning Series Arising From the Expansion of Factors.&quot;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The English translation by Ian Bruce of the &lt;em&gt;Introductio&lt;/em&gt; is in the public domain, and I'm guessing that you have found that file. If not, the full file can be found on Ian Bruce's site, &lt;a href=&quot;http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;17centurymaths.com - EULER&amp;#39;S INTRODUCTIO IN ANALYSIN INFINITORUM&quot;&gt;17centurymaths.com&lt;/a&gt; (or just &lt;a href=&quot;http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch15vol1.pdf&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;17centurymaths.com - Euler&amp;#39;s Introduction to Analysis: Vol. 1 Chapter XV&quot;&gt;click here for Chapter XV&lt;/a&gt;). The identity is specifically found on &lt;strong&gt;pages 470 and 471&lt;/strong&gt; of that translation, &lt;strong&gt;§273&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;The identity is first explicity stated with &lt;span class=&quot;math-container&quot;&gt;$s = 1$&lt;/span&gt; in §273. §274, however, is where he states the more generalised identity for case where &lt;span class=&quot;math-container&quot;&gt;$s \neq 1$&lt;/span&gt; (referring to &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; instead), investigating this further in subsequent sections. Be warned, however, that Euler's notation, even post-translation, is very different from the modern standard (e.g. using simply &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; sans parentheses to denote the natural logarithm function).&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="12674" LastActivityDate="2020-06-16T16:12:59.857" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11912" PostTypeId="2" ParentId="11902" CreationDate="2020-06-16T16:36:14.440" Score="2" Body="&lt;p&gt;Merton and Scholes received the 1997 Nobel prize in Economics for the famous Black–Scholes–Merton model, which is a mathematical model for the dynamics of a financial market containing derivative investment instruments and the foundation of mathematical finance.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-02-26T20:24:52.597" LastActivityDate="2021-02-26T20:24:52.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11913" PostTypeId="1" CreationDate="2020-06-16T16:40:24.410" Score="9" ViewCount="717" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_von_Neumann&quot; rel=&quot;nofollow noreferrer&quot;&gt;John von Neumann&lt;/a&gt; wrote the following in his essay &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Mathematician&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from &amp;quot;reality&amp;quot; it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much &amp;quot;abstract&amp;quot; inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical. In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, did John von Neumann hate pure mathematics that became too abstract? In many ways, he seems to be the complete opposite of G.H Hardy.&lt;/p&gt;&#xA;" OwnerUserId="6459" LastEditorUserId="4251" LastEditDate="2023-12-05T22:04:35.980" LastActivityDate="2023-12-05T22:04:35.980" Title="Did John von Neumann hate pure mathematics that became too abstract?" Tags="|mathematics|mathematicians|biographical-details|quote|von-neumann|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11915" PostTypeId="1" AcceptedAnswerId="11916" CreationDate="2020-06-16T22:40:31.743" Score="3" ViewCount="273" Body="&lt;p&gt;When and where did Legendre first publish or write about his &lt;a href=&quot;https://en.wikipedia.org/wiki/Legendre%27s_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;conjecture&lt;/a&gt; that there is a prime between consecutive square numbers?&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n^2 &amp;lt; p &amp;lt; (n+1)^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&#xA;&lt;p&gt;I have looked through edition 1 and 2 of his &lt;em&gt;Essai sur la Théorie des Nombres&lt;/em&gt; but I can't find any reference. &lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="4251" LastEditDate="2022-10-19T16:27:41.450" LastActivityDate="2022-10-19T16:27:41.450" Title="When and where was Legendre's Conjecture first published?" Tags="|mathematics|reference-request|number-theory|legendre|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11916" PostTypeId="2" ParentId="11915" CreationDate="2020-06-17T00:09:21.010" Score="8" Body="&lt;p&gt;Actually Legendre proved, starting from a false theorem, that between &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L+2\sqrt{L}$&lt;/span&gt; there is always a prime number.  See the second edition of &lt;em&gt;Essai sur la Théorie des Nombres&lt;/em&gt; at page 406 (paragraph 409).&#xA;From the same theorem &lt;a href=&quot;http://www.numdam.org/article/NAM_1855_1_14__281_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Desboves&lt;/a&gt; proved in 1855 as a corollary (p. 290, Corollary II) that there is always a prime number between two consecutive squares, and actually that there are at least two primes between two consecutive squares (Theorem II). Desboves was well aware that Legendre's proof of the main proposition was wrong, and in the article assumes Legendre's result as a postulate. Also notice that Desboves explicitly states that Legendre did not enunciate Corollary II, but only &amp;quot;Corollary I&amp;quot;, i.e., that there is a prime number between &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n+2\sqrt{n}+1$&lt;/span&gt; (this is not exact as Legendre discarded the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$+1$&lt;/span&gt;&amp;quot;), but it is clear that the result is the same.&#xA;Summarizing: Legendre stated the conjecture (as a theorem) in a slightly different form in his &lt;em&gt;Essai&lt;/em&gt;, while Desboves was the first to explicitly consider the problem of finding primes between two consecutive squares.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="72" LastEditDate="2022-07-02T17:46:00.270" LastActivityDate="2022-07-02T17:46:00.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11918" PostTypeId="1" AcceptedAnswerId="11919" CreationDate="2020-06-17T18:52:25.090" Score="-1" ViewCount="1038" Body="&lt;p&gt;Specifically, is there any research paper or record of where someone renowned in their field performed experiments by using the properties of physics, engineering or mathematics to develop a musical instrument that created the best sound and had negligible chance to to go off key or to lose its pitch? The piano and violin are two instruments that come to mind for this question. If there are any papers that speak of efforts such as this, please post an answer below. Any help is tremendously appreciated. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="10106" LastEditDate="2020-06-18T14:53:53.427" LastActivityDate="2020-06-18T14:53:53.427" Title="Was there ever a mathematician or physicist who spearheaded a concerted effort to create a flawless musical instrument?" Tags="|mathematicians|experimental-physics|engineering|physicists|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11919" PostTypeId="2" ParentId="11918" CreationDate="2020-06-17T19:48:32.290" Score="4" Body="&lt;p&gt;There is probably nothing &amp;quot;flawless&amp;quot; in this world. But mathematicians and physicists indeed took part in the improvement of musical instruments. See, for example Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Well_temperament&quot; rel=&quot;nofollow noreferrer&quot;&gt;Well temperament&lt;/a&gt;&#xA;which mentions the invention of the musical scale for modern keyboard instruments. Many people, scientists and musicians participated in this development. Some historians credit the invention to &lt;a href=&quot;https://en.wikipedia.org/wiki/Simon_Stevin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simon Stevin&lt;/a&gt; (physicist), others to &lt;a href=&quot;https://en.wikipedia.org/wiki/Zhu_Zaiyu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zhu Zaiyu&lt;/a&gt; (mathematician). Anyway, a modern grand piano is probably a &amp;quot;perfect instrument&amp;quot;, and several mathematicians and physicists took active part in the development of the theory on which its design is based. French Wikipedia gives much more detail on this history and&#xA;mentions &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert&quot; rel=&quot;nofollow noreferrer&quot;&gt;d'Alembert&lt;/a&gt; (mathematician) in this connection. Russian Wikipedia mentions mathematicians&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Henricus_Grammateus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henricus Grammateus&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Marin_Mersenne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marin Mersenne&lt;/a&gt; who contributed.&lt;/p&gt;&#xA;&lt;p&gt;Hellenistic physicist/engineer &lt;a href=&quot;https://en.wikipedia.org/wiki/Ctesibius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ctesibius&lt;/a&gt; is credited with the invention of&#xA;the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pipe_organ&quot; rel=&quot;nofollow noreferrer&quot;&gt;pipe organ&lt;/a&gt;&#xA;on which the invention of piano was later based.&lt;/p&gt;&#xA;&lt;p&gt;Let me add that the very first person who investigated the musical scale from the point of view of mathematics was Pythagoras himself. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_tuning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pythagorean tuning&lt;/a&gt;. So applications to music are as old as mathematics itself.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-18T11:52:41.657" LastActivityDate="2020-06-18T11:52:41.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11920" PostTypeId="1" AcceptedAnswerId="11922" CreationDate="2020-06-18T01:41:45.583" Score="4" ViewCount="235" Body="&lt;p&gt;Does anyone here know what technology or instrument was used to typeset the first edition of the well-received textbook &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387902791&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Number Fields by Daniel A. Marcus&lt;/a&gt;? I ask because the original edition looked like it was typeset using a toaster (forgive my exaggeration). A preview of the first edition can be viewed on &lt;a href=&quot;https://books.google.ca/books?id=kmnlBwAAQBAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt;. It was published in 1995 by Springer, whereas I have read papers from much earlier where the typesetting was pretty good (for example, this &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.bams/1183496203&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper of Lehmer from 1932&lt;/a&gt;), so I can't fathom why the typesetting was so bad. It looks like it was written on a typewriter, though given the usage of notation from algebraic number theory, it probably was not an actual typewriter.&lt;/p&gt;&#xA;&#xA;&lt;p&gt;(Thankfully, a LaTeX-ed &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3319902326&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;second edition&lt;/a&gt; was finally published by Springer in 2018.)&lt;/p&gt;&#xA;" OwnerUserId="12679" LastEditorUserId="26" LastEditDate="2021-02-20T15:11:27.227" LastActivityDate="2021-02-20T15:11:27.227" Title="Why was the original typesetting of Number Fields by Marcus so horrible?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11921" PostTypeId="2" ParentId="11918" CreationDate="2020-06-18T03:49:57.663" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Leon_Theremin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leon Theremin&lt;/a&gt;, born in St. Petersburg, Russia in 1896, was a youthful enthusiast of both music and physics.  As a young man working at the Physical Technical Institute in Petrograd, he noticed that something odd happened when he hooked up audio circuits to an electrical device called an oscillator in a certain configuration. The oscillator produced an audible tone when he held his hands near it, and he could shift the tone just by waving his hands back and forth. He conceived the idea of an electronic musical instrument: &lt;strong&gt;not just an acoustic instrument embellished or amplified electronically, but an instrument that would produce purely electronic music&lt;/strong&gt;.&lt;/p&gt;&#xA;&#xA;&lt;blockquote&gt;&#xA;  &lt;p&gt;While adapting the dielectric device by adding circuitry to generate&#xA;  an audio tone, Theremin noticed that the pitch changed when his hand&#xA;  moved around. In October 1920 he first demonstrated this to&#xA;  Ioffe who called in other professors and students to hear.&#xA;  Theremin recalled trying to find the notes for tunes he remembered&#xA;  from when he played the cello, such as the Swan by&#xA;  Saint-Saëns. &lt;strong&gt;By November 1920 Theremin had given his first&#xA;  public concert with the instrument&lt;/strong&gt;, now modified with a horizontal&#xA;  volume antenna replacing the earlier foot-operated volume&#xA;  control. He named it the &quot;etherphone&quot;, to be known as the&#xA;  Терменвокс (Termenvox) in the Soviet Union, as the Thereminvox in&#xA;  Germany, and later as the &quot;&lt;strong&gt;theremin&lt;/strong&gt;&quot; in the United States.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;&lt;p&gt;&lt;strong&gt;Theremin’s invention is still the only musical instrument, which is played without any physical contact.&lt;/strong&gt; The sound it can produce runs from a drone to a whine, but is always somewhat otherworldly, and is best recognised for its prominence in the soundtrack to various early Hollywood sci-fi movies, like “&lt;strong&gt;The Day the Earth Stood Still&lt;/strong&gt;.”  However, the theremin does make a memorable pop music appearance in the background of the Beach Boys’ “&lt;strong&gt;Good Good&lt;/strong&gt;.”&lt;/p&gt;&#xA;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8FNvg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8FNvg.jpg&quot; alt=&quot;enter image description here&quot;&gt;&lt;/a&gt;&#xA;&lt;sub&gt; Musician Alexandra Stepanoff plays the theremin on NBC radio in 1930. &lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-06-18T03:59:07.480" LastActivityDate="2020-06-18T03:59:07.480" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11922" PostTypeId="2" ParentId="11920" CreationDate="2020-06-18T06:27:55.287" Score="6" Body="&lt;p&gt;The book came out in 1977, not 1995; look at the year at the end of the Foreword that Marcus wrote. My copy has a copyright from 1977 too. LaTex or even TeX was not an option in 1977 since that was the year before the original version of TeX was released. The typesetting in the original version of Marcus was clearly done on a typewriter.&lt;/p&gt;&#xA;&lt;p&gt;I agree it is unusual for a professionally published book from the 1970s not to be typeset better, but if you look at Springer  Lecture Notes in Mathematics from that period you will find volumes that were prepared on typewriters too. Among math books, Serre’s Lie Algebras and Lie Groups (1965), Fulton’s Algebraic Curves (1969), the Proc. Symp. Pure Math volume on Mathematical Developments from the Hilbert Problems (1974), and Mumford’s Red Book (1976) were all prepared on typewriters. So I’d say Marcus was in good company.  Around 10 years after &amp;quot;Number Fields&amp;quot; first came out, Cambridge Univ. Press published &amp;quot;Local Fields&amp;quot; by Cassels in 1986 and it was quite clearly prepared on a typewriter.&lt;/p&gt;&#xA;&lt;p&gt;That there are math symbols (Greek letters, doubled Z and R for &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;, etc.) does not mean it could not be done on a typewriter. Typewriters had special mechanisms to type a range of technical symbols. Articles  were submitted to journals back then as typewriter copy.&lt;/p&gt;&#xA;&lt;p&gt;Look &lt;a href=&quot;https://mathoverflow.net/questions/19930/writing-papers-in-pre-latex-era&quot;&gt;here&lt;/a&gt; for a discussion of mathematical typing before LaTeX.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2020-06-25T00:29:03.737" LastActivityDate="2020-06-25T00:29:03.737" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11923" PostTypeId="2" ParentId="11918" CreationDate="2020-06-18T11:01:56.380" Score="2" Body="&lt;p&gt;Maybe it does not have a direct relationship to your question, but I remember having a teacher in college for a Mechanics course, which developed a musical instrument himself in order to have more bass-sounds in a folk Catalan musical ensemble. Unfortunately, I haven't found any documentation related to it in English. The only information I have found is in Catalan in this page:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.iec.cat/recerca/projecte1.asp?codi=PT2008-S0301-AGULLO01&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.iec.cat/recerca/projecte1.asp?codi=PT2008-S0301-AGULLO01&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The abstract of the project seems semi-related to your question, as basically it is the design of a new instrument with all the physics in mind:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Design of a xeremia barítona (barytone), non-existent until now, which is a type of wind&#xA;instrument that has been designed to perform Catalan chamber music.&#xA;This musical instrument will be used with other traditional Catalan&#xA;instruments such as the tenora and the tible (type of shawm).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hope it helps!&lt;/p&gt;&#xA;" OwnerUserId="12680" LastActivityDate="2020-06-18T11:01:56.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11925" PostTypeId="1" CreationDate="2020-06-19T08:17:36.903" Score="2" ViewCount="62" Body="&lt;p&gt;The Rydberg formula on the wavelengths of a spectral line in chemical elements was first stated empirically in 1888 by Johannes Rydberg. A theoretical explanation of the formula wouldn't arive until Niels Bohr's quantum-mechanical model of the atom in 1913. A quick search tells me that Rydberg became severely ill precisely in 1913, only to pass away six years later. It seems not unlikely that Rydberg learned of Bohr's explanation. Are there any accounts of this?&lt;/p&gt;&#xA;" OwnerUserId="12703" OwnerDisplayName="Mr. Palomar" LastEditorUserId="26" LastEditDate="2021-02-20T15:11:20.557" LastActivityDate="2021-02-20T15:11:20.557" Title="Did Rydberg ever learn of Bohr's quantum-mechanical explanation of his formula?" Tags="|physics|biographical-details|quantum-mechanics|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11926" PostTypeId="1" CreationDate="2020-06-19T11:33:21.777" Score="0" ViewCount="185" Body="&lt;p&gt;In this MathEducator StackExchange article, &amp;quot;&lt;a href=&quot;https://matheducators.stackexchange.com/questions/5999/notation-of-points-with-coordinates&quot;&gt;Notation of points with coordinates&lt;/a&gt;&amp;quot;, it's posed the question about what is the best notation for geometrical points and their coordinates: &lt;span class=&quot;math-container&quot;&gt;$P(3, 4)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$P = (3, 4)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Peharps History of Mathematics could help with this discussion: what is the first known written register of such association point &amp;lt;-&amp;gt; coordinates? Did &lt;span class=&quot;math-container&quot;&gt;$P(3, 4)$&lt;/span&gt; come first, for instance?&lt;/p&gt;&#xA;&lt;p&gt;Descartes' &amp;quot;La Géométrie&amp;quot; has no such explicit association.&lt;/p&gt;&#xA;&lt;p&gt;Somehow, this question is related to the posts &amp;quot;&lt;a href=&quot;https://hsm.stackexchange.com/questions/2378/when-do-we-see-for-the-first-time-the-use-of-the-cartesian-coordinates/2382#2382&quot;&gt;When do we see for the first time the use of the Cartesian coordinates?&lt;/a&gt;&amp;quot; and &amp;quot;&lt;a href=&quot;https://hsm.stackexchange.com/questions/3312/cartesian-coordinate-system-in-newtons-work&quot;&gt;Cartesian coordinate system in Newton's work&lt;/a&gt;&amp;quot;. But, in the references given, I've found no explicit association either.&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="4251" LastEditDate="2020-06-22T07:12:31.493" LastActivityDate="2021-11-13T14:06:00.577" Title="History of points with coordinates notation" Tags="|mathematics|notation|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11927" PostTypeId="2" ParentId="11926" CreationDate="2020-06-19T12:47:24.443" Score="2" Body="&lt;p&gt;IMO, the notation &lt;span class=&quot;math-container&quot;&gt;$P(x,y)$&lt;/span&gt; is &amp;quot;more correct&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;See e.g. Felix Klein, &lt;a href=&quot;https://books.google.it/books?id=JazDAgAAQBAJ&amp;amp;pg=PA10&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elementary Mathematics from an Advanced Standpoint: Geometry (1908)&lt;/a&gt;, page 10.&lt;/p&gt;&#xA;&lt;p&gt;As you have already noted, the usual representation of the coordinate plane is quite late: I suppose that we cannot find it in any 19th Century treatise regarding Analytic Geometry.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2020-06-19T12:47:24.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11928" PostTypeId="1" CreationDate="2020-06-16T19:37:56.433" Score="25" ViewCount="1336" Body="&lt;p&gt;How did André Weil react when Pierre Deligne finally solved the most important and hardest of the Weil conjectures ?&lt;/p&gt;&#xA;&lt;p&gt;Is there any written account on this ?&lt;/p&gt;&#xA;&lt;p&gt;I guess Serre's and Grothendieck's (unfortunate) reaction are well-known, but I have never seen anything written about Weil's own experience on the whole matter.&lt;/p&gt;&#xA;" OwnerDisplayName="THC" LastEditorUserId="2324" LastEditDate="2021-09-26T05:30:18.280" LastActivityDate="2021-09-26T05:30:18.280" Title="What was Weil's reaction to Deligne's proof?" Tags="|mathematicians|reference-request|weil|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11929" PostTypeId="2" ParentId="11928" CreationDate="2020-06-16T21:13:52.810" Score="24" Body="&lt;p&gt;My recollection is that, when Deligne completed the proof of the Weil conjectures, Weil was in Paris and Deligne went to his apartment and explained the proof to him personally. Deligne was on good terms with Weil. Grothendieck was also in Bures around that time, and I remember seeing Messing explaining the proof to him. I think both Grothendieck and Weil reacted positively, although Grothendieck was disappointed that Deligne hadn't proved his standard conjectures, which remain open to this day. As the quote in Merel's answer indicates, Weil also was surprised that Deligne hadn't deduced it from some general positivity statement in algebraic geometry.&lt;/p&gt;&#xA;" OwnerDisplayName="anon" LastActivityDate="2020-06-16T21:13:52.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11930" PostTypeId="2" ParentId="11928" CreationDate="2020-06-16T20:40:18.963" Score="29" Body="&lt;p&gt;In the commentary on his 1954 paper &lt;em&gt;&amp;quot;Abstract versus classical algebraic geometry&amp;quot;&lt;/em&gt; [1954h], p. 539 in volume II of his &amp;quot;Œuvres scientifiques&amp;quot;, Weil wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Quant à mes commentaires sur l'hypothèse de Riemann en dimension &amp;gt; 1, cf. [1941]. Mon espoir qu'on la tirerait d'un théorème de positivité valable en géométrie algébrique abstraite ne s'est pas trouvé vérifié, malgré une ingénieuse suggestion de Serre (Ann. of Math. 71 (1960), pp. 392–394); la démonstration de Deligne est toute arithmétique.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I read the last sentence 30 years ago, and never forgot it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(English translation: &lt;em&gt;As for my comments on the Riemann hypothesis in dimension &amp;gt; 1, cf. [1941]. My hope that it would be obtained from a positivity theorem in abstract algebraic geometry has not been realised, despite an ingenious suggestion of Serre (Ann. of Math. 71 (1960), pp. 392–394); Deligne’s proof is entirely arithmetic.&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="Loïc Merel" LastActivityDate="2020-06-17T12:46:39.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11931" PostTypeId="1" AcceptedAnswerId="11932" CreationDate="2020-06-20T01:15:14.627" Score="2" ViewCount="222" Body="&lt;p&gt;Compactness for first-order predicate logic was first proven as a corollary of (Gödel 1930). Does anyone know a reference for the first proof of the compactness of propositional logic?&lt;/p&gt;&#xA;&lt;p&gt;Some proofs from around 1915 (e.g. Lowenheim's proof of the Lowenheim-Skolem theorem) seem to&#xA;invoke it implicitly and I am wondering whether it was considered so obvious as to never have received a formal proof.&lt;/p&gt;&#xA;" OwnerUserId="10497" LastActivityDate="2021-04-14T17:56:58.480" Title="When was compactness theorem for propositional logic first proven?" Tags="|mathematical-logic|logic|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11932" PostTypeId="2" ParentId="11931" CreationDate="2020-06-20T08:43:09.853" Score="5" Body="&lt;p&gt;For a general overview of the history of first order logic see &lt;a href=&quot;https://plato.stanford.edu/entries/logic-firstorder-emergence/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, The Emergence of First-Order Logic&lt;/a&gt;. On the history of compactness theorem more specifically see &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01445349308837208&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dawson, The compactness of first-order logic: from Gödel to Lindström&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/books?id=5ya4A0w62skC&amp;amp;pg=PA44&amp;amp;lpg=PA44&amp;amp;dq=van+Heijenoort++Dreben+introductory+note+to+Godel&amp;amp;source=bl&amp;amp;ots=FnM20j9bmq&amp;amp;sig=ACfU3U14qyGggtSCsrVqm8N6CBaxyCBnFQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjL6dvc9I_qAhUKRK0KHQuwD8MQ6AEwBHoECAUQAQ#v=onepage&amp;amp;q=van%20Heijenoort%20%20Dreben%20introductory%20note%20to%20Godel&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;van Heijenoort, Dreben, Introductory note on 1929, 1930 and 1930a to Kurt Gödel: Collected Works: Volume I&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What these sources note is that neither the notion of first order logic nor the semantic/syntactic distinction essential to the formulation of the compactness theorem appear in the literature before Gödel's 1929 dissertation. So it is fair to say that the compactness theorem was not only first proved, but even first stated, by Gödel in 1930, who inferred it from a generalization of his completeness theorem. None of the sources credits Löwenheim even with loosely anticipating the compactness theorem (a set of sentences is satisfiable if and only if every finite subset of it is satisfiable), in contrast to the downward Löwenheim-Skolem theorem (if a set of sentences is satisfiable it is satisfiable on a countable model).&lt;/p&gt;&#xA;&lt;p&gt;The credit for anticipating compactness goes rather to Skolem's &lt;em&gt;Einige Bemerkungen zur axiomatischen Begriindung der Mengenlehre&lt;/em&gt; (1923). The 1923 paper is known for giving a proof of a version of the downward Löwenheim-Skolem theorem that did not appeal to the axiom of choice, which he previously used in &lt;em&gt;Logisch-kombinatorische Untersuchungen iiber die Erfiillbarkeit oder Beweisbarkeit mathematischer Satze nebst einem Theoreme über dichte Mengenwas&lt;/em&gt; (1920). In its turn, this prior proof fixed Löwenheim's 1915 proof, which is considered faulty because it implicitly used the König's lemma not known at the time.&lt;/p&gt;&#xA;&lt;p&gt;Here is Dawson:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Gödel's proofs employed&#xA;Skolem's methods; but, unlike Skolem, Gödel carefully distinguished between&#xA;syntactic and semantic notions. The relation between the works of the two men has&#xA;been examined by Vaught (1974, 157-159) and, in great detail, by van Heijenoort&#xA;and Dreben 1986. All three commentators agree that both the completeness and&#xA;compactness theorems were implicit in Skolem 1923, but that no one before Gödel&#xA;drew them as conclusions, not even after Hilbert and Ackermann, in their 1928 book&#xA;Grundzüge der theoretischen Logik singled out first-order logic for attention and&#xA;explicitly posed the question of its completeness.&lt;/em&gt;&#xA;&lt;br&gt; &lt;br&gt;&#xA;&lt;em&gt;Vaught attributes the delay in the&#xA;enunciation of the completeness theorem to 'the lack of able logicians who knew and&#xA;appreciated both the notion of model and the notion of logistic system', but he notes&#xA;that such an excuse does not apply in the case of the compactness theorem, since it is&#xA;a purely semantic statement. Rather, he opines that perhaps 'the compactness&#xA;theorem was not [...] inferred by Skolem or others' at that time simply because,&#xA;'when viewed as a theorem of pure model theory [... it] appears wholly unlikely'. Alternatively, Godel himself attributed the 'blindness of logicians' toward the&#xA;completeness theorem (and, by extension, the compactness theorem as well) to the&#xA;'widespread lack, at that time, of the required epistemological attitude', not only&#xA;'toward metamathematics' but 'toward non-finitary reasoning'.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Van Heijenoort and Dreben add the following (Theorem IX is a generalization of the completeness theorem):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1930 (below, page 119) the generalization is labeled Theorem IX&#xA;and is obtained immediately, by means of the completeness theorem,&#xA;from Theorem X, which does not appear in 1929 and is known today as&#xA;the compactness theorem... The proof sketched for Theorem&#xA;X differs essentially from the 1929 sketch for the generalization (Theorem IX) only in one regard... But, since provability in a formal system&#xA;is now discarded, Gödel's argument for compactness comes very close&#xA;to Skolem's (suggested) argument in 1923a for his generalization of the&#xA;Löwenheim-Skolem Theorem.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-04-14T17:56:58.480" LastActivityDate="2021-04-14T17:56:58.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11933" PostTypeId="2" ParentId="11931" CreationDate="2020-06-20T10:03:23.347" Score="4" Body="&lt;p&gt;See John Dawson, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01445349308837208&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Compactness of First-order Logic From Gödel to Lindström (HPL,1993)&lt;/a&gt;, page 18:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After 1934, [...] the only person who seems to have recognized the importance of&#xA;compactness was the Russian A. I. Maltsev. Beginning in 1936, he published 'a&#xA;seminal run' of papers in what would now be called model-theoretic algebra, papers in which he 'obtained applications [of logic] to group theory of greater technical virtuosity than the possibly more basic applications to algebra later on found [...] by Henkin and Abraham Robinson' (Sabbagh 1991). Three of those papers (1936,1940 and 1941) are of interest here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In his first published work, written in German, Maltsev 1936 [Maltsev, A.I. 1936 '&lt;em&gt;Untersuchungen aus dem Gebiete der mathematischen Logik&lt;/em&gt;', Matematicheskii Sbornik, n.s., 1, 323-336.] devoted his efforts to generalizing two theorems, one for the propositional calculus and the other for the restricted functional calculus. The theorems in question were Godel's compactness theorem and Skolem's result that no denumerable set of formulas of first-order logic can completely characterize the structure of the natural numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See English translation into: A.I. Mal'cev, &lt;a href=&quot;https://books.google.it/books?id=GeS2uhg74cAC&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Metamathematics of Algebraic Systems: Collected Papers 1936-1967 (North Holland, 1971)&lt;/a&gt;, page 1:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This article is devoted to generalizing two theorems, one for propositional calculus (PC) and the other for first-order predicate logic (FOPL).&#xA;The first theorem is due to Gödel [K. Gödel, &lt;em&gt;Die Vollsthdigkeit der Axiome des logischen Funktionenkalkuls&lt;/em&gt; (1936),] and can be formulated as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For any countable system of formulas of PC to be consistent, it&#xA;is sufficient that every finite part of the system be consistent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, see also &lt;em&gt;On a General Method for Obtaining Local Theorems in Group Theory by A. Mal'cév&lt;/em&gt;, Review by Leon Henkin and Andrzej Mostowski (Jsl, 1959):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Historical note&lt;/em&gt;. The formulation and proof of the &amp;quot;general local theorem&amp;quot; [&lt;em&gt;If every finite subset of a given (possibly non-denumerable) set of first-order sentences is satisfiable, then so is the whole set&lt;/em&gt;] for denumerable sets of first-order sentences is of course due to Gödel in 1930. The corresponding result for non-denumerable sets of formulas of propositional calculus was given by Mal'cev in 1936.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The result can be easily derived from A.Tarski's 1930 paper (&lt;a href=&quot;https://books.google.it/books?id=2uhra9PEFZsC&amp;amp;pg=PA30&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On some fundamental concepts of Metamathematics&lt;/em&gt;&lt;/a&gt;, published in German in 1931) were it is stated (without proof) as Theorem 11, expressing the Finiteness property of the consequence relation: thus, it was &amp;quot;obviously&amp;quot; applicable to propositional calculus.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2020-06-21T11:38:20.933" LastActivityDate="2020-06-21T11:38:20.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11934" PostTypeId="5" CreationDate="2020-06-21T09:42:52.013" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-21T09:42:52.013" LastActivityDate="2020-06-21T09:42:52.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11935" PostTypeId="4" CreationDate="2020-06-21T09:42:52.013" Score="0" Body="Jean-Baptiste Joseph Fourier (1768-1830) was a French mathematician and physicist best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-22T11:52:00.917" LastActivityDate="2020-06-22T11:52:00.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11936" PostTypeId="5" CreationDate="2020-06-21T10:40:51.243" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-21T10:40:51.243" LastActivityDate="2020-06-21T10:40:51.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11937" PostTypeId="4" CreationDate="2020-06-21T10:40:51.243" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-21T10:40:51.243" LastActivityDate="2020-06-21T10:40:51.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11938" PostTypeId="1" CreationDate="2020-06-21T12:37:07.507" Score="0" ViewCount="372" Body="&lt;p&gt;Enumeration of all positive fractions recently has gained renewed interest (see the list below).&lt;/p&gt;&#xA;&lt;p&gt;By translation invariance we can be sure that in all intervals (n, n+1] of the real axis, there are the same number of fractions:&#xA;#(n, n+1] = #(m, m+1] for all natural numbers n and m.&lt;/p&gt;&#xA;&lt;p&gt;That is a matter of symmetry, independent of the method used for counting them.&lt;/p&gt;&#xA;&lt;p&gt;Cantor's famous sequence&lt;/p&gt;&#xA;&lt;p&gt;1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...&lt;/p&gt;&#xA;&lt;p&gt;however has half of its hits in the first unit interval and less than 1/n in the interval (n, n+1]. This raises two important questions: (1) What are the precise values of the function f(n) = #(n, n+1]/#(0, 1] in the limit? (2) how can the deviation from f(n) = 1 be interpreted?&lt;/p&gt;&#xA;&lt;p&gt;None of these question could be answered in MathOverflow or Math.StackExchange.&lt;/p&gt;&#xA;&lt;p&gt;My question: Can it be that this topic has never been treated in the literature or in lessons on set theory, for instance in the first years of set theory? I don't believe this but could not yet find any evidence.&lt;/p&gt;&#xA;&lt;p&gt;List of sites&lt;/p&gt;&#xA;&lt;p&gt;What fraction of fractions does Cantor's famous sequence enumerate?&#xA;&lt;a href=&quot;https://mathoverflow.net/questions/362791/what-fraction-of-fractions-does-cantors-famous-sequence-enumerate&quot;&gt;https://mathoverflow.net/questions/362791/what-fraction-of-fractions-does-cantors-famous-sequence-enumerate&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Relative abundance of rationals in Cantor's bijection?&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/3708845/relative-abundance-of-rationals-in-cantors-bijection&quot;&gt;https://math.stackexchange.com/questions/3708845/relative-abundance-of-rationals-in-cantors-bijection&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Translation invariance&#xA;&lt;a href=&quot;https://groups.google.com/forum/#!topic/sci.logic/OIrleZcHXW0%5B1-25%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://groups.google.com/forum/#!topic/sci.logic/OIrleZcHXW0%5B1-25%5D&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Enumerating the rationals&#xA;&lt;a href=&quot;https://groups.google.com/forum/#!topic/sci.math/D2UYGbt9Qh8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://groups.google.com/forum/#!topic/sci.math/D2UYGbt9Qh8&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Translation invariance&#xA;&lt;a href=&quot;https://groups.google.com/forum/#!topic/scivszfc/tt_qy_ymAgo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://groups.google.com/forum/#!topic/scivszfc/tt_qy_ymAgo&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Enumerations of the Rationals - two methods&#xA;&lt;a href=&quot;https://groups.google.com/forum/#!topic/sci.math/uudISZF3An4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://groups.google.com/forum/#!topic/sci.math/uudISZF3An4&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12699" LastEditorUserId="4251" LastEditDate="2020-06-23T13:24:27.837" LastActivityDate="2022-06-25T13:06:08.427" Title="Has Cantor's irregular enumeration of rationals ever been discussed?" Tags="|mathematics|set-theory|cantor|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11939" PostTypeId="5" CreationDate="2020-06-21T13:52:44.917" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-21T13:52:44.917" LastActivityDate="2020-06-21T13:52:44.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11940" PostTypeId="4" CreationDate="2020-06-21T13:52:44.917" Score="0" Body="Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-22T11:51:28.977" LastActivityDate="2020-06-22T11:51:28.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11941" PostTypeId="5" CreationDate="2020-06-22T12:15:16.517" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-06-22T12:15:16.517" LastActivityDate="2020-06-22T12:15:16.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11942" PostTypeId="4" CreationDate="2020-06-22T12:15:16.517" Score="0" Body="René Descartes (1596-1650) was a French philosopher, mathematician, and scientist." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-06-23T12:06:07.290" LastActivityDate="2020-06-23T12:06:07.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11943" PostTypeId="1" AcceptedAnswerId="11944" CreationDate="2020-06-22T15:11:47.427" Score="3" ViewCount="1459" Body="&lt;p&gt;I'm a high school teacher, and I was just wondering why complex numbers are called 'complex'. I have read that Gauss coined the term. But I couldn't find any reference where it was explained.&lt;/p&gt;&#xA;&lt;p&gt;I also know that 'komplex' in German comes from Latin 'complexus' which means 'complex' but also 'intertwined'. And I thought to myself, 'Well, complex multiplication does intertwine the real and imaginary parts'. But that's just a guess.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have a reference on this?&lt;/p&gt;&#xA;" OwnerUserId="12704" LastEditorUserId="4251" LastEditDate="2021-09-08T06:05:07.870" LastActivityDate="2021-09-08T06:05:07.870" Title="Why are complex numbers called 'complex'?" Tags="|mathematics|terminology|reference-request|complex-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11944" PostTypeId="2" ParentId="11943" CreationDate="2020-06-22T20:41:47.120" Score="4" Body="&lt;p&gt;Complex numbers were used long before Gauss. They appeared for the first time in 16th century when people found a formula for solving cubic equations. One problem with this formula is that even for simplest equations like &lt;span class=&quot;math-container&quot;&gt;$x^3-x=0$&lt;/span&gt; which have 3 real solutions, square roots of negative numbers occur in the formula (they cancel in the end, when you do calculation correctly). So use of the formula requires calculations with complex numbers, and people started investigating the rules of such calculations. They were called various names, &amp;quot;imaginary&amp;quot; numbers, &amp;quot;impossible&amp;quot; numbers, all these terms reflecting peoples confusion with them lasting till the beginning of 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Gauss found how to represent them geometrically, but even here he was not the first.&#xA;Predecessors were Jean-Robert Argand and Caspar Wessel. Various authors combined these three names with words such as plane, diagram etc.&#xA;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Complex_plane#Argand_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Argand diagram&lt;/a&gt; in Wikipedia.&lt;/p&gt;&#xA;&lt;p&gt;Eventually the modern terminology emerged in 19th century: &amp;quot;complex numbers&amp;quot;, meaning that they consist of two parts, real and imaginary. &amp;quot;Imaginary number&amp;quot; is used sometimes&#xA;to denote a complex number which is not real, or more frequently a number whose real part is zero (a.k.a &amp;quot;pure imaginary&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Gauss also investigated numbers of the form &lt;span class=&quot;math-container&quot;&gt;$m+ni$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; are integers. These are still called &amp;quot;Gauss integers&amp;quot;, they have applications in questions about ordinary integers (number theory). The reason is that some primes can be factored using Gauss integers, like &lt;span class=&quot;math-container&quot;&gt;$5=(1+2i)(1-2i)$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-22T20:58:36.963" LastActivityDate="2020-06-22T20:58:36.963" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11945" PostTypeId="1" CreationDate="2020-06-23T03:05:20.867" Score="1" ViewCount="252" Body="&lt;p&gt;I was reading about the history of Newton's Method. Newton used a cubic equation, &lt;span class=&quot;math-container&quot;&gt;$x^3 - 2x - 5 = 0$&lt;/span&gt;, to show the efficacy of his method around 1670. I was wondering that why Newton would choose this particular depressed cubic equation, one could have used the already existing &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubic_equation#Cardano%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cardano's formula&lt;/a&gt;, an analytical/symbolic method, for a depressed cubic equations to get the real value of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. There are two additional roots of the given cubic equation which are complex and complex numbers hadn't entered the mainstream at that time.&lt;/p&gt;&#xA;&lt;p&gt;It seems like Newton was more interested in coming up with an alternative way of solving problems and hence he decided to use an equation which was solvable using another existing method. Previously, the numerical techniques were used more in an ad hoc manner or out of necessity - either the solution was not obtainable using symbolic techniques or they lacked the proper mathematical tools at that time to come up with a symbolic formula. In this regard, Newton could be considered the first or one of the pioneers who laid down the foundation of numerical analysis, the &amp;quot;new analysis&amp;quot;, as an alternative approach.&lt;/p&gt;&#xA;&lt;p&gt;Do you agree with me to some extent? I'd appreciate it if you could correct me.&lt;/p&gt;&#xA;&lt;p&gt;Later, in 1687 in his book &lt;em&gt;Philosophiae NaturalisPrincipia Mathematica&lt;/em&gt;, Newton used his method of approximation to solve the &lt;a href=&quot;https://www.math.uni-bielefeld.de/documenta/vol-ismp/13_deuflhard-peter.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;nonpolynomial equation&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$x-e \sin(x)=z$&lt;/span&gt;, also known as Kepler's problem, which doesn't have an analytical solution. In the equation, &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is the known mean anomaly, &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; the eccentricity, and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; the eccentric anomaly to be determined.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;10.2.6 Algebraic Nonintegrability of Ovals&lt;/p&gt;&#xA;&lt;p&gt;Section 6, Book 1, of the Principia is devoted to the solution of the&#xA;so-called Kepler problem. The problem consists in finding the area of&#xA;a focal sector of the ellipse and is equivalent to the solution for x&#xA;of the equation &lt;span class=&quot;math-container&quot;&gt;$x — e \sin x = z$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; given). ...In Lemma 28,&#xA;Section 6, Newton demonstrated that this problem cannot be resolved in&#xA;finite algebraic terms (§13.3). In Proposition 30, Section 6, he&#xA;showed that the determination of the position of a body orbiting in a&#xA;parabolic trajectory (such that the area law is valid for the focus)&#xA;is instead algebraic (see chapter 11). But to deal with the Kepler&#xA;problem for elliptic trajectories, polynomial equations are not&#xA;enough. In Section 6, Newton therefore illustrated how the roots of&#xA;the Kepler’s equation can be determined via his method of successive&#xA;approximations, which is to say, via infinite series (see §7.5,&#xA;figure 7.10).2' Infinite series occur throughout the Principia&#xA;(especially in the final Scholium to Section 13, Book 1: Proposition&#xA;45, Book 1: and Proposition 10, Book 2), and &lt;strong&gt;Newton might well have&#xA;had this in mind when he stated that his work was based on &lt;em&gt;new&#xA;analysis&lt;/em&gt;&lt;/strong&gt;. He also employed quadrature techniques, another key&#xA;element of his new analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[Isaac Newton on Mathematical Certainty and Method, Issue 4, By Niccolò Guicciardini, 2009; page #248]&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="10106" LastEditDate="2020-06-23T16:06:22.260" LastActivityDate="2020-07-31T02:08:15.880" Title="Newton as the first one to establish numerical analysis as a new field of study" Tags="|mathematics|mathematicians|newton|computation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11946" PostTypeId="1" CreationDate="2020-06-23T17:11:18.297" Score="5" ViewCount="565" Body="&lt;p&gt;After Uranus was discovered and its orbit calculated, its future orbit was calculated, and its future positions as seen from Earth were calculated.&lt;/p&gt;&#xA;&lt;p&gt;And observers of Uranus began to notice that Uranus was deviating from its calculated positions. Eventually the possibility that an undiscovered planet was perturbing the orbit of Uranus was used by Adams and Le Verrier to calculate the orbit of Neptune and led to the discovery of Neptune.&lt;/p&gt;&#xA;&lt;p&gt;So how large in angles and/or in miles/kilometers were the deviations in the apparent position of Uranus that were used to calculate the orbit and apparent position of Neptune?&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="26" LastEditDate="2021-02-20T15:10:28.183" LastActivityDate="2021-02-20T15:10:28.183" Title="How large were the differences in the orbit of Uranus which led to the calculation of the existence of Neptune?" Tags="|astronomy|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11947" PostTypeId="2" ParentId="11946" CreationDate="2020-06-23T23:59:37.330" Score="4" Body="&lt;p&gt;Starting off with a crude order-of-magnitude estimate, the anomalous acceleration of Uranus (&lt;span class=&quot;math-container&quot;&gt;$m_1$&lt;/span&gt;) due to Neptune (&lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt;) should be on the order of &lt;span class=&quot;math-container&quot;&gt;$Gm_1m_2/a^2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, a scale for the distance between the two planets, can be taken to equal the radius of Uranus's orbit. The secular trend probably vanishes, so let's take the time of interaction to be on the order of Uranus's orbital period, which is given by Kepler's laws. Then ignoring universal factors of order unity, the displacement of Uranus should be &lt;span class=&quot;math-container&quot;&gt;$\Delta x\sim (m_2/m_S)a$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$m_S$&lt;/span&gt; is the mass of the sun. This results in an angular displacement  &lt;span class=&quot;math-container&quot;&gt;$\Delta\theta\sim\Delta x/a\sim m_2/m_S\sim 10''$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In terms of actual observation, here is a graph of the residuals from &lt;a href=&quot;http://articles.adsabs.harvard.edu/pdf/1946C%26T....62..369D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Danjon, 1946&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ADBh5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ADBh5.jpg&quot; alt=&quot;graph of residual error in observations of Uranus&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The simple order-of-magnitude estimate agrees reasonably well with the observations. Presumably people at the time did some kind of estimate similar to this in order to estimate &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt;, and the fact that it was on the right order of magnitude for a planet encouraged them to hypothesize a new planet.&lt;/p&gt;&#xA;&lt;p&gt;The relative orbital period &lt;span class=&quot;math-container&quot;&gt;$1/(1/T_1-1/T_2)$&lt;/span&gt; is 170 years, which seems to equal, to within a factor 2, the time-scale for the observed oscillations in the residuals. This could have been estimated from Bode's law, which would further support the hypothesis of a new planet.&lt;/p&gt;&#xA;&lt;p&gt;What seems to be a much more difficult problem is to estimate the new planet's orbital elements and the uncertainty of those elements, hence the controversy between supporters of Adams and Le Verrier.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-06-24T00:14:21.677" LastActivityDate="2020-06-24T00:14:21.677" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11948" PostTypeId="2" ParentId="11946" CreationDate="2020-06-24T10:21:53.670" Score="3" Body="&lt;p&gt;This topic is extensively discussed by astronomer, historian, and Marxist theorist Anton Pannekoek in his 1961 book &lt;em&gt;A History of Astronomy&lt;/em&gt; on pp. 359-363 and his 1953 article &amp;quot;&lt;a href=&quot;https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.1600-0498.1953.tb00525.x&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Discovery of Neptune&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As he explains, the deviation in the calculated orbit and the observed orbit was 30&amp;quot; in 1835 and 70&amp;quot; by 1840.&lt;/p&gt;&#xA;&lt;p&gt;LeVerrier and Adams both used these calculations and observations to predict an orbit of the hypothetical planet, but it is important to remember that these predictions contained many unknown variables that depended on the other variables, like mass of the hypothetical planet, its average distance from the sun and its eccentricity. To nevertheless compute an orbit, they used the at-the-time well-known Titius-Bode law that stated that the orbits of planets in AU followed the equation &lt;span class=&quot;math-container&quot;&gt;$$a = 0.4 + 0.3 \times 2^m$$&lt;/span&gt;&#xA;with m being -∞ for Mercury, 0 for Venus, 1 for Earth, 2 for Mars, 3 for Ceres, Pallas, Juno, and Vesta (which were considered planets), 4 for Mars, 5 for Jupiter, 6 for Saturn, and 7 for Uranus. It would therefore make sense that the new planet should have a semimajor axis of 38.8 AU (m=8).&lt;/p&gt;&#xA;&lt;p&gt;Following LeVerrier's predictions, Galle of the Berlin Observatory indeed found a unmarked star in the region where LeVerrier had predicted the new planet would be. After tracking the star for months, however, astronomers quickly realized that the star was a planet but that it followed a different orbit than predicted. It turned to be smaller and nearer than the calculations predicted. Pannekoek illustrated this with a diagram:&lt;a href=&quot;https://i.stack.imgur.com/pxjW6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pxjW6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This realization that the orbit was completely different than predicted led to a whole new discussion on whether LeVerrier and Adams had indeed discovered the planet, or that this was a new planet was a different object altogether.&lt;/p&gt;&#xA;&lt;p&gt;Pannekoek goes into this discussion from a Marxist perspective and attempts to explain it by invoking the social circumstances of the various astronomers involved. France is in the middle of the 1848 revolutions and the bourgeoisie want to illustrate the power of the natural sciences against received wisdom, which explains why LeVerrier is so adamant at prompting Galle to find the new planet and why the French resist the argument that the calculated and the observed planets are different objects. Britain, meanwhile, already had a revolution in the seventeenth century and has the bourgeoisie already entrenched in government, which explains why Adams was only interested in theoretical astronomy and not so much the observational verification. The United States on the other hand had started as a bourgeois and democratic society where debate was important, so it made sense that American astronomers would challenge the idea that the calculated and observed planets are the same.&lt;/p&gt;&#xA;&lt;p&gt;For discussions on Pannekoek's interpretations, see:&lt;/p&gt;&#xA;&lt;p&gt;Robert W. Smith, &amp;quot;&lt;a href=&quot;https://www.journals.uchicago.edu/doi/abs/10.1086/355082&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cambridge network in action: The discovery of Neptune&lt;/a&gt;&amp;quot;, &lt;em&gt;Isis: A Journal of the History of Science&lt;/em&gt; 80 (1989): 395-422&lt;/p&gt;&#xA;&lt;p&gt;John G. Hubbell and Robert W. Smith, &amp;quot;&lt;a href=&quot;https://journals.sagepub.com/doi/abs/10.1177/002182869202300402&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neptune in America: Negotiating a Discovery&lt;/a&gt;&amp;quot;, &lt;em&gt;Journal for the History of Astronomy&lt;/em&gt; 23 (1992): 261-291&lt;/p&gt;&#xA;&lt;p&gt;Bart Karstens, &amp;quot;&lt;a href=&quot;https://www.jstor.org/stable/j.ctvp7d57c.12&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anton Pannekoek as a Pioneer in the Sociology of Knowledge&lt;/a&gt;&amp;quot; in: Chaokang Tai, Bart van der Steen, and Jeroen van Dongen (eds.), &lt;em&gt;Anton Pannekoek: Ways of Viewing Science and Society&lt;/em&gt; (Amsterdam University Press, 2019): 197-218&lt;/p&gt;&#xA;" OwnerUserId="11059" LastActivityDate="2020-06-24T10:21:53.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11949" PostTypeId="1" CreationDate="2020-06-24T12:34:04.307" Score="3" ViewCount="458" Body="&lt;p&gt;A comment on another site brought the article &lt;a href=&quot;https://www.theatlantic.com/education/archive/2017/04/racist-math-education/524199/&quot; rel=&quot;nofollow noreferrer&quot;&gt;How Does Race Affect a Student's Math Education?&lt;/a&gt; to my attention. In the article, the author observes (emphasis mine),&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But she’s also constrained by the institutional aspects of whiteness in her classroom that exist outside her teaching methods—not simply the how of teaching, but what the state standards value. She and her students share a culture that isn’t reflected in the way she’s expected to teach math. &lt;strong&gt;Required to rely on what she calls a “western white lens,” other sources of math knowledge that would be relevant to her students remain untapped. “What are the theorems that we have known here in America before colonization?&lt;/strong&gt; What indigenous mathematicians have we had? We’re not a written society, so we don't have these books that say, ‘Here’s this Ojibwe person’s knowledge.’...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Do we know anything about what mathematical theorems were known in the Americas &lt;em&gt;prior&lt;/em&gt; to European contact and settlement? It's become somewhat of a trend to point out that the Maya civilization of present-day Mexico had the concept of zero, but little else seems to be said about the theoretical mathematical knowledge of native civilizations of the Americas - that is, whether they conceived of math simply as a set of basic operations (+, -, etc.), with more advanced math coming only through European educators, or whether or not they had developed an advanced conceptual knowledge of numbers, geometry, etc. in a way analogous to how Europeans at the beginning of the Age of Sail and colonization were studying and developing theorems at universities.&lt;/p&gt;&#xA;&lt;p&gt;To be clear, the article mentions that native peoples of the Americas may not have expressed mathematical truths in the same way that Europeans do in terms of peer-reviewed mathematical journals, textbooks, specific written notations developed by European mathematicians, etc., so I'm not expecting to see that. This doesn't mean that they couldn't have known about such truths - maybe they had an oral tradition/story/ritual that clearly demonstrates knowledge that the square root of two is irrational, or maybe some archaeologist found a twelfth-century totem pole somewhere in Oregon that appears to demonstrate knowledge of how the Pythagorean Theorem works.&lt;/p&gt;&#xA;&lt;p&gt;In response to comments, I'm not looking for a specific type of pre-Columbian mathematical paper, publication, journal, dissertation, or presentation that would meet modern Eurocentric standards of academic scholarship. I'm looking for evidence of a pre-contact understanding of mathematics - that is, whether natives of the Americas conceived of math as a theoretical discipline that could be studied intentionally or whether it was only ever a &lt;em&gt;practical&lt;/em&gt; endeavor (e.g. &amp;quot;I had three tents yesterday. I built another one, so now I have four tents.&amp;quot; or &amp;quot;I had five children, one died, so now I have, uhh, one, two, three, four children.&amp;quot;).&lt;/p&gt;&#xA;" OwnerUserId="6976" LastEditorUserId="6976" LastEditDate="2020-06-25T10:43:54.140" LastActivityDate="2020-06-25T11:48:45.067" Title="What mathematical theorems were known in the Americas prior to European contact?" Tags="|mathematics|mathematics-social-history|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="11950" PostTypeId="2" ParentId="11949" CreationDate="2020-06-25T11:19:39.607" Score="4" Body="&lt;p&gt;The Greek word &amp;quot;theorem&amp;quot; has a precise meaning &amp;quot;a statement for which a mathematical proof exists&amp;quot;. As far as we know these notions were invented only once: in ancient Greece in 6 century BC. More precisely in the Greek city of Miletus on the territory of modern Turkey. This is what the Greek tradition says. From what we know, this discovery was never made independently in any other place and time. This does not mean that other civilizations had no mathematical knowledge. For example, Babylonian and Chinese civilizations accumulated a lot of sophisticated mathematical knowledge, but they had no &amp;quot;theorems&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The knowledge that (for example) a triangle with sides 3,4,5 has a right angle is DIFFERENT from the Pythagorean theorem, because Pythagorean theorem is proved. One of the theorems credited to Thales says that &amp;quot;vertical angles obtained by intersection of two straight lines are equal&amp;quot;. Or that &amp;quot;all right angles are equal&amp;quot;. They say that Thales was ridiculed for such trivial statements. Of course, everyone understands that this is so.&lt;/p&gt;&#xA;&lt;p&gt;The unique contribution of the Greeks is that such things can be PROVED, by a very specific procedure which is called a &amp;quot;mathematical proof&amp;quot;. Many civilizations knew the empirical fact that there are 5 regular polytopes. But the theorem saying that a) these 5 really exist (=can be constructed by a very specific procedure using a compass and a ruler), and b) there are only 5 of them and no more, is a Greek achievement, and&#xA;as far as we know, nobody even tried to give a precise definition of&#xA;a regular polytope, and to prove these exact statements.&lt;/p&gt;&#xA;&lt;p&gt;There is no evidence that someone somewhere else at some other time proposed this idea independently.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-25T11:48:45.067" LastActivityDate="2020-06-25T11:48:45.067" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11951" PostTypeId="2" ParentId="11946" CreationDate="2020-06-25T17:57:15.820" Score="0" Body="&lt;p&gt;It is possible that some persons here might not understand the scale of deviations in the answers, So I decided to post an answer myself using the data from the two answers so far to explain the scale a little bit more.&lt;/p&gt;&#xA;&lt;p&gt;The answer by Ben Crowell makes a rough calculation that the displacement should be on the order of ten arc seconds.  It also has a plot of the variation in longitude of the observed and predicted positions of Uranus over time, which seem to differ by up to a few tens of arc seconds from predicted positions.&lt;/p&gt;&#xA;&lt;p&gt;The answer by cktai states that according to his source the deviation between the calculated and the observed orbit was 30 arc seconds in 1835 and 70 arc seconds by 1840.&lt;/p&gt;&#xA;&lt;p&gt;An arc second is one in 1,296,000, or 0.0000007, of a full circle.&lt;/p&gt;&#xA;&lt;p&gt;At the average distance between the Sun and Uranus, about 19.22 Astronomical units or AU, a full circle is about 120.76271 AU. Since the orbital period of Uranus is 84.01 years, Uranus travels at an orbital speed of about 1.4374801 AU per year, or about 0.0039356 AU per day, or about 0.0001639 AU per hour, or about 0.0000027 AU per minute, or about 0.000000045 AU per second.&lt;/p&gt;&#xA;&lt;p&gt;0.000000045 AU is 6.81433 kilometers, so the average orbital speed of Uranus should be about 6.81433 kilometers per second.  The average orbital speed of Uranus is listed as 6.8 kilometers per second.&lt;/p&gt;&#xA;&lt;p&gt;Earth is at an average distance of 1.0 Astronomical Units from the Sun, so the distance between Earth and Uranus varies between about 17.33 and 21.11 AU, and the average distance is about 19.22 AU.&lt;/p&gt;&#xA;&lt;p&gt;So since there are 360 degrees in a full circle, at the average distance between Earth and Uranus a degree of arc would be about 0.33545 AU, and an arc minute would be about 0.00559 AU, and an arc second would be about 0.00009318 AU or 13,939.61 kilometers.&lt;/p&gt;&#xA;&lt;p&gt;Being off position by 30 arc seconds would equal being off position by 418,188.38 kilometers, and being off position by 70 arc seconds would equal being off position by 975,772.88 kilometers.  And that would be approximately the distance traveled by Uranus in about a day.&lt;/p&gt;&#xA;&lt;p&gt;And I guess that gives some idea of the scale of the deviations which were considered a problem and which led to the discovery of Neptune.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2020-06-25T17:57:15.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11952" PostTypeId="1" CreationDate="2020-06-25T18:36:11.080" Score="1" ViewCount="181" Body="&lt;p&gt;I'm reading an article about the &lt;a href=&quot;https://plus.maths.org/content/dynamic-logarithms&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of logarithms&lt;/a&gt; and it says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One problem that was plaguing people at the time, especially&#xA;astronomers, was arithmetic. Astronomical calculations required the&#xA;multiplication and division of very large numbers, something that’s&#xA;pretty hard to do without a calculator&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd like to know some specific astronomy examples that would require the multiplication of large numbers by then.&lt;/p&gt;&#xA;" OwnerUserId="12712" LastEditorUserId="26" LastEditDate="2021-02-20T15:10:21.097" LastActivityDate="2021-02-20T15:10:21.097" Title="What astronomy efforts required multiplication of large numbers around 1600?" Tags="|astronomy|logarithm|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11953" PostTypeId="2" ParentId="11952" CreationDate="2020-06-26T03:10:55.383" Score="5" Body="&lt;p&gt;The main calculation in practical astronomy is solving a spherical triangle, from three elements one wants to find the rest, using formulas of spherical trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;This calculation is always performed when&#xA;for example one transfers between the coordinate systems.&#xA;Positions of celestial bodies are usually described in ecliptic coordinates, while observations are made in equatorial coordinates with stationary telescopes,&#xA;and in practical astronomy (navigation) horizontal coordinates must be used. A typical formula used in navigation looks like this:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sin h=\sin\phi\sin\delta+\cos\phi\cos\delta\cos t,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; is latitude, &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; is the declination &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is the hour angle, and &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; the altitude of a celestial body. To find the ship position with the most common method, using a chronometer, one has to use this formula at least twice (for two stars, for example). Without a chronometer, a more complicated calculation is required.&lt;/p&gt;&#xA;&lt;p&gt;Under ordinary circumstances,&#xA;for example in navigation one computes each element of a triangle in degrees, minutes and fractions of a minute. (In the past they used seconds, but in 20th century decimal fractions of a minute were more commonly used in navigation.) Since 90 degrees=5400 minutes=324000 seconds,&#xA;one needs 5-6 digits for the needs of practical astronomy/navigation.&#xA;Naked eye and practical astronomy do not allow measuring of angles with higher accuracy. In scientific astronomy, the accuracy is higher, since with the invention of telescope they measured angles to fractions of a second, which means already 7-8 digits.&lt;/p&gt;&#xA;&lt;p&gt;Now try to multiply two random 5 digit numbers by hand, and you will see why logarithms were such a big discovery. And a typical astronomical calculation involves many such multiplications.&lt;/p&gt;&#xA;&lt;p&gt;By the way the first log tables (of Napier) tabulated not &lt;span class=&quot;math-container&quot;&gt;$\log x$&lt;/span&gt; but &lt;span class=&quot;math-container&quot;&gt;$\log\sin x$&lt;/span&gt;, that is they were specially designed for trigonometric calculations in astronomy/geodesy.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-06-26T03:40:33.117" LastActivityDate="2020-06-26T03:40:33.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11954" PostTypeId="1" AcceptedAnswerId="15086" CreationDate="2020-06-26T17:55:39.943" Score="3" ViewCount="283" Body="&lt;p&gt;Some &lt;a href=&quot;https://medium.com/however-mathematics/understanding-cantors-mathematical-infinity-bc9ffe7465da#:%7E:text=He%20showed%20us%20that%20an,smaller%20than%20another%20infinity%20set.&amp;amp;text=Infinity%20would%20swallow%20everything.,Aristotle%27s%20idea%20with%20set%20theory.&quot; rel=&quot;nofollow noreferrer&quot;&gt;media sources&lt;/a&gt; say that &amp;quot;Cantor claimed that there would only be potential infinity, not actual infinity&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In addition, the following link claims that Hilbert, Poincaré, and Cantor were fully aware of the distinction between &amp;quot;potential infinity&amp;quot; and the actual one.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/7028/is-there-a-formal-distinction-between-potential-and-actual-infinities/7040#7040&quot;&gt;Is there a formal distinction between potential and actual infinities?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course all those mathematicians talked a lot about infinity in their publications; however I am having trouble finding an exact quote on &amp;quot;potential infinity&amp;quot; from them. In which chapters did they mention the exact term &amp;quot;potential infinity&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;I did notice that Fraenkel mentioned &amp;quot;potential infinity&amp;quot; a lot, but not for Hilbert or Poincaré.&lt;/p&gt;&#xA;" OwnerUserId="12713" LastEditorUserId="17306" LastEditDate="2023-01-08T10:32:58.893" LastActivityDate="2023-01-20T09:59:04.037" Title="In which article/book chapter did Cantor, Hilbert, and Poincaré formally define or directly discusse the term “potential infinity”?" Tags="|mathematicians|reference-request|mathematical-logic|set-theory|infinity|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11955" PostTypeId="2" ParentId="11954" CreationDate="2020-06-27T04:16:13.570" Score="6" Body="&lt;p&gt;You have misinterpreted &lt;a href=&quot;https://medium.com/however-mathematics/understanding-cantors-mathematical-infinity-bc9ffe7465da&quot; rel=&quot;nofollow noreferrer&quot;&gt;the article you refer to&lt;/a&gt;; nowhere does it say that &lt;em&gt;&amp;quot;Cantor claimed that there would only be potential infinity, not actual infinity&amp;quot;&lt;/em&gt;. In fact, it says the opposite:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Furthermore, Cantor claimed that we could add and multiply infinity sets. Until that time, humans had followed Aristotle’s ideas about infinity. According to Aristotle, if we multiply the number 3 by infinity, it would be infinite again. Infinity would swallow everything. Based on this, he claimed that there would only be potential infinity, not actual infinity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's &lt;em&gt;Aristotle&lt;/em&gt; claiming that there is only potential infinity, and we have since learned that in many ways, Aristotle was just a big windbag.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A potential infinity is the kind we work with all the time in calculus: the limit of something as something else controlling it it is allowed to grow with out bound. There is no &amp;quot;largest&amp;quot; natural number, you can always add one and get a bigger one. In a set-theoretical framework with basic axioms, any &lt;a href=&quot;https://en.wikipedia.org/wiki/Class_(set_theory)#:%7E:text=A%20class%20that%20is%20not,classes%20in%20many%20formal%20systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;proper class&lt;/a&gt; is a potential infinity.&lt;/p&gt;&#xA;&lt;p&gt;An absolute infinity is a &amp;quot;completed&amp;quot; one. As Cantor later showed, you need a whole new axiom to make it work formally. The usual way to do this is to assert the existence of a complete set of natural numbers (&lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;) as an object you can manipulate. It's this &lt;a href=&quot;https://en.wikipedia.org/wiki/Axiom_of_infinity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Axiom of Infinity&lt;/a&gt; that formally defines an absolute infinity.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A number of years ago I was led to the infinite real whole numbers &lt;em&gt;[die unendlichen realen ganzen Zahlen]&lt;/em&gt; without having realized that they were concrete numbers of real significance.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cantor would &lt;em&gt;&lt;strong&gt;never&lt;/strong&gt;&lt;/em&gt; have claimed there was only potential infinity. His transfinite numbers are all completed ones.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=leI9DwAAQBAJ&amp;amp;pg=PA97&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Warren Dauben's biography of Cantor&lt;/a&gt; discusses this concept in Chapter 5, which is all about Cantor's 1883&#xA;&lt;em&gt;Grundlagen einer allgemeinen Mannigfaltigskeitlehre&lt;/em&gt;.This title may cause you to do a double-take, as it transtates as &amp;quot;Foundations of a Theory of Manifolds&amp;quot;. Here is the quote you're looking for:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As far as the mathematical infinite is concerned: to  the extent that it has found justifiable use in science and made a useful contribution, the mathematical infinite has principally occurred in the meaning of a variable magnitude, either growing beyond all limits or diminishing to an arbitrary smallness, always,  however remaining finite. I call this infinite the non-genuine-infinite [das Uneigentlich-unendliches].&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There was great opposition to Cantor's ideas by people who couldn't wrap their heads around them. There's a remarkable quote later on in the book, criticizing &amp;quot;proofs&amp;quot; that completed infinities couldn't exist, because they assumed there were only potential infinities to begin with:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on General grounds as well. It is their &lt;em&gt;πρώτον ψεύδος&lt;/em&gt; [first lie] that from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they're to be considered in any form at all, must (in their contrast of the finite numbers) constitute &lt;strong&gt;an entirely new kind of number&lt;/strong&gt;, whose nature is entirely dependent upon the nature of things and is an object of research but not of our arbitrariness or prejudices.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;-Cantor to Eneström&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;:&lt;em&gt;Georg Cantor: His Mathematics and Philosophy of the Infinite&lt;/em&gt;, by Joseph Warren Dauben, Princeton University Press, 1990, quotation, p. 57&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;Cantor, G. &lt;em&gt;Grundlagen einer allgemeinen Mannigfaltigskeitlehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen.&lt;/em&gt; Leipzig, B.G. Teubner, 1883. Later in &lt;em&gt;Mathematische Annalen&lt;/em&gt;. Translated by Uwe Parpart, first appearing in &lt;em&gt;The Campaigner&lt;/em&gt;,  January-February 1976, Vol 9 Nos. 1-2, P. 70.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; Dauben, quotation, p. 125&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2020-06-30T20:25:22.253" LastActivityDate="2020-06-30T20:25:22.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11956" PostTypeId="1" CreationDate="2020-06-27T20:39:10.497" Score="2" ViewCount="112" Body="&lt;p&gt;My question refers to a 2-pages fragment of Gauss, entitled: &amp;quot;Note on the resolution of a special system of linear equations&amp;quot;, which is found on pages 30-31 of volume 8 of his works. In this fragment, Gauss expresses the solution vector &lt;span class=&quot;math-container&quot;&gt;$(P,Q,R,S,T)$&lt;/span&gt; of the following inhomogenous system of &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; linear equations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ aP + bQ + cR + dS + eT = A $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ bP + cQ + dR + eS + aT = B $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ cP + dQ + eR + aS + bT = C $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ dP + eQ + aR + bS + cT = D $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ eP + aQ + bR + cS + dT = E $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(note that the coefficents of each row are a cyclic shift of the previous row) by  five auxillary parameters &lt;span class=&quot;math-container&quot;&gt;$p,q,r,s,t$&lt;/span&gt; which solve another system of &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; linear equations:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ ap + bq + cr + ds + et = 1 $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ bp + cq + dr + es + at = 0 $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ cp + dq + er + as + bt = 0 $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ dp + eq + ar + bs + ct = 0 $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ ep + aq + br + cs + dt = 0 $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Gauss then expresses &lt;span class=&quot;math-container&quot;&gt;$P,Q,R,S,T$&lt;/span&gt; through a bilinear expression in &lt;span class=&quot;math-container&quot;&gt;$p,q,r,s,t, A,B,C,D,E$&lt;/span&gt;. To find &lt;span class=&quot;math-container&quot;&gt;$p,q,r,s,t$&lt;/span&gt; Gauss employs a certain algebraic trick involving the fifth root of unity &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\epsilon^5 = 1$&lt;/span&gt;).According to the short comment of Fricke on Gauss's result,  the final form of Gauss's formulas for &lt;span class=&quot;math-container&quot;&gt;$p,q,r,s,t$&lt;/span&gt; involves an expression for the norm of the cyclotomic number &lt;span class=&quot;math-container&quot;&gt;$a +b\epsilon + c\epsilon^2+d\epsilon^3+e\epsilon^4$&lt;/span&gt; in the quintic (cyclotomic) field &lt;span class=&quot;math-container&quot;&gt;$Q[\epsilon]$&lt;/span&gt; (see also this post: &lt;a href=&quot;https://math.stackexchange.com/questions/24840/finding-the-norm-in-the-cyclotomic-field-mathbbqe2-pi-i-5&quot;&gt;https://math.stackexchange.com/questions/24840/finding-the-norm-in-the-cyclotomic-field-mathbbqe2-pi-i-5&lt;/a&gt; ). The whole Gaussian method looks like an hybrid between techniques of linear algebra and techniques fron alegraic number theory (roots of unity), and therefore it aroused my curiosity (this complicated procedure suggests that Gauss had an idea here...). Here i omitted the final formulas, because they are long and the best way to gain insight into them is simply to look into Gauss's note.&lt;/p&gt;&#xA;&lt;p&gt;I understand that trying to give a sound interpretation of such a complicated note (and without supporting context) is a very difficult task, but maybe someone who is familiar with advanced algebraic techniques will find familiar patterns in Gauss's fragment (honestly i have no clue or idea about how to dechiper this note).&lt;/p&gt;&#xA;&lt;p&gt;Therefore, my question is about giving a rough outline and interpretation of the meaning of Gauss's statements in this note.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-06-29T10:15:31.537" LastActivityDate="2020-06-29T10:15:31.537" Title="Interpretation of a short note of Gauss on the resolution of a special system of inhomogeneous linear equations by roots of unity" Tags="|mathematics|gauss|linear-algebra|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11957" PostTypeId="1" AcceptedAnswerId="11960" CreationDate="2020-06-28T14:09:35.143" Score="4" ViewCount="1286" Body="&lt;p&gt;According to &lt;a href=&quot;http://scienceworld.wolfram.com/physics/ThinLensFormula.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weisstein&lt;/a&gt;'s webpage it was Halley in 1693 (quoting Steinhaus); but I've also seen it attributed to Cotes, Huygens, even Gauss (eg Britannica). Wikipedia's History of Optics does not give an originator.&lt;/p&gt;&#xA;&lt;p&gt;A related question would be: Who invented the 2-ray method of geometrical optics of thin lenses, now taught in schools? Gauss? The proof of the lens equation follows by similar triangles from these lens diagrams. The earliest online book I can find that has something similar is &lt;a href=&quot;https://archive.org/details/leonhardieulerid027742mbp/page/n45/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler&lt;/a&gt;'s (eg page 36).&lt;/p&gt;&#xA;&lt;p&gt;Add: I know Euclid and Hero had some geometric optics, but it is not near proving the lens formula. Kepler drew the first geometric optics diagrams of this sort, but they are still not close.&lt;/p&gt;&#xA;" OwnerUserId="9714" LastActivityDate="2020-06-29T06:53:22.537" Title="Who discovered the thin lens equation $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$?" Tags="|euclidean-geometry|gauss|optics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11958" PostTypeId="1" CreationDate="2020-06-28T17:59:20.973" Score="2" ViewCount="160" Body="&lt;p&gt;In a few places, such as &lt;a href=&quot;http://self.gutenberg.org/articles/eng/Nicolas_Bourbaki?View=embedded&quot; rel=&quot;nofollow noreferrer&quot;&gt;this web page&lt;/a&gt;, I have read the following statement about &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Dieudonné&lt;/a&gt;, who was a founding member of the French &amp;quot;secret society&amp;quot; of mathematicians, &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicolas_Bourbaki&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nicolas Bourbaki&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dieudonné stated the view that most workers in mathematics were doing ground-clearing work, in order that a future Riemann could find the way ahead intuitively open.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have two questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there a reliably-sourced quote that can be directly attributed to Dieudonné where he expresses this sentiment? I have been unable to find such a quote. If such a quote exists, it might be in French.&lt;/li&gt;&#xA;&lt;li&gt;What did Dieudonné mean by this? That is, what did he expect that a future mathematician would do that was reminiscent of Riemann's work? If a direct quote is found as in the last question, its surrounding context might shed light on this matter.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12679" LastActivityDate="2020-06-28T17:59:20.973" Title="Where is this statement of Bourbaki's Dieudonné from and what does it mean?" Tags="|mathematics|quote|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11959" PostTypeId="1" CreationDate="2020-06-28T12:02:40.773" Score="1" ViewCount="90" Body="&lt;p&gt;I have a question that is more related to the &lt;em&gt;history&lt;/em&gt; of &lt;strong&gt;evolutionary biology&lt;/strong&gt; rather than the science &lt;em&gt;itself&lt;/em&gt;, namely I am interested in knowing what might have been considered the '&lt;em&gt;orthodoxy&lt;/em&gt;' of the field during the period 1880-1890, predominantly in the United States.&lt;/p&gt;&#xA;&lt;p&gt;If it makes any difference for anyone interested in contributing an answer, I am interested in the particular chronology in order to understand what was the backdrop for Thorstein Veblen's article &amp;quot;&lt;em&gt;&lt;a href=&quot;http://dx.doi.org/10.2307/1882952&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why is Economics not an Evolutionary Science?&lt;/a&gt;&lt;/em&gt;&amp;quot; published in 1898 (&lt;a href=&quot;https://www.jstor.org/stable/1882952&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I won't mind having long bibliographical references, but I will be satisfied with short comprehensive accounts too.&lt;/p&gt;&#xA;" OwnerUserId="12723" OwnerDisplayName="user42582" LastEditorUserId="26" LastEditDate="2021-02-20T15:10:12.050" LastActivityDate="2021-02-20T15:10:12.050" Title="What was considered Evolutionary Science or Biology proper in 1880-1890 in the US?" Tags="|biology|evolution|19th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11960" PostTypeId="2" ParentId="11957" CreationDate="2020-06-29T06:53:22.537" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=Ye_1AAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Darrigol's History of Optics&lt;/a&gt; names Barrow (1669) and Huygens (1653) as giving semi-verbal case by case variants of the lens formula before Halley, with Molyneux being the first to publish in 1692. Halley (1693) is credited as the first who wrote the formula algebraically. &lt;a href=&quot;https://ris.utwente.nl/ws/portalfiles/portal/6074921/t0000003.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dijksterhuis in Lenses and Waves&lt;/a&gt; gives detailed description of Huygens's &lt;em&gt;Dioptrica&lt;/em&gt; (which he calls &lt;em&gt;Tractatus&lt;/em&gt;), with reproductions of his diagrams that look roughly modern, and remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the twentieth proposition of &lt;em&gt;Tractatus&lt;/em&gt;, Huygens extends the results for thin lenses to non-parallel rays... Huygens had to treat all cases of positive and negative lens sides separately, but the result comes down to the modern formula &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{p'}+\frac{1}{p}=\frac{1}{f}$&lt;/span&gt;&lt;/em&gt;&amp;quot; [p.18].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A systematic development of modern Gaussian/paraxial optics that the formula naturally belongs to first appeared in print in Hamilton's works of 1828-33. Gauss did it earlier but his memoir was only published in 1840. Here are the relevant excerpts from Darrigol:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Barrow, who treated optics as a mathematical playground, did not dwell on the paraxial&#xA;(Gaussian optics) that is most useful to the theory of optical instruments. He nonetheless&#xA;gave relations equivalent to the various cases of the modern formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{n}{p'}+\frac{1}{p}=\frac{n-1}{R}$$&lt;/span&gt;&#xA;relating the algebraic distances &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p'$&lt;/span&gt;&#xA;(from the intersection of the axis with the refracting&#xA;surface) of two conjugate points of a spherical refracting sphere with the index &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and the&#xA;radius &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, as well as unnecessarily complicated relations of conjugation for thin lenses. In&#xA;contrast, Huygens’s early private treatises were aimed at improving the telescopes that he was&#xA;designing with his brother. In his Dioptrica of 1653, he derived a variant of the now familiar&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{p'}+\frac{1}{p}=\frac{1}{f}\text{ with }\frac{1}{f}=(n-1)\left(\frac{1}{R}-\frac{1}{R'}\right)$$&lt;/span&gt;&#xA;for a thin lens of index &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, radiuses &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$R'$&lt;/span&gt;, and focal length &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;. This formula first appeared&#xA;in print forty years later: case by case in Molyneux’s treatise, and in algebraic form in a&#xA;memoir by Edmond Halley also including thick lenses.&lt;/em&gt;&amp;quot; [pp. 75-6]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;With these powerful tools, Hamilton developed the study of caustics, foci, and various&#xA;kinds of aberrations. In a memoir read at the British Association meeting of 1833, he&#xA;briefly indicated how to use the &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; function in the case of optical instruments with axial symmetry... The &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; function thus leads to the basic concepts and relations of paraxial optics, although Carl Friedrich Gauss was first to define them in a general manner&#xA;in a theory conceived in his youth and published belatedly in 1840. In his demonstrations,&#xA;Gauss relied on the combination of the linear transformations connecting the parameters&#xA;(inclination and transverse coordinates in a fixed plane) of incoming and outgoing rays for&#xA;each refracting surface, as is most commonly done nowadays.&lt;/em&gt;&amp;quot; [pp. 266-7]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Footnotes to the quoted text give references to the original sources.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-06-29T06:53:22.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11961" PostTypeId="1" AcceptedAnswerId="11963" CreationDate="2020-06-30T08:39:30.723" Score="0" ViewCount="91" Body="&lt;p&gt;I was reading Stillwell's &lt;a href=&quot;http://dx.doi.org/10.1007/978-1-4419-6053-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics and its History&lt;/a&gt;, where Lagrange's theory of quadratic forms is synoptically presented, and I was wondering of what are the direct predecessors of the theory.&lt;/p&gt;&#xA;&lt;p&gt;Specifically, I am interested in the possible results that inspired him to investigate the discriminant, equivalency and the composition of quadratic forms in order to resolve number-theoretic conjectures.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2020-07-01T03:51:29.060" LastActivityDate="2020-07-01T03:51:29.060" Title="What are the direct predecessors of Lagrange's theory of quadratic forms?" Tags="|mathematics|biographical-details|number-theory|lagrange|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11962" PostTypeId="1" CreationDate="2020-06-30T11:28:51.973" Score="4" ViewCount="361" Body="&lt;p&gt;Volume 5 of Gauss's works contains a section which includes &amp;quot;Essays on various objects in mathematical physics&amp;quot;. Just to emphasize the importance of this section, i'll mention that it includes, for example, his theory of falling objects on the rotating earth (precursor to the theory of Coriolis force), as well as his treatise about the achromatic double lenses, which was the subject of a previous unanswered post (&lt;a href=&quot;https://hsm.stackexchange.com/questions/7484/what-was-gausss-theoretical-work-related-to-his-invention-of-the-double-gauss&quot;&gt;What was Gauss&amp;#39;s theoretical work related to his invention of the &amp;quot;Double-Gauss lens&amp;quot;?&lt;/a&gt;).  The same section includes a 2-pages &amp;quot;letter to Brandes on the same subject&amp;quot; (the subject of achromatic double lenses). This letter is of great interest to me, because it includes uncommon and interesting formulas in optical theory - it involves, for example, certain integrals (so this letter doesn't seem to deal with the traditional aspects of optics). An almost complete english translation of Gauss's letter is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The circumstances to which you briefly allude at the end of your letter has caused me to devote a spare hour to the essay which you mention. The proper meaning of my remarks there doesn't appear to have been comprehended by all; but the essay itself requires some modification. I now find, as a result of a more laborious investigation, that the indistictness which depends upon the value of the fourth power of the distance of the incindent rays from the axis in the expression for the longitudinal aberration has the least possible influence when the objective is so constructed that those rays, which are incident infinitely near the axis, and also those incident at a distance = &lt;span class=&quot;math-container&quot;&gt;$R\sqrt{\frac{6}{5}}$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is the radius of the objective) are combined in one point A, and the eyepiece will be so placed as to observe that point in the axis where the rays combine whose distances = &lt;span class=&quot;math-container&quot;&gt;$(\frac{3}{5} - \frac{\sqrt{6}}{10})R$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\frac{3}{5} + \frac{\sqrt{6}}{10})R$&lt;/span&gt; from the axis. If you imagine a plane, perpendicular to the axis, and cutting this at this point, the image upon it would be the more indistinct, the greater the circle around A, which is filled by rays from the object through the object glass, and the intensity of the rays upon each part of the circle must also be noticed. It may seem rather empirical; but i think the most advisable course to go upon the same principles as lie the root of least squares. If &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; be an element of this circle, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; the distance of this element from A, and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; the intensity of the ray, I assume that &lt;span class=&quot;math-container&quot;&gt;$\int zy^2dx$&lt;/span&gt; will be the value of the total indistinctness and make this a minimum. I arrived at the following results:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If the object-glass be constructed in such a manner that the term in the longuitudinal aberration which depends upon the square of the distance from the axis = 0, and the eyepiece be so placed that A falls at that point where these rays cut it, the value of this integral = &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If, with the same arrangement, the eyepiece be placed in such a manner as to make this integral as small as possible (when A becomes the point of union of rays whose distance is &lt;span class=&quot;math-container&quot;&gt;$R(\frac{1}{2})^{\frac{1}{4}}$&lt;/span&gt;), the integral is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}E$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On the other hand, with the above arrangement and the best position for the eyepiece, the integral = &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{100}E$&lt;/span&gt;, the absolute minimum.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The above result, namely, that the purely imaginary image produced by those rays which lie infinitely near the axis, should be combined at the point of union with rays of a greater distance from the axis than the semi-diameter of the aperture appears at the first very surprising and paradoxical; but the reason is soon seen with a little study. The so-called principal image (by rays lying near the axis) is of the least importance, as it were, on acount of its feeble intensity, and its of more consequence that those rays from the parts which lie nearer the borders of the glass, should be kept near together among each other, so to speak, which is best done in the above manner. I am sorry the the limits of the letter will not allow me to enlarge upon this subject: an accurate calculation is needed - in loose talk, an important point is easily overlooked; but these hints will suffice for those who understand. I generally find that with the best position for the eyepiece, the integral is always equal to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}E(1-\frac{8}{5}\mu^2 + \frac{2}{3}\mu^4)$&lt;/span&gt;, when the object-glass is so constructed that rays at a distance &lt;span class=&quot;math-container&quot;&gt;$\mu R$&lt;/span&gt; are united in one point with the so-called principal image. This is a minimum, for &lt;span class=&quot;math-container&quot;&gt;$\mu = \sqrt{\frac{6}{5}}$&lt;/span&gt; and it then &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{100}E$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$\mu = 1$&lt;/span&gt; it would only equal to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{60}E$&lt;/span&gt;, and for &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; infinitely small = &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}E$&lt;/span&gt;. Not only is Bohnenberger wrong in this, but I was also wrong myself then, because I have not differed from him as much as I ought to have done. I had then only considered the whole magnitude of the indistinct image, without having sufficiently noticed the intensity of each separate part.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I searched in Clemens Schaefer's treatise for information about this, and in article 84, it states that his letter to Brandes is a &amp;quot;result of a new calculation, which takes the intensity of light into account&amp;quot;. Another useful piece of information which i inffered from Schaefer's treatise is that the calculation Gauss made to arrive at his results is reproduced in J.C.E Schmidt's &amp;quot;textbook on analytical optics&amp;quot; (published in 1834).&lt;/p&gt;&#xA;&lt;p&gt;Therefore:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What was the specific problem which Gauss solved in this letter (Gauss only gives final results)? unfortunately, i was not able to understand much from the section on optics in Schaefer's treatise, as it immediately throws the reader into the details of Gauss's occupation with optics and doesn't give much background and supporting motivation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;While my posted answer clarified a little bit the context of writing of Gauss's letter, it still doesn't give a direct answer to the title question. Therefore, in order to futher eludicate the answer, i'll focus now more on the details of Gauss's letter.&lt;/p&gt;&#xA;&lt;p&gt;Gauss writes in his letter:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I now find through a more in-depth investigation that the indistinctness, which depends on the fourth power of the distance of the striking rays in the expression for the length deviation, has the smallest possible total influence, if one constructs the objective in such a way...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There, the main riddle to dechiper now is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;what is the meaning of the term  &amp;quot;indistinctness&amp;quot; which Gauss uses? why does it depend on the fourth power of the distance of the striking rays in the expression for the length deviation? (and what is &amp;quot;the expression for the length deviation&amp;quot;?)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2021-03-17T00:22:06.920" LastActivityDate="2021-03-17T00:22:06.920" Title="What was the optical problem discussed and solved by Gauss in his 1831 letter to Brandes?" Tags="|gauss|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11963" PostTypeId="2" ParentId="11961" CreationDate="2020-06-30T11:47:34.130" Score="3" Body="&lt;p&gt;The motivation for the theory of quadratic forms was representation of integers by sums of squares. This business was started by Fermat and continued by Euler, Lagrange and Legendre, then Gauss. Generalization from sums of squares to arbitrary quadratic forms is natural. Two forms represent the same integers when they are equivalent. Discriminant is the natural invariant of this equivalence relation, so it gives certain classification&#xA;of quadratic forms.&lt;/p&gt;&#xA;&lt;p&gt;Ref. André Weil, Number theory. An approach through history from Hammurapi to Legendre, Birkhäuser Boston, Inc., Boston, MA, 2007.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-06-30T11:47:34.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11965" PostTypeId="2" ParentId="11945" CreationDate="2020-07-01T01:21:09.227" Score="1" Body="&lt;p&gt;Cardano's formula is useless if you want to really solve the cubic equation.  In our culture we say that &amp;quot;solving an equation&amp;quot; means writing its solution in some closed form using a certain set of mathematical symbols. But this is just an agreement, and it was not always the case. If you need to solve an equation not to pass an exam, but for some practical purposes, you need to find an approximation of your solution. If you are going to use Cardano's formula you need to take cubic and square roots. But how do you take - say - a cubic root of 5 without a calculator? The easiest way is to use Newton's method to solve the equation &lt;span class=&quot;math-container&quot;&gt;$x^3=5$&lt;/span&gt;. Now to use Cardano formula you need to take two cubic roots and one square root, i.e. to use Newton methods three times. Why not just to use it for the original equation?&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-07-01T01:21:09.227" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11966" PostTypeId="2" ParentId="7957" CreationDate="2020-07-01T20:02:59.283" Score="0" Body="&lt;p&gt;The OP wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any good sources (references, books, videos (most preferable) etc.) that provide a walkthrough to help &amp;quot;discover/create&amp;quot; the findings of Fourier by yourself?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think chapters 8 and 9 of &lt;a href=&quot;https://www.goodreads.com/book/show/18686876-17-equations-that-changed-the-world&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;17 Equations That Changed the World&lt;/em&gt;&lt;/a&gt; by Ian Stewart provide such a walkthrough.&lt;/p&gt;&#xA;" OwnerUserId="12735" LastActivityDate="2020-07-01T20:02:59.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11968" PostTypeId="2" ParentId="11959" CreationDate="2020-07-04T14:20:29.243" Score="1" Body="&lt;p&gt;I encountered an insightful article that may address the origins of Veblen's work. Here is an excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A number of economists and other social scientists have addressed evolutionary explanations. There is a large and valuable literature that considers how humans have evolved in groups and how human propensities for altruism and cooperation have emerged, alongside more selfish and competitive tendencies.........But Veblen made an even more radical point, which is still difficult for many economists to swallow. He argued that the idea of the utility-maximizing individual is inconsistent with the principle of evolutionary explanation. This point remains pertinent, because the idea of a fixed utility function – even if it is one that has “social” or “altruistic” preferences – lacks a clear evolutionary and causal explanation of its origins. It is simply assumed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is the link to the article: &lt;a href=&quot;https://evonomics.com/imagine-economics-evolutionary-science/#:%7E:text=In%201898%2C%20the%20great%20American,the%20social%20and%20behavioral%20sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://evonomics.com/imagine-economics-evolutionary-science/#:~:text=In%201898%2C%20the%20great%20American,the%20social%20and%20behavioral%20sciences&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2020-07-04T14:20:29.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11969" PostTypeId="1" AcceptedAnswerId="12038" CreationDate="2020-07-05T02:29:43.073" Score="6" ViewCount="267" Body="&lt;p&gt;Nowadays, when one searches for little-known trigonometric functions, one usually finds a list containing the versine, coversine, vercosine, and covercosine. When using this list, &lt;span class=&quot;math-container&quot;&gt;$1+\cos(x)$&lt;/span&gt; is given the name vercosine. However, I have not found any references to this before the year 2009, when it was added to Wikipedia.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, I can easily find references to the versine and coversine back to at least Cauchy's 1821 &amp;quot;Analyse Algébrique&amp;quot;. Cauchy calls the coversine the &lt;em&gt;cosinus versus&lt;/em&gt;, which formula is &lt;span class=&quot;math-container&quot;&gt;$1-\sin(x)$&lt;/span&gt;. A more detailed history can be found in &lt;a href=&quot;http://jeff560.tripod.com/v.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Miller's Earliest Known Uses&lt;/a&gt;. I have seen this function also called the vercosine, for instance in &lt;a href=&quot;https://www.google.com/books/edition/MathHorizons/JBYrAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=vercosine&amp;amp;dq=vercosine&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathHorizon (2006)&lt;/a&gt; and the &lt;a href=&quot;https://www.google.com/books/edition/Science_Vocabulary_Builder/mZTWAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=vercosine&amp;amp;dq=vercosine&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Science Vocabulary builder by Johnson O'Connor in 1956&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Is there any source before 2009 where the function &lt;span class=&quot;math-container&quot;&gt;$1 + \cos(x)$&lt;/span&gt; was given a name?&lt;/p&gt;&#xA;" OwnerUserId="12742" LastEditorUserId="12742" LastEditDate="2020-07-05T15:33:03.407" LastActivityDate="2020-07-19T03:32:39.033" Title="When was the function 1 + cos(x), aka the vercosine, given a name?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11970" PostTypeId="1" CreationDate="2020-07-05T12:59:30.460" Score="3" ViewCount="107" Body="&lt;p&gt;According to this &lt;a href=&quot;https://plato.stanford.edu/entries/intuitionistic-logic-development/#Kolm1925&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; Kolmogorov published a paper in 1925 in which he attempted to formalize Brouwer’s intuitionistic mathematics.&#xA;In that paper there are the following logical formulas:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align} \tag{1} A &amp;amp; \rightarrow(B \rightarrow A)\\ \tag{2} (A&#xA;&amp;gt; \rightarrow(A \rightarrow B)) &amp;amp; \rightarrow (A \rightarrow B)\\&#xA;&amp;gt; \tag{3} (A \rightarrow(B \rightarrow C)) &amp;amp; \rightarrow (B&#xA;&amp;gt; \rightarrow(A \rightarrow C))\\ \tag{4} (B \rightarrow C) &amp;amp;&#xA;&amp;gt; \rightarrow((A \rightarrow B) \rightarrow(A \rightarrow C))\\ \tag{5}&#xA;&amp;gt; (A \rightarrow B) &amp;amp; \rightarrow((A \rightarrow \neg B) \rightarrow&#xA;&amp;gt; \neg A)\\ \end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Were there any preceding concepts that he used to come up with the above?&lt;/p&gt;&#xA;&lt;p&gt;Is it possible to reconstruct a plausible thought process and preceding conceptual resources that he used to come up with these formulas?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2020-07-05T23:24:25.527" LastActivityDate="2020-07-05T23:24:25.527" Title="How did Kolmogorov came up with his formalization of intuitionistic logic?" Tags="|mathematics|mathematicians|discoveries|mathematical-logic|kolmogorov|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11971" PostTypeId="1" AcceptedAnswerId="11974" CreationDate="2020-07-06T08:54:45.403" Score="6" ViewCount="215" Body="&lt;p&gt;An incident in the negotations of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sixth_Crusade&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sixth Crusade&lt;/a&gt; is described as &lt;a href=&quot;https://en.wikipedia.org/wiki/Frederick_II,_Holy_Roman_Emperor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frederick II&lt;/a&gt; asking help from Arab scholars with some mathematical problems:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... and the sultan graciously allowed Frederick to submit a list of difficult mathematical problems for the consideration of the finest scholars at the Ayyubid court.&lt;br /&gt;&#xA;&lt;sub&gt;—Jones &lt;a href=&quot;https://books.google.ee/books?id=NTSGDwAAQBAJ&amp;amp;pg=PT194&amp;amp;lpg=PT194&amp;amp;dq=%22and%20the%20sultan%20graciously%20allowed%22&amp;amp;source=bl&amp;amp;ots=ndEg7aFOm0&amp;amp;sig=ACfU3U1jGjd9iZJJLAKR-rCphGp8HLpY8w&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjnw8bxnrjqAhUyyKYKHecKCGYQ6AEwAHoECAEQAQ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Crusaders: The Epic History of the Wars for the Holy Lands'&lt;/a&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Apparently the negotiations over Jerusalem were at a stalemate when, as the fifteenth-century Egyptian historian al-Maqrizi recounted, &amp;quot;Frederick sent several difficult questions pertaining to the science of mathematics to the Sultan, who gave them to men of great learning for appropriate answers. ...&amp;quot;&lt;br /&gt;&#xA;&lt;sub&gt;—Ahmed, &lt;a href=&quot;https://books.google.ee/books?id=LpALDgAAQBAJ&amp;amp;pg=PA137&amp;amp;lpg=PA137&amp;amp;dq=frederick%20al-kamil%20negotiations%20math&amp;amp;source=bl&amp;amp;ots=v0zriaA41b&amp;amp;sig=ACfU3U2m8VR5oibq0zBOHNfeOa3pbbbJWw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwilnL2gnrjqAhVizqYKHVX7DzAQ6AEwCnoECAoQAQ#v=onepage&amp;amp;q=frederick%20al-kamil%20negotiations%20math&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Journey into Europe: Islam, Immigration, and Identity'&lt;/a&gt;&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;What mathematical problems did the Sicilians need Arab help with in 1229, and did the Arabs solve these problems?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5957" LastEditorUserId="55" LastEditDate="2020-07-07T07:57:44.150" LastActivityDate="2020-07-07T19:56:01.273" Title="What mathematical problems did the Sicilians need Arab help with in 1229? Did the Arabs solve these problems?" Tags="|mathematics|debunking|middle-ages|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11973" PostTypeId="1" AcceptedAnswerId="11976" CreationDate="2020-07-07T06:24:51.667" Score="5" ViewCount="1363" Body="&lt;p&gt;I understand that they invented calculus independently at roughly the same time, but why do we use Leibniz's terminology/notation rather than Newton's?&lt;/p&gt;&#xA;&lt;p&gt;For example, why don't we use &amp;quot;fluxion&amp;quot; and &amp;quot;fluent&amp;quot;? Instead, we use Leibniz's derivatives.&lt;/p&gt;&#xA;&lt;p&gt;What are the historical reasons for this?&lt;/p&gt;&#xA;" OwnerUserId="12751" LastEditorUserId="4251" LastEditDate="2020-07-07T07:06:29.647" LastActivityDate="2020-07-07T19:51:05.493" Title="Why do we use Leibniz's “version” of calculus instead of Newton's?" Tags="|mathematics|calculus|newton|leibniz|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11974" PostTypeId="2" ParentId="11971" CreationDate="2020-07-07T07:56:50.873" Score="2" Body="&lt;p&gt;This is a semi-legendary story with shifty details told to showcase Frederick's enlightened ways with Muslims and intellectual prowess. There might be some historical basis to it, although Muslim historians had reasons to exaggerate and embellish considering how friendly Frederick was compared to other Christian crusaders. &lt;strong&gt;In any case, the original sources give no details on the content of the problems.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;According to Ibn-Wasil, a contemporary chronicler, the problems were sent as a challenge to ease tensions during negotiations over the status of Jerusalem during Frederick's siege of it, see &lt;a href=&quot;https://erenow.net/postclassical/muslims-and-crusaders-christianitys-wars-in-the-middle-east-1095-1382/32.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Two sources on the handover of Jerusalem to Frederick II&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The emir Fakhr al-Din ibn al-Shaykh carried the messages between him and the king-emperor, and between them there were discussions on various topics. During these the emperor sent al-Malik al-Kamil difficult philosophical, geometrical and mathematical questions, in order to put the learned men who were with him to the test. Al-Malik al-Kamil showed the mathematical questions that had come to him to the shaykh ‘Alam al-Din Qaysar ibn Abi’l-Qasim, who was a leader in the field, and showed the remainder to a group of learned men, and they answered the lot. Then the sultan al-Malik al-Kamil swore to abide by what they had agreed on, as did the emperor, and they concluded a peace agreement for a fixed term.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A similar account is given by Maqrīzī, see &lt;a href=&quot;https://www.tandfonline.com/eprint/9uXyTqBMeSJazHNMNpEg/full&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frederick II's crusade by Takayama&lt;/a&gt;. Unfortunately, neither of them gives any details on the problems. &lt;a href=&quot;https://en.wikipedia.org/wiki/Alam_al-Din_al-Hanafi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alam al-Din Qaysar al-Katib al-Hanafi&lt;/a&gt;, nicknamed Ta'asif, is an Egyptian born scholar of the time who designed a &lt;a href=&quot;https://www.metmuseum.org/art/collection/search/660539&quot; rel=&quot;nofollow noreferrer&quot;&gt;celestial globe&lt;/a&gt; that al-Kamil sent to Frederick as a diplomatic gift c. 1225. He also wrote a geometric treatise on Euclid's postulates, but is better known for engineering feats and astronomy. &lt;a href=&quot;https://www.jstor.org/stable/1425493&quot; rel=&quot;nofollow noreferrer&quot;&gt;Architects and Artists in Mamluk Society by Rabbat&lt;/a&gt; describes him as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He was a katib (a scribe or clerk serving in the diwan, or chancery, in Cairo), a faqih (jurist, though not a refined one), a geometrician, mathematician, and astronomer, and a muhandis, which seems here to mean a civil and military engineer. He is credited with building a mill on the Orontes for the Ayyubid king of Hama, al-Muzzafar Mahmud (d. 1244), and an unspecified number of towers around the city, for which he invented a number of engineering devices, or perhaps designs (hiyal handasiyya). He is also said to have built a wooden globe for the same ruler on which he marked all the stars in the celestial sphere (preserved at the Museo Nazionale in Naples).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There also exist multiple other instances where Frederick is described as having sent questions on various subjects to Muslim scholars. One source with more details is the well-known work of &lt;a href=&quot;https://en.wikipedia.org/wiki/Sicilian_Questions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ibn-Sab'in, Sicilian Questions&lt;/a&gt;, where the answers to Frederick's questions are supposedly given. But Ibn-Sab'in only answers philosophical ones (on the eternity of the world, soul, and the like), and Frederick's involvement is most likely his fabrication, see &lt;a href=&quot;https://d-nb.info/1098310926/34&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weltecke, Emperor Frederick II&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;One of the sources Abulafia relies on is a famous Muslim text reporting on Frederick sending philosophical questions to the Arab world. These so called Sicilian Questions had always been a classical topos in the Orientalist construction. They were regarded as evidence for Frederick's close contacts to Muslim scholars and his interest in intricate philosophical problems...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;At the time nobody could anticipate that Anna Akasoy would today very&#xA;convincingly suggest that the Sicilian Questions were not authentic: Frederick&#xA;could not have been acquainted with Ibn Sab'in because at the time the latter&#xA;was not a known scholar at all. In fact, Ibn Sab'in was in his early twenties when he wrote the Sicilian Questions, which was his first book. It is therefore highly unlikely that Frederick sent him these philosophical questions and, likewise, that the Arab answered him in this fashion in a personal letter. Instead, Ibn Sab'in invented the questions himself and probably used the emperor's famous name and anti-Christian slander to sell his book in Morocco&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-07-07T18:37:51.077" LastActivityDate="2020-07-07T18:37:51.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11975" PostTypeId="1" CreationDate="2020-07-07T09:02:53.750" Score="5" ViewCount="218" Body="&lt;p&gt;In general relativity, it is common to use the comma notation for partial derivatives&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\partial g_{\mu\nu}}{\partial x_\rho} = g_{\mu\nu_,\rho}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where did this notation first appear? Was it known in general mathematics before Einstein?&lt;/p&gt;&#xA;&lt;p&gt;I don't know much about mathematics books, but I traced it back to Synge's book on General Relativity(1960). Could it have arisen earlier?&lt;/p&gt;&#xA;" OwnerUserId="11051" LastEditorUserId="4251" LastEditDate="2020-07-07T10:08:51.647" LastActivityDate="2020-07-07T10:08:51.647" Title="Who introduced the comma notation for partial derivatives?" Tags="|mathematics|physics|notation|calculus|theoretical-physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11976" PostTypeId="2" ParentId="11973" CreationDate="2020-07-07T09:36:08.060" Score="2" Body="&lt;p&gt;There are many notations for derivatives as the concept has been expanded in many different ways. For example,  there is also Heaviside's operational D. This is also used for the Frechet and Gateaux derivative (which is implicitly used in the notation for tangent bundles in differential geometry).&lt;/p&gt;&#xA;&lt;p&gt;Newton chose a notation for ease of use. As a physicist he was mostly interested in the first and second derivatives of time. Since the dependent variable implicitly understood, there is no need for the notation to reflect this. Hence he needed only to indicate the degree of the derivative. This is an integer. Buy since we are only interested in the first two,  1 &amp;amp;  2, we don't have to indicate the degree by a numerical prefix (as they do in some notations), we can simply indicate it by a single or double dot. It's quicker a d more convenient.&lt;/p&gt;&#xA;&lt;p&gt;When one is interested in the calculus for its own sake, then a more comprehensive notation is necessary. This should indicate the degree, the dependent and independent variables. Thus Liebniz's notation is more natural here.&lt;/p&gt;&#xA;&lt;p&gt;Had Newton been more interested in geometry than physics, and Liebniz more interested in physics than geometry it would have been likely we would have seen their notations being swapped. In other words, the  notation associated with their names reflected their interests.&lt;/p&gt;&#xA;&lt;p&gt;(It's worth adding that doing differential geometry with intuitionistic logic allows the introduction of infinitesimals that is much closer to how Newton envisaged them, his fluxions, rather than the traditional epsilon-delta techniques of traditional analysis. Moreover, they generalise to the infinite-dimensional context seamlessly, unlike the usual calculus for which there are many differing techniques).&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-07-07T19:51:05.493" LastActivityDate="2020-07-07T19:51:05.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11977" PostTypeId="1" AcceptedAnswerId="15209" CreationDate="2020-07-07T14:24:45.783" Score="5" ViewCount="129" Body="&lt;p&gt;I was reading a wikipedia article about &lt;a href=&quot;https://en.wikipedia.org/wiki/Communication_complexity&quot; rel=&quot;nofollow noreferrer&quot;&gt;communication complexity&lt;/a&gt; and it seems to me that it bears some resemblance to &lt;a href=&quot;https://en.wikipedia.org/wiki/Kolmogorov_complexity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kolmogorov complexity&lt;/a&gt;.&#xA;Was the founder of communication complexity influenced by Kolmogorov complexity?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="26" LastEditDate="2021-02-20T15:10:02.387" LastActivityDate="2023-03-04T05:56:09.087" Title="Did Kolmogorov complexity influence the development of communication complexity?" Tags="|biographical-details|computer-science|kolmogorov|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="11978" PostTypeId="2" ParentId="11973" CreationDate="2020-07-07T15:38:53.123" Score="1" Body="&lt;p&gt;Many textbooks in English &lt;em&gt;did&lt;/em&gt; use Newton's notation and terminology for a long time. For example, see Hutton, 1807, &lt;a href=&quot;https://archive.org/details/acoursemathemat02huttgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/acoursemathemat02huttgoog&lt;/a&gt; , which uses the dot notation and terms like &amp;quot;fluent.&amp;quot; We still do use elements of Newton's notation in many fields. For example, in physics, if you have a function of position and time, it's common to use dots for time derivatives and primes for spatial derivatives.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz's notation has some objective advantages. Unlike Newton's notation, it makes it easy to do dimensional analysis, and it works well when you have lots of different variables that you might be differentiating or integrating with respect to. It works whether you want to think in terms of variables or functions, limits or infinitesimals.&lt;/p&gt;&#xA;&lt;p&gt;Newton's notation was unclearly presented, and many people didn't understand how he intended it to be used. He had a notation &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; for an infinitesimal change in the independent variable, so that if &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, then what in Leibniz notation would be written as &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; would be notated in Newton's notation as &lt;span class=&quot;math-container&quot;&gt;$\dot{x}o$&lt;/span&gt;. But he had a shorthand convention of writing this as &lt;span class=&quot;math-container&quot;&gt;$\dot{x}$&lt;/span&gt;, omitting the &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt; when context made it clear that what was intended was an infinitesimal change in &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. This confused his readers. There is a discussion of this in Boyer, &lt;a href=&quot;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment&lt;/a&gt; at p. 201.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastActivityDate="2020-07-07T15:38:53.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11979" PostTypeId="2" ParentId="3726" CreationDate="2020-07-07T16:58:54.377" Score="1" Body="&lt;p&gt;Not strictly an archive, but some related materials to Martin Gardner and his history can be found at &lt;a href=&quot;https://www.gathering4gardner.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.gathering4gardner.org/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12754" LastActivityDate="2020-07-07T16:58:54.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11980" PostTypeId="2" ParentId="11971" CreationDate="2020-07-07T19:56:01.273" Score="1" Body="&lt;p&gt;This sounds to me like an apocryphal story rather like the story that over Plato's Academy there was a sign saying 'let no-one ignorant of geometry enter here'.&lt;/p&gt;&#xA;&lt;p&gt;It's most likely a story dreamt up by mathematicians to promote the study of mathematics. I wouldn't put much credence in it.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-07-07T19:56:01.273" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="11981" PostTypeId="2" ParentId="11977" CreationDate="2020-07-07T20:09:20.463" Score="1" Body="&lt;p&gt;The first link says that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;communication complexity studies the amount of &lt;em&gt;communication&lt;/em&gt; required to solve a problem when the problem is &lt;em&gt;distributed&lt;/em&gt; between two or more parties.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Whilst the second link says that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kolmogorov complexity of an object, such as a piece of text, is the length of the shortest computer program... that produces the object as output. It is a measure of the computational resource...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Had you read further down in the first link you would have discovered:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Note that unlike computational complexity theory, communication complexity is not concerned with the amount of computation performed by Alice or Bob, or the size of the memory used, as we assume nothing about the computational power of Alice or Bob.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So,  no, the two concepts are not at all related other than sharing the word 'complexity' in their names. And that connection is spurious.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-07-07T20:09:20.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11982" PostTypeId="1" AcceptedAnswerId="13063" CreationDate="2020-07-07T20:32:33.760" Score="4" ViewCount="626" Body="&lt;p&gt;I recently stumbled upon &lt;a href=&quot;https://archive.org/details/oeloglogvalgorit691yaoa/page/n11/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;this text&lt;/a&gt; about Yao's algorithm for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Minimum_spanning_tree&quot; rel=&quot;nofollow noreferrer&quot;&gt;minimum spanning tree&lt;/a&gt; (MST) and I was wondering if there are some preceding algorithms (other than &lt;a href=&quot;https://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sollin's algorithm&lt;/a&gt;) that were published shortly before Yao's algorithm and that may have assisted (maybe some algorithms that grouped edges in a logarithmic number of groups) him in obtaining this result.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2021-04-10T15:03:21.003" LastActivityDate="2023-05-30T09:24:16.790" Title="How did Yao come up with his minimum spanning tree algorithm?" Tags="|mathematics|biographical-details|computer-science|algorithm|computer-scientists|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11983" PostTypeId="2" ParentId="3726" CreationDate="2020-07-08T10:50:36.417" Score="-1" Body="&lt;p&gt;Try here: &lt;a href=&quot;https://z-lib.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://z-lib.org/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this site, just by typing 'Martin Gardner', you'll find almost all the books and papers written by Gardner, in particular, all the 'Mathematical Games' articles published in 'Scientific American'.&lt;/p&gt;&#xA;" OwnerUserId="6222" LastEditorUserId="6222" LastEditDate="2020-07-09T20:05:13.053" LastActivityDate="2020-07-09T20:05:13.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11984" PostTypeId="2" ParentId="3767" CreationDate="2020-07-08T11:20:13.507" Score="1" Body="&lt;p&gt;Set theory was already around before Cantor used it to theorise his notion of cardinals. It's a mistake then to associate his name solely with set theory, and personally I think he,  himself would be horrified to be so honoured. Like almost all great mathematicians, he knew how much he owed to both his colleagues and his predeccessors.&lt;/p&gt;&#xA;&lt;p&gt;It's also a mistake to think of mathematics as a competition or a race. To do so is to turn it into a kind of parlour game, which it emphatically is not. More politically speaking it's symptomatic of neoliberalism -  a horizon under which science,  as a whole, does not succeed as it should -  which perhaps is the point of neoliberal disciplinary systems.&lt;/p&gt;&#xA;&lt;p&gt;So the answer to your question is -  it's the wrong question.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-07-08T11:20:13.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11985" PostTypeId="1" CreationDate="2020-07-08T12:55:23.823" Score="1" ViewCount="210" Body="&lt;p&gt;It seems to me that the ancient Greeks knew that geographic location affected the apparent position of the sun in sky but given the lack of rapid travel or communications or reliable clocks, how was this ever discovered?&lt;/p&gt;&#xA;&lt;p&gt;I am guessing that travelers might have noticed that their own internal clocks became out of synch with the local time but again, travel was so slow that I assume there really was no &amp;quot;jet lag&amp;quot; -- moving at most, aboard a ship which I think would have been the fastest was to move large distances, about 100 miles a day the change in local time from that of the traveler's origin would have been too slow for any jet lag-like phenomenon.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I understand that once it had been concluded that the Earth was round and revolved, an unusually educated person would understand, perhaps, that this implied that the sun would be in a different position in the sky at the same time to people in two far-apart cities.&lt;/p&gt;&#xA;&lt;p&gt;But I would suggest that very few people could visualize this idea. There was no way to communicate to the person in the distant city -- the idea of simultaneity might have been very hard to grasp, again, since time and solar position were so tightly coupled.&lt;/p&gt;&#xA;&lt;p&gt;There was a very clever (I assert) inventor who devised a time-keeper for court in ancient Greece and he made provisions for the time-keeper to run differently at different times of the year -- he did not have, I think, the idea of an hour but rather some equal division of daylight. Abstract time may not have been a thing to the ancients and therefore simultaneity in two distant locations may not have been a thing.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="4251" LastEditDate="2020-07-09T20:17:05.980" LastActivityDate="2020-07-09T20:17:05.980" Title="When and how was it discovered that the sun was in different positions depending upon longitude?" Tags="|time|archaeoastronomy|navigation|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="11986" PostTypeId="5" CreationDate="2020-07-08T21:15:45.510" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-08T21:15:45.510" LastActivityDate="2020-07-08T21:15:45.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11987" PostTypeId="4" CreationDate="2020-07-08T21:15:45.510" Score="0" Body="Andrey Nikolaevich Kolmogorov (1903-1987) was a Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-09T20:43:40.950" LastActivityDate="2020-07-09T20:43:40.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11988" PostTypeId="2" ParentId="11985" CreationDate="2020-07-09T01:49:01.267" Score="3" Body="&lt;p&gt;This is an immediate logical consequence of sphericity of the Earth. Greek tradition credits sphericity to Pythagoras, but modern historians even doubt that he ever existed. So the question has no exact answer. Those Greek writers on astronomy and geography whose work survived take this fact as evident.&lt;/p&gt;&#xA;&lt;p&gt;Chinese mathematicians who thought that the Earth is flat (in 2nd century AD) but the distance to the Sun is moderate (comparable with the size of the Earth) also concluded that the time will be different at different places. The only model which would give you the same time at all places is flat Earth and&#xA;enormous distance to the Sun (so that the Earth size is negligible in comparison with this distance). As far as I know, nobody proposed this model.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-07-09T11:52:52.080" LastActivityDate="2020-07-09T11:52:52.080" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="11990" PostTypeId="2" ParentId="3174" CreationDate="2020-07-09T15:53:35.250" Score="0" Body="&lt;p&gt;I guess the initial discovery and subsequent justification may have been done just by trying some calculations, like below. Nowadays we can use Excel for this kind of investigation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kXWi7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kXWi7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4730" LastEditorUserId="4730" LastEditDate="2020-07-10T12:42:22.107" LastActivityDate="2020-07-10T12:42:22.107" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="11991" PostTypeId="2" ParentId="3174" CreationDate="2020-07-09T17:36:51.743" Score="2" Body="&lt;p&gt;Formally, &lt;span class=&quot;math-container&quot;&gt;$(\sin x)'=\cos x$&lt;/span&gt; is true only if you measure x in radians. So you need to use radians for this fact anyway. If you will to ignore this, you can say that Euclid knew that the limit of  &lt;span class=&quot;math-container&quot;&gt;$\frac{\sin x}{x}$&lt;/span&gt; is 1, at least when &lt;span class=&quot;math-container&quot;&gt;$x=2\pi/n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n\rightarrow \infty$&lt;/span&gt;. Of course Euclid did not use radians, sin and lim, but he proved that the difference between the area of a circle and the area of an inscribed right polygon will become arbitrarily small as number of sides becomes large (by method of exhaustion). The area of right n-gon inscribed in a circle of radius 1 is &lt;span class=&quot;math-container&quot;&gt;$\frac 12 n\sin (2\pi/n)$&lt;/span&gt;. So using our notation, we can say that Euclid proved that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{ n\rightarrow \infty} \frac 12 n\sin (2\pi/n) = \pi.$$&lt;/span&gt;&#xA;This is equivalent to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{n\rightarrow \infty}\frac{ \sin (2\pi/n)}{2\pi/n} = 1.$$&lt;/span&gt;&#xA;Btw, that was used by Archimed to estimate &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;: he approximated &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; by area of 96-gon. That is, he calculated &lt;span class=&quot;math-container&quot;&gt;$\sin(2\pi/96)$&lt;/span&gt; and used that &lt;span class=&quot;math-container&quot;&gt;$2\pi/96 &amp;gt;\sin(2\pi/96)$&lt;/span&gt;. And to prove the upper bound he used &lt;span class=&quot;math-container&quot;&gt;$x&amp;lt; \tan x$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="6788" LastEditDate="2020-07-09T17:42:38.620" LastActivityDate="2020-07-09T17:42:38.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11992" PostTypeId="5" CreationDate="2020-07-09T21:25:58.133" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-09T21:25:58.133" LastActivityDate="2020-07-09T21:25:58.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11993" PostTypeId="4" CreationDate="2020-07-09T21:25:58.133" Score="0" Body="Alan Mathison Turing (1912-1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-11T19:08:10.020" LastActivityDate="2020-07-11T19:08:10.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11994" PostTypeId="1" CreationDate="2020-07-10T14:54:50.477" Score="1" ViewCount="67" Body="&lt;p&gt;According to wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler&lt;/a&gt; came up with Venn-like diagrams well before Venn but &lt;a href=&quot;https://en.wikipedia.org/wiki/Venn_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lull and Leibnitz&lt;/a&gt; came up with pictorial representations of set relations even before that.&#xA;Was Lull the first who is known to have come up with set diagrams?&#xA;It seems to me that the &lt;a href=&quot;https://mountainstoic.com/2015/03/30/on-the-grounding-of-ethics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;circles&lt;/a&gt; of &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oikei%C3%B4sis&quot; rel=&quot;nofollow noreferrer&quot;&gt;oikiosis&lt;/a&gt;&amp;quot; are an even earlier example.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-07-10T14:54:50.477" Title="When was the earliest usage of diagrams to represent set relations?" Tags="|mathematics|biographical-details|philosophy-of-science|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="11995" PostTypeId="1" AcceptedAnswerId="12019" CreationDate="2020-07-10T19:00:54.490" Score="4" ViewCount="240" Body="&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_numeral_systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of numeral systems&lt;/a&gt; lists only &lt;span class=&quot;math-container&quot;&gt;$10,20,60$&lt;/span&gt; as having been used in history. There are about twenty-five sets of symbols there used by different groups of people, but only three different bases. I'm a little surprised that, even if some of these civilizations learned about positional notation from others and did not independently discover it, they changed up the symbols without exploring whether a different base could be used. I would have thought that base &lt;span class=&quot;math-container&quot;&gt;$30$&lt;/span&gt; would be a decent candidate since it multiplies the first three primes (there are advantages with respect to the quantity of terminating rational numbers and convenient divisibility rules) while having only half the number of distinct symbols as &lt;span class=&quot;math-container&quot;&gt;$60,$&lt;/span&gt; which has been used by Babylonians, and the number of days in a lunar month is 29-30 days (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Islamic_calendar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arabic calendar&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Question: &lt;strong&gt;Has there been a large group of people that worked with a base other than &lt;span class=&quot;math-container&quot;&gt;$10,20,60$&lt;/span&gt; for day-to-day activities?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have read &lt;a href=&quot;https://hsm.stackexchange.com/questions/106/have-numbering-systems-other-than-base-ten-ever-been-used-or-popular&quot;&gt;this thread&lt;/a&gt; and it also mentions bases &lt;span class=&quot;math-container&quot;&gt;$10,20,60,$&lt;/span&gt; but no other significant ones. There is a &lt;a href=&quot;https://www.youtube.com/watch?v=cZH0YnFpjwU&quot; rel=&quot;nofollow noreferrer&quot;&gt;TED-Ed talk&lt;/a&gt; about on this subject as well, but it doesn't mention any others. excluding the usage of bases that are powers of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; for purposes related to computers.&lt;/p&gt;&#xA;" OwnerUserId="12679" LastEditorUserId="26" LastEditDate="2021-02-20T15:09:45.357" LastActivityDate="2021-02-20T15:09:45.357" Title="Has any large group of people used a base other than 10, 20 and 60 for ordinary purposes?" Tags="|mathematics|notation|number-theory|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="11996" PostTypeId="1" CreationDate="2020-07-11T05:10:54.537" Score="0" ViewCount="49" Body="&lt;p&gt;During the 19th century and before, bad water made drinking alcoholic drinks almost a necessity -- in any case, people drank by modern standards a tremendous amount.&lt;/p&gt;&#xA;&lt;p&gt;I am guessing that Fetal Alcohol Syndrome, despite the large number of cases that no doubt existed, was not well understood if only because it was so common; I know FAS is a fairly modern term.&lt;/p&gt;&#xA;&lt;p&gt;Delirium Tremens (DTs) -- is it possible that the chronic usage of alcohol made this ironically a rare condition, maybe observed on ships when the rum ran out and only became really common during prohibition?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2020-07-11T05:10:54.537" Title="How well was alcoholism understood at a time when people drank chronically?" Tags="|medicine|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="11997" PostTypeId="5" CreationDate="2020-07-11T12:37:59.857" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-11T12:37:59.857" LastActivityDate="2020-07-11T12:37:59.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11998" PostTypeId="4" CreationDate="2020-07-11T12:37:59.857" Score="0" Body="Norbert Wiener (1894-1964) was an American mathematician who did important work on probability and established the science of cybernetics." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-17T14:09:56.683" LastActivityDate="2020-07-17T14:09:56.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="11999" PostTypeId="1" CreationDate="2020-07-11T14:03:49.637" Score="4" ViewCount="494" Body="&lt;p&gt;For a long time now, I've been thinking about this: when was the first painting, illustration or depiction of any kind where our planet Earth is seen as a sphere in space? Possibly even from the Moon?&lt;/p&gt;&#xA;&lt;p&gt;Please note that I'm not talking about the famous photograph actually &lt;em&gt;taken&lt;/em&gt; from the Moon when they went there in the late 1960s. Nor am I talking about the black-and-white photo (which has also been colorized) taken a couple of decades earlier from the first camera-fitted weather balloon, which is far too late in history and also doesn't show the entire Earth.&lt;/p&gt;&#xA;&lt;p&gt;I'm talking about some kind of illustration, preferably ancient, displaying our planet as an orb/sphere surrounded by emptiness and maybe the moon and other planets, but the point is that it's &amp;quot;based on modern scientific understanding&amp;quot; rather than a freeform religious interpretation of what might lay &amp;quot;beyond the big sky dome with painted-on stars&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I hope I'm being clear. Basically, &amp;quot;modern society&amp;quot;, if you want to call it that, such as in the 1920s, did not have any space craft nor weather balloon photos, but obviously they at that point knew that the Earth isn't flat and that there is such a thing as space, etc. Oddly enough, I've never seen any such illustrations or paintings of any kind, or, if I have, I certainly cannot remember them.&lt;/p&gt;&#xA;" OwnerUserId="12763" LastActivityDate="2020-07-11T15:26:29.673" Title="What was the first &quot;scientific&quot;, non-photographic depiction of Earth as seen from outside of Earth?" Tags="|discoveries|19th-century|18th-century|16th-century|17th-century|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12000" PostTypeId="2" ParentId="11999" CreationDate="2020-07-11T15:26:29.673" Score="5" Body="&lt;p&gt;I don't know, but Images of earth from outside had been made for more than 500 years.&lt;/p&gt;&#xA;&lt;p&gt;A globe of the Earth would seem to count as a &amp;quot;depiction of any kind&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The sphericity of the Earth was established by Greek astronomy in the 3rd century BC, and the earliest terrestrial globe appeared from that period. The earliest known example is the one constructed by Crates of Mallus in Cilicia (now Çukurova in modern-day Turkey), in the mid-2nd century BC.&lt;/p&gt;&#xA;&lt;p&gt;No terrestrial globes from Antiquity or the Middle Ages have survived. An example of a surviving celestial globe is part of a Hellenistic sculpture, called the Farnese Atlas, surviving in a 2nd-century AD Roman copy in the Naples Archaeological Museum, Italy.[5]&lt;/p&gt;&#xA;&lt;p&gt;Early terrestrial globes depicting the entirety of the Old World were constructed in the Islamic world.[6][7] According to David Woodward, one such example was the terrestrial globe introduced to Beijing by the Persian astronomer, Jamal ad-Din, in 1267.[8]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The earliest extant terrestrial globe was made in 1492 by Martin Behaim (1459–1537) with help from the painter Georg Glockendon.[5] Behaim was a German mapmaker, navigator, and merchant. Working in Nuremberg, Germany, he called his globe the &amp;quot;Nürnberg Terrestrial Globe.&amp;quot; It is now known as the Erdapfel. Before constructing the globe, Behaim had traveled extensively. He sojourned in Lisbon from 1480, developing commercial interests and mingling with explorers and scientists. He began to construct his globe after his return to Nürnberg in 1490.&lt;/p&gt;&#xA;&lt;p&gt;Another early globe, the Hunt–Lenox Globe, ca. 1510, is thought to be the source of the phrase Hic Sunt Dracones, or “Here be dragons”. A similar grapefruit-sized globe made from two halves of an ostrich egg was found in 2012 and is believed to date from 1504. It may be the oldest globe to show the New World. Stefaan Missine, who analyzed the globe for the Washington Map Society journal Portolan, said it was “part of an important European collection for decades.”[9] After a year of research in which he consulted many experts, Missine concluded the Hunt–Lenox Globe was a copper cast of the egg globe.[9]&lt;/p&gt;&#xA;&lt;p&gt;A facsimile globe showing America was made by Martin Waldseemueller in 1507. Another &amp;quot;remarkably modern-looking&amp;quot; terrestrial globe of the Earth was constructed by Taqi al-Din at the Constantinople Observatory of Taqi ad-Din during the 1570s.[10]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Globe#History%5B1%5D&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Globe#History[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So there are surviving globes of the Earth dating to about 1500, and mentions of earlier globes of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to an illustration of the geocentric model o f the universe, dated to 1568.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Geocentric_model#/media/File:Bartolomeu_Velho_1568.jpg%5B2%5D&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Geocentric_model#/media/File:Bartolomeu_Velho_1568.jpg[2]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that it not only shows Earth as a globe in the center, but also shows continents.  Since the Ptolemaic model of the geocentric universe was published in the 2nd century, there could be surviving diagrams of the Ptolemaic model much older than 1568.  And those diagrams would depict Earth as a globe or circle, and perhaps sometimes put some features on it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The globus cruciger (Latin for &amp;quot;cross-bearing orb&amp;quot;), also known as &amp;quot;the orb and cross&amp;quot;, is an orb (Latin: globus) surmounted (Latin: gerere, to wear) by a cross (Latin: crux). It has been a Christian symbol of authority since the Middle Ages, used on coins, in iconography, and with a sceptre as royal regalia.&lt;/p&gt;&#xA;&lt;p&gt;Holding the world in one's hand, or, more ominously, under one's foot, has been a symbol since antiquity. To citizens of the Roman Empire, the plain spherical globe held by the god Jupiter represented the world or the universe, as the dominion held by the Emperor. A 2nd-century coin from the reign of Emperor Hadrian shows the Roman goddess Salus with her foot upon a globus, and a 4th-century coin from the reign of Emperor Constantine I shows him with a globus in hand. The orbis terrarum was central to the iconography of the Tetrarchy, in which it represented the Tetrarchs' restoration of security to the Roman world&lt;/p&gt;&#xA;&lt;p&gt;With the growth of Christianity in the 5th century, the orb (in Latin works orbis terrarum, the 'world of the lands', whence &amp;quot;orb&amp;quot; derives) was surmounted with a cross, hence globus cruciger, symbolizing the Christian God's dominion of the world. The Emperor held the world in his hand to show that he ruled it on behalf of God.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So a &lt;em&gt;globus cruciger&lt;/em&gt; is an image of the Earth or of the universe with a cross on top for Christianity.  Actual physical imperial and royal orbs were made beginning some time in the middle ages.&lt;/p&gt;&#xA;&lt;p&gt;the present imperial orb of the Holy Roman Empire is dated to western Germany, around the end of the 12th century (c. 1151-1200), but orbs were used earlier. I once read that in an imperial coronation in the 11th century (1001-1100) the emperor's orb was filled with soil from different parts of his realm, showing that the orb symbolized the Earth and not the heavens.&lt;/p&gt;&#xA;&lt;p&gt;So Earth has been depicted more or less as known to science for at least 500 years and possibly for over 2,000 years.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2020-07-11T15:26:29.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12001" PostTypeId="5" CreationDate="2020-07-11T17:20:27.677" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-11T17:20:27.677" LastActivityDate="2020-07-11T17:20:27.677" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12002" PostTypeId="4" CreationDate="2020-07-11T17:20:27.677" Score="0" Body="Stefan Banach (1892-1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th century mathematicians. He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics.&#xD;&#xA;" OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-12T11:32:18.093" LastActivityDate="2020-07-12T11:32:18.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12003" PostTypeId="5" CreationDate="2020-07-11T17:22:14.903" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-11T17:22:14.903" LastActivityDate="2020-07-11T17:22:14.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12004" PostTypeId="4" CreationDate="2020-07-11T17:22:14.903" Score="0" Body="David Hilbert (1862-1943) was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-12T11:31:55.917" LastActivityDate="2020-07-12T11:31:55.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12005" PostTypeId="5" CreationDate="2020-07-11T17:49:12.490" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-11T17:49:12.490" LastActivityDate="2020-07-11T17:49:12.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12006" PostTypeId="4" CreationDate="2020-07-11T17:49:12.490" Score="0" Body="John von Neumann (1903-1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-09-17T14:33:57.570" LastActivityDate="2021-09-17T14:33:57.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12007" PostTypeId="2" ParentId="2078" CreationDate="2020-07-11T18:57:05.193" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Halmos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Halmos&lt;/a&gt;&lt;span class=&quot;math-container&quot;&gt;$^\color{magenta}{\star}$&lt;/span&gt; on &lt;a href=&quot;https://en.wikipedia.org/wiki/John_von_Neumann&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Neumann&lt;/a&gt;'s prodigious memory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Even in the city and in the time that produced Szilárd (1898), Wigner (1902), and Teller (1908), von Neumann's brilliance stood out, and the legends about him started accumulating in his childhood. Many of the legends tell about his memory. His love of history began early, and, since he remembered what he learned, he ultimately became an expert on Byzantine history, the details of the trial of Joan of Arc, and minute features of the battles of the American Civil War.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He could, it is said, memorize the names, addresses, and telephone numbers in a column of the telephone book on sight. Some of the later legends tell about his wit and his fondness for humor, including puns and off-color limericks. Speaking of the Manhattan telephone book he said once that he knew all the numbers in it - the only other thing he needed, to be able to dispense with the book altogether, was to know the names that the numbers belonged to.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, &lt;a href=&quot;https://en.wikipedia.org/wiki/John_von_Neumann&quot; rel=&quot;nofollow noreferrer&quot;&gt;von Neumann&lt;/a&gt; was still prone to absent-mindedness:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Von Neumann was definitely not the caricatured college professor. He was a round, pudgy man, always neatly, formally dressed. There are, to be sure, one or two stories of his absentmindedness. Klári told one about the time when he left their Princeton house one morning to drive to a New York appointment, and then phoned her when he reached New Brunswick to ask: &amp;quot;Why am I going to New York?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{magenta}{\star}$&lt;/span&gt; Paul Richard Halmos, &lt;a href=&quot;http://dx.doi.org/10.2307/2319080&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Legend of John von Neumann&lt;/a&gt;, The American Mathematical Monthly, Volume 80, Number 4, April 1973.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;/questions/tagged/20th-century&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;20th-century&amp;#39;&quot; aria-label=&quot;show questions tagged &amp;#39;20th-century&amp;#39;&quot; rel=&quot;tag&quot; aria-labelledby=&quot;tag-20th-century-tooltip-container&quot;&gt;20th-century&lt;/a&gt; &lt;a href=&quot;/questions/tagged/mathematicians&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;mathematicians&amp;#39;&quot; aria-label=&quot;show questions tagged &amp;#39;mathematicians&amp;#39;&quot; rel=&quot;tag&quot; aria-labelledby=&quot;tag-mathematicians-tooltip-container&quot;&gt;mathematicians&lt;/a&gt; &lt;a href=&quot;/questions/tagged/von-neumann&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;von-neumann&amp;#39;&quot; aria-label=&quot;show questions tagged &amp;#39;von-neumann&amp;#39;&quot; rel=&quot;tag&quot; aria-labelledby=&quot;tag-von-neumann-tooltip-container&quot;&gt;von-neumann&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2023-12-05T20:44:08.517" LastActivityDate="2023-12-05T20:44:08.517" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12008" PostTypeId="1" AcceptedAnswerId="12018" CreationDate="2020-07-11T20:00:54.297" Score="1" ViewCount="115" Body="&lt;p&gt;While it is known that William Herschel was both a composer and an astronomer, could it be that he ever wrote music based on astronomy, or how he felt his research was proceeding? Just as Beethoven wrote Moonlight Sonata by being inspired by the Moon, did he ever compose symphonies, movements, or operas based on his work or his outlook of it? For example, if he felt sad, or depressed over how his research may have been encountering impediments, is there any music written by him that may be characterized as &amp;quot;sad/gloomy&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;If anyone has any reference to these potential works, please leave an answer below. Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2020-07-14T06:06:39.180" Title="Was Herschel ever inspired to write music through his astronomy research?" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12009" PostTypeId="2" ParentId="17" CreationDate="2020-07-12T02:41:41.603" Score="1" Body="&lt;p&gt;There is no known record of any personal encounter between Banach and Hilbert. But the  not-so-random connection between the two  was  Hugo Steinhaus (Banach's discoverer, and later  collaborator and colleague), who was a PhD student of Hilbert in  Göttingen.  Steinhaus's thesis, titled &lt;em&gt;Neue Anwendungen des Dirichlet'schen Prinzips&lt;/em&gt;, defended in 1911, was still  rather traditional in its approach to variational problems for second-order partial differential equations.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, Banach's PhD thesis &lt;em&gt;O operacjach na zbiorach abstrakcyjnych z zastosowaniami do równań  cał kowych&lt;/em&gt; [On operations on abstract sets with applications to integral equations] defended in Lwów in 1920, introduced fundamental notions and properties of linear normed complete spaces (in an axiomatic way), and applied them to integral operators defined by kernels. The actual thesis of Banach and its defense became  stuff of legends, but at least there is a publication based on the thesis, S. Banach, &lt;em&gt;Sur les opérations dans les ensembles abstracts et leur application aux équations intégrales&lt;/em&gt;},&#xA;&lt;a href=&quot;http://kielich.amu.edu.pl/Stefan_Banach/pdf/oeuvres2/305.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fundamenta Mathematicae 3 (1922),  pp. 133-181&lt;/a&gt;.&#xA;In the introduction Banach mentions preceding work on ``functional operations'' by Volterra, Fréchet, Hadamard, F. Riesz, Pincherle, Steinhaus, Weyl, Lebesgue and others. He particularly credits the works of Hilbert, which according to him enabled treatment of (the spaces of) square-integrable functions, not only smooth functions.  This is also evidence that Banach studied works of Hilbert before visiting Paris.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, in 1917 Banach and Steinhaus both lived in Krakow and took part in meetings of an informal mathematical society. Other members of this group were mathematicians Wlodzimierz Stozek, Wladyslaw Slebodzinski, Leon Chwistek (also a philosopher and a painter) and a physicist Jan Norbert Kroo, all of whom spent some time studying in Göttingen.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastEditorUserId="48" LastEditDate="2022-03-05T16:26:58.430" LastActivityDate="2022-03-05T16:26:58.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12010" PostTypeId="2" ParentId="9785" CreationDate="2020-07-12T11:55:46.810" Score="4" Body="&lt;p&gt;I remember Prof. Ahlfors lecturing in our complex analysis class.  One day he covered Bloch's inequality.  He calmly remarked: &amp;quot;Bloch&lt;sup&gt;1&lt;/sup&gt; is the only well-known mathematician convicted of mass murder.&amp;quot;  Then he continued with the lecture.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9_Bloch_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;André Bloch&lt;/a&gt; &amp;quot;...killed three of his family members, for which he was institutionalized in a mental asylum for 31 years, during which all of his mathematical output was produced.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2024-03-06T14:31:30.090" LastActivityDate="2024-03-06T14:31:30.090" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12011" PostTypeId="5" CreationDate="2020-07-12T12:52:14.850" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-12T12:52:14.850" LastActivityDate="2020-07-12T12:52:14.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12012" PostTypeId="4" CreationDate="2020-07-12T12:52:14.850" Score="0" Body="Charles Robert Darwin (1809-1882) was an English naturalist whose scientific theory of evolution by natural selection became the foundation of modern evolutionary studies." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-17T14:09:50.887" LastActivityDate="2020-07-17T14:09:50.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12014" PostTypeId="1" AcceptedAnswerId="12015" CreationDate="2020-07-12T18:26:46.790" Score="7" ViewCount="5710" Body="&lt;p&gt;I stumbled upon the following napkin (yes, a napkin) and was wondering who the lady on the picture is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/XhPrP.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XhPrP.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;She is surrounded by some basic mathematical functions, as well as physical equations which relate to some drawings (mostly mechanics, and specifically dynamics).&lt;/p&gt;&#xA;&lt;p&gt;The napkin is not related to any particular event (a dinner linked to some historical events in science for instance) so it may very well be that this is something a graphical artist came with - without any links to science.&lt;/p&gt;&#xA;" OwnerUserId="4337" LastActivityDate="2020-07-14T14:15:38.823" Title="Who is the lady on the image?" Tags="|physics|depiction-of-scientists|" AnswerCount="2" CommentCount="5" ClosedDate="2020-07-15T06:00:17.147" ContentLicense="CC BY-SA 4.0" />
  <row Id="12015" PostTypeId="2" ParentId="12014" CreationDate="2020-07-13T00:14:49.093" Score="19" Body="&lt;p&gt;This is a stylized picture of a generic pretty lady of the mid 1700's,&#xA;created in the mid 1800's to illustrate a sentimental poem in a book collection of such  for the amusement of genteel young ladies.  It is not a portrait of any particular person.  A modern equivalent would be the cover illustration on an airport novel.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=nyp.33433000134712&amp;amp;view=1up&amp;amp;seq=139&amp;amp;size=175&quot; rel=&quot;nofollow noreferrer&quot;&gt;original&lt;/a&gt; of this image, captioned &amp;quot;Lady of the Court of Louis XVth&amp;quot;,  appears in &lt;em&gt;Fisher's drawing room scrap-book 1848&lt;/em&gt;&#xA;illustrating the poem &amp;quot;The Message&amp;quot; by  Miss A. D. Reynett.&#xA;See &lt;a href=&quot;https://catalog.hathitrust.org/Record/000063998&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; for bibliographic info.  Another version (in &lt;em&gt;The Gallery of Engravings&lt;/em&gt;, a.k.a. &lt;em&gt;The Peoples Gallery of Engravings, Second Series&lt;/em&gt;, ed. by Mrs Milner ) appears &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=uc2.ark:/13960/t0ms40s04&amp;amp;view=2up&amp;amp;seq=152&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  Reynett's initials are given &lt;a href=&quot;http://poetessarchive.org/hootman/drawingroom1848.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;; she is probably the Reynett  described &lt;a href=&quot;https://littleprofessor.typepad.com/the_little_professor/2006/11/this_weeks_acqu_3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;..&#xA;The fine print at the bottom  shows the engraving was by &lt;a href=&quot;http://www.walterscott.lib.ed.ac.uk/portraits/engravers/robinson.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;H. Robinson&lt;/a&gt; (see &lt;a href=&quot;https://www.britishmuseum.org/collection/term/BIOG247160&quot; rel=&quot;nofollow noreferrer&quot;&gt;also here&lt;/a&gt;), after a drawing by &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Jewell_Penstone&quot; rel=&quot;nofollow noreferrer&quot;&gt;J.J. Penstone&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think it is unlikely this is Marie Antoinette, who was famously a person of the court of Louis XVI, or of Émilie du Châtelet.&lt;/p&gt;&#xA;&lt;p&gt;For what it's worth, the poem itself says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;LADY OF THE COURT OF LOUIS XV.&lt;/p&gt;&#xA;&lt;p&gt;THE MESSAGE.&lt;/p&gt;&#xA;&lt;p&gt;BY MISS REYNETT.&lt;/p&gt;&#xA;&lt;p&gt;When thou, dear Bird, art privileged to stand&lt;br /&gt;&#xA;On the white fingers of that fairy hand,&lt;br /&gt;&#xA;Forget not then, thy Master's fond behest,&lt;br /&gt;&#xA;Who pines far off, and knows nor joy nor rest.&lt;br /&gt;&#xA;Say that I trust her, who was borne away&lt;br /&gt;&#xA;By nearer friends, to shine in conquest's ray;&lt;br /&gt;&#xA;Say, though the tinsel chain of Fashion bind&lt;br /&gt;&#xA;Her outward form, I know her heart and mind;&lt;br /&gt;&#xA;Say that I love her, tho' we meet no more;&lt;br /&gt;&#xA;Say that I prize her, tho' hope's dream is o'er ;&lt;br /&gt;&#xA;Say that I think of her — where'er I go ;&lt;br /&gt;&#xA;Say that I sigh for her, in joy or wo;&lt;br /&gt;&#xA;Say when I stand amidst the gayest throng ;&lt;br /&gt;&#xA;Say when I listen to the sweetest song;&lt;br /&gt;&#xA;Say when I feel soft music's magic tone,&lt;br /&gt;&#xA;I hear her, see her, think of her, alone;&lt;br /&gt;&#xA;Say, ( tho’ I dare not breathe my thoughts aloud,&lt;br /&gt;&#xA;A solitary soul - amidst the crowd;)&lt;br /&gt;&#xA;That her loved form, which fills at once my heart,&lt;br /&gt;&#xA;Holds me, for ever, from that crowd apart:&lt;br /&gt;&#xA;My guardian Angel breaks all earthly spells,&lt;br /&gt;&#xA;And purifies the temple — where she dwells!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note:  I had just learned about Google image search, and on my&#xA;first try-out got numbers of  hits, some with captions citing &amp;quot;Gallery of engravings, vol. 1&amp;quot; (in English, on some Spanish- and Russian-language equivalents of Pintarest).   Once I realized &amp;quot;Gallery of engravings&amp;quot; could be a publication  title I went to Hathi Trust.&lt;/p&gt;&#xA;&lt;p&gt;As for the larger question of &amp;quot;so what&amp;quot;?  This image, a sentimental drawing of a stylized un-named pretty lady of the mid 1700s, was created and published in the mid 1800s, came into the public domain, and entered collections of stock images in the internet era.  Now, in the last decade, a napkin seller has used a formula-defaced version of this image as a napkin design, declaring it to be an image of Marie Antoinette. It is possibly evidence of a kind of geek-chic&#xA;mentality of the 2010's, perhaps meant to be evocative of the napkins at a place like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scottish Cafe&lt;/a&gt; where Borsuk and Ulam caroused (but I suspect &lt;em&gt;that&lt;/em&gt; cafe had cloth napkins, not paper ones).&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-07-14T14:15:38.823" LastActivityDate="2020-07-14T14:15:38.823" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12016" PostTypeId="2" ParentId="8062" CreationDate="2020-07-13T01:02:41.263" Score="2" Body="&lt;p&gt;Perhaps the following papers may help clarifying the issue:&#xA;Hoyrup, Jens :&#xA;i) “The Formation of Islamic Mathematics”: Sources and Conditions. Preprints og Reprints 1987 Nr 1. (available online)&#xA;ii) “Algebraic Traditions behind Ibn Turk and al-Khwārizmī'. In Acts of the International Symposium on Ibn Turk, Khwarezmī, Farabī, Beyronī and Ibn Sīna&#xA;(Ankara 9-12 september 1985). Available online.&lt;/p&gt;&#xA;&lt;p&gt;I am working on a paper related to the subject and it will published on Slideshare within a few days&lt;/p&gt;&#xA;" OwnerUserId="12767" LastActivityDate="2020-07-13T01:02:41.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12017" PostTypeId="2" ParentId="12014" CreationDate="2020-07-13T03:15:58.027" Score="5" Body="&lt;p&gt;Possibly &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89milie_du_Ch%C3%A2telet&quot; rel=&quot;noreferrer&quot;&gt;Émilie du Châtelet&lt;/a&gt; who was certainly of the right historical period (1706-1740, compared with Louis XV's reign of 1715-1774).&lt;/p&gt;&#xA;&lt;p&gt;Her best remembered achievement was a translation of Newton's &lt;em&gt;Principia&lt;/em&gt; which (according to Wikipedia) is still considered to be the standard French translation.&lt;/p&gt;&#xA;" OwnerUserId="10039" LastActivityDate="2020-07-13T03:15:58.027" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12018" PostTypeId="2" ParentId="12008" CreationDate="2020-07-13T06:40:04.660" Score="4" Body="&lt;p&gt;It is hard to say a definitive no to &amp;quot;ever&amp;quot;, but the facts are not there. First, as on can see from chronology in &lt;a href=&quot;http://www.radioswissclassic.ch/en/music-database/musician/157620f3db1699fcba0221df1ab15400e086c1/biography&quot; rel=&quot;nofollow noreferrer&quot;&gt;RadioSwiss database&lt;/a&gt;, Herschel wrote the bulk of his musical compositions in 1760-1769, before developing interest in astronomy. Second, we have plenty of first hand testimony about his life, much of it reviewed in &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4123665/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philomaths, Herschel, and the myth of the self-taught man by Winterburn&lt;/a&gt;. There are several autobiographies, including a &amp;quot;memorandum of life&amp;quot;, and contemporary accounts by his sister Caroline, who lived with him, but none of them mention music inspired by astronomy. This is a particularly odd omission considering the traditional associations with the &amp;quot;music of the spheres&amp;quot; that go back to Pythagoreans. Moreover, we have positive evidence that Herschel was more &amp;quot;mathematical&amp;quot; and pragmatic than romantic in his composition of music, and gave it up with relief when offered a position of royal astronomer, see &lt;a href=&quot;http://adsabs.harvard.edu/full/1962PASP...74...55D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir William Herschel as a Composer by Duckles&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There&#xA;was nothing mystical or metaphysical in his approach to the art&#xA;of music. He was a practical musician of his time, and he lived&#xA;in a time when musicians were too much occupied with the problems&#xA;of survival to indulge in cosmic or sentimental views about&#xA;their role in society... One gets the impression from this concerto of a strong intellect&#xA;at work, a mind that takes willful control of the materials&#xA;and shapes them, even forces them, into line. He has a structural approach toward composition; he uses musical ideas as building&#xA;blocks and the joints by which they are connected are not always&#xA;smooth... It remains for the listener to decide whether or not the world&#xA;lost a great composer when it gained a great astronomer. Herschel&#xA;himself had no doubts in the matter.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even the popular story about Haydn getting inspired to write his oratorio &lt;em&gt;The Creation&lt;/em&gt; after conversing with Herschel at his observatory in 1792 (Herschel's own music is often characterized as &amp;quot;Haydnesque&amp;quot;) turns out to be a fable. Herschel was away on the day of Haydn's visit and they never met, see &lt;a href=&quot;http://www.charlespetzold.com/blog/2019/03/Haydn-and-Herschel-in-the-Clair-de-Lune.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Petzold, Haydn and Herschel in the Clair de Lune&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is, on the other hand, plenty of evidence of the converse - that Herschel's experience with music led him to develop interest first in mathematics, and then in optics and astronomy. His evolution is documented in detail by Winterburn. The stimulus was Robert Smith's book Harmonics, or the Philosophy of Musical Sounds that inspired Herschel to write his own Treatise on Music (1764, unpublished), where it is cited for the &amp;quot;mathematics of harmonics&amp;quot;, and &amp;quot;mathematical division of intervals&amp;quot;. And the catalyst was Herschel's involvement with the Philomaths society that promoted new mathematics and natural philosophy from 1766 on.&lt;/p&gt;&#xA;&lt;p&gt;What followed was reading Maclaurin's Treatise on Fluxions and Smith's also Compleat System of Opticks that included telescope designs. With the help of Caroline and brother Alexander, who joined him in Bath in 1772, Herschel started constructing his own telescopes. But by then he stopped composing, although kept performing to earn a living (he was an organist at the Octagon Chapel in Bath since 1767). In the late 1770s he was writing on natural history, electricity and, of course, astronomy, and regularly presenting his research to Bath Philosophical Society.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;William had learnt music through concentrated repetitive solitary practice from an early age and here, described in Smith, presented as an application of Locke's ideas, was that same practice applied to sight... By practising observing as he had practised music, William became an excellent observer. His instrument-making skills also improved as he practised and perfected the art of mirror making, experimenting with different alloy compositions and polishing techniques... By the end of 1779 William had successfully tested the quality of his telescopes and... began to report on observations made with his telescopes. Two of these papers—one on a star in Collo Ceti, the other on lunar mountains—were later read to the Royal Society... When he came to discover the planet Uranus in 1781, his involvement with the society meant he knew just how to word his announcement and whom to tell.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When Lalande informed Herschel about calculations performed on his new planet and inquired about his biography, the latter gave him a glossed version that he was &amp;quot;&lt;em&gt;brought up to Music&lt;/em&gt;&amp;quot;, but &amp;quot;&lt;em&gt;my leisure hours were generally devoted to mathematics and other studies&lt;/em&gt;&amp;quot;. In 1782 he was offered a position as Astronomer Royal for King George III, as well as funds for building bigger telescopes, and left his musical posts in Bath. His transition from music to astronomy was complete.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-07-14T06:06:39.180" LastActivityDate="2020-07-14T06:06:39.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12019" PostTypeId="2" ParentId="11995" CreationDate="2020-07-13T23:30:44.647" Score="4" Body="&lt;p&gt;Wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Languages using &lt;a href=&quot;https://en.wikipedia.org/wiki/Duodecimal&quot; rel=&quot;nofollow noreferrer&quot;&gt;duodecimal&lt;/a&gt; number systems are uncommon. Languages in the Nigerian Middle Belt such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Janji_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Janji&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Gbiri-Niragu_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gbiri-Niragu&lt;/a&gt; (Gure-Kahugu), &lt;a href=&quot;https://en.wikipedia.org/wiki/Piti_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Piti&lt;/a&gt;, and the Nimbia dialect of &lt;a href=&quot;https://en.wikipedia.org/wiki/Gwandara_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gwandara&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Chepang_language&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chepang&lt;/a&gt; language of Nepal are known to use duodecimal numerals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is thought that Nimbia, which is isolated from the rest of Gwandara, acquired its duodecimal system from neighboring &lt;a href=&quot;https://en.wikipedia.org/wiki/East_Kainji_languages&quot; rel=&quot;nofollow noreferrer&quot;&gt;East Kainji&lt;/a&gt; languages. It is duodecimal even to powers of base twelve.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It also appears that many ancient progenitors of Indo-European languages were based on 4 or 12, due to remnant number words that don't line up with a purely decimal system:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they come from Proto-Germanic *ainlif and *twalif (meaning, respectively one left and two left), suggesting a decimal rather than duodecimal origin.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Wikipedia's primary source appears to be &lt;a href=&quot;https://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/%7EP_aflang/TEXTS/oct98/decimal.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; by Shuji Matsushita.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2020-08-23T20:15:56.570" LastActivityDate="2020-08-23T20:15:56.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12021" PostTypeId="2" ParentId="3038" CreationDate="2020-07-15T05:59:32.673" Score="1" Body="&lt;p&gt;One contemporary vanguard is the environmental movement and one aspect of this is anthropogenic climate change.&lt;/p&gt;&#xA;&lt;p&gt;Global Witness, a London based rights organisation have documented more than 1500 environmentalists who have been killed whilst defending the environment in 50 countries between 2002 and 2017. During those fifteen years the rate of killings increased from 2 per week to 4 per week with the death toll in Brazil, Honduras, the Phillipines and Columbia accounted for just over 70%.&lt;/p&gt;&#xA;&lt;p&gt;Amongst them is Lei Yang, an environmental scientist in China, who died whilst in police custody. The authorities stated that no charges would be brought against the officers involved. The case in China provoked a huge protest accusing the police of brutality.&lt;/p&gt;&#xA;&lt;p&gt;According to John Knox, a former UN special Rapporteur on Human Rights and the Environment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;murder is not the only way enviromental defenders are persecuted. For every 1 killed, there are 20 to 100 others who are harassed, unlawfully and lawfully persecuted, and sued for defamation amongst other intimidations.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would expect that there are many more educated and working in the environmental sciences amongst the list.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-07-15T05:59:32.673" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12022" PostTypeId="2" ParentId="11397" CreationDate="2020-07-15T06:20:27.033" Score="1" Body="&lt;p&gt;James Clerk Maxwell for coming up with his eponymous equations. According to Freeman Dyson:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the year 1865, James Clerk Maxwell published his paper 'a dynamical theory of the electromagnetic field' in &lt;em&gt;The Philosophical Transactions of the Royal Society&lt;/em&gt;...we, with the benefit of hindsight can see that Maxwells paper was the most important event in the history of the physical sciences... but the importance of Maxwell's work was not obvious to his contemporaries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For more than twenty years his work on electromagnetism was mostly ignored. Physicists found it hard to understand because his equations were complicated. Mathematicians found it hard to understand because he used physical language to explain them. It was regarded as an obscure speculation without much experimental evidence to support it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Twenty years after publication would be 1885. He passed away in 1879.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-07-15T06:20:27.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12023" PostTypeId="2" ParentId="188" CreationDate="2020-07-15T06:24:28.180" Score="-1" Body="&lt;p&gt;The turboprops of the Tupolev Tu-95 are the most powerful ever built, even more powerful than those fitted to the Airbus A400.&lt;/p&gt;&#xA;&lt;p&gt;The design of the turboprops for the Tu-95 was due to a team of (ex-Junkers) German technicians more or less forcefully 'imported' from the vanquished Germany. One of the leading German engineers of that group recounted his experience in  a Swiss technical journal, in the 1950s:&lt;/p&gt;&#xA;&lt;p&gt;Schweizerische Bauzeitung, issues 32 &amp;amp; 33. 'Die Propellerturbinen-Entwicklung in der Sowjetunion', by Ferdinand Brandner&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=sbz-002:1957:75::463&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.e-periodica.ch/digbib/view?pid=sbz-002:1957:75::463&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2020-07-15T06:24:28.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12024" PostTypeId="2" ParentId="188" CreationDate="2020-07-15T07:41:31.020" Score="2" Body="&lt;p&gt;According to Boris Chertoks, &lt;em&gt;Rockets and People&lt;/em&gt;, a four volume memoir on the Soviet rocket programme and the most cited text in this area, several hundred Nazi era German scientists were brought into the programme and whilst he mentions contributions by them during their first year there in 1946, they hardly figure in subsequent years.&lt;/p&gt;&#xA;&lt;p&gt;The official record is not any better. For example, the 1969 edition of &lt;em&gt;The Soviet Encyclopedia of Space Flight&lt;/em&gt; does not mention a single German scientist or engineer whilst giving broad credit to German scientists employed in the American space programme. Most likely this was due to national pride. We will only know more once more honest and careful  appraisals of the period are researched and published - I mean who did what and why and when.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-07-15T07:52:58.577" LastActivityDate="2020-07-15T07:52:58.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12025" PostTypeId="1" AcceptedAnswerId="12028" CreationDate="2020-07-16T01:03:00.033" Score="3" ViewCount="194" Body="&lt;p&gt;Human beings keeping count is as old as recorded history; however, how did humans first think about keeping documentation and counting numbers? Is there any defined period in history where numbers were written about, where early scholars and scientists discoursed about the origins of counting? Specifically, how did we as a species come to realize that counting is possible and useful in everyday life?&lt;/p&gt;&#xA;&lt;p&gt;If any links or references exist to answer this question, please post an answer below. Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2020-07-17T06:38:33.933" Title="How did human beings come to realize the concept of counting numbers?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ClosedDate="2020-07-16T18:27:19.037" ContentLicense="CC BY-SA 4.0" />
  <row Id="12028" PostTypeId="2" ParentId="12025" CreationDate="2020-07-16T14:36:14.000" Score="4" Body="&lt;p&gt;The simple answer is that we don't know how human beings came to realize the concept of counting numbers. The discovery of counting is lost in prehistory. Most likely humans are born with the capacity to count, given that scientists have established that other species like apes or parrots can count. This would be the numerical analogue to Chomsky's notion of an innate grammar.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="12330" LastEditDate="2020-07-17T06:38:33.933" LastActivityDate="2020-07-17T06:38:33.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12029" PostTypeId="1" CreationDate="2020-07-16T22:59:01.070" Score="0" ViewCount="75" Body="&lt;p&gt;The wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_experiments&quot; rel=&quot;nofollow noreferrer&quot;&gt;page&lt;/a&gt; about experimental research history cites Ibn al-Haytham as an early pioneer of experimental research.&#xA;Since Galen was the last prominent ancient greek philosophers/doctor and according to his wikipedia page he conducted some &lt;a href=&quot;https://en.wikipedia.org/wiki/Galen&quot; rel=&quot;nofollow noreferrer&quot;&gt;medical&lt;/a&gt; demonstrations/experiments on animals I am wondering if he philosophized on the theoretical aspects of his demonstrations.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-07-18T09:06:39.730" LastActivityDate="2020-07-18T09:06:39.730" Title="Did Galen make any theoretical contributions to the experimental science?" Tags="|biographical-details|philosophy-of-science|medicine|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12031" PostTypeId="1" CreationDate="2020-07-17T15:41:01.837" Score="5" ViewCount="558" Body="&lt;p&gt;Raymond Boyce was one of the first pioneers of the relational database and a co-developer of SQL, yet it is almost impossible to find any pictures of him:&#xA;&lt;a href=&quot;https://www.google.com/search?q=raymond+f+boyce&amp;amp;tbm=isch&amp;amp;ved=2ahUKEwiBrvvZwdTqAhXlgHMKHUyTDPIQ2-cCegQIABAA&amp;amp;oq=raymond+f+boyce&amp;amp;gs_lcp=CgNpbWcQAzIECCMQJzIECCMQJzICCAAyBggAEAcQHjIECAAQGDIECAAQGDIECAAQGFCPLVj_WGC5WWgAcAB4AIABMIgBMJIBATGYAQCgAQGqAQtnd3Mtd2l6LWltZ8ABAQ&amp;amp;sclient=img&amp;amp;ei=1LcRX4F05YHOA8ymspAP&amp;amp;bih=937&amp;amp;biw=1920&amp;amp;rlz=1C1CHBF_enGB859GB859&quot; rel=&quot;noreferrer&quot;&gt;https://www.google.com/search?q=raymond+f+boyce&amp;amp;tbm=isch&amp;amp;ved=2ahUKEwiBrvvZwdTqAhXlgHMKHUyTDPIQ2-cCegQIABAA&amp;amp;oq=raymond+f+boyce&amp;amp;gs_lcp=CgNpbWcQAzIECCMQJzIECCMQJzICCAAyBggAEAcQHjIECAAQGDIECAAQGDIECAAQGFCPLVj_WGC5WWgAcAB4AIABMIgBMJIBATGYAQCgAQGqAQtnd3Mtd2l6LWltZ8ABAQ&amp;amp;sclient=img&amp;amp;ei=1LcRX4F05YHOA8ymspAP&amp;amp;bih=937&amp;amp;biw=1920&amp;amp;rlz=1C1CHBF_enGB859GB859&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;All the results show a man who seems to be about 60+ years old but Boyce died at the age of 27. The only image that I could find of him was in this slideshow on Database Management Systems:&#xA;&lt;a href=&quot;https://slideplayer.com/slide/13584514/&quot; rel=&quot;noreferrer&quot;&gt;https://slideplayer.com/slide/13584514/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The picture quality looks like in could be from the 70s (when Boyce was in his 20s) and the person in the photo looks to be roughly the same age. But a google search for similar images produces absolutely nothing.&lt;/p&gt;&#xA;&lt;p&gt;I get that it might be difficult to find pictures of somebody from the 70s who died before they became truly famous. But co-founding SQL is a big deal. Even if there are reasonable answers for...&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why are there no pictures of Raymond Boyce?&#xA;OR&lt;/li&gt;&#xA;&lt;li&gt;Where on earth did the producers of that slideshow get that other image?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;... I haven't been able to derive any sensible answers to the strangest question ...&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How come they are no threads on forums asking where they can get a genuine picture of him?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is honestly the first question I've typed into google in years that nobody else has asked before, or something similar. Any thoughts?&lt;/p&gt;&#xA;" OwnerUserId="12796" LastActivityDate="2020-07-19T05:49:24.613" Title="Why so hard to find references to pictures of Raymond Boyce?" Tags="|computer-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12032" PostTypeId="5" CreationDate="2020-07-17T22:33:39.347" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-17T22:33:39.347" LastActivityDate="2020-07-17T22:33:39.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12033" PostTypeId="4" CreationDate="2020-07-17T22:33:39.347" Score="0" Body="Nicolas Bourbaki was the pseudonym chosen by a group of young mathematicians in France in the mid 1930s." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-24T00:09:35.763" LastActivityDate="2020-07-24T00:09:35.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12034" PostTypeId="2" ParentId="8071" CreationDate="2020-07-18T14:41:17.440" Score="1" Body="&lt;p&gt;The gear from Olbia has been confirmed as a legitimate find by the archaeologists who discovered it and later restored it; the same archaeologists, not Pastore, established its dating. I do not know on what basis Jones rejects the dating, but if it is only on the idea that the gear &amp;quot;seems too complex&amp;quot; for its dating, that would be the same reason to reject the Antikythera mechanism dating as a &amp;quot;contamination&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Regarding Pastore book, his arguments are summed up in the YouTube video (&lt;a href=&quot;https://www.youtube.com/watch?v=hp-FXnjxNfw&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=hp-FXnjxNfw&lt;/a&gt;), I do not believe there is more than that&lt;/p&gt;&#xA;" OwnerUserId="12801" LastActivityDate="2020-07-18T14:41:17.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12035" PostTypeId="2" ParentId="12031" CreationDate="2020-07-18T17:01:59.260" Score="7" Body="&lt;p&gt;I think the answer to the main question is clear from his Wikipedia biography: Boyce died in 1974 aged 27, the same year SQL was released. He was a young researcher working for IBM in a pre-digital age, so there would have been very few reasons why photographs of him would have been taken and published during his tragically short life. And his input to computer science, while important, is not significant enough to prompt his few short biographical notes to make the effort to seek out personal photographs.&lt;/p&gt;&#xA;&lt;p&gt;That doesn’t explain why some online biographies seem to use clearly incorrect photographs, though these may just be a result of bad web searching or scraping.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: I suspect this is &lt;a href=&quot;https://www.findagrave.com/memorial/175650158/raymond-francis-boyce/photo&quot; rel=&quot;noreferrer&quot;&gt;Boyce’s grave&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update #2&lt;/strong&gt;: I contacted Professor Kimelfeld, who confirmed that the photo used in the slides was obtained directly from colleagues at IBM Almaden who had worked with Boyce.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2020-07-19T05:49:24.613" LastActivityDate="2020-07-19T05:49:24.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12038" PostTypeId="2" ParentId="11969" CreationDate="2020-07-19T03:32:39.033" Score="1" Body="&lt;p&gt;The historical name of this function is the &lt;em&gt;suversed sine&lt;/em&gt;, &lt;em&gt;suversine&lt;/em&gt;, or &lt;em&gt;susinus versus&lt;/em&gt;, and is abbreviated &lt;span class=&quot;math-container&quot;&gt;$\operatorname{suvers}(x)$&lt;/span&gt;. (Similarly, the function &lt;span class=&quot;math-container&quot;&gt;$1 + \sin(x)$&lt;/span&gt; is called the &lt;em&gt;cosuversine&lt;/em&gt; or &lt;em&gt;sucoversine&lt;/em&gt;.) The earliest use of this name may be in 1801 by Joseph de Mendoza y Rios.&lt;/p&gt;&#xA;&lt;p&gt;We see in &lt;a href=&quot;https://www.google.com/books/edition/Mathematics_for_Practical_Men/jqFfD3S-eJAC?hl=en&amp;amp;gbpv=1&amp;amp;dq=suversine&amp;amp;pg=PA137&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gregory and Law's 1862 &lt;em&gt;Mathematics for Practical Men&lt;/em&gt;&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;12&quot;&gt;&#xA;&lt;li&gt;The &lt;em&gt;suversine&lt;/em&gt; of an arc is the versed sine of its supplement, as &lt;span class=&quot;math-container&quot;&gt;$A'D$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.google.com/books/edition/The_elements_of_plane_trigonometry/x1MEAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=suversine&amp;amp;pg=PA9&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snowball's 1837 &lt;em&gt;The elements of plane trigonometry&lt;/em&gt;&lt;/a&gt;, we see&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;8&quot;&gt;&#xA;&lt;li&gt;The versed sine of the supplement of the &lt;span class=&quot;math-container&quot;&gt;$\angle BAC$&lt;/span&gt; is called the &lt;em&gt;suversine&lt;/em&gt; of the &lt;span class=&quot;math-container&quot;&gt;$\angle BAC$&lt;/span&gt;;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;or, &lt;span class=&quot;math-container&quot;&gt;$\operatorname{suversin}\angle BAC = \operatorname{versin}(180^\circ -\angle BAC).$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Curiously, this source does not reference the coversine (or sucoversine) at all.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/The_Complete_Mathematical_and_General_Na/_ZtDAAAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=versed+sine+supplement&amp;amp;pg=PA58&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Kerigan (1828)&lt;/a&gt; uses the term &amp;quot;versed sine supplement&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.google.com/books/edition/The_Monthly_Review/Z-RKAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22suversed%22&amp;amp;pg=PA133&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Monthly Review, For October, 1806. Art. II&lt;/a&gt;, it is suggested that Joseph de Mendoza y Rios is the originator of the name, from his &lt;em&gt;Tables for Navigation&lt;/em&gt; (1806):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have mentioned certain terms, suversed, sucoversed, &amp;amp;c. which are novel in mathematical language; and M. Mendoza is, we believe, the author of the &amp;quot;&lt;em&gt;callida junctura&lt;/em&gt;.&amp;quot;— We subjoin the values of these lines, from which our readers may easily discern the reason for their denomination. Suppose the radius 1&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;\text{then} \operatorname{versin.} A &amp;amp;= 1 - \operatorname{cos.} A \\&#xA;\operatorname{suvers.} A &amp;amp;= 1 + \operatorname{cos.} A \\&#xA;\operatorname{covers.} A &amp;amp;= 1 - \operatorname{sin.} A \\&#xA;\operatorname{sucovers.} A &amp;amp;= 1 + \operatorname{sin.} A&#xA;\end{align}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Actually, the suversed sine and sucoversed sine are mentioned in &lt;a href=&quot;https://www.google.com/books/edition/Tables_for_Facilitating_the_Calculations/FzIAAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=suversed&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mendoza's earlier 1801 set of tables&lt;/a&gt;.&#xA;However, no justification for the terms is given, which suggests that they were either already in common use or had sufficiently obvious names.&lt;/p&gt;&#xA;" OwnerUserId="12742" LastActivityDate="2020-07-19T03:32:39.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12039" PostTypeId="1" CreationDate="2020-07-19T08:24:30.077" Score="2" ViewCount="93" Body="&lt;p&gt;F. Klein mentions a certain Blumenthal in his famous book on the history of mathematics in 19th century with the following verse(when he was talking about the curvature in space):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Menschen fassen kaum es&lt;/p&gt;&#xA;&lt;p&gt;Das Krümmungsmaß des Raumes&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; I am wondering whom this Blumenthal Klein referred to. Is he Otto Blumenthal or somebody else?&lt;/p&gt;&#xA;" OwnerUserId="6450" LastEditorUserId="6450" LastEditDate="2020-07-19T09:51:35.837" LastActivityDate="2020-07-19T17:03:55.673" Title="A Reference in Klein's Book" Tags="|reference-request|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12041" PostTypeId="2" ParentId="5537" CreationDate="2020-07-19T15:03:35.923" Score="1" Body="&lt;p&gt;@Ben,&lt;/p&gt;&#xA;&lt;p&gt;Thank you for asking:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Here are some questions to ask a creationist or climate change denier:&lt;br&gt;&#xA;(1) If they don't want to be guided by scientific consensus in matters of science, what alternative guide would they like to use?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Evidence. The divide between &amp;quot;consensus&amp;quot; and &amp;quot;deniers&amp;quot; seems to be along what evidence they find acceptable. No one will argue that &amp;quot;numbers of believers&amp;quot; makes evidence more compelling. In fact, consensus-holders' resorting to name-calling the unconvinced, &amp;quot;deniers&amp;quot;, can be offered as its own evidence that they've run out of arguments. (However, the more likely explanation is that it arises from frustration at not being able to eliminate dissenting opinions which too now appear to be &amp;quot;guiding people who are not scientists.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Persuade &lt;em&gt;me&lt;/em&gt;. Don't &lt;em&gt;order&lt;/em&gt; me to follow &lt;strong&gt;because&lt;/strong&gt; there's a herd. If it's difficult, and you don't want to bother, and deem it not worth the effort, I understand, but my response to being written off and called names won't be warm acceptance.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;(2) What standard of proof do they think is appropriate, and why?&amp;quot;&lt;br&gt;&#xA;Seems to suggest that the consensus position has been proved, in which case there would be no room for alternate interpretations of evidence, only &amp;quot;denial&amp;quot; of proof. There is no proof, only interpretations of the evidence, more or less persuasive.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;(3) On the issue being discussed, is there an alternative theory that fits the evidence?&amp;quot;&lt;br&gt;&#xA;Yes, but being already persuaded by an alternative explanation, you would likely &amp;quot;deny&amp;quot; its arguments. ;^)&lt;/p&gt;&#xA;&lt;p&gt;=====&lt;br&gt;&#xA;I agree that it should be harder to form consensus around a bad theory today, but people are still people and are not immune to wanting to be in the &amp;quot;in&amp;quot; group, esp. if there's monetary incentive (as in research dollars, resulting in increased fame and salaries).&lt;/p&gt;&#xA;&lt;p&gt;I could even agree with your statement:&lt;br&gt;&#xA;&amp;quot;...even if it's only a 99% chance that it's right, it seems to me to make sense to take strong action to keep from driving our planet off a cliff.&amp;quot;&lt;br&gt;&#xA;if we could agree on the definition of the word &amp;quot;strong.&amp;quot; I would be averse for example to capital punishment (exc. for the people on what W.S. Gilbert called &amp;quot;my little list&amp;quot; [must see Mikado]) or to replacing the U.S. Constitution with something more flexible for implementing government policies (unless I managed to get into the Senate by that time).&lt;/p&gt;&#xA;&lt;p&gt;What makes me a skeptic?&lt;/p&gt;&#xA;&lt;p&gt;Given the track record of experts, I simply cannot accept that they can predict the future. Given my personal association with scientists, I cannot accept that they are, some how, more &amp;quot;pure&amp;quot; than, say, bankers.  People are people.  Greed is greed.  The money is going to research proving human-caused climate change.&lt;/p&gt;&#xA;&lt;p&gt;The consensus do their work in institutions, like the one I did, that are air-conditioned and with refrigerated drinking water, and the most energy-hogging supercomputers on the planet (which subsequently require extra cooling/heat-disposal measures). Once a year they preside over piling up waste in &amp;quot;spend-it-or-lose-it&amp;quot; end-of-budget-year orgies and then plead poverty. Been there. Seen it. Participated myself under orders from superiors.&lt;/p&gt;&#xA;&lt;p&gt;They have not persuaded themselves.&lt;/p&gt;&#xA;&lt;p&gt;When I see the consensus wearing sackcloth and ashes and moving out of their upscale lifestyles, I'll start shopping for a similar outfit on Amazon.&lt;/p&gt;&#xA;&lt;p&gt;The story of J. Harlen Bretz was first to cause me to question consensus-as-truth. He was sorely treated by establishment scientists in his day, but doggedly pressed on.&lt;/p&gt;&#xA;&lt;p&gt;Even if we disagree, let us agree to try to understand each other. I hope we can agree that, even if things go the way of the consensus, that &amp;quot;denial&amp;quot; doesn't have to become a capital crime; you know, for the &amp;quot;sake of the consensus.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nationalgeographic.com/magazine/2015/03/science-doubters-climate-change-vaccinations-gmos/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nationalgeographic.com/magazine/2015/03/science-doubters-climate-change-vaccinations-gmos/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Atheism is the elephant in the scientific room: &amp;quot;There is no god but fact.&amp;quot; I have also not been persuaded that there is nothing more to humanity than what we can sense. However if atheism is true, I cannot understand the angst of atheistic scientists over the disappearance of meaningless, after all, human life. Why not party on to the end?&lt;/p&gt;&#xA;&lt;p&gt;Here's the thing, Adrian Monk would begin, I'm willing to accept that the earth is now warming, now cooling, ...changing. I'm willing to accept that mankind can do better at being stewards of their home (remember that Genesis-era concept before growth-fueled consumerism?). If we have waited too long to be able to do enough to reverse our doom--short of draconian measures--so be it, and R.I.P. human race.&lt;/p&gt;&#xA;&lt;p&gt;Respectfully, I very much hope, and have labored to try to assure,&lt;/p&gt;&#xA;" OwnerUserId="12805" LastActivityDate="2020-07-19T15:03:35.923" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12043" PostTypeId="1" AcceptedAnswerId="12044" CreationDate="2020-07-19T23:10:01.460" Score="2" ViewCount="814" Body="&lt;p&gt;Richard Feynman is considered to be one of the greatest minds in physics, and has won many accolades as a result of his research in areas such as quantum mechanics and particle physics. However, I am interested in knowing about how Feynman first originated what is known as Feynman's Trick in Integral Calculus. What specifically inspired him to formulate this technique? What steps did he take to create his trick? I am particularly interested in any proof or verification he wrote for justifying this new invention. If there are any papers, articles or journals that document the steps of this endeavor, please leave links in an answer below. Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="4251" LastEditDate="2020-07-25T12:22:26.790" LastActivityDate="2020-07-25T12:22:26.790" Title="What steps did Richard Feynman take to devise his Integral Trick?" Tags="|mathematics|calculus|feynman|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12044" PostTypeId="2" ParentId="12043" CreationDate="2020-07-20T01:40:15.713" Score="6" Body="&lt;p&gt;According to the history given &lt;a href=&quot;https://medium.com/cantors-paradise/richard-feynmans-integral-trick-e7afae85e25c&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, he didn't.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2020-07-20T01:40:15.713" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12045" PostTypeId="2" ParentId="5240" CreationDate="2020-07-20T05:12:31.877" Score="1" Body="&lt;p&gt;The shift to other countries being the STEM leaders these days is already happening.&lt;/p&gt;&#xA;&lt;p&gt;China, by 2017, put in 49% of all patent applications of that year, but also more than every other country in most categories. Meanwhile, US dominance has never been as strong since at least 2000 with different fields generally dominated by different countries (incl. Germany, South Korea, and Japan). Of course, this doesn't mean there can't be reversions, but there is at least a trend for Asian countries to innovate more. [&lt;a href=&quot;https://vdata.nikkei.com/en/newsgraphics/patent-wars/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Patent Wars, Nikkei&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After analyzing patent application data in 10 categories, including artificial intelligence and quantum computing, Nikkei has concluded that China will reign supreme in nine categories. Chinese tech giants Baidu and Alibaba Group will be the major innovators. Meanwhile, 64 of the global top 100 companies in terms of patent quality will be American.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/npRhu.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/npRhu.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5957" LastActivityDate="2020-07-20T05:12:31.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12046" PostTypeId="5" CreationDate="2020-07-20T11:57:36.893" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-20T11:57:36.893" LastActivityDate="2020-07-20T11:57:36.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12047" PostTypeId="4" CreationDate="2020-07-20T11:57:36.893" Score="0" Body="Photonics is the science and engineering of light generation, detection and manipulation." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-24T00:09:28.560" LastActivityDate="2020-07-24T00:09:28.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12048" PostTypeId="5" CreationDate="2020-07-20T11:58:48.207" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-20T11:58:48.207" LastActivityDate="2020-07-20T11:58:48.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12049" PostTypeId="4" CreationDate="2020-07-20T11:58:48.207" Score="0" Body="Electronics is the science and engineering dealing with the emission, flow and control of electrons in vacuum and matter." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-24T00:09:46.310" LastActivityDate="2020-07-24T00:09:46.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12051" PostTypeId="1" AcceptedAnswerId="12053" CreationDate="2020-07-21T07:48:02.723" Score="4" ViewCount="334" Body="&lt;p&gt;I am not sure in what bio I read this but it was not a new bio -- not Isaacson, for example. The anecdote was simply that he wanted to see an old professor/teacher (as many of us I think might do if we were near our old college or high school) and the teacher, an old man by then, was concerned that he wanted to borrow money and the reunion did not go congenially or was very short.&lt;/p&gt;&#xA;&lt;p&gt;I wonder when this might have happened since after 1905, I would guess Einstein was starting already to become known. And I am wondering who it might have been -- I am pretty sure in the bio he was not named and that implies to me that he was more likely a high school teacher rather than a college professor. Of course, this may now be lost to history but I can think of few people whose past has been more thoroughly researched (as in the case of the haunting book Einstein's Daughter where the author managed to meet people who had known the family of his first wife or even people who had known his first wife).&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2021-02-20T22:12:08.453" Title="Einstein visited former teacher/professor who thought he wanted to borrow money: who might that have been?" Tags="|biographical-details|einstein|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12052" PostTypeId="1" CreationDate="2020-07-21T21:47:15.333" Score="3" ViewCount="186" Body="&lt;p&gt;I had asked the following question on MSE &lt;a href=&quot;https://math.stackexchange.com/questions/3764826/newtons-evaluation-of-1-1-3-1-5-1-7-1-9-1-11&quot;&gt;here&lt;/a&gt;, and I was directed to this exchange.  There is a nice solution in the comments there, but perhaps someone here can add some additional insight?&lt;/p&gt;&#xA;&lt;p&gt;How might have Newton evaluated the following series?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{2} \, \frac{\pi}{4} = 1 + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \cdots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The method of the &lt;a href=&quot;https://math.stackexchange.com/questions/1462803/showing-pi-2-sqrt3-1-1-51-7-1-111-13-1-171-19-cdots&quot;&gt;this thread&lt;/a&gt; applies by setting &lt;span class=&quot;math-container&quot;&gt;$x=\pi/4$&lt;/span&gt; in the Fourier series for &lt;span class=&quot;math-container&quot;&gt;$f(x) = \pi/2 - x/2$&lt;/span&gt; and then subtracting the extraneous terms (which are a multiple of the Gregory-Leibniz series for &lt;span class=&quot;math-container&quot;&gt;$\pi/4$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I read that this series appears in a letter from Newton to Leibniz.  However, I do not have access the letter which appears in &lt;a href=&quot;https://www.cambridge.org/ec/academic/subjects/physics/history-philosophy-and-foundations-physics/correspondence-isaac-newton-volume-2?format=PB&amp;amp;isbn=9780521085991&quot; rel=&quot;nofollow noreferrer&quot;&gt;this volume&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12811" LastActivityDate="2020-07-21T21:47:15.333" Title="Newton's evaluation of 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11" Tags="|calculus|newton|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12053" PostTypeId="2" ParentId="12051" CreationDate="2020-07-22T07:27:54.950" Score="6" Body="&lt;p&gt;So far I have been unable to find the original source for the story mentioned in the question. However, there are multiple mentions of this incident in the general literature about Einstein. All accounts agree that the teacher was Dr. Ferdinand Ruess, who taught at the Luitpold-Gymnasium in Munich, the high school that Einstein &lt;a href=&quot;https://de.wikipedia.org/wiki/Albert_Einstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;attended&lt;/a&gt; from 1888 to 1894.&lt;/p&gt;&#xA;&lt;p&gt;Denis Brian, &amp;quot;&lt;a href=&quot;http://catalogimages.wiley.com/images/db/pdf/0471718408.excerpt.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Unexpected Einstein&lt;/a&gt;&amp;quot;, Hoboken, NJ: Wiley 2005&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nor to the one teacher he really liked, a Dr. Ferdinand Ruess, who taught history and German. [...] Many years later, probably in 1909, Einstein returned to the school to call on Dr. Ruess, who did not remember him. Einstein was poorly dressed and assumed that Ruess suspected that he was a stranger down on his luck who had come to beg, borrow, or steal from him. Einstein ended the embarrassing encounter with a quick exit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Carlos I. Calle, &amp;quot;&lt;a href=&quot;https://www.google.com/books/edition/Einstein_For_Dummies/EDM0fx8N3rAC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Another+teacher+there,+Dr.+Ferdinand+Ruess,+was+different+from+the+rest.%22&amp;amp;pg=PA21&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein For Dummies&lt;/a&gt;&amp;quot;, Hoboken, NJ: Wiley 2005&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another teacher there, Dr. Ferdinand Ruess, was different from the rest. Instead of emphasizing memorization and passive acceptance of facts, he made the students think for themselves. He inspired in them a love for German literature and for the study of ancient civilizations. Einstein had a great appreciation for Dr. Ruess. Later in life, when Einstein was famous, he decided to pay his old teacher a visit. As often happens, Ruess didn’t recognize his former student. Seeing Einstein in his usual baggy and worn-out clothes, Ruess mistook him for a beggar and had his maid throw him out.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ronald W. Clark, &amp;quot;Einstein: The Life and Times&amp;quot;, 1971 [&lt;em&gt;Thanks to @kimchi lover for the pointer in comments&lt;/em&gt;]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the Gymnasium there appears to have been, as there frequently is in such schools, one master who stood apart, the odd man out going his nonconformist way. His name was Reuss. [...] In later life Einstein would recall how Reuss had tried to spark alive a real interest in ancient civilizations and their influences which could still be seen in the contemporary life of southern Germany.  There was to be an unexpected footnote to Einstein's memory. For after his first work had began to pass a disturbing electric shock through the framework of science, he himself visited Munich and called on his old teacher, then living in retirement. But the worn suit and baggy trousers which had already become the Einstein hallmark among his colleagues merely suggested poverty. Reuss had no recollection of Einstein's name and it became clear that he thought his caller was on a begging errand. Einstein left hurriedly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gerald Holton, &amp;quot;&lt;a href=&quot;https://books.google.com/books?id=gcbbAAAAQBAJ&amp;amp;pg=PA74&amp;amp;lpg=PA74&amp;amp;dq=%22+the+supervision+of+his+only+beloved+teacher,+Ferdinand+Ruess,+poems+by+Uhland%22&amp;amp;source=bl&amp;amp;ots=s0lfuA-FoM&amp;amp;sig=ACfU3U3yvRX6hGZN7LIWaH-vmndvAVeuJA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwik-Y6hl-DqAhVDKH0KHRB5DC8Q6AEwAHoECAgQAQ#v=onepage&amp;amp;q=%22%20the%20supervision%20of%20his%20only%20beloved%20teacher%2C%20Ferdinand%20Ruess%2C%20poems%20by%20Uhland%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein and the Cultural Roots of Modern Science.&lt;/a&gt;&amp;quot; In:  Peter Galison, Michael Gordin, David Kaiser, eds. &amp;quot;Making Special Relativity&amp;quot;, Routledge 2001&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;then, under the supervision of his only beloved teacher, Ferdinand Ruess, poems by Uhland, Schiller, Goethe, and others;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The curriculum notes in the &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol1-doc/419&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein papers at Princeton&lt;/a&gt; suggest that in addition to German and history, Dr. Ruess also taught Latin and Greek and was Einstein's Ordinarius (a function analogous to an American home room teacher) in 1893/1894.&lt;/p&gt;&#xA;&lt;p&gt;If we accept the dating of the incident to 1909, it is likely that it happened during a stopover in Munich as Einstein traveled from Switzerland to Salzburg to give a talk at the 81st Meeting of German Scientists and Physicians (Versammlung Deutscher Naturforscher und Ärzte) in Salzburg on September 21, 1909: &amp;quot;Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung&amp;quot; (On the Development of our Views Concerning the Nature and Constitution of Radiation).&lt;/p&gt;&#xA;&lt;p&gt;I was able to find the &lt;a href=&quot;https://books.google.com/books?id=SuASAAAAYAAJ&amp;amp;pg=RA11-PA14&amp;amp;lpg=RA11-PA14&amp;amp;dq=%22Ruess%22+%22Ferdinand%22+%221853%22&amp;amp;source=bl&amp;amp;ots=7loBLgkSiT&amp;amp;sig=ACfU3U1B7bqwbGJ5ivnCtVDbBYwwoudE-w&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwj4367AkeDqAhWRHjQIHWqUBHE4FBDoATADegQIChAB#v=onepage&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan of an unnamed publication&lt;/a&gt; in Google Books that contains some biographical data on Dr. Ruess:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;RUESS, Ferdinand, Gymn.-Prof., Dr. — *30 V 1853, Augsburg. — V: Matthias R.; M: Viktoria gb. Bach. — Gymn. Augsburg; Univ. München, Würzburg. — Verh.: 82 m. Karoline gb. Schmid T v. Georg S., Fabrikant München — K. Hermann *80. — 75 Assist. am Realgymn. Würzburg; 80 Gymnlehr. in Neuburg a. d. Donau; 93 Gymn.-Professor am Luitpold-Gymn. München — W: Tachygr. d. Röm. 79; Griech. Tachygr. 82; Progr. üb. Tiron. Noten 83 u. 89; Stenogr. Wörterb. 96, 00, 03, 06; Lehrb. d. Gab. Stenogr. 04 — L-B: Alte Tachygr.; Gab. Stenogr. — 1. Vors. d. Gab. Sten.-Zentralver.; Mitgl. d. Prüfgskomm. f. d. Lehramt d. Sten. — Ehrenmitgl. mehr. Sten,-Ver. — München, Steinsdorfstr. 6 III&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ferdinand Ruess was born on May 30, 1853 in Augsburg, attended the local high school there and subsequently studied at the universities of Munich and Würzburg. He started working as a teacher in the latter city in 1875. He married the daughter of a Munich industrialist in 1882. He had a keen interest in shorthand and published several books on ancient Greek and Roman shorthand as well as modern shorthand and was a member of the Munich stenographer's association (Stenographenverein). The Internet Archive provides a &lt;a href=&quot;https://archive.org/details/uebergriechische00ruesuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;high-quality scan&lt;/a&gt; of his 1882 book &amp;quot;Ueber griechische Tachygraphie&amp;quot; (On Greek Tachygraphy).&lt;/p&gt;&#xA;&lt;p&gt;Some additional biographical details and a picture of Dr. Ruess are provided in a publication related to the Gabelsberger stenography association:&lt;/p&gt;&#xA;&lt;p&gt;K. Heck, &amp;quot;&lt;a href=&quot;https://books.google.com/books?id=znxQAAAAYAAJ&amp;amp;pg=PA11&amp;amp;lpg=PA11&amp;amp;dq=%22Ferdinand+Ruess%22&amp;amp;source=bl&amp;amp;ots=pqJ9tKxao8&amp;amp;sig=ACfU3U31338_h7rv--5wFmiRA-KBeL5K4g&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjTvMffnuDqAhWUGTQIHVCQD4E4MhDoATAJegQICRAB#v=onepage&amp;amp;q=%22Ferdinand%20Ruess%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geschichte der Schule 'Gabelsberger', Zweiter Teil&lt;/a&gt;,&amp;quot; Wolfenbüttel: Hecknersche Druckerei 1902.&lt;/p&gt;&#xA;&lt;p&gt;From this we learn that he started working at the Luitpold-Gymnasium in 1887 and was promoted there to the rank of Gymnasialprofessor in 1893.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A curious side note, which to me sheds some doubt on the story in the question, which may thus be apocryphal.&lt;/p&gt;&#xA;&lt;p&gt;In 2018 some postcards sent to Einstein were put up for auction. Among these is a postcard by Dr. Ruess in Munich, written in shorthand and dated September 23, 1900. This would seem to indicate that the teacher was corresponding with Einstein after he left high-school, making it less likely that he wouldn't have recognized him by name in 1909. The &lt;a href=&quot;https://www.christies.com/img/LotImages/2018/CKS/2018_CKS_16447_0009_004(postcards_to_the_young_einstein_zurich_1898-1900).jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan of the postcard provided by the auction house&lt;/a&gt; is of sufficient quality that someone with knowledge of Gabelsberger shorthand could decipher and transcribe it.&lt;/p&gt;&#xA;&lt;p&gt;The postcard is potentially a reply to a letter by Einstein. In a letter to Mileva dated September 19, 1900 &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol1-doc/330&quot; rel=&quot;nofollow noreferrer&quot;&gt;preserved in the Princeton Einstein papers&lt;/a&gt; he mentions writing a letter to a former teacher in Munich, who may well have been Dr. Ruess:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gestern hab ich in einem launigen Stündchen einem meiner früheren Lehrer in München, den ich besonders gern hatte, einen Brief geschrieben, will sehen, ob er mir vielleicht was drauf antwortet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Addendum: Thanks to &lt;a href=&quot;https://mathematica.stackexchange.com/users/1598/andreas-lauschke&quot;&gt;Andreas Lauschke&lt;/a&gt; and a second person, each of whom supplied me with a partial transcription of the postcard, it is clear that this is indeed Dr. Ruess's reply to Einstein's letter. Dr. Ruess writes that he was quite surprised to receive the letter, states that he read about the developments in Einstein's life with interest, and congratulates him on the recent conclusion of his studies (Einstein had received his teaching degree, &lt;a href=&quot;https://de.wikipedia.org/wiki/Albert_Einstein#/media/Datei:Einstein_Lehrerdiplom_Bauzeitung.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Diplom als Fachlehrer in mathematischer Richtung&lt;/em&gt;&lt;/a&gt;, from Polytechnikum Zürich in late July 1900).&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-02-20T22:12:08.453" LastActivityDate="2021-02-20T22:12:08.453" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12054" PostTypeId="5" CreationDate="2020-07-22T09:35:49.287" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-07-22T09:35:49.287" LastActivityDate="2020-07-22T09:35:49.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12055" PostTypeId="4" CreationDate="2020-07-22T09:35:49.287" Score="0" Body="Euclid of Alexandria was a Greek mathematician and the &quot;father of geometry&quot;. " OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-07-24T00:09:22.733" LastActivityDate="2020-07-24T00:09:22.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12056" PostTypeId="1" CreationDate="2020-07-22T14:51:52.603" Score="4" ViewCount="533" Body="&lt;p&gt;The universal trigonometric substitution converts a rational function in &lt;span class=&quot;math-container&quot;&gt;$\sin(x), \cos(x) $&lt;/span&gt;  into a rational function of a new variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; by the substitution &lt;span class=&quot;math-container&quot;&gt;$t = \tan(x/2) $&lt;/span&gt;. It therefore enables to solve the integral of any function in &lt;span class=&quot;math-container&quot;&gt;$R[\sin x,\cos x] $&lt;/span&gt; by the methods of decomposition into partial fractions.&lt;/p&gt;&#xA;&lt;p&gt;According to Wikipedia, this method is named after Weierstrass although It was used much earlier by Euler. It seems to me very elementary, and even Euler himself calculated far more difficult integrals, so i'd like to know why the name of a mathematician as modern as Weierstrass is attached to such an elementary technique; did he contribute something to it? did he just popularize it?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="5989" LastEditDate="2020-07-25T08:54:26.223" LastActivityDate="2020-07-25T08:54:26.223" Title="Why is the &quot;universal trigonometric substitution&quot; called &quot;Weierstrass's substitution&quot;?" Tags="|calculus|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12057" PostTypeId="1" CreationDate="2020-07-23T15:45:08.080" Score="16" ViewCount="1013" Body="&lt;p&gt;Recently, I've been looking into the life of the brilliant &lt;a href=&quot;https://en.wikipedia.org/wiki/Norbert_Wiener&quot; rel=&quot;noreferrer&quot;&gt;Norbert Wiener&lt;/a&gt; and the field he spawned, &lt;a href=&quot;https://en.wikipedia.org/wiki/Cybernetics&quot; rel=&quot;noreferrer&quot;&gt;cybernetics&lt;/a&gt;. Before reading into it, I thought &amp;quot;cybernetics&amp;quot; was a pseudo-science-new-wave-steam-punk-post-human-cyborg-thing, so I was surprised to discover it's really just a generalization of control theory from industrial engineering to both animal and digital systems.&lt;/p&gt;&#xA;&lt;p&gt;Wanting to find out more about the subject, I started looking for resources but found almost nothing dated in the last 20 years.&lt;/p&gt;&#xA;&lt;p&gt;What's happened to this field? Was it simply too broad and the community fragmented and specialized? Did it die with the original creators–– there are no people left with the wide breadth of interdisciplinary knowledge left to maintain it? Or did the term just suffer a semantic creep that led researchers away?&lt;/p&gt;&#xA;&lt;p&gt;Appreciate any insights.&lt;/p&gt;&#xA;" OwnerUserId="12816" LastEditorUserId="9523" LastEditDate="2021-12-01T07:58:21.450" LastActivityDate="2022-05-25T23:05:50.717" Title="What happened to cybernetics?" Tags="|computer-science|engineering|cybernetics|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12058" PostTypeId="2" ParentId="11962" CreationDate="2020-07-23T19:49:50.170" Score="1" Body="&lt;p&gt;The article &amp;quot;Is lens design legal?&amp;quot; gives some details about Gauss's letter to Brandes; although it doesn't explain much about Gauss's formulas and calculations, it describes very well the historical significance and meaning of this letter. The article discusses &amp;quot;figures of merit&amp;quot; of optical systems; a kind of quantitive measure of the quality of an image formed by a lens system. The following passage is taken from this article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1831 Gauss balanced first-order and second-order spherical aberrations in the presence of defocusing. In his published &amp;quot;letter to Brandes&amp;quot; Gauss wrote: &amp;quot;Based on your letter with regard to my recent paper on lens design I decided to spend one more hour on this topic, which apparently has been misunderstood by some readers.&amp;quot; And: &amp;quot;Originally I minimized the total size of the blurred image without weighting its parts&amp;quot; - Apparently Gauss had used Chebyshev's worst case rule to optimize transversal aberrations -  &amp;quot;But that is somewhat arbitrary. I consider it now most appropriate to apply the principle of least mean squares, which i call &lt;strong&gt;total-undeutlichkeit&lt;/strong&gt; (total blur) in this context&amp;quot;. Obviously, Gauss used the second Gauss moment of transversal aberrations as the figure of merit for lens design. The concluding sentence reads: &amp;quot;The limits of a letter forbid a detailed proof, but my hints will suffice for the expert.&amp;quot; (This was written nearly 160 years ago.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The principle used by Gauss to quantify &amp;quot;blurriness&amp;quot; (the principle of least mean squares) is at the foundation of current theories related to image proccesing (like algorithms for removing blurriness of an image), and this letter suggests a possible reason why &amp;quot;&lt;strong&gt;Gaussian blur&lt;/strong&gt;&amp;quot; is called by this name (although i'm not quite sure about the naming of &amp;quot;Gaussian blur&amp;quot;). The article explains more about the consequences of Gauss's calculation but unfortunately i'm not familiar enough with optical theory for understanding it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;More detailed explanation:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In Gauss's letter he gives the following definition for the &amp;quot;total blur&amp;quot; of an image:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int i \rho^2 ds$$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; is the intensity of light falling on a point of the image plane, &lt;span class=&quot;math-container&quot;&gt;$ds$&lt;/span&gt; is the area element, and &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; is the distance of that point from a certain point &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; which Gauss defines (i didn't understand the definition of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;). Gauss definition is therefore coinciding with the algorithm of &amp;quot;&lt;strong&gt;weighted least-squares&lt;/strong&gt;&amp;quot; - here the weights are the intensities of Light. Note that according to this definition, even the image of a point source might have non-zero blur - exactly because of the spherical aberration! More generally, the quantification of the quality of a lens system is done by substracting the blur of the source from the blur of it's image (that definition is relevant for extended sources).&lt;/p&gt;&#xA;&lt;p&gt;Since the modern approach to identify high-contrast images (images with low blur) is to subject it to a Fourier transform and see if there are significant high-frequency components, one can state affirmatively that Gauss's integral formula captures the essence of the modern approach. Gauss gives in his letter several formulas that describe the conditions needed to minimize the total blur.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: this is a very partial answer that articulates only one aspect of Gauss's letter, and a further examination of it is needed. Any informative and usefull comments will be blessed!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-07-25T19:14:25.057" LastActivityDate="2020-07-25T19:14:25.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12059" PostTypeId="1" AcceptedAnswerId="12060" CreationDate="2020-07-24T05:01:15.063" Score="5" ViewCount="1282" Body="&lt;p&gt;It is stated in innumerous introductory histology texts that there are four basic types of tissue - epithelial, connective, muscular, and nervous. I have never come across one that provides a reference, however.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Tissue_(biology)#History_of_the_concept&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; on this subject says that Xavier Bichat identified 21 types of elementary human tissue, and that this number was later reduced by other authors. But no description is given as to who those authors might be, nor any reference to consult.&lt;/p&gt;&#xA;&lt;p&gt;I would like to know who first classified animal tissue into four types.&lt;/p&gt;&#xA;&lt;p&gt;In fact, what I really want to know is: &amp;quot;How did the scientific community come to think that 4 is a necessary and sufficient number of tissue types to describe animal microanatomy?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;For example, why don't we put nervous tissue under the category of epithelial tissue, since nervous tissue develops from the neuroepithelium? Or why don't we consider blood as a distinct tissue type, given its obvious peculiarities compared to other types of connective tissue?&lt;/p&gt;&#xA;&lt;p&gt;Of course, I could come up with some answers of my own to the questions I posed above. But I want to know how &lt;em&gt;past scientists&lt;/em&gt; have answered them. So I would like to read about the conceptual evolution of animal tissue classification in general, if there is any material to read at all.&lt;/p&gt;&#xA;" OwnerUserId="12817" LastEditorUserId="4251" LastEditDate="2020-07-25T20:39:31.733" LastActivityDate="2020-10-06T07:16:49.727" Title="Who identified the four basic types of animal tissue?" Tags="|biology|medicine|biologists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12060" PostTypeId="2" ParentId="12059" CreationDate="2020-07-24T06:31:34.233" Score="6" Body="&lt;p&gt;This is only a partial answer. A good source on the history of animal morphology is &lt;a href=&quot;http://www.gutenberg.org/ebooks/20426&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's Form and Function&lt;/a&gt;, freely available on Project Gutenberg.&lt;/p&gt;&#xA;&lt;p&gt;It seems that the crucial step from Bichat's 21 types to 4, not just numerically but also methodologically, was taken by &lt;a href=&quot;https://en.wikipedia.org/wiki/Theodor_Schwann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theodor Schwann&lt;/a&gt;, one of the founders of modern histology, in &lt;a href=&quot;https://archive.org/details/b29287455&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mikroskopische Untersuchungen über die Uebereinstimmung in der Struktur und dem Wachstume der Tiere und Pflanzen&lt;/em&gt; (1839)&lt;/a&gt;. However, he had 5 rather than 4 types, and they do not match the modern ones. Nervous and muscular tissues fall under the same type, and he took the OP suggestion to separate blood and lymph into a separate type, distinct from other connective tissues. Bichat's classification &amp;quot;&lt;em&gt;was macroscopical and physiological; he relied upon texture and function in distinguishing them rather than upon microscopical structure&lt;/em&gt;&amp;quot;. In contrast, Schwann came to his types based on studying the emergence of the tissues from the ovum, and used cellular structure as a consistent criterion. An intermediate figure between Bichat ans Schwann was von Baer (1828), who already emphasized the connection between differentiation of germ layers and emerging types of tissues, although, like Bichat, he used textural rather than cellular characteristics in his classification.&lt;/p&gt;&#xA;&lt;p&gt;Here is Russell's retelling of Schwann's reasoning:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Five classes of tissue can be distinguished, according to the extent and manner of the modifications which the cells composing them have undergone. There are first of all independent and isolated cells, such as the corpuscles of the blood and lymph, not forming a coherent tissue in the ordinary sense. Next there are the assemblages of cells lying in contiguity with one another, but not in any way fused; examples of this class are the epidermal tissues and the lens of the eye. In the third class come tissues the cells of which have fused by their walls, but whose cell-cavities are not in continuity, such as osseous tissue and cartilage. In the tissues of the fourth class, comprising the most highly specialised of all, not only are the cell-walls continuous but also the cell-cavities; to this class belong muscle, nerve and capillary vessels. A fifth class, of rather a special nature, includes the fibrous tissues of all kinds. This is the first classification of tissues upon a cellular basis, and it marks the foundation of a new histology which took the place of the &amp;quot;general anatomy&amp;quot; of Bichat...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Schwann found it easy to demonstrate the cellular nature of the tissues of his first three classes. With the other two classes he had more difficulty. Fibres of all kinds, he considered, arose by an elongation of cells, which afterwards split longitudinally into long strips, forming as the case might be white or elastic fibrous tissue. Muscle-fibres and nerve-fibres were formed in a totally different way, by coalescence of cells; each separate muscle-fibre and nerve-fibre was thus a compound cell. Capillaries, Schwann held, were formed by cells hollowed out like drain-pipes, and 179set end to end—a mistaken view soon corrected by Vogt (Embryologie des Salmones, p. 206, 1842).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Further studies of blastoderm differentiation (so-called germ-layer theory), which led to better understanding of epithelial tissues and distinguishing nervous and muscular ones, were undertaken, among others, by &lt;a href=&quot;https://books.google.com.br/books?id=Kknf7ZRHzagC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Remak in &lt;em&gt;Untersuchungen über die Entwickelung der Wirbelthiere&lt;/em&gt; (1850-5)&lt;/a&gt;, and further by &lt;a href=&quot;https://books.google.com/books?id=UPkzAQAAMAAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;O. and R. Hertwig in &lt;em&gt;Studien zur Blättertheorie&lt;/em&gt; (1879-80)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-06T07:16:49.727" LastActivityDate="2020-10-06T07:16:49.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12061" PostTypeId="1" AcceptedAnswerId="12062" CreationDate="2020-07-24T07:51:40.947" Score="10" ViewCount="1454" Body="&lt;p&gt;In an interview of Heisenberg by Thomas Kuhn:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When Pauli had died, I was asked to write this memorial volume. Weisskopf had asked me. Then, actually, originally I had written &lt;strong&gt;an article on Pauli's philosophical views&lt;/strong&gt;, but this article was not accepted. Weisskopf said, &amp;quot;Well, this article is very nice, but you know we don't like to discuss this side of Pauli so much. We want to see Pauli as a physicist.&amp;quot; So actually I was a bit angry about Weisskopf, but, well, I had to take his opinion, and apparently other people agreed. &lt;strong&gt;Afterwards I did publish my article of Pauli's philosophical views.&lt;/strong&gt; I first then &lt;strong&gt;published it in German&lt;/strong&gt;. May I give you a copy? &lt;strong&gt;Later on it did appear in a rather obscure periodical in the United States&lt;/strong&gt; because there were still some people in America who were still interested in it, but not the physicists. These were people of a different structure. Still, I like this article on Pauli's philosophical views. I think that I had succeeded in describing very accurately how Pauli's mind was constructed. I also hoped that I had made it clear to many people that I liked this kind of mind, and that my own mind is not so very different from that of Pauli. I may just have it here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any detail of this article (title, name of journal of publication,...) would be really appreciated.&lt;/p&gt;&#xA;" OwnerUserId="12821" OwnerDisplayName="Sha" LastEditorUserId="4251" LastEditDate="2021-09-08T06:26:29.583" LastActivityDate="2021-09-08T06:26:29.583" Title="Heisenberg's Obituary to Pauli" Tags="|physics|physicists|heisenberg|20th-century|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12062" PostTypeId="2" ParentId="12061" CreationDate="2020-07-24T10:13:02.543" Score="11" Body="&lt;p&gt;I think the article in question is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Heisenberg, W. &lt;em&gt;Wolfgang Paulis philosophische Auffassungen.&lt;/em&gt; Naturwissenschaften 46, 661–663 (1959). &lt;a href=&quot;https://doi.org/10.1007/BF00594803&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1007/BF00594803&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I found it via citation in &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.491.9616&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="18333" OwnerDisplayName="Vadim" LastActivityDate="2020-07-24T10:13:02.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12063" PostTypeId="2" ParentId="12061" CreationDate="2020-07-24T11:42:22.083" Score="11" Body="&lt;p&gt;The English translation by Kurt Leidecker is &lt;a href=&quot;ftp://tug.ctan.org/pub/bibnet/authors/h/heisenberg-werner.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;referenced as&lt;/a&gt; &amp;quot;Wolfgang Pauli's philosophical views&amp;quot;, in &lt;em&gt;Main Currents in Modern Thought&lt;/em&gt;, volume 17, pages 51-54 (1961).&#xA;The paper is also &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00709378&quot; rel=&quot;nofollow noreferrer&quot;&gt;cited as&lt;/a&gt; &amp;quot;Wolfgang Pauli's philosophical views. Towards a new understanding of the unitary order of the cosmos,&amp;quot; pp.38-41. The journal is indeed quite obscure and I'm not able to locate a copy of that issue in any library.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="12825" LastEditDate="2020-07-28T06:47:24.847" LastActivityDate="2020-07-28T06:47:24.847" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12064" PostTypeId="1" AcceptedAnswerId="12065" CreationDate="2020-07-25T02:05:26.967" Score="4" ViewCount="334" Body="&lt;p&gt;Isaac Barrow was one of the professors who taught Isaac Newton at Cambridge. According to &lt;a href=&quot;http://users.wfu.edu/kuz/Stamps/Newton/Newton.htm#:%7E:text=Isaac%20Barrow%2C%20Newton%27s%20teacher%20at,the%20Fundamental%20Theorem%20of%20Calculus.&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt;, he is said to have made contributions to the Fundamental Theorem of Calculus that was devised by Newton.&lt;/p&gt;&#xA;&lt;p&gt;What specific contributions did Barrow make to this theorem? What formulae, conjectures, or hypotheses did Barrow develop that assisted in the formulation of the theorem?&lt;/p&gt;&#xA;&lt;p&gt;If there are any articles or citations that document Barrow's contributions, please post an answer with the appropriate links below. Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="4251" LastEditDate="2020-07-25T20:39:39.830" LastActivityDate="2020-07-25T20:39:39.830" Title="What did Newton's teacher contribute to the Fundamental Theorem of Calculus?" Tags="|mathematics|mathematicians|calculus|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12065" PostTypeId="2" ParentId="12064" CreationDate="2020-07-25T03:06:54.520" Score="4" Body="&lt;p&gt;Isaac Barrow had given a series of lectures in 1664-65. These lectures dealt with the general concepts of space, time and motion.&lt;/p&gt;&#xA;&lt;p&gt;With Newton assisting in their preparation, these lectures were compiled and published in 1670 as &lt;em&gt;&lt;a href=&quot;https://books.google.co.in/books/about/Lectiones_Geometricae.html?id=PKhY2CfFe4AC&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lectiones Geometricae&lt;/a&gt;&lt;/em&gt;. It is in the Lecture X, Proposition 11 of &lt;em&gt;Lectiones Geometricae&lt;/em&gt;, that a geometrical theorem is mentioned, which in modern-day terminology is called the Fundamental Theorem of Calculus.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-07-25T06:49:37.070" LastActivityDate="2020-07-25T06:49:37.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12066" PostTypeId="2" ParentId="710" CreationDate="2020-07-25T09:05:20.880" Score="2" Body="&lt;p&gt;In 2006 a fragment of a cogwheel was discovered in Olbia (Italy); the layer where the fragment was found has been dated to the second quarter of the 2nd century BCE (&lt;a href=&quot;https://www.academia.edu/35795869/Un_frammento_del_Planetario_di_Archimede_da_Olbia&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.academia.edu/35795869/Un_frammento_del_Planetario_di_Archimede_da_Olbia&lt;/a&gt;, p. 1788). The striking characteristic of this cogwheel, however, is that its teeth are not triangular but have an involuted profile, which is a strong indicator that the concept of cogwheel had already been in use for some time.&lt;/p&gt;&#xA;&lt;p&gt;One Italian engineer, Giovanni Pastore, claims the Olbia cogwheel actually belongs to a planetarium built by Archimedes (&lt;a href=&quot;https://www.youtube.com/watch?v=hp-FXnjxNfw&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=hp-FXnjxNfw&lt;/a&gt;), but I do not know how scientifically strong this claim is.&lt;/p&gt;&#xA;" OwnerUserId="12801" LastActivityDate="2020-07-25T09:05:20.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12067" PostTypeId="2" ParentId="7299" CreationDate="2020-07-25T16:20:07.533" Score="8" Body="&lt;p&gt;I first ran across this story at an exhibit in Los Angeles' Museum of Science and Industry some time before 1970, maybe as early as 1964. In the version I read there, a magistrate was involved, and the person questioning the baker's methods had some sort of civic power connection.&lt;/p&gt;&#xA;&lt;p&gt;When you suggest that no Frenchman might have taken a loaf of bread home and weighed it, don't forget two things: a) nerds have always lived among us, and b) Paris is the home of the &amp;quot;Grand Prix de la Baguette de la Ville de Paris.&amp;quot; This is a competition in which the winner, the top baguette-maker of the year, gets the honor of supplying the French presidential residence with baguettes every morning for the entire year. Mightn't it seem totally reasonable for a competitor to weigh his adversaries' loaves?&lt;/p&gt;&#xA;" OwnerUserId="12835" LastActivityDate="2020-07-25T16:20:07.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12068" PostTypeId="1" AcceptedAnswerId="12072" CreationDate="2020-07-25T21:35:16.247" Score="4" ViewCount="227" Body="&lt;p&gt;(I asked this question in the general Mathematics forum, but I have been advised to post it here instead -- or as well.)&lt;/p&gt;&#xA;&lt;p&gt;In David Wells's &amp;quot;Curious and Interesting Puzzles&amp;quot;, Penguin, 1992, his Puzzle 38 is taken from a work (unspecified) by Abu'l-Wafa Al-Buzjani (but I suspect it could be &amp;quot;A Book on Those Geometric Constructions Which Are Necessary for a Craftsman&amp;quot;), reproduced, apparently, in J.L. Berggren: &amp;quot;Episodes in the Mathematics of Medieval Islam&amp;quot;, Springer, 1986.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Construct an equilateral triangle inside a square, so that one vertex is at a corner of the square and the other two vertices are on the opposite sides.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This is one of the three constructions provided in the solutions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZvA9j.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZvA9j.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; be the midpoint of &lt;span class=&quot;math-container&quot;&gt;$CD$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Construct &lt;span class=&quot;math-container&quot;&gt;$MB$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Construct an arc centre &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; to cut &lt;span class=&quot;math-container&quot;&gt;$MB$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Produce &lt;span class=&quot;math-container&quot;&gt;$DN$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$DH$&lt;/span&gt; is then one side of the equilateral triangle, where &lt;span class=&quot;math-container&quot;&gt;$DG = DH$&lt;/span&gt; is one of the other sides.&lt;/p&gt;&#xA;&lt;p&gt;Except it's not. &lt;span class=&quot;math-container&quot;&gt;$GH$&lt;/span&gt; is longer than &lt;span class=&quot;math-container&quot;&gt;$DH$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Analysing the angles, it turns out that &lt;span class=&quot;math-container&quot;&gt;$\angle CDH = \arctan \frac {3 - \sqrt 5} 4$&lt;/span&gt;, which is about &lt;span class=&quot;math-container&quot;&gt;$10.8$&lt;/span&gt; degrees.&lt;/p&gt;&#xA;&lt;p&gt;So clearly this is a mistake. (Wonderful though Wells's books are, they are often riddled with errors, from simple typos and misattributions through to bad mathematics.)&lt;/p&gt;&#xA;&lt;p&gt;I have been unable to find online copy of either Abu'l-Wafa Al-Buzjani's work or J.L. Berggren's (and at this stage I am unable to hunt it down in a library, and unwilling to get a copy of my own), so I have not been able to find out whether the mistake is Wells's (mistranscribing the construction), or whether it has been sitting there all this time in Abu'l-Wafa Al-Buzjani and nobody has noticed it, or halfway between the two.&lt;/p&gt;&#xA;&lt;p&gt;Is anyone able to throw any light on what is shown in those source works -- and if the error is in there as well, has anyone else ever noticed this?&lt;/p&gt;&#xA;&lt;p&gt;Or even: am &lt;em&gt;I&lt;/em&gt; the one to analyse this all wrong?&lt;/p&gt;&#xA;&lt;p&gt;(Also, can anyone suggest the appropriate tags to place this under? I've hunted around for good ones, but I can only find the one. Please bear with me, I'm new here, and I'm afraid I'm an autodidact.)&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="26" LastEditDate="2021-02-20T15:09:33.267" LastActivityDate="2021-02-20T15:09:33.267" Title="Inscribing equilateral triangle in square — mistake in historical work by Abu'l-Wafa Al-Buzjani?" Tags="|euclidean-geometry|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12069" PostTypeId="2" ParentId="12068" CreationDate="2020-07-25T23:12:14.847" Score="3" Body="&lt;p&gt;Denote &lt;span class=&quot;math-container&quot;&gt;$\beta=\angle MBC,\;\alpha=\angle DMN,\;\gamma=\angle CDH$&lt;/span&gt;. We have:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\tan\beta=1/2,\quad\cos\alpha=-\sin\beta=-1/\sqrt{5},\quad\sin\alpha=\cos\beta=2/\sqrt{5}.$$&lt;/span&gt;&#xA;Take &lt;span class=&quot;math-container&quot;&gt;$DM=1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$MN=\sqrt{5}-2$&lt;/span&gt;, and let &lt;span class=&quot;math-container&quot;&gt;$x=DN$&lt;/span&gt;. By the rule of cosines,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^2=1^2+(\sqrt{5}-2)^2+2(\sqrt{5}-2)/\sqrt{5}=12-24/\sqrt{5}.$$&lt;/span&gt;&#xA;Then by the rule of sines,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sin\gamma=\frac{(\sqrt{5}-2)(2/\sqrt{5})}{\sqrt{12-24/\sqrt{5}}}\approx0.1875,$$&lt;/span&gt;&#xA;and the calculator gives &lt;span class=&quot;math-container&quot;&gt;$\gamma\approx 10.8$&lt;/span&gt; degrees.&lt;/p&gt;&#xA;&lt;p&gt;Remark.&#xA;An internet search finds a Russian translation of this book of Abu'l Wafa,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Абу-л-Вафа ал-Бузджани. Книга о том, что необходимо ремесленнику из геометрических построений. &lt;a href=&quot;https://books.google.com/books/about/%D0%A4%D0%B8%D0%B7%D0%B8%D0%BA%D0%BE_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8.html?id=FXQLAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Физико-математические науки в странах Востока&lt;/a&gt;, 1966, 1, 56-140.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a collection of translations. But this book I could not find on Internet.&lt;/p&gt;&#xA;&lt;p&gt;Remark: since the true solution involves a construction of &lt;span class=&quot;math-container&quot;&gt;$2-\sqrt{3}$&lt;/span&gt; with a compass and ruler, which is easy, it is plausible that Abu'l Wafa had a correct construction which was distorted by some scribe, or translator, or by Wells himself. But there is no answer to this unless we get some translation of Abu'l Wafa.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-07-26T00:16:41.187" LastActivityDate="2020-07-26T00:16:41.187" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12072" PostTypeId="2" ParentId="12068" CreationDate="2020-07-26T10:35:41.563" Score="3" Body="&lt;p&gt;Wells' book is &lt;a href=&quot;https://www.pdfdrive.com/the-penguin-book-of-curious-and-interesting-puzzles-e176233842.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;available for download&lt;/a&gt; on PdfDrive. Puzzle #38 appears in the section of problems attributed to Abū al-Wafā and is preceeded by the comment that  &amp;quot;&lt;em&gt;he is best&#xA;known for his study of geometrical dissections and of constructions&#xA;with a rusty compass, meaning a compass which is so stiff that it can&#xA;be used with only one opening&lt;/em&gt;&amp;quot;. In the solution section he adds &amp;quot;&lt;em&gt;Abul Wafa gave five different solutions. Here are three of them&lt;/em&gt;&amp;quot;. There is no citation except for &amp;quot;[Berggren, 1986]&amp;quot; at the bottom of the page after the three solutions. As indicated in the OP, this apparently refers to Berggren's Episodes in the Mathematics of Medieval Islam. Similar citation appears under #45. Some others (#41,#42) have &amp;quot;[Wells, 1975]&amp;quot;, which is his own article On Gems and Generalisations in Games and Puzzles magazine.&lt;/p&gt;&#xA;&lt;p&gt;Berggren's book is available on &lt;a href=&quot;https://www.researchgate.net/publication/322080297_Episodes_in_the_Mathematics_of_Medieval_Islam_J_L_Berggren&quot; rel=&quot;nofollow noreferrer&quot;&gt;ResearchGate&lt;/a&gt; (one can download from it after registering). In it section 8 of chapter 3 is entitled Geometry with a Rusty Compass and features five problems from Abū al-Wafā's On Those Parts of Geometry Needed by Craftsmen on pp. 107-111. They are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;To construct at the endpoint A of a segment AB a perpendicular to that&#xA;segment, without prolonging the segment beyond A.&lt;br&gt;&lt;/li&gt;&#xA;&lt;li&gt;To divide a line segment into any number of equal parts.&lt;br&gt;&lt;/li&gt;&#xA;&lt;li&gt;To bisect a given angle.&lt;br&gt;&lt;/li&gt;&#xA;&lt;li&gt;To construct a square in a given circle.&lt;br&gt;&lt;/li&gt;&#xA;&lt;li&gt;To construct in a given circle a regular pentagon with a compass&#xA;opening equal to the radius of the circle.&lt;br&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;At the end he adds &amp;quot;&lt;em&gt;Abū al-Wafā”s treatise contains a wealth of beautiful constructions for regular ngons,&#xA;including exact constructions for n = 3, 4, 5, 6, 8, 10. It also gives a verging&#xA;construction for n = 9 which goes back to Archimedes and the approximation for&#xA;n = 7 that gives the side of a regular heptagon in a circle as equal to half the side of&#xA;an inscribed equilateral triangle.&lt;/em&gt;&amp;quot; Some of these do appear as Wells's &amp;quot;puzzles&amp;quot; (#44,#45), but apparently not #38 or several others. Nor does it appear as an exercise to the chapter 3, or in chapter 5 on trigonometry, where Abū al-Wafā is also featured. Presumably, Wells got the missing ones from somewhere, perhaps even directly from Geometry Needed by Craftsmen, but he does not say.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-07-26T10:35:41.563" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12073" PostTypeId="1" CreationDate="2020-07-26T11:08:05.107" Score="0" ViewCount="85" Body="&lt;p&gt;I was wondering if Lagrange was the first to use generalized coordinates as defined by their &lt;a href=&quot;https://en.wikipedia.org/wiki/Generalized_coordinates&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt; article.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2020-07-26T18:39:40.723" LastActivityDate="2020-07-26T18:39:40.723" Title="Was Lagrange the first to have used generalized coordinates?" Tags="|mathematics|physics|biographical-details|lagrange|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12074" PostTypeId="1" AcceptedAnswerId="12075" CreationDate="2020-07-26T15:52:12.060" Score="4" ViewCount="332" Body="&lt;p&gt;Galileo is known to have studied kinematics through his work with projectiles. How did he first consider researching motion and velocity? Was he inspired by previous work done by earlier scientific minds such as Archimedes, or was this work of his own volition?&lt;/p&gt;&#xA;&lt;p&gt;If there are any links that highlight the source of Galileo's motivations, please leave an answer below.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="4251" LastEditDate="2020-07-26T17:07:25.477" LastActivityDate="2020-07-27T01:08:21.220" Title="Who studied kinematics before Galileo? Did Galileo base his kinematic research on the previous work of any other scientist?" Tags="|mathematics|physics|classical-mechanics|galileo|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12075" PostTypeId="2" ParentId="12074" CreationDate="2020-07-27T00:00:40.253" Score="7" Body="&lt;p&gt;Kinematics was distinguished from dynamics by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Oxford_Calculators&quot; rel=&quot;noreferrer&quot;&gt;Merton school (a.k.a. Oxford calculators)&lt;/a&gt; of scholastics in 14th century, who worked out kinematics of uniformly accelerated motion. In particular, they formulated the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mean_speed_theorem&quot; rel=&quot;noreferrer&quot;&gt;mean speed theorem (a.k.a. Merton rule)&lt;/a&gt; (distance traveled is half the sum of the initial and final velocities, times the elapsed time), which was proved by Oresme using geometry and simple function graphs.&lt;/p&gt;&#xA;&lt;p&gt;The influence on Galileo is controversial. He seemed to insist on his originality in a letter to Sarpi, and he initially thought that speed is proportional to distance traveled despite being likely familiar with the Merton rule from the university years. What he wanted, unlike Mertonians or Oresme, who explicitly described his work as &lt;em&gt;secundum imaginationem&lt;/em&gt;, was an experimental demonstration one way or the other. In Dialogue Concerning the Two Chief World Systems (1632) he finally accepted Oresme's postulate that speed is proportional to time traveled, and in Two New Sciences (1638) stated a version of the mean speed theorem, along with the &amp;quot;law of odd numbers&amp;quot; (consecutive distances traveled in equal times are in the same ratios as consecutive odd numbers) that follows from it. Both the statement and the demonstration are reminiscent of Oresme's.&lt;/p&gt;&#xA;&lt;p&gt;Here is from &lt;a href=&quot;https://www.nagaitoshiya.com/en/2013/mean-speed-theorem/#Did_Oresme.E2.80.99s_theory_influence_Galileo.E2.80.99s_physics.3F&quot; rel=&quot;noreferrer&quot;&gt;How original is Galileo’s work on kinematics? by Nagai&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There is evidence of Galileo knowing the Merton rule and Oresme’s theory. We can find Heytesbury, Oxford Calculators, the mean speed theorem, “uniformly difform (uniformiter difformis)&amp;quot; quality and “Parisian doctors (Doctores Parisienses)&amp;quot; in manuscripts that Favaro, chief editor of the National Edition of the works of Galileo Galilei, called Youthful Writings[17]. Judging from the contents of the manuscript, the “Parisian doctors&amp;quot; must include not only Jean Buridan (1295 – 1358), who developed the concept of impetus close to the modern concept of inertia, but also his disciple, Oresme. Their theories were made so popular by Albert of Saxony (Albertus de Saxonia; ca. 1320 – 1390）and Domingo de Soto (1494 – 1560) that it is natural that Galileo could hear of them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;He finally left the University in 1585. His later works show his contempt for and hostility toward the Scholastics and the Aristotelians. It is no wonder that he did not accept all of the theories by the “Parisian doctors&amp;quot; that he considered Scholastic and Aristotelian. His contempt for and hostility toward the Scholastics and the Aristotelians is the key to the riddle of Galileo’s theory of falling bodies... we can find Oresme’s influence on Galileo, but he never mentioned his name, although he often heaped the highest praise on the works of Archimedes. Maybe he did not want to credit the achievement to the Aristotelian Scholastics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For more on Oresme and Merton school see &lt;a href=&quot;https://www.jstor.org/stable/41133336&quot; rel=&quot;noreferrer&quot;&gt;Medieval Quantifications of Qualities by Sylla&lt;/a&gt;. For a broader perspective see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-94-007-4132-4_34&quot; rel=&quot;noreferrer&quot;&gt;The Case of Kinematics, the Genesis of a Discipline by Koetsier&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-07-27T00:00:40.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12077" PostTypeId="1" CreationDate="2020-07-27T18:51:49.003" Score="5" ViewCount="118" Body="&lt;p&gt;I've been doing some research for a cosmology series and I'm struck by how many physicists and philosophers, from Newton to Einstein, had a notion that the Universe should be static and eternal. Why was this view so widespread?&lt;/p&gt;&#xA;" OwnerUserId="12851" LastEditorUserId="10106" LastEditDate="2020-07-27T19:39:39.057" LastActivityDate="2020-09-23T02:02:25.230" Title="Why is it that so many early astronomers and cosmologists wanted to believe in a static/infinite/eternal Universe?" Tags="|physics|philosophy-of-science|cosmology|astrophysics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12078" PostTypeId="1" CreationDate="2020-07-27T18:52:21.023" Score="2" ViewCount="38" Body="&lt;p&gt;In biology and botany, there is extensive research concerning the nature of different plant types and their methods of respiration and nutrient intake.&lt;/p&gt;&#xA;&lt;p&gt;How were &lt;a href=&quot;https://en.wikipedia.org/wiki/Crassulacean_acid_metabolism&quot; rel=&quot;nofollow noreferrer&quot;&gt;CAM (Crassulacean Acid Metabolism) plants&lt;/a&gt; first studied? Who first developed the methods to observe and study CAM plants, and how did they utilize these methods to discover respiration, chemical properties and other facets that constitute what a CAM plant is?&lt;/p&gt;&#xA;&lt;p&gt;If there are any links regarding the origins of such methods, please leave an answer below.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="12487" LastEditDate="2020-07-27T22:39:38.303" LastActivityDate="2020-07-27T22:39:38.303" Title="What were the original techniques used to study CAM (Crassulacean acid metabolism) plants? Who first employed these methods?" Tags="|biology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12079" PostTypeId="1" CreationDate="2020-07-28T19:09:40.827" Score="2" ViewCount="182" Body="&lt;p&gt;In my opinion, one of Isaac Newton's greatest achievements in the &amp;quot;purer&amp;quot; aspects of mathematics was his discovery of Puiseux series; power series with fractional exponents. According to p.6 of the book &amp;quot;A singular mathematical promenade&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Strictly speaking, Newton did not provide proofs, but he did understand that locally an analytic curve consists of a finite number of branches, which are “graphs” of formal power series with&#xA;rational exponents.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In his published work, Newton shows how the general method of power series with fractional exponents enables to present algebraic planar curves, which are more often given in implicit form &lt;span class=&quot;math-container&quot;&gt;$F(x,y) = 0$&lt;/span&gt;, in an explicit form &lt;span class=&quot;math-container&quot;&gt;$y = f(x)$&lt;/span&gt;. In a letter to Oldenburg he also describes the use of &amp;quot;Newton polygon&amp;quot; as a graphical aid in the calculation of Puiseux series. These spectacular achievements were made about 150 years before the developements by Laurent and Puiseux, and i think the generallity and usefulness of these methods somehow hides Newton's remarkable thought process behind what is apparently a practical motivation.&lt;/p&gt;&#xA;&lt;p&gt;I have to say that the same book i quoted also says that Newton's results were the first important results toward understanding of the topology of an algebraic (or analytic) curve in the neighbourhood of a singular point. I have to admit that i don't really understand this matter (i.e development in Puiseux series), so the sentence i quoted might already give a kind of answer to the title question, but i need to see and understand it in my eyes in order to see the answer (i hope the answer to this question will give me some idea about those matters). Also, i'm not familiar with the orginization and chronology of Newton's works, so i find it hard to search for the interconnections in Newton's mathematical works.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, my questions are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Were Newton's discoveries related to his work on the classification of cubic curves? i suggest it because some cubic curves have &amp;quot;cusp singularities&amp;quot; so this might be a possible source of inspiration for Newton's discoveries.&lt;/li&gt;&#xA;&lt;li&gt;If the answer to the previous question is no, then what was Newton's road to these discoveries?&lt;/li&gt;&#xA;&lt;li&gt;What are the modern uses and connections of Puiseux series in mathematics?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-07-29T19:02:22.773" LastActivityDate="2020-07-29T19:02:22.773" Title="What was Newton's road to his discovery of &quot;Puiseux series&quot; and &quot;Newton polygon&quot;?" Tags="|mathematics|mathematicians|biographical-details|newton|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12081" PostTypeId="1" CreationDate="2020-07-30T12:57:47.873" Score="2" ViewCount="78" Body="&lt;p&gt;Out of sheer curiosity:&lt;br /&gt;&#xA;When was the first discovery/estimate of the rotation rate of our Galaxy?&lt;/p&gt;&#xA;&lt;p&gt;(To my understanding: in astronomy the angular velocity of the Solar System with respect to the Milky Way's center is used to convert to Galactocentric coordinates, hence it is common to have that angular velocity represent the overall galaxy rotation rate.)&lt;/p&gt;&#xA;&lt;p&gt;It could well be, I don't know, that estimates of rotation rate of other galaxies were obtained first, as other galaxies can be observed as a whole.&lt;/p&gt;&#xA;&lt;p&gt;I tried search engine search, of course, but this one is hard to search for because the word 'Galaxy' is both used for other galaxies and our Galaxy. &amp;quot;Milky way&amp;quot; is exclusively our Galaxy, but that name is used far less, it appears.&lt;/p&gt;&#xA;&lt;p&gt;The reason I'm curious:&lt;br /&gt;&#xA;Milky way rotation rate is too slow to be measured directly.&lt;/p&gt;&#xA;&lt;p&gt;Once astronomers had inferred from observation that the Milky Way is a spiral galaxy it followed that it must have an overall rotation rate (contrary to, say, a globular galaxy). I'm curious what kind of astronomic observation can be done to infer the Milky Way rotation rate.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Later edit:&lt;br /&gt;&#xA;I found a review article by Yoshiaki Sofue:&lt;br /&gt;&#xA;&lt;a href=&quot;https://academic.oup.com/pasj/article/69/1/R1/2632658&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rotation and mass in the Milky Way and spiral galaxies&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The earliest article listed in that review article is from 1954.&lt;/p&gt;&#xA;&lt;p&gt;Kwee K. K., Muller C. A., Westerhout G. 1954, Bull. Astron. Inst. Ned.&lt;br /&gt;&#xA;&lt;a href=&quot;https://openaccess.leidenuniv.nl/handle/1887/5974&quot; rel=&quot;nofollow noreferrer&quot;&gt;The rotation of the inner parts of the Galactic System&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;By the looks of it: by that time the astronomers were already in the nitty-gritty of trying to obtain observations for the rotation rate of the inner parts.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Later later edit:&lt;br /&gt;&#xA;Progress report: I found historical information in a review article by Gianfranco Bertone and Dan Hooper &lt;a href=&quot;https://ned.ipac.caltech.edu/level5/Sept16/Bertone/Bertone2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;A history of dark matter&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In 1884 Lord Kelvin gave a series of lectures, in one of them he presented some theoretical explorations of galaxy dynamics.&lt;br /&gt;&#xA;&amp;quot;Lord Kelvin was among the first to attempt a dynamical estimate [...] His argument was simple yet powerful: if stars in the Milky Way can be described as a gas of particles, acting under the influence of gravity, then one can establish a relationship between the size of the system and the velocity dispersion of the stars&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(Copyright on the lecture notes is dated to 1884, Date of publication is given as 1904)&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;This material was known to Henri Poincaré. There is a 1906 article by Poincaré, in which he writes about some further theoretical exploration he had undertaken.&lt;/p&gt;&#xA;&lt;p&gt;Poincaré:&lt;br /&gt;&#xA;&amp;quot;Here a first remark presents itself; the result of Lord Kelvin which I have just verified by an approximate calculation obviously agrees with the estimates with the observers have been able to make with their telescopes; so that we must concede that we have very nearly pierced the Milky Way.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Henri Poincare, 1906, Popular Astronomy, vol. 14, pp.475-488 &lt;a href=&quot;http://articles.adsabs.harvard.edu/pdf/1906PA.....14..475P&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Milky Way and the Theory of Gases&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Poincaré did of course begin with pointing out 'agrees with' should be understood as 'to within a factor of 10')&lt;/p&gt;&#xA;&lt;p&gt;Lord Kelvin and Poincaré had simplified the model to a spherical galaxy (knowing the Milky Way isn't actually spherical). According to Bertone and Hooper the first to model the Milky Way as a flattened structure was the astronomer Kapteyn, 1922.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="7244" LastEditDate="2020-07-31T14:00:34.360" LastActivityDate="2020-07-31T14:00:34.360" Title="History of: Galaxy rotation rate estimates" Tags="|astronomy|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12082" PostTypeId="2" ParentId="2223" CreationDate="2020-07-31T23:13:44.580" Score="1" Body="&lt;p&gt;You say &amp;quot;there are no random variables in Cardano, de Moivre or Laplace&amp;quot;, but there are at least in Laplace. Not in the sense of maps from a sample space to a measurable space, but in the intuitive sense of &lt;em&gt;variable quantities which can assume different values with different probabilities&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the following Laplace clearly treats &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; as a random variable in this intuitive sense. From  &lt;em&gt;Mémoire sur les probabilités&lt;/em&gt;, 1778 (&lt;a href=&quot;https://archive.org/details/uvrescompltesde04fragoog/page/n399/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;French&lt;/a&gt;, &lt;a href=&quot;https://www.semanticscholar.org/paper/ME%CC%81MOIRE-SUR-LES-PROBABILITE%CC%81S%E2%88%97-Laplace/7af5d303ea76eecb8a49f8c7abb1c9e2a9bdeef9&quot; rel=&quot;nofollow noreferrer&quot;&gt;English&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;II.&lt;/p&gt;&#xA;&lt;p&gt;We suppose that two players &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, of whom the respective skills are unknown, play at any game, and we propose to determine the probability that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; will win the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; games.&lt;/p&gt;&#xA;&lt;p&gt;If the question were only of a single game, it is clear that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; must necessarily win it, these two events are equally probable, in such a way that the probability of the first is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;; whence, by following the ordinary rule of the analysis of chances, we conclude that the probability of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; winning the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; games is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2^n}$&lt;/span&gt;. This consequence will be exact if the probability &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; were based on an absolute equality between the possibilities of the two events of which there is question; but there is equality only relatively to the ignorance which we have of the skills of two players, and this equality does not prevent that the one is able to be much stronger than the other. We suppose consequently that &lt;span class=&quot;math-container&quot;&gt;$\frac{1+\alpha}{2}$&lt;/span&gt; represents the probability of the strongest player to win a game, and &lt;span class=&quot;math-container&quot;&gt;$\frac{1-\alpha}{2}$&lt;/span&gt; that of the weakest; by naming &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; the probability that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; will win the first &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; games, we will have &lt;span class=&quot;math-container&quot;&gt;$$P= \frac{1}{2^n}(1+\alpha)^n \quad \text{ or }\quad P = \frac{1}{2^n}(1-\alpha)^n,$$&lt;/span&gt; according as &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; will be the strongest orthe weakest: now, as we have no reason to suppose the one rather than the other, it is clear that, in order to have a true value of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, we must take the mean of the sum of the two preceding values, which gives &lt;span class=&quot;math-container&quot;&gt;$$ P = \frac{1}{2^{n+1}}[(1+\alpha)^n +(1-\alpha)^n].$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, if we would know the limit and the law of possibility of the values of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; [the probability distribution of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;], nothing would be more easy than to solve this problem exactly; because, if we name &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; this limit and if we represent by &lt;span class=&quot;math-container&quot;&gt;$\psi(\alpha)$&lt;/span&gt; the probability [density] of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, we see first that, &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; having&#xA;necessarily to fall between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, the function &lt;span class=&quot;math-container&quot;&gt;$\psi(\alpha)$&lt;/span&gt; must be such that we have &lt;span class=&quot;math-container&quot;&gt;$$\int d\alpha \psi(\alpha) = 1,$$&lt;/span&gt; the integral being taken from &lt;span class=&quot;math-container&quot;&gt;$\alpha = 0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\alpha = q$&lt;/span&gt;. We will multiply therefore by &lt;span class=&quot;math-container&quot;&gt;$d\alpha\psi(\alpha)$&lt;/span&gt; the probabilities determined by that which precedes, and, by integrating these products from &lt;span class=&quot;math-container&quot;&gt;$\alpha = 0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\alpha = q$&lt;/span&gt;, we will have the sought probabilities; we will find in this manner, for the value of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; in article II, &lt;span class=&quot;math-container&quot;&gt;$$ P = \int d\alpha\psi(\alpha) \frac{1}{2^{n+1}}[(1+\alpha)^n+(1-\alpha)^n].$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;More examples of such random variables can be found in the same article. Also Chebyshev has random variables of this type.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2020-07-31T23:13:44.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12083" PostTypeId="1" CreationDate="2020-08-01T14:08:49.967" Score="-1" ViewCount="102" Body="&lt;p&gt;It is a well-known fact now that the sun is one of the multiple stars in the universe.&lt;/p&gt;&#xA;&lt;p&gt;But in which year people at least hypothesized the fact that we are living in a universe with &lt;em&gt;multiple stars&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastEditorUserId="2692" LastEditDate="2020-08-02T05:49:52.027" LastActivityDate="2020-08-02T05:49:52.027" Title="In which year, people found that sun is not the only star in the universe?" Tags="|astronomy|" AnswerCount="1" CommentCount="0" ClosedDate="2020-08-02T12:15:19.083" ContentLicense="CC BY-SA 4.0" />
  <row Id="12085" PostTypeId="2" ParentId="12083" CreationDate="2020-08-01T17:15:09.183" Score="2" Body="&lt;p&gt;This belief is known as  &lt;a href=&quot;https://en.wikipedia.org/wiki/Cosmic_pluralism&quot; rel=&quot;nofollow noreferrer&quot;&gt;cosmic pluralism&lt;/a&gt; and was found among the ancient Greek philosophers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After Anaximander opened the door to an infinite universe, a strong pluralist stance was adopted by the atomists, notably Leucippus, Democritus, and Epicurus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Though this was the belief that stars had planets -- that is, were, like the sun, not just points of light but with satellites.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2020-08-01T17:15:09.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12087" PostTypeId="1" AcceptedAnswerId="12088" CreationDate="2020-08-01T15:54:17.593" Score="6" ViewCount="597" Body="&lt;p&gt;It is occasionally remarked that Einstein was unhappy that SR became referred to as a ‘theory of relativity’, when in his eyes it was, much more importantly, a theory of invariants (&lt;em&gt;Invariantentheorie&lt;/em&gt;).  I'm trying to find a quotable source for this, but can't.  Can anyone point me in the right direction?&lt;/p&gt;&#xA;&lt;p&gt;I believe that part of his disquiet was to do with the mischief (potential or actual?) caused by linking SR with philosophical or ethical relativism (which it of course has nothing to do with), but probably more significantly that calling it ‘relativity’ muddles the problem (the frame dependence of coordinates) with the solution (the invariance of the interval).&lt;/p&gt;&#xA;&lt;p&gt;I can find &lt;a href=&quot;https://hrcak.srce.hr/file/19107&quot; rel=&quot;noreferrer&quot;&gt;an essay&lt;/a&gt; (see p.270) which says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We know from his biographies that Einstein was not content with associations which his expression “Principle of Relativity” (or “Theory of Relativity”) had provoked, so he would prefer that his theory was named &lt;em&gt;die Invariantentheorie&lt;/em&gt; (Theory of Invariance), following Felix Klein’s term, but it was already too late to rename it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As well, I've found a rather hand-waving &lt;a href=&quot;https://motanomics.com/category/quantum-theory-and-relativity/&quot; rel=&quot;noreferrer&quot;&gt;remark&lt;/a&gt; that ‘The historian of science Gerald Holton reports that Einstein was unhappy with the label ‘relativity theory’ and in his correspondence referred to it as &lt;em&gt;Invariantentheorie…&lt;/em&gt;’.  Neither of these authors, however, seems to quote an actual source (grr).&lt;/p&gt;&#xA;&lt;p&gt;I feel sure there must be a mention of this somewhere in the Pais book, but I've been unable to find it anywhere the Table of Contents or the Index suggests I try.  I can't find anything relevant in the &lt;em&gt;Autobiographical Notes&lt;/em&gt;, either.&lt;/p&gt;&#xA;&lt;p&gt;I have a teaching purpose here.  I believe it's illuminating to stress that the frame-dependence is the problem, and the invariance both the solution and the clearest link to GR.  It would be at least tactically useful to quote the man himself in support of this position.&lt;/p&gt;&#xA;" OwnerUserId="12870" OwnerDisplayName="Norman Gray" LastEditorUserId="4251" LastEditDate="2020-08-02T10:23:11.860" LastActivityDate="2020-08-02T11:11:28.060" Title="Einstein: 'SR is a theory of invariants, not relativity' -- source?" Tags="|physics|relativity-theory|einstein|quote|physicists|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12088" PostTypeId="2" ParentId="12087" CreationDate="2020-08-01T21:24:50.683" Score="4" Body="&lt;p&gt;For sure 'Invariance Theory' would have been a better name.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein papers project&lt;/a&gt; is set up to be the definitive source of historical information.&lt;/p&gt;&#xA;&lt;p&gt;The content is available online, and, by the looks of it, has been indexed by Google (and presumably other search engines too).&lt;/p&gt;&#xA;&lt;p&gt;For search engine search one can use:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;invariantentheorie site:einsteinpapers.press.princeton.edu&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Einstein papers project website itself has a search field too, which is what I used just now. I used the german word, as in german it is a single word.&lt;/p&gt;&#xA;&lt;p&gt;The search term 'invariantentheorie' gives multiple locations, but not all are about physics. It appears 'invariantentheorie' is also a name for a specific area of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;This one seems the strongest candidate:&lt;/p&gt;&#xA;&lt;p&gt;From a &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol12-doc/372&quot; rel=&quot;nofollow noreferrer&quot;&gt;letter to Eberhard Zschimmer september 30, 1921&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sehr geehrter Herr Kollege! Zunächst versichere ich Sie, dass ich von&#xA;Herrn Norbert Einstein in keiner Weise gezupft worden bin. Nun zum&#xA;Namen Relativitäts-Theorie. Ich gebe zu, dass dieser nicht glücklich&#xA;ist und zu philosophischen Missverständnissen Anlass gegeben hat. Der&#xA;Name Invarianz-Theorie würde die Forschungsmethode der Theorie&#xA;bezeichnen, leider aber nicht den materiellen Inhalt der Theorie&#xA;(Konstanz der Lichtgeschwindigkeit, Wesensgleichheit von Trägheit und&#xA;Schwere). Trotzdem wäre die von Ihnen vorgeschlagene Bezeichnung&#xA;vielleicht besser, ich glaube aber, dass es Verwirrung anrichten&#xA;würde, den allgemein akzeptierten Namen nachträglich zu verändern.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems to me this letter is not enough evidence one way or the other. It could be, for instance, that Einstein was just being gracious in writing to Zschimmer that maybe the name 'Invarianz-Theorie' would have been better.&lt;/p&gt;&#xA;&lt;p&gt;(I translated the letter: scroll to the end of this answer)&#xA;&lt;br&gt;&lt;/p&gt;&#xA;&lt;p&gt;The search found this location because in the (english) commentary it is noted that Felix Klein had proposed 'invariententheorie'. That is how the search found that; the actual letter has the expression 'invarianz-Theorie'.&lt;/p&gt;&#xA;&lt;p&gt;Doing additional search, with variations, is worth a try, by the looks of it.&lt;/p&gt;&#xA;&lt;p&gt;I noticed an expression 'Minkowskischen invariantentheorie', which presumably refers to the invariant spacetime interval. This was in the course of a series of 4 lectures, delivered in 1922.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Additional historical information:&lt;/p&gt;&#xA;&lt;p&gt;Volume 2 of the Einstein Collected Papers Project presents Einstein's scientific publications from 1900 to 1909&lt;/p&gt;&#xA;&lt;p&gt;In the paragraph below ( &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol2-doc/290&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 254&lt;/a&gt; ) the editors discuss some history of naming. While volume 2 covers the years 1900-1909, they look ahead to 1915, mentioning that it was only then that Einstein started using the expression 'special theory' so as to differentiate it from 'general theory'. They do mention that in 1910 Felix Klein proposed the name 'invariance theory', but the editors give no indication that at the time Einstein was aware of that proposal, or that Einstein had independently thought of that as a possible name.&lt;/p&gt;&#xA;&lt;p&gt;The editors write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Strictly speaking, it is anachronistic to use the term &amp;quot;the theory of&#xA;relativity&amp;quot; in discussing Einstein's first papers on the subject. In&#xA;them he referred to the &amp;quot;principle of relativity&amp;quot; (&amp;quot;Prinzip der&#xA;Relativität&amp;quot; or &amp;quot;Relativitätsprinzip&amp;quot;). Max Planck used the term&#xA;&amp;quot;Relativtheorie&amp;quot; in 1906 to describe the Lorentz-Einstein equations of&#xA;motion for the electron, and this expression continued to be used from&#xA;time to time for several years. Bucherer seems to have been the first&#xA;person to use the term &amp;quot;Relativitätstheorie&amp;quot; in the discussion&#xA;following Planck's lecture. The term was used in an article by&#xA;Ehrenfest and adopted by Einstein in 1907, in his reply. Although&#xA;Einstein used the term from time to time thereafter, for several years&#xA;he continued to employ &amp;quot;Relativitätsprinzip&amp;quot; in the titles of his&#xA;articles. In 1910 the mathematician Felix Klein suggested the name&#xA;&amp;quot;Invariantentheorie,&amp;quot; but this suggestion does not seem to have been&#xA;adopted by any physicist. In 1915 Einstein started to refer to his&#xA;earlier work as &amp;quot;the special theory of relativity&amp;quot; (&amp;quot;die spezielle&#xA;Relativitätstheorie&amp;quot;) to contrast it with his later &amp;quot;general theory&amp;quot;&#xA;(&amp;quot;allgemeine Theorie&amp;quot;). In Einstein 1907j (Doc. 47) he does refer to&#xA;the need for generalizing the &amp;quot;principle of relativity&amp;quot; in order to&#xA;include gravitation in the theory, but throughout the present volume&#xA;the phrase &amp;quot;the theory of relativity&amp;quot; is used to denote the special&#xA;theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Translation of the letter to Eberhard Zschimmer.&lt;br /&gt;&#xA;Note: since Einstein refers to 'equality of gravity and inertia' he must have the general theory in mind.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Now to the name relativity theory. I admit that it is unfortunate and has given rise to philosophical misunderstandings. The name invariance theory would describe the research method of theory, but unfortunately not the material content of the theory (constancy of the speed of light, equality of inertia and gravity). Still, the term you suggested might be better, but I think it would be confusing to change the commonly accepted name later.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;About naming:&lt;br /&gt;&#xA;While 'Invariance theory' would be better for the special theory, it is, it seems to me, a poor fit for the general theory.&lt;/p&gt;&#xA;&lt;p&gt;Newtonian dynamics is a theory of motion.&lt;br /&gt;&#xA;Invariance theory is a theory of motion.&lt;br /&gt;&#xA;The general theory is a &lt;em&gt;unification&lt;/em&gt; of two kinds of theory: theory of &lt;em&gt;motion&lt;/em&gt; and theory of &lt;em&gt;gravity&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7244" OwnerDisplayName="Cleonis" LastEditorUserId="7244" LastEditDate="2020-08-02T11:11:28.060" LastActivityDate="2020-08-02T11:11:28.060" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12089" PostTypeId="1" CreationDate="2020-08-02T12:28:20.653" Score="3" ViewCount="269" Body="&lt;p&gt;Volume 3 of Gauss's werke contains an unpublished treatise with the title &amp;quot;&lt;em&gt;&lt;strong&gt;Theory of new transcendents&lt;/strong&gt;&lt;/em&gt;&amp;quot; (p.433-481 of the same volume), whose writing is dated, according to the editor Schering, to 1808; Schering states that those investigations prompted a communication to Schumacher. On page 441 of the same volume appears an interesting identity on theta functions which i suspect is a result of Gauss which i tried to locate in his writings for a very long time. The simplest way to describe it is to quote Gauss:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The division of seven leads to the following  equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$b = (\frac{1-2x+2x^4-...}{1+2x+2x^4+...})^2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A = (\frac{1+2x^7+2x^{28}+...}{1+2x+2x^4+...})^2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$B = (\frac{1-2x^7+2x^{28}-...}{1+2x+2x^4+...})^2$$&lt;/span&gt;&#xA;then:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A^8-\frac{4}{7}A^6+\frac{16-32b^2}{49}A^5-\frac{30}{343}A^4+\frac{32-64b^2}{2401}A^3-\frac{2+768b^2-768b^4}{16807}A^2+\frac{48-2144b^2+6144b^4-4096b^6}{823543}A -\frac{1}{823543} = 0$$&lt;/span&gt;&#xA;If we replace &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$1-b^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$-A$&lt;/span&gt;, the expression for &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is unchanged.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gauss's fragment contains a similar result for &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; but i didn't want to write it (it's very similar to the result for &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;). Note also that the denominators in all of the terms are powers of &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Rewriting Gauss's indentities in modern notation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$b = (\frac{\theta_4(0,x)}{\theta_3(0,x)})^2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A = (\frac{\theta_3(0,x^7)}{\theta_3(0,x)})^2$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$B = (\frac{\theta_4(0,x^7)}{\theta_3(0,x)})^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$\theta_3(0,x)=1+2\sum_{n=1}^{\infty} x^{n^2},\theta_4(0,x) = 1+2\sum_{n=1}^{\infty}(-1)^n x^{n^2}$&lt;/span&gt; are Jacobi's theta functions. Then Gauss found an inhomogenous algebraic (polynomial) expression &lt;span class=&quot;math-container&quot;&gt;$P(A,b) = 0$&lt;/span&gt;. This long presentation of Gauss's fragment was intended to present his results as accurate as possible. In a previous post (&lt;a href=&quot;https://hsm.stackexchange.com/questions/11563/question-about-hermites-1858-solution-to-the-quintic-equation-using-elliptic-mo/11564#11564&quot;&gt;Question about Hermite&amp;#39;s 1858 solution to the quintic equation using elliptic modular functions and it&amp;#39;s relation to Gauss&amp;#39; and Jacobi&amp;#39;s work&lt;/a&gt;), one of my questions was where Gauss's modular transformation of order 7 (according to some books, modular transformations of order 3,5 and 7 were known to Gauss since 1808). This fragment of Gauss deals with theta functions (not elliptic integrals), but since elliptic integrals and theta functions are related, i suspect this is the identity i searched for.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, my questions are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Can anyone familiar with the theory of elliptic integrals (and theta functions) confirm that is the so-called modular transformation of order 7?&lt;/li&gt;&#xA;&lt;li&gt;does this identity contain arithmetical information? i ask that because on another previous post (&lt;a href=&quot;https://hsm.stackexchange.com/questions/6256/did-gauss-know-jacobis-four-squares-theorem&quot;&gt;Did Gauss know Jacobi&amp;#39;s four squares theorem?&lt;/a&gt;) i refered to another identity of Gauss (on p.445 in the same treatise) on an expansion of &lt;span class=&quot;math-container&quot;&gt;$\theta_3^4(0,x)$&lt;/span&gt; in infinite series, and this identity does yield arithmetic information (namely Jacobi's four squares theorem).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And i apologize once again about my endless questioning about Gauss (i simply can't find references to many results in his writings...)!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-10-23T19:42:41.297" LastActivityDate="2020-10-25T10:16:39.357" Title="Meaning of certain identities of Gauss on theta functions" Tags="|mathematics|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12090" PostTypeId="1" CreationDate="2020-08-03T06:03:33.667" Score="0" ViewCount="143" Body="&lt;p&gt;In Thm. 4, Prop. 4 of Galileo's 'Two New Sciences' (pg. 187, Crew Translation), Galileo says the following: &amp;quot;From a single point &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; draw the planes &lt;span class=&quot;math-container&quot;&gt;$BA$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt;, having the same length but different inclinations; let &lt;span class=&quot;math-container&quot;&gt;$AE$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$CD$&lt;/span&gt; be horizontal lines drawn to meet the perpendicular &lt;span class=&quot;math-container&quot;&gt;$BD$&lt;/span&gt;; and let &lt;span class=&quot;math-container&quot;&gt;$BE$&lt;/span&gt; represent the height of the plane &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$BD$&lt;/span&gt; the height of &lt;span class=&quot;math-container&quot;&gt;$BC$&lt;/span&gt;; also let &lt;span class=&quot;math-container&quot;&gt;$BI$&lt;/span&gt; be a mean proportional to &lt;span class=&quot;math-container&quot;&gt;$BD$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$BE$&lt;/span&gt;; then the ratio of &lt;span class=&quot;math-container&quot;&gt;$BD$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$BI$&lt;/span&gt; is equal to the square root of the ratio of &lt;span class=&quot;math-container&quot;&gt;$BD$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$BE$&lt;/span&gt;.&amp;quot; (See figure)&lt;/p&gt;&#xA;&lt;p&gt;The claim struck me as odd and I experimented many times with various geometrical figures to see if I could reproduce this result in at least one instance, but I couldn't. What's going on here?&#xA;&lt;a href=&quot;https://i.stack.imgur.com/s4Sds.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/s4Sds.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12872" LastEditorUserId="26" LastEditDate="2021-02-20T15:09:26.243" LastActivityDate="2021-02-20T15:09:26.243" Title="Did Galileo use an erroneous geometrical result in 'Two New Sciences'?" Tags="|physics|geometry|classical-mechanics|galileo|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12091" PostTypeId="2" ParentId="12089" CreationDate="2020-08-03T11:12:25.440" Score="1" Body="&lt;p&gt;The formulas mentioned in Gauss's note are not the modular equation of order 7, but a very close one; the formula for &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is in fact the &amp;quot;&lt;strong&gt;Multiplier equation&lt;/strong&gt;&amp;quot; .The point is that confirming that &lt;span class=&quot;math-container&quot;&gt;$P(B,b) = 0$&lt;/span&gt; (not &lt;span class=&quot;math-container&quot;&gt;$P(A,b)$&lt;/span&gt;; the equation &lt;span class=&quot;math-container&quot;&gt;$P(B,b) = 0$&lt;/span&gt; was not written by me in the posted question) is the modular equation of order 7 would be much easier, and the answer would be yes, if only Gauss would define &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; to be &lt;span class=&quot;math-container&quot;&gt;$(\frac{\theta_4(0,x^7)}{\theta_3(0,x^7)})^2$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$(\frac{\theta_4(0,x^7)}{\theta_3(0,x)})^2$&lt;/span&gt;. To show the equivalency with the so-called &amp;quot;Transformation problem&amp;quot; of elliptic integrals, let us recall a few definitions and facts.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Definitions&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The nome &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; of a theta function is denoted &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in Gauss fragment.&lt;/li&gt;&#xA;&lt;li&gt;The modulus &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; of a complete elliptic integral of the first kind is the constant parameter in&#xA;&lt;span class=&quot;math-container&quot;&gt;$K(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1-k^2sin^2(\phi)}} = \int_0^1 \frac{dt}{\sqrt{(1-t^2)(1-kt^2)}} $&lt;/span&gt;. The complementary modulus is defined to be &lt;span class=&quot;math-container&quot;&gt;$k' = \sqrt{1-k^2}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;A complete elliptic integral of the first kind can be written as quotient of theta functions. The complementary elliptic modulus &lt;span class=&quot;math-container&quot;&gt;$k'$&lt;/span&gt; can be written as: &lt;span class=&quot;math-container&quot;&gt;$(\frac{\theta_4(0,q)}{\theta_3(0,q)})^2$&lt;/span&gt; where the nome &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$e^{-\pi\frac{K(k')}{K(k)}}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The transformation problem, or the problem of modular equation determination, is for one modulus &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, to find a second modulus &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\frac{K(l')}{K(l)} = n\frac{K(k')}{K(k)}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;The modular equation of order n&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Using the third fact, one can write &lt;span class=&quot;math-container&quot;&gt;$(\frac{\theta_4(0,q^n)}{\theta_3(0,q^n)})^2 = (\frac{\theta_4(0,e^{-n\pi\frac{K(k')}{K(k)}})}{\theta_3(0,e^{-n\pi\frac{K(k')}{K(k)}})})^2$&lt;/span&gt;. Rewriting &lt;span class=&quot;math-container&quot;&gt;$n\frac{K(k')}{K(k)}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\frac{K(l')}{K(l)}$&lt;/span&gt;, one gets that &lt;span class=&quot;math-container&quot;&gt;$(\frac{\theta_4(0,q^n)}{\theta_3(0,q^n)})^2$&lt;/span&gt; is nothing else than &lt;span class=&quot;math-container&quot;&gt;$l'$&lt;/span&gt;. Therefore an implicit relation between &lt;span class=&quot;math-container&quot;&gt;$k' = (\frac{\theta_4(0,q)}{\theta_3(0,q)})^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$l' = (\frac{\theta_4(0,q^n)}{\theta_3(0,q^n)})^2$&lt;/span&gt; is indeed the modular equation of order &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;$P(k',l')=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The connection to Gauss's fragment:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, as i remarked in the begining, Gauss defines &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; to be &lt;span class=&quot;math-container&quot;&gt;$(\frac{\theta_4(0,x^7)}{\theta_3(0,x)})^2$&lt;/span&gt; so the whole argument doesn't correspond to Gauss's note. However, since he states the results for both &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, and since &lt;span class=&quot;math-container&quot;&gt;$\frac{B}{A} = (\frac{\theta_4(0,x^7)}{\theta_3(0,x^7)})^2$&lt;/span&gt;, one can say that at least numericaly, Gauss's results enable to determine &lt;span class=&quot;math-container&quot;&gt;$l'$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$k'$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The relation of Gauss's fragment to the &amp;quot;multiplier equation&amp;quot;:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;According to p.512 volume 2 of the book series &amp;quot;Die elliptischen Funktionen und ihre Anwendungen&amp;quot; by Robert Fricke, the correct &amp;quot;Multiplikator gleichungen&amp;quot; (german for: &amp;quot;Multiplier equation&amp;quot;) of order 7 is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$M^8 - 28M^6 + 112(1-2k^2)M^5 - 210M^4 + 224(1-2k^2)M^3 - (140+5376k^2k'^2)M^2 + (1-2k^2)(48-2048k^2k'^2)M - 7 = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subsituting &lt;span class=&quot;math-container&quot;&gt;$M = 7A$&lt;/span&gt; in the correct equation and dividing by &lt;span class=&quot;math-container&quot;&gt;$7^8 = 7\cdot 823543$&lt;/span&gt;, one gets Gauss's equation. So it's 99% certain that Gauss's formula is in fact the multiplier equation. The only thing i don't understand is why &lt;span class=&quot;math-container&quot;&gt;$M = 7A$&lt;/span&gt; - according to all sources i found &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is supposed to be equal to &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-10-25T10:16:39.357" LastActivityDate="2020-10-25T10:16:39.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12093" PostTypeId="2" ParentId="12090" CreationDate="2020-08-03T13:38:38.697" Score="2" Body="&lt;p&gt;As &lt;span class=&quot;math-container&quot;&gt;$BI$&lt;/span&gt; is &lt;a href=&quot;https://www.quora.com/What-is-mean-proportion&quot; rel=&quot;nofollow noreferrer&quot;&gt;mean proportional&lt;/a&gt; to &lt;span class=&quot;math-container&quot;&gt;$BD$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$BE$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{array}{l}&#xA;\Rightarrow \frac{B D}{B I}=\frac{B I}{B E} \\&#xA;\Rightarrow \frac{B D}{B I} \times B D=\frac{B I}{B E} \times B D \\&#xA;\Rightarrow \frac{B D^{2}}{B I}=\frac{B I \times B D}{B E} \\&#xA;\Rightarrow \frac{B D^{2}}{B I^{2}}=\frac{B D}{B E}&#xA;\end{array}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$Q.E.D$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user10963" LastEditorDisplayName="user10963" LastEditDate="2020-08-05T11:23:29.640" LastActivityDate="2020-08-05T11:23:29.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12094" PostTypeId="1" CreationDate="2020-08-03T22:12:43.113" Score="3" ViewCount="341" Body="&lt;p&gt;The history of how the concept of complex numbers developed is convoluted.&lt;/p&gt;&#xA;&lt;p&gt;On physics.stackexchange questions about complex numbers keep recurring. It seems to me this indicates that when authors of physics textbooks introduce complex numbers there is a need (in the students) that is not met.&lt;/p&gt;&#xA;&lt;p&gt;Let me make a comparison.&lt;br /&gt;&#xA;When non-euclidean geometry was introduced (spherical geometry and hyperbolic geometry) the development to robust branches of geometry was fairly quick.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;In the case of complex numbers: from the first inklings to comfortable understanding took centuries.&lt;/p&gt;&#xA;&lt;p&gt;There is a theorem in the &lt;em&gt;theory of equations&lt;/em&gt; that states that the field of complex numbers is algebraically closed.&lt;/p&gt;&#xA;&lt;p&gt;Quoting the wikipedia article about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;fundamental theorem of algebra&lt;/a&gt;:&lt;br /&gt;&#xA;Every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.&lt;/p&gt;&#xA;&lt;p&gt;This property of the field of complex numbers is sometimes presented as the most important property of complex numbers. I agree with that statement, but it took mathematicians &lt;em&gt;centuries&lt;/em&gt; to come to that realization.&lt;/p&gt;&#xA;&lt;p&gt;It appears to me that for centuries complex numbers had quite a nebulous status. I'm using the name 'complex number', but of course throughout those centuries, throughout the stages of understanding, there were various names for the concept.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Histories of complex numbers describe that Descartes coined the name 'imaginary number'.&lt;/p&gt;&#xA;&lt;p&gt;I tried to find whether Isaac Newton had ever discussed the imaginary unit. The contributions of Isaac Newton to mathematics are wide and varied, but it appears Newton never discussed the concept of the imaginary unit, or complex numbers, in whatever form. (Arguably that's a deafening silence.)&lt;/p&gt;&#xA;&lt;p&gt;Next in the history of mathematics is the explorations of the exponential function, and the sine and cosine function, by Euler.&lt;/p&gt;&#xA;&lt;p&gt;As we know, the exponential function is its own derivative. The sine function &lt;em&gt;is its own derivative&lt;/em&gt; too, only it goes through a 4-step cycle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d(\sin(\alpha))}{d\alpha} = \cos(\alpha) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d(\cos(\alpha))}{d\alpha} = -\sin(\alpha) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d(-\sin(\alpha))}{d\alpha} = -\cos(\alpha) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d(-\cos(\alpha))}{d\alpha} = \sin(\alpha) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As we know, the imaginary unit &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; provides an efficient way to express the connection between the exponential function and the sine function&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ e^{ix} = \cos(x) + i \sin(x) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;At this point in time this formula, Euler's formula, is not yet particularly important. But not long after that physicists start recognizing that many phenomena involve oscillation, and many phenomena involve propagation of transversal waves.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that it is only around here that things start to come together for complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;It appears to me that until then complex numbers were, if used at all, used in an ad hoc manner. Starting with the work of d'Alembert and Cauchy I get a sense of systemization.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;I'm curious whether people will agree with this view.&lt;/p&gt;&#xA;&lt;p&gt;The history of how the concept of complex numbers developed is among the most convoluted in the history of mathematics. Compared to other concepts complex numbers remained nebulous for a very long time.&lt;br /&gt;&#xA;Do you agree? Do you disagree?&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="4251" LastEditDate="2020-08-04T10:26:25.197" LastActivityDate="2020-08-04T18:33:01.430" Title="On the history of development of the concept of complex numbers" Tags="|mathematics|complex-analysis|" AnswerCount="2" CommentCount="3" ClosedDate="2020-08-05T15:05:01.737" ContentLicense="CC BY-SA 4.0" />
  <row Id="12095" PostTypeId="2" ParentId="12094" CreationDate="2020-08-04T00:21:02.190" Score="2" Body="&lt;p&gt;This is a reply to some of the points you have made rather than an answer to the very broad question you have posed, which is essentially &amp;quot;yes, it is a very convoluted and tortured history&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;A reliable source on the subject of the history of complex numbers is the mathematical historian Leo Corry.&lt;/p&gt;&#xA;&lt;p&gt;According to Corry and contra your source, Descartes used the term “false” rather than imaginary, as Cardano had also done much earlier, though Descartes did refer to them as existing “only in our imagination.”  While Descarte seemed to understand the basic idea behind the fundamental theorem, namely that if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is a root of the polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; can be divided exactly by the factor &lt;span class=&quot;math-container&quot;&gt;$(x-a)$&lt;/span&gt;, this had already been stated formally by &lt;a href=&quot;https://en.wikipedia.org/wiki/Albert_Girard&quot; rel=&quot;nofollow noreferrer&quot;&gt;Albert Girart&lt;/a&gt; in his 1629 work &lt;em&gt;L’invention en algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Corry writes that Newton may have been the first to formulate a version of the fundamental theorem of algebra, to be found in his early lecture notes from Cambridge. It asserts that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[T]he number of roots of a polynomial equation cannot surpass the highest order of the unknown in the equation, but these roots may be positive, negative, or “impossible” (rather than “imaginary”).  And what he meant by impossible he explained by reference to the equation &lt;span class=&quot;math-container&quot;&gt;$$x^2 - 2ax +b^2 = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here, we obtain two roots, namely &lt;span class=&quot;math-container&quot;&gt;$$a+\sqrt{a^2-b^2}$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$ a - \sqrt{a^2-b^2}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, when &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; is greater than &lt;span class=&quot;math-container&quot;&gt;$b^2$&lt;/span&gt; - Newton wrote - the roots are “real”. In the opposite case [...] the root “impossible”.  But, interestingly, Newton nevertheless went on to stress that both expressions are roots of the polynomial for the simple reason that when they are introduced into the equation in place of the unknown, the equation is satisfied because “their factors eliminate each other.”  In other words, a square root of a negative number is an impossibility and hence does not represent a number in the proper sense of the word, but expressions containing such impossible entities are legitimate roots of an equation and allow for an appealing formulation of the fundamental theorem of algebra, as Newton conceived of it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gauss proved the fundamental theorem in 1799 and appears to have been working with a geometric interpretation of complex numbers.  However, efforts to better understand the nature of imaginary (and indeed negative) numbers continued.  Success started with the work of mathematicians such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Caspar_Wessel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Caspar Wessel&lt;/a&gt; (largely ignored), &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean-Robert_Argand&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean-Robert Argand&lt;/a&gt;, &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Francais_Jacques/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacques Français&lt;/a&gt;, and several others including a brilliant interpretation by &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Buee/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adrien Quentin Buée&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-08-04T00:21:02.190" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12096" PostTypeId="1" CreationDate="2020-08-04T00:50:28.817" Score="-1" ViewCount="326" Body="&lt;p&gt;In the current era, we label the Cartesian plane in x and y as our basis vectors, but what did the Greeks use to label their axis? The Greeks were around long before Descartes, so did they even use the same axis system? Or how did they graph things?&lt;/p&gt;&#xA;" OwnerUserId="12877" LastEditorUserId="10106" LastEditDate="2020-08-04T03:41:32.070" LastActivityDate="2020-08-05T01:23:44.920" Title="How did the Greeks label their axes?" Tags="|mathematics|ancient-greece|naming-conventions|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12097" PostTypeId="2" ParentId="12094" CreationDate="2020-08-04T18:33:01.430" Score="1" Body="&lt;p&gt;This is not a self-answer, I'm using this space to present my views as they have evolved to this point.&lt;/p&gt;&#xA;&lt;p&gt;I want to recapitulate the following:&lt;/p&gt;&#xA;&lt;p&gt;The exponential function is it's own derivative:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{d(e^x)}{dx} = e^x $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The hyperbolic sine and the hyperbolic cosine are each others derivative; a 2-step cycle.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sinh = \tfrac{1}{2}e^x - \tfrac{1}{2}e^{-x} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \cosh = \tfrac{1}{2}e^x + \tfrac{1}{2}e^{-x} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That is, we can think of the &lt;span class=&quot;math-container&quot;&gt;$\cosh$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sinh$&lt;/span&gt; functions as constructed out of the exponential function in such way that you get a sign flip resulting in a 2-step cycle.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;For the trigonometric functions the taking-the-derivative cycle is a 4-step cycle, with sign flips.&lt;/p&gt;&#xA;&lt;p&gt;It doesn't matter how that 4-step cycle is implemented, but the more efficient the better of course.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ e^{ix} = \cos(x) + i \sin(x) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have now convinced myself that this formula is &lt;em&gt;exclusively&lt;/em&gt; about the connection between the &lt;em&gt;exponential function&lt;/em&gt; and the &lt;em&gt;sine function&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Using complex number notation is purely a matter of implementation here; it so happens that it fits the bill.&lt;/p&gt;&#xA;&lt;p&gt;In physics the exponential-function-in-4step-cycle is ubiquitous of course. the exponential-function-in-4step-cycle allows efficient manipulation of quantities such as frequency, amplitude, phase.&lt;/p&gt;&#xA;&lt;p&gt;If I were to write a physics textbook I would treat Euler's formula as a standalone formula, emphasizing that it is about the connection between the exponential function and the sine function.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2020-08-04T18:33:01.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12099" PostTypeId="2" ParentId="12096" CreationDate="2020-08-05T01:23:44.920" Score="1" Body="&lt;p&gt;The ancient Greeks did not use rectangular coordinate systems in their mathematics, so it is unclear what are you asking about. The method of coordinates was introduced to mathematics by Descartes.&lt;/p&gt;&#xA;&lt;p&gt;However the Greeks used spherical coordinates in astronomy and geography, and they labeled the coordinate circles the same way as we do (or more precisely we do the same as they did). I mean equator, ecliptic, meridians etc. In practical astronomy and geography we use the same coordinates as Ptolemy did, with essentially the same names and labels.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-08-05T01:23:44.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12100" PostTypeId="1" AcceptedAnswerId="12101" CreationDate="2020-08-05T11:36:55.323" Score="0" ViewCount="356" Body="&lt;p&gt;I've desperately searched the Internet, to no avail, to find a citation of how the &amp;quot;Wheel of Theodorus&amp;quot; became known as the &amp;quot;Wheel of Einstein&amp;quot; as claimed by &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Spiral_of_Theodorus&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Wikipedia&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;https://mathworld.wolfram.com/EinsteinSpiral.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Wolfram&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12885" LastEditorUserId="12885" LastEditDate="2020-08-05T21:15:45.447" LastActivityDate="2020-08-05T21:15:45.447" Title="How did the &quot;Wheel of Theodorus&quot; become known as the &quot;Wheel of Einstein&quot;?" Tags="|geometry|einstein|naming-conventions|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12101" PostTypeId="2" ParentId="12100" CreationDate="2020-08-05T11:58:57.920" Score="1" Body="&lt;p&gt;Considering that &amp;quot;The Wheel of Theodorus is also called the Spiral of Theodorus,&#xA;the Square Root Spiral, Einstein Spiral, or Pythagorean Spiral,&amp;quot;&lt;a href=&quot;https://www.geogebra.org/m/BwKQyu69&quot; rel=&quot;nofollow noreferrer&quot;&gt; [W.Lombard]&lt;/a&gt; I doubt you'll find anything other than random common usage as a reason for naming.&#xA;( also  “Wurzel Spirale&amp;quot; )&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-08-05T11:58:57.920" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12102" PostTypeId="1" CreationDate="2020-08-05T12:24:36.587" Score="1" ViewCount="144" Body="&lt;p&gt;The background of this question is as follows. &lt;a href=&quot;https://en.wikipedia.org/wiki/Sean_M._Carroll&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sean Carroll&lt;/a&gt; is in the process of giving a series of talks recorded in his home office and published on YouTube. The setup is intentionally very informal. The overall name of the series is &lt;em&gt;Biggest ideas in the Universe&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Recently, he threw out an on-the-side remark that really annoyed me. Sean Carroll was talking about Ludwig Boltzmann, and about opposition from some of his contemporaries.&lt;/p&gt;&#xA;&lt;p&gt;Transcript &lt;a href=&quot;https://youtu.be/mDUgPJrXqRc?t=3155&quot; rel=&quot;nofollow noreferrer&quot;&gt;52:35&lt;/a&gt; into the video:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] a lot of people in germany led by Ernst Mach of Mach's principle&#xA;fame didn't want to believe in the existence of atoms. They thought&#xA;that things like the second law of thermodynamics were not&#xA;probabilistic. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;So, at the time, what was actually going on? What, in truth, was the attitude of Ernst Mach towards the existence of atoms?&lt;/p&gt;&#xA;&lt;p&gt;The Stanford Encyclopedia of Philosophy has an &lt;a href=&quot;https://seop.illc.uva.nl/entries/atomism-modern/&quot; rel=&quot;nofollow noreferrer&quot;&gt;article about atomism&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The striking thing about atomism in the 19th century: on one hand there is circumstantial evidence in favor of atomism, in the form of elements tending to combine in particualar ratios, and from kinetic theory of gases. On the other hand, for both lines of circumstantial evidence there were also anomalies. Cases were known where the chemical reaction doesn't show a preferred ratio, cases were known where certain experimental results could not be accounted for in terms of the existing kinetic theory of gases.&lt;/p&gt;&#xA;&lt;p&gt;In the Stanford Encyclopedia article Ostwald is mentioned. Ostwald was of the opinion that the case for atomism was not conclusive.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;&lt;strong&gt;Mach's philosophy of science&lt;/strong&gt;&lt;br /&gt;&#xA;Mach's philosophy of science was very austere. At the time there was the school of thought called 'Logical Positivism'. As I understand it: in terms of logical positivism only experimental results and the formulas that express the structure of a theory of physics are regarded to fall in realm of science.&lt;/p&gt;&#xA;&lt;p&gt;In this austere view even &lt;em&gt;interpretation&lt;/em&gt; of a theory of physics isn't necessarily science, in the sense that this interpretation may be in part speculative.&lt;/p&gt;&#xA;&lt;p&gt;So, I can see Ernst Mach arguing that the evidence available at the time does not prove atomism beyond doubt.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Specific question: of the people who opposed the views of Ludwig Boltzman, what was their opinion on the status of atomism?&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="4251" LastEditDate="2020-08-07T13:56:17.730" LastActivityDate="2020-08-07T13:56:17.730" Title="On atomism and philosophy of science" Tags="|physics|philosophy-of-science|atomic-theory|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12103" PostTypeId="1" AcceptedAnswerId="12105" CreationDate="2020-08-07T07:02:24.717" Score="4" ViewCount="621" Body="&lt;p&gt;A lot of phenomena in radiative transfer are named after a person who studied them (Rayleigh scattering, Mie scattering, Bragg diffraction, Kikuchi lines, Tyndall effect,...). Others are designated by a description of sorts of what is going on (geometric scattering, photon diffusion,...) But some are just the German word for what is going on, like gegenschein and bremsstrahlung. Sometimes &amp;quot;back-scatter&amp;quot; or &amp;quot;braking radiation&amp;quot; is used, but just as often, the German word is used. Why is it that the German words stuck? Why German rather than another language, say Italian or Dutch?&lt;/p&gt;&#xA;" OwnerUserId="10806" LastEditorUserId="10806" LastEditDate="2021-07-26T07:07:48.353" LastActivityDate="2021-07-26T07:07:48.353" Title="Why are there so many German terms in the field of radiative transfer?" Tags="|physics|terminology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12104" PostTypeId="2" ParentId="12103" CreationDate="2020-08-07T07:27:06.773" Score="3" Body="&lt;p&gt;German was the main language of Physics, Chemistry, and Medicine during the second half of the 19th century and early 1900s. Besides being the language of the German Reich, it was also spoken and written across the very large Danubian monarchy. At that time, in Prague, Bucarest or Budapest, and even in Lemberg (Lvov/Lwiw today) the language of educated people was mostly German...&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2020-08-07T07:27:06.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12105" PostTypeId="2" ParentId="12103" CreationDate="2020-08-07T07:52:37.413" Score="5" Body="&lt;p&gt;German was the language of chemistry, physics and mathematics until the 1940s. Just today, I had to request two German translated papers in chemistry from the library. Anyone interested in doing a PhD in sciences had to show a reading knowledge of German, French or Russian and this requirement was determined by the field of study in American, British or Canadian universities. The term &lt;em&gt;bremsstrahlung&lt;/em&gt; was coined by Sommerfeld, a German physicist. Since a majority of the natural scientists knew some functional German it was not a big deal to use this terminology occasionally. Another reason for keeping German words was that an English equivalent did not exist. Another famous term is convolution, for ages it remained as &lt;em&gt;Faltung&lt;/em&gt; (folding).&lt;/p&gt;&#xA;&lt;p&gt;In the late 60s and 70s, universities began to relax these conditions and by late 90s foreign language requirements vanished from the scene in the sciences. However, it is always beneficial to learn one more classical European language besides English in the sciences-if you wish to read original articles.  Perhaps, that is why I think Harvard still requires a functional knowledge of one these languages: Chinese, French, German or Russian for a doctorate in mathematics. Fortunately, machine translation has become so good in the last two three years that scientific texts in German and French can be easily rendered into English even with extensive symbolism.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2020-08-07T07:52:37.413" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12107" PostTypeId="1" CreationDate="2020-08-07T12:02:45.550" Score="3" ViewCount="114" Body="&lt;p&gt;Since Lagrange published his &lt;em&gt;Reflections&lt;/em&gt; papers during the early 1770s — around 30 years before Ruffini took up and extended the subject — I was wondering if there were any results that were discovered in the meantime that may have aided Ruffini in making his group-theoretic discoveries related to his work on the quintic equation.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="26" LastEditDate="2021-02-20T15:09:19.513" LastActivityDate="2021-02-20T15:09:19.513" Title="How did Ruffini manage to extend the methods of Lagrange in order to &quot;prove&quot; the insolvability of the general quintic equation?" Tags="|mathematics|biographical-details|abstract-algebra|lagrange|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12108" PostTypeId="1" CreationDate="2020-08-08T07:07:18.543" Score="3" ViewCount="650" Body="&lt;p&gt;I remember hearing once that the first &amp;quot;proof&amp;quot; that the angle to maximise projectile range gave the correct answer, 45 degrees, but was later found that the proof was wrong. I can't remember many other details, but I believe the name was Italian. Does anybody know a reference or a name that I could use to find out more about this?&lt;/p&gt;&#xA;" OwnerUserId="12901" LastEditorUserId="55" LastEditDate="2020-08-08T12:31:29.347" LastActivityDate="2020-08-10T04:30:16.467" Title="Who made the first derivation of the angle to maximise projectile range, which turned out to be wrong?" Tags="|physics|reference-request|classical-mechanics|mathematical-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12109" PostTypeId="2" ParentId="12108" CreationDate="2020-08-08T12:25:18.853" Score="9" Body="&lt;p&gt;This probably refers to Galileo's &amp;quot;derivation&amp;quot; of Tartaglia's observation that cannon balls achieve maximal range when fired at 45°. &lt;a href=&quot;https://en.wikipedia.org/wiki/Niccol%C3%B2_Fontana_Tartaglia#Ballistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tartaglia's theory of projectile motion&lt;/a&gt; was wrong, he assumed that fired balls follow a line segment going up, then an arc of a circle to change direction, and finally fall vertically down, but the observation was experimental. Galileo obtained the same angle from his theory of accelerated motion in &lt;em&gt;Discorsi intorno a Due Nuove Scienze&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But, as he came to realize, this only applied to slow projectiles launched without gunpowder (&lt;a href=&quot;https://en.wikipedia.org/wiki/Explosive#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;black powder&lt;/a&gt; different from  gunpowder of today), e.g from bows and catapults, or even with it from mortars, but not from cannons. Because for fast projectiles air resistance alters trajectories considerably. And Galileo had no theory to take it into account. For example, in modern guns maximal range is attained at angles nearer to 30° rather than 45°.&lt;/p&gt;&#xA;&lt;p&gt;Ironically, fast projectiles have trajectories shaped more like Tartaglia's than like Galileo's parabolas. Nonetheless, Galileo fudged the difference to make his result look more general and significant, see &lt;a href=&quot;https://www.jstor.org/stable/4025144&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rose, Galileo's Theory of Ballistics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In his letter of March I637, Galileo had enthusiastically proclaimed a table usable for both cannon and mortar shooting. Between writing that letter and submitting the final version of the Fourth Day to Elsevir three months later, Galileo had come to realize that because of air- resistance a general solution to the ballistic problem of ranging was beyond him; in the Discorsi he had to be content with a partial solution applying to mortars only.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;How then has the notion arisen that the table in the Discorsi represents a general solution? Undoubtedly part of the blame can be attached to Galileo's obliqueness about the exact purpose of the table that he does print. This equivocation was perhaps born from a desire to seem to have discovered the long sought after general solution and yet still to retain his integrity. This equivocation was in fact noticed by Descartes... Other readers were less perceptive than Descartes, and rushed to hasty optimistic conclusions. Torricelli and Mersenne, both friends of Galileo, though aware of the existence of air-resistance, still thought that the resistance was so slight as to be disregarded in the preparation of general range-tables.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When Torricelli's tables were tested by Renieri in 1646 he expressed astonishment that &amp;quot;&lt;em&gt;such a well-founded theory should respond so poorly in practice&lt;/em&gt;&amp;quot;. In response, Torricelli invoked Galileo's authority. The error came to be commonly attributed to Galileo himself. Descartes was quite derisive in his commentary:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It ought to be noticed that in proposing his suppositions he excepts from them artillery so that he may demonstrate them more easily. But that, all the same, towards the end it is principally to artillery that he applied his conclusions. In a word, il a tout basti en l'air&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The last phrase is a pun, &amp;quot;he did everything in the air&amp;quot;, as in, he screwed everything up.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-10T04:30:16.467" LastActivityDate="2020-08-10T04:30:16.467" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12110" PostTypeId="1" CreationDate="2020-08-08T14:32:11.250" Score="2" ViewCount="61" Body="&lt;p&gt;Robert Millikan's oil drop experiment help to find the charge of electrons using oil. An x-ray was aimed at the oil drops to give them a charge, and the droplets were attracted to the negative plate. How did Millikan think of oil droplets; could he not have considered any other substance to carry a charge?&lt;/p&gt;&#xA;" OwnerUserId="12487" LastEditorUserId="4251" LastEditDate="2020-08-08T21:28:17.583" LastActivityDate="2020-08-08T21:28:17.583" Title="How did Millikan consider utilizing oil to help determine the charge of the electron?" Tags="|physics|experimental-physics|atomic-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12112" PostTypeId="1" CreationDate="2020-08-09T13:29:28.430" Score="3" ViewCount="264" Body="&lt;p&gt;It seems pretty strange to me that the idea of infinity and infinite sets was initially opposed by many prominent Mathematicians, even driving Cantor into depression. However, in modern days, everyone seems to accept infinity without any problem, despite the existence of many paradoxes like the Banach-Tarski paradox and the prisoner hat paradoxes.&lt;/p&gt;&#xA;&lt;p&gt;I understand that modern Mathematics is largely built on the concept of infinity, and Mathematics has already become a &amp;quot;game of ideal world,&amp;quot; and I am not here to oppose the idea. What I am confused about is that historically once upon a time the majority of Mathematicians opposed the idea (at a time where Mathematics was not yet built on infinity), even driving Cantor himself into craziness. It seems pretty strange to me why and how the world has changed to this current state, where anyone opposing the idea is labeled a &amp;quot;crank&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="9748" LastEditorUserId="12724" LastEditDate="2020-09-27T17:38:31.193" LastActivityDate="2020-09-27T17:38:31.193" Title="History of Infinity: How does the idea of infinite set become widely accepted?" Tags="|mathematicians|infinity|" AnswerCount="2" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12113" PostTypeId="2" ParentId="3837" CreationDate="2020-08-09T19:52:51.510" Score="1" Body="&lt;p&gt;According to one biography I read, Einstein didn't learn much from his university courses in Electromagnetism. Presumably this was simply because he was already looking through the works of Maxwell, Hertz and so on. Also, according to Freeman Dyson, mainstream physics didn't really take to Maxwell Equations - no matter how they are lauded today. There was a lapse of at least two decades before they really began to be appreciated. And presumably this would have shown in the curriculum.&lt;/p&gt;&#xA;&lt;p&gt;This is understandable in a way. Maxwell notation is not as compact as we have today. When he wrote down his equations there twenty of them. It took the efforts of Heaviside and Gibbs to reduce them to the four vector equations we have now, plus the Lorentz force equation. Nowadays, using the Cartan formalism, aka differential forms, Maxwell equation has been reduced to a single equation (this doesn't include the Lorentz force law)! This might seem like a marvel of compactness, but here all the complications have been hidden away in the formalism (one has to learn about manifolds, bundles and so on).&lt;/p&gt;&#xA;&lt;p&gt;One thing that he would have learnt from Hertz which I only learnt of recently, and which isn't emphasised enough, is that Hertz held it to be a defect of Newtonian physics that there were two basic notions - inertia and force. For philosophical reasons he wanted a force less mechanics and to that end introduced additional variables which refer to 'hidden' dimensions of space.&lt;/p&gt;&#xA;&lt;p&gt;This is obviously interesting in two ways. The first, a forceless gravity, is what GR is. Hidden dimensions of space manifesting themselves as force is exactly what Kaluza-Klein showed. In fact, the latter notion goes back to Riemann, who spoke of the possibility of metrical change, not in the large as one might expect - but in the small, that is the infinitesimal.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-09T19:52:51.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12114" PostTypeId="1" AcceptedAnswerId="12115" CreationDate="2020-08-10T06:18:59.263" Score="1" ViewCount="1139" Body="&lt;p&gt;&lt;a href=&quot;https://www.azquotes.com/quote/573998&quot; rel=&quot;nofollow noreferrer&quot;&gt;The quote&lt;/a&gt; is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When asked ... [about] an underlying quantum world, Bohr would answer,&#xA;'There is no quantum world. There is only an abstract quantum physical&#xA;description. It is wrong to think that the task of physics is to find&#xA;out how nature is. Physics concerns what we can say about Nature.'&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What is the source of this quote?&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerDisplayName="user10963" LastEditorUserId="5989" LastEditDate="2020-08-13T12:00:39.887" LastActivityDate="2020-08-13T12:00:39.887" Title="What is the source of a quotation by Niels Bohr?" Tags="|quantum-mechanics|quote|physicists|bohr|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12115" PostTypeId="2" ParentId="12114" CreationDate="2020-08-10T07:23:25.133" Score="4" Body="&lt;p&gt;The source is Aage Petersen's article &lt;a href=&quot;https://books.google.pt/books?id=DgcAAAAAMBAJ&amp;amp;lpg=PA36&amp;amp;pg=PA8&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Philosophy of Niels Bohr&lt;/em&gt;&lt;/a&gt;, published in 1963. According to &lt;a href=&quot;https://en.wikipedia.org/wiki/N._David_Mermin&quot; rel=&quot;nofollow noreferrer&quot;&gt;N. David Mermin&lt;/a&gt;, in his article &lt;a href=&quot;https://physicstoday.scitation.org/doi/10.1063/1.1688051&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;What's Wrong with this Quantum World&lt;/em&gt;&lt;/a&gt;, “Bohr, who took writing very seriously indeed, never published such an assertion in any of his writings, although he repeatedly refined, reformulated and often simply repeated his position on the philosophical foundations of the quantum theory”.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-08-10T07:23:25.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12116" PostTypeId="2" ParentId="12112" CreationDate="2020-08-10T10:10:19.237" Score="1" Body="&lt;p&gt;Quoting numerous sources,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The German physicist Max Planck said that science advances one funeral&#xA;at a time. Or more precisely: “A new scientific truth does not triumph&#xA;by convincing its opponents and making them see the light, but rather&#xA;because its opponents eventually die, and a new generation grows up&#xA;that is familiar with it.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That's not 100% applicable here, since mathematics simply accepts anything which results from a valid proof.  Remember that math does not necessarily hew to the physical world, so the existence of &amp;quot;infinities&amp;quot; of different measure is perfectly acceptable, just as is nonEuclidean geometry,  the Cantor set, and the Discrete Metric, to name some famous examples.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2020-08-10T10:10:19.237" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12117" PostTypeId="1" AcceptedAnswerId="12122" CreationDate="2020-08-11T00:21:46.503" Score="2" ViewCount="237" Body="&lt;p&gt;Watching the video &lt;a href=&quot;https://www.youtube.com/watch?v=0sfvQ_fsil4&quot; rel=&quot;nofollow noreferrer&quot;&gt;So much universe, so little time&lt;/a&gt;, I was surprised to hear Fred Hoyle talked about in glowing terms.&lt;/p&gt;&#xA;&lt;p&gt;According to the narrator, by the late 1950’s the big bang model was losing favour; by 1961 Lemaître was lamenting that his ideas had become old-fashioned; steady state theory was getting a lot of attention and Hoyle was :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;being held in a dogmatic veneration similar to Aristotle in the middle ages.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I’m not sure what to make of this comment, which occurs at about 22:40, other than it seems doubtful.  On the other hand, many were reluctant to give up the static, eternal model so this may have amplified  enthusiasm for the steady state model with a knock-on effect for Hoyle's reputation.&lt;/p&gt;&#xA;&lt;p&gt;My question is, to what extent did Hoyle’s reputation in the scientific community gain currency at this time, and did his “fame” reach into popular culture to any degree?&lt;/p&gt;&#xA;&lt;p&gt;Hoyle was knighted, which suggests that in Britain he would have been a known popular figure of sorts.  His reputation appears to have suffered considerably in subsequent years, to the point that he was snubbed by the Nobel committee regarding his original work on stellar nucleosynthesis.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2020-08-12T06:51:23.010" LastActivityDate="2020-08-23T23:32:02.903" Title="Was Fred Hoyle’s career due, in part, to the popularity of the static universe in 1950s?" Tags="|biographical-details|cosmology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12118" PostTypeId="1" CreationDate="2020-08-11T07:11:01.613" Score="1" ViewCount="149" Body="&lt;p&gt;There are stories that famous physicists made predictions of a list of problems; problems that the future generation of physicists will be busy with or revolutionize the future of physics in the upcoming years. Any idea?&lt;/p&gt;&#xA;" OwnerUserId="9960" LastActivityDate="2021-05-22T20:38:16.787" Title="What are the stories of physicists's prediction of problems that future generation will be busy with/topics that revolutionize future of physics?" Tags="|physics|theoretical-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12119" PostTypeId="1" CreationDate="2020-08-11T09:32:15.060" Score="8" ViewCount="2885" Body="&lt;p&gt;From the Wikipedia page about &lt;a href=&quot;https://en.wikipedia.org/wiki/Vieta_jumping&quot; rel=&quot;noreferrer&quot;&gt;Vieta jumping&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Emanouil Atanassov, Bulgaria, solved the problem [assumed to be the most difficult one on the 1988 International Mathematics Olympiad] in a paragraph and received a special prize.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The reference links to &lt;a href=&quot;https://www.imo-official.org/participant_r.aspx?id=1586&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;. It does not mention anything about that paragraph. Does anyone have a copy of the actual paragraph written by him to solve the problem?&lt;/p&gt;&#xA;" OwnerUserId="12901" LastEditorUserId="4251" LastEditDate="2021-09-09T06:17:57.007" LastActivityDate="2022-12-02T00:05:20.673" Title="What paragraph was written by Emanouil Atanassov to solve problem 6 on the International Mathematical Olympiad in 1988?" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12120" PostTypeId="1" CreationDate="2020-08-12T00:43:32.197" Score="3" ViewCount="881" Body="&lt;p&gt;I am unaware if someone has asked this before, but I am studying classical mechanics and I don’t know why do we use &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; for potential energy. I have read that Rankine used it first, but I can’t find any explanation for this.&lt;/p&gt;&#xA;" OwnerDisplayName="René Quijada" LastEditorUserId="4251" LastEditDate="2020-08-12T23:36:22.383" LastActivityDate="2020-08-12T23:36:22.383" Title="Why do we use $U$ for potential energy in classical mechanics?" Tags="|physics|notation|classical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12121" PostTypeId="2" ParentId="12120" CreationDate="2020-08-12T06:06:03.960" Score="2" Body="&lt;p&gt;It is possible that &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; stands for &amp;quot;utility&amp;quot; (in mechanics), and it is due to Hamilton and not to Rankine. Possibly, Rankine wished to be consistent with Hamilton's notation, but, in any case, I suspect hat the modern use in mechanics has more to do with following Hamilton than with following Rankine, whose musings are very abstract and metaphysical, and are not even directed at mechanics. He does not even mention kinetic energy, for example.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786445308647205&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rankine's paper On the general law of the transformation of energy (1853)&lt;/a&gt; that introduces potential energy explicitly gives no explanation, &amp;quot;&lt;em&gt;Let &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; denote this potential energy&lt;/em&gt;&amp;quot; is all he wrote. It should be noted that the concept of potential energy is much older. It appears, in a rudimentary form, in Leibniz's writings under the name of &amp;quot;motive force&amp;quot;, along with a prototype of mechanical conservation law, which Rankine spelled out explicitly two centuries later, see &lt;a href=&quot;https://www.jstor.org/stable/228997&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz and the Vis Viva Controversy by Iltis&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Lagrange used &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; for potential energy in &lt;a href=&quot;https://archive.org/details/mcaniqueanalyti02lagrgoog/page/n14/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mécanique analytique&lt;/em&gt; (1788-9)&lt;/a&gt;, possibly from Latin &lt;em&gt;vis&lt;/em&gt;, and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; for kinetic energy. Hamilton kept Lagrange's &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;, but, for whatever reason, switched from &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; in &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;On a General Method in Dynamics (1834)&lt;/a&gt;, which introduces what we now call Hamiltonian dynamics. Of course, neither of them called it &amp;quot;potential energy&amp;quot;, but the letters occupy the familiar places in deriving equations of motion from a variational principle. Hamilton introduces&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;A function &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; of the masses and mutual distances of the several points of the system, of which the form depends on the laws of their mutual actions, by the equation &lt;span class=&quot;math-container&quot;&gt;$U=\sum. mm_{'}f(r)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; being the distance between any two points &lt;span class=&quot;math-container&quot;&gt;$m,m_{'}$&lt;/span&gt;, and the function &lt;span class=&quot;math-container&quot;&gt;$f(r)$&lt;/span&gt; being such that the derivative or differential coefficient &lt;span class=&quot;math-container&quot;&gt;$f'(r)$&lt;/span&gt; expresses the law of their repulsion, being negative in the case of attraction. The function which has been here called &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; may be named the force-function of a system: it is of great utility in theoretical mechanics, into which it was introduced by Lagrange, and it furnishes the following elegant forms for the differential equations of motion... the second members of these equations being the partial differential coefficients of the first order of the function &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt;&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The only word in this passage that suggests a reason for denoting the &amp;quot;force-function&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; is &amp;quot;utility&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-12T22:32:06.450" LastActivityDate="2020-08-12T22:32:06.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12122" PostTypeId="2" ParentId="12117" CreationDate="2020-08-12T06:49:37.973" Score="4" Body="&lt;p&gt;The quoted passage is a bit exaggerated, but the gist is right, at least for a segment of scientists. The main driving force was anti-theism, the Big Bang cosmology looked uncomfortably close to the Biblical creation, and Lemaître's vocation as a Catholic priest did not help, see e.g. &lt;a href=&quot;https://books.google.com/books/about/Losing_the_Nobel_Prize_A_Story_of_Cosmol.html?id=lu1CDwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keating's book Losing the Nobel Prize&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Lemaître's model... upset the millennia-old orthodoxy of an eternal, unchanging cosmos. It clearly implied that everything had been smaller and denser in the past, and that the universe must itself have had a birth at a finite time in the past... Many atheist scientists were repulsed by the Big Bang's creationist overtones. According to Hoyle, it was cosmic chutzpah of the worst kind: &amp;quot;The reason why scientists like the 'big bang' is because they are overshadowed by the Book of Genesis.&amp;quot; In contrast, the Steady State model was the rightful heir to the Copernican principle. It combined the banality of space with humanity's mediocrity in time. Thanks to Hoyle, humanity had humility.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The steady state model of Bondi-Gold-Hoyle (1948), with its perpetual creation of new matter, &amp;quot;&lt;em&gt;sure as hell didn't look like the creation narrative in Genesis 1:1&lt;/em&gt;&amp;quot;. And of the three it was Hoyle who brought the narrative to the masses, jokingly nicknaming Lemaître's primeval atom &amp;quot;Big Bang&amp;quot;, and mocking it on the radio, see &lt;a href=&quot;https://www.nytimes.com/2001/08/22/world/fred-hoyle-dies-at-86-opposed-big-bang-but-named-it.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;NYT obituaries&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In a series of popular radio talks in Britain in the 1940's, he coined ''big bang'' to ridicule the rival concept of an explosive origin of the universe, but the term is now widely used and the explosion theory is generally accepted. In recent years Sir Fred joined those arguing for a universe that -- while eternal -- expands and contracts.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, pop-sci politics was not the sole source of Hoyle's success, he really was a superb astronomer and cosmologist. The publication in 1957 (with Fowler and Burbidges) of a realisitic scenario of how the heavy elements  formed in star's interiors under the steady state model, which also explained the prevalence of light elements, contributed to the popularity of the model aside from its pop-sci fame. In 1958 Hoyle was appointed to the Plumian professor of astronomy chair, previously held by Eddington, among others. There is more discussion and references in &lt;a href=&quot;https://plato.stanford.edu/entries/cosmology-theology/#3&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, Cosmology and Theology&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Hoyle referred to big-bang cosmology as “a form of religious fundamentalism” (Hoyle 1994, 413)... It was widely assumed in the 1950s that the steady-state universe was contrary to theism or at least made God superfluous as a creator of the cosmos. After all, how can God have created a universe which has existed in an infinity of time? According to the astronomer, science popularizer and non-believer Carl Sagan, “this is one conceivable finding of science that could disprove a Creator—because an infinitely old universe would never have been created” (1997, 265).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-12T06:55:57.833" LastActivityDate="2020-08-12T06:55:57.833" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12123" PostTypeId="1" CreationDate="2020-08-12T15:31:30.627" Score="1" ViewCount="88" Body="&lt;p&gt;The Ancient Civilizations of Earth are highly advanced than accredited for.&#xA;Their respect for numerology throughout various parts of Earth is interesting.&lt;/p&gt;&#xA;&lt;p&gt;It baffles me how they did it with such limited technology. Forcing humanity to rely on their intelligence.&lt;/p&gt;&#xA;&lt;h2&gt;Question&lt;/h2&gt;&#xA;&lt;p&gt;What are the most peculiar discoveries in Ancient Civilizations that baffle mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="9773" LastActivityDate="2020-08-12T15:31:30.627" Title="What are some archaeological discoveries from Ancient Civilizations that baffled mathematicians?" Tags="|mathematics|ancient|" AnswerCount="0" CommentCount="2" ClosedDate="2020-08-14T16:27:49.850" ContentLicense="CC BY-SA 4.0" />
  <row Id="12124" PostTypeId="1" AcceptedAnswerId="12125" CreationDate="2020-08-12T18:27:21.933" Score="4" ViewCount="188" Body="&lt;p&gt;A 1959 paper written by J. Hubbard called &amp;quot;Calculation of Partition Functions&amp;quot; and published in Physical Review Letters contains the following identity (Equation 2):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int\limits_{-\infty}^{\infty} \exp\{ - \pi x^2 - 2 \pi^{1/2} a x\} \,dx = \exp\left\{a^2\right\}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Which can also be written as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{\sqrt{\pi}} \int\limits_{-\infty}^{\infty} \exp\left\{2ax-x^2\right\} \,dx = \exp\left\{a^2\right\}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am curious if anyone knows what the exact origin of this identity is, e.g., when was it first published and by whom? Hubbard does not provide any reference in his paper.&lt;/p&gt;&#xA;" OwnerUserId="12916" LastEditorUserId="7205" LastEditDate="2022-09-28T13:36:16.640" LastActivityDate="2022-09-28T13:36:16.640" Title="Origin of identity: $\int\limits_{-\infty}^{\infty} \exp\{ - \pi x^2 - 2 \pi^{1/2} a x\} \,da = \exp\left\{a^2\right\}$" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12125" PostTypeId="2" ParentId="12124" CreationDate="2020-08-12T22:26:45.940" Score="11" Body="&lt;p&gt;This is a variation on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaussian_integral&quot; rel=&quot;noreferrer&quot;&gt;Gaussian integral&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\int\limits_{-\infty}^{\infty} \exp\left\{-x^2\right\} \,dx=\sqrt{\pi}$&lt;/span&gt;, a.k.a. Poisson or Euler-Poisson integral, to which it reduces by completing the square in the exponent:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\int\limits_{-\infty}^{\infty} \exp\left\{2ax-x^2\right\}\exp\left\{-a^2\right\} \,dx\\ = \int\limits_{-\infty}^{\infty} \exp\left\{-(x-a)^2\right\}\,dx=\sqrt{\pi}.&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;De Moivre discovered it in 1733, Gauss published a proof in 1809, Poisson gave a neat trick for evaluating it using polar coordinates, popularized in &lt;a href=&quot;https://archive.org/details/ecolepolytechni02sturrich/page/n35/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Sturm's &lt;em&gt;Cours d’Analyse de l’ecole polytechnique&lt;/em&gt; (1877)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But the first proof appears to be due to Laplace in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k77596b/f32&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mémoire sur la probabilité des causes par les évènements&lt;/em&gt; (1774)&lt;/a&gt;. See &lt;a href=&quot;https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf&quot; rel=&quot;noreferrer&quot;&gt;The probability integral&lt;/a&gt; for a list of other tricks by various authors with citations to the original sources.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-12T22:40:41.780" LastActivityDate="2020-08-12T22:40:41.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12126" PostTypeId="1" AcceptedAnswerId="12128" CreationDate="2020-08-13T11:36:00.733" Score="3" ViewCount="1254" Body="&lt;p&gt;Etymology dictionaries mention the word science coming from the latin word scientia from the XII century, but they don't reference any written piece where it was recorded.&lt;/p&gt;&#xA;&lt;p&gt;What's the first recorded use of the word scientia?&lt;/p&gt;&#xA;" OwnerUserId="4512" LastEditorUserId="55" LastEditDate="2020-08-13T20:07:03.477" LastActivityDate="2020-08-13T20:07:03.477" Title="What is the first recorded use of the word &quot;scientia&quot;?" Tags="|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12127" PostTypeId="2" ParentId="26" CreationDate="2020-08-13T18:12:17.263" Score="1" Body="&lt;p&gt;The best example I can think of is steady-state versus big bang cosmology.&lt;/p&gt;&#xA;&lt;p&gt;In steady-state models the universe is homogeneous in space and in time. In big bang models it's homogeneous in space but not in time. Big bang models have far more parameters than steady-state models because there are so many things that plausibly could have been different in earlier eras.&lt;/p&gt;&#xA;&lt;p&gt;Models of both types were taken seriously until the early 1990s when &lt;a href=&quot;https://en.wikipedia.org/wiki/Cosmic_Background_Explorer&quot; rel=&quot;nofollow noreferrer&quot;&gt;COBE&lt;/a&gt; found anisotropies in the cosmic microwave background. ΛCDM, a big-bang model, has enough parameters to fit the &lt;a href=&quot;http://www.astro.ucla.edu/%7Ewright/CMB-LCDM-5yWMAP-03Mar08-125.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;CMB power spectrum&lt;/a&gt; (which, it must be noted, is vaguely &lt;a href=&quot;https://www.johndcook.com/blog/2011/06/21/how-to-fit-an-elephant/&quot; rel=&quot;nofollow noreferrer&quot;&gt;elephant&lt;/a&gt; shaped). Steady-state models can't reproduce it, so they're wrong.&lt;/p&gt;&#xA;" OwnerUserId="9390" LastActivityDate="2020-08-13T18:12:17.263" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12128" PostTypeId="2" ParentId="12126" CreationDate="2020-08-13T20:03:36.763" Score="3" Body="&lt;p&gt;&lt;em&gt;Scientia&lt;/em&gt; is the Latin translation of Greek &lt;a href=&quot;https://en.wikipedia.org/wiki/Episteme&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;épistémè&lt;/em&gt;&lt;/a&gt;, it goes back to Plato and Aristotle. And it did not mean &amp;quot;science&amp;quot; in the modern sense. Theology was &lt;em&gt;scientia&lt;/em&gt; too, as in a systematic body of provable knowledge, see &lt;a href=&quot;https://link.springer.com/referenceworkentry/10.1007%2F978-3-319-02848-4_266-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Scientia&lt;/em&gt; by Demeter-Lang-Schmal&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The ambiguity and heterogeneity of the modern term “science” is undeniable. Some put more emphasis on the instrumentality of potential technological output, some on the nature-controlling aspect, some on the universal laws, some on experimental foundations and empirical methodology, others on the institutional settings involving laboratories, funding schemes, professionalization, and academies. None of these approaches is helpful in understanding what scientia meant in the period between 1350 and 1650. Scientia in the first part of this period referred to any body of certain theoretical knowledge which approached its subject (not necessarily nature) in a systematic way and relied on apodictic demonstration. Thus scientia, unlike today, involved theology as well, because it was demonstrable knowledge.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The term “scientia” is a long-standing heritage of Aristotelian logic and denotes an epistemic ideal pursued through several centuries. According to Aristotle’s Posterior Analytics, knowledge that conforms to this ideal must consist of propositions that are universally and necessarily true. This necessity can be demonstrated through syllogistic inferences that proceed from premises containing the cause of the conclusion. This procedure leads to certain knowledge and understanding of the necessary cause of the phenomena and an explanation of why it is the way it is and cannot be otherwise.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.academia.edu/1668906/Aquinas_Scientia_and_a_Medieval_Misconstruction_of_Aristotles_Posterior_Analytics_Revised_Greek_font_fixed_&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hall's Aquinas, Scientia and a Medieval Misconstruction of Aristotle’s Posterior Analytics&lt;/a&gt; gives references to the 12th century Latin translations of Aristotle's Posterior Analytics where &amp;quot;&lt;em&gt;scientia&lt;/em&gt;&amp;quot; appears:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first and most widely read translation was produced sometime in the second quarter of the twelfth century by James of Venice (Iacobus Veneticus), of whom little is known. The second translation came out some time before 1159, when it is cited in John of Salisbury’s Metalogicon, and is likely a recensio of James’s. Less still is known of its translator, whose name may have been John (Ioannes). Finally, there is the translation of William of Moerbeke (Guillelmum de Moerbeka), which was produced around 1269, and adopted by Aquinas around 1271.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-08-13T20:03:36.763" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12129" PostTypeId="1" CreationDate="2020-08-13T20:40:50.447" Score="1" ViewCount="115" Body="&lt;p&gt;Who and when began the writing of &lt;a href=&quot;https://en.wikipedia.org/wiki/Pointwise_product&quot; rel=&quot;nofollow noreferrer&quot;&gt;function multiplication&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)×g(x)$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$(f×g)(x)$&lt;/span&gt; and of &lt;a href=&quot;https://en.wikipedia.org/wiki/Function_composition&quot; rel=&quot;nofollow noreferrer&quot;&gt;function composition&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$f\big(g(x)\big)$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$(f∘g)(x)$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="12923" LastEditorUserId="12923" LastEditDate="2020-08-15T21:23:00.130" LastActivityDate="2020-08-15T21:23:00.130" Title="Origin of (f×g)(x) and (f∘g)(x) notations" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12130" PostTypeId="1" AcceptedAnswerId="12131" CreationDate="2020-08-13T22:12:45.310" Score="17" ViewCount="4366" Body="&lt;p&gt;Some years ago, I read that Kolmogorov was so enraged that Karatsuba disproved one of his conjectures that he terminated his seminar shortly thereafter.&lt;/p&gt;&#xA;&lt;p&gt;This &lt;a href=&quot;https://en.wikipedia.org/wiki/Karatsuba_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page&lt;/a&gt; claims that Kolmogorov was just &amp;quot;very excited&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Do we have any testimony of what really happened?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2021-01-31T17:59:55.330" LastActivityDate="2021-01-31T17:59:55.330" Title="Was Kolmogorov enraged after learning about the Karatsuba multiplication algorithm?" Tags="|mathematicians|biographical-details|abstract-algebra|algorithm|kolmogorov|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12131" PostTypeId="2" ParentId="12130" CreationDate="2020-08-13T23:45:08.923" Score="26" Body="&lt;p&gt;Karatsuba's own report can be found in his 1995 paper &lt;a href=&quot;http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;amp;jrnid=tm&amp;amp;paperid=1120&amp;amp;option_lang=rus&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Сложность вычислений&lt;/em&gt; (Complexity of computations)&lt;/a&gt;. The phrasing he uses is &amp;quot;сильно взволновалo&amp;quot;, which Google does translate as &amp;quot;very excited&amp;quot; or &amp;quot;greatly agitated&amp;quot;. But we can see the usual dynamic of an anecdote in the making on the &lt;a href=&quot;https://www.reddit.com/r/programming/comments/mhbjy/kolmogorov_was_very_upset_about_the_discovery/&quot; rel=&quot;noreferrer&quot;&gt;Reddit thread&lt;/a&gt;, where Kolmogorov is &amp;quot;very upset&amp;quot;, and writing on the &lt;a href=&quot;https://i.stack.imgur.com/YWK39.jpg&quot; rel=&quot;noreferrer&quot;&gt;image&lt;/a&gt; linked by one of the users already has him &amp;quot;angry&amp;quot;. In any case, the difference between &amp;quot;excited&amp;quot;, &amp;quot;upset&amp;quot; and &amp;quot;enraged&amp;quot; is a matter of opinion even for those observing first hand.&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov did do something unusual afterwards: abruptly ended the seminar and two years later published a paper with the result under Karatsuba's and Ofman's names (Kolmogorov's student) without letting Karatsuba know, see &lt;a href=&quot;https://cstheory.stackexchange.com/q/21564&quot;&gt;Why did Kolmogorov publish Karatsuba's algorithm? on CS SE&lt;/a&gt;.  The joint authorship led to a confusion about the credit for Karatsuba's algorithm, although it was clearly credited to him in the paper. As many commenters noted, Kolmogorov's was the opposite of the usual misbehavior where people take somebody else's result and publish it under &lt;em&gt;their own&lt;/em&gt; name.&lt;/p&gt;&#xA;&lt;p&gt;Why Kolmogorov did what he did one can only speculate. Perhaps, he was embarassed that his conjecture of four years was disproved so easily, and not by him. Perhaps he wanted to bundle Karatsuba's result with Ofman's with which he was more involved. Or the discovery made his plans for the seminar obsolete. Or perhaps he felt he misjudged the nature of the problems to be discussed: they were not safely publicizable long shots but doable problems that &amp;quot;&lt;em&gt;like parables, are only explained to the own disciples in private&lt;/em&gt;&amp;quot; (as &lt;a href=&quot;http://mat.net.ua/mat/Arnold-Geom-Metod-Difuri.htm&quot; rel=&quot;noreferrer&quot;&gt;Arnold put it&lt;/a&gt;, alluding to the gospel of Mark).&lt;/p&gt;&#xA;&lt;p&gt;Here is Karatsuba's own telling in edited Google translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the fall of 1960 the Department of Mechanics and Mathematics of Moscow University opened a seminar on mathematical issues of cybernetics began under the leadership of A.N. Kolmogorov, where A.N. Kolmogorov formulated the &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; conjecture and a number of problems were posed on the estimation of the complexity of solutions to linear systems of&#xA;equations and other similar calculations. I actively began to reflect on the &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; conjecture and exactly one week later I discovered that the algorithm with which I was hoping to get a lower bound for &lt;span class=&quot;math-container&quot;&gt;$M(n)$&lt;/span&gt; gives an estimate of the form...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;After a regular session of the seminar I told A.N. Kolmogorov about the new multiplication algorithm and refutation of the &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; conjecture. &lt;strong&gt;This greatly agitated A.N. Kolmogorov, since it contradicted his rather plausible conjecture.&lt;/strong&gt; On the next session of the seminar my method of multiplication was reported by A.N. Kolmogorov,&#xA;and at this the seminar stopped its work. Later, in 1962, A.N. Kolmogorov wrote (perhaps with the participation of Yu.P. Ofman) a short article and published it in the Reports of the Academy of Sciences of the USSR. The article was titled as follows: A. Karatsuba, Yu. Ofman,&#xA;Multiplication of Multidigit Numbers by Automata (Dokl. AN SSSR. 1962. V. 145, No. 2. P. 293-294). I learned about this article only when I was given its reprints. The unusual way of publication is also emphasized by the fact that both articles [5] and [8] were presented by A.N. Kolmogorov for publication simultaneously on November 13, 1962.&lt;/em&gt;&amp;quot; [emphasis added]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[5] is the Karatsuba-Ofman article, and [8] is an article &lt;em&gt;Об алгоритмической сложности дискретных функций&lt;/em&gt; (On algorithmic complexity of discrete functions) by Ofman alone. Here is the Russian original:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Осенью 1960 г. в Московском университете на механико-математическом факультете&#xA;начал работать семинар по математическим вопросам кибернетики под&#xA;руководством А.Н. Колмогорова, где А.Н. Колмогоровым была сформулирована гипотеза&#xA;&lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; и поставлен ряд задач об оценке сложности решений линейных систем&#xA;уравнений и других сходных вычислений. Я активно стал размышлять над гипотезой&#xA;&lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; и ровно через неделю обнаружил, что алгоритм, которым я надеялся получить&#xA;нижнюю оценку величины &lt;span class=&quot;math-container&quot;&gt;$M(n)$&lt;/span&gt;, дает оценку вида...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;После очередного заседания семинара я сообщил А.Н. Колмогорову о новом алгоритме&#xA;умножения и об опровержении гипотезы &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt;. &lt;strong&gt;Это сильно взволновало А.Н. Колмогорова,&#xA;так как противоречило его довольно правдоподобной гипотезе.&lt;/strong&gt; На следующем&#xA;заседании семинара мой метод умножения был рассказан самим А.Н. Колмогоровым,&#xA;и на этом семинар прекратил свою работу. Позднее, в 1962 г., А.Н. Колмогоров&#xA;написал (может быть, при участии Ю.П. Офмана) небольшую статью и опубликовал&#xA;ее в Докладах АН СССР. Статья называлась так: А. Карацуба, Ю. Офман,&#xA;Умножение многозначных чисел на автоматах (Докл. АН СССР. 1962. Т. 145, № 2.&#xA;С. 293-294). Об этой статье я узнал только тогда, когда мне были даны ее оттиски.&#xA;Необычность способа публикации подчеркивается и тем, что обе статьи [5] и [8]&#xA;представлены А.Н. Колмогоровым к опубликованию одновременно 13.11 1962 г.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-14T06:48:35.740" LastActivityDate="2020-08-14T06:48:35.740" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12135" PostTypeId="1" AcceptedAnswerId="12182" CreationDate="2020-08-14T20:12:14.493" Score="5" ViewCount="376" Body="&lt;p&gt;Please explain why the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dirac_delta_function&quot; rel=&quot;noreferrer&quot;&gt;Dirac delta&lt;/a&gt; function was named after Dirac, who lived in the 20th century, and what was so special about it.&lt;/p&gt;&#xA;&lt;p&gt;I ask this because it is used in &lt;a href=&quot;https://en.wikipedia.org/wiki/Green%27s_function&quot; rel=&quot;noreferrer&quot;&gt;Green's function&lt;/a&gt; which pre-dated Dirac by 200 years.  I find Green's function more insightful and so wonder why it is not called &lt;em&gt;Green's delta&lt;/em&gt;.  Fourier and Cauchy are also  cited on the Wikipedia page for the Dirac delta as having used the impulse concept.&lt;/p&gt;&#xA;&lt;p&gt;I am therefore hoping to find out what Dirac's special contribution was since the maths was already two centuries old.  I don't see this contribution on the Wikipedia page even though its implicitly referred to as Dirac's invention.&lt;/p&gt;&#xA;" OwnerUserId="6396" LastEditorUserId="4251" LastEditDate="2020-08-14T21:11:57.550" LastActivityDate="2020-08-24T14:47:27.483" Title="Why is Dirac delta named after Dirac when the concept was already over two centuries old?" Tags="|mathematics|mathematics-social-history|dirac|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12137" PostTypeId="1" AcceptedAnswerId="12139" CreationDate="2020-08-16T11:59:08.870" Score="7" ViewCount="240" Body="&lt;p&gt;I have found this in David Wells' &amp;quot;Curious and Interesting Puzzles&amp;quot; (1992), Penguin books, page 36:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The next problem occurs first in Urbino d'Aviso's treatise on the sphere (1682):&lt;/p&gt;&#xA;&lt;ol start=&quot;126&quot;&gt;&#xA;&lt;li&gt;A strip of paper can be transformed into a pentagon. How?&amp;quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Yes, I know what the answer is, you tie it into a trefoil knot and press it flat, the question is a chestnut. This is not what I am asking.&lt;/p&gt;&#xA;&lt;p&gt;What I am asking is: who was Urbino d'Aviso? I can find no reference to such a person anywhere on the internet.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="55" LastEditDate="2020-08-16T13:34:20.927" LastActivityDate="2020-08-17T09:29:57.610" Title="Who was &quot;Urbino d'Aviso&quot; that described how to fold paper into a pentagon?" Tags="|mathematicians|17th-century|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12138" PostTypeId="1" CreationDate="2020-08-16T12:07:18.047" Score="5" ViewCount="90" Body="&lt;p&gt;In 1849, &lt;a href=&quot;https://www.economist.com/books-and-arts/2020/08/01/how-hand-washing-explains-economic-expansion&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Economist&lt;/em&gt;&lt;/a&gt; declared that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The belief in contagion, like the belief in astrology and witchcraft, seems destined to die out; and as we have got rid of all regulations for consulting the starts or attending to omens before we begin any undertakings, and of all the laws against feeding evil spirits and punishing witches, so we shall no doubt in time get rid of the quarantine regulations that were established from the old belief in contagion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How widespread in England in the 19&lt;sup&gt;th&lt;/sup&gt; century was the idea that contagion is just an “old belief”? Is anyone aware of more texts written around the same time in a similar spirit?&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-08-16T12:07:18.047" Title="Contagion, astrology and witchcraft" Tags="|medicine|19th-century|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12139" PostTypeId="2" ParentId="12137" CreationDate="2020-08-16T13:33:13.583" Score="6" Body="&lt;p&gt;Wells misspelled him a little. Urbano d'Aviso (1618-1685) was a little known student of Cavalieri. Aside from the pentagon folding and disputed authorship of &lt;em&gt;Trattato della Sfera&lt;/em&gt;, originally published in 1656 under a pseudonym, he is sometimes mentioned as a Catholic cleric who suggested that water vapor consisted of miniscule bubbles of water filled with fire in 1666 (e.g. by Middleton in &lt;a href=&quot;https://books.google.com/books/about/A_history_of_the_theories_of_rain_and_ot.html?id=_jNRAAAAMAAJ&quot; rel=&quot;noreferrer&quot;&gt;A History of the Theories of Rain&lt;/a&gt;), an idea later explored by Halley.&lt;/p&gt;&#xA;&lt;p&gt;In 1883 he was brought out of obscurity by Lucas's reference to &lt;em&gt;Trattato della Sfera&lt;/em&gt; in connection with this very pentagon construction. Ironically, the &lt;em&gt;Trattato&lt;/em&gt; was basically an edited edition of Galileo's lectures, and that construction was out of place there. Sharp gives a facsimile of the relevant page with translation in &lt;a href=&quot;https://www.tandfonline.com/doi/10.1080/17498430.2016.1162402&quot; rel=&quot;noreferrer&quot;&gt;Folding the regular pentagon&lt;/a&gt;. Friedman's book &lt;a href=&quot;https://books.google.com/books?id=HHJdDwAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;noreferrer&quot;&gt;A History of Folding in Mathematics, p.307ff&lt;/a&gt; gives more details:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;However, Lucas’s knotted folded pentagon was gaining more popularity than his&#xA;folded stamps. Already in 1883, Ferdinand Jacoli published a paper reviewing the&#xA;works of Urbano d’Aviso (1618–1685), called: “Intorno al problema “Le Noeud de&#xA;cravate” e ad alcune opere di Urbano d’Aviso Romano.” This paper mentions and cites completely Lucas’s work on the knotted pentagon in Récréations&#xA;mathématiques, and then surveys d’Aviso’s work. As Jacoli writes, two hundred&#xA;years before Lucas’s book, d’Aviso, in the 1682 version of the book Trattato della&#xA;Sfera, already described how to form a regular pentagon (and a regular hexagon)&#xA;using the knotting technique.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In fact, Jacoli’s paper is not about d’Aviso’s method&#xA;of knotting a pentagon, but about claiming that the additions to the 1656 original&#xA;version of the book Trattato della Sfera (which afterwards had two further editions,&#xA;in 1682 and in 1690, where, in these two editions, a part called “Prattiche&#xA;Astronomiche. Intorno alli circoli della Sfera” was added by d’Aviso) were&#xA;written by Urbano d’Aviso and not by his teacher Bonaventura Cavalieri&#xA;(1598–1647). The book Trattato della Sfera, in fact, comments on and explains&#xA;works and lectures by Galileo Galilei, and, as John Sharp comments, it is not known&#xA;why d’Aviso decided to insert the constructions of polygons on the last page of&#xA;Prattiche Astronomiche, as the construction of a regular pentagon is not needed or&#xA;mentioned in the previous pages.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;On the question of the authorship of this manuscript, see: Cioffarelli (1987). Cioffarelli ends his&#xA;paper with no definite conclusion regarding which additions were written by whom. However, it is&#xA;clear that d’Aviso wrote the part regarding the construction of the pentagon (ibid., p. 33), although&#xA;the concluding paragraph of Cioffarelli’s paper presents d’Aviso as an ambiguous character:&#xA;“Urban Daviso reveals himself as an ambiguous figure for these faulty omissions in the publication&#xA;of Cavalieri’s Tractatus, as well as for the awkward attempt to attribute to himself the edited version&#xA;of the work.”&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-08-16T13:33:13.583" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12140" PostTypeId="2" ParentId="12137" CreationDate="2020-08-16T14:31:50.290" Score="6" Body="&lt;p&gt;If you can read Italian, &lt;a href=&quot;http://www.treccani.it/enciclopedia/urbano-giovan-francesco-davisi_(Dizionario-Biografico)/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; you can find full biographic details of his life. What follows is a short summary.&lt;/p&gt;&#xA;&lt;p&gt;As was customary in Italy (and not only in Italy) until at least the 18th century, his name was written in many different ways, like Urbano (Giovan Francesco) Davisi, or D'Aviso, Avvisi, De Aviso, De Avisus and in the Latinized form Avisius, and also under the anagram &amp;quot;Buonardo Savi&amp;quot; used for Galileo's &lt;em&gt;Trattato della sfera&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;He was born in Rome on 25 May 1618, the fifth son of Giovanni Andrea, know as  &amp;quot;berrettaro&amp;quot; (beret maker).&#xA;He entered the &lt;a href=&quot;https://en.wikipedia.org/wiki/Jesuati&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jesuati Order&lt;/a&gt; (not the Jesuits) in 1636, where he studied philosophy and theology. From Rome he moved to Bologna to study mathematics under Cavalieri (also a Jesuat).&#xA;In 1650 he took care of the reprinting of Cavalieri's &lt;em&gt;Specchio Ustorio&lt;/em&gt;, &amp;quot;to satisfy the intentions&amp;quot; of the master (died three years before, in 1647).&#xA;He returned to Rome in 1650 and in 1656 became General Procurator and Prior of the convent of Saints John and Paul. In 1656, under the name of &amp;quot;Buonardo Savi&amp;quot;, he printed the Galileo's manuscript of the &lt;em&gt;Trattato della Sfera&lt;/em&gt; together with some astronomical teachings &amp;quot;taught by Cavalieri to his pupils&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The two letters (&lt;em&gt;Due lettere scritti dal rev. padre fra Urbano Davisi...&lt;/em&gt;, Bologna 1667) can be considered a short treatise on meteorology. In the first, to give reasons for various meteorological effects (rain, snow, ...), he places the element of fire at the center of the earth and not in the concave of the Moon as Aristotle wanted. The subject of the letter to &lt;a href=&quot;https://en.wikipedia.org/wiki/Geminiano_Montanari&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geminiano Montanari&lt;/a&gt; is another classic question of meteorology: the origin of the sources and rivers.&lt;/p&gt;&#xA;&lt;p&gt;He was interested in applied hydraulics and, among other things, in the navigation of the Tiber, as evidenced by the title of one of his manuscripts (&lt;em&gt;Tracratus de Tyberis navigatione ...&lt;/em&gt;), now lost.&lt;/p&gt;&#xA;&lt;p&gt;In 1682, another &lt;em&gt;Trattato della Sfera&lt;/em&gt; (different the Galileo's one) came out. Much has been discussed, but without reaching a definitive conclusion, on the authorship of this work, since in 1690, after its dead, it was reprinted under the modified title &lt;em&gt;Sfera astronomica&lt;/em&gt;, which indicated as author Cavalieri and relegated Davisi to the role of simple publisher.&lt;/p&gt;&#xA;&lt;p&gt;In reality, the treatise is largely by Cavalieri: divided into two parts - one setting out the doctrine of the sphere, the second made up of astronomical practices (many of which Davisi had already published in 1656) - and preceded by a life of Cavalieri, it is a partial translation, with additions and changes to &lt;em&gt;Sphaera seu doctrinae sphaericae tractatus... authore F. Bonaventura Cavalerio&lt;/em&gt;, dated 1642 and preserved in the Library of the University of Bologna. It is still unclear why Davisi, always ready to acknowledge his debts to the master (I think this is why Cioffarelli speaks about &amp;quot;the awkward attempt to attribute to himself the edited version of the work&amp;quot;), did not do so in this case as well. Nor it is possible to establish whether the &amp;quot;reprint&amp;quot; of the 1690 is the result of an initiative of the bookseller or it represents the fulfilment of a will of Davisi.&lt;/p&gt;&#xA;&lt;p&gt;Davisi died after a long and painful infirmity on 17th September 1686; he was buried in the church of &lt;a href=&quot;https://en.wikipedia.org/wiki/San_Giovanni_della_Malva_in_Trastevere&quot; rel=&quot;nofollow noreferrer&quot;&gt;S. Giovanni della Malva&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Here the two &lt;strong&gt;different&lt;/strong&gt; &lt;em&gt;Trattato della Sfera&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://repository.ou.edu/uuid/6053b820-d595-5605-8711-fbe73547dbbd#page/1/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trattato della sfera di Galileo Galilei&lt;/a&gt;, Roma 1656. The authors are Galileo and &amp;quot;Buonardo Savi&amp;quot;: the first part is taken from lessons given by Galileo, the second part (&amp;quot;Prattiche Astronomiche&amp;quot;) are based on the astronomical teachings of Cavalieri with Davisi's additions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/sferaastronomic00avisgoog/page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sfera Astronomica del P. Bonaventura Cavalieri&lt;/a&gt;, Roma 1690. The work is preceded by a life of Cavalieri, the first of its kind and the principal document for the mathematician's biography, definitely due to Davisi.&#xA;The construction of the pentagon and of the hexagon at &lt;a href=&quot;https://archive.org/details/sferaastronomic00avisgoog/page/n291/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 255&lt;/a&gt; is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Con l’occasione di questo disegnare le figure, ti voglio dare il modo di descrivere, e formare mechanicamente un Pentagono, che è una delle più difficili figure da disegnare, e pure è la più facile, che si facci in natura, perché non è altro, che un semplice nodo. Prenderai per tanto una striscia di carta della larghezza, che tu vorrai, e che habbi li lati paralleli, e con quella procura di fare un nodo, come se fosse una corda, auertendo però che la carta resti sempre stesa nelle piegature, che stringendola tanto che resti ben tirata, se taglierai con le forbici li capi che auanzano, hauerai fatto un Pentagono giustissimo.&lt;/p&gt;&#xA;&lt;p&gt;Farai anco la figura Esagona se prenderai due striscie di carta di eguale lunghezza, e con li lati paralleli, e procurerai di fare con esse un nodo, facendo che le punte dell’incuruatura, che hauerai fatta di una striscia, passino per l’aperto dell’incuruatura dell’altra, che stringendole adattamente, e che mantenghino sempre la loro larghezza, tagliando l’auanzi delle punte, hauerai fatto un Esagono perfettissimo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As we are drawing some figures, I want to give you the way to describe, and mechanically form, a Pentagon, which is one of the most difficult figures to draw, and also the easiest that there is in nature, because it is nothing more than a simple knot. You will therefore take a strip of paper of the width that you want, and that has parallel sides, and with it you will make a knot, as if it were a rope, but being careful that the paper remains always stretched during the folding, and tightening it so that it remains well pulled, then, if you cut with scissors the ends that advance, you will have made a perfect Pentagon.&lt;/p&gt;&#xA;&lt;p&gt;You will also make a figure in the shape of a hexagon if you take two strips of paper of equal length, and with the sides parallel, and make a knot with them, making the tips of the curvature of one strip, pass through the opening of the curvature of the other, so that by tightening them properly, always keeping their width and cutting the leftovers of the tips, you will have made a perfect Hexagon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-08-17T09:29:57.610" LastActivityDate="2020-08-17T09:29:57.610" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12141" PostTypeId="2" ParentId="5453" CreationDate="2020-08-16T15:34:27.770" Score="1" Body="&lt;p&gt;In there philosophy of chance and probability theory,&#xA;there is a type of instability which does not depends to initial conditions and trajectories, called structural instability which referes to changes in dynamical models, laws and equations. There is a Mathematical définition of structural instability by Pontryagin and Andronov.&#xA;See the Book of Philosophy of Complex Systems,&lt;/p&gt;&#xA;&lt;p&gt;Static instability which is based on initial conditions.&lt;/p&gt;&#xA;&lt;p&gt;Poincaré said chance events are such as arise from the action of causes in which small differences produce great differences in effect, it follows immediately that chance effects must be random&lt;/p&gt;&#xA;&lt;p&gt;For instability  he gives an example in his paper about chance,1908&lt;/p&gt;&#xA;&lt;p&gt;Great differences in the cause and small differences in the effect Flammarion once imagined an observer moving away from the earth at a velocity greater than that of light. For him time would have its sign changed, history would be reversed, and Waterloo would come before Austerlitz. Well, for this observer effects and causes would be inverted, unstable equilibrium would no longer be the exception; on account of the universal irreversibility, everything would seem to him to come out of a kind of chaos in unstable equilibrium, and the whole of nature would appear to him to be given up to chance.&lt;/p&gt;&#xA;" OwnerDisplayName="user12766" LastActivityDate="2020-08-16T15:34:27.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12142" PostTypeId="1" CreationDate="2020-08-16T18:04:00.427" Score="2" ViewCount="77" Body="&lt;p&gt;I would like to know if there is any data on this topics out of curiosity. I know that telling everyone which textbooks they used isn't a common habit, but I am certain there are textbooks that some of the greatest (most famous, most revolutionary and/or most important in the development of physics and/or mathematics) physicists and mathematicians of the history recommended.&lt;/p&gt;&#xA;&lt;p&gt;Is there any article or something like that on this topic?&lt;/p&gt;&#xA;&lt;p&gt;Feel free to mention any book by the greatest physicists and mathematicians of history that contains a recommended bibliography or references to textbooks too.&lt;/p&gt;&#xA;" OwnerUserId="12280" LastActivityDate="2020-08-16T18:04:00.427" Title="Which books did famous physicists and mathematicians use to study in university?" Tags="|mathematicians|physicists|" AnswerCount="0" CommentCount="6" ClosedDate="2020-09-19T21:56:28.390" ContentLicense="CC BY-SA 4.0" />
  <row Id="12143" PostTypeId="1" AcceptedAnswerId="12155" CreationDate="2020-08-17T08:25:45.660" Score="3" ViewCount="582" Body="&lt;p&gt;I have been analysing the rise and fall of the various centres of mathematical excellence, and following how, for example, Ancient Greece gave way to the Arab and Persian world, which inherited from the Indian world, in turn giving rise to the Italian renaissance on the collapse of the Byzantine Empire, and so on.&lt;/p&gt;&#xA;&lt;p&gt;Most nations feature a significant number of mathematicians in their history, but there are some significant exceptions: most notably, to my mind, being Spain and Portugal. While there are plenty of contemporary active mathematicians, there are scarce few from the 19th century and before.&lt;/p&gt;&#xA;&lt;p&gt;What are the historical and sociological factors that caused Spain and Portugal to miss out on the drive to increase mathematical knowledge.&lt;/p&gt;&#xA;&lt;p&gt;The same can be applied to the Balkans as well, which if anything is more puzzling -- Spain is to a certain extent isolated from the rest of Europe by the Pyrenees, but this does not apply to the Balkans, which I would have thought was on the main trade route between Europe and the Arab world, but for the sea route to Italy.&lt;/p&gt;&#xA;&lt;p&gt;Has there been any formal historical research investigating this interesting void?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="26" LastEditDate="2021-02-20T15:09:11.590" LastActivityDate="2021-02-20T15:09:11.590" Title="Why are there so few Iberian mathematicians in history?" Tags="|mathematics-social-history|geography|" AnswerCount="2" CommentCount="11" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12144" PostTypeId="1" CreationDate="2020-08-17T10:51:36.097" Score="1" ViewCount="45" Body="&lt;p&gt;In logic we are taught the following definition.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$A\to B$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;that yields false whenever &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is true &lt;strong&gt;and&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is false.&lt;/p&gt;&#xA;&lt;p&gt;Who defined this and why was it defined that way? What inspired him/her?&lt;/p&gt;&#xA;" OwnerUserId="2903" LastEditorUserId="2903" LastEditDate="2020-08-17T11:41:57.580" LastActivityDate="2020-08-17T11:41:57.580" Title="Who defined $A\to B$ that yields false whenever $A$ is true and $B$ is false and why was it defined that way?" Tags="|mathematical-logic|" AnswerCount="0" CommentCount="6" ClosedDate="2020-08-17T12:30:19.660" ContentLicense="CC BY-SA 4.0" />
  <row Id="12145" PostTypeId="2" ParentId="12143" CreationDate="2020-08-17T12:48:20.530" Score="11" Body="&lt;p&gt;Unlike North Africa and the Near East, the Iberian Peninsula missed out on the Hellenic influence that dominated &lt;strong&gt;classical mathematics&lt;/strong&gt;; hence the lack of an Iberian Euclid. Compared to the Greeks, the Romans produced very little mathematics.&lt;/p&gt;&#xA;&lt;p&gt;During the &lt;strong&gt;medieval period&lt;/strong&gt;, there were in fact a fair number of internationally significant Iberian mathematicians: Pedro Nunes from Portugal, Abraham bar Hiyya from Catalonia, Al-Qalaṣādī and Ibn Muʿādh from Al-Andalus, and many others.&lt;/p&gt;&#xA;&lt;p&gt;However, following the &lt;strong&gt;reconquista&lt;/strong&gt; (and colonisation of the New World), Spain and Portugal developed a martial, aristocratic culture rather than one of artisans and merchants like in France or Italy. The &lt;strong&gt;Counter-Reformation&lt;/strong&gt; resulted in the stagnation of universities, and though the region did undergo a period of Enlightenment in the 18th century, it remained scientifically backwards compared to the rest of Western Europe. In modern history, Spain and Portugal's long periods of dictatorship and economic stagnation didn't help either. There have been just 2 scientific Nobel Laureates from Spain and 1 from Portugal, compared to 13 from Italy, 39 from France and 92 from Germany.&lt;/p&gt;&#xA;" OwnerUserId="1912" LastEditorUserId="1912" LastEditDate="2020-08-18T15:29:15.317" LastActivityDate="2020-08-18T15:29:15.317" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12146" PostTypeId="1" CreationDate="2020-08-17T17:49:23.373" Score="8" ViewCount="474" Body="&lt;p&gt;I am wondering if anyone knows any more on the history of the term 'co-domain' as it relates to functions.&lt;/p&gt;&#xA;&lt;p&gt;Two sources I found:&lt;/p&gt;&#xA;&lt;p&gt;Russell and Whitehead, Principia Mathematica, 1915, page 34 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the class of all terms to which something or other has the relation &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is called the &lt;em&gt;converse domain&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;; it is the same as the domain of the converse of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cassius Keyser, Mathematical Philosophy, 1922, page 168:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A &lt;em&gt;relation&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; has what is called a &lt;em&gt;domain&lt;/em&gt;, - the class of all the terms such that each of them has the relation to something or other, - and also a &lt;em&gt;codomain&lt;/em&gt; - the class of all the terms such that, given any one of them, something has the relation to it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems to me that when Keyser talks about a 'codomain', he is talking about the same thing as Russell and Whitehead's 'converse domain'. So, it looks like we went from 'converse domain' to 'codomain' .... to 'co-domain'? That would seem to make sense.&lt;/p&gt;&#xA;&lt;p&gt;Also, both texts talk about relations, not functions. But, a function is of course a special kind of relation. So ... it still makes sense.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;However!&lt;/strong&gt; (and this is really why I am asking this question): the way these two texts talk about the 'converse domain' and 'codomain' is (when applied to functions) what we nowadays call the 'range' or 'image' of the function, and &lt;em&gt;not&lt;/em&gt; what we nowadays call its 'co-domain'.&lt;/p&gt;&#xA;&lt;p&gt;Concrete example:&lt;/p&gt;&#xA;&lt;p&gt;Take a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; whose domain is defined as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} - \{ 0 \}$&lt;/span&gt;, whose co-domain is defined as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, and whose mapping is defined as &lt;span class=&quot;math-container&quot;&gt;$f(x) =1/x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For this function, the range or image is &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} - \{ 0 \}$&lt;/span&gt;, and that is what (again, if we see this function as a relation) Russell &amp;amp; Whitehead would consider its 'converse domain' what Keyser would call its 'codomain'.&lt;/p&gt;&#xA;&lt;p&gt;But the 'co-domain' of this function was defined as &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R} - \{ 0 \}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So I think there has been a shift in the use of the term ... That is, it seems like we got:&lt;/p&gt;&#xA;&lt;p&gt;'converse domain' -&amp;gt; 'codomain' -&amp;gt; 'range'&lt;/p&gt;&#xA;&lt;p&gt;... while 'co-domain' is something different!&lt;/p&gt;&#xA;&lt;p&gt;This is weird! What happened? Does anyone have some insight into any of this?&lt;/p&gt;&#xA;" OwnerUserId="12942" LastEditorUserId="12942" LastEditDate="2020-08-18T02:04:06.437" LastActivityDate="2023-02-08T02:00:40.100" Title="What is the history on the term 'co-domain'?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="12147" PostTypeId="1" AcceptedAnswerId="12149" CreationDate="2020-08-17T18:25:10.923" Score="1" ViewCount="96" Body="&lt;p&gt;The &lt;em&gt;OED&lt;/em&gt; defines &lt;a href=&quot;https://oed.com/view/Entry/76395&quot; rel=&quot;nofollow noreferrer&quot;&gt;galvanism&lt;/a&gt; (coined ~1792) as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Electricity developed by chemical action&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and &lt;a href=&quot;https://oed.com/view/Entry/224500&quot; rel=&quot;nofollow noreferrer&quot;&gt;voltaic&lt;/a&gt; (coined ~1813) as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Used in producing electricity by chemical action after the method discovered by Volta&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Of electricity: Generated by chemical action.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are &amp;quot;galvanic&amp;quot; and &amp;quot;voltaic&amp;quot; synonyms?&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;cf. &lt;a href=&quot;https://books.google.com/ngrams/graph?content=voltaic%2C%20galvanic&amp;amp;year_start=1792&amp;amp;year_end=2019&amp;amp;corpus=26&amp;amp;smoothing=0&amp;amp;case_insensitive=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Ngram graph of &amp;quot;voltaic&amp;quot; vs. &amp;quot;galvanic&amp;quot;&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-08-18T18:41:59.110" LastActivityDate="2020-08-22T07:05:17.660" Title="Are &quot;galvanic&quot; and &quot;voltaic&quot; synonymous?" Tags="|physics|terminology|electricity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12148" PostTypeId="1" CreationDate="2020-08-17T23:34:19.777" Score="5" ViewCount="148" Body="&lt;p&gt;Often people talk of Leonardo da Vinci as an example of multi-disciplinary or inter-disciplinary successful approach towards the three fields - art, science and technology. This seems an uninformed habit because after all I think we drew lines between fields so just to plausibly build the human body of knowledge and we perhaps did so only recently.&lt;/p&gt;&#xA;&lt;p&gt;When were the three disciplines of art, science, and technology separated into distinct fields? Why is there a return these days to multidisciplinary approach even though surely the amount of knowledge increases if not explodes?&lt;/p&gt;&#xA;" OwnerUserId="10142" LastEditorUserId="26" LastEditDate="2021-02-20T15:09:02.233" LastActivityDate="2021-02-20T15:09:02.233" Title="When did the disciplinary distinction between art, science, and technology occur?" Tags="|technology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12149" PostTypeId="2" ParentId="12147" CreationDate="2020-08-18T08:56:11.273" Score="2" Body="&lt;p&gt;Not really, although they broadly referred to the same phenomena. Galvanism was associated with a particular view that ascribed special powers to living organisms in producing &amp;quot;animal electricity&amp;quot;. Some, although not Galvani himself, even thought that there was a special &amp;quot;galvanic fluid&amp;quot; separate from electric fluid, perhaps an embodiment of &lt;em&gt;élan vital&lt;/em&gt;. Volta attempted to dissociate &amp;quot;galvanic&amp;quot; from that context, and for a while &amp;quot;galvanic electricity&amp;quot; was treated as a special phenomenon whose nature was to be established.&lt;/p&gt;&#xA;&lt;p&gt;Oersted delivered lectures on Electricity, Galvanism, and Magnetism in 1819-20, Ampére named a measuring device &amp;quot;galvanometer&amp;quot; in 1823, and Ohm titled his seminal paper &lt;em&gt;Maassbestimmungen über die Galvanische Kette&lt;/em&gt; (Measurements of the Galvanic Circuit). After the acceptance of Ohm's theory the name persisted mostly in popular literature and &lt;em&gt;Naturphilosophie&lt;/em&gt; standing for a &amp;quot;force of nature&amp;quot;, like gravity but more animalistic. Mary Shelley's Frankenstein, Schelling, Hegel and Schopenhauer (&amp;quot;&lt;em&gt;galvanism overcomes chemical affinity... and so entirely suspends the laws of chemistry&lt;/em&gt;&amp;quot;), among others, mentioned Galvanism, see &lt;a href=&quot;https://philosophy.stackexchange.com/a/67668/9148&quot;&gt;How influential was “galvanism” on philosophy?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In contrast, there was no &amp;quot;Voltaism&amp;quot;, after Volta's demystification of &amp;quot;animal electricity&amp;quot; it was gradually absorbed into electrochemistry, with major contrinutions from Ampére and Ohm, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5150/55&quot;&gt;What is the history of electric current and resistance?&lt;/a&gt; &amp;quot;Voltaic&amp;quot; referred to electricity generated by voltaic piles and similar processes. A good idea of the use of both terms can be gotten from e.g.  &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich/page/66/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whittaker's classic History Of The Theories Of Aether And Electricity, ch. III&lt;/a&gt;, &lt;a href=&quot;https://www.jstor.org/stable/27757377&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Galvanism to Electrodynamics by Caneva&lt;/a&gt; and &lt;a href=&quot;https://www.jstor.org/stable/27757295&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Electric Current in Early Nineteenth-Century French Physic by Brown&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Although there were few, if any, relevant papers in the French journals in the 1780's, by the 1790's members of the Institut had started to consider the multiplying and perplexing phenomena of &amp;quot;le Galvanisme.&amp;quot; Reports on new aspects of &amp;quot;l'influence de l'application des metaux sur l'irritabilité et la sensibilité&amp;quot; were frequent in 1796 and 1797, and during 1798 a regularly functioning &amp;quot;Commission du Galvanisme&amp;quot; consistently reported new discoveries and experiments. By 1800, the variations and complexities of galvanic irritability were still pursued as popular and respectable research topics.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;But in 1800 the discussion of &amp;quot;animal electricity&amp;quot; entered a new phase in France as in the rest of Europe. Volta, a professor of physics who had been supporting a rival theory to Galvani's since 1792, announced his invention of a device, the &amp;quot;pile,&amp;quot; with which he claimed to overthrow once and for all the theory of the &amp;quot;animal electricians&amp;quot;... All that the animal electricians really see in their so-called &amp;quot;galvanic&amp;quot; phenomena, he argued, are the physiological effects of contact-induced and pile-multiplied &amp;quot;voltaic&amp;quot; current.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-22T07:05:17.660" LastActivityDate="2020-08-22T07:05:17.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12150" PostTypeId="1" CreationDate="2020-08-18T14:00:23.107" Score="5" ViewCount="212" Body="&lt;p&gt;I want to understand why, according to several sources, Gauss's discovery of Gaussian quadrature in his 1814 article was &amp;quot;the most significant event of the 19th century in the field of numerical integration and perhaps in all of numerical analysis&amp;quot; (this quotation is taken from the article &amp;quot;Construction and applications of Gaussian quadratures with nonclassical and exotic weight functions&amp;quot;).The level of my knowledge at numerical analysis is very low, and the most advanced method that i know is Newton-Cotes method for approximate integration by using an &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; degree interpolating polynomial (where &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is the number of sample points on the function).&lt;/p&gt;&#xA;&lt;p&gt;According to several sources, Gauss proved that for an unknown function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;, it's better (in the sense of the size of remainder) to integrate the function in the interval &lt;span class=&quot;math-container&quot;&gt;$(-1,+1)$&lt;/span&gt; by using a weighted sum of it's values at abscisas which are roots of Legendre-polynomials, rather than uniformly spaced abscisaas. Gauss originally proved this by using his method of continued fractions derived from hypergeometric series, and later (1826) Jacobi gave an elegant interpretation using orthogonal polynomials.&lt;/p&gt;&#xA;&lt;p&gt;Therefore my questions are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Did Gauss prove that for arbitrary function, the best way to numerically integrate it (i.e the difference between the area under the function and the weighted sum will be minimal) is to use Gauss-Legendre quadrature? obviously for some functions this will be a better method while for others this will  be less-suitable, so if this statement is true, than it must be true &amp;quot;in average&amp;quot;. This seems to me like a very general functional statement, and a very difficult theorem to prove... so do i understand Gauss's statement correctly?&lt;/li&gt;&#xA;&lt;li&gt;If the answer to the previous question is yes, than obviously i understand the historic significance of Gauss's article on numerical integration. If the answer is no, than i'd also like to know what was so significant in Gauss's article?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2020-08-18T14:30:58.807" LastActivityDate="2024-01-04T18:42:18.620" Title="Question about the significance of &quot;Gauss-Legendre quadrature&quot;" Tags="|mathematics|real-analysis|gauss|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12151" PostTypeId="1" AcceptedAnswerId="12153" CreationDate="2020-08-18T04:47:55.150" Score="3" ViewCount="203" Body="&lt;p&gt;The BIPM specifies twenty prefixes for the International System of Units (SI). All ten of the fractional prefixes are lowercase. However, only seven of the multiplicative prefixes are uppercase, the exceptions being, da, h, and k, that is, the first three.&lt;/p&gt;&#xA;&lt;p&gt;The fact that these prefixes seem to be older than the larger ones is a possible reason.&#xA;The conflict between k (kilo) and K (kelvin) could also be a reason, but it seems unlikely to be the case as m (milli) and m (metro) are the same letter.&lt;/p&gt;&#xA;&lt;p&gt;Is there any reliable source about why those three (and only those three) prefixes are lowercase?&lt;/p&gt;&#xA;" OwnerUserId="12950" OwnerDisplayName="Bernardo Sulzbach" LastEditorUserId="26" LastEditDate="2021-02-20T15:08:56.487" LastActivityDate="2021-02-20T15:08:56.487" Title="Why are the first three multiplicative SI prefixes lowercase?" Tags="|terminology|notation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12153" PostTypeId="2" ParentId="12151" CreationDate="2020-08-18T22:13:33.873" Score="5" Body="&lt;p&gt;In &lt;a href=&quot;https://books.google.nl/books?id=RSoPAAAAQAAJ&amp;amp;pg=PA86&amp;amp;lpg=PA86&amp;amp;dq=centigramme%20gc.&amp;amp;source=bl&amp;amp;ots=kEqbbNGWT5&amp;amp;sig=nXunsLTDyBNEuYuyspqcF37CXwE&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjE6aPr2sbOAhVRFMAKHQmTAOAQ6AEIHDAA#v=onepage&amp;amp;q=centigramme%20gc.&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;1795&lt;/a&gt; lower case abbreviations were proposed for the prefixes myria, kilo, hecto, deca, deci, centi, milli: m, k, h, d, d, c, m. They were rarely used until after 1840, when the temporary mesures usuelles were replaced by the original unit names of the metric system. By then capitals were often, but not always, used for the multiples (myria, kilo, hecto, deca: M, K, H, D) and lower case for the submultiples (deci, centi, milli: d, c, m).&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.bipm.org/utils/common/pdf/CIPM-PV-OCR/CIPM1879.pdf&quot; rel=&quot;noreferrer&quot;&gt;1879&lt;/a&gt;, a few years after the introduction of the CGS system, lower case abbreviations were internationally adopted for &lt;em&gt;all&lt;/em&gt; common prefixes, because a combination of upper- and lower-case would be too susceptible for errors, and it would curb the ease of writing. Myria was excluded from the recommendation, because it would otherwise get the same abbreviation as milli. Initially hecto and deka/deca were also excluded, hecto because h was objectionable for Italy (the prefix is &amp;quot;etto&amp;quot; in Italian), and deka/deca because dk was unacceptable for France. A few decades later h and dk/da were included in lowercase. &lt;a href=&quot;https://books.google.nl/books?id=9cXsAoDALkEC&amp;amp;pg=PA13&amp;amp;dq=%22in%20lower%20case%20and%20without%20period%22&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=0ahUKEwjSvYWj3sbOAhUIGhQKHRcBDEsQ6AEIHjAA#v=onepage&amp;amp;q=%22in%20lower%20case%20and%20without%20period%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, at the end of the 19th century, all common prefixes were lower case for the convenience of ordinary people, commerce and trade. Capitalized prefixes were reserved for scientists who needed higher prefixes.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastActivityDate="2020-08-18T22:13:33.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12154" PostTypeId="1" CreationDate="2020-08-19T00:38:37.100" Score="0" ViewCount="109" Body="&lt;p&gt;Riccardo Giacconi (1931-2018) was an Italian astrophysicist who was awarded with the Nobel prize in physics back in 2002 for his important contributions to astrophysics.&lt;/p&gt;&#xA;&lt;p&gt;Since he was an astrophysicist, he must have heard about the multiverse hypothesis. Still, I have not found a single paper or article or interview from him where he indicates his opinion (neither positive nor negative thoughts) about the possibility of the existence of multiple universes.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, does anybody know what did Mr. Giacconi thought about the multiverse hypothesis? Did he like it? Or, on the contrary, he thought it was nonsense and not worthy of any discussion or study?&lt;/p&gt;&#xA;" OwnerUserId="10956" LastEditorUserId="10106" LastEditDate="2021-08-26T03:50:51.440" LastActivityDate="2021-08-26T03:50:51.440" Title="What did the Italian Nobel laureate Riccardo Giacconi think about the Multiverse hypothesis?" Tags="|physics|biographical-details|theoretical-physics|nobel-prize|astrophysics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12155" PostTypeId="2" ParentId="12143" CreationDate="2020-08-19T09:05:11.277" Score="10" Body="&lt;p&gt;As I was uncertain about the premise of this question—i.e., it looks as if there are fewer Spanish and Portuguese mathematicians than others—but without the numbers to prove it, I went in for some research on &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Map/&quot; rel=&quot;noreferrer&quot;&gt;the OP's Cambridge website&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In short, while there is some basis for the claims, I don't think the situation is as dire as claimed. Below is a table I compiled based on only the mathematicians on that website; I checked it for duplicates (because their sorting is horrible so &amp;quot;De L'Hôpital&amp;quot; was there thrice as, &amp;quot;De L'Hôpital&amp;quot;, &amp;quot;L'Hôpital, De&amp;quot;, and &amp;quot;Hôpital, De L'&amp;quot; and as this is relatively frequent in Dutch, French, and Italian names then there were quite a few duplicates).&lt;/p&gt;&#xA;&lt;p&gt;The second thing to note is that this is based on the country of birth &lt;em&gt;and&lt;/em&gt; century of death (the former, from the database, and the latter, make things simpler for myself).&lt;/p&gt;&#xA;&lt;p&gt;I included Argentina, Brazil, Chile, Cuba, Mexico, Uruguay, and Venezuela to see if any then-Spanish imperial territories produced any mathematicians of note. Mexico in the 17th century is the only really relevant one for our purposes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yzGJi.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yzGJi.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Overall, Spain is notable for coming in early—due to the Islamic centers of learning in &lt;a href=&quot;https://en.wikipedia.org/wiki/Al-Andalus&quot; rel=&quot;noreferrer&quot;&gt;al-Andalus&lt;/a&gt; while also managing to hold its own up to the 16th/17th century. It is notable that the large expansion of English, French, German, and Italian mathematicians in the 16th to 17th century is not replicated in Spain or Portugal (Portugal, notably, also missed out on large cities and centres of learning during the Islamic period).&lt;/p&gt;&#xA;&lt;p&gt;It is clear that this coincides with vast Spanish and Portuguese expansion across Central and South America and in the Indian Ocean. The Counter-Revolution already mentioned in @UriGranta's answer was clearly relevant in trying to stem the tide of Enlightenment and reason. The Iberian nobility's opportunity to enrich itself in the plentiful colonies was also relevant in directing possible investments away from engineering schools to those &lt;a href=&quot;https://en.wikipedia.org/wiki/School_of_Salamanca&quot; rel=&quot;noreferrer&quot;&gt;of theology&lt;/a&gt; (to justify expansion and train missionaries to evangelize in the colonies).&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Economic_history_of_Spain#Bourbon_reforms&quot; rel=&quot;noreferrer&quot;&gt;Bourbon Reforms section from Wikipedia&lt;/a&gt; is instructive:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A slow economic recovery began in the last decades of the 17th century under the Habsburgs. Under the Bourbons, government efficiency was improved, especially under Charles III's reign. The Bourbon reforms, however, resulted in no basic changes in the pattern of property holding. The nature of bourgeois class consciousness in Aragon and Castile hindered the creation of a middle-class movement. At the instance of liberal thinkers including Campomanes, various groups known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Sociedad_Econ%C3%B3mica_de_los_Amigos_del_Pa%C3%ADs&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Economic Societies of friends of the Country&amp;quot;&lt;/a&gt; were formed to promote economic development, new advances in the sciences, and Enlightenment philosophy. However, despite the development of a national bureaucracy in Madrid, the reform movement could not be sustained without the patronage of Charles III, and it did not survive him.&lt;/p&gt;&#xA;&lt;p&gt;... &lt;strong&gt;Castile's exchequer still used Roman numerals and there was no proper accounting.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In other words, the monarch who ruled was extremely important to maintain any forward momentum. Yet, after the &lt;a href=&quot;https://en.wikipedia.org/wiki/Spanish_Golden_Age&quot; rel=&quot;noreferrer&quot;&gt;Siglo de Oro&lt;/a&gt;, late 17th century Spain essentially languished under &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_II_of_Spain&quot; rel=&quot;noreferrer&quot;&gt;Carlos II&lt;/a&gt; until the Bourbon monarchs were instated—Felipe V was not a great improvement, and only &lt;a href=&quot;https://en.wikipedia.org/wiki/Ferdinand_VI&quot; rel=&quot;noreferrer&quot;&gt;Fernando V&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_III_of_Spain&quot; rel=&quot;noreferrer&quot;&gt;Carlos III&lt;/a&gt; managed to carry out some &lt;a href=&quot;https://en.wikipedia.org/wiki/Enlightenment_in_Spain&quot; rel=&quot;noreferrer&quot;&gt;much needed reforms&lt;/a&gt;. Nevertheless, what no king managed to reform was the society in which nobility and feats of arms were honoured more than learnedness. Similarly, the merchant class stayed relatively weak. It is, of course, up for discussion whether the monarchs actually wanted to promote sciences over merely preferring the &lt;em&gt;result&lt;/em&gt; of science (and engineering) (such as monumental architecture, etc...).&lt;/p&gt;&#xA;&lt;p&gt;Friar Benito Jerónimo Feijóo y Montenegro gave this some thought in the mid-18th century, as described below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If Philip V took the unprecedented step of allowing foreign academicians to enter his overseas domain , it was in part because , by the early eighteenth century, Spain had acquired a reputation abroad for intellectual retardation . Once feared for its intransigent Catholicism , the Peninsula was now scorned for its cultural inertia . The revolution in science associated with the names of Copernicus, Galileo, Kepler, Boyle, Leibniz and Newton was a European movement in which Spaniards were conspicuous only by their absence. &lt;strong&gt;As the Benedictine abbot, Benito Jerónimo Feijoo (1676–1764), admitted in his &lt;em&gt;Teatro critico universal&lt;/em&gt; (1726–39), 'physics and mathematics are almost foreigners in Spain'.&lt;/strong&gt; Indeed, he confessed that in the study of nature Spanish universities still relied on Aristotle, pinned like faded butterflies to the level of 1600. A devoted reader of the memoirs of the French Academy of Science and of other learned societies, Feijoo sought to transmit to a broad public some notion of the recent advances in knowledge, at all points emphasising the primacy of observation and experiment over argument and authority. ...&lt;br /&gt;&#xA;&lt;sup&gt;—Brading, &lt;a href=&quot;https://books.google.ee/books?id=vQdyHIKrqJYC&amp;amp;pg=PA423&amp;amp;lpg=PA423&amp;amp;dq=%22physics,%20and%20mathematics%20are%20almost%20foreigners%20in%20Spain%22&amp;amp;source=bl&amp;amp;ots=R8OPjHLclw&amp;amp;sig=ACfU3U2-90kg47u1IQ4BIeq1KO5WXVPqPA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiR6PLZ8abrAhXtlYsKHW5OCYYQ6AEwAXoECAEQAQ#v=onepage&amp;amp;q=%22physics%2C%20and%20mathematics%20are%20almost%20foreigners%20in%20Spain%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;'The First America'&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Admittedly, Brading doesn't cite the reasons why Feijoo thought so, however, even when the Crown started to sponsor scientific expeditions the interest did not seem to immediately extend to mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5957" LastActivityDate="2020-08-19T09:05:11.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12156" PostTypeId="1" CreationDate="2020-08-19T08:59:17.633" Score="0" ViewCount="76" Body="&lt;p&gt;All the questions are closely related to each other therefore, in my opinion, it wouldn't make much sense to ask them separately. I'm trying to find the answers in historical context. I appreciate your help and time!&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was quickly noted after the discovery of radioactivity in 1897,&#xA;that &lt;strong&gt;the total energy due to radioactive processes is about one&#xA;million times greater than that involved in any known molecular&#xA;change&lt;/strong&gt;. However, it raised the question where this energy is coming&#xA;from. &lt;strong&gt;After eliminating the idea of absorption and emission of some&#xA;sort of Lesagian ether particles&lt;/strong&gt;, the existence of a huge amount of&#xA;latent energy, stored within matter, was proposed by Ernest Rutherford&#xA;and Frederick Soddy in 1903. Rutherford also suggested that this&#xA;internal energy is stored within normal matter as well. He went on to&#xA;speculate in 1904:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;If it were ever found possible to control at will the rate of&#xA;disintegration of the radio-elements, an enormous amount of energy&#xA;could be obtained from a small quantity of matter.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Radioactivity_and_nuclear_energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mass–energy_equivalence#Radioactivity_and_nuclear_energy&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Where does this huge amount of energy come from? For a quarter of a&#xA;century scientists were racking their brains over this question, all&#xA;in vain. Opinions were divided: Crookes, Pierre and Marie Curie, and&#xA;some other scientists, were inclined to believe that radium atoms work&#xA;as energy transformers, i.e., they at first absorb the energy of waves&#xA;of unknown nature that permeate the entire universe, like the ether,&#xA;and then re-emit the accumulated energy. Others likened the ejection&#xA;of alpha particles to the process of evaporation of molecules. But&#xA;then the particles should have different energies, and William Bragg&#xA;in 1904 definitely established that this is not so: all the alpha&#xA;particles coming from radium have the same energy. Rutherford insisted&#xA;on the subatomic origin of radium's energy and was shown to be&#xA;absolutely right by later research.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://imagizer.imageshack.com/img923/4051/drMaT0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;imagizer&lt;/a&gt;, &lt;a href=&quot;https://i.stack.imgur.com/F863x.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;imgur&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Around 1900 they had already discovered alpha particles, beta particles, and gamma rays.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1:&lt;/strong&gt; Radium was found to be emitting alpha particles and gamma rays. Were they thinking that radium was absorbing all of these alpha particles and gamma rays from the surrounding atmosphere, and then later shooting them out as projectiles of higher energy? I think that they did know that material particles were originating from within radium and as a result radium should be losing its mass. Just like water loses its mass when some of its molecules evaporate but in radium's case the problem was that they didn't understand where all the energy for 'evaporation' was coming from and it was a huge amount of energy compared to the amount released in any known exothermic chemical reaction. Am I thinking along the right lines given the historical evolution of subject?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most rational explanation of this strange phenomenon is given by&#xA;J. J. Thomson in his corpuscle theory, which assumes the existence of&#xA;particles of matter called ions, thousands of times smaller than&#xA;atoms. It is supposed that myriads of these corpuscles are thrown off&#xA;at such a tremendous velocity as to generate the force which is known&#xA;as radioactive energy. Seguin's idea, we may recall, gives&#xA;reinforcement to the Thomsonian corpuscle theory. The incessant&#xA;bombardment occasioned by the particles thrown off from radium may be&#xA;demonstrated by the Crookes spinthariscope.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://imagizer.imageshack.com/img922/6923/9XMgjQ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;imagizer&lt;/a&gt;, &lt;a href=&quot;https://i.stack.imgur.com/pP5vu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;imgur&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2:&lt;/strong&gt; Why didn't they think that some kind of bond breaking is taking place? They could have thought that the corpuscles are held together with bonds and in some elements such as radium these bonds are unstable. Once bonds are broken, energy is released which propels the freed corpuscles outward like projectiles just like in an exothermic chemical reaction gas is released.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The fact that certain bodies possess the power of emitting invisible&#xA;and penetrating rays was first brought forward by Seguin. Many years&#xA;ago it was noticed that if a coin or a print were shut up in a box in&#xA;close proximity to a sheet of blank paper, an image was made, even in&#xA;the absence of solar light. These mysterious functions were formerly&#xA;accounted for as the product of superhuman agency ; but Seguin&#xA;suggests that the bodies capable of producing this singular phenomenon&#xA;did so by giving off a continuous stream of impalpable particles,&#xA;projected at such a velocity as to make an impression on the substance&#xA;which they bombarded. This hypothesis was rejected by physicists, who&#xA;declared that such a radiation of matter going on forever without loss&#xA;of weight was impossible. Though, indeed, almost inconceivable, yet&#xA;this hypothesis is the best explanation so far offered to account for&#xA;the strange behavior of the latest scientific wonder, radium.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://imagizer.imageshack.com/img922/6923/9XMgjQ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;imagizer&lt;/a&gt;, &lt;a href=&quot;https://i.stack.imgur.com/qrkNv.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;imgur&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 3:&lt;/strong&gt; Who was that Seguin? Is it &lt;a href=&quot;https://en.wikipedia.org/wiki/Marc_Seguin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marc Seguin&lt;/a&gt;? But Marc Seguin died in 1875 and radioactivity was found around 1897. Could you please help me to find more context for the mentioned Seguin's hypothesis that the bodies are capable of giving off a continuous stream of impalpable particles?&lt;/p&gt;&#xA;" OwnerUserId="9341" OwnerDisplayName="PG1995" LastEditorUserId="10106" LastEditDate="2020-08-19T11:24:23.710" LastActivityDate="2020-08-19T11:24:23.710" Title="Some questions about radioactivity around 1900" Tags="|physics|radiation|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="12157" PostTypeId="1" AcceptedAnswerId="12159" CreationDate="2020-08-19T22:22:34.840" Score="6" ViewCount="171" Body="&lt;p&gt;I'm investigating the conics in ancient Greece, I have the works of Apollonius, Diocles and Euclid, written with great commentary (both explaining the math and the historical context) that make them very readable and enjoyable.&lt;/p&gt;&#xA;&lt;p&gt;I have not had that much luck with Pappus.&#xA;I'm particularly interested in the works of Pappus on conics, and while I don't &lt;strong&gt;need&lt;/strong&gt; a &amp;quot;direct translation&amp;quot; I'm looking for something close (think Heath's version of &lt;em&gt;Conics&lt;/em&gt; of Apollonius) that adresses it and also has commentary to make it understandable and provide historical context.&lt;/p&gt;&#xA;&lt;p&gt;Any sources or book recommendations are welcome.&lt;/p&gt;&#xA;" OwnerUserId="12887" LastEditorUserId="10106" LastEditDate="2020-08-20T10:44:20.220" LastActivityDate="2020-08-20T10:44:20.220" Title="Are there commented English translations of Pappus's works on conics?" Tags="|reference-request|ancient-greece|translations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12158" PostTypeId="1" AcceptedAnswerId="12161" CreationDate="2020-08-20T01:47:10.073" Score="2" ViewCount="117" Body="&lt;p&gt;Is there an english translation of Xu Yue's Shushu Jiyi? This is the work, around 190 CE, often described as containing the first description of the abacus. It is often associated with China's &amp;quot;Ten Computational Canons&amp;quot;, but usually isn't listed as one of the 10 officially.&lt;/p&gt;&#xA;&lt;p&gt;I understand that it describes 14 different computational mechanisms/strategies, one of which is the abacus. Even if there is no full translation, is there perhaps a paper or book chapter in english which describes each of the 14 techniques?&lt;/p&gt;&#xA;&lt;p&gt;I am a mathematician, but I know basically nothing about ancient chinese mathematics, so maybe there are easy references that I'm not aware of. Thanks for any help!&lt;/p&gt;&#xA;" OwnerUserId="12954" LastEditorUserId="10106" LastEditDate="2020-08-20T10:45:44.153" LastActivityDate="2020-08-20T11:22:49.857" Title="English translation of Xu Yue's Shushu Jiyi?" Tags="|mathematics|reference-request|translations|ancient-china|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12159" PostTypeId="2" ParentId="12157" CreationDate="2020-08-20T08:04:49.700" Score="6" Body="&lt;p&gt;The first translation into a modern European language of Pappu's &lt;em&gt;Collection&lt;/em&gt; is the 1933 French edition by &lt;a href=&quot;https://fr.wikipedia.org/wiki/Paul_ver_Eecke&quot; rel=&quot;noreferrer&quot;&gt;Paul Ver Eecke&lt;/a&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Federico_Commandino&quot; rel=&quot;noreferrer&quot;&gt;Commandino&lt;/a&gt; had put it in Latin in 1588). &lt;a href=&quot;https://www.ams.org/journals/bull/1934-40-01/S0002-9904-1934-05766-5/S0002-9904-1934-05766-5.pdf&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; you can find a review of that translation.&lt;/p&gt;&#xA;&lt;p&gt;There no complete English version, but the only books of Pappus dealing with conics are the fourth and the seventh and, quite luckily, you can find the &lt;a href=&quot;https://www.springer.com/gp/book/9781849960045&quot; rel=&quot;noreferrer&quot;&gt;fourth&lt;/a&gt; and &lt;a href=&quot;https://www.springer.com/gp/book/9780387962573&quot; rel=&quot;noreferrer&quot;&gt;seventh&lt;/a&gt; book with a literal translation and a commentary on textual, historical, and mathematical aspects of the Collection published by Springer.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;You don't ask for the textual sources of Pappus work, anyway &lt;a href=&quot;https://www.persee.fr/doc/scrip_0036-9772_1957_num_11_2_2937&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; you can find an old but very interesting paper about the manuscript tradition of the &lt;em&gt;Collection&lt;/em&gt;, from which one can understand the tremendous work needed to provide a full edition and translation of the work of Pappus.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-08-20T08:29:39.870" LastActivityDate="2020-08-20T08:29:39.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12160" PostTypeId="1" CreationDate="2020-08-20T08:24:05.497" Score="2" ViewCount="221" Body="&lt;p&gt;I read in &lt;a href=&quot;https://www.quantamagazine.org/string-theorys-strange-second-life-20160915/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; Quanta article that although many critics say that it is far removed from nature, it has developed many powerful tools. Furthermore some don’t care if it’s a theory of everything and just apply it to real physical problems and get results.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Once the elementary things we’re probing spaces with are strings instead of particles,” said Beem, the strings “see things differently.” If it’s too hard to get from A to B using quantum field theory, reimagine the problem in string theory, and “there’s a path,” Beem said.&lt;/p&gt;&#xA;&lt;p&gt;In cosmology, string theory “packages physical models in a way that’s easier to think about,” Silverstein said.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To me, this seems reminiscent of the early days of the atomic theory where the likes Maxwell, Boltzmann and Gibbs developed tools of statistical mechanics that reinterpreted thermodynamics as a statistical outcome of atomic interactions.&lt;/p&gt;&#xA;" OwnerUserId="12956" LastEditorUserId="10106" LastEditDate="2020-08-20T10:48:16.747" LastActivityDate="2020-09-22T22:01:40.167" Title="Is string theory today facing the same backlash that atomic theory faced during its inception?" Tags="|physics|atomic-theory|string-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12161" PostTypeId="2" ParentId="12158" CreationDate="2020-08-20T11:01:43.743" Score="3" Body="&lt;p&gt;I'm afraid that there no English translation.&#xA;There is a Russian translation, see &lt;a href=&quot;https://books.google.it/books?id=NfxdDwAAQBAJ&amp;amp;pg=PA115&amp;amp;lpg=PA115&amp;amp;dq=Shushu+Jiyi+translation&amp;amp;source=bl&amp;amp;ots=jPAsYENtxK&amp;amp;sig=ACfU3U0KX2sZpnqVBRkI4cq5d2gm2k0gFw&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwjKkIGUx6nrAhUFCuwKHenUB_wQ6AEwCXoECAEQAQ#v=onepage&amp;amp;q=Shushu%20Jiyi%20translation&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, where it is also affirmed that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The secon part contains the descriptions of 13 [not 14] methods of representation of numbers with counting devices; one of them, the &amp;quot;computation with pearls&amp;quot;, refers to a form of abacus similar to the Roman abacus. There is an additional part [...] concerning some particularities of operations with counting rods.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.it/books?id=ACK1jreKgCoC&amp;amp;pg=PA99&amp;amp;lpg=PA99&amp;amp;dq=Shushu+Jiyi+translation&amp;amp;source=bl&amp;amp;ots=ggQMDHDyV2&amp;amp;sig=ACfU3U2Wi9ftV-HqnqkawAFLBhvGcdfyKQ&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwjKkIGUx6nrAhUFCuwKHenUB_wQ6AEwA3oECAMQAQ#v=onepage&amp;amp;q=Shushu%20Jiyi%20translation&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; you can find the English translation of a short  excerpt, more in the paper&#xA;of Alexei Volkov &lt;a href=&quot;https://www.persee.fr/doc/oroc_0754-5010_1994_num_16_16_991&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Large Numbers and Counting Rods&lt;/em&gt;&lt;/a&gt; mainly devoted to the problem of the &amp;quot;finitess of numbers&amp;quot; raised in the first part of the &lt;em&gt;Shushu Jiyi&lt;/em&gt; (with longer excerpts), while you are interested in the second part, devoted to the &amp;quot;counting devices&amp;quot;, that Volkov briefly discuss in its eighth paragraph.&lt;/p&gt;&#xA;&lt;p&gt;Anyway:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yet this theoretical construction (whether we understood it properly or not) has one essential defect: it could not insure that there exists in reality something corresponding to very large numbers; a priori, one cannot be sure that in this world there is something corresponding to 104096 (the largest number in the superior system).&lt;/p&gt;&#xA;&lt;p&gt;Xu Yue's descriptions are very brief, and &lt;strong&gt;the modern reconstructions are due to abundant commentaries of Zhen Luan who systematically interpreted each &amp;quot;method&amp;quot; as a real counting device (which probably does not correspond to the original meaning of these descriptions.)&lt;/strong&gt; [emphasis added] For the most part the described &amp;quot;devices&amp;quot; have been organized as linear sequences of decimal positions; the techniques of representation of a digit in one position are diverse and some of them are based on the traditional connotations between colors, orientations, etc.&lt;/p&gt;&#xA;&lt;p&gt;The first &amp;quot;device&amp;quot; in this list is that of the counting rods. They are not described as if they were well-known to the reader. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-08-20T11:22:49.857" LastActivityDate="2020-08-20T11:22:49.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12162" PostTypeId="1" CreationDate="2020-08-22T06:53:30.547" Score="3" ViewCount="214" Body="&lt;p&gt;I'm currently working on a thesis about Zermelo's axioms. In my first chapter I'm giving an introduction to the numerical treatment that Cantor gave to infinity.&lt;/p&gt;&#xA;&lt;p&gt;When I was writing something about transfinite numbers as an expansion of natural numbers, I realized that I needed to include some reference to the formalization of numbers. I know that von Neumann gave the formal definition of natural number, but I couldn't find who formalized Integers (&lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know who formalized integers, rationals, and reals? I know the construction as the palm of my hand but I don't know who did the great job.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance and may the health be with you!&lt;/p&gt;&#xA;" OwnerUserId="12964" LastEditorUserId="2324" LastEditDate="2020-08-30T00:04:12.750" LastActivityDate="2020-08-30T10:40:28.070" Title="Who formalized integer numbers?" Tags="|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12165" PostTypeId="2" ParentId="5860" CreationDate="2020-08-23T21:22:20.197" Score="0" Body="&lt;p&gt;The basic picture to hold in mind is how a manifold is described in terms of an atlas. This is more or less exactly the same as an atlas that one would have seen in a book before technology came along and did this for us.&lt;/p&gt;&#xA;&lt;p&gt;Recall, each page in an atlas covers a portion of the landscape to be described and moreover, there is an area of overlap on the four sides of each page with other pages. Matching these overlaps on the atlas describes the whole landscape.&lt;/p&gt;&#xA;&lt;p&gt;Grothendieck took this description of an atlas as a novel description of a topology, the so-called Grothendieck topology which was flexible enough to describe generalisations of manifolds in contexts where the classical description doesn't apply.&lt;/p&gt;&#xA;&lt;p&gt;The term 'etale' comes from this description. Envisage the landscape and each page of the atlas above the landscape, then they lie 'flat' and 'spreading out' over the landscape'. In fact, what one does is take the disjoint union of all these pages, and the same picture applies, it (rather than 'they' since we have disjointly glued the pages together) lies 'flat' and is 'spread out'. Hence the term etale morphism, a morphism that lies flat and spreading out across the space.&lt;/p&gt;&#xA;&lt;p&gt;Another way of looking at the same picture is to consider covering spaces. Here, the cover lies flat and 'spread out' over the base. The notion of an etale morphism is a generalisation of this. A vast generalisation, but nevertheless, a generalisation.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-08T05:50:52.463" LastActivityDate="2021-03-08T05:50:52.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12166" PostTypeId="2" ParentId="9875" CreationDate="2020-08-23T21:44:26.553" Score="2" Body="&lt;p&gt;No, he didn't.&lt;/p&gt;&#xA;&lt;p&gt;In fact he is known for coming up with his own interpretation of QM that relies on a non-local pilot wave. It's usually referred to as de Broglie-Bohmian mechanics as de Broglie, earlier,  had similar ideas.&lt;/p&gt;&#xA;&lt;p&gt;He is also known for developing a notion of implicate and explicate order in QM, writing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I propose that each moment in time is a projection of the total implicate order.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This new insight can be best described as Undivided Wholeness in Flowing Movement. This view implies that Flow is in some sense prior to that of things which seem to dissolve in this flow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is basically a kind of ontological holism and emphasises becoming over being (though perhaps not of Being itself, given Bohms interest in Advaita Vedantism).&lt;/p&gt;&#xA;&lt;p&gt;Basil Hiley, a coworker of Bohm said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Bohm says in his quantum theory book, the original one, that quantum mechanics is a mis-nomer. It should be called quantum non-mechanics .... it's nothing like that. It's not mechanism. It's organicism. Nature is more organic than we think it is. And then you can understand why life arose, because if nature is organic, it has the possibility of life in it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This itself is an old idea. According to Bertrand Russell, this is one reason why Thales conceived the world as water, as water has the possibility of life - it is seminal. Moreover, the idea of nature, that is the universe, is organic and has the nature of an organism was explicitly mentioned in Aristotle.&lt;/p&gt;&#xA;&lt;p&gt;All this shows that Bohm was far from the notion of a multiverse - he was thinking in terms of an entirely different category to what is common today - at least in physics - but perhaps, not elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-08-30T04:40:44.997" LastActivityDate="2020-08-30T04:40:44.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12167" PostTypeId="2" ParentId="12160" CreationDate="2020-08-23T21:59:19.337" Score="2" Body="&lt;p&gt;Lots of mathematical physicists don't actually care about the physics and focus on the mathematics. This is not new with string theory. What is new with string theory is that a great deal of mathematics which did not seem to have direct physical applications were found to have such an interpretation. This in itself isn't particularly surprising since a great deal of mathematics has been discovered by 'upping the dimension'. For example, Fourier analysis is basically harmonic analysis on a circle. The circle is the simplest compact &lt;em&gt;1d&lt;/em&gt; Lie group. Now see how the theory changes when you exchange this parameter for say a higher dimensional Lie group. String theory, in one interpretation, is simply taking physical theory for the relativistic point, and exchanging it for the relativistic line. One can see immediately that lines can close up (which points can't) hence closed strings. That strings can carry vibrations (which again points can't). Then we up the dimension again by arguing for membranes rather than just strings and so on and on.&lt;/p&gt;&#xA;&lt;p&gt;It's actually much closer to the really early days of atomic theory (or 2,500 years ago) rather than the renaissance of atomic theory in the modern era, simply because the Planck scale is so unimaginably far from what we can probe with today's technology. It's no accident that quantum mechanics was discovered both theoretically and experimentally. It's one of those auspicious times when theory and experiment develop together.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-23T21:59:19.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12168" PostTypeId="2" ParentId="12118" CreationDate="2020-08-23T22:34:36.260" Score="1" Body="&lt;p&gt;Atomic theory was predicted 2,500 years ago by Democritus and Leucippus, and it did revolutionise physics once that scale could be probed. In fact, its revolutionary potential was such that Feynman wrote in his lectures that if all scientific knowledge was forgotten, and one sentence was to be preserved for future generations, then he would pick 'all things are made up of atoms.' Actually, this did occur but we didn't have just a single sentence turn up in the early modern period, but a whole book - Lucretious's &lt;em&gt;On the Nature of Things&lt;/em&gt;. Not only did it describe atoms, it also described what was later taken to be one of the irrefutable proofs of atoms - Brownian motion with essentially the same explanation!&lt;/p&gt;&#xA;&lt;p&gt;The critique of atomic theory was also done 2,500 years ago by Aristotle. This particular revolution is still ongoing as people are still struggling to come to terms with what is known as wave-particle duality. In other words, it revolutionised our understanding of what constitutes being: atoms are not permanent being-like entities but are also involved in becoming.&lt;/p&gt;&#xA;&lt;p&gt;The main critique of Newtonian Gravity was that it acted at a distance with no intervening mechanism. When this was solved by Einstein in his theory of general relativity, it revolutionised our ideas of space and time. This geometric picture of space and time was later revolutionised our understanding of classical field theories so they too became geometric.&lt;/p&gt;&#xA;&lt;p&gt;Still, one might say that this was all predicted by Plato, who was channeling the Pythagoreans, when he suggested that physical nature at bottom was geometry, and geometry was the visible manifestation of necessity in nature.&lt;/p&gt;&#xA;&lt;p&gt;The main prediction for revolutionary physics in the future is Quantum Gravity with a lot of physicists in search for it with maybe half a dozen major programmes from LQG, causal dynamical triangulations, string theory, asymptotic safety and so on.&lt;/p&gt;&#xA;&lt;p&gt;On a more applied level, a global infrastructure of renewable energy will undoubtably revolutionise the global economy once all the problems with generation, storage, transmission and manufacturing have been sorted out. Especially when one realises how much of geo-politics is predicated on securing energy supplies, which in today's world, still means fossil fuels.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="12724" LastEditDate="2021-05-22T20:38:16.787" LastActivityDate="2021-05-22T20:38:16.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12169" PostTypeId="2" ParentId="12162" CreationDate="2020-08-23T22:43:40.650" Score="-3" Body="&lt;p&gt;There's two notions of the formalisations of integers that I know of. The better known tradition follows Dedekind and Peano, culminating in Peanos axioms (this actually was developed under the impetus of then newly developing set theory. However, it turns out that it's possible to lift Peanos axioms into the context of topos theory, a rival to set theory whose underlying logic is intuitionistic rather than classical - in that context it's called the Natural Numbers Object).&lt;/p&gt;&#xA;&lt;p&gt;The less well-known tradition, but older one, is one instigated by Gauss: his Gaussian integers. The modern interpretation is constructing a ring of integers for every number field.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;@Bachfold: it would be bad practise in a journal article. However, this is not a journal. Moreover, we live in the age of search engines and SE is hosted on the net and one can assume that if they are browsing SE then they also have access to the net. Why can't I then assume that if they have the wit to read what I wrote and are interested in what I have written and want to know more, or verify its accuracy that they simply can't type 'Gaussian Integers' or 'Natural Numbers Object' into a search engine? For example, the top three results after typing the former into Google are articles on such from NLab, Wikipedia and Springers Encyclopedia of Mathematics - any of these three will do as standard references. It seems to me that readers should put in a little work themselves - especially when writers are unpaid for their time and trouble.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-08-30T10:40:28.070" LastActivityDate="2020-08-30T10:40:28.070" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12170" PostTypeId="2" ParentId="12112" CreationDate="2020-08-23T22:56:12.443" Score="-2" Body="&lt;p&gt;It's not that surprising when all things are considered. Most people do not like their cherished ideas over-turned. Take, for instance, calculus.  Newton expected opposition to his new ideas, which is why he couched his discoveries not in calculus, but in the mathematical language of the day:  Euclidean geometry. Given his mathematical sophistication, no doubt he actually enjoyed coming up with geometric proofs of what he discovered by use of the calculus.&lt;/p&gt;&#xA;&lt;p&gt;Set theory was still a new idea when Cantor came up with his new theory of the infinite. So for most mathematicians, this was newness piled upon newness - so probably one newness too many. Still, most mathematicians came round to Cantor's point of view, since his reasoning was sound. The main problem with it is that it is a theory with very few real applications.&lt;/p&gt;&#xA;&lt;p&gt;In other words, it's widely accepted but also widely useless.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="12724" LastEditDate="2020-09-23T13:18:56.737" LastActivityDate="2020-09-23T13:18:56.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12171" PostTypeId="2" ParentId="11120" CreationDate="2020-08-23T23:18:24.913" Score="1" Body="&lt;p&gt;Proportion is the key concept that underlies most of mathematics. In its modern guise, it's described as the straight line or linearity.&lt;/p&gt;&#xA;&lt;p&gt;Now consider that the epitome of motion in Newtons theory is straight line motion. Further consider that Einstein then described motion in GR as straight lines on a curved surface. More, consider that calculus is simply the linearisation of a nonlinear function - in none variable, in many variables, in curved surfaces, in curved manifolds and in curved higher dimensional manifolds.&lt;/p&gt;&#xA;&lt;p&gt;Even more, consider that quantum mechanics - mathematically speaking, is a linear theory.&lt;/p&gt;&#xA;&lt;p&gt;Once one has understood the many different ways mathematicians have leveraged the concept of linearity, and hence proportionality into many areas and fields of mathematics, one might think that mathematics is simply the unfoldingness of one very simple idea: that of proportion.&lt;/p&gt;&#xA;&lt;p&gt;This is an idea that every child can easily understand. It's not occult, nor is it a secret. It's one that every mathematician knows. Perhaps it's a secret to the average man in the street who is impressed by the occult looking symbols. This is the same impulse undoubtably that has driven numerology and astrology. Rest easy, there is only one single idea here - if one can only see it.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-23T23:18:24.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12172" PostTypeId="2" ParentId="12117" CreationDate="2020-08-23T23:32:02.903" Score="0" Body="&lt;p&gt;One perspective of Big Bang cosmology that often gets lost, is its sheer unbelievability. It's not the idea of creation per se, but simply the notion of the whole universe, all of its billions of lights years, it's billions of galaxies all shrunk into one very small vanishing point.&lt;/p&gt;&#xA;&lt;p&gt;It sounds crazy.&lt;/p&gt;&#xA;&lt;p&gt;So it's perhaps no surprise that without convincing evidence otherwise people opted for the steady state theory. In fact, Hoyles theory was not really 'steady state' since he postulated a very small breaking of one of the cardinal principles of physics - the conservation of energy.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-23T23:32:02.903" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12173" PostTypeId="2" ParentId="12146" CreationDate="2020-08-23T23:45:24.217" Score="1" Body="&lt;p&gt;It's an early recognition of duality in set theory. Domain vs Codomain suggests a relationship that is missing from domain and range.&lt;/p&gt;&#xA;&lt;p&gt;This is hidden in set theory as functions are biased in that they are not symmetrically defined. Nor is it easy to conceptualise one to many functions naturally, and dually to many to one functions, which they do naturally.&lt;/p&gt;&#xA;&lt;p&gt;This is fixed in category theory where duality is made explicit, rather than in the secret, furtive way it's done in set theory. Moreover, category theory is the correct conceptualisation of covariance as in the notion of general covariance that Einstein used heuristically in his investigations into the general character of physical law.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, one of the major discoveries of string theory is the role that dualities play in physics. (In ordinary physics, we see duality manifest itself in the duality between the electric and magnetic fields). It wouldn't surprise me if at bottom this had the same root as dualities in category theory.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-23T23:45:24.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12174" PostTypeId="1" AcceptedAnswerId="12179" CreationDate="2020-08-24T00:22:56.920" Score="1" ViewCount="5077" Body="&lt;p&gt;A simple question: Where did the contour integral sign appear for the first time?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kCkg7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kCkg7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia says that it was introduced by physicist Arnold Sommefield in 1917 ( &lt;a href=&quot;https://en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date&quot; rel=&quot;nofollow noreferrer&quot;&gt;Table of mathematical symbols by introduction date&#xA;&lt;/a&gt;  ) but there are not any references to prove this&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know if this is true? Is there a scientific paper or a book where this symbol was introduced?&lt;/p&gt;&#xA;" OwnerUserId="12968" LastEditorUserId="5957" LastEditDate="2020-08-26T07:47:23.600" LastActivityDate="2020-08-26T07:47:23.600" Title="Where did the contour integral sign appear for the first time?" Tags="|mathematics|physics|notation|calculus|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12175" PostTypeId="2" ParentId="11312" CreationDate="2020-08-24T01:11:05.597" Score="0" Body="&lt;p&gt;Simply because the Romans understood nothing as a concept does not mean that they need recognise it as a number per se. After all, every society understands the distinction between having something and having nothing.&lt;/p&gt;&#xA;&lt;p&gt;Moreover adding nothing to anything leaves that unchanged. This is unlike every other addition, for example, adding two or seven to something. It's perhaps then not suprising that the notion of nothing as a number was slow to arise. It's a question here, not of the mathematical properties of nothing, but of the different ontological status of nothing and something.&lt;/p&gt;&#xA;&lt;p&gt;Thus, whilst it's commonly said that the Romans didn't have the concept of zero as a number due to their numeral system not being positional, I'd rather say it's simply because of the special ontological status of zero as a number which signifies no thing as opposed to some thing.&lt;/p&gt;&#xA;&lt;p&gt;We can, for example, easily imagine a Roman numeral system with a zero. But unlike the decimal system, where zero is used positionally, to signify for example, twenty or three hundred and six; a Roman zero would only have a single use, that is to signify nothing and only nothing.&lt;/p&gt;&#xA;&lt;p&gt;It's worth stressing that this ontological difference between nothing and something disappears once the notion of geometrising number appears. Then zero is a position, as is the number five or twelve. All numbers are then alike, as all positions are alike. This then makes the notion of zero (as well as negative numbers) much more intuitive than that of counting - which measures what is, and what is not. However, Wikipedia suggests that the geometric number line was a relatively recent discovery - which I find odd, given that geometry has been quantified from even before Babylonian times. The very term - geometry - is named after measuring the earth, that is the ground beneath us, after all. It's possibly suggestive of where new research might be worthwhile.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-08-31T17:26:25.573" LastActivityDate="2020-08-31T17:26:25.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12176" PostTypeId="2" ParentId="12077" CreationDate="2020-08-24T01:23:21.783" Score="1" Body="&lt;p&gt;Personally I'm surprised that this is the case given that for hundreds of years Europe was Christian and the biblical Genesis quite matter of factly said that the universe was created. It's strange that this didn't factor in their speculations about the universe and its history.&lt;/p&gt;&#xA;&lt;p&gt;Still, it's understandable. After all, if the universe was created then it would have been created all at once and it's difficult to see just how that can be the case when the universe appears spread over such an enormous scale. One way around this, which LeMaitre took, is to suggest that the universe was created as a kind of cosmic egg, which cracked open (symmetry breaking!) and which then expanded. But then one has to imagine the whole, vast expanse of the universe compressed into something very small and compact and that also beggars the imagination.&lt;/p&gt;&#xA;&lt;p&gt;No doubt that's why astronomers took the simplest, safest and most cautious option and just declared the universe to have been as it always was. Undoubtably this was easy to do since the sky at night, apart from the sun and the moon doesn't appear to change very much from year to year and in fact was taken as the epitome of a permanent and changeless divine sphere: the supralunary as opposed to the sublunary, so to speak.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-24T01:23:21.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12179" PostTypeId="2" ParentId="12174" CreationDate="2020-08-24T08:34:36.663" Score="2" Body="&lt;p&gt;Jeff Miller's very valuable collection of the origins of mathematical expressions has the entrie &amp;quot;&lt;a href=&quot;http://jeff560.tripod.com/calculus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Integration around a closed path&lt;/a&gt;&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dan Ruttle, a reader of this page, has found a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868-1951) in 1917 in Annalen der Physik, &amp;quot;Die Drudesche Dispersionstheorie vom Standpunkte des Bohrschen Modelles und die Konstitution von H2, O2 und N2.&amp;quot; This use is earlier than the 1923 use shown by Cajori. Ruttle reports that J. W. Gibbs used only the standard integral sign in his Elements of Vector Analysis (1881-1884), and that and E. B. Wilson used a small circle below the standard integral symbol to denote integration around a closed curve in his Vector Analysis (1901, 1909) and in Advanced Calculus (1911, 1912).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even if Cauchy actually used contour integrals, he didn't use the symbol: what we know as &amp;quot;Cauchy's integral formula&amp;quot; was proved in Cauchy, &lt;em&gt;Sur les integrales qui s'etendent a tous les points d'une courbe fermee&lt;/em&gt;, Camp. Rend., 23 (1846), 251-255, you can see &lt;a href=&quot;https://www.e-rara.ch/zut/content/zoom/5704829&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that Cauchy simply used the standard integration symbol.&lt;/p&gt;&#xA;&lt;p&gt;Until at least 1905 that symbol was not in use, see this &lt;a href=&quot;http://archive.numdam.org/article/AFST_1905_2_7_3_265_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; of Dimitrie Pompeiu where Cauchy's formula is stated using the standard integration symbol.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-08-24T08:56:24.253" LastActivityDate="2020-08-24T08:56:24.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12180" PostTypeId="1" AcceptedAnswerId="12181" CreationDate="2020-08-24T12:16:55.967" Score="1" ViewCount="80" Body="&lt;p&gt;I am reading H.T.H. Piaggio's &amp;quot;An Elementary Treatise on Differential Equations and their Applications&amp;quot;, published 1920 originally; I have the 1952 edition.&lt;/p&gt;&#xA;&lt;p&gt;The first chapter &amp;quot;Historical Introduction&amp;quot; goes:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The geometrical interpretation [of differential equations] was given by Lagrange in 1774, but the theory in its present form was not given until much later by Cayley (1872) and M.J.M. Hill (1888).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And again:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;These theories [of partial differential equations] still remain in an unfinished state; contributions have been made by Chrystal (1892) and Hill (1917).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Who is this &amp;quot;M.J.M. Hill&amp;quot;? Piaggio cites him a few times during the course of the text, citing papers in Philosophical Transactions and Proceedings of the London Mathematical Society, etc., but I can't find anything more than his footprint in the journals.&lt;/p&gt;&#xA;&lt;p&gt;Anyone with any information about him?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2020-08-24T12:45:11.927" Title="Who is the &quot;M.J.M. Hill&quot; as mentioned in the historical introduction to Piaggio's &quot;Differential Equations&quot;?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12181" PostTypeId="2" ParentId="12180" CreationDate="2020-08-24T12:29:59.887" Score="5" Body="&lt;p&gt;There's a Wikipedia page: &lt;a href=&quot;https://en.wikipedia.org/wiki/Micaiah_John_Muller_Hill&quot; rel=&quot;noreferrer&quot;&gt;Micaiah John Muller Hill&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hill is ofted cited in Ramanujan's biographies, &lt;a href=&quot;https://books.google.it/books?id=Of5G0r6DQiEC&amp;amp;pg=PA208&amp;amp;lpg=PA208&amp;amp;dq=M.J.M.+Hill+differential+equations+professor+University+College&amp;amp;source=bl&amp;amp;ots=Pv7yCg5Mc8&amp;amp;sig=ACfU3U1r8dL3ZmjZ30XNRhrf5gwwfbxatw&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwjQjq-767PrAhUCC-wKHepiBzE4ChDoATAEegQICRAB&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; there is something more than in the Wikipedia page:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] Hill was born at Berhapore, Bengal (narrowly escaping death during the Indian mutiny) and was Professor of Mathematics at University College, London, from 1884 until 1926. He had a reputation as a sympathetic and painstaking teacher. He had been at Peterhouse in Cambridge as an undergraduate, and was classed as 4th Wrangler (equal) in the Mathematical Tripos in 1879. Hill did research on hydrodynamics and differential equations, but at the time of Griffith's letter [Griffith sent some Ramanujan works to Hill, see &lt;a href=&quot;https://www.imsc.res.in/%7Erao/ramanujan/newnow/enggcollege.htm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for something about him] was primarly interested in Euclid's axioms [...]. In 1907 and 1908, Hill published two papers giving an asymptotic formula for partial sums of certain hypergeometric series [...]. It is unfortunate that no one at that time knew of this mutual interest of Hill and Ramanujan [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2020-08-24T12:45:11.927" LastActivityDate="2020-08-24T12:45:11.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12182" PostTypeId="2" ParentId="12135" CreationDate="2020-08-24T14:47:27.483" Score="2" Body="&lt;p&gt;It's because of Diracs use of it in QM. After QM was a revolutionary new theory of physics and so had immense visibility because of this.&lt;/p&gt;&#xA;&lt;p&gt;This is very similar to how Einstein popularised the study of non-Euclidean geometry by his use of such in his revolutionary theory of space and time. After all, non-Euclidean geometry had been known since Gauss's time but it simply didn't have the glamour or cachet that it began to have once Einstein arrived on the scene.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-24T14:47:27.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12183" PostTypeId="1" CreationDate="2020-08-24T19:46:22.753" Score="1" ViewCount="89" Body="&lt;p&gt;What was the first Trigonometry book ever to appear containing most (if not all) things we study nowadays in Trigonometry?&lt;/p&gt;&#xA;&lt;p&gt;I have heard that Pythagoras never wrote any book and that we have very few fragments of his writings. So which was the first book that contained most of the trigonometry as it is known nowadays?&lt;/p&gt;&#xA;" OwnerUserId="12280" LastActivityDate="2020-08-24T19:46:22.753" Title="What was the first trigonometry book ever to appear containing most (or all) things we study nowadays in trigonometry" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12184" PostTypeId="2" ParentId="12148" CreationDate="2020-08-25T02:52:12.133" Score="2" Body="&lt;p&gt;Plato already distinguished between different professions and also different intellectual capacities in his &lt;em&gt;Republic&lt;/em&gt;. His master discipline was the statesman who placed all professions in the right place in the city - read republic - so that they can best flourish. This is his conception of the philosopher-king, he who marries well both thought and action.  This was in counterpoint to the politics of his day which tended to benign tyranny at best. Although it's common today to praise Athenian democracy, Plato himself didn't think much of it - until faced with worse. In today's democracies, this role is not played by persons, but by institutions.&lt;/p&gt;&#xA;&lt;p&gt;This of course, is not the kind of multi-disciplinary work that you mention. And I think it's important to distinguish this from inter-disciplinary work which builds bridges between fields, rather than simply being fluent in two, very different fields.&lt;/p&gt;&#xA;&lt;p&gt;Of course Plato also thought it important to focus on the educational policy of a republic to build a republic and part of his &lt;em&gt;The Republic&lt;/em&gt; is devoted to this subject. Music was an important branch of this - so one might say here, the arts. As well as mathematics, and so science. Given his views on the professions, and supposing that technology, ie &lt;em&gt;techne&lt;/em&gt;, is the focus of the professions then this captures all three of the divisions you have pointed out. One key aspect that is often forgotten in Platos discussion of education is physical activity. It's not enough to train the mind, one ought to train the body. So the real division is not merely three ways - but four.&lt;/p&gt;&#xA;&lt;p&gt;It's a good question to ask what kind of physical activity Leonardo got upto. And it's possibly a reflection on our own cultural/educational matrix that we don't think of asking - as though physical education is somehow missing the point of education. It's worth reflecting on this next time the medical profession points out the epidemic of obesity that is afflicting affluent countries.&lt;/p&gt;&#xA;&lt;p&gt;In early medieval Europe, around the 6th C,  liberal education drew on Plato to establish the &lt;em&gt;trivium&lt;/em&gt;, the three fields of rhetoric, logic and grammar; this as preparatory for the &lt;em&gt;quadrivium&lt;/em&gt;, which was composed of arithmetic, geometry, music and astronomy. These were distinguished from the practical - read technological skills - of medicine and architecture.&lt;/p&gt;&#xA;&lt;p&gt;As one can see, the disciplinary differences were established early. Notably, CP Snow, on his essays on the two cultures of the arts and sciences wasn't about the disciplinary differences as such. After all, even in the sciences there are distinctions. For example, between mathematics and physics. His point was that these two large fields were no longer on speaking terms. Worse, that they failed to comprehend each other. One might think that in an earlier era, this role was played by philosophy or theology - the then unifying disciplines. This role, after the break with Christianity in the West can no longer played by it - at least in the same fashion. Nevertheless, for the &lt;em&gt;episteme&lt;/em&gt; of a people, a nation or a civilisation to flourish, this conversation has to somehow be begun, and then sustained. Hence, perhaps - though given I've never worked in a university - and so am not in a position to say,  the renewed emphasis on multidisciplinarity/inter-disciplinarity today.&lt;/p&gt;&#xA;&lt;p&gt;The 'increasing' or 'explosive' increase of knowledge has been noted by many thinkers, with Simone Weil, for example, criticising the 'useless piling up of knowledge' as though that constitutes progress. Da Vinci, was alive when the renaissance of science was just beginning and one can judge that his famous drawing of &lt;em&gt;The Vitruvian Man&lt;/em&gt;, of man encircled by a geometric figure, was his 'prophecy' of a science in harmony with that of man. Whereas on the cusp of the 20th C, Francis Bacon replied with a painting of man enslaved and eviscerated within a matrix of cubic satanic lines: the development of technological 'progress' rather than freeing man, has made of him a slave.&lt;/p&gt;&#xA;&lt;p&gt;Simone Weil, considered that the critique of religion begun by Marx and his contemporaries (and which does not neccessarily mean it's overthrowing) had to be accompanied or replaced by a critique of technology and science. In the secular west, she considered this as the replacement of religion as the new opium of the people.&lt;/p&gt;&#xA;&lt;p&gt;This cannot be done without surveying the whole and is another motivational reason for inter-disciplinary as well as multi-disciplinary studies.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-08-30T20:33:00.627" LastActivityDate="2020-08-30T20:33:00.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12185" PostTypeId="1" CreationDate="2020-08-26T12:33:57.017" Score="0" ViewCount="779" Body="&lt;p&gt;I stumbled across the following quote and couldn't understand how one wouldn't use the factor of 1/2 without completely disrupting the work-energy principle. Though, informal, energy is defined as the ability to do work, it still captures the essence of definition and the relationship. You remove the factor of 1/2 from kinetic energy formula, you wrongly end up with more work.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Preston and De Pretto, following Le Sage, imagined that the universe&#xA;was filled with an ether of tiny particles that always move at speed&#xA;c. Each of these particles has a kinetic energy of mc^2 up to a small&#xA;numerical factor. &lt;strong&gt;The nonrelativistic kinetic energy formula did not&#xA;always include the traditional factor of 1/2, since Leibniz introduced&#xA;kinetic energy without it, and the 1/2 is largely conventional in&#xA;prerelativistic physics.&lt;/strong&gt;[74] By assuming that every particle has a&#xA;mass that is the sum of the masses of the ether particles, the authors&#xA;concluded that all matter contains an amount of kinetic energy either&#xA;given by E = mc^2 or 2E = mc^2 depending on the convention.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Others&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mass–energy_equivalence#Others&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Both Preston and De Pretto used the kinetic energy formula without the 1/2 factor in 1875 and 1903 respectively.[&lt;a href=&quot;https://arxiv.org/ftp/physics/papers/0611/0611300.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Leibniz introduced the vis viva concept back in 1686.&lt;/p&gt;&#xA;&lt;p&gt;Source: The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering 2013, Page #5&lt;/p&gt;&#xA;&lt;p&gt;It was Coriolis who introduced the kinetic energy formula with factor 1/2 and the notion of work around 1829.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Coriolis introduces the word work (travail) to indicate what Carnot&#xA;called moment of activity and others moment, mechanical power,&#xA;quantity of action, energy, or even simply force.&lt;/p&gt;&#xA;&lt;p&gt;These various and quite vague expressions were not suitable to spread&#xA;easily. We propose the appellation dynamical work, or simply work, for&#xA;the quantity ∫Pds [...]. This name will not be confused with any&#xA;other mechanical denomination. It seems suitable to give the right&#xA;idea of the thing, by maintaining its common meaning of physical work&#xA;[...] this name is then suitable to designate the union of these two&#xA;concepts, displacement and force [78].10 (A. 16.8)&lt;/p&gt;&#xA;&lt;p&gt;Coriolis uses the term work also in subsequent studies, particularly&#xA;in the Mémoire sur la manière d’differéns établi les principes pour&#xA;des systèmes de mécanique des corps, comme en des assemblages de&#xA;considérant the molecules of 1835 [79]. It is a use that he definitely&#xA;will consolidate with his work Mécanique des corps solides of 1844&#xA;where, in the preface, he writes:&lt;/p&gt;&#xA;&lt;p&gt;I employed in this work some new nomenclature: I name work the&#xA;quantity usually named puissance mécanique, quantité d'action ou effet&#xA;dynamique, and I propose the name dy- namode for the unity of measure&#xA;of this quantity. I introduced also one more little innovation by&#xA;naming living force the product of the weight times the height&#xA;associated to the height. This living force is one half of the product&#xA;that today is associated to this name, i.e. the mass times the square&#xA;of speed [80].11 (A. 16.9)&lt;/p&gt;&#xA;&lt;p&gt;Notice Coriolis is introducing the factor 1/2 in front of the&#xA;expression of living force (i.e. kinetic energy), because he suggests&#xA;measuring the living force of a body of mass m and velocity v with the&#xA;product mh, with h the height the body can reach if thrown upward&#xA;with an initial velocity v (h = v^2/2).&lt;/p&gt;&#xA;&lt;p&gt;In a note to the passage quoted above, Coriolis writes:&lt;/p&gt;&#xA;&lt;p&gt;This term work is so natural in the sense that I use it, which, though&#xA;it has never been either proposed or approved as a technical&#xA;expression, nevertheless it was used accidentally by Mr. Navier in his&#xA;notes on Belidor and Prony in his Mémoire sur les expériences de la&#xA;machine du Gros-Caillou [80].12 (A. 16.10) Although Coriolis’s texts&#xA;were fundamental to the spread of the term work, again, at the end of&#xA;the XIX century propositions like: principle of virtual velocities,&#xA;principle of moments and principle of virtual work, co-existed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: History of Virtual Work Laws, A History of Mechanics Prospective By Danilo Capecchi • 2012, Page #368&lt;/p&gt;&#xA;&lt;p&gt;My question is that how come scientists, like Preston and De Pretto, were using the 'wrong' formula for kinetic energy without any 1/2 factor. Was work-energy relationship not really established until the beginning of 20th century, or did they not really care about the relationship? Personally, I doubt that the relationship wasn't well establish or they didn't care about it.&lt;/p&gt;&#xA;&lt;p&gt;I'd appreciate it if you could help me to find the answer.&lt;/p&gt;&#xA;&lt;p&gt;By the way, Wikipedia cites this article &lt;em&gt;Prentiss, J.J. (August 2005), &amp;quot;Why is the energy of motion proportional to the square of the velocity?&amp;quot;, American Journal of Physics&lt;/em&gt; but unfortunately it's behind the paywall and I cannot pay for it.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="9341" LastEditDate="2020-08-27T00:02:05.257" LastActivityDate="2020-08-27T00:02:05.257" Title="kinetic energy formula written as mv^2" Tags="|physics|terminology|classical-mechanics|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12186" PostTypeId="1" CreationDate="2020-08-26T14:14:03.353" Score="3" ViewCount="725" Body="&lt;p&gt;Regarding the Islamic mathematician and astronomer Omar Khayyam, known (among other things) for his accurate calculation of the year, quoted from &lt;a href=&quot;https://www.famousscientists.org/omar-khayyam/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.famousscientists.org/omar-khayyam/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Khayyam found that 1,029,983 days made 2,820 years. This gives a tropical year length of 365.2422 days to seven significant figures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I haven't been able to find out how he did such a thing.  Clearly he did not wait 2820 years and count the days.  So what was the method?&lt;/p&gt;&#xA;" OwnerUserId="12978" LastEditorUserId="4931" LastEditDate="2020-08-27T21:29:48.017" LastActivityDate="2020-08-28T18:01:14.393" Title="How did Khayyam calculate the year so accurately?" Tags="|astronomy|archaeoastronomy|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12188" PostTypeId="1" CreationDate="2020-08-26T21:05:41.587" Score="1" ViewCount="52" Body="&lt;p&gt;I was looking at this &lt;a href=&quot;https://www.jstor.org/stable/1576418?origin=crossref&amp;amp;seq=1#metadata_info_tab_contents&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; about &amp;quot;Low complexity art&amp;quot; which involves kolmogorov complexity and I was wondering of when did that complexity theory began being applied to other things that aren't purely mathematical.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-08-26T21:05:41.587" Title="When did Kolmogorov complexity begin being applied to real world objects and ideas?" Tags="|mathematics|biographical-details|kolmogorov|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12189" PostTypeId="1" AcceptedAnswerId="12190" CreationDate="2020-08-27T09:30:08.653" Score="6" ViewCount="646" Body="&lt;p&gt;For quite some time I had the wrong impression that classical Greek mathematicians didn't use fractions.  (I don't remember where I had this from.)  But I recently looked into Heath's book about Archimedes and apparently Archimedes used fractions pretty much like we use them today - except for utilizing a different notation.&lt;/p&gt;&#xA;&lt;p&gt;My question now is: How did Archimedes and his contemporaries view fractions?&lt;/p&gt;&#xA;&lt;p&gt;To be a bit more clear about what I mean: In our &amp;quot;modern&amp;quot; view, fractions are simply numbers.  We can add or multiply them to get other fractions.  Natural numbers are just special fractions in the sense that &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; is identified with &lt;span class=&quot;math-container&quot;&gt;$7/1$&lt;/span&gt;.  It makes sense to compare natural numbers and fractions as in &lt;span class=&quot;math-container&quot;&gt;$1 &amp;lt; 6/5 &amp;lt; 2$&lt;/span&gt;, you can &amp;quot;mix&amp;quot; natural numbers and fractions in computations, and the sum or the product of two fractions can be a natural number.&lt;/p&gt;&#xA;&lt;p&gt;Is this essentially also how Archimedes saw it or were fractions completely different from natural numbers in the sense that natural numbers were the only &amp;quot;real&amp;quot; numbers while fractions were just a convenient device to express ratios between geometric magnitudes?  (If the latter were the case, it wouldn't make much sense to add fractions, for example.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In searching for the source of my misconception I found this&#xA;sentence in Dantzig's &amp;quot;Number - The Language of Science&amp;quot;:&#xA;&amp;quot;Diophantus was the first Greek mathematician who frankly&#xA;recognized fractions as numbers.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="3555" LastEditorUserId="4251" LastEditDate="2021-09-24T08:00:20.937" LastActivityDate="2021-09-24T08:00:20.937" Title="Did Archimedes view fractions as &quot;numbers&quot;?" Tags="|mathematics|ancient-greece|archimedes|antiquity|arithmetic|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12190" PostTypeId="2" ParentId="12189" CreationDate="2020-08-27T10:24:10.133" Score="7" Body="&lt;p&gt;No, Archimedes, and ancient Greeks generally, did not see fractions as numbers, and they did not use fractions as we use them today, they did not use them at all. What they used was ratios of magnitudes. Despite some superficial similarities, ratios were not fractions, and they were not single entities, numbers or otherwise, they were relations between magnitudes, numbers, lines, areas, volumes, etc. Both relata were preserved in the ratio, 7:1 was not identified with 7, inscribed sphere:cylinder was not identified with 2:3, even though Archimedes proved them equal. Equality was not identity, areas and volumes were not numbers attached to geometric objects, they &lt;em&gt;were&lt;/em&gt; the objects.&lt;/p&gt;&#xA;&lt;p&gt;Ratios could be compared using Eudoxus's trick, multiplied only when it made sense geometrically, the product of line ratios was an area ratio, but not added or subtracted. See &lt;a href=&quot;https://hsm.stackexchange.com/a/1931/55&quot;&gt;What was the aftermath of the proof of irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; for the Greeks?&lt;/a&gt; for more details and references, and &lt;a href=&quot;https://hsm.stackexchange.com/a/7885/55&quot;&gt;Did Eudoxus really set out to present irrationals as Dedekind cuts?&lt;/a&gt; on how Eudoxian ratio theory compares to modern real numbers. Here is from  &lt;a href=&quot;http://www.lamath.org/journal/vol5no2/Ratio_Proportion.pdf&quot; rel=&quot;noreferrer&quot;&gt;Ratio and Proportion in Euclid by Madden&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We think of a ratio as a number obtained from other numbers by division. A proportion, for us, is a statement of equality between two “ratio‐numbers”. When we write a proportion such as a/b=c/d, the letters refer to numbers, the slashes are operations on numbers and the expressions on either side of the equals sign are numbers (or at least become numbers when the numerical values of the letters are fixed). This was not the thought pattern of the ancient Greeks. When Euclid states that the ratio of A to B is the same as the ratio of C to D, the letters A, B, C and D do not refer to numbers at all, but to segments or polygonal regions or some such magnitudes. The ratio itself, according to Definition V.3, is just “a sort of relation in respect of size” between magnitudes.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If we wish to compare two magnitudes, the first thing about them that we observe is their relative size. They may be the same size, or one may be smaller than the other. If one is smaller, we acquire more information by finding out how many copies of the smaller we can fit inside the larger. We can get even more information if we look at various multiples of the larger, and for each multiple, determine how many copies of the smaller fit inside. So, a ratio is implicitly a comparison of all the potential multiples of one magnitude to all the potential multiples of the other.  (Two magnitudes are incommensurable exactly when no multiple of one is ever exactly equal to any multiple of the other.) To compare two ratios, A:B and C:D, then, we ought to be prepared to compare the array of all possible (whole‐number) multiples of the first pair with the array of all possible (whole‐number) multiples of the second.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is this sort of comparison that Archimedes uses in his Measurement of the Circle, and which is today reinterpreted as producing fractional &amp;quot;estimates&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, see &lt;a href=&quot;https://hsm.stackexchange.com/a/1868/55&quot;&gt;Who was the first to calculate &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-08-27T10:34:13.573" LastActivityDate="2020-08-27T10:34:13.573" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12191" PostTypeId="2" ParentId="2135" CreationDate="2020-08-27T11:40:06.800" Score="2" Body="&lt;p&gt;I agree with Patrice Ayme on the topic of Buridan and will add that one major role of those employed in a teaching role at a university should be to provide comments and ideas that are helpful to the progress of thought. In this he was clearly successful in spite of religious leaders who had immense power to destroy those who opposed them. The fact that those leaders of society were religious leaders enables people in the modern world to easily criticise but religious viewpoints aren't the only way that free thought may be discriminated against.&lt;/p&gt;&#xA;&lt;p&gt;Philosophy isn't as highly regarded as it was a few decades ago but I wish it was. Modern thought on many topics including science, economic and perhaps even political theory may leave us with many conflicting ideas but sometimes a lack of real direction towards problem solving. Theories have the value of creating a framework that can hopefully be tested but conversely can be held back by the need to exist within the framework of pre-existing theories. What about laws rather than theories of explanation, like the laws of thermodynamics, for example? Perhaps it's possible to explain gravity, for example, by using the laws of thermodynamics, but such an explanation is limited in its use to science unless it exists within a framework of measurable results. To provide the latter may well require a total rethink on certain ideas, definitions or theories. Perhaps to continue the thinking on what controls movement of mass we need philosophers to question those ingrained ideas on basic notions like &amp;quot;What is a force?&amp;quot;, for example and is it worth having another look at the relationship between energy and momentum.&lt;/p&gt;&#xA;&lt;p&gt;Einstein created a new framework to connect mass to the phenomenon that we comprehend as gravity. He also used F = ma in creating a new way to define energy in terms of mass and the speed of light yet he didn't so much say what caused these properties to exist. It was more about explaining what we already knew was happening in another measurable way, which in turn led to numerous conclusions of immense value. Things like relativity were already on the discussion table at least as far back as Galileo but mathematically minded scientists like Newton just didn't have a known framework for approaching the necessary measurements, let alone calculations.&lt;/p&gt;&#xA;" OwnerUserId="12980" LastActivityDate="2020-08-27T11:40:06.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12193" PostTypeId="2" ParentId="12186" CreationDate="2020-08-27T18:56:12.480" Score="-1" Body="&lt;p&gt;He used the same principle as Hipparchus and all other astronomers, including those more ancient than Hipparchus.&lt;/p&gt;&#xA;&lt;p&gt;The idea is the following: if you can determine some fixed moment in the year (like a solstice or an equinox) with error E, then by doing this N years apart, you divide the number of days between two successive events on N, and your error will be E/N. So larger the N is better your accuracy is. For example, if you know the only the number of days from one equinox to another 1000 years later, your can determine the length of a year with accuracy 1/1000 of a day.&lt;/p&gt;&#xA;&lt;p&gt;We don't know exactly what ancient observations Hipparchus used (but it is almost certain that he had access to some ancient Babylonian observations few hundred years before him. Hipparchus own observations are preserved by Ptolemy.&lt;/p&gt;&#xA;&lt;p&gt;But Omar Khayyam had a much longer period at his disposal, at least from Hipparchus to his own time. (And more, since Ptolemy also preserved more ancient observations). Accordingly his accuracy was higher.&#xA;Speaking of the techniques of fixing the exact moment of a solstice or equinox,&#xA;there was no much improvement from the times of Hipparchus to the time of Khayyam.&#xA;So using my notation, their E was of the same order of magnitude, while Khayyam's N was much larger.&lt;/p&gt;&#xA;&lt;p&gt;You can obtain even better accuracy by computing the time from the earliest observation known to you to an equinox which you can observe yourself.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-08-27T18:56:12.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12194" PostTypeId="2" ParentId="12186" CreationDate="2020-08-28T08:57:17.747" Score="4" Body="&lt;p&gt;Khayyam simply did not achieve this precision. It is an urban myth, typically lacking any proper sources and references. Only occasionaly one finds mentions such as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;the Persian calendar using the 2820-year arithmetic algorithm, as&#xA;suggested by Ahmad Birashk and others (and mistakenly attributed to&#xA;Omar Khayyam)&amp;quot;..&#xA;&lt;a href=&quot;http://persian-computing.org/archives/Sharif-FarsiWeb-Inc/Iranian_calendar.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://arxiv.org/pdf/astro-ph/0409620.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;a paper at the arxiv the author states&lt;/a&gt; that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The length of the year in the Iranian calendar, as conceived by&#xA;Khayyam et al., is 365.2424.. days (Youschkevitch &amp;amp; Rosenfeld 1973), a&#xA;logical consequence of the intercalation system: 365 + 8 / 33 = 365.2424.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Section 7 of the paper is &lt;em&gt;&amp;quot;The putative 2820-year cycle&amp;quot;&lt;/em&gt; where  he elaborates:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;recently an extremely precise value for the duration of the year&#xA;(365.24219858156 days) has been attributed to Khayyam (O’Connor &amp;amp;&#xA;Robertson 1999), although it is supported by no historical source, as&#xA;far as we know. We think that this may be a spurious value resulting&#xA;from the erroneous 2820-year cycle suggested recently.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In an other paper at the arxiv there is just a hint at the issue mentioning &lt;em&gt;The  researches  of  two  modern  Iranian  scholars. These  two  scholars  are  Zabīh Behrūz and  Ahmad  Birashk&lt;/em&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;They  have  accepted a  system  of  intercalation  based on a 2820&#xA;years principal cycle with its own 128 years sub- cycles, 29 or 33&#xA;years sub-sub-cycles, “tetraennial”s, and “ pentaennial”s.&lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1111/1111.4926.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[2]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it appears that they relied on modern 19th c. data to obtain a cycle with an integer number of days and next proceeded to implement an intercalation scheme that matches older practices. (For the record &lt;a href=&quot;https://en.wikipedia.org/wiki/Metonic_cycle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Meton&lt;/a&gt; solved an analogous problem for the luni-solar calendar). There is no good explanation how to  explain the 4 years added at the end of a series of 22 repetions of 128 years sub cycles, except that the number 2820 is known in advance. The precision needed to arrive at it does not seem possible without sophisticated time keeping devices and observational instruments as the equinox should be clocked with an error less than a half of second. (Averaging data could produce by chance the decimals needed.)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2020-08-28T18:01:14.393" LastActivityDate="2020-08-28T18:01:14.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12195" PostTypeId="1" CreationDate="2020-08-28T10:35:12.083" Score="6" ViewCount="181" Body="&lt;p&gt;When I read Dickson's &lt;em&gt;History Of The Theory Of Numbers Vol-2&lt;/em&gt;, I found that there seems to be a mistake in the approximation of partition numbers p(200). For this reason, I found the original text according to the literature. In the original text, the theorem gives the asymptotic formula of partition number.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$p(n)\sim\sum^v_{q=1}A_q\phi_q+\mathcal{o}\left(n^{-\frac{1}{4}}\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{matrix}&#xA;\begin{array}{ll}&#xA;A_1=1&amp;amp;A_2=\cos(n\pi)\\&#xA;A_3=2\cos\left(\frac{2}{3}n\pi-\frac{\pi}{18}\right)&amp;amp;&#xA;A_4=2\cos\left(\frac{1}{2}n\pi-\frac{\pi}{8}\right)\\&#xA;A_5=2\cos\left(\frac{2}{5}n\pi-\frac{\pi}{5}\right)+2\cos\left(\frac{4}{5}n\pi\right)\qquad&amp;amp;A_6=2\cos\left(\frac{1}{3}n\pi-\frac{5\pi}{18}\right)\\&#xA;\end{array}\\&#xA;\begin{array}{l}&#xA;A_7=2\cos\left(\frac{2}{7}n\pi-\frac{5\pi}{14}\right)+2\cos\left(\frac{4}{7}n\pi-\frac{\pi}{14}\right)+2\cos\left(\frac{6}{7}n\pi+\frac{\pi}{14}\right)\qquad\\&#xA;A_8=2\cos\left(\frac{1}{4}n\pi-\frac{7\pi}{16}\right)+2\cos\left(\frac{3}{4}n\pi-\frac{\pi}{16}\right)\\&#xA;\end{array}&#xA;\end{matrix}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\phi_q&amp;amp;=\dfrac{\sqrt{q}}{2\pi\sqrt{2}}\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\frac{C\lambda_n}{q}\right)}{\lambda_n}\right)\\&#xA;&amp;amp;=\dfrac{\sqrt{q}}{2\pi\sqrt{2}}\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\pi\sqrt{\frac{2}{3}}\cdot\frac{1}{q}\cdot\sqrt{\frac{1}{n}-\frac{1}{24}}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\\&#xA;&amp;amp;=\dfrac{\sqrt{q}}{2\pi\sqrt{2}}\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\frac{\pi}{q}\sqrt{\frac{2}{3}\left(\frac{1}{n}-\frac{1}{24}\right)}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/e9LJA.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/e9LJA.png&quot; alt=&quot;Approximation formulas&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the original text, the first six terms of series were used to calculate p(100), and the first eight terms of series were used to calculate p(200). But Wikipedia[2] says the first five terms of the series are used to calculate p(200). &lt;strong&gt;However, in the process of calculating p(200), are some values marked in red incorrect?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;A_1\phi_1(200)&amp;amp;=\dfrac{1}{2\pi\sqrt{2}}\left.\left\{\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\pi\sqrt{\frac{2}{3}\left(\frac{1}{n}-\frac{1}{24}\right)}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\right\}\right|_{n=200}\\&#xA;&amp;amp;\approx3972998993185.896\\&#xA;A_2\phi_2(200)&amp;amp;=\dfrac{1}{2\pi}\left.\left\{\cos\left(n\pi\right)\dfrac{\mathrm{d}}{\mathrm{d}n}\left(\dfrac{\exp\left(\frac{\pi}{2}\sqrt{\frac{2}{3}\left(\frac{1}{n}-\frac{1}{24}\right)}\,\right)}{\sqrt{\frac{1}{n}-\frac{1}{24}}}\right)\right\}\right|_{n=200}\\&#xA;&amp;amp;\approx36282.978\\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;&amp;amp;&amp;amp;A_3\phi_3(200)&amp;amp;\approx-87.584&amp;amp;&amp;amp;A_4\phi_4(200)\approx5.147&amp;amp;&amp;amp;A_5\phi_5(200)\approx1.424\\&#xA;&amp;amp;&amp;amp;A_6\phi_6(200)&amp;amp;\approx0.071&amp;amp;&amp;amp;A_7\phi_7(200)=0&amp;amp;&amp;amp;A_8\phi_8(200)\approx0.044\\&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\text{My Vertical Calculation}&amp;amp;\qquad&amp;amp;\text{Original Vertical Calculation}\\&#xA;\begin{array}{r}&#xA;3972998993185.896\\&#xA;+\,\,36282.978\\&#xA;-\,\,87.584\\&#xA;+\,\,5.147\\&#xA;+\,\,1.424\\&#xA;+\,\,0.071\\&#xA;+\,0\qquad\\&#xA;+\,\,0.044\\&#xA;\hline&#xA;3972999029387.975&#xA;\end{array}&amp;amp;\qquad&amp;amp;&#xA;\begin{array}{r}&#xA;3972998993185.896\\&#xA;+\,\,36282.978\\&#xA;-\,\,{\color{red}{87.\boxed{555}}}\\&#xA;+\,\,5.147\\&#xA;+\,\,1.424\\&#xA;+\,\,0.071\\&#xA;+\,0\qquad\\&#xA;+\,\,{\color{red}{0.\boxed{043}}}\\&#xA;\hline&#xA;3972999029388.004&#xA;\end{array}&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally, I searched the results (3972999029387.975) in &lt;em&gt;Google&lt;/em&gt; and found the lecture notes [5].&lt;/p&gt;&#xA;&lt;hr/&gt;&#xA;&lt;p&gt;[1] History Of The Theory Of Numbers Vol-2 (Leonard Eugene Dickson)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/HistoryOfTheTheoryOfNumbersVolII/page/n187/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/HistoryOfTheTheoryOfNumbersVolII/page/n187/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[2] Partition function [Wikipedia]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Approximation_formulas&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Approximation_formulas&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[3] &lt;em&gt;Asymptotic Formulæ In Combinatory Analysis&lt;/em&gt; (G. H. Hardy &amp;amp; S. Ramanujan) [Original edition]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/hardy-ramanujan-1918-asymptotic-formulaae-in-combinatory-analysis/page/84/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/hardy-ramanujan-1918-asymptotic-formulaae-in-combinatory-analysis/page/84/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[4] &lt;em&gt;Asymptotic Formulæ In Combinatory Analysis&lt;/em&gt; (G. H. Hardy &amp;amp; S. Ramanujan) [Reset edition BY LaTeX]&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://ramanujan.sirinudi.org/Volumes/published/ram36.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://ramanujan.sirinudi.org/Volumes/published/ram36.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[5] &lt;em&gt;Lectures on Integer Partitions&lt;/em&gt; (Herbert S. Wilf)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.math.upenn.edu/%7Ewilf/PIMS/PIMSLectures.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.math.upenn.edu/~wilf/PIMS/PIMSLectures.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12982" LastEditorUserId="10106" LastEditDate="2020-08-29T04:03:02.507" LastActivityDate="2020-08-29T04:03:02.507" Title="Did Hardy and Ramanujan miscalculate these values?" Tags="|mathematics|number-theory|ramanujan|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12196" PostTypeId="1" AcceptedAnswerId="12197" CreationDate="2020-08-29T20:58:29.400" Score="7" ViewCount="517" Body="&lt;p&gt;I recently came across the term &amp;quot;triangle given in species&amp;quot; in Hatton's &lt;a href=&quot;https://archive.org/details/cu31924060184045/page/n367/mode/2up?q=species&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Projective Geometry&lt;/em&gt;&lt;/a&gt;.  Searching in &lt;a href=&quot;https://archive.org/search.php?query=%22given%20in%20species%22&amp;amp;sin=TXT&amp;amp;page=3&quot; rel=&quot;noreferrer&quot;&gt;archive.org&lt;/a&gt; turned up other examples (&lt;a href=&quot;https://archive.org/details/elementsgeometr02leslgoog/page/n380/mode/2up?q=species&quot; rel=&quot;noreferrer&quot;&gt;such as this&lt;/a&gt;) of 19th century texts, and it is also used in Euclid.  But I don't see it defined anywhere.  By context, I think it refers to similar triangles, but it would be nice to see a definition somewhere.  Also, does it have a meaning wider than &amp;quot;similar&amp;quot; in geometry?&lt;/p&gt;&#xA;" OwnerUserId="11204" LastActivityDate="2020-08-30T02:04:42.243" Title="What does &quot;given in species&quot; mean in old geometry textbooks?" Tags="|mathematics|terminology|geometry|ancient-greece|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12197" PostTypeId="2" ParentId="12196" CreationDate="2020-08-29T22:24:12.273" Score="9" Body="&lt;p&gt;Such terms as “given in species” are defined in Euclid’s &lt;em&gt;Data&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=vLM2AAAAMAAJ&amp;amp;pg=PA2#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Greek&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=LwE_vwEACAAJ&amp;amp;pg=PA367#v=onepage&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;English&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;III. Rectilineal figures are said to be given in species, which have each of their angles given, and the ratios of their sides given. (English version, R. Simpson, 1810, p. 367)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[&lt;em&gt;Species&lt;/em&gt; is the translation of &lt;em&gt;eidos&lt;/em&gt;, shape or form; see LSJ, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/morph?l=ei%29%3Ddos&amp;amp;la=greek#lexicon&quot; rel=&quot;noreferrer&quot;&gt;εἶδος&lt;/a&gt;, def. A.2.b.]&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2020-08-30T02:04:42.243" LastActivityDate="2020-08-30T02:04:42.243" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12198" PostTypeId="1" CreationDate="2020-08-29T22:59:32.513" Score="1" ViewCount="67" Body="&lt;p&gt;Many mathematical objects and even theories are named after people, and most of the people with mathematical things named after them did not name said things after themselves. There is the famous anecdote that in a seminar David Hilbert had to ask what a &amp;quot;Hilbert space&amp;quot; was. Presumably Hilbert called it something else.&lt;/p&gt;&#xA;&lt;p&gt;In fact, mathematics is full of things named after people. A very short and nowhere near complete list include the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Tate_module&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tate modules&lt;/a&gt; ;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Cartier_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cartier divisors&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Selberg_sieve&quot; rel=&quot;nofollow noreferrer&quot;&gt;Selberg sieve&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/G-ring&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grothendieck ring&lt;/a&gt;;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hardy_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hardy space&lt;/a&gt;, etc.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Of course, there are also many mathematical objects with very mundane names. What are some examples of mathematical objects, preferably those involved in active areas of research, which retained the names given to them by their discoverers?&lt;/p&gt;&#xA;" OwnerUserId="2681" LastActivityDate="2020-08-29T22:59:32.513" Title="Which mathematical objects retained the names given by their discoverer(s)?" Tags="|mathematicians|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12199" PostTypeId="2" ParentId="12196" CreationDate="2020-08-29T23:09:05.810" Score="1" Body="&lt;p&gt;This terminology does appear in English 100+ years ago, and is arguably archaic, but just means &amp;quot;types&amp;quot; or &amp;quot;kinds&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2020-08-29T23:09:05.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12200" PostTypeId="1" CreationDate="2020-08-29T23:10:46.287" Score="15" ViewCount="354" Body="&lt;p&gt;This is my first time posting on HSM, so please bear with me if it's off-topic. I can move it to Stats.SE or Mathematics.SE if necessary.&lt;/p&gt;&#xA;&lt;p&gt;A &lt;a href=&quot;https://nvlpubs.nist.gov/nistpubs/jres/70C/jresv70Cn4p263_A1b.pdf&quot; rel=&quot;noreferrer&quot;&gt;widely cited 1966 paper&lt;/a&gt; (with currently 1030 citations) mentions the &amp;quot;law of propagation of error&amp;quot; but does not actually state it or give any citation directly.&lt;/p&gt;&#xA;&lt;p&gt;That 1966 paper does open though, with a citation to a 1939 paper by Raymond Birge which never mentions the &amp;quot;law of propagation of error&amp;quot; but Birge did write &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.40.207&quot; rel=&quot;noreferrer&quot;&gt;a 1932 paper&lt;/a&gt; which refers to what he calls &amp;quot;the well-known law of propagation of error&amp;quot;, but again with no statement of the law, or citation to where that term was used previously.&lt;/p&gt;&#xA;&lt;p&gt;I assume that the &amp;quot;law&amp;quot; to which they are referring is this one:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9tHM0.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9tHM0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;which is quite ubiquitous, and is given for example in &lt;a href=&quot;https://en.wikipedia.org/wiki/Propagation_of_uncertainty&quot; rel=&quot;noreferrer&quot;&gt;this Wikipedia article&lt;/a&gt;. Surprisingly though, the entire Wikipedia article does not discuss the origins of this formula. &lt;em&gt;Furthermore,&lt;/em&gt; if I search &amp;quot;law of propagation of error&amp;quot; on Google, I basically only find the above papers over and over again, which is quite frustrating. Both of Raymond Birge's papers heavily cite a 1894 book called &amp;quot;Method of Least Squares&amp;quot; of which I was able to &lt;a href=&quot;https://archive.org/details/textbookonmethod00merruoft/page/62/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;find&lt;/a&gt; the 1910 edition.&lt;/p&gt;&#xA;&lt;p&gt;A similar question has been asked here at HSM.SE but it was about when this formula became &amp;quot;prominent&amp;quot; not about when the &amp;quot;law&amp;quot; was first stated: &lt;a href=&quot;https://hsm.stackexchange.com/q/6092/8052&quot;&gt;When did error propagation become prominent in physics?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What is the origin of the &amp;quot;Law of propoagation of error&amp;quot;?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;Update: &lt;a href=&quot;https://aapt.scitation.org/doi/10.1119/1.1991484&quot; rel=&quot;noreferrer&quot;&gt;The 1939 paper&lt;/a&gt; by Raymond Birge does actually state the &amp;quot;law&amp;quot; and it is indeed what I thought it was:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Z62kG.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z62kG.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The reason I didn't find it earlier was because Adobe's optical character recognition was unable to find any instances of &amp;quot;law&amp;quot; or &amp;quot;propagation&amp;quot; in the text.&lt;/p&gt;&#xA;" OwnerUserId="8052" LastEditorUserId="8052" LastEditDate="2020-08-31T22:18:30.150" LastActivityDate="2020-08-31T22:18:30.150" Title="When was the &quot;Law of Propagation of Error&quot; first stated?" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12201" PostTypeId="2" ParentId="12189" CreationDate="2020-08-29T23:44:40.663" Score="0" Body="&lt;p&gt;Whilst Archimedes may not have viewed fractions as we view them, this doesn't mean that they didn't have the concept of a fraction. This is in part, because there is more than one way to view the concept of a number.&lt;/p&gt;&#xA;&lt;p&gt;One of the viewpoints put forward by category theory is that the natural integers (the strictly positive integers, that is not including zero) are a decategorification of &lt;em&gt;FinSet&lt;/em&gt;, the category of all finite sets. This is different from von Neumanns definition of a natural integer, which is to take the quotient by the natural equivalence relation that identifies all sets of the same cardinality.&lt;/p&gt;&#xA;&lt;p&gt;To reassert &lt;em&gt;FinSet&lt;/em&gt;, in place of the natural integers is to say that we categorify the positive integers. That this is a natural thing to do is shown by the fact that in certain places where we would use the natural integers, it is more natural to parametrise by finite sets.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, we can categorify fractions as we use them today, and find that we have the fractions that Archimedes used. In that language, whilst 8/1 is &lt;em&gt;not&lt;/em&gt; 8, they are most assuredly isomorphic - and this means here, equivalent.&lt;/p&gt;&#xA;&lt;p&gt;We might then say that Archimedes is being more modern than the most modern conceptions of number ...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-29T23:44:40.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12202" PostTypeId="2" ParentId="2135" CreationDate="2020-08-30T00:02:04.427" Score="2" Body="&lt;p&gt;Newton is 'venerated' for his synthesis of the laws of motion, the theory of gravity and the calculus. Virtually everything he did had some antecedent in work earlier than his, but he put it all together.&lt;/p&gt;&#xA;&lt;p&gt;Buridans name 'has faded into obscurity' not because of the lack of the importance of his work, but simply because the history of science is not taken as seriously as science itself. What is presented as such in textbooks is merely the barest smattering of such a history.&lt;/p&gt;&#xA;&lt;p&gt;By the way, it's not true that the notion of 'inertia' was 'unknown' to Aristotle. This is a common misconception. He would not have phrased it that way. Nevertheless, he did develop the notion of 'natural motion' which is the antecedent notion from which inertia developed. Moreover, it's also incorrect to say that he wrote 'a body is only maintained in motion so long as a continuous force is applied'. Or rather, the interpretation is incorrect.&lt;/p&gt;&#xA;&lt;p&gt;What he said, was that real change occurs only and for as long as force is applied (and the body is potentially capable of changing). The converse, of course, is that when no force is applied, no change occurs.&lt;/p&gt;&#xA;&lt;p&gt;The mistake here is to consider change of position to be actual change. The actual change is not change in position, but change in velocity - that is acceleration.&lt;/p&gt;&#xA;&lt;p&gt;What we then see is that Aristotle elaborated a general theory of change, and then Newton as well as many others preceding him, specialised this for actual physical motion.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-30T00:02:04.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12203" PostTypeId="2" ParentId="6092" CreationDate="2020-08-30T00:24:48.057" Score="4" Body="&lt;p&gt;I came across this question while trying to figure out when the &amp;quot;law of propagation of error&amp;quot; was first stated, which resulted in this question: &lt;a href=&quot;https://hsm.stackexchange.com/q/12200/8052&quot;&gt;When was the &amp;quot;Law of Propagation of Error&amp;quot; first stated?&lt;/a&gt; which now contains a reference to your question.&lt;/p&gt;&#xA;&lt;p&gt;Your question is about when error propagation become prominent (rather than about when it first appeared). That part I can answer by saying that Raymond Birge &lt;a href=&quot;https://aapt.scitation.org/doi/10.1119/1.1991484&quot; rel=&quot;nofollow noreferrer&quot;&gt;in 1939 said&lt;/a&gt; that the question of how to assign an uncertainty has been discussed for decades but the subject matter of error propagation is one for which &amp;quot;many scientists still fail to avail themselves&amp;quot; and that &amp;quot;others frequently use the theory [of error propagation] incorrectly and thus arrive at quite misleading conclusions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Raymond Birge was a prominent enough physicist, as he was the head of Berkeley's Physics Department and was regularly in contact with people like Gilbert Lewis, Robert Oppenheimer, and various other famous Nobel laureates or future Nobel Laureates. Therefore if there was people widely doing error propagation in a way that was &amp;quot;correct&amp;quot; in his eyes, he would have known about it or had access to the people who could advise him of the truth on the matter. In fact it was his frustration with inconsistencies in how people made measurements and reported fundamental physics constants that lead him to publish a series of papers about error propagation. People making measurements of fundamental physics constants would have been elite physicists (having enough money to do such experiments, and having also enough education to do such experiments), so if they were doing error propagation wrong, you can imagine that they were also teaching it wrong to the next generation of physics student.&lt;/p&gt;&#xA;&lt;p&gt;As my above linked question describes, the chronologically next paper which comes up most often is &lt;a href=&quot;https://nvlpubs.nist.gov/nistpubs/jres/70C/jresv70Cn4p263_A1b.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 1966 paper&lt;/a&gt; which says in the first sentence of the abstract:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;law of propagation of error&amp;quot; is a tool that physical scientists&#xA;have conveniently and frequently used in their work for many years&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So by 1966, error propagation seems to have been quite popular. Perhaps Birge's papers in the 1920s and 1930s were what made error propagation popular. I already mentioned that Birge was the head of Physics at Berkeley and therefore taught many students that became prominent future scientists and had influence on a lot of other people too, but if that argument is not convincing enough you can see that &lt;a href=&quot;https://scholar.google.com/scholar?hl=en&amp;amp;as_sdt=2005&amp;amp;sciodt=0%2C5&amp;amp;cites=1510594166718718924&amp;amp;scipsc=&amp;amp;as_ylo=1930&amp;amp;as_yhi=1940&quot; rel=&quot;nofollow noreferrer&quot;&gt;between 1930 and 1940&lt;/a&gt; his paper &amp;quot;Probable values of the general physical constants&amp;quot; was cited in Physical Review or Reviews in Modern Physics papers by I.I. Rabi (from Rabi Oscillations), Marie Curie (who needs not further words), Harold Urey (from the Miller-Urey experiment), Irving Langmuir (the namesake of the ACS journal &lt;em&gt;Langmuir&lt;/em&gt;), Robert Millikan (from the Millikan oil drop experiment), E.E. Witmer (from the Wigner-Witmer rules), Walther Gerlach (from the Stern-Gerlach experiment), and about 70 other papers between 1930 and 1940 alone: this I think is enough to say that Birge was influential in science and probably were his papers about error propagation in the 1920s and 1930s.&lt;/p&gt;&#xA;&lt;p&gt;Birge himself said in the 1939 paper that the idea of error propagation was not new, and he frequently cited a 1894 book by Mansfield Merriman called &amp;quot;Method of least Squares&amp;quot; for which a 1910 version can be found &lt;a href=&quot;https://archive.org/details/textbookonmethod00merruoft/page/228/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. So perhaps Merriman (a civil engineer) popularized error propagation enough for it to be noticed by Birge (but insufficiently enough for Birge to be complaining about most people either not being aware of the theory or using it wrong), and then Birge popularized it to many of the prominent or to-become-prominent scientists of his time.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion:&lt;/strong&gt; &lt;em&gt;The mathematics of error propagation was discussed earlier by people like Mansfield Merriman, but even in 1939 it was not well-known by many people and was used incorrectly by many people, but Raymond Birge published several papers in the 1920s and 1930s on the topic of uncertainties in measurements, which were cited by many of the top scientists of the time. By 1966 error propagation was considered to be quite common.&lt;/em&gt; &lt;/p&gt;&#xA;" OwnerUserId="8052" LastEditorUserId="8052" LastEditDate="2020-08-31T12:52:52.440" LastActivityDate="2020-08-31T12:52:52.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12204" PostTypeId="2" ParentId="6092" CreationDate="2020-08-30T01:16:02.007" Score="1" Body="&lt;p&gt;I find this question baffling. Why would anyone be interested in 'propagating errors?'! Surely the question ought to be on estimating errors? Experimental physicists are only interested in how errors propagate in order to estimate errors - it's not an end in itself, but a means to an end and even then, the end here is to establish the level of confidence we have in the predicted value.&lt;/p&gt;&#xA;&lt;p&gt;As soon as people developed the notion of a calculation - lost in time - and as soon as they developed the notion of one calculation depending upon another - again lost in time - they will have noticed as a matter of fact that 'errors propagate'. The likelihood is this will have been in astronomy, or land measurement as these were the earliest of sciences.&lt;/p&gt;&#xA;&lt;p&gt;Archimedes used the method of exhaustion to calculate the area of a circle by an inscribed polygon and also provided an error estimate.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-08-30T04:07:02.550" LastActivityDate="2020-08-30T04:07:02.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12205" PostTypeId="1" CreationDate="2020-08-30T03:54:58.677" Score="1" ViewCount="55" Body="&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a, b$&lt;/span&gt; are positive and coprime integers, then the set of linear combinations of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; with nonnegative coefficients is all integers past &lt;span class=&quot;math-container&quot;&gt;$(a - 1)(b - 1)$&lt;/span&gt;; i.e. &lt;span class=&quot;math-container&quot;&gt;$\{ \lambda_1 a + \lambda_2 b : \lambda_1, \lambda_2 \in \mathbb{Z}_{\geq 0} \} = (a - 1)(b - 1) + \mathbb{Z}_{\geq 0}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On Wikipedia (&lt;a href=&quot;https://en.wikipedia.org/wiki/Coin_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Coin_problem&lt;/a&gt;), it says &amp;quot;This formula was discovered by James Joseph Sylvester in 1882,[6] although the original source is sometimes incorrectly cited as,[7] in which the author put his theorem as a recreational problem[8] (and did not explicitly state the formula for the Frobenius number).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Citation [6] is: Sylvester, J. J. (1882). On subvariants, ie semi-invariants to binary quantics of an unlimited order. &lt;em&gt;American Journal of Mathematics&lt;/em&gt;, 5(1), 79-136.&lt;/p&gt;&#xA;&lt;p&gt;Citation [7] is: Sylvester, J. J. (1884). Question 7382. &lt;em&gt;Mathematical questions from the educational times&lt;/em&gt;, 37, 26.&lt;/p&gt;&#xA;&lt;p&gt;Citation [8] is: Alfonsín, J. L. R., &amp;amp; Alfonsín, J. L. R. (2005). &lt;em&gt;The diophantine Frobenius problem&lt;/em&gt;. Oxford University Press.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Where in Sylvester's paper does he prove this formula? I tried finding it but had no luck.&lt;/li&gt;&#xA;&lt;li&gt;Did Frobenius have anything to do with this proof? These types of problems are called &amp;quot;Frobenius problems,&amp;quot; so it's a little strange that Sylvester is the pioneer.&lt;/li&gt;&#xA;&lt;li&gt;Did anyone famous other than Sylvester independently discover their own proof of this formula? It seems that W. J. Curran Sharp responded to Sylvester's recreational problem in the &lt;em&gt;Educational Times&lt;/em&gt;. In other words, should Sylvester get all the credit?&lt;/li&gt;&#xA;&lt;li&gt;What is a timeline for the first results on the Frobenius problem?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The preface of [8] says that Frobenius brought up these problems in his lectures. [8] also says that Sylvester proved in 1882 that the number of positive integers that are not expressible as nonnegative linear combinations of &lt;span class=&quot;math-container&quot;&gt;$a, b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}(a - 1)(b - 1)$&lt;/span&gt;. Then [8] mentions citation [7]: &amp;quot;in 1884, in the &lt;em&gt;Educational Times&lt;/em&gt; journal, Sylvester posed (as a recreational problem) the question of finding such a formula.&amp;quot; The book [8] has citations, but unfortunately I only have online access to the preface.&lt;/p&gt;&#xA;" OwnerUserId="12710" LastActivityDate="2020-08-30T03:54:58.677" Title="Timeline for the earliest work on Frobenius problems" Tags="|mathematics|mathematicians|discoveries|number-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12207" PostTypeId="2" ParentId="6092" CreationDate="2020-08-30T16:15:57.560" Score="1" Body="&lt;p&gt;A few musings (at least one of which, I think, qualifies this to be an answer rather than a comment):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Strictly speaking, the quantities about which OP is asking are &amp;quot;uncertainties&amp;quot;, not &amp;quot;errors&amp;quot; (&amp;quot;uncertainty&amp;quot; means the standard deviation of the forward probability distribution of a measurement given a fixed true value of the measurand, &amp;quot;error&amp;quot; means the difference between a single sample from that probability distribution and the true value - see the BIPM &lt;a href=&quot;https://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Guide to the expression of uncertainty in measurement&lt;/em&gt;&lt;/a&gt;.  However, I realize this piece of pedantry is unhelpful in a question about the early history of the concepts, because those quantities &lt;em&gt;were&lt;/em&gt; often called random errors in older publications.&lt;/li&gt;&#xA;&lt;li&gt;The formal justification for propagating uncertainties through a calculation by multiplying by the modulus of the relevant partial derivative, and for combining uncertainties by adding in quadrature, is that these procedures emerge from the leading-order Laplace's method approximation to the integral that defines the standard deviation of the forward probability distribution.  Hence, the procedures can't (or at least, shouldn't) have become prominent before Laplace's method was devised: according to the Wikipedia article on Laplace's method, that was in 1774.&lt;/li&gt;&#xA;&lt;li&gt;The earliest example I've been able to find of someone distinguishing clearly between systematic errors and random errors in measurements is by Evelyn (1798, &lt;em&gt;Philos. Trans. R. Soc. Lond.&lt;/em&gt; &lt;strong&gt;88&lt;/strong&gt;:133-182).  Again, it wouldn't have made any sense for anyone to try to propagate uncertainties before that distinction was made.&lt;/li&gt;&#xA;&lt;li&gt;The familiar (to us) formula for propagation and combination of uncertainties is stated and derived in section 149 of Galloway (1839, &lt;em&gt;A treatise on probability&lt;/em&gt;, Adam and Charles Black).  That source appears to be a republication as a book of an article from the 7th edition of Encyclopaedia Britannica, which was published in 1827.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12994" LastEditorUserId="12994" LastEditDate="2020-08-30T21:38:36.370" LastActivityDate="2020-08-30T21:38:36.370" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12208" PostTypeId="1" AcceptedAnswerId="12222" CreationDate="2020-08-30T17:30:16.723" Score="3" ViewCount="180" Body="&lt;p&gt;What was known about the properties of the nucleus (its shape, its density etc) and the nuclear forces before the Liquid drop model was proposed? I believe that some empirical knowledge must be out there already which motivated that the nuclei behave like a drop of liquid.&lt;/p&gt;&#xA;" OwnerUserId="9960" LastActivityDate="2020-09-02T17:42:15.433" Title="What was known about the properties of the nucleus before the Liquid drop model was proposed?" Tags="|physics|experimental-physics|nuclear-physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12209" PostTypeId="2" ParentId="12200" CreationDate="2020-08-30T19:32:54.957" Score="7" Body="&lt;p&gt;The law of propagation of error is discussed at length in&#xA;&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=uc1.$b45951&amp;amp;view=1up&amp;amp;seq=230&amp;amp;q1=209&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Text-book on geodesy and least squares, prepared for the use of civil engineering students&lt;/em&gt;&lt;/a&gt; by Charles L. Crandall, 1907.&lt;/p&gt;&#xA;&lt;p&gt;There is also a paper &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=nyp.33433069089963&amp;amp;view=1up&amp;amp;seq=305&amp;amp;q1=propagation%20of%20error&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;The essentials of error theory for practical engineers&amp;quot;&lt;/a&gt; by L. B. Tuckerman, in the &lt;em&gt;Nebraska Blue Print&lt;/em&gt;, v. 13 (1914), pp.61-84 that discusses the &amp;quot;The law of ''propagation of error,'' i. e. the law expressing the relation between the error of the result, &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, and the errors . . .&amp;quot;) This zinger appears on p.63, but the essay is largely on this subject.&lt;/p&gt;&#xA;&lt;p&gt;It is hard to believe either the concept or name originated in 1907.  Both works I mention above seem pretty sound-minded to me.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2020-08-30T19:50:05.553" LastActivityDate="2020-08-30T19:50:05.553" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12210" PostTypeId="1" CreationDate="2020-08-30T18:41:40.387" Score="0" ViewCount="98" Body="&lt;p&gt;Since it is a strictly physics-related quote i thought this was the best place to ask. It goes something like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;There are no particles, only fields&amp;quot;, to which X replied &amp;quot;there are no fields, but just operators.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not trying to open a discussion about it's phenomenology, just looking for sources. I remember reading it on the web: either from a paper or a site, so no help there (yes, i've googled any key word and searched every page).&lt;/p&gt;&#xA;" OwnerUserId="12997" OwnerDisplayName="physika" LastEditorUserId="26" LastEditDate="2021-02-20T15:08:30.743" LastActivityDate="2021-02-20T15:08:30.743" Title="Looking for the source of a quote in QFT history" Tags="|particle-physics|quantum-field-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12211" PostTypeId="2" ParentId="12200" CreationDate="2020-08-30T20:25:19.243" Score="12" Body="&lt;p&gt;That formula was stated (albeit in a rather different notation) and derived in section 149 of Galloway (1839, &lt;em&gt;A treatise on probability&lt;/em&gt;, Adam and Charles Black), of which Google Books has the full text available. That work appears to be a republication as a book of an article from the 7th edition of Encyclopaedia Britannica, which was published in 1827.&lt;/p&gt;&#xA;&lt;p&gt;I can't be &lt;em&gt;sure&lt;/em&gt; that that's the earliest appearance of it, but as I argued in my &lt;a href=&quot;https://hsm.stackexchange.com/a/12207/12994&quot;&gt;answer&lt;/a&gt;  to the above-mentioned related question, it can't have appeared very much earlier, because the formula relies on a conceptual understanding of errors that was first clearly described in 1798, and an approximation method for integrals which was invented in 1774.&lt;/p&gt;&#xA;" OwnerUserId="12994" LastEditorUserId="8052" LastEditDate="2020-08-31T16:45:24.607" LastActivityDate="2020-08-31T16:45:24.607" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12212" PostTypeId="2" ParentId="12210" CreationDate="2020-08-30T20:27:05.887" Score="1" Body="&lt;p&gt;@Oбжорoв answered this in the PSE: a source is&#xA;Robert J. Sciamanda, Am. J. Phys. 81, 645 (2013); &lt;a href=&quot;http://dx.doi.org/10.1119/1.4812316&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi: 10.1119/1.4812316&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By extending some of Hobson’s ideas, I arrive at the conclusion that in addition to there being no particles, there are not even fields!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7076" LastActivityDate="2020-08-30T20:27:05.887" CommentCount="1" CommunityOwnedDate="2020-08-30T20:27:05.887" ContentLicense="CC BY-SA 4.0" />
  <row Id="12214" PostTypeId="2" ParentId="12154" CreationDate="2020-08-31T00:56:37.520" Score="1" Body="&lt;p&gt;I can confirm that there isn't anything on the net.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, Giacconni has written two books, one on X-Ray astronomy, first published in 1974 and although he mentions the cosmic X-Ray background which would be relevant in the inflationary hypothesis and which gave birth to the multiverse hypothesis (so a hypothesis squared, so to speak) he does not mention the multiverse - which is no surprise - as the earliest mention of multiverses in Steinhardts paper in 1982 in the context of Inflation, and in the following year by Vilenkin in a paper published in &lt;em&gt;Physics Review D&lt;/em&gt;, called &lt;em&gt;Birth of Inflationary Universes&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Nor does he mention either hypothesis in his later, more autobiographical book, &lt;em&gt;The Secrets of the Hoary Deep&lt;/em&gt; and which was published in 2008. According to an Internet search he was not a man to shy away from controversies, which suggests that he simply didn't think that this hypothesis squared was worth his trouble, or anybody else's and that its value as a scientific idea was over-inflated (pun intended).&lt;/p&gt;&#xA;&lt;p&gt;Moreover, it seems in this he's been joined by Steinhardt, who as mentioned above was one of the key originators of the theory and who has since then become a vocal critic of the idea.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-08-31T00:56:37.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12216" PostTypeId="1" CreationDate="2020-08-31T08:34:15.670" Score="1" ViewCount="27" Body="&lt;p&gt;In modern view any quantum field theory that has any reason (mathematical or physical) to not be tenable at some short length scale is assumed to be replaced by a more fundamental theory, that we call &amp;quot;UV completion&amp;quot;.&#xA;Such name is given because it completes the previous theory at some higher energy (UV is an abuse of nomenclature to say higher energy than the relevant scale for which I had formulated the first theory).&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that this is a modern concept (few decades old). For instance I cannot find any trace of such need to complete the model of Fermi contact interactions ( those responsible for muon decay and all &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; decays) even in Enrico Fermi's own speech about the possibility to build a &lt;a href=&quot;https://www.fnal.gov/pub/today/archive/archive_2007/today07-02-16.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;particle collider around the globe&lt;/a&gt; (Also in &lt;code&gt;&amp;quot;Fermi Remembered&amp;quot;  by  James W. Cronin (Editor) - Chapter 9 - What can we learn from high energy accelerators?&lt;/code&gt; ). He gave this speech in 1954 and thought about a fixed target collider (beam on target) which could reach center of mass energy in the TeV range.&lt;/p&gt;&#xA;&lt;p&gt;In modern view the theory of Fermi interactions has a limited energy range of applicability, because it stops making sense from a mathematical point of view, giving probabilities larger than one if one sticks to perturbation theory. Very importantly, the maximum range of applicability is predictable from perturbation theory itself in the same way the Taylor series of any function can be anticipated to break down if we use it too far away from the point around which it has been expanded. Armed with this knowledge, at the master degree level you already know that, carrying out experiments in the TeV energy range, the &amp;quot;globatron&amp;quot; collider Fermi was discussing would have been the perfect machine to discover the UV completion of Fermi theory of weak interactions. Despite this, Fermi does not make any mention of that in this speech.&lt;/p&gt;&#xA;&lt;p&gt;Therefore I have a number of questions, among which:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;in the 1950s was there a sense of how Fermi theory of weak interactions should have become untenable at higher energies?&lt;/li&gt;&#xA;&lt;li&gt;at what point in time it became clear that quantum field theories can be &amp;quot;effective low energy&amp;quot; theories (and there is no harm in that)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thanks for your inputs&lt;/p&gt;&#xA;" OwnerUserId="5863" LastActivityDate="2020-08-31T08:34:15.670" Title="When the concept of &quot;Ultraviolet completion&quot; of a low energy effective theory first arose?" Tags="|particle-physics|quantum-field-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12218" PostTypeId="1" CreationDate="2020-08-31T13:18:42.143" Score="1" ViewCount="61" Body="&lt;p&gt;Just making sure of this.&lt;/p&gt;&#xA;&lt;p&gt;I've just found this statement in the Collins Dictionary of Mathematics (E.J. Borowski and J.M. Borwein, 1989) in the entry for &amp;quot;Lucas Numbers&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;(Named after the English mathematician &lt;em&gt;Henry Lucas&lt;/em&gt; (died 1663).)&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This &lt;em&gt;is&lt;/em&gt; rubbish, isn't it? Every single other reference to Lucas numbers attributes them to Edouard Lucas (1842 - 1891) (or whatever his dates were).&lt;/p&gt;&#xA;&lt;p&gt;My information has it that Henry Lucas was not even a mathematician, &amp;quot;just&amp;quot; a clergyman and politician who bequeathed the Lucasian Chair of Mathematics in his legacy.&lt;/p&gt;&#xA;&lt;p&gt;A simple conflation?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2020-08-31T13:18:42.143" Title="It was Edouard Lucas who is the eponym of Lucas Sequence, and not Henry Lucas, yes?" Tags="|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12219" PostTypeId="2" ParentId="639" CreationDate="2020-08-31T16:07:18.210" Score="0" Body="&lt;p&gt;Fundamentally speaking, there is no distinction.&lt;/p&gt;&#xA;&lt;p&gt;However, innovation is the word adopted by business change management in management studies to both study and manage the process of change which comes with the various and diverse inventions of industrialised capitalism and to theorise a typology of invention with respect to how influential it is - hence the various kinds mentioned in the post -&#xA;sustaining, evolutionary, revolutionary and disruptive. This also explains its adoption as a term by government funding agencies and the like.&lt;/p&gt;&#xA;&lt;p&gt;One important thing to note is that managing innovation is not the same as innovation itself (even if one innovates in the field of managing innovation). This is just as important a distinction in other fields - we don't expect an art critic to neccessarily be any good at painting or drawing after all; and generally speaking, they usually aren't. Likewise with managing innovation and innovation itself. Whilst art can live without art critics and for a long time didn't; art critics cannot live without artists and their making of art. Likewise, we can view the field of innovation studies as an ephiphenenon of the higher pace of innovation in society as it becomes more literate and more skilled in the various arts, that is &lt;em&gt;techne&lt;/em&gt; - ie crafts.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-09-02T11:36:25.943" LastActivityDate="2020-09-02T11:36:25.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12220" PostTypeId="1" CreationDate="2020-08-31T16:54:11.153" Score="-1" ViewCount="193" Body="&lt;p&gt;It's often said - and more often written - and perhaps, even more spoken of - that modern physics began with Galileo due to his application of mathematics to motion. This is the position taken by Einstein in his semi-popular book on the evolution of physics with Infield. Whereas motion had been theorised before, it was Galileos paradigm-shattering idea to quantify motion and hence discover actual quantitative relationships between the various factors of motion, that is distance, mass, speed and acceleration.&lt;/p&gt;&#xA;&lt;p&gt;However, more broadly speaking mathematics had been applied much earlier in mechanics. For example, Archimedes law of the lever. And although I haven't checked, I expect that Hero of Alexandria had a quantitative idea of how block and tackle pulleys worked.&lt;/p&gt;&#xA;&lt;p&gt;Thus, the key paradigmatic idea that separates the Renaissance of mechanics in 16th C Italy and the Hellenic Greece of Antiquity seems not be as paradigmatic as one might suppose.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Why then the focus, as exemplified by Einstein, on the history of mechanics or physics as beginning in Renaissance Europe, when the evidence points to a much broader and historically, much longer period of time?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-09-05T00:02:10.970" LastActivityDate="2020-09-05T00:02:10.970" Title="Why isn't the history of mechanics dated from Archimedes time?" Tags="|mathematics|physics|ancient-greece|classical-mechanics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12221" PostTypeId="2" ParentId="12220" CreationDate="2020-08-31T17:25:03.987" Score="1" Body="&lt;p&gt;Because otherwise historians had to admit that Christianity, by destroying the ancient civilization, caused a regress. Even if after the fall of the Roman Empire the population experienced some relief, ultimately the whole Western world stagnated and sunk culturally. Quantitative data is available through Ian Morris project (&lt;em&gt;Why the West Rules - For now&lt;/em&gt;, 2010 and &lt;a href=&quot;https://web.archive.org/web/20120330231438/http://www.ianmorris.org/docs/social-development.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;additional material&lt;/a&gt;). The renewed  access to Aristotle, it is universally admitted, caused a &lt;a href=&quot;https://en.wikipedia.org/wiki/Renaissance_of_the_12th_century&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;medieval renaissance&amp;quot;&lt;/a&gt; which is a further proof that the general intellectual level has become lower than 1500 years ago. The whole catastrophic cultural episode brought by church ideology has been denied by religiously minded historians but also by marxists who insisted that history &lt;em&gt;is progress&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-08-31T17:25:03.987" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12222" PostTypeId="2" ParentId="12208" CreationDate="2020-08-31T22:29:49.770" Score="3" Body="&lt;p&gt;Sizes and masses were roughly known since Rutherford's 1911 experiments. Gamow references some more precise measurements from 1920s in &lt;a href=&quot;https://www.jstor.org/stable/95297&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mass Defect Curve and Nuclear Constitution&lt;/a&gt;. For example, size measurements of Bieler from 1924 and Hardmeier from 1927 for light elements, and of Houtermans with Atkinson and with him from 1928-9 for heavy elements. The data on mass defects was taken from &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0106&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aston's 1927 Bakerian Lecture on mass-spectrograph&lt;/a&gt; published in the Proceedings of Royal Society, which summarizes data accumulated since 1921.&lt;/p&gt;&#xA;&lt;p&gt;Gamow also describes the interplay of attraction and repulsion forces known to exist between &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-particles in the nucleus as motivating his analogy to surface tension. His goal was to theoretically reproduce the experimental mass defect curve, but the motivation came from a theoretical analogy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the present paper I shall attempt to treat the problem more closely, analysing from the theoretical point of view the experimental facts concerning the nuclear energy. To begin with we shall treat a nucleus built from a certain number of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-particles only... The modern quantum theory of interaction between two complex particles gives a rather complicated expression for the mutual potential energy. We have here two kind of forces corresponding to symmetrical/antisymmetrical solutions of the wave equation. Both solutions show a strong repulsion at the distances compared with the dimension of particles. At greater distances the symmetrical solution gives an attraction decreasing exponentially with distance, the antisymmetrical one a repulsion of the same type...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Examining the behaviour of a collection of particles attracting one another with the forces very rapidly decreasing with distance (we neglect at first the Coulomb forces which are comparatively small at nuclear distances) we can introduce the well-known ideas made use of in the theory of capillarity&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-01T17:46:52.327" LastActivityDate="2020-09-01T17:46:52.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12223" PostTypeId="2" ParentId="11312" CreationDate="2020-08-31T23:56:11.300" Score="1" Body="&lt;p&gt;Because, very simply, their number system was evolved to fit the abacus device, in any of its forms, as they used it.&lt;/p&gt;&#xA;&lt;p&gt;They had an upper and lower register, the upper being half the register value, whatever that might be, and the lower the &amp;quot;ones&amp;quot; for the register. This allowed them to use fewer single item markers in each column of the abacus making its use both easier and less prone to error (pushing a stone for 5 toward the line between the upper and lower halves of the register and two stones for 1 upward to it was less prone to error than moving seven stones for 1. Also faster.&lt;/p&gt;&#xA;&lt;p&gt;Accordingly, they had a symbol for one in each power of ten register (yes, like us, they used positions (yes, their system was indeed positional:  while one COULD write IIMX for 1012, no one ever would for two reasons that will momentarily be obvious) in the form of columns for each power ten) and a symbol for half the value of the register.&lt;/p&gt;&#xA;&lt;p&gt;The symbols for half the value of the register were LITERALLY the symbol for the next register, halved in some obvious way. So the value for half a register in a &amp;quot;ones&amp;quot; column was &amp;quot;V&amp;quot;, the upper half of &amp;quot;X&amp;quot;. This is actually hugely more obvious if one looks at the symbols used before they stopped differentiating them from letters and just used the letters they most looked like.&lt;/p&gt;&#xA;&lt;p&gt;So why write them in particular order (remember IIMX, above)? So that one 1) Wrote them right off the abacus, left (highest) to right (lowest), not in some mixed way that would only confuse and would be prone to the error of forgetting a column. And in the logical manner of usually the half a register symbol, then the ones for the column, unless one had 9 in the column in which case, they seem to have felt it easier to write, say, XC as in &amp;quot;one less&amp;quot; than a full column (as a full column there (the tens) would equal 10 tens, or 100) and this is the second reason why to not mix as in my example above, would he &amp;quot;II&amp;quot; mean two pebbles in the ones column or combine with the thousand to mean &amp;quot;two less than 1000 (998) instead, and 2) Could therefore &amp;quot;write&amp;quot; them right back onto the abacus in the same exact way.&lt;/p&gt;&#xA;&lt;p&gt;(We are used to working right to left, &amp;quot;carrying&amp;quot; as we call it, to a final answer. They could work either direction more easily than we as one would usually be snapping the new number in (something being added, say) and if overrunning the 9 they could represent, snapping them all outward and snapping one more in the ones half of the next higher register. That would sometimes run into lots of that going left thing and they did seem to love efficiency, so my bet is on loading right to left usually.)&lt;/p&gt;&#xA;&lt;p&gt;But since they used different symbols for each column (tens position), reading them out left to right (highest to lowest going rightward) did not inhibit them loading right to left as seeing LXX meant activity in the tens column, no question for anyone, and not activity in the ones or ten thousands column. No ambiguity at all.&lt;/p&gt;&#xA;&lt;p&gt;So, with that background, &lt;strong&gt;the need for a &amp;quot;zero&amp;quot; in their number writing DID NOT EXIST and would have served no purpose.&lt;/strong&gt; An absence of symbols for a column's value meant nothing goes in that column. No need at all for a special symbol to note that:  one just simply skips over it/them moving on to the column for the next set of symbols.&lt;/p&gt;&#xA;&lt;p&gt;Did this mean they never had a need for zero? No, as noted in all the other answers and even the question. Just not in the simple use of the numbers in calculations. Their number system was not positional in its written form, though in practice, they did keep things in order like we would. But the abacus WAS utterly positional and that was where they calculated, not on paper or with a calculator that has to have a way to know a column is empty:  their calculator had that in that they just skipped a column as needed.&lt;/p&gt;&#xA;&lt;p&gt;To make sure references to the abacus are not misunderstood, they mgiht normally have a tray of sand which they'd smooth out when needed, then draw the column lines with a finger, and the upper and lower separator line as well, then place back their sets of pebbles. Nicer &amp;quot;models&amp;quot; might have a paddle to do the smoothing rather than a palm and fingers, a stylus for the lines, and sets of pebbles that were color-coded. Think chess set and expensive chess set. How simple is a tray with sand and pebbles? A more involved setup might be a large sand area where several to many abacus sets might be drawn and pebbled. But they also could have beads or stones on strings in the setup we think of for &amp;quot;abacus.&amp;quot; Indeed any arrangement they liked:  imagine Alice playing &amp;quot;abacus&amp;quot; rather than &amp;quot;chess&amp;quot;...&lt;/p&gt;&#xA;&lt;p&gt;One pictures mathemeticians either choosing to work on problems tools existed for, as nowadays, or inventing their own methods, as needed, as nowadays, but no matter their needs and inventiveness, the vast majority of users of numbers would have had no need for or knowledge of, them, as nowadays.&lt;/p&gt;&#xA;&lt;p&gt;The reason the abacus, and therefore Roman numerals, passed away for most uses is because actual paper came along and gradually became cheap enough to do things like bookkeeping. Mankind is very much driven to choose practical things (outside the field of women's shoes). The Roman numeral system worked for 2,000 years before paper made something different more practical (and cheap enough to be worth doing). It did not get supplanted because it wasn't awfully good at what it did but rather because something better became possible. And the something better (paper) offered a much easier set of methods for doing arithmetic, methods that made the abacus more of a specialty tool. Roman numerals that showed their positional value in their very symbols were no longer needed then either. (Never really noted is that we enlarged the symbol set one needed from seven symbols to 10.)&lt;/p&gt;&#xA;" OwnerUserId="13003" LastActivityDate="2020-08-31T23:56:11.300" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12224" PostTypeId="2" ParentId="12220" CreationDate="2020-09-01T08:13:22.877" Score="3" Body="&lt;p&gt;Dugas’ &lt;a href=&quot;https://books.google.it/books?id=kdDDAgAAQBAJ&amp;amp;pg=PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Mechanics (French ed 1958)&lt;/a&gt; starts with Aristotle followed by Archimedes.&lt;/p&gt;&#xA;&lt;p&gt;Then Hellenistic and Arabic science, followed by Middle Age and Renaissance.&lt;/p&gt;&#xA;&lt;p&gt;Thus it is old but quite comprehensive.&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2020-09-02T12:02:58.953" LastActivityDate="2020-09-02T12:02:58.953" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12227" PostTypeId="2" ParentId="12208" CreationDate="2020-09-02T17:42:15.433" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What was known about the properties of the nucleus (its shape, its density etc) and the nuclear forces before the Liquid drop model was proposed?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1897 Thomson discovered the negatively charged electron.  This prompted a revolution in thinking about the structure of the atom which before then had hardly been theorised. Thomsons discovery suggested that the neutral atom must contain a positively charged element.  In 1903, Thomson proposed his Plum-Pudding Model - that the atom was a sphere of uniform positive electrification, with electrons scattered through it like plums in a pudding, giving rise to name.&lt;/p&gt;&#xA;&lt;p&gt;In the following year, the Japanese physicist, Hantaro Nagaoka, rejected this model argung that charged elements must be impenetrable. In it's stead, he proposed a planetary model,  in which a positively charged center is surrounded by a number of revolving electrons, in the manner of Saturn and its rings - prompting th name - The Saturnian Model. His model made two predictions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;a very massive atomic center (in analogy to a very massive planet)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;electrons revolving around the nucleus, bound by electrostatic forces (in analogy to the rings revolving around Saturn, bound by gravitational forces).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Both of these predictions were successfully confirmed by Ernest Rutherford in his 1911 experiments and who mentions Nagaoka's model in his paper of that year - where the model of an atomic nucleus is mentioned.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-09-02T17:42:15.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12228" PostTypeId="1" CreationDate="2020-09-02T19:21:54.233" Score="1" ViewCount="199" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sylvester_James_Gates&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sylvester James Gates&lt;/a&gt; was one of the co-discoverers of Adinkras. These are graphical representations of susy (supersymmetry) algebras.&lt;/p&gt;&#xA;&lt;p&gt;They are named after a West African people — the Akan of Ghana and the Gyamen of the Côte d'Ivoire. They are visual symbols called by the same name and represent concepts or aphorisms, which in itself sounds like a fascinating idea.&lt;/p&gt;&#xA;&lt;p&gt;This suggested to me that perhaps Gayes had some direct link to the Akan or Gyamen people. However, I couldn't find any information on this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Why did Sylvester James Gates choose this name for these graphs?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Obviously this is to celebrate his African heritage. But why this as opposed to many other things he could have chosen?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2022-09-16T12:47:40.590" LastActivityDate="2022-09-16T12:47:40.590" Title="Why did Sylvester Gates choose the name Adinkra?" Tags="|terminology|notation|physicists|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12232" PostTypeId="1" CreationDate="2020-09-04T00:07:33.723" Score="2" ViewCount="1064" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Oceanography&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page on Oceanography&lt;/a&gt; says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Humans first acquired knowledge of the waves and currents of the seas and oceans in pre-historic times. Observations on tides were recorded by Aristotle and Strabo in 384-322 BC. Early exploration of the oceans was primarily for cartography and mainly limited to its surfaces and of the animals that fishermen brought up in nets, though depth soundings by lead line were taken.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But it gives no specific examples of when ocean currents were discovered. It seems plausible to me that currents like, for example, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Canary_Current&quot; rel=&quot;nofollow noreferrer&quot;&gt;Canary Current&lt;/a&gt; could have been discovered and described by Mediterranean sailors in ancient times. Likewise the currents of the Indian and north Pacific Oceans by those living near them.&lt;/p&gt;&#xA;&lt;p&gt;What is the earliest clear mention or description of ocean currents?&lt;/p&gt;&#xA;" OwnerUserId="3830" LastEditorUserId="4251" LastEditDate="2020-09-07T06:28:42.363" LastActivityDate="2020-09-07T06:28:42.363" Title="What is the first clear mention of ocean currents?" Tags="|earth-sciences|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12233" PostTypeId="2" ParentId="12232" CreationDate="2020-09-04T05:39:08.923" Score="6" Body="&lt;p&gt;Wikipedia gives no examples of written mentions from ancient times because there are none. And by the time written records arrive what they record is not really &amp;quot;discovered&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Ancient seafarers likely did have operational knowledge of ocean currents, i.e. they made use of them, but they left nothing in writing about it, at least nothing that survived. For example, it is often inferred from the pattern of colonization of Polynesian islands that &lt;a href=&quot;https://divediscover.whoi.edu/history-of-oceanography/polynesian-seafarers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;pre-historic Polynesians&lt;/a&gt; had working knowledge of currents. But they had no written language and only made stick charts. Here is another typical inference in &lt;a href=&quot;https://www.washingtonpost.com/wp-srv/inatl/daily/sept99/noah26.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gugliotta's Trailing Ancient Mariners&lt;/a&gt; from artifact similarities in Synope (modern Turkey) and Crimea (modern Ukraine) c. 5000 BC:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Along the coast, whether in Synope or modern Ukraine or Russia, artifacts showed remarkable similarities. Roof tiles in the Crimea were stamped with the Greek word &amp;quot;Synope,&amp;quot; and studies of ocean currents and winds showed that sailors could travel the 180-mile south-north route across the sea from Synope to the Crimea, and probably did.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That ancient Phoenicians knew of the Canary current is similarly inferred, see e.g. &lt;a href=&quot;https://www.jstor.org/stable/3984323&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ecohistory of the Canary Islands by Crosby&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As far as the written record, timelines typically start with Ben Franklin naming Gulf Stream that and mapping it in 1769, see &lt;a href=&quot;http://core.ecu.edu/geology/woods/HISTOCEA.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;HISTOCEA's History of Oceanography&lt;/a&gt;, or even with US Navy officer Maury starting a systematic data gathering on currents in 1842, see &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/B9780124159907000028?via%3Dihub&quot; rel=&quot;nofollow noreferrer&quot;&gt;The History of Measuring Ocean Currents by Joseph&lt;/a&gt;. Maury was extracting information from already existing ship logs, and was slightly predated by Dana, who sailed under captain Wilkes in 1838-42 and undertook systematic studies of Antarctic currents. So it was a surprise that &lt;em&gt;Carta Marina&lt;/em&gt; published by Olaus Magnus in 1539 already had an accurate depiction of currents near Iceland, see &lt;a href=&quot;http://news.bbc.co.uk/2/hi/science/nature/3683087.stm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient map captures ocean front by Kettlewell&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Most of the northeast Atlantic is drawn using more-or-less straight lines. However, things change off the east coast of Iceland, where the lines suddenly morph into a large group of whorls. &amp;quot;Their location, size and spacing seem too deliberate to be purely artistic expression,&amp;quot; Tom Rossby told BBC News Online. &amp;quot;Nowhere else on the chart do these whorls appear in such a systematic fashion.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;When oceanographers from the UK's Plymouth Marine Laboratory and the University of Rhode Island compared these whorls with thermal images from an Earth observation satellite, they saw something surprising.&#xA;Magnus's whorls corresponded almost perfectly with the Iceland-Faroes Front - where the Gulf Stream meets cold waters coming down from the Arctic. &amp;quot;They are the earliest known description of large-scale eddies in the ocean - these are huge bodies of water, 100km in diameter, that turn slowly,&amp;quot; Professor Rossby said.&#xA;&amp;quot;It seems the lines were deliberately drawn to aid navigation.&amp;quot;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-05T04:48:20.663" LastActivityDate="2020-09-05T04:48:20.663" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12234" PostTypeId="1" CreationDate="2020-09-05T00:22:05.110" Score="1" ViewCount="280" Body="&lt;p&gt;The Vietnamese Mathematical Society was set up in 1965 by Le Van Thiem and Hoang Tuy. Both had studied in Europe, the former in Paris and Germany and the latter in Moscow. By 1965, the Vietnam War, which in Vietnam is called the Resistance War against America, or simply the American War had been on progress for ten years and Hanoi University had been closed for four years.&lt;/p&gt;&#xA;&lt;p&gt;How in such difficult circumstances had van Thiem and Hoang Tuy set up the Journal and what support did they have from the larger international mathematical community?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-08T05:46:18.877" LastActivityDate="2021-09-09T06:41:16.660" Title="How did the Vietnamese manage set up the Vietnamese Mathematical Society during the Vietnam War?" Tags="|mathematics|mathematicians|social-context|20th-century|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12235" PostTypeId="2" ParentId="12234" CreationDate="2020-09-05T17:07:44.410" Score="5" Body="&lt;p&gt;Following the advice of Dave Renfo in comments, I looked up the 1990 interview with Tụy in &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt;.  The article is fascinating, and I'd recommend it.&lt;/p&gt;&#xA;&lt;p&gt;Sadly, it gives a pretty brief account of the founding of the Society, and none of the journal.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Koblitz&lt;/strong&gt;: And you were able to do mathematics during all&#xA;this? &lt;br&gt;&lt;br&gt;&#xA;&lt;strong&gt;Tụy&lt;/strong&gt;: Yes, in fact our general morale was so high that we continued our seminars on a regular basis during this whole period. The Mathematical Society, which was founded in 1965 by Lê Van Thiêm (I was general secretary), organized joint seminars in optimization, probability, functional analysis, algebra, numerical analysis. People from Hanoi University, the Pedagogical Institute, and the Polytechnic Institute participated (the Hanoi MathematicaI Institute was not formed until 1970). Since the three institutions had been evacuated in different directions from Hanoi, we held the seminars in Hanoi. They met twice a month. I must say, people were very diligent about attending. Many of us would take advantage of the opportunity to visit our families. Since my wife and children were on the opposite side of Hanoi from the university, it was much more convenient for me to visit them after the seminars held in Hanoi.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He has a longer description of what it was like trying to run the University at the time, after it had been evacuated to a spot in the forest some 170km outside of Hanoi, having to teach classes near trenches they could jump into if the Americans started a bombing run, etc., and some amusing anecdotes about Grothendieck visiting the society and having to be evacuated due to bombing.&lt;/p&gt;&#xA;" OwnerUserId="1731" LastEditorUserId="4251" LastEditDate="2021-09-09T06:41:16.660" LastActivityDate="2021-09-09T06:41:16.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12236" PostTypeId="1" AcceptedAnswerId="12237" CreationDate="2020-09-06T14:12:37.027" Score="3" ViewCount="679" Body="&lt;p&gt;This &lt;a href=&quot;http://galileo.phys.virginia.edu/classes/252/energy_p_reln.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;web page&lt;/a&gt; shows how to derive energy-momentum relationship, &lt;span class=&quot;math-container&quot;&gt;$E_{total}^2=p^{2}c^{2}+\left( mc^{2}\right) ^{2}$&lt;/span&gt;, given the following equations. Please note that some sources make a distinction between &lt;span class=&quot;math-container&quot;&gt;$m_{0}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$m_{0}$&lt;/span&gt; is taken to be rest mass.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$K=\frac{mc^{2}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}-mc^{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$E_{total}=K+mc²=\frac{mc^{2}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://youtu.be/msLSb8g2Zy4?t=516&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt;, from 08:34 up to 10:23, it is shown that how energy-momentum relation could be shown to be equivalent to &lt;span class=&quot;math-container&quot;&gt;$E_{total}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Both relationships, &lt;span class=&quot;math-container&quot;&gt;$E=mc^{2}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K=\frac{mc^{2}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}-mc^{2}$&lt;/span&gt;, were found around 1905. [&lt;a href=&quot;https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Mass%E2%80%93velocity_relationship&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;p&gt;Now coming to the question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Energy–momentum relation was first established by Paul Dirac in&#xA;1928 under the form &lt;span class=&quot;math-container&quot;&gt;$E=\root{2}\of{p^{2}c^{2}+\left( mc^{2}\right) ^{2}}+V$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is the amount of potential energy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation#Origins_and_derivation_of_the_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Energy–momentum_relation#Origins_and_derivation_of_the_equation&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have read on some other web pages as well that the energy-momentum relationship was derived by Dirac. I understand that science doesn't progress and evolve as is described in textbooks. What was the problem that it had to wait until 1928 to get energy-momentum relationship when it could have been easily derived using two fundamental relativistic equations much earlier? It might be that I'm overthinking. If my confusion is legitimate, please try to keep the answer simple.&lt;/p&gt;&#xA;" OwnerUserId="9341" OwnerDisplayName="PG1995" LastEditorUserId="55" LastEditDate="2020-09-06T23:01:33.797" LastActivityDate="2020-09-07T15:01:21.620" Title="Why did energy-momentum relationship have to wait until 1928 to be established?" Tags="|physics|relativity-theory|energy|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12237" PostTypeId="2" ParentId="12236" CreationDate="2020-09-06T22:51:01.253" Score="3" Body="&lt;p&gt;It is not that it had to wait to be &amp;quot;established&amp;quot;, it is obtainable from what was known by trivial algebra, but rather that it had to wait for a reason to write it that way. In the early years of relativity the concept of the &amp;quot;electromagnetic mass&amp;quot; of electron was prominent, which suggested that said mass is velocity dependent. It was at odds with Einstein's kinematic approach in special relativity, but he reflected it nonetheless in what came to be called &amp;quot;relativistic mass&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. So it was more natural to relate total energy to that mass rather than to the rest mass &lt;span class=&quot;math-container&quot;&gt;$m_0$&lt;/span&gt;, which made for a simpler formula &lt;span class=&quot;math-container&quot;&gt;$E_{total}=mc^2$&lt;/span&gt;, see &lt;a href=&quot;https://hsm.stackexchange.com/q/2498/55&quot;&gt;Why is Einstein's mass-energy relation usually written as &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt;, and not &lt;span class=&quot;math-container&quot;&gt;$\Delta E=\Delta m c^2$&lt;/span&gt;?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The notion of relativistic momentum &lt;span class=&quot;math-container&quot;&gt;$p=\frac{m_0v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$&lt;/span&gt; was introduced by Planck in 1906, but it is only natural in Minkowski's spacetime (4-vector) context, which Einstein disfavored for a long time as too fanciful, see &lt;a href=&quot;https://hsm.stackexchange.com/a/9977/55&quot;&gt;What was the relationship between Einstein and Minkowski?&lt;/a&gt; Apparently, he changed his mind around 1921, as reflected in Stafford Little Lectures. Incidentally, the &amp;quot;relativistic mass&amp;quot; does not fit well with the 4-vectors (it is not Lorentz-invariant), see &lt;a href=&quot;https://hsm.stackexchange.com/q/2465/55&quot;&gt;When and why did the concept of relativistic mass become outdated?&lt;/a&gt;, so it became reasonable to relate the total energy to the rest mass and momentum instead, as Dirac did. Einstein only disclaimed the &amp;quot;relativistic mass&amp;quot; explicitly in a 1948 letter to Barnett, where he also endorsed Dirac's form of the momentum-energy relation. Here is from &lt;a href=&quot;http://sites.fas.harvard.edu/%7Ephys191r/References/b5/Adler1987.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adler's Does mass really depend on velocity, dad?&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The electromagnetic world view that occupied much of&#xA;the first quarter of this century has been extensively and&#xA;elegantly discussed elsewhere. The general idea was to&#xA;construct an electromagnetic model of the extended, as opposed&#xA;to the point, electron. The properties derived in that&#xA;way were assumed to be extendable to bodies other than the&#xA;electron. One result of this work was to predict a velocity dependent&#xA;mass... Einstein's relativistic&#xA;mass had its origin in the kinetics of his special theory&#xA;and not in the structure of the particle. In fact he observes&#xA;that &amp;quot;with a different definition of force and&#xA;acceleration we should naturally obtain other values for the&#xA;masses (meaning, longitudinal and transverse masses)&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Whatever Einstein's precise early views were on the subject,&#xA;his view in later life appears clear. In a 1948 letter to&#xA;Lincoln Barnett, he wrote&#xA;&amp;quot;It is not good to introduce the concept of the mass&#xA;&lt;span class=&quot;math-container&quot;&gt;$M=\frac{m}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$&lt;/span&gt;&#xA;of a body for which no clear definition&#xA;can be given. It is better to introduce no other mass&#xA;than 'the rest mass' &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. Instead of introducing &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, it is better&#xA;to mention the expression for the momentum and energy of&#xA;a body in motion&amp;quot;. The question naturally arises as to what motivated Einstein to this new view given his earlier use of the concept. The answer, I believe, is that by at least 1922 he had adopted&#xA;Minkowski's 1908 space-time (four-vector) approach to&#xA;special relativity.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-06T23:02:22.333" LastActivityDate="2020-09-06T23:02:22.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12238" PostTypeId="1" AcceptedAnswerId="12241" CreationDate="2020-09-07T02:42:49.887" Score="8" ViewCount="1403" Body="&lt;p&gt;In what book/letter did Gottfried Wilhelm Leibniz explore the product rule as part of differential calculus?&lt;/p&gt;&#xA;" OwnerUserId="8219" LastEditorUserId="4251" LastEditDate="2020-09-08T02:53:12.323" LastActivityDate="2020-09-08T02:53:12.323" Title="Where did Leibniz explore the product rule of differential calculus?" Tags="|mathematics|mathematicians|calculus|leibniz|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12239" PostTypeId="2" ParentId="12238" CreationDate="2020-09-07T09:36:48.773" Score="6" Body="&lt;p&gt;Leibniz states the product rule in his first paper on the calculus (1684). It's in the middle of the fist page (page 467) as can be seen here: &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-differential-calculus&quot; rel=&quot;noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-differential-calculus&lt;/a&gt; and also in English translation (top of page 2) here: &lt;a href=&quot;http://www.17centurymaths.com/contents/Leibniz/nova1.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://www.17centurymaths.com/contents/Leibniz/nova1.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2020-09-07T09:36:48.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12240" PostTypeId="1" CreationDate="2020-09-07T09:59:50.640" Score="0" ViewCount="75" Body="&lt;p&gt;It seems to me that very few if any scientists/philosophers stated that science could be used for predictions before the early 1900s. In the scientific context I have only encountered the use of the word prediction only in relation to astronomy.&#xA;I suppose this may be due to the few predictive theories outside astronomy back then&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-09-10T07:06:07.500" LastActivityDate="2020-09-10T07:06:07.500" Title="Who was the first to state that scientific theories could be used to make predictions?" Tags="|biographical-details|discoveries|astronomy|philosophy-of-science|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12241" PostTypeId="2" ParentId="12238" CreationDate="2020-09-07T11:02:31.240" Score="14" Body="&lt;p&gt;It is discussed in multiple manuscripts, letters and publications from 1675 to 1701.&lt;/p&gt;&#xA;&lt;p&gt;According to Fracois Ziegler's post on MO &lt;a href=&quot;https://mathoverflow.net/a/181429/51484&quot;&gt;Did Leibniz really get the Leibniz rule wrong?&lt;/a&gt;, Leibniz originally thought &lt;span class=&quot;math-container&quot;&gt;$d(uv)=du\,dv$&lt;/span&gt; in a special case, but corrected his mistake the same month in the manuscript &lt;a href=&quot;https://books.google.com/books?id=QG_QAAAAMAAJ&amp;amp;pg=PA93#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Methodi tangentium inversae exempla&lt;/em&gt; (November 11, 1675)&lt;/a&gt;. Later the same month the correct general rule &lt;span class=&quot;math-container&quot;&gt;$d\overline xy = d\overline{xy}-xd\overline y$&lt;/span&gt; appears in &lt;a href=&quot;https://books.google.com/books?id=QG_QAAAAMAAJ&amp;amp;pg=PA107#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Pro methodo tangentium inversa et aliis tetragonisticis specimina et inventa&lt;/em&gt; (November 27, 1675)&lt;/a&gt;, where he calls it &amp;quot;&lt;em&gt;a really noteworthy theorem and a general one for all curves&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;It reappears in &lt;em&gt;Elementa calculi novi...&lt;/em&gt; (1680), which was a draft of the &lt;em&gt;Acta Eruditorum&lt;/em&gt; paper &lt;em&gt;Nova Methodus pro Maximis et Minimis...&lt;/em&gt; (October 1684) linked in Victor Blasjo's answer, his first publication on the new calculus. Even there he does not prove the product rule, or other propositions, but justifications appears e.g. in a letter to Wallis from March 30, 1699, based on what we now call the &amp;quot;method of exhaustion&amp;quot;, and in &lt;em&gt;Cum  prodiisset atque increbuisset Analysis mea infinitesimalis&lt;/em&gt; (1701), based on Leibniz's signature &amp;quot;continuity principle&amp;quot;:&amp;quot;&lt;em&gt;In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included&lt;/em&gt;&amp;quot;. Both methods are used in &lt;em&gt;Justification du Calcul des infinitesimales&lt;/em&gt; (1701).&lt;/p&gt;&#xA;&lt;p&gt;Leibniz's evolution on infinitesimals, and the product rule in particular, is described in &lt;a href=&quot;https://www.jstor.org/stable/40694040&quot; rel=&quot;noreferrer&quot;&gt;On the Attempts made by Leibniz to Justify his Calculus by Horvath&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;From about 1680, the leibnizian calculus step by step becomes more developed, and more consolidated, and the variable feature of the infinitely small quantity comes into the foreground. From that time on we may observe in Leibniz's works the more conscious attempts to outline the concept, and the use of the infinitely small quantities. For example, in his manuscript Elementa, the quantities &lt;span class=&quot;math-container&quot;&gt;$_1D_2C,\,_2C_3D, ...$&lt;/span&gt; are conceived as &amp;quot;incrementa momentanea&amp;quot; of the line segment BC8. In the same manuscript he states that &lt;span class=&quot;math-container&quot;&gt;$d\overline{xy}$&lt;/span&gt;, that is &lt;span class=&quot;math-container&quot;&gt;$d(xy)$&lt;/span&gt;, is the difference between two proximate terms of the finite, two-dimensional, geometrical variable &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;. One of these terms is the variable xy, and the other is &lt;span class=&quot;math-container&quot;&gt;$(x+dx)(y+dy)$&lt;/span&gt;, so that &lt;span class=&quot;math-container&quot;&gt;$d(xy)$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$(x+dx)(y+dy) - xy$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...]  &lt;em&gt;The above-mentioned manuscript Elementa is a preliminary draft of Leibniz's famous article entitled Nova Methodus pro maximis et minimis (see [4]). It is worth observing that there is a sharp distinction between the two conceptions of infinitely small quantities used by Leibniz in these two papers. In the draft, (see [3]), as we have seen, the line segments &lt;span class=&quot;math-container&quot;&gt;$DC$&lt;/span&gt; were thought by Leibniz to be infinitely small, that is &amp;quot;incrementa momentanea&amp;quot;. In the final version [4], however, Leibniz does not use differentials but only differences in the sense of fixed, small, finite quantities. Leibniz supposedly does not use the term &amp;quot;infinitely small&amp;quot; in his article [4] in order to avoid controversies which most likely would have arisen in connection with this notion&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;It is known that Leibniz did not prove the propositions which occurred in his first publication of his calculus (see [4]). Now let's turn to the discussion how Leibniz justifies, for example, the rule of differentiation for a product &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt; using incomparable quantities, that is alluding to the method of Archimedes. The quantity &lt;span class=&quot;math-container&quot;&gt;$d(xy)$&lt;/span&gt; is equal to the quantity &lt;span class=&quot;math-container&quot;&gt;$(x+dx) (y+dy) - xy$&lt;/span&gt;. (So &lt;span class=&quot;math-container&quot;&gt;$d(xy)$&lt;/span&gt; is the same as the difference between two adjacent &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;, of which let one be &lt;span class=&quot;math-container&quot;&gt;$xy$&lt;/span&gt;, the other &lt;span class=&quot;math-container&quot;&gt;$(x+dx)(y+dy)$&lt;/span&gt;.) During calculations the quantity &lt;span class=&quot;math-container&quot;&gt;$dxdy$&lt;/span&gt; can be omitted since this term is incomparably smaller with respect to the others, thus we obtain the sound result &lt;span class=&quot;math-container&quot;&gt;$d(xy) = xdy + ydx$&lt;/span&gt;. That reasoning can be treated in archimedian style, stating that the error will be smaller than any given positive quantity. Leibniz points out ([30], p. 63) that he does not discuss the question of whether or not inassignable quantities are fictitious concepts or not, because it is sufficient that these quantities serve for the shortening of the reasoning.&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Horvath also describes the continuity principle justification, but it is more involved, so I will not reproduce it here. Leibniz's manuscipts on calculus are collected in &lt;a href=&quot;https://books.google.com/books/about/Historia_et_origo_calculi_differentialis.html?id=UOM3AAAAMAAJ&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Historia et origo calculi differentialis a G. G. Leibnitio conscripta&lt;/em&gt; (1846) edited by Gerhardt&lt;/a&gt;, see also &lt;a href=&quot;https://archive.org/details/historiaetorigo00gerhgoog/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Archive's version&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-07T11:10:46.173" LastActivityDate="2020-09-07T11:10:46.173" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12242" PostTypeId="2" ParentId="12236" CreationDate="2020-09-07T15:01:21.620" Score="2" Body="&lt;p&gt;Actually, it wasn't Dirac who first found that relation. It was already used by Planck as early as in 1906 while deriving the Hamiltonian equations of motion&lt;/p&gt;&#xA;&lt;p&gt;Planck: &lt;a href=&quot;https://en.wikisource.org/wiki/Translation:The_Principle_of_Relativity_and_the_Fundamental_Equations_of_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Principle of Relativity and the Fundamental Equations of Mechanics (1906).&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;He first gave the Lagrangian function&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (1)\quad L={\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+{\dot {y}}{\frac {\partial H}{\partial {\dot {y}}}}+{\dot {z}}{\frac {\partial H}{\partial {\dot {z}}}}-H={\frac {mc^{2}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}+const. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;then momentum&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ (2)\quad \xi ^{2}+\eta ^{2}+\zeta ^{2}=\varrho ^{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \xi ={\frac {\partial H}{\partial {\dot {x}}}}={\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}},\ etc.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;He obtained the energy-momentum relation by combining (1) and (2):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ L=mc^{2}{\sqrt {1+{\frac {\varrho ^{2}}{m^{2}c^{2}}}}}+const $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Setting the constant to zero, the previous relation can be written as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ L^{2}=m^{2}c^{4}+\varrho^{2}c^{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="7601" LastActivityDate="2020-09-07T15:01:21.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12243" PostTypeId="1" CreationDate="2020-09-07T18:03:57.653" Score="12" ViewCount="392" Body="&lt;p&gt;Who was the (Japanese?) mathematician who said in a lecture that a certain conjecture still remained to be proved, and one of the students present told him that he (the mathematician) had already proved that conjecture decades earlier?&lt;/p&gt;&#xA;" OwnerUserId="13028" LastEditorUserId="4251" LastEditDate="2020-09-11T10:34:34.783" LastActivityDate="2021-07-13T18:08:42.420" Title="Who was that forgetful mathematician?" Tags="|mathematicians|conjecture|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12244" PostTypeId="1" CreationDate="2020-09-07T19:40:47.163" Score="7" ViewCount="260" Body="&lt;p&gt;I hope to know who first used the name &amp;quot;Hawaiian Earrings.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Barratt, Milnor(1962) says &amp;quot;This example was suggested by Steenrod&amp;quot; in its Introduction:&#xA;&lt;br&gt;&#xA;&lt;a href=&quot;https://www.ams.org/journals/proc/1962-013-02/S0002-9939-1962-0137110-9/S0002-9939-1962-0137110-9.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.ams.org/journals/proc/1962-013-02/S0002-9939-1962-0137110-9/S0002-9939-1962-0137110-9.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And this is Steenrod(1949):&#xA;&lt;br&gt;&#xA;&lt;a href=&quot;https://www.jstor.org/stable/1969589&quot; rel=&quot;noreferrer&quot;&gt;https://www.jstor.org/stable/1969589&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But these two articles don't use the term &amp;quot;Hawaiian Earrings.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I posted this question at Reddit and got a recommendation for this forum.&#xA;&lt;br&gt;&#xA;Also kindly proposed was Kaminker, Schochet(1977):&#xA;&lt;br&gt;&#xA;&lt;a href=&quot;https://www.ams.org/journals/tran/1977-227-00/S0002-9947-1977-0431136-9/S0002-9947-1977-0431136-9.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.ams.org/journals/tran/1977-227-00/S0002-9947-1977-0431136-9/S0002-9947-1977-0431136-9.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Any additional suggestions will be appreciated.&lt;/p&gt;&#xA;" OwnerDisplayName="user12897" LastActivityDate="2020-09-07T19:40:47.163" Title="Who coined the Hawaiian Earrings?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12245" PostTypeId="1" AcceptedAnswerId="12249" CreationDate="2020-09-08T17:43:03.170" Score="1" ViewCount="314" Body="&lt;p&gt;I am speculating about the value of Tycho Brahe’s - for his time - accurate observations and Kepler’s calculations.&lt;/p&gt;&#xA;&lt;p&gt;Might it have been possible for Kepler to formulate his laws based on Ptolemy’s system, assuming he adapted the Earth’s position in a heliocentric system?&lt;/p&gt;&#xA;&lt;p&gt;For example you might expect that the period of the planets might be known relatively accurately also from Ptolemy’s data.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, if you were ready to accept elliptical orbits, planets such as Mercury and Mars might give some input to the second law. I realize that there is a component of circular reason involved here, in that you must be open to the laws from the beginning in order to want to make such tests using  Ptolemy’s system.&lt;/p&gt;&#xA;" OwnerUserId="5602" OwnerDisplayName="Mikael Jensen" LastEditorUserId="4251" LastEditDate="2021-03-01T03:34:01.620" LastActivityDate="2021-03-01T03:34:01.620" Title="How critical were Tycho Brahe’s accurate observations?" Tags="|physics|astronomy|16th-century|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12246" PostTypeId="2" ParentId="12245" CreationDate="2020-09-08T18:09:32.930" Score="3" Body="&lt;p&gt;Brahe's observations were crucial. More than the other things that you mentioned: Copernicus system and knowledge of periods. Copernicus system indeed simplifies calculations for planets a lot, but everyone understood how to transfer in principle calculations in one system into another. Periods+Copernicus system would be enough for the discovery of the Third Law,&#xA;Copernicus system is needed to find heliocentric distances.)&lt;/p&gt;&#xA;&lt;p&gt;But the most important discovery of Kepler was the First Law, and for this one really needs observed positions at various times and with high accuracy. Kepler used Mars observations of Brahe. Without these high accuracy observations, one could not guess that the true trajectory is an ellipse rather than a circle (or a combination of circles). The deviation of this ellipse from a circle is quite small. And astronomy, since the time of Aristotle was dominated by the dogma that all motions have to be expressed in terms of uniform motions on circles. This is what everyone from Hipparchus to Copernicus and Galileo accepted without any doubts. It is demolishing of this dogma which constitutes the truly revolutionary discovery of Kepler.&lt;/p&gt;&#xA;" OwnerUserId="64" OwnerDisplayName="Alexandre Eremenko" LastActivityDate="2020-09-08T18:09:32.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12247" PostTypeId="1" CreationDate="2020-09-09T17:31:27.260" Score="1" ViewCount="174" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Parity_(mathematics)#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page on parity&lt;/a&gt; currently says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why didn't they consider 1 as odd? (I am assuming they already had the concept of odd and even numbers.) What did they consider it as?&lt;/p&gt;&#xA;" OwnerUserId="10623" LastEditorUserId="10106" LastEditDate="2020-09-09T19:07:28.827" LastActivityDate="2020-09-09T19:07:28.827" Title="Why didn't the ancient Greeks consider 1 to be odd?" Tags="|mathematics|ancient-greece|number-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12248" PostTypeId="1" CreationDate="2020-09-10T09:45:57.733" Score="3" ViewCount="87" Body="&lt;p&gt;We know that Newton's hair samples showed high level of mercury and of course he used mercury like crazy in his alchemy experiments (as did many/all alchemists not just in the west but also China) and in fact Newton had some concern about this or at least thought it affected his hair color (he made some comment about this specifically). The dangers of mercury were not unknown then and by the 19th century they must have been fairly well established although mercury continued to be prescribed as a medicine (and dental filling material as an amalgam) during the US civil war and even into the 20th century for syphilis although by then it was used more topically. Even now I think Mercurochrome is still used as a topical disinfectant in some countries.&lt;/p&gt;&#xA;&lt;p&gt;We know also that Faraday used mercury so much that he believed, quite reasonably, that it was affecting his cognition/memory. Now, mercury used medicinally is quite different than using it in electrical experiments as Faraday used it -- I believe the first electrical motor of Faraday involved a pool of the element.&lt;/p&gt;&#xA;&lt;p&gt;Mercury is a good conductor and it also is probably at that time the only conductor that was liquid at room temperature. Was mercury's wide use in experiments due to its physical properties or did it retain some of the glamour from the days of alchemy -- did scientists like Faraday believe it had some special properties beyond the measurable physical properties just as Newton did?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2020-09-10T12:04:32.027" LastActivityDate="2020-09-10T12:04:32.027" Title="Was the wide use of mercury in experiments in the 19th century related to alchemy?" Tags="|physics|chemistry|electricity|alchemy|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12249" PostTypeId="2" ParentId="12245" CreationDate="2020-09-10T18:30:54.380" Score="4" Body="&lt;p&gt;In process of finding his laws Kepler tried to construct circular motions for planet orbits. The best fit disagree with observations by 8 minutes. Kepler knew that Brahe's observations could not have such errors, so he had to reject his hypothesis. Ptolemy's observation do not have such accuracy. For stars he give coordinates rounded to 20' or 15'. So yes, Brahe's data were crucial&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-09-10T18:30:54.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12250" PostTypeId="2" ParentId="2520" CreationDate="2020-09-10T21:50:34.723" Score="2" Body="&lt;p&gt;Although I've already accepted one answer, I added this answer in order to clarify what is known about Gauss's work towards the general Gauss-Bonnet theorem and what is a matter of speculation; this distinction is not clear in Mark Yasuda's answer, and one might get from it incorrect impression about the roots of Gauss differential geometric ideas.&lt;/p&gt;&#xA;&lt;p&gt;Gauss probably did know the global statement of the Gauss-Bonnet theorem, at least in the case of a simply-connected surface with boundary (like in the case of a spherical surface cutted along a small circle, see figure below; this case requires both the curvature surface integral of the piece of sphere and the line integral along the boundary). However, the Gauss-Bonnet theorem for a general surface requires topological notions like the genus &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; of a surface and it's Euler characteristic &lt;span class=&quot;math-container&quot;&gt;$\chi$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$\chi = 2 - 2g $&lt;/span&gt; for oriented surface), so confirming Gauss's knowledge of it is merely a matter of speculation about the topological ideas that were slowly crystallizing in Gauss's mind.&lt;/p&gt;&#xA;&lt;p&gt;My indications are based upon reading Oscar Bolza's treatise on Gauss's contibution to the calculus of variations and differential geometry; a section of it is devoted to &amp;quot;the geodetic curvature and the theorem of total curvature&amp;quot;. In this section, Bolza analyzes Gauss's unpublshed manuscript entitled &amp;quot;&lt;strong&gt;Die Sietenkrummung&lt;/strong&gt;&amp;quot; (&amp;quot;the side curvature&amp;quot;), which dates back to the years before his publication on differential geometry (the years 1822-1825). Gauss introduces in this manuscript the &amp;quot;extrinsic&amp;quot; definition of the geodesic curvature &lt;span class=&quot;math-container&quot;&gt;$K_g$&lt;/span&gt; at a given point of a curve embedded in the surface , and gives an &amp;quot;intrinsic&amp;quot; formula for it in term of the coefficients of the first fundamental form &lt;span class=&quot;math-container&quot;&gt;$E,F,G$&lt;/span&gt; and their first and second derivatives with respect to the curvilinear coordinates &lt;span class=&quot;math-container&quot;&gt;$p,q$&lt;/span&gt; - this formula proves that the geodesic curvature is an isometric invariant of the surface, in contrast to the apparently extrinsic definition of it (like in the case of the &amp;quot;Theorema Egregium&amp;quot; on Gauss curvature). This result was rediscovered by Ferdinand Minding about a decade later.&lt;/p&gt;&#xA;&lt;p&gt;In addition, Bolza makes the following remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the last section of the work mentioned previously, Gauss gives a strange transformation for the expression he found for the geodesic curvature, which is the actual core not only of Gauss's theorem about total curvature, but also of Bonnet's later generalization (1848). The expression given by Gauss has the following form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$K_gds = \frac{\sqrt{EG-F^2}(p'dq'-q'dp')}{Ep'^2 + 2Fp'q' + Gq'^2}+\frac{\Phi(p,q,p',q')dt}{(Ep'^2 + 2Fp'q' + Gq'^2)\sqrt{EG-F^2}}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; is a certain cubic form in variables &lt;span class=&quot;math-container&quot;&gt;$p', q'$&lt;/span&gt; whose coefficients are rational functions of &lt;span class=&quot;math-container&quot;&gt;$E,F,G$&lt;/span&gt; and their first derivatives with respect to &lt;span class=&quot;math-container&quot;&gt;$p,q$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bolza continues and describes Gauss's derivation of the following formula (I didn't include the derivation):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(*) K_gds - d\theta = Pdp + Qdq$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$ds$&lt;/span&gt; is an element of length of a curve, &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle between the positive p-direction and the tangent to the curve, and &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt; are the following functions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P = \frac{EF_p - \frac{1}{2}EE_q - \frac{1}{2}FE_p}{E\sqrt{EG-F^2}},Q = \frac{\frac{1}{2}EG_p - \frac{1}{2}FE_q}{E\sqrt{EG-F^2}} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Taking a line integral of both sides of equation (*) along a closed loop and applying Green's theorem on the right side enables to write the right side in the form:&lt;span class=&quot;math-container&quot;&gt;$\int\int(\frac {\partial P}{\partial q}-\frac{\partial Q}{\partial p})dpdq$&lt;/span&gt;. Apparently the term &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial P}{\partial q}-\frac{\partial Q}{\partial p}$&lt;/span&gt; is none other than the Gaussian curvature &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Bolza preceeds and says that integrating these equations in geodesic polar coordinates yields Bonnet's generalization of the theorem of total curvature (the famous Gauss-Bonnet theorem):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{\Omega} Kd\sigma = \int_{d\Omega}d\theta - \int_{d\Omega}K_gds$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since the boundary of the surface is a closed regular loop,  &lt;span class=&quot;math-container&quot;&gt;$\int_{d\Omega}d\theta = 2\pi$&lt;/span&gt;, so this theorem agrees with the fact that the Euler characteristic of a simply-connected surface with boundary (such a surface is topologically equivalent to a closed disk) is &lt;span class=&quot;math-container&quot;&gt;$\chi = 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The final step of integrating the expressions is not made explicitly in Gauss's manuscript.&lt;/p&gt;&#xA;&lt;p&gt;Bolza cocludes with the following words:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It can hardly be assumed that this obvious connection could have escaped from Gauss, and so one can probably assume that Gauss was already familiar with Bonnet's generalization of his theorem, as R.V. Lilienthal and P. Stackel have concluded from the fact that Gauss wrote the expression &lt;span class=&quot;math-container&quot;&gt;$K_gds$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$df$&lt;/span&gt;, and reffered to it as &amp;quot;the differential of the curvature of the side&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, there is one more piece of evidence to consider - Gauss in his published treatise on differential geometry (&amp;quot;General investigations of curved surfaces&amp;quot;, 1828) anounced the publication of further investigations on the &amp;quot;curvature integral&amp;quot; (such a memoir, however, didn't appear); in light of this unpublished work, it's not difficult to guess what he had in mind when he made the anouncement.&lt;/p&gt;&#xA;&lt;p&gt;As an illustration of several ideas expressed in this answer, I've added this figure.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SmNkI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SmNkI.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;em&gt;A sphere cutted along a small circle is an example of a curved and simply-connected surface with a non-geodesic boundary (boundary with geodesic curvature). The case of Gauss-Bonnet theorem which was probably covered in Gauss's investigations assures that for all geometries (including those with  metric, Gauss curvature and/or geodesic curvature different than a punctured sphere) with this topology (i.e, homeomorphic to a sphere with a hole) the sum of the total curvature of the surface and the total geodesic curvature of it's boundary is always &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2023-09-16T15:18:52.973" LastActivityDate="2023-09-16T15:18:52.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12252" PostTypeId="1" CreationDate="2020-09-11T03:14:49.580" Score="3" ViewCount="101" Body="&lt;p&gt;My question is basically about four different versions of mass from the early years of special relativity when the concept of relativistic mass was acceptable. I'd appreciate it if you try to keep your answer as simple as possible.&lt;/p&gt;&#xA;&lt;p&gt;Rest mass: &lt;span class=&quot;math-container&quot;&gt;$m_{0}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Relativistic mass: &lt;span class=&quot;math-container&quot;&gt;$m_{rel}=\frac{m_{0}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Longitudinal mass: &lt;span class=&quot;math-container&quot;&gt;$m_{L}=\frac{m_{0}}{\left( \root{2}\of{1-\frac{v^{2}}{c^{2}}}\right) ^{3}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Transverse mass: &lt;span class=&quot;math-container&quot;&gt;$m_{T}=\frac{m_{0}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I was under the impression that the concepts of longitudinal mass and transverse mass are no longer valid but it seems like I was wrong. The text indirectly points out that the concepts are okay but there might be some other reasons for not using them.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sometimes people like to use the terminology ‘longitudinal mass’ &lt;span class=&quot;math-container&quot;&gt;$\gamma ^{3}m_{0}$&lt;/span&gt; and&#xA;‘transverse mass’ &lt;span class=&quot;math-container&quot;&gt;$\gamma m_{0}$&lt;/span&gt;. This can be useful, but we will not adopt it.&#xA;The main point is that there is a greater inertial resistance to&#xA;velocity changes (whether an increase or a decrease) along the&#xA;direction of motion, compared to the inertial resistance to picking up&#xA;a velocity component transverse to the current motion (and both exceed&#xA;the inertia of the rest mass).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: Relativity Made Relatively Easy By Andrew M. Steane · 2012&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus, at the turn of the century, there arose, through, as we now&#xA;understand, the incorrect use of nonrelativistic equations to describe&#xA;relativistic objects, a family of “masses” that increase with the&#xA;energy of the body:&lt;/p&gt;&#xA;&lt;p&gt;“relativistic mass” &lt;span class=&quot;math-container&quot;&gt;$m = \frac{E}{c^{2}}$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;“transverse mass” &lt;span class=&quot;math-container&quot;&gt;$m_{t}=m\gamma $&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;“longitudinal mass” &lt;span class=&quot;math-container&quot;&gt;$m_{l}=m\gamma ^{3}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Note that for m ≠ 0 the relativistic mass is equal to the transverse&#xA;mass, but, in contrast to the latter, it also exists for massless&#xA;bodies, for which m = 0. We here use the letter m in the usual sense,&#xA;since we used it in the first part of this paper. But all physicists&#xA;during the first five years of this century, i.e., before the creation&#xA;of the theory of relativity, and many after the creation of that&#xA;theory called the relativistic mass the mass and denoted it by the&#xA;letter m, as did Poincaré in his 1900 paper. And then there must&#xA;necessarily arise, and did arise, a further, fourth term: the “rest&#xA;mass,“ which was denoted by &lt;span class=&quot;math-container&quot;&gt;$m_{0}$&lt;/span&gt;. The term “rest mass“ came to be&#xA;used for the ordinary mass, which, in a consistent exposition of the&#xA;theory of relativity, is denoted m.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: Energy And Mass In Relativity Theory 2009&lt;/p&gt;&#xA;&lt;p&gt;The pages #305 and #306 from &amp;quot;Classical Dynamics (1977)&amp;quot; by Donald T. Greenwood are also relevant to this discussion.&lt;/p&gt;&#xA;&lt;p&gt;Einstein also used longitudinal mass and transverse mass in his 1905 special relativity paper. Longitudinal mass is greater than the transverse mass, and transverse mass is equal to relativistic mass. When Force, &lt;strong&gt;F&lt;/strong&gt;, and velocity, &lt;strong&gt;v&lt;/strong&gt;, are in the same direction, longitudinal mass would be used and acceleration would be in the direction of &lt;strong&gt;v&lt;/strong&gt;. When &lt;strong&gt;F&lt;/strong&gt; and &lt;strong&gt;v&lt;/strong&gt; are perpendicular to each other, transverse mass would come into play, and &lt;strong&gt;v&lt;/strong&gt; changes in direction but not in magnitude.&lt;/p&gt;&#xA;&lt;p&gt;Now coming to the question.&lt;/p&gt;&#xA;&lt;p&gt;I had though that relativistic mass is a combination of some sort of transverse mass and longitudinal mass but I think I was wrong. Assuming that the relativistic mass and transverse mass are the same, I'd say that longitudinal mass should have been used more frequently because force and velocity are mostly in the same direction. Phrasing it differently, if longitudinal mass is (or, was) real (&lt;em&gt;please check edit #1 below&lt;/em&gt;) then it should have been used more than the transverse or relativistic mass but in actuality relativistic/transverse mass was the one which was mostly used in calculations. Personally, I have never seen longitudinal mass being used. Could you please help me with it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit #1:&lt;/strong&gt;&#xA;I'd say that using the word '&lt;em&gt;real&lt;/em&gt;' was a poor of choice of words. I was trying to say that if it is true that more force is required to accelerate a mass in the direction of velocity, then using longitudinal mass would have been a better choice for doing calculations. I think that if you are trying to accelerate an electron, which is already moving toward positive x-axis at constant velocity, then one would rather choose its longitudinal mass to calculate acceleration for a given force.&lt;/p&gt;&#xA;" OwnerUserId="9341" OwnerDisplayName="PG1995" LastEditorUserId="9341" LastEditDate="2020-09-11T08:51:08.843" LastActivityDate="2020-09-11T08:51:08.843" Title="Different versions of mass during early years of special relativity" Tags="|physics|relativity-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12254" PostTypeId="1" CreationDate="2020-09-12T00:23:29.230" Score="8" ViewCount="421" Body="&lt;p&gt;It is claimed &lt;a href=&quot;https://youtu.be/I3zgWE8z8Cw?t=3274&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that toward the end of his life Werner Heisenberg worked on a non-linear broadening or generalization of quantum mechanics. What work was that? Was it published?&lt;/p&gt;&#xA;&lt;p&gt;Is it listed in Cassidy's &lt;a href=&quot;https://archive.org/details/trent_0116400254532&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Werner Heisenberg: A Bibliography of His Writings&lt;/em&gt;&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2020-09-17T20:04:59.313" LastActivityDate="2021-08-25T11:24:31.357" Title="Heisenberg's last work on a non-linear generalization of quantum mechanics?" Tags="|reference-request|quantum-mechanics|heisenberg|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12256" PostTypeId="2" ParentId="12254" CreationDate="2020-09-12T09:27:16.703" Score="8" Body="&lt;p&gt;This likely refers to Heisenberg's non-linear spinor theory of elementary particles, on which he worked from 1953 to the end of his life. There was a prominent write-up in an unpublished 1958 preprint with Pauli, from which Pauli withdrew his name. Heisenberg later gave a lecture &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-642-61742-3_61&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Electrodynamics in the Nonlinear Spinor Theory (1966)&lt;/a&gt; and published a book &lt;a href=&quot;https://archive.org/details/introductiontoun0000heis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;An   Introduction   to   the   Unified   Theory   of   Elementary   Particles&lt;/em&gt;&lt;/a&gt; (1967) based on his Munich collaborations with Durr et al. In the book Heisenberg proposes that all properties of matter should follow from a highly symmetrical field equation that plays a role of &amp;quot;Plato's form&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Heisenberg's publications are listed on &lt;a href=&quot;https://academictree.org/physics/publications.php?pid=1932&quot; rel=&quot;nofollow noreferrer&quot;&gt;Academictree&lt;/a&gt;. The latest on the spinor theory is from 1974, &lt;a href=&quot;https://link.springer.com/article/10.1007%2FBF00602885&quot; rel=&quot;nofollow noreferrer&quot;&gt;The unified field theory of elementary particles: Some recent advances&lt;/a&gt;. Considering his health problems in 1973-6 it is unlikely that he worked much on it &amp;quot;in the last year&amp;quot;, see &lt;a href=&quot;https://www.heisenberg-gesellschaft.de/7-the-munich-years-1958-ndash-1976.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Munich Years (1958 – 1976)&lt;/a&gt;, the 1975-6 publications are more historical/philosophical.&lt;/p&gt;&#xA;&lt;p&gt;Here is &lt;a href=&quot;https://arxiv.org/abs/physics/0503235&quot; rel=&quot;nofollow noreferrer&quot;&gt;Todorov's description&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Heisenberg tries to build up a theory of a (not directly observable!) spinor field, which obeys a non-linear partial differential equation, with a coupling constant of dimension length square. The proton and the neutron, the building blocks of  the atomic nucleus, are not identified with the quanta of the basic field (as it is the case   with   electrons  in   quantum  electrodynamics), but  arise   as   its   bound  states. Heisenberg does not follow the beaten tracks (not even those that brought him success in the old days), he is looking for something really new...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In fact, Heisenberg is conscientious,that there are questions and “details”, which are not yet clear, but he is an optimist.For less emotionally engaged theorists the remaining difficulties and open problems set doubt on the whole enterprise. Shortly before his death later the same year (1958) Pauli   withdraws   his   name   from  their   joint   work   and  expresses   publicly   his disapproval. Heisenberg is affected and disappointed but he continues steadfastly his work (with his young collaborator H.P. Dürr). He works, like most great physicists of the  past   century   in   their   later   years,   in   virtual   isolation,  with   the  world   of   science ignoring his latest ideas.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;With   the  hindsight  of   subsequent   development   we  can   now  point  to   both   a prophetic vision and some shortcomings in this last attempt of the great scientist at anew breakthrough in fundamental physics... The quarks’ field in what we now call the standard model can be viewed as a realization of Heisenberg’s idea. (He discusses himself such a possibility after the first success of the quark model – some 10 years after his controversial 1958 paper.) In both cases the observable (strongly interacting) hadrons are bound states of the basic field whose quanta do not correspond to free particles. The idea of a spontaneously broken symmetry, associated today with the name of Higgs, plays a prominent part inthe Heisenberg model: it originates in his treatment of ferromagnetism of the 1930’s.The concept of a fundamental length is present in the now fashionable “superstring theory”...  Heisenberg   is   well   aware   of   the  non-renormalizability   of   his   model   but  he hopes that the gauge invariant equations of quantum electrodynamics will follow from his theory at distances large compared to the fundamental length.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="232" LastEditDate="2020-09-17T22:47:01.360" LastActivityDate="2020-09-17T22:47:01.360" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12258" PostTypeId="1" CreationDate="2020-09-13T20:33:25.540" Score="-2" ViewCount="170" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GeAKn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GeAKn.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Hi guys, I've been reading about the wave-particle duality of matter, and I came across this statement that said de Broglie was a prince. Is that true?&lt;/p&gt;&#xA;" OwnerUserId="13050" LastActivityDate="2020-09-19T02:40:12.277" Title="Was de Broglie, the physicist who suggested matter behaves like waves, a prince?" Tags="|physics|" AnswerCount="1" CommentCount="2" ClosedDate="2020-09-14T18:09:27.637" ContentLicense="CC BY-SA 4.0" />
  <row Id="12260" PostTypeId="2" ParentId="12258" CreationDate="2020-09-14T16:12:44.040" Score="5" Body="&lt;p&gt;Wikipedia describes the House of Broglie:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Broglia [it] (pronounced [ˈbrɔʎʎa]) was the name of an old Piedmontese family, from which were descended the counts of Casalborgone, Mombello and Revello, and the lords of Arignano, Cortandone, Fontanetto Po, Chieri, Cocconato, Monale, Montaldo,[8] Pont Canavese and Santena. The first reference to the name is dated 1245, mentioning one Ardizzone Broglia, father of Guglielmo, decurione of Chieri.[9]&lt;/p&gt;&#xA;&lt;p&gt;The founder of the French de Broglie line was Francesco Maria, count of Revello, of the Broglia di Chieri family. Born in 1611 in Piedmont, he took service in the French army in the Thirty Years' War and was naturalized in France after 1643. He is now known as François-Marie, comte de Broglie. After distinguishing himself as a soldier, he died, a lieutenant-general, at the siege of Valenza on 2 July 1656.[7]&lt;/p&gt;&#xA;&lt;p&gt;His son, Victor-Maurice, comte de Broglie (1647–1727), served under Condé, Turenne and other great commanders of the age of Louis XIV. He became maréchal de camp in 1676, lieutenant-general in 1688, and finally marshal of France in 1724.[7]&lt;/p&gt;&#xA;&lt;p&gt;His grandson, François-Marie, was made duc de Broglie and a peer of France in 1742. His great-grandson, Victor-François, 2nd duc de Broglie, was created prince of the Holy Roman Empire in 1759 by Emperor Francis I.&lt;/p&gt;&#xA;&lt;p&gt;All junior members of the House of Broglie bear the title of prince de Broglie, while the head of the family is the duc de Broglie. A junior line used the title of prince de Broglie-Revel, after one of its lordships.[7]&lt;/p&gt;&#xA;&lt;p&gt;Louis, 7th duc de Broglie (1892–1987), a physicist and Nobel laureate, was one of the founders of quantum theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/House_of_Broglie%5B1%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/House_of_Broglie[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Louis de Broglie (1892-1987), 7th Duke of Broglie, was the youngest child of Louis-Alphonse-Victor, (1846-1906), 5th Duke of Broglie, and the younger brother of Maurice de Broglie (1875-1960) 6th Duke of Broglie.&lt;/p&gt;&#xA;&lt;p&gt;So he didn't become the 7th Duke of Broglie until he was 68 years old.  So when he was a student he was not yet the Duke of Broglie.&lt;/p&gt;&#xA;&lt;p&gt;What does the Wikipedia quote above say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All junior members of the House of Broglie bear the title of prince de Broglie, while the head of the family is the duc de Broglie.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So apparently the title of Duc de Broglie descends by primogeniture, while the lesser title of Prince of Broglie is shared by all male members of the family.  The second Duc de Broglie was made a Prince of the Holy Roman Empire by Emperor Francis I in 1759, and possibly that title is shared by all of his male line descendants.&lt;/p&gt;&#xA;&lt;p&gt;The English word prince has several different meanings, as did the Latin word &lt;em&gt;princeps&lt;/em&gt; which it is based on.  In fact, the German language has two different words which are translated into English as prince - &lt;em&gt;Fürst&lt;/em&gt; and &lt;em&gt;Prinz&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the Holy Roman Empire, the word &lt;em&gt;Fürst&lt;/em&gt; itself had two meanings, one a subset of the other meaning.  A &lt;em&gt;Fürst&lt;/em&gt; was the &amp;quot;first&amp;quot; man in in his principality, and collectively the &lt;em&gt;Fürsten&lt;/em&gt; were the first men in the empire.  The princes of the Empire included rulers who used a number of different titles.  From the lowest to the highest the titles used by princes or &lt;em&gt;Fürsten&lt;/em&gt; of the empire included princely count, landgrave, margrave, count palatine, prince (&lt;em&gt;Fürst&lt;/em&gt;), duke, and the rare titles of grand duke and archduke.  Thus the princes of the Empire whose German title was &lt;em&gt;Fürst&lt;/em&gt; (prince) were a subcategory of the princes of the Empire.&lt;/p&gt;&#xA;&lt;p&gt;The German word &lt;em&gt;Prinz&lt;/em&gt; is used for the sons or grandsons,etc. of a king, royal princes.&lt;/p&gt;&#xA;&lt;p&gt;The Emperor also granted titles, such as &lt;em&gt;Fürst&lt;/em&gt;, to persons who did not rule fiefs or principalities in the Empire and thus did not have votes in the imperial Reichstag.  The de Broglie title of Prince of the Holy Roman Empire would thus be one of those titular prince titles without a functioning principality.&lt;/p&gt;&#xA;&lt;p&gt;The title of Prince of Broglie used by junior members of the Broglie family is obviously a French or Holy Roman Empire noble title that was granted to all male descendants of the orginal grantee, and thus would be &lt;em&gt;Fürst von Broglie&lt;/em&gt; in German, while the title of Duke of Broglie is a French title and descends to only the head of the Broglie family.&lt;/p&gt;&#xA;&lt;p&gt;So The 7th Duke of Broglie was apparently entitled to the title of Prince of Broglie all his life if that title was granted to all members of the Broglie family, while he didn't inherit the title of Duke of Broglie until his older brother died without sons.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="2905" LastEditDate="2020-09-19T02:40:12.277" LastActivityDate="2020-09-19T02:40:12.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12261" PostTypeId="1" CreationDate="2020-09-15T02:34:27.600" Score="1" ViewCount="85" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Antimatter#Conceptual_history&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; Schuster apparently discussed matter and antimatter annihilating each other.&lt;/p&gt;&#xA;&lt;p&gt;Had anyone suggested before this that matter could be transformed into energy?&#xA;(If indeed that is what is implied by annihilation or perhaps he meant just disappearing with no energy release.)&lt;/p&gt;&#xA;&lt;p&gt;Or had others seen matter and energy being convertible, and what Einstein did was quantify this conversion?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="12724" LastEditDate="2020-09-15T12:39:49.190" LastActivityDate="2020-09-15T12:39:49.190" Title="Did Arthur Schuster anticipate the equivalence between matter and energy in his speculations about antimatter?" Tags="|relativity-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12262" PostTypeId="1" AcceptedAnswerId="12263" CreationDate="2020-09-15T11:12:12.807" Score="6" ViewCount="471" Body="&lt;p&gt;Richard Bellman said :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The 1950s were not good years for mathematical research. We had a&#xA;very interesting gentleman in Washington named Wilson. He was&#xA;Secretary of Defense, and he actually had a pathological fear and&#xA;hatred of the word research.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Basically he said army and politicians hated the maths and research. That's why he named it : dynamic programming.&lt;/p&gt;&#xA;&lt;p&gt;But why ? According to my knowledge in WW2 had a lot of progress in maths and research : linear programming (simplex method), conditional probability (enigma machine), AI(it was basically maths research).  So even with those achievements, why did they hate maths in 1950?&lt;/p&gt;&#xA;&lt;p&gt;Honestly I didn't found this information on web (or maybe I don't know how ask to google)&lt;/p&gt;&#xA;&lt;p&gt;Best regards&lt;/p&gt;&#xA;" OwnerUserId="13052" LastEditorUserId="4251" LastEditDate="2021-08-30T22:00:17.977" LastActivityDate="2021-08-30T22:00:17.977" Title="Why did Richard Bellman, creator of dynamic programing say: 1950s were not good years for maths?" Tags="|mathematics-social-history|cold-war|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12263" PostTypeId="2" ParentId="12262" CreationDate="2020-09-15T14:14:04.307" Score="9" Body="&lt;p&gt;The quote appears to be from:  &lt;em&gt;Stuart Dreyfus,  (2002) Richard Bellman on the Birth of Dynamic Programming. Operations Research 50(1):48-51. &lt;a href=&quot;https://doi.org/10.1287/opre.50.1.48.17791&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1287/opre.50.1.48.17791&lt;/a&gt;&lt;/em&gt;.  In context, it's clear Bellman is talking about DoD spending on basic math/research specifically, not math in general.&lt;/p&gt;&#xA;&lt;p&gt;As to why the DoD was cutting spending, Eisenhower spent much of the period between the end of the Korean War and Sputnik trying to cut the large Defense budget left from the Truman admin., and Wilson was something of a hatchet man in this effort.  Basic research, which wasn't directly related to the military's basic mission, was an obvious target.  (Famously, Eisenhower would reverse course on this after Sputnik was launched.)&lt;/p&gt;&#xA;" OwnerUserId="1731" LastEditorUserId="12330" LastEditDate="2020-09-15T18:14:58.540" LastActivityDate="2020-09-15T18:14:58.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12264" PostTypeId="2" ParentId="12245" CreationDate="2020-09-15T23:56:04.267" Score="1" Body="&lt;p&gt;Tycho'  observations impacted on numerous levels. First, it presented an accurate, detailed calculation of the motion of the heavenly bodies.  At the time, 2 theories existed; helio- and earth-centric. The debate, via religion gravitated to the latter.  Yet, facts are stubborn things. Given data, Kepler formulated his 3 Laws. The accurate observations  where the prodigy of mathematical,  astronomical. Theoretical, observational, astrophysical, cosmological,terrestrial. And yes, theological sciences. To answer your question; inestimable.&lt;/p&gt;&#xA;&lt;p&gt;See: Feynman Lectures on Physics : Law of Gravitation&lt;/p&gt;&#xA;" OwnerUserId="13054" LastActivityDate="2020-09-15T23:56:04.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12265" PostTypeId="1" AcceptedAnswerId="12266" CreationDate="2020-09-16T16:50:10.147" Score="3" ViewCount="227" Body="&lt;p&gt;In its most basic form the &lt;a href=&quot;https://en.wikipedia.org/wiki/Turnstile_antenna&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turnstile antenna&lt;/a&gt; is two half-wave dipole antennas that are perpendicular and driven 90 degrees out of phase. For a recent review see &lt;a href=&quot;https://ieeexplore.ieee.org/document/7293591&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crossed Dipole Antennas: A review&lt;/a&gt; (also available &lt;a href=&quot;http://ael.cbnu.ac.kr/lectures/graduate/antenna-engineering/lab-11-crossed-dipole/ta-crossed%20dipole%20review%20%EA%B3%B5%EB%B6%80%EC%9E%90%EB%A3%8C.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Usually when used on Earth's surface the &amp;quot;X&amp;quot; is in the horizontal plane and the radiation linearly polarized, but out-of-plane (up and down) they also radiate, but with circular polarization.&lt;/p&gt;&#xA;&lt;p&gt;I see reference to the inventor's publication as&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;G. Brown, “The turnstile,” Electron., pp. 14–17, Apr. 1936.&lt;/li&gt;&#xA;&lt;li&gt;Brown, George (April 1936). &amp;quot;The Turnstile Antenna&amp;quot;. Electronics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;and the 1935 patent is &lt;a href=&quot;https://patents.google.com/patent/US2086976&quot; rel=&quot;nofollow noreferrer&quot;&gt;US2086976&lt;/a&gt;, but when I type those into a search engine I can not find the original paper, presumably in a journal or magazine titled simply &amp;quot;Electronics&amp;quot; in 1936.&lt;/p&gt;&#xA;&lt;p&gt;I don't even know how to reference the journal, because &amp;quot;electronics journal&amp;quot; returns quite a lot of results.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How can I find the full, complete, and unambiguous citation for the journal paper that I believe Brown published in 1936 that explains the Turnstile antenna?&lt;/p&gt;&#xA;&lt;p&gt;Through the university's library system I may be able to read the article online once I can zero in on the specific journal, and perhaps with the help of a librarian, but there weren't too many engineering schools here in 1936 so I may not be able to get a hold of a bound copy.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="10106" LastEditDate="2020-09-17T04:26:17.273" LastActivityDate="2020-09-17T13:55:00.690" Title="Where exactly did George Brown publish the first paper about Turnstile antennas?" Tags="|physics|reference-request|electromagnetism|electronics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12266" PostTypeId="2" ParentId="12265" CreationDate="2020-09-16T21:48:20.507" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Electronics_(magazine)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Electronics&lt;/a&gt;, (subtitled 'radio, communication, industrial applications of electron tubes ... engineering and manufacture') was published by McGraw Hill. Not a journal per se, more of an industry magazine.  One place to find old copies is through &lt;a href=&quot;https://worldradiohistory.com/Electronics%20_Master_Page.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;World Radio History&lt;/a&gt;, which includes a pdf of the April 1936 issue in question. Indeed, on page 14 is the article The &amp;quot;Turnstile&amp;quot; by George H. Brown of RCA.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/v2gD3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/v2gD3.png&quot; alt=&quot;article header&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Never underestimate the desire of radio enthusiasts, including ham folks, of archiving old issues of various magazines. And, if you are interested in the history of radio, be prepared to spend a looooong time looking at stuff on that site (as one example, the lead article in this issue discusses Pan Am's radio system including how that helped the Clippers fly across the Pacific).&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2020-09-17T13:55:00.690" LastActivityDate="2020-09-17T13:55:00.690" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12267" PostTypeId="2" ParentId="12262" CreationDate="2020-09-16T22:23:32.440" Score="5" Body="&lt;p&gt;simplicio's answer certainly provides the correct historical context. Adding to his answer, the quote is also found in Bellman's autobiography, p.159 - which seems to be the original source. Immediately after the sentences you quoted, Bellman continues:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] hatred of the word research. I'm not using the term lightly; I'm&#xA;using it precisely. His face would suffuse, he would turn red, and he&#xA;would get violent if people used the term, research, in his presence.&#xA;You can imagine how he felt, then, about the term, mathematical. The&#xA;RAND Corporation was employed by the Air Force, and the Air Force had&#xA;Wilson as its boss, essentially. Hence, I felt I had to do something&#xA;to shield Wilson and the Air Force from the fact that I was really&#xA;doing mathematics inside the RAND Corporation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From Bellman's description, it seems that in particular Wilson as a person had a very adverse attitude towards basic research. Wilson was only a hatchet man for Eisenhower's mission, but he seems to have been very passionate about cutting down basic research.&lt;/p&gt;&#xA;&lt;p&gt;Source:&#xA;Richard Bellman, Eye of the hurricane. An autobiography, World Scientific, 1984.&lt;/p&gt;&#xA;" OwnerUserId="9523" LastActivityDate="2020-09-16T22:23:32.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12268" PostTypeId="1" AcceptedAnswerId="12275" CreationDate="2020-09-17T20:54:38.867" Score="3" ViewCount="182" Body="&lt;p&gt;There are many schematics and many photographs of tubes used later but I cannot find a single photograph of the original apparatus.  Even the sketches that were made in the late 19th century are missing important details (like the fact that the electrodes need to be in a vacuum vessel).&lt;/p&gt;&#xA;" OwnerUserId="13061" LastActivityDate="2020-09-20T07:26:25.707" Title="Are there any photographs of the original apparatus used by Hertz to demonstrate the photoelectric effect?" Tags="|reference-request|discoveries|experimental-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12269" PostTypeId="1" AcceptedAnswerId="12270" CreationDate="2020-09-17T22:33:53.337" Score="3" ViewCount="2352" Body="&lt;p&gt;I'm aware that it was known to the ancient Greeks that the planets were somehow different from the stars. But in what depth did they know the solar system? In particular, did they know:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;that the sun was an ordinary star?&lt;/li&gt;&#xA;&lt;li&gt;the motion of the planets?&lt;/li&gt;&#xA;&lt;li&gt;the different sizes of the planets (&lt;em&gt;this is the part that started the whole question: Jupiter is big, while Mercury is small and fast. I suppose they chose the names knowing this, but the calculations aren't that simple&lt;/em&gt;)?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If any of the answers is yes, how did they? What math did they use?&lt;/p&gt;&#xA;" OwnerUserId="10750" LastEditorUserId="5788" LastEditDate="2022-12-28T20:30:23.970" LastActivityDate="2022-12-29T00:11:46.573" Title="What did the ancient Greeks know about the solar system, and how?" Tags="|astronomy|ancient-greece|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12270" PostTypeId="2" ParentId="12269" CreationDate="2020-09-18T01:35:49.663" Score="10" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;No they did not know this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The motion of the Sun, Moon and planets (as seen from the Earth) was known, in the sense that it could be predicted with reasonable accuracy.&#xA;To do this, they used an ingenious model, approximating the periodic motions by combinations of circular motions, the same principle that is used nowadays for predictions of celestial motions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The sizes of the planets and distances to them were not known, except for the Moon but the order (Mercury, Venus, Mars, Jupiter, Saturn was known, or better to say &amp;quot;correctly guessed&amp;quot;). About the Moon, they new approximate distance, so they could compute the size. They had no means to estimate the distance to the Sun&#xA;and planets, thus no means to estimate their sizes. There were attempts to do this but they came with wrong answers by the orders of magnitude. But Ptolemy understood that &amp;quot;The Earth is like a point in comparison to&#xA;distances to planets and stars&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;They understood that stars and planets must be at an enormous distance since they experience no visible parallax. But they could not imagine how really enormous these distances are. (The absence of parallax was the strongest argument against&#xA;the heliocentric system which they proposed and then rejected.)&lt;/p&gt;&#xA;&lt;p&gt;All this reflect the knowledge at the time of Ptolemy, 2-nd century AD, Egypt, Roman empire. He is considered a Greek because he wrote in Greek, as all other astronomers in the Roman empire. Most of the earlier work is lost, and we know about it only from mentioning in the secondary sources&#xA;and Ptolemy himself.&lt;/p&gt;&#xA;&lt;p&gt;What math did they use? Arithmetic (they did complicated calculations in sexagesimal system), Geometry and Trigonometry.&#xA;Trigonometry was actually invented for this very purpose, and the earliest surviving source on trigonometry is also Ptolemy. (Ptolemy mentions Hipparchus, but the work of Hipparchus did not survive. There is also strong indirect evidence that the tables of sines existed before Ptolemy.)&lt;/p&gt;&#xA;&lt;p&gt;Important remarks. You are asking about &amp;quot;ancient Greeks&amp;quot;. I interpret this in a broad sense. Very little survives of Greek astronomy before Ptolemy.&#xA;Ptolemy was a &amp;quot;Greek&amp;quot; only in the sense that he wrote in Greek (as all other astronomers of the Roman empire). He was born at the time of Trajan and died at the time of Marcus, and probably was a Roman citizen. He probably spent most of his active life in Alexandria, in the Roman Egypt. Almost all that we know about Greek astronomy comes from his work.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-12-29T00:11:46.573" LastActivityDate="2022-12-29T00:11:46.573" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12271" PostTypeId="1" AcceptedAnswerId="12272" CreationDate="2020-09-19T01:08:05.723" Score="2" ViewCount="777" Body="&lt;p&gt;Wikipedia says he discovered that the electron was different than an atom or molecule. His line of reasoning is not shown. I additionally searched stack exchange and unless I missed it.....&lt;/p&gt;&#xA;&lt;p&gt;I am lost as to what tag to use on this or if it is &amp;quot;off topic&amp;quot;. Perhaps there is a good resource on the J. J. Thomson discovery. It is interesting that he was quoted as saying the electron was too small to ever be useful for anything. But since he lived until 1940 he was probably a very happy camper to know how useful his discovery turned out to be.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4251" LastEditDate="2020-09-19T21:53:53.387" LastActivityDate="2021-08-07T18:15:46.317" Title="How did J.J. Thomson learn that what he discovered was different than an atom or a molecule?" Tags="|physics|discoveries|electricity|physicists|atomic-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12272" PostTypeId="2" ParentId="12271" CreationDate="2020-09-19T03:14:46.953" Score="9" Body="&lt;p&gt;Thomson drew his conclusion based on his calculation of the charge to mass ratio of the &amp;quot;corpuscular carriers of negative charge in cathode rays&amp;quot;. Quoting Thomson's &lt;a href=&quot;https://web.lemoyne.edu/giunta/EA/THOMSONann.HTML&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nobel Lecture of 1906&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This constant value, when we measure &lt;span class=&quot;math-container&quot;&gt;$e/m$&lt;/span&gt; in the c.g.s. system of magnetic units, is equal to about &lt;span class=&quot;math-container&quot;&gt;$1.7\times10^7$&lt;/span&gt;. If we compare this with the value of the ratio of the mass to the charge of electricity carried by any system previously known, we find that it is of quite a different order of magnitude. Before the cathode rays were investigated, the charged atom of hydrogen met with in the electrolysis of liquids was the system which had the greatest known value of &lt;span class=&quot;math-container&quot;&gt;$e/m$&lt;/span&gt;, and in this case the value is only &lt;span class=&quot;math-container&quot;&gt;$10^4$&lt;/span&gt;, hence for the corpuscle in the cathode rays the value of &lt;span class=&quot;math-container&quot;&gt;$e/m$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$1700$&lt;/span&gt; times the value for the corresponding quantity for the charged hydrogen atom. This discrepancy must arise in one or other of two ways; either the mass of the corpuscle must be very small compared with that of the atom of hydrogen, which until quite recently was the smallest mass recognized in physics, or else the charge on the corpuscle must be very much greater than that on the hydrogen atom. Now it has been shown by a method which I shall shortly describe, that the electric charge is practically the same in the two cases; hence we are driven to the conclusion that the mass of the corpuscle is only about &lt;span class=&quot;math-container&quot;&gt;$1/1700$&lt;/span&gt; of that of the hydrogen atom. Thus the atom is not the ultimate limit to the subdivision of matter; we may go further and get to the corpuscle, and at this stage the corpuscle is the same from whatever source it may be derived.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See the linked transcript for details of Thomson's method of calculation.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2021-08-07T18:15:46.317" LastActivityDate="2021-08-07T18:15:46.317" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12273" PostTypeId="1" CreationDate="2020-09-19T18:46:05.687" Score="3" ViewCount="407" Body="&lt;p&gt;Riazuddin was a Pakistani theoretical physicist, his name looks a little bit different to me from others because he doesn't have a last name. There are some others Pakistani theoretical physicists that have a last name, like Abdus Salam. Why is it that Riazuddin doesn't have one?&lt;/p&gt;&#xA;&lt;p&gt;It seems that his original name was رياض الدين. Why does the Occident translates this as just one name when there is clearly a space between the two words?&lt;/p&gt;&#xA;" OwnerUserId="12968" LastEditorUserId="6085" LastEditDate="2020-09-19T23:22:49.690" LastActivityDate="2020-09-20T20:10:20.730" Title="Why does Riazuddin not have a last name?" Tags="|physics|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12274" PostTypeId="2" ParentId="12273" CreationDate="2020-09-19T20:25:25.513" Score="6" Body="&lt;p&gt;I believe you are referring to the Nobel laureate's Abdus Salam's PhD student who himself became a particle physicist like his advisor. &lt;a href=&quot;https://en.wikipedia.org/wiki/Riazuddin_(physicist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riazuddin, physicist&lt;/a&gt;. A quick search of Google Scholar shows his early papers in particle physics as M. Riazuddin, which is &lt;em&gt;mostly likely&lt;/em&gt; an abbreviated version of Mohammad  Riazuddin. It is very common in South Asia to skip the first name and call by the middle name if the first name starts with Mohammad (out of respect).&lt;/p&gt;&#xA;&lt;p&gt;You are right, in Urdu, Riaz is an independent name which is plural of &amp;quot;garden&amp;quot; , but &amp;quot;Uddin&amp;quot; cannot exist independently.&lt;/p&gt;&#xA;&lt;p&gt;So to parse his name M. Riazuddin, it could be Mohammad Riaz Uddin but his joint spelling Riazuddin is also common.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, it seems that associating &amp;quot;M.&amp;quot; with Riazuddin is a mistake of CERN / Google Scholar.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; If we check Cambridge University library for his PhD thesis, his only name appears as Riazuddin as the author. In CERN scanned papers, the databases list him as M. Riazuddin but actually only Riazuddin appears in the actual pdf of the paper. See for example: &lt;a href=&quot;https://cds.cern.ch/record/143116&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cds.cern.ch/record/143116&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My (pure) speculation is that both last names of the twin brothers rhyme, so if they &lt;em&gt;ever had&lt;/em&gt; first names, they just dropped them for convenience. It is very likely that he never had or used M. before his name. He just had a single name which just means &amp;quot;gardens of faith&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2020-09-20T20:10:20.730" LastActivityDate="2020-09-20T20:10:20.730" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12275" PostTypeId="2" ParentId="12268" CreationDate="2020-09-20T07:14:22.177" Score="4" Body="&lt;p&gt;Appleyard's paper &lt;a href=&quot;https://worldradiohistory.com/Archive-ITT/20s/ITT-Vol-06-1927-02.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pioneers of Electrical Communication part 5 – Heinrich Rudolph Hertz&lt;/a&gt; (Electrical Communication, 6 (1927) no.2, pp. 63-77) has a number of photographs of Hertz's experimental devices, including the apparatus he used to study the photoelectric effect in 1887 (Hertz did not use the name) after discovering the influence of light on production of sparks in his experiments on reception of electromagnetic waves. A reproduction of the image and the accompanying text are below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He observed that the sparks at the gaps of the exploring resonators were influenced by light from any&#xA;neighbouring sparks - for example, by light from&#xA;the originating spark at the induction coil. He&#xA;was in this manner led to discover the influence&#xA;of ultra-violet light upon electric discharges.&#xA;The original apparatus employed by Hertz at&#xA;Karlsruhe to demonstrate this effect is illustrated&#xA;in Figure 10. A vacuum is first formed&#xA;in the receiver by an air-pump, and the spark gap&#xA;is adj u sted so as to be somewhat too long to allow the discharge, with ordinary illumination,&#xA;to take place. Ultra-violet light, from&#xA;another spark or from some other source, is&#xA;then allowed to fall upon the gap, ionization&#xA;consequently occurs, and the spark passes.&#xA;Screening of the spark-gap from ultra-violet&#xA;light consequently diminishes the maximum&#xA;spark-length in a resonator corresponding to&#xA;any given arrangement of the resonator. These&#xA;results were published by him in June, 1887.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Figure 10: Original Apparatus Used by Hertz at&#xA;Karlsruhe, to illustrate the effect of ultra-violet light&#xA;on electric discharge&lt;/em&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Zq7Mk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Zq7Mk.png&quot; alt=&quot;Hertz Karlsruhe 1887&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-20T07:26:25.707" LastActivityDate="2020-09-20T07:26:25.707" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12277" PostTypeId="1" CreationDate="2020-09-20T12:40:21.503" Score="2" ViewCount="110" Body="&lt;p&gt;The book &amp;quot;The Road to Relativity&amp;quot; by Gutfreund and Renn annotates Einstein's original handwritten manuscript from 1916 - &amp;quot;The Foundation of the General Theory of Relativity.&amp;quot; I can find the original German published version as well as several translations but not the handwritten manuscript. Is there an electronic version available of the handwritten manuscript?&lt;/p&gt;&#xA;" OwnerUserId="13072" OwnerDisplayName="Rob Dimeo" LastEditorUserId="10106" LastEditDate="2020-09-20T13:37:02.053" LastActivityDate="2020-09-20T13:37:02.053" Title="Einstein's handwritten manuscript on General Relativity" Tags="|physics|reference-request|einstein|physicists|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12278" PostTypeId="1" AcceptedAnswerId="12280" CreationDate="2020-09-21T10:52:58.733" Score="3" ViewCount="739" Body="&lt;p&gt;I would like to hear about the historical &lt;em&gt;and&lt;/em&gt; technical reasons for why Zermelo-Fraenkel set theory with the axiom of Choice became the dominant standard for the foundations of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;The system has certainly gained a lot of momentum in the academic cirlce since its inception a century ago, but what are the details of the whole story?&lt;/p&gt;&#xA;&lt;p&gt;I have not been able to find any answer online.&lt;/p&gt;&#xA;" OwnerUserId="13076" LastEditorUserId="13076" LastEditDate="2020-09-27T17:37:11.217" LastActivityDate="2020-09-27T17:37:11.217" Title="How did ZFC become the standard foundations of mathematics?" Tags="|mathematics|set-theory|mathematics-social-history|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12279" PostTypeId="1" CreationDate="2020-09-21T21:31:43.040" Score="5" ViewCount="157" Body="&lt;p&gt;HR diagrams show in which of several sequences individual stars fall, each respecting the rough principle that hotter stars are of higher luminosity. (Sequences other than the main sequence &lt;a href=&quot;https://en.wikipedia.org/wiki/Stellar_classification#/media/File:HR-diag-no-text-2.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;may bend or be close to level&lt;/a&gt;, but let's put that aside.) Much about such diagrams takes some getting used to:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We convey temperature on the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis and luminosity and the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis, and so might expect the main sequence to slope up. In practice it slopes down, because the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis is &amp;quot;running backwards&amp;quot;: higher temperatures are on the left.&lt;/li&gt;&#xA;&lt;li&gt;Of course, the fact that the main sequence slopes down in this convention tells us the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis is &amp;quot;the right way up&amp;quot;: the highest parts of the diagram have greater luminosity. But if you glance at the numbers on the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis, you might get the erroneous impression this isn't so, because the absolute magnitude &lt;span class=&quot;math-container&quot;&gt;$M_V$&lt;/span&gt; falls toward the top. This is because it is a decreasing function of luminosity (a fact you'd never guess from the variable's name), specifically a negative logarithm of a nondimensionalised luminosity.&lt;/li&gt;&#xA;&lt;li&gt;Finally (this last concern is much lesser than the others), just in case the uninitiated's head isn't spinning enough, many values of &lt;span class=&quot;math-container&quot;&gt;$M_V$&lt;/span&gt; are negative, which can confuse people trying to order the numbers too.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Needless to say, if we just put Kelvin on a logarithmic &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-axis, increasing to the right, and wattage on a logarithmic &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-axis (note that log-wattage will be positive even for very dim stars), increasing upwards, this would be a lot easier on everyone.&lt;/p&gt;&#xA;&lt;p&gt;Now, part of the history behind this is that some decisions were made when we had a poorer understanding of what we were measuring. But this doesn't tell the whole story; as standards developed, one would expect rival diagram designs to be considered, and each option's pros and cons reviewed. How did astronomers settle on the status quo?&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2020-09-22T11:19:42.630" LastActivityDate="2020-09-22T11:19:42.630" Title="How did Hertzsprung-Russell diagrams end up so confusing?" Tags="|astronomy|notation|astrophysics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12280" PostTypeId="2" ParentId="12278" CreationDate="2020-09-23T07:46:11.827" Score="8" Body="&lt;p&gt;First, &amp;quot;foundations&amp;quot; are not what they once used to be. The idea of &amp;quot;one true logic&amp;quot; and &amp;quot;one true mathematics&amp;quot; justifiable from self-evident truths does not have much currency these days. So interest in real foundations, and belief in their existence or necessity, has been consistently waning, see &lt;a href=&quot;https://as.tufts.edu/philosophy/sites/all/themes/asbase/assets/documents/azzouniStillASense.pdf&quot; rel=&quot;noreferrer&quot;&gt;Azzouni, Is there still a Sense in which Mathematics can have Foundations?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Frege-Russell's project of transforming mathematical practice into formal enterprise did not come to fruition either. &lt;span class=&quot;math-container&quot;&gt;$ZFC$&lt;/span&gt; is getting textbook lip service as a check on paradoxes and a source of mathematical logic curiae, but the actual proofs are still given essentially in the vernacular of naive set theory, see comments in the &lt;a href=&quot;https://mathoverflow.net/q/94021/51484&quot;&gt;MO thread&lt;/a&gt; for current reactions. It is more prominent in mathematical logic and higher set theory, where the plethora of technical results on first order logic, on independence and consistency strength, and the fact that more complex theories are transparently modeled in it, turned &lt;span class=&quot;math-container&quot;&gt;$ZFC$&lt;/span&gt; into a convenient common yardstick, a &lt;em&gt;lingua franca&lt;/em&gt; of the field. But that, as with Latin or English, is, in part, a historical contingency.&lt;/p&gt;&#xA;&lt;p&gt;Now, how it happened, &lt;a href=&quot;http://math.bu.edu/people/aki/16.pdf&quot; rel=&quot;noreferrer&quot;&gt;Kanamori's The Mathematical Development of Set Theory from Cantor to Cohen&lt;/a&gt; is a detailed source. After the loud controversy over his axiom of choice Zermelo in 1908 set up a system &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; of seven axioms (AC not included) that &amp;quot;&lt;em&gt;started from set theory as it is historically given... to exclude all contradictions&lt;/em&gt;&amp;quot; and &amp;quot;&lt;em&gt;to retain all that is valuable&lt;/em&gt;&amp;quot;. The authoritative summary of set theory, Hausdorff's &lt;em&gt;Grundzüge der Mengenlehre&lt;/em&gt; (1914), which was to become Bourbaki's inspiration, did not incorporate it, see &lt;a href=&quot;https://blog.degruyter.com/set-theory-evolved-hausdorff-today/&quot; rel=&quot;noreferrer&quot;&gt;How Set Theory Evolved From Hausdorff Until Today&lt;/a&gt;. Hausdorff considered axiomatizations premature, and used refined naive set theory instead. But in 1910-1913 Russell and Whitehead published their &lt;em&gt;Principia Mathematica&lt;/em&gt;, which accomplished a formidable task (with largely unacknowledged help from Schröder's &lt;em&gt;Algebra der Logik&lt;/em&gt; (1890-1905)): it convinced the initiated that all of mathematics known to date could, in principle, be fully formalized.&lt;/p&gt;&#xA;&lt;p&gt;Over the course of 1920-s two major developments took place: von Neumann and Fraenkel added axioms of regularity and foundation to &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt;, and the primality of first order logic started emerging. The latter is often credited to Skolem and Hilbert, see &lt;a href=&quot;https://philosophy.stackexchange.com/a/2618/9148&quot;&gt;How did first-order logic come to be the dominant formal logic?&lt;/a&gt;, but it was solidified by Gödel's theorems that displayed its technical virtues. Ironically, Gödel originally proved incompleteness in &lt;span class=&quot;math-container&quot;&gt;$PM$&lt;/span&gt;, which was not first order, and Zermelo, who endorsed &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; with additional axioms in 1930, the modern &lt;span class=&quot;math-container&quot;&gt;$ZF$&lt;/span&gt;, advocated its second order reading. How &lt;span class=&quot;math-container&quot;&gt;$PM$&lt;/span&gt; was gradually washed out of circulation due to its maze of ramified types and clumsy notation, can be seen, in part, from &lt;a href=&quot;https://hsm.stackexchange.com/a/11608/55&quot;&gt;Who superseded Peano's dot notation in symbolic logic and when?&lt;/a&gt; So the early alternatives fell by the wayside. When Bourbaki started putting out their &lt;em&gt;Éléments de mathématique&lt;/em&gt; in 1939 their axioms were not quite Zermelo's, but the system was equivalent to &lt;span class=&quot;math-container&quot;&gt;$ZFC$&lt;/span&gt; minus foundation, see &lt;a href=&quot;https://www.jstor.org/stable/24021615&quot; rel=&quot;noreferrer&quot;&gt;On Bourbaki's axiomatic system for set theory&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What about later alternatives? Gödel showed that &lt;span class=&quot;math-container&quot;&gt;$PM$&lt;/span&gt;'s theory of types was equivalent in consistency strength and expressive power to &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt;, which was simpler and closer to the vernacular. Bernays, anticipated by von Neumann, proposed a set theory with classes, &lt;span class=&quot;math-container&quot;&gt;$NBG$&lt;/span&gt;, adopted by Gödel in 1940, which was proved to be a conservative extension of &lt;span class=&quot;math-container&quot;&gt;$ZFC$&lt;/span&gt;. Quine, another influential advocate of first order logic, proposed New Foundations in 1937, later &lt;span class=&quot;math-container&quot;&gt;$NFU$&lt;/span&gt;, which also turned out to be bi-interpretable with &lt;span class=&quot;math-container&quot;&gt;$ZFC$&lt;/span&gt;. By 1960-s it became clear that genuine alternatives (see &lt;a href=&quot;https://plato.stanford.edu/entries/settheory-alternative/&quot; rel=&quot;noreferrer&quot;&gt;SEP survey&lt;/a&gt;) concern themselves with matters that ordinary mathematicians need not concern themselves with. And &lt;span class=&quot;math-container&quot;&gt;$ZFC$&lt;/span&gt; had the advantages of simplicity and familiarity. For comparisons with later non-set-theoretic &amp;quot;foundational&amp;quot; alternatives, such as category theory or recent univalent foundations, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s40961-016-0082-6&quot; rel=&quot;noreferrer&quot;&gt;Dzamonja, Set Theory and its Place in the Foundations of Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-09-26T19:04:11.653" LastActivityDate="2020-09-26T19:04:11.653" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12281" PostTypeId="1" CreationDate="2020-09-23T10:29:54.407" Score="3" ViewCount="501" Body="&lt;p&gt;The title says it all.  The irrationality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; was proved by Lambert in the 18th century, but the Greeks at the time of Pythagoras already knew that &lt;span class=&quot;math-container&quot;&gt;$\sqrt2$&lt;/span&gt; and the golden ratio were irrational.  Did the classical Greek mathematicians already suspect that &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is also irrational or did nobody think about this before Lambert?  Did he settle an old question or did his result come more or less out of the blue?&lt;/p&gt;&#xA;" OwnerUserId="3555" LastEditorUserId="5989" LastEditDate="2020-09-23T15:51:09.043" LastActivityDate="2020-09-24T13:25:49.717" Title="When did mathematicians suspect that $\pi$ is irrational?" Tags="|mathematics|ancient-greece|18th-century|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12282" PostTypeId="2" ParentId="12281" CreationDate="2020-09-23T11:18:44.680" Score="3" Body="&lt;p&gt;&lt;em&gt;&amp;quot;suspect&amp;quot; is hard to track&lt;/em&gt;...&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;François Viète&lt;/a&gt; in &lt;em&gt;Variorum de rebus responsorum mathematics liber VIII&lt;/em&gt; (1593) discovered the first infinite product in the history of mathematics by giving an expression of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; with what is now called &lt;a href=&quot;https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Viète's formula&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/John_Wallis&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Wallis&lt;/a&gt;, like Viète, expressed &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; in the form of an infinite formula, but involving only rational operations.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Brouncker,_2nd_Viscount_Brouncker#Brouncker%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Brouncker&lt;/a&gt; transformed Wallis' formula into a continued fraction.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonhard Euler&lt;/a&gt; produced a lot of formulas involving &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; without founding any &amp;quot;periodicty&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Thus, at least from Euler's time, the suspect that it is irrational must be quite common, and a new issue emerged: what type of number &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is, algebraic or transcendental ?&lt;/p&gt;&#xA;&lt;p&gt;But see &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Gregory_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Gregory (1638 – 1675)&lt;/a&gt;'s &lt;a href=&quot;http://www-groups.mcs.st-andrews.ac.uk/%7Eedmund/GpsStAnd/Groups05.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vera circuli et hyperbolae quadratura(1667)&lt;/a&gt; that attempts&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;to prove that π and e are transcendental but contains a subtle error.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also: Lennart Berggren &amp;amp; Peter Borwein, &lt;a href=&quot;https://books.google.it/books?id=QhbrBwAAQBAJ&amp;amp;pg=PA97&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pi: A Source Book (Springer, 2004)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gregory attempted to show that the area of a general sector of an ellipse, circle or hyperbola could not be expressed in terms of the areas of the inscribed  and circumscribed triangle and quadrilateral using arithmetical operations and root extractions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2020-09-24T13:25:49.717" LastActivityDate="2020-09-24T13:25:49.717" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12283" PostTypeId="1" CreationDate="2020-09-23T13:17:07.843" Score="2" ViewCount="391" Body="&lt;p&gt;I am seeking an understanding of the scientific term &amp;quot;matter&amp;quot;. My research tells me that &amp;quot;hyle&amp;quot; and &amp;quot;materia&amp;quot; were both used.  I am trying to create something of a timeline for the development of the idea of matter and struggling a bit with the etymology aspect.  I am simply a homeschool parent.  Don't judge me too harshly.  Thanks!&lt;/p&gt;&#xA;" OwnerUserId="13083" LastActivityDate="2020-10-23T14:03:10.293" Title="What are the origins of the term &quot;matter&quot;?" Tags="|terminology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12284" PostTypeId="1" AcceptedAnswerId="12286" CreationDate="2020-09-23T13:17:10.833" Score="1" ViewCount="131" Body="&lt;p&gt;Besides the Peano axioms which perhaps brought him fame but which are considered a refinement of similar previous axioms Peano seems to have done relatively little original work.&#xA;Does the rest of his work compare favourably to the totality of the works of the other great mathematicians of the same era?&#xA;It seems to me that whether this can be answered objectively enough or not may be determined by his &amp;quot;significance distance&amp;quot; from the great mathematicians of that era.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-09-25T00:38:02.397" LastActivityDate="2020-09-25T00:38:02.397" Title="Was Giuseppe Peano one of the greatest mathematicians of his era?" Tags="|mathematics|mathematicians|biographical-details|" AnswerCount="2" CommentCount="2" ClosedDate="2020-09-25T07:07:11.923" ContentLicense="CC BY-SA 4.0" />
  <row Id="12285" PostTypeId="2" ParentId="12283" CreationDate="2020-09-23T13:24:44.970" Score="1" Body="&lt;p&gt;It will be very difficult to trace who came up with the concept of matter. Why don't you start with the concept of atom for educational purposes. This idea is well researched.&lt;/p&gt;&#xA;&lt;p&gt;The unabridged Oxford Dictionary (subscription based) traces the word origin of matter as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Etymology: &amp;lt; Anglo-Norman matier, matere, matire, mateire, Old French matere, matiere (12th cent.; French matière ) &amp;lt; classical Latin māteria (also māteriēs ) wood, timber, building material, material of which a thing is made, purulent matter, subject of discourse or consideration, also (in philosophical use) ‘matter’ in contradistinction to ‘mind’ or to ‘form’ &amp;lt; māter mother n.1 + -ia -ia suffix1 (usually explained as originally denoting the trunk of a tree regarded as the ‘mother’ of its offshoots). Compare Italian materia (12th cent.), Spanish materia (1220–50), Portuguese matéria (14th cent.), Old Occitan materia (14th cent.), Romanian materie, etc.&#xA;The sense development of Latin māteria was influenced by that of ancient Greek ὕλη hyle n., of which it was the accepted equivalent in philosophical use.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The origin of hyle is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Etymology: &amp;lt; medieval Latin hȳlē, &amp;lt; Greek ὕλη wood, timber, material, by Aristotle and in later Greek ‘matter’.(Show Less)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2020-09-23T13:24:44.970" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12286" PostTypeId="2" ParentId="12284" CreationDate="2020-09-23T13:31:27.307" Score="8" Body="&lt;p&gt;Actually, Peano did much more original work than that. Among other things, he&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;gave the modern definition of vector spaces and linear maps;&lt;/li&gt;&#xA;&lt;li&gt;defined the first space-filling curve;&lt;/li&gt;&#xA;&lt;li&gt;stated and proved the Peano existence theorem (on the existence of solutions to certain initial value problems); and&lt;/li&gt;&#xA;&lt;li&gt;developed the logical notation that would be later used by Bertrand Russell and Alfred North Whitehead to write &lt;em&gt;Principia Mathematica&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, although he was not on the same level as Henri Poincaré (who was four years older than he) or as David Hilbert (who was four years younger), he was a very gifted mathematician, whose work was important and concerned several distinct areas of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="12330" LastEditDate="2020-09-24T20:53:45.130" LastActivityDate="2020-09-24T20:53:45.130" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12287" PostTypeId="1" CreationDate="2020-09-23T18:01:42.850" Score="1" ViewCount="150" Body="&lt;p&gt;Greek geometry was confronted with problems such as squaring the circle. Straightedge and compass constructions were unable to provide a solution, but other mechanical curves, such as the quadratix, were introduced to deal with these difficult problems.&lt;/p&gt;&#xA;&lt;p&gt;(a) Was there a general consensus/awareness that it is &lt;strong&gt;impossible&lt;/strong&gt; (as opposed to: “not yet knowing how to”) to solve certain geometrical problems with straightedge and compass constructions?&lt;/p&gt;&#xA;&lt;p&gt;(b) Who was the first person in history to give an &amp;quot;impossibility proof&amp;quot; that some geometrical problems cannot be solved with such-and-such construction principle. I know Gauss found a proof for the constructibility of regular polygons, but I am looking for earlier mathematical contributions.&lt;/p&gt;&#xA;" OwnerUserId="13085" LastEditorUserId="55" LastEditDate="2020-09-23T22:25:17.320" LastActivityDate="2020-09-23T22:25:17.320" Title="Were there impossibility proofs for constructions in Greek geometry?" Tags="|euclidean-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12288" PostTypeId="2" ParentId="12287" CreationDate="2020-09-23T22:24:17.770" Score="5" Body="&lt;p&gt;As in most such cases, nobody was &amp;quot;first&amp;quot;. One can present already the Pythagorean proof of incommensurability of the side and the diagonal of a square as an &amp;quot;impossibility proof&amp;quot;. It amounts to saying that it is impossible to split the side into a whole number of equal segments so that the diagonal can be assembled from a whole number of segments of the same length. The proof (in Euclid) is by &lt;em&gt;reductio&lt;/em&gt;, as are some other proofs that can be similarly presented. And Gauss did not prove the impossibility part, it was &lt;a href=&quot;https://en.wikipedia.org/wiki/Constructible_polygon#Conditions_for_constructibility&quot; rel=&quot;noreferrer&quot;&gt;Wantzel in 1837&lt;/a&gt;, probably the first geometric impossibility proof recognized as &amp;quot;proper&amp;quot; by modern standards. But by then there were &amp;quot;flawed&amp;quot; 17th century arguments, Euler's proof of irrationality of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, Lambert's of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, and Abel's of insolvability of the quintic in radicals.&lt;/p&gt;&#xA;&lt;p&gt;But generally this sort of thinking was alien to the constructive &amp;quot;classical conception of proof and knowledge&amp;quot; in mathematics, as Detlefsen called it. For that matter, Greeks lacked &amp;quot;metamathematical&amp;quot; perspective or methods for proving impossibility on a grander scale, like the impossibility of solving &amp;quot;the three construction problems of antiquity&amp;quot; (duplication, trisection and quadrature) with straightedge and compass. Such tools were part of first order discourse about geometric objects, there was no second order discourse &lt;em&gt;about tools&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But there was practical awareness. The three problems have been solved by other tools (quadratrix, intersecting solids, conic sections) since Pythagoreans (Archytas, Hippias), and continued to be so solved even after the straightedge and compass priority was codified by Euclid. &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Pappus/&quot; rel=&quot;noreferrer&quot;&gt;Pappus in the &lt;em&gt;Collection&lt;/em&gt;&lt;/a&gt; even classifies problems according to tools needed and advocates solving them by &amp;quot;appropriate&amp;quot; tools. It is doubtful he'd make the classification if he thought that straightedge and compass constructions were just not yet known (that Euclid, Archimedes and Apollonius did not find them was a strong clue):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There are, we say, three types of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems. Those that can be solved with straight line and circle are properly called 'plane' problems, for the lines by which such problems are solved have their origin in a plane. Those problems that are solved by the use of one or more sections of the cone are called 'solid' problems. For it is necessary in the construction to use surfaces of solid figures, that is to say, cones. There remain the third type, the so-called 'linear' problem. For the construction in these cases curves other than those already mentioned are required, curves having a more varied and forced origin and arising from more irregular surfaces and from complex motions. Of this character are the curves discovered in the so-called 'surface loci' and numerous others even more involved...&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Second order discourse only emerged in 17th century, &lt;a href=&quot;https://hal.archives-ouvertes.fr/tel-01098493/document&quot; rel=&quot;noreferrer&quot;&gt;Crippa's dissertation Impossibility Results&lt;/a&gt; is a comprehensive study. Descartes, Gregory and Leibniz gave some geometric impossibility arguments, but they were flawed from modern perspective:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The conviction on the impossibility of squaring the circle, duplicating&#xA;the cube and trisecting the angle by ruler and compass was in fact grounded on&#xA;arguments which, although they appeared either flawed or lacking in rigour if examined&#xA;from our viewpoint, enjoyed some circulatation during the second half of XVIIth century,&#xA;and were studied and discussed. In particular, my examination will consider some&#xA;of the earliest instances of impossibility results and correlated arguments given by René&#xA;Descartes (1596-1650) in La Géométrie (1637), then an important impossibility result&#xA;about the quadrature of the circle by algebraic or analytical methods, argued by James&#xA;Gregory (1638-1675) in his Vera Circuli et Hyperbolae Quadratura (1667), and finally,&#xA;the same impossibility result discussed by G.W. Leibniz (1646-1716), in an unpublished&#xA;work, which had however a vast resonance among contemporary mathematicians: De&#xA;quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria&#xA;sine tabulis (ultimated in 1676).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-09-23T22:24:17.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12289" PostTypeId="1" CreationDate="2020-09-24T00:51:20.420" Score="0" ViewCount="50" Body="&lt;p&gt;As I understand schemes for sending different symbols over different wires were implemented but the simplicity of Morse's dot or dash made it the easiest to read and/or implement. It seems similar to early computer design have instead of binary digits decimal representation which ultimately were impractical because the precise machining necessary was too expensive.&lt;/p&gt;&#xA;&lt;p&gt;(Perhaps the Jacquard loom was another example but I do not think there were competing designs -- I am interested in binary winning out over competing approaches.)&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2020-09-24T00:51:20.420" Title="Was Morse code the first practical application of binary encoding of information?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12290" PostTypeId="2" ParentId="12284" CreationDate="2020-09-24T01:04:14.313" Score="2" Body="&lt;p&gt;See also&lt;/p&gt;&#xA;&lt;p&gt;The forgotten mathematical legacy of Peano&#xA;Volume 537 / 2019&#xA;Szymon Dolecki, Gabriele H. Greco Dissertationes Mathematicae 537 (2019), 1-77 MDOI: 10.4064/dm769-4-2018 Published online: 28 February 2019 (free download)&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2020-09-24T01:04:14.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12291" PostTypeId="1" CreationDate="2020-09-24T04:01:41.530" Score="1" ViewCount="186" Body="&lt;p&gt;In what paper did Dyson derive the Dyson equation, as shown on &lt;a href=&quot;https://en.wikipedia.org/wiki/Self-energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="8219" LastEditorUserId="4251" LastEditDate="2021-09-22T08:42:31.570" LastActivityDate="2021-09-22T08:42:31.570" Title="Original derivation of the Dyson equation" Tags="|physics|quantum-mechanics|theoretical-physics|quantum-field-theory|dyson|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12293" PostTypeId="1" AcceptedAnswerId="12338" CreationDate="2020-09-24T07:39:54.317" Score="1" ViewCount="376" Body="&lt;p&gt;The logo on the &lt;a href=&quot;https://www.bipm.org/en/about-us/&quot; rel=&quot;nofollow noreferrer&quot;&gt;BIPM's (International Bureau of Weights and Measures) site&lt;/a&gt; is this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NzOqA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NzOqA.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, when I visit the English (and, most importantly, French) Wikipedia articles on BIPM, and also when I google &amp;quot;BIPM logo&amp;quot;, I mostly get the following image:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/jmQ65.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jmQ65.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;So, which one is the current BIPM's logo?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I suspect that the second image is BIPM's first logo, and the modern logo is now on their official site, though I'm not 100% sure.&lt;/p&gt;&#xA;" OwnerUserId="13076" LastEditorUserId="26" LastEditDate="2021-02-20T15:08:20.033" LastActivityDate="2021-02-20T15:08:20.033" Title="What is the official logo of the BIPM?" Tags="|measurement|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12295" PostTypeId="2" ParentId="5645" CreationDate="2020-09-25T10:04:46.933" Score="2" Body="&lt;p&gt;Emanuel Lasker is better known as a world chess champion, but he was a mathematician too. David Hilbert was one of his doctoral advisors and he published at least four mathematical articles. And he also wrote a drama, &lt;em&gt;Vom Menschen die Geschichte&lt;/em&gt; (together with his brother Berthold), as well as several philosophical books.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-09-25T10:04:46.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12296" PostTypeId="1" AcceptedAnswerId="12297" CreationDate="2020-09-26T02:31:18.167" Score="2" ViewCount="213" Body="&lt;p&gt;I would like to ask how Isaac Newton specifically applied mathematics to reducing forgeries and improving coinage while he was Master of the Royal Mint.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="18820" LastActivityDate="2020-09-26T13:05:20.830" Title="Mr. Newton at the Royal Mint" Tags="|mathematicians|newton|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12297" PostTypeId="2" ParentId="12296" CreationDate="2020-09-26T13:05:20.830" Score="5" Body="&lt;p&gt;This is not mathematics, but Newton invented the ridges on coins to prevent theft. I am not sure if he invented it while he was director of the Royal Mint, but I suspect so.&#xA;In good ol times, coins were made of valuable metals (in fact, they were worth their value). Thieves would scrape the coins around the edge, thus making them smaller (but hardly noticeable the difference), to obtain a bit of that valuable metal. By making ridges on the sides, it was clear if a coin had been scraped.&lt;br /&gt;&#xA;Always a genius.&lt;/p&gt;&#xA;" OwnerUserId="11040" LastActivityDate="2020-09-26T13:05:20.830" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12300" PostTypeId="1" CreationDate="2020-09-27T13:11:37.867" Score="1" ViewCount="110" Body="&lt;p&gt;There must have been a moment when most people had heard of telescopes, but they were only affordable by rich people or ship's captains. When did they first become affordable by the general public?&lt;/p&gt;&#xA;&lt;p&gt;After a few hours of research, I have not found an answer to this online.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="13099" LastEditorUserId="26" LastEditDate="2021-02-20T15:06:25.820" LastActivityDate="2021-02-20T15:06:25.820" Title="When did telescopes become affordable?" Tags="|technology|telescope|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12301" PostTypeId="1" CreationDate="2020-09-27T18:19:56.913" Score="1" ViewCount="125" Body="&lt;p&gt;&amp;quot;1+2+3+4=10&amp;quot; is an arithmetical triviality that is popular as an example of Pythagoreanism.  There seems to be no mention of it in ancient Greek texts prior to Speusippus (e.g. ca. 350BCE; see Zhmud's &lt;em&gt;Arithmology&lt;/em&gt;), so  it seems rightly considered to be &lt;em&gt;pseudo&lt;/em&gt;-pythagorean. There is however a possibility that it could have been noted and/or commented  at earlier times in some other accessible culture, e.g. Egypt or India, etc (probably not Babylon where they used hexadecimal numbers). Refs to papers or sources mentioning such early occurences would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2020-09-27T18:19:56.913" Title="What would be some earlier comments on &quot;1+2+3+4=10&quot;?" Tags="|reference-request|ancient-india|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12302" PostTypeId="1" CreationDate="2020-09-27T20:59:07.573" Score="0" ViewCount="198" Body="&lt;p&gt;I read a dramatic statistic something like 3 out of 5 living humans owe their existence to food that could be grown due to Haber's work. But was there a time when anyone said, we need to be able to dramatically increase the world population? Was it understood in fact that greater food production capacity would have this effect?&lt;/p&gt;&#xA;&lt;p&gt;I could see business leaders perhaps wanting more workers and more people to sell products to although I would guess it would be the food producers themselves who wanted cheaper fertilizer (which prior to Haber-Bosch was obtained from sources like bat guano).&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-02-10T16:22:25.587" Title="Fritz Haber and nitrogen fixing: Was there a movement to increase the population?" Tags="|chemistry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12305" PostTypeId="1" AcceptedAnswerId="12311" CreationDate="2020-09-29T14:28:02.320" Score="5" ViewCount="113" Body="&lt;p&gt;In John Von Neumann's &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_1/&quot; rel=&quot;noreferrer&quot;&gt;The Mathematician&lt;/a&gt; one can read that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[&lt;span class=&quot;math-container&quot;&gt;$\dots$&lt;/span&gt;] even after the reign of rigour was essentially re-established with Cauchy, a very peculiar relapse into semi-physical methods took place with Riemann.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I skimmed through the rest of this short text and found no details. So, I wonder what are these semi-physical methods of Riemann alluded to.&lt;br /&gt;&#xA;Some details and references would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="13109" LastActivityDate="2020-09-29T20:59:02.353" Title="What were Riemann's &quot;semi-physical&quot; methods?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12306" PostTypeId="1" CreationDate="2020-09-29T14:38:36.210" Score="1" ViewCount="214" Body="&lt;p&gt;I recently saw a paper where there are presented some rules on how to learn mathematics (and do research) which were firstly articulated by &lt;a href=&quot;https://link.springer.com/article/10.1007/s40329-014-0038-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt;.&#xA;Are there any similar rules that were expressed before early 1800s by other mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-11-25T21:20:05.343" LastActivityDate="2020-11-28T10:37:39.087" Title="Are there any mathematicians who expressed non-obvious sets of rules on how to do research?" Tags="|mathematics|mathematicians|biographical-details|discoveries|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12307" PostTypeId="1" CreationDate="2020-09-29T15:37:00.113" Score="3" ViewCount="663" Body="&lt;p&gt;In the proceedings of the &lt;em&gt;XIth International Congress of Mathematical Physics&lt;/em&gt; Edward Witten wrote (p. 704)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[&lt;span class=&quot;math-container&quot;&gt;$\dots$&lt;/span&gt;] when a mathematical result is really relevant to a physics problem it often happens that, turning things around, the result can be deduced from the behavior of the physics problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Witten's own work perfectly exemplifies this method. This method can be traced back to Archimedes, famous for his mechanical style.&lt;br /&gt;&#xA;I would be interested in learning about any piece of work in this vein that happened between Archimedes and Witten. In particular, is there any example from the work of natural philosophers in the XVIIIth or XIXth century? Is there any striking example from the first half of the XXth century with a mathematical result whose proof had been first sketched by physicists, using a heuristic proof originating in general relativity or quantum mechanics, and later made rigorous by mathematicians, possibly using very different methods?&lt;br /&gt;&#xA;Please provide references either to original pieces of work or to historical accounts.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: as it should be clear from Witten's quote the question is not so much about physical insights leading to new fields of mathematics, but rather about physical insights leading to sketches of proofs of conjectures in well established mathematical fields or leading to new results altogether in a well established mathematical theory.&lt;/p&gt;&#xA;" OwnerUserId="13109" LastEditorUserId="13109" LastEditDate="2020-09-30T08:04:54.083" LastActivityDate="2020-11-13T05:28:31.307" Title="What are some historical examples in physics of heuristic proofs of mathematical results?" Tags="|mathematics|physics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12308" PostTypeId="1" AcceptedAnswerId="12312" CreationDate="2020-09-29T18:01:43.433" Score="3" ViewCount="1944" Body="&lt;p&gt;Who was the first mathematician/physicist to derive the heat equation &lt;span class=&quot;math-container&quot;&gt;$u_t=\Delta u$&lt;/span&gt; and when? Was it already known to explain most diffusion phenomena? How much time passed between the first appearance of the heat equation and Fourier's solution? What were the previous attempts?&lt;/p&gt;&#xA;&lt;p&gt;I'm not looking for a very detailed monograph of this part of history but rather a short account that may be condensed in an answer.&lt;/p&gt;&#xA;" OwnerUserId="10086" LastEditorUserId="10106" LastEditDate="2020-09-29T18:11:34.297" LastActivityDate="2020-09-29T23:29:19.143" Title="History behind the heat equation" Tags="|physics|differential-equations|fourier|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12309" PostTypeId="2" ParentId="12308" CreationDate="2020-09-29T18:39:55.840" Score="1" Body="&lt;p&gt;The first mathematician who did this accurately and with all detail was Joseph Fourier. Though he had predecessors. (One can almost never answer the question &amp;quot;who did this first&amp;quot;, so on my opinion, it is not even worth asking. &amp;quot;Nothing is new under the Moon&amp;quot;, as Ecclesiastes wrote). Fourier wrote a very influential book&#xA;titled&#xA;&amp;quot;Analytic theory of heat&amp;quot; (there is an English translation), where he accurately derived the heat equation and proposed a method of its solution. For this he invented what is called &amp;quot;Fourier Analysis&amp;quot; today, though in this invention he also had predecessors.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-09-29T18:39:55.840" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12310" PostTypeId="2" ParentId="12307" CreationDate="2020-09-29T18:44:00.660" Score="6" Body="&lt;p&gt;The first and most famous example was the book of Archimedes which is usually called The Method (more complete title is The Method of mechanical theorems), where he uses&#xA;mechanics (statics) to compute volumes of various bodies. Unfortunately, this book was lost and found again only in the beginning of 20th century.&#xA;Meanwhile what he did there was rediscovered in 17 century by people&#xA;like Stevin, Fermat, Kepler and  Cavalieri.&lt;/p&gt;&#xA;&lt;p&gt;Archimedes (a pure mathematician of the highest rank) writes very clearly that the method is not rigorous. It took&#xA;2 centuries of development of Calculus/Analysis in 17-19 centuries to make it rigorous.&lt;/p&gt;&#xA;&lt;p&gt;Another example is Maxwell's Treatise on electricity and magnetism, where he anticipates a lot of 20th century mathematics, including such things as differential forms, cohomology theories and extremal length. Unfortunately, mathematicians of 19th century did not appreciate Maxwell. (There is a nice paper about this, by Freeman Dyson, titled &amp;quot;Lost opportunities&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;For example, Maxwell's discussion of electric resistance (Ch. VIII, art. 306-309) of conductors contains a method of estimating this resistance.&#xA;Maxwell mentions Rayleigh as the author of the idea.&#xA;This method was rediscovered by Ahlfors and Bers in 1950s&#xA;under the name &lt;a href=&quot;https://en.wikipedia.org/wiki/Extremal_length&quot; rel=&quot;nofollow noreferrer&quot;&gt;Extremal length&lt;/a&gt;&#xA;which became one of the main working tools in the theory of conformal mappings. They do not refer on Rayleigh or Maxwell: the earliest predecessor they refer to is Courant, who wrote in 20th century.&lt;/p&gt;&#xA;&lt;p&gt;Examples from 20th century are abundant: whole new areas of mathematics&#xA;were developed to lay a rigorous foundation for the insights&#xA;of Maxwell, Boltzmann and Gibbs in statistical mechanics. 20th century mathematicians are more inclined to talk to physicists and read their writings.&lt;/p&gt;&#xA;&lt;p&gt;For example see Wikipedia articles &lt;a href=&quot;https://en.wikipedia.org/wiki/Ergodic_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ergodic hypothesis&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Ergodic_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ergodic theory&lt;/a&gt;.&#xA;All statistical mechanics was developed by Maxwell, Boltzman and Gibbs on&#xA;the &amp;quot;pysical level of rigor&amp;quot;, and mathematicians are still busy with converting their &amp;quot;laws&amp;quot; into theorems. There is still a large gap between the laws of statistical mechanics and rigorously proved results. On the other hand, considerations from statistical mechanics led to discovery of new mathematical theorems not related to physics directly. See for example,&#xA;&lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.bams/1183554642&quot; rel=&quot;nofollow noreferrer&quot;&gt;D. Ruelle, Is our mathematics natural?&lt;/a&gt; or &lt;a href=&quot;https://www.math.purdue.edu/~eremenko/dvi/hamilt.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;.&#xA;Such examples are really numerous.&lt;/p&gt;&#xA;&lt;p&gt;Finally let me mention Fourier and his remarkable book Analytic theory of heat, whose main points were that a) every periodic function can be expanded&#xA;into Fourier series,&#xA;and b) every reasonable function on the real line can be represented by Fourier integral. He gives all kinds of ingenious arguments in favor of these statements (including experimental evidence with heated metal rings!). It took more than a century to mathematicians to rigorously state and justify his main assertions. Some of them were proved only recently,&#xA;MR1769725&#xA;Ki, Haseo and  Kim, Young-One,&#xA;On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture, Duke Math. J. 104 (2000), no. 1, 45–73.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-09-30T23:14:21.610" LastActivityDate="2020-09-30T23:14:21.610" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12311" PostTypeId="2" ParentId="12305" CreationDate="2020-09-29T20:59:02.353" Score="7" Body="&lt;p&gt;There is a detailed account of Riemann's physical thinking in &lt;a href=&quot;https://openlibrary.org/works/OL3783589W/Development_of_mathematics_in_the_19th_century&quot; rel=&quot;noreferrer&quot;&gt;Klein's Development of mathematics in the 19th century&lt;/a&gt;, see also &lt;a href=&quot;https://archive.schillerinstitute.com/fid_02-06/2004/044_riemann_dirichlet.html&quot; rel=&quot;noreferrer&quot;&gt;Bernhard Riemann’s ‘Dirichlet’s Principle’&lt;/a&gt;. One of the most prominent examples is the use of potential theory in his dissertation (1854) and in Theory of Abelian Functions (1857) to prove theorems about algebraic functions on Riemann surfaces, particularly what Riemann called the &amp;quot;Dirichlet principle&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The idea is that if one imagines a surface as conducting electricity, with batteries at the poles, then the resulting flow has to satisfy the least action principle and be harmonic, which produces the (real part) of a holomorphic function with given singularities. The original application was to the Dirichlet boundary value problem for the Laplace equation, but Riemann went far beyond that. Dirichlet principle was later criticized by Weierstrass as unjustified, he gave a counterexample to the existence of a least action minimizer for a different functional, see &lt;a href=&quot;https://mathoverflow.net/q/42176/51484&quot;&gt;What was Weierstrass's counterexample to the Dirichlet Principle? on MO&lt;/a&gt;. But for the original context the principle was eventually vindicated by Schwarz and Hilbert.&lt;/p&gt;&#xA;&lt;p&gt;Here is Klein's description:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Riemann then significantly generalized this principle: instead of a bounded portion of the plane, he took a portion of a Riemann surface or even&#xA;a whole closed Riemann surface, and instead of the boundary values of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; he&#xA;took arbitrary relations that may exist between the boundary values of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;. It is impossible here to pursue this principle in all its generality.&#xA;Rather, I can only cite the results it first yielded for the theory of algebraic&#xA;functions and their integrals. According to Riemann's own statement, he found&#xA;them right at the start, in the winter of 1851/52, in connection with his&#xA;dissertation.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The fundamental thought experiment is: to think of the Riemann surface&#xA;as uniformly conducting electricity. This can be realized very simply: One&#xA;covers the surface with tinfoil, and, for an isolated interpenetration of the&#xA;sheets, one has combs meshing into one another along the branch cuts in such a&#xA;way that the electrical resistance [Leitungswiderstand] in the teeth of the&#xA;combs is the same as in the homogeneous tinfoil cover. At two points &lt;span class=&quot;math-container&quot;&gt;$A_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A_2$&lt;/span&gt; one places the poles of a galvanic battery of appropriate strength. A flow&#xA;develops, whose potential &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; on the surface is everywhere single-valued and continuous and satisfies the equation &lt;span class=&quot;math-container&quot;&gt;$\Delta u = 0$&lt;/span&gt;, except that at &lt;span class=&quot;math-container&quot;&gt;$A_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$A_2$&lt;/span&gt;&#xA;it becomes discontinuous like &lt;span class=&quot;math-container&quot;&gt;$\log r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-\log r_2$&lt;/span&gt;, respectively.&#xA;With this we have won another existence theorem, which can be formulated&#xA;thus: On every closed Riemann surface there exists a continuous potential&#xA;function &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; which at two prescribed places becomes logarithmically infinite in a definitely prescribed way.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;With this first existence theorem, we now have a won game. One can form&#xA;&amp;quot;integrals of the second kind&amp;quot;, i.e., integrals whose only infinities single&#xA;poles of the form &lt;span class=&quot;math-container&quot;&gt;$1/( z-a)$&lt;/span&gt;; then &amp;quot;integrals of the first kind&amp;quot;, which are&#xA;finite everywhere. Moreover, one can construct &amp;quot;algebraic functions&amp;quot; on the&#xA;surface in various ways: by combining integrals of the second or first kinds so that all the moduli of peridocity become zero, or else by simply differentiating--&#xA;thus &lt;span class=&quot;math-container&quot;&gt;$d\Pi/dz$&lt;/span&gt;, and so on.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-09-29T20:59:02.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12312" PostTypeId="2" ParentId="12308" CreationDate="2020-09-29T23:29:19.143" Score="7" Body="&lt;p&gt;Heat equation was first formulated by Fourier in a manuscript presented to Institut de France in 1807, followed by his book &lt;em&gt;Theorie de la Propagation de la Chaleur dans les Solides&lt;/em&gt; the same year, see &lt;a href=&quot;https://link.springer.com/article/10.1007/BF02842327&quot; rel=&quot;noreferrer&quot;&gt;Narasimhan, Fourier’s heat conduction equation: History, influence, and connections&lt;/a&gt;. &amp;quot;Diffusion phenomena&amp;quot; were not studied until much later, when atomic theory was accepted, Fourier succeeded exactly by ignoring microscopic physics.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Essentially, Fourier moved away from discontinuous&#xA;bodies and towards continuous bodies. Instead of&#xA;starting with the basic equations of action at a&#xA;distance, Fourier took an empirical, observational&#xA;approach to idealize how matter behaved macroscopically.&#xA;In this way he also avoided discussion of the&#xA;nature of heat... In formulating heat conduction in terms of a partial&#xA;differential equation and developing the methods for&#xA;solving the equation, Fourier initiated many innovations.&#xA;He visualized the problem in terms of three&#xA;components: heat transport in space, heat storage&#xA;within a small element of the solid, and boundary&#xA;conditions. The differential equation itself pertained&#xA;only to the interior of the flow domain. The interaction&#xA;of the interior with the exterior across the&#xA;boundary was handled in terms of &amp;quot;boundary conditions,&amp;quot;&#xA;conditions assumed to be known a priori.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the book Fourier presented a solution in terms of trigonometric series. Trigonometric series were suggested for solving other equations earlier by Bernoulli, but accepting them as valid solutions was controversial because of the prevailing wisdom of treating functions as analytic expressions. A committee of luminaries, consisting of Laplace, Lagrange, Lacroix, Monge and Poisson, initially dismissed Fourier's solution as unsound. Fourier's approach led to a &amp;quot;crisis&amp;quot; and reconsideration of the foundations of calculus described by &lt;a href=&quot;https://www.macalester.edu/aratra/&quot; rel=&quot;noreferrer&quot;&gt;Bressoud in A Radical Approach to Real Analysis&lt;/a&gt;, which resulted in the more general modern concept of functions and rigorous analysis that goes with it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The crisis struck four days before Christmas 1807. The edifice of calculus was shaken&#xA;to its foundations. In retrospect, the difficulties had been building for decades. Yet while&#xA;most scientists realized that something had happened, it would take fifty years before the&#xA;full impact of the event was understood...&#xA;Here was the heart of the crisis. Infinite sums of trigonometric functions had appeared before.&#xA;Daniel Bernoulli (1700-1782) proposed such sums in 1753 as solutions to the problem&#xA;of modeling the vibrating string. They had been dismissed by the greatest mathematician&#xA;of the time, Leonhard Euler (1707-1783). Perhaps Euler scented the danger they presented&#xA;to his understanding of calculus... Well into the 1820s, Fourier&#xA;series would remain suspect because they contradicted the established wisdom about the&#xA;nature of functions.&#xA;Fourier did more than suggest that the solution to the heat equation lay in his trigonometric&#xA;series. He gave a simple and practical means of finding those coefficients, the ai, for&#xA;any function. In so doing, he produced a vast array of verifiable solutions to specific&#xA;problems.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-09-29T23:29:19.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12313" PostTypeId="2" ParentId="12307" CreationDate="2020-09-30T09:19:04.063" Score="1" Body="&lt;p&gt;&lt;a href=&quot;http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/dyson-quasicrystal.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dyson proposed&lt;/a&gt; that a good understanding of quasicrystals would help solving the Riemann hypothesis.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My suggestion is the following. Let us pretend that we do not know&#xA;that the Riemann Hypothesis is true. Let us tackle the problem from&#xA;the other end. Let us try to obtain a complete enumeration and&#xA;classification of one-dimensional quasicrystals. That is to say, we&#xA;enumerate and classify all point distributions that have a discrete&#xA;point spectrum...We shall then find the well-known quasi-crystals&#xA;associated with PV numbers, and also a whole universe of other&#xA;quasicrystals, known and unknown. Among the multitude of other&#xA;quasi-crystals we search for one corresponding to the Riemann&#xA;zeta-function and one corresponding to each of the other&#xA;zeta-functions that resemble the Riemann zeta-function. Suppose that&#xA;we find one of the quasi-crystals in our enumeration with properties&#xA;that identify it with the zeros of the Riemann zeta-function. Then we&#xA;have proved the Riemann Hypothesis and we can wait for the telephone&#xA;call announcing the award of the Fields Medal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the same vein Salvatore Torquato, who is studying &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperuniformity&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperuniformity&lt;/a&gt;, has outlined a proposal for the distribution of primes.&#xA;His work is  presented in &lt;a href=&quot;https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quanta&lt;/a&gt; with refs., most  of them in the &lt;a href=&quot;https://arxiv.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;arxiv&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2020-09-30T17:32:35.503" LastActivityDate="2020-09-30T17:32:35.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12314" PostTypeId="1" CreationDate="2020-09-30T18:15:15.140" Score="4" ViewCount="151" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Space-filling_curve&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; cites an earlier result of Cantor as an inspiration but I wonder if there are any previous results of some kind of recursive curve constructions that may have also &amp;quot;inspired&amp;quot; him to come up with that.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="55" LastEditDate="2020-10-04T18:58:43.553" LastActivityDate="2020-10-04T18:58:43.553" Title="How did Peano come up with his space filling curve?" Tags="|mathematics|biographical-details|" AnswerCount="0" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="12315" PostTypeId="1" AcceptedAnswerId="12316" CreationDate="2020-10-01T03:13:13.140" Score="3" ViewCount="694" Body="&lt;p&gt;It is my understanding that Cauchy was the first to incorporate the notion of a &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;-&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; limit in his proofs, although a definition was not formulated until Weierstrass did so.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;How far back does the concept of &lt;em&gt;limit&lt;/em&gt; go and unto what purpose(s)?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Also, is there any evidence that the ancient Babylonians were aware of the concept?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="18820" LastEditorUserId="18820" LastEditDate="2021-08-20T22:20:23.133" LastActivityDate="2021-08-28T12:07:35.393" Title="When did the Notion of &quot;Limit&quot; Arise and for What Purpose?" Tags="|mathematics|calculus|real-analysis|ancient|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12316" PostTypeId="2" ParentId="12315" CreationDate="2020-10-01T22:12:00.623" Score="7" Body="&lt;p&gt;Rigorous notion of limit for special cases arose in the work of Eudoxus and Archimedes, when determining the length of a circle, volume of the pyramid etc. (The work of Eudoxus did not survive, we know about it from Euclid&#xA;and ancient historians of mathematics).&lt;/p&gt;&#xA;&lt;p&gt;The argument to find&#xA;these limits was called the &amp;quot;method of exhaustion&amp;quot;. It is equivalent to the modern notion of limit, in the special cases they studied, but does not coincide with it exactly. And applies only to special cases, they had no general definition.&lt;/p&gt;&#xA;&lt;p&gt;The length of the circle for example is DEFINED as the limit of perimeters&#xA;of inscribed polygons, and Euclid and Archimedes were dealing with it&#xA;absolutely rigorously from the modern point of view, except in the cases where Archimedes himself clearly says that his arguments are heuristic.&lt;/p&gt;&#xA;&lt;p&gt;Same applies to many other works of Archimedes on areas and volumes. They are all defined as limits, and Archimedes was able to find them, and&#xA;his arguments are rigorous from the modern point of view.&lt;/p&gt;&#xA;&lt;p&gt;The modern definition of a limit was systematically used by Cauchy in the early 19 century.&lt;/p&gt;&#xA;&lt;p&gt;But many people in between were talking about the limits, and were able to compute them, with various degrees of details and rigor. The degree of rigor in calculus similar to that of Archimedes was achieved only in the second half of 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Like many other mathematical notions, one can say that&#xA;the general notion of limit &amp;quot;slowly evolved&amp;quot; rather than was &amp;quot;invented&amp;quot; by some person at some definite time. Cauchy, who played an important role in formalization of calculus still made &amp;quot;elementary mistakes&amp;quot; from the point of view of a modern student. (For example he claimed and &amp;quot;proved&amp;quot; that the limit of continuous functions is continuous. His proof is valid if by the &amp;quot;limit&amp;quot; one means &amp;quot;uniform limit&amp;quot;, but he did not state the definition of uniform limit carefully).&lt;/p&gt;&#xA;&lt;p&gt;EDIT. Ancient Babylonians had an algorithm which approximated square roots of integers&#xA;by rational numbers with any given accuracy. So one can say that they were able to compute these limits. However they gave no definitions and did not prove anything (at least no traces of proofs survive).&lt;/p&gt;&#xA;&lt;p&gt;All this intends to show that the limit was not invented at some definite time, but that people just studied it, and the modern notion slowly evolved from these studies; the evolution took about 2500 years or more.&lt;/p&gt;&#xA;&lt;p&gt;Your second question: &amp;quot;for what purpose?&amp;quot; has an evident answer: to define and compute the things like the length of a circle or the volume of a pyramid. (It is a remarkable theorem of Max Dehn, solving a problem of Hilbert, that the volume of a pyramid cannot be DEFINED without the notion of a limit. Unlike the area of a triangle which can be).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-28T12:07:35.393" LastActivityDate="2021-08-28T12:07:35.393" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12317" PostTypeId="1" AcceptedAnswerId="12318" CreationDate="2020-10-02T10:41:38.413" Score="11" ViewCount="476" Body="&lt;p&gt;Blum's &lt;a href=&quot;https://en.wikipedia.org/wiki/Blum%27s_speedup_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;speedup theorem&lt;/a&gt; seems to me that bears at least some superficial resemblance  to Gödel's research on the length of proofs under certain axiomatic systems.&#xA;Does Blum's speedup theorem have any conceptual predecessors?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="1880" LastEditDate="2023-10-05T09:12:46.670" LastActivityDate="2023-10-05T09:12:46.670" Title="Does Blum's speedup theorem have any conceptual predecessors?" Tags="|mathematics|biographical-details|discoveries|computer-science|computation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12318" PostTypeId="2" ParentId="12317" CreationDate="2020-10-02T17:50:05.093" Score="12" Body="&lt;p&gt;The resemblance is not superficial. There is a precise relation between computer programs and formal proofs known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Curry–Howard correspondence&lt;/a&gt; that took shape in 1960s. And Gödel's results on proof length were a direct inspiration for speed-up theorems starting with Blum's 1967 result, see &lt;a href=&quot;https://www.jstor.org/stable/2322023&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dawson, The Godel Incompleteness Theorem from a Length-of-Proof Perspective&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In one year (1936) three important papers appeared, all based on considerations of the length or complexity of proofs in formal systems. One, Rosser's improvement of the Incompleteness Theorem, has already been mentioned. The others were Gentzen's consistency proof for arithmetic ([7], translated in [23, pp. 132-213]), the first of its kind, employing a transfinite induction on the complexity of derivations; and a brief and little-noticed article by Gödel himself ([10], translated in [5, pp. 82-83]). The latter, titled simply &amp;quot;On the Length of Proofs,&amp;quot; pointed out that, by passing to systems of &amp;quot;higher type&amp;quot; (allowing sets of integers, sets of sets of integers, etc.), not only can new theorems be proved, but &amp;quot;it becomes possible to shorten extraordinarily infinitely many of the proofs already available.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Translated only in 1965, Gödel's length-of-proof paper was largely ignored until after the advent of the computer revolution, when concern for efficiency of computation led to Manuel Blum's creation of computational complexity theory, and Gödel's proof-shortening result was resurrected to become the progenitor of a whole class of speed-up theorems (see [1, pp. 253 and 261-263]).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[1] is &lt;a href=&quot;https://www.academia.edu/25318719/xiii_412_pp_Michael_A_Arbib_Theories_of_Abstract_Automata_1969_Prentice_Hall_Providence_RI&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arbib, Theories of Abstract Automata (1969)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="1880" LastEditDate="2023-10-04T23:03:03.440" LastActivityDate="2023-10-04T23:03:03.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12321" PostTypeId="1" CreationDate="2020-10-04T16:06:49.640" Score="1" ViewCount="66" Body="&lt;p&gt;What are the roots of the Intelligence question, i.e. since when do &amp;quot;humans&amp;quot; think about what &amp;quot;Intelligence&amp;quot; means and where it actually comes from?&lt;/p&gt;&#xA;&lt;p&gt;I did some digging in the greek philosophy around the topic of &amp;quot;Nous&amp;quot;, though it seems the roots go way deeper - unfortunately not finding any proper sources. Hence: Does anyone know about the &amp;quot;Origin of the Intelligence question?&amp;quot;? (&lt;em&gt;Preferably with sources&lt;/em&gt;)&lt;/p&gt;&#xA;" OwnerUserId="13121" LastActivityDate="2020-10-04T16:06:49.640" Title="Origin of the Intelligence question?" Tags="|philosophy-of-science|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12322" PostTypeId="1" AcceptedAnswerId="12324" CreationDate="2020-10-05T16:36:43.100" Score="6" ViewCount="562" Body="&lt;p&gt;The mathematician Vladimir Arnold claimed that mathematics is a part of physics.&lt;/p&gt;&#xA;&lt;p&gt;I am aware of Arnold's &lt;a href=&quot;https://www.uni-muenster.de/Physik.TP/%7Emunsteg/arnold.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Teaching Mathematics&lt;/a&gt; where he stated this view, but is there any piece of writing where Arnold, or someone else, elaborated on it?&lt;/p&gt;&#xA;" OwnerUserId="13109" LastEditorUserId="4251" LastEditDate="2022-08-16T14:42:13.203" LastActivityDate="2022-08-16T14:42:13.203" Title="Some references for Vladimir Arnold's thesis &quot;Mathematics is a part of physics&quot;?" Tags="|mathematics|mathematicians|reference-request|quote|vladimir-arnold|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12323" PostTypeId="1" AcceptedAnswerId="12325" CreationDate="2020-10-06T00:48:43.227" Score="9" ViewCount="1962" Body="&lt;p&gt;I was recently doing research on the &amp;quot;Rutherford model&amp;quot; of the atom. I found that there seem to be three different accounts of Ernest Rutherford's theory circulating online:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Electrons move move randomly in the space around the nucleus. Like &lt;a href=&quot;https://slideplayer.com/slide/3866473/13/images/1/Models+of+the+Atom+-+Dalton%E2%80%99s+model+%281803%29+Greek+model+%28400+B.C.%29.jpg&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Electrons orbit the nucleus in rings each containing many electrons. This model appears to be identical to that proposed by Japanese physicist Hantaro Nagaoka (&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4989051/&quot; rel=&quot;noreferrer&quot;&gt;This paper&lt;/a&gt; argues that Nagaoka's model has been widely mischaracterized as &amp;quot;Saturnian&amp;quot; when in fact it features a small nucleus and allows for the possibility of rings not lying in the same plane.).&lt;/li&gt;&#xA;&lt;li&gt;The number of electrons in a neutral atom is equal to the atom's atomic number. Electrons orbit the nucleus along solitary paths (or with a partner) like planets orbiting the sun. Like &lt;a href=&quot;https://i1.wp.com/www.compoundchem.com/wp-content/uploads/2016/10/The-History-of-the-Atom-%E2%80%93-Theories-and-Models.png?fit=1323%2C935&amp;amp;ssl=1&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. This model is very similar to the Bohr model, except that it lacks Bohr's requirement of fixed, discrete orbits. It is what most people think of when they hear the term and is often anachronistically represented as having featured a nucleus containing separate protons and neutrons, even though the structure of the nucleus had not yet been determined.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Which of these is the &amp;quot;Rutherford model?&amp;quot; Did Rutherford propose all three of them (or some subset thereof) at different points in his career? If not, then why is he being given credit for a model/models which he didn't come up with?&lt;/p&gt;&#xA;" OwnerUserId="13125" LastEditorUserId="55" LastEditDate="2020-10-06T09:28:49.840" LastActivityDate="2020-11-15T01:34:56.310" Title="What exactly was the Rutherford model of the atom?" Tags="|physics|debunking|atomic-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12324" PostTypeId="2" ParentId="12322" CreationDate="2020-10-06T07:16:18.167" Score="13" Body="&lt;p&gt;First, on the question in the narrow sense the answer is in the negative, I am afraid, although there are some other places where Arnold expresses his views on mathematics: &lt;a href=&quot;https://iopscience.iop.org/article/10.1070/RM1996v051n01ABEH002734/pdf&quot; rel=&quot;noreferrer&quot;&gt;An apologia for Applied Mathematics&lt;/a&gt; in his 1996 survey, a short paper &lt;a href=&quot;https://www.math.ru.nl/%7Emueger/arnold.pdf&quot; rel=&quot;noreferrer&quot;&gt;The antiscientifical revolution and mathematics&lt;/a&gt;, an &lt;a href=&quot;https://www.ocf.berkeley.edu/%7Elekheng/interviews/VladimirArnold.pdf&quot; rel=&quot;noreferrer&quot;&gt;interview with Liu&lt;/a&gt; for Mathematical Notices, a &lt;a href=&quot;http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;amp;jrnid=kvant&amp;amp;paperid=324&amp;amp;option_lang=rus&quot; rel=&quot;noreferrer&quot;&gt;short note Why do we study mathematics&lt;/a&gt; for Russian undergraduate physics/mathematics magazine Kvant, etc.&lt;/p&gt;&#xA;&lt;p&gt;From these one can surmise that he was deeply concerned with the social perception of mathematics, the trajectory of mathematical teaching, opposed &amp;quot;Bourbakization&amp;quot;, etc. But none of them adds much to the thesis from the OP linked note, except, perhaps, to suggest that it is deliberately polemical and sociologically motivated. For example, in the &lt;em&gt;Apologia&lt;/em&gt; he writes that &amp;quot;&lt;em&gt;the difference between pure and applied mathematics is not scientific but only&#xA;social&lt;/em&gt;&amp;quot;, and:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;As a result, there came about a divorce of 'pure' mathematics from all sciences, a system of mathematical education, criminal against those taught, and the image of mathematics in the common mind was of a dangerous parasitic sect on the body of science and technology, consisting of priests of a dying religion like the druids.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Taking a broader view, and aside from Arnold's social causes, the attitude is shared, in part. For example in &lt;a href=&quot;https://hsm.stackexchange.com/q/6157/55&quot;&gt;Are there well-known mathematicians who shared Arnold's view about mathematics as natural science?&lt;/a&gt; Newton and Kronecker are named as expressing something vaguely similar. As we shall see, Gauss can be added to the list. Quine opined in &lt;a href=&quot;https://www.theologie.uzh.ch/dam/jcr:ffffffff-fbd6-1538-0000-000070cf64bc/Quine51.pdf&quot; rel=&quot;noreferrer&quot;&gt;Two Dogmas&lt;/a&gt; that logic and mathematics are of a kind with natural science, more &amp;quot;entrenched&amp;quot; but still subject to revision based on the sum total of observations and experiments. &lt;a href=&quot;https://en.wikipedia.org/wiki/Proofs_and_Refutations&quot; rel=&quot;noreferrer&quot;&gt;Lakatos's Proofs and Refutations&lt;/a&gt; explicitly assimilates mathematics to the &amp;quot;hypothetico-deductive method&amp;quot; (the title is a play on Popper's Conjectures and Refutations). &lt;a href=&quot;http://andrewlias.blogspot.com/2004/08/is-mathematics-science.html&quot; rel=&quot;noreferrer&quot;&gt;Magidin in Is Mathematics a Science?&lt;/a&gt; suggests that mathematics &amp;quot;&lt;em&gt;follows the scientific method&lt;/em&gt;&amp;quot;, etc.&lt;/p&gt;&#xA;&lt;p&gt;However, taken literally, the thesis is hard to defend. Perhaps, mathematics of the early days of calculus can be seen as a science of bold speculation and cheap experiments, but typical works do not present conjectures confirmed by calculations and simulations, as one would expect were it true. As Arnold's own publications would attest, he was well aware of the singular role of rigorous proofs as mathematical confirmations, which is dissimilar to what we have in experimental sciences.&lt;/p&gt;&#xA;&lt;p&gt;One author that particularly comes to mind as defending a nuanced version of Arnold's thesis, long before him, is C.S. Peirce, see e.g. his &lt;a href=&quot;https://uberty.org/wp-content/uploads/2015/02/Peirce-Philosophy-of-Mathematics-Selected-Writings.pdf&quot; rel=&quot;noreferrer&quot;&gt;Philosophy of Mathematics, sec.10&lt;/a&gt;. He talks of mathematics (and logic) as staging &amp;quot;ideal experiments&amp;quot; on &amp;quot;diagrams&amp;quot; (formal models), from which general rules are inductively surmised, and describes the function of proofs employing them as checks against human error. In other words, they serve as reliable shorthands for ascertaining semantic consequence in formal models. The distinction with physics then is not just that the &amp;quot;ideal experiments&amp;quot; are cheap, but also that we exercise full control over the formation of their subjects:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Now it is plainly not an essential part of this&#xA;method in general that the tests were made by the observation of natural objects. For&#xA;the immense progress which modern mathematics has made is also to be explained by&#xA;the same intense interest in testing general propositions by particular cases — only the&#xA;tests were applied by means of particular demonstrations. This is observation, still, for&#xA;as the great mathematician Gauss has declared — algebra is a science of the eye,&#xA;only it is observation of artificial objects and of a highly recondite character.&lt;/em&gt; [CP 1.34]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and  physical  research.  Chemists  have  ere  now,  I  need  not  say,  described experimentation  as  the  putting  of  questions  to  Nature.  Just  so,  experiments upon  diagrams  are  questions  put  to  the  Nature  of  the  relations  concerned...&lt;/em&gt; [CP 4.530]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Not only is it true that by experimentation upon some diagram an experimental proof can be obtained of every necessary conclusion from any given Copulate of Premisses, but, what is more, no “necessary” conclusion is anymore  apodictic  than  inductive  reasoning  becomes  from  the  moment  when experimentation  can  be  multiplied  ad  libitum  at  no  more  cost  than  a  summons before the imagination...  It is true that what must be is not to be learned by simple inspection of anything. But when we talk of deductive  reasoning  being  necessary,  we  do  not  mean,  of  course,  that  it  is infallible. But precisely what we do mean is that the conclusion follows from the  form  of  the  relations  set  forth  in  the  premiss.&lt;/em&gt; [CP 4.531]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It would be a great mistake to suppose that ideal experimentation can be&#xA;performed without danger of error; but by the exercise of care and industry this&#xA;danger may be reduced indefinitely. In sensible experimentation, no care can always&#xA;avoid error... Thus, the necessary reasoning of mathematics is performed by means of&#xA;observation and experiment, and its necessary character is due simply to the&#xA;circumstance that the subject of this observation and experiment is a diagram of our&#xA;own creation, the conditions of whose being we know all about.&lt;/em&gt;&amp;quot; [CP 3.528, 560]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-07T21:29:45.407" LastActivityDate="2020-10-07T21:29:45.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12325" PostTypeId="2" ParentId="12323" CreationDate="2020-10-06T09:18:47.427" Score="12" Body="&lt;p&gt;&amp;quot;The Rutherford planetary model&amp;quot; was not exactly... anything, it was a vague umbrella term.  The inspection of &lt;a href=&quot;https://www.lawebdefisica.com/arts/structureatom.pdf&quot; rel=&quot;noreferrer&quot;&gt;Rutherford's 1911 paper&lt;/a&gt; shows that Rutherford did not propose even one of the three OP versions. What he did propose, that there is a positively charged nucleus, does not distinguish among them. He did assume that the electron cloud was spherical, but that was not set in stone. That was it, and he was not much more specific in the book &lt;a href=&quot;https://archive.org/details/radioactivesubst00ruthuoft/page/422/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Radioactive substances and their radiations (1913)&lt;/a&gt; two years later. But one can hardly expect historical accuracy from the loose pop-sci snippets such as the ones linked. For a more accurate account see &lt;a href=&quot;https://www.tandfonline.com/doi/pdf/10.1080/00033796700203356&quot; rel=&quot;noreferrer&quot;&gt;Rutherford, Nagaoka, and the nuclear atom&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Most of the 1911 paper is devoted to ruling out the then dominant &amp;quot;plum pudding&amp;quot; model of Thomson. Continuous distribution of the positive charge over the entire atom was inconsistent with the scattering experiments, and Rutherford had to shrink it to the central nucleus to explain them. He does not say much about electrons because their influence on scattering (beyond balancing the charge) is negligible. Indeed, he does not even say definitively that the nucleus is positively charged:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Consider an atom which contains a charge &lt;span class=&quot;math-container&quot;&gt;$±Ne$&lt;/span&gt; at its centre surrounded by a sphere of electrification containing a charge &lt;span class=&quot;math-container&quot;&gt;$±Ne$&lt;/span&gt; [N.B. in the original publication, the second plus/minus sign is inverted to be a minus/plus sign] supposed uniformly distributed throughout a sphere of radius &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is the fundamental unit of charge, which in this paper is taken as &lt;span class=&quot;math-container&quot;&gt;$4.65\times10^{-10}$&lt;/span&gt; E.S. unit. We shall suppose that for distances less than &lt;span class=&quot;math-container&quot;&gt;$10^{-12}$&lt;/span&gt; cm. the central charge and also the charge on the alpha particle may be supposed to be concentrated at a point. It will be shown that the main deductions from the theory are independent of whether the central charge is supposed to be positive or negative. For convenience, the sign will be assumed to be positive. The question of the stability of the atom proposed need not be considered at this stage, for this will obviously depend upon the minute structure of the atom, and on the motion of the constituent charged parts.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Near the end Rutherford did approvingly quote Nagaoka's &amp;quot;Saturnian model&amp;quot; (he knew of Perrin's 1901 planetary model as well, mentioned in the 1913 book). But here again he is non-committal as to how electrons are distributed in it, experiments could not resolve that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is of interest to note that Nagaoka has mathematically considered the properties of the Saturnian atom which he supposed to consist of a central attracting mass surrounded by rings of rotating electrons. He showed that such a system was stable if the attracting force was large. From the point of view considered in his paper, the chance of large deflexion would practically be unaltered, whether the atom is considered to be disk or a sphere.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The last remark concerns the only supposed difference between &amp;quot;the Rutherford model&amp;quot; and Nagaoka's. That Nagaoka was committed to a disk is indeed disputable, but then it is not like Rutherford was really committed to a sphere. As to why he gets the credit, it is similar to Newton being credited for the inverse square law, which many speculated about before him. Nagaoka's model was a mathematical speculation, and there was a &lt;a href=&quot;https://skullsinthestars.com/2008/05/27/the-gallery-of-failed-atomic-models-1903-1913/&quot; rel=&quot;noreferrer&quot;&gt;host of other models besides&lt;/a&gt;, including planetary ones. Rutherford convinced people that something like it was real, at least as far as having a nucleus was concerned.&lt;/p&gt;&#xA;&lt;p&gt;Bohr in his &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786441308634955?journalCode=tphm17&quot; rel=&quot;noreferrer&quot;&gt;1913 paper&lt;/a&gt; refers to that same 1911 paper for &amp;quot;the Rutherford's model&amp;quot; and confirms this attitude. As can be seen from his remarks, what mattered was only that the model was nuclear, and discussing what electrons were doing there was moot because any version of it was classically unstable. Contrary to what Rutherford thought about Nagaoka's model, and in contrast to Thomson's model:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;According to this theory, the atoms consist of a  positively charged  nucleus surrounded by a system of electrons kept together  by attractive forces from tile nucleus; the total :negative charge of the electrons is equal to the positive charge of the nucleus. Further, the nucleus is assumed to be the seat of the essential part of the mass of the atom, and to have linear dimensions exceedingly small compared with the linear dimensions of the whole atom. The number of electrons in an atom is deduced to be approximately equal to half the atomic weight. Great interest is to be attributed to this atom-model; for, as Rutherford has shown, the assumption of the existence of nuclei, as those in question, seems to be necessary in order to account for the results of the experiments on large angle scattering of the &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;  rays.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;...The principal difference between the atom-models proposed by Thomson and Rutherford consists in the circumstance that the forces acting on the electrons in the atom-model of Thomson allow of certain configurations and motions of the electrons for which the system is in a stable equilibrium; such configurations, however, apparently do not exist for the second atom-model.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-06T19:52:15.977" LastActivityDate="2020-10-06T19:52:15.977" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12327" PostTypeId="1" AcceptedAnswerId="12333" CreationDate="2020-10-08T02:08:50.587" Score="1" ViewCount="292" Body="&lt;p&gt;I saw this quote today...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Some subjects are so serious that one can only joke about them.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is attributed to Niels Bohr.&lt;/p&gt;&#xA;&lt;p&gt;What was the subject referred to?&lt;/p&gt;&#xA;" OwnerUserId="13131" LastActivityDate="2020-10-09T02:04:28.803" Title="context of a Niels Bohr quote &quot;some subjects are so serious that one can only joke about them&quot;" Tags="|bohr|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12328" PostTypeId="1" CreationDate="2020-10-08T11:42:21.730" Score="4" ViewCount="166" Body="&lt;p&gt;Rebecca Barlow is the discoverer of an interesting &lt;a href=&quot;https://en.wikipedia.org/wiki/Barlow_surface&quot; rel=&quot;nofollow noreferrer&quot;&gt;surface&lt;/a&gt; in algebraic geometry. Is anybody aware of a full list of her contributions? Has she continued working in mathematics in the 21st century?&lt;/p&gt;&#xA;" OwnerUserId="13133" LastActivityDate="2020-10-08T13:28:33.803" Title="Complete list of publications of Rebecca Barlow" Tags="|mathematics|mathematicians|biographical-details|differential-geometry|algebraic-geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12329" PostTypeId="1" AcceptedAnswerId="12331" CreationDate="2020-10-08T13:01:35.067" Score="3" ViewCount="263" Body="&lt;p&gt;Where comes the word root when talking about the points when a polynomial gets the value &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="13134" LastActivityDate="2020-10-08T13:33:21.063" Title="Where does the word root comes from?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12330" PostTypeId="2" ParentId="12328" CreationDate="2020-10-08T13:28:33.803" Score="3" Body="&lt;p&gt;Math Sci Net lists 6 publications, dating from 1984-1999.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2020-10-08T13:28:33.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12331" PostTypeId="2" ParentId="12329" CreationDate="2020-10-08T13:33:21.063" Score="3" Body="&lt;p&gt;The usage of &amp;quot;root&amp;quot; is so old that it is hard to pin-point the first usage and as to why we call them roots. At in English, the usage dates back to 14th century, and if this &amp;quot;root&amp;quot; concept was borrowed from Arabic, as suggested by Jeff Miller of the &amp;quot;Earliest Known Uses of Some of the Words of Mathematics&amp;quot;, then it will be even more difficult to rationalize why a root is called a root. Google Translate also shows that the Arabic word for root (of a polynomial) is &lt;em&gt;jazr&lt;/em&gt;, which literally means a root.&lt;/p&gt;&#xA;&lt;p&gt;According to the Earliest known uses of the word of mathematics&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;RADIX, ROOT, UNKNOWN, SQUARE ROOT. Late Latin writers used res for the&#xA;unknown. This was translated as cosa in Italian, and the early Italian&#xA;writers called algebra the Regola de la Cosa, whence the German Die&#xA;Coss and the English cossike arte (Smith vol. 2, page 392).&lt;/p&gt;&#xA;&lt;p&gt;Other Latin terms used in the Middle Ages for the uknown quantity and&#xA;its square were radix, res, and census. The term root was used by&#xA;al-Khowarizmi; the word is rendered radix in Robert of Chester’s Latin&#xA;translation of the algebra of al-Khowarizmi. Radix also is used in&#xA;translations from Arabic to Latin by John of Seville, Gerard of&#xA;Cremona, and Leonardo of Pisa. For an early English use of root, see&#xA;addition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2020-10-08T13:33:21.063" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12332" PostTypeId="1" CreationDate="2020-10-09T01:57:31.767" Score="0" ViewCount="498" Body="&lt;p&gt;I think this is the appropriate SE to ask about the following, but feel free to tell me otherwise.&lt;/p&gt;&#xA;&lt;p&gt;As I remember it, the meter is a &amp;quot;human&amp;quot; or &amp;quot;natural&amp;quot; length, because it was choosen to be exactly 1/40000th of Earth's circumference.&lt;/p&gt;&#xA;&lt;p&gt;A few minutes ago, I was thinking about the centimeter and how unnatural it is. It is quite small and even though I am French, I think the centimeter does not come naturally as a unit of measurement.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I had this idea. What if the meter had been defined based on the diameter of the planet instead of its circumference?&lt;/p&gt;&#xA;&lt;p&gt;I had a strange intuition about this.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, if the circumference is taken to be exactly 40000 meters (real circumference is slightly longer), diameter is 40000 divided by pi, i.e. 12732.39 meters.&lt;/p&gt;&#xA;&lt;p&gt;And now, let say I use this diameter to define my new meter. For example, I could say the diameter is exactly 10000 new meters.&lt;/p&gt;&#xA;&lt;p&gt;That would make the new centimeter slightly longer, i.e. 1.2732 centimeter.&lt;/p&gt;&#xA;&lt;p&gt;And I find this VERY intriguing because, if I simply multiply this by 2, I get 2.54 centimeters which is... an inch!&lt;/p&gt;&#xA;&lt;p&gt;Is this a mere coincidence or was the circumference choosen over the diameter on purpose to avoid getting back to the imperial system?&lt;/p&gt;&#xA;&lt;p&gt;Edit: Jean Picard determined Earth radius in 1669. So for the ones arguing it was obvious to use the circumference, as obvious as anything looked in retrospect, I need better arguments.&lt;/p&gt;&#xA;" OwnerUserId="9955" LastEditorUserId="9955" LastEditDate="2020-10-09T19:09:12.923" LastActivityDate="2020-10-09T19:09:12.923" Title="On the choice of Earth circumference vs diameter to define the meter" Tags="|measurement|" AnswerCount="1" CommentCount="3" ClosedDate="2020-10-16T21:51:00.973" ContentLicense="CC BY-SA 4.0" />
  <row Id="12333" PostTypeId="2" ParentId="12327" CreationDate="2020-10-09T02:04:28.803" Score="3" Body="&lt;p&gt;This quote has a single attestation, p.24 of the book &amp;quot;The genius of science : a portrait gallery&amp;quot; (2000) by Abraham Pais, a physicist who would be in a position to know, having been Bohr's assistant.  I have examined an online library copy of the book.  Pais mentions the quote in a discussion of Bohr's philosophical beliefs, specifically his codification of &amp;quot;complementarity&amp;quot; (pp.21-24), the idea that a full description of an entity may entail properties that are mutually exclusive to each other (such as the wave-particle duality of photons).  To be specific, it seems the quote relates to Bohr's characterization of an &amp;quot;expert&amp;quot; as one who hones their knowledge in a specific area until they know &amp;quot;everything about nothing,&amp;quot; and a &amp;quot;philosopher&amp;quot; as one who broadens their ken until they know &amp;quot;nothing about everything.&amp;quot;  Pais's introduction of the quote comes right after he relates a favorite story of Bohr's, concerning a student who is certain the lecture he just attended was the &amp;quot;finest&amp;quot; ever, because neither he nor the lecturer understood the topic at all.  So the quote doesn't seem to have been directed at any specific subject.  I hope that helps.&lt;/p&gt;&#xA;" OwnerUserId="13136" LastActivityDate="2020-10-09T02:04:28.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12335" PostTypeId="2" ParentId="12332" CreationDate="2020-10-09T03:15:44.293" Score="6" Body="&lt;p&gt;If one decides to choose some measurement of the Earth as a standard of length, circumference&#xA;has one serious advantage: it is easier to measure directly.&lt;/p&gt;&#xA;&lt;p&gt;The Earth is not an exact sphere. And the meter was originally defined as 1/40,000&#xA;of Paris meridian (not of the equator, not of some other meridian). And this meridian was carefully measured. How would you propose to measure a &amp;quot;diameter&amp;quot;?&#xA;Which diameter? The distance, say from Paris to the antipodal point by a straight line through the Earth? How would you measure this?&lt;/p&gt;&#xA;&lt;p&gt;Speaking of what units are &amp;quot;human&amp;quot; or convenient is a matter of opinion.&#xA;Americans think that old British are &amp;quot;human&amp;quot; and convenient, other people, including the British think otherwise.&lt;/p&gt;&#xA;&lt;p&gt;And that 4/&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is approximately 1/2 inches in centimeters is a pure coincidence, of course. The ratio of inch to centimeter has no relation to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-10-09T03:26:56.620" LastActivityDate="2020-10-09T03:26:56.620" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12336" PostTypeId="1" AcceptedAnswerId="12337" CreationDate="2020-10-10T10:05:10.547" Score="8" ViewCount="364" Body="&lt;p&gt;I wonder whether the questions on a calculus exam at university were easier or harder 100 years ago. Nowadays we have all these aids and different learning methods.&lt;/p&gt;&#xA;&lt;p&gt;I would love to see an old exam.&lt;/p&gt;&#xA;" OwnerUserId="13144" LastEditorUserId="26" LastEditDate="2021-02-20T15:06:18.657" LastActivityDate="2021-02-20T15:06:18.657" Title="What does an 100 year old calculus exam look like?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12337" PostTypeId="2" ParentId="12336" CreationDate="2020-10-10T13:44:32.793" Score="9" Body="&lt;p&gt;It depends on what you call &amp;quot;calculus&amp;quot; exactly. A nice large sample of exam problems (of slightly older age&#xA;120-130 years ago) in Cambridge can be seen in the book by Whittaker and Watson, Course of Modern Analysis. The first edition was published in 1902 and the last in 1927.&lt;/p&gt;&#xA;&lt;p&gt;In the end of each chapter it contains exam problems from previous years&#xA;with indication where and in what year each problem was offered. The book is easily available, since it is still in print (and available on the Internet) and is widely used (not for exams, I suppose).&lt;/p&gt;&#xA;&lt;p&gt;But you have to take into account that Cambridge was one of the top schools in the world at that time, and the level at an average US university could be lower.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-10-11T00:21:05.707" LastActivityDate="2020-10-11T00:21:05.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12338" PostTypeId="2" ParentId="12293" CreationDate="2020-10-10T21:37:45.613" Score="2" Body="&lt;p&gt;The BIPM website contains copies of several memoranda of understanding between BIPM and other organisations, which indicate that there are &lt;em&gt;two&lt;/em&gt; distinct pictorial symbols that represent BIPM: one called the &amp;quot;emblem&amp;quot; and one called the &amp;quot;seal&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The first of the two images presented by OP (the one with the three vertical lines and the sans serif text) has been registered as a trade mark in the US, Singapore, and Canada.  The Singaporean registration record describes it as the &amp;quot;emblem&amp;quot;.  Hence, I deduce that the second image presented by OP (the circular one with the three classical humanoid figures) is the &amp;quot;seal&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The lexico.com thesaurus lists both &amp;quot;emblem&amp;quot; and &amp;quot;seal&amp;quot; as synonyms of &amp;quot;logo&amp;quot;, so as far as the original question goes, take your pick.&lt;/p&gt;&#xA;" OwnerUserId="13150" LastActivityDate="2020-10-10T21:37:45.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12339" PostTypeId="1" CreationDate="2020-10-11T10:55:31.683" Score="9" ViewCount="352" Body="&lt;p&gt;I really liked Hermann Weyl's mathematical books and would like to get accustomed to general relativity from his perspective, but wonder if it's still relevant after almost 100 (!) years?&lt;br /&gt;&#xA;Can this book be used to actually learn something useful about GR or should it be read only as investigation of historical perspective? Won't it cause any misconceptions?&lt;/p&gt;&#xA;" OwnerUserId="13153" LastEditorUserId="12724" LastEditDate="2020-10-12T06:56:36.077" LastActivityDate="2020-11-24T12:30:04.067" Title="Is Hermann Weyl's book “Space, Time, Matter” (1923) on General Relativity still relevant?" Tags="|relativity-theory|mathematical-physics|resource-recommendation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12340" PostTypeId="2" ParentId="6157" CreationDate="2020-10-12T08:44:43.043" Score="4" Body="&lt;p&gt;Gregory Chaitin openly holds this position (though he might dispute the &amp;quot;experiment is cheap&amp;quot; part). He believes that the incompleteness results of Gödel, Turing and himself show that the mathematical knowledge has limits, and (according to him) that the limits are quite near. From that observation, he constructs an interesting argument (interesting in more meanings than one).&lt;/p&gt;&#xA;&lt;p&gt;A Newtonian physicist, Chaitin says, would, after experimenting with the electro-magnetic field, be led to phenomena that cannot be explained through Newtonian mechanisms and eventually to Maxwell's equations, that is to say to new fundamental principles of physics that cannot be reduced to Newtonian physics. Likewise, the experimental investigation of the microscopic world would led a physicist who has mastered Newton's and and Maxwell's contributions to quantum mechanics and ultimately Schrödinger's equation; another principle that cannot be reduced to former principles.&lt;/p&gt;&#xA;&lt;p&gt;Chaitin believes that mathematicians should do the same thing. When faced with a problem they cannot solve or with a new phenomenon, they should adopt new axioms, see where these new axioms lead them &lt;em&gt;experimentally&lt;/em&gt; and advance in this way, backpedaling if necessary if one axiom turns out to lead to a contradiction. He is perfectly clear that these new axioms should absolutely not be axioms in the Euclidean sense, that is to say evident truths or foundational principles. They may be, indeed should be in his mind, complex, non-obvious propositions, like Schrödinger's equation. As an explicit example of a new axiom he recommend mathematicians adopt and whose consequences he recommend they experimentally explore, he mentions the Riemann hypothesis.&lt;/p&gt;&#xA;&lt;p&gt;Chaitin doesn't deny that proofs are important. &amp;quot;It's nice if you can prove [a conjecture]&amp;quot; he writes &amp;quot;especially if the proof is short. And if you have different proofs from different viewpoints, that's very good. But sometimes you can't find a proof and you can't wait for someone else to find a proof&amp;quot;. In that case, &amp;quot;[mathematics] should be pursued more in the spirit of experimental science, and [mathematicians] should be willing to adopt new principles&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Reference: &lt;em&gt;Randomness in Arithmetic and The Decline and Fall of Reductionism in Pure Mathematics&lt;/em&gt;. Gregory Chaitin (1993).&lt;/p&gt;&#xA;" OwnerUserId="189" LastActivityDate="2020-10-12T08:44:43.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12341" PostTypeId="1" CreationDate="2020-10-12T14:02:54.063" Score="1" ViewCount="171" Body="&lt;p&gt;I could not find any descriptions of the first motors of W. H. Taylor in Wikipedia of any reference, history or circumstances surrounding its invention. Apparently the motor was never used for any applications?&lt;/p&gt;&#xA;&lt;p&gt;It is used today in I believe the Tesla electric cars but was curious what this animal looked like when Taylor put it together. I assume the motor can use both AC and DC and was versatile but was never worked on? Not sure of the tags on this one or if the question is off topic. Maybe more of an electrical engineering question? I don't know.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="12330" LastEditDate="2020-10-13T10:14:15.910" LastActivityDate="2020-10-13T10:14:15.910" Title="Are there any descriptions of the reluctance motor invented by W.H. Taylor in the 1800's?" Tags="|electromagnetism|electricity|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12342" PostTypeId="2" ParentId="6157" CreationDate="2020-10-12T15:42:32.197" Score="3" Body="&lt;p&gt;George Pólya gave a similar statement (see &lt;a href=&quot;https://www.jstor.org/stable/2690404?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The same general point of view is developed in his two-volume book &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics_and_Plausible_Reasoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics and Plausible Reasoning&lt;/a&gt; (suggested read if you're interested in this point of view on Mathematics).&lt;/p&gt;&#xA;" OwnerUserId="13159" LastActivityDate="2020-10-12T15:42:32.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12343" PostTypeId="1" AcceptedAnswerId="12344" CreationDate="2020-10-12T20:02:23.657" Score="2" ViewCount="310" Body="&lt;p&gt;Arno Allan Penzias and Robert Woodrow Wilson were awarded (one half of) the 1978 Nobel prize in Physics, &amp;quot;for their discovery of cosmic microwave background radiation&amp;quot;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.nobelprize.org/prizes/physics/1978/summary/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nobelprize.org/prizes/physics/1978/summary/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The story about their discovery is usually told with an anecdote about the pigeon droppings (allegedly described by Penzias as a &amp;quot;white dielectric&amp;quot;) in their horn antenna, which was a cause of unwanted noise. I cannot find the origin of this story, presumably told by Penzias himself. Was this part of some interview or does it appear somewhere in a published (paper) form?&lt;/p&gt;&#xA;" OwnerUserId="10760" LastActivityDate="2020-10-18T08:26:28.523" Title="Cosmic microwave background and &quot;white dielectric&quot;" Tags="|nobel-prize|anecdote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12344" PostTypeId="2" ParentId="12343" CreationDate="2020-10-12T23:27:46.017" Score="3" Body="&lt;p&gt;Wilson tells the story in his &lt;a href=&quot;https://www.nobelprize.org/uploads/2018/06/wilson-lecture-1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nobel lecture of 1978&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus we seemed to be left with the antenna as the source of our extra&#xA;noise. We calculated a contribution of &lt;span class=&quot;math-container&quot;&gt;$0.9 \text{K}$&lt;/span&gt; from its resistive loss using&#xA;standard waveguide theory. The most lossy part of the antenna was its&#xA;small diameter throat, which was made of electroformed copper. We had&#xA;measured similar waveguides in the lab and corrected the loss calculations&#xA;for the imperfect surface conditions we had found in those waveguides.&#xA;The remainder of the antenna was made of riveted aluminum sheets, and&#xA;although we did not expect any trouble there, we had no way to evaluate&#xA;the loss in the riveted joints. A pair of pigeons was roosting up in the small&#xA;part of the horn where it enters the warm cab. They had covered the inside&#xA;with a white material familiar to all city dwellers. We evicted the pigeons&#xA;and cleaned up their mess, but obtained only a small reduction in antenna&#xA;temperature.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2020-10-12T23:27:46.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12345" PostTypeId="1" CreationDate="2020-10-13T14:17:54.517" Score="2" ViewCount="201" Body="&lt;p&gt;Recently, I came across a book about Grace Hopper where it is stated that Howard Aiken wanted her to become familiar with papers by Ada Lovelace and Charles Babbage as he considered some of Babbage's ideas interesting and worth implementing.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I have always suspected that the &lt;em&gt;von Neumann scheme&lt;/em&gt; is very similar to a design of Babbage's analytical engine - there are input and output devices, a controller, memory, and ALU (a &lt;em&gt;mill&lt;/em&gt; in Babbage's words).&lt;/p&gt;&#xA;&lt;p&gt;So, my question is if any other of modern computing pioneers - like von Neumann, Mauchly, Eckert, Bush, Atanasoff, Zuse to name few - also claimed that they were inspired by Babbage's and Lovelace's work. Any link to literature/articles/papers will be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="12316" LastEditorUserId="12724" LastEditDate="2020-10-19T04:42:17.540" LastActivityDate="2020-11-13T04:56:35.997" Title="Was a work of Ada Lovelace and Charles Babbage inspiration for pioneers of modern computing?" Tags="|computer-science|computers|computer-scientists|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12346" PostTypeId="1" CreationDate="2020-10-14T23:04:15.417" Score="1" ViewCount="120" Body="&lt;p&gt;Here is a fragment from Anders Hald's &lt;a href=&quot;https://books.google.com.br/books?id=GvFRPGCEoZAC&amp;amp;pg=PA82&amp;amp;lpg=PA82&amp;amp;dq=%22Statistics%20was%20used%20in%20this%20sense%20in%20Italy%20in%20the%2016th%20century%22&amp;amp;source=bl&amp;amp;ots=vAXQj-zXDX&amp;amp;sig=ACfU3U3TbOL7Brz54WJHo260vYfzW83kAg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiSq_XUkLXsAhVOIbkGHfGRBe0Q6AEwAHoECAEQAg#v=onepage&amp;amp;q=%22Statistics%20was%20used%20in%20this%20sense%20in%20Italy%20in%20the%2016th%20century%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Probability and Statistics and Their Applications before 1750&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The original meaning of statistics is thus a collection of facts of interest to a statesman. Statistics was used in this sense in Italy in the 16th century, and from there it spread to France, Holland, and Germany, &lt;em&gt;&lt;strong&gt;where it was taught at the universities in the 17th and 18th centuries&lt;/strong&gt;&lt;/em&gt;. It embraced the political constitutions of states and mainly verbal descriptions of the important characteristics of states, such as population, economy, and geography. The use of the word “statistics” in this sense died out in the beginning of the 19th century. [emphasis added]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would like to have details on the bold part but I couldn't find it. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What were said universities?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Were they located only in Germany?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For what students were the classes intended to?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2611" LastEditorUserId="4251" LastEditDate="2021-09-16T11:56:23.777" LastActivityDate="2021-09-16T11:56:23.777" Title="Where was statistics taught in the 17th and 18th centuries?" Tags="|statistics|teaching|18th-century|17th-century|" AnswerCount="0" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12347" PostTypeId="1" CreationDate="2020-10-15T18:43:27.760" Score="1" ViewCount="99" Body="&lt;p&gt;I have a vague memory of a (possibly-apocryphal) quote by a physicist (I remember it as being Giorgio Parisi, which could be wrong), saying something to the effect of&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;any function which can be evaluated by a computer in under 1 second should be considered as `analytically available'&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is: can anybody corroborate the existence of such a quote, and if so, is there a reference for it?&lt;/p&gt;&#xA;" OwnerUserId="13169" LastEditorUserId="12316" LastEditDate="2020-10-16T10:47:55.017" LastActivityDate="2020-10-16T10:47:55.017" Title="Quotation reference: &quot;functions which can be evaluated under 1 sec are as good as analytically available&quot;" Tags="|physics|reference-request|quote|computation|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12348" PostTypeId="1" AcceptedAnswerId="12352" CreationDate="2020-10-16T22:06:08.140" Score="0" ViewCount="139" Body="&lt;p&gt;I'm reading Dickson's History on the Theory of Numbers, and find this on page 34:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The third &lt;span class=&quot;math-container&quot;&gt;$P_3$&lt;/span&gt; [that is, triperfect number], discovered by André Jumeau, Prior of Sainte-Croix&amp;quot;, is &lt;span class=&quot;math-container&quot;&gt;${P_3}^{(3)} = 523776 = 2^9 3 \cdot 11 \cdot 31$&lt;/span&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I can find no biographical details of this &amp;quot;André Jumeau&amp;quot;. There seems to be nothing out there.&lt;/p&gt;&#xA;&lt;p&gt;I'd be grateful for any information about who he was.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2020-10-17T12:13:03.813" Title="Who was the 17th century mathematician &quot;André Jumeau, Prior of Sainte-Croix&quot;?" Tags="|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12350" PostTypeId="2" ParentId="11496" CreationDate="2020-10-16T22:30:47.020" Score="4" Body="&lt;p&gt;There's an important point that seems to have been missed here: it was initially proposed that the metre would be the length of a pendulum whose period would be equal to one second, when let swing at the surface of Earth.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, it was realised only too late that in fact because acceleration due to gravity varies slightly depending on where you stand on the surface of Earth that of course a pendulum that has a swing period of 1 second in one place will have a different swing period in another.&lt;/p&gt;&#xA;&lt;p&gt;However, it was noticed that this length was close enough to the 1/10,000,000 the distance from the North Pole to the Equator through Paris, that the latter was what was ultimately chosen.&lt;/p&gt;&#xA;&lt;p&gt;It is only so much more recently that the exact length is now established by defining it in terms of the speed of light, about which I don't need to comment because someone has already said it.&lt;/p&gt;&#xA;&lt;p&gt;But if you've ever wondered at the coincidence that acceleration due to gravity at Earth's surface measured in metres per second squared is interestingly close to pi squared --- that's why.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="12839" LastEditDate="2023-04-21T08:24:28.393" LastActivityDate="2023-04-21T08:24:28.393" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12351" PostTypeId="1" AcceptedAnswerId="12375" CreationDate="2020-10-17T00:47:05.763" Score="3" ViewCount="566" Body="&lt;p&gt;Whilst looking at the Wikipedia pages for some well known mathematicians I was surprised at how many of them were advised by other recognisable mathematicians. In some cases I could back track 5 or more mathemaicians I recognised just through Advisors.&lt;/p&gt;&#xA;&lt;p&gt;Is there a website or database etc that has collated this in some sort of &amp;quot;family tree&amp;quot; style format? I think it would be interesting to see visually how interconnected the influential mathematicians of the past were.&lt;/p&gt;&#xA;" OwnerUserId="13175" LastEditorUserId="10106" LastEditDate="2020-10-17T03:51:55.267" LastActivityDate="2020-10-25T12:54:18.167" Title="&quot;Family tree&quot; of mathematicians and their PhD advisors and students" Tags="|mathematics|mathematicians|mathematics-social-history|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12352" PostTypeId="2" ParentId="12348" CreationDate="2020-10-17T09:03:42.497" Score="7" Body="&lt;p&gt;Adrien Baillet &lt;em&gt;La vie de M. Descartes&lt;/em&gt; vol.1 Ch.11, p. 146 (year 1626) in google books&#xA;&lt;a href=&quot;https://i.stack.imgur.com/UgYRd.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UgYRd.jpg&quot; alt=&quot; Adrien Baillet La vie de M. Descartes vol.1&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;M. De Sainte Croix was another distinguished arithmetician, but even&#xA;more - an intimate  friend of M. Descartes. I believe that he is the same one&#xA;that we find called by other people André Jumeau, who was prior of the&#xA;Sainte Croix (Holy Cross), and who had been tutor to M. Le Duc De&#xA;Verneuïl. M. Descartes testified that he esteemed very particularly&#xA;the profound knowledge which M. De Sainte Croix had of arithmetic and&#xA;algebra: and he took a singular pleasure in answering his questions,&#xA;because he found almost as much satisfaction in them as M. De Sainte&#xA;Croix testified to this for his answers. He died before M Descartes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is some ambiguity as he is most often named by contemporaries &amp;quot;Monsieur de Sainte Croix&amp;quot;. His name appears frequently enough in connection with Descartes and his circle (Mersenne, Fermat, etc.). 1588-1651 are the dates of his life as given by notes to editions of the letters exchanged. He has not published anything, Mersenne notes, but passing remarks are easily found; search has not discovered some  encyclopaedical entry about him.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2020-10-17T12:13:03.813" LastActivityDate="2020-10-17T12:13:03.813" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12353" PostTypeId="1" AcceptedAnswerId="12354" CreationDate="2020-10-17T09:22:55.670" Score="0" ViewCount="151" Body="&lt;p&gt;I will be glad if someone who has seen Peirce's paper could summarily describe here  Peirce's axioms and describe their relation to Peano's.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="26" LastEditDate="2021-02-20T15:06:11.737" LastActivityDate="2021-02-20T15:06:11.737" Title="What are the Peirce's axioms of arithmetic and how do they relate to the Peano axioms?" Tags="|mathematics|mathematicians|mathematical-logic|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12354" PostTypeId="2" ParentId="12353" CreationDate="2020-10-17T17:10:27.170" Score="2" Body="&lt;p&gt;C.S.Peirce, &lt;a href=&quot;https://www.jstor.org/stable/pdf/2369151.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the logic of number (1881)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Peirce's system is equivalent to Peano axiomatization.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A discrete system is one in which every quantity greater than another is next greater than some quantity (that is, greater than without being greater than something greater than).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An infinite system is one in which any quantity greater than x can be reaclhed from x by successive steps to the next greater (or less) quantity than the one already arrived at.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;we may say that an infinite class is one in which if it is true that every quantity next greater than a quantity of a given class itself belongs to that class, then it is true that every quantity greater than a quantity of that class belongs to that class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The minimum number is called one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The language is cumbersome but the basic &amp;quot;ingredients&amp;quot; are there: an initial number; the successor function; induction.&lt;/p&gt;&#xA;&lt;p&gt;Then the standard recursive definitions of &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt; follow:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By &lt;span class=&quot;math-container&quot;&gt;$x + y$&lt;/span&gt; is meant, in case &lt;span class=&quot;math-container&quot;&gt;$x = 1$&lt;/span&gt;, the number next greater than &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; [i.e. &lt;span class=&quot;math-container&quot;&gt;$s(y)$&lt;/span&gt;, the &lt;em&gt;successor&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;]; and in other cases, the number next greater than &lt;span class=&quot;math-container&quot;&gt;$x' + y$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$x'$&lt;/span&gt; is the number next smaller than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; [i.e. &lt;span class=&quot;math-container&quot;&gt;$s(x)+y=s(x+y)$&lt;/span&gt;].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Induction is explicitly used in the proof of theorems, starting from the associative principle of addition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The proof in each case will consist in showing, first, that the proposition is true of the number one, and second, that if true of the number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; it is true of the number &lt;span class=&quot;math-container&quot;&gt;$1 + n$&lt;/span&gt;, next larger than &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2020-10-17T17:21:25.237" LastActivityDate="2020-10-17T17:21:25.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12355" PostTypeId="1" AcceptedAnswerId="12357" CreationDate="2020-10-18T08:04:25.330" Score="15" ViewCount="2943" Body="&lt;p&gt;I remember a video, perhaps a &lt;a href=&quot;https://www.youtube.com/user/numberphile/videos&quot; rel=&quot;noreferrer&quot;&gt;Numberphile&lt;/a&gt; episode where a mathematician was described who would simply move in to the home of a collaborator with which they were engaged to so that they could work more closely together, essentially sleeping on their sofa and going to work with them and keeping a temporary office nearby.&lt;/p&gt;&#xA;&lt;p&gt;They would travel from one city to another basically following the math, in order to maximize focus on the problem.&lt;/p&gt;&#xA;&lt;p&gt;Is it possible to identify this mathematician and the general field(s) in which they worked?&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;bonus points&amp;quot; for help finding the video (Numberphile?) in which they were mentioned and these stories described.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2020-10-18T17:35:13.610" Title="Which mathematician traveled to and moved in with each collaborator?" Tags="|mathematics|mathematicians|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12356" PostTypeId="2" ParentId="12343" CreationDate="2020-10-18T08:18:16.350" Score="2" Body="&lt;p&gt;With &lt;a href=&quot;https://space.meta.stackexchange.com/a/1636/12102&quot;&gt;this help&lt;/a&gt; I found &lt;a href=&quot;https://chat.stackexchange.com/transcript/message/33153481#33153481&quot;&gt;this old chat discussion&lt;/a&gt; from which I relocated the YouTube video &lt;a href=&quot;https://youtu.be/Sq_KshBxqDM?t=265&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arno Penzias &amp;amp; Robert Wilson from BBC movie &amp;quot;Hawking&amp;quot; (2004)&lt;/a&gt; in which they recount the story, including both the dielectric euphemism and more literal (guttural) term.&lt;/p&gt;&#xA;&lt;p&gt;Note that these are actors, this is historical fiction, so it's not a primary answer to your question, but let's assume the BBC at least tried to reflect some aspects of reality.&lt;/p&gt;&#xA;&lt;p&gt;The reenactment is quite humorous in the telling and then quite compelling (closed captions available)&lt;/p&gt;&#xA;&lt;p&gt;I still think that &lt;em&gt;The Pigeons Were Innocent&lt;/em&gt; would be a great title for a film, but they'd probably use that for the working title and call it &lt;em&gt;The His Was Still There&lt;/em&gt; upon release.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2020-10-18T08:26:28.523" LastActivityDate="2020-10-18T08:26:28.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12357" PostTypeId="2" ParentId="12355" CreationDate="2020-10-18T09:02:41.763" Score="27" Body="&lt;p&gt;The man is Paul Erdős. As pointed out by the OP in the comments, his &amp;quot;nomadic&amp;quot; lifestyle is briefly mentioned in the &lt;a href=&quot;https://www.youtube.com/watch?v=-Djj6pfR9KU&amp;amp;feature=youtu.be&quot; rel=&quot;noreferrer&quot;&gt;Brown Numbers - Numberphile video, 3.05-3.30&lt;/a&gt;. It is also concisely described in Currey's book Daily Rituals read in this &lt;a href=&quot;https://www.youtube.com/watch?v=E1O2pJiHhrY&quot; rel=&quot;noreferrer&quot;&gt;YouTube video&lt;/a&gt; that refers to Hoffman's popular biography &lt;a href=&quot;https://www.maa.org/press/maa-reviews/the-man-who-loved-only-numbers-the-story-of-paul-erdos-and-the-search-for-mathematical-truth&quot; rel=&quot;noreferrer&quot;&gt;The Man Who Loved Only Numbers&lt;/a&gt;, which is featured in &lt;a href=&quot;https://www.youtube.com/watch?v=9634A0iBw7w&quot; rel=&quot;noreferrer&quot;&gt;Hoffman's own video&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Despite Hoffman's title, Erdős's interests were not confined to number theory but covered also combinatorics and graph theory (where he pioneered probabilistic methods), and analysis (approximation of functions), and he made sporadic contributions to other areas, such as general topology (an example of a totally disconnected space that is not zero-dimensional). AMS Notices &lt;a href=&quot;https://www.ams.org/notices/199801/vertesi.pdf&quot; rel=&quot;noreferrer&quot;&gt;surveyed Erdős's mathematics&lt;/a&gt; in some detail in 1998, 2 years after his death.&lt;/p&gt;&#xA;&lt;p&gt;Here is Currey's excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He was also, as Paul Hoffman documents in&#xA;his book The Man Who Loved Only Numbers, a true eccentric — a&#xA;“mathematical monk” who lived out of a pair of suitcases, dressed in&#xA;tattered suits, and gave away almost all the money he earned,&#xA;keeping just enough to sustain his meager lifestyle... Erdős liked to work in short, intense collaborations with other&#xA;mathematicians, and he crisscrossed the globe seeking fresh talent,&#xA;often camping out in colleagues’ homes while they worked on a&#xA;problem together. One such colleague remembered an Erdős visit&#xA;from the 1970s:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;…he only needed three hours of sleep. He’d get up early&#xA;and write letters, mathematical letters. He’d sleep downstairs.&#xA;The first time he stayed, the clock was set wrong. It said 7:00,&#xA;but it was really 4:30 A.M. He thought we should be up&#xA;working, so he turned on the TV full blast. Later, when he&#xA;knew me better, he’d come up at some early hour and tap on&#xA;the bedroom door. “Ralph, do you exist?” The pace was&#xA;grueling. He’d want to work from 8:00 A.M. until 1:30 A.M.&#xA;Sure we’d break for short meals but we’d write on napkins&#xA;and talk math the whole time. He’d stay a week or two and&#xA;you’d collapse at the end.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The colleague is Ralph Faudree, and the quote is taken from Hoffman's book, p.256.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-18T17:35:13.610" LastActivityDate="2020-10-18T17:35:13.610" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12358" PostTypeId="1" CreationDate="2020-10-18T19:24:59.690" Score="7" ViewCount="387" Body="&lt;p&gt;This is more a soft question, and I am not sure where I should place it so I ask for excuses if this is not the right place, but I could not find any other that seemed more suitable.&lt;/p&gt;&#xA;&lt;p&gt;I would like to know which Universities in Europe are as of 2020 having or offering a PhD in history of mathematics. Of course, I do not expect a full list but at least a list of some Math Departments having a certain &amp;quot;tradition&amp;quot; on the subject.&lt;/p&gt;&#xA;&lt;p&gt;Also, I do not know what skills (at any level) are desirable to study History of Mathematics (speaking/understanding several different languages is something useful? what about knowing latin or greek?).&lt;/p&gt;&#xA;&lt;p&gt;Thanks advance.&lt;/p&gt;&#xA;" OwnerUserId="13186" LastActivityDate="2020-10-20T10:01:27.157" Title="Where to pursue a PhD in history of mathematics in Europe?" Tags="|reference-request|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12360" PostTypeId="2" ParentId="12358" CreationDate="2020-10-20T10:01:27.157" Score="8" Body="&lt;p&gt;According to the Mathematics Genealogy Project (&lt;a href=&quot;https://genealogy.math.ndsu.nodak.edu/search.php&quot; rel=&quot;noreferrer&quot;&gt;https://genealogy.math.ndsu.nodak.edu/search.php&lt;/a&gt;) database, the following universities have awarded three or more mathematics PhDs for dissertations with a history subject classification since 2000:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Germany&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Bergische Universität Wuppertal&lt;/p&gt;&#xA;&lt;p&gt;Friedrich-Schiller-Universität Jena&lt;/p&gt;&#xA;&lt;p&gt;Ludwig-Maximilians-Universität München&lt;/p&gt;&#xA;&lt;p&gt;Martin-Luther-Universität Halle-Wittenberg&lt;/p&gt;&#xA;&lt;p&gt;Philipps-Universität Marburg&lt;/p&gt;&#xA;&lt;p&gt;Technische Universität Berlin&lt;/p&gt;&#xA;&lt;p&gt;Universität Hamburg&lt;/p&gt;&#xA;&lt;p&gt;Universität Stuttgart&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;France&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Université Paris Diderot - Paris 7&lt;/p&gt;&#xA;&lt;p&gt;Université Pierre-et-Marie-Curie - Paris VI&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Netherlands&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Universiteit Utrecht&lt;/p&gt;&#xA;&lt;p&gt;Universiteit van Amsterdam&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;UK&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Open University&lt;/p&gt;&#xA;&lt;p&gt;University of St. Andrews&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;US&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Brown University&lt;/p&gt;&#xA;&lt;p&gt;Harvard University&lt;/p&gt;&#xA;&lt;p&gt;The Johns Hopkins University&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Brazil&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Universidade Federal do Espírito Santo&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2020-10-20T10:01:27.157" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12362" PostTypeId="1" AcceptedAnswerId="12364" CreationDate="2020-10-21T00:53:55.183" Score="3" ViewCount="1149" Body="&lt;p&gt;I was reading about &lt;em&gt;&lt;strong&gt;The Drag Equation&lt;/strong&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ F_D = \frac{1}{2} \rho v^2 C_D A $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ F_D $&lt;/span&gt; is the drag force&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \rho $&lt;/span&gt; is the mass density of the fluid&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ v $&lt;/span&gt; is the flow velocity relative to the object&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ A $&lt;/span&gt; is the reference area&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ C_D  $&lt;/span&gt; is the drag coefficient&lt;/p&gt;&#xA;&lt;p&gt;It is the equation responsible for explaining the terminal velocity of a falling object within a fluid&lt;/p&gt;&#xA;&lt;p&gt;I remember seeing it for the 1st time in a book called &lt;em&gt;&lt;strong&gt;Fundamentals of Physics&lt;/strong&gt;&lt;/em&gt; by &lt;em&gt;&lt;strong&gt;David Halliday&lt;/strong&gt;&lt;/em&gt; and &lt;em&gt;&lt;strong&gt;Robert Resnick&lt;/strong&gt;&lt;/em&gt;. I even remember solving some problems using it&lt;/p&gt;&#xA;&lt;p&gt;I have just read that this equation is attributed to &lt;em&gt;&lt;strong&gt;Lord Rayleigh&lt;/strong&gt;&lt;/em&gt;, but I have been unable to find where did he publish it? Does anyone know where does this equation appeard for the 1st time?&lt;/p&gt;&#xA;" OwnerUserId="12968" LastEditorUserId="26" LastEditDate="2021-02-20T15:06:04.423" LastActivityDate="2021-02-20T15:06:04.423" Title="Was Rayleigh the first to derive the drag equation?" Tags="|reference-request|debunking|fluid-dynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12363" PostTypeId="1" CreationDate="2020-10-21T07:09:52.997" Score="5" ViewCount="144" Body="&lt;p&gt;As a student in applied maths, I can remember being told that in the 1940s there were early attempts of parallel computing not using any machine but only human calculation power.&lt;/p&gt;&#xA;&lt;p&gt;I believe the story ran that students were sitting in a classroom and exchanging piece of papers between neighbours. I also believe I was told it was led by Kolmogorov. I have no idea whether this has any true basis and couldn't find a lead on the web.&lt;/p&gt;&#xA;" OwnerUserId="13210" LastEditorUserId="4251" LastEditDate="2021-11-05T17:51:01.683" LastActivityDate="2021-11-05T17:51:01.683" Title="Early parallel computing with human exchanging messages: is this story true?" Tags="|computer-science|kolmogorov|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12364" PostTypeId="2" ParentId="12362" CreationDate="2020-10-21T07:39:29.763" Score="5" Body="&lt;p&gt;Rayleigh &amp;quot;derived&amp;quot; the drag equation in &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786447608639132?journalCode=tphm16&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Resistance of Fluids&lt;/a&gt;, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser.5, v.2 (1876) no. 13, 430-441. But it is hard to tell why Wikipedia decided to attribute it to him in particular, when it was an intermediate entry in a centuries long controversy over the nature of drag that focused on a side issue surrounding oblique flows, and was based on a &lt;a href=&quot;https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox#Inviscid_separated_flow:_Kirchhoff_and_Rayleigh&quot; rel=&quot;nofollow noreferrer&quot;&gt;theory of separated flow&lt;/a&gt; that is now discarded. This seems like a prime example of May's &amp;quot;&lt;em&gt;priority chasing&lt;/em&gt;&amp;quot; and &amp;quot;&lt;em&gt;the same unnatural tidiness [that] often afflicts the potted histories of textbooks, which offer students a version of the past cleansed of its natural chaos&lt;/em&gt;&amp;quot;. Buchanan highlights it in &lt;a href=&quot;https://guava.physics.uiuc.edu/news/articles/Buchanan_low_Re.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The messy truth about drag&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Consider fluid dynamics, for example, and its ‘simplest’ problem — that of finding the drag on a sphere moving at fixed speed through a viscous fluid. Many texts give the impression that George Stokes worked out a good solution in 1851, at least for slow or ‘creeping’ flow (with Reynold’s number, &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, close to zero), finding that&#xA;the drag coefficient goes as &lt;span class=&quot;math-container&quot;&gt;$C_D\sim 1/R$&lt;/span&gt;.&#xA;Yet the truth about Stokes’ solution&#xA;turns out to be a little messier. Physicists John Veysey II and Nigel Goldenfeld tell the surprising story  of how it took more than 100 years before&#xA;experiments could even measure&#xA;the drag with appreciable accuracy,&#xA;and 150 years of confusion before&#xA;theorists learned how to calculate the&#xA;drag, for a small Reynold’s number,&#xA;in a systematic and accurate way.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;When applied to a cylinder, rather&#xA;than a sphere, his mathematical&#xA;technique gives singularities. Later&#xA;theorists — including Lord Rayleigh&#xA;and Horace Lamb — pointed out why Stokes’ approximations&#xA;broke down and recognized the&#xA;importance of the boundary layer.&#xA;Yet incredibly, only in the 1950s did&#xA;physicists finally produce the first&#xA;legitimate approximate solution to&#xA;lowest order in &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That the drag is proportional to fluid density was suggested already in the 17th century by Galileo, and that it is proportional to speed squared by Marriotte (the exponent was known to vary from under &lt;span class=&quot;math-container&quot;&gt;$1.83$&lt;/span&gt; to over &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; by the end of 19th century, see &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.80893/page/n91/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Forsyth, Newton's Problem&lt;/a&gt;). One finds a drag formula of this general form in book II of Newton's &lt;em&gt;Principia&lt;/em&gt; (1687), where it is derived from his unrealistic model of collisions with uniformly arranged particles. It might work, at best, in a very rare medium, but is applied to ships in water nonetheless (&lt;a href=&quot;https://en.wikipedia.org/wiki/Newton%27s_minimal_resistance_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's &amp;quot;solid of least resistance&amp;quot;&lt;/a&gt; known from calculus of variations).&lt;/p&gt;&#xA;&lt;p&gt;All of this happened even before the theory of continuous fluid flow was developed, and led first to the &lt;a href=&quot;https://en.wikipedia.org/wiki/D%27Alembert%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;d'Alembert's paradox (1752)&lt;/a&gt;, that a body in a potential flow of a perfect fluid encounters zero drag (because pressures on the face are canceled by equal and opposite ones on the rear), and later to Helmholtz's idea about discontinuity along a separation surface, and Stokes's &amp;quot;solution&amp;quot;. A good post-Stokes account that discusses Rayleigh's 1876 contribution, and Kelvin's criticism of it based on dynamic instability of the flow near the separation surface, is &lt;a href=&quot;https://www.annualreviews.org/doi/10.1146/annurev.fl.01.010169.000245&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fluid Mechanics in the First Half of this Century by Goldstein&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Kelvin seems to have been more and more unconvinced. In 1894 he published&#xA;four notes on the question of resistance in Nature, which are reproduced&#xA;in Volume 4 of his Mathematical and Physical Papers, with a note by&#xA;the editor, Sir Joseph Larmor, that &amp;quot;These communications formed the subject of a prolonged playful controversy between Lord Kelvin and his intimate&#xA;friend Sir George Stokes, in a series of letters which have been preserved.&amp;quot;&#xA;Kelvin showed that the results of the surfaces-of-discontinuity theory for a&#xA;flat plate were not in agreement with the experiments of Dines, published in&#xA;the Proceedings of the Royal Society in 1890.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Rayleigh, for his part, after discussing d'Alembert's paradox and bringing up Helmholtz's suggestion that the resistance is due to &amp;quot;&lt;em&gt;slipping between contiguous layers&lt;/em&gt;&amp;quot;, attributes the rest to &amp;quot;&lt;em&gt;the old theory of resistances&lt;/em&gt;&amp;quot; (of separated flow, presumably) that &amp;quot;&lt;em&gt;was not very wide of&lt;/em&gt; [sic!] &lt;em&gt;the mark&lt;/em&gt;&amp;quot; (it is considered unsatisfactory by modern lights, except for cavity flows). He then recounts it as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Behind the lamina the fluid is at rest&#xA;under a pressure equal to that which prevails at a distance,&#xA;the region of rest being bounded by a surface of separation or&#xA;discontinuity which joins the lamina tangentially, and is determined&#xA;mathematically by the condition of constant pressure.&#xA;On the anterior surface of the lamina there is an augmentation&#xA;of pressure corresponding to the loss of velocity...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; be the velocity of the stream, the increment of pressure&#xA;due to the loss of velocity is &lt;span class=&quot;math-container&quot;&gt;$\frac12\rho u^2-\frac12\rho v^2$&lt;/span&gt;, and can never exceed&#xA;&lt;span class=&quot;math-container&quot;&gt;$\frac12\rho u^2$&lt;/span&gt;, which value corresponds to a place of rest where the&#xA;whole of the energy, originally kinetic, has become potential.&#xA;The old theory of resistances went on the assumption&#xA;that the velocity of the stream was destroyed over the&#xA;whole of the anterior face of the lamina, and therefore led&#xA;to the conclusion that the resistance amounted to &lt;span class=&quot;math-container&quot;&gt;$\frac12\rho u^2$&lt;/span&gt; for&#xA;each unit of area exposed.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;It is evident at once that this is&#xA;an overestimate, since it is only near the middle of the anterior&#xA;face that the fluid is approximately at rest; towards the&#xA;edge of the lamina the fluid moves outwards with no inconsiderable velocity, and at the edge itself retains the full velocity&#xA;of the original stream. Nevertheless the amount of error involved&#xA;in the theory referred to is not great, as appears from&#xA;the result of Kirchhoff's calculation of the case of two dimensions,&#xA;from which it follows that the resistance per unit of area&#xA;is &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}{\pi+4}\rho u^2$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\frac12\rho u^2$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-28T13:16:55.727" LastActivityDate="2020-10-28T13:16:55.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12365" PostTypeId="1" CreationDate="2020-10-21T14:42:22.547" Score="3" ViewCount="58" Body="&lt;p&gt;Who are the person or people who really discovered field emission in the first place? According to this slide &lt;a href=&quot;https://www.researchgate.net/publication/261273784_Tutorial_Lecture_on_Field_Electron_Emission_Theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/publication/261273784_Tutorial_Lecture_on_Field_Electron_Emission_Theory&lt;/a&gt; the field emission gas discharge was first reported by Johann Heinrich Winkler around 1744-45 and Joseph Priestley in his 1767 textbook. Another source said it was Robert Williams Wood who the first to observe field emission in 1897. The usual context of field emission is electron are emitted from a solid surface, such as metal, to vacuum. For example, the electrons that are generated attracted to the positive anode in vacuum. But didn't Robert Boyle already discovered that electric attraction can happen in vacuum? So, somehow we can say people already have the understanding that things such as field emission are possible as electric attraction can happen in vacuum?&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, whoever actually discovered field emission it could be anyone who spent their time experimenting with metals with an understanding of electrostatics somewhere around 1700 to 1800 or even earlier. Because many people with engineering background said the inventor of this and that is not really the first person to create or invent that something, as people always been copying each others' idea and then attempted to improve it by trial and error.&lt;/p&gt;&#xA;" OwnerUserId="13212" LastActivityDate="2020-10-21T14:42:22.547" Title="Was Johann Heinrich Winkler really the first person who discovered field emission?" Tags="|electricity|engineering|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12366" PostTypeId="2" ParentId="5402" CreationDate="2020-10-22T06:52:32.617" Score="1" Body="&lt;p&gt;I know I'm a few years late to this party, but I stumbled across your post in search of a related question.  After reading your question and the responses below, I figured I could fill in a few gaps left unanswered.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How did these diagrams help Gaileo reason about uniform motion?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It may help you to see the context of where your passages are pulled from.  The first passage you quote is Uniform Motion Theorem VI, Proposition VI.  The second passage you quote is Uniform Motion Theorem III, Proposition III.&lt;/p&gt;&#xA;&lt;p&gt;What Galileo is doing in Theorems I-VI is illustrating the consequences of his proposed definition of uniform motion.  You need to consider that Galileo's &amp;quot;Two Sciences&amp;quot; is written as a sequential series of proofs that builds up in logical order starting from a set of assumed true and simple axioms. Simply put, in the section of Uniform Motion, the order of his argument is as follows:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The definition of Uniform Motion.&lt;/li&gt;&#xA;&lt;li&gt;An emphasis (i.e. &amp;quot;Caution&amp;quot;) on how the definition provided is an improvement upon older definitions of uniform motion.&lt;/li&gt;&#xA;&lt;li&gt;The required Axioms I-IV that allow the definition to hold true.&lt;/li&gt;&#xA;&lt;li&gt;Logical deductions (i.e., Theorems I-VI) resulting from the definition and its required axioms.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So to answer your question about the theorem's diagrams, Galileo is using them to provide Eucleadian (geometrical) arguments in comparing magnitude lengths corresponding to the time, distance, and speed qualities of the motion of two particles.  He starts off investigating how the difference (or more accurately, ratio) in one quality of motion results in a difference of another while holding all others equal, and then builds up to show how the ratio of a quality results when all the qualities are different.  In other words, Theorems I-VI (1-6 for formating reasons) demonstrate how a quality of motion is proportions related:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;time and distance for equal speeds (using definition and axioms);&lt;/li&gt;&#xA;&lt;li&gt;distance and speed for equal times (using definition and axioms);&lt;/li&gt;&#xA;&lt;li&gt;time and speed for equal distances (using Theorems 1 and 2);&lt;/li&gt;&#xA;&lt;li&gt;distances for differing speeds and times (using Theorems 1 and 2);&lt;/li&gt;&#xA;&lt;li&gt;times for differing speeds and distances (using Theorems 1 and 3); and&lt;/li&gt;&#xA;&lt;li&gt;speeds for different distances and times (using Theorems 2 and 3).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So you can see he covered all the scenarios for differing variables.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some of these propositions follow readily from his definition of uniform motion:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That's literally what Galileo is arguing in his section of Uniform Motion: that these relationships are the result of the definition of uniform motion.  Notice how the logic of Theorems 1-6 above can each be traced back to the fundamental definition of uniform motion and its associated axioms.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is a bit of an idealization, no? Even a car starts and stops and&#xA;exhibits small amounts of acceleration driving up a hill or down a&#xA;road. Perhaps on a very smooth highway and for short periods of time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Yes, it's idealization, but the purpose is to apply a mathematical model that accurately describes reality with an acceptable degree of error.  Are you able to draw a perfect circle or a parallel set of lines?  Of course not, but we can still deduce further truths from these idealization from our mind's eye and then apply them to reality.  Air resistance, friction, and road bumps are only inhibitions to the true undisturbed motion, but these can be added later.  We want to know how motion works within a metaphorical and literal vacuum.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One could argue hardly anything moves in the way he is decribing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The philosophy and the mathematics of motion have been lengthly discussed long before Galileo.  The &amp;quot;Mean Speed Theorem&amp;quot; (aka The Merton Rule) was mathematically demonstrated by 14th-century philosopher Nicole Oresme, who predates Galileo by about 300 years, and Galileo originally set out to prove him wrong, only to realize that Oresme was correct (much to his frustration, as discussed in his &amp;quot;Two Sciences&amp;quot;).  If you read the rest of his book, you will see that Galileo was able to build upon the definition of uniform motion along with the definition of uniform acceleration to further deduce a hypothesis that objects fall with a distance proportional to a square of their times.&lt;/p&gt;&#xA;&lt;p&gt;What makes Galileo unique over his predecessors is that he took it a step further and experimentally measured it to show that data matched his hypothesis, which is something Oresme never bothered to do, and that's what makes Galileo the &amp;quot;father of modern science.&amp;quot;  This is why the scientific method has us develop a logically reasoned hypothesis first, then experimentally test a measurable deduction from our hypothesis to see if the data matches our theory.  If the data doesn't match our reasoning, then there's a flaw somewhere in our initial assumptions that must be routed out.  This is also why Flat Earthers fail at performing legitimate science: they develop &amp;quot;theory&amp;quot; AFTER any flat/round earth experiment is performed.  No deduction is made nor is it further tested, and any data that doesn't conform to their model is tossed out.  For example, a scientist would say, if we assume the earth is spherical with poles and equator, and if we assume Newton's laws are true, then we would expect that an oscilating pendulum will precess at the poles (with a period of 24-hrs, no less or more) but none at all at the equator -- &amp;quot;Foucault's Pendulum&amp;quot; then confirms this.  Conversely, a Flat Earther will see the precession of Foucault's Pendulum first and then argue that it's because of &amp;quot;invisible vortices&amp;quot; with no further deduction and experimentation.   To turn this back around to your comment, Galileo effectively demonstrated that the definitions of uniform motion and acceleration are accurate based on his supported by data, so really, it does &amp;quot;move in the way he is describing.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Though in your defense, you COULD argue it doesn't reflect reality to some degree.  As we've learned from Einstein, we don't live in Euclidean space, so our axioms in Euclidean geometry are called into question. For instance, under special relativity, two different observers moving at different speeds won't agree on the speed of a third object (e.g., speed of light is always constant for all observers).  This is because time and space measurements will differ for each observer.  Does that make everything Galileo and Newton found to be garbage?  No, because it's more practical to use these Newtonian/Galilean assumptions when the errors caused by relativity are small.  The take away here is that it's important to understand the underlying assumptions for a particular theory in order to understand its limitations.&lt;/p&gt;&#xA;&lt;p&gt;As the last point to wrap things up, I saw some of your other comments that particularly stuck out to me as being the real question behind your post:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Unfortunately his proofs are in arcane language. So I am asking, how did he prove it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm going to make two guesses in you mean by &amp;quot;it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;First, if &amp;quot;it&amp;quot; you mean the definition of uniform motion, then look to his axioms for support.  As an axiom, they're generally so fundamental, they're assumed to be undeniable -- they're the initial assumptions.  They don't necessarily need to be true, but they are assumed to be true for argument's sake.  Here are my interpreation of his axioms for uniform motion:&lt;/p&gt;&#xA;&lt;p&gt;For a particle moving with velocity v through space s and time t such that time intervals Δt1 and Δt2 correspond to displacements Δs1 and Δs2, wherein such intervals may take on any size, and wherein such intervals can be chosen at any point during the motion, then&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;if v is constant and Δt1 &amp;gt; Δt2, then Δs1 &amp;gt; Δs2;&lt;/li&gt;&#xA;&lt;li&gt;if v is constant and Δs1 &amp;gt; Δs2, then Δt1 &amp;gt; Δt2;&lt;/li&gt;&#xA;&lt;li&gt;If v1 &amp;gt; v2 and Δt1 = Δt2, then Δs1 &amp;gt; Δs2;&lt;/li&gt;&#xA;&lt;li&gt;If Δs1 &amp;gt; Δs2 and Δt1 = Δt2, then v1 &amp;gt; v2.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;From there, considering all the inequalities above, it's safe to deduce that a special condition exists where simultaneously Δt1= Δt2 and Δs1= Δs2, which is defined to be &amp;quot;Uniform Motion.&amp;quot; With axioms as simple as that, it's hard to argue against it without tossing out all of Euclidean geometry.&lt;/p&gt;&#xA;&lt;p&gt;If &amp;quot;it&amp;quot; you mean the theorems, then walk through the theorems in sequential order and follow the proofs he provides within.  Theorem 1 was hard for me because I didn't understand the Theory of Ratios.  For that, I had to look up the mathematical definition of a ratio.  If I recall correctly, Galileo borrows a proof from Euclid's Elements Book 5, definition 5.&lt;/p&gt;&#xA;&lt;p&gt;For each Theorem, he first states the theorem, followed by a geometric statement of the theorem, and lastly followed by the proof.  For example, to match the diagram you originally provided (i.e., Theorem III):&lt;/p&gt;&#xA;&lt;p&gt;General Statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In the case of unequal speeds, the time-intervals required to&#xA;traverse a given space are to each other inversely as the speeds.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a just a broad conclusory statment of which he will elaborate.&lt;/p&gt;&#xA;&lt;p&gt;Geometric statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let the larger of the two unequal speeds be indicated by A; the smaller, by B; and let the motion corresponding to both traverse the given space CD.  Then I say the time required to traverse the distance CD at speed A is to the time required to traverse the same distance at speed B, as the speed B is to the speed A.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, the first sentence sets the stage, then the second sentence details the theorem in context of the figure.&lt;/p&gt;&#xA;&lt;p&gt;Proof:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For let CD be to CE as A is to B; then, from the preceding (he's refering to theorems 1 and 2), it follows that the time required to complete the distance CD at speed A is the same as the time necessary to complete CE at speed B; but the time needed to traverse the distance CE at speed B is to the time required to traverse the distance CD at the same speed as CE is to CD, that is, as speed B is to speed A.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Decyphering this can be a challenge, but if you understand how ratios are written in prose (i.e., &amp;quot;let CD be to CE as A is to B&amp;quot; means CD/CE = A/B), and if you follow it step by step, you'll see his argument unfold.  Also, you can see in his proof above that he references prior proofs.  Sometimes he won't even say why a relationship is invoked simply because it just &amp;quot;is&amp;quot; because he showed it earlier in the section.  This is why it's important to understand the context of each theorem in relation to prior thoerems, definitions, and axioms.  Jumping straight into a random theorem in a book like this is much like jumping to the middle of the novel: you won't understand what's happening in the plot or why.  I learned the hard way when I tried doing that with Newton's Principia and I was absolutely baffled as to what the hell he was doing until I understood the context of the theorem.&lt;/p&gt;&#xA;&lt;p&gt;Anyways, I know it's been 4 years since you posted this, but I hope you ended up finding your answer since then, and I hope that maybe this'll be useful for anyone else who stumbles here.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2020-10-22T06:52:32.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12367" PostTypeId="1" CreationDate="2020-10-22T19:59:51.293" Score="3" ViewCount="173" Body="&lt;p&gt;Turing Test has often been referred to both in academia and outside it when talking about AI. Yet there were I suppose many more thinkers except for Turing, possibly Descartes among them, that dealt with the question &amp;quot;Can machines think.&amp;quot; If so, who were the major ones? And why did Turing's Test became the Influential one among his predecessors?&lt;/p&gt;&#xA;&lt;p&gt;Additionally, is there any historical link between Turing's choice of test and the psychological theory of behaviorism?&lt;/p&gt;&#xA;" OwnerUserId="10142" LastEditorUserId="4251" LastEditDate="2021-10-22T13:28:28.477" LastActivityDate="2021-10-22T13:28:28.477" Title="Who were the major thinkers before Turing to ask whether machines can think? How come Turing's Test lasted as the influential one?" Tags="|computer-science|artificial-intelligence|computer-scientists|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12368" PostTypeId="1" AcceptedAnswerId="12371" CreationDate="2020-10-23T20:36:27.693" Score="3" ViewCount="158" Body="&lt;p&gt;Nathan Jacobson was one of the leading algebraists in the last century. Several mathematical objects are named after him such as the Jacobson radical. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Nathan_Jacobson&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nathan_Jacobson&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;I wonder what is the right pronunciation of his name.&#xA;Most often his name is pronunced as an american name but it sounds like a  German name.&#xA;I am asking this because two people at (different) conferences told me to pronounce his name in the right way. One said the american way is right and the other said that it is a German name. I do not know what is right, but I heard that his students used his american name.&lt;/p&gt;&#xA;" OwnerUserId="13221" LastActivityDate="2020-10-24T17:14:54.753" Title="How to pronounce Jacobson" Tags="|mathematicians|" AnswerCount="1" CommentCount="3" ClosedDate="2020-10-28T18:09:41.593" ContentLicense="CC BY-SA 4.0" />
  <row Id="12369" PostTypeId="1" AcceptedAnswerId="12381" CreationDate="2020-10-24T14:36:18.127" Score="5" ViewCount="210" Body="&lt;p&gt;I have been reading about the Lebesgue differentiation theorem from Terence Tao's book and came across a bunch of things. In his book, Tao uses the Vitali Covering lemma (finite), Hardy-Littlewood maximal inequalities, and rising sun lemma to prove the Lebesgue differentiation theorem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/7752/were-there-proofs-of-the-lebesgue-differentiation-theorem-without-using-maximal&quot;&gt;This&lt;/a&gt; answer gives a chronology of various proofs of the differentiation theorem. What I am interested in is the context under which the covering lemma, say appears.&lt;/p&gt;&#xA;&lt;p&gt;Were Vitali, Riesz, Hardy and Littlewood developing their theorems and inequalities in order to prove the differentiation theorem, or do they fit in some larger picture?&lt;/p&gt;&#xA;&lt;p&gt;I have seen the maximal function being used in the context of harmonic analysis, although I don't understand how. So, was Hardy working on harmonic analysis when he defined his maximal function?&lt;/p&gt;&#xA;&lt;p&gt;In short, what were these mathematicians working on when they discovered the theorems above? Also, how do other covering lemmas (the infinite version or the Besicovitch covering theorem) fit into this picture?&lt;/p&gt;&#xA;" OwnerUserId="13222" LastEditorUserId="55" LastEditDate="2020-10-27T06:45:09.887" LastActivityDate="2020-10-27T18:50:30.500" Title="Was Lebesgue differentiation theorem the motivation for Vitali's, Riesz's and Hardy-Littlewood's results used to prove it?" Tags="|mathematics|mathematicians|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12370" PostTypeId="1" AcceptedAnswerId="12374" CreationDate="2020-10-24T17:01:35.790" Score="2" ViewCount="208" Body="&lt;p&gt;In which article did Glashow introduce (1961?) a unified description of the electromagnetic and weak interactions, i.e., the electroweak interaction that earned him the Nobel prize in physics?&lt;/p&gt;&#xA;" OwnerUserId="13109" LastEditorUserId="10106" LastEditDate="2020-10-26T02:44:36.503" LastActivityDate="2020-10-26T07:24:33.563" Title="In which article did the physicist Sheldon Glashow introduce his electroweak theory?" Tags="|physics|reference-request|physicists|particle-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12371" PostTypeId="2" ParentId="12368" CreationDate="2020-10-24T17:14:54.753" Score="4" Body="&lt;p&gt;When I was a sort of postdoc at Yale in the mid-to-late 1970s, Nate Jacobson was still very much alive and doing stuff. His name was always pronounced &amp;quot;in the American style&amp;quot;, &lt;em&gt;not&lt;/em&gt; with &amp;quot;J&amp;quot; as &amp;quot;Y&amp;quot; as in German. (... though I imagine the situation was different in his parents or earlier generations, for obvious reasons).&lt;/p&gt;&#xA;&lt;p&gt;By the way, the style of pronouncing Jacobi's name when I was in grad school ages ago, which I still use, is &amp;quot;J&amp;quot; as &amp;quot;Y&amp;quot;, in German style, not American style. This surely was partly influenced by the fact that in those days the first language of many of the faculty was not English, etc.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2020-10-24T17:14:54.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12373" PostTypeId="1" CreationDate="2020-10-25T05:53:43.840" Score="2" ViewCount="64" Body="&lt;p&gt;In graph theory, it is a well-known (I guess folklore) result that the following three conditions are equivalent for a maximal planar graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; (definitions are given at the end).&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Every vertex in &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; has even degree (in other words, &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; in&#xA;Eulerian)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is 3-colorable.&lt;/li&gt;&#xA;&lt;li&gt;The dual graph of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is bipartite (ie, 2-colorable)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It is reported that Heawood proved equivalence of (2) and (3) in &amp;quot;On the four-color map problem&amp;quot; (Quart. J. Pure Math, 1898). It seems to me that this paper is not available (at least not online). Ore's book &lt;a href=&quot;https://www.elsevier.com/books/the-four-color-problem/ore/978-0-12-374571-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;The four-color problem&lt;/a&gt; gave a simple proof of (2) implies (3) (see Theorem 7.4.3, it is on page 100). &lt;strong&gt;I would like to know who came up with this proof first.&lt;/strong&gt; I think Heawood was aware of this simple proof. Or &lt;strong&gt;was his proof of (2) implies (3) different?&lt;/strong&gt; If so, what was (the basic idea of) his proof?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Definitions:&lt;/em&gt;&lt;br /&gt;&#xA;A (simple) graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is said to be &lt;em&gt;maximal planar&lt;/em&gt; (or triangulated) if &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; has a plane drawing without edge crossings such that every face boundary is a triangle (adding one more edge would make the graph non-planar; hence the name maximal planar). A &lt;em&gt;coloring&lt;/em&gt; of a graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; on the vertex set of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; such that endpoints of each edge get different 'colours' (ie, &lt;span class=&quot;math-container&quot;&gt;$f(u)\neq f(v)$&lt;/span&gt; for every edge &lt;span class=&quot;math-container&quot;&gt;$\{u,v\}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;). For &lt;span class=&quot;math-container&quot;&gt;$k\in\mathbb{N}$&lt;/span&gt;, a &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-coloring is a coloring that uses at most &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; colors.&lt;/p&gt;&#xA;" OwnerUserId="13224" LastActivityDate="2020-10-25T05:53:43.840" Title="What was Heawood's proof of maximal planar graph 3-colorable implies its dual is bipartite?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12374" PostTypeId="2" ParentId="12370" CreationDate="2020-10-25T09:18:27.637" Score="4" Body="&lt;p&gt;Glashow did not introduce it in any article, in 1961 or ever. &lt;a href=&quot;https://inspirehep.net/literature/4328&quot; rel=&quot;nofollow noreferrer&quot;&gt;Glashow's Partial Symmetries of Weak Interactions&lt;/a&gt; (Nucl. Phys. 22 (1961) 579-588) did something more modest, it proposed &lt;span class=&quot;math-container&quot;&gt;$SU(2)×U(1)$&lt;/span&gt; model that showed the possibility of symmetry between electromagnetic and weak interactions and predicted the &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; boson. Since the prediction did not match any experiments it went largely unnoticed.&lt;/p&gt;&#xA;&lt;p&gt;To get to the electroweak theory, one needed to get a mechanism of spontaneous symmetry breaking, proposed by Higgs in 1964, and to put it together with the &lt;span class=&quot;math-container&quot;&gt;$SU(2)×U(1)$&lt;/span&gt; model, which was only done three years later, and not by Glashow. Weinberg published his version in 1967, and Salam gave his, which he called &amp;quot;the electroweak theory&amp;quot;, in lectures at the Imperial College later the same year. The lectures were only published in 1968 in the Proceedings of a Nobel symposium. Here is from &lt;a href=&quot;https://arxiv.org/abs/1502.06276&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kibble's History of electroweak symmetry breaking&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This led Schwinger [9] to suggest a gauge theory of weak interactions mediated by &lt;span class=&quot;math-container&quot;&gt;$W^+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$W^−$&lt;/span&gt; exchange. He even asked: could there be a unified theory of weak and electromagnetic interactions, involving three gauge bosons, &lt;span class=&quot;math-container&quot;&gt;$W^+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$W^−$&lt;/span&gt;, and the photon &lt;span class=&quot;math-container&quot;&gt;$γ$&lt;/span&gt;? But this idea immediately ran into difficulty. If there is in fact a symmetry between these three gauge bosons, it clearly must be severely broken, because there are major differences between them... There was another key difference: it was known that the weak interactions do not conserve parity — they violate mirror symmetry — whereas the electromagnetic interactions are parity-conserving. So how could there be a symmetry between the two?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This latter problem was solved in 1961 by Glashow [10], who proposed an extended model with a larger symmetry group, &lt;span class=&quot;math-container&quot;&gt;$SU(2)×U(1)$&lt;/span&gt;, and a fourth gauge boson &lt;span class=&quot;math-container&quot;&gt;$Z^0$&lt;/span&gt;. He showed that by an intriguing mixing mechanism between the two neutral gauge bosons, one could end up with one boson (&lt;span class=&quot;math-container&quot;&gt;$γ$&lt;/span&gt;) with parity-conserving interactions and three that violate parity, &lt;span class=&quot;math-container&quot;&gt;$W^+$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$W^−$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z^0$&lt;/span&gt;. In 1964, Salam and his long-term collaborator John Ward, apparently unaware of Glashow’s work, proposed a very similar model also based on &lt;span class=&quot;math-container&quot;&gt;$SU(2)×U(1)$&lt;/span&gt; [11]. But in all these models, the symmetry breaking, giving the &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; bosons masses, had to be inserted by hand, and theories of spin-1 bosons with explicit masses were well known to be non-renormalizable and thus unphysical.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-26T07:24:33.563" LastActivityDate="2020-10-26T07:24:33.563" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12375" PostTypeId="2" ParentId="12351" CreationDate="2020-10-25T12:54:18.167" Score="2" Body="&lt;p&gt;Yes, this database is called &amp;quot;Math Genealogy&amp;quot;, genealogy.math.ndsu.nodak.edu.&#xA;Its value and reliability has been much discussed, but on my opinion, it is useful&#xA;and entertaining. AMS is probably of the same opinion: it includes links to it&#xA;in MathScinet.&lt;/p&gt;&#xA;&lt;p&gt;Some points of criticism are the following: the notions of PhD degree and PhD adviser is relatively recent; it did not exist in all countries at all times, though the database covers&#xA;some people back to 15 century.&lt;/p&gt;&#xA;&lt;p&gt;Until 18 century mathematics was not a profession, so inevitably the database contains many people who cannot be called mathematicians, or mathematicians who&#xA;earned a degree for other things.&lt;/p&gt;&#xA;&lt;p&gt;The data about 19-20 centuries PhD does not cover various countries uniformly.&lt;/p&gt;&#xA;&lt;p&gt;With all these drawbacks, one still can learn something interesting from this database.&lt;/p&gt;&#xA;&lt;p&gt;BTW it is not surprising that well known mathematicians come from well known advisers:-) Exceptions from this rule are more surprising.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-10-25T12:54:18.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12376" PostTypeId="1" AcceptedAnswerId="12377" CreationDate="2020-10-25T20:06:07.113" Score="6" ViewCount="265" Body="&lt;p&gt;I was teaching my students about Delaunay Triangulation which is a method for dividing a surface into triangles. This triangulation method is the basis of most computer calculations that require a surface to be approximated as small elements. Here is an example&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0TWF8.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0TWF8.png&quot; alt=&quot;Example of triangulation&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I pointed out that Delaunay's contribution was made in 1934. One student asked why Delaunay would be working on a computer method well before computers were invented. This is a good question that I was wondering if anyone on this site could answer.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Boris_Delaunay&quot; rel=&quot;noreferrer&quot;&gt;Wickipedia article&lt;/a&gt; on Boris Delaunay does mention that he worked on crystal structures so this may be a connection but that is just my speculation.&#xA;It is interesting that his ideas should be so widely used in areas well outside his interests.&lt;/p&gt;&#xA;&lt;p&gt;Why did Boris Delaunay work on the very modern topic of triangulation of surfaces when there were no computer methods in his day?&lt;/p&gt;&#xA;" OwnerUserId="13226" LastEditorUserId="55" LastEditDate="2020-10-26T09:00:25.740" LastActivityDate="2020-10-27T05:00:44.843" Title="What did Delaunay invent Delaunay triangulations for before computers were developed?" Tags="|geometry|computation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12377" PostTypeId="2" ParentId="12376" CreationDate="2020-10-26T08:53:07.573" Score="11" Body="&lt;p&gt;Delaunay (Gallicized version of Russian Delone) did not invent them, they were used long before 1934. Delaunay triangulations, or more generally tesselations, are dual to &lt;a href=&quot;https://en.wikipedia.org/wiki/Voronoi_diagram#History_and_research&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voronoi diagrams&lt;/a&gt;, the circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. One can infer the motivation from the very title of Delaunay's 1934 paper: &lt;a href=&quot;http://mi.mathnet.ru/eng/izv4937&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sur la sphere vide. A la memoire de  Georges Voronoi&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Dirichlet used such diagrams for regular lattices to study 2D and 3D quadratic forms in 1850 (although they were occasionally used before), and Voronoy introduced a multidimensional generalization in 1908, the year of his death, along with the dual Delaunay tessellations. Delaunay met Voronoy at the University of Warsaw as a teenager, and was strongly influenced. So much so that &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=74408&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Genealogy Project&lt;/a&gt; lists Voronoy as his thesis advisor, even though he defended well after Voronoy’s death. Delaunay's contribution was to study Voronoi diagrams and their duals for irregularly placed points or sites, but even he used them before 1934, e.g. in &lt;a href=&quot;https://doi.org/10.1524/zkri.1933.84.1.109&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Neue Darstellung der geometrischen Kristallographie&lt;/em&gt;&lt;/a&gt;, &lt;em&gt;Z. Kristallograph.&lt;/em&gt;, 84, 109–149 (1933).&lt;/p&gt;&#xA;&lt;p&gt;Voronoi diagrams were sporadically used even before Dirichlet, as far back as Kepler and Descartes. Kepler even used Delaunay tessellations to study the shapes of snowflakes and the sphere packing problem. They were also discovered and rediscovered in gold mining, crystallography, metallurgy and meteorology long before computers, e.g. Wigner and Seitz introduced Voronoi diagrams induced by the atoms of a metallic crystal in 1933, without any knowledge of Delaunay's work. See &lt;a href=&quot;https://www.math.uni-bielefeld.de/documenta/vol-ismp/60_liebling-thomas.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voronoi diagrams and Delaunay triangulations by Liebling and Pournin&lt;/a&gt; for details of applications and a historical sketch.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-27T05:00:44.843" LastActivityDate="2020-10-27T05:00:44.843" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12381" PostTypeId="2" ParentId="12369" CreationDate="2020-10-27T06:37:05.733" Score="5" Body="&lt;p&gt;Vitali and Riesz were working on differentiation specifically, Hardy and Littlewood were far from it. Vitali writes in &lt;a href=&quot;https://archive.org/stream/attidellarealeac43real#page/228/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sui gruppi di punti e sulle funzioni di variabili reali&lt;/em&gt; (1908)&lt;/a&gt; (On sets of points and functions of real variables) after proving the covering lemma:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In my note &amp;quot;On integral functions&amp;quot;, published in&#xA;1905 by the Royal Academy of Sciences of Turin I gave the necessary and sufficient condition for a function of one real variable to be an integral. The demonstration methods I used in that note cannot be extended to cases of two or more variables. In this chapter, I modify those methods to make them also applicable to those cases. The results of the previous chapter are useful for this purpose. Because it is easy to understand how to extend the result of my cited note to the functions of several variables, I will only indicate this extension to functions of two variables in the last § of this chapter.&lt;/em&gt;&amp;quot; [edited Google translation]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Riesz's paper's title is self-explanatory: &lt;a href=&quot;http://pub.acta.hu/acta/showCustomerArticle.action?id=5332&amp;amp;dataObjectType=article&amp;amp;returnAction=showCustomerVolume&amp;amp;sessionDataSetId=3309077869c34c64&amp;amp;style=&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sur l'existence de la dérivée des fonctions monotones et sur quelques problèmes qui s'y rattachent&lt;/em&gt; (1932)&lt;/a&gt; (On the existence of the derivative of monotone functions and some problems associated with it).&lt;/p&gt;&#xA;&lt;p&gt;Hardy and Littlewood state at the beginning of their paper &lt;a href=&quot;https://projecteuclid.org/euclid.acta/1485887846&quot; rel=&quot;nofollow noreferrer&quot;&gt;A maximal theorem with function-theoretic applications (1930)&lt;/a&gt; that they were trying to prove an integral inequality for analytic functions, very far from the differentiation theorem. The maximal function came up in the middle, like things do sometimes, and they motivate it with averaging in cricket games, not measure theoretic generalities.&lt;/p&gt;&#xA;&lt;p&gt;After Riesz it became something of a sport to come up with proofs of the theorem of some desirable form, see &lt;a href=&quot;https://hsm.stackexchange.com/a/7754/55&quot;&gt;Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-10-27T18:50:30.500" LastActivityDate="2020-10-27T18:50:30.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12382" PostTypeId="1" CreationDate="2020-10-29T09:08:32.953" Score="3" ViewCount="513" Body="&lt;p&gt;[&lt;strong&gt;Note:&lt;/strong&gt; &lt;a href=&quot;https://math.stackexchange.com/questions/763998/eulers-errors&quot;&gt;This&lt;/a&gt; is the same question on MSE, but it has only five answers, one of which doesn't give an exact answer (it tells that a conjecture of Euler was wrong, but a conjecture isn't a mistake) and the question is unlikely to get answers since it was active 2 years ago.]&lt;br /&gt;&#xA;Euler was one of the greatest and smartest mathematician in the history. He contributed to many fields of mathematics. But no one is perfect, and this holds for Euler also. For example, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.2007.11920416&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; tells that Euler thought that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{ab}=\sqrt{a}\sqrt{b}$$&lt;/span&gt;&#xA;whether or not &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are positive or negative. So this is one of Euler's mistake. I want to see his some more mistakes. So my question is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are some of Euler's mistakes?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Like most big list questions, please give only one mistake per answer.&lt;/p&gt;&#xA;" OwnerDisplayName="user13238" LastEditorDisplayName="user13238" LastEditDate="2020-10-29T09:14:13.223" LastActivityDate="2020-11-01T02:49:29.543" Title="What are some of Euler's mistakes?" Tags="|mathematics|euler|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12383" PostTypeId="1" CreationDate="2020-10-29T21:57:58.747" Score="5" ViewCount="106" Body="&lt;p&gt;In the book &lt;br&gt;Mathews, Walker: Mathematical Methods of Physics, Addison-Wesley(1969),&lt;br&gt;&#xA;there is a pictorial notation of the solution found by Fredholm about an integral equation.&lt;br&gt;&lt;a href=&quot;https://www.dropbox.com/s/fuictheof7xzwlh/a1.pdf?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;p.304&lt;/a&gt;, &lt;a href=&quot;https://www.dropbox.com/s/jrmelpm96kupffe/a2.pdf?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;p.305&lt;/a&gt;&lt;br&gt;&lt;br&gt;This circle and cross notation might be the authors' original idea inspired by the Feynman diagrams.  But I'm not sure.&lt;br&gt;In mathematics, such a pictorial notation was popular before Feynman?&lt;br&gt;&lt;br&gt;edit: I fancied some mathematicians were making pictorial representations of equations in the 19th or 18th century as an appendix to the rigorous proof.&lt;/p&gt;&#xA;" OwnerDisplayName="user12897" LastEditorUserId="55" LastEditDate="2020-10-30T10:11:49.813" LastActivityDate="2020-10-30T10:11:49.813" Title="Were pictorial notations like Feynman diagrams for integrals used before Feynman?" Tags="|mathematics|mathematical-physics|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12384" PostTypeId="1" CreationDate="2020-10-30T06:10:34.200" Score="1" ViewCount="23" Body="&lt;p&gt;Who are the people actually 1st to mention about undetermined forms like 0^∞ or 0/0 or ∞/∞ etc .&#xA;In which books these forms are mentioned and if the people were not known specifically , mention the region where these forms are mentioned in that region's scientific works.&lt;/p&gt;&#xA;" OwnerUserId="13012" LastActivityDate="2020-10-30T06:10:34.200" Title="Founders of undetermined forms" Tags="|mathematicians|calculus|scientists|" AnswerCount="0" CommentCount="0" ClosedDate="2020-10-30T15:12:02.383" ContentLicense="CC BY-SA 4.0" />
  <row Id="12385" PostTypeId="1" AcceptedAnswerId="12386" CreationDate="2020-10-30T08:12:07.333" Score="5" ViewCount="165" Body="&lt;p&gt;Is there any indication that Charles Peirce was aware of Babbage and his work on mechanical computing?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-11-01T20:43:00.573" Title="Was Charles Sanders Peirce aware of Charles Babbage's difference engine?" Tags="|mathematicians|biographical-details|computer-science|computation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12386" PostTypeId="2" ParentId="12385" CreationDate="2020-10-30T09:45:26.393" Score="7" Body="&lt;p&gt;Not only was Peirce aware of the difference engine, he was also aware of the analytic engine, that was never built, of Jevons's 1870 machine, and was later personally involved with designing its improved version by his student Marquand in 1880-s at Johns Hopkins (a diagram of an electrical 'logical machine' possibly drawn by him was found among Marquand's papers by Church). It was at that time that Peirce invented relational calculus with quantifiers independently of Frege.&lt;/p&gt;&#xA;&lt;p&gt;Peirce wrote Babbage's obituary published in The Nation on November 9, 1871 that shows uncanny familiarity with the details of Babbage's project and gives a glowing appraisal of it (&amp;quot;&lt;em&gt;beyond question, the most stupendous work of human invention&lt;/em&gt;&amp;quot;). At the same time, Peirce saw severe limitations to what such machines can do, and did not subscribe to the &amp;quot;computational theory of mind&amp;quot;, see &lt;a href=&quot;http://www.commens.org/bibliography/journal_article/ketner-kenneth-l-1988-peirce-and-turing-comparisons-and-conjectures&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ketner, Peirce and Turing&lt;/a&gt; for details. In 1894 Peirce wrote (some see it as an anticipation of Turing's limiting result concerning the halting problem):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Some mathematical results, doubtless could have been worked out with Babbage's&#xA;analytical engine; but did anyone ever suppose that the subject could at all really&#xA;be advanced by such a machine?... Even syllogistic reasoning in its higher&#xA;varieties as they appear in the logic of relatives, requires a living act of choice&#xA;based on discernment, beyond the powers of any conceivable machine; and this&#xA;sufficiently refutes the idea that man is a mere mechanical automaton endowed&#xA;with an idle consciousness.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Peirce's other thoughts on such machines are expressed in a 1886 letter to Marquand that suggested the use of relays to implement AND and OR, 1887 paper Logical Machines, 1900 manuscript Our Senses as Reasoning Machines, and in 1902 notes. Further references on Peirce's involvement with &amp;quot;reasoning machines&amp;quot; and his possible influence on the development of digital computers are collected in this &lt;a href=&quot;https://peirce-l.lyris.ttu.narkive.com/EV0ilOoH/juliette-s-correspondence-now-imaged-online&quot; rel=&quot;nofollow noreferrer&quot;&gt;Narkive thread&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The original source of Charles's familiarity was likely his father Benjamin, a well-known mathematician with a keen interest in logical machines that Charles shared. Benjamin was a leader of a small group of New England scientists, known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientific_Lazzaroni&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Scientific Lazzaroni&amp;quot;&lt;/a&gt;, a member of which, astronomer Gould, personally met with Babbage in 1845, and had the difference engine explained to him. In 1853 American Association for the Advancement of Science, at a meeting presided over by Benjamin, passed a resolution expressing &amp;quot;deep interest&amp;quot; in completing the analytic engine. In 1857 Sheutz's difference engine, built at Babbage's encouragement, was purchased by the Dudley observatory with Gould's mediation for $5000, leading to a major row with the Dudley Board of Trustees.&lt;/p&gt;&#xA;&lt;p&gt;From Peirce's obituary of Babbage, which is quoted in full by Ketner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;About 1822, he made&#xA;his first model of a calculating machine. It was a 'difference engine', that is, the&#xA;first few numbers of a table being supplied to it, it would go on and calculate the&#xA;others successively according to the same law... In 1833, a portion of the engine,&#xA;sufficient to illustrate the working of the whole, was put together. It was a&#xA;wonderful piece of workmanship, of a precision then unknown, and still&#xA;unrivalled. To make it, it had been necessary not only to contrive new tools, but to&#xA;lay a scientific foundation of the principles of tools, and to educate the mechanics&#xA;who were to use them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1833, Mr. Babbage declined to continue&#xA;this system, and, in consequence, the engineer discontinued the construction of the&#xA;engine, dismissed the workmen, and took away all the tools. During the&#xA;suspension of the work caused by this circumstance, the great misfortune of his&#xA;life befell Mr. Babbage. He discovered the possibility of a new analytical engine, to&#xA;which the difference engine was nothing; for it would do all the arithmetical work&#xA;that that would do, but infinitely more; it would perform the most complicated&#xA;algebraical processes, elimination, extraction of roots, integration, and it would&#xA;find out for itself what operations it was necessary to perform; and the principle of&#xA;this machine was such as immensely to simplify the means of attaining the object&#xA;of a difference machine.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;One would suppose that, finding himself so unlucky as to have thought of such a thing, Babbage would at least have had the sense to keep it&#xA;strictly to himself. Instead of that, he wrote immediately and communicated it to&#xA;the Government! Before that, all was going smoothly; after that, they never would&#xA;advance another penny... But the analytical engine is, beyond question, the most&#xA;stupendous work of human invention. It is so complicated that no man's mind&#xA;could trace the manner of its working through drawings and descriptions, and its&#xA;author had to invent a new notation to keep account of it. This mechanical&#xA;notation has been found very serviceable for simpler cases.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-01T20:43:00.573" LastActivityDate="2020-11-01T20:43:00.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12387" PostTypeId="2" ParentId="7716" CreationDate="2020-10-30T20:46:51.883" Score="2" Body="&lt;p&gt;Bocher, the author of the 1892 article cited in Wikipedia, published a &lt;a href=&quot;https://www.jstor.org/stable/1967346&quot; rel=&quot;nofollow noreferrer&quot;&gt;follow-up note&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I find that in the Educational Times for March, &lt;strong&gt;1864&lt;/strong&gt;, Clifford refers&#xA;incidentally, in the solution of a problem set by Prof. Sylvester, to&#xA;the &amp;quot;nine-point conic;&amp;quot; thus showing that this conic, to which I&#xA;called attention in the March number of the ANNALS, was then familiar&#xA;to English mathematicians. I have not been able, however, to find any&#xA;earlier mention of the subject.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it seems to have been in the &amp;quot;folklore&amp;quot; as early as the &lt;strong&gt;1860&lt;/strong&gt;'s.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;: It's referred to in &lt;a href=&quot;https://archive.org/details/atreatiseonconi00unkngoog/page/n158/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Salmon, &lt;em&gt;Conic Sections&lt;/em&gt;, 3rd Edition (&lt;strong&gt;1855&lt;/strong&gt;), Art. 154, pg 137, Ex. 4.&lt;/a&gt;.  It is a special case (shown later in the book) of the theorem that the locus of a pole of a fixed line with respect to the conics through four points is a conic.&lt;/p&gt;&#xA;&lt;p&gt;It also comes up in the &lt;a href=&quot;https://archive.org/details/treatiseoncon2ed00salmuoft/page/308/mode/2up?&quot; rel=&quot;nofollow noreferrer&quot;&gt;Second Edition (&lt;strong&gt;1850&lt;/strong&gt;), pg 309, Ex. 10&lt;/a&gt;, but only six of the nine points are mentioned.&lt;/p&gt;&#xA;&lt;p&gt;It's worth pointing out, as Bôcher does in the note cited above, that although the &amp;quot;nine point conic of a quadrangle&amp;quot; was known much earlier, his &amp;quot;nine point conic of a triangle and a point&amp;quot; connects it more closely with the nine point circle of a triangle.&lt;/p&gt;&#xA;" OwnerUserId="11204" LastEditorUserId="11204" LastEditDate="2021-07-19T03:12:49.080" LastActivityDate="2021-07-19T03:12:49.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12389" PostTypeId="2" ParentId="12382" CreationDate="2020-10-31T03:58:22.663" Score="5" Body="&lt;p&gt;There is &lt;a href=&quot;https://en.wikipedia.org/wiki/Betteridge%27s_law_of_headlines&quot; rel=&quot;nofollow noreferrer&quot;&gt;Betteridge's law of headlines&lt;/a&gt;: &amp;quot;Any headline that ends in a question mark can be answered by the word no.&amp;quot; The article you quote (&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.2007.11920416&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's “Mistake”?&lt;/a&gt;) is not an exception. &lt;span class=&quot;math-container&quot;&gt;$\sqrt{ab}=\sqrt{a}\sqrt{b}$&lt;/span&gt; is not a mistake. It is true if you understand square root as a &lt;a href=&quot;https://en.wikipedia.org/wiki/Multivalued_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;multivalued function&lt;/a&gt;. Another common understanding of √ is positive square root, but it is defined only for positive numbers. The author of the paper gave two weird interpretations of square root, under which this equality indeed is not true. However there is no evidence that these interpretations were common or even used in Euler time. So,  this is not a mistake at all, at least until someone proves that Euler meant one of these interpretations.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="4931" LastEditDate="2020-11-01T02:49:29.543" LastActivityDate="2020-11-01T02:49:29.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12390" PostTypeId="1" CreationDate="2020-10-31T11:15:47.710" Score="1" ViewCount="81" Body="&lt;p&gt;I found this claim in the book &amp;quot;How many grapes went into the wine&amp;quot;, in the Artorga section:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1956 I devised a game for solving simultaneous linear equations in two variables. The theory used the properties of groups to simplify the arithmetic: the game worked in homomorphic transformations modulo 5. It was played by unsophisticated persons, such as children, who could not be expected to know everything about simultaneous equations. They competed with each other to make correct selections from sets of five, and then to synthesize the selections and select therefrom, by means of a simple machine. This simulated algedonic feedback by means of coloured lights which announced that 'pleasure' or 'pain' was being experienced.&lt;br /&gt;&#xA;The idea of this was to demonstrate the possibility of intelligence amplification. Intelligence, in the sense of an ability to steer one's way from pain to pleasure within a simple game language, was certainly amplified in that (unknown to the competitors) the equations were actually solved. Was any particular solution implicit in the game language? I think not; because the game could and did handle actual equations which had not been previously studied by the inventor. This series of experiments was entirely successful; unfortunately the opportunity to write them up for publication has now been lost.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does anybody knows how this could be made (the mechanics of the method of solving the systems) or has any idea about it, and if other similar researches were made (and link material them)?&#xA;I'm very curious about it&lt;/p&gt;&#xA;" OwnerUserId="13245" LastEditorUserId="10106" LastEditDate="2020-10-31T13:17:52.583" LastActivityDate="2020-10-31T13:17:52.583" Title="Research about Stafford Beer's claim about a method for solving simultaneous equations unknowingly via a game by kids?" Tags="|mathematics|linear-algebra|algebra|cybernetics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12392" PostTypeId="1" CreationDate="2020-10-31T21:15:07.943" Score="0" ViewCount="69" Body="&lt;p&gt;I am interested in any historical account of the following attributes of Peirce:character,cognition and appearance.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="131" LastEditDate="2020-11-02T08:47:23.637" LastActivityDate="2020-11-02T08:47:23.637" Title="Are there any accounts of Charles Peirce's character,cognition and physical appearance?" Tags="|mathematicians|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12394" PostTypeId="1" CreationDate="2020-11-02T01:47:54.420" Score="7" ViewCount="386" Body="&lt;p&gt;Donald Knuth credits Pierre-Arnoul de Marneffe's idea of &amp;quot;Holon Programming&amp;quot; as the main influence on Literate Programming. See page 13 of &lt;a href=&quot;http://literateprogramming.com/knuthweb.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Literate Programming&amp;quot;&lt;/a&gt;, Knuth's paper introducing the concept. Knuth calls de Marneffe's work &amp;quot;pioneering&amp;quot; and says that it &amp;quot;has not received the attention it deserves&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Knuth cites the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;P.A. de Marneffe, &lt;em&gt;Holon Programming&lt;/em&gt;. Univ. de Liege, Service D'Informatique (December, 1973).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;P.A. de Marneffe and D. Ribbens, Holon Programming, in A. Günther et al. (eds.), &lt;em&gt;International Computing Symposium&lt;/em&gt; 1973, Amsterdam, North-Holland (1974).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also cited as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proceedings of the International Computing Symposium 1973, Davos, Switzerland, 4–7 September 1973. Edited by Alfred Gunther, Bernard Levrat and H. Lipps&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Literate_programming#cite_note-19&quot; rel=&quot;nofollow noreferrer&quot;&gt;Literate Programming refers to the first source, but adds that it was &amp;quot;Report PMAR 73–23&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately these sources are nowhere to be found. I don't mean that there's just no free PDF online; I mean that if you type &lt;code&gt;&amp;quot;holon programming&amp;quot; PMAR 73-23&lt;/code&gt; into Google, you get fewer than 100 results, many of which are suppressed because Google identified them as essentially duplicates of other pages. None of the results are helpful. If you search for &lt;code&gt;PMAR 73-23&lt;/code&gt; alone, then you get many more results, but they are all irrelevant. You might be asking &amp;quot;What if Wikipedia's extra information is wrong?&amp;quot; Sadly there's still no luck searching for the titles or for the publications or for both.&lt;/p&gt;&#xA;&lt;p&gt;Although it would be great to get ahold of these documents, I don't really mind if I can't. What I really want to know is simply this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;What exactly is Holon Programming?&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know it has something to do with the concept of Holon (outside of programming), but that isn't very useful. Was there an actual implementation of de Marneffe's ideas? If so, then is there any information about it?&lt;/p&gt;&#xA;" OwnerUserId="13246" LastEditorUserId="4931" LastEditDate="2021-12-05T01:01:13.327" LastActivityDate="2021-12-05T01:01:13.327" Title="What is Holon Programming?" Tags="|computer-science|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12395" PostTypeId="1" CreationDate="2020-11-02T16:44:29.967" Score="1" ViewCount="197" Body="&lt;p&gt;Before learning any topic, I always find the reason of &amp;quot;Why definitions are created for that particular topic?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In fluid statistics, I encountered a term called &amp;quot;pressure&amp;quot;.&#xA;But I can't find any reference.&lt;/p&gt;&#xA;&lt;p&gt;Where can I find who first defined it or why was it defined that way that time?&lt;/p&gt;&#xA;&lt;p&gt;I mean I found the definition on Wikipedia:&lt;/p&gt;&#xA;&lt;p&gt;Pressure is a scalar quantity. It relates the vector area element (a vector normal to the surface) with the normal force acting on it.&#xA;The pressure is the scalar proportionality constant that relates the two normal vectors:&#xA;&lt;span class=&quot;math-container&quot;&gt;$${\displaystyle d\mathbf {F} _{n}=-p\,d\mathbf {A} =-p\,\mathbf {n} \,dA}$$&lt;/span&gt;&#xA;Where did that come from?&lt;/p&gt;&#xA;&lt;p&gt;For other things such as Torque and Centre of Mass,I found reference to Newton and Archimedes but I can't find the reference for pressure.&lt;/p&gt;&#xA;&lt;p&gt;Any help would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="13256" LastEditorUserId="13257" LastEditDate="2020-11-03T04:50:12.017" LastActivityDate="2020-11-03T04:50:12.017" Title="Who first defined pressure?" Tags="|physics|discoveries|depiction-of-scientists|fluid-dynamics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12396" PostTypeId="1" CreationDate="2020-11-02T18:02:33.233" Score="2" ViewCount="179" Body="&lt;p&gt;Differential homological algebra in its initial formulation is due to Eilenberg and Moore, who first published the homological version of the Eilenberg–Moore spectral sequence in 1965 (and the cohomological version never), sufficiently long after its date of discovery that other accounts had been published in the meantime.&lt;/p&gt;&#xA;&lt;p&gt;Paul Baum's 1963 thesis gives an account, but Moore had spoken on this theory at least as early as the 1959–60 &lt;em&gt;Seminaire Henri Cartan&lt;/em&gt;, and in that appearance, says he and Eilenberg had worked it out and that it had appeared in notes &lt;em&gt;multigraphié&lt;/em&gt; from the 1957–8 Princeton seminar on algebraic topology, which thus may be the earliest written source.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Are these still extant&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;p&gt;If so, where can they be found?&lt;/p&gt;&#xA;" OwnerUserId="1775" LastEditorUserId="10106" LastEditDate="2020-11-03T02:08:20.650" LastActivityDate="2021-04-12T07:00:40.083" Title="The origins of differential homological algebra" Tags="|mathematics|reference-request|algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12397" PostTypeId="1" AcceptedAnswerId="12398" CreationDate="2020-11-03T02:57:07.810" Score="1" ViewCount="428" Body="&lt;p&gt;I was studying calculus and the following question came to my mind: Who was the first person to use or suggest the use of the Completeness Axiom of the Real Numbers?&lt;/p&gt;&#xA;" OwnerUserId="13017" LastEditorUserId="55" LastEditDate="2020-11-03T09:55:05.373" LastActivityDate="2021-07-05T05:43:12.470" Title="Who first used the Completeness Axiom for real numbers?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12398" PostTypeId="2" ParentId="12397" CreationDate="2020-11-03T09:54:48.340" Score="6" Body="&lt;p&gt;Nobody. Those who were first did not have a clear idea of real numbers or completeness, and by the time the concepts took shape those who used them were no longer first, see &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Real_numbers_2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor, The real numbers: Stevin to Hilbert&lt;/a&gt;. The first to &lt;em&gt;state&lt;/em&gt; completeness as an axiom, to back up his prior axiomatization of geometry, was Hilbert in &lt;a href=&quot;https://eudml.org/doc/144659&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Über den Zahlbegriff&lt;/em&gt; (1900)&lt;/a&gt;, see &lt;a href=&quot;https://plato.stanford.edu/entries/proof-theory/appendix-a.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, Formal Axiomatics&lt;/a&gt; for commentary. But he sure was not the first to &lt;em&gt;use&lt;/em&gt; it.&lt;/p&gt;&#xA;&lt;p&gt;The intuitive notion of &amp;quot;real numbers&amp;quot; was implicit in Stevin's use of infinite decimals that became increasingly common in 17-18th centuries, and completeness was &amp;quot;used&amp;quot; when assuming that graphs of functions intersect  where it looks like they do, etc. But the concepts remained vague and implicit until late 19th century. Cauchy &amp;quot;used&amp;quot; completeness in &lt;em&gt;Cours d'Analyse&lt;/em&gt; (1821), but as Grabiner writes in The Origins of Cauchy's Rigorous Calculus:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...though Cauchy implicitly assumed several forms of the completeness axiom for the real numbers, he did not fully understand the nature of completeness or the related topological properties of sets of real numbers or of points in space... Cauchy did not have explicit formulations for the completeness of the real numbers. Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bolzano was more explicit with his definition of real numbers as convergent sequences of rationals from 1817 on, but historians disagree as to whether his development was logically satisfactory. Even if it was, it was ahead of his time and went largely unnoticed.&lt;/p&gt;&#xA;&lt;p&gt;In any case, in Bolzano's treatment, as well as in Weierstrass's, Cantor's, Dedekind's and Heine's later into the century, completeness appears not as an axiom but as a theorem, a consequence of their constructions of real numbers in terms of rational sequences or cuts, see &lt;a href=&quot;https://projecteuclid.org/euclid.rml/1081173837&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snow, Views on the real numbers and the continuum&lt;/a&gt; for details. Frege complained that such constructions do not guarantee that the resulting systems are consistent, but, as we now know from Goedel's results, no such guarantees can be given in principle, whether by constructions or by Hilbert style axiomatizations.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-07-05T05:43:12.470" LastActivityDate="2021-07-05T05:43:12.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12399" PostTypeId="1" AcceptedAnswerId="12400" CreationDate="2020-11-04T11:09:41.960" Score="7" ViewCount="1259" Body="&lt;p&gt;I opened the same &lt;a href=&quot;https://physics.stackexchange.com/questions/591515/why-did-galileo-use-ratios-in-his-experiment-with-inclined-planes/591605#591605&quot;&gt;question&lt;/a&gt; on Physics Stack Exchange, but it seems more suited for this site.&lt;/p&gt;&#xA;&lt;p&gt;I've been reading about Galileo's experiment with inclined planes, and he ends up saying something along the lines of &amp;quot;the ratio of distances is equal to the ratio of the times squared&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;My initial thought is that, with initial velocity zero. A first distance can be defined as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ s_{1} = \frac{1}{2} a t_{1}^2 $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And a second distance as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ s_{2} = \frac{1}{2} a t_{2}^2 $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where I can take the ratio of the distances and end up with:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ \frac{s_{1}}{s_{2}} = \frac{t_{1}^2}{t_{2}^2} = (\frac{t_{1}}{t_{2}})^2 $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So one doesn't need to know what is the constant of proportionality but can know there's a proportionality if the data matches the previous equation.&lt;/p&gt;&#xA;&lt;p&gt;However, I'm not sure if this is all there is to it. Is there any other reason for looking at the data of this experiment as ratios? It was customary, back then, to speak of ratios as geometry was the most common way of expressing mathematics?&lt;/p&gt;&#xA;" OwnerUserId="12712" LastEditorUserId="55" LastEditDate="2020-11-04T14:44:38.253" LastActivityDate="2020-11-05T06:17:07.477" Title="Why did Galileo express himself in terms of ratios when describing laws of accelerated motion?" Tags="|physics|classical-mechanics|galileo|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12400" PostTypeId="2" ParentId="12399" CreationDate="2020-11-04T14:42:44.907" Score="19" Body="&lt;p&gt;Galileo followed a venerable tradition of distinguishing numbers, magnitudes of different kinds (lengths, times, areas, etc.) and ratios. This is somewhat analogous to the strictures of modern dimensional analysis used in physics, but even stricter, and ancient Greeks did not have dimensional constants to bridge the gaps. They did not even have enough dimensionless numbers, only positive integers were admitted, not even rationals. Geometry was far ahead of arithmetic and algebra in the level of sophistication. And so lengths and areas were not numbers assigned to geometric figures, as we think today, they were literally the &lt;em&gt;figures themselves&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ratios were defined for both numbers and magnitudes, and were the only &amp;quot;legitimate&amp;quot; way to connect numbers to magnitudes, or magnitudes of different kinds to each other, as their ratios could be equated (being dimensionless), see &lt;a href=&quot;https://hsm.stackexchange.com/a/5958/55&quot;&gt;What did the ratio of two magnitudes mean to ancient Greek mathematicians?&lt;/a&gt; And so Euclid does not say that the area of a circle is a constant times diameter squared, but says &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookXII/propXII2.html&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;&lt;em&gt;circles are to one another as the squares on their diameters&lt;/em&gt;&amp;quot;&lt;/a&gt;. Archimedes does not say that a weight balancing a lever is a constant divided by the length of the leg it is on, but says that the balanced weights are in the opposite ratio to that of the legs, etc.&lt;/p&gt;&#xA;&lt;p&gt;The distinction was eroding since the late antiquity, as more and more entities were admitted as numbers, but it was still influential in Galileo's time. And on accelerated motion he had a direct predecessor, Oresme (1320-1382), see &lt;a href=&quot;https://www.jstor.org/stable/10.4169/002557010x479965&quot; rel=&quot;noreferrer&quot;&gt;Nicodemi Galileo and Oresme&lt;/a&gt;. Oresme called it &amp;quot;uniformly difform&amp;quot; motion, and developed a theory of it, which included graphing velocities (he used bar graphs, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2382/55&quot;&gt;When do we see for the first time the use of the Cartesian coordinates?&lt;/a&gt;).  In The Geometry of Qualities and Motions Oresme expresses himself in the same fashion:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The universal rule is this, that the measure or ratio of any two linear or surface qualities or velocities is as that of the figures by which they are comparatively and mutually imagined... Therefore, in order to have measures and ratios of qualities and velocities one must have recourse to geometry.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In contrast, Galileo in Two New Sciences (1638) is already one step away from geometry. But not from the ratio language:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If a moveable descends from rest in uniformly accelerated motion,the spaces run through in any times whatever are to each other as the duplicated ratio of their times; that is, are as the squares of those times.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-05T06:17:07.477" LastActivityDate="2020-11-05T06:17:07.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12401" PostTypeId="1" AcceptedAnswerId="12404" CreationDate="2020-11-04T19:58:57.333" Score="7" ViewCount="250" Body="&lt;p&gt;One problem in enumerative geometry consists in counting the number of rational curves of degree &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; in the plane going through &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; general points. If &lt;span class=&quot;math-container&quot;&gt;$n = 3d-1$&lt;/span&gt;, this number, denoted &lt;span class=&quot;math-container&quot;&gt;$N_d$&lt;/span&gt;, is finite and the problem is to find a formula for &lt;span class=&quot;math-container&quot;&gt;$N_d$&lt;/span&gt;.&lt;br /&gt;&#xA;Now, one can ask the same question but for rational curves not on the plane but on a quintic hypersurface given by the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x_1^5 + x_2^5 + x_3^5 + x_4^5 + x_5^5 = 0$$&lt;/span&gt;&#xA;in projective &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-space. In this last case, a classical result from the XIXth century gives &lt;span class=&quot;math-container&quot;&gt;$N_1 = 2875$&lt;/span&gt;, &lt;em&gt;i.e.&lt;/em&gt; there are &lt;span class=&quot;math-container&quot;&gt;$2875$&lt;/span&gt; lines on the quintic. The next number &lt;span class=&quot;math-container&quot;&gt;$N_2$&lt;/span&gt; counts the number of conics in the quintic and was found around 1980. The number of twisted cubics, &lt;span class=&quot;math-container&quot;&gt;$N_3 = 317 206 375$&lt;/span&gt;, was the result of a complicated computer program.&lt;br /&gt;&#xA;I remember a mathematican telling me there was a bug in the program and mathematicians were not aware of it initially. Actually, physicists noticed an error in the mathematicians's computation due to some discrepancy with their own result on the same problem calculated with very different methods (not the result of a computer program). As a result, physicists called the mathematicians's attention to a possible bug in their computer program. As it happens, the physicists were right and a bug was hidden in the computer program of the mathematicians.&lt;br /&gt;&#xA;Can someone confirm this story or correct it? If the story is true, please provide details and references.&lt;/p&gt;&#xA;" OwnerUserId="13109" LastEditorUserId="13109" LastEditDate="2020-11-08T17:55:39.120" LastActivityDate="2020-11-08T17:55:39.120" Title="Did physicists correct an error of mathematicians in counting twisted cubics in the quintic?" Tags="|mathematics|physics|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12402" PostTypeId="1" AcceptedAnswerId="12411" CreationDate="2020-11-04T21:30:50.090" Score="5" ViewCount="231" Body="&lt;p&gt;Newton was known to keep manuscripts of his thoughts and workings on math/physics (and even more related to religion) which are kept in Cambridge I believe.&lt;/p&gt;&#xA;&lt;p&gt;My question is, are there examples of the contrary; mathematicians who are thought not to have made extensive use of physical media (parchment/paper/computers) to do their work? Instead, they may have had strong mental &amp;quot;note-keeping&amp;quot; skills so to speak, as well as strong mental imagery.&lt;/p&gt;&#xA;&lt;p&gt;They could still publish their results of course.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I originally posted this question on math.stackexchange, but closed it to post it on HSM. Some of the comments I received were broadly:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Euler continued to work after becoming blind. In fact, half of his work was published after becoming blind.&lt;/li&gt;&#xA;&lt;li&gt;Ramanujan mostly wrote only his final results in his notebooks.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;One other mathematician who was blind that I've come across since originally asking this question is Lev Pontryagin.&lt;/p&gt;&#xA;" OwnerUserId="13262" LastEditorUserId="13262" LastEditDate="2020-11-04T21:37:27.273" LastActivityDate="2020-11-09T00:32:06.177" Title="Mathematicians who often didn't use notebooks/devices to work?" Tags="|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12403" PostTypeId="1" CreationDate="2020-11-05T03:54:16.593" Score="1" ViewCount="99" Body="&lt;p&gt;I will be giving a lecture soon (on Friday) about continued fractions.  One of the historical compressed notations for continued fractions uses &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; (from the German word for continued fraction):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;a_1 + \underset{k=2}{\overset{\infty}{\sf K}} \frac{1}{a_k} = &#xA;a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + \cfrac{1}{{\ddots}}}}}.&#xA;$$&lt;/span&gt;&#xA;This usage is analogous to &lt;span class=&quot;math-container&quot;&gt;$\sum$&lt;/span&gt; for sums and &lt;span class=&quot;math-container&quot;&gt;$\prod$&lt;/span&gt; for products.&lt;/p&gt;&#xA;&lt;p&gt;In many places, including the Wikipedia page on continued fractions, the &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-notation is attributed to Gauss. My question: what is the first paper in which Gauss used this notation, or in fact &lt;em&gt;any&lt;/em&gt; paper where he used it? I found a couple of papers by Gauss involving continued fractions and in none of them is the &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-notation used.&lt;/p&gt;&#xA;&lt;p&gt;In 2017 the same question was asked by someone else &lt;a href=&quot;https://math.stackexchange.com/questions/2280938/reference-in-the-literature-for-the-first-use-of-the-k-notation-continued-fract&quot;&gt;here&lt;/a&gt;, and that OP had gone through (quickly) all papers by Gauss that appeared related to continued fractions without seeing the  &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;-notation in them. That question has not yet received an  answer. I hope by posting this on a site more focused on the history of (science and) math it may lead to a good answer to this historical question on notation, whether or not it was really due to Gauss.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2020-11-05T03:54:16.593" Title="First use of $K$ in notation for continued fractions" Tags="|notation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12404" PostTypeId="2" ParentId="12401" CreationDate="2020-11-05T09:27:55.203" Score="9" Body="&lt;p&gt;Indeed they did, the OP story is told by Yau in &lt;a href=&quot;https://zr9558.files.wordpress.com/2014/11/shing-tung_yau_nadis_s-_the_shape_of_inner_space.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Shape of Inner Space, pp.169-70&lt;/a&gt;. Calculating the number of twisted cubics in the quintic hypersurface was a big early coup for mirror symmetry that attracted mathematicians' attention to string theory and revitalized enumerative geometry. In &lt;a href=&quot;https://webspace.science.uu.nl/%7Ebeuke106/HypergeometricFunctions/COGP.pdf&quot; rel=&quot;noreferrer&quot;&gt;A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory (1991)&lt;/a&gt; Candelas, De La Ossa, Green and Parkes managed to compute what was beyond the means of mathematicians at the time. As Katz remarked in   &lt;a href=&quot;http://www.numdam.org/article/CM_1986__60_2_151_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;On the finiteness of rational curves on quintic threefolds (1986)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It would be very interesting to know what the number of&#xA;twisted cubics on a generic quintic is. A more detailed understanding of&#xA;the moduli space of twisted cubics is necessary. In [9], we see that this&#xA;number is divisible by 5, by studying the degeneration of the quintic to a&#xA;union of 5 hyperplanes, along the lines of [8].&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In fact, COGP computed the number of rational curves of degree up to &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;. The computer error story began at the 1991 MSRI conference, where Candelas reported their results which left most mathematicians skeptical. In addition to  unfamiliarity and non-rigorous nature of the new methods, the number &lt;span class=&quot;math-container&quot;&gt;$317,206,375$&lt;/span&gt; did not match the number &lt;span class=&quot;math-container&quot;&gt;$2,682,549,425$&lt;/span&gt; obtained by Norwegian mathematicians Ellingsrud and Strømme using a computer.&lt;/p&gt;&#xA;&lt;p&gt;However, a month later they noticed an error in the program, and in &lt;a href=&quot;https://www.jstor.org/stable/24491856&quot; rel=&quot;noreferrer&quot;&gt;The Number of Twisted Cubic Curves on the General Quintic threefold (1995)&lt;/a&gt; rederived the COGP result for twisted cubics theoretically. This moved the attitudes. Morrison, one of the most vocal critics of COGP at MSRI, became one of the leading lights in mirror symmetry and mathematical string theory. Kontsevich won the Fields medal in 1998 in part for confirming the degree four count &lt;span class=&quot;math-container&quot;&gt;$242,467,530,000$&lt;/span&gt; in 1995, and turning semi-physical heuristics of mirror symmetry into a system of precise formulations that could be rigorously proved.&lt;/p&gt;&#xA;&lt;p&gt;Here is Yau's account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Fortuitously, at about this time—May 1991, to be exact — I had organized a&#xA;conference at the Mathematical Sciences Research Institute (MSRI) at Berkeley&#xA;for mathematicians and physicists to talk about mirror symmetry... Candelas discussed his result for the Schubert problem there, but his number&#xA;disagreed with the number obtained through ostensibly more rigorous techniques&#xA;by the Norwegian mathematicians Geir Ellingsrud and Stein Arild&#xA;Strømme, who had arrived at 2,682,549,425. The algebraic geometers in the&#xA;crowd tended to be arrogant, assuming the physicists must have made the mistake.&#xA;For one thing, explains University of Kaiserslautern mathematician Andreas&#xA;Gathmann, “mathematicians didn’t understand what the physicists were&#xA;doing, because they [the physicists] were using completely different methods —&#xA;methods that didn’t exist in math and didn’t always appear to be justified.”&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Candelas and Greene worried that they might have made an error, but they&#xA;couldn’t see where they had gone wrong. I spoke with both of them at the time,&#xA;especially Greene, wondering whether something might have gone awry during&#xA;the process of integrating over an infinite-dimensional space that then has to&#xA;be reduced to finite dimensions. Choices have to be made in the course of doing&#xA;that, and none of the options are perfect. While that made Candelas and Greene&#xA;somewhat uneasy, we couldn’t find any flaws in their reasoning, which was based&#xA;on physics rather than an outright mathematical proof. Moreover, they remained&#xA;confident about mirror symmetry in general, despite the mathematicians’&#xA;criticism.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This all became clear a month or so later, when Ellingsrud and Strømme&#xA;found an error in their computer program. Upon rectifying it, they got the same&#xA;answer that Candelas et al. had. The Norwegian mathematicians showed great&#xA;integrity in rerunning their program, checking their results, and publicizing&#xA;their mistake. Many people in the same position try to conceal a mistake for as&#xA;long as possible, but Ellingsrud and Strømme did the opposite, promptly informing&#xA;the community of their error and subsequent correction&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-05T11:40:31.433" LastActivityDate="2020-11-05T11:40:31.433" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12405" PostTypeId="1" AcceptedAnswerId="12406" CreationDate="2020-11-05T15:01:53.007" Score="6" ViewCount="503" Body="&lt;p&gt;Richard K. Guy refers to him as &amp;quot;F. Göbel&amp;quot; in his Unsolved Problems of Number Theory but that's as close as I can get.&lt;/p&gt;&#xA;&lt;p&gt;For reference:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathworld.wolfram.com/GoebelsSequence.html&quot; rel=&quot;noreferrer&quot;&gt;https://mathworld.wolfram.com/GoebelsSequence.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2020-11-05T15:56:42.193" Title="Who is the Göbel who is the eponym of Göbel's Sequence?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12406" PostTypeId="2" ParentId="12405" CreationDate="2020-11-05T15:50:54.170" Score="7" Body="&lt;p&gt;The first thing I tried was searching for &amp;quot;Gobel&amp;quot; at the &lt;a href=&quot;https://www.mathgenealogy.org/search.php&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt;, which I figured wouldn't be successful because for something like this it's likely the person doesn't have a math Ph.D. (or any Ph.D.), and it's also something I would assume you tried. However, there was a single &amp;quot;F. Göbel&amp;quot; listed, &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=24121&quot; rel=&quot;noreferrer&quot;&gt;Frits Göbel&lt;/a&gt;, and his thesis title also somewhat suggests that this might be the correct person. (A title such as &amp;quot;&lt;em&gt;Enriched Categories and Quasi-uniform Spaces&lt;/em&gt;&amp;quot; would, for example, have had me thinking otherwise.)&lt;/p&gt;&#xA;&lt;p&gt;Then I tried the google search &lt;a href=&quot;https://www.google.com/search?q=%22G%C3%B6bel%27s+sequence%22+%22Fritz%22&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Göbel's sequence&amp;quot; + &amp;quot;Fritz&amp;quot;&lt;/a&gt; and got exactly one hit, this &lt;a href=&quot;https://oeis.org/A005169/a005169_6.pdf&quot; rel=&quot;noreferrer&quot;&gt;25 September 1986 letter from Richard K. Guy to Neil J. A. Sloan&lt;/a&gt;. The sequence &lt;span class=&quot;math-container&quot;&gt;$1,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$3,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$5,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$10,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$28,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$154,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$3520,\, \ldots$&lt;/span&gt; at the bottom of the first page is identified with the name Fritz Göbel. The google search &lt;a href=&quot;https://www.google.com/search?q=%22Fritz+G%C3%B6bel%22+%22sequence%22&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Fritz Göbel&amp;quot; + &amp;quot;sequence&amp;quot;&lt;/a&gt; appears to confirm that this is the Göbel you want. I don't know whether he has a middle name or his birth year, but I believe he is still alive.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2020-11-05T15:56:42.193" LastActivityDate="2020-11-05T15:56:42.193" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12407" PostTypeId="1" CreationDate="2020-11-06T14:34:51.857" Score="0" ViewCount="168" Body="&lt;p&gt;This is one of those questions which has confused a lot of students like me and I know similar questions have been asked on Physics Stack exchange but I literally want to know what was the reason behind defining it towards the axis of rotation and not any other ? Why did scientists used this convention ?&lt;/p&gt;&#xA;" OwnerUserId="13274" LastEditorUserId="10106" LastEditDate="2020-11-06T15:56:49.120" LastActivityDate="2020-11-20T04:43:15.447" Title="The reason behind defining the direction of angular velocity towards the axis of rotation?" Tags="|physics|terminology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12408" PostTypeId="1" CreationDate="2020-11-06T14:49:31.917" Score="4" ViewCount="129" Body="&lt;p&gt;In Wikipedia, I found that Lagrange did not state Lagrange's Theorem in its general form. He stated&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;If a Polynomial in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; variables has its variables permuted in all &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; ways, the number of different polynomials that are obtained is always a factor of &lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Where can I find the proof he provided to this statement?&lt;/p&gt;&#xA;" OwnerUserId="13275" LastEditorUserId="13275" LastEditDate="2020-11-06T16:13:44.477" LastActivityDate="2020-11-06T16:13:44.477" Title="Lagrange's Theorem as he stated it" Tags="|mathematics|group-theory|lagrange|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12409" PostTypeId="1" CreationDate="2020-11-06T22:33:11.407" Score="5" ViewCount="141" Body="&lt;p&gt;I am writing a text on the Theory of Computation.  I am looking for a photo of the mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Wilhelm_Ackermann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilhelm Ackermann&lt;/a&gt;.  He is well-known in the field, was a student of one of the most famous mathematicians of his time, and developed the important Ackermann function.  So he is reasonably prominent in the history of mathematics in the early to mid 1900's, and I'd expect there to be a number of high quality pictures.&lt;/p&gt;&#xA;&lt;p&gt;But there seems to be only one picture available (it is in the Wikipedia article linked-to earlier).  It is terrible quality-- it looks like a photocopied high school yearbook, although I don't actually know where it comes from-- and I don't think even his mom or wife could like it.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know of another?  My text is Freely available, so I need it to be free of license restrictions, but after all this time I can hope that licenses are not a problem anyway.&lt;/p&gt;&#xA;" OwnerUserId="13276" LastEditorUserId="13276" LastEditDate="2020-11-10T14:46:35.903" LastActivityDate="2020-11-10T14:46:35.903" Title="Photo of Wilhelm Ackermann" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12410" PostTypeId="2" ParentId="12407" CreationDate="2020-11-07T19:40:32.680" Score="4" Body="&lt;p&gt;Not really discussing history, but since you asked ...&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jctcP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jctcP.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Bold is vector, normal is magnitude)&lt;br&gt;&#xA;The position vector is &lt;span class=&quot;math-container&quot;&gt;$\mathbf r$&lt;/span&gt;, the velocity vector is &lt;span class=&quot;math-container&quot;&gt;$\mathbf  v$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf \omega$&lt;/span&gt; is the angular velocity vector.&lt;/p&gt;&#xA;&lt;p&gt;We know that the angular velocity and the velocity are related through &lt;span class=&quot;math-container&quot;&gt;$v = R\omega$&lt;/span&gt;, and, if you look carefully, you will see that &lt;span class=&quot;math-container&quot;&gt;$R = r\sin\alpha$&lt;/span&gt;, hence &lt;span class=&quot;math-container&quot;&gt;$v = r\omega\sin\alpha$&lt;/span&gt;. Looks like the magnitude of a cross product!&lt;/p&gt;&#xA;&lt;p&gt;This might very well give one the idea of defining an angular velocity vector such that &lt;span class=&quot;math-container&quot;&gt;$\mathbf v=\mathbf\omega\times\mathbf r$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbf r\times\mathbf\omega$&lt;/span&gt;. Since the right hand rule is a thing, it seems best to define &lt;span class=&quot;math-container&quot;&gt;$\mathbf v=\mathbf\omega\times\mathbf r$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\mathbf\omega$&lt;/span&gt; would be in the direction your thumb points at when you close your hands in the direction of rotation.&lt;/p&gt;&#xA;&lt;p&gt;Why is this useful? We can use any theorem that we have found for the cross product. :)&lt;/p&gt;&#xA;" OwnerUserId="9968" LastActivityDate="2020-11-07T19:40:32.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12411" PostTypeId="2" ParentId="12402" CreationDate="2020-11-08T07:42:12.933" Score="5" Body="&lt;p&gt;Lagrange is one example. For instance, W. W. Rouse Ball wrote in his book &amp;quot;A Short Account of the History of Mathematics&amp;quot; that Lagrange:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;... always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another example is Shiing-Shen Chern.  From the tribute article for Chern in the &lt;a href=&quot;https://www.ams.org/notices/201109/rtx110901226p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;October 2011 AMS Notices&lt;/a&gt;, Robert Greene recalled that Chern&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...  would come in every day and fill the blackboards with the&#xA;long calculations that are needed in treating the subject in terms&#xA;of differential forms. He never brought any notes, never paused&#xA;to pursue any elusive recollection, and never made mistakes. The&#xA;whole subject unfolded as smoothly and gracefully as if he&#xA;were reading from a perfectly written book.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finally, one of the more courageous students asked him after&#xA;class how this was possible. He replied quietly that in fact he had&#xA;developed the whole subject to begin with without writing anything down. He said it was as if he had a blackboard in his mind on which things could&#xA;be written and stayed forever.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2020-11-09T00:32:06.177" LastActivityDate="2020-11-09T00:32:06.177" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12413" PostTypeId="2" ParentId="5011" CreationDate="2020-11-09T06:53:54.797" Score="-2" Body="&lt;p&gt;I will admit to the breaking of a rule in this case and offering an opinion because we are referring to a controversy that set England and Continental Europe against each other, only matched in personal ferocity by the argument between Jacob and Johann Bernoulli about which of them first successfully applied the Calculus (which they learned through correspondence with Leibniz) to derive the function that accurately described the catenary curve (how suspension bridges and necklaces hang). Opinion is really all we have.&lt;/p&gt;&#xA;&lt;p&gt;The Two crucial points would seem to be--in my view: 1) His introduction of differential notation. 2) His principle of the Identity of Indiscernibles, which implies a (finite) limit of discernibility. The two together indicate that long before Weierstrass (1872) had to point it out, commencing the Great Crisis of Mathematics and the separation of intuitionism from realism, Liebnitz understood that differential calculus was about an ultimately fine-grained finite algebra--not actually the infinitely small. Assuming this superior insight, he had a truer grasp of what the Calculus was about than Newton.&lt;/p&gt;&#xA;&lt;p&gt;[The universe is quantized after all, and not infinitely divisible. Thus, differential and integral calculus are not intuitionist infinite step constructs but approximations to the finite step constructs of science and engineering, and actually, work.]&lt;/p&gt;&#xA;" OwnerUserId="13283" LastEditorUserId="13283" LastEditDate="2020-11-09T06:59:15.503" LastActivityDate="2020-11-09T06:59:15.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12414" PostTypeId="1" CreationDate="2020-11-09T11:48:54.310" Score="-1" ViewCount="140" Body="&lt;p&gt;It is well known that the theory of types, first introduced by Bertrand Russell in 1903 and developed with Whitehead in their &lt;em&gt;Principia Mathematica&lt;/em&gt; (1910), was a way to deal with paradoxes in set theory. Type theory was later developed by Church, Per Martin-Löf and many others.&lt;/p&gt;&#xA;&lt;p&gt;In type theory, every mathematical object belongs to a type. One could link this with the fact that in physics every quantity has a unit. Knowing that units were introduced in physics certainly well before types found their way explicitly in logic, did units in physics play any role in the introduction of types in logic and the foundations of mathematics?&lt;/p&gt;&#xA;" OwnerUserId="13109" LastEditorUserId="4251" LastEditDate="2021-09-14T17:32:06.957" LastActivityDate="2021-09-14T17:32:06.957" Title="Do units in physics have any link to the origins of type theory in logic?" Tags="|physics|mathematical-logic|logic|type-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12415" PostTypeId="1" CreationDate="2020-11-09T16:25:18.687" Score="1" ViewCount="61" Body="&lt;p&gt;I'd like to have a broad understanding of how various societies in the world developed pre-scientific understanding.  There are a lot of resources for ancient Greece.  There are even a few for what immediately preceded the origins of an early scientific understanding around 300 BC.  Trying to go a little further back and broader, there are some resources on ancient Egyptian and Babylonian science and mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I know that what I'm searching for may not have any records, written or otherwise.  But I at least wanted to ask if anyone knows of accounts from 1) other societies, or 2) any information on earlier peoples and their knowledge.  More specific questions include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Which tools were used (by which people, when, and where)?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What did people know about astronomy, what medicinal practices did people have, did they have some understanding of &amp;quot;chemistry&amp;quot; in the form of metallurgy for instance?  How did they discover these very early discoveries?  How would someone stumble onto metallurgy?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How did people explain the world through religion or other concepts?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How was society structured so that knowledge could be gathered and processed and transmitted?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In trying to reach farther back in time, I'd be interested to know about say the Sumerians, the Nufians, hunter-gatherers, perhaps even anything we might know about Neanderthals.  In trying to go geographically broader, I'd be interested to know about what the earliest people were up to in the lands of India, China, Europe, sub-Saharan Africa, Meso- and South-America.&lt;/p&gt;&#xA;&lt;p&gt;In general I'd like to have as much understanding as is possible about what context gave rise to ancient science.  I fully recognize how difficult the task is given the sparse (and sometimes utterly non-existent) writing available from the time.  But if anyone has any resources I'd appreciate it!&lt;/p&gt;&#xA;" OwnerUserId="2732" LastActivityDate="2020-11-09T16:25:18.687" Title="The earliest origins of science and math" Tags="|reference-request|ancient|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12416" PostTypeId="1" CreationDate="2020-11-09T22:22:37.107" Score="1" ViewCount="675" Body="&lt;p&gt;I am asking how did Galileo Galilei derive the equation for height as a function of time, which we now write as &lt;span class=&quot;math-container&quot;&gt;$y=\frac{1}{2}gt^2$&lt;/span&gt;. We now know that the equation is the solution of Newton's second law when the force is a constant &lt;span class=&quot;math-container&quot;&gt;$mg$&lt;/span&gt;, which can be found by integration. However, Galileo lived between 1564 and 1642, he died before the birth of Newton and Leibniz, which are credited for the discovery of calculus, so he could not have used it. I have heard that he used the so-called Merton rule for deriving it, but how exactly did he used it? Can someone give a modern version of what Galileo did?&lt;/p&gt;&#xA;" OwnerUserId="10492" LastEditorUserId="10106" LastEditDate="2020-11-10T01:18:51.100" LastActivityDate="2020-11-13T02:10:39.383" Title="How did Galileo derive the law for a falling body without calculus?" Tags="|physics|galileo|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12417" PostTypeId="2" ParentId="12416" CreationDate="2020-11-10T00:18:53.417" Score="1" Body="&lt;p&gt;Galileo did not derive this law. He discovered it from his experiments with inclined planes. And the law for falling body would make possible to discover Newton's second law, not other way around.&lt;/p&gt;&#xA;&lt;p&gt;But the fact that constant acceleration (i.e. gaining equal amounts of velocity in equal amounts of time) meant that path is proportional to time squared was known even to Babylonian. You don't need full calculus for that. You can see it by summing arithmetic progression. Babylonian approximate movement of Son and planets by a motion with piecewise linear speed. E.g. they assumed that a Sun moves &lt;span class=&quot;math-container&quot;&gt;$1^\circ$&lt;/span&gt; per day for a half year and a little bit less for another half, so it will go &lt;span class=&quot;math-container&quot;&gt;$360^\circ$&lt;/span&gt; for 365 days. And they knew formula for calculating position of the Sun in a given day.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-11-10T00:18:53.417" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12418" PostTypeId="1" AcceptedAnswerId="12421" CreationDate="2020-11-10T06:01:27.073" Score="2" ViewCount="366" Body="&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: I didn't write the word &amp;quot;ancient&amp;quot; in the title because I want to see the notation from 1400 A.D. to 1700 A.D.&lt;br /&gt;&#xA;Mathematical notation has changed very much from the past millennium, and new notations are made for new concepts. For example, there was a time when the notations for representing numbers was different in different regions. I saw somewhere a piece of a page from one of Euler's books or articles:&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ABunR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ABunR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;This is very close to modern notation, but not same as it (for example, the root symbols are larger). After seeing this page, a question cam in my mind:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are some major differences in today's notation and notation of 1400 A.D. to 1700 A.D. era?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These differences are to be mentioned:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Differences in symbols representing numbers, for example, somewhere in an old article (probably of Euler's) the number &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; was written as &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt; (I thought that it was roman, but 2 was written normally)&lt;/li&gt;&#xA;&lt;li&gt;Major differences in &amp;quot;minor notations&amp;quot;. For example, while reading Ramanujan's handwritten manuscripts, I found that He wrote &lt;span class=&quot;math-container&quot;&gt;$\&amp;amp;c$&lt;/span&gt; instead of the normal &lt;span class=&quot;math-container&quot;&gt;$\dots$&lt;/span&gt; in infinite sum. He would have written:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1^2}+\frac{1}{2^2}+\&amp;amp;c$$&lt;/span&gt;&#xA;instead of:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1^2}+\frac{1}{2^2}+\dots$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;Differences in the &amp;quot;texture&amp;quot; of the symbols. For example, in the image I have given above, it can be noticed that the integral symbols and the root symbols are a little bit different.&lt;/li&gt;&#xA;&lt;li&gt;Differences in terminology. Many examples are in the first answer, and I think there are very few that he didn't mention. If there are any more, please also mention them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Like most big list questions, please mention only one difference per answer. If I missed any other difference in symbols case, please mention in the comments or edit the question.&lt;/p&gt;&#xA;" OwnerUserId="13286" LastEditorUserId="13286" LastEditDate="2020-11-11T02:27:53.370" LastActivityDate="2020-11-11T02:27:53.370" Title="Differences between modern and old mathematical notations" Tags="|mathematics|notation|big-list|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12419" PostTypeId="1" CreationDate="2020-11-10T11:55:34.577" Score="2" ViewCount="129" Body="&lt;p&gt;Feynman diagrams representing the interactions of subatomic particles were introduced by Richard Feynman in 1948. The &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.76.769&quot; rel=&quot;nofollow noreferrer&quot;&gt;first published Feynman diagram&lt;/a&gt; appeared in Physical Review in 1949.&lt;br /&gt;&#xA;Did Feynman diagrams play a key role in the creation of the electroweak theory developed between 1949 and 1967 in the work of Yang and Mills, Glashow, Higgs, Weinberg, Salem and their collaborators?&lt;br /&gt;&#xA;If so, please explain how (for instance these diagrams might have rendered some difficult calculations more feasible).&lt;/p&gt;&#xA;" OwnerUserId="13109" LastActivityDate="2020-11-13T02:57:25.230" Title="Were Feynman diagrams important for the creation of the electroweak theory?" Tags="|physics|particle-physics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12420" PostTypeId="1" CreationDate="2020-11-10T14:54:07.777" Score="1" ViewCount="83" Body="&lt;p&gt;I have often heared that Newton compared the centripetal acceleration of an (imaginary) cannonball revolving around the earth just above the surface with the centripetal acceleration of the moon and discovered that a 60 times larger distance results in a &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3600} = \frac{1}{60^2}$&lt;/span&gt; fraction of the centripetal acceleration. From this he concluded the hypothesis of the inverse square law for gravity.&lt;/p&gt;&#xA;&lt;p&gt;What is true about this story? Do you have references to Newtons writings where he describes this idea?&lt;/p&gt;&#xA;&lt;p&gt;Did Newton also have other &amp;quot;derivations&amp;quot; to the inverse square law? If so, which idea was most important from his point of view?&lt;/p&gt;&#xA;&lt;p&gt;Is it also true that he abandoned the idea at first time because he didn't have an accurate value for the earth diameter and then got back to it over 15 years later when a better value was available?&lt;/p&gt;&#xA;" OwnerUserId="5670" LastActivityDate="2020-11-10T14:54:07.777" Title="Is it true that Newton used the gedankenexperiment of a cannonball revolving around the earth to find the inverse square law of gravity?" Tags="|physics|newton|gravity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12421" PostTypeId="2" ParentId="12418" CreationDate="2020-11-10T16:04:12.680" Score="7" Body="&lt;h1&gt;I. Notes on notation and English terms in 1800s literature (primarily related to series)&lt;/h1&gt;&#xA;&lt;p&gt;The following is from the introductory comments for something involving conditionally convergent series I was working on several years ago.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;The logarithm function&lt;/strong&gt; is often denoted by a lower case L, sometimes italicized (e.g. &lt;a href=&quot;https://archive.org/details/traitlmentaired01catagoog&quot; rel=&quot;noreferrer&quot;&gt;Catalan's book&lt;/a&gt;) and sometimes not italicized (e.g. &lt;a href=&quot;https://books.google.com/books?id=hQfj8Rx_sc4C&quot; rel=&quot;noreferrer&quot;&gt;Laurent's book&lt;/a&gt;). Incidentally, this leads to many instances of &lt;em&gt;ln&lt;/em&gt; appearing, but these denote the logarithm of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; rather than the often now used logarithm function &lt;span class=&quot;math-container&quot;&gt;${\ln}.$&lt;/span&gt; Iterated logarithms were denoted either by consecutive L's or the use of superscripts (e.g. &lt;span class=&quot;math-container&quot;&gt;$\ln \ln \ln n$&lt;/span&gt; might be denoted by &lt;em&gt;llln&lt;/em&gt; or &lt;em&gt;lll n&lt;/em&gt; or &lt;span class=&quot;math-container&quot;&gt;${l^3}n,$&lt;/span&gt; or by similar expressions in which the lower case L is NOT italicized).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$n!$&lt;/span&gt; &lt;strong&gt;(factorial)&lt;/strong&gt; was often denoted by placing &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; within a &amp;quot;partial box&amp;quot; formed by using only the left and bottom sides of a rectangular box. Anytime you are unsure how factorials are denoted in a book, simply look in places where power series expansions of exponential and trigonometric functions appear. However, most of the time a &amp;quot;factorial notation&amp;quot; was not used, and instead the first few terms (and the general term, if not obvious) were written.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Multiplication&lt;/strong&gt; was often denoted by a period (easier to typeset than our current centered dot notation), and thus the expression &lt;span class=&quot;math-container&quot;&gt;$1.2.3.4$&lt;/span&gt; represents &lt;span class=&quot;math-container&quot;&gt;$4! = 24.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sometimes an &lt;strong&gt;overline bar&lt;/strong&gt; (called a &lt;em&gt;vinculum&lt;/em&gt;) is used instead of parentheses for grouping purposes.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Miscellaneous terms:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;em&gt;vulgar fraction&lt;/em&gt; = a quotient of integers written in vertical fraction format&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;impossible number&lt;/em&gt; = non-real complex number&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;possible number&lt;/em&gt; = real number&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;finite value/quantity&lt;/em&gt; = (especially before late 1800s) nonzero value/quantity&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;absolute term&lt;/em&gt; =  constant term (e.g. of a polynomial)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;numerically greater&lt;/em&gt; =  absolute value is greater&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;algebraically greater&lt;/em&gt; = greater as signed numbers&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;contrary signs&lt;/em&gt; = opposite signs&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;vanishing fraction&lt;/em&gt; = &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; indeterminate form&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;integral rational function&lt;/em&gt; = polynomial&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;integral function&lt;/em&gt; = power series&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;development&lt;/em&gt; = series expansion&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;line&lt;/em&gt; = curve (e.g. not necessarily a straight line)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;right line&lt;/em&gt; or &lt;em&gt;straight line&lt;/em&gt; = line&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;curve of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;em&gt;dimensions&lt;/em&gt; = curve of algebraic degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;rectilinear asymptote&lt;/em&gt; =  linear/oblique asymptote&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;invariable quantity&lt;/em&gt; = constant quantity&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;squared paper&lt;/em&gt; = graph paper&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;the term be wanting&lt;/em&gt; = the term is absent&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;equation is rigorous&lt;/em&gt; = exact equality holds&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;steadily towards&lt;/em&gt; = monotonically towards&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;rigid proof&lt;/em&gt; = rigorous proof&lt;/p&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;&lt;strong&gt;Colloquial journal names:&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;em&gt;Annales de Gergonne&lt;/em&gt; = Annales de Mathématiques Pures et Appliquées&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Crelle's Journal&lt;/em&gt; and &lt;em&gt;Borchardt's journal&lt;/em&gt; = Journal für die reine und angewandte Mathematik&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Darboux's Bulletin&lt;/em&gt; = Bulletin des Sciences Mathématiques et Astronomiques [“et Astronomiques” was later dropped]&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Grunert's Archiv&lt;/em&gt; = Archiv der Mathematik und Physik&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Hoffmann's Journal&lt;/em&gt; = Zeitschrift fur Mathematischen und Naturwissenschaftlichen Unterricht&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Liouville's Journal&lt;/em&gt; = Journal de Mathématiques Pures et Appliquées&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Terquem's journal&lt;/em&gt; = Nouvelles Annales de Mathématiques&lt;/p&gt;&#xA;&lt;h1&gt;II. Glossary of early to mid 20th century terms in some nowhere differentiable continuous functions literature:&lt;/h1&gt;&#xA;&lt;p&gt;The following list is from near the beginning of &lt;a href=&quot;https://mathoverflow.net/a/162605/15780&quot;&gt;this mathoverflow answer&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;enumerable&lt;/em&gt; = countable&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;unenumerable&lt;/em&gt; = uncountable&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;progressive derivative&lt;/em&gt; = right derivative&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;regressive derivative&lt;/em&gt; = left derivative&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;non-dense&lt;/em&gt; = nowhere dense&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;everywhere dense&lt;/em&gt; = dense in the smallest interval containing the set (sometimes “dense”, without being prefaced by “everywhere”, means in context “everywhere dense”)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;incrementary ratio&lt;/em&gt; = the ordinary difference quotient that a derivative is the limit of&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;line of invariability&lt;/em&gt; (of a function) = an interval on which the function is constant&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;the set exists&lt;/em&gt; = the set is not empty.&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2020-11-10T16:10:57.417" LastActivityDate="2020-11-10T16:10:57.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12422" PostTypeId="1" CreationDate="2020-11-12T14:52:18.963" Score="6" ViewCount="261" Body="&lt;p&gt;Musing about the historical evolution of the notation for the gravitational constant (&lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\kappa$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\kappa^2$&lt;/span&gt;), I found myself digging for the first time in my life into Newton's &lt;em&gt;Principia&lt;/em&gt;, looking for the data he could have used (had he thought algebrically) to estimate his constant (see &lt;a href=&quot;https://physics.stackexchange.com/questions/331459/did-newton-estimate-the-gravitational-constant-g&quot;&gt;this question&lt;/a&gt;). In the process, I noticed a funny calculation error in a simple division in Book III, Proposition XIX, Problem III (e.g. p.406 of Motte's translation, but I could find it in every version of the book I checked). Here it is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the circumference of the earth is 123,249,600 feet and its semidiameter 19615800 Paris feet, upon the supposition that the earth is of a spherical figure. [...] A body in every sidereal day of 23h 56 4s uniformly revolving in a circle at the distance of 19615800 feet from the centre, in one second of time describes an arc of 1433.46 feet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dividing 123,249,600 feet by the 86,164 seconds of the sideral day, he should have found 1430.407! The error was acknowledged e.g. by &lt;a href=&quot;https://books.google.fr/books?id=t77juT326MYC&amp;amp;pg=PA172&amp;amp;lpg=PA172&amp;amp;dq=newton%201433.46&amp;amp;source=bl&amp;amp;ots=qeWUS-OZIS&amp;amp;sig=ACfU3U0QoD1k19bm4sD5Y5ndk1C2dHSoUg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjs8oXBiv3sAhVCzhoKHY9pBWYQ6AEwAHoECAIQAg#v=onepage&amp;amp;q=newton%201433.46&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;MacDougal in his undergrads lecture in 2012 p.172&lt;/a&gt; where he noted in a footnote that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Using Newton’s numbers for circumference and time, our calculations show the velocity value to be 1,430.4 Paris feet per second, about 3 feet per second less than Newton’s value. &lt;strong&gt;It is unknown why there is this small discrepancy against Newton’s value.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can also find it in &lt;a href=&quot;https://books.google.fr/books?id=pDUfqo-q0fkC&amp;amp;pg=PA254&amp;amp;lpg=PA254&amp;amp;dq=newton+1433.46&amp;amp;source=bl&amp;amp;ots=uQI-NgmBPc&amp;amp;sig=ACfU3U1Dm4H9qptI9u1eeW9aUFC2Z2en1g&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjKuen6zvzsAhUM_RQKHVzZBIcQ6AEwBHoECAMQAg#v=onepage&amp;amp;q=newton%201433.46&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Harper's book (2011) p.254 footnote 74&lt;/a&gt;. Harper made no comments about Newton's value, and had the wrong number of seconds in a day (86,160) so his result doesn't count, but he made me wonder if this calculation is mathematically cursed. At least in that case the mistake is obvious. But, and that is my question here, &lt;strong&gt;what happened with Isaac's division?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I find doubtful the possibility that 1433.46 instead of 1430.40(7) is a typo (2 non-adjacent digits are involved). I tried to play with the inputs, but again you need to have more that one digit wrong to get Newton's answer. (I'm making the implicit assumption that Isaac did at most one silly mistake.) So there may be an obvious calculation error somewhere, but I can't find anything plausible, so any hint is welcome.&lt;/p&gt;&#xA;&lt;p&gt;I know it is not really an interesting physical question (neither mathematical, nor historical) but I am kind of haunted by it now, and I needed to do something to get rid of it.&lt;/p&gt;&#xA;" OwnerUserId="13299" LastActivityDate="2020-12-21T08:18:07.263" Title="Can anyone find Newton's calculation error in Principia, Book III, Proposition XIX?" Tags="|newton|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12423" PostTypeId="2" ParentId="12414" CreationDate="2020-11-13T02:03:30.263" Score="1" Body="&lt;p&gt;And even before units were introduced in physics, people knew, for example, not to add oranges to say, apples. At least this is what I was taught in primary school, and I imagine that this goes back a long way.&lt;/p&gt;&#xA;&lt;p&gt;Unless some evidence turns up showing that Russell or Whitehead were directly influenced by the notion of dimension in physics, I'd say this is just as likely to be an independent rediscovery - as so often happens in physics and mathematics, one of the most notorious example being the independent discovery of calculus by Newton and Liebniz.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In type theory every mathematical object belongs to a type&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Proofs, as mathematical objects, also denote types. However, this doesn't obtain in physics. In physics, we don't tend to have proofs, rather we have arguments. This is something that the more mathematically orientated physicists have a hard time understanding...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-13T02:03:30.263" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12424" PostTypeId="2" ParentId="12416" CreationDate="2020-11-13T02:10:39.383" Score="1" Body="&lt;p&gt;You don't need calculus to show the relationship that you are pointing out. One only needs to plot distance wrt time.&lt;/p&gt;&#xA;&lt;p&gt;Of course one needs instruments that can accurately measure time which is perhaps where Galileo was helped by his discovery that the pendulum can act as an excellent clock.&lt;/p&gt;&#xA;&lt;p&gt;Calculus is required to demonstrate that relationship follows from Newtons laws of motion, most of which, I should wasn't discovered by Newton, but his predecessors, as he freely acknowledged.&lt;/p&gt;&#xA;&lt;p&gt;I should add that it's not neccessary that calculus is 'required' for these demonstrations as Newton himself did without it in his &lt;em&gt;Principia&lt;/em&gt;. In this, Newton was also preceded by Archimedes who also presented geometrical proofs of results which were derived by him through his 'mechanical method', as the geometry was the lingua fraca of mathematicians then, as it is now.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-13T02:10:39.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12425" PostTypeId="2" ParentId="12419" CreationDate="2020-11-13T02:30:45.537" Score="1" Body="&lt;p&gt;Julian Schwinger said of Feynman diagrams:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;that they brought QFT to the masses.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, he along with Sin-Itiro Tomonaga  independently developed an alternative formalism that is non-perturbative from the beginning, in contrast to Feynmans perturbative approach.&lt;/p&gt;&#xA;&lt;p&gt;Feynmans approach, being perturbative, misses a great deal. One does not need to know QFT to see this.&lt;/p&gt;&#xA;&lt;p&gt;After all, the functions that are perturbative on the real line are exactly the analytic functions. But these are subsumed by the smooth functions. That is there are many more smooth functions than there are analytic ones. And this is before one begins to examine the possibility that non-smooth functions also play a role - as they do with say, Diracs delta function and their derivatives.&lt;/p&gt;&#xA;&lt;p&gt;More physically speaking, instantons are twists in vector bundles, and are global phenomena. And being global they are missed by physics which only probe locally, as Feynman diagrams do.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-13T02:57:25.230" LastActivityDate="2020-11-13T02:57:25.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12426" PostTypeId="2" ParentId="12407" CreationDate="2020-11-13T02:41:11.527" Score="0" Body="&lt;p&gt;Angular velocity is not a vector and not is it a pseudo-vector. These are in fact abbreviations for the correct notion.&lt;/p&gt;&#xA;&lt;p&gt;A rotation in 3d has an axis of rotation. However, when we look at rotations in higher dimensions, say for example in 4d or 5d, the notion of an axis of rotation does not generalise (we can find one in odd dimensions but not in even ones).&lt;/p&gt;&#xA;&lt;p&gt;However, rotations in 3D occurs in an invariant plane and its this plane that generalises to rotations in higher dimensions.&lt;/p&gt;&#xA;&lt;p&gt;Such planes are indicated by bivectors. This is what it sounds like - two chosen vectors which are not linearly independent and so they span a plane.&lt;/p&gt;&#xA;&lt;p&gt;A bivector in 3d is equivalent to a pseudo-vector, which is where this latter notion comes in, in discussions of angular velocity or momentum and the like, for example - torque.&lt;/p&gt;&#xA;&lt;p&gt;I'd also add, given some of the nonsense said both for and against higher dimensions, that here is an example of thinking in higher dimensions which is helpful in thinking about our own spatial 3d. But nonetheless, it says nothing about the &lt;em&gt;actual&lt;/em&gt; existence of higher dimensions.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-20T04:43:15.447" LastActivityDate="2020-11-20T04:43:15.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12428" PostTypeId="2" ParentId="12396" CreationDate="2020-11-13T04:16:20.900" Score="-1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;The origins of differential homological algebra&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to Weibel, who has written one of the standard texts on homological algebra:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Homological algebra had its origins in the 19th century, via the work of Riemann (1857) and Betti (1871) on 'homology numbers' and the rigorous development of homology numbers by Poincare in 1895. An observation of Emmy Noether in 1925 shifted attention to homology groups of a space ... [later] Cartan and Eilenbergs book crystallised and completely redirected the field ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd also add, which is often not noted, that the term 'differential' for the morphisms between the chain modules actually harks back to calculus. After all, the differential &lt;em&gt;d&lt;/em&gt;, satisfies &lt;em&gt;d^2=0&lt;/em&gt;, which is the signature relation for the fluxions of Newton and the infinitesimals of Liebniz.&lt;/p&gt;&#xA;&lt;p&gt;It's also worth noting, in line with this connection, that the differential calculus on a manifold (and which physicists would recognise as vector analysis) assembles itself into the de Rham complex with the de Rham differential.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-13T04:23:45.680" LastActivityDate="2020-11-13T04:23:45.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12429" PostTypeId="2" ParentId="12345" CreationDate="2020-11-13T04:35:22.280" Score="1" Body="&lt;p&gt;According to Wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ada is a programming language designed a team led by Jean Ichbiah at Honeywell under contract to the US Department of Defense from 1977-1983 to supersede over 450 programming languages then used by the DoD then.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Given the name, we can certainly see that people were inspired by Ada's example. In fact, the Military Standard reference manual was approved on 10th Dec, 1980 - which is Ada's birthday - or to give her full title: Augusta Ada King, Countess of Lovelace, nee Byron.&lt;/p&gt;&#xA;&lt;p&gt;(Although the Wikipedia article rather sneers at it for being a bulky and cumbersome language, at the time it was an innovative language and compact given just how many languages it was designed to supersede. I might add, one might want to take a look under the hood of Windows, or Visual Basic to see a modern bulky language. They also sneer at it for being designed by committee. Of course, if they used their eyes and heads they ought to notice that committees are rife everywhere. In modern commercial parlance they are not called committee's - they are called teams).&lt;/p&gt;&#xA;&lt;p&gt;It's worth noting, given its military origins, that it was adopted as an &lt;em&gt;open&lt;/em&gt; standard by ANSI in 1983 - not much later after the design and implementation work on the language had finished. Moreover, given its design objectives it was successful as the number of ad-hoc languages dropped from 450 to just 37 by the late 90s.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-13T04:56:35.997" LastActivityDate="2020-11-13T04:56:35.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12430" PostTypeId="2" ParentId="12339" CreationDate="2020-11-13T05:04:37.640" Score="-1" Body="&lt;p&gt;It's not a book of physics per se, but a book on metaphysics, that is &lt;em&gt;meta&lt;/em&gt;-&lt;em&gt;physics&lt;/em&gt;, where the term means &lt;em&gt;on&lt;/em&gt; or &lt;em&gt;about&lt;/em&gt; physics. It's a general disquisition on the most general notions that physics contemplates: space, time and matter.&lt;/p&gt;&#xA;&lt;p&gt;In fact, Aristotle had a similar division in his book on metaphysics. Having looked at both, I would say that Aristotle is the easier read. But then again, Weyl was not noted for his literary style, whereas Aristotle was: how one writes is important if one is to be understood by both the few and the many.&lt;/p&gt;&#xA;&lt;p&gt;If you want to read a more upto date work on similar material and with a similar slant, try &lt;em&gt;Space, Time &amp;amp; Matter&lt;/em&gt;, ed. Majid with contributions from Polkinghorne and Connes and others.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;@slaus: Try reading my answer again to understand what I'm saying ...&lt;/p&gt;&#xA;&lt;p&gt;@slaus: It's strange that you say that Weyl's book mentions &lt;em&gt;gauge&lt;/em&gt;, since, although Weyl was in part responsible for it; it's &lt;strong&gt;not&lt;/strong&gt; mentioned in the book - which perhaps is not entirely surprising since it took around 50 years for the notion to be understood fully and Weyls book was first published in 1921.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-24T12:30:04.067" LastActivityDate="2020-11-24T12:30:04.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12431" PostTypeId="2" ParentId="12323" CreationDate="2020-11-13T05:12:28.813" Score="1" Body="&lt;p&gt;The Nagaokan model was suggested by Nagoaka and who called it the Saturnian model. He proposed it in 1904, contra Thomsons 'plum pudding' model. It might have been more accurately called the planetary model.&lt;/p&gt;&#xA;&lt;p&gt;In fact, Rutherford &lt;em&gt;credited&lt;/em&gt; Nagaoka with the model in his paper of 1911 where he elaborated on his experimental work that showed Nagaoka was &lt;em&gt;justified&lt;/em&gt; in his theoretical description of the atomic structure, as far as it was known then.&lt;/p&gt;&#xA;&lt;p&gt;Neither Nagoaka nor Rutherford called the model after themselves. It was upto their contemporaries and later generations to assign credit where credit is due.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-15T01:34:56.310" LastActivityDate="2020-11-15T01:34:56.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12432" PostTypeId="2" ParentId="12307" CreationDate="2020-11-13T05:28:31.307" Score="1" Body="&lt;p&gt;Witten gave a spinorial proof of the positive energy theorem in GR. This was originally conjectured by Arnowitt, Deser and Misner in the early 60s. Special cases were then shown by a great many people with the general theorem finally established by Schoen and Yau.&lt;/p&gt;&#xA;&lt;p&gt;Witten also gave a super-symmetric physics proof of the Atiyah-Singer index theorem. This had already been established by Atiyah &amp;amp; Singer.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, so far, all experiments have shown that supersymmetry is not an option taken by the real universe as opposed to the physically speculative universes dreamt up by physically orientated mathematicians. After all, Witten win a Fields prize for mathematics and not a Nobel prize for physics ...&lt;/p&gt;&#xA;&lt;p&gt;As for string theory - well that was mostly Schwartz and Green's work who showed that various anomalies cancelled as so string theory was a viable project. Witten hadn't bothered to work on string theory until then.&lt;/p&gt;&#xA;&lt;p&gt;Rather like Picasso, who looked over other people's work, deciding what he could 'steal' and work on himself. After all, Picasso did state:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;good artists borrow, great artists steal ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As you might gather from this description, I tend to think of Wittens work as being over-hyped as far as its significance to actual living and breathing physics is concerned.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-13T05:28:31.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12433" PostTypeId="2" ParentId="12306" CreationDate="2020-11-13T05:37:42.527" Score="-2" Body="&lt;p&gt;Plato had a simple rule: learn geometry. This of course is well before the 1800s.&lt;/p&gt;&#xA;&lt;p&gt;This is tempered by the time of al-Jabr who taught us to learn algebra.&lt;/p&gt;&#xA;&lt;p&gt;Another heuristic is one by Newton who said 'he stood on the shoulders of giants'. This implies that one ought to study the works of the great masters of the subject thoroughly.&lt;/p&gt;&#xA;&lt;p&gt;Another useful heuristic is by Solomon Lefschetz. Unfortunately this is after your cut off date, being in the early to mid 20C.&lt;/p&gt;&#xA;&lt;p&gt;He had one simple rule: he said he wasn't interested in prettifying already established results, what he was interested was in original work that showed something new.&lt;/p&gt;&#xA;&lt;p&gt;This is what he told the graduate students who were under his supervision.&lt;/p&gt;&#xA;&lt;p&gt;Another heuristic, again unfortunately well after your cut-off date is by Atiyah who reminded students that there were a lot of gems in the work of older mathematicians and which had been forgotten. This, however, ties in with what Newton thought.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-28T10:37:39.087" LastActivityDate="2020-11-28T10:37:39.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12435" PostTypeId="1" CreationDate="2020-11-13T16:04:45.187" Score="1" ViewCount="48" Body="&lt;p&gt;Lowenheim (1915) is credited with the first proof of the decidability of semantic validity over the monadic fragment of first-order predicate logic. However, I find no reference in his proof to an upper bound on the size of the finite model whose construction is guaranteed if the formula is satisfiable. This is surprising since the bound is not hard to guess - it is equal to &lt;span class=&quot;math-container&quot;&gt;$2^k \cdot r$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the number of distinct (monadic) predicate letters and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; the number of individual variables. Does anyone know who the first person to discover this bound was?&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;p&gt;LÖWENHEIM, LEOPOLD&lt;/p&gt;&#xA;&lt;p&gt;[1915] On Possibilities in the Calculus of Relatives, in van Heijenoort (ed.) [1967], pp. 228–251.&lt;/p&gt;&#xA;&lt;p&gt;VAN HEIJENOORT, JEAN (ED.)&lt;/p&gt;&#xA;&lt;p&gt;[1967] From Frege to Gödel; a source book in mathematical logic, 1879-1931. Cambridge, Harvard University Press.&lt;/p&gt;&#xA;" OwnerUserId="10497" LastEditorUserId="26" LastEditDate="2021-02-20T15:05:55.590" LastActivityDate="2021-02-20T15:05:55.590" Title="Who first estimated the minimal size of a model for a satisfiable formula of monadic first-order logic?" Tags="|mathematical-logic|logic|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12437" PostTypeId="1" CreationDate="2020-11-14T13:31:16.740" Score="0" ViewCount="531" Body="&lt;p&gt;It's one of the first things learned in Physics.&lt;/p&gt;&#xA;&lt;p&gt;However, what's the history behind deriving the SUVAT equations?&lt;/p&gt;&#xA;&lt;p&gt;And, there were other equivalent equations used before SUVAT came into the curriculum?&lt;/p&gt;&#xA;" OwnerUserId="12712" LastActivityDate="2020-11-14T13:31:16.740" Title="When were the SUVAT equations derived for the first time?" Tags="|physics|classical-mechanics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12438" PostTypeId="1" CreationDate="2020-11-15T02:16:06.927" Score="7" ViewCount="737" Body="&lt;p&gt;When and why did Mathematicians saw a need to define Tensor Products?&lt;/p&gt;&#xA;&lt;p&gt;I want to know the historical development of the idea &amp;quot;Tensor Product&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="13275" LastActivityDate="2021-12-12T02:03:01.680" Title="Origin of Tensor Product" Tags="|mathematics|linear-algebra|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12439" PostTypeId="2" ParentId="12438" CreationDate="2020-11-15T02:47:17.107" Score="-2" Body="&lt;p&gt;The tensor product is actually a very simple concept.&lt;/p&gt;&#xA;&lt;p&gt;It goes back to Babylonian times when people realised that two edges describes an area. Intuitively they realised that geometric area was bilinear but all this wasn't formalised until the twentieth century.&lt;/p&gt;&#xA;&lt;p&gt;Mathematicians, being mathematicians, generalised by allowing the edges to take values in any vector space, as well as the attendant geometric area.&lt;/p&gt;&#xA;&lt;p&gt;That formalisation took so long shouldn't be such a surprise. After arithmetic was invented well before Babylonian times and yet was only formalised with Peano's axioms in the early 20th C. I'd also add that it has taken the subject so far away from its geometric roots that its difficult to see the geometry of a tensor.&lt;/p&gt;&#xA;&lt;p&gt;Its worth adding that the tensors of general relativity, which made the subject famous, are actually fields of tensors, that is tensor fields. Moreover, the tensors they use are of a special kind, comprising of a tensor power of a vector space tensored with a tensor power of its dual space.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-12-12T02:03:01.680" LastActivityDate="2021-12-12T02:03:01.680" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="12441" PostTypeId="1" CreationDate="2020-11-16T04:21:07.893" Score="4" ViewCount="147" Body="&lt;p&gt;I am curious about the history of the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Whitehead_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whitehead's Conjecture&lt;/a&gt;, as this was the &lt;em&gt;first&lt;/em&gt; natural mathematical statement, in the sense that mathematicians were actually interested in the answer, that was shown to be independent of ZFC. That is, in Gödel's language, it is undecidable. Before then, all statements shown to be undecidable were artificial, in the sense that no mathematician was actually interested in the statement itself. They were merely examples showing that such statements actually existed.&lt;/p&gt;&#xA;&lt;p&gt;John Constatine Henry Whitehead, to give him his full name, and who also happens to be the nephew of Alfred North Whitehead who collaborated with Bertrand Russell on their Principia Mathematica, was lead to this question by his investigation of the Cousain problems in complex analysis/geometry. The linked article is clear enough for those with some mathematical maturity. It's expressed in the language of homological algebra but it's explained what this means in terms in a more prosaic manner. To rephrase:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An abelian group &lt;em&gt;G&lt;/em&gt;, is called a Whitehead group when every surjective morphism, &lt;em&gt;f&lt;/em&gt;, to it from any other abelian group splits. That is, it has a right inverse.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, it turns out that every free group is a Whitehead group. J. C. H. Whitehead asked whether the converse is true. That is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is every Whitehead group also a free group?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ekloff, in an article in &lt;em&gt;The American Mathematical Monthly (vol.83, no.10, Dec.'76)&lt;/em&gt; points out that this has a natural interpretation in terms of topological groups by taking advantage of Pontryagin duality: that is, is every path connected abelian group, a power of the circle group, that is a product of a number of copies of the circle group.&lt;/p&gt;&#xA;&lt;p&gt;Stein actually showed that Whiteheads conjecture follows when the group is countable. But otherwise progress was slow until Saharon Shelah in '74, completely unexpectedly showed it was undecidable from ZFC. In fact more, he showed it was undecidable if one also assumes GCH, the Generalised Continuum Hypothesis.&lt;/p&gt;&#xA;&lt;p&gt;Equally interestingly, he also showed that if one assumes that V=L, that the von Neumann universe is equivalent to Gödel's constructible universe then it does happen to be true!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Have any mathematicians postulated that V=L is a natural axiom to add to ZFC as it resolves Whiteheads question in a natural way? More, is it also evidence that one ought to consider a plurality of set theories in the same way that we have a plurality of number systems (of many kinds: groups, modules etc).&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-09-26T10:19:13.837" LastActivityDate="2021-09-26T10:19:13.837" Title="What role has Whitehead's Conjecture played in the thinking on the plurality of set theories?" Tags="|mathematics|set-theory|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12442" PostTypeId="1" AcceptedAnswerId="12443" CreationDate="2020-11-16T17:33:54.520" Score="1" ViewCount="127" Body="&lt;p&gt;I enjoy reading manuscripts written by mathematicians of old, and I would love to read the famous De Thiende by Simon Steven. I've done some research online here and there, but I have not found any English translations.&lt;/p&gt;&#xA;" OwnerUserId="10682" LastEditorUserId="10106" LastEditDate="2020-11-17T05:24:25.723" LastActivityDate="2020-11-17T06:30:38.510" Title="An English translation of Simon Stevin's De Thiende" Tags="|mathematics|mathematicians|translations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12443" PostTypeId="2" ParentId="12442" CreationDate="2020-11-16T17:56:40.390" Score="2" Body="&lt;p&gt;An old English translation is available on &lt;a href=&quot;https://books.google.nl/books?id=aVdIAQAAMAAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt; (or in transcribed form &lt;a href=&quot;https://adcs.home.xs4all.nl/stevin/telconst/10ths.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). The &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/DSB/Stevin.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dictionary of Scientific Biography entry for Stevin&lt;/a&gt; lists several translations in its bibliography.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2020-11-16T17:56:40.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12444" PostTypeId="1" CreationDate="2020-11-16T20:38:07.893" Score="0" ViewCount="103" Body="&lt;p&gt;I am not sure if it's new although it may be an easy consequence of some theorem or lemma.The result is as follows:&#xA;By choosing a set of numbers between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;(for any &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;) picking each number at most once and summing them you can create any number between 0 and &lt;span class=&quot;math-container&quot;&gt;$n(n+1) / 2$&lt;/span&gt; .&#xA;The above result can be easily proven by induction.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-11-16T20:50:26.953" LastActivityDate="2020-11-17T06:24:16.673" Title="Does this mathematical result have a specific name?" Tags="|number-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12447" PostTypeId="1" CreationDate="2020-11-17T17:37:29.570" Score="2" ViewCount="112" Body="&lt;p&gt;Somewhere in the world is housed what is thought to be the largest complete history of physics. I recall it being of some ridiculous length, something like hundreds or thousands of volumes. I cannot, however, recall the name of this &amp;quot;book&amp;quot;, nor any further details.&lt;/p&gt;&#xA;&lt;p&gt;Where is this collection held? What is its name?&lt;/p&gt;&#xA;" OwnerUserId="13250" LastEditorUserId="26" LastEditDate="2021-02-20T15:05:47.190" LastActivityDate="2021-02-20T15:05:47.190" Title="What is the name of the &quot;largest complete history&quot; of physics?" Tags="|mathematics|physics|theoretical-physics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12448" PostTypeId="1" CreationDate="2020-11-17T18:58:28.997" Score="13" ViewCount="2404" Body="&lt;p&gt;I recently read the following quote (unfortunately, I copied it down without attribution):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You may be surprised to know that Fourier analysis played a role in the early development of set theory. In fact, it was a Fourier-analytic question that led to Cantor's introduction of ordinal numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How was Fourier analysis important to the development of set theory? This does indeed seem rather surprising to me.&lt;/p&gt;&#xA;" OwnerUserId="13250" LastEditorUserId="4251" LastEditDate="2020-11-18T00:41:43.360" LastActivityDate="2020-11-19T14:21:10.323" Title="How was Fourier analysis important to the development of set theory?" Tags="|mathematics|set-theory|cantor|harmonic-analysis|" AnswerCount="4" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12449" PostTypeId="2" ParentId="12448" CreationDate="2020-11-17T21:53:41.213" Score="5" Body="&lt;p&gt;Unfortunately, I do not have a concrete reference... but I seem to recall that Cantor's earliest work was about &amp;quot;sets of uniqueness&amp;quot; for Fourier series (I think not Fourier transforms, but I could easily be mistaken).&lt;/p&gt;&#xA;&lt;p&gt;This would be similar to other late 19th century &amp;quot;constructive&amp;quot; analysis projects, where limits of limits of ... continuous... functions were taken. Even into the early 20th century, there were attempts to &amp;quot;classify&amp;quot; subsets of the real line, but/and, to my understanding, it turned out that there are tooooo many, and toooo complicated, such sets.&lt;/p&gt;&#xA;&lt;p&gt;Part of that classification attempt involved transfinite limits... though not using that word. Long ago, I had some Dover reprints of some archaic &amp;quot;real analysis&amp;quot; texts that took that approach.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2020-11-17T21:53:41.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12450" PostTypeId="2" ParentId="12448" CreationDate="2020-11-17T22:14:01.073" Score="5" Body="&lt;p&gt;paul garrett has the idea.  &lt;span class=&quot;math-container&quot;&gt;$E \subseteq \mathbb R$&lt;/span&gt; is a &lt;strong&gt;set of uniqueness&lt;/strong&gt;&#xA;if:  given a trigonometric series &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=-\infty}^\infty c_n e^{int}$&lt;/span&gt;, if it converges to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; except possibly on &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$c_n = 0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Set_of_uniqueness&quot; rel=&quot;noreferrer&quot;&gt;Here&lt;/a&gt; is a description.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cantor's research (I think) went like this.  A finite set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is a set of uniqueness.  A set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; with finitely many limit points is a set of uniquness.  A set whose limit points have finitely many limit points is a set of uniqueness.  Continuing with this, Cantor was led to transfinite ordinals...  Of course, when he was doing this, &amp;quot;countable&amp;quot; and &amp;quot;closed&amp;quot; were not standard notions.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2020-11-17T22:20:59.870" LastActivityDate="2020-11-17T22:20:59.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12451" PostTypeId="2" ParentId="12448" CreationDate="2020-11-18T00:22:46.237" Score="20" Body="&lt;p&gt;It was Fourier series rather than Fourier transform. Considering that the sets where Fourier series converge can be very intricate it is not that surprising that they led Cantor to develop set theory for subsets of real numbers. But at some point he took a turn into the abstract (for which he is best known today) that was not really motivated by the initial problem, but matched his more metaphysical interests, see &lt;a href=&quot;https://philarchive.org/archive/TERGCv1&quot; rel=&quot;noreferrer&quot;&gt;Ternullo, Gödel’s Cantorianism&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The specific problem posed to Cantor by Heine was the following, see &lt;a href=&quot;https://www.ias.ac.in/article/fulltext/reso/019/11/0977-0999&quot; rel=&quot;noreferrer&quot;&gt;Srivastava, How did Cantor Discover Set Theory and Topology?&lt;/a&gt; Suppose a trigonometric series converges to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; pointwise, must all of its coefficients be &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; as well? Essentially, it is the uniqueness problem for the Foureier series. Dirichlet, Heine, Lipschitz and Riemann tried their hand at it, but were only able to prove the result under heavy restrictions (e.g. Heine assumed uniform convergence).&lt;/p&gt;&#xA;&lt;p&gt;Cantor not only was able to prove it in full generality (1870), but noticed that the assumption of pointwise convergence everywhere could be relaxed. He introduced a &amp;quot;set of uniqueness&amp;quot; as such a set that the uniqueness was assured by pointwise convergence outside of it. Utilizing Heine's notion of ‘condensation point’ (now limit or accumulation point) Cantor defined the &lt;em&gt;derived set&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt; of a set &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; as the set of its condensation points. He then proved first (1871) that &lt;span class=&quot;math-container&quot;&gt;$P'=\emptyset$&lt;/span&gt; was sufficient for &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to be a set of uniqueness, and later that even &lt;span class=&quot;math-container&quot;&gt;$P^{(n)}=\emptyset$&lt;/span&gt; for some finite &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; sufficed. It is the derived set construction that served as a springboard for Cantor's set theoretic and point set topological breakthroughs.&lt;/p&gt;&#xA;&lt;p&gt;The proof required, in particular, making the notion of real numbers more precise, and led to Cantor's construction of them in terms of Cauchy sequences of rationals. After that Cantor's interests shifted from the trigonometric series to the more abstract properties of point sets of reals and then to abstract sets in general. He introduced denumerable (now countable) sets, identified rationals and algebraic numbers as denumerable, and then proved non-denumerability of open real intervals. This led to the idea of comparing &amp;quot;sizes&amp;quot; of infinite sets, their cardinalities, and eventually to the diagonal argument and the continuum hypothesis.&lt;/p&gt;&#xA;&lt;p&gt;In another line of inquiry, after finding examples of sets such that &lt;span class=&quot;math-container&quot;&gt;$P^{(n)}\neq\emptyset$&lt;/span&gt; for any finite &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, Cantor extended the recursion into the transfinite (his set of uniqueness result continues to hold when &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is replaced by any countable ordinal). Transfinite ordinals were a new notion that he had to introduce, develop, and defend against the Aristotelian preconceptions about actual infinity, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2419/55&quot;&gt;Why did Cantor (and others) use c&#xA;for the continuum?&lt;/a&gt; The transfinite induction, as well as the notions of dense and perfect sets came out of this circle of ideas. But the ultimate solution to the uniqueness problem for the Fourier series was no longer in Cantor's view, and had to wait until the introduction of Lebesgue measure theory. The sets of uniqueness turned out to be the sets of Lebesgue measure zero.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-19T01:34:36.863" LastActivityDate="2020-11-19T01:34:36.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12452" PostTypeId="1" CreationDate="2020-11-18T11:49:51.920" Score="3" ViewCount="180" Body="&lt;p&gt;In his answer to &lt;a href=&quot;https://hsm.stackexchange.com/questions/12307/what-are-some-historical-examples-in-physics-of-heuristic-proofs-of-mathematical&quot;&gt;a previous question&lt;/a&gt; Alexandre Eremenko pointed out that Joseph Fourier in his book &lt;em&gt;Analytic Theory of Heat&lt;/em&gt; gave all kinds of arguments in favor of the following mathematical discoveries,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Every periodic function can be expanded into a Fourier series.&lt;/li&gt;&#xA;&lt;li&gt;Every reasonable function on the real line can be represented by a Fourier integral.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;including experimental evidence with heated metal rings.&lt;br /&gt;&#xA;Can someone provide more details on these heated metal rings and their link with the statements above, as well as a precise reference in Fourier's book?&lt;/p&gt;&#xA;" OwnerUserId="13109" LastActivityDate="2021-05-06T02:06:26.183" Title="Did Fourier use heated metal rings as experimental evidence to justify his mathematical discoveries?" Tags="|mathematics|mathematicians|fourier|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12456" PostTypeId="1" CreationDate="2020-11-19T01:47:38.847" Score="8" ViewCount="1203" Body="&lt;p&gt;Whilst researching science in the ancient world, I came across an observation, which unfortunately I did not make a note of, and so cannot credit, that Ptolemy's epicycles were an early form of Fourier analysis. This made immediate sense to me. After all, we know that smooth functions can be arbitrarily well approximated by Fourier series. Hence any motion can also be.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Has anyone actually made a study of Ptolemy's epicycles from this angle?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="4251" LastEditDate="2021-12-27T10:45:29.277" LastActivityDate="2021-12-27T10:45:29.277" Title="Has anyone explored Ptolemy's epicycles as an early form of Fourier analysis?" Tags="|mathematics|astronomy|antiquity|harmonic-analysis|ptolemy|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12457" PostTypeId="2" ParentId="12456" CreationDate="2020-11-19T07:43:39.077" Score="2" Body="&lt;p&gt;Yes. That was done by Giovanni Schiaparelli in a book called &lt;em&gt;Scritti sulla storia dell'astronomia antica&lt;/em&gt;, published in 1926 and reprinted in 1997.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2020-11-19T07:43:39.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12458" PostTypeId="2" ParentId="12456" CreationDate="2020-11-19T08:54:56.067" Score="3" Body="&lt;p&gt;This is very popular myth, but it is not true: Ptolemy's epicycles are not Fourier analysis!&#xA;Fourier series can indeed approximate an arbitrary periodic function. And you can approximate an arbitrary motion of period &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; by the series of epicycles. The first is just a circular motion of period &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;, the second is an epicycle with period &lt;span class=&quot;math-container&quot;&gt;$T/2$&lt;/span&gt; and so on.&#xA;But apparent planetary motion is not periodic: it is sum of two motions with different periods: one is proper planet motion and another is Earth motion. So Ptolemy's epicycles account just for these two motions. It's not Fourier approximation.&lt;/p&gt;&#xA;&lt;p&gt;However because planets move not uniformly by circles but by Kepler's laws, the simple model of epicycles would give a very rough approximation. Here Fourier approximation would help: you can add additional epicycle to account for that. But Ptolemy's system did not do that. Ptolemy is most famous for inventing the model of equant. In this system a planet (or actually a center of epicycle) is moving on circle but not with uniform speed, it's moving with uniform angular speed with respect to some imaginary point that lies off center.&lt;/p&gt;&#xA;&lt;p&gt;But there is some grain of truth in this myth. People indeed used epicycles that you would get Fourier's analysis of planetary motion. But it was done not by Ptolemy, but by... Copernicus. We know Copernicus for inventing heliocentric system. Because of this system Copernicus did not need Ptolemy's epicycles. But this is only one part of Copernicus work. The part that Copernicus was most proud was that he abolished Ptolemy's equant, but introduced new epicycles instead. These new epicycles were exactly the ones that we would get from Fourier's analysis. They have period &lt;span class=&quot;math-container&quot;&gt;$T/2$&lt;/span&gt; as you would expect. This system had the same order of error as Ptolemy's: Ptolemy equant system had error &lt;span class=&quot;math-container&quot;&gt;$\approx 1/2e^2$&lt;/span&gt;, and Copernicus epicycles had errors &lt;span class=&quot;math-container&quot;&gt;$\approx 3/2e^2$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is eccentricity, but Copernicus system was a little bit simpler, because it was sum of two uniform circular motions.&lt;/p&gt;&#xA;&lt;p&gt;So, to recup: Copernicus got rid of Ptolemy's epicycles (which are unrelated to Fourier's analysis), because he put Sun in the center of Universe. But introduced new epicycles (which are the first Fourier approximation) instead of Ptolemy's equant system (which had the same purpose: approximate unknown Kepler's motion).&lt;/p&gt;&#xA;&lt;p&gt;BTW, another legend that usually goes with this myth says that people before Copernicus added more and more epicycles to Ptolemy's system to make it more accurate. This is not true. No one before Kepler made a more accurate system that Ptolemy's. Fourier's epicycles was indeed added by Copernicus, but not to improve accuracy, but to make it simpler. And it was just first approximation (if we consider uniform motion as zeroth approximation). There was no epicycles that correspond to second term in Fourier series.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2020-11-19T08:54:56.067" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12459" PostTypeId="1" AcceptedAnswerId="12465" CreationDate="2020-11-19T13:26:49.667" Score="11" ViewCount="325" Body="&lt;p&gt;I have just stumbled across &lt;a href=&quot;https://hsm.stackexchange.com/questions/5656/who-invented-the-divisibility-symbol-and-why-is-it-backwards&quot;&gt;this post&lt;/a&gt; and became curious about the same question, namely the part regarding the origin/history of the vertical bar symbol &lt;span class=&quot;math-container&quot;&gt;$a\vert b$&lt;/span&gt; that we use to denote &amp;quot;a divides b&amp;quot; (I don't care at all about why it is written &amp;quot;backward&amp;quot; in the sense asked there).&lt;/p&gt;&#xA;&lt;p&gt;While the OP of that post seems satisfied with the answer, the part about the origin of this symbol was still left out. In one of the comments there, there was a suggestion that the answer could be found in Florian Cajori's book &lt;a href=&quot;https://books.google.it/books?id=bT5suOONXlgC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;A History of Mathematical Notations&lt;/a&gt;. I have a copy of that book but I found nothing directly related to the history of the symbol &lt;span class=&quot;math-container&quot;&gt;$\vert$&lt;/span&gt; , unfortunately.&lt;/p&gt;&#xA;&lt;p&gt;I'd highly appreciate if anyone could point me to a good resource about this subject matter, be it a book or an article. More specifically, I want to know the time period the notation &lt;span class=&quot;math-container&quot;&gt;$\vert$&lt;/span&gt; was introduced and the names of the mathematicians associated with its development.&lt;/p&gt;&#xA;" OwnerUserId="3179" LastEditorUserId="26" LastEditDate="2021-02-20T15:05:41.307" LastActivityDate="2021-02-20T15:05:41.307" Title="Who introduced the divisibility symbol $a\vert b$ (&quot;$a$ divides $b$&quot;) and when?" Tags="|mathematics|terminology|reference-request|notation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12460" PostTypeId="2" ParentId="12456" CreationDate="2020-11-19T14:03:52.570" Score="6" Body="&lt;p&gt;As @Andrei Kopylov noticed, epicycle theory is not the theory of Fourier series of a periodic function. Still this is called (generalized) Fourier analysis. Such functions are called almost periodic or quasi-periodic, and they expand into generalized Fourier series&#xA;of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum c_k e^{i\lambda_k t},$$&lt;/span&gt;&#xA;with arbitrary real &lt;span class=&quot;math-container&quot;&gt;$\lambda_k$&lt;/span&gt;, which is also called a (generalized) Fourier series. From the modern point of view such series occupy an intermediate position between the ordinary Fourier series and Fourier integrals. But the general idea is the same: to represent a motion as a superposition of uniform circular motions, and the idea was clearly stated for the first time in ancient Greece.&#xA;Mathematical theory of such series was systematically developed only in 20 century&#xA;(H. Bohr, H. Weyl, N. Wiener, ..., A. Kolmogorov, V. Arnold...).&#xA;A historical survey can be found here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/abs/chao-dyn/9907004&quot; rel=&quot;noreferrer&quot;&gt;Quasi periodic motions from Hipparchus to Kolmogorov&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Remark. In the earliest models of Apollonius and Ptolemy the periods were commensurable. The first person who clearly realized that the motions of planets have to be modeled with non-commensurable periods was Nicole Oresme (1320-1382).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-11-19T14:28:28.247" LastActivityDate="2020-11-19T14:28:28.247" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12461" PostTypeId="2" ParentId="12459" CreationDate="2020-11-19T14:12:16.280" Score="1" Body="&lt;p&gt;I think that the history of how we write fractions is helpful here. Although fractions were known in ancient times - the Babylonians and Egyptians used them - the modern notation for them began with the system of &lt;em&gt;bhinnarasi&lt;/em&gt; by Aryabhatta around 5th Century AD and then Brahmagupta and (c. 626) and Bhaskara (c. 1150).&lt;/p&gt;&#xA;&lt;p&gt;In their works, they formed fractions by placing the numerators (&lt;em&gt;amsa&lt;/em&gt;) over the denominators (&lt;em&gt;cheda&lt;/em&gt;) without a separating line. From there it is an easy step to put this in to emphasise the separation of the two numbers and this is first attested to in the work of al-Hassar (c. 1200), a Muslim mathematician working in Fez, Morroco.&lt;/p&gt;&#xA;&lt;p&gt;The same notation then appeared soon after in Europe, for example in the work of Fibonnaci (c. 1300).&lt;/p&gt;&#xA;&lt;p&gt;Obviously, it's not easy to write or print numbers in such a fashion, especially with the advent of algebra, and lengthy expressions in either the numerator or denominator; and so the next obvious step is to write them horizontally as a/b, with the separating bar now positioned vertically.&lt;/p&gt;&#xA;&lt;p&gt;This explains how we have the vertical bar for division. As your linked post then explains, it would be sensible to them express divisibility with a similar notation and hence the introduction of the vertical bar with the terms arranged in order of how we say them: a divides b as a|b.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I'd like to add, that in modern notation, we express divisibility both ways: a divides b, can be written as a\b and b/a. We see this freedom of expression when expressing quotients of groups, rings when dividing by ideals, modules or algebras, for example. We &lt;strong&gt;don't&lt;/strong&gt; however, generally see this freedom with numbers.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-24T12:09:29.653" LastActivityDate="2020-11-24T12:09:29.653" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12462" PostTypeId="2" ParentId="12448" CreationDate="2020-11-19T14:21:10.323" Score="3" Body="&lt;p&gt;Here are the references to the original papers of Cantor:&lt;/p&gt;&#xA;&lt;p&gt;On a theorem concerning trigonometric series. (Ueber einen die trigonometrischen Reihen betreffenden Lehrsatz.)&#xA;Borchardt J. LXXII, 130-138 (1870).&lt;/p&gt;&#xA;&lt;p&gt;Proof that a function given for every real value of    by a trigonometric series has only one representation in this form. (Beweis, dass eine für jeden reellen Werth von    durch eine trigonometrische Reihe gegebene Funktion    sich nur auf eine einzige Weise in dieser Form darstellen lässt.)&#xA;Borchardt J. LXXII, 139-142 (1870).&lt;/p&gt;&#xA;&lt;p&gt;On trigonometric series. (Ueber trigonometrische Reihen.)&#xA;Clebsch Ann. IV, 139-143 (1871).&lt;/p&gt;&#xA;&lt;p&gt;On the extension of a theorem from the theory of trigonometric series. (Ueber die Ausdehnung eines Satzs aus der Theorie der trigonometrischen Reihen.)&#xA;Clebsch Ann. V, 123-133 (1872).&lt;/p&gt;&#xA;&lt;p&gt;Borchardt J. = Journal fur die reine und angewandte Mathematik,&lt;/p&gt;&#xA;&lt;p&gt;Clebsch Ann. = Mathematische Annalen.&lt;/p&gt;&#xA;&lt;p&gt;Both journals are available on Internet.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-11-19T14:21:10.323" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12464" PostTypeId="2" ParentId="12452" CreationDate="2020-11-19T19:43:13.790" Score="1" Body="&lt;p&gt;The essential part of the answer (page references) is contained in the comment of @Conifold. However his general conclusion is plain wrong, and I would like to put the things straight.&lt;/p&gt;&#xA;&lt;p&gt;A scientist makes ASSUMPTIONS. Then develops a theory. And then compares with observations/experiments. If this comparison works, this CONFIRMS his assumptions.&lt;/p&gt;&#xA;&lt;p&gt;For example, Newton (and others) assume the inverse square law. Then Newton (and others) derived a lot of consequences from it, which can be tested by observations. And the agreement with observations proves the initial conjecture. This is how science works.&lt;/p&gt;&#xA;&lt;p&gt;Returning to Fourier. Of course, he could not prove mathematically the statement&#xA;that &amp;quot;arbitrary periodic function has a Fourier expansion&amp;quot;, for the simple reason&#xA;that the modern notion of &amp;quot;arbitrary function&amp;quot; did not exist at that time.&#xA;It was first stated by Dirichlet, whose purpose was to give a mathematical justification of Fourier discoveries.&lt;/p&gt;&#xA;&lt;p&gt;(Further attempts in this direction led to further evolution of the notion of function: &amp;quot;generalized functions&amp;quot; or &amp;quot;distributions&amp;quot; were also introduced ith the purpose to justify Fourier analysis.&lt;/p&gt;&#xA;&lt;p&gt;Fourier himself was a scientist, first of all. And he lived at the time when science&#xA;was not separated from mathematics (it is still not separated completely). So his approach is that of a scientist: he makes assumptions, develops a theory and then tries to test it.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2020-11-19T19:43:13.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12465" PostTypeId="2" ParentId="12459" CreationDate="2020-11-20T09:34:45.693" Score="10" Body="&lt;p&gt;This is a case where it seems that the symbol should be old, from Euler's or Gauss's time at least, but it is not. It does not appear in &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Dickson's History of the theory of numbers (1919)&lt;/a&gt;, whose entire first volume is dedicated to divisibility, nor in Cajori's comprehensive &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp&quot; rel=&quot;noreferrer&quot;&gt;History Of Mathematical Notations (1928)&lt;/a&gt;, and not even in &lt;a href=&quot;https://archive.org/details/modernalgebrainp0001waer/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;van der Waerden's &lt;em&gt;Moderne Algebra&lt;/em&gt; (1930)&lt;/a&gt;, which became a blueprint for modern algebra textbooks.&lt;/p&gt;&#xA;&lt;p&gt;The earliest use I found is in Hall's &lt;a href=&quot;https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/s1-8.3.162&quot; rel=&quot;noreferrer&quot;&gt;Slowly increasing arithmetic series (1933)&lt;/a&gt;, where it is introduced in a footnote thus:&amp;quot;&lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$x|y$&lt;/span&gt; means &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; divides &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;&amp;quot;&lt;/em&gt;&amp;quot;, no comment. Hall's references, &lt;a href=&quot;https://www.jstor.org/stable/1968235&quot; rel=&quot;noreferrer&quot;&gt;Lehmer's An Extended Theory of Lucas' Functions (1930)&lt;/a&gt; and &lt;a href=&quot;https://www.ams.org/journals/tran/1931-033-01/S0002-9947-1931-1501585-5/S0002-9947-1931-1501585-5.pdf&quot; rel=&quot;noreferrer&quot;&gt;Engstrom's On sequences defined by linear recurrence relations (1931)&lt;/a&gt;, still use words or congruences for the task. On the other hand, Hall and Ward use &lt;span class=&quot;math-container&quot;&gt;$|$&lt;/span&gt; extensively in their 1936-38 publications on linear divisibility sequences.&lt;/p&gt;&#xA;&lt;p&gt;After graduating from Yale in 1932 Hall worked with Hardy at Cambridge for a year before returning to Yale in 1936. And the first book occurrence seems to be Hardy-Wright's classic &lt;a href=&quot;https://archive.org/details/introductiontoth0004hard&quot; rel=&quot;noreferrer&quot;&gt;An introduction to the theory of numbers&lt;/a&gt; (first edition came out in 1938), where we read on the very first page:&amp;quot;&lt;em&gt;We express the fact that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is a divisor of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, by &lt;span class=&quot;math-container&quot;&gt;$b|a$&lt;/span&gt;&lt;/em&gt;&amp;quot;.&#xA;Vinogradov's Elements of Number Theory (first Russian edition came out in 1936, &lt;a href=&quot;https://archive.org/details/elementsofnumber0000vino&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt; in 1954) uses &lt;span class=&quot;math-container&quot;&gt;$b\backslash a$&lt;/span&gt; instead, suggesting that the notation was not established yet. Hall's notation was adopted in &lt;a href=&quot;https://www.springer.com/gp/book/9783540007067&quot; rel=&quot;noreferrer&quot;&gt;Bourbaki's Algebra II, chapitre VI&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;All of these authors are very matter-of-factly and laconic when introducing the symbol, and neither motivate it nor refer to anyone, including each other, for it. Not even Hardy-Wright, who have a special note on notations, or Bourbaki, who have extensive historical notes. So it is hard to say who came up with it (it could have been Hall or Hardy) and why. But the shapes suggest that it was simply a variation on the division symbol &lt;span class=&quot;math-container&quot;&gt;$/$&lt;/span&gt;, and Hardy-Wright explicitly introduce logical symbols in their Remarks on Notation, and use &lt;span class=&quot;math-container&quot;&gt;$|$&lt;/span&gt; to illustrate their use. It seems that the turn towards abstraction in algebra and number theory, and proliferation of symbolism from foundational studies in mathematical logic in 1930s made symbolizing a relation that was previously expressed in words or congruences timely.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-21T07:59:16.570" LastActivityDate="2020-11-21T07:59:16.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12467" PostTypeId="1" AcceptedAnswerId="12474" CreationDate="2020-11-20T21:11:13.263" Score="3" ViewCount="456" Body="&lt;p&gt;According to &lt;a href=&quot;https://physicstoday.scitation.org/doi/full/10.1063/1.3273015&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Peirce was the first to experimentally tie a unit, the meter, to an&#xA;absolute standard, the wavelength of a spectral line&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Did C.S Peirce make any other noteworthy contributions to physics or chemistry?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-11-20T21:38:33.153" LastActivityDate="2020-11-20T22:26:41.590" Title="Did C.S Peirce make any noteworthy contributions to physics or chemistry?" Tags="|physics|biographical-details|philosophy-of-science|chemistry|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12468" PostTypeId="1" CreationDate="2020-11-20T21:12:37.693" Score="1" ViewCount="195" Body="&lt;p&gt;In Isaac Deutscher's, &lt;em&gt;The Prophet Armed: Trotsky 1879-1921&lt;/em&gt;, he writes of the youthful Trotsky:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;His stay in Odessa ended in 1896. A &lt;em&gt;realschule&lt;/em&gt; normally had seven forms but St. Paul's had only six so had to attend a similar school at Nikolayev to matriculate. He was now nearly seventeen but no political ideal had so far appealed to him. The year before Friedrich Engels had died; the event did not register in his mind of the future revolutionary - even the name of Karl Marx had not yet come to his ears. He was, in his own words, 'poorly equipped politically for a boy of seventeen at that time.' He was attracted by literature and was preparing for a university course in pure mathematics. These two approaches to life, the imaginative, and the abstract, lured him - later, he would strive to unite them in his writings ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the summer of 1896 he arrived at Nikolayev to complete his secondary education. He was lodging with a family whose sons had already been touched by socialist ideals. They drew him into argument and tried to impress their ideas on him. For several months they seemed to make no headway. He supercilious lay dismissed their 'Socialist Utopia'. Assailed with arguments he would adopt the posture of a somewhat conservative young man, not devoid of sympathy with the people but distrusting 'mob ideology' and 'mob rule'. His passion was for pure mathematics and he had no time or taste for politics ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. There's no indication in Deutscher of what kind of direction Trotsky took in his thinking about mathematics. Is there any indication elsewhere?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. How did Trotsky attempts to unite the 'abstract' and the 'imaginative' in his writings?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've read only a third of the first book and so it's possible that the answer may be later in the trilogy - but I think it's most likely that it isn't.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-30T14:15:17.413" LastActivityDate="2020-11-30T14:15:17.413" Title="Is there any indication of what mathematics Trotsky studied?" Tags="|biographical-details|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12469" PostTypeId="1" AcceptedAnswerId="12482" CreationDate="2020-11-20T21:22:07.800" Score="4" ViewCount="675" Body="&lt;p&gt;The unit of electric charge is the coulomb, named after Charles Augustin de Coulomb. This makes sense because Coulomb's law talks about the force between two charges. Likewise, it also makes sense to name unit of electric current after André-Marie Ampère, who too was a contemporary of Coulomb. But I have seen that the SI unit of capacitance is the farad, named after Michael Faraday. Why was Michael Faraday's name given to the unit of capacitance instead of any other electromagnetic unit? Likewise, why is the unit of electrical resistance or magnetic reactance named after Ohm, given that Ohm's law talks about the relationship between current and resistance, and electrical resistance can exist irrespective of Ohm's law. Similarly, I do not understand why the SI unit of magnetic flux density is named after Tesla (see a related post &lt;a href=&quot;https://hsm.stackexchange.com/questions/2085/why-is-magnetic-flux-density-named-after-nikola-tesla&quot;&gt;here&lt;/a&gt;), since he most contributed to electrical engineering. Further examples include the volt, siemens, weber, and henry (for electromotive force, electric conductance, magnetic flux, and inductance, respectively). Why were their names prioritized over other famous scientists, like Pierre Curie or Hippolyte Fizeau or Emil Lenz?&lt;/p&gt;&#xA;&lt;p&gt;More generally: How were the names of the SI units relevant to electromagnetism decided on, and why were certain scientists prioritized over others?&lt;/p&gt;&#xA;" OwnerUserId="13344" LastEditorUserId="13344" LastEditDate="2023-05-17T16:45:15.587" LastActivityDate="2023-05-17T16:45:15.587" Title="Names of the electromagnetic units in SI" Tags="|physics|electromagnetism|physicists|units|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12470" PostTypeId="2" ParentId="12447" CreationDate="2020-11-20T21:23:26.270" Score="2" Body="&lt;p&gt;This sounds like the physics equivalent of the Library of Babel by the Argentine author, Jorge Luis Borges and similarly, as fictive.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps in the future we can imagine some such 'Encyclopadia Principae Physicae' hyperlinked to every journal ever printed and instantly accessible to everyone and running to millions of pages and many thousands of volumes and updated in real-time by students and researchers alike, as well as the interested layman.&lt;/p&gt;&#xA;&lt;p&gt;There is no such thing now - though, perhaps, Wikipedia comes closest.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-20T21:23:26.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12472" PostTypeId="2" ParentId="12467" CreationDate="2020-11-20T22:05:29.597" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;https://physicstoday.scitation.org/doi/full/10.1063/1.3273015&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Physics Today&lt;/em&gt; article&lt;/a&gt; shows that Michelson knew of C. S. Peirce's unpublished results on measuring the wavelength of sodium emission:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Michelson realized that the interferometer he and Morley had developed and were just then using to detect ether drift could also be used to make precise wavelength measurements. In June 1887, after getting initial results in their epochal experiment on the speed of light, Michelson and Morley conducted preliminary spectroscopic metrology measurements. Their paper “On a Method of making the Wave-length of Sodium Light the actual and practical Standard of Length” begins, “The first actual attempt to make the wave-length of sodium light a standard of length was made by Peirce.”&lt;sup&gt;15&lt;/sup&gt; But, they pointed out, Peirce’s measurements, “which have not as yet been published” (and never would be), had many systematic errors.&lt;hr&gt;&lt;sub&gt;15. A. Michelson, and E. Morley, Am. J. Sci. &lt;strong&gt;34&lt;/strong&gt;, 427 (1887).&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Peirce's surveying work required determining the length of pendula accurately, which is related to his contributions in geodesy (&lt;a href=&quot;https://www.encyclopedia.com/people/philosophy-and-religion/philosophy-biographies/charles-sanders-peirce#articleWrapper-961088&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Complete Dictionary of Scientific Biography&lt;/em&gt; article on Peirce&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1879 Peirce succeeded in determining the length of the meter from a wavelength of light. Benjamin Peirce described this feat, an adumbration of the work of Michelson, as “the only sure determination of the meter, by which it could be recovered if it were to be lost to science.” By 1882 Peirce was engaged in a mathematical study of the relation between the variation of gravity and the figure of the earth. He claimed that “divergencies from a spherical form can at once be detected in the earth’s figure by this means,” and that “this result puts a new face on the relation of pendulum work to geodesy.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2020-11-20T22:05:29.597" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12473" PostTypeId="2" ParentId="12467" CreationDate="2020-11-20T22:06:29.963" Score="2" Body="&lt;p&gt;Pierce was educated as a chemist and employed as a scientist. He considered himself primarily as a logician in the American school of analytic philosophy.&lt;/p&gt;&#xA;&lt;p&gt;His biographer Joseph Brent, called him 'he was at first, almost stupefied, and then aloof, cold, depressed, extremely suspicious, impatient of the slightest crossing and subject to violent outbursts of temper.' And according to Wikipedia, it's consequences might have led to his later social isolation.&lt;/p&gt;&#xA;&lt;p&gt;Given that his main employment, intermittently, was by the US Coast Survey and its successor, the US Coast and Geodetic Survey, it's not surprising that he was responsible for some innovations in metrology.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Physics Today&lt;/em&gt;, confirms that Pierce 'was the first to experimentally tie a unit, the meter, to an absolute standard, a spectral line' but they go on to add&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For several reasons that contribution has not received much attention. First, he never finished it to his satisfaction and only left fragmentary reports in his 12 000 published pages, and 80,000 hand-written pages of notes and letters - mostly on logic, mathematics, science and philosophy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They go onto say that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pierce's chaotic personal and professional life has hindered a comprehensive assessment of contributions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So I expect, we will find out - eventually ...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-20T22:06:29.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12474" PostTypeId="2" ParentId="12467" CreationDate="2020-11-20T22:13:39.077" Score="3" Body="&lt;p&gt;Not really. Peirce received formal training in chemistry at Harvard, and wrote some manuscripts and papers on the subject in 1860s. One of them, &lt;a href=&quot;https://www.google.com/books/edition/Writings_of_Charles_S_Peirce_1867_1871/yrHk57CxGNYC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;The  Pairing  of  the Elements (Chemical News, 1869)&lt;/a&gt;, published the same year as Mendeleev's celebrated work, even anticipated the periodic law, but in a way common to chemists of the time (Hinrich, Odling, de Chancourtois, Meyer, and Newlands). At the same time, contra Mendeleev, he supported the &lt;a href=&quot;https://en.wikipedia.org/wiki/Prout%27s_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prout’s hypothesis&lt;/a&gt; that elements were aggregates of  a  single &amp;quot;protyle&amp;quot; (the hydrogen atom), and attempted to test it experimentally.&lt;/p&gt;&#xA;&lt;p&gt;Peirce's originality rather manifested in his using of the chemical background as a springboard for analysis of relational logic, diagrammatic reasoning and various classifications he was so fond of. What some characterize as Peirce's philosophical proto-structuralism is directly motivated by his analysis of chemical graphs, see &lt;a href=&quot;https://www.academia.edu/17198277/Peirces_Chemistry_of_Concepts&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bellucci, Peirce’s Chemistry of Concepts&lt;/a&gt;. It was a major contribution to the philosophy of chemistry, and science generally, that conceptualized the hypothetico-deductive method, the use of abduction and the function of diagrams in science, and chemistry in particular, but not to chemistry itself. The same goes for physics. Aside from methodology, he was one of the first to go against the grain of times (and Kant) to argue from physics for metaphysical indeterminism (&amp;quot;tychism&amp;quot;), but did not do much in physics itself.&lt;/p&gt;&#xA;&lt;p&gt;Campbell's recent thesis, &lt;a href=&quot;https://www.semanticscholar.org/paper/The-chemistry-of-relations-%3A-the-periodic-table-the-Campbell/1f558de33b9967262c7ca51af420921995e25a29?p2df&quot; rel=&quot;nofollow noreferrer&quot;&gt;The chemistry of relations&lt;/a&gt;, has a chapter dedicated to Peirce's manuscripts and papers on chemistry proper, it is the first study of this kind:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I examine Peirce’s research interest in developing a system for classifying and grouping  the  chemical  elements  according  to  atomic  weight  is  common  to  nineteenth century chemistry. In this respect Peirce’s interest is again main-stream for  the  period.In fact,  as has already  been  discussed  in chapter one, Peirce’s interests were also shared by his tutor Josiah Cooke... The  arrangement  of  the  chemical elements  that  Peirce achieved  is described by Nathan Houser (1982:xx) as going ‘far in Mendeleev’s direction, before Mendeleev’s announcement of the [periodic] law’ and before Mendeleev’s work ‘became known in Western Europe and America’. Here again Peirce’s chemical researches mirror  the  concerns  of  other  researches  in  both  North  America  and  in  Europe.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;...Whilst not wishing to challenge Fisch’s view of Peirce as ‘most original and versatile intellect’, I have highlighted that as a chemist, Peirce operated within the boundaries of what Thomas Kuhn (1996) described  as normal  science.  Where  Peirce   differed  was  in  his  ready willingness  to engage with the metaphysical issues that  many chemists,  including his tutor Josiah  Cooke,  chose  to  ignore. During  the  1860s  most  scholars  agree  that  Peirce  almost ‘outsources’ his metaphysics from  Kant  when  forming  his  philosophical  position.  What  I believe is an addition to this scholarship is my claim that Peirce’s Kantian metaphysics emerges in his chemistry.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Of  significance  is  that  in1869  Peirce,  by  employing  inductive  reasoning,  argues  for  an orderly arrangement of the chemical elements. That same year Peirce publishes a series of three  papers  which  includes  a  justification  for  inductive  reasoning  which denies J.S Mill’s defence in terms of the orderliness of nature. I would argue that Peirce does not deny the discoverability  of  regularities  in  the  world;  rather  the  insufficiency  of  such  an  appeal  to order to justify inductive reasoning. As we have seen Peirce’s justification turns on the likelihood that in the longer term inquirers –such as chemists seeking a ordered system for the  chemical  elements –would be ‘fated’ on occasion to be successful  in their inductions, inductive reasoning  being, at least at this stage in Peirce’s philosophy,the  only  form  of ‘synthetic reasoning’ inquirers possess.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-20T22:26:41.590" LastActivityDate="2020-11-20T22:26:41.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12475" PostTypeId="1" CreationDate="2020-11-20T22:57:23.307" Score="0" ViewCount="44" Body="&lt;p&gt;In some of his online biographies Peirce is described as having showed early signs of precocity.&#xA;Can someone describe some of these signs of precocity?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-11-20T23:49:42.093" Title="What are some facts about Peirce's early precocity?" Tags="|biographical-details|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12477" PostTypeId="2" ParentId="12475" CreationDate="2020-11-20T23:49:42.093" Score="1" Body="&lt;p&gt;Pierce's father, Benjamin Pierce, was professor of mathematics at Harvard and was a founder, and for a while, a director the US Coast and Geoduetuc Survey, for whom Pierce worked for much of his adult life.&lt;/p&gt;&#xA;&lt;p&gt;The SEP has this to say about Pierce's early life:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pierce recieved much of the substance of his early education as well as much intellectual encouragement and stimulation from his father. Benjamin's didactic technique consisted mostly of setting interesting problems for his son ... in this challenging instructional atmosphere Pierce acquired his lifelong habit of thinking through philosophical and scientific questions entirely on his own. To this habit, perhaps, is to be attributed Pierce's considerable originality ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Personally, I think this is true; intelligence and originality is not thinly spread amongst people; but widely spread and merely takes the right kind of stimulation to seed, to flower and then bloom ...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-20T23:49:42.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12478" PostTypeId="1" CreationDate="2020-11-21T00:13:18.130" Score="6" ViewCount="324" Body="&lt;p&gt;I was surprised to learn that the note published by Emmy Noether in 1925 that suggested that Homology was better thought through as actual groups rather than numerically as Betti numbers consisted of &lt;em&gt;fourteen&lt;/em&gt; lines. This is mentioned in Weibels &lt;em&gt;History of Homological Algebra&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What was the content of this note?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I had originally said two lines but checking back with Weibel I noted that I had made a mistake and it should have been fourteen.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-21T23:37:50.283" LastActivityDate="2020-11-21T23:37:50.283" Title="What was the content of Emmy Noether's 'two-line' note on Homology?" Tags="|geometry|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12479" PostTypeId="1" CreationDate="2020-11-21T00:35:06.123" Score="2" ViewCount="111" Body="&lt;p&gt;Satyendra Nath Bose who discovered boson statistics and is credited by Einstein for also discovering the Bose-Einstein condensate was born in Calcutta, was educated there at Presidency College and then at the University of Calcutta. After graduating he taught there at Rajbazar Science College.&lt;/p&gt;&#xA;&lt;p&gt;In 1921 he moved to the newly founded Dhaka university as a Reader at the Physics Department helping to set up 'whole new departments including laboratories' and teaching both undergraduate and graduate courses. He worked mainly in theoretical physics and X-Ray crystallography. It's whilst there he published his famous paper in 1924, &lt;em&gt;Plancks Law and the Light Quantum Hypothesis&lt;/em&gt;, a copy of which he had sent to Einstein and who responded enthusiastically.&lt;/p&gt;&#xA;&lt;p&gt;What prompted his move to a newly founded university with, presumably, no reputation to speak of?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-23T05:29:25.027" LastActivityDate="2020-11-23T05:29:25.027" Title="What prompted Bose's move to Dhaka University in 1921?" Tags="|physics|social-context|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12482" PostTypeId="2" ParentId="12469" CreationDate="2020-11-21T23:56:38.780" Score="7" Body="&lt;p&gt;Actually, the farad was the term used for a unit of charge by Latimer Clark and Charles Bright in 1861 in honour of Michael Faraday.&lt;/p&gt;&#xA;&lt;p&gt;But by 1873, it had become the unit of capacitance and was adopted as such by The British Association for the Advancement of Science by the first report of their Committee for the Selection and Nomenclature of Dynamic and Electrical Units chaired by Sir W Thomson in their 43rd meeting.&lt;/p&gt;&#xA;&lt;p&gt;More, it was then adopted as such by the International Congress of Electricians held in Paris  in 1881 as reported by &lt;em&gt;The Electrician&lt;/em&gt; of that year.&lt;/p&gt;&#xA;&lt;p&gt;The Tesla as the unit of magnetic induction was adopted much later. It was adopted by the General Conference on Weights and Measures held in 1960 and was proposed by the electrical engineer, France Avcin.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-22T21:24:58.803" LastActivityDate="2020-11-22T21:24:58.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12483" PostTypeId="1" CreationDate="2020-11-22T00:01:11.020" Score="3" ViewCount="232" Body="&lt;p&gt;The following text from &lt;a href=&quot;https://en.wikipedia.org/wiki/Ars_Magna_(Cardano_book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ars magna&lt;/a&gt; (1545) by Girolamo Cardano is known as the inception of complex numbers: &amp;quot;imaginaberis ℞ m 15&amp;quot; (You will imagine the square root of  minus 15):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LXf97m.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LXf97m.jpg&quot; alt=&quot;Ars magna, complex numbers inception&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;R&amp;quot; with the stroke on its leg could read as ℞ (Rx), an abbreviation for &lt;em&gt;radix&lt;/em&gt;, or root in Latin. Was this notation frequent, and does it have a known origin?&lt;/p&gt;&#xA;&lt;p&gt;Note: this appears on the page &lt;em&gt;Cap. XXXVII. De Regula fals. 287, Demonstratio&lt;/em&gt; of this Latin version of &lt;a href=&quot;http://www.filosofia.unimi.it/cardano/testi/operaomnia/vol_4_s_4.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Artis magnae sive de regulis algebraicis, liber unus&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Cd2PVb.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Cd2PVb.jpg&quot; alt=&quot;Artis magnae sive de regulis algebraicis, liber unus, Cap. XXXVII. De Regula fals. 287, Demonstratio&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4130" LastEditorUserId="4130" LastEditDate="2020-11-23T11:33:29.460" LastActivityDate="2020-11-23T11:33:29.460" Title="Origin of notation &quot;R with a stroke on the leg&quot; for the square-root (℞)" Tags="|mathematics|reference-request|notation|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12484" PostTypeId="1" AcceptedAnswerId="12490" CreationDate="2020-11-22T12:51:51.667" Score="5" ViewCount="464" Body="&lt;p&gt;In the second part of his fourth proof of the fundamental theorem of the algebra (published in 1849 [1]), entitled &amp;quot;A contribution to the theory of algebraic equations&amp;quot;, Gauss gives a new method for numerical solution of trinomial equations, i.e equations of the form &lt;span class=&quot;math-container&quot;&gt;$x^n + fx^m +g = 0$&lt;/span&gt;, which also includes the case when &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are complex. For example, the Bring radical &lt;span class=&quot;math-container&quot;&gt;$x^5+px+q = 0$&lt;/span&gt; belongs to this class of equations.&lt;/p&gt;&#xA;&lt;p&gt;Numerical methods for solving trinomial equations date back to the time of J.H. Lambert and L. Euler, who gave a solution in the form of a series expansion. However, Gauss remarks in his 1849 publication that these series converge very slowly in the case when two roots are very close to each other, and that in the extreme case when there are multiple roots (i.e when two roots are equal) the convergence is weaker than that of any geometric progression. He says that his method is better suited for these cases, and mentions that he had already applied this method in 1843 for the solution of a certain cubic equation occurring in astronomy [2] (the last fact is interesting, since cubic equations need not be numericaly solved - they can be solved by radicals).&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, as is usually the case with Gauss, he goes directly to describing his algorithm without revealing his actual thought process. So, I just wanted to check if anyone can tell something about Gauss's method. Maybe someone who is familiar with numerical methods for solving equations in the complex plane can help here.&lt;/p&gt;&#xA;&lt;p&gt;EDIT&lt;/p&gt;&#xA;&lt;p&gt;[1]&#xA;&amp;quot;Beitraege zur Theorie der algebraischen Gleichungen.&amp;quot; Abhandlungen der Koeniglichen Gesellschaft der Wissenschaften zu Goettingen 4. 1849. &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=nyp.33433089737823&amp;amp;view=1up&amp;amp;seq=85&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[2]&#xA;&amp;quot;Astronomische Nachrichten No. 474.&amp;quot; Astronomische Nachrichten 20. Akademie-Verlag, 1843. &lt;a href=&quot;https://archive.org/details/bub_gb_e1c_AAAAcAAJ/page/n153/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link.&lt;/a&gt; Gauss's letter appears on pp. 299-300.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-11-23T10:56:52.620" LastActivityDate="2021-02-22T15:25:59.673" Title="What was Gauss's method for solving the &quot;trinomial equation&quot;?" Tags="|gauss|algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12485" PostTypeId="1" CreationDate="2020-11-22T16:33:46.980" Score="2" ViewCount="118" Body="&lt;p&gt;In mathematical logic, where does the adjective &amp;quot;geometric&amp;quot; comes from, in terms like &amp;quot;geometric theories&amp;quot; and &amp;quot;geometric logic&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;These terms come up in fields like topos theory, e.g. in works such as the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/pdf/1006.3930.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The unification of Mathematics via Topos Theory&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6208" LastActivityDate="2020-11-24T12:13:14.337" Title="Where does the name &quot;geometric theory&quot; come from?" Tags="|mathematics|terminology|mathematical-logic|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12486" PostTypeId="2" ParentId="12485" CreationDate="2020-11-22T21:39:55.733" Score="3" Body="&lt;p&gt;&lt;em&gt;This answer is CW, so I don't gain reputation for someone else's work.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;To quite &lt;a href=&quot;https://math.stackexchange.com/a/3911599/28111&quot;&gt;Kevin Arlin's answer&lt;/a&gt; to the same question on math.stackexchange:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Geometric logic constitutes the logic, models of whose theories are preserved by geometric &lt;em&gt;morphisms&lt;/em&gt; between topoi. Geometric morphisms are those appropriate to toposes viewed as generalized spaces, for instance, identifying the topos of sheaves on a topological space, or on a locale, with the space itself. Historically, toposes were first introduced by Grothendieck's school to model generalized &lt;em&gt;algebro-geometric&lt;/em&gt; spaces, which explains why these morphisms are called geometric, rather than topological.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="8037" LastActivityDate="2020-11-22T21:39:55.733" CommentCount="0" CommunityOwnedDate="2020-11-22T21:39:55.733" ContentLicense="CC BY-SA 4.0" />
  <row Id="12487" PostTypeId="1" CreationDate="2020-11-22T22:41:34.260" Score="2" ViewCount="119" Body="&lt;p&gt;I am reading some random bits from Peirce's collected works and they give me the impression that Peirce tried to integrate every(or nearly every) major scientific and mathematical concept which he was aware of(or some conceptual negation that he derived from it) in his philosophy.&#xA;Were there any descriptions of a close enough concept to abductive reasoning in the works of any previous mathematician?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-11-23T15:26:39.500" LastActivityDate="2020-12-26T00:03:30.400" Title="Was C.S Peirce in his abductive reasoning influenced by the generalization from special cases that is used by mathematicians?" Tags="|biographical-details|philosophy-of-science|mathematical-logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12488" PostTypeId="2" ParentId="12485" CreationDate="2020-11-22T23:45:49.857" Score="0" Body="&lt;p&gt;The geometry in geometric theories, as in topoi, comes from topology and hence also the name topos. They, that is the topoi, are seen as generalisation of a topology. This generalisation was by Grothendieck and generalises that of a topological covering which is, in his language, a sieve. And in this way we get elementary topoi, which was lawvere's axiomatisation of Grothendieck topoi, that is of sheaves on the poset of opens of underlying topological space, reverse ordered by inclusion.&lt;/p&gt;&#xA;&lt;p&gt;Every topos has an internal language as generally seen in the underlying language of logic and it turns out that each topos has an intuitionistic logic.&lt;/p&gt;&#xA;&lt;p&gt;Generally, in mathematics, geometry refers to the metric as this is used to measure distances and angles; here, it refers to the aspect of continuity of geometry and hence the geometric moniker.&lt;/p&gt;&#xA;&lt;p&gt;Given that each topos has a logic, this is why these logics are termed geometric logic.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-24T12:13:14.337" LastActivityDate="2020-11-24T12:13:14.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12489" PostTypeId="2" ParentId="12487" CreationDate="2020-11-22T23:56:05.017" Score="-2" Body="&lt;p&gt;Poincare mentions that inductive reasoning, in his &lt;em&gt;Geometry and the Imagination&lt;/em&gt;, that is, generalising from the special to the general case as being the essence of mathematics, and that of science.&lt;/p&gt;&#xA;&lt;p&gt;I don't know if Pierce was familiar with this work, but I assume it is likely that he was ...&lt;/p&gt;&#xA;&lt;p&gt;(It's worth adding that this is not special to science, it's a general feature of the rational faculty of the human mind and seen very much in evidence in everyday life).&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-25T23:08:58.170" LastActivityDate="2020-11-25T23:08:58.170" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12490" PostTypeId="2" ParentId="12484" CreationDate="2020-11-23T02:45:25.747" Score="7" Body="&lt;p&gt;I think I can shed some sort of light on the other methods in use for solving trinomials (and general polynomials) from Lambert (1758) to Langrange (1770) and Euler (1776), as well as the (more specialized) method of Gauss.&lt;/p&gt;&#xA;&lt;h2&gt;BACKGROUND&lt;/h2&gt;&#xA;&lt;p&gt;In 1758, Johann Heinrich Lambert published &lt;em&gt;Observationes Variae in Mathesin Puram&lt;/em&gt; (Various observation on pure mathematics) in &lt;em&gt;Acta Helvetica Physico-Mathematico-Botanico-Medica&lt;/em&gt; (specifically, Acta Helvetica Vol III). In &lt;span class=&quot;math-container&quot;&gt;$\S$&lt;/span&gt;29 he considers the general polynomial,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$0 = x^m - Ax^{m-1}+Bx^{m-2}-...+Hx^2-Ix+K$$&lt;/span&gt;&#xA;Where &lt;span class=&quot;math-container&quot;&gt;$x=\alpha, \beta, \gamma, \delta, etc$&lt;/span&gt; are the &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; roots of the equation.&lt;/p&gt;&#xA;&lt;p&gt;Now take the sums of the roots, their squares, their cubes, and so on, and denote these sums by &lt;span class=&quot;math-container&quot;&gt;$\Sigma r$&lt;/span&gt; for the sum of roots, &lt;span class=&quot;math-container&quot;&gt;$\Sigma r^2$&lt;/span&gt; for the sum of the squares of the roots, etc. So,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;\alpha+\beta+\gamma+\delta+... &amp;amp;= \Sigma r \\&#xA;\alpha^2+\beta^2+\gamma^2+\delta^2+... &amp;amp;= \Sigma r^2 \\&#xA;\alpha^3+\beta^3+\gamma^3+\delta^3+... &amp;amp;= \Sigma r^3 \\&#xA;&amp;amp;...&#xA;\end{align*}$$&lt;/span&gt;&#xA;Next, substitute the roots into the original expression as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;0 &amp;amp;= \alpha^m - A \alpha^{m-1} + B\alpha^{m-2}-...+H\alpha^2-I\alpha+K. \\&#xA;0 &amp;amp;= \beta^m - A \beta^{m-1} + B\beta^{m-2}-...+H\beta^2-I\beta+K. \\&#xA;0 &amp;amp;= \gamma^m - A \gamma^{m-1} + B\gamma^{m-2}-...+H\gamma^2-I\gamma+K. \\&#xA;0 &amp;amp;= \delta^m - A \delta^{m-1} + B\delta^{m-2}-...+H\delta^2-I\delta+K. \\&#xA;&amp;amp;...&#xA;\end{align*}&#xA;$$&lt;/span&gt;&#xA;Add all of these equations together, to obtain&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0 = \Sigma r^m - A \Sigma r^{m-1} + B\Sigma r^{m-2}-...+H\Sigma r^2-I\Sigma r+mK.&#xA;$$&lt;/span&gt;&#xA;And thus, we can solve for &lt;span class=&quot;math-container&quot;&gt;$\Sigma r^m$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Sigma r^m = A \Sigma r^{m-1} - B\Sigma r^{m-2}+...-H\Sigma r^2+I\Sigma r-mK.$$&lt;/span&gt;&#xA;Take &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; to be &lt;span class=&quot;math-container&quot;&gt;$1, 2, 3, ...$&lt;/span&gt;, and we have expressions for the different sums,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;\Sigma r &amp;amp;= A \\&#xA;\Sigma r^2 &amp;amp;=  A \Sigma r - 2B\\&#xA;\Sigma r^3 &amp;amp;=  A \Sigma r^2 - B \Sigma r + 3C\\&#xA;\Sigma r^4 &amp;amp;=  A \Sigma r^3 - B \Sigma r^2 + C \Sigma r - 4D\\&#xA;\Sigma r^5 &amp;amp;=  A \Sigma r^4 - B \Sigma r^3 + C \Sigma r^2 - D \Sigma r + 5E &#xA;\end{align*}&#xA;$$&lt;/span&gt;&#xA;We can substitute recursively, as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;\Sigma r &amp;amp;= A \\&#xA;\Sigma r^2 &amp;amp;= A^2-2B \\&#xA;\Sigma r^3 &amp;amp;= A^3-3AB+3C \\&#xA;\Sigma r^4 &amp;amp;= A^4-4A^2 B + 2B^2 + 4AC - 4D \\&#xA;&amp;amp;...&#xA;\end{align*}$$&lt;/span&gt;&#xA;But these expressions still do not give us the roots directly.&lt;/p&gt;&#xA;&lt;p&gt;Instead, Lambert makes a critical assumption: assume there exists a root which is purely real, and which has a larger real part than all other roots' real parts. (Lambert's assumption becomes the condition for convergence of his series solution. ) Then, taking the ratio of two sums, in the limit as &lt;span class=&quot;math-container&quot;&gt;$m\rightarrow \infty$&lt;/span&gt;, all terms go to zero but the leading powers.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{m\rightarrow \infty} \frac{\alpha^m + \beta^m + \gamma^m + \delta^m + ...}{\alpha^{m-1}+\beta^{m-1}+\gamma^{m-1}+\delta^{m-1}+...}  = \frac{\alpha^m}{\alpha^{m-1}} = \alpha$$&lt;/span&gt;&#xA;The series representation of this root can be found as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$S = \frac{A \Sigma r^{m-1} - B\Sigma r^{m-2}+...-H\Sigma r^2+I\Sigma r-mK}{A \Sigma r^{m-2} - B\Sigma r^{m-3}+...+H\Sigma r^2-I\Sigma r+(m-1)K}$$&lt;/span&gt;&#xA;In using this to solve for the trinomial,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^m + px = q$$&lt;/span&gt;&#xA;Lambert obtains the series solution,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = \frac{q}{p} - \frac{q^m}{p^{m+1}} + m \frac{q^{2m-1}}{p^{2m+1}} - m \frac{3m-1}{2!} \frac{q^{3m-2}}{p^{3m+1}} + m \frac{(4m-1)(4m-2)}{3!} \frac{q^{4m-3}}{p^{4m+1}} - ...$$&lt;/span&gt;&#xA;Which I've put into summation-notation form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$S = \sum_{k=0}^\infty (-1)^k \frac{q^{k(m-1)+1}} {p^{km+1}} \frac{1}{k!} \prod_{j=0}^{k-2}(km-j)$$&lt;/span&gt;&#xA;Details of the actual derivation can be found in &lt;em&gt;Observationes&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\S$&lt;/span&gt;38.&lt;/p&gt;&#xA;&lt;p&gt;Lambert's solution was used by Euler in 1776 to obtain his series solution for a symmetric trinomial, that is, a trinomial symmetric in parameters &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; of the form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^\alpha - x^\beta = (\alpha-\beta)vx^{\alpha+\beta}$$&lt;/span&gt;&#xA;But his solution technique was the same as Lambert's, so he doesn't go into it, and neither will I. (I did try to derive my own series solution, but obtained a slightly different answer than Euler; I wrote to him regarding the discrepancy, but have yet to hear back).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In 1770, six years prior to Euler's &lt;em&gt;De Serie Lambertina&lt;/em&gt;, J-L. Lagrange published &lt;em&gt;Nouvelle méthode pour résourdre les équation littérales par le moyen des séries.&lt;/em&gt; (Mémoires de l'Académie royale des Sciences et Belles-Letters de Berline, t. XXIV, 1770). In it, he describes a new method of series expansion for the roots of a general polynomial, of any degree, and suitable for calculating &lt;em&gt;any&lt;/em&gt; roots (and their powers).&lt;/p&gt;&#xA;&lt;p&gt;The first method employed appears to have been known for some time, and it is distinct still from Lambert's series. It uses logarithms, and though it gets more involved (I have a translation of the first section on my History of Math and Science blog here) it begins in a fairly straightforward way.&lt;/p&gt;&#xA;&lt;p&gt;Take as our general equation,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ a - bx + cx^2 - dx^3 +... $$&lt;/span&gt;&#xA;Whose roots are &lt;span class=&quot;math-container&quot;&gt;$p, q, r, ...$&lt;/span&gt;, so that, from elementary algebra, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ a-bx+cx^2-dx^3+...=a\left(1-\frac{1}{p}\right)\left(1-\frac{x}{q}\right)\left(1-\frac{x}{r}\right)...$$&lt;/span&gt;&#xA;Divide by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, and take the log of both sides,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\log \left(1-\frac{bx-cx^2+dx^3-...}{a}\right) = \log \left(1-\frac{x}{p}\right) + \log \left(1-\frac{x}{q}\right) + \log \left(1-\frac{x}{r}\right) + ...&#xA;$$&lt;/span&gt;&#xA;Now Lagrange uses the fact that &lt;span class=&quot;math-container&quot;&gt;$\log(1-u) = -u - \frac{u^2}{2} - \frac{u^3}{3}-...$&lt;/span&gt; to obtain&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;    \log &amp;amp;\left(1-\frac{x}{p}\right) + \log \left(1-\frac{x}{q}\right) + \log \left(1-\frac{x}{r}\right) + ... \\&#xA;    = &amp;amp;- x\left(\frac{1}{p} + \frac{1}{q} + \frac{1}{r}+...\right) \\&#xA;    &amp;amp;-\frac{x^2}{2}\left(\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}+...\right)\\&#xA;    &amp;amp;-\frac{x^3}{3}\left(\frac{1}{p^3} + \frac{1}{q^3} + \frac{1}{r^3}+...\right)\\ &#xA;    &amp;amp;.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\&#xA;    \end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Expand: &lt;span class=&quot;math-container&quot;&gt;$-\log \left(1-\frac{bx-cx^2+dx^3-...}{a}\right)$&lt;/span&gt; into a power series &lt;span class=&quot;math-container&quot;&gt;$Ax + Bx^2+Cx^3$&lt;/span&gt; to obtain:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;    A &amp;amp;= \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + ...,\\&#xA;    2B &amp;amp;= \frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2} + ...,\\&#xA;    3C &amp;amp;= \frac{1}{p^3} + \frac{1}{q^3} + \frac{1}{r^3} + ...,\\&#xA;    &amp;amp;.\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\&#xA;    \end{align*}&#xA;    $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, since we assumed a power series exists in the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$-\log \left(1-\frac{bx-cx^2+dx^3-...}{a}\right) = Ax + Bx^2 + Cx^3 + ...,$$&lt;/span&gt;&#xA;Taking the differential of both sides, and divide by &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{b-2cx+3dx^2-...}{a-bx+cx^2-dx^3+...} = A + 2Bx + 3Cx^2 + ...,$$&lt;/span&gt;&#xA;And finally cross multiplying, we have formulas which, Lagrange says, were known even to Newton.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align*}&#xA;    A &amp;amp;=\frac{b}{a}, \\&#xA;    2B &amp;amp;= \frac{Ab-2c}{a},\\&#xA;    3C &amp;amp;= \frac{2Bb-Ac+3d}{a}&#xA;    \end{align*}$$&lt;/span&gt;&#xA;Lagrange goes on, obtaining (after much manipulation) a general expression, and as a particular case, an expression for the general trinomial.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;GAUSS'S METHOD&lt;/h2&gt;&#xA;&lt;p&gt;We can compare these two methods with what Gauss is doing in his work. However, it will soon become clear that Gauss's work is rather unique in its approach.&lt;/p&gt;&#xA;&lt;p&gt;He begins with the trinomial,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^{m+n} \pm e x^m \pm f = 0$$&lt;/span&gt;&#xA;Under the conditions that the desired root is real, and &lt;span class=&quot;math-container&quot;&gt;$m,n,e,f &amp;gt; 0$&lt;/span&gt;, and (because it is sufficient) &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; have no common divisor. Doing away with the case where &lt;span class=&quot;math-container&quot;&gt;$x^{m+n} + ex^m + f = 0$&lt;/span&gt; (which cannot have positive real roots) we take the first case,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^{m+n} + e x^m - f = 0$$&lt;/span&gt;&#xA;I'll fill in some of the gaps with the reasoning here. With a bit of rearranging, we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{x^{m+n}}{f} + \frac{ex^m}{f} = 1$$&lt;/span&gt;&#xA;Noting the similarities between this expression and the Pythogorean identity,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \sin^2 \theta + \cos^2 \theta = 1$$&lt;/span&gt;&#xA;We introduce the quantity &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;, under the constraint that &lt;span class=&quot;math-container&quot;&gt;$0 \le \theta \le 90^o$&lt;/span&gt;, such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sin^2 \theta = \frac{x^{m+n}}{f},\ \ \ \cos^2 \theta = \frac{ex^m}{f}$$&lt;/span&gt;&#xA;Note that the existence of &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is guaranteed by the Pythagorean identity. We can use &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; to solve for &lt;span class=&quot;math-container&quot;&gt;$x^{m+n}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$x^m$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;. The first two are given immediately:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^{m+n} = f \sin^2 \theta,\ \ \ x^m = \frac{f \cos^2 \theta}{e}$$&lt;/span&gt;&#xA;To find &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;, start with the equation for &lt;span class=&quot;math-container&quot;&gt;$x^{m+n}$&lt;/span&gt; and multiply throughout by &lt;span class=&quot;math-container&quot;&gt;$x^{-m}$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^{-m}\frac{x^{m+n}}{f} = x^{-m}\sin^2 \theta$$&lt;/span&gt;&#xA;From the equation for &lt;span class=&quot;math-container&quot;&gt;$x^m$&lt;/span&gt; we know&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^{-m} = \frac{e}{f \cos^2 \theta}$$&lt;/span&gt;&#xA;Finally giving&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{x^n}{f} = \frac{e}{f} \frac{\sin^2 \theta}{\cos^2 \theta}$$&lt;/span&gt;&#xA;Or equivalently,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^n = e \tan^2 \theta$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now we wish to solve for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; in terms of the coefficients &lt;span class=&quot;math-container&quot;&gt;$e,f$&lt;/span&gt;. To eliminate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, we must find a way to solve for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in two equations so that the expressions can be set equal. In fact, if we have &lt;span class=&quot;math-container&quot;&gt;$x^{m+n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x^n$&lt;/span&gt;, then we can make &lt;span class=&quot;math-container&quot;&gt;$x^{m(m+n)}$&lt;/span&gt; from both; the first by raising it to the power &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, the other by raising to &lt;span class=&quot;math-container&quot;&gt;$m+n$&lt;/span&gt;. Note, this is just one way, and it's convenient because it only involves positive powers (useful for Gauss's hand-calculations with tables). We get:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;&amp;amp;(x^{m})^{m+n} = \left(\frac{f}{e}\cos^2 \theta\right)^{m+n}\\&#xA;&amp;amp;x^{m(m+n)} = \frac{f^{m+n}}{e^{m+n}}\cos^{2m+2n} \theta\\&#xA;&amp;amp;(x^{m+n})^m = \left( f \sin^2 \theta \right)^m\\&#xA;&amp;amp;x^{m(m+n)} = f^m \sin^{2m} \theta&#xA;\end{align*}&#xA;$$&lt;/span&gt;&#xA;Now set these equal and we get our solution,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{f^n}{e^{m+n}} = \frac{sin^{2m}\theta}{\cos^{2m+2n}\theta} $$&lt;/span&gt;&#xA;Because the left hand side appears in other derivations, Gauss denotes it by &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In order to solve the trinomial, then, solve for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; in the above equation. Gauss assures that the right hand side is monotonically increasing for &lt;span class=&quot;math-container&quot;&gt;$0 \le \theta \le 90^o$&lt;/span&gt;, and so the root is uniquely determined.&lt;/p&gt;&#xA;&lt;p&gt;This case, and the case of &lt;span class=&quot;math-container&quot;&gt;$x^{m+n}-ex^m-f=0$&lt;/span&gt;, are comparatively simple, while the third case, &lt;span class=&quot;math-container&quot;&gt;$x^{m+n}-ex^m+f=0$&lt;/span&gt;, has some more considerations. But they all follow this general procedure. In the remainder of the paper, these solutions are put into a form using logarithms, which is where the utility becomes more apparent, as extensive log tables were available.&lt;/p&gt;&#xA;&lt;h2&gt;CONCLUSION&lt;/h2&gt;&#xA;&lt;p&gt;Gauss saw the issue with the existing methods (slow to converge, difficult to compute by hand) and provided a new method which, while more specialized, is very efficient for hand calculation, approximating a purely real root of the given trinomial. There's more to be learned from his solution, e.g. which root does it actually give? And then, can we use polynomial long division to find the remaining roots? But these I'll leave for a post e.g. on Math.StackExchange.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2021-02-22T15:25:59.673" LastActivityDate="2021-02-22T15:25:59.673" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12491" PostTypeId="2" ParentId="12485" CreationDate="2020-11-23T10:20:38.933" Score="2" Body="&lt;p&gt;Negri and collaborators call &amp;quot;geometric&amp;quot; theories that can be axiomatized by FOL formulas of a special sort, namely universal closures of &lt;span class=&quot;math-container&quot;&gt;$A\to B$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; do not contain &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$∀$&lt;/span&gt;, see e.g. &lt;a href=&quot;https://link.springer.com/article/10.1007/s001530100124&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contraction-free sequent calculi for geometric&#xA;theories with an application to Barr’s theorem&lt;/a&gt;. The terminology goes back to, at least, &lt;a href=&quot;https://www.google.com/books/edition/Constructivism_in_Mathematics_Vol_1/-tc2qp0-2bsC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Constructivism in Mathematics by Troelstra and van Dalen (1988)&lt;/a&gt;, where it is used in exercise 2.6.14. Such theories have  particularly nice proof theory, e.g. a classical proof  of a geometric implication in a geometric theory can be  transformed  into a constructive proof (Barr's theorem). This was originally proved in what they call &amp;quot;classical geometric theories&amp;quot; modeled in Grothendieck topoi, hence the name:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Examples of geometric theories are given by Robinson arithmetic, the theory of&#xA;constructive plane affine geometry, the theory of ordered fields and of real closed&#xA;fields... In the last section we apply our method to a general result on geometric theories. The result states that if a geometric implication is provable classically&#xA;in a geometric theory, then it is provable intuitionistically.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This result is proved in topos theory by using a completeness theorem for geometric theories in&#xA;Grothendieck topoi and the construction of a suitable Boolean topos out of a Grothendieck topos.&#xA;By our method, the result reduces to a proof-theoretical triviality: A classical proof of a geometric implication in a geometric theory formulated as a sequent&#xA;system with rules is an intuitionistic proof already.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The name is doubly apt since a recent &lt;a href=&quot;https://arxiv.org/abs/0810.4315&quot; rel=&quot;nofollow noreferrer&quot;&gt;formalization of Euclidean geometry by Avigad, Dean and Mumma&lt;/a&gt;, that attempts (unlike Hilbert's) to mirror Euclid's proofs as closely as possible, is (nearly) geometric in the above sense, and their proof of its completeness uses Negri's results:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Those geometric formulas with only a single disjunct in the consequent (i.e. geometric formulas in which &lt;span class=&quot;math-container&quot;&gt;$∨$&lt;/span&gt; does not appear) are called regular. Note that, on our modeling, Euclid’s propositions are almost of this latter form, the difference being that arbitrary literals (negated atomic formulas as well as atomic formulas) are allowed in the antecedent and consequent. Sara Negri [42], building on earlier joint work with Jan von Plato [43], has established a cut-elimination theorem for geometric theories that we can put to use in our completeness proof.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-11-23T10:20:38.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12492" PostTypeId="1" CreationDate="2020-11-22T10:30:26.663" Score="5" ViewCount="543" Body="&lt;p&gt;The first, known to me, drawing of a simulated view of Earth from outside—roughly passable* by modern standards—is this pic from 1834 attributed to Henry De la Beche&#xA;&lt;a href=&quot;https://i.stack.imgur.com/m4ei0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m4ei0.jpg&quot; alt=&quot;drawing of the Eastern hemishpere&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;sub&gt;(source: &lt;a href=&quot;https://www.abroadintheyard.com/wp-content/uploads/1834-Researches-in-Theoretical-Geology-Henry-de-la-Beche.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;abroadintheyard.com&lt;/a&gt;)&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;Was he the first who tried hard? Or you know earlier artworks presenting the scene using the best scientific knowledge available?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt; * De la Beche drew a large dark spot somewhere around Lake Chad in Africa, but completely missed a permanent (and anyway much larger) Lake Victoria. And the limb doesn’t look impressive – the most striking error is overextended Sea of Okhotsk. Otherwise the image appears to have a fairly good accuracy for a distant (a hundred megametres or more) view of the planet.&lt;/p&gt;&#xA;" OwnerUserId="13360" OwnerDisplayName="Incnis Mrsi" LastEditorUserId="2277" LastEditDate="2022-12-26T00:14:33.823" LastActivityDate="2022-12-26T00:14:33.823" Title="Early geographically accurate drawings of Earth" Tags="|earth-sciences|cosmology|geography|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12493" PostTypeId="2" ParentId="12492" CreationDate="2020-11-23T15:22:38.027" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Early_world_maps&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; has a good list of such maps. The first one for me that shows the continents roughly as we see today (Minus Antarctica) seems to be the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kunyu_Wanguo_Quantu&quot; rel=&quot;noreferrer&quot;&gt;Kunyu Wanguo Quantu 1602&lt;/a&gt; map, included below, which is only missing Australia and Antarctica, plus a fair bit of the North American continent. Another candidate, included below, is the &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Orbis_Terrarum_Nova_et_Accuratissima_Tabula_by_Nicolaes_Visscher,_1658.jpg&quot; rel=&quot;noreferrer&quot;&gt;Orbis Terrarum 1658&lt;/a&gt; map which shows even Australia. Lastly, the first with the outlines of everything except for Antarctica more or less correct is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Early_world_maps#/media/File:1794_Samuel_Dunn_Wall_Map_of_the_World_in_Hemispheres_-_Geographicus_-_World2-dunn-1794.jpg&quot; rel=&quot;noreferrer&quot;&gt;Samuel Dunn's map of the world (1794)&lt;/a&gt;, which is the final one below.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7f0eL.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7f0eL.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6ws3b.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6ws3b.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/35sIK.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/35sIK.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13355" LastActivityDate="2020-11-23T15:22:38.027" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12494" PostTypeId="1" AcceptedAnswerId="12508" CreationDate="2020-11-23T19:33:49.480" Score="0" ViewCount="197" Body="&lt;p&gt;Hoping a philomath can help - I'm searching how Archimedes calculates the centroid of a hemisphere without calculus  as behind the below sentence in the  wikipedia entry on &lt;em&gt;&lt;strong&gt;&amp;quot;the Method of Mechanical Theorems&amp;quot;&lt;/strong&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;This type of method* &lt;em&gt;[leverage argument - please see wikipedia]&lt;/em&gt; &amp;quot;can be used to find the area of an arbitrary section of a parabola, and similar arguments can be used to find the integral of any power of x, although higher powers become complicated without algebra. &lt;em&gt;&lt;strong&gt;Archimedes only went as far as the integral of x3, which he used to find the center of mass of a hemisphere, and in other work, the center of mass of a parabola.&amp;quot;&lt;/strong&gt;&lt;/em&gt; ...&#xA;&amp;quot;Other propositions in the palimpsest&#xA;&amp;quot;A series of propositions of geometry are proved in the palimpsest by similar arguments. One theorem is that the location of a center of mass of a hemisphere is located 5/8 of the way from the pole to the center of the sphere. This problem is notable, because it is evaluating a cubic integral.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&lt;/a&gt;&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I found reference to Proposition 12 in Archimedes, the Center of Gravity, and the First Law of Mechanics&#xA;2nd edition The Law of the Lever Andre K.T. Assis which is &amp;quot;If any number of straight lines drawn from the origin to meet the spiral make equal angles with one another, the lines will be in arithmetical progression.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I am intrigued how Archimedes could use the integral of a cube by the leverage argument to determine the centre of mass of a hemisphere and a parabola?  I know the proofs by calculus which Archimedes did not have full access to - although he seems to have used some of its concepts.  Archimedes intuitive and elegant method of leverage has much didactic benefit - but this part escapes me.&lt;/p&gt;&#xA;&lt;p&gt;Maths stackexchange refers to Pappus but also has footnotes referring to Archimedes but again no explanation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/questions/387640/compute-the-centroid-of-a-semicircle-without-calculus&quot;&gt;https://math.stackexchange.com/questions/387640/compute-the-centroid-of-a-semicircle-without-calculus&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Rupert&lt;/p&gt;&#xA;" OwnerUserId="6396" LastActivityDate="2020-11-25T11:29:17.863" Title="Archimedes - method of mechanical theorems - centroid of hemisphere" Tags="|archimedes|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12495" PostTypeId="1" CreationDate="2020-11-23T22:20:30.557" Score="1" ViewCount="59" Body="&lt;p&gt;I remember reading something to the effect that the moral and aesthetic preferences influence scientific belief in his collected works.&#xA;Was C.S Peirce the first philosopher to posit that value belief serves as some sort of guide for scientific belief?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-11-23T22:49:28.027" LastActivityDate="2020-11-24T05:08:15.167" Title="Was C.S Peirce the first philosopher to posit that value belief serves as some sort of guide for scientific belief?" Tags="|biographical-details|discoveries|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12498" PostTypeId="2" ParentId="12495" CreationDate="2020-11-24T04:35:45.047" Score="-1" Body="&lt;p&gt;In his &lt;em&gt;Metaphysics&lt;/em&gt;, Aristotle says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To say of what is, that it is not; or if what is not, that it is, is false; whilst to say of what is, that it is; and that what is not, is not, is truth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would not normally have suggested that truth was a value, assuming that all held to this; but given the disinformation climate of our times, it seems that truth is a value, and one not held by all. And in this situation, it seems that Aristotle was one of the first to say so. But of course he was simply paraphrasing Parmenides here; for example, in Parmenides poem, he says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Come now, I will tell thee - and do thou hearken to my saying, the only two ways of searching that can be thought of: the first, that it is, and that it is impossible for it not to be - is the way of conviction, for truth is it companion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The other, namely that it is not, and that something needs not be - that I tell you is a wholly untrustworthy path. For you cannot know what is not - that is impossible - nor utter it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and possibly Parmenides was paraphrasing Zoroaster - given that there was commerce between the Greeks and Persians - and who is known for taking truth to be a religious value. For example, one of of his &lt;em&gt;gathas&lt;/em&gt; (hymns) has:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I approach you with Good Thought, O Mazda Ahura, so that you may grant me the (blessing) of two existences, the material and that of thought. The blessing emanating from Truth, which one can put support in with comfort.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-24T05:08:15.167" LastActivityDate="2020-11-24T05:08:15.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12500" PostTypeId="5" CreationDate="2020-11-24T10:45:17.223" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-11-24T10:45:17.223" LastActivityDate="2020-11-24T10:45:17.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12501" PostTypeId="4" CreationDate="2020-11-24T10:45:17.223" Score="0" Body="Ptolemy was an astronomer, mathematician, and geographer of Greek descent who flourished in Alexandria during the 2nd century CE." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2020-12-17T11:10:49.503" LastActivityDate="2020-12-17T11:10:49.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12502" PostTypeId="1" CreationDate="2020-11-24T16:12:45.680" Score="0" ViewCount="102" Body="&lt;p&gt;It seems to me that once black holes were theorised then the obvious singularity at its centre - though only ratified after Hawking and Penrose's singularity theorems - one would have to ask where the matter and energy went to, if we are to save the conservation of mass-energy. After Einstein &amp;amp; Rosen theorised the Einstein-Rosen Bridge, the otherwise known, wormhole, the obverse of a black hole could be thought: that is a white hole. Here, matter or light can't enter it, but they can escape.&lt;/p&gt;&#xA;&lt;p&gt;Was this the earliest conceptualisation of such, or was there an earlier conceptualisation? There's very little history about this in the usual sources.&lt;/p&gt;&#xA;&lt;p&gt;**@Ben Crowell: Although I dreamt up the question, I don't think of it as a 'nice' question as its a little too simple-minded for that. It's no mistake on my part, as I'm referring to the original notion of a black hole by John Michell in the late 19th C though he called them dark stars. And the insight that gravity would produce a singularity was the insight of Roger Boscovitch if there was no countervailing pressure - though for him - this led to the notion of atoms and interatomic forces that push back against gravity in order that matter is stable - and this was well before atoms became fashionable.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-26T01:26:20.077" LastActivityDate="2020-11-26T01:26:20.077" Title="When were white holes theorised?" Tags="|physics|einstein|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12503" PostTypeId="1" AcceptedAnswerId="12507" CreationDate="2020-11-24T18:01:39.850" Score="3" ViewCount="135" Body="&lt;p&gt;Since Euler's result on the Seven Bridges of Königsberg problem appears to be the first known graph theoretical result I was wondering if any mathematicians after Euler and before Hamilton commented on it or investigated any other similar results. Perhaps prominent mathematicians like Lagrange, Laplace or Gauss.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="55" LastEditDate="2020-11-25T08:44:21.593" LastActivityDate="2020-11-25T08:44:21.593" Title="Was there a continuation to the Seven Bridges of Königsberg problem between Euler and Hamilton?" Tags="|biographical-details|graph-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12504" PostTypeId="1" CreationDate="2020-11-24T19:47:00.107" Score="1" ViewCount="175" Body="&lt;p&gt;Do we have any evidence on how Charles Sanders Peirce reacted to his stigmatization by various academics? By stigmatization, I mean denying him tenure at Johns Hopkins, leaving the house when they learned that he was staying for the night, etc.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2021-03-09T09:55:28.750" LastActivityDate="2021-03-09T09:55:28.750" Title="How did Charles Sanders Peirce react to his stigmatization?" Tags="|biographical-details|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12505" PostTypeId="2" ParentId="12502" CreationDate="2020-11-24T21:49:39.167" Score="1" Body="&lt;p&gt;This is a nice question. As a preliminary, it contains a couple of physics mistakes.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It seems to me that once black holes were theorised then the obvious singularity at its centre&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The singularity of a black hole is not a point at its center, it's a spacelike singularity.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;one would have to ask where the matter and energy went to, if we are to save the conservation of mass-energy&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Conservation of mass-energy is not a principle that can be formulated in GR for all spacetimes, although for black hole spacetimes it can be, because they're asymptotically flat. There is no difficulty with explaining where the mass-energy ends up. Given any event P, a spacelike surface can be constructed that includes P and that intersects the world-line of all infalling matter.&lt;/p&gt;&#xA;&lt;p&gt;The Penrose singularity theorem is not particularly relevant. White holes are not expected to form as the result of gravitational collapse, for reasons independent of the reasons that originally led people to disbelieve the formation of black holes by gravitational collapse, until the Penrose singularity theorem.&lt;/p&gt;&#xA;&lt;p&gt;So anyway, about the history: --&lt;/p&gt;&#xA;&lt;p&gt;Finkelstein proposed the one-way membrane interpretation of the Schwarzschild spacetime in a 1958 paper. In the same paper, he references a communication from Kruskal describing the maximal extension of this spacetime, which includes a region referred to as region IV or, later, the &amp;quot;white hole&amp;quot; region.&lt;/p&gt;&#xA;&lt;p&gt;The name &amp;quot;black hole&amp;quot; was probably current among some relativists as early as the early 60's (Herdeiro and Lemos), but was popularized in a lecture by Wheeler in 1967, which was published in written form in 1968. A google ngrams search shows that both &amp;quot;black hole&amp;quot; and &amp;quot;white hole&amp;quot; start to rise above background noise around 1968. Although it was clear immediately that white holes should not arise as the result of gravitational collapse, and were not expected to exist as astronomical objects, people did publish speculative/flaky ideas to the contrary as early as 1971. Examples are Robert Hjellming in 1971 and Narlikar in 1975.&lt;/p&gt;&#xA;&lt;p&gt;So the conceptual apparatus of black holes and white holes appeared at the same time, in 1958, and both words began appearing in print at the same time, around 1968.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;References&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finkelstein, Phys. Rev. 110, 965–967 (1958), &amp;quot;Past-Future Asymmetry of the Gravitational Field of a Point Particle,&amp;quot; downloadable from his web page at &lt;a href=&quot;https://www.physics.gatech.edu/user/david-finkelstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.physics.gatech.edu/user/david-finkelstein&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Herdeiro and Lemos, &lt;a href=&quot;https://arxiv.org/abs/1811.06587&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/1811.06587&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-11-24T21:56:40.497" LastActivityDate="2020-11-24T21:56:40.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12506" PostTypeId="2" ParentId="2126" CreationDate="2020-11-24T21:50:07.230" Score="8" Body="&lt;p&gt;Its first appearance in particle physics was in Pais, A., and S. B. Treiman (1975), &amp;quot;How Many Charm Quantum Numbers are There?&amp;quot; &lt;em&gt;Physical Review Letters&lt;/em&gt; &lt;strong&gt;3&lt;/strong&gt;5, no. 23, p.  1556. The point was to stop repeating &amp;quot;what we all agree on, the SSB gauge theory of SU(2)xU(1) of...&amp;quot;, dropping the names of the contributors, an increasing list that would then quickly  slip off to &amp;quot;me too / him too&amp;quot; bibliographical tangents. Worse yet, people would champion their favorite contributors with emphasizing different aspects of the model, declaring them as more conceptually crucial. It was thus a brilliant argument stopper, adopted very quickly.&lt;/p&gt;&#xA;&lt;p&gt;To my knowledge,  the name was spread to cosmology by particle cosmologists in the 1980s.&lt;/p&gt;&#xA;&lt;p&gt;I am not, however, familiar with its usage in other, more remote fields.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2020-11-25T01:15:48.687" LastActivityDate="2020-11-25T01:15:48.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12507" PostTypeId="2" ParentId="12503" CreationDate="2020-11-25T07:44:13.100" Score="5" Body="&lt;p&gt;Yes, although Euler's work initially got so little attention that when Hierholzer proved existence of an Eulerian path in a graph with none or two vertices of odd degree (published posthumously in 1873) he was not aware of it. Euler himself stated the result without proof, only proving the converse. For the background on the original problem see &lt;a href=&quot;https://hsm.stackexchange.com/a/5849/55&quot;&gt;What was the origin of the Seven Bridges of Königsberg problem before Euler?&lt;/a&gt; and the &lt;a href=&quot;https://www.routledgehandbooks.com/pdf/doi/10.1201/b16132-3&quot; rel=&quot;noreferrer&quot;&gt;history section of Routledge's Handbook of Graph Theory&lt;/a&gt;. A comprehensive reference is &lt;a href=&quot;https://link.springer.com/chapter/10.1007/BFb0070360&quot; rel=&quot;noreferrer&quot;&gt;200 years of graph theory by Wilson&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The aftermath is described in detail in &lt;a href=&quot;https://www.google.com/books/edition/Graph_Theory_1736_1936/XqYTk0sXmpoC?hl=en&quot; rel=&quot;noreferrer&quot;&gt;Graph Theory, 1736-1936 by Biggs, Lloyd and Wilson, ch.1&lt;/a&gt;, with translated excerpts from the original sources. The Königsberg's remembrance is especially endearing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Fortunately, Euler's work had not been completely forgotten, for in 1851 a French translation [2] by E. Coupy was published in the Nouvelles Annales de Mathematiques, a journal intended primarily for the students at the Ecole Polytechnique in Paris. Coupy also applied the methods of Euler to the analogous problem of the bridges over the River Seine. Nor was the problem forgotten in Königsberg; in 1875. L. Saalschütz [3] reported that a new bridge had been constructed there, joining the land areas denoted by B and C, and that the citizens' perambulation was now theoretically possible.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Coupy's interest in the work of Euler stemmed from two references to&#xA;it in the early nineteenth century. The first of these occurred in a&#xA;textbook of algebra [4], written by the Swiss mathematician Simon Antoine-Jean Lhuilier in 1804, and the second appeared in a memoir on polygons and polyhedra [5], written by Louis Poinsot in 1809... Poinsot proved, using arguments similar to those of Euler, that an Eulerian path in &lt;span class=&quot;math-container&quot;&gt;$K_n$&lt;/span&gt;&lt;/em&gt; [complete graph on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; vertices] &lt;em&gt;is impossible when &lt;span class=&quot;math-container&quot;&gt;$n = 4, 6, 8, ...$&lt;/span&gt;, because in these cases there are more&#xA;than two vertices with odd valency. He also gave an ingenious method for constructing an Eulerian path in &lt;span class=&quot;math-container&quot;&gt;$K_n$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is odd.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Poinsot was interested in solving diagram-tracing puzzles, i.e. tracing a graph &amp;quot;&lt;em&gt;in a simple continuous stroke&lt;/em&gt;&amp;quot;. It is possible if and only if the graph has an Eulerian path. &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1002/asna.18440211402&quot; rel=&quot;noreferrer&quot;&gt;Clausen in &lt;em&gt;De linearum tertii ordinis proprietatibus&lt;/em&gt; (1844)&lt;/a&gt; and Listing were interested in them too. The latter even included them in his book &lt;a href=&quot;https://archive.org/details/vorstudienzurto00listgoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Vorstudien zur Topologie&lt;/em&gt; (1847)&lt;/a&gt;, which was Hierholzer's source and the first appearance of &amp;quot;topology&amp;quot; in print (Poincare will still call the subject by Leibniz's name &lt;em&gt;Analysis Situs&lt;/em&gt; later into the century).&lt;/p&gt;&#xA;&lt;p&gt;But Listing mentioned neither Euler nor Poinsot. Ironically, Hierholzer called Listing's treatise &amp;quot;&lt;em&gt;unfortunately little-known&lt;/em&gt;&amp;quot; as well, but credited Listing with &amp;quot;&lt;em&gt;incomplete proof&lt;/em&gt;&amp;quot; of his result. A commentary on Poinsot's memoir penned by Terquem appeared in the &lt;em&gt;Nouvelles Annales de Mathematiques&lt;/em&gt; (1849), and included an interpretation of Poinsot's Eulerian path in &lt;span class=&quot;math-container&quot;&gt;$K_7$&lt;/span&gt; as a complete game of dominoes. A general count of such games was published by Reiss in 1871, and a general count of Eulerian paths by Tarry in 1886. But the obliviousness continued, lengthy &lt;a href=&quot;https://archive.org/details/ontraversinggeo00wilsgoog&quot; rel=&quot;noreferrer&quot;&gt;Wilson's On the traversing of geometrical figures (1905)&lt;/a&gt;, published in Oxford, shows no awareness of any of the above. The subject is prone to independent rediscovery.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-11-25T07:44:13.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12508" PostTypeId="2" ParentId="12494" CreationDate="2020-11-25T11:29:17.863" Score="0" Body="&lt;p&gt;I confirm that Archimedes did  get the centroid of a hemisphere.&#xA;This can be done - without calculus -using the leverage argument in the wikipedia article relying on the integral of a cube which follows from the leverage argument after using Archimedes method for the centroid of a parabolic segment which Archimedes go in Proposition 8 of Book 2 (see&#xA;The Works Of Archimedes : Heath,T.L.).&#xA;Take a hemisphere of radius 1.&#xA;So for the hemisphere, volume of hemisphere is 2/3pi.&#xA;Here Archimedes proves the centroid of a parabola is 3/4 which shows the integral of the cube is 1/4 - by geometry and the leverage argument! (start with the center of gravity of parabolic segment and go from there).&#xA;Using leverage this balances the integral of pi.x(1-x^2)=pi(x-x^3)=pi(1/2-1/4)=pi/4&#xA;So if centroid of hemisphere is &amp;quot;x&amp;quot; then&#xA;x.2/3pi =pi/4&#xA;so x=3/8 QED&#xA;Hats off to Archimedes! Eureka!! Wikipedia is right on this!!!&lt;/p&gt;&#xA;" OwnerUserId="6396" LastActivityDate="2020-11-25T11:29:17.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12509" PostTypeId="1" CreationDate="2020-11-25T12:39:07.640" Score="2" ViewCount="388" Body="&lt;p&gt;By far, looking at several curriculums of High School/First-Year Undergraduate physics. The curriculum always starts with Mechanics, moves into Electricity/Electromagnetism and ends with Modern Physics (Nuclear, Quantum).&lt;/p&gt;&#xA;&lt;p&gt;I'm wondering how was this order determined? Was there any historical standardization of the Physics curriculum where it was pedagogically decided that this order was the most appropriate to learn or is it just a product of the evolution of physics through time?&lt;/p&gt;&#xA;" OwnerUserId="12712" LastActivityDate="2020-11-25T16:27:48.457" Title="How did the current physics curriculum came to be?" Tags="|physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12510" PostTypeId="1" AcceptedAnswerId="12512" CreationDate="2020-11-21T23:43:43.773" Score="4" ViewCount="171" Body="&lt;ol&gt;&#xA;&lt;li&gt;Sheldon Glashow, Abdus Salam, and Steven Weinberg propose the unification of the weak and electromagnetic interaction between elementary particles.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Glashow, Salam, and Weinberg proposed the model during 1964-1967.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They were awarded the 1979 Nobel Prize in Physics for their contributions to the unification of the weak and electromagnetic&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Higgs particle plays an important role for the Weinberg–Salam theory, in order to explain the gauge symmetry breaking &lt;span class=&quot;math-container&quot;&gt;$$SU(2)_{weak} \times U(1)_{hypercharge} \to U(1)_{electromangetism}.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Peter Higgs proposed the Higgs model in 1964.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(In 1964 by three groups by Robert Brout and François Englert; Peter Higgs; Gerald Guralnik, C. R. Hagen, and Tom Kibble)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;But Peter Higgs and François Englert had been awarded only much later for the 2013 Nobel Prize in Physics.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So the historical timeline is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1959, Glashow, S &amp;quot;The renormalizability of vector meson interactions.&amp;quot; Nucl. Phys. 10, 107.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1959, Salam, A.; Ward, J. C.. &amp;quot;Weak and electromagnetic interactions&amp;quot;. Nuovo Cimento. 11 (4): 568–577.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1961, S.L. Glashow. &amp;quot;Partial-symmetries of weak interactions&amp;quot;. Nuclear Physics. 22 (4): 579–588.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1964, Higgs et all work on Higgs mechanism, &amp;quot;Broken symmetries and the masses of gauge bosons&amp;quot;. Physical Review Letters. 13 (16): 508–09.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1967, Weinberg work on  &amp;quot;A Model of Leptons&amp;quot;. Phys. Rev. Lett. 19 (21): 1264–66&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1979, Glashow, Salam, and Weinberg - Nobel Prize&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2013, Higgs et all - Nobel Prize&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;: since Weinberg–Salam theory heavily relies on the Higgs mechanism and Higgs work --- why had Glashow, Salam, and Weinberg been awarded Nobel prizes (1979) much earlier than Higgs (2013)? What are the crucial ingredients in Weinberg–Salam theory independent from the Higgs model? Although that Higgs et all needs to be awarded only after the discovery of Higgs particle in CERN LHC on 8 October 2013, but Weinberg–Salam theory can be confirmed earlier (even the Higgs is not required)?&lt;/p&gt;&#xA;&lt;p&gt;p.s. The Weinberg's weak mixing angle &lt;span class=&quot;math-container&quot;&gt;$\theta_W$&lt;/span&gt; again requires the Higgs mechanism, even though   &lt;span class=&quot;math-container&quot;&gt;$\theta_W$&lt;/span&gt; can be verified as a parameter.&lt;/p&gt;&#xA;" OwnerUserId="2162" OwnerDisplayName="wonderich" LastEditorUserId="26" LastEditDate="2021-02-20T15:05:29.280" LastActivityDate="2021-02-20T15:05:29.280" Title="Significance of Higgs model used in Glashow-Salam-Weinberg theory" Tags="|experimental-physics|particle-physics|" AnswerCount="3" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12511" PostTypeId="2" ParentId="12510" CreationDate="2020-11-22T05:37:37.447" Score="6" Body="&lt;p&gt;It is called verification from experimental results.&lt;/p&gt;&#xA;&lt;p&gt;The GSW theories fitted mathematically the observed particle zoo symmetries and explained the approximate symmetries seen in the data.&lt;/p&gt;&#xA;&lt;p&gt;The Higgs model  at the time of award was not the only model for getting the results of symmetry breaking, it could be a different process  inducing the breaking , as in &lt;a href=&quot;https://en.wikipedia.org/wiki/Technicolor_(physics)&quot; rel=&quot;noreferrer&quot;&gt;techincolor&lt;/a&gt;, where a composite particle was proposed for the higgs.&lt;/p&gt;&#xA;&lt;p&gt;The specific higgs model used in GSW theories had a field for the Higgs, which means in quantum field theory  that an elementary  particle &lt;strong&gt;should&lt;/strong&gt; exist , and it needed its discovery in order to uniquely define the Higgs mechanism with nature.&lt;/p&gt;&#xA;&lt;p&gt;All experiments were looking for a Higgs &lt;em&gt;particle&lt;/em&gt; , but their energy at the time was too low.The LEP accelerator just missed the mass found in LHC.&lt;/p&gt;&#xA;" OwnerUserId="3848" OwnerDisplayName="anna v" LastActivityDate="2020-11-22T05:59:00.840" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12512" PostTypeId="2" ParentId="12510" CreationDate="2020-11-22T15:21:57.857" Score="8" Body="&lt;p&gt;This is precisely why this question belongs to &lt;a href=&quot;https://hsm.stackexchange.com&quot;&gt;HSM.SE&lt;/a&gt; with both feet! Your vision of what happened is deeply misleading, possibly requiring time travel. Recall the GWS 79 prize citation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Glashow's cornerstone  Electroweak &lt;a href=&quot;https://doi.org/10.1016/0029-5582(61)90469-2&quot; rel=&quot;noreferrer&quot;&gt;unification 1961 paper&lt;/a&gt; does not know, or &lt;em&gt;care&lt;/em&gt; about the 1964 Higgs work. It is mostly on requiring neutral currents for consistency. Salam and Weinberg tack the Higgs mechanism onto it as an extra bell-whistle, and it remains largely irrelevant (fine print) until 't Hooft's 1971 renormalizability work. The (badly &lt;em&gt;misnamed&lt;/em&gt;) &amp;quot;Weinberg&amp;quot; angle does not require a Higgs model!&lt;/p&gt;&#xA;&lt;p&gt;Glashow, as a Schwinger student, believed in some type of dynamical SSB picture, not a Higgs model. Like anybody else, he routinely used sigma models to exemplify the group theory of SSB nonlinear realizations, but, at that time, these were only &lt;em&gt;&lt;strong&gt;models&lt;/strong&gt;&lt;/em&gt;, only to be taken seriously in terms of their current and amplitude consequences.&lt;/p&gt;&#xA;&lt;p&gt;The breathtaking achievement of GWS is the consistent realization of all extant chiral symmetries including the presence of fermion masses induced by Yukawa couplings, given short shrift in modern introductions anxious to sell the Higgs mechanism to novices.  It virtually amounts to a magic trick to distract attention.&lt;/p&gt;&#xA;&lt;p&gt;So, no,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Higgs particle plays an important role for the Weinberg–Salam theory&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is &lt;em&gt;not&lt;/em&gt; the sine-qua-non you are making it out to be. It's nice to know the exact, original, minimal &amp;quot;Weinberg model&amp;quot; incorporating the elementary Higgs is now promoted to a full &lt;em&gt;theory&lt;/em&gt;, beyond model; and the discovery of the remnant boson, (the Higgs boson, proposed in 1966), was a good last chance to celebrate most of the 1964 Higgsers.&lt;/p&gt;&#xA;&lt;p&gt;But, really, with or without the elementary Higgs boson, the Higgs mechanism should have been celebrated anyway, long before, as 't Hooft's renormalizability proof made it a darling of theorists, and led to the triumph of QFT, celebrated in the 1999 NP. From that point on, a NP for the Higgsers was overdue, waiting for the cherry on the cake, the Higgs boson.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;NB. Edit on crucial features beyond Higgs.&lt;/em&gt; Higgsless models survived right up to the discovery of the Higgs, precisely because the Higgs isn't really a live-or-die feature of the SM, but merely the simplest and actual historical option. Even now, &lt;em&gt;SSB chiral gauge invariance&lt;/em&gt; with the suitable groups and representations is basically the standard EW &lt;em&gt;theory&lt;/em&gt;. To put it dramatically, the elementary Higgs is the &lt;em&gt;only&lt;/em&gt; &amp;quot;model&amp;quot; part of the SM, and that's why its confirmation took so long: the rest of the &lt;em&gt;theory&lt;/em&gt; doesn't much care about it!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7076" OwnerDisplayName="Cosmas Zachos" LastActivityDate="2020-11-23T20:28:55.303" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12513" PostTypeId="1" CreationDate="2020-11-23T19:59:14.990" Score="2" ViewCount="127" Body="&lt;p&gt;Although I cannot find the related Q/A, I remember that someone here notified me that the so-called &lt;em&gt;Einstein's light clock&lt;/em&gt; was first introduced by Max von Laue or someone but Einstein. Does anybody have any precise information/reference regarding this matter?&lt;/p&gt;&#xA;" OwnerUserId="13363" OwnerDisplayName="Mohammad Javanshiry" LastEditorUserId="26" LastEditDate="2021-02-20T15:03:41.847" LastActivityDate="2021-02-20T15:03:41.847" Title="Does &quot;Einstein's Light clock&quot; really belong to Einstein?" Tags="|einstein|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12514" PostTypeId="2" ParentId="12509" CreationDate="2020-11-25T14:43:54.977" Score="4" Body="&lt;p&gt;Here in the US, public high schools date to 1820, and physics instruction began in high schools. Only later was it adopted as a subject in colleges and universities.  The original mode of instruction was that there was no laboratory. Students attended lectures, memorized facts from a textbook, and recited those facts from memory. Physics was seen as a practical subject, and textbooks are full of pictures of butter churns, explanations of why gasoline is hazardous, and so on. There was not as much theoretical material as we expect today, and very little math. The vector-scalar system hadn't been invented yet, and the alternative would have been quaternions, which high school students didn't know about.&lt;/p&gt;&#xA;&lt;p&gt;An early and influential text in the US was the English translation of a French text by Ganot. The first edition of the translation was in 1863. Electricity and magnetism was in a primitive state at that time. Maxwell's equations were cutting-edge science. So if you consider the state of the art at that time, it's pretty natural that these books started with mechanics, fluids, and thermodynamics, and then at the end put in some electromagnetism. Later texts such as Millikan and Gale show a pretty similar order of topics, although Millikan puts molecular motion and molecular forces pretty early. The main difference from the modern order of topics is that there is a lot more material early on in the course that is not Newtonian mechanics. This was possible because there was almost none of the math and problem solving that today causes us to spend 10 weeks on Newton's laws. The steam engine was the latest technology, so it was natural to want to do a lot of thermodynamics early on.&lt;/p&gt;&#xA;&lt;p&gt;Later books seem to have mostly just tacked on later developments in historical order, while also moving almost everything but Newtonian mechanics out of the first-semester course. Following the historical sequence seems reasonable, since the historical development does relate to the difficulty and abstraction of the subject. Electromagnetism is less concrete than mechanics, and relativity and quantum mechanics are even more remote from human experience.&lt;/p&gt;&#xA;&lt;p&gt;A notable effort to make radical changes in the sequence of topics was the Sputnik-era PSSC Physics, which started with waves and only later covered mechanics. This innovation did not catch on. Another change that made limited headway was to put relativity very early, and then use relativity explicitly in the development of electromagnetism. This was done in the text by Purcell, which is still being used and is in a third edition.&lt;/p&gt;&#xA;&lt;p&gt;Today, very few high school students take physics, so it's mainly a college class, and it's extremely difficult to change the order of topics in a freshman physics sequence, because there is an expectation that it will be standardized in order to allow students to transfer from one school to another. Community college students in a large city in the US may randomly take classes at several different community colleges before transferring to a four-year school.&lt;/p&gt;&#xA;&lt;p&gt;References&lt;/p&gt;&#xA;&lt;p&gt;Meltzer and Otero, A brief history of physics education in the United States, DOI: 10.1119/1.4902397&lt;/p&gt;&#xA;&lt;p&gt;Holbrow, Archaeology of a Bookstack: Some Major Introductory Physics Texts of the&#xA;Last 150 Years,  &lt;a href=&quot;http://dx.doi.org/10.1063/1.882613&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://dx.doi.org/10.1063/1.882613&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Millikan, &lt;a href=&quot;https://archive.org/details/afirstcourseinp01galegoog/page/n12/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/afirstcourseinp01galegoog/page/n12/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Ganot-Atkinson, &lt;a href=&quot;https://catalog.hathitrust.org/Record/101717020&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://catalog.hathitrust.org/Record/101717020&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-11-25T15:29:12.100" LastActivityDate="2020-11-25T15:29:12.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12515" PostTypeId="2" ParentId="12504" CreationDate="2020-11-25T16:09:14.877" Score="1" Body="&lt;p&gt;Pierce was a difficult man. A &lt;em&gt;Physics World&lt;/em&gt; article implied that it's was his character that alienated him from his fellow academics: He was 'aloof', 'suspicious' and subject to 'fits of temper'.&lt;/p&gt;&#xA;&lt;p&gt;There was nothing in the article about him being 'stigmatised'. If so, this might say much more about the culture and sociology of academia, and specifically that of the physical sciences than the man himself. After all, even Einstein was marginalised within the physics community, with even Freeman Dyson avoiding him after arranging a meeting with him ...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-25T16:09:14.877" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12516" PostTypeId="2" ParentId="12513" CreationDate="2020-11-25T16:11:35.083" Score="-1" Body="&lt;p&gt;It would be odd that Laue would have thought up the light clock rather than Einstein as this relied upon his physical insight into the nature of gravity - the equivalence principle.&lt;/p&gt;&#xA;&lt;p&gt;I'd also say that the idea is not 'owned' by Einstein - got ideas are free; but that it is &lt;em&gt;credited&lt;/em&gt; to him.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;@Ben Crowell: I said it seems 'odd' not that it's impossible. Hence, seeing that you're a man for logic, this is not a contradiction ... as it is, I was actually thinking not of the light clock but light travelling in an elevator - hence the reference to the equivalence principle. As that's the famous thought experiment showing that light travels in a curved trajectory in the presence of gravity.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-25T23:53:07.407" LastActivityDate="2020-11-25T23:53:07.407" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12517" PostTypeId="2" ParentId="12509" CreationDate="2020-11-25T16:27:48.457" Score="0" Body="&lt;p&gt;It's simply in its natural order of discovery: Cosmology &amp;amp; Mechanics, Electromagnetism and then Relativity and Quantum Mechanics. It's determined by what physicists believe are the most prominent discoveries and theories are.&lt;/p&gt;&#xA;&lt;p&gt;In my own schooling, which will more or less reflect my generation (in Britain), this pedagogical order is reinforced several times, particular beginning with middle school. It's no surprise that cosmology is first since primary school kids love stars, the moon and the sun.&lt;/p&gt;&#xA;&lt;p&gt;One point to note - at least for me in my masters - is that experiment which is tied with physics all along this pedagogical route begins to Peter out there which is why I think most of the students in my class on String Theory were dismayed by how things were just pulled out of a hat. It's only much later when I investigated the historical antecedents of string theory that I began to understand it as physics. Importantly, it connected to the experimental study of hadronic physics - that is the strong force. Personally, I think this implies that the teaching of string theory is too heavy on the theory side and it could be tempered with historical and experimental insights - exactly how the rest of the physics curriculum is construed.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-25T16:27:48.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12518" PostTypeId="1" AcceptedAnswerId="12519" CreationDate="2020-11-25T21:16:35.357" Score="2" ViewCount="106" Body="&lt;p&gt;From the questions on some of Gauss' philosophical ideas here at HSM stackexchange it's clear that Gauss had some major philosophical ideas that despite their profundity don't seem to have had much influence in the later developments of philosophy(perhaps philosophers with a mathematical inclination like Peirce were aware of some of them but didn't cite them).&#xA;Is there any book or site where Gauss' collected philosophical writings are presented?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-11-27T12:47:23.370" LastActivityDate="2020-11-27T19:16:45.670" Title="Is there any book or site where Gauss' collected philosophical writings are presented?" Tags="|mathematics|biographical-details|philosophy-of-science|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12519" PostTypeId="2" ParentId="12518" CreationDate="2020-11-25T22:34:46.853" Score="1" Body="&lt;p&gt;I'm not aware of any writings of Gauss that touched upon philosophical concerns. That doesn't mean that Gauss wasn't philosophically aware. He's been said to have read Kants &lt;em&gt;Critique&lt;/em&gt; five times which probably isn't surprising given the esteem that Kant was held in the German consciousness of the time - rather like Hawking is held in ours.&lt;/p&gt;&#xA;&lt;p&gt;And to me, it's clear that there was a possible influence of Kants analysis of space on his discovery of non-Euclidean geometry.&lt;/p&gt;&#xA;&lt;p&gt;In fact, in Dunningtons &lt;em&gt;Gauss: A Titan of Mathematics&lt;/em&gt;, we have:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schleiden told the philosopher Rudolf Eucken that Gauss read Kant's &lt;em&gt;Critique of Pure Reason&lt;/em&gt; five times. The fifth time he is reported to have said:&amp;quot;now it's dawning on me&amp;quot;...Gauss was interested in the manner in which Kant founded his concept of space and pioneered in the field of non-Euclidean geometry ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Although Dunnington adds he also disagreed with Kant, where Gauss said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The distinction between right and left cannot be defined, but only shown ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But this is wrong as Kant in his 1768 essay, &lt;em&gt;Concerning the Ultimate Foundation of the Differentiation of the Regions in Space&lt;/em&gt;, did exactly that - the essay is notoriously hard to understand, and Gauss may have just misunderstood it (we don't know if he had read this fives times!)&lt;/p&gt;&#xA;&lt;p&gt;Donnington goes on to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss and Kant thus differed in the manner of founding a theory of space, but there is scarcely any difference in their views about space. Gauss did not deny the &lt;em&gt;a priori&lt;/em&gt; character of space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Given the preceding, the difference is even slimmer than Donnington assumes here.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-27T19:16:45.670" LastActivityDate="2020-11-27T19:16:45.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12520" PostTypeId="2" ParentId="12513" CreationDate="2020-11-25T23:30:54.330" Score="1" Body="&lt;p&gt;The earliest presentation of this idea that I know of was by Lewis and Tolman in 1909:&lt;/p&gt;&#xA;&lt;p&gt;Proceedings of the American Academy of Arts and Sciences, 1909, 44: 709–726&#xA;&lt;a href=&quot;https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikisource.org/wiki/The_Principle_of_Relativity,_and_Non-Newtonian_Mechanics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Another early presentation of this concept seems to have been by von Laue in 1917:&lt;/p&gt;&#xA;&lt;p&gt;von Laue, M. (1917). Jahrbuch der Radioaktivitat und Elektronik, 14,263.&lt;/p&gt;&#xA;&lt;p&gt;online: &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015075057870&amp;amp;view=1up&amp;amp;seq=309&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://babel.hathitrust.org/cgi/pt?id=mdp.39015075057870&amp;amp;view=1up&amp;amp;seq=309&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I don't read German, and the paper is long, but it actually appears to be about Nordström's theory of gravitation. On p. 293, section 4a is titled &amp;quot;Der Gang der Lichtuhr,&amp;quot; which google translate says means &amp;quot;the course of the light clock.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The following 1976 paper by Alex Harvey states that &amp;quot;the use of a shuttling light signal as a timekeeper was introduced by von Laue,&amp;quot; giving the reference above.&lt;/p&gt;&#xA;&lt;p&gt;Harvey, A. (1976). Photon clocks. General Relativity and Gravitation, 7(11), 891–893. doi:10.1007/bf00771021&lt;/p&gt;&#xA;&lt;p&gt;Either Harvey didn't know about the earlier presentation by Lewis, or he only intended to talk about presentations in the context of gravity.&lt;/p&gt;&#xA;&lt;p&gt;Von Laue used mirrors attached to the ends of a rigid rod, while a later presentation by Marzke and Wheeler used mirrors moving along geodesics. Harvey criticizes both as having &amp;quot;serious conceptual difficulties,&amp;quot; and estimates the rate at which the clock drifts due to recoil from the reflections of the light.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-11-26T00:02:18.293" LastActivityDate="2020-11-26T00:02:18.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12521" PostTypeId="2" ParentId="5950" CreationDate="2020-11-26T01:14:38.217" Score="1" Body="&lt;p&gt;Lemaitre, a Belgian Diocesan priest and also a scientist was a pioneer in applying GR to cosmology. He derived Hubbles law two years before Hubble actually experimentally demonstrated it to be true.&#xA;He was also the first to give an estimate for Hubbles constant.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, Nature has reported that after a vote in 2018, the International Astronomical Union had proposed that Hubbles law be renamed the Lemaitre-Hubble law to pay tribute to Lemaitre.&lt;/p&gt;&#xA;&lt;p&gt;Einstein was initially sceptical about Lemaitre's claims at the Solvay Conference and pointed out that Friedman had also come up with a similar solution a few years earlier (and which he had criticised but had later withdrawn).&lt;/p&gt;&#xA;&lt;p&gt;Both Friedman and Lemaitre had proposed an expanding universe but  he was the first to say that this explains the observable redshift. It was not a purely theoretical calculation but one with observable consequences.&lt;/p&gt;&#xA;&lt;p&gt;It was also Lemaitre that first came up with the Big Bang hypothesis calling it the 'hypothesis of the primeval atom.'&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia article on Lemaitre points out that his work is under-appreciated in the USA. It figures.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-12-05T21:44:41.770" LastActivityDate="2020-12-05T21:44:41.770" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12522" PostTypeId="2" ParentId="12510" CreationDate="2020-11-26T02:07:02.487" Score="1" Body="&lt;p&gt;In the development for the theory of weak interactions one important papers was the 1967 paper of Weinbergs where he combined a theory of leptons with a spontaneous breaking of the SU(2) x U(1) local or gauge symmetry. A dis Salam had done very similar work.&lt;/p&gt;&#xA;&lt;p&gt;It was not immediately taken notice of because of the lack of experimental verification. However, an accelerator in Chicago was being under construction that would push the boundaries of what energies could be probed.  By 1971 work on it had finished and it discovered the neutral currents.&lt;/p&gt;&#xA;&lt;p&gt;As we can see here, what was important was not only theory - but also experiment. The experimental verification of the Higgs was way out of reach, taking until the second decade of this century before it was shown that it existed. Hence the delay in awarding Higgs a Nobel prize (with others).&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-26T02:07:02.487" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12523" PostTypeId="2" ParentId="2126" CreationDate="2020-11-26T02:27:08.817" Score="1" Body="&lt;p&gt;The word 'standard' has been long in use in mathematics and science as a reference point. In fact, other terms are used as synonyms such as 'normal' or 'characteristic' and even 'universal'.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia points out that the term 'standard model' was a coinage by Abraham Pais and Sam Treiman in 1975 published in &lt;em&gt;Physical Review&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It took some time for the term to migrate into standard usage (pun intended) as I wasn't aware of it studying physics at college in the late eighties although I had heard of the Glashow-Salam-Weinberg theory.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-26T03:16:21.783" LastActivityDate="2020-11-26T03:16:21.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12524" PostTypeId="1" CreationDate="2020-11-26T03:41:58.700" Score="3" ViewCount="186" Body="&lt;p&gt;We now all “know” that visible light has a wavelength between about 380 nanometres and 700 nanometres, and that shorter wavelength are the domain of UV, X-rays, and gamma rays; and longer wavelengths for infrared, microwave, and radio.&lt;/p&gt;&#xA;&lt;p&gt;My question—and I have Googled far and wide and read quite few books to find the answer, but I guess I’m asking the wrong question—is: “How did we figure out the wavelengths of specific rays?”&lt;/p&gt;&#xA;&lt;p&gt;I mean, for example, how did we figure out that yellow has a wavelength of 580 nm? (or any other color/band has any other wavelength)&lt;/p&gt;&#xA;" OwnerUserId="13368" LastEditorDisplayName="user466" LastEditDate="2020-11-26T12:58:59.877" LastActivityDate="2020-11-26T12:58:59.877" Title="How were wavelengths in the electromagnetic spectrum first found?" Tags="|physics|optics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12525" PostTypeId="2" ParentId="12524" CreationDate="2020-11-26T04:19:23.517" Score="2" Body="&lt;p&gt;It should not take googling far and wide, a standard college book on physics will give you the details.&lt;/p&gt;&#xA;&lt;p&gt;Basically, Snells law on refraction, tells us that light is a wave.&lt;/p&gt;&#xA;&lt;p&gt;Then as diffraction occurs when diffraction gratings are around the wavelength of the incident light, experiments with known diffraction gratings then tells us the wavelength of specific light - say that of purple light.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-26T04:19:23.517" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12527" PostTypeId="1" CreationDate="2020-11-26T23:56:08.370" Score="2" ViewCount="92" Body="&lt;p&gt;Biology nowadays is filled with mathematics. Indeed, the field of mathematical biology is huge, and shows no sign of decay. But the mathematisation of biology is, to my knowledge, a recent phenomenon - dating to the early 20th century with the contributions of Ronald Fisher and his contemporaries.&lt;/p&gt;&#xA;&lt;p&gt;Any resources detailing the history of this quantification would be fascinating to read; in particular, something which comments on the birth of mathematical biology and its relationship to biology done in the field or in the lab. But I can't find much! I'm looking for recommendations - books, articles, anything - on this aspect of the history and philosophy of biology.&lt;/p&gt;&#xA;" OwnerUserId="6867" LastEditorUserId="10106" LastEditDate="2020-11-27T06:24:54.560" LastActivityDate="2020-11-27T19:46:28.437" Title="History of Mathematical Biology - Resource Recommendations" Tags="|mathematics|biology|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12528" PostTypeId="2" ParentId="12527" CreationDate="2020-11-27T05:51:52.747" Score="3" Body="&lt;p&gt;It is not &lt;em&gt;that&lt;/em&gt; recent (apart from the name), before Fisher there were Bernoulli (1760), Malthus (1789) and Verhulst (1838), to name a few, although one might view them as pre-history. Here is a &lt;a href=&quot;https://web.archive.org/web/20060901224506/http://www1.appstate.edu/%7Emarland/math_bio/Sunday_intro.ppt&quot; rel=&quot;nofollow noreferrer&quot;&gt;PowerPoint&lt;/a&gt; that has the main highlights with dates. Rashevsky is generally credited with establishing the field, see &lt;a href=&quot;https://arxiv.org/abs/1511.01455&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hoffman, The Dawn of Mathematical Biology&lt;/a&gt;. For book length treatments see &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B00BYPK4E2&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essays on the history of mathematical biology by Israel&lt;/a&gt;, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0226437280&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Modeling Nature by Kingsland&lt;/a&gt; (it focuses on population ecology), and &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Mathematics_of_Life&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stewart's The Mathematics of Life&lt;/a&gt; for a mathematician's take.&lt;/p&gt;&#xA;&lt;p&gt;For some philosophical treatments see books like &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1108727859&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Otsuka's The Role of Mathematics in Evolutionary Theory&lt;/a&gt; or &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0226727262&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Instrumental Biology, or, the Disunity of Science by Rosenberg&lt;/a&gt;, and papers like &lt;a href=&quot;https://philpapers.org/rec/DORMBA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dorato's Mathematical biology and the existence of biological laws&lt;/a&gt; or &lt;a href=&quot;https://www.researchgate.net/publication/268068126_The_mathematical_theory_of_categories_in_biology_and_the_concept_of_natural_equivalence_in_Robert_Rosen&quot; rel=&quot;nofollow noreferrer&quot;&gt;The mathematical theory of categories in biology by Varenne&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-11-27T08:31:00.193" LastActivityDate="2020-11-27T08:31:00.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12529" PostTypeId="2" ParentId="12527" CreationDate="2020-11-27T06:55:25.380" Score="1" Body="&lt;p&gt;One of the first application of mathematics and logic to biology was classification. This was done by Aristotle and his terms still apply today - ie genus, species. These are logical terms taken from his &lt;em&gt;Organon&lt;/em&gt; which detail the Aristotelian understanding of logic.&lt;/p&gt;&#xA;&lt;p&gt;A good reference for this is the book by Rene Thom, &lt;em&gt;Semiophysics&lt;/em&gt; where he also details the work of his own school to biological morphology and other aspects of Aristotelian biology. (This work he called catastrophe theory which Vladimir Arnold took umbrage at, and which he called singularity theory. And with good reason as many mathematical laymen misunderstood the term &lt;em&gt;catastrophe&lt;/em&gt; - it's a change of form - nevertheless it did publicise the subject).&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-27T19:46:28.437" LastActivityDate="2020-11-27T19:46:28.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12530" PostTypeId="1" CreationDate="2020-11-27T08:52:30.980" Score="3" ViewCount="76" Body="&lt;p&gt;I was just thinking how the BOM in files makes a lot of sense. I mean the fact that by default numbers grow to the right. And then I started wondering why our numbers grow to the left, if columns of numbers could be compared far easier if the ordering went from the smallest number on the left to the largest number on the right.&lt;/p&gt;&#xA;&lt;p&gt;I believe that's exactly why the ordering is like that. Just that it was created with Arabic writing in mind, which is from the right to the left.&lt;/p&gt;&#xA;&lt;p&gt;But then I wondered. Is that accident of history really it or are there amazing advantages of having the biggest number on the left I just cannot think of at the moment?&lt;/p&gt;&#xA;" OwnerUserId="13376" LastActivityDate="2020-11-27T08:56:14.333" Title="Advantages of numbers growing to the left (ordered Arabic numbers)?" Tags="|mathematics|" AnswerCount="0" CommentCount="7" ClosedDate="2020-11-30T21:12:44.293" ContentLicense="CC BY-SA 4.0" />
  <row Id="12531" PostTypeId="2" ParentId="7826" CreationDate="2020-11-27T08:52:57.280" Score="0" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.pbs.org/wgbh/nova/newton/principia.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nova&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A more recent blow was the departure from Cambridge of John Wickens, Newtons room-mate for twenty years. Wickens became the minister of the parish church at Stoke Edith, married and fathered a son called Nicholas. Though they had through much together, the two friends would never meet again, and exchanged no more than a letter or two in the coming years.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus there is not likely to be much. Though see &lt;a href=&quot;https://www.heritage-history.com/index.php?c=read&amp;amp;author=gibson&amp;amp;book=scientists&amp;amp;story=more&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; for some anecdotal details.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-27T11:34:39.810" LastActivityDate="2020-11-27T11:34:39.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12534" PostTypeId="1" CreationDate="2020-11-27T12:14:12.683" Score="0" ViewCount="52" Body="&lt;p&gt;I was wondering if the psychophysicists and brain anatomists from 1800 up until 1860s made any novel, noteworthy philosophical speculations about the emotions and cognition.If they did is there any evidence that they influenced later philosophers like Peirce and William James.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2021-01-30T12:59:44.233" LastActivityDate="2021-01-30T12:59:44.233" Title="Did the brain anatomists and psychophysicists of the early to mid 1800s make any philosophical speculations on the nature of emotions and cognition?" Tags="|biographical-details|philosophy-of-science|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12536" PostTypeId="1" CreationDate="2020-11-27T17:26:33.793" Score="1" ViewCount="79" Body="&lt;p&gt;In &lt;em&gt;Why are special functions special&lt;/em&gt; [&lt;a href=&quot;https://doi.org/10.1063/1.1372098&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Physics Today&lt;/em&gt; &lt;strong&gt;54&lt;/strong&gt;, 11 (2007)&lt;/a&gt;; &lt;a href=&quot;https://michaelberryphysics.files.wordpress.com/2013/07/berry326.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;eprint&lt;/a&gt;], Michael Berry makes the following observations:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This continuing and indeed increasing reliance on special functions is a surprising development in the sociology of our profession. One of the principal applications of these functions was in the compact expression of approximations to physical problems for which explicit analytical solutions could not be found. But since the 1960s, when scientific computing became widespread, direct and “exact” numerical solution of the equations of physics has become available in many cases. &lt;strong&gt;It was often claimed that this would make the special functions redundant.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, this makes a lot of sense, and given that Berry was around as a researcher since 1965, it is reasonable to take this as a primary-source eyewitness account, but I would like to get a bit more flavour for the zeitgeist that Berry depicts in this paragraph (i.e., that the emergence of electronic computers would make special functions obsolete, with direct numerical solution of the problems replacing the special-function corpus). Is there a good primary source from the time showing this view, or (even better) a good review of the development of the viewpoint?&lt;/p&gt;&#xA;" OwnerUserId="272" LastActivityDate="2020-11-27T17:26:33.793" Title="Finding sources for &quot;computers will become so powerful that special functions will become obsolete&quot; as a zeitgeist" Tags="|mathematics|reference-request|mathematical-physics|computers|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12539" PostTypeId="1" CreationDate="2020-11-27T22:13:04.060" Score="0" ViewCount="77" Body="&lt;p&gt;According to Feynman, there was a quota for Jews at American Universities and when he was accepted on a graduate programme - I think at Princeton - they were told &amp;quot;he's Jewish but he doesn't act like one&amp;quot;, with the impression that this was said approvingly. I think this was part of anecdote in one of his popular books. (When I read that as a teenager, I sat up and took notice, because it made me think about the so-called 'freedoms' of the United States in the modern era. After all, I knew about the slavery and how it had been gotten rid of by courageous men and women. But this showed me that there was still a dark underbelly where one might least expect it).&lt;/p&gt;&#xA;&lt;p&gt;I'm under the impression that Feynman was more or less agnostic to athiest. So what prompted his, I take, late soul searching in his lecture &lt;em&gt;On the Meaning of it All&lt;/em&gt;, where he explains that he saw the world in a new way and that Christianity was a great faith? This lecture never seems to be mentioned by his fan club - which I take to be atheist to a man and woman, generally speaking. If Feynman was alive, and given his views, I expect him to walk right pass them.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. When was this quota imposed - presumably it was an informal one. And when was this lifted?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. What led to his religious awakening? Is this clarified in any of his later writings or conversations with friends?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've written the above from memory, so some of the details may be a bit skewy, but the main thrust of it is correct.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-28T16:59:50.083" LastActivityDate="2020-11-28T16:59:50.083" Title="Do we know how Feynmans religious views changed throughout his life?" Tags="|physics|biographical-details|social-context|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12540" PostTypeId="1" AcceptedAnswerId="12541" CreationDate="2020-11-28T00:54:22.863" Score="3" ViewCount="134" Body="&lt;p&gt;The quicksort algorithm is based on recursively choosing an element to partition the array. In every modern exposition that I've seen, this element is called the &amp;quot;pivot&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;However, as far as I can tell, the &lt;a href=&quot;https://web.archive.org/web/20200614000936/http://rabbit.eng.miami.edu/class/een511/quicksort.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;inventor of the algorithm used the term &amp;quot;bound&amp;quot;&lt;/a&gt; circa 1960, and Robert Sedgewick used the term &lt;a href=&quot;https://web.archive.org/web/20200619041621/https://www.cs.princeton.edu/%7Ers/papers/Quicksort.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;partitioning element&amp;quot; in his thesis&lt;/a&gt; circa 1980.&lt;/p&gt;&#xA;&lt;p&gt;So, where did the term &amp;quot;pivot&amp;quot; come from? Was it borrowed from numerical linear algebra (via LU decomposition with pivoting)?&lt;/p&gt;&#xA;" OwnerUserId="2835" LastActivityDate="2020-11-28T06:33:13.520" Title="Where does the term &quot;pivot&quot; come from in the quicksort algorithm?" Tags="|computer-science|algorithm|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12541" PostTypeId="2" ParentId="12540" CreationDate="2020-11-28T06:33:13.520" Score="4" Body="&lt;p&gt;Wegner writes of &amp;quot;pivot element&amp;quot; that partitions the array in &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0020019082901181&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sorting a linked list with equal keys (1982)&lt;/a&gt;, while none of his references (Sedgewick, Rivest, Loeser, Motzkin) does, as far as I can tell. But it does not pick up until 1986, when multiple authors start using it, including &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01939369&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bing-Chao and Knuth in A one-way, stackless quicksort algorithm&lt;/a&gt;. Bentley still does not use it in his &lt;a href=&quot;https://openlibrary.org/works/OL4986385W/Programming_pearls&quot; rel=&quot;nofollow noreferrer&quot;&gt;Programming Pearls (1986)&lt;/a&gt;, that introduced Lomuto's version of Quicksort, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Introduction_to_Algorithms&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Big White Book (Introduction to Algorithms by Cormen et al., 1990)&lt;/a&gt; does.&lt;/p&gt;&#xA;&lt;p&gt;Wegner gives no reasons for it, or suggests any relation to LU decomposition. But since &amp;quot;pivot elements&amp;quot; partition rows/columns of a matrix in Gaussian elimination and the simplex method (the use &lt;a href=&quot;http://jeff560.tripod.com/p.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;dates back to Whittaker (1923)&lt;/a&gt;) its appropriation is not particularly surprising, as these things go. &amp;quot;Bound&amp;quot; is the value and not the element, and &amp;quot;partitioning element&amp;quot; is a mouthful.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-11-28T06:33:13.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12542" PostTypeId="1" CreationDate="2020-11-28T09:39:08.690" Score="0" ViewCount="89" Body="&lt;p&gt;The Chinese were using matrices ( &lt;em&gt;fengcheng&lt;/em&gt; in the &lt;em&gt;Nine Chapters of the Mathematical Art&lt;/em&gt;), long before they were used in Europe which suggests that possibly they were introduced by way of them. For example, Leibniz was well known as a sinophile. What's the earliest article he wrote using matrices or &lt;em&gt;fengcheng&lt;/em&gt; (the latter is sometimes translated as rectangular array).&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-30T13:06:32.907" LastActivityDate="2020-11-30T13:06:32.907" Title="What is the earliest article in which Leibniz used 'matrices'?" Tags="|geometry|abstract-algebra|linear-algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12545" PostTypeId="1" CreationDate="2020-11-29T12:26:06.090" Score="4" ViewCount="128" Body="&lt;p&gt;Robert Peary writes in his book &lt;em&gt;The North Pole&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The instruments used in taking&#xA;observations for latitude may be either a sextant and an artificial&#xA;horizon, or a small theodolite. Both these instruments were taken on the&#xA;sledge journey; but the theodolite was not used, owing to the low altitude&#xA;of the sun. Had the expedition been delayed on the return until May or&#xA;June, the theodolite would then have been of value in determining position&#xA;and variation of the compass.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why would the low elevation of the Sun prevent Peary from using the theodolite? He arrived to the pole around April 6th, when the elevation was just under 7 degrees. He states that the theodolite would have been useful had he arrived later. In May the Sun would have risen by approx. another 10 degrees. Where is the difference, i.e., what principle made the sextant useful as opposed to the theodolite?&lt;/p&gt;&#xA;&lt;p&gt;Peary's exact quotations and full texts are available from &lt;a href=&quot;https://books.google.cz/books?id=G2enZ3_f2mUC&amp;amp;pg=PA369#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt; or &lt;a href=&quot;http://www.gutenberg.org/files/18975/18975-h/18975-h.htm#Footnote_1_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Gutenberg&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13393" LastEditorUserId="12330" LastEditDate="2020-11-29T19:20:11.887" LastActivityDate="2020-11-29T19:20:11.887" Title="Why couldn't Robert Peary use his theodolite at the North pole" Tags="|discoveries|geometry|optics|geography|navigation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12548" PostTypeId="1" AcceptedAnswerId="12551" CreationDate="2020-11-29T19:10:02.420" Score="4" ViewCount="298" Body="&lt;p&gt;Was there a specific reason that prevented researchers in Boolean algebra to invent such quantifiers in the flexible format that are known today earlier?&lt;/p&gt;&#xA;&lt;p&gt;Since the compact symbols for multiplication and sum were all there, it's surprising that they weren't used immediately after Boole's invention in logic.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2021-09-08T07:27:53.393" LastActivityDate="2021-09-08T07:27:53.393" Title="After the invention of Boolean algebra, why did it take so long for the existential and universal quantifiers to become formalized into symbols?" Tags="|mathematics|philosophy-of-science|mathematical-logic|logic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12549" PostTypeId="1" AcceptedAnswerId="12621" CreationDate="2020-11-30T10:48:47.380" Score="5" ViewCount="2598" Body="&lt;p&gt;Since merge sort is the first &lt;span class=&quot;math-container&quot;&gt;$O(n \log n)$&lt;/span&gt; time general purpose sorting algorithm I find it rather surprising that it was discovered without having any obvious conceptual predecessors. Are there any probable conceptual predecessors to &lt;a href=&quot;https://en.wikipedia.org/wiki/Merge_sort&quot; rel=&quot;nofollow noreferrer&quot;&gt;merge sort&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2021-09-20T13:43:41.420" LastActivityDate="2021-09-20T13:43:41.420" Title="How did von Neumann come up with his merge sort algorithm?" Tags="|computer-science|algorithm|von-neumann|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12550" PostTypeId="1" CreationDate="2020-11-30T11:45:15.710" Score="1" ViewCount="199" Body="&lt;p&gt;I was wondering if von Neuman has any philosophical writings beside his writings on the relation of computers to the human mind and vice versa.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4251" LastEditDate="2021-09-20T13:41:48.430" LastActivityDate="2021-09-20T13:41:48.430" Title="Is there any book or site where John von Neumann's collected philosophical writings are presented?" Tags="|biographical-details|reference-request|philosophy-of-science|von-neumann|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12551" PostTypeId="2" ParentId="12548" CreationDate="2020-11-30T13:10:38.630" Score="1" Body="&lt;p&gt;This is likely to be for the same reason as to why arithmetic wasn't formalised until long after arithmetic was discovered: there was no point to it. Peano Arithmetic was formalised when the question of formalising the foundations of mathematics became a successful research direction, likewise with the propositional and predicate logical calculi once Boolean algebra showed how to formalise logic.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-30T13:10:38.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12552" PostTypeId="2" ParentId="12550" CreationDate="2020-11-30T13:18:53.020" Score="-3" Body="&lt;p&gt;Is this the same von Neumann who said he could only envisage two or three computers required in the world - one of which would be used for weather forecasting? There is volume 5 of his collected works on the design of computers and theory of automata. There might be something in there ...&lt;/p&gt;&#xA;&lt;p&gt;... It doesn't appear to contain his lectures, left incomplete shortly before his death, to be given at Yale, on consciousness and the computer and which was published as &lt;em&gt;The Brain &amp;amp; the Computer&lt;/em&gt;. There's a question as to just how seriously von Neumann took them, since anecdotally, when von Neumann was faced with his own death, a friend of his reported to that he had never seen him so scared. After all, if the mind is a computer, to switch it off and then back on again - as we see everyday - is of no consequence.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-11-30T16:01:37.840" LastActivityDate="2020-11-30T16:01:37.840" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12553" PostTypeId="2" ParentId="12549" CreationDate="2020-11-30T13:31:16.497" Score="-3" Body="&lt;p&gt;Merge sort is an algorithm based upon 'divide and conquer' with it first bring announced in 1945 by von Neumann and a detailed report three years later by himself &amp;amp; Goldstine. As divide and conquer is an obvious strategy - it's what Britain did to India, from all reports (and Europe to the Rest of the World) -  I don't see this as being the conceptual novelty here. The naming of a concept is not as important as the substance of it - this is the matter of it and what matters.&lt;/p&gt;&#xA;&lt;p&gt;What was new was the notion of an automaton but this had already been proposed by Charles Babbage (and even earlier by Aristotle) but without the technology precise enough to carry it out, it had to await the understanding of electromagnetic machines rather than mechanical machines. It's the actual practical novelty then, that was the key precursor that made it possible to start thinking about computers and their algorithms in a progressive manner with the results we see today.&lt;/p&gt;&#xA;&lt;p&gt;Ref: Wikipedia&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-12-02T17:53:35.850" LastActivityDate="2020-12-02T17:53:35.850" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12555" PostTypeId="1" CreationDate="2020-11-30T17:41:20.317" Score="-2" ViewCount="100" Body="&lt;p&gt;Stephen Smale, an American mathematician and Jerry Rubin, who was at Berkeley before dropping out to organise around left wing causes, set up the Vietnam Day Committee in 1965 during a 35 hour anti-Vietnam war protest.&lt;/p&gt;&#xA;&lt;p&gt;Why was the reason for the 'Day' in its name?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-11-30T20:47:42.990" Title="Why was the Vietnam Day Committee, begun by Stephen Smale and Jerry Rubin, named as it was?" Tags="|mathematics|social-context|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12556" PostTypeId="2" ParentId="12555" CreationDate="2020-11-30T20:47:42.990" Score="5" Body="&lt;p&gt;According to the &lt;a href=&quot;https://oac.cdlib.org/ark:/13030/kt3h4n99mj/?brand=oac4&quot; rel=&quot;noreferrer&quot;&gt;Online Archive of California&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Vietnam Day Committee, Berkeley, was created out of Vietnam Day, the 35-hour anti- Vietnam war teach-in which brought 35,000 people to the University of California campus on May 21 and 22, 1965.&#xA;Immediately after Vietnam Day, members of the VDC met and determined the following as priorities for the peace movement...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, there was a day (plus 11 hours) of protest in May, and out of that came the committee that planned further protests.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2020-11-30T20:47:42.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12558" PostTypeId="1" CreationDate="2020-11-30T22:58:29.300" Score="1" ViewCount="142" Body="&lt;p&gt;This question was sitting on my &lt;em&gt;to do&lt;/em&gt; list for sometime.&#xA;So, as I was reading a book on history of science, I came across of a paragraph where the author attempted to give a historical development about &amp;quot;how we got the symbols that we use today for the numbers 1 to 9?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He started with &lt;code&gt;1&lt;/code&gt; saying that humans would represent &lt;code&gt;1&lt;/code&gt; with a stick on the ground which eventually has been drawn, let's say, in the sand, and reached us in the form we know it.&lt;/p&gt;&#xA;&lt;p&gt;Next, &lt;code&gt;2&lt;/code&gt; was represented with two horizontal sticks (ex: =) and, by drawing it, the fast hand connected the right end of the upper stick with the beginning of the left end of the lower stick, giving as the &lt;code&gt;2&lt;/code&gt; that we know.&lt;/p&gt;&#xA;&lt;p&gt;Something with &lt;code&gt;3&lt;/code&gt;...&lt;code&gt;3&lt;/code&gt; parallel sticks...fast hand drawing transformed it into our &lt;code&gt;3&lt;/code&gt;. And that's where the author stopped the story!&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know the rest of the story for the rest of the symbols, if there is one?&lt;/p&gt;&#xA;&lt;p&gt;PS: Feel free to change the title of the post if you think that is not suggestive enough!&lt;/p&gt;&#xA;" OwnerUserId="1714" LastEditorUserId="26" LastEditDate="2021-02-20T15:03:29.237" LastActivityDate="2021-02-20T15:03:29.237" Title="How were number symbols derived/shaped up?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12559" PostTypeId="2" ParentId="12558" CreationDate="2020-12-01T00:11:49.203" Score="-2" Body="&lt;p&gt;It's easy to see that 1, 2 &amp;amp; 3 represent these numbers. In fact, this is so obvious a way to represent numbers that both Egyptian hieratic - from which our system of representing numbers came from - and Chinese &lt;em&gt;xiaoxie&lt;/em&gt; numerals use the same symbols.&lt;/p&gt;&#xA;&lt;p&gt;However, the eye doesn't easily distinguish higher numbers of parallel strokes and hence it made sense to change how we represent the higher numbers because of counting the strokes, which takes time, we merely recognise the symbol. In the Egyptian hieratic they do this at five and higher, whilst in Chinese &lt;em&gt;xiaoxie&lt;/em&gt;, it's at four and higher; it's likely that these came out through trial and error and also tradition, once it set in.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-12-01T22:25:08.577" LastActivityDate="2020-12-01T22:25:08.577" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12560" PostTypeId="1" CreationDate="2020-12-01T09:45:56.417" Score="4" ViewCount="349" Body="&lt;p&gt;I can separate this into two questions at some point if necessary, but it's possible that sources for the answer to one will provide the answer to the other at the same time.&lt;/p&gt;&#xA;&lt;p&gt;I learned about &lt;a href=&quot;https://en.wikipedia.org/wiki/Eisenstein_integer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eisenstein integers&lt;/a&gt; after studying &lt;a href=&quot;https://math.stackexchange.com/a/3829536/284619&quot;&gt;this answer&lt;/a&gt; to a mathematics problem I'd asked about. Briefly they are represented by a hexagonal lattice on the complex plane, the distance of the six closest points to the origin are all unit length from it. With integers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; they are&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a + bu$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$u = \frac{1+ i \sqrt{3}}{2}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then I learned about &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaussian_integer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaussian integers&lt;/a&gt; which are represented by a square lattice of length one on the complex plane. With integers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; they are of the form&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a + bi.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Eisenstein integers are named after &lt;a href=&quot;https://en.wikipedia.org/wiki/Gotthold_Eisenstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gotthold Eisenstein&lt;/a&gt; and I assume Gaussian integers are named after &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Friedrich Gauss&lt;/a&gt;, but who gave these names to these number sets in the complex plane?&lt;/p&gt;&#xA;&lt;p&gt;Or at least how did consensuses for their names arise?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;The linked answer uses that expression for &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; because that's how the question was formulated. In wikipedia (and likely elsewhere) it is the oblique form (120°) rather than the acute (60°):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$u = \frac{-1 + i \sqrt{3}}{2}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;To make a hexagonal lattice one can use any two of the three unit vectors.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uMw3g.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uMw3g.png&quot; alt=&quot;Eisenstein integer lattice&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/File:Eisenstein_integer_lattice.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-11-20T23:41:19.770" LastActivityDate="2023-01-24T05:34:47.457" Title="How did Gaussian and Eisenstein integers get their names?" Tags="|mathematics|mathematicians|terminology|naming-conventions|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12561" PostTypeId="2" ParentId="12560" CreationDate="2020-12-01T11:55:45.063" Score="4" Body="&lt;p&gt;The article you linked to gives some historical background: It's whilst Gauss was investigating reciprocity laws that he discovered the Eisenstein and Gaussian integers. The former are the natural domain to study cubic reciprocity and the latter for quartic. He also notes that the integers in higher extensions would help prove higher reciprocity laws.&lt;/p&gt;&#xA;&lt;p&gt;I don't know who gave them their names but it would be later than 1832 when Gauss introduces both types of numbers in his second monograph on quartic, that is biquadratic, reciprocity.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-12-01T11:55:45.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12562" PostTypeId="1" CreationDate="2020-12-01T17:48:21.927" Score="1" ViewCount="134" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Golden_Gate_Bridge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Golden Gate Bridge&lt;/a&gt; was built in the 1930's across the the Golden Gate strait. The conditions in this straight are pretty challenging from an environmental perspective with strong tides.&lt;/p&gt;&#xA;&lt;p&gt;In modern construction, nearly every project is accompanied by extensive geophysical and geotechnical surveys in order to understand local geological conditions. Yet, many of these techniques have been developed relatively lately and would not have been accessible in the 1930's, marine geophysical investigations especially so. I've been investigating what were the survey techniques and methods used prior to the building of the Golden Gate Bridge which nevertheless had to have significant geological input into its design.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.goldengate.org/exhibits/engineering-the-design/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The bridge's official website&lt;/a&gt; notes a geologist, Andrew C. Lawson, conducting experiments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The geology of the south tower location was investigated before construction could begin. The south tower was planned for construction over 1,100 feet (335 meters) offshore on serpentine rock. Consulting geologist Andrew C. Lawson oversaw a load test performed by placing weight equivalent to a fully loaded railroad boxcar on an area of serpentine rock only 20 inches (508 millimeters) square. The rock was more than strong enough.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The website includes this letter from Lawson explaining these results:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pb5h6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pb5h6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some mentions of land-bound rock boring are also made next to these as well as some marine cores.&lt;/p&gt;&#xA;&lt;p&gt;The geological model that Lawson designed is illustrated as well by one of the attachments on that site:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/65k4N.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/65k4N.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The above-mentioned site also links to a &lt;a href=&quot;https://www.goldengate.org/assets/1/6/willis_geology_report3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;1934 South Pier Geology Report&amp;quot;&lt;/a&gt; which is illuminating. This mentions both hydrographic surveying (as well as the wonderful comment that &amp;quot;A comparison between the results of modern surveys and those made many years ago has always been open to question on account of the lower standards prevailing in the past.&amp;quot;) as well as boring, but not the details of these surveys (i.e., number and/or locations of cores or equipment). The most detail is given on one of the cores (the only one?) with:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The drilling proceeded to a depth of 251.62 feet below sea level and 159.37 feet below the base of the pier.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The only specific mention on hydrographic methodology is &amp;quot;Sextant angles taken by observers at fixed intervals were used to locate certain soundings * * * *&amp;quot; but this is only regarding the positioning of the samples and does not relate to how the samples were acquired. Geotechnical methodology is described in the smallest length &lt;a href=&quot;https://www.goldengate.org/assets/1/6/engineering-the-design-borings-started-for-gate-span.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; as &amp;quot;diamond-tipped drills used in boring&amp;quot; without specifying anything else.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What else do we know about the hydrographic or geotechnical methodology that was utilized? Do we have any information on the extent of the survey campaign?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5957" LastActivityDate="2020-12-01T17:48:21.927" Title="What did the pre-construction survey campaign of the Golden Gate Bridge involve?" Tags="|engineering|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12563" PostTypeId="1" CreationDate="2020-12-02T14:26:07.693" Score="1" ViewCount="51" Body="&lt;p&gt;C.S Peirce is credited by wikipedia with the invention of randomized experiments.&#xA;What was C.S Peirce's contribution to randomized experiments exactly if there were such experiments before Peirce?&#xA;How did the psychophysicists whose results Peirce was improving when he discovered the principles of randomized experiments conducted their experiments?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2020-12-02T15:38:48.853" LastActivityDate="2020-12-02T15:38:48.853" Title="What was the first randomized experiment that was ever conducted?" Tags="|biographical-details|philosophy-of-science|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12564" PostTypeId="1" AcceptedAnswerId="12566" CreationDate="2020-12-02T20:56:43.477" Score="0" ViewCount="2798" Body="&lt;p&gt;I know about &lt;a href=&quot;https://en.wikipedia.org/wiki/Manjul_Bhargava&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manjul Bhargava&lt;/a&gt;, a mathematician and a Fields Medal winner who is a tabla player.&#xA;Are there any other mathematicians who have had such a musical background?&lt;/p&gt;&#xA;&lt;p&gt;By &amp;quot;musical background&amp;quot;, I mean those who have had played musical instruments.&lt;/p&gt;&#xA;" OwnerUserId="10835" LastEditorUserId="10106" LastEditDate="2022-08-10T01:56:44.897" LastActivityDate="2022-08-15T02:17:35.690" Title="Who were some mathematicians who have had a musical background?" Tags="|mathematicians|" AnswerCount="5" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12565" PostTypeId="2" ParentId="12564" CreationDate="2020-12-02T22:59:35.257" Score="4" Body="&lt;p&gt;Einstein was a violinist, though I think he'd be the first to say he wasn't a musician.&lt;/p&gt;&#xA;&lt;p&gt;Eugenia Cheng is a mathematician who specialises in higher category theory. She is also a concert pianist specialising in Lieder and art song. She was awarded the Sheila Mossman Memorial award from the Associated Board of the Royal School of Music and was the first recipient of the Brughtin &amp;amp; Hove Arts Council award for the musician of the year.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, according to Wikipedia, Iannis Xanakis, a French-Greek  musician who incorporated mathematical modelling, such as stochastic processes and game and set theory into his compositions. Though, personally, I think this goes way back to Duchamp and his notion of 'found art'.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-12-02T22:59:35.257" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12566" PostTypeId="2" ParentId="12564" CreationDate="2020-12-03T03:31:04.293" Score="7" Body="&lt;p&gt;The MacTutor Article &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Archibald_music_1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematicians and Music&lt;/a&gt; provides an overview of this relationship.&lt;/p&gt;&#xA;&lt;p&gt;Amongst those mathematicians who were accomplished players and had performed in public, the article includes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Grassmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grassmann&lt;/a&gt; (piano, singer and choir master)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Farkas_Bolyai&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bolyai&lt;/a&gt; (violin)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Louis_Maupertuis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maupertius&lt;/a&gt; (flageolet and guitar)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Augustus_De_Morgan&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Morgan&lt;/a&gt; (flute)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Mathematicians who were also composers with published works include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Herschel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Herschel&lt;/a&gt; (also an accomplished violinist, hautboyist, organist, and conductor)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Mathematicians who wrote on the subject of music and its analysis include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/James_Joseph_Sylvester&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sylvester&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_von_Helmholtz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helmholtz&lt;/a&gt; (really a physicist)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The article comments on the common belief that mathematicians tend to make good musicians, noting recent statistical analysis which suggests that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... in only about 2 per cent of the cases considered was there any appreciable correlation between talent for music and talent for mathematics; they found also that the percentage of males lacking in talent for music but showing a talent for mathematics was comparatively high, about 13 per cent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT 30, July 2022&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Boltzmann, a mathematical physicist if not a mathematician, was a trained classical pianist who studied piano with none other than &lt;a href=&quot;https://en.wikipedia.org/wiki/Anton_Bruckner&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anton Bruckner&lt;/a&gt;.  Although he never performed for a paying audience he would frequently entertain friends, students, and colleagues by performing &lt;a href=&quot;https://en.wikipedia.org/wiki/Franz_Liszt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liszt's&lt;/a&gt; piano transcriptions of the Beethoven symphonies.  Whenever living in Vienna he was in regular attendance at the concert halls and theatres.&lt;/p&gt;&#xA;&lt;p&gt;Here is Boltzmann's decidedly musical, if somewhat corny reaction to Maxwell's 1866 paper, &lt;em&gt;On the Dynamical Theory of Gases&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;First the variations in velocity develop majestically, then the equations of state enter on one side, the equations of motion on the other; ever higher surges the chaos of formulas. Suddenly, four words sound out: 'Put &lt;span class=&quot;math-container&quot;&gt;$\text{N} = 5.$&lt;/span&gt;' The evil demon &lt;span class=&quot;math-container&quot;&gt;$\text{V}$&lt;/span&gt; vanishes, just as in music a disruptive figure in the bass abruptly falls silent...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2022-07-30T21:37:33.213" LastActivityDate="2022-07-30T21:37:33.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12568" PostTypeId="1" AcceptedAnswerId="12577" CreationDate="2020-12-03T05:35:46.567" Score="4" ViewCount="147" Body="&lt;p&gt;The Nobel Prize was awarded to David Gross, Frank Wilczek and Hugh David Politzer. I believe David Gross worked with Frank Wilczek and was his thesis advisor.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, I couldn't find much detail of the role of Hugh Politzer played in this discovery. It would also be interesting to know who proposed the method of theory validation. Again the same question would apply. I am lost on which tag to use so I may be in the incorrect area.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="4931" LastEditDate="2020-12-05T04:26:29.870" LastActivityDate="2020-12-05T04:26:29.870" Title="Did Hugh David Politzer work independently for his contribution to asymptotic freedom?" Tags="|quantum-mechanics|theoretical-physics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12569" PostTypeId="2" ParentId="3230" CreationDate="2020-12-03T06:01:28.213" Score="2" Body="&lt;p&gt;Though the idea of combining space and time dimensions was already tacit in Galileo's Relativity Principle, which expands the set of classical symmetries (translations and rotations for space, translations for time) to include a symmetry transform (the &amp;quot;Galilean&amp;quot; boost) that involves both space and time dimensions, it was not explicitly recognized as such before the 20th century. So the marriage of the two was an elopement that was not consummated until a few hundred years later.&lt;/p&gt;&#xA;&lt;p&gt;The idea that the two should be combined with the &lt;em&gt;specific&lt;/em&gt; metric that Minkowski advocated was certainly not present before Einstein. Though Lorentz introduced a set of symmetries to try and recover the &amp;quot;stationary&amp;quot; form of Maxwell's isotropic constitutive laws, the equations he actually published were &lt;em&gt;not&lt;/em&gt; relativistic (contrary to a claim made by an earlier respondent), but actually covariant with respect to the Galilean transform.&lt;/p&gt;&#xA;&lt;p&gt;The idea that Lorentz's theory was relativistic is a widespread folklore myth present, even in the Physics community. But, in fact, the non-relativistic nature of Lorentz's theory was a point that Einstein, himself, made shortly after the publication by Minkowski and later by Einstein and Laub of a relativistic theory for moving media in 1908.&lt;/p&gt;&#xA;&lt;p&gt;Minkowski's paper was, in fact, where he also introduced his geometry. The note about Lorentz's theory was &amp;quot;Comment on the paper of D. Mirimanoff 'On the Fundamental Equations...'&amp;quot; Annalen der Physik 28 (1909): 885-888, contained in the compilation below. And, I'll say a few words on it here, below, as well.&lt;/p&gt;&#xA;&lt;p&gt;The Einstein-Laub papers are discussed in further detail here &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol2-doc/539&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://einsteinpapers.press.princeton.edu/vol2-doc/539&lt;/a&gt;&#xA;along with some notes on their relation to Minkowski's &amp;quot;moving media&amp;quot; paper (&amp;quot;Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern&amp;quot;, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 53–111). As you can see in the discussion in the attached link: initially Einstein and Laub were not keen on the idea of fitting Relativity into a Minkowski geometry.&lt;/p&gt;&#xA;&lt;p&gt;Minkowski's paper actually went a bit further, by the way, than just laying out a set of equations for moving media: bringing up notions of transformation groups, for instance. I also notice that he appears to have used some kind of 5th coordinate (which he denoted by ϑ). I haven't read the paper in enough detail, yet, to see why he did that, but it's something that's of interest to me, for reasons that will shortly become clear.&lt;/p&gt;&#xA;&lt;p&gt;The Maxwell equations for moving media with what's now known as the Maxwell-Minkowski constitutive laws are&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    ∇· = 0, ∇× + ∂/∂t = &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;for electric force  and magnetic induction ,&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    ∇· = ρ, ∇× - ∂/∂t = &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;for electric induction  and magnetic force , with charge density ρ and current density , and&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;     + α × = ε( + ×),  - α × = μ( - ×)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;for the Maxwell-Minkowski constitutive law, for moving media with a reference velocity .&lt;/p&gt;&#xA;&lt;p&gt;What Minkowski and Einstein-Laub published was the version of the equations equivalent to the above where α &amp;gt; 0, with light speed identified as c = 1/√α. For the vacuum, εμ = α, and provided that || &amp;lt; c, the constitutive equations are equivalent to the &amp;quot;stationary&amp;quot; isotropic form for  = .&lt;/p&gt;&#xA;&lt;p&gt;In contrast, what Lorentz published (as Einstein also pointed out in 1909) is equivalent to the non-relativistic version of the constitutive law, which is the case where α = 0. The resulting equation set, in turn, is consistent with those used by Heaviside and Hertz. All of them are non-relativistic.&lt;/p&gt;&#xA;&lt;p&gt;What Lorentz did in his papers was perform a Galilean transform on the equations to get the non-relativistic (and incorrect) constitutive law with α = 0; and then to handle the case of the vacuum, posed a second ad hoc transform to recover the &amp;quot;stationary&amp;quot; form of the law. That second transform is equivalent to composing a reverse Galilean transform and forward Lorentz transform. So, in this way, he stumbled onto what we now call the Lorentz transforms.&lt;/p&gt;&#xA;&lt;p&gt;But instead of characterizing it as a fundamental property of space-time and then &lt;em&gt;most importantly&lt;/em&gt; making the relativistic corrections to the constitutive law, he kept the constitutive law intact in non-relativistic form and pushed the inverse-Galilean + forward-Lorentz fix as something that probably arose from the mechanical properties of the electromagnetic field, itself. His equations are non-relativistic and Galilean-covariant; and, because of that, even with his extra fix, his constitutive equations for the moving media were simply wrong.&lt;/p&gt;&#xA;&lt;p&gt;The equations, when α &amp;gt; 0, are associated naturally with a geometry which has - as its invariants -&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    dt² - α (dx² + dy² + dz²),&#xA;    ((∂/∂x)² + (∂/∂y)² + (∂/∂z)²) - α (∂/∂t)²,&#xA;    dt (∂/∂t) + dx (∂/∂x) + dy (∂/∂y) + dz (∂/∂z),&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;which is the same as what characterizes Minkowski geometry.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, the equations published by Lorentz -- and others pre-dating Einstein -- (as well as the equations describing the fundamentals of Newtonian physics) accord with the geometry that has these invariants&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    dt²,&#xA;    (∂/∂x)² + (∂/∂y)² + (∂/∂z)²,&#xA;    dt (∂/∂t) + dx (∂/∂x) + dy (∂/∂y) + dz (∂/∂z).&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;And this is what people pre-dating the 20th century would have been faced with, if they wanted to devise a unified chrono-geometry for space-time ... rather than Minkowski geometry. So, for that reason it never really got anywhere; and also for that reason, no notion of Minkowski geometry would have arisen before the 20th century in connection with electromagnetic theory or any part of Newtonian Physics.&lt;/p&gt;&#xA;&lt;p&gt;In fact, as you're about to see, there would have been a strong disincentive against putting forth a four dimensional geometry for one very simple reason - a reason that wasn't given a proper account until well into the 20th century. And it may also be what lay behind Einstein and Laub's initial resistance to Minkowski's 4D geometry idea even if they themselves didn't realize it at the time.&lt;/p&gt;&#xA;&lt;p&gt;The reason is this: mass, kinetic energy and momentum transform together as a 5 vector, not a 4 vector! Before Relativity it would have been natural to consider the mass-momentum-energy 5-vector (m, , H). Einstein, in his earlier papers treated H and m separately and didn't write anything like E = Mc², but rather M = m + H/c² for &amp;quot;moving mass&amp;quot; and kinetic energy. Under a Galilean boost transform, with boost velocity , the 5 vector transforms as (m, , H) → (m,  ‒ m, H ‒ · + ½ w² m). Under an infinitesimal boost , it transforms as Δ(m, , H) = (0, ‒m, ‒·).&lt;/p&gt;&#xA;&lt;p&gt;For relativistic theory, what Einstein showed is that the infinitesimal transform has to be modified to Δ(m, , H) = (0, ‒(m + αH), ‒·), taking α = 1/c² &amp;gt; 0, in place of α = 0. Hence, the earliest papers asserted only that &amp;quot;energy has inertia, too&amp;quot; and identified the &amp;quot;moving mass&amp;quot; of a body as M = m + αH.&lt;/p&gt;&#xA;&lt;p&gt;After Minkowski's paper, the consensus fell on strengthening the statement &amp;quot;energy has inertia, too&amp;quot; to &amp;quot;mass is energy&amp;quot; and combining the kinetic energy and mass into the total energy E = mc² + H; shifting the focus by writing E = Mc², instead of M = E/c²; and then considering only the 4-vector (, E) instead of the 5-vector, since it is closed under transforms, with the infinitesimal boost given by Δ(, E) = (‒αE, ‒·). That's the energy-momentum 4-vector.&lt;/p&gt;&#xA;&lt;p&gt;Probably the most important feature of this 4-vector is that it forms an invariant with the coordinate differentials (dx, dy, dz, dt) = (d, dt), since their boost transform is given infinitesimally by Δ(d, dt) = (‒ dt, ‒α·d). Consequently, together they yield the invariant&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    ·d ‒ E dt.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;No such invariant can be formed - either relativistically or non-relativistically - with the momentum and kinetic energy. In fact, what you find is that&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    Δ(·d ‒ H dt) = (‒(m + αH))·d ‒ (‒·) dt + ·(‒ dt) ‒ H(‒α·d) = ‒m(·d)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;so you lose one of the main advantages of the 4D geometric approach used in Relativity. It wasn't until well into the 20th century - the 1950's in fact - that this issue was resolved - to associate another coordinate (u) with the mass m, possessing the infinitesimal transform Δ(du) = ·d, and then one recovers an invariant for the 5-vector:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    Δ(·d ‒ H dt + m du) = 0.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;This applies to both the non-relativistic case α = 0 &lt;em&gt;and&lt;/em&gt; the relativistic case α &amp;gt; 0! In the former case, the geometry is known as a Bargmann geometry and the symmetry group that goes with it is called the Bargmann group. It is equivalently described as the central extension of the Galilei group.&lt;/p&gt;&#xA;&lt;p&gt;This suffices, in fact, to give you the basis for the Newtonian analogue to General Relativity, as discovered much more recently in the 1980's(!) &amp;quot;Bargmann Structures and Newton-Cartan Theory&amp;quot; &lt;a href=&quot;https://inspirehep.net/literature/202550&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://inspirehep.net/literature/202550&lt;/a&gt;. The non-relativistic version of the Schwarzschild solution is given by&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    dx² + dy² + dz² + 2 dt du ‒ 2U dt² = 0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;where U = -GM/r is the gravitational potential per unit mass. The geodesics are the orbits described by Newton's Law of gravity.&lt;/p&gt;&#xA;&lt;p&gt;In the relativistic case, the group and geometry have no name that I am aware of; but it is closely linked to a variety of disparate issues that have never really fit well Relativity under the framework provided by Minkowski geometry (e.g. complexification of the Dirac algebra, the &amp;quot;Stuckelberg trick&amp;quot;, as well as a host of other issues). It also arises in 5D cosmology, and the extra coordinate can be re-interpreted as &amp;quot;historical time&amp;quot; s = t + αu, since the coordinate (s) transforms as an invariant. Ironically, that both recovers and gives geometric realization to Lorentz's &amp;quot;absolute time&amp;quot; versus &amp;quot;local time&amp;quot;. So, you might say, on account of this, that Lorentz was already heading in this direction.&lt;/p&gt;&#xA;&lt;p&gt;The symmetry group for the geometry is just the 10 generator inhomogeneous Lorentz group (= Poincaré group), itself, trivially extended with a translation symmetry for the u coordinate as an 11th generator.&lt;/p&gt;&#xA;&lt;p&gt;The Schwarzschild solution, itself, can be written equivalently in this geometry as&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    dx² + dy² + dz² + 2 dt du + α du² ‒ 2U dt² ‒ 2αU dr²/(1 + 2αU) = 0&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;where r = √(x² + y² + z²) and dr = (x dx + y dy + z dz)/r; with the s coordinate giving you the proper time for the Schwarzschild metric.&lt;/p&gt;&#xA;&lt;p&gt;This is what people before the 20th century would have been faced with, if they tried to unify space and time into a single chrono-geometry. With α &amp;gt; 0, using a 5D geometry is optional, since you can get away with 4D. But with α = 0, it virtually mandatory to have the extra (u) coordinate in place, too, to get things to work consistently. Like Michael Fox would have said, after traveling back in time to steam-punk days, I don't think they would have been ready for that.&lt;/p&gt;&#xA;" OwnerUserId="13406" LastEditorUserId="13406" LastEditDate="2020-12-03T07:07:38.147" LastActivityDate="2020-12-03T07:07:38.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12570" PostTypeId="2" ParentId="12564" CreationDate="2020-12-03T07:25:14.743" Score="2" Body="&lt;p&gt;There's always &lt;a href=&quot;https://en.wikipedia.org/wiki/Tom_Lehrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tom Leher&lt;/a&gt;.  I think he counts as a musician and mathematician.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=UQHaGhC7C2E&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=UQHaGhC7C2E&lt;/a&gt; (&amp;quot;Lobachevsky&amp;quot;.  It's satire, by the way)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=yhuMLpdnOjY&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=yhuMLpdnOjY&lt;/a&gt; (Poisoning Pigeons in the Park)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=zGM-wSKFBpo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=zGM-wSKFBpo&lt;/a&gt; (The Elements)&lt;/p&gt;&#xA;" OwnerUserId="13407" LastEditorUserId="13407" LastEditDate="2020-12-03T07:34:25.480" LastActivityDate="2020-12-03T07:34:25.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12571" PostTypeId="2" ParentId="12564" CreationDate="2020-12-03T11:12:59.987" Score="3" Body="&lt;p&gt;&lt;strong&gt;Per Enflo&lt;/strong&gt;  (see &lt;a href=&quot;https://en.wikipedia.org/wiki/Per_Enflo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; for a longer summary)&lt;/p&gt;&#xA;&lt;p&gt;concert pianist, mathematician (solutions for the basis problem, the approximation problem, the invariant subspace problem in Banach space theory )&lt;/p&gt;&#xA;&lt;p&gt;At a certain point in his life, he had to choose between the two.&#xA;He reasoned that the likelihood of ever reaching the top as a concert pianist was small, compared to the same thing in mathematics.  He chose mathematics.  But throughout his life, despite being a professor of mathematics, he continued to do piano performances.  When I knew him (maybe 1980's) he still spent two hours a day practicing on the piano.&lt;/p&gt;&#xA;&lt;p&gt;The basis problem was placed in the &lt;em&gt;Scottish Book&lt;/em&gt; by Banach.  The prize offered was a live goose.  Enflo did, indeed, receive the prize.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2020-12-03T11:18:46.537" LastActivityDate="2020-12-03T11:18:46.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12577" PostTypeId="2" ParentId="12568" CreationDate="2020-12-04T16:55:25.793" Score="4" Body="&lt;p&gt;I a not sure I understand what you mean by &amp;quot;theory validation&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In any case, you may learn Politzer's part of the story from his Nobel lecture paper &lt;a href=&quot;https://www.worldscientific.com/doi/pdf/10.1142/S0217751X05029010?casa_token=2RGRxuctlVkAAAAA:fRIPg_6vg3JhwtkL91YsVvJj1nZkF5ATn5jDl_oHDaDKADUoeOOocXq95Rx0M0f6qqTxLjVjfwA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Politzer, H.D., 2005. &lt;em&gt;The dilemma of attribution&lt;/em&gt;,  &lt;em&gt;International Journal of Modern Physics&lt;/em&gt; &lt;strong&gt;A&lt;/strong&gt; 20(25), pp.5741-5751&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;His advisor was the legendary &lt;a href=&quot;https://en.wikipedia.org/wiki/Sidney_Coleman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sidney Coleman&lt;/a&gt;, who was visiting Princeton from Harvard for a semester, Winter 1972-Spring 1973. Politzer visited him at Princeton a couple of times, and conferred with him and Gross. His unofficial mentor was Erick Weinberg, a more advanced graduate student with &lt;em&gt;β&lt;/em&gt; - function experience. He worked independently, and was actually coming from a different direction: generalizing Coleman-Weinberg EW breaking to non-abelian gauge theories: he refocussed his discovery to the strong interactions himself. Unlike Gross &amp;amp; Wilczek, Politzer never wavered about the negative sign of that  &lt;em&gt;β&lt;/em&gt; - function.&lt;/p&gt;&#xA;&lt;p&gt;If by validation you meant evidence in support, most theorists had accepted the point within the year of publication (1973), as described in &lt;a href=&quot;https://www.pnas.org/content/102/26/9099&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gross&lt;/a&gt;'s and Politzer's recollections. The experimental results of DIS, etc were already at hand, so it was the interpretation and acceptance by theorists that was paramount at that stage. By year's end, it had been accepted as the enabling element of QCD, speculated about in 1972 by Gell-Mann, Bardeen and Fritzsch. Few theorists failed to instinctively connect asymptotic freedom to its obverse, infrared slavery, and arguments for confinement, proffered in 1974 by Wilson.&lt;/p&gt;&#xA;&lt;p&gt;Experimentally, scaling violations predicted by it emerged convincingly in 1974. &lt;span class=&quot;math-container&quot;&gt;$e^+e^-$&lt;/span&gt;-collision jets were observed in 1975, and 3-jets in 1979 at PETRA. By that time there were hardly any doubters...&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2020-12-04T21:27:16.243" LastActivityDate="2020-12-04T21:27:16.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12578" PostTypeId="2" ParentId="2488" CreationDate="2020-12-05T03:50:27.700" Score="0" Body="&lt;p&gt;It is not quite clear, even from the more explicit post on physics stack exchange, which of the two games you are interested in.  They are different.  One is the idea of the various bundles involved, e.g., that a 1-form is not (as the earlier mathematicians thought of it) &amp;quot;an expression of the form pdX + qdY+ rdZ&amp;quot; (plus, they well knew how the expresion transforms under changes of variable), but rather a section of a vector bundle.  The next generation (Ehresmann, Chevalley, Lichnerowicz, H. Cartan, Bourbaki in general) learned the essence of fibre bundles from Cartan's work, and developed their better way of expressing it from their attempts to clarify his very difficult to read papers.&#xA;The other is the exterior calculus, i.e., the union of the Grassmann calculus with Riemannian geometry, all the rules for calculating with tensors, d, curvature 2-form with values in a Lie algebra, etc.  Of this, Cartan's only truly original contribution was the definition of the exterior derivative, d, for forms of higher degree.  And its notation.  But d for higher degree forms does not seem to be what Kosyakov was really referring to.  He seems to be referring to Cartan's systematisation of ideas that do go back to Hamilton, Grassmann, Stokes, Frobenius, Pfaff, Riemann, and Ricci.  Cartan's sytematisation and notation was better, but&#xA;the ideas were there earlier.&lt;br /&gt;&#xA;In particular, &lt;span class=&quot;math-container&quot;&gt;$dx_i$&lt;/span&gt; as a notation was early, but as a section of&#xA;the cotangent bundle, that is due to Cartan (or perhaps Ricci). And I do not see that Cartan used &lt;span class=&quot;math-container&quot;&gt;$\partial_x$&lt;/span&gt; early on.&lt;/p&gt;&#xA;" OwnerUserId="13415" LastEditorUserId="13461" LastEditDate="2020-12-12T07:15:19.787" LastActivityDate="2020-12-12T07:15:19.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12580" PostTypeId="1" AcceptedAnswerId="12581" CreationDate="2020-12-05T11:10:09.603" Score="29" ViewCount="7783" Body="&lt;p&gt;Einstein dedicated his time between 1905-1915 to develop &lt;a href=&quot;https://en.wikipedia.org/wiki/General_relativity&quot; rel=&quot;noreferrer&quot;&gt;general relativity&lt;/a&gt; (GR). It seems strange to me that no other physicists attempted to tackle this problem in this ten-year period. After all, developing a relativistic theory of gravity superseding Newton's should have been the holy grail of physics at that period. &lt;strong&gt;How come none gave it a shot?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Contrast this with &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_electrodynamics&quot; rel=&quot;noreferrer&quot;&gt;quantum electrodynamics&lt;/a&gt; where a lot of physicists (Feynman, Dyson, Schwinger, Tomonaga, etc.) rushed/raced to develop the theory. Even other scientists/mathematicians (e.g., Marcel Grossmann) intended only to help/guide Einstein, but not compete with him.&lt;/p&gt;&#xA;&lt;p&gt;No one independently tackled the problem except for Hilbert who successfully discovered the Einstein field equations in 1915 simultaneously with Einstein. But this is very late, given that Einstein had already been working on it for 10 years. Moreover, this suggests that other eminent physicists might have been able to develop GR &lt;em&gt;if only they attempted to work on it in&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So are there any (failed) attempts at developing GR in 1905-1915 by other scientists? If not, how come no one gave it a shot in ten years given the importance of the problem?&lt;/p&gt;&#xA;" OwnerUserId="2423" LastEditorUserId="2423" LastEditDate="2020-12-07T10:25:32.910" LastActivityDate="2020-12-07T10:25:32.910" Title="Why did no one else, except Einstein, work on developing General Relativity between 1905-1915?" Tags="|physics|theoretical-physics|relativity-theory|einstein|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12581" PostTypeId="2" ParentId="12580" CreationDate="2020-12-05T11:33:28.090" Score="42" Body="&lt;p&gt;From the rough understanding of the physics of the era I have, I would say there are at least two main reasons as to why it is actually quite expected that finding a relativistic theory of gravity wouldn't be the priority of the majority of physicists.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The holy grail of physics was quantum mechanics since the mid-1900s if not from the beginning of the century itself. Planck proposes the light-quanta based solution to the blackbody radiation problem in 1900, kicking off the quantum era. Thompson proposes the so-called plum-pudding model of the atom in 1904. Einstein comes up with his explanation of the photoelectric effect in 1905, elevating the status of photons from a mathematical trick in Planck's idea to a real physical thing. Milikan shows the quantization of charge in 1907 with the famous oil-drop experiment. Rutherford does the gold-foil experiment showing that the atom has a nucleus in 1911. Finally, Bohr introduces the historic Bohr model of the atom in 1913.  As you can see, the world of quantum mechanics is continuously action-packed in the era with which we are concerned.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On the other hand, special relativity wasn't noticed at all in 1905. The acceptance and awareness of special relativity began around 1909 when Planck compared it to be a revolutionary step of the order of the Copernican revolution. A major step in the acceptance of special relativity was the development of the geometric formulation of special relativity by Minkowski in 1907 which also took some time to get acceptance (funnily enough, Einstein himself wasn't quite kind to his old teacher Minkowski about his funny four-dimensional diagrams). While most physicists accepted special relativity by around 1911-1912, it still was a controversial subject, to say the least. Einstein was nominated (along with Lorentz) for the Nobel prize in physics for special relativity but the Nobel committee had decided against it, in part because the old guard of physics still believed in the aether theory.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In addition to it being unlikely that finding a relativistic theory of gravity is the top priority of many physicists, the first physical insight which led to general relativity was put in its furnished form only in 1911 when Einstein laid out the equivalence principle in detail which he had introduced in 1907 earlier. In any case, just like Einstein, all other physicists were completely unaware of the machinery of differential geometry which was needed to mathematically formulate a theory that utilized the equivalence principle. So, it makes sense that it was a mathematician who actually tried to do it, namely, Hilbert. Also, given how much of an esoteric and a purely theoretical subject general relativity was considered even after it was &amp;quot;completed&amp;quot; in 1915 (until the 60s and the 70s when the likes of John Wheeler revived it), one can imagine how esoteric it would have been considered before 1915. People had much more tangible work to do in quantum mechanics.&lt;/p&gt;&#xA;&lt;p&gt;Despite all these reasons, Hilbert was by no means the only person other than Einstein who worked on it. As is famous, Emmy Noether (another mathematician) worked on it, helping Hilbert. In fact, Poincare tried to create his own relativistic theory of gravity by trying to modify Newton's law of gravity in analogy with the relativistic Maxwell's equations as early as 1905. In around 1912, Abraham and Mile formulated a scalar theory of gravity which was Lorentz invariant but failed to satisfy the equivalence principle. The most impressive effort was by Nordstrom who actually formulated a self-consistent scalar theory of gravity in 1912-1913, consistent with both Lorentz invariance and the equivalence principle, but it violated the requirement of general covariance which Einstein thought was crucial. In the end, as we know, Einstein's general theory of relativity which was a tensor theory was consistent with Lorentz invariance, equivalence principle, as well as general covariance--and most importantly, has matched in its predictions with all the experiments so far.&lt;/p&gt;&#xA;" OwnerUserId="4257" LastEditorUserId="4257" LastEditDate="2020-12-06T05:21:03.243" LastActivityDate="2020-12-06T05:21:03.243" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="12582" PostTypeId="2" ParentId="12580" CreationDate="2020-12-05T15:42:45.510" Score="14" Body="&lt;p&gt;Your claim is simply not true: there was work published by Einstein, Hilbert, Nordstrom, and (slightly later) Whitehead. It was a very hard problem, so undoubtedly for every person who came up with something publishable, there were many others who tried but who we don't know about.&lt;/p&gt;&#xA;&lt;p&gt;This was also very early for anyone to be working on such a theory, based on the state of the art. There was no motivation for working on a relativistic theory of gravity unless you already believed that special relativity was correct. But SR was a new and almost completely untested. The evidence in its favor was indirect and could be interpreted in other ways. For example, Michelson and Morley believed that their original experiment had been a failure because it didn't produce the &amp;quot;right&amp;quot; result, and as late as 1925 their successors were trying to get a correct, positive result by redoing it on Mt. Wilson. A &lt;a href=&quot;https://www.nature.com/articles/116049a0&quot; rel=&quot;noreferrer&quot;&gt;1925 paper&lt;/a&gt; in Nature by Miller describing a positive (!) Mt. Wilson result doesn't even mention special relativity, and Miller was still publishing this type of material in 1933. Detailed, unambiguous experimental evidence for SR didn't come until experiments like Ives-Stilwell in 1938.&lt;/p&gt;&#xA;" OwnerDisplayName="user466" LastEditorDisplayName="user466" LastEditDate="2020-12-05T15:56:24.360" LastActivityDate="2020-12-05T15:56:24.360" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12583" PostTypeId="2" ParentId="12580" CreationDate="2020-12-05T17:26:00.043" Score="-2" Body="&lt;p&gt;People have been working on extending Newtonian gravity ever since Newton discovered it. The major problem of the theory, as admitted by Newton, was its 'action at a distance' character and for which he was unable to come up with a solution. As various people worked on those extensions, the generic term for the medium of transmission was called &lt;a href=&quot;https://en.wikipedia.org/wiki/Aether_theories&quot; rel=&quot;nofollow noreferrer&quot;&gt;the aether&lt;/a&gt;. Notably, given the prominence of mechanics, the aether was hypothesised mechanically.&lt;/p&gt;&#xA;&lt;p&gt;It's only after electromagnetism was discovered with Faraday introduced his lines of force that another way of conceptualising the aether came to the fore, that is by the field. A number of authors were working on aetherial theory of gravity, the most important of which, according to Einstein, was Nordstroms scalar-tensor theories. His second theory even satisfied Einsteins weak equivalence principle!&lt;/p&gt;&#xA;&lt;p&gt;However, it failed to predict light deflection and also the anomalous &lt;a href=&quot;https://en.wikipedia.org/wiki/Apsidal_precession#General_relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;precession of Mercury&lt;/a&gt;, which Einsteins 1915 theory did.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="8151" LastEditDate="2020-12-07T06:45:23.810" LastActivityDate="2020-12-07T06:45:23.810" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12584" PostTypeId="2" ParentId="11450" CreationDate="2020-12-05T19:07:15.687" Score="6" Body="&lt;p&gt;In Aristotles &lt;em&gt;Meteorology&lt;/em&gt;, we have:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, the sun, moving as it does, sets up processes of change and becoming and decay, and by it's agency the finest and the sweetest water is carried up and is dissolved into vapour and rises to the upper region, where it is condensed again by the cold and so returns the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And he notes this again in his &lt;em&gt;Physics&lt;/em&gt; that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Zeus does not send the rain in order to make the corn grow: it comes of neccessity. The stuff which has been drawn up is bound to cool, and having cooled, turns to water and comes down. It is merely concurrent that this having happened, the corn grows.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hence, the hydrological or water cycle, at least in Europe, was known by Aristotles time, and so by 350 BCE.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-04-16T17:23:55.123" LastActivityDate="2021-04-16T17:23:55.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12585" PostTypeId="2" ParentId="3230" CreationDate="2020-12-05T19:22:26.650" Score="-1" Body="&lt;p&gt;Einstein was a mathematical physicist rather than a pure mathematician, that is one interested in mathematics for its own sake. It was likely that he was aware of non-Euclidean geometry given its prominence in mathematical circles, but given that it hadn't been shown to be useful in physics, he knew nothing about it until Minkowski's reformulation brought it to his attention. Mathematicians were already interested in this space as it was an instance of hyperbolic space as opposed to our usual Euclidean space; the other main example of non-Euclidean geometry being spherical or elliptical space.&lt;/p&gt;&#xA;&lt;p&gt;This was of some consequence, as it was in this language that he formulated General Relativity - however, he always insisted on the primacy of the physical; for example, when a colleague said that motion in GR should be understood geometrically via the geodesic equation, he insisted otherwuse; to him, that equation was to be understood as a unification of inertia and gravitation. In this sense, it harks back to Hertz who was looking at a way of unifying physics by expressing everything through constraining forces.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-12-05T19:22:26.650" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12589" PostTypeId="1" CreationDate="2020-12-06T20:05:46.047" Score="1" ViewCount="59" Body="&lt;p&gt;My understanding that among the key technologies that allowed manned flight was a powerful enough engine that was also lightweight. Steam was attempted but the gasoline engine turned out to be key.&lt;/p&gt;&#xA;&lt;p&gt;Two other technologies besides steam and gasoline occur to me: rockets and compressed air, both of which were well understood in the 19th century.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2020-12-06T23:36:17.217" LastActivityDate="2020-12-06T23:36:17.217" Title="Why was using rockets to propel a winged vehicle not proposed in the 19th century?" Tags="|steam-engines|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12591" PostTypeId="1" CreationDate="2020-12-07T08:52:38.310" Score="4" ViewCount="383" Body="&lt;p&gt;What we often refer to as &lt;a href=&quot;https://en.wikipedia.org/wiki/Snell%27s_law#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snell's law&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n_1 \sin(\theta_1) - n_2 \sin(\theta_2) = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;has quite a bit of history behind it. It can be demonstrated in several ways, one of which is by asserting that along the boundary there is no discontinuity in phase from one side to the other, or if there is, that it's constant.&lt;/p&gt;&#xA;&lt;p&gt;What I usually call &amp;quot;the grating equation&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$n_1 \sin(\theta_1) - n_2 \sin(\theta_2) = \frac{m \lambda}{d}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the integer order and &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; are the wavelength and periodic grating spacing can be demonstrated by asserting that &lt;em&gt;at a periodic array of points&lt;/em&gt; spaced &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; apart there is no discontinuity in phase, but what happens between those points is now unconstrained. That's how multiple non-zero orders are now possible.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Proper, periodic diffraction gratings are much more recent items than glass lenses. Where does the grating equations come from? Who first wrote it, did it it precede actual diffraction grating experiments, &lt;em&gt;and does it have another name?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-01-17T23:50:52.077" LastActivityDate="2022-03-27T11:31:17.113" Title="Where does &quot;the grating equation&quot; come from? Does it have a another name?" Tags="|physics|terminology|optics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12592" PostTypeId="2" ParentId="12591" CreationDate="2020-12-07T15:30:26.790" Score="4" Body="&lt;p&gt;Your two equations are very different in fact: Snell's law can be derived using geometric optics, while the grating equation requires wave theory of light.&#xA;The grating (as an optical instrument) was invented by Joseph von Fraunhofer in 1823,&#xA;when the wave theory was already available but not universally recognized. Apparently the equation is due to him as well.&lt;/p&gt;&#xA;&lt;p&gt;Source: S. Sternberg, A history of 19th century spectroscopy, Appendix F to his book&#xA;Group Theory and Physics, Cambridge UP, 1994.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-12-12T13:33:54.093" LastActivityDate="2020-12-12T13:33:54.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12593" PostTypeId="2" ParentId="6256" CreationDate="2020-12-07T15:43:11.807" Score="5" Body="&lt;p&gt;I just wanted to summarize several useful facts about Gauss's notation and methodology which i infered from Gauss's original writings after a lot of effort (this doesn't constitute an answer to the title question).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Notation&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In all of Gauss's posthomous papers on elliptic and theta function (which are notoriously difficult to read!) he employs several definitions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On p.440 of volume 3 of Gauss's werke,  Gauss defines the infinite product &lt;span class=&quot;math-container&quot;&gt;$\prod_{n=1}^{\infty}(1-x^n)$&lt;/span&gt; (in modern terminology this is called Euler function) and denotes it &lt;span class=&quot;math-container&quot;&gt;$[x]$&lt;/span&gt;. Gauss makes a lot of use of this infinite product.&lt;/li&gt;&#xA;&lt;li&gt;On p. 465 of this volume Gauss defines functions which are of essential importance for his theory, so he designates them with special symbols. The functions are:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P(x,y)=1+x(y+\frac{1}{y})+x^4(y^2+\frac{1}{y^2})+x^9(y^3+\frac{1}{y^3})+...$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Q(x,y)= 1-x(y+\frac{1}{y})+x^4(y^2+\frac{1}{y^2})-x^9(y^3+\frac{1}{y^3})+...$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$R(x,y)=x^{\frac{1}{4}}(y^{\frac{1}{2}}+y^{-\frac{1}{2}})+x^{\frac{9}{4}}(y^{\frac{3}{2}}+y^{-\frac{3}{2}})+x^{\frac{25}{4}}(y^{\frac{5}{2}}+y^{-\frac{5}{2}})+...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;These functions include the usual theta functions as special cases, for example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P(x,1)=1+2x+2x^4+2x^9+...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorems discovered by Gauss&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On page 470 appears the most &amp;quot;general theorem&amp;quot; of Gauss (Gauss writes it's date of writing as August 6, 1827):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P(x,ty)\cdot P(x,\frac{y}{t}) = P(x^2,t^2)P(x^2,y^2) + R(x^2,t^2)R(x^2,y^2) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;On p. 471 Gauss defines:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P(x^3,1)=P , P(x,1)=p$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Q(x^3,1)=Q , Q(x,1)=q$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$R(x^3,1)=R , R(x,1)= r$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And on p. 476 appears the definitions &lt;span class=&quot;math-container&quot;&gt;$P^0=P(x^{\frac{1}{3}},1), Q^0=Q(x^{\frac{1}{3}},1)$&lt;/span&gt;&#xA;The identities mentioned in the posted question, which were used by Karel Petr for the determination of the number of times that certain quaternary quadratic forms represent a given integer &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, appear in the same passage (article [10]) on p. 476:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(\frac{3P^2-P^0\cdot P^0}{2})^2= p^4-4(\frac {pqr}{2})^{\frac{4}{3}}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(\frac{3Q^2-Q^0\cdot Q^0}{2})^2=q^4+4(\frac{pqr}{2})^{\frac{4}{3}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Significance of the remarks in passage [10]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I also had to write up also some very useful remarks by the editor of Gauss's analytic papers (Ludwig Schlesinger). On p.186 of Schlesinger's essay on Gauss's contributions to analysis, Schlesinger has the following things to say about this part of the posthomous paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... This is followed (article [7]) by the modular transformation of 7th order, and, on August 29, also the 5th order modular transformation. The passage in article [10] (p.476), where the general transformation of odd order is mentioned, &lt;em&gt;&lt;strong&gt;is remarkable&lt;/strong&gt;&lt;/em&gt;; so Gauss really has, as he writes to Schumacher, the theorem contained in the second letter of Jacobi... very easily derived from his own research on the transcendents. The article [12] relating to the theory of the modular function has already been discussed above.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That's all up to now. If anyone has useful comments on the meaning of the functions mentioned here, it will blessed!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2020-12-16T08:02:45.693" LastActivityDate="2020-12-16T08:02:45.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12596" PostTypeId="1" CreationDate="2020-12-08T09:02:59.493" Score="4" ViewCount="122" Body="&lt;p&gt;I really should know this, but ...&lt;/p&gt;&#xA;&lt;p&gt;When/where/by whom was first-order Peano Arithmetic first clearly and explicitly formulated in a recognizably modern form (perhaps exact notation apart) -- with the usual recursion axioms for successor, addition, multiplication and the schema for induction axioms?&lt;/p&gt;&#xA;" OwnerUserId="343" LastActivityDate="2020-12-08T09:02:59.493" Title="Where is First-Order Peano Arithmetic first clearly formulated?" Tags="|number-theory|mathematical-logic|arithmetic|" AnswerCount="0" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="12597" PostTypeId="1" CreationDate="2020-12-10T00:48:09.247" Score="0" ViewCount="156" Body="&lt;p&gt;Since Leibniz is known to have expressed opinions about Descartes' and Newton's abilities maybe he did the same for the prominent thinkers of antiquity.&#xA;Did Leibniz ever express any opinion about Aristotle, Plato or Archimedes?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-12-10T00:48:09.247" Title="Did Leibniz ever express any opinion about Aristotle, Plato or Archimedes?" Tags="|biographical-details|leibniz|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12598" PostTypeId="1" AcceptedAnswerId="12599" CreationDate="2020-12-10T02:46:59.387" Score="1" ViewCount="681" Body="&lt;p&gt;Ancient Greek had a large impact on scientific and mathematical language and thinking leading into the Enlightenment period and beyond. But &amp;quot;ancient Greek&amp;quot; could refer to any dialect spanning a period of hundreds of years. Today, the Ancient Greek taught e.g. in Greece is Koine Greek (I believe), and this aligns with e.g. Aristotle who was obviously hugely influential in Medieval and Enlightenment thought, so I would assume this is the dialect that would have been taught to budding scientists and mathematicians like Newton. But I'm looking for some type of reference which can clarify what was taught and when?&lt;/p&gt;&#xA;&lt;p&gt;=Edit= Correction: Koine Greek was Roman, Attic Greek was used in Athens so that seems to align more with Aristotle and Enlightenment ideals.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="2324" LastEditDate="2020-12-13T07:07:14.673" LastActivityDate="2020-12-13T08:05:44.150" Title="What dialect of ancient Greek was taught to natural philosophers?" Tags="|ancient-greece|education|language|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12599" PostTypeId="2" ParentId="12598" CreationDate="2020-12-10T17:36:19.463" Score="9" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;Attic&lt;/strong&gt;&lt;/em&gt;, the highest prestige dialect, hands down. For centuries, this has been the mainstay of Greek language education in Europe.&lt;/p&gt;&#xA;&lt;p&gt;There is no better reminder of this than &lt;a href=&quot;http://cudl.lib.cam.ac.uk/view/MS-ADD-03996/12&quot; rel=&quot;noreferrer&quot;&gt;Newton's Trinity College Notebooks&lt;/a&gt; taken down in serviceable, not perfect, classic Attic. (Note he uses Attic syntax, verbs in 3rd person singular, matching a &lt;em&gt;plural&lt;/em&gt; subject.)&lt;/p&gt;&#xA;&lt;p&gt;Of course, works of Italiot authors, like Archimedes, were written in Doric, but have not survived, and all extant excerpts in our possession were &amp;quot;translated&amp;quot; to Attic in antiquity.  Aristotle's books and Euclid's elements are basically in  Attic, virtually fully overlapping with Koine, and serving as its cynosure; and fully comprehensible to educated Koine readers, then and now. Even Diophantus, 3rd century AD, writes in Classical language, the language of all mathematical texts he'd read, and not in Koine, the language he probably spoke at home.&lt;/p&gt;&#xA;&lt;p&gt;The Ancient Greek taught in Greece today is Classical (Attic), with some Homeric. Koine need not be taught as &amp;quot;Ancient Greek&amp;quot; there, as Testament (funny Koine) Greek is used in churches every day, and underlies the reconstructed formal language, Καθαρεύουσα, used for higher education and state functions for close to two hundred years; much of this language was taught as &amp;quot;modern&amp;quot; Greek. Panegyric school speeches were given in it, arguably comprehensible to students. It was abandoned only recently in the mainstream, but  still lingers in the culture; as English incorporates Shakespeare maxims, so does colloquial Greek run on  Testament maxims. Wonderful 150 year old works of literature in Koine/Καθαρεύουσα   have started being translated to colloquial Greek only recently.&lt;/p&gt;&#xA;&lt;p&gt;In Western Europe, and the US, &amp;quot;Ancient Greek&amp;quot; textbooks have, almost exclusively, been covering Classical (Attic) Greek,  given the bulk of the corpus of Greek authors; with a smattering of Homeric. I think Biblical scholars in the last century have splintered off and are hyper focussing on Koine, but this is a side phenomenon.&lt;/p&gt;&#xA;&lt;p&gt;(I know I might irritate people to scream with this, but you might think of Homeric as Robert of Gloucester, Classical as Chaucer, and Koine as Shakespeare.)&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2020-12-12T15:01:10.017" LastActivityDate="2020-12-12T15:01:10.017" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12600" PostTypeId="1" AcceptedAnswerId="12601" CreationDate="2020-12-10T20:15:57.603" Score="0" ViewCount="164" Body="&lt;p&gt;Why theology play a great role  in  becoming  great mathematicians?&lt;/p&gt;&#xA;&lt;p&gt;I have some confusion about theology with mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I read the biography of Euler and Bernhard Riemann.&lt;/p&gt;&#xA;&lt;p&gt;Both were studying theology in order to become a pastor, and after that they become great mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;Even if you see the  other great mathematicians like Blaise pascal, Newton etc they all studied theology.&lt;/p&gt;&#xA;" OwnerUserId="10835" LastEditorUserId="217" LastEditDate="2020-12-11T09:14:34.380" LastActivityDate="2021-12-21T03:39:52.367" Title="Why theology plays a great role in becoming great mathematicians?" Tags="|biographical-details|mathematicians|" AnswerCount="1" CommentCount="4" ClosedDate="2020-12-11T11:54:29.650" ContentLicense="CC BY-SA 4.0" />
  <row Id="12601" PostTypeId="2" ParentId="12600" CreationDate="2020-12-11T03:19:09.127" Score="3" Body="&lt;p&gt;An historian could tell the tale more fully but back before Galileo and Newton there were no sciences to speak of. There was medicine of Galen from even further back. The practice of using Greek logic to solve mysteries of nature was not yet popular, at least in Europe.&lt;/p&gt;&#xA;&lt;p&gt;The major form of scholarship there was of the Bible and histories. Anyone whose abilities were more intellectual than physical would tend to move in that direction. This would invite them to become theologians of some repute, or related professions, before their other interests in matters as novel as science started to be of some value.  The Church kept intellectuals and intellectualism alive for centuries, even up into the 19th century.&lt;/p&gt;&#xA;&lt;p&gt;In earlier days European mathematics had crept along with much the same surroundings as Cardano could tell you.&lt;/p&gt;&#xA;&lt;p&gt;I look forward to informative if scathing corrections.&lt;/p&gt;&#xA;" OwnerUserId="7448" LastEditorUserId="7448" LastEditDate="2021-12-21T03:39:52.367" LastActivityDate="2021-12-21T03:39:52.367" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12602" PostTypeId="2" ParentId="3569" CreationDate="2020-12-11T07:28:02.773" Score="1" Body="&lt;p&gt;Okay, this also came up in an interesting blog, Galileo unbound, by a physics prof at&#xA;Purdue.  &lt;a href=&quot;https://galileo-unbound.blog&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://galileo-unbound.blog&lt;/a&gt;  In particular, his bit about Grassmann. &lt;a href=&quot;https://galileo-unbound.blog/2019/12/02/hermann-grassmanns-nimble-wedge-product/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://galileo-unbound.blog/2019/12/02/hermann-grassmanns-nimble-wedge-product/&lt;/a&gt;  Thx, too, for your pointer to Ehresmann.  So I looked at the Sem. Math. Julia 1936-1937 at numdam.org.  According to the obituary for Cartan published in the Bull AMS, written by Chern and Chevalley, it was Anre Weil who got that semester's seminar devoted to Cartan's work, which was still under-appreciated, and Weil kicked it off.  &lt;a href=&quot;http://www.numdam.org/article/SMJ_1936-1937__4__A1_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.numdam.org/article/SMJ_1936-1937__4__A1_0.pdf&lt;/a&gt; Weil mostly uses juxtaposition for the exterior product, but remarks that he will use ^ when advisable to avoid confusion.  Then come lectures by Henri Cartan, the son, who uses wedge systematically.  Then comes Ehresmann, ditto.&#xA;So we see it there, and I do not see it earlier.  But I noticed something interesting.  Boulignand, an associate of Cartan's, wrote a book meant as a prerequisite for general relativity.  I can't access the book, but numdam.org has a correction &lt;a href=&quot;http://www.numdam.org/article/NAM_1924_5_3__245_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.numdam.org/article/NAM_1924_5_3__245_1.pdf&lt;/a&gt; he published.  In that note, he uses ^, but only in a context where it can simply mean the vector cross product in three dimensions. And one sees from Cajori that both [vw] and v^w were common notations for the vector cross product.  And, of course, Cartan's wedge product is, for three dimensions, the vector cross product when it gets applied to one-forms.  So this seems to me to be the evolution of the exterior product from the older cross product, and why it was sometimes juxtaposition, sometimes (in Cartan) [&lt;span class=&quot;math-container&quot;&gt;$\omega \eta$&lt;/span&gt;], and with his son and others in the next generation, ^.&lt;/p&gt;&#xA;" OwnerUserId="13461" LastEditorUserId="13461" LastEditDate="2020-12-11T22:24:24.270" LastActivityDate="2020-12-11T22:24:24.270" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12603" PostTypeId="1" CreationDate="2020-12-11T15:48:14.263" Score="2" ViewCount="49" Body="&lt;p&gt;How close were the slits in the Young's double slit experiment in 1804 (the second one) and how were they realized?&lt;/p&gt;&#xA;&lt;p&gt;This is not a duplicate of &lt;a href=&quot;https://hsm.stackexchange.com/questions/2629/how-did-young-perform-his-double-slit-experiment&quot;&gt;this question&lt;/a&gt;: I am not asking about how the experiemnts were performed, I am asking the measure of the distance of the slits.&lt;/p&gt;&#xA;&lt;p&gt;The answer to the &lt;a href=&quot;https://hsm.stackexchange.com/questions/2629/how-did-young-perform-his-double-slit-experiment&quot;&gt;question linked&lt;/a&gt; doesn't provide the information I am asking.&lt;/p&gt;&#xA;" OwnerUserId="3983" LastEditorUserId="3983" LastEditDate="2020-12-18T15:47:59.390" LastActivityDate="2020-12-18T15:47:59.390" Title="How close were the slits in Young's experiment?" Tags="|physics|light|" AnswerCount="0" CommentCount="3" ClosedDate="2020-12-17T04:09:16.150" ContentLicense="CC BY-SA 4.0" />
  <row Id="12604" PostTypeId="1" CreationDate="2020-12-11T16:10:19.163" Score="1" ViewCount="152" Body="&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_gravitational_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; wiki article, the history of gravity starts with Aristotle. However, what were the ideas about Earth's gravity and motion of planets before Aristotle?&lt;/p&gt;&#xA;" OwnerUserId="12670" LastEditorUserId="4251" LastEditDate="2021-09-14T17:36:33.607" LastActivityDate="2021-09-14T17:36:33.607" Title="Gravity before Aristotle" Tags="|physics|classical-mechanics|gravity|antiquity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12605" PostTypeId="1" AcceptedAnswerId="12608" CreationDate="2020-12-11T20:46:50.883" Score="1" ViewCount="274" Body="&lt;p&gt;The website &lt;a href=&quot;http://ucum.org/ucum.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ucum.org/ucum.html&lt;/a&gt; tells us whether every unit is metric or not. Metric units can be multiplied by a power of 10 and can be combined with a prefix. One ångström is defined as 0.1 nanometers or &lt;span class=&quot;math-container&quot;&gt;$10^{-10}$&lt;/span&gt; meters. It is the unit closest to the Bohr radius and the diameter of a hydrogen atom. But the website above says that the ångström is not a metric unit. What are the factors affecting whether units should be classified as a metric or not? And why is the ångström not classified as a metric unit?&lt;/p&gt;&#xA;" OwnerUserId="13344" LastEditorUserId="13344" LastEditDate="2022-09-29T21:08:29.820" LastActivityDate="2022-09-29T21:08:29.820" Title="The unit Ångström" Tags="|physics|units|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12606" PostTypeId="2" ParentId="12604" CreationDate="2020-12-12T04:59:15.880" Score="-4" Body="&lt;p&gt;The earliest is simply observational. We walk on the surface of the earrh and things fall down. This is so obvious a thing that it passed without comment. We know this is the case because when stories talk about flying things like say pegasus or the like, it was seen as fabulous. That is against nature, against the norm.&lt;/p&gt;&#xA;&lt;p&gt;I'd say Aristotle is unlikely to be the originator of the idea of gravity as you mention. This is a common mistake. He was writing about was known of physics then, in the same way that Feynman did with his lectures. But no-one makes the mistake that Feynman discovered everything he wrote about.&lt;/p&gt;&#xA;&lt;p&gt;The motion of the sun is referred to in an ancient egyptian myth about how the sun god Ra travels in a golden boat above the sky from east to west, then disembarks onto a night boat on which he travels through the underworld. This goes back at least 2500 BCE. We can see thos as progress as he us also said to have bern born every day in the east and died every noght in the west. That is the sun, far from being created every day in the east and destroyed at the end of every day in the west, travels around the world.&lt;/p&gt;&#xA;&lt;p&gt;Much later, people realised that this meant the earth must hang upon nothing, so to speak, and given that both the sun and the moon are round, it couldn't have been much of a leap to think that so must the earth. This was realised by the time of early Greek natural philosophy in Miletus, on the coast of what is now modern day Turkey.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-12-12T04:59:15.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12608" PostTypeId="2" ParentId="12605" CreationDate="2020-12-13T01:45:04.977" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;The UCUM website tells us whether every unit is metric or not&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No. That website specifies which units are metric &lt;em&gt;according to UCUM&lt;/em&gt;. In other contexts the distinction metric-or-not is less clear for some of the units. I suppose many physicsts would consider the Angstrom to be a metric unit, although non-SI.&lt;/p&gt;&#xA;&lt;p&gt;The Unified Code for Units of Measure (UCUM) is a system of codes for unambiguously representing measurement units. Its primary purpose is machine-to-machine communication. Some of the UCUM base units are different from the SI base units, such as gram instead of kilogram.&lt;/p&gt;&#xA;&lt;p&gt;Each unit represented in UCUM is identified as either &amp;quot;metric&amp;quot; or &amp;quot;non-metric&amp;quot;. Metric units can accept metric prefixes (as in milligram), and non-metric units can not.&lt;/p&gt;&#xA;&lt;p&gt;In UCUM, nanometer and picometer are metric, whereas Angstrom is not (it is not a base unit with a prefix). So milliAngstrom is not allowed in UCUM.&lt;/p&gt;&#xA;" OwnerUserId="2667" LastActivityDate="2020-12-13T01:45:04.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12609" PostTypeId="2" ParentId="12598" CreationDate="2020-12-13T07:04:26.077" Score="0" Body="&lt;p&gt;The ancient greek language was spoken in many different dialects. Given that philosophy emerged in dialogue, that is spoken speech; after all, that is how Plato presented it as (there's only one dialogue where a document is presented and this is in &lt;em&gt;Parmenides&lt;/em&gt; where he and Zeno meet Socrates), we must assume philosophy was spoken in many different dialects.&lt;/p&gt;&#xA;&lt;p&gt;However, the earliest Greek natural philosophy arose on Asia Minor, that is in Miletus. Here, they spoke Aeolic. In fact, it's considered as a subdialect of Aeolic. Note, that they would not have used the term 'Greek'.&lt;/p&gt;&#xA;&lt;p&gt;The case of Parmenides is interesting as he wrote his major work as a poem in traditional greek epic, that is, derived from Homer &amp;amp; Hesiod and this is &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Homeric_Greek&quot; rel=&quot;nofollow noreferrer&quot;&gt;written&lt;/a&gt; and this was written in a combination of Ionic &amp;amp; Aeolic, and whilst it's literary form was influenced by Attic,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Homeric Greek is like Ionic Greek and quite unlike Classical Attic Greek in shifting the long a to [e=eta]. So for example attic troia became Ionic/Aeolic troie.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-12-13T08:05:44.150" LastActivityDate="2020-12-13T08:05:44.150" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12610" PostTypeId="1" AcceptedAnswerId="12614" CreationDate="2020-12-13T11:20:28.213" Score="8" ViewCount="256" Body="&lt;p&gt;I was watching a TV series (&lt;em&gt;Godless&lt;/em&gt;) which is set in the Wild West in 1884. At some point they sing &lt;a href=&quot;https://www.youtube.com/watch?v=x3RBDGRqUFg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Don't forget the girls of LaBelle&lt;/em&gt;&lt;/a&gt;, which includes the following verses:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Creole girl is a one-eyed queen&lt;/p&gt;&#xA;&lt;p&gt;With a kiss as addictive as nicotine&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It shocked me to see a depiction of a popular song in 1884 mention the addictive properties of nicotine. I know tobacco was known to be addictive from at least the 16th century, and that nicotine was first isolated from the tobacco plant in 1828. But that doesn't necessarily mean that it was known that it was nicotine in particular which was responsible for the addictive properties.&lt;/p&gt;&#xA;&lt;p&gt;Is that verse a historical blunder or were the addictive properties of nicotine known by 1885?&lt;/p&gt;&#xA;" OwnerUserId="13468" LastEditorUserId="13468" LastEditDate="2020-12-14T10:52:20.187" LastActivityDate="2020-12-14T13:15:25.957" Title="When was nicotine identified as an addictive substance?" Tags="|chemistry|psychology|neuroscience|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12611" PostTypeId="1" AcceptedAnswerId="12613" CreationDate="2020-12-13T13:34:35.837" Score="2" ViewCount="162" Body="&lt;p&gt;The case of these triangles is special and close to the Pythagorean theorem, namely &lt;span class=&quot;math-container&quot;&gt;$c^2=a^2+b^2\pm ab$&lt;/span&gt;. This is a particular case of trigonometric relations and the law of cosines, and remains quite easy without it, using similar triangles and a few tricks.&lt;/p&gt;&#xA;&lt;p&gt;So my question is: is it known when this result was studied/written for the first time and was it long before the study of trigonometry? I think it would be strange if it has not been found before understanding the general case. This question was first asked on &lt;a href=&quot;https://math.stackexchange.com/questions/3946854/triangles-with-angle-of-60-degrees-and-120-degrees&quot;&gt;Math Stack Exchange&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13469" LastEditorUserId="55" LastEditDate="2020-12-13T20:23:19.133" LastActivityDate="2020-12-13T21:01:55.360" Title="When was the special relation between sides in a triangle with an angle of 60 or 120 degrees written for the first time?" Tags="|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12613" PostTypeId="2" ParentId="12611" CreationDate="2020-12-13T21:01:55.360" Score="4" Body="&lt;p&gt;Geometric analogs of the law of cosines are the propositions &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookII/propII12.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;II.12&lt;/a&gt; and &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookII/propII13.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;II.13&lt;/a&gt; of Euclid's &lt;em&gt;Elements&lt;/em&gt; (c. 300 BC) for obtuse and acute angles, respectively. Here is Euclid's formulation for the acute case:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In acute-angled triangles the square on the side opposite the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides about the acute angle, namely that on which the perpendicular falls, and the straight line cut off within by the perpendicular towards the acute angle.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Geometric constructions needed to establish numerical values for the cosines of &lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$120^\circ$&lt;/span&gt; and some other angles, e.g. &lt;span class=&quot;math-container&quot;&gt;$36^\circ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$72^\circ$&lt;/span&gt;, also occur in the &lt;em&gt;Elements&lt;/em&gt;, when Euclid is inscribing regular triangles, pentagons and hexagons into a circle. Ptolemy in Almagest (c. 150 AD) uses such constructions and additional geometric theorems to derive an entire table of chord values for such angles and their subdivisions (chord trigonometry is equivalent to the modern one). See &lt;a href=&quot;https://www.google.com/books/edition/Episodes_From_the_Early_History_of_Astro/kZnxBwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aaboe, Episodes From the Early History of Astronomy&lt;/a&gt; for a modern exposition of his methods.&lt;/p&gt;&#xA;&lt;p&gt;The modern formulation of the law of cosines is usually attributed to Al-Kashi (c. 1400 AD), see &lt;a href=&quot;https://mir.kashanu.ac.ir/article_88765.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Overview of Mathematical Contributions of Ghiyath al-Din Jamshid Al-Kashi by Azarian&lt;/a&gt;, he also refined Ptolemy's tables and expressed them in terms of sines and cosines. In Europe the law was popularized by Viete. None of these authors singled out &lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$120^\circ$&lt;/span&gt; triangles for special treatment.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-12-13T21:01:55.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12614" PostTypeId="2" ParentId="12610" CreationDate="2020-12-14T01:56:55.223" Score="12" Body="&lt;p&gt;It is a blunder, but there is a saving grace to it, even two. &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicotiana&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Nicotiana&lt;/em&gt;&lt;/a&gt; is the Latin name of tobacco, and it was in use as early as 16th century, and not just by botanists. Moreover, for the first 50 years tobacco was seen as a medicinal herb (after its primary use by American Indians) rather than a recreational drug. &lt;a href=&quot;https://pubs.acs.org/doi/10.1021/ed018p303&quot; rel=&quot;noreferrer&quot;&gt;Fletcher in The History of Nicotine&lt;/a&gt; quotes one John Frampton the merchant, who wrote in 1577:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Nicotiane [the plant], although it bee not long since it hath beene knowen in Fraunce, not-withstanding it deserveth Palme and Price emong all other Medicinable herbes, it deserveth to standc in the first rancke, by reason of his singular vertues, and as it were almost to be had in admiration, as hereafter you shall understande.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As mentioned in the OP, the addictiveness (in a loose sense of the word) of tobacco was realized soon after it started to be used recreationally. Francis Bacon wrote in &lt;a href=&quot;https://archive.org/details/hin-wel-all-00001727-001&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Historia vitae et mortis&lt;/em&gt; (1623)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In our time the use of tobacco is growing greatly and conquers men with a certain secret pleasure, so that those who have once become accustomed thereto can later hardly be restrained therefrom.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So if we charitably interpret the reference to &amp;quot;nicotine&amp;quot; as a reference to the herb Godless would be fine. It is unlikely that they had &lt;em&gt;that&lt;/em&gt; in mind though.&lt;/p&gt;&#xA;&lt;p&gt;But there is more room for charity. After a near miss with discovering nicotine the alkaloid by Vauquelin in 1809 (he attributed its alcalinity to ammonia contamination) in 1828 the University of Heidelberg offered a prize for the best paper on the active principle of tobacco and its physiological action. The prize was won by a paper written by two young students, Posselt and Reimann (&lt;em&gt;Chemische Untersuchungen des Tabaks und Darstellung des eigenhumlichen wirksamen Principes dieser Pflanze&lt;/em&gt;. &lt;em&gt;Geigers Magazin der Pharmazie&lt;/em&gt; 1828;24: 138-61). The paper was met with skepticism that a pure alkaloid could be liquid, and the result was only accepted in 1837, when Henry and Boutron-Charlard confirmed it. Since tobacco was known to be addictive and the nicotine was its &amp;quot;active principle&amp;quot;, applying &amp;quot;addictive&amp;quot; to nicotine directly was not out of bounds by 1885. Although the adjective more commonly used with &amp;quot;nicotine&amp;quot; in the 19th century was &amp;quot;poisonous&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, addiction and addictiveness, of nicotine in particular, in the modern technical sense (going beyond mere bad habit) is appropriately recent. Here is from &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3193858/#S3title&quot; rel=&quot;noreferrer&quot;&gt;Historical and Current Perspective on Tobacco use and Nicotine Addiction by Dani and Balfour&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Scientific progress indicating that tobacco smoke is addictive post-dates the first evidence in the early 1950s that smoking was associated with a substantially increased incidence of lung cancer. Initially, tobacco smoking was seen as a habit, but by 1971 researchers were beginning to recognize that many smokers were addicted to nicotine present in tobacco smoke. Much of the work focused on the evidence that chronic or repeated exposure to nicotine results in a withdrawal effect if the drug is precipitously withheld. By the 1980s, the validity of the nicotine dependence hypothesis was becoming more widely accepted, and the concept of using nicotine replacement therapy to aid successful cessation attempts had moved to the fore. This period also saw the development of tests that could measure the degree of tobacco dependence.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-12-14T04:18:42.997" LastActivityDate="2020-12-14T04:18:42.997" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12615" PostTypeId="2" ParentId="12610" CreationDate="2020-12-14T05:19:20.517" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.discovermagazine.com/planet-earth/native-americans-smoked-tobacco-long-before-europeans-arrived&quot; rel=&quot;nofollow noreferrer&quot;&gt;discover&lt;/a&gt; magazine:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Tobacco kills hundreds of thousands of people every year but the addictive substance has a complex history. For many Native Americans tobacco has been ritually for hundreds of years. Now researchers have shown that they were smoking tobacco a thousand years before European fur-traders arrived ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For many native people, tobacco use was historically associated with sacred ceremonies and only certain tribal members smoked a limited amount of the plant. Now 34% of Idaho American Indians smoke tobacco according to a 2015 survey by the Idaho Department of Work &amp;amp; Health.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is confirmed in a 2018 &lt;a href=&quot;https://www.pnas.org/content/115/46/11742&quot; rel=&quot;nofollow noreferrer&quot;&gt;report&lt;/a&gt;  in the &lt;em&gt;Proceedings of the National Academy of Sciences of the United States&lt;/em&gt; by Tushingham et al in 2018, where they report&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dried trade tobacco, much more potent than any native smoke variety, was conveyed in sizable, easy to carry bundles, known as 'twists' or 'carrots'. Its use spread like wildfire ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This shows that an important factor was the commodification of tobacco by the migration of Europeans, a highly commercialised and mercantile civilisation into the Americas. This is one reason that one Native American site,  &lt;a href=&quot;https://keepitsacred.itcmi.org/tobacco-and-tradition/traditional-tobacco-use/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keeping it Sacred&lt;/a&gt; states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In many teachings, the smoke from burned tobacco has the purpose of carrying thoughts to the Spirit World or the Creator and when used appropriately traditional tobacco is not associated with addiction or with adverse health impacts&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A useful comparison is to see how alcohol, a traditional drink within European communities, circumscribed by social practises and used sacramentally within Roman Catholic communities has impacted Native American communities. According to the &lt;a href=&quot;https://americanaddictioncenters.org/alcoholism-treatment/native-americans&quot; rel=&quot;nofollow noreferrer&quot;&gt;American Addiction Centre&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Native Americans experience [alcohol] abuse and addiction at much higher rates than other ethnic communities ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's likely that Native American communities did not have the social &amp;amp; cultural knowledge in how to handle alcohol: It was not part of their ceremonies nor in how they relaxed.&lt;/p&gt;&#xA;&lt;p&gt;All this shows that an important factor that has been neglected in the study of nicotine addiction is its social, cultural &amp;amp; religious context and the commodification of tobacco to a potent drug - think of how drug cartels now with cannabis have bred skunk, a much stronger version of traditional cannabis.&lt;/p&gt;&#xA;&lt;p&gt;The actual isolation of nicotine as a chemical was in 1828 by Posselt &amp;amp; Reimann who believed it to be a poison. There are trace quantities (millionths of a percent) also in potatoes, eggplants and tomatoes.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-12-14T13:15:25.957" LastActivityDate="2020-12-14T13:15:25.957" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12617" PostTypeId="1" CreationDate="2020-12-14T15:12:56.163" Score="1" ViewCount="235" Body="&lt;p&gt;What is the description of the gravitational interaction in the philosophy of Plato? Are there some drastic differences from Aristotle's physics?&lt;/p&gt;&#xA;" OwnerUserId="12670" LastEditorUserId="10106" LastEditDate="2020-12-14T18:03:49.500" LastActivityDate="2020-12-14T18:14:56.213" Title="Gravity of Plato" Tags="|physics|ancient-greece|gravity|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12618" PostTypeId="2" ParentId="12617" CreationDate="2020-12-14T18:14:56.213" Score="3" Body="&lt;p&gt;I don't think he discussed anything like gravity. This does not mean he lacked interest in physics. For example, his elemental atoms were platonic solids whose sides were broken down into triangles. So in a basic way, he was saying the world was geometric. And this a long time before Minkowski.&lt;/p&gt;&#xA;&lt;p&gt;The key difference between him and Aristotle is that Plato is an atomist whereas Aristotle is a thinker on the continuum. According to Rene Thom, the topologist, Aristotle was more or less the only thinker on the continuum for two millenia.&lt;/p&gt;&#xA;&lt;p&gt;Whereas a key similarity is that both were Pythagoreans. For Plato,  mathematics was the first step on a dialectic where one progressively understands the nature of the One or the monad in Pythagorean language and which derives from the word &lt;em&gt;monas&lt;/em&gt; (alone). Aristotle, does something similar via his notion of cause tracing the All back to a unitary first cause and mover but which is unmoved itself.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-12-14T18:14:56.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12619" PostTypeId="1" CreationDate="2020-12-14T21:02:16.457" Score="0" ViewCount="68" Body="&lt;p&gt;Any Wightman-based approach to Axiomatical Quantum Field Theory states that quantum fields are (operator-valued) distributions. Is there a first rigorous proof of this fact which became trivial as early as 1970? Maybe the name of the person is K. Friedrich or A. Wightman, but I want the exact reference for the proof.&lt;/p&gt;&#xA;" OwnerUserId="7069" LastActivityDate="2020-12-14T21:02:16.457" Title="Who was the first one to rigorously show that quantum fields are operator-valued distributions?" Tags="|mathematics|reference-request|quantum-field-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12621" PostTypeId="2" ParentId="12549" CreationDate="2020-12-15T07:50:36.807" Score="7" Body="&lt;p&gt;Donald Knuth's The Art of Computer Programming (TAOCP) Vol. 3 &amp;quot;Sorting and Searching&amp;quot; gives a detailed account on the history of ideas, including the sorting by merging, in Chapter 5.5. Quoting from the 2nd Edition, pp. 383-385:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A search for the origin of today's sorting techniques takes us back to the nineteenth century, when the first machines for sorting were invented.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Towards the end of the century, the young &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Herman_Hollerith&quot; rel=&quot;noreferrer&quot;&gt;Hermann Hollerith&lt;/a&gt; invented several generations of card sorting machines, for helping to accelerate the census of all U.S. American citizens.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hollerith's card sorting machine [i.e., the later one patented in 1901 and 1904] is, of course, the basis for radix sorting methods now used in digital computers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So radix sort can be seen as a predecessor of von Neumann's algorithm. Observe that radix sort is faster than Mergesort if the numbers to be sorted are guaranteed to be within a fixed range.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The idea of merging goes back to another card-walloping machine, the collator, which was a much later invention (1938). With its two feeding stations, it could merge two sorted decks of cards into one, in only one pass; the technique for doing this was clearly explained in the first IBM collator manual (April 1939). [See James W. Bryce, U.S. Patent 2189024 (1940).]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The chapter is definitely worth a read; Knuth lays out some more of the early developments, and for example also explains von Neumann's motivation for designing the merge sort algorithm. Von Neumann definitely did not make his algorithm out of thin air.&lt;/p&gt;&#xA;&lt;p&gt;Hat Tip: The article by Knuth that kimchi lover mentioned in their comment is probably also worth reeading. In fact that comment made me look into TAOCP.&lt;/p&gt;&#xA;" OwnerUserId="9523" LastEditorUserId="9523" LastEditDate="2020-12-20T09:30:07.187" LastActivityDate="2020-12-20T09:30:07.187" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12622" PostTypeId="1" CreationDate="2020-12-15T09:07:02.113" Score="0" ViewCount="38" Body="&lt;p&gt;In my study of Peirce I came across a number statements that he extended De Morgan's relation algebra.&#xA;What exactly are his contributions to relation algebra?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2020-12-15T09:07:02.113" Title="What were C.S Peirce's contributions to theory of relations?" Tags="|biographical-details|philosophy-of-science|mathematical-logic|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12623" PostTypeId="1" AcceptedAnswerId="12628" CreationDate="2020-12-15T09:19:15.030" Score="5" ViewCount="596" Body="&lt;p&gt;Antonius van den Broek was the first person to suggest that the atomic number (which at that time only referred to an element's position in the periodic table and had no physical interpretation) of an element was exactly equal to the charge of its nucleus, then later Henry Moseley found evidence that the atomic number is a more fundamental quantity (Moseley's Law), thus making Van Den Broek's statement more likely to be true.&lt;/p&gt;&#xA;&lt;p&gt;My question is how did Broek even come up with his hypothesis? He published his hypothesis in 1911. The idea of 'individual' protons of unit charge (hydrogen nuclei) making up the nucleus was discovered in 1919. in 1911, the model of the atom was essentially 'positive dense nucleus in center surrounded by electrons.'&lt;/p&gt;&#xA;&lt;p&gt;According to rutherford's model (1911), most of the mass is concentrated in the center and is positively charged, so I think it makes sense for scientists at that time to believe that the charge of a nucleus increases with atomic weight, as heavier atoms would likely have heavier nuclei and therefore likely to have more charge. But how did Van Den Broek think that the value of the charge (in electron units) was EXACTLY equal to its atomic number (or roughly half its atomic weight, why not 1/4th, or twice, etc)&lt;/p&gt;&#xA;&lt;p&gt;Were there experiments (in 1911 or prior) that showed that the charge of a nucleus is roughly half its atomic weight?&lt;/p&gt;&#xA;" OwnerUserId="13473" LastEditorUserId="55" LastEditDate="2020-12-16T08:00:31.300" LastActivityDate="2020-12-16T08:00:31.300" Title="How did Antonius Van den Broek come up with his hypothesis that the atomic number matches the charge of the nucleus?" Tags="|atomic-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12626" PostTypeId="1" AcceptedAnswerId="12627" CreationDate="2020-12-15T20:04:43.230" Score="8" ViewCount="2093" Body="&lt;p&gt;It seems that all stars as bright or brighter than Polaris have names dating to ancient times, including many stars &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_brightest_stars&quot; rel=&quot;nofollow noreferrer&quot;&gt;dimmer than Polaris&lt;/a&gt;. So what was Polaris called, in any language, before it became primarily known for its proximity to the celestial pole?&lt;/p&gt;&#xA;" OwnerUserId="261" LastEditorUserId="261" LastEditDate="2021-12-18T11:55:39.137" LastActivityDate="2022-10-27T14:28:19.947" Title="Name of Polaris before it became a pole star?" Tags="|astronomy|naming-conventions|" AnswerCount="4" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12627" PostTypeId="2" ParentId="12626" CreationDate="2020-12-16T07:01:41.680" Score="4" Body="&lt;p&gt;Polaris is a very well known star and named in many different languages. For example, in Inuit astronomy it was known as &lt;em&gt;Niquirtsuituq&lt;/em&gt; and is depicted in the flag and coat of arms of the Canadian Inuit territory of Nunavut as well on the flag of Alaska. In fact, the Inuits called the three stars, Polaris, Kochab &amp;amp; Pherkad, &lt;em&gt;Nuutuitut&lt;/em&gt;, the never-moving.&lt;/p&gt;&#xA;&lt;p&gt;It is part of the constellation Ursa Minor. Since constellations are fixed despite precession it would have been first known as belonging to that constellation. In fact, at first the entire constellation was used for navigation.&lt;/p&gt;&#xA;&lt;p&gt;Polaris is actually a triple star system with it's most prominant star being a yellow gas giant. Planets have been detected around four of the stars in the constellation and the constellation also includes an isolated neutron star, Calvera, as well as the hottest white dwarf star ever found with a surface temperature of 200,000 K.&lt;/p&gt;&#xA;&lt;p&gt;In Europe, it was named Polaris in the Renaissance from the Latin &lt;em&gt;Stella Polaris&lt;/em&gt; or pole star when it approached tge celestial pole to within a few degrees. In Antiquity it was not close enough to the celestial pole to be used for navigaton and the whole constellation was used. In Old English it was called &lt;em&gt;scip steorra&lt;/em&gt;, or ship star.&lt;/p&gt;&#xA;&lt;p&gt;In the Babylonian star catalogues Ursa Minor was known as the Wagon of Heaven, &lt;em&gt;Mul.mar.gid.da.an.na&lt;/em&gt; associated with the goddess &lt;em&gt;Damkina&lt;/em&gt;. It's named in the &lt;em&gt;Mul.apin&lt;/em&gt; catalogue whose earliest recorded copy is dated to 686 BCE but thought to have been first compiled around 1000 BCE.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Ursa_Minor&quot; rel=&quot;nofollow noreferrer&quot;&gt;According&lt;/a&gt; to Diogenes Laertius and citing Callimachus, Thales of Miletus 'measured the stars of the wagon by which the Phoenicians sail'. Diogenes identified this with the constellation Ursa Minor. The Greeks also reported that the Phoenicians used this star for navigation at sea and for this reason called it also &lt;em&gt;Phoineke&lt;/em&gt;. The tradition of naming the constellation as a bear appears to be genuinely Greek although Homer speaks of only one bear.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The original bear is thus Ursa Major, and Ursa Minor was admitted as a second or Phoenician Bear (Ursa Phoenica) only later, and according to Strabo, this was due to a suggestion by Thales that it be used as a navigation aid by the Greeks who had been using Ursa Major in this way. Given the above, this is unlikely to have been an independent discovery by Thales, despite how Strabo reported it, but he was simply reporting on how the Phoenicians were already using this constellation in navigation.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2020-12-16T08:08:36.917" LastActivityDate="2020-12-16T08:08:36.917" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12628" PostTypeId="2" ParentId="12623" CreationDate="2020-12-16T07:59:14.663" Score="4" Body="&lt;p&gt;Van den Broek's train of thought is described in detail by &lt;a href=&quot;https://link.springer.com/article/10.1007/s11224-017-0948-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scerri in The Gulf between chemistry and philosophy of chemistry&lt;/a&gt;. His starting points in the 1907 article were Rutherford's remarks on the nature of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; particles creatively combined with the old &lt;a href=&quot;https://en.wikipedia.org/wiki/Prout%27s_hypothesis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prout's hypothesis&lt;/a&gt; of 1815 that all atoms are made from &amp;quot;protyles&amp;quot; (from Greek &lt;em&gt;proto&lt;/em&gt;+&lt;em&gt;hyle&lt;/em&gt;, prime matter), i.e. copies of hydrogen atom.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; came from the experiments of Rutherford and Barkla, who suggested the approximate proportional relationship between atomic charges and weights with this coefficient. But van den Broek made a further connection with the ordering of elements in the periodic table that his more eminent sources missed. Although the idea is already present in a 1911 paper, it is entangled there with his &amp;quot;cubic periodic table&amp;quot; and a modification of Prout's hypothesis, both of which he soon abandoned. The clearest formulation appears in a 1913 paper cited by Bohr:&amp;quot;&lt;em&gt;The serial number of every element in the sequence ordered by increasing atomic weight equals half the atomic weight and therefore the intra-atomic charge&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, after examining experimental data of Geiger and Marsden published the same year, van den Broek noticed that the ratio of &lt;span class=&quot;math-container&quot;&gt;$α$&lt;/span&gt;-particle scattering per atom to the square of the charge showed much better constancy than its ratio to the square of atomic weight. So he took the final step and dropped the shaky middle term of atomic weight:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If now in these values the number M of the place each element occupies in Mendeléeff’s series is taken instead of A, the atomic weight, we get a real constant (18.7 +/− 0.3); the hypothesis proposed holds good for Mendeléeff’s series, but the nuclear charge is not equal to half the atomic weight&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here are some relevant excerpts from Scerri's account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;One of Rutherford’s three suggestions was that an α-particle might be half of a helium atom with a charge of twice the hydrogen atom. This idea appealed to van den Broek who coined the term alphon to describe such a particle. He then proposed that such an alphon particle might better serve the role that Prout had intended for the hydrogen atom in his famous hypothesis of 1815. According to van den Broek, the atoms of the elements should therefore consist of a series corresponding to the even whole numbers from 2 up to 240 such that there should be a total of 120 elements, each one made up of a whole number of alphons each with a weight of 2 units.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Van den Broek’s reason for this proposal rested on experiments by Rutherford and Charles Barkla who had independently concluded that the charge of any atom is approximately half of its atomic weight. Van den Broek’s article of 1907 does not yet show any direct signs of the concept of atomic number, unless one divides each of the atomic weights in van den Broek’s table by two to obtain a sequence of values from one to 120.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1911... van den Broek published a remarkably short and suggestive statement in Nature magazine... Stated otherwise, van den Broek is suggesting that since the charge on an atom is half of its atomic weight, and since the weights of successive elements differ by 2 units in a stepwise fashion, the charge on an atom defines its position in the periodic table. Neither Rutherford, Barkla, nor anybody else had considered the elements in the periodic table as a whole, and consequently, they had missed this key point, whereas Rutherford and Barkla recognized that &lt;span class=&quot;math-container&quot;&gt;$\textrm{charge}\approx A/2$&lt;/span&gt; van den Broek went further in seeing that, &lt;span class=&quot;math-container&quot;&gt;$\textrm{charge}\approx A/2=\textrm{atomic number}$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1913... the most significant development took place in another article that van den Broek placed in Nature magazine in which he abandoned the connection with atomic weight altogether. Van den Broek began with a set of experiments by Hans Geiger and Ernest Marsden, aimed at examining the ratio of scattering of α-particles per atom in several elements. According to Rutherford, this ratio needed to be constant, but this is not what Geiger and Marsden found when they divided the scattering by the atomic weight of each element. Nevertheless, Geiger and Marsden were not concerned about this discrepancy, believing that the error was small and could be ignored. Meanwhile, van den Broek got to work trying to make the ratio more constant. He achieved this by dividing the amount of scattering for each element by its charge rather than by its atomic weight.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2020-12-16T07:59:14.663" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12629" PostTypeId="5" CreationDate="2020-12-16T09:27:11.830" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2020-12-16T09:27:11.830" LastActivityDate="2020-12-16T09:27:11.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12630" PostTypeId="4" CreationDate="2020-12-16T09:27:11.830" Score="0" Body="For questions asking for and about translations of text associated with History of Science and Maths" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2020-12-17T11:12:28.170" LastActivityDate="2020-12-17T11:12:28.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12631" PostTypeId="2" ParentId="12626" CreationDate="2020-12-16T14:18:02.347" Score="7" Body="&lt;p&gt;Except for very few conspicuous stars (like Sirius), most stars had no individual names in the ancient Greece and Rome. In particular, in Ptolemy catalog, no star in Ursa Minor has a name.&#xA;Polaris is described in this catalog as &amp;quot;the star at the end of the tail&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Same applies to the earliest surviving Greek text on astronomy: the poem Phenomena of Aratus.&lt;/p&gt;&#xA;&lt;p&gt;It was in the medieval time when stars were given individual names. And &amp;quot;stella polaris&amp;quot; obtained this name as a conspicuous star close to the pole.&lt;/p&gt;&#xA;&lt;p&gt;Remark. Aratus gives an alternative name for the Little Bear: Κυνόσουραν,&#xA;and mentions that it is used for navigation since it is close to the pole, but does not mention any individual stars in it.&lt;/p&gt;&#xA;&lt;p&gt;Remark 2. Another reason why it had no name during classical antiquity is that it was not so close to the pole (10 degrees away), so there was no special reasons to give it a name.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-10-27T14:28:19.947" LastActivityDate="2022-10-27T14:28:19.947" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12632" PostTypeId="2" ParentId="12626" CreationDate="2020-12-16T20:01:41.530" Score="3" Body="&lt;p&gt;I have a copy of Richard Hinckley Allen's &lt;em&gt;Star Names: Their Lore and Meaning&lt;/em&gt;, 1899, 1963.&lt;/p&gt;&#xA;&lt;p&gt;The section on Polaris, Alpha Ursae Minoris, extends from page 453 to page 458.&lt;/p&gt;&#xA;&lt;p&gt;It says the early Greek name for Polaris was Phoenice.&lt;/p&gt;&#xA;&lt;p&gt;Pages 456 to 457 give its names in China (Pih Keih and others), in India (Grahadara or Dhruva), among the Arabs (Al Kiblah and Al Kaukab al Shamalyy), and the Turks (Yilduz).&lt;/p&gt;&#xA;&lt;p&gt;This book should definitely be consulted by anyone interested in the names of stars.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Stars of Jade: Astronomy and Star Lore of Very Ancient Imperial China&lt;/em&gt; Julius H.W. Staal, 1984, has a lot of information about Chinese constellations and star names.&lt;/p&gt;&#xA;&lt;p&gt;And I suspect that there are other sources for star names in many other cultures.&lt;/p&gt;&#xA;&lt;p&gt;According to Wikipedia, the Inuit call Polaris &lt;em&gt;Niqirtsuitug&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Polaris#Names%5B1%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Polaris#Names[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But I don't know when those various names were adopted.  As the north celestial pole got closer and closer to Polaris, it became more likely that a particular culture would give the star a name referring to its position on the celestial sphere.&lt;/p&gt;&#xA;&lt;p&gt;In some cases those &amp;quot;polar&amp;quot; names might have been the first names given to Polaris by a particular culture.  But in some cultures, it might have been the practice to name almost every star visible to the naked eye, presumably by their position with an asterism or constellation, and many cultures used asterisms and constellations different from modern ones.  Thus Polaris might have had names in some cultures which referred to it being, for example,  at the end of the handle or tail of an asterism or constellation, before that culture named it the pole star.&lt;/p&gt;&#xA;&lt;p&gt;Since Polaris is very close - about 0.75 of a degree - to the celestial north pole, it is only visible in the Northern hemisphere - and a tiny distance south of the equator at some times.&lt;/p&gt;&#xA;&lt;p&gt;So the native cultures of Australia, southern Africa, and most of South America would never have had any names for Polaris until people from the northern hemisphere arrived.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="7205" LastEditDate="2022-10-26T10:07:27.070" LastActivityDate="2022-10-26T10:07:27.070" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12633" PostTypeId="1" AcceptedAnswerId="12634" CreationDate="2020-12-17T05:37:00.390" Score="4" ViewCount="543" Body="&lt;p&gt;One of the algorithms for finding Eulerian paths and circuits in graphs that have them is due to Fleury.  Lucas mentioned this in his 1892 &lt;a href=&quot;https://archive.org/details/rcrationsmathma02delagoog/page/n148/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;recreational mathematics collection&lt;/a&gt;, referring to &amp;quot;M. Fleury, chef d'institution à Marseille.&amp;quot;  The citation for Fleury's 1883 article is below.  Looking elsewhere in the journal, I wonder if &amp;quot;M.&amp;quot; stands for Monsieur rather than an initial. Does anyone know more about Fleury?  In particular, what was his first name?&lt;/p&gt;&#xA;&lt;p&gt;M. Fleury, Deux problèmes de géométrie de situation, &lt;em&gt;Journal de Mathématiques Élémentaires&lt;/em&gt;, 257-261, 1883.&lt;/p&gt;&#xA;" OwnerUserId="3429" LastEditorUserId="4201" LastEditDate="2020-12-17T11:10:46.343" LastActivityDate="2020-12-17T18:06:12.677" Title="Who was Fleury? And what was his first name?" Tags="|biographical-details|mathematicians|euler|19th-century|graph-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12634" PostTypeId="2" ParentId="12633" CreationDate="2020-12-17T18:06:12.677" Score="8" Body="&lt;p&gt;A Google scan of Fleury's article &amp;quot;Deux problèmes de géométrie de situation&amp;quot; can be found &lt;a href=&quot;https://books.google.com/books?id=l-03AAAAMAAJ&amp;amp;pg=PA257#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, and identifies the author as &amp;quot;M. Fleury, chef d'institution&amp;quot;, where &amp;quot;M.&amp;quot; is the abbreviation for &amp;quot;Monsieur&amp;quot; in French literature of that time. Interestingly, per the introductory sentence, Fleury's article is inspired by the first volume of Édouard Lucas's &amp;quot;Récréations mathématiques&amp;quot;, published in Paris in 1882.&lt;/p&gt;&#xA;&lt;p&gt;Searching for &amp;quot;chef d'institution a Marseille&amp;quot; we find one Henri Fleury as author of the &lt;a href=&quot;https://www.google.com/books/edition/Nouvelle_th%C3%A9orie_des_fonctions_qui_se_p/BHlaAAAAcAAJ?gbpv=1&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt; &amp;quot;Nouvelle théorie des fonctions qui se présentent sous une forme dit indéterterminée&amp;quot;, Paris: Noblet et Baudry 1866. The author is described as &amp;quot;chef d'institution a Marseille, licencié ès sciences mathématiques&amp;quot;, that is, head of a school in Marseille with a degree in mathematics. It seems likely that Henri Fleury is the same person as the Monsieur Fleury whose article is referenced in the fourth volume of Lucas's book, but we cannot be certain.&lt;/p&gt;&#xA;&lt;p&gt;The BnF (Bibliothèque nationale de France) has a &lt;a href=&quot;https://data.bnf.fr/fr/12462978/pierre-henry_fleury/&quot; rel=&quot;noreferrer&quot;&gt;list of books&lt;/a&gt; published by Henri Fleury, which shows that he also published under the names Henry Fleury and Pierre-Henry Fleury, with the BnF treating the last of these as the principal name and the other ones as alternates. The author is described as &amp;quot;professeur de mathématiques&amp;quot;, that is, a (high-school) math teacher, with a birth year of 1833. Looking at available scans of some of Fleury's books, most appear to cover high school-level math.&lt;/p&gt;&#xA;&lt;p&gt;Fleury &lt;em&gt;may&lt;/em&gt; have been in correspondence with other mathematicians of his time, as Jeanne Peiffer, &amp;quot;Joseph Liouville (1809-1882): ses contributions à la théorie des fonctions d'une variable complexe&amp;quot;, &lt;em&gt;Revue d'histoire des sciences&lt;/em&gt;, vol. 36, no. 3-4, 1983, pp. 209-248, &lt;a href=&quot;https://www.persee.fr/doc/rhs_0151-4105_1983_num_36_3_1939&quot; rel=&quot;noreferrer&quot;&gt;mentions&lt;/a&gt; a &amp;quot;lettre du 16 janvier 1866 à M. Fleury de Marseille&amp;quot;. Again, we cannot be certain that this pertains to the same person.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2020-12-17T18:06:12.677" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12635" PostTypeId="2" ParentId="2279" CreationDate="2020-12-18T05:41:31.967" Score="10" Body="&lt;p&gt;I was one of those secretaries that knew how to use those beasts.  In fact, I used a manual typewriter (not a Selectric) to type these formulas for the 30 chemists I worked for at General Tire in the U.S.  Our typewriters were outfitted with a holder.  I had boxes of individual keys for each of those Greek letters, etc.  To create the formula, I picked out the key I needed, inserted it into the holder, struck the X key.  I typed reports of hundreds of these formulas, on a manual, in draft, and then in final on multilith paper to be run off on a machine.  They didn't pay me enough or appreciate the work.  It really was a piece of art when we finished.&lt;/p&gt;&#xA;" OwnerUserId="13488" LastActivityDate="2020-12-18T05:41:31.967" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12636" PostTypeId="1" AcceptedAnswerId="12640" CreationDate="2020-12-18T15:26:49.353" Score="-3" ViewCount="999" Body="&lt;p&gt;Albert Einstein was a genius, Kurt Gödel was a genius, John Von Neumann was a genius (I hope that people over here will share the meaning of &amp;quot;genius&amp;quot; which I understand, for me I always find Von Neumann being described as &amp;quot;genius&amp;quot; that is very highly intelligent and very much above  an average species' mind), Alan Turing was considered a genius, Wolfgang Pauli was considered a genius etc.&lt;/p&gt;&#xA;&lt;p&gt;We hardly find Rene Descartes, Galilee Galileo, even Isaac Newton and James Clerk Maxwell being described as geniuses, they are simply termed as good theorists. Philosophers are almost never described as geniuses, Bertrand Russell's (an example of 20th century) philosophies are not celebrated much (the answer that he was proven wrong by Gödel is almost same as saying Catholics were proven wrong by Protestants and hence doesn't tell us anything).&lt;/p&gt;&#xA;&lt;p&gt;There is a difference between the way Aristotle is revered and the way Einstein is revered, what has caused that have been making us to refer post 19th century scientists/mathematicians as geniuses? Why there number has increased almost like a &amp;quot;step-function&amp;quot; after 19th century?&lt;/p&gt;&#xA;&lt;p&gt;Pierre De Laplace, Karl Frederich Gauss, and few others of 18th-19th centuries were prodigies (similar to today's &amp;quot;genius&amp;quot;) but then also they weren't called as very much above the human species' mind, all they did was that they learned things quite early in their age.&lt;/p&gt;&#xA;" OwnerUserId="13489" LastActivityDate="2020-12-18T20:07:52.237" Title="Why we got so many geniuses after 19th century?" Tags="|mathematicians|physicists|scientists|" AnswerCount="2" CommentCount="5" ClosedDate="2020-12-19T16:24:40.490" ContentLicense="CC BY-SA 4.0" />
  <row Id="12637" PostTypeId="2" ParentId="12636" CreationDate="2020-12-18T17:26:07.320" Score="4" Body="&lt;p&gt;The obvious answer is that the tiny percentage of persons who are geniuses didn't change.&lt;/p&gt;&#xA;&lt;p&gt;But the number of persons did change, as world propulation multiplied several times between 1800 and 2020, thus multiplying the population of geniuses.  And also the percentage of geniuses who didn't have to work hard at manual labor to earn a living but could have the leisure to think about stuff increased.  And also with the spread of universal enducation, the percentage of geniuses who got a good education and knew enough facts to build on multiplied several times.&lt;/p&gt;&#xA;&lt;p&gt;Since the question is about scientific geniuses, it should be noted that probably more professional scientists have worked as scientists within the last 50 years or so than all the scientists or philiosophers in all of previous history.&lt;/p&gt;&#xA;&lt;p&gt;Remember that for hundreds of thousands of years all humans lived as hunter gatherers, and anyone who was born then with the personality to beome a scientific genius never got a chance to use their scientific potential but had to spend their time chasing animals, picking plants, chipping stones, and weaving baskets, etc., even if they lived long enough to be an adult, or even a teenager.  The majority of persons born before the last century or two died as young children, so perhaps we should change our mental picture of a typical human to be a child instead of an adult.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2020-12-18T17:26:07.320" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12640" PostTypeId="2" ParentId="12636" CreationDate="2020-12-18T20:07:52.237" Score="1" Body="&lt;p&gt;Genius is a loaded word. Richard Feynman was keen to point out that his IQ was no more than a few points above average. Plus a lot of the writing on Einstein, Newton, Maxwell and so on verge on the hagiographical. In fact, I'd say a lot of writing on 'genius' in physics goes quite a bit beyond that. As you've pointed the word is rarely used in other fields in the same way.&lt;/p&gt;&#xA;&lt;p&gt;It's not just talent that distinguished them from others, but also determination and opportunity.&lt;/p&gt;&#xA;&lt;p&gt;Take Feynman for instance, the opportunity wouldn't have been there if mass public education wasn't the norm. Nor would it not have been there if the states wasn't an advanced industrialised economy. The determination speaks for itself: he was building crystal radios when he was a young teenager and fixing them, and having read calculus was important he was determined to read up on it as much as he could. It's not surprising then he was ahead of the class.&lt;/p&gt;&#xA;&lt;p&gt;Another thing to ponder is that there are many fields of endeavour. For example, physics is now split into many sub-fields so that there are many more to succeed in.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2020-12-18T20:07:52.237" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12641" PostTypeId="1" CreationDate="2020-12-19T02:05:02.363" Score="0" ViewCount="51" Body="&lt;p&gt;From what I read the expansion of the universe predicted by QFT is said to be &lt;a href=&quot;https://www.space.com/38911-worst-theoretical-prediction-physics-history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;the worst theoretical prediction in the history of physics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is also lots of buzz in the astronomy press about the &lt;a href=&quot;https://www.nasa.gov/feature/goddard/2019/new-hubble-constant-measurement-adds-to-mystery-of-universe-s-expansion-rate&quot; rel=&quot;nofollow noreferrer&quot;&gt;latest Hubble expansion measurements&lt;/a&gt; being askew from predicted values.&lt;/p&gt;&#xA;&lt;p&gt;Does the new crisis in the Hubble constant measurement refer to the QFT prediction or something else? Perhaps the crisis refers to discrepencies in more than one way to measure the constant and not more than one way to predict.&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="55" LastEditDate="2020-12-19T06:42:50.503" LastActivityDate="2020-12-19T06:42:50.503" Title="How do the latest measurements of the Hubble expansion match the QFT prediction?" Tags="|cosmology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12642" PostTypeId="1" CreationDate="2020-12-18T17:52:43.090" Score="4" ViewCount="118" Body="&lt;p&gt;I am trying to write a paper on how Einstein's special relativity (SR) essentially saved electromagnetism for high school and am confused on how the following two problems were solved.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;How did physicists explain the (theoretical) disappearance of the magnetic force on a moving charge by changing the frame of reference?&lt;/strong&gt;&#xA;Special relativity saves this by essentially showing that changing frames changes the electric force too, as well as the magnetic, but I am unsure about how this was tackled before&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Michelson &amp;amp; Morley disproved the existence of luminiferous ether in 1887, 18 years before SR. Was the world of physics in complete chaos for the next 18 years with the apparent breaking of electromagnetism, or was there a temporary solution?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerDisplayName="OVERWOOTCH " LastEditorUserId="10106" LastEditDate="2020-12-19T17:06:26.513" LastActivityDate="2020-12-19T17:06:26.513" Title="How did physicists deal with the variance of electromagnetism before special relativity?" Tags="|physics|electromagnetism|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12643" PostTypeId="2" ParentId="12642" CreationDate="2020-12-18T23:50:56.630" Score="6" Body="&lt;p&gt;There certainly WERE formulae that correctly described parts of the electromagnetic&#xA;puzzle, including Fitzgerald-Lorentz contraction of distance, and even&#xA;some medium-drag light delay experiments with transparent media instead&#xA;of the fictitious ether.&lt;/p&gt;&#xA;&lt;p&gt;What special relativity did was to incorporate all the various 'it-works-like-this'&#xA;rules into a single coherent picture, including generating all magnetic forces&#xA;from the behaviors of charge; it cut the four Maxwell equations&#xA;for electricity and magnetism down to two equations, and a principle&#xA;of relativity...   while simultaneously predicting a variety of phenomena&#xA;regarding space and time.   It revolutionized both electricity, AND&#xA;mechanics, with one concept.&lt;/p&gt;&#xA;" OwnerUserId="5111" OwnerDisplayName="Whit3rd" LastActivityDate="2020-12-18T23:50:56.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12644" PostTypeId="2" ParentId="12642" CreationDate="2020-12-19T01:25:14.087" Score="1" Body="&lt;p&gt;Special Relativity(SR) didn't save electromagnetism, not even remotely. In fact, all of SR can essentially be derived from classical electromagnetism. That is essentially what Lorentz Ether Theory(LET) is. For example, before SR came into the picture, Lorentz showed that electromagnetic systems, such as matter, would experience length contraction in the direction of movement at speeds close to the speed of light. LET essentially assumes the entire universe consists of electromagnetic systems in an ether(which in this case means a homogeneous medium with a given electric permittivity and magnetic permeability). If you have a moving charge in LET, a Lorentz transformation can be done so that the charge is stationary and the magnetic field disappears. In case you're wondering, LET an SR give the exact same experimental predictions.&lt;/p&gt;&#xA;" OwnerDisplayName="Laff70" LastActivityDate="2020-12-19T01:25:14.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12648" PostTypeId="1" AcceptedAnswerId="12651" CreationDate="2020-12-20T09:24:24.633" Score="5" ViewCount="344" Body="&lt;p&gt;Why 1 is not prime? I read in &lt;a href=&quot;https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/#:%7E:text=The%20confusion%20begins%20with%20this,are%20not%20two%20distinct%20factors.&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; that G.H Hardy explicitly included 1 as a prime in the first 6 editions of &amp;quot;A Course in Pure Mathematics&amp;quot;, published between 1908-1933. He updated the definition in 1938 to make &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; the smallest prime. What was the reason behind this?&lt;/p&gt;&#xA;&lt;p&gt;Also from todays Ring Theory course we get the definition of prime to be &lt;span class=&quot;math-container&quot;&gt;$$\text{if}\  p|ab \ \text{then either} \ p|a \ \text{or} \ p|b.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although in a &lt;span class=&quot;math-container&quot;&gt;$\operatorname{PID}$&lt;/span&gt; primes and irreducibles are equivalent.&lt;/p&gt;&#xA;&lt;p&gt;What changes in mathematics resulted in the change of the definition of primes and exclude 1?&lt;/p&gt;&#xA;&lt;p&gt;I mean how the definition of prime evolved with time?&lt;/p&gt;&#xA;" OwnerUserId="13275" LastEditorUserId="55" LastEditDate="2020-12-20T23:48:29.530" LastActivityDate="2020-12-23T21:47:39.397" Title="What changes in mathematics resulted in the change of the definition of primes and exclusion of 1?" Tags="|terminology|number-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12650" PostTypeId="1" CreationDate="2020-12-20T17:43:17.130" Score="4" ViewCount="171" Body="&lt;p&gt;The system of calculating area in terms of square units is pretty philosophical and not very intuitive. It must have taken a great amount of time for humanity to arrive at such a convention and to spread it across different societies.&#xA;My question is about finding basics of such a convention and should the person who first thought of calculating areas in square units be regarded a great philosopher equivalent to Newton and Einstein?&#xA;Moreover, could we have evolved a different method of calculating areas?&lt;/p&gt;&#xA;" OwnerUserId="13511" LastEditorUserId="13511" LastEditDate="2020-12-20T19:07:52.957" LastActivityDate="2020-12-20T19:07:52.957" Title="What is the basic idea behind calculation of area?" Tags="|mathematics|geometry|mathematical-logic|mathematics-social-history|" AnswerCount="0" CommentCount="3" FavoriteCount="0" ClosedDate="2020-12-21T15:31:08.933" ContentLicense="CC BY-SA 4.0" />
  <row Id="12651" PostTypeId="2" ParentId="12648" CreationDate="2020-12-20T23:46:40.123" Score="10" Body="&lt;p&gt;As to the primality of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, nothing really evolved. This is an illustration that some things in mathematics are not right or wrong, they are just a matter of taste. And some fraction of the population prefers a definition that does not make an explicit exception for &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. The prime/irreducible shift in terminology, on the other hand, does reflect changes in mathematics, namely the exploration of number rings without unique factorization in the 19th century, first by Kummer and then by Dedekind.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Caldwell et al. in The History of the Primality of One&lt;/a&gt; collected source quotes concerning &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; being prime (and a number) from Plato to 2012, and they show that the disarray on this issue goes back to antiquity. While Plato and Aristotle did not even consider &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to be a number, let alone prime, Plato's nephew and successor Speusippus took it to be both, although he was unique in this regard at the time. Over the centuries he got company (Waring, Gregory, Legendre, etc.), but  &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-is-a-prime party has always been in the minority. Hardy was its last prominent representative, but some schoolbooks count &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; among primes as late as 2012. In &lt;a href=&quot;https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why Isn't 1 a Prime Number? Evelyn Lamb&lt;/a&gt; speculates that the exclusion of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; is due to the desire to have uniqueness in the prime factorization theorem, and hence to separate primes from units. But even if this is a factor now the widespread exclusion much predates Gauss's formulation of the theorem and the times when people paid attention to such abstractions.&lt;/p&gt;&#xA;&lt;p&gt;But unique factorization did play a direct role in the definitional shift in ring theory. In &lt;a href=&quot;https://math.stackexchange.com/a/813827/152568&quot;&gt;History of the terms “prime” and “irreducible” in Ring Theory on Math SE&lt;/a&gt; @mweiss speculated that the shift occurred due to first reproducing the traditional definition of (integer) primes at the level of ideals, and then descending it to the level of elements. This leads to the modern convention in ring theory that turns &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_lemma&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid's lemma&lt;/a&gt; into the definition of prime quoted in the OP.&lt;/p&gt;&#xA;&lt;p&gt;As neat as this theory is, it is not how it happened. The shift can be traced to &lt;a href=&quot;http://www.numdam.org/article/BSMA_1877_2_1_1_144_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dedekind's &lt;em&gt;Sur la théorie des nombres entiers algébriques&lt;/em&gt; (1877)&lt;/a&gt;, and Dedekind makes it &lt;em&gt;before&lt;/em&gt; introducing ideals, when considering divisibility in the ring &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}(\sqrt{-5})$&lt;/span&gt; singled out by Kummer. Moreover, Dedekind makes clear that he considers Euclid's lemma, not the traditional definition, to be the &amp;quot;characteristic&amp;quot; property of primes, because it is what gives unique factorization. Accordingly, he renames what the traditional definition defines into &amp;quot;&lt;em&gt;irréductible&lt;/em&gt;&amp;quot;, &amp;quot;irreducible&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;A nice expose of Dedekind's 1877 work, with English translation of extensive excerpts and commentary, is &lt;a href=&quot;https://digitalcommons.ursinus.edu/triumphs_abstract/1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barnett's Richard Dedekind and the Creation of an Ideal: Early Developments in Ring Theory&lt;/a&gt;. Here are some Dedekind's quotes in Barnett's translation (he translates &amp;quot;&lt;em&gt;irréductible&lt;/em&gt;&amp;quot; as &amp;quot;indecomposable&amp;quot;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Each positive number, different&#xA;from unity, is either a prime number, that is, a number divisible only by itself and unity, or else&#xA;a composite number. In the latter case we can always express it as a product of prime numbers&#xA;and -- which is the most important thing -- in only one way. That is, the system of prime&#xA;numbers occurring as factors in this product is completely determined by giving the number of&#xA;times a designated prime number occurs as factor. This property depends essentially on the&#xA;theorem that a prime divides a product of two factors only when it divides one of the factors.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;A number (different from zero and &lt;span class=&quot;math-container&quot;&gt;$\pm1$&lt;/span&gt;) is called decomposable when it is the product of two factors, neither of which is a unit. In the contrary case the number is called indecomposable... However, despite the indecomposability of these fifteen numbers, there are numerous relations between their products... In each of these ten relations, the same number is represented in two or three different ways&#xA;as a product of indecomposable numbers. Thus one sees that an indecomposable number&#xA;may very well divide a product without dividing any of its factors. Such an indecomposable&#xA;number therefore does not possess the property which, in the theory of rational numbers, is&#xA;characteristic of a prime number.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-12-23T21:47:39.397" LastActivityDate="2020-12-23T21:47:39.397" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12653" PostTypeId="1" CreationDate="2020-12-21T18:31:54.520" Score="6" ViewCount="119" Body="&lt;p&gt;my question is very simple. Physicists use this notation in order to write a (for example) plane wave:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\xi(z) = \xi^+ \mathrm{e}^{+\mathrm{i}kz} + \xi^- \mathrm{e}^{-\mathrm{i}kz},&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\xi^+$&lt;/span&gt; is the progressive wave and &lt;span class=&quot;math-container&quot;&gt;$\xi^-$&lt;/span&gt; is the regressive wave.&lt;/p&gt;&#xA;&lt;p&gt;Engineers adopt a different notation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\xi(z) = \xi^+ \mathrm{e}^{-\mathrm{i}kz} + \xi^- e^{+\mathrm{i}kz}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Basically you can move from one notation to the other by choosing &lt;span class=&quot;math-container&quot;&gt;$\mathrm{i} \to -\mathrm{i}$&lt;/span&gt;, or &lt;em&gt;vice versa&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Obviously both are correct, but I find a little more sense in the engineers' notation, because &lt;span class=&quot;math-container&quot;&gt;$\xi(z)$&lt;/span&gt; can be interpreted as a phasor, and in order to come back in the time domain you have to multiply for &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}^{\mathrm{i}\omega t}$&lt;/span&gt;, and this is the same thing that you do when you do the inverse Fourier transform, in order to get a function in the time domain:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(t) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty} F(\omega) \ \mathrm{e}^{\mathrm{i} \omega t} \mathrm{d}\omega,&#xA;$$&lt;/span&gt;&#xA;while the physicists have to multiply for &lt;span class=&quot;math-container&quot;&gt;$\mathrm{e}^{-\mathrm{i}\omega t}$&lt;/span&gt;, in order to have &lt;span class=&quot;math-container&quot;&gt;$\xi^+$&lt;/span&gt; as the coefficient of the progressive wave and &lt;span class=&quot;math-container&quot;&gt;$\xi^-$&lt;/span&gt; as the coefficient of the regressive wave.&lt;/p&gt;&#xA;&lt;p&gt;Obviously this is not fundamental, because the only thing important is to have a &lt;span class=&quot;math-container&quot;&gt;$f(kz-\omega t)$&lt;/span&gt; as the progressive wave and &lt;span class=&quot;math-container&quot;&gt;$f(kz+\omega t)$&lt;/span&gt; as the regressive wave, but I find more natural the engineers' notation. Is there a reason why physicists use another notation?&lt;/p&gt;&#xA;&lt;p&gt;A friend of mine told me that in origin engineer chose another imaginary root, but I'm not convinced. He basically told me that since the equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x^2 = -1 \Longrightarrow \left| x \right| = \sqrt{-1}&#xA;$$&lt;/span&gt;&#xA;has two roots, namely: &lt;span class=&quot;math-container&quot;&gt;$x_1 = \sqrt{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_2 = -\sqrt{-1}$&lt;/span&gt;, physicists and mathematicians chose &lt;span class=&quot;math-container&quot;&gt;$\mathrm{i} = \sqrt{-1}$&lt;/span&gt;, while engineers chose &lt;span class=&quot;math-container&quot;&gt;$j = -\sqrt{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The reason why I'm not convinced is that they write other quantities in the same way, for example you can find &lt;span class=&quot;math-container&quot;&gt;$Z_L = \mathrm{i}\omega L$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Z_L = j\omega L$&lt;/span&gt; in physics and engineering books.&lt;/p&gt;&#xA;&lt;p&gt;Do you have any idea why this discrepancy does exists? Can you cite some source about the notation discrepancy? I didn't find anything.&lt;/p&gt;&#xA;" OwnerUserId="13517" LastActivityDate="2020-12-21T18:31:54.520" Title="What is the origin in the discrepancy between engineers' and physicists' notation of waves?" Tags="|mathematics|physics|notation|mathematical-physics|engineering|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12654" PostTypeId="2" ParentId="11598" CreationDate="2020-12-21T18:51:09.113" Score="5" Body="&lt;p&gt;Based on the discussion of Eratosthenes' method in Daniel Špelda's &lt;em&gt;Astronomie v Antice&lt;/em&gt; (Astronomy in Antiquity, ISBN 80-7225-210-0), the distance of 5000 stadia was estimated by Eratosthenes based on the time traders took to travel between the cities. A round value was used, similar to other values Eratosthenes used in his computation, suggesting that even he himself was aware that the numbers were just estimations.&lt;/p&gt;&#xA;" OwnerUserId="13393" LastActivityDate="2020-12-21T18:51:09.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12656" PostTypeId="1" AcceptedAnswerId="12658" CreationDate="2020-12-23T16:01:49.780" Score="10" ViewCount="393" Body="&lt;p&gt;I've been thinking about the history of the irreducible polynomials and why they were introduced. I found &lt;a href=&quot;https://hsm.stackexchange.com/q/2504/13336&quot;&gt;What is the origin of polynomials and notation for them?&lt;/a&gt;, but it's about polynomials in general.&lt;/p&gt;&#xA;&lt;p&gt;Could anyone please describe the historical motivation for introducing and studying irreducible polynomials? I would love to get some references for that.&lt;/p&gt;&#xA;" OwnerUserId="13336" LastEditorUserId="13336" LastEditDate="2020-12-24T12:10:19.987" LastActivityDate="2020-12-24T12:10:19.987" Title="History of irreducible polynomials and motivation for them" Tags="|mathematics|reference-request|abstract-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12658" PostTypeId="2" ParentId="12656" CreationDate="2020-12-24T09:14:14.413" Score="11" Body="&lt;p&gt;I will skip the pre-history of solving polynomial equations and factoring polynomials. Let me mention that the analogy between long division of numbers and polynomials goes back to medieval Islamic mathematician al-Samawal, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2108/55&quot;&gt;Who invented short and long division?&lt;/a&gt;, and the Euclidean algorithm for polynomials was optimized by Hudde, a younger contemporary of Descartes, see &lt;a href=&quot;https://www.maa.org/programs/maa-awards/writing-awards/the-lost-calculus-1637-1670-tangency-and-optimization-without-limits&quot; rel=&quot;noreferrer&quot;&gt;Suzuki, The Lost Calculus&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The history proper of irreducibles starts with cyclotomic polynomials in Gauss's &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; (1801). His motivation was related to inscribing regular polygons into a circle with straightedge and compass, and a cryptic remark pointed to a generalization to the lemniscate. The early theory was developed in the context of &amp;quot;higher congruences&amp;quot;, polynomial equations modulo primes and their powers, see &lt;a href=&quot;https://www.maa.org/programs/maa-awards/writing-awards/why-eisenstein-proved-the-eisenstein-criterion-and-why-sch-nemann-discovered-it-first&quot; rel=&quot;noreferrer&quot;&gt;Cox's Why Eisenstein Proved the&#xA;Eisenstein Criterion&lt;/a&gt; and &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick/page/232/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Dickson's History of the theory of numbers, ch. VIII&lt;/a&gt;. The study of general number rings by Kummer and Dedekind came from the same source.&lt;/p&gt;&#xA;&lt;p&gt;Gauss proved that cyclotomic polynomials with prime indices are irreducible (he did not use such terminology). In the course of it he proved the first general result on irreducibility, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomial)&quot; rel=&quot;noreferrer&quot;&gt;Gauss's lemma&lt;/a&gt;. Even more relevant was the unpublished section 8 of &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt;, titled &lt;em&gt;Disquisitiones generales de congruentiis&lt;/em&gt;, where Gauss studied &amp;quot;polynomial congruences&amp;quot; modulo &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, i.e. polynomials in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_p[x]$&lt;/span&gt; in modern terms, see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-540-34720-0_6&quot; rel=&quot;noreferrer&quot;&gt;Frei, The Unpublished Section Eight&lt;/a&gt;. He counted the number of irreducible monic polynomials in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{F}_p[x]$&lt;/span&gt;, and proved a case of &lt;a href=&quot;https://en.wikipedia.org/wiki/Hensel%27s_lemma&quot; rel=&quot;noreferrer&quot;&gt;Hensel's lemma&lt;/a&gt; in the course of it. But all of this only became available after Dedekind published section 8 in 1863 (full version in 1876), and was rediscovered by others in the meantime, especially Schönemann and Dedekind himself.&lt;/p&gt;&#xA;&lt;p&gt;But even the published parts were inspiration enough for Abel and Galois. &lt;a href=&quot;https://en.wikipedia.org/wiki/Abel%27s_irreducibility_theorem&quot; rel=&quot;noreferrer&quot;&gt;Abel's irreducibility theorem&lt;/a&gt;, not so formulated, appeared in his &lt;em&gt;Mémoire sur une classe particulière d'équations résolubles algébriquement&lt;/em&gt; (1829). Abel was led to it by his earlier extension to the lemniscate of Gauss's result on subdividing a circle into equal parts, per Gauss's remark. In Galois' note &lt;em&gt;Sur la theorie des nombres&lt;/em&gt; (1830, it appears with English translation in &lt;a href=&quot;https://uberty.org/wp-content/uploads/2015/11/Peter_M._Neumann_The_Mathematical_Writings.pdf&quot; rel=&quot;noreferrer&quot;&gt;The mathematical writings of Évariste Galois&lt;/a&gt;) we see the term &amp;quot;&lt;em&gt;irréductible&lt;/em&gt;&amp;quot;, although it is applied to congruences rather than polynomials, and a related construction of finite fields.&lt;/p&gt;&#xA;&lt;p&gt;But Schönemann in a two part paper &lt;em&gt;Grundzuge einer allgemeinen Theorie der hohern Congruenzen&lt;/em&gt; (1845) and &lt;em&gt;Von denjenigen Moduln, welche&#xA;Potenzen von Primzahlen sind&lt;/em&gt; (1846) independently rediscovered both Gauss's and Galois' results and went much further. In particular, he applies &amp;quot;irreducible&amp;quot; to polynomials, and states a general problem:&amp;quot;&lt;em&gt;To investigate, whether the power of an irreducible polynomial modulo&#xA;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is or is not irreducible modulo &lt;span class=&quot;math-container&quot;&gt;$p^m$&lt;/span&gt;&lt;/em&gt;&amp;quot;, which he solves using a version of what is now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eisenstein%27s_criterion&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Eisenstein criterion&amp;quot;&lt;/a&gt; of irreducibility (largely due to van der Waerden's oversight). Eisenstein rediscovered the criterion when reproving Abel's theorem on subdividing the lemniscate, and shared the idea in a letter to Gauss in 1847, but the published version only appeared in &lt;em&gt;Uber die Irreductibilitat und einige andere Eigenschaften der Gleichung&lt;/em&gt; (1850). A number of authors worked on higher congruences from that point on, Mathieau, Serret, Dedekind, Kronecker, Jordan, Weber, etc.&lt;/p&gt;&#xA;&lt;p&gt;In the hands of Dedekind, after his &lt;em&gt;Abriß einer Theorie der hoheren Kongruenzen in bezug auf einen reellen Primzahl-Modulus&lt;/em&gt; (1857), the story took a more abstract turn that led to modern ring theory. Later Dedekind synthesized the work of Gauss, Galois, Schönemann and Kummer by introducing rings and ideals, and developing unified terminology of primes and irreducibles, see &lt;a href=&quot;https://hsm.stackexchange.com/a/12651/55&quot;&gt;What changes in mathematics resulted in the change of the definition of primes and exclusion of 1?&lt;/a&gt; In a more concrete vein, Kronecker gave a general algorithm for completely factoring&#xA;a rational integer polynomial into a product of irreducibles&#xA;in 1882, see &lt;a href=&quot;https://www.jstor.org/stable/2301357&quot; rel=&quot;noreferrer&quot;&gt;Dorwart, Irreducibility of Polynomials&lt;/a&gt;. The Schönemann-Eisenstein criterion was extended by Konigsberger (1895), Netto (1896) Bauer and Perron (1905). Dumas developed the now popular Newton polygon method to study irreducibility in &lt;em&gt;Sur quelques cas d’irreductibilite des polynomes a coefficients rationnels&lt;/em&gt; (1906), see &lt;a href=&quot;https://arxiv.org/abs/1304.0874&quot; rel=&quot;noreferrer&quot;&gt;Schönemann-Eisenstein-Dumas-type irreducibility conditions by Bonciocat&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-12-24T09:28:51.537" LastActivityDate="2020-12-24T09:28:51.537" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12659" PostTypeId="1" CreationDate="2020-12-25T13:09:04.247" Score="2" ViewCount="90" Body="&lt;p&gt;I was wondering if anyone has read '' alhazen book of optics'' and has understood his solution of Alhazen's problem.&lt;/p&gt;&#xA;&lt;p&gt;I know a modern solution of the problem by Dörrie-100 great problems of mathematics, but I'm interested really in ibn alhaytham's solution. I'm just curious about how he went about solving the problem without the modern tools. Also I couldn't find the problem in his book, maybe I didn't look carefully for it, but it would be nice if someone could reference me to it in his book also.&lt;/p&gt;&#xA;" OwnerUserId="9863" LastActivityDate="2020-12-25T13:09:04.247" Title="Alhazen's problem history" Tags="|optics|mathematics-social-history|physics-social-history|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12660" PostTypeId="1" CreationDate="2020-12-26T00:10:43.390" Score="0" ViewCount="516" Body="&lt;p&gt;Is it true that ancient Greek knowledge of mathematics, science, astronomy, engineering, architecture, civilization came from the ancient Minoan civilization and ancient Mycenaean civilizations and not from the ancient Mesopotamians and ancient Egyptians? If this is true, then why are the ancient Egyptians and ancient Mesopotamians credited as the cradles of mathematics, science, astronomy, engineering, architecture, civilization, etc.?&lt;/p&gt;&#xA;&lt;p&gt;Does this mean that ancient Mesopotamians and ancient Egyptians aren't cradles of mathematics, science, astronomy, engineering, architecture, civilization, etc.?&lt;/p&gt;&#xA;&lt;p&gt;Seriously, what contributions did ancient Mesopotamians and ancient Egyptians really had on ancient Greeks during the time of Pre-Socratic philosophers such as Thales? Or should we just conclude that ancient Egypt and ancient Mesopotamia had no contributions to ancient Greece?&lt;/p&gt;&#xA;&lt;p&gt;Evidence suggests that ancient Minoans and ancient Mycenaeans were capable of advanced engineering, including four-story palaces with drainage and beehive tombs. Ancient Minoan hydro-engineering is also generally considered comparable with that of 18th century Europe CE.&lt;/p&gt;&#xA;&lt;p&gt;If this is really true then it means that the ancient Greeks already possessed knowledge of mathematics, science, astronomy, engineering, and architecture that the ancient Egyptians and ancient Mesopotamians have or even better so they don't have to borrow anything from the ancient Egyptians and ancient Mesopotamians.&lt;/p&gt;&#xA;&lt;p&gt;Seriously, what's so special about ancient Mesopotamia and ancient Egypt for them to be called as the cradles of mathematics, science, astronomy, engineering, architecture, civilization, etc. compared to other ancient civilizations such as China, India, Indus Valley, Mesoamerica, Hittites, the Minoans, the Mycenaeans, etc.? Are the ancient Egyptians and ancient Mesopotamians just overrated?&lt;/p&gt;&#xA;&lt;p&gt;Please help me. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="13535" LastActivityDate="2020-12-26T16:12:47.473" Title="Are the ancient Egyptians and the ancient Mesopotamians really the cradles of civilization?" Tags="|mathematics|philosophy-of-science|ancient-greece|mathematics-social-history|ancient-egypt|" AnswerCount="1" CommentCount="2" ClosedDate="2020-12-27T06:47:57.860" ContentLicense="CC BY-SA 4.0" />
  <row Id="12661" PostTypeId="1" CreationDate="2020-12-26T04:43:58.710" Score="1" ViewCount="47" Body="&lt;p&gt;The Hawking-Hartle no boundary condition is a well known model that tries to explain how did the universe begin. The authors considered a &amp;quot;many worlds/histories&amp;quot; model considering a sum over all compact euclidean metrics.&lt;/p&gt;&#xA;&lt;p&gt;But are there any models or theories that consider a sum over all possible metrics or boundaries? Have any physicists ever proposed such model or theory?&lt;/p&gt;&#xA;" OwnerUserId="10956" LastActivityDate="2020-12-26T04:43:58.710" Title="Has someone ever proposed a &quot;many worlds/histories&quot; model where all possible boundary conditions are considered?" Tags="|physics|quantum-mechanics|theoretical-physics|cosmology|" AnswerCount="0" CommentCount="0" ClosedDate="2020-12-28T00:05:45.050" ContentLicense="CC BY-SA 4.0" />
  <row Id="12662" PostTypeId="2" ParentId="12660" CreationDate="2020-12-26T15:58:36.610" Score="3" Body="&lt;p&gt;All these assessments about &amp;quot;cradle of civilization&amp;quot; are based on the archeological records that reached us and on surviving texts.&lt;/p&gt;&#xA;&lt;p&gt;The oldest surviving samples of writing come from Mesopotamia (Sumer)&#xA;and Egypt. There was a long discussion whether writing was invented in these regions independently or there was some transmission. As I understand, the opinion of majority of scientists is that writing was invented in Mesopotamia first, and then the idea was transmitted to Egypt.&lt;/p&gt;&#xA;&lt;p&gt;The earliest advanced mathematical and astronomical texts that we possess also come from Mesopotamia. We do not have Minoan or Micenaean mathematical or astronomical texts. Of course this does not prove that they did not exist. (Mesopotamian texts were written on a very durable material, and for this reason they reached us).&lt;/p&gt;&#xA;&lt;p&gt;Ancient Greek themselves thought that their science comes from Egypt. This is not confirmed by modern science. Roughly speaking, there is nothing in the Egyptian records that we know, that they could teach Thales or Pythagoras. On the other hand the influence of Babylonian astronomy on the Greek astronomy is confirmed by modern research. However, this transmission&#xA;which we know happened after Macedonian conquests, that is at the much later time than Thales and Pythagoras.&lt;/p&gt;&#xA;&lt;p&gt;Greek writing comes from the Middle Eastern Phoenician writing, not from the earlier Minoan writing. And Phoenician writing is a remote descendent of Babylonian cuneiform.&lt;/p&gt;&#xA;&lt;p&gt;It is true that Minoan and Micenaean civilizations existed on the territory of later Greece, but they are not as old as civilizations of Sumer and Egypt. Then, in 12 century BC, all East Mediterranean civilizations, except Egyptian, collapsed. To such extent that writing was forgotten. When it was recovered, it had to be recovered from another source.&lt;/p&gt;&#xA;&lt;p&gt;Refs. O. Neugebauer, A history of ancient mathematical astronomy,&lt;/p&gt;&#xA;&lt;p&gt;O. Neugebauer, Exact sciences in antiquity,&lt;/p&gt;&#xA;&lt;p&gt;Ch. Woods, ed. Visible language.  Inventions of writing in the Middle East and beyond.&lt;/p&gt;&#xA;&lt;p&gt;E. Cline, 1177 B. C. The year Civilization collapsed.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2020-12-26T16:12:47.473" LastActivityDate="2020-12-26T16:12:47.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12663" PostTypeId="1" CreationDate="2020-12-27T16:03:55.787" Score="1" ViewCount="157" Body="&lt;p&gt;&lt;strong&gt;tl;dr&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why has the array paradigm, which emerged in the 1950s and 60s amongst mathematicians, and which underpins certain programming languages, apparently failed to capture and maintain interest amongst mathematicians as a foundational innovation in their field, despite tacit applications of the principles being widespread?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The long read&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;As an analyst and a programmer working in industry, conventionally I would be considered an IT professional, but I tend to consider my real profession to be that of &lt;em&gt;administration&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Administration has a longer history than computing, both as an occupation, and as a focus for systematic study. A bread and butter activity in administration is what would nowadays be expressed as the storage, maintenance, and processing of data.&lt;/p&gt;&#xA;&lt;p&gt;In my view, it's a peculiar feature of administration (despite its extensive handling of numbers as well as other kinds of information) that the mathematics involved are rarely regarded as difficult, whether by mathematicians or anybody else. IT professionals are rarely expected to have more than a basic knowledge of conventional mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Complexity arises in administration largely because mathematics as practised here, is a mathematics where calculations often consist of a long series of simple operations, carried out by multiple people in different places, and on a schedule that is often ill-defined and unreliable.&lt;/p&gt;&#xA;&lt;p&gt;These challenges of calculation are unfamiliar to the academic mathematician who works at the blackboard, and who does not routinely do a third of the calculation now, a third over the road, and a third next week. Indeed these challenges are alien to students, in almost all educational settings where mathematical knowledge is reproduced.&lt;/p&gt;&#xA;&lt;p&gt;Another feature of mathematics as practised in administration, much more pertinent to the question I will arrive at, is that it is frequently concerned with operating on multiple &amp;quot;values&amp;quot; - that is, multiple items of data, or put another way, arrays of data. Exactly how such arrays of data are arranged for storage and movement, as part of what is essentially a process of constant calculation, is a central concern.&lt;/p&gt;&#xA;&lt;p&gt;There is however no standardised treatment of arrays in conventional mathematics. No standard notation. No standard battery of operations. No algebra.&lt;/p&gt;&#xA;&lt;p&gt;Don't get me wrong, I'm aware of statistics, and calculus, and matrix multiplication and it's associated notation, but these areas are not regarded as uniform, and they are not regarded as elementary (that is, not in a way that is commensurate with how administration is considered to have elementary aspects that all citizens of a society like ours ought to be familiar with, and handling arrays of values are undoubtedly elementary in administration).&lt;/p&gt;&#xA;&lt;p&gt;The notation for the &amp;quot;binary iterated operations&amp;quot;, like sigma for summation and pi for the combined product, are probably the closest one comes, and the notation is in a class of its own. Again also, it is not regarded as elementary mathematics (as would be on a school curriculum with addition and multiplication), despite the ubiquity of summation in daily life.&lt;/p&gt;&#xA;&lt;p&gt;For reasons that are certainly obvious once one explicitly acknowledges the close connection between computers and the field of administration, all the innovations in this area have been in computer science.&lt;/p&gt;&#xA;&lt;p&gt;The SQL programming language is underpinned by &amp;quot;relational algebra&amp;quot;, work done in the 1960s at IBM by British mathematician E.F. Codd, and published in 1970, which (amongst other things) defines a simple array model consisting of &amp;quot;tables&amp;quot;, a set of operators which accept tables as operands and manipulate them to produce a table as an output, and which operators obey algebraic principles. Codd himself used a maths-like notation for his work, but SQL then resulted from work by ergonomics and usability experts to ease the notation.&lt;/p&gt;&#xA;&lt;p&gt;SQL remains in ubiquitous use today, and many IT professionals who use it routinely will know of Codd.&lt;/p&gt;&#xA;&lt;p&gt;It's worth mentioning in passing however, that amongst syllabuses I've seen on the &amp;quot;relational model&amp;quot;, many revert to using Codd's notation (and some of his awkward vocabulary too) which is completely superseded by SQL, and such syllabuses do not succeed in giving a general introduction to arrays and array operators.&lt;/p&gt;&#xA;&lt;p&gt;There was also a period in the past 20 years, though now seemingly over, when SQL became the subject of a lot of negative hype about its own complexity and user-unfriendliness, and even in the 90s it was often regarded as mind-bending by many programmers of conventional languages.&lt;/p&gt;&#xA;&lt;p&gt;One imagines that this could be related to consistent poor tuition of the SQL language, and failure to reproduce the mathematical concepts which underpin it (since it is definitely a form of mathematics, but one which is specifically designed to operate on arrays of values).&lt;/p&gt;&#xA;&lt;p&gt;A now much-lesser-known figure from the same era as Codd, and publishing some years prior, is the Canadian mathematician and later computer scientist, Kenneth E. Iverson.&lt;/p&gt;&#xA;&lt;p&gt;People familiar with mathematics may be most familiar with the notation of the ceiling and floor functions which Iverson devised: &lt;span class=&quot;math-container&quot;&gt;$\lceil x \rceil$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lfloor x \rfloor$&lt;/span&gt;, respectively.&lt;/p&gt;&#xA;&lt;p&gt;Most of Iverson's academic and industrial career however was dedicated to a programming language called APL. Devised originally by him in the 1950s as a terse, pen-and-paper notation for describing algorithms, and intended to augment and improve upon the standard mathematical notation available, some of it's main innovations were a large set of brand-new mathematical symbols, and it's incorporation of an array model (with operators and notation).&lt;/p&gt;&#xA;&lt;p&gt;In the 1960s, it found an immediate niche for interactive use on early computers based around a teletype machine. For the unfamiliar, a teletype is a computer terminal which resembles a standard typewriter, having a keyboard for entry, and a computer-controlled printer mechanism for output, but no monitor. Interactive use means you type input and immediately receive output, via the printer - step-by-step like a retail till printing items on receipt paper.&lt;/p&gt;&#xA;&lt;p&gt;Being so terse allowed a teletype operator to do a lot of ad-hoc numerical computation quickly with such limited hardware, and IBM made a significant investment in APL at the time, primarily to cater for financial applications. The symbol set was tailored to what a daisy-wheel printer of the time could output, and a custom keyboard was designed for input of the same symbols.&lt;/p&gt;&#xA;&lt;p&gt;What distinguishes APL from SQL (with the latter's basic tenet of a 2-d tabular model), is that the arrays in APL need not take the form of &amp;quot;tables&amp;quot;, and there are thus a more extensive complement of operators in APL which are not designed to process the values in arrays, but instead to change the structural arrangement of those values within the array, or to influence the order or the axis of the array which an operator acts upon (for example, the distinction between summing columns or summing rows).&lt;/p&gt;&#xA;&lt;p&gt;Only a sniff of such complex functionality has been introduced into SQL, with its so-called window/analytic functions. But one never sees the implied, non-tabular intermediary results which these operators must generate internally, and they cannot be explained within the tenets of Codd's relational algebra. They can only be described within a more general array model.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, the symbology and the way in which APL notated and parsed expressions, in conjunction with its other innovations, became infamous in computer science. Influential Dutch computer scientist, Edsger Dijkstra, ridiculed the APL language as &amp;quot;a mistake, carried through to perfection&amp;quot;. Others condemned it as &amp;quot;write-only code&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;By the end of his career in the late 80s, Iverson had unequivocally recognised the strategic mistake of the non-standard symbology, which made it difficult by then to represent APL on standard PC hardware, and imposed a learning curve in terms of being able to pronounce and remember the functionality of utterly unfamiliar symbols. The terseness also became less important than it had been with a teletype interface.&lt;/p&gt;&#xA;&lt;p&gt;But adjustments to the notation failed to reinvigorate the APL language, which exists now only in small niches. Only SQL has emerged vital in the modern day, and like I say, not without enduring a period of negative hype, and not without most users struggling with its conceptual underpinnings.&lt;/p&gt;&#xA;&lt;p&gt;For those who have tolerated this tome of historical context so far, my question is this.&lt;/p&gt;&#xA;&lt;p&gt;Why has the array paradigm, which emerged in the 1950s and 60s amongst mathematicians, apparently failed to capture and maintain interest amongst mathematicians as a foundational innovation in their field, despite tacit applications of the principles being widespread?&lt;/p&gt;&#xA;" OwnerUserId="6847" LastEditorUserId="6847" LastEditDate="2020-12-27T18:33:04.647" LastActivityDate="2020-12-27T18:33:04.647" Title="The mathematics of multiple values" Tags="|mathematics|computer-science|" AnswerCount="0" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="12664" PostTypeId="1" CreationDate="2020-12-27T16:15:33.367" Score="2" ViewCount="203" Body="&lt;p&gt;Suppose an object (or a concept or ...) is named after the person &lt;code&gt;X&lt;/code&gt;, in honor of Mr. or Mrs. X in mathematics: &lt;code&gt;X&lt;/code&gt;-ian objects/ &lt;code&gt;X&lt;/code&gt;-ic objects/ &lt;code&gt;X&lt;/code&gt; objects.&lt;/p&gt;&#xA;&lt;p&gt;It is natural for me to write the first letter of his/ her name in English in capital letters when referring to that object/ concept. For instance: Gaussian curvature, Newtonian mechanics, Riemannian geometry, Archimedean valuation, Eulerian graph, Platonic solids, Jacobian matrix, Noetherian rings, Artinian rings, Hamiltonian path, Hermitian matrix, Hessian matrix, ...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But why do we usually write &lt;code&gt;abelian&lt;/code&gt; group instead of &lt;code&gt;Abelian&lt;/code&gt; group? Or why do we usually write &lt;code&gt;abelian&lt;/code&gt; variety instead of &lt;code&gt;Abelian&lt;/code&gt; variety?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My strong suspicion is that perhaps the importance of abelian groups has reached us through mathematicians in languages other than English. But since I do not have a background in mathematical history, it is very likely that my guess is wrong.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is this related to the French school of mathematics? (I know that &amp;quot;variety&amp;quot; is the French equivalent of &amp;quot;Manifold&amp;quot;.) If yes, then why do we write Galois extensions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Are there any exceptions other than abelian groups?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, I do not know what is the suitable tag for my question.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My answer:&lt;/p&gt;&#xA;&lt;p&gt;I thought that perhaps the word abelian had entered the &amp;quot;ordinary human language&amp;quot;, and had been &lt;strong&gt;accepted as an adjective&lt;/strong&gt;. (Why did I use the term &amp;quot;&lt;strong&gt;accepted&lt;/strong&gt;&amp;quot;? Because &lt;strong&gt;without losing accuracy&lt;/strong&gt;, we can &lt;strong&gt;easily&lt;/strong&gt; translate this word into &amp;quot;ordinary human language&amp;quot;. In some sense, it has &lt;strong&gt;replaced&lt;/strong&gt; the word commutative in some areas of the mathematical literature.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;That's why I checked &lt;a href=&quot;https://www.merriam-webster.com/dictionary/abelian&quot; rel=&quot;nofollow noreferrer&quot;&gt;ablian's definition&lt;/a&gt; from Merriam-Webster's dictionary, and this hypothesis was reinforced for me. (It states that: &amp;quot;adjective, often capitalized&amp;quot;) Also, the status of the word &lt;a href=&quot;https://www.merriam-webster.com/dictionary/euclidian&quot; rel=&quot;nofollow noreferrer&quot;&gt;euclidean&lt;/a&gt; is the same as the status of the word abelian. (It is mentioned again that: adjective, often capitalized) Also, see &lt;a href=&quot;https://www.merriam-webster.com/dictionary/platonic&quot; rel=&quot;nofollow noreferrer&quot;&gt;platonic&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When I came here to add my observations to my questions, I saw &lt;a href=&quot;https://hsm.stackexchange.com/questions/12664/why-do-we-write-abelian-group-instead-of-abelian-group?noredirect=1#comment25190_12664&quot;&gt;Professor K. Conrad's comment&lt;/a&gt;, in which he introduced an &lt;a href=&quot;https://mathoverflow.net/questions/44946/why-is-abelian-infrequently-capitalized&quot;&gt;MO post (Why is “abelian” infrequently capitalized?)&lt;/a&gt;, which is exactly the subject. I saw the answers there, and the discussions there were very convincing to me.&lt;/p&gt;&#xA;" OwnerUserId="6407" LastEditorUserId="6407" LastEditDate="2020-12-28T10:19:58.553" LastActivityDate="2020-12-28T10:19:58.553" Title="why do we write `abelian` group instead of `Abelian` group?" Tags="|mathematics|language|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12667" PostTypeId="1" CreationDate="2020-12-28T08:18:06.107" Score="10" ViewCount="3243" Body="&lt;p&gt;Why are letters &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; used for axes names in mechanics while letters &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; are used in mathematics?&lt;/p&gt;&#xA;&lt;p&gt;Why these dimensions weren't called A, B and C or F, G and H?&lt;/p&gt;&#xA;" OwnerUserId="13549" LastEditorUserId="12316" LastEditDate="2020-12-31T10:51:07.547" LastActivityDate="2020-12-31T10:51:07.547" Title="Why do I , J and K in mechanics represent X , Y and Z in maths?" Tags="|notation|classical-mechanics|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12668" PostTypeId="2" ParentId="12667" CreationDate="2020-12-28T19:10:53.103" Score="17" Body="&lt;p&gt;This usage of &lt;span class=&quot;math-container&quot;&gt;$\mathbf i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf j$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf k$&lt;/span&gt; is not specific to physics. It is also used in mathematics, specifically when teaching linear algebra or multivariable calculus in &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^3$&lt;/span&gt; as well as group theory for the quaternion group &lt;span class=&quot;math-container&quot;&gt;$Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The notation came from the use of the letters &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; in quaternions, which were created by Hamilton (a physicist). They are a basis of the 3-dimensional space of &amp;quot;pure quaternions&amp;quot; or &amp;quot;imaginary quaternions&amp;quot;  &lt;span class=&quot;math-container&quot;&gt;$\{bi + cj + dk : b, c, d, \in \mathbf R\}$&lt;/span&gt;, and conjugation on that subspace by all nonzero  quaternions (or just by the unit sphere &lt;span class=&quot;math-container&quot;&gt;$S^3$&lt;/span&gt;) provides a quaternionic description of proper rotations in 3-dimensional space.  This made the space of pure quaternions a natural model of physical 3-dimensional space on which you could not only add elements but also describe their rotations in a novel (at the time) way through conjugation by a 4-dimensional quaternion.&lt;/p&gt;&#xA;&lt;p&gt;Hamilton called the pure quaternion part of a quaternion &lt;span class=&quot;math-container&quot;&gt;$a+bi+cj+dk$&lt;/span&gt; a “vector” and the real part of a quaternion a &amp;quot;scalar&amp;quot;. This was in the 1840s, before linear algebra was well-developed, and this usage by Hamilton is how we got the labels &amp;quot;vector&amp;quot; and “scalar” in linear algebra. Search for vector on &lt;a href=&quot;https://jeff560.tripod.com/v.html&quot; rel=&quot;noreferrer&quot;&gt;this page&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The first group theory paper that mentions the quaternion group is Cayley's &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786445908642716&quot; rel=&quot;noreferrer&quot;&gt;1859 paper&lt;/a&gt;, where he writes the elements as&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;1, \vartheta, \alpha, \beta, \gamma, \vartheta\alpha, \vartheta\beta, \vartheta\gamma.&#xA;$$&lt;/span&gt;&#xA;He says it looks simpler if he writes &lt;span class=&quot;math-container&quot;&gt;$\vartheta$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; and that  &lt;span class=&quot;math-container&quot;&gt;$\alpha, \beta$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; &amp;quot;combine according to the laws of the quaternion symbols &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In Dedekind's &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002256258&amp;amp;physid=phys565#navi&quot; rel=&quot;noreferrer&quot;&gt;1897 paper&lt;/a&gt; on nonabelian finite groups in which every subgroup is normal (&lt;span class=&quot;math-container&quot;&gt;$Q_8$&lt;/span&gt; is the first example), Dedekind did not use the &lt;span class=&quot;math-container&quot;&gt;$i, j, k$&lt;/span&gt; notation. He called the group &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$Q_8$&lt;/span&gt;, and wrote &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, and he wrote &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$Q = \{1, \varepsilon, \alpha, \varepsilon\alpha, \beta, \varepsilon\beta, \gamma, \varepsilon\gamma\}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2020-12-28T22:09:34.863" LastActivityDate="2020-12-28T22:09:34.863" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12669" PostTypeId="2" ParentId="12667" CreationDate="2020-12-29T10:34:09.180" Score="1" Body="&lt;p&gt;A reference to deep pre-history and quaternions is interesting but it can be very misleading to imply the present tense. Quaternions are &lt;strong&gt;not&lt;/strong&gt; used in anything remotely contemporary - math or physics. They have some residual use in game development but much more in pseudo-theoretic quibbles (to claim that Cartesian coordinates have intrinsic problems - problems are lazy Devs that change coordinates partially instead of calculating at least good, if not optimal paths for camera motions).&lt;/p&gt;&#xA;&lt;p&gt;Historically speaking, Hamilton's quaternions stifled development of physics for about 50 years, until Gibbs introduced normal vector spaces and accompanying notation and then Heaviside used it to rewrite Maxwell's electrodynamics equations in a normal, vector form (Maxwell was a staunch quaternion supporter, that's how influential Hamilton was). Maxwell is so highly regarded as superior mathematician among physicists that to this day there are claims that he's the one who was &amp;quot;meant&amp;quot; to write Relativity and even Quantum Field Theory (he essentially invented the concept of fields) if he didn't die so early (at 48). One thing that gets easily forgotten is that he was completely stuck with quaternions.&lt;/p&gt;&#xA;&lt;p&gt;In today's books &lt;span class=&quot;math-container&quot;&gt;$\mathbf i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf j$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf k$&lt;/span&gt; by and large denote just unit vectors &lt;span class=&quot;math-container&quot;&gt;$\hat{\mathbf{x}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hat{\mathbf{y}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\hat{\mathbf{z}}$&lt;/span&gt; or in a more general form &lt;span class=&quot;math-container&quot;&gt;$e_\mathbf{x}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e_\mathbf{y}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e_\mathbf{z}$&lt;/span&gt; and then  &lt;span class=&quot;math-container&quot;&gt;$e_\mathbf{1}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e_\mathbf{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e_\mathbf{3}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$e^\mathbf{1}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e^\mathbf{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e^\mathbf{3}$&lt;/span&gt;  to make it clear very early that there's more to come - N coordinates, no particular names, N getting big, going to infinity when it's convenient ... To that extent that might have been the reason why someone used i,j,k as well - to nudge the reader to detach conceptually from the all too familiar symbols. In &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^3$&lt;/span&gt; you'll have to go to &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{\hat{\mathbf{r}}}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{\hat{\mathbf{\phi}}}$&lt;/span&gt;,  &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{\hat{\mathbf{\theta}}}$&lt;/span&gt;, or  &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{\hat{\mathbf{\rho}}}$&lt;/span&gt;, pretty much on the next page and latter you may end up with something like &lt;span class=&quot;math-container&quot;&gt;$\mathbf{q_\mathbf{i}}$&lt;/span&gt;, and then &lt;span class=&quot;math-container&quot;&gt;$q_{i}$&lt;/span&gt; where it's assumed that unit vectors follow the names which just denote n-th coordinate.&lt;/p&gt;&#xA;&lt;p&gt;In game development you are usually in &lt;span class=&quot;math-container&quot;&gt;$\mathbf R^4$&lt;/span&gt; and the 4-th is usually denoted by &lt;span class=&quot;math-container&quot;&gt;$\mathbf{w}$&lt;/span&gt; thereby &lt;span class=&quot;math-container&quot;&gt;$\hat{\mathbf{w}}$&lt;/span&gt; - for no particular reason - just to have a different symbol for a coordinate that doesn't automatically mean that you have a Minkowski space of any form (the one used in Special Relativity has a very particular metric). Minkowski himself had a variant in which the 4-th coordinate was simply named &lt;span class=&quot;math-container&quot;&gt;$\mathbf t$&lt;/span&gt;. So the general lesson from history is - don't get attached to the names of coordinates.&lt;/p&gt;&#xA;" OwnerUserId="13553" LastActivityDate="2020-12-29T10:34:09.180" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12672" PostTypeId="1" CreationDate="2020-12-30T22:31:25.073" Score="4" ViewCount="556" Body="&lt;p&gt;1860 Manual of Algebra &lt;a href=&quot;https://books.google.com/books?id=hGlaAAAAcAAJ&amp;amp;pg=PA89&quot; rel=&quot;nofollow noreferrer&quot;&gt;describes&lt;/a&gt; a method which is now taught in upper secondary schools worldwide:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To rationalize the denominators of fractions which consist of binomial quadratic surds, use the following&#xA;RULE:&#xA;Multiply the numerator and denominator by a binomial surd, conjugate in form to that which appears in the denominator.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I wasn't able to find an attestation of the term &lt;em&gt;conjugate&lt;/em&gt; in this context earlier (there are a lot of &lt;a href=&quot;https://en.wikipedia.org/wiki/Conjugate_diameters&quot; rel=&quot;nofollow noreferrer&quot;&gt;conjugate diameters&lt;/a&gt; noising the search results though, and I could miss something), but plenty of people seem to be using multiplication by conjugate &lt;a href=&quot;https://en.wiktionary.org/wiki/ante_litteram&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;ante litteram&lt;/em&gt;&lt;/a&gt;, cf. these &lt;a href=&quot;https://books.google.com/books?id=lL02AAAAMAAJ&amp;amp;pg=PA72&quot; rel=&quot;nofollow noreferrer&quot;&gt;1813&lt;/a&gt; and even &lt;a href=&quot;https://books.google.com/books?id=B29bAAAAQAAJ&amp;amp;pg=PA111&quot; rel=&quot;nofollow noreferrer&quot;&gt;1702&lt;/a&gt; examples (the notation in the latter one is different from the modern one, but it seems that the technique is the same).&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately prior to the turn of the 18th entury the math literature was written and published mainly in Latin, which I don't know. However, a single &lt;a href=&quot;https://books.google.com/books?id=mNsrHTlVHXIC&amp;amp;pg=PA243&quot; rel=&quot;nofollow noreferrer&quot;&gt;1673 source&lt;/a&gt; on a related topic which doesn't appear to explicitly state this rule refers to 10th Book of Euclid's &lt;em&gt;Elements&lt;/em&gt;, is the method actually as early as that? I tried to read Euclid as well as &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/031508609290029B&quot; rel=&quot;nofollow noreferrer&quot;&gt;two&lt;/a&gt; &lt;a href=&quot;https://paperity.org/p/84954535/book-x-of-the-elements-ordering-irrationals&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern&lt;/a&gt; retellings, but didn't find a formulation of this technique: the classic seems to describe related properties of irrational expressions rather than instructing how to solve problems here.&lt;/p&gt;&#xA;" OwnerUserId="13566" LastEditorUserId="55" LastEditDate="2020-12-31T08:25:46.977" LastActivityDate="2020-12-31T08:25:46.977" Title="Who invented multiplying by the conjugate to rationalize denominators and when?" Tags="|mathematics|elementary-algebra|algebra|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12676" PostTypeId="2" ParentId="12672" CreationDate="2020-12-31T08:04:29.577" Score="6" Body="&lt;p&gt;As with many &amp;quot;who was first&amp;quot; questions there is no straightforward answer. This is what May described as priority chasing coming to grief in &lt;a href=&quot;https://core.ac.uk/download/pdf/82610716.pdf&quot; rel=&quot;noreferrer&quot;&gt;Historiographic vices&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The hope of finding a 'first' comes to grief because of the historically dynamic character of ideas... If we describe a result with sufficient vagueness, there seems to be an endless sequence of those who had something within the vague specifications&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Book X of Elements is a geometric theory attributed to &lt;a href=&quot;https://en.wikipedia.org/wiki/Theaetetus_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Theaetetus (c. 400 BC)&lt;/a&gt; of what we now express as quadratic irrationals. With some creativity one can read conjugates into it, specifically into &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookX/propX112.html&quot; rel=&quot;noreferrer&quot;&gt;X.112&lt;/a&gt;, but it requires a lot of modern rewriting as one can see by looking at the original. For the rewriting see e.g. &lt;a href=&quot;https://www.google.com/books/edition/A_History_of_Mathematics/V6RUDwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=counterparts&quot; rel=&quot;noreferrer&quot;&gt;History of Math by Merzbach and Boyer, p. 105&lt;/a&gt; and &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookX/propX112.html&quot; rel=&quot;noreferrer&quot;&gt;Heath's commentary on X.112, p.246&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;X.112 is the equivalent of rationalising the denominators of the fractions &lt;span class=&quot;math-container&quot;&gt;$\frac{c^2}{\sqrt{A}+\sqrt{B}}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{c^2}{a+\sqrt{B}}$&lt;/span&gt;&#xA;by multiplying numerator and denominator by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{A}-\sqrt{B}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a-\sqrt{B}$&lt;/span&gt; respectively. Euclid proves that &lt;span class=&quot;math-container&quot;&gt;$\frac{\sigma^2}{\rho+\sqrt{k}\rho}=\lambda\rho-\sqrt{k}\,\lambda\rho$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$k &amp;lt; 1$&lt;/span&gt;), and his method enables us to see that &lt;span class=&quot;math-container&quot;&gt;$\lambda=\sigma^2/(\rho^2-k\rho^2)$&lt;/span&gt;.&#xA;The proof is a remarkable instance of the dexterity of the Greeks in using geometry as the equivalent of our algebra. Like so many proofs in Archimedes and Apollonius, it leaves us completely in the dark as to how it was evolved. That the Greeks must have had some analytical method which suggested the steps of such proofs seems certain; but what it was must remain apparently an insoluble mystery.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Heath proceeds to rewrite the proof of X.112 in modern algebraic notation. What &amp;quot;seemed certain&amp;quot; to him is no longer believed by historians, but the use of conjugates did emerge in the algebraization of book X by Islamic commentators, e.g. Berggren credits &lt;a href=&quot;https://en.wikipedia.org/wiki/Abu_Kamil&quot; rel=&quot;noreferrer&quot;&gt;Abu-Kamil (c. 900 AD)&lt;/a&gt; with it in&#xA;&lt;a href=&quot;https://www.springer.com/gp/book/9781461246084#&quot; rel=&quot;noreferrer&quot;&gt;Episodes in the Mathematics of Medieval Islam, p.133&lt;/a&gt;. But even his rationalizing is verbal, modern algebraic notation is a much later development.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2020-12-31T08:23:05.657" LastActivityDate="2020-12-31T08:23:05.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12678" PostTypeId="1" CreationDate="2021-01-01T02:49:26.973" Score="2" ViewCount="208" Body="&lt;p&gt;When making notes recently I felt like using a symbol for &amp;quot;a large amount of&amp;quot;, and it occurred to me that surely others before me must have experienced the need for such a symbol.&lt;/p&gt;&#xA;&lt;p&gt;What symbols have been used for &amp;quot;many&amp;quot; (or &amp;quot;a large amount of&amp;quot;) or &amp;quot;a few&amp;quot; (or &amp;quot;a small amount of&amp;quot;) in the history of mathematics or in other fields of theoretical endeavour, or practical work or experimentation, such as medicine, chemistry, alchemy, or physics?&lt;/p&gt;&#xA;&lt;p&gt;Edit&lt;/p&gt;&#xA;&lt;p&gt;The concept of &amp;quot;many&amp;quot; or &amp;quot;a large quantity&amp;quot; is surely an anthropological invariant. The apparent absence of well-known symbols for it is intriguing. I would have expected there to be symbols for it in alchemy and medicine, denoting a quantity of one substance that's easily sufficient to flood out another. And is there no symbol for a &amp;quot;trace amount&amp;quot;?&lt;/p&gt;&#xA;" OwnerDisplayName="user13571" LastEditorDisplayName="user13571" LastEditDate="2021-05-18T14:45:28.983" LastActivityDate="2021-05-18T14:45:28.983" Title="What symbols have been used for &quot;many&quot; (or &quot;a large amount of&quot;) or &quot;a few&quot; (or &quot;a small amount of&quot;) in the history of mathematics or other fields?" Tags="|notation|anthropology|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12681" PostTypeId="1" CreationDate="2021-01-01T20:06:12.037" Score="1" ViewCount="107" Body="&lt;p&gt;My understanding is that before the epsilon-delta definition of a limit, the rigor and soundness of the definition of a limit was not good enough.&lt;/p&gt;&#xA;&lt;p&gt;So, how did the definitions of a limit vary before the epsilon-delta definition?&lt;/p&gt;&#xA;&lt;p&gt;I've read the &lt;a href=&quot;https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page&lt;/a&gt; for the definition, so I understand past mathematicians couldn't define an infinitesimal. But, I was just hoping for further explanation and elaboration on what they did and didn't know compared to now. Was there no use of formal statements, quantifiers, conditionals back then, like seen in the epsilon-delta definition?&lt;/p&gt;&#xA;&lt;p&gt;I was kind of hoping to see some notation and statements, equations, etc.&lt;/p&gt;&#xA;" OwnerUserId="13555" LastEditorUserId="10106" LastEditDate="2021-01-10T03:56:03.467" LastActivityDate="2021-01-10T03:56:03.467" Title="How did definitions of a limit vary before the epsilon-delta definition?" Tags="|mathematics|calculus|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12683" PostTypeId="1" CreationDate="2021-01-02T14:04:17.473" Score="1" ViewCount="54" Body="&lt;p&gt;As I searched history of nomenclature of charges I got this..,&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Ben Franklin imagined electricity as a type of invisible fluid that could build up or be absent from a material, or at least certain materials. He believed that when this invisible fluid built up the object was positively charged. When there was an absence of this fluid he called that material negatively charged.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Benjamin Franklin thought of electricity as a fluid, and said when there is abundance of fluid it is positive, stating positively charged fluid flows in electricity.&lt;/p&gt;&#xA;&lt;p&gt;Further, Thomson found that negatively charged particles flow and not positive ones, as the cathode rays exhibited characteristics of negatively charged particles. But how we first clarified what is what?&lt;/p&gt;&#xA;&lt;p&gt;How we distinguished between two types of charges and established their properties? Positive charges repel positive charges and attract negative ones. But from there we can't say whether the particle we are trying to understand is positive or negative. So how Thomson said electricity is flow of negative charges?&lt;/p&gt;&#xA;" OwnerUserId="13219" LastEditorUserId="26" LastEditDate="2021-02-20T15:03:19.417" LastActivityDate="2021-02-20T15:03:19.417" Title="How we distinguished charges and named them?" Tags="|electricity|naming-conventions|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12684" PostTypeId="1" AcceptedAnswerId="12689" CreationDate="2021-01-02T14:47:48.423" Score="32" ViewCount="14490" Body="&lt;p&gt;Constructing houses, telescopes, and most other important projects requires shaping pieces to precise size, at perfect right angles, or to have flat surfaces.  People today have all kinds of ways of accomplishing similar precision, but only because they have tools which were themselves made with impressive precision.  Levels and rulers come from the factory, and if you tried to make your own perfectly straight or right-angled object from scratch, I couldn't imagine doing it.  So how did people before industrial technology accomplish this?&lt;/p&gt;&#xA;" OwnerUserId="2732" LastActivityDate="2021-08-19T21:40:08.710" Title="How did people make things perfectly straight?" Tags="|technology|" AnswerCount="10" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="12685" PostTypeId="2" ParentId="12684" CreationDate="2021-01-02T15:18:00.010" Score="32" Body="&lt;p&gt;Indeed, it is much easier to manufacture an arc of circle than a straight ruler.&lt;/p&gt;&#xA;&lt;p&gt;When you rub two rulers against each other you obtain an arc of a circle (or a straight line) as their common edge. This is because the only curves of constant curvature are straight lines and circles. So nothing else will slide smoothly, without gaps.&lt;/p&gt;&#xA;&lt;p&gt;But there is a simple way to obtain a straight line by a similar process.&#xA;You need 3 rulers. When you rub rulers A and B against each other, their common edge&#xA;becomes a circle (or a line). Suppose that the edge of A becomes convex and the edge of B concave. Then you rub A and B against ruler C, and repeat this process of rubbing each pair of the triple, until all three edges will slide against&#xA;each other smoothly. The only possibility is that all three edges are straight lines.&lt;/p&gt;&#xA;&lt;p&gt;The process is described in the book M. Berger, Geometry revealed (Springer 2009), on p. 270. He claims that perfect rulers are made by this process to this day.&lt;/p&gt;&#xA;&lt;p&gt;Similar process can be used to make flat 2-dimensional surfaces. It is used to make spherical lenses and mirrors. If two pieces of glass&#xA;slide nicely against each other in all directions, then they must be&#xA;spherical. By using three pieces you can make a flat surface. There is no such simple method to make a parabolic mirror, for example.&lt;/p&gt;&#xA;&lt;p&gt;Another pure mathematical method is based on the use of &lt;a href=&quot;https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paucellier-Lipkin inversor&lt;/a&gt; (this is a hinge mechanism, of which one of the points moves along a straight line). For long time people thought that such a mechanism is impossible, until it was invented by&#xA;Paucellier&#xA;in 1864. (The legend says that Chebychev wept when he was shown a working model:-) Lipkin was Chebyshev's student who proved that the Paucellier's mechanism has the required property. Ref.&#xA;Philip Davis, The Thread (Birkhauser, 1983). But I've never heard of any practical use of the Paucelier-Lipkin inversor.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-15T13:55:48.793" LastActivityDate="2021-08-15T13:55:48.793" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12686" PostTypeId="2" ParentId="12684" CreationDate="2021-01-02T15:30:03.200" Score="12" Body="&lt;p&gt;To construct a right angle, you take three cords, one of which is three of whatever unit you use, the second four of it, and the third five.  Put them together in a triangle and the angle between the three and the four ones will be a right angle.  This was used for a long time in ancient Egypt and Babylon.  (Alternatively, you can use some other of the Pythagorean triples.  There are many ancient lists of them.)&lt;/p&gt;&#xA;&lt;p&gt;Straight lines can likewise be measured with nothing more than a weight and a cord:  the plumb bob, which was certainly in use in ancient Egypt.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2021-01-02T15:30:03.200" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12687" PostTypeId="1" CreationDate="2021-01-02T19:44:33.390" Score="3" ViewCount="73" Body="&lt;p&gt;Ibn al-Haytham is considered one of the first experimenters. Since he was also a kind of philosopher did he write anything close in meaning and generality to the scientific method?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastActivityDate="2021-01-02T19:44:33.390" Title="Did Ibn al-Haytham write anything about his experiments that is general enough to be seen as prefiguring the scientific method?" Tags="|biographical-details|discoveries|philosophy-of-science|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12689" PostTypeId="2" ParentId="12684" CreationDate="2021-01-03T02:51:46.910" Score="42" Body="&lt;p&gt;The easiest way to check if something is &amp;quot;straight&amp;quot; is to just look. Humans can pretty much tell if something is not straight, to a certain point. Another easy way is to use a piece of string. If you pull their ends apart, it will provide you a &amp;quot;straight&amp;quot; reference which you can use to compare to other things. That way you can, for example, draw a line beside it on paper, or make a mold to build a straight object, or hammer down imperfections.&lt;/p&gt;&#xA;&lt;p&gt;Talking about strings, you can make a square angle (on paper) with a string acting as compass. Use one end as the center of the circle and a pen in the other end.&lt;/p&gt;&#xA;&lt;p&gt;See this image below. Already having a straight line (black line) as starting point, pick any two points (red dots) in the line. Then draw two circles (with compass, or string, or whatever you have like that) with a radius that makes them overlap (you can even use different radii!). Take the points where the circles touch (blue dots) and draw a line through them (blue line). There you have: a line that is in right angle to the original line. If you can draw that on paper, you can also carve the angle in a piece of wood (for example), comparing it to the drawing.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tgrlg.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tgrlg.png&quot; alt=&quot;Draw a right angle&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Always remember that there is not a single real &lt;em&gt;perfect&lt;/em&gt; object. Perfection is reserved for abstractions. Real objects respect &lt;em&gt;tolerances&lt;/em&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Metrology&quot; rel=&quot;noreferrer&quot;&gt;Metrology&lt;/a&gt;, the science of measuring, makes it very clear that you simply &lt;em&gt;can't&lt;/em&gt; know how perfect something is because &amp;quot;measuring&amp;quot; is always comparing to something that also has some tolerance. You can only try to reduce imperfections until you get to your technological limits or to your requirements. Luckily technology evolves through careful iteration and breakthroughs (only up to a certain point too!).&lt;/p&gt;&#xA;&lt;p&gt;By the way, the latest version of the &lt;a href=&quot;https://www.bipm.org/en/measurement-units/&quot; rel=&quot;noreferrer&quot;&gt;International System of Units (SI)&lt;/a&gt; tries very hard to make measurements absolute at least in conceptual terms, linking units to absolute values in physics. However, keep in mind that the instruments that will measure those values, and then be reference to other, simpler, instruments, will also have tolerances. Remember that it isn't just fabrication that affects an instrument, but ambient temperature, pressure, humidity, operator skills and all sorts of other factors.&lt;/p&gt;&#xA;" OwnerUserId="13577" LastActivityDate="2021-01-03T02:51:46.910" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12690" PostTypeId="2" ParentId="12684" CreationDate="2021-01-03T08:02:32.767" Score="6" Body="&lt;p&gt;There are quite a few methods. If you take three almost straight objects and grind pairs against each other, they will approach a straight edge. That is, given objects A, B, and C, you grind A and B against each other, then B and C, then C and A, etc. You need a material that is soft enough that it grinds away reasonably quickly, but is strong enough that it doesn't fall apart.&lt;/p&gt;&#xA;&lt;p&gt;You can also use string to get a straight line, either by stretching it taut, or using it to suspend a weight. Given its high density, lead was often for the weight, and the Latin word for lead is &amp;quot;plumbum&amp;quot;, giving us the term &amp;quot;plumb line&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;For short distances, a crystal that cleaves along straight lines can be used.&lt;/p&gt;&#xA;" OwnerUserId="9468" LastActivityDate="2021-01-03T08:02:32.767" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12691" PostTypeId="1" AcceptedAnswerId="12693" CreationDate="2021-01-03T08:48:28.560" Score="4" ViewCount="637" Body="&lt;p&gt;I was told it was Joseph von Fraunhofer who made the first observation that shining sun rays through a slit created faint rainbow fringes to the sides of the outgoing ray. Is there any evidence of this historical claim? Is there anyone earlier who made similar observations? Also, what were the follow-up investigations after Fraunhofer regarding the single-slit?&lt;/p&gt;&#xA;" OwnerUserId="6965" LastEditorUserId="6965" LastEditDate="2021-01-03T11:25:16.430" LastActivityDate="2021-01-05T20:43:11.233" Title="When was single-slit diffraction first observed?" Tags="|optics|light|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12692" PostTypeId="2" ParentId="12684" CreationDate="2021-01-03T16:28:35.097" Score="10" Body="&lt;p&gt;From Quine's &lt;em&gt;Quiddities&lt;/em&gt;, s.v. &amp;quot;Lines&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My own heart leaps up a little when I behold what radically dissimilar criteria all attest to one and the same trite trait, the straightness of a line. I think of four.&lt;/p&gt;&#xA;&lt;p&gt;One way of testing straightness is by use of a taut string. This test recalls indeed the origin of our word &lt;em&gt;line&lt;/em&gt;, Latin &lt;em&gt;linea&lt;/em&gt;; it is related to &lt;em&gt;linen&lt;/em&gt; and &lt;em&gt;lint&lt;/em&gt;. Sailors use the word in its primary sense in calling their ropes lines. The same test is implicit in our word &lt;em&gt;straight&lt;/em&gt;, which is idientical historically with &lt;em&gt;stretched&lt;/em&gt;. To the Greeks, more effete, a line was &lt;em&gt;grammē&lt;/em&gt;, something drawn. Still the taut string is what embodies Euclid's characterization of the line as the shortest path between two points.&lt;/p&gt;&#xA;&lt;p&gt;A strikingly different and independent way of checking the straightness of a line is by sighting along it.  If one speck blocks your view of another, your eye and the two specks are all in line.…&lt;/p&gt;&#xA;&lt;p&gt;A third embodiment of straightness is the crease of a folded sheet of stiff cardboard. Here we have a manifestation of Euclid's proposition that planes intersect in a straight line.…&lt;/p&gt;&#xA;&lt;p&gt;A fourth criterion of straightness is afforded by a sliding edge. Thus suppose we are testing the straightness of of a long mark on the floor. We pass a card along the line. If the mark and the edge of the card are both in fact straight, then we can preserve full contact of the edge of the card with the mark throughout the length of the mark. This test is not yet conclusive; the mark and the edge might be curved, rather, with an equal and constant curvature. But if we repeat the test passing the card along the other side of the mark, then the test is conclusive.… It would take quite a substantial chapter of geometry to show why the taut string, the crease, and the sliding edge all attest to the same simple quality. The remaining criterion, the line of sight, we leave in the lap of the physicist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This doesn't directly answer your question of how people &lt;em&gt;did&lt;/em&gt; measure straightness, but it gives you some ideas of how they &lt;em&gt;could have&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="214" LastActivityDate="2021-01-03T16:28:35.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12693" PostTypeId="2" ParentId="12691" CreationDate="2021-01-04T12:37:13.773" Score="6" Body="&lt;p&gt;Fraunhofer is credited not with discovering the single slit diffraction, but with explaining the far-field diffraction pattern from a single slit. Fresnel explained the near-field pattern before him, in 1815, but he was not the discoverer either.&lt;/p&gt;&#xA;&lt;p&gt;The discovery of diffraction and the introduction of the term are due to Grimaldi, specifically his &lt;em&gt;Physico-Mathesis de Lumine&lt;/em&gt; posthumously published in Bologna in 1665. In the more commonly cited experiment he placed a sharp knife into the cone of light passed into a dark room through a small hole, and observed knife's anomalously wide shadow along with the colored bands which he called &lt;em&gt;seriae lucidae&lt;/em&gt;. But in a second experiment he put the cone through a second small hole, and observed what is now called &amp;quot;single slit&amp;quot; diffraction, see &lt;a href=&quot;https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=80496&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crosswell, Diffraction: the first recorded observation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Grimaldi's book was little known, but his experiments were popularized in Fabri's &lt;em&gt;Dialogi Physici&lt;/em&gt; (1669), which found its way into Newton's hands in 1672, when Collins sent him a copy, see &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rsnr.1990.0002&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hall's Beyond the Fringe: Diffraction as seen by Grimaldi, Fabri, Hooke and Newton&lt;/a&gt;. Newton read the book, but did not pay much attention to it even after Hooke presented to the Royal Society in 1675 his own observation of  &amp;quot;&lt;em&gt;a Penumbra or darker Ring encompassing the lighter Circle&lt;/em&gt;&amp;quot; obtained by passing sunlight through a small hole, and noted that &amp;quot;&lt;em&gt;‘it could not proceed from a Penumbra caused by the bigness of the Hole upon the common Principles, that is, from the Supposition of the Rays from every point of the Sun proceeding in strait Lines&lt;/em&gt;&amp;quot;. Hooke did not mention anything about colors.&lt;/p&gt;&#xA;&lt;p&gt;Some time between then and the publication of Opticks (1704) Newton did perform detailed experiments extending those of Grimaldi (c. 1692 in Hall's estimation). He renamed diffraction into &amp;quot;inflexion&amp;quot; (Fabri did not use Grimaldi's term, its return is due to Fresnel), &lt;em&gt;seriae lucidae&lt;/em&gt; into &amp;quot;fringes&amp;quot;, and came up with a theory of light repulsion by matter to reconcile the effect with his corpuscular theory. He did not seem to know about Grimaldi's second experiment, but one observation concerning the knife's shadow led him  to a &amp;quot;single slit&amp;quot; variant anyway. And he did observe &amp;quot;&lt;em&gt;three Parallel Fringes or&#xA;Bands of Colour’d Light&lt;/em&gt;&amp;quot;, for which he did not really have a good explanation. &lt;a href=&quot;https://www.jstor.org/stable/229974&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stuewer in Critical Analysis of Newton's Work on Diffraction&lt;/a&gt; writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This keen observation (which was understood only after the work of Young and Fresnel) led Newton to further experiments which we would now classify as &amp;quot;single slit diffraction&amp;quot; experiments. Thus he placed two knives with their edges opposite each other and gradually decreased the space between them. When they were quite close together Newton observed a bright line in the center of the pattern with three colored fringes on either side of it. At smaller knife separations, these fringes began to spread out... Closing the gap still further, Newton found that beginning at separations somewhat greater than 1/400 inch the fringes began to disappear, first the two outer fringes, then the two middle fringes, and finally the two inner fringes.&lt;/em&gt; &lt;br&gt;&#xA;[...] &lt;em&gt;Thus Newton - in a beautifully detailed description - clearly reported observations on light bending into the geometric shadow. His explanation for his observations rested on his belief that each of the two knives repelled the rays as they passed by, thereby producing two sets of three exterior fringes&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-05T20:43:11.233" LastActivityDate="2021-01-05T20:43:11.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12694" PostTypeId="2" ParentId="12684" CreationDate="2021-01-04T19:34:42.590" Score="1" Body="&lt;p&gt;Mostly, gravity. To get things perfectly flat people would use water. a tube with 2 graduated cylinders on either end. Raise/lower one end until the water is at the same level in both cylinders and you have your level. In terms of getting things perfectly vertical, they would use (and we still use this today), a piece of string with a weight on the end. Once the weight has stopped moving you know that the point under the string is directly below the top point of the string. Things haven't really changed that much.&lt;/p&gt;&#xA;" OwnerUserId="13604" LastActivityDate="2021-01-04T19:34:42.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12695" PostTypeId="1" CreationDate="2021-01-04T19:40:38.087" Score="1" ViewCount="67" Body="&lt;p&gt;People living in certain regions throughout history produced little to none recorded intellectual achievements.&#xA;Assuming that highly creative individuals existed in these regions, are there any historical accounts of their lives?&lt;/p&gt;&#xA;&lt;p&gt;If we have such accounts are there any similarities in their lives?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2021-01-13T21:39:12.817" LastActivityDate="2021-01-13T21:39:12.817" Title="What were the likely lives of the highly creative or intellectually gifted people in declining or primitive civilizations?" Tags="|biographical-details|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12696" PostTypeId="1" CreationDate="2021-01-04T20:45:41.197" Score="5" ViewCount="138" Body="&lt;p&gt;I can write out the following CNF in various different ways:&lt;/p&gt;&#xA;&lt;p&gt;In mathematical textbook notation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$(A \land B \land C) \lor (\lnot A \land B \land \lnot C) $&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In C-like programming notation:&#xA;&lt;code&gt;(A &amp;amp;&amp;amp; B &amp;amp;&amp;amp; C) || (!A &amp;amp;&amp;amp; B &amp;amp;&amp;amp; !C)&lt;/code&gt;&lt;/p&gt;&#xA;&lt;p&gt;In engineering logic notation: &lt;span class=&quot;math-container&quot;&gt;$(ABC) + (\overline{A}B\overline{C})$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, I guess my question is in the title: why do we have so many different systems of boolean logic notation when boolean logic as a field is quite a recent invention?&lt;/p&gt;&#xA;" OwnerUserId="10448" LastEditorUserId="609" LastEditDate="2021-01-05T12:58:11.320" LastActivityDate="2021-01-05T13:02:16.830" Title="Why are there so many different, and widely accepted, notational systems for boolean logic?" Tags="|notation|mathematical-logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12699" PostTypeId="1" AcceptedAnswerId="12703" CreationDate="2021-01-04T23:14:03.190" Score="2" ViewCount="156" Body="&lt;p&gt;I saw a problem asking to use Leonardo's table of chords to find the arc length cut off by a chord with length 8 rods, 3 ft, and 16 2/7 unciae in a circle of diameter 10.&lt;/p&gt;&#xA;&lt;p&gt;I do not understand at all how to read Leonardo's table. Can anyone help explain?&lt;/p&gt;&#xA;&lt;p&gt;This is a picture of Leonardo's table of chords: &lt;a href=&quot;https://i.stack.imgur.com/2h4gD.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2h4gD.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13427" LastEditorUserId="13427" LastEditDate="2021-01-05T02:11:52.300" LastActivityDate="2021-01-05T20:15:10.460" Title="Using Leonardo of Pisa's table of Chords" Tags="|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12701" PostTypeId="2" ParentId="12696" CreationDate="2021-01-05T13:02:16.830" Score="5" Body="&lt;p&gt;The software part is easy:  it uses standard ASCII characters so the source code can be universally applied.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Engineering Logic&amp;quot; has its own variations,but notice that &amp;quot;addition&amp;quot; is similar to &amp;quot;OR&amp;quot; and &amp;quot;multiplication&amp;quot; is similar to &amp;quot;AND&amp;quot; when dealing with base-2 numbers -- and that Boolean logic is fundamentally binary.&lt;/p&gt;&#xA;&lt;p&gt;As to tilde vs. exclamation point vs. overbar, my guess is that different groups or locales got started independently.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-01-05T13:02:16.830" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12702" PostTypeId="2" ParentId="3131" CreationDate="2021-01-05T18:02:37.890" Score="4" Body="&lt;p&gt;No one mentioned John and Alicia Nash (if Alicia was a theoretical physicist, then &lt;a href=&quot;https://scienceisbeauty.tumblr.com/post/47396338287/fields-arranged-by-reality-based-upon-the-fields&quot; rel=&quot;nofollow noreferrer&quot;&gt;I think she counts&lt;/a&gt;), even though they died earlier in 2015, the year the question was posted.&lt;/p&gt;&#xA;&lt;p&gt;They died in a car accident.&lt;/p&gt;&#xA;" OwnerUserId="3338" LastEditorUserId="4251" LastEditDate="2022-04-26T16:13:51.067" LastActivityDate="2022-04-26T16:13:51.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12703" PostTypeId="2" ParentId="12699" CreationDate="2021-01-05T20:15:10.460" Score="2" Body="&lt;p&gt;Here is my guess. Consider a circle of the diameter 42. It's circumference would be 132. (Note that &lt;span class=&quot;math-container&quot;&gt;$\pi \approx 132/42$&lt;/span&gt;, also note that 132 feet is 8 rods, if we use &lt;a href=&quot;https://en.wikipedia.org/wiki/Measuring_rod#Middle_Ages&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's value&lt;/a&gt; for rod as 8.5 feet). Then the table shows how long is a chord of an arc with a given length. That is, what is &lt;span class=&quot;math-container&quot;&gt;$f(x)=42\sin(\pi\frac{x}{132})$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;The first column is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. The second column is just a supplementary angle of x (i.e. 132-x).  The rest columns are the value of &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; in some units. It looks like the units that were used are feet (3rd column), feet/6 (3th column), feet/6/18 (4th column). I'm not sure about the last column, maybe it feet/6/18/18 or feet/6/18/20. While third and fourth column is calculated correctly (except obvious typographical mistakes in 3rd column in 55th and 56th rows, where it should be 40 instead of 41 - otherwise the values do not even monotonically increase), the last two columns contain errors.&#xA;&lt;a href=&quot;https://docs.google.com/spreadsheets/d/1c598VOXNBVePfL7eihkQ_pvMo8sPYJpphu8Hf__D5w8/edit?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is my recalculation of this table in google docs.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2021-01-05T20:15:10.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12704" PostTypeId="2" ParentId="6256" CreationDate="2021-01-05T22:10:24.903" Score="1" Body="&lt;p&gt;The first formula of Gauss comes from the&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean&quot; rel=&quot;nofollow noreferrer&quot;&gt;arithmetic-geometric mean&lt;/a&gt;&#xA;(AGM) algorithm. More precisely, if we are given three numbers&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ a_1,b_1,c_1 \qquad \text{ such that }\qquad&#xA;a_1^2=b_1^2+c_1^2\tag{1} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;then define another such triple with&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ a_2 = (a_1+b_1)/2, \;\; b_2 = \sqrt{a_1\,b_1},&#xA;\;\; c_2 = (a_1-b_1)/2. \tag{2} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This process can be iterated to produce &lt;span class=&quot;math-container&quot;&gt;$\,a_m,b_m,c_m\,$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\,m = 2^n.$&lt;/span&gt; This is the quadratic AGM&#xA;studied by Gauss which can be parametrized&#xA;using theta functions by&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ a_m = \theta_3(q^m)^2,\;\; b_m = \theta_4(q^m)^2,&#xA;\;\; c_m = \theta_2(q^m)^2. \tag{3} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus, equation &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; in terms of theta functions is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \theta_3(q)^4 = \theta_4(q)^4 + \theta_2(q)^4. \tag{4} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The three &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-series are the generating functions of OEIS&#xA;sequences &lt;a href=&quot;https://oeis.org/A000118&quot; rel=&quot;nofollow noreferrer&quot;&gt;A000118&lt;/a&gt;,&#xA;&lt;a href=&quot;https://oeis.org/A096727&quot; rel=&quot;nofollow noreferrer&quot;&gt;A096727&lt;/a&gt;, and&#xA;&lt;a href=&quot;https://oeis.org/A129588&quot; rel=&quot;nofollow noreferrer&quot;&gt;A129588&lt;/a&gt; respectively. See also OEIS&#xA;sequence &lt;a href=&quot;https://oeis.org/A008438&quot; rel=&quot;nofollow noreferrer&quot;&gt;A008438&lt;/a&gt;. These &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-series&#xA;all have &lt;a href=&quot;https://en.wikipedia.org/wiki/Lambert_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lambert series&lt;/a&gt;&#xA;expansions such as the one given for &lt;span class=&quot;math-container&quot;&gt;$\,\theta_3(q)^4.\,$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;More details are in the David A. Cox article&#xA;&lt;em&gt;The Arithmetic-Geometric Mean of Gauss&lt;/em&gt;. Among which is that&#xA;Gauss computed the AGM of &lt;span class=&quot;math-container&quot;&gt;$\,1\,$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\,\sqrt{2}\,$&lt;/span&gt; in 1799.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The 98th entry, dated May 30, 1799, reads as follows:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have established that the arithmetic-geometric mean between&#xA;&lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\pi/\varpi$&lt;/span&gt; to the eleventh decimal&#xA;place; the demonstation of this fact will surely open an entirely&#xA;new field of analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cox also writes that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This algorithm first appeared in a paper of Lagrange, but it was&#xA;Gauss who really discovered the amazing depth of this subject.&#xA;Unfortunately, Gauss published little on the agM (his abbreviation&#xA;for the arithmetic-geometric mean) during his lifetime. It was&#xA;only with the publication of his collected works [12] between&#xA;1868 and 1927 that the full extent of his work became apparent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="13622" LastEditDate="2021-01-06T01:56:47.593" LastActivityDate="2021-01-06T01:56:47.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12705" PostTypeId="2" ParentId="3131" CreationDate="2021-01-06T01:17:32.047" Score="2" Body="&lt;p&gt;These mathematicians (or very mathematical non-mathematicians) died young (&amp;lt;50) and/or from non-natural causes; I did not see these names:&lt;/p&gt;&#xA;&lt;p&gt;Yom Tov Lipman Lipkin (c.1846-c.1876), smallpox.  Codiscoverer of the Peaucellier-Lipkin linkage.  A son of the famed Salanter rabbi, founder of the Mussar movement.&lt;/p&gt;&#xA;&lt;p&gt;Hermann Minkowski (1864-1909), appendicitis.  Number theorist, but best known for his work in relativity.&lt;/p&gt;&#xA;&lt;p&gt;Janina Hosiasson (1899-1942), murdered by Gestapo.  Foundations of probability and inference.  Wife of:&lt;/p&gt;&#xA;&lt;p&gt;Adolf Lindenbaum (1904-1941), murdered by Gestapo.  Mathematical logic, published little, but collaborated widely, especially with Tarski.&lt;/p&gt;&#xA;&lt;p&gt;Duro Kurepa (1907-1993), robbed and murdered during the post-Yugoslavia wars.&lt;/p&gt;&#xA;&lt;p&gt;Carol Karp (1926-1972), cancer.  Major founder of infinitary logic.&lt;/p&gt;&#xA;&lt;p&gt;John Nash (1928-2015), car crash, wife also killed.  Career cut short by schizophrenia.&lt;/p&gt;&#xA;&lt;p&gt;Karel de Leeuw (1930-1978), murdered in his Stanford office by his perpetual grad student.  Analyst.&lt;/p&gt;&#xA;&lt;p&gt;Richard Montague (1930-1971), robbed and murdered at home, apparently by strangers he picked up cruising.  Logic, philosophy, linguistics.&lt;/p&gt;&#xA;&lt;p&gt;Robert Hamilton Boyer (1932-1966), murdered, a Texas tower massacre victim.  General relativity, Boyer-Lindquist coordinates.&lt;/p&gt;&#xA;&lt;p&gt;Frank Adams (1930-1989), car crash.  Major topologist.&lt;/p&gt;&#xA;&lt;p&gt;Alan Mekler (1947-1992), cancer.  Applications of set theory and model theory to algebra.&lt;/p&gt;&#xA;&lt;p&gt;Boris Weisfeiler (1941-?1985), disappeared while hiking in Chile during the Pinochet regime.  Assumed kidnapped, tortured and murdered.&lt;/p&gt;&#xA;&lt;p&gt;Zoltan Balogh (1953-2002), sudden illness (with history, also father Tibor died young).  Set-theoretic topologist.&lt;/p&gt;&#xA;&lt;p&gt;Elizabeth Gardner (1957-1988), cancer.  Theoretical physicist specializing in disordered systems, her most mathematical work was the statistical physics of neural networks.&lt;/p&gt;&#xA;&lt;p&gt;Moez Alimohamed (1967-1994), robbed and murdered outside his apartment.  A UPenn math grad student, he had just finished his first paper/thesis.  Published and degreed posthumously.&lt;/p&gt;&#xA;&lt;p&gt;Michael B. Cohen (1992-2017), sudden illness.  Up and coming superstar in CS, especially algorithms.&lt;/p&gt;&#xA;" OwnerUserId="13624" LastActivityDate="2021-01-06T01:17:32.047" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12706" PostTypeId="1" CreationDate="2021-01-06T15:11:16.670" Score="1" ViewCount="87" Body="&lt;h2&gt;Introduction&lt;/h2&gt;&#xA;&lt;p&gt;The Bessel function of the first kind &lt;span class=&quot;math-container&quot;&gt;$J_n(x)$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$n \in \mathbb{Z},\ x \in \mathbb{R}$&lt;/span&gt;) appeared early among other topics, in Celestial Mechanics, in the series expression of the &lt;a href=&quot;https://en.wikipedia.org/wiki/True_anomaly&quot; rel=&quot;nofollow noreferrer&quot;&gt;true anomaly&lt;/a&gt; in function of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mean_anomaly&quot; rel=&quot;nofollow noreferrer&quot;&gt;mean anomaly&lt;/a&gt;, used for the description of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Elliptic_orbit&quot; rel=&quot;nofollow noreferrer&quot;&gt;elliptic motion&lt;/a&gt; as a function of time.&lt;/p&gt;&#xA;&lt;p&gt;Such developments seemed to have found the series solution for &lt;span class=&quot;math-container&quot;&gt;$J_n(x)$&lt;/span&gt; by favoring the use of &lt;a href=&quot;https://en.wikipedia.org/wiki/Laurent_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laurent series&lt;/a&gt; expansion and the generating function approach. See for instance Tisserand's &lt;em&gt;Treatise on Celestial Mechanics&lt;/em&gt; p.206-207 &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k29031t/f218.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;These developments do not use complex analysis and the residue theorem (that otherwise makes for a good proof), but despite their elegance I find them lackluster on the mathematical side, since they seem to omit 'obvious' steps.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What I would like to see&lt;/strong&gt;, and could not achieve yet, &lt;strong&gt;is a robust proof along these lines, with no other tool&lt;/strong&gt; (since these methods seemed, again, 'obvious', and convincing enough back in the day).&lt;/p&gt;&#xA;&lt;p&gt;I looked at a good amount of literature from the 1800s, early 1900s, and then 1960s and up (after Celestial Mechanics was studied again due to the Space Race), but could not find any clear proof along these lines, cf the Notes at the the end.&lt;/p&gt;&#xA;&lt;h2&gt;Historical developments&lt;/h2&gt;&#xA;&lt;p&gt;So these proofs usually go as follows: let us consider the following function &lt;span class=&quot;math-container&quot;&gt;$f(x,z)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x \in \mathbb{R},\ z \in \mathbb{C}$&lt;/span&gt; (the generating function):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tag{*}&#xA;f(x,z) = e^ {\frac{x}{2}\left(z-\frac{1}{z}\right)}.&#xA;$$&lt;/span&gt;&#xA;This function being holomorphic on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^*$&lt;/span&gt;, which is an annulus, it admits an expansion in Laurent series on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^*$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tag{**}&#xA;f(x,z) = \sum_{n=-\infty}^{+\infty} J_n(x) z^n.&#xA;$$&lt;/span&gt;&#xA;The &lt;span class=&quot;math-container&quot;&gt;$J_n(x)$&lt;/span&gt; coefficients being what is to be determined (i.e. the connection to the Bessel function is unknown so far).&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, we also have that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x,z) = e^{\frac{xz}{2}} e^{\frac{-x}{2z}}= \sum_{q=0}^{+\infty} \frac{1}{q!} \left( \frac{x z}{2} \right)^q \times \sum_{p=0}^{+\infty} \frac{(-1)^p}{p!} \left( \frac{x}{2z} \right)^p,&#xA;$$&lt;/span&gt;&#xA;and these two series being absolutely convergent, their product is also convergent and equal to the following double sum:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tag{1}&#xA;f(x,z) = \sum_{q=0}^{+\infty}\sum_{p=0}^{+\infty} \left( \frac{x}{2} \right)^{q+p}\frac{(-1)^{p}}{q!p!}z^{q-p}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;h2&gt;The specific problems and my attempt&lt;/h2&gt;&#xA;&lt;p&gt;This is where for me the problems arise. If one follows Tisserand for instance, he sets &lt;span class=&quot;math-container&quot;&gt;$q=p+n$&lt;/span&gt; and says without further explanation that one directly obtains by identification the following usual expression for &lt;span class=&quot;math-container&quot;&gt;$J_n(x)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$n \geq 0$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tag{2}&#xA;J_n(x) = \sum_{p=0}^{+\infty}\frac{(-1)^p}{p!(p+n)!}\left( \frac{x}{2} \right)^{2p+n}.&#xA;$$&lt;/span&gt;&#xA;The case for &lt;span class=&quot;math-container&quot;&gt;$n&amp;lt;0$&lt;/span&gt; yielding an identical expression is solved by realizing that &lt;span class=&quot;math-container&quot;&gt;$J_{-n}(x) = (-1)^n J_n(x)$&lt;/span&gt;, which is easily seen by doing &lt;span class=&quot;math-container&quot;&gt;$z \to -\frac{1}{z}$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$(*)$&lt;/span&gt; and seeing that &lt;span class=&quot;math-container&quot;&gt;$(*)$&lt;/span&gt; does not change, and then comparing the two expressions using &lt;span class=&quot;math-container&quot;&gt;$(**)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From my point of view, to be able to say &lt;em&gt;directly by identification&lt;/em&gt; that we have &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; can only come from the fact that &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; &lt;em&gt;should be able to be recast as&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tag{3}&#xA;f(x,z) = \sum_{n=-\infty}^{+\infty}\sum_{p=0}^{+\infty} \epsilon(n) \left( \frac{x}{2} \right)^{2p+n}\frac{(-1)^{p}}{(p+n)!p!}z^n&#xA;$$&lt;/span&gt;&#xA;with &lt;span class=&quot;math-container&quot;&gt;$\epsilon(n) = 1$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$n \geq 0$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(-1)^n$&lt;/span&gt; otherwise.&lt;/p&gt;&#xA;&lt;p&gt;So, since &lt;span class=&quot;math-container&quot;&gt;$(1) \to (2)$&lt;/span&gt; did not seem obvious at all to me, I did the actual calculation by changing the indices in the double sum &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt;, this yields:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\tag{4}&#xA;f(x,z) = \sum_{n=-p}^{+\infty}\sum_{p=0}^{+\infty} \left( \frac{x}{2} \right)^{2p+n}\frac{(-1)^{p}}{(p+n)!p!}z^n&#xA;$$&lt;/span&gt;&#xA;and this is for me where the real problem is because:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The first sum starts at &lt;span class=&quot;math-container&quot;&gt;$-p$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$-\infty$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;There is no &lt;span class=&quot;math-container&quot;&gt;$\epsilon(n)$&lt;/span&gt; function anywhere&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So this is where I am, I cannot find a nice change of indices, or a nice way to reorder &lt;span class=&quot;math-container&quot;&gt;$(4)$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$(3)$&lt;/span&gt;. Now, I am unsure of the actual rules to switch infinite double sums, which may or may not be one of the cruxes to what my problem boils down to...but even by assuming everything is fine with switching I cannot find a way.&lt;/p&gt;&#xA;&lt;h2&gt;Notes&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I know that Watson in his reference treaty on the subject &lt;em&gt;Treatise on the theory of Bessel functions&lt;/em&gt; (p. 14-16, see &lt;a href=&quot;https://archive.org/details/treatiseontheory00watsuoft/page/14/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) solves the problem by looking at each term yielding a &lt;span class=&quot;math-container&quot;&gt;$z^n$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$z^{-n}$&lt;/span&gt; and collects all of them to yield &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt;. This is pretty convincing, but nevertheless, he somewhat masks the double sums by doing so, and I feel that there is a way of solving the problem more robustly as I described above, by some nice reordering or change of indices.&lt;/li&gt;&#xA;&lt;li&gt;There are also a number of proofs in earlier works (that don't even use the summation sign), that claim that by 'looking' at the sums and extracting the good terms, you can 'see' that you have &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt;, but that does not cut it for me.&lt;/li&gt;&#xA;&lt;li&gt;Maybe there is no way exactly as I described, but I of course welcome any other convincing solution not hiding anything, and at the same time not using complex analysis, or other kinds of tools.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;N.B.: I posted &lt;a href=&quot;https://math.stackexchange.com/questions/3792217/historical-proofs-of-the-series-expression-for-the-bessel-function-of-the-first&quot;&gt;this question&lt;/a&gt; some months ago on Math.SE which got one comment that did not really help, but I feel like this is a more appropriate place given the historical context at play, so I hope to get more answers here.&lt;/p&gt;&#xA;" OwnerUserId="10608" LastEditorUserId="10608" LastEditDate="2022-11-22T09:46:19.840" LastActivityDate="2022-11-22T09:46:19.840" Title="Historical proofs of the series expression for the Bessel function of the first kind" Tags="|mathematics|physics|reference-request|astronomy|celestial-mechanics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12707" PostTypeId="1" CreationDate="2021-01-06T17:07:22.550" Score="1" ViewCount="48" Body="&lt;p&gt;Who suggested to him the idea that charge wasn't really some excess or defect of &amp;quot;electric fluid&amp;quot;, but rather something carried by a particle?&lt;/p&gt;&#xA;" OwnerUserId="13636" LastEditorUserId="26" LastEditDate="2021-02-20T15:03:12.563" LastActivityDate="2021-02-20T15:03:12.563" Title="How and for what purpose did Lorentz come out with his &quot;electron model&quot;?" Tags="|electromagnetism|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12708" PostTypeId="1" AcceptedAnswerId="12710" CreationDate="2021-01-06T18:46:55.953" Score="4" ViewCount="496" Body="&lt;p&gt;I think this question is useful to know how sir J.J.Thomson inferred from his experiment that the electron had to be 1700 lighter than hydrogen, as he declared in his Nobel Lecture of 1906:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Before the cathode rays were investigated, the charged atom of hydrogen met with in the electrolysis of liquids was the system which had the greatest known value of e/m, and in this case the value is only 104, hence for the corpuscle in the cathode rays the value of e/m is 1700 times the value for the corresponding quantity for the charged hydrogen atom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In some nuclear physics notes I've found that Thomson compared his ratio to another calculated by H.A.Lorentz starting from electrolysis products. He was attempting to give an explanation for the&#xA;Zeeman Effect and measured the charge to mass ratio of the hydrogen.&lt;/p&gt;&#xA;&lt;p&gt;Is this true? Are there any reliable sources?&lt;/p&gt;&#xA;" OwnerUserId="13636" LastActivityDate="2021-01-06T20:16:55.053" Title="How was charge to mass ratio measured via electrolysis in 19th century?" Tags="|electromagnetism|physics-social-history|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12710" PostTypeId="2" ParentId="12708" CreationDate="2021-01-06T20:16:55.053" Score="3" Body="&lt;p&gt;The determination of mass to charge ratio of the hydrogen ion came from electrolysis experiments and Faraday's law of electrolysis, which were well established in J. J. Thomson's time. I wish modern textbooks also included this experiment, but a 1913 &lt;em&gt;The Chemical Engineer&lt;/em&gt; (monthly publication of applied/analytical chemistry) gives an account (Vol XX, No. 2, pg. 28). Google Books has it.&lt;/p&gt;&#xA;&lt;p&gt;It was established that if 96,540 Coulombs were passed through HCl solution, one gram equivalent or 1.008 g of hydrogen ions &lt;span class=&quot;math-container&quot;&gt;${H^+}$&lt;/span&gt; will be reduced to hydrogen gas. The number of Coloumbs could be determined by silver coulometry or current/time relationship. I do not exactly how Faraday determined 96,540 in the beginning. &lt;em&gt;His law is exact&lt;/em&gt; and no exception has been found so far! However, measuring the mass of electrolyzed gas was easy.&lt;/p&gt;&#xA;&lt;p&gt;Here is an excerpt from &lt;em&gt;The Chemical Engineer&lt;/em&gt;.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/JOwxv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JOwxv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-01-06T20:16:55.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12711" PostTypeId="1" AcceptedAnswerId="12713" CreationDate="2021-01-07T18:27:38.800" Score="5" ViewCount="1076" Body="&lt;p&gt;So, I came here from &lt;a href=&quot;https://math.stackexchange.com&quot;&gt;Mathematics StackExchange&lt;/a&gt; where I posted &lt;a href=&quot;https://math.stackexchange.com/questions/3976346/why-do-we-need-polar-coordinates&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, I want to know why polar coordinates came into existence. Why exactly did the mathematician who introduced them...introduce them? Was it because he encountered a problem where defining the position of a point in terms of the distance of that point from the origin and it's &amp;quot;direction&amp;quot; gave rise to easier calculations? If so, what was the problem that he/she encountered and what type of calculations are we talking about here? If not, how did the idea of polar coordinates originate?&lt;/p&gt;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="13642" LastActivityDate="2021-01-09T01:04:12.083" Title="How did polar coordinates come into existence?" Tags="|mathematics|geometry|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12712" PostTypeId="1" CreationDate="2021-01-07T20:07:30.827" Score="1" ViewCount="161" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Faraday%27s_laws_of_electrolysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is the reference. I'm wondering how he experimentally proceeded to derive the fact the if we pass 96485 Coulombs through a solution, then 1 gram equivalent of substance is electrolyzed.&lt;/p&gt;&#xA;" OwnerUserId="13636" LastEditorUserId="26" LastEditDate="2021-02-20T15:02:55.343" LastActivityDate="2021-02-20T15:02:55.343" Title="How did Faraday determine the Faraday's constant?" Tags="|physics|experimental-physics|chemistry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12713" PostTypeId="2" ParentId="12711" CreationDate="2021-01-07T22:19:45.267" Score="8" Body="&lt;p&gt;The following information is found &lt;a href=&quot;https://jeff560.tripod.com/p.html&quot; rel=&quot;noreferrer&quot;&gt;HERE&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;According to Daniel L. Klaasen in &lt;em&gt;Historical Topics for the Mathematical Classroom:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Isaac Newton was the first to think of using polar coordinates. In a treatise &lt;em&gt;Method of Fluxions&lt;/em&gt; (written about 1671), which dealt with curves defined analytically, Newton showed ten types of coordinate systems that could be used; one of these ten was the system of polar coordinates. However, this work by Newton was not published until 1736; in 1691 Jakob Bernoulli derived and made public the concept of polar coordinates in the &lt;em&gt;Acta eruditorum&lt;/em&gt;. The polar system used for reference a point on a line rather than two intersecting lines. The line was called the &amp;quot;polar axis,&amp;quot; and the point on the line was called the &amp;quot;pole.&amp;quot; The position of any point in a plane was then described first by the length of a vector from the pole to the point and second by the angle the vector made with the polar axis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to Smith (vol. 2, page 324), &amp;quot;The idea of polar coordinates seems due to Gregorio Fontana (1735-1803), and the name was used by various Italian writers of the 18th century.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Polar co-ordinates&amp;quot; is found in English in 1816 in a translation of Lacroix’s &lt;em&gt;Differential and Integral Calculus:&lt;/em&gt; &amp;quot;The variables in this equation are what Geometers have called polar co-ordinates&amp;quot; (OED2).&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-01-07T22:19:45.267" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12714" PostTypeId="1" CreationDate="2021-01-07T22:25:20.417" Score="3" ViewCount="97" Body="&lt;p&gt;I am looking for a book about the history of gravitational theory. It should obviously include discussions of Newton, Einstein, and their theories, and hopefully it would include the work of other famous scientists (philosophers) like Aristotle and Galileo that were part of the development. It would be great to find a book that also discusses lesser known theories, like Le Sage's, Descartes's, or Nordstrom's and theories like MOND.&lt;/p&gt;&#xA;&lt;p&gt;I have come across three books that might fit this but am looking for recommendations of which of these, or another related book, is best.&lt;/p&gt;&#xA;&lt;p&gt;The Ascent of Gravity by Marcus Chown (&amp;quot;the force that explains everything&amp;quot;.. ehh)&lt;/p&gt;&#xA;&lt;p&gt;Gravity's Arc by David Darling&lt;/p&gt;&#xA;&lt;p&gt;Gravity by George Gamow&lt;/p&gt;&#xA;" OwnerUserId="13643" LastEditorUserId="10106" LastEditDate="2021-01-08T02:55:10.517" LastActivityDate="2021-05-11T05:04:55.473" Title="Books about the development/history of gravitational theory" Tags="|physics|reference-request|relativity-theory|gravity|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12715" PostTypeId="2" ParentId="12684" CreationDate="2021-01-08T04:07:22.243" Score="3" Body="&lt;p&gt;There are good answers here. I would like to add that a great deal of pre-industrial construction is not straight at all. Cathedrals often have walls that bend or kink from one end to the other or became kinked or curved from top to bottom.&lt;/p&gt;&#xA;&lt;p&gt;I once looked at an apartment in a building built shortly before 1500 (in England). The ceiling of the bedroom can only be described as wonky, being at least two feet lower in one corner than another and with no apparent even curvature across it.&lt;/p&gt;&#xA;&lt;p&gt;The standards of some craftsmen were very high, eg, armorers. Nothing they did depended on having a straight line. Armor must be curved properly to deflect weapons.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes straightness and levelness are desirable but these were rarely necessary in the past. When they were the methods of using gravity with water or strings to make straight lines were well known to the building trades, as were methods of making a right angle.&lt;/p&gt;&#xA;&lt;p&gt;The required standards of accuracy greatly increased with the advent of navigation across oceans and industrial standardization.&lt;/p&gt;&#xA;" OwnerUserId="10759" LastActivityDate="2021-01-08T04:07:22.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12716" PostTypeId="2" ParentId="7865" CreationDate="2021-01-08T16:08:15.303" Score="1" Body="&lt;p&gt;The first prize winning essays are available in full via Google Books in &lt;em&gt;Pieces qui ont remporte les deux prizes de l'Academie Royale des Sciences&lt;/em&gt; (Paris 1721). Hope your 18th century French is up to scratch!&lt;/p&gt;&#xA;" OwnerUserId="13647" LastActivityDate="2021-01-08T16:08:15.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12717" PostTypeId="2" ParentId="12711" CreationDate="2021-01-08T20:39:01.783" Score="5" Body="&lt;p&gt;It depends what you consider by &amp;quot;introducing polar coordinates&amp;quot;. Polar (and spherical) coordinates were used (without explicitly naming them) long before Cartesian one. Ptolemy gave spherical coordinates of celestial bodies (two angles and distance from Earth). It is the most natural way to describe position of a point relative to an observer.&#xA;Of course Ptolemy did not described curves analytically in polar coordinates as Newton. So you can say that he did not &amp;quot;introduce&amp;quot; them as in contemporary textbooks. But he certainly used them.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="6229" LastEditDate="2021-01-09T01:04:12.083" LastActivityDate="2021-01-09T01:04:12.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12719" PostTypeId="1" CreationDate="2021-01-09T17:57:31.370" Score="0" ViewCount="124" Body="&lt;p&gt;I have just started learning about trigonometric ratios of complex arguments but I couldn't find any justification or derivation for extending trigonometric ratios to complex field. Also the Euler's relationship is for &lt;span class=&quot;math-container&quot;&gt;$exp(x+iy)$&lt;/span&gt;&#xA;where x and y are real so putting &lt;span class=&quot;math-container&quot;&gt;$exp(iz)$&lt;/span&gt; where z is complex like that and defining ratios in complex arguments , i wasn't able to digest. Can anyone please tell how,when and why were trigonometric ratios of complex arguments defined and their development in course of time. There would be some reason to do this and also to not change the names because these names where historically defined for triangles. I think the intuition for its historical development may help me in grasping the concept.&lt;/p&gt;&#xA;" OwnerUserId="13653" LastActivityDate="2021-01-11T21:05:40.173" Title="History of complex trigonometric ratios" Tags="|complex-analysis|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12720" PostTypeId="2" ParentId="3131" CreationDate="2021-01-10T05:25:28.957" Score="4" Body="&lt;p&gt;Sonya Kovalevskaya died at the age of 41, of the 'coronavirus' of her time. &lt;a href=&quot;https://en.wikipedia.org/wiki/Sofya_Kovalevskaya#Last_years_in_Germany_and_Sweden&quot; rel=&quot;nofollow noreferrer&quot;&gt;From Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1889, Kovalevskaya fell in love with Maxim Kovalevsky, a distant relation of her deceased husband, but insisted on not marrying him because she would not be able to settle down and live with him.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kovalevskaya died of epidemic influenza complicated by pneumonia in 1891 at age forty-one, after returning from a vacation in Nice with Maxim...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kovalevskaya's mathematical results, such as the Cauchy–Kowalevski theorem, and her pioneering role as a female mathematician in an almost exclusively male-dominated field...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also her husband, paleontologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Kovalevsky_(paleontologist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vladimir Kovalevsky&lt;/a&gt;, had committed suicide 8 years earlier.&lt;/p&gt;&#xA;" OwnerUserId="3338" LastEditorUserId="3338" LastEditDate="2021-01-10T05:42:31.797" LastActivityDate="2021-01-10T05:42:31.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12721" PostTypeId="1" CreationDate="2021-01-10T08:40:58.197" Score="2" ViewCount="110" Body="&lt;p&gt;I am looking for a book that describes the historical development of mathematics from the ancient times to our advanced developed concept. Along with the derivation and mathematical concepts as introduced in the respective time. For example if it was on history of number theory of gauss , then the proofs as given by him would also be stated. It doesn't has to be a single book. Any help would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="13653" LastActivityDate="2021-01-11T04:11:24.483" Title="Book to know about development of mathematical concepts" Tags="|reference-request|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12722" PostTypeId="2" ParentId="12721" CreationDate="2021-01-10T10:59:38.073" Score="0" Body="&lt;p&gt;Your question is too broad, I think.&lt;/p&gt;&#xA;&lt;p&gt;One nice scholarly book (which covers only the changes around 1850 to 1910, as mathematics went from a mishmash of physics and psychology, and became an abstract deductive science):&lt;/p&gt;&#xA;&lt;p&gt;Jeremy Gray, &lt;em&gt;Plato's Ghost: The Modernist Transformation of Mathematics&lt;/em&gt;, Princeton University Press, 2008&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/I4Nwo.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I4Nwo.jpg&quot; alt=&quot;image&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-01-10T10:59:38.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12723" PostTypeId="1" CreationDate="2021-01-10T12:18:42.307" Score="13" ViewCount="4373" Body="&lt;p&gt;I am interested in very early inventions that allowed energy to be stored and released after a delay even it's just a short time. With &amp;quot;invention&amp;quot; I mean a novelty that is the result of humans' intellectual creativity. Someting that cannot just be done like throwing a stone but requires ingenuity and/or experimentation. A bow is better. A fly wheel better still.&lt;/p&gt;&#xA;&lt;p&gt;I am aware that it is unlikely to find a proven first use of such inventions. Still, I am interested in what history knows about it's early uses.&lt;/p&gt;&#xA;&lt;p&gt;My current assumption is that the first invention to store energy was the fly wheel. And one of the first uses or probably THE first use of a fly wheel is pottery. &lt;a href=&quot;https://www.tomorrowsworldtoday.com/2018/08/01/a-brief-history-of-pottery/&quot; rel=&quot;noreferrer&quot;&gt;One site&lt;/a&gt; states without providing sources.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first potter’s wheel is believed to have come from Sumer in 3129 BC, although there is evidence that points to other places of origin. Precursors to the wheel started appearing as early as 4500 BC though, so an exact date is likely impossible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Though the date 3129 BC seems questionable (how could it be determined so precisely?) the rough date and origin are confirmed by the &lt;a href=&quot;https://www.britannica.com/art/pottery/Western-pottery#ref73835&quot; rel=&quot;noreferrer&quot;&gt;Encyclopaedia Britannica&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Handmade pottery has been found at Ur, in Mesopotamia [...] dating from a time soon after the Flood (about 3000 BCE), was wheelmade decorated pottery of a type usually called Al ’Ubaid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Are these sources correct and are there better ones?&lt;/li&gt;&#xA;&lt;li&gt;What about other regions of this world? E.g. the Chinese were famous for their pottery for a very long time and it only seems reasonable to assume that fly wheels were invented in Asia independently from teh middle east.&lt;/li&gt;&#xA;&lt;li&gt;Are there any other uses of fly wheels that date back that far?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="12814" LastEditorUserId="12814" LastEditDate="2021-01-11T18:54:26.633" LastActivityDate="2021-01-14T05:50:41.843" Title="What are the earliest inventions to store and release energy (e.g. fly wheels)?" Tags="|physics|discoveries|energy|ancient|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12724" PostTypeId="2" ParentId="9789" CreationDate="2021-01-10T12:54:01.680" Score="1" Body="&lt;p&gt;In Gert Schubring's translation from 2016, the exact statement is:&#xA;&amp;quot;Every person believes that he knows what a curve is until he has learned so much mathematics that the countless possible abnormalities confuse him.&amp;quot; (p. 199).&lt;/p&gt;&#xA;" OwnerUserId="13656" LastActivityDate="2021-01-10T12:54:01.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12725" PostTypeId="2" ParentId="12723" CreationDate="2021-01-10T15:09:02.653" Score="30" Body="&lt;p&gt;One early invention for storing energy was a basin above the level of the river.&#xA;It was filled with water when the water in the river was high, and then, when the water in the river was low, it was allowed to flow to the fields from the basin.&#xA;Such a basin could also be filled manually. Such devices were used in ancient Egypt, as a part of their irrigation system even before the &amp;quot;beginning of history&amp;quot; (by which many understand the invention of writing).&lt;/p&gt;&#xA;&lt;p&gt;Another pre-historic device for storing energy, which is even older is a bow. It stores the energy when you draw it, to release it quickly when you shoot.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-01-10T15:09:02.653" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12726" PostTypeId="2" ParentId="12721" CreationDate="2021-01-10T15:14:51.153" Score="0" Body="&lt;p&gt;There is a nice book called Princeton Companion to Mathematics (written by many authors, edited by Gowers) which explains many parts of modern mathematics, and has&#xA;a chapter on development of principal concepts.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-01-10T15:14:51.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12727" PostTypeId="2" ParentId="12723" CreationDate="2021-01-11T02:35:01.127" Score="21" Body="&lt;p&gt;This is probably not what you were thinking of but &amp;quot;the earliest invention that allowed energy to be stored and released after a delay even it's just a short time&amp;quot; was a stone.&lt;/p&gt;&#xA;&lt;p&gt;I can store energy in it briefly by flinging it at your head (kinetic). I can store energy in it even longer by putting it on a cliff above your head (potential).&lt;/p&gt;&#xA;&lt;p&gt;Stone based inventions might not be sexy but we did have an entire age named after them.&lt;/p&gt;&#xA;" OwnerUserId="13661" LastActivityDate="2021-01-11T02:35:01.127" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12728" PostTypeId="2" ParentId="12721" CreationDate="2021-01-11T04:11:24.483" Score="2" Body="&lt;p&gt;There are many histories of mathematics. These two might be what you're looking for:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Carl B. Boyer, A History of Mathematics&lt;/li&gt;&#xA;&lt;li&gt;W. W. Rouse Ball, &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.167204/&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Short Account Of The History Of Mathematics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user13662" LastActivityDate="2021-01-11T04:11:24.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12729" PostTypeId="2" ParentId="12714" CreationDate="2021-01-11T04:52:02.417" Score="1" Body="&lt;p&gt;The first two chapters of this book discuss the history of gravitational theory:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Robert Grant, &lt;a href=&quot;https://archive.org/details/historyphysical00grangoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of physical astronomy from the earliest ages to&#xA;the middle of the nineteenth century&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user13662" LastActivityDate="2021-01-11T04:52:02.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12731" PostTypeId="1" AcceptedAnswerId="12733" CreationDate="2021-01-11T05:18:34.887" Score="3" ViewCount="245" Body="&lt;p&gt;In mathematics, 1760 Plateau's problem were solved, but it was only in 1930 that general&#xA;solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó.&lt;/p&gt;&#xA;&lt;p&gt;1760 - 1930 = &lt;strong&gt;170 years&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is the oldest open question solved in mathematics?&lt;/p&gt;&#xA;" OwnerUserId="13665" LastActivityDate="2021-01-12T14:49:38.070" Title="What is the oldest open question solved in mathematics?" Tags="|mathematics|scientists|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12732" PostTypeId="2" ParentId="12723" CreationDate="2021-01-11T05:19:41.517" Score="3" Body="&lt;p&gt;How about good-old fashioned pit traps? If 'stored energy' is something where you input work ahead of time and then it is expended at a later date for a use, a pit trap is a perfect example.&lt;/p&gt;&#xA;&lt;p&gt;While digging the pit, you are removing a great deal of material, and effectively 'storing' gravitational potential energy in the difference between the height of the bottom of the pit, and the top. When an animal falls into the pit, you use that laboriously stored energy to kill it.&lt;/p&gt;&#xA;&lt;p&gt;Of course, you then need to expend additional energy to drag out the carcass, and 'reset' the trap, the same way you would with a snare.&lt;/p&gt;&#xA;&lt;p&gt;Putting a date on the earliest pit trap is likely impossible, but was &amp;gt; 50,000 years ago.&lt;/p&gt;&#xA;" OwnerUserId="13664" LastActivityDate="2021-01-11T05:19:41.517" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12733" PostTypeId="2" ParentId="12731" CreationDate="2021-01-11T13:18:31.347" Score="9" Body="&lt;p&gt;The Delian problem (doubling the cube) and the angle trisection problem are probably the oldest ones, though their exact age is not known. (Same applies to all problems of constructions with ruler and compass). They were certainly around in 4 bc, and existing sources show that by that time they were already old. Since both problems were solved only in 19th century, we conclude that it took at least 22 centuries to solve them, probably longer. Of course there are also unsolved problems going back to the same times, like the problem about odd perfect numbers.&lt;/p&gt;&#xA;&lt;p&gt;Some authors speculate that these problems are near the origin of mathematics, &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00327767&quot; rel=&quot;noreferrer&quot;&gt;A. Seidenberg, The ritual origin of geometry&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-01-12T14:49:38.070" LastActivityDate="2021-01-12T14:49:38.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12734" PostTypeId="2" ParentId="12723" CreationDate="2021-01-11T13:30:59.660" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.sappohill.com/pilot.asp?pg=Activated_Charcoal&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt;, charcoal was in use circa 3750 BCE.  That's an energy storage medium, although perhaps not the class of energy-release you were thinking of.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-01-11T13:30:59.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12735" PostTypeId="2" ParentId="12723" CreationDate="2021-01-11T15:55:32.533" Score="10" Body="&lt;p&gt;&lt;strong&gt;Agriculture&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The purpose of farming is to harness the freely available energy from the sun, and convert it into a form that can fuel the human body. Agriculture enabled significant food surpluses, which is simply a form of energy storage. Similar to how a flywheel stores mechanical energy which can be used later, food surpluses store chemical energy which can be used later. The invention of agriculture over 10,000 years ago solved an energy storage problem, allowing early humans to effectively store the abundant energy of the growing season for use at a later time when energy was not as plentiful.&lt;/p&gt;&#xA;" OwnerUserId="5256" LastActivityDate="2021-01-11T15:55:32.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12736" PostTypeId="2" ParentId="12731" CreationDate="2021-01-11T18:22:19.727" Score="1" Body="&lt;p&gt;It's hard to say, but here are my two completely different suggestions.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Is &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; rational?&lt;/strong&gt; The squaring of circle was already mentioned. But people (Egyptians and Babylonians) tried to calculate &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; long before invention of ruler-and-compass constructions. Even if they didn't know the notion of irrational numbers, they probably would be interested in a problem whether a circumference of a unit circle could be described as &amp;quot;number&amp;quot; (e.g. sexagesimal number for Babylonians). So, you can argue that this problem is much older than squaring a circle. And, even if irrationality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;  was proved earlier (in 1760s) than transcendence of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and impossibility of squaring a circle (in 1882), the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; was probably much older open problem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But maybe the oldest open problem was a very simple one. For example, &lt;strong&gt;how to calculate the area of a given quadrilateral?&lt;/strong&gt; Babylonians used an incorrect formula &lt;span class=&quot;math-container&quot;&gt;$\frac{a+c}2 \cdot \frac{b+d}2$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; are the sides. I'm not sure when the correct formula was discovered, but it needs trigonometry, which was invented about 2nd century BC (depending on definition of invention of trigonometry). It's impossible to say when people first interested in calculating area of quadrilateral, but it is possible that it was more than two milleniums before asking a question about &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. So it may be the oldest open question at the time when it was solved.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="6788" LastEditDate="2021-01-11T18:27:31.590" LastActivityDate="2021-01-11T18:27:31.590" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12737" PostTypeId="2" ParentId="12723" CreationDate="2021-01-11T19:16:57.617" Score="3" Body="&lt;p&gt;Tree limb.  Before humans became ground bound, they used tree limbs to catapult themselves to the next tree.  But seriously, probably a stone, then a stone attached to a stick, then a stone attached to a stick shot from a string attached to a stick.  Do I get voted off the island?  I just got here.&lt;/p&gt;&#xA;" OwnerUserId="13675" LastActivityDate="2021-01-11T19:16:57.617" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12738" PostTypeId="2" ParentId="12719" CreationDate="2021-01-11T21:05:40.173" Score="1" Body="&lt;p&gt;I am reasonably certain that the answer to your question about&#xA;complex trigonometric functions such as &lt;span class=&quot;math-container&quot;&gt;$\,\sin(z)\,$&lt;/span&gt; goes&#xA;back to the late 1800s when complex analysis was brought to&#xA;an advanced state. The idea is that some well behaved real&#xA;functions can be extended uniquely using the analytic continuation&#xA;process to the whole complex plane (with some possible poles&#xA;allowed). These functions include polynomial and rational&#xA;functions, circular and hyperbolic trigonometric functions,&#xA;and many special functions including the gamma function. One of&#xA;the simplest elementary functions is the exponential function&#xA;&lt;span class=&quot;math-container&quot;&gt;$\,\exp(z)\,$&lt;/span&gt; which has an imaginary period of &lt;span class=&quot;math-container&quot;&gt;$\,2\pi i.\,$&lt;/span&gt;&#xA;The circular or hyperbolic trigonometric functions are then&#xA;defined as certain rational functions of &lt;span class=&quot;math-container&quot;&gt;$\,\exp(iz)\,$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\,\exp(z).$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13622" LastActivityDate="2021-01-11T21:05:40.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12739" PostTypeId="1" CreationDate="2021-01-12T06:54:49.713" Score="3" ViewCount="79" Body="&lt;p&gt;Modern experiments, especially in fields like particle physics, often collect far more data than they can process and save. Similarly, I can imagine that historical experimenters were limited by how much they could write down.&lt;/p&gt;&#xA;&lt;p&gt;Experimenters have to carefully consider which data captured by their sensors to save, and which to throw away, roughly dependent on the theory they are testing.&lt;/p&gt;&#xA;&lt;p&gt;Are there known examples of instances in which it was later noticed that data which was collected, but thrown away, could have been used? Including instances where a new theory might have been able to be tested by historical data that was thrown away?&lt;/p&gt;&#xA;" OwnerUserId="13682" LastEditorUserId="10106" LastEditDate="2021-01-12T07:03:19.163" LastActivityDate="2021-01-12T07:03:19.163" Title="Experimenters regretting throwing away data?" Tags="|physics|experimental-physics|big-list|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12740" PostTypeId="2" ParentId="12723" CreationDate="2021-01-13T15:26:12.053" Score="1" Body="&lt;p&gt;I agree to candied_orange that it might be a stone, but for different reasons.&lt;/p&gt;&#xA;&lt;p&gt;There still is a practice among nomadic people to heat stones in a fire [aka store energy] and then transfer them into a container (e.g. the fur of an animal filled with water) to cook/boil the contents [release the energy].&lt;/p&gt;&#xA;&lt;p&gt;Here are two sources I could find:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://roadsandkingdoms.com/2013/bodog/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Travel report from Mongolia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mongolfood.info/en/recipes/khorkhog.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nomadic recipe&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13693" LastEditorUserId="13693" LastEditDate="2021-01-14T05:50:41.843" LastActivityDate="2021-01-14T05:50:41.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12741" PostTypeId="1" CreationDate="2021-01-13T18:24:17.043" Score="4" ViewCount="143" Body="&lt;p&gt;It must have been observed, for example, that there was no element lighter than hydrogen or that there were not different-charged electrons (I believe the electron was discovered prior to Planck's Constant) but only one charge and that was the smallest charge. So was imagining that energy itself came in a smallest packet a huge surprise or was it confirmation of something that had already been theorized?&lt;/p&gt;&#xA;" OwnerUserId="10856" OwnerDisplayName="releseabe" LastEditorUserId="10106" LastEditDate="2021-01-14T16:11:26.833" LastActivityDate="2021-01-14T18:14:10.120" Title="Was Planck's Constant a huge leap conceptually?" Tags="|physics|quantum-mechanics|" AnswerCount="2" CommentCount="5" ClosedDate="2021-01-21T07:22:03.863" ContentLicense="CC BY-SA 4.0" />
  <row Id="12742" PostTypeId="2" ParentId="12741" CreationDate="2021-01-13T18:48:58.793" Score="6" Body="&lt;p&gt;The quantization of energy of light of a given frequency into photons was a truly enormous leap from classical physics.&lt;/p&gt;&#xA;&lt;p&gt;First, the ideas of atoms and elementary particles including electrons were being developed at the roughly same time as quantum mechanics. So the premise of the question is not quite right; the electron was discovered by JJ Thompson in 1897, and the charge by Millikan in 1913, while Planck's paper on black body radiation was published in 1900.&lt;/p&gt;&#xA;&lt;p&gt;But even if we take for granted the existence of an electron with a unit charge, the quantization of energy is yet another major conceptual leap. Maxwell's classical theory of electromagnetism can very easily incorporate point charges&lt;span class=&quot;math-container&quot;&gt;$^\star$&lt;/span&gt;, and even if it does not explain why charges should come in units of the electron charge, there is no logical problem with there being a unit charge.&lt;/p&gt;&#xA;&lt;p&gt;However Maxwell's equations make a very firm prediction that the energy of an electromagnetic wave is proportional to the amplitude of the wave, and the amplitude is a continuous quantity. There is no way to reproduce &lt;span class=&quot;math-container&quot;&gt;$E=\hbar \omega$&lt;/span&gt; within classical electromagnetism; since accelerating charges produce electromagnetic waves, an charge oscillating at a given frequency can produce larger or smaller waves by increasing the amplitude of its oscillation. In classical physics, particles can oscillate with any amplitude. Thus there really is something extra and (from the point of view of classical physics) extreme that is needed to explain the black body radiation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^\star$&lt;/span&gt; modulo issues regarding the self force, which I will ignore.&lt;/p&gt;&#xA;" OwnerDisplayName="Andrew" LastActivityDate="2021-01-13T18:48:58.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12743" PostTypeId="1" CreationDate="2021-01-14T15:17:27.233" Score="0" ViewCount="95" Body="&lt;p&gt;I was recently reading a paper on the historical development of probability theory where Cardano is presented as having discovered some elementary laws of probability in the 1550s. But he is described as confusing the multiplication rule for independent events with the multiplication of the odds of these events.&#xA;How well did he really understand probability and what is the most likely way that he derived these results?&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="4931" LastEditDate="2021-01-14T19:29:27.933" LastActivityDate="2021-01-14T19:29:27.933" Title="How well did Cardano understand elementary probabilities, as shown from his writings?" Tags="|mathematics|biographical-details|probability|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12744" PostTypeId="2" ParentId="12741" CreationDate="2021-01-14T17:57:34.657" Score="3" Body="&lt;p&gt;In early modern philosophy and science there was a &lt;a href=&quot;https://en.wikipedia.org/wiki/Corpuscular_theory_of_light&quot; rel=&quot;nofollow noreferrer&quot;&gt;corpuscular theory of light&lt;/a&gt; and Newton's authority was also behind it. At the end of the 19th c. electromagnetic phenomena demonstrated that light is a wave, but Planck's work seemed to vindicate the alternative view. Actually the constant was introduced as a mathematical trick - an integral diverges to infinity, while the corresponding sum is finite. Planck resorted to it in exasperation and later even said he regretted inventing this.   But this view explains a lot more - something which could not and did not happen at once: there was no leap, only a gradual realization.&lt;/p&gt;&#xA;&lt;hr/&gt;&#xA;see  Jammer M., *The Conceptual Development of Quantum Mechanics.* NY, 2nd ed. 1989. (There is also a scale problem : it took a generation and more to admit that quantum theory will stay, but from a distance it is a revolution.) &#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2021-01-14T18:14:10.120" LastActivityDate="2021-01-14T18:14:10.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12745" PostTypeId="1" CreationDate="2021-01-14T23:37:49.387" Score="3" ViewCount="311" Body="&lt;p&gt;I've recently seen the term &amp;quot;arm's-length recursion&amp;quot; for a recursive method with a check that short-circuits the method's true or intended base case. What's the origin of this term? How did it get this name?&lt;/p&gt;&#xA;" OwnerUserId="9499" OwnerDisplayName="Jason Hemann" LastEditorUserId="10106" LastEditDate="2021-01-15T06:21:22.813" LastActivityDate="2022-10-06T05:15:13.687" Title="Where does the term &quot;arm's-length recursion&quot; come from?" Tags="|terminology|computer-science|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12746" PostTypeId="1" CreationDate="2021-01-15T16:30:24.760" Score="6" ViewCount="215" Body="&lt;p&gt;Hermann Weyl is widely reported as making the following observation about a compact space:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;This was reported in a paper by Edwin Hewitt in a 1960 paper in AMM, and hence more widely propagated since. (I first encountered it in W.A. Sutherland's &amp;quot;Introduction to Metric and Topological Spaces&amp;quot; (Oxford Science Publications, 1975).)&lt;/p&gt;&#xA;&lt;p&gt;However, I am having difficulty tracking down the precise origin. (Weyl wrote prolifically.)&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know where he first made that remark?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="205" LastEditDate="2021-01-16T16:05:49.027" LastActivityDate="2021-01-16T16:05:49.027" Title="What is the source of Hermann Weyl's remark about &quot;near-sighted policemen&quot; with respect to compact spaces?" Tags="|mathematics|quote|topology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12747" PostTypeId="1" CreationDate="2021-01-15T18:22:14.367" Score="6" ViewCount="505" Body="&lt;p&gt;I've seen that there is some information in the first volume of Polya's &amp;quot;Mathematics and Plausible Reasoning&amp;quot;. Also, the following paper deals with the usage in probability - &amp;quot;The historical development of the use of generating functions in probability theory&amp;quot; by Seal (&lt;a href=&quot;https://www.e-periodica.ch/cntmng?pid=msa-001:1949:49::299&quot; rel=&quot;noreferrer&quot;&gt;https://www.e-periodica.ch/cntmng?pid=msa-001:1949:49::299&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;However, I am interested to see whether anybody know any other place where one may find information on the topic. Is there any other literature on the topic?&lt;/p&gt;&#xA;" OwnerUserId="13183" LastActivityDate="2023-01-06T15:26:17.340" Title="History of generating functions" Tags="|mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12748" PostTypeId="2" ParentId="12747" CreationDate="2021-01-15T21:46:59.793" Score="6" Body="&lt;p&gt;Seal covers the probabilistic side pretty well. A more recent source with a lot of material on generating functions is  &lt;a href=&quot;https://www.springer.com/gp/book/9780387878560&quot; rel=&quot;noreferrer&quot;&gt;Fisher's History of the Central Limit Theorem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For the combinatorial side, &lt;a href=&quot;https://www.google.com/books/edition/Enumerative_Combinatorics_Volume_1/EvJg1VjIGyMC?hl=en&quot; rel=&quot;noreferrer&quot;&gt;Stanley's Enumerative Combinatorics, v1&lt;/a&gt; has extensive historical notes at the end of each chapter, chapters 4 and 5 are on rational generating functions. The notes to them begin as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The basic theory of rational generating functions in one variable belongs to the calculus of finite differences. Charles Jordan [4.25] ascribes the origin of this calculus to Brook Taylor in 1717 but states that the real founder was James Stirling in 1730. The first treatise on the subject was written by Euler in 1755, where the notation &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;&#xA;for the difference operator was introduced.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The compositional formula (Theorem 5.1.4) and the exponential formula (Corollary&#xA;5.1.6) had many precursors before blossoming into their present form. A&#xA;purely formal formula for the coefficients of the composition of two exponential&#xA;generating functions goes back to Faa di Bruno [23][24] in 1855 and 1857, and&#xA;is known as Faa di Bruno's formula. For additional references on this formula,&#xA;see [2.3, p. 137]. An early precursor of the exponential formula is due to Jacobi&#xA;[38]. The idea of interpreting the coefficients of &lt;span class=&quot;math-container&quot;&gt;$e^{F(x)}$&lt;/span&gt; combinatorially was considered in certain special cases by Touchard [ 69] and by Riddell and Uhlenbeck [56]... It was not until the early 1970s that a general combinatorial interpretation of&#xA;&lt;span class=&quot;math-container&quot;&gt;$e^{F(x)}$&lt;/span&gt; was developed independently by Foata and Schiltzenberger [26], Bender and&#xA;Goldman [3.3], and Doubilet, Rota, and Stanley [3.12].&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/Combinatorial_Enumeration/ufiJAwAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Combinatorial Enumeration by Goulden and Jackson&lt;/a&gt; also has notes and references at the end of chapters, but far less extensive than Stanley's.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-15T22:02:42.047" LastActivityDate="2021-01-15T22:02:42.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12750" PostTypeId="2" ParentId="12747" CreationDate="2021-01-16T03:49:06.623" Score="1" Body="&lt;p&gt;An excellent source on the history of generating functions (in the broader context of Harmonic Analysis) are the papers of&#xA;George Mackey:&lt;/p&gt;&#xA;&lt;p&gt;Harmonic analysis as the exploitation of symmetry - a historical survey. Zbl 0437.43001&#xA;Bull. Am. Math. Soc., New Ser. 3, 543-698 (1980).&lt;/p&gt;&#xA;&lt;p&gt;The scope and history of commutative and noncommutative harmonic analysis. Zbl 0766.43001&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-01-16T03:49:06.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12751" PostTypeId="2" ParentId="12746" CreationDate="2021-01-16T08:31:40.957" Score="9" Body="&lt;p&gt;The closest match I could find is in H. Weyl, &amp;quot;Harmonics on homogeneous manifolds.&amp;quot; &lt;em&gt;Annals of Mathematics&lt;/em&gt;, Second Series, Vol. 35, No. 3, July 1934, pp. 486-499, as &lt;a href=&quot;https://www.google.com/books/edition/Gesammelte_Abhandlungen/vzMKd3IlqswC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=manifolds&quot; rel=&quot;noreferrer&quot;&gt;reproduced&lt;/a&gt; in K. Chandrasekharan (ed.), &amp;quot;Hermann Weyl: Gesammelte Abhandlungen Band III,&amp;quot; Springer 1968. On page 389 of &amp;quot;Gesammelte Abhandlungen&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A &amp;quot;finite&amp;quot; country can be watched by a finite number of policemen, however small the radius of action of the single policeman may be!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-01-16T08:37:44.927" LastActivityDate="2021-01-16T08:37:44.927" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12752" PostTypeId="2" ParentId="7879" CreationDate="2021-01-16T09:24:57.830" Score="3" Body="&lt;p&gt;I found a useful reference that might help answering this question (this doesn't constitute an answer to the questions in post). The reference is from p. 46-47 of the book &amp;quot;From Riemann to Differential Geometry and Relativity&amp;quot; - in those pages the author cites a 1863 letter from Enrico Betti to another mathematician:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What gave Riemann the idea of cuts was that Gauss defined them to him, talking about other matters, in a private conversation. In his writings one finds that analysis situs, that is, the consideration of quantities independently from their measure, is important; in the last years of his life he has been much concerned with a problem in analysis situs, namely: given a winding thread and knowing, at every one of its self-intersections, which part is above and which below, to find whether it can be unwound without making knots; this problem he didn't succeed in solving except in special cases...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I guess the &amp;quot;thread problem&amp;quot; which is mentioned in Betti's letter is a topological problem (on braids) in the modern sense. So understanding the meaning of the problem which Gauss tried to solve might shed some light about the note mentioned in Epple's article - in which Gauss made the first mathematical discussion of a certain braid.&lt;/p&gt;&#xA;&lt;p&gt;In addition, there is a letter from Mobius  to Gauss on February 2, 1847:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As i heard from Wilhelm  Weber,  you found it nessecary, a few years ago, as an introduction or preperation for the theory of electrical or magnetic currents, to write a treatise on all possible wraps (configurations) of a thread. Is it not to be hoped that this essay will appear soon? The fulfillment of this hope will be very much appreciated by me and certainly also by many others.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2021-01-17T11:25:13.017" LastActivityDate="2021-01-17T11:25:13.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12755" PostTypeId="1" CreationDate="2021-01-17T16:11:41.493" Score="1" ViewCount="65" Body="&lt;p&gt;This terminology seems to be referred to a notion invented in an old book of Oliver Heaviside [1]. What is it and why the Biot-Savart law is not unique unless it is applied to a Heaviside rational current element as claimed in this paper [2]? Moreover, why such direct-current element is a physical impossibility?&lt;/p&gt;&#xA;&lt;p&gt;[1] Oliver, Heaviside. Electrical Papers, Vol.2, Article 48. 1892&#xA;[2] Hammond, P. &amp;quot;Electric and magnetic images.&amp;quot; Proceedings of the IEE-Part C: Monographs 107, no. 12 (1960): 306-313.&lt;/p&gt;&#xA;" OwnerUserId="13720" LastActivityDate="2021-01-17T16:11:41.493" Title="What is a Heaviside rational current element?" Tags="|electromagnetism|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12756" PostTypeId="1" CreationDate="2021-01-17T16:44:43.303" Score="3" ViewCount="248" Body="&lt;p&gt;I am looking for the earliest published source that gives and perhaps proves the identity &lt;span class=&quot;math-container&quot;&gt;$\det(I+AB)=\det(I+BA)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; are just rectangular matrices of finite dimensions (as opposed to possibly infinite-dimensional operators of 'trace class', etc.).&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia article for this identity attributed it to J. J. Sylvester without citation of any original work of Sylvester, until June 2019 when the page was &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Weinstein%E2%80%93Aronszajn_identity&amp;amp;oldid=903027375&quot; rel=&quot;nofollow noreferrer&quot;&gt;edited&lt;/a&gt;&#xA;and now attributes this identity to Nathan Aronszajn and Alexander Weinstein. A few months later, Terence Tao edited a 2013 blog post changing the attribution from Sylvester to Aronszajn–Weinstein, presumably following the Wikipedia article.&lt;/p&gt;&#xA;&lt;p&gt;A. Chervov, G. Falqui and V. Rubtsov. &lt;em&gt;Algebraic properties of Manin matrices 1.&lt;/em&gt; (2009), Remark 23 tells me that the name &amp;quot;Aronszajn–Weinstein formula&amp;quot; comes from Tosio Kato. &lt;em&gt;Perturbation Theory for Linear Operators&lt;/em&gt; (1966), chapter 4 section 6. Browsing through Kato, I found lines like the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For any &lt;span class=&quot;math-container&quot;&gt;$\zeta\in P(T)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$A(T-\zeta)^{-1}$&lt;/span&gt; is then a degenerate operator belonging to &lt;span class=&quot;math-container&quot;&gt;${\scr B}({\sf X})$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\omega(\zeta)&#xA;&amp;amp;=\omega(\zeta;T,A)&#xA;=\det(1+A(T-\zeta)^{-1})\\&#xA;&amp;amp;=\det[(T+A-\zeta)(T-\zeta)^{-1}]&#xA;\end{align*}&lt;/span&gt;&#xA;is defined...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It definitely seems to have some resemblances to the identity I am pursuing, but I couldn't find the actual identity in any form in his book. This may very well be due to my lack of any knowledge of the topic; perhaps it is simple for the expert to extract from this abstraction the identity in question. For what it's worth, casually browsing the references Kato provides to the original papers (c. 1940s) of Aronszajn and Weinstein also did not help me. Also, this whole Aronszajn–Weinstein stuff appears very general to me compared to the case for &lt;span class=&quot;math-container&quot;&gt;$m\times n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n\times m$&lt;/span&gt; matrices, with &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; finite — surely even if Aronszajn and Weinstein did prove the identity in their general case, someone should have proven the simpler version before them?&lt;/p&gt;&#xA;&lt;p&gt;An earlier edit of the Wikipedia page for this identity stated without any explicit citation that Sylvester had stated the result without proof in a paper of his of 1857. That said, I've looked through volume 2 of &lt;em&gt;The Collected Mathematical Papers of James Joseph Sylvester&lt;/em&gt;, which is supposed to contain all his papers from 1857, and did not find anything useful. It's worth noting however that there's an identity by the name &amp;quot;Sylvester's determinant identity&amp;quot; that stems from J. J. Sylvester. &lt;em&gt;On the relation between the minor determinants of linearly equivalent quadratic functions&lt;/em&gt; (1851).&lt;/p&gt;&#xA;&lt;p&gt;In summary, I would like to see a paper of J. J. Sylvester, or perhaps someone earlier, containing the identity &lt;span class=&quot;math-container&quot;&gt;$\det(I+AB)=\det(I+BA)$&lt;/span&gt;. It would be nice if the paper contained a proof, but it isn't necessary. As an aside, though not as important, it would also be nice to see where the first usage of the term &amp;quot;Weinstein–Aronszajn identity&amp;quot; as referring to just the identity &lt;span class=&quot;math-container&quot;&gt;$\det(I+AB)=\det(I+BA)$&lt;/span&gt; showed up, preferably from before 2009.&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="13719" LastActivityDate="2021-01-17T16:44:43.303" Title="The origins of $\det(I+AB)=\det(I+BA)$" Tags="|mathematics|reference-request|linear-algebra|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12757" PostTypeId="1" AcceptedAnswerId="12758" CreationDate="2021-01-17T19:30:08.970" Score="0" ViewCount="351" Body="&lt;p&gt;Who were the mathematicians who hated teaching in class or disliked teaching?&lt;/p&gt;&#xA;&lt;p&gt;From  this &lt;a href=&quot;https://www.josephdagnese.com/blog/2019/5/10/24-things-you-didnt-know-about-carl-friedrich-gauss#:%7E:text=He%20hated%20teaching%2C%20and%20believed,the%20coming%20of%20the%20euro.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Article&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss&lt;/a&gt; hated teaching, and believed some students robbed him of his time.&lt;/p&gt;&#xA;&lt;p&gt;From this  &lt;a href=&quot;https://www.newyorker.com/magazine/2006/08/28/manifold-destiny&quot; rel=&quot;nofollow noreferrer&quot;&gt;Article&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Grigori_Perelman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Perelman&lt;/a&gt; hated teaching in class.  He decided to work alone same like Isaac Newton so he left a U.S. university and went to his native country Russia, where he was paid less than a hundred dollars per month.&lt;/p&gt;&#xA;&lt;p&gt;Are there any others?&lt;/p&gt;&#xA;" OwnerUserId="10835" LastEditorUserId="12330" LastEditDate="2021-01-18T07:52:47.567" LastActivityDate="2021-01-29T08:29:21.970" Title="Who were the mathematicians who hated teaching in class or disliked teaching?" Tags="|biographical-details|mathematicians|teaching|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12758" PostTypeId="2" ParentId="12757" CreationDate="2021-01-18T01:23:41.403" Score="2" Body="&lt;p&gt;One example is &lt;a href=&quot;https://en.wikipedia.org/wiki/Sidney_Coleman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sidney Coleman&lt;/a&gt;, though he was a theoretical physicist and not a mathematician. In an interview, he said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I hate [teaching]. You do it as part of the job. Well, that's of course false ... or maybe more true than false when I say I hate it. ... But I certainly would be just as happy if I had no graduate students. ... Occasionally there is a graduate student who is a joy to collaborate with. Both David Politzer and Erick Weinberg were of this kind, but they were essentially almost mature physicists. They were very bright by the time they came to me. In general, working with a graduate student is like teaching a course. It's tedious, unpleasant work. A pain in the neck. You do it because you're paid to do it. If I weren't paid to do it I certainly would never do it.&lt;a href=&quot;http://www.aip.org/history-programs/niels-bohr-library/oral-histories/31234&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Interview of Sidney Coleman by Katherine Sopka on 1977 January 18,&#xA;Niels Bohr Library &amp;amp; Archives, American Institute of Physics,&#xA;College Park, MD USA,&#xA;&lt;a href=&quot;http://www.aip.org/history-programs/niels-bohr-library/oral-histories/31234&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.aip.org/history-programs/niels-bohr-library/oral-histories/31234&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10710" LastActivityDate="2021-01-18T01:23:41.403" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12759" PostTypeId="1" AcceptedAnswerId="12760" CreationDate="2021-01-18T17:35:17.970" Score="7" ViewCount="432" Body="&lt;p&gt;This question concerns Fatou's Lemma:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$(f_n)$&lt;/span&gt; be a sequence of nonnegative measurable functions on a measure space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;. Define &lt;span class=&quot;math-container&quot;&gt;$f: X\to [0,+\infty]$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$f(x) :=\liminf_{n\to\infty} f_n(x)$&lt;/span&gt;, for every &lt;span class=&quot;math-container&quot;&gt;$x\in X$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is measurable and &lt;span class=&quot;math-container&quot;&gt;$\int_X f\,d\mu \le \liminf_{n\to\infty} \int_X f_n\,d\mu$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This result is, of course, very famous now, but I'm having a hard time tracing back its genesis. After his thesis on integration theory, Fatou has worked in several areas, including complex analysis and iterated functions; so a natural guess would be that this Lemma originates from his PhD thesis, but I couldn't find any precise quote. In fact, it seems that the first occurrence of the name &amp;quot;Fatou's Lemma&amp;quot; (par excellence) can be found in a 1932 &lt;a href=&quot;https://www.jstor.org/stable/1989506?casa_token=fNtkJZBACXwAAAAA%3AEwhNAltGxG1lVK8jfipAjBgOhyvKiDLm6QTlx9D24w7Lu5_7gOAIG8gOJPVkH-LCIHq6n_OwMXi9WS-FeCoN_6VMb_pOpvDIkLgrrURISxeBxwoDtTo&amp;amp;seq=1#metadata_info_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; by Doob, although I'm not completely sure Doob was referring to precisely what today goes under the same name.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have closer information?&lt;/p&gt;&#xA;" OwnerUserId="3022" LastActivityDate="2021-01-18T20:00:48.020" Title="History of Fatou's Lemma" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12760" PostTypeId="2" ParentId="12759" CreationDate="2021-01-18T19:21:52.530" Score="8" Body="&lt;p&gt;Yes, Fatou formulated the lemma the modern way that Doob refers to. It appears in Fatou's paper &lt;a href=&quot;https://projecteuclid.org/euclid.acta/1485887163&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Series trigonometriques et series de Taylor&lt;/em&gt;, p. 375&lt;/a&gt; (Acta Math., 30 (1906) 335-400), which he presented as his doctoral thesis. It was a side result in proving that if &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is bounded and analytic for &lt;span class=&quot;math-container&quot;&gt;$|z|&amp;lt;1$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\lim_{r\nearrow1} f(re^{it})$&lt;/span&gt; exists almost everywhere. This, in turn, was part of his proof of a general form of Parseval's identity for trigonometric series of summable functions. Fatou's research was personally encouraged and aided by Lebesgue himself.&lt;/p&gt;&#xA;&lt;p&gt;The details are described in &lt;a href=&quot;https://www.maa.org/press/maa-reviews/lebesgues-theory-of-integration-its-origins-and-development&quot; rel=&quot;noreferrer&quot;&gt;Lebesgue's Theory of Integration: Its Origins and Development by Hawkins, pp. 168-172&lt;/a&gt;. Theorem 6.6 in the quote below is what we now call the Fatou's lemma:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Theorem 6.6 is similar to the theorem of Beppo Levi referred to in&#xA;5.3. Fatou proved his theorem as follows: Let &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; denote a positive number, and consider the set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; of all &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f(x) &amp;lt; L$&lt;/span&gt;. A sequence of functions &lt;span class=&quot;math-container&quot;&gt;$g_n$&lt;/span&gt; is constructed by setting &lt;span class=&quot;math-container&quot;&gt;$g_n(x) = f_n(x)$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$f_n(x) &amp;lt;L$&lt;/span&gt; and&#xA;&lt;span class=&quot;math-container&quot;&gt;$g_n(x)=f(x)$&lt;/span&gt; otherwise. Then for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;g_n(x)&amp;lt;L$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$g_n(x)$&lt;/span&gt; converges&#xA;to &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;. Lebesgue's Theorem 5.3 applies and shows that &lt;span class=&quot;math-container&quot;&gt;$\int_E f=&#xA;\lim \int_E g_n &amp;lt;\liminf \int_a^b f_n$&lt;/span&gt;. The lemma then follows, since the right-hand side of the inequality is independent of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The theorem 5.3 referred to is Lebesgue's extension of&#xA;the theorems of Arzela and Osgood to convergence of bounded measurable functions. The theorem of Levi was that if &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; is a non-decreasing sequence of summable functions such that &lt;span class=&quot;math-container&quot;&gt;$\lim \int f_n$&lt;/span&gt; is finite, then &lt;span class=&quot;math-container&quot;&gt;$f(x) = \lim f_n (x)$&lt;/span&gt; is finite almost everywhere and summable with &lt;span class=&quot;math-container&quot;&gt;$\int f_n= \lim \int f_n$&lt;/span&gt;. It was published by Levi the same year in &lt;em&gt;Sopra l'integrazione delle serie&lt;/em&gt; (Milano Ist. Lomb.&#xA;Rend., (2) 39 (1906) 775-80) and used by Fubini to prove his theorem on exchanging the order of integration. Here is some broader context from Hawkins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Parseval's equality for any summable function, bounded or not, was&#xA;obtained shortly thereafter by Pierre Fatou (1878-1929). As his doctoral&#xA;thesis Fatou [1906] presented a paper on &amp;quot;&lt;em&gt;Series trigonometriques&#xA;et series de Taylor&lt;/em&gt;&amp;quot; with the principal aim &amp;quot;to show the advantage that&#xA;can be obtained in these questions from the new notions of the measure&#xA;of sets and the generalized definite integral&amp;quot; [1906: 335]. He was particularly&#xA;encouraged in this undertaking by his &amp;quot;friend H. Lebesgue&#xA;who has not ceased to interest himself with my researches and whose&#xA;advice has been very useful to me&amp;quot; [1906: 338].&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-18T20:00:48.020" LastActivityDate="2021-01-18T20:00:48.020" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12761" PostTypeId="1" AcceptedAnswerId="12791" CreationDate="2021-01-19T03:45:03.737" Score="3" ViewCount="145" Body="&lt;p&gt;When some fundamental new theory is published, derived work will be done &amp;quot;instantly&amp;quot;, example from quantum mechanics:&lt;/p&gt;&#xA;&lt;p&gt;1926 Schrödinger equation&lt;/p&gt;&#xA;&lt;p&gt;1927 The first application of quantum mechanics to the diatomic hydrogen molecule&lt;/p&gt;&#xA;&lt;p&gt;1928 The concept of the Bloch state&lt;/p&gt;&#xA;&lt;p&gt;1928 Relativistic quantum mechanics&lt;/p&gt;&#xA;&lt;p&gt;1940s Quantum electrodynmics.&lt;/p&gt;&#xA;&lt;p&gt;But wave guide theory is only published in 1897 &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786449708620969&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Passage of Electric Waves through Tubes&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The above QM theories are much more &amp;quot;harder&amp;quot; than the theory of wave guide, then why is it published after such a long timer after Maxwell's equations were published?&lt;/p&gt;&#xA;" OwnerUserId="13727" LastEditorUserId="10106" LastEditDate="2021-01-19T05:44:16.520" LastActivityDate="2021-11-22T13:07:25.773" Title="Why is wave guide theory developed so long after Maxwell's work was published?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12762" PostTypeId="1" CreationDate="2021-01-19T09:22:26.780" Score="2" ViewCount="355" Body="&lt;p&gt;Here &lt;a href=&quot;https://en.wikipedia.org/wiki/Narinder_Singh_Kapany&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiki&lt;/a&gt; says Narinder Singh Kapany&#xA;is &amp;quot;father of fiber optics&amp;quot;. Here &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3146387/&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;&#xA;says Charles Kuen Kao is &amp;quot;father of fiber optics&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Who is the father of fiber optics: Charles Kuen Kao or Narinder Singh Kapany?&lt;/p&gt;&#xA;" OwnerUserId="13728" LastActivityDate="2021-01-19T14:28:44.093" Title="Who is the father of fiber optics?" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12763" PostTypeId="1" CreationDate="2021-01-19T11:00:34.737" Score="2" ViewCount="77" Body="&lt;p&gt;How did the historical development of the notion of homeomorphism take place?&lt;/p&gt;&#xA;&lt;p&gt;At first, I think the most important thing to study were continuous functions and metric structures.&lt;/p&gt;&#xA;&lt;p&gt;When did the focus of mathematicians shift from metric to open sets?&lt;/p&gt;&#xA;" OwnerUserId="13275" LastEditorUserId="12330" LastEditDate="2021-01-20T17:03:31.533" LastActivityDate="2021-01-20T17:03:31.533" Title="Development of the Idea of Homeomorphism?" Tags="|topology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12764" PostTypeId="2" ParentId="12762" CreationDate="2021-01-19T14:28:44.093" Score="2" Body="&lt;p&gt;It depends on, what they did.&lt;/p&gt;&#xA;&lt;p&gt;Neither of them found the concept, but both made a lot of inventions/developments to make optical cables working.&lt;/p&gt;&#xA;&lt;h3&gt;Narinder Singh Kapany&lt;/h3&gt;&#xA;&lt;p&gt;He was the &lt;a href=&quot;https://web.archive.org/web/20050731094716/http://www.soe.ucsc.edu/administration/dean/council/Kapany_Bio.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;founder and chairman K2 Optronics Inc&lt;/a&gt;, and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;published over 100 scientific papers and four books on opto-electronics and entrepreneurship.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h3&gt;Charles Huen Kao&lt;/h3&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;created various methods to combine glass fibres with lasers in order to transmit digital data, which laid the groundwork for the evolution of the Internet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Fibre optics was not a single, revolutionary development, instead it was the evolution of many small results. Both of them made many essential steps. This technology seems to have many fathers.&lt;/p&gt;&#xA;&lt;p&gt;The possibility that it could work, was probably well known about from the development of the LEDs.&lt;/p&gt;&#xA;" OwnerUserId="217" LastActivityDate="2021-01-19T14:28:44.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12765" PostTypeId="1" AcceptedAnswerId="12766" CreationDate="2021-01-20T08:00:59.760" Score="2" ViewCount="68" Body="&lt;p&gt;I read in Wikipedia that Arend Heyting, logician, died in Lugano while all the biographies focus on his career in Amsterdam and &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Heyting/&quot; rel=&quot;nofollow noreferrer&quot;&gt;nearby&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Do we know if he lived in Lugano or not, and why died there?&lt;/p&gt;&#xA;" OwnerUserId="7165" LastEditorUserId="10106" LastEditDate="2021-01-20T08:47:49.800" LastActivityDate="2021-01-20T11:29:54.597" Title="Heyting and Lugano" Tags="|biographical-details|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12766" PostTypeId="2" ParentId="12765" CreationDate="2021-01-20T11:29:54.597" Score="5" Body="&lt;p&gt;A.S. Troelstra, &lt;a href=&quot;https://www.dwc.knaw.nl/DL/levensberichten/PE00000845.pdf&quot; rel=&quot;noreferrer&quot;&gt;Levensbericht A. Heyting&lt;/a&gt;, in: Jaarboek, 1980, Amsterdam, pp. 172-175:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hoewel hij zich na zijn hartaanval wel ontzien moest, genoot hij toch een redelijke gezondheid tot vlak voor zijn dood. Hij overleed tijdens een vakantie te Lugano, Zwitserland, op 9 juli 1980, na een kortstondig ziekbed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;With Google translator:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He died while on vacation in Lugano, Switzerland, on July 9 1980, after a short illness.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2021-01-20T11:29:54.597" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12767" PostTypeId="2" ParentId="12684" CreationDate="2021-01-20T18:59:51.447" Score="2" Body="&lt;p&gt;Just to add to the posts that talk of the need to define tolerance, two examples that I am fond of:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The British surveys of northern India during the 19th century, when Mt. Everest was discovered, it was also discovered that a plumb bob in that region measurably misses the center of the earth due to the gravitational attraction of the Himalayas.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;See a brief mention in Wikipedia: &lt;a href=&quot;https://en.wikipedia.org/wiki/Great_Trigonometrical_Survey&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Great_Trigonometrical_Survey&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;That piece includes this reference:&#xA;Pratt, John Henry (1855). &amp;quot;On the Attraction of the Himalaya Mountains, and of the Elevated Regions beyond Them, upon the Plumb-Line in India&amp;quot;. Philosophical Transactions of the Royal Society of London. 145: 53–100. doi:10.1098/rstl.1855.0002. JSTOR 108510.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Google &amp;quot;Palmdale Bulge&amp;quot;. An apparent uplift north of Los Angeles during the 1970's detected by leveling lines- crews that went out with measured rods and accurately level sights to measure the elevation of the topography. After much controversy, and corrections for the curvature of sight lines caused by atmospheric density gradients caused by surface heating by the sun, the bulge largely disappeared.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;As also discussed previously, these effects don't matter when you're building a house. And if you're trying to level a rice paddy, the water surface follows local gravity so variations in the gravitational field don't matter.&lt;/p&gt;&#xA;" OwnerUserId="13734" LastEditorUserId="13734" LastEditDate="2021-08-19T21:40:08.710" LastActivityDate="2021-08-19T21:40:08.710" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12768" PostTypeId="2" ParentId="5943" CreationDate="2021-01-20T19:29:50.167" Score="0" Body="&lt;p&gt;Richard Feynman - Certainly you are joking.&lt;/p&gt;&#xA;&lt;p&gt;I would suggest the biography of J D R Tata: beyond the blue mountains by R M Lala.&lt;/p&gt;&#xA;&lt;p&gt;He is not a physicist but it a good biography&lt;/p&gt;&#xA;" OwnerUserId="13736" LastActivityDate="2021-01-20T19:29:50.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12769" PostTypeId="2" ParentId="12684" CreationDate="2021-01-21T10:51:28.387" Score="1" Body="&lt;p&gt;I'm surprised no one mentioned the &lt;strong&gt;Whitworth Three Plates Method&lt;/strong&gt;, a process that were used to make surfaces very flat using gravity, which you can read about in great detail here: &lt;a href=&quot;https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-three-plates-method&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-three-plates-method&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13747" LastActivityDate="2021-01-21T10:51:28.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12770" PostTypeId="2" ParentId="12684" CreationDate="2021-01-21T15:50:05.773" Score="1" Body="&lt;p&gt;Since the beginning of time, people discovered that if you rub two stones together with some abrasives in between they form a flat surface. They also discovered that if you rub 2 stones together the surface may come concave, i.e we identical curvature on both stones. The solution is to use 3 stones. This method is still valid now and still in use for producing very flat surfaces for scientific and industrial applications.&lt;/p&gt;&#xA;&lt;p&gt;Here are few references from google:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-three-plates-method&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ericweinhoffer.com/blog/2017/7/30/the-whitworth-three-plates-method&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13750" LastEditorUserId="13750" LastEditDate="2021-01-22T22:01:53.670" LastActivityDate="2021-01-22T22:01:53.670" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12771" PostTypeId="1" AcceptedAnswerId="12773" CreationDate="2021-01-21T20:20:09.353" Score="1" ViewCount="457" Body="&lt;p&gt;There is a famous image that pretends to explain the origin of the shapes of the digits by the number of angles in them.&lt;/p&gt;&#xA;&lt;p&gt;I do know that it is erroneous but I would like to know if this is an old legend or a recent one. After all, maybe centuries ago, someone spread this false etymology, maybe ten years ago.&lt;a href=&quot;https://i.stack.imgur.com/fRmjV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fRmjV.png&quot; alt=&quot;image&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13754" LastEditorUserId="13754" LastEditDate="2021-01-21T20:35:31.373" LastActivityDate="2021-01-22T01:28:30.143" Title="How old are the shape of the numbers based on the number of angles?" Tags="|notation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12773" PostTypeId="2" ParentId="12771" CreationDate="2021-01-22T00:40:14.867" Score="3" Body="&lt;p&gt;In the book Florian Cajori &amp;quot;A History of Mathematical Notations&amp;quot; on &lt;a href=&quot;https://www.google.com/books/edition/A_History_of_Mathematical_Notations/rhEh8jPGQOcC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA65&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp 65-66&lt;/a&gt; says: &amp;quot;A French writer, P. Voizot&amp;lt;a reference to 1899 work&amp;gt;, entertained the theory that originally a numeral contained as many angles as it represents units. ... He did not claim credit for this explanation, but ascribed it to a writer in the Genova Catholico Militarite&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="6788" LastEditDate="2021-01-22T01:28:30.143" LastActivityDate="2021-01-22T01:28:30.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12774" PostTypeId="1" AcceptedAnswerId="12775" CreationDate="2021-01-22T15:03:13.303" Score="12" ViewCount="3297" Body="&lt;p&gt;A number of years ago, I read a (non-fiction) story about a young mathematician whose professor wrote an open problem on the board and the student went ahead and wrote down a solution after a moment of thought. The writer goes on to say something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From that moment on, I was afraid of [this student].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think the article mentioned that, when asked questions in class, many students would offer ideas without thought but that this particular student would think quietly before saying what was correct. Would someone know where this story can be found, probably in an article, and who the student was? I would like to say that the student was a young Grothendieck but my Google searches in that direction were fruitless.&lt;/p&gt;&#xA;" OwnerUserId="12679" LastEditorUserId="4251" LastEditDate="2022-09-08T10:07:20.567" LastActivityDate="2022-09-08T10:07:20.567" Title="Story of a student who solves an open problem" Tags="|mathematics|mathematicians|open-problems|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12775" PostTypeId="2" ParentId="12774" CreationDate="2021-01-22T19:21:06.560" Score="35" Body="&lt;p&gt;That's John von Neumann, about whom George Pólya wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I have confirmed that the passage above can be found at &lt;em&gt;The Pólya picture album&lt;/em&gt;, edited by G. L. Alexanderson (Birkhäuser, 1987, p. 154). The sentences quoted above are part of the caption of a picture of von Neumann and &lt;a href=&quot;https://en.wikipedia.org/wiki/Abraham_H._Taub&quot; rel=&quot;noreferrer&quot;&gt;Abraham Taub&lt;/a&gt;, which begins with:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is my only picture of von Neumann. He is the only student of mine I was ever intimidated by. He was so quick.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2021-02-16T11:39:35.577" LastActivityDate="2021-02-16T11:39:35.577" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12777" PostTypeId="1" CreationDate="2021-01-22T10:20:06.110" Score="3" ViewCount="96" Body="&lt;p&gt;The importance of Rolle's theorem lies in the fact that it's used to prove the mean value theorem, which is a central result of analysis, eventually leading to the fundamental theorem of calculus. But if I compare the usual proof of the mean value theorem to other typical proofs, I come to the conclusion that the theory is built up differently than usual. A typical proof in analysis might start off with: &amp;quot;Without loss of generality, assume &lt;span class=&quot;math-container&quot;&gt;$f(0)=0$&lt;/span&gt;, otherwise consider &lt;span class=&quot;math-container&quot;&gt;$f(x)-f(0)$&lt;/span&gt;&amp;quot;, or similar statements. The mean value theorem could go exactly the same way: &amp;quot;without loss of generality, assume &lt;span class=&quot;math-container&quot;&gt;$f(a)=f(b)$&lt;/span&gt;, otherwise consider &lt;span class=&quot;math-container&quot;&gt;$f(x)-(x-a)\frac{f(b)-f(a)}{b-a}$&lt;/span&gt;&amp;quot;, and then we'd just use the proof of Rolle's theorem.&lt;/p&gt;&#xA;&lt;p&gt;But in the &amp;quot;canonical&amp;quot; development of analysis, Rolle's theorem gets to be its own theorem, instead of just being part of the proof of the mean value theorem. Why is that? Does it have any other significant applications I'm unaware of? For instance, the special case of the identity theorem in complex analysis which says that zeros of holomorphic functions &lt;span class=&quot;math-container&quot;&gt;$\neq0$&lt;/span&gt; are isolated is important on its own because it's used in treating isolated singularities of fractions of holomorphic functions. This makes it a reasonable thing to declare as a theorem of its own, even though it could also be a lemma on the way to the identity theorem. I don't know of any such important application of Rolle's theorem. Or is it a pedagogical choice, aiming to make the structure of the argument for the MVT clearer? Any other possible, maybe historical reasons?&lt;/p&gt;&#xA;" OwnerDisplayName="Vercassivelaunos" LastActivityDate="2021-01-23T00:56:20.290" Title="Why does Rolle get its own theorem?" Tags="|real-analysis|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12778" PostTypeId="1" CreationDate="2021-01-23T01:08:33.143" Score="2" ViewCount="104" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ko46P.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ko46P.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;On a forum I frequent this strange device was shown with a query as to its purpose. It is in France just at the North edge of Paris in the St Denis area( it is actually currently the site of Olympic site infrastructure.  The device is incomplete and the signs in front of it (in French) do not refer to the actual device. The clearrish part in the center of the &amp;quot;drum&amp;quot; is a ground glass plate which has regular markings (perhaps for measuring something?)  A forum member was at a tour and the guide mentions that it is the &amp;quot;vestiges of equipment  for research and development&amp;quot;.  The site was previously part of a research park of the French energy company Engie.&lt;/p&gt;&#xA;" OwnerUserId="13764" LastEditorUserId="55" LastEditDate="2021-01-27T00:29:37.007" LastActivityDate="2021-01-27T00:29:37.007" Title="What is the purpose of large drum-like device displayed in Saint-Denis, Paris?" Tags="|earth-sciences|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12779" PostTypeId="1" CreationDate="2021-01-24T11:40:55.290" Score="1" ViewCount="114" Body="&lt;p&gt;I'm reading a 1939 paper by the great and famous J. von Neumann, &amp;quot;On infinite direct products&amp;quot; (of vector spaces), available here &lt;a href=&quot;http://www.numdam.org/item/?id=CM_1939__6__1_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.numdam.org/item/?id=CM_1939__6__1_0&lt;/a&gt;, legally I presume.&lt;/p&gt;&#xA;&lt;p&gt;What he calls the &lt;em&gt;direct product&lt;/em&gt; is clearly what we understand today as a &lt;em&gt;tensor product&lt;/em&gt;, while our understanding of the &lt;em&gt;direct product&lt;/em&gt; is the full (possibly infinite) &lt;em&gt;Cartesian product&lt;/em&gt; with some &lt;em&gt;projections&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I'm interested what is the background for this, was this general usage in the dim and distant past, when did the terminology change and why?&lt;/p&gt;&#xA;&lt;p&gt;(copied here from &lt;a href=&quot;https://math.stackexchange.com/q/3857681&quot;&gt;https://math.stackexchange.com/q/3857681&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="13775" LastEditorUserId="10106" LastEditDate="2021-01-24T15:13:50.733" LastActivityDate="2021-01-24T15:13:50.733" Title="History: Direct Product became Tensor Product?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12780" PostTypeId="1" AcceptedAnswerId="12785" CreationDate="2021-01-24T15:21:51.737" Score="0" ViewCount="214" Body="&lt;p&gt;I wonder if would be right to say that while most ancient astronomical activities, such as building and improvement of observational structures, had a strong astrological component, the Greek philosopher’s or mathematiscians' deduction of the spherical shape of the moon, the Earth and the calculation of the Earth’s dimensions constitutes what we might call basic astronomical research, in that the endpoint was not related to the ability to make horoscopes or to any other astrological activity?&lt;/p&gt;&#xA;&lt;p&gt;Is it therefore right to call these activities - in the above sense - one of the first purely astronomical research activities?&lt;/p&gt;&#xA;" OwnerUserId="5602" LastActivityDate="2021-01-26T22:10:57.143" Title="Greek astronomy vs astrology" Tags="|astronomy|ancient-greece|archaeoastronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12781" PostTypeId="1" CreationDate="2021-01-24T16:15:35.570" Score="3" ViewCount="75" Body="&lt;p&gt;I wanted to ask about how they arrived to the rule of addition of vectors. How did they know that if we add the X's and Y's of two vectors they would get a third vector which has exactly the same direction and magnitude of the force that could replace these two vectors or forces.&lt;/p&gt;&#xA;&lt;p&gt;I'm convinced that it's correct and can feel or see the direction a point will accelerate in if two certain forces are applied to it. And can feel how the resultant force tends to get closer to the bigger force and how if two equal forces are applied with the same angle the resultant force is going to be exactly between them.&lt;/p&gt;&#xA;&lt;p&gt;So I'm sure who invented the vectors had the same feelings and visions too but how did he arrive at this simple method to get such fascinating and exact results, not only did he manage to get the direction but the magnitude of the resultant too!&lt;/p&gt;&#xA;" OwnerUserId="13779" LastEditorUserId="10106" LastEditDate="2021-01-24T16:56:22.947" LastActivityDate="2021-01-24T16:56:22.947" Title="How did we arrive at the rule of addition of vectors?" Tags="|physics|mathematical-physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12782" PostTypeId="1" CreationDate="2021-01-24T18:22:57.817" Score="3" ViewCount="150" Body="&lt;p&gt;I'm self-learning about Model Theory and I just got to the proof of Hilbert's 17th Problem via Model Theory of Real Closed Fields. The 17th problem asks to show that a non-negative rational function must be the sum of squares of rational functions.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that I lack a strong enough understanding of the context of the problem to understand its significance. Why did Hilbert think it was important enough to include it as one of his 23 problems?&lt;/p&gt;&#xA;" OwnerUserId="13780" LastActivityDate="2021-01-24T18:22:57.817" Title="Why is Hilbert's Seventeenth Problem important?" Tags="|mathematics|logic|hilbert|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12784" PostTypeId="1" AcceptedAnswerId="12795" CreationDate="2021-01-25T00:30:18.263" Score="3" ViewCount="396" Body="&lt;p&gt;Roger Apery, a French mathematician most remembered for proving that &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$&lt;/span&gt; (irrational) in &lt;span class=&quot;math-container&quot;&gt;$1979$&lt;/span&gt;. In this video posted by Numberphile: &lt;a href=&quot;https://www.youtube.com/watch?v=ur-iLy4z3QE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=ur-iLy4z3QE&lt;/a&gt; the person in the video briefly stated that during a seminar for which Apery &lt;em&gt;unexpectedly&lt;/em&gt; proved the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt;, mathematicians who attended that seminar asked him about how he came up with his binomial expansion of &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt;, he replied:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Oh they grow in my garden&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I do acknowledge the sense of humor that Apery had but what did he really meant/implied by his reply?&lt;/p&gt;&#xA;" OwnerUserId="13782" LastEditorUserId="10106" LastEditDate="2021-01-25T01:34:20.657" LastActivityDate="2021-01-26T11:08:38.823" Title="What did Roger Apery mean by &quot;Oh they grow in my garden&quot;" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12785" PostTypeId="2" ParentId="12780" CreationDate="2021-01-25T01:05:20.747" Score="5" Body="&lt;p&gt;There was a link between astronomy and astrology, as @Mauro Allegranza notes but the two things were strictly separated. Ptolemy's Almagest (and his many other scientific books) contains no single word, not even a hint of astrology. On astrology he wrote a separate book, Tetrabiblos. Same applies to other Greek writers on astronomy. No one mixed the two subjects.&lt;/p&gt;&#xA;&lt;p&gt;To a lesser extent this applies to Muslim astronomers, Brahe and Kepler, but still the two things were usually strictly separated. When one compares&#xA;&amp;quot;Astronomia Nova&amp;quot; with astrological writings of Kepler, one has a strong impression that these are works of different authors. Same applies to Ptolemy's books.&lt;/p&gt;&#xA;&lt;p&gt;I suppose, an explanation of occupation of astronomers by astrology lies in their need to make a living somehow. There was little public demand for pure astronomy,&#xA;and very few or no positions where you could earn your living by practicing  pure science.&#xA;For the same reason, many mathematicians and astronomers until 17 century&#xA;practiced medicine (or law). This is my own conjecture, please do not ask for confirming references.&lt;/p&gt;&#xA;&lt;p&gt;One confirmation of this is an inspection of the web site which is called &lt;a href=&quot;https://genealogy.math.ndsu.nodak.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math genealogy&lt;/a&gt; You see a lot of people whom we remember as scientists&#xA;and mathematicians, and who obtained their degree in medicine.&lt;/p&gt;&#xA;&lt;p&gt;Edit. There is a strong indication, that some of the astronomy and all astrology came to the Greeks from Babylon, after Macedonian conquests.&#xA;(Astrology did not exist in Greece before that, and astronomy was rudimentary). How well they were separated in Babylonia, is not clear.&#xA;But it seems that they were. At least modern scientists are able to make a clear distinction between astronomical and astrological texts that survive.&lt;/p&gt;&#xA;&lt;p&gt;Edit 2. In the late Roman empire, public interest to science vanished almost completely and the word &amp;quot;mathematicus&amp;quot; at that time meant &amp;quot;astrologer&amp;quot;. Eventually this occupation was simply prohibited by the ban of the Emperor.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-01-26T22:10:57.143" LastActivityDate="2021-01-26T22:10:57.143" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12786" PostTypeId="2" ParentId="2332" CreationDate="2021-01-25T01:44:42.920" Score="1" Body="&lt;p&gt;Though not nearly as ubiquitous as digital computing, analog computing is still an existing technology and is an active research domain, particularly with respect to novel hardware approaches for AI applications.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ibm.com/blogs/research/2019/06/analog-ai-vlsi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;An example of the use of Analog computing from IBM Research&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13783" LastActivityDate="2021-01-25T01:44:42.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12787" PostTypeId="1" AcceptedAnswerId="12788" CreationDate="2021-01-25T02:33:09.853" Score="3" ViewCount="262" Body="&lt;p&gt;I've heard rumors in a couple of places that the modern presentation of a topology as &lt;span class=&quot;math-container&quot;&gt;$(X, \tau)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is our topological space and &lt;span class=&quot;math-container&quot;&gt;$\tau$&lt;/span&gt; is our set of open sets is &amp;quot;somewhat&amp;quot; new and a simplification of how topologies used to be presented. We can also pick simple variants of this such as &lt;span class=&quot;math-container&quot;&gt;$(X, B)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is a basis or subbase and identify them with the modern presentation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;What are the different ways that topologies have been presented or defined over time?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This &lt;a href=&quot;https://www.youtube.com/watch?v=HOGaBNl0VWc&amp;amp;t=339s&quot; rel=&quot;nofollow noreferrer&quot;&gt;youtube video&lt;/a&gt; is a review of an old book called &lt;em&gt;Elements of Point Set Topology&lt;/em&gt;, which defines a system of neighborhoods at each point. The neighborhood presentation is sort of confusing, but I think the way to map it back to a more modern presentation is to note that the union of all neighborhoods for every point forms a basis, &lt;a href=&quot;https://proofwiki.org/wiki/Topology_Defined_by_Neighborhood_System&quot; rel=&quot;nofollow noreferrer&quot;&gt;based on this article on ProofWiki&lt;/a&gt;, but I am a little confused as to the intuition behind a neighborhood system.&lt;/p&gt;&#xA;&lt;p&gt;My question is similar &lt;a href=&quot;https://hsm.stackexchange.com/questions/82/what-examples-led-to-the-modern-definition-of-a-topological-space&quot;&gt;to this one&lt;/a&gt;, but is specifically about the old definitions that were &lt;em&gt;out-competed&lt;/em&gt; in a sense.&lt;/p&gt;&#xA;" OwnerUserId="9243" LastActivityDate="2021-02-11T17:54:51.073" Title="Historical ways of *presenting* or axiomatizing the notion of a topological space" Tags="|topology|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12788" PostTypeId="2" ParentId="12787" CreationDate="2021-01-25T14:35:58.837" Score="6" Body="&lt;p&gt;An early definition of topology is given in the book Kuratowski, Topology I&#xA;(first edition, 1933).&#xA;It is defined in terms of a &amp;quot;closure operator&amp;quot;, &lt;span class=&quot;math-container&quot;&gt;$X\mapsto\overline{X}$&lt;/span&gt;&#xA;acting on the set of all subsets of a set, which satisfies 3 axioms:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\overline{(X\cup Y)}=\overline{X}\cup\overline{Y},\quad \overline{\overline{X}}=\overline{X},$$&lt;/span&gt;&#xA;and &lt;span class=&quot;math-container&quot;&gt;$\overline{X}=X$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is one point or empty.&lt;/p&gt;&#xA;&lt;p&gt;The modern definition was popularised by Bourbaki, and in English by Kelley, in his book General topology, in the early 1950s,&#xA;but it was already in use the 1920s, in the work of P. Aleksandrov and Urysohn, for example.&lt;/p&gt;&#xA;&lt;p&gt;Aleksandrov, in his comments to the Russian edition of his work with Urysohn, credits the modern axioms to Hausdorff, Grundzuge der Mengenlehre, 1914, though his set of axioms included&#xA;the &amp;quot;Hausdorff axiom&amp;quot; which nowadays is not included in the general definition of a topological space. He writes (my translation from the Russian):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In subsequent development (after Hausdorff) of axioms of topological spaces, various notions were chosen as basic: closure (Kuratowski), open set (Aleksandrov), closed set (Sierpinski). All these approaches led to the same now universally accepted notion of a topological space (&lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;-space), and undisputable credit for this belongs to Kuratowski.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastEditorUserId="12330" LastEditDate="2021-01-25T21:12:25.500" LastActivityDate="2021-01-25T21:12:25.500" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12789" PostTypeId="2" ParentId="55" CreationDate="2021-01-25T22:39:10.637" Score="4" Body="&lt;p&gt;I'm not a historian, but I have to answer this because the previous answers have gotten it completely wrong. Leibniz's &lt;span class=&quot;math-container&quot;&gt;$\displaystyle \frac{df}{dx}$&lt;/span&gt; is &lt;em&gt;not equal&lt;/em&gt; to &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I think there are two main differences between Newton's calculus and Leibniz's:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Newton's calculus is about functions. Leibniz's calculus is about relations defined by constraints.&lt;/li&gt;&#xA;&lt;li&gt;In Newton's calculus, there is (what would now be called) a limit built into every operation. In Leibniz's calculus, the limit is a separate operation.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Both points, and the second one especially, seem to be poorly understood today. When people write what looks like Leibniz notation today, they're almost always doing Newtonian functional calculus in a weird and inappropriate notation. Then they complain that it's inconsistent and blame Leibniz. It isn't Leibniz's fault.&lt;/p&gt;&#xA;&lt;p&gt;In actual Leibniz calculus, when you write &lt;span class=&quot;math-container&quot;&gt;$y=x^3$&lt;/span&gt;, that isn't a definition of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, it's a constraint on valid &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; tuples. The locus of valid points may happen to be the graph of a function, but nonfunctional constraints like &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2=1$&lt;/span&gt; are equally acceptable.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; applied to an expression gives you the difference between its value at the current point and at some other point that satisfies the constraints, where the other point depends only on the current point and not on the expression. In other words, if &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; satisfies the constraints then so does &lt;span class=&quot;math-container&quot;&gt;$(x+dx,y+dy)$&lt;/span&gt;. Supposing we use the constraint &lt;span class=&quot;math-container&quot;&gt;$y=x^3$&lt;/span&gt;, we therefore have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{eqnarray}&#xA;   y+dy &amp;amp;=&amp;amp; (x+dx)^3&#xA;\\ dy &amp;amp;=&amp;amp; 3x^2dx+3x\,dx^2+dx^3&#xA;\\ dy/dx &amp;amp;=&amp;amp; 3x^2+3x\,dx+dx^2&#xA;\end{eqnarray}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This holds for any two points satisfying the constraint, if &lt;span class=&quot;math-container&quot;&gt;$dx\ne 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$dy/dx$&lt;/span&gt; is not equal to &lt;span class=&quot;math-container&quot;&gt;$3x^2$&lt;/span&gt; (unless &lt;span class=&quot;math-container&quot;&gt;$dx=-3x$&lt;/span&gt;). But they are equal in an appropriate limit. Leibniz wrote &lt;span class=&quot;math-container&quot;&gt;$dy/dx \mathrel{{}_{\ulcorner\!\urcorner}} 3x^2$&lt;/span&gt; for this. (I've seen this relation called &lt;a href=&quot;https://en.wikipedia.org/wiki/Adequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;adequality&lt;/a&gt;. LaTeX (and MathJax) apparently doesn't have a symbol for it. See the source for the terrible hack I used to write it, which isn't my invention.)&lt;/p&gt;&#xA;&lt;p&gt;Because &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; are defined everywhere the constraint is satisfied, we also have&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{eqnarray}&#xA;   \frac{dy}{dx} + d\Big(\frac{dy}{dx}\Big) &amp;amp;=&amp;amp; 3(x{+}dx)^2+3(x{+}dx)(dx{+}ddx)+(dx{+}ddx)^2&#xA;\\ \frac{d(dy/dx)}{dx} &amp;amp;=&amp;amp; 6x + 6dx + 5ddx + 3x\,ddx/dx + (ddx)^2/dx&#xA;\end{eqnarray}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This holds for any &lt;em&gt;three&lt;/em&gt; points satisfying the constraint. If you question the legitimacy of this, one way of looking at it is to imagine that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are functions of an arbitrary parameter and &lt;span class=&quot;math-container&quot;&gt;$(df)(t)=f(t{+}δt)-f(t)$&lt;/span&gt;. The expression is then a correct statement about &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; evaluated at &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$t{+}δt$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$t{+}2δt$&lt;/span&gt;, and these are arbitrary points since the parameter is arbitrary.&lt;/p&gt;&#xA;&lt;p&gt;Clearly it's a PITA to carry all these terms around, especially when you know they're going to disappear in a limit. But if you &lt;em&gt;know&lt;/em&gt; that they're going to disappear then you can drop them early. That doesn't make the calculus nonrigorous, any more than omitting tedious intermediate algebraic steps (as I just did) makes it nonrigorous. What does make it nonrigorous is failing to observe a distinction between equality and equality-in-the-limit, because then you can prove nonsense like &lt;span class=&quot;math-container&quot;&gt;$dx=-3x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz wrote &lt;span class=&quot;math-container&quot;&gt;$d^2x$&lt;/span&gt; as a shorthand for &lt;span class=&quot;math-container&quot;&gt;$ddx=d(dx)$&lt;/span&gt;, but he didn't write &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\frac{d^2y}{dx^2}$&lt;/span&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz%27s_notation#Leibniz_notation_for_higher_derivatives&quot; rel=&quot;nofollow noreferrer&quot;&gt;at least according to Wikipedia&lt;/a&gt;). I suspect he would've considered it an abomination, because it doesn't mean &lt;span class=&quot;math-container&quot;&gt;$(d^2y)/(dx^2)$&lt;/span&gt;, even though that is a meaningful quantity in his notation. It means &lt;span class=&quot;math-container&quot;&gt;$d(dy/dx)/dx$&lt;/span&gt;. In effect, the scope of the top left &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; operator extends out of the &amp;quot;numerator&amp;quot;, into the &amp;quot;denominator&amp;quot;, and stops &lt;em&gt;halfway through&lt;/em&gt; the product represented by &lt;span class=&quot;math-container&quot;&gt;$dx^2$&lt;/span&gt;. It's as though &lt;span class=&quot;math-container&quot;&gt;$fx^2$&lt;/span&gt; meant not &lt;span class=&quot;math-container&quot;&gt;$f(x^2)$&lt;/span&gt;, not &lt;span class=&quot;math-container&quot;&gt;$f(x)^2$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$f(x)\cdot x$&lt;/span&gt;. I assume that this notation was invented by people who thought that &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\frac{d}{dx}$&lt;/span&gt; was just a weird way of writing a functional derivative.&lt;/p&gt;&#xA;&lt;p&gt;I blame the modern misunderstanding of Leibniz's notation on the lack of a standard way to write anonymous functions. If you define &lt;span class=&quot;math-container&quot;&gt;$f(x)=x^3$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$\dot f$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; is concise and convenient, but people don't like having to name everything. Instead of inventing a sensible, composable notation like &lt;span class=&quot;math-container&quot;&gt;$D(λx.x^3)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(x\mapsto x^3)'$&lt;/span&gt;, they decided to write &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\frac{d}{dx} x^3 = 3x^2$&lt;/span&gt;, which is inconsistent with Leibniz's notation and which still left them with no way of writing an anonymous function unless they happened to be taking the derivative of it.&lt;/p&gt;&#xA;" OwnerUserId="9390" LastActivityDate="2021-01-25T22:39:10.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12790" PostTypeId="1" AcceptedAnswerId="12793" CreationDate="2021-01-25T23:07:11.417" Score="2" ViewCount="131" Body="&lt;p&gt;Entries &lt;span class=&quot;math-container&quot;&gt;$93$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$94$&lt;/span&gt; in Steen and Seebach's &amp;quot;Counterexamples in Topology&amp;quot; are named &amp;quot;Thomas's Plank&amp;quot; and &amp;quot;Thomas's Corkscrew&amp;quot;. These are apparently named for a certain &amp;quot;John Thomas&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Who is this &amp;quot;John Thomas&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;The reference given in the back is to &amp;quot;A regular space, not completely regular&amp;quot; from AMM v 76 (1969) p. 181, by &amp;quot;J. Thomas&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Beyond that I can find nothing else about him apart from a few other sundry papers on topological matters.&lt;/p&gt;&#xA;&lt;p&gt;Who is/was he? Date and place of birth (and death if he's died), and nationality would be a good start, but anything else to start the ball rolling, so to speak, would be useful.&lt;/p&gt;&#xA;&lt;p&gt;Can we even be certain that the author of the paper given above is even the same Thomas whose name is attached to &amp;quot;Thomas's Plank&amp;quot; and &amp;quot;Thomas's Corkscrew&amp;quot; in the first place?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="12839" LastEditDate="2021-01-26T08:29:48.820" LastActivityDate="2021-09-06T00:14:48.300" Title="Who is the John Thomas of &quot;Thomas's Plank&quot; and &quot;Thomas's Corkscrew&quot;?" Tags="|mathematicians|topology|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12791" PostTypeId="2" ParentId="12761" CreationDate="2021-01-26T02:47:35.883" Score="5" Body="&lt;p&gt;This is not an answer but a viewpoint to examine: EM theory as we now know it was largely developed by Oliver Heaviside in the 1880s and 1890s from Maxwell's somewhat awkward formulation (it's tedious). Heaviside began as a telegraphy technician without much formal education, none in science or math. But he showed great physical intuition from experiments with long distance telegraphy. He also taught himself the math (extremely difficult in those days compared to the present); he eventually became a frequent and respected contributor to telegraphy and EE journals. Josiah Willard Gibbs transformed (contributed to?) Heaviside's mathematical approach via vector calculus. See the Wikipedia bio on Heaviside (I'm a physicist and found it eye opening: we were never taught anything about him). The other part to your question is that waveguides depend on EM wave generation, which was not so well understood in the 1880s and 1890s until after these mathematical transformations began and better test instruments were invented. Remember that Heinrich Hertz did not prove that EM waves actually did exist until about 1887 based on his interpretation of Maxwell's work. Also note that non-trivial math was not so widely used in professional science and engineering circles as they are today. Penetration of mathematical ideas (on which waveguides depend) was much slower in the late 19th century compared to just fifty years later. For example, electrical engineers were not universally taught calculus until about World War II (I've seen some of the textbooks). Yes, some practical understanding was achieved by analog reasoning and experimentation but a full appreciation and exploitation eventually required the math. HTH.&lt;/p&gt;&#xA;" OwnerUserId="13790" LastEditorUserId="6229" LastEditDate="2021-10-23T12:24:08.750" LastActivityDate="2021-10-23T12:24:08.750" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12792" PostTypeId="1" AcceptedAnswerId="12794" CreationDate="2021-01-26T07:40:30.817" Score="19" ViewCount="2943" Body="&lt;p&gt;Arxiv website is very good at collecting mathematical/science work updates. However, where were mathematical/science work updates posted before arxiv website existed?&lt;/p&gt;&#xA;" OwnerUserId="13795" LastEditorUserId="10106" LastEditDate="2021-01-26T13:01:18.103" LastActivityDate="2021-02-07T14:09:19.587" Title="Where were mathematical/science works posted before the arxiv website?" Tags="|mathematics|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12793" PostTypeId="2" ParentId="12790" CreationDate="2021-01-26T09:34:20.717" Score="4" Body="&lt;p&gt;This answer assumes that the person we are looking for is identical with the author of the following paper:&lt;/p&gt;&#xA;&lt;p&gt;John Thomas, &amp;quot;A regular space, not completely regular.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 76 (1969), No. 2, pp. 181-182.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.math.wustl.edu/%7Efreiwald/4181RegnotCReg.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; gives the author's affiliation as New Mexico State University. I found two mentions of Dr. Thomas in AMS publications from around the time of publication:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.ams.org/journals/notices/196902/196902FullIssue.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notices of the American Mathematical Society&lt;/a&gt;&lt;/em&gt;, Vol. 16, No. 2, February 1969, p. 394:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dr. John D. Thomas, Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88001.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.ams.org/journals/notices/197010/197010FullIssue.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notices of the American Mathematical Society&lt;/a&gt;&lt;/em&gt;, Vol. 17, No. 6, October 1970, p. 872:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dr. JOHN D. THOMAS of New Mexico State University will be on sabbatical leave during the fall semester of 1970. He will spend his leave at Los Alamos Scientific Laboratory, New Mexico.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We also find Dr. Thomas mentioned in various annual reports of the mathematical sciences department of NMSU.&lt;/p&gt;&#xA;&lt;p&gt;From the &lt;a href=&quot;https://math.nmsu.edu/files/2019/10/65-66.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1965-1966&lt;/a&gt; report we can infer that Dr. Thomas joined the faculty of NMSU around 1960:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While the new staff added for next year are of very high quality and will further strengthen the group, it should be pointed out that of the twenty-four staff members presently hired for next fall, only Drs. Giever, E. Walker, Thomas and Ader will have been on the staff more than four years&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;http://emmy.nmsu.edu/alumni.orig/Annual6667.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;1966-1967&lt;/a&gt; report he is is listed as a conference attendee:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dr. John D. Thomas, Second Annual Conference on Pure and Applied Mathematics, (State).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://math.nmsu.edu/files/2019/10/68-69.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1968-1969&lt;/a&gt; report he is listed (p. 221) as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;John D. Thomas, Project Directors of NSF Summer Institutes Meeting (national). Holiday Mathematics Symposium (national).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;http://emmy.nmsu.edu/alumni.orig/DeptReport71.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1970-1971&lt;/a&gt; report he is listed (p.184) as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thomas, John D., Pi Mu Epsilon Faculty Sponsor, Reviewer for Zentralblatt fur Mathematik&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://math.nmsu.edu/files/2019/10/72-73.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1972-1973&lt;/a&gt; report he is listed (p. 182) as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;John D. Thomas, Reviewer for Zentralblatt fur Mathematik. Chairman, Departmental Committee for Applied Mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://math.nmsu.edu/files/2019/10/77-78.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1977-1978&lt;/a&gt; report he is listed (p. 206) as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;J. D. Thomas, Reviewer, Zentralblatt für Mathematik. Faculty advisor, Alpha Tau Omega Fraternity. Director, intercollegiate regional bridge tournament, NMSU Bridge Club. Chairman, Computer Advisory Group.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://math.nmsu.edu/files/2019/10/83-84.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1983-1984&lt;/a&gt; report, he is listed (p. 264) as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;J. D. Thomas, Chairman, Department of Mathematical Sciences Undergraduate Majors Committee (spring), [...] Chairman, doctoral committee, mathematics (one). [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=11318&amp;amp;fChrono=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt; we learn that Dr. Thomas advised nine Ph.D. students at New Mexico State University between 1965 and 1988, and that his full name is John David Thomas. We further see that he received his Ph.D. from the University of Oklahoma in 1959. The title of his dissertation was &amp;quot;A Local Coefficient Cohomology Theory for Lattices&amp;quot;. This is confirmed by a &lt;a href=&quot;https://primoa.library.unsw.edu.au/primo-explore/fulldisplay?vid=UNSWS&amp;amp;tab=default_tab&amp;amp;docid=UNSW_ALMA21128201000001731&amp;amp;lang=en_US&amp;amp;context=L&quot; rel=&quot;nofollow noreferrer&quot;&gt;catalog entry&lt;/a&gt; at the library of the University of New South Wales in Sydney, Australia, which holds a copy of this thesis on microfilm.&lt;/p&gt;&#xA;&lt;p&gt;There are two mentions of Dr. Thomas in minutes of the meetings of the Board of Regents of the University of Oklahoma.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://digital.libraries.ou.edu/regents/minutes/1955_06_09.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;June 9, 1955&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;GRADUATE ASSISTANTS [...] John David Thomas, Department of Mathematics, \$1,500 for 9 months, September 1, 1955&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://digital.libraries.ou.edu/regents/minutes/1957_01_17.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;January 17, 1957&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;John David Thomas, title changed from Graduate Assistant to Instructor in&#xA;Mathematics, salary changed from rate of \$1,500 for 9 months, 1/2 time,&#xA;to rate of \$3,624 for 9 months, full time, January 16 to June 1, 1957.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So far I have identified the following publications by Dr. Thomas:&lt;/p&gt;&#xA;&lt;p&gt;John D. Thomas, H. Ralph Lewis, and K. J. Melendez, &amp;quot;Computation of Fourier integrals of exponentials of truncated Fourier series.&amp;quot; &lt;em&gt;Journal of Computational Physics&lt;/em&gt;, Vol. 12, No. 4, August 1973, pp. 462-470&lt;/p&gt;&#xA;&lt;p&gt;John D. Thomas and Joseph D. Zund, &amp;quot;A note on the theory of immanants.&amp;quot; &lt;em&gt;Rendiconti del Circolo Matematico di Palermo&lt;/em&gt; &lt;strong&gt;28&lt;/strong&gt;, 143-150 (1979)&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-01-26T22:19:40.710" LastActivityDate="2021-01-26T22:19:40.710" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12794" PostTypeId="2" ParentId="12792" CreationDate="2021-01-26T10:39:09.220" Score="24" Body="&lt;p&gt;Before 1991 - nowhere, there were no platforms to &amp;quot;post&amp;quot; preprints on. Some were distributed by mail, and even collected and catalogued by large libraries. US National Institutes of Health even briefly supported Information Exchange Groups that circulated preprints, from 1961 to 1967. It was shut down because journals refused to publish shared memos, see &lt;a href=&quot;https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.2003995&quot; rel=&quot;noreferrer&quot;&gt;The prehistory of biology preprints by Cobb&lt;/a&gt;. In 1969 this pushback was enshrined by applying the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ingelfinger_rule&quot; rel=&quot;noreferrer&quot;&gt;Ingelfinger rule&lt;/a&gt; of non-duplication to preprints. Nonetheless, the practice spread informally in some fields, especially physics. As &lt;a href=&quot;https://science.sciencemag.org/content/357/6358/1344/tab-e-letters&quot; rel=&quot;noreferrer&quot;&gt;Klein reported in a letter to Science&lt;/a&gt;, it was established by 1980s:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;When I was a graduate student in particle physics in the 1980’s, preprints were well established. When we prepared a manuscript for Physical Review, we would also prepare a preprint version, which we would then mail to colleagues, and to a list of the major particle physics research centers around the world. Larger institutions had instituted a numbering system, providing a modicum of citability.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;We also made preprints for write-ups of conference talks. Since bound printed conference proceedings took so long to appear, this was the main channel for propagating preliminary results. Preprints were standard for all of the large experimental collaborations, and for many, if not most theoretical groups. The process had some drawbacks – it was somewhat clunky and labor intensive, and it was elitist. If you weren’t on the right mailing list, you would miss important results.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1990 preprints began to be shared by email, starting with Cohn. In 1991 she inspired Ginsparg to create a central repository at Los Alamos that could be accessed remotely. First by FTP, then by Gopher, and from 1993 over the Web, see &lt;a href=&quot;http://physicsbuzz.physicscentral.com/2020/07/arxiv-ing-history-of-preprints-in.html&quot; rel=&quot;noreferrer&quot;&gt;arXiv-ing the History of Preprints in Physics by Pell&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-27T18:45:36.090" LastActivityDate="2021-01-27T18:45:36.090" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12795" PostTypeId="2" ParentId="12784" CreationDate="2021-01-26T10:52:35.200" Score="6" Body="&lt;p&gt;Who knows, and who knows if it even happened. This anecdote is repeated almost verbatim after &lt;a href=&quot;https://prespacetime.com/index.php/pst/article/view/40&quot; rel=&quot;noreferrer&quot;&gt;Gibbs's “Crackpots” Who Were Right II&lt;/a&gt; published in Prespacetime Journal, not exactly a standard of historical scholarship. Other sources omit the colorful phrase, see e.g. &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Apery/&quot; rel=&quot;noreferrer&quot;&gt;MacTutor on Apery&lt;/a&gt;. Gibbs gives no explanations:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1978 he presented a lecture on his proof at the Journées Arithmétiques de Marseille which was&#xA;greeted with doubt and disbelief. Each step he wrote on the blackboard appeared to be a&#xA;remarkable identity that his audience considered unlikely to be true. When someone asked him&#xA;“where do these identities come from?” he replied “They grow in my garden.” obviuosly&lt;/em&gt; [sic!] &lt;em&gt;this did not&#xA;boost anyone’s confidence.&#xA;Nevertheless, a few mathematicians recognised that there was something significant in the&#xA;proposed proof. They checked the identities numerically and found that they did indeed seem to&#xA;hold. It was not long before the full validity of Apéry’s work was confirmed and the skeptics were&#xA;forced to eat their words.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gardening metaphors are not uncommon among mathematicians, see e.g. &lt;a href=&quot;https://www.scienceandnonduality.com/article/the-secrets-of-ramanujans-garden&quot; rel=&quot;noreferrer&quot;&gt;The Secrets of Ramanujan’s Garden&lt;/a&gt;. Plato himself uses it &lt;a href=&quot;http://harpermcalpineblack.blogspot.com/2013/08/the-gardening-metaphor-in-plato.html&quot; rel=&quot;noreferrer&quot;&gt;in Theaetetus&lt;/a&gt; to describe growth of knowledge, albeit not directly applied to mathematics. One could speculate that Apery referred to some kind of platonic &amp;quot;garden&amp;quot; of mathematical curiae he was keeping, but that would just be adding to a likely fable.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-01-26T11:08:38.823" LastActivityDate="2021-01-26T11:08:38.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12796" PostTypeId="2" ParentId="12792" CreationDate="2021-01-26T12:55:17.600" Score="21" Body="&lt;p&gt;I remember those days.  (I am in mathematics.)  When I completed a paper, I would mail (not email) photocopies (called &amp;quot;preprints&amp;quot;) to researchers I knew that I hoped would be interested in it.  Then I might attend a conference or visit another university, where I would give a talk on my work, and bring a stack of preprints for interested participants to take home.  An abstract might be published (in the publicity about a meeting, or a separate publication &lt;em&gt;Abstracts of the American Mathematical Society&lt;/em&gt;) and interested mathematicians would send mail to me asking for a preprint.  Sometimes (if it caught on) some of those recipients would make photocopies for their own interested colleagues.&lt;/p&gt;&#xA;&lt;p&gt;Back then, photocopying was a real expense, and researchers would pay for it from their research grants.&lt;/p&gt;&#xA;&lt;p&gt;All this would take place during the many months I waited for the journal to decide to publish the paper, and the many more months before that issue of the journal came out in hard copy.&lt;/p&gt;&#xA;&lt;p&gt;When the paper was published, the journal would provide me (or sell me) copies of the printed paper.  These were called &amp;quot;reprints&amp;quot; or &amp;quot;offprints&amp;quot; and the whole process continues: interested mathematicians, sometimes years later, who see the paper in the journal, would mail and ask me to send a reprint to them.  (Since that was cheaper than making photocopies themselves.)&lt;/p&gt;&#xA;&lt;p&gt;I had a large file drawer where I kept reprints and preprints of papers by others on topics that I was interested in.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2021-01-27T17:08:08.470" LastActivityDate="2021-01-27T17:08:08.470" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12797" PostTypeId="1" CreationDate="2021-01-26T16:07:44.033" Score="2" ViewCount="434" Body="&lt;p&gt;As a high school physics teacher, I like to motivate all concepts and terminology with how they were first developed historically. Recently I did some research on the motivation behind introducing the concept of gravitational potential, but I could not really find any clear story on the history of it. Why did people have a need for defining this? I can understand the reasoning for defining a gravitational field, but I do not understand the necessity of introducing gravitational potential. Any explanations at high school physics level would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="13802" LastActivityDate="2022-07-23T02:01:51.753" Title="What was the historical motivation for introducing the concept 'gravitational potential'?" Tags="|physics|gravity|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12798" PostTypeId="2" ParentId="12792" CreationDate="2021-01-26T17:55:05.357" Score="5" Body="&lt;p&gt;Depending on what you mean by &amp;quot;science work updates&amp;quot;, bear in mind that this might be covered by different types of journal: shorter works might be submitted to &amp;quot;&lt;em&gt;Rapid Communications in...&lt;/em&gt;&amp;quot; with a quick turnaround, whereas the definitive write-up would eventually appear in the more prestigious &amp;quot;&lt;em&gt;Journal of...&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Also, many institutions would formally issue &amp;quot;Technical Reports&amp;quot; (as a printed and bound series) that could by ordered either by individual researchers or a library (for example, here is a randomly-chosen example of a &lt;a href=&quot;https://archive.org/details/quarterlytechnic1966univ&quot; rel=&quot;nofollow noreferrer&quot;&gt;progress report&lt;/a&gt; from UIUC in 1966). These might only be reviewed internally, and covered the role of modern preprints.&#xA;NASA used to have a service - &amp;quot;Center for Aerospace Information&amp;quot; - that would let one order microfiche copies of any report prepared for them.&lt;/p&gt;&#xA;&lt;p&gt;The obvious other way of keeping people abreast of what one was up to was simply to talk. My personal impression is that the number of small, local (national/regional) one-day and half-day meetings has significantly fallen in the past couple of decades, possibly because of difficulty getting funding in academia for attendance at anything that doesn't result in published proceedings.&lt;/p&gt;&#xA;&lt;p&gt;(There's also the option of writing letters, as in &lt;a href=&quot;https://academia.stackexchange.com/questions/161935/what-type-of-letter-is-he-talking-about-and-why-did-he-write-a-letter-to-other-m&quot;&gt;What type of letter is he talking about and why did he write a letter to other mathematicians?&lt;/a&gt;, though this doesn't scale well!)&lt;/p&gt;&#xA;" OwnerUserId="13804" LastEditorUserId="13804" LastEditDate="2021-02-07T14:09:19.587" LastActivityDate="2021-02-07T14:09:19.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12800" PostTypeId="2" ParentId="12797" CreationDate="2021-01-26T20:01:54.607" Score="3" Body="&lt;p&gt;The theory of potentials has over a century of history from conception to present status, so the following is a brief resume, and may contains errors due to the summation (besides the errors due to my ignorance, of course).&lt;/p&gt;&#xA;&lt;p&gt;I- The mathematical theory of potentials (info from M. Kline, Mathematical thought from ancient to modern times, vol. 2)&lt;/p&gt;&#xA;&lt;p&gt;The earliest use of potentials, including the name, is derived from Daniel Bernoulli's Hydrodinamica (1738). Bernoulli introduced the concept by noting that in the equations of motion of a fluid one could obtain the forces by differentiation of a scalar function.&lt;/p&gt;&#xA;&lt;p&gt;Subsequently the concept was co-opted to the study of gravitational fields of solids of revolution by Euler, Lagrange, Laplace and Legendre (and others). The idea was that it was easy to calculate the gravitational force produced by a sphere (as done by Newton), but just using geometric arguments it was difficult for ellipsoids. So they introduced the potential as a mathematical tool to calculate first the potential, and from it the force, of various solids of revolution. This was spurred from research into the shape of Earth,since Newton had suggested that the figure should be an oblate spheroid but did not prove it.&lt;/p&gt;&#xA;&lt;p&gt;After that the concept was developed by Green and Gauss in early 19th century. Both of them were now extending the theory in order to tackle electric problems. I'm (sadly) omitting lots of other names for brevity, but the gist is that from middle 18th century to middle 19th century mathematicians introduced the potential as an aide to calculation of the forces for more complex geometrical figures.&lt;/p&gt;&#xA;&lt;p&gt;II- The Physical Theory of Potentials (info from Silvanus P. Thompson, Elementary Lessons in Electricity and Magnetism(1884))&lt;/p&gt;&#xA;&lt;p&gt;The principle of conservation of energy was only established in the second half of the 19th century by Helmholtz, Thomson, Joules and Mayer. Before that conservation of energy was known, but would be established individually for the systems at hand, only after this point it was taken for granted the convertibility of all forms of mechanical and thermal energy. Silvanus in fact notes that as he is writing his book it is still a matter of contention whether electrical energy is also conserved in general.&lt;/p&gt;&#xA;&lt;p&gt;I mention this because in Silvanus' book he devotes a chapter to the electrostatic potential, which he defines as &amp;quot;The potential at any point is the work that must be spent upon a unit of positive electricity in bringing it up to that point from an infinite distance&amp;quot;. In a footnote in page 193 writes &amp;quot;In its widest meaning the term &amp;quot;potential&amp;quot; must be understood as &amp;quot;power to do work&amp;quot;.(....) If we lift a pound five feet high against the force of gravity, the weight of the pound can in turn do five foot-pounds of work in falling back to the ground. See the Lesson on Energy in Professor Balfour Stewart's Lessons in Elementary Physics.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;s&gt; Now, Silvanus' book is not a work on history, but rather an early, and influential, textbook on electricity, but nevertheless it allows us the inference that by late 19th century the gravitational potential, initially introduced as a calculation tool, was already been taught in terms of &amp;quot;power to do work&amp;quot;. This, presumably, because of the desire to provide physical discussions in terms of work/conservation of energy, as well as the fruitfulness of derived notions such as equipotential lines that might simplify some discussions. &lt;/s&gt;&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I have completely forgotten about the best resource for the history of mechanics, R. Dugas &amp;quot;A History of Mechanics&amp;quot;. The book does not have a subject index, but from reading the table of contents, and by memory of having read it in the past, the first mention of potential in general appears in chapter 10 part 1, aptly named &amp;quot;Helmholtz and the Energetic Thesis&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;According to Dugas, Helmholtz introduced the potential energy for a central force (whether attractive or repulsive) in a paper called On the Conservation of Force(1847), although the name Helmholtz credits to Rankine. Helmholtz says in the article that the advantage of introducing the potential is that all of mechanics is reduces to a single &amp;quot;almost popular rule&amp;quot;, which is that all motion of a body that interacts with another there is a loss of potential energy proportional to gained kinetic energy (and consequently velocity). So one of Helmholtz primary motivations seems to simplify mechanical discussions, instead of having Newton's laws and vector algebra, you could do everything with a single axiom.&lt;/p&gt;&#xA;&lt;p&gt;Helmholtz also notes that another advantage is that the potential extends directly to solid bodies and perfectly elastic fluids. The idea here is that for extended bodies, solid of fluid, the Newtonian treatment requires you to analyse the forces at each infinitesimal part of the body and then integrate everything to find out what is the movement, but using potential you can avoid this. One example would be Bernoulli's principle, which is trivial when thinking from a potential vs kinetic point of view, but would be very hard to prove using forces.&lt;/p&gt;&#xA;&lt;p&gt;III - Conclusion &lt;s&gt;(and Apologies)&lt;/s&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think is clear and established that the gravitational potential was at first introduced as a mathematical tool. I will take Kline's omission of the physical interpretation as sign that the physical interpretation was not emphasized at first (I do this because Kline is a very thorough fellow, and other history of math books, like Howard Eve's, who ties more with physics at points, also do not comment on that). At some point in late 19th century we have records of people using the physical interpretation of &amp;quot;power to do work&amp;quot;. This &lt;s&gt;seems to be&lt;/s&gt; was under the influence of the principle of conservation of energy, so that mechanical discussions that started focused on analyzing forces gave way to a focus on analyzing movement in terms of potential lines and work, following Helmholtz. &lt;s&gt; A wild speculation would be that in the present day we introduce the gravitational potential early in high school less for it's actual value, but more as a parallel to help students better grasp the electric potential, which is much more used.&lt;/s&gt;&lt;/p&gt;&#xA;&lt;p&gt;Concerning high school explanations, from the physical point of view I would stress the ability of using potentials to avoid having to do vector analysis via Newton's Laws. For instance it should be much easier to show that the gravitational potential allows stable circular orbits this way. Also for discussing fluid mechanics.&lt;/p&gt;&#xA;&lt;p&gt;Since the potential first appeared as a calculation tool, but your students don't know calculus, you could give them a taste using an analogous model with electrostatics. Say you want to figure out the gravitational forces produced by a non-spherical object, like a oblate spheroid. Since the gravitational potential functions just like the electric potential you could assemble an object of the desired shape and charge it electrically. Then you would use a voltimeter to measure the potential across the surface of the object,and voila you would know the relative distribution of the gravitational potential, and thus the force, exerted by an arbitrary shape, without having to do any math.&lt;/p&gt;&#xA;&lt;p&gt;&lt;s&gt; I stress that neither of the sources claim any of this. I have taken the facts they provided and construed a theory regarding the motivation for the usage of the gravitational potential, as is done in modern high school, by way of two assumptions, that early mathematicians did not concerned themselves with the physical interpretation, and that later physicists emphasized said interpretation in light of conservation of energy. So I apologize for having done so, which devalues this answer as it pertains to the history of things, but I hope it helps in your primary concern, or as a starting point to an actual, sourced, answer.&lt;/s&gt;&lt;/p&gt;&#xA;" OwnerUserId="5904" LastEditorUserId="5904" LastEditDate="2021-02-12T19:27:09.817" LastActivityDate="2021-02-12T19:27:09.817" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12801" PostTypeId="1" CreationDate="2021-01-27T00:46:45.753" Score="3" ViewCount="186" Body="&lt;p&gt;I am reading this book on trigonometric series, &amp;quot;Тригонометрические ряды от Эйлера до Лебега&amp;quot; (Trigonometric series from Euler to Lebesgue) , it is in Russian, and my Russian is abysmal. But there is a very interesting formula, it is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{1-r\cos(x)}{1-2r\cos(x)+r^2}=1+r\cos(x)+r^2\cos(2x)+r^{3}\cos(3x)...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{r\sin(x)}{1-2r\cos(x)+r^2}=r\sin(x)+r^2\sin(2x)+r^{3}\sin(3x)...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As &lt;span class=&quot;math-container&quot;&gt;$r=1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$r=-1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(The title of the book is above, it is on page 12)&lt;/p&gt;&#xA;&lt;p&gt;We have: &lt;span class=&quot;math-container&quot;&gt;$$-\dfrac{1}{2}=\cos(x)+\cos(x)+\cos(3x)+...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{1}{2}=\cos(x)-\cos(x)+\cos(3x)+...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;He integrated the second series and obtain the trigonometric series:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\dfrac{x}{2}=\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The first and second series are divergent, so I think the result doesn't make sense, although the formal method of deriving it may seem correct.&lt;/p&gt;&#xA;&lt;p&gt;The third series is correct on the interval &lt;span class=&quot;math-container&quot;&gt;$[\pi, -\pi]$&lt;/span&gt;, because the series expansion converges to that function (Correct me if I am wrong here, I haven't studied Fourier series).&lt;/p&gt;&#xA;&lt;p&gt;I wish to ask how did he obtain the above formula? Are these formula incorrect because the issue of convergence? Also, which papers contain these results?&lt;/p&gt;&#xA;" OwnerUserId="13807" LastEditorUserId="13807" LastEditDate="2021-01-28T23:16:27.683" LastActivityDate="2021-01-28T23:16:27.683" Title="How did Euler obtain this formula from a paper/work in 1748?" Tags="|mathematics|mathematicians|calculus|real-analysis|euler|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12802" PostTypeId="1" CreationDate="2021-01-27T02:19:38.620" Score="2" ViewCount="193" Body="&lt;p&gt;With the benefit of hindsight we tend to look back at the the decision to bring ~2000 Nazi Scientists to the US after WW2 with a well deserved level of cynicism. But reading into some of the history it seems as though this was not a remotely controversial decision at its time, both by the political establishment and the scientific community.&lt;/p&gt;&#xA;&lt;p&gt;On the political side, I am specifically referring to some internal memorandum in the War Department, including the letter sent to Truman by Acheson.&lt;/p&gt;&#xA;&lt;p&gt;My only evidence for the reaction of the Scientific Community inside of the US (who had just dedicated 5 years to fighting the Nazis), is their loud apathy towards the matter, no one said anything. This is particularly noticeable because many in the Scientific community were proudly anti-Soviet. Any refference to the reaction of scientists during this time would be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="13800" LastActivityDate="2021-01-28T02:17:48.853" Title="Was 'Operation Paperclip' controversial at the time?" Tags="|world-war-2|nazi-germany|cold-war|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12803" PostTypeId="1" CreationDate="2021-01-27T14:40:53.890" Score="1" ViewCount="104" Body="&lt;p&gt;My question is about linear forms, quadratic forms, n-linear forms, differential forms and so on. The first term of these names seem clear to me, but I cannot make a link between these mathematical objects and the usual meanings of the word &amp;quot;form&amp;quot;.&#xA;I was unable to find an answer on the Internet. Knowing the first occurence of this use of &amp;quot;form&amp;quot; in mathematics would probably help.&lt;/p&gt;&#xA;" OwnerUserId="13818" LastEditorUserId="10106" LastEditDate="2021-01-27T15:15:22.407" LastActivityDate="2021-01-27T15:15:22.407" Title="Why are linear forms called &quot;forms&quot;?" Tags="|mathematics|terminology|linear-algebra|algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12804" PostTypeId="1" AcceptedAnswerId="12807" CreationDate="2021-01-27T15:52:22.373" Score="0" ViewCount="301" Body="&lt;p&gt;I have a reference to &amp;quot;Theory and Examples of Point Set Topology&amp;quot; by J. Greever (1967: Brooks/Cole).&lt;/p&gt;&#xA;&lt;p&gt;I searched for biographical information on J. Greever and I identified:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;John Greever, b. 30-Jan-1934&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;John Jay Greever Jr, 1912-1989.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I believe the two are conflated in what can be found online. The general consensus is that John Greever is the son of John Jay Greever Jr., and it was John Greever who wrote &amp;quot;Theory and Examples of Point Set Topology&amp;quot;, as the latter was a mathematics professor at Virginia.&lt;/p&gt;&#xA;&lt;p&gt;But it is possible to find links that suggest &amp;quot;John Jay Greever&amp;quot; wrote TaEoPST. So while I &lt;em&gt;suspect&lt;/em&gt; that this is propagation of a mistake, there are a few compelling reasons to suggest that &amp;quot;John Greever&amp;quot; may &lt;em&gt;also&lt;/em&gt; be a &amp;quot;John Jay Greever&amp;quot;, as he can be found under that name in Mathematics Genealogy Project. Prabook gives him as &amp;quot;John Greever&amp;quot; and identifies him as the son of &amp;quot;John Jay Greever&amp;quot; (whence the confusion in the first place), but if &amp;quot;John Greever&amp;quot; really &lt;em&gt;is&lt;/em&gt; &amp;quot;John Jay Greever&amp;quot;, then my understanding is that he would properly be styled &amp;quot;John Jay Greever III&amp;quot; or some such.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any inside information as to exactly how the Greever who wrote this book is actually named?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="4931" LastEditDate="2021-01-31T06:54:50.700" LastActivityDate="2021-01-31T06:54:50.700" Title="Did John Greever or John Jay Greever write &quot;Theory and Examples of Point Set Topology&quot;?" Tags="|mathematicians|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12806" PostTypeId="1" CreationDate="2021-01-27T17:11:37.467" Score="2" ViewCount="80" Body="&lt;p&gt;I'm a mathematician.&lt;br /&gt;&#xA;I'm required to give a lecture on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stone-von Neumann theorem&lt;/a&gt;. I already have all the mathematical details figured out, but I wish to make the lecture more interesting by giving the physical connection to the theorem.&lt;/p&gt;&#xA;&lt;p&gt;Problem is, I don't recall much of the quantum physics course I took as an electrical engineering student 10 years ago. Can you please give me a fair background on the theorem, in simple words, I can present as an intro to the mathematical details (Historical anecdotes, the relation to the wave function etc.)?&lt;/p&gt;&#xA;" OwnerUserId="13821" LastEditorUserId="13821" LastEditDate="2021-01-28T07:33:19.120" LastActivityDate="2021-01-28T07:33:19.120" Title="Background on the Stone-von Neumann theorem" Tags="|quantum-mechanics|mathematical-physics|hilbert|von-neumann|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12807" PostTypeId="2" ParentId="12804" CreationDate="2021-01-27T18:32:07.637" Score="5" Body="&lt;p&gt;Did you try the &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=15617&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt; website? I use this so often that it's one of the 7 links I have on my top google toolbar. Sometimes their information is not exactly correct (and I inform them when I find an error), so I double check with a &lt;a href=&quot;https://dissexpress.proquest.com/search.html&quot; rel=&quot;noreferrer&quot;&gt;ProQuest search&lt;/a&gt; (another one of my 7 google toolbar links), using last name and a dissertation title word (as needed), which resulted in &lt;a href=&quot;https://dissexpress.proquest.com/dxweb/results.html?QryTxt=&amp;amp;By=Greever&amp;amp;Title=Transformations&quot; rel=&quot;noreferrer&quot;&gt;John Jay Greever, III&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, to make sure the ProQuest data is correct (FYI, in looking up hundreds of names in 15+ years, I don't believe I've ever encountered a name spelling or dissertation title wording/spelling that I found to be incorrect), I'll &lt;a href=&quot;https://www.google.com/search?q=Greever+%22Fixed+Points+of+Finite+Groups+of+Transformations%22&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;do a search for the dissertation title and last name in google-books&lt;/a&gt; to make sure that the various publications that publish the dissertation title agree. Sometimes you can find the dissertation online by using a regular google search. If you can find an actual copy of the dissertation, then you can check the name in the title page. Much more significant is that you can sometimes find an author biography/vitae at the end, especially in older dissertations. And if there is no biography/vitae, then the &lt;em&gt;Acknowledgments&lt;/em&gt; section (or similar such page) at the beginning will often have useful information, such as sibling/parent names that you can then use for additional google searches. In the case of Greever, I wasn't able to find a copy of his dissertation that is freely available.&lt;/p&gt;&#xA;&lt;p&gt;Additionally, you can search the university library where Greever got his Ph.D. Sometimes more author information can be found there, such as a birth year. Also, on rare occasions you might find that the dissertation is available in .pdf form there even through a google search didn't turn up this possibility. In the case of Greever, however, I didn't find anything more &lt;a href=&quot;https://search.lib.virginia.edu/sources/uva_library/items/u1348864&quot; rel=&quot;noreferrer&quot;&gt;at the library info. page for his dissertation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Given that his Ph.D. was awarded in 1958, the 1934 birth date is almost certainly correct. To confirm it, try searching in google-books for things like &lt;a href=&quot;https://www.google.com/search?q=%22Greever%2C+John%22+math&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Greever, John&amp;quot; + math&lt;/a&gt; and look for &amp;quot;American Men &amp;amp; Women of Science&amp;quot; (or in the West, MidWest, etc.) and/or various &amp;quot;Who's Who&amp;quot; book entries, where the google-snippet views will often give you a birth date. Note that I specifically did NOT put 'math' in quotes. This is because you want various cognates of 'math' to be included in the search, not just the exact 4-character word 'math'.&lt;/p&gt;&#xA;&lt;p&gt;In the case of Greever, the google-books search &lt;a href=&quot;https://www.google.com/search?q=%22Greever%2C+John%22+math+%22American+Men%22&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Greever, John&amp;quot; + math + &amp;quot;American Men&amp;quot;&lt;/a&gt; confirms several times that 1934 is correct. There's also a &amp;quot;Who's Who in the West&amp;quot; book entry you can find using &lt;a href=&quot;https://www.google.com/search?q=%22Greever%2C+John%22+math+%22Who%27s+Who%22&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Greever, John&amp;quot; + math + &amp;quot;Who's Who&amp;quot;&lt;/a&gt;. The birth year is missing in the snippet view I see (but it's a very close miss), so in these cases (i.e. when you have a close miss) try clicking on the entry itself and see what shows up (takes you &lt;a href=&quot;https://books.google.com/books?id=voRmAAAAMAAJ&amp;amp;q=%22Greever,+John%22+math+%22Who%27s+Who%22&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;) (sometimes you'll get the birth date just by doing this), then delete the words &amp;quot;math&amp;quot; and &amp;quot;Who's Who&amp;quot; in an attempt to get a snippet view that shows the birth date. However, &lt;a href=&quot;https://books.google.com/books?id=voRmAAAAMAAJ&amp;amp;q=%22Greever%2C+John%22&quot; rel=&quot;noreferrer&quot;&gt;it doesn't work in this case&lt;/a&gt;. Another trick is to go back to the google-books snippet part and copy a phrase at the end before the snippet part ends. I tried &lt;a href=&quot;https://www.google.com/search?q=%22GREEVER+%2C+JOHN+%2C+mathematics+educator+%3B+b%22&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;GREEVER , JOHN , mathematics educator ; b&amp;quot;&lt;/a&gt;, but it didn't give me anything more this time. However, sometimes doing this last step will give you words in a new snippet view that appear after the words you were previously able to see. Although the birth year is definitely 1934 based on the &amp;quot;American Men ...&amp;quot; entries, I've included the details about various follow-ups to the &amp;quot;Who's Who in the West&amp;quot; search to show what you can attempt if you get a snippet view that seems to just miss showing you a birth date.&lt;/p&gt;&#xA;&lt;p&gt;As for whether he's still alive, you can try searches such as {&amp;quot;John J. Geever&amp;quot; + city-he-seems-to-live-in} and maybe include &amp;quot;mylife&amp;quot; (a web source that is one of the best for giving such information). Also try searches for various versions of his name, city lived in, and birth year, along with words such as &amp;quot;obituary&amp;quot; or &amp;quot;died&amp;quot;. As for cities a mathematician has lived in, do a google-books search for {&amp;quot;last name, first name&amp;quot; + math + &amp;quot;Combined Membership&amp;quot;}. This will give you some of the various AMS/MAA/SIAM Combined Membership entries for him in snippet views (these used to be books mailed out, or maybe purchased cheaply, I forgot -- I have several from the mid 1970s to early/mid 1990s), along with where the person lived for particular years. Of course, some of where the person has lived will also be in the &amp;quot;American Men ...&amp;quot; and &amp;quot;Who's Who ...&amp;quot; entries, but even when you can't find any such &amp;quot;American Men ...&amp;quot; entries for a mathematician, you can still often find some information about where they've lived and taught in a &amp;quot;Combined Membership&amp;quot; search.&lt;/p&gt;&#xA;&lt;p&gt;By the way, matching the Greever above with the author of the topology book is easy, since the author of the topology book taught at Harvey Mudd College, and the 1934 Greever &lt;a href=&quot;https://www.google.com/search?q=%22Theory+and+Examples+of+Point+Set+Topology%22+%22Harvey+Mudd+College%22&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;is easily identified with the Harvey Mudd College Greever&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2021-01-27T18:32:07.637" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12810" PostTypeId="2" ParentId="12787" CreationDate="2021-01-28T01:51:31.953" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What are the different ways that topologies have been presented or defined over time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Topology is about a structure of cohesion on the points of a space. The question of what constitutes cohesion has been asked since earliest times, essentially arising from the paradoxes of Zeno. Aristotle thought of the physical continuum in terms of potentiality and actuality.&lt;/p&gt;&#xA;&lt;p&gt;The same question was tackled by Hegel who thought of a moving point as being here and there at the same time. This is worth considering since an open set in pointless topology, is here and also over there. I mean by that, it is not localised to a point. The question of what constitutes the physical continuum was tackled in Weyls &lt;em&gt;Space, Time &amp;amp; Matter&lt;/em&gt;, first published in 1918.&lt;/p&gt;&#xA;&lt;p&gt;This sense of a 'motion' culminating in a 'limit' is axiomatised via nets and which generalises the definition of a limit of a sequence. This is exactly equivalent to the traditional notion of a topology by open or closed sets and deserves to be better known that it is now, especially given that limits are usually introduced at school and hence, pedagogically speaking, generalised limits - aka nets -  are a natural way to introduce topology.&lt;/p&gt;&#xA;&lt;p&gt;We have the classical notion of a topology in terms of nets (or filters), opens (or closeds) and interior (or closure operators). There are  more general notion of topologies such as pseudo-topologies and convergence spaces. They aim to have better categorical properties. For example, Top, the category of topological spaces does not have exponentials, that is a natural topology on mapping spaces. Whereas Conv, the category of convergence spaces, being a quasi-topos, does; and moreover, Top embeds in it as a full subcategory. This means Conv extends the notion of topology.&lt;/p&gt;&#xA;&lt;p&gt;But there is more to the question of cohesion than simpe topology. We can also ask how smoothly do they cohere. This of course is differentiability and this has also been axiomatised.  For example, there is diffeology, which was put forward by Souriau in the 1980s, and which aims to do for smoothness what topology does for continuity. There is also the notion of bornology, which does the same for boundedness - in fact, one mathematician has said that in functional analysis, bornology was more natural than topology. In fact, there is also the notion of convexology , a neologism for a similar notion for convex structures. This might seem to be far from the notion of cohesion but it's possible that the fine structure of a space may be convex as in locally convex spaces and where the convexity is crucial to define a well-behaved notion of differentiability and hence of smoothness.&lt;/p&gt;&#xA;&lt;p&gt;It's worth adding that categories can be topologised. Here, what is used is an abstraction of the notion of covering from topology and what is called a Grothendieck topology. This was formulated by Grothendieck in the early 1960s.&lt;/p&gt;&#xA;&lt;p&gt;It's probably also worth saying that uniform continuity had a separate axiomatic development with the development of uniform spaces and uniformities. And also that completeness, which is about the fine structure of points; in particular, that there are no 'missing' points and hence 'complete' also has a separate axiomatix treatment as Cauchy spaces. The name here obviously derives from Cauchy sequences or nets which are traditionally used to define the completeness of a metric space, or more generally, of a metrisable topological space.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-02-11T17:54:51.073" LastActivityDate="2021-02-11T17:54:51.073" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12811" PostTypeId="2" ParentId="12802" CreationDate="2021-01-28T02:17:48.853" Score="3" Body="&lt;p&gt;According to one history that I've read, denazification was done more thoroughly by the former Soviet Union than by the Western powers, although mainly on class lines. The Weatern Powers were concerned more with the growing 'threat' of communism hence it's likely Operation Paperclip was seen as non-controversial as they saw it simply as harnessing German scientific expertise.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-01-28T02:17:48.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12813" PostTypeId="1" AcceptedAnswerId="12817" CreationDate="2021-01-29T00:27:59.163" Score="2" ViewCount="290" Body="&lt;p&gt;If we divide a positive number by another positive number, the result becomes greater as the divisor becomes smaller. If we continue this logic, division by a negative number should be greater than any division by a positive number.&lt;/p&gt;&#xA;&lt;p&gt;So, I wonder whether the idea that the result of such division should be negative was ever controversial, or other ideas were ever proposed as alternatives (for instance, counting such result as infinitely large value or otherwise non-real number)?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="12330" LastEditDate="2024-02-20T02:08:38.270" LastActivityDate="2024-02-20T02:08:38.270" Title="Has the idea that the result of division of positive number by negative number should be negative ever been controversial?" Tags="|algebra|elementary-algebra|numbers|arithmetic|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12815" PostTypeId="2" ParentId="5078" CreationDate="2021-01-29T04:25:59.697" Score="3" Body="&lt;p&gt;&lt;strong&gt;About vector potential there are different situations,&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;(1) Neumann vector potential &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{A}_{N}$&lt;/span&gt;(1845):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{A}_{N}=\frac{\mu_{0}}{4\pi}\iiint_{V}\frac{\boldsymbol{J}(x',t)}{r}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;r=|\boldsymbol{x}-\boldsymbol{x}'|&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This formula should be credit to Neumann. Perhaps Neumann did not&#xA;written the above formula, but Neumann has the formula,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;U=II'\oint\oint\frac{d\boldsymbol{l}\cdot d\boldsymbol{l}'}{r}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; is the energy move the two coils from distance &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; to infinity.&#xA;From the above it is easy to write,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{A}=\oint\frac{I'}{r}d\boldsymbol{l}'&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;U=\oint\boldsymbol{A}\cdot d\boldsymbol{l}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(2)Weber vector potential &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{A}_{W}$&lt;/span&gt;(1846)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{A}_{W}=\frac{\mu_{0}}{4\pi}\iiint_{V}\frac{(\boldsymbol{J}(x',t)\cdot\boldsymbol{r})\boldsymbol{r}}{r}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{r}=\boldsymbol{x}-\boldsymbol{x}'&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(3) William Thomson (Lord Kelvin) 1847&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{B}=\nabla\times\boldsymbol{A}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{A}$&lt;/span&gt; is Neumann or Weber vector potential.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{B}=\frac{\mu_{0}}{4\pi}\iiint_{V}\boldsymbol{J}(x',t)\times\frac{\boldsymbol{r}}{r^{3}}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;=\frac{\mu_{0}}{4\pi}\iiint_{V}\nabla(\frac{1}{r})\times\boldsymbol{J}(x',t)dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;=\frac{\mu_{0}}{4\pi}\iiint_{V}\nabla\times\frac{\boldsymbol{J}(x',t)}{r}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;=\nabla\times\boldsymbol{A}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(4) Maxwell (1856)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{E}=-\frac{\partial\boldsymbol{A}_{N}}{\partial t}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(5) Gustav Robert Kirchhoff (1857)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{J}=-\sigma(\nabla\phi+\frac{\partial\boldsymbol{A}_{W}}{\partial t})&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(6) Gustav Robert Kirchhoff gauge condition 1857,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\nabla\cdot\boldsymbol{A}_{W}=\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\phi&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\phi(\boldsymbol{x},t)=\frac{1}{4\pi\epsilon_{0}}\iiint_{V}\frac{\rho(\boldsymbol{x}',t)}{r}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kirchhoff got his result by using the current continuity equation,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\nabla\cdot\boldsymbol{J}=-\frac{\partial}{\partial t}\rho&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Continuity equation is very important, it was the reason Maxwell introduced the displacement current&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\frac{\partial\boldsymbol{D}}{\partial t}&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(7) Retarded potential Lorenz (1867)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\iiint_{V}\frac{\boldsymbol{J}(x',t-r/c)}{r}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\phi(\boldsymbol{x},t)=\frac{1}{4\pi\epsilon_{0}}\iiint_{V}\frac{\rho(x',t-r/c)}{r}dV&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the speed of light. In the same time Lorenz offers the Lorenz&#xA;gauge condition.&lt;/p&gt;&#xA;&lt;p&gt;(8) Lorenz gauge condition. (1867)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\nabla\cdot\boldsymbol{A}_{N}=-\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\phi&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kirchhoff gauge condition (1857) derivation has shown that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\nabla\cdot\boldsymbol{A}_{N}-\nabla\cdot\boldsymbol{A}_{W}=-2\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\phi&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lorenz get the Lorenz gauge condition is too easy. I thought Lorenz&#xA;gauge perhaps should be referred as Kirchhoff-Lorenz gauge. Any way&#xA;Lorenz gauge is very important. It is a is only bridge to get correct&#xA;retarded potential and advanced potential (for both vector and scale&#xA;potential).&lt;/p&gt;&#xA;&lt;p&gt;Summary:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Neumann and Weber vector potential are the most important work to contributed for Maxwell equations, but even in the Maxwell equations this work has been credited to as Faraday induction law.&lt;/li&gt;&#xA;&lt;li&gt;Lorenz gauge should be referred as Kirchhoff-Lorenz gauge. Kirchhoff did most work for Lorenz gauge.  Without Lorenz gauge it is not possible to obtained correct solution of Maxwell's equations (retarded potential and advanced potential)&lt;/li&gt;&#xA;&lt;li&gt;Lorenz retarded potential is equivalent to the theorem of Maxwell (retarded potential = displacement current). He follows the Kirchhoff's thought, however he has replaced the Weber vector potential as Neumann vector potential and obtained the important results of the retarded potential, which is the solution of Maxwell's equations.&lt;/li&gt;&#xA;&lt;li&gt;If Kirchhoff had applied Neumann vector potential in 1857 instead of Weber vector potential, He would replace Maxwell as the greatest scientist in electromagnetic field theory. Kirchhoff first utilized the current continuity equation to obtained his gauge condition. Lorenz following Kirchhoff and got the correct results, but a few years later than Maxwell. Any way, Kirchhoff together with Lorenz, should get the same respect as Maxwell.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13831" LastActivityDate="2021-01-29T04:25:59.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12816" PostTypeId="1" CreationDate="2021-01-29T07:16:04.060" Score="0" ViewCount="35" Body="&lt;p&gt;I am making sure that my information on certain mathematicians is up to date and accurate. To this end I am looking at Robert Earl MacKenzie, who (according to my records) was born March 17, 1920. I have scoured the internet for news of his passing, but the only results I have found are inconclusive: PraBook does not record a death date, and there are 2 obituaries: one for a 2011 date and one for a 2016 date (so possibly neither one). Unfortunately it is a not uncommon name, so there may be plenty of records that refer to different people.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone out there have any inside knowledge? He is listed in this book I have as being in the dept of mathematics at Indiana at some stage.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2021-01-29T07:16:04.060" Title="Is Robert Earl MacKenzie (co-author of Introduction to Differential Manifolds) still alive?" Tags="|mathematicians|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12817" PostTypeId="2" ParentId="12813" CreationDate="2021-01-29T07:53:26.787" Score="12" Body="&lt;p&gt;Much has been written about various roadblocks to the acceptance of negative numbers, and I have a folder containing photocopies of a few such papers, but I don't have time now to look for that folder. However, I found the following citation and excerpt in a manuscript of mine. The paper is behind a paywall (which I don't have access to, but I have a photocopy from the original library journal volume in that folder), so I'm including the excerpt I have.&lt;/p&gt;&#xA;&lt;p&gt;George Abram Miller (1863−1951), &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1949-8594.1933.tb10676.x&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A crusade against the use of negative numbers&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;School Science and Mathematics&lt;/strong&gt; 33 #9 (December 1933), 959−964.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from pp. 959−960)&lt;/strong&gt; It appears that the greatest opposition to the use of negative numbers in algebra developed just before the dawn of clear light on this subject near the beginning of the nineteenth century. Among the well known opponents to this use was Robert Simson who was professor of mathematics in the University of Glasgow for fifty years beginning with 1711 and edited a well known edition of Euclidís &lt;em&gt;Elements&lt;/em&gt;. One of the serious obstacles in the way towards accepting the negative numbers as real numbers is that this acceptance implies that we cannot thereafter continue to support the view that the ratio of a smaller number to a larger number must differ from the ratio of a larger number to a smaller since it is obvious, for instance that &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3} = -2/-3$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; is smaller than &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt; is larger than &lt;span class=&quot;math-container&quot;&gt;$-3.$&lt;/span&gt; The example of greatest historical interest in this connection is &lt;span class=&quot;math-container&quot;&gt;$1/-1 = -1/1.$&lt;/span&gt; In dealing only with positive numbers it is obvious that the ratio of a smaller number to a larger one is always less than unity while the ratio of a larger number to a smaller number is always greater than unity, and hence two such ratios cannot be equal to each other in this restricted number field. This difficulty is the more serious in view of the fact that the concepts of ratio and proportion are among the earliest mathematical concepts and that in Greek mathematics the latter largely replaced our modern concept of equation. In all these early uses of the concept of ratio, including the golden period of Greek mathematical developments, the ratio of a smaller number to a larger number was always assumed to be less than the ratio of a larger number to a smaller number. The former corresponds to a proper fraction while the latter corresponds to an improper fraction, but the division of fractions into these two classes was inaugurated after the middle ages although the Hindus calculated already with fractions whose numerators are larger than their denominators. I. Newton and others defined a number as the ratio between two line segments. These examples of the early use of the concept of ratio are cited here in order to exhibit more clearly an explanation of the reluctance with which mathematicians of the seventeenth and of the eighteenth centuries abandoned the idea that the ratio of a larger number to a smaller number is not always greater than the ratio of a smaller number to a larger one. In 1768 W. J. G. Karsten, who was then professor of logic and later professor of mathematics in a German university, published an article under the title &lt;a href=&quot;https://books.google.com/books?id=pWdKRWfzj2gC&quot; rel=&quot;nofollow noreferrer&quot;&gt;“Von den Logarithmen verneinten Grösen”&lt;/a&gt; in which he argued that a negative number cannot be less than zero because if this were true it would follow from the correct proportion &lt;span class=&quot;math-container&quot;&gt;$1:-1=-1:1$&lt;/span&gt; that the larger of two numbers may have the same ratio to the smaller as the smaller to the larger, which seemed to him to be a contradiction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;(Additional reference added nearly 3 years later)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;George Abram Miller (1863−1951), &lt;a href=&quot;https://archive.org/details/dli.ernet.524180/page/1/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Vague historical views relating to the negative numbers&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;The Mathematics Student&lt;/strong&gt; 2 #1 (March 1934), pp. 1−6.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(first paragraph of the paper, on p. 1)&lt;/strong&gt; In the history of negative numbers it is desirable to distinguish sharply between the following three points: their correct use, their correct theory and their introduction as a permanent element of mathematics. It is well-known that the Hindus used these numbers correctly and that this is the earliest such use thereof that has thus far been definitely established. On the other hand, the eighteenth century European mathematicians did not yet generally possess a correct theory of these numbers so that J. Tropfke could truly say in the third edition of his noted &lt;em&gt;Geschichte der Elementar-Mathematik&lt;/em&gt;, Volume 2 (1933), page 101, that the eighteenth century suffered because a generally satisfactory introduction of these numbers was then lacking. The symbols &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt; were employed in calculations and were then used at a bound to represent positive and negative numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(first two complete paragraphs on p. 2)&lt;/strong&gt; It should be emphasized in this connection that the use of negative numbers is not fully justified from their analogy to debt. The product of two debts is not a credit. When debts are multiplied, they are multiplied by positive numbers and not by negative numbers so as to become a credit. The history of negative numbers becomes clear only by noting that the analogy of negative numbers to debt relates only to some of the properties of these numbers while it is foreign to other properties thereof. Even such an eminent mathematician as H. Cardan tried to prove in the latter half of the sixteenth century that it is incorrect to say that &lt;span class=&quot;math-container&quot;&gt;$(-a).(-b)=+ab,$&lt;/span&gt; and in the noted algebra of C. Clavius (1608) the author ascribes to the weakness of the human mind the impossibility of understanding this rule, but he did not doubt it correctness in view of the fact that it had been verified by many examples. The theory of negative numbers with respect to the operations of addition and subtraction is much more simple than their theory with respect to the operations of multiplication and division. In the former theory, the analogy of these numbers to debt when the corresponding positive numbers represent credit and to distances in the opposite direction from those represented by the corresponding positive numbers are illuminating. In trigonometry and in analytic geometry, negative numbers are especially useful with respect to the operations of addition and subtraction. Hence it is perhaps natural that some mathematical historians have unduly emphasized this analogy in presenting the extension of the number concept so as to include negative numbers. This extension, however, implied also a correct theory of these numbers with respect to the operations of multiplication and division, and the latter theory does not seem to have been satisfactorily presented before about the beginning of the nineteenth century and it seems to be entirely due to European mathematicians who, therefore, should have the credit of having completed the introduction of the negative numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(last two paragraphs of the paper, on pp. 5−6)&lt;/strong&gt; The three points to which we referred at the opening of this article can now be briefly answered as follows. The Hindus used negative numbers correctly at least as early as the times of Brahmagupta, who was born in 598; and they also possessed a correct theory of these numbers as regards addition and subtraction, but there is no definite evidence that a correct theory as regards the multiplication and division of these numbers was developed before about the beginning of the nineteenth century. The latter theory is a special case of the corresponding theory relating to complex numbers and it was developed simultaneously with this more general theory. The introduction of negative numbers as a permanent element of mathematics was inaugurated by the use of these numbers by the Hindus and was later inspired by the many simplifications which were effected by this use in various fields of mathematical developments. This introduction met with considerable opposition from time to time until a satisfactory theory of the negative numbers was developed around the beginning of the nineteenth century and thus completed their introduction more than a thousand years after this introduction was inaugurated. The introduction of the negative numbers belongs to the extensive field of mathematical developments which are due to a failure to state all the restrictions involved in certain early proofs. The history of these developments has never been written and involves many difficulties; but the general history of mathematics is much enriched by noting that many of the early proofs apply to wider fields than the authors of these proofs had in mind. When Diophantus first stated that the product of a subtracted number by a subtracted number is an added number, he naturally had no idea of the great generality of this rule, just as J. L. Lagrange had no idea of the generality of the so-called Lagrange theorem in group theory. Many mathematical reputations are largely due to the later growth of theorems and theories rather than to their observed generality at the time when they were first announced by those to whom they are now credited. Negative numbers entered mathematics as useful servants and later were accepted as staunch citizens after their real merits became known.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2023-12-10T11:56:26.497" LastActivityDate="2023-12-10T11:56:26.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12818" PostTypeId="2" ParentId="12757" CreationDate="2021-01-29T08:29:21.970" Score="1" Body="&lt;p&gt;Charles Dodgson (aka Lewis Carroll) taught at school for one term and while he did not hate it, he found it very challenging. Here are some diary entries:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;26 February 1856&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The class again noisy and inattentive; it is very disheartening, and I almost think I had better give up teaching there for the present.&lt;/p&gt;&#xA;&lt;p&gt;29 February 1856&lt;/p&gt;&#xA;&lt;p&gt;[I] left word at the school that I shall not be able to come again for the present. I doubt if I shall try again next term, the good done does not seem worth the time and trouble.&lt;/p&gt;&#xA;&lt;p&gt;Taken from here:&#xA;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Dodgson_teaching/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Extras/Dodgson_teaching/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9523" LastActivityDate="2021-01-29T08:29:21.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12819" PostTypeId="1" CreationDate="2021-01-29T14:28:21.290" Score="1" ViewCount="174" Body="&lt;p&gt;When the Gregorian calendar was implemented in 1582, one of its adjustments was the &lt;em&gt;lunar correction&lt;/em&gt;---to increase the epact by 1 eight times within a span of 2500 years; specifically, seven times after each interval of 300 years followed by an increment of 1 after 400 more years---thus completing the 2500 year cycle.&lt;/p&gt;&#xA;&lt;p&gt;An &lt;em&gt;epact&lt;/em&gt; is any integer between 0 and 29 which represents the phase of the moon on January 1st of a given year. Knowing this, one can then calculate the date of the ecclesiastical full moon---whence, the date of Easter.&lt;/p&gt;&#xA;&lt;p&gt;The first Gregorian lunar correction occurred in the year 1800---which was also the year in which Gauss produced his celebrated algorithm for calculating the date of Easter.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;MY QUESTION IS:&lt;/strong&gt;  Does anyone know if this was (or may have been) the motivating reason why Gauss produced the algorithm when he did. This, as well as any other factual details as to the motivation underlying his famous algorithm is appreciated.&lt;/p&gt;&#xA;" OwnerUserId="18820" LastEditorUserId="18820" LastEditDate="2021-02-02T01:37:23.790" LastActivityDate="2021-02-02T23:11:09.383" Title="Did the 1800 Gregorian Lunar Correction Motivate Gauss' Computus Algorithm?" Tags="|gauss|algorithm|calendar|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12820" PostTypeId="1" AcceptedAnswerId="12822" CreationDate="2021-01-29T19:28:37.290" Score="5" ViewCount="447" Body="&lt;p&gt;As far as I understand, Einstein’s original formulation of Special Relativity did not include the Clock Hypothesis, though it was implicitly assumed.&lt;/p&gt;&#xA;&lt;p&gt;The modern formulation of SR adds a formal statement of the Clock Hypothesis to Einstein’s original set of two postulates.&lt;/p&gt;&#xA;&lt;p&gt;When was the need to formalize the Clock Hypothesis identified and when did a three postulate presentation of SR first appear in a standard text?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="55" LastEditDate="2021-01-30T07:28:56.350" LastActivityDate="2021-06-01T10:01:45.207" Title="What is the history of adding the Clock Hypothesis to Special Relativity?" Tags="|physics|relativity-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12821" PostTypeId="1" AcceptedAnswerId="12823" CreationDate="2021-01-29T23:59:57.917" Score="4" ViewCount="139" Body="&lt;p&gt;The quote in question goes something like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is a quote of which I am very fond but I have just noticed that I actually can't tell where it was that Proclus brought it up... Do you know where it was that he expressed that opinion of his about mathematics?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advanced for your replies.&lt;/p&gt;&#xA;" OwnerUserId="6596" LastActivityDate="2021-01-30T19:33:33.570" Title="Source of a quote typically attributed to Proclus Lycaeus" Tags="|debunking|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12822" PostTypeId="2" ParentId="12820" CreationDate="2021-01-30T07:27:55.837" Score="8" Body="&lt;p&gt;Introduction of the clock hypothesis is usually associated with Rindler's &lt;em&gt;Special Relativity&lt;/em&gt; (1960), and the current debates over it were sparked after Brown defended its necessity in his book &lt;em&gt;Physical Relativity: Spacetime structure from a dynamical perspective&lt;/em&gt; (2005). According to &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-642-13538-5_7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arthur's chapter Minkowski’s Proper Time and the Status of the Clock Hypothesis&lt;/a&gt;, one can trace the hypothesis to a remark Einstein made to Sommerfeld (published in 1913):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In passing I note that, in their insistence that the clock hypothesis is a separate assumption in SR, Rindler and Brown could appeal to the authority of Einstein. For in the notes he made on Minkowski’s original “Raum und Zeit” paper, Arnold Sommerfeld attributes a remark to Einstein that may indicate the origin of the idea that the clock hypothesis is needed – at any rate it is the first statement of it that I have been able to find. Right after mentioning Minkowski’s remark that &lt;span class=&quot;math-container&quot;&gt;$d\tau$&lt;/span&gt; is not a complete differential, and noting that this shows that the proper times of two motions connecting two world points will generally differ (as discussed in Sect. 2&#xA;above), Sommerfeld adds:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This assertion is based, as Einstein has stressed, on the (unprovable) assumption that the&#xA;moving clock actually indicates the proper time, i.e. that at each instant it gives the time that&#xA;corresponds to the instantaneous state of velocity, regarded as constant. The moving clock&#xA;must naturally have been moved with acceleration (with changes of velocity or direction)&#xA;in order to be compared with the stationary clock at the world-point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Arthur and others (see e.g. &lt;a href=&quot;https://www.jstor.org/stable/26514394&quot; rel=&quot;nofollow noreferrer&quot;&gt;Valente, What do light clocks say to us regarding the so-called clock hypothesis?&lt;/a&gt;) do not consider the clock hypothesis to be on a par with the standard two postulates of special relativity, but classify it with what philosophers call coordinating principles. Those relate theoretical notions to measured quantities, proper time to clock readings in this case, and are not part of the theory proper. They are needed for any theory to make it testable and not just a mathematical abstraction. Here is Arthur's summary of the positions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Rindler claims it is an assumption that it is necessary to make in order to get from “purely kinematic laws about acceleration” to the dynamics of really accelerated systems (1966, 28) and Harvey Brown claims something similar in his recent book (Brown 2005, 9)... On the other hand, Jim Hartle holds that Minkowski’s formula for the proper&#xA;time holds “even for accelerating clocks, i.e., when the velocity is dependent on the&#xA;time” (Hartle 2003, 62), and he makes no use of the clock hypothesis in his textbook.&#xA;Roberto Torretti allows that the clock hypothesis “may be viewed as a conventional&#xA;definition of what we mean by clock accuracy, and hence by physical time” (1996,&#xA;96), but argues that “Special Relativity would doubtless have been rejected or, at&#xA;any rate, deeply modified, if the clock hypothesis were not fulfilled – to a satisfactory&#xA;approximation – by the timepieces actually used in physical laboratories”...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;On these latter views, provided a given process approximates well enough an&#xA;ideal clock, the clock hypothesis seems to amount to little more than the desideratum&#xA;that, with the metric locally Minkowskian, the predictions of SR should agree&#xA;with experimental fact. So the question is, why should it be necessary to state it as an independent hypothesis? Two main sets of considerations have been adduced. As&#xA;we have seen, one kind of justification has been that, since many natural clocks are&#xA;subject to accelerations which result in their failing to satisfy the clock hypothesis,&#xA;we need to appeal to the hypothesis in passing from the kinematics of acceleration&#xA;of ideal clocks to the dynamics of really moving clocks. A second kind of justification&#xA;has to do with the different status of Special Relativity within the General&#xA;Relativistic context, where, as is shown by the example of Weyl’s unified theory of&#xA;electromagnetism and gravitation, the metric could be locally Minkowskian and yet&#xA;the rate of clocks could be path-dependent in such a way that the instantaneous rate&#xA;would depend on the way the clock had been accelerated hitherto, contrary to the&#xA;clock hypothesis. This, it is argued, proves the independence of the clock hypothesis&#xA;from the assumption that spacetime is locally Minkowskian.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-01T18:59:37.987" LastActivityDate="2021-02-01T18:59:37.987" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12823" PostTypeId="2" ParentId="12821" CreationDate="2021-01-30T12:38:53.237" Score="6" Body="&lt;p&gt;The common source of all modern quotes is probably Kline, &lt;em&gt;Mathematical Thought from Ancient to Modern Times&lt;/em&gt; (1972), or Terence Tao, &lt;em&gt;Solving Mathematical Problems, a personal perspective&lt;/em&gt; (1992), anyway, it is a genuine quote from Proclus' &lt;em&gt;Commentary on the First Book of Euclid's &amp;quot;Elements&amp;quot;&lt;/em&gt;, last lines from the first part of the Prologue.&#xA;Unfortunately, I don't have an English translation, the original Greek can be found &lt;a href=&quot;https://archive.org/details/proclidiadochiin00procuoft/page/46/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, page 46 starting from line 15, in English it would sound like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;This is therefore the &amp;quot;mathesis&amp;quot;&lt;/strong&gt; [we can translate it by general learning or education]&lt;strong&gt;: reminiscence&lt;span class=&quot;math-container&quot;&gt;$^*$&lt;/span&gt; of the eternal ideas that are in the soul&lt;/strong&gt;; and it is the reason why the study that helps us so well for the reminiscence of those ideas has taken the name of mathematics. And the function of this science is revealed by its name [which means to learn, as Proclus says, it is the study by which it is reached the general education]; that is, &lt;strong&gt;it sets in motion our innate knowledge, awakens the Intellect, purifies the thought, highlights the concepts that are in us in essence, eliminates the oblivion and ignorance we have from birth&lt;/strong&gt;, loosens the bonds deriving from irrationality; following in this the god [maybe Hermes Trismegistus] who is the actual superintendent of this science, who brings to light our intellectual gifts, fills all things with divine reasons, moves souls towards the Intellect and awakens them as from a deep sleep, by means of research makes them reflect on themselves, perfects them through maieutic art, and with the discovery of the pure Intellect it also leads to a happy life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Anyway, I suggest you to read the entire Prologue (and the whole Commentary, of course), which discusses in details the ideas on mathematics of Plato, Aristotle and the so called school of Pythagoras, an English translation is &lt;a href=&quot;https://www.degruyter.com/princetonup/view/title/588770&quot; rel=&quot;noreferrer&quot;&gt;available&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^*$&lt;/span&gt; About mathematics as reminiscence, at page 45 Proclus cites Plato, &lt;em&gt;Phaedo&lt;/em&gt;, section 73b (&amp;quot;When people are questioned, if you put the questions well, they answer correctly of themselves about everything; and yet if they had not within them some knowledge and right reason, they could not do this. And that this is so is shown most clearly if you take them to mathematical diagrams or anything of that sort&amp;quot;) and 72e and &lt;em&gt;Meno&lt;/em&gt;, section 82a.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2021-01-30T19:33:33.570" LastActivityDate="2021-01-30T19:33:33.570" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12824" PostTypeId="1" CreationDate="2021-01-31T03:55:01.650" Score="1" ViewCount="37" Body="&lt;p&gt;A classical theorem on harmonic functions states that singularities of bounded harmonic functions are removable if the singular set is of null capacity. This theorem is sometimes attributed to Bouligand, and sometimes to Wiener. I have seen it proved in Wiener's 1924 paper, but I cannot find any source related to Bouligand. Can you please help me locate the original paper of Bouligand where it is proved, or a secondary source that claims Bouligand's priority?&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="4251" LastEditDate="2021-09-07T18:42:28.710" LastActivityDate="2021-09-07T18:42:28.710" Title="Singularities on null capacity sets are removable — Wiener or Bouligand?" Tags="|mathematics|discoveries|potential|wiener|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12825" PostTypeId="1" CreationDate="2021-01-31T13:39:14.423" Score="2" ViewCount="124" Body="&lt;p&gt;Gauss's results on the interconnection between the different values of the arithmetic-geometric mean of two complex numbers as recorded in his private notebooks led him to introduce foundational aspects of the theory of modular functions, and is perhaps the deepest part of his work in this area (as far as &amp;quot;modular forms&amp;quot; could be concieved in the early 1800s). This work is the only trace of such mathematical activity for about 4-5 decades (Abel and Jacobi were more concerned in the algebraic developement of the theory and not in the basic conception of modular forms), and is therefore very interesting historically.&lt;/p&gt;&#xA;&lt;p&gt;In connection with this work of Gauss, I found an &lt;a href=&quot;http://science.larouchepac.com/gauss/ceres/PDF/SourceBook/GaussWorks/LF_Gauss_agM.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation of some of Gauss's fragments&lt;/a&gt; - just google &amp;quot;&lt;strong&gt;Posthumous Fragments on the Theory of the Arithmetic-Geometric Mean and the Modulus Function&lt;/strong&gt;&amp;quot; and it will appear as one of the first search results. These translated fragments appear to be very deep, especially the parts on the &amp;quot;&lt;em&gt;&lt;strong&gt;Construction of the continuous fraction&lt;/strong&gt;&lt;/em&gt;&amp;quot; and &amp;quot;&lt;em&gt;&lt;strong&gt;The Quadratic Form. The Fundamental Domain&lt;/strong&gt;&lt;/em&gt;&amp;quot;. In addition, the translation refers to the exact pages in Gauss's werke from which they were translated.&lt;/p&gt;&#xA;&lt;p&gt;However, going into the same pages which the english translation refers to reveals that those fragments are not there, and in fact are not extant at all in Gauss's collected works (for example, the figure on p.25 of the translation cannot be found)! Therefore, i'd like to know what is the reference which those translated fragments are based on.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="55" LastEditDate="2021-02-01T18:55:23.347" LastActivityDate="2021-02-01T18:55:23.347" Title="Reference request for some fragments of Gauss with dubious origin" Tags="|reference-request|gauss|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12826" PostTypeId="1" CreationDate="2021-01-31T17:39:48.123" Score="1" ViewCount="293" Body="&lt;p&gt;I have recently been reading Gunter Ludwig's book wave mechanics to get a better understanding of quantum mechanics and in reading through the book I came across the relation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$m\sum_s \{|q_{rs}|^2\omega_{sr}-\omega_{rs}|q_{rs}|^2\}=\hbar$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$\omega_{nm}$&lt;/span&gt; is the quantised frequency &lt;span class=&quot;math-container&quot;&gt;$\nu\omega\to\omega_{nm}$&lt;/span&gt;, satisfying &lt;span class=&quot;math-container&quot;&gt;$\hbar\omega_{nm}=E_n-E_m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q_{nm}$&lt;/span&gt; are the hamiltonian coordinates.&lt;/p&gt;&#xA;&lt;p&gt;To justify the expression, he points to the equation &lt;span class=&quot;math-container&quot;&gt;$J_r=r\hbar+J_0$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; is the (quantised) action variable, although I am not sure how it follows.&lt;/p&gt;&#xA;&lt;p&gt;To try and get a better insight, I turned to the paper by Heisenberg, but that just referenced other papers by Khun and Thomas which I could not seem to find online.&lt;/p&gt;&#xA;&lt;p&gt;Might anyone be able to offer any insight on how this equation was arrived at and explain how experiments were carried out to justify it?&lt;/p&gt;&#xA;" OwnerUserId="13843" LastEditorUserId="10106" LastEditDate="2021-02-01T03:23:51.807" LastActivityDate="2021-02-23T14:03:27.420" Title="Origins of the canonical commutation relation" Tags="|physics|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12828" PostTypeId="2" ParentId="5937" CreationDate="2021-02-01T08:16:14.577" Score="0" Body="&lt;p&gt;According to Abraham Pais biography of Einstein, &lt;em&gt;Subtle Is the Lord: The Science and the Life of Albert Einstein&lt;/em&gt;, and who he knew both personally and professionally, Einstein was not so much bothered by the indeterminancy of quantum mechanics, but by it's non-locality, this was the crux of his paper with Podolsky &amp;amp; Rosen and which goes by the name, the EPR paradox. This is particularly pertinent to Einstein especially, as he discovered the local form of Newtonian gravity, and which Newton himself regarded as  a failure of his theory - that it acted at a distance. This is General relativity. Here it is spacetime itself that acts as the bearer of the metric field and which is gravity.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-02-01T08:16:14.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12830" PostTypeId="1" CreationDate="2021-02-01T22:00:55.107" Score="2" ViewCount="205" Body="&lt;p&gt;The concept of entropy in classical thermodynamics has its roots in attempts to improve the design and efficiency of steam engines; through Sadi Carnot's work, scientists eventually came to realize the significance of the notion of &amp;quot;reversible process&amp;quot; for estimation of the theoretical performance of heat engines (with the most efficient thermodynamic cycle being the &amp;quot;Carnot cycle&amp;quot;). This in turn led to the conception of the &amp;quot;second law of thermodynamics&amp;quot; and to the most important &amp;quot;state function&amp;quot; connected with it -- the entropy. If i understand its history correctly, classically the entropy was an artificial measure of the tendency of thermodynamics events (flow of heat from hot to cold bodies, spontaneous expansion when gas is allowed to make free expansion, etc.) to occur spontaneously; if &lt;span class=&quot;math-container&quot;&gt;$\Delta S &amp;gt;0$&lt;/span&gt; for a process that it should naturally occur, and, with the introduction of Helmholtz and Gibbs free energies, the entropy enabled to express &amp;quot;available energy&amp;quot; for work in different conditions.&lt;/p&gt;&#xA;&lt;p&gt;Later, Boltzmann gave the statistical interpretation of entropy, &lt;span class=&quot;math-container&quot;&gt;$S = k_B \mathbb{log \Omega}$&lt;/span&gt; , as a measure of the number of microstates corresponding to the same macro-state. This gave a new way of looking into the second law of thermodynamics (which states the entropy of the universe always rises):  since macrostates that correspond to greater amount of micro-states are more probable, the second law states that the universe tends to advance from the less probable to the more probable state, which&#xA;sounds quite logical.&lt;/p&gt;&#xA;&lt;p&gt;However, as with all great scientific theories, the final form of the theory somehow hides the evolution of ideas behind it, and renders it more difficult to gain insight into the thought process behind it. As a case example for which I lack both the statistical understanding and the historical understanding, I'd like to ask about the Clausius-Clapeyron relation for characterising the coexistence curve between two phases of matter.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I always found it quite tricky to gain intuition for Clausius-Clapeyron relation, &lt;span class=&quot;math-container&quot;&gt;$\frac{dP}{dT} = \frac {L}{T\Delta v}$&lt;/span&gt; , and although I understand its modern derivation (using entropy and chemical potential), I still feel like I don't really understand it. Therefore, I'd like to know how those authors (Emile Clapeyron and Rudolf Clausius) originally thought and derived their relation. This question is interesting since Clapeyron derived this relation before the notion of entropy was introduced.&lt;/li&gt;&#xA;&lt;li&gt;In addition, and this question is more physical than historical, I'd like to know how one can give a mechanical statistics interpretation of the change of entropy between two phases of matter, especially between liquid and gas; it's clear to me why there is a difference in the &amp;quot;level of disorder&amp;quot; between solid and liquid (since solid tends to be crystalline), but less clear in the case of liquid-gas phase transition.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="12330" LastEditDate="2021-02-02T06:40:26.157" LastActivityDate="2021-02-02T06:40:26.157" Title="Historical evolution of the ideas behind Clausius-Clapeyron equation and the classical/statistical notion of entropy" Tags="|thermodynamics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12832" PostTypeId="1" CreationDate="2021-02-02T17:23:23.053" Score="3" ViewCount="125" Body="&lt;p&gt;As some of you may recall, President James A. Garfield published at some point of his life a proof of the celebrated Pythagorean proposition.&lt;/p&gt;&#xA;&lt;p&gt;I am interested in acquiring a pdf copy of President Garfield's paper containing  his proof. If I recall correctly, the journal it was published in is &amp;quot;The New England Journal of Education&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Does any of you happen to possess a copy of the said document or know where it is that one may obtain one?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for your replies.&lt;/p&gt;&#xA;" OwnerUserId="6596" LastEditorUserId="16128" LastEditDate="2022-03-26T17:06:31.610" LastActivityDate="2022-03-26T17:06:31.610" Title="President James A. Garfield and the Pythagorean proposition" Tags="|reference-request|pythagorean-proposition|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12833" PostTypeId="1" CreationDate="2021-02-02T17:56:15.750" Score="8" ViewCount="12759" Body="&lt;p&gt;I recall reading in several sources the story about the letter &amp;quot;template&amp;quot; with which Edmund Landau used to answer individuals that sent him their &amp;quot;proofs&amp;quot; of Fermat's Last Theorem... According to what we read in Simon Singh's &amp;quot;Fermat's Last Theorem&amp;quot;, the template read something like this (the  poor translation into German is mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Herr/Frau ...&lt;/p&gt;&#xA;&lt;p&gt;Danke für der Brief mit Seinen Beweis der Große Fermatscher Satz. Wir&#xA;findet die Erste Fehler im Seite __, Linie ___. Deshalb der Beweis ist nicht&#xA;richtig.&lt;/p&gt;&#xA;&lt;p&gt;MfG,&lt;/p&gt;&#xA;&lt;p&gt;Doktor E. Landau.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Do you know if Landau actually sent one such letter once?  In case the answer is a resounding YES, do you know if copies of the template have been preserved somewhere?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for your knowledgeable replies...&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="55" LastEditDate="2021-02-04T08:55:40.837" LastActivityDate="2021-02-06T03:01:16.313" Title="Are there any extant letters backing up the famous anecdote about Edmund Landau and Fermat's Last Theorem?" Tags="|debunking|social-context|nachlasse|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="12834" PostTypeId="2" ParentId="12819" CreationDate="2021-02-02T23:11:09.383" Score="2" Body="&lt;p&gt;The motivation for producing the algorithm was for a convenient way of calculating Easter by arithmetical methods, without requiring reference to extensive tables which was the established method at the time.&lt;/p&gt;&#xA;&lt;p&gt;In that era, the reproduction of printed material was expensive, and paper itself was considerably more expensive than today. The church, at least at the time of the earlier Gregorian reform, had also prohibited the production of such tables without a license, apparently to avoid the propagation of errors by careless publishers (a reasonable fear, given the confusion which not only preceded the Julian reform in 45BC, but followed for many decades after).&lt;/p&gt;&#xA;&lt;p&gt;Informal reproduction of tables by hand for personal use, would also have been labour-intensive and error-prone, and asking for access may possibly have raised eyebrows amongst the clergy who held the master copies of such tables produced under licence from the church, and they may have viewed the knowledge in the tables as rightfully the property of the church.&lt;/p&gt;&#xA;&lt;p&gt;As an aside, when the British finally adopted the Gregorian reforms commencing in 1752, a tabular calculation (which differed in its mechanical steps from how the Catholic Church performed the calculation, but produced the same results) was required to be published publicly in prayer books.&lt;/p&gt;&#xA;&lt;p&gt;The obvious function of this British approach was to disseminate knowledge of the calendar change throughout society, but (in conjunction with the Catholic Church's attitude to licensing reproduction of tables) it also suggests that hitherto it may not have been the established practice of the church to make such comprehensive information about the calendar easily and publicly available to the laity. Even then, the British tabular calculation covers more than a dozen printed pages, and is not easy to apply in my experience.&lt;/p&gt;&#xA;&lt;p&gt;So reducing it to an algorithm with an economical number of sequential steps, and in which the essential elements of the calculation could be described and verified briefly (covering maybe two dozen lines of free text, instead of exceeding a dozen pages of typeset tables), could be seen as a real boon for ease of use and reproduction.&lt;/p&gt;&#xA;&lt;p&gt;In terms of broadly explaining the &lt;em&gt;timing&lt;/em&gt; of Gauss's devising of the algorithm, it should be noted that the upcoming application of the first lunar correction could well have invalidated any existing tables to which people had been referring, and that probably caught Gauss's attention.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics had also been developing, and applications in commerce meant that many more people in society had a familiarity with arithmetic, so it was feasible to devise and publish such an algorithm and it have some social usefulness.&lt;/p&gt;&#xA;&lt;p&gt;I believe Gauss did in fact make an error in his first publishing of the algorithm, confirming why the church was itself highly conservative about such matters, and he was forced to publish a second corrected version.&lt;/p&gt;&#xA;&lt;p&gt;It's not actually clear how widely the algorithm was employed for any practical purpose in Gauss's time however, as opposed to being an intellectual curiosity. I imagine those compiling calendars and almanacs for widespread public consumption (something that only really became common in the 19th century) would often have preferred to rely on the tabular method which carried the sanction of authority, and most people with a need for the information would then have referred to such calendars and almanacs rather than performing their own calculations.&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2021-02-02T23:11:09.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12835" PostTypeId="2" ParentId="8332" CreationDate="2021-02-03T02:09:29.803" Score="3" Body="&lt;p&gt;Whilst researching Noethers theorems I noticed that an early case of the Noethers first theorem was proven by the Cosserat brothers in 1909 in a paper titled, &lt;em&gt;Theorie des corps deformables&lt;/em&gt;. To give them their proper names, they were Eugene &amp;amp; Francois Cosserat. One was a mathematician, and the other, an engineer.&lt;/p&gt;&#xA;&lt;p&gt;As the paper hints at, they were known for their work in elasticity, and which in modern terms would now be phrased via the theory of fibre bundles, or more precisely, G-bundles.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-02-03T15:46:08.403" LastActivityDate="2021-02-03T15:46:08.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12837" PostTypeId="1" CreationDate="2021-02-03T11:24:35.977" Score="1" ViewCount="155" Body="&lt;p&gt;There is a famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Cutting_the_Stone&quot; rel=&quot;nofollow noreferrer&quot;&gt;painting&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Hieronymus_Bosch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hieronymus Bosch&lt;/a&gt; in which someone is being operated on to remove the &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stone_of_madness&quot; rel=&quot;nofollow noreferrer&quot;&gt;stone of folly&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/8/82/Hieronymus_Bosch-Removing_the_Rocks_from_the_Head-Detail.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/8/82/Hieronymus_Bosch-Removing_the_Rocks_from_the_Head-Detail.jpg&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Detail from Hieronymus Bosch's painting titled &amp;quot;Extraction of the Stone of Folly&amp;quot;. It depicts a surgeon, who is wearing a funnel hat and is cutting into the skull of a patient by &lt;a href=&quot;https://en.wikipedia.org/wiki/Trepanning&quot; rel=&quot;nofollow noreferrer&quot;&gt;trepanation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is whether &lt;em&gt;all&lt;/em&gt; of the following is possible:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;someone exhibits cognitive difficulties due to pressure from a tumor, perhaps one so large that it deformed the skull (not sure if that can happen);&lt;/li&gt;&#xA;&lt;li&gt;this suggested that cutting into the head might be indicated, and&lt;/li&gt;&#xA;&lt;li&gt;assuming the patient did not die, perhaps sometimes some improvement was indeed observed because the pressure was relieved.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="10856" OwnerDisplayName="releseabe" LastEditorUserId="14365" LastEditDate="2021-05-17T18:28:40.153" LastActivityDate="2021-05-23T19:55:48.207" Title="&quot;Stone of Folly&quot; based on maybe finding and removing brain tumors?" Tags="|biology|medicine|middle-ages|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12838" PostTypeId="2" ParentId="12837" CreationDate="2021-02-03T14:22:29.087" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Trepanation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trephination&lt;/a&gt; has an ancient history.  In the right hands, it may have been a useful tool to deal with depressed skull fractures and subdural hematomas.  Many of the patients &lt;em&gt;did&lt;/em&gt; survive, rather amazingly when one considers it is at least nominally brain surgery done in 6500 B.C.  (Also note that many ancient cultures, armed with opium, nightshade, cannabis, and more dubious agents, were more skilled with anaesthesia than witch-fearing Medieval Europe)&lt;/p&gt;&#xA;&lt;p&gt;A subdural hematoma is a somewhat solid object (which is to say, clotted blood) not otherwise found in the body, which can cause strange symptoms (Richard Feynman told a great story - &lt;a href=&quot;https://www.latimes.com/archives/la-xpm-1986-04-20-tm-1265-story.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here's one version&lt;/a&gt;).  You can see from &lt;a href=&quot;https://www.dovepress.com/cr_data/article_fulltext/s193000/193548/img/ndt-193548-f01.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this set of images of a modern trephination&lt;/a&gt; that the operation essentially consists of the removal of a solid, unwanted body (panel E) that could cause behavioral symptoms.  One can argue this way it &lt;em&gt;ought&lt;/em&gt; to have been counted as a &amp;quot;stone of madness&amp;quot;, but proving that point historically is another matter.&lt;/p&gt;&#xA;" OwnerUserId="13864" OwnerDisplayName="Mike Serfas" LastEditorUserId="13864" LastEditDate="2021-05-23T19:55:48.207" LastActivityDate="2021-05-23T19:55:48.207" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12839" PostTypeId="2" ParentId="7083" CreationDate="2021-02-03T16:44:16.750" Score="0" Body="&lt;p&gt;Both Newton and Liebniz used the notion of infinitesimals which Newton called fluxions. Given the philosopher, Bishop Berkleys acid criticisms of 'the ghosts of departed quantities' mathematicians began searching for a more rigorous approach culminating in Cauchy's definition of a limit. However, Liebniz already had this idea. Today, the pendulum has turned full circle with approaches much closer to Newton and Liebniz's notion of an infinitesimal now finding a rigorous foundation with synthetic differential geometry.&lt;/p&gt;&#xA;&lt;p&gt;According to the Encyclopedia Brittanica, both Newton and Liebniz wete aware of the chain rule. However, what was essential was their understanding of infinitesimals as moments of change.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-02-03T16:44:16.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12840" PostTypeId="1" CreationDate="2021-02-04T03:55:11.070" Score="0" ViewCount="58" Body="&lt;p&gt;In Numerical root solving technique,  which comes first in history - Newton or Secant - and each one is named after whom?&lt;/p&gt;&#xA;" OwnerUserId="13872" OwnerDisplayName="Vemula Ramakrishna Reddy" LastEditorUserId="10106" LastEditDate="2021-03-01T04:42:31.857" LastActivityDate="2021-03-01T04:42:31.857" Title="Which one goes first - Secant or Newton - in Numerical root finding technique?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12841" PostTypeId="1" CreationDate="2021-02-05T01:47:10.017" Score="2" ViewCount="271" Body="&lt;p&gt;I am reading Euler's &amp;quot;&lt;a href=&quot;http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to analysis of the infinite&lt;/a&gt;&amp;quot;, chapter 8, page at the end of page 208, beginning of page 209 and came across his derivation of the power series for &lt;span class=&quot;math-container&quot;&gt;$\sin(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(x)$&lt;/span&gt;. I try hard to follow his steps but fail flat.&lt;/p&gt;&#xA;&lt;p&gt;He wrote as followed:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(nz)=\dfrac{n}{1}(\cos(z))^{n-1}\sin(z)+\dfrac{n(n-1)(n-2)}{3!}\cos(z)^{n-3}(\sin(z))^{3}+\dfrac{n(n-1)(n-2)(n-3)(n-4)}{5!}(\cos(z))^{n-5}(\sin(z)^5 .etc$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(nz)=\cos(z)^{n}+\dfrac{n(n-1)}{2!}(\cos(z))^{n-2}(\sin(z))^{2}+\dfrac{n(n-1)(n-2)(n-3)}{4!}\cos(z)^{n-4}(\sin(z)^4$&lt;/span&gt; .etc&lt;/p&gt;&#xA;&lt;p&gt;He stated:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let the arc of z be infinitely small; there will be &lt;span class=&quot;math-container&quot;&gt;$\sin(z)=z$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(z)=1$&lt;/span&gt;; moreover &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; shall be an infinitely great number, so that the arc &lt;span class=&quot;math-container&quot;&gt;$nz$&lt;/span&gt; shall be of finite magnitude, for example &lt;span class=&quot;math-container&quot;&gt;$nz=v$&lt;/span&gt;; on account of &lt;span class=&quot;math-container&quot;&gt;$\sin(z)=z=\dfrac{v}{n}$&lt;/span&gt; there becomes&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(v)=1-\dfrac{v^2}{2!}+\dfrac{v^4}{4!}-\dfrac{v^6}{6!}+.etc$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(v)=v-\dfrac{v^3}{3!}+\dfrac{v^5}{5!}-\dfrac{v^7}{7!}+.etc$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have a couple of questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What does it mean when he says &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; be infinitely small and &lt;span class=&quot;math-container&quot;&gt;$sin(z)=z$&lt;/span&gt;? On the interval &lt;span class=&quot;math-container&quot;&gt;$(0,\dfrac{\pi}{2})$&lt;/span&gt;, we have the equality &lt;span class=&quot;math-container&quot;&gt;$\sin(z)&amp;lt;z$&lt;/span&gt;. Besides &lt;span class=&quot;math-container&quot;&gt;$-1\leq\sin(z)\leq1$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\forall z$&lt;/span&gt;, does it mean that &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; approach &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, because obviously &lt;span class=&quot;math-container&quot;&gt;$\sin(z)=z$&lt;/span&gt; only when &lt;span class=&quot;math-container&quot;&gt;$z=0$&lt;/span&gt;. This makes sense when &lt;span class=&quot;math-container&quot;&gt;$\cos(0)=1$&lt;/span&gt;, as he stated. Also, what does he mean when he said &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is infinitely large?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;I've tried to substitute &lt;span class=&quot;math-container&quot;&gt;$\sin(z)=z=\dfrac{v}{n}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos(z)=1$&lt;/span&gt; but I could not eliminate the whole bunch of &lt;span class=&quot;math-container&quot;&gt;$n(n-1)(n-2)$&lt;/span&gt; (for &lt;span class=&quot;math-container&quot;&gt;$\sin(nz)$&lt;/span&gt;) or &lt;span class=&quot;math-container&quot;&gt;$n(n-1)$&lt;/span&gt; (for &lt;span class=&quot;math-container&quot;&gt;$\cos(nz)$&lt;/span&gt;) or the likes to arrive at the final power series for both functions. Here are what I've got.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sin(nz)=\dfrac{n\cdot z}{1}-\dfrac{n(n-1)(n-2)}{3!}z^3+\dfrac{n(n-1)(n-2)(n-3)(n-4)}{5!}z^5+ .etc$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Obviously the first term is &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, but what about the rest of the terms?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cos(nz)=1-\dfrac{n(n-1)}{2!}z^2+\dfrac{n(n-1)(n-2)(n-3)}{4!}z^4- .etc$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Obviously because I don't know what is the value of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; so I can't make them varnish. Could you help me here? Thank you so much!&lt;/p&gt;&#xA;" OwnerUserId="13807" LastActivityDate="2021-11-02T21:06:00.753" Title="How to derive the power series of $\sin(x)$ and $\cos(x)$ followed the footstep of Euler" Tags="|calculus|real-analysis|euler|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12843" PostTypeId="2" ParentId="12841" CreationDate="2021-02-05T19:05:25.327" Score="3" Body="&lt;p&gt;Concerning your question &amp;quot;Obviously the first term is &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt;, but what about the rest of the terms?&amp;quot;. Since he considers &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to be infinitely large he probably has &lt;span class=&quot;math-container&quot;&gt;$n(n-1)(n-2)\cdots (n-(k-1))=n^k$&lt;/span&gt; so that &lt;span class=&quot;math-container&quot;&gt;$n(n-1)(n-2)\cdots (n-(k-1))z^k=v^k$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2021-02-05T19:05:25.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12844" PostTypeId="1" CreationDate="2021-02-05T23:25:15.920" Score="11" ViewCount="367" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Muon_neutrino#Further_reading&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article on the muon neutrino&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The muon neutrino is a lepton, an elementary subatomic particle which has the symbol &lt;span class=&quot;math-container&quot;&gt;$\nu_\mu$&lt;/span&gt; and no net electric charge. Together with the muon it forms the second generation of leptons, hence the name muon neutrino. It was first hypothesized in the early 1940s by several people, and was discovered in 1962 by Leon Lederman, Melvin Schwartz and Jack Steinberger.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But it doesn't say who hypothesized it, and I haven't been able to find that out.&lt;/p&gt;&#xA;&lt;p&gt;So: who predicted the existence of the muon neutrino, and in what papers did they do it?&lt;/p&gt;&#xA;" OwnerUserId="13875" LastEditorUserId="2324" LastEditDate="2021-02-06T04:10:53.680" LastActivityDate="2021-02-09T20:54:18.903" Title="Who predicted the existence of the muon neutrino?" Tags="|physics|biographical-details|particle-physics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12845" PostTypeId="2" ParentId="12844" CreationDate="2021-02-06T00:22:49.393" Score="13" Body="&lt;p&gt;Nobody in particular, it was what is called &amp;quot;folklore&amp;quot;. The idea came up naturally when muon decay was observed by several groups in 1948, see &lt;a href=&quot;https://arxiv.org/abs/physics/0503172&quot; rel=&quot;noreferrer&quot;&gt;Anicin's The Neutrino - Its Past, Present and Future&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;When in 1948&#xA;the electron spectrum from muon decay was found to be continuous it became&#xA;obvious that not one but two neutrinos are emitted along with the electron.&#xA;Pontecorvo witnesses that at that time everybody felt that the two neutrinos&#xA;should be different. They were even named differently, the &amp;quot;neutrino&amp;quot; and the&#xA;&amp;quot;neutretto&amp;quot;, but with time the idea seem to have been forgotten and it was only&#xA;in 1962, when the difference between the two neutrinos has been clearly&#xA;demonstrated in the first of a long series of important accelerator neutrino&#xA;experiments, that the electron and the muon neutrino were finally given life.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Pontecorvo reference is to his &lt;a href=&quot;https://inspirehep.net/literature/178813&quot; rel=&quot;noreferrer&quot;&gt;The infancy and youth of neutrino physics: some recollections&lt;/a&gt;, where we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Several groups, among which J.&#xA;Steinberger, E. Hincks and I, and others were investigating the (cosmic) muon decay.&#xA;The result of the investigations was that the decaying muon emits 3 particles: one&#xA;electron (this we found by measuring the electron bremsstrahlung) and two neutral&#xA;particles, which were called by various people in different ways: two neutrinos,&#xA;neutrino and neutretto, &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\nu'$&lt;/span&gt;, etc. I am saying this to make clear that for people&#xA;working on muons in the old times, the question about different types of neutrinos&#xA;has always been present. True, later on many theoreticians forgot all about it, and&#xA;some of them &amp;quot;invented&amp;quot; again the two neutrinos (for example M. Markov), but for&#xA;people like Bernardini, Steinberger, Hincks and me ... the two neutrino question was&#xA;never forgotten... How to perform the decisive experiment I was able to formulate&#xA;/40/ clearly enough (the use of muon neutrino beams). At the time the idea of&#xA;the experiment was not obvious, although the statement may be strange today: one&#xA;must search for electrons and muons produced in matter by muon neutrinos&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;/40/ is the reference to Pontecorvo's 1959 paper in JETP, &lt;a href=&quot;http://www.jetp.ac.ru/cgi-bin/e/index/e/9/5/p1148?a=list&quot; rel=&quot;noreferrer&quot;&gt;The Universal Fermi Interaction and Astrophysics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-06T00:51:57.037" LastActivityDate="2021-02-06T00:51:57.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12846" PostTypeId="2" ParentId="12833" CreationDate="2021-02-06T00:51:52.197" Score="6" Body="&lt;p&gt;The context of this anecdote is the Wolfskehl Prize for the first correct proof of Fermat's Last Theorem. At its creation, the prize was endowed with the enormous sum of 100,000 German marks in gold. The Königliche Gesellschaft der Wissenschaften at Göttingen (the Göttingen Royal Society of Science), was charged with determining the eventual recipient. For a comprehensive description see the &lt;a href=&quot;http://www.ams.org/notices/199710/barner.pdf&quot; rel=&quot;noreferrer&quot;&gt;following paper&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Klaus Barner, &amp;quot;Paul Wolfskehl and the Wolfskehl Prize.&amp;quot; &lt;em&gt;Notices of the AMS&lt;/em&gt;, Vol. 44, No. 10, Nov. 1997, pp. 1294-1303.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On June 27, 1908, the day exactly eighty-nine years prior to the presentation of the prize to Wiles, the conditions for the Wolfskehl Prize endowment were laid down by the Göttingen Royal Society of Science. These were published in several journals, for instance in the &lt;em&gt;Jahresbericht der DMV&lt;/em&gt; 17 (1908), 111–113 (with commentary by Felix Klein); the &lt;em&gt;Mathematische Annalen&lt;/em&gt; 66 (1909), 143–144; and the &lt;em&gt;Acta Mathematica&lt;/em&gt; 31 (1908), at the end of the volume. The prize was, according to the testament, to be valid until September 13, 2007.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A high-quality &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0066?tify=%7B%22pages%22:%5B149%5D,%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt; of the notice about the prize from &lt;em&gt;Mathematische Annalen&lt;/em&gt; can be found at the Göttingen Digitization Center.&lt;/p&gt;&#xA;&lt;p&gt;Unsurprisingly, the prize elicited a huge number of submissions, which were examined by &amp;quot;Assistenten&amp;quot; (literally: assistants; I do not know whether this equates to postdocs, assistant lecturers, or assistant professors) at the Mathematical Institute at Göttingen, where Edmund Landau worked from 1909 to 1934. According to the &lt;a href=&quot;https://de.wikipedia.org/wiki/Wolfskehl-Preis&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; about the Wolfskehl Prize in the German Wikipedia, which cites Paulo Ribenboim's &lt;em&gt;Fermat´s Last Theorem&lt;/em&gt;, there were 621 submissions of supposed proofs in the first year alone. Citing an employee of the institute in the mid-1970s, there were still 3 to 4 submissions per month at that time.&lt;/p&gt;&#xA;&lt;p&gt;A fairly extensive online search for Edmund Landau, reminiscences of people who knew him personally during the 1930s, and mathematics at Göttingen between 1909 and 1938 (the year of Landau's death) did not provide any evidence of the anecdote. The story is therefore very likely apocryphal. The earliest mention in print that I could find is &lt;a href=&quot;https://books.google.com/books?id=uiPpu1jPJW4C&amp;amp;pg=PA17&amp;amp;lpg=PA17&amp;amp;dq=%22When+Edmund+Landau+was+head+of+the+mathematics+department%22&amp;amp;source=bl&amp;amp;ots=MPD1Pb7oqg&amp;amp;sig=ACfU3U1lSJe3_dCByYfVreI3g__iOdt7Dw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiasoLshNTuAhXiHDQIHYwzD4EQ6AEwAHoECAEQAg#v=onepage&amp;amp;q=%22When%20Edmund%20Landau%20was%20head%20of%20the%20mathematics%20department%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Nelson Stephens, &amp;quot;The prize puzzle in mathematics&amp;quot;, &lt;em&gt;New Scientist&lt;/em&gt;, Vol. 102, No. 1409, May 10, 1984, pp. 16-19.&lt;/p&gt;&#xA;&lt;p&gt;Referring to the Georg August University of Göttingen, the author writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When Edmund Landau was head of the mathematics department there from 1909 to 1934, he prepared preprinted letters beginning something along the lines of &amp;quot;Dear  , Thank you for your manuscript on the proof of Fermat's Last Theorem. The first mistake is on line  , page  , and invalidates the proof.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It might be interesting to trace the history of this anecdote back through the literature; I am marking this answer as Community wiki for that purpose.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-02-06T03:01:16.313" LastActivityDate="2021-02-06T03:01:16.313" CommentCount="0" CommunityOwnedDate="2021-02-06T00:51:52.197" ContentLicense="CC BY-SA 4.0" />
  <row Id="12848" PostTypeId="1" CreationDate="2021-02-06T12:41:15.620" Score="2" ViewCount="117" Body="&lt;p&gt;In the Einstein's 1905 paper on light quanta, he wrote the following about Boltzmann's notion of probability:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The word probability is used in a sense that does not confirm to its definition as it given in the theory of probability. In particular &amp;quot;cases of equal probability&amp;quot; are often hypothetically defined in istances where the theoretical pictures used are sufficiently definite to give a deduction rather than a hypothetical assertion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What does this mean?&lt;/p&gt;&#xA;" OwnerDisplayName="user13853" LastEditorUserId="4251" LastEditDate="2021-02-19T10:41:39.327" LastActivityDate="2021-02-19T10:41:39.327" Title="What is the difference between Einstein's and Boltzmann's definitions of probability?" Tags="|mathematics|einstein|probability|statistical-mechanics|boltzmann|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12849" PostTypeId="2" ParentId="12848" CreationDate="2021-02-06T14:54:05.717" Score="1" Body="&lt;p&gt;He's saying that the hypothesis of &amp;quot;cases of equal probability&amp;quot; is taken in situations (&amp;quot;instances&amp;quot;) where the &amp;quot;theoretical picture&amp;quot; used is &amp;quot;definite&amp;quot; enough that we can deduce this hypothesis and not merely assert that this is the case.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-02-06T14:54:05.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12850" PostTypeId="1" CreationDate="2021-02-06T19:58:57.177" Score="0" ViewCount="96" Body="&lt;p&gt;Im doing this presentation for school and we have to find out who exactly first used independent and dependent variables. I've searched everywhere but no answer was found its either John radford young or Marshall.&lt;/p&gt;&#xA;" OwnerUserId="13880" LastActivityDate="2021-02-06T22:34:22.260" Title="Independent and dependent variables" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12851" PostTypeId="2" ParentId="12850" CreationDate="2021-02-06T22:34:22.260" Score="2" Body="&lt;p&gt;Have you seen the Earliest Known Uses of the Word of Mathematics.&#xA;&lt;a href=&quot;https://jeff560.tripod.com/d.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://jeff560.tripod.com/d.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The point is that this might be first English usage? What about German or French?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;DEPENDENT/INDEPENDENT VARIABLE. Independent variable appears in 1813&#xA;in Memoirs of the Analytical Society (Cambridge): “The method of&#xA;Laplace for reducing an equation of the first order, where the&#xA;difference of the independent variable is any function of the variable&#xA;itself, to one wherein that difference is constant, is well known.”&#xA;[Google print search by James A. Landau]&lt;/p&gt;&#xA;&lt;p&gt;Subordinate variable and independent variable appear in English in the&#xA;1816 translation of Differential and Integral Calculus by Lacroix:&#xA;&amp;quot;Treating the subordinate variables as implicit functions of the&#xA;indepdndent [sic] ones&amp;quot; [OED].&lt;/p&gt;&#xA;&lt;p&gt;Dependent variable appears in in 1831 in the second edition of&#xA;Elements of the Differential Calculus (1836) by John Radford Young:&#xA;&amp;quot;On account of this dependence of the value of the function upon that&#xA;of the variable the former, that is y, is called the dependent&#xA;variable, and the latter, x, the independent variable&amp;quot; [James A.&#xA;Landau].&lt;/p&gt;&#xA;&lt;p&gt;In Statistics R. A  Fisher used the terms dependent and independent in&#xA;his presentation of regression analysis: see Section 25 of Statistical&#xA;Methods for Research Workers (1925) on &amp;quot;Regression coefficients&amp;quot;. For&#xA;Fisher these terms qualified variate but later writers have generally&#xA;favoured variable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-02-06T22:34:22.260" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12854" PostTypeId="1" CreationDate="2021-02-07T15:05:54.220" Score="3" ViewCount="108" Body="&lt;p&gt;In the book &amp;quot;Ludwig Boltzmann, the man who trusted atoms&amp;quot;by C.Cercignani, I read about the thesis according to which it was Boltzmann, not Gibbs who first introduced the concept of &amp;quot;statistical ensemble&amp;quot;, which however he named in a different way (&amp;quot;monodes&amp;quot;). The passage in question  is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;As a matter of fact a statistical ensemble (in Gibbs's terminology) is called a monode by Boltzmann. The question posed in the above-mentioned paper [ 1] is the following: what statistical families of steady distributions have the property that, when an infinitesimal change is made in their parameters, the infinitesimal changes in the average total energy of the system E, of the pressure p, and of the volume V are such that (dE + p dV)/ T (where T is the average kinetic energy per particle) is an exact differential (at least in the thermodynamic limit. when V &lt;span class=&quot;math-container&quot;&gt;$\rightarrow \infty$&lt;/span&gt; ,N &lt;span class=&quot;math-container&quot;&gt;$\rightarrow \infty$&lt;/span&gt; , whereas N / V remains bounded)? These families are called orthodes by Boltzmann. The answer given by Boltzmann to his own question is that there are at least two ensembles of this kind, the ergode (Gibbs's microcanonical ensemble) and the holode (Gibbs's canonica/ ensemble).&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are there historical studies about the Boltzmann contribution to this area of research?&lt;/p&gt;&#xA;" OwnerDisplayName="user13853" LastEditorUserId="4251" LastEditDate="2021-03-09T07:29:46.410" LastActivityDate="2021-03-09T07:29:46.410" Title="What was Boltzmann's contribution to the theory of &quot;statistical ensemble&quot;?" Tags="|physics|statistical-mechanics|boltzmann|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12855" PostTypeId="2" ParentId="12844" CreationDate="2021-02-07T21:54:47.413" Score="5" Body="&lt;p&gt;I have compiled some answers to my question here:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://johncarlosbaez.wordpress.com/2021/02/07/neutrinos-and-neutrettos/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neutrinos and neutrettos&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The history is indeed complex, but a lot of credit must be given to &lt;a href=&quot;https://en.wikipedia.org/wiki/Shoichi_Sakata&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sakata&lt;/a&gt; and Inoue, who published in 1942 a paper on a theory involving two mesons and two neutrinos.   It was translated into English and published here in 1946:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Shoichi Sakata and Takesi Inoue, &lt;a href=&quot;https://academic.oup.com/ptp/article/1/4/143/1846220&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the correlations between mesons and Yukawa particles&lt;/a&gt;, &lt;em&gt;Progress of Theoretical Physics&lt;/em&gt; &lt;strong&gt;1&lt;/strong&gt; (4) (1946), 143–150.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It's open-access but I haven't read it yet.  The webpage says &amp;quot;the content of this paper was read before the symposium of the meson theory held on September, 1913,&amp;quot; but that can't be right: it must have been 1943.&lt;/p&gt;&#xA;" OwnerUserId="13875" LastEditorUserId="13875" LastEditDate="2021-02-09T20:54:18.903" LastActivityDate="2021-02-09T20:54:18.903" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12856" PostTypeId="2" ParentId="12787" CreationDate="2021-02-08T12:54:27.043" Score="1" Body="&lt;p&gt;&lt;strong&gt;intuition behind a neighborhood system.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A version of this that I remember, leading up to Hausdorff (1914) defining his neighborhood axioms.&lt;/p&gt;&#xA;&lt;p&gt;There was a notion (nowadays called a &amp;quot;manifold&amp;quot;) where you specify a system of &amp;quot;charts&amp;quot; ... subsets of your space, and each of these subsets is homeomorphic to an open set in Euclidean space &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt;.  Then you define &amp;quot;continuous&amp;quot; (as well as &amp;quot;dfferentiable&amp;quot; etc.) using the corresponding property in &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt;.  But at some point Hausdorff noticed that to define &amp;quot;continuous&amp;quot; you could ignore the structure of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt;: if you allow arbitrarily small &amp;quot;neighborhoods&amp;quot; then you can define continuity just in terms of whether certain neighborhoods of your domain map into certain other neighborhoods of your codomain, without referring to the &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^n$&lt;/span&gt; structure at all.  Writing this abstractly led to the &amp;quot;Hausdorff space&amp;quot;, entirely specified in terms of &amp;quot;neighborhoods&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Later, when other abstract notions of topology were investigated, it turned out in many cases that Hausdorff space was the same thing, if you add the one extra axiom, now known as the Hausdorff axiom.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-02-08T12:54:27.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12857" PostTypeId="2" ParentId="12826" CreationDate="2021-02-08T17:41:12.070" Score="2" Body="&lt;p&gt;Indeed, Heisenberg's 1925 paper got the ball rolling, but was not recognized as matrix mechanics and transcribed/streamlined in the standard matrix language we all recognize today until later (merely 60 days!) that year in the seminal paper by Born and Jordan &lt;a href=&quot;https://doi.org/10.1007/BF01328531&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Quantenmechanik. &lt;em&gt;Z. Physik&lt;/em&gt;  &lt;strong&gt;34&lt;/strong&gt;  858–888 (1925)&lt;/a&gt;, detailed in Jeremy Bernstein's must-read article &lt;a href=&quot;https://doi.org/10.1119/1.2060717&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max Born and the quantum theory, &lt;em&gt;American Journal of Physics&lt;/em&gt; &lt;strong&gt;73&lt;/strong&gt; 999 (2005)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Max Born had learned about matrices in his study under Jakob Rosanes at Breslau University. His assistant and former student Pascual Jordan had been an assistant to mathematician Richard Courant at Göttingen, helping in the formulation of Hilbert space! Born got the Nobel prize in 1954 for this work, and for complex and fraught reasons Jordan was skipped over.&lt;/p&gt;&#xA;&lt;p&gt;Your stated relation, for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Matrix_mechanics#Harmonic_oscillator&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantum oscillator&lt;/a&gt;, maps to Born's central relation  &lt;span class=&quot;math-container&quot;&gt;$\hat x \hat p - \hat p \hat x = i \hbar I$&lt;/span&gt;, accessed &lt;a href=&quot;https://hsm.stackexchange.com/questions/7081/how-did-born-come-up-with-the-canonical-commutation-relation-hat-x-hat-p-ha&quot;&gt;here&lt;/a&gt;; cf details &lt;a href=&quot;https://physics.stackexchange.com/a/388014/66086&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If, on the other hand, you are asking not what it means, in today's terms, but how did Heisenberg intuit it, the word &lt;a href=&quot;https://en.wikipedia.org/wiki/Heisenberg%27s_entryway_to_matrix_mechanics#The_breakthrough_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;magic&amp;quot; has been proffered&lt;/a&gt;. He is composing radiation intensities in a manner consistent with Bohr's quantization rule, and guessing how &lt;span class=&quot;math-container&quot;&gt;$\hat x^2$&lt;/span&gt; and (essentially) &lt;span class=&quot;math-container&quot;&gt;$\hat p^2$&lt;/span&gt; behave w.r.t. each other, his &amp;quot;reinterpretation&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;A final note added, since, despite all this, it really lies at the heart of the question. The modern understanding and interpretation of the essence of Heisenberg's invention, albeit in the language &lt;em&gt;it&lt;/em&gt; created and which the world has adopted since (including the generous term &amp;quot;Heisenberg's equations of motion&amp;quot;), actually got its start in Dirac's 1925 equally breathtaking leap of &amp;quot;reinterpretation&amp;quot; of Heisenberg's paper: &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1925.0150?casa_token=e2GxgU84gfIAAAAA:12M1t4SIul1ZxeE_VZhks-PwryNS1yL76gkjvSir60TYiSHOj6rQPirFScvXR88MRdXBt9fGoJIAkg&quot; rel=&quot;nofollow noreferrer&quot;&gt;The fundamental equations of quantum mechanics. &lt;em&gt;Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character&lt;/em&gt;, &lt;strong&gt;109&lt;/strong&gt; (752), 642-653  (1925)&lt;/a&gt;, &lt;a href=&quot;https://www.jstor.org/stable/94441&quot; rel=&quot;nofollow noreferrer&quot;&gt;else here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-02-23T14:03:27.420" LastActivityDate="2021-02-23T14:03:27.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12858" PostTypeId="1" CreationDate="2021-02-09T02:15:37.537" Score="1" ViewCount="48" Body="&lt;p&gt;The full title:&#xA;RECHERCHES PHYSIQUES MÉCANIQUES ET ANALYTIQUES SUR LE SON ET SUR LE TONS DES TUYAUX D’ORGUES DIFFÉREMMENT CONSTRUITS&lt;/p&gt;&#xA;&lt;p&gt;Link to a pdf: &lt;a href=&quot;https://catalog.hathitrust.org/Record/011812546&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://catalog.hathitrust.org/Record/011812546&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some excerpts are translated in the 1890 treatise &amp;quot;The Flute&amp;quot; by R. S. Rockstro (Part I, Ch. 4):&#xA;&lt;a href=&quot;https://www.cs.dartmouth.edu/%7Ewbc/rockstro/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cs.dartmouth.edu/~wbc/rockstro/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thanks for any help!&lt;/p&gt;&#xA;" OwnerUserId="13896" LastActivityDate="2021-02-09T02:15:37.537" Title="Has Daniel Bernoulli's 1764 paper on organ pipe acoustics been translated into English?" Tags="|physics|acoustics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12860" PostTypeId="1" CreationDate="2021-02-10T16:47:14.730" Score="1" ViewCount="99" Body="&lt;p&gt;It is well known that Banach's fixed-point theorem was initially conceived as a fixed-point theorem&#xA;for applications defined in normed spaces (see [1]).&lt;/p&gt;&#xA;&lt;p&gt;This theorem was conceived in 1922 by Stefan Banach (1892 - 1945) in a famous effort to unify several techniques that forced the convergence of recursive sequences whose usefulness was to prove the existence of solutions for differential equations.&lt;/p&gt;&#xA;&lt;p&gt;According to [2] it was Felix Hausdorff (1868 - 1942) who in 1914 conceived the notion of metric space&#xA;in his publication Grundzüge der Mengenlehreem (see [3]).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Question: Who extended the Banach fixed point theorem from the context of normed spaces to the context of metric spaces? What is the original reference that proves the answer?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note. Some authors say that a primary early example of an extension of Banach's principle is a theorem of Caccioppoli [4] which asserts that the Picard iterates of a mapping &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; converge in a complete metric space &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; provided for each &lt;span class=&quot;math-container&quot;&gt;$n \geq 1,$&lt;/span&gt; there exists a constant &lt;span class=&quot;math-container&quot;&gt;$c_{n}$&lt;/span&gt; such that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;d\left(T^{n}(x), T^{n}(y)\right) \leq c_{n} d(x, y)&#xA;$$&lt;/span&gt;&#xA;for all &lt;span class=&quot;math-container&quot;&gt;$x, y \in M,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^{\infty} c_{n}&amp;lt;\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But, unless I am mistaken, the access I had to the publication [4] does not allow to conclude this statement. Although it is in Italian, the reference seems to be limited to differential equations in normed spaces.&lt;/p&gt;&#xA;&lt;p&gt;[1] Banach, Stefan. &amp;quot;Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.&amp;quot; Fund. math 3.1 (1922): 133-181.&lt;/p&gt;&#xA;&lt;p&gt;[2] &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Hausdorff/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Biographies/Hausdorff/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[3] Hausdorff, Felix. Grundzüge der mengenlehre. Vol. 7. von Veit, 1914.&lt;/p&gt;&#xA;&lt;p&gt;[4] R. Caccioppoli, Una teorema generale sull'esistenza di elementi uniti in una transformazione funzionale, Ren. Accad. Naz Lincei 11(1930), 794-799 .&lt;/p&gt;&#xA;" OwnerUserId="1846" LastEditorUserId="26" LastEditDate="2021-02-20T15:02:40.210" LastActivityDate="2021-02-20T15:02:40.210" Title="Who extended the Banach fixed point theorem from the context of normed spaces to the context of metric spaces?" Tags="|reference-request|topology|functional-analysis|banach|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12861" PostTypeId="1" CreationDate="2021-02-11T06:56:00.363" Score="1" ViewCount="392" Body="&lt;p&gt;I’m looking for an English translation of Newton’s &lt;em&gt;De analysi&lt;/em&gt;. (Alas, my Latin is weak.) I’m rather dismayed by the fact that I can’t appear to find one. How is it possible that one of the most important texts in the history of science has gone untranslated into English for the public domain? (The Latin is &lt;a href=&quot;http://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00204&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;I’m aware D. T. Whiteside has volumes that collect Newton’s work: there is &lt;em&gt;Mathematical Works&lt;/em&gt; published by Johnson Reprint Corp. in 1964, and then also a multivolume series with Cambridge University Press, &lt;em&gt;Mathematical Papers&lt;/em&gt;. But these volumes cost hundreds of dollars and are hard to access. And again dismayingly, there is no preview available on Google Books: I cannot tell whether the &lt;em&gt;De Analysi&lt;/em&gt; that appears in these volumes is in Latin or English.&lt;/p&gt;&#xA;" OwnerUserId="1812" LastEditorUserId="10106" LastEditDate="2022-03-26T12:52:47.030" LastActivityDate="2022-03-26T12:52:47.030" Title="Is there an English translation of Newton’s De Analysi?" Tags="|mathematics|reference-request|calculus|newton|translations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12862" PostTypeId="2" ParentId="12861" CreationDate="2021-02-11T09:48:57.780" Score="3" Body="&lt;p&gt;See &lt;a href=&quot;https://books.google.it/books?id=noQ_AAAAcAAJ&amp;amp;pg=PA344&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir Isaac Newton's two treatises of the quadrature of curves and analysis by equations of an infinite number of terms (1745)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2021-02-11T09:48:57.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12863" PostTypeId="1" AcceptedAnswerId="12864" CreationDate="2021-02-11T09:58:00.980" Score="2" ViewCount="186" Body="&lt;p&gt;A nephroid is an epycloid that can be generated by rolling a circle on the outside of a circle with doubled radius.&#xA;It was called by Richard Proctor (1878) because its shape looks like a kidney (see &lt;a href=&quot;https://mathworld.wolfram.com/Nephroid.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathworld.wolfram.com/Nephroid.html&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;A nephroid of Freeth (called from Thomas Jacob Freeth, 1879) is a strophoid of the circle and has nothing to do with Proctor's nephroid (from what I undertand) and doesn't look like a kidney, nor Proctor's nephroid (see &lt;a href=&quot;https://mathworld.wolfram.com/FreethsNephroid.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathworld.wolfram.com/FreethsNephroid.html&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;So: why is Freeth's nephroid called a nephroid ?&lt;/p&gt;&#xA;&lt;p&gt;Note: I could not find the original paper of Freeth where he may have explained the name.&lt;/p&gt;&#xA;" OwnerUserId="13909" LastEditorUserId="26" LastEditDate="2021-02-20T15:02:30.983" LastActivityDate="2021-02-20T15:02:30.983" Title="Why is Freeth's nephroid called a nephroid?" Tags="|geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12864" PostTypeId="2" ParentId="12863" CreationDate="2021-02-11T20:59:27.473" Score="10" Body="&lt;p&gt;Doesn't look like a kidney?&lt;/p&gt;&#xA;&lt;p&gt;Freeth's nephroid&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/BrCSe.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BrCSe.gif&quot; alt=&quot;Freeth&quot; /&gt;&lt;/a&gt;&lt;span class=&quot;math-container&quot;&gt;$\qquad$&lt;/span&gt;&lt;a href=&quot;https://i.stack.imgur.com/lhdbF.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lhdbF.jpg&quot; alt=&quot;kidney&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-02-11T20:59:27.473" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12865" PostTypeId="1" CreationDate="2021-02-13T09:42:19.340" Score="6" ViewCount="148" Body="&lt;p&gt;Dyck's theorem in topology is sometimes stated as follows: the connected sum of a torus and projective plane is homeomorphic to the connected sum of three projective planes.&lt;/p&gt;&#xA;&lt;p&gt;Certainly, this is the modern formulation of his theorem, given that Dyck proved his result in 1888 (the citation that I have seen for this theorem is usually given as: W. Dyck (1888), Beiträge zur Analysis situs. I. &lt;em&gt;Math. Ann.&lt;/em&gt; &lt;strong&gt;32&lt;/strong&gt;, 457–512). For my own interest, I have been trying to read the original source of this theorem, but I'm having trouble mostly because Walther Dyck wrote in German whereas I am not fluent in the language. So, I was hoping to receive some help here.&lt;/p&gt;&#xA;&lt;p&gt;Could someone explain to me how (and where) Dyck stated and proved his theorem in his 1888 &lt;em&gt;Math. Ann.&lt;/em&gt; paper? This paper seems to be freely available at the following DOI: &lt;a href=&quot;https://doi.org/10.1007/BF01443580&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/BF01443580&lt;/a&gt;. At over 50 pages long, I'm finding it quite a struggle to go through without any pointers, given my language barrier.&lt;/p&gt;&#xA;&lt;p&gt;I am also curious whether this result also appeared earlier in any of his 3 papers published in &lt;em&gt;Leipz. Ber.&lt;/em&gt; between 1885 and 1887 with the same title. The impression I have from the &lt;a href=&quot;https://zbmath.org/?q=ti%3A%22Beitr%C3%A4ge+zur+Analysis+situs%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;reviews on zbMATH Open&lt;/a&gt; is that the &lt;em&gt;Math. Ann.&lt;/em&gt; papers expand upon the papers in &lt;em&gt;Leipz. Ber.&lt;/em&gt;, but I am not certain. I am also not able to locate the journal &lt;em&gt;Leipz. Ber.&lt;/em&gt; anywhere, neither its full form nor its ISSN (if it has one), let alone the papers themselves.&lt;/p&gt;&#xA;" OwnerDisplayName="user13915" LastEditorDisplayName="user13915" LastEditDate="2021-02-23T05:13:16.690" LastActivityDate="2022-02-01T08:25:27.713" Title="How did Dyck originally state and prove his theorem in topology about the connected sum of a torus and projective plane?" Tags="|mathematics|topology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12866" PostTypeId="1" CreationDate="2021-02-13T13:08:00.163" Score="13" ViewCount="5163" Body="&lt;p&gt;Is this simply because of marketing, hype, etc?&lt;/p&gt;&#xA;&lt;p&gt;The bicomplex numbers (&lt;strong&gt;especially&lt;/strong&gt; tessarines) look just great being commutative and all.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/cOmQT.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cOmQT.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5Yzwk.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5Yzwk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Images source:&lt;a href=&quot;https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.527.356&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.527.356&amp;amp;rep=rep1&amp;amp;type=pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="4251" LastEditDate="2022-09-08T06:20:54.633" LastActivityDate="2022-09-08T06:20:54.633" Title="Why are quaternions more popular than tessarines despite being non-commutative?" Tags="|mathematics|complex-analysis|algebra|numbers|quaternions|" AnswerCount="3" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="12867" PostTypeId="2" ParentId="12866" CreationDate="2021-02-13T20:41:15.360" Score="11" Body="&lt;p&gt;Hamilton expected that the quaternions would be of physical interest. In this, he was right. But he was too early. He had discovered them in 1843, it was almost a century later, in 1928, when Dirac discovered his equation involving the Pauli matrices, that it was seen that the quaternions were naturally implicated in quantum field theory. (Here, the Pauli matrices are a representation of the quaternions on the space of 2 complex dimensional space).&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-02-13T20:41:15.360" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12868" PostTypeId="2" ParentId="12797" CreationDate="2021-02-13T20:56:55.593" Score="0" Body="&lt;p&gt;Whilst vectors are great, scalars are simpler. The gravitational potential is a scalar rather than a vector field and this helped manipulating expressions involving them.&lt;/p&gt;&#xA;&lt;p&gt;Whilst potentials were seen as a valuable nathematical reformulation, physicists wete inclined not to see them as directly of physical relevance and were thought to be mere mathematical artifice.&lt;/p&gt;&#xA;&lt;p&gt;This was up until the Ahronov-Bohm effect first suggested in a paper by Ahronov &amp;amp; Bohm in 1959. In fact, as they later credited, the effect had already been suggested a decade earlier by Ehrenburg &amp;amp; Siday in 1949. This effect shows that a electromagetic potential has a visible physical effect on quantum phase and this has now been experimentally verified. The first claimed verification was by Chambers in 1960 but this was contested. The first unambiguous confirmation was by Tonamura et al in 1986.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-02-13T20:56:55.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12869" PostTypeId="2" ParentId="12866" CreationDate="2021-02-13T23:07:28.413" Score="42" Body="&lt;p&gt;Commutativity is over-rated: in fact, it holds back bicomplex numbers:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It prevents your number system characterising non-commuting operations, e.g. rotations in &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-dimensional space, Hamilton's original focus.&lt;/li&gt;&#xA;&lt;li&gt;Since &lt;span class=&quot;math-container&quot;&gt;$0=i^2-j^2=(i-j)(i+j)$&lt;/span&gt; in bicomplex numbers, you have zero divisors, so it's not a normed division algebra; no convenient conjugates, no general definition of &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; even with &lt;span class=&quot;math-container&quot;&gt;$b\ne0$&lt;/span&gt;, which hinders the description of invertible processes - again, such as rotations. (Apart from conjugation, these are also issues with &amp;quot;hyperbolic&amp;quot; or split-complex numbers, viz. &lt;span class=&quot;math-container&quot;&gt;$0=1-j^2=(1-j)(1+j)$&lt;/span&gt;.) In this respect, your number system runs into difficulties even octonions don't face, though sedenions do. I can't offer a comprehensive overview of how this has stymied historical use, but uses of split-complex numbers have never become widespread (even when they arguably deserved to be, e.g. &lt;a href=&quot;https://physics.stackexchange.com/a/615119&quot;&gt;you can use split-complex numbers in relativity&lt;/a&gt;, but ironically complex numbers &lt;a href=&quot;https://physics.stackexchange.com/questions/155762/complex-numbers-in-quantum-mechanics-and-in-special-relativity&quot;&gt;have proved more popular&lt;/a&gt; for this).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2021-09-22T09:29:07.543" LastActivityDate="2021-09-22T09:29:07.543" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="12870" PostTypeId="1" CreationDate="2021-02-14T02:24:33.620" Score="5" ViewCount="262" Body="&lt;p&gt;I have a strong emotional reaction when I read the works of Euler. I have seen many extremely beautiful and intriguing identities in the notebook of Ramanujan, so much so that I think he is indeed a Euler of th 20th century.&lt;/p&gt;&#xA;&lt;p&gt;But what I notice is that mathematics have changed since the 19th century. Math seems to develop more theory and become much more rigorous due to the development of logic (think of Gottlob Frege). I realize that math during the day of Euler, Lagrange and d'Alembert is very calculational, computational. They &lt;a href=&quot;https://hsm.stackexchange.com/questions/2035/was-18th-century-algebra-more-symbolic-formal-than-the-modern-conception&quot;&gt;trust algebraic symbols and are much more formal&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From Littlewood's Miscellany (page 95, 96):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;But the great day of formulae seems to be over.&lt;/strong&gt; No-one, if we are again to take the highest standpoint, seems to be able to discover a radically new type, though Ramanujan comes near in his work of partition series; it is futile to multiply examples in the sphere of Cauchy's theorem and elliptic function theory,  and some general theory dominates, if in a less degree, every other field. A hundred years or so ago, his power would have had ample scope. Discoveries alter the general mathematical atmosphere and have very remote effects, and we are not prone to attach great weight to rediscoveries, however independent they seem. How much are we allow for this, how great a mathematician might Ramanujan have been 100 or 150 years ago; and what would have happened if he had come in touch with Euler at the right moment?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the article &amp;quot;&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086004000916&quot; rel=&quot;noreferrer&quot;&gt;Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem&lt;/a&gt;&amp;quot;, the author also wrote that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From the mid-18th century to the mid-19th century, the style of mathematical research in analysis underwent changes from a formula-centered approach epitomized by L. Euler (1707–1783) to a concept-centered style presented in works of G. P. L. Dirichlet (1805–1859) and G. F. B. Riemann (1826–1866). The transition manifested itself in multiple aspects of the mathematical enterprise including notations, questions, results, methods, and techniques. It was felt by the active and creative mathematicians of the nineteenth century who spotted a difference between the computational machinery associated with the formula-centered approach and the decidedly mental analysis belonging to the concept-centered approach. We find this distinction seized upon for instance in Dirichlet's obituary of C. G. J. Jacobi (1804–1851) where Dirichlet noticed1&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[…] the constantly increasing tendency of the new analysis to put thoughts in the place of calculations, […],2&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;in the methodological principle attributed by D. Hilbert (1862–1943) to Riemann near the end of the century,&#xA;I have tried to avoid the large computational apparatus of Kummer such that also here Riemann's principle should be observed, according to which one should conquer proofs not by computations but solely through thoughts,3&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;or in what H. Minkowski (1864–1909) called the “second Dirichlet principle” heralding the modern times in mathematics, according to which problems should be conquered&#xA;with a minimum of blind calculations and a maximum of enlightening thoughts.4&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is it true that formulae is no longer in its heyday as compared to the 18th, early 19th century? After all, we have Roger Apéry proving that &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt; is an irrational number and probably cannot be expressed in closed form (I think I am wrong here).&lt;/p&gt;&#xA;" OwnerUserId="13807" LastEditorUserId="5989" LastEditDate="2021-02-14T08:31:44.743" LastActivityDate="2021-02-14T22:56:31.253" Title="Has the heyday of mathematical formulae ended?" Tags="|calculus|real-analysis|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12871" PostTypeId="2" ParentId="12866" CreationDate="2021-02-14T02:26:04.497" Score="27" Body="&lt;p&gt;Your description &amp;quot;total uselessness of quaternions&amp;quot; in a comment above is poorly chosen, and reflects more on your interests than on the real state of knowledge of mathematics. The Hamilton quaternions are the simplest nontrivial example of a &lt;em&gt;quaternion algebra&lt;/em&gt;, which has turned out to be a really important concept in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;It is useful to think of the Hamilton quaternions &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt; as being analogous to the ring of &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; real matrices &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf R)$&lt;/span&gt;. At first these might seem quite different: one is a division ring and the other is not. But here are analogies between them: both are 4-dimensional with center the real numbers and in both of them there is no 2-sided ideal other than &lt;span class=&quot;math-container&quot;&gt;$(0)$&lt;/span&gt; and the whole ring. A ring that is 4-dimensional over a subfield &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, has &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; as its center, and has no 2-sided ideals other than &lt;span class=&quot;math-container&quot;&gt;$(0)$&lt;/span&gt; and the whole ring is called a quaternion algebra over &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;.&#xA;The matrix ring &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(K)$&lt;/span&gt; is a quaternion algebra over &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, and it is the only kind that is not a division ring.&lt;/p&gt;&#xA;&lt;p&gt;Sometimes &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(K)$&lt;/span&gt; is the only quaternion algebra over &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, e.g., when &lt;span class=&quot;math-container&quot;&gt;$K = \mathbf C$&lt;/span&gt; and when &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is a finite field. There are two quaternion algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbf R$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf R)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt;. Similarly, there are two quaternion algebras over the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers for each prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;. For &lt;span class=&quot;math-container&quot;&gt;$p = 2$&lt;/span&gt; the two examples are &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf Q_2)$&lt;/span&gt; and the Hamilton quaternions with &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_2$&lt;/span&gt;-coefficients, but for odd primes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; you need to do something different to get the &amp;quot;nontrivial&amp;quot; example because the Hamilton quaternions with &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_p$&lt;/span&gt; coefficients is isomorphic to &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf Q_p)$&lt;/span&gt; rather than being a division ring.&lt;/p&gt;&#xA;&lt;p&gt;Things get really interesting for fields like &lt;span class=&quot;math-container&quot;&gt;$K = \mathbf Q$&lt;/span&gt;, because&#xA;there are &lt;em&gt;infinitely many&lt;/em&gt; different quaternion algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q$&lt;/span&gt;. The ring &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf Q)$&lt;/span&gt; and the rational Hamilton quaternions are merely two examples out of a huge list of rational quaternion algebras. These &amp;quot;new&amp;quot; rational quaternion algebras show up in various places in mathematics: (supersingular) elliptic curves, Brauer groups, Shimura curves, and hyperbolic geometry in 2 and 3 dimensions.  Take a look at John Voight's &lt;a href=&quot;https://math.dartmouth.edu/%7Ejvoight/quat-book.pdf&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt; in order to appreciate that you should not dismiss quaternions as being a &amp;quot;totally useless&amp;quot; concept.  If you look at Ribet's &lt;a href=&quot;https://math.berkeley.edu/%7Eribet/Articles/invent_100.pdf&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt; that showed Fermat's Last Theorem is a consequence of modularity of elliptic curves over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q$&lt;/span&gt;, you'll see a role there for rational quaternion algebras that are division rings and are not the rational Hamilton quaternions.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2021-02-14T02:26:04.497" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12872" PostTypeId="2" ParentId="12870" CreationDate="2021-02-14T03:15:27.633" Score="6" Body="&lt;p&gt;There are plenty of formulas in the new maths. Littlewood is right that the great &amp;quot;heyday&amp;quot; of certain &lt;em&gt;types&lt;/em&gt; of formulas is over. But what this really means that mathematical taste has both changed and have widened.&lt;/p&gt;&#xA;&lt;p&gt;One formula that I like is Stokes' theorem. It's a formula of vector analysis. By finding the right context for it, that is on manifolds, the formula is now valid for any &lt;em&gt;curved&lt;/em&gt; space (the original formulas was only valid in flat 3d space), and for curved spaces in higher dimensions as well as lower. In fact, in 1d the formula becomes the fundamental theorem of calculus first shown by Newton. Moreover, the formula is systematic.&lt;/p&gt;&#xA;&lt;p&gt;What's not to like about this?&lt;/p&gt;&#xA;&lt;p&gt;One aim of the new maths is to make sure that all the formulas found by Euler, Littlewood et al. find similar contexts that explain why they take the form they do and relate them.&lt;/p&gt;&#xA;&lt;p&gt;Another aim is that proofs become mathematised into formulae. This is one of the great successes of category theory and why it's talked about as a 'language' of mathematics. It helps discern proof strategies in theories and makes formulae out of them.&lt;/p&gt;&#xA;&lt;p&gt;For example, Emily Riehl, in one of her books on category theory, showed that singular homology is the composition of five functors. This is as much of a formula as the identities beloved by Euler or Littlewood.&lt;/p&gt;&#xA;&lt;p&gt;What's not to like about that either?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="12330" LastEditDate="2021-02-14T16:42:43.540" LastActivityDate="2021-02-14T16:42:43.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12873" PostTypeId="2" ParentId="12870" CreationDate="2021-02-14T22:56:31.253" Score="2" Body="&lt;p&gt;It seems, you are talking about new formulas in analysis and algebra. I think, the formula age is not yet over there, and yes, I think, many of new formulas will be related to zeta function and differentiated gamma functions.&lt;/p&gt;&#xA;&lt;p&gt;We still do not understand zeta function well.&lt;/p&gt;&#xA;&lt;p&gt;Regarding &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt; is not closed form, I disagree with you. How closer do you want it? As an elementary function of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;? It is already as much closed form as closed form can be.&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2021-02-14T22:56:31.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12874" PostTypeId="1" CreationDate="2021-02-15T05:55:43.197" Score="2" ViewCount="76" Body="&lt;p&gt;I'm looking for a well-written biographical book about Kurt Gödel. Any titles you'd recommend?&lt;/p&gt;&#xA;" OwnerUserId="3006" LastEditorUserId="4251" LastEditDate="2022-09-23T06:01:32.843" LastActivityDate="2022-09-23T06:01:32.843" Title="Kurt Gödel: a biography" Tags="|mathematicians|biographical-details|book-recommendation|logicians|goedel|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12875" PostTypeId="1" CreationDate="2021-02-15T15:47:34.570" Score="3" ViewCount="315" Body="&lt;p&gt;If you look at a work such as Bertrand Russel's Principia Mathematica there are no brackets at all. So are brackets a recent invention?&lt;/p&gt;&#xA;&lt;p&gt;Newton used to draw a line above long expressions to group terms. (We still use this notation for things like roots).&lt;/p&gt;&#xA;&lt;p&gt;How long have they been used?&lt;/p&gt;&#xA;&lt;p&gt;When was the first time parenthesis have been used in mathematical expressions?&lt;/p&gt;&#xA;" OwnerUserId="7722" LastActivityDate="2021-02-15T15:47:34.570" Title="How long have parenthesis (brackets) been used?" Tags="|notation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12878" PostTypeId="1" AcceptedAnswerId="12879" CreationDate="2021-02-16T14:23:56.770" Score="5" ViewCount="244" Body="&lt;p&gt;The dates of various physical implementations of planimeters are pretty well known.  I'm interested in discovering when formal mathematical proofs were published that any given design does calculate exactly (or with analytic error) the area of a curve or figure.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Planimeter#Mathematical_derivation&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia page&lt;/a&gt; uses Green's Theorem of line integrals, but I noticed that &lt;a href=&quot;https://en.wikipedia.org/wiki/Green%27s_theorem#History&quot; rel=&quot;noreferrer&quot;&gt;Green himself&lt;/a&gt; is not credited with proving this theorem (Riemann takes that, in 1851  in his inaugural dissertation), and that even Green appears to have come up with the conjecture after the very first planimeter was built.&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="55" LastEditDate="2021-02-16T20:10:15.703" LastActivityDate="2021-02-17T10:38:32.633" Title="What are the earliest known proofs that planimeters 'work'?" Tags="|mathematics|geometry|engineering|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12879" PostTypeId="2" ParentId="12878" CreationDate="2021-02-16T21:41:40.617" Score="11" Body="&lt;p&gt;Using planimeters to illustrate Green's theorem is a relatively recent didactic development. Neither Green, nor Cauchy, nor Riemann had any interest in the instruments, and vice versa, planimeter developers did not involve something so abstract in explaining their &amp;quot;principle&amp;quot; mathematically.&lt;/p&gt;&#xA;&lt;p&gt;The kinds of mathematical proofs given can be seen in &lt;a href=&quot;https://archive.org/details/mechanicalintegr00shaw/page/22/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Shaw's Mechanical integrators, including the various forms of planimeters (1886)&lt;/a&gt;. They were not formal, but then no proofs were before formal mathematics was introduced at the end of 19th century. The idea was to directly reduce the wheel movements to what we would now call a Riemann sum. One finds proofs of the same sort in modern textbooks. Shaw seems to take Amsler as his principal source on mathematics. Amsler was a Swiss mathematician who invented the polar planimeter in 1854, and was likely familiar with Gauss's work on differential geometry from his studies at Königsberg in 1840-s. He is explicitly credited by Shaw for extending the idea to non-flat surfaces:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;As the Amsler planimeter alone, so far as the author is aware, has been modified to measure the area of any nondevelopable surface, this modification may be here noticed... The theory of the action&#xA;of this instrument has been fully explained by Professor Amsler, in an article in which the theory of the relations between measurement upon a spherical surface and upon a plane surface is discussed.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But the more basic planar proofs are certainly older. Maxwell ventured into constructing his own planimeter (&amp;quot;platometer&amp;quot;, as he called it after Sang) after seeing them displayed at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Great_Exhibition&quot; rel=&quot;noreferrer&quot;&gt;Great Exhibition of 1851&lt;/a&gt; in the Crystal Palace, London, which generally gave them prominence. In &lt;a href=&quot;https://archive.org/details/scientificpapers01maxw/page/230/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Description of a New Form of Platometer an Instrument for Measuring the Areas of Plane Figures (1855)&lt;/a&gt; he notes that &amp;quot;&lt;em&gt;a very able exposition of the principle of such instruments will be found in the article on Planimeters in the Reports of the Juries of the Great Exhibition, 1851&lt;/em&gt;&amp;quot;, and gives a geometric argument presumably modeled on it. Here is an excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Therefore, if we have a machine with an index of any kind, which, while the generating line moves one inch downwards, moves forward as many degrees as the generating line is inches long, and if the generating line be alternately moved an inch and altered in length, the index will mark the number of square inches swept over during the whole&#xA;operation. By the ordinary method of limits, it may be shown that, if these changes be made continuous instead of sudden, the index will still measure the area of the curve traced by the extremity of the generating line.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Earlier publications by the original inventors are hard to track, but one can expect arguments of the same sort there, perhaps without mentioning limits. The earliest known one is by Gonella of the University of Florence, who reinvented a wheel-and-cone planimeter in 1824 and published almost immediately. He also quickly came up with the wheel-and-disc design, which was the one he presented at the Great Exhibition of 1851.&lt;/p&gt;&#xA;&lt;p&gt;Hermann, a Munich land surveyor, who invented the earliest known planimeter in 1814, not only did not publish anything, but even used it only privately. It is first described in Bauernfeind's &lt;em&gt;Zur Geschichete der Planimeter&lt;/em&gt; (1855) published in Dingler’s &lt;em&gt;Polytechnische Journal&lt;/em&gt;. There might have been earlier private inventions, now lost, and independent reinvention continued until 1851, when Sang invented his platometer, see &lt;a href=&quot;https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.522.4918&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;Care, Illustrating the History of the Planimeter&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-17T10:38:32.633" LastActivityDate="2021-02-17T10:38:32.633" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12880" PostTypeId="1" CreationDate="2021-02-17T19:05:35.037" Score="2" ViewCount="316" Body="&lt;p&gt;Newton explained the inverse square law in &lt;em&gt;Principia&lt;/em&gt;. On looking through an English translation, though, I'm having difficulty pulling out a single quote that is Newton's clearest statement of the idea that point particles attracts all point particles with a force that is inversely proportional to the square of the distance and proportional to each of the masses.&lt;/p&gt;&#xA;&lt;p&gt;I see proofs of the shell theorem that state that the force between a point and a sphere obeys this law, and I see proofs that if we start by assuming an inverse square law, we get conic section orbits. But the book is very large and difficult to read; I simply can't find the best passage stating the universal gravitational law in Newton's own words. What is the best passage to use?&lt;/p&gt;&#xA;" OwnerUserId="2238" LastEditorUserId="2238" LastEditDate="2021-02-17T19:18:45.480" LastActivityDate="2021-02-17T22:52:02.777" Title="What was Newton's statement of the universal law of gravitation?" Tags="|physics|newton|gravity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12882" PostTypeId="1" AcceptedAnswerId="12885" CreationDate="2021-02-17T20:26:18.707" Score="8" ViewCount="509" Body="&lt;p&gt;The Cavendish experiment was equivalent to measuring &lt;span class=&quot;math-container&quot;&gt;$G,$&lt;/span&gt; Newton's gravitational constant. However, because physicists at the time did not write equations in the same way we do now, Cavendish didn't have the concept of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; and reported the results of his experiment as a measurement of the mass of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;When did physicists begin using &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="2238" LastActivityDate="2022-08-12T13:09:50.840" Title="When did physicists begin using the symbol $G$ for Newton's gravitational constant?" Tags="|physics|notation|gravity|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12883" PostTypeId="2" ParentId="12880" CreationDate="2021-02-17T22:52:02.777" Score="3" Body="&lt;p&gt;Chrandrasekhar's &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=4189&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Newton's&lt;/em&gt; Principia &lt;em&gt;for the Common Reader&lt;/em&gt;&lt;/a&gt; p. 370:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h2&gt;104. Proposition VII: the universal law of gravitation&lt;/h2&gt;&#xA;&lt;p&gt;We have now reached the climactic point of &lt;em&gt;Philosphie naturalis Principia Mathematica&lt;/em&gt;. After establishing the preceding propositions, particularly, Propositions IV and VI, Newton, at long last, is ready to enunciate his law of gravitation. Let him speak.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Proposition VII. Theorem VII&lt;/em&gt;&lt;br&gt;&lt;em&gt;That there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they contain.&lt;/em&gt;&lt;/blockquote&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&#xA;" OwnerUserId="232" LastActivityDate="2021-02-17T22:52:02.777" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12884" PostTypeId="2" ParentId="12882" CreationDate="2021-02-17T23:22:20.873" Score="4" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; was invented after a system of units that included force was devised.&lt;/p&gt;&#xA;&lt;p&gt;Jungnickel &amp;amp; McCormmach's &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=6044&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Cavendish: The Experimental Life&lt;/em&gt;&lt;/a&gt; &lt;a href=&quot;https://www.mprl-series.mpg.de/studies/7/19/index.html#74&quot; rel=&quot;nofollow noreferrer&quot;&gt;pt. 2, §16&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Cavendish experiment today is often called the experiment to determine &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, which is correct given that the experiment is the common possession of physics. It is often said that Cavendish’s object was to determine &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, which as a historical statement is incorrect but understandable given that the constant is more significant than the density of the Earth. In Cavendish’s time, there was no independent unit of force, such as our dyne and Newton. The strength of any force was expressed in terms of an equivalent gravitational attraction, and weight was the measure of mass. The universal gravitational constant did not come up, though we can easily calculate it from Cavendish’s data.&lt;sup&gt;89&lt;/sup&gt; We find implicit in his work two of the three principal universal constants, the velocity of light &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; (Planck’s constant &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is the third), but Cavendish did not think of &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; as necessarily having a constant value, and it was the better part of a century after Cavendish’s experiment before &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; entered physics.&lt;/p&gt;&#xA;&lt;hr&gt;89. Cavendish did not write an equation for the force of universal gravitation, as we do: &lt;span class=&quot;math-container&quot;&gt;$F=\frac{GMm}{R^2}$&lt;/span&gt;. He could have calculated &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; without having a unit of force, but he had no need for it, and it would not have occurred to him. Clotfelter (1987, 213).&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clotfelter, B. E. “&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/I6G97S27/Clotfelter%20-%201987%20-%20The%20Cavendish%20experiment%20as%20Cavendish%20knew%20it.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Cavendish Experiment as Cavendish Knew It&lt;/a&gt;.” &lt;em&gt;American Journal of Physics&lt;/em&gt; 55, no. 3 (March 1, 1987): 210–13.&lt;br&gt;p. 213:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If Cavendish, or one of his contemporaries, had wished to calculate a gravitational constant, how could it have been expressed? The system of units in use did not include a unit for force; &lt;strong&gt;no unit of force was proposed until 1873&lt;/strong&gt;, when the dyne was introduced. Cavendish expressed distances in feet and inches and weights or masses in grains. One might express a gravitational constant without using a unit for force, but surely the idea of measuring such a constant is less likely to occur to an experimenter when no unit for force is available.&lt;br&gt;    We may conclude that Cavendish did precisely what his paper says—he measured the density of the Earth. His report of the experiment gives no hint that he thought in terms of gravitational constant, and his analysis neither needs nor suggests a constant in the equation expressing universal gravitation. It is, of course, not difficult to take the data Cavendish gave and derive from them a value of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, but he did not do that and he did not suggest that he knew that it might be desirable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2021-02-18T19:38:55.227" LastActivityDate="2021-02-18T19:38:55.227" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12885" PostTypeId="2" ParentId="12882" CreationDate="2021-02-18T01:38:19.023" Score="9" Body="&lt;p&gt;The constant was introduced in &lt;a href=&quot;https://archive.org/details/meandensityeart00poyngoog/page/n26/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poynting's essay The Mean Density of the Earth&lt;/a&gt; that won the Adams prize at the University of Cambridge in 1893 on the subject &amp;quot;The Methods of determining the absolute and relative value of gravitation and the mean density of the earth&amp;quot;. It is introduced matter of factly on p. 2 of the essay:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We are therefore led to conclude that the law is general, or that&#xA;if we have any two bodies, of masses &lt;span class=&quot;math-container&quot;&gt;$M_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$M_2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; apart, the force on either is &lt;span class=&quot;math-container&quot;&gt;$$G\frac{M_1M_2}{d^2}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is a constant, the constant of gravitation.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He proceeds to describe experiments that determine &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; (including density of the Earth measurements &lt;em&gt;a la&lt;/em&gt; Cavendish). Later (p. 5) he also calls it &amp;quot;&lt;em&gt;the constant of attraction&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The next year Boys made a bigger deal out of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; at a &lt;a href=&quot;https://www.google.com/books/edition/Proceedings/ZrloHemOmUEC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA353&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;June 1894 meeting of the Royal Institution of Great Britain&lt;/a&gt;, with reference to Poynting's essay. He names it &amp;quot;&lt;em&gt;the Newtonian Constant of Gravitation&lt;/em&gt;&amp;quot;, reinterprets Cavendish's experiment as measuring it, and illustrates its significance with passion. Here is how it is introduced:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Newton showed that to complete his law and to put in a numerical constant (the Newtonian Constant of Gravitation) that would convert&#xA;his proportion into an equality, two methods are available: we may&#xA;either make observations on the disturbance of the earth's gravitation&#xA;by the action of isolated parts of it, we may either find the relative&#xA;attraction of an isolated mountain or the strata above the bottom of a&#xA;deep mine, or we may make an artificial planet of our own and find&#xA;the attraction which it exerts.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The Newtonian Constant will be known if we know the force of&#xA;attraction between two bodies which we can completely measure and&#xA;weigh. Employing tho C. G. S. system of measurement, the Newtonian&#xA;Constant is equal to the force of attraction in dynes between&#xA;two balls weighing a gramme each, with their centres one centimeter&#xA;apart. Of course it may be referred to pounds and inches or ton.a&#xA;and yards, but as soon as all the quantities but &lt;span class=&quot;math-container&quot;&gt;$\textrm{G}$&lt;/span&gt; in Newton's&#xA;equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\textrm{Force} =\textrm{G}\frac{\textrm{Mass}\times \textrm{Mass}}{\textrm{Distance}^2}$$&lt;/span&gt;&#xA;are known, no matter in what units the quantities are measured, &lt;span class=&quot;math-container&quot;&gt;$\textrm{G}$&lt;/span&gt; is&#xA;known. The conversion of its numerical value from one system of&#xA;measarement to another is of course a mere matter of arithmetic.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-20T05:09:05.917" LastActivityDate="2021-02-20T05:09:05.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12887" PostTypeId="1" AcceptedAnswerId="13551" CreationDate="2021-02-20T00:00:25.187" Score="10" ViewCount="1622" Body="&lt;p&gt;In discussions of Sobolev spaces one often sees the &lt;em&gt;Japanese bracket&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\langle x \rangle = (1+|x|^2)^{1/2},$$&lt;/span&gt; as useful shorthand.&lt;/p&gt;&#xA;&lt;p&gt;I was not easily able to find information about this term.&lt;/p&gt;&#xA;&lt;p&gt;(1) What was the first systematic usage of something like this shorthand?&lt;/p&gt;&#xA;&lt;p&gt;(2) What is the origin of the specific phrase &amp;quot;Japanese bracket&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;(3) Why is it called the &amp;quot;Japanese bracket&amp;quot;? The answer to this may be obvious from the answers to either (1) or (2).&lt;/p&gt;&#xA;" OwnerUserId="13950" LastEditorUserId="13950" LastEditDate="2021-02-20T11:31:48.803" LastActivityDate="2021-08-12T03:09:08.203" Title="What is the origin of the &quot;Japanese bracket&quot;?" Tags="|mathematics|terminology|notation|differential-equations|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12888" PostTypeId="1" CreationDate="2021-02-19T16:42:25.893" Score="0" ViewCount="120" Body="&lt;p&gt;What is the origin of the term &amp;quot;planet&amp;quot; and how did astronomers first define the term?&lt;/p&gt;&#xA;" OwnerDisplayName="Anna Hamilton" LastEditorUserId="10106" LastEditDate="2021-02-20T14:15:38.120" LastActivityDate="2021-02-20T14:15:38.120" Title="How did the first astronomers define what a planet is?" Tags="|physics|terminology|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12889" PostTypeId="2" ParentId="12888" CreationDate="2021-02-19T18:24:02.857" Score="7" Body="&lt;p&gt;It was defined as what the name states: πλάνητες ἀστέρες (Planetes asteres) means 'wandering stars' in Greek. By this, any light on the nocturnal firmament that didn't move along with the other stars was called a wandering star while the other stars were called 'fixed stars' since they remained always on the same location on the firmament relative to the other fixed stars, while wandering stars move on their own paths. Wandering stars (planets) included the Sun and the Moon as well. They didn't include the Earth as yet.&lt;/p&gt;&#xA;" OwnerDisplayName="Greenhorn" LastActivityDate="2021-02-19T18:24:02.857" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12890" PostTypeId="5" CreationDate="2021-02-20T15:04:25.797" Score="0" Body="&lt;p&gt;Please do not use this tag; it is useless. -The moderators&lt;/p&gt;&#xA;" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2021-02-20T15:04:25.797" LastActivityDate="2021-02-20T15:04:25.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12891" PostTypeId="4" CreationDate="2021-02-20T15:04:25.797" Score="0" Body="Please do not use this tag. It is useless, since every question that is on-topic on this site is about history. -The moderators" OwnerUserId="26" LastEditorUserId="26" LastEditDate="2021-02-20T15:04:25.797" LastActivityDate="2021-02-20T15:04:25.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12892" PostTypeId="1" CreationDate="2021-02-20T17:17:37.660" Score="5" ViewCount="898" Body="&lt;p&gt;Why and when was the hour divided into exactly &lt;strong&gt;60&lt;/strong&gt; minutes (and not for example &lt;strong&gt;70&lt;/strong&gt; or &lt;strong&gt;80&lt;/strong&gt;)?&lt;/p&gt;&#xA;" OwnerUserId="12313" LastEditorUserId="10106" LastEditDate="2021-02-21T04:11:06.873" LastActivityDate="2021-02-21T10:27:34.043" Title="Why historically the hour was divided into 60 minutes and when it had started?" Tags="|physics|units|time|" AnswerCount="1" CommentCount="0" ClosedDate="2021-02-21T13:35:27.253" ContentLicense="CC BY-SA 4.0" />
  <row Id="12893" PostTypeId="2" ParentId="12892" CreationDate="2021-02-20T17:50:25.880" Score="14" Body="&lt;p&gt;It comes from the ancient Babylonian numeration system which had base 60. (The reason for the choice of such a base is simplicity of calculation: 60 is divisible by 2,3,4,5,6,10,12,15,20,30. Much more convenient than base 10, whose only justification is the number of fingers on both hands).&lt;/p&gt;&#xA;&lt;p&gt;It was used mainly in astronomy (ancient people had little need in dividing an hour into smaller parts, and if necessary they used 1/2 or 1/4 of an hour). Then with the spread of clocks, it started to be used in daily life.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-02-20T17:50:25.880" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12894" PostTypeId="2" ParentId="3045" CreationDate="2021-02-21T01:46:58.480" Score="1" Body="&lt;p&gt;It's not at all clear what this question is asking. The fact that the author posted the question 5-6 years ago, and only just replied to a comment, is also very odd. It probably doesn't deserve much consideration, but anyway, here are some notes.&lt;/p&gt;&#xA;&lt;p&gt;(Adapted from &lt;a href=&quot;https://hsm.stackexchange.com/questions/12484/what-was-gausss-method-for-solving-the-trinomial-equation/12490#12490&quot;&gt;my previous answer here&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;In 1758, J. H. Lambert published a series solution for a real root of a general polynomial, in &lt;em&gt;Observationes Variae in Mathesin Puram&lt;/em&gt; (Various observations on pure mathematics) in &lt;em&gt;Act Helvetica Physico-Mathematico-Botanico-Medica&lt;/em&gt; Vol III. Considering as a special case the trinomial,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x^m + px = q $$&lt;/span&gt;&#xA;He obtains:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ x = \frac{q}{p} - \frac{q^m}{p^{m+1}} + m \frac{q^{2m-1}}{p^{2m+1}} - m \frac{3m-1}{2!} \frac{q^{3m-2}}{p^{3m+1}} + m \frac{(4m-1)(4m-2)}{3!} \frac{q^{4m-3}}{p^{4m+1}} - ... $$&lt;/span&gt;&#xA;Or, in summation notation,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$S = \sum_{k=0}^\infty (-1)^k \frac{q^{k(m-1)+1}} {p^{km+1}} \frac{1}{k!} \prod_{j=0}^{k-2}(km-j)$$&lt;/span&gt;&#xA;It might seem like Lambert's solution is to a more specific problem, but in fact it has broad (enough) generality, with appropriate change of variables. I haven't checked, but it seems very plausible that the two series solutions are equivalent.&lt;/p&gt;&#xA;&lt;p&gt;As for the fact that a 5th degree polynomial has five roots, Peter Roth published &lt;em&gt;Arithmetica Philosophica&lt;/em&gt; (1608) which states that a polynomial of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; with real coefficients may have &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; solutions (this information is available directly from the Wikipedia page for &lt;em&gt;Fundamental theorem of algebra&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2021-02-21T01:46:58.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12896" PostTypeId="1" CreationDate="2021-02-22T08:08:34.973" Score="9" ViewCount="210" Body="&lt;p&gt;I am a playwright working on a play about Sophie Germain. When Sophie was competing for the prix extraordinaire to find effective formulas to describe the vibrations of elastic surfaces, she believed a combination of formulas from Lagrange and Euler would work. During her efforts, she discovered that an important equation of Euler's was incorrect. I've been trying to find out what that equation was. Does anyone know what it might be? Or perhaps where I might look to find that information?&lt;/p&gt;&#xA;" OwnerUserId="13967" LastEditorUserId="3869" LastEditDate="2022-04-23T02:53:43.230" LastActivityDate="2022-04-23T02:53:43.230" Title="Did Sophie Germain find a flaw in Euler's equations for elastic vibrations?" Tags="|physics|mathematical-physics|euler|sophie-germain|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12898" PostTypeId="1" CreationDate="2021-02-22T16:40:05.357" Score="0" ViewCount="60" Body="&lt;p&gt;It is been some years since I completed my graduate studies in mathematics at a Spanish university. I remember one of the most pleasant and enriching moments I experienced was when reading Euclid´s Elements. This book taught me how modern maths is written and what classic geometry is all about. Also taught me a new way of thinking about maths.&lt;/p&gt;&#xA;&lt;p&gt;Since I started learning maths, I have been curious about how maths developed in the Indian and Chinese cultures. This is why I wanted to ask the following question: Is there a book (or collections of books) that have been essential when developing mathematics in the Indian or Chinese cultures (similar to what Euclid´s Elements did in ancient Greece and Europe)?&lt;/p&gt;&#xA;&lt;p&gt;PS: I have heard that Asians were a very advanced culture when talking about combinatory. Is there any reference for that?&lt;/p&gt;&#xA;" OwnerUserId="13976" LastEditorUserId="10106" LastEditDate="2021-02-23T03:02:45.853" LastActivityDate="2021-02-23T03:02:45.853" Title="Old Indian and Chinese references" Tags="|mathematics|reference-request|mathematics-social-history|ancient-india|ancient-china|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12899" PostTypeId="1" CreationDate="2021-02-22T19:07:30.903" Score="1" ViewCount="32" Body="&lt;p&gt;Here is Dirac's argument about one reason to motivate quantum mechanics:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;It is usually assumed that, by being careful, we may cut down the disturbance accompanying our observation to any desired extent. The concepts of big and small are then purely relative and refer to the gentleness of our means of observation as well as to the object being described. In order to give an absolute meaning to size, such as is required for any theory of the ultimate structure of matter, it becomes necessary to assume that there is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance – a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And next paragraph on causality:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A consequence of the preceding discussion is that we must revise&#xA;our ideas of causality. Causality applies only to a system which is&#xA;left undisturbed. If a system is small, we cannot observe it without&#xA;producing a serious disturbance and hence we cannot expect to find&#xA;any causal connexion between the results of our observations.&#xA;Causality will still be assumed to apply to undisturbed systems and&#xA;the equations which will be set up to describe an undisturbed system&#xA;will be differential equations expressing a causal connexion between&#xA;conditions at one time and conditions at a later time. These equations&#xA;will be in close correspondence with the equations of classical&#xA;mechanics, but they will be connected only indirectly with the results&#xA;of observations. There is an unavoidable indeterminacy in the calcu-&#xA;lation of observational results, the theory enabling us to calculate in&#xA;general only the probability of our obtaining a particular result when&#xA;we make an observation.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Is Dirac already signaling(?), asserting(?) that QM is to be epistemic? How else do I buy the argument that because &lt;em&gt;humans/human-sized-experiments&lt;/em&gt; have investigative limits, that reality must have a notion of size and smallness. Isn't the observation that our measurements cause disturbances at most a claim about what &lt;em&gt;we&lt;/em&gt; can investigate, and not about what &lt;em&gt;is&lt;/em&gt;? Maybe the photon is a constituent particle after all.&lt;/p&gt;&#xA;&lt;p&gt;To me, his argument must either be epistemic or circular. There could be a smaller reality beyond the photon and quantum mechanics would still describe what we see right?&lt;/p&gt;&#xA;&lt;p&gt;But then why is this a new kind of epistemism requiring us to modify our notion of causality? Surely scientists have always limited their theories to what we can observe? Or was physics ontic to Dirac until this point? Is the uniqueness of quantum mechanic's epistemism that it is part of the physical dynamics? The measurements cause a disturbance. Not just that we are limited to epistemic theories, but by measuring we are partaking in an epistemic physical theory/reality? Did we never have other physical/dynamical epistemic theories until the 1930s? And if there were, why didn't they need to revise our notion of causality too?&lt;/p&gt;&#xA;&lt;p&gt;Is Dirac arguing quantum mechanics is epistemic not because we are limited to epistemic knowledge as all of science is, but quantum mechanics itself along with our measurements &lt;em&gt;creates&lt;/em&gt; some kind of epistemic reality through a physical process involving measurement and collapse of the wavefunction? Is there even a difference between the two ideas of epistemism? Or he is arguing somehing slightly different?&lt;/p&gt;&#xA;&lt;p&gt;I do not expect an answer to explicitly answer all of my rambling questions. Thanks&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/stream/1quantumMechanicsByPamDirac/1quantum%20mechanics%20by%20pam%20dirac_djvu.txt&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/stream/1quantumMechanicsByPamDirac/1quantum%20mechanics%20by%20pam%20dirac_djvu.txt&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13053" LastEditorUserId="13053" LastEditDate="2021-02-22T19:16:54.457" LastActivityDate="2021-02-22T19:16:54.457" Title="Dirac's non-neglibile disturbances argument and epistemic quantum mechanics" Tags="|physics|quantum-mechanics|dirac|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12901" PostTypeId="1" AcceptedAnswerId="12907" CreationDate="2021-02-23T00:57:31.177" Score="2" ViewCount="336" Body="&lt;p&gt;In analytic number theory it is traditional to write a complex variable as &lt;span class=&quot;math-container&quot;&gt;$s = \sigma + it$&lt;/span&gt;, with the letter &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; going back to Riemann's paper on the zeta-function (1859) and the letter &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; having a more complicated explanation that was asked about on Mathoverflow &lt;a href=&quot;https://mathoverflow.net/questions/87505/origin-of-the-notation-s-sigmait-in-analytic-number-theory&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It recently came to my attention that in electrical engineering the notation &lt;span class=&quot;math-container&quot;&gt;$s = \sigma + j\omega$&lt;/span&gt; is widely used, where &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; describes a frequency. Engineers don't want to use &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; as the imaginary part since &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; always means time, and &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; is a commonly used alternative to &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; since &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; in electrical engineering often means current.  My question is: who first introduced the notation &lt;span class=&quot;math-container&quot;&gt;$s = \sigma + j\omega$&lt;/span&gt;?  I am wondering in particular if this use of &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; can be directly tied to the use of &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; as the real part of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; in analytic number theory. Edit: I am &lt;em&gt;not&lt;/em&gt; wondering about the use of &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, so please don’t focus on that or refer me to &lt;a href=&quot;https://hsm.stackexchange.com/questions/5082/introduction-of-imath-and-jmath-notations-for-the-imaginary-unit&quot;&gt;this&lt;/a&gt; page, which does not answer my question.&lt;/p&gt;&#xA;&lt;p&gt;One of the settings where the notation &lt;span class=&quot;math-container&quot;&gt;$s = \sigma + j\omega$&lt;/span&gt; is used is in Bode plots, and I looked at several papers of Bode from the 1930s. In none of them did I find &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; being used to describe complex numbers and their real parts. Complex numbers were introduced to electrical engineering by Charles Steinmetz, and in his 1893 paper &amp;quot;Complex Quantities and Their Use in Electrical Engineering&amp;quot; he introduced &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\tilde{\omega}$&lt;/span&gt; as a phase angle, but does not use &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; in the above sense.  He writes &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; for completely different purposes: &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; denotes inductance (a real quantity) and &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; denotes the EMF (electromotive force) of self-induction as a fraction of total EMF.&lt;/p&gt;&#xA;&lt;p&gt;Note: This question is 1 day old and has 4 votes to close. I am baffled and would appreciate it if someone voting to close could please explain their reason for judging this question unsuitable to be asked here.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2021-02-24T07:17:20.560" LastActivityDate="2021-02-25T00:32:57.947" Title="Origin of the notation $s = \sigma + j\omega$ in electrical engineering/control theory" Tags="|notation|engineering|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12902" PostTypeId="2" ParentId="12896" CreationDate="2021-02-23T02:01:16.970" Score="11" Body="&lt;p&gt;There were two episodes with Germain and Euler's mistake in elasticity, but neither one of them is of Germain discovering Euler's mistake. The first one is from 1811 when she was starting to work on prix extraordinaire and corresponded with Legendre on Euler's paper on elastic rods. Euler wrote an incorrect solution for certain eigenmodes, but it was Legendre who spotted that in one of his letters to her. The second episode is of Germain using an incorrect equation from Euler to derive boundary conditions in the 1816 memoir &lt;a href=&quot;https://www.google.com/books/edition/Recherches_sur_la_th%C3%A9orie_des_surfaces/23eSa2KMttEC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Recherches sur la théorie des surfaces élastiques&lt;/em&gt;&lt;/a&gt; (published with corrections in 1821) that won the prize, which apparently led to a disagreement with Chladni's experiments.&lt;/p&gt;&#xA;&lt;p&gt;A lot happened in between and after the two episodes, as described in detail in Hill's thesis &lt;a href=&quot;https://core.ac.uk/download/pdf/36683994.pdf&quot; rel=&quot;noreferrer&quot;&gt;Sophie Germain: a mathematical biography&lt;/a&gt;. The Academy extended the contest several times. It was dissatisfied with Germain's 1811 submissions because &amp;quot;&lt;em&gt;the true equations of the movement were not established&lt;/em&gt;&amp;quot;, they were also littered with technical mistakes. Lagrange derived the correct equation for vibrating thin plates based on Germain's hypothesis before passing away in 1813, but did not engage with her on the subject. Germain's second, 1813, submission, which was also deemed unsatisfactory.&lt;/p&gt;&#xA;&lt;p&gt;The prize was finally awarded for the lengthy 1816 memoir, the first ever awarded to a woman, but could not attend because the Academy did not allow women at its meetings (other than members' wives). It was still dissatisfied with Germain's derivation because of technical flaws, not that it informed her of that. All of this took place against the background of the Academy members shunning Germain, and not communicating their responses to her work adequately; unethical behavior of Poisson, who took advantage of his position as a judge with access to Germain's 1813 submission, and presented her equation for thin plates with a technically correct derivation without crediting her; etc.&lt;/p&gt;&#xA;&lt;p&gt;The first episode is described in Langton's talk The Reception of Euler’s Elasticity: Letters from Legendre to Sophie Germain at the 2007 workshop &lt;a href=&quot;https://www.mfo.de/document/0733a/OWR_2007_38.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Reception of the Work of Leonhard Euler (1707-1783)&lt;/a&gt;. Euler was solving a special case of the partial differential equation for a vibrating elastic rod &lt;span class=&quot;math-container&quot;&gt;$$\sigma\frac{\partial^2 y}{\partial t^2}+B\frac{\partial^4 y}{\partial s^4}=0$$&lt;/span&gt; by separating variables, and for a rod pinned at both ends and at the midpoint the equation for the eigenfrequencies he gets is &lt;span class=&quot;math-container&quot;&gt;$$\sin\frac12\omega\left(\tan\frac12\omega-\tanh\frac12\omega\right)=0.$$&lt;/span&gt; Sophie asked Legendre, her patron, to clarify the derivation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the course of solving his &lt;span class=&quot;math-container&quot;&gt;$8 × 8$&lt;/span&gt; system, Euler has to divide by &lt;span class=&quot;math-container&quot;&gt;$\sin\frac12\omega$&lt;/span&gt;. Later, when he works out the solutions corresponding to &lt;span class=&quot;math-container&quot;&gt;$\sin\frac12\omega=0$&lt;/span&gt; and finds that some&#xA;of them have &lt;span class=&quot;math-container&quot;&gt;$\sin\frac12\omega$&lt;/span&gt; in the denominator, he multiplies that factor out again. Legendre, in his first letter to Germain, asserts that this is an error on Euler’s part. Later, in his second letter, Legendre has understood better what Euler did, but he now discovers an actual error in Euler’s work. The solution that Euler writes&#xA;down for the case &lt;span class=&quot;math-container&quot;&gt;$\sin\frac12\omega=0$&lt;/span&gt; is incorrect.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gray's note is behind a paywall, but the essence is reproduced almost verbatim &lt;a href=&quot;https://en.wikipedia.org/wiki/Sophie_Germain#Subsequent_attempts_for_the_Prize&quot; rel=&quot;noreferrer&quot;&gt;in Wikipedia&lt;/a&gt;, with some details added. Gray herself is not specific as to which incorrect equation of Euler was at fault in 1816:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;A third competition was announced, to which Sophie,  after consulting with the mathematician Poisson, submitted a memoir in her own name. This time she won the  prize in spite of the fact that the equation was not rigorously demonstrated and that the agreement with experimental data was not too close because of her employment of an incorrect equation of Euler.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It could have been one of Euler's boundary conditions for the vibrating rod extended to plates. But, according to Hill, already in 1813 the Academy was impressed enough with Germain's predictions of nodal lines and frequency ratios in Chladni's experiments for square and rectangular plates, and she does not mention a boundary value issue in the 1816 memoir.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-23T06:32:02.920" LastActivityDate="2021-02-23T06:32:02.920" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12903" PostTypeId="1" CreationDate="2021-02-23T12:07:09.020" Score="2" ViewCount="37" Body="&lt;p&gt;I am wondering who actually invented the (magnetic) quadrupole lens or who used it for the first time. I have found some papers from the 50s and 60s about more complicated systems based on Quadrupoles, but i am interested, who came up with idea to use quadrupoles for focusing of charged articles and who built the first focusing doublet. Here are some publications and patents that I found so far, that are not satisfying me, because there must be earlier work:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Ernest D. Courant, M. Stanley Livingston, and Hartland S. Snyder,&#xA;&amp;quot;The Strong-Focusing Synchroton - A New High Energy Accelerator&amp;quot;,&#xA;Phys. Rev. 88 (1952), pp. 1190--1196.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Nicholas Christofilos, &amp;quot;Focussing system for ions and electrons&amp;quot;&#xA;(1956), US2736799A.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Walter Glaser, &amp;quot;Electron Lens&amp;quot; (1959), US2919381.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am wondering whether the use of quadrupoles for (1D) focusing of charged particles is just to straight forward, to publish an article about it, but still it should be quite noteworthy when some one built the first quadrupole doublet for (anasticmatic) imaging with charged particles.&lt;/p&gt;&#xA;" OwnerUserId="13981" LastEditorUserId="13981" LastEditDate="2021-02-24T18:49:49.793" LastActivityDate="2021-02-24T18:49:49.793" Title="Who invented the quadrupole lens for charged particles?" Tags="|physics|electromagnetism|experimental-physics|optics|particle-physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12904" PostTypeId="2" ParentId="11298" CreationDate="2021-02-24T03:49:50.847" Score="4" Body="&lt;p&gt;Sue Simkin was my sister (she recently passed away). I have a suggestion for another explanation (rather than Detroit electricity), based on family experiences, that might have been how she could recognize 30 Hz pulsing. When we were children, our family visited a great aunt every summer. The aunt lived in Scotland, Ontario (near Brantford). The lighting there was pulsing. It might have been that what Sue remembered was the pulsing of the light in our aunt's house. I'll  try asking our brother (who is an electrical engineer) if he remembers details about the lighting there.&lt;/p&gt;&#xA;" OwnerUserId="13982" LastActivityDate="2021-02-24T03:49:50.847" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12905" PostTypeId="1" CreationDate="2021-02-24T10:32:26.863" Score="5" ViewCount="52" Body="&lt;p&gt;Ada Lovelace is well known as an translator of Luigi's Menabrea article on Babbage's Analytical engine. She also added notes to the translation which are in the end longer than the translation itself. Somebody even considers these notes to be basics of modern computing.&lt;/p&gt;&#xA;&lt;p&gt;In the book &lt;em&gt;Grace Hopper: Admiral of Cyber Sea&lt;/em&gt; by Kathleen B. Williams, the author stated that Howard Aiken wanted Grace Hopper to study Babbage's and Lovelace's articles to better understnad computing. This leads me to question &lt;strong&gt;if there are another Ada's articles than the Notes on Analytical Engine&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12316" LastActivityDate="2021-02-24T10:32:26.863" Title="Are there other articles by Ada Lovelace than translation of Menabrea's notes on Analytical Engine?" Tags="|computer-science|computers|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12906" PostTypeId="1" AcceptedAnswerId="13005" CreationDate="2021-02-24T16:16:11.887" Score="4" ViewCount="156" Body="&lt;p&gt;I've been looking for the origin of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Particle_zoo&quot; rel=&quot;nofollow noreferrer&quot;&gt;'particle zoo'&lt;/a&gt; expression but so far failed to track down who first used it or at least who popularized it.&lt;/p&gt;&#xA;" OwnerUserId="13985" LastEditorUserId="13985" LastEditDate="2021-03-19T18:19:06.403" LastActivityDate="2021-03-19T18:19:06.403" Title="Who coined the 'particle zoo' expression?" Tags="|physics|terminology|particle-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12907" PostTypeId="2" ParentId="12901" CreationDate="2021-02-24T17:21:57.590" Score="4" Body="&lt;p&gt;I'll be answering my own question, since an anonymous person reached out to me with the following information about the likely origin of the notation &lt;span class=&quot;math-container&quot;&gt;$s = \sigma + j\omega$&lt;/span&gt; in electrical engineering.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Deakin's paper &amp;quot;The Ascendancy of the Laplace Transform and how it Came About&amp;quot; (Archive for History of Exact Sciences 44 (1992), 265-286) says that Gustav Doetsch wrote the first textbook treatment of the Laplace transform with his 1937 book &lt;em&gt;Theorie und Anwendung der Laplace-transformation&lt;/em&gt; (Springer-Verlag). In this book, which was written for a mathematical audience rather than engineers and was never translated into English, Doetsch writes complex numbers as &lt;span class=&quot;math-container&quot;&gt;$\sigma + iy$&lt;/span&gt; (and &lt;span class=&quot;math-container&quot;&gt;$x+iy$&lt;/span&gt;). His use of &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; is not a surprise, since Doetsch's PhD advisor was  Edmund Landau (see &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=47733&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) and Landau was responsible for making &lt;span class=&quot;math-container&quot;&gt;$\sigma + it$&lt;/span&gt; the standard notation for complex numbers in analytic number theory.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Gardner and Barnes wrote the book &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.74407/page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Transients In Linear Systems Studied By The Laplace Transformation&lt;/em&gt;&lt;/a&gt; in 1942, aimed directly at electrical engineers. They use the notation &lt;span class=&quot;math-container&quot;&gt;$s := \sigma + j\omega$&lt;/span&gt; for complex numbers starting from page 12, where &lt;span class=&quot;math-container&quot;&gt;$j = \sqrt{-1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; is real. This book was largely responsible for the widespread adoption of Laplace transforms in electrical engineering education in the English-speaking world (replacing Heaviside operational calculus), so this answers my question about why the notation &lt;span class=&quot;math-container&quot;&gt;$\sigma + j\omega$&lt;/span&gt; (especially the &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; part) is used in electrical engineering.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Widder's 1946 book &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.201038&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Laplace Transform&lt;/em&gt;&lt;/a&gt; uses the notation &lt;span class=&quot;math-container&quot;&gt;$s = \sigma + i\tau$&lt;/span&gt; for complex numbers, and Widder writes in the preface that he learned about the Laplace transform from Hardy, who like Landau was an analytic number theorist. This book is aimed at mathematicians, not engineers, but again we see that the choice of notation &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;${\rm Re}(s)$&lt;/span&gt; here can be related to its use in analytic number theory.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2021-02-25T00:32:57.947" LastActivityDate="2021-02-25T00:32:57.947" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12909" PostTypeId="1" CreationDate="2021-02-24T20:33:21.657" Score="7" ViewCount="111" Body="&lt;p&gt;From a lot of googling, it seems like the answer might be &amp;quot;Mizar&amp;quot;, but I am not completely sure.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What was (or is?) the first automated theorem prover (i.e. not necessarily active right now)?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="13076" LastEditorUserId="9523" LastEditDate="2021-02-26T22:55:43.923" LastActivityDate="2021-02-26T22:55:43.923" Title="What was the first automated theorem prover?" Tags="|mathematics|computer-science|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12910" PostTypeId="1" CreationDate="2021-02-25T02:30:30.813" Score="1" ViewCount="77" Body="&lt;p&gt;Were Maxwells equations (in their usual common form) assumed to be valid only in ether frame,in the past? If someone would ask them &amp;quot;In what frame are you writing these equations &amp;quot; ,what would they say?&lt;/p&gt;&#xA;&lt;p&gt;If that is true they assumed that the equations were in ether frame then how come they assumed this?&#xA;Knowing that the constituent equations that make up the Maxwell equations were experimentally proved on the earth, which was known to be moving with respect to the ether?&lt;/p&gt;&#xA;" OwnerUserId="13989" LastActivityDate="2021-02-25T02:30:30.813" Title="Maxwell equations and aether" Tags="|physics|relativity-theory|maxwell|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12911" PostTypeId="1" AcceptedAnswerId="12912" CreationDate="2021-02-25T11:11:47.120" Score="6" ViewCount="509" Body="&lt;p&gt;Back in the past, was the sun assumed to be at rest in the luminous aether frame?&lt;/p&gt;&#xA;" OwnerUserId="13989" LastActivityDate="2023-02-13T06:00:22.107" Title="Was the sun assumed to be at rest aether frame?" Tags="|physics|maxwell|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12912" PostTypeId="2" ParentId="12911" CreationDate="2021-02-25T12:39:48.863" Score="7" Body="&lt;p&gt;Not necessarily. They did not know the velocity of the Sun with respect to aether. What they tried to do is to measure the speed of the Earth with respect to&#xA;aether. If any velocity were detected, one could infer the speed of the Sun. But no speed was detected. And the idea that the Earth is at rest while everything else moves looked absolutely implausible in 19th century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-02-25T12:39:48.863" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12913" PostTypeId="2" ParentId="11590" CreationDate="2021-02-25T16:16:05.477" Score="3" Body="&lt;p&gt;Heisenberg's &amp;quot;attitude&amp;quot; about the wave formulation of QM is vastly exaggerated. By 1929 he was lecturing in great detail on  Schrödinger's equation at the University of Chicago, and next year he published his classic introductory book &lt;em&gt;The Physical Principles of the Quantum Theory&lt;/em&gt; covering it extensively. He understood that, as the virtual initiator of the field, he had to cover its huge span, and was in close touch with both Jordan and Dirac who explored the link between the two formulations much more thoroughly than Schrödinger. This mature attitude is already reflected in the advice he gave to graduate student Dirac in 1925.&lt;/p&gt;&#xA;&lt;p&gt;Quoting from &lt;em&gt;The Strangest Man -— The Hidden Life of Paul Dirac, Mystic of the Atom&lt;/em&gt; by Graham Farmelo (Basic Books, 2009),&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At first, Dirac was annoyed by Schrödinger’s theory, as he resented even the thought of suspending work on the new quantum mechanics and starting afresh. But in late May, as he was finishing the writing of his Ph.D. thesis, he received a persuasive letter from Heisenberg urging him to take Schrödinger’s work seriously. This wise advice was ironic coming from Heisenberg, an opponent of the rival theory, who had written to Wolfgang Pauli in early June, ‘The more I reflect on the physical portion of Schrödinger’s theory the more disgusting I find it. What Schrödinger writes on the visualizability of his theory is probably not quite right. In other words, it’s crap.’ Schrödinger gave as good as he got, dismissing the mathematical arcana of Heisenberg’s theory and the idea of quantum jumps. The two theorists clashed unpleasantly when they first met a month later at a packed seminar in Munich, the first skirmish in what was to be a long and acrimonious dispute.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dirac ignored Schrödinger’s theory in his Ph.D. thesis, ‘Quantum Mechanics’, the first to be submitted anywhere on the subject. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In very short order, Dirac would go on to make trenchant contributions  to formulating the time-dependent  Schrödinger equation.&lt;/p&gt;&#xA;&lt;p&gt;Heisenberg's &amp;quot;attitude&amp;quot; is not entirely unwarranted. The community at the time was far more conversant with differential equations than with matrix techniques, an accidental  mathematics education shortcoming, giving the wave formulation an instant popular advantage. Nevertheless,  matrix techniques have ultimately triumphed over a century; connections to classical mechanics, such as phase-space quantization are quite closer to the matrix mechanics spirit than to wave functions. In fact, Dirac's fusion language has made the two formulations virtual &amp;quot;dialects&amp;quot; of the same formal structure, routinely used side-by-side.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-02-25T16:53:48.390" LastActivityDate="2021-02-25T16:53:48.390" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12914" PostTypeId="2" ParentId="12911" CreationDate="2021-02-25T20:40:54.330" Score="4" Body="&lt;p&gt;Short Answer:&lt;/p&gt;&#xA;&lt;p&gt;19th centry astronomers believed that the Sun and the Solar System revolved around the central point of the Milky Way Galaxy.  Thus 19th centry physicists should have believed that the Sun moved with respect to the aether.&lt;/p&gt;&#xA;&lt;p&gt;Long Answer:&lt;/p&gt;&#xA;&lt;p&gt;By the beginning of the 19th century, astronomers were certain that the Sun was a star in the Milky Way Galaxy, which most astronomers assumed was the entire universe, though there were speculations that other galaxies might exist beyond the Milky Way.&lt;/p&gt;&#xA;&lt;p&gt;According to Newton's law of gravity all objects in the Milky Way Galaxy must revolve around the barycenter, the mathematical center of mass, of the Milky Way Galaxy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Star Names:  Their Lore and Meaning&lt;/em&gt;  Richard Hincley Allen, 1899, 1963, The section on constellation Hercules - star Lambda Heruclis says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Athough Johann Tobias Mayer of Gottingen seems to have been the pioneer, in 1760, in the efforts to ascertain the direction of the sun's movement among the stars, yet Sir William Herschell was thes first successful investigator as to this, about 1806, and he settled upon the vicinity of Lambda as the objective point of our solar system, the &lt;strong&gt;Apex of the Sun's Way&lt;/strong&gt;; and his determination was, in a great measure, confirmed by later astronomers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And Wikpedia says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1783, English-German astronomer William Herschel described the solar apex, the point in sky towards which the Solar System is moving; using data from double stars, he identified this position as close to Lambda Herculis. Today it is known the solar apex is not so close to this star, however it is only 10° away from the position currently accepted (in Hercules, southwest of Vega).[14][15][16]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lambda_Herculis%5B1%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Lambda_Herculis[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Astronomers knew that the distances between one star and its nearest neighbor must be immense, to say nothing of the distance from one one edge of the Milky Way Galaxy accross the center point to the opposite edge.  But they didn't know what the scale of interstellar space was until the distances to three stars were measured in the late 1830s and the distances to several more were measured in following decades.&lt;/p&gt;&#xA;&lt;p&gt;I happen to own an astronomy book written by Ormsby M. Mitchell (1810-1862),&lt;em&gt;The Planetary and Stellar Worlds&lt;/em&gt;, 1851.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ormsby_M._Mitchel%5B2%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ormsby_M._Mitchel[2]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lecture X the Motions and Revolutions of the Fixed Stars, pages 318-319, describes Madler's theory that Alcyone in the Pleiades star cluster is the central sun at the center of gravity of the Milky Way Galaxy (then assumed to be the entire universe).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After a profund examination, Madler reached the conclusion &lt;em&gt;that Alcyone, the principal star in the group of the Pleiades, now occupies the centre of gravity, and is at present the sun about which the universe of stars comprising our central system are all revolving.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Star Names:  Their Lore and Meaning&lt;/em&gt;  Richard Hincley Allen, 1899, 1963, the section on Canis Major - Sirius says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The celebrated Kant thught that Sirius was the central sun of the Milky Way;...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If Sirus was the central sun, other stars would revolve around it.  Emmanuel Kant lived from 1724 to 1804.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Star Names:  Their Lore and Meaning&lt;/em&gt;  Richard Hincley Allen, 1899, 1963, the section on Taurus - The Pleidades - Alcyone says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Madler located in Alcyone the center of the universe, but his theory has been shown to be fallacius.  There is no satisfactory reason for his conclusions, and little more for Miss Clerke's remarks about the probable size and distance of Alcyone,--that it shines to its fellow stars with a luster eighty three times that of Sirius in terrestrial skies, while its intrinsic briliancy, as compared with that of the sun is 1000 times greater.  All this rests upon the extremely doubtful assumption of a parallax of 0&amp;quot;.013 deduced from the star's proper motion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If Alcyone was in the center of the universe all the other stars would have to revolve around it.&lt;/p&gt;&#xA;&lt;p&gt;Actually Alcyone is even farther than Miss Clerke said, since it now listed as having about 2,400 times the luminosity of the Sun.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alcyone_(star)%5B3%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Alcyone_(star)[3]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;At the present time the Sun is believed to orbit the gravitational center of the Milky Way Galaxy in a roughly circular orbit at a distance of about 27,140 plus or minus 460 light years from the central point.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It takes the Solar System about 240 million years to complete one orbit of the Milky Way (a galactic year),[16] so the Sun is thought to have completed 18–20 orbits during its lifetime and 1/1250 of a revolution since the origin of humans.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Milky_Way#Sun%27s_location_and_neighborhood%5B4%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Milky_Way#Sun's_location_and_neighborhood[4]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If the Sun has an perfectly circular orbit of exactly 27,140 light years radius, and thus exactly 185,415.052 light years circumference, and it takes exactly 240,000,000 years to make one orbit, the line between the Sun and the center of the Galaxy would change its angle very slightly during any time period which is understandable  by humans.&lt;/p&gt;&#xA;&lt;p&gt;There are 360 degrees of arc in a full circle, so it would take 666,666.666 years for the angle to the Sun to move by one degree as seen from the galaxtic center.&lt;/p&gt;&#xA;&lt;p&gt;There are 60 minutes of arc in a degree, so it would take 11,111.1111 years for the angle to the Sun to move by one minute as seen from the galactic center.&lt;/p&gt;&#xA;&lt;p&gt;There are 60 seconds of arc in a minute of arc, so it would take 185.185 years for the angle to the Sun to move by one second as seen from the galactic center.&lt;/p&gt;&#xA;&lt;p&gt;In Edgar Allan Poe's story &amp;quot;Mellonta Tauta&amp;quot; (1849), a journel written in the year 2848, it is said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;April 6.—Last night had a fine view of Alpha Lyrae, whose disk, through our captain's spy-glass, subtends an angle of half a degree, looking very much as our sun does to the naked eye on a misty day. Alpha Lyrae, although so very much larger than our sun, by the by, resembles him closely as regards its spots, its atmosphere, and in many other particulars. It is only within the last century, Pundit tells me, that the binary relation existing between these two orbs began even to be suspected. The evident motion of our system in the heavens was (strange to say!) referred to an orbit about a prodigious star in the centre of the galaxy. About this star, or at all events about a centre of gravity common to all the globes of the Milky Way and supposed to be near Alcyone in the Pleiades, every one of these globes was declared to be revolving, our own performing the circuit in a period of 117,000,000 of years! We, with our present lights, our vast telescopic improvements, and so forth, of course find it difficult to comprehend the ground of an idea such as this. Its first propagator was one Mudler. He was led, we must presume, to this wild hypothesis by mere analogy in the first instance; but, this being the case, he should have at least adhered to analogy in its development. A great central orb was, in fact, suggested; so far Mudler was consistent. This central orb, however, dynamically, should have been greater than all its surrounding orbs taken together. The question might then have been asked—&amp;quot;Why do we not see it?&amp;quot;- we, especially, who occupy the mid region of the cluster—the very locality near which, at least, must be situated this inconceivable central sun. The astronomer, perhaps, at this point, took refuge in the suggestion of non-luminosity; and here analogy was suddenly let fall. But even admitting the central orb non-luminous, how did he manage to explain its failure to be rendered visible by the incalculable host of glorious suns glaring in all directions about it? No doubt what he finally maintained was merely a centre of gravity common to all the revolving orbs—but here again analogy must have been let fall. Our system revolves, it is true, about a common centre of gravity, but it does this in connection with and in consequence of a material sun whose mass more than counterbalances the rest of the system. The mathematical circle is a curve composed of an infinity of straight lines; but this idea of the circle—this idea of it which, in regard to all earthly geometry, we consider as merely the mathematical, in contradistinction from the practical, idea—is, in sober fact, the practical conception which alone we have any right to entertain in respect to those Titanic circles with which we have to deal, at least in fancy, when we suppose our system, with its fellows, revolving about a point in the centre of the galaxy. Let the most vigorous of human imaginations but attempt to take a single step toward the comprehension of a circuit so unutterable! I would scarcely be paradoxical to say that a flash of lightning itself, travelling forever upon the circumference of this inconceivable circle, would still forever be travelling in a straight line. That the path of our sun along such a circumference—that the direction of our system in such an orbit—would, to any human perception, deviate in the slightest degree from a straight line even in a million of years, is a proposition not to be entertained; and yet these ancient astronomers were absolutely cajoled, it appears, into believing that a decisive curvature had become apparent during the brief period of their astronomical history—during the mere point—during the utter nothingness of two or three thousand years! How incomprehensible, that considerations such as this did not at once indicate to them the true state of affairs—that of the binary revolution of our sun and Alpha Lyrae around a common centre of gravity!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikisource.org/wiki/Mellonta_Tauta&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikisource.org/wiki/Mellonta_Tauta&lt;/a&gt; &lt;a href=&quot;https://en.wikisource.org/wiki/Mellonta_Tauta&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Poe also wrote similarly about the Sun's orbit in his 1848 essay &lt;em&gt;Eureka: a Prose Poem&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Eureka:_A_Prose_Poem%5B6%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Eureka:_A_Prose_Poem[6]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I note that if it took the Sun 117,000,000 years to make a complete circle, it would take it 325,000 years for its path to curve by a degree, and 5,416.666 years for its path to curve by an arc minute, and 90.2777 years for its path to curve by one arc second.  Astronomers could already measure angles of an arc second or smaller in Poe's era.  So the claim that the curvature of the Sun's path would undetectable after a million years is false - a circular orbit taking 117,00,000 years would curve by 3.076 degrees in a million years.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I read many Poe stories, including &amp;quot;Mellonta Tauta,&amp;quot; when I was 11 or 12.  So if someone asked me about the subject when I was 12 years old, I could have told them that 19th century astronomers believed that the Sun orbited around the center of the galaxy, and thus 19th century physicists should have believed that the Sun moved with respect to the aether.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2023-02-13T06:00:22.107" LastActivityDate="2023-02-13T06:00:22.107" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12915" PostTypeId="1" AcceptedAnswerId="12920" CreationDate="2021-02-25T20:45:42.317" Score="4" ViewCount="212" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre-Simon_Laplace#Stability_of_the_Solar_System&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; it states,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1776, Laplace published a memoir in which he first explored the possible influences of a purported luminiferous ether or of a law of gravitation that did not act instantaneously. He ultimately returned to an intellectual investment in Newtonian gravity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is it known whether or not Einstein was aware of Laplace's memoir (and the calculations within) when he himself considered the propagation of gravity as not being instantaneous?&lt;/p&gt;&#xA;&lt;p&gt;More specifically, was Laplace's &amp;quot;latency problem&amp;quot; merely an artifact of his approach? Or was it pointing to something deep about gravity that was explained by general relativity? Was it influential on Einstein? Or was it already obsolete by the time of Einstein?&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="7638" LastEditorUserId="7638" LastEditDate="2021-02-27T19:06:24.143" LastActivityDate="2021-02-27T21:47:38.593" Title="Did Einstein know about Laplace's query into whether gravity is instantaneous?" Tags="|biographical-details|einstein|gravity|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12916" PostTypeId="1" AcceptedAnswerId="12917" CreationDate="2021-02-25T21:40:42.803" Score="18" ViewCount="3671" Body="&lt;p&gt;Is there any evidence that Fermat wrote on the margin of a book &amp;quot;I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.&amp;quot; Everyone repeats this, but is there an image of the note? And did people write on margins of books in the 17th century when books must have been truly expensive? Has any author looked at these questions?&lt;/p&gt;&#xA;" OwnerUserId="7571" OwnerDisplayName="Wynne" LastEditorUserId="4251" LastEditDate="2021-02-26T22:55:51.427" LastActivityDate="2021-02-26T22:55:51.427" Title="Is the story about Fermat's writing on a margin true?" Tags="|mathematics|mathematicians|number-theory|17th-century|fermat|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12917" PostTypeId="2" ParentId="12916" CreationDate="2021-02-25T22:22:30.543" Score="33" Body="&lt;p&gt;Yes, it is true. Fermat's own copy was used in the publication&#xA;of Diophantus by Fermat's son Samuel,&#xA;and he included Fermat's notes. The original with Fermat's handwriting is lost.&#xA;&lt;a href=&quot;https://www.joh.cam.ac.uk/library/special_collections/early_books/fermat.htm#:%7E:text=When%20reviewing%20his%20copy%20of,to%20fit%20in%20the%20margin&quot; rel=&quot;noreferrer&quot;&gt;https://www.joh.cam.ac.uk/library/special_collections/early_books/fermat.htm#:~:text=When%20reviewing%20his%20copy%20of,to%20fit%20in%20the%20margin&lt;/a&gt;.&#xA;(The page with his most famous marginal note is reproduced, &amp;quot;Observatio Domini Petri de Fermat&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;Fermat made many remarks on the margin of Diophantus. These remarks inspired Euler and others, and eventually all his statements were proved,&#xA;though the proof of the last one took a while.&lt;/p&gt;&#xA;&lt;p&gt;For the general question: yes people wrote on margins. I guess this was the original purpose of margins. Many old books were even bound with several&#xA;blank pages in the beginning or more frequently at the end, to give more space for people to write their notes. Yes, books were expensive. But a book, especially a scientific one, was considered not a decoration object but rather a tool for work. And once one had some substantial thoughts or remarks when reading the text, the most natural place to record them was the margins, or sheets attached to the book. Please notice that paper for writing was also expensive.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-02-26T11:32:28.093" LastActivityDate="2021-02-26T11:32:28.093" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="12919" PostTypeId="1" CreationDate="2021-02-26T23:24:13.330" Score="1" ViewCount="116" Body="&lt;p&gt;I was reading a chapter of Fourier's seminal work &amp;quot;Analytic Theory of Heat&amp;quot;. The third chapter of this book was translated by the famous Stephen Hawking in his book &amp;quot;God created the integers&amp;quot;. I fail to find a reference for how Fourier determine the coefficients of the Fourier series. Fourier, in my opinion, should be ranked as one the greatest mathematicians in the 19th century for he laid a great foundation on the development of trigonometric series, an essential area of modern mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Can you summarize his findings and point out which part of his work contains his derivation? I am greatly interested in how he was able to determine the coefficients of Fourier series, a task that Daniel Bernoulli failed to achieve.&lt;/p&gt;&#xA;&lt;p&gt;I hope my shallow knowledge of mathematics do not offend and repel you from this question.&lt;/p&gt;&#xA;" OwnerUserId="13807" LastEditorUserId="4251" LastEditDate="2021-02-27T07:41:46.393" LastActivityDate="2021-02-27T07:41:46.393" Title="How did Fourier determine the coefficients of Fourier series?" Tags="|mathematics|calculus|real-analysis|harmonic-analysis|fourier|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12920" PostTypeId="2" ParentId="12915" CreationDate="2021-02-27T01:26:54.480" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Speed_of_gravity#Laplace&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laplace in 1799 introduced&lt;/a&gt; velocity dependence in a particular way, modifying the inverse square law so that the force was directed towards the position retarded according to the relative velocity &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; and the distance was computed accordingly. The consequence of that was that the correction to the Newtonian force was of the order &lt;span class=&quot;math-container&quot;&gt;$v/c$&lt;/span&gt;, and, according to Laplace's calculations, the planets would fly off of their orbits in a hurry if gravity propagated at the speed of light. It had to be at least &lt;span class=&quot;math-container&quot;&gt;$7×10^6$&lt;/span&gt; times faster than light to match observations.&lt;/p&gt;&#xA;&lt;p&gt;By the turn of 19th century velocity dependent gravity was instead modeled on Maxwell's electromagnetism, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2395/55&quot;&gt;What 19th century developments contributed to the General theory of Relativity?&lt;/a&gt; Gerber in 1898 proposed such a theory in a paper deemed unintelligible, and Lorentz proposed a coherent one in 1900 based on Mossotti's 1830 idea that electric attraction and repulsion do not balance each other exactly and the difference is gravity. They were (implicitly) special relativistic since the Maxwell equations are Lorentz invariant, and in them the correction was of the order &lt;span class=&quot;math-container&quot;&gt;$v^2/c^2$&lt;/span&gt;. So Laplace's fly off problem did not arise, but they still did not predict the correct precession of the perihelion of Mercury, and did not satisfy the more general relativity principle that Einstein favored. It is these theories that were discussed by contemporary authors like Mach and Poincare, and that Einstein was familiar with. Laplace's theory was not discussed at the time due to its absurd consequences.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-02-27T21:47:38.593" LastActivityDate="2021-02-27T21:47:38.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12922" PostTypeId="2" ParentId="7957" CreationDate="2021-02-27T19:44:29.447" Score="1" Body="&lt;p&gt;I know a book that is completely devoted to the history and development of trigonometric series (including of course Fourier series). But it is not in English, it is in Russian. It gives a rather detailed treatment on the method of various mathematicians who involved in this field. The title is &amp;quot;A.B Paplaukas Trigonometric series from Euler to Lebesgue&amp;quot; (Паплаускас А.Б. Тригонометрические ряды от Эйлера до Лебега). It has a free electronic version, so if you can read Russian, you can download it.&lt;/p&gt;&#xA;&lt;p&gt;I like Russian math books. They are quite rigorous while stay away from heavy formalism. The book I recommend do not spend times on chatting about irrelevant stories. It is very direct in style.&lt;/p&gt;&#xA;" OwnerUserId="13807" LastActivityDate="2021-02-27T19:44:29.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12923" PostTypeId="1" AcceptedAnswerId="12924" CreationDate="2021-02-28T09:29:46.430" Score="5" ViewCount="362" Body="&lt;p&gt;The famed &lt;a href=&quot;https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem&quot; rel=&quot;noreferrer&quot;&gt;Wallace–Bolyai–Gerwien theorem&lt;/a&gt; has got its name from three mathematicians who proved it independently. More precisely speaking&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Farkas Bolyai first formulated the question. Gerwien proved the theorem in 1833, but in fact Wallace had proven the same result already in 1807. According to other sources, Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The odd thing that came to my attention was, there is no trace of a mathematician named Paul Gerwien anywhere on the web or history books as far as I know. This sounds strange since Gerwien has proved the theorem much later than Wallace, and I think that in order to get one's name labeled on an already proved theorem, one must be a famous and established mathematician.&lt;/p&gt;&#xA;&lt;p&gt;Out of curiosity, I have searched quite thoroughly and yet, found no biography of a 19th century Paul Gerwien. There are no other contributions from this mysterious mathematician other than WBG theorem. So my question is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Who was Paul Gerwien and does he have any other contributions in mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14015" LastEditorUserId="10106" LastEditDate="2021-02-28T09:35:24.637" LastActivityDate="2021-09-15T08:14:08.913" Title="Who was Paul Gerwien?" Tags="|mathematics|mathematicians|biographical-details|19th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12924" PostTypeId="2" ParentId="12923" CreationDate="2021-03-01T02:55:15.173" Score="12" Body="&lt;p&gt;I have not been able to find any primary sources that confirm that Gerwien's first name was Paul. Gerwien published three papers in Crelle's &amp;quot;Journal für die reine und angewandte Mathematik&amp;quot;, where he is listed in the table of contents as P. Gerwien. High-quality scans of these publications are available from the &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Göttingen Digitization Center&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;P. Gerwien, &amp;quot;Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke.&amp;quot; &lt;em&gt;Journal für die reine und angewandte Mathematik&lt;/em&gt;, 1833 (10), pp. 228-234. (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0010?tify=%7B%22pages%22:%5B234%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;P. Gerwien, &amp;quot;Beweise einiger auf der Kugel Statt findenden Sätze.&amp;quot; &lt;em&gt;Journal für die reine und angewandte Mathematik&lt;/em&gt;, 1834 (11), pp. 130-135 (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0011?tify=%7B%22pages%22:%5B136%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;P. Gerwien, &amp;quot;Einige geometrische Sätze.&amp;quot; &lt;em&gt;Journal für die reine und angewandte Mathematik&lt;/em&gt;, 1834 (11), pp. 264-271 (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0011?tify=%7B%22pages%22:%5B274%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;All three articles identify the author as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Gerwien&lt;/em&gt;, Pr. Lieuten. im Königlich. Preußs. 22sten Inf. Reg.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This tells us that Gerwien was a 1st lieutenant in the Royal Prussian 22nd Infantry Regiment at the time of publication. In 1831-1832 he also published, as co-author with a fellow officer, a &lt;a href=&quot;https://www.google.com/books/edition/Aufgaben_systeme_und_sammlungen_aus_der/1HMLAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Aufgaben-Systeme+und+Sammlungen+aus+der+ebenen+Geometrie&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbook&lt;/a&gt; on geometry in two volumes:&lt;/p&gt;&#xA;&lt;p&gt;H. v. Holleben and P. Gerwien, &amp;quot;Aufgaben-Systeme und Sammlungen aus der ebenen Geometrie, zu einem selbstständigen Unterricht in der Analysis geordnet und durch Gesetze vorbereitet&amp;quot;, Berlin: Reimer 1831 &amp;amp; 1832&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0009?tify=%7B%22pages%22:%5B318%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;announcement for this work&lt;/a&gt; in Crelle's journal identifies H. v. Holleben as a lieutenant in the 21st Infantry Regiment, P. Gerwien as a lieutenant in the 22nd Infantry Regiment, and both as teachers at the Royal Prussian Cadet Corps. There is a published history of the 22nd Infantry Regiment that lists an officer by the name of Gerwien who is very likely identical with the author of the above publications:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Geschichte des 1. Oberschlesischen Infanterie-Regiments Nr. 22&amp;quot;. Berlin: Ernst Siegfried Mittler und Sohn 1884 (&lt;a href=&quot;https://books.google.com/books?id=B78QAAAAYAAJ&amp;amp;pg=PP6&amp;amp;lpg=PP6&amp;amp;dq=%22Oberschlesisches+Infanterie-Regiment%22+%22no.+22%22&amp;amp;source=bl&amp;amp;ots=rARrCkaoL6&amp;amp;sig=ACfU3U1Jo5VBk-YV0-fnt8CTukHqWmcB0A&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjkwdq54o3vAhWVjp4KHZknAI8Q6AEwDnoECBAQAw#v=onepage&amp;amp;q=Gerwien&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gerwien, geb. in Ostpreußen. Früher: im Garde-Ulan. Regt. Eintritt: vor 1822 als Sek. Lt. Austritt: 1845 als Maj. in den Gen. St. versetzt. 1830 Lebensalter 30 J. Dienstzeit 15. J. 2. Sek. Lt. und komm. als Lehrer beim Kad. Korps Berlin. 1840 4. Kapt. Starb als General beim Gr. Gen. St. in Berlin&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This tells us that Gerwien was born in East Prussia around 1800, entered military service around 1815, started service with the 22nd Infantry Regiment prior to 1822 as 2nd lieutenant, and was posted as a teacher to the Cadet Corps in Berlin. He left the unit in 1845 as a major, transferring to the General Staff. He died as a general while working in the Great General Staff in Berlin.&lt;/p&gt;&#xA;&lt;p&gt;Some evidence for this biographical sketch is provided in the following four publications:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Militair-Wochenblatt&lt;/em&gt;, 18. Jahrgang 1833, No. 892 (&lt;a href=&quot;https://books.google.com/books?id=AzixC0Lf7SAC&amp;amp;pg=RA2-PA4973&amp;amp;lpg=RA2-PA4973&amp;amp;dq=Gerwien+ulan&amp;amp;source=bl&amp;amp;ots=P6lC8vy4Yu&amp;amp;sig=ACfU3U0AaxarY0GkI2udG0LmKXC7ogCVMQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjm9MK28Y3vAhXSsZ4KHS0DBFIQ6AEwAHoECAIQAw#v=onepage&amp;amp;q=Gerwien%20ulan&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;12&quot;&gt;&#xA;&lt;li&gt;Juli 1833&lt;br /&gt;&#xA;Gerwien, Pr. Lt., com. beim Cadett. Corps, tritt ins 22. Inf. Regt. zurück&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This tells us that Gerwien, with the rank of 1st lieutenant, previously posted to the Cadet Corps, was assigned back to the 22nd Infantry Regiment on July 12, 1833.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Rang- und Quartierliste der Königlich-Preußsischen Armee für das Jahr 1847&lt;/em&gt; (&lt;a href=&quot;https://books.google.com/books?id=D2BDAAAAcAAJ&amp;amp;pg=PA348&amp;amp;lpg=PA348&amp;amp;dq=Generalstab+Gerwien&amp;amp;source=bl&amp;amp;ots=fTuvg5CRFC&amp;amp;sig=ACfU3U1gL_uUOQkmbBWIFotd-pvC_GBRTg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiU8OOU7I3vAhVFnJ4KHV0xDIgQ6AEwD3oECAYQAw#v=onepage&amp;amp;q=Generalstab%20Gerwien&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gerwien, Gen.-Stab, Major 12. Juni 1843&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This tells us that Gerwien was promoted to major on June 12, 1843 and that in 1847 he served in the General Staff.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Militair-Wochenblatt&lt;/em&gt;, 33. Jahrgang, 1849, No. 28 (&lt;a href=&quot;https://books.google.com/books?id=4PGkjkh1y8wC&amp;amp;pg=PA141&amp;amp;lpg=PA141&amp;amp;dq=Gerwien+ulan&amp;amp;source=bl&amp;amp;ots=Vj4WU_s_Ca&amp;amp;sig=ACfU3U0yKAgf3BQNJBE3oLILV1wiGNHmMw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjm9MK28Y3vAhXSsZ4KHS0DBFIQ6AEwAnoECAMQAw#v=onepage&amp;amp;q=Gerwien%20ulan&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;26&quot;&gt;&#xA;&lt;li&gt;Juni 1849&lt;br /&gt;&#xA;Gerwien, Major vom gr. Generalstabe, zum Abtheilungs-Vorsteher&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This tells us that Gerwien, a major serving in the Great General Staff, was promoted to department head effective June 26, 1849.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Königlich Preußischer Staats-Anzeiger&lt;/em&gt;, 1856 (&lt;a href=&quot;https://opacplus.bsb-muenchen.de/Vta2/bsb10486471/bsb:4173777?queries=Gerwien&amp;amp;language=de&amp;amp;c=default&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan by Bavarian State Library&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Berlin, 4. Juli. Seine Majestät der König haben den nachbenannten Offizieren und Beamten die Erlaubnis zur Anlegung der von des Kaisers von Rußlands Majestät ihnen verliehenen Orden Allergnädigst zu ertheilen geruht. Es haben erhalten:&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;der Oberst Gerwien im großen Generalstabe;&lt;br /&gt;&#xA;den St.-Wladimir-Orden vierter Klasse&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On July 4, 1856, Gerwien, then colonel in the Great General Staff, received royal permission to wear a Russian order he had received, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Order_of_Saint_Vladimir&quot; rel=&quot;nofollow noreferrer&quot;&gt;Order of Saint Vladimir&lt;/a&gt;, 4th class.&lt;/p&gt;&#xA;&lt;p&gt;In the Berlin address directory for 1855 we &lt;a href=&quot;https://digital.zlb.de/viewer/image/34111729_1855/190/&quot; rel=&quot;nofollow noreferrer&quot;&gt;find&lt;/a&gt; colonel P. Gerwien, department head in the Great General Staff, residing in Berlin at Bernburger Straße 30 (&lt;a href=&quot;https://www.google.com/maps/place/Bernburger+Str.+30,+10963+Berlin,+Germany/@52.5049,13.377296,527m/data=!3m2!1e3!4b1!4m5!3m4!1s0x47a851ccfdfc38c5:0xdd5d6d86814973!8m2!3d52.5049!4d13.37949&quot; rel=&quot;nofollow noreferrer&quot;&gt;this address in Google Maps today&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gerwien, P., Oberst und Abtheilungs-Vorsteher im großen General-Stabe, Bernburgerstr. 30&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have some tentative information that Gerwien had a daughter Clotilda probably born around 1831, and that he had already passed away when she &lt;a href=&quot;https://books.google.com/books?id=tcql-Pq7g7UC&amp;amp;pg=PA7&amp;amp;lpg=PA7&amp;amp;dq=%22gerwien%22+%22Clotilda%22&amp;amp;source=bl&amp;amp;ots=41mA-qeEMq&amp;amp;sig=ACfU3U3TRRSajWA4MTQdmBmZUeLZBDpKZw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiPq_u-qY7vAhUDGTQIHUZzDV8Q6AEwBHoECAUQAw#v=onepage&amp;amp;q=%22gerwien%22%20%22Clotilda%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;married&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Peter_Bayne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peter Bayne&lt;/a&gt; on April 26, 1858. The following article mentions a second daughter by the name of Anna:&lt;/p&gt;&#xA;&lt;p&gt;R. Buchholz, &amp;quot;Das Blutegel-Palais.&amp;quot; &lt;em&gt;Monatsblatt der Gesellschaft für Heimatkunde der Provinz Brandenburg zu Berlin&lt;/em&gt;, Vol. 21 (1913) pp. 92-93 (&lt;a href=&quot;https://digital.ub.uni-potsdam.de/periodical/pageview/405447?query=Gerwien&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Fräulein Anna Gerwien, eine Tochter des im Jahre 1858 verstorbenen Generals Gerwien, hatte die Freundlichkeit, mir ein Manuskript, die Lebensgeschichte ihres Vaters darstellend, zur Einsicht zu leihen. Darin finden sich mancherlei interessante Episoden berichtet aus den Jahren 1847 bis 1856, in denen der General als Stabsoffizier in Berlin wohnte; er war unter andern im Generalstab und Taktiklehrer des 18jährigen Kornprinzen und hatte dabei Gelegenheit, in Hofkreisen und mit berühmten Zeitgenossen in Verkehr zu treten.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The author reports that he had temporary access to a manuscript chronicling the life of General Gerwien, and that during his time in Berlin (1847-1856) he had been a member of the General Staff as well as an instructor in military tactics for the crown prince, making contacts at court and with various famous contemporaries.&lt;/p&gt;&#xA;&lt;p&gt;A poorly cropped &lt;a href=&quot;https://www.google.com/books/edition/Milit%C3%A4r_Zeitung/6LHi-Lrsn-EC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=Oberst%20Gerwien%20starb%201858&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt; of the &lt;em&gt;Militärzeitung&lt;/em&gt; from 1858 reports on the death of Gerwien, who was a major general and the commander of the 15th Infantry Brigade in &lt;a href=&quot;https://en.wikipedia.org/wiki/M%C3%BCnster&quot; rel=&quot;nofollow noreferrer&quot;&gt;Münster&lt;/a&gt; at that time. It should be noted that in the Prussian army of that time a Generalmajor (major general) was the lowest rank of general, with the modern equivalent being the Brigadegeneral (brigadier general). Using other news items for temporal context, his death &lt;em&gt;seems to&lt;/em&gt; have occurred in late April of 1858:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Zu Münster verschied in der letzten Woche des v. Mts. nach längerem Leiden&#xA;der k. preußische Generalmajor und Kommandant der 15. Infanteriebrigade Gerwien, ein hochverdienter Offizier, der sich nicht allein in der preußischen, sondern fast in jeder europäischen Armee sich eines höchst ehrenvollen Rufes erfreute.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally found a notice detailing the date of death:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Zeitschrift für Kunst, Wissenschaft und Geschichte des Krieges&lt;/em&gt; (1859) (&lt;a href=&quot;https://books.google.com/books?id=fMBLAAAAYAAJ&amp;amp;pg=PA100&amp;amp;lpg=PA100&amp;amp;dq=%22M%C3%BCnster%22+%22gerwien%22&amp;amp;source=bl&amp;amp;ots=Q1tQhAotX3&amp;amp;sig=ACfU3U&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Während des Jahres 1858 starben im Preußischen Staate folgende, noch im Dienste befindliche oder zur Disposition gestellte und verabschiedete Generale:&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;Gerwien, General-Major und Kommandeur der 26sten Infanterie-Brigade, 59 Jahre alt, am 24sten April in Münster&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gerwien, major general and commander of the 26th Infantry Brigade died on April 24, 1858 in Münster at the age of 59. This suggests he was born in 1799.&lt;/p&gt;&#xA;&lt;p&gt;In his autobiography &amp;quot;Ein Leben voller Abenteuer&amp;quot;, in the chapter covering his life as a young cadet, &lt;a href=&quot;https://en.wikipedia.org/wiki/Otto_von_Corvin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Otto von Corvin&lt;/a&gt; (1812-1886) &lt;a href=&quot;https://www.google.com/books/edition/Ein_Leben_voller_Abenteuer/j-ArAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Leutnant+Gerwien%22&amp;amp;pg=PA99&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;describes&lt;/a&gt; Gerwien as a young lieutenant with blonde curls. Corvin feared him as a strict math teacher, but also states that he was a well-meaning person who simply took his job very seriously.&lt;/p&gt;&#xA;&lt;p&gt;I noticed belatedly an entry in the German Wikipedia for &lt;a href=&quot;https://de.wikipedia.org/wiki/Karl_Ludwig_Gervien&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl Ludwig Gervien&lt;/a&gt; (born May 13, 1799 in Landsberg; died April 24, 1858 in Münster) whose biographical details match the information collected above very closely. The source cited by Wikipedia is Kurt von Priesdorff, &lt;em&gt;Soldatisches Führertum, Band 6&lt;/em&gt;. Hamburg: Hanseatische Verlagsanstalt 1938.&lt;/p&gt;&#xA;&lt;p&gt;I have been able to find other references to Karl Ludwig Ger&lt;strong&gt;w&lt;/strong&gt;ien (e.g. in the list of &lt;a href=&quot;http://genwiki.genealogy.net/26._Infanterie-Brigade&quot; rel=&quot;nofollow noreferrer&quot;&gt;commanders of the 26th Infantry Brigade&lt;/a&gt;), so that Ger&lt;strong&gt;v&lt;/strong&gt;ien is almost certainly an errant spelling. In particular, I found &lt;a href=&quot;https://www.cs.purdue.edu/homes/gnf/book/Booknews/ch14.html#MPG&quot; rel=&quot;nofollow noreferrer&quot;&gt;this research&lt;/a&gt; by Greg N. Frederickson:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Karl Ludwig Gerwien was born in 1799 in Landsberg in East Prussia. He entered the Garde-Ulanen-Regiment in 1815, then climbed the military ladder: becoming a second lieutenant in 22nd infantry regiment in 1818, a first lieutenant in 1829, a captain and company commander in 1835, then major in 1843, lieutenant colonel in 1851, colonel in 1852, major general in 1857. He died in Münster, Westfalen, in 1858.  Gerwien commanded a general war school 1820-1823, was a teacher in the division school of the 12th division 1823-26, was a teacher at the cadet institute in 1826-1833, shifted to the general staff in 1849, and became commander of the 26th infantry brigade in 1856. [...] his dates and positions are so close a match that he is almost certainly the author listed as &amp;quot;P. Gerwien&amp;quot; in Crelle's Journal&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-09-15T08:14:08.913" LastActivityDate="2021-09-15T08:14:08.913" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12925" PostTypeId="1" CreationDate="2021-03-01T03:20:59.633" Score="5" ViewCount="99" Body="&lt;p&gt;I am reading Carlslaw's &amp;quot;&lt;a href=&quot;https://books.google.com/books/about/Introduction_to_the_Theory_of_Fourier_s.html?id=fHdVAAAAYAAJ&amp;amp;printsec=frontcover&amp;amp;source=kp_read_button&amp;amp;newbks=1&amp;amp;newbks_redir=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Introduction to the Theory of Fourier's Series and Integrals&lt;/a&gt;&amp;quot;, first chapter on the history of Fourier series, page 3. The author asserts that Clairaut and Euler did derive in parts the coefficients of Fourier series. He said Euler wrote:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$f(x)=a_0+2a_1\cos(x)+2a_2\cos(2x)+...2a_n(\cos(n)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Euler multipled both sides &lt;span class=&quot;math-container&quot;&gt;$\cos(nx)$&lt;/span&gt; and integrated the series from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; term by term to obtain:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a_n=\int_{0}^{\pi}f(x)\cos(nx)dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;His reference is Petrop.N.Acta, 11, 1793(1798), page 94 (May 1777).&lt;/p&gt;&#xA;&lt;p&gt;I checked it and find &amp;quot;&lt;a href=&quot;https://scholarlycommons.pacific.edu/cgi/viewcontent.cgi?article=1702&amp;amp;context=euler-works&quot; rel=&quot;noreferrer&quot;&gt;Methodus facilis inveniendi series per sinus sosinusve angulorum multiplorum procedentes, quarum usus in universa theoria astronomiae est amplissimus&lt;/a&gt;&amp;quot;. I think this is the correct paper, but I cannot find the reference that the author made.&lt;/p&gt;&#xA;&lt;p&gt;In that article, I only see on page 101, this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$S=1+\cos(x)+\cos(2x)+\cos(3x)+...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Multiply by &lt;span class=&quot;math-container&quot;&gt;$2\sin(\frac{x}{2})$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;One get:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2\sin(\frac{x}{2})S=2\sin(\frac{x}{2})+2\sin(\frac{x}{2})\cos(x)+2\sin(\frac{x}{2})\cos(2x)+2\sin(\frac{x}{2})\cos(3x)+...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2\sin(\frac{x}{2})S=2\sin(\frac{x}{2})+\sin(\frac{3}{2}x)-\sin(\frac{3}{2}x)+\sin(\frac{5}{2}x)-\sin(\frac{5}{2}x)...+\sin(n+\frac{1}{2}x)-\sin(n-\frac{1}{2}x)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is on page 101.&lt;/p&gt;&#xA;&lt;p&gt;I couldn't find the reference for this integral, can you help me to find it?&lt;/p&gt;&#xA;" OwnerUserId="13807" LastActivityDate="2023-02-27T04:05:38.797" Title="Where could I find the reference for the statement of Euler finding out the coefficients of Fourier series?" Tags="|real-analysis|trigonometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12927" PostTypeId="1" CreationDate="2021-03-02T10:40:08.087" Score="1" ViewCount="81" Body="&lt;p&gt;If Maxwell's equations were experimentally found on Earth such as Faradays law, coulombs law , and worked in the Earth's frame of reference, then why did people believe them to be valid in the aether's reference frame?&lt;/p&gt;&#xA;" OwnerUserId="13989" LastEditorUserId="13989" LastEditDate="2021-03-04T05:16:43.000" LastActivityDate="2021-03-04T05:16:43.000" Title="Validity of Maxwell's equations in the aether's reference frame" Tags="|physics|theoretical-physics|electromagnetism|light|maxwell|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12929" PostTypeId="1" CreationDate="2021-03-02T15:00:08.953" Score="3" ViewCount="108" Body="&lt;p&gt;While Marie Skłodowska's husband, Pierre Curie, died in a traffic accident, she died of cancer. &lt;a href=&quot;https://hsm.stackexchange.com/q/745/217&quot;&gt;Various sources&lt;/a&gt; claim that the radiation she got during their experiments might have contributed significantly to that. Obviously, the effects of radioactivity were not yet known at the time (not even much later).&lt;/p&gt;&#xA;&lt;p&gt;Today, the acquired dose of the human body is measured in &lt;a href=&quot;https://en.wikipedia.org/wiki/Sievert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sieverts&lt;/a&gt;, which is essentially the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gray_(unit)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gray&lt;/a&gt; (which is the mass-specific radioactive energy), weighted with the sensitivity of the organs.&lt;/p&gt;&#xA;&lt;p&gt;Is it known, or at least estimated, how many Sieverts the Curies did get?&lt;/p&gt;&#xA;" OwnerUserId="217" LastEditorUserId="4251" LastEditDate="2021-03-02T17:28:55.193" LastActivityDate="2021-03-02T17:28:55.193" Title="How much radiation did the Curies get while synthetising Radium?" Tags="|physics|radiation|radioactivity|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12930" PostTypeId="1" CreationDate="2021-03-02T17:30:38.970" Score="3" ViewCount="529" Body="&lt;p&gt;Who came up with the name &amp;quot;Manhattan distance&amp;quot; (for the distance between two points as measured by the sum of the horizontal and vertical distances, as opposed to the length of the straight line between them)?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Zd3sj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Zd3sj.png&quot; alt=&quot;illustration of Manhattan and Euclidean distance&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10773" LastActivityDate="2021-03-03T01:44:03.743" Title="Who came up with the name &quot;Manhattan distance&quot;?" Tags="|mathematics|terminology|geometry|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12931" PostTypeId="2" ParentId="12930" CreationDate="2021-03-03T01:10:06.553" Score="5" Body="&lt;p&gt;I am capturing information from a comment as a starting point for a community wiki answer to which others can add earlier citations.&lt;/p&gt;&#xA;&lt;p&gt;Harvey L. Garner and Jon S. Squire, &amp;quot;Iterative Circuit Computers&amp;quot;, in &lt;em&gt;Proceedings of the 1962 Workshop on Computer Organization&lt;/em&gt;, Baltimore, 1962, pp. 156-181. (&lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=WssmAAAAMAAJ&amp;amp;oi=fnd&amp;amp;pg=PA156&amp;amp;dq=%22Manhattan+distance%22&amp;amp;ots=Tpr5qLpxbS&amp;amp;sig=gqCZcoWWfTFA3YXdeAe4-F0mDow#v=snippet&amp;amp;q=Manhattan&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Google scan&lt;/a&gt;). On page 165:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$W(p)$&lt;/span&gt; is the number of distinct pairs of modules at Manhattan distance, &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, and does not refer to the number of ways of connecting a pair of modules.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paper is about a computer arranged as a two-dimensional grid of tiles, which explains the use of Manhattan distance. The term is not explained anywhere in the paper, so its use appears to have been well established by 1962.&lt;/p&gt;&#xA;&lt;p&gt;By collating many overlapping snippets from Google's snippet view, I was able to extract the following quotation from a document dating to the same year that explains the term:&lt;/p&gt;&#xA;&lt;p&gt;Rodolfo Gonzales and Sandra Palais, &amp;quot;A Path-Building Procedure for Iterative Circuit Computers&amp;quot;, University of Michigan Information System Laboratory Technical Note, 1962. On page 22:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] two senses are to be followed when tracing the segments in each of the horizontal and vertical directions. This means that once the vertical and horizontal priorities are established, for example vertical down and horizontal to the right, all the path segments have to be traced in either of the two specified senses exclusively. It is to be noted that this restriction on the allowable class of paths has the advantage of eliminating the&#xA;need for an algorithm capable of tracing a minimum length path, since all non-regressive paths have the same length when measured in terms of the number of segments needed to connect the modules. This method of measuring&#xA;distance has been called &amp;quot;Manhattan distance.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-03-03T01:44:03.743" LastActivityDate="2021-03-03T01:44:03.743" CommentCount="0" CommunityOwnedDate="2021-03-03T01:10:06.553" ContentLicense="CC BY-SA 4.0" />
  <row Id="12932" PostTypeId="1" CreationDate="2021-03-03T04:54:14.597" Score="3" ViewCount="76" Body="&lt;p&gt;In an introductory course on PDE's I got as a project to prove and present a version of &lt;a href=&quot;https://en.wikipedia.org/wiki/Hopf_lemma&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hopf's (boundary) lemma&lt;/a&gt;. Namely:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\Omega \subset R^{d}$&lt;/span&gt; be an non-empty open connected set with a&#xA;twice differentiable boundary. Let &lt;span class=&quot;math-container&quot;&gt;$u\in C^{2}(\Omega) \cap&#xA;&amp;gt; C^{1}(\bar{\Omega})$&lt;/span&gt; be a subharmonic function (&lt;span class=&quot;math-container&quot;&gt;$\Delta u \geq 0)$&lt;/span&gt;. If&#xA;there exists a &lt;span class=&quot;math-container&quot;&gt;$M\geq 0$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$u(x) &amp;lt; M$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$x\in \Omega$&lt;/span&gt;&#xA;and &lt;span class=&quot;math-container&quot;&gt;$u(x_{0}) = M$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$x_{0}$&lt;/span&gt; on the boundary of &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; then&#xA;the outward normal derivative at &lt;span class=&quot;math-container&quot;&gt;$x_{0}$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$(\nabla u(x_{0})\cdot n)$&lt;/span&gt; is&#xA;stricly positive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Where can I find the original source of Hopf's lemma? I have looked trough Evans - Partial Differential Equations (second edition) and some notes written by professor John K. Hunter at UC Davis, but have been unable to make any progress.&lt;/p&gt;&#xA;" OwnerDisplayName="user14031" LastEditorUserId="10106" LastEditDate="2021-03-03T07:08:19.670" LastActivityDate="2021-03-03T07:08:19.670" Title="What is the source of Hopf's (boundary) lemma?" Tags="|mathematics|reference-request|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12933" PostTypeId="1" CreationDate="2021-03-03T05:26:31.283" Score="2" ViewCount="308" Body="&lt;p&gt;What are Steven Weinberg's contributions to physics other than the Nobel-winning work on electroweak unification? Why is he so famous and respected among particle physicists compared to other Nobel laureates?&lt;/p&gt;&#xA;" OwnerUserId="9960" LastEditorUserId="4251" LastEditDate="2021-03-08T07:38:19.493" LastActivityDate="2021-03-08T09:34:02.583" Title="What are Steven Weinberg's contribution to physics other than electroweak unification?" Tags="|physics|physicists|scientists|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12935" PostTypeId="1" AcceptedAnswerId="12937" CreationDate="2021-03-03T06:23:38.537" Score="3" ViewCount="141" Body="&lt;p&gt;Donald E. Knuth reports in his: TAOCP: Volume 1: Fundamental Algorithms (3rd ed.) in &lt;span class=&quot;math-container&quot;&gt;$\S1.2.4$&lt;/span&gt;: Integer Functions and Elementary Number Theory: Exercise 38 that this result:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{0 \mathop \le k \mathop &amp;lt; y} \left\lfloor {x + \dfrac k y}\right\rfloor = \lfloor {x y + \lfloor {x + 1} \rfloor (\lceil y \rceil - y) }$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;was the work of &lt;strong&gt;E. Busche&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This result apparently appeared in Crelle's v. 136, 39-57 (1909).&lt;/p&gt;&#xA;&lt;p&gt;I have found out that &amp;quot;E. Busche&amp;quot; was &amp;quot;Conrad Heinrich Edmund Friedrich Busche&amp;quot;, but apart from the fact that he died fighting in WWI, I have not been able to identify any biographical details about him.&lt;/p&gt;&#xA;&lt;p&gt;Can anybody help?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2021-03-03T11:58:47.313" Title="Who was E. Busche?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12936" PostTypeId="1" CreationDate="2021-03-03T09:12:24.773" Score="5" ViewCount="186" Body="&lt;p&gt;I am interested in looking up P. G. Tait's flawed proof of the four-colour theorem, published in 1880. The citation that I have seen is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;P. G. Tait, &lt;em&gt;On the colouring of maps&lt;/em&gt;, Proc. Roy. Soc. Edinburgh &lt;strong&gt;10&lt;/strong&gt; (1879–80), no. 106, 501–503, &lt;a href=&quot;https://doi.org/10.1017/S0370164600044229&quot; rel=&quot;noreferrer&quot;&gt;doi:10.1017/S0370164600044229&lt;/a&gt;. &lt;a href=&quot;https://zbmath.org/?q=an:12.0408.02&quot; rel=&quot;noreferrer&quot;&gt;JFM 12.0408.02&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, this is just an abstract of the full paper; several details are omitted, and it is indicated that these will be given in full detail somewhere else, presumably the full paper.&lt;/p&gt;&#xA;&lt;p&gt;I am not familiar with how this particular journal works; perhaps it is not unusual for only abstracts to be published, and for the full paper to appear elsewhere? If so, where can I find the full paper of an abstract published in this journal?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;What follows is a summary of my search for details regarding this. Perhaps those who are knowledgeable on this matter can confirm or deny how close my guesses are.&lt;/p&gt;&#xA;&lt;p&gt;In this case, I found the following paper by Tait (again, actually an abstract) that was published later in the same volume of this journal:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;P. G. Tait, &lt;em&gt;Remarks on the previous communication&lt;/em&gt;, Proc. Roy. Soc. Edinburgh &lt;strong&gt;10&lt;/strong&gt; (1879–80), no. 107, 729, &lt;a href=&quot;https://doi.org/10.1017/S0370164600044643&quot; rel=&quot;noreferrer&quot;&gt;doi:10.1017/S0370164600044643&lt;/a&gt;. &lt;a href=&quot;https://doi.org/?q=an:12.0409.02&quot; rel=&quot;noreferrer&quot;&gt;JFM 12.0409.02&lt;/a&gt;.&lt;sup&gt;[1]&lt;/sup&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In this paper, Tait writes,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But a remark made incidentally in the abstract of my former paper has led me to a totally different mode of attacking the question, which puts its nature in clearer light. I have therefore withdrawn my former paper, as in great part superseded by the present one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, perhaps the first paper never appeared in full anywhere, since Tait says he is withdrawing it? Then, the question becomes whether the paper &amp;quot;Remarks on the previous communication&amp;quot; was published in full anywhere else. I cannot find any other paper by Tait with the same title, but admittedly this is a very unenlightening title. If the full paper appeared elsewhere, possibly it would have a different title attached to it: with this guess, a search among other papers of Tait in the same year turned up the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;P. G. Tait, &lt;em&gt;Note on a theorem in geometry of position&lt;/em&gt;, Trans. Roy. Soc. Edinburgh &lt;strong&gt;29&lt;/strong&gt; (1880), part 2, 657–660, plate 16, &lt;a href=&quot;https://doi.org/10.1017/S008045680002617X&quot; rel=&quot;noreferrer&quot;&gt;doi:10.1017/S008045680002617X&lt;/a&gt;. &lt;a href=&quot;https://zbmath.org/?q=an:12.0409.01&quot; rel=&quot;noreferrer&quot;&gt;JFM 12.0409.01&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This paper references the paper in Proc. Roy. Soc. Edinburgh &lt;strong&gt;10&lt;/strong&gt;, no. 106, and concerns the proof (albeit flawed) of the remark that was noted in Proc. Roy. Soc. Edinburgh &lt;strong&gt;10&lt;/strong&gt;, no. 107, but it does not mention the latter paper. However, from the content it does appear that this might be the one that contains the full details of the latter; another reason for this suspicion is that both of these were read on July 19, 1880.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To conclude, these are the questions that I am specifically looking to be answered:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;For an abstract published in Proc. Roy. Soc. Edinburgh, is the complete paper then published in Trans. Roy. Soc. Edinburgh?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is Tait's paper in Trans. Roy. Soc. Edinburgh &lt;strong&gt;29&lt;/strong&gt; the full paper whose abstract was published in Proc. Roy. Soc. Edinburgh &lt;strong&gt;10&lt;/strong&gt;, no. 107?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What is the correct citation for Tait's 1880 flawed proof of the four-colour theorem? Is it the &lt;em&gt;Remarks on …&lt;/em&gt; paper, or the &lt;em&gt;Note on …&lt;/em&gt; paper, or something else altogether?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;[1]: The &amp;quot;previous communication&amp;quot; mentioned in the title is the following paper by Frederick Guthrie: F. Guthrie, &lt;em&gt;Note on the colouring of maps&lt;/em&gt;, Proc. Roy. Soc. Edinburgh &lt;strong&gt;10&lt;/strong&gt; (1879–80), no. 107, 727–728, &lt;a href=&quot;https://doi.org/10.1017/S0370164600044631&quot; rel=&quot;noreferrer&quot;&gt;doi:10.1017/S0370164600044631&lt;/a&gt;. This paper in turn references the first paper by Tait mentioned above, and provides some further background on the problem.&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user13915" LastActivityDate="2022-06-24T08:08:14.947" Title="Where can I find the complete papers of abstracts published by P. G. Tait in Proc. Roy. Soc. Edinburgh in 1880?" Tags="|mathematics|reference-request|geometry|graph-theory|" AnswerCount="1" CommentCount="4" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12937" PostTypeId="2" ParentId="12935" CreationDate="2021-03-03T11:35:21.673" Score="7" Body="&lt;p&gt;Edmund Busche was killed on the Western Front in WW1, on May 2, 1916. This was reported by his colleague &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Riebesell&quot; rel=&quot;noreferrer&quot;&gt;Paul Riebesell&lt;/a&gt; in the annual report of the &lt;a href=&quot;https://en.wikipedia.org/wiki/German_Mathematical_Society&quot; rel=&quot;noreferrer&quot;&gt;German Mathematical Society&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung&lt;/em&gt;, Vol. 25, 1917, p. 283 (&lt;a href=&quot;https://www.google.com/books/edition/Jahresbericht_der_Deutschen_Mathematiker/WCcTH55FpSQC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=Edmund%20Busche&quot; rel=&quot;noreferrer&quot;&gt;Google snippet&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;E. Busche †&lt;br /&gt;&#xA;Von P. Riebesell in Hamburg&lt;br /&gt;&#xA;Am 2. Mai 1916 fiel als Hauptmann der Landwehr a.D. an der Westfront Edmund Busche im Alter von 55 Jahren. Der Verstorbene war seit 1888 Mitglied der Mathematischen Gesellschaft in Hamburg, seit 1891 auch Mitglied der Deutschen Mathematiker-Vereinigung. Nachdem er etwa 20 Jahre als Oberlehrer an der Hansaschule in Bergedorf tätig gewesen war, wirkte er in den letzten zehn Jahren als Professor an der Oberrealschule in Hamburg-Eimsbüttel und als Dozent [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From this we learn that at the time of death he held the rank of captain in the militia. Edmund Busche had been a member of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hamburg_Mathematical_Society&quot; rel=&quot;noreferrer&quot;&gt;Hamburg Mathematical Society&lt;/a&gt; since 1888 and a member of the &lt;a href=&quot;https://en.wikipedia.org/wiki/German_Mathematical_Society&quot; rel=&quot;noreferrer&quot;&gt;German Mathematical Society&lt;/a&gt; (founded in 1890 by Georg Cantor) since 1891. He had worked for about twenty years as a senior teacher at a high-school in the Hamburg suburb of Bergedorf (now a borough within the City of Hamburg), and thereafter for ten years as a professor at a high-school in Hamburg-Eimsbüttel.&lt;/p&gt;&#xA;&lt;p&gt;The Hamburg Mathematical Society honored Edmund Busche in volume 5, issue 6 of their journal &lt;em&gt;Mitteilungen der Mathematischen Gesellschaft in Hamburg&lt;/em&gt;. A high-quality &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=njp.32101075456671&amp;amp;view=1up&amp;amp;seq=233&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt; of the issue with a &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=njp.32101075456671&amp;amp;view=1up&amp;amp;seq=231&quot; rel=&quot;noreferrer&quot;&gt;picture&lt;/a&gt; of Busche is provided by the HathiTrust Digital Library.&lt;/p&gt;&#xA;&lt;p&gt;The issue starts out with an article by E. Hoppe in Hamburg &amp;quot;Edmund Busche zum Gedächtnis&amp;quot; (In Memory of Edmund Busche) which provides a biographical sketch of Busche's life. He was born on May 2, 1861 in Neuland in the district of &lt;a href=&quot;https://en.wikipedia.org/wiki/Kehdingen&quot; rel=&quot;noreferrer&quot;&gt;Kehdingen&lt;/a&gt; which is on the Elbe river north of Hamburg. He was the second son of a minor official. Soon after his birth the family moved to the small town of &lt;a href=&quot;https://en.wikipedia.org/wiki/Drochtersen&quot; rel=&quot;noreferrer&quot;&gt;Drochtersen&lt;/a&gt; where his father died in 1868. His mother then moved the family to &lt;a href=&quot;https://en.wikipedia.org/wiki/Walsrode&quot; rel=&quot;noreferrer&quot;&gt;Walsrode&lt;/a&gt;, where they lived for three years, before moving to &lt;a href=&quot;https://en.wikipedia.org/wiki/Uelzen&quot; rel=&quot;noreferrer&quot;&gt;Uelzen&lt;/a&gt; in 1871. In 1875 they moved to &lt;a href=&quot;https://en.wikipedia.org/wiki/Hanover&quot; rel=&quot;noreferrer&quot;&gt;Hanover&lt;/a&gt; where Busche completed high-school.&lt;/p&gt;&#xA;&lt;p&gt;He then attended the University of Göttingen to study mathematics. After only four years he completed his dissertation under &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Christian_Julius_Schering&quot; rel=&quot;noreferrer&quot;&gt;Ernst Schering&lt;/a&gt;, finishing his doctoral studies on February 27, 1883. A few days later he passed the teaching exam, allowing him to work as &amp;quot;Oberlehrer&amp;quot; (senior teacher). He then served in the military, where he ultimately achieved the rank of lieutenant. Due to a glut of teachers he initially had trouble finding work and had to work as a private tutor for a while before finding permanent employment with the Hansaschule in Bergedorf in 1886.&lt;/p&gt;&#xA;&lt;p&gt;He got married in 1901 and had two children, a girl and a boy. According to &lt;a href=&quot;https://baronsdebode.wordpress.com/2017/06/16/1-1-von-bode-von-halem-illustrated-descendancy/&quot; rel=&quot;noreferrer&quot;&gt;genealogical information&lt;/a&gt; I found on the internet for whose accuracy I cannot vouch, his wife Ottilie Fastenau was born in 1876, his daughter Friederike in 1902 and the son Reinhard in 1904.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=njp.32101075456671&amp;amp;view=1up&amp;amp;seq=246&quot; rel=&quot;noreferrer&quot;&gt;next article&lt;/a&gt; in the commemorative issue is by H. v. Mangoldt in Danzig, &amp;quot;Edmund Busches wissenschaftliches Lebenswerk&amp;quot; (The Scientific Œuvre of Edmund Busche). Including his Ph.D thesis, it lists thirty-five publications by Busche. The author points out that it is clear from the topics of the publications that Busche's interests lay entirely within pure mathematics, with a focus on number theory and projective geometry. He mentions that since Busche gave a talk on the theory of relativity at the Hamburg Mathematical Society in 1911 he must have also had a passing interest in mathematical physics.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=njp.32101075456671&amp;amp;view=1up&amp;amp;seq=253&quot; rel=&quot;noreferrer&quot;&gt;next article&lt;/a&gt; is by P. Riebesell in Hamburg, &amp;quot;Eine Verallgemeinerung des Pascalschen Satzes für beliebige Sechsecke nach E. Busche&amp;quot; (A Generalization of Pascal's Theorem for Arbitrary Hexagons, based on work by E. Busche), in which the author attempts to complete Busche's final, unfinished, work. In the introductory paragraph he mentions that Busche served on the front in Belgium. He last saw him when he was furloughed from the front in January of 1916, at which time Busche asked him to do some background research on this topic.&lt;/p&gt;&#xA;&lt;p&gt;Busche's 1883 Ph.D. thesis is entitled: &amp;quot;Ueber eine Beweismethode in der Zahlentheorie und einige Anwendungen derselben, insbesondere auf das Reciprocitätsgesetz in der Theorie der quadratischen Reste&amp;quot; (On a method of proof in number theory and some applications of the same, in particular to the reciprocity law in the theory of quadratic residues) and is dedicated to his mother. A &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN837594464?origin=/collection/mathematica?filter%255B0%255D%255Bfacet_place_publish%255D%3DG%25C3%25B6ttingen%26filter%255B1%255D%255Bfacet_creator_personal%255D%3DBusche%252C%2520Edmund&amp;amp;collection=mathematica&quot; rel=&quot;noreferrer&quot;&gt;high-quality scan&lt;/a&gt; is available from the Göttingen Digitization Center. At the &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN837594464?origin=/collection/mathematica?filter%255B0%255D%255Bfacet_place_publish%255D%3DG%25C3%25B6ttingen%26filter%255B1%255D%255Bfacet_creator_personal%255D%3DBusche%252C%2520Edmund&amp;amp;collection=mathematica&amp;amp;tify=%7B%22pages%22:%5B58%5D,%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;end&lt;/a&gt; of the dissertation there is a brief biographical sketch that confirms information from other sources: Busche was born on May 2, 1861 in Neuland in the district of Kehdingen. His mother's name was Friederike. In 1879 he finished high-school in Hanover. From Easter 1879 through Easter 1883 he studied mathematics in Göttingen. On March 3, 1883 he passed the teaching exam and subsequently departed for mandatory military service to Berlin. He thanks his Ph.D. advisor, professor E. Schering, for much valuable advice.&lt;/p&gt;&#xA;&lt;p&gt;An &lt;a href=&quot;https://books.google.com/books?id=3SZGAAAAYAAJ&amp;amp;pg=PA73&amp;amp;lpg=PA73&amp;amp;dq=%22Lehrer%22+%22E.+Busche%22&amp;amp;source=bl&amp;amp;ots=2uuLDPjemY&amp;amp;sig=ACfU3U3NqEvQwyRYAkgOkwlLFobfXdRd1g&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiryuXl5pPvAhWMu54KHVGHBrQQ6AEwDXoECBIQAw#v=onepage&amp;amp;q=%22Lehrer%22%20%22E.%20Busche%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;annual report&lt;/a&gt; of the Hansaschule from 1889 and a &lt;a href=&quot;https://books.google.com/books?id=__XzftKUCKwC&amp;amp;pg=PA44&amp;amp;lpg=PA44&amp;amp;dq=%22Edmund+Busche%22+%22Mai%22&amp;amp;source=bl&amp;amp;ots=liPHvU602i&amp;amp;sig=ACfU3U2fHsUT1Iodia2mAg9pZk4kjPTMkg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwje1_Kj6pPvAhW8GDQIHW3bDE4Q6AEwBHoECAUQAw#v=onepage&amp;amp;q=%22Edmund%20Busche%22%20%22Mai%22&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;festschrift&lt;/a&gt; published by the school in 1908 confirm some of the biographical details already mentioned above.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-03-03T11:58:47.313" LastActivityDate="2021-03-03T11:58:47.313" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12938" PostTypeId="2" ParentId="12933" CreationDate="2021-03-03T15:33:00.880" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;https://biography.yourdictionary.com/steven-weinberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt;, Steven Weinberg also worked in plasma physics and other fields within particle physics. The same page mentions the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Though his Nobel Prize was for work in unification, Weinberg made significant contributions in a wide range of areas in particle physics and even in plasma physics. Among colleagues he was known more for versatility than for mathematical strength.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In an &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5146&quot; rel=&quot;nofollow noreferrer&quot;&gt;interview&lt;/a&gt;, he mentioned that he also wrote a book about cosmology and gravitation. Here is an excerpt of the interview:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was just amusing to me that at a time when I was deeply involved in the late 1960s in exploring the implications of broken symmetries for elementary particle physics an example like this would crop up in plasma physics, it didn't affect what I did in elementary particle physics. I don't think the JASON work had much effect on my work in elementary particle physics, but it did have a big effect on my work, such as it is, in cosmology and in writing my book about gravitation and cosmology. Because, you know, for example, in an article I wrote in 1977 Ben Lee and I solved the Boltzmann equation in a situation where you have heavy neutrinos colliding and gradually disappearing. And I wouldn't have known what the Boltzmann equation was if it hadn't been for JASON.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I hope this helps.&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2021-03-03T15:33:00.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12939" PostTypeId="1" CreationDate="2021-03-03T22:14:41.617" Score="1" ViewCount="115" Body="&lt;p&gt;How did mathematicians interpret quartic equations and fourth powers before Descartes propose to perform elementary arithmetic on line segments?&lt;/p&gt;&#xA;&lt;p&gt;I ask this because it seems strange to me that multiplication before him was essentially a geometric operation (as far as I know), and still mathematicians could study methods for finding roots of quartic equations, for instance.&lt;/p&gt;&#xA;&lt;p&gt;Since I am new to History of Mathematics, I would also be glad to know about references in which I can learn more about it.&lt;/p&gt;&#xA;" OwnerUserId="10126" LastEditorUserId="10106" LastEditDate="2021-03-04T00:32:39.267" LastActivityDate="2021-03-04T00:32:39.267" Title="Fourth powers and quartic equations before Descartes" Tags="|mathematics|reference-request|geometry|elementary-algebra|descartes|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12940" PostTypeId="1" AcceptedAnswerId="12942" CreationDate="2021-03-03T22:16:52.903" Score="1" ViewCount="1076" Body="&lt;p&gt;The double slit experiment is usually given as the foremost example of a physical experiment that requires quantum mechanics to satisfactorily explain. However, every account i've seen of it (such as in, eg, Feynman's famed QED book) can be perfectly justified using classical wave theory.&lt;/p&gt;&#xA;&lt;p&gt;So what actual example of physical phenomena can be given to undergraduate students that conclusively demonstrates the insufficiency of classical theory and thus motivates the introduction of quantum mechanics?&lt;/p&gt;&#xA;" OwnerUserId="14037" LastEditorUserId="10106" LastEditDate="2021-03-04T00:35:12.457" LastActivityDate="2021-03-05T04:26:41.847" Title="What are the definitive experiments/phenomena which motivate quantum mechanics?" Tags="|physics|quantum-mechanics|examples|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12941" PostTypeId="1" CreationDate="2021-03-03T18:56:04.340" Score="0" ViewCount="240" Body="&lt;p&gt;The mathematical concept of distance is fundamental in all mathematics and, since Bernard Riemann’s definition of manifolds, is also foundational in geometry and geometry of physics.&lt;/p&gt;&#xA;&lt;p&gt;Contrary to a widespread belief, probably encouraged by that particular distance known as Euclidean Distance,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I could not find a mathematical definition of distance in Euclid's Elements and&lt;/li&gt;&#xA;&lt;li&gt;I could not find any significant use of the word or of the concept except for Postulate 3, Book 1 (&amp;quot;To produce any circle with any centre and distance&amp;quot;) – postulate in which the word is used just in the common sense of “separation” (notes on postulate 3, p.199, “Euclid’s the Elements”, Sir T. L. Heath, Dover)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Euclid, instead, speaks and deals with mathematical length – a particular magnitude.&lt;/p&gt;&#xA;&lt;p&gt;I am not interested in the empirical and practical equivalence of length and distance.&#xA;I am concerned with their significant difference with respect to the foundations of geometry.&lt;/p&gt;&#xA;&lt;p&gt;I would appreciate some help on the following questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there specific references to the concept of mathematical distance in the Elements that I have been missing – may be because they are disguised in other wordings?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there any explicit discussions, in the Elements, that help to understand Euclid’s point of view on mathematical distance?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there any scholarly work in which the difference between distance and length has been analyzed with respect to the foundations of modern geometry – including historical analysis?&#xA;(I am aware some geometers work on “Length Geometry” but I am not aware if and where they have done any analysis of the kind above)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="14036" OwnerDisplayName="massimo" LastActivityDate="2021-03-08T06:21:06.457" Title="No distance in Euclid" Tags="|euclidean-geometry|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="12942" PostTypeId="2" ParentId="12940" CreationDate="2021-03-03T23:44:50.610" Score="2" Body="&lt;p&gt;I will describe the most important experimental data which led to&#xA;creation of quantum mechanics, in the chronologic order of their explanations, not the order of experiments.&lt;/p&gt;&#xA;&lt;p&gt;The idea of quanta was initially motivated by the theory of black body radiation. Plank derived his formula generalizing other formulas which came from experiments. To explain his empiric formula he had to invent the energy quanta (1900).&lt;/p&gt;&#xA;&lt;p&gt;Another phenomenon which was unexplained by classical physics was photoelectric effect. (This is probably the simplest experiment to explain to beginning students. Understanding this work of Einstein requires almost no background in physics or mathematics, unlike other work mentioned here). Einstein explained what we observe in photoelectric effect by using Plank's idea of quanta, and extending it to electromagnetic radiation (1905).&lt;/p&gt;&#xA;&lt;p&gt;Another experimental data which defied classical explanation were spectra of atoms. Bohr used the idea of quanta and explained the Balmer lines of hydrogen. Balmer discovered his empiric formula for the spectral lines in 1885 and Bohr &amp;quot;explained&amp;quot; it in 1913.&lt;/p&gt;&#xA;&lt;p&gt;These are three kinds of experimental data from which quantum mechanic was born, historically. All three named physicists (Planck, Einstein and Bohr) were eventually awarded Nobel prizes for these discoveries, but it took some time, until 1930s before quantum mechanics obtained its modern form&#xA;(Heisenberg, Born, Jordan, Dirac, Schrodinger and von Neumann).&lt;/p&gt;&#xA;&lt;p&gt;Slit experiments played a role later than the three pieces of experimental data mentioned above. There was nothing unusual for physicists, since 19th century in the slit experiments with light.&#xA;But discovery of electron diffraction in 1924 confirmed quantum mechanics.&lt;/p&gt;&#xA;&lt;p&gt;Even more important was the Stern-Gerlach experiment (1922) which led to the discovery of spin. This eventually led to the explanation of the most important experimental fact of all mentioned: the Periodic Table.&lt;/p&gt;&#xA;&lt;p&gt;Added on 3.4.2021. An outstanding exposition of the history of spectral lines is the article by S. Sternberg, A history of 19th century spectroscopy (one of the very best articles on history of science that I know). It is published as Appendix F to his book &amp;quot;Group theory and physics&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;On Stern-Gerlach (and Einstein-Podolsky-Rosen) there is a nice book by&#xA;Jim Baggott, &amp;quot;The meaning of quantum theory&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately I don't know any good exposition of Planck's discovery.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-03-04T20:59:46.897" LastActivityDate="2021-03-04T20:59:46.897" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12943" PostTypeId="2" ParentId="8360" CreationDate="2021-03-04T15:02:40.273" Score="4" Body="&lt;p&gt;Israel in the 60's: these experiments were probably from Gerald Stanhill (one of the discovers of global dimming). The &amp;quot;tubs&amp;quot; were probably class A &lt;a href=&quot;https://en.wikipedia.org/wiki/Pan_evaporation&quot; rel=&quot;nofollow noreferrer&quot;&gt;evaporation pans&lt;/a&gt; (a fancy name for tubs!). Some of his publications on this topic include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The control of field irrigation practice from measurements of evaporation (1962)&lt;/li&gt;&#xA;&lt;li&gt;The use of Class A evaporation pan data to estimate the irrigation water requirements of the cotton crop (1963)&lt;/li&gt;&#xA;&lt;li&gt;A field test of the control of cotton irrigation practice from Class A pan data (1964)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But all three were published in the &lt;em&gt;Israel journal of agricultural research&lt;/em&gt; and seem very hard to find online... You might have to try to get a paper copy from a science library, or maybe send him a request via &lt;a href=&quot;https://www.researchgate.net/profile/Gerald-Stanhill&quot; rel=&quot;nofollow noreferrer&quot;&gt;ResearchGate&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14043" LastActivityDate="2021-03-04T15:02:40.273" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12944" PostTypeId="2" ParentId="12940" CreationDate="2021-03-05T04:26:41.847" Score="0" Body="&lt;p&gt;There's no easy answer of course, things moved quite quickly at the turn of the century. Below, I'll give an overview of what was going on and what experents were being performed, but I'll have to defer to more substantial sources for any details.&lt;/p&gt;&#xA;&lt;p&gt;Experimental evidence begins, I would argue, with Hertz and Wallachs in the 1890s. (Experiments on black body radiation would of course be a more sound foundation, but the results of those experiments were less startling, although they may have been, if investigated more carefully.) Hertz discovered an effect of the ultraviolet light from a spark upon the intensity of a second spark near by, which his lab assistant (and successor of sorts) Wallachs further investigated. This phenomenon was originally called the Wallachs effect, but today we recognize it as early evidence for the photoelectric effect. Lenard would push the experimental evidence supporting this phenomenon further, and from there came the behavior which Einstein was attempting to explain in his 1905 paper.&lt;/p&gt;&#xA;&lt;p&gt;Spectroscopy, the investigation of the spectral lines of elements, was one of the most exciting areas of research. By taking a sample of e.g. hydrogen, excited to emitting visible light, we have our source; the light from the source is then targeted at a diffraction grating, and the resulting spectrum is viewed on a color scale of sorts. Such experiments were popular, and they were the source of much theoretical work. Particularly notable is the development of the Balmer series for the locations of spectral lines of hydrogen, but other phenomena like spectral line splitting (Zeeman effect) would be highly influential as well.&lt;/p&gt;&#xA;&lt;p&gt;Experiments on cathode rays were the basis of many important discoveries, but most significant is J. J. Thompson's experiment demonstrating the fact that the &amp;quot;rays&amp;quot; in fact contained electric charge, and thus were the flow of matter (electrons). The study of x-rays also developed from cathode ray tube experiments, and combined with the newly discovered phenomena of radioactivity, this opened the door to far more sophosticated models of atomic structure. Such models are essential to the development of quantum theory.&lt;/p&gt;&#xA;&lt;p&gt;At around the same time (late 1800s to 1900-1901) experimental data regarding black body radiation revealed limits to the models of the period, particularly Wein's law, and in response to this, Planck developed his famous theory of quanta. This is an instance where rash assumptions had to be made to obtain a suitable model of reality, but Planck made no claims of the physical 'truth' of his quanta, he only demonstrated a derivation which was far more successful than existing theories, with semi-heuristic reasoning. Even still, Einstein's 1905 paper on the photoelectric effect drew a lot from Planck's ideas.&lt;/p&gt;&#xA;&lt;p&gt;Experiments targeting the structure of matter allowed Bohr (among many others) to formulate his theories of the atom. Chief among them was Rutherford's gold foil experiment, an experiment made possible by the studies of radioactivity performed in the decade leading up. Spectroscopic experiments were also most influential for Bohr and his contemporaries. But early efforts to explain the behavior of the atom suffered from paradoxes. For example, the laws of electromagnetism forbade a charged particle to revolve in orbit around a nucleus without constantly radiating energy. The so-called 'old quantum theory' was built by embracing quantization in energy to explain this fact, working both from Planck's theory, and from experimental and theoretical work in spectroscopy.&lt;/p&gt;&#xA;&lt;p&gt;As the 1900s turned into the 1910s, very careful experiments on the photoelectric effect were verifying key aspects of the corpuscule-wave debate. Applications of classical mechanics, modified to allow for quantization, revealed some inadequacies in the working theories of Bohr et al. Efforts to combine Einstein's (highly successful but still developing) theory of the interaction between radiation and matter, with the known wave-like phenomena associated with light (particularly dispersion) inspired more abstract work, including the famous BKS theory, as well as important theoretical work from Kramers, Heisenberg, Born, Jordan, Dirac, and Schrodinger. But I digress, we've departed from the experiments.&lt;/p&gt;&#xA;&lt;p&gt;Many further experiments shaped the concepts of quantum mechanics. Some eliminated classical ideas, like Michelson-Morely; others demonstrated examples of quantization beyond energy, like the Stern-Gerlach experiment. A key piece of supporting evidence for Einstein's theory of photons was Compton's work on x-ray diffraction. Discoveries of protons, muons and pions, and eventually neutrons, began the development of the standard model of particle physics. There are really endless paths to follow, I'm sure I'm forgetting things, but this provides a rough chronology.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References and Suggested Reading&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Jammer, &lt;em&gt;The Conceptual Development of Quantum Mechanics&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Waerden, &lt;em&gt;Sources of Quantum Mechanics&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Segre, &lt;em&gt;From X-Rays to Quarks&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2021-03-05T04:26:41.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12945" PostTypeId="1" CreationDate="2021-03-06T08:28:03.063" Score="4" ViewCount="230" Body="&lt;p&gt;A question about the history of the maths classroom (which I hope isn't off topic).&lt;/p&gt;&#xA;&lt;p&gt;The idea of using a chalky stone to write graffiti diagrams on a dark plastered wall (marks that could be washed off and replaced) surely MUST have occurred early to ancient teachers of geometry in the academies. So much easier, you'd have thought, than drawing in the sand ....!&lt;/p&gt;&#xA;&lt;p&gt;OK, so when do we first hear about something &lt;em&gt;like&lt;/em&gt; blackboards??&lt;/p&gt;&#xA;" OwnerUserId="343" LastEditorUserId="2277" LastEditDate="2021-10-10T14:29:49.783" LastActivityDate="2021-10-10T14:29:49.783" Title="Blackboards as math tech!" Tags="|education|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="12946" PostTypeId="2" ParentId="12945" CreationDate="2021-03-06T21:00:42.930" Score="5" Body="&lt;p&gt;&lt;strong&gt;If we ignore the &amp;quot;wall&amp;quot; part&lt;/strong&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/XBRVf.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XBRVf.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&amp;quot;Pastel of Boy with Slate&amp;quot; 1822&lt;br /&gt;&#xA;Schoolboys would use these, since paper was too expensive to waste on practice or scratch work.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The exact origins of the writing slate remain unclear. References to its use can be found in the fourteenth century&lt;br /&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Slate_(writing)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;for the wall version&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the first mounted classroom chalkboard was pioneered in Scotland in the early 1800s,&lt;br /&gt;&#xA;...&#xA;Teachers would usually have to individually transcribe the problems onto each individual slate, which took up a lot of time. All of this changed when teachers began mounting bigger boards onto classroom walls.&lt;br /&gt;&#xA;&lt;a href=&quot;https://www.wisegeek.com/what-is-the-history-of-the-chalkboard.htm&quot; rel=&quot;noreferrer&quot;&gt;wisegeek&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This leap forward is widely believed to have first happened in a geography classroom in Edinburgh, Scotland; that teacher, James Pillans, is said to have taken a rough piece of raw slate and mounted it himself up on the wall behind his desk.&lt;br /&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Slate_(writing)&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2021-03-06T21:14:58.313" LastActivityDate="2021-03-06T21:14:58.313" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12947" PostTypeId="1" AcceptedAnswerId="12951" CreationDate="2021-03-07T04:26:03.950" Score="4" ViewCount="228" Body="&lt;p&gt;In the 17th century Netherlands, could the astronomers, or sailors trained in stellar navigation, predict either total or partial (at least 40% obscured) solar eclipses over the the town of Aardenburg (about halfway between Bruges and Vlissingen), in the Dutch province of Zeeland), six weeks (or more) in advance?&lt;/p&gt;&#xA;&lt;p&gt;While I've seen Edmond Halley being credited for being the first to accurately predict an eclipse in Europe, &amp;quot;Halley wasn’t the first to map a solar eclipse, either, with such charts having appeared since the 17th century.&amp;quot; &lt;a href=&quot;https://www.theguardian.com/science/the-h-word/2015/may/03/halleys-eclipse-newtonian-selling-science-history#:%7E:text=Halley%20wasn%27t%20the%20first,his%20use%20of%20Newton%27s%20theory.&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Guardian&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14055" LastEditorUserId="14055" LastEditDate="2021-03-07T17:32:24.470" LastActivityDate="2021-06-06T02:25:48.650" Title="Could 17th century astronomers in the Netherlands predict solar eclipses a few months in advance?" Tags="|physics|astronomy|17th-century|navigation|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="12948" PostTypeId="1" CreationDate="2021-03-07T06:55:38.677" Score="3" ViewCount="116" Body="&lt;p&gt;The interesting &lt;a href=&quot;http://www.cosmicelk.net/cosmicelk.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cosmic Elk&lt;/a&gt; article &lt;a href=&quot;http://www.cosmicelk.net/eclipsesinSiam.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eclipses in Siam (now Thailand) History and Legends&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While the Siamese ambassadors and their entourage were visiting ship yards&#xA;and armouries and making careful notes of all they saw, preparations were being made for the return journey in October.&lt;/p&gt;&#xA;&lt;p&gt;Uniforms for 500 soldiers were provided, and 1800 pairs of shoes for them. The number of troops was 600, with 160 cannon – including those requested by King Narai plus more cannon used as ballast on the ships, together with 2,000 cannon balls, 2,000 grenades, gunpowder for cannons and for 300 muskets, and this was only part of the armaments sent with the ships. The expedition was led by Simon de la Loubère a diplomat and Claude Céberet du Boullay a director of the Compagnie des Indes Orientales (CIO). Tachard was going back carrying out instructions from Phaulkon.&lt;/p&gt;&#xA;&lt;p&gt;The ships were also laden with gifts for King Narai, including carpets, elephant harnesses and decorations, a celestial and a terrestrial globe both specially printed in Siamese, telescopes and other scientific instruments and 4,264 mirrors. Phaulkon was to be made a French Count and given the Order of St. Michel. His wife was to receive a harpsichord and three musicians – equipped with the scores of Lully, Louis XIV's favourite composer.&lt;/p&gt;&#xA;&lt;p&gt;1,361 passengers in addition to crews and soldiers were loaded onto the Oiseau, Maligne, Gallard and Normande, Loire and Dromadaire, plus three store ships. Including General Desfarges, his two sons, his officers and his bombardiers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;They finally left France in March 1687. On the 11th May there was an eclipse of the sun, observed by the Siamese and the Jesuits, and they passed the Tropic of Capricorn. The voyage did not go well.&lt;/strong&gt; Because of the Revocation of the Edict of Nantes the French were not welcome at the Dutch ports on which they depended for stop-overs, and there was a lot of sickness on board the overcrowded ships. The Maligne was sent back from the Cape and the other five ships went on to Siam, arriving late September.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This sounds like quite a big deal and observing a solar eclipse must have been ancillary, assuming that they even knew about it.&lt;/p&gt;&#xA;&lt;p&gt;Below I show a screenshot of (probably) the path of totality; NASA warns that the plot is for &amp;quot;development purposes only&amp;quot; :-) But it would have been visible as a partial solar eclipse and viewable/enjoyable by pinhole or eyepiece projection or anyone &lt;a href=&quot;https://www.youtube.com/watch?v=CCqwJ7E45W8&quot; rel=&quot;nofollow noreferrer&quot;&gt;unwie enough&lt;/a&gt; to look directly at the Sun.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Was observation of the May 11, 1687 solar eclipse &lt;em&gt;expected&lt;/em&gt; by the Jesuits and Siamese dignitaries aboard this ship? Was observing the eclipse in some scientific way part of the plan, or was this &amp;quot;just one of those things&amp;quot; that happen?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://eclipse.gsfc.nasa.gov/SEsearch/SEsearchmap.php?Ecl=16870511&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://eclipse.gsfc.nasa.gov/SEsearch/SEsearchmap.php?Ecl=16870511&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_solar_eclipses_in_the_17th_century&quot; rel=&quot;nofollow noreferrer&quot;&gt;List of Solar Eclipses in the 17th Century&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xm6rT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xm6rT.jpg&quot; alt=&quot;NASA solar eclipse May 11, 1687&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-03-07T07:06:27.873" LastActivityDate="2021-03-07T07:06:27.873" Title="Did a boat full of Louis XIV's Jesuits and some Siamese dignitaries plan on seeing a solar eclipse on May 17, 1687" Tags="|astronomy|17th-century|france|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12950" PostTypeId="2" ParentId="12947" CreationDate="2021-03-07T18:10:01.830" Score="-1" Body="&lt;p&gt;In short, I don't know.&lt;/p&gt;&#xA;&lt;p&gt;One problem in the 17th century was that astronomers didn't know the scale of the Sun-Earth-Moon system.  The approximate sizes of the Earth and the Moon and their approximate average distance had been known since antiquity, but I doubt whether the exact distance between the Moon and a point on the Earth's surface at any specific time could be precdicted very accurately.  And more important, it was known that the Sun was many times farther than the Moon, but there was not yet any reasonably accurate measurement of the Astronomical Unit, the distance between the Sun and the Earth.  This meant that astronomers couldn't use geometry to predict the width of the Moon's shadow on Earth.&lt;/p&gt;&#xA;&lt;p&gt;They could use reports of the width of the Moon's shadow in previous eclipses, but that varied according to the exact distances between the Earth, hte Moon, and the Sun in each eclipse, and those distances vary due to the ellipical orbits of the Earth and the Moon.&lt;/p&gt;&#xA;&lt;p&gt;Some information about the history of eclipse predicitons.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the first accounts of an eclipse—though one should note that this might be an old wives tale—dates back to 2136 B.C. Legend has it that the Emperor Chung K’ang executed his royal astronomers, Hi and Ho, for failing to predict an eclipse.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.history.com/news/how-5-ancient-cultures-explained-solar-eclipses%5B1%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.history.com/news/how-5-ancient-cultures-explained-solar-eclipses[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The eclipse of Thales was a solar eclipse that was, according to The Histories of Herodotus, accurately predicted by the Greek philosopher Thales of Miletus. If Herodotus's account is accurate, this eclipse is the earliest recorded as being known in advance of its occurrence. Many historians believe that the predicted eclipse was the solar eclipse of 28 May 585 BC.&lt;a href=&quot;https://en.wikipedia.org/wiki/Eclipse_of_Thales#:%7E:text=If%20Herodotus%27s%20account%20is%20accurate%2C%20this%20eclipse%20is,assert%20the%20eclipse%20was%20never%20predicted%20at%20all.&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; How exactly Thales predicted the eclipse remains uncertain; some scholars assert the eclipse was never predicted at all.&lt;a href=&quot;https://en.wikipedia.org/wiki/Antikythera_mechanism&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/March_1504_lunar_eclipse&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt; Others have argued for different dates,[6] but only the eclipse of 28 May 585 BC matches the conditions of visibility necessary to explain the historical event.[7]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Eclipse_of_Thales#:%7E:text=If%20Herodotus%27s%20account%20is%20accurate%2C%20this%20eclipse%20is,assert%20the%20eclipse%20was%20never%20predicted%20at%20all.%5B2%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Eclipse_of_Thales#:~:text=If%20Herodotus%27s%20account%20is%20accurate%2C%20this%20eclipse%20is,assert%20the%20eclipse%20was%20never%20predicted%20at%20all.[2]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Answers to this question discuss the history of eclipse predictions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://history.stackexchange.com/questions/16959/what-was-the-first-solar-eclipse-that-was-demonstrably-predicted-in-advance%5B4%5D&quot;&gt;https://history.stackexchange.com/questions/16959/what-was-the-first-solar-eclipse-that-was-demonstrably-predicted-in-advance[4]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Antikythera mechanism (/ˌæntɪkɪˈθɪərə/ AN-tih-kih-THEER-ə) is an ancient Greek hand-powered orrery, described as the first analogue computer,&lt;a href=&quot;https://www.history.com/news/how-5-ancient-cultures-explained-solar-eclipses&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; the oldest known example of such a device&lt;a href=&quot;https://history.stackexchange.com/questions/16959/what-was-the-first-solar-eclipse-that-was-demonstrably-predicted-in-advance&quot;&gt;2&lt;/a&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/March_1504_lunar_eclipse&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt;[6][7][8][9][10][excessive citations] used to predict astronomical positions and eclipses for calendar and astrological purposes decades in advance.[11][12][13] It could also be used to track the four-year cycle of athletic games which was similar to an Olympiad, the cycle of the ancient Olympic Games.[14][15][16]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Antikythera_mechanism%5B3%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Antikythera_mechanism[3]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A total lunar eclipse occurred on 1 March 1504, visible at sunset for the Americas, and later over night over Europe and Africa, and near sunrise over Asia.&lt;/p&gt;&#xA;&lt;p&gt;Christopher Columbus, in an effort to induce the natives of Jamaica to continue provisioning him and his hungry men, successfully intimidated the natives by correctly predicting a total lunar eclipse for 1 March 1504 (visible on the evening of 29 February in the Americas). Some have claimed that Columbus used the Ephemeris of the German astronomer Regiomontanus,&lt;a href=&quot;https://www.history.com/news/how-5-ancient-cultures-explained-solar-eclipses&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; but Columbus himself attributed the prediction to the Almanach by Abraham Zacuto.&lt;a href=&quot;https://en.wikipedia.org/wiki/Eclipse_of_Thales#:%7E:text=If%20Herodotus%27s%20account%20is%20accurate%2C%20this%20eclipse%20is,assert%20the%20eclipse%20was%20never%20predicted%20at%20all.&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On 30 June 1503, Christopher Columbus beached his two last caravels and was stranded in Jamaica. The indigenous people of the island welcomed Columbus and his crew and fed them, but after six months, the natives halted the food supply.&lt;a href=&quot;https://history.stackexchange.com/questions/16959/what-was-the-first-solar-eclipse-that-was-demonstrably-predicted-in-advance&quot;&gt;4&lt;/a&gt; Columbus had on board an almanac authored by Abraham Zacuto of astronomical tables covering the years 1475–1506.&lt;a href=&quot;https://en.wikipedia.org/wiki/March_1504_lunar_eclipse&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt;[6] Upon consulting the book, he noticed the date and the time of an upcoming lunar eclipse. He was able to use this information to his advantage. He requested a meeting for that day with the Cacique, the leader, and told him that God was angry with the local people's treatment of Columbus and his men[citation needed]. Columbus said God would provide a clear sign of displeasure by making the rising full Moon appear &amp;quot;inflamed with wrath&amp;quot;[citation needed].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/March_1504_lunar_eclipse%5B5%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/March_1504_lunar_eclipse[5]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Of course this famous story is about a lunar eclipse visible from an entire hemisphere at any one moment, and not a solar eclipse visible only over a long, narrow path. Predicting where a solar eclipse will be visible is much more difficult.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2021-03-07T18:10:01.830" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12951" PostTypeId="2" ParentId="12947" CreationDate="2021-03-08T01:19:56.513" Score="2" Body="&lt;p&gt;The short answer is yes. Under the following conditions:&lt;/p&gt;&#xA;&lt;p&gt;Let me state the problem more exactly: suppose that a solar eclipse of at least 40% WILL occur within a year from now. Could some person in the second half of 17th century make this prediction with 100% probability?&lt;/p&gt;&#xA;&lt;p&gt;a) prediction for short period (few months, perhaps up to a year).&lt;/p&gt;&#xA;&lt;p&gt;b) possession of a good angle measuring instrument (they were not common in 17th century, they started to make them in the middle of 17 century, and they were custom made. (The Royal observatory was founded in 1675, and this was the first such establishment in Europe). And certainly few people possessed such instruments. (They started using them in navigation only in the middle of 18th century).&lt;/p&gt;&#xA;&lt;p&gt;c) very few people had necessary knowledge, certainly not common navigators, not speaking of astrologists. One of such people was Edmund Halley, by the way, I can name few others: Newton, Flamsteed (Astronomer Royal), Thomas Harriot (1560-1621), perhaps 10 or 20 others in Europe by the end of 17th century. There is no doubt that a few of them were in Netherlands.&lt;/p&gt;&#xA;&lt;p&gt;In principle, the motion of Sun and Moon with necessary accuracy for such predictions was known&#xA;at the time of Ptolemy, but his parameters were not exact. One needed a good instrument to measure these parameters. Published tables of sufficient accuracy were not available yet. (They appeared only in 18th century, for example the tables produced by the same Royal observatory, or analogous tables in France).&lt;/p&gt;&#xA;&lt;p&gt;So the answer is yes, this was possible in principle, in the second half of the century, but required a very special person with special knowledge, possessing rare and expensive instruments.&lt;/p&gt;&#xA;&lt;p&gt;Remark. It is very different for Lunar eclipses, which could be predicted since the times of Ptolemy.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-03-08T12:14:29.330" LastActivityDate="2021-03-08T12:14:29.330" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12952" PostTypeId="2" ParentId="2460" CreationDate="2021-03-08T05:15:00.370" Score="2" Body="&lt;p&gt;Everyone makes mistakes. Look at Newton, even though he knew perfectly well that action at a distance was philosophically speaking, nonsense, he still went with it because he could see no way past this impasse. It was only after Einstein we see where the mistake is.&lt;/p&gt;&#xA;&lt;p&gt;Neverthless, we don't belittle Newton for not inventing general relativity, and celebrate what he did get right: his three laws of motion, his gravitational theory and the calculus.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, rather than belittling Aristotle for what he got wrong, we should celebrate what he got right. Carlo Rovelli, in his book, &lt;em&gt;Quantum Gravity&lt;/em&gt;, stated that it was Descartes that should be honoured for forcefully enunciating the principle of locality in physics.&lt;/p&gt;&#xA;&lt;p&gt;He's right on that this notion is central to physics, but wrong about who should get credit for this notion. Aristotle, long before Descartes, stated that change only occurs on the application of a force, and only then if the substance on which the force is applied has the potential to change and actually does change. His careful language around this principle has all the hallmarks of careful scientific thinking.&lt;/p&gt;&#xA;&lt;p&gt;This, by the way, is more or less the essence of Newtons first law of motion: that an object will carry on in straight line motion unless a force is applied.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-08T05:15:00.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12953" PostTypeId="2" ParentId="5221" CreationDate="2021-03-08T05:43:38.753" Score="0" Body="&lt;p&gt;I recently looked at Grothendieck's Kansas report on the general structure theory of fibre spaces. I don't recall there being much use of categorical thinking there. It may have been used in passing or in phrasing, if that.&lt;/p&gt;&#xA;&lt;p&gt;We don't know where exactly Grothendieck discovered the language of category theory. It has to be before 1965 when he gave a lecture course on &amp;quot;Introduction au langage functorial&amp;quot; in Algiers, of which Max Karoubi took down typed notes. The name of the course speaks for itself.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-08T05:43:38.753" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12954" PostTypeId="2" ParentId="12941" CreationDate="2021-03-08T06:21:06.457" Score="1" Body="&lt;p&gt;The concept of distance is far from being fundamental to mathematics, in the sense that fundamental is foundational. After all, nothing in set theory presupposes distance, and topology dispenses with the notion altogether.&lt;/p&gt;&#xA;&lt;p&gt;Let us say that the notion of distance has been important in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;In Euclid, the theory of measurement, ratio and proportion are in the fifth, sixth, seventh and tenth books. The fifth book, in particular,  is devoted to theory of ratio and proportion of magnitude in general and the sixth book applies this theory to plane geometry.&lt;/p&gt;&#xA;&lt;p&gt;According to the commentator Proclus, the theory of ratio and proportion was actually pioneered by Eudoxus a century before Euclid.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-08T06:21:06.457" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12955" PostTypeId="2" ParentId="5852" CreationDate="2021-03-08T09:19:32.230" Score="0" Body="&lt;p&gt;It's probably best not to see science and physics in nationalist terms. What's more important was the revival of science in the modern era in Europe. This was preceded by a renaissance in science in the Islamic world. In particular by Ibn Rushd (Averroes) and Ibn Sina (Avicenna) who wrote widely on the philosophy of Aristotle. There was in fact a movement called Averroism in Europe which persisted up until the sixteenth century which shows just how widespread their influence was (Averroes himself lived in the 12th C).&lt;/p&gt;&#xA;&lt;p&gt;Whilst both Newton &amp;amp; Liebniz both discovered the calculus, it was only Newton that came up with a quantifiable theory of gravity. Before him, there was only a qualitative theory, for example Kepler noted the universality of gravitational attraction. As Newton was the first to come up with a quantified law of a physical force this helped propel physics into a newfounded maturity in Europe.&lt;/p&gt;&#xA;&lt;p&gt;Whilst Einstein himself dated the scientific revolution from Galileo, it's clear from the book, &lt;em&gt;The Forgotten Revolution&lt;/em&gt; by Russo, that the modern science owed a great deal from ancient science. This isn't often acknowledged in modern physics courses but I imagine this will change in the future as a more inclusive history of science is written.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-12T05:50:29.310" LastActivityDate="2021-03-12T05:50:29.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12956" PostTypeId="2" ParentId="12933" CreationDate="2021-03-08T09:34:02.583" Score="0" Body="&lt;p&gt;You make it sound as though Stephen Weinberg's work on unifying the electromagnetic force with the weak force was a piece of cake and hardly worth winning the Nobel prize along with Abdus Salam and Sheldon Lee Glashow in 1979.&lt;/p&gt;&#xA;&lt;p&gt;Is that not enough?&lt;/p&gt;&#xA;&lt;p&gt;Abdus Salam was known for also pioneering supersymmetry and also for establishing the Trieste Centre of Physics to help develop the physics in the developing countries.&lt;/p&gt;&#xA;&lt;p&gt;Along with James Bjorken, Sheldon Glashow predicted the existemce of a fourth quark, the charmed quark. Also along with Howard Georgi, he came up with first example of a grand unified theory.&lt;/p&gt;&#xA;&lt;p&gt;Weinberg is known for pioneering renormalisation methods in physics and also the one example of a prediction based on the anthropic principle that worked, the anthropic bound on the cosmological constant.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-08T09:34:02.583" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12957" PostTypeId="1" CreationDate="2021-03-09T09:48:48.517" Score="0" ViewCount="87" Body="&lt;p&gt;&lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=140927&quot; rel=&quot;nofollow noreferrer&quot;&gt;Derek Oppen&lt;/a&gt; was a student of Stephen Cook known for the &lt;a href=&quot;https://www.lsi.upc.edu/%7Eoliveras/TDV/NO.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nelson-Oppen combination of decision procedures method&lt;/a&gt;. Yet, there is no Wikipedia entry dedicated to him and his &lt;a href=&quot;https://dblp.org/pid/36/2612.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;publication list&lt;/a&gt; extends during a single decade.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering if there are any references about his academic career. To be precise, I'm interested on how the collaboration between Nelson and Oppen happened. My understanding is that &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=87404&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nelson&lt;/a&gt; was still a PhD student and &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=140927&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oppen&lt;/a&gt; had recently graduated.&lt;/p&gt;&#xA;" OwnerUserId="7165" LastActivityDate="2021-03-09T09:48:48.517" Title="Derek Oppen and the Nelson-Oppen combination procedure" Tags="|computer-science|computer-scientists|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12958" PostTypeId="1" CreationDate="2021-03-09T22:30:04.670" Score="2" ViewCount="53" Body="&lt;p&gt;This question comes from a point of view of a student in current times. The general curriculum for a physics student presents that a static analysis of a simple system requires that the forces in each direction sum to zero (i.e. the object does not move). The explanation that comes generally is that this is simple implementation of Newton's first law. However, it is often times, when considering truss systems for example, that Newton's first law is not sufficient to provide enough equations and we must supplement the law of moments: around every point within a non-rotating object, the sum of moments should be equal to zero.&lt;/p&gt;&#xA;&lt;p&gt;I understand that this was essentially known by Archimedes (and I wonder, why don't we call it Archimedes' law?). However, this does not get the same reputation, in my eyes, as Newton's first law (which seems a foundation of modern physics).&lt;/p&gt;&#xA;&lt;p&gt;I was wondering whether the principle of moments, as a tool in engineering and in mathematics, was known as a &amp;quot;law&amp;quot; (equivalent and on the same standing as we regard Newton's laws)?&lt;/p&gt;&#xA;" OwnerUserId="14064" LastEditorUserId="10106" LastEditDate="2021-03-10T02:36:53.410" LastActivityDate="2021-03-10T02:36:53.410" Title="Did Newton discuss and/or use the principle of moments?" Tags="|physics|theoretical-physics|newton|archimedes|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12959" PostTypeId="1" CreationDate="2021-03-07T07:32:55.410" Score="10" ViewCount="241" Body="&lt;p&gt;It seems G. H. Hardy once wrote that &amp;quot;The problem of deciding whether two given limit operations are commutative is one of the most important in mathematics&amp;quot;. I was wondering what led to the &lt;em&gt;need&lt;/em&gt; to formulate the question of when the limit operations commute.&lt;/p&gt;&#xA;" OwnerDisplayName="A.D." LastActivityDate="2021-03-14T11:50:45.330" Title="Historically, what led to the question of the validity of interchange of limit operations?" Tags="|real-analysis|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12960" PostTypeId="2" ParentId="12959" CreationDate="2021-03-07T14:44:48.503" Score="4" Body="&lt;p&gt;Once you're working with &lt;span class=&quot;math-container&quot;&gt;$\ge 2$&lt;/span&gt; independent variables, it's necessary anytime you need to be able to commute differential/integral operators. With differentials, for example, we need it in order to change the order in which we take partial derivatives:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \frac{\partial }{\partial x}\left( \frac{\partial f}{\partial y}\right) = \frac{\partial }{\partial y}\left( \frac{\partial f}{\partial x}\right) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;With integrals, it's necessary anytime we want to take double (or larger) integrals: &lt;span class=&quot;math-container&quot;&gt;$$ \int_{X \times Y} f(x,y) \ d(x,y) = \int_X \left( \int_Y f(x,y) \ dy \right) dx = \int_Y \left( \int_X f(x,y) \ dx \right) dy $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It also plays a role when working with limits of derivatives or limits of integrals, as for example in Hardy's &lt;a href=&quot;https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;own work&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14065" OwnerDisplayName="Amaan M" LastActivityDate="2021-03-07T16:03:39.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12961" PostTypeId="2" ParentId="12959" CreationDate="2021-03-07T17:22:11.403" Score="8" Body="&lt;p&gt;G. H. Hardy wrote in &lt;em&gt;&lt;a href=&quot;https://www.maa.org/press/maa-reviews/a-course-of-pure-mathematics-centenary-edition&quot; rel=&quot;noreferrer&quot;&gt;A Course in Pure Mathematics&lt;/a&gt;&lt;/em&gt;  in Appendix II, section &lt;em&gt;A note on double limit problems&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Let us consider some special instances. In &lt;span class=&quot;math-container&quot;&gt;$\S\ 213$&lt;/span&gt; we proved that&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\log(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\cdots&#xA;\end{align*}&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$-1&amp;lt;x\leq 1$&lt;/span&gt;, by integrating the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;1/(1+t)=1-t+t^2-\cdots&#xA;\end{align*}&lt;/span&gt;&#xA;between the limits &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. What we proved amounted to this, that&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\int_{0}^{x}\frac{dt}{1+t}=\int_{0}^{x}\,dt-\int_{0}^{x}t\,dt+\int_{0}^{x}t^2\,dt-\ldots&#xA;\end{align*}&lt;/span&gt;&#xA;or in other words that &lt;em&gt;the integral of the sum of the infinite series &lt;span class=&quot;math-container&quot;&gt;$1-t+t^2-\cdots$&lt;/span&gt; taken between the limits &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, is equal to the sum of the integrals of its terms taken between the same limits.&lt;/em&gt; Another way of expressing this fact is to say that the operations of summation from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;, and of integration from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, are &lt;strong&gt;commutative&lt;/strong&gt; when applied to the function &lt;span class=&quot;math-container&quot;&gt;$(-1)^nt^n$&lt;/span&gt;, i.e. that it does not matter in what order they are performed on the function.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;he continues with another important example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;... we proved that the differential coefficient of the exponential function&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\exp x=1+x+\frac{x^2}{2!}+\cdots&#xA;\end{align*}&lt;/span&gt;&#xA;is itself equal to &lt;span class=&quot;math-container&quot;&gt;$\exp x$&lt;/span&gt;, or that&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;D_x\left(1+x+\frac{x^2}{2!}+\cdots\right)=D_x1+D_xx+D_x\frac{x^2}{2!}+\cdots;&#xA;\end{align*}&lt;/span&gt;&#xA;that is to say that &lt;em&gt;the differential coefficient of the sum of the series is equal to the sum of the differential coefficients of its terms&lt;/em&gt;, or that the operations of summations from &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; and of differentiation with respect to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; are &lt;strong&gt;commutative&lt;/strong&gt; when applied to &lt;span class=&quot;math-container&quot;&gt;$x^n/n!$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The author accompanies these examples with some more and concludes then in convincing manner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The really important case (as is suggested by the instances which we gave from Ch. IX) is that in which each operation is one which involves a passage to the limit, such as a differentiation or the summation of an infinite series: such operations are called &lt;em&gt;limit operations&lt;/em&gt;. The general question as to the circumstances in which two given limit operations are &lt;strong&gt;commutative&lt;/strong&gt; is one of the most important in all mathematics. ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We can already defer from the examples above that exchanging limits is of fundamental importance and seemingly ubiquituous in analysis. So, the question for historical origin of the need of exchanging limits is rather hard to answer.&lt;/p&gt;&#xA;&lt;p&gt;Maybe one could trace back some lines of history by looking at the history of &lt;em&gt;named theorems&lt;/em&gt; which are based upon the exchange of limits, like &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fubini%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;Fubini's Theorem&lt;/a&gt;&lt;/em&gt;&#xA;,&lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives&quot; rel=&quot;noreferrer&quot;&gt;Theorem of Schwarz&lt;/a&gt;&lt;/em&gt;, etc.&lt;/p&gt;&#xA;" OwnerUserId="4084" OwnerDisplayName="Markus Scheuer" LastActivityDate="2021-03-07T17:22:11.403" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12962" PostTypeId="2" ParentId="12959" CreationDate="2021-03-07T19:33:30.583" Score="1" Body="&lt;p&gt;Here is one example. The question of whether limits commute arises when we ask whether the limit of a sequence of continuous functions is continuous. If each &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; is continuous, and &lt;span class=&quot;math-container&quot;&gt;$f_n \to f$&lt;/span&gt; pointwise, then it’s not necessarily true that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous. But the result is true if the convergence is uniform. With uniform convergence, we can interchange limits.&lt;/p&gt;&#xA;&lt;p&gt;From Walter Rudin’s &lt;em&gt;Principles of Mathematical Analysis&lt;/em&gt;, 3rd edition, p.144:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;      The main problem which arises is to determine whether important properties of functions are preserved under the limit operations (1) and (2). For instance, if the functions &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; are continuous, or differentiable, or integrable, is the same true of the limit function? What are the relations between &lt;span class=&quot;math-container&quot;&gt;$f_n’$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f’$&lt;/span&gt;, say, or between the integrals of &lt;span class=&quot;math-container&quot;&gt;$f_n$&lt;/span&gt; and that of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;      To say that &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is continuous at &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; means&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{t \to x} f(t) = f(x).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence, to ask whether the limit of a sequence of continuous functions is continuous is the same as to ask whether&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{t \to x} \lim_{n \to \infty} f_n(t) = \lim_{n \to \infty} \lim_{t \to x} f_n(t),&#xA;$$&lt;/span&gt;&#xA;i.e., whether the order in which the limit processes are carried out is immaterial.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the Wikipedia page on uniform convergence, there is a historical note:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="twosigma" LastActivityDate="2021-03-07T19:33:30.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12963" PostTypeId="1" CreationDate="2021-03-10T03:22:45.547" Score="4" ViewCount="127" Body="&lt;p&gt;I was recently reading an essay by Churchill titled “Fifty Years Hence” from 1932, wherein he discusses the inevitability that scientists will &amp;quot;spark a nuclear bonfire&amp;quot;, he was apparently taught the basic Physics by his friend Frederick Lindemann. He also discussed it in his essay &amp;quot;Shall We All Commit Suicide&amp;quot; from 1927 in much more vague terms; about how scientific supremacy in the future could destroy liberal societies, with no mention of the science itself.&lt;/p&gt;&#xA;&lt;p&gt;Was this the first instance of a Politician(or any Non-Scientist) coming to terms with the reality/inevitability of weapons of mass destruction? I can't seem to think of anyone who would have had such foresight in the 1920s; when the idea of another world conflict was beyond the pale.&lt;/p&gt;&#xA;" OwnerUserId="13800" LastEditorUserId="13800" LastEditDate="2021-03-10T17:40:25.457" LastActivityDate="2021-03-10T18:00:01.950" Title="First Recognition of the Atom Bomb" Tags="|world-war-2|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12964" PostTypeId="2" ParentId="12963" CreationDate="2021-03-10T13:01:31.067" Score="1" Body="&lt;p&gt;I read that column, and noticed there is no mention of fission, only fusion:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nuclear energy is incomparably greater than the molecular energy which&#xA;we use today. The coal a man can get in a day can easily do five&#xA;hundred times as much work as the man himself. Nuclear energy is at&#xA;least one million times more powerful still. If the hydrogen atoms in&#xA;a pound of water could be prevailed upon to combine together and form&#xA;helium, they would suffice to drive a thousandhorsepower engine for a&#xA;whole year. If the electrons, those tiny planets of the atomic&#xA;systems, were induced to combine with the nuclei in the hydrogen the&#xA;horsepower liberated would be 120 times greater still.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Further, my take is that he's not primarily warning of war, but rather commenting on the pace of technological development and its effect on peacetime society.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-03-10T13:01:31.067" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="12965" PostTypeId="2" ParentId="12963" CreationDate="2021-03-10T18:00:01.950" Score="2" Body="&lt;p&gt;You might want to look up answers to:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scifi.stackexchange.com/questions/241706/first-atomic-powered-transportation-in-science-fiction%5B1%5D&quot;&gt;https://scifi.stackexchange.com/questions/241706/first-atomic-powered-transportation-in-science-fiction[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scifi.stackexchange.com/questions/2742/earliest-story-about-civil-nuclear-power/43647#43647%5B2%5D&quot;&gt;https://scifi.stackexchange.com/questions/2742/earliest-story-about-civil-nuclear-power/43647#43647[2]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When scientists realized how much binding energy was in atoms, and when Einstein's theory of specal relativity included the equation E = mc2, showing that mass and energy were the same, it became natural to speculate whether it could be possible to release the energy of atoms somehow and produce vast amounts of energy.&lt;/p&gt;&#xA;&lt;p&gt;So there were early science fiction stories using types of atomic energy, which seem more or less plausible - often much less plausible - according to modern knowledge, since the early 1900s at least.&lt;/p&gt;&#xA;&lt;p&gt;One example is H.G. Wells's 1914 Novel &lt;em&gt;The World Set Free&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wells's knowledge of atomic physics came from reading William Ramsay, Ernest Rutherford, and Frederick Soddy; the last discovered the disintegration of uranium. Soddy's book Wealth, Virtual Wealth and Debt praises The World Set Free. Wells's novel may even have influenced the development of nuclear weapons, as the physicist Leó Szilárd read the book in 1932, the same year the neutron was discovered. In 1933 Szilárd conceived the idea of neutron chain reaction, and filed for patents on it in 1934.[8]&lt;/p&gt;&#xA;&lt;p&gt;Wells's &amp;quot;atomic bombs&amp;quot; have no more force than ordinary high explosive[dubious – discuss] and are rather primitive devices detonated by a &amp;quot;bomb-thrower&amp;quot; biting off &amp;quot;a little celluloid stud.&amp;quot;[9] They consist of &amp;quot;lumps of pure Carolinum&amp;quot; that induce &amp;quot;a blazing continual explosion&amp;quot; whose half-life is seventeen days, so that it is &amp;quot;never entirely exhausted,&amp;quot; so that &amp;quot;to this day the battle-fields and bomb fields of that frantic time in human history are sprinkled with radiant matter, and so centres of inconvenient rays.&amp;quot;[10]&lt;/p&gt;&#xA;&lt;p&gt;Never before in the history of warfare had there been a continuing explosive; indeed, up to the middle of the twentieth century the only explosives known were combustibles whose explosiveness was due entirely to their instantaneousness; and these atomic bombs which science burst upon the world that night were strange even to the men who used them.[11]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_World_Set_Free%5B3%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/The_World_Set_Free[3]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;With the coming of pulp science fiction magazines beginning in 1926, stories about peaceful or warlike use of atomic power became rather common, such as E.E. Smith's &lt;em&gt;The Skylark of Space&lt;/em&gt; (1928), begun in 1915.&lt;/p&gt;&#xA;&lt;p&gt;So the phrase &amp;quot;atomic bomb&amp;quot; was used in some early stories.&lt;/p&gt;&#xA;&lt;p&gt;For example, atomic bombs are used in Stanley G. Weibaum's short story &amp;quot;Redemption Cairn&amp;quot;, &lt;em&gt;Astounding Stories&lt;/em&gt;, March, 1936.  Those were about the size of hand grenades, and much less powerful than real atomic bombs, and apparently were available for civilian as well as military use.&lt;/p&gt;&#xA;&lt;p&gt;But nobody had any idea how to make atomic bombs in real life, until uranium fission was discoverd in 1938, and in the next few years various governments began projects to develop fission reactors for energy production and/or atomic bombs for war.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2021-03-10T18:00:01.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12966" PostTypeId="2" ParentId="5977" CreationDate="2021-03-12T06:49:53.837" Score="1" Body="&lt;p&gt;It is an apocraphal story, so it's difficult to know just how much credence one should place on it. There is a similar story of when Grothendieck was asked to give a prime number, he suggested 57 - which isn't prime.&lt;/p&gt;&#xA;&lt;p&gt;These stories may have been true but they equally may not have been true. What they're pointing to is a mismatch between Grothendiecks heavily abstract mathematics and the nuts and bolts of the everyday maths that we use.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-12T06:49:53.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12967" PostTypeId="2" ParentId="12959" CreationDate="2021-03-12T07:02:41.673" Score="1" Body="&lt;p&gt;The finite case is the simplest and is easily seen by observation. If we set out an array of pebbles with differing column and rows sizes, we can see just by observation that the total number of pebbles must be the same whether we count by rows or by columns.&lt;/p&gt;&#xA;&lt;p&gt;This would be appear to generalise to the infinite case where we take a pear and infinitely slice in the vertical and horizontal case. Then adding up the infinitesimal cubes first in the horizontal direction and then in the vertical should gives us back tge pear; lilewise when the two operations are swapped, that is we first add the infinitesinal cubes in the vertical and then the horizontal directions.&lt;/p&gt;&#xA;&lt;p&gt;However, this is in fact no longer true in the infinite case, at least without qualification. This is where analysis comes in, and specifically, the Fubini-Tonelli theorem on swapping limits. Since Hardy was an analyst, it may have been this theorem he was thinking off.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-14T11:50:45.330" LastActivityDate="2021-03-14T11:50:45.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12968" PostTypeId="1" CreationDate="2021-03-13T10:50:12.020" Score="4" ViewCount="232" Body="&lt;p&gt;Infinitesimal calculus and the introduction of derivatives is often linked to Newton and Leibniz.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering, when and why the idea of studying formal derivatives (e.g., of a formal polynomial) emerged?&lt;/p&gt;&#xA;&lt;p&gt;For instance, what were the motivations to introduce derivatives to study things over finite fields and their extensions (where the visual intuition of infinitesimal as linked to Newton and Leibniz is lost)?&lt;/p&gt;&#xA;" OwnerUserId="14079" LastEditorUserId="10106" LastEditDate="2021-03-13T18:10:07.643" LastActivityDate="2023-07-21T17:06:00.040" Title="How did the idea of a formal derivation emerge?" Tags="|mathematics|calculus|newton|leibniz|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12969" PostTypeId="2" ParentId="12968" CreationDate="2021-03-13T14:38:04.620" Score="10" Body="&lt;p&gt;The motivation for applying derivatives to polynomials over general fields is their use in detecting multiple roots: if &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is a general field, a polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$K[x]$&lt;/span&gt; has no repeated roots (edit: in a field extension of &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; over which &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; splits completely) if and only if &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is relatively prime to &lt;span class=&quot;math-container&quot;&gt;$f'(x)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$K[x]$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Formal derivatives on polynomials in &lt;span class=&quot;math-container&quot;&gt;$K[x]$&lt;/span&gt; for a general field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; were introduced by Steinitz in Section 9 of his big 1910 paper on &lt;a href=&quot;https://eudml.org/doc/149323&quot; rel=&quot;noreferrer&quot;&gt;the algebraic theory of fields&lt;/a&gt; and used in Section 10 in order to determine when a polynomial (especially an irreducible polynomial) over a general field has a multiple root. (Perfect fields are defined in Section 11 and separable field extensions are defined in Section 13 as extensions of the &amp;quot;first kind&amp;quot;.)&lt;/p&gt;&#xA;&lt;p&gt;Bourbaki writes in the Historical Note after the section Commutative Fields in his book Algebra II that in the 19th century, Dedekind &amp;quot;sensed&amp;quot; the general notion of a derivation (look at his paper &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235693928?tify=%7B%22pages%22:%5B416%5D,%22panX%22:0.52,%22panY%22:0.676,%22view%22:%22info%22,%22zoom%22:0.486%7D&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and see the footnote by Emmy Noether on the 2nd page) and that Galois &amp;quot;is fully aware of the formal character&amp;quot; of differentiation of polynomials and used this to show an irreducible polynomial in &lt;span class=&quot;math-container&quot;&gt;$\mathbf F_p[x]$&lt;/span&gt; has no multiple roots in his development of finite fields. So if you are looking for the very first use of derivatives outside the setting of real or complex polynomials, then it goes back to Galois for the case of polynomials over a finite field.&lt;/p&gt;&#xA;&lt;p&gt;Going back even farther, the use of derivatives as a formal operation (no limits) to detect multiple roots of real polynomials in some sense precedes calculus by Newton and Leibniz! This is called Hudde’s rule and you can read about it &lt;a href=&quot;https://www.maa.org/book/export/html/451228&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-06-20T07:21:51.637" LastActivityDate="2023-06-20T07:21:51.637" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12970" PostTypeId="1" AcceptedAnswerId="12983" CreationDate="2021-03-13T19:59:42.910" Score="8" ViewCount="219" Body="&lt;p&gt;In his recollections Newton describes how he made his greatest discovery:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In August 1665, Sir I., who was not then not 24 brought at Sturbridge fair a Prism to try some experiments&#xA;upon Descartes's book on colors &amp;amp; when he came home he made a hole in his shutter &amp;amp; darkened the room &amp;amp; put his Prism between that &amp;amp; the wall... etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question: who and for what reason was making prisms and selling them in the market? There must have been some demand. Who was buying them them and for what purpose?&lt;/p&gt;&#xA;&lt;p&gt;Edit. My earlier naive conjecture was that there were some scientific equipment makers catering to Cambridge scientists:-) But this is refuted by two facts: a) First of all this did not happen in Cambridge. Newton was at that time on quarantine in his mother's estate, far from Cambridge. And b) scientists in Cambridge were not interested at that time in scientific experiments: they studied mostly theology and Aristotle's writings.&lt;/p&gt;&#xA;&lt;p&gt;Westfall in his most comprehensive book about Newton, &amp;quot;Never at rest&amp;quot;, writes: &amp;quot;Newton's recollection of Sturbridge Fair may have been mistaken,&#xA;there was also annual Midsummer Fair which managed to escape both plagues.&#xA;If he purchased the prism there in 1665, he could have taken it home with him and performed there basic, though perhaps crude, experiments connected with his initial insight&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;He also mentions that &amp;quot;prisms seem an unlikely item of commerce for a small market town&amp;quot; but does not elaborate on this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-03-14T23:43:48.597" LastActivityDate="2021-03-14T23:49:48.683" Title="Who was buying prisms in 1665?" Tags="|newton|17th-century|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12973" PostTypeId="1" AcceptedAnswerId="12975" CreationDate="2021-03-14T02:46:18.163" Score="1" ViewCount="152" Body="&lt;p&gt;&lt;strong&gt;Q.&lt;/strong&gt; Is it true that Euler's proof of infinite primes was the first since Euclid's which was from around 300BC?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; By Euler's proof, I mean the use of his Euler product formula for the zeta function &lt;span class=&quot;math-container&quot;&gt;$\zeta(s)$&lt;/span&gt; which diverges as &lt;span class=&quot;math-container&quot;&gt;$s\rightarrow\infty$&lt;/span&gt;, implying an infinitude of factors in the product, which means an infinite of primes.&lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="5989" LastEditDate="2021-03-14T10:19:32.240" LastActivityDate="2021-03-14T19:07:44.557" Title="Euler's proof of infinite primes first since Euclid?" Tags="|number-theory|euler|euclid|primes|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12974" PostTypeId="1" CreationDate="2021-03-14T07:33:51.033" Score="2" ViewCount="179" Body="&lt;p&gt;Consider a uniform random variable &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; which is an integer in the interval &lt;span class=&quot;math-container&quot;&gt;$2^{1023} &amp;lt; n &amp;lt; 2^{1024}$&lt;/span&gt;. What is the oldest algorithm capable of determining whether or not &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is a probable prime and which could be executed in a reasonable amount of time in pen and paper by mathematicians of the day?&lt;/p&gt;&#xA;&lt;p&gt;The oldest example I know of is Fermat's little theorem from 1640. Is there anything older?&lt;/p&gt;&#xA;" OwnerUserId="9646" LastEditorUserId="9646" LastEditDate="2021-03-14T07:42:40.270" LastActivityDate="2021-03-15T00:49:16.887" Title="History of primality testing" Tags="|mathematics|algorithm|primes|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12975" PostTypeId="2" ParentId="12973" CreationDate="2021-03-14T07:58:30.007" Score="6" Body="&lt;p&gt;There is a proof by Goldbach, which appears in a letter he wrote to Euler in 1730 (a few years before Euler published his product formula for the zeta function). It is as follows: if &lt;span class=&quot;math-container&quot;&gt;$F_n=2^{2^n}+1$&lt;/span&gt; (that is, if &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt; is the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&lt;sup&gt;th&lt;/sup&gt; Fermat number), then&lt;span class=&quot;math-container&quot;&gt;$$F_n=F_1F_2\ldots F_{n-1}+2.\tag1$$&lt;/span&gt;Since each &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt; is odd, it follows from &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; that, if &lt;span class=&quot;math-container&quot;&gt;$n\ne m$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$F_m$&lt;/span&gt; are relatively prime. So, take a prime factor &lt;span class=&quot;math-container&quot;&gt;$p_n$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt;, and the set &lt;span class=&quot;math-container&quot;&gt;$\{p_1,p_2,p_3,\ldots\}$&lt;/span&gt; will be an infinite set of prime numbers.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2021-03-14T19:07:44.557" LastActivityDate="2021-03-14T19:07:44.557" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12976" PostTypeId="1" CreationDate="2021-03-14T11:18:02.700" Score="4" ViewCount="509" Body="&lt;p&gt;It is an Indian (Kerala) text, it would be the first modern proof (based on earlier knowledge) of&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$$&lt;/span&gt;&#xA;The integral would be a Riemann sum.&lt;/p&gt;&#xA;&lt;p&gt;3 references, giving 3 different dates:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Yuktibh%C4%81%E1%B9%A3%C4%81&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia/Yuktibhāṣā&lt;/a&gt; 1530&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/23497280?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;The First Textbook of Calculus: &amp;quot;Yuktibhāṣā&amp;quot;&lt;/a&gt; 1555&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://users.uoa.gr/%7Eapgiannop/Sources/Roy-pi.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Discovery of the Series Formula for π by&#xA;Leibniz, Gregory and Nilakantha&lt;/a&gt; circa 1600&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Additionally to the date problem, can we decipher at least a small part of the original text to make ourselves an idea of its content?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It bears repetition that, as in the excerpted passages, so in the entire text, no&#xA;symbols are employed to represent the mathematical objects being manipulated, no formal notation for relations among them and operations on them,&#xA;no diagrammatic guide to the geometric constructions invoked.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is an English translation of the text, but it is really hard to follow, not made for mathematicians: Ganita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva.&lt;/p&gt;&#xA;&lt;p&gt;Edit: I can't find the proof of &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}$&lt;/span&gt; and of &lt;span class=&quot;math-container&quot;&gt;$\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$&lt;/span&gt; in this translation. So &lt;strong&gt;this may be the main question&lt;/strong&gt;: where is the proof? See this excerpt, it is more a cooking recipe as an attempt to define the Leibniz series than rigorous mathematics&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/10Kl6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/10Kl6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the translation Rsine means &lt;span class=&quot;math-container&quot;&gt;$R \sin(\theta)$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; the radius) and Rversine means &lt;span class=&quot;math-container&quot;&gt;$R(1-\cos(\theta))$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="14086" LastEditorUserId="14086" LastEditDate="2021-03-15T12:47:48.453" LastActivityDate="2022-05-18T14:25:26.337" Title="Yuktibhāṣā, 16th century, first modern proof of $\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$" Tags="|mathematics|calculus|ancient-india|" AnswerCount="2" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="12977" PostTypeId="2" ParentId="12976" CreationDate="2021-03-14T12:55:32.640" Score="2" Body="&lt;p&gt;Since the &lt;em&gt;Yuktibhasa&lt;/em&gt; (The Rationale) was written in Malayam in 1530, by Jyesthadeva, a Keralan astronomer, it will be easily translatable into English. The question is finding someone to finance such a venture.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I think that as India becomes more aware of its own scientific heritage instead of just catching up with the West (and according to Mathilde Marcolli, a geometer, the best of their institutes compete with the best of Western institutes)  then it will become more likely that we will see such translations.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-14T12:55:32.640" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12978" PostTypeId="1" AcceptedAnswerId="12981" CreationDate="2021-03-14T14:34:50.033" Score="4" ViewCount="136" Body="&lt;p&gt;Samuel Glasstone, in The Book of Mars (NASA, 1968, p. 109), wrote&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;In 1934, the German-born astronomer Rupert Wildt suggested that the&#xA;bright areas on Mars were composed 'of strongly oxidized sandy formations, with iron almost completely in the form of the oxide Fe2O3 [ferric oxide].'&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Does someone know the original paper by Wildt? Or an earlier sugestion that the red hue of the dust is due to iron oxides?&lt;/p&gt;&#xA;" OwnerUserId="9778" LastActivityDate="2021-03-15T05:23:23.193" Title="Martian dust as ferric oxide and Rupert Wildt" Tags="|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12979" PostTypeId="1" CreationDate="2021-03-14T17:26:11.433" Score="4" ViewCount="326" Body="&lt;p&gt;I am looking for the original (&lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;)-definition of limit by Weierstrass, but I cannot find an exact quote or a reference. I saw that somewhere it was claimed that this definition was actually published in Heine's notes of Weierstrass' lectures, but I did not find any exact reference either.&lt;/p&gt;&#xA;&lt;p&gt;Where could I check the exact formulation of Weierstrass' definition? (I guess I know the definition, I just want to be sure that this is how he formulated it.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(To explain my interest, I am trying to find out who was that evil person who decided to use the condition &lt;span class=&quot;math-container&quot;&gt;$0 &amp;lt; |x - a| &amp;lt;\delta\Rightarrow|f(x) - b| &amp;lt;\varepsilon$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$|x - a| &amp;lt;\delta\Rightarrow|f(x) - b| &amp;lt;\varepsilon$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="3030" LastEditorUserId="3030" LastEditDate="2021-03-16T09:08:18.837" LastActivityDate="2021-03-16T09:08:18.837" Title="($\varepsilon$, $\delta$)-definition of limit by Weierstrass" Tags="|mathematics|reference-request|19th-century|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="12981" PostTypeId="2" ParentId="12978" CreationDate="2021-03-14T20:06:40.273" Score="6" Body="&lt;p&gt;Rupert Wildt, &amp;quot;Ozon und Sauerstoff in den Planetenatmosphären.&amp;quot; Veröffentlichungen der Universitäts-Sternwarte zu Göttingen 3 (1933): 151-159 (&lt;a href=&quot;http://adsabs.harvard.edu/pdf/1933VeGoe...3..151W&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Will man die Annahme gelten lassen, daß die Zufuhr juvenilen Kohlendioxyds in der Entstehungsgeschichte des Mars von der gleichen Größenordnung wie bei Erde und Venus war, so folgt aus dem heutigen spärlichen Vorkommen von freiem Sauerstoff, daß in der Vergangenheit das Kohlendioxyd sehr vollständig zur Photosynthese verbraucht und der abgespaltene Sauerstoff wieder fossil geworden ist. Im Einklang damit würde die Beobachtung stehen, daß ausgedehnte Teile der Marsoberfläche eine stark rötliche Tönung aufweisen; diese lassen sich am einfachsten als stark oxydierte Sandsteinformationen, deren Eisen fast nur als Fe&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;3&lt;/sub&gt; vorläge, deuten. Bemerkenswert ist in diesem Zusammenhang, daß die Venusoberfläche keine Farbtönung aufweist;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;If one accepts the assumption that the supply of juvenile carbon dioxide during the formation history of Mars was on the same order of magnitude as on Earth and Venus, the conclusion to be drawn from the rarity of free oxygen at present is that the carbon dioxide was completely used up by photosynthesis in the past and that the oxygen generated from it has become fossil again. Consistent with that would be the observation that extensive parts of Mars' surface show a strong reddish hue; the most straightforward interpretation is that these are formations of highly oxidized sandstone, in which iron occurs almost exclusively as Fe&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;3&lt;/sub&gt;. In this context it is noteworthy that the surface of Venus does not show any color hues;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Above I cited Wildt's publication as it is currently being cited by Google Scholar. His notice is dated December 23, 1933, but it was presented in a meeting on January 12, 1934. Various publications citing Wildt use the following style:&lt;/p&gt;&#xA;&lt;p&gt;Rupert Wildt, &amp;quot;Ozon und Sauerstoff in den Planetenatmosphären.&amp;quot; Veröffentlichungen der Universitäts-Sternwarte zu Göttingen, No. 38 (1934)&lt;/p&gt;&#xA;&lt;p&gt;Best I could determine from a quick review of the literature, Wildt was the first to suggest that the reddish color of Mars could be due to iron oxide in sandstone by analogy with terrestrial varieties of sandstone of a reddish color.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-03-15T05:23:23.193" LastActivityDate="2021-03-15T05:23:23.193" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12982" PostTypeId="1" CreationDate="2021-03-14T22:54:24.607" Score="4" ViewCount="117" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What's the antecedent of &amp;quot;&lt;strong&gt;its meaning&lt;/strong&gt;&amp;quot;? I'm guessing &lt;em&gt;fisike&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Can you please expound on this shift that I embolded? The author didn't.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;physics [16] &lt;em&gt;Physics&lt;/em&gt; comes ultimately from&lt;/h3&gt;&#xA;&lt;p&gt;Greek &lt;em&gt;phúsis&lt;/em&gt; ‘nature’, a derivative of &lt;em&gt;phúein&lt;/em&gt;&#xA;‘bring forth, cause to grow’. The science of&#xA;studying the natural world was hence &lt;em&gt;phusiké&#xA;epistémē&lt;/em&gt; ‘knowledge of nature’, and &lt;em&gt;phusiké&lt;/em&gt;,&#xA;turned into a noun, passed into English via Latin&#xA;&lt;em&gt;physica&lt;/em&gt; and Old French &lt;em&gt;fisique&lt;/em&gt; as &lt;em&gt;fisike&lt;/em&gt;. B&lt;strong&gt;y now&#xA;its meaning had shifted from ‘natural science’ to&#xA;‘medicine’&lt;/strong&gt;, a sense preserved in the now archaic&#xA;&lt;em&gt;physic&lt;/em&gt; [13] and in the derivative &lt;em&gt;physician&lt;/em&gt; [13],&#xA;and the modern plural form, which restores the&#xA;original meaning, was a direct translation of&#xA;Greek &lt;em&gt;tà phusiká&lt;/em&gt; ‘the physics’, the title of&#xA;Aristotle’s writings on natural science. &lt;em&gt;Physique&lt;/em&gt;&#xA;[19] was borrowed from French.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Word Origins&lt;/em&gt; (2005 2e) by John Ayto, p 376.&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastEditorDisplayName="user3478" LastEditDate="2021-03-14T23:03:59.930" LastActivityDate="2021-03-15T11:32:29.443" Title="How did &quot;fisike&quot; shift from meaning &quot;natural science&quot; to &quot;medicine&quot;?" Tags="|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12983" PostTypeId="2" ParentId="12970" CreationDate="2021-03-14T23:43:21.823" Score="4" Body="&lt;p&gt;Maybe some of Newton's prisms were preserved. An &lt;a href=&quot;https://royalsocietypublishing.org/doi/abs/10.1098/rsnr.1981.0002&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; by A.A. Mills (1981) shows images. He writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton appears to have purchased all his prisms: there is no intimation that he made any of them, although he was obviously skilled at grinding lenses and mirrors. This availability of ready-made glass prisms is rather puzzling, for the period in question appears to pre-date the earliest recorded chandeliers, which in any case contained facetted droplets rather than triangular prisms. Presumably there was sufficient demand for prisms to be made and sold purely as curiosities or toys, the optical quality being correspondingly low.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But also:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;None of the extant prisms claiming an association with Newton can be identified with any of his recorded prisms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6784" LastEditorUserId="6784" LastEditDate="2021-03-14T23:49:48.683" LastActivityDate="2021-03-14T23:49:48.683" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12984" PostTypeId="2" ParentId="12974" CreationDate="2021-03-15T00:49:16.887" Score="2" Body="&lt;p&gt;According to this &lt;a href=&quot;http://www.math.stonybrook.edu/%7Emoira/mat331-spr10/papers/2002%20MollinA%20Brief%20History%20of%20Factoring.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; for the paper &amp;quot;A Brief History of Factoring and Primality Testing B. C. (Before Computers)&amp;quot;, Cataldi created a method for primality testing which was verified by Fermat. The following is an excerpt from the paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In particular, his work &amp;quot;&lt;em&gt;Trattato del modo brevissimo di trovar la radice quadra delli numeri&lt;/em&gt;&amp;quot;&#xA;published in 1613, represents a significant contribution to the development of continued fractions. His work on perfect numbers was also considerable. Among his thirty&#xA;20 MATH EMATICS MAGAZI N E&#xA;VOL. 75, NO. 1, FEBRUARY2002 21&#xA;books, he wrote on military applications of algebra, and even published an edition of&#xA;Euclid's Elements. Cataldi proved that the fifth, sixth, and seventh perfect numbers&#xA;are:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wW2om.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wW2om.png&quot; alt=&quot;Perfect Numbers&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is uncertain whether Cataldi was the first to discover these&#xA;perfect numbers, but his are the first known proofs of these facts.&#xA;Cataldi was also the first to observe that if 2n _ 1 is prime then n&#xA;must be prime. In fact, the following result was known to Cataldi,&#xA;though proved by Fermat.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bbBX4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bbBX4.png&quot; alt=&quot;Theorem 1&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Long after Cataldi, Euler showed that every even perfect number has the form given&#xA;in Theorem 1. It is unknown whether there are any odd perfect numbers and the search&#xA;for them has exceeded the bound 10^300. Moreover, if such a beast exists, then it is&#xA;known that it must have at least twenty-nine (not necessarily distinct) prime factors&#xA;(see Guy [6, B l, p. 44]).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I hope this helps!&lt;/p&gt;&#xA;" OwnerUserId="12487" LastActivityDate="2021-03-15T00:49:16.887" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12985" PostTypeId="2" ParentId="12979" CreationDate="2021-03-15T07:21:02.190" Score="3" Body="&lt;p&gt;The issue is that most of W's foundational works was unpublished or scattered into his lectures.&lt;/p&gt;&#xA;&lt;p&gt;See Pierre Dugac, &lt;a href=&quot;https://www.jstor.org/stable/41133363?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eléments d'analyse de Karl Weierstrass (AHES, 1973)&lt;/a&gt;, Appendice II, page 119, with some extracts from &lt;em&gt;Ausarbeitung der Vorlesung an dem Konig. Gewerbeinstitut zu Berlin im Sommersem. 1861&lt;/em&gt; von H.A. SCHWARZ (Institut Mittag-Leffler)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ist &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; eine Funktion von &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; und &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; ein bestimmter Wert, so wird sich die&#xA;Funktion, wenn &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$x + h$&lt;/span&gt; ubergeht, in &lt;span class=&quot;math-container&quot;&gt;$f(x + h)$&lt;/span&gt; andern; die Differenz &lt;span class=&quot;math-container&quot;&gt;$f(x + h) f(x)$&lt;/span&gt; nennt man die Veranderung, welche die Funktion dadurch erfahrt, dass das&#xA;Argument von &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$x + h$&lt;/span&gt; ubergeht. Ist es nun moglich, fur heine Grenze &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; zu bestimmen, sodass fur alle Werte von &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, welche ihrern absoluteu Betrage noch kleiner&#xA;als &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; sind, &lt;span class=&quot;math-container&quot;&gt;$f(x + h) - f(x)$&lt;/span&gt; kleiner werde als irgendeine noch so kleine Grosse &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastActivityDate="2021-03-15T07:21:02.190" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12986" PostTypeId="2" ParentId="12982" CreationDate="2021-03-15T11:32:29.443" Score="4" Body="&lt;p&gt;The meaning did not shift as described in the OP quote. The Old French (which is the source of English) use of &lt;em&gt;fysike&lt;/em&gt; for healing potions, and, accordingly, &lt;a href=&quot;https://www.etymonline.com/word/physician&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;fisiciien&lt;/em&gt;&lt;/a&gt; for healer, dates back to 12th century, and predates its &lt;a href=&quot;https://www.etymonline.com/word/physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;later use&lt;/a&gt; for the study of nature from 1580-s. More narrow modern use as in &lt;a href=&quot;https://www.etymonline.com/word/physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;physics&lt;/a&gt; (that replaced natural philosophy) is even older, c. 1715, and the term &lt;a href=&quot;https://www.etymonline.com/search?q=physicist&quot; rel=&quot;nofollow noreferrer&quot;&gt;physicist&lt;/a&gt; was only introduced by Whewell in 1836 with explicit reference to already established use for healers:&amp;quot;&lt;em&gt;As we cannot use physician for a cultivator of physics, I have called him a physicist.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Of course, the original Greek use of &amp;quot;physics&amp;quot;, as in the title of Aristotle's treatise, goes back to ancient times, but it did not filter through to medieval Europe until translations of Aristotle became widely available in 16-th century, and then the shift happened &lt;em&gt;the other way&lt;/em&gt;. By the way, Greek &lt;a href=&quot;https://en.wikipedia.org/wiki/Physis&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;physis&lt;/em&gt;&lt;/a&gt; has always had an ambiguous meaning, with nature as in cause of things being prominent, and this idea of natural causes was actively emphasized in &lt;a href=&quot;https://en.wikipedia.org/wiki/Hippocratic_Corpus#Natural_vs._divine_causality&quot; rel=&quot;nofollow noreferrer&quot;&gt;corpus Hippocraticum&lt;/a&gt;:&amp;quot;&lt;em&gt;It is not sufficient to learn simply that cheese is a bad food, as it gives a pain to one who eats a surfeit of it; we must know what the pain is, the reasons for it [dia ti], and which constituent of man is harmfully affected&lt;/em&gt;&amp;quot;. This may explain the medieval European use prior to the re-introduction of Aristotle.&lt;/p&gt;&#xA;&lt;p&gt;Nonetheless, Greeks themselves used a different word (&lt;em&gt;iatros&lt;/em&gt;) for healers, including in Aristotle's reference to Hippocrates as a &amp;quot;great physician&amp;quot; (as it is typically translated) in Politics VII, 1326a, see &lt;a href=&quot;https://www.degruyter.com/document/doi/10.1515/apeiron-2019-0052/html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Baker, Aristotle on the Nature and Politics of Medicine&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;By way of preface, I here note that the ancient Greek word iatros (ἰατρός), though variously translated as “healer”, “doctor”, or “physician”, is not simply ambiguous. Instead, there seems to have been, even in the time of Aristotle, a close semantic connection in iatros between “healer” and “medical doctor” since of course a medical doctor is someone skilled at healing. Something similar may be said of the corresponding verb iatreuō (ἰατρεύω; “to heal”, “to treat medically”) as well as the phrase technē iatrikē (τεχνὴ ἰατρική; “the medical art”, “the art of healing”). So while I will consistently translate iatros as “doctor” and (technē) iatrikē as the “medical art,” let us bear in mind that the former would have been heard as something like “expert healer,” and the latter as something like the “the expertise of healing”.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-03-15T11:32:29.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12987" PostTypeId="2" ParentId="6447" CreationDate="2021-03-15T20:02:33.663" Score="2" Body="&lt;p&gt;Full details in my paper: &lt;a href=&quot;https://cedar.wwu.edu/cgi/viewcontent.cgi?referer=https://scholar.google.co.uk/&amp;amp;httpsredir=1&amp;amp;article=1093&amp;amp;context=math_facpubs&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Historical Development of the Newton-Raphson Method&lt;/em&gt;&lt;/a&gt;, published in SIAM Review in 1995. Deuflhard's paper is basically a summary of that work.&lt;/p&gt;&#xA;" OwnerUserId="14093" LastEditorUserId="5659" LastEditDate="2021-03-15T20:25:22.417" LastActivityDate="2021-03-15T20:25:22.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12988" PostTypeId="1" AcceptedAnswerId="12990" CreationDate="2021-03-16T04:23:57.973" Score="5" ViewCount="663" Body="&lt;p&gt;By &amp;quot;algebra&amp;quot;, I'm assuming the solving of polynomial equations like every school child does. I accept that many kids loathe algebra, and that solving equations can be dreary and weariful, especially if you have to derive &lt;a href=&quot;https://mathworld.wolfram.com/CubicFormula.html&quot; rel=&quot;noreferrer&quot;&gt;the Cubic Formula&lt;/a&gt;!&#xA;But I don't understand the diction. What are the &amp;quot;broken parts&amp;quot;? What's being reunited?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;algebra [16]&lt;/h3&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Algebra symbolizes the debt of&#xA;Western culture to Arab mathematics, but&#xA;ironically when it first entered the English&#xA;language it was used as a term for the setting of&#xA;broken bones, and even sometimes for the&#xA;fractures themselves (‘The helpes of Algebra&#xA;and of dislocations’, Robert Copland, &lt;em&gt;Formulary&#xA;of Guydo in surgery&lt;/em&gt; 1541). This reflects the&#xA;original literal meaning of the Arabic term &lt;em&gt;al&#xA;jebr&lt;/em&gt;, ‘the reuniting of broken parts’, from the&#xA;verb &lt;em&gt;jabara&lt;/em&gt; ‘reunite’. The anatomical&#xA;connotations of this were adopted when the&#xA;word was borrowed, as &lt;em&gt;algebra&lt;/em&gt;, into Spanish,&#xA;Italian, and medieval Latin, from one or other of&#xA;which English acquired it. In Arabic, however, it&#xA;had long been applied to the solving of algebraic&#xA;equations (the full Arabic expression was ’&lt;em&gt;ilm&#xA;aljebr wa’lmuqābalah&lt;/em&gt; ‘the science of reunion&#xA;and equation’, and the mathematician al-Khwarizmi used &lt;em&gt;aljebr&lt;/em&gt; as the title of his treatise&#xA;on algebra – see ALGORITHM), and by the end of&#xA;the 16th century this was firmly established as&#xA;the central meaning of &lt;em&gt;algebra&lt;/em&gt; in English.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Word Origins&lt;/em&gt; (2005 2e) by John Ayto, p 16 Left column.&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2021-04-15T01:36:27.373" Title="Why did al-Khwarizmi use al jebr, ‘the reuniting of broken parts’ to signify algebra?" Tags="|elementary-algebra|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="12989" PostTypeId="2" ParentId="12988" CreationDate="2021-03-16T06:20:04.790" Score="3" Body="&lt;p&gt;I have also wondered about the term &lt;em&gt;al-jabr&lt;/em&gt; which ultimately became algebra. Re Please see Solomon Gandz (1926) The Origin of the Term “Algebra”, The American Mathematical Monthly, 33:9, 437-440, DOI: 10.1080/00029890.1926.11986615.&lt;/p&gt;&#xA;&lt;p&gt;The author proposes an alternative and exactly notes your concern that al-jabr meant joining the broken parts. Gandz states that this word is not of Arabic origin and most likely it is Assyrian word.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;here are still remnants in the mathematical literature suggesting that in olden times the term al-jabr alone was used for the science of equations, and the term al-jabriyyun was taken for the masters of algebra. &lt;span class=&quot;math-container&quot;&gt;${ }^{1}$&lt;/span&gt; On the other hand the term al-muqabalah alone, according to its real meaning of &amp;quot;putting face to face, confronting, equation,' seems to be the most appropriate name for equations in general. With these difficulties in mind, the writer undertook to search out the real meaning of jabara in the related Semitic languages. Now the Assyrian name gabr\hatu-mahâru means to be equal, to correspond, to confront, or to put two things face to face; see Delitzsch, Assyrisches Handwöterbuch, under gabru and maharu, pp. &lt;span class=&quot;math-container&quot;&gt;$193,401,$&lt;/span&gt; and Muss-Arnolt, Assyrian Dictionary, under gabru and maxaru, &lt;span class=&quot;math-container&quot;&gt;$^{2}$&lt;/span&gt; pp. &lt;span class=&quot;math-container&quot;&gt;$210,525 .$&lt;/span&gt; From the first of these we have the etymology of the Hebrew geber and gibbor. Geber is the mature man leaving the state of boyhood and being equal in rank and ralue to the other men of the assembly or army. Gibbôr is the hero who is strong enough to fight and overcome his equals and rivals in the hostile army. Gabara &lt;span class=&quot;math-container&quot;&gt;$=j$&lt;/span&gt; abar &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, in its original Assyrian meaning, is therefore the corresponding name for the Arabic qabala (verbal noun muqabalah), and an appropriate name for equations in general.&#xA;The Egyptians &lt;span class=&quot;math-container&quot;&gt;$^{3}$&lt;/span&gt; knew and wrote books on algebra as early as 1600 B.C., and it would be very strange if the Assyrians, having the same level of culture as the Egyptians and having close political and economic relations with them, were quite ignorant of this art. Gabr must have been the original Assyrian form of the word. The Arabs received this ancient science, with its original Assyrian name (in Arabic pronunciation al-jabr) from the Aramaeans and Syrians, who lived on Assyrian territory, and added the Arabic name al-muqabalah, which is nothing else than the literal Arabic translation of &lt;span class=&quot;math-container&quot;&gt;$a l$&lt;/span&gt; jabr. This took place many hundreds of years before Mohammed ibn Musâ al-Kowârizmi. Later on the real meaning of the word was forgotten, and the simple meaning seems not to have been regarded as scholarly enough for good usage. The scholastic method at that time was common in both the philosophical and the theological schools. The scholars tried to find in the Bible, the Koran, and the old philosophical texts everything but the simple, plain meaning. The same method was followed towards these two mathematical terms. The masters simply declared them to signify the first two operations of algebra, namely the removal of the negative and positive quantities, without worrying much about philological reasons. This conception of the two words was already well known at the time of Mohammed ibn Musâ al Khowârizmî, and the latter used the terms in the traditional way without any further explanation, as shown by both Rosen and Ruska. &lt;strong&gt;For more than a thousand years this scholastic interpretation prevailed in the Arabic and European worlds. In reality, however, it would seem that the expression 'Ilm al-jabr w'al-muqabalah ought to be rendered simply as Science of equations, al-jabr being the Assyrian and al-muqabalah the Arabic name for equation.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His version seems plausible.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-03-16T06:20:04.790" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12990" PostTypeId="2" ParentId="12988" CreationDate="2021-03-16T15:25:50.210" Score="4" Body="&lt;p&gt;See Roshdi Rashed, &lt;a href=&quot;https://books.google.it/books?id=qfRTBAAAQBAJ&amp;amp;pg=PA107&quot; rel=&quot;noreferrer&quot;&gt;Classical Mathematics from Al-Khwarizmi to Descartes (Routledge, 2014)&lt;/a&gt;, page 107 [but I've not seen: R. Rashed (editor), &lt;a href=&quot;https://books.google.it/books?id=nphePgAACAAJ&amp;amp;dq=rashed+Al-Khw%C4%81rizm%C4%AB:+The+Beginnings+of+Algebra&quot; rel=&quot;noreferrer&quot;&gt;Al-Khwārizmī: The Beginnings of Algebra (2009)&lt;/a&gt;]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term &lt;em&gt;al-jabr&lt;/em&gt; (algebra) is indeed an Arabic term, a name for the&#xA;action of the verb (&lt;em&gt;a maṣdar&lt;/em&gt;, according to the grammarians), the root of&#xA;which has the general meaning of rectifying or correcting something using&#xA;some form of constraint, such as setting a broken bone, for example. It is&#xA;thus an ordinary-language term that can take on multiple meanings. It was devoid of technical meaning before al-Khwārizmī for the first time gave it&#xA;a two-fold technical signification. When associated with the word &lt;em&gt;almuqābala&lt;/em&gt;,&#xA;it designates both a discipline and an operation. The successors&#xA;of al-Khwārizmī will quickly give pride of place to the first word,&#xA;‘algebra’, to name the discipline, and derive from this single word the name&#xA;of the professional, ‘algebraist’. This usage already appears in Thābit ibn&#xA;Qurra (826–901). But this word also designates an operation: that of&#xA;‘restoring’ an equation, that is, adding to its two members the subtracted&#xA;terms. For example, in&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 + c – bx = d$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$c &amp;gt; d$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the operation ‘algebra’ consists in adding &lt;span class=&quot;math-container&quot;&gt;$bx$&lt;/span&gt; to each side,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 + c = bx + d$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;and the operation &lt;em&gt;al-muqābala&lt;/em&gt;, that is, ‘opposition’ or ‘reduction’ amounts&#xA;to&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$x^2 + (c – d) = bx$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The point of the two related operations is to bring the equation back to&#xA;one of the canonical types that al-Khwārizmī defined &lt;em&gt;a priori&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But see also: George Saliba, &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1600-0498.1973.tb00193.x&quot; rel=&quot;noreferrer&quot;&gt;The Meaning of &lt;em&gt;al-jabr wa'l-muqabalah&lt;/em&gt; (1973)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The word &lt;em&gt;jabr&lt;/em&gt; (root &lt;em&gt;jubara&lt;/em&gt;) has a number of meanings in ordinary&#xA;Arabic, two of which are relevant here. The first is “to reduce a fracture”,&#xA;while the second, which is more general, is “to force, to compel”. The&#xA;traditional interpretation has connected &lt;em&gt;jabr&lt;/em&gt; with the fist meanings for&#xA;two main reasons. [...] We believe, on the contrary, that the root &lt;em&gt;jabara&lt;/em&gt; was employed by the medieval algebraists in its second sense, “to compel”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hence the science &lt;em&gt;al-jabr wa’l-muqabalah&lt;/em&gt; is the science by which one&#xA;forces the unknown to acquire certain values and checks the appropriate&#xA;value by comparing the results with the original conditions of the problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2021-03-16T15:48:34.113" LastActivityDate="2021-03-16T15:48:34.113" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="12992" PostTypeId="1" AcceptedAnswerId="12994" CreationDate="2021-03-16T23:45:26.717" Score="3" ViewCount="348" Body="&lt;p&gt;&lt;sub&gt;A version of this question first appeared in &lt;a href=&quot;https://mathoverflow.net/questions/380132/when-and-where-did-gauss-say-this&quot;&gt;MathOverflow&lt;/a&gt; but did not receive satisfactory answers.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This quote is often attributed to Gauss:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The earliest source for this quote is the book by Wolfgang Sartorius von Waltershausen &amp;quot;Gauss zum Gedächtniss&amp;quot; published a year after Gauss' death.  The book contains some other stories about Gauss as well. For example this is the place where the story of the sum 1+2+...+100 first appeared. That story is now viewed as incorrect.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Should we believe that Gauss indeed is the author of the quote? The use of &amp;quot;Konigin&amp;quot; is somewhat suspicious because women had no power in any part of Germany at that time and before. To become powerful, a German princess should have married a foreign prince (French, Russian, etc.).&lt;/p&gt;&#xA;&lt;p&gt;In particular, I would also like to know what exactly this quote means. German speaking mathematicians gave me different explanations. In that regards, why is it &amp;quot;queen&amp;quot; and not &amp;quot;king&amp;quot;? Is it about power or something else (beauty perhaps)?&lt;/p&gt;&#xA;" OwnerDisplayName="user14095" LastEditorUserId="4251" LastEditorDisplayName="user14095" LastEditDate="2021-09-22T08:40:33.270" LastActivityDate="2021-09-22T08:40:33.270" Title="A quote attributed to Gauss" Tags="|mathematics|gauss|quote|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="12993" PostTypeId="1" CreationDate="2021-03-17T03:10:47.757" Score="1" ViewCount="131" Body="&lt;p&gt;I am currently preparing a presentation about the value of more complex (specically: non-Gaussian) statistical inference. I thought it might be interesting to start the presentation with a small real-world example about disasters or mistakes caused by over-ambitious simplification.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, it quickly turned out that there do not seem to be that many stories about statistics disasters, which came as quite a suprise to me. One story I found &lt;a href=&quot;https://harvardpolitics.com/how-restructuring-our-economy-couldve-prepared-us-for-this-pandemic/&quot; rel=&quot;nofollow noreferrer&quot;&gt;relates the 2008 financial crash to the oversight of fat-tailed interactions&lt;/a&gt; caused by using Gaussian copula models. That's already not bad, but there must be better anecdotes. Do you know of any other entertaining/interesting stories in which statistical simplifications or errors led to unexpected results?&lt;/p&gt;&#xA;" OwnerUserId="14103" LastEditorUserId="10106" LastEditDate="2021-03-17T05:50:38.027" LastActivityDate="2021-03-17T11:50:51.503" Title="Stories about the consequences of statistical simplification?" Tags="|mathematics|reference-request|statistics|probability|anecdote|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12994" PostTypeId="2" ParentId="12992" CreationDate="2021-03-17T07:38:29.997" Score="4" Body="&lt;p&gt;The statement attributed to Gauss apparently first appeared in a biography of Gauss by the geologist Wolfgang Sartorius von Waltershausen, published shortly after Gauss's death:&lt;/p&gt;&#xA;&lt;p&gt;W. Sartorius v. Waltershausen, &amp;quot;Gauss zum Gedächtniss.&amp;quot; Leipzig: Salomon Hirzel 1856&lt;/p&gt;&#xA;&lt;p&gt;On page 79, we &lt;a href=&quot;https://www.google.com/books/edition/Gauss/TQo_AAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=der%20Wissenschaften&quot; rel=&quot;nofollow noreferrer&quot;&gt;find&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Mathematik hielt Gauss um seine eigenen Worte zu gebrauchen, für die Königin der Wissenschaften und die Arithmetik für die Königin der Mathematik. Diese lasse sich dann öfter herab der Astronomie und anderen Naturwissenschaften einen Dienst zu erweisen, doch gebühre ihr unter allen Verhältnissen der erste Rang.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Gauss considered, to use his own words, mathematics to be the queen of sciences and arithmetic to be the queen of mathematics. While she would stoop to serve astronomy and other natural sciences now and then, she would be due first rank in all contexts.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;No further information is provided as to when Gauss said or wrote this. The comparison with royalty is clearly based on the supreme rank mathematics is accorded among the sciences, and queen specifically may have been chosen to match the female grammatical gender of mathematics and arithmetic in German: &lt;em&gt;die Mathematik, die Arithmetik&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, this is not the first mention of mathematics as the queen of sciences one can find in the literature. This occurred (at least) two decades earlier in a text book for high-schools:&lt;/p&gt;&#xA;&lt;p&gt;Andreas Neubig, &amp;quot;Grundriss der reinen Mathematik mit einer kurzen Anweisung zum Feldmessen,&amp;quot; Bayreuth: Grau 1835&lt;/p&gt;&#xA;&lt;p&gt;There we &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=2k41AAAAcAAJ&amp;amp;oi=fnd&amp;amp;pg=PA1&amp;amp;dq=Grundriss+der+reinen+Mathematik+mit+einer+kurzen+Anweisung+zum+Feldmessen&amp;amp;ots=Ldlm1LD4HD&amp;amp;sig=EjEIvpHySHWs4bVUj-IS-OAWL2c#v=onepage&amp;amp;q=K%C3%B6nigin%20der%20Wissenschaften&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;find&lt;/a&gt; the following in the introduction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Mathematik, sagt Thiersch, ist die Königin der Wissenschaften.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation: &amp;quot;Mathematics, Thiersch says, is the queen of sciences&amp;quot;. I note that this is from the third edition of Neubig's book. I was able to establish that the second edition was published in 1828. After a lengthy search, I was able to locate a scan of this edition at a &lt;a href=&quot;http://www.manuscriptorium.com/apps/index.php?direct=record&amp;amp;pid=NKCR__-NKCR__75_B_000055_3EZR890-cs#search&quot; rel=&quot;nofollow noreferrer&quot;&gt;Czech website&lt;/a&gt;, and was able to verify that the same sentence already appeared in it, on &lt;a href=&quot;https://imagines.manuscriptorium.com/loris/NKCR__-NKCR__75_B_000055_3EZR890-cs/IMG00000022/full/full/0/default.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 8&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Andreas Neubig was born on May 6, 1780 in &lt;a href=&quot;https://en.wikipedia.org/wiki/Kulmbach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kulmbach&lt;/a&gt;. His dissertation was printed in 1811 by Adolf Ernst Junge in &lt;a href=&quot;https://en.wikipedia.org/wiki/Erlangen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erlangen&lt;/a&gt;. From 1821 to 1851 he worked as a &lt;em&gt;Lyceal-Professor&lt;/em&gt; (professor at a high school) in &lt;a href=&quot;https://en.wikipedia.org/wiki/Bayreuth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bayreuth&lt;/a&gt;. He died in November 1861.&lt;/p&gt;&#xA;&lt;p&gt;The most likely match for the Thiersch in Neubig's book is &lt;a href=&quot;https://en.wikipedia.org/wiki/Friedrich_Thiersch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedrich Wilhelm Thiersch&lt;/a&gt;. He was a philologist and educator born on June 17, 1784 in &lt;a href=&quot;https://en.wikipedia.org/wiki/Kirchscheidungen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kirchscheidungen&lt;/a&gt; and died on February 25, 1860 in Munich. He first worked as a professor at a high school before becoming a university professor.&lt;/p&gt;&#xA;&lt;p&gt;Neubig and Thiersch are of the same generation and both resided in the Kingdom of Bavaria during their professional careers. The religious affiliation of both men appears to have been Lutheran. I haven't been able to find out whether they knew each other personally or whether they corresponded with each other, so it is unclear from what source Neubig quotes Thiersch. In 1825 the Bavarian king put Thiersch in charge of higher education, so he was presumably well known among Bavarian educators.&lt;/p&gt;&#xA;&lt;p&gt;A potential source for Neubig's quote is the following publication by Friedrich Wilhelm Thiersch:&lt;/p&gt;&#xA;&lt;p&gt;Friedrich Thiersch, &amp;quot;Ueber gelehrte Schulen. Zweiter Band,&amp;quot; Stuttgart and Tübingen:  J. G. Cotta 1827&lt;/p&gt;&#xA;&lt;p&gt;On &lt;a href=&quot;https://books.google.com/books?id=e2lP38XKep4C&amp;amp;pg=PA188&amp;amp;lpg=PA188&amp;amp;dq=%22Thiersch%22+%22Mathematik%22&amp;amp;source=bl&amp;amp;ots=-kWe3cIKWd&amp;amp;sig=ACfU3U1-xyiON_N_mHYQbED925WC6iuLMw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiC3Z3ZkLfvAhV6HDQIHUotBxQ4ChDoATAIegQIBxAD#v=onepage&amp;amp;q=%22der%20Wissenschaften%20kennen%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 63&lt;/a&gt; (my emphasis):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mißlicher noch wird die Anordnung der zweiten Section, in der die  mathematischen Wissenschaften, die physikalischen und also wohl auch die Naturgeschichte eingeschichtet wird, die zu der Mathematik in keinem näheren Verhältnis als zu irgend einer anderen Wissenschaft steht. &lt;strong&gt;Die Mathematik allein in der dürftigen Gestalt, in der unsre Universitäten diese Königinn der Wissenschaften kennen&lt;/strong&gt;, gab nicht Stoff genug, eine Section zu füllen, und die Kunden der Natur, Physik, Mineralogie, Botanik, Zoologie wurden um so bereitwilliger zu ihr aufgenommen, weil sie an den übrigen Sectionen&#xA;der heiligen Vierzahl umherirrten, ohne darin Unterkommen zu finden.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Even more unfortunate is the composition of the second department, which includes the mathematical sciences, the physical ones and therefore likely also natural history, which is not in a closer relationship to mathematics than any of the other sciences. &lt;strong&gt;Mathematics alone, in the paltry form in which our universities know this queen of the sciences&lt;/strong&gt;, did not provide enough material to fill a department, and the subjects dealing with the natural world, physics, mineralogy, botany, zoology were accepted all the more readily in it as they strayed through the other departments of the holy quaternity without finding a place in it.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It is likely impossible to establish whether Gauss (or von Walterhausen, for that matter) was aware of what Thiersch had stated earlier.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-09-22T06:46:41.393" LastActivityDate="2021-09-22T06:46:41.393" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="12995" PostTypeId="2" ParentId="12993" CreationDate="2021-03-17T11:50:51.503" Score="1" Body="&lt;p&gt;&amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gambler%27s_ruin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gambler's Ruin&lt;/a&gt;&amp;quot; would seem to be the one which pervades both Las Vegas and Wall Street to this day.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-03-17T11:50:51.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="12997" PostTypeId="2" ParentId="2687" CreationDate="2021-03-17T17:54:35.857" Score="1" Body="&lt;p&gt;The derivation from solid state physics is quite simple. At equilibrium, atoms are at distances where the first derivative of potential vanishes. So for small deviations, a series expansion is dominated by the quadratic term.&lt;/p&gt;&#xA;&lt;p&gt;Force is the derivative of potential. Far a harmonic potential, the force increases linearly with distance from the equilibrium position. That is Hooke's law.&lt;/p&gt;&#xA;&lt;p&gt;Hooke's law depends on the harmonic approximation of interatomic potential. He did not really know about atoms, but it seems that he was on to something. Moyer says that Hooke was wrong, but maybe Hooke was just mostly confused or lacking the terminology to express his intuition. It was before Brook Taylor introduced series expansions in 1715.&lt;/p&gt;&#xA;" OwnerUserId="6784" LastActivityDate="2021-03-17T17:54:35.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12998" PostTypeId="2" ParentId="2687" CreationDate="2021-03-17T20:41:23.013" Score="1" Body="&lt;p&gt;Once you have calculus and if we can assume that the tension is a &lt;em&gt;smooth&lt;/em&gt; function of displacement,  it is easy enough to discover Hookes law, it's simply the statement that to a good approximation that the tension is proportional to the displacement for small displacements. The physics comes in determining the constant of proportionality.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-17T20:41:23.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="12999" PostTypeId="2" ParentId="12988" CreationDate="2021-03-17T22:24:31.340" Score="0" Body="&lt;p&gt;Around 820 CE, the polymath Muhammed Ibn Musa al-Khwarizmi published his book, &lt;em&gt;al-Kitab al-Mukhtasar fi Hisab al-Jabr wal Muqabalah&lt;/em&gt; in Abbasid Baghdad. This was translated into Latin as &lt;em&gt;Liber Algebrae er Almucabola&lt;/em&gt; and was referred to as &lt;em&gt;al-Jabr&lt;/em&gt;. It's full title is translated as &lt;em&gt;The Encyclopedic Treatise on Calculation by Balancing and Completion&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The term &lt;em&gt;al-jabr&lt;/em&gt; refers to the balancing where terms in an algebraic identity are transposed from one side to the other as is taught in many schools, even now. Presumably the term &lt;em&gt;al-jabr&lt;/em&gt;, in Abbasid Baghdadi Arabic meant balancing in some sense and thos was transposed to the mathematical setting of &lt;em&gt;al-jabr&lt;/em&gt;'s treatise.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-04-15T01:36:27.373" LastActivityDate="2021-04-15T01:36:27.373" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13001" PostTypeId="2" ParentId="2506" CreationDate="2021-03-18T03:53:47.960" Score="0" Body="&lt;p&gt;Whilst given our cosmology we can see that a year is the time taken for the earth to orbit the sun, when the year was first understood it would have been by the cycle of the seasons. Later, through astronomical observations a more precise idea of how long a year should be would have been theorised. This is one reason for the importance of early astronomical observations, say by the Babylonian astronomers.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-03-18T03:53:47.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13002" PostTypeId="2" ParentId="293" CreationDate="2021-03-18T04:01:43.963" Score="0" Body="&lt;p&gt;Newton called himself a natural philosopher and this is how the term was understood since ancient Greece. In fact, Plato called Aristotle a daemon of nature because of his theorisation of nature amd hos advocacy of a philosophy of nature through his books on physics and meteorology.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, even though the term 'scientist' had been advocated by William Whewell in 1834 to identify people doing 'science', we find as late as the early 20C, physicists such as Rutherford publishing results in &lt;em&gt;The Philosophical Magazine&lt;/em&gt; - and he wasn't the only one: Maxwell, Joule, Faraday and Humphrey Davy also published in it.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-18T07:42:47.393" LastActivityDate="2021-03-18T07:42:47.393" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13003" PostTypeId="1" CreationDate="2021-03-18T05:15:03.337" Score="0" ViewCount="118" Body="&lt;p&gt;Years ago I stumbled upon a Wikipedia article on two logicians debating whether exact synonyms could exist. (was curious since an English professor emphatically stated they cannot exist, while I believe it is logically possible, but highly unlikely to actually encounter) They had a life long feud over this, and the last of them had his response set to be published by his estate after his opponent died. From there no one really cared about synonymy, and it was declared by the supporters of the other dude that it is impossible. Does anyone know which two people I am referring to? Which books this references?&lt;/p&gt;&#xA;&lt;p&gt;I can't seem to find it again, and doing searches on Google with &amp;quot;Synonymy&amp;quot; brings up hundreds of thousands of dictionary pages just by including that world.&lt;/p&gt;&#xA;&lt;p&gt;I can't remember much more. I think one was a logical positivist, and synonymy came up in an attempt to poke holes in it. It might be Quine and Carnap, but I believe the two feuding were actually minor figures.&lt;/p&gt;&#xA;" OwnerUserId="14113" LastEditorUserId="14113" LastEditDate="2021-03-20T07:19:29.820" LastActivityDate="2021-03-20T07:19:29.820" Title="Who were the two logicians that argued over whether exact synonyms exist?" Tags="|logic|logicians|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13004" PostTypeId="1" CreationDate="2021-03-18T08:33:50.353" Score="3" ViewCount="340" Body="&lt;p&gt;I agree that logarithms are numbers, and Napier chose &lt;em&gt;arithmós&lt;/em&gt; reasonably. But why did Napier pick Greek &lt;em&gt;logós&lt;/em&gt; ‘ratio’? I don't see a ratio in this definition of the logarithm in James Stewart, &lt;em&gt;Calculus&lt;/em&gt; 2011 7e (not the Early Transcendentals version), p 404.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d6arT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d6arT.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Word Origins&lt;/em&gt; (2005 2e) by John Ayto. p 314 Right column.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;logarithm [17]&lt;/h3&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Greek &lt;em&gt;lógos&lt;/em&gt; had a remarkably&#xA;wide spread of meanings, ranging from ‘speech,&#xA;saying’ to ‘reason, reckoning, calculation’, and&#xA;‘ratio’. The more ‘verbal’ end of its spectrum has&#xA;given English the suffixes &lt;em&gt;-logue&lt;/em&gt; and &lt;em&gt;-logy&lt;/em&gt; (as in&#xA;&lt;em&gt;dialogue, tautology&lt;/em&gt;, etc), while the ‘reasoning’&#xA;component has contributed logic [14] (from the&#xA;Greek derivative &lt;em&gt;logiké&lt;/em&gt;), &lt;em&gt;logistic&lt;/em&gt; [17] (from the&#xA;Greek derivative &lt;em&gt;logistikós&lt;/em&gt; ‘of calculation’), and&#xA;logarithm, coined in the early 17th century by&#xA;the English mathematician John Napier from&#xA;Greek &lt;em&gt;logós&lt;/em&gt; ‘ratio’ and &lt;em&gt;arithmós&lt;/em&gt; ‘number’&#xA;(source of English &lt;em&gt;arithmetic&lt;/em&gt; [13]).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't know if &lt;a href=&quot;https://www.funtrivia.com/askft/Question101467.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;zbeckabee's quotation here&lt;/a&gt; matches Napier's rationale.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It comes from the Greek words 'logos' and 'arithmes.' The second word means 'number'. The first word has a broad range of meanings, starting from 'word' or 'speech' (thus our word 'dialog'), going into 'thought' and 'reason' (thus 'logical'), and on to 'proportion'. In this range of meanings, it resembles the Latin 'ratio', which means both 'reason' and 'proportion' (thus the two meanings of 'rational': reasonable, and expressable as a ratio or proportion of two integers).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus, a 'logarithm' is a 'proportion number'. I suppose the thinking was that logarithms help in the calculation of proportions. Instead of multiplying a number by a fraction, you just add or subtract the logarithm of the fraction to the logarithm of the number.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastEditorDisplayName="user3478" LastEditDate="2021-03-18T09:13:39.507" LastActivityDate="2022-04-30T23:15:41.757" Title="Why did John Napier use logós ‘ratio’, to coin &quot;logarithm&quot;?" Tags="|logarithm|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13005" PostTypeId="2" ParentId="12906" CreationDate="2021-03-18T17:57:42.383" Score="1" Body="&lt;p&gt;I suspect if the witticism had been associated to a specific individual, we'd be hearing more and more, and also competitive attributions to it. Oppenheimer (below) had a yen for adopting catchy expressions.&lt;/p&gt;&#xA;&lt;p&gt;It was an obvious one, in the 1950s, after the discovery of mesons, hadronic resonances, hyperons, and on and on. The new accelerators and technique made possible the sudden proliferation of particles, so that there was an acute classification issue developing, akin to that of the Zoologists--or Mendeleev's.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;Strange Beauty: Murray Gell-Mann and the Revolution in Twentieth-Century Physics&lt;/em&gt; (1999) by George Johnson, p 222, you can read&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“If you run out of Greek letters, you can always use the names of Pullman cars,” Von Neumann said at the time.  Oppenheimer provided the name that stuck: the subnuclear zoo.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;p 755&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[footnote 108]: Oppenheimer coined the term &amp;quot;subnuclear zoo&amp;quot; in a public lecture at the Rochester 6 conference, [Sec VIII p1 of the proceedings]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;p 337&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finally, as the 1950s drew to a close, there was a deep feeling that an understanding of how the forces were related would point the way toward categorizing the dozens of different particles that continued to pour from the sky and accelerators in a random spray.  Oppenheimer talked about the subatomic zoo, but physicists had hardly begun putting the beast in labeled cages ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Rochester 6 took place in 1956.&lt;/p&gt;&#xA;&lt;p&gt;Of course, the biography deals with the &amp;quot;Mendeleev&amp;quot; &lt;strong&gt;and&lt;/strong&gt;, remarkably, in short order!, the &amp;quot;Henry Moseley&amp;quot; of hadrons, among other things, namely the tabulator, classifier, and then &lt;em&gt;explainer of the reason&lt;/em&gt; behind the pattern: quarks.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-03-18T19:16:40.470" LastActivityDate="2021-03-18T19:16:40.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13007" PostTypeId="1" CreationDate="2021-03-19T20:55:18.927" Score="0" ViewCount="199" Body="&lt;p&gt;I am interested in Lev Landau. I was trying to find what was his approach to the interpretation of quantum mechanics, I am unable to find any particular source were this is discussed.&lt;/p&gt;&#xA;&lt;p&gt;Was he an advocate of any particular interpretation? or was he like Dirac that was purposely never interested in this problem? I hope there is some information about it.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="4251" LastEditDate="2021-09-06T15:23:29.770" LastActivityDate="2021-09-06T15:23:29.770" Title="Was Landau interested in the interpretations of quantum mechanics?" Tags="|physics|biographical-details|quantum-mechanics|physicists|landau|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13009" PostTypeId="1" CreationDate="2021-03-20T17:15:50.387" Score="2" ViewCount="701" Body="&lt;p&gt;I was browsing the book &lt;em&gt;Isotopes: Principles and Applications&lt;/em&gt; by Faure and Mensing and I would like to know what is the history of the use of the word daughter for a decay product. It seems to me that the book gives no explanation about the terminology so I think it is a well established convention for a long time.&lt;/p&gt;&#xA;&lt;p&gt;A quick search gives me this Historical perspective:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A radionuclide generator is a concept defined as an effective&#xA;radiochemical separation of decaying parent and daughter radionuclides&#xA;such that the daughter is obtained in a pure radionuclidic and&#xA;radiochemical form. Radionuclide generators were historically called&#xA;‘cows’ since the daughter radioactivity was ‘milked’ (i.e. removed)&#xA;from its precursor and the parent then generated a fresh supply of the&#xA;daughter.&lt;sup&gt;1&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but even if I understand the cow and milk metaphor I do not understand the use of the word daughter, contrasted for example with the word son; moreover, I do not get the asymmetry between daughter, which has a specific gender, and parent, which it seems to me gender neutral.&lt;/p&gt;&#xA;&lt;p&gt;I know that maybe this question is more suitable for &lt;a href=&quot;https://english.stackexchange.com/&quot;&gt;https://english.stackexchange.com/&lt;/a&gt; but I am interested in the context of physics: maybe someone received an explanation about the terminology during a formal nuclear physics university lesson.&lt;/p&gt;&#xA;&lt;p&gt;In my bookshelf, an Italian author&lt;sup&gt;2&lt;/sup&gt; uses the word &lt;em&gt;discendente&lt;/em&gt; (descendant) that I would consider almost gender neutral; another Italian author&lt;sup&gt;3&lt;/sup&gt;, translating from the English, uses the word &lt;em&gt;figlio&lt;/em&gt; (son).&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;Radionuclide Generators. In : &lt;em&gt;Handbook of Nuclear Chemistry&lt;/em&gt;. Boston, MA : Springer US, 2004. p. 1379–1415. ISBN 978-0-387-30682-7. Available from: &lt;a href=&quot;https://doi.org/10.1007/0-387-30682-X_32&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/0-387-30682-X_32&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt;Paolo Chiorboli, &lt;em&gt;Fondamenti di Chimica&lt;/em&gt;, UTET, Torino, 1975 (II edizione, 1980).&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;3&lt;/sup&gt;Cornelis Klein, Anthony R. Philpotts&#xA;&lt;em&gt;Mineralogia e petrografia&lt;/em&gt;&#xA;Prima edizione italiana condotta sulla seconda edizione inglese&#xA;A cura di Giorgio Gasparotto, Roberto Braga. Zanichelli, Bologna, 2018.&#xA;&lt;/p&gt;&#xA;" OwnerUserId="5516" LastEditorUserId="10106" LastEditDate="2021-03-21T09:48:11.863" LastActivityDate="2023-08-12T15:44:53.763" Title="What is the history of the use of the word daughter for a decay product in nuclear physics?" Tags="|physics|terminology|nuclear-physics|radioactivity|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13012" PostTypeId="1" AcceptedAnswerId="13015" CreationDate="2021-03-21T17:59:48.277" Score="2" ViewCount="163" Body="&lt;p&gt;I am searching for Newton's original formulation of the Law of Cooling. In his &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.449479/page/n141/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; (p.140-143), Newton said:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;For the heat which the hot iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;For thus equal parts of the air are warmed in equal times and carry away a heat proportional to the heat of the iron.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, he does not really state that the change in temperature is proportional to the difference in temperatures between the body and the environment. Therefore I wonder, where/how did Newton originally formulate his law of cooling or do the quotes above imply the law and I am just quibbling?&lt;/p&gt;&#xA;" OwnerUserId="14130" LastEditorUserId="10106" LastEditDate="2021-03-21T19:35:21.697" LastActivityDate="2021-03-22T09:42:24.797" Title="How did Newton originally formulate the Law of Cooling?" Tags="|physics|newton|thermodynamics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13014" PostTypeId="1" CreationDate="2021-03-22T08:35:55.423" Score="3" ViewCount="111" Body="&lt;p&gt;&lt;a href=&quot;https://www.etymonline.com/search?q=temperature&quot; rel=&quot;nofollow noreferrer&quot;&gt;Etymonline&lt;/a&gt;  avouches that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sense of &amp;quot;degree of heat or cold&amp;quot; first recorded 1670 (Boyle), from Latin &lt;em&gt;temperatura&lt;/em&gt;, used in this sense by Galileo.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But &amp;quot;degree of heat or cold&amp;quot; doesn't semantically appertain to &lt;em&gt;temperatura&lt;/em&gt;, which originally signified &amp;quot;mixture&amp;quot; as quoted below. Why did Galileo pick this unrelated noun of &lt;em&gt;temperatura&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;temper [OE]&lt;/h3&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The verb &lt;em&gt;temper&lt;/em&gt; was borrowed into&#xA;Old English from Latin &lt;em&gt;temperāre&lt;/em&gt; ‘mix, blend’.&#xA;This seems originally to have meant ‘mix in due&#xA;proportion’, and so may have been derived from&#xA;Latin &lt;em&gt;tempus&lt;/em&gt; ‘time, due time’ (source of English&#xA;&lt;em&gt;temporary&lt;/em&gt;). The noun &lt;em&gt;temper&lt;/em&gt; was derived from&#xA;the verb in the 14th century in the sense ‘mixture&#xA;of elements’, and this led on in the 17th century&#xA;to ‘set of mental traits’ (a meaning that has now&#xA;largely passed to the derivative &lt;em&gt;temperament&lt;/em&gt;&#xA;[15]). The modern sense ‘ill humour’ emerged&#xA;from this in the 19th century. Another meaning&#xA;of Latin &lt;em&gt;temperāre&lt;/em&gt; was ‘restrain oneself’, which&#xA;has come through into the derivatives&#xA;&lt;em&gt;temperance&lt;/em&gt; [14] and &lt;em&gt;temperate&lt;/em&gt; [14]. Other&#xA;relatives include &lt;em&gt;distemper&lt;/em&gt; and &lt;em&gt;temperature&lt;/em&gt;.&lt;br /&gt;&#xA;       &lt;em&gt;Tamper&lt;/em&gt; probably originated as an alteration&#xA;of &lt;em&gt;temper&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;temperature [16]&lt;/h3&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Like its relatives &lt;em&gt;temper&lt;/em&gt; and&#xA;&lt;em&gt;temperament&lt;/em&gt;, &lt;em&gt;temperature&lt;/em&gt; originally meant&#xA;‘mixture’ (Philemon Holland in 1601 wrote of ‘a&#xA;temperature of brass and iron together’). The&#xA;modern sense ‘degree of heat’ emerged in the&#xA;late 17th century, and seems to have evolved&#xA;from another early and now obsolete sense,&#xA;‘mild weather’. This reflected the ‘restraint’&#xA;strand of meaning in the word’s ultimate source,&#xA;Latin &lt;em&gt;temperāre&lt;/em&gt;, which also survives in English&#xA;&lt;em&gt;temperance&lt;/em&gt; and &lt;em&gt;temperate&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Word Origins&lt;/em&gt; (2005 2e) by John Ayto. p 500 Left column.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.merriam-webster.com/words-at-play/origin-of-temperament-and-temperature&quot; rel=&quot;nofollow noreferrer&quot;&gt;Temperament and Temperature: Are They Related? | Merriam-Webster&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although &lt;em&gt;temperature&lt;/em&gt; is rarely if ever used in these ways today, the implication of moderation likely influenced people to apply it as a word for the mildness of weather or climate,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastEditorDisplayName="user3478" LastEditDate="2021-03-23T05:51:32.573" LastActivityDate="2021-03-23T05:51:32.573" Title="Why did Galileo pick &quot;temperatura&quot; to signify 'degree of heat or cold'?" Tags="|terminology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13015" PostTypeId="2" ParentId="13012" CreationDate="2021-03-22T09:42:24.797" Score="2" Body="&lt;p&gt;This explanation helped me understand Newton's language and answered my question:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;the excess of the degrees of the heat … were in geometrical progression when the times are in an arithmetical progression (by 'degree of heat' Newton meant what we now call 'temperature', so that 'excess of the degrees of the heat' means ‘temperature difference').&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20150614122639/http://paginas.fisica.uson.mx/laura.yeomans/tc/Sci-Edu-Springer-2010.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Besson's article&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14130" LastActivityDate="2021-03-22T09:42:24.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13017" PostTypeId="1" CreationDate="2021-03-22T20:00:00.077" Score="2" ViewCount="79" Body="&lt;p&gt;Forces like the gravitational and the electric force are in the direction of the force field. Why is it that the magnetic force is orthogonal to the velocity of the charge and the magnetic field? Was this derived or empirically verified? If the latter, then what experiments were performed, and how did the cross product perfectly fit the direction of the orthogonal vector?&lt;/p&gt;&#xA;" OwnerUserId="14039" LastEditorUserId="10106" LastEditDate="2021-03-23T05:26:02.513" LastActivityDate="2021-03-23T05:26:02.513" Title="How was it discovered that the Magnetic force was orthogonal to the B-field and the velocity of the charged particle?" Tags="|physics|magnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13018" PostTypeId="1" AcceptedAnswerId="13019" CreationDate="2021-03-23T07:01:41.863" Score="5" ViewCount="131" Body="&lt;p&gt;Blyth writes in his 2006 book &lt;em&gt;Lattices&lt;/em&gt; (p. 65):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;It is a curious historical fact that it was originally thought that every&#xA;lattice was distributive!&lt;/strong&gt; That this is not so is shown by the following  example [M3]&#xA;of a modular lattice that is not distributive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I'll spare you the Hasse diagram of M3 here since it's well-know/obvious and &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Distributive_lattice&amp;amp;oldid=1013366038#Characteristic_properties&quot; rel=&quot;noreferrer&quot;&gt;found&lt;/a&gt; on Wikipedia too.)&lt;/p&gt;&#xA;&lt;p&gt;What I want to ask here is: what does Blyth mean by the first sentence highlighted? Did people at one point &lt;em&gt;define&lt;/em&gt; a lattice in a way that made it distributive, or was M3 (or N5) really that non-obvious as a counter-example to some early writers on the topic even under a more general defintion?&lt;/p&gt;&#xA;" OwnerUserId="1885" LastEditorUserId="1885" LastEditDate="2021-03-23T07:15:08.443" LastActivityDate="2021-03-24T04:31:14.670" Title="In what sense was it thought (at one point) that every lattice was distributive?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13019" PostTypeId="2" ParentId="13018" CreationDate="2021-03-23T10:04:22.250" Score="7" Body="&lt;p&gt;The original approach to lattices, following Boole, was algebraic rather than graphic or order-theoretic, and simple examples that quickly come to mind today were not really in the offering. The pioneers, Boole, Peirce and Schröder, operated with a fluid set of algebraic rules and logic related principles, and examples of logical systems they operated with were hard to visualize and distinguish from each other. Blyth likely refers to the episode with Peirce's casual remark in On the algebra of logic (1880) that was disputed by Schröder. It was instrumental in sorting out which sets of rules lead to which structures, and coining the lattice concept in particular. The relations between the syllogistic, propositional logic and &amp;quot;the logic of relatives&amp;quot; (predicate calculus that Peirce discovered independently of Frege) were clarified around the same time.&lt;/p&gt;&#xA;&lt;p&gt;Ma and Pietarinen describe the episode as follows in &lt;a href=&quot;https://www.researchgate.net/publication/311363435_Peirce%27s_Sequent_Proofs_of_Distributivity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peirce’s Sequent Proofs of Distributivity&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the 1880 paper, Peirce stated the following distributive laws... He casually mentions that “they are easily proved . . ., but the proof is too tedious&#xA;to give” [11, p. 33]. This passage provoked a challenge by Schröder, who took the&#xA;laws of distributivity to be independent from the lattice axioms. A rejoinder and&#xA;a lively discussion ensued and Peirce’s lost and subsequently recovered version&#xA;of the proof was finally added to Huntington’s 1904 paper... Huntington published in 1904 the proof Peirce had sent to him,&#xA;including Peirce’s footnote about it. In the published proof, the axiom — Huntington’s “postulate” — number 9 is crucial. Without it, the axioms 1–8 only&#xA;define uniquely complemented lattices which need not be distributive.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is Peirce's own description from the follow-up paper On the algebra of logic (1885), where he admits slipping in a &amp;quot;non-syllogistic&amp;quot; principle (dilemmatic argument) into his proof of the &amp;quot;wrong&amp;quot; direction of distributivity (&lt;span class=&quot;math-container&quot;&gt;$-\!\!\!&amp;lt;$&lt;/span&gt; is Peirce's symbol for the material implication):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The dilemma was only introduced into logic from rhetoric&#xA;by the humanists of the renaissance; and at that time logic was studies with so&#xA;little accuracy that the peculiar nature of this mode of reasoning escaped notice.&#xA;I was thus led to suppose that the whole non-relative logic was derivable from the&#xA;principles of the ancient syllogistic, and this error is involved in Chapter II of my paper in the third volume of this Journal [the 1880 paper]. My friend, Professor&#xA;Schröder, detected the mistake and showed that the distributive formulae&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x + y)z\ -\!\!\!&amp;lt;\ xz + yz$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$(x + z)(y + z)\ -\!\!\!&amp;lt;\ xy + z$$&lt;/span&gt;&#xA;could not be deduced from syllogistic principles. I had myself independently discovered&#xA;and virtually stated the same thing. (Studies in Logic, p. 189.)&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Schröder developed what we would now call theory of lattices in &lt;em&gt;Vorlesungen Über die Algebra der Logik&lt;/em&gt; (1891), the first systematic exposition of modern logic that laid the foundation of Russell's &lt;em&gt;Principia&lt;/em&gt;. There he gave the first example of a non-distributive lattice, but it was not an elementary one one would think of today. Here is from &lt;a href=&quot;https://pdfs.semanticscholar.org/9afd/c33b44792492d01024a6026bf5d05594a049.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peckhaus's Mathematical Origins of 19th Century Algebra of Logic&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the sections dealing with statements&#xA;without negation he proves one direction of the distributivity law for&#xA;logical addition and logical multiplication, but shows that the other side can not be proved, he rather shows its independence by formulating a model in&#xA;which it does not hold, the so-called “logical calculus with groups, e. g. functional&#xA;equations, algorithms or calculi”. He thereby found the first example&#xA;of a non-distributive lattice.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-03-24T04:31:14.670" LastActivityDate="2021-03-24T04:31:14.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13022" PostTypeId="1" AcceptedAnswerId="13025" CreationDate="2021-03-24T20:09:27.053" Score="4" ViewCount="1730" Body="&lt;p&gt;&lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4684-5772-8_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/chapter/10.1007/978-1-4684-5772-8_2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Using Newton's recently formulated laws of motion, Brook Taylor&#xA;(1685–1721) discovered the wave equation by means of physical insight&#xA;alone&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Wave_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Wave_equation&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;says:&lt;/p&gt;&#xA;&lt;p&gt;In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Who discovered the basic 1D wave equation?&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="7211" LastActivityDate="2021-03-26T09:37:30.277" Title="Who discovered the wave equation?" Tags="|physics|mathematical-physics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13023" PostTypeId="2" ParentId="13004" CreationDate="2021-03-25T13:49:32.333" Score="3" Body="&lt;p&gt;Your specific question is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I don't see a ratio in this definition of the logarithm in James Stewart, [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The ratio is implicit in that definition, but it is explicit in the change of base formula&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ \log_a(x) = \ln(x)/\ln(a) $$&lt;/span&gt;&#xA;which equates a logarithm to a given base to the quotient (or ratio) of&#xA;natural logarithms. In fact, logarithms to a given base are exactly similar&#xA;to measurement of a quantity in a given unit of measure. That is, they are&#xA;the ratio of a given quantity relative to the unit quantity. That is how&#xA;you can conceptualize logarithms as the relative ratio of the growth rate&#xA;of two geometric progressions. One based on the given growth rate and the&#xA;other based on the standard unit growth rate. For example, the geometric&#xA;progression &lt;span class=&quot;math-container&quot;&gt;$\,1,a^2,a^4,a^6,\dots\,$&lt;/span&gt; grows at twice the rate of the&#xA;geometric progression &lt;span class=&quot;math-container&quot;&gt;$\,1,a,a^2,a^3,\dots\,$&lt;/span&gt; in the sense that the&#xA;first takes &lt;span class=&quot;math-container&quot;&gt;$\,1/2\,$&lt;/span&gt; as many steps to grow to a given size than the other.&#xA;This interpretation agrees with the result &lt;span class=&quot;math-container&quot;&gt;$\,\log_a(a^2) = 2.\,$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="13622" LastEditDate="2022-04-30T23:15:41.757" LastActivityDate="2022-04-30T23:15:41.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13024" PostTypeId="1" CreationDate="2021-03-25T17:52:21.220" Score="3" ViewCount="123" Body="&lt;p&gt;I enjoyed Asimov's &amp;quot;The new intelligent man's guide to science&amp;quot; very much when I was a teenager. I noticed that he later published &amp;quot;Asimov's New Guide to Science&amp;quot;, however that was also nearly 40 years ago. My question is, are there more recent books that are more or less equivalent (in term of range and depth) to Asimov's book?&lt;/p&gt;&#xA;" OwnerUserId="14169" LastActivityDate="2021-03-25T17:52:21.220" Title="Is there a modern equivalence of Asimov's book &quot;The new intelligent man's guide to science&quot;?" Tags="|reference-request|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13025" PostTypeId="2" ParentId="13022" CreationDate="2021-03-25T20:58:23.687" Score="6" Body="&lt;p&gt;On the traditional account, it was D’Alembert. Taylor did investigate the vibrating string earlier (c.1713), but he treated it as a collection of individually vibrating points and did not write down the PDE. He did not even use derivatives and gave geometric arguments instead, but he did guess the shape of the fundamental mode correctly &amp;quot;by means of physical insight&amp;quot;. Struik wrote even more charitably:&amp;quot;&lt;em&gt;This means that for small vibrations Taylor has in principle&#xA;[the wave equation, although] there is no evidence that he had any notion of partial derivatives&lt;/em&gt;&amp;quot;. Perhaps, Robinson, quoted in the OP, shared the sentiment.&lt;/p&gt;&#xA;&lt;p&gt;Here is from &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4613-9461-7_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cannon's chapter on Taylor&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Brook Taylor presented his work on the vibrating string to the Royal&#xA;Society in September of 1712. His paper appeared in 1714 in the volume&#xA;of the Philosophical Transactions for 1713; 1 and a revised version was&#xA;included in his book on the calculus/ published in 1715. This work introduces&#xA;the pendulum condition. All elements of the string are supposed to&#xA;undergo small vibrations as simple pendulums all of the same period; hence&#xA;they are restored by harmonic forces of the same intensity... He sees each element&#xA;&lt;span class=&quot;math-container&quot;&gt;$\rho\ ds$&lt;/span&gt; as a simple particle oscillating in one dimension with the same period&#xA;&lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; no matter which the element and what the amplitude of the oscillation.&#xA;Now, presumably thinking of this from the perspective of dynamics in a&#xA;single degree of freedom, Taylor concludes that each element &lt;span class=&quot;math-container&quot;&gt;$\rho\ ds$&lt;/span&gt; moves&#xA;as a simple pendulum... Taylor doesn't use differential notation... Taylor doesn't introduce a notation for the mass density &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt;;&#xA;geometrically Taylor's &lt;span class=&quot;math-container&quot;&gt;$pP$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$ds$&lt;/span&gt; and physically it is &lt;span class=&quot;math-container&quot;&gt;$\rho\ ds$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;D’Alembert did build on Taylor's work, and &lt;em&gt;he&lt;/em&gt; first used partial derivatives and wrote down the equation, see &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007%2F978-1-4612-1766-4_2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Garber, Vibrating Strings and Eighteenth-Century Mechanics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;D'Alembert was not the first mathematician to focus on establishing mathematically,&#xA;then solving, the equation of motion for a string under tension. In&#xA;examining the problem, d'Alembert exploited the newly developed partial differential&#xA;equations... What d' Alembert proposed to demonstrate for a string under tension,&#xA;using Taylor's conditions, was that &amp;quot;there are infinitely many curves other than&#xA;the companion of the elongated cycloid [sine curve] that satisfy the problem in&#xA;question&amp;quot;. Starting from Brooke Taylor's work, d'Alembert focussed on Taylor's&#xA;expression for the accelerating force on an element of the string...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-03-26T09:29:57.897" LastActivityDate="2021-03-26T09:29:57.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13026" PostTypeId="2" ParentId="13022" CreationDate="2021-03-26T09:06:13.863" Score="2" Body="&lt;p&gt;Mersenne's law were first proposed by Marin Mersenne in his 1636 work, &lt;em&gt;Harmonie Universalis&lt;/em&gt;. It described the frequency of oscillation of a stretched string in relation to its length, its density and the stretching force and gave an equation for the fundamental frequency. This was all established on the basis of experiments and not on the basis of  mechanics which was only then beginning to gather momentum with the work of Galileo, Huygens and Newton with Newton publishing his &lt;em&gt;Principia&lt;/em&gt; in 1687.&lt;/p&gt;&#xA;&lt;p&gt;Although Mersenne's law was correct, his measurements were not very precise and they were greatly improved by Joseph Saveur. Incidentally, he is credited with coining the term &lt;em&gt;acoustique&lt;/em&gt; which he derived from the Greek word meaning able to be heard. Moreover, in 1701 he proposed the superposition principle. Again, all of this work was experimental with no theoretical mechanics involved.&lt;/p&gt;&#xA;&lt;p&gt;Brook Taylor of Taylor series fame was a fervant Newtonian and  probably one of mathematics most artistically gifted members, he learned in his adolescence to play the harpsichord and to paint and which later found mathematical expression through his pioneering theoryof the vibrating string and also of perspective. He was the first mathematician to deduce Mersenne's expression from the geometry of the string and its mechanics by considering it as a row of rigid particles.&lt;/p&gt;&#xA;&lt;p&gt;His interest in oscillations is apparent in his first important paper which was published in &lt;em&gt;The Philosophical Transactions of the Royal Society&lt;/em&gt; in 1714 but which was actually written in 1708, according to his correspondance with John Keill, a mathematician at St John's, Oxford, where he had matriculated in 1701. Here he tackled the centre of oscillation of a body. The work led to a priority dispute with Bernouilli.&lt;/p&gt;&#xA;&lt;p&gt;The following year he published &lt;em&gt;Methodus Incrementorum Directa et Inversa&lt;/em&gt; in London. Here, he moves on to the theory of the transverse vibrations of strings. He wasn't attempting to find the equation of motion for a string, but was considering the oscillations of a flexible string in terms of the isochrony of a pendulum. He showed that the resultant of forces acting on the extremities of a string was proportional to the curvature and directed along its normal .&lt;/p&gt;&#xA;&lt;p&gt;Newton had approached curvature as the determination of the centre of curvature as the limit of the intersection of two normals. Although this was not published until 1736, Taylor was familiar with this work but preferred, at least here, to concieve the radius of curvature as Newtons continental rival, Liebniz, did, taking it to be centre of a limiting circle through three points and associated curvature with the angle of contact dating back to Euclid.&lt;/p&gt;&#xA;&lt;p&gt;He then used these notions to describing the fundamental mode of a vibrating string. In propositions XXII &amp;amp; XXIII he showed that under his conditions each point would vibrate as a cycloidal pendulum, and determined the period in terms of the length of the string and its tension. In deriving this formula he assumed that every point of the string passes through its equilibrium position at the same time, an assumption that D'Alembert showed later to be unnecessary.&lt;/p&gt;&#xA;&lt;p&gt;Although Taylor was also able to describe the form of the string at any instant it was nevertheless a static solution, the element of time was not really introduced. This was the achievement of D'Alembert who gave the equation of motion of the string as the one dimensional wave equation in 1747 in his &lt;em&gt;Recherches sur la courbe qui forme une corde tendue mise en vibration&lt;/em&gt; in which he obtained the PDE of a vibrating string by combining the restoring force of Taylor's with Newton's acceleration law.&lt;/p&gt;&#xA;&lt;p&gt;It's probably worth adding that in 1751, David Bernouilli gave a theoretical justification of the superposition principle that had been proposed by Joseph Saveur in 1701 and that in the late 1740s, a bitter dispute broke out between d'Alembert and Euler over the solution of the wave equation. The issues were, the nature of mathematical functions, that is, what is meant by their continuity and differentiability and which highlighted the relations between mathematics, metaphysics and experimental natural philosophy in the 18C.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Elizabeth Garber, &lt;em&gt;The Language of Physics&lt;/em&gt;&#xA;Wikipedia.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-03-26T09:37:30.277" LastActivityDate="2021-03-26T09:37:30.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13027" PostTypeId="2" ParentId="13009" CreationDate="2021-03-26T10:38:47.440" Score="4" Body="&lt;p&gt;According to Wikipedia, a decay product is also referred to as a daughter product, a daughter isotope, radio-daughter or daughter nuclide.&lt;/p&gt;&#xA;&lt;p&gt;The term possibly arose from cell biology where it is common to call the cells after division daughter cells, possibly because they too can go on to divide. In fact, it is documented that the term fission was borrowed from cell biology:&lt;/p&gt;&#xA;&lt;p&gt;In 1938, Lise Meitner spent Christmas at the home of her friend Eva von Bahr and whilst there, she received a letter from Otto Hahn describing his chemical proof that some of the products of the bombardment of uranium with neutrons were barium. No previous known method of radioactive decay could account for such a large product and she wrote back to Hahn to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the moment, the assumption of such a thoroughgoing breakup seems very difficult to me, but in nuclear physics we have experienced so many surprises, that one cannot unconditionally say: &amp;quot;it is impossible&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Later, after a walk in the woods with her nephew Otto Frisch, in fact in the midst of one, Meitner &amp;amp; Frisch correctly interpreted Hahn's results as the splitting of the uranium atom. Frisch decided that he needed a name for this new process and spoke to William Arnold, an American biologist and asked him what biologists called the process by which living cells divided into two cells. Arnold informed him that the term was &lt;em&gt;fission&lt;/em&gt; and Frisch duly used the term in his joint paper with Meitner submitted to &lt;em&gt;Nature&lt;/em&gt; describing the phenomenon of uranium fission. As the term fission was new, he used it in quotes. However, it was the first and last time that the term fission was used in quotes as the term was quickly assimilated into the vocabulary of nuclear physics.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="7205" LastEditDate="2023-01-25T04:37:44.413" LastActivityDate="2023-01-25T04:37:44.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13028" PostTypeId="1" CreationDate="2021-03-26T21:09:00.057" Score="0" ViewCount="74" Body="&lt;p&gt;Do you know of some books or papers dealing with the history of the Euler line?&lt;/p&gt;&#xA;&lt;p&gt;Was L. Euler the first mathematician that &lt;em&gt;notice&lt;/em&gt; its existence? Are there any interesting paragraphs out there discussing the fact that apparently it was not widely known in Ancient Greece?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for your replies.&lt;/p&gt;&#xA;" OwnerUserId="6596" LastActivityDate="2021-03-26T21:09:00.057" Title="On the Euler line" Tags="|reference-request|euclidean-geometry|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13029" PostTypeId="1" CreationDate="2021-03-26T21:33:23.267" Score="1" ViewCount="182" Body="&lt;p&gt;Do you know if  there is a pdf file containing President George Washington's notes on geometry and surveying somewhere in the Internet?&lt;/p&gt;&#xA;&lt;p&gt;I recall reading a few weeks ago that those notes had been recently made publicly available by the Library of Congress but I could not find a single file containing all of them in the relevant site:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.loc.gov/resource/mgw1a.002/?sp=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.loc.gov/resource/mgw1a.002/?sp=2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If it is actually not possible to have these notes in a single pdf file, do you know of an expert on President Washington that has written about what it is that we are to find in those manuscripts?&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="6596" LastEditorUserId="6596" LastEditDate="2021-03-26T21:39:30.943" LastActivityDate="2021-04-12T19:29:48.767" Title="G. Washington notes on geometry" Tags="|reference-request|euclidean-geometry|nachlasse|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13030" PostTypeId="1" AcceptedAnswerId="13032" CreationDate="2021-03-26T22:23:40.880" Score="1" ViewCount="136" Body="&lt;p&gt;In his letter to Hansen (december 11, 1825), Gauss responds to several questions asked by Hansen about his recent 1822 publication (&amp;quot;&lt;em&gt;&lt;strong&gt;A general solution to the problem of mapping the parts of a given surface onto another surface such that the image and the mapped part are similar in the smallest parts&lt;/strong&gt;&lt;/em&gt;&amp;quot;), and writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You are quite right, that the essential condition in every map-projection is the infinitesimal similarity, a condition which should be discarded only in very special cases of need.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Reading more into this letter, i saw that Gauss gives some details about this remark, and says that the exceptions mentioned by him arise when there are singular points on the surfaces (i should have included a translation of Gauss's letter, but google translate gives too bad translation). Unfortunately, and i know this might look like a basic question to some readers here, i don't understand why exceptions arise near singular points.&lt;/p&gt;&#xA;&lt;p&gt;In my reading of articles about the ideas of Gauss and his followers, I had seen that this letter to Hansen is mentioned and cited many times, not in the context of my question, but as a kind of historical document that traces his own perspective of many of his discoveries (issues mentioned in this letter include the meaning of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; and the metaphysics of space). Since this letter belongs to a period of intensive occupation with differential geometry, and since it was around this time that Gauss tried to classify &amp;quot;tract figures&amp;quot; (singular curves that arise as projections of knots into the plane) in plane, i suspect there might be interesting connections of this remark to his other activities. Anyway, my last statement is entirely speculation, and and i'd rather prefer to concentrate at giving an explanation to Gauss's remark in the context of conformal mappings.&lt;/p&gt;&#xA;&lt;p&gt;Any explanation\usefull comment will be blessed!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2021-03-27T14:01:30.640" Title="Explanation of a remark of Gauss on exceptional cases of conformal mappings" Tags="|gauss|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13031" PostTypeId="2" ParentId="13029" CreationDate="2021-03-27T02:43:13.403" Score="4" Body="&lt;p&gt;The historian of mathematics Fred Rickey (West Point) may have some information of use to you. He is a scholar who has published &lt;a href=&quot;https://www.researchgate.net/profile/Fred-Rickey&quot; rel=&quot;nofollow noreferrer&quot;&gt;multiple papers on George Washington's notes&lt;/a&gt;. If a complete version of Washington's notes is available online, Fred Rickey might well know how to find it.&lt;/p&gt;&#xA;" OwnerUserId="14176" LastEditorUserId="26" LastEditDate="2021-04-12T19:29:48.767" LastActivityDate="2021-04-12T19:29:48.767" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13032" PostTypeId="2" ParentId="13030" CreationDate="2021-03-27T14:01:30.640" Score="3" Body="&lt;p&gt;First of all, one would need to define the meaning of conformality of a map &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; from a nonregular surface &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; to the plane at points &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; of where &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is non-regular. This is actually a nontrivial issue. One possibility is to use the &amp;quot;tangent cone&amp;quot; (if it exists, which is not always the case!) &lt;span class=&quot;math-container&quot;&gt;$C_p(S)$&lt;/span&gt;, and then require for the map &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; to preserve angles between the directions. In the example of the cone that I gave you  in my comment and , &lt;span class=&quot;math-container&quot;&gt;$C_p(S)$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; itself. Hence, there is no chance  for a conformal map to the plane because the total angle of the cone at &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;2\pi$&lt;/span&gt;. There are other examples, where &lt;span class=&quot;math-container&quot;&gt;$C_p(S)$&lt;/span&gt; is actually 1-dimensional (cuspidal singularities). One can give examples  of non-regular surfaces with tangent cones isometric to the plane. Could Gauss handle these? No: His achievement was actually quite spectacular, he proved that for &lt;strong&gt;real-analytic&lt;/strong&gt; regular surfaces, the metric is locally conformally flat. However, he could not even handle the case of infinitely-differentiable surfaces (or metrics).&lt;/p&gt;&#xA;&lt;p&gt;Gauss' work set off a long chain of developments, documented for instance in &lt;a href=&quot;https://en.wikipedia.org/wiki/Isothermal_coordinates&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; Wikipedia article, which culminated in the &lt;strong&gt;Measurable uniformization theorem&lt;/strong&gt;. The modern viewpoint on this issue is to work within a suitable Sobolev space (&lt;span class=&quot;math-container&quot;&gt;$W^{2,1}$&lt;/span&gt;). But Gauss simply did not have the tools needed to overcome the difficulties that one has to deal with in the singular case.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2021-03-27T14:01:30.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13033" PostTypeId="1" CreationDate="2021-03-28T07:34:44.137" Score="2" ViewCount="66" Body="&lt;p&gt;In the wikipedia article about &lt;a href=&quot;https://en.wikipedia.org/wiki/Snell%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snell's law&lt;/a&gt; of indexes of refraction there is a demonstration that shows the interconnection with Fermat's law of least time.&lt;/p&gt;&#xA;&lt;p&gt;The strategy of that demonstration:&lt;br /&gt;&#xA;The overall travel time is a function of the variation of the path. The aspect that is to be compared is not the travel time itself, but the travel time &lt;em&gt;derivative&lt;/em&gt; with respect to the path variation.&lt;/p&gt;&#xA;&lt;p&gt;There is medium 1 and medium 2, and there is the interface between the two media. In the general case that interface is not at right angles to the line connecting the start point and the end point.&lt;/p&gt;&#xA;&lt;p&gt;There is the travel time &lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt; for the path from start point to the interface, and &lt;span class=&quot;math-container&quot;&gt;$t_2$&lt;/span&gt; for the path from the interface to the end point.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt; has an individual minimum, and &lt;span class=&quot;math-container&quot;&gt;$t_2$&lt;/span&gt; has an individual minimum. The respective minima do not coincide, so there is an interval in the variation space where &lt;span class=&quot;math-container&quot;&gt;$t_1$&lt;/span&gt; is still descending whereas  &lt;span class=&quot;math-container&quot;&gt;$t_2$&lt;/span&gt; is alraady ascending again (or vice versa). The global minimum of the travel time occurs at the &lt;em&gt;crossover point&lt;/em&gt; of one still descending and the other already ascending. At that crossover point the two &lt;em&gt;derivatives&lt;/em&gt; match.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;In physics derivations are bidirectional.&lt;/p&gt;&#xA;&lt;p&gt;(There is a general discussion of &lt;a href=&quot;https://physics.stackexchange.com/a/525265/17198&quot;&gt;derivation in physics&lt;/a&gt; by stackexchange contributor knzhou, writing: &amp;quot;in physics, you can often run derivations in both directions: you can use X to derive Y, and also Y to derive X. That isn't circular reasoning, because the real support for X (or Y) isn't that it can be derived from Y (or X), but that it is supported by some experimental data D. This two-way derivation then tells you that if you have data D supporting X (or Y), then it also supports Y (or X).&amp;quot;&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;The thrust of my question is the following:&lt;br /&gt;&#xA;Snell's law describes refraction as something that occurs &lt;em&gt;locally&lt;/em&gt;. Light is propagating, and when it encounters an interface to a medium with a different index of refraction the direction of propagation changes right there, right at that instant in time.&lt;/p&gt;&#xA;&lt;p&gt;Fermat's law of least time obtains the change of direction as a consequence of a &lt;em&gt;global&lt;/em&gt; criterium; the global travel time.&lt;/p&gt;&#xA;&lt;p&gt;So: at first sight the two approaches appear to be in some way the opposite of each other.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;The derivation shows that the the two ways to obtain the angle of refraction are mathematically equivalent. So: the point is moot.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;I'm very curious: when was this first recognized?&lt;br /&gt;&#xA;Was Fermat already aware of it?&#xA;Huygens?&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing Newton was not aware of it. Newton favored a theory of light where propagation of light is thought of as motion of particles. My understanding is that Newton's theory of refraction required light to be faster in glass than in air.&lt;/p&gt;&#xA;&lt;p&gt;(In general, theory of refraction is about &lt;em&gt;ratios&lt;/em&gt;. A refraction experiment finds a ratio. It wasn't until the time of Fizeau that the technology was available to measure actual speed of light.)&lt;/p&gt;&#xA;&lt;p&gt;(It could be, I haven't verified, that Fermat's law can be restated as a law of &lt;em&gt;most&lt;/em&gt; time, under the newtonian assumption that in glass light propagates faster. That would mean that distinction between Fermat least time and Fermat most time is moot.)&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;With the above context in place:&lt;br /&gt;&#xA;It seems to me that awareness of the mathematical connection between Snell's law and Fermat's time law makes a lot of difference. So I think it's higly relevant to know when it was first recognized.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2021-03-28T07:34:44.137" Title="When was the mathematical relation between Snell's law and Fermat time law first recognized?" Tags="|theoretical-physics|optics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13034" PostTypeId="2" ParentId="2901" CreationDate="2021-03-29T05:59:46.577" Score="0" Body="&lt;p&gt;I'm surprised no one has mentioned the story which I have commonly heard. Yes &amp;quot;calculus&amp;quot; means pebble or stone in Latin. But just saying this...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Calculus has always had, and still has, a more general meaning than&#xA;the differentiation and integration. It literally means a method of&#xA;calculating, and the etymology goes back to small stones used as counters&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;... misses a more interesting story. &lt;a href=&quot;https://skeptics.stackexchange.com/questions/8841/were-metered-taxis-busy-roaming-imperial-rome&quot;&gt;A &amp;quot;calculus,&amp;quot; or calculi, essentially enabled Roman taxi cabs to figure a fare.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The odometer of Vitruvius was based on chariot wheels of 4 feet (1.2 m)&#xA;diameter turning 400 times in one Roman mile (about 1400 m). For each&#xA;revolution a pin on the axle engaged a 400 tooth cogwheel thus turning&#xA;it one complete revolution per mile. This engaged another gear with holes&#xA;along the circumference, where pebbles (calculus) were located, that were&#xA;to drop one by one into a box. The distance traveled would thus be given&#xA;simply by counting the number of pebbles. Whether this instrument was ever&#xA;built at the time is disputed. Leonardo da Vinci later tried to build&#xA;it himself according to the description, but failed. However, in 1981 engineer&#xA;Andre Sleewyck built his own replica, replacing the square-toothed gear designs&#xA;of da Vinci with the triangular, pointed teeth found in the Antikythera&#xA;mechanism. With this modification, the Vitruvius odometer functioned perfectly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14181" LastActivityDate="2021-03-29T05:59:46.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13035" PostTypeId="1" CreationDate="2021-03-29T08:41:45.847" Score="0" ViewCount="552" Body="&lt;p&gt;Are there any famous/notorious Romani physicist?&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="14182" LastEditorUserId="14182" LastEditDate="2021-04-02T15:58:12.067" LastActivityDate="2024-03-04T06:03:00.320" Title="Famous gypsy/romani physicists?" Tags="|physics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13036" PostTypeId="1" CreationDate="2021-03-29T09:36:38.830" Score="0" ViewCount="137" Body="&lt;p&gt;There is an interesting extension of Einstein's Special Relativity, known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Doubly_special_relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Doubly Special Relativity&lt;/a&gt; (presented after 2002) in which there is not only an observer-independent maximum velocity (i.e., the speed of light), but also, an observer-independent maximum energy scale (i.e., the Planck energy). Generalizing this idea in the presence of gravity led to the &lt;a href=&quot;https://arxiv.org/abs/1206.4823&quot; rel=&quot;nofollow noreferrer&quot;&gt;Doubly General Relativity&lt;/a&gt; (introduced 2004) which is generally referred to as &lt;a href=&quot;https://iopscience.iop.org/article/10.1088/0264-9381/21/7/001/meta&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gravity's Rainbow&lt;/a&gt; (in addition, there is &lt;a href=&quot;http://Gravity%E2%80%99s%20rainbow:%20Limits%20for%20the%20applicability%20of%20the%20equivalence%20principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;another paper in 1995&lt;/a&gt;, before the discovery of Doubly Special Relativity!, in which the authors have been used the term &amp;quot;Gravity’s rainbow&amp;quot;). Interestingly, the novelist &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Pynchon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Pynchon&lt;/a&gt; had already written a dense, complex and fictional novel entitled &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gravity%27s_Rainbow&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gravity's Rainbow in 1973&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I strongly feel there is a connection between these nominal similarities in science (physics) and literature. &lt;strong&gt;Does anyone know anything about this?&lt;/strong&gt; Please inform me if you know anything.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Thank you very much in advance ...&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="14138" LastEditorUserId="14138" LastEditDate="2021-03-29T10:27:01.090" LastActivityDate="2021-03-29T10:27:01.090" Title="Gravity's Rainbow: An accidental nominal similarity in science and literature?" Tags="|physics|theoretical-physics|relativity-theory|gravity|physics-social-history|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13038" PostTypeId="1" AcceptedAnswerId="13041" CreationDate="2021-03-29T18:06:56.347" Score="5" ViewCount="280" Body="&lt;p&gt;An often used definition of &amp;quot;Faraday's Law of Induction&amp;quot; goes something like this (found in Wikipedia)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This law is often attributed to Faraday's &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rstl.1832.0006&quot; rel=&quot;nofollow noreferrer&quot;&gt;1832 Report to the Royal Society&lt;/a&gt; published under the name &amp;quot;Experimental Researches in Electricity&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;However, in that document, I was unable to find any reference to an &lt;em&gt;equality&lt;/em&gt; between a quantified electromotive force with a rate of change of a quantified magnetic flux. Rather, in section 42 of that document, he refers to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a law hereafter to be described (114)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and in section 114, describes a qualitative, but not quantitative law which he refers to as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The relation which holds between the magnetic pole, the moving wire&#xA;or metal, and the direction of the current evolved&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I will not repeat everything Faraday wrote about this relation. Suffice it to say, that the explication of this law turns out to be a far cry from a quantitative law equating the e.m.f. and the rate of change of magnetic flux.&lt;/p&gt;&#xA;&lt;p&gt;Now, there is &lt;a href=&quot;http://www.gutenberg.org/ebooks/14986&quot; rel=&quot;nofollow noreferrer&quot;&gt;another published work by Faraday with the same name&lt;/a&gt; i.e. &amp;quot;Experimental Researches in Electricity&amp;quot;, published in three volumes from 1839 to 1855. The first part of volume 1 of this larger work appears to be identical to that published in 1832. I did not read it all. However, a search of the words &amp;quot;flux&amp;quot; and &amp;quot;electromotive&amp;quot; shows they only appear 3 times each in the first volume, and none in conjunction with each other.&lt;/p&gt;&#xA;&lt;p&gt;So, my question is, when was Faraday's Law of Induction, in the form of a quantitative equality, first expressed. And was it even first expressed by Faraday? Was it by Maxwell? Heaviside? Or did it come earlier than those two scientists? Is there something that Faraday wrote that entails a mathematical equality, even though it may have been expressed with words other than &amp;quot;flux&amp;quot; or &amp;quot;electromotive force&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="14184" LastEditorUserId="10106" LastEditDate="2021-03-30T03:50:25.290" LastActivityDate="2021-03-30T08:22:00.483" Title="When was &quot;Faraday's Law of Induction&quot; first expressed in a quantified form?" Tags="|physics|electromagnetism|19th-century|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13039" PostTypeId="1" CreationDate="2021-03-30T03:03:23.017" Score="4" ViewCount="325" Body="&lt;p&gt;Throughout the history of mathematics and physics, there has been many examples where mathematics was discovered first prior to its application in physics. Consider &lt;span class=&quot;math-container&quot;&gt;$i=\sqrt {-1}$&lt;/span&gt; as an example, among many others.&lt;/p&gt;&#xA;&lt;p&gt;I am interested to know about discoveries in physics, for which a mathematical counterpart did not exist at the time and for which new mathematics had to be created. Are there such examples? If so, a few examples would be appreciated (preferably with sources).&lt;/p&gt;&#xA;" OwnerUserId="14188" LastActivityDate="2021-04-09T21:50:51.197" Title="Examples of Physical Discoveries with no Counterpart in Mathematics" Tags="|mathematical-physics|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13040" PostTypeId="2" ParentId="13039" CreationDate="2021-03-30T06:18:57.497" Score="6" Body="&lt;p&gt;Paul Dirac developed the Dirac Delta for physics. Mathematicians said it made no sense. It took the development of the theory of distributions to show they were wrong.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2021-03-30T06:18:57.497" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13041" PostTypeId="2" ParentId="13038" CreationDate="2021-03-30T08:07:01.507" Score="4" Body="&lt;p&gt;Faraday stated the law more quantitatively in the report titled On Lines of Magnetic Force read to the Royal Society in 1851. He does not specify the direction of the force (and he originally got it wrong in 1832), but the sign was already established by &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k151161/f499.item.r=lenz.langEN&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lenz in &lt;em&gt;Ueber die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme&lt;/em&gt; (1834)&lt;/a&gt;. However, Faraday's formulation is still verbal, and the lack of mathematical gloss was generally an obstacle to the reception of his ideas. They were assimilated only after Maxwell gave them an explicitly mathematical formulation in 1861-2, although practical applications, like electric generators and telegraph, go back to 1830-s, see &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rsta.2014.0208&quot; rel=&quot;nofollow noreferrer&quot;&gt;The birth of the electric machines&lt;/a&gt;. In particular, Maxwell reformulated the law of induction as a partial differential equation, which Heaviside originally dubbed Faraday's law, but which is &lt;a href=&quot;https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;now called the Maxwell–Faraday equation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Faraday's report is printed in &lt;a href=&quot;https://archive.org/details/B-001-002-720&quot; rel=&quot;nofollow noreferrer&quot;&gt;Experimental Researches in Electricity, vol.3&lt;/a&gt;, here is the relevant passage:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;3114. It is also evident, by the results of the rotation of the wire and magnet (3097. 3 106.), that when a wire is moving amongst equal lines (or in a field of equal magnetic force), and with an uniform motion, then the current of electricity produced is proportionate&#xA;to the time; and also to the velocity of motion. 3115. They also prove, generally, that &lt;strong&gt;the quantity of electricity thrown into a current is directly as the amount of curves intersected.&lt;/strong&gt;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;quantity of electricity thrown into a current&amp;quot; is essentially the electromotive force, the &amp;quot;curves&amp;quot; are Faraday's lines of force, and &amp;quot;the amount of curves intersected&amp;quot; is proportional to the intensity of the magnetic field in modern terms. Whittaker explains the context in &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich/page/190/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of the Theories of Aether and Electricity, pp. 190ff&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Faraday conceived the idea of partitioning all space into&#xA;compartments by tubes, each tube being such that this product&#xA;has the same definite value. For simplicity, each of these&#xA;tubes may be called a &amp;quot;unit line of force&amp;quot;; the strength of&#xA;the field is then indicated by the separation or concentration of&#xA;the unit lines of force, so that the number of them which&#xA;intersect a unit area placed at right angles to their direction at any point measures the intensity of the magnetic field at that point.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He then paraphrases the boldface passage into a more recognizable form:&amp;quot;&lt;em&gt;&lt;strong&gt;The induced electromotive force is, in fact, simply proportional to the number of the unit lines of magnetic force intersected by the wire per second&lt;/strong&gt;&lt;/em&gt;&amp;quot;. Symbolically, with Lenz's sign rule, &lt;span class=&quot;math-container&quot;&gt;$\mathcal{E}\sim-\frac{d\Phi_B}{dt}$&lt;/span&gt;, as written in modern textbooks.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-03-30T08:22:00.483" LastActivityDate="2021-03-30T08:22:00.483" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13042" PostTypeId="1" CreationDate="2021-03-30T16:35:48.257" Score="5" ViewCount="96" Body="&lt;p&gt;Is there an historical tradition in languages read right-to-left (Arabic, Hebrew, Urdu, etc.)&#xA;to display mathematical equations in some right-to-left form?&#xA;So, instead of&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$&lt;/span&gt;&#xA;perhaps swapping expressions across the equals-sign,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = x$$&lt;/span&gt;&#xA;or even more thorough reversal inside expressions, e.g.,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\sqrt{b^2 - 4ac} \pm -b}{2a}= x$$&lt;/span&gt;&#xA;?&lt;/p&gt;&#xA;" OwnerUserId="606" LastActivityDate="2021-03-30T16:35:48.257" Title="Equations in right-to-left languages" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13043" PostTypeId="2" ParentId="13039" CreationDate="2021-03-30T19:29:02.743" Score="10" Body="&lt;p&gt;Potential theory (Green's formulas, Green's function etc.) was discovered by George Green who was doing physics. His work was called &amp;quot;An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism&amp;quot;.&#xA;Laplace equation was first written in a paper on Saturn rings.&lt;/p&gt;&#xA;&lt;p&gt;Eigenvalues, eigenvectors and adjoint operators were discovered by mathematicians who&#xA;were doing physics (namely celestial mechanics). Fourier transform was&#xA;discovered by Fourier who was doing physics (in his studies of heat flow).&#xA;Same applies to Bessel functions and theta functions which appear for the first time&#xA;in Fourier's book on heat. Fourier series were discovered by Daniel Bernoulli who was doing physics (oscillations of a string).&lt;/p&gt;&#xA;&lt;p&gt;Vector analysis was invented by physicists Herz and Gibbs.&#xA;Maxwell's book on electromagneitsm is literally packed with mathematical discoveries.&lt;/p&gt;&#xA;&lt;p&gt;Calculus of operators (a.k.a. operational calculus) was discovered&#xA;by physicist/electrical engineer Oliver Heaviside, who was studying the&#xA;&amp;quot;telegrapher's equation&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The theories of unbounded operators, operator algebras and quantum logic were developed by von Neumann to give mathematical foundations of quantum mechanics.&lt;/p&gt;&#xA;&lt;p&gt;Lee and Yang Theorem was discovered by two physicists who studied phase transitions.&lt;/p&gt;&#xA;&lt;p&gt;Yang-Mills equations were discovered by physicists Yang and Mills working on quantum field theory.&lt;/p&gt;&#xA;&lt;p&gt;The list can be made almost infinite.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-04-01T00:52:38.770" LastActivityDate="2021-04-01T00:52:38.770" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13044" PostTypeId="1" CreationDate="2021-03-30T20:04:49.717" Score="1" ViewCount="134" Body="&lt;p&gt;In &lt;a href=&quot;https://archive.org/details/eassayonthepsych006281mbp/page/n71/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The psychology of invention in the mathematical field&lt;/em&gt;, p. 53&lt;/a&gt;, Hadamard makes the following claim:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iOTLm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iOTLm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is his point that there must, in any axiomatic theory, be undefined terms, and if you write the parallel postulate entirely expanded, i.e., with definienda substituted for definiens, then you will see that its not a truth of logic? But that seems completely off. You may still &amp;quot;see&amp;quot; that its a truth for perceptual space (let us grant, for the sake of the argument), and moreover, you may still &lt;em&gt;fail to see&lt;/em&gt; that it isn't implied by the other axioms+logic. Thus, while you &lt;em&gt;may&lt;/em&gt; arguably conceive of the possiblity of non-Euclidean axiom systems, you might still not yet perceive that of non-Euclidean &lt;em&gt;geometries&lt;/em&gt; (qua sciences of space); and in fact, without a proof of relative consistency of these systems, e.g. by an inner model in Euclidean geometry, or by a model in &lt;strong&gt;R&lt;/strong&gt;, you aren't technically allowed to even assume so much as the possibility of these systems, since (lacking such proof), the PP might -- for all you know -- after all be a consequence of the other axioms, just that you haven't found the proof yet (which &lt;em&gt;we&lt;/em&gt; know in fact doesn't exist).&lt;/p&gt;&#xA;&lt;p&gt;Am I completely misunderstanding Hadamard's point here?&lt;/p&gt;&#xA;&lt;p&gt;Edit: Upon further reflection, perhaps his point is simply that the notion of formal axiomatics is implicit in Pascal's ideas? But what is all this business with &amp;quot;the true meaning of the celebrated Euclid postulate&amp;quot; about, then? Is there another postulate that could be meant?&lt;/p&gt;&#xA;&lt;p&gt;Crosspost: &lt;a href=&quot;https://math.stackexchange.com/questions/4083340/why-does-hadamard-claim-that-pascal-could-have-discovered-non-euclidean-geometry&quot;&gt;https://math.stackexchange.com/questions/4083340/why-does-hadamard-claim-that-pascal-could-have-discovered-non-euclidean-geometry&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14192" LastEditorUserId="55" LastEditDate="2021-04-01T11:52:05.973" LastActivityDate="2021-04-01T12:08:09.213" Title="Does Hadamard claim that Pascal could have discovered non-Euclidean geometry?" Tags="|geometry|logic|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13045" PostTypeId="1" CreationDate="2021-03-30T22:39:11.833" Score="3" ViewCount="371" Body="&lt;p&gt;The currently standard matrix multiplication is isomorphic to &lt;a href=&quot;https://en.wikipedia.org/wiki/Split-quaternion&quot; rel=&quot;nofollow noreferrer&quot;&gt;split-quaternions&lt;/a&gt; (for 2x2 matrices, and similar for higher ranks, maybe this is called Clifford algebras). As such, the sets of square matrices is not commutative, has zero divisors, idempotents and nilpotents.&lt;/p&gt;&#xA;&lt;p&gt;I wonder, why other matrix multiplication rules were not chosen. For instance, the tessarine-like that would be commutative?&lt;/p&gt;&#xA;&lt;p&gt;Yes, usual quaternions or tessarines or split-numbers of any dimension can be represented by the matrices, but since the multiplication rules differ, one would need the matrices of squared rank to represent them.&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="2203" LastEditDate="2022-02-17T07:14:18.143" LastActivityDate="2022-02-17T07:14:18.143" Title="Why were the matrix multiplication rules chosen this way?" Tags="|mathematics|linear-algebra|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13046" PostTypeId="2" ParentId="13045" CreationDate="2021-03-31T05:36:15.373" Score="3" Body="&lt;p&gt;Matrices were discovered in China in the 2nd century BCE as documented in the &lt;em&gt;Nine Chapters of the Mathematical Art&lt;/em&gt;. Here, the method of Gaussian Elimination  is used to solve matrix equations - and which suggests that the term Gaussian elimination is misnamed. However, the &lt;em&gt;Nine Chapters&lt;/em&gt; is an anonymous work, or rather, the author's name has not come down to us, so there is no easy way to refer to the method, other than Gaussian elimination.&lt;/p&gt;&#xA;&lt;p&gt;This can be generalised to matrix multiplication. More modernly, matrix multiplication is simply the multiplication, or rather composition, of linear transformations in a given basis.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-04-03T12:39:16.637" LastActivityDate="2021-04-03T12:39:16.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13047" PostTypeId="2" ParentId="13045" CreationDate="2021-03-31T06:33:31.403" Score="8" Body="&lt;p&gt;As with &lt;a href=&quot;https://hsm.stackexchange.com/questions/12866/why-are-quaternions-more-popular-than-tessarines-despite-being-non-commutative&quot;&gt;your previous question&lt;/a&gt; about why tessarines' commutativity doesn't win them more interest, &lt;strong&gt;commutativity is not mathematicians' goal; problem-solving is&lt;/strong&gt;. We're interested in vector spaces, and in linear operations from such spaces to the same or other spaces (i.e. matrices), and in the composition of such linear operations (per @Conifold's comment) viz.&lt;span class=&quot;math-container&quot;&gt;$$(AB)v:=A(Bv)\implies(AB)_{ij}v_j=A_{ik}(Bv)_k=A_{ik}B_{kj}v_j\implies(AB)_{ij}=A_{ik}B_{kj}.$$&lt;/span&gt;You're welcome to consider other operations of the form &lt;span class=&quot;math-container&quot;&gt;$(A\circ B)_{ij}:=A_{mn}B_{pq}C_{ijmnpq}$&lt;/span&gt;; the example above is &lt;span class=&quot;math-container&quot;&gt;$C_{ijmnpq}:=\delta_{im}\delta_{jq}\delta_{np}$&lt;/span&gt;.&lt;span class=&quot;math-container&quot;&gt;${}^\dagger$&lt;/span&gt; You're welcome to determine which &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; obtain &lt;span class=&quot;math-container&quot;&gt;$A\circ B=B\circ A$&lt;/span&gt;. But the irony is this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;These matrices comprise a vector space. We can halve the index count with suitable relabelling that squares the dimension.&lt;/li&gt;&#xA;&lt;li&gt;Each &lt;span class=&quot;math-container&quot;&gt;$\circ$&lt;/span&gt; comprises a two-vectors-to-one-vector bilinear operation, analogous to the cross product on &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;-dimensional space. (Does it bother you that that's anticommutative?)&lt;/li&gt;&#xA;&lt;li&gt;Such operations also have a natural characterization in terms of matrices as they're defined above. To wit:&lt;span class=&quot;math-container&quot;&gt;$$(v\circ w)_i:=K_{ijk}v_jw_k\implies(v\circ)_{ik}=K_{ijk}v_j,\,(\circ w)_{ij}=K_{ijk}w_k.$$&lt;/span&gt;In other words, matrices are still the natural software you're running on.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${}^\dagger$&lt;/span&gt; No others will obey &lt;span class=&quot;math-container&quot;&gt;$(AB)v=A(Bv)$&lt;/span&gt;, so you'll just be treating &amp;quot;matrices&amp;quot; as &amp;quot;arrays of numbers I can do anything I want with&amp;quot; rather than quantities that represent an underlying something in a suitable basis, but I suppose mathematics doesn't force you to do something &lt;em&gt;useful&lt;/em&gt; if you don't want to. On the other hand, it doesn't force people to be interested in what you find elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2022-02-16T20:18:51.983" LastActivityDate="2022-02-16T20:18:51.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13048" PostTypeId="1" CreationDate="2021-03-31T09:56:25.287" Score="0" ViewCount="32" Body="&lt;p&gt;In which year, does the fact: &lt;strong&gt;every star will ultimately die&lt;/strong&gt; came in to existence?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastEditorUserId="10106" LastEditDate="2021-03-31T12:46:15.897" LastActivityDate="2021-03-31T12:46:15.897" Title="In which year the notion of star death started?" Tags="|physics|astronomy|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13049" PostTypeId="1" CreationDate="2021-03-31T14:51:18.833" Score="4" ViewCount="70" Body="&lt;p&gt;Well this is really a silly question. However I am curious to know about this topic.&lt;/p&gt;&#xA;&lt;p&gt;Concept of each of the quantities discussed in physics have come into existence based on some basic human thoughts. For example if we are talking about force, it might have been termed based on the pull and push concept and then people discussed about it rigorously. Again displacement might have been termed based on our change in position.(I don't know for sure if these were the ways how their concept emerged, presented these just as examples)&lt;/p&gt;&#xA;&lt;p&gt;Now how did the concept of work emerge? Why did the one who first thought about it formulated work as the dot product of force and the displacement caused by the force?&lt;/p&gt;&#xA;&lt;p&gt;If we are at the initial stage of learning and understanding the nature we might have thought that standing for a long period of time has drained something from our body. Someone might have termed that imaginary thing as &amp;quot;energy&amp;quot;. Again some have termed this as &amp;quot;work&amp;quot;. So if we start to think from that scenario we might consider standing for a long period is work. So what did it cause the enthusiast to exclude these type of cases where dot product of force and displacement yields zero while defining &amp;quot;work&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;I am apologising for asking such a wierd and a baseless question. I could not get rid of this question from my mind so thought of asking it here.&lt;/p&gt;&#xA;&lt;p&gt;*Anyways is there any book regarding the history of physics and math? If there is any kindly suggest some,that would be a great help.&lt;/p&gt;&#xA;" OwnerUserId="14195" LastEditorUserId="14195" LastEditDate="2021-03-31T20:04:57.620" LastActivityDate="2021-03-31T20:04:57.620" Title="How did the concept of work being the dot product of force and displacement come into existence?" Tags="|physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13050" PostTypeId="1" CreationDate="2021-03-31T20:18:56.227" Score="3" ViewCount="38" Body="&lt;p&gt;What where the crucial experiments which showed  the identity of voltaic and static electricity? How did the experiments work in detail and what are the original references for it?&lt;/p&gt;&#xA;&lt;p&gt;In &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/A_History_of_the_Theories_of_Aether_and_Electricity&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of the Theories of Aether and Electricity - The Classical Theories&lt;/a&gt;&amp;quot; by E.T.Whittaker, I found the follwing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But the connection of the different branches of electric science with each other was still not altogether clear. Although Wollaston’s experiments of 1801 had in effect proved the identity in kind of the currents derived from frictional and voltaic sources, the question was still regarded as open thirty years afterwards,no satisfactory explanation being forthcoming of the fact that frictional electricity appeared to be a surface-phenomenon, whereas voltaic electricity was conducted within the interior substance of bodies. To this question Faraday now applied himself; and in 1833 he succeeded in &lt;strong&gt;showing that every known effect of electricity—physiological, magnetic, luminous, calorific, chemical and mechanical—may be obtained indifferently either with the electricity which is obtained by friction or with that obtained from a voltaic battery&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(bold mine)&lt;/p&gt;&#xA;&lt;p&gt;In this text Whittaker seems to give a reference about what I am looking for, but I didn't find a source for it: &amp;quot;Exp. Ran, series iii&amp;quot; (what magazine is &amp;quot;Exp. Ran&amp;quot;?). In the case of Wollastons exeriment there is also a reference (Phil. Trans. xci (1801), p. 4-27). The magazine seems to be &amp;quot;Philosophical Transactions&amp;quot;, but I didn't find a link to the paper as well.&lt;/p&gt;&#xA;" OwnerUserId="2380" LastEditorUserId="2380" LastEditDate="2021-03-31T21:00:14.117" LastActivityDate="2021-03-31T21:00:14.117" Title="History of experimental verification of the identity of voltaic and static electricity and references" Tags="|physics|electricity|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13051" PostTypeId="1" AcceptedAnswerId="13573" CreationDate="2021-03-31T21:55:53.343" Score="2" ViewCount="553" Body="&lt;p&gt;For various reasons already discussed in other stackexchange posts, we implicitly use radians in trigonometric functions by convention. For example, one period of &lt;span class=&quot;math-container&quot;&gt;$sin(x)$&lt;/span&gt; lies in &lt;span class=&quot;math-container&quot;&gt;$0 \leq x &amp;lt; 2\pi$&lt;/span&gt;, when we could have instead used &lt;span class=&quot;math-container&quot;&gt;$\sin_{\text{degree}}(d)=\sin(\frac{\pi}{180}d)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\sin_{\text{cycle}}(c)=\sin(2 \pi c)$&lt;/span&gt; where one period lies in &lt;span class=&quot;math-container&quot;&gt;$0 \leq d &amp;lt; 360$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$0 \leq c &amp;lt; 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So how did we still end up using degrees and cycles? Angles are measured in degrees more often than radians in basic geometry content, where &lt;span class=&quot;math-container&quot;&gt;$\sin$&lt;/span&gt; may even mean &lt;span class=&quot;math-container&quot;&gt;$\sin_{\text{degree}}$&lt;/span&gt; with no warning. Ordinary frequency (cycles per second a.k.a &lt;em&gt;hertz&lt;/em&gt;) is often used instead of angular frequency (radians per second).&lt;/p&gt;&#xA;" OwnerUserId="14196" LastEditorUserId="2905" LastEditDate="2021-04-10T22:49:55.103" LastActivityDate="2021-08-22T05:13:26.300" Title="Why do we still use degrees and cycles if we all implicitly use radians in trigonometric functions?" Tags="|mathematics|trigonometry|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13052" PostTypeId="1" AcceptedAnswerId="13062" CreationDate="2021-03-31T22:12:20.160" Score="2" ViewCount="991" Body="&lt;p&gt;In teaching introductory mechanics classes, I like to give my students a much-abridged sketch of some history of mechanics, starting with Aristotle and ending with Galileo and Newton. Most of the introductory physics texts I've seen contain at least a cursory treatment of this story.&lt;/p&gt;&#xA;&lt;p&gt;However, I've never seen similar discussion of the history of rotational kinematics or dynamics anywhere. I once had a student ask whether what we call &amp;quot;Newton's law for rotation&amp;quot; was due to Newton, and I had to admit that I had no idea. I'm hoping that someone can give me an outline of the development of rotational mechanics, including things like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Kinematic description of rotational motion in terms of angular displacement, angular velocity, and angular acceleration&lt;/li&gt;&#xA;&lt;li&gt;Torque&lt;/li&gt;&#xA;&lt;li&gt;The relationship between angular acceleration and torque&lt;/li&gt;&#xA;&lt;li&gt;Moment of inertia&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'd also be interested to know how much of this &amp;quot;Physics 101&amp;quot; treatment of rigid body motion preceded the more sophisticated description of rigid body mechanics with Euler equations, moment of inertia tensor, etc.&lt;/p&gt;&#xA;" OwnerUserId="14197" LastActivityDate="2021-04-05T12:18:18.830" Title="History of rotational / rigid body mechanics" Tags="|physics|classical-mechanics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13053" PostTypeId="2" ParentId="13051" CreationDate="2021-04-01T00:49:13.327" Score="7" Body="&lt;p&gt;We do not use radians &amp;quot;by convention&amp;quot;. Radian is a necessary intrinsic measure of an angle, which is related to the &amp;quot;fact of nature&amp;quot; that the length of a unit circle is &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;. Turns is also a natural measure, which is sometimes preferred to radians, when it is convenient.&lt;/p&gt;&#xA;&lt;p&gt;Degrees are different, the reason of using them is purely historical, related to&#xA;the ancient Babylonian numeration system, with base 60. So the use of degrees, and other similar things is motivated by history tradition and convenience. Like hours,&#xA;grads and other measures of angles of historic/cultural origin.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-04-01T00:49:13.327" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13054" PostTypeId="1" CreationDate="2021-04-01T05:23:33.597" Score="3" ViewCount="235" Body="&lt;p&gt;There is a lot of information on the history of the Pythagorean theorem, but not much on its closely related cousin; the &lt;a href=&quot;https://en.wikipedia.org/wiki/Inverse_Pythagorean_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Inverse Pythagorean Theorem&lt;/a&gt;. Would appreciate any resources on the the history of this theorem, not exclusive to &amp;quot;The first time it was written&amp;quot; (although that would be cool).&lt;/p&gt;&#xA;" OwnerUserId="13800" LastActivityDate="2021-04-01T11:12:33.517" Title="Origin of &quot;Inverse Pythagorean Theorem&quot;" Tags="|mathematics|geometry|trigonometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13055" PostTypeId="2" ParentId="13054" CreationDate="2021-04-01T11:12:33.517" Score="2" Body="&lt;p&gt;Seems to be pretty commonplace as well as having multiple names.  Here's an &lt;a href=&quot;https://www.maa.org/sites/default/files/Nelsen2009-316026.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; which references some other article, both relatively recent.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iJWMT.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iJWMT.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-04-01T11:12:33.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13056" PostTypeId="1" AcceptedAnswerId="13058" CreationDate="2021-04-01T11:13:01.317" Score="4" ViewCount="234" Body="&lt;p&gt;As is known, &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss&lt;/a&gt; came to similar conclusions as &lt;a href=&quot;https://en.wikipedia.org/wiki/Nikolai_Lobachevsky&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lobachevskii&lt;/a&gt; in the problem of &lt;a href=&quot;https://en.wikipedia.org/wiki/Parallel_postulate&quot; rel=&quot;nofollow noreferrer&quot;&gt;fifth postulate&lt;/a&gt;, but I don't know whether Gauss had any impact to the contemporaries in that story. Can anybody enlighten me? Will it be correct to say that Gauss' opinion had any role in the disputes of XIX century about non euclidean geometry? When were his diaries (with the results on non-euclidean geometry) published? And what exacly was contained there?&lt;/p&gt;&#xA;" OwnerUserId="14212" LastEditorUserId="10106" LastEditDate="2021-04-01T11:51:03.060" LastActivityDate="2021-04-02T11:16:16.707" Title="What was the influence of Gauss on the understanding of the importance of non-Euclidean geometry?" Tags="|mathematics|mathematicians|biographical-details|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13057" PostTypeId="2" ParentId="13044" CreationDate="2021-04-01T11:47:15.630" Score="3" Body="&lt;p&gt;Hadamard's proposes that if Pascal juxtaposed two of his ideas he could have confronted the &amp;quot;great problem of logic&amp;quot; and understood the &amp;quot;true meaning&amp;quot; of the parallel postulate, i.e. its meaning under the modern axiomatic method. Before addressing Hadamard's proposal let me explain the differences between modern understanding of mathematics and Pascal's, which, I believe, Hadamard overlooks.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1) Formal revolution&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;People today tend to underestimate the magnitude of the Copernican revolution in mathematics initiated by Hilbert. Instead of centering mathematics on concrete objects sketched out definitionally, with axioms as mere shortcuts for streamlining arguments about them, Hilbert turned &lt;em&gt;them&lt;/em&gt; into the new center. The objects then became formal shells circumscribed by the axioms only, so &amp;quot;lines&amp;quot; and &amp;quot;points&amp;quot; can be seamlessly replaced by &amp;quot;desks&amp;quot; and &amp;quot;chairs&amp;quot;. The revolution came with a quick and thorough memory wipe of what came before, possibly because it was not opposed by powerful cultural institutions, as in the case of Copernicus or Darwin.&lt;/p&gt;&#xA;&lt;p&gt;The modern idea of pre-Hilbertian mathematics is of deducing theorems from axioms much like now, only with axioms vague and incomplete and proofs informal and gappy, but with steady tightening up of the rigor until axiomatic ideal was finally reached. That this was not the case is witnessed by Frege's objections to Hilbert's approach in the &lt;a href=&quot;https://plato.stanford.edu/entries/frege-hilbert/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frege-Hilbert correspondence (1899-1900)&lt;/a&gt;, and even by Russell's &lt;em&gt;Principia&lt;/em&gt; (1910-13) that still followed the old paradigm in Frege's footsteps. The second part of Russell's quip — &amp;quot;&lt;em&gt;The method of &amp;quot;postulating&amp;quot; what we want has many advantages; they are the same as the advantages of theft over honest toil&lt;/em&gt;&amp;quot; — is often quoted, out of context, but its meaning is largely forgotten. What Frege and Russell wanted was &amp;quot;honest toil&amp;quot; of constructing mathematical objects by &amp;quot;genetic&amp;quot; definitions (like Dedekind's construction of real numbers) instead of now familiar &amp;quot;theft&amp;quot; of formal axioms treated as &amp;quot;implicit&amp;quot; definitions. On the original, Euclid's, understanding of axioms and their role in geometry see &lt;a href=&quot;https://hsm.stackexchange.com/a/7104/55&quot;&gt;What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2) Definitional mathematics of 17th century&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The conception of mathematics at the time of Pascal's writing is described in &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-19863-7_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Risi, Leibniz on the Parallel Postulate and the Foundations of Geometry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Contrary to our modern understanding of the role of an axiom as an indemonstrable statement, they claimed that geometrical principles were self-evident truths that required no proof to be accepted, but nevertheless were not incapable of proof... These demonstrations of the axioms depend on the definitions of the mathematical terms, and the underlying epistemology seems to be that a statement whose proof follows immediately (whatever this means) from definitions alone is an axiom, whereas a statement that requires further means in order to be proved (other axioms, constructions, theorems) is a theorem in the proper sense.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;2) Hadamard's proposal&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A look at Pascal's remarks that Hadamard refers to shows that he works squarely within this conception, see &lt;a href=&quot;https://en.wikisource.org/wiki/Blaise_Pascal/Of_the_Geometrical_Spirit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pascal, Of the Geometrical Spirit&lt;/a&gt;. What Hadamard wishes Pascal to realize, apparently, is that once all definitions are dissolved into primitive ideas there is nothing left to give substance to those ideas other than the axioms. And that would reveal the &amp;quot;true meaning&amp;quot; of the axioms,  which to Hadamard, naturally, is Hilbert's conception of them as implicit definitions.&lt;/p&gt;&#xA;&lt;p&gt;However, &lt;em&gt;pace&lt;/em&gt; Hadamard, Pascal &lt;em&gt;did&lt;/em&gt; juxtapose those two statements, but the result is too alien for Hadamard's ear to register. Likely because he, like many others, views mathematics of the past as a cloudy anticipation of what modern times set straight and right. That there could be forks in the road, with different &amp;quot;right&amp;quot; outcomes that were equally &amp;quot;anticipated&amp;quot;, is rarely pondered.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;4) Pascal's alternative&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In any case, Pascal's solution is not formal axioms but &lt;em&gt;il lume naturale&lt;/em&gt;, the &amp;quot;natural light&amp;quot; that Descartes and other 17th century mathematicians inherited from scholastics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For there is one, and it is that of geometry, which is in truth inferior in that it is less convincing, but not in that it is less certain. It does not define every thing and does not prove every thing, and it is in this that it is inferior; but it assumes nothing but things clear and constant by natural enlightenment, and this is why it is perfectly true, nature sustaining it in default of discourse... We see clearly enough from this that there are some words incapable of being defined; and, if nature had not supplied this defect by a corresponding idea which she has given to all mankind, all our expressions would be confused; whilst we use them with the same assurance and the same certainty as though they were explained in a manner perfectly exempt from ambiguities; because nature herself has given us, without words, a clearer knowledge of them than art could acquire by our explanations.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Pascal did not effect the &amp;quot;profound revolution&amp;quot; because there was no need for it at the time. Mathematics was taken for something like physics of the abstract. This conception is limited, and it would not accommodate much of modern mathematics, but it was self-coherent, and Pascal did not fail to connect the dots, which then has to be explained by his focus being elsewhere.&lt;/p&gt;&#xA;&lt;p&gt;Geometry had to be reconceptualized as a study of plane/space as a whole rather than of diagrams (this was done by Leibniz), and the nascent movement to leave no &amp;quot;immediate&amp;quot; deliverances of the &amp;quot;natural light&amp;quot; unstated had to reach its pinnacle in the work of Pasch and Frege before Hilbert could effect his Copernican turn. Indeed, non-Euclidean geometries were not originally discovered in the formal manner Hadamard envisioned even in the 19th century. Gauss, Helmholtz and later Poincare interpreted them in the spirit of Kant's pure intuition, which replaced the natural light, but with a caveat that said intuition was not precise enough to zero in on a single geometry, the Euclidean one, thus leaving the determination (of curvature) to empirical studies, see &lt;a href=&quot;http://www.rehseis.cnrs.fr/IMG/pdf/Friedman_RelativizedAPriori.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedman's Einstein, Kant, and the Relativized A Priori&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-04-01T12:08:09.213" LastActivityDate="2021-04-01T12:08:09.213" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13058" PostTypeId="2" ParentId="13056" CreationDate="2021-04-02T00:30:05.057" Score="6" Body="&lt;p&gt;There was not much of an influence. Gauss chose to keep his thoughts out of public view to avoid controversy (&amp;quot;&lt;em&gt;uproar of the Boeotians&lt;/em&gt;&amp;quot;, as he put it poetically) and only shared them with a few correspondents in private letters, Bolyai senior, Gerling, Olbers, Taurinus and Bessel. English translations of relevant excerpts are collected by Burris in &lt;a href=&quot;https://www.math.uwaterloo.ca/%7Esnburris/htdocs/noneucl.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss and Non-Euclidean Geometry&lt;/a&gt;. Only brief sketches of hyperbolic geometry appear in Gauss's letters, no systematic developments, and they were published only posthumously and long after Beltrami's Euclidean models of the hyperbolic plane (1868) led to its acceptance. Most in volume 8 of his &lt;em&gt;Werke&lt;/em&gt; published by Stäckel in 1900, a summary appeared earlier in &lt;em&gt;Die Theorie der Parallellinien von Euklid bis auf Gauss&lt;/em&gt; (1895) edited by Engel and Stäckel, see &lt;a href=&quot;https://www.researchgate.net/publication/243025019_Gauss_and_the_Definition_of_the_Plane_Concept_in_Euclidean_Elementary_Geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zormbala, Gauss and the Definition of the Plane Concept in Euclidean Elementary Geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Gauss's Kantian route towards non-Euclidean geometry, and its philosophical relevance, was also independently arrived at by Helmholtz and others. As Gauss wrote to Taurinus, the idea was that physical geometry is not settled &lt;em&gt;a priori&lt;/em&gt;, and might require empirical input:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It seems to me that in spite of the word-mastery of the metaphysicians, we know really too little, or even nothing at all, about the true nature of space to be able to confuse something that seems unnatural with absolutely impossible. If non-Euclidean geometry is the real one and the constant is comparable to the magnitudes that we encounter on earth or in the heavens then it can be determined a posteriori.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One might expect some Gaussian influence filtering through to Bolyai junior, but then his father explicitly and passionately discouraged him from pursuing the matter. When Schweikart sent Gauss his sketch of hyperbolic geometry all Gauss did was ask Gerling to convey to him his &amp;quot;very best&amp;quot;. He did write to Schweikart's nephew Taurinus, which apparently encouraged the latter to pursue the parallel lines further. This might have been Gauss's biggest influence. But he did nothing publicly when Bolyai's and Lobachevsky's work appeared in print. Perhaps, Gauss's acceptance played some positive role after his letters were finally published, perhaps he influenced Riemann's outlook, although Riemann credits Gauss's work on surfaces for inspiration in his famous lecture, not his hyperbolic musings. For a discussion of historical context and Gauss's thought process on hyperbolic geometry see &lt;a href=&quot;https://anewdifferentworld.wordpress.com/2015/07/16/gauss-kastner-and-kant/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Baker, Gauss, Kästner, and Kant&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Gauss explained his attitude in a 1832 letter to Bolyai senior, after his son already published on hyperbolic geometry on his own:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If I start by saying “I cannot praise it” then you will most likely be taken aback; but I cannot do otherwise; to praise it would be to praise myself; the entire contents of the work, the path that your son has taken and the results to which it leads, are almost perfectly in agreement with my own meditations, some going back 30 – 35 years. In truth I am astonished. My intention was not to release any of my own work in my lifetime. Most people don’t have a true sense of what is involved, and I have found very few who are particularly interested. To appreciate what is going on one must first of all have a real grasp of what is missing, and on this point most are in the dark. On the other hand it was my intention to write everything down so that it didn’t perish with me.&#xA;So I am truly surprised that I am now spared this effort, and it is the greatest joy for me that precisely the son of my old friend is the one who preceded me in such a remarkable manner.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-04-02T11:16:16.707" LastActivityDate="2021-04-02T11:16:16.707" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13059" PostTypeId="1" CreationDate="2021-04-02T02:17:34.580" Score="2" ViewCount="75" Body="&lt;p&gt;I found a reference to the following article ;&lt;/p&gt;&#xA;&lt;p&gt;O B Sheynin, The appearance of Dirac's delta functions in the works of P S Laplace (Russian), Istor.-Mat. Issled. Vyp. 20 (1975), 303-308, 381.&lt;/p&gt;&#xA;&lt;p&gt;I don't have access to this nor can I read in Russian. Can anyone summarize this paper? I'm fairly familiar with the Laplace and other integral transforms, so a brief statement of and explanation of any nonstandard notation in the central formula or formulas as presented by Laplace would probably suffice.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastActivityDate="2021-04-02T02:17:34.580" Title="Appearance of the Dirac delta operator in Laplace's work" Tags="|mathematics|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="13060" PostTypeId="1" CreationDate="2021-04-02T15:29:20.620" Score="0" ViewCount="75" Body="&lt;p&gt;In nearly all areas of mathematics, logarithmic function applied to various objects where it is defined, produces either direction data or scale data of the object.&lt;/p&gt;&#xA;&lt;p&gt;Are there any examples of mathematical objects that have logarithmic functions defined on them, that produce other characteristic of the object besides the direction and the scale?&lt;/p&gt;&#xA;&lt;p&gt;I think of something exotic, like knots, singularities, polyhedrons, graphs, etc.&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2021-04-02T15:29:20.620" Title="Historically, was logarithmic function ever used to determine anything other than direction and scale of a vector?" Tags="|logarithm|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13061" PostTypeId="1" CreationDate="2021-04-03T19:54:37.930" Score="1" ViewCount="80" Body="&lt;p&gt;Throughout the centuries the label 'principle' has been applied liberally. That blurs the lines, it devaluates the word 'principle'.&lt;/p&gt;&#xA;&lt;p&gt;As an example of a good use of the word 'principle' the &lt;em&gt;Principle of relativity of inertial motion&lt;/em&gt;  comes to mind.&lt;/p&gt;&#xA;&lt;p&gt;An example of improper use of the word 'principle', I submit, is d'Alembert's principle of virtual work.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Of course, the designations 'principle' and 'law' have never been sharply defined (nor will they ever be). In effect it is used as a general way of saying: this is important, this has widespread validity.&lt;/p&gt;&#xA;&lt;p&gt;But the designation 'theorem', I feel, expresses a certain deference. In mathematics, when there is an axiomatic system, you have the axioms and everything that can be derived from those axioms is referred to as 'theorem'. I assume that using the designation 'theorem' carries a message: acknowledgement that the Work-Energy relation derives from &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt;. The choice of name expresses how the Work-Energy relation is perceived.&lt;/p&gt;&#xA;&lt;br&gt;&#xA; &#xA;Someone must have been the first to use the name 'Work-Energy theorem'. Is it known who was the first? Did it gain wide adoption quickly?&#xA;&lt;p&gt;A google search reported 170 million hits for 'work-energy principle' and 40 million hits for 'work-energy theorem'.&lt;/p&gt;&#xA;&lt;p&gt;Have the two names always coexisted? Or has there been, in the past decades, a shift in the naming?&lt;/p&gt;&#xA;&lt;p&gt;It may be that a shift in naming indicates a general loss of physics understanding. I just checked a prominent educational website. It is stated there that &amp;quot;the work-energy principle is derivable from conservation of energy.&amp;quot;&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;I feel this is an area where knowledge of &lt;em&gt;history&lt;/em&gt; of physics can be used as an instrument to try and counteract erosion of physics understanding.&lt;/p&gt;&#xA;&lt;p&gt;For completeness I give the derivation:&lt;/p&gt;&#xA;&lt;p&gt;the following two relations will be used:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ ds = v \ dt  \qquad (1)  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ dv = a \ dt  \qquad (2)   $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;The integral for acceleration from a starting point &lt;span class=&quot;math-container&quot;&gt;$s_0$&lt;/span&gt; to a final point &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_{s_0}^s a \ ds \qquad (3)  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Intermediary step: use (1) to change the differential from &lt;em&gt;ds&lt;/em&gt; to &lt;em&gt;dt&lt;/em&gt;. Since the differential is changed the limits change accordingly.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_{t_0}^t a \ v \ dt \qquad (4) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Rearrange the order.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_{t_0}^t v \ a \ dt  \qquad (5) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Use (2) to change the differential, the limits change accordingly.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_{v_0}^v v \ dv  \qquad (6)  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Putting everything together:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_{s_0}^s a \ ds = \tfrac{1}{2}v^2 - \tfrac{1}{2}v_0^2  \qquad (7) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Multiplying both sides with &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and applying &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt; gives:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \int_{s_0}^s F \ ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2  \qquad (8) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Work-Energy theorem is narrower in scope than the principle of conservation of Energy. Kinetic energy is galilean invariant, so the Work-Energy theorem is applicable only for forces such that the change of velocity is independent of the current velocity.&lt;/p&gt;&#xA;&lt;p&gt;(Also, the Work-Energy theorem is applicable only if  an unambiguous integration of the force exists.)&lt;/p&gt;&#xA;&lt;p&gt;The principle of conservation of Energy, on the other hand, is an unconditional assertion. The scope is unlimited.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="7244" LastEditDate="2021-07-18T18:30:11.330" LastActivityDate="2021-07-18T18:30:11.330" Title="During which period did the name Work-Energy theorem enter physics discourse" Tags="|energy|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13062" PostTypeId="2" ParentId="13052" CreationDate="2021-04-03T22:41:10.420" Score="3" Body="&lt;p&gt;In my opinion: Euler did so much already that the contributions from later physicists/mathematicians are at a level of abstraction that is beyond the scope of standard physics courses.&lt;/p&gt;&#xA;&lt;p&gt;Among the subjects not already covered by Euler, I assume, is the quite unique case of the intermediate axis theorem. Arguably awareness of that phenomenon took off only because of the video recorded on a space station showing a &lt;a href=&quot;https://www.youtube.com/watch?v=1n-HMSCDYtM&quot; rel=&quot;nofollow noreferrer&quot;&gt;demonstration of the motion of a rigid body with three different moments of inertia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The equations for that case are known, numerical simulations reproduce the motion. But will a textbook author include a section about the intermediate axis theorem? I doubt it; even the minimum mathematics to treat the case is already very abstract.&lt;br /&gt;&#xA;(Also, in order to move according to the idealized model the object must be &lt;em&gt;perfectly rigid&lt;/em&gt;. But in the real world there is no such thing as perfect rigidity. Any mechanism that &lt;em&gt;can&lt;/em&gt; dissipate kinetic energy &lt;em&gt;will&lt;/em&gt; dissipate kinetic energy.)&lt;/p&gt;&#xA;&lt;p&gt;So, in the end: the part of rotational mechanics that makes it into physics textbooks has a short history: it's Euler.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;As to development of rotational mechanics prior to Euler:&lt;/p&gt;&#xA;&lt;p&gt;Newton had inferred that since the Earth is rotating it will have an equatorial bulge. Because of that non-spherical shape the center of gravitational attraction does not coincide with the center of mass. (The two coincide only in the case of a perfectly spherical celestial body.) As a consequence the gravity of the Sun exerts a torque on the Earth, and so does the gravity of the Moon. Newton had arrived at the view that the precession of the equinoxes was due to this combined torque effect from Sun and Moon.&lt;/p&gt;&#xA;&lt;p&gt;The interpretation must be that Newton had anticipated the concept of gyroscopic precession. Not a generalized concept of gyroscopic precession, but a concept specific for the Earth precession. In the Principia Newton offers a calculation.&lt;/p&gt;&#xA;&lt;p&gt;The outcome of Newton's calculation matches the actual precession of the equinoxes quite well, but historians of science state that the accuracy of the data that Newton was working with was certainly not good enough to be that succesful. There is a question here on HSM about that &lt;a href=&quot;https://hsm.stackexchange.com/questions/7881/was-newtons-successful-calculation-of-precession-of-equinoxes-a-fluke&quot;&gt;Was Newton's succesful calculation of precession of equinoxes a fluke?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Most of the Principia was later restated in the form of differential calculus. It was that more accessible form that spread throughout the physics community. It would appear: on the subject of the precession of the equinoxes Newton's description in the Principia was not enough to reconstruct his thought process. So it appears that on that subject  there was no follow-up.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2021-04-03T22:41:10.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13063" PostTypeId="2" ParentId="11982" CreationDate="2021-04-05T09:48:49.290" Score="4" Body="&lt;p&gt;I don't have a fully satisfactory answer, but maybe this helps.&lt;/p&gt;&#xA;&lt;p&gt;First one should note that Tarjan came up with an &lt;span class=&quot;math-container&quot;&gt;$O(m \log \log n)$&lt;/span&gt;-algorithm roughly at the same time. It's in this technical report:&#xA;&lt;a href=&quot;http://www2.eecs.berkeley.edu/Pubs/TechRpts/1975/ERL-m-501.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www2.eecs.berkeley.edu/Pubs/TechRpts/1975/ERL-m-501.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yao says in his paper that Tarjan developed an &lt;span class=&quot;math-container&quot;&gt;$O(m \sqrt{\log n})$&lt;/span&gt; algorithm, but it was not published. Maybe some of Tarjan's ideas influenced him, who knows.&lt;/p&gt;&#xA;&lt;p&gt;I think Yao's algorithm is natural once a linear-time median-finding algorithm is available. Selection Sort is slow because we look at all remaining elements in each pass in order to determine the minimum. Boruvka's (Sollin's) algorithm does something very similar, it determines the minimum outgoing edge per component in every phase. If we find some &amp;quot;approximate&amp;quot; ordering of the outgoing edges that allows us to ignore some elements until later on, we can speed up the algorithm.&#xA;Now there are two avenues that are &amp;quot;orthogonal&amp;quot; to each other:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Either partition the edges into blocks whose content is unsorted, but the blocks are &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt;&amp;quot;-ordered. This can be done using an algorithm for finding medians (in a quicksort with optimal pivots that stops the recursion once the desired block size is reached). Then we only look for the minimum edge in the first block until it is used up, then continue with the next block and so on.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Or partition the adjacency lists into &amp;quot;packets&amp;quot; that have no order among them, but each of packet is sorted (what you get when stopping the Merge Sort recursion once the desired packet size is reached). Then compare the currently smallest elements from all packets. The packet whose edge is the smallest then moves on to the second-largest, and so on. This is a viable alternative to Yao's approach, which uses 1., and is in fact used to turn the Fredman-Tarjan algorithm into the Gabow-Galil-Spencer-Tarjan algorithm, (but more complicated since Fibonacci Heaps are used for the packets in order to make them meldable).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;By the way, I think Yao made a small mistake in the case that &lt;span class=&quot;math-container&quot;&gt;$m &amp;lt; n \log n$&lt;/span&gt;. He should use a &lt;strong&gt;contractive&lt;/strong&gt; variant of Boruvka to reduce the number of vertices and avoid the union-find structure, which could incur a runtime of &lt;span class=&quot;math-container&quot;&gt;$n \log n$&lt;/span&gt; or more even in the &lt;span class=&quot;math-container&quot;&gt;$\log \log n$&lt;/span&gt; phases, depending on the UF structure. One can contract in every of the &lt;span class=&quot;math-container&quot;&gt;$\log \log n$&lt;/span&gt; phases in linear time. Then use Yao's trick with the blocks. (Once might even first run the simplest variant of the algorithm where the components are determined naively in every phase and contract only once after the &lt;span class=&quot;math-container&quot;&gt;$\log \log n$&lt;/span&gt; Phases are finished.)&lt;/p&gt;&#xA;" OwnerUserId="14224" LastEditorUserId="14224" LastEditDate="2023-05-30T09:24:16.790" LastActivityDate="2023-05-30T09:24:16.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13064" PostTypeId="2" ParentId="13052" CreationDate="2021-04-05T12:18:18.830" Score="2" Body="&lt;p&gt;The notion of moment is important in rigid body dynamics. Historically speaking, this was pioneered by Archimedes in his &lt;em&gt;Method&lt;/em&gt; where he outlined his theory of the lever.&lt;/p&gt;&#xA;&lt;p&gt;It's worth adding that the parallelogram of forces/velocities was pioneered in the work known as &lt;em&gt;Mechanica&lt;/em&gt;, traditionally thought to be Aristotle, but now thought to be by a student of his school, possibly Achytas. It was also an influence on Archimedes.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-04-05T12:18:18.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13065" PostTypeId="2" ParentId="11982" CreationDate="2021-04-06T11:09:23.267" Score="2" Body="&lt;p&gt;Adding to Flowi's excellent answer, it seems that the main new ingredient in Yao's algorithm is a linear-time selection algorithm, which was new at the time.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I think Yao's algorithm is natural once a linear-time median-finding algorithm is available.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I found two sources confirming this. The first source is Chapter 6 &amp;quot;Minimum Spanning Trees&amp;quot; in Tarjan's book &amp;quot;Data Structures and Network Algorithms&amp;quot;, SIAM 1983:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By judiciously implementing a Boruvka-like algorithm using the appropriate data structures, we can obtain a method that is faster on sparse graphs than is any of the classical algorithms. Yao [29] was the first to propose such a method. His algorithm runs in&#xA;O(m log log n) time but needs a linear-time selection algorithm [2], [24] and is thus not very practical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second source is Ronald L. Graham, Pavol Hell:&#xA;On the History of the Minimum Spanning Tree Problem. IEEE Ann. Hist. Comput. 7(1): 43-57 (1985)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A straightforward implementation of Algorithm 3 would run in time O(e log u), because each time the rule defining Algorithm 3 is applied, the number of fragments decreases by at least one-half. Yao [Yao75] has discovered an implementation of Algorithm 3 in time 0( e log log u) using the linear selection algorithm [BFPRT73].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9523" LastActivityDate="2021-04-06T11:09:23.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13066" PostTypeId="1" CreationDate="2021-04-06T14:10:55.063" Score="1" ViewCount="63" Body="&lt;p&gt;What is the middle name of &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Gr%C3%A4tzer&quot; rel=&quot;nofollow noreferrer&quot;&gt;George A. Grätzer&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerDisplayName="user14234" LastEditorUserId="10106" LastEditDate="2021-04-06T14:17:42.387" LastActivityDate="2021-04-06T14:17:42.387" Title="What is the middle name of George A. Grätzer?" Tags="|mathematics|logicians|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13067" PostTypeId="1" CreationDate="2021-04-06T17:12:32.453" Score="2" ViewCount="214" Body="&lt;p&gt;A black hole is a compact region of spacetime where gravity is so strong that nothing, even light, can escape from its event horizon, and, interestingly, Einstein didn't accept this viewpoint. Now, I'm curious:  who named &amp;quot;Black Hole&amp;quot;? And when? I know that John Archibald Wheeler is best known for using the term &amp;quot;black hole&amp;quot;, but I heard that he adopted this term from the other physicists. Does anyone knows extra formal information about this? For example, when was the term &amp;quot;black hole&amp;quot; used for the first time in a manuscript?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Thanks you very much in advance ...&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="14138" LastEditorUserId="12724" LastEditDate="2021-04-07T02:59:10.250" LastActivityDate="2021-04-07T12:13:08.117" Title="Who named Black Hole?" Tags="|physics|relativity-theory|physics-social-history|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13068" PostTypeId="1" AcceptedAnswerId="13070" CreationDate="2021-04-06T19:12:48.193" Score="2" ViewCount="144" Body="&lt;p&gt;Why was there an approximately thirty year gap between the discovery of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Compactness_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;compactness theorem&lt;/a&gt; (for countable theories) in 1930 and Robinson's elucidation of &lt;a href=&quot;https://en.wikipedia.org/wiki/Nonstandard_analysis&quot; rel=&quot;nofollow noreferrer&quot;&gt;nonstandard analysis&lt;/a&gt; in the early 1960s? Perhaps my question is naive, but my reasoning is thus: Once one has the compactness theorem, the existence of nonstandard models of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; follows straightforwardly: Add a new constant symbol &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; and an axiom &lt;span class=&quot;math-container&quot;&gt;$c &amp;gt; n$&lt;/span&gt; for each natural number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (the full details can be found in chapter 1 of Richard Kaye's book &lt;a href=&quot;https://global.oup.com/academic/product/models-of-peano-arithmetic-9780198532132&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Models of Peano Arithmetic&lt;/em&gt;&lt;/a&gt;). Building up a thorough, polished theory of NSA once the existence of nonstandard models of the reals has been established is of course hardly a trivial task, but I wonder why such a project wasn't already undertaken in the 1930s.&lt;/p&gt;&#xA;&lt;p&gt;(Note this tangentially relevant &lt;a href=&quot;https://hsm.stackexchange.com/questions/3723/&quot;&gt;question&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="12735" LastActivityDate="2021-04-07T13:54:47.130" Title="Why was there a thirty year gap between the compactness theorem and nonstandard analysis?" Tags="|mathematical-logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13069" PostTypeId="2" ParentId="13067" CreationDate="2021-04-07T12:13:08.117" Score="3" Body="&lt;p&gt;Space.com claims that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term &amp;quot;black hole&amp;quot; was coined many years later in 1967 by American&#xA;astronomer John Wheeler.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which, given Conifold's references via Wikipedia, lead me to propose that Wheeler gets credit for making the name &amp;quot;go viral.&amp;quot;   As with many names or inventions in science (e.g., who gets credit for television), fame goes to the person who is at the 'tipping point' rather than the original inventor/discoverer.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-04-07T12:13:08.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13070" PostTypeId="2" ParentId="13068" CreationDate="2021-04-07T13:54:47.130" Score="5" Body="&lt;p&gt;The understanding of the &amp;quot;obvious&amp;quot; corollary of &lt;a href=&quot;https://plato.stanford.edu/entries/goedel/#ComThe&quot; rel=&quot;noreferrer&quot;&gt;Gödel Completeness Theorem&lt;/a&gt; now called Compactness Theorem and its consequences was a &amp;quot;slow&amp;quot; process.&lt;/p&gt;&#xA;&lt;p&gt;See John Dawson, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/01445349308837208&quot; rel=&quot;noreferrer&quot;&gt;The Compactness of First-order Logic From Gödel to Lindström (HPL,1993)&lt;/a&gt;, page 18:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After 1934, [...] the only person who seems to have recognized the importance of&#xA;compactness was the Russian A. I. Maltsev. Beginning in 1936, he published 'a&#xA;seminal run' of papers in what would now be called model-theoretic algebra, papers in which he 'obtained applications [of logic] to group theory of greater technical virtuosity than the possibly more basic applications to algebra later on found [...] by Henkin and Abraham Robinson' (Sabbagh 1991). Three of those papers (1936,1940 and 1941) are of interest here.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In his first published work, written in German, Maltsev 1936 [Maltsev, A.I. 1936 '&lt;em&gt;Untersuchungen aus dem Gebiete der mathematischen Logik&lt;/em&gt;', Matematicheskii Sbornik, n.s., 1, 323-336.] devoted his efforts to generalizing two theorems, one for the propositional calculus and the other for the restricted functional calculus. The theorems in question were Godel's compactness theorem and Skolem's result that no denumerable set of formulas of first-order logic can completely characterize the structure of the natural numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See English translation into: A.I. Mal'cev, &lt;a href=&quot;https://books.google.it/books?id=GeS2uhg74cAC&amp;amp;pg=PA1&quot; rel=&quot;noreferrer&quot;&gt;The Metamathematics of Algebraic Systems: Collected Papers 1936-1967 (North Holland, 1971)&lt;/a&gt;, page 1:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This article is devoted to generalizing two theorems, one for propositional calculus (PC) and the other for first-order predicate logic (FOPL).&#xA;The first theorem is due to Gödel [K. Gödel, &lt;em&gt;Die Vollsthdigkeit der Axiome des logischen Funktionenkalkuls&lt;/em&gt; (1936),] and can be formulated as follows:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For any countable system of formulas of PC to be consistent, it&#xA;is sufficient that every finite part of the system be consistent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, see also &lt;em&gt;On a General Method for Obtaining Local Theorems in Group Theory by A. Mal'cév&lt;/em&gt;, Review by Leon Henkin and Andrzej Mostowski (Jsl, 1959):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Historical note&lt;/em&gt;. The formulation and proof of the &amp;quot;general local theorem&amp;quot; [&lt;em&gt;If every finite subset of a given (possibly non-denumerable) set of first-order sentences is satisfiable, then so is the whole set&lt;/em&gt;] for denumerable sets of first-order sentences is of course due to Gödel in 1930. The corresponding result for non-denumerable sets of formulas of propositional calculus was given by Mal'cev in 1936.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See Abraham Robinson, &lt;a href=&quot;https://projecteuclid.org/search?term=Abraham+Robinson+Nonstandard+arithmetic+&quot; rel=&quot;noreferrer&quot;&gt;Nonstandard arithmetic (1967)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In 1934 it was pointed out by Thoralf Skolem that there exist proper extensions of the natural number system which have, in some sense, &amp;quot;the same properties&amp;quot; as the natural numbers. [...] he did not concern himself further with the properties of the structures whose existence he had established. n due course these and similar structures became known as nonstandard models of arithmetic and papers concerned with them appeared in the literature (e.g. Henkin, Kemeny, Mendelson, Robinson). Beginning in the fall of 1960, the application of similar ideas to analysis led to a rapid development in which nonstandard models of arithmetic played an auxiliary but vital part.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;IMO it is important to observe that in Abraham Robinson, &lt;em&gt;On the Metamathematics of Algebra&lt;/em&gt; (1951) there is no Compactness Th, while in the revised and enlarged edition: Abraham Robinson, &lt;em&gt;Introduction to Model Theory and the Metamathematics of Algebra&lt;/em&gt; (1963) the theorem is there (page 21).&lt;/p&gt;&#xA;&lt;p&gt;What happened in between?: the &amp;quot;slow&amp;quot; process described above that occurred during the 50s and 60s.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2021-04-07T13:54:47.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13071" PostTypeId="1" CreationDate="2021-04-07T14:21:07.880" Score="3" ViewCount="57" Body="&lt;p&gt;In Hardy's &lt;em&gt;A Course of Pure Mathematics&lt;/em&gt; (§117 in the 10th edition), in a discussion of differentiation of polynomials, he introduces what he calls the &amp;quot;binomial form&amp;quot; of a polynomial:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;a_0x^n + \pmatrix{n \\ 1}a_1x^{n-1} + \pmatrix{n\\2}a_2x^{n-2}&#xA; + \ldots + a_n&#xA;$$&lt;/span&gt;&#xA;and states that this is often written symbolically as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(a_0, a_1, \ldots, a_n)\!\!(x, 1)^n&#xA;$$&lt;/span&gt;&#xA;This is a notation that I have not encountered before. I would be very interested in any information about this notation and its history. Also I have one specific question: what does the &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$)\!\!(x, 1)$&lt;/span&gt; mean?&lt;/p&gt;&#xA;" OwnerUserId="6443" LastActivityDate="2021-04-07T14:21:07.880" Title="Notation for the &quot;binomial form&quot; of a polynomial" Tags="|notation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13072" PostTypeId="2" ParentId="2892" CreationDate="2021-04-07T15:14:34.843" Score="3" Body="&lt;p&gt;al-Jazari (and Ridwan too) explicitly says that Archimedes is the source for several of the components of his clock(s); he also says that one component of Archimedes' clock (the flow regulator) does not work, and proposes a solution where the regulator is a full circle rather than a semi-circle.&lt;/p&gt;&#xA;&lt;p&gt;The Arab treatise on the construction of water clocks that is attributed to Archimedes describes a clock that is very similar, from a mechanical, pneumatic and hydraulic point of view, to al-Jazari's and Ridwan's clocks, so it is very probable that both those engineers based their own designs on this treatise (or on the tradition it is based onto).&lt;/p&gt;&#xA;&lt;p&gt;Scholars such as Hill believe that the work might have been originally written in Greek, but that the treatise was heavily reworked by Islamic authors; others, such as Drachmann, believe that it might be a compilation of several Greek authors.&lt;/p&gt;&#xA;&lt;p&gt;More recently, Anette Schomberg recognised the treaty as (mostly) a translation of a treatise on water clocks by Philo of Byzantium; the clock described by Philo, therefore, should be imaged as similar to the Gaza clock described by Procopius, or the water clock described by Cassiodorus, which later were adopted by the Islamic engineers (see al-Jazari Castle clock for a very similar example).&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, Schomberg recognizes both the clock and the flow regulator in the description of Ctesibius' clock written by Vitruvius: this flow regulator is the same as that described by al-Jazari, so the basic idea should be:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Philo of Byzantium invents a water clock based on a floater sinking in a vessel that provides the motion for several mechanical, hydraulic and pneumatic automata (including the hour indication). Philo's flow regulator is semi-circular&lt;/li&gt;&#xA;&lt;li&gt;Ctesibius improves Philo's clock on a couple of points, including a circular flow regulator&lt;/li&gt;&#xA;&lt;li&gt;This clock is know to Vitruvius and some implementations are known to other authors (Procopius of Gaza, Cassiodorus and Boethisu, maybe Teophanes)&lt;/li&gt;&#xA;&lt;li&gt;Arab authors translate a treatise that describes Philo's clock (including the semi-circular flow regulator)&lt;/li&gt;&#xA;&lt;li&gt;al-Jazari and Ridwan receive this tradition and build several clocks on the same principles; they either receive the tradition that links directly back to Philo (maybe the same Arab treatise we have), and re-invent the circular flow regulator, or they receive Ctesibius' tradition and present this innovation as their own (the second possibility has a higher probability, due to the presence of another of Ctesibius' improvements into al-Jazari's and Ridwan's clocks)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So to sum it up:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;What were Archimedes' innovations to the water clock?&lt;/strong&gt;&#xA;Probably none, he did not build any as far as we know.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What were Philo's innovations to the water clock?&lt;/strong&gt;&#xA;He probably built the original one, with all the automata and the sounds, and also wrote the treatise that is currently attributed to pseudo-Archimedes.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What were Ctesibius' innovations to the water clock?&lt;/strong&gt;&#xA;He definitely produced a better outflow device (Vitruvius explicitly says so) and he probably invented the circular flow regulator. &amp;quot;Ctesibius clock&amp;quot; was, in any major sense, similar to al-Jazari's Castle clock&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;How did the flow regulator work?&lt;/strong&gt;&#xA;It was composed by two parts, one to make the inflow to the tank regular (through a cone-shaped floater that matches a cone-shaped spout) and one to make it vary with the day of the year (the semi-circular/circular stuff I mentioned above)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What about the syphon in Ctesibius' clock?&lt;/strong&gt;&#xA;Very probably never existed, it was probably due to the imagination of Perrault, who produced a lavish and accurate edition of Vitruvius' in the 17th century, but proposed the wrong reconstruction for Ctesibius' clock&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source:  Schomberg, A. (2017): &amp;quot;To amaze the world&amp;quot; - A contribution to the shape and mean-ing of the water clock in antiquity. In: Wellbrock, K. (ed.): Schriften der Deutschen Wasserhistorischen Gesellschaft, Band 27-1, Siegburg, 2017, 301-340. (you can find it on academia.edu)&lt;/p&gt;&#xA;" OwnerUserId="12801" LastActivityDate="2021-04-07T15:14:34.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13073" PostTypeId="1" AcceptedAnswerId="13074" CreationDate="2021-04-08T16:35:52.997" Score="3" ViewCount="235" Body="&lt;p&gt;(originally asked at M.SE: &lt;a href=&quot;https://math.stackexchange.com/questions/4094361/first-time-the-reals-were-axiomatized-as-the-unique-complete-ordered-field&quot;&gt;https://math.stackexchange.com/questions/4094361/first-time-the-reals-were-axiomatized-as-the-unique-complete-ordered-field&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;I'm looking for historical references on the history of the axiomatization of real numbers. I found out that Tarski had an axiomatization of the reals as in &lt;a href=&quot;https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals&lt;/a&gt;. However, the usual approach to the reals is as the &amp;quot;unique complete ordered field&amp;quot;, which is somewhat different.&lt;/p&gt;&#xA;&lt;p&gt;Who first described the reals as the &amp;quot;unique complete ordered field&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Any other axiomatizations which have appeared throughout history are also welcome.&lt;/p&gt;&#xA;" OwnerUserId="14260" LastEditorUserId="10106" LastEditDate="2021-04-09T03:41:18.707" LastActivityDate="2021-04-10T03:15:41.910" Title="First time the real numbers were axiomatized as the &quot;unique complete ordered field&quot;" Tags="|mathematics|real-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13074" PostTypeId="2" ParentId="13073" CreationDate="2021-04-08T18:06:54.243" Score="3" Body="&lt;p&gt;After Hilbert published a paper on complete ordered field axioms &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0008?tify=%7B%22view%22:%22info%22,%22pages%22:%5B184%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Über den Zahlbegriff&amp;quot;&lt;/a&gt;  in 1900, a major paper that laid the foundation of abstract field theory was &lt;a href=&quot;http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Algebraische Theorie der Körper&amp;quot;&lt;/a&gt; published by Ernst Steinitz in 1910. It contains axioms and proofs for field theory that are (very) closed to modern algebra texts. The topics covered include abstract field axioms, isomorphism, integral domain, prime field, quotient field, finite field (with characteristics of p), function field, field extension, algebraic complete field, order field (in order type and set), and more. This paper also greatly influenced Emmy Noether for her subsequent work in modern algebra.&lt;/p&gt;&#xA;&lt;p&gt;Dedekind defined reals based on rationals known as Dedekind cut in his 1872's paper &lt;a href=&quot;http://www.math.ubc.ca/%7Ecass/courses/m446-05b/dedekind-book.pdf#page=15&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Continuity and Irrational Numbers&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-04-10T03:15:41.910" LastActivityDate="2021-04-10T03:15:41.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13075" PostTypeId="2" ParentId="13039" CreationDate="2021-04-08T18:42:14.740" Score="0" Body="&lt;p&gt;Joseph Fourier created Fourier series to study the propagation of heat in solid bodies initially but it became an important field in pure mathematics later.&lt;/p&gt;&#xA;&lt;p&gt;Oliver Heaviside developed Heaviside calculus to investigate electric circuits, which was popular among electrical engineers but was initially rejected by mathematicians for lacking mathematical rigor. Later, Heaviside calculus was reworked by Bromwich into what is known as Laplace transform today.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-04-09T21:50:51.197" LastActivityDate="2021-04-09T21:50:51.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13076" PostTypeId="1" AcceptedAnswerId="13078" CreationDate="2021-04-08T18:58:53.993" Score="6" ViewCount="905" Body="&lt;p&gt;I'm currently in a class on formal mathematics/formal logic/axiomatic set theory. Someone asked, &amp;quot;At the end of the day, as mathematicians, why do we care about rigor?&amp;quot; My professor gave an example of a mathematician who did not believe in the importance of rigor (implicitly refering to a time in which mathematics was rigorous by modern standards) and developed a number of results which others took up, following in his tradition of non-rigor, which eventually formed and became recognized as a branch or subfield of mathematics on its own. He said that the whole field was non-rigorous, and when outside mathematicians eventually bothered to look at their results rigorously it was found that the whole field was unfounded due to inconsistencies in some founding axioms. I asked what the name of the field or of the mathematician were, but he did not remember details. He only said that he thought it was a group of french mathematicians in the early 1900's studying a subfield of topology.&lt;/p&gt;&#xA;&lt;p&gt;My question is this: Does anyone else recognize this example, and if so, could you provide me with details/references? I have already dome some searching using a variety of search terms, as well as poked around on various mathematics-related SE sites and didn't find anything that looked like what my professor was talking about.&lt;/p&gt;&#xA;" OwnerUserId="14261" LastActivityDate="2021-05-12T17:16:46.920" Title="A branch of mathematics which refused to be rigorous?" Tags="|mathematics|mathematicians|reference-request|mathematical-logic|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13077" PostTypeId="1" CreationDate="2021-04-08T20:23:40.370" Score="0" ViewCount="82" Body="&lt;p&gt;Both the constant &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and the constant &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; have clear algebraic roles in complex numbers and in differential calculus.&lt;/p&gt;&#xA;&lt;p&gt;But did anyone ever propose an algebraic role for Euler-Mascheroni constant &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2021-04-09T00:00:39.523" Title="Did anyone ever try to determine or propose the algebraic role of Euler-Mascheroni constant?" Tags="|calculus|algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13078" PostTypeId="2" ParentId="13076" CreationDate="2021-04-08T20:42:24.240" Score="10" Body="&lt;p&gt;You might be looking for the Italian School of Algebraic Geometry. It has become the canonical example of problems with a lack of rigour.&lt;/p&gt;&#xA;&lt;p&gt;The short summary is that the school started with some unfounded postulates, that they used to derive a wide number of results. This must be understood in the context of an incipient field, where no rigorous foundation was available and occurred concurrently with the struggle to give analysis a rigorous foundation (late 19th century) so it is not even clear what debate about rigour could be had at this point.&lt;/p&gt;&#xA;&lt;p&gt;In any case, the first results could be proven by different methods in a much more laborious way, so that confidence grew in their methods. Eventually, some results of the Italian school were disproven, foundational problems were found that shown their methods inconsistent, and the whole tradition was abandoned in the early twentieth century.&lt;/p&gt;&#xA;&lt;p&gt;This summary is likely unjust, in the sense that it does not credit the members of the Italian school for advancing the field at its Naissance, and does not discuss that some members were more careless than others, and further that standards of rigour have evolved during that time, so it is not fair to judge a whole mathematical scene on some wrong results from some of the involved.&lt;/p&gt;&#xA;" OwnerUserId="5904" LastEditorUserId="14422" LastEditDate="2021-05-12T17:16:46.920" LastActivityDate="2021-05-12T17:16:46.920" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13079" PostTypeId="2" ParentId="13077" CreationDate="2021-04-08T23:48:21.157" Score="3" Body="&lt;p&gt;The answer is yes:&#xA;&lt;a href=&quot;https://www.ams.org/journals/bull/2013-50-04/S0273-0979-2013-01423-X/S0273-0979-2013-01423-X.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. Lagarias, Euler's constant: Euler's work and modern developments&lt;/a&gt;, BAMS,  50 (2013), no. 4, 527–628.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-04-09T00:00:39.523" LastActivityDate="2021-04-09T00:00:39.523" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13080" PostTypeId="2" ParentId="13076" CreationDate="2021-04-09T00:05:27.300" Score="1" Body="&lt;p&gt;Fuzzy set theory that was first introduced in 1965 as an extension of the classical notion of sets could be such a field you are looking for. In general, fuzzy logic resembles multi-valued logic but lacks a rigorous mathematical foundation. As the result, its conclusions are plausible but inexact (or fallacious).&lt;/p&gt;&#xA;&lt;p&gt;However, the uncertainty issues tackled by fuzzy set theory are often important real-world problems beyond the realm of probability. For example, one of the most active applications of fuzzy sets is to model linguistic expressions involving adjectives and adverbs because there are no rigorous mathematical models for them so far. As the result, a lot of researchers love working in fuzzy sets because they are attracted by its plausible appeal and choose to ignore (completely) the non-rigorous fact of it.&lt;/p&gt;&#xA;&lt;p&gt;As a result, fuzzy set theory has become a fast growing field (of mathematics) since it is introduced and boasts the most number of research papers among all fields of mathematics nowadays. Many people in practical fields such as electrical engineering, artificial intelligence and so on, like to work in fuzzy set theory.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-04-09T20:48:53.750" LastActivityDate="2021-04-09T20:48:53.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13081" PostTypeId="1" AcceptedAnswerId="13083" CreationDate="2021-04-09T09:12:39.697" Score="10" ViewCount="1983" Body="&lt;p&gt;I remember that I read somewhere that Lorentz remained an ether advocate till his death despite the empirical successes of Einstein's relativity in rejecting any kind of ether. However, I cannot remember the source. Can anybody please give me a possible reference regarding this?&lt;/p&gt;&#xA;" OwnerUserId="13363" LastActivityDate="2021-04-11T00:14:11.213" Title="Did Lorentz remain an ether advocate till his death?" Tags="|theoretical-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13083" PostTypeId="2" ParentId="13081" CreationDate="2021-04-09T11:14:48.493" Score="22" Body="&lt;p&gt;Lorentz did remain sympathetic to ether to the end. However, it was not necessarily to his &lt;em&gt;original&lt;/em&gt; conception of ether, and it was not &amp;quot;despite the empirical successes&amp;quot; of special relativity. Special relativity and Lorentz's ether theory not only produce identical empirical predictions but are equivalent in an even stronger sense, see e.g. &lt;a href=&quot;https://arxiv.org/abs/physics/0308033&quot; rel=&quot;noreferrer&quot;&gt;Szabo&lt;/a&gt;. So no empirical successes could distinguish between them, a fact recognized by both Einstein and Lorentz. Their diverging decisions were made on conceptual grounds. And Einstein showed similar ether loss remorse in his views after general relativity, so they were not that far apart.&lt;/p&gt;&#xA;&lt;p&gt;For Lorentz's evolution on the subject see &lt;a href=&quot;https://aapt.scitation.org/doi/pdf/10.1119/1.1975220&quot; rel=&quot;noreferrer&quot;&gt;Goldberg, The Lorentz Theory of Electrons and Einstein's Theory of Relativity&lt;/a&gt; and &lt;a href=&quot;https://aapt.scitation.org/doi/pdf/10.1119/1.1975220&quot; rel=&quot;noreferrer&quot;&gt;Kox, Hendrik Antoon Lorentz, the ether, and the general theory of relativity&lt;/a&gt;, where we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is well known that until his death in 1928 LORENTZ kept insisting on the usefulness of an ether. In spite of his often-expressed admiration for EINSTEIN'S&#xA;special theory of relativity, he preferred his own ether-based 'theory of electrons'. LORENTZ admitted that his theory and the special theory of relativity&#xA;had the same empirical consequences and that the ether could not be experimentally detected, but he maintained that some kind of ether was needed as carrier&#xA;of the electromagnetic field.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Lorentz's position shows considerable nuance and awareness of factors in theory choice that go beyond empirical consequences. In his Lectures on Theory of Electrons (1909) he admitted intuitive attachment, for example:&amp;quot;&lt;em&gt;I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter.&lt;/em&gt;&amp;quot; In Lectures on Theoretical Physics (1915) he suggested that the choice was, in part, a matter of taste, and that some ether substitute was needed either way, an opinion shared by Einstein (see below):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This, however, is a question of taste and of words.&#xA;For whether there is an ether or not, electromagnetic&#xA;fields certainly exist, and so also does the energy of&#xA;the electrical oscillations. If we do not like the name&#xA;of &amp;quot;ether&amp;quot; we must use another word as a peg to&#xA;hang all of these things upon. It is not certain&#xA;whether &amp;quot;space&amp;quot; can be so extended as to take care&#xA;not only of the geometrical properties but of the&#xA;electrical ones.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And in Problems of Modern Physics (1922) Lorentz treated it as a &amp;quot;grand scheme of things&amp;quot; choice, and even suggested that Einstein's decision to dispense with the electromagnetic ether was right &lt;em&gt;for Einstein&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;All our theories help us to form pictures or&#xA;images of the world around us, and we try to do this&#xA;in such a way that the phenomena may be coordinated&#xA;as well as possible, and that we may see&#xA;clearly the way in which they are connected... My notion&#xA;of time is so definite that I clearly distinguish in my&#xA;picture what is simultaneous and what is not...&#xA;As to the ether... though the conception of it has&#xA;certain advantages, it must be admitted that if&#xA;Einstein had maintained it he certainly would not&#xA;have given us his theory, and so we are grateful to&#xA;him for not having gone along old fashioned roads.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Einstein explained his differences with Lorentz, as well their points of convergence, in his May 1920 address &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Einstein_ether/&quot; rel=&quot;noreferrer&quot;&gt;Ether and Relativity (published in 1922)&lt;/a&gt; at Lorentz's University of Leiden:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;As to the mechanical nature of the Lorentzian ether, it may be said of it, in a somewhat playful spirit, that immobility is the only mechanical property of which it has not been deprived by H A Lorentz. It may be added that the whole change in the conception of the ether which the special theory of relativity brought about, consisted in taking away from the ether its last mechanical quality, namely, its immobility... More careful reflection teaches us however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, i.e. we must by abstraction take from it the last mechanical characteristic which Lorentz had still left it.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;As to the part which the new ether is to play in the physics of the future we are not yet clear. We know that it determines the metrical relations in the space-time continuum, e.g. the configurative possibilities of solid bodies as well as the gravitational fields; but we do not know whether it has an essential share in the structure of the electrical elementary particles constituting matter. Nor do we know whether it is only in the proximity of ponderable masses that its structure differs essentially from that of the Lorentzian ether; whether the geometry of spaces of cosmic extent is approximately Euclidean.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-04-11T00:14:11.213" LastActivityDate="2021-04-11T00:14:11.213" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13087" PostTypeId="1" CreationDate="2021-04-11T11:34:30.083" Score="1" ViewCount="185" Body="&lt;p&gt;My question is a mix of mathematical and historical, if you consider my question will be better answered in the mathematics community, please tell me.&lt;br /&gt;&#xA;I want to know the historical roots of Stone's representation theorem. Here is what I have found:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In &lt;a href=&quot;https://plato.stanford.edu/entries/boolalg-math/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stanford Encyclopedia of Philosophy&lt;/a&gt;, it says that one of the roots and applications of the duality is in the &amp;quot;algebra of projections (Functional analysis)&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;In &amp;quot;The theory of representations for Boolean algebras&amp;quot; M.H.Stone says: &lt;br /&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The writer's interest in the subject, for example,&#xA;arose in connection with the spectral theory of symmetric transformations in&#xA;Hilbert space and certain related properties of abstract integrals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have tried to look in &amp;quot;Linear transformations in Hilbert Space and their application to analysis&amp;quot; by M.H.Stone, in the chapter IX &amp;quot;Symmetric transformations&amp;quot; but my knowledge in functional analysis is not very advanced and I'm not used to the old notation that is used in it.&lt;br /&gt;&#xA;What I'm trying to find is where do the boolean algebras and booleans rings appear in functional analysis. &lt;br /&gt;&#xA;Again, if you think could be better answered in mathematics section, i would be grateful if someone could tell how to redirect it.&lt;/p&gt;&#xA;" OwnerUserId="14296" LastEditorUserId="14296" LastEditDate="2021-04-12T12:52:22.540" LastActivityDate="2021-04-12T12:52:22.540" Title="Origins of Stone duality" Tags="|mathematics|mathematicians|mathematical-logic|functional-analysis|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13088" PostTypeId="1" CreationDate="2021-04-13T09:46:35.590" Score="21" ViewCount="6666" Body="&lt;p&gt;The Wikipedia page on &lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_Galilei&quot; rel=&quot;noreferrer&quot;&gt;Gallileo Galilei&lt;/a&gt; mentions, among other things:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;His multiple interests included the study of astrology, which at the time was a discipline tied to the studies of mathematics and astronomy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is - did he actually believe that astrology worked? Is there any evidence on his attitude towards astrology?&lt;/p&gt;&#xA;" OwnerUserId="14326" LastEditorUserId="55" LastEditDate="2021-04-13T11:43:34.983" LastActivityDate="2021-04-15T05:41:27.947" Title="Did Galileo Galilei believe in astrology?" Tags="|astronomy|social-context|galileo|" AnswerCount="5" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13089" PostTypeId="2" ParentId="13088" CreationDate="2021-04-13T11:41:38.397" Score="26" Body="&lt;p&gt;Galileo not only believed but taught and actively practiced astrology, like Ptolemy and Kepler before him. His primary source might have been Porphyry’s commentary on Ptolemy's Tetrabiblos. In particular, he constructed and interpreted horoscopes, including ones for himself and his daughters that are extant. His discovery of the moons of Jupiter was followed by making a 1610 horoscope of Cosimo de Medici, where Jupiter is prominently featured. Astrology was taken as a matter of course at the time, so believing it carried no more weight than believing that the Sun goes around the Earth by someone never forced to confront contrary evidence or consider alternatives, and going about their everyday business on that assumption.&lt;/p&gt;&#xA;&lt;p&gt;The subject was intertwined with what we now call astronomy, and taught at top universities by &lt;em&gt;mathematici&lt;/em&gt; like Galileo. Ironically, the Church was on the opposite side on this one, as it opposed &amp;quot;fatalistic&amp;quot; interpretations of astrological predictions. Galileo was investigated for practicing &amp;quot;fatalistic astrology&amp;quot; by the Inquisition in 1604, but the charges were dropped. The seeds of modern skepticism about astrology only emerged in the later part of 17th century, after Galileo's death.&lt;/p&gt;&#xA;&lt;p&gt;Galileo's practice was long overlooked and/or misinterpreted by earlier historians, which is also reflected in the more literary works like Brecht's play Galileo and Koestler's Sleepwalkers, which anticipated some themes from Kuhn's Scientific Revolutions. For a comprehensive review of primary sources and commentary see &lt;a href=&quot;https://www.jstor.org/stable/10.1353/ren.2008.0133&quot; rel=&quot;noreferrer&quot;&gt;Galileo’s Astrology volume (2003)&lt;/a&gt;. Here is from &lt;a href=&quot;https://journals.sagepub.com/doi/abs/10.1177/0021828618793218?journalCode=jhaa&quot; rel=&quot;noreferrer&quot;&gt;Rutkin's review Galileo as Practising Astrologer&lt;/a&gt; of Ernst's critical edition (2017) of Galileo's astrological manuscript no.81, that was only recently made available to scholars in full:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It contains horoscopes for 19 people (some with two slightly different&#xA;figures) in Galileo’s own hand and was gathered together into one manuscript from&#xA;originally separated folia. Although it has been known for over a century (since 1881&#xA;when Antonio Favaro wrote about it in a very brief article) and although its attribution&#xA;has never been seriously challenged, MS Gal. 81 and the essentially normal astrological&#xA;practices that it reveals have not yet been taken fully into account in the literature seeking&#xA;to understand and assess Galileo’s life and works, and it has never been published in its&#xA;entirety...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;While teaching at the University of Padua, Galileo practised astrology by constructing&#xA;and interpreting horoscopes for a range of patrons, students and family members,&#xA;seeking, among other things, to raise extra money as we can see in his contemporary&#xA;account books for payments made to him “per sortem.” This evidence dates some of&#xA;these horoscopes to 1603, that is, soon after the death of one of his early patrons, Gian&#xA;Vincenzo Pinelli (1535–1601)...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;We possess other evidence for Galileo’s astrological practice around this time, including&#xA;his first investigation by the Venetian Inquisition in 1604, following the official&#xA;denunciation made by his former amenuensis, Silvestro Pagnoni from Pesaro, who had&#xA;accused Galileo of practising a deterministic astrology. We know about this revealing&#xA;incident in some detail from Antonino Poppi’s valuable researches, including that the&#xA;case was dropped because Galileo was practising normal, not deterministic astrology. As&#xA;we now know without any doubt, throughout the 16th and well into the 17th century&#xA;astrology was still taught at the finest Italian universities, including at both Padua and&#xA;Bologna, and was a normal part of the study, practice and teaching of an early modern&#xA;“mathematicus.”...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In addition to setting out all of the astronomical and astrological data as well as the&#xA;numerical calculations used in MS 81 for erecting and interpreting these horoscopes,&#xA;perhaps the most valuable feature of Ernst’s edition is the transcription of Galileo’s astrological&#xA;interpretations, which were all written in Latin in the five horoscopes that include&#xA;them, some of which are very difficult to read in the manuscript. These interpretations&#xA;are very interesting, in that we can thereby catch a glimpse of Galileo doing the normal&#xA;work of both casting and interpreting horoscopes... The construction&#xA;of his own horoscope (in two variant versions, but without an interpretation) and those&#xA;for his two daughters (with their interpretations), as well as for other family members,&#xA;belie assertions claiming that Galileo himself personally rejected astrology or that he&#xA;only used it cynically for patronage purposes.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-04-14T06:57:20.727" LastActivityDate="2021-04-14T06:57:20.727" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13090" PostTypeId="1" AcceptedAnswerId="13208" CreationDate="2021-04-13T16:22:07.497" Score="1" ViewCount="283" Body="&lt;p&gt;I noticed that we use &lt;span class=&quot;math-container&quot;&gt;$\sum$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\prod$&lt;/span&gt; for summation and infinite product (I don't know why it does not have a name like the other two), but we use &lt;a href=&quot;https://en.wikipedia.org/wiki/Tetration#Notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;different looking notation&lt;/a&gt; for tetration. Is there some sort of historical reason why not, or did no-one just do it.&lt;/p&gt;&#xA;" OwnerUserId="14329" LastActivityDate="2021-05-19T23:54:52.923" Title="Why is there no notation for tetration similar to summation?" Tags="|notation|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13091" PostTypeId="1" CreationDate="2021-04-13T17:01:32.680" Score="0" ViewCount="283" Body="&lt;p&gt;I know that problems caused by radiation are pretty much random and unexpected at low levels of exposition but this randomness begins to be more probable when the exposition grows and this makes me think on the particular case of Ève Curie, who died 102. How could she be so healthy under the circumstances of her childhood? A good person to compare with would be her sister, Irène, who sadly died young because of leukaemia. However, this does not imply directly that her illness was caused by the pre-born exposition, as she also worked with radioactive materials. The longest lived, and most common, isotope of radium, &lt;span class=&quot;math-container&quot;&gt;$^{226}$&lt;/span&gt;Ra decays usually in mode &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; particles can be stopped by the body so Marie's body could protect her daughter’s during pregnancy from &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; particles. But what about the gamma emissions and sporadic &lt;span class=&quot;math-container&quot;&gt;$\beta^{-}$&lt;/span&gt; decays? How was Ève protected against these? Was she just lucky or is there some explanation?&lt;/p&gt;&#xA;&lt;p&gt;I knew the story of Marie and Pierre and about their daughters but these questions came to my mind after watching the (very cute) romantic movie dedicated to their scientific-personal relation.&lt;/p&gt;&#xA;" OwnerUserId="13404" LastActivityDate="2021-04-13T17:01:32.680" Title="How is it possible that Pierre and Marie Curie's daughter Ève was so healthy?" Tags="|physics|biographical-details|chemistry|nobel-prize|radiation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13092" PostTypeId="2" ParentId="12988" CreationDate="2021-04-14T02:28:43.043" Score="2" Body="&lt;p&gt;I published an article on the meaning of the terms al-jabr and al-muqabala in Arabic algebra, which you can download from my Academia.edu page:&#xA;&amp;quot;Simplifying equations in Arabic algebra&amp;quot; Historia Mathematica 34 (2007), 45-61.&#xA;To be brief, in an equation that we would write as 10 - x = 4x, the &amp;quot;10 - x&amp;quot; (ten dirhams less a thing&amp;quot;) was regarded as a deficient 10. One must &amp;quot;restore&amp;quot; it (al-jabr means &amp;quot;to restore&amp;quot;) to a full 10, and then also add x (&amp;quot;a thing&amp;quot; to the other side to balance the equation.&lt;/p&gt;&#xA;" OwnerUserId="14331" LastEditorUserId="26" LastEditDate="2021-04-14T06:28:48.453" LastActivityDate="2021-04-14T06:28:48.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13094" PostTypeId="1" AcceptedAnswerId="13096" CreationDate="2021-04-14T11:44:21.933" Score="3" ViewCount="299" Body="&lt;p&gt;I am curious to understand the history of QM. In particular, how did the physicists conclude that observables can be treated as operators, or the use of complex state spaces, or that Eigenvalues of the operators are the only values that can be observed. Can somebody recommend a book that discusses this historically?&lt;/p&gt;&#xA;" OwnerUserId="14337" LastEditorUserId="10106" LastEditDate="2021-04-14T18:00:00.797" LastActivityDate="2021-04-14T18:00:00.797" Title="A technical and historical introduction to quantum mechanics" Tags="|physics|reference-request|quantum-mechanics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13095" PostTypeId="2" ParentId="13094" CreationDate="2021-04-14T13:46:49.267" Score="3" Body="&lt;p&gt;I think the best book for this is B. L. van der Waerden's &amp;quot;Sources of Quantum Mechanics&amp;quot;. In short, the book explains several articles of early quantum mechanics in a way that a modern reader can follow, starting from an Einstein paper from 1916 up to  a Dirac paper from 1926, together with a reprint of all the original papers translated to english.&lt;/p&gt;&#xA;&lt;p&gt;This work is remarkable because by following the papers one can see how Heisenberg's introduction of matrix operators was very natural given the classical and semi-classical results previously published.&lt;/p&gt;&#xA;&lt;p&gt;One note, the book concerns the operator approach, and as best as I can remember, says nothing of the concurrent approach that led to Schrodinger's wave theory&lt;/p&gt;&#xA;" OwnerUserId="5904" LastActivityDate="2021-04-14T13:46:49.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13096" PostTypeId="2" ParentId="13094" CreationDate="2021-04-14T13:59:12.737" Score="5" Body="&lt;p&gt;There are two very comprehensive books covering the whole story&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Jagdish Mehra and Helmut Rechenberg, The historical development of quantum mechanics,&#xA;in 3 volumes,&#xA;and&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mitsuo Taketani, Formation and logic of quantum mechanics, in 3 volumes, World Sci. 2001 (translated from the Japanse).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some good shorter books are&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Jim Baggott, The Quantum story. A history in 40 moments, Oxford, 2011,&#xA;and other books of the same author, for example The Quantum cookbook, Oxford, 2020.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is  a nice collection of English translations of the most important original papers, with commentaries:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;B. L. Van der Waerden, Sources of quantum mechanics, North Holland 1967.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are also books and papers that cover history specific parts or aspects of QM:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sin-Itiro Tomonaga, The story of spin, U. Chicago, 1997 (translated from Japanese), unlike other listed books, this is written by one of the key developers of the subject. And the article by&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Shlomo Sternberg, A history of 19th century spectroscopy, Appendix F to his book Group theory and physics, Cambridge, 1994. (This is one of the best texts on history of science that I ever read).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-04-14T14:15:46.670" LastActivityDate="2021-04-14T14:15:46.670" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13097" PostTypeId="2" ParentId="13088" CreationDate="2021-04-14T16:27:58.037" Score="9" Body="&lt;p&gt;The Galileo's astrology activities are well summarized in &lt;a href=&quot;https://hsm.stackexchange.com/a/13089/14341&quot;&gt;Conifold's answer&lt;/a&gt;. But, in relevance to the question at hand (did he believe it worked) I'd like to make a couple of points:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The fact that Galileo was making horoscopes of people of power and wealth doesn't really mean he believed that they actually meant anything. He could have been motivated by money and whatever political favors he could have gained a the time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Making a horoscope of oneself and the people you know best is the way a skeptical person would go about if he wanted to check the validity of astrology.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Galileo's views on astrology could have evolved with time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As the evidence for Galileo's later disdain for astrology I have two things of notice.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;One is the quote from &lt;a href=&quot;https://it.wikisource.org/wiki/Dialogo_sopra_i_due_massimi_sistemi_del_mondo_tolemaico_e_copernicano/Giornata_quarta&quot; rel=&quot;noreferrer&quot;&gt;Dialogo sopra i due massimi sistemi del mondo&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ma tra tutti gli uomini grandi che sopra tal mirabile effetto di natura hanno filosofato, piú mi meraviglio del Keplero che di altri, il quale, d’ingegno libero ed acuto, e che aveva in mano i moti attribuiti alla Terra, abbia poi dato orecchio ed assenso a predominii della Luna sopra l’acqua, ed a proprietà occulte, e simili fanciullezze.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Galileo essentially states that taking about &amp;quot;occult properties&amp;quot; and actions of planets (the Moon in this case) - &amp;quot;fanciullezze&amp;quot; - a childish and unserious endeavors.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Second is the letter from Ascanio Piccolomini, in whose house Galileo was hosted after his arrest in 1633. Claim is that Ascanio stated that Galileo derided astrology and made fun of it as a &amp;quot;&lt;em&gt;profession founded on the most uncertain, if not false, foundations&lt;/em&gt;&amp;quot;. &lt;a href=&quot;https://books.google.fr/books?id=VF22DwAAQBAJ&amp;amp;pg=PA42&amp;amp;lpg=PA42&amp;amp;dq=Ascanio+II+Piccolomini+astrology&amp;amp;source=bl&amp;amp;ots=YSbQEPC0J_&amp;amp;sig=ACfU3U2xH0BP0FXFVSy2dzBV62k2N6YLrA&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwi66YbOkfzvAhX1TxUIHdjDB8MQ6AEwEnoECBIQAw#v=onepage&amp;amp;q=Ascanio%20II%20Piccolomini%20astrology&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;I can only provide this source for that quote&lt;/a&gt; although I'd love to see the precise text of the letter someday...&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="14341" LastActivityDate="2021-04-14T16:27:58.037" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13098" PostTypeId="1" CreationDate="2021-04-14T16:42:19.217" Score="2" ViewCount="230" Body="&lt;p&gt;When was the soundness theorem for first order predicate logic (quantification theory) first proven?&lt;/p&gt;&#xA;&lt;p&gt;Is there any evidence that soundness was presupposed or taken as self-evident prior to 1930?&lt;/p&gt;&#xA;" OwnerUserId="10497" LastActivityDate="2021-12-07T23:05:44.930" Title="Earliest proof of the soundness of first order predicate logic" Tags="|mathematical-logic|logic|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13099" PostTypeId="2" ParentId="13088" CreationDate="2021-04-14T20:32:29.303" Score="2" Body="&lt;p&gt;Rutkin D., &lt;em&gt;Galileo Astrologer&lt;/em&gt;, Galileana II 2005 p 107-143&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Favaro (in vol.19 of the Ed.N) transcribed and published the&#xA;astrological judgments written in Latin of Galileo's two daughters&#xA;which along with the construction of his own horoscope undercuts the&#xA;argument that Galileo only practiced astrology toward patronage ends.&#xA;In addition to those of his family and himself there are 20&#xA;additional horoscopes..&amp;quot; P.117&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2021-04-14T20:32:29.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13100" PostTypeId="2" ParentId="13098" CreationDate="2021-04-14T21:18:55.357" Score="0" Body="&lt;p&gt;Gödel first proved the soundness theorem in his completeness theorem in 1930 based on proof theory (&amp;quot;Die Vollständigkeit der Axiome des logischen Funktionenkalküls&amp;quot;). The model-theoretic proof of the soundness theorem, i.e. a set of sentences is consistent if and only if they have a model, was first given by Henkin in 1949. However, the completeness theorem is also a corollary of results in Skolem's lecture note titled &amp;quot;On mathematical logic&amp;quot; in 1928.&lt;/p&gt;&#xA;&lt;p&gt;A good sourcebook for mathematical logic is &amp;quot;From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931&amp;quot; edited by van Heijenoort in 1967, which contains the English translation of both Gödel's paper (p582--591) and Skolem's note (p508-524) listed above.&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="2014" LastEditDate="2021-12-07T23:05:44.930" LastActivityDate="2021-12-07T23:05:44.930" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13101" PostTypeId="1" CreationDate="2021-04-14T21:25:52.267" Score="1" ViewCount="97" Body="&lt;p&gt;This was closed as off-topic on math.se, and it was suggested I post this here, so here goes.&lt;/p&gt;&#xA;&lt;p&gt;Firstly, I am aware that &lt;a href=&quot;https://math.stackexchange.com/questions/1021519/examples-of-mathematics-in-court&quot;&gt;this thread&lt;/a&gt; exists, and I'll definitely be ordering a copy of the book, &amp;quot;Math on Trial...&amp;quot;, in the top answer. My question, while related, is more specific.&lt;/p&gt;&#xA;&lt;p&gt;I imagine there are many, many examples of the normal distribution being useful in decision-making of processes, such as whether or not to modify a bolt-making factory machine depending on whether it passes quality control standards, and this is found out by taking a sample of bolts and approximating the mean and variance of the length of the bolt, and seeing if the mean is off or if the variance is too large.&lt;/p&gt;&#xA;&lt;p&gt;Another example could be in medicine: if the gaps between heart beats follows some sort of modified normal distribution; that could tell doctors if someone's heart-rate variability is high enough so that medical intervention is required.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But my question here today is about how statistical distributions -&#xA;Binomial, Normal, Poisson or otherwise was critical in the decision of&#xA;the ruling of a law, be it in a court case or otherwise.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Something I have in mind is if an individual, or a gambling site/entity, was accused of cheating and a hypothesis test with the normal distribution was a key factor in a trial. If it was the individual accused of cheating, then maybe the evidence was that it was that their winnings were statistically too unlikely to be coincidental. On the other hand, if it was the gambling company accused of being rigged, then maybe statistical evidence was found that that the probabilities in whatever game was played, was found to not be equal to the probabilities if the game were truly fair.&lt;/p&gt;&#xA;&lt;p&gt;But also, it doesn't have to be a court case: it could be how calculations from statistical distributions affected government policy (law), e.g. decision making when it came to the placement of speed cameras on a road or the height of a building or whatever decision that policy-makers have to make.&lt;/p&gt;&#xA;" OwnerUserId="14344" LastActivityDate="2021-04-14T22:12:31.853" Title="Examples of when statistical distributions like Binomial or Normal distribution was critical in a law/policy decision, in a court case or otherwise" Tags="|statistics|mathematics-social-history|big-list|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13102" PostTypeId="2" ParentId="13101" CreationDate="2021-04-14T22:06:08.923" Score="1" Body="&lt;p&gt;Although the applications are not universally accepted, statistical distributions have found use in employment discrimination cases. The gist of the usage is that in some cases, US courts have found that statistical evidence can count as &lt;em&gt;prima facie&lt;/em&gt; evidence of discrimination, i.e. the statistical evidence is obvious or stark enough that it warrants further examination.&lt;/p&gt;&#xA;&lt;p&gt;Quoting from this paper:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://repository.uchastings.edu/cgi/viewcontent.cgi?article=3699&amp;amp;context=hastings_law_journal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ikuta, Ben. &amp;quot;Why Binomial Distributions Do Not Work as Proof of Employment Discrimination.&amp;quot; &lt;em&gt;Hastings LJ&lt;/em&gt; 59 (2007): 1235.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A. &lt;em&gt;INTERNATIONAL BROTHERHOOD OF TEAMSTERS V. UNITED STATES&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In Teamsters, the central claim was that the employer had engaged in a pattern or practice of discriminating against minorities in &amp;quot;line driving&amp;quot; positions. The company had three main employee positions: &amp;quot;line drivers,&amp;quot; &amp;quot;servicemen,&amp;quot; and &amp;quot;city operations.&amp;quot; Although none of these positions required a formal education, &amp;quot;line drivers&amp;quot; were paid a higher salary than the other positions. In analyzing the statistical data, the Court observed that of 571 minority employees, less than 3% held &amp;quot;line driving&amp;quot; positions and only 39% of the non minority employees held the two lower paying positions. Rejecting the defendant's argument that &amp;quot;statistics can never in and of themselves prove the existence of a pattern or practice of discrimination, or even establish a prima facie case,&amp;quot; the Court held that discrimination should be inferred &amp;quot;where it reached proportions comparable to those in this case.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;However, the most widely followed section is the analysis in footnote seventeen, which compares the ratio of minorities to non minorities in the &amp;quot;line driver&amp;quot; positions to the ratio of minorities to non minorities in the general population of the cities where each of the terminals were located. The Court reasoned that if 17.88% of Los Angeles is African-American, then the percentage of African-Americans in a low-education job like &amp;quot;line driving&amp;quot; should be comparable to 17.88%. Although these comparisons were extremely significant in illustrating what the appropriate statistical comparisons should be, the Court did not address the degree of statistical discrepancy needed to find discrimination because the statistical evidence was so one-sided.&lt;/p&gt;&#xA;&lt;p&gt;B. &lt;em&gt;HAZELWOOD SCHOOL DISTRICT V. UNITED STATES&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem left open in Teamsters concerning the amount of statistical discrepancy needed to raise an inference of discrimination was&amp;quot;solved&amp;quot; in Hazelwood. Hazelwood involved the hiring of minorities for teaching positions in a public school. Similarly to Teamsters, the Court in Hazelwood used the percentage of minorities in teaching positions in the surrounding areas as a basis of comparison. The Court recognized that the percentage of minority teachers in the surrounding areas could either be 5.7% or 15.4%, depending on whether the court thought that it was appropriate to include one surrounding school district that had chosen to pursue a goal of hiring 50% minority teachers. Observing that the school at issue in the case had only a 3.7% hiring rate for minority teachers, the Court relied on Castaneda when it held that using &amp;quot;statistical methodology.., involving the calculation of the standard deviation as a measure of predicted fluctuations [shows] the difference between using 15.4% and 5.7% as the area-wide figure would be significant. However, Hazelwood did not fully describe the method or the pitfalls of using standard deviation based on binomial distributions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;More broadly there is a long history of using statistics in court cases.There is a long history of using statistics in court cases (not necessarily normal or binomial distributions). &lt;a href=&quot;https://en.wikipedia.org/wiki/Shelby_County_v._Holder&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shelby County v. Holder&lt;/a&gt; is a more recent example.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Two other articles that might be of interest:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Rubinfeld, Daniel L. &amp;quot;Econometrics in the Courtroom.&amp;quot; &lt;em&gt;Columbia Law Review&lt;/em&gt; 85, no. 5 (1985): 1048-1097.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Meier, Paul, Jerome Sacks, and Sandy L. Zabell. &amp;quot;What happened in Hazelwood: Statistics, employment discrimination, and the 80% rule.&amp;quot; &lt;em&gt;Law &amp;amp; Social Inquiry&lt;/em&gt; 9, no. 1 (1984): 139-186.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'm sure more recent sources are available too.&lt;/p&gt;&#xA;" OwnerUserId="14345" LastEditorUserId="14345" LastEditDate="2021-04-14T22:12:31.853" LastActivityDate="2021-04-14T22:12:31.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13103" PostTypeId="2" ParentId="13088" CreationDate="2021-04-15T00:48:28.887" Score="1" Body="&lt;p&gt;Nick Kollerstrom in his online Galileo's Astrology points out:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1609 Galileo moved to Florence. In 1610, his revolutionary best-seller, &lt;em&gt;Sidereus Nuncius (The Message of the Stars)&lt;/em&gt;, appearing in March 1610, opened with an eloquent account of the traditional qualities assigned to Jupiter and he states:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Who, I say, does not know that these qualities, according to the providence of God, from whom all good things come, emanate from the benign star of Jupiter?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This shows that he believed in astrology as he believed in the Christianity of his time which also put credence in astrology. The roots of this is intermingled with the emanationist philosophy of Plato.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-04-15T00:48:28.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13104" PostTypeId="2" ParentId="13090" CreationDate="2021-04-15T01:49:20.737" Score="0" Body="&lt;p&gt;Knuth arrow notation can be used to notate the hyper-operations, the fourth of which is tetration and the fifth, pentation and so on. These operations were first defined by Goodstein in 1947.&lt;/p&gt;&#xA;&lt;p&gt;Here, the number of up arrows minus two give the degree of the hyperoperation. Thus&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a \uparrow \uparrow b$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; tetrated to &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That this is not common is simply these operations aren't common.&lt;/p&gt;&#xA;&lt;p&gt;This notation could be lifted to a symbol expessing an infinite tetration. That is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\uparrow \uparrow a_i$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;meaning&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$a_0 \uparrow \uparrow a_1 \uparrow \uparrow a_2 ...$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where left assosciatevity is assumed.&lt;/p&gt;&#xA;&lt;p&gt;Such a lifted operation is not common - if even used.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-04-15T01:49:20.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13105" PostTypeId="1" CreationDate="2021-04-15T02:36:20.453" Score="0" ViewCount="57" Body="&lt;p&gt;I am looking to see whether historic mathematicians did any work to explain the slightly unexpected distribution of prime gaps?&lt;/p&gt;&#xA;&lt;p&gt;I would have expected Gauss, who studied lists of primes and proposed a form for their average density &lt;span class=&quot;math-container&quot;&gt;$1/\log(n)$&lt;/span&gt;, to have considered this question too?&lt;/p&gt;&#xA;&lt;p&gt;The following chart plots the log of prime gap counts for each gap in the number range 1 to 500 million. It is unusual to see such a linear and tightly bounded shape.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/DdOpo.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DdOpo.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I believe, but am not certain, that no results have been proven to explain the shape of this distribution.&lt;/p&gt;&#xA;" OwnerUserId="12633" LastActivityDate="2021-04-15T02:36:20.453" Title="Any historical work on the distribution of prime gaps?" Tags="|number-theory|gauss|primes|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13106" PostTypeId="2" ParentId="13088" CreationDate="2021-04-15T05:34:29.700" Score="1" Body="&lt;p&gt;What we call now astronomy and what we call astrology have been one and the same thing for millennia, and it was so at Galileo's time. The reason for studying the motion of planets relative to the stars was to predict their position for oracular responses, or to pinpoint their past position at birth's time for compiling horoscopes.&lt;/p&gt;&#xA;&lt;p&gt;The fact that (as stated in other responses) Galileo made horoscopes for his own family members is proof enough that he attributed at least some value to them.&lt;/p&gt;&#xA;&lt;p&gt;The fact that (&lt;a href=&quot;https://hsm.stackexchange.com/a/13097/14348&quot;&gt;Tziolkovski's answer&lt;/a&gt;) he was skeptical about some specific topics does not necessarily mean that he rejected astrology. I'd say that is is analogous to being skeptical today about string theory, which does not mean that you don't believe in modern physics.&lt;/p&gt;&#xA;" OwnerUserId="14348" LastEditorUserId="14348" LastEditDate="2021-04-15T05:41:27.947" LastActivityDate="2021-04-15T05:41:27.947" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13107" PostTypeId="1" CreationDate="2021-04-15T05:43:32.023" Score="0" ViewCount="101" Body="&lt;p&gt;Who invented the first pronunciation system for binary numbers that is roughly analogous to how we pronounce everyday decimal numbers (by which I mean how 220 is pronounced 'two hundred twenty')?&lt;/p&gt;&#xA;&lt;p&gt;Joshua Stern in 25th June 1958 has a paper in a journal. That is the earliest one I have been able to find. Stern (1958, Science 128(3324):594-596).&lt;/p&gt;&#xA;&lt;p&gt;There is also Franklin Steuart McFeely almost at the same time. McFeely (1959, Math. Teach. 52(5):356-357).&lt;/p&gt;&#xA;&lt;p&gt;I have read the Stern paper, so I know how his system works. It is a usable system for binary numbers. I know nothing about McFeely's system except what the title of the paper reveals. The title is 'One plus one is toon'. I anyone can tell me where I can see the the McFeely paper, I would appreciate that very much.&lt;/p&gt;&#xA;&lt;p&gt;I also would like to know who was the first to think of pronouncing binary in a way analogous to how we pronounce decimal numbers, whether or not he or she devised a usable system together with a set of names. For example, who first pronounced binary 110 as 'four two' or the equivalent in French or some other human language?&lt;/p&gt;&#xA;" OwnerUserId="14349" LastEditorUserId="4251" LastEditDate="2021-10-22T13:28:58.393" LastActivityDate="2021-10-22T13:28:58.393" Title="Who invented the first pronunciation system for binary numbers that is roughly analogous to how we pronounce everyday decimal numbers?" Tags="|education|mathematics-social-history|numbers|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13108" PostTypeId="2" ParentId="13098" CreationDate="2021-04-15T06:14:24.823" Score="4" Body="&lt;p&gt;See D.Hilbert &amp;amp; W.Ackermann, &lt;a href=&quot;https://en.wikipedia.org/wiki/Principles_of_Mathematical_Logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Principles of Mathematical Logic&lt;/a&gt;: the 1950 American translation of the 1938 second edition of &lt;em&gt;Grundzüge der theoretischen Logik&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The 1928 first edition is considered the first elementary text exposing the formalism now known as first-order logic.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://www.google.it/books/edition/Principles_of_Mathematical_Logic/45ZGMjV9vfcC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;§9 The Consistency and Independence of the System of Axioms&lt;/strong&gt;&lt;/a&gt;, page 87-on.&lt;/p&gt;&#xA;&lt;p&gt;The consistency proof is based on an arithmetical interpretation in the universe &lt;span class=&quot;math-container&quot;&gt;$\{ 0,1 \}$&lt;/span&gt; showing that all the axioms always yield the value &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and, in addition, that if one or more formulas always have the value &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, any other formula obtained from them by means of the rules [of inference] likewise always gives the value &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2021-04-15T06:14:24.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13109" PostTypeId="1" AcceptedAnswerId="13110" CreationDate="2021-04-15T14:40:23.420" Score="2" ViewCount="234" Body="&lt;p&gt;I have learned about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dottie_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dottie number&lt;/a&gt;, though I am unsure to whom it is attributed to and why it is named as so.&lt;/p&gt;&#xA;" OwnerUserId="3120" LastEditorUserId="3120" LastEditDate="2021-04-16T11:48:04.420" LastActivityDate="2023-09-16T03:45:02.087" Title="Who is the Dottie number named after?" Tags="|mathematics-social-history|numbers|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13110" PostTypeId="2" ParentId="13109" CreationDate="2021-04-15T14:40:23.420" Score="1" Body="&lt;p&gt;The Dottie number is named in &lt;a href=&quot;https://www.maa.org/sites/default/files/Kaplan2007-131105.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Samuel Kaplan&lt;/a&gt;'s (2007) paper after the women who discovered it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Dottie number was the nickname among my graduate school friends&#xA;for the unique real root of cos(x) = x.  The story goes that Dottie, a&#xA;professor of French, noticed that whenever she put a number in the&#xA;calculator and hit the cos button over and over again, the number on&#xA;the screen always went to the same value, about 0.739085.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3120" LastActivityDate="2021-04-15T14:40:23.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13112" PostTypeId="1" CreationDate="2021-04-16T09:00:13.483" Score="1" ViewCount="150" Body="&lt;p&gt;Basically the title, I found it weird that we use two different names for each fields. What were the reasons for doing so, historically speaking?&lt;/p&gt;&#xA;" OwnerDisplayName="user12582" LastEditorUserId="12330" LastEditDate="2021-04-18T05:29:49.843" LastActivityDate="2021-04-19T15:06:00.260" Title="Why is the constant of magnetic fields called permeability and that of electric fields called permittivity?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="3" ClosedDate="2021-04-20T06:46:30.840" ContentLicense="CC BY-SA 4.0" />
  <row Id="13113" PostTypeId="2" ParentId="8171" CreationDate="2021-04-16T16:55:40.463" Score="-1" Body="&lt;p&gt;The demarcation problem began with Parmenides who distinguished discursive knowledge based on truth orientated procedures from mere opinion. This is the original sense of the notion of the term science. It meant a body of knowledge butressed by rules of evidence, reasoning and justification which is peculiar to the field in question.  In this sense, history and politics are sciences and its why in Islam there is the term, the science of hadith or the Qu'ranic sciences - which sounds odd for anyone brought up on a diet of modern science (as it did to me once).&lt;/p&gt;&#xA;&lt;p&gt;The Parmenidian thesis was then taken up by Plato. For example, his dialogue, &lt;em&gt;Georgias&lt;/em&gt;, has Socrates arguing against rhetoricians or as he called them, sophists and who traded in opinion said well - though he admits the utility of things said well and again in his late dialogue, &lt;em&gt;The Sophist&lt;/em&gt;. Then through Plato it fed into the broader stream of Western philosophical thinking.&lt;/p&gt;&#xA;&lt;p&gt;This tradition was inverted by Popper who declared there was no truth procedures but only procedures that can be falsified. This position has become popular in the modern analytical philosophy if science.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I prefer Parmenides.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-04-16T16:55:40.463" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13114" PostTypeId="2" ParentId="13112" CreationDate="2021-04-16T17:08:10.907" Score="2" Body="&lt;p&gt;Whilst we now call it &lt;em&gt;electromagnetism&lt;/em&gt;, it was originally electricity and magnetism. They were seen as two separate phenomena. Hence the terms permeability and permittivity arose from two distinct fields of study that were only later unified.&lt;/p&gt;&#xA;&lt;p&gt;It was Oersted in 1820 who first published a link between the two. He had discovered that a magnetic compass needle was deflected by a nearby electric current. This motivated research into electrodynamics, and, forty years later, in 1862, Maxwell published his equations that unified electricity and magnetism. It was one of the earliest unifications. Incidentally, Oersted is often said to gave discovered the link accidentally. This is simply myth. The truth is he was researching for a link between electricity and magnetism for around two years before he made the above discovery.&lt;/p&gt;&#xA;&lt;p&gt;Whilst one could argue that the two terms - permeability and permittivity - should be unified into a single term, the terms remained as they were, as they had already been established for some time. In fact, recently standards organisations have moved to calling them the electric and magnetic constants.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-04-18T21:32:12.763" LastActivityDate="2021-04-18T21:32:12.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13115" PostTypeId="1" CreationDate="2021-04-17T11:17:46.833" Score="2" ViewCount="143" Body="&lt;p&gt;In 1786, Coulomb announced his law that showed that the electromagnetic force between two charged sources followed an inverse square law.&lt;/p&gt;&#xA;&lt;p&gt;Given that gravity followed a similar law, did anyone suggest a unified description?&lt;/p&gt;&#xA;&lt;p&gt;It's worth noting that Priestley twenty years earlier, carrying out an experiment suggested by Benjamin Franklin, and on noting the absence of electrical influence inside a hollow charged sphere said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;May we infer from the experiment that the atteaction of electricity follows the same laws as that of gravity and is therefore according to the inverse square of the distances, since it is easily demonstrated if the earth is in the shape of a shell, a body inside it would not be attracted to one side of it more than the ither side.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, Priestley wasn't saying that electricity and gravity had a unified description.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-04-17T21:12:58.580" Title="When was the earliest speculation that gravity and electromagnetism had a unified description?" Tags="|electromagnetism|gravity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13116" PostTypeId="2" ParentId="13115" CreationDate="2021-04-17T18:05:11.333" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clifford 1876&lt;/a&gt; conjectured all motion results from a curvature of space, in which the unification of the OP's two specific examples is implicit. I'll hereafter discuss subsequent attempts to mathematize the details of electromagnetism/gravity unification.&lt;/p&gt;&#xA;&lt;p&gt;An attempt to explain inertial mass in terms of electromagnetism &lt;a href=&quot;https://en.wikisource.org/wiki/On_the_Electric_and_Magnetic_Effects_produced_by_the_Motion_of_Electrified_Bodies&quot; rel=&quot;nofollow noreferrer&quot;&gt;began in 1881&lt;/a&gt;. &lt;a href=&quot;https://en.wikisource.org/wiki/Considerations_on_Gravitation&quot; rel=&quot;nofollow noreferrer&quot;&gt;In 1900&lt;/a&gt;, it was suggested this could also cause gravity, with inertial mass equaling gravitational mass.&lt;/p&gt;&#xA;&lt;p&gt;The earliest attempt to explain both forces from a common origin (rather than as one resulting from the other) was Nordström, Gunnar (1914). &amp;quot;On the possibility of unifying the gravitational and electromagnetic fields&amp;quot;. Phys. Zeitschr. 15: 504.. You can find the original German version &lt;a href=&quot;http://www.netti.fi/%7Eborgbros/nordstrom/GN_PZ_XV_1914.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and a translation &lt;a href=&quot;https://arxiv.org/pdf/physics/0702221.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. The latter provides context for modern readers, in the form of an abstract and a notation-modernizing appendix.&lt;/p&gt;&#xA;&lt;p&gt;Nordström discusses no prior interest in unifying electromagnetism and gravity, prior suspicion such a unification is possible or should be attempted, or prior work to that end. What he does mention is work the previous year by him and Mie on a special relativistic (but pre-GR) theory of gravity, lacking unification with electromagnetism. Nor can I find any discussion of the history of such unification research that implies any work beyond Nordström's predates the much more famous, slightly more recent work of Kaluza &amp;amp; Klein.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastEditorUserId="5659" LastEditDate="2021-04-17T21:12:58.580" LastActivityDate="2021-04-17T21:12:58.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13117" PostTypeId="1" CreationDate="2021-04-17T21:34:38.210" Score="1" ViewCount="92" Body="&lt;p&gt;Around 1870, William Clifford published a short note in &lt;em&gt;The Proceedings of the Cambridge Philosophical Society&lt;/em&gt; describing all forces as an aspect of the curvature of space.&lt;/p&gt;&#xA;&lt;p&gt;This is the basis of General Relativity as well as the gauge theory of Electromagnetism. Whilst it's well known that curvature is the manifestation of gravity, it is less well-known that this is also the case for electromagnetism (also for the weak and strong force).&lt;/p&gt;&#xA;&lt;p&gt;(It's also possible to read Cliffords note as also stating that matter is also an aspect of curvature, and this is how supergravity theorises matter).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Is there any documented evidence that Einstein knew of Cliffords bold speculation?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-04-25T15:07:50.577" LastActivityDate="2021-04-25T15:07:50.577" Title="Is there any evidence that Einstein knew of Cliffords speculation on the nature of gravity and electromagnetism?" Tags="|physics|einstein|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13118" PostTypeId="1" AcceptedAnswerId="13120" CreationDate="2021-04-18T01:37:48.863" Score="4" ViewCount="211" Body="&lt;p&gt;I have been puzzled for a long time at the way psychologists and medical researchers state that they have 'significant' results, and at the way this statement is relayed to the public who are misled into thinking the results are in some way &lt;em&gt;important&lt;/em&gt;. I also started to wonder why the 'significant' correlation was supposedly all the more remarkable when a very large number of people comprised the sample, since that made it almost inevitable that an effect would be detected, either positive or negative, no matter how weak.&lt;/p&gt;&#xA;&lt;p&gt;While researching this, I browsed the thread &lt;a href=&quot;https://datascience.stackexchange.com/questions/89308/p-value-and-effect-size&quot;&gt;https://datascience.stackexchange.com/questions/89308/p-value-and-effect-size&lt;/a&gt;, and in the answer there  &lt;a href=&quot;https://datascience.stackexchange.com/users/43000/desertnaut&quot;&gt;desertnaut&lt;/a&gt; recommended to read the paper &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3444174/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Using Effect Size—or Why the P Value Is Not Enough&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The paper makes a very good point. But it seems to be a very obvious point, so I am wondering &lt;em&gt;how long the point has been getting made&lt;/em&gt;. When was this point first made? And why does it seem to be ignored?&lt;/p&gt;&#xA;" OwnerUserId="14349" LastEditorUserId="14349" LastEditDate="2021-04-19T02:24:05.667" LastActivityDate="2021-04-19T02:24:05.667" Title="How long have people been debunking the P value (statistical significance) as commonly used in the human sciences: medicine, psychology and so on?" Tags="|terminology|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13120" PostTypeId="2" ParentId="13118" CreationDate="2021-04-18T09:58:49.797" Score="6" Body="&lt;p&gt;Depending on how narrowly the point is pinpointed the dating can be spread out, but it is old, some of it predates the official introduction of NHST by Fisher, see &lt;a href=&quot;https://web.archive.org/web/20170812054959/http://psych.colorado.edu/%7Ewillcutt/pdfs/Nickerson_2000.pdf&quot; rel=&quot;noreferrer&quot;&gt;Nickerson, Null Hypothesis Significance Testing: A Review of an Old and Continuing Controversy&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Criticism of the method, which essentially began with the introduction of the technique (Pearce, 1992), has waxed and waned over the years; it has been intense in the recent past. Apparently, controversy regarding the idea of NHST more generally extends back more than two and a half centuries (Hacking, 1965).&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That &amp;quot;statistical significance&amp;quot; is misleading as to significance in the usual sense of the word was spelled out e. g. by Eysenck in 1960:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Eysenck (1960) made a case for not using the term significance in reporting the results of research. C. A. Clark (1963) argued that statistical significance tests do not provide the information scientists need and that the null hypothesis is not a sound basis for statistical investigation.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The more recent flare ups are associated with APA's near banishment of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-values in 1999 (some psychology journals did banish them), see &lt;a href=&quot;https://hsm.stackexchange.com/a/3186/55&quot;&gt;Hypothesis testing: Fisher vs. Popper vs. Bayes&lt;/a&gt;, and the 2010-2014 unrest that culminated in &lt;a href=&quot;https://www.nature.com/news/scientific-method-statistical-errors-1.14700&quot; rel=&quot;noreferrer&quot;&gt;Nuzzo's 2014 article in Nature&lt;/a&gt; that became one of the most highly viewed in its history. Along with spreading sentiments, also captured by Siegfried's contemporaneous quip &amp;quot;&lt;em&gt;statistical techniques for testing hypotheses… have more flaws than Facebook’s privacy policies&lt;/em&gt;&amp;quot;, it prompted an unprecedented &lt;a href=&quot;https://www.stat.berkeley.edu/%7Ealdous/Real_World/ASA_statement.pdf&quot; rel=&quot;noreferrer&quot;&gt;ASA's 2016 policy statement on &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-values&lt;/a&gt;, where principle #5 reads:&amp;quot;&lt;em&gt;A &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value, or statistical significance, does not measure the size of an effect or the importance of a result&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;One explanation as to why the point has to be made over and over again can be found in Leek's post subtitled &lt;a href=&quot;https://simplystatistics.org/2014/02/14/on-the-scalability-of-statistical-procedures-why-the-p-value-bashers-just-dont-get-it/&quot; rel=&quot;noreferrer&quot;&gt;why the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-value bashers just don't get it&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Despite their flaws, from a practical perspective it is and oversimplification to point to the use of P-values as the critical flaw in scientific practice. The problem is not that people use P-values poorly it is that the vast majority of data analysis is not performed by people properly trained to perform data analysis... By scientific standards, the growth of data came on at a breakneck pace. Over a period of about 40 years we went from a scenario where data was measured in bytes to terabytes in almost every discipline. Training programs haven’t adapted to this new era... Since most people performing data analysis are not statisticians there is a lot of room for error in the application of statistical methods. This error is magnified enormously when naive analysts are given too many “researcher degrees of freedom”.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;P-values can and are misinterpreted, misused, and abused both by naive analysts and by statisticians. Sometimes these problems are due to statistical naiveté, sometimes they are due to wishful thinking and career pressure, and sometimes they are malicious. The reason is that P-values are complicated and require training to understand. Critics of the P-value argue in favor of a large number of the procedures to be used in place of P-values. But when considering the scale at which the methods must be used to address the demands of the current data rich world, many alternatives would result in similar flaws. This is in no way proves the use of P-values is a good idea, but it does prove that coming up with an alternative is hard.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-04-18T09:58:49.797" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13121" PostTypeId="1" CreationDate="2021-04-18T10:44:22.977" Score="0" ViewCount="92" Body="&lt;p&gt;I am reading the Essay of the title written by Beltrami in Italian and I found a specific point of the essay which in my opinion could be fully clarified only if compared with its translations. At the moment, I do not have access to Stillwell's translation for example, so I am asking here in case someone has it.&lt;/p&gt;&#xA;&lt;p&gt;I am looking for the translation of the little paragraph just after Beltrami computes the formula for the geodetic arch &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;, which you can find at page 7 in its Italian version &lt;a href=&quot;http://www.caressa.it/pdf/beltrami01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14370" LastEditorUserId="10106" LastEditDate="2021-04-18T13:05:35.557" LastActivityDate="2021-04-18T13:05:35.557" Title="Beltrami's Essay on the Interpretation of non-Euclidean Geometry" Tags="|mathematics|geometry|translations|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13123" PostTypeId="1" CreationDate="2021-04-19T00:45:07.400" Score="1" ViewCount="71" Body="&lt;p&gt;Kants theory of space, time and causality is that they are the very forms of experience and ground all the other phenomena that we can experience. He called them the forms of sensibility.&lt;/p&gt;&#xA;&lt;p&gt;Kant studied at Konigsberg under Martin Knutzen who introduced him to Newtons natural philosophy and cosmology and to which he contributed original theses on tidal locking and the origins of the solar system. Newton notably also wrote on light.&lt;/p&gt;&#xA;&lt;p&gt;It's startling to first realise that what we experience as the qualia of colour is purely a mental phenomena and does not inhere in the object itself.&lt;/p&gt;&#xA;&lt;p&gt;This is suggestively close to Kants notion of the qualia of space, time and causality that it is tempting to posit that he may have been inspired by it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Is there anything which Kant wrote that backs this up, or has this been posited by any commentator on Kant?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's also worth noting that this may not have been wholly original with Kant. Pythagoras is said to have suggested that we see by something emitted by the eye that illuminates our surroundings. This theory was refuted by Aristotle who wrote that had this been true then we should be able to see in the dark. This is so obvious that one is compelled to rethink what Pythagoras was aiming at with his suggestion and I can only think he means that it is our own sensibility that goes out from us to meet the world and form it for us, in a sense, illuminating it for us. That is, the eye is not a window onto the world through which a vision of the world enters but that our vision of the world is a synthesis of our sensibility which goes out to the phenomenal world and shapes it so that it is rendered intelligible to the mind.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-04-19T00:58:28.793" LastActivityDate="2021-04-19T00:58:28.793" Title="Was Kants theory of space and time inspired by that of colour?" Tags="|physics|philosophy-of-science|newton|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13124" PostTypeId="1" CreationDate="2021-04-19T01:32:44.647" Score="5" ViewCount="190" Body="&lt;p&gt;I found that the term &amp;quot;&lt;em&gt;&lt;strong&gt;Digraph&lt;/strong&gt;&lt;/em&gt;&amp;quot; was coined in 1955 by Frank Harary in &lt;a href=&quot;https://www.jstor.org/stable/1993073?seq=1&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;The number of linear, directed, rooted, and connected graphs&amp;quot;&lt;/a&gt;, and that it was a term actually suggested by George Pólya as Harary later tells in 1989 in &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.1989.11972278?journalCode=uamm20&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;The Pólya Picture Album—Encounters of a Mathematician&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now, the problem is that in some places like in this one: &lt;a href=&quot;https://jeff560.tripod.com/d.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics (D)&lt;/a&gt; it is said that the term &amp;quot;&lt;em&gt;&lt;strong&gt;Directed Graph&lt;/strong&gt;&lt;/em&gt;&amp;quot; was also introduced in that same paper&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;DIGRAPH was used in 1955 by F. Harary in Transactions of the American Mathematical Society. The term directed graph also occurs there [OED].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But if you check older texts like in the first textbook about graph theory: &lt;a href=&quot;http://www.gbv.de/dms/ilmenau/toc/160367581.PDF&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Theorie der endlichen und unendlichen Graphen&amp;quot;&lt;/a&gt; by Dénes Kőnig published in 1936, the Chapter 7 is called: &amp;quot;Basisprobleme für &lt;em&gt;&lt;strong&gt;Gerichtete Graphen&lt;/strong&gt;&lt;/em&gt;&amp;quot; which is translated as: &amp;quot;Basis Problems for &lt;em&gt;&lt;strong&gt;Directed Graphs&lt;/strong&gt;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So, when did the term &amp;quot;&lt;em&gt;&lt;strong&gt;Directed Graph&lt;/strong&gt;&lt;/em&gt;&amp;quot; actually appear?&lt;/p&gt;&#xA;" OwnerUserId="12968" LastActivityDate="2021-04-19T01:32:44.647" Title="Who coined the term: &quot;Directed Graph&quot;?" Tags="|mathematicians|terminology|graph-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13127" PostTypeId="2" ParentId="8232" CreationDate="2021-04-19T15:27:36.893" Score="2" Body="&lt;p&gt;I believe batteries were used to generate several hundred volts in the early days. They were called Voltaic piles. Davy used a 2000-pair pile made for the Royal Institution in 1808. I believe that would produce a voltage in excess of 2000 volts.  Voltaic piles were the source of choice until dynamos and alternators were invented.&lt;/p&gt;&#xA;" OwnerUserId="14375" LastActivityDate="2021-04-19T15:27:36.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13128" PostTypeId="1" CreationDate="2021-04-19T18:14:39.740" Score="0" ViewCount="464" Body="&lt;p&gt;By looking at the portraits of prominent mathematicians and scientists of the previous centuries it seems like they have a higher incidence of strabismus than the general population of today.&#xA;Are there any papers on the rate of strabismus among (prominent) scientists that compare it to that of the rest of the population?&lt;/p&gt;&#xA;&lt;p&gt;Here is kind of an update on this:&#xA;A gene mutation that causes blindness also appears to be associated with above-average intelligence according to this &lt;a href=&quot;https://neurosciencenews.com/cord7-gene-iq-20550/&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2022-05-30T12:38:46.717" LastActivityDate="2022-05-30T12:38:46.717" Title="Is there any relation between strabismus and mathematical or scientific ability?" Tags="|biographical-details|mathematics-social-history|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13131" PostTypeId="1" AcceptedAnswerId="13132" CreationDate="2021-04-21T08:20:03.173" Score="0" ViewCount="93" Body="&lt;p&gt;What other institutes such as these two exist for a visit by a scientist for an undisturbed period of short research? Ideally with a good landscape.&#xA;Dagstuhl is another one I found.&lt;/p&gt;&#xA;" OwnerUserId="14383" LastActivityDate="2021-04-21T11:50:17.410" Title="Mathematical research institutes similar to Banff and Oberwolfach" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ClosedDate="2021-04-21T22:54:26.823" ContentLicense="CC BY-SA 4.0" />
  <row Id="13132" PostTypeId="2" ParentId="13131" CreationDate="2021-04-21T11:45:15.433" Score="0" Body="&lt;p&gt;7 of them in the USA are listed here:&#xA;&lt;a href=&quot;https://aimath.org/news/postdoc/seveninstitutes.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://aimath.org/news/postdoc/seveninstitutes.html&lt;/a&gt;&#xA;and there are more, for example, &lt;a href=&quot;http://www.math.stonybrook.edu/institute-mathematical-sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.math.stonybrook.edu/institute-mathematical-sciences&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Not all of them have beautiful nature environment but MSRI certainly does.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-04-21T11:50:17.410" LastActivityDate="2021-04-21T11:50:17.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13133" PostTypeId="2" ParentId="7468" CreationDate="2021-04-21T23:35:31.097" Score="1" Body="&lt;p&gt;Yes, they did meet. Their meeting(s) are recounted by Bernard Carr in &lt;a href=&quot;https://www.theguardian.com/science/2012/jan/01/stephen-hawking-at-70&quot; rel=&quot;nofollow noreferrer&quot;&gt;the following article from The Guardian&lt;/a&gt;. The article, however, does not go into detail on the topics that were discussed in their meetings.&lt;/p&gt;&#xA;" OwnerUserId="14389" LastEditorUserId="26" LastEditDate="2021-04-22T08:49:02.383" LastActivityDate="2021-04-22T08:49:02.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13134" PostTypeId="1" CreationDate="2021-04-22T02:04:58.460" Score="2" ViewCount="107" Body="&lt;p&gt;I'm currently writing my thesis about ZFC origins, and I need a source from where to know who gives the actual first order formulation of ZFC. I got to the point where Bernays writes about it and establishes them, but it seems like a background of NBG. I'm trying also to get access to Skolem's paper of 1929 called Über einige Grundlagenfragen der Mathematik, but I cannot find it anywhere so came here for some help.&lt;/p&gt;&#xA;&lt;p&gt;Thanks&lt;/p&gt;&#xA;" OwnerUserId="12964" LastEditorUserId="12330" LastEditDate="2021-04-23T04:13:43.847" LastActivityDate="2021-04-23T04:13:43.847" Title="Where can I find the actual first order formulation of ZFC Axioms?" Tags="|set-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13135" PostTypeId="1" CreationDate="2021-04-24T00:38:49.447" Score="9" ViewCount="262" Body="&lt;p&gt;This year (2021) marks the 100th anniversary of Emmy Noether's 1921 paper in which she introduced Noetherian rings and proved the primary ideal decomposition for them. The original version of her paper is &lt;a href=&quot;https://eudml.org/doc/158855&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; and an English translation is on the arXiv &lt;a href=&quot;https://arxiv.org/abs/1401.2577&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. She of course does not call rings &amp;quot;Noetherian&amp;quot; but instead refers to rings satisfying &amp;quot;the finiteness condition&amp;quot; (die Endlichkeitsbedingung).&#xA;When referring to results of Hilbert that we'd call Hilbert's basis theorem and Hilbert's Nullstellensatz at the start of Section 10 of the paper, she writes &amp;quot;Hilbert’s Module Basis Theorem&amp;quot; (Hilbertschen Theorem von der Modulbasis) and &amp;quot;a famous theorem of Hilbert’s&amp;quot; (... eines bekannten Hilbertschen Satzes).  In two footnotes on this page she also refers to &amp;quot;Hilbert's theorem&amp;quot; instead of the Nullstellensatz.&lt;/p&gt;&#xA;&lt;p&gt;The Nullstellensatz was published by Hilbert in 1893 &lt;a href=&quot;https://eudml.org/doc/157652&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; (see the theorem starting on the bottom of p. 320), and that was nearly 30 years earlier. I am surprised that Noether refers to that result simply as &amp;quot;Hilbert's Theorem&amp;quot;.  Thus my question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What is the earliest known reference to Hilbert's theorem as the Nullstellensatz or Hilbert's Nullstellensatz (in German, presumably)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The answer is no later than 1927, when a paper by van der Waerden &lt;a href=&quot;https://eudml.org/doc/159164&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; refers to Hilbert's Nullstellensatz on the top of p. 202. This is the earliest of two papers mentioned in Zentralblatt that has &amp;quot;Nullstellensatz&amp;quot; in a &lt;em&gt;review&lt;/em&gt;. The other earliest paper in Zentralblatt is by Grete Hermann from 1926 &lt;a href=&quot;https://eudml.org/doc/159153&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, which refers to Hentzelt's Nullstellensatz in the 2nd and 3rd paragraphs of the third page (Hentzelt's Nullstellensatz also appears in van der Waerden's paper on p. 203). I didn't find her mentioning Hilbert's Nullstellensatz. Hermann's paper is in English translation &lt;a href=&quot;https://dl.acm.org/doi/pdf/10.1145/307339.307342&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; -- see the first column of the second page. The paper by Hentzelt is from 1923 &lt;a href=&quot;https://eudml.org/doc/158977&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; (it was edited by Noether since Hentzelt went missing in Belgium in 1914 during WW1), and I did not find the term &amp;quot;Hilbert's Nullstellensatz&amp;quot; used in it.&lt;/p&gt;&#xA;&lt;p&gt;For what it's worth, the earliest article in Zentralblatt with Nullstellensatz in a &lt;em&gt;title&lt;/em&gt; is the paper by Rabinowitsch &lt;a href=&quot;https://eudml.org/doc/159393&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; (1930) in which he used his trick to prove the Nullstellensatz.&#xA;That is also the earliest paper in MathSciNet's records with Nullstellensatz appearing anywhere. That's not from a real MathSciNet review since it predates the birth of MathSciNet (well, Mathematical Reviews) in 1940. The earliest real review in MathSciNet mentioning the Nullstellensatz is Zariski's paper from 1947 with his new proof of the Nullstellensatz by &amp;quot;Zariski's lemma&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2021-04-24T04:02:01.647" LastActivityDate="2021-04-24T04:02:01.647" Title="First use of term &quot;Hilbert's Nullstellensatz&quot;" Tags="|terminology|algebraic-geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13137" PostTypeId="1" CreationDate="2021-04-24T23:12:33.143" Score="1" ViewCount="123" Body="&lt;p&gt;The real number axis is asymmetric against zero: for instance, multiplication of two negative or two positive numbers will produce a positive number, a square root of a negative number is not real, natural exponent of real numbers is always positive, etc.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the imaginary axis is totally symmetric against zero: every truth about &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; is also true about &lt;span class=&quot;math-container&quot;&gt;$-i$&lt;/span&gt; up to isomorphism. The same is true for other axes in hypercomplex number systems I know, including split-complex, dual numbers, bicomplex numbers, tessarines, quaternions, split-quaternions...&lt;/p&gt;&#xA;&lt;p&gt;I also know that in physical space we also have one anisotropic axis (time) and three symmetric spatial axes. This is true both for classical spacetime or Minkowski spacetime (whose metric is isomorphic to the metric of 4-dimensional split numbers).&lt;/p&gt;&#xA;&lt;p&gt;As such, a question emerged, whether the real axis is naturally linked to the time dimension, because it is the only non-symmetric axis? Is the existence of no more than one asymmetric axis determined by the laws of mathematics?&lt;/p&gt;&#xA;&lt;p&gt;Or it is possible to invent a ring over reals that would naturally include more than one asymmetric axis?&lt;/p&gt;&#xA;&lt;p&gt;Did anyone propose such system?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="2203" LastEditDate="2021-04-25T07:48:46.653" LastActivityDate="2021-04-25T07:48:46.653" Title="Did anyone ever propose a hypercomplex numbers system with more than one anisotropic axis?" Tags="|mathematics|physics|complex-analysis|algebra|numbers|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13138" PostTypeId="2" ParentId="13137" CreationDate="2021-04-25T06:55:54.213" Score="1" Body="&lt;p&gt;This won't be much of a &amp;quot;historical&amp;quot; answer (although my external links probably have interesting history in them), but hopefully it'll be mathematically useful, at least if construed as a long comment. I'll use &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein notation&lt;/a&gt; throughout, except in blue expressions.&lt;/p&gt;&#xA;&lt;p&gt;Consider a system of arithmetic on a finite-dimensional vector space over &lt;span class=&quot;math-container&quot;&gt;$\Bbb R$&lt;/span&gt; (or your favourite subfield, e.g. &lt;span class=&quot;math-container&quot;&gt;$\Bbb Q$&lt;/span&gt;) with basis &lt;span class=&quot;math-container&quot;&gt;$e_i$&lt;/span&gt; satisfying &lt;span class=&quot;math-container&quot;&gt;$a_ie_i+b_ie_i=(a_i+b_i)e_i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_ie_ib_je_j=u_{ijk}a_ib_je_k$&lt;/span&gt; for coefficients &lt;span class=&quot;math-container&quot;&gt;$u_{ijk}$&lt;/span&gt;. We can write this as &lt;span class=&quot;math-container&quot;&gt;$ab=c$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$c_k=u_{ijk}a_ib_j$&lt;/span&gt;. Axis &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; exhibits the symmetry you describe iff &lt;span class=&quot;math-container&quot;&gt;$a_i^\prime=\color{blue}{(1-2\delta_{il})a_l}$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$(ab)^\prime=a^\prime b^\prime$&lt;/span&gt;, i.e.&lt;span class=&quot;math-container&quot;&gt;$$\color{blue}{[(1-2\delta_{il})(1-2\delta_{jl})-1+2\delta_{kl}])u_{ijk}}=0.$$&lt;/span&gt;You've noted the &lt;span class=&quot;math-container&quot;&gt;$u_{ijk}$&lt;/span&gt; for complex numbers satisfy this condition for &lt;span class=&quot;math-container&quot;&gt;$l=1$&lt;/span&gt; but not &lt;span class=&quot;math-container&quot;&gt;$l=0$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$e_0:=1,\,e_1:=i$&lt;/span&gt;. @Conifold's point is &lt;span class=&quot;math-container&quot;&gt;$u_{ijk}=\color{blue}{\delta_{ij}\delta_{ik}}$&lt;/span&gt; satisfies the condition for all &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I also know that in physical space we also have one anisotropic axis (time) and three symmetric spatial axes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I may not be able to walk into the past, but the equations describing nature are (&lt;a href=&quot;https://en.wikipedia.org/wiki/CP_violation&quot; rel=&quot;nofollow noreferrer&quot;&gt;largely&lt;/a&gt;) time-symmetric. (If that seems like a contradiction, bear in mind &lt;a href=&quot;https://en.wikipedia.org/wiki/Arrow_of_time&quot; rel=&quot;nofollow noreferrer&quot;&gt;this is a big topic&lt;/a&gt;, sadly not addressed by the line of analysis you tried).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a question emerged, whether the real axis is naturally linked to the time dimension&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;em&gt;real&lt;/em&gt; mathematical reason time is dissimilar to space is the signs of real eigenvalues of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)&quot; rel=&quot;nofollow noreferrer&quot;&gt;metric tensor&lt;/a&gt;. (There are two conventions regarding &lt;em&gt;which&lt;/em&gt; sign time gets; in the linked article, the chosen convention makes time's eigenvalue negative, but I'll use the opposite convention hereafter in explaining the relevance.) Infinitesimal paths have squared length &lt;span class=&quot;math-container&quot;&gt;$ds^2=dt^2-dx^2-dy^2-dz^2$&lt;/span&gt; in special relativity, if we use units where &lt;span class=&quot;math-container&quot;&gt;$c=1$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$\sqrt{ds^2}$&lt;/span&gt; is real for paths that don't exceed the speed of light, i.e. are causal. In more general geometries than Minkowski, the details become more complicated, but the &lt;em&gt;signs&lt;/em&gt; of metric tensor eigenvalues are unchanged, and effectively allow us to count time and space dimensions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is the existence of no more than one asymmetric axis determined by the laws of mathematics?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As noted above, no. As for &lt;a href=&quot;https://en.wikipedia.org/wiki/Multiple_time_dimensions&quot; rel=&quot;nofollow noreferrer&quot;&gt;multiple time dimensions&lt;/a&gt;, that's also doable, albeit for unrelated reasons you can guess it in light of the paragraph above.&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2021-04-25T06:55:54.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13139" PostTypeId="1" CreationDate="2021-04-25T15:57:12.253" Score="8" ViewCount="344" Body="&lt;p&gt;I'm aware that the choice of &amp;quot;Z&amp;quot; comes from German &lt;em&gt;zahlen&lt;/em&gt; (for &amp;quot;numbers&amp;quot;); however, I was curious to know when the dedicated font, which I believe is called &amp;quot;blackboard bold&amp;quot; was introduced?&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Blackboard_bold#:%7E:text=Blackboard%20bold%20in%20fact%20originated,the%20point%20of%20a%20chalk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; on blackboard bold is poorly documented on this point (&lt;em&gt;i.e.&lt;/em&gt; specifically the introduction and wide adoption, in print; indeed one of the two links provided in support of their claims is broken). Does anyone have suggestions for a better source on this topic?&lt;/p&gt;&#xA;" OwnerUserId="5127" LastEditorUserId="26" LastEditDate="2021-04-26T15:21:26.357" LastActivityDate="2021-04-27T16:10:10.433" Title="Origin / first use of $\mathbb{Z}$ (blackboard bold Z)?" Tags="|notation|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13140" PostTypeId="1" AcceptedAnswerId="13141" CreationDate="2021-04-25T20:17:27.760" Score="11" ViewCount="2173" Body="&lt;p&gt;It seems many sources&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; attribute the use of the letter &amp;quot;Q&amp;quot; to represent the rationals to the N. Bourbaki group (in the 1930's); however, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Rational_number&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia entry on rational numbers&lt;/a&gt; claims that Giuseppe Peano introduced the notation in 1895 (unfortunately, no link to support that claim &amp;amp; no mention of the title).&lt;/p&gt;&#xA;&lt;p&gt;The Peano etymology suggests it is from the Italian &lt;em&gt;quoziente&lt;/em&gt; (see the Wikipedia article quoted above) whereas the Bourbaki etymology is traced to the German &lt;em&gt;quotient&lt;/em&gt; (see Wolfram Mathworld link, below).&lt;/p&gt;&#xA;&lt;p&gt;I suspect that the letter 'Q' was in use prior to Bourbaki, and that Bourbaki was responsible for the introduction of the double struck blackboard bold &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;, and that this was conflated with the introduction of the letter Q, itself, but have not found any sources to confirm this.&lt;/p&gt;&#xA;&lt;p&gt;Any suggestions for sources would be greatly appreciated.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;:&#xA;For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://jeff560.tripod.com/nth.html&quot; rel=&quot;noreferrer&quot;&gt;https://jeff560.tripod.com/nth.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathworld.wolfram.com/RationalNumber.html&quot; rel=&quot;noreferrer&quot;&gt;https://mathworld.wolfram.com/RationalNumber.html&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Screenshot of Wikipedia paragraph referring to Peano:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/kwHxj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/kwHxj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Based on @user6530's citations (at least of the &lt;em&gt;Formulario Mathematico&lt;/em&gt;), it would seem that the claim that Peano introduced the use of 'Q' &lt;em&gt;for the rationals&lt;/em&gt; is inaccurate.&lt;/p&gt;&#xA;" OwnerUserId="5127" LastEditorUserId="5127" LastEditDate="2021-04-27T12:52:56.940" LastActivityDate="2021-04-27T19:31:20.583" Title="Origin of Q for the set of rational numbers?" Tags="|notation|set-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13141" PostTypeId="2" ParentId="13140" CreationDate="2021-04-26T21:06:56.877" Score="12" Body="&lt;p&gt;tl;dr&lt;/p&gt;&#xA;&lt;p&gt;Dedekind was the first to use a letter (R) for sets of rational numbers in 1872, then, starting from 1895, Peano began to use the letter r (lowercase) to denote the same set (and, from 1889, R for the set of positive rationals). Other authors proposed different letters, and only in the early forties Bourbaki introduced the letter Q (not the blackboard bold &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;). Possibly, the blackboard bold version was introduced (maybe) in the late fifties or (more likely) in the early sixties, but this is &lt;a href=&quot;https://hsm.stackexchange.com/q/13139/6545&quot;&gt;another question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Now, some references.&lt;/p&gt;&#xA;&lt;p&gt;Dedekind used the letter R (uppercase) for the set of rational numbers in&#xA;&lt;a href=&quot;https://archive.org/details/stetigkeitundir00dedegoog/page/n20/mode/2up?q=Erscheinungen&quot; rel=&quot;noreferrer&quot;&gt;Stetigkeit und irrationale Zahlen&lt;/a&gt; (1872), &lt;span class=&quot;math-container&quot;&gt;$\S 3$&lt;/span&gt;, page 16 (&amp;quot;die Gerade L ist unendlich viel reicher an Punkt-Individuen, als das Gebiet R der rationalen Zahlen an Zahl-Individuen&amp;quot;, i.e. &amp;quot;the straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals&amp;quot;, &lt;a href=&quot;https://archive.org/details/cu31924001586282/page/n21/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; the English translation).&lt;/p&gt;&#xA;&lt;p&gt;About Peano, Wikipedia &amp;quot;clearly&amp;quot; refers to somewhere in the &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Formulario_mathematico&quot; rel=&quot;noreferrer&quot;&gt;Formulaire de mathématiques/Formulaire Mathématique/Formulario Mathematico&lt;/a&gt;&lt;/em&gt;, where Peano actually used extensively letters to denote sets (&lt;em&gt;classe&lt;/em&gt;) of numbers. There's no doubt about this, as this is the only work of Peano concerning the subject for the year 1895 (see &lt;a href=&quot;https://iris.unito.it/retrieve/handle/2318/52525/373970/All3.14PeanoCronologia.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for the complete bibliography of Peano, the works for the year 1895 start at page 45). The problem is that (see the wiki page)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the five editions of the Formulario [are not] editions in the usual sense of the word. Each is essentially a new elaboration, although much material is repeated. Moreover, the title and language varied: the first three, titled Formulaire de Mathématiques, and the fourth, titled, Formulaire Mathématique, were written in French, while Latino sine flexione, Peano's own invention, was used for the fifth edition, titled Formulario Mathematico. ... Ugo Cassina lists no less than twenty separately published items as being parts of the 'complete' Formulario!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and moreover the &lt;em&gt;Formulario&lt;/em&gt; was written by many Peano's collaborators, such as Giovanni Vailati, Mario Pieri, Alessandro Padoa, Giovanni Vacca, Vincenzo Vivanti, Gino Fano and Cesare Burali-Forti, so when one writes &amp;quot;in the Formulario Peano says...&amp;quot; one have to understand that actually one is referring to Peano or to one of his collaborators.&lt;/p&gt;&#xA;&lt;p&gt;The following quotes are taken from the &lt;em&gt;Formulario Mathematico&lt;/em&gt; edited in 1908, for the reason that it is the clearer and more complete exposition of the subject. One can read the 1908 &lt;em&gt;Formulario&lt;/em&gt; &lt;a href=&quot;https://archive.org/details/formulairedemat04peangoog&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;, while different editions of the &lt;em&gt;Formulaire de mathématiques&lt;/em&gt; are on Gallica, for example &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k84142s/f130.item.texteImage.zoom&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;First Peano writes (in I.&lt;span class=&quot;math-container&quot;&gt;$\S 1$&lt;/span&gt;, of course) the symbol &lt;span class=&quot;math-container&quot;&gt;$N_0$&lt;/span&gt;, together with &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; as &amp;quot;idea primitivo&amp;quot; [sic], i.e. undefined primitive ideas used to define all the other symbols of the &amp;quot;Arithmetica&amp;quot;, and whose sense is determined by a system of proposition, the first is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cdot 0\quad N_0\ \varepsilon\ \text{Cls}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;that he read &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$N_0$&lt;/span&gt; is a class&amp;quot;, and so on. In II.&lt;span class=&quot;math-container&quot;&gt;$\S 5$&lt;/span&gt; we find &lt;span class=&quot;math-container&quot;&gt;$N_1=N_0+1$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$N_1$&lt;/span&gt; is the set of strictly positive natural numbers, and in II.&lt;span class=&quot;math-container&quot;&gt;$\S 6$&lt;/span&gt; he uses &lt;span class=&quot;math-container&quot;&gt;$+N_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-N_0$&lt;/span&gt; for positive and negative numbers, and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; for the union (so &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; stands for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;). Then  in III.&lt;span class=&quot;math-container&quot;&gt;$\S 8$&lt;/span&gt; we read&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cdot 2 \quad R=N_1/N_1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$R=$&lt;/span&gt; &amp;quot;Numero rationale&amp;quot;. Illo es omni expressione de forma &lt;span class=&quot;math-container&quot;&gt;$b/a$&lt;/span&gt;, ubi &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; es numero naturale...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e., &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; denotes a &lt;strong&gt;rational&lt;/strong&gt; number, and it is any expression of the for &lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; are natural numbers; clearly Peano also introduces &lt;span class=&quot;math-container&quot;&gt;$r=+R\cup -R\cup \iota 0$&lt;/span&gt; (III.&lt;span class=&quot;math-container&quot;&gt;$\S 9 1\cdot 0$&lt;/span&gt;). Moreover he adds&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cdot 01\quad r=n/N_1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$r=$&lt;/span&gt; numero rationale relativo&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and so &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; stands for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;.&#xA;Finally, in III.&lt;span class=&quot;math-container&quot;&gt;$\S 12$&lt;/span&gt; we find&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$5\cdot 0\quad Q=l'`[\text{Cls}` R \cap u\ni(\exists u.\exists R=\eta u)])$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;that Peano reads&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, lege &amp;quot;quantitate reale positivo&amp;quot; es omni limite supero de aliquo classe &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; de rationale, existente, et tale que existe aliquo rationale maiore de omni &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e., &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;, that reads as &amp;quot;real positive numbers&amp;quot;, is any supremum of some existing set (classe) &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; of rationals, such that there exists some ratonal number greater then all elements in &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;; clearly in III.&lt;span class=&quot;math-container&quot;&gt;$\S 13$&lt;/span&gt; Peano defines &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$Q\cup -Q\cup \iota 0$&lt;/span&gt;, ans so &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; stands for our &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the &lt;em&gt;Formulaire de mathématiques&lt;/em&gt; one can read more or less the same things (in French, of course), in particular in the index at page 56 we find &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; nombre rationnel&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, Peano used letters for sets of numbers before 1895, see &lt;em&gt;&lt;a href=&quot;https://archive.org/details/arithmeticespri00peangoog/page/n12/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Arithmetices principia, nova methodo exposita&lt;/a&gt;&lt;/em&gt; (1889), the &lt;em&gt;Signorum Tabula&lt;/em&gt; at page 13 (&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; quantitas, sive numerus realis positivus&amp;quot;, i.e. &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; [stands for] quantity, that is, a real positive number&amp;quot;, while &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is used for &lt;em&gt;positive&lt;/em&gt; rationals, no symbol for the rationals, positive or negative).&lt;/p&gt;&#xA;&lt;p&gt;Summing up: yes, Peano uses letters to denote classes of numbers, but no, his use is different (indeed, antithetical) from the modern one; moreover, capital letters are used for set of only positive or only negative numbers of some kinds, while lowercase letters for either positive or negatives numbers (&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; [sic]).&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, I cannot find any source independent of Wikipedia about the use of letter &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; from the Italian &amp;quot;quoziente&amp;quot;, and moreover, in Italian the elements of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; are called &amp;quot;frazioni&amp;quot; (fractions) or &amp;quot;numeri razionali&amp;quot; (rational numbers, from the Latin word &amp;quot;ratio&amp;quot;), while &amp;quot;quoziente&amp;quot; is the result of an operation. The choice of letters made by Peano is transparent: &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is for &amp;quot;numerus&amp;quot; (whole number, &amp;quot;numero&amp;quot; in Italian), &amp;quot;r&amp;quot; for &amp;quot;(numero) rationale&amp;quot; (rational number, but also &amp;quot;rapporto&amp;quot; in Italian, ratio in English), &amp;quot;q&amp;quot; for &amp;quot;quantitas&amp;quot; (Italian &amp;quot;quantità&amp;quot;, English quantity).&lt;/p&gt;&#xA;&lt;p&gt;Now, some words about Bourbaki. Yes, &amp;quot;they&amp;quot; uses &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; for rational numbers, and no, they does not use blackboard bold &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; (at least in 1940s papers). An early occurence (maybe the earliest printed on paper) of &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; to denote the set of rational numbers is &lt;a href=&quot;http://sites.mathdoc.fr/archives-bourbaki/PDF/nbt_006.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; at page 3 in the number 5 (7-10 December 1940) of &lt;em&gt;La Tribu&lt;/em&gt;, the Bourbaki's internal newsletter. We read&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; est ordonné [...] Topologie de &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; [...] Complétion de &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; : nombres réels&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;so there is no doubt that here &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; refers to our &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt;. Clearly, we find &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; for rational numbers in the 1942 &lt;em&gt;&lt;a href=&quot;http://sites.mathdoc.fr/archives-bourbaki/PDF/033_iecnr_040.pdf&quot; rel=&quot;noreferrer&quot;&gt;Algèbre&lt;/a&gt;&lt;/em&gt; (from page 29).&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2021-04-27T19:31:20.583" LastActivityDate="2021-04-27T19:31:20.583" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13142" PostTypeId="1" AcceptedAnswerId="13143" CreationDate="2021-04-26T21:15:31.230" Score="0" ViewCount="129" Body="&lt;p&gt;It seems to me that Kepler's Laws necessitate some definition of an ellipse in terms of a coordinate system. I am wondering whether Kepler's Laws mathematically defined what an ellipse is, or if he used an already defined shape in his laws.&lt;/p&gt;&#xA;&lt;p&gt;The reason I got curious about this in the first place was that I realized that Descartes may not have invented the Cartesian coordinate system by the time that Kepler discovered these laws and I have only seen proofs of Kepler's Laws in a Cartesian system.&lt;/p&gt;&#xA;" OwnerUserId="14406" LastActivityDate="2021-04-26T21:23:32.770" Title="Were Kepler's Laws of Planetary Motion the first formal definition of an ellipse?" Tags="|astronomy|geometry|euclidean-geometry|mathematical-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13143" PostTypeId="2" ParentId="13142" CreationDate="2021-04-26T21:23:32.770" Score="6" Body="&lt;p&gt;No. Conic sections were studied in Greece since the IV&lt;sup&gt;th&lt;/sup&gt; century BC. And Apollonius of Perga (c. 240 BC – c. 190 BC), who coined the term ellipse, wrote a treatise on conic sections (&lt;em&gt;Conics&lt;/em&gt;) in eight books.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2021-04-26T21:23:32.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13145" PostTypeId="2" ParentId="13139" CreationDate="2021-04-27T16:10:10.433" Score="3" Body="&lt;p&gt;Part of @user6530's answer to &lt;a href=&quot;https://hsm.stackexchange.com/questions/13140/origin-of-q-for-the-set-of-rational-numbers/13141?noredirect=1#comment27525_13141&quot;&gt;this related question&lt;/a&gt; about the use of Q and &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Q}$&lt;/span&gt; for the rationals discussed the use of blackboard bold, so I thought I'd share it, here, for those interested:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, some words about Bourbaki. Yes, &amp;quot;they&amp;quot; uses  for rational&#xA;numbers, and no, they does not use blackboard bold ℚ (at least in&#xA;1940s papers). An early occurence (maybe the earliest printed on&#xA;paper) of  to denote the set of rational numbers is here at page 3&#xA;in the number 5 (7-10 December 1940) of La Tribu, the Bourbaki's&#xA;internal newsletter. We read&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt; est ordonné [...] Topologie de  [...] Complétion de  : nombres&#xA;réels&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;so there is no doubt that here  refers to our ℚ. Clearly, we find &#xA;for rational numbers in the 1942 Algèbre (from page 29).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5127" LastActivityDate="2021-04-27T16:10:10.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13146" PostTypeId="1" AcceptedAnswerId="13163" CreationDate="2021-04-28T08:54:20.900" Score="0" ViewCount="194" Body="&lt;p&gt;This is the paper:&lt;/p&gt;&#xA;&lt;p&gt;“Variétés. Essai de dialectique rationnelle”. Annales de Mathématiques pures et appliquées, tome 7 (1816-1817), p. 189-228&lt;/p&gt;&#xA;&lt;p&gt;(“Varieties. Essay about rational dialectic”, By J.D. Gergonne).&#xA;Between else, in this paper we find the first use of the symbols “C” and “Ɔ”, which later evolved into the modern symbols “&lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt;” and “&lt;span class=&quot;math-container&quot;&gt;$\supset$&lt;/span&gt;”, used in set theory and, in older texts, for implication.&lt;/p&gt;&#xA;&lt;p&gt;You can find it &lt;a href=&quot;http://www.numdam.org/article/AMPA_1816-1817__7__189_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, in French.&lt;/p&gt;&#xA;&lt;p&gt;If there isn't any, I believe it deserves one. I would certainly do it, if I could.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Copied from &lt;a href=&quot;https://math.stackexchange.com/questions/4115262/is-there-any-english-translation-of-this-gergonne-paper&quot;&gt;Math StackExchange&lt;/a&gt;, after sugestion from the user Rob Arthan.&lt;/p&gt;&#xA;" OwnerUserId="14415" LastEditorUserId="14415" LastEditDate="2021-04-28T18:22:47.490" LastActivityDate="2021-05-31T15:08:34.917" Title="Is there any English translation of this Gergonne paper?" Tags="|mathematics|translations|logic|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13147" PostTypeId="2" ParentId="13146" CreationDate="2021-04-28T11:37:26.893" Score="4" Body="&lt;p&gt;I doubt that there is an English traslation, and old papers that quote it are in French (e.g. Peano, &lt;em&gt;Origine du signe &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt;&lt;/em&gt; in &lt;em&gt;L'Intermédiaire des Mathématiciens&lt;/em&gt;, 1903, or Giard, &lt;em&gt;La « dialectique rationnelle » de Gergonne&lt;/em&gt; in &lt;em&gt;Revue d'histoire des sciences&lt;/em&gt;, 1972, this paper in particular is very interesting, as it gives a detailed analysis of Gergonne's original paper that is not always so clear). Anyway, small parts are reported in the invaluable collection of &lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%B3zef_Maria_Boche%C5%84ski&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bocheński&lt;/a&gt; &lt;em&gt;Formale Logik&lt;/em&gt; (1956), translated into English as &lt;em&gt;A history of formal logic&lt;/em&gt; (1961). The translation is easy accessible (&lt;a href=&quot;https://circulosemiotico.files.wordpress.com/2012/10/historyofformall00boch.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; a digitalized copy). You will find some excerps at page 277 (&lt;span class=&quot;math-container&quot;&gt;$\S$&lt;/span&gt; 38.15 and 38.16), with this comment of Bocheński &amp;quot;This text is not altogeter clear; Gergonne seems to equate the (Aristotelian) use of variables with formalism. But we can see the idea of formalism becoming clearer&amp;quot;, and at page 304 (&lt;span class=&quot;math-container&quot;&gt;$\S$&lt;/span&gt; 40.12).&lt;/p&gt;&#xA;&lt;p&gt;Maybe some other author translated wider excerpt of Gergonne's paper in English, but I haven't been able to find any of them.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2021-04-28T11:37:26.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13148" PostTypeId="1" AcceptedAnswerId="13149" CreationDate="2021-04-28T23:33:28.840" Score="4" ViewCount="149" Body="&lt;p&gt;In which paper/book (most likely) by either Sobolev or Schwartz is the Dirac function properly and explicitly substantiated as a functional (tempered distribution), preferably quoting Dirac's name? I need to know the oldest (the original) source, in order to quote it as a remark in an article that I'm writing on the history of mathematical foundations of Dirac's inventions (Delta function and bra-kets).&lt;/p&gt;&#xA;" OwnerUserId="7069" LastActivityDate="2021-04-29T18:09:57.440" Title="Original mathematical foundation of Dirac's function" Tags="|reference-request|functional-analysis|dirac|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13149" PostTypeId="2" ParentId="13148" CreationDate="2021-04-29T12:15:59.883" Score="4" Body="&lt;p&gt;I found another, &lt;a href=&quot;https://core.ac.uk/download/pdf/82190991.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;later paper&lt;/a&gt; which also references the Schwartz item that @sand1 commented as well as another paper by Schwartz from 1945.&#xA;The reference paper is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Historia Mathematics 10 (1983) 149-183  DISTRIBUTIONS: THE EVOLUTION&#xA;OF A MATHEMATICAL THEORY  BY JOHN SYNOWIEC  INDIANA UNIVERSITY&#xA;NORTHWEST, GARY, INDIANA 46408&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The references listed therein include&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schwartz, L. 1944. Sur certaines families non fondamentales de&#xA;fonctions continues.  Bulletin de la Société Mathematique de France&#xA;72, 141-145.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&#xA;&lt;ol start=&quot;1945&quot;&gt;&#xA;&lt;li&gt;Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathematiques et&#xA;physiques. Annales de l'Université de Grenoble 21, 57-74.&#xA;-- 1947-1948. Theorie des distributions et transformations de Fourier. Annales  de l'université de Grenoble 23, 7-24.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&#xA;&lt;ol start=&quot;1948&quot;&gt;&#xA;&lt;li&gt;Généralisation de la notion de fonction et de dérivation. Théorie de  distributions. Annales Télécommunications 3, 135-140.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and many more.&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorUserId="26" LastEditDate="2021-04-29T18:09:57.440" LastActivityDate="2021-04-29T18:09:57.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13150" PostTypeId="1" AcceptedAnswerId="13154" CreationDate="2021-05-01T15:45:58.870" Score="10" ViewCount="2715" Body="&lt;p&gt;A fairly well-known story is Fulton offering to build steam ships for The Emperor and Napoleon replying something like, &amp;quot;A ship that sails by bonfires under its decks?? Away with you, visionist!&amp;quot; (This story is alluded to in &lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt; when Roosevelt and an advisor are discussing the prospect of nuclear power/weapons.)&lt;/p&gt;&#xA;&lt;p&gt;I find the story hard to believe in that I would guess that steam power was hardly a new thing in the early 1800s -- steam pumps had been used in mines for almost a century and in fact a steam-powered boat was exhibited as early as 1787. So Napoleon would have sounded very unsophisticated indeed if he spoke of steam engines as almost something magical any time after 1800 if not earlier.&lt;/p&gt;&#xA;&lt;p&gt;Is this story real?  What is the evidence about it?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="12724" LastEditDate="2021-05-02T21:00:03.953" LastActivityDate="2021-05-03T16:06:12.423" Title="Napoleon I and Fulton: Steamship rejection story real?" Tags="|steam-engines|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13152" PostTypeId="1" CreationDate="2021-05-01T20:13:34.800" Score="1" ViewCount="95" Body="&lt;p&gt;A man in Tennessee has written to me, asking for a reference about the method in my answer to &lt;a href=&quot;https://math.stackexchange.com/questions/81405/anti-curl-operator&quot;&gt;https://math.stackexchange.com/questions/81405/anti-curl-operator&lt;/a&gt;&#xA;He is writing an article on finite element methods for the unit cube. His main reference is Ciarlet, &lt;em&gt;The Finite Element Method for Elliptic Problems&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The version I gave there is called the Poincare Lemma by Spivak (&lt;em&gt;Calculus on Manifolds&lt;/em&gt;). The integration is along a line segment beginning at the origin.&lt;/p&gt;&#xA;&lt;p&gt;My friend Dmitry says that most texts use &amp;quot;partial integration,&amp;quot;  which is how he describes three integrals, along segments parallel to the axes.&lt;/p&gt;&#xA;&lt;p&gt;A very different approach, and likely the earliest, is due to Helmholtz, who wrote an integral over all of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^3.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Anyway, I am still uncertain what to tell him. He wants a reference as part of an article he is submitting soon. He said he does not recognize the Poincare version, so he is more familiar with the other two methods. He plans to include his own short proof in any case.&lt;/p&gt;&#xA;&lt;p&gt;from my answer:  Given your divergence-free vector field &lt;span class=&quot;math-container&quot;&gt;$(F_1(x,y,z), \; F_2(x,y,z), \; F_3(x,y,z)),$&lt;/span&gt; the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-coordinate of the new vector field &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ G_1(x,y,z) = \int_0^1 \; \left( \; t z F_2(tx, ty,tz) - t y F_3(tx, ty,tz) \; \right) \; dt, $$&lt;/span&gt;&#xA;the &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;-coordinate is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ G_2(x,y,z) = \int_0^1 \; \left( \;  t x F_3(tx, ty,tz) - t z F_1(tx, ty,tz) \; \right) \; dt, $$&lt;/span&gt;&lt;br /&gt;&#xA;with &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;-coordinate&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ G_3(x,y,z) = \int_0^1 \; \left( \;  t y F_1(tx, ty,tz) - t x F_2(tx, ty,tz) \; \right) \; dt. $$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="401" LastActivityDate="2021-05-01T20:13:34.800" Title="history of curl and inverting the curl" Tags="|reference-request|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13153" PostTypeId="2" ParentId="13150" CreationDate="2021-05-01T20:43:43.237" Score="8" Body="&lt;p&gt;The thing that Fulton tried to sell to the French (and after their refusal, to the British) was not a steamship but a submarine equipped with a &amp;quot;torpedo&amp;quot;. What Napoleon really said, and whether Fulton spoke personally to Napoleon is not recorded. (In France he dealt with the Minister of Marine,&#xA;and obtained funds for an experimental submarine. Another experimental submarine he built in England. It was demonstrated in the presence of the&#xA;prime minister (W. Pitt)). His submarines were propelled my muscular force.&lt;/p&gt;&#xA;&lt;p&gt;And you are right: he was not the first person to propose a steamship of&#xA;a submarine. Steamships were already in use at that time, mainly as tugs.&lt;/p&gt;&#xA;&lt;p&gt;Failing in this Fulton came back to the US and made a fortune by building steamships.&lt;/p&gt;&#xA;&lt;p&gt;Reference: N. Mostert, The line upon a wind. The great war at sea, 1793-1815, W. Norton, NY, 2007, chapter 44 is on Fulton's submarines, torpedoes&#xA;and steamships.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-05-03T00:45:19.047" LastActivityDate="2021-05-03T00:45:19.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13154" PostTypeId="2" ParentId="13150" CreationDate="2021-05-01T21:39:51.270" Score="13" Body="&lt;p&gt;As Alex correctly noted, the story is about a submarine, not a steamboat (your source is just wrong).&lt;/p&gt;&#xA;&lt;p&gt;The story is discussed in detail in the Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Nautilus_(1800_submarine)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nautilus&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here are the key paragraphs:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Through friends like Gaspard Monge and Pierre-Simon Laplace, Fulton obtained an interview with Napoleon, but was unable to garner support for his vessel; however, Fulton's friends pushed the Minister of Marine into appointing a scholarly panel, to consist of Volney, Monge, and Laplace, to assess the submarine.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, Fulton was taken seriously.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In September, Napoleon expressed interest in seeing Nautilus, only to find that, as it had leaked badly, Fulton had her dismantled and the more important bits destroyed at the end of the tests. Despite the many reports of success by reliable witnesses, like the Prefect Marine of Brest, Napoleon decided Fulton was a swindler and charlatan. The French navy had no enthusiasm for a weapon they considered suicidal for the crews even though Fulton had had no problems(*) and despite evidence it would be overwhelmingly destructive against conventional ships.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(*) I think, Wikipedia article is too kind to Fulton here: It's clear from the above that Fulton had some pretty serious reliability issues with his design.&lt;/p&gt;&#xA;&lt;p&gt;My take is that Fulton (and his design) failed at the worst possible time  and that Napoleon decided to believe his eyes rather than arguments from his experts and subordinates (who disagreed among each other). What were his exact words, I do not know, my suggestion is to check the references given by the Wikipedia article:&lt;/p&gt;&#xA;&lt;p&gt;Robert Forrest Burgess. &amp;quot;Ships Beneath the Sea.&amp;quot; McGraw-Hill.  (1975)&lt;/p&gt;&#xA;&lt;p&gt;William Barclay Parsons. &amp;quot;Robert Fulton and the submarine.&amp;quot; Columbia University Press.  (1922)&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2021-05-03T16:06:12.423" LastActivityDate="2021-05-03T16:06:12.423" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13155" PostTypeId="1" CreationDate="2021-05-02T11:33:23.917" Score="5" ViewCount="604" Body="&lt;p&gt;I have been trying to get my head around how Roger Cotes first discovered &lt;a href=&quot;https://en.wikipedia.org/wiki/Roger_Cotes#cite_note-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler Formula&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I knew how Euler did it, but I wanted a new perspective, especially from someone who discovered it earlier.&lt;/p&gt;&#xA;&lt;p&gt;Cotes used geometry to explain his discovery, but it was very terse (or at least to my preference). I tried to draw the diagram after his explanation, but I still can't understand how he suddenly conclude that &lt;span class=&quot;math-container&quot;&gt;$\ln{\frac{EX+XC\sqrt{-1}}{CE}}=\text{arc length}\cdot\sqrt{-1}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;He used many old keywords such as &amp;quot;measurement of ratio&amp;quot; to represent natural log. How is natural log a &amp;quot;measurement of ratio&amp;quot;, and why does taking &amp;quot;measurement of ratio&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$\cos{\theta}+i\sin{\theta}$&lt;/span&gt; equate to arc length of circle intuitively?&lt;/p&gt;&#xA;" OwnerUserId="14431" LastEditorUserId="16985" LastEditDate="2022-11-17T16:50:40.880" LastActivityDate="2022-12-02T15:35:39.270" Title="How did Roger Cotes come up with logarithm form of Euler formula?" Tags="|mathematics|geometry|complex-analysis|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13157" PostTypeId="1" AcceptedAnswerId="13166" CreationDate="2021-05-03T00:27:59.220" Score="2" ViewCount="158" Body="&lt;p&gt;The Cramér probabilistic model of primes is built on the assumption that the probability of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; being prime is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\Pr(n)=\frac{1}{\log (n)}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is not a big leap from the Prime Number Theorem which tells us that the density of primes is approximately &lt;span class=&quot;math-container&quot;&gt;$1/\log(x)$&lt;/span&gt;, an approximation that is asymptotically more accurate for larger &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question is, was Harald Cramér the first to use this density as a probability, in the 1930s?&lt;/p&gt;&#xA;&lt;p&gt;His famous paper was published 1936/7 (&lt;a href=&quot;http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="12633" LastEditDate="2021-05-03T04:46:03.000" LastActivityDate="2021-05-06T05:59:33.983" Title="Was Cramér the first to interpret the PNT's $1/\log(x)$ as probability of primes?" Tags="|number-theory|probability|primes|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13158" PostTypeId="1" AcceptedAnswerId="13159" CreationDate="2021-05-03T03:39:54.230" Score="3" ViewCount="472" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, Euler (1707-1783) &amp;quot;mastered Russian and settled into life in Saint Petersburg&amp;quot; in 1727. Did he produce any text in Russian, either mathematical or personal?&lt;/p&gt;&#xA;&lt;p&gt;On &lt;a href=&quot;http://eulerarchive.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Euler Archive&lt;/a&gt;, I can only find writings in Latin, German, and French (maybe I just did not search hard enough).&lt;/p&gt;&#xA;" OwnerUserId="7439" LastEditorUserId="4251" LastEditDate="2021-09-16T06:39:37.743" LastActivityDate="2021-09-16T06:39:37.743" Title="Did Euler produce any Russian text?" Tags="|mathematics|mathematicians|euler|18th-century|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13159" PostTypeId="2" ParentId="13158" CreationDate="2021-05-03T11:53:37.410" Score="13" Body="&lt;p&gt;Yes, Euler wrote letters in Russian:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/IcuVv.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IcuVv.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Euler's letter to Nartov, June 18, 1743)&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, he signed papers as &amp;quot;Леонгардъ Эулеръ&amp;quot;, a direct transliteration of his name, not how he is called in Russian today (imitating pronunciation).&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="2203" LastEditDate="2021-05-03T12:08:10.320" LastActivityDate="2021-05-03T12:08:10.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13160" PostTypeId="1" CreationDate="2021-05-03T12:36:01.780" Score="1" ViewCount="79" Body="&lt;p&gt;There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.&lt;/p&gt;&#xA;&lt;p&gt;Still, in my view there is fundamental difference between divergent integrals or series that diverge to infinity, monotonously growing as opposed to those which have infinite parts of opposite signs that cancel each other or can be averaged so to arrive at finite values. I would call them strongly divergent and weakly divergent. If we apply an absolute value to a function under a weakly divergent integral, it becomes strongly divergent.&#xA;The concept is similar to absolute and non-absolute convergence.&lt;/p&gt;&#xA;&lt;p&gt;Examples of integrals that &amp;quot;diverge&amp;quot; but can be averaged (weakly divergent):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_0^\infty\sin x dx=1$$&lt;/span&gt;&#xA;(Cesaro average)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-1}^1 \frac1xdx=0$$&lt;/span&gt;&#xA;(Cauchy mean value)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_0^\infty \left(x-\frac2{x^3}\right)dx=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(applying area-preserving operator &lt;span class=&quot;math-container&quot;&gt;$\mathcal{L}_t[t f(t)](x)$&lt;/span&gt; to one term gives the other with opposite sign, so they cancel each other)&lt;/p&gt;&#xA;&lt;p&gt;Examples of strongly-divergent integrals:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-1}^1 \frac1{x^2}dx=2\pi\delta(0)-2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using Hadamard finite part can be regularized to &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_0^\infty 1 dx=\pi\delta(0)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can be obtained from the previous one using the area-preserving operator, regularizes to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_0^1 \frac1x dx$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using Ramanujan regularization of its Riemann sum (and other methods), can be regularized to &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{k=0}^\infty k$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(can be regularized to &lt;span class=&quot;math-container&quot;&gt;$-\frac1{12}$&lt;/span&gt; using Ramanujan's, Dirichlet's or Zeta regularization)&lt;/p&gt;&#xA;&lt;p&gt;Thus, all summation methods can be divided into two very distinct parts: those that somehow average the result, attempting to cancel the infinite parts and those that do not make such attempts, but extract the finite part, dropping the infinite part.&lt;/p&gt;&#xA;&lt;p&gt;Was such classification of summation methods ever been proposed?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2021-05-03T12:36:01.780" Title="Did anyone ever propose the distinction between &quot;divergent to infinity&quot; as opposed to &quot;divergent but with finite average&quot;?" Tags="|mathematics|calculus|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13161" PostTypeId="2" ParentId="376" CreationDate="2021-05-03T12:58:46.030" Score="1" Body="&lt;p&gt;Initially numbers were used for accounting. Thus, it seems, the numbering systems initially appeared around the time of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Neolithic_Revolution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Neolithic revolution&lt;/a&gt;. Neolithic revolution signifies the change in society, which is characterised by the appearance of organized agriculture, as well as state power. With new technology the food could be produced in excess, stored and redistributed.&lt;/p&gt;&#xA;&lt;p&gt;This led to the need for counting and accounting.&lt;/p&gt;&#xA;&lt;p&gt;Linguistics confirms that languages got numerals around this change as well. The Neolithic revolution happened first in Fertile Crescent around 10000 BC. This is roughly when integers were invented.&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2021-05-03T12:58:46.030" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13163" PostTypeId="2" ParentId="13146" CreationDate="2021-05-04T14:00:32.217" Score="4" Body="&lt;p&gt;Because I'm fond of translating old math and science papers, particularly from French and Latin, I decided to start a new translation project for this work into English. You can check the progress at &lt;a href=&quot;https://histomathsci.blogspot.com/2021/05/translation-of-j-d-gergonnes-varities.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;my website here&lt;/a&gt;, where I post translations. This project could take some time, if only because the text requires careful reading to ensure the meaning is properly conveyed. I'm hoping to finish by this July.&lt;/p&gt;&#xA;&lt;p&gt;--Update--&lt;/p&gt;&#xA;&lt;p&gt;Finished rough draft of the translation May 30, far ahead of schedule (hooray!). The translation is &lt;a href=&quot;https://histomathsci.blogspot.com/2021/05/translation-of-j-d-gergonnes-varities.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;available here&lt;/a&gt;. Typesetting problems from a previous version are now fixed, but I also posted the original .tex for anyone to experiment with.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2021-05-31T15:08:34.917" LastActivityDate="2021-05-31T15:08:34.917" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13164" PostTypeId="2" ParentId="13157" CreationDate="2021-05-04T18:21:13.280" Score="2" Body="&lt;p&gt;As Professor K. Conrad mentions, the short answer to this question is NO.&lt;/p&gt;&#xA;&lt;p&gt;Around 1792 Gauß already &lt;em&gt;knew&lt;/em&gt; that the &amp;quot;frequency [of the primes] is on the average inversely proportional to the logarithm, so that the number of primes below a given bound &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is approximately equal to &lt;span class=&quot;math-container&quot;&gt;$\int \frac{1}{\log n} \, dn,$&lt;/span&gt; where the given logarithm is understood to be hyperbolic&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Gauß mentioned this investigations of his to Johann Franz Encke in a  letter dated December 24, 1849. The said letter can be found here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://cims.nyu.edu/%7Etschinke/princeton/.gauss/Briefe-B.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cims.nyu.edu/~tschinke/princeton/.gauss/Briefe-B.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If I recall correctly, a complete translation into English of this letter appeared at the end of the following article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;L. J. Goldstein, A history of the prime number theorem, Amer. Math.&#xA;Monthly 80 (1973), 599–615. MR 313171, &lt;a href=&quot;https://doi.org/10.2307/2319162&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.2307/2319162&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If you don't have access to the above article, you may want to take a look at &lt;a href=&quot;https://www.ams.org/journals/bull/2006-43-01/S0273-0979-05-01096-7/S0273-0979-05-01096-7.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 2005 note&lt;/a&gt; of Y. Tschinkel in BAMS.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2021-05-05T00:50:03.407" LastActivityDate="2021-05-05T00:50:03.407" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13165" PostTypeId="2" ParentId="2389" CreationDate="2021-05-05T15:14:39.860" Score="0" Body="&lt;p&gt;At least till the end of the 19th century this was not common knowledge yet as many scientists thought the space was filled with ether.&lt;/p&gt;&#xA;&lt;p&gt;This was finally ruled out in the Michelson–Morley experiment in 1887. This is usually considered the key point when the theory of physical vacuum started.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the OP possibly wants to know when the idea of air filling space was abandoned.&lt;/p&gt;&#xA;&lt;p&gt;There are numerous science-fiction stories from all epochs that involve traveling to the Moon via air, but many of them are based on the &amp;quot;True story&amp;quot; by Lucian (such as the adventures of Münchhausen), and as such, are satirical in nature.&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2021-05-05T15:14:39.860" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13166" PostTypeId="2" ParentId="13157" CreationDate="2021-05-06T05:59:33.983" Score="3" Body="&lt;p&gt;A good place to look is Granville’s paper “Harald Cramér and the distribution of primes numbers.” It is on Granville’s  website &lt;a href=&quot;https://dms.umontreal.ca/%7Eandrew/PDF/cramer.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. He brings in Cramér’s work starting on the bottom of page 19. The reason Cramér deserves to have this probabilistic model for the primes named after him is that he went much further than anyone before him in exploring consequences of the basic &lt;span class=&quot;math-container&quot;&gt;$1/\log n$&lt;/span&gt; heuristic to primes. He applied nontrivial theorems from probability theory to make predictions about the primes in a style nobody had ever done before. This did not directly &lt;em&gt;prove&lt;/em&gt; anything about the primes, and Cramér acknowledged this, but simply having an idea of what might be true is often very important in trying to understand how mathematical patterns fit together.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2021-05-06T05:59:33.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13167" PostTypeId="1" CreationDate="2021-05-06T17:31:55.260" Score="1" ViewCount="425" Body="&lt;p&gt;I'm asking who was the first to use and define velocity and acceleration in the modern, now standard way, with velocity being the first derivative of position and acceleration being the second derivative of the position vector. I have heard that Galileo Galilei, but he was born before the invention of calculus. I am also aware that vector analysis as we know it did not start until the 19th century.&lt;/p&gt;&#xA;" OwnerUserId="10492" LastActivityDate="2021-05-06T17:31:55.260" Title="Who first defined velocity and acceleration?" Tags="|mathematics|physics|classical-mechanics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13168" PostTypeId="1" CreationDate="2021-05-08T11:35:48.593" Score="4" ViewCount="150" Body="&lt;h1&gt;Gibbs' inequality&lt;/h1&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$-\sum\limits_{i=1}^n p_{i} \cdot \log{p_{i}} \le -\sum\limits_{i=1}^n p_{i} \cdot \log{q_{i}}$$&lt;/span&gt;&#xA;is such a popular thing that I cannot find where it was introduced.&lt;/p&gt;&#xA;&lt;h1&gt;My findings&lt;/h1&gt;&#xA;&lt;p&gt;I looked at &amp;quot;Elementary principles in statistical mechanics&amp;quot; (1902), and the most similar thing I've found is the theorem about the maximum entropy of uniform distributions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Theorem IX&lt;/em&gt;. A uniform distribution of a given number of systems within given limits of phase gives a less average index of probability of phase than any other distribution.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;meaning (&lt;em&gt;index of probability&lt;/em&gt; is the natural logarithm of the probability)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int\limits_{x \in X} \left(\Delta\eta\left( x \right) + \eta \right) \cdot e^{\Delta\eta\left( x \right) + \eta} dx&#xA;&amp;gt; \int\limits_{x \in X} \eta \cdot e^\eta dx,$$&lt;/span&gt;&#xA;where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int\limits_{x \in X} e^{\Delta\eta\left( x \right) + \eta} dx = \int\limits_{x \in X} e^\eta dx = 1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In &amp;quot;On the equilibrium of heterogeneous substances&amp;quot; (1874) I see&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the first place, if the system is in a state in which its entropy is greater than in any other state of the same energy, it is evidently in equilibrium, as any change of state must involve either a decrease of entropy or an increase of energy, which are alike impossible for an isolated system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but I can't find formulae looking like the Gibbs' inequality.&lt;/p&gt;&#xA;&lt;h1&gt;Question&lt;/h1&gt;&#xA;&lt;p&gt;Had Gibbs actually shown the inequality or it's just called after his name as a sign of respect?&lt;/p&gt;&#xA;&lt;p&gt;If he had, in which work can it be found and how is it formulated?&lt;/p&gt;&#xA;" OwnerUserId="14478" LastEditorUserId="4931" LastEditDate="2021-05-09T14:53:11.377" LastActivityDate="2021-05-09T14:53:11.377" Title="In which work was Gibbs' Inequality introduced?" Tags="|reference-request|mathematical-physics|statistics|entropy|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13170" PostTypeId="1" CreationDate="2021-05-09T18:53:34.203" Score="25" ViewCount="1812" Body="&lt;p&gt;(Disclaimer: apologies for any incorrect usage of mathematical terminology throughout this question.)&lt;/p&gt;&#xA;&lt;p&gt;In modern mathematical notation, a variable with a subscript can represent a couple of different concepts relating to the notion of index.&lt;/p&gt;&#xA;&lt;p&gt;For example, we can define an integer sequence such as the triangle numbers as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$T_n = \frac{n(n+1)}{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Or, we can write an infinite series as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{i=1}^\infty a_i = a_1+a_2+a_3+\ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And we can label the elements of a matrix or vector like so:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mathbf{A} = &#xA;\begin{bmatrix}&#xA;a_{11} &amp;amp; a_{12} &amp;amp; a_{13}\\&#xA;a_{21} &amp;amp; a_{22} &amp;amp; a_{23}\\&#xA;a_{31} &amp;amp; a_{32} &amp;amp; a_{33}&#xA;\end{bmatrix}&#xA;,&#xA;\mathbf{v} = &#xA;\begin{bmatrix}&#xA;v_1\\&#xA;v_2\\&#xA;v_3&#xA;\end{bmatrix}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But, when did this notation first arrive in mathematics? I've been trying to track down the origins of where this notation comes from through various sources, to no avail. The closest place I've come to an answer was through the book &lt;em&gt;A History of Mathematical Notations&lt;/em&gt;, by Florian Cajori. One interpretation of the book may hint that they seem to have emerged around the time determinants were beginning to be studied, before the modern day invention of matrices, possibly by Leibniz? However, this is just from my reading, and the book doesn't acknowledge this directly, or make any note of it. And they could have very easily originated from later or earlier than this. Is there an original field of maths they come from, that their use proliferated out from?&lt;/p&gt;&#xA;" OwnerDisplayName="thBeam" LastActivityDate="2021-05-12T23:35:04.297" Title="When was the first recorded use of subscript in mathematics to represent index?" Tags="|notation|" AnswerCount="6" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13171" PostTypeId="2" ParentId="13170" CreationDate="2021-05-09T19:16:46.153" Score="4" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.stephenwolfram.com/publications/mathematical-notation-past-future/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article Frege used it in 1879:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/lBCNu.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lBCNu.jpg&quot; alt=&quot;Frege works&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although you may also be interested in &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_mathematical_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article which says &lt;a href=&quot;https://en.wikipedia.org/wiki/Georg_Cantor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georg Cantor&lt;/a&gt; used subscripts in his notation for cardinal numbers, and he was born in 1845.&lt;/p&gt;&#xA;&lt;p&gt;Whilst this is just my opinion, I suspect them to be much older due to the use of symbols to vary letters in many languages and variations of characters in previous cultures however I cannot find a source for this.&lt;/p&gt;&#xA;" OwnerUserId="14509" OwnerDisplayName="Henry Lee" LastActivityDate="2021-05-09T19:16:46.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13172" PostTypeId="2" ParentId="13170" CreationDate="2021-05-10T11:33:02.550" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Grassmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Grassmann&lt;/a&gt; used subscripts in his &lt;em&gt;&lt;a href=&quot;https://books.google.no/books?id=XgcHAAAAcAAJ&amp;amp;printsec=frontcover&amp;amp;source=gbs_ge_summary_r&amp;amp;cad=0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lineale Ausdehnungslehre&lt;/a&gt;&lt;/em&gt; from 1844. For example, on page 71 he sets up a system of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; linear equations in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; unknowns on the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$a_1 x_1 + a_2 x_2 +\dots + a_n x_n = a_0 \\&#xA;b_1 x_1 + b_2 x_2 +\dots+ b_n x_n = b_0 \\&#xA;\vdots \\&#xA;s_1 x_1 + s_2 x_2 + \dots + s_n x_n = s_0$$&lt;/span&gt;&#xA;Note that he does not use double subscripts, as we would today.&lt;/p&gt;&#xA;" OwnerUserId="40" OwnerDisplayName="Per Manne" LastActivityDate="2021-05-10T11:33:02.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13173" PostTypeId="2" ParentId="13170" CreationDate="2021-05-10T12:11:42.870" Score="23" Body="&lt;p&gt;Looking in &lt;a href=&quot;https://jeff560.tripod.com/matrices.html&quot; rel=&quot;noreferrer&quot;&gt;Mathsym&lt;/a&gt;, I find:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bullet \;$&lt;/span&gt;Use of &lt;span class=&quot;math-container&quot;&gt;$\;{}^1a,\;{}^2a,\;{}^3a$&lt;/span&gt; in Laplace, 1772.&lt;br /&gt;&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k35711&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Histoire de l'Académie royale des sciences&lt;/em&gt;&lt;/a&gt;,  p. 294&lt;br /&gt;&#xA;It didn't catch on.&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/AX8SI.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AX8SI.gif&quot; alt=&quot;Laplace&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\bullet \;$&lt;/span&gt;Then Cauchy, 1815, with subscripts.&lt;br /&gt;&#xA;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k90193x&quot; rel=&quot;noreferrer&quot;&gt;Œuvres complètes d'Augustin Cauchy, Serie 2, tome 1&lt;/a&gt;, p. 130&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AGs0C.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AGs0C.gif&quot; alt=&quot;Cauchy&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" OwnerDisplayName="Gerald Edgar" LastEditorUserId="229" LastEditDate="2021-05-11T17:57:32.940" LastActivityDate="2021-05-11T17:57:32.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13174" PostTypeId="1" CreationDate="2021-05-11T00:32:05.110" Score="1" ViewCount="702" Body="&lt;p&gt;I have always heard this claim that, out of all disciplines, Mathematicians have the shortest 'shelf-life'. More specifically, that they peak early in their twenties &lt;em&gt;(i.e. present their most important contributions / most of their work in their early twenties)&lt;/em&gt; and then decline fast. As opposed to Biologists, for example, who are said to peak much later in their lives and keep contributing well into their 70s or beyond.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;How historically &lt;em&gt;(statistically?)&lt;/em&gt; accurate is this claim?&lt;/li&gt;&#xA;&lt;li&gt;And if it is true, what would be the reason behind it? Could it be because of the decline in fluid intelligence &lt;em&gt;(pure reasoning)&lt;/em&gt; that has been empirically observed to also peak in the early twenties and start declining by late twenties?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14510" LastActivityDate="2021-05-11T00:32:05.110" Title="Is it true that Mathematicians have a short shelf-life / peak early?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13175" PostTypeId="2" ParentId="13170" CreationDate="2021-05-11T09:15:41.437" Score="6" Body="&lt;p&gt;There is an interesting passage in&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dodgson, Charles Lutwidge. An Elementary Treatise on Determinants:&#xA;With Their Application to Simultaneous Linear Equations and&#xA;Algebraical Geometry. Vol. 13. Macmillan and Company, 1867:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;New words and symbols are always a most unwelcome addition to a&#xA;Science, especially to one already burdened with an enormous&#xA;vocabulary, yet I think the Definitions given of them will be found to&#xA;justify their  introduction, as the only way of avoiding tedious&#xA;periphrasis.  The symbols employed to represent the single elements of&#xA;a Determinant, require perhaps a word of apology, and it may be well&#xA;to enumerate those already in use, and to point out what seem to be&#xA;their chief defects.  We may commence with&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/OTTka.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/OTTka.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;, where the change of letter indicates a change of&#xA;column, and the change of subscript a change of row. Now the&#xA;properties of Determinants, relating to columns, being always&#xA;convertible into properties relating to rows, and vice versa, it was a&#xA;sufficient objection to this system of notation, that it represented&#xA;things distinctly analogous by methods so different, and it was&#xA;properly superseded by the notation introduced by Leibnitz,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Nz0um.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Nz0um.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;, both of column and row, are alike&#xA;denoted by subscripts.  But it seems a fatal objection to this system&#xA;that most of the space is occupied by a number of a's, which are&#xA;wholly superfluous, while the only important part of the notation is&#xA;reduced to minute subscripts, alike difficult to the writer and the&#xA;reader.  It was almost an obvious improvement on this system to raise&#xA;the subscripts into the line, and omit the a's altogether, as&#xA;suggested by Baltzer, thus -&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/owZC1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/owZC1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;, and this system, though tedious for writing, might serve very well,&#xA;were it not for its liability to be confused with the notation, common&#xA;in Plane Algebraical Geometry, by which &lt;span class=&quot;math-container&quot;&gt;$(1,1)$&lt;/span&gt; denotes the Point&#xA;&lt;span class=&quot;math-container&quot;&gt;$x=1,\ y=1$&lt;/span&gt;.  The symbol&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/pIgqY.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pIgqY.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;, which I have ventured  to suggest as an emendation on this last, will&#xA;be found, I have great hopes,  sufficient'y simple, distinct, and easy&#xA;to be written. I have turned the symbol towards the left,    in&#xA;order to avoid all chance of confusion with &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt;, the symbol for&#xA;integration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The reference to the&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=lSNrc9eMjUE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hidden Math in Alice in Wonderland&lt;/a&gt;&#xA;is due since that is where I've learned about that published math book.&lt;/p&gt;&#xA;" OwnerUserId="14513" LastEditorUserId="14513" LastEditDate="2021-05-12T06:28:17.050" LastActivityDate="2021-05-12T06:28:17.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13177" PostTypeId="2" ParentId="13170" CreationDate="2021-05-12T17:57:18.093" Score="3" Body="&lt;p&gt;Cajori's &lt;em&gt;History of Mathematical Notations&lt;/em&gt;, which you mention, also describes the use of subscripts for terms of Taylor series by William Emerson in &lt;em&gt;Method of Increments&lt;/em&gt; (1763), which are introduced in the section &amp;quot;Notation&amp;quot; on page 2.&lt;/p&gt;&#xA;&lt;p&gt;Indeed, you can see this in Google Books: &lt;a href=&quot;https://www.google.com/books/edition/The_Method_of_Increments/2N9JAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA2&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Method of Increments&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2077" LastActivityDate="2021-05-12T17:57:18.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13178" PostTypeId="2" ParentId="13170" CreationDate="2021-05-12T23:35:04.297" Score="2" Body="&lt;p&gt;Comments above have centered on the sequential notation used in matrix assessment, or expansion of determinants, in particular.  There is a thread that seems to trace the collective history on the use of subscripts, but key aspects regarding dates seem elusive.  This is for good reason: access to original materials is rare.  Nevertheless, we may glean from the writings of others some information regarding the circumstances and earliest dates when subscripts were first used.  When, exactly, was the first use of the subscript as an index?&lt;/p&gt;&#xA;&lt;p&gt;From the comments above, g.kov mentions Dodgson's commentary noting Leibnitz as having introduced the notation we currently use.  However, Dodgson gives no date to establish when Leibnitz introduced this notation.  A valuable clue is nevertheless given by Paul H. Hanus in &lt;em&gt;An Elementary Treatise on the Theory of Determinants,&lt;/em&gt; published in 1886 by Ginn and Company, Boston.  Hanus mentions in Chapter 1, Preliminary Notions and Definitions, Article 1, &lt;em&gt;Discovery of Determinants,&lt;/em&gt; the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a letter dated April 28, 1693, Leibnitz communicates his discovery to L'Hospital; and later, in another letter, expresses the conviction that the functions will develop remarkable and very important properties, - a conviction which time has abundantly verified.  Leibnitz, however, never pursued the subject himself, and his discovery lay dormant till the middle of the eighteenth century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hanus continues that in 1750, Gabriel Cramer, rediscovered determinants while working upon the analysis of curves, wherein Cramer had to solve sets of linear equations and naturally encountered the same functions that had attracted the attention of Leibnitz.  Hanus further states in Chapter 1, articles 2 thru 7, &lt;em&gt;Determinants produced by eliminating the unknowns from a system of simultaneous equations,&lt;/em&gt; that the notation is attributable to Laplace, namely:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...the terms in which the subscripts occur in their natural order are positive, while in the negative terms there is an inversion of the natural order in the subscripts...  It has been agreed to denote them, following Laplace, by writing the letters involved in regular succession, affecting each with the subscripts in order, and enclosing the whole expression within parentheses, thus,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(a_1 b_2) \equiv a_1 b_2 - a_2 b_1 ; (a_2 c_3) \equiv a_2 c_3 - a_3 c_2, etc. $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is, of course, a second order determinant, the first part of which, expressed in current, or modern, notation, would appear as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ &#xA;\begin{bmatrix}&#xA;a_{1} &amp;amp; b_{1}\\&#xA;a_{2} &amp;amp; b_{2}\\&#xA;\end{bmatrix} = a_1 b_2 - a_2 b_1 .$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This general result is known as Laplace's expansion as noted in C. R. Wylie, Jr, &lt;em&gt;Advanced Engineering Mathematics,&lt;/em&gt; Second Edition, McGraw-Hill Book Company, Inc., New York, 1960.&lt;/p&gt;&#xA;" OwnerUserId="14530" LastActivityDate="2021-05-12T23:35:04.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13179" PostTypeId="1" AcceptedAnswerId="13183" CreationDate="2021-05-14T18:33:41.433" Score="1" ViewCount="198" Body="&lt;p&gt;in Gauss's Section in 'The World of Mathematics' Gauss is said to have declared that :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is this from one of Gauss's letters? the section on Gauss is written by Eric Temple Bell, and as far as I can see there is no Indication throughout the text as to what is the source of this quote so I am doubting whether the quote is true or not. there are many other instances like this in the book where the author does not give a source for the quote. Eric Temple Bell seems to be careful about the authenticity of his claims, for example when he quotes something from Gauss's surviving relatives he says whether they were old and whether the data is reliable or not. so it seems odd to me that he repeatedly quotes many figures without giving a source, is there a specific reason for this?&lt;/p&gt;&#xA;" OwnerUserId="14536" LastActivityDate="2021-05-15T06:06:27.283" Title="Authenticity of Gauss's Quote &quot;If others reflect on mathematical truths as deeply as I...&quot;" Tags="|gauss|quote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13180" PostTypeId="1" CreationDate="2021-05-14T19:50:18.487" Score="0" ViewCount="87" Body="&lt;p&gt;I would guess that occasionally pots of Greek fire did explode and if so, one would have seen how far shards were sent. Assuming this is true, did anyone suggest something like a cannon or a bomb meant to blast through a wall, not just spread fire?&lt;/p&gt;&#xA;&lt;p&gt;Or maybe Greek fire does not explode.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="14365" LastEditDate="2021-11-29T07:16:19.227" LastActivityDate="2021-11-29T07:16:19.227" Title="Greek fire explosive I assume -- did an accident lead any inventor to consider using it as a propellant?" Tags="|ancient-greece|technology|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13181" PostTypeId="2" ParentId="13179" CreationDate="2021-05-14T21:12:57.127" Score="0" Body="&lt;p&gt;That chapter on Gauss in &lt;em&gt;The World of Mathematics&lt;/em&gt; is taken from Bell's book &lt;em&gt;Men of Mathematics&lt;/em&gt;.&lt;br /&gt;&#xA;You are right, it has no references.  It is a &amp;quot;popular&amp;quot; account, not a &amp;quot;scholarly&amp;quot; one.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-05-14T21:12:57.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13182" PostTypeId="1" CreationDate="2021-05-15T00:58:14.683" Score="2" ViewCount="265" Body="&lt;p&gt;Did the Inquisition threaten Galileo with torture and, if so, did he know he was not actually in danger of being tortured?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_affair&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Galileo_affair&lt;/a&gt; says:&#xA;&amp;quot;Galileo was interrogated while threatened with physical torture.[46] A panel of theologians, consisting of Melchior Inchofer, Agostino Oreggi and Zaccaria Pasqualigo, reported on the Dialogue. Their opinions were strongly argued in favour of the view that the Dialogue taught the Copernican theory.[52]&lt;/p&gt;&#xA;&lt;p&gt;Galileo was found guilty, and the sentence of the Inquisition, issued on 22 June 1633,[53] was in three essential parts:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Galileo was found &amp;quot;vehemently suspect of heresy&amp;quot;, namely of having held the opinions that the Sun lies motionless at the center of the universe, that the Earth is not at its centre and moves, and that one may hold and defend an opinion as probable after it has been declared contrary to Holy Scripture. He was required to &amp;quot;abjure, curse, and detest&amp;quot; those opinions.[54]&#xA;He was sentenced to formal imprisonment at the pleasure of the Inquisition.[55] On the following day this was commuted to house arrest, which he remained under for the rest of his life.&#xA;His offending Dialogue was banned; and in an action not announced at the trial, publication of any of his works was forbidden, including any he might write in the future.[56]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;According to popular legend, after his abjuration Galileo allegedly muttered the rebellious phrase &amp;quot;and yet it moves&amp;quot; (Eppur si muove), but there is no evidence that he actually said this or anything similar.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.iitaly.org/node/52894&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.iitaly.org/node/52894&lt;/a&gt; says:&#xA;&amp;quot;On April 12, 1633, Galileo was summoned to Rome by the Inquisition, the Holy Office (Santi Uffizi). In an article for the journal Church History last year, Prof. Henry Kelly of UCLA wrote that, at that trial, Galileo said that after 1616 he had never considered heliocentrism to be possible. “Galileo was clearly stretching the truth... Admitting otherwise would have increased the penance he was given, but would not have endangered his life, since he agreed to renounce the heresy.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;From the inquisitors came no threat of torture, says Prof. Kelly. Still, Galileo abjured but was found guilty of heresy on June 22, 1633. Ever since then writers have invented words for him, such as this version by Italian writer Primo Levi (1919 - 1987), the author of &amp;quot;Se questo è un uomo&amp;quot;: &amp;quot;I have had to bow down and say that I did not see what I saw.&amp;quot; Galileo was first held in the home of the Archbishop of Siena and then under house arrest at Arcetri on the Florentine Hills, where he died on January 8, 1642.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://newsroom.ucla.edu/releases/the-truth-about-galileo-and-his-conflict-with-the-catholic-church&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://newsroom.ucla.edu/releases/the-truth-about-galileo-and-his-conflict-with-the-catholic-church&lt;/a&gt; says:&#xA;&amp;quot;Kelly also noted that by the practice of the time, Galileo’s guilty plea, which denied actual belief in the heresy, triggered an automatic examination of his private beliefs under torture, a new procedure adopted by the church around the turn of the 17th century. Galileo was never tortured, however. The pope decreed that the interrogation should stop short with the mere threat of torture. This was a routine kind of limitation for people of advanced age and ill health like Galileo, and it should not be attributed to the influence of the scientist’s supporters.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.cambridge.org/core/journals/church-history/article/galileos-nontrial-1616-pretrial-16321633-and-trial-may-10-1633-a-review-of-procedure-featuring-routine-violations-of-the-forum-of-conscience/F5BB9632FA3F4C34B33C119A401EB3BC&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cambridge.org/core/journals/church-history/article/galileos-nontrial-1616-pretrial-16321633-and-trial-may-10-1633-a-review-of-procedure-featuring-routine-violations-of-the-forum-of-conscience/F5BB9632FA3F4C34B33C119A401EB3BC&lt;/a&gt; , which is the actual paper by Henry Ansgar Kelly about the interactions between the Inquisition and Galileo, says:&#xA;&amp;quot;In Galileo's case, the examination on intention was to be carried out not with actual torture but only with the threat thereof. Whether Galileo was told of this limitation is not known. It may well have been part of the plea bargain that he made with Maculano, or it may have been the result of standard rules for defendants of advanced age or ill heath.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So Kelly does not know whether Galileo knew that he was in no danger of being tortured.&lt;/p&gt;&#xA;" OwnerUserId="14349" LastEditorUserId="14349" LastEditDate="2021-05-15T13:17:26.993" LastActivityDate="2021-05-15T13:17:26.993" Title="Did the Inquisition threaten Galileo with torture and if so, did he know he that there was no danger of that happening?" Tags="|galileo|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13183" PostTypeId="2" ParentId="13179" CreationDate="2021-05-15T06:06:27.283" Score="2" Body="&lt;p&gt;the Book&#xA;&lt;em&gt;Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work&lt;/em&gt;&#xA;written by G. Waldo Dunnington, is the definitive biography of Gauss. the quote mentioned in my question is present in this book. the writer devoted 30 years to writing this book and it is evident from the preface that he did it in an academic and scholarly manner. in a part of the preface the author says :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The present volume is derived almost entirely from contemporary&#xA;sources, both manuscript and printed works. The letters and collected&#xA;works of Gauss have been a rich mine of information. Reminiscences of&#xA;his friends and students have been useful. His descendants have been&#xA;very kind in allowing me to use letters, pictures, and other material. I&#xA;have drawn freely from what I found. The nearest approximation to a&#xA;biography of Gauss is the memorial monograph by his friend Sartorius&#xA;A856). Any biographer must use it. For some aspects of his life it is the&#xA;only source. I have profited by the learning of those who have covered&#xA;the ground before me.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;in another part of the preface the book reads as follows :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is hoped that the Appendixes will serve as a place of reference to&#xA;primary sources. The reader will observe that Chapter XXIV and&#xA;Appendix I are largely quotation, since it seems advisable to allow Gauss&#xA;and certain authorities to speak for themselves. An effort has been made&#xA;to keep footnotes to a minimum. In a work of this character it has not&#xA;been deemed advisable to cite in footnotes bibliographical references to&#xA;published letters. These are arranged chronologically in the published&#xA;correspondence and can be easily found by anyone who is interested.&#xA;The published correspondence is listed completely in the bibliography.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;other similar remarks in the preface make it quite clear(at least to me) that the data in this book is genuine and reliable. the original quote of my question is on page 231 :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was his custom to tell his friends that if others would&#xA;meditate as long and as deeply as he did on mathematical truths, they&#xA;would be able to make his discoveries. He said that often he&#xA;meditated for days on a piece of research without finding a solution, which finally became clear to him after a sleepless night. His conversation with friends sometimes was interrupted for meditation; sometimes the conversation would be continued after a gap of several days.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;as a side note, there is an excellent site which contains all of of Gauss's known letters, completely sorted and listed. Currently the site provides scans, digitized printed editions and/or transcriptions of about 83% of the letters. the link is : &lt;a href=&quot;https://gauss.adw-goe.de/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gauss.adw-goe.de/&lt;/a&gt; the site is currently maintained by the university of gottingen.&lt;/p&gt;&#xA;" OwnerUserId="14536" LastActivityDate="2021-05-15T06:06:27.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13184" PostTypeId="1" CreationDate="2021-05-15T10:24:14.547" Score="2" ViewCount="295" Body="&lt;p&gt;The idea of using Lie groups in physics can be easily understood intuitively, but what are the origins of the use of representation theory of Lie groups and Lie algebras in physics?&lt;/p&gt;&#xA;&lt;p&gt;We mathematicians use representation theory as a tool (among others, such as cohomology) to get information about groups and other algebraic structures that could not be gathered otherwise. But I guess physicists do not have the same use of it?&lt;/p&gt;&#xA;" OwnerUserId="14542" LastEditorUserId="9430" LastEditDate="2021-05-19T03:51:08.300" LastActivityDate="2021-05-19T03:51:08.300" Title="Representation theory in physics" Tags="|theoretical-physics|group-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13185" PostTypeId="1" CreationDate="2021-05-15T11:18:50.030" Score="2" ViewCount="182" Body="&lt;p&gt;Is the proof of the Cayley-Hamilton theorem given by Serge Lang in &lt;a href=&quot;https://www.springer.com/gp/book/9780387953854&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Algebra&lt;/em&gt;&lt;/a&gt; (page 561) an original one, or has it been borrowed from some earlier sources? Who came up with it first?&lt;/p&gt;&#xA;&lt;p&gt;(Lang's proof is closely followed by the current version of &lt;a href=&quot;https://ncatlab.org/nlab/show/characteristic+polynomial&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;characteristic polynomial&lt;/em&gt;&lt;/a&gt; page of nLab wiki.)&lt;/p&gt;&#xA;&lt;p&gt;I am asking because I &lt;a href=&quot;https://arxiv.org/abs/2105.09285&quot; rel=&quot;nofollow noreferrer&quot;&gt;reformulated&lt;/a&gt; it recently, and I am looking for a proper attribution.&lt;/p&gt;&#xA;" OwnerUserId="3030" LastEditorUserId="3030" LastEditDate="2022-11-16T21:51:13.497" LastActivityDate="2022-11-16T21:51:13.497" Title="Origin of Lang's proof of the Cayley-Hamilton theorem" Tags="|mathematics|abstract-algebra|linear-algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13186" PostTypeId="2" ParentId="13184" CreationDate="2021-05-15T14:46:41.030" Score="8" Body="&lt;p&gt;Explicit applications of group representation to physics start with&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;E. Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1918: 235–257.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where the celebrated &lt;a href=&quot;https://en.wikipedia.org/wiki/Noether%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;Noether's Theorem&lt;/a&gt; was proved.&lt;/p&gt;&#xA;&lt;p&gt;Applications to quantum mechanics were systematically developed by H. Weyl and E. Wigner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;H. Weyl,&#xA;The theory of groups in quantum mechanics, first edition 1928&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;E. Wigner,&#xA;Group theory and its application to the quantum mechanics of atomic spectra,&#xA;first edition 1931.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are many good modern books with &amp;quot;group theory&amp;quot; and &amp;quot;physics&amp;quot; or &amp;quot;quantum mechanics&amp;quot; in the title,&#xA;my favorite is S. Sternberg, Group theory and physics, Cambridge, 1994.&lt;/p&gt;&#xA;&lt;p&gt;Remark. The word &amp;quot;explicit&amp;quot; in the beginning of my answer is important.&#xA;Considerations based on symmetry are as old as physics itself: they were used already by Aristotle. A major 19&lt;sup&gt;th&lt;/sup&gt; century application was crystallography.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="5989" LastEditDate="2021-05-16T07:01:17.480" LastActivityDate="2021-05-16T07:01:17.480" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13187" PostTypeId="1" CreationDate="2021-05-15T16:37:00.817" Score="3" ViewCount="324" Body="&lt;p&gt;Why did Zariski feel the need to define his famous topology? Was this notion used in one form or another prior to him in algebraic geometry?&lt;/p&gt;&#xA;" OwnerUserId="14542" LastEditorUserId="10106" LastEditDate="2021-05-31T02:11:50.393" LastActivityDate="2021-05-31T02:11:50.393" Title="Origins of Zariski topology" Tags="|mathematics|algebraic-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13188" PostTypeId="1" CreationDate="2021-05-15T21:24:49.743" Score="2" ViewCount="112" Body="&lt;p&gt;Euler developed the Euler Product Formula which shows that the Riemann zeta function encodes information about the prime.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\zeta(s)=\sum_{n}\frac{1}{n^{s}}=\prod_{p}(1-\frac{1}{p^{s}})^{-1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Riemann is considered to have been the first to extend the domain &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; from the real numbers  &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; to the complex domain &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1&lt;/strong&gt;: what motivated the need to widen the domain from the reals to the complex domain?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2&lt;/strong&gt;: what is the history around extending the domain from integers &lt;span class=&quot;math-container&quot;&gt;$s\in\mathbb{Z}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$s\in\mathbb{R}$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;For clarity, I am not asking about the analytic continuation of the zeta function.&lt;/p&gt;&#xA;" OwnerUserId="12633" LastActivityDate="2021-05-15T21:24:49.743" Title="First motivation for extending Riemann Zeta to complex domain?" Tags="|number-theory|euler|primes|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13189" PostTypeId="1" CreationDate="2021-05-16T03:13:48.327" Score="3" ViewCount="338" Body="&lt;p&gt;I've done my B.S. in Electrical Engineering as well as mathematics but I'd like to get a proper, or complete history of Maxwell and the history of his derivation of the equations and the newness of the methods he employed. Freeman Dyson's description of how mathematics parted from physics in the 19th century has me looking into what details he was describing. As always your input is greatly valued.&lt;/p&gt;&#xA;" OwnerUserId="7448" LastEditorUserId="16128" LastEditDate="2022-01-28T17:36:11.293" LastActivityDate="2022-01-28T17:36:11.293" Title="Best history of Maxwell and his equations" Tags="|mathematics|physics|mathematical-physics|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13190" PostTypeId="2" ParentId="13189" CreationDate="2021-05-16T13:03:50.660" Score="4" Body="&lt;p&gt;On my opinion, the best source is Maxwell's book itself, Treatise on electricity and magnetism. But there is also a comprehensive history:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;E. Whittaker, A History of the Theories of Aether and Electricity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-05-17T19:35:51.153" LastActivityDate="2021-05-17T19:35:51.153" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13192" PostTypeId="2" ParentId="13187" CreationDate="2021-05-17T04:03:19.020" Score="8" Body="&lt;p&gt;Zariski introduced his topology in this paper: &lt;em&gt;The compactness of the Riemann manifold of an abstract field of algebraic functions&lt;/em&gt;, Bull. Amer. Math. Soc., 50 (1944), 683-691.  You can read it online &lt;a href=&quot;https://www.ams.org/journals/bull/1944-50-10/S0002-9904-1944-08206-2/S0002-9904-1944-08206-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. See also p. 79 of his talk at the 1950 ICM: &lt;em&gt;The fundamental ideas of abstract algebraic geometry&lt;/em&gt;, pp. 77-89 in &amp;quot;Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950&amp;quot;, vol. 2, Amer. Math. Soc., Providence, R. I., 1952. Scroll down to the volume 2 link for the 1950 ICM &lt;a href=&quot;https://www.mathunion.org/icm/proceedings&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some pages about the importance of the Zariski topology in algebraic geometry are &lt;a href=&quot;https://mathoverflow.net/questions/21502/what-is-the-zariski-topology-good-bad-for/21529&quot;&gt;this MO page&lt;/a&gt; and MSE pages &lt;a href=&quot;https://math.stackexchange.com/questions/161884/why-zariski-topology&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/questions/55259/did-zariski-really-define-the-zariski-topology-on-the-prime-spectrum-of-a-ring&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/questions/53852/is-there-a-way-of-working-with-the-zariski-topology-in-terms-of-convergence-limi/53931&quot;&gt;here&lt;/a&gt;, and &lt;a href=&quot;https://math.stackexchange.com/questions/1798667/why-was-sheaf-cohomology-invented&quot;&gt;here&lt;/a&gt;.&#xA;The idea that maximal ideals (not yet prime ideals) in a commutative ring could be profitably viewed as the points in a space can be seen in Gelfand's work in the 1930s, and whether or not that might have had an influence in algebraic geometry was asked on the MO page &lt;a href=&quot;https://mathoverflow.net/questions/20268/did-gelfands-theory-of-commutative-banach-algebras-influence-algebraic-geometer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2021-05-17T06:14:42.810" LastActivityDate="2021-05-17T06:14:42.810" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13193" PostTypeId="2" ParentId="13189" CreationDate="2021-05-17T04:45:39.953" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.encyclopedia.com/topic/Pierre_Maurice_Marie_Duhem.aspx#1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre Duhem&lt;/a&gt;, &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=4976&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;The Electric Theories of J. Clerk Maxwell&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;, trans. of 1902 &lt;a href=&quot;https://archive.org/details/lesthoriesle00duheuoft/page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Les théories électriques de J. Clerk Maxwell&lt;/em&gt;&lt;/a&gt; (&lt;a href=&quot;https://archive.org/details/LesTheoriesElectriquesDeMaxwell_re-typeset&quot; rel=&quot;nofollow noreferrer&quot;&gt;2015 re-typset ed.&lt;/a&gt;), vol. 314, &lt;em&gt;Boston Studies in the Philosophy and History of Science&lt;/em&gt; (Cham: Springer International Publishing, 2015).&lt;/li&gt;&#xA;&lt;li&gt;Sarkar, Tapan. 2006. &lt;a href=&quot;https://isidore.co/calibre#panel=book_details&amp;amp;book_id=5099&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;History of Wireless&lt;/em&gt;&lt;/a&gt;. Hoboken, N.J.: Wiley-Interscience.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="232" LastActivityDate="2021-05-17T04:45:39.953" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13195" PostTypeId="2" ParentId="13189" CreationDate="2021-05-17T17:13:42.743" Score="5" Body="&lt;p&gt;Just quickly adding some more references for you. The analytical theory of EM starts with Biot and Poisson at the turn of the 19th century. The works by Poisson were particularly influential to George Green, who in turn was influential to Maxwell. Faraday was also critical to Maxwell's conceptions of the electromagnetic field, particularly his first few researches on electricity and magnetism, and the last few (if I recall correctly). They're good reads, and readily available.&lt;/p&gt;&#xA;&lt;p&gt;I wrote a translation of Poisson's first memoir on electricity, posted on my &lt;a href=&quot;https://histomathsci.blogspot.com/2021/04/translation-of-s-d-poissons-premiere.html?m=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;website here&lt;/a&gt;, and Biot's paper is there too. Green's works are available in English online. You could also look into Ampere's writing and contemporary reviews of it. That'll set you up for reading Maxwell, and understanding the views he expresses.&lt;/p&gt;&#xA;&lt;p&gt;The most direct resource that I personally have experience with would be Simpson's &lt;em&gt;Maxwell on the Electromagnetic Field&lt;/em&gt;, which is an annotated walk-through of three of Maxwell's most important papers. It's written for a non-technical audience, but it doesn't shy away from details; it's filled with notes and context and I like it quite a bit.&lt;/p&gt;&#xA;&lt;p&gt;Then after reading Maxwell's first few papers, you can read his Treatise, which is the master work. You'll learn things about electromagnetism there that they stopped teaching a long time ago, but it's still very valuable. Good luck!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Biot, J-B. &lt;em&gt;Sur un problem physique, relatif a l'electricite&lt;/em&gt; (1801). &lt;a href=&quot;https://histomathsci.blogspot.com/2020/11/one-of-earliest-analytical-papers-on.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Poisson, S-D. &lt;em&gt;Memoire sur la distribution de l'electricite a la surfaces des corps conducteurs&lt;/em&gt; (1812-13). &lt;a href=&quot;https://histomathsci.blogspot.com/2021/04/translation-of-s-d-poissons-premiere.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Green, George. &lt;em&gt;An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism&lt;/em&gt; (1828). &lt;a href=&quot;https://books.googleusercontent.com/books/content?req=AKW5Qac-W8iQ8y71OAQEhcsXjHcyEYTiuK6XlYOkrtm99Xql_49w6-zRfDJHVsYLeU04-Avqa4tDe-SeYGYnF2yJm48ypKqdxAByZBy-7wc2p-cRS3GJSz7J0aOTRgsRidBaIQzRexIp2SFww1oSeyYrONXIb9JB5zkZoJCwAreVbJaH11kSITSTqWl9f5sP1fvkq6Yfhv4OuEY2SE2GqTdnDbUpnKv61SForFwP9FmKKS6z9zG9mivKMRTMzjdu_wGNZ1AC3LBpmokYIIPAyaHGobQ38wk55KfMJ-5E6sS699m-Us3CI7w&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Faraday, Michael. &lt;em&gt;Experimental Researches in Electricity&lt;/em&gt; (1831-54). Series I, II, III, IX, XI, XIV, XIX, XXVIII, XXIX.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Assis &amp;amp; Chaib. &lt;em&gt;Ampere's Electrodynamics: Analysis of the Meaning and Evolution of Ampere’s Force between Current Elements&lt;/em&gt; (2015).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Simpson, Thomas K. &lt;em&gt;Maxwell on the Electromagnetic Field&lt;/em&gt; (1997).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maxwell, J. C. &lt;em&gt;On Faraday's Lines of Force&lt;/em&gt; (1855). Transactions of the Cambridge Philosophical Society, Vol X. part I.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maxwell, J. C. &lt;em&gt;On Physical Lines of Force&lt;/em&gt; (1861). Philosophical Magazine, Vol XXI.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maxwell, J. C. &lt;em&gt;A Dynamical Theory of the Electromagnetic Field&lt;/em&gt; (1864). Royal Society Transactions, Vol. CLV.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Maxwell, J. C. &lt;em&gt;A Treatise on Electricity and Magnetism&lt;/em&gt; (1873).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2021-05-18T14:34:49.577" LastActivityDate="2021-05-18T14:34:49.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13196" PostTypeId="1" CreationDate="2021-05-17T22:54:27.847" Score="1" ViewCount="141" Body="&lt;p&gt;From &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:William_Herschel_Museum_-_40-foot_telescope_model_1.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EXcxj.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EXcxjm.jpg&quot; alt=&quot;Herschel 40 foot telescope model&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;https://i.stack.imgur.com/MA1ZA.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MA1ZAm.jpg&quot; alt=&quot;Herschel 40 foot telescope model&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;click for larger&lt;/p&gt;&#xA;&lt;p&gt;Herschel's &lt;a href=&quot;https://en.wikipedia.org/wiki/40-foot_telescope&quot; rel=&quot;nofollow noreferrer&quot;&gt;40 foot telescope&lt;/a&gt; had a diameter of 48 inches (122 cm) and was f/10. It was not a good mirror (cast in metal) and apparently not used that much.&lt;/p&gt;&#xA;&lt;p&gt;But my question is about where the eyepiece was planned to be held during normal use (which may not have happened). The Wikipedia article says only:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The telescope was located on the grounds of Observatory House, Herschel's house in Slough, between 1789 and 1840. The first observation with the telescope was on 19 February 1787, when Herschel pointed the then-incomplete telescope towards the Orion nebula, which he observed by crawling into the telescope and using a hand-held eyepiece: &amp;quot;The apparatus for the 40-foot telescope was by this time so far completed that I could put the mirror into the tube and direct it to a celestial object; but having no eye-glass fixed, not being acquainted with the focal length which was to be tried, I went into the tube, and laying down near the mouth of it I held the eye-glass in my hand, and soon found the place of the focus. The object I viewed was the nebula in the belt of Orion, and I found the figure of the mirror, though far from perfect, better than I had expected. It showed four small stars in the nebula and many more. The nebula was extremely bright.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above quote describes some tests of the mirror when the telescope was not yet complete, I'm asking about the plan for the completed configuration.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Where did &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Herschel&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Herschel&lt;/a&gt; plan to put their head when looking through the famous 40-foot telescope? Was the plan to peer over the edge of the tube or try to get less off-axis; closer to the center and basically partially block the aperture with the upper body? Or was there going to be a diagonal mirror like a &lt;a href=&quot;https://en.wikipedia.org/wiki/Newtonian_telescope&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newtonian telescope&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-08-07T04:41:34.243" LastActivityDate="2021-08-07T04:41:34.243" Title="Where did Wm. Herschel plan to put their head? Peer over the edge of the famous 40-foot telescope or try to get less off-axis; closer to the center?" Tags="|astronomy|optics|telescope|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13197" PostTypeId="1" CreationDate="2021-05-18T08:06:30.263" Score="7" ViewCount="316" Body="&lt;p&gt;The standard use of the minus sign is such that &lt;span class=&quot;math-container&quot;&gt;$A-B$&lt;/span&gt; means you subtract B from A. Thus &lt;span class=&quot;math-container&quot;&gt;$$5-2 = 3.$$&lt;/span&gt; Has any culture used a symbol (let's call it &lt;span class=&quot;math-container&quot;&gt;$\oplus$&lt;/span&gt;) where &lt;span class=&quot;math-container&quot;&gt;$A \oplus B$&lt;/span&gt; means you subtract A from B? Thus you would get &lt;span class=&quot;math-container&quot;&gt;$$2\oplus5=3.$$&lt;/span&gt;&#xA;Note that in languages written from right to left, this question has to run in that direction. What I am interested in is the case when reading symbols written in the standard direction (whichever direction that is) you would say &amp;quot;A [symbol] B&amp;quot; and get the answer &lt;span class=&quot;math-container&quot;&gt;$B-A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That 3 is said to be equal to &lt;span class=&quot;math-container&quot;&gt;$$5 \;[\text{symbol for subtraction}] \;2$$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$$2 \;[\text{symbol for subtraction}] \;5,$$&lt;/span&gt; and more generally we write &lt;span class=&quot;math-container&quot;&gt;$$[\text{minuend}]\;[\text{symbol for subtraction}]\;[\text{subtrahend}], $$&lt;/span&gt; is of course convention. I am interested in whether any cultures have used a symbol for &amp;quot;reverse subtraction&amp;quot;; and if so, how widespread its use has been, both in absolute terms and relative to the use of a minus symbol with the usual meaning. Perhaps there have been cases where the &amp;quot;reverse&amp;quot; symbol has been the standard one, or the only one; or perhaps two symbols have been in more or less equally widespread use, one for one meaning and one for the other?&lt;/p&gt;&#xA;" OwnerDisplayName="user13571" LastEditorDisplayName="user13571" LastEditDate="2021-05-18T14:42:38.970" LastActivityDate="2021-05-25T11:06:49.573" Title="Reverse subtraction: has any culture had a symbol (call it $\oplus$) where $A \oplus B$ (read in the same direction as in the language) $:= B - A$?" Tags="|mathematics|notation|anthropology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13198" PostTypeId="2" ParentId="13197" CreationDate="2021-05-18T12:01:05.843" Score="3" Body="&lt;p&gt;It may remain a mystery in some cases.  &lt;a href=&quot;https://www.britannica.com/science/mathematics/Mathematics-in-ancient-Egypt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Britannica&lt;/a&gt; states that the ancient Egyptians wrote right-to-left but also states that their algorithm for subtraction is not known.&lt;br /&gt;&#xA;Then again, &lt;a href=&quot;https://www.math.wichita.edu/history/topics/arithmetic.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wichita&lt;/a&gt;   says (sadly, without a picture),&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In one example, from the Rhind Papyrus, addition and subtraction&#xA;signs were represented through figures which resemble the legs of a&#xA;person advancing for addition, and departing for subtraction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They did have a horizontal &amp;quot;equals&amp;quot; glyph.&lt;/p&gt;&#xA;" OwnerUserId="609" LastEditorDisplayName="user13571" LastEditDate="2021-05-23T06:39:10.353" LastActivityDate="2021-05-23T06:39:10.353" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13199" PostTypeId="5" CreationDate="2021-05-18T14:58:25.313" Score="0" Body="&lt;p&gt;For questions about the history of science or mathematics that concern anthropology, defined as the study of culture, itself defined as the beliefs (including morality), actions (including customs and the making of rules) and abilities (including the use of language and symbols) that people acquire or carry out as a result of belonging to the society they live in.&lt;/p&gt;&#xA;" OwnerDisplayName="user13571" LastEditorDisplayName="user13571" LastEditDate="2021-05-19T07:33:32.950" LastActivityDate="2021-05-19T07:33:32.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13200" PostTypeId="4" CreationDate="2021-05-18T14:58:25.313" Score="0" Body="For questions about the history of science or mathematics that concern anthropology." OwnerDisplayName="user13571" LastEditorDisplayName="user13571" LastEditDate="2021-09-06T17:34:53.257" LastActivityDate="2021-09-06T17:34:53.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13201" PostTypeId="1" CreationDate="2021-05-18T15:04:32.817" Score="3" ViewCount="137" Body="&lt;p&gt;While it is clear that there are several conventions for Fourier transforms of intragrable functions on &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, I don't think I have ever seen anything different from the three following conventions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\hat{f}(\xi)    =                        \int f(x) e^{-i 2 \pi \xi x} dx \\&#xA;\hat{f}(\omega) =                        \int f(x) e^{-i \omega x} dx    \\&#xA;\hat{f}(\omega) = \frac{1}{\sqrt{2 \pi}} \int f(x) e^{-i \omega x} dx    \\&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, it seems to me (please correct me if I am wrong) that the characteristic function of a random variable / probability measure is &lt;em&gt;always&lt;/em&gt; (or very close to &lt;em&gt;always&lt;/em&gt;) defined as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\hat{\mu}(\omega) = \int e^{i \omega x} d \mu(x)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;All this seems to be confirmed by Wikipedia:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fourier_transform#Other_conventions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Fourier_transform#Other_conventions&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So the main difference between analysis / signal processing / electrical engineering on one side and probability on the other is mostly the mius sign.&lt;/p&gt;&#xA;&lt;p&gt;Even though I do not think there is anything deep underneath this, i.e. just another convention (please confirm), I would be interested in understanding how we ended up with this confusion from a historical perspective.&lt;/p&gt;&#xA;&lt;p&gt;Maybe there is a reason why the characteristic function was defined without a minus and nobody ever questioned that? Weren't the people who introduced the characteristic function in probability (who was that btw?) looking at analysis books/papers?&lt;/p&gt;&#xA;&lt;p&gt;Note that my question is admittedly similar to, e.g.,&#xA;&lt;a href=&quot;https://math.stackexchange.com/questions/2306738/fourier-transform-with-a-different-sign-convention&quot;&gt;https://math.stackexchange.com/questions/2306738/fourier-transform-with-a-different-sign-convention&lt;/a&gt;&#xA;but I am more insterested in the history.&lt;/p&gt;&#xA;" OwnerUserId="14563" LastEditorUserId="14563" LastEditDate="2021-05-20T20:29:02.677" LastActivityDate="2021-05-20T20:29:02.677" Title="Different conventions between Fourier Transform and Characteristic Function" Tags="|mathematics|probability|fourier|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13202" PostTypeId="2" ParentId="13201" CreationDate="2021-05-19T03:11:57.330" Score="2" Body="&lt;p&gt;There is a fourth convention&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\hat{f}(s)=\frac{1}{2\pi}\int e^{-its}f(t)dt,$$&lt;/span&gt;&#xA;which is used sometimes.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-05-19T15:02:21.450" LastActivityDate="2021-05-19T15:02:21.450" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13203" PostTypeId="1" AcceptedAnswerId="14145" CreationDate="2021-05-19T03:48:54.337" Score="4" ViewCount="285" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/a/2059/3869&quot;&gt;This answer to &lt;em&gt;When was the first true Gregorian telescope built?&lt;/em&gt;&lt;/a&gt; explains:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newtonian uses one concave and one flat mirror (or just one concave).&lt;/p&gt;&#xA;&lt;p&gt;Gregorian uses two concave mirrors, and&lt;/p&gt;&#xA;&lt;p&gt;Cassegrain uses one concave and one convex mirror.&lt;/p&gt;&#xA;&lt;p&gt;All these mirrors should be parabolic in an ideal situation. First Gregorian was made by Hooke in 1673.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;This was 5 years before John Hadley was born. Hadley made a parabolic mirror in 1721, for a Newtonian telescope btw.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There was certainly known that for Newtonian telescope optics, at some point when the f/no becomes small enough &lt;a href=&quot;https://en.wikipedia.org/wiki/Spherical_aberration&quot; rel=&quot;nofollow noreferrer&quot;&gt;spherical aberration&lt;/a&gt; will limit the resolution rather than diffraction, and that a more parabolic shape can avoid this particular problem.&lt;/p&gt;&#xA;&lt;p&gt;When grinding optical surfaces, near-spheres are the easiest, the tend to come naturally. There are ways to make them aspherical known to present-day amateur telescope makers.&lt;/p&gt;&#xA;&lt;p&gt;But I don't know anything about how Hadley's first parabolic mirror was made parabolic, nor how it was tested to verify its asphericity, nor what it was used for; a Newtonian telescope or a more complex system with two or more curved surfaces.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How did Hadley do it and &lt;em&gt;why exactly?&lt;/em&gt; How did Hadley make the first parabolic telescope mirror, verify its asphericity, and for what kind of telescope was this work done?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="10106" LastEditDate="2022-01-31T17:28:15.310" LastActivityDate="2022-01-31T22:06:02.397" Title="How was the first parabolic telescope mirror made, and how was it used, and for what kind of telescope was this work done?" Tags="|physics|astronomy|optics|telescope|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13204" PostTypeId="1" CreationDate="2021-05-19T07:58:49.260" Score="6" ViewCount="1873" Body="&lt;p&gt;I'm interested in the history of the concept of tidal locking but haven't been able to find any articles presenting a timeline of its development. I'm hoping to have a look at the first published paper on the subject, if possible.&lt;/p&gt;&#xA;&lt;p&gt;Any leads would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="14568" LastActivityDate="2021-05-19T16:10:53.130" Title="Who first proposed the theory of tidal locking?" Tags="|physics|gravity|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13205" PostTypeId="2" ParentId="13204" CreationDate="2021-05-19T12:51:05.830" Score="9" Body="&lt;p&gt;Here's an upper bound, so to speak, from a &lt;a href=&quot;https://arxiv.org/pdf/2011.10833.pdf&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This was first “predicted” by Immanuel Kant in 1754, when he proposed&#xA;that the gravitational force of the Moon would slow down Earth’s&#xA;rotation until the two objects would be tidally locked.22 While Kant’s&#xA;argument was merely verbal, it did provide a correct rationale.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which implies that tidal locking was well-known by his time.  That paper seems to state that Newton at least recognized frictional forces due to tides and the implications of this, as they reference his work&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;3 Sir Isaac Newton,Philosophiæ Naturalis Principia Mathematica (1687).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-05-19T12:51:05.830" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13206" PostTypeId="2" ParentId="13201" CreationDate="2021-05-19T16:10:18.757" Score="2" Body="&lt;p&gt;I agree: the main difference is the minus sign in the exponent.  The other differences are cosmetic.&lt;br /&gt;&#xA;For a probability measure, we normalize&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\hat{\mu}(\omega) = \int e^{i \omega x} d \mu(x)&#xA;$$&lt;/span&gt;&#xA;because &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; has total mass &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For a Fourier series we normalize&#xA;&lt;span class=&quot;math-container&quot;&gt;$$c_n = \frac{1}{L}\int_0^L e^{-i(2\pi/L)nx}\;dx$$&lt;/span&gt;&#xA;to make the exponentials into an orthonormal system.&lt;/p&gt;&#xA;&lt;p&gt;For the Fourier transform, we have neither of these reasons for the normalization.  Which leads to the various conventions.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-05-19T16:10:18.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13207" PostTypeId="2" ParentId="13204" CreationDate="2021-05-19T16:10:53.130" Score="11" Body="&lt;p&gt;The concept of tidal locking appeared, at least qualititively, in proposition 38 of book III of Newton's principia. Newton wrote there:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hence it is that the same face of the moon always respects the earth; nor can the body of the moon possibly rest in any other position, but would return always by a libratory motion to this situation; but those librations, however, must be exceedingly slow, because of the weakness of the forces which excite them; so that the face of the moon, which should be always obverted to the earth, may, for the reason assigned in Prop. XVII be turned towards the other focus of the moon's orbit, without being immediately drawn back, and converted again towards the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Proposition 38 is in the subsection entitled &amp;quot;forces to move the sea&amp;quot;; this fact shows Newton placed the &amp;quot;tidal locking&amp;quot; effect within the theory of tidal forces.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2021-05-19T16:10:53.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13208" PostTypeId="2" ParentId="13090" CreationDate="2021-05-19T23:54:52.923" Score="1" Body="&lt;p&gt;Mostly, it comes down to usefulness.&lt;/p&gt;&#xA;&lt;p&gt;Pick up any mathematics textbook more advanced than elementary arithmetic, and you're probably going to find summation in there somewhere (it's especially common in statistics).  Iterated multiplication is less common, having fewer real-world applications, but it's a natural counterpart to summation.&lt;/p&gt;&#xA;&lt;p&gt;Tetration?  It's rarely seen outside of number theory and pop-science articles.&lt;/p&gt;&#xA;&lt;p&gt;There's also the matter of age: sigma notation dates back to &lt;a href=&quot;https://en.wikipedia.org/wiki/Leonhard_Euler#Mathematical_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler&lt;/a&gt;, who first used it around 1755, and the concept itself is considerably older.  Pi notation is more elusive, but appears to date to the &lt;a href=&quot;https://hsm.stackexchange.com/a/7127/3186&quot;&gt;early-mid 1800s&lt;/a&gt;.  In contrast, tetration appears to date back no further than &lt;a href=&quot;https://en.wikipedia.org/wiki/Tetration#Notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;1901&lt;/a&gt;, with most work being post-1950.&lt;/p&gt;&#xA;" OwnerUserId="3186" LastActivityDate="2021-05-19T23:54:52.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13209" PostTypeId="1" CreationDate="2021-05-20T15:00:41.303" Score="2" ViewCount="78" Body="&lt;p&gt;In Simachew's &lt;a href=&quot;https://algant.eu/documents/theses/simachew.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;A Survey on Euclidean Number Fields&amp;quot;&lt;/a&gt;, he said that Gauss used the existence of a Euclidean algorithm in Gaussian integers to prove that it has unique factorization. Also, he said that Wantzel observed that norm-Euclideanity implies unique factorization in cyclotomic integers. And Kummer used this to prove some stuff about some cyclotomic integers, which then paved the way to the desire of determining Euclideanity of number fields. In Lemmermeyer's survey, &lt;a href=&quot;http://www.rzuser.uni-heidelberg.de/~hb3/publ/survey.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Euclidean Algorithm in Algebraic Number Fields&amp;quot;&lt;/a&gt;, he said that Dirichlet was &amp;quot;the first mathematician who emphasized that the existence of a Euclidean algorithm implied unique factorization&amp;quot; in 1847. However, Wantzel's observation was also around 1847.&lt;/p&gt;&#xA;&lt;p&gt;So my question is, in this case, was it Dirichlet who emphasized that it is true &amp;quot;in general&amp;quot;? Like not just norm-Euclideanity, and not just on a specific ring of integers?&lt;/p&gt;&#xA;" OwnerUserId="14578" LastActivityDate="2021-05-20T15:00:41.303" Title="Who first proved that the existence of a Euclidean algorithm implies unique factorization?" Tags="|mathematics|mathematicians|number-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13210" PostTypeId="2" ParentId="7957" CreationDate="2021-05-20T17:51:07.147" Score="0" Body="&lt;p&gt;For me, the &amp;quot;a-ha&amp;quot; moment with Fourier transforms was when I related it to resolving a vector into components.  If you have a vector in some arbitrary direction and for convenience of calculation or representation you want to express it as the sum of scalar multiples of another pair of vectors (let's call them the &amp;quot;basis pair&amp;quot;), you use the dot product of that vector with each of the basis vectors (divided by the basis vector's magnitude) to find something that could be called &amp;quot;the shadow of that vector projected along the line of the basis vector&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In a simple case, the basis pair could be the horizontal unit vector and the vertical unit vector--that's what you're most familiar with--but any two vectors that don't lie along the same line can be used, you just have to project the vector you're looking at along the line of each vector in turn and see what &lt;strong&gt;a&lt;/strong&gt; and &lt;strong&gt;b&lt;/strong&gt; you need to use as a multiple to get &lt;strong&gt;a&lt;/strong&gt; times {one vector} plus &lt;strong&gt;b&lt;/strong&gt; times {the other vector} to add up to your original vector.  Fourier transforms use a similar idea--&amp;quot;I've got this complicated wave form, let me see if I can break it up into a sum of multiples of simpler wave forms.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Think about the Fourier transform on a sound wave as an example. What you're basically doing is projecting the sound wave onto a pure 440Hz signal and saying &amp;quot;it's about this much middle A&amp;quot; and then projecting the wave onto a 466Hz signal says &amp;quot;and now about this much B flat&amp;quot;.  Continuing in this way you would have the tones on a piano as your &amp;quot;basis vectors&amp;quot; and you could reproduce the sound (approximately) by playing each key harder or softer according to what the calculation gave you at each frequency.&lt;/p&gt;&#xA;&lt;p&gt;To me, this takes some of the &amp;quot;well how would anyone ever come up with this?&amp;quot; mystery out of it.  The idea of representing something as &lt;strong&gt;a&lt;/strong&gt; times this basis vector plus &lt;strong&gt;b&lt;/strong&gt; times that basis vector is just being generalized to &amp;quot;I can represent this sound wave as &lt;strong&gt;a&lt;/strong&gt; times this pure sine wave plus &lt;strong&gt;b&lt;/strong&gt; times this pure sine wave, plus &lt;strong&gt;c&lt;/strong&gt; times this pure sine wave&amp;quot; and so on.&lt;/p&gt;&#xA;&lt;p&gt;I don't know if this is historically where the realization came from, but I think it answers your question in the sense of &amp;quot;how would I ever come up with this idea myself?&amp;quot;.  You come up with it by thinking of a curve as being the sum of components of an infinite-dimensional vector space.  Don't let the &amp;quot;infinite dimensions&amp;quot; mess with your head--in practice, you just get it as close as you need it to be, like with digital music you take the natural sound and sample it over enough different frequencies (basis vectors) that it sounds ok when you play it back as x times this frequency plus y times that frequency plus z times that frequency, etc.&lt;/p&gt;&#xA;&lt;p&gt;(This is how a Fourier transform on an input wave could be used to digitize music--you go from a natural wave to &amp;quot;play frequency x [a number, stored digitally] at volume y [a number, stored digitally] and, simultaneously, frequency z [a number, stored digitally] at volume v [a number, stored digitally]&amp;quot;.  You're just &amp;quot;resolving&amp;quot; the music into numerical volume multipliers of numerical frequencies.)&lt;/p&gt;&#xA;&lt;p&gt;You don't get an exact fit for every possible curve but by using more and more &amp;quot;basis vectors&amp;quot; you get closer and closer to the exact curve.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;piano keys as basis vectors&amp;quot; idea is not new to me--one of my physics professors showed us how you could yell into an open piano and the sympathetic vibration of the strings would make an approximate Fourier transform of your voice.  See also this experiment along those lines in a YouTube video (narration in German but the piano &amp;quot;talks&amp;quot; in English): &lt;a href=&quot;https://www.youtube.com/watch?v=muCPjK4nGY4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=muCPjK4nGY4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other concepts in math have a similar basis (pardon the pun).  Taylor series expansion and Bernoulli polynomials are similar things where you get close to some function or curve by approximating it with a bunch of scalar multipliers to something that approximates an infinite-dimensional vector space.  The process of figuring out what the scalar multipliers are for a given curve usually looks something like the Fourier transform.&lt;/p&gt;&#xA;" OwnerUserId="10560" LastActivityDate="2021-05-20T17:51:07.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13211" PostTypeId="1" CreationDate="2021-05-20T21:16:14.207" Score="2" ViewCount="83" Body="&lt;p&gt;When the Nobel prize or Fields Medal gets awarded, I believe one of the criteria is the impact the discoveries have been. My question is, whether there are any objective ways to measure the impact? It could be the impact on science or on the society as a whole. I guess the number of citations or which journals have published an article is one way, but are there other ways? Can the objective ways be applied to compare between different fields of science? Are there any such comparisons where I could get a list of the most significant scientific discoveries ordered by impact?&lt;/p&gt;&#xA;&lt;p&gt;Here is a related question:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/7626/quantitative-measures-of-rise-and-decline-of-scientific-fields-over-time-beyond&quot;&gt;Quantitative measures of rise and decline of scientific fields over time (beyond number of publications)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14582" LastActivityDate="2021-05-20T21:16:14.207" Title="Is there an objective way to measure how significant a discovery has been?" Tags="|philosophy-of-science|academia|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13212" PostTypeId="1" CreationDate="2021-05-22T19:41:50.853" Score="4" ViewCount="294" Body="&lt;p&gt;The Clebsch-Gordan coefficients &lt;span class=&quot;math-container&quot;&gt;$C_{\pm }(J,M)$&lt;/span&gt; arise in quantum mechanics in the problem of addition of angular momentum. They also arise in mathematics in the more theoretical (but related) problem of the decomposition of the tensor product of irreducible representations of a group into irreducible representations. Originally, Clebsch and Gordan discovered the coefficients while working on a problem in invariant theory (which in modern language is equivalent to the decomposition problem of representations mentioned above).&lt;/p&gt;&#xA;&lt;p&gt;It is well known, as can be found in any quantum mechanics textbook, that the Clebsch-Gordan coefficients satisfy the recurrence relation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$C_{\pm }(J,M)\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,(M\pm 1)\rangle =C_{\pm }(j_{1},m_{1}\mp 1)\langle j_{1}\,(m_{1}\mp 1)\,j_{2}\,m_{2}|J\,M\rangle +C_{\pm }(j_{2},m_{2}\mp 1)\langle j_{1}\,m_{1}\,j_{2}\,(m_{2}\mp 1)|J\,M\rangle$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This relation (when supplemented by a rather trivial initial condition) allows us to calculate all of the Clebsch-Gordan coefficients, a problem which is very difficult to solve without this relation.&lt;/p&gt;&#xA;&lt;p&gt;I'm interested in who first discovered this recurrence relation, and what line of thought led them to it (was it considerations in the representation theory of Lie algebras?) Wikipedia says it was first discovered by Giulio Racah in 1941, but this is clearly not true, as the relation appears explicitly in Condon and Shortley's classic &amp;quot;The Theory of Atomic Spectra&amp;quot; from 1935 (in chapter 3, section 14). In fact Racah himself cites Condon and Shortley for proof of the relation, in his 1942 paper &amp;quot;Theory of Complex Spectra II&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, Condon and Shortley in their preface explain that they deliberately hide the group-theoretic arguments and methods that underlie their treatment, in an attempt to make the book more readable to the physicist, for whom Lie group theory was an obscure part of mathematics at the time. For this reason I'm interested in where the relation first appeared? I could not find an earlier reference to them, however I did find a similar recurrence relation for the related Lande-g factor in Weyl's 1929 &amp;quot;The Theory of Group and Quantum Mechanics&amp;quot; (in chapter IV, section A.4).&lt;/p&gt;&#xA;" OwnerUserId="11209" LastEditorUserId="14548" LastEditDate="2021-09-26T13:30:13.837" LastActivityDate="2022-02-23T14:09:20.950" Title="Origin of the recurrence relation for Clebsch-Gordan coefficients" Tags="|quantum-mechanics|theoretical-physics|group-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13213" PostTypeId="1" AcceptedAnswerId="13215" CreationDate="2021-05-23T17:57:11.533" Score="-1" ViewCount="107" Body="&lt;p&gt;Given that Euclid's work is titled &amp;quot;The Elements&amp;quot;, it is safe to imply that it is a compendium of elementary results of Ancient Greek Geometry. Other works, such as those of Apollonius, Archimedes and others of Euclid's can be considered the 'achievements' of the Ancient Greek Geometry.&lt;/p&gt;&#xA;&lt;p&gt;Is there any knowledge of extant, or evidence of, other ancient Greek books for introducing mathematics?&lt;/p&gt;&#xA;" OwnerUserId="14592" LastActivityDate="2021-05-23T23:43:05.330" Title="Do we know of any ancient 'Introduction to Mathematics' in Ancient Greece, besides &quot;The Elements&quot;?" Tags="|mathematics|ancient-greece|" AnswerCount="2" CommentCount="1" ClosedDate="2021-05-23T22:29:49.587" ContentLicense="CC BY-SA 4.0" />
  <row Id="13215" PostTypeId="2" ParentId="13213" CreationDate="2021-05-23T18:44:34.787" Score="0" Body="&lt;p&gt;Quoting from &lt;em&gt;A History of Mathematics&lt;/em&gt; (3&lt;sup&gt;rd&lt;/sup&gt; edition), by Victor J. Katz:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although there were earlier versions of &lt;em&gt;Elements&lt;/em&gt; before that of Euclid, his is the only one to survive, perhaps because it was the first one written after both the foundations of proportion theory and the theory of irrationals had been developed and the careful distinctions always to be made between number and magnitude had been proponded by Aristotle. It was therefore both “complete” and well organized. Since the mathematical community as a whole was of limited size, once Euclid's work was recognized for its general excellence, there was no reason to keep another inferior work in circulation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2021-05-23T18:44:34.787" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13216" PostTypeId="2" ParentId="13213" CreationDate="2021-05-23T20:05:40.000" Score="0" Body="&lt;p&gt;Diophantus wrote one of the first number theory books &amp;quot;Arithmetica&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14594" LastEditorUserId="14594" LastEditDate="2021-05-23T23:43:05.330" LastActivityDate="2021-05-23T23:43:05.330" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13217" PostTypeId="1" AcceptedAnswerId="13218" CreationDate="2021-05-24T03:51:39.633" Score="1" ViewCount="77" Body="&lt;p&gt;The Langevin function &lt;span class=&quot;math-container&quot;&gt;$\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$&lt;/span&gt; is named after the French physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Langevin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Langevin&lt;/a&gt; (1872-1946).&#xA;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; on the Brillouin and Langevin functions does not cite a publication by Langevin and other sources I consulted merely indicate that it originates in Langevin's work on paramagnetism.&lt;/p&gt;&#xA;&lt;p&gt;In which publication by Langevin did this function make its first appearance?&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2021-07-07T23:30:55.933" Title="First occurrence of the Langevin function in scientific literature" Tags="|physics|reference-request|magnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13218" PostTypeId="2" ParentId="13217" CreationDate="2021-05-24T03:51:39.633" Score="2" Body="&lt;p&gt;Best I could determine after searching for a couple of hours, this function was first introduced in the following publication, a &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k34935p/f68.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt; of which is made available by the Bibliothèque nationale de France:&lt;/p&gt;&#xA;&lt;p&gt;P. Langevin, &amp;quot;Magnétisme et théorie des électrons,&amp;quot; &lt;em&gt;Annales de chimie et de physique&lt;/em&gt;, 8:5 (1905), 70-127.&lt;/p&gt;&#xA;&lt;p&gt;On &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k34935p/f115.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 117&lt;/a&gt; one finds:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Le moment magnétique total des &lt;span class=&quot;math-container&quot;&gt;$\mathrm{N}$&lt;/span&gt; molécules est évidemment&#xA;dirigé parallèlement au champ et est égal à la somme des projections&#xA;sur cette direction des moments composants. Pour l'unité de volume&#xA;supposée contenir &lt;span class=&quot;math-container&quot;&gt;$\mathrm{N}$&lt;/span&gt; molécules, ce moment résultant&#xA;représente l'intensité d'aimantation &lt;span class=&quot;math-container&quot;&gt;$\mathrm{I}$&lt;/span&gt; :&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathrm{I}=\int{\mathrm{M}\cos \alpha}{\ dn} = \int_{-1}^{+1}{2\pi\mathrm{MK\ }x\ e^{ax}\ {dx}}.$$&lt;/span&gt;&#xA;Or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int_{-1}^{+1}{x\ }{e^{ax}}{\ dx} = 2 \left(\frac{\mathrm{Ch\ } {a}}{a}-\frac{\mathrm{Sh\ }a}{a^{2}}\right),$$&lt;/span&gt;&#xA;d'où &lt;span class=&quot;math-container&quot;&gt;$$\mathrm{I}=\mathrm{MN}\left(\frac{\mathrm{Ch\ }a}{\mathrm{Sh\&#xA; }{a}}-\frac{1}{a}\right).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Ch}$&lt;/span&gt; represents the hyperbolic cosine and &lt;span class=&quot;math-container&quot;&gt;$\mathrm{Sh}$&lt;/span&gt; the hyperbolic sine, thus &lt;span class=&quot;math-container&quot;&gt;$\mathrm{I}=\mathrm {MN}\left(\mathrm{coth}(a) - \frac{1}{a}\right) = \mathrm{MN}\mathcal{L}(a).$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-07-07T23:30:55.933" LastActivityDate="2021-07-07T23:30:55.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13219" PostTypeId="1" CreationDate="2021-05-24T08:34:25.073" Score="0" ViewCount="107" Body="&lt;p&gt;I recently found out that a lot of Plato's work can be drawn geometrically. See the Cerritos College YouTube video &lt;a href=&quot;https://www.youtube.com/watch?v=TKPCzRdzsWA&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Platos Divided Line&amp;quot;&lt;/a&gt; with the description&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cerritos College Professor Rodney Swearengen talks about &amp;quot;The Mundane Geometry that Constitutes the Analogy of Plato's Divided Line and The Relevance of The Socratic Philosophy of Science Embodied Therein&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are there any books, papers detailing this kind of work that you know of? I don't mind older sources.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="14598" LastEditorUserId="131" LastEditDate="2021-05-24T11:43:04.430" LastActivityDate="2021-05-24T11:43:04.430" Title="Platonian geometry illustrated" Tags="|reference-request|geometry|euclidean-geometry|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13220" PostTypeId="2" ParentId="606" CreationDate="2021-05-25T00:51:22.913" Score="3" Body="&lt;p&gt;I've come to the conclusion that the Wikipedia article is simply wrong, and I've completely rewritten it (&lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Principle_of_permanence&amp;amp;oldid=937725516&quot; rel=&quot;nofollow noreferrer&quot;&gt;permalink to original&lt;/a&gt;). It described the &lt;a href=&quot;https://en.wikipedia.org/wiki/Identity_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Identity Theorem&lt;/a&gt;, not any kind of &amp;quot;Principle of Permanence&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia article was clearly based on &lt;a href=&quot;https://mathworld.wolfram.com/PrincipleofPermanence.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this MathWorld article&lt;/a&gt;; the Wikipedia article's other citations were either about Peacock's principle of permanence, or were about mathematical facts that were unrelated to whether the described facts should be called &amp;quot;the identity theorem&amp;quot; or &amp;quot;the principle of permanence&amp;quot;. The MathWorld article was not written by a regular contributor to MathWorld; &amp;quot;Principle of Permanence&amp;quot; appears to be &lt;a href=&quot;https://mathworld.wolfram.com/topics/Asimov.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;the only MathWorld article that person has written&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The MathWorld article has only one citation, a footnote from Stephen Wolfram's &lt;em&gt;A New Kind of Science&lt;/em&gt; that says nothing like what the MathWorld article says, and instead supports the book you quoted (&lt;em&gt;Beginning Algebra for College Students&lt;/em&gt;), &lt;a href=&quot;https://hsm.stackexchange.com/a/3303/14472&quot;&gt;the extensively-sourced top answer to this question&lt;/a&gt;, and &lt;a href=&quot;https://math.stackexchange.com/a/751658/924867&quot;&gt;the approachable and well-written top answer to your earlier question&lt;/a&gt;. To quote &lt;a href=&quot;https://www.wolframscience.com/nks/notes-12-9--generalization-in-mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the footnote&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Systems that have evolved from the basic notion of numbers provide a characteristic example of the process of progressive generalization in mathematics. The main such systems and their dates of earliest known reasonably formalized use have been [...]: positive integers (before 10,000 BC), rationals (3000 BC), square roots (2000 BC), other roots (1800 BC), all integers (600 AD, 1600s), [...]. New systems have usually been introduced in connection with extending the domains of particular existing operations. But in almost all cases the systems are set up so as to preserve as many theorems as possible—a notion that was for example made explicit in the Principle of Permanence discussed by George Peacock in 1830 and extended by Hermann Hankel in 1869.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Wolfram is clearly describing the same Principle of Permanence described by &lt;em&gt;Beginning Algebra for College Students&lt;/em&gt; and the top answers to both this question and your earlier question. What the Wikipedia and MathWorld articles described is the Identity Theorem, which is unrelated to the Principle of Permanence.&lt;/p&gt;&#xA;" OwnerUserId="14472" LastEditorUserId="14472" LastEditDate="2021-05-25T19:04:09.010" LastActivityDate="2021-05-25T19:04:09.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13221" PostTypeId="2" ParentId="13197" CreationDate="2021-05-25T07:47:13.663" Score="1" Body="&lt;p&gt;The closest thing I can think of are symbols that mean &amp;quot;(the absolute value of) the difference between&amp;quot;, a sort of &amp;quot;commutative subtraction&amp;quot; operator:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$4 \sim 5 = 5 \sim 4 = 1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have access to a copy of &lt;em&gt;Webster's New International Dictionary, Second Edition&lt;/em&gt; with a (publication?) date of 1949 on the title page, copyright date of 1934, and new words section copyright 1939 and 1945. The appendix &amp;quot;Arbitrary Signs and Symbols&amp;quot;, section &amp;quot;Mathematics&amp;quot;, sub-section &amp;quot;Relations and Operations&amp;quot; on page 3005 contains the following entries:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; &lt;em&gt;or&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathrel{-}:$&lt;/span&gt;   Difference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; ~ &lt;em&gt;or&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\bumpeq$&lt;/span&gt;   The difference between ; used to indicate the difference between two quantities without designating which is the greater ; as, &lt;span class=&quot;math-container&quot;&gt;$a \sim b$&lt;/span&gt; ; that is, the difference between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\mathrel{-}:$&lt;/span&gt;   The difference between; excess.   &lt;em&gt;Rare.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nGw1g.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nGw1g.png&quot; alt=&quot;difference symbol entries&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the Unicode &lt;a href=&quot;https://www.unicode.org/charts/PDF/U2200.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematical Operators&lt;/a&gt; block, these symbols are called&lt;/p&gt;&#xA;&lt;p&gt;∼ U+223C TILDE OPERATOR (&amp;quot;= difference between&amp;quot;)&lt;br /&gt;&#xA;≏ U+224F DIFFERENCE BETWEEN&lt;br /&gt;&#xA;∹ U+2239 EXCESS&lt;/p&gt;&#xA;" OwnerUserId="14604" LastEditorUserId="14604" LastEditDate="2021-05-25T11:06:49.573" LastActivityDate="2021-05-25T11:06:49.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13222" PostTypeId="1" AcceptedAnswerId="13224" CreationDate="2021-05-25T16:56:36.923" Score="2" ViewCount="178" Body="&lt;p&gt;If one studies the paper written by Banach in which he first proved his fixed point theorem one would find that he did not prove the theorem for abstract complete metric spaces. He proved it for the special case of complete normed vector spaces.&lt;/p&gt;&#xA;&lt;p&gt;My question is - who extended his result to complete metric spaces?&lt;/p&gt;&#xA;" OwnerUserId="14609" LastEditorUserId="14609" LastEditDate="2021-05-27T04:57:19.317" LastActivityDate="2021-05-27T04:57:19.317" Title="Who proved Banach fixed point theorem in abstract metric spaces for the first time?" Tags="|topology|functional-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13223" PostTypeId="2" ParentId="606" CreationDate="2021-05-25T19:38:41.373" Score="2" Body="&lt;p&gt;Yes, there are at least four somewhat-different things that have either been called a &amp;quot;principle of permanence&amp;quot; historically, or might be called so currently.&lt;/p&gt;&#xA;&lt;p&gt;The easiest to explain (and legitimize) is in complex analysis, more often nowadays called the Identity Principle. It had also been called &amp;quot;The Principle of Permanence of Analytic Relations&amp;quot; (though, sadly, I am not able to easily find a source for the latter). The literal assertion is that two holomorphic (=complex-analytic=complex-differentiable) functions on a connected, non-empty set which agree on a subset with an accumulation point in the set must agree everywhere. In practice, this can be used to show that a formula that holds for real values of a parameter holds for complex values... where the formula for real values can be proven by changes of variables, etc., that would be harder to justify with complex values.&lt;/p&gt;&#xA;&lt;p&gt;There certainly is Robinson's &amp;quot;transfer principle&amp;quot; in non-standard analysis, as mentioned already by @Mikhail Katz. Depending on one's tastes, this might be construed as going in the opposite direction... namely, very roughly, that if something holds in the large world that includes (legitimate) infinitesimals and such, and the assertion does not directly mention them, then the assertion holds for &lt;em&gt;standard&lt;/em&gt; real numbers. EDIT: for example, &lt;em&gt;existence&lt;/em&gt;... is interesting, in this context.&lt;/p&gt;&#xA;&lt;p&gt;Also using some model theory and other things, the theory of motivic integration and transfer ideas can move results in the function field case to the number field case (culminating in Ngo's work on the trace formula, etc.) I am not competent to explain why this works. :) See papers of Francois Loeser, for example, for background.&lt;/p&gt;&#xA;&lt;p&gt;The older &amp;quot;purely algebraic&amp;quot; principle going back to Peacock or earlier has an easy modern interpretation in terms of Zariski closures, or universal mapping properties. The most trivial case is that, if an algebraic identity can be proven &amp;quot;for integers&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$x_1,\ldots,x_n$&lt;/span&gt; in a fashion that does not use anything other than associativity, commutativity, and distributivity of the operations, then it is true in any commutative ring, EDIT: because the proof actually gives the result for the polynomial ring &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z[x_1,\ldots,x_n]$&lt;/span&gt;. By its universal property, it maps forward to &lt;em&gt;any&lt;/em&gt; commutative ring generated by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; things. This was simply not possible to say directly in Peacock's day. Subtler versions, e.g., that sufficiently many integer points determine a polynomial, or such stuff, verge into Zariski-closure issues. Although I know nothing about it, I would imagine that &amp;quot;universal algebra&amp;quot; (in the modern sense) could formalize broad versions of this.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastEditorUserId="1932" LastEditDate="2021-05-25T20:30:12.213" LastActivityDate="2021-05-25T20:30:12.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13224" PostTypeId="2" ParentId="13222" CreationDate="2021-05-26T06:26:14.097" Score="8" Body="&lt;p&gt;Pages 97-107 of the book &lt;em&gt;Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles&lt;/em&gt; is an article by Guerraggio about Caccioppoli. On p. 100, Guerraggio says the contraction mapping theorem for complete metric spaces is due to Caccioppoli, who is said to have been almost alone in publishing in Banach-style functional analysis in the 1930s. Specifically, Guerraggio says that in Caccioppoli's short paper &amp;quot;Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale&amp;quot;, Rend. Accad. Naz. Lincei, 11 (1930), 794–799 the fixed point theorem for contractions on complete metric spaces appears and all that is written there for a proof is the statement that the proof is obvious. Locate a copy of this paper to see if it answers your question.&lt;/p&gt;&#xA;&lt;p&gt;On p. 176 of &lt;em&gt;Italian Mathematics Between the Two World Wars&lt;/em&gt; by  Guerraggio and Nastasi, Caccioppoli's formulation of the fixed point theorem on complete metric spaces appears in Italian near the bottom of the page and a translation into English is in a footnote on that page.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2021-05-26T14:04:01.000" LastActivityDate="2021-05-26T14:04:01.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13226" PostTypeId="1" CreationDate="2021-05-26T13:00:27.953" Score="6" ViewCount="377" Body="&lt;p&gt;Back to 1868, Mendeleev's periodic table has not been published yet, but we are quite there.&#xA;As a scientist, you're still struggling to identify very clearly these elements with limited means.&#xA;Especially, you feel and observe that there are some kind of cycles, and would like to classify elements highlighting these suspected periods.&lt;/p&gt;&#xA;&lt;p&gt;Then &lt;a href=&quot;https://en.wikipedia.org/wiki/Lothar_Meyer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Meyer&lt;/a&gt;, still in 1868 apparently, got the idea to plot atomic volumes of known elements, depending on their relative atomic masses.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IfyUd.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IfyUd.jpg&quot; alt=&quot;Meyer&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, relative atomic masses were known (more or less since Dalton), and improved by Stanislao Cannizzaro just before Meyer used them.&lt;/p&gt;&#xA;&lt;p&gt;But, &lt;strong&gt;how did they get the atomic volumes of these elements then&lt;/strong&gt; ?&#xA;I suspect something like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In a solid state, if they already knew that atoms were touching each other, we can get the density of elements using for example a standard mold of 1cm3, and then I don't know!&lt;/li&gt;&#xA;&lt;li&gt;In a gaseous state, some computing with the Avogadro constant which was already known, but same, I don't know.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Since some elements are usually gas, some are usually solid, I guess we need to normalize the data so that they are actual atomic volumes and not raw densities.&#xA;If you can take one or two concrete examples to explain, it'll be appreciated!&lt;/p&gt;&#xA;" OwnerUserId="14617" OwnerDisplayName="user1556814" LastEditorUserId="4251" LastEditDate="2021-09-09T06:12:36.967" LastActivityDate="2021-09-09T06:12:36.967" Title="How did Meyer know atomic volumes to plot them just before Mendeleev's periodic table?" Tags="|chemistry|19th-century|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13227" PostTypeId="2" ParentId="13226" CreationDate="2021-05-27T04:55:03.347" Score="6" Body="&lt;p&gt;Good question. I had to look up his original paper which is freely available on Google Books and Hathi Trust. The paper is &amp;quot;Die Natur der chemischen Elemente als Function ihrer Atomgewichte; von Lothar Meyer.&amp;quot; (The nature of chemical elements as a function of their atomic weights)&lt;/p&gt;&#xA;&lt;p&gt;So you are right atomic weights were well known then. So what is atomic volume? It turns out to be relatively simple. You know very well&lt;/p&gt;&#xA;&lt;p&gt;Density= Mass / volume;&lt;/p&gt;&#xA;&lt;p&gt;and then&lt;/p&gt;&#xA;&lt;p&gt;Atomic volume= Atomic weight/ Density of the (solid) element.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Eine der Eigenschaften, welche mit dem Atomgewicht ziemlich regelmäßig sich ändert, ist die Raumerfüllung der Elemente, das Atomvolumen. Tafel III. gibt eine graphische Darstellung seiner Änderungen in ihrer Abhängigkeit von den Änderungen der Atomgewichte. Als Abscissen einer Kurve sind den Atomgewichten proportionale Längen, als Ordinaten solche, welche den zugehörigen Atomvoluminibus der Elemente im festen Zustande (nur für das Chlor im flüssigen), also den Quotienten aus Atomgewicht und Dichte proportional sind, aufgetragen. Als Einheiten sind das Atomgewicht des Wasserstoffs und die Dichte des Wassers genommen.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;One of the properties which changes rather regularly with the atomic weight is the space filling of the elements, the atomic volume. Table III gives a graphical representation of its changes in their dependence on the changes of the atomic weights. As abscissas of a curve are plotted heights proportional to the atomic weights, as ordinates those which are proportional to the corresponding atomic volumes of the elements in the solid state (only for chlorine in the liquid state), i.e. the quotients of atomic weight and density. The atomic weight of hydrogen and the density of water are taken as units.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translated with &lt;a href=&quot;http://www.DeepL.com/Translator&quot; rel=&quot;noreferrer&quot;&gt;www.DeepL.com/Translator&lt;/a&gt; with some manual changes.&lt;/p&gt;&#xA;&lt;p&gt;So all the comparison is being made, it is relative to hydrogen.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-05-27T04:55:03.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13228" PostTypeId="1" CreationDate="2021-05-27T12:36:19.903" Score="2" ViewCount="729" Body="&lt;p&gt;The Indian word for what we call sine was 'jyā'.&#xA;The Surya-siddhanta mentions 'jyā' many many times.&#xA;As does the Aryabhatiya.&#xA;In contrast they mention the synonym 'jivā' only once or twice.&#xA;So why was 'jiva' transliterated into Arabic rather than 'jyā'?&#xA;It makes no sense; 'Jyā' is written all over these texts and 'jivā' is not.&#xA;Did later Indian mathematicians perhaps use 'jivā' more often?&#xA;Was it one particular Arabic translator who translated a particular Indian text which happened to use 'jivā' rather than 'jyā'?&lt;/p&gt;&#xA;" OwnerUserId="14625" LastActivityDate="2021-05-29T03:57:46.643" Title="Etymology of 'sine'. The Arabic 'jayb' and 'jiba' derive from the Sanskrit 'jivā' ... but why not from the more common Sanskrit 'jyā'?" Tags="|trigonometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13229" PostTypeId="2" ParentId="13228" CreationDate="2021-05-27T15:28:33.147" Score="4" Body="&lt;p&gt;The unabridged Oxford English Dictionary, which is quite authentic does not trace sine to any Sanskrit words&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Etymology: &amp;lt; Latin sinus a bend, bay, etc.; also, the hanging fold of&#xA;the upper part of a toga, the bosom of a garment, and hence used to&#xA;render the synonymous Arabic jaib , applied in geometry as in sense 2.&#xA;Compare French sinus, Spanish seno, Italian seno.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;However, a book by A. Bello, Origins of mathematical words: a comprehensive dictionary of Latin, Greek, and Arabic roots states the following&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vQ5Cr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vQ5Cr.png&quot; alt=&quot;Book image&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Regardless the sad thing nobody has ever produced even images of original documents of Sanskrit, or even the authors. So take everything as a grain of salt especially the &lt;em&gt;stories&lt;/em&gt; on Wikipedia. I am curious to see any original source and the name of the Sanskrit speaking author who first used those words.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2021-05-29T03:57:46.643" LastActivityDate="2021-05-29T03:57:46.643" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13231" PostTypeId="2" ParentId="180" CreationDate="2021-05-28T08:37:08.280" Score="0" Body="&lt;p&gt;Even today, I suspect making a space mirror capable of reflecting sunlight accurately onto a region of Earth as a weapon or for agricultural would be extremely difficult. A group of astronauts working together to construct a large mirror I think is unprecedented; if we tried it today, it would be done robotically which would require AI that only in the past 20 years or less has become available and it would still be a hard problem.&lt;/p&gt;&#xA;&lt;p&gt;Before 1945, the multiple manned space craft coordinating efforts to assemble the mirror were easily half a century away. I don't think any kind of space-based weapon even today would be very effective against Earth, and if we wanted one, it would likely be just a missile base in space or a killer satellite. One thing about a space mirror -- impossible to hide and easily destroyed by a single projectile perhaps fired from the Earth's surface. Also: I do not think &amp;quot;space lasers&amp;quot; will ever work due to beam spreading and power loss due to atmospheric absorption -- I do not think we have lasers with anywhere near that kind of range. (In 2021, a crazed congress member will suggest that space lasers are deployed and used to set wildfires -- but lasers as weapons are only effective at ranges of a mile or less.)&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2021-05-28T08:37:08.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13232" PostTypeId="1" AcceptedAnswerId="13263" CreationDate="2021-05-29T21:16:56.803" Score="4" ViewCount="318" Body="&lt;p&gt;I'm interested in reading more about how Newton developed his method of interpolation and also the proofs he developed to this topic.  I'm currently reading &amp;quot;Analysis by its history&amp;quot; which gives a good first insight and mentions &amp;quot;The Mathematical Papers of Isaac Newton - volume 8&amp;quot; as a source, this seems, however, not be available as a pdf or ebook and only as a paperback version. Can anyone recommend me some resources where I can read more about this topic which are eventually accessible for free?&lt;/p&gt;&#xA;" OwnerUserId="14637" LastActivityDate="2021-07-09T10:02:30.287" Title="History of interpolation methods - Newton" Tags="|calculus|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13233" PostTypeId="1" AcceptedAnswerId="13235" CreationDate="2021-05-30T18:03:40.023" Score="16" ViewCount="3311" Body="&lt;p&gt;I am currently reading Oliver Bryne's edition of Euclid's &lt;em&gt;Elements&lt;/em&gt;, and in &lt;em&gt;The Elements&lt;/em&gt; many arithmetic propositions are proved geometrically, and it feels to me that numbers are always treated as something that is tied to a geometrical object (be it the length of a segment, the area of a shape, or the angle between two lines) and never as entities independent of a geometrical realisation.&#xA;For example, Euclid never seems to bring up the product of more than three quantities, which would indicate that his idea of the product was rooted in its geometric interpretation (as well as that he was probably limiting himself to only three dimensions).&lt;/p&gt;&#xA;&lt;p&gt;Euclid also categorises quantities as being of one of four &amp;quot;kinds&amp;quot; : &lt;em&gt;linear&lt;/em&gt;, &lt;em&gt;superficial&lt;/em&gt;, &lt;em&gt;solid&lt;/em&gt;, and &lt;em&gt;angular&lt;/em&gt;, and he never adds two quantities of different kinds together, indicating that he really thought of them as distinct in nature. To me, this is very reminiscent of dimensional analysis, but it leads me to wonder if Euclid thought that quantities consisted of a number and a kind (or a number and a unit, in modern terms), or if the number itself was of a certain kind. In other words, if presented with a segment of length &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and a square of area &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, would both &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;s have been considered the same or different?&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, I know that Pythagorean scholars represented integers as dots arranged in triangles, rectangles, and other shapes, which allowed to think about them in simple geometrical ways.&lt;/p&gt;&#xA;&lt;p&gt;This leads me to ask myself if Greek mathematicians of the Antiquity ever considered numbers as independent mathematical objects, which were not necessarily geometrical quantities. The use of dots by Pythagoreans makes me think that it could be the case, but still they did arrange them in geometrical shapes, so I am still unclear.&lt;/p&gt;&#xA;&lt;p&gt;(I am not just asking if Euclid or Pythagoreans ever did consider numbers as such, but if there was a point in the vast period that is Greek Antiquity where numbers were considered to be separable from geometry.)&lt;/p&gt;&#xA;" OwnerUserId="14641" LastEditorUserId="12724" LastEditDate="2021-06-01T06:28:05.923" LastActivityDate="2023-07-15T22:58:27.797" Title="Did ancient Greek mathematicians consider numbers independently of geometry?" Tags="|philosophy-of-science|geometry|number-theory|euclidean-geometry|euclid|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13234" PostTypeId="2" ParentId="13233" CreationDate="2021-05-30T23:37:57.560" Score="5" Body="&lt;p&gt;If you consider Diophantus &amp;quot;ancient&amp;quot; then the answer is &amp;quot;no&amp;quot;. In his &amp;quot;Arithmetic&amp;quot; numbers are not necessarily related to geometry or physics. For Pythagoras, indeed, numbers did not exist without geometry or physics interpretation.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-05-30T23:37:57.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13235" PostTypeId="2" ParentId="13233" CreationDate="2021-05-31T00:41:10.403" Score="27" Body="&lt;p&gt;The answer is yes. There was a split. First of all, for the Greek mathematics (and very long after them)&#xA;&amp;quot;numbers&amp;quot; were integers. &amp;quot;Rational numbers&amp;quot; were called fractions, and no concept of&#xA;real number existed. Therefore, mathematics was essentially split into two independent areas: arithmetic and geometry. Diophantus wrote on arithmetic, he never mentions geometric interpretation of his problems, and it is not known whether he was aware of any such interpretation. (By modern nomenclature his research subject belongs to algebraic geometry). Apollonius wrote on geometry, and never mentions numbers. (From the modern point of view, he is another founder of algebraic geometry).&#xA;Euclid wrote on both subjects, but his arithmetic books are separate from his geometric books, and there is little interaction.&lt;/p&gt;&#xA;&lt;p&gt;People like Euclid and Archimedes had of course a good intuitive grasp of the concept of real number, and they had a theory of proportions when discussing such thing as lengths and areas, and what we call irrational numbers. This theory of proportions was used up to 18th century.&lt;/p&gt;&#xA;&lt;p&gt;Remark. I recommend to read Euclid with a good modern commentary written by a mathematician, rather than an historian. The best one on my point of view is by Robin Hartshorne.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-05-31T00:41:10.403" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13237" PostTypeId="1" CreationDate="2021-06-02T03:08:12.493" Score="0" ViewCount="165" Body="&lt;p&gt;We know that Sir Isaac Newton thought about calculus when he tried to efficiently describe his physical laws but what made Sir Gottfried Leibniz think about something which we know today as calculus?&lt;/p&gt;&#xA;" OwnerDisplayName="user14661" LastActivityDate="2021-06-02T03:08:12.493" Title="What made Leibniz think about calculus?" Tags="|calculus|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13238" PostTypeId="1" CreationDate="2021-06-02T14:17:55.457" Score="6" ViewCount="431" Body="&lt;p&gt;(This question toes the line between belonging here and belonging on the Retrocomputing Stack Exchange.)&lt;/p&gt;&#xA;&lt;p&gt;Here is the quote; sometimes the first sentence is omitted:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most important thing in a programming language is the name. A language will not succeed without a good name. I have recently invented a very good name, and now I am looking for a suitable language.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is always attributed to Donald Knuth. Some instances date it to 1967.&lt;/p&gt;&#xA;&lt;p&gt;What is the original source? I can't find it anywhere. The earliest reference I can find online is in &lt;a href=&quot;http://www.cs.virginia.edu/%7Eevans/cs655-S00/manifests/apr18.html&quot; rel=&quot;noreferrer&quot;&gt;a manifest for a programming languages course&lt;/a&gt; from 2000. Even if the ultimate source is not a published document, but rather a statement in a conversation or in some other form that was not recorded, we should expect earlier references, especially if the quote really is from 1967.&lt;/p&gt;&#xA;" OwnerUserId="13246" LastActivityDate="2021-06-02T14:17:55.457" Title="What is the source of Donald Knuth's remark about naming programming languages?" Tags="|computer-science|quote|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13239" PostTypeId="1" CreationDate="2021-06-03T18:17:04.327" Score="1" ViewCount="619" Body="&lt;p&gt;The first famous measurement of the Earth's circumference was made by Eratosthenes. But he wasn't the only one to carry out this measurement. For instance Posidonius provided the estimate of the circumference that was used by Christopher Columbus. Did other people provide other estimates during the Antiquity?&lt;/p&gt;&#xA;" OwnerUserId="10806" LastActivityDate="2021-06-04T17:56:47.927" Title="Who else measured the circumference of the Earth in the Antiquity?" Tags="|astronomy|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13241" PostTypeId="1" AcceptedAnswerId="13245" CreationDate="2021-06-04T02:33:37.277" Score="3" ViewCount="181" Body="&lt;p&gt;Kindly see the embolded sentence below.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;      To write Cauchy’s definitions down precisely takes a bit more work. This&#xA;was especially true for Cauchy himself, who had not quite phrased the ideas in&#xA;their clean, modern form.* &lt;strong&gt;(In mathematics, you very seldom get the clearest&#xA;account of an idea from the person who invented it.) [Emphasis mine]&lt;/strong&gt; Cauchy was an&#xA;unwavering conservative and a royalist, but in his mathematics he was proudly&#xA;revolutionary and a scourge to academic authority. Once he understood how&#xA;to do things without the dangerous infinitesimals, he unilaterally rewrote his&#xA;syllabus at the École Polytechnique to reflect his new ideas. This enraged&#xA;everyone around him: his mystified students, who had signed up for freshman&#xA;calculus, not a seminar on cutting-edge pure mathematics; his colleagues, who felt that the engineering students at the École had no need for Cauchy’s level&#xA;of rigor; and the administrators, whose commands to stick to the official&#xA;course outline he completely ignored. The École imposed a new curriculum&#xA;from above that emphasized the traditional infinitesimal approach to calculus,&#xA;and placed note takers in Cauchy’s classroom to make sure he complied.&#xA;Cauchy did not comply. Cauchy was not interested in the needs of engineers.&#xA;Cauchy was interested in the truth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;*If you've ever taken a math course that uses epsilons and deltas, you've seen the descendants of Cauchy's formal definitions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ellenberg, &lt;em&gt;How Not to Be Wrong&lt;/em&gt; (2014), p 49.&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2021-06-04T18:25:31.387" Title="Why don't inventors clearly describe or expound their ideas?" Tags="|mathematicians|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13242" PostTypeId="1" CreationDate="2021-06-02T18:33:26.390" Score="0" ViewCount="138" Body="&lt;p&gt;Gauss' law for electricity and magnetism, Ampere's law and Biot-Savart's law — all of these laws are crucial for understanding electromagnetism and Maxwell's equations.&lt;/p&gt;&#xA;&lt;p&gt;But I want to know how each of them derived their laws. If anyone could give me a link to any article or suggest me any kind of books then that would be big help too. I want to learn their methods for deriving these laws in a rigorous way.&lt;/p&gt;&#xA;" OwnerUserId="14670" OwnerDisplayName="AYM Shahriar Rahman" LastEditorUserId="10106" LastEditDate="2022-01-31T11:55:06.737" LastActivityDate="2022-01-31T11:55:06.737" Title="How did Gauss, Ampere and Biot-Savart derive their laws?" Tags="|physics|electromagnetism|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13243" PostTypeId="2" ParentId="13241" CreationDate="2021-06-04T14:26:55.767" Score="2" Body="&lt;p&gt;Inventing new things is difficult. Explaining your inventions to other people is difficult as well. In many cases, the way you think about your solution is quite different from how you explain it to other people so that they can understand. Sometimes, it requires a significant amount of effort and ingenuity to come up with the &amp;quot;right&amp;quot; way to explain it. Do you have any experience writing a computer code? If you do, try to explain it to somebody else in words. See how easy it is to give a clear explanation.  If you never wrote a code, ask a friend who did to explain their code to you.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2021-06-04T14:48:18.020" LastActivityDate="2021-06-04T14:48:18.020" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13244" PostTypeId="2" ParentId="13239" CreationDate="2021-06-04T17:56:47.927" Score="3" Body="&lt;p&gt;I don't know if it counts as &amp;quot;the Antiquity&amp;quot;, but medievalMuslim scholars did also calulate the circumference of Earth.&lt;/p&gt;&#xA;&lt;p&gt;This paper about Al-Biruni (973-&amp;gt;1050) says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In fact, it was long before Al-Biruni’s time that Muslim scholars started their researches on&#xA;mathematical geography and geodesy. A success obtained by them was the measurement of the&#xA;meridian arc, measurement made by astronomers of Caliph Al-Mamun. From this measure, Earth’s&#xA;circumference had been deduced and compared to the value given by ancient Greeks. Al-Biruni,&#xA;eager to find his own value, proposed a new method based on sine trigonometric functions. Using it,&#xA;he measured the Earth’s circumference finding a value quite close to the modern one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-02264631/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://hal.archives-ouvertes.fr/hal-02264631/document&lt;/a&gt; &lt;a href=&quot;https://search.yahoo.com/search?fr=mcafee&amp;amp;type=E211US105G0&amp;amp;p=Al-Biruni&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Al-Ma'mun was born in 786 and reigned from 813-833.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Al-Ma'mun conducted, in the plains of Mesopotamia, two astronomical operations intended to achieve a degree measurement (al-Ma'mun's arc measurement). The crater Almanon on the moon is named in recognition of his contributions to astronomy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2021-06-04T17:56:47.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13245" PostTypeId="2" ParentId="13241" CreationDate="2021-06-04T18:25:31.387" Score="4" Body="&lt;p&gt;The &amp;quot;clean, modern form&amp;quot; of Cauchy's ideas were arrived at after many generations of refinement. This process necessarily takes time: the ideas need to propagate throughout the research community, have to be understood, assimilated, and put to use. Then, simplifications, generalizations and variations of the idea are found, often by incorporating insights from fields other than the one where the idea originally sprung from. As this process keeps iterating, the original ideas get distilled thoroughly and we are left with their &amp;quot;clean, modern form&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;This entire process is far too complex for any one person to complete before sending their idea out into the world. I am not even sure that this is somehow unique to mathematics. Look at what is considered standard undergraduate fare in any subject of your choice. At some point, these ideas were complicated, mysterious, or ground-breaking ideas in that field. But those ideas have now been distilled into something that we can explain to a teenager today and expect them to understand. The form and presentation of the idea will almost certainly be very different from how it looked when it was first propounded.&lt;/p&gt;&#xA;" OwnerUserId="14365" LastActivityDate="2021-06-04T18:25:31.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13246" PostTypeId="1" CreationDate="2021-06-05T12:28:13.360" Score="5" ViewCount="203" Body="&lt;p&gt;Before Abel embarked to proving the insolubility of the quintic with radicals, he thought he had found a solution and sent his work in a letter to the Danish mathematician Carl Ferdinand Degen in 1821.&#xA;Is this letter in existence today? Do we have any evidence of this purported &amp;quot;solution&amp;quot; and what it looked like? It's great fun to be looking at the blunders of the greatest mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="13809" LastEditorUserId="13809" LastEditDate="2021-06-05T22:16:04.087" LastActivityDate="2021-06-05T22:16:04.087" Title="Abel's &quot;solution&quot; of the quintic" Tags="|abstract-algebra|algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13247" PostTypeId="1" AcceptedAnswerId="13248" CreationDate="2021-06-05T13:37:57.227" Score="6" ViewCount="424" Body="&lt;p&gt;The ordinal multiplication &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt; can be defined recursively via ordinal addition &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt; for any ordinal &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; as follows:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\alpha\cdot 0=0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\alpha\cdot (\beta+1)=\alpha\cdot \beta+\alpha$&lt;/span&gt; for any ordinal &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\alpha\cdot \lambda=\bigcup \{\alpha\cdot\beta: \beta\in \lambda\}$&lt;/span&gt; for any limit ordinal &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;According to this widely accepted definition, &lt;span class=&quot;math-container&quot;&gt;$\alpha\cdot \beta$&lt;/span&gt; can be informally interpreted as the &amp;quot;length&amp;quot; of the concatenation of &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;. For instance,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\omega\cdot 2$&lt;/span&gt; is the length of the concatenation of 2 copies of &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;, which is strictly larger than &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$2\cdot \omega$&lt;/span&gt; is the length of the concatenation of &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$2=\{0,1\}$&lt;/span&gt;, which is still equal to &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Since &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt; is not commutative as can be seen from the above example, &lt;span class=&quot;math-container&quot;&gt;$\alpha\cdot \beta$&lt;/span&gt; cannot be interpreted as the &amp;quot;length&amp;quot; of the concatenation of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;.&#xA;However, I personally think that &amp;quot;the length of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; copies of &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;&amp;quot; would have been a more natural way to informally interpret &lt;span class=&quot;math-container&quot;&gt;$\alpha\cdot\beta$&lt;/span&gt;. Therefore, I wonder if there are any historical reasons for this widely accepted definition instead of defining &lt;span class=&quot;math-container&quot;&gt;$\cdot$&lt;/span&gt; the other way around, i.e., for any ordinal &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$0\cdot \beta=0$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$(\alpha+1)\cdot \beta=\alpha\cdot \beta+\beta$&lt;/span&gt; for any ordinal &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\lambda\cdot \beta=\bigcup \{\alpha\cdot\beta: \alpha\in \lambda\}$&lt;/span&gt; for any limit ordinal &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7937" LastEditorUserId="10106" LastEditDate="2022-02-02T08:22:57.920" LastActivityDate="2022-02-02T08:22:57.920" Title="Definition of ordinal multiplication" Tags="|mathematics|terminology|set-theory|definition|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13248" PostTypeId="2" ParentId="13247" CreationDate="2021-06-05T16:45:52.937" Score="5" Body="&lt;p&gt;If you look at Cantor's writing you will find the same idea.  He first defined multiplication the way you suggest.  But later he switched to the definition we use today.  Reference:&lt;br /&gt;&#xA;Joseph W. Dauben. &lt;em&gt;Georg Cantor, his mathematics and philosophy of the infinite.&lt;/em&gt; Harvard University Press, 1979&lt;/p&gt;&#xA;&lt;p&gt;Why?  My guess: When defining &lt;span class=&quot;math-container&quot;&gt;$\alpha + \beta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\alpha^\beta$&lt;/span&gt; we do it by induction in the second variable.  So to be consistent, do it that way also for the definition of &lt;span class=&quot;math-container&quot;&gt;$\alpha\beta$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-06-05T16:45:52.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13249" PostTypeId="1" AcceptedAnswerId="13253" CreationDate="2021-06-06T00:44:50.723" Score="3" ViewCount="479" Body="&lt;p&gt;Please see the embold phrases below. I'm just a laywoman, and I'm just seeking simple answers. I last took math when I was 17.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I read  &lt;a href=&quot;https://hsm.stackexchange.com/q/7907&quot;&gt;Has the standard of mathematical proofs changed over time?&lt;/a&gt;, but how did the standards of mathematical proof change &amp;quot;in the millennia separating Euclid from Hilbert&amp;quot;? I don't understand how they can change, when logic is forevermore permanent and objective.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why  are &amp;quot;standards of mathematical proof&amp;quot; &amp;quot;still not quite settled&amp;quot;?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;RATIONALITY IN FLUX&lt;/h3&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Economic analyses have been based on Plato’s notion of a skillful weigher: To evaluate a plan, one first assigns utilities to every type of pain or pleasure that may result from the plan, at every point in time; then weights the utilities by probabilities to take account of uncertainty; and, finally, evaluates the plan in terms of such a weighted sum. My arguments indicate the need for a broader standard of rationality. Why should the tandard not change? &lt;strong&gt;Even standards of mathematical proof changed in the millennia separating Euclid from Hilbert, and they are still not quite settled. [Emphasis mine]&lt;/strong&gt; I see no reason why Plato’s standard of rationality—or its modern economic versions, for that matter—should not also change in the face of increased knowledge of decision making, especially knowledge about choice construction and context-dependent social goals.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Paul Slovic, &lt;em&gt;The Irrational Economist&lt;/em&gt; (2010), p 70.&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2021-06-18T10:05:11.613" Title="Why are the standards of mathematical proof still unsettled?" Tags="|mathematics|" AnswerCount="4" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13250" PostTypeId="2" ParentId="12947" CreationDate="2021-06-06T02:25:48.650" Score="0" Body="&lt;p&gt;I found that a prediction of a solar eclipse was made Nicholas Culpeper for the April 8, 1652 eclipse in England. At first I was confused by the date, thinking the prediction was wrong, then realized it is the Julian Date which corresponds to the Georgian date in April.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://collation.folger.edu/2017/08/black-monday-great-solar-eclipse-1652/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Black Monday: the Great Solar Eclipse of 1652&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p0x0P.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p0x0P.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14055" LastActivityDate="2021-06-06T02:25:48.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13252" PostTypeId="2" ParentId="13249" CreationDate="2021-06-06T15:01:44.270" Score="9" Body="&lt;p&gt;The statement you cite is somewhat misleading. If you compare Euclid or Archimedes with modern mathematical publications, there is almost no difference in the standard of rigor.&lt;/p&gt;&#xA;&lt;p&gt;When they say that &amp;quot;standard of rigor changes&amp;quot;, they usually mean the epoch from 17 to 19 centuries when calculus was invented, and its practitioners were impatient to obtain new results, instead of working on foundations. But even at that time, best practitioners of calculus understood, that the rigor of their proofs was inferior to that of Archimedes.&lt;/p&gt;&#xA;&lt;p&gt;Same happens in modern times. When a big discovery is made, people sometimes care less about rigor and hurry to obtain new results. If these results are important, other people work on making the proof rigorous. For example, it took about 200 years to develop rigorous foundations of calculus (which would satisfy Archimedes).&lt;/p&gt;&#xA;&lt;p&gt;By the way, when we are speaking of Archimedes, he also obtained non-rigorous results, based on physical intuition. But he had a clear understanding of this, and carefully separated his rigorous results from non-rigorous ones.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-06-06T15:01:44.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13253" PostTypeId="2" ParentId="13249" CreationDate="2021-06-06T21:06:52.643" Score="5" Body="&lt;p&gt;First of all,  Slovic is a psychologist, not a mathematician. &lt;em&gt;Most likely,&lt;/em&gt; his knowledge of modern mathematics is quite limited. (Modern here means, roughly, developed in the last 100 years.) I would be very surprived if he were able to read any papers published, say, by &lt;em&gt;Annals of Mathematics&lt;/em&gt; (which is widely regarded as the best math journal) in the last 80 years.&#xA;Hence, he should not be regarded as an authority in math matters.&lt;/p&gt;&#xA;&lt;p&gt;Standards of mathematical proof (at least in pure math) remained essentially the same for the last 100 years. (David Hilbert was probably the one most responsible for the current standards.) There are several important exceptions here:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Computer-aided proofs mentioned by sand1 in a comment. The first computer-aided proof I know, appeared in the solution of the 4-color problem (mid-1970s). Another, more recent, example is the solution (in mid-1990s) of Kepler's problem on optimal ball packing in the 3-dimensional Euclidean space. There is no commonly accepted protocol on when to regard such proofs as solid. (The issue is that it's hard to check correctness of long computer codes.) As for &lt;em&gt;why&lt;/em&gt;, the answer is relatively clear: Technological developments outpace our capacity to formulate the standards.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some proofs are incredibly long, complex and verifications require specialized knowledge that very few mathematicians have. When to regard such proofs as solid is still unsettled. (Relatively) recent work of Shinichi Mochizuki on Inter-universal Teichmüller theory is one such example. (I am not going to take sides in this controversy, this is too far from my area of research.) As to &lt;em&gt;why&lt;/em&gt;, a partial answer is the increase in specialization, when even best experts working in the same area of math do not completely understand each other. Another part of the answer is that it is only recently that we started to see proofs which (when written in detail) are over 500 page-long.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;However, keep in mind that (so far) these exceptions apply only to a very  small fraction of mathematical results. Hence, &lt;strong&gt;overall&lt;/strong&gt; our standards of rigor in math are currently settled.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit. I looked up David H. Krantz. Yes, he indeed was a math major at Yale. He published just one paper in pure math (after getting his BA from Yale) and after that, all his math-related papers  are in applied math (I counted 19, in applied mathematics), where standards of rigor are quite different from pure math. My remarks regarding Slovic, apply to him as well.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2021-06-13T23:10:36.413" LastActivityDate="2021-06-13T23:10:36.413" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13254" PostTypeId="1" AcceptedAnswerId="13256" CreationDate="2021-06-07T16:19:18.133" Score="32" ViewCount="8405" Body="&lt;p&gt;Consider a fundamental concept in maths that was created to 'solve' a problem that simply couldn't be solved by any other approach (or maybe for some other reason). Now let's assume that this concept made no sense to the contemporary mathematicians whatsoever that they simply brushed it aside as a purely mathematical tool with &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; physical significance. &lt;em&gt;like the square root of &lt;span class=&quot;math-container&quot;&gt;$-1&#xA;, $&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$ i$&lt;/span&gt; or the Euler number maybe?&lt;/em&gt;. Anything that makes no sense to a present-day mathematical outsider really.&lt;/p&gt;&#xA;&lt;p&gt;A couple of decades later after its birth, baam, a scientist realises that the only way to solve a paradox in physics was to use this very mathematical concept that had to seem to be abstract at the time. Is there any good example of this in history?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Inspiration&lt;/p&gt;&#xA;&lt;p&gt;This is with respect to &lt;a href=&quot;https://matheducators.stackexchange.com/questions/20962/response-to-students-who-say-this-is-not-important&quot;&gt;the idea of maths being seen as a subject&lt;/a&gt; with no real-life application. I wanted to disprove by saying that many mathematical concepts do have a real-life application (if we look back at the history), but it's just that we haven't found any use so far.&lt;/p&gt;&#xA;&lt;p&gt;I am slightly aware that e and &lt;a href=&quot;https://en.wikipedia.org/wiki/Complex_number&quot; rel=&quot;noreferrer&quot;&gt;complex numbers&lt;/a&gt; like &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; makes it into the world of electrical engineering though I don't remember where exactly. I also vaguely recollect topology space and an array of mathematical concepts coming into the world by the theory of &lt;a href=&quot;https://en.wikipedia.org/wiki/General_relativity&quot; rel=&quot;noreferrer&quot;&gt;general relativity&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_mechanics&quot; rel=&quot;noreferrer&quot;&gt;quantum mechanics&lt;/a&gt;. What I am really looking for is&lt;/p&gt;&#xA;&lt;p&gt;(1) a concept in maths that initially seems to have had no use &lt;br /&gt;&#xA;(2) 70 years later it is been used in a formula (please do add the formula too; that's really the reason I came here)&lt;/p&gt;&#xA;" OwnerUserId="14225" LastEditorUserId="6229" LastEditDate="2022-02-02T18:18:11.987" LastActivityDate="2022-02-02T18:25:33.947" Title="Has physics ever given a physical significance to a mathematically abstract idea?" Tags="|mathematics|mathematical-physics|" AnswerCount="11" CommentCount="23" ContentLicense="CC BY-SA 4.0" />
  <row Id="13255" PostTypeId="2" ParentId="13254" CreationDate="2021-06-07T17:18:04.063" Score="21" Body="&lt;p&gt;&lt;strong&gt;fractal&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The so-called Cantor set was described by&lt;br /&gt;&#xA;&lt;strong&gt;Georg Cantor&lt;/strong&gt;, 1884 (or &lt;strong&gt;H. J. S. Smith&lt;/strong&gt;, 1875?)&lt;br /&gt;&#xA;Sets with &amp;quot;fractional&amp;quot; dimension were described by&lt;br /&gt;&#xA;&lt;strong&gt;Felix Hausdorff&lt;/strong&gt;, 1918&lt;br /&gt;&#xA;Investigated thoroughly by&lt;br /&gt;&#xA;&lt;strong&gt;A. S. Besicovitch&lt;/strong&gt;, 1930s - 1950s&lt;br /&gt;&#xA;But these were only mathematical abstractions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Benoit B. Mandelbrot&lt;/strong&gt;, 1960s and later&lt;br /&gt;&#xA;claimed relevance of these sets in natural sciences.&lt;br /&gt;&#xA;Mandelbrot reported that, in the early days, when he would submit a paper&#xA;to a physics journal with a Cantor set in it, that it would be rejected out-of-hand as &amp;quot;unphysical&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Largely due to Mandelbrot's efforts, that situation has been completely reversed.&lt;br /&gt;&#xA;In 1992, I counted that physics papers with the word &amp;quot;fractal&amp;quot; in the title were appearing more frequently than once a week.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2021-06-07T17:26:53.947" LastActivityDate="2021-06-07T17:26:53.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13256" PostTypeId="2" ParentId="13254" CreationDate="2021-06-07T20:37:34.347" Score="33" Body="&lt;p&gt;Physics cannot help giving physical significance to things. But, yes, the &lt;em&gt;first&lt;/em&gt; item on your list should have been&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lie_group#History&quot; rel=&quot;noreferrer&quot;&gt;Lie Groups&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Developed as a mathematical &amp;quot;sudoku&amp;quot; game generalizing rotations, in the late 19th century, by Sophus Lie, Felix Klein, Friedrich Engel, Henri Poincare, and, as  associated  applied mathematical structures by James Joseph Sylvester and Arthur Cayley,&#xA;they were recognized by physicists in the early 20th century to apply to the new-fangled science of &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics&quot; rel=&quot;noreferrer&quot;&gt;Quantum mechanics&lt;/a&gt;, the actual cornerstone of modern physics, by Eugene Wigner and Hermann Weyl, firstly through the rotation group SO(3) ~ SU(2).&lt;/p&gt;&#xA;&lt;p&gt;In the 1960s, Murray Gell-Mann appreciated that SU(3), both classified and further described, in fact, elementary particle interactions, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eightfold_way_(physics)&quot; rel=&quot;noreferrer&quot;&gt;Eightfold Way&lt;/a&gt;; whence  the full-fledged Lie Group theory entered as &lt;em&gt;the&lt;/em&gt; mainstay of Quantum Field Theory describing fundamental particles and forces in &lt;em&gt;all&lt;/em&gt; of High Energy physics. Now &lt;em&gt;that's&lt;/em&gt; an example of abstract math underlying all of nature at its most fundamental, a century before that was recognized!&lt;/p&gt;&#xA;&lt;p&gt;Today, hardly anything describing electroweak interactions, strong interactions, or gravity is thinkable without crucial and &lt;em&gt;irreplaceable&lt;/em&gt; reliance on &lt;a href=&quot;https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory&quot; rel=&quot;noreferrer&quot;&gt;Lie Group representation theory&lt;/a&gt;. &lt;em&gt;&lt;strong&gt;The fundamental laws of nature run on Lie groups&lt;/strong&gt;&lt;/em&gt;. Theoretical HEP theorists are part-time Lie-Group applied mathematicians. &lt;a href=&quot;http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.126.1581&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;noreferrer&quot;&gt;This&lt;/a&gt; is the sort of thing they have to master in school.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-06-08T19:19:45.440" LastActivityDate="2021-06-08T19:19:45.440" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13257" PostTypeId="1" CreationDate="2021-06-07T21:47:09.737" Score="4" ViewCount="424" Body="&lt;p&gt;I have encountered a user on Math Stack Exchange with &lt;a href=&quot;https://math.stackexchange.com/users/698573/erik-satie&quot;&gt;writing in his bio&lt;/a&gt; that &lt;a href=&quot;https://en.wikipedia.org/wiki/Empress_Elisabeth_of_Austria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Empress Elisabeth of Austria&lt;/a&gt; (&amp;quot;Sisi&amp;quot;) did some math and she was famous for an unsolvable integral:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{0}^{1}\operatorname{si}(x)\operatorname{si}(x)\;\text{d}x$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am unsure if this is the precise integral that I saw in his bio. Anyways, I have did google searches and didn't find anything about her as a mathematician. I hope someone can provide any historical context.&lt;/p&gt;&#xA;" OwnerUserId="14690" LastEditorUserId="6229" LastEditDate="2022-02-02T18:17:39.627" LastActivityDate="2024-02-19T18:24:55.120" Title="Is it true that Empress Elisabeth of Austria did math?" Tags="|mathematics|debunking|19th-century|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13258" PostTypeId="2" ParentId="13254" CreationDate="2021-06-08T03:59:00.097" Score="18" Body="&lt;p&gt;The following quote is from &lt;a href=&quot;https://en.wikipedia.org/wiki/Yang_Chen-Ning&quot; rel=&quot;noreferrer&quot;&gt;C. N. Yang&lt;/a&gt;, delivered at a 1979 symposium dedicated to the geometer &lt;a href=&quot;https://en.wikipedia.org/wiki/Shiing-Shen_Chern&quot; rel=&quot;noreferrer&quot;&gt;Chern&lt;/a&gt;: &amp;quot;&lt;em&gt;When I met Chern, I told him that I finally understood the beauty of the theory of fibre bundles and the elegant Chern​–Weil theorem.  I was struck that gauge fields, in particular connections on fibre bundles, were studied by mathematicians without any appeal to physical realities.  I added that it is mysterious and incomprehensible how you mathematicians could think this up out of nothing.  To this Chern immediately objected. 'No, no, this concept is not invented — it is natural and real&lt;/em&gt;.'&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2021-06-10T02:23:53.737" LastActivityDate="2021-06-10T02:23:53.737" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13259" PostTypeId="2" ParentId="13254" CreationDate="2021-06-08T16:49:24.593" Score="1" Body="&lt;p&gt;Not really physics, but: I think that number theory (edit: specifically research on primes) was an entirely idle mathematical pastime until cryptography became a thing.&lt;/p&gt;&#xA;" OwnerUserId="9476" LastEditorUserId="9476" LastEditDate="2021-06-10T16:49:40.293" LastActivityDate="2021-06-10T16:49:40.293" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13260" PostTypeId="2" ParentId="94" CreationDate="2021-06-08T20:41:07.960" Score="1" Body="&lt;p&gt;The term 'phase' appears to have been coined by J. Willard Gibbs:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In considering the different homogeneous bodies which can be formed out of any set of component substances, it is convenient to have a term which shall refer solely to the composition and thermodynamic state of any such body without regard to its size or form. The word &lt;em&gt;phase&lt;/em&gt; has been chosen for this purpose.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;J. Willard Gibbs, &amp;quot;On the Equilibrium of Heterogeneous Substances&amp;quot;, Transactions of the Connecticut Academy of Arts and Sciences, 3, pp. 108-248 and 343-524.&lt;/p&gt;&#xA;&lt;p&gt;Available on-line at &lt;a href=&quot;https://archiv.ub.uni-heidelberg.de/volltextserver/13220/1/repertorium_gibbs2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archiv.ub.uni-heidelberg.de/volltextserver/13220/1/repertorium_gibbs2.pdf&lt;/a&gt;, where this statement appears on p. 306.&lt;/p&gt;&#xA;" OwnerUserId="14703" LastEditorUserId="6609" LastEditDate="2024-01-04T14:18:49.347" LastActivityDate="2024-01-04T14:18:49.347" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13261" PostTypeId="2" ParentId="13254" CreationDate="2021-06-09T04:20:05.090" Score="8" Body="&lt;p&gt;Abelian and non-abelian group theory -&amp;gt; quantum chromodynamics&lt;/p&gt;&#xA;&lt;p&gt;Noneuclidean geometry -&amp;gt; general relativity&lt;/p&gt;&#xA;&lt;p&gt;Sorry, I cannot write you any equations as examples.&lt;/p&gt;&#xA;" OwnerUserId="14713" LastActivityDate="2021-06-09T04:20:05.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13262" PostTypeId="2" ParentId="13254" CreationDate="2021-06-09T05:06:05.067" Score="4" Body="&lt;p&gt;Spinors were introduced by E.Cartan and found applications in quantum theory 15 years later.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2021-06-09T05:06:05.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13263" PostTypeId="2" ParentId="13232" CreationDate="2021-06-09T09:25:23.447" Score="1" Body="&lt;p&gt;After a bit of research I stumbled across &lt;a href=&quot;https://www.jstor.org/stable/41137708?refreqid=excelsior%3A0cbf6a33e7b2d5c845e68465ba549c09&amp;amp;seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; paper which might be a good starting point. As often in mathematics, Newton discovered his method of &lt;em&gt;divided differences&lt;/em&gt; through pattern finding whereby the paper gives a slightly adjusted approach after some examination of Newton's thoughts which will give you a good idea why his method works.&lt;/p&gt;&#xA;" OwnerUserId="14638" LastActivityDate="2021-06-09T09:25:23.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13264" PostTypeId="2" ParentId="13254" CreationDate="2021-06-09T11:36:58.543" Score="8" Body="&lt;p&gt;I don't know if this can be counted for as physics, but to my knowledge the radon transformation was mostly something mathematicians thought about without any application. Now, it is widely used (and necessary) to transform images in tomography.&lt;/p&gt;&#xA;&lt;p&gt;1917 - Radon transform introduced by Johann Radon&lt;/p&gt;&#xA;&lt;p&gt;1970s - First Computer Tomograph Scan&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Radon_transform&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Radon_transform&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14719" LastEditorUserId="14719" LastEditDate="2021-08-17T12:27:50.780" LastActivityDate="2021-08-17T12:27:50.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13265" PostTypeId="2" ParentId="13254" CreationDate="2021-06-09T18:02:00.770" Score="6" Body="&lt;p&gt;There are many examples of a physical significance given to abstract mathematical ideas.&lt;br /&gt;&#xA;One of the most prominent is the spacetime as introduced in general relativity. In general relativity a mathematical framework is given that informs you about the structure of space. It even relates space with time, thereby introducing new physical stuff, i.e., spacetime. The theory is inspired by real physical things (space and time) but the extra quality that came up was a connection between those two, given by the math. Matter and energy already existed before the theory but general relativity proposed a connection between these and the new kind of spacetime that it introduced. This connection was absent in the earlier theory of Newton who said that space and time are not connected and are not influenced by the matter and energy that is contained in space.&lt;br /&gt;&#xA;New things like spacetime curvature, an absolute limit on speed (which actually gave rise to the concept of relativity), the equivalence of mass and energy, etc., were introduced as mathematical expressions, changing the physical aspects of the universe.&lt;br /&gt;&#xA;Einstein, for formulating his ideas, made use of in his time already existing mathematical ideas introduced by mathematicians like Riemann, Ricci, Levi-Civita, and Killing.&lt;br /&gt;&#xA;New discoveries were made, inspired by Einstein's ideas. Though physical things like space and time already existed, the mathematics of Riemannian manifolds was used to make a connection between them, thereby postulating the new physical entity, i.e., spacetime.&lt;br /&gt;&#xA;So, a physical significance was indeed given to the ideas of real manifolds and tensor calculus.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-06-09T18:02:00.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13267" PostTypeId="2" ParentId="13254" CreationDate="2021-06-10T12:41:43.720" Score="5" Body="&lt;p&gt;I think one of the best example is the binary system, the representing of numbers in 0 and 1 by diving them by 2. This is how every processor, ram, HDD, SSD,.. etc keep data and how digital communications work. Also the whole Boole algebra is the basics of how computer processors calculate numbers.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1605 Francis Bacon discussed a system whereby letters of the&#xA;alphabet could be reduced to sequences of binary digits, which could&#xA;then be encoded as scarcely visible variations in the font in any&#xA;random text.[16] Importantly for the general theory of binary&#xA;encoding, he added that this method could be used with any objects at&#xA;all: &amp;quot;provided those objects be capable of a twofold difference only;&#xA;as by Bells, by Trumpets, by Lights and Torches, by the report of&#xA;Muskets, and any instruments of like nature&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Binary_number&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Binary_number&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14737" LastActivityDate="2021-06-10T12:41:43.720" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13268" PostTypeId="2" ParentId="13254" CreationDate="2021-06-10T14:49:55.797" Score="6" Body="&lt;p&gt;Riemann geometry: Einstein had to look around for the mathematics that could describe General Relativity. Before that nobody would have guessed that curved geometry has anything to do with space-time and gravity.&lt;/p&gt;&#xA;" OwnerUserId="14738" LastActivityDate="2021-06-10T14:49:55.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13270" PostTypeId="1" CreationDate="2021-06-10T19:03:56.950" Score="3" ViewCount="394" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Copenhagen_interpretation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Copenhagen interpretation&lt;/a&gt; of quantum mechanics is nowadays the interpretation that prevails over other interpretations (or theories). Basically, it states that all processes in Nature are inherently probabilistic. Contrary to, for example, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pilot_wave_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;pilot wave theory&lt;/a&gt;, of which Louis Victor Pierre Raymond, 7th Duc de Broglie and David Bohm (who revived the theory) are proponents. This theory sees Nature as inherently probabilistic. Quantum stuff is seen as governed by the deterministic motion of a wave accompanying the stuff (like a Brownian particle is accompanied by the fluid it moves in). So, such a theory (it's not an interpretation, I guess) explains the strange behavior of stuff in quantum mechanics by the introduction of maybe even stranger things as a pilot wave or hidden variables (introduced by Bohm).&lt;/p&gt;&#xA;&lt;p&gt;Now I can imagine that this vision could be taught in present-day schools or where ever. But this isn't the case. The Copenhagen interpretation rules supreme, so to speak. Why is this? Is it because a bunch of physicists (like Bohr, Heisenberg, and Born) decided in Copenhagen that this view should be &lt;strong&gt;the&lt;/strong&gt; view? Einstein struggled against the view (god doesn't play dice) but to no avail. Was it powerplay that decided the faith of quantum mechanics?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4251" LastEditDate="2021-06-10T19:14:20.490" LastActivityDate="2021-07-18T17:03:57.700" Title="Why did the Copenhagen interpretation of quantum mechanics got the upperhand?" Tags="|physics|quantum-mechanics|einstein|probability|bohr|" AnswerCount="1" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="13271" PostTypeId="1" CreationDate="2021-06-11T04:22:14.363" Score="3" ViewCount="300" Body="&lt;p&gt;Is there any collection or list of artworks both literary and visual which make homage to Scientists(Mathematicians and Physicists specifically)? I am not necessarily thinking of portraits, instead paintings made in their honor or odes written in their memory.&lt;/p&gt;&#xA;&lt;p&gt;Some examples of what I am speaking of:&lt;/p&gt;&#xA;&lt;h2&gt;&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Anything from &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Newton_poetry/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Newton, William Blake&lt;/li&gt;&#xA;&lt;li&gt;The Refusal of Archimedes, Sebastiano Ricci&lt;/li&gt;&#xA;&lt;li&gt;Pythagoras, Girolamo Mazzola Bedoli&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;&lt;/h2&gt;&#xA;&lt;p&gt;I would appreciate any suggestions of art, or resources that could help me in this regard.&lt;/p&gt;&#xA;" OwnerUserId="13800" LastEditorUserId="10106" LastEditDate="2022-02-02T08:26:20.523" LastActivityDate="2022-02-02T08:26:20.523" Title="References to Scientists in Art and Poetry" Tags="|scientists|art|" AnswerCount="4" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13272" PostTypeId="2" ParentId="13271" CreationDate="2021-06-11T07:32:23.857" Score="0" Body="&lt;p&gt;&amp;quot;&lt;a href=&quot;https://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf77&quot; rel=&quot;nofollow noreferrer&quot;&gt;Math a mort&lt;/a&gt;&amp;quot; is a famous murder-mystery about the French Institute of Advance Studies (IHES) by Margot Bruyère where several permanent members mathematicians in IHES play important roles. The Web site above also lists other math mystery books. Truth, this is not poetry or &amp;quot;art&amp;quot;. Of course, Prof.  Moriarty was also a mathematician in Sherlock Holmes stories but he did not have a real life mathematician prototype.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-06-16T20:32:23.133" LastActivityDate="2021-06-16T20:32:23.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13274" PostTypeId="2" ParentId="13226" CreationDate="2021-06-12T14:30:09.033" Score="1" Body="&lt;p&gt;Following M. Farooq recommendation, I had a look at the original paper and the extreme left of the table was indeed unknown in the solid state. They apparently got some densities by &amp;quot;transpiration&amp;quot; (ie. in the gaseous state) which should be explained in a previous version of the book (Ann. Chem. Pharm. 1867, Supplem. V, 129 if.). I will try and find the reference later on.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[DE] Da die Atomvolumina von H, N, O, F und Ti in festen Zustande unbekannt sind, so bleibt der Abschnitt I. und die zweite Hälfte von II. vor der Hand unbestimmt. Aus dem durch die Transpiration bestimmten Molecularvolumen der drei ersten dieser Elemente im gasförmigen Zustande (Ann. Chem. Pharm. 1867, Supplem. V, 129 if.) und aus dem der festen Fluor- und Titanverbindungen, verglichen mit dem verwandter Stoffe, läßt sich aber mit ziemlicher Sicherheit folgern, daß die Curve der Atomvolumina der Elemente im festen Zustande im ersten und zweiten Abschnitte ungefähr den auf Taf. III gezeichneten Verlauf haben wird. Aehnliches gilt vom Ende des Abschnittes V. und dem Anfange von VI., zwischen denen die Curve unzweifelhaft im Cs ein Maximum erreichen wird.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[EN] Since the atomic volumes of H, N, O, F and Ti in the solid state are unknown, section I. and the second half of II. remain undetermined before hand. From the molecular volume of the first three of these elements in the gaseous state, determined by transpiration (Ann. Chem. Pharm. 1867, Supplem. V, 129 if.), and from that of the solid fluorine and titanium compounds, compared with that of related substances, it can be concluded with a fair degree of certainty that the curve of the atomic volumes of the elements in the solid state in the first and second sections is approximately the same as that shown in Table I. and Table II. The first and second sections of the curve will have approximately the shape shown in Taf. III. The same applies to the end of section V and the beginning of VI, between which the curve will undoubtedly reach a maximum in Cs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14617" LastEditorUserId="3223" LastEditDate="2021-06-12T16:50:28.810" LastActivityDate="2021-06-12T16:50:28.810" CommentCount="4" CommunityOwnedDate="2021-06-12T14:30:09.033" ContentLicense="CC BY-SA 4.0" />
  <row Id="13275" PostTypeId="1" CreationDate="2021-06-13T08:38:42.030" Score="1" ViewCount="274" Body="&lt;p&gt;I have been looking for a list of Nobel laureates classified by their majors, e.g., electrical engineering, mechanical engineering, etc. Since some of them may have changed the major from master course or doctor course, it would be preferable that the list is listed with all majors in BSc, MSc, DSc.&lt;/p&gt;&#xA;&lt;p&gt;Given my current lack of Google-ability, I've been unable to found it. Should I have to know it checking each of the laureates by Wikipedia?&lt;/p&gt;&#xA;&lt;p&gt;At least I want to know the laureates who have got degrees in electrical engineering or kind of like that.&lt;/p&gt;&#xA;" OwnerUserId="14765" LastEditorUserId="4251" LastEditDate="2021-09-25T15:19:48.870" LastActivityDate="2021-09-25T15:19:48.870" Title="List of Nobel laureates classified by majors" Tags="|nobel-prize|big-list|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13276" PostTypeId="2" ParentId="13275" CreationDate="2021-06-13T16:20:36.683" Score="2" Body="&lt;p&gt;Google gives:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Jack St. Clair Kilby, a 1950 masters degree graduate in electrical engineering at the University of Wisconsin–Madison, has been named a co-winner of the 2000 Nobel Prize in physics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-06-13T16:20:36.683" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13277" PostTypeId="2" ParentId="3707" CreationDate="2021-06-14T09:58:05.713" Score="2" Body="&lt;p&gt;Those Wikipedia and Wikiversity pages (&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_Lorentz_transformations&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Lorentz transformations&lt;/a&gt; and &lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Topics in Special Relativity&lt;/a&gt;)  have lots of information on the mathematical history of the Lorentz transformations preserving:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\pm\left[x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}\right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Starting as early as 1800 by Gauss and many others, quadratic forms equivalent to the spacetime interval and their transformations were studied in the theory of indefinite quadratic forms, elliptic functions, hyperbolic geometry, long before Lorentz, Einstein or Minkowski used them in physics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(general)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Most general Lorentz transformations&lt;/a&gt;&#xA;Lagrange (1773), Gauss (1798–1818), Jacobi (1827, 1833/34), Lebesgue (1837), Bour (1856), Somov (1863), Killing (1878–1893), Poincaré (1881), Cox (1881–1883), Hill (1882), Picard (1882-1884), Callandreau (1885), Lie (1885-1890), Gérard (1892), Hausdorff (1899), Woods (1901-05), Liebmann (1904–05)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(imaginary)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via imaginary orthogonal transformation&lt;/a&gt;&#xA;Euler (1771), Wessel (1799), Cauchy (1829), Lie (1871), Minkowski (1907–1908), Sommerfeld (1909)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(hyperbolic)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via hyperbolic functions&lt;/a&gt;&#xA;Riccati (1757), Lambert (1768–1770), Taurinus (1826), Cayley (1859-84), Beltrami (1868), Klein (1871), Laisant (1874), Escherich (1874), Glaisher (1878), Günther (1880/81), Schur (1885/86, 1900/02), Lindemann (1890–91), Gérard (1892), Killing (1893,97), Woods (1903), Whitehead (1897/98), Liebmann (1904–05), Herglotz (1909/10)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(velocity)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via velocity&lt;/a&gt;&#xA;Euler (1735), Beltrami (1868), Schur (1885/86, 1900/02), Lipschitz (1885–86), Voigt (1887), Heaviside (1888), Thomson (1889), Searle (1896), Lorentz (1892, 1895), Larmor (1897, 1900), Lorentz (1899, 1904), Poincaré (1900, 1905), Einstein (1905), Minkowski (1907–1908), Sommerfeld (1909), Herglotz (1909/10), Varićak (1910), Ignatowski (1910), Noether (1910), Klein (1910), Conway (1911), Silberstein (1911), Ignatowski (1910/11), Herglotz (1911), Borel (1913–14), Gruner (1921)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(conformal)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via conformal, spherical wave, and Laguerre transformation&lt;/a&gt;&#xA;Lie (1871), Klein &amp;amp; Pockels &amp;amp; Bôcher (1871-91), Laguerre (1882), Stephanos (1883), Darboux (1887), Scheffers (1899), Smith (1900), Bateman and Cunningham (1909–1910)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(Cayley-Hermite)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via Cayley–Hermite transformation&lt;/a&gt;&lt;br /&gt;&#xA;Euler (1771), Cayley (1846–1884), Hermite (1853, 1854), Bachmann (1869), Laguerre (1882), Darboux (1887), Smith (1900), Borel (1913–14)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(M%C3%B6bius)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via Cayley–Klein parameters, Möbius and spin transformations&lt;/a&gt;&#xA;Lagrange (1773), Gauss (1800), Cayley (1854), Klein (1871–97), Selling (1873–74), Poincaré /1881-86), Bianchi (1888-93), Fricke (1891–97), Woods (1895), Herglotz (1909/10)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(Quaternions)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via quaternions and hyperbolic numbers&lt;/a&gt;&#xA;Euler (1771), Hamilton (1844/45), Cayley (1845), Cockle (1848), Cox (1882), Stephanos (1883),   Buchheim (1884–85), Lipschitz (1885/86), Vahlen (1901/02), Noether (1910), Klein (1910), Conway (1911), Silberstein (1911)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(trigonometric)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via trigonometric functions&lt;/a&gt;&#xA;Bianchi (1886–1893), Darboux (1881/94), Scheffers (1899), Eisenhart (1905), Varićak (1910), Gruner (1921)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(squeeze)&quot; rel=&quot;nofollow noreferrer&quot;&gt;LT via squeeze mappings&lt;/a&gt;&#xA;Laisant (1874), Lie (1879-84), Günther (1880/81), Laguerre (1882), Darboux (1883–1891), Lipschitz (1885/86), Bianchi (1886–1893), Lindemann (1890/91), Smith (1900), Eisenhart (1905)&lt;/p&gt;&#xA;" OwnerUserId="7601" LastActivityDate="2021-06-14T09:58:05.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13279" PostTypeId="1" AcceptedAnswerId="13289" CreationDate="2021-06-15T14:42:27.593" Score="4" ViewCount="321" Body="&lt;p&gt;Ok so if the Romans did not use things like IX and IV and XC etcetera then addition and subtraction would be almost as instant as it is in our number system.&lt;/p&gt;&#xA;&lt;p&gt;However with the new system it seems to me like it isn't that quick, the only benefit that I can find is that it makes numbers shorter.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any insights as to why they introduced this concept (I think that it must have not been in use originally as I believe the system originally came from some sort of tally system).&lt;/p&gt;&#xA;&lt;p&gt;Also if anyone has some reasoning as to why it doesn't really make addition harder that would be great.&lt;/p&gt;&#xA;" OwnerDisplayName="user14780" LastActivityDate="2021-06-19T06:33:20.043" Title="Why did the romans use IV and why doesn't it overcomplicate things?" Tags="|roman-empire|numbers|arithmetic|" AnswerCount="2" CommentCount="19" ContentLicense="CC BY-SA 4.0" />
  <row Id="13280" PostTypeId="1" CreationDate="2021-06-16T07:31:41.730" Score="0" ViewCount="109" Body="&lt;p&gt;I want to know the mathematician who named PSL(2,Z) as the modular group first. Also, I would be pleased if someone suggest to me some papers about the history of the modular group. Thanks a lot in advance for your help.&lt;/p&gt;&#xA;" OwnerUserId="14782" LastEditorUserId="10106" LastEditDate="2022-02-03T18:12:53.710" LastActivityDate="2022-02-03T18:12:53.710" Title="History of PSL(2,Z)" Tags="|mathematics|terminology|group-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13281" PostTypeId="1" CreationDate="2021-06-16T12:06:15.817" Score="1" ViewCount="232" Body="&lt;p&gt;I am reposting a thread from &amp;quot;physics stack exchange&amp;quot; :&#xA;I was wondering -&#xA;Why do we use the word &amp;quot;field&amp;quot; to describe a vector field? i.e., a field is &amp;quot;an expanse of open or cleared ground, especially a piece of land suitable or used for pasture or tillage&amp;quot; (from dictionary.com).&#xA;So, why historically this was the name chosen for the description of a function &lt;span class=&quot;math-container&quot;&gt;$F:\mathbb{R}^n \to \mathbb{R}^n$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerUserId="14785" LastEditorUserId="4251" LastEditDate="2021-06-24T07:59:14.860" LastActivityDate="2021-06-24T07:59:14.860" Title="Origin of the term &quot;field&quot; (in &quot;vector field&quot;)" Tags="|mathematics|terminology|quantum-field-theory|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13282" PostTypeId="2" ParentId="13281" CreationDate="2021-06-16T12:35:00.247" Score="3" Body="&lt;p&gt;From &lt;a href=&quot;https://jeff560.tripod.com/v.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathword&lt;/a&gt; &amp;quot;Earliest Known Uses of&#xA;Some of the Words of Mathematics&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;VECTOR FIELD is found in 1905 in “The Present Problems of Geometry” by Dr. Edward Kasner in &lt;em&gt;Congress of Arts and Science, Universal Exposition, St. Louis, 1904:&lt;/em&gt; “The vector field deserves to be introduced as a standard form into geometry.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We could investigate whether this was before or after the corresponding terms in physics, like  &amp;quot;magnetic field&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-06-16T12:35:00.247" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13283" PostTypeId="1" CreationDate="2021-06-16T13:37:31.493" Score="2" ViewCount="160" Body="&lt;p&gt;I have seen many times before that Hamilton started off believing he would need a three-dimensional system over the reals in order to describe 3D rotations. He considered numbers of the form &lt;span class=&quot;math-container&quot;&gt;$a + bi + cj$&lt;/span&gt;, defined summation component-wise and tried to find a suitable multiplication.&lt;/p&gt;&#xA;&lt;p&gt;My question is: how did he conclude these numbers could not be divided and how did he, from this point, conclude there would need to be four dimensions for the quaternions to work? Any references for further reading are also greatly appreciated!&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="14787" LastEditorUserId="4251" LastEditDate="2022-09-08T06:20:41.080" LastActivityDate="2022-09-08T06:20:41.080" Title="How did Hamilton conclude the quaternions had to be four dimensional?" Tags="|mathematics|mathematicians|abstract-algebra|quaternions|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13284" PostTypeId="2" ParentId="13279" CreationDate="2021-06-16T14:47:39.130" Score="2" Body="&lt;p&gt;It seems like the &amp;quot;subtractive&amp;quot; convention was not as commonly used as I initially thought.&lt;/p&gt;&#xA;&lt;p&gt;As is said &lt;a href=&quot;https://www.quora.com/How-did-the-Romans-do-multiplication-with-Roman-numerals/answer/B%C3%A9a-Tremblay&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; the additive convention was in many times utilized when the number could be further manipulated.&lt;/p&gt;&#xA;&lt;p&gt;I believe the &amp;quot;subtractive&amp;quot; convention was mostly used for shortening.&lt;/p&gt;&#xA;&lt;p&gt;I would like to add that I am not completely sure, but I think I have more chances of someone correcting me if I write out this answer.&lt;/p&gt;&#xA;" OwnerDisplayName="user14780" LastActivityDate="2021-06-16T14:47:39.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13285" PostTypeId="1" CreationDate="2021-06-17T23:24:43.637" Score="2" ViewCount="75" Body="&lt;p&gt;In 1925, Frenkel wrote a paper titled &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01331692&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Elektrodynamik punktförmiger Elektronen&lt;/a&gt;, which used complex analysis to treat an electron as a point, and its corresponding potential function as an isolated singularity at its position; hence eliminating the problems of an infinite self-energy and further problems for an extended electron with an internal structure. For me, the idea is profoundly elegant and simple, yet the paper is currently only cited by 57 other papers, which suggests to me that Frenkel may not have been the first to use this technique.&lt;/p&gt;&#xA;&lt;p&gt;If not Frenkel, who was the first to use complex analysis to deal with isolated singularities in field theory?&lt;/p&gt;&#xA;" OwnerUserId="2461" LastActivityDate="2021-06-17T23:24:43.637" Title="Who first used complex analysis to account for singularities in field theory?" Tags="|theoretical-physics|mathematical-physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13286" PostTypeId="1" CreationDate="2021-06-18T03:08:20.240" Score="0" ViewCount="120" Body="&lt;p&gt;I have heard it said that even if Fuchs had not provided details which must have been invaluable — just knowing, for example, that implosion and explosive lenses were employed without any further details would have been huge — the Soviets would have had a huge advantage over allied scientists.&lt;/p&gt;&#xA;&lt;p&gt;This would have come from the fact that they, after Hiroshima, knew that such a device could be constructed.&lt;/p&gt;&#xA;&lt;p&gt;That certainly rings true to me — the top soviet physicists would not have had to wonder if the whole idea was crazy and that would have saved a tremendous amount of physic energy. I am looking for a quote where one of the theoreticians actually credits this as having helped them.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="4251" LastEditDate="2021-09-16T13:39:24.427" LastActivityDate="2021-09-16T13:39:24.427" Title="Soviet atomic bomb development — advantage of knowing it could be done, was this ever said by any Soviet scientist?" Tags="|physics|nuclear-physics|cold-war|soviet-union|20th-century|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13287" PostTypeId="2" ParentId="13249" CreationDate="2021-06-18T04:09:39.890" Score="0" Body="&lt;p&gt;The reason is that there is no analogue of Church–Turing thesis in logic.&lt;/p&gt;&#xA;&lt;p&gt;Let me explain. In computational theory there is well-known Church–Turing thesis that states that all reasonable definitions of computation are equivalent to each other. That is, all possible definitions of computation (like Turing machine, lambda-calculus, Minsky machine) turn out to be equivalent: they define exсctly the same set of computable functions. So there is one natural definition of computation, you don't need to pay attention to details.&lt;/p&gt;&#xA;&lt;p&gt;This is somehow misterius. But what is much more misterius, is that there is no such thing in logic! There is no one way to define what proofs are and what is provable. Every historical attempt definitions of proofs turned out to define different set of theorems. There are constructive proofs and non constructive proofs (and there are many flavors of constructivism). But this is not all. You can use set theory or type theory or some more exoctic way to define mathematics. They are not equivalent! Suppose you decide to pick one - say set theory. But there are many set theories like ZF, ZFC and so on. Even if you settle on your favorite theory, like ZFC, then there still many variants in details of definion that may lead to different set of theorems.&lt;/p&gt;&#xA;&lt;p&gt;So there in no any natural universan definition of mathematical proof.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2021-06-18T04:09:39.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13288" PostTypeId="2" ParentId="13249" CreationDate="2021-06-18T10:05:11.613" Score="1" Body="&lt;p&gt;I do not completely agree with Alexander Eremenko's answer (though I understand his point - just hope the following does not sound too provocative).&lt;/p&gt;&#xA;&lt;p&gt;Was Italian algebraic geometry at the beginning of the XXth century providing proofs of their statement? Yes, certainly, though a number of them turned out to be wrong, therefore not rigorous. And yet their arguments revealed a good amount of informations on &amp;quot;true&amp;quot; algebraic varieties once some additional hypothesis were added. Certainly at the time there were published there was wide consensus on the fact that those proofs were rigorous.&lt;/p&gt;&#xA;&lt;p&gt;Were papers published in many URSS journals between the 50ies and 80'ies rigorous? Proofs usually were given in the form of very brief hints on the line of thought of the proof and details were abundantly left to the reader. Still there was a widespread consensus on the fact that those can be considered proofs, sketchy as they were (and some of them turned out to be not completely correct after all). Would they have been accepted in a Western math journal without questioning their correctness at the time? Probably not, so that one should say that the standard of rigor was at least country-dependent.&lt;/p&gt;&#xA;&lt;p&gt;A proof is a couple of different things. A completely clear and rigorous set of steps from some hypothesis (and under some agreed upon set of rules which, as Kopylov remarked, were never  completely agreed upon, see the dispute around Axiom of Choice). But in this sense proofs exists only in dreams - or rather, at present, in computer-verified proofs. Then, also, they are a way to communicate to other mathematicians a line of thought, more or less detailed, around which a community agrees that it can be turned into a rigorous and complete proof. And here you see my choice of words: &amp;quot;communicate&amp;quot;, &amp;quot;community&amp;quot;,  &amp;quot;agreement&amp;quot; - it is partly a social issue and as such it changes with time and place.&lt;/p&gt;&#xA;&lt;p&gt;Is a &amp;quot;proof&amp;quot; what it is supposed to be if there is no one on Earth able to comprehend it? Not so sure about it. Mochizuki's proof of the &amp;quot;ABC&amp;quot; conjecture comes to mind here. But what about the classification of finite simple groups? Are we sure it is proved if after its &amp;quot;proof&amp;quot; so many different mistakes were found and fixed, or do we simply just believe in it?&lt;/p&gt;&#xA;&lt;p&gt;Is the authority of who states the proof really, in practice, not relevant? Are we really ready to accept that an argument is a proof if it comes from an unknown amateur researcher with no affiliation and no previous research record than we are ready to accept it from a Fields medalist? And how it comes so, if proofs are carved in marble for eternity?&lt;/p&gt;&#xA;&lt;p&gt;How it comes that we feel safer about a statement in math if we know that it can be proved with different techniques? A statement is true or false, after all: but in our research experience we remember way too well of that many times that we strongly believed that some math statement was true (as a consequence of very convincing arguments) and later realized it was wrong.&lt;/p&gt;&#xA;&lt;p&gt;So my answer to this question is: a math proof is always, partly, a &amp;quot;social contract&amp;quot; and from this perspective no doubts that its standards changed and will keep on changing.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2021-06-18T10:05:11.613" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13289" PostTypeId="2" ParentId="13279" CreationDate="2021-06-18T10:39:07.077" Score="6" Body="&lt;p&gt;I'd like to add my comment as an answer to have memory of a side comment.&lt;/p&gt;&#xA;&lt;p&gt;As I was saying the subtractive notation was a way of sparing characters in carving and this is the reason behind it becoming popular.&lt;/p&gt;&#xA;&lt;p&gt;@paul garret made a very interesting comment on the fact that with Latin numerals computations were not reported on paper/parchment and computations with the abacus were effective since they did not require keeping track of intermediate steps.&lt;/p&gt;&#xA;&lt;p&gt;This was, in fact, one of the reasons of a quarrel after Fibonacci introduced Indian digits in his Liber Abbaci (&lt;em&gt;Book of Calculations&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;At first computations with what we nowadays call Arabic digits were considered to be cumbersome because they required keeping track of intermediate steps. But, at the time, paper was not widespread in Europe (paper is another invention that arrived in Europe from China through Arabs) and was very expensive.&lt;/p&gt;&#xA;&lt;p&gt;Abachists kept preferring abacus computations for years since it was cheaper and still effective when compared to computations made by &amp;quot;algoritmists&amp;quot; - such were termed those in favour of arab numerals.&lt;/p&gt;&#xA;&lt;p&gt;Only after some years, when starting from Fabriano, Italian production of paper became quite standard and costs dropped, algoritmists won their war.&lt;/p&gt;&#xA;&lt;p&gt;So at least twice in this story the material required for writing numbers played a role.&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="12316" LastEditDate="2021-06-19T06:33:20.043" LastActivityDate="2021-06-19T06:33:20.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13291" PostTypeId="1" CreationDate="2021-06-19T23:30:02.457" Score="1" ViewCount="222" Body="&lt;p&gt;When in history did the notion of space, geometric space appear? I. e. when in history geometric space was treated or thought of as a whole, as the site in which all geometric objects exist? When I think of Euclid's Elements, I have the impression that it just treats relations between segments, areas; straight lines, planes (?), but doesn't treat space as a whole, as the site in which all geometric objects exist. (I am assuming the whole content in Euclid's Elements is what I said it is, but I have never read it thoroughly.)&lt;/p&gt;&#xA;" OwnerUserId="14802" LastEditorUserId="10106" LastEditDate="2022-01-30T05:36:19.803" LastActivityDate="2022-01-30T05:36:19.803" Title="When did the notion of space, geometric space appear?" Tags="|mathematics|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13292" PostTypeId="2" ParentId="13291" CreationDate="2021-06-20T03:12:46.370" Score="1" Body="&lt;p&gt;It is the opposite: for Euclid there is nothing but the (2- or 3-)space. All the other things (points, lines, planes, etc.) live there.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-06-20T03:37:12.937" LastActivityDate="2021-06-20T03:37:12.937" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13293" PostTypeId="1" CreationDate="2021-06-20T08:42:28.967" Score="5" ViewCount="311" Body="&lt;p&gt;According to the Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89variste_Galois&quot; rel=&quot;noreferrer&quot;&gt;Évariste Galois&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He submitted his memoir on equation theory several times, but it was&#xA;never published in his lifetime due to various events. Though his&#xA;first attempt was refused by Cauchy, in February 1830 following&#xA;Cauchy's suggestion he submitted it to the Academy's secretary Joseph&#xA;Fourier, to be considered for the Grand Prix of the Academy.&#xA;&lt;strong&gt;Unfortunately, Fourier died soon after, and the memoir was lost.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Abel's memoir was &lt;a href=&quot;https://hsm.stackexchange.com/questions/5341/did-cauchy-forget-or-lose-mathematical-papers-aside-from-abels-and-galoiss&quot;&gt;lost in a similar circumstance&lt;/a&gt; by Cauchy and Legendre some years prior to this incident and was only recovered partially, &lt;a href=&quot;https://www.abelprize.no/nedlastning/litteratur/centina_parisan.pdf&quot; rel=&quot;noreferrer&quot;&gt;until 2002 when the last few missing pages were also found&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question: Similar to this, has Galois' lost memoir ever been found?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="5189" LastEditorUserId="4251" LastEditDate="2021-06-21T04:54:05.260" LastActivityDate="2021-06-21T04:54:05.260" Title="Lost memoir of Évariste Galois" Tags="|mathematics|mathematicians|reference-request|galois-theory|galois|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13294" PostTypeId="2" ParentId="13293" CreationDate="2021-06-21T03:22:28.787" Score="4" Body="&lt;p&gt;There have been several versions of the memoir. The last version was sent by Galois to his friend  &lt;a href=&quot;https://www.ias.ac.in/article/fulltext/reso/004/10/0093-0100&quot; rel=&quot;nofollow noreferrer&quot;&gt;Auguste Chevallier&lt;/a&gt;.   just before the duel and was published by Camille Jordan. For details see the &lt;a href=&quot;https://ems-ph.org/books/book.php?proj_nr=137&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-06-21T03:36:20.197" LastActivityDate="2021-06-21T03:36:20.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13295" PostTypeId="1" AcceptedAnswerId="13296" CreationDate="2021-06-21T18:44:23.083" Score="20" ViewCount="4596" Body="&lt;p&gt;Is it OK to write Emmy Nöther instead of Emmy Noether?  I always assumed that the oe was an anglicization of ö (o with an umlaut).  But when I look her up, I never see Nöther.&lt;/p&gt;&#xA;" OwnerUserId="13029" LastActivityDate="2021-06-24T10:59:23.433" Title="Emmy Noether or Emmy Nöther?" Tags="|terminology|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13296" PostTypeId="2" ParentId="13295" CreationDate="2021-06-21T20:55:02.643" Score="34" Body="&lt;p&gt;Taking a look at &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF01452845.pdf&quot; rel=&quot;noreferrer&quot;&gt;her 1933 paper in Mathematische Annalen&lt;/a&gt; one sees:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CHgEg.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CHgEg.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Similarly for a &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF01458137.pdf&quot; rel=&quot;noreferrer&quot;&gt;1923 paper&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/glvtW.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/glvtW.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;From a glance at a few other papers, she (or all the journals) used &amp;quot;Noether&amp;quot; for her last name. Further, since the titles clearly indicate that the journals are quite happy with umlauts it is not by accident.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2021-06-21T21:03:04.783" LastActivityDate="2021-06-21T21:03:04.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13297" PostTypeId="2" ParentId="13295" CreationDate="2021-06-22T09:14:02.670" Score="11" Body="&lt;p&gt;“Goethe” is not an “anglicization” of “Göthe” or indeed of “Göte”; it is the way the poet spelt his own name. The overriding principle is that everyone is entitled to spell his or her name as he or she likes.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2021-06-22T09:14:02.670" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13298" PostTypeId="2" ParentId="13281" CreationDate="2021-06-22T09:56:19.527" Score="0" Body="&lt;p&gt;Consider a magnet. By playing around with it for a short time, you'll observe that the magnet exerts an influence on other magnetic objects, which is stronger when they're closer. This influence is outside the physical body of the magnet itself - it extends into the space around it, whether that space is occupied or not. We can form a concept of the &lt;em&gt;magnetic field&lt;/em&gt;, and assign a quantity of some kind to different points in space, representing the strength and direction of the attraction.&lt;/p&gt;&#xA;&lt;p&gt;In a similar way, by looking closely at the sun and the planets, we can theorize that they exert a force on each other, which extends out into the empty space between. (We might, or might not, imagine that the force is carried by some substance called &amp;quot;the aether&amp;quot;, but in any case we have a way to assign a quantity to each point in space.) This is a &lt;em&gt;gravitational field&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Again, considering water flowing in a pipe, you can assign a speed and direction to each tiny bit of water, as Leonhard Euler did in his work on hydrodynamics (&lt;em&gt;Mechanica sive motus scientia analytice exposita&lt;/em&gt;, 1736).&lt;/p&gt;&#xA;&lt;p&gt;The mathematical formulation of a vector field is a way of giving formal structure to these notions. The term &lt;em&gt;field&lt;/em&gt; was used before the formalization of &lt;em&gt;vector field&lt;/em&gt;, &lt;em&gt;scalar field&lt;/em&gt;, etc., to talk about the area of influence around a body. It took some time to develop the unified mathematical theory, and to sort out various competing notions of what was going on physically - so people like Kepler or Newton might not have had the same mental picture as we do in modern times. But there was certainly some understanding of bodies exerting a sort of influence on the surrounding space, even if a particular bit of that space is not occupied.&lt;/p&gt;&#xA;&lt;p&gt;One of the critical papers for this development is Maxwell's paper &lt;em&gt;On physical lines of force&lt;/em&gt; (1861). He spends some time talking about scattering iron filings near a magnet to see that they form lines, and trying to justify from similar observations that it is right to model magnetism by giving a magnitude and a direction (i.e., a vector) for each point surrounding the magnetic object. Later in the paper, he mentions &amp;quot;a field of magnetic force such as that of the earth&amp;quot;, and generally &amp;quot;the magnetic field&amp;quot;. What he's doing here is giving a new mathematical intepretation (in modern words, a vector field) of the intuitive concept that magnets have a field of influence around them.&lt;/p&gt;&#xA;&lt;p&gt;The English word &lt;em&gt;field&lt;/em&gt; is suitable for the intuitive concept, because of its sense as a marked-off physical area that's dedicated for some common reason - in agriculture, or as a battlefield, that kind of thing. Its abstract meanings, as in &amp;quot;a field of study&amp;quot;, are also well-established. So it's a plausible term for the area of influence. In other European languages we use similar words; German &lt;em&gt;Feld&lt;/em&gt;, French &lt;em&gt;champ&lt;/em&gt;, Italian &lt;em&gt;campo&lt;/em&gt;, Russian поле.&lt;/p&gt;&#xA;" OwnerUserId="14810" LastActivityDate="2021-06-22T09:56:19.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13301" PostTypeId="1" AcceptedAnswerId="13501" CreationDate="2021-06-23T00:25:05.213" Score="5" ViewCount="161" Body="&lt;p&gt;I was watching &lt;a href=&quot;https://www.youtube.com/watch?v=3JNn-vggUH4&quot; rel=&quot;noreferrer&quot;&gt;this series of lectures on universal algebra&lt;/a&gt; on YouTube and the instructor, Charlotte Aten, mentioned that Øystein Ore studied lattices with the goal of using lattices as a unifying abstraction for mathematical structures. I don't remember which exact lecture she said this in, but she compares the goals of the lattice-based approach with category theory, which has been successful as a unifying abstraction.&lt;/p&gt;&#xA;&lt;p&gt;I'm curious what the &amp;quot;lattice program&amp;quot; (so to speak) achieved, how it was perceived at the time, where it got stuck, and how it ended.&lt;/p&gt;&#xA;" OwnerUserId="9243" LastActivityDate="2021-08-01T00:36:59.173" Title="Why did Øystein Ore's lattice-based approach disappear?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13302" PostTypeId="2" ParentId="13271" CreationDate="2021-06-23T07:20:46.570" Score="1" Body="&lt;p&gt;Though by no means exhaustive, there are a number of literary sources in the anthology &lt;em&gt;Literature and Science in the Nineteenth Century&lt;/em&gt; edited by Laura Otis. The book contains the writings of novelists, poets and scientists.&lt;/p&gt;&#xA;&lt;p&gt;The list of contents can be found on Google Books: &lt;a href=&quot;https://books.google.co.uk/books?id=b8YUDAAAQBAJ&amp;amp;lpg=PP1&amp;amp;ots=LsIxN-FwlH&amp;amp;dq=literature%20and%20science%20in%20the%20nineteenth%20century%20contents&amp;amp;lr&amp;amp;pg=PP1#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.co.uk/books?id=b8YUDAAAQBAJ&amp;amp;lpg=PP1&amp;amp;ots=LsIxN-FwlH&amp;amp;dq=literature%20and%20science%20in%20the%20nineteenth%20century%20contents&amp;amp;lr&amp;amp;pg=PP1#v=onepage&amp;amp;q&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14819" LastActivityDate="2021-06-23T07:20:46.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13303" PostTypeId="1" AcceptedAnswerId="13310" CreationDate="2021-06-23T17:35:11.557" Score="3" ViewCount="369" Body="&lt;p&gt;It was not before the early fifties of the last century that the shape of the Milky Way was figured out. Uptill then only other galaxies were known to have certain shapes&lt;/p&gt;&#xA;&lt;p&gt;It's always easier to find out the&#xA;shape of a form if you are looking at it from a distance. But in the case of the Milky Way there is no way to get outside of it.&lt;/p&gt;&#xA;&lt;p&gt;This seems like a big obstacle. And we can't even move in it significantly ( wrt the stars in it). On top of that the Milky Way looks pretty static too (but looks can deceive).&lt;/p&gt;&#xA;&lt;p&gt;How is this obstacle surmounted? Was it indeed that big that it took so long to figure out its shape? It looks as if its size (or the extent) is easier to find. Were the shapes of other galaxies helpfull?&lt;/p&gt;&#xA;&lt;p&gt;Why did it take so long to find out the shape of the Milky Way?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-06-23T17:41:06.253" LastActivityDate="2021-06-29T07:11:38.613" Title="Why did it take so long to discover the shape of the Milky Way?" Tags="|astronomy|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13304" PostTypeId="1" AcceptedAnswerId="13309" CreationDate="2021-06-23T21:35:25.300" Score="1" ViewCount="133" Body="&lt;p&gt;I know the Watt, as a unit of measuring power was defined somewhere in early 1900's.&lt;/p&gt;&#xA;&lt;p&gt;But who was the first physicist to consider &amp;quot;work over time&amp;quot;? Perhaps Newton?&lt;/p&gt;&#xA;&lt;p&gt;Problem with searching something like this is that it keeps coming up with &amp;quot;Who invented electricity&amp;quot; - but I think power, as form of &amp;quot;energy conversion&amp;quot; or &amp;quot;work over time&amp;quot;, surely existed before we were researching electricity!&lt;/p&gt;&#xA;" OwnerUserId="14828" LastEditorUserId="14828" LastEditDate="2021-06-24T16:17:36.237" LastActivityDate="2021-06-24T16:17:36.237" Title="When was the concept of “Power” defined?" Tags="|physics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13306" PostTypeId="2" ParentId="13295" CreationDate="2021-06-24T10:04:42.243" Score="4" Body="&lt;p&gt;As you can read on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Max_Noether&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article about Emmy's father, Max Noether&lt;/a&gt;, the family only received their name in 1809. The generation of the great-grandfather of Emmy.&lt;/p&gt;&#xA;&lt;p&gt;The first spelling I found of Elias Samuel, is &amp;quot;Netter&amp;quot;, but &amp;quot;Noether&amp;quot; was already used as alternative: &lt;a href=&quot;https://www.geni.com/people/Elias-Noether/6000000006433734670&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geni.com/people/Elias-Noether/6000000006433734670&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;His son, Hermann (Herz Elias) Noether seems to have used &amp;quot;Nöther&amp;quot; as a surname. &lt;a href=&quot;https://www.geni.com/people/Hermann-Herz-N%C3%B6ther/6000000002955209487&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geni.com/people/Hermann-Herz-N%C3%B6ther/6000000002955209487&lt;/a&gt;, it's also spelled Nöther on his tombstone: &lt;a href=&quot;https://images.findagrave.com/photos/2021/141/138798362_02a7afc7-e049-4bf2-9185-973df3f39bf9.jpeg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://images.findagrave.com/photos/2021/141/138798362_02a7afc7-e049-4bf2-9185-973df3f39bf9.jpeg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;From Emmy's father, Max, onwards, the family seems to standardise on &amp;quot;Noether&amp;quot;, perhaps because they already had more international contacts: &lt;a href=&quot;https://www.geni.com/people/Max-Noether/6000000002955209501&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.geni.com/people/Max-Noether/6000000002955209501&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even on &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Schild,_Geburtshaus_Emmy_Noether_-_Hauptstra%C3%9Fe_23,_Erlangen.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;German signs&lt;/a&gt;, Emmy's surname is spelled as Noether. So it's safe to assume that this was the preferred spelling by the time Emmy lived. Though other spellings have been used in their family.&lt;/p&gt;&#xA;" OwnerUserId="12794" LastActivityDate="2021-06-24T10:04:42.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13307" PostTypeId="2" ParentId="13295" CreationDate="2021-06-24T10:59:23.433" Score="3" Body="&lt;p&gt;Speaking as a German I wouldn't change a oe to an ö when it's about a name. Also in Dutch (which can seem quite similar to German) oe is not pronounced like an ö but more like a German u. So I would say it's safer to stay away from changing the spelling :)&lt;/p&gt;&#xA;" OwnerUserId="14833" LastActivityDate="2021-06-24T10:59:23.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13309" PostTypeId="2" ParentId="13304" CreationDate="2021-06-24T13:02:59.233" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://physics.info/power/&quot; rel=&quot;noreferrer&quot;&gt;This source&lt;/a&gt; says the following about James Watt,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Watt is most famous for inventing an improved steam engine in the&#xA;years around 1770 and slightly less famous for inventing the concept&#xA;of power shortly thereafter.  Watt wouldn't have thought about power&#xA;they same way we do today. The concept of energy wasn't invented until&#xA;after he died. For him, power was the product of force and velocity.&lt;/p&gt;&#xA;&lt;p&gt;P = Fv&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That page references a &lt;a href=&quot;https://books.google.com/books?id=kmUJAAAAQAAJ&amp;amp;pg=324&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt;,&#xA;&amp;quot;The Edinburgh Review, Oct 1808&amp;quot;  containing a treatise by Olinthus Gregory on steam engines.&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-06-24T13:02:59.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13310" PostTypeId="2" ParentId="13303" CreationDate="2021-06-24T22:03:05.087" Score="4" Body="&lt;p&gt;The short answer is that we are inside the Milky Way instead of looking at it from outside.&lt;/p&gt;&#xA;&lt;p&gt;And it is not exactly true that it was not until the 1950s that the shape of the Milky Way was figured out.&lt;/p&gt;&#xA;&lt;p&gt;As soon as Galilleo pointed a telescope at the Milky Way and resolved it into many stars, it has been known that the light of the Milky Way is composed of the light of many, many stars blended together.&lt;/p&gt;&#xA;&lt;p&gt;There were three possible asumptions about the Milky Way stars.&lt;/p&gt;&#xA;&lt;p&gt;One) that they were smaller than the naked eye stars, and so could be at the same distance as the naked eye stars.&lt;/p&gt;&#xA;&lt;p&gt;Two) that they were so much smaller than the naked eye stars that they were much closer to Earth than the naked eye stars.&lt;/p&gt;&#xA;&lt;p&gt;Three) that they were as large as the naked eye stars,and thus were so much fainter than the naked eye stars because they were very much father away than the naked eye stars.&lt;/p&gt;&#xA;&lt;p&gt;Starting in the 17th century astronomers tried to measure the parallaxes of stars to find their distance.  In the Hellenistic age, it was realized that if the Earth revolved around  the Sun, the positions of the stars should move during the year as the Earth moved.  Since no such movement could be detected in ancient times, that was one of the arguments used against the ancient heliocentric theory.&lt;/p&gt;&#xA;&lt;p&gt;When early modern astronomers accepted the Heliocentry theory, they realized that detecting the parallax of stars would be a major piece of evidence for the heliocentric theory and would help figure out the size and shape of the universe.&lt;/p&gt;&#xA;&lt;p&gt;So attempts to measure the parallaxes of stars were made and failed for over 150 years.  If any astrnomers believed that the Milky Way stars were closer to Earth and chose to measure their parallaxes, the failure to do so would prove that the Milky Way stars must not be any closer than the naked eye stars.  Finally, more advanced instruments in the 1830s enabled the parallaxes of 61 Cygni, Alpha centauri, and Vega, to be measured, followed by those of a few other stars in the following decades.&lt;/p&gt;&#xA;&lt;p&gt;Those proved, among other things, that stars were not all at the same distance, so they were not all on the surface of a spherical shell around Earth.  They also showed that, when the apparent magnitudes of stars were compared to their distances, the absolute magnitudes of stars could be calaculated, and those absolute magnitudes varied greatly.  So the apparent magnitude of a star was a very poor indicator of the distance of that star.&lt;/p&gt;&#xA;&lt;p&gt;Back in the 18th century, the Milky Way was assumed to be either a disc or a torus shape with the Sun and Earth inside it.  Thus the stars of the Milky Way appeared as a band across the entire sky as seen from Earth, inside the Milky Way.&lt;/p&gt;&#xA;&lt;p&gt;By then number of tiny hazy patches of light - &amp;quot;nebulae&amp;quot; - in the sky have been discovered, and some people speculated that since they looks like tiny bits of the Milky Way, the Milky Way might not be the entire universe,  and there could be other things like the Milky Way at great distances from it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proof of the Milky Way consisting of many stars came in 1610 when Galileo Galilei used a telescope to study the Milky Way and discovered that it is composed of a huge number of faint stars.[224][225] In a treatise in 1755, Immanuel Kant, drawing on earlier work by Thomas Wright,[226] speculated (correctly) that the Milky Way might be a rotating body of a huge number of stars, held together by gravitational forces akin to the Solar System but on much larger scales.[227] The resulting disk of stars would be seen as a band on the sky from our perspective inside the disk. Wright and Kant also conjectured that some of the nebulae visible in the night sky might be separate &amp;quot;galaxies&amp;quot; themselves, similar to our own. Kant referred to both the Milky Way and the &amp;quot;extragalactic nebulae&amp;quot; as &amp;quot;island universes&amp;quot;, a term still current up to the 1930s.[228][229][230]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Milky_Way#Astronomical_history%5B1%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Milky_Way#Astronomical_history[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I note that the German astronomer Johann Heinrich von Maedler (1794-1874) believed that all the stars in the Milky Way revolved around a center of mass (correct), and that there was a central Sun at that point (not so correct), and that point was at or near Alcyone in the Pleaides star cluster (which much too close to Earth and in the opposite direction from the real galactic center).&lt;/p&gt;&#xA;&lt;p&gt;My copy of &lt;em&gt;Men of Other Planets&lt;/em&gt; by Kenneth Heuer, 1951, 1954, says on page 134:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And there are estimated to be upwards of 10,000,000,000 stars alone in the great system of stars we call the Milky Way.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, the sun has not received any special privilege, although it is suspected to be larger and brighter than the average star.  Comparing the shape of the Milky Way Galaxy to a wheel,the sun is located about 3/5 of the way from the hub to the edge, while the entire star system has a diameeter of approximatley 100,000 light years.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And there is a rough diagram of the shape of the Milky Way on page 134.&lt;/p&gt;&#xA;&lt;p&gt;So by 1951, or 1954 at the very latest, the shape of the Milky Way Galaxy was known fairly accurately by contempoarary standards.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first project to describe the shape of the Milky Way and the position of the Sun was undertaken by William Herschel in 1785 by counting the number of stars in different regions of the sky. He produced a diagram of the shape of the galaxy with the Solar System close to the center.[33][34] Using a refined approach, Kapteyn in 1920 arrived at the picture of a small (diameter about 15 kiloparsecs) ellipsoid galaxy with the Sun close to the center. A different method by Harlow Shapley based on the cataloguing of globular clusters led to a radically different picture: a flat disk with diameter approximately 70 kiloparsecs and the Sun far from the center.[31] Both analyses failed to take into account the absorption of light by interstellar dust present in the galactic plane, but after Robert Julius Trumpler quantified this effect in 1930 by studying open clusters, the present picture of our host galaxy, the Milky Way, emerged.[35]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Galaxy#Milky_Way%5B2%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Galaxy#Milky_Way[2]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So this shows that the shape and size of the galaxy, and the Sun's position in it, was determined by astroomers sometime during the period of 1930 to 1951.&lt;/p&gt;&#xA;&lt;p&gt;I note that with the developement of radio astronomy after WWII, astronomers traced the radio waves emitted by interstellar clouds of hygrogen and began to use them to trace the spiral arms of the Milky Way.  This was already being done by the 1960s.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12724" LastEditDate="2021-06-29T07:11:38.613" LastActivityDate="2021-06-29T07:11:38.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13311" PostTypeId="1" CreationDate="2021-06-24T23:15:28.557" Score="2" ViewCount="123" Body="&lt;p&gt;First please enjoy &lt;a href=&quot;https://hsm.stackexchange.com/a/6035/3869&quot;&gt;this wonderful answer to &lt;em&gt;How did Newton explain his interference rings without wave optics?&lt;/em&gt;&lt;/a&gt; which describes the story behind what we now call &lt;a href=&quot;https://en.wikipedia.org/wiki/Newton%27s_rings&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's rings&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Below I show an example of the rings using a low pressure therefore fairly monochromatic sodium lamp; if angular spread is managed the number of fringes potentially viewable can be dozens or more. Then I show some examples of fringes using white light. These give some impression of what one would see, but the actual contrast we would see (or in photograph) in color depends on the spectrum of light and the spectral responses of the &amp;quot;color channels&amp;quot; of our vision system or the color camera used.&lt;/p&gt;&#xA;&lt;p&gt;What Newton could do with a narrow beam of sunlight in a darkened room continues to floor me! He made ample use of prisms to separate sunlight into colors and he might have viewed &amp;quot;thin plate&amp;quot; or thin film or thin gap interference patterns with wavelength-restricted beams via prism dispersion.&lt;/p&gt;&#xA;&lt;p&gt;But I wonder if he also looked at such interference effects by using color filters.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Did Newton ever use filtered or prism-dispersed colored light to view &amp;quot;Newton's rings&amp;quot; or other thin-film interference effects?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MgBvO.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MgBvOm.jpg&quot; alt=&quot;Newton's Rings as observed through a microscope. The lens used is a 20cm convex lens and the light source is a sodium lamp.&quot; /&gt;&lt;/a&gt; (click for larger)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton's Rings as observed through a microscope. The lens used is a 20cm convex lens and the light source is a sodium lamp. &lt;a href=&quot;https://en.wikipedia.org/wiki/File:20cm_Air_1.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NZSdf.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NZSdfm.jpg&quot; alt=&quot;Newton's rings in white light; Fundamental problems of 17th Century Physics The Nature&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;https://i.stack.imgur.com/Z047S.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Z047S.jpg&quot; alt=&quot;ewtons Ring, Engineering Physics by Dr. Amita Maurya, Peoples University, Bhopal&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;left:&lt;/strong&gt; Source: &lt;a href=&quot;https://slidetodoc.com/fundamental-problems-of-17-thcentury-physics-the-nature/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fundamental problems of 17th Century Physics The Nature&lt;/a&gt; (click for larger) &lt;strong&gt;right:&lt;/strong&gt; Source: &lt;a href=&quot;https://sites.google.com/site/puenggphysics/home/Unit-II/newtons-ring&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newtons Ring, Engineering Physics by Dr. Amita Maurya, Peoples University, Bhopal&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-08-07T04:42:05.683" LastActivityDate="2023-07-02T04:20:42.160" Title="Did Newton ever use filtered or prism-dispersed colored light to view &quot;Newton's rings&quot; or other thin-film interference effects?" Tags="|newton|optics|technology|light|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13312" PostTypeId="1" AcceptedAnswerId="13313" CreationDate="2021-06-25T11:41:05.040" Score="4" ViewCount="641" Body="&lt;p&gt;From what I understand musical isomorphism is the formal way to speak about raising the index of a vector field (&lt;a href=&quot;https://en.wikipedia.org/wiki/Musical_isomorphism&quot; rel=&quot;nofollow noreferrer&quot;&gt;refer here&lt;/a&gt;). But what I don't understand is why is it called musical isomorphism! In the Wikipage, the following is said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The exact origin of this notation is not known, but the term musicality in this context would be due to Marcel Berger&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Was there any documentation on what exactly was musical in this to Marcel? BTW I am using the word musical in the following sense:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Musical, also called musical comedy, theatrical production that is characteristically sentimental and amusing in nature, with a simple but distinctive plot, and offering music, dancing, and dialogue.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user12582" LastEditorUserId="10106" LastEditDate="2022-02-03T18:17:38.483" LastActivityDate="2022-02-03T18:17:38.483" Title="What makes the musical isomorphism, musical?" Tags="|mathematics|differential-geometry|naming-conventions|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13313" PostTypeId="2" ParentId="13312" CreationDate="2021-06-25T12:29:37.790" Score="2" Body="&lt;p&gt;The linked &lt;a href=&quot;https://mathoverflow.net/questions/69074/the-origin-of-the-musical-isomorphisms&quot;&gt;MathOverflow&lt;/a&gt;  page (from the Wiki page) says that Berger himself doesn't remember how the name came about.  The ♭: V → V* and ♯: V* → V  notation uses musical symbols, which to me suggests  an infinite time-travel loop between naming and selection of symbols :-) .&lt;/p&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-06-25T12:29:37.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13315" PostTypeId="1" CreationDate="2021-06-25T14:35:56.840" Score="0" ViewCount="81" Body="&lt;p&gt;Anyone know the history of using hyperbolic functions to parameterize the parabola?  Wolfram hasn't really helped. Anyone know where to look?&lt;/p&gt;&#xA;" OwnerUserId="14845" LastEditorUserId="14845" LastEditDate="2021-06-25T15:42:34.897" LastActivityDate="2021-06-25T15:44:15.570" Title="Who first started parameterizing hyperbolas with hyperbolic functions?" Tags="|algebraic-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13316" PostTypeId="2" ParentId="13315" CreationDate="2021-06-25T15:39:10.150" Score="1" Body="&lt;p&gt;For the old version of the question, who first used the terms:&lt;/p&gt;&#xA;&lt;p&gt;I got this at &lt;a href=&quot;https://jeff560.tripod.com/h.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;HYPERBOLIC SINE and HYPERBOLIC COSINE. Vincenzo Riccati (1707-1775) introduced hyperbolic functions in volume I of his &lt;em&gt;Opuscula ad Res Physicas et Mathematicas pertinentia&lt;/em&gt; of 1757. Presumably he used these terms, since he used the notation Sh &lt;em&gt;x&lt;/em&gt; and Ch &lt;em&gt;x&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2021-06-25T15:44:15.570" LastActivityDate="2021-06-25T15:44:15.570" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13317" PostTypeId="1" AcceptedAnswerId="13331" CreationDate="2021-06-25T23:03:28.627" Score="1" ViewCount="286" Body="&lt;p&gt;Tarski's truth theorem asserts that a truth definition for a (reasonably strong) theory cannot be formalized within that theory.&lt;/p&gt;&#xA;&lt;p&gt;I have seen that Tarski's theory of truth has received a lot of criticism. For examples, Hilary Putnam commented in “A comparison of something with something else”: As a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail; Hartry Field in an influential discussion and diagnosis of what is lacking in Tarski’s account, pointed out that Tarski’s theory does not offer an account of reference and satisfaction at all. Rather, it only offers a number of disquotation clauses such as &amp;quot;‘Snow’ refers to snow&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So I wonder what is the status of Tarski's theory of truth in mathematical and philosophical logic. Is it widely accepted or not?&lt;/p&gt;&#xA;" OwnerUserId="2014" LastEditorUserId="4251" LastEditDate="2021-06-27T14:49:08.320" LastActivityDate="2021-07-07T18:55:05.650" Title="The status of Tarski's theory of truth" Tags="|philosophy-of-science|mathematical-logic|logic|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13318" PostTypeId="1" AcceptedAnswerId="13325" CreationDate="2021-06-26T17:35:41.220" Score="0" ViewCount="271" Body="&lt;p&gt;I have been reading an introductory text in mathematical logic (Holden, 1995).  The final chapter presents the resolution of Hilberts's tenth problem concerning the integer roots of an arbitrary polynomial over &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The resolution (to the negative) follows from a theorem of &lt;a href=&quot;https://en.wikipedia.org/wiki/Yuri_Matiyasevich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yuri Matiyasevich&lt;/a&gt; which tells us that every recursively enumerable relation/function is Diophantine.&lt;/p&gt;&#xA;&lt;p&gt;On the one hand, it is not surprising that a resolution comes via mathematical logic - after all, it is a decidability problem.  On the other hand, it is a very natural question with a long history whose resolution is of fundamental importance to Diophantine analysis.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt;  What are some examples of mathematical logic being successfully applied elsewhere?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Rightly or wrongly, I consider set theory to be a part of mathematical logic, so, for example, the CH independence proofs of Cohen and Gödel I am considering to be part of logic.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2021-06-28T14:37:21.280" Title="What are some notable contributions of mathematical logic to mathematics (outside of mathematical logic)?" Tags="|mathematics|mathematical-logic|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13319" PostTypeId="1" AcceptedAnswerId="13335" CreationDate="2021-06-26T19:23:54.567" Score="8" ViewCount="357" Body="&lt;p&gt;I've been reading about 17th and 18th century geometers and their research into plane curves, especially algebraic curves.  A term that comes up frequently is &amp;quot;organic&amp;quot;. By context it seems to pertain to the mechanical construction of curves, such as &lt;a href=&quot;https://www.geogebra.org/m/hj2Cnmcd&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's &amp;quot;organic&amp;quot; construction of a conic&lt;/a&gt;.  Some names I see in connection with this are van Schooten, Newton, and MacLaurin.&lt;/p&gt;&#xA;&lt;p&gt;I cannot find a definition.  Apparently it is a term that was widely understood by geometers before 1900, like &amp;quot;species&amp;quot; and &amp;quot;right line&amp;quot;, but has become obsolete in modern times.&lt;/p&gt;&#xA;&lt;p&gt;Newton is thought to have fabricated sophisticated &amp;quot;organic rulers&amp;quot; to support his investigations into curves (see &lt;a href=&quot;https://pearl.plymouth.ac.uk/bitstream/handle/10026.1/3273/2015bloye209589phd.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloye, &lt;em&gt;The Geometrical Thought of Isaac&#xA;Newton&lt;/em&gt;&lt;/a&gt; and &lt;a href=&quot;http://www.stephenhuggett.com/Newton.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloye and Huggett, &lt;em&gt;Newton, the Geometer&lt;/em&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;A few archive.org links: &lt;a href=&quot;https://archive.org/details/proceedings3619151916roya/page/86/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tweedie&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/geometriaorgani00maclgoog/page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacLaurin&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/sirisaacnewtons00newtgoog/page/n33/mode/2up.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I'm looking for more information on the meaning and usage of this&#xA;term.  I'm also interested in more information on Newton's &amp;quot;organic&#xA;rulers.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm used to thinking that these days we are spoiled by being able to draw curves at will with Geogebra, but I suspect that we would be surprised at the forgotten tricks and technology used by the geometers of yore (the best term I can think of for the ancient and pre-1800 geometers) to visualize their subject.&lt;/p&gt;&#xA;" OwnerUserId="11204" LastEditorUserId="11204" LastEditDate="2021-06-27T15:31:08.163" LastActivityDate="2021-07-05T17:07:49.990" Title="What does &quot;organic&quot; mean in old texts when describing plane curves and their construction?" Tags="|reference-request|geometry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13320" PostTypeId="2" ParentId="13319" CreationDate="2021-06-26T20:10:47.560" Score="7" Body="&lt;p&gt;For historical usages of words, the Unabridged Oxford Dictionary is the best resource. Usages from 15th century can also be found. Under the entry &amp;quot;organic&amp;quot;, there is a historical usage.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mathematics. Of the drawing of a curve: performed by means of instruments; mechanical. Chiefly in organic description. Cf. earlier organical adj. 3c. historical.&#xA;Chiefly with reference to Isaac Newton.&lt;/p&gt;&#xA;&lt;p&gt;1860   C. R. M. Talbot tr. I. Newton Enumeration Lines of Third Order&#xA;vi. 26 (heading)    Of the Organic Description of Curves.&lt;/p&gt;&#xA;&lt;p&gt;1949   Amer. Math. Monthly 56 46   When only 21 years of age Maclaurin&#xA;wrote a work called Geometria Organica which was inspired..by the&#xA;organic description of a conic which we quoted from Newton's&#xA;Principia.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2021-06-26T20:42:51.433" LastActivityDate="2021-06-26T20:42:51.433" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13322" PostTypeId="2" ParentId="13318" CreationDate="2021-06-28T03:35:12.190" Score="5" Body="&lt;p&gt;The van den Dries-Wilkie &lt;a href=&quot;https://core.ac.uk/download/pdf/82511175.pdf&quot; rel=&quot;noreferrer&quot;&gt;proof&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Gromov%27s_theorem_on_groups_of_polynomial_growth#:%7E:text=Gromov%27s%20theorem%20states%20that%20a,that%20is%20of%20finite%20index.&quot; rel=&quot;noreferrer&quot;&gt;Gromov polynomial growth theorem&lt;/a&gt; is a non-trivial and very fruitful application of logic (model theory to be precise) to group theory and geometry.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-06-28T03:35:12.190" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13323" PostTypeId="2" ParentId="13312" CreationDate="2021-06-28T12:59:05.390" Score="7" Body="&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$\flat$&lt;/span&gt; map in music lowers the pitch of a note (by one half) and the corresponding map lowers indices, the &lt;span class=&quot;math-container&quot;&gt;$\sharp$&lt;/span&gt; map raises it, as well as the indices. And of course adding a flat after a sharp goes back to the original pitch, i.e. gives you the identity map.&lt;/p&gt;&#xA;&lt;p&gt;Isn't this all that should be said about it?&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2021-06-28T12:59:05.390" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13324" PostTypeId="1" CreationDate="2021-06-28T13:50:17.213" Score="5" ViewCount="191" Body="&lt;p&gt;For the umpteenth time, I'm reading in some book (no intention to pillory the authors here) that, say, 500 years ago people believed in a flat Earth, since, in the author's opinion, there is no apparent evidence of it being spherical. This is of course false, especially speaking about 500 years ago, when people built globes, circumnavigated Earth and so on. But it was, more or less, common-ish knowledge well before that: Dante describes how he and Virgil rotated their posture around the center of Earth, Aristotle wrote about it, ancient Greeks in general observed masts before hulls and measured the radius of Earth, and so on.&lt;/p&gt;&#xA;&lt;p&gt;My question is: assuming that that author referred not to humanity as a whole but to the “average” person, would they be right? Is it possible to estimate which fraction of humanity knew about Earth being spherical in different ages?&lt;/p&gt;&#xA;" OwnerUserId="634" LastActivityDate="2021-06-29T07:11:51.050" Title="How many people knew that Earth is spherical in different ages?" Tags="|geography|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13325" PostTypeId="2" ParentId="13318" CreationDate="2021-06-28T14:37:21.280" Score="3" Body="&lt;p&gt;There is a survey of such applications in&lt;/p&gt;&#xA;&lt;p&gt;Ehud Hrushovski,&#xA;Geometric model theory.&#xA;Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998).&#xA;Doc. Math. 1998, Extra Vol. I, 281–302.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-06-28T14:37:21.280" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13326" PostTypeId="2" ParentId="13324" CreationDate="2021-06-28T17:03:22.013" Score="5" Body="&lt;p&gt;In Medieval European and Muslim societies the educated believed in a spherical Earth, and there were very few geographical works which took biblical descriptins of a flat Earth seriously.&lt;/p&gt;&#xA;&lt;p&gt;However, it is of course uncertain how much that knowledge penetrated to the lesser educated members of society.&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, nobody in the Americas, not even in the most advanced civilizations, believed in a spherical Earth until the coming of Europeans.&lt;/p&gt;&#xA;&lt;p&gt;I lack knowledge of whether the educated elite people in African societies, except for Muslim and Christian societies, believed in a spherical Earth.&lt;/p&gt;&#xA;&lt;p&gt;I lack knowledge of whether south Asians believed in a spherical Earth.&lt;/p&gt;&#xA;&lt;p&gt;In East Asia, apparently there was no knowledge of a spherical Earth until Jesuit missionaries introduced the concept at the Ming dynasty court about 1600.&lt;/p&gt;&#xA;&lt;p&gt;In recent centuries education has spread so that at the present almost every child gets some education.  And in recent centuries Europeans travelled everywhere and colonised most of the world and had big influences on uncolonized countries, which included spreading modern technological and scientific knowledge.&lt;/p&gt;&#xA;&lt;p&gt;So if I had to guess, I might guess that in the year 1800 a minority of the world's population knew of a spheriodal Earth, and in the year 2000 the vast majority of the world population know of a spherical Earth.&lt;/p&gt;&#xA;&lt;p&gt;And see answers at: &lt;a href=&quot;https://history.stackexchange.com/questions/54196/when-did-the-spherical-shape-of-the-earth-become-common-knowledge%5B1%5D&quot;&gt;https://history.stackexchange.com/questions/54196/when-did-the-spherical-shape-of-the-earth-become-common-knowledge[1]&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="2905" LastEditDate="2021-06-29T07:11:51.050" LastActivityDate="2021-06-29T07:11:51.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13328" PostTypeId="1" CreationDate="2021-06-30T15:00:30.163" Score="1" ViewCount="116" Body="&lt;p&gt;All the logical notations I've seen, from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Begriffsschrift&quot; rel=&quot;nofollow noreferrer&quot;&gt;Begriffschrift notation&lt;/a&gt; on, place quantifiers before the proposition containing the variables the quantifier binds. For example, in modern notation we have well-formed formulas like the following.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \forall x \mathop. \exists y \mathop. P(f(x), y) $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think it took a while for the choice of &lt;span class=&quot;math-container&quot;&gt;$\forall$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\exists$&lt;/span&gt; as universal and existential quantifiers to stabilize, but I'm curious about their order relative to the formula associated with the quantifier. I'm wondering if there's historical usage of postfix quantifiers (strawman syntax shown below).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ P(f(x), y) \;\; \exists[y] \; \forall[x] $$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="9243" LastActivityDate="2021-06-30T15:00:30.163" Title="Usage of postfix notation for quantifiers" Tags="|notation|logic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13329" PostTypeId="1" CreationDate="2021-07-01T08:39:55.740" Score="1" ViewCount="164" Body="&lt;p&gt;Are there historians of science that are systematically recording current activity in research mathematics ?&lt;/p&gt;&#xA;&lt;p&gt;For example, a decade ago there was a lot of stuff through blogs and Mathoverflow, nowadays there at least sociological interactions on Twitter. All this will probably be easy to browse in the future thanks to the Wayback Machine.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;but what about private emails of researchers with colleagues and with journals, are there efforts to archive this very valuable material?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14874" LastEditorUserId="5989" LastEditDate="2021-07-02T06:44:19.957" LastActivityDate="2021-07-02T06:44:19.957" Title="Recording current development of research mathematics" Tags="|mathematics|mathematics-social-history|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13330" PostTypeId="2" ParentId="13329" CreationDate="2021-07-01T10:21:32.340" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;but what about private emails of researchers with colleagues and with journals, are there efforts to archive this very valuable material?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Private emails are supposed to be private and any efforts to archive them would be illegal.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-07-01T10:21:32.340" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13331" PostTypeId="2" ParentId="13317" CreationDate="2021-07-02T04:17:01.143" Score="1" Body="&lt;p&gt;&lt;strong&gt;EDIT 07/07/21&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I thought I would add more detail to my overly simplistic and perhaps somewhat misleading account of the philosophical criticism of Tarski’s theory of truth.&lt;/p&gt;&#xA;&lt;p&gt;There is criticism of Tarski’s T-Scheme (and recursive definition) and there is criticism of Tarski’s philosophical view that our intuitive notion of truth is self-contradictory.&lt;/p&gt;&#xA;&lt;p&gt;Historically, the earliest criticism of Tarski’s T-Scheme has its roots in the writing of the polymath &lt;a href=&quot;https://en.wikipedia.org/wiki/Frank_Ramsey_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;F.P. Ramsey&lt;/a&gt; and goes under the name &lt;em&gt;deflationism&lt;/em&gt;.  Ramsey identified the &lt;em&gt;equivalence principle&lt;/em&gt; which is present in Tarski’s T-Scheme as leading to the view that the idea of a truth predicate is redundant.  Thus, when we say “snow is white” is equivalent to saying “it is true that snow is white”, the difference between them is merely stylistic and the truth predicate is redundant.&lt;/p&gt;&#xA;&lt;p&gt;Other deflationists were quick to follow.  For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/P._F._Strawson&quot; rel=&quot;nofollow noreferrer&quot;&gt;P.F. Strawson&lt;/a&gt; argued that while “snow is white” does express a proposition, “it is true that snow is white” does not express a proposition about a proposition, as Tarski would contend.  Rather, it should be treated as a type of &lt;em&gt;endorsement&lt;/em&gt; of the proposition “snow is white”.&lt;/p&gt;&#xA;&lt;p&gt;Next to criticize  Tarski was &lt;a href=&quot;https://en.wikipedia.org/wiki/Willard_Van_Orman_Quine&quot; rel=&quot;nofollow noreferrer&quot;&gt;W.V.O Quine&lt;/a&gt; with a view called &lt;em&gt;quotationalism&lt;/em&gt;.  For Quine, when Tarski intends&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;”Snow is white” is true if and only if snow is white.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;the quotation marks turn the sentence &lt;em&gt;Snow is white&lt;/em&gt; into something noun-like.  This leads to a type of circularity since adding the truth predicate gives us back a sentence whose truth we understand as well or poorly as we understood the original sentence itself.  Quine gives his own analysis of truth predicates.&lt;/p&gt;&#xA;&lt;p&gt;Another philosophical issue with Tarski’s T-Scheme goes under the name of &lt;em&gt;indeterminacy&lt;/em&gt;.  Problems here include, for example, problems of &lt;em&gt;vagueness&lt;/em&gt;.  Natural languages include many &lt;em&gt;vague&lt;/em&gt; predicates - e.g., &lt;em&gt;is bald&lt;/em&gt;, &lt;em&gt;is tall&lt;/em&gt;, &lt;em&gt;is a heap&lt;/em&gt;.  This makes it hard, if not impossible to determine the truth of the axiom:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;”Augustin-Louis Cauchy was bald” is true if and only if Augustin-Louis Cauchy was bald.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a taste of some of the early criticisms of Tarski’s philosophical theory.  The &lt;a href=&quot;https://plato.stanford.edu/entries/truth/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stanford Encyclopedia of Philosophy&lt;/a&gt; provides a detailed account.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(&lt;strong&gt;Original answer&lt;/strong&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Regarding mathematical logic and the status of Tarski's semantic definition of truth and his theorem that there is no 1-ary relation that identifies the true formulae of formal arithmetic, these are fundamental to first-order logic.  I'm certainly not aware of any controversy amongst mathematical logicians.&lt;/p&gt;&#xA;&lt;p&gt;Regarding Tarski's philosophical ideas it is important to keep in mind that no philosophical theory is without controversy.  Any metaphysical theory will have to confront the realist/anti-realist divide.  In the case of Tarski's theory of truth, a &amp;quot;realist&amp;quot; will believe that a proposition is either true of false regardless of whether it is decidable, while an &amp;quot;anti-realist&amp;quot; will believe that an undecidable proposition has no truth value.&lt;/p&gt;&#xA;&lt;p&gt;At first glance Tarski's semantic definition of truth may appear to suggest that Tarski was a realist, but this is not clear because, after all, this is philosophy so the idea of being clear is out of the question.  Regardless of whether Tarski was a realist or anti-realist his theory would be subject to criticism from the other side.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;TL/DR (and probably not relevant)&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure how well acquainted you are with philosophy, and I'm certainly no expert, but it may be helpful to keep in mind that Tarski's use of the word &lt;em&gt;semantic&lt;/em&gt; is an archaic usage, prevalent amongst positivist philosophers in the 1930s. (Tarski's T-Schema make clear that Tarski had a positivist attitude.)  The positivists used a trio of terms: &lt;em&gt;syntactic, semantic, and pragmatic&lt;/em&gt;.  Since this answer is so far unreferenced, I'll slip in:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A notion was called &lt;em&gt;syntactic&lt;/em&gt; if it pertains to relations among words, &lt;em&gt;semantic&lt;/em&gt; if it pertains to relations among words and things, and &lt;em&gt;pragmatic&lt;/em&gt; if it pertains to relations among words, things, and people.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[Source: &lt;em&gt;Truth&lt;/em&gt;, by Burgess &amp;amp; Burgess, Princeton Press, 2011].&lt;/p&gt;&#xA;&lt;p&gt;So Tarski's use of the word &lt;em&gt;semantic&lt;/em&gt; had nothing to do with its current use in linguistics as &amp;quot;meaning&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Snow is white&amp;quot; is true iff snow is white&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is &lt;em&gt;semantic&lt;/em&gt; because it involves words and a thing (snow), but no people.&lt;/p&gt;&#xA;&lt;p&gt;Since Tarski was a mathematician doing philosophy it is difficult to say whether he was a realist or anti-realist.  As a mathematician he was most likely a realist (i.e., a Platonist) while as a philosopher he should be an anti-realist.  So here we have another source of controversy: the mathematicians will tend to poo-poo the philosophers while the philosophers will tend to laugh at the mathematicians naivety.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2021-07-07T18:55:05.650" LastActivityDate="2021-07-07T18:55:05.650" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13332" PostTypeId="1" CreationDate="2021-07-02T14:16:13.400" Score="0" ViewCount="60" Body="&lt;p&gt;Should Occam's razor be used in making preference between cosmological theories, which in order to fit in new experimental results, introduce hypothetical concepts, such as dark matter and dark energy, and cosmological theories (such as, for example, &lt;a href=&quot;https://themindguild.com/new-theory-claims-universe-works-without-dark-matter-dark-energy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://themindguild.com/new-theory-claims-universe-works-without-dark-matter-dark-energy/&lt;/a&gt; &lt;a href=&quot;https://phys.org/news/2021-06-dark-real-misunderstood-gravity.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://phys.org/news/2021-06-dark-real-misunderstood-gravity.html&lt;/a&gt;), which try to modify existent theories without introducing new hypothetical concepts?&lt;/p&gt;&#xA;" OwnerUserId="14876" LastActivityDate="2021-07-02T14:16:13.400" Title="Should Occam's razor be used in judging hypothetical cosmological theories?" Tags="|philosophy-of-science|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13333" PostTypeId="1" AcceptedAnswerId="13339" CreationDate="2021-07-03T15:00:13.430" Score="10" ViewCount="261" Body="&lt;p&gt;I am really curious about and struggling with finding when the word &amp;quot;Real number&amp;quot; began to be used as an official terminology to refer to both rational and irrational numbers.&lt;/p&gt;&#xA;&lt;p&gt;In Wiki, it says the adjective &amp;quot;Real&amp;quot; was introduced by Descartes, and Leo Corry says, after Descartes's introduction&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;terms like &amp;quot;imaginary&amp;quot; and &amp;quot;real&amp;quot; were sweepingly adopted by mathematicians of the following generations&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, I cannot find any pieces of literature where &amp;quot;Real number&amp;quot; refers to both rational and irrational numbers before Dedekind's time. I think they just used &amp;quot;numbers&amp;quot;, not &amp;quot;real numbers.&amp;quot; So, is Dedekind the one who officially used that term when he gave a rigorous definition of the real number system?&lt;/p&gt;&#xA;&lt;p&gt;I am also aware that even before Dedekind gave the definition, many mathematicians, such as Newton and Euler, didn't differentiate irrationals from rationals when they were using them. I am really curious when this specific term &amp;quot;Real numbers&amp;quot; started to be used as an official title to refer to rationals and irrationals before Dedekind.&lt;/p&gt;&#xA;&lt;p&gt;I'd really appreciate it if you'd share your knowledge and information!&lt;/p&gt;&#xA;" OwnerUserId="14881" LastActivityDate="2021-07-06T00:02:12.550" Title="When did the word &quot;Real number&quot; begin to be used as an official terminology to refer to both rational and irrational numbers?" Tags="|newton|euler|cantor|numbers|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13335" PostTypeId="2" ParentId="13319" CreationDate="2021-07-03T19:17:16.950" Score="1" Body="&lt;p&gt;While in the present day the word &amp;quot;organic&amp;quot; has connotations of &amp;quot;natural&amp;quot; and &amp;quot;holistic&amp;quot;, back in Newton's day it was much closer to the Greek word oργανoν (&lt;em&gt;organon&lt;/em&gt;: tool, instrument).&lt;/p&gt;&#xA;&lt;p&gt;For an exploration of organic methods and their place in 17th century geometry I recommend Guicciardini, &lt;em&gt;Isaac Newton on Mathematical Certainty and Method&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From pg 6:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another line of research concerned the so-called organic description&#xA;(or generation) of curves. This was an important topic, since in order&#xA;to determine the point of intersection of curves in the construction&#xA;of geometrical solutions, it was natural to think of the curves as&#xA;generated by a continuous motion driven by some instrument (an&#xA;oργανoν). It is the continuity of the motion generating the curves&#xA;that guarantees a point of intersection can be located exactly.&#xA;Descartes had devised several mechanisms for generating curves. In &lt;em&gt;De&#xA;Organica Conicarum Sectionum in Plano Descriptione Tractatus&lt;/em&gt; (1646),&#xA;which Newton read in &lt;em&gt;Exercitationum Mathematicarum&lt;/em&gt; (1657), van&#xA;Schooten had presented several mechanisms for generating conic&#xA;sections. This research field was connected with practical&#xA;applications, for instance, lens grinding and sundial design, but it&#xA;was also sanctioned by classical tradition and motivated the highly&#xA;abstract needs underlined by Descartes. Newton was able to devise a&#xA;mechanism for generating conics and to extend it to higher-order&#xA;curves.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One theme found throughout the book is Newton's criticism of &amp;quot;speculative&amp;quot; methods such as Cartesian analysis, and the definition of conics as planar sections of a cone.  Philosophically he much preferred the mechanical (organic).&lt;/p&gt;&#xA;&lt;p&gt;From pgs 106-107:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] Newton criticized the lack of elegance and ease in Cartesian&#xA;analysis of indeterminate problems and believed Cartesian synthesis&#xA;was based on “spurious” postulates: point-wise constructions, motion&#xA;of curves, construction with threads, and intersection of cone and&#xA;plane (barren speculation). He favored expressing the contents of&#xA;known and unknown segments that have a given ratio to one another, and&#xA;using projective properties to compose the locus by organic&#xA;descriptions based on acceptable postulates, namely, rotating rulers.&#xA;Nevertheless, Newton was a master in Cartesian algebra, the “common&#xA;analysis of the moderns,” and when he had to deal with a challenging&#xA;problem, the classification of cubic curves, he made use of algebraic&#xA;techniques that were at odds with his program of restoration of&#xA;ancient analysis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The book describes many organic mechanisms for generating curves.  Newton invented a mechanism for determining tangents, and one that could be used to draw the conic defined by five points &lt;span class=&quot;math-container&quot;&gt;$A,B,C,D,E$&lt;/span&gt; (pg 94, letter to John Collins, 1672,see below).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Yvly4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Yvly4.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11204" LastActivityDate="2021-07-03T19:17:16.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13336" PostTypeId="2" ParentId="5782" CreationDate="2021-07-04T16:26:16.980" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Aufbau_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article &lt;/a&gt;clearly states :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[n+l rule] was suggested by Charles Janet in 1928...Janet &amp;quot;adjusted&amp;quot; some of the actual n + ℓ values of the elements, since they did not accord with his energy ordering rule, and he considered that the discrepancies involved must have arisen from measurement errors. In the event, the actual values were correct and the n + ℓ energy ordering rule turned out to be an approximation rather than a perfect fit...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user14884" LastActivityDate="2021-07-04T16:26:16.980" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13337" PostTypeId="1" CreationDate="2021-07-04T19:03:50.910" Score="7" ViewCount="333" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;This is a question that I have for a long time, Maybe it is something silly, but I really want to know. Why is the number of elements in a group called &amp;quot;order&amp;quot;? I mean, the word &amp;quot;order&amp;quot; in Spanish (which is my language) has a very strong meaning in terms of &amp;quot;ordering&amp;quot;, but it does not refer to quantities. What was the motivation for this?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I asked this question &lt;a href=&quot;https://math.stackexchange.com/questions/4189729/why-is-the-number-of-elements-in-a-group-called-order?noredirect=1#comment8689362_4189729&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14885" LastEditorUserId="10106" LastEditDate="2021-07-05T07:23:08.453" LastActivityDate="2021-07-19T12:05:47.600" Title="Why is the number of elements in a group called &quot;order&quot;?" Tags="|mathematics|terminology|abstract-algebra|group-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13338" PostTypeId="2" ParentId="13319" CreationDate="2021-07-05T17:07:49.990" Score="1" Body="&lt;p&gt;Of interest may be the paper &amp;quot;Newton, the Geometer&amp;quot; (N Bloye and S A Huggett, &lt;em&gt;Newsletter of the European Mathematical Society&lt;/em&gt;, number 82, 19-27, December 2011); preprint &lt;a href=&quot;http://www.stephenhuggett.com/Newton.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;PDF available here&lt;/a&gt;.  The authors abstract:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We describe some of Newton’s most profound geometrical discoveries, arguing that by thinking&#xA;of him as a geometer we gain a deep insight into his peculiar genius. We pay particular attention to&#xA;Newton’s work on the organic construction, which deserves to be better known, being a classical geometrical&#xA;construction of the Cremona transformation (1862). Newton was aware of its importance in geometry, using&#xA;it to generate algebraic curves, including those with singularities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1723" LastActivityDate="2021-07-05T17:07:49.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13339" PostTypeId="2" ParentId="13333" CreationDate="2021-07-06T00:02:12.550" Score="6" Body="&lt;p&gt;It is hard to say what &amp;quot;official&amp;quot; means exactly, it is not like there was a bureau of terminological standards. But &amp;quot;real numbers&amp;quot;, &amp;quot;real values&amp;quot; and &amp;quot;real quantities&amp;quot; were certainly widely used long before Dedekind, and the erasure of the distinction between rationals and irrationals where it is not relevant is even older, it goes back to Stevin (1585), see &lt;a href=&quot;https://hsm.stackexchange.com/a/3213/55&quot;&gt;How were irrational numbers accepted by mathematicians?&lt;/a&gt; It was supported by Stevin's normalization of arbitrary decimals and the idea of generation of &amp;quot;quantities&amp;quot; by continuous motion, upon which Newton, Euler and Cauchy founded calculus. Both served as an early substitute for the arithmetical constructions of real numbers &lt;em&gt;a la&lt;/em&gt; Bolzano, Meray, Dedekind, Cantor, etc.&lt;/p&gt;&#xA;&lt;p&gt;Descartes did not employ &amp;quot;real numbers&amp;quot; as a term, but he did start the tradition of inserting the adjective to distinguish them from imaginary numbers. Here is the relevant quote from &lt;em&gt;La Geometrie&lt;/em&gt; (1637):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For the rest, neither the false nor the true roots are always real, sometimes they are only imaginary, that is to say one may imagine as many as I said in each equation, but sometimes there exists no quantity corresponding to those one imagines.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since by &amp;quot;roots&amp;quot; he means intersections of arbitrary polynomial curves with the axis he clearly did not distinguish rational from irrational. Newton, Leibniz, Euler and others continued in the same vein, with &amp;quot;real&amp;quot; becoming a term. Leibniz's unpublished 1675 paper (see &lt;a href=&quot;https://www.jstor.org/stable/2299067&quot; rel=&quot;noreferrer&quot;&gt;A Contribution of Leibniz to the History of Complex Numbers by McClennon&lt;/a&gt;) discusses the cubic as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have three real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible&lt;/em&gt;&amp;quot;; &amp;quot;&lt;em&gt;How can it be that a real quantity, a root of the proposed equation, is expressed by the intervention of the imaginary?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Euler's &lt;em&gt;Introductio in analysin infinitorum&lt;/em&gt; (1748) uses &amp;quot;real numbers&amp;quot;, &amp;quot;real values&amp;quot; and so on routinely. In chapter I (&lt;a href=&quot;http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm&quot; rel=&quot;noreferrer&quot;&gt;Bruce's translation&lt;/a&gt;) we read, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Thus, even if this function &lt;span class=&quot;math-container&quot;&gt;$\sqrt{(9-zz)}$&lt;/span&gt; at no time is able to receive real numbers substituted in place of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; greater than the number three, yet imaginary values of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; may be attributed&lt;/em&gt;&amp;quot;; &amp;quot;&lt;em&gt;The three values of &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; for that one corresponding value of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; either are all real or one will be real, while the two remaining are imaginary.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Considering the general influence of Euler's textbooks on the subsequent notation and terminology his use can be said to make it &amp;quot;official&amp;quot;. By the way, when Euler uses &amp;quot;numbers&amp;quot; without qualification he means integers, not real numbers, see e.g. chapter XVI.&lt;/p&gt;&#xA;&lt;p&gt;Cauchy's &lt;em&gt;Cours d'Analyse&lt;/em&gt; (1821), another influential textbook, adds &amp;quot;real&amp;quot; in the same context, but Cauchy typically talks about (variable) &amp;quot;quantities&amp;quot; or &amp;quot;functions&amp;quot; and their values, not &amp;quot;numbers&amp;quot;. In section 7.1 (&lt;a href=&quot;https://www.springer.com/us/book/9781441905482&quot; rel=&quot;noreferrer&quot;&gt;Bradley-Sandifer's translation&lt;/a&gt;) he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In general, we call an imaginary expression any symbolic expression of the form &lt;span class=&quot;math-container&quot;&gt;$\alpha+\beta\sqrt{-1}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; denote real quantities&lt;/em&gt;&amp;quot;; &amp;quot;&lt;em&gt;Given this, any imaginary equation is just the symbolic representation of two equations involving real quantities&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Note III he even uses &amp;quot;real&amp;quot; without direct relation to the imaginary, in the intermediate value theorem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If the two quantities &lt;span class=&quot;math-container&quot;&gt;$f(x_0)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f(X)$&lt;/span&gt; have opposite signs we can satisfy the equation &lt;span class=&quot;math-container&quot;&gt;$f(x)=0$&lt;/span&gt; with one or several real values of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; contained between &lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-07-06T00:02:12.550" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13340" PostTypeId="1" AcceptedAnswerId="13487" CreationDate="2021-07-06T14:06:08.847" Score="2" ViewCount="194" Body="&lt;p&gt;I am currently researching on all primality tests deriving from Lucas' original paper &lt;em&gt;Théorie des Fonctions Numériques Simplement Périodiques&lt;/em&gt;, which is of course known for its great deal of confusion. Still, I managed to make my way trough basically all of the recent tests (LLehmer, LLehmerRiesel, etc.) using publications from the authors and various sources - some of which I'll leave below, more on them in the appropriate paragraph. Note that there are various revisions from Lucas himself of the article, but every work of Lucas is a perfect source to me. I will not differentiate between them.&lt;/p&gt;&#xA;&lt;p&gt;Back to us: time came I had to write about the original Lucas test in order to further develop it into the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lucas_pseudoprime#Lucas_probable_primes_and_pseudoprimes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lucas probable prime test&lt;/a&gt; and then the &lt;a href=&quot;https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bailie-PSW test&lt;/a&gt;. Of course those two are of no concern; there are plenty of sources. However I hardly managed to find something about the original formulation Lucas gave; lots of tests have the name &lt;em&gt;&amp;quot;Lucas&amp;quot;&lt;/em&gt; in them, but I'm interested in that original one. It is my understanding that Lucas did not properly formulate it, but he just used it in some confused example and left it there. Still, I could not find a great deal of info.&lt;/p&gt;&#xA;&lt;p&gt;So in short I'm trying to understand what Lucas did on its original paper - no reworks as in Carmichael; I am interested in the original for historical reasons. The article is known to be kind of a mess, and what's more it's in French -- I can't understand it nor find a translation. I do not know what version of the test he used; from what I read I suppose it was just a one-time procedure on an example which was then further developed in most of the &amp;quot;Lucas&amp;quot; tests. &lt;strong&gt;Specifically, my question is&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;does anyone have a better idea on how he exactly used his sequences in primality testing on the original paper?&lt;/li&gt;&#xA;&lt;li&gt;are there some examples, especially from his paper, you could share?&lt;/li&gt;&#xA;&lt;li&gt;are there some better sources? of every kind: papers, videos, etc...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thank you all in advance and sorry for the long post / weird formatting, it's my first time on SE!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h4&gt;Duplicate(?)&lt;/h4&gt;&#xA;&lt;p&gt;While this is a well discussed topic, I hardly found discussions about the original test by Lucas and I thought it would be useful for the community to put here all my sources and ask for more insight on the original work - not the latter ones. That I did not find, and I'd like to help out people in my situation.&lt;/p&gt;&#xA;&lt;p&gt;For example, I encountered &lt;a href=&quot;https://math.stackexchange.com/questions/509518/original-lucas-sequences-primality-test&quot;&gt;this discussion&lt;/a&gt; which gave a book by Hans Riesel itself as an interesting source. Fact is, I do not have access to that book.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;I also posted &lt;a href=&quot;https://math.stackexchange.com/questions/4190716/confusion-on-the-original-lucas-test&quot;&gt;this same question&lt;/a&gt; on Mathematics stackexchange&lt;/em&gt;, and was advised to cross-post it here due to its historical focus. I modified my post a little since it seemed kinda ambiguous on the nature of the test I'm interested in.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h4&gt;Useful sources&lt;/h4&gt;&#xA;&lt;p&gt;The original article can be found in French on JSTOR, divided into two parts:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;American Journal of Mathematics , Vol. 1, No. 2 (1878), pp. 184-196. &lt;a href=&quot;http://www.jstor.org/stable/2369308&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/2369308&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;American Journal of Mathematics , Vol. 1, No. 4 (1878), pp. 289-321. &lt;a href=&quot;http://www.jstor.org/stable/2369373&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/2369373&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;and I leave for posterity an English translation and revision of the first part -- it could be of great help for someone. If anyone manages to find the second part I'll be glad to add it too -- I couldn't and I'm not sure it does exist.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Sidney Kravitz, Douglas Lind: &lt;em&gt;“The theory of simply periodic numerical functions”&lt;/em&gt;, Fibonacci Association (1969). &lt;a href=&quot;https://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathstat.dal.ca/FQ/Books/Complete/simply-periodic.pdf&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;[?]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Last but not least, the famous Carmichael did a great job revising the original article; sadly he modified it to make it work better, which is awesome but not what I need. As before, I leave it here for posterity - it's a life saver.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;R. D. Carmichael. “On the Numerical Factors of the Arithmetic Forms &lt;span class=&quot;math-container&quot;&gt;$α^n±β^n$&lt;/span&gt;”, Annals of Mathematics15 1/4 (1913), pp.30–48. &lt;a href=&quot;http://www.jstor.org/stable/1967797&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/1967797&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;R. D. Carmichael. “On the Numerical Factors of the Arithmetic Forms &lt;span class=&quot;math-container&quot;&gt;$α^n±β^n$&lt;/span&gt;”, Annals of Mathematics15 1/4 (1913), pp. 49–70. &lt;a href=&quot;http://www.jstor.org/stable/1967798&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/1967798&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;" OwnerUserId="14893" LastEditorUserId="14893" LastEditDate="2021-07-29T21:48:31.553" LastActivityDate="2021-07-29T21:49:43.563" Title="Confusion on the original article by Lucas" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="13341" PostTypeId="1" CreationDate="2021-07-07T00:23:20.890" Score="2" ViewCount="75" Body="&lt;p&gt;In their amazing paper &lt;a href=&quot;https://www.dcs.warwick.ac.uk/people/academic/Mike.Paterson/papers/overhang.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Overhang&lt;/a&gt; (&lt;i&gt;Amer. Math. Monthly&lt;/i&gt; &lt;b&gt;116&lt;/b&gt; (2009), 19–44), Mike Paterson and Uri Zwick revisited the old chestnut of how much of an overhang one can achieve by stacking bricks at the edge of a table. Crucial to their new results was the idea that a brick can have more than one other brick placed directly on top of it, thus serving as a counterweight.&lt;/p&gt;&#xA;&lt;p&gt;I am curious about earlier examples of people who considered the possibility of stacking more than one brick on top of a given brick.  Paterson and Zwick mention a few such precursors on the second page of their paper, but I have found one other instance not mentioned by them.  In the book &lt;a href=&quot;http://alum.mit.edu/www/tchow/overhang.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;i&gt;Ingenious Mathematical Problems and Methods&lt;/i&gt;&lt;/a&gt; by L. A. Graham, there are several pictures of arrangements that stack more than one brick on another brick.  These were sent in by Graham's readers; curiously, Graham did not bother to analyze them, but breezily dismissed them all as &amp;quot;marvels of graceful instability.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Are there other precursors of Paterson and Zwick's work?&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-07T00:23:20.890" Title="Counterweights in the overhang problem" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13342" PostTypeId="1" AcceptedAnswerId="13343" CreationDate="2021-07-07T04:11:19.253" Score="2" ViewCount="111" Body="&lt;p&gt;Wikipedia says that Schwarz (b. 1843) marrid Kummer in 1912 at the age of 69, had six children, then died in 1921 at the age of 78.  I'm surprised (not against!).&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://sprague.one-name.net/getperson.php?personID=I447193&amp;amp;tree=Sprague&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sprague.one-name.net/getperson.php?personID=I447193&amp;amp;tree=Sprague&lt;/a&gt; , their first child was born 1862, last child 1883, but again marriage 1912.  The source includes wikipedia.&lt;/p&gt;&#xA;&lt;p&gt;I think something could be wrong with this data.  So is there another source independent of Wikipedia?&lt;/p&gt;&#xA;" OwnerUserId="629" LastEditorUserId="629" LastEditDate="2021-07-07T15:10:38.893" LastActivityDate="2021-07-09T09:28:16.773" Title="When were Hermann Amandus Schwarz and Marie Elizabeth Kummer married?" Tags="|biographical-details|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13343" PostTypeId="2" ParentId="13342" CreationDate="2021-07-07T04:30:00.120" Score="3" Body="&lt;p&gt;OK, I digged out a biography written by Carathéodory (&lt;a href=&quot;https://archive.org/details/DeutschesBiographischesJahrbuchBd031921/page/n233/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Deutsches biographisches Jahrbuch III (1921), 236-238.&lt;/a&gt;).  The marriage was in 1868.&lt;/p&gt;&#xA;&lt;p&gt;@njuffa in the comment find that the exact date of marriage was 15 Apr. 1868, and the location was Berlin.&lt;/p&gt;&#xA;" OwnerUserId="629" LastEditorUserId="629" LastEditDate="2021-07-09T09:28:16.773" LastActivityDate="2021-07-09T09:28:16.773" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13344" PostTypeId="1" CreationDate="2021-07-07T12:07:00.203" Score="3" ViewCount="293" Body="&lt;p&gt;Reading Dante's &lt;em&gt;Comedy&lt;/em&gt;, I was interested in understand a little bit how medieval geography worked, so I decided to have a look at&#xA;Edward Luther Stevenson's translation of Ptolemy's &lt;em&gt;Geographia&lt;/em&gt;. I've also read the review of this book that Audrey Diller wrote in 1935, so I know it's not precisely the best translation of Ptolemy's work, but it's better than nothing.&lt;/p&gt;&#xA;&lt;p&gt;In particular, in the XXVII &lt;em&gt;canto&lt;/em&gt; of the &lt;em&gt;Paradiso&lt;/em&gt;, Dante mentions the first clime, so I was curious to understand what climes exactly were.&lt;/p&gt;&#xA;&lt;p&gt;Comparing the explanation of parallels used in the delineation of maps and the reproduction of maps that are at the end of the book, which in fact are not the maps of Ptolemy, but the 27 Ptolemaic maps as revised by Donnus Nicolaus Germanus in the XV century (in particular, they are designed using a different projection), I have seen the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The first clime is located on the parallel through the Meroe island.&lt;/li&gt;&#xA;&lt;li&gt;The second clime is located on the parallel through Syene.&lt;/li&gt;&#xA;&lt;li&gt;The third clime is located on the parallel with a latitude of 30° 20'.&lt;/li&gt;&#xA;&lt;li&gt;The fourth clime is located on the parallel drawn through Rhodes.&lt;/li&gt;&#xA;&lt;li&gt;The fifth clime is located on the parallel with a latitude of 40° 55'.&lt;/li&gt;&#xA;&lt;li&gt;The sixth clime is located on the parallel with a latitude of 45°.&lt;/li&gt;&#xA;&lt;li&gt;The seventh clime is located on the parallel with a latitude of 48° 30'.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I've also read that the part of the earth that was considered as inhabited by humans was bounded on the south by a land which was unknown which encompasses some part of Africa and the Indian Ocean (the &amp;quot;Indian sea&amp;quot; at that time). To delineate maps, the southern boundary of the inhabited earth was defined by the parallel that has a latitude of 16° 25' south of the equator. Now, my question is: was this parallel considered the southern limit of the first clime?&lt;/p&gt;&#xA;&lt;p&gt;At this &lt;a href=&quot;https://www.dropbox.com/s/gbw4jjz9lotoih7/carta_limite_sud_africa_quattrocento.pdf?dl=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; you will find the Latin inscription on such parallel on the map of Africa: I can't understand what it says.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've found this illustration in the book, but I have no idea where it comes from, but I think it's probably from the Renaissance, since the source of Stevenson's work seems to be Renaissance Latin translations of the &lt;em&gt;Geographia&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zjBw3.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zjBw3.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So I now suspect that the southern limit of the first clime was the equator and that the parallel with a latitude of 16° 25' south of the equator was defined  at Renaissance (?) as the Antidiameroes, that is, the boundary of some kind of first clime in the southern hemisphere. Is that correct?&lt;/p&gt;&#xA;" OwnerUserId="10392" LastEditorUserId="4251" LastEditDate="2021-09-17T06:28:53.147" LastActivityDate="2023-01-04T02:09:53.217" Title="On Ptolemy climes" Tags="|middle-ages|geography|ptolemy|antiquity|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13345" PostTypeId="1" CreationDate="2021-07-07T12:45:15.693" Score="2" ViewCount="370" Body="&lt;p&gt;The third and fourth definitions in Euclid's Elements say:&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;The ends of a line are points.&lt;/li&gt;&#xA;&lt;li&gt;A straight line is a line which lies evenly with the points on itself.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The fourth definition is usually regarded by modern mathematicians as meaningless or puzzling at best. A good definition we are told is that a straight line is the shortest distance between two points. However, Euclid's third, sixth and seventh definitions can provide some guidance on how to read the fourth definition.&lt;/p&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;The edges of a surface are lines.&lt;/li&gt;&#xA;&lt;li&gt;A plane surface is a surface which lies evenly with the straight lines on itself.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I would argue the proper way to appreciate seventh definition is through the eye of a builder or surveyor. The straight line in this context is a builder's ideal straight edge which is used to gauge the flatness of a surface. As an ideal straight edge slides over and around the surface one looks from the edge of the surface for gaps between the edge and the surface. If no gaps are seen then the &amp;quot;plane surface lies evenly with straight lines on itself&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;By analogy with the surface, the straight line in the fourth definition should be viewed from its end points. From this perspective if no part of the line lies away from the end points, then the line &amp;quot;lies evenly with the points on itself&amp;quot; which makes it a straight line.&lt;/p&gt;&#xA;&lt;p&gt;Source for definitions:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/java/elements/bookI/bookI.html#defs&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14904" LastActivityDate="2021-07-12T13:33:19.167" Title="What did Euclid mean by a straight line in his time?" Tags="|euclidean-geometry|definition|" AnswerCount="2" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="13346" PostTypeId="2" ParentId="13291" CreationDate="2021-07-07T13:39:19.450" Score="0" Body="&lt;p&gt;In general, Euclid's definitions talk about points and lines having a relationship to a plane but he does not say they need a plane in order to live. The only line he says needs a plane is the circle.&lt;/p&gt;&#xA;&lt;p&gt;I think he had a geometric conception of space as a whole, but it differs from the modern conception of space where a plane or surface are needed for points or lines to live. The latter are called manifolds, but Euclidean geometry, when understood properly, is not a flat manifold because it is not a manifold.&lt;/p&gt;&#xA;" OwnerUserId="14904" LastEditorUserId="14904" LastEditDate="2021-07-07T13:59:27.747" LastActivityDate="2021-07-07T13:59:27.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13347" PostTypeId="1" CreationDate="2021-07-08T05:36:17.897" Score="0" ViewCount="186" Body="&lt;p&gt;I looked into Carl. B. Boyer and Morris Kline books of math history, some calculus books like Apostol and Swokowski, many pages on the internet and even the Tractatus de Seriebus Infinitis of Jacobi Bernoulli with no sucess to find out the origin of the term &amp;quot;Telescoping Series&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;On Math Exchange Sammy Black explained the meaning, but I still have no reference to the history of the term.&lt;/p&gt;&#xA;&lt;p&gt;May someone help?&lt;/p&gt;&#xA;" OwnerUserId="14923" LastActivityDate="2021-07-08T05:36:17.897" Title="What is the origin of the term “Telescoping Series”?" Tags="|mathematics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13348" PostTypeId="1" CreationDate="2021-07-09T09:27:42.350" Score="0" ViewCount="61" Body="&lt;p&gt;William Herschel thought that the Sun was inhabited as did others at that time.  Presumably there was no knowledge of its surface temperature, so when was it first realised that the surface was far too hot for Life to exist there?&lt;/p&gt;&#xA;" OwnerUserId="14932" LastActivityDate="2021-07-09T09:27:42.350" Title="When was the surface temperature of the Sun first measured?" Tags="|astronomy|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13349" PostTypeId="2" ParentId="13345" CreationDate="2021-07-09T13:16:48.363" Score="4" Body="&lt;p&gt;These &amp;quot;definitions&amp;quot; in Euclid are not really definitions in the modern sense. They are rather attempts to explain the intuitive meaning of the terms. And they are never used in his proofs.&lt;/p&gt;&#xA;&lt;p&gt;What we would call axioms, is called in Euclid &amp;quot;Postulates&amp;quot; and &amp;quot;Common notions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Some historians argue that &amp;quot;definitions&amp;quot; are not a part of the original text;&#xA;they are later inserts. See, for example,&lt;/p&gt;&#xA;&lt;p&gt;Lucio Russo, The Definitions of Fundamental Geometric Entities Contained in Book I of Euclids Elements,&#xA;Archive for history of exact sciences, 52 (1998) 3, 195-219.&lt;/p&gt;&#xA;&lt;p&gt;The best modern analysis of Euclid, on my opinion is&lt;/p&gt;&#xA;&lt;p&gt;Robin Hartshorne, Geometry: Euclid and beyond, Springer 1997.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-07-09T13:16:48.363" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13350" PostTypeId="1" CreationDate="2021-07-09T15:07:03.723" Score="0" ViewCount="205" Body="&lt;p&gt;It is well-known that Galileo was the first one to state that motions in different dimensions (or components) are independent of one another. Where is this in Galileo's writings?&lt;/p&gt;&#xA;" OwnerUserId="14933" LastEditorUserId="10106" LastEditDate="2022-02-03T18:18:47.697" LastActivityDate="2022-02-03T18:18:47.697" Title="Galileo's discovery of the independence of motion" Tags="|physics|classical-mechanics|physicists|galileo|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13351" PostTypeId="2" ParentId="2181" CreationDate="2021-07-09T20:53:50.650" Score="0" Body="&lt;p&gt;See Forslund, Robert R., &amp;quot;A logical alternative to the existing positional number system.&amp;quot; (&lt;a href=&quot;https://eudml.org/doc/228435&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&#xA;He uses 1 to b (the base) instead of 0 to (b-1). For digits, he chooses &amp;quot;A&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="14933" LastActivityDate="2021-07-09T20:53:50.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13352" PostTypeId="1" CreationDate="2021-07-11T01:58:56.267" Score="0" ViewCount="101" Body="&lt;p&gt;On page 44 of [1] Flanders writes &amp;quot;A treatment from this point of view of exterior calculus&#xA;which is not quite completely satisfactory and which unfortunately is embellished&#xA;with historical comments often in bad taste is found in Blaschke.&amp;quot; His reference is Baschke's book [2] as listed on page 197.&lt;/p&gt;&#xA;&lt;p&gt;What comments did Flanders think were in bad taste in [2] and what/why did he consider Blaschke's presentation an unsatisfactory treatment when compared to his?&lt;/p&gt;&#xA;&lt;p&gt;[1] Flanders: Differential Forms with Applications to the Physical Sciences, Dover Pubs. 1989,&lt;/p&gt;&#xA;&lt;p&gt;[2] Blaschke: Einfuhrung in die Differentialgeometrie, Berlin 1950.&lt;/p&gt;&#xA;" OwnerUserId="7908" LastActivityDate="2021-07-11T01:58:56.267" Title="historical comments in bad taste" Tags="|differential-geometry|mathematics-social-history|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13353" PostTypeId="1" AcceptedAnswerId="13371" CreationDate="2021-07-11T10:50:21.383" Score="6" ViewCount="3224" Body="&lt;p&gt;The following is a formula for the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th prime number (&lt;span class=&quot;math-container&quot;&gt;$[\,]$&lt;/span&gt; represents the floor function). Who was the first person to discover it?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\newcommand{\lowerbrack}[2]{%&#xA;  \raise{-#1}{\left[\raise{#1}{#2}\right]}}&#xA;p(n) = 1 + \sum_{k=1}^{2^n} \lowerbrack{9pt}{ \sqrt[n]{\frac{n}{\sum\limits_{i=1}^k \left[ \cos^2 \frac{(i-1)! + 1}{i} \pi \right]}} }&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The value of this formula: people have been exploring the general term of the prime number sequence, and this is an answer to it.&lt;/p&gt;&#xA;" OwnerUserId="14936" LastEditorUserId="14365" LastEditDate="2022-11-02T07:48:20.500" LastActivityDate="2022-11-02T07:48:20.500" Title="Who discovered this closed form formula for the n-th prime number?" Tags="|mathematics|number-theory|primes|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13354" PostTypeId="2" ParentId="7807" CreationDate="2021-07-11T18:29:18.570" Score="2" Body="&lt;p&gt;I think the short answer is that problems for which it is easier to use umbral calculus rather than a more &amp;quot;standard&amp;quot; technique don't arise all that frequently in practice.&lt;/p&gt;&#xA;&lt;p&gt;My favorite reference for the umbral calculus is Ira Gessel's paper, &lt;a href=&quot;https://doi.org/10.1007/s00012-003-1813-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Applications of the classical umbral calculus&lt;/a&gt;, &lt;i&gt;Algebra universalis&lt;/i&gt; &lt;b&gt;49&lt;/b&gt; (2003), 397–434. After a brief explanation of umbral notation, Gessel goes on to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I first encountered umbral notation it seemed to me that this was all there was to it; it was simply a notation for dealing with exponential generating functions, or to put it bluntly, it was a method for avoiding the use of exponential generating functions when they really ought to be used.  The point of this paper is that my first impression was wrong: none of the results proved here (with the exception of Theorem 7.1, and perhaps a few other results in section 7) can be easily proved by straightforward manipulation of exponential generating functions.  The sequences that we consider here are defined by exponential generating functions, and their most fundamental properties can be proved in a straightforward way using these exponential generating functions.  What is  surprising is that these sequences satisfy additional relations whose proofs require other methods. The classical umbral calculus is a powerful but specialized tool that can be used to prove these more esoteric formulas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the umbral calculus definitely has value as an enumerative tool and is not superseded by other methods. On the other hand, Gessel does use the words &lt;i&gt;specialized&lt;/i&gt; and &lt;i&gt;esoteric&lt;/i&gt; to describe the situations in which the umbral calculus outperforms more standard techniques.&lt;/p&gt;&#xA;&lt;p&gt;Corroboration of the claim that the umbral calculus has a rather narrow range of application may be found by examining some of the standard references for enumerative combinatorics.  Richard Stanley's encyclopedic two-volume work &lt;i&gt;Enumerative Combinatorics&lt;/i&gt; mentions umbral notation just once in passing.  I couldn't find any mention of umbral calculus at all in the table of contents or index of the &lt;a href=&quot;https://www.routledge.com/Handbook-of-Enumerative-Combinatorics/Bona/p/book/9781482220858&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;i&gt;Handbook of Enumerative Combinatorics&lt;/i&gt;&lt;/a&gt;. The same could be said of just about any other graduate text on combinatorics, such as Goulden and Jackson's classic book &lt;i&gt;Combinatorial Enumeration&lt;/i&gt; or  Sagan's recent book &lt;i&gt;Combinatorics: The Art of Counting&lt;/i&gt;.  Evidently, all these authors felt that the umbral calculus was too specialized to deserve space in their books.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-11T18:29:18.570" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13355" PostTypeId="2" ParentId="5507" CreationDate="2021-07-11T20:42:04.803" Score="3" Body="&lt;p&gt;I don't think that the undergraduate curriculum has changed very much in the past generation (for the sake of concreteness, let's say that a generation is 40 years).&lt;/p&gt;&#xA;&lt;p&gt;Suppose we take a look at the requirements for a mathematics major at a top liberal arts college such as &lt;a href=&quot;https://math.williams.edu/courses/requirements-for-the-major/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Williams College&lt;/a&gt;. We see discrete mathematics, differential equations and vector calculus, mathematical methods for scientists, statistics, linear algebra, real analysis, and abstract algebra. Contra Stella Biderman, I don't think that there have been &amp;quot;massive changes&amp;quot; in these topics at the undergraduate level caused by ultrafilters, category theory, the PCP theorem, the regularity lemma, the triangle removal lemma, or the classification of finite simple groups.  At most, there might be some category theory in the abstract algebra course which would not have been in there 40 years ago.  The other topics would merit &lt;i&gt;at most&lt;/i&gt; a passing mention in an undergraduate course.&lt;/p&gt;&#xA;&lt;p&gt;All the above topics, except possibly for discrete mathematics, are classical, and the core content would likely be familiar to your hypothetical grandfather.  The one thing that I think has changed substantially in the past 40 years is the importance of computers. Discrete mathematics is now a common (though not ubiquitous) requirement for an undergraduate mathematics major, whereas it would have been unusual 40 years ago, and clearly it is computers that have made discrete mathematics more important. Many of the other topics will likely show some influence of the computer revolution, especially linear algebra, differential equations, and statistics; algorithms for handling large problems would probably not have been discussed 40 years ago at the undergraduate level, but might get a significant amount of class time now.  Real analysis and abstract algebra are less likely to have changed much, although even abstract algebra may show some influence from the existence of computer algebra packages.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-11T20:42:04.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13356" PostTypeId="1" AcceptedAnswerId="13359" CreationDate="2021-07-12T01:51:29.853" Score="38" ViewCount="6928" Body="&lt;p&gt;Famous conjectures whose solutions took decades or centuries were usually resolved with the help of sophisticated theories and techniques unknown at the time the conjecture was first claimed. Is there any example where mathematicians didn't see a solution that after the fact was recognized as simple, basic or not very elaborated but people failed to see for a long time?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastActivityDate="2024-02-16T09:03:59.220" Title="Is there any example of a long-standing mathematical conjecture whose resolution did not require advanced knowledge?" Tags="|mathematics|conjecture|" AnswerCount="9" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13357" PostTypeId="2" ParentId="13356" CreationDate="2021-07-12T04:23:20.463" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hanna_Neumann_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Hanna Neumann conjecture&lt;/a&gt; was open for 40+ years and was solved in 2011 by relatively elementary methods independently and almost simultaneously by Friedman and Mineyev. The existing proofs actually proved the strengthened Hanna Neumann conjecture formulated in 1990 by Walter Neumann. All tools used in Mineyev's proof certainly existed in 1990 and much earlier.&lt;/p&gt;&#xA;&lt;p&gt;Just this year the 80 years old &lt;a href=&quot;https://www.sciencetimes.com/articles/30628/20210413/researcher-disproves-80-year-old-kaplansky-unit-conjecture.htm&quot; rel=&quot;noreferrer&quot;&gt;Kaplansky unit conjecture&lt;/a&gt; was disproved by Giles Gardam, also by quite elementary methods.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-07-12T04:23:20.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13358" PostTypeId="1" CreationDate="2021-07-12T10:37:56.730" Score="0" ViewCount="78" Body="&lt;p&gt;Google just won't tell me about any other than &amp;quot;the grand hotel&amp;quot;. I know of the Turing paradox but did he propose any other? If yes, which?&lt;/p&gt;&#xA;" OwnerUserId="14938" LastEditorUserId="10106" LastEditDate="2022-02-02T19:06:23.203" LastActivityDate="2022-02-02T19:06:23.203" Title="What paradoxes did David Hilbert propose?" Tags="|mathematics|mathematicians|hilbert|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13359" PostTypeId="2" ParentId="13356" CreationDate="2021-07-12T12:46:32.050" Score="27" Body="&lt;p&gt;Sylvester's line problem (1893) was to prove that there exists no finite configuration of points in real projective plane such that every line through&#xA;two points actually contains at least 3 points, unless all points are on the same line. Sylvester himself found such a configuration in the complex projective plane.&lt;/p&gt;&#xA;&lt;p&gt;This problem was quite famous, and was considered very difficult:&#xA;people compared its difficulty with the 4-color problem.&lt;/p&gt;&#xA;&lt;p&gt;It was solved in 1944 by Grunwald, and a simple solution was obtained by Kelly in 1948. This solution occupies less than 1 page, and can be understood by any&#xA;high school student.&lt;/p&gt;&#xA;&lt;p&gt;Edit. Another somewhat similar example is &amp;quot;Gehring's problem on linked curves&amp;quot; described in my answer to: &lt;a href=&quot;https://mathoverflow.net/questions/8247/one-step-problems-in-geometry/112076#112076&quot;&gt;this question&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Second edit. The so called &lt;a href=&quot;https://en.wikipedia.org/wiki/Stahl%27s_theorem&quot; rel=&quot;noreferrer&quot;&gt;BMV conjecture&lt;/a&gt; was proposed in 1975, a lot of sophisticated mathematics was used in attempts to prove it. It was proved by H. Stahl in 2011, and then I simplified his proof to &lt;a href=&quot;https://arxiv.org/abs/1312.6003&quot; rel=&quot;noreferrer&quot;&gt;6 pages using nothing that was not well-known in 1900&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-17T13:07:40.470" LastActivityDate="2021-07-17T13:07:40.470" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13361" PostTypeId="2" ParentId="13345" CreationDate="2021-07-12T13:33:19.167" Score="0" Body="&lt;p&gt;If you place two lines parallel wrt to each other then the very concept of being parallel becomes quaky. When you have two infinite and non-evenly lines they will intersect at many points. You could define a lnon-even line and say the non-evenness has to be the same for all lines. But in that case you have to put the lines next to each other in a very special way to be parallel (equal distance everywhere). I don't think that Euclid had this in mind. There are many ways to define non-even lines. There is just one line where eveness is evenly divided over the line. In contrast to non-even lines, even lines show no diversions.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-12T13:33:19.167" CommentCount="26" ContentLicense="CC BY-SA 4.0" />
  <row Id="13362" PostTypeId="1" CreationDate="2021-07-12T13:44:54.643" Score="0" ViewCount="107" Body="&lt;p&gt;When was the imaginary number, i, introduced? Why was it called imaginary? Isn't it just as imaginary as a negative number? Aren't all numbers imaginary, for that matter?&lt;/p&gt;&#xA;&lt;p&gt;I am not interested in the notation of the number (i or j). I'm interested in why it is called imaginary.&lt;/p&gt;&#xA;&lt;p&gt;The two questions of which this is supposed to be a copy don't ask why it is called imaginary.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-13T04:47:50.447" LastActivityDate="2021-07-13T04:47:50.447" Title="What is the origin of the &quot;imaginary&quot; in imaginary numbers?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ClosedDate="2021-07-13T02:49:07.107" ContentLicense="CC BY-SA 4.0" />
  <row Id="13363" PostTypeId="1" CreationDate="2021-07-12T13:51:29.580" Score="0" ViewCount="77" Body="&lt;p&gt;I've been researching the motivation behind the invention of the Cartesian Co-ordinate system. But I was wondering, which coordinate system was the first to be invented? Was it the Spherical Co-ordinate system? And if yes, why was it needed?&lt;/p&gt;&#xA;" OwnerDisplayName="user14943" LastActivityDate="2021-07-12T13:51:29.580" Title="Which was the first coordinate system? And why was it used?" Tags="|geometry|descartes|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13364" PostTypeId="1" CreationDate="2021-07-12T13:53:54.280" Score="0" ViewCount="154" Body="&lt;p&gt;Einstein made intensive use of non-Eucliden geometry already developed by mathematicians. The difference being that the real manifolds (well, one of them) as imagined by the mathematicians were supposed by Einstein to correspond to real spacetime.&lt;/p&gt;&#xA;&lt;p&gt;Was Einstein influenced in forming his ideas by the ideas of people like Riemann? Was he the first to consider the spaces or manifolds as corresponding to real spacetime?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-12T13:53:54.280" Title="Was Einstein motivated by Riemannian geometry, or did he merely use it?" Tags="|einstein|riemannian-geometry|" AnswerCount="0" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="13365" PostTypeId="1" CreationDate="2021-07-12T14:05:07.630" Score="1" ViewCount="56" Body="&lt;p&gt;Born came up with his probability interpretation of quantum mechanics. But why did he think about such an interoretation? Why not thinking that a deterministic process has to lay at its foundation? All known processes were known to be fundamentally deterministic. A dice thrown, the motion of the stars, the motions of atoms in a macroscopic gas, etc. Why did he think that the processes in QM are inherently probabilistic? A notion hard to digest (you can say QM is hard to digest but this begs the question).&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-12T14:05:07.630" Title="What is the origin of the probability interpretation of quantum mechanics?" Tags="|quantum-mechanics|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13366" PostTypeId="2" ParentId="13356" CreationDate="2021-07-12T15:36:38.263" Score="10" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pentagonal_tiling&quot; rel=&quot;noreferrer&quot;&gt;Tiling of the plane by pentagons.&lt;/a&gt;&lt;br /&gt;&#xA;After the problem was discussed in Martin Gardner's &amp;quot;Mathematical Games&amp;quot; column in Scientific American magazine of July 1975, Marjorie Rice (an amateur mathematician) discovered four new examples.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-07-12T15:36:38.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13367" PostTypeId="1" AcceptedAnswerId="13370" CreationDate="2021-07-12T15:50:58.980" Score="1" ViewCount="109" Body="&lt;p&gt;When the quark model wasn' t there yet, was there evidence that hadrons are composite structures? In hindsight of course yes. But at the time back then, were there signs? Maybe not noticed, but which could have made people thinking this could be the case, without having compositeness in mind a priori?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-12T16:24:18.777" LastActivityDate="2021-07-13T00:34:17.400" Title="Before the advent of the quark model, was there evidence that the hadrons were composite structures?" Tags="|physics|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="13368" PostTypeId="2" ParentId="13356" CreationDate="2021-07-12T16:23:49.373" Score="6" Body="&lt;p&gt;Four high school students here in the Netherlands constructed a &lt;span class=&quot;math-container&quot;&gt;$12\times12$&lt;/span&gt; Franklin square with very extra symmetries. It wasn't completely Franklin though. The half horizontals didn't add up to half the whole value. I went on to develop a &lt;span class=&quot;math-container&quot;&gt;$16\times16$&lt;/span&gt; completely symmetrical and magical &lt;span class=&quot;math-container&quot;&gt;$16\times16$&lt;/span&gt; square. More magical (with the numbers &lt;span class=&quot;math-container&quot;&gt;$1-256$&lt;/span&gt;) it can't get. I made a &lt;span class=&quot;math-container&quot;&gt;$32\times32$&lt;/span&gt; one too. With paper, pencil, and a calculator.&lt;/p&gt;&#xA;&lt;p&gt;After they were on the news, a considerable wind was blowing in the magic square country. It was proven by direct calculation that a &lt;span class=&quot;math-container&quot;&gt;$12\times12$&lt;/span&gt; Franklin square (like the one the students made) doesn't exist. If perfect &lt;span class=&quot;math-container&quot;&gt;$12\times12$&lt;/span&gt; Franklin squares existed was a riddle up till then.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4296" LastEditDate="2024-02-16T09:03:59.220" LastActivityDate="2024-02-16T09:03:59.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13369" PostTypeId="1" CreationDate="2021-07-12T16:37:28.490" Score="1" ViewCount="61" Body="&lt;p&gt;The Standard Model is a sophisticated mathematical framework to capture the behavior of elementary particles. It is a dream castle which is built on loose sand and every change in its construction or shifting of the sand will lead to a collapse of the castle. At the same time, there are many possibilities to keep the castle intact and to prevent the loose sand from shifting.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering. When the castle was constructed, were aestetical features for the castle taken into consideration? Like a beautifull unification of the weak and electromagnetic force, which is part of the model but can easily be proven to be an artificial unification, based on math only and with no relation to reality whatsoever.&lt;/p&gt;&#xA;&lt;p&gt;Now, in constructing a real castle beauty considerations are certainly present. Were they too in constructing the Standard Model?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-12T16:37:28.490" Title="Did aesthetics play a role in the construction of the Standard Model?" Tags="|physics|theoretical-physics|particle-physics|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13370" PostTypeId="2" ParentId="13367" CreationDate="2021-07-12T23:34:39.140" Score="3" Body="&lt;p&gt;As @Jon Custer points out, &lt;a href=&quot;https://en.wikipedia.org/wiki/Quark_model#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;WP&lt;/a&gt; covers the history of such models pretty well. Once the classification of hadrons was completed by  Gell-Mann and Ne'eman, in 1961, the patterns involved, underlain by flavor SU(3), begged  the question of compositeness. A black box rattles and squeals (spectra, couplings) in orderly, non-random ways: this is catnip for theorists. So, discussions by Sakata, and   Peterman, succeeded in the end by  Gell-Mann and, independently, Zweig's successful introduction of quarks in 1964.&lt;/p&gt;&#xA;&lt;p&gt;The point is systematic patterns of atomic properties in the periodic table classification of Mendeleev (1870) were ultimately understood by Moseley in 1913 in terms of nuclear charges, building on Rutherford's van den Broeck, and Bohr's work, in a virtually constituent model of the nucleus, let us say. Since then, nuclear physicists cleaned  the structure  up in detail.&lt;/p&gt;&#xA;&lt;p&gt;The leap from hadron classification to constituent structure took 3, instead of 45 years, and involved partially the same investigator, MGM, who landed on &amp;quot;mathematical quarks&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$^\natural$&lt;/span&gt; through SU(3) group theory (!). George Zweig, a continent apart from MGM, and conceptually so as well, was unique in that he was not driven by expediting SU(3) representation theory, but, instead, dynamics (long life of the φ meson), so he introduced &amp;quot;nuclear physicists' quarks&amp;quot; (constituent quarks; very harshly received at the time: an astounding &lt;a href=&quot;https://authors.library.caltech.edu/18969/1/Origins_of_the_Quark_Model_Final_Zweig%5B1%5D.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;saga&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;For a long time, both types of quarks were highly speculative entities, to be considered or not by the mainstream of the field, certainly not by experimentalists; until, finally, Feynman's &lt;a href=&quot;https://en.wikipedia.org/wiki/Parton_(particle_physics)#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;parton&lt;/a&gt; model (1969) explained scaling through them, and appreciated them as DIS essentially phenomenological objects.&lt;/p&gt;&#xA;&lt;p&gt;The book by A Pais, &lt;em&gt;Inward Bound&lt;/em&gt;, (1986;  ISBN13:9780198519713) explains the saga quite wonderfully.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\natural$&lt;/span&gt; &lt;sub&gt;&amp;quot;A search for stable quarks of charge -1/3  or +2/3 and/or stable di-quarks of charge -2/3  or +1/3  or +4/3 at the highest energy accelerators would help to reassure us of the non-existence of real quarks.&amp;quot; MGM, 1964&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-07-13T00:34:17.400" LastActivityDate="2021-07-13T00:34:17.400" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13371" PostTypeId="2" ParentId="13353" CreationDate="2021-07-13T04:36:45.940" Score="9" Body="&lt;p&gt;The formula is derived in &lt;a href=&quot;https://www.jstor.org/stable/3611701&quot; rel=&quot;noreferrer&quot;&gt;Willans, On Formulae for the nth Prime Number (1964)&lt;/a&gt; (Mathematical Gazette&#xA;vol. 48, no. 366, pp. 413-415), who references &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick/page/412/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Dickson's History of the Theory of Numbers, ch. XVIII&lt;/a&gt; &amp;quot;&lt;em&gt;for references to other formulae of this nature&lt;/em&gt;&amp;quot;. He does not quite mean explicit formulas for &lt;span class=&quot;math-container&quot;&gt;$p_n$&lt;/span&gt;, but rather explicit formulas for functions that take different types of values on primes and non-primes (and can be adapted to the task by simple manipulations that Willans sketches). Explicit formulas of a somewhat different nature were published earlier by Mills &lt;a href=&quot;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-53/issue-6/A-prime-representing-function/bams/1183510803.full&quot; rel=&quot;noreferrer&quot;&gt;A prime-representing function (1947)&lt;/a&gt; and Wright &lt;a href=&quot;https://www.jstor.org/stable/2306356&quot; rel=&quot;noreferrer&quot;&gt;A Prime-Representing Function (1951)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Dickson gives a couple of examples, including Pocklington's 1911 example based on Wilson's theorem that seems to be Willans's inspiration. Later variations on the theme are reviewed in &lt;a href=&quot;https://www.jstor.org/stable/2690261&quot; rel=&quot;noreferrer&quot;&gt;Dudley's Formulas for Primes (1982)&lt;/a&gt;, who remarks:&amp;quot;&lt;em&gt;Although Wilson's Theorem has proved to be notoriously unuseful in finding primes, this has not stopped the production of formulas in which it is essential&lt;/em&gt;&amp;quot;. Interestingly, Willans shared the sentiment, and already felt the need to justify the exercise:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;While the formulae in this article are unsuitable for application to problems in prime number theory, they at least provide definite answers to the questions (see e.g. Hardy and Wright, An Introduction to the Theory of Numbers, § 1.5):&#xA;&lt;br&gt; Is there a formula for the nth prime number?&#xA;&lt;br&gt; Is there a formula for a prime, given the preceding prime?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Connes, the Fields medalist best known for developing noncommutative differential geometry, wrote a recent follow-up &lt;a href=&quot;https://arxiv.org/abs/1809.02832&quot; rel=&quot;noreferrer&quot;&gt;Around Wilson's theorem (2018)&lt;/a&gt; that quotes Willans's related formula for the prime number function. &lt;a href=&quot;https://arxiv.org/abs/0710.3217&quot; rel=&quot;noreferrer&quot;&gt;Rowland's A natural prime-generating recurrence (2017)&lt;/a&gt; quotes Willans's formula from the OP noting that it and its relatives are &amp;quot;&lt;em&gt;generally infeasible to compute in practice&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-07-13T21:28:19.887" LastActivityDate="2021-07-13T21:28:19.887" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13373" PostTypeId="1" CreationDate="2021-07-13T10:51:06.757" Score="5" ViewCount="234" Body="&lt;ul&gt;&#xA;&lt;li&gt;A recent &lt;a href=&quot;https://hsm.stackexchange.com/questions/13293/lost-memoir-of-%c3%89variste-galois&quot;&gt;question&lt;/a&gt; asked whether Galois’ “lost” memoir has been since recovered (similar to Abel’s lost memoir). The memoir referenced in the question is the one Galois submitted to the French “Académie des Sciences” in competition for the “Grand Prix” (in Mathematics), (probably) in Feb. 1830.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The interesting ramification of this question is that, would our perception of Galois’ relationship with his contemporary mathematicians be different, had we been able to read this lost memoir?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;From Peter Neumann’s compilation &lt;a href=&quot;https://uberty.org/wp-content/uploads/2015/11/Peter_M._Neumann_The_Mathematical_Writings.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;“The writings of Evariste Galois”&lt;/a&gt;, we have the following relevant dates (my short list for this question):&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;. Oct 1811: Birth&lt;/p&gt;&#xA;&lt;p&gt;. Apr 1829: published first article on continued fractions. Manuscript available.&lt;/p&gt;&#xA;&lt;p&gt;. May &amp;amp; June 1829: Submitted two articles to the Académie on algebraic researches. Not found in archives (a theory runs that it was Cauchy who lost it).&lt;/p&gt;&#xA;&lt;p&gt;.  Feb. 1830 (probably): re-submitted (at Cauchy’s recommendation) his work to the Académie for the “Grand Prix”. Not found in archives. The Grand Prix was awarded to Abel (posthumously).&lt;/p&gt;&#xA;&lt;p&gt;. Jan. 1831: Submitted to the Académie his memoir on the resolvability of equations (now referred to as the “First Memoir”).&lt;/p&gt;&#xA;&lt;p&gt;. July 1831: Negative assessment report by the Académie, by a committee chaired by Poisson (and Lacroix).&lt;/p&gt;&#xA;&lt;p&gt;. 29th May 1832: Letter (in a form of a testament) to his friend Auguste Chevallier. Summary by himself of his work with recommendations on where it should be published and to whom it should be transmitted for further review.&lt;/p&gt;&#xA;&lt;p&gt;. 30th May 1832: Shot in a duel (died on 31st).&lt;/p&gt;&#xA;&lt;p&gt;. 1846: Publication by Liouville, first recognition of Galois' exceptional contribution to the progress of Mathematics.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;My Question: Was Poisson (and Lacroix) professional and correct in his assessment of the Galois “First Memoir”, with the knowledge at that time and the material submitted by Galois?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14689" LastEditorUserId="14689" LastEditDate="2021-07-14T09:08:12.107" LastActivityDate="2023-02-05T09:01:05.157" Title="Continuation on Galois’ lost memoir(s): was Poisson right in his review of Galois’ memoir?" Tags="|mathematics|group-theory|algebraic-geometry|galois|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13374" PostTypeId="1" CreationDate="2021-07-13T13:46:45.397" Score="0" ViewCount="210" Body="&lt;p&gt;Was Picasso inspired by the back then new fourth dimension of space? A 3D observer in 2D space can see a 2D object in its entirety. It is assumed that a 4D perspective can do the same for 3D objects. In Picasso's paintings we can see frequently faces that are seen from various directions at once.&lt;/p&gt;&#xA;&lt;p&gt;Was he curious about 4D space?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2022-04-09T17:07:00.633" Title="Was Picasso inspired by physics?" Tags="|art|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13375" PostTypeId="2" ParentId="13374" CreationDate="2021-07-13T17:00:14.393" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Pablo Picasso not only looked at the projections of four-dimensional&#xA;cubes in a mathematics book when he invented cubism, he also read the&#xA;text, embracing not just the images but also the ideas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Tony Robbin, &lt;em&gt;Shadows of Reality The Fourth Dimension in&#xA;Relativity, Cubism, and Modern Thought&lt;/em&gt;, Yale UP 2006. (pref. p.X)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2021-07-13T17:00:14.393" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13377" PostTypeId="2" ParentId="12243" CreationDate="2021-07-13T18:08:42.420" Score="3" Body="&lt;p&gt;This kind of thing happens surprisingly often, so there is probably more than one &amp;quot;right answer.&amp;quot;  I am reminded of William Durfee's anecdote about J. J. Sylvester (&lt;a href=&quot;https://todayinsci.com/S/Sylvester_James/SylvesterJames-Quotations.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;as quoted by Florian Cajori&lt;/a&gt; in &lt;i&gt;Teaching and History of Mathematics in the United States&lt;/i&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I remember once submitting to Sylvester some investigations that I had been engaged on, and he immediately denied my first statement, saying that such a proposition had never been heard of, let alone proved. To his astonishment, I showed him a paper of his own in which he had proved the proposition; in fact, I believe the object of his paper had been the very proof which was so strange to him.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above anecdote matches your account in several details: Sylvester was a mathematician; he insisted that a certain result was unproved; someone else pointed out that Sylvester himself had proved the result.  But the setting was not a lecture, and it was Durfee, not Sylvester, who stated the proposition first.  Here's another anecdote about a different person (not a mathematician) taken from the book &lt;i&gt;The Dynamics of Modern Society&lt;/i&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Nobel laureate, Otto Loewi, reports having waked in the middle of the night, jotting down some notes on what he sensed to be a momentous discovery, going back to sleep, awaking to find that he could not possibly decipher his scrawl, spending the day in a miserable and unavailing effort to remember what he had had in mind, being again aroused from his slumber at three the next morning, racing to the laboratory, making an experiment and two hours later conclusively proving the chemical transmission of nervous impulse. So far, so good; another case, evidently, of the pattern of subconscious creativity unforgettably described by Poincare.  But some years later, when Loewi, upon request, reported all this to the International Physiological Congress, he was reminded by a former student that, eighteen years before that nocturnal discovery, he had fully reported his basic idea. &amp;quot;This,&amp;quot; says Loewi, &amp;quot;I had entirely forgotten.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note, however, that there are different accounts of the Loewi incident; here's what &lt;a href=&quot;https://www.encyclopedia.com/people/medicine/medicine-biographies/otto-loewi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia.com&lt;/a&gt; says about it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In fact, there is an element of mystery and drama in the way Loewi came to demonstrate experimentally the chemical transmission of nervous impulses. By the time he did so, in 1921, the hypothesis of chemical transmission was nearly twenty years old. Credit for the hypothesis is usually given to Elliott, who published the suggestion in 1904. In 1929, after Loewi’s work had become well known, another Cambridge physiologist, Walter Fletcher, recalled that Loewi had independently proposed the chemical transmission hypothesis in 1903, in a private conversation with Fletcher, who was then working in Loewi’s laboratory at Marburg. Before this reminder from Fletcher, Loewi had completely forgotten the conversation; but he thereafter attached considerable significance to it, undoubtedly to emphasize the independence of his own work. However, as Dale suggested more than once, it is hard to believe that Loewi’s mind had not been at least somewhat prepared for the idea by his meeting with Elliott and in general by his visit to Cambridge, where the meaning of the neuromimetic effects of drugs was then a topic of intense interest and discussion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Finally, I can't resist mentioning a case where Vaughan Jones asked a question on MathOverflow and &lt;a href=&quot;https://mathoverflow.net/questions/23989/are-the-norms-of-graphs-dense-in-any-interval#comment50174_23993&quot;&gt;someone pointed out&lt;/a&gt; that the answer could be found in a book by…Vaughan Jones.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-13T18:08:42.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13379" PostTypeId="2" ParentId="3234" CreationDate="2021-07-13T18:33:52.323" Score="8" Body="&lt;p&gt;The history of the cup product is described on pages 135–136 of &lt;i&gt;Never a Dull Moment: Hassler Whitney, Mathematics Pioneer&lt;/i&gt; by Keith Kendig.&lt;/p&gt;&#xA;&lt;p&gt;A key event was the 1934 International Conference in Topology, held in Moscow. Andrey Kolmogoroff and James Alexander independently presented the basic idea, but it apparently wasn't quite right.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whitney heard Kolmogoroff and Alexander's definitions of product, and on the one hand the idea seemed tremendously significant, but at the same time something didn't smell quite right.  Whitney had made friends with the Czech mathematician Edouard Čech, and the two of them agreed that the multiplication idea seemed powerful and promising. … Nevertheless, there seemed to be a problem: If you multiply two sets like a disk and an interval, the result is a solid cylinder—the dimensions add: &lt;span class=&quot;math-container&quot;&gt;$2+1=3$&lt;/span&gt;. … However, Kolmogoroff and Alexander's definition of product didn't do that, instead giving a dimension that was larger than the sum by one. Whitney and Čech puzzled over this problem, and within a few months they had ironed out the difficulty. … By that time, Alexander had already written a paper using his original definition and submitted it to the &lt;i&gt;Annals&lt;/i&gt;. Upon learning of the Whitney–Čech revision, Alexander immediately saw its advantages. He rewrote and resubmitted his paper, which appeared in 1936.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-13T18:33:52.323" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13380" PostTypeId="1" CreationDate="2021-07-13T19:00:38.627" Score="1" ViewCount="125" Body="&lt;p&gt;It is welll known that there was something like &amp;quot;Entartete Kunst&amp;quot; in Nazi Germany. Degenerate art was all art of Jewish artists and art that didn't fit the nazi worldview.&lt;/p&gt;&#xA;&lt;p&gt;Was all science conneted with jewishness also considered degenerate? Were there other forms of science that were considered degenerate (so not connected with jewishness)?&lt;/p&gt;&#xA;&lt;p&gt;What was the attitude towards Einsteins general relativity? Did they really think this was &amp;quot;entarted&amp;quot;? Was jewish science in principle degenerate (though maybe used in the production of whatever, maybe even in the production of Zyclon B...)?&lt;/p&gt;&#xA;&lt;p&gt;I think it is rather, eeeh, irrational, to put it in rather friendly words, to make the distinction, but then again the whole era was rather irrational, to put it mildly (and seen from our ratio).&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-13T23:34:31.447" Title="Was there something like &quot;Entartete Wissenschaft&quot; in Nazi Germany?" Tags="|nazi-germany|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13382" PostTypeId="1" CreationDate="2021-07-13T19:55:31.920" Score="1" ViewCount="82" Body="&lt;p&gt;Gauge theories in physics are theories that introduce extra fields, the mediating fields, after the non-mediated fields are gauged.&lt;/p&gt;&#xA;&lt;p&gt;Now to gauge something means that you set some reference point from which measurements are made. But what is set in non-interacting fields? Why were the (arbitrary or non-arbitrary) phase transformations on matter/energy fields seen as gauges? Who came up with that idea?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-13T19:55:31.920" Title="What inspired the use of the term &quot;gauge&quot; in gauge theories?" Tags="|physics|particle-physics|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13383" PostTypeId="1" AcceptedAnswerId="13393" CreationDate="2021-07-13T20:09:16.257" Score="2" ViewCount="234" Body="&lt;p&gt;In the early sixties of the previous century, the quark model of hadrons was conceived. In 1969 Richard Feynman introduced the parton model of the proton.&lt;/p&gt;&#xA;&lt;p&gt;Wasn't his parton model simply the same as the quark model? Didn't he &amp;quot;just&amp;quot; put some flesh on the skeletal conception of quarks being present in hadrons?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-16T13:50:58.203" Title="Wasn't Feynman's parton model the same as the quark model?" Tags="|nuclear-physics|feynman|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13384" PostTypeId="2" ParentId="13380" CreationDate="2021-07-13T23:34:31.447" Score="1" Body="&lt;p&gt;Yes, there was definitely &amp;quot;Jewish science&amp;quot; in Nazi thought.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Deutsche_Physik&quot; rel=&quot;nofollow noreferrer&quot;&gt;Deutsche Physik&lt;/a&gt; attempted to purge Jewish influence from science.  It was not entirely successful, in that while many Jews fled Germany and could not, of course, teach physics, they were unable to purge Heisenberg, a German but deeply involved in the physics they denounced.&lt;/p&gt;&#xA;&lt;p&gt;Freud also came under attack for being Jewish, and &lt;a href=&quot;https://encyclopedia.ushmm.org/content/en/article/sigmund-freud&quot; rel=&quot;nofollow noreferrer&quot;&gt;his books were burned&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2021-07-13T23:34:31.447" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13386" PostTypeId="1" CreationDate="2021-07-14T06:18:54.137" Score="2" ViewCount="51" Body="&lt;p&gt;We can read in Wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mach's principle, in cosmology, hypothesis that the inertial forces experienced by a body in nonuniform motion are determined by the quantity and distribution of matter in the universe. It was so called by Albert Einstein after the 19th-century Austrian physicist and philosopher Ernst Mach.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does this lay at the center of his view on spacetime? Einstein basically says the same in his equivalence principle. Is the main difference that Mach saw spacetime being Newtonian while Einstein used it to formulate GR from SR?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-14T06:18:54.137" Title="How did Mach's view on spacetime differ from Einstein's?" Tags="|physics|relativity-theory|einstein|physicists|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13389" PostTypeId="2" ParentId="13356" CreationDate="2021-07-14T10:19:42.837" Score="9" Body="&lt;p&gt;Euler's conjecture stated in 1769 &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture&lt;/a&gt;&#xA;was &lt;em&gt;resolved&lt;/em&gt; as disproven in 1966 using one basic line&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$27^5 + 84^5 +110^5 + 133^5= 144^5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf&lt;/a&gt;&#xA;There may be more similar cases from that time-period of counter-examples that are trivial to check, but hard to find.&lt;/p&gt;&#xA;" OwnerUserId="14960" LastActivityDate="2021-07-14T10:19:42.837" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13390" PostTypeId="2" ParentId="11274" CreationDate="2021-07-14T13:11:51.860" Score="2" Body="&lt;p&gt;Richard Stanley's article &lt;a href=&quot;http://math.mit.edu/~rstan/papers/hip.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hipparchus, Plutarch, Schröder, and Hough&lt;/a&gt; (&lt;i&gt;Amer. Math. Monthly&lt;/i&gt; &lt;b&gt;104&lt;/b&gt; (1997), 344–350) describes a possible example. Among Plutarch's writings appears the following enigmatic statement.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a  million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These numbers mystified generations of historians, until in 1994, David Hough noticed that 103,049 is the tenth &lt;a href=&quot;http://oeis.org/A001003&quot; rel=&quot;nofollow noreferrer&quot;&gt;(little) Schroeder number&lt;/a&gt;.  Moreover, although Stanley's article does not offer an explanation of the number 310,952, &lt;a href=&quot;https://www.jstor.org/stable/3109806&quot; rel=&quot;nofollow noreferrer&quot;&gt;it was noticed soon afterward&lt;/a&gt; that the number 310,954 (off by 2) is the tenth element of a &lt;a href=&quot;http://oeis.org/A010683&quot; rel=&quot;nofollow noreferrer&quot;&gt;closely related sequence&lt;/a&gt;.  Thus the circumstantial evidence is overwhelming that these numbers are what Plutarch and Hipparchus were referring to.&lt;/p&gt;&#xA;&lt;p&gt;Along the lines of Conifold's remark, now that we know this, the mathematics is no longer lost, right?  Well, it's still not clear exactly what is meant by a &amp;quot;compound proposition,&amp;quot; nor is it clear why 310,952 is off by 2 from 310,954.  Perhaps more importantly, the &lt;i&gt;method&lt;/i&gt; by which Hipparchus computed these numbers has been lost.  We do know of recurrences that Hipparchus could have used to calculate these numbers, and it is a plausible conjecture that he used such a recurrence, but unless someone discovers further corroborating documents from that time period, this conjecture remains unconfirmed.  In any case, computing these numbers requires considerable sophistication in combinatorics; as far as I know, we have no other evidence from that far back of any other combinatorial calculation of comparable difficulty.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2021-07-14T13:24:35.593" LastActivityDate="2021-07-14T13:24:35.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13391" PostTypeId="1" CreationDate="2021-07-14T13:37:49.127" Score="3" ViewCount="114" Body="&lt;p&gt;According to a translation of the letter of Collatz to Professor Mays in 1980, Collatz mentions that he hasn't figured out whether the number n = 80 resulted in a cycle or not, concerning the collatz function. But this doesn't make sense, since the number 80 is easily computed under the collatz function. I've found the probable copy of the letter in german, but after carefully analyzing it, it isn't clear what is the number, it seems like to be 30 (which would make more sense, but still, its orbit only reaches 160 at most), or 80. My guess is that it should be 27, because this number has a really long orbit almost reaching 10000 and Collatz probably lost his patience to compute it.&lt;/p&gt;&#xA;&lt;p&gt;The alegated copy of the letter in german can be seen in the link:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cecm.sfu.ca/organics/papers/lagarias/paper/goodies/letter2.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cecm.sfu.ca/organics/papers/lagarias/paper/goodies/letter2.gif&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And the english translation version can be seen in:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.cecm.sfu.ca/organics/papers/lagarias/paper/goodies/ubersetzung/html/ubersetzung.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.cecm.sfu.ca/organics/papers/lagarias/paper/goodies/ubersetzung/html/ubersetzung.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can someone confirm that 27 is the number, by exhibiting an explicit copy of the letter different from the one that I have? Or do you think the number is either 30 or 80?&lt;/p&gt;&#xA;&lt;p&gt;I would appreciate any clarifying answer.&#xA;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="14963" LastActivityDate="2021-07-14T13:37:49.127" Title="Collatz letter to Professor Mays" Tags="|mathematics|translations|conjecture|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13392" PostTypeId="1" CreationDate="2021-07-14T15:23:54.043" Score="3" ViewCount="323" Body="&lt;p&gt;I have noticed a tendency among some historians and scholars of mathematics to regard the mathematics of antiquity as a less developed version of modern mathematics. This view reminds me of the belief that evolution is directional and was bound to produce the species we know today.&lt;/p&gt;&#xA;&lt;p&gt;A corollary to this belief, in my opinion, is that if the body of knowledge we now call modern mathematics had not emerged, then the mathematics of antiquity would have stagnated.&lt;/p&gt;&#xA;" OwnerUserId="14904" LastEditorUserId="4251" LastEditDate="2021-12-17T19:47:26.957" LastActivityDate="2021-12-18T02:37:43.093" Title="Does the practice of viewing modern mathematics as the necessary direction in which ancient mathematics must have evolved have a name?" Tags="|mathematics|mathematics-social-history|" AnswerCount="2" CommentCount="33" ContentLicense="CC BY-SA 4.0" />
  <row Id="13393" PostTypeId="2" ParentId="13383" CreationDate="2021-07-14T20:52:50.650" Score="7" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Wasn't his parton model simply the same as the quark model? Didn't he [Feynman] &amp;quot;just&amp;quot; put some flesh on the skeletal concept of quarks present inside hadrons?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not at all! Before the discovery of asymptotic freedom (1973; and hence envisioning of confinement), the absence of free quarks and the understanding of how they behaved was all speculation: a bag of intuitive mathematical tricks to produce relations experimentalists could use, without quite believing in quarks. You may think of Gell-Mann, Zweig, and Feynman all landing on completely different parts of a continent, and only barely appreciating they were on the same one. (Of course, all three knew each other: MGM and RPF were competitive/collaborative colleagues sparring with each other every day, and GZ was RPF's PhD student, and ultimately colleague.) But they did not quite align in their vision of the big picture, and all three were subtly off the ultimate one.&lt;/p&gt;&#xA;&lt;p&gt;1964; Four advances not fully fitting with each other:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;MGM appreciates the &amp;quot;missing&amp;quot; &lt;strong&gt;3&lt;/strong&gt; irreducible representation of the flavor SU(3) he had invented three years before would tensor with itself and its conjugate to constitute the extant irreps of hadrons, &lt;strong&gt;8&lt;/strong&gt;, &lt;strong&gt;10&lt;/strong&gt;, etc, provided their fractional electric charges be 2/3, -1/3, so far unobservable. He boldly, and correctly, wagers that such charges will not be observed in the lab or cosmic rays &lt;em&gt;directly&lt;/em&gt;, (&amp;quot;physical non-existence of quarks&amp;quot;). He thus ushers in an understanding of his &amp;quot;eightfold-way&amp;quot; flavor-SU(3) hadron classification scheme. Today, these quarks would be understood as constituent ones. (In ignorance of color, difficulties with their statistics and confinement are left in abeyance.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/abs/1007.0494&quot; rel=&quot;nofollow noreferrer&quot;&gt;GZ introduces&lt;/a&gt; 3 (4!) quarks, which he calls &amp;quot;aces&amp;quot;. These, beyond reproducing SU(3) mass formulas and coupling relations much more simply and perspicuously, explain a dynamical mystery: why the φ vector meson is far more stable than its peers, by dint of his (Zweig's) rule:  quark-antiquark pairs in a hadron are reluctant to annihilate themselves. (Ultimately 2, 3, 4, ..., other names attach themselves to the &amp;quot;rule&amp;quot;.) He is less savvy in anticipating criticism, and not yet established, so he is attacked harshly by established physicists, muttering about &amp;quot;charlatan&amp;quot; stuff...&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Gürsey &amp;amp; Radicatti combine SU(3) with spin SU(2) in an SU(6) which is even more spectacularly successful in classifying baryons than flavor SU(3).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;MGM introduces current algebra, a sub-rosa field theory, whose current commutators greatly expedite and clarify electromagnetic &amp;amp; weak amplitudes. Today, we'd recognize the quarks in the currents' bilinears as &amp;quot;current quarks&amp;quot;, the constituent quark cores (constituent quarks with no covering) of  valence quarks.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For five years, confusion reigns supreme (Mellosh transform, etc..), despite increasing rewards, fits, and easy computability if one took the quark model seriously. The conservatives would just call them &amp;quot;mathematical&amp;quot; to fob off insistent observability questions; while MGM would describe the QFT currents he studied as a mere crutch to get answers. His analog was a haute-cuisine dish involving cooking a piece of pheasant meat   between two slices of veal, which are then discarded.  (The veal is the QFT machinery!)&lt;/p&gt;&#xA;&lt;p&gt;1969&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Apprised of Bjorken's conformal arguments to explain scaling in deep inelastic scattering of hadrons by photons, RPF envisions hadrons as swarms of bees (partons) where a projectile (photon given off by an electron) hits just &lt;em&gt;one&lt;/em&gt; bee and leaves a characteristic relativistic kinematic distribution signature. He explains scaling and its central significance in simple, intuitive terms, and effectively impresses the experimental community to focus on questions that matter. &lt;em&gt;This marks the beginning of a productive experimental program investigating quarks&lt;/em&gt;. (Abortive searches for free quarks merely confirm their confinement.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;At this point, everyone thinks of parton as quarks, but doesn't say so, so as to avoid unanswerable questions (at that stage, once an identification with quarks were made). Eventually, with careful measurements and additional theoretical work (parton sum rules) it is appreciated that, indeed, the charged partons are zillions of soft quarks, but half the momentum of the baryon is carried by chargeless partons (now identified as gluons: RPF's presciently noncommittal name made sense.)&lt;/p&gt;&#xA;&lt;p&gt;[A small aside, personally witnessed. MGM was so frustrated by RPF's refusal to call his partons &amp;quot;quarks&amp;quot;, that he often referred to them as &amp;quot;put-ons&amp;quot;, since &amp;quot;of course they are quarks!!&amp;quot;.  With intensity.]&lt;/p&gt;&#xA;&lt;p&gt;But at that point, everyone, including experimentalists, is sold on quarks, and it is evident they are behind motivating and explaining a spate of DIS experiments. They just know there are mysteries about them, and theorists  shift to overdrive to resolve them.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It all finally &amp;quot;clicks&amp;quot;  in 1973. First, Gross &amp;amp; Wilczek, and, independently, Politzer, discover asymptotic freedom of nonabelian gauge theories, (weakening of the coupling with increasing energy), thereby explaining DIS scaling in a QFT setting. So QFT is veal no more. Then (even though the work was started in 1972) Bardeen, Gell-Mann and Fritzsch argue for the color degree of freedom, homing in on QCD, the strong interactions' QFT.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Along the way, an awful lot of mistakes were made, misconceptions came and went, and lots of incremental advances and clarifications firmed up the correct, effective, super-accurate QFT theory underlying half of the Standard Model. A highly collective enterprise.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-07-16T13:50:58.203" LastActivityDate="2021-07-16T13:50:58.203" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13394" PostTypeId="2" ParentId="3001" CreationDate="2021-07-15T01:37:25.500" Score="7" Body="&lt;p&gt;An example whose &amp;quot;great impact&amp;quot; is debatable but which was definitely anonymous (and recent) is the &lt;a href=&quot;https://www.wired.com/story/how-an-anonymous-4chan-post-helped-solve-a-25-year-old-math-puzzle/&quot; rel=&quot;nofollow noreferrer&quot;&gt;anonymous 4chan poster&lt;/a&gt; who made significant progress on the problem of finding optimal superpermutations.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;b&gt;EDIT:&lt;/b&gt; Along similar lines, a recent anonymous contribution with some impact, but not &amp;quot;great impact,&amp;quot; was an &lt;a href=&quot;https://web.archive.org/web/20160416025436/http://note.chiebukuro.yahoo.co.jp/detail/n365238&quot; rel=&quot;nofollow noreferrer&quot;&gt;elementary geometric solution by aerile_re&lt;/a&gt; of certain&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Langley%27s_Adventitious_Angles&quot; rel=&quot;nofollow noreferrer&quot;&gt;adventitious quadrangles problems&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2022-04-10T02:11:48.833" LastActivityDate="2022-04-10T02:11:48.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13395" PostTypeId="2" ParentId="7576" CreationDate="2021-07-15T01:47:34.940" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Religious_views_of_Isaac_Newton&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isaac Newton&lt;/a&gt; devoted much of his later life to religious speculations that even most religious people regard as having dubious value.  In particular, he spent a lot of time on what are now generally termed &lt;a href=&quot;https://en.wikipedia.org/wiki/Isaac_Newton%27s_occult_studies&quot; rel=&quot;nofollow noreferrer&quot;&gt;occult studies&lt;/a&gt; such as alchemy and mystical interpretations of the Bible.  Martin Gardner wrote an entertaining &lt;i&gt;Skeptical Inquirer&lt;/i&gt; article on this topic, entitled &amp;quot;Isaac Newton: Alchemist and Fundamentalist.&amp;quot; The article is reprinted in the anthology &lt;i&gt;Did Adam and Eve Have Navels?&lt;/i&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;b&gt;EDIT:&lt;/b&gt; Here are a few quotes from Martin Gardner's article, to give some sense of how much effort Newton devoted to these studies.&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For a large part of his life Newton's time and energy were devoted to fruitless alchemy experiments and efforts to interpret Biblical prophecy. His handwritten manuscripts on those topics far exceed his writings about physics. They constitute several million words now scattered in rare book rooms of libraries and in private collections. … [Newton] read all the old books on alchemy he could find, accumulating more than 150 for his library. He built furnaces for endless experiments and left about a million words on the topic. …&lt;br&gt;&lt;br&gt;&#xA;Newton's passion for alchemy was exceeded only by his passion for Biblical prophecy. Incredible amounts of intellectual energy were spent trying to interpret the prophecies of Daniel in the Old Testament and the Book of Revelation in the New. He left more than a million words on these subjects, seeing himself as one who for the first time was correctly judging both books. …&lt;br&gt;&lt;br&gt;&#xA;Newton's writings on Biblical prophecy are so huge an embarrassment to his admirers that to this day they are downplayed or ignored. … [John Maynard] Keynes spoke of having gone through some million of Newton's words on alchemy and found them &amp;quot;wholly devoid of scientific value.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2021-07-19T04:25:21.863" LastActivityDate="2021-07-19T04:25:21.863" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13396" PostTypeId="1" CreationDate="2021-07-15T05:52:15.527" Score="3" ViewCount="586" Body="&lt;p&gt;For example the &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Fourier_analysis&amp;amp;oldid=1032517308&quot; rel=&quot;nofollow noreferrer&quot;&gt;current version of the Fourier analysis article&lt;/a&gt; on Wikipedia says the study is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[…] named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And as mentioned in e.g. &lt;a href=&quot;https://hsm.stackexchange.com/questions/11744/origin-of-the-fourier-transform-1878&quot;&gt;Origin of the Fourier transform (1878)&lt;/a&gt;, the book often cited re. Fourier series has a title that translates to &lt;em&gt;Analytical Theory of Heat&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But I don't see any sines or cosines in the (rather simple-looking) &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Heat_equation&amp;amp;oldid=1028898554#Statement_of_the_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;heat equation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays it seems that techniques based on Fourier's work are particularly famous through their use in Digital Signal Processing, in which FFTs/DFTs are often applied to data series that are directly related to sine waves. When I think of Fourier transforms I think of analyzing data that's full of signals and noise and rapid changes in levels, not something like a hot spot slowly averaging itself out through a chunk of metal.&lt;/p&gt;&#xA;&lt;p&gt;There's another related question here, &lt;a href=&quot;https://hsm.stackexchange.com/questions/7957/what-are-some-good-references-elucidating-the-discovery-creation-of-fourier-seri&quot;&gt;What are some good references elucidating the discovery/creation of Fourier Series?&lt;/a&gt;, but as that asked for &lt;em&gt;references&lt;/em&gt; the answers there don't focus on specifically answering this question per se, and in some ways could be considered &lt;a href=&quot;https://meta.stackexchange.com/questions/8231/are-answers-that-just-contain-links-elsewhere-really-good-answers&quot;&gt;link-only answers&lt;/a&gt; relative to &lt;em&gt;this&lt;/em&gt; question.&lt;/p&gt;&#xA;" OwnerUserId="9552" LastEditorUserId="9552" LastEditDate="2021-07-15T18:58:15.990" LastActivityDate="2021-07-15T18:58:15.990" Title="What does the Fourier transform have to do with heat?" Tags="|mathematics|physics|differential-equations|harmonic-analysis|fourier|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13397" PostTypeId="2" ParentId="13396" CreationDate="2021-07-15T05:58:20.660" Score="5" Body="&lt;p&gt;Not to sway others from providing a better answer from more interesting sources, but I think one clue can be found in the &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Joseph_Fourier&amp;amp;oldid=1031174965#The_Analytic_Theory_of_Heat&quot; rel=&quot;noreferrer&quot;&gt;current version&lt;/a&gt; of the Wikipedia article for Joseph Fourier himself:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There were three important contributions in this work [i.e. &lt;em&gt;The Analytic Theory of Heat&lt;/em&gt;], one purely mathematical, two essentially physical. In mathematics, Fourier claimed that any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. Though this result is not correct without additional conditions, Fourier's observation that some discontinuous functions are the sum of infinite series was a breakthrough. […]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Further simplifying and adding emphasis: &amp;quot;&lt;strong&gt;any&lt;/strong&gt; function can be expanded into a series of sines&amp;quot;. So even though heat transfer might not seem very &amp;quot;wavelike&amp;quot;, functions representing it could still be analyzed in terms of trigonometric series. And perhaps doing so was useful to find or analyze solutions of the underlying differential equation.&lt;/p&gt;&#xA;" OwnerUserId="9552" LastActivityDate="2021-07-15T05:58:20.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13398" PostTypeId="1" AcceptedAnswerId="15064" CreationDate="2021-07-15T14:47:00.950" Score="4" ViewCount="131" Body="&lt;p&gt;In &amp;quot;&lt;em&gt;On the Revolutions of the Heavenly Spheres&lt;/em&gt;&amp;quot; Book 5. chapter 16, Copernicus appears to acknowledge a discrepancy between the old Ptolemy observation and the model. &lt;a href=&quot;https://translate.google.com/translate?hl=&amp;amp;sl=de&amp;amp;tl=en&amp;amp;u=https%3A%2F%2Fde.wikisource.org%2Fwiki%2FNicolaus_Coppernicus_aus_Thorn_%25C3%25BCber_die_Kreisbewegungen_der_Weltk%25C3%25B6rper%2FF%25C3%25BCnftes_Buch_Teil_B%23C16&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here is&lt;/a&gt; a Google Translate from a German translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;if one calculates back, it turns out that the location of the apogeum of the eccentric circle was in 119 ° 40 ′ of the fixed star sphere at this last observation. Ptolemy found this place at the time of Antoninus at 108 ° 50 ′; so it has moved forward by 10 ° 50 ′ except for us. We have found the distance between the centers to be 40 such parts smaller, of which 10,000 come to the radius of the eccentric circle. Not as if Ptolemy or we were wrong, but to prove that the center of the earth's orbit has approached the center of the orbit of Mars while the sun has remained immobile...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Copernicus seems to be saying neither he nor Ptolemy were in error, but that the &amp;quot;center of the Earth's orbit&amp;quot; has moved. Did Copernicus ever try to explain how the &amp;quot;center of the Earth's orbit&amp;quot; moved? or to investigate the nature of the movement?&lt;/p&gt;&#xA;" OwnerUserId="14954" LastActivityDate="2022-12-27T17:12:32.117" Title="How does Copernicus explain the discrepancy from &quot;On the Revolutions of the Heavenly Spheres&quot; Book 5. chapter 16" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13400" PostTypeId="1" CreationDate="2021-07-16T22:44:29.483" Score="0" ViewCount="49" Body="&lt;p&gt;Proponents of classical thermodynamics believed the subject matter to be continuous, while the statistical thermodynamics proponents' subject matter was made up of atoms and variations of collectives of them.&lt;/p&gt;&#xA;&lt;p&gt;Were they in those days (19th century) each other's rival, or did they get along fine?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="12330" LastEditDate="2021-07-19T06:58:53.753" LastActivityDate="2021-07-19T06:58:53.753" Title="Did proponents of classical and statistical thermodynamics get along well with each other?" Tags="|physics|thermodynamics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13401" PostTypeId="1" AcceptedAnswerId="13405" CreationDate="2021-07-17T02:31:57.030" Score="0" ViewCount="135" Body="&lt;p&gt;I know &lt;a href=&quot;https://en.wikipedia.org/wiki/One-time_pad&quot; rel=&quot;nofollow noreferrer&quot;&gt;One-time-pad&lt;/a&gt; were used during ww2 and, I want to know how the random numbers were generated.&lt;/p&gt;&#xA;" OwnerUserId="14765" LastActivityDate="2021-07-17T12:41:59.970" Title="How the random numbers were generated which were used during world war 2(ww2)?" Tags="|cryptology|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13403" PostTypeId="2" ParentId="13392" CreationDate="2021-07-17T10:50:24.997" Score="0" Body="&lt;p&gt;You can call it whatever you like, depending on your attitude towards it. From a fallacy to an absolutely objective Platonic realism.&lt;/p&gt;&#xA;&lt;p&gt;I think it is a naive realistic view on history. It is the same realism as expressed by many physicists. A reality is assumed which pulls our thoughts in the right direction. Now this can of course not be denied. It is the question though how we know that our theories are indeed rightly pulled. Just stating that this happens automatically in the course of time is, well, naive. History shows that this is not the case and ignoring history or even subject it to a suppose rule is, well, naive.&lt;/p&gt;&#xA;&lt;p&gt;This view inhibits progress as new theories, mathematical or not, are excluded. Views that diverge are seen as not real and will consequently be put aside, ignored, or even laughed at and ridiculed.&lt;/p&gt;&#xA;&lt;p&gt;So one can call the view a conservative anti-revolutionary Platonic realism, in favor of the existing modes of thinking.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-19T00:24:17.637" LastActivityDate="2021-07-19T00:24:17.637" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="13404" PostTypeId="1" CreationDate="2021-07-17T11:36:31.513" Score="1" ViewCount="52" Body="&lt;p&gt;Higgs invented a mechanism to endow quarks and leptons an W- and Z-particles with mass. If a particle is massless though then how can it be slowed down by interacting with another particle? Maybe the particle gets an effective mass (like a photon traveling through a medium though its velocity can't change contrary to that of quarks and leptons) but the mass of the particle itself will not change. Its motion will at the most be retarded and thus make it seem the particle has a speed smaller than the speed of light and thus (effective) mass.&lt;/p&gt;&#xA;&lt;p&gt;What was Higgs' attitude towrds his own mechanism? Did he think in terms of an effective mass?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-17T11:36:31.513" Title="What was Higgs' attitude towards his own mechanism?" Tags="|particle-physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13405" PostTypeId="2" ParentId="13401" CreationDate="2021-07-17T12:41:59.970" Score="2" Body="&lt;p&gt;US Patent &lt;a href=&quot;https://patentimages.storage.googleapis.com/dc/40/14/2ec5d3d4b6bb03/US2406031.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;2406031&lt;/a&gt;, filed on 21 May 1942, issued on 20 Aug 1946, describes one such technology:  A hopper feeds a jumble of transparent and opaque beads past  a photocell array.  The beads are fed back into the hopper, and the data read from the photocells ends up on a punched tape.&lt;/p&gt;&#xA;&lt;p&gt;I don't know if this particular invention was used in practice, but it seems likely to me that something along these lines was used to produce the one-time keying material (the bulk of which was for use with teleprinters) for US forces in the war.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2021-07-17T12:41:59.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13408" PostTypeId="1" CreationDate="2021-07-18T00:43:31.667" Score="1" ViewCount="80" Body="&lt;p&gt;It is a well known fact that he admired Dostoyevsky even more than scientists. I'm looking for things like, Einstein's opinions about his religious discourse? What did he think of his bold statements on the natural sciences and how it's progress removed the &amp;quot;sacred?&amp;quot; Was Einstein enamored on the themes of organized religion in Dostoyevsky's fiction or more so his philosophical deism?&lt;/p&gt;&#xA;" OwnerUserId="14984" LastActivityDate="2021-07-18T00:43:31.667" Title="Is there any literature or articles which expand more on Einstein's thoughts about Dostoyevsky's novels?" Tags="|physics|einstein|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13409" PostTypeId="1" AcceptedAnswerId="13433" CreationDate="2021-07-18T06:18:57.357" Score="25" ViewCount="6376" Body="&lt;p&gt;Cosmic microwave background radiation was discovered after Penzias &amp;amp; Wilson couldn't get rid of the noise generated by their horn. In fact, the noise was their discovery.&lt;/p&gt;&#xA;&lt;p&gt;The strings in string theory were introduced in the context of the strong interaction to make the back-then newly discovered interaction between quarks visible. The strings turned out to be applicable (in closed form) even to gravity, certainly not what they were intended for.&lt;/p&gt;&#xA;&lt;p&gt;What are other examples of serendipity in the sciences?&lt;/p&gt;&#xA;&lt;p&gt;(If someone has a tag suggestion it is welcome)&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="12330" LastEditDate="2023-02-08T00:16:19.773" LastActivityDate="2024-01-12T14:24:19.003" Title="What are examples of serendipity in the history of the sciences and math?" Tags="|physics|big-list|" AnswerCount="14" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13410" PostTypeId="2" ParentId="13409" CreationDate="2021-07-18T07:10:04.670" Score="15" Body="&lt;p&gt;One example is radioactivity. In 1896, Henri Becquerel was working on an experiment involving a uranium-enriched crystal. He believed that sunlight was the reason that the crystal would burn its image on a photographic plate. But then, on a day, the weather was bad, with dark clouds blocking the light from the Sun. Becquerel packed up his stuff and decided to continue his research on another sunny day.&lt;/p&gt;&#xA;&lt;p&gt;Some few days later, he retrieved the crystal from a darkened drawer, but the image burned on the plate was, as he described, “fogged.” The crystal emitted rays that fogged a plate, but were dismissed as weaker rays compared to William Roentgen's X-rays. Becquerel wouldn't go on to put a name to the phenomenon. He left that for a couple of fellow scientists: Pierre and Marie Curie.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2021-07-18T07:10:04.670" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13411" PostTypeId="1" AcceptedAnswerId="13422" CreationDate="2021-07-18T08:11:13.960" Score="1" ViewCount="192" Body="&lt;p&gt;Can the following point of view be traced back to a particular publication by Poincaré? In &lt;a href=&quot;https://archive.org/details/GravitationMisnerThorneWheeler/page/n95/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Section 3.1 of Gravitation&lt;/a&gt;, by Misner, Thorne and Wheeler, the authors write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Here and elsewhere in science, as stressed not least by Henri Poincare, that view is out of date which used to say, “Define your terms before you proceed.” All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts they use (here B and E) and make statements about these concepts. Contrariwise, the absence of some body of theory, law, and principle deprives one of the means properly to define or even to use concepts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The reason I'm so interested in this: I do still encounter that &amp;quot;out of date&amp;quot; view.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Looking for clues I checked the &lt;a href=&quot;https://plato.stanford.edu/entries/poincare/#GeoConIntAes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stanford encyclopedia of Philosophy entry for Poincaré&lt;/a&gt;, which has&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Poincaré and Hilbert propose a radically new view of geometry. They hold that all that we can say about the meaning of ‘point’, ‘straight line’, ‘distance’, etc. is that which we have stated in the axioms or principles of the system, and that geometry is not a set of truths about some previously known objects. Thus Poincaré formulates a new view of geometric theories, that geometry does not express true or false propositions and that there are no special objects which geometry studies. Rather, geometry is just a system of relations that can be applied to many kinds of objects.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It comes close to what is raised in MTW, but it's not quite there, it seems. It may well be, of course, that more was attributed to Poincaré than was ever in Poincaré's actual writings.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For context: the same quote with paragraphs leading up to it included: from 'Gravitation', by Misner, Thorne and Wheeler's Section 3.1, The Lorentz force and the electromagnetic field tensor:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the opposite extreme from an impulsive change of momentum in a collision (the last topic of Chapter 2) is the gradual change in the momentum of a charged particle under the action of electric and magnetic forces (the topic treated here).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let electric and magnetic fields act on a system of charged particles. The accelerations of the particles reveal the electric and magnetic field strengths. In other words, the Lorentz force law, plus measurements on the components of acceleration of test particles, can be viewed as defining the components of the electric and magnetic fields. Once the field components are known from the accelerations of a few test particles, they can be used to predict the accelerations of other test particles (Box 3.1). Thus the Lorentz force law does double service (1) as definer of fields and (2) as predicter of motions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Here and elsewhere in science, as stressed not least by Henri Poincare, that view is out of date which used to say, “Define your terms before you proceed.” All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts they use (here B and E) and make statements about these concepts. Contrariwise, the absence of some body of theory, law, and principle deprives one of the means properly to define or even to use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concept, law, and method of measurement—forever inseparable—are born into the world in union.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Later in the book this point is raised again: that laws and theories of physics are both laws/theories of physics and &lt;em&gt;operational definitions&lt;/em&gt; of the concepts that the laws make statements about. See &lt;a href=&quot;https://archive.org/details/GravitationMisnerThorneWheeler/page/n318/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Section 12.3&lt;/a&gt; in 'Gravitation'.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="7244" LastEditDate="2023-01-22T11:44:59.083" LastActivityDate="2023-01-22T11:44:59.083" Title="Did Poincaré argue against defining terms before you proceed?" Tags="|philosophy-of-science|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13412" PostTypeId="1" CreationDate="2021-07-18T11:20:38.837" Score="2" ViewCount="99" Body="&lt;p&gt;Becquerel was awarded the Nobel prize for his discovery of radioactivity. He was researching phosphoresence and decided one day to place the stuff he used in a drawer to keep it out of the sun. He put it on top of a photographic plate (coincidentally) and saw that on the plate the form of the stuff he put on top. Radioactivity was discovered.&lt;/p&gt;&#xA;&lt;p&gt;Now in Wikipedia one can resd:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As often happens in science, radioactivity came close to being discovered nearly four decades earlier in 1857, when Abel Niépce de Saint-Victor, who was investigating photography under Michel Eugène Chevreul, observed that uranium salts emitted radiation that could darken photographic emulsions. By 1861, Niepce de Saint-Victor realized that uranium salts produce &amp;quot;a radiation that is invisible to our eyes&amp;quot;. Niepce de Saint-Victor knew Edmond Becquerel, Henri Becquerel's father. In 1868, Edmond Becquerel published a book, La lumière: ses causes et ses effets (Light: Its causes and its effects). On page 50 of volume 2, Edmond noted that Niepce de Saint-Victor had observed that some objects that had been exposed to sunlight could expose photographic plates even in the dark. Niepce further noted that on the one hand, the effect was diminished if an obstruction were placed between a photographic plate and the object that had been exposed to the sun, but &amp;quot; … d'un autre côté, l'augmentation d'effet quand la surface insolée est couverte de substances facilement altérables à la lumière, comme le nitrate d'urane … &amp;quot; ( ... on the other hand, the increase in the effect when the surface exposed to the sun is covered with substances that are easily altered by light, such as uranium nitrate ... )&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So Becquerel's father was already aware of the effect. The effect on photographic plates was enhanced if a uranium salt was placed between the sun-stuff and the plates. How was this interpreted? How could uranium salt enhance this effect on the plates, instead of decreasing it? I guess not in the way his son did. I am sure that Becquerel jr. read his father's book. Was he influenced by this observation his father made? At first sight you would say no, because then he would have done experiments with salts without sunlight from the start. To examine what his father described in his book. Or would he have done these anyhow? Was he unaware of the book maybe? I cannot imagine. Why didn't he try to reproduce the observation made already in 1861 by de Saint Victor? Now of course people back then were still in the dark and it is always east talk in hindsight. Was Bequerel trying to explain it all by electromagnetic radiation only? So the uranium salt put on the phosphoresence stuff was somehow activated by that stuff? Did he think from the start that the guy his father wrote about saw radioactivity? Did he need some incident to make it clear that the uranium salt radiates on his own? This is pure soeculation of course. If he thought uranium salt radiates on its own he might just as well have had put it in the dark on a photographic plate. Did he maybe think a coincidental discovery was more spectacular? On the other hand, doing a conscious experiment to examine if your theory is right demonstrates the fertility of the mind. Was he aware of his fathers writings (testimonies)?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="6085" LastEditDate="2021-07-25T20:16:38.200" LastActivityDate="2021-07-25T20:16:38.200" Title="Was Becquerel's discovery of radioactivity inspired by his father?" Tags="|physics|biographical-details|discoveries|physicists|radioactivity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13413" PostTypeId="1" CreationDate="2021-07-18T12:49:55.417" Score="0" ViewCount="139" Body="&lt;p&gt;Ptolemy invented a system to describe the periodic motion of the planets by epicycles. Fourier did something similar for periodic motion in mechanics. Every such motion can be thought of as constructed of out of different kinds of epicycles, namely sine functions with different amplitudes and frequencies.&lt;/p&gt;&#xA;&lt;p&gt;Was Fourier inspired by Ptolemy?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4251" LastEditDate="2021-07-19T12:42:36.483" LastActivityDate="2021-07-19T13:57:58.133" Title="Was Fourier inspired by Ptolemy?" Tags="|mathematicians|19th-century|fourier|harmonic-analysis|ptolemy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13414" PostTypeId="2" ParentId="13413" CreationDate="2021-07-18T13:18:55.637" Score="5" Body="&lt;p&gt;Fourier was not inspired by Ptolemy or by mechanics. His work where he developed Fourier series and invented Fourier transform was about heat. He never mentions Ptolemy (or any astronomer). The variable in his Fourier series is&#xA;always a space variable, not the time.&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy and other astronomers were concerned not only with periodic motions but with what we would call&#xA;almost periodic motions, &lt;span class=&quot;math-container&quot;&gt;$\sum c_k e^{i\lambda_k t}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\lambda_k$&lt;/span&gt; are in general non-commensurable. Later (in 20th century) such series were called&#xA;'non-harmonic Fourier series&amp;quot;, or almost periodic functions, and Fourier analysis was generalized to include them.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-19T13:57:58.133" LastActivityDate="2021-07-19T13:57:58.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13416" PostTypeId="2" ParentId="13409" CreationDate="2021-07-18T13:30:55.483" Score="29" Body="&lt;p&gt;One famous example is that of &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Fleming#Discovery_of_penicillin&quot; rel=&quot;noreferrer&quot;&gt;Alexander Fleming&lt;/a&gt;. He left a cup of staphylococci on his desk and later discovered that it was contaminated with some fungus. Instead of disposing this cup, he started to investigate and the result was the discovery of penicillin, the first antibiotic. This is how he later described this himself:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One sometimes finds, what one is not looking for. When I woke up just after dawn on September 28, 1928, I certainly didn't plan to revolutionize all medicine by discovering the world's first antibiotic, or bacteria killer. But I suppose that was exactly what I did.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-07-18T13:30:55.483" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13417" PostTypeId="2" ParentId="13270" CreationDate="2021-07-18T17:03:57.700" Score="1" Body="&lt;p&gt;As suggested in a &lt;a href=&quot;https://hsm.stackexchange.com/questions/13270/why-did-the-copenhagen-interpretation-of-quantum-mechanics-got-the-upperhand#comment27857_13270&quot;&gt;comment&lt;/a&gt;, this question may not be objectively answerable, but for what it is worth, here is a quote from the book &lt;i&gt;What is Real?&lt;/i&gt; by Adam Becker (mentioned in &lt;a href=&quot;https://hsm.stackexchange.com/questions/13270/why-did-the-copenhagen-interpretation-of-quantum-mechanics-got-the-upperhand#comment27855_13270&quot;&gt;another comment&lt;/a&gt;). Becker proposes the following narrative, which argues (as mentioned in &lt;a href=&quot;https://hsm.stackexchange.com/questions/13270/why-did-the-copenhagen-interpretation-of-quantum-mechanics-got-the-upperhand#comment27911_13270&quot;&gt;yet another comment&lt;/a&gt;) that John von Neumann played an important role.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yet Einstein’s pleas for a more complete theory went unheard, in part because of John von Neumann’s proof that no such theory was possible. Von Neumann was arguably the greatest mathematical genius alive. … His colleagues at Princeton said, only half-joking, that von Neumann could prove anything—and anything he proved was correct.&lt;br&gt;&lt;br&gt;Von Neumann published his proof as part of his textbook on quantum physics in 1932. There’s no evidence that Einstein was even aware of this proof, but many other physicists were—and for them, merely the idea of a proof from the mighty von Neumann was enough to settle the debate. The philosopher Paul Feyerabend experienced this firsthand after attending a public talk given by Bohr: “At the end of the lecture [Bohr] left, and the discussion proceeded without him. Some speakers attacked his qualitative arguments—there seemed to be lots of loopholes. The Bohrians did not clarify the arguments; they mentioned the alleged proof by von Neumann and that settled the matter…like magic, the mere name of ‘von Neumann’ and the mere word ‘proof’ silenced the objectors.”&lt;br&gt;&lt;br&gt;&#xA;At least one person did notice a problem with von Neumann’s proof shortly after it was published. Grete Hermann, a German mathematician and philosopher, published a paper in 1935 criticizing von Neumann’s proof. Hermann pointed out that von Neumann failed to justify a crucial step, and thus the whole proof was flawed. But nobody listened to her, partly because she was an outsider to the physics community—and partly because she was a woman.&lt;br&gt;&lt;br&gt;&#xA;Despite the flaw in von Neumann’s proof, the Copenhagen interpretation remained totally dominant. Einstein was painted as an old man out of touch with the rest of the world, and questioning the Copenhagen interpretation became tantamount to questioning the massive success of quantum physics itself. And so quantum physics continued for the next twenty years, piling success upon success, without any further questions about the hole at its heart.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As an aside, I do not think that the Copenhagen interpretation “reigns supreme” nowadays.  Textbooks that spend any time discussing the philosophy of quantum mechanics typically mention that the topic is controversial, and briefly describe several different approaches.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-18T17:03:57.700" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13418" PostTypeId="2" ParentId="13356" CreationDate="2021-07-18T18:04:49.290" Score="11" Body="&lt;p&gt;Perhaps the &lt;a href=&quot;https://en.wikipedia.org/wiki/Poster_child&quot; rel=&quot;noreferrer&quot;&gt;poster child&lt;/a&gt; for this kind of thing is Apéry's theorem that ζ(3) is irrational, as described for example in Alf van der Poorten's paper, &lt;a href=&quot;http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;A proof that Euler missed&lt;/a&gt; (&lt;i&gt;Mathematical Intelligencer&lt;/i&gt; &lt;b&gt;1&lt;/b&gt; (1979), 195–203). Apéry's proof was highly ingenious and original, and not easy to grasp even after the fact, but it was short and used only techniques that Euler would have had no difficulty understanding (hence the title of Van der Poorten's paper).&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2021-07-18T18:16:30.500" LastActivityDate="2021-07-18T18:16:30.500" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13419" PostTypeId="2" ParentId="13356" CreationDate="2021-07-18T18:35:09.490" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Robbins_algebra&quot; rel=&quot;noreferrer&quot;&gt;Robbins conjecture&lt;/a&gt; was posed in the 1930s but not proved until 1996.  The proof was achieved by William McCune with the assistance of the automated theorem prover EQP. Unlike some other computer-assisted proofs, this proof was digestible by a human being, especially after simplifications found by Bernd Dahn.&lt;/p&gt;&#xA;&lt;p&gt;Dahn's version of the proof is short and completely elementary, but still rather mysterious.  It shows that a certain identity is a formal consequence of the axioms, but via a sequence of manipulations that is difficult for a human to find, not so much because the proof is infeasibly long or laborious, but because there does not seem to be a clear conceptual idea to guide one's thinking from beginning to end.  The proof is simple enough that it could have been discovered without computers, but at the same time, it is not too surprising that it was first discovered with the aid of a computer.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-18T18:35:09.490" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13420" PostTypeId="1" CreationDate="2021-07-18T20:15:40.980" Score="3" ViewCount="788" Body="&lt;p&gt;Fourier stated that every function can be decomposed into sine and cosine functions. Was he referring to periodic functions only? To a certain class only? I ask, because it seems clear (at least to me) that most functions &lt;em&gt;cannot&lt;/em&gt; be so decomposed. One simple example is the n-th degree polynomial. A small portion can be approximated, but the whole function finds no place in the function space spanned by the sines and cosines.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4251" LastEditDate="2021-07-19T12:43:44.137" LastActivityDate="2021-07-19T14:20:54.797" Title="What did Fourier mean by stating that every function can be decomposed into sine and cosine functions?" Tags="|mathematics|fourier|harmonic-analysis|" AnswerCount="1" CommentCount="16" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13421" PostTypeId="2" ParentId="13409" CreationDate="2021-07-18T21:57:51.323" Score="4" Body="&lt;p&gt;The principle of the microwave oven was discovered by accident.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1946, the engineer Dr. Percy LeBaron Spencer, who worked for the Raytheon Corporation, was working on magnetrons. One day at work, he had a candy bar in his pocket, and found that it had melted. He realized that the microwaves he was working with had caused it to melt. After experimenting, he realized that microwaves would cook foods quickly - even faster than conventional ovens that cook with heat.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.enchantedlearning.com/inventors/page/m/microwaveoven.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zoom Inventors and Inventions&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10187" LastActivityDate="2021-07-18T21:57:51.323" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13422" PostTypeId="2" ParentId="13411" CreationDate="2021-07-18T23:51:16.113" Score="4" Body="&lt;p&gt;A good account on genesis of the modern view of definitions (as opposed to the definitions-first Aristotelian one) is &lt;a href=&quot;https://journals.openedition.org/philosophiascientiae/1827&quot; rel=&quot;noreferrer&quot;&gt;Fontanella, Axioms as Definitions: Revisiting Poincaré and Hilbert&lt;/a&gt;. Perhaps the clearest expression of it from Poincare comes from his response to Russell &lt;em&gt;Sur les principes de la géométrie: réponse à M. Russell, Revue de Métaphysique et de Morale&lt;/em&gt;, 8(1) (1900) 73–86:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If one wants to isolate a term and exclude its relations with other terms, nothing will remain. This term will not only become indefinable, it will become devoid of meaning.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Science and Hypothesis (1902), when speaking of geometry, he links the idea to his famous conventionalism:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The axioms of geometry are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free and is only limited by the necessity of avoiding every contradiction [...]. In other words, the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Although it is an implication, I doubt that Poincare explicitly stated in the digested didactic manner of MTW that we shoud stop asking for terms to be defined before proceeding.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-07-18T23:51:16.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13424" PostTypeId="2" ParentId="13409" CreationDate="2021-07-19T08:07:30.100" Score="10" Body="&lt;p&gt;The subject of &lt;strong&gt;Graph Theory&lt;/strong&gt; in mathematics, via the &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;noreferrer&quot;&gt;Seven Bridges of Königsberg&lt;/a&gt; and involving Euler of all people!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/0C78d.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/0C78d.png&quot; alt=&quot;Konigsberg Bridges&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;&amp;quot;The theory of graphs is one of the few fields of mathematics with a&#xA;definite birth date.&amp;quot;&lt;/strong&gt; -Oystein Ore, graph theorist &amp;amp; number theorist&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Namely, that birthdate is in 1736 when Leonard Euler refers to &amp;quot;geometry of position&amp;quot; that we've come to know and love as Graph Theory.&lt;/p&gt;&#xA;&lt;p&gt;Quotes below are from Chapter 3 of &lt;em&gt;Graphs &amp;amp; Digraphs&lt;/em&gt;, 5th Edition, by Chartrand, Lesniak, and Zhang:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Early in the 18th century, the East Prussian city of Konigsberg (now called&#xA;Kaliningrad and located in Russia) occupied both banks of the River Pregel&#xA;and the island of Kneiphof, lying in the river at a point where it branches into&#xA;two parts. There were seven bridges that spanned various sections of the river.&#xA;A popular puzzle, called the Konigsberg Bridge Problem, asked whether&#xA;there was a route that crossed each of these bridges exactly once. Although such&#xA;a route was long thought to be impossible, the first mathematical verification of&#xA;this was presented by the famed mathematician Leonhard Euler (1707–1783) at&#xA;the Petersburg Academy on 26 August 1735. Euler’s proof was contained in a paper that would turn out to be the beginning of graph theory. This paper&#xA;appeared in the 1736 volume of the proceedings of the Petersburg Academy.&#xA;Euler’s paper, written in Latin, started as follows (translated into English):&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In addition to that branch of geometry which is concerned with magnitudes, and which has always received the greatest attention, there&#xA;is another branch, previously almost unknown, which Leibniz first&#xA;mentioned, calling it &lt;strong&gt;the geometry of position&lt;/strong&gt;. This branch is concerned only with the determination of position and its properties; it&#xA;does not involve measurements, nor calculations made with them.&#xA;&lt;strong&gt;It has not yet been satisfactorily determined what kind of problems&#xA;are relevant to this geometry of position, or what methods should&#xA;be used in solving them.&lt;/strong&gt; Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did&#xA;not require the measurement of distances, nor did calculation help&#xA;at all, I had no doubt that it was concerned with the geometry of&#xA;position – especially as its solution involved only position, and no&#xA;calculation was of any use. I have therefore decided to give here the&#xA;method which I have found for solving this kind of problem, as an&#xA;example of the geometry of position.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Euler goes on to say (see an excerpt from this &lt;a href=&quot;https://youtu.be/fEWj93XjON0?t=1095&quot; rel=&quot;noreferrer&quot;&gt;amazing lecture on Euler's genius&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;&amp;quot;...this solution bears little relationship to mathematics, and I do not understand why to expect a mathematician to produce it, rather than anyone else, for the solution is based on logic alone.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I guess I find this story so serendipitous, because you had to have so many things fall into place:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the rivers naturally flowing in a unique way&lt;/li&gt;&#xA;&lt;li&gt;people to settle and&#xA;build just the right bridges in just the right places to make the&#xA;underlying problem unsolvable&lt;/li&gt;&#xA;&lt;li&gt;an arbitrary human constraint to want&#xA;to cross bridges exactly once on a Sunday afternoon walk&lt;/li&gt;&#xA;&lt;li&gt;petitioning&#xA;one of the most prolific and insightful minds of mathematics for all&#xA;time (Euler)...&lt;/li&gt;&#xA;&lt;li&gt;...who proceeded to solve it &lt;em&gt;despite the fact&lt;/em&gt; that he&#xA;didn't think it had any applications in math, nor did he think a&#xA;mathematician was needed to solve this problem!&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I find it hard to believe Graph Theory would not have been developed eventually by someone else for another of its many applications, but to think that &lt;em&gt;this&lt;/em&gt; was the fashion that it was introduced makes me consider it a serendipitous outcome.&lt;/p&gt;&#xA;" OwnerUserId="285" LastActivityDate="2021-07-19T08:07:30.100" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13425" PostTypeId="2" ParentId="13337" CreationDate="2021-07-19T12:05:47.600" Score="4" Body="&lt;p&gt;I will promote my comment to an answer. The theory of permutations and permutation groups was the original (abstract) setting of group theory, and so the term originated there. I believe the reason for it was as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The order of a permutation is the least number such that the ordering of the index set is preserved.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I looked in both Lagrange's &lt;a href=&quot;https://books.google.co.uk/books?id=_-U_AAAAYAAJ&amp;amp;pg=PA138&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1771 memoir&lt;/a&gt; and Cauchy's &lt;a href=&quot;https://books.google.co.uk/books?id=-c3fxufDQVEC&amp;amp;pg=PA151#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;1844 memoir&lt;/a&gt;, which both address &lt;a href=&quot;https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange's theorem&lt;/a&gt; on orders of subgroups. Lagrange barely uses the word order (well, &amp;quot;ordre&amp;quot; as he is writing in French), and doesn't use it in a way which influences out discussion here (the only relevant use is on p201, when he changes the orders of variables in an equation).&lt;/p&gt;&#xA;&lt;p&gt;Cauchy's memoir is more relevant. He goes from talking about, basically, &amp;quot;permutations change the order of the index set&amp;quot; to defining the order of a permutation. He doesn't justify this name, but writing like this does suggest the reason is that it is the least number such that the ordering of the index set is preserved, as per my claim above.&lt;/p&gt;&#xA;" OwnerUserId="9504" LastActivityDate="2021-07-19T12:05:47.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13426" PostTypeId="2" ParentId="5875" CreationDate="2021-07-19T13:05:52.867" Score="1" Body="&lt;p&gt;From Wikipedia;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although &lt;span class=&quot;math-container&quot;&gt;$\gamma ^{5}$&lt;/span&gt; uses the letter gamma, it is not one of the gamma matrices of Cl1,3(R). The number 5 is a relic of old notation in which &lt;span class=&quot;math-container&quot;&gt;$\gamma ^{0}$&lt;/span&gt; was called &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\gamma ^{4}$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-19T13:05:52.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13427" PostTypeId="2" ParentId="13420" CreationDate="2021-07-19T14:13:04.870" Score="7" Body="&lt;p&gt;First of all, one has to notice that the work of Fourier predates all modern notions of function. The notion of function was first clarified by Dirichlet, in his attempt to justify Fourier's arguments. What Fourier probably meant was that&lt;/p&gt;&#xA;&lt;p&gt;a) any reasonable periodic function, say with period &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt; can be expanded into&#xA;a series of the form &lt;span class=&quot;math-container&quot;&gt;$\sum a_k\cos kx+b_k\sin kx$&lt;/span&gt;. And&lt;/p&gt;&#xA;&lt;p&gt;b) that many non-periodic functions on the real line can be represented by Fourier integrals&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int \cos(sx)\phi(s)ds,\quad\int\sin(sx)\psi(s)ds.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Fourier was solving specific problems of mathematical physics.&#xA;Even his contempories understood that many of his arguments are not mathematically rigorous (and he had difficulty with publication of his book for this reason; it was very much criticized).&lt;/p&gt;&#xA;&lt;p&gt;He did not specify the exact class of functions for which these statements are true, and it took mathematicians many decades to arrive at exact statements.&#xA;The very notion of function evolved in this process, beginning with Dirichlet's definition, and proceeding to &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$L^2$&lt;/span&gt;-functions&amp;quot; (which are not exactly functions in the Dirichlet sense), and then to distributions and hyperfunctions. All these notions were invented with the purpose of making precise sense of Fourier ideas.&lt;/p&gt;&#xA;&lt;p&gt;To take your specific example of polynomials, only in the 1950s it became clear how to represent polynomials as Fourier integrals; this requires the theory of &amp;quot;tempered distributions&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-19T14:20:54.797" LastActivityDate="2021-07-19T14:20:54.797" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13428" PostTypeId="2" ParentId="13409" CreationDate="2021-07-19T16:17:40.533" Score="8" Body="&lt;p&gt;There's always Archimedes ''eureka' moment, sitting in a bath tub and realizing that a given volume of a material will displace the same amount of water when fully submerged regardless of the object's shape.&lt;/p&gt;&#xA;" OwnerUserId="13957" LastActivityDate="2021-07-19T16:17:40.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13429" PostTypeId="2" ParentId="13409" CreationDate="2021-07-19T17:23:42.727" Score="28" Body="&lt;p&gt;In 1961, Edward Lorenz was running a numerical simulation of weather systems.  To restart a run half-way, he copied the initial values from a previous printout — but saw that the simulation began to diverge from the previous run.&lt;/p&gt;&#xA;&lt;p&gt;This turned out to be the result of the printout showing values that were rounded to 3 decimal places, without the full precision used in the calculations.  He expected the resulting errors to stay small; but instead, they grew exponentially, doubling roughly every four simulated days, until after two simulated months the weather scenario was completely different.&lt;/p&gt;&#xA;&lt;p&gt;He thus encountered what is now known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Butterfly_effect&quot; rel=&quot;noreferrer&quot;&gt;butterfly effect&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="13776" LastEditorUserId="13776" LastEditDate="2021-07-20T20:13:45.420" LastActivityDate="2021-07-20T20:13:45.420" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13431" PostTypeId="1" CreationDate="2021-07-19T19:56:08.177" Score="3" ViewCount="162" Body="&lt;p&gt;After he read Mobius's 1827 treatise on the &amp;quot;barycentric calculus&amp;quot; (according to Gauss's own testimony, he read this treatise only in 1843), Gauss wrote down several unpublished notes on what he called &amp;quot;resultant calculus&amp;quot; (I don't know if this a general term or just Gauss's own term), and then he used this in the context of his studies of the &amp;quot;pentagramma myrificum&amp;quot;. To make it easier to offer explanation of the main points of this calculus, I added a (google) translation of Gauss's fragment, which is very short:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[1] The barycentric calculus finds its counterpart in another (presumably still comprehensive) calculus, which one might call the resultant calculus. Just as the first deals with points in which one assumes heavy masses; so the latter would have as its object lines in which forces act. If &lt;span class=&quot;math-container&quot;&gt;$a,b,c,d$&lt;/span&gt; and so on are such lines in which, in each of them in a certain sense, forces act which are proportional to the numbers &lt;span class=&quot;math-container&quot;&gt;$\alpha,\beta,\gamma,\delta$&lt;/span&gt; and so on, then the equation becomes &lt;span class=&quot;math-container&quot;&gt;$$a\alpha+b\beta+c\gamma+d\delta+\cdots = 0$$&lt;/span&gt;, and it means that these forces keep each other in balance.&#xA;[2] If &lt;span class=&quot;math-container&quot;&gt;$B,C$&lt;/span&gt; are two points in &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, one can quite appropriately put &lt;span class=&quot;math-container&quot;&gt;$B-C = ai$&lt;/span&gt;.&#xA;[3] The main question now is what kind of force one has to apply, baycentrically to the angular points, or resultantically to the sides of the pentagon, so that the simplest possible relationships take place between them and their equivalents at the inner pentagon.&#xA;[4] The co-linearity of two systems of points or straight lines can be best defined by the fact that one linear equation can always be divided between five points (lines) of one system and the five corresponding ones of the other.&#xA;[5] The characteristic of colineation seems to be expressed most simply like this: let &lt;span class=&quot;math-container&quot;&gt;$A,B,...$&lt;/span&gt; be straight lines, &lt;span class=&quot;math-container&quot;&gt;$P,Q,...$&lt;/span&gt; points of the one system to which the lines &lt;span class=&quot;math-container&quot;&gt;$a,b...$&lt;/span&gt; and the points &lt;span class=&quot;math-container&quot;&gt;$p,q...$&lt;/span&gt; correspond in the other. If now one expresses the (vertical) distance of the point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; from the straight line &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; using &lt;span class=&quot;math-container&quot;&gt;$PA$&lt;/span&gt;, then in general: &lt;span class=&quot;math-container&quot;&gt;$$\frac{PA\cdot QB\cdot pb \cdot qa}{pa \cdot qb\cdot PB \cdot QA}=1$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, I have many misunderstandings concerning this fragment:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;strong&gt;The basic idea&lt;/strong&gt;&lt;/em&gt; - While the conception of &amp;quot;barycentric coordinates&amp;quot; is clear to me, the general conception of the &amp;quot;resultant calculus&amp;quot;, as a kind of dual-equivalent to Mobius's calculus, is unclear to me. Therefore, I'd like to know if anyone can understand from the translation what is the basic idea behind it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;&lt;strong&gt;Notation clarification&lt;/strong&gt;&lt;/em&gt; -  Does Gauss refer to complex sizes in what he denotes &lt;span class=&quot;math-container&quot;&gt;$\alpha,\beta,\gamma,\delta$&lt;/span&gt; or just to real sizes?  I ask because in note [2] Gauss uses &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; (the imaginary unit).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have several other questions on this fragment and its relation to the pentagramma myrificum, but as a start I just want to clarify the notation and the basic idea.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="12330" LastEditDate="2021-07-19T20:38:20.453" LastActivityDate="2021-07-19T20:38:20.453" Title="Explanation of the main points in Gauss's resultant calculus" Tags="|geometry|gauss|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13432" PostTypeId="2" ParentId="13409" CreationDate="2021-07-19T22:37:47.280" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/a/13410/3869&quot;&gt;José Carlos Santos' answer&lt;/a&gt; mentions X-rays but not their discovery.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Nature&lt;/em&gt; 53,–276 (1896) By W. C. Röntgen. Translated by Arthur Stanton from the Sitznngsbcrichte der Würzburger Physik-medic. Gesellschaft, 1895: &lt;a href=&quot;https://www.nature.com/articles/053274b0.pdf&quot; rel=&quot;noreferrer&quot;&gt;On a New Kind of Rays&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;The Guardian has interesting background and mentions &lt;a href=&quot;https://www.google.com/doodles/discovery-of-x-rays&quot; rel=&quot;noreferrer&quot;&gt;this google doodle&lt;/a&gt; as well &lt;a href=&quot;https://www.theguardian.com/technology/blog/2010/nov/08/google-doodle-x-ray-115-year-anniversary&quot; rel=&quot;noreferrer&quot;&gt;Google doodle celebrates 115 years of X-rays&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From the American Physical Society's &lt;a href=&quot;https://www.aps.org/publications/apsnews/200111/history.cfm&quot; rel=&quot;noreferrer&quot;&gt;This Month in Physics History; November 8, 1895: Roentgen's Discovery of X-Rays&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Roentgen's scientific career was one beset with difficulties.&lt;/strong&gt; As a student in Holland, he was expelled from the Utrecht Technical School for a prank committed by another student. His lack of a diploma initially prevented him from obtaining a position at the University of Würzburg even after he received his doctorate, although he eventually was accepted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://columbiasurgery.org/news/2015/09/17/history-medicine-dr-roentgen-s-accidental-x-rays&quot; rel=&quot;noreferrer&quot;&gt;History of Medicine: Dr. Roentgen’s Accidental X-Rays&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wilhelm Roentgen, Professor of Physics in Wurzburg, Bavaria, &lt;strong&gt;discovered X-rays in 1895—accidentally—while testing whether cathode rays could pass through glass&lt;/strong&gt;. His cathode tube was covered in heavy black paper, &lt;strong&gt;so he was surprised when an incandescent green light nevertheless escaped and projected onto a nearby fluorescent screen.&lt;/strong&gt; Through experimentation, he found that the mysterious light would pass through most substances but leave shadows of solid objects. Because he did not know what the rays were, he called them ‘X,’ meaning ‘unknown,’ rays.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://en.wikipedia.org/wiki/X-ray#Discovery_by_R%C3%B6ntgen&quot; rel=&quot;noreferrer&quot;&gt;X-ray; Discovery by Röntgen&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On November 8, 1895, German physics professor Wilhelm Röntgen stumbled on X-rays while experimenting with Lenard tubes and Crookes tubes and began studying them. He wrote an initial report &amp;quot;On a new kind of ray: A preliminary communication&amp;quot; and on December 28, 1895 submitted it to Würzburg's Physical-Medical Society journal. This was the first paper written on X-rays. Röntgen referred to the radiation as &amp;quot;X&amp;quot;, to indicate that it was an unknown type of radiation. The name stuck, although (over Röntgen's great objections) &lt;strong&gt;many of his colleagues suggested calling them Röntgen rays. They are still referred to as such in many languages, including German, Hungarian, Ukrainian, Danish, Polish, Bulgarian, Swedish, Finnish, Estonian, Turkish, Russian, Latvian, Lithuanian, Japanese, Dutch, Georgian, Hebrew and Norwegian.&lt;/strong&gt; Röntgen received the first Nobel Prize in Physics for his discovery.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2021-07-19T22:55:10.900" LastActivityDate="2021-07-19T22:55:10.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13433" PostTypeId="2" ParentId="13409" CreationDate="2021-07-20T02:37:36.420" Score="16" Body="&lt;p&gt;Teflon (Polytetrafluoroethylene) was discovered by Roy Plunkett while working on new refrigerant.  There was a residue at the bottom of the bottle he was using, a bottle which should have been emptied of gas and yet still registered extra weight.  He discovered that the milky new substance was stable and extremely slippery and, voila, we can no longer use metal implements on our non-stick pans!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.sciencehistory.org/historical-profile/roy-j-plunkett&quot; rel=&quot;noreferrer&quot;&gt;Roy J Plunkett; Science History Institute&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Another example is the discovery in 2006 that tarantulas have silk-producing spinnerets on their feet which they use for traction.  Adam Summers, a research assistant at University of California, Irvine normally placed tarantulas on a glass plate tilted at an angle to store them temporarily. Tarantulas have a fear of falling and would simply cling to the glass without moving, a fact long-established.  One day he noticed that there were trails of web from the tarantulas' feet where the tarantulas had been slowly sliding down the angled glass plate.  Prior to this observation, it was not known, apparently, that any spiders possessed spinnerets on their feet, a fact with evolutionary implications. A chance observation led to the discovery.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.sciencedaily.com/releases/2006/09/060927201410.htm&quot; rel=&quot;noreferrer&quot;&gt;Tarantulas Produce Silk From Feet, Science Daily&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14997" LastEditorUserId="14997" LastEditDate="2021-07-22T11:32:17.990" LastActivityDate="2021-07-22T11:32:17.990" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13434" PostTypeId="1" CreationDate="2021-07-20T07:44:11.133" Score="-1" ViewCount="264" Body="&lt;p&gt;In landmark experiment in physics Wu discovered that Nature has a preference for right or left. It were the proposers of this asymmetry though who ran away with the the Nobel prize. Madam Wu wasn't awarded one.&lt;/p&gt;&#xA;&lt;p&gt;Now proposing such a theory isn't that difficult. Showing that this is actually the case is. The experiment was proposed by the ones who proposed the idea. They ran away with the prize.&lt;/p&gt;&#xA;&lt;p&gt;Why wasn't she awarded the Nobel Prize? Because she was female and Asian? Or what?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-21T08:24:28.157" LastActivityDate="2021-07-22T00:35:45.193" Title="Why wasn't madam Wu awarded a Nobel Prize?" Tags="|particle-physics|nobel-prize|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13435" PostTypeId="2" ParentId="13409" CreationDate="2021-07-20T19:37:22.603" Score="22" Body="&lt;p&gt;It was well known that when you iterate &lt;span class=&quot;math-container&quot;&gt;$x\mapsto \lambda x(1-x)$&lt;/span&gt;, the following happens: when &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; is small, &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; is an attracting fixed point; as &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; grows, at some moment &lt;span class=&quot;math-container&quot;&gt;$\lambda_1$&lt;/span&gt; the attracting point becomes repelling, but an attracting &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-cycle is born nearby. (&lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;-cycle is an fixed point of &lt;span class=&quot;math-container&quot;&gt;$f\circ f$&lt;/span&gt;). As &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; grows further, this attracting cycle becomes repelling at &lt;span class=&quot;math-container&quot;&gt;$\lambda_2&amp;gt;\lambda_1$&lt;/span&gt;, but a&#xA;new attracting &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;-cycle is born nearby, so we have a sequence of &amp;quot;doubling bifurcations&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\lambda_1&amp;lt;\lambda_2&amp;lt;\ldots$&lt;/span&gt;, and this sequence converges to some&#xA;&lt;span class=&quot;math-container&quot;&gt;$\lambda_\infty$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In 1978, &lt;a href=&quot;https://en.wikipedia.org/wiki/Mitchell_Feigenbaum&quot; rel=&quot;noreferrer&quot;&gt;Mitchell Feigenbaum&lt;/a&gt; studied this sequence using a hand calculator. He used Newton method to compute &lt;span class=&quot;math-container&quot;&gt;$\lambda_k$&lt;/span&gt;. Newton's method requires a good initial guess. Since the computation was slow, Feigenbaum started to think what the best initial guess will be, that is how to extrapolate the obtained sequence one step further. And he discovered that&#xA;the ratio &lt;span class=&quot;math-container&quot;&gt;$(\lambda_{n}-\lambda_{n-1})/(\lambda_{n+1}-\lambda_{n})$&lt;/span&gt; tends to a constant. To his great surprise, when he tried another function, &lt;span class=&quot;math-container&quot;&gt;$x\mapsto \lambda\sin x$&lt;/span&gt;, he found that this ratio tends to the SAME constant!&lt;/p&gt;&#xA;&lt;p&gt;This great discovery is called the Feigenbaum Universality, and the constant the&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Feigenbaum_constants&quot; rel=&quot;noreferrer&quot;&gt;Feigenbaum constant&lt;/a&gt;. It is a universal mathematical constant (like &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;),&#xA;and it is approximately equal to 4.669201609102990671853203820466...&lt;/p&gt;&#xA;&lt;p&gt;He said himself once, that the discovery was made because&#xA;he had no programmable calculator at his disposal.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-02-05T14:32:21.023" LastActivityDate="2023-02-05T14:32:21.023" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13437" PostTypeId="2" ParentId="13434" CreationDate="2021-07-21T05:43:16.830" Score="1" Body="&lt;p&gt;I think it is clear by now. As Cosmas Zachos pointed out, the theory was not just proposed. The symmetry between left and right was regarded almost as sacred.  Nature was written not right- and not left-handed. Allah, JHWH, or God was not prejudiced or discriminating. So proposing a model that went against this was almost blasphemy. It could cost your career (which is why it is better not to have one maybe). So the Nobel Prize for Lee and Yang was well-deserved. If the creator truly showed a preference remains to be seen. In our part of the universe yes but maybe there is another part were only anti particles exist (mirror universe).&lt;/p&gt;&#xA;&lt;p&gt;Now at the time madame Wu did her experiment there were three other experiments done. They didn't involve nuclear beta radiation but different stuff to look for the asymmetry. Shortly after one another (just days) these were finished. Ledermann, working at the same university as Wu, heard about her experiment and proposed another experiment. He and some collegues work the whole weekend like mad scientists. Wu could not be the first! Did she refuse to go out with Ledermann maybe? Who knows. The result was that he got arrived later than madame Wu's.&lt;/p&gt;&#xA;&lt;p&gt;So there were four results in a few days. To whom assign the Nobel Prize. To nobody, and that included Wu.There you go...&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-22T00:35:45.193" LastActivityDate="2021-07-22T00:35:45.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13438" PostTypeId="1" AcceptedAnswerId="13444" CreationDate="2021-07-21T10:49:55.620" Score="2" ViewCount="642" Body="&lt;p&gt;At university I learned quantum mechanics in &lt;em&gt;or&lt;/em&gt; position &lt;em&gt;or&lt;/em&gt; momentum variables. Later (in fact on this site), I learned about the phase space formulation in which both are used at the same time.&lt;/p&gt;&#xA;&lt;p&gt;The Dutchman Groenewold (hiding from the Nazis in the Dutch polder landscape beneath the sea), Moyal, Wigner, and more contributed. But matrix mechanics and path integrals (both formulated in one of the two variables) got the upper hand.&lt;/p&gt;&#xA;&lt;p&gt;Now why was that? I had instead learned the combined view (I am doing that right now though; I never went to university anyhow, except the last year in which you could choose nice subjects yourself (I did a course on quantum fields)), but IMO the phase space approach stands much closer to everyday experience (though I'm not sure if a path integral can be defined in both position and momentum at once).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Phase-space_formulation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; are some nice examples of the Wigner quasi-probability functions to be found. They are much closer to macroscopic reality and thus give an easier view, I guess.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4931" LastEditDate="2021-07-21T23:39:44.360" LastActivityDate="2021-07-24T15:23:09.970" Title="Why didn't the phase space formulation of Quantum Mechanics get the upper hand?" Tags="|physics|quantum-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13440" PostTypeId="2" ParentId="8265" CreationDate="2021-07-21T13:54:18.037" Score="8" Body="&lt;p&gt;He may simply not have cared. A reasonably small distance along the east-west axis will only lead to a very small error in the distance along the north-south axis.&lt;/p&gt;&#xA;&lt;p&gt;E.g. an azimuth of 190° instead of 180° when looking from Alexandria to Syene will only lead to a &lt;a href=&quot;https://www.calculator.net/triangle-calculator.html?vc=10&amp;amp;vx=100&amp;amp;vy=&amp;amp;va=90&amp;amp;vz=&amp;amp;vb=&amp;amp;angleunits=d&amp;amp;x=78&amp;amp;y=16&quot; rel=&quot;noreferrer&quot;&gt;1.5% difference&lt;/a&gt; between straight distance and the north-south distance. This is probably considerably less than the error in the distance measurement.&lt;/p&gt;&#xA;" OwnerUserId="15011" LastActivityDate="2021-07-21T13:54:18.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13441" PostTypeId="2" ParentId="13409" CreationDate="2021-07-21T14:40:04.747" Score="3" Body="&lt;p&gt;The eponymous &lt;em&gt;&lt;strong&gt;Serendipity&lt;/strong&gt;&lt;/em&gt; &lt;em&gt;&lt;strong&gt;elements&lt;/strong&gt;&lt;/em&gt; used in the finite element method for solving PDEs fit the bill.&lt;/p&gt;&#xA;&lt;p&gt;These elements ignore the internal nodes (unlike their Lagrange counterparts), reducing the number of degrees of freedom of the system while (and this is the key) retaining the &lt;em&gt;same&lt;/em&gt; element order.&lt;/p&gt;&#xA;&lt;p&gt;Serendipity elements are therefore less computationally expensive than Lagrange elements but just as accurate and herein lies the &amp;quot;happy chance&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="15012" LastActivityDate="2021-07-21T14:40:04.747" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13442" PostTypeId="2" ParentId="11598" CreationDate="2021-07-21T16:17:20.733" Score="1" Body="&lt;p&gt;You might have heard of something called geometry.&lt;/p&gt;&#xA;&lt;p&gt;Geometry is from the Greek for &amp;quot;earth measurement&amp;quot;, and ancient people use it for measuring land, or surveying.&lt;/p&gt;&#xA;&lt;p&gt;The ancient Egyptians needed geometry to re survey property boundries after each annual Nile flood.  So the ancient Egyptians had been using geometry for at least 3,000 years before the Ptolomaic dynasty gained the throne.  And the Ptolomies ran Egypt like a money making business.  So it seems pretty obvious that Egypt would have been surveyed rather well during the Ptolomaic dynasty.&lt;/p&gt;&#xA;&lt;p&gt;Of course the Nile river twisted and turned like most rivers do.  If property lines were surveyed at right angles to the river's direction at a spot, instead of all going straight east to west, those property lines would not be all equally spaced from north to south and counting them would not give an accurate north-south distance.&lt;/p&gt;&#xA;&lt;p&gt;Eqypt would have been mapped for military purposes also.  The Ptolomies might want to invade the countries to the south as earlier Egyptian dynasties had, and the Nubian kingdoms might want to invade Egypt - there had been a Nubian conquest that put a Nubian dynasty on the Egyptian throne.  And the native Egyptians sometimes revolted.&lt;/p&gt;&#xA;&lt;p&gt;And obviously there was a military benefit to knowing all the sortcuts through the desert that bypassed the meandering loops of the Nile.  So any caravan routes through the desert which were shortcuts between loops would have been surveyed to find all the waterholes and estimate how large an army could use those waterholes.&lt;/p&gt;&#xA;&lt;p&gt;So it is quite likey that Eqypt was surveyed down to Syene, modern Aswan, a frontier town with a Ptolemaic garrison.&lt;/p&gt;&#xA;&lt;p&gt;And no doubt the government files had distances and routes between various places measured in case of need.&lt;/p&gt;&#xA;&lt;p&gt;Aswan/Cyene is at 24 degrees, 05 minutes, 20 seconds north, and Alexandria is at 31 degrees, 12 minutes north, or a little more than 7 degrees of latitude farther north.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aswan%5B1%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Aswan[1]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alexandria%5B2%5Do&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Alexandria[2]o&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;An ancient Greek &lt;em&gt;stadion&lt;/em&gt; was 600 greek feet, but different foot lengths were used in different Greek regions.  A &lt;em&gt;stadion&lt;/em&gt; was probably about 157.7 meters or about 172.5 yards.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stadion_(unit)#Calculations%5B3%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Stadion_(unit)#Calculations[3]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The length of a degree of latitude is about 67 miles or 111 kilometers, which is 111,000 meters.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.johndcook.com/how_big_is_a_degree.html%5B4%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.johndcook.com/how_big_is_a_degree.html[4]&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So a latitude difference of about 7 degrees is about 7 X 111,000 meters, or about 777,000 meters.  And that equals about 4,927.07 stadia.&lt;/p&gt;&#xA;&lt;p&gt;The actual latitude difference between Alexandria and Syene is about 7 degrees and 7 minutes.  A minute of latitude is one sixtieth of a degree, or about 1,850 meters.  7 minutes of latitude are about 12,950 meters, 777,000 meters plus 12,950 meters is 789,950 meters.&lt;/p&gt;&#xA;&lt;p&gt;So the actual latitude difference between Alexandria and Syene is closer to about 789,950 meters, or about 5,009.1946 stadia, than to 4,927.07 stadia.&lt;/p&gt;&#xA;&lt;p&gt;So maybe Eratosthenes made a lucky guesstimate, as Frigo suggests, that was rather close to the accurate distance by chance.&lt;/p&gt;&#xA;&lt;p&gt;Or maybe Egypt was survevyed well enough that the latitude difference between Alexandria and Aswan/Cyene was known pretty accurately to people in the government survey office.  Educated people who didn't work in such an office wouldn't remember the exact figure without looking it up, but would remember that it was close to a round 5,000 stadia, especially allowing for various inevitable errors in the surveys which would make more precision meaningless.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2021-07-21T16:24:02.410" LastActivityDate="2021-07-21T16:24:02.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13443" PostTypeId="1" CreationDate="2021-07-21T17:44:45.430" Score="0" ViewCount="392" Body="&lt;p&gt;Does anyone know about Prof. Abdul Aziz on whose name Aziz's theorem is named?&lt;/p&gt;&#xA;&lt;p&gt;Aziz's theorem is a theorem about the location of zeros of polynomials.&lt;/p&gt;&#xA;" OwnerUserId="15014" LastActivityDate="2024-03-27T23:35:22.023" Title="Aziz of Complex Analysis" Tags="|mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13444" PostTypeId="2" ParentId="13438" CreationDate="2021-07-21T18:46:13.680" Score="10" Body="&lt;p&gt;This is what they call a &amp;quot;good question&amp;quot;, as the expression goes. I and my collaborators had it in the mid 1990s, and about a 3rd of the audience in my colloquia ask it to themselves and me, rarely getting satisfactory answers.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The short answer must be that there are no &amp;quot;killer apps&amp;quot; of it, namely problems which this formulation solves dramatically better than the other two formulations.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The &amp;quot;gold medal&amp;quot; one, operators in  Hilbert space, is what is normally used in atomic, molecular, solid state, and subnuclear physics, by dint of practicality.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;silver medal&amp;quot; one, functional integrals, languished likewise for two dozen years, until it was used to quantize gauge theories (ghosts), organize their Green's functions, and ultimately have them effectively solved by lattice computer simulations.&lt;/p&gt;&#xA;&lt;p&gt;But the &amp;quot;bronze medal&amp;quot; one you are asking about has merely helped several researchers, including ourselves, to think more clearly about problems they solved; it cannot be argued, in retrospect, that the solution would have been impractical in Hilbert space or path integrals, only harder. So this formulation, so far, is mostly good for providing intuition on subtle problems, including visualization of the classical limit and killing rampant misconceptions on quantization schemes. Its tasteful math is also useful in Lie Algebras (* product, Moyal Brackets, large N limit), but the mainstream regards such things as mathematical physics diversions.&lt;/p&gt;&#xA;&lt;p&gt;I know some of the history of the subject; here is &lt;a href=&quot;https://www.hep.anl.gov/czachos/APPNqmps12.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;our review&lt;/a&gt; of it. We conjecture there that, in the late 40s, the formulation was eclipsed by path integrals, as well as the formulation of QED by Feynman (who, like Dirac, was intensely interested in it), and which &amp;quot;stole the show&amp;quot;. By looking at the literature over the years, I've confirmed the impression of H Corben (quoted in Moyal's &lt;a href=&quot;https://press.anu.edu.au/publications/maverick-mathematician&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography&lt;/a&gt; by his widow, Ann Moyal, now also deceased) that its misapplication in the quantization problem by wrong-minded adherents of Weyl's paradigmatic quantization dream damaged the formulation significantly. Frustratingly, it took the smoke to clear about 50 years, if that.&lt;/p&gt;&#xA;&lt;p&gt;A corollary to this is the ignorance of Groenewold's * product and its significance, and Groenewold's epochal thesis/paper at large. Even today, see below, it looks subdominant in citations to Wigner's and Moyal's, even though it is conceptually co-equal, if not over-arching. And to think that Hip worked so hard to have it published in the Physical Review, and failed.&lt;/p&gt;&#xA;&lt;p&gt;However, let me flesh out your picture by including three citation (&amp;quot;knowledge&amp;quot;) graphs versus time, for the three seminal papers of the subject. They all exhibit the same peculiar pattern. I'll use the plots of INSPIRE, which focusses on HEP, but has ready plots, which I (perhaps unjustifiably) don't know how to access in Google Scholar, far more complete. The sampling by INSPIRE, however, follows the time development of the larger GS set.&lt;/p&gt;&#xA;&lt;p&gt;Wigner, E., 1932: On the Quantum Correction For Thermodynamic Equilibrium,  &lt;em&gt;Phy Rev&lt;/em&gt;   &lt;strong&gt;40&lt;/strong&gt;(5), p.749; Wigner's 1932 paper has 11367 cites in GS, and 1414 in ISPIRE, evolving as &lt;a href=&quot;https://i.stack.imgur.com/XKqij.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/XKqij.png&quot; alt=&quot;Wigner&quot; /&gt;&lt;/a&gt;&#xA;Note the pitiful performance for half a century.&lt;/p&gt;&#xA;&lt;p&gt;Groenewold, H.J., 1946: On the principles of elementary quantum mechanics, &lt;em&gt;Physica&lt;/em&gt; &lt;strong&gt;12&lt;/strong&gt;(7), pp.405-460; Groenewold's 1946 thesis has  1528 in GS, and 524 in INSPIRE, and evolves as &lt;a href=&quot;https://i.stack.imgur.com/sHzBJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sHzBJ.png&quot; alt=&quot; Groenewold&quot; /&gt;&lt;/a&gt; , again bizarrely neglected for half a century.&lt;/p&gt;&#xA;&lt;p&gt;Moyal, J.E., 1949: Quantum mechanics as a statistical theory,  &lt;em&gt;Mathematical Proceedings of the Cambridge Philosophical Society&lt;/em&gt;  &lt;strong&gt;45&lt;/strong&gt; (1)  pp. 99-124;  Moyal's 1949 (well, really 1941 or so...) paper has 3868 in GS, and 1078 in INSPIRE, &lt;a href=&quot;https://i.stack.imgur.com/zpx4C.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zpx4C.png&quot; alt=&quot; Moyal&quot; /&gt;&lt;/a&gt;.  The scale, for whatever reason, only goes back to 1953, but note in its midpoint year, 1987, it only has 13 citations, and the next year, 1988 only 4, the dip you can identify in the middle. Again, it is languishing for half a century.&lt;/p&gt;&#xA;&lt;p&gt;The awareness &amp;amp; appeal of that field only started crawling up in the  1990s, and then got going. I suspect a professional historian of science should have insights on that shift, but my gut feeling is that it's just generational: an entire generation of physicists with the wrong (paradigm) focus (&amp;quot;what is the correct quantization path from classical mechanics?&amp;quot;) had to die, before the modern viewpoint (&amp;quot;what more can we learn about QM from this equivalent formulation?&amp;quot;) could thrive. Just a thought.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-07-24T15:23:09.970" LastActivityDate="2021-07-24T15:23:09.970" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13445" PostTypeId="2" ParentId="13443" CreationDate="2021-07-21T19:58:30.177" Score="6" Body="&lt;p&gt;As per the obituaries, the person behind Aziz's theorem is Prof. Aziz ul Azeem. He was a professor of mathematics at the University of Kashmir (located in South Asia). This is not another well known mathematics professor from Howard University, Abdul-Aziz Yakubu, (the name is quite similar). From the obituary of Prof. Azeem &lt;a href=&quot;https://www.greaterkashmir.com/todays-paper/veteran-mathematician-prof-aziz-ul-azeem-demise-condoled&quot; rel=&quot;nofollow noreferrer&quot;&gt;News link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Prof Aziz-ul-Azeem was a distinguished mathematician who contributed significantly to the growth and development of the university. His contributions will continue to serve as a guiding force for young scholars to draw inspiration from,” Prof Talat said. Prof Aziz-ul-Azeem was globally known for ‘Aziz Theorem’ while leading several path-breaking research projects in mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2021-07-22T02:50:33.847" LastActivityDate="2021-07-22T02:50:33.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13446" PostTypeId="2" ParentId="13409" CreationDate="2021-07-21T20:21:56.620" Score="3" Body="&lt;p&gt;Alfred Wilm discovered the age hardening effect of aluminium, after delaying his testing of some freshly annealed samples (I believe to take the weekend off to go sailing). This is now used in the design of aluminum alloys with applications from aircraft to bicycles to keychain bottle openers, with his alloy used in &amp;quot;Duralinium&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.researchgate.net/publication/279898292_Aluminium_Alloys_-_A_Century_of_Age_Hardening&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/publication/279898292_Aluminium_Alloys_-_A_Century_of_Age_Hardening&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15015" LastActivityDate="2021-07-21T20:21:56.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13447" PostTypeId="1" CreationDate="2021-07-22T07:01:10.227" Score="1" ViewCount="103" Body="&lt;p&gt;Let me state firstly that I don't think that the nervous system is a computer.&#xA;Not down up, nor top down.  On top of that, brains cannot truly be separated from the body.&lt;/p&gt;&#xA;&lt;p&gt;I am curious still.  Who were the first people to think that the brain is a  computer?&#xA;Today this view can be found in mind philosophers like Hofstadter.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4251" LastEditDate="2021-10-02T14:00:25.293" LastActivityDate="2021-10-02T14:00:25.293" Title="When were brains seen as computers for the first time?" Tags="|neuroscience|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13448" PostTypeId="1" CreationDate="2021-07-22T14:40:30.330" Score="1" ViewCount="101" Body="&lt;p&gt;I was reading this &lt;a href=&quot;https://en.wikipedia.org/wiki/Egyptian_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page&lt;/a&gt; searching for the Egyptian area of circle and there is a following picture there:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bsrG2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bsrG2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Trying to understand what is meant by this since it is under the &amp;quot;Area&amp;quot; section. I've read some other pieces regarding area of circle where Egyptians usually inscribed the circle inside of a square instead of drawing it like this. There is not much description on the page regarding this picture so just trying to make sense of it. Anyone have any idea what the picture is trying to explain?&lt;/p&gt;&#xA;" OwnerUserId="15021" LastActivityDate="2021-07-22T14:40:30.330" Title="Help understanding Egyptian circle" Tags="|mathematics|geometry|ancient-egypt|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13449" PostTypeId="1" CreationDate="2021-07-22T20:38:01.683" Score="0" ViewCount="140" Body="&lt;p&gt;Suppose that, for whatever reason, neither Copernicus nor anyone else had discovered the heliocentric model, and cosmology remained stuck in the geocentric model. (&amp;quot;For whatever reason&amp;quot; could mean that the Church violently and successfully suppressed this line of research; funding for astronomy dried up; scientists just lost interest &amp;amp; worked on other fields; etc.).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;However&lt;/strong&gt;, suppose that all other fields of science, technology, engineering, and invention proceeded apace.&lt;/p&gt;&#xA;&lt;p&gt;Which technologies/inventions of our modern world could exist even if people still thought the Sun revolved around the Earth, and which couldn't?&lt;/p&gt;&#xA;&lt;p&gt;I'm assuming, for example, that space probes sent to other planets wouldn't exist; you couldn't properly calculate the trajectory based on a fundamental misunderstanding of cosmology. But could you have artificial satellites in Earth orbit? Travel to the Moon?&lt;/p&gt;&#xA;&lt;p&gt;And what about non-obviously-space-related inventions? Do you need heliocentric cosmology to invent/design the steam engine? The cotton gin? The automobile? The airplane? Antibiotics? Skyscrapers? The nuclear bomb? The IBM PC? The Internet? The iPhone? Or could most/all of these exist even if everyone mistakenly believed the Earth was the center of the universe?&lt;/p&gt;&#xA;&lt;p&gt;What I'm really asking about, I think, is about path-dependence. My instinct is that most if not all modern terrestrial technologies could have been invented with the wrong cosmology, but maybe I'm wrong...?&lt;/p&gt;&#xA;" OwnerUserId="15024" LastEditorUserId="12724" LastEditDate="2021-07-23T06:53:23.967" LastActivityDate="2021-07-23T16:04:54.627" Title="Which modern sciences/technologies were contingent on the Copernican Revolution, &amp; which could have developed even while believing in geocentrism?" Tags="|engineering|technology|cosmology|" AnswerCount="2" CommentCount="7" ClosedDate="2021-07-23T15:17:59.777" ContentLicense="CC BY-SA 4.0" />
  <row Id="13450" PostTypeId="1" AcceptedAnswerId="13451" CreationDate="2021-07-23T00:24:35.570" Score="3" ViewCount="194" Body="&lt;p&gt;Neither &lt;a href=&quot;https://en.wikipedia.org/wiki/Equivalence_class&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt; or the first few pages of Google are showing me much about the history of the development of &lt;a href=&quot;https://mathworld.wolfram.com/EquivalenceClass.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;equivalence classes&lt;/a&gt;. When was this notion first formalized?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Footnote: I originally asked this on &lt;a href=&quot;https://math.stackexchange.com/questions/4205152/when-were-equivalence-classes-formalized&quot;&gt;math.SE&lt;/a&gt; but it was recommended that I ask here on hsm.SE.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2021-07-26T18:03:03.847" Title="When were equivalence classes formalized?" Tags="|mathematics|definition|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13451" PostTypeId="2" ParentId="13450" CreationDate="2021-07-23T01:01:58.113" Score="4" Body="&lt;p&gt;The answer depends on how much of &amp;quot;formalization&amp;quot; counts, and of what exactly. &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/s11229-018-1674-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;Asghari's Equivalence: an attempt at a history of the idea&lt;/a&gt; gives a detailed account of the genesis. Implicit use goes back to Euclid and before him, Dedekind cuts are more explicit. But outside of specific contexts we need to go all the way to the beginning of 20th century. Equivalence relations under another name are defined in Jourdain's paper (1912), &amp;quot;equivalence relations&amp;quot; with defined classes that are not named &amp;quot;equivalence classes&amp;quot; - in Hasse's &lt;em&gt;Höhere Algebra&lt;/em&gt; (1926), and both together - in the third edition of Birkhoff's Lattice Theory (1948):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I call a relation which is reflexive, symmetrical, and transitive an isoid relation. Jourdain (1912) presented his article On isoid  relations  and  theories  of  irrational number to the Fifth International Congress of Mathematics, starting with the sentence above. At the time, the process of “definition by abstraction” (Russell 1903, pp. 219-220) was quite well established but the term “equivalence” was mainly attached to the context of cardinal numbers.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Jourdain was one of the first who suggested a decontextualized term for what we now know as “equivalence relation”. However, he was not successful at popularizing the term. Even Russell, one of his closest correspondences, refers to such relations with no name and only qualify them as an “important kind of relation”, noting that, “similarity is one of this kind of relations” (Russell 1919, p. 16). By 1919, neither the combination “equivalence relation” nor “equivalence class”was in use. The necessity of naming was first felt with the relation, not with the classes that are formed by that relation... Hasse (1926) wrote:&amp;quot;We call such a decomposition a partition of M,and the subsets thereby determined its classes.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Interestingly it seems that “equivalence class” was&#xA;enjoying a path of its own for a long time. Von Neumann (1926,1929), Hopf (1930), and Seifert and Threlfall (1934) used the term “Äquivalenzklasse”, Von Neumann (1936) “equivalence-class” (with a hyphen) and Solomon Lefschetz (1938,1942) “equivalence class” (without a hyphen). The terminology was for denoting the classes constructed by certain contextualized equivalences, none of which was referred to as an “equivalence relation”...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The third edition&lt;/em&gt; [of Birkhoff's Lattice Theory] &lt;em&gt;is a typical example of what we observe in the texts written by mathematicians for mathematicians after 1950. Before 1950, we might find all different sorts of the use of terminology: no mention at all (e.g. Russell 1903), alternative terms (e.g. Jourdain 1912; Carnap 1928), only “equivalence relation” with no mention of the term “equivalence class” (e.g. Weyl 1949) or an alternative (e.g. Birkhoff 1935). Generally, the combination “equivalence class” became common much later than the “equivalence relation” while closer to 1950 we can observe clear uses of both combinations alongside each other.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-07-23T02:49:21.137" LastActivityDate="2021-07-23T02:49:21.137" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13452" PostTypeId="2" ParentId="13449" CreationDate="2021-07-23T07:09:38.970" Score="-2" Body="&lt;p&gt;A lot would have changed. General relativity would have been discovered sooner. But this time with a velocity of light that was direction dependent. The universe would be rotating around the Earth. Relativity says that it &lt;em&gt;could&lt;/em&gt; be viewed the other way round as Copernicus did. Newton's laws might not even have been discovered firstly. There would be no need for them. Relativity was already there! The heliocentric view of Copernicus &lt;em&gt;could&lt;/em&gt; be considered. But the prevailing view would be that the Earth is in the center of it all. In the center of the &amp;quot;heelal&amp;quot;, Dutch for &amp;quot;universe&amp;quot; (&amp;quot;whole all&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;All other forms of science, apart from the heavenly, would have developed in the same way and probabky much sooner. But by other people. So thanks to the church science would have evolved more as it has now.&lt;/p&gt;&#xA;&lt;p&gt;So maybe we would have lived in a world with much more advanced science and technology, a healthy planet, a less reductionist, and less materialistic view (which is not to say that material is less valid, only less contentive). Maybe the vision that the mind is an executed program working on data didn't took hold as it did in the present world (dataism). Maybe the brain was viewed as it is. An ongoing process in an ongoing body in an ongoing world that cannot be turned on and off like a computer. It could have been established that quarks and leptons are not fundamental and the Higgs particle is just a particle made up of the more (most) elementary stuff and that the Higgs mechanism is just an imaginary one.&lt;/p&gt;&#xA;&lt;p&gt;Now I could write a lot more about how it could have been (for example that the gene-centered view on evolution could have been replaced by a view which puts the whole organism in the first place in which genes have a pure altruistic and secondary function or that maybe a deterministic view on quantum mechanics was the accepted view) but it will always be my speculation and airing of my personal view. Who knows for sure how the world would have looked like? Nobody.&lt;/p&gt;&#xA;&lt;p&gt;So to answer your question, all of science could have seen the light. All kinds of technology too. One can even argue that the heliocentric view has been a holdback. Sticking to the geocentric view would have taken us many steps ahead from where the sciences are today.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-23T16:04:54.627" LastActivityDate="2021-07-23T16:04:54.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13453" PostTypeId="1" CreationDate="2021-07-23T08:54:45.383" Score="0" ViewCount="129" Body="&lt;p&gt;Scientists use the story of Galilei to advocate the glory of the sciences and the stupidity of the church. This is obvious from ironic remarks made by them that criticize its attitude. In fact, this is exactly the same attitude to the church as the church had towards the science of the heavens (that is, the heliocentric view). The difference being though that the church was rational in it's attitude. So they were not &lt;em&gt;exactly&lt;/em&gt; the same.&lt;/p&gt;&#xA;&lt;p&gt;Gallilei is idolized and given the status of a hero. The hero of science who dared to fight against the irrational views of the church in the name of Reason and Enlightment.&lt;/p&gt;&#xA;&lt;p&gt;But what is true of all the stories? Was he thrown in jail? Ot maybe something worse? Did he believe in God himself?&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, Galileo Galilei is called Galilei in Holland. I don't know about other countries but it could be they call him Galileo there.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-23T21:04:53.040" LastActivityDate="2021-07-23T21:04:53.040" Title="What's the true story about Galilei?" Tags="|galileo|" AnswerCount="1" CommentCount="6" ClosedDate="2021-08-14T04:09:19.693" ContentLicense="CC BY-SA 4.0" />
  <row Id="13455" PostTypeId="2" ParentId="5645" CreationDate="2021-07-23T12:30:57.327" Score="1" Body="&lt;p&gt;Norbert Wiener wrote the 1959 novel &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B000OK0CNQ&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Tempter&lt;/em&gt;&lt;/a&gt;, which I have a vague memory of having unsuccessfully tried to read in one point in my life.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastActivityDate="2021-07-23T12:30:57.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13456" PostTypeId="2" ParentId="13453" CreationDate="2021-07-23T13:52:00.063" Score="6" Body="&lt;p&gt;As Mauro Allegranza mentioned, the literature is enormous. To put it short, Galileo was censored. He was given a warning not to advocate Copernican theory as an established fact but was permitted to discuss it as a mathematical hypothesis. (The main reason was that the Bible contains a sentence which seems to imply that Sun rotates around the Earth. Galileo dared to propose his own interpretation of this sentence. But interpretation of the Bible was strictly the prerogative of the Church.)&lt;/p&gt;&#xA;&lt;p&gt;He violated this warning, and published a book for a broad audience, where some passages could be interpreted as a ridicule of the arguments stated by the Pope himself.&lt;/p&gt;&#xA;&lt;p&gt;Then he was summoned to Rome, and forced to abdicate his views (under the threat of torture, and worse). He publicly abdicated, and after that was kept under some kind of home arrest.&#xA;He lived with a friend, and was prohibited to travel or to publish anything on the topic. Copernicus book was banned. Notice that it was banned 74 years after publication, and only in connection with Galileo's process.&lt;/p&gt;&#xA;&lt;p&gt;One can add, that a) Galileo's main argument &amp;quot;proving&amp;quot; revolution of the Earth was wrong (it was based on his incorrect explanation of tides), and that at that time there was no conclusive experiment which should allow to prove or disprove it. b) He was in priority conflict with some Jesuit scientists who could possibly cause his persecution, and c) There were many scientists&#xA;at that time who shared the heliocentric views, and had good relations with the church establishment. So some researchers argue that he simply provoked the Church to ban Copernicus theory. On this I can cite two sources which are critical with respect to Galileo:&lt;/p&gt;&#xA;&lt;p&gt;A. Koestler, The sleepwalkers, and&lt;/p&gt;&#xA;&lt;p&gt;S. Gindikin, Tales of mathematicians and physicists.&lt;/p&gt;&#xA;&lt;p&gt;Their point of view is that Galileo unnecessarily provoked the Church.&#xA;They unfavorably compare him with his contemporary Kepler who never had any conflict with Church. (It is true that Kepler was a Protestant, but Protestants also have a record of burning heretics).&lt;/p&gt;&#xA;&lt;p&gt;And to answer the last question: yes Galileo believed in God, and was a devote Catholic.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-23T14:26:10.090" LastActivityDate="2021-07-23T14:26:10.090" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="13457" PostTypeId="2" ParentId="13449" CreationDate="2021-07-23T14:35:16.180" Score="2" Body="&lt;p&gt;Copernicus theory was crucial for the development of all sciences. It made possible Kepler's discovery of the true planet orbits, and this lead to the discovery of the Law of the Universal gravitation. The Law of universal gravitation was a real scientific revolution. Among its immediate practical consequences were establishing of the true shape of the Earth, explanation of tides, explanation of the motion of the Moon, and later the discovery of new planets.&lt;/p&gt;&#xA;&lt;p&gt;More generally, and more importantly, it established a &amp;quot;paradigm&amp;quot; for the whole future development of&#xA;science until the middle of the 19 century. One can say that the work from Galileo and Kepler to Newton showed convincingly for the first time the power of application of mathematics to the real world. This is the basis of all modern exact science. And the work of Galileo, Kepler and Newton would be impossible without Copernicus discovery.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-23T15:43:04.110" LastActivityDate="2021-07-23T15:43:04.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13458" PostTypeId="2" ParentId="5166" CreationDate="2021-07-23T16:11:31.847" Score="5" Body="&lt;p&gt;This is really a comment rather than a full answer.&lt;/p&gt;&#xA;&lt;p&gt;Florian Cajori's encyclopedic tome &lt;i&gt;A History of Mathematical Notations&lt;/i&gt; does not seem to discuss this symbol specifically; the closest thing is the following remark in Section 269:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;L. Gustave du Pasquier (&lt;i&gt;Comptes Rendus du Congrès International des Mathématicians&lt;/i&gt; (Strasbourg, 22–30 Septembre 1920), p. 164) in discussing general complex numbers employs the sign of&#xA;double equality &lt;span class=&quot;math-container&quot;&gt;$\overset{\displaystyle =}{=}$&lt;/span&gt; to signify “equal by definition.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I checked MathSciNet, and the review of the 1953 paper &amp;quot;Zwei Klassen von Flächen, deren Bestimmung von einem Integral der Telegraphengleichung abhängt&amp;quot; by Hans Jonas (MR0058259) uses the notation &lt;span class=&quot;math-container&quot;&gt;$=_{\text{def}}$&lt;/span&gt; (but note that Jonas's paper itself does not use that notation). This seems to be the earliest occurrence of &lt;span class=&quot;math-container&quot;&gt;$=_{\text{def}}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\overset{\text{def}}=$&lt;/span&gt; in MathSciNet.  But I do not have access to a print version to verify that the notation was actually used back in the 1953 edition of Mathematical Reviews.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2021-07-23T16:11:31.847" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13459" PostTypeId="1" CreationDate="2021-07-23T16:27:57.337" Score="0" ViewCount="254" Body="&lt;p&gt;I read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Suddenly – boink! -an apple hits him on the head. “Aha!” he shouts, or perhaps, “Eureka!” In a flash he understands that the very same force that brought the apple crashing toward the ground also keeps the moon falling toward the Earth and the Earth falling toward the sun: gravity&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now this story is not litterally true. It is written though that Newton sat in the garden where he spent his his childhood and saw an apple falling. He pondered why apples don't fall sideways but straight down instead.&lt;/p&gt;&#xA;&lt;p&gt;How did this lead him to his laws? He imagined the apple being the moon that fell (or the moon being the apple). And because of the Earth's curvature it would fall forever, so orbiting. But what is the connection with his universal law of gravitation? Didn't he think before that the same force that makes the apple fall can be applied to the moon, and in fact &lt;em&gt;is&lt;/em&gt; the force responsible for the moon's motion. How were the orbits of the planets accounted for before his law? By the planets moving through invisible tubes?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-23T19:35:48.190" LastActivityDate="2021-10-21T23:44:27.513" Title="About Newton's apple" Tags="|physics|newton|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13460" PostTypeId="1" CreationDate="2021-07-23T19:12:49.430" Score="5" ViewCount="606" Body="&lt;p&gt;Many famous scientists are connected by their doctorate advisor-student relations. I wonder if we know which is the largest continuous chain of famous researchers. I have checked pages like academictree.org but tracing back the genealogy can rapidly get you into dead ends. Also sometimes senior academic collaborators are counted as advisor but I wanted to find a genealogy of purely doctorate advisors or equivalent degrees that is confirmed.&lt;/p&gt;&#xA;&lt;p&gt;Update: &lt;a href=&quot;https://genealogy.math.ndsu.nodak.edu/extrema.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://genealogy.math.ndsu.nodak.edu/extrema.php&lt;/a&gt; has some indications of the extrema of their tree but seems mainly focused on mathematics.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-19T17:03:51.753" LastActivityDate="2024-02-18T17:05:50.650" Title="What is the largest academic genealogy tree that we know of?" Tags="|biographical-details|academia|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="13461" PostTypeId="2" ParentId="6157" CreationDate="2021-07-23T20:13:15.697" Score="0" Body="&lt;p&gt;Tegmark is a present-day example. You can't find them more Platonic (with a big difference though so maybe he's exactly the opposite). Let's not hope his lovelife is like his view on reality.&lt;/p&gt;&#xA;&lt;p&gt;He states that the universe &lt;em&gt;is&lt;/em&gt; math. All that there is to discover are mathematical formulae. Contrary to Plato though he thinks we stare the mathematical formulae and structures right in the face. They don't merely throw shadows in the material world we live in, like Plato thought. The material world &lt;em&gt;is&lt;/em&gt; the world of math and can be known.&lt;/p&gt;&#xA;&lt;p&gt;Max Tegmark even thinks that there are multitudes of different universes each showing different mathematical systems which can't be compared or translated among each other. A person living in our universe, being a math structure himself, will never be able to understand the other kinds of math.&lt;/p&gt;&#xA;&lt;p&gt;This view is an extreme one and obviously overlooks many aspects of reality. But if he is happy with it... It is a legitimate view.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/David_Deutsch&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Deutsch&lt;/a&gt; is a similarily minded physicist. Also he thinks the are four levels of parallel universes whose Nature is math and that in different universes different mathematics can be encountered. See &lt;a href=&quot;https://www.scientificamerican.com/article/multiverse-the-case-for-parallel-universe/#:%7E:text=When%20talking%20about%20parallel%20universes,of%20space%20where%20even%20the&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-24T09:01:01.657" LastActivityDate="2021-07-24T09:01:01.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13462" PostTypeId="1" CreationDate="2021-07-23T20:23:54.880" Score="1" ViewCount="58" Body="&lt;p&gt;Einstein was the first who saw spacetime as a mallable substance to be malled with mass. As far as I know there were no people who suggested such a view of spacetime before him. But I could be wrong.&lt;/p&gt;&#xA;&lt;p&gt;I could find no info on the net. Did maybe the founders of the math that lays at the base of general relativity adhered to such a view of spacetime?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-07-23T20:23:54.880" Title="Where there scientists before Einstein who viewed space(time) as being elastic?" Tags="|physics|einstein|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13463" PostTypeId="1" CreationDate="2021-07-23T21:44:28.110" Score="1" ViewCount="91" Body="&lt;p&gt;Mendel's laws (like the laws of segregation or inheritance) became known to a wider publis decades after he discovered these laws. He assumed trait particles in operation that were responible for his observed laws.&lt;/p&gt;&#xA;&lt;p&gt;Darwin is known for his rheory of evolution. DNA was not known to him. He considered organisms as a whole that evolved. Later his views were reduced to the level of genes (Dawkins). The genes are seen as the (selfish) units around which evolution takes place. All evolution is directed towards the prolongation of the genes and organisms are seen as a means to an end. We have sex to pass through our genes. This is a far cry from the organism centered view in which the genes perform a secondary (altruistic) role. We have sex to make a new organism live. Obviously this also results in the passing through of genes but the meaning of (objective) evolution is drastically changed. When questioning this paradigm of the selfish gene the reaction many molecular biologists is an irrational one. Their view can't be questioned. I asked three questions about this on the biology site and they were all downvoted massively and I couldn't comment anymore. My last question didn't even make it and I was suspended within an hour. Which only goes to show how rigid and conservative and irrational they are.&lt;/p&gt;&#xA;&lt;p&gt;But that aside. What were the basis entities of Mendel and Darwin in the context of evolution. They certainly knew each other. But they never met. Was Mendel even in favour of evolution?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-24T07:24:02.757" LastActivityDate="2021-07-24T07:24:02.757" Title="What were Gregor Mendel's and Darwin's views on evolution?" Tags="|biology|evolution|darwin|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13464" PostTypeId="2" ParentId="13271" CreationDate="2021-07-24T08:16:39.347" Score="0" Body="&lt;p&gt;I am not that good in remembering names. But I can remember looking in a book of art and being surprised to see a few paintings of a female painter that showed scientific diagrams and other means of scientific representations. It gave a strange feeling to see them taken out of their scientific context. They made it clear (at last to me) that science is just one culture amongst many. I think it is a pity that I don't remember her name. I can't find her on the internet.&lt;/p&gt;&#xA;&lt;p&gt;There is a painting &amp;quot;De Lijkschouwer&amp;quot;. It shows the disection of a dead body by physicians to know more about the inner workings of people.&lt;/p&gt;&#xA;&lt;p&gt;Walking the streets of Amsterdam one day (near the Spiegelgracht, where a lot of antiquity shops and art galleries are situated) I saw a painting that attracted me. It showed, in a magical, almost surreallistically, and dramatically lighted scene, a huge telescope directed towards the heavens (which reminds right now of a book by a Dutch author: &amp;quot;The Discovery of Heaven&amp;quot;, in which a physics professor discovers a negatively curved space in which heaven is situated). Around the telescope human activity can be seen. People are looking at observation results and one stares in awe at the sky. I don't remember who has painted it.&lt;/p&gt;&#xA;&lt;p&gt;And of course there is a huge portrait gallery. The images of almost all famous scientists are put on linen. But I don't think you are interested in this.&lt;br /&gt;&#xA;Maybe one artist is worth mentioning. The Dutchman Escher made a lot of prints of geometric figures and transformations thereof. He combined geometry with life in appealing and aestethically pleasing pictures. I am not aware that he made paintings. He made pencil drawings and prints.&lt;/p&gt;&#xA;&lt;p&gt;I have a small poetry bundle with poems glorifying science. It is written by a Dutch physicist and the poems try to glorify mainly by showing the stupidity of the non-scientific views. Not really my cup of tea. The language used is colorfull though and I think he could better have put it in another practice or context. I will look it up to give an example (though translating poetry is always difficult).&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-24T08:25:44.310" LastActivityDate="2021-07-24T08:25:44.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13465" PostTypeId="1" CreationDate="2021-07-24T09:30:27.577" Score="2" ViewCount="131" Body="&lt;p&gt;Max Ernst was a painter belonging to the Dadaistic movement. One of his paintings shows Euclid in a somewhat, well, let's say Dadaistic fashion (although fashion and Dadaism don't go along well). We can read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Max Ernst displayed interest in Science and Mathematics that is certainly traceable in his artistic yield, both on the level of technique and on that of content. It is reported that, after a visit at the Henri Poincare Institute in Paris, he pointed out to Christian Zervos (1889 – 1970) , a Greek art critic and publisher of the Parisian “Cahiers d’ Art” magazine, a series of models on display, representing mathematical surfaces, that struck his imagination. This encounter brought these surfaces into the artistic surrealist foreground of the time and the American artist Man Ray (1890 – 1976), after a suggestion of Zervos, produced a series of photographs of these exhibits, under the title “Mathematical Objects”. Man Ray later described these surfaces as “so unusual, as revolutionary as anything that is being done today in painting or in sculpture”, though he admitted that he understood nothing of their mathematical nature. As curious objects, reminiscent of algebraic surfaces, appear in several of Ernst’s paintings, it is quite possible that he himself made attempts to incorporate them in his works. In “Design in Nature” (1947), a painting heavily influenced by Geometry and carried out using geometric technique, Ernst provides an allusion on the mathematical nature of the world’s design. In the background, a variety of shapes and curves give birth to living forms among alternating areas of light and dark. A mathematical solid, drenched in light, dominates the foreground topped by two horn – like structures, bringing to mind the “horn” or Dupin cyclides,  two types of mathematical surfaces , forming thus an abstract ibex head.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and it goes on further. It seems a bit contrary that a Dadaist adheres to the rigor and programs of math but then again, not to follow any program allows for following a program. I could not find what Ernst's pictured math entails (excepgt that the structures he depicted resemble alebraic curves, as can be read above). How are his paintings seen by Max himself? Does he just admire math or does he want to contribute? Or is he maybe even criticizing math?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-25T06:00:32.643" LastActivityDate="2021-07-25T06:00:32.643" Title="What kind of mathematics had Max Ernst in mind?" Tags="|mathematics|art|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13467" PostTypeId="1" CreationDate="2021-07-25T07:29:29.170" Score="6" ViewCount="959" Body="&lt;p&gt;Quaternions were made up by Hamilton. They are an extension of complex numbers. It is said that he first introduced &amp;quot;3d tertions&amp;quot;. He was thinking what the relation between &lt;span class=&quot;math-container&quot;&gt;$\bf i$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\bf j$&lt;/span&gt; had to be (in &lt;span class=&quot;math-container&quot;&gt;$t = a + b {\bf i} + c {\bf j}$&lt;/span&gt;), walking on a bridge in his hometown. On the bridge it struck him: &lt;span class=&quot;math-container&quot;&gt;${\bf i} \times {\bf j} = {\bf k}$&lt;/span&gt; (and &lt;span class=&quot;math-container&quot;&gt;${\bf j} \times {\bf i} = -{\bf k}$&lt;/span&gt; as holds for all pairs of different bases). So the quaternion was born&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ q = a + b {\bf i} + c {\bf j} + d {\bf k}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It didn't get a grip on physics though. Hamilton invented them for the sake of rotation in classical mechanics. Rotation in two dimensions can be described by complex numbers (which are two dimensional). That's why Hamilton thought to extend the complex number to a tertion. A tertion could be represented by a vector in the complex 3D space. It turned out that he needed one more dimension. To incorporate &lt;em&gt;k&lt;/em&gt;. This was because although the square of &lt;em&gt;i&lt;/em&gt; and &lt;em&gt;j&lt;/em&gt; are both -1, the product &lt;em&gt;ij&lt;/em&gt; could not be accounted for in a tertion. The multiplication has to be closed. Which requires a &lt;em&gt;k&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Whatever happened to these guys? Are they just mathematical curiosities instead of the heroes they were supposed to be by Hamilton? I read that the whole of special relativity could be defined by them. Only to give rise to a positive metric of Minkowski space. Is that all? Whatever happened to them heroes?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4251" LastEditDate="2022-09-08T06:17:41.643" LastActivityDate="2022-09-08T06:17:41.643" Title="Whatever happened to quaternions?" Tags="|mathematics|physics|quaternions|" AnswerCount="5" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13469" PostTypeId="1" AcceptedAnswerId="14157" CreationDate="2021-07-25T10:06:00.210" Score="2" ViewCount="156" Body="&lt;p&gt;I dimly remember watching a popular sciencie movie that mentioned an experiment conducted by a pair of great physisicists (Einstein was one of them, unless my memory is playing tricks on me) that had surprising results, because they were both heavy smokers and their tobacco smoke entered an unforseen reaction with the subject of study.&lt;/p&gt;&#xA;&lt;p&gt;Very sadly, I forgot all other details.&lt;/p&gt;&#xA;&lt;p&gt;Was there such a situation? What were they experimenting on? What was this reaction that gave misleading results?&lt;/p&gt;&#xA;" OwnerUserId="15028" LastActivityDate="2022-02-07T15:00:08.563" Title="Did tobacco smoke confound the results of an experiment by entering a reaction with the subject of study?" Tags="|experimental-physics|chemistry|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13470" PostTypeId="2" ParentId="13467" CreationDate="2021-07-25T15:44:43.243" Score="11" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Are they mathematical curiosities instead of the heroes they were supposed to be by Hamilton?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Neither. It is true that there was a great hype around them, after their discovery. The hype passed. And they took their modest place among the rest of mathematics. Quaternions indeed give a very convenient way to calculate with rotations, and widely used in applied mathematics, wherever calculations with rotations are necessary (for example in computer graphics, robotics, aircraft dynamics). They give a convenient rational parametrization of the orthogonal groups in dimensions 3 and 4 (actually guessed by Euler, without proof).&lt;/p&gt;&#xA;&lt;p&gt;There were hopes that the monumental 19 century building called Functions of Complex Variables will be generalized to quaternions, but these hopes waned. Functions of a complex variable remain one of the central research areas of mathematics, while quaternions occupy their small place, as a useful tool.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-07-25T15:44:43.243" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13471" PostTypeId="1" CreationDate="2021-07-26T17:39:28.847" Score="1" ViewCount="88" Body="&lt;p&gt;De Broglie proposed a pilot wave in connection with quantum mechanics. He was more or less forced to abandon this wave in favor of the Copenhagen view. Bohm furthered his approach in the fifties but he also met resistance. Maybe because of his early communist sympathies. And maybe these same sympathies didn't make him succomb to the Copenhagen view.&lt;/p&gt;&#xA;&lt;p&gt;When I sat with my feet in an electric foot bath I noticed that the water surface surrounding my feet exhibited a standing wave pattern. The smaller the wavelengths were the stabler it got. I could move my feet around without disturbing the pattern at a certain high frequency point. I observed all kinds of motions when I changed the vibrating frequency. Things got out of hand when my girfriend added too much soap...&lt;/p&gt;&#xA;&lt;p&gt;By accident I learned that what I saw were Faraday waves. We can read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Faraday wave and its wavelength is analogous to the de Broglie wave with the de Broglie wavelength in De Broglie–Bohm theory in the field of quantum mechanics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I dont get this. I can see that if you put a small object on the waving surface, the object will stay put in crests and it will move if the wave moves. But I see no erratic behavior of the small mass. Then why is it said that de Broglie (Bohm) waves are Faraday waves? Did Faraday have some hidden structure view on the waves named after him?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-26T19:16:48.663" LastActivityDate="2021-07-26T19:16:48.663" Title="Why are pilot waves in quantum mechanics said to be Faraday waves?" Tags="|quantum-mechanics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13473" PostTypeId="1" CreationDate="2021-07-26T20:17:39.187" Score="4" ViewCount="289" Body="&lt;p&gt;E.g. &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Weil_prison/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; it says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the Second World War André Weil was first imprisoned in Finland, accused of spying, then after being returned to France he was put into Rouen prison convicted of being a deserter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And &lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9_Weil&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Weil was in Finland when World War II broke out; he had been traveling in Scandinavia since April 1939. His wife Éveline returned to France without him. Weil was mistakenly arrested in Finland at the outbreak of the Winter War on suspicion of spying;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I got curious about the purpose of his trip to the north. Was he just travelling for leisure, or was he visiting Nordic universities as a scholar? Wikipedia doesn't give a very clear timeline of his career, but based on it, I guess he had a teaching position in Strassbourg until the war, or had it already ended?&lt;/p&gt;&#xA;" OwnerUserId="13422" LastEditorUserId="4251" LastEditDate="2021-09-23T12:36:24.737" LastActivityDate="2021-10-05T07:55:38.313" Title="Why was André Weil in Finland?" Tags="|mathematicians|biographical-details|world-war-2|weil|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13475" PostTypeId="2" ParentId="13467" CreationDate="2021-07-26T23:13:06.637" Score="5" Body="&lt;p&gt;Hamilton introduced the terms &amp;quot;vector&amp;quot;, &amp;quot;scalar&amp;quot; and many others.&#xA;The Wikipedia article&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Classical Hamiltonion quaternions&lt;/a&gt;&#xA;states&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A quaternion can be represented as the sum of a scalar and a vector.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By the middle of the nineteenth century the concept of vector&#xA;spaces was not yet developed fully although linear algebra with its&#xA;systems of equations goes back to antiquity. Thus, Hamilton and his followers, used quaternions as a substitute for three&#xA;dimensional vector analytic geometry. That is, the three&#xA;quaternion units &lt;strong&gt;i, j, k&lt;/strong&gt; represented the three unit&#xA;vector basis of ordinary space. After the development of&#xA;vector spaces, the three unit vectors lost their association&#xA;with quaternions and stood on their own. Same with the real&#xA;scalars. The multiplication of two vector quaternions is given&#xA;by &lt;span class=&quot;math-container&quot;&gt;$\,u v = -u\cdot v+u\times v.\,$&lt;/span&gt; Later, the dot product&#xA;and the cross product became independent and also lost their&#xA;association with quaternions.&lt;/p&gt;&#xA;&lt;p&gt;Almost all of the machinery, notation and terminology that&#xA;Hamilton developed for quaternions as mentioned in the&#xA;Wikipedia article, is now obsolete, or else repurposed.&lt;/p&gt;&#xA;&lt;p&gt;More about the history of this development&#xA;is in the Wikipedia article&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_quaternions&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Quaternions&lt;/a&gt;&#xA;For example, this excerpt&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since quaternions demand their readers to imagine four dimensions, there is a metaphysical aspect to their invocation. Quaternions are a philosophical object. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of dot products and cross products in three-dimensional space, for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus Willard Gibbs and Oliver Heaviside made this accommodation, for pragmatism, to avoid the distracting superstructure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In answer to the question&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whatever happened to quaternions?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;the Wikipedia article also states&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions and the sedenions; they are also a useful tool in number theory, particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="13622" LastEditDate="2021-08-19T13:33:53.443" LastActivityDate="2021-08-19T13:33:53.443" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13476" PostTypeId="1" CreationDate="2021-07-27T06:48:15.183" Score="1" ViewCount="86" Body="&lt;p&gt;Felix Ehrenhaft was married to Lilly Rona. Einstein and Ehrenhaft regularly met and discussed about magnetic poles or fractional charges. Einstein didn't agree and used his authority to keep Ehrenhaft from doing things like being a director of a science iinstitute. Their friendship fell apart because of a silly thing like a monopole or a fractional charge (about which Eherenhaft turned out to be right later, after quarks were found).&lt;/p&gt;&#xA;&lt;p&gt;We can read (&lt;a href=&quot;https://www.researchgate.net/publication/231872289_A_debate_on_magnetic_current_The_troubled_Einstein-Ehrenhaft_correspondence&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The unconventional correspondence between physicists Albert Einstein and Felix Ehrenhaft, especially at the height of the alleged production by the latter of magnetic monopoles, is examined in the following paper. Almost unknown by the general public, it is sometimes witty, yet it can be pathetic, and certainly bewildering. At one point the arguments they exchanged became a poetic duel between Einstein and Ehrenhaft's wife. Ignored by conventional Einstein biographies, this episode took place during the initial years of the Second World War, but was rooted in disputes dating back to the early years of the twentieth century. The interesting intersection of a series of scientific controversies also highlights some aspects of the personal dramas involved, and after so many years the whole affair in itself is still intriguing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Both Felix and Lilly asked Einstein not to show up again. I'm curious about the poems that were used in the duel with words between Ehrenhaft's wife and Einstein. What were they about?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-27T14:45:26.900" LastActivityDate="2021-07-27T14:45:26.900" Title="What was the poetic duel Einstein had with Lilly Rona about?" Tags="|biographical-details|einstein|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13477" PostTypeId="2" ParentId="13467" CreationDate="2021-07-27T22:28:09.397" Score="8" Body="&lt;p&gt;Your question reminds me of another question on this site &lt;a href=&quot;https://hsm.stackexchange.com/questions/12866/why-are-quaternions-more-popular-than-tessarines-despite-being-non-commutative&quot;&gt;here&lt;/a&gt;, which was in a different direction: why quaternions are more popular than another algebraic system.  I gave an answer there, which I will edit and include here to address your question of &amp;quot;what happened&amp;quot; to quaternions.&lt;/p&gt;&#xA;&lt;p&gt;Hamilton's quaternions are far from being just a &amp;quot;mathematical curiosity&amp;quot;.  They are the simplest case of a very general construction that people who didn't study abstract algebra far enough may never have heard about: a &lt;em&gt;quaternion algebra&lt;/em&gt;.  This is a really significant idea within the areas of (mainstream) mathematics that make use of it.&lt;/p&gt;&#xA;&lt;p&gt;The Hamilton quaternions &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; real matrices &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf R)$&lt;/span&gt;, are both 4-dimensional noncommutative rings, with their center equal to the real numbers. In both of them there is no 2-sided ideal other than &lt;span class=&quot;math-container&quot;&gt;$(0)$&lt;/span&gt; and the whole ring. A ring that is 4-dimensional over a field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, has &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; as its center, and has no 2-sided ideals other than &lt;span class=&quot;math-container&quot;&gt;$(0)$&lt;/span&gt; and the whole ring is called a &lt;em&gt;quaternion algebra over&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;. It turns out that the matrix ring &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(K)$&lt;/span&gt; is always a quaternion algebra over &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; and every other example (if it exists) is a division ring: all nonzero elements have a left and right multiplicative inverse, just like in Hamilton's quaternions.&lt;/p&gt;&#xA;&lt;p&gt;There are two quaternion algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbf R$&lt;/span&gt;: &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf R)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt;. There is only one quaternion algebra over &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\mathbf C$&lt;/span&gt; or a finite field, namely &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(K)$&lt;/span&gt;. If you have heard of the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_p$&lt;/span&gt; (if you haven't, skip the rest of this paragraph) then you'll be amused to find out there are two quaternion algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_p$&lt;/span&gt; for each prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, just like over the real numbers. For &lt;span class=&quot;math-container&quot;&gt;$p = 2$&lt;/span&gt; the two examples are the &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_2$&lt;/span&gt; and the Hamilton quaternions with &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_2$&lt;/span&gt;-coefficients, but for odd primes &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; you need to do something different to get the quaternion algebra over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_p$&lt;/span&gt; that is a division ring because the Hamilton quaternions with &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q_p$&lt;/span&gt;-coefficients is isomorphic to &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf Q_p)$&lt;/span&gt; rather than being a division ring.&lt;/p&gt;&#xA;&lt;p&gt;Things get really interesting for quaternion algebras over fields like &lt;span class=&quot;math-container&quot;&gt;$K = \mathbf Q$&lt;/span&gt;, because&#xA;there are &lt;em&gt;infinitely many&lt;/em&gt; different (that is, nonisomorphic) quaternion algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q$&lt;/span&gt;. The ring &lt;span class=&quot;math-container&quot;&gt;${\rm M}_2(\mathbf Q)$&lt;/span&gt; and the rational Hamilton quaternions are merely two examples out of a huge list of quaternion algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q$&lt;/span&gt;. The other rational quaternion algebras show up in various places in mathematics: (supersingular) elliptic curves, Brauer groups, Shimura curves, and hyperbolic geometry in 2 and 3 dimensions.  Take a look at John Voight's &lt;a href=&quot;https://math.dartmouth.edu/%7Ejvoight/quat-book.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt; in order to appreciate that quaternion algebras other than Hamilton's one example have a lot of interesting math associated to them.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-08-20T04:47:59.563" LastActivityDate="2022-08-20T04:47:59.563" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13478" PostTypeId="1" CreationDate="2021-07-28T09:26:27.377" Score="1" ViewCount="93" Body="&lt;p&gt;Einstein invented, on his own, a statistical mechanics, presupposing the existence of atoms. I can't imagine he was unaware of the shortly-before-invented statistical approach by Boltzmann and others, but then again the internet days hadn't arrived yet. &lt;a href=&quot;https://www.ans.org/news/article-969/albert-einstein-and-the-most-elemental-atomic-theory/#:%7E:text=Einstein%20also%20in%201905%20mathematically,use%20of%20statistics%20and%20probability.&amp;amp;text=Atomic%20theory%20says%20that%20any,always%20in%20random%2C%20ceaseless%20motion.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein and the most elemental atomic theory&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Einstein pointed to the motion of Brownian particles, an effect already seen in 1827:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1827, the Scottish botanist Robert Brown looked through a microscope at pollen grains suspended in water and discovered what we now call Brownian Motion. It was an unintentional discovery. He was not looking for the effect that now bears his name, but was, rather, curious about reproduction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am not sure if this was already interpreted as atoms hitting the grains. Atomic theory existed since Democritus (and maybe even before, but we are unaware of that). So maybe it could have been already interpreted by using atoms (even if these atoms were very different from the present ones, being geometric forms with hooks).&lt;/p&gt;&#xA;&lt;p&gt;Anyhow, Mach was an opponent of the atomic view. The thermodynamics before the statistical approach imagined continuous substances. These were supposed to be divisible until the end. People hadn't yet actually tried to do this. And this hasn't been done until the present day. The evidence for atoms was (and still is) indirect. A substance hasn't been actually cut up to single atoms.&lt;/p&gt;&#xA;&lt;p&gt;The existence of atoms was later confirmed by other experiments (culminating in experiments showing the existence of quarks and recently maybe even the existence of sub-leptons as is maybe seen in the recent muon g2 experiment at CERN). A detailed model came into being.&lt;/p&gt;&#xA;&lt;p&gt;How did the opponents of Einstein react to the evidence proposed by Einstein (the Brownian motion)? How would Mach have countered the experiments performed later on in the 20th century? Could they still stick to the view that atoms are not there?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="12330" LastEditDate="2021-07-30T01:37:07.350" LastActivityDate="2021-07-30T01:37:07.350" Title="What was the attitude towards atoms around the turn of the nineteenth century?" Tags="|physics|atomic-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13480" PostTypeId="1" CreationDate="2021-07-28T18:33:40.027" Score="2" ViewCount="126" Body="&lt;p&gt;In late 1868, Norman Lockyer discovered the existence of the D3 emission line of helium in the solar spectrum. What was the earliest published example of a numeric value for the wavelength of this line, and what was the value?&lt;/p&gt;&#xA;&lt;p&gt;By 1895, when Ramsay isolated helium, the line was sufficiently well-known that Lockyer could refer to it as &amp;quot;the 587.59 line&amp;quot;, but I want to know how soon after its discovery the numeric value was established.&lt;/p&gt;&#xA;" OwnerUserId="7183" LastActivityDate="2021-07-31T16:11:25.990" Title="Earliest numeric value for helium D3 line" Tags="|physics|astronomy|experimental-physics|astrophysics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13481" PostTypeId="1" CreationDate="2021-07-28T19:45:13.863" Score="1" ViewCount="66" Body="&lt;p&gt;Dirac said the magnetic g-factor (a measure of the electron's wiggling in a magnetic field) is exactly 2. Quantum field theory introduced a g-factor that diverged a bit from 2. It needs an increasing amount of Feynmann diagrams to calculate its value. Every diagram is a calculation an Sich. The fine structure constant is involved in this. I have never quite understood how the g-factor can be calculated without a precise value of this structure constant but that aside. How were the calculations done (the first calculation was done in 1947)? There were no mainframes in sight yet but there &lt;em&gt;was&lt;/em&gt; this room-filling behemoth. Was this used? Were the calculations done by hand (maybe with an advanced abacus)? I think that we can say one thing is certain: they werent't done from the head.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-28T20:45:48.767" LastActivityDate="2021-07-28T20:45:48.767" Title="How was the first calculation of the g-factor done?" Tags="|physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13482" PostTypeId="1" AcceptedAnswerId="13491" CreationDate="2021-07-29T06:30:42.170" Score="0" ViewCount="273" Body="&lt;p&gt;Einstein once got involved in true experimental physics. That is, he designed his little machine (Maschinchen, in German, a word that makes me laugh; Einstein's Maschinchen...) to detect small potentials. Voltages, which are so easy to detect nowadays.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For as long as the name Einstein is remembered, it will surely be as a conjurer of great ideas, relativity being foremost among them. But the most celebrated theoretical physicist of all time also tried his hand once at designing an experimental instrument. It didn't amount to much, but that doesn't make it any less fascinating to physicists, who have now reconstructed the device that Einstein called his &amp;quot;little machine.&amp;quot;&#xA;In his biography of Einstein, the late physicist Abraham Pais asserted that the old man's biggest gift lay in an unprecedented understanding of statistical phenomena, which, for one thing, allowed Einstein to prove the existence of atoms by analyzing the zigzag Brownian motion of particles in water. Two years after that breakthrough, in 1907, Einstein applied a similar statistical perspective to electrons in a capacitor, a simple device for storing electrical charge, says physicist Danny Segers, director of the Museum for the History of Sciences in Ghent, Belgium. Einstein reasoned that when a capacitor is allowed to discharge, it should still contain a small voltage in the vicinity of one half millivolt. Today a voltmeter from the hardware store will tick off fractions of millivolts, but a century ago it was impossible to measure voltages that small, Segers explains.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So from 1907 to 1910 Einstein worked on a design he devised for amplifying a voltage--his Maschinchen, or little machine, as he would later refer to it. His friends Paul Habicht, an engineer and instrument maker, and Paul's brother Conrad helped him perfect the device, three copies of which survive in museums.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;When you look through the literature you don't find any measurements done on it, so we had doubts whether the machine ever worked,&amp;quot; Segers says. Wanting to test one directly, Segers and his colleague Jos Uyttenhove built their own Maschinchen, based on a schematic published by the Habichts and an interior view of one of the museum pieces. They assembled a stack of six stages, or groups of metal plates, housed in a cylinder. In the first stage a plate slides past the discharged test capacitor, picking up a charge induced by any residual voltage left in the capacitor, and deposits this amplified charge in the next stage, where the process repeats to create a measurable signal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The workings of his Maschinchen remind me of the workings of a &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Photomultiplier_tube&quot; rel=&quot;nofollow noreferrer&quot;&gt;photon multiplicator tube&lt;/a&gt; (in 1930 invented by the Russian physycist Kubetski). Are these related? Did Kubetski or people around him know about the little machine (there was no internet yet)? Or is the construction for amplifying small amounts bound by certain rules that lead to the desired outcome? Then again, I can't imagine that Einstein knew of such rules. How did he arrive at his idea?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="6609" LastEditDate="2021-07-30T06:28:20.660" LastActivityDate="2021-08-01T14:13:11.373" Title="Was Einstein's &quot;Maschinchen&quot; a predecessor of the photomultiplier tube?" Tags="|physics|experimental-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13483" PostTypeId="2" ParentId="13467" CreationDate="2021-07-29T10:13:34.850" Score="1" Body="&lt;p&gt;As a side note I want to add that Hans Georg Küssner (&lt;a href=&quot;https://en.wikipedia.org/wiki/Hans_Georg_K%C3%BCssner&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hans_Georg_Küssner&lt;/a&gt;) published two books (in 1946 &amp;quot;Principia Physica&amp;quot; and 1976 &amp;quot;Grundlagen einer einheitlichen Theorien der physikalischen Teilchen und Felder&amp;quot;) in which he tried to build a new foundation of physics and where Quaternions play an integral role.&lt;/p&gt;&#xA;" OwnerUserId="6911" LastActivityDate="2021-07-29T10:13:34.850" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13484" PostTypeId="1" CreationDate="2021-07-29T16:58:49.223" Score="2" ViewCount="490" Body="&lt;p&gt;Why was periodic motion started to be represented as a sine function at the beginning? Is it because it is perhaps the simplest periodic function? Is there certainly any reason behind this representation?&lt;/p&gt;&#xA;" OwnerUserId="14195" LastActivityDate="2021-07-30T01:47:13.800" Title="Why is periodic motion always represented as a sine function" Tags="|physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13485" PostTypeId="2" ParentId="13484" CreationDate="2021-07-29T17:17:29.987" Score="2" Body="&lt;p&gt;All periodic motions can be represented by (finite or infinite) sums of sines and cosines (Fourier).&lt;/p&gt;&#xA;&lt;p&gt;You &lt;em&gt;could&lt;/em&gt; have started with other basis functions. Like zigzag functions with different frequencies. The fact is though that most waves in Nature are caused by vibrations obeying Hooke's law, which has sine or cosine solutions.&lt;/p&gt;&#xA;&lt;p&gt;There are other kinds of periodic processes which obey different (non-linear) equations. These processes can't be approximated by a superposition of sines or cosines ( no superposition). Like the waveforms resembling a curved mountain with a sharp peak. I forgot their name.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure if this is a history question but it's a fact that physics stayed in the linear realm a long time. Non-linear phenomena are much more interesting. But much more complex. If mechanics had started out non-linearly then maybe today you would have learned about non-linear signals already at high school. Although... Computers were not available in the time of Newton and Huygens. And you need these to do the intensive numerical calculations going along with non-linearity. To do this by hand was even for Newton too much.(sorry for eventual spelling mistakes; I type on a phone...).&lt;/p&gt;&#xA;&lt;p&gt;By the way, the non-linear &amp;quot;mountain&amp;quot; waves are called &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Peakon&quot; rel=&quot;nofollow noreferrer&quot;&gt;peakons&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Already in the 17th century Hooke invented &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Hooke%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;his law&lt;/a&gt;. It is a linear law (pull a spring twice as far and the opposing spring force gets twice that big). Solutions to this law are sines and cosines. They can both involve time and distance so they can represent waves, which were thought to be linear phenomena (which they are not in reality but the appriximation showed to hold very well).&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-29T19:18:55.343" LastActivityDate="2021-07-29T19:18:55.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13486" PostTypeId="2" ParentId="13484" CreationDate="2021-07-29T18:42:28.197" Score="4" Body="&lt;p&gt;Periodic functions can be also represented by cosine functions, linear combinations of sine and cosine, and more generally by linear combinations&#xA;of exponential functions &lt;span class=&quot;math-container&quot;&gt;$e^{int}$&lt;/span&gt; which represents a circular motion with fixed angular speed.&lt;/p&gt;&#xA;&lt;p&gt;Of course, they can be also represented differently, like for example zig-zag functions used in ancient Babylon. The main mathematical advantage of exponentials come from the fundamental fact that they are eigenfunctions of the shift operator:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$e^{in(t+a)}=e^{ina}e^{int}.$$&lt;/span&gt;&#xA;This fundamental property implies all other useful properties.&lt;/p&gt;&#xA;&lt;p&gt;The idea that &amp;quot;every (celestial) motion can be represented in terms of uniform circular motions&amp;quot; was promoted by Greek philosophers in 4th century BC, and it was implemented in practice by Ptolemy and later astronomers.&lt;/p&gt;&#xA;&lt;p&gt;Added. In my earlier answer I said that Fourier was not influenced by astronomy. He was solving different problems, related to heat equation. But the prominent role of sine and cosine in his work is explained by the SAME fundamental reason: they are eigenfunctions of the shift operator. And this fundamental property explains their omnipresence in mathematical physics and other areas of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-30T01:47:13.800" LastActivityDate="2021-07-30T01:47:13.800" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13487" PostTypeId="2" ParentId="13340" CreationDate="2021-07-29T21:36:14.570" Score="0" Body="&lt;p&gt;User @Conifold provided a perfect source, I shall briefly report the information it contains to close the question. I hope answering myself is fine in these situations, feel free to correct or post another answer.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick/page/392/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dickson's History of the theory of numbers, chapter XVII&lt;/a&gt; for the full content.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;After defining his sequences &lt;span class=&quot;math-container&quot;&gt;$u_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v_n$&lt;/span&gt;, Lucas studied relations between their terms and gave some properties of purely algebraic nature. These were then properly demonstrated.&lt;/p&gt;&#xA;&lt;p&gt;What I was searching for can be found starting from page 397:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lucas stated that, if &lt;span class=&quot;math-container&quot;&gt;$4m+3$&lt;/span&gt; is prime, &lt;span class=&quot;math-container&quot;&gt;$p = 2^{4m+3} — 1$&lt;/span&gt; is prime if the first term of the series 3, 7, 47,..., defined by &lt;span class=&quot;math-container&quot;&gt;$r_{n+1} = r_n^2 — 2$&lt;/span&gt;, which is divisible by &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is of rank &lt;span class=&quot;math-container&quot;&gt;$4m+2$&lt;/span&gt;; but &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is composite if no one of the first &lt;span class=&quot;math-container&quot;&gt;$4m +2$&lt;/span&gt; terms is divisible by &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This will of course be later reformulated in the famous Lucas-Lehmer test. The formulation Lucas gave, albeit weaker than Lehmer's, can find some divisors if the number is composite.&lt;/p&gt;&#xA;&lt;p&gt;Lucas then proceeds giving some examples: see page 399 for a primality test for numbers in the form &lt;span class=&quot;math-container&quot;&gt;$p = 2^{4q+1} — 1$&lt;/span&gt;, note that more can be found. Important paragraphs are conveniently marked with a pencil.&lt;/p&gt;&#xA;&lt;p&gt;The chapter then proceeds with more applications and examples of historical relevance, like exchanges with other mathematicians of the time.&lt;/p&gt;&#xA;" OwnerUserId="14893" LastEditorUserId="14893" LastEditDate="2021-07-29T21:49:43.563" LastActivityDate="2021-07-29T21:49:43.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13488" PostTypeId="1" CreationDate="2021-07-29T22:04:49.703" Score="2" ViewCount="584" Body="&lt;p&gt;The term &amp;quot;sandwich&amp;quot; and the verb &amp;quot;to sandwich&amp;quot; appear pretty common but informally in quantum mechanics. Generally when describing some kind of inner product of the form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\langle L | H | L'\rangle $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$\langle L|$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|L'\rangle $&lt;/span&gt; are some (quantum) vectors (the loaf), generally for the same state &lt;span class=&quot;math-container&quot;&gt;$L=L'$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is some operator.  The &lt;span class=&quot;math-container&quot;&gt;$\langle L | H | L'\rangle $&lt;/span&gt; is called the sandwich, and to take the expectation value is &amp;quot;to sandwich&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would have thought that the terminology appeared at least as early as Dirac notation, because the &amp;quot;sandwiching&amp;quot; would be less obvious with integrals.&lt;/p&gt;&#xA;&lt;p&gt;For some examples, check these questions in Physics Stack Exchange: &lt;a href=&quot;https://physics.stackexchange.com/questions/168056/solving-for-the-density-operator-in-the-quantum-brownian-motion-master-equation&quot;&gt;Solving for the density operator in the quantum Brownian motion master equation&lt;/a&gt;, &lt;a href=&quot;https://physics.stackexchange.com/questions/644057/manipulations-of-the-wavefunction-are-hard-to-physically-understand&quot;&gt;Manipulations of the wavefunction are hard to physically understand&lt;/a&gt; and &lt;a href=&quot;https://physics.stackexchange.com/questions/513332/is-there-any-relationship-between-s-matrix-elements-and-the-path-integral&quot;&gt;Is there any relationship between S-matrix elements and the path integral?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know that sandwich can be a very coloquial term for other stuff in science. It is even sometimes used formally in mathematics for theorems like the squeeze theorem (AKA sandwich theorem) and the ham sandwich theorem. However I would like to find the earliest attestation of this term used in quantum mechanics.&lt;/p&gt;&#xA;&lt;p&gt;I searched some of my books, but I could only find the term &amp;quot;sandwich&amp;quot; in R. Shankar &lt;em&gt;Principles of Quantum Mechanics&lt;/em&gt; and DJ Griffiths' &lt;em&gt;Introduction to Quantum Mechanics&lt;/em&gt;, which are relatively new (1990s).&lt;/p&gt;&#xA;&lt;p&gt;I had sometimes joked with the idea that the term came from expectations values of a Hamiltonian (abbreviated as Ham) but maybe that's stretch.&lt;/p&gt;&#xA;&lt;p&gt;The earliest paper I could find is &lt;a href=&quot;https://journals-aps-org.scd-rproxy.u-strasbg.fr/pr/abstract/10.1103/PhysRev.179.1499&quot; rel=&quot;nofollow noreferrer&quot;&gt;Non-Lagrangian models of current algebra, KG Wilson, Phys. Rev., 1969&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In a nutshell, what's the earliest recorded use of sandwich in quantum mechanics?&lt;/p&gt;&#xA;&lt;p&gt;Bonus: this terminology has even crossed language boundaries, see for example &lt;a href=&quot;https://www.fisica.unam.mx/personales/romero/IFC2018/Resumen-Mecanica-Cuantica.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Resumen de mecánica cuántica, VR. Rochin UNAM&lt;/a&gt;, &lt;a href=&quot;https://cel.archives-ouvertes.fr/cel-00136189/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantique: éléments F. Balibar et al.&lt;/a&gt;, &lt;a href=&quot;https://www.ph.tum.de/academics/bsc/break/2013s/fk_PH0007_01_course.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ferienkurs Quantenmechanik&#xA;Sommersemester 2013 Technische Universität München&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-07-31T09:35:13.853" LastActivityDate="2021-07-31T09:35:13.853" Title="On the origin of &quot;sandwiches&quot; in quantum mechanics" Tags="|physics|terminology|quantum-mechanics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13489" PostTypeId="1" CreationDate="2021-07-29T22:21:23.040" Score="-2" ViewCount="246" Body="&lt;p&gt;I'm not sure if this question can be answered objectively but I still ask.&lt;/p&gt;&#xA;&lt;p&gt;Once upon a time there were people like Galileo, Newton, Einstein, Feynman, Curie, Tesla, Dirac, Schrödinger, maybe Chopra, and all I don't mention. More recently maybe Hawking (about whom even a film was made, but maybe he had his wheelchair playing in favor of him). In Amsterdam and many other cities streets are named after these guys and girls.&lt;/p&gt;&#xA;&lt;p&gt;Though I have my personal present hero, it seems that the past people don't have a counterbalance in recent or present day, again Hawing being the exception.&lt;/p&gt;&#xA;&lt;p&gt;Why is it that almost all people know Einstein, Curie, Newton, or Hawking, but almost nobody knows Smolin, Witten, Verlinde (a Dutch physicist who invented an emergent gravity theory, who was even hailed on the Dutch news as the inventor of a completely new gravity), or Sabine what's-her-name (the German physicist who is popular on the internet)?&lt;/p&gt;&#xA;&lt;p&gt;What was different in the past? Was the world still to be discovered (everybody knows Columbus or othe discovery travelers who were heading for the new world)? Like the invention (discovery) of the major fundaments?&lt;/p&gt;&#xA;&lt;p&gt;Of course there are TV celebrities but I can hardly call them the heroes of physics. Maybe the advocates.&lt;/p&gt;&#xA;&lt;p&gt;Truly coincidentally, I just read a piece in a Dutch newspaper, about the history of science (my translation):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was a time of titans. It was the time of scientists like Eduard Meyers, Jan-Hendrik Oort, Pieter Zeeman, and Johan Huizinga, individuals who blew their faculties to great heights. It was a time when Leiden university could imagine itself to be the best in Holland.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is exactly what I mean. No more heroes anymore...&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-07-31T13:14:46.573" LastActivityDate="2021-07-31T13:14:46.573" Title="How come that the &quot;heroes&quot; in physics are gone?" Tags="|physics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13490" PostTypeId="1" AcceptedAnswerId="13494" CreationDate="2021-07-29T22:41:56.897" Score="7" ViewCount="479" Body="&lt;p&gt;While researching on Mersenne numbers, I often stumble upon statements of this nature (it is not verbatim):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Huldaricus Regius in 1536 proved that &lt;span class=&quot;math-container&quot;&gt;$2^{11}-1$&lt;/span&gt; is not prime, providing a factorisation and shattering the common misconception that all numbers of the form &lt;span class=&quot;math-container&quot;&gt;$2^p-1$&lt;/span&gt; are primes for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; prime.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is of course an important step in the history of Mersenne Primes -- and this made me wonder the following. You can find his name in different variations (&lt;em&gt;Huldalrichus Regius, Hudalrichus Regius, Ulricus Regius, etc&lt;/em&gt;) and in multiple sources: &lt;a href=&quot;https://wwwhomes.uni-bielefeld.de/achim/mersenne_history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this table (English)&lt;/a&gt;, &lt;a href=&quot;http://www.bitman.name/math/article/288/167&quot; rel=&quot;nofollow noreferrer&quot;&gt;in this brief history of Mersenne primes (Italian)&lt;/a&gt; and &lt;a href=&quot;http://www.batmath.it/matematica/a_primi/pg5.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;in this one too (Italian)&lt;/a&gt;. Although I can't read French, &lt;a href=&quot;https://books.google.it/books?id=0BtsDwAAQBAJ&amp;amp;pg=PT70&amp;amp;lpg=PT70&amp;amp;dq=huldaricus+regius&amp;amp;source=bl&amp;amp;ots=fIKDxJSSp5&amp;amp;sig=ACfU3U1-OjaZQR09EHjvGgycEcZhRtlQeg&amp;amp;hl=it&amp;amp;sa=X&amp;amp;ved=2ahUKEwjE-u_UqYnyAhXchP0HHb1CDY0Q6AEwEHoECBAQAw#v=onepage&amp;amp;q=huldaricus%20regius&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book (French)&lt;/a&gt; could also come in handy.&lt;/p&gt;&#xA;&lt;p&gt;He seems an interesting figure, but I could find little to no information about him. Was he real? Does anyone know more about him? I'm asking this of course for historical interest, but also because Huldaricus Regius is ominously similiar to &amp;quot;Huldaricus Regius [x]&amp;quot;, which in Latin could possibly hint to a title or a group of people -- depending on the truncated [x].&lt;/p&gt;&#xA;" OwnerUserId="14893" LastEditorUserId="6229" LastEditDate="2021-07-30T11:03:22.997" LastActivityDate="2021-07-30T11:03:22.997" Title="$2^{11} - 1$ and the mystery of Huldaricus Regius" Tags="|number-theory|primes|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13491" PostTypeId="2" ParentId="13482" CreationDate="2021-07-29T23:03:49.933" Score="2" Body="&lt;p&gt;No, Einstein's maschinchen is not much like a photomultiplier tube. It's much more like a Van de Graaff generator.&lt;/p&gt;&#xA;&lt;p&gt;A photomultiplier is an example of an exponential cascade amplifier. A photon hitting the first surface will (typically with a probability of about 10 to 50%) knock loose an electron. The electron is accelerated toward the next stage, where it knocks loose more than one electron (on average). The secondary emission ratio, as this is known, can be in the range of 10 to 20. The process continues, with each stage increasing the number of electrons knocked free, and hence the current. At the end of the cascade, a relatively large number of electrons is released, which are enough for ready measurement.&lt;/p&gt;&#xA;&lt;p&gt;The maschinchen seems to more like a Van de Graaff generator, where a moving element picks up electrons and transfers them to a different location. Multiple elements produce a higher voltage, but there is no change in the number of electrons. In a Van de Graaff generator, this is done in a single pass, using a flexible belt to transfer electrons. The accumulation of electrons produces a high voltage. The Van de Graaff was used to provide high voltage for early particle acccelerators (called &amp;quot;atom smashers&amp;quot; in the popular press).&lt;/p&gt;&#xA;" OwnerUserId="7183" LastEditorUserId="7183" LastEditDate="2021-08-01T14:13:11.373" LastActivityDate="2021-08-01T14:13:11.373" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13493" PostTypeId="2" ParentId="13490" CreationDate="2021-07-30T00:14:49.223" Score="3" Body="&lt;p&gt;Hudalrichus Regius (English spelling?) is perhaps best known as the discoverer of the first perfect number since Euclid, who had identified 6, 28, 496 and 8128.&lt;/p&gt;&#xA;&lt;p&gt;Euclid knew that &lt;span class=&quot;math-container&quot;&gt;$2^{n-1}(2^n - 1)$&lt;/span&gt; is perfect when &lt;span class=&quot;math-container&quot;&gt;$(2^n - 1)$&lt;/span&gt; is prime and applied this to &lt;span class=&quot;math-container&quot;&gt;$n = 2, 3, 5, 7$&lt;/span&gt;.  Hudalrichus Regius tried &lt;span class=&quot;math-container&quot;&gt;$n = 11$&lt;/span&gt; and found that &lt;span class=&quot;math-container&quot;&gt;$2^{11}-1$&lt;/span&gt; was not prime, however &lt;span class=&quot;math-container&quot;&gt;$n = 13$&lt;/span&gt; does yield a prime and the corresponding perfect number &lt;span class=&quot;math-container&quot;&gt;$33350336$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This also refuted a claim of Nicomachus that the &lt;span class=&quot;math-container&quot;&gt;$n'\text{th}$&lt;/span&gt; perfect number had &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; digits.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Cataldi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source : MacTutor entry on Pietro Antonio Cataldi.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Perfect_numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source : MacTutor entry on perfect numbers.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2021-07-30T00:29:24.780" LastActivityDate="2021-07-30T00:29:24.780" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13494" PostTypeId="2" ParentId="13490" CreationDate="2021-07-30T02:02:44.900" Score="9" Body="&lt;p&gt;Huldalrichus Regius seems to be the same person as Ulrich Regius, otherwise known as Ulrich Rieger, according to the &lt;a href=&quot;https://www.cerl.org/&quot; rel=&quot;noreferrer&quot;&gt;Consortium of European Research Libraries&lt;/a&gt; (CERL)  &lt;a href=&quot;https://data.cerl.org/thesaurus/cnp01109708&quot; rel=&quot;noreferrer&quot;&gt;entry&lt;/a&gt; for him.&lt;/p&gt;&#xA;&lt;p&gt;His book, &lt;em&gt;VTRIVSQVE  ARITHMETICES Epitome ex uarijs authoribus concinnata per HVdalrichum  Regium&lt;/em&gt; was published in 1536, with reissues in  1543 and 1550.   You can see a digital version &lt;a href=&quot;https://www.digitale-sammlungen.de/en/view/bsb10186797?page=4,5&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.  The &lt;a href=&quot;https://www.worldcat.org/search?q=ti%3AVTRIVSQVE+ARITHMETICES&amp;amp;qt=advanced&amp;amp;dblist=638&quot; rel=&quot;noreferrer&quot;&gt;Worldcat entry&lt;/a&gt; also gives details of editions and locations of copies.  I'm guessing that the title means something like &lt;em&gt;Summary of both kinds of arithmetic, compiled from various experts by Ulrich Regius&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It was common in that era for scholars to use latinate forms of their birth names, and it is sometimes hard to us to match the forms of the names. According to &lt;a href=&quot;https://www.mathgenealogy.org/&quot; rel=&quot;noreferrer&quot;&gt;https://www.mathgenealogy.org/&lt;/a&gt; one of my doctoral ancestors is &amp;quot;Regiomontanus&amp;quot;, who was the same person as Johannes Müller, from Königsberg.  If you know a tiny bit of Latin and German you see that his scholarly name is a Latin calque of the name of his birthplace.  Similarly, one of Euler's doctoral ancestors was Philip Melanchthon, whose name is Greek calque of his birth name, Philipp Schwartzerdt.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2021-07-30T11:02:00.630" LastActivityDate="2021-07-30T11:02:00.630" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13495" PostTypeId="2" ParentId="13489" CreationDate="2021-07-30T02:28:22.923" Score="5" Body="&lt;p&gt;There are several reasons for this phenomenon.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What you are talking about is not really history of science, but history of mass media. That some physicists are household names while others&#xA;are in relative obscurity. (For example you do not mention Stevin, Huygens or Hooke, Laplace or Hamilton, or even Maxwell (!!), and they are not among the names recognizable by everybody). This is a property of mass media to choose one &amp;quot;hero&amp;quot; at the expense of others. Public attention cannot concentrate on too many &amp;quot;heroes&amp;quot;. (Some self-promotion can also play the role, sometimes).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Some time should pass before the really great discoveries are properly evaluated. Newton's greatness was not so much recognized in his lifetime as it was after. (Jonathan  Swift once wrote about Newton to his friend: &amp;quot;They say ridiculous things, for example that there is an&#xA;an instrument maker in London, whole fame will be greater than mine!&amp;quot;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;But the main reason is that physics is nowadays much larger than it used to be. There are much more great physicists and much more discoveries than there&#xA;used to be at the time of Newton. And public attention has only very limited ability to digest this.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;As someone said: &amp;quot;The system of the world can be discovered only once&amp;quot;. In some sense, the most fundamental discoveries are made at earlier times, and later generations can only build on them. So as time goes, the fame of&#xA;people like Newton, Maxwell, Einstein and Bohr increases all the time so the later discoverers can hardly compete. At the time when Newton lived, Archimedes and Descartes were more famous than he.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Newton's main discoveries are easy to explain to the general public: everyone can see comets, planets, rainbows and tides, and even imagine the shape of Earth. Electrical phenomena are not so clearly visible in the nature, but their discoveries immediately led to new technologies which everyone uses in daily life. Same with Quantum theory, it eventually led to many new technologies, including atomic energy. But modern theoretical physics is more remote from our daily life: nothing in our ordinary life is related to quarks, black holes or Higgs boson.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-07-30T18:38:18.013" LastActivityDate="2021-07-30T18:38:18.013" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13496" PostTypeId="1" CreationDate="2021-07-30T06:20:01.093" Score="4" ViewCount="154" Body="&lt;p&gt;In 1957 Crick propounded the central dogma of molecular biology:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Central Dogma. This states that once &amp;quot;information&amp;quot; has passed into protein it cannot get out again. In more detail, the transfer of information from nucleic acid to nucleic acid, or from nucleic acid to protein may be possible, but transfer from protein to protein, or from protein to nucleic acid is impossible. Information means here the precise determination of sequence, either of bases in the nucleic acid or of amino acid residues in the protein.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why is it so important? What was the evidence Crick based his dogma upon?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="20856" LastEditDate="2024-03-19T07:52:16.697" LastActivityDate="2024-03-19T07:52:16.697" Title="What was the basis for Crick's proposal of the central dogma of molecular biology?" Tags="|biology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13499" PostTypeId="2" ParentId="13480" CreationDate="2021-07-31T15:16:10.860" Score="2" Body="&lt;p&gt;The earliest references that I can find about the measurement of helium D3 line is&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;D'Arrest, H. L. (1872).&lt;/strong&gt; &lt;em&gt;Über die Position der Lichtlinie D3 im Protuberanzspectrum&lt;/em&gt;. Astronomische Nachrichten, 79, 1. (On the position of D3 line in the prominence spectrum). For serious academic work, the 2 paged article need to be fully translated. You also need to see &amp;quot;Schellen's Spectralanalysis, 1871 p.380,&amp;quot; cited there. DeepL is a very good translator, but you might need a lot of text pre-processing to get a decent translation. The machine translation methodology has been published in Journal of Chemical Information and Modeling.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AyQ2d.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AyQ2d.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For the wavelength of the light in which &lt;span class=&quot;math-container&quot;&gt;$D_{3}$&lt;/span&gt; appears, quite a lot of observations are available, in that the observers have always related the position itself consequently to the near sodium lines visible at the same time as &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{D}_{\mathbf{3}}$&lt;/span&gt;. If it is now assumed that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{l}&#xA;D_{1}=589,50 \text { angstr. }=1002,8  \mathrm{Kirchh} \\&#xA;D_{2}=588,90 \quad=\quad=1006,8 \quad=&#xA;\end{array}&#xA;$$&lt;/span&gt;&#xA;we get from Young's, Rayet's and other data&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;D_{3}=587,50 \text { Angstr. }=1016,1 \text { Kirchh }&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Translated with &lt;a href=&quot;http://www.DeepL.com/Translator&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.DeepL.com/Translator&lt;/a&gt; (free version)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Sadly he does not cite Young's paper but there is a footnote&lt;/p&gt;&#xA;&lt;p&gt;*) Rayet has e.g. &lt;span class=&quot;math-container&quot;&gt;$1016,8 \boldsymbol{K}$&lt;/span&gt; which leads to &lt;span class=&quot;math-container&quot;&gt;$587,4 \boldsymbol{A} .$&lt;/span&gt;. If, on the other hand, in Schellen's Spectralanalysis, 1871 p.380, &lt;span class=&quot;math-container&quot;&gt;$D_{3}=588,27$&lt;/span&gt; is set, this large deviation comes from the fact that all three &lt;span class=&quot;math-container&quot;&gt;$\boldsymbol{D}$&lt;/span&gt; - lines appear at the place obviously about 0,93 too large.&lt;/p&gt;&#xA;&lt;p&gt;In older times it appears that people were often unaware of previous works so measurements appear again.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Maunder, E. W. (1889).&lt;/strong&gt; Spectroscopic observations of sundry stars and comets made at the Royal Observatory, Greenwich, chiefly in the years 1887 and 1888. &lt;em&gt;Monthly Notices of the Royal Astronomical Society&lt;/em&gt;, 49, 300.&lt;/p&gt;&#xA;&lt;p&gt;This seems to be the first because this paper does not give any reference to previous works.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iEMSG.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iEMSG.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2021-07-31T16:11:25.990" LastActivityDate="2021-07-31T16:11:25.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13500" PostTypeId="2" ParentId="3131" CreationDate="2021-07-31T19:58:49.317" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Blaise_Pascal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blaise Pascal&lt;/a&gt; a French mathematician, physicist, and philosopher died at the age of 39.&lt;/p&gt;&#xA;" OwnerUserId="15087" LastEditorUserId="4251" LastEditDate="2023-12-03T09:14:37.720" LastActivityDate="2023-12-03T09:14:37.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13501" PostTypeId="2" ParentId="13301" CreationDate="2021-08-01T00:36:59.173" Score="7" Body="&lt;p&gt;I'm the same Charlotte Aten you were watching on YouTube. I had intended to make this a comment and not an answer (since I don't think I can provide a complete one), but since I just joined this StackExchange community for the purpose of responding to this I don't have that ability here yet.&lt;/p&gt;&#xA;&lt;p&gt;In any case, the main source I have for that story is the book «Modern Algebra and the Rise of Mathematical Structures» by Leo Corry. I don't remember how much depth the author goes into about the short-lived program of Ore's. I think that it was mainly just presented to set the stage for a much more detailed discussion of the history of category theory and as a kind of segue away from abstract/universal algebra. You may be able to find more information in the copious references in that book.&lt;/p&gt;&#xA;" OwnerUserId="15088" LastActivityDate="2021-08-01T00:36:59.173" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13502" PostTypeId="1" CreationDate="2021-08-01T10:56:15.763" Score="0" ViewCount="75" Body="&lt;p&gt;I can remember after attending a philosophy of science class, I put forward the thought ancient Greek had about the motion of the planets. I had read about them thinking there are wooden circular structures placed on the heavenly sphere in which the planets are guided. Like a train on a rail.&lt;/p&gt;&#xA;&lt;p&gt;I am sure I read it, which is the reason I brought it up. I cant find anything about it and I cant remember where I read it. Maybe the book on the history of astronomy by Dyksterhuis.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-08-01T10:56:15.763" Title="Is it true that there were &quot;wooden guiding structures&quot; proposed, by the ancient Greek, to account for the motions of the planets?" Tags="|astronomy|ancient-greece|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13503" PostTypeId="1" CreationDate="2021-08-01T12:23:16.513" Score="5" ViewCount="262" Body="&lt;p&gt;The Medieval Islamic world, between 8th to the mid 13th century, is known for its developments in academic knowledge. Particularly in astronomy. Many stellar objects still derive their name from Arabic names. There were also advances in optics (e.g. al-Haytham/Alhazen) and kinematics, which were later explored or rediscovered during European renaissance science. Mathematics (e.g. al-Khwarizmi), alchemy (e.g. Jabir ibn Hayyan/Geber), engineering (e.g al-Jazari) and fluid dynamics (e.g. al-Biruni) took a boost too.&lt;/p&gt;&#xA;&lt;p&gt;What I have never seen are particular advances in thermodynamics or electromagnetic theory from this period. I am putting aside here developments in optics and philosophical discussions on atomism.&lt;/p&gt;&#xA;&lt;p&gt;Early Greek hypotheses, included the concept of the pre-Socratic 4 elements (later expanded into chemistry by Geber). As stated in Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_thermodynamics&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of thermodynamics&lt;/a&gt;, the concept of fire element is like the precursor of phlogiston and caloric fluid theories. The name &amp;quot;electron&amp;quot; is related to Greek name for amber, as they were the first to record the concept electric forces (same for magnetic forces). The Ancient Greeks also first developed ideas about the void and atomism.&lt;/p&gt;&#xA;&lt;p&gt;I know Medieval Islamic scholars advanced much in alchemy/chemistry. But were there any particular advances in the thermodynamics and electromagnetism, concepts like heat, pressure, microscopic origin of temperature/pressure, perpetual motion, current, magnets, charges and so on?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-01-25T19:49:34.873" LastActivityDate="2023-01-25T19:49:34.873" Title="Were there any contributions to thermodynamics and electromagnetism from Medieval Islamic science?" Tags="|electromagnetism|thermodynamics|islamic-science|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13504" PostTypeId="1" CreationDate="2021-08-01T18:29:14.503" Score="17" ViewCount="7657" Body="&lt;p&gt;Ramanujan's story is a well known story of the Indian young man who turned out to be a mathematical genius without a scholarly education. He was &amp;quot;discovered&amp;quot; by the mathematician Hardy at the beginning of last century and came to England where he was appointed a university job.&lt;/p&gt;&#xA;&lt;p&gt;Are there similar examples in the realm of physics?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4251" LastEditDate="2021-09-08T06:38:50.317" LastActivityDate="2021-09-08T06:38:50.317" Title="Has there been an equivalent in physics to Ramanujan in maths?" Tags="|physics|physicists|" AnswerCount="7" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13505" PostTypeId="2" ParentId="13504" CreationDate="2021-08-02T05:13:28.420" Score="29" Body="&lt;p&gt;Michael Faraday would meet your criteria as a physicist of historic significance without a formal education.  Notably, he spent most of his teenage years working as an apprentice bookbinder. Somewhat like Ramanujan, his research career began following his initiation of an unsolicited correspondence (in his case, to the chemist Humphry Davy).&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2021-08-02T05:28:28.307" LastActivityDate="2021-08-02T05:28:28.307" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13506" PostTypeId="2" ParentId="13504" CreationDate="2021-08-02T05:32:39.577" Score="38" Body="&lt;p&gt;George Green may be considered a physicist or a mathematician. (The titles of his papers suggest a physicist). He was a son of a miller, and a miller himself,&#xA;in Nottingham, self educated. At the age of 35 he rented out his mill, and wrote &amp;quot;An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism&amp;quot; where he discovered what is called now Green's formula, Green function and potential, and established the foundation of potential theory. He published it by himself, as a little booklet, supported by subscription of his friends. And only after that he went to Cambridge where he obtained a formal education, wrote few more papers on mathematical physics (his first work is by far the most famous) and died at the age of 48.&lt;/p&gt;&#xA;&lt;p&gt;Edit. Later historians investigated what books were available to him in Nottingham. They found one appropriate entry in the local gentlemen's club library: Laplace, Traité de mécanique céleste. If this was the source of his self-education, then it is not clear where he learned French. He never attended any school before he wrote his opus magnum.&lt;/p&gt;&#xA;&lt;p&gt;Edit 2. The mill has been restored and still functions as a museum of George Green.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-02T17:35:06.580" LastActivityDate="2021-08-02T17:35:06.580" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="13508" PostTypeId="2" ParentId="13504" CreationDate="2021-08-02T15:28:35.867" Score="16" Body="&lt;p&gt;Other possibilities might be James Joule and Oliver Heaviside.&lt;/p&gt;&#xA;&lt;p&gt;Joule was a brewer by trade and worked on science entirely as an amateur (as far as I know).  He was one of the few examples of a physicist who made fundamental contributions to the field despite working in isolation from the community of scientists.  In fact, he was very much at odds with the community of his time and was embroiled in some controversies and priority disputes with Mayer.  He performed ground-breaking experiments on heat which ultimately led to the law of conservation of energy and the first law of thermodynamics.&lt;/p&gt;&#xA;&lt;p&gt;Heaviside made fundamental contributions to physics and electrical engineering despite being self-taught and isolated from the scientific community for most of his life.  Heaviside re-wrote Maxwell's equations in the compact form which is now known to all physics students.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastEditorUserId="10345" LastEditDate="2021-09-02T13:28:49.250" LastActivityDate="2021-09-02T13:28:49.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13509" PostTypeId="1" AcceptedAnswerId="13510" CreationDate="2021-08-02T21:32:43.477" Score="0" ViewCount="236" Body="&lt;p&gt;In astronomy the parsec is a unit of length:&lt;/p&gt;&#xA;&lt;p&gt;A parsec (abbreviated pc) is a unit of distance used by astronomers, cosmologists, and astrophysicists.&lt;/p&gt;&#xA;&lt;p&gt;1 parsec is equal to &lt;span class=&quot;math-container&quot;&gt;$3.08567758\times10^{16}$&lt;/span&gt; meters, or &lt;span class=&quot;math-container&quot;&gt;$3.26163344$&lt;/span&gt; light years&lt;/p&gt;&#xA;&lt;p&gt;I think this is a rather arbitrary number. Somewhat like a (land)mile being about &lt;span class=&quot;math-container&quot;&gt;$1.5\dots$&lt;/span&gt; kilometers.&lt;/p&gt;&#xA;&lt;p&gt;What is the ratio behind this number?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="229" LastEditDate="2021-08-03T13:40:12.187" LastActivityDate="2021-08-03T13:40:12.187" Title="What is the origin of the parsec?" Tags="|astronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13510" PostTypeId="2" ParentId="13509" CreationDate="2021-08-02T21:54:12.540" Score="2" Body="&lt;p&gt;This word was coined by H. H. Turner.&lt;/p&gt;&#xA;&lt;p&gt;par- (in parallax n.) + sec- (in second)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A unit of length equal to the distance at which a star would have a&#xA;heliocentric parallax of one second of arc, approximately equivalent&#xA;to 3·086 × 10^13 kilometres (19·2 × 1012 miles, 3·26 light years).&lt;/p&gt;&#xA;&lt;p&gt;1913   F. W. Dyson in Monthly Notices Royal Astron. Soc. 73 342&lt;br /&gt;&#xA;There is need for a name for this unit of distance... Professor [H.&#xA;H.] Turner suggests Parsec, which may be taken as an abbreviated form&#xA;of ‘a distance corresponding to a parallax of one second’.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ref: &lt;em&gt;Unabridged Oxford English Dictionary&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;BTW, This is an excellent resource for word origins but it is not public. Good libraries have subscription or you can buy your own annually.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-08-02T21:54:12.540" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13511" PostTypeId="2" ParentId="13504" CreationDate="2021-08-03T07:59:18.113" Score="8" Body="&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Sorry for the hype in this answer. But trying to convey how singular - how truly remarkable Ramanujan was, as a mathematician and innovator, is truly difficult without writing a book on him... so please forgive the hype in this answer. I'm trying to give a feel of something that makes hairs stand up on the necks of world class mathematicians even a century later....&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Realistically there has &lt;strong&gt;never&lt;/strong&gt; been a physicist equivalent of Ramanujan. Physics requires observation and not just conceptual structures, while mathematics is its own edifice, so to speak.&lt;/p&gt;&#xA;&lt;p&gt;When you discuss Ramanujan, there are two aspects to his story that are signular - the story of the person (self taught Indian genius who despite this became a world leader in his field), and the story of his capabilities (for example his familiarity with numbers, his almost incomprehensible abilities with arcane and infinite nested formulae, his sheer almost inhuman &lt;strong&gt;intuition&lt;/strong&gt; for a formulaic answer to a problem ...)&lt;/p&gt;&#xA;&lt;p&gt;There have been self taught physicists who came from non physics backgrounds yet upended the physics world. I am extremely doubtful that there has ever been one - perhaps never will or could be one - who has shown within physics anytning that could be considered reasonably equivalent to the jawdropping capabilities Ramanujan showed within number theory and mathematics.&lt;/p&gt;&#xA;&lt;p&gt;We are still trying to comprehend &lt;strong&gt;how&lt;/strong&gt; he could work out what he knew, and not merely make use of and develop &lt;strong&gt;what&lt;/strong&gt; he knew and discovered. Even with a century's hindsight he could do things that seem implausible today.&lt;/p&gt;&#xA;&lt;p&gt;You'd be talking about the kind of person who could develop much of the entire body of modern and near future quantum physics as it stands in 2021 ~ 2040 on the back of an envelope, starting from the knowledge of a 1915 indian villager with a few old textbooks, or something like that, to stand a chance of being equivalent. Maybe modern superconductor theory on the side too. And deduce the masses and charges of bosons that wont be seen for decades or even suspected yet, solving seemingly impossibly tangled quantum wave equations and operator equations (which he also invented or greatly developed) often on pure intuition, seemingly as easily as most physicists can solve school physics problems . and did it all, with the help of some old envelopes for writing out, and a few physics textbooks from 1850-1910, and much of this by age 26.&lt;/p&gt;&#xA;&lt;p&gt;A Ramanujan would be someone who, in 1920, could be notionally standing in front of Dirac, Schroedinger, Feynman, Wienberg, Higgs and a hundred others (if alive then), and say &lt;em&gt;&amp;quot;It was obviously a group, and it came to me that all but gravity are governed by SU(3) × SU(2) × U(1), including a mass-generating field and mechanism, and it predicts a particle i call a neutron as well as these others I call bosons, quarks, leptons, and their properties and interactions.... &amp;quot;&lt;/em&gt;, having also derived or extended the tools needed on the way, such as much of quantum mechanics, quantum field theory, and anything needed on the way as related to groups and operators, including the calculations/arithmetic needed for actual numeric predictions - and then left the sketchiest of densely written notebooks to guide researchers after he died at 32, as to what else he theorised and was working on, and the mathematical and intuitional tools and techniques that had achieved this&lt;/p&gt;&#xA;&lt;p&gt;That would be a Ramanujan.&lt;/p&gt;&#xA;&lt;p&gt;I don't think physics works that way, pure and simple.&lt;/p&gt;&#xA;" OwnerUserId="15118" LastEditorUserId="15118" LastEditDate="2021-08-03T08:51:04.850" LastActivityDate="2021-08-03T08:51:04.850" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13512" PostTypeId="2" ParentId="13504" CreationDate="2021-08-03T15:25:32.217" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BE%D0%BB%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B9,_%D0%9A%D0%BE%D0%BD%D1%81%D1%82%D0%B0%D0%BD%D1%82%D0%B8%D0%BD_%D0%AD%D0%B4%D1%83%D0%B0%D1%80%D0%B4%D0%BE%D0%B2%D0%B8%D1%87&quot; rel=&quot;noreferrer&quot;&gt;Konstantin Tsiolkovski&lt;/a&gt; (or Ciolkovskij, or whatever you favourite transliteration...) was an autodidact, worked as a high school teacher, his first major work was a kinetic theory of gases (25 years too late), later he focused on rocketry and astronautics (where his work - often self-published - has been mostly ignored for many years). He became really famous only towards the end of his life, and mostly in the USSR, less in the rest of the world.&lt;/p&gt;&#xA;" OwnerUserId="15121" LastActivityDate="2021-08-03T15:25:32.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13513" PostTypeId="2" ParentId="13504" CreationDate="2021-08-03T17:08:01.823" Score="5" Body="&lt;p&gt;Archimedes comes to mind.&lt;/p&gt;&#xA;&lt;p&gt;I suppose I can't count Aristotle, since he attended Plato's school.&lt;/p&gt;&#xA;" OwnerUserId="10312" LastActivityDate="2021-08-03T17:08:01.823" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13514" PostTypeId="2" ParentId="13504" CreationDate="2021-08-03T20:21:26.163" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Freeman_Dyson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Freeman Dyson&lt;/a&gt; could be considered a more recent example. He became professor of physics at Cornell in 1951 despite never having earned a PhD. He had earned a BA at Cambridge just after WW2, and I believe he obtained a Master's degree at some point too. While these two degrees definitely count as 'formal education', it is generally thought that it's the PhD that does the most towards preparing someone for academic life. In this regard, the granting of a professorship without a PhD is highly unusual.&lt;/p&gt;&#xA;&lt;p&gt;Freeman Dyson went on to contribute to a multitude of areas, with numerous papers on mathematics before he reached the age of 20. He worked on many high concept space-related ideas, such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Orion&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyson_sphere&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dyson sphere&lt;/a&gt;. While he was never involved in the Manhattan Project, he worked with many who did, and was later involved in writing papers on strategy and use of nuclear weapons. He died 18 months ago.&lt;/p&gt;&#xA;" OwnerUserId="10028" LastEditorUserId="10028" LastEditDate="2021-08-03T22:35:55.397" LastActivityDate="2021-08-03T22:35:55.397" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13516" PostTypeId="1" CreationDate="2021-08-05T04:50:11.780" Score="1" ViewCount="332" Body="&lt;p&gt;The wikipedia article about Folium of Descartes says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Descartes challenged Pierre de Fermat to find the tangent line to the&#xA;curve at an arbitrary point since Fermat had recently discovered a&#xA;method for finding tangent lines. Fermat solved the problem easily,&#xA;something Descartes was unable to do.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I want to know how Fermat found the tangent line and where it is written.&lt;/p&gt;&#xA;" OwnerDisplayName="user14400" LastEditorDisplayName="user14400" LastEditDate="2021-08-05T14:53:10.567" LastActivityDate="2021-08-05T14:53:10.567" Title="How did Fermat find tangent lines of Folium of Descartes?" Tags="|mathematics|mathematicians|fermat|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13517" PostTypeId="1" CreationDate="2021-08-05T08:26:58.500" Score="1" ViewCount="101" Body="&lt;p&gt;It's commonly encountered in in textbooks nowadays, e.g. in Jowett &amp;amp; O'Donnell, &lt;em&gt;Propaganda &amp;amp; Persuasion&lt;/em&gt;. But I'm not sure who coined this distinction. I found it mentioned quite explicitly in a 1949 &lt;a href=&quot;https://www.jstor.org/stable/2086855&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; of Howard [P.] Becker, so it's quite an old typology, but Becker neither seems to claim to have devised nor cites someone else for it...  (In fact, he criticized the notion of &amp;quot;gray&amp;quot; propaganda in that article, which strongly suggests he didn't originate it.)&lt;br /&gt;&#xA;Becker's Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Howard_P._Becker&quot; rel=&quot;nofollow noreferrer&quot;&gt;bio&lt;/a&gt; doesn't mention this typology as a contribution of his, so I'm guessing he wasn't the one who came up with it. He, in fact, writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The division between the two was recognized during the recent war; the Office of War Information (OWI) was restricted to white, and the Office of Strategic Services (OSS) to black.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which suggests it might have originated in the division of labor between these services, but again, that's not terribly clear. So, is there a traceable authoriship or &amp;quot;priority claim&amp;quot; for this typology of propaganda, either in these precise terms or similar ones?&lt;/p&gt;&#xA;" OwnerUserId="1885" LastEditorUserId="1885" LastEditDate="2021-08-05T08:42:31.717" LastActivityDate="2021-08-05T08:42:31.717" Title="Who coined the distinction, or at least the terminology of white/black/grey propaganda?" Tags="|social-sciences|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13518" PostTypeId="1" AcceptedAnswerId="13519" CreationDate="2021-08-05T16:58:56.307" Score="21" ViewCount="2607" Body="&lt;p&gt;There is one historic black and white portrait of Évariste Galois that is often used [1]. However there are a number of more recent colorful portraits that imagine him to be anything from straw blond[2], things in between[3], brown[4] and black[5].&lt;/p&gt;&#xA;&lt;h3&gt;Is there any historical source that describes his hair color?&lt;/h3&gt;&#xA;&lt;h3&gt;Are there arguments to that would suggest that in [1] his hair is not only black because of the medium such as that other hair colors like brown and blond were historically portrayed differently in that medium?&lt;/h3&gt;&#xA;&lt;p&gt;A sourced answer to any of these two questions would suffice. If there is indirect information (such as the color of the hair of his parent is verifiable to a similar standard of proof as asked for above) that is welcome too.&lt;/p&gt;&#xA;&lt;p&gt;I am aware that modern recoloration are not a reliable source but the variety of hair color he is portrayed with suggests this fact is not know or easily googleable.&lt;/p&gt;&#xA;&lt;p&gt;[1] &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Evariste_galois.jpg&quot; rel=&quot;noreferrer&quot;&gt;https://commons.wikimedia.org/wiki/File:Evariste_galois.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[2] &lt;a href=&quot;https://www.deviantart.com/selunec/art/Evariste-Galois-189586719&quot; rel=&quot;noreferrer&quot;&gt;https://www.deviantart.com/selunec/art/Evariste-Galois-189586719&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[3] &lt;a href=&quot;https://pl.mathigon.org/timeline/galois&quot; rel=&quot;noreferrer&quot;&gt;https://pl.mathigon.org/timeline/galois&lt;/a&gt; or &lt;a href=&quot;https://www.deviantart.com/autumn-sacura/art/Evariste-Galois-168556089&quot; rel=&quot;noreferrer&quot;&gt;https://www.deviantart.com/autumn-sacura/art/Evariste-Galois-168556089&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[4] &lt;a href=&quot;https://fineartamerica.com/featured/evariste-galois-granger.html?product=yoga-mat&quot; rel=&quot;noreferrer&quot;&gt;https://fineartamerica.com/featured/evariste-galois-granger.html?product=yoga-mat&lt;/a&gt; (this seems to be an coloration of [5]) or &lt;a href=&quot;https://www.deviantart.com/daga-ra/art/Evariste-Galois-283582290&quot; rel=&quot;noreferrer&quot;&gt;https://www.deviantart.com/daga-ra/art/Evariste-Galois-283582290&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[5] &lt;a href=&quot;https://www.alamy.de/stockfoto-evariste-galois-1811-1832-franzosischer-mathematiker-leider-starb-er-in-einem-duell-im-alter-von-20-jahren-datum-1832-105254851.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.alamy.de/stockfoto-evariste-galois-1811-1832-franzosischer-mathematiker-leider-starb-er-in-einem-duell-im-alter-von-20-jahren-datum-1832-105254851.html&lt;/a&gt; (again a black and white picture in a different style) or &lt;a href=&quot;https://www.assignmentpoint.com/arts/biography/biography-of-evariste-galois.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.assignmentpoint.com/arts/biography/biography-of-evariste-galois.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15140" LastEditorUserId="15140" LastEditDate="2021-08-05T17:58:45.807" LastActivityDate="2022-07-13T09:20:51.840" Title="What hair color did Évariste Galois have?" Tags="|biographical-details|galois|france|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13519" PostTypeId="2" ParentId="13518" CreationDate="2021-08-05T20:27:35.253" Score="42" Body="&lt;p&gt;Galois was imprisoned twice, at Sainte-Pélagie prison in 1831 and at La Force prison in 1832. The prison records are apparently retained in French archives.&lt;/p&gt;&#xA;&lt;p&gt;His prison record from Sainte-Pélagie is reproduced (as text only) in the following article, a &lt;a href=&quot;http://www.numdam.org/item/10.24033/asens.427.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt; of which is available from NUMDAM:&lt;/p&gt;&#xA;&lt;p&gt;P. Dupuy, &amp;quot;La vie d’Évariste Galois&amp;quot;, &lt;em&gt;Annales scientifiques de l’É.N.S. 3e série&lt;/em&gt;, tome 13 (1896), p. 197-266.&lt;/p&gt;&#xA;&lt;p&gt;On page 264 of the article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cheveux ..... chatains&lt;br /&gt;&#xA;Sourcils .... id.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The color of his hair is described as chestnut, and the color of his eyebrows identical to the hair color. An &lt;a href=&quot;https://images.math.cnrs.fr/Galois-a-La-Force.html?lang=fr&quot; rel=&quot;noreferrer&quot;&gt;online article&lt;/a&gt; shows a cropped picture of Galois's second prison record:&lt;/p&gt;&#xA;&lt;p&gt;Olivier Courcelle, &amp;quot;Galois à La Force&amp;quot;, &lt;em&gt;CNRS Images des mathématiques&lt;/em&gt;, January 22, 2017&lt;/p&gt;&#xA;&lt;p&gt;In the picture shown at the top of the webpage:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;cheveux et sourcils  chatains-bruns&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Both hair and eyebrows are described as having chestnut-brown color.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2021-08-05T20:27:35.253" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13520" PostTypeId="1" AcceptedAnswerId="13522" CreationDate="2021-08-05T22:00:19.893" Score="13" ViewCount="940" Body="&lt;p&gt;Just curious: when (approximately) was the first scientific conference which contained a poster session? Wikipedia does not give an answer.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2022-02-27T15:21:57.823" LastActivityDate="2022-02-27T15:21:57.823" Title="History of poster sessions" Tags="|untagged|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13521" PostTypeId="2" ParentId="13520" CreationDate="2021-08-06T00:43:29.267" Score="1" Body="&lt;p&gt;The earliest scientific conference with a poster session I have been able to find so far is summarized in:&lt;/p&gt;&#xA;&lt;p&gt;Lee P. Herrington and Gordon M. Heisler, &amp;quot;Conference summary&amp;quot;, &lt;em&gt;Bulletin of the American Meteorological Society&lt;/em&gt;, Vol. 57, No. 6., June 1976, pp. 696-699 (&lt;a href=&quot;https://journals.ametsoc.org/downloadpdf/journals/bams/57/6/1520-0477-57_6_696.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;This is a summary of: Conference on metropolitan physical environment: vegetation, space, and structure for human amenities, 25-29 August 1975, U.S. Department of Agriculture Syracuse, N.Y.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;About one-third of the papers presented at the conference were &amp;quot;poster&amp;quot; presentations. Rather than losing time in coffee breaks, part of each halfday session was devoted to this type of presentation. Authors prepared posters (3 ft X 5 ft) outlining the major points of their work. The &amp;quot;audience&amp;quot; moved from poster to poster as individuals, entering into discussions with the poster authors as interest dictated. Both participants and audience found this to be an effective way to exchange information. In our opinion, all AMS conferences should consider this highly efficient means of encouraging scientific interchange.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2021-08-06T00:43:29.267" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13522" PostTypeId="2" ParentId="13520" CreationDate="2021-08-06T03:34:08.540" Score="9" Body="&lt;p&gt;The paper &lt;a href=&quot;https://teca.unibo.it/article/download/13137/12727/47897&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Lire debout. Le poster comme pratique de lecture dans le monde scientifique.&lt;/em&gt;&lt;/a&gt; writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Les «premiers» posters ou des produits fort&#xA;proches furent présentés dès 1967 pour le moins au Medical Research&#xA;Center (Carshalton), institut majeur de la recherche britannique. C’étaient&#xA;des «feuilles préparées» (prepared cards) sur lesquelles des chercheurs&#xA;regroupaient des résultats, surtout des graphiques et des tableaux – en&#xA;fait, le matériel qui était usuellement montré par des diapositives; les&#xA;feuilles étaient placées sur des panneaux et les auteurs se tenaient à leurs&#xA;côtés pour une discussion. Ce type de présentation, né dans le domaine de&#xA;la biochimie, gagna toute la discipline, puis dans les années 1975 et&#xA;suivantes, fut introduit dans les congrès internationaux et nationaux de&#xA;toutes les sciences de la vie et de la matière.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first posters appeared at least as early as 1967 at the &lt;a href=&quot;https://academic.oup.com/toxsci/article/62/1/4/1720006&quot; rel=&quot;noreferrer&quot;&gt;Carshalton Medical Research Centre&lt;/a&gt; in Great Britain. These were 'prepared cards' on which researchers&#xA;gathered results, mostly graphs and tables – in fact, the material was usually shown by &lt;a href=&quot;https://en.wikipedia.org/wiki/Slide_show&quot; rel=&quot;noreferrer&quot;&gt;slides&lt;/a&gt;; the leaves were placed on panels and the authors stood at their sides for a discussion. This type of presentation, born in the field of&#xA;biochemistry, won the whole discipline, then in the year 1975 it was introduced at the international and national congresses of all life and matter sciences.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(The &lt;a href=&quot;https://academic.oup.com/toxsci/article/62/1/4/1720006&quot; rel=&quot;noreferrer&quot;&gt;Carshalton Medical Research Centre&lt;/a&gt; is now part of The United Kingdom Medical Research Council)&lt;/p&gt;&#xA;&lt;p&gt;The paper goes on to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Le succès du poster dans le monde scientifique s’explique aisément:&#xA;le nouveau format répondait tout à la fois aux attentes des organisateurs&#xA;de congrès, des auteurs de communications et des participants.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The success of the poster in the scientific world can be easily explained:&#xA;the new format met both the expectations of the organizers conference, authors of papers and participants.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paper continues by noting that from the 1960s, scientific meetings had to face a&#xA;considerable increase in the number of communications related to the strong&#xA;growth of the scientific population and the poster session format provided a suitable response to these increased demands.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2021-08-06T03:58:13.883" LastActivityDate="2021-08-06T03:58:13.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13523" PostTypeId="1" CreationDate="2021-08-06T12:42:24.733" Score="0" ViewCount="508" Body="&lt;p&gt;The atomic theory was already present in ancient Greece. In the nineteenth century it was disputed only to return with a vengeance later on.&lt;br /&gt;&#xA;Likewise the geocentric view was replaced by the helio centric one, to be replaced by Einstein's view which stated that both are equally valid.&lt;br /&gt;&#xA;Are there examples of theories that suffered the same faith? Or only the first part? I'm interested mainly in theories that were introduced from the time the present view had it's main predecessors ready, say from the time of Maxwell. Later FR and QM were introduced but within these confines for sure non-realities were upheld.&lt;/p&gt;&#xA;&lt;p&gt;I ask because I am very conspicious about the Higgs mechanism. It was introduced in the context of the unification of the weak and electromagnetic force. It gives mass to the vector bodons governing the weak force and leaves the photon massles. Four massless particles are thought to exist before the spontaneous symmetry breaking of the Higgs field. And this field has some very weird properties. Its value is zero for non-zero field variables! This is reflected in the Mexican hat potential. The theory introduces ghost particles on top of that. If that doesn't make one frown... Nevertheless it is considered an integral (integrating) psrt of the standard model.In the future, this theory will be seen as another kind of phlogiston, for sure.&lt;/p&gt;&#xA;&lt;p&gt;The theory can be replaced by a more fundamental though. There are equal amounts of matter and antimatter (only an asymmetric distribution) in that theory and only two elementary particles (you can't have less). There is no need for some magic mechanism and the weak force is just a residue one.&lt;/p&gt;&#xA;&lt;p&gt;So. It would be comforting to know if the Higgs mechanism (not the particle) could turn out to be not true.&lt;/p&gt;&#xA;&lt;p&gt;I mention the Higgs mechanism because it's a present day phlogiston and contrary to phlogiston unprovable.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="6229" LastEditDate="2022-02-02T18:15:57.660" LastActivityDate="2022-02-02T18:15:57.660" Title="What are examples of theories in recent history of science and mathematics that were hold pretty long but turned out non-true?" Tags="|philosophy-of-science|" AnswerCount="4" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="13524" PostTypeId="1" CreationDate="2021-08-06T13:38:16.847" Score="4" ViewCount="173" Body="&lt;p&gt;Time has been mechanized already by sundials. These nails and chains that are banged in and put around time have been increasingly refined to accumulate in the advent of the atomic clock. So-called Natural units were even invented. Without mechanized time the world would have looked very different. Maybe we have some kind of marshmallow minds. Maybe we could better wear aikhornnuts on our wrists as Krznaric puts it.&lt;/p&gt;&#xA;&lt;p&gt;When were the first mechanized timekeepers used in science? Was Galileo the first one? Did this use further science a lot? So newer means and more precise keepers could be constructed?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-08-07T00:12:01.207" LastActivityDate="2021-08-27T00:15:47.993" Title="When were clocks used for the first time in science?" Tags="|time|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13525" PostTypeId="2" ParentId="13524" CreationDate="2021-08-06T14:47:42.983" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Water_clock&quot; rel=&quot;nofollow noreferrer&quot;&gt;Water clock&lt;/a&gt; is also a kind of &amp;quot;mechanized clock&amp;quot;, and it can be very sophisticated. There is a strong evidence that in ancient Babylon&#xA;and Greece they were used in science, namely for astronomy. In Ptolemy, we can&#xA;read records of some astronomical events, like lunar eclipses, which happened during a night, and their time is recorded to a fraction of an hour (like 1/2 or 1/3 of an hour). This indicates that a water clock probably had to be used to time them. What else? Ptolemy mentions water clock only once in his book, (Chap. V, sect 14) only to say&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Of all methods of this investigation... we rejected those using water clock...since these methods are not sufficiently precise&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This seem to indicate that other investigators did use them.&lt;/p&gt;&#xA;&lt;p&gt;Edit. From the Wikipedia article cited above, I learned that &lt;a href=&quot;https://en.wikipedia.org/wiki/Herophilos&quot; rel=&quot;nofollow noreferrer&quot;&gt;Herophilos&lt;/a&gt; used a clepsydra to measure the pulse rate of his patients,&#xA;which can be also considered as &amp;quot;scientific purpose&amp;quot;. The same article mentions several astronomers who constructed and used clepsydras.&lt;/p&gt;&#xA;&lt;p&gt;As I said, Ptolemy was not satisfied with the accuracy of clepsydras for astronomical purposes. Weight driven clocks of the Middle age were not more accurate. The first breakthrough in accuracy came when Galileo invented the first clock regulator, that is the pendulum.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-27T00:15:47.993" LastActivityDate="2021-08-27T00:15:47.993" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13526" PostTypeId="2" ParentId="13523" CreationDate="2021-08-06T16:21:01.140" Score="6" Body="&lt;p&gt;Massless neutrinos. Mostly certainly thought as massless since the 50s (even if some like scientists like Pontecorvo were against it since the beginning). Many people were even describing neutrino physics using Weyl equation (the equation for massless fermions).&#xA;Neutrinos were definetely proved to be not massless in 1998 with Super Kamiokande.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-08-06T19:34:11.413" LastActivityDate="2021-08-06T19:34:11.413" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13527" PostTypeId="2" ParentId="13523" CreationDate="2021-08-06T21:18:19.200" Score="4" Body="&lt;p&gt;The  Cosmological constant, according to Wikipedia with refs there in:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From the 1930s until the late 1990s, most physicists assumed the&#xA;cosmological constant to be equal to zero.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the 1990's&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;astrophysicists carried out measurements on distant supernovae and&#xA;showed that the speed of galaxies recession in relation to the Milky&#xA;Way increases over time. The universe is in accelerated expansion,&#xA;which requires having a strictly positive Λ.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15147" LastActivityDate="2021-08-06T21:18:19.200" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13528" PostTypeId="2" ParentId="13523" CreationDate="2021-08-07T02:51:01.227" Score="2" Body="&lt;p&gt;The fine structure constant &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; which was believed to be &lt;span class=&quot;math-container&quot;&gt;$\sim \frac{1}{137}$&lt;/span&gt; (and was taught this way in most physics courses till very recently) is now believed to vary with time. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Fine-structure_constant#Potential_Time-Variation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Fine-structure_constant#Potential_Time-Variation&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-08-07T02:51:01.227" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13529" PostTypeId="1" CreationDate="2021-08-07T07:33:03.327" Score="4" ViewCount="277" Body="&lt;p&gt;I find quite interesting the choice for the shapes drawn into the sphere that resembles continents.&lt;/p&gt;&#xA;&lt;p&gt;Was this choice arbitrary or do we know if there is some justification behind it?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qjgkv.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qjgkv.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12712" LastEditorUserId="12712" LastEditDate="2021-08-07T07:41:17.273" LastActivityDate="2021-10-21T23:03:49.680" Title="In Newton's cannon image, where do the shapes drawn into the sphere come from?" Tags="|physics|astronomy|newton|gravity|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13530" PostTypeId="1" CreationDate="2021-08-07T17:25:35.133" Score="1" ViewCount="110" Body="&lt;p&gt;I recall there being two Russian math prodigies who published a joint paper in the Crelle's journal at the age of 18 or so. I think they lived in the XIX century. What were their names?&lt;/p&gt;&#xA;&lt;p&gt;I can't remember and my description may be a little off.&lt;/p&gt;&#xA;" OwnerUserId="15151" LastEditorUserId="4251" LastEditDate="2021-09-16T06:43:40.357" LastActivityDate="2021-09-16T06:43:40.357" Title="XIX century Russian math prodigies who published in Crelle" Tags="|mathematics|mathematicians|19th-century|russia|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13531" PostTypeId="1" CreationDate="2021-08-07T23:11:52.763" Score="-1" ViewCount="226" Body="&lt;p&gt;We can read Wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s, but the terms fractal and fractal dimension were coined by mathematician Benoit Mandelbrot in 1975.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am always fascinated by fractal dimensions. Infinity in a finite plane. An infinite line (plane?) fit into a small area (volume?). It is always said that Mandelbrot invented them in 1975. Now I read that already in the 17th century they were given thought. But where and who gave it thought?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-08-08T06:37:30.377" LastActivityDate="2022-03-12T17:21:21.570" Title="Did fractals already exist in the 17th century?" Tags="|mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13532" PostTypeId="1" AcceptedAnswerId="13547" CreationDate="2021-08-07T23:39:53.373" Score="4" ViewCount="268" Body="&lt;p&gt;The classical Schwarz lemma from one-variable complex analysis states that a holomorphic map &lt;span class=&quot;math-container&quot;&gt;$f : \Delta(r) \to \Delta(R)$&lt;/span&gt; between two disks in the complex plane such that &lt;span class=&quot;math-container&quot;&gt;$f(0)=0$&lt;/span&gt; satisfies &lt;span class=&quot;math-container&quot;&gt;$$|f(z)| \leq \frac{R}{r} | z |$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have access to the original paper (I assume is due to Hermann Schwarz) that presents this? I'd be interested in the original proof of the result.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://www.ams.org/notices/199908/fea-osserman.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Osserman&lt;/a&gt; The proof is given in &lt;em&gt;Gesammelte Mathematische Abhandlungen. II, Springer-Verlag, Berlin, 1890.&lt;/em&gt; I've checked the reference, and used a translator on a number of the pages Osserman specifically references, but can't find the exact place where the Schwarz lemma is stated (and proved).&lt;/p&gt;&#xA;" OwnerUserId="15153" LastEditorUserId="136" LastEditDate="2022-11-18T23:44:53.690" LastActivityDate="2022-11-18T23:44:53.690" Title="Original proof of the Schwarz lemma" Tags="|mathematics|reference-request|complex-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13533" PostTypeId="2" ParentId="13531" CreationDate="2021-08-08T02:44:19.043" Score="6" Body="&lt;p&gt;The term &amp;quot;fractal&amp;quot; was coined by B. Mandelbrot (1970s).&#xA;But these objects were known to mathematicians for long time. The first definition of dimension which can take non-integer values is due to Hausdorff,&lt;/p&gt;&#xA;&lt;p&gt;Hausdorff, F.&#xA;Dimension und äußeres Maß. (German) JFM 46.0292.01&#xA;Math. Ann. 79, 157-179 (1918).&lt;/p&gt;&#xA;&lt;p&gt;Some first mathematical examples of what is called fractals nowadays are Weierstrass continuous nowhere differentiable function, which was discovered in 1872 but not published (Weierstrass rarely published his results, preferring to communicate them in lectures and talks).&lt;/p&gt;&#xA;&lt;p&gt;Another early example was a &lt;a href=&quot;https://en.wikipedia.org/wiki/Cantor_set#Historical_remarks&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cantor-like set&lt;/a&gt; occurring in the work&#xA;of &lt;a href=&quot;https://en.wikipedia.org/wiki/Friedrich_Schottky&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schottky&lt;/a&gt; (some time in 1880s, before Cantor).&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Julia_set&quot; rel=&quot;nofollow noreferrer&quot;&gt;Julia sets&lt;/a&gt; were introduced independently by Fatou and Julia approximately in 1918.&#xA;Of course nobody knew that these examples have fractional dimension until&#xA;Hausdorff introduced his notion of dimension.&lt;/p&gt;&#xA;&lt;p&gt;By the way, exact value of dimension of the graph of Weierstrass function&#xA;is still unknown.&lt;/p&gt;&#xA;&lt;p&gt;I do not think that any sets of fractional dimension occurred in mathematics&#xA;before this example of Weierstrass. And to my knowledge no mathematician discussed fractional dimensions before Hausdorff.&#xA;Of course some fractal-like patterns that exist in the nature drew attention of physicists, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Lichtenberg_figure&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lichtenberg figure&lt;/a&gt; and philosophers, for example there is a famous description&#xA;of an (imaginary) fractal &lt;a href=&quot;https://en.wikipedia.org/wiki/Indra%27s_Pearls_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;in an old Buddhist text&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="12330" LastEditDate="2022-03-12T17:21:21.570" LastActivityDate="2022-03-12T17:21:21.570" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13535" PostTypeId="2" ParentId="13523" CreationDate="2021-08-08T09:13:58.383" Score="0" Body="&lt;p&gt;An example from mathematics is that it was generally believed that the logarithmic integral was always greater than the prime counting function until Littlewood proved that &lt;span class=&quot;math-container&quot;&gt;$\operatorname{li}(x)−\pi(x)$&lt;/span&gt; changed signs infinitely often. The first change occurs for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; approximately &lt;span class=&quot;math-container&quot;&gt;$10^{316}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thanks to @Nick who answered in a comment. That was before I included math. It's an example which shows that quite the opposite can be true of what was thought. It shows the hugeness of infinity. I wonder how it could be actually has been seen that for such a huge number it's&#xA;untrue.&lt;/p&gt;&#xA;&lt;p&gt;The steady-state universe was a model still believed to be viable in the fifties. Hoyle thought it to be true. But it isn't.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="6229" LastEditDate="2021-08-09T02:14:49.517" LastActivityDate="2021-08-09T02:14:49.517" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13537" PostTypeId="2" ParentId="13531" CreationDate="2021-08-08T16:51:22.750" Score="3" Body="&lt;p&gt;Maybe this is what Wikipedia meant by &amp;quot;1600s&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;B. B. Mandelbrot, in his book &lt;em&gt;The Fractal Geometry of Nature&lt;/em&gt;,&#xA;has some historical remarks in the last chapter, including:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The idea of fractional differentiaton had occurred to Leibniz as soon as he had developed his version of calculus and invented the notations &lt;span class=&quot;math-container&quot;&gt;$\frac{d^k F}{dx^k}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\frac{d}{dx})^kF$&lt;/span&gt; ... Leibniz's letter to de l'Hospital dated September 30, 1693 ... “John Bernoulli seems to have told you of my having mentioned to him a marvelous analogy which makes it possible to say in a way that successive differentials are in geometric progression.  One can ask what would be a differential having as its exponent a fraction.  You see that the result can be expressed by an infinite series.  Although this seems removed from Geometry, which does not yet know of such fractional exponents, it appears that one day these paradoxes will yield useful consequences, since there is hardly a paradox without utility.  Thoughts that mattered little in themselves may give occasion to more beautiful ones.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-08-08T16:51:22.750" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13538" PostTypeId="2" ParentId="3131" CreationDate="2021-08-08T18:04:02.873" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Kingdon_Clifford&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Kingdom Clifford&lt;/a&gt; suffered from a nervous breakdown and exhaustion on at least two occasions, almost certainly due to overwork, before passing away at the age of 33 from tuberculosis on the island of Medeira, Portugal,  while recovering from a collapse.&lt;/p&gt;&#xA;&lt;p&gt;Clifford produced the correct algebraic description of three dimensional Cartesian vectors by combining the rotational algebra of complex numbers and Hamilton's quaternions with Gibbs and Heaviside's reworking of quaternions, and by building on the ideas of Grassman.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2021-08-08T18:04:02.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13539" PostTypeId="1" CreationDate="2021-08-09T04:20:39.567" Score="1" ViewCount="131" Body="&lt;p&gt;At what time people understood or realized that there are changes in times between countries and continental, and when it's a day in one place it's a night in another place? Allegedly, It makes sense that in the past people thought it's the same time everywhere.&lt;/p&gt;&#xA;" OwnerUserId="12313" LastActivityDate="2021-08-10T16:49:12.070" Title="At what time people relized the changes in times between countries?" Tags="|time|" AnswerCount="1" CommentCount="1" ClosedDate="2021-08-09T20:41:24.460" ContentLicense="CC BY-SA 4.0" />
  <row Id="13540" PostTypeId="1" AcceptedAnswerId="13542" CreationDate="2021-08-09T10:10:20.390" Score="2" ViewCount="104" Body="&lt;p&gt;I remember having read a statement, I think by one of the founding fathers of quantum mechanics, that we will probably never be able to control (or manipulate or see or isolate...?) a single quantum system (or atom).&lt;/p&gt;&#xA;&lt;p&gt;The quote was used to highlight the fast progress experimental quantum physics has made over the last few decades (seeing that single-particle control is routinely achieved in experiments).&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know the quote and its source?&lt;/p&gt;&#xA;" OwnerUserId="15156" LastEditorUserId="10106" LastEditDate="2022-02-01T10:30:09.117" LastActivityDate="2022-02-01T10:30:09.117" Title="Need quote from early 20th century about humanity probably never being able to control (manipulate, see?) a single quantum system" Tags="|physics|reference-request|quantum-mechanics|quote|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13541" PostTypeId="1" CreationDate="2021-08-09T11:01:16.087" Score="1" ViewCount="260" Body="&lt;p&gt;This problem is giving me headaches for quite some time now I guess mainly because I am not a mathematician, but I would very much like to know if there is anyone here that has any deeper insight into this.&lt;/p&gt;&#xA;&lt;p&gt;So, the Papyrus translation is given &lt;a href=&quot;https://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrus#Two_geometry_problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Additionally there is a discussion presented &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086009000305&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; on whether the translation of the object in question is correct. They say it could possibly refer to the area of a semi-cylinder.&lt;/p&gt;&#xA;&lt;p&gt;That said, I am wondering if anyone knows how could these ancient Egyptians have derived this algorithm? There is a fraction omnipresent throughout this algorithm (1/9). What could that fraction represent? Is there any way we could explain this algorithm intuitively? I guess we can 'assume' that they didn't know calculus, so how?&lt;/p&gt;&#xA;&lt;p&gt;P.S. I have asked this question already on Stack Exchange, but without satisfying results.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;So to be more specific. The algorithm obviously works ignoring the small error compared to our current formula. They were very practical and didn't have sophisticated mathematical knowledge. So this is why I essentially want that whoever answers this forgets about our modern mathematics and thinks exclusively in the context of this algorithm and its operations to somehow make sense of this.&lt;/p&gt;&#xA;" OwnerUserId="15021" LastEditorUserId="15021" LastEditDate="2021-08-13T11:39:22.140" LastActivityDate="2021-08-13T11:39:22.140" Title="Analysis of Moscow Mathematical Papyrus problem no. 10" Tags="|geometry|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13542" PostTypeId="2" ParentId="13540" CreationDate="2021-08-09T12:28:34.520" Score="3" Body="&lt;p&gt;Found it:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;[...] we never experiment with just one electron or atom or (small) molecule. In thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences [...]&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In: Schrödinger, E. (1952). Are There Quantum Jumps? Part II. The British Journal for the Philosophy of Science, 3(11), 233-242. Retrieved August 9, 2021, from &lt;a href=&quot;http://www.jstor.org/stable/685266&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/685266&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15156" LastActivityDate="2021-08-09T12:28:34.520" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13543" PostTypeId="2" ParentId="13539" CreationDate="2021-08-09T15:37:13.927" Score="0" Body="&lt;p&gt;Different people disovered or realized that time changes with longitude at different times.  For most of history most people never traveled more than a few dozen miles from their birth places.  The rare people who travelled hundreds or thousands of miles travelled so slowly they wouldn't notice time differences.  So the idea of time zoness only became well known in recent times.&lt;/p&gt;&#xA;&lt;p&gt;When people discovered the earth was spehrical - knowledge that took centuries or millennia to become known to the majority of people - they realized that there were different times of day at different longitudes.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to a similar question and its answers.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://history.stackexchange.com/questions/64471/when-were-time-zones-or-at-least-time-difference-discovered/64487#64487%5B1%5D&quot;&gt;https://history.stackexchange.com/questions/64471/when-were-time-zones-or-at-least-time-difference-discovered/64487#64487[1]&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2021-08-10T16:49:12.070" LastActivityDate="2021-08-10T16:49:12.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13544" PostTypeId="1" CreationDate="2021-08-09T23:06:20.260" Score="1" ViewCount="542" Body="&lt;p&gt;Le Sage's theory described gravity by means of corpuscules, small particles pervading space that hit masses all around. Between two masses a shadow comes about. None of them are present there. Masses form an umbrella for these particles, and this umbrella forms the shadow. As a result, the masses are hit more on one side so the masses gain momentum and accelerate towards each other. It's similar somehow to the workings of gravitons.&lt;/p&gt;&#xA;&lt;p&gt;For a long time the theory was taken seriously. It could compete with Newtonian gravity and even supplied an explanation of gravity. Even modern theories don't have an explanation, merely a description.&lt;/p&gt;&#xA;&lt;p&gt;What was the deathblow for the theory? I can remember vaguely that Lorenz (or someone else, I am not sure) made calculations regarding the impact of solar energy and the impact needed for these corpuscules to deflect Earth around the Sun.&lt;/p&gt;&#xA;&lt;p&gt;What can be said about this? Can the theory still be viable?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="4120" LastEditDate="2021-08-09T23:18:21.213" LastActivityDate="2023-08-04T13:51:48.367" Title="What was the decisive argument against Le Sage's theory of gravity?" Tags="|physics|gravity|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13545" PostTypeId="1" CreationDate="2021-08-10T01:49:31.163" Score="4" ViewCount="82" Body="&lt;p&gt;I'm looking for a vignette about von Neumann. My recollection is as follows but my google-fu is not bringing up anything.&lt;/p&gt;&#xA;&lt;p&gt;von Neumann, having prodigious mathematical capacity even of the most mundane kind, could effortlessly interconvert between binary and decimal numbers — in his head!&lt;/p&gt;&#xA;&lt;p&gt;His students were of course not thus endowed and struggled.&#xA;One of them finally bit the bullet and wrote software that would do this.&lt;/p&gt;&#xA;&lt;p&gt;JvN got angry and said (something like) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Computers are super expensive equipment. And you use them for such mundane clerical affairs?!  Under my (lab/supervision/whatever) this will not be allowed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clearly for JvN computers were meant for computing exclusively in the classical numerical sense.&lt;/p&gt;&#xA;" OwnerUserId="12727" LastEditorUserId="12727" LastEditDate="2021-08-10T02:03:59.857" LastActivityDate="2021-08-10T02:03:59.857" Title="von Neumann vignette on early computer usage" Tags="|reference-request|computers|von-neumann|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13546" PostTypeId="1" CreationDate="2021-08-10T08:17:50.167" Score="2" ViewCount="351" Body="&lt;p&gt;I was interested in reading his work, but I couldn't find a translation in google, is there any? I can understand Spanish and English&lt;/p&gt;&#xA;" OwnerUserId="15158" LastActivityDate="2021-08-10T08:17:50.167" Title="Is there a translation to Lagrange's Réflexions sur la résolution algébrique des équations?" Tags="|mathematics|translations|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13547" PostTypeId="2" ParentId="13532" CreationDate="2021-08-10T09:44:13.580" Score="6" Body="&lt;p&gt;The following paragraph in &lt;a href=&quot;https://arxiv.org/abs/1001.0559&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Julius and Julia: mastering the art of the Schwarz lemma&amp;quot;&lt;/a&gt; by Professor Harold P. Boas (&lt;em&gt;Amer. Math. Monthly&lt;/em&gt;, &lt;strong&gt;117&lt;/strong&gt; (2010), no. 9, pp. 770-785) might be relevant:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now the special case of the lemma that Schwarz handled appears in the notes [41, pp. &lt;strong&gt;109-111&lt;/strong&gt;] from his lectures during 1869-70, but as far as I know, these notes did not see print until the publication of Schwarz's collected works in 1890...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Naturally, item 41 in the list of references of this paper is the second volume of Schwarz's &lt;em&gt;Gesammelte Mathematische Abhandlungen&lt;/em&gt; (Springer, Berlin, 1890).&lt;/p&gt;&#xA;" OwnerUserId="136" LastActivityDate="2021-08-10T09:44:13.580" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13548" PostTypeId="1" CreationDate="2021-08-10T10:38:18.143" Score="2" ViewCount="301" Body="&lt;p&gt;In &lt;a href=&quot;https://www.leydesdorff.net/vonneumann/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article, about an unfinished book by von Neumann (he died in 1957 before he could finish it), one can read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After stating that he himself is neither a neurologist nor a psychiatrist, but a mathematician, Von Neumann embarks upon the first part of this essay by explaining the components of a computer in the language of a computer scientist. First, the difference between analog and digital computers is explained. This distinction will have some relevance for the discussion of the brain because—as stated in the second part—the brain can prima facie be considered as a digital computer. However, upon further reflection, some elements of analog computing (e.g., the chemistry) will also become relevant in understanding the functioning of the brain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is he saying the brain is an analogue computer? What does he even mean with that?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-08-10T23:24:47.560" Title="Did von Neumann consider the brain an analogue computer?" Tags="|computers|von-neumann|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13549" PostTypeId="1" CreationDate="2021-08-10T23:01:13.537" Score="1" ViewCount="109" Body="&lt;p&gt;LeSage proposed a theory of gravity that describes as well as explains gravity. Space is filled with corpuscles that push masses on all side. As a result two masses will start to move to each other because they are pushed on one side more than on the other.&lt;/p&gt;&#xA;&lt;p&gt;How did he see these particles himself? Like worldly particles? Like extramundane particles? Like gravitons maybe, the source being the masses themselves?&lt;/p&gt;&#xA;" OwnerUserId="4120" LastEditorUserId="12330" LastEditDate="2021-08-11T01:34:01.837" LastActivityDate="2021-08-11T01:34:01.837" Title="How did LeSage look at corpuscles?" Tags="|gravity|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13550" PostTypeId="2" ParentId="13548" CreationDate="2021-08-10T23:24:47.560" Score="1" Body="&lt;p&gt;Von Neumann makes the distinction between chemistry and, well, non-chemistry. In fact, the &lt;em&gt;whole&lt;/em&gt; brain is chemistry. There are no power sources like that of a digital computer. All power is inherent to the processes. There is no external source pushing the events. Nor is there an external program. The fact that there are pulsed currents is the only reason he makes the comparison with digital computers. It &lt;em&gt;looks&lt;/em&gt; as if zeros and ones are manipulated. In a digital cimputer though the ones and zeros are, again, pushed by external potentials. Declaring himself to be no expert on neurons it is understandable he makes the comparison. The spikey potentials in the brain run completely different though from those in a computer.&lt;/p&gt;&#xA;&lt;p&gt;I read in a comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It doesn't matter what it is, it matters how it functions. Digital and analog computers function differently, but can be combined.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This comment shows the same naive view. The processes in the brain &lt;em&gt;all&lt;/em&gt; function differently from a digital computer. The brain fuctions like an analog computer as a whole. An analog of external processes in the external world is internally given form inside huge chemical processes. Huge numbers of neurons play in concert to internally form representations of the world outside. Von Neumann came close. His genius had not enough knowledge of the brain to extend the notion of an analog computer to the whole brain.&lt;/p&gt;&#xA;" OwnerUserId="4120" LastActivityDate="2021-08-10T23:24:47.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13551" PostTypeId="2" ParentId="12887" CreationDate="2021-08-12T03:09:08.203" Score="7" Body="&lt;p&gt;I can provide a partial response to this question. As noted in my earlier comment, the term &amp;quot;Japanese bracket&amp;quot; appears in the 1999 paper&#xA;&lt;a href=&quot;https://www.researchgate.net/profile/Changxing-Miao/publication/233549699_Global_existence_of_small_solutions_to_the_generalized_derivative_nonlinear_Schrodinger_equation/links/567ea25c08ae1e63f1e7d38a/Global-existence-of-small-solutions-to-the-generalized-derivative-nonlinear-Schroedinger-equation.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&amp;quot;Global existence of small solutions to the generalized derivative nonlinear Schrödinger equation&lt;/em&gt;”&lt;/a&gt;'' (p. 135, journal pagination).&lt;/p&gt;&#xA;&lt;p&gt;I contacted two of the authors of this paper,&#xA;and received one response regarding the notation. The terminology predates the paper. The author who responded commented that he had&#xA;heard the term used at several conferences prior to the drafting of the 1999 paper, and that the notation was widely used by Japanese&#xA;analysts at that time. The inference is that the answer to your third question is &amp;quot;due to widespread use by the Japanese mathematical community&amp;quot;.  His response also provides a temporal bound for your first question: no later than the late 1990s.&lt;/p&gt;&#xA;&lt;p&gt;If you're interested in more specifics, I'd recommend pulling on this thread a bit more by directly contacting early users of the notation / terminology.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2021-08-12T03:09:08.203" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13552" PostTypeId="2" ParentId="13544" CreationDate="2021-08-12T14:14:31.123" Score="1" Body="&lt;p&gt;I considered this theory, and concluded it would result in a drag force opposite to the direction of motion so, for example, a planet orbiting the Sun would spiral inward towards the Sun.&lt;/p&gt;&#xA;" OwnerUserId="14904" LastEditorUserId="14904" LastEditDate="2021-08-13T00:55:41.330" LastActivityDate="2021-08-13T00:55:41.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13553" PostTypeId="1" CreationDate="2021-08-13T01:44:43.263" Score="1" ViewCount="61" Body="&lt;p&gt;I understand that every &lt;strong&gt;energy&lt;/strong&gt; in our universe can act as a &lt;em&gt;wave&lt;/em&gt; or as a &lt;em&gt;mass&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If I understand correctly, the scientific and engineeric field of &lt;em&gt;signal processing&lt;/em&gt; generally deals with &lt;strong&gt;signals from&lt;/strong&gt; &lt;em&gt;wave energy&lt;/em&gt;, but were signals from &lt;em&gt;matter energy&lt;/em&gt; ever learnt and if so what would be a typical example?&lt;/p&gt;&#xA;" OwnerUserId="15179" LastActivityDate="2021-08-13T01:44:43.263" Title="Signal Processing research: Was matter ever learnt?" Tags="|experimental-physics|" AnswerCount="0" CommentCount="2" ClosedDate="2021-08-14T08:46:03.427" ContentLicense="CC BY-SA 4.0" />
  <row Id="13554" PostTypeId="1" CreationDate="2021-08-13T04:48:16.903" Score="3" ViewCount="796" Body="&lt;p&gt;I want to know who was the first scholar or mathematician to have introduced and formulate the concept of recurrence relations, that is finding a function given the how one value of a sequence is related to the previous ones.&lt;/p&gt;&#xA;" OwnerUserId="10492" LastEditorUserId="55" LastEditDate="2021-08-13T09:10:26.407" LastActivityDate="2021-08-14T02:57:25.643" Title="Who introduced recurrence relations and sequences?" Tags="|mathematics|functional-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13555" PostTypeId="2" ParentId="13554" CreationDate="2021-08-13T09:09:42.557" Score="10" Body="&lt;p&gt;Depends on what &amp;quot;introduced&amp;quot; means. If we skip arithmetic and geometric series as too simple to suggest something general the idea goes back to Fibonacci and his rabbit breeding problem that sets up a recurrence in &lt;em&gt;Liber Abaci&lt;/em&gt; (1202). Jacob (1564) and Kepler (1618) pointed out that the limit ratios of consecutive terms approach the golden ratio, see &lt;a href=&quot;https://www.bachnetwork.org/ub1/tatlow.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tatlow, The Use and Abuse of Fibonacci Numbers&lt;/a&gt; for the early pre-history. Girard wrote down the recurrence explicitly in 1634, and De Moivre (c. 1718) and Bernoulli (1726) solved it explicitly, i.e. derived what is now called the Binet formula, see &lt;a href=&quot;https://hsm.stackexchange.com/q/2067/55&quot;&gt;With what kind of proof was the Binet formula derived for the first time?&lt;/a&gt; But this was developing general ideas on a concrete example rather than introducing a more abstract concept explicitly.&lt;/p&gt;&#xA;&lt;p&gt;De Moivre's treatise &lt;em&gt;Miscellanea analytica de seriebus et quadraturis&lt;/em&gt; (1730) is sometimes credited as the first systematic study of linear recurrences. He also introduced the idea of a generating function, and derived it for the Fibonacci sequence. But in a different guise, as solving finite difference equations, a general approach appears already in Taylor's treatise &lt;em&gt;Methodus Incrementorum&lt;/em&gt; (1715), see &lt;a href=&quot;https://www.jstor.org/stable/41133765&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feigenbaum, Brook Taylor and the Method of Increments&lt;/a&gt;. Propositions 6-9 cover some methods for solving difference equations, including by infinite series. &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-booles-calculus-of-finite-differences&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boole's Treatise on the Calculus of Finite Differences&lt;/a&gt; (1859) was a continuation of this, but like other literature on finite differences it did not focus on topics usually associated with recurrent sequences as such today.&lt;/p&gt;&#xA;&lt;p&gt;Judging by the historical survey in &lt;a href=&quot;https://archive.org/details/historyoftheoryo01dick/page/392/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dickson's History of the Theory of Numbers, ch. XVII&lt;/a&gt;, many of the modern themes and techniques, especially number theoretic ones, were first put together and systematized in a series of works by Lucas in 1876-78, as applied to the second order sequences. They became the template for treating higher order sequences in later works. A little later, Poincaré initiated a general study of the asymptotics of solutions to difference equations in &lt;em&gt;Sur les equations lineaires aux differentieles et aux difference finies&lt;/em&gt; (1885), see &lt;a href=&quot;http://ramanujan.math.trinity.edu/tumath/research/reports/report58.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abu-Saris et al., Poincaré Types Solutions of Systems of Difference Equations&lt;/a&gt;. In particular, he proved a first general result on the limit ratios of higher order sequences, which generalize the golden ratio.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-08-14T02:57:25.643" LastActivityDate="2021-08-14T02:57:25.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13556" PostTypeId="2" ParentId="13529" CreationDate="2021-08-14T06:40:24.833" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EzIjc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EzIjc.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There seem to be alternative versions of this illustration: this one from page 6 of the Second Edition of On the System of the World [from the ResearchGate site].  I've always just taken this to be a &amp;quot;generic&amp;quot; planet for illustrative purposes.  I've been looking for information on the engraver, in case anything was ever said about the intent of the image.  It is supposed to be based on &amp;quot;the original sketch&amp;quot;, but it wasn't clear if that is Newton's own.&lt;/p&gt;&#xA;&lt;p&gt;For what it's worth, here's an example of a map Newton might have seen as a student.  My own guess is that he may not have wanted to refer to Earth specifically, since he was describing what he took to be a &amp;quot;universal&amp;quot; result.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ejPQC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ejPQC.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user15168" LastEditorDisplayName="user15168" LastEditDate="2021-08-14T06:53:23.930" LastActivityDate="2021-08-14T06:53:23.930" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13557" PostTypeId="1" CreationDate="2021-08-14T11:45:57.817" Score="0" ViewCount="151" Body="&lt;p&gt;Looking at the pefsu problem of the Moscow Mathematical Papyrus &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Moscow_Mathematical_Papyrus#Pefsu_problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; I don't understand why the algorithm takes &lt;span class=&quot;math-container&quot;&gt;$1/2$&lt;/span&gt; of the calculated grain measure to produce beer. Why aren't the &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; heqats multiplied by &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; to get &lt;span class=&quot;math-container&quot;&gt;$20$&lt;/span&gt; quantities of beer which would be a better deal in exchange?&lt;/p&gt;&#xA;" OwnerUserId="15021" LastActivityDate="2021-08-20T12:53:15.027" Title="Egypt pefsu problem" Tags="|algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13558" PostTypeId="1" CreationDate="2021-08-17T03:40:58.770" Score="1" ViewCount="81" Body="&lt;p&gt;I am looking at Euclidean domains for the first time and it is a subject that has caught my attention. I wonder, who was the first to talk about this? I've been reading a little and so far the oldest definition I've seen is that of Van der Waerden, but I don't really know who introduced these ideas and definitions. In advance I appreciate any information you can give me&lt;/p&gt;&#xA;" OwnerUserId="14885" LastActivityDate="2021-08-17T03:40:58.770" Title="Who first spoke of Euclidean domains?" Tags="|reference-request|abstract-algebra|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13560" PostTypeId="2" ParentId="13544" CreationDate="2021-08-17T17:08:47.120" Score="3" Body="&lt;p&gt;Technically, there never was a &amp;quot;deathblow&amp;quot;, since the LeSage Theory is still alive and kicking - but with a very small audience. This should not be too surprising, since there are still strong proponents of the luminiferous ether.&lt;/p&gt;&#xA;&lt;p&gt;However, the most obvious, and strongest problem, is the viscous drag which such a mechanism produces. If the flux has an effective rest condition, in which the net force on an object is zero, then any relative velocity will produce more collisions on the &amp;quot;leading edge&amp;quot; of the body than on the &amp;quot;trailing edge&amp;quot;. This is because the slower particles which would normally impact on the trailing edge will not catch up, so no impact. Likewise, the impacts on the leading edge will be stronger, due to the increased velocity of impact. The result is that any moving object will be slowed down and eventually come to rest. The consequences of this to any planetary system should be obvious.&lt;/p&gt;&#xA;&lt;p&gt;Richard Feynman talks about this in one of his lectures, and this forms the first 4 minutes of &lt;a href=&quot;https://www.youtube.com/watch?v=h53aRqoQDWY&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt; However, note that the rest of the video discusses why the maker thinks Feynman is wrong - the theory is not dead.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=rksKb5T7AFA&quot; rel=&quot;nofollow noreferrer&quot;&gt;Another video&lt;/a&gt; addresses other issues and how physicists considered working around them.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;obvious&amp;quot; concept of the LeSage theory states that each mass reduces the number of flux particles by absorbing them. Maxwell pointed out that the energy released by this absorption would heat the body, and we don't see any such heating. As a matter of fact, all matter in the universe should be quickly vaporized.&lt;/p&gt;&#xA;&lt;p&gt;You can get around this by assuming that the particles are tiny and very fast - but other constraints require that the particles travel much faster than light, and this is not a popular position post-Einstein.&lt;/p&gt;&#xA;&lt;p&gt;You can get around this by assuming that the collisions are only partly inelastic, but then ...&lt;/p&gt;&#xA;&lt;p&gt;And you get around this by ...&lt;/p&gt;&#xA;&lt;p&gt;Well, you see how this goes. The situation is very similar to the ether theory, in which the Michelson-Morley experiment is explained by the concept of ether drag, in which something very much like viscous drag causes the ether at the surface of a planet (hint: the Earth) to become stationary at the surface so no velocity of the ether is observed. That this causes other issues can be dealt with by the application of great ingenuity, and the struggle goes on for those who favor the ether.&lt;/p&gt;&#xA;&lt;p&gt;It should be noted that, as the technology has improved, the Michelson-Morley experiment, or its equivalent, has been &lt;a href=&quot;https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;repeatedly performed&lt;/a&gt;, each time providing a lower and lower limit on the value of the ether wind at the Earth's surface. The current value is phenomenally small - on the order of 3 nanometers/second. But it's not zero.&lt;/p&gt;&#xA;" OwnerUserId="7183" LastActivityDate="2021-08-17T17:08:47.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13561" PostTypeId="1" CreationDate="2021-08-17T17:12:55.297" Score="2" ViewCount="65" Body="&lt;p&gt;If I throw an object, I can derive its trajectory using the forces acted on it. I can also do the same using conservation of energy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Since when is it commonplace to use conservation of energy for dynamics problems?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Possibly related:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The first law of thermodynamics, ie the realization that when you account for heat, energy is conserved, was not realized until the 19th century, if I remember well.&lt;/li&gt;&#xA;&lt;li&gt;The first derivation of energy conservation from a quantum mechanical point of view happened much later, probably around the middle of the 20th century.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15201" LastEditorUserId="10106" LastEditDate="2021-08-18T02:54:13.430" LastActivityDate="2021-08-18T02:54:13.430" Title="Since when is it commonplace to use conservation of energy for dynamics problems?" Tags="|physics|thermodynamics|energy|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13562" PostTypeId="1" CreationDate="2021-08-18T04:49:10.060" Score="0" ViewCount="86" Body="&lt;p&gt;I would like to know whether &lt;a href=&quot;http://www.math.lsa.umich.edu/%7Eabloch/&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Anthony Bloch's homepage&quot;&gt;Anthony Bloch&lt;/a&gt; is a descendant of &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Bloch&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Felix Bloch - Wikipedia&quot;&gt;Felix Bloch&lt;/a&gt; or not. The &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=33933&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Anthony Bloch - The Mathematics Genealogy Project&quot;&gt;Anthony Bloch's page on The Mathematics Genealogy Project&lt;/a&gt; only shows his advisory relationships, not kinships.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;Meta: &lt;a href=&quot;https://hsm.meta.stackexchange.com/q/371/65&quot;&gt;Is it ok to ask whether a person is a descendants of a scientist?&lt;/a&gt; &lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="4251" LastEditDate="2021-09-16T10:12:44.307" LastActivityDate="2021-09-16T10:12:44.307" Title="Does Anthony Bloch have a kinship with Felix Bloch?" Tags="|mathematicians|physicists|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13563" PostTypeId="1" CreationDate="2021-08-18T08:16:12.870" Score="5" ViewCount="147" Body="&lt;p&gt;I hope that the slightly abrasive title is forgivable, as the judgement on this poetry is not mine, but Hans Freudenthal's. Here is the background: in the Dictionary of Scientific Biography, &lt;a href=&quot;https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/augustin-louis-cauchy&quot; rel=&quot;noreferrer&quot;&gt;there is an entry on Augustin-Louis Cauchy&lt;/a&gt; written by Freudenthal.  It is an excellent biography, but there is one line in there which surprised me:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Cauchy] also produced French and Latin poetry, which, however, is better forgotten.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Freudenthal was &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Freudenthal/&quot; rel=&quot;noreferrer&quot;&gt;apparently&lt;/a&gt; quite interested in and well-read on the subject of poetry, so this judgement might well be sound -- but I'd like to find out for myself! I am unable to find any trace of this harshly sentenced poetry of Cauchy's online, so my question is precisely as stated in the title!&lt;/p&gt;&#xA;" OwnerUserId="12439" LastActivityDate="2021-08-18T08:16:12.870" Title="Where can I read Cauchy's terrible poems?" Tags="|biographical-details|reference-request|cauchy|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13564" PostTypeId="1" CreationDate="2021-08-19T00:05:55.460" Score="2" ViewCount="82" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;To wit, what's &amp;quot;Sensitive&amp;quot; about True Positive Rates, and &amp;quot;Specific&amp;quot; about True Negative Rates?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why weren't these &lt;a href=&quot;https://en.wikipedia.org/wiki/Semantic_change#Typology_by_Blank_(1999)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Metaphors&lt;/a&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Loanword&quot; rel=&quot;nofollow noreferrer&quot;&gt;Imports&lt;/a&gt; reversed? Why wasn't &amp;quot;Sensitive&amp;quot; termed to signify True Negative Rates, and &amp;quot;Specific&amp;quot;  True Positive Rates instead?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I learn best visually. Can these graphs on &lt;a href=&quot;https://en.wikipedia.org/wiki/Sensitivity_and_specificity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; answer my questions?&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/oMDcK.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://i.stack.imgur.com/tqSHh.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2021-08-19T00:05:55.460" Title="Why's the true positive rate termed Sensitivity and true negative rate Specificity, not vice versa?" Tags="|probability|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13565" PostTypeId="1" AcceptedAnswerId="13566" CreationDate="2021-08-19T02:36:26.077" Score="0" ViewCount="80" Body="&lt;p&gt;Comments under the Space SE question &lt;a href=&quot;https://space.stackexchange.com/q/54651/12102&quot;&gt;How do orbits around Jacobi ellipsoids behave?&lt;/a&gt; include:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://articles.adsabs.harvard.edu/full/1978CeMec..17...37K&quot; rel=&quot;nofollow noreferrer&quot;&gt;Periodic orbits around a rotating ellipsoid&lt;/a&gt; &amp;quot;This paper extends results obtained during the completion of the author’s doctoral dissertation under the supervision of &lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%BCrgen_Moser&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jürgen Moser&lt;/a&gt;. (Kammeyer, P. C. 1974, Thesis, New York University)&amp;quot; which would probably make for excellent reading!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;..and &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Ludwig_Siegel#Biography&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Ludwig Siegel&lt;/a&gt;'s &amp;quot;best student was Jürgen Moser, one of the founders of &lt;a href=&quot;https://en.wikipedia.org/wiki/KAM_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;KAM theory&lt;/a&gt; (&lt;a href=&quot;https://en.wikipedia.org/wiki/Andrey_Kolmogorov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kolmogorov&lt;/a&gt;–&lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Arnold&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arnold&lt;/a&gt;–&lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%BCrgen_Moser&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moser&lt;/a&gt;), which lies at the foundations of chaos theory.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%BCrgen_Moser#Work&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jürgen Moser; work&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Moser completed his undergraduate education at and received his Dr. rer. nat. from the University of Göttingen in 1952, studying under Franz Rellich. After his thesis, he came under the influence of &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Ludwig_Siegel#Biography&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Ludwig Siegel&lt;/a&gt;, with whom he coauthored the second and considerably expanded English language edition of a monography on celestial mechanics. Having spent the year 1953 at the Courant Institute of New York University as a Fulbright scholar, he emigrated to the United States in 1955 becoming a citizen in 1959.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What monograph on celestial mechanics did Jürgen Moser coauthor the 2nd and considerably expanded English language edition of with Carl Ludwig Siegel?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2021-08-19T04:01:45.583" Title="What monograph on celestial mechanics did Jürgen Moser coauthor the 2nd and considerably expanded English language edition of with Carl Ludwig Siegel?" Tags="|mathematics|mathematicians|reference-request|celestial-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13566" PostTypeId="2" ParentId="13565" CreationDate="2021-08-19T04:01:45.583" Score="3" Body="&lt;p&gt;Siegel's original monograph of 1956 is available as &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3540020160&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Vorlesungen uber Himmelsmechanik (Grundlehren der mathematischen Wissenschaften)&lt;/em&gt;&lt;/a&gt;, Springer, June 11, 1956.&lt;/p&gt;&#xA;&lt;p&gt;The expanded 1971 edition, authored with Moser is available as &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3540586563&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Lectures on Celestial Mechanics&lt;/em&gt;&lt;/a&gt;, Springer, January 31, 1971.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2021-08-19T04:01:45.583" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13567" PostTypeId="1" CreationDate="2021-08-19T14:46:54.733" Score="1" ViewCount="98" Body="&lt;p&gt;I was looking for an historical approach to learn about the initiation of trigonometry but got stuck in a part where I have been trying to undestand but couldn't till now.&lt;/p&gt;&#xA;&lt;p&gt;By far, I could learn that trigonometry has basically evolved from calculations regarding chords of a circle of unit radius. And the development of mathematics till then mainly developed due to establishment of relation of sides of two congruent triangles, that is ratio of similar sides of two similar triangles are equal. Perhaps trigonometry evolved from that relation. But how come anvient mathematicians establish relation between angles and sides of a triangle? I have assumed that they have been carrying out experiments through brute measurements of angles and their relation to sides(ratio of sides) but again stumbled onto were there such precise instruments to measure angles at that time or not. I am pretty much puzzled about this.&lt;/p&gt;&#xA;&lt;p&gt;I am looking forward to get a satisfactory explanation about this topic  from here&lt;/p&gt;&#xA;" OwnerUserId="14195" LastActivityDate="2021-08-20T13:11:33.937" Title="Establishment of relation between angles and sides of a triangle" Tags="|trigonometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13569" PostTypeId="2" ParentId="13557" CreationDate="2021-08-20T01:54:11.287" Score="1" Body="&lt;p&gt;The short answer, it seems, is that the type of beer referenced in the problem (&amp;quot;1/2 1/4 malt-date beer&amp;quot;) is a weaker beer, and thus only uses 1/2 as much grain as &amp;quot;regular&amp;quot; beer.  So instead of 20 jars of regular beer using 5 heqat of grain (equaling 4 pefsu), you get 10 jars of weaker beer using 2.5 heqat (still equaling 4 pefsu).&lt;/p&gt;&#xA;&lt;p&gt;Clagget discusses this in his book &lt;a href=&quot;https://books.google.com/books?id=8c10QYoGa4UC&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;lpg=PP1&amp;amp;pg=PP1#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ancient Egyptian Science: Ancient Egyptian Mathematics&lt;/a&gt; (free in Google Books).  &lt;a href=&quot;https://books.google.com/books?id=8c10QYoGa4UC&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;lpg=PA229&amp;amp;pg=PA229#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;See page 229, note 7&lt;/a&gt; regarding Problem 5, which is also a pefsu problem from the MMP.  Clagget says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...the name of the weak beer only serves to indicate that this is a weaker beer like that of beer made from a mixture of malt and dates whose components are somehow represented by the fractions 1/2 and 1/4. The usual pefsu of stronger beer is 2.  Hence the beer of pefsu 4 needs only half the quantity of that of pefsu 2...Therefore, in determining the jugs of beer here, the author takes 1/2 the heqat measure of meal that produced the 100 loaves of bread with pefsu 20 in order to produce 10 jugs of beer of pefsu 4.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clagget's translation of &lt;a href=&quot;https://books.google.com/books?id=8c10QYoGa4UC&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;lpg=PA214&amp;amp;pg=PA214#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Problem 5 (page 214 in the same book)&lt;/a&gt; sheds a little light on this as well.  The last two lines indicate you'd solve the problem for &amp;quot;regular&amp;quot; beer, then take half that answer for the weaker malt-date variety.&lt;/p&gt;&#xA;&lt;p&gt;So, in Problem 8, the scribe calculates that 100 loaves of bread of pefsu 20 must use 5 heqat of grain. As you note in your question, using 5 heqat of grain would yield 20 jars of &amp;quot;regular&amp;quot; beer. But the problem concerns the weaker malt-date beer, so the scribe cuts the heqat in half to get 2.5 heqat.  To maintain the overall pefsu of 4, only 10 jars of this weaker beer can be made.&lt;/p&gt;&#xA;" OwnerUserId="7074" LastEditorUserId="7074" LastEditDate="2021-08-20T12:53:15.027" LastActivityDate="2021-08-20T12:53:15.027" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13570" PostTypeId="1" AcceptedAnswerId="13605" CreationDate="2021-08-20T12:27:15.793" Score="0" ViewCount="558" Body="&lt;p&gt;I am aware of the fact that Euclid's &lt;em&gt;Elements&lt;/em&gt; is a compilation of the works of previous Greek mathematicians like Thales, Pythagoras (his school), Eudoxus, Theaetetus, etc. However, I want to know the fate of their original texts. What happened to them? Did the papyrus in which they wrote their texts rot? Did the papyrus rolls burn as they were kept in the Library of Alexandria? Or were they even kept there in the first place? Why don't we have even a papyrus fragment from these mathematicians before Euclid?&lt;/p&gt;&#xA;&lt;p&gt;Also, please don't be like Euclid. Please mention your sources.&lt;/p&gt;&#xA;" OwnerDisplayName="user14943" LastEditorUserId="4251" LastEditDate="2021-12-08T15:07:53.710" LastActivityDate="2021-12-08T15:07:53.710" Title="What happened to the original sources of Euclid's Elements?" Tags="|mathematics|ancient-greece|antiquity|euclid|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13571" PostTypeId="2" ParentId="13570" CreationDate="2021-08-20T12:56:27.443" Score="6" Body="&lt;p&gt;Multiple copies of the works of Euclid's predecessors probably existed. Including similar compilations called Elements. On the opinion of people who did mathematics at the time and after Euclid, his Elements were superior, and they did not care to copy his predecessors. This was sufficient for their work to be lost. There are many ways for a papyrus scroll to be lost (some of them you listed). Papyrus, unlike clay table, is not a durable material. Euclid's original writings were also lost by the way: what we have is much later copies.&#xA;By the estimates of archeologists, 90% of the literature of Ancient Greece is lost, in the sense that we do not have even later copies. And 99.9% of what survived, survived only in the form of later copies. Few scraps of papyrus found in Egypt, where they were miraculously preserved due to very unusual conditions, do not change this picture.&lt;/p&gt;&#xA;&lt;p&gt;Ref.: &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_sources_1/&quot; rel=&quot;noreferrer&quot;&gt;How do we know about Greek mathematics?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-22T05:00:32.760" LastActivityDate="2021-08-22T05:00:32.760" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13572" PostTypeId="2" ParentId="13567" CreationDate="2021-08-20T13:11:33.937" Score="3" Body="&lt;p&gt;Trigonometry was developed for the needs of astronomy. So originally this was not the plane trigonometry that we study at school but spherical trigonometry.&#xA;The main (and almost only) source that survived to our days from antiquity is the book of Ptolemy, Mathematical synthesis (a.k.a. Almagest, available in English translation). Improvements and developments&#xA;were slowly made in this business in the Middle age, mainly by Muslim astronomers. It is they, who introduced our main trigonometric functions; Ptolemy had only one: the chord.&lt;/p&gt;&#xA;&lt;p&gt;Then this knowledge penetrated to the West, and trigonometry acquired more&#xA;or less modern form in the work of Johannes Muller (&lt;a href=&quot;https://en.wikipedia.org/wiki/Regiomontanus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Regiomontanus&lt;/a&gt;).&#xA;For example, he proved the cosine rule.&lt;/p&gt;&#xA;&lt;p&gt;The conjectural picture of &amp;quot;development of mathematics&amp;quot; that you describe is&#xA;incorrect. First of all, similar triangles did not play important role in this development, since the primary subject was spherical triangles, and there are no similar spherical triangles. Second, mathematicians usually do not discover their theorems by carrying on &amp;quot;brute measurements&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-08-20T13:11:33.937" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13573" PostTypeId="2" ParentId="13051" CreationDate="2021-08-20T21:58:07.947" Score="1" Body="&lt;p&gt;I ran across an answer to my satisfaction, but it sort of turns the question on its head: &amp;quot;Why do we implicitly use radians in trigonometric functions if we still use degrees and cycles?&amp;quot; It's not as historical as I imagined, but there is a bit.&lt;/p&gt;&#xA;&lt;p&gt;Each choice may arise more naturally in different contexts:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Degrees: integers for a wide variety of angles, early astronomy (days in year), chords geometry&lt;/li&gt;&#xA;&lt;li&gt;Cycles: rotations, describing periodic signals as periods per unit time&lt;/li&gt;&#xA;&lt;li&gt;Radians: arc geometry, limits scaling, derivatives scaling&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Each choice lets you do the same math, but once we hit the Computer Age, the radian's advantage with limits became indispensible.&lt;/p&gt;&#xA;&lt;p&gt;I'm not going to describe the entirety of the iterative CORDIC algorithm or its applications, just an overview for sine and cosine. With each iteration &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, it adjusts the angle of a unit vector by &lt;span class=&quot;math-container&quot;&gt;$\arctan (2^{-k})$&lt;/span&gt; toward the target angle, so that the unit vector's position (sine and cosine) can be adjusted with cheap bit-shifts instead of generic multiplication. &lt;span class=&quot;math-container&quot;&gt;$\arctan$&lt;/span&gt; isn't cheap, of course, but &lt;span class=&quot;math-container&quot;&gt;$\arctan (2^{-k})$&lt;/span&gt;  can be precomputed for a natural number sequence &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; and stored in a lookup table. It is the lookup table where the choice of degree/cycle/radian is made.&lt;/p&gt;&#xA;&lt;p&gt;With radians, &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to 0^+} \arctan (x)/x = 1$&lt;/span&gt;, so at large enough &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, you can take &lt;span class=&quot;math-container&quot;&gt;$\arctan (2^{-k}) = 2^{-k}$&lt;/span&gt; and don't need to precompute further. Adjustment by &lt;span class=&quot;math-container&quot;&gt;$2^{-k}$&lt;/span&gt; only requires changing a single bit. By contrast, &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to 0^+} \arctan_{cycle} (x)/x = 1 / (2 \pi) $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\lim_{x \to 0^+} \arctan_{degree} (x)/x = 180 / \pi $&lt;/span&gt;. Adjustment by &lt;span class=&quot;math-container&quot;&gt;$2^{-k}/ (2 \pi)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$2^{-k}180 / \pi$&lt;/span&gt; requires full-on addition.&lt;/p&gt;&#xA;" OwnerUserId="14196" LastEditorUserId="14196" LastEditDate="2021-08-21T03:33:07.083" LastActivityDate="2021-08-21T03:33:07.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13574" PostTypeId="1" AcceptedAnswerId="13576" CreationDate="2021-08-21T07:45:30.673" Score="2" ViewCount="810" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Yet the naive definition,&#xA;even when inappropriate, has a powerful hold on people's intuitions. When Marilyn&#xA;vos Savant presented a correct solution to the Monty Hall problem in her column for &lt;em&gt;Parade&lt;/em&gt; magazine in 1990, she received thousands upon thousands of letters from&#xA;readers (even mathematicians) insisting that she was wrong.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Blitzstein. &lt;em&gt;Introduction to Probability&lt;/em&gt; (2019 2 ed). pp 70, 71.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://priceonomics.com/the-time-everyone-corrected-the-worlds-smartest/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Time Everyone “Corrected” the World’s Smartest Woman&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Though her answer was correct, a vast swath of academics responded with outrage. In the proceeding months, vos Savant received more than &lt;a href=&quot;http://marilynvossavant.com/game-show-problem/&quot; rel=&quot;nofollow noreferrer&quot;&gt;10,000 letters&lt;/a&gt; -- including a pair from the Deputy Director of the Center for Defense Information, and a Research Mathematical Statistician from the National Institutes of Health -- all of which contended that she was entirely incompetent:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://facweb1.redlands.edu/fac/jim_bentley/Data/Math%20311/MontyHall/MontyHall.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Monty Hall Problem&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I put my solution of the problem on the bulletin board in the physics department office at the Naval Academy, following it with a declaration that you were right. All morning I took a lot of criticism and abuse from my colleagues, but by late in the afternoon most of them came around. I even won a free dinner from one overconfident professor. – Eugene Mosca, Ph.D., U.S. Naval Academy, Annapolis, Maryland&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://answers.google.com/answers/threadview?id=510729&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Answers&lt;/a&gt;, solely &lt;a href=&quot;https://science.gmu.edu/directory/robert-sachs&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Sachs&lt;/a&gt; &amp;quot;&lt;a href=&quot;https://www.nytimes.com/1991/07/21/us/behind-monty-hall-s-doors-puzzle-debate-and-answer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;was one of the few with the grace to concede his mistake.&lt;/a&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But Aaron Brown &lt;a href=&quot;https://www.linkedin.com/in/aaron-brown-a0137313/&quot; rel=&quot;nofollow noreferrer&quot;&gt;SB Applied Mathematics (Harvard 1974-1978), MBA Finance (Booth School of Business, Univ. Chicago 1980-1984)&lt;/a&gt; wrote a different side of the story on &lt;a href=&quot;https://www.quora.com/Why-do-some-PhDs-argue-against-Marilyn-Vos-Savant-s-conclusion-on-the-Monty-Hall-problem-https-youtu-be-4Lb-6rxZxx0/answer/Aaron-Brown-165&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quora&lt;/a&gt;. Who's correct?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You have to have lived through this ancient travesty to care about it. It’s probably best forgotten.&lt;/p&gt;&#xA;&lt;p&gt;Marilyn Vos Savant published an incorrect answer to the Monte Hall problem. She got the correct answer from a lot of readers, including me. She then published a second incorrect answer claiming that all the letters she got—including from math PhDs were wrong. Finally she got the answer right—the one she undoubtedly got from any math PhD or intelligent person who wrote her—but continued to claim (a) that her previous answers were correct and (b) that all the letters were wrong.&lt;/p&gt;&#xA;&lt;p&gt;As a result, the public is mostly misinformed about this simple problem.&lt;/p&gt;&#xA;&lt;p&gt;The set-up is simple. There are three doors, one of which has a valuable prize behind it, and two of which have worthless prizes. You select one door. The host opens one of the other two doors to show you a worthless prize, and offers you the opportunity to switch your choice.&lt;/p&gt;&#xA;&lt;p&gt;The key to this problem, which Marilyn finally saw in her third solution, is the knowledge and intentions of the host. If the host doesn’t know which door has the prize, there’s no reason to switch or not switch, it’s the same either way. If the host knows the door and is trying to hurt you, don’t switch. If the host knows the door and is trying to help you, switch.&lt;/p&gt;&#xA;&lt;p&gt;Marilyn’s first answer was to always switch, regardless of the host’s intentions. This is also the answer that most people now believe. It’s a common error that people make all the time. That’s why people wrote in to correct her.&lt;/p&gt;&#xA;&lt;p&gt;In order to justify her incorrect first answer, she claimed it was obvious that the host used the strategy of always opening a non-prize door and offering you the chance to switch. This does indeed justify switching. But that assumption was missing from her first and second answers, along with any discussion of the host’s knowledge and intentions mattering.&lt;/p&gt;&#xA;&lt;p&gt;Her justification that the assumptions were too obvious to need stating was the actual television program Let’s Make a Deal, hosted by Monte Hall, which eventually gave the name to the problem. But anyone who watched the show knew that was not Monte’s strategy. Sometimes he opened another door and gave a chance to switch, sometimes he didn’t. He was well aware of the problem if he followed Marilyn’s assumed strategy, and was scrupulous about not giving any advantage to the contestant. You can tell this from statistics on the show—on average it didn’t pay to switch—and his published statements.&lt;/p&gt;&#xA;&lt;p&gt;Clearly there are much bigger problems in the world, and it doesn’t pay to get upset about misinformation put out by popular vain people. If was very frustrating at the time, but it seems quaint that such things upset us in the 90s when there is so much worse information put out today by even more popular and more vain people.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2021-08-21T22:34:42.657" Title="Is the number of PhDs who berated Marilyn von Savant's answer to the Monty Hall problem exaggerated?" Tags="|probability|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13576" PostTypeId="2" ParentId="13574" CreationDate="2021-08-21T22:29:38.633" Score="2" Body="&lt;p&gt;To add just a bit to the great documentation in the question, and to Dave Renfro's and Conifolds accurate/helpful comments:&lt;/p&gt;&#xA;&lt;p&gt;First, as in the question, the actual conduct of Monty Hall in the TV show was different from the iconic statement of the problem. (I remember seeing it, live, long ago...) That is, as in the question, he did &lt;em&gt;not&lt;/em&gt; always open a door... thus, could &lt;em&gt;not&lt;/em&gt; be counted on to impart information... I didn't think about it at the time (though I was a mathematically inclined kid), but, in hindsight, he was either deliberately or subliminally trying to prevent &amp;quot;gaming&amp;quot; his game.&lt;/p&gt;&#xA;&lt;p&gt;It is not clear to me whether peoples' reading of the &amp;quot;Monty Hall Problem&amp;quot; accounts for the twist that Monty &lt;em&gt;always&lt;/em&gt; opens a door.&lt;/p&gt;&#xA;&lt;p&gt;But now let's I would not be at all surprised if tens of thousands of fairly serious math people either misread or jump to a conclusion... I do remember being caught off guard when I first heard of this &amp;quot;Problem&amp;quot;, perhaps in the 1990s. My first reaction was, indeed, that &amp;quot;the probabilities changed&amp;quot;... sorta subconsciously based on the &amp;quot;principle&amp;quot; that Laplace already spoke against 200 years ago: ignorance is not a proof of equal probabilities. :)&lt;/p&gt;&#xA;&lt;p&gt;In my intro to cryptology classes in the late 1990s and early 2000's, many, many students could not be persuaded that the probabilities did not change, despite a variety of explanations, and despite extensive simulations. :)&lt;/p&gt;&#xA;&lt;p&gt;Apart from the misreading or the (over-) simplification of the game from the reality, I think this story is powerful evidence for an innate tendency of human beings to believe in that false principle already denounced by Laplace (and probably understood by earlier people who wanted to gamble relatively successfully...)&lt;/p&gt;&#xA;&lt;p&gt;But, yeah, in my direct observation, lotta otherwise-serious math people &amp;quot;jump&amp;quot; at the naive/wrong conclusion. :)&lt;/p&gt;&#xA;&lt;p&gt;(Now that I think further, I do suspect that a certain number of math people have taken offence at the &amp;quot;presumption&amp;quot; of an outsider to have public opinions on math things... Even apart from gender bias and such.)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastEditorUserId="1932" LastEditDate="2021-08-21T22:34:42.657" LastActivityDate="2021-08-21T22:34:42.657" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13578" PostTypeId="2" ParentId="13051" CreationDate="2021-08-22T05:13:26.300" Score="0" Body="&lt;p&gt;Cycles is pretty straightforward, at least in some cases.&lt;/p&gt;&#xA;&lt;p&gt;When measuring the rate of rotation of a rotating object, the most common way to do it is to provide some sort of indexing mark, and observe how frequently the mark passes by a reference point. For instance, a flat on a shaft and a switch to detect the flat. Or a bright spot on the shaft, and an optical sensor detects every time the spot passes the sensor.&lt;/p&gt;&#xA;&lt;p&gt;Or when measuring the frequency of an electrical (sine wave) oscillator, a level detector is used which produces a square wave output, and either the rising or falling edge is detected.&lt;/p&gt;&#xA;&lt;p&gt;In these cases, one event/count is provided per rotation. Converting that to radians/sec is very often not a useful exercise. A simple cam on a shaft, for instance, will provide one activation of a mechanism per revolution of the shaft. If the shaft speed were measured in rad/sec, it would be necessary to divide by two pi to get the activation frequency of the mechanism. Using cycles is much easier.&lt;/p&gt;&#xA;&lt;p&gt;An everyday example would be engine rpms vs spark plug firing rate in an automotive engine. If a V4 is operating at 6000 rpm (100 rps), it is trivial to determine the spark plug firing rate: 400 Hz. Adding a conversion factor of two pi to get 628 rps (radians per second rather than revolutions) would not do anybody any favors.&lt;/p&gt;&#xA;" OwnerUserId="7183" LastActivityDate="2021-08-22T05:13:26.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13579" PostTypeId="1" CreationDate="2021-08-23T08:54:47.277" Score="0" ViewCount="91" Body="&lt;p&gt;Sir Isaac Newton led the foundation of his famous laws of motion during the 17th Century but at that time SI system hadn't existed. So in which units did he define force? Did he define it in some other units or was later redefined by scientists who made the SI system?&lt;/p&gt;&#xA;" OwnerUserId="15224" LastActivityDate="2021-08-23T09:14:25.077" Title="In which units did Sir Isaac Newton define force at that time as SI system didn't existed then?" Tags="|mathematics|philosophy-of-science|newton|" AnswerCount="1" CommentCount="1" ClosedDate="2021-08-23T17:04:21.673" ContentLicense="CC BY-SA 4.0" />
  <row Id="13580" PostTypeId="2" ParentId="13579" CreationDate="2021-08-23T09:14:25.077" Score="2" Body="&lt;p&gt;The definitions related to Newton's laws of motion do not require the use of any units. The concepts of force, velocity, distance and time are expressed in general terms. Units are unnecessary.&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2021-08-23T09:14:25.077" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13581" PostTypeId="1" CreationDate="2021-08-22T19:29:50.563" Score="9" ViewCount="900" Body="&lt;p&gt;Sir Isaac Newton led the foundation of his famous laws of motion during the 17th Century but at that time SI system hadn't existed. So in which units did he define force? Did he define it in some other units or was later redefined by scientists who made the SI system?&lt;/p&gt;&#xA;" OwnerUserId="15223" OwnerDisplayName="INFOTECH  GAMING" LastEditorUserId="10106" LastEditDate="2022-02-02T17:22:08.497" LastActivityDate="2022-09-22T07:50:24.410" Title="In which units did Sir Isaac Newton define force at that time as SI system didn't exist then?" Tags="|physics|units|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13582" PostTypeId="2" ParentId="13581" CreationDate="2021-08-22T22:48:25.153" Score="10" Body="&lt;p&gt;If you read the article &lt;a href=&quot;https://www.readcube.com/articles/10.1007%2Fs11191-008-9183-1?author_access_token=IEr_xY1Utg_UolinMfL3__e4RwlQNchNByi7wbcMAY6lkz5MwIOXRNWEELyyYBPNl_uWZ9lvRceW5onOV00FqrtHFJ0Pl6NdjlUobMDWn-OQE1UYZPhmeJSZ3Lk3YKNQ8u13sMxF8jRYZk0eRK1ZoA==&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Concept of Force: How Understanding its History can Improve Physics Teaching&lt;/a&gt; you will realise that Newton's ideas about force are not the same as those of today. Indeed Newton thought that there were different types of force&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;.&lt;br /&gt;&#xA;No unit of force was proposed until &lt;a href=&quot;https://www.biodiversitylibrary.org/page/29853094&quot; rel=&quot;nofollow noreferrer&quot;&gt;1873&lt;/a&gt;, when the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyne&quot; rel=&quot;nofollow noreferrer&quot;&gt;dyne&lt;/a&gt; was introduced. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces cancelled each other out. The strength of any force was expressed in terms of an equivalent gravitational attraction, with  weights or masses measured in &lt;a href=&quot;https://en.wikipedia.org/wiki/Grain_(unit)&quot; rel=&quot;nofollow noreferrer&quot;&gt;grains&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; I have added an amplification as a result of a comment by @terry-s.&lt;br /&gt;&#xA;What @terry-s has written is certainly true as explained in The - Stanford Encyclopedia of Philosophy, Newton’s Philosophiae Naturalis Principia Mathematica, &lt;a href=&quot;https://plato.stanford.edu/entries/newton-principia/#DefAbsSpaTimMot&quot; rel=&quot;nofollow noreferrer&quot;&gt;4. “Definitions” and absolute space, time, and motion&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Newton distinguishes among three ways of quantifying centripetal forces: the absolute quantity, which corresponds to what we would call the field strength of a central force field; the accelerative quantity, which “is the measure of this force that is proportional to the acceleration generated in a given time;” and the motive quantity, which is the measure of the force proportional to what we would call the change in linear momentum in a given time.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://vdoc.pub/download/the-principia-preceded-by-a-guide-to-newtons-principia-by-i-bernard-cohen-mathematical-principles-of-natural-philosophy-11lp60qrugco&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Principia. Preceded By A Guide To Newton's Principia By I. Bernard Cohen: Mathematical Principles Of Natural Philosophy&lt;/a&gt; there is the following paragraph:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Another significant novelty of the second edition was the introduction of a conclusion to the great work, the celebrated General Scholium that appears at the conclusion of book 3. The original edition ended rather abruptly with a discussion of the orbits of comets, a topic making up about a third of book 3. Newton had at first essayed a conclusion, but later changed his mind. His intentions were revealed in 1962 by A. Rupert Hall and Marie Boas Hall, who published the original drafts. In these texts, Newton shows that he intended to conclude the Principia with a discussion of the forces between the particles of matter, but then thought better of introducing so controversial a topic. While preparing the second edition, Newton thought once again of an essay on &amp;quot;the attraction of the small particles of bodies,&amp;quot; but on &amp;quot;second thought&amp;quot; he chose &amp;quot;rather to add but one short Paragraph about that part of Philosophy.&amp;quot; The conclusion he finally produced is the celebrated General Scholium, with its oft-quoted slogan &amp;quot;Hypotheses non fingo.&amp;quot; This General Scholium ends with a paragraph about a &amp;quot;spirit&amp;quot; which has certain physical properties, but whose laws have not as yet been determined by experiment. Again thanks to the researches of A. Rupert Hall and Marie Boas Hall, we now know that while composing this paragraph, Newton was thinking about the new phenomena of electricity.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;em&gt;forces on particles of matter&lt;/em&gt; is elaborated on:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In a draft of the General Scholium (ULC MS Add. 3995, fols. 351-352, published by Rupert and Marie Hall), Newton outlined, but did not give details concerning, the attraction among “very small particles,” explaining that it is “of the electrical kind.” He then lists briefly some of the properties of what he calls “the electric spirit.” Since that publication in 1962, there has been no doubt that the “spirit” mentioned in the final paragraph of the General Scholium is electrical.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="6351" OwnerDisplayName="Farcher" LastEditorUserId="6351" LastEditDate="2022-02-02T12:09:17.970" LastActivityDate="2022-02-02T12:09:17.970" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13583" PostTypeId="1" AcceptedAnswerId="13584" CreationDate="2021-08-23T19:39:04.917" Score="15" ViewCount="4610" Body="&lt;p&gt;The Large Hadron Collider (LHC) in CERN is well known for being one of the largest international collaborations to look for fundamental particles. Other big collaborations include the International Space Station, the Human Genome Project, the Manhattan Project, LIGO, etc.&lt;/p&gt;&#xA;&lt;p&gt;Were there any pre-modern large international scientific collaborations? Some kind of quest for El Dorado or the philosopher's stone but in a collaborative way. If there is no such pre-modern collaboration and if most of the large collaborations were from the last century, which was the earliest?&lt;/p&gt;&#xA;&lt;p&gt;I am looking for a collaboration, where many scientists (let's say more than 100) from different nationalities go to the same place to share research insight about a topic or build an experiment.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="15251" LastEditDate="2021-08-26T15:56:19.430" LastActivityDate="2021-09-02T21:22:16.637" Title="What are some early examples of large international scientific collaborations?" Tags="|scientists|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13584" PostTypeId="2" ParentId="13583" CreationDate="2021-08-23T20:28:55.787" Score="23" Body="&lt;p&gt;The conversion to the metric system in Continental Europe in the 1870s was a coordinated process, involving scientists and politicians from a multitude of nations. Apart from the processes in the various individual nations, the International Bureau of Weights and Measures was established in 1875. The initial period of the Bureau, with much laboratory work, later called the heroic years, culminated in 1889 with the completion of the prototype of the meter and the kilogram.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2021-08-23T20:28:55.787" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13585" PostTypeId="2" ParentId="13583" CreationDate="2021-08-24T06:22:33.237" Score="17" Body="&lt;p&gt;They didn't all go to the same place, but many, hundreds?, of scientists took part in expeditions all over the world to observe the &lt;a href=&quot;https://en.wikipedia.org/wiki/Transit_of_Venus#1761_and_1769&quot; rel=&quot;noreferrer&quot;&gt;1761 and 1769 transits of Venus&lt;/a&gt;. Perhaps the most famous of these was the Banks/Cook voyage to the Pacific&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In 1761 numerous expeditions were made to various parts of the world so that precise observations of the transit could be made in order to make the calculations as described by Halley—an early example of international scientific collaboration. This collaboration was, however, underpinned by competition, the British, for example, being spurred to action only after they heard of French plans from Joseph-Nicolas Delisle. In an attempt to observe the first transit of the pair, astronomers from Britain (William Wales and Captain James Cook), Austria (Maximilian Hell) and France (Jean-Baptiste Chappe d'Auteroche and Guillaume Le Gentil) traveled to destinations around the world, including Siberia, Newfoundland and Madagascar... For the 1769 transit (taking place on 3–4 June N.S., 23 May O.S.), scientists traveled to Tahiti, Norway, and locations in North America including Canada, New England, and San José del Cabo (Baja California, then under Spanish control).&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15231" LastEditorUserId="55" LastEditDate="2021-08-24T07:13:45.010" LastActivityDate="2021-08-24T07:13:45.010" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13586" PostTypeId="1" AcceptedAnswerId="13588" CreationDate="2021-08-24T07:11:24.597" Score="5" ViewCount="209" Body="&lt;p&gt;The first coordinate system was used by Eratosthenes to locate things using 2 points. Did this influence the development of the Cartesian coordinate system in any way?&lt;/p&gt;&#xA;" OwnerDisplayName="user14943" LastEditorUserId="55" LastEditDate="2021-08-24T11:47:33.023" LastActivityDate="2021-08-24T12:44:40.147" Title="Did the coordinate system used by Eratosthenes influence the development of the Cartesian coordinate system?" Tags="|geography|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13587" PostTypeId="2" ParentId="13583" CreationDate="2021-08-24T07:58:21.860" Score="9" Body="&lt;p&gt;Here are two older examples. The first is ancient Mouseion (Musaeum) founded c. 300 BC after the Alexandrian conquests that briefly unified the Hellenistic world, a prototype of modern academies. However it had broad scope not restricted to a specific research project. Its classical age ended with the expulsion of intellectuals from Alexandria in 145 BC by Ptolemy VIII:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The Musaeum was an institution founded, according to Johannes Tzetzes, by Ptolemy I Soter (c. 367 BC – c. 283 BC) at Alexandria... Rather than simply a museum in the sense that has developed since the Renaissance, it was an institution that brought together some of the best scholars of the Hellenistic world, as Germain Bazin compared it, &amp;quot;analogous to the modern Institute for Advanced Study in Princeton or to the Collège de France in Paris.&amp;quot;... More than 1,000 scholars lived in the Mouseion at a given time. Staff members and scholars were salaried by the Mouseion and paid no taxes.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second is&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Maragheh_observatory#History&quot; rel=&quot;noreferrer&quot;&gt;Maragheh observatory&lt;/a&gt; founded by al-Tusi in 1259, who justified the undertaking to Hulagu, Genghis Khan's grandchild, by the need to compile new astronomical tables adapted to Persian locations:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Men of mathematics, science, and astronomy came to the Maragheh Observatory from across the Islamic world and further. According to historical texts recovered from the observatory, the site had a reputation so widespread it had reached as far as China... A number of other prominent astronomers worked with Tusi at the observatory, such as Muhyi al-Din al-Maghribi, Mu'ayyid al-Din al-'Urdi, from Damascus, Qutb al-Din al-Shirazi, and Hulagu's Chinese astronomer Fao Munji, whose Chinese astronomical experience brought improvements to the Ptolemaic system used by Tusi. After 12 years of intense work by Tusi and other scientists... the tables were compiled in the Zij-i Ilkhani.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The obsevatory lost its funding after 1282, and became inactive by c. 1300.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-08-24T08:06:10.310" LastActivityDate="2021-08-24T08:06:10.310" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13588" PostTypeId="2" ParentId="13586" CreationDate="2021-08-24T11:46:44.973" Score="4" Body="&lt;p&gt;It seems like it should have, but not really. &lt;a href=&quot;https://press.uchicago.edu/books/HOC/HOC_V1/HOC_VOLUME1_chapter18.pdf&quot; rel=&quot;noreferrer&quot;&gt;Woodward in Medieval Mappaemundi&lt;/a&gt; tries to draw the connection thus:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the words of a modern historian of mathematics: &amp;quot;The development of graphical representation forged a link between the intuitive concepts of continuously varying quantities arising from physical phenomena and the geometry of the Greeks. The connection between the graphing concept and cartography&#xA;is seen in Oresme's use of the terms &amp;quot;longitude&amp;quot;&#xA;and &amp;quot;latitude&amp;quot; for the independent and dependent variables plotted on a graph. Once these ideas were associated with algebraic symbolism, the seventeenth-century mathematicians Rene Descartes and Pierre de&#xA;Fermat were able to formulate analytical geometry in&#xA;the form familiar today.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is true that Oresme drew bar graphs of distances and velocities in the 14th century and called the axes longitude and latitude. But there is no evidence that Fermat and Descartes, or their immediate successors, were aware of his work and plenty of evidence that they came to coordinates from a very different tradition, unrelated to cartography. They did not even draw two axes, only one, the second one was introduced later by van Schooten. Even Boyer, who is more open to it, writes in &lt;a href=&quot;https://www.jstor.org/stable/24945951&quot; rel=&quot;noreferrer&quot;&gt;The Invention of Analytic Geometry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Because Descartes carefully avoided any reference to his predecessors, one cannot say with assurance that he was familiar with the work of Oresme. It seems quite probable that he was. Yet the differences&#xA;between Descartes' system, called Cartesian geometry, and the graphical representation of the latitude of forms are so great as to make questionable any decisive influence.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Instead, the main source of inspiration was Apollonius's Conica (c. 200 BC), where he anticipates the use of coordinate equations (&amp;quot;symptoms&amp;quot;) to study conic sections, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2382/55&quot;&gt;When do we see for the first time the use of the Cartesian coordinates?&lt;/a&gt; The names &amp;quot;abscissa&amp;quot; and &amp;quot;ordinate&amp;quot;, instead of longitudes and latitudes, come from translations of Apollonius's terms, see &lt;a href=&quot;https://scholarship.claremont.edu/jhm/vol5/iss1/14/&quot; rel=&quot;noreferrer&quot;&gt;Pierce, Abscissas and Ordinates&lt;/a&gt;. The association of geometry to algebraic symbolism, on which Descartes and Fermat did rely, was done by Vieta in &lt;em&gt;Supplementum Geometriae&lt;/em&gt; (1593), where he systematically assigns letters to segments and areas and solves the resulting equations, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2408/55&quot;&gt;Viète's Relevance and his Connection to Euler&lt;/a&gt;. But again, so far as we can tell, Vieta was not aware of Oresme or motivated by cartography.&lt;/p&gt;&#xA;&lt;p&gt;Was there some vague influence through some general cultural osmosis, as Woodward suggests? Maybe. Descartes and Fermat probably saw maps even if they did not read Oresme. But why wasn't there more definitive influence? A speculation in the accepted answer to &lt;a href=&quot;https://math.stackexchange.com/a/157872/152568&quot;&gt;Was there some prior idea that inspired both Fermat &amp;amp; Descartes to invent coordinates? on Math SE&lt;/a&gt; suggests a possible answer. Coordinates in cartography and geometry were used for two very different purposes, at least in the early stages. In cartography they were used to identify locations of individual points. In geometry, since Apollonius, the idea was to study &lt;em&gt;curves&lt;/em&gt; by relating them to coordinate &lt;em&gt;equations&lt;/em&gt;. Ancient and medieval cartographers did not write equations for curves on a map, and mathematicians had little interest in associating points to pairs of numbers as such. So the two fields were of little help to each other.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-08-24T11:46:44.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13589" PostTypeId="1" CreationDate="2021-08-24T12:39:32.837" Score="2" ViewCount="81" Body="&lt;p&gt;The need for &lt;a href=&quot;https://en.wikipedia.org/wiki/Dark_matter#Early_history&quot; rel=&quot;nofollow noreferrer&quot;&gt;dark matter&lt;/a&gt; was recognized in the 1980's, after Rubin and Ford presented their results on the rotation curves of galaxies. However, as early as the 30's, Fritz Zwicky inferred that there was some dark mass from the movements of galaxies in the Coma cluster.&lt;/p&gt;&#xA;&lt;p&gt;Why was dark matter ignored between 1940 and the 1970's? What kept it from being recognized before the 80's?&lt;/p&gt;&#xA;" OwnerUserId="10806" LastActivityDate="2021-08-24T12:39:32.837" Title="Why wasn't the need for dark matter recognized until the 80's?" Tags="|astrophysics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13590" PostTypeId="2" ParentId="13586" CreationDate="2021-08-24T12:44:40.147" Score="4" Body="&lt;p&gt;Just an extended remark. There is nothing special about Eratosphenes here. Coordinates were used in astronomy since its very beginning, to describe stars positions. At least three of them were used since antiquity: ecliptic, equatorial&#xA;and horizontal. Astronomers routinely did analytic geometry in these coordinates;&#xA;it is called spherical trigonometry. Whether this influenced Descartes, it is hard to say; one cannot penetrate Descartes mind.&lt;/p&gt;&#xA;&lt;p&gt;Another remark is that probably neither Descartes nor Fermat could read Erathosphenes, since all his writings were already lost by that time. They could learn about coordinate systems from Ptolemy or later astronomers and geographers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-08-24T12:44:40.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13592" PostTypeId="2" ParentId="13583" CreationDate="2021-08-24T17:13:56.050" Score="12" Body="&lt;p&gt;&lt;a href=&quot;https://whc.unesco.org/en/list/1187/&quot; rel=&quot;noreferrer&quot;&gt;Struve Geodetic Arc&lt;/a&gt; was a series of geodetic measurements conducted in (according to borders at that time) Russia, Sweden and Norway between 1816 and 1855.&lt;/p&gt;&#xA;&lt;p&gt;The purpose was to more accurately determine the shape and size of the Earth.  From &lt;a href=&quot;https://fig.net/organisation/perm/hsm/history_of/measurements/struve/struve_arc_smith_2005.pdf&quot; rel=&quot;noreferrer&quot;&gt;the 2005 summary by J. R. Smith&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The work on the Russian part of the arc was carried out jointly by Imperial staff officers, the Dorpat Observatory and the Central Observatory of Poulkowa. That on the Scandinavian sector was carried out by Swedish and Norwegian experts, with the assistance of astronomers and equipment from Poulkowa Observatory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3054" LastActivityDate="2021-08-24T17:13:56.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13593" PostTypeId="2" ParentId="7626" CreationDate="2021-08-24T19:17:04.167" Score="2" Body="&lt;p&gt;While there is not likely to be a single source which addresses your query, direction and suggestions may be had from several differing sources:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Machlup's 1962 &lt;em&gt;The Production And Distribution Of Knowledge In The United States&lt;/em&gt; focuses on the patent system but expands widely beyond that. &lt;a href=&quot;https://www.google.com/books/edition/The_Production_and_Distribution_of_Knowl/kp6vswpmpjoC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/books/edition/The_Production_and_Distribution_of_Knowl/kp6vswpmpjoC?hl=en&amp;amp;gbpv=0&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;A global decline in research productivity? Evidence from China and Germany&lt;/em&gt; &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S0165176520304067#&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencedirect.com/science/article/abs/pii/S0165176520304067#&lt;/a&gt;! In a recent paper, Bloom et al. (2020) find evidence for a substantial decline in research productivity in the U.S. economy during the last 40 years. In this paper, we replicate their findings for China and Germany, using detailed firm-level data spanning three decades. Our results indicate that diminishing returns in idea production are a global phenomenon, not just confined to the U.S.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cauwels and Sornette, &lt;em&gt;Are ‘Flow of Ideas’ and ‘Research Productivity’ in secular decline?&lt;/em&gt; &lt;a href=&quot;https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3716939&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3716939&lt;/a&gt; It is widely held true that fundamental scientific knowledge has been accelerating exponentially over the past centuries and will continue to do so for the foreseeable future. Moreover, endogenous growth theory postulates that this exponential accumulation of knowledge is the main source of the ubiquitous exponential economic growth. We test these claims by constructing two new series of knowledge indices, one representing the historical evolution of the Flow of Ideas, the other of the Research Productivity, for the time period between 1750 and 1988. Three different geographical regions are covered: 1) Continental Europe, 2) the United Kingdom, and 3) the United States; and two disciplines: a) the physical sciences, and b) the life sciences. Our main result is that scientific knowledge has been in clear secular decline since the early 1970s for the Flow of Ideas and since the early 1950s for the Research Productivity. We also observe waves coinciding with the three industrial and technological revolutions, in particular in the United Kingdom. Overall, our results support the Kuhnian theory of knowledge creation through scientific revolutions, punctuation and paradigm shifts and falsify the gradualism that lies at the basis of the currently prevailing economic paradigm of endogenous growth.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2021-08-24T19:17:04.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13594" PostTypeId="1" AcceptedAnswerId="13595" CreationDate="2021-08-25T04:17:41.583" Score="0" ViewCount="421" Body="&lt;p&gt;Can someone give a history of coordinates? When did coordinates first appear, which was the first ever coordinate system? In which field were they used?&lt;/p&gt;&#xA;" OwnerDisplayName="user14943" LastActivityDate="2021-08-26T01:45:20.963" Title="What is the History of Coordinates?" Tags="|geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13595" PostTypeId="2" ParentId="13594" CreationDate="2021-08-25T04:40:21.280" Score="2" Body="&lt;p&gt;The coordinates on a surface (sphere) first appear in the Greek astronomy, in the very beginning of Hellenistic period (~300s BC). Since almost no sources from that time survive, it is speculated that one purpose was to make celestial globes (One of them is credited to Archimedes. This is attested by some Roman authors, but nobody describes it in detail). Hipparchus is said to make the first star catalog, so he must have used some coordinates in it, unfortunately most works of Hipparchus are lost. (There is a conjecture that he used some mixture of ecliptic and equatorial coordinates).&lt;/p&gt;&#xA;&lt;p&gt;At the same time Eratosthenes used coordinates in geography. Unfortunately no works of Eratosthenes survive.&#xA;We know about the work of Hipparchus and Eratosthenes from later sources which cite them.&lt;/p&gt;&#xA;&lt;p&gt;Our main source on Greek astronomy is the book of Ptolemy, Mathematical Synthesis (better known its Arabic name Almagest), which is written in the second century AD. In it he describes the work of his predecessors, in particular Hipparchus.&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy and other Greek astronomers made use of 3 different coordinate systems for the sky: ecliptic (to describe the motion of Sun, Moon and planets), equatorial (to describe stars) and horizontal (to describe everything as seen by the observer), and the familiar to us geographic coordinates, which are similar to equatorial coordinates in the sky. All these coordinate systems are still widely used in astronomy and geography, almost without change.&lt;/p&gt;&#xA;&lt;p&gt;On the pure mathematical side, Apollonius used some rudimentary coordinates&#xA;in his work on conic section. There is no apparent connection of his work with contemporary work with spherical coordinates in astronomy and geography. He was a contemporary of Eratosthenes, slightly younger.&lt;/p&gt;&#xA;&lt;p&gt;Remark. The first of the two ecliptic coordinates, longitude, comes from Babylon. The Greeks took from Babylonians the idea to divide the ecliptic into 12 &amp;quot;signs&amp;quot; and each sign into 30 degrees. So if we include the coordinate on a circle to coordinate systems, then the priority should be probably assigned to Babylonians. Neugebauer in his&#xA;book &amp;quot;The exact sciences in antiquity&amp;quot; estimates that division of the circle into signs and degrees was introduced between 500 and 400 BC.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-08-26T01:45:20.963" LastActivityDate="2021-08-26T01:45:20.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13596" PostTypeId="2" ParentId="12254" CreationDate="2021-08-25T11:24:31.357" Score="2" Body="&lt;p&gt;As Conifold already mentioned, there were several problems with this initial approach: spinors where observable physical objects due to their dimension of &lt;span class=&quot;math-container&quot;&gt;$-\frac{3}{2}$&lt;/span&gt;, the introduction of a dimensioned coupling constant lead to non-renormalizability, etc. I have been following this topic for a while now and just recently came across this video proposing changes to this initial approach, leading to some quite interesting results. I think it is worth to watch:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=BDiuQqfVm3Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;A non-linear spinor theory of elementary particles&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15239" LastActivityDate="2021-08-25T11:24:31.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13597" PostTypeId="1" CreationDate="2021-08-25T20:35:41.240" Score="1" ViewCount="54" Body="&lt;p&gt;I'm looking for primary sources in which the author(s) commit &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientism&quot; rel=&quot;nofollow noreferrer&quot;&gt;scientistic&lt;/a&gt; fallacies when speaking about Native Americans.&lt;/p&gt;&#xA;&lt;p&gt;I'm working my way through S.G. Morton's &lt;em&gt;Crania Americana&lt;/em&gt;, which is close, but the appendix in this, regarding physiognomy, by George Combe is even closer.&lt;/p&gt;&#xA;&lt;p&gt;Could anyone tell me any more examples of where a border-crossing violation (Peterson, &lt;em&gt;Demarcation and the Scientistic Fallacy&lt;/em&gt;), that is to say when someone tries to expand their field of science onto something that isn't actually covered by it, occurs.&lt;/p&gt;&#xA;&lt;p&gt;Thanks :)&lt;/p&gt;&#xA;" OwnerUserId="15240" LastEditorUserId="10106" LastEditDate="2021-08-26T03:53:07.280" LastActivityDate="2021-08-26T03:53:07.280" Title="Need cases of *scientistic* fallacies in 19th century anthropological/ethnological studies of Native Americans" Tags="|philosophy-of-science|anthropology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13599" PostTypeId="1" CreationDate="2021-08-26T10:22:35.227" Score="8" ViewCount="456" Body="&lt;p&gt;Are there any examples in the history of mathematics of a mathematical proof that was found by accident, in the sence that in the effort of proving it, ending up proving something intuitively different?&#xA;Of course, I do not mean trying to prove that a proposition holds and proving that indeed it does not.&#xA;Are there any branch of mathematics born as a result of such an &amp;quot;accident&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="15249" OwnerDisplayName="asknohope" LastEditorUserId="15245" LastEditDate="2021-08-26T13:05:34.283" LastActivityDate="2021-08-29T17:00:30.100" Title="Proof by &quot;accident&quot;" Tags="|mathematics|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13601" PostTypeId="2" ParentId="13271" CreationDate="2021-08-26T12:15:27.883" Score="0" Body="&lt;p&gt;There are so many possible examples of artistic production's overlap with science that it's hard to know where to begin.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Martin Kemp, British art historian, for many years contributed a regular column to Nature magazine on artistic visualizations in science, e.g., &lt;em&gt;Visualizations: The nature book of art and science&lt;/em&gt; is one source.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Clarke and Henderson's book &lt;em&gt;From Energy to Information&lt;/em&gt; explicitly examines how 19th c scientific discoveries wrt energy, entropy and thermodynamics fed creativity in art and literature in the 20th c.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;David Wooton's &lt;em&gt;The Invention of Science: A new history of the scientific revolution&lt;/em&gt; reviews, in depth, the overlap between emerging conceptions of science and its analogues in the arts, with a particular focus on poets such as John Donne.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Karen Barad's &lt;em&gt;Meeting the Universe Halfway&lt;/em&gt; contains an extended discussion of Michael Frayn's play &lt;em&gt;Copenhagen&lt;/em&gt; about a meeting between Werner Heisenberg and Nils Bohr during WWII.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2021-08-26T12:15:27.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13605" PostTypeId="2" ParentId="13570" CreationDate="2021-08-28T09:14:17.037" Score="0" Body="&lt;p&gt;While I fully agree about competition of new &lt;a href=&quot;https://en.wikipedia.org/wiki/Epitome&quot; rel=&quot;nofollow noreferrer&quot;&gt;epitomes&lt;/a&gt; of earlier works driving original, more detailed works to extinction (that's how we lost many historical sources and lexicographical works), I would add that the driving force of this competition was economical. Medieval books were a luxury item &lt;a href=&quot;https://aidanc.net/2010/09/16/the-price-of-a-book-in-the-middle-ages&quot; rel=&quot;nofollow noreferrer&quot;&gt;costing&lt;/a&gt; to an order of magnitude as much as an average annual income of a worker, so most secular books that were copied had practical applications, and among those works we could consider arts and sciences most popular were those from the school cirriculum. Euclid had got included in it, but those older authors had not. And BTW, even modern authors don't include much historical details in school textbooks on math.&lt;/p&gt;&#xA;&lt;p&gt;Regarding the lost works, when stored properly and protected from moisture, papyrus is about as strong as parchment, and in Ancient Egypt 250-year-old unneeded papyri &lt;a href=&quot;https://books.google.com/books?id=b-x5DwAAQBAJ&amp;amp;pg=PA311&quot; rel=&quot;nofollow noreferrer&quot;&gt;were sometimes&lt;/a&gt; considered preserved enough to use again. However, unlike parchment it becomes very fragile if low and high humidity alternate, which is typical for European climate. Fungi (like molds) and insects (like ants) may also destroy the papyrus for the sake of calories. As for the parchment, even though it's more easy to store in European climate, being untanned skin, it's no less nutritiuos to living beings from the size of mice all the way down to bacteria.&lt;/p&gt;&#xA;&lt;p&gt;Ancient Alexandria in particular is now &lt;a href=&quot;https://www.theguardian.com/travel/2007/jul/29/escape.egypt&quot; rel=&quot;nofollow noreferrer&quot;&gt;below sea level&lt;/a&gt; due to a &lt;a href=&quot;https://en.wikipedia.org/wiki/365_Crete_earthquake&quot; rel=&quot;nofollow noreferrer&quot;&gt;very powerful earthquake&lt;/a&gt;, but the library &lt;a href=&quot;https://en.wikipedia.org/wiki/Library_of_Alexandria#Roman_Period_and_destruction&quot; rel=&quot;nofollow noreferrer&quot;&gt;dwindled&lt;/a&gt; due to lack of financing a century before that.&lt;/p&gt;&#xA;&lt;p&gt;Also as &lt;a href=&quot;https://books.google.com/books?id=b-x5DwAAQBAJ&amp;amp;pg=PA313&quot; rel=&quot;nofollow noreferrer&quot;&gt;noted&lt;/a&gt; by historians, during the early period of archeology a great deal of excavated papyri were destroyed due to multiple factors.&lt;/p&gt;&#xA;" OwnerUserId="13566" LastEditorUserId="13566" LastEditDate="2021-12-06T13:07:10.973" LastActivityDate="2021-12-06T13:07:10.973" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13606" PostTypeId="1" CreationDate="2021-08-25T23:04:05.057" Score="3" ViewCount="243" Body="&lt;p&gt;I was reading Wikipedia article on Bohr-Kramers-Slater (BKS) theory, &lt;a href=&quot;https://en.wikipedia.org/wiki/BKS_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/BKS_theory&lt;/a&gt;. I encountered two interesting points and need your help to understand the reasons behind them.&lt;/p&gt;&#xA;&lt;p&gt;Einstein was awarded Nobel Prize in 1921 [&lt;a href=&quot;https://www.nobelprize.org/prizes/physics/1921/einstein/facts/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;] primarily for his work on photoelectric effect which used photon model to explain the phenomenon.&lt;/p&gt;&#xA;&lt;p&gt;BKS theory was put forward around 1924.&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia article says the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nevertheless, Bohr and Kramers had two objections to Slater's&#xA;proposal:&lt;/p&gt;&#xA;&lt;p&gt;1: The assumption that photons exist. Even though Einstein's photon&#xA;hypothesis could explain in a simple way the photoelectric effect, as&#xA;well as conservation of energy in processes of de-excitation of an&#xA;atom followed by excitation of a neighboring one, Bohr had always been&#xA;reluctant to accept the reality of photons, his main argument being&#xA;the problem of reconciling the existence of photons with the&#xA;phenomenon of interference;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Niels Bohr was one the leading physicist at that time; around 1924. I'm sure there would be others prominent physicists like him who didn't really believe in photons. I understand there are always some persons who tend to oppose or doubt even well established theories or models. In this case, the mention of Bohr really fascinated me. I know that Einstein and Bohr had totally different interpretation of quantum mechanics.&lt;/p&gt;&#xA;&lt;p&gt;The article also says the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, physically the most provocative element of the theory, that&#xA;momentum and energy would not necessarily be conserved in each&#xA;interaction but only overall, statistically, was soon shown to be in&#xA;conflict with experiment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You could check the Wikipedia article for little more details. Anyway, I don't really understand what it means. I don't think they were saying that the laws of energy conservation and momentum conservation don't hold at all because these two laws, in my view, were very firmly established by then. I think they were saying that energy conservation doesn't hold when you look at individual interactions but it only holds when you look at the complete system. Could you please help me with it? I'm just a learner so please try to keep it simple if you  can.&lt;/p&gt;&#xA;&lt;p&gt;I also find some articles which I cannot access and even if I were able to, I don't think I would be able to understand!&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://www.nature.com/articles/137298a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dirac, Paul Adrien Maurice. &amp;quot;Does conservation of energy hold in atomic processes?.&amp;quot; Nature 137.3460 (1936)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-540-70626-7_19&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kragh, Helge. &amp;quot;Bohr—Kramers—Slater Theory.&amp;quot; Compendium of Quantum Physics. Springer (2009)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="9341" OwnerDisplayName="PG1995" LastEditorUserId="6609" LastEditDate="2021-08-28T22:28:10.570" LastActivityDate="2021-08-30T02:10:43.053" Title="Bohr-Kramers-Slater (BKS) theory and energy conservation only on statistically basis" Tags="|quantum-mechanics|electromagnetism|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13607" PostTypeId="1" CreationDate="2021-08-28T19:48:52.503" Score="1" ViewCount="105" Body="&lt;p&gt;Ecliptic longitude may be expressed in degrees; my understanding is that prior to the 19th century, expressions of ecliptic longitude contained zodiacal signs. What contemporaneous accounts describe the transition away from use of zodiacal signs in expressions of ecliptic longitude and is there a specific occasion that may demarcate the advent of contemporary expressions of ecliptic longitude as standard practice?&lt;/p&gt;&#xA;&lt;p&gt;Edit: I recognize that the equivalence of mathematical and zodiacal expressions of ecliptic longitude has long been recognized; as use of zodiacal notation fell out of favor in recent centuries, was there an official statement or authoritative discourse that hastened or indicated its demise?&lt;/p&gt;&#xA;" OwnerUserId="9312" LastEditorUserId="9312" LastEditDate="2021-08-30T21:55:20.573" LastActivityDate="2021-09-03T18:25:08.670" Title="When did contemporary practices for indicating ecliptic longitude supplant those containing zodiacal signs?" Tags="|mathematics|astronomy|notation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13608" PostTypeId="2" ParentId="13599" CreationDate="2021-08-28T22:01:19.127" Score="4" Body="&lt;p&gt;One can argue that most parts of mainstream mathematics are borne &amp;quot;by accident&amp;quot;. That is we start with a very particular but still important example, then generalizations are discovered and a large theory emerges completely unexpected by the person who started all that. For example Max Dehn studied tessellations of hyperbolic plane and discovered an algorithm to solve the word problem in surface groups. This eventually lead to the small cancelation theory, solutions of bounded Burnside problem and many other group theoretic problems, Gromov hyperbolic groups and the whole geometric group theory.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-08-28T22:19:22.573" LastActivityDate="2021-08-28T22:19:22.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13609" PostTypeId="2" ParentId="13599" CreationDate="2021-08-29T02:48:01.940" Score="2" Body="&lt;p&gt;Perhaps something closer to the question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;in the effort of proving it, ending up proving something intuitively different&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One can count here numerous attempts to prove well known open conjectures which resulted in proving weaker and seemingly different statements. For example, Tao's &lt;a href=&quot;https://arxiv.org/abs/1909.03562&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;approximation&amp;quot;&lt;/a&gt; of Collatz problem is interesting by itself but is a result of an attempt to solve the problem, &lt;a href=&quot;https://arxiv.org/abs/0905.1118&quot; rel=&quot;nofollow noreferrer&quot;&gt;Moore's theorem&lt;/a&gt; about Folner sets in the R.Thompson group &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is a result of his attempt to resolve the amenability question for this group.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-08-29T02:48:01.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13610" PostTypeId="2" ParentId="13599" CreationDate="2021-08-29T17:00:30.100" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Squaring_the_circle&quot; rel=&quot;noreferrer&quot;&gt;Squaring the circle&lt;/a&gt; is the age-old attempt to construct a square the same size as a circle, using a compass and straightedge, in a finite number of steps.  It was finally proven impossible in 1882.&lt;/p&gt;&#xA;&lt;p&gt;However, Menaechmus, in his study of the issue, devised the &lt;a href=&quot;http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/emat6690/Insunit/conicsunit.html&quot; rel=&quot;noreferrer&quot;&gt;conic sections&lt;/a&gt;, beginning a whole branch of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2021-08-29T17:00:30.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13611" PostTypeId="1" AcceptedAnswerId="13612" CreationDate="2021-08-29T23:53:19.937" Score="3" ViewCount="501" Body="&lt;p&gt;It is sometimes claimed that Kirchhoff's Voltage Law (KVL) is equivalent to the statement that a given electric field is conservative. Or, put another way, is is sometimes claimed that KVL is equivalent to the equation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\oint_{\Gamma}\vec{E}\cdot d\vec{\ell} = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;According to this claim, if a given electric field is not conservative, then KVL is said not to apply and/or it is violated.&lt;/p&gt;&#xA;&lt;p&gt;How did Kirchhoff actually explain KVL?&lt;/p&gt;&#xA;" OwnerUserId="14184" LastEditorUserId="14184" LastEditDate="2021-12-17T16:43:48.570" LastActivityDate="2022-09-28T00:51:00.180" Title="How did Kirchhoff express his voltage law (KVL)" Tags="|physics|electricity|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13612" PostTypeId="2" ParentId="13611" CreationDate="2021-08-29T23:53:19.937" Score="4" Body="&lt;p&gt;Kirchhoff's Voltage Law appears in his 1845 paper &lt;a href=&quot;https://zenodo.org/record/2422851&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Ueber den Durchgang eines electrischen stromes etc.&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;wenn die Draehte &lt;span class=&quot;math-container&quot;&gt;$1,2,...\nu$&lt;/span&gt; eine geschlossene Figur bilden,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$I_1\omega_1 + I_2\omega_2+ ... +I_{\nu}\omega_{\nu}$$&lt;/span&gt;&#xA;= der Summe alle elektromotorischen Kraefte die sich auf Wege:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which I translate as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;when the wires &lt;span class=&quot;math-container&quot;&gt;$1,2,...n$&lt;/span&gt; form a closed figure,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$I_1R_1 + I_2R_2 + ... I_nR_n$$&lt;/span&gt;&#xA;= the sum of all electromotive forces that are on the way:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&lt;p&gt;The original question states that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is sometimes claimed that Kirchhoff's Voltage Law (KVL) is equivalent to the statement that a given electric field is conservative.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sredni Vashtar, in an lengthy answer, defends exactly that claim of equivalence. S.V. asserts that.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;in modern notation the original form of KVL can also be written as:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\oint_{\Gamma}\vec{E}\cdot d\ell = 0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;S.V. supports his case by citing texts such as the second volume of Berkeley Physics: Electricity and Magnetism 3rd edition by Purcell and Morin, and  Ramo Whinney VanDuzer, &amp;quot;Fields and Waves for Communication Electronics&amp;quot; 3rd edition, p. 179. The are numerous other texts that S.V. could have cited as well.&lt;/p&gt;&#xA;&lt;p&gt;It is agreed upon by all sides, that this re-cast version of Kirchhoff's voltage law is physically inaccurate in the presence of a time varying magnetic field. But I believe that the text of Kirchhoff's formulation makes it clear that the modern version is a &lt;em&gt;re-casting&lt;/em&gt;, and not simply the original formulation of KVL by Kirchhoff.&lt;/p&gt;&#xA;&lt;p&gt;It thus behooves us to examine whether or not this recasting is &lt;em&gt;faithful&lt;/em&gt; to the original, or whether it introduces some innovation, that cannot be properly ascribed to the original. In particular, we need to examine in what ways we may re-cast the terms &lt;span class=&quot;math-container&quot;&gt;$I_CR_C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathscr{E}_C$&lt;/span&gt;, where the latter term is electromotive force along the path &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;S.V. re-arranges Kirchhoff's equation into the form:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(R1 I1 - emf1) + (R2 I2 - emf2) + ... + (Rn In - emfn) = 0&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which is completely justifiable. However, he then proceeds to claim that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the twentyfirst century we recognize that each term in parenthesis represents the path integral of the total electric field along each of the segments 1, 2,... n, in which the closed path (the loop) has been partitioned. Therefore, in modern terms, the original formulation of KVL says&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sum of path integral of Etot.dl along the branches of our closed figure = 0&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Although this is in agreement with modern texts, one must ask if there is any justification for re-casting in this way.&lt;/p&gt;&#xA;&lt;p&gt;I do not take issue with recasting &lt;span class=&quot;math-container&quot;&gt;$I_CR_C$&lt;/span&gt; in the form of an &lt;span class=&quot;math-container&quot;&gt;$\int_C\vec{E}\cdot d\ell$&lt;/span&gt; term. Indeed, I think such a recasting is an improvement over Kirchhoff. It removes the dependence of KVL in it's original formulation on Ohm's law.&lt;/p&gt;&#xA;&lt;p&gt;However, the recasting of &lt;span class=&quot;math-container&quot;&gt;$\mathscr{E}$&lt;/span&gt; in the form of an &lt;span class=&quot;math-container&quot;&gt;$\int_C\vec{E}\cdot d\ell$&lt;/span&gt; term is objectionable.&lt;/p&gt;&#xA;&lt;p&gt;Gauss's law tells us that an electric field has a non-zero divergence only in the presence of a net non-zero charge density. It is thus reasonable to take the EMF produced by a time varying magnetic field along some curve, not as the &lt;em&gt;total&lt;/em&gt; electric field, but as the &lt;em&gt;rotational&lt;/em&gt; or &lt;em&gt;solenoidal&lt;/em&gt; component of the electric field. [See Helmholz decomposition for an understanding that a 3D vector field can be decomposed into a rotational/solenoidal/divergence-free component and an irrotational/conservative/curl-free component.] That is,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathscr{E}_{induced} = \int_C\vec{E}_{rotational} \cdot d\ell$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that for &lt;em&gt;closed loops&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathscr{E}_{induced} = \oint_C\vec{E}_{rotational} \cdot d\ell = \oint_C\vec{E}_{total} \cdot d\ell$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;but the above does not apply for &lt;em&gt;open&lt;/em&gt; paths.&lt;/p&gt;&#xA;&lt;p&gt;Although the change in subscript appears minor, it means the difference between a law which is correct in general, and a law which is correct only in the absence of time varying magnetic fields.&lt;/p&gt;&#xA;&lt;p&gt;[It is easy to see that if &lt;span class=&quot;math-container&quot;&gt;$\mathscr{E}_{induced}$&lt;/span&gt; is recast as &lt;span class=&quot;math-container&quot;&gt;$\mathscr{E}_{induced} = \int_C\vec{E}_{rotational} \cdot d\ell$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$\mathscr{E}_{induced} = \int_C\vec{E}_{total} \cdot d\ell$&lt;/span&gt; then the resulting KVL is correct, even in the presence of time-varying magnetic fields. &lt;span class=&quot;math-container&quot;&gt;$\vec{E}_{total} = \vec{E}_{rotational} + \vec{E}_{irrotational}$&lt;/span&gt;. So, if one subtracts &lt;span class=&quot;math-container&quot;&gt;$\vec{E}_{total}$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$\vec{E}_{rotational}$&lt;/span&gt; one is left with an irrotational, i.e. conservative vector field, and therefore the sum of any closed loop contour integral of this latter field will always be 0.]&lt;/p&gt;&#xA;&lt;p&gt;Is there any justification for re-casting KVL in a form that is broken, when it could have been re-cast in a form that is &lt;em&gt;not&lt;/em&gt; broken? Is there any evidence that the broken form is more faithful to Kirchhoff than the unbroken form?&lt;/p&gt;&#xA;&lt;p&gt;S.V. makes the case that Kirchhoff did not base his formulation of KVL on experiments that employed induction of EMF via a time-varying magnetic field. The unstated implication seems to be that Kirchhoff was perhaps ignorant of this source of EMF. Perhaps a more egregious unstated implication is that, being unaware of this source of EMF, the more faithful re-casting of KVL is the one which is broken.&lt;/p&gt;&#xA;&lt;p&gt;I do not have at hand evidence that Kirchhoff was aware of induction of EMF via a time-varying magnetic field in 1845. However, Maxwell wrote in &amp;quot;A Treatise on Electricity and Magnetism&amp;quot; Volume 2, 1873, chapter XVIII p358, section 759, that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first determination of a resistance of a wire in electromagnetic measure was made by Kirchhoff*. He employed two coils...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;'Bestimmung der Constanten von Welchar die Intensitat inducirter elektrischer Stromme abhungt' Pogg. Ann. lxxvi (April 1849)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;including an illustration of Kirchhoff's apparatus:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7U4Vd.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7U4Vd.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, we know that no later than 4 years after publishing KVL, Kirchhoff was not only aware of Faraday induction, but actively employed it in his experiments.&lt;/p&gt;&#xA;&lt;p&gt;If, we suppose, as S.V. seems to, that Kirchhoff did not learn of Faraday induction at the University of Konigsberg, then it surely would have come as a major surprise to him, no more than 4 years later. If he didn't consider induced EMF in formulating KVL, would it not be surprising that he didn't reconsider or amend KVL upon learning of it's existence?&lt;/p&gt;&#xA;&lt;p&gt;One final comment regarding S.V.'s reply.&lt;/p&gt;&#xA;&lt;p&gt;I agree with Sredni Vashtar that Kirchhoff's voltage law and Faraday's law of induction are different. KVL says nothing about the origin of EMFs, Faraday's law of induction does. KVL expresses the relationship between EMFs and resistive voltage drops in any arbitrary loop in any arbitrary circuit, but Faraday's does not. So, we can all agree that Kirchhoff's voltage law, and Faraday's law of induction are different. Where I disagree is with S.V.'s claim that Faraday's law (in it's original form) is an &lt;em&gt;extension&lt;/em&gt; of KVL (in it's original form).&lt;/p&gt;&#xA;&lt;p&gt;Edit:&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://electronics.stackexchange.com/a/598968/268467&quot;&gt;another stack exchange answer&lt;/a&gt; S.V. calls the electric field induced by a time varying magnetic field &lt;span class=&quot;math-container&quot;&gt;$E_{ind}$&lt;/span&gt; and uses the following formula and caption:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yT0F0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yT0F0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;voltage = induced voltage + scalar potential difference&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From which, one can safely infer that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathscr{E}_{ind} = \int_{\gamma_{A\rightarrow B}} \vec{E}_{ind} \cdot d\vec{\ell}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(or possibly the negative of that).&lt;/p&gt;&#xA;&lt;p&gt;S.V.'s &lt;span class=&quot;math-container&quot;&gt;$\vec{E}_{ind}$&lt;/span&gt; is the same as my &lt;span class=&quot;math-container&quot;&gt;$\vec{E}_{rotational}$&lt;/span&gt;, that is, the rotational or solenoidal component of the total electric field.&lt;/p&gt;&#xA;&lt;p&gt;So, by rights, S.V. ought to use&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathscr{E}_{ind} = \int_{\gamma_{A\rightarrow B}} \vec{E}_{ind} \cdot d\vec{\ell}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;for the magnetically induced EMF when recasting KVL. That is, he should acknowledge that his claim that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the twentyfirst century we recognize that each term in parenthesis represents the path integral of the total electric field along each of the segments 1, 2,... n, in which the closed path (the loop) has been partitioned. Therefore, &lt;strong&gt;in modern terms, the original formulation of KVL says&lt;/strong&gt; [emphasis added]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sum of path integral of Etot.dl along the branches of our closed figure = 0&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is wrong. The original formulation uses the specific term &amp;quot;emf&amp;quot;, rather than &amp;quot;voltage&amp;quot;, and by S.V.'s own account, the &amp;quot;emf&amp;quot; induced along a branch of a circuit by a time-varying magnetic field is &lt;em&gt;not&lt;/em&gt; the path integral of Etot.dl, but the path integral of Eind.dl&lt;/p&gt;&#xA;" OwnerUserId="14184" LastEditorUserId="14184" LastEditDate="2021-12-17T16:37:01.980" LastActivityDate="2021-12-17T16:37:01.980" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13613" PostTypeId="2" ParentId="13606" CreationDate="2021-08-30T02:10:43.053" Score="2" Body="&lt;p&gt;From the BKS paper itself (emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On the one hand, the phenomena of interference, on which the action of all optical instruments essentially depends, claim an aspect of continuity of the same character as that involved in the wave theory of light, especially developed on the basis of the laws of classical electrodynamics. On the other hand, the exchange of energy and momentum between matter and radiation, on which the observation of optical phenomena ultimately depends, claims essentially discontinuous features. These have even led to the introduction of the theory of light-quanta, which in its &lt;strong&gt;most extreme form&lt;/strong&gt; denies the wave constitution of light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So BKS here give a good hint that at that moment there were a variety o views regarding the existence of photons. Still regarding the photons:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although the great heuristic value of this hypothesis is shown by the confirmation of Einstein's predictions concerning the photoelectric phenomenon, still &lt;strong&gt;the theory of light-quanta can obviously not be considered as a satisfactory solution of the problem of light propagation&lt;/strong&gt;. This is clear even from the fact that the radiation 'frequency' &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; appearing in the theory is defined by experiments on interference phenomena which apparently demand for their interpretation a wave constitution of light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As wikipedia states, the interference phenomena really were a block to accepting the photon, at least for the people involved. In the the second section they start explaining their model of the matter-radiation interaction with a focus on the transistions between stationary states in an atom as described by Bohr's model.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As regards the occurrence of transitions, which is the essential feature of the quantum theory, we abandon on the other hand any attempt at a causal connexion between the transitions in distant atoms, and &lt;strong&gt;especially a direct application of the principles of conservation of energy and momentum&lt;/strong&gt;, so characteristic for the classical theories. The application of these principles to the interaction between individual application principles atomic systems is, on our view, limited to interactions which take place when the atoms are so close that the forces which would be connected with the radiation field on the classical theory are small compared with the conservative parts of the fields of force originating from the electric charges in the atom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and later:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The cause of the observed statistical conservation of energy and momentum we shall not seek in any departure from the electrodynamic theory of light as regards the laws of propagation of radiation in free space, but in the peculiarities of the interaction between the virtual field of radiation and the illuminated atoms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So yes, the paper assumes that conservation of energy/momentum would be violated in individual atomic transitions involving atoms far away from each other, being maintained only statistically on macroscopic matter.&lt;/p&gt;&#xA;" OwnerUserId="5904" LastActivityDate="2021-08-30T02:10:43.053" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13614" PostTypeId="1" AcceptedAnswerId="13618" CreationDate="2021-08-30T13:06:21.237" Score="17" ViewCount="1326" Body="&lt;p&gt;In &lt;a href=&quot;https://link.springer.com/book/9783540287346&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yesterday and Long Ago&lt;/a&gt; (2007), mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Arnold&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vladimir Arnold&lt;/a&gt;  wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I resided at Cambridge as a senior fellow of Trinity College,Indian colleagues told  me  some  details  of  &lt;a href=&quot;https://en.wikipedia.org/wiki/Srinivasa_Ramanujan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramanujan&lt;/a&gt;'s life which are  not well  known.&lt;/p&gt;&#xA;&lt;p&gt;Once  an  Indian  physicist  who  arrived  from  the  USA  visited Ramanujan in  Trinity.  The  room  of  his  friend  seemed  too  cold for  him,  but  Ramanujan  explained  that  he  felt  cold  only  at night - when  there  is  sometimes  frost  in  Cambridge!  The visitor  asked  to  be  shown  how  Ramanujan  slept,  and  found that  he  slept on blankets  and never  suspected  that he should have  covered  himself  with  them  (people in  Madras  never do this!).  That  is  why  he  felt  cold,  and  that  is  why  he  got ill  (it  seems  that  first  it  was  pneumonia  and  then tuberculosis)  and  died  at  a very  young  age.&lt;/p&gt;&#xA;&lt;p&gt;I  think  that  &lt;a href=&quot;https://en.wikipedia.org/wiki/G._H._Hardy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hardy&lt;/a&gt;'s snobbery  and  his  inhuman  behavior  did not let him visit  his  sick student, who  lived  in the  same house, and  did not let him give  elementary practical  advice. While telling me  this  story,  my  Indian  colleagues  tactfully avoided  a discussion  of  British customs and  connected the  reason for Rarnanujan's  death  with  an  Indian  custom,  according to  which his  wife  did  not  follow  him,  but  stayed  in  Madras.  She  was supposed  to  take  care  of  her  mother-in-law,  Ramanujan's mother; this  duty  was  more  important  than  to  take  care  of her husband!  Since  that  time  Indian  students in Cambridge have been telling each other  how to make a bed and they do not feel cold anymore.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I have never heard this story before and was really surprised. How reliable are these facts and Arnold's explanations? Are there any other sources that corroborate these facts? Possibly that Indian  physicist mentioned by Arnold did write something about this?&lt;/p&gt;&#xA;&lt;p&gt;I asked this question on MathOverflow but it was &lt;a href=&quot;https://mathoverflow.net/questions/402838/does-arnold-say-that-hardy-is-responsible-for-ramanujans-untimely-death?noredirect=1#comment1030254_402838&quot;&gt;closed&lt;/a&gt;, because I had very little reputation they ruled the question is not suitable for their site.&lt;/p&gt;&#xA;" OwnerUserId="15271" LastEditorUserId="4251" LastEditDate="2022-08-17T17:14:41.317" LastActivityDate="2022-08-17T17:14:41.317" Title="Does Arnold say that Hardy is responsible for Ramanujan's untimely death?" Tags="|mathematicians|biographical-details|mathematics-social-history|ramanujan|vladimir-arnold|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13616" PostTypeId="1" CreationDate="2021-08-31T01:09:39.113" Score="5" ViewCount="1337" Body="&lt;p&gt;How, a century ago, could Stern and/or Gerlach KNOW that they had created single silver atoms?&lt;/p&gt;&#xA;&lt;p&gt;How were they moved, or accelerated?&lt;/p&gt;&#xA;" OwnerUserId="15273" LastActivityDate="2021-08-31T18:17:28.263" Title="How did Stern or Gerlach, of Stern-Gerlach experiment, create individual silver atoms? How were they accelerated?" Tags="|physics|quantum-mechanics|electromagnetism|experimental-physics|magnetism|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13617" PostTypeId="2" ParentId="13616" CreationDate="2021-08-31T04:07:27.387" Score="9" Body="&lt;p&gt;Atomic spectroscopy was very advanced 100 years ago (1920s) and we must appreciate their intelligence. If a metal like silver is being heated to the extent of boiling in high vacuum, all you get is silver atoms. Just like when liquid water is strongly heated, one would get gaseous water molecules- each molecule is separate.&lt;/p&gt;&#xA;&lt;p&gt;Stern and Gerlach write in  &lt;em&gt;Der experimentelle Nachweis des magnetischen Moments des Silberatoms.&lt;/em&gt;  (The experimental proof of magnetic moments of silver atoms), Zeitschrift fur Physik 8.1 (1922): 110-111.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ein Silberatomstrahl von &lt;span class=&quot;math-container&quot;&gt;$1 / 20 \mathrm{~mm}$&lt;/span&gt; Durchmesser geht in hohem Vakuum &lt;span class=&quot;math-container&quot;&gt;$(10^{-4}$&lt;/span&gt; bis &lt;span class=&quot;math-container&quot;&gt;$10^{-5} \mathrm{~mm} \mathrm{Hg})$&lt;/span&gt; hart an der Kante des schneidenförmigen Polschuhs eines Elektromagneten [Halbringelektromagnet nach du Bois&lt;sup&gt;2&lt;/sup&gt;] vorbei. Der Strahl kommt aus einem kleinen &lt;span class=&quot;math-container&quot;&gt;$(1 / 2 \mathrm{~cm}^{3}$&lt;/span&gt; Inhalt&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt;, elektrisch geheizten, stählernen Öfchen durch eine im Deckel befindliche, &lt;span class=&quot;math-container&quot;&gt;$1 \mathrm{~mm}^{2}$&lt;/span&gt; große, kreisförmige Offnung. Der Ofen ist von einem wassergekühlten Mantel umgeben. Etwa &lt;span class=&quot;math-container&quot;&gt;$1 \mathrm{~cm}$&lt;/span&gt; vom Ofenloch entfernt passiert er die erste kreisförmige Blende &lt;span class=&quot;math-container&quot;&gt;$(1 / 20 \mathrm{~mm}$&lt;/span&gt; Durchmesser&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt; in einem Platinblech. &lt;span class=&quot;math-container&quot;&gt;$3 \mathrm{~cm}$&lt;/span&gt; hinter dieser passiert er eine zweite, ebensolche Blende, die sich am vorderen Ende des Schneidenpols des Elektromagneten befindet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A silver atom beam of &lt;span class=&quot;math-container&quot;&gt;$1 / 20 \mathrm{~mm}$&lt;/span&gt;  diameter passes in high vacuum &lt;span class=&quot;math-container&quot;&gt;$(10^{-4}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10^{-5} \mathrm{~mm} \mathrm{ Hg})$&lt;/span&gt;  very close by the edge of the blade-shaped pole shoe of an electromagnet (half-ring electromagnet according to du Bois). The stream comes out of a small &lt;span class=&quot;math-container&quot;&gt;$(1 / 2 \mathrm{~cm}^{3}$&lt;/span&gt; capacity&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt;, electrically heated, steel furnace through a &lt;span class=&quot;math-container&quot;&gt;$1 \mathrm{~mm}^{2}$&lt;/span&gt; sized circular opening located in the lid. The furnace is surrounded by a water-cooled jacket.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translated with &lt;a href=&quot;http://www.DeepL.com/Translator&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.DeepL.com/Translator&lt;/a&gt; (free version) with some editing by @Hobbamok (native German speaker).&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="10563" LastEditDate="2021-08-31T18:17:28.263" LastActivityDate="2021-08-31T18:17:28.263" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="13618" PostTypeId="2" ParentId="13614" CreationDate="2021-08-31T07:40:41.157" Score="27" Body="&lt;p&gt;I think it is fairly obvious from the quote that Arnold is blaming Hardy's neglect for Ramanujan's  death. The real question is whether his charge is supported by evidence. Arnold's evidence comes from unnamed &amp;quot;Indian colleagues&amp;quot; who got it from also unnamed &amp;quot;Indian physicist&amp;quot;, accompanied by his personal &amp;quot;it seems&amp;quot; diagnosis of  Ramanujan's ailment, and complete with also his &amp;quot;I think&amp;quot; theory about Hardy's behavior.&lt;/p&gt;&#xA;&lt;p&gt;As it happens, documents concerning the final years of Ramanujan's life and his medical condition have been collected by &lt;a href=&quot;https://link.springer.com/article/10.1007/BF02840652&quot; rel=&quot;noreferrer&quot;&gt;Rankin in Ramanujan as a patient&lt;/a&gt; and later analyzed by &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rsnr.1994.0009&quot; rel=&quot;noreferrer&quot;&gt;Young in Ramanujan’s illness&lt;/a&gt;. And there is plenty in them to tell that Arnold's anecdote is just out there.&lt;/p&gt;&#xA;&lt;p&gt;For one, at Cambridge Hardy saw Ramanujan almost every day. As one can see from his correspondence with Ram and Thomson, Hardy was also closely following minutiae of Ramanujan's circumstances during his illness, including his stays, diet and specifics of diagnoses and changes in medical condition. He also advocated for his recognition by the college to provide moral support and remedy his depressions.&lt;/p&gt;&#xA;&lt;p&gt;For two, Ramanujan spent more time at various nursing homes and other medical establishments than at Trinity. Even assuming that he could not figure out how to use blankets to stay warm, and could not get tips from anyone other than Hardy at Trinity, it is hard to believe that doctors would have missed that.&lt;/p&gt;&#xA;&lt;p&gt;For three, there are plenty of other, well documented, factors that contributed to Ramanujan's medical problems, like malnourishment and sleep deprivation. His chronic depressions even led to a suicide attempt by falling in front of a train in a London subway station (the train stopped in time):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is clear that there were times when, because of the unavailability of acceptable Indian dishes, he was reduced to a very restricted diet of bread, milk and fried rice cakes (aplams). There is ample evidence that as a patient his recovery was retarded by his obstinacy in dietary and other matters. There is confirmation for this dating back to 1909 [10, p. 74], before he left India, throughout his time as a patient in Matlock (see below) and after his return to India in 1919.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is probable that Ramanujan's fits of depression were the result of the rather solitary life he lived while in Cambridge and in the various nursing homes where he was a patient. For this his working habits and vegetarianism were largely to blame. B M Wilson [W 107C, 11 (1-6)], who was an undergraduate in Trinity at the time, notes that Ramanujan was very rarely seen in Cambridge and that he sometimes worked for 30 or so hours at a stretch and then slept for 20.&lt;/em&gt;&amp;quot; [quoted from Rankin]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And finally, the &amp;quot;first pneumonia then tuberculosis&amp;quot; has always been shaky:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It appears that the first diagnosis of gastric ulcer was later changed, on not very firm evidence, to tuberculosis. By the end of 1918 his health had improved, and his election to the Royal Society and a Fellowship at Trinity may have been instrumental in this... Evidently, by this time the diagnosis of tuberculosis had been abandoned by his English doctors. On his return to India his health did not improve further, as had been hoped. Tuberculosis was again the preferred diagnosis.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Young speculates, based on detailed analysis of Ramanujan's medical history and symptoms, that Ramanujan had hepatic amebiasis, an aggressive infection of the large intestine that goes back to his 1906 bout of dysentery, of which his English doctors were unaware.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-08-31T23:05:58.560" LastActivityDate="2021-08-31T23:05:58.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13619" PostTypeId="1" AcceptedAnswerId="13665" CreationDate="2021-08-31T10:59:53.170" Score="0" ViewCount="133" Body="&lt;p&gt;I am doing a marathon of data visualizations with real world datasets. I am interested in historical war datasets.&lt;/p&gt;&#xA;&lt;p&gt;Does anybody know about the historical data on The Bombe Machine cracking the codes of enigma and how did the &amp;quot;minimization actions&amp;quot; of the machine were used to avoid Nazi to realize what was happening and why they were losing the battles?&lt;/p&gt;&#xA;" OwnerUserId="10607" LastEditorUserId="4251" LastEditDate="2021-09-08T06:13:33.463" LastActivityDate="2021-09-08T06:13:33.463" Title="Where can I find the historical information or the stats related to Winning WW2 with the minimum actions using The Bombe Machine of Alan Turing?" Tags="|reference-request|statistics|world-war-2|turing|cryptology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13620" PostTypeId="2" ParentId="13616" CreationDate="2021-08-31T17:41:22.953" Score="4" Body="&lt;p&gt;For the second part of the question: they weren't accelerated in any way other than by heating, they formed what is known as a thermal beam. Atoms in the gaseous phase are always moving, with an average speed determined by their temperature. At higher pressures the atoms all collide with each other about every 100 nanometres, but under high vacuum they can travel long distances in straight paths. If you heat a sample (under vacuum) to the point that it produces a significant amount of vapour, and then let the vapour pass through two slits or apertures (called collimators) to select only the atoms going in a certain direction, you form a (low intensity) beam as shown in this sketch of the apparatus (with Cs instead of Ag), from &lt;a href=&quot;https://cronodon.com/Atomic/quantum_angular_momentum.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/AlKR7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/AlKR7.png&quot; alt=&quot;Stern-Gerlach apparatus showing two apertures&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As long as the distances aren't too large and the temperature of the beam isn't extremely low, the effect of gravity on the trajectory of the beam is negligible and it can be treated as travelling in a straight line.&lt;/p&gt;&#xA;" OwnerUserId="12826" LastActivityDate="2021-08-31T17:41:22.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13621" PostTypeId="2" ParentId="13607" CreationDate="2021-09-01T09:08:06.247" Score="2" Body="&lt;p&gt;Apparently ever since precision in astronomy became really important professionals did use ecliptic coordinates. Hipparchus had demonstrated the precession of the vernal point and its position against the stars as the Zero of ecliptic became problematic. Already Babylonians had two values for it, either as 10 or 8 of Aries; Eudoxus said it was 15 but Hipparchus took it to be 0 (Neugebauer, &lt;em&gt;Sel. Essays&lt;/em&gt; p.251) Initially it was thought that the vernal point does not drift through the zodiac but merely oscillates (this was called &lt;a href=&quot;https://en.wikipedia.org/wiki/Trepidation&quot; rel=&quot;nofollow noreferrer&quot;&gt;trepidation&lt;/a&gt;). Establishing positions with reference to the vernal point has been preferred as less contenious than the use of constellation limits. Precise values for them were adopted by International  Astronomical Union only in 1930.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2021-09-03T18:25:08.670" LastActivityDate="2021-09-03T18:25:08.670" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13622" PostTypeId="2" ParentId="130" CreationDate="2021-09-01T16:53:00.347" Score="5" Body="&lt;p&gt;Charles Babbage (the grandfather of the computer) in the &lt;em&gt;Reflections on the Decline of Science in England, and on Some of Its Causes&lt;/em&gt; (1830) writes about misconduct in science and criticizes British science for lack of disciplined data recollection. He provides an insight of how common it was.&lt;/p&gt;&#xA;&lt;p&gt;He divided fraud in four practices:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Hoaxing: falsifying the object of study, creating creatures or phenomena that do not exist. Not so commonly practiced.&lt;/li&gt;&#xA;&lt;li&gt;Forging: fabricating data. Not so widespread, but much more damaging because it can be difficult to notice.&lt;/li&gt;&#xA;&lt;li&gt;Trimming: taking bits of data out that did not fit. More extensively used.&lt;/li&gt;&#xA;&lt;li&gt;Cooking: cherry-picking observations, conclusions that do not follow from the facts, removing unfavorable results. Widespread among reputable scientists, who make a name out of their results.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;He also list various examples of fraud throughout the book. He emphatically calls out problems with &lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Sabine&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward Sabine&lt;/a&gt;'s extraordinary accuracy with pendulum calculations for which Sabine had received some grant. However, Babbage's accusations did not affect Sabine's career as he became later the president of the Royal Society.&lt;/p&gt;&#xA;&lt;p&gt;Babbage also cites an example of data forgery from the 18th century&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The observations of the second comet of 1784, which was only seen by the Chevalier D'Angos, were long suspected to be a forgery, and were at length proved to be so by the calculations and reasonings of Encke. The pretended observations did not accord amongst each other in giving any possible orbit. But M. Encke detected an orbit, belonging to some of the observations, from which he found that all the rest might be almost precisely deduced, provided a mistake of a unity in the index of the logarithm of the radius vector were supposed to have been made in all the rest of the calculations. ZACH. CORR. ASTRON. Tom. IV. p. 456.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-09-04T13:49:29.917" LastActivityDate="2021-09-04T13:49:29.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13623" PostTypeId="1" CreationDate="2021-09-02T16:22:03.307" Score="1" ViewCount="133" Body="&lt;p&gt;I have a question about ancient Greek astronomy. We know certainly that the likes of Pythagoras, Aristotle, Anaximander, et al had much to say about the motions of the stars, planets, comets, etc,  mythology notwithstanding.&lt;/p&gt;&#xA;&lt;p&gt;My question is a rather simple one, yet I have not been able to find an answer anywhere.&lt;/p&gt;&#xA;&lt;p&gt;Just how much of this knowledge would the ‘man in the street’ been aware of?? For example, how likely would they be to know of Pythagoras’ explanation that the morning star and evening star were both Venus, or that Aristotle thought comets were sublunar objects?&lt;/p&gt;&#xA;&lt;p&gt;Surely these and other early Greek philosophers did not publish in scientific journals or publish popular literature. So, how was their thinking spread and how far? Did it reach only those select few who  attended their lectures? Or, if the ‘common man' was aware, how would they know? Were the early Greeks more likely to accept that the sun’s daily motion was down to Helios driving his golden chariot rather than a more ’scientific' explanation?&lt;/p&gt;&#xA;" OwnerUserId="15287" LastActivityDate="2021-09-05T21:30:09.980" Title="To what extent was astronomical knowledge in ancient Greece common knowledge?" Tags="|astronomy|ancient-greece|social-context|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13624" PostTypeId="1" CreationDate="2021-09-02T17:57:43.707" Score="1" ViewCount="1195" Body="&lt;p&gt;Wikipedia says he died of natural causes. I recall going through a N. Wildberger lecture on the history of mathematics and got a different account. He was stabbed by a Roman soldier while Rome was busy conquering Greece. There was a model of the solar system in his room. Etc...What is the fact here?&lt;/p&gt;&#xA;&lt;p&gt;Edit: Yeah I did confuse Archimedes with Aristotle. My bad.&lt;/p&gt;&#xA;" OwnerUserId="6084" LastEditorUserId="6084" LastEditDate="2021-09-03T02:24:20.980" LastActivityDate="2021-09-03T02:24:20.980" Title="How did Aristotle die?" Tags="|aristotle|" AnswerCount="1" CommentCount="4" ClosedDate="2021-09-03T14:53:15.390" ContentLicense="CC BY-SA 4.0" />
  <row Id="13625" PostTypeId="2" ParentId="13624" CreationDate="2021-09-02T18:33:48.780" Score="2" Body="&lt;p&gt;There were no Roman soldiers in Greek lands when Aristotle died. Probably he died of natural causes, though I've heard of a speculation that he was murdered on the order of Alexander the great. It is true that at some point Alexandre went crazy&#xA;and started killing his friends, for example Callisthenes, a great nephew of Aristotle. But I think this speculation about ordering to kill Aristotle is  unjustified.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-09-02T18:33:48.780" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13626" PostTypeId="2" ParentId="13623" CreationDate="2021-09-02T18:43:51.930" Score="3" Body="&lt;p&gt;This question is impossible to answer with any certainty in principle, since we know too little about common people. All literature which reached us was written by people of certain class, and they wrote for each other. Most people were probably barely literate.&lt;/p&gt;&#xA;&lt;p&gt;There is however some indirect evidence. At some point Greek science and philosophy, spread and became fashionable in Rome (also among upper classes, of course). However from what survived of Roman literature we know that they understood very little about Greek mathematics or astronomy. They were not interested. They were much more interested in Babylonian astrology.&lt;/p&gt;&#xA;&lt;p&gt;Approximately as much as modern general public understand about modern mathematics and astronomy, perhaps less, since nowadays we have some kind of education for all, and all now that &amp;quot;science is important&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-09-02T18:43:51.930" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13627" PostTypeId="2" ParentId="13583" CreationDate="2021-09-02T21:22:16.637" Score="1" Body="&lt;p&gt;I would think that the so-called Magnetischer Verein, initiated and led by Gauß and Weber in Göttingen, qualifies. It formed a network of magnetic observatories that measured the elements of the geomagnetic field for several years in the 1830s according to a common protocol and involved participants especially from the German states, Austria, and Russia (including Finland, which was a part of the Russian empire at that time). The Verein also had its own report series.&lt;/p&gt;&#xA;" OwnerUserId="15289" LastActivityDate="2021-09-02T21:22:16.637" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13628" PostTypeId="1" AcceptedAnswerId="13630" CreationDate="2021-09-03T11:26:03.997" Score="2" ViewCount="112" Body="&lt;p&gt;While preparing trigonometric exercises for my students, I learned that, in 1771, French astronomers determined the distance of the Moon from the Earth by measuring the appropriate angles from both Berlin and Cape Town at the same time. I would like to know more about this such as who these astronomers were and where were the results published?&lt;/p&gt;&#xA;&lt;p&gt;My Google-fu skills were obviously not black belt since I couldn't find anything. This could be because I searched with the date 1771 whereas I suppose a publication would come at a later date.&lt;/p&gt;&#xA;" OwnerUserId="13640" LastEditorUserId="4251" LastEditDate="2021-09-09T06:43:27.557" LastActivityDate="2021-09-09T06:43:27.557" Title="Lunar distance measurement reference" Tags="|reference-request|astronomy|18th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13630" PostTypeId="2" ParentId="13628" CreationDate="2021-09-03T12:47:34.477" Score="3" Body="&lt;p&gt;I can describe what they probably did, and why this measurement became possible in the end of 18th century, but to find the reference on this specific work, you give insufficient information. Names of these astronomers would help to find a reference.&lt;/p&gt;&#xA;&lt;p&gt;Measuring a distance to the moon has a long history beginning at least from Hipparchus (2nd century BC) who obtained the right order of magnitude. It was improved by Ptolemy (2nd century AD). And was always improved since then, so the measurement that you mention is just one step in the long progress.&lt;/p&gt;&#xA;&lt;p&gt;From your description, what they used is the most straightforward procedure, which leads to a very precise result, but the use of this procedure was technically impossible until the second half of 18th century.&lt;/p&gt;&#xA;&lt;p&gt;If you have two observers A and B in Berlin and Paris, you can measure two angles at A and B of the triangle ABM, where M is a point on the Moon. Two angles determine the triangle up to similarity, so if you know the distance AB, you obtain the distance to the moon. The difference between these two angles is called parallax. There are two problems involved:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The distance AB. In 18th century this was determined with very high accuracy by triangulation. This is exactly when they started to make accurate maps based on land surveys using triangulation nets.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Clock synchronization in Berlin and Paris: the measurement had to be made simultaneously, since the Moon moves. This was a major scientific problem since XVI century. And the second half of 18th century is the time when it was finally solved. One of the solutions was the invention of an accurate mechanical clock which can be transported long distance while keeping exact time.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;These two things made such a measurement possible.&lt;/p&gt;&#xA;&lt;p&gt;All previous measurements were based on more primitive technology but on a much more sophisticated theory. Namely the theory which can predict the Moon&#xA;motion, as seen from the center of the Earth. Then measuring the actual position of the Moon in the sky from a single place, you find the difference which is called parallax, and this gives you the distance. Interestingly, you can use the theory of Moon motion for time measurement, by a kind of reverse procedure.&lt;/p&gt;&#xA;&lt;p&gt;Refs.&#xA;&lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/2007/2007.14504.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/ftp/arxiv/papers/2007/2007.14504.pdf&lt;/a&gt;&#xA;and also &lt;a href=&quot;https://faculty.humanities.uci.edu/bjbecker/ExploringtheCosmos/lecture16.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://faculty.humanities.uci.edu/bjbecker/ExploringtheCosmos/lecture16.html&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Added: the exact reference was finally found: see comments to this answer.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-04T12:08:49.750" LastActivityDate="2021-09-04T12:08:49.750" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13631" PostTypeId="1" AcceptedAnswerId="13632" CreationDate="2021-09-03T19:01:28.657" Score="7" ViewCount="1012" Body="&lt;p&gt;I'm looking for a source (of which I've only ever read a quote) discussing the then seven known planets. In particular, I'm looking for the part where the author explains why that number seven makes sense with reference to (I think a Christian) God. His explanation goes something like this: the planets are holes in a celestial sphere; God mirrors those in the sphere of our own heads; and so just as there are seven holes in the head -- one mouth, plus two each of nostrils, eyes, and ears -- so are there seven planets in the heavens.&lt;/p&gt;&#xA;&lt;p&gt;I &lt;em&gt;think&lt;/em&gt; the writer was one of the Bacons, and probably Roger, in which case a likely candidate is his Opus Majus. But so far I haven't been able to find what I'm after.&lt;/p&gt;&#xA;" OwnerUserId="15291" LastEditorUserId="55" LastEditDate="2021-09-03T22:43:11.660" LastActivityDate="2021-09-03T22:43:11.660" Title="Did Bacon analogize planets to holes in the head to explain why their number was (believed to be) seven?" Tags="|astronomy|social-context|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13632" PostTypeId="2" ParentId="13631" CreationDate="2021-09-03T22:32:44.653" Score="14" Body="&lt;p&gt;This probably refers to the &amp;quot;argument&amp;quot; of the Florentine astronomer Francesco Sizzi against the validity of Galileo's discovery of the moons of Jupiter. His &amp;quot;argument&amp;quot; was &lt;a href=&quot;https://www.stephenhicks.org/2017/03/11/why-there-can-be-only-seven-planets/&quot; rel=&quot;noreferrer&quot;&gt;apparently quoted&lt;/a&gt; from the introduction to &lt;a href=&quot;https://archive.org/details/francisbaconsele00baco/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Francis Bacon, A Selection of His Works&lt;/a&gt; (where it illustrates the context of the times), and thereby entered the internet, but it is (thankfully) not Bacon's. Here is the full quote as translated in &lt;a href=&quot;https://archive.org/details/galileohislifewo00fahiuoft/page/102/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Fahie, Galileo, his life and work, p.103&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There are seven windows given to animals&#xA;in the domicile of the head, through which the&#xA;air is admitted to the tabernacle of the body, to&#xA;enlighten, to warm, and to nourish it. What&#xA;are these parts of the microcosmos? Two nostrils,&#xA;two eyes, two ears, and a mouth. So in the&#xA;heavens, as in a macrocosmos, there are two&#xA;favourable stars, two unpropitious, two luminaries,&#xA;and Mercury undecided and indifferent.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;From this and many other similarities in nature, such as the seven metals, etc., which it were tedious&#xA;to enumerate, we gather that the number of&#xA;planets is necessarily seven. Moreover, these&#xA;satellites of Jupiter are invisible to the naked&#xA;eye, and therefore can exercise no influence on&#xA;the earth, and therefore would be useless, and&#xA;therefore do not exist. Besides, the Jews and&#xA;other ancient nations, as well as modern&#xA;Europeans, have adopted the division of the&#xA;week into seven days, and have named them&#xA;after the seven planets. Now, if we increase&#xA;the number of the planets, this whole and&#xA;beautiful system falls to the ground.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;logic&amp;quot; is ironclad.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-09-03T22:42:18.330" LastActivityDate="2021-09-03T22:42:18.330" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13633" PostTypeId="2" ParentId="3131" CreationDate="2021-09-04T04:36:05.223" Score="1" Body="&lt;p&gt;Ada Lovelace, the 1st computer programmer, died at 36yo. From &lt;a href=&quot;https://en.wikipedia.org/wiki/Ada_Lovelace#Death&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiki&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lovelace died at the age of 36 on 27 November 1852,[48] from uterine cancer probably exacerbated by bloodletting by her physicians.[49] The illness lasted several months, in which time Annabella took command over whom Ada saw, and excluded all of her friends and confidants. Under her mother's influence, Ada had a religious transformation and was coaxed into repenting of her previous conduct and making Annabella her executor.[50] She lost contact with her husband after confessing something to him on 30 August which caused him to abandon her bedside. It is not known what she told him.[51] She was buried, at her request, next to her father at the Church of St. Mary Magdalene in Hucknall, Nottinghamshire. A memorial plaque, written in Latin, to her and her father is in the chapel attached to Horsley Towers.[citation needed]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3338" LastActivityDate="2021-09-04T04:36:05.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13634" PostTypeId="1" CreationDate="2021-09-04T10:16:51.650" Score="13" ViewCount="9584" Body="&lt;p&gt;Inspired by &lt;a href=&quot;https://hsm.stackexchange.com/questions/3131/which-mathematicians-died-very-young-or-in-a-tragic-way&quot;&gt;Which mathematicians died very young or in a tragic way?&lt;/a&gt; , I wonder which physicists had similar fates.&lt;/p&gt;&#xA;&lt;p&gt;A quick search lead me to Heinrich Hertz who died from a malignant bone condition at 36. Ludwig Boltzmann is also known for committing suicide before seeing how important his contribution was to prove the existence of atoms. Paul Ehrenfest had also a very tragic story that I am unable to write down here. What are some examples? Be free to expand on the above cited. Was there some Galois kind of figure?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-09-05T10:23:55.553" LastActivityDate="2024-03-07T18:45:53.080" Title="Which physicists died very young or in a tragic way?" Tags="|physics|physicists|death|" AnswerCount="12" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13635" PostTypeId="2" ParentId="13634" CreationDate="2021-09-04T12:57:44.480" Score="18" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Moseley#:~:text=Moseley%20was%20assigned%20to%20the,at%20the%20age%20of%2027&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henry Moseley&lt;/a&gt; was KIA at Gallipoli at the age of 27&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ren%C3%A9_Gateaux&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rene Gateaux&lt;/a&gt; was KIA in the Battle of Grand Couronné at the age of 25. (France lost many young mathematicians in WWI. The list can be continued).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Frank_Ramsey_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frank Ramsey&lt;/a&gt;, mathematician who died at the age of 27. He was the founder of&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ramsey_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramsey theory&lt;/a&gt;, a hot topic in todays research. Wikpedia lists his important contributions to other sciences, but not to pure math! It is truly amazing how much he accomplished by the age of 27.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Friedmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexander Friedmann&lt;/a&gt; died at the age of 37 (of poisoning and/or poor treatement).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Matvei_Petrovich_Bronstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;Matvei Bronstein&lt;/a&gt; was executed by Soviet communists at the age of 32.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lev_Shubnikov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lev Shubnikov&lt;/a&gt; was executed by Soviet communists at the age of 36.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%A8%D1%83%D0%B1%D0%B8%D0%BD,_%D0%A1%D0%B5%D0%BC%D1%91%D0%BD_%D0%9F%D0%B5%D1%82%D1%80%D0%BE%D0%B2%D0%B8%D1%87&quot; rel=&quot;nofollow noreferrer&quot;&gt;Semyon Shubin&lt;/a&gt; [ru] died in Soviet prison at the age of 30.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hans_Hellmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hans Hellman&lt;/a&gt; was executed by Soviet communists at the age of 35.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%92%D0%B8%D1%82%D1%82,_%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80_%D0%90%D0%B4%D0%BE%D0%BB%D1%8C%D1%84%D0%BE%D0%B2%D0%B8%D1%87&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexander Witt&lt;/a&gt; [ru] died in a Soviet prison camp at the age of 36.&lt;/p&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A4%D0%B8%D0%B7%D0%B8%D0%BA%D0%B8,_%D1%80%D0%B5%D0%BF%D1%80%D0%B5%D1%81%D1%81%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D0%B2_%D0%A1%D0%A1%D0%A1%D0%A0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russian Wikipedia&lt;/a&gt;&#xA;has a list of physicists killed or imprisoned by Soviet authorities; I selected from this list those who were killed at the age less than 40, and gave links to English Wikipedia when available.)&#xA;I don't have a list of mathematicians, but the number of victims of Soviet regime was very large.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-07T18:45:53.080" LastActivityDate="2024-03-07T18:45:53.080" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13637" PostTypeId="2" ParentId="3147" CreationDate="2021-09-04T16:51:17.150" Score="3" Body="&lt;p&gt;Here's a link to Jordan Bell's English translation of Euler's original paper, &amp;quot;&lt;a href=&quot;https://arxiv.org/abs/1009.1439&quot; rel=&quot;nofollow noreferrer&quot;&gt;Various observations on angles proceeding in geometric progression&lt;/a&gt;&amp;quot;. Euler begins exactly as the OP outlines, starting from the double-angle formula for sine. By the second page he has given the following version of the OP's formula:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Therefore the arc &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; itself can be very prettily defined by its sine and the&#xA;cosines of arcs continually diminished in double ratio, as &lt;span class=&quot;math-container&quot;&gt;$$s = \frac{\sin. s}{\cos. \frac{1}{2}s \cos. \frac{1}{4}s \cos. \frac{1}{8}s \cos. \frac{1}{16}s \cos. \frac{1}{32}s \cdot \operatorname{etc.}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, &lt;a href=&quot;https://hsm.stackexchange.com/users/64/alexandre-eremenko&quot;&gt;Alexandre Eremenko&lt;/a&gt;'s &lt;a href=&quot;https://hsm.stackexchange.com/a/3151/&quot;&gt;answer&lt;/a&gt; makes a lot of sense, especially since from a modern perspective, we know the infinite product formulas&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\cos \frac{x}{2} = &amp;amp;\prod_{n=0}^\infty \left( 1 - \frac{x^2}{\pi^2 (2n+1)^2} \right), \\&#xA;\frac{\sin x}{x} = &amp;amp;\prod_{k=0}^\infty \left( 1 - \frac{x^2}{\pi^2 k^2} \right). \&#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Clearly in the second product, every integer &lt;span class=&quot;math-container&quot;&gt;$k \geq 1$&lt;/span&gt; can be expressed as a product &lt;span class=&quot;math-container&quot;&gt;$k = (2n+1) 2^m$&lt;/span&gt; for unique nonnegative integers &lt;span class=&quot;math-container&quot;&gt;$n, m \geq 0,$&lt;/span&gt; so we find&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{align*}&#xA;\frac{\sin x}{x} = &amp;amp;\prod_{k=0}^\infty \left( 1 - \frac{x^2}{\pi^2 k^2} \right) \\&#xA; = &amp;amp;\prod_{m=0}^\infty \prod_{n=0}^\infty \left( 1 - \frac{x^2}{\pi^2 (2^m (2n+1))^2} \right) \\&#xA; = &amp;amp;\prod_{m=0}^\infty \cos \left( \frac{x}{2^{m+1}}\right) \\&#xA; =&amp;amp; \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cos \frac{x}{16}... \  &#xA;\end{align*}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's not clear to me why &amp;quot;one should be mad&amp;quot; to discover or prove the identity this way.&lt;/p&gt;&#xA;&lt;p&gt;In fact, Euler &lt;em&gt;did&lt;/em&gt; know the infinite product formula &lt;span class=&quot;math-container&quot;&gt;$$\frac{\sin x}{x} = \prod_{k=0}^\infty \left( 1 - \frac{x^2}{\pi^2 k^2} \right),$$&lt;/span&gt; although per &lt;a href=&quot;https://math.stackexchange.com/users/118193/heropup&quot;&gt;heropup&lt;/a&gt;'s &lt;a href=&quot;https://math.stackexchange.com/a/674790/&quot;&gt;answer&lt;/a&gt; linking to &lt;a href=&quot;http://eulerarchive.maa.org/hedi/HEDI-2004-03.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;an old Ed Sandifer article&lt;/a&gt;, Euler never actually proved this formula rigorously.&lt;br /&gt;&#xA;In his paper &amp;quot;&lt;a href=&quot;https://arxiv.org/abs/math/0506415&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the sums of series of reciprocals&lt;/a&gt;&amp;quot; (again translated by Jordan Bell), Euler used it to calculate various specific series of the form &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty \frac{1}{n^{2p}}$&lt;/span&gt;, including the first solution to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Basel_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Basel problem&lt;/a&gt;.&lt;br /&gt;&#xA;In Euler's notation below (from p. 6 of Bell's translation), &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is an angle (or &amp;quot;arc&amp;quot;), and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; represents &lt;span class=&quot;math-container&quot;&gt;$\sin s$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/gYGYl.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/gYGYl.png&quot; alt=&quot;Euler compares the infinite series expansion for sin s divided by s to the infinite product expansion which has zeroes at positive and negative multiples of &amp;quot;p&amp;quot;, i.e. pi.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And on the next page, Euler states that &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p648s.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p648s.png&quot; alt=&quot;The sum of the reciprocals of the positive integer squares equals pi squared, divided by 6.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12930" LastActivityDate="2021-09-04T16:51:17.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13638" PostTypeId="2" ParentId="9" CreationDate="2021-09-04T17:29:03.123" Score="6" Body="&lt;p&gt;My copy of &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3764354100&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laura Toti Rigatelli's biography of Galois&lt;/a&gt; (&lt;em&gt;Vita Mathematica,&lt;/em&gt; Volume 11), gives the following account of Galois's end in Section 4, &amp;quot;To Louis-Philippe!&amp;quot; and Section 5, &amp;quot;A Pointless Death&amp;quot;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Galois was a political prisoner during a period of unrest against the French monarchy under Louis-Philippe (he was one of the republicans seeking to incite a revolution). In the course of a cholera epidemic, he was transferred from the prison at Sainte-Pelagie to a hospital, where he fell in love with the daughter, Stephanie, of one of the doctors there.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Stephanie rejected him, or didn't feel the same way (much of their correspondence was destroyed by Galois, so the exact circumstances were unclear). In a fit of depression over romantic rejection, as well as his mathematical discoveries being variously overlooked, mislaid, or rejected by the Paris academic establishment, he resolved to martyr himself for the Republican cause, hoping that his death would be the inciting incident for an armed revolution:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The need for an armed uprising was immediately accepted. All that was missing was a pretext to provoke the fury of the crowds, and a date. One idea, that was not at first taken very seriously, was that a corpse to be revenged would be very useful. A hero was needed, in whose name the people of Paris would fight, a name to shout, while firing on Louis-Philippe's police, a name on the lips of the dying. The discussion soon heated up, and, as his companions expressed their views, Galois, who had not spoken up to that point, became more and more excited.&lt;/p&gt;&#xA;&lt;p&gt;He soon asked to speak, in such an authoritative tone, that all the others immediately fell silent. He explained, movingly, that his life had become pointless. All that was left for him was to offer it to the only thing he still loved: France. The corpse they needed would be his.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In the coming days, Galois wrote a number of letters designed to create a cover story that he was &amp;quot;challenged to a duel by two patriots&amp;quot; (i.e. monarchists). In fact, he had arranged with his friend L.D. to stage a duel in the course of which Galois would be fatally shot. The duel, and Galois's death, proceeded as planned. At his funeral,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;roughly 3,000 people were present, ready to attack the police, as soon as the coffin had been lowered into the grave.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;However, ironically, General Lamarque had recently died, and this news was made known during Galois's funeral. Expecting that Lamarque's funeral would be a better occasion, with a larger crowd, for an armed uprising, the decision was made to put off the revolution for a couple days, and Galois' funeral &amp;quot;came to a hasty, silent end.&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The specific provenance of these details is unclear. Rigatelli's book gives an extensive historical bibliography, but doesn't provide in-line citations; that would be the obvious starting point for further investigation, although several of the sources seem to be early-1800s works in French, and might not be readily accessible online (or to English speakers).&lt;/p&gt;&#xA;" OwnerUserId="12930" LastActivityDate="2021-09-04T17:29:03.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13639" PostTypeId="1" CreationDate="2021-09-04T22:35:25.283" Score="1" ViewCount="130" Body="&lt;p&gt;It's usually believed that Fermat's claim that he had a proof for the Last Theorem is false, and that it might have been more of a conjecture. Or considering it took many centuries and advanced mathematics to prove it, Fermat may have been mistaken in thinking he did have a proof.&lt;/p&gt;&#xA;&lt;p&gt;But how did he come up with that theorem in the first place?&lt;/p&gt;&#xA;" OwnerUserId="15299" LastActivityDate="2021-09-04T22:35:25.283" Title="How did Fermat come up with his Last Theorem?" Tags="|mathematics|number-theory|fermat|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13640" PostTypeId="2" ParentId="13634" CreationDate="2021-09-05T00:05:13.277" Score="15" Body="&lt;p&gt;A few more.  At the time of their deaths ...&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alexis_Th%C3%A9r%C3%A8se_Petit&quot; rel=&quot;noreferrer&quot;&gt;Alexis Thérèse Petit&lt;/a&gt; was 28 (tuberculosis): known for work in thermodynamics, including the Dulong–Petit law&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Walther_Ritz&quot; rel=&quot;noreferrer&quot;&gt;Walther Ritz&lt;/a&gt; was 31 (tuberculosis): known primarily for his numerical eigenvalue approximation technique (the Rayleigh–Ritz method) but wikipedia mentions that he also made an empirical discovery regarding the spectral lines for atoms.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Otto_Sackur&quot; rel=&quot;noreferrer&quot;&gt;Otto Sackur&lt;/a&gt; was 34 (chemical explosion): known for the Sackur–Tetrode equation&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hugo_Tetrode&quot; rel=&quot;noreferrer&quot;&gt;Hugo Tetrode&lt;/a&gt; was 35 (tuberculosis): known for the Sackur–Tetrode equation&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot&quot; rel=&quot;noreferrer&quot;&gt;Nicolas Léonard Sadi Carnot&lt;/a&gt; was 36 (cholera): well known pioneer in thermodynamics (e.g., the Carnot cycle)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Also, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ettore_Majorana&quot; rel=&quot;noreferrer&quot;&gt;Ettore Majorana&lt;/a&gt; disappeared at the age of 31. He introduced the concept of Majorana fermions, among other things. In his well known biography on Fermi, Emilio Segrè mentions some important work Majorana did in nuclear physics which went unpublished.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2021-09-05T11:44:28.653" LastActivityDate="2021-09-05T11:44:28.653" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13641" PostTypeId="2" ParentId="13634" CreationDate="2021-09-05T03:41:18.710" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://fr.wikipedia.org/wiki/Louis_Cartan&quot; rel=&quot;noreferrer&quot;&gt;Louis Cartan&lt;/a&gt;, son of Élie Cartan and brother of Henri Cartan, was a member of the French Resistance and was executed by the Germans in 1943, when he was 33 years old.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2021-09-05T03:41:18.710" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13642" PostTypeId="1" CreationDate="2021-09-05T09:59:14.583" Score="7" ViewCount="4803" Body="&lt;p&gt;In physics there is this famous history facts about &lt;a href=&quot;https://en.wikipedia.org/wiki/Ettore_Majorana&quot; rel=&quot;noreferrer&quot;&gt;Ettore Majorana&lt;/a&gt;, brilliant student of Enrico Fermi that predicted the neutron and developed the idea of Majorana fermions, that suddenly disappeared. Are there similar disappearance stories in the history of mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastActivityDate="2021-09-06T17:34:25.243" Title="What are some famous mathematicians that disappeared?" Tags="|mathematicians|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13643" PostTypeId="2" ParentId="13642" CreationDate="2021-09-05T11:19:47.490" Score="18" Body="&lt;p&gt;There are some other examples of physicists and mathematicians disappeared after Majorana.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Alexandrov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vladimir Alexandrov&lt;/a&gt; disappeared in 1985 at 47. He created a mathematical model for the nuclear winter theory and the circumstances of his disappearance have never been clarified (maybe they were related to his work).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Boris_Weisfeiler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boris Weisfeiler&lt;/a&gt; was a US algebrist who disappeared in Chile in 1985; it is possible that for some unknown reason he was seized by a Chilean army patrol and taken into a secretive Germanic agricultural commune and then killed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aleksandr_Khazanov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aleksandr Khazanov&lt;/a&gt; was a PhD student and a child prodigy, one of the youngest to get perfect scores at an International Mathematical Olympiad. Despite his young age (he disappeared in 2001 at 20), he authored a paper on a variant of Fermat's Last Theorem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Tseng&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Tseng&lt;/a&gt; was a professor in applied mathematics (he was a major researcher in optimization). He disappeared in 2009 at 50 while kayaking in the Yantze, and probably died.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="-1" LastEditDate="2021-09-06T17:34:25.243" LastActivityDate="2021-09-06T17:34:25.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13644" PostTypeId="2" ParentId="13642" CreationDate="2021-09-05T13:44:31.183" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ted_Kaczynski#Mathematics_career&quot; rel=&quot;noreferrer&quot;&gt;Ted Kazynski&lt;/a&gt;&lt;br /&gt;&#xA;Ph.D. Univ. Michigan, 1967&lt;br /&gt;&#xA;specialty: complex analysis&lt;br /&gt;&#xA;Prize-winning dissertation, his advisor Alan Shields called it the best thesis he had ever directed.&lt;br /&gt;&#xA;1968, appointed Assistant Professor at Berkeley&lt;br /&gt;&#xA;Resigned suddenly, June 30, 1969.&lt;br /&gt;&#xA;Disappeared.  Not seen for more than 25 years.&lt;/p&gt;&#xA;&lt;p&gt;We now know he spent many years &amp;quot;off the grid&amp;quot; in a remote cabin in Montana that he built himself.&lt;/p&gt;&#xA;&lt;p&gt;From 1978 to 1995 he carried out a series of bombings, becoming known as the &amp;quot;Unabomber&amp;quot;.&lt;br /&gt;&#xA;Arrested, April 3, 1996.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="6229" LastEditDate="2021-09-05T14:35:30.307" LastActivityDate="2021-09-05T14:35:30.307" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13646" PostTypeId="2" ParentId="13634" CreationDate="2021-09-05T21:38:32.680" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Schwarzschild&quot; rel=&quot;noreferrer&quot;&gt;Karl Schwarzschild&lt;/a&gt; died aged 42 from the autoimmune disease pemphigus after returning from the Russian front in World War I. While not as young as many of the examples in other answers, a particularly tragic aspect is that he wrote his most influential papers on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schwarzschild_metric&quot; rel=&quot;noreferrer&quot;&gt;Schwarzschild metric&lt;/a&gt; during his service when he was already suffering from pemphigus.&lt;/p&gt;&#xA;" OwnerUserId="3584" LastActivityDate="2021-09-05T21:38:32.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13647" PostTypeId="2" ParentId="13634" CreationDate="2021-09-05T22:38:27.773" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Harry_Daghlian&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harry Daghlian&lt;/a&gt;, 24&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Slotin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Louis Slotin&lt;/a&gt;, 35&lt;/p&gt;&#xA;&lt;p&gt;Both killed at Los Alamos during separate reckless criticality accidents with the same plutonium bomb core (afterwards nicknamed the &lt;a href=&quot;https://en.wikipedia.org/wiki/Demon_core&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Demon Core&amp;quot;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Daghlian died 25 days after the accident after suffering severe radiation poisoning and falling into a coma.  Slotin died after nine days of progressive and agonizing deterioration of his bodily functions.&lt;/p&gt;&#xA;" OwnerUserId="15305" LastEditorUserId="15305" LastEditDate="2021-09-08T10:22:52.697" LastActivityDate="2021-09-08T10:22:52.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13648" PostTypeId="2" ParentId="12790" CreationDate="2021-09-06T00:14:48.300" Score="5" Body="&lt;p&gt;I can't speak to whether my father (John David Thomas II) is the author of Thomas' Plank and Thomas'Corkscrew, but he DEFINITELY is the John Thomas you reference that taught at NMSU.  I am grateful to you for your efforts to give credit to my father (who passed away on June 11th of this year), if he indeed is the person who wrote those papers.&lt;/p&gt;&#xA;&lt;p&gt;A bit more about my father that you might find interesting is that prior to his teaching at NMSU, he was involved in testing of nuclear weapons.  He was closely enough involved with it that he would pat the bombs on the side and say, &amp;quot;Now be a good little bomb and don't go off until we tell you to.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If you give me the dates of the two papers you cite, I can discuss it with my mother  and we can discuss what Dad was doing during that time.  That might provide a clue as to whether he is the John Thomas you are seeking.&lt;/p&gt;&#xA;" OwnerUserId="15293" LastActivityDate="2021-09-06T00:14:48.300" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13649" PostTypeId="2" ParentId="13634" CreationDate="2021-09-06T11:04:23.457" Score="5" Body="&lt;p&gt;A famous example of not very young, but still a tragic death is &lt;a href=&quot;https://en.wikipedia.org/wiki/Marie_Curie&quot; rel=&quot;noreferrer&quot;&gt;Marie Curie&lt;/a&gt;, who died (at 66) of &amp;quot;aplastic anemia from exposure to radiation&amp;quot; while studying radioactivity. I'm guessing she's not the only one, since the dangers of radioactivity were very long not known, and it was by some even touted as some kind of wonder cure-all, and they even put in things like toothpaste. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Henri_Becquerel&quot; rel=&quot;noreferrer&quot;&gt;Becquerel&lt;/a&gt; died at age 55 from &amp;quot;unknown causes, but was reported that &amp;quot;he had developed serious burns on his skin, likely from the handling of radioactive materials.&amp;quot;&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Marie Curie's husbad, &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Curie&quot; rel=&quot;noreferrer&quot;&gt;Pierre Curie&lt;/a&gt;, did die younger, at the age of 46, when he fell under a horse-drawn cart. According to Wikipedia, &amp;quot;statements made by his father and lab assistant imply that Curie's characteristic absent-minded preoccupation with his thoughts contributed to his death.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="15309" LastActivityDate="2021-09-06T11:04:23.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13650" PostTypeId="1" CreationDate="2021-09-06T11:42:03.120" Score="4" ViewCount="152" Body="&lt;p&gt;In my linear-algebra and numerics courses, I frequently heard an anecdote about some professor betting – literally, with money – that there would never be any application where computing the actual inverse of a matrix was the best option. (Obviously, some fineprint needs to be added to that bet, but that’s not pertinent here.)&lt;/p&gt;&#xA;&lt;p&gt;Who was this and what was the actual bet? Or is it just a myth? Given that there is no shortage of myths, I only care about anything coming with a trustworthy reference.&lt;/p&gt;&#xA;" OwnerUserId="13" LastEditorUserId="4251" LastEditDate="2021-09-06T16:21:51.653" LastActivityDate="2021-09-06T16:21:51.653" Title="Who bet against the usefulness of matrix inversion – or is it a myth?" Tags="|mathematics|reference-request|linear-algebra|computation|anecdote|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13651" PostTypeId="2" ParentId="11771" CreationDate="2021-09-06T11:47:18.103" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/11771/who-was-heinrich-kornblum/11778#11778&quot;&gt;The answer given by njuffa&lt;/a&gt; is very convincing and attractive to me, but just in case if someone is interested to know exactly and &lt;strong&gt;briefly&lt;/strong&gt; what Kornblum's mathematical contribution was, then she/ he can read page 130 of the book &amp;quot;Number Theory 2: Introduction to Class Field Theory&amp;quot; (by Kazuya Kato, Nobushige Kurokawa, Takeshi Saito) (I can not be sure about the writers' historical comments):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9c75Y.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9c75Y.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6407" LastActivityDate="2021-09-06T11:47:18.103" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13652" PostTypeId="2" ParentId="13634" CreationDate="2021-09-06T13:54:25.627" Score="1" Body="&lt;p&gt;Ettore Majorana, of which the presumed death was in 1938, at age of 32&lt;/p&gt;&#xA;" OwnerUserId="15314" LastActivityDate="2021-09-06T13:54:25.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13653" PostTypeId="5" CreationDate="2021-09-06T15:25:34.527" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-09-06T15:25:34.527" LastActivityDate="2021-09-06T15:25:34.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13654" PostTypeId="4" CreationDate="2021-09-06T15:25:34.527" Score="0" Body="Lev Davidovich Landau (1908-1968) was a Soviet physicist of Jewish ancestry who made fundamental contributions to many areas of theoretical physics." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-09-07T00:26:41.803" LastActivityDate="2021-09-07T00:26:41.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13655" PostTypeId="2" ParentId="13634" CreationDate="2021-09-06T21:12:23.837" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bogle%E2%80%93Chandler_case&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gilbert Stanley Bogle&lt;/a&gt;, born in 1924, was a physicist who worked at the Commonwealth Scientific and Industrial Research Organisation (CSIRO) on the campus of the University of Sydney. Married with several children, he was considered to be a brilliant scientist and had been a Rhodes Scholar.&#xA;...&#xA;Bogle's body was discovered near Fullers Bridge by two youths searching for golf balls[1 January 1963]. They saw his body and presumed him to be drunk. When they returned an hour later to find that he had not moved and that his face had turned blue, they went to fetch help.&lt;/p&gt;&#xA;&lt;p&gt;When police arrived at the scene they discovered that Bogle's body was half-undressed. Somebody had placed his trousers over the back of his legs in such a way that he appeared to be dressed, but was not. A piece of carpet was also laid on top of his back underneath his jacket, which was laid perfectly on his back.&lt;/p&gt;&#xA;&lt;p&gt;Shortly after this, Mrs Chandler's body was discovered a short distance away. ...&lt;/p&gt;&#xA;" OwnerUserId="15315" LastActivityDate="2021-09-06T21:12:23.837" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13656" PostTypeId="2" ParentId="2804" CreationDate="2021-09-06T21:24:54.773" Score="2" Body="&lt;p&gt;In 1932 Landau speculated that the conservation of energy is not valid in neutron stars and appealed to the authority of Niels Bohr[1]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Following a beautiful idea of Professor Niels Bohr’s we are able to&#xA;believe that the stellar radiation is due simply to a violation of the&#xA;law of energy, which law, as Bohr has first pointed out, is no longer&#xA;valid in the relativistic quantum theory, when the laws of ordinary&#xA;quantum mechanics break down..&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A few years later there was some ideological witch hunting announced by Friedman's article in the journal &lt;em&gt;Under the banner of Marxism&lt;/em&gt; with the title &amp;quot;Against Denials of the Law of conservation and transformation of Energy&amp;quot;.[2] Landau found himself imprisoned and some details of the story were recounted recently[3]; as its authors report:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;in his first letter to Stalin on 28 April 1938, the very day of the&#xA;Landau’s arrest, Kapitsa argued that Landau “. . . published one&#xA;remarkable paper, where he was the first to show a new source of&#xA;stellar radiation.”  Thus Landau’s paper was used as an argument to&#xA;save his life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr/&gt;&#xA;Refs.[1] L. D. Landau, Phys. Z. Sowjetunion, 1(1932), 285. &#xA;1932 &quot;On the theory of stars&quot;; reprinted in *Neutron &#xA;Stars, Black Holes and Binary X-ray Sources*, ed. H Gursky, R Ruffini, Dordrecht: Reidel, 1974) p. 271&#xA;[2] Ф. Г. Фридман, Под знаменем марксизма, 1937, 12, С. 192-200.&lt;a href=&quot;http://sceptic-ratio.narod.ru/po/fridman-vavilov.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt; Против отрицания закона сохранения и превращения энергии&lt;/a&gt; &#xA;[3]Yakovlev D., et al., arxiv.org (2012) *&lt;a href=&quot;https://arxiv.org/pdf/1210.0682.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt; Lev Landau and the conception of neutron stars&lt;/a&gt;*&#xA;" OwnerUserId="3329" LastActivityDate="2021-09-06T21:24:54.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13657" PostTypeId="5" CreationDate="2021-09-07T05:16:28.937" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-09-07T05:16:28.937" LastActivityDate="2021-09-07T05:16:28.937" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13658" PostTypeId="4" CreationDate="2021-09-07T05:16:28.937" Score="0" Body="Évariste Galois (1811-1832) was a French mathematician who produced a method of determining when a general polynomial equation could be solved by radicals and is famous for his development of early group theory. He died very young after fighting a duel." OwnerUserId="4251" LastEditorUserId="6229" LastEditDate="2021-10-10T21:16:11.993" LastActivityDate="2021-10-10T21:16:11.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13659" PostTypeId="2" ParentId="13634" CreationDate="2021-09-07T11:18:41.643" Score="3" Body="&lt;p&gt;Bruno Renner known for the Gell-Mann-Oakes-Renner relation died at age 32 in a mountaineering accident. Here is &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/pdf/10.1002/phbl.19730290907&quot; rel=&quot;nofollow noreferrer&quot;&gt;a link&lt;/a&gt; to an obituary in German.&lt;/p&gt;&#xA;" OwnerUserId="15317" LastActivityDate="2021-09-07T11:18:41.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13660" PostTypeId="2" ParentId="13634" CreationDate="2021-09-07T14:32:36.503" Score="3" Body="&lt;p&gt;I think perhaps the closest we can get to a romantic ''Galois figure'' in physics is maybe Henry Moseley.  Isaac Asimov argued (probably correctly) that Moseley would have been a very strong contender to win the Nobel Prize for his groundbreaking spectroscopy experiments if he had not been sadly killed.&lt;/p&gt;&#xA;&lt;p&gt;If I remember correctly, Rutherford tried to protect Moseley by making the case that he was an essential research worker and that he would be of more benefit in a civilian role working on military research, but Moseley was very patriotic and insisted on going to fight on the frontlines as a soldier.&lt;/p&gt;&#xA;&lt;p&gt;Robert Boyer (widely known for Boyer-Lindquist coordinates in general relativity) was killed at age 33 in the 1966 University of Texas tower shooting.  At the time this was the deadliest shooting by a lone gunman in US history, although it has now sadly been surpassed several times.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastEditorUserId="10345" LastEditDate="2021-09-07T15:25:54.913" LastActivityDate="2021-09-07T15:25:54.913" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13661" PostTypeId="2" ParentId="13634" CreationDate="2021-09-07T17:00:27.137" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://fr.m.wikipedia.org/wiki/F%C3%A9lix_Esclangon&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Félix Esclangon&quot;&gt;Félix Esclangon&lt;/a&gt; died at the age of 51 during a lecture. He got electrocuted while he was performing an experiment with X-rays in front of the students.&lt;/p&gt;&#xA;&lt;p&gt;There is now an Esclangon building at Sorbonne Université, close to where the dramatic accident took place.&lt;/p&gt;&#xA;" OwnerUserId="15319" LastEditorUserId="4251" LastEditDate="2021-09-08T03:29:13.640" LastActivityDate="2021-09-08T03:29:13.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13663" PostTypeId="2" ParentId="13634" CreationDate="2021-09-07T21:14:08.157" Score="1" Body="&lt;p&gt;Alfred Hill in 1988 at age 29. He was a passenger on PanAM 103, which was brought down by a bomb.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://physicstoday.scitation.org/doi/abs/10.1063/1.2810472&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://physicstoday.scitation.org/doi/abs/10.1063/1.2810472&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15323" LastActivityDate="2021-09-07T21:14:08.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13664" PostTypeId="1" AcceptedAnswerId="13666" CreationDate="2021-09-07T21:36:31.743" Score="6" ViewCount="126" Body="&lt;p&gt;I am fascinated to read about the evolution of cephalopod intelligence while I understand why cephalopods are mollusks (e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Cephalopod&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cephalopod&lt;/a&gt;), but I cannot easily find when zoologists finally recognize that cephalopods are mollusks.&lt;/p&gt;&#xA;&lt;p&gt;For example, did the assignment occur before geneticists could examine molecular phylogenies while the zoologists deducted the taxonomy based merely on morphology?&lt;/p&gt;&#xA;&lt;p&gt;When were cephalopods assigned to the taxonomic phylum Mollusca?&lt;/p&gt;&#xA;" OwnerUserId="15322" LastEditorUserId="15322" LastEditDate="2021-09-07T22:32:41.160" LastActivityDate="2021-09-07T23:44:48.840" Title="When were cephalopods assigned to the taxonomic phylum Mollusca?" Tags="|reference-request|biology|taxonomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13665" PostTypeId="2" ParentId="13619" CreationDate="2021-09-07T21:40:09.923" Score="0" Body="&lt;p&gt;One of the best authoritative sources regarding the use of the Bombe and methods to determine most efficient settings for the Bombe for decoding, is given in the following source:&lt;/p&gt;&#xA;&lt;p&gt;HUGH SEBAG-MONTEFIORE&#xA;&lt;em&gt;Enigma: The Battle for the Code&lt;/em&gt;&#xA;John Wiley &amp;amp; Sons, Incorporated&#xA;April 2001&#xA;448 Pages&#xA;ISBN:   0471407380&#xA;ISBN13: 9780471407386&lt;/p&gt;&#xA;&lt;p&gt;Several editions of this book are available electronically &lt;a href=&quot;https://b-ok.cc/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  Because the Bombe was ultimately arrayed, the source will provide insight into the processes used.  The whole of activity at Bletchley Park was a mathematical and statistical-probabilistic tour de force.  This book may not answer all of your questions, but the appendices give &lt;em&gt;excellent&lt;/em&gt; insights into the methods used. Be advised: The book is directed toward an audience of general interests, not specifically toward persons with a mathematics background. You may have already reviewed this book.&lt;/p&gt;&#xA;" OwnerUserId="15324" LastActivityDate="2021-09-07T21:40:09.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13666" PostTypeId="2" ParentId="13664" CreationDate="2021-09-07T23:44:48.840" Score="4" Body="&lt;p&gt;Ironically, Aristotle originally reserved the term &amp;quot;mollusk&amp;quot; (&lt;em&gt;malakia&lt;/em&gt;) only for cephalopods and slugs without shells. So &lt;em&gt;some&lt;/em&gt; of them were grouped with &lt;em&gt;some&lt;/em&gt; other mollusks (as we use the term) from the beginning. And so it remained for two millenia.  Linneaus, for example, scattered mollusks among orders &lt;em&gt;intestina&lt;/em&gt;, &lt;em&gt;mollusca&lt;/em&gt; (Aristotle's &lt;em&gt;malakia&lt;/em&gt;) and &lt;em&gt;testacea&lt;/em&gt; as late as 1758, see &lt;a href=&quot;https://zse.pensoft.net/article/1087/&quot; rel=&quot;noreferrer&quot;&gt;Vinarski, The birth of malacology. When and how?&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In 1795 Cuvier finally proposed a new taxonomical principle based on internal anatomy rather than looks as the deciding factor. He had precursors, like da Costa in 1776, who considered anatomical taxonomy to be more &amp;quot;scientific&amp;quot;, but they still saw shell-based taxonomy to be the only useful one in practice, because specimen were hard to obtain, and anatomical studies were &amp;quot;&lt;em&gt;too abstruse and unentertaining&lt;/em&gt;&amp;quot; (sic!). Even Lamarck advocated shell-based taxonomy for such reasons as late as 1792. Cuvier's was quite a paradigm shift towards modern biology, which Foucault in The Order of Things described as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;One day, towards the end of the eighteenth century, Cuvier was to topple the glass jars of the Museum, smash them open and dissect all the forms of animal visibility that the Classical age had preserved in them&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cuvier's taxonomy changed the meaning of &amp;quot;mollusk&amp;quot; to the modern one, and united cephalopods with the rest under it. Here is from Vinarski:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In his famous “Historia animalium”, Aristotle gave a collection of his own thorough observations on external morphology, anatomy, reproduction, ecology and behavior of many soft-bodied animals that&#xA;are classified today within the phylum Mollusca... The group of bloodless creatures included&#xA;at least four gene megista (great families): insects, crustaceans, testaceans, and mollusks (“Historia animalium”, I, 6). The first two families correspond approximately to the contemporary taxa, whereas testaceans (Ostracoderma) and mollusks (Malakia) are merely two divisions of that large group known today as the phylum Mollusca. Aristotle’s Malakia included mostly cephalopods, while Ostracoderma embraced shelled gastropods and bivalves, as well as some other shelled organisms. Hence, the modern&#xA;concept of mollusks did not exist in Aristotle’s time, and no serious emendations were made until the end of the 18th century.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1795, Georges Cuvier proposed a new classification of invertebrate animals based on anatomical data. He created a new concept of mollusks as representatives of a unique type of morphological organization of animals. Before Cuvier, the name “mollusks” was used only for cephalopods without external shells and slugs, whereas all shelled mollusks were placed in another taxon, Testacea. The Cuvier’s works (1795a, b, 1798) are considered here as the starting point of transformation of classical conchology (= study of shells) into modern malacology (= study of molluscous animals as whole organisms). This process ended in 1825 when the very term “malacology” was finally established by Ducrotay de Blainville.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-09-07T23:44:48.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13667" PostTypeId="2" ParentId="3131" CreationDate="2021-09-08T02:23:38.403" Score="1" Body="&lt;p&gt;Misha Alekhnovich (1978-2006), a very talented mathematician in the area of Complexity Theory:&#xA;&lt;a href=&quot;http://people.cs.uchicago.edu/%7Erazborov/files/misha_sigact.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://people.cs.uchicago.edu/~razborov/files/misha_sigact.pdf&lt;/a&gt;&#xA;See also: &lt;a href=&quot;http://people.cs.uchicago.edu/%7Erazborov/files/foreword.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://people.cs.uchicago.edu/~razborov/files/foreword.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2076" LastActivityDate="2021-09-08T02:23:38.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13668" PostTypeId="1" CreationDate="2021-09-08T04:41:02.710" Score="0" ViewCount="44" Body="&lt;p&gt;Bernard Stiegler argued that technics was a form of memory and was always progressing. Yet, philosophy isn’t the right thing to rely on when dealing with science. Science doesn’t have a mind and technology doesn’t have a mind in reality. I know about Moore’s Law but are there any other scientists who have found some proof in speculating that technological progression is infinite?&lt;/p&gt;&#xA;" OwnerUserId="15325" LastActivityDate="2021-09-08T04:41:02.710" Title="Is there any indication throughout history that technological progression will end?" Tags="|philosophy-of-science|technology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13672" PostTypeId="1" CreationDate="2021-09-09T19:07:35.517" Score="1" ViewCount="137" Body="&lt;p&gt;I am looking for books on the history of biochemistry. Searching online there are some short articles that can be found, which just mention a few highlights, but I haven't really found more comprehensive works.&lt;/p&gt;&#xA;&lt;p&gt;I am hoping for books that explain the history of biochemistry, with the aim of understanding how we have reached our current knowledge of the different biomolecules and their structure and biological function, and different biochemical pathways.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="15336" LastEditorUserId="15336" LastEditDate="2021-09-10T07:04:20.227" LastActivityDate="2021-09-10T07:04:20.227" Title="Books on history of biochemistry" Tags="|chemistry|biology|book-recommendation|biochemistry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13674" PostTypeId="1" CreationDate="2021-09-10T19:50:54.917" Score="1" ViewCount="73" Body="&lt;p&gt;The origin of consciousness has been a major scientific and philosophic debate since ever. Sometimes the origin is considered a philosophical issue, while others consider it a physics issue.&lt;/p&gt;&#xA;&lt;p&gt;Most modern physical views on consciousness are materialistic. These include, for example, the integrate information theory, where some mathematical quantity &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; would quantify connectivity, complexity and somehow consciousness.&lt;/p&gt;&#xA;&lt;p&gt;Modern science is also based on this idea of emergence, that some physics can only be explained as a collective phenomena described by the subigredients. Who was the first science to produce such an hypothesis for consciousness? I would guess that some knowledge of statistics and physical phenomena would be required.&lt;/p&gt;&#xA;&lt;p&gt;I have found that Thomas Henry Huxley had some theories on how consciousness is an epiphenomena that originates from neural activity. Was he the first or was somebody before him? Or was somebody else after with a more precise emergence hypothesis?&lt;/p&gt;&#xA;&lt;p&gt;Similar question: &lt;a href=&quot;https://hsm.stackexchange.com/questions/13447/when-were-brains-seen-as-computers-for-the-first-time&quot;&gt;When were brains seen as computers for the first time?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-09-12T20:59:57.290" LastActivityDate="2021-09-12T20:59:57.290" Title="Who was the first person to propose the idea that consciousness arises from complexity?" Tags="|statistics|neuroscience|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13675" PostTypeId="1" CreationDate="2021-09-11T06:00:31.390" Score="1" ViewCount="102" Body="&lt;p&gt;I am doing research on the Ottoman Engineer and Mathematician Aram Margossian (1853-?) (in Turkish sources Margosyan is also used) He published a book, De l’ordonnance des nombres dans les carrés magiques, on Magic Squares in 1908 in France. He also published various papers, in France, on &amp;quot;recreational mathematics&amp;quot; on Euler Squares, Latin Squares etc. I have found the titles of 3 papers up until now, dated 1912, 1931,1936. I did some research on BnF Gallica, Zentralblatt, Mathscinet. Not much is known about his life; there is a paper in Turkish that summarizes a calculus book that he has written in Ottoman Turkish and there is some biographical information there. In his 1936 paper he gives Vienna, Austria, as his address. He has been cited by well-known mathematicians of the time like Maurice Kraitchik.&lt;/p&gt;&#xA;&lt;p&gt;I am interested in finding out the state of the art about Magical Squares at the beginning of the 20th century.&lt;/p&gt;&#xA;&lt;p&gt;Also I am trying to find out the PhD advisor of&#xA;Edward Fleisher&#xA;PhD thesis N. Y. University 1935&#xA;On Euler Squares&#xA;Advisor unknown ( Math Genealogy Project)&lt;/p&gt;&#xA;&lt;p&gt;In his thesis Fleisher refers to the 1912 paper of Margossian.&lt;/p&gt;&#xA;&lt;p&gt;Alp Eden, Turkey&lt;/p&gt;&#xA;" OwnerUserId="15344" LastEditorUserId="12330" LastEditDate="2021-09-12T06:51:51.517" LastActivityDate="2021-09-12T06:51:51.517" Title="Magic Squares at the beginning of 20th century" Tags="|mathematicians|resource-recommendation|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13676" PostTypeId="1" CreationDate="2021-09-11T09:03:51.577" Score="0" ViewCount="242" Body="&lt;p&gt;Problem no. 12 from Moscow Mathematical Papyrus:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;Example of calculation of &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; heqats of grain&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;If someone says to you: Take &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; heqats of grain to make them into &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; jugs of beer&lt;/li&gt;&#xA;&lt;li&gt;Note that the amount of grain for &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; jug is &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{6}$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Reckon with &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{6}$&lt;/span&gt; in order to find &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The result is &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; times.&lt;/li&gt;&#xA;&lt;li&gt;Reckon with &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; to find &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The result is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; pefsu and this is the solution.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The only ratio that is sufficient for solving these problems is the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\text{pefsu} = \frac{\text{number of jugs of beer}}{\text{number of heqats of grain}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The problem is essentially looking for the pefsu of those &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; beers that need to be made. I am stuck at the line &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; where I don't understand why the author divides &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; to get a pefsu of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;. With that logic it would be that not all &lt;span class=&quot;math-container&quot;&gt;$13$&lt;/span&gt; initial heqats of grain were used to produce &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; jugs of beer since the pefsu formula would then be &lt;span class=&quot;math-container&quot;&gt;$$3 = \frac{18}{6}$$&lt;/span&gt;, i.e. only &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; heqats were used to produce the needed number of beer. Anyone can explain?&lt;/p&gt;&#xA;&lt;p&gt;Here is a reference from a &lt;a href=&quot;https://books.google.hr/books?id=8c10QYoGa4UC&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;lpg=PA214&amp;amp;hl=hr&amp;amp;pg=PA220#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;book&lt;/a&gt;. The book specifies this as a simple problem, so I am looking for an answer, because it makes no sense.&lt;/p&gt;&#xA;" OwnerUserId="15021" LastEditorUserId="4251" LastEditDate="2021-09-12T06:24:24.953" LastActivityDate="2024-01-09T15:06:46.310" Title="Pefsu problem explanation" Tags="|mathematics|ancient-egypt|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13677" PostTypeId="1" AcceptedAnswerId="13679" CreationDate="2021-09-11T14:22:19.793" Score="6" ViewCount="1576" Body="&lt;p&gt;For a (sufficiently smooth) function &lt;span class=&quot;math-container&quot;&gt;$f\colon \Bbb R^n\to\Bbb R$&lt;/span&gt;, the Laplacian of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined to be &lt;span class=&quot;math-container&quot;&gt;$\sum_{j=1}^n \frac{\partial^2 f}{\partial x_j^2}$&lt;/span&gt;. There are two notations for the Laplacian that I have seen being commonly used, viz., &lt;span class=&quot;math-container&quot;&gt;$\nabla^2 f$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Delta f$&lt;/span&gt;. The first one is mostly used by physicists, whereas mathematicians tend to prefer the latter one (at least in my area of study, where &lt;span class=&quot;math-container&quot;&gt;$\nabla^2 f$&lt;/span&gt; is reserved for the Hessian of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I would like to ask if anyone knows the origin of these two usages, in particular regarding the people who first introduced the symbols and the people who popularized them.&lt;/p&gt;&#xA;" OwnerUserId="3179" LastEditorUserId="12330" LastEditDate="2021-09-12T06:52:06.770" LastActivityDate="2021-09-12T07:36:04.977" Title="Notations for Laplacian: $\nabla^2$ vs. $\Delta$" Tags="|mathematics|notation|calculus|differential-equations|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13678" PostTypeId="2" ParentId="13677" CreationDate="2021-09-11T15:45:17.600" Score="3" Body="&lt;p&gt;I believe the notation &lt;span class=&quot;math-container&quot;&gt;$\nabla^2 f$&lt;/span&gt; comes from&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\nabla^2 f = \nabla\cdot\nabla f = &#xA;\operatorname{div}(\operatorname{grad} f)&#xA;$$&lt;/span&gt;&#xA;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Vector_calculus_identities#Laplacian&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-09-11T15:45:17.600" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13679" PostTypeId="2" ParentId="13677" CreationDate="2021-09-11T18:40:30.163" Score="5" Body="&lt;p&gt;What follows is from &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486679101&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A History of Vector Analysis&lt;/strong&gt;&lt;/a&gt; by Michael John Crowe (1967; 1985 Dover corrected reprint).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from middle of p. 167)&lt;/strong&gt; In Heaviside's later papers of 1883 and 1884 use was made of vectors, but no new principles were &lt;span class=&quot;math-container&quot;&gt;$\text{introduced.}^{35}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from middle of p. 180)&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;${}^{35}$&lt;/span&gt;This should perhaps be qualified by the statement that [the symbol] &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt; made its first appearance in an 1884 paper. See (5,I; 338). No comment was given by Heaviside about its meaning. He seems to use it as &lt;span class=&quot;math-container&quot;&gt;$+\left(\frac{d^2}{dx^2} + \frac{d^2}{dy^2} + \frac{d^2}{dz^2}\right)$&lt;/span&gt; rather than, as Maxwell and Tait had done, as the negative of the above.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have no idea what specific reference (paper/book title) Crowe's cryptic code refers to. There is no bibliography (by chapter or for the entire book). After several minutes of looking through the book (preface, beginning of chapter notes, various things at end of the book, etc.), I gave up trying to figure out what it means. I suspect the reference is to Volume I of Heaviside's collected works, and this is probably mentioned somewhere in pages before this, but I don't have time now to keep looking. &lt;em&gt;(rant)&lt;/em&gt; I think there's an important lesson to be learned here for anyone wishing to write something like Crowe's book: Don't use your own private bibliographic code unless you &lt;strong&gt;clearly&lt;/strong&gt; indicate how to decipher it, in a location that a casual user can reasonably be expected to find without much difficulty.&lt;/p&gt;&#xA;&lt;p&gt;I think the reason some people used a negative sign is because of the influence of quaternions, where squares of &lt;strong&gt;i&lt;/strong&gt;, &lt;strong&gt;j&lt;/strong&gt;, &lt;strong&gt;k&lt;/strong&gt; are negative.&lt;/p&gt;&#xA;&lt;p&gt;Possibly useful to look over is &lt;a href=&quot;https://archive.org/details/vectoranalysiste00gibbiala&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Vector Analysis&lt;/strong&gt;&lt;/a&gt; by Gibbs/Wilson (1901), which I believe is the primary text that popularized and spread vector ideas beyond the relatively few researchers that had up to that time been using them.&lt;/p&gt;&#xA;&lt;p&gt;Finally, &lt;a href=&quot;https://www.biodiversitylibrary.org/item/93055#page/332/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;see p. 2&lt;/a&gt; of Peter Guthrie Tait, [untitled address to the &lt;em&gt;Mathematics and Physics&lt;/em&gt; section], pp. 1-8 in the 2nd paging of &lt;a href=&quot;https://www.biodiversitylibrary.org/item/93055#page/7/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Report of the Forty-First Meeting of the British Association for the Advancement of Science&lt;/strong&gt;&lt;/a&gt; (Edinburgh, 2−9 August 1871), John Murray (London), 1872, cv + 207 + 281 + iv + 83 pages. (If anyone is curious as to how I happen to know about Tait's comments, see &lt;a href=&quot;https://hsm.stackexchange.com/questions/3810/what-is-the-history-of-linear-vs-logarithmic-scales/3811#comment30259_3811&quot;&gt;these 2 comments&lt;/a&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2021-09-12T07:36:04.977" LastActivityDate="2021-09-12T07:36:04.977" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13680" PostTypeId="2" ParentId="13677" CreationDate="2021-09-12T02:00:17.567" Score="6" Body="&lt;p&gt;Jeff Miller's site gives the &lt;a href=&quot;https://jeff560.tripod.com/calculus.html&quot; rel=&quot;noreferrer&quot;&gt;first occurrence of &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;&lt;/a&gt; as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; for the Laplacian operator (also represented by &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt;) was introduced by Robert Murphy in 1833 in &lt;em&gt;Elementary Principles of the Theories of Electricity&lt;/em&gt;. (Kline, page 786)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://jeff560.tripod.com/l.html&quot; rel=&quot;noreferrer&quot;&gt;first use of the term &lt;em&gt;Laplace's Operator&lt;/em&gt;&lt;/a&gt; is given as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The term LAPLACE’S OPERATOR (for the differential operator &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt;) was used in 1873 by James Clerk Maxwell in &lt;em&gt;A Treatise on Electricity and Magnetism&lt;/em&gt; (p. 29): &amp;quot;...an operator occurring in all parts of Physics, which we may refer to as Laplace’s Operator&amp;quot; (OED).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first use by physicist of the term &lt;em&gt;Laplacian&lt;/em&gt; for the &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt; operator is given on the same page as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;LAPLACIAN (as a noun, for the differential operator &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt;) was used in 1935 by Pauling and Wilson in Introd. Quantum Mech. (OED).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, these entries do not make clear when the &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt; notation was first introduced.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="12330" LastEditDate="2021-09-12T06:23:15.150" LastActivityDate="2021-09-12T06:23:15.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13682" PostTypeId="1" CreationDate="2021-09-14T00:18:27.767" Score="3" ViewCount="41" Body="&lt;p&gt;The two main forms of (animal) learning are operant conditioning and classical conditioning, both established only in the 20th century. However, before they were established, surely there have been alternative theories of how animals learn. What were they?&lt;/p&gt;&#xA;" OwnerUserId="11081" LastEditorUserId="4251" LastEditDate="2021-09-14T17:53:28.800" LastActivityDate="2021-09-14T17:53:28.800" Title="What alternative theories of learning were there before classical and operant conditioning?" Tags="|biology|psychology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13683" PostTypeId="1" CreationDate="2021-09-14T00:29:39.453" Score="2" ViewCount="71" Body="&lt;p&gt;In the Biblical account of creation, G-d divided the waters. The rabbi and Biblical commentator &lt;a href=&quot;https://en.wikipedia.org/wiki/Nachmanides&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nachmanides&lt;/a&gt; wrote about this that he believes the Greek demonstration that the earth is mostly covered with water.&lt;/p&gt;&#xA;&lt;p&gt;I've been thinking about this statement. At first blush, the portion of the earth covered with water obviously seems to be an empirical question which could not be resolved by any logical argument. But I wonder whether Nachmanides was thinking of an ancient computation about the size of the earth (based on sun angles establishing the degrees of arc between two places at a known distance), coupled with the assumption that someone sailing west from Europe would encounter no land for most of the way to Japan, as apparently anyone would have thought who was unaware of the Americas.&lt;/p&gt;&#xA;&lt;p&gt;(He would also need some opinion about the size of the known world, but apparently this is the easy part.)&lt;/p&gt;&#xA;&lt;p&gt;Is there any way of identifying this argument with any greater specificity? What works of Greek geography (or what would the word be?) were available to Nachmanides, and which of these dealt with the question of the amount of the earth's surface covered by water? How widely shared was this opinion?&lt;/p&gt;&#xA;" OwnerUserId="9545" LastEditorUserId="131" LastEditDate="2021-09-14T06:08:46.520" LastActivityDate="2021-09-14T06:08:46.520" Title="Nachmanides and the Greek proof that the earth is mostly covered with water" Tags="|ancient-greece|middle-ages|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13684" PostTypeId="1" CreationDate="2021-09-14T03:35:41.817" Score="2" ViewCount="364" Body="&lt;p&gt;A calculus textbook I’m using references a calculus book of &lt;strong&gt;L’Hôpital&lt;/strong&gt; in which he illustrates his rule, which is taught in many calculus classes.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have a source as a scanned PDF? I’d love to show my students. Thank you!&lt;/p&gt;&#xA;" OwnerUserId="5973" LastEditorUserId="12330" LastEditDate="2021-09-14T23:30:23.450" LastActivityDate="2022-07-19T00:05:50.550" Title="Source of L’Hôpital’s 1696 Calculus textbook" Tags="|mathematics|mathematicians|reference-request|calculus|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13685" PostTypeId="1" CreationDate="2021-09-14T09:18:57.960" Score="0" ViewCount="75" Body="&lt;p&gt;We have characters for letters and numbers. We typically view them as a discrete sets. Has any culture ever mangled the two? To some degree it seems almost logical if your letters are already conceptually ordinal,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;I have a pear.&lt;/em&gt; becomes &lt;em&gt;I have 1 pear&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;I have b pear.&lt;/em&gt; becomes &lt;em&gt;I have 2 pears&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15360" LastEditorUserId="4251" LastEditDate="2021-09-14T17:52:31.983" LastActivityDate="2021-09-14T17:52:31.983" Title="Has any culture in history ever not invented digits, and just used letters?" Tags="|numbers|linguistics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13686" PostTypeId="1" CreationDate="2021-09-14T14:21:14.273" Score="2" ViewCount="1166" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Mercury_(element)#Etymology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page&lt;/a&gt; for mercury says that it was named after the Roman god because of his speed and mobility. When did the name mercury start to be used to designate the metal?&lt;/p&gt;&#xA;" OwnerUserId="10806" LastEditorUserId="4251" LastEditDate="2021-09-14T19:37:51.843" LastActivityDate="2021-09-15T07:02:14.670" Title="When was mercury given its name?" Tags="|terminology|chemistry|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13687" PostTypeId="2" ParentId="13686" CreationDate="2021-09-14T15:29:41.297" Score="2" Body="&lt;p&gt;Mercury the metal (a.k.a quicksilver).&lt;/p&gt;&#xA;&lt;p&gt;Usage in &lt;strong&gt;English&lt;/strong&gt; ... the &lt;a href=&quot;https://www.oed.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford English Dictionary&lt;/a&gt; (subscription required) has an example in Chaucer (~1395):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And in amalgamyng and calcenyng Of quyk siluer, yclept Mercurie crude... Oure orpiment and sublymed Mercurie.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Probably we need to investigate this in other languages; the English usage may come from Latin?&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-09-14T15:29:41.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13688" PostTypeId="2" ParentId="13686" CreationDate="2021-09-14T16:55:40.900" Score="6" Body="&lt;p&gt;Mercury was initially known &lt;em&gt;Hydrargyros&lt;/em&gt; (in the times of Aristotle) and later known in its liquid state as &lt;em&gt;Argenturn Vivurn&lt;/em&gt; translated “alive silver” or in English as “quicksilver&amp;quot; in the 4th century BCE.&lt;/p&gt;&#xA;&lt;p&gt;The association of traditional metals with the planets dates already back to the 2th century CE (in a passage of Celsus preserved by Origenes [ContraCelsum, VI, 22] mercury gets associated with iron).&lt;/p&gt;&#xA;&lt;p&gt;The first known mention of planet Mercury in association to &lt;em&gt;Hydrargyros&lt;/em&gt; is found in a list by &lt;a href=&quot;https://en.wikipedia.org/wiki/Stephanus_of_Alexandria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stephanus of Alexandria&lt;/a&gt; from the 7th century: Sun-gold, Moon-silver,&#xA;Mercury-quicksilver, Venus-copper, Mars-iron, Jupiter-tin, Saturn-lead.&lt;/p&gt;&#xA;&lt;p&gt;Tracking exactly when &amp;quot;mercury&amp;quot; became more popular than its alternative name quicksilver is hard, as one will have to search for meaning. You can look at tables of &lt;a href=&quot;https://en.wikipedia.org/wiki/Alchemical_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;alchemical symbols&lt;/a&gt; where elements are always associated with some astronomical body and quicksilver is represented by ☿ (Mercury). As pointed in the comments, mercury was one of the most important alchemical elements thought to be one of the main ingredients for the philosopher's stone, which allegedly turned normal metals into gold.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Pseudo-Geber&quot; rel=&quot;nofollow noreferrer&quot;&gt;pseudo-Geber&lt;/a&gt; in the 13-14th century (sometimes [mis]attributed to Arabic alchemist Jabir ibn Hayyan or Geber from 9th century) in his &lt;em&gt;Summa perfectionis magisterii&lt;/em&gt; refers to it as both mercury (&lt;em&gt;Mercurio&lt;/em&gt;) and quicksilver (&lt;em&gt;argento uiuo&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/e/ee/Affinity-table.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;This table&lt;/a&gt; by &lt;em&gt;Étienne François Geoffroy&lt;/em&gt; from 1718 of alchemical elements already uniquely refers to quicksilver  as &lt;em&gt;mercure&lt;/em&gt;  (mercury in French).&#xA;During the Chemistry Revolution, many chemical names were standardised. In 1787 &lt;em&gt;Méthode de nomenclature chimique&lt;/em&gt;, Antoine Lavoisier, &lt;em&gt;et al&lt;/em&gt; fixed the name of many elements and they chose to keep only the name &lt;em&gt;mercure&lt;/em&gt; for the liquid metal.&lt;/p&gt;&#xA;&lt;p&gt;Englishman John Dalton, seems to have adopted the French standard. When writing a list of elements in the early 1800, he referred to the liquid metal simply as mercury. Note that in some European languages, that did not catch on (in German is still called &lt;em&gt;Quecksilber&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Some sources: &lt;a href=&quot;https://doi.org/10.1002/jps.3080070513&quot; rel=&quot;nofollow noreferrer&quot;&gt;Otto Raubenheimer, &lt;em&gt;History of mercury&lt;/em&gt; (1912)&lt;/a&gt;  and &lt;a href=&quot;https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.493.6897&amp;amp;rep=rep1&amp;amp;type=pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yannis Almirantis &lt;em&gt;The Paradox of the Planetary Metals&lt;/em&gt; (2005)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-09-15T07:02:14.670" LastActivityDate="2021-09-15T07:02:14.670" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="13689" PostTypeId="2" ParentId="13686" CreationDate="2021-09-15T02:33:44.437" Score="-1" Body="&lt;p&gt;Mercury is not the not only element which is named after the heavenly bodies, we have Uranium, Neptunium, Plutonium, Tellurium (Earth), Selenium (moon). It is not suprising then that ancient and modern chemists were fascinated by the objects in the sky. For a serious historical debate, one should not then focus only one element, we have to look at the whole picture. There are several excellent articles in the &lt;em&gt;Journal of Chemical Education&lt;/em&gt;, see, &lt;em&gt;Origin of the Names of Chemical Elements&lt;/em&gt; by Vivi Ringnes (Available from Google Scholar).&lt;/p&gt;&#xA;&lt;p&gt;I show one Table from there.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/DSiJP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/DSiJP.png&quot; alt=&quot;Alchemists&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Historically then, mercury is the only element which retained  its name, for &lt;em&gt;unknown&lt;/em&gt; reasons. One may then ask, why copper is associated with Venus and lead with Saturn. Unfortunately, Wikipedia quality is really bad when it comes to chemical and mathematical history, so don't rely on Wikipedia explanations because it is highly biased by the writer. The problem is that it is open to editing by everyone (which is good and bad both ways). As you can see, Wikipedia's explanation for the name of mercury has no standing whatsoever. It is a conjecture.&lt;/p&gt;&#xA;&lt;p&gt;For instance, we have a list,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{ll}&#xA;\text { Cerium } &amp;amp; \text { the asteroid Ceres } \\&#xA;\text { Helium } &amp;amp; \text { Gr., helios, the sun } \\&#xA;\text { Mercury } &amp;amp; \text { the planet Mercury } \\&#xA;\text { Neptunium } &amp;amp; \text { the planet Neptune } \\&#xA;\text { Palladium } &amp;amp; \text { the asteroid Pallas } \\&#xA;\text { Phosphorus } &amp;amp; \text { Gr., phosphoros, light-bearing; a name applied to the } \\&#xA;\text { planet Venus when appearing as a morning star } \\&#xA;\text { Plutonium } &amp;amp; \text { the planet Pluto } \\&#xA;\text { Selenium } &amp;amp; \text { Gr., Selene, moon } \\&#xA;\text { Tellurium } &amp;amp; \text { L., tellus, the earth } \\&#xA;\text { Uranium } &amp;amp; \text { the planet Uranus }&#xA;\end{array}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-09-15T02:33:44.437" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13690" PostTypeId="5" CreationDate="2021-09-15T05:23:22.280" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-09-15T05:23:22.280" LastActivityDate="2021-09-15T05:23:22.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13691" PostTypeId="4" CreationDate="2021-09-15T05:23:22.280" Score="0" Body="Carl Friedrich Gauss (1777-1855) was a German mathematician and physicist who worked in number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. His contributions were immense." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-09-17T14:34:09.083" LastActivityDate="2021-09-17T14:34:09.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13693" PostTypeId="1" CreationDate="2021-09-16T02:22:09.970" Score="1" ViewCount="189" Body="&lt;p&gt;I’m pretty interested in the history of mathematics, and it has always been my belief that the great pioneers of Differential Geometry were Gauss and Riemann, and the father of topology was mostly accredited to Poincare. However, now I am taking a differential topology/geometry sequence and we basically learn two semesters of Differential Topology as a prerequisite for Differential Geometry, and this confuses me since Riemann and Gauss were both dead before Poincare could’ve done anything with topology. So was Differential Geometry created before Differential Topology? If so how were Gauss and Riemann able to do it before what seems to be the necessary definitions of topological spaces, and smooth manifolds?&lt;/p&gt;&#xA;&lt;p&gt;Also if anyone has any sources on a history of modern mathematics (like 18-21st century) that they could recommend, that would be very cool since I really can’t seem to find any that really get into the weeds as much as I would like.&lt;/p&gt;&#xA;" OwnerUserId="15369" LastActivityDate="2021-09-16T23:14:46.413" Title="Did the formation of Differential Geometry come before Differential Topology/ Topology in general?" Tags="|mathematics|differential-geometry|gauss|topology|riemannian-geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13695" PostTypeId="1" CreationDate="2021-09-16T18:58:07.873" Score="-1" ViewCount="197" Body="&lt;p&gt;Did the statistician Egon Pearson have a PhD? If not, to what extent did he write a dissertation?&lt;/p&gt;&#xA;" OwnerUserId="15375" LastEditorUserId="10106" LastEditDate="2022-01-28T14:59:11.427" LastActivityDate="2022-01-28T14:59:11.427" Title="Did Egon Pearson have a PhD?" Tags="|mathematics|mathematicians|statistics|statisticians|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13696" PostTypeId="2" ParentId="13695" CreationDate="2021-09-16T22:37:30.003" Score="1" Body="&lt;p&gt;A &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rsbm.1981.0017&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography of Egon Pearson&lt;/a&gt; from the Royal Society covers his education this way:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After finishing his school education at the Dragon School, Oxford, and&#xA;Winchester College, E. S. Pearson would have liked to join up in the&#xA;summer of 1914, but was considered by his father hardly fit enough&#xA;physically, and went somewhat reluctantly to Trinity College, Cambridge,&#xA;being admitted as an Entrance Scholar on 25 June of that year.&#xA;His tutor was Mr Whetham (afterwards Sir William Dampier), and he&#xA;obtained a First class in Part I of the Mathematical Tripos at the end of&#xA;his first year, in spite of being quite ill for a time. Still anxious to do&#xA;something useful for the war effort, he obtained a post at the Admiralty&#xA;and then the Ministry of Shipping. This war service enabled him to&#xA;qualify for his B.A. in 1920 (and M.A. in 1924) by taking a further&#xA;approved course of study.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is no mention of a doctorate, or any thesis work. This was not uncommon in much of the 20th century. A PhD was not a requirement to be a professor for quite some time.&lt;/p&gt;&#xA;&lt;p&gt;Now, as noted by @njuffa in comments, his tagline in publications changed from M.A. to D.Sc between 1925 and 1926, several years after getting his masters. By 1926 he had published 8 papers, which is a pretty good output for a Ph.D student (note njuffa's pointer to what a D.Sc. represents). And it is a bit curious that the Royal Society biography would not mention it.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2021-09-17T13:45:19.290" LastActivityDate="2021-09-17T13:45:19.290" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13697" PostTypeId="2" ParentId="13693" CreationDate="2021-09-16T23:14:46.413" Score="2" Body="&lt;p&gt;Yes, differential geometry is older. Though he had predecessors, Gauss can be considered the founding father of differential geometry with his book General investigation of curved surfaces, 1827. This was long before the subject of topology was born as a subject (though some isolated results in both differential geometry and topology are much older). Systematic development of topology is usually credited to a series of papers of Poincare published in 1893-1904. This is about systematic development of the two disciplines.&lt;/p&gt;&#xA;&lt;p&gt;But when you look at the earliest isolated results, the situation is similar: Differential Calculus was applied to geometry (of curves) since its very beginning of calculus, while the first recorded theorem of topology is probably Euler's Konigsberg bridges problem.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-09-16T23:14:46.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13699" PostTypeId="1" AcceptedAnswerId="13701" CreationDate="2021-09-18T14:54:00.197" Score="3" ViewCount="260" Body="&lt;p&gt;In a letter to Kepler, his former professor Michael Mästlin mentioned that he had calculated the golden section to the 9&lt;sup&gt;th&lt;/sup&gt; decimal place. Why was it considered notable? What method&lt;span class=&quot;math-container&quot;&gt;$^\dagger$&lt;/span&gt; did Mästlin use?&lt;/p&gt;&#xA;&lt;p&gt;I assume they were not aware of the Fibonacci sequence, but they must have been aware that the answer was correct when the digital expansions after the decimal point were the same for &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\varphi^2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\varphi^{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;span class=&quot;math-container&quot;&gt;$\dagger$&lt;/span&gt; In schoolboy maths if possible!&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="1763" LastEditorUserId="6229" LastEditDate="2022-01-30T15:54:25.977" LastActivityDate="2022-01-30T15:54:25.977" Title="How did Kepler's teacher Mästlin calculate the Golden section?" Tags="|mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13700" PostTypeId="2" ParentId="3360" CreationDate="2021-09-18T16:36:15.130" Score="7" Body="&lt;p&gt;(Almost certainly) Paul Dirac&lt;span class=&quot;math-container&quot;&gt;$^\dagger$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Generations learned the algebraic derivation of the oscillator spectrum from  Dirac's landmark book, &lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt;. I am confident it is featured in his first edition (1930), but I only have access to his 3rd (1947), in which it is detailed in classic Diracese in Chapter IV,  §34, p 136 et seq.&lt;/p&gt;&#xA;&lt;p&gt;G Farmelo's 2009 biography (&lt;em&gt;The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius&lt;/em&gt;) describes the invention having been made in the winter of 1926-27 in Copenhagen, and you may see it in p 251 (the &amp;quot;q-number&amp;quot; operators &lt;em&gt;b&lt;/em&gt;) of his (1927)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1098/rspa.1927.0039&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The quantum theory of the emission and absorption of radiation&lt;/em&gt;&lt;/a&gt;,   &lt;em&gt;Proceedings of the Royal Society of London&lt;/em&gt;  Ser A,  &lt;strong&gt;114&lt;/strong&gt;(767), 243-265;&lt;/p&gt;&#xA;&lt;p&gt;although he is steering the reader to his  §8 of his (1926)  &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1926.0068&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The elimination of the nodes in quantum mechanics&lt;/em&gt;&lt;/a&gt;, &lt;em&gt;Proceedings of the Royal Society of London&lt;/em&gt;&#xA;Ser A &lt;strong&gt;111&lt;/strong&gt; (757), 281-305.&lt;/p&gt;&#xA;&lt;p&gt;It is evident he kept on streamlining and refining the concept/method for quite a while.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt; &lt;span class=&quot;math-container&quot;&gt;$\dagger$&lt;/span&gt; Also in G Farmelo's celebrated biography of PAMD, p 118, it is stated that &lt;/sub&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the end of January [1927], as he was preparing to leave Copenhagen, Dirac posted his paper to the Royal Society. It turned out that he was the first to introduce the mathematics of creation and annihilation into quantum theory, though his results had been reached independently by John Slater, studying in Cambridge with Fowler. Slater was one of the many who admired Dirac’s paper for its content but found its presentation perversely complicated: ‘his paper was a typical example of what I very much distrusted, namely one in which a great deal of seemingly unnecessary mathematical formalism is introduced'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sub&gt;This is meant to be referenced in &lt;em&gt;Slater, J. (1975) Solid-State and Molecular Theory: A Scientific Biography, New York:&#xA;John Wiley and Sons.&lt;/em&gt;  The subtext is that Slater anticipated the radiation picture in parallel, however &lt;em&gt;without&lt;/em&gt; the glorious ladder operator formalism asked about!&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-09-19T22:40:05.757" LastActivityDate="2021-09-19T22:40:05.757" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13701" PostTypeId="2" ParentId="13699" CreationDate="2021-09-19T00:58:59.813" Score="5" Body="&lt;p&gt;This episode is discussed in&#xA;&lt;a href=&quot;https://fibonacci.com/golden-number-herz-fischler/&quot; rel=&quot;noreferrer&quot;&gt;A Mathematical History of the Golden Number by Herz-Fischler&lt;/a&gt;. The 1597 letter was not from Mästlin to Kepler but from Kepler to Mästlin. In it Kepler described a construction of a right triangle whose sides are in geometric progression, now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler_triangle&quot; rel=&quot;noreferrer&quot;&gt;Kepler triangle&lt;/a&gt;, which he attributes to &amp;quot;Magirus&amp;quot;, who &amp;quot;&lt;em&gt;pleased me among others with a new enthusiasm for geometry&lt;/em&gt;&amp;quot;. Who that was is unknown.&lt;/p&gt;&#xA;&lt;p&gt;A simple application of the Pythagorean theorem shows that the quotient of the geometric progression is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\varphi}$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; is the golden ratio, or the extreme and mean ratio, as Kepler calls it after Euclid. Considering Kepler's preoccupation with both mystical geometry and numerology his pleasure with this triangle is not surprising.&lt;/p&gt;&#xA;&lt;p&gt;In his annotation of the letter, Mästlin put some numbers on the sides and calculated numerical values in terms of square roots, which gave him an approximation of the reciprocal golden ratio and its square root, but not to 9 places, only 7. He did not say how he calculated the square roots, but quickly converging iterative methods were known since &lt;a href=&quot;https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method&quot; rel=&quot;noreferrer&quot;&gt;Heron&lt;/a&gt;, if not Babylonians.&lt;/p&gt;&#xA;&lt;p&gt;Btw, Kepler rediscovered the relation between the golden ratio and the Fibonacci numbers (noted by Jacob prior to 1564, and by an unknown Italian, who left an undated note in Pacioli's 1509 edition of &lt;em&gt;Elements&lt;/em&gt;), but later. It is described in his Six-Cornered Snowflake (1611), where the sequence comes up in approximating regular pentagons, see &lt;a href=&quot;https://www.bachnetwork.org/ub1/tatlow.pdf&quot; rel=&quot;noreferrer&quot;&gt;Tatlow, The Use and Abuse of Fibonacci Numbers&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is from Kepler's letter translated by Herz-Fischler:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Now furthermore so that the Royal professor may achieve something even worthy of his title he proposes to you most renowned D. [Doctor] a geometrical problem; to build a right angled triangle all of whose three sides are mutually and continuously proportional so that just as the lesser side is to the greater around the right triangle so is the latter to the one subtended by the right right angle [i.e., the hypotenuse]. I do not know the distinctive use of this discovery except that it more&#xA;greatly enhances geometry.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This discovery is due to Magirus wherefore I courteously greet him and show myself thankful to him because of this, that as by his very pleasant theorem he has pleased me among others with a new enthusiasm for geometry. The proof is easy on the basis of his discovery; because I have changed it into another form such that I think I will easily persuade even Magirus himself to think that it is entirely mine.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Next is Mästlin's annotation with Herz-Fischler's insertions referring to the image reproduced from his book. This is the first known decimal approximation, although sexagesimal ones appear already in Ptolemy's table of chords (but he was not specifically interested in &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt;). In the quote below &lt;span class=&quot;math-container&quot;&gt;$S()$&lt;/span&gt; is the square, and &lt;span class=&quot;math-container&quot;&gt;$R()$&lt;/span&gt; the square root:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If EA is 10, ED will be R(125) - 5 and EF 15-R(125). And since [S(AE)] = 100 and [S(ED)] = 150 - R(12500), [S(AD)] will be R(12500) - 50. Its [i.e., S(AD)] side is AD. Therefore the following things are proportional: 1) ED [=] R(125) - 5, AD [=] R(R(12500) - 50). 3) AE [=] 10. Thus just as EA = 10000000 so ED is about 6180340 and AD is about 7861514.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LWYKm.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LWYKm.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-09-19T04:35:25.010" LastActivityDate="2021-09-19T04:35:25.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13706" PostTypeId="1" CreationDate="2021-09-21T05:54:01.190" Score="0" ViewCount="43" Body="&lt;p&gt;The general consensus is that Euclid was  a real historical figure. Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Euclid&lt;/a&gt; concludes on the  hypothesis that Euclid was not a real person,&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This hypothesis is not well accepted by scholars and there is little evidence in its favor&amp;quot;&lt;/em&gt;.&#xA;Recently I came across a paper by C. K. Raju &lt;a href=&quot;http://ckraju.net/papers/Euclid.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://ckraju.net/papers/Euclid.pdf&lt;/a&gt;. In this paper Raju opines that&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If one discounts later Arab sources, as Heath does, our belief in the historicity of Euclid rests wholly and solely on a single remark attributed to Proclus. In this remark, Proclus is  not  particularly definite  about  Euclid,  for his  language admittedly shows that he is the  first  to speak of Euclid, and  is proceeding  on speculative inferences  about events some seven  centuries before his time.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Same argument is also advanced later in his book &amp;quot;Euclid and Jesus&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;If Raju is right then it casts serious doubts about historicity of Euclid given how influential  Euclid's Elements has been in Greek mathematics and culture. I am surprised that no other scholar has taken note of this and has accepted Euclid as real historical figure.&lt;/p&gt;&#xA;&lt;p&gt;My question : is Raju's above comment correct? Is there no reference of Euclid before Proclus other than later Arabic sources?&lt;/p&gt;&#xA;" OwnerUserId="15386" LastEditorUserId="15386" LastEditDate="2021-09-21T06:07:43.667" LastActivityDate="2021-09-21T06:07:43.667" Title="Historicity of Euclid. Looking for the references of Euclid in ancient texts that has survived" Tags="|mathematics|mathematicians|geometry|ancient-greece|euclid|" AnswerCount="0" CommentCount="3" ClosedDate="2021-09-21T17:53:06.613" ContentLicense="CC BY-SA 4.0" />
  <row Id="13708" PostTypeId="1" CreationDate="2021-09-21T11:24:23.150" Score="4" ViewCount="588" Body="&lt;p&gt;In an unrelated question, some users started arguing about the existence of &lt;a href=&quot;https://en.wikipedia.org/wiki/Jabir_ibn_Hayyan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geber&lt;/a&gt; (Jabir ibn Hayyan - جابر بن حيّان) from 806−816 AD  and pseudo-Geber (probably from 13th-14th century). This discussion led to the manipulation of some Wikipedia entries adding the claim that Geber may have or have never existed.&lt;/p&gt;&#xA;&lt;p&gt;To clear up the issue, I wanted to ask, is there some evidence about his historicity (as a person)? To what extent do we know about him?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-09-23T08:53:20.923" LastActivityDate="2021-09-23T08:53:20.923" Title="What is the evidence for the existence of Geber?" Tags="|chemistry|debunking|islamic-science|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13709" PostTypeId="2" ParentId="13708" CreationDate="2021-09-21T14:27:21.663" Score="6" Body="&lt;p&gt;See Lawrence Principe, &lt;a href=&quot;https://www.google.it/books/edition/The_Secrets_of_Alchemy/cs9_mXyN0OsC&quot; rel=&quot;noreferrer&quot;&gt;The Secrets of Alchemy (2013, The University of Chicago Press)&lt;/a&gt;, page 33-on for &lt;em&gt;Jabir and the Jabirian Corpus&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;a person who played as large a role in Arabic alchemy as Zosimos did in the Greco-Egyptian—one Jabir ibn-Ḥayyan. Or, to speak more accurately, several Jabir ibn-Ḥayyans. Or perhaps none at all. A persistent problem facing historians of alchemy is figuring out if an author really is who he says he is, and if he lived when and where he claims.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And see page 55:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;the most influential of these 13th-century Latin&#xA;alchemical compositions appeared under a very familiar name, that of&#xA;Jabir, rendered in medieval Latin spelling as Geber. Thus, the “Jabir&#xA;problem” discussed in the previous chapter had yet another dimension:&#xA;whether the Latin books known under the name of Geber were translations&#xA;of Jbir, or whether they were native Latin productions. Historians&#xA;of science argued vociferously over whether Geber was really Jabir.&#xA;Recent scholarship has settled the issue: he was not. Geber was a late&#xA;thirteenth-century Latin author.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The author concealed behind the pseudonym of Geber is probably&#xA;an Italian Franciscan friar and lecturer named Paul of Taranto. [&lt;em&gt;Footnote&lt;/em&gt;. We owe this identification, and the solution to the “Jabir-Geber” problem,to the painstaking studies of William R. Newman. For a detailed treatment of Geber’s identity, see Newman, “New Light on the Identity of Geber,” &lt;em&gt;Sudhoffs Archiv&lt;/em&gt; 69 (1985): 79–90, and “Genesis of the Summa perfectionis,” &lt;em&gt;Archives internationales d’histoire des&#xA;sciences&lt;/em&gt; 35 (1985): 240–302. For an edition, translation, and historical contextualization of the &lt;em&gt;Summa&lt;/em&gt;, see Newman’s &lt;em&gt;The Summa Perfectionis of Pseudo-Geber&lt;/em&gt;.]&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2021-09-21T15:22:57.463" LastActivityDate="2021-09-21T15:22:57.463" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13710" PostTypeId="2" ParentId="12291" CreationDate="2021-09-21T14:36:09.713" Score="3" Body="&lt;p&gt;It is equation (63) in his epochal &lt;a href=&quot;https://doi.org/10.1103/PhysRev.75.1736&quot; rel=&quot;nofollow noreferrer&quot;&gt;second paper&lt;/a&gt;, Dyson, F  J  (1949). &lt;em&gt;&amp;quot;The S matrix in quantum electrodynamics&amp;quot;&lt;/em&gt;, &lt;em&gt;Phys Rev&lt;/em&gt;  &lt;strong&gt;75&lt;/strong&gt; (11), 1736.&lt;/p&gt;&#xA;&lt;p&gt;He refers to his first one, but the systematics in momentum space of the reducible self energy Σ are there.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastActivityDate="2021-09-21T14:36:09.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13711" PostTypeId="1" CreationDate="2021-09-21T21:43:24.290" Score="3" ViewCount="135" Body="&lt;p&gt;In Interchapter VII of his biography of Gauss, W.K. Buhler describes Gauss's discovery of one of the important properties that characterize the &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; invariant (Klein's absolute invariant; Gauss called it &amp;quot;Summatorische Function&amp;quot;): that it assumes every complex value exactly once when calculated on a fundamental domain for the modular group action. He mentions that this result was rediscovered much later by Dedekind.&lt;/p&gt;&#xA;&lt;p&gt;In other words, this result says that &lt;span class=&quot;math-container&quot;&gt;$j(\tau)$&lt;/span&gt; is a surjective function (&amp;quot;onto function&amp;quot;) from the fundamental domain into the complex plane. And it also says that &lt;span class=&quot;math-container&quot;&gt;$j(\tau)$&lt;/span&gt; is injective, since it maps the fundamental domain to the entire complex plane in a one-to-one correspondence.&lt;/p&gt;&#xA;&lt;p&gt;I was trying to search for this result in Gauss's &lt;em&gt;werke&lt;/em&gt;, and I did find a result that comes close to this statement, but not exactly; in p. 478 of volume 3 of his works Gauss says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The equation &lt;span class=&quot;math-container&quot;&gt;$$\left(\frac{Q(t)}{P(t)}\right)^2 = A$$&lt;/span&gt; always has one and exactly one solution in a certain domain.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gauss doesn't specify any restriction on &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and therefore i assume it means &amp;quot;an arbitrary complex number&amp;quot;. This remark is accompanied by a drawing of a certain fundamental domain. &lt;span class=&quot;math-container&quot;&gt;$Q(t),P(t)$&lt;/span&gt; are Jacobi's theta functions, and therefore the left side of the equation coincides with the definition of the square root of the &amp;quot;modular lambda function&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$\lambda(\tau)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Therefore Gauss says that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\lambda(\tau)}$&lt;/span&gt; maps the fundamental domain onto the entire complex plane. Now, I read that &lt;span class=&quot;math-container&quot;&gt;$\lambda(\tau)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$j(\tau)$&lt;/span&gt; are intimately related, but does the result written by Gauss implies the result discovered later by Dedekind? and if not, can someone refer to the exact place in Gauss's werke where he states this result?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="4251" LastEditDate="2021-09-22T05:46:53.350" LastActivityDate="2021-09-22T17:33:01.190" Title="Reference request for Gauss's original discovery of the special property of the $j$ function" Tags="|mathematics|reference-request|gauss|complex-analysis|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13712" PostTypeId="2" ParentId="13708" CreationDate="2021-09-22T04:30:37.337" Score="0" Body="&lt;p&gt;All serious scholars of science or history of science should realize that Wikipedia has a long way to go to become a reliable source of scientific history, especially if it is about very ancient personalities. Article written by people under nicknames or pseudonyms and of sometimes of dubious credibility have little value for researchers. Afterall, who are the authors of those entries? We have no clue. If we start believing whatever appears on Wiki or Stack Exchange all academic history departments in universities will become redundant and they should shut down.&lt;/p&gt;&#xA;&lt;p&gt;Besides other historians, consult the work of Syed Nomanul Haq, &lt;em&gt;Names, Nature and Things&lt;/em&gt;, &lt;a href=&quot;https://link.springer.com/book/10.1007/978-94-011-1898-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer Link&lt;/a&gt; from the Center for Middle Eastern Studies, Harvard University. Another highly respected historian of Jabir is P. Kraus, who is extensively referenced in his book. Kraus also learned Arabic and consulted original bodies of Arabic works.  I am not sure if the chemist turned historian, Lawrence Principe, learned ancient Arabic to consult original works. There is no evidence from his CV.&lt;/p&gt;&#xA;&lt;p&gt;Haq agrees that not all the 3000 pieces of can be attributed to one person. Of course, there is no doubt that &lt;em&gt;one&lt;/em&gt; Jabir Ibn Hayyan existed. Just like botanical and zoological nomenclature of species, it is hard of match trinomial or tetranomial ancient Arabic names, so the point of having &lt;em&gt;many&lt;/em&gt; Jabirs is useless.&lt;/p&gt;&#xA;&lt;p&gt;The problem is what is attributed to him as his body of work (Stigler's law of eponymy) is true or not. Likewise we can question if Pythagorous theorem is his actually his or not? Did Euclid write all the books? Common problem with ancient texts, but should not doubt if Euclid existed or not. Otherwise, skepticism has no limits.&lt;/p&gt;&#xA;&lt;p&gt;The author writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;${ }^{1}$&lt;/span&gt; In the Jabirian treatises which have either been published&#xA;or read by me in manuscripts, as well as in the traditional&#xA;biographies, the patronymic part of the author's name (kunya) appears&#xA;frequently as Abū Mūsa, but sometimes also as Abū 'AbdAllāh. Often&#xA;attached to it is the epithet al-Süfi, the tribal name (nisba) al-Azdĩ&#xA;and names indicating Jäbir's place of origin al-Kūfi or al-Tūsī. Ibn&#xA;Khallikān (d. &lt;span class=&quot;math-container&quot;&gt;$681 / 1282)$&lt;/span&gt; reports in his Biographical Dictionary&#xA;that Jäbir was from Tarsūs (see de Slane tr. [1842-71], I, p. 300&lt;span class=&quot;math-container&quot;&gt;$)$&lt;/span&gt;&#xA;and this is confirmed by another standard source, the Kashf al-Zunūn&#xA;of Häjji Khalifa (d. &lt;span class=&quot;math-container&quot;&gt;$1069 / 1658)$&lt;/span&gt;. However, this latter biographer -&#xA;according to whom Jäbir died in &lt;span class=&quot;math-container&quot;&gt;$160 / 777-$&lt;/span&gt; calls him al-Tarsūsi at&#xA;one place, but at another place says that Jäbir was from Tarsūs and is&#xA;called al-Tūsi (!) (see Flügel ed. [1835-1858], p. 34 and p. 79). Ibn&#xA;al-Nadim, who in naming Jäbir wavers between both kunyas, Abū Mūsa and&#xA;Abū 'AbdAllāh, mentions the belief that Jäbir was originally from&#xA;Khurāsān (see Flügel ed. [1871], pp. 354-358).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-09-22T04:30:37.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13713" PostTypeId="5" CreationDate="2021-09-22T08:44:18.740" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-09-22T08:44:18.740" LastActivityDate="2021-09-22T08:44:18.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13714" PostTypeId="4" CreationDate="2021-09-22T08:44:18.740" Score="0" Body="Freeman John Dyson (1923-2020) was a British-American theoretical and mathematical physicist, mathematician, and statistician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum mechanics, condensed matter physics, nuclear physics, and engineering." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-09-22T16:00:15.287" LastActivityDate="2021-09-22T16:00:15.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13717" PostTypeId="1" CreationDate="2021-09-22T17:18:40.743" Score="1" ViewCount="144" Body="&lt;p&gt;My old linear-algebra teacher, whom I can no longer ask,  wrote on a black board an exclamation mark over the binary symbol of a logical formula, the main symbol of which is that binary symbol, to say that the formula is yet unproven but is about to be proven next. For example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We now turn to &lt;span class=&quot;math-container&quot;&gt;$\varphi \stackrel{!}{\Leftrightarrow} \psi$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“&lt;span class=&quot;math-container&quot;&gt;$\Leftarrow$&lt;/span&gt;”: … [here comes a proof of &lt;span class=&quot;math-container&quot;&gt;$\varphi \Leftarrow \psi$&lt;/span&gt;] …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“&lt;span class=&quot;math-container&quot;&gt;$\Rightarrow$&lt;/span&gt;”: … [here comes a proof of &lt;span class=&quot;math-container&quot;&gt;$\varphi \Rightarrow \psi$&lt;/span&gt;] …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Who was the/a first to introduce this kind of convention and with which meaning exactly? References (ideally, more substantial than „L. Euler, Private communication“ or “Emmy Noether, lectures on algebra in 1934 in Bryn Mawr” :-). )?&lt;/p&gt;&#xA;" OwnerDisplayName="user14234" LastEditorDisplayName="user14234" LastEditDate="2021-09-29T20:56:29.207" LastActivityDate="2021-09-29T20:56:29.207" Title="The exclamation mark over a relation symbol" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13718" PostTypeId="2" ParentId="13711" CreationDate="2021-09-22T17:33:01.190" Score="2" Body="&lt;p&gt;I have found a reference for the injectivity property of the &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; function when it acts on a fundamental domain, and this constitutes the first part of the &amp;quot;special property&amp;quot;. However, regarding the surjectivity of &lt;span class=&quot;math-container&quot;&gt;$j(\tau)$&lt;/span&gt;, I'm still searching.&lt;/p&gt;&#xA;&lt;p&gt;In a short (unpublished) paragraph in p.386 of vol.3, Gauss introduces a mapping from the set of binary quadratic forms of discriminant &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; and the upper half plane &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;. To each form &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)$&lt;/span&gt; Gauss associates a (complex) point &lt;span class=&quot;math-container&quot;&gt;$t = \frac{\sqrt{d}+bi}{a}$&lt;/span&gt;. This mapping connects the arithmetical theory of binary quadratic forms (which Gauss has exhaustively dealt with in chapter V of the D.A) with its function-theoretic aspects. Gauss than describes a &amp;quot;summatory function&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; (which Klein later called &lt;span class=&quot;math-container&quot;&gt;$j(\tau)$&lt;/span&gt;), as a function that remains constant under the action of the full modular group. He says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In particular, if the forms &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(A,B,C)$&lt;/span&gt; are equivalent, so then set &lt;span class=&quot;math-container&quot;&gt;$f(t)=f(u)$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$\frac{t-u}{i}$&lt;/span&gt; is a whole number or if &lt;span class=&quot;math-container&quot;&gt;$t=\frac{1}{u}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gauss says here that &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; remains constant when applying the two generators of the modular group (and therefore doesnot alter under any combination of these generators). Note that he works with the right half plane (not the upper half), but the idea is similar.&lt;/p&gt;&#xA;&lt;p&gt;Although this is not explicity stated in Gauss's passage, his arithmetical definition of the unique reduced quadratic form (for a given equivalency class) as one with &lt;span class=&quot;math-container&quot;&gt;$2b\le a &amp;lt; c$&lt;/span&gt;, corresponds under Gauss's mapping to the fundamental domain for the modular group (and therefore for &lt;span class=&quot;math-container&quot;&gt;$j(\tau)$&lt;/span&gt;). From the uniqueness of the reduced form one concludes that &lt;span class=&quot;math-container&quot;&gt;$f(t)$&lt;/span&gt; is indeed injective on a fundamental domain; this is not surprising, as this was the initial motivation for the construction of a fundamental domain. Gauss drew fundamental domains for several subgroups of &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt; (the modular group), but I didn't find the one for the full &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;; however, from his familiarity with this matters, one can conclude that he definitely knew it.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2021-09-22T17:33:01.190" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13719" PostTypeId="2" ParentId="13473" CreationDate="2021-09-23T07:07:56.433" Score="2" Body="&lt;p&gt;Since no answers are forthcoming, I'll write up what I've found online without digging too deep.&lt;/p&gt;&#xA;&lt;p&gt;In summary, I get the following impression:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;He wanted to leave France to escape the war, primary by going to the USA&lt;/li&gt;&#xA;&lt;li&gt;Before managing to travel to the USA he visited Finland with at least the intention of visiting the USSR.&lt;/li&gt;&#xA;&lt;li&gt;He did not expect war between Finland and the USSR, at least so soon.&lt;/li&gt;&#xA;&lt;li&gt;It's unclear to me whether he visited the USSR or not before being arrested in Finland.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In a comment to my question, Alexandre Eremenko referred to &lt;a href=&quot;https://hsm.stackexchange.com/questions/317/what-are-some-scientific-breakthroughs-that-have-been-done-during-jail-time/318#318&quot;&gt;another answer of his&lt;/a&gt;, which says he had visited the USSR and had come (back?) to Finland, citing Weil's memoirs.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;he left France to visit....Soviet Union:-) After Soviet Union, he visited Finland... just few days before Soviet Union attacked it&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All this is from Weil's memoirs &amp;quot;The apprenticeship of a mathematician&amp;quot;, Translated from the 1991 French original by Jennifer Gage. Birkhäuser Verlag, Basel, 1992.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to this &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Weil/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor biography&lt;/a&gt;, Weil was in Finland visiting Rolf Nevanlinna and Lars Ahlfors, but this article does not mention him ever being in the USSR. It quotes a letter from Weil himself, but the quote only mentions a letter about a potential visit:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There was also a letter in Russian, from Pontryagin, I believe, in response to a letter I had written at the beginning of the summer regarding a possible visit to Leningrad&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The MacTutor article does not cite a source for that particular paragraph. The next paragraph about his imprisonment in France cites: A Borel, André Weil, &lt;em&gt;Bull. Amer. Math. Soc.&lt;/em&gt; &lt;strong&gt;46&lt;/strong&gt; (4) (2009), 661–666. That paper only says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In summer 1939, A. Weil and his wife, Eveline, were in Finland when the Second World War broke out. He decided not to return to France to joint the army, though it was his legal obligation, of course. While his wife went back to France, he remained temporarily in Finland, before deciding which course of action to take.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've found the following citation which seems authorative:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Osmo Pekonen: L'affaire Weil à Helsinki en 1939, &lt;em&gt;Gazette des mathématiciens&lt;/em&gt; &lt;strong&gt;52&lt;/strong&gt; (avril 1992), pp. 13–20. With an afterword by André Weil.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but unfortunately I cannot read French well enough to follow it up. In any case a footnote in &lt;a href=&quot;https://www.math.stonybrook.edu/%7Eaknapp/pdf-files/weil.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this biographical article&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An article by O. Pekonen (L’affaire Weil à Helsinki en 1939, Gaz. Math., No. 52 (1992), 13–20) gives further historical background but disputes that Weil was about to be executed. In a postscript at the end of the Pekonen article, Weil points out that Pekonen provides no facts that contradict Weil’s version.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13422" LastEditorUserId="13422" LastEditDate="2021-10-05T07:55:38.313" LastActivityDate="2021-10-05T07:55:38.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13720" PostTypeId="1" CreationDate="2021-09-23T12:01:55.743" Score="1" ViewCount="98" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Why, then, did Miramontes remain an obscure figure? He was a student at a time when science was especially hierarchical. And the fact that Searle beat Syntex to market cut Mexico — and Miramontes — out of the narrative. Perhaps renown was not Miramontes’ preference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;strong&gt;Source:&lt;/strong&gt; Carmen Drahl, &lt;a href=&quot;https://www.sciencenews.org/article/luis-miramontes-chemistry-birth-control-pill-ingredient-sexual-revolution&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luis Miramontes helped enable the sexual revolution. Why isn’t he better known?&lt;/a&gt;, Science News, September 21, 2021.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I liked the conclusion, but does the fact that he is Mexican, at a time when Latin Americans were (and still are) marginalized in the USA, didn't contribute to this story being erased? I think that an American's voice has always been more powerful than anyone else's, that may have contributed as well. Not to mention the fact that Djerassi himself, one of the directors of the company responsible for the creation of the hormone norethindrone, canceled the participation of the chemist Miramontes from all his writings on the hormone and its creation, citing only him as &amp;quot;a young Mexican chemist&amp;quot; (part cited in the article as well), and what intrigues me is also the fact that many people (mainly feminists who study on the subject) do not know him, or have not heard of his name, even though he played a very important role in history of feminism with the sexual revolution (as stated in the title of the article).&lt;/p&gt;&#xA;" OwnerUserId="15404" LastEditorUserId="6609" LastEditDate="2021-09-23T13:05:03.017" LastActivityDate="2021-09-23T13:05:03.017" Title="Luis Miramontes helped enable the sexual revolution. Why isn’t he better known?" Tags="|social-context|chemists|social|" AnswerCount="0" CommentCount="4" ClosedDate="2021-09-23T17:33:52.440" ContentLicense="CC BY-SA 4.0" />
  <row Id="13721" PostTypeId="1" AcceptedAnswerId="13722" CreationDate="2021-09-23T14:10:57.213" Score="1" ViewCount="133" Body="&lt;p&gt;Some sources (modern sources, and Kepler himself) claim that in his Geo-Heliocentric (Tychonic) model, Tycho Brahe saw that the orbs of the Sun and Mars intersect, and this was one of the reasons which led him to reject the Solid Orbs:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When Tycho first devised his system, he was still thinking of celestial bodies being imbedded in solid orbs. However he found that in his geo-heliocentric system, the orb of Mars intersected the orb of the sun. This led Tycho to [reject the solid orbs]. (&lt;a href=&quot;https://books.google.com/books?id=4b8DCgAAQBAJ&amp;amp;pg=PA70&amp;amp;lpg=PA70&amp;amp;dq=solid%20orbs%20mars%20tycho&amp;amp;source=bl&amp;amp;ots=blL1NjAclc&amp;amp;sig=ACfU3U0qZ4w2rgzVt5l-1STbR_unfAIIQQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;redir_esc=y#v=onepage&amp;amp;q=solid%20orbs%20mars%20tycho&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Oxford Handbook of Early Modern European History, 1350-1750. p.70&lt;/a&gt; ).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have two questions (I would appreciate even reply to one of those questions):&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Why this could not be shown earlier using the traditional Ptolemy form. Was that Geo-Heliocentric model that was necessary, or merely new numbers(?) that Tycho plugged in?. For on first sight I can't see why; since, if there is a point in the orb of Mars that is closer to the Earth, than some other point in the orb of the Sun, why couldn't this be seen also in Ptolemy system where the epicycle of Mars must also intersect the eccentric of the Sun (If we would plug the same numbers to make the models equivalent with respect to the vision from Earth).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In Kepler's &lt;em&gt;Astronomia Nova&lt;/em&gt; he thus writes in chapter 6 (Translation William H. Donahue):&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nevertheless, in the theory of Mars, the planet's eccentric is so small in proportion to the Sun's eccentric that Mars's eccentric and the point O and F are nearer to the Earth C than is the sun S, which was one of the reasons why Brahe denied the solidity of the orbs&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not quite sure I understand what Kepler means by &amp;quot;planet's eccentric is so small in proportion to the Sun's eccentric&amp;quot; because how could it be that the eccentric of Mars be smaller than the Sun's?&lt;/p&gt;&#xA;" OwnerUserId="14954" LastEditorUserId="14954" LastEditDate="2021-09-24T07:05:57.753" LastActivityDate="2021-09-24T07:05:57.753" Title="The Rejection of Solid Orbs by Geo-Heliocentric (Tychonic) model of the solar system" Tags="|astronomy|ptolemy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13722" PostTypeId="2" ParentId="13721" CreationDate="2021-09-23T23:28:48.563" Score="2" Body="&lt;p&gt;I can explain the principle (question 1). In Ptolemy's theory (in Almagest), the sizes of&#xA;planetary orbits are not specified. They can be arbitrary, since scaling of&#xA;any single orbit does not change anything from the point of view of an observer at the center. In his later work (Planetary Hypotheses) Ptolemy decided to fit the orbits as tightly as possible, so that they did not intersect. This hypothesis is not justified by any observation. Later people speculated of solid spheres&#xA;which support planets.&lt;/p&gt;&#xA;&lt;p&gt;In Brahe (and Copernicus) systems where planets revolve around the Sun, the  ratios of the orbits are not arbitrary, they are dictated by the observed motion. So for example Brahe's theory gives a definite ratio of the radius of the orbit of&#xA;Mars, to the radius of the orbit of Sun (I simplify, assuming that all orbits are circles, and the circles are in the same plane.) This ratio tells us&#xA;that the orbits intersect. So there cannot be any solid spheres.&lt;/p&gt;&#xA;&lt;p&gt;Copernicus and Brahe theories are equivalent. And they permit to determine&#xA;the ratios of sizes of all orbits correctly. Ptolemy theory did not allow this, as exposed in Almagest these ratios can be arbitrary, and his later additional hypothesis is not justified. The main difference is that in Copernicus and Brage theory the observer (on Earth) is at a place&#xA;different from the planets orbits center (Sun).&lt;/p&gt;&#xA;&lt;p&gt;Of course, in Brahe's theory Mars never actually collides with Mars,&#xA;since his theory is geometrically equivalent to the Copernicus, and thus it is essentially correct. But their orbits intersect, and this&#xA;excludes any physical existence of &amp;quot;supporting celestial spheres&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-09-23T23:33:59.327" LastActivityDate="2021-09-23T23:33:59.327" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13723" PostTypeId="1" CreationDate="2021-09-24T10:34:41.213" Score="2" ViewCount="107" Body="&lt;p&gt;Who established the current standard(*) demonstration of Euler-Lagrange equations in calculus of variations, that is, &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial \dot y} = 0 $&lt;/span&gt;?. As far as I know, the closest author could be JACOBI (&lt;em&gt;On the theory of the Calculus of Variations and of differential equations&lt;/em&gt;, 1837(**). Really, he starts writing the exact equation and after tries to demonstrate it.&lt;/p&gt;&#xA;&lt;p&gt;So, he starts writing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Let us consider the simplest case; let the integral is to have a maximum or minimum value be&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\displaystyle\int f(x, y, y') dx,$$&lt;/span&gt;&#xA;&lt;em&gt;where &lt;span class=&quot;math-container&quot;&gt;$y'$&lt;/span&gt; is put &lt;span class=&quot;math-container&quot;&gt;$\displaystyle\frac{dy}{dx}$&lt;/span&gt;. Then we know that &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is to be found from the differential equation&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\displaystyle\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial \dot y} = 0$$&lt;/span&gt;&#xA;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since then, the demonstrations Jacobi made have nothing to do with modern usuals ones, yet cited. I think they are more complex and, actually, I don't understand them at all.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, why or who put the so called name 'Euler-Lagrange' to the equation, authors whose works in calculus of variations are older and haven't nor to do with it? (***)&lt;/p&gt;&#xA;&lt;p&gt;(*) E.g., Goldstein, Marion/Thornton, Taylor, etc.&lt;/p&gt;&#xA;&lt;p&gt;(**) A english translation in Todhunter, &lt;em&gt;History of the calculus of Variation&lt;/em&gt;s, 1861. There's a recent text about this question: Goldstine, &lt;em&gt;A History of the Calculus of Variations from the 17th through the 19th Century&lt;/em&gt;, 2011&lt;/p&gt;&#xA;&lt;p&gt;(***) Euler, &lt;em&gt;Elementa calculi variationum&lt;/em&gt;, 1766(there's english translation); Lagrange, &lt;em&gt;Méthode de maximis e minimis&lt;/em&gt;, Miscellanea taurinensiea, T.I, 1759 (french translation in Ouvres de Lagrange, T.I); &lt;em&gt;Sur la methode des variations&lt;/em&gt;, Miscellanea taurinensiea, t.IV, 1766(french translation in Ouvres de Lagrange, T.IV)&lt;/p&gt;&#xA;" OwnerDisplayName="follonic" LastEditorUserId="10106" LastEditDate="2021-09-25T13:29:26.240" LastActivityDate="2021-09-25T13:29:26.240" Title="Who established the current standard demonstration of Euler-Lagrange equation in calculus of variations?" Tags="|mathematics|mathematical-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13724" PostTypeId="1" CreationDate="2021-09-24T15:21:01.433" Score="6" ViewCount="134" Body="&lt;p&gt;The shape of the blackbody radiation spectrum was known in the 19th century from experimental measurements, and before the theoretical discovery of Planck's law.&lt;/p&gt;&#xA;&lt;p&gt;At those times, how did people manage to measure the intensity distribution of the blackbody spectrum in its different frequency windows? What sort of technology available at that time was being used to measure intensities of the UV, IR, and visible part of the spectrum? Thanks!&lt;/p&gt;&#xA;" OwnerUserId="9960" LastActivityDate="2021-09-24T15:21:01.433" Title="What technology was used to determine the shape of the blackbody spectrum at the 19th century?" Tags="|physics|quantum-mechanics|experimental-physics|technology|radiation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13725" PostTypeId="1" AcceptedAnswerId="13732" CreationDate="2021-09-25T14:22:07.417" Score="1" ViewCount="150" Body="&lt;p&gt;I read recently that there exists an Abel-Runge lemma. What is it? Google does not give an answer. I know about Abel's lemma (the summation by parts) and the Runge-Kutta method but I have never heard about Abel-Runge lemma before.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update.&lt;/strong&gt; This is a history of mathematics question. Apparently there was a result of Abel which was later modified (generalized?) by Runge. I do not know the work of Runge well enough to find out what it was. Hence the question.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-09-25T22:46:47.667" LastActivityDate="2021-09-26T20:38:57.683" Title="Abel-Runge lemma" Tags="|mathematics|" AnswerCount="1" CommentCount="8" ClosedDate="2021-09-29T23:06:59.403" ContentLicense="CC BY-SA 4.0" />
  <row Id="13726" PostTypeId="2" ParentId="13275" CreationDate="2021-09-25T15:18:29.807" Score="1" Body="&lt;p&gt;Paul Dirac is a well known physics Nobel laureate for his work in the foundations of quantum mechanics (in 1933 alongside Erwin Schrödinger and Werner Heisenberg). His biography in the Nobel foundation reads &lt;a href=&quot;https://www.nobelprize.org/prizes/physics/1933/dirac/biographical/&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paul Adrien Maurice Dirac was born on 8th August, 1902, at Bristol, England, his father being Swiss and his mother English. He was educated at the Merchant Venturer’s Secondary School, Bristol, then went on to Bristol University. Here, he studied electrical engineering, obtaining the B.Sc. (Engineering) degree in 1921.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6609" LastActivityDate="2021-09-25T15:18:29.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13727" PostTypeId="5" CreationDate="2021-09-25T15:22:04.923" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-09-25T15:22:04.923" LastActivityDate="2021-09-25T15:22:04.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13728" PostTypeId="4" CreationDate="2021-09-25T15:22:04.923" Score="0" Body="Alexander Grothendieck (1928-2014) was a French mathematician (born in Germany) and Fields medal winner.  He made important contributions in topology, algebra and logic." OwnerUserId="4251" LastEditorUserId="5989" LastEditDate="2021-09-26T08:27:54.233" LastActivityDate="2021-09-26T08:27:54.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13729" PostTypeId="2" ParentId="6358" CreationDate="2021-09-26T05:16:53.537" Score="-1" Body="&lt;p&gt;Theoretical mathematicians and physicists often make calculational errors. And it seems that this is the case here. The numbers that theoretical mathematicians often use are one or two with the rest having a vanishing small probability of occuring. Hence, it's not surprising they make mistakes.&lt;/p&gt;&#xA;&lt;p&gt;Without actually being comfortable with using prime numbers on a daily basis, I would have said 57 was prime. Like Grothendieck did. Though of course, I'm not comparimg myself to him in any way. Another example is a theoretical physicist who is interested space and geometry but who often mixes up his left and right - like I do - and have just done so in the last hour or so.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-09-26T05:16:53.537" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13730" PostTypeId="2" ParentId="317" CreationDate="2021-09-26T05:26:14.340" Score="3" Body="&lt;p&gt;Jean Leray, a French mathematician, was a prisoner-of-war in Edelbach, Austria from 1940-45. He concealed his expertise in differential equations for fear he might be put in the service of the Nazi war work and hence concentrated on more theoretical topics, in this case, algebraic topology. His work was innovative in sheaf theory and laid the foundations for what is known as spectral sequences.&lt;/p&gt;&#xA;&lt;p&gt;The Leray spectral sequence is named in his honour and was actually introduced by him in 1946. Today, it's now seen as a special case of the Grothendieck spectral sequence and which shows its seminal importance. It is important in the study of cohomology and sheaf theory was a strong motivation for Topos theory.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-09-26T10:20:36.953" LastActivityDate="2021-09-26T10:20:36.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13732" PostTypeId="2" ParentId="13725" CreationDate="2021-09-26T20:24:42.813" Score="2" Body="&lt;p&gt;I found two papers by Runge about solving polynomial equations (one published in 1885, Runge, C. On the solvable equations of the form &lt;span class=&quot;math-container&quot;&gt;$x^5+ u x + v = 0$&lt;/span&gt; (German) Acta Math. 7 (1885), no. 1, 173-186, and one published in 1963, about 40 years after Runge's death, by Ostrovski, Runge, Carl&#xA;Eine Vorzeichenregel in der Theorie der algebraischen Gleichungen. (German)&#xA;Jber. Deutsch. Math.-Verein. 66 (1963/64), Abt. 1, 52–66). None of them mentions Abel. So I guess the Abel-Runge lemma does not exist and this answers my question.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-09-26T20:38:57.683" LastActivityDate="2021-09-26T20:38:57.683" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13733" PostTypeId="1" CreationDate="2021-09-27T21:41:49.173" Score="5" ViewCount="6626" Body="&lt;p&gt;In the memoirs of the Polish mathematician Hugo Steinhaus (&amp;quot;Mathematician for all seasons&amp;quot;, vol. I, English translation, Springer, 2015) he recollects a conversation with Henri Lebesgue in 1938:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;He told us that it was a mathematician&#xA;at the Sorbonne in the thirteenth century who had first observed that the two feet&#xA;of a normal human being are not congruent by means of an orientation-preserving&#xA;Euclidean transformation, and only then did shoemakers begin making different&#xA;left and right shoes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Can this be true?? I remember seeing a few surviving samples of ancient shoes in museums, but did not pay attention on this detail.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-09-29T01:11:11.547" Title="When did they begin to make left and right shoes?" Tags="|mathematics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13734" PostTypeId="2" ParentId="13733" CreationDate="2021-09-27T22:48:29.437" Score="11" Body="&lt;p&gt;While some sources claim it is a modern invention, Encyclopedia Brittanica claims that Romans already differentiated right from left footwear, see &lt;a href=&quot;https://www.britannica.com/topic/shoe&quot; rel=&quot;noreferrer&quot;&gt;shoe&lt;/a&gt;. Look also at &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/e/e7/Egyptian_sandals%2C_vegetable_fiber_-_Bata_Shoe_Museum_-_DSC00009.JPG&quot; rel=&quot;noreferrer&quot;&gt;these sandals&lt;/a&gt; from Ancient Egypt.&lt;/p&gt;&#xA;&lt;p&gt;Looking on the web, it is a mess as is full of blog/news articles without sources pointing to somebody in the 1800s. For English sources it is sometimes claimed that left/right shoes were invented somewhere in Philadelphia, Pennsylvania. Some French sources claim that it was &lt;a href=&quot;https://fr.wikipedia.org/wiki/Alexis_Godillot&quot; rel=&quot;noreferrer&quot;&gt;Alexis Godillot&lt;/a&gt;, an industrial man that introduced some famous &lt;em&gt;&lt;a href=&quot;https://fr.wikipedia.org/wiki/Godillot&quot; rel=&quot;noreferrer&quot;&gt;godillot&lt;/a&gt;&lt;/em&gt; boots that differentiated left from right.&lt;/p&gt;&#xA;&lt;p&gt;The closest source I got so far was from German Wikipedia (which also confusingly) claims that it was a German academic called  &lt;a href=&quot;https://de.wikipedia.org/wiki/Georg_Hermann_von_Meyer&quot; rel=&quot;noreferrer&quot;&gt;Georg Hermann von Meyer&lt;/a&gt; who apparently advocated for chiral shoes under a work (yet to be found) called &lt;em&gt;Über die richtige Gestalt des Schuhes (1858)&lt;/em&gt;. Some of von Meyer's work is freely available (translated) in Wikisource, &lt;a href=&quot;https://en.wikisource.org/wiki/Why_the_Shoe_Pinches&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. This observation implies that about 1858, chiral shoes  were at least as popular as symmetrical shoes.&lt;/p&gt;&#xA;&lt;p&gt;Anyway most of the stories seem like oversimplified anecdotes (including Lebesgue's story) and I don't think there is a clear source but I will update this answer if something more concrete comes up.&lt;/p&gt;&#xA;&lt;p&gt;Edit: Conifold provided this academic paper by A. Swallow &lt;a href=&quot;https://doi.org/10.1016/S0950-3579(87)80011-0&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;History of Shoes&lt;/em&gt; (1987)&lt;/a&gt;, which does not cite any of the men above but clearly states that chirality in shoes has been reintroduced many times in history depending on the kind of footwear.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-09-28T10:11:15.513" LastActivityDate="2021-09-28T10:11:15.513" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="13735" PostTypeId="1" AcceptedAnswerId="13737" CreationDate="2021-09-28T19:52:11.803" Score="1" ViewCount="195" Body="&lt;p&gt;Safe human travel into space had to have become scientifically accepted as plausible long before any nations managed to successfully pull it off (and unsafe or one-way human travel likely earlier than that), but I'm having trouble finding any documentation of scientific discussion of the concept in terms of when it went from being considered as &amp;quot;probably impossible&amp;quot; or &amp;quot;currently unknown&amp;quot; to &amp;quot;probably possible in the future&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I've found that the idea using rockets to reach space was conceptualized in the 19th century and the early 20th century by scientists such as William Leitch, Konstantin Tsiolkovsky, and others, but I'm unsure how seriously these were taken by the scientific communities at the time, or if it was thought possible earlier, just without knowing a plausible mechanism to do so yet.&lt;/p&gt;&#xA;" OwnerUserId="15428" LastEditorUserId="4251" LastEditDate="2021-09-29T04:39:59.213" LastActivityDate="2021-10-03T01:58:32.343" Title="Roughly when did it become accepted by the western scientific community that human space flight could be possible in the future?" Tags="|physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13736" PostTypeId="2" ParentId="3131" CreationDate="2021-09-28T20:33:30.073" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Takeo_Nakasawa&quot; rel=&quot;nofollow noreferrer&quot;&gt;Takeo Nakasawa&lt;/a&gt; was a Japanese mathematician who died at age 33. He published 4 papers introducing the subject of matroid theory. At the same time this subject was independently discovered elsewhere by Hassler Whitney, leading to Nakasawa's contributions being lost until much later. After this, he became a bureaucrat in Japanese-controlled Manchuria in 1938. Quoting wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With the defeat of Japan in 1945, the Soviets took control of Manchuria and Nakasawa was carted off to Siberia. He died of dystrophia at the age of 33 in Khabarovsk in the Soviet Union in 1946.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15429" LastActivityDate="2021-09-28T20:33:30.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13737" PostTypeId="2" ParentId="13735" CreationDate="2021-09-28T22:42:00.643" Score="5" Body="&lt;p&gt;From the &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D1%81%D0%BC%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BF%D0%BE%D0%BB%D1%91%D1%82&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russian Wikipedia article&lt;/a&gt; (my translation): Perhaps the first person&#xA;who pointed to the possibility of using rockets for travel in space was W. Leitch. In 1861 he wrote an article &amp;quot;A Journey Through Space&amp;quot; which was published in 1862 in his book &amp;quot;God’s Glory in the Heavens&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In 1881, Kibaltchich, while imprisoned [for  revolutionary activity], put up an idea of a flying apparatus with jet propulsion. A few days before his execution, Kibaltchich developed a detailed project of such an apparatus for space travel. It was published only in 1918.&lt;/p&gt;&#xA;&lt;p&gt;First person who studied theoretical aspects of space travel was Tsiolkovsky who developed, in particular, the Tsiolkovsy formula (that, in particular, formed the basis of Soviet space program).&lt;/p&gt;&#xA;&lt;p&gt;A little later, German scientist H. Obert also found, in 1923, the main equation of the rocket technology and showed that using multi-stage rocket (as suggested by Tsiolkovsky) one can move an arbitrary large object  into a desired orbit.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Konstantin_Tsiolkovsky&quot; rel=&quot;nofollow noreferrer&quot;&gt;English Wikipedia article&lt;/a&gt; about Tsiolkovsky includes a part about his influence on the scientists who made actual rockets for space travel (von Braun, Korolev, others):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although many called his ideas impractical, Tsiolkovsky influenced later rocket scientists throughout Europe, like Wernher von Braun. Soviet search teams at Peenemünde found a German translation of a book by Tsiolkovsky of which &amp;quot;almost every page...was embellished by von Braun's comments and notes.&amp;quot;  Leading Soviet rocket-engine designer Valentin Glushko and rocket designer Sergey Korolev studied Tsiolkovsky's works as youths, and both sought to turn Tsiolkovsky's theories into reality.In particular, Korolev saw traveling to Mars as the more important priority, until in 1964 he decided to compete with the American Project Apollo for the Moon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="643" LastEditDate="2021-10-01T22:40:44.497" LastActivityDate="2021-10-01T22:40:44.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13738" PostTypeId="2" ParentId="13733" CreationDate="2021-09-29T01:11:11.547" Score="1" Body="&lt;p&gt;I would speculate that the first shoes were chiral and that symmetrical shoes are a relatively recent invention.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;I believe that ancient cobblers would have made the shoe custom for the patron's foot.  They would not have even had to think about it, the pair of shoe would be chiral.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Symmetrical shoes would have been introduced with mass production.  It must be easier to tool a production line to produce one type of shoe than two types of shoes.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;That is my speculation.  In support of this consider &lt;a href=&quot;https://bellatory.com/fashion-industry/The-History-of-Shoes-Early-Footwear&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://bellatory.com/fashion-industry/The-History-of-Shoes-Early-Footwear&lt;/a&gt;, which says that the earliest shoes were sandals.  Just try and imagine symmetrical sandals.  It does not make any sense.&lt;/p&gt;&#xA;" OwnerUserId="15431" LastActivityDate="2021-09-29T01:11:11.547" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13739" PostTypeId="2" ParentId="13735" CreationDate="2021-09-29T12:33:20.430" Score="5" Body="&lt;p&gt;Among the early proponents of space travel, a Ukrainian engineer &lt;a href=&quot;https://en.wikipedia.org/wiki/Yuri_Kondratyuk&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yuri Vasilievich Kondratyuk&lt;/a&gt; should be mentioned. In 1925 he published his book Conquest of Interplanetary Space with theoretical justification of space travel. His research could not continue since he was arrested and imprisoned by the Spoviet authorities. But his book made an influence of the later developers, and his contribution is well recognized: for example a crater on the Moon and an asteroid are named after him. His ideas were used in planning of Apollo missions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-10-03T01:58:32.343" LastActivityDate="2021-10-03T01:58:32.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13742" PostTypeId="1" CreationDate="2021-10-02T05:43:18.500" Score="1" ViewCount="125" Body="&lt;p&gt;Social sciences, especially sociology, embrace a distinction between interpretivism &lt;sup&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Antipositivism&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/sup&gt; and (post-)positivism &lt;sup&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Postpositivism&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/sup&gt;. Some social scientist embrace one method over the other and some speak in favor of both. Are there any findings discovered by interpretivist methodology which were also confirmed by (post-)positivist methodology?&lt;/p&gt;&#xA;" OwnerUserId="5483" LastEditorUserId="4251" LastEditDate="2021-10-02T08:08:53.017" LastActivityDate="2021-10-04T08:51:19.693" Title="Are there examples where interpretivist findings have been confirmed by (post-)positivist methodology?" Tags="|philosophy-of-science|scientific-method|social-sciences|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13744" PostTypeId="1" CreationDate="2021-10-03T02:32:53.460" Score="1" ViewCount="50" Body="&lt;p&gt;It's often stated that Tycho Brahe objected to Copernicus' heliocentric model on the basis that, if Earth were revolving around the Sun, stellar parallax would be observable, due to the changing position of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;In describing what model Copernicus and Brahe used for stellar parallax, most things I've seen give an explanation of the more common modern model, in which stars believed to be in the 'foreground' relative to Earth are observed against stars contended to be much further away (For the purposes of the formula, unmoving stars).&lt;/p&gt;&#xA;&lt;p&gt;This couldn't be the the model meant by Copernicus and Brahe, however;  Both believed the stars to be set upon an outer sphere (Although Brahe believed this sphere revolved around Earth and Copernicus believed it to be unmoving), and as such neither would have suggested the version of the stellar parallax model which sets stars contended to be in the 'foreground' against 'background' stars.&lt;/p&gt;&#xA;&lt;p&gt;My question is, what was actually being contended by Brahe and by Copernicus, respectively?  What form of stellar parallax model were they holding to with regards to this?&lt;/p&gt;&#xA;" OwnerUserId="15441" LastActivityDate="2021-10-03T02:32:53.460" Title="What did Copernicus and Tycho Brahe actually contend about stellar parallax?" Tags="|astronomy|" AnswerCount="0" CommentCount="5" ClosedDate="2021-10-03T06:13:45.820" ContentLicense="CC BY-SA 4.0" />
  <row Id="13748" PostTypeId="2" ParentId="6543" CreationDate="2021-10-05T20:12:41.447" Score="3" Body="&lt;p&gt;Actually, ancient, medieval, and early modern astronomers mostly believed that the stars were lights attached to the inside of a hollow sphere centered on Earth and that the hollow sphere rotated 360 degrees every day.&lt;/p&gt;&#xA;&lt;p&gt;Those who believed in the heliocentry theory believed that the stars were on the inside of a hollow sphere centered on the Sun.  They believed that the hollow sphere that the stars were stuck to stood still and the Earth rotated 360 degrees every day.&lt;/p&gt;&#xA;&lt;p&gt;Thus they almost all believed that all the stars were at the same distance from the center of the star sphere.  So they didn't believe that some stars were nearer than other stars and didn't think that you could measure shifts in the relative poitions of the nearer stars compared to the farther stars, because they didn't believe in nearer and farther stars.&lt;/p&gt;&#xA;&lt;p&gt;In the heliocentric theory Earth would orbit around the Sun once a year, and every six months it would be on the opposite side of the Sun from where it was 6 months before and where it would be six months later.&lt;/p&gt;&#xA;&lt;p&gt;Nobody knew what the distance between Earth and Sun - which is called the Astronomical Unit, or AU - was in ancient times.  Hellenistic philosophers calculated the size of the Earth and the relative size and distance of the Moon.  Since the Moon sometiems eclipsed the Sun, they knew that the Sun was farther than the Moon, but didn't know how many times farther.&lt;/p&gt;&#xA;&lt;p&gt;So if the orbit of the Earth was 1 AU (however long an AU was) from the Sun, Earth's position at any one moment would be two AU away from where it would be 6 months before or after that moment.&lt;/p&gt;&#xA;&lt;p&gt;Ancient astronomers measured the directions to stars.  They measured the angle of how far above the horizon a star was at a specific moment, and the angle of how far horizontally it was from a landmark at a specific moment.  And they did that for hundreds and even a few thousand stars visible without telescopes.  So they made maps of the celestial sphere, which they imagined was an actual physical sphere, and lists of the coordinates of stars on that sphere.&lt;/p&gt;&#xA;&lt;p&gt;If the heliocentric theory was true,  the Earth would travel to points 2 AU apart in 6 months. If the sphere of the stars was 20 AU from the Sun, ancient astronomers could have measured shifts in the positions of all the stars every six months.  They didn't&lt;/p&gt;&#xA;&lt;p&gt;If the sphere of the stars was 200 AU from the Sun, ancient astronomers culd have measured shifts in the positions of all the stars every six months.  They didn't.&lt;/p&gt;&#xA;&lt;p&gt;So ancient astronomers knew that either the heliocentric theory was wrong, or the distance to the sphere of the starswas many times as great as the distance between Earth and the Sun - and they knew that the distance between Earth and the Sun was at least as great as the distance between Earth and the Moon, which they knew was about 250,000 modern miles.&lt;/p&gt;&#xA;&lt;p&gt;Most ancient astronomers who knew of the heliocentric theory couldn't believe that the sphere of the stars could be as far away as it would have to be if they couldn't measure any annual parallax.  So they believed in the geocentric theory instead of the heliocentric theory.&lt;/p&gt;&#xA;&lt;p&gt;The precession of the equinoxes is a long term wobble in the Earth's axis of rotation, taking about 25,000 years to make one full circle.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The ancient Greek astronomer Hipparchus (c. 190–120 BC) is generally accepted to be the earliest known astronomer to recognize and assess the precession of the equinoxes at about 1° per century (which is not far from the actual value for antiquity, 1.38°),[6] although there is some minor dispute about whether he was.[7] In ancient China, the Jin-dynasty scholar-official Yu Xi (fl. 307–345 AD) made a similar discovery centuries later, noting that the position of the Sun during the winter solstice had drifted roughly one degree over the course of fifty years relative to the position of the stars.[8]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Precession&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Precession&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The heliocentric theory was revived and expanded by Coppernicus in the early 1500s and became widely known.  Copernicus was able to deduce the relative sizes of the planetary orbits, so that if the width of any orbit could be determined the sizes of all would be known.&lt;/p&gt;&#xA;&lt;p&gt;In 1609 Galleleo began his telescopic observantions, making many discoveries.  Johannes Kelper later discovered that the planets orbit in ellipical orbits, getting near to and farther from the Sun during each orbit, thus convincing those who believed in the heliocentric theory to give up the idea that the planets were attached to giant transparent spheres. If the planetary spheres didn't exist, maybe the sphere of the stars didn't exist.&lt;/p&gt;&#xA;&lt;p&gt;In 1672 the parallax of Mars was observed from 2 different spots on Earth, thus providing the first reasonably accurate measurement of the AU.  More and more accurate measurements hae been made in the following centuries.&lt;/p&gt;&#xA;&lt;p&gt;Precession results in all the stars appearing to move in the same direction and distance. But there is also proper motion of stars, very slow apparent motion independent of precession.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proper motion was suspected by early astronomers (according to Macrobius, AD 400) but a proof was not provided until 1718 by Edmund Halley, who noticed that Sirius, Arcturus and Aldebaran were over half a degree away from the positions charted by the ancient Greek astronomer Hipparchus roughly 1850 years earlier.[23]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proper_motion#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Proper_motion#History&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Sun and the other stars are all orbiting the center of the galaxy with somewhat different speeds and directions, thus resulting in a slow change in the direction to each star as seen from another star, such as the Sun.&lt;/p&gt;&#xA;&lt;p&gt;Once astronomers accepted proper motion, they had to give up the idea that all the stars were stuck to the inner surface of a hollow sphere, and had to accept that they were flying around in three dimensional space.&lt;/p&gt;&#xA;&lt;p&gt;Proper motion means that if Earth didn't orbit around the Sun, a star's apparent motion across the sky over many years would appear to be a straight line.  But because Earth does orbit the Sun, a star's apparent motion will actually be a sort of wobbling line acorss the sky.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a treatise in 1755, Immanuel Kant, drawing on earlier work by Thomas Wright,[229] speculated (correctly) that the Milky Way might be a rotating body of a huge number of stars, held together by gravitational forces akin to the Solar System but on much larger scales.[230] The resulting disk of stars would be seen as a band on the sky from our perspective inside the disk. Wright and Kant also conjectured that some of the nebulae visible in the night sky might be separate &amp;quot;galaxies&amp;quot; themselves, similar to our own. Kant referred to both the Milky Way and the &amp;quot;extragalactic nebulae&amp;quot; as &amp;quot;island universes&amp;quot;, a term still current up to the 1930s.[231][232][233]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Milky_Way&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Milky_Way&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So by then astronomers seemed to have abandoned the idea that all stars were at the same distance.&lt;/p&gt;&#xA;&lt;p&gt;Searching for stellar parallaxes, James Bradley discovered another tiny apparent motion of stars, caused by the abberation of light due to Earth's motion around the Sun inn1725-1727.  Observing stars for 20 years from 17271747, Bradley proved that abberation existed, and also discovered another woblbe in stellar posiitons due to nutation, a wobble of the Earth over an 18.6 year period (there are other smaller amounts of nutatoion with different periods).&lt;/p&gt;&#xA;&lt;p&gt;But the abberation of light and nutation make much larger changes in the apparent positons of stars than stellar parallax does.&lt;/p&gt;&#xA;&lt;p&gt;Finally, in the 1830s, astronomical instruments and techniques adavanced enough for the prallaxes of stars to be measure.  Despite al lthe evidence over the centuries of greater and greater minimum possible distances to the stars, the distances measured were shockingly to astronomers.&lt;/p&gt;&#xA;&lt;p&gt;Thomas Henderson, at the Cape of Good HOpe, measured the parallax of Alpha Centauri, but couldn't believe how distant it was and didn't publish his results.  Struve measured the parallax of Vega fairly accurately and published his results. And then Friderich Wilhelm Bessel measured the parallax of 61 Cygni. Since 61 cygni was much closer than Vega, Struve chickened out and published a revised parallax for Vega that was closer to the parallax of 61 Cygni and not nearly as accurate.  And so Bessel is usually credited with making the first stellar parallax measurement.&lt;/p&gt;&#xA;&lt;p&gt;So by 1840 astronomers knew that stars had different distances and were not attached to ine inside of a sphere.&lt;/p&gt;&#xA;&lt;p&gt;In the later 19th century astronomical photography became more and more common and it became used for parallax measurements.&lt;/p&gt;&#xA;&lt;p&gt;I believe that one technique became to air a telescope at a star believed to be many times father than the target star, and ppearing lcose to it in the sky.  They would take photos of the sky around the distant star over time, and then measure how much the position of the target star moved compared to the position of the star the telescope was aimed at.&lt;/p&gt;&#xA;&lt;p&gt;In modern times electronic light detecting devices have replaced photography for many astronomical purposes.&lt;/p&gt;&#xA;&lt;p&gt;And at the present time many telescopes are capable of detecting very faint and distant galaxies. Those faint background galaxies can be seen all over the sky except in the plane of the Mikly Way Galaxy where all he stars and lcouds of gas and dust blot them out.&lt;/p&gt;&#xA;&lt;p&gt;So it might be the modern technique to aim the telescope at a distant background galaxy thousands or millions of times farther than the target star to see how much the target star moves compared to the distant galaxy and measure its parallax.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;em&gt;Hipparchos&lt;/em&gt; satellite automatically measured the positions, proper motions, and parallaxes of hundreds of thousands of stars from 1889-1993.  The &lt;em&gt;Gaia&lt;/em&gt; satallite from 2013-c.2022 has been measuring the parallaxes of millions of stars with even greater precision.&lt;/p&gt;&#xA;&lt;p&gt;And I guess that the logical next step in parallax precision would be to develop satellites even more precise and accurate than &lt;em&gt;Gaia&lt;/em&gt; and put one in Jupiter's L4 posiiton and one in Jupiter's L5 position, where they will always be about 10 AU apart, 5 times the maximum baseline for Earthbound observatiories, &lt;em&gt;Hipparchos&lt;/em&gt;, and &lt;em&gt;Gaia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;And the next step after that would be to put observatories in the L4 and L5 positions of Neptune, where they will always be about 60 AU apart, 30 times teh baseline on Earth.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2021-10-05T20:12:41.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13749" PostTypeId="1" AcceptedAnswerId="13750" CreationDate="2021-10-06T08:00:45.833" Score="0" ViewCount="191" Body="&lt;p&gt;What is the difference between caloric and heat? Is there a difference?&lt;/p&gt;&#xA;" OwnerUserId="15449" LastActivityDate="2021-10-06T13:42:30.127" Title="What is the difference between caloric and heat?" Tags="|physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13750" PostTypeId="2" ParentId="13749" CreationDate="2021-10-06T09:12:59.967" Score="0" Body="&lt;p&gt;Heat is related to energy transfer (usually dissipative). Caloric is an obsolete hypothetical fluid that explained how matter was modified when heated.&lt;/p&gt;&#xA;&lt;p&gt;Caloric theory was popular in the at the end of 18th century and the beginning of 19th century, it was a refined version of phlogiston theory (from early 18th century) which was an hypothetical fluid that entered or left matter during combustion. For example, phlogiston flow explained why some containers gained mass when the contents inside burned. Phlogiston theory was abandoned when Antoine Lavoisier demonstrated the conservation of mass. The gain of mass was explained by oxidation (gaining oxygen atoms from the air). But Lavoisier himself refined the theory and invented caloric theory. Under the new theory, caloric was weightless, but still flowed from hot objects to cool objects increasing the temperature of the latter. Two of the problems with caloric theory are that it could not explain heat coming from friction (as it heats both the rubbing and the rubbed object) nor the heat coming from radiation (like that coming from the sun). The caloric theory was eventually superseded by mechanical theories of heat and the discovery of energy conservation.&lt;/p&gt;&#xA;&lt;p&gt;Modern thermodynamic theory is much more subtle and it is based on the microscopic description of matter. Matter is composed of atoms which are constantly jiggling or moving around. When you heat molecular oxygen gas, for example, you increase the average speed of the molecules. If this gas A interacts with a gas B that is colder (lower average speed of the molecules), the molecules of gas A will bump with the molecules of gas B transferring them energy (heat) until the two gases are in average at the same temperature. Radiative heat is explained in the same way but it is explained by the energy that can be transferred by electromagnetic waves (photons).&lt;/p&gt;&#xA;&lt;p&gt;I recommend this answer from physics SE:&lt;a href=&quot;https://physics.stackexchange.com/a/316155/168640&quot;&gt;Why was caloric theory accepted despite observations that heat was produced by friction?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-10-06T13:42:30.127" LastActivityDate="2021-10-06T13:42:30.127" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13751" PostTypeId="1" CreationDate="2021-10-06T11:50:26.570" Score="1" ViewCount="192" Body="&lt;p&gt;How far away did Tycho Brahe estimate that the fixed stars were? In multiples of our sun I imagine!&lt;/p&gt;&#xA;" OwnerUserId="1763" LastEditorUserId="4251" LastEditDate="2021-10-09T09:29:53.847" LastActivityDate="2021-10-09T09:29:53.847" Title="Tycho Brahe and the distance to the fixed stars" Tags="|astronomy|brahe|astronomers|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13754" PostTypeId="1" CreationDate="2021-10-08T10:09:10.253" Score="-1" ViewCount="106" Body="&lt;p&gt;For instance, force was discovered to be proportional to mass and acceleration.&lt;/p&gt;&#xA;&lt;p&gt;How are these proportional relationships discovered and proven to be true?&lt;/p&gt;&#xA;" OwnerUserId="13555" LastActivityDate="2021-10-08T10:37:41.930" Title="In physics, how are two quantities found to be proportional to each other?" Tags="|physics|" AnswerCount="1" CommentCount="1" ClosedDate="2021-10-10T14:11:37.873" ContentLicense="CC BY-SA 4.0" />
  <row Id="13755" PostTypeId="2" ParentId="13754" CreationDate="2021-10-08T10:37:41.930" Score="2" Body="&lt;p&gt;I think that the problem is that people think that &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt; is a law (and needs a proof) while it is a definition of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, so does not require a proof.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2021-10-08T10:37:41.930" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13758" PostTypeId="1" CreationDate="2021-10-09T15:18:45.110" Score="0" ViewCount="49" Body="&lt;p&gt;I would be grateful if anyone could shed some light on these questions.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Which scientist(s) first measured the elementary charge? If it wasn't measured in a straightforward experimental setting, and was posited a priori or derived from some prior equation formally, what was the underlying reasoning?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What experiment or a set of experiments led to the empirical confirmation of the value of the elementary charge?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15244" LastEditorUserId="4251" LastEditDate="2021-10-09T17:59:58.277" LastActivityDate="2021-10-09T17:59:58.277" Title="Which scientist(s) first measured the elementary charge and how did they do it?" Tags="|physics|electromagnetism|experimental-physics|electricity|physicists|" AnswerCount="0" CommentCount="3" ClosedDate="2021-10-10T01:43:13.253" ContentLicense="CC BY-SA 4.0" />
  <row Id="13759" PostTypeId="1" CreationDate="2021-10-10T03:46:23.070" Score="9" ViewCount="410" Body="&lt;p&gt;From reading a few accounts of the unsolvability of the quintic, I am told that, e.g., there were public contests in which people competed to solve polynomial equations, and that over the course of 200-300 years so much effort went into trying to find a solution for the quintic that people started to doubt that a solution even existed (which did eventually lead to such a result).&lt;/p&gt;&#xA;&lt;p&gt;Generally, I get the impression that a lot of brainpower went into trying to solve polynomial equations (both trying to come up with general solution formulas and solving particular instances) and that people were really interested in this problem.&lt;/p&gt;&#xA;&lt;p&gt;Why? What was so interesting about solving polynomial equations that people spent so much effort on the problem?&lt;/p&gt;&#xA;" OwnerUserId="13141" LastEditorUserId="4251" LastEditDate="2021-10-10T06:58:53.080" LastActivityDate="2021-10-26T11:00:41.447" Title="Why was solving polynomial equations historically considered so interesting?" Tags="|mathematics|abstract-algebra|elementary-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13760" PostTypeId="1" CreationDate="2021-10-10T07:23:57.440" Score="2" ViewCount="138" Body="&lt;p&gt;Who first proposed the idea of &lt;a href=&quot;https://en.wikipedia.org/wiki/Borel_functional_calculus#Resolution_of_the_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;resolution of the identity&amp;quot;&lt;/a&gt; as used in the functional calculus of self-adjoint operators? Was it  von Neumann?&lt;/p&gt;&#xA;&lt;p&gt;In Japanese, it translates as &amp;quot;resolution of the unit&amp;quot;. To figure out why it was translated that way, I did some research and now I'd like to know who came up with the idea in the first place.&lt;/p&gt;&#xA;" OwnerUserId="15467" LastEditorUserId="55" LastEditDate="2021-10-12T06:55:07.893" LastActivityDate="2021-10-12T06:55:07.893" Title="Who first proposed the idea of &quot;resolution of the identity&quot;?" Tags="|physics|quantum-mechanics|theoretical-physics|mathematical-physics|von-neumann|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13761" PostTypeId="5" CreationDate="2021-10-10T09:33:34.337" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-10-10T09:33:34.337" LastActivityDate="2021-10-10T09:33:34.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13762" PostTypeId="4" CreationDate="2021-10-10T09:33:34.337" Score="0" Body="André Weil (1906-1998) was a French mathematician who worked on algebraic geometry and number theory." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-10-10T21:16:04.517" LastActivityDate="2021-10-10T21:16:04.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13763" PostTypeId="2" ParentId="13760" CreationDate="2021-10-10T10:13:10.373" Score="4" Body="&lt;p&gt;It was Hilbert's way to generalize spectral theory to infinite dimensional operators when he discovered continuous spectrum in 1906. Hilbert used bilinear forms when writing integrals, and the more recognizable operator version was introduced by Riesz in &lt;em&gt;Les systemes d'equations lineaires a une infinite d'inconnues&lt;/em&gt; (1913), see &lt;a href=&quot;https://books.google.hu/books?id=mg7r4acKgq0C&amp;amp;lpg=PP1&amp;amp;hl=de&amp;amp;pg=PP1#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Dieudonne, History of functional analysis, VII.2&lt;/a&gt;. The name &amp;quot;resolution of the identity&amp;quot; was popularized by Stone in 1930s, see &lt;a href=&quot;https://www.jstor.org/stable/2319079&quot; rel=&quot;noreferrer&quot;&gt;Steen, Highlights in the History of Spectral Theory&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hilbert and Riesz only dealt with bounded operators, Stone and von Neumann extended the theory to unbounded self-adjoint operators in 1930s. Riesz's form of the spectral theorem remained dominant until around 1950 (according to Dieudonne). After the study of Banach algebras in 1940s, it started to be gradually eclipsed by the multiplicative form spelled out in passing by Stone in his 1932 book (based on von Neumann's ideas from a year before), see &lt;a href=&quot;https://mathoverflow.net/q/165113/51484&quot;&gt;MathOverflow, Who first used the multiplication operator version of spectral theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-10-10T10:22:15.370" LastActivityDate="2021-10-10T10:22:15.370" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13764" PostTypeId="1" CreationDate="2021-10-10T17:29:51.493" Score="14" ViewCount="4844" Body="&lt;p&gt;Is there any example of major controversy in the scientific community caused due to poor wording and/or misinterpretation of words?&lt;/p&gt;&#xA;" OwnerUserId="15468" LastEditorUserId="6229" LastEditDate="2022-01-30T15:53:53.383" LastActivityDate="2022-01-30T15:53:53.383" Title="Historical example of research papers being misinterpreted due to poor wording and creating controversy?" Tags="|mathematics|physics|social-context|language|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13765" PostTypeId="2" ParentId="13764" CreationDate="2021-10-10T18:48:55.420" Score="33" Body="&lt;p&gt;&lt;a href=&quot;http://scihi.org/giovanni-schiaparelli-martian-canals/&quot; rel=&quot;noreferrer&quot;&gt;Giovanni Schiaparelli&lt;/a&gt; ...&lt;br /&gt;&#xA;He wrote in 1877 about his telescopic observations of Mars.  He described some features using the Italian word &lt;em&gt;canali&lt;/em&gt;.  English translation would be &lt;em&gt;channels&lt;/em&gt;.  But the term was mistranslated as English &lt;em&gt;canals&lt;/em&gt;, which was taken to mean they were artificial constructions.  This led to much controversy about whether Mars was inhabited!&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-10-10T18:48:55.420" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13766" PostTypeId="1" CreationDate="2021-10-10T19:21:10.920" Score="0" ViewCount="151" Body="&lt;p&gt;Functional analysis and signal processing have a strong bond and I am trying to understand how and when it all started. Technically, signal processing is heavily based on Fourier analysis, which helps model natural phenomena using signals. Eventually, functional analysis became somehow interconnected to signal processing and communications but it is not clear to me how this type of bond originated.&lt;/p&gt;&#xA;" OwnerUserId="14690" LastEditorUserId="4251" LastEditDate="2021-10-10T19:23:46.760" LastActivityDate="2021-10-10T19:23:46.760" Title="When and how did signal processing reach the core of functional analysis?" Tags="|mathematics|functional-analysis|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13768" PostTypeId="2" ParentId="13764" CreationDate="2021-10-11T04:48:25.483" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paxos_(computer_science)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paxos Algorithm&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Leslie_Lamport&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leslie Lamport&lt;/a&gt; comes to my mind.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I submitted the paper to TOCS in 1990.  All three referees said that&#xA;the paper was mildly interesting, though not very important, but that&#xA;all the Paxos stuff had to be removed.  I was quite annoyed at how&#xA;humorless everyone working in the field seemed to be, so I did nothing&#xA;with the paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See paper [123] from &lt;a href=&quot;http://lamport.azurewebsites.net/pubs/pubs.html#lamport-paxos&quot; rel=&quot;nofollow noreferrer&quot;&gt;his webpage&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15469" LastEditorUserId="14365" LastEditDate="2021-10-11T07:36:57.190" LastActivityDate="2021-10-11T07:36:57.190" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13769" PostTypeId="1" CreationDate="2021-10-11T13:09:01.020" Score="2" ViewCount="140" Body="&lt;p&gt;In a fragment entitled &amp;quot;inversion of the elliptic integral of the first genus&amp;quot; (Gauss's werke, volume 8, p. 96-97), Gauss inverts the general elliptic integral of the first kind: he writes &lt;span class=&quot;math-container&quot;&gt;$\int\frac{dx}{\sqrt{(1-x^2)(1-\mu x^2)}} = \varphi$&lt;/span&gt;, and by a certain algebraic developement derives the inverse function&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = f(\varphi) = \frac{P(\varphi)}{Q(\varphi)} = \frac{\varphi+A_1\varphi^3+A_2\varphi^5+A_3\varphi^7+...}{1+B_2\varphi^4+B_3\varphi^6+B_4\varphi^8+...}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where the coefficients &lt;span class=&quot;math-container&quot;&gt;$A_1,A_2,A_3...$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B_2,B_3,B_4...$&lt;/span&gt; are polynomials in &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Initially, I thought this result is only one of many others that Gauss obtained on inversion of elliptic integrals, but than i saw that in his remarks on Gauss's fragment, Fricke writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This note, the date of which is presumed to be based on a diary from May 6, 1806..., is one of the most interesting Gauss made in the field of elliptical functions... Objectively, the present developement is illuminated in a very interesting way by the letter from Gauss to Bessel printed above. There Gauss repeatedly emphasizes the importance of entire transcendent functions. Here the function &lt;span class=&quot;math-container&quot;&gt;$f\varphi$&lt;/span&gt;... is represented in an elegant developement as the quotient of two functions &lt;span class=&quot;math-container&quot;&gt;$P\varphi,Q\varphi$&lt;/span&gt;, which Gauss undoubtedly knew its property as complete transcendent functions. It may also be remarked that the functions &lt;span class=&quot;math-container&quot;&gt;$P,Q$&lt;/span&gt;, which Gauss introduces here, are none other than those which Weierstrass later called &lt;span class=&quot;math-container&quot;&gt;$Al(\varphi)_1,Al(\varphi)_0$&lt;/span&gt;, in connection with Abel's work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I also checked in volume 1 of Weierstrass's werke, and in a paper &amp;quot;on the developement of modular functions&amp;quot;, Weierstrass also derives &lt;span class=&quot;math-container&quot;&gt;$A_1,A_2,A_3,B_2,B_3,B_4$&lt;/span&gt; (and he adds more coefficients). The coefficients he writes down are exactly those which Gauss wrote, so i didn't misunderstand Fricke's remark.&lt;/p&gt;&#xA;&lt;p&gt;I tried to search for information on the so-called &amp;quot;Weierstrass's Al functions&amp;quot; and i didn't find much material in the internet on those functions (except the lecture notes &lt;a href=&quot;https://www.wias-berlin.de/workshops/weierstrass200/slides/09_Ullrich.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Karl Weierstraß&#xA;and the theory of Abelian and elliptic functions&amp;quot;&lt;/a&gt;, and two other sources). In a &lt;a href=&quot;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-9/issue-7/Gausss-Collected-Works/bams/1183417390.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;synopsis of volume 8 of Gauss's werke&lt;/a&gt;, the author describes Gauss's derivation as &amp;quot;nothing more nor less than Weierstrass's celebrated expression of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as the quotient of two entire transcendental functions, viz, &lt;span class=&quot;math-container&quot;&gt;$Al_1(u),Al_0(u)$&lt;/span&gt;&amp;quot;, so it seems that there is really an important point here in the work of both Gauss and Weierstrass.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, my question is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;From the modern framework of elliptic functions and analysis, what is the fundamental point of significance in this developement?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2021-10-11T13:14:55.643" LastActivityDate="2021-10-11T13:14:55.643" Title="What is the significance of Gauss-Weierstrass's derivation of &quot;Al functions&quot;?" Tags="|gauss|complex-analysis|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13771" PostTypeId="1" CreationDate="2021-10-11T19:28:19.827" Score="0" ViewCount="92" Body="&lt;p&gt;In magnetism, there are two different magnetic fields, that I will simply call &lt;span class=&quot;math-container&quot;&gt;$\mathbf B$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt;. The distinction is not always clear. In experiments &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt; is the applied field while &lt;span class=&quot;math-container&quot;&gt;$\mathbf B$&lt;/span&gt; is the total field that includes the magnetization &lt;span class=&quot;math-container&quot;&gt;$\mathbf M$&lt;/span&gt; of matter. The three fields are unified under the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbf H=\frac{1}{\mu_0}\mathbf B-\mathbf M.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Who was the first to make this distinction?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2021-10-11T19:37:03.567" LastActivityDate="2021-10-11T19:37:03.567" Title="Who introduced the difference between magnetic fields in matter? $\mathbf H$ vs $\mathbf B$" Tags="|physics|physicists|magnetism|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13773" PostTypeId="2" ParentId="13764" CreationDate="2021-10-13T17:03:40.003" Score="8" Body="&lt;p&gt;&lt;em&gt;Airborne Contagion and Air Hygiene&lt;/em&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/William_F._Wells&quot; rel=&quot;noreferrer&quot;&gt;William Firth Wells&lt;/a&gt; was one of the earliest works on disease transmission through air. There was research in there showing that droplets &lt;em&gt;in general&lt;/em&gt; will tend to stay airborne the longest if they are around 100 microns in size. There was also a specific study included which showed that tuberculosis &lt;em&gt;in particular&lt;/em&gt; is infectious if the droplets are under 5 microns (note that tuberculosis specifically needs to reach a deep section of the lungs in order to infect someone--deeper than other diseases). But apparently people skimmed over the 100 microns part and fixated on the 5 microns part. Fast forward to 2020 and now we have an &lt;a href=&quot;https://www.wired.com/story/the-teeny-tiny-scientific-screwup-that-helped-covid-kill/&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; explaining how this misunderstanding lead to COVID guidance such as &amp;quot;3-6 feet apart&amp;quot;. (Further reading: &lt;a href=&quot;https://en.wikipedia.org/wiki/Wells_curve&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What must have happened, she thought, was that after Wells died, scientists inside the CDC conflated his observations. They plucked the size of the particle that transmits tuberculosis out of context, making 5 microns stand in for a general definition of airborne spread.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13194" LastActivityDate="2021-10-13T17:03:40.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13774" PostTypeId="1" CreationDate="2021-10-13T17:46:54.817" Score="4" ViewCount="251" Body="&lt;p&gt;What were some back-of-the-envelope estimates that scientists made where the result of the estimate was important to either the intellectual development of the field or had significant social consequences?&lt;/p&gt;&#xA;&lt;p&gt;For example, Helmholtz made some simple estimates to show that chemical energy could not possibly fuel the sun for its known lifespan, then argued that the only remaining energy source was gravitational, and used this to estimate an (incorrect) upper bound for the age of the sun. A detailed model or extensive calculation wasn't necessary; his points could be made just as well with order of magnitude numbers.&lt;/p&gt;&#xA;&lt;p&gt;This is different from, for example, Fermi's famous estimation of the yield of the first atomic weapon test by dropping slips of paper. While his estimation was impressive, there was (I presume) a team of scientists collecting far more data and crunching through more detailed models to calculate the yield to much greater accuracy than Fermi a short time later, so while his estimation was interesting and amusing it had little real consequence. On the other hand, Taylor's estimate of the Trinity test yield based on photos from a magazine might count because he was able to reveal important classified information with this estimate, and he came to with a very simple method, not needing the more detailed approaches he had been considering.&lt;/p&gt;&#xA;" OwnerUserId="2238" LastActivityDate="2022-11-13T15:39:36.357" Title="What were some historically important Fermi estimates in the history of science?" Tags="|examples|big-list|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13775" PostTypeId="1" CreationDate="2021-10-14T07:52:55.163" Score="13" ViewCount="1879" Body="&lt;p&gt;Was it Einstein? Or was there someone before him to hypothesise this?&lt;/p&gt;&#xA;" OwnerDisplayName="Shannon T" LastEditorUserId="4251" LastEditDate="2021-10-14T21:09:32.560" LastActivityDate="2021-10-16T03:35:33.803" Title="Who was the first to hypothesise that gravity from one mass causes the spacetime around another mass to curve?" Tags="|physics|relativity-theory|physicists|gravity|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13776" PostTypeId="2" ParentId="13775" CreationDate="2021-10-14T11:23:30.287" Score="28" Body="&lt;p&gt;The first metric theory of gravitation, in which the effects of gravitation are treated entirely in terms of the geometry of curved spacetime, was published by Finnish physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Gunnar_Nordstr%C3%B6m&quot; rel=&quot;noreferrer&quot;&gt;Gunnar Nordström&lt;/a&gt; in 1913. Einstein published his own theory of general relativity in 1915.&lt;/p&gt;&#xA;&lt;p&gt;The measurement of the deflection of light passing close to the Sun during the solar eclipse of 1919 (the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eddington_experiment&quot; rel=&quot;noreferrer&quot;&gt;Eddington experiment&lt;/a&gt;) produced results that agreed with Einstein's theory, whereas Nordström's theory predicted no deflection. Nordström subsequently nominated Einstein for the Nobel Prize in Physics.&lt;/p&gt;&#xA;" OwnerUserId="9542" OwnerDisplayName="gandalf61" LastActivityDate="2021-10-14T11:23:30.287" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13777" PostTypeId="2" ParentId="13775" CreationDate="2021-10-14T21:40:45.623" Score="2" Body="&lt;p&gt;The first person that I know of was the mathematician and physicist William Clifford who, on reading the &lt;em&gt;Habilitationsschrift&lt;/em&gt; of Riemann and, almost forty years before Einstein developed general relativity, hypothesised that not only all motion was due to the curvature of space but so was matter.&lt;/p&gt;&#xA;&lt;p&gt;This is amazingly prescient considering that Einstein only showed that gravity could be interpreted as the curvature of spacetime. However, modern unified theory also shows that the electromagnetic, the weak and the strong force can also be interpreted via curvature - not now of spacetime - but of a bundle over spacetime. Moreover, the hypothetical supersymmetry which assumes a symmetry between bosons and fermions, show that matter ought to be, if supersymmetry is correct, also an aspect of curvature.&lt;/p&gt;&#xA;&lt;p&gt;It's interesting to speculate how much of Clifford's work was known by Einstein. It would be extraordinary if had not at least known about it, given his own preoccupations. Also, it is interesting to speculate how much of his unified field work also went down the same path as Clifgord - that is to imagine all forces and all matter was an aspect of matter.&lt;/p&gt;&#xA;&lt;p&gt;Riemann's Habilitationsschrift was published in 1854 essay and was titled, &lt;em&gt;On the Hypotheses which lie at the Bases of Geometry&lt;/em&gt;. Clifford translated the essay in 1867 and three years later, he reported to the Cambridge Philosophical Society the curved space concept of Riemann's including his hypothesis that both space and matter were the effects of curvature.  A further three years later, his translation was published in &lt;em&gt;Nature&lt;/em&gt;. In 1876, his report to the society was published as &lt;em&gt;The Space Theory of Matter&lt;/em&gt; in their &lt;em&gt;Proceedings&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Riemann himself suggested something similar, but his was a more general comment in the microstructure of space, closer in spirit to Wheelers spacetime foam.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-10-16T03:35:33.803" LastActivityDate="2021-10-16T03:35:33.803" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13778" PostTypeId="1" AcceptedAnswerId="13779" CreationDate="2021-10-16T20:25:45.827" Score="2" ViewCount="554" Body="&lt;p&gt;The English language has more than one word for 12 (twelve, dozen) and 12^2 (one hundred and forty-four, gross), but has there ever been a word for 2*12 other than twenty-four?&lt;/p&gt;&#xA;&lt;p&gt;I would also be interested in whether other languages developed special words for the number twenty-four.&lt;/p&gt;&#xA;" OwnerUserId="5906" LastEditorUserId="55" LastEditDate="2021-10-17T04:47:12.030" LastActivityDate="2021-10-17T04:47:12.030" Title="Was there ever a word for 24 like &quot;dozen&quot; for 12?" Tags="|language|numbers|" AnswerCount="1" CommentCount="10" ClosedDate="2021-10-18T05:53:37.347" ContentLicense="CC BY-SA 4.0" />
  <row Id="13779" PostTypeId="2" ParentId="13778" CreationDate="2021-10-17T04:46:23.240" Score="4" Body="&lt;p&gt;I am not aware of any current or archaic English word for 24 other than the obvious combinations like &amp;quot;double dozen&amp;quot; or &amp;quot;score and four&amp;quot; (&amp;quot;score&amp;quot; is 20). German style wording like &amp;quot;four-and-twenty&amp;quot; was occasionally used until 19th century, and survives in some idioms, see &lt;a href=&quot;https://english.stackexchange.com/q/3903&quot;&gt;English SE&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One of the two formations (2x12 or 20+4) is typical in most languages. &lt;a href=&quot;https://www.etymonline.com/word/dozen&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Dozen&amp;quot;&lt;/a&gt; derives from Latin &lt;em&gt;duodecim&lt;/em&gt; (2+10). Analogously, base 24 system is now called &lt;a href=&quot;https://www.tapatalk.com/groups/dozensonline/acrobatic-tetravigesimal-base-24-t780.html&quot; rel=&quot;noreferrer&quot;&gt;tetravigesimal&lt;/a&gt; (biunqual or binilimal are rarely used) from Latin &lt;em&gt;tetra&lt;/em&gt; (4) and &lt;em&gt;vigesimus&lt;/em&gt; (20). But no old French or English vulgarization emerged.&lt;/p&gt;&#xA;&lt;p&gt;Some interpret &lt;a href=&quot;https://lingweb.eva.mpg.de/channumerals/Umbu-Ungu.htm&quot; rel=&quot;noreferrer&quot;&gt;Umbu-Ungu&lt;/a&gt;, the counting system of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kaugel_language&quot; rel=&quot;noreferrer&quot;&gt;Kaugel&lt;/a&gt; in Papua New Guinea, as base 24, although ethnologists call it 4-cycle system, see &lt;a href=&quot;https://www.google.com/books/edition/History_of_Number/rnQ7DwAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Owens et al., History of Number: Evidence from Papua New Guinea and Oceania, p.117ff&lt;/a&gt;. The word for 24 in it is &lt;em&gt;tokapu&lt;/em&gt;, and it is not 2x12, which would be &lt;em&gt;rurepo talu&lt;/em&gt;, or 20+4 (20 is &lt;em&gt;supu&lt;/em&gt; and 4 is &lt;em&gt;kise&lt;/em&gt;). It is used as a root to form larger numerals.&lt;/p&gt;&#xA;&lt;p&gt;Of course, &lt;em&gt;day&lt;/em&gt; in English is 24 hours, if one chooses to see it that way. &lt;a href=&quot;https://en.wikipedia.org/wiki/Nychthemeron&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Nychthemeron&lt;/em&gt;&lt;/a&gt; (from Greek &lt;em&gt;núx&lt;/em&gt;, night, and &lt;em&gt;hēméra&lt;/em&gt;, day) is occasionally used for periods of 24 hours to remove the ambiguity, and there are similar disambiguating words in other Western languages.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-10-17T04:46:23.240" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13780" PostTypeId="1" CreationDate="2021-10-17T12:30:18.803" Score="0" ViewCount="125" Body="&lt;p&gt;For me the concept of energy in physics is the least intuitive one as in all the books on Physics that I have read so far. The concept of energy have been the least intuitive one as I cannot understand How these things actually came into existence.&lt;/p&gt;&#xA;&lt;p&gt;Even the definition of Energy is a bit vague, many books define &lt;em&gt;Energy as the ability to do work&lt;/em&gt; but then the question comes that what is work ? then they say that &lt;em&gt;work is the process of transfer of energy&lt;/em&gt; or &lt;em&gt;to make it even more boring they will just read out the formula for work done as&lt;/em&gt; ( except Fundamentals of Physics by Resnick Halliday which admits that it is quite difficult to write a definition of energy).&lt;/p&gt;&#xA;&lt;p&gt;So to get a better understanding of this concept of energy I started reading about the origin of this concept of energy in Physics like the Vis Viva controversy, answers on this site related to my questions like &lt;a href=&quot;https://physics.stackexchange.com/questions/132754/how-was-the-formula-for-kinetic-energy-found-and-who-found-it&quot;&gt;How was the formula for kinetic energy found, and who found it?&lt;/a&gt; , &lt;a href=&quot;https://physics.stackexchange.com/questions/128598/how-was-the-definition-and-the-formula-of-work-derived-is-it-the-best-possible&quot;&gt;How was the definition and the formula of work derived?&lt;/a&gt; and a few more material that I could get, I even tried to find the research papers related to all this couldn't find them. And by my experience of reading all this material I felt that this is not helpful to any curious student because as a student I am particularly interested in the experiments that lead to the concept of energy, the observations made by the genius minds that leads to result that we study ( for example, Leibnitz observed that &lt;span class=&quot;math-container&quot;&gt;$\Sigma m_{i} a_{i} = constant$&lt;/span&gt; fore systems with no friction in case of elastic collision ) but these articles and answers are more focused on the history of who did what, who said what rather focusing on How did they do that ?&lt;/p&gt;&#xA;&lt;p&gt;The timeline of all the system is very confusing ( more than the plots of Christopher Nolan's movies ) and more importantly useless to a student ( since student, it maybe interesting to a history student ).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;To summarise:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What I am looking for is answer/ article/ website anything that can help me to understand How this concept of Energy works ? How are the results that we study like &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{l}&#xA;d W=\vec{F} \cdot \overrightarrow{d s} \\&#xA;K \cdot E=\frac{1}{2} m v^{2}&#xA;\end{array}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;came into existence, what experiments lead to these conclusions?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In a nutshell&lt;/strong&gt;, I want to know what experiment and observations lead to the modern concept of energy I am not interested in knowing the names of the scientist rather I am interested in there experiments, observations and mathematics&lt;/p&gt;&#xA;&lt;p&gt;I know all the above is very lengthy but it was important to prove that this is not a homework type question.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;This is my question :&lt;/strong&gt;&#xA;What were the experiments and observations that lead to the formula &lt;span class=&quot;math-container&quot;&gt;$ W=\vec{F} \cdot \overrightarrow{d s}$&lt;/span&gt;, if you know any derivation (except the one using Kinetic Energy ) of this please let me know.&lt;/p&gt;&#xA;&lt;p&gt;Hope this will clarify my question to you.&lt;/p&gt;&#xA;" OwnerUserId="15512" LastEditorUserId="15512" LastEditDate="2021-10-17T13:51:04.377" LastActivityDate="2021-10-17T17:55:59.673" Title="What led to the formula $W=\vec{F} \cdot \overrightarrow{d s}$?" Tags="|classical-mechanics|newton|energy|" AnswerCount="1" CommentCount="5" ClosedDate="2021-10-18T03:51:09.447" ContentLicense="CC BY-SA 4.0" />
  <row Id="13781" PostTypeId="2" ParentId="13780" CreationDate="2021-10-17T17:50:42.890" Score="1" Body="&lt;p&gt;Mirowski P., &lt;em&gt;More heat than light&lt;/em&gt; (1989 Cambr.UP), Chap.2. pp.11-98 &lt;em&gt;The history of the energy concept&lt;/em&gt;.&amp;lt; It's a great book for various other reasons.&amp;gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For Ernst Mach, energy was more or less defined as the ability to do&#xA;work. Although &amp;quot;work&amp;quot; could mean many things to many people, Mach felt&#xA;it was merely an historical accident that the term was defined&#xA;mechanically in early physics: that is, &lt;strong&gt;lifting a known weight through&#xA;a certain distance&lt;/strong&gt;. (p.15)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, here is the key observation, some details and derivation are given in the text of this chapter.&#xA;The problem &lt;a href=&quot;https://philosophy.stackexchange.com/questions/85899/is-the-idea-that-everything-is-energy-even-coherent/85902#85902&quot;&gt;https://philosophy.stackexchange.com/questions/85899/is-the-idea-that-everything-is-energy-even-coherent/85902#85902&lt;/a&gt; was recently discussed and the book mentioned there /Coopersmith J., (Oxford UP, 2010)  &lt;em&gt;Energy, the Subtle Concept&lt;/em&gt;. / has also historical details.&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="3329" LastEditDate="2021-10-17T17:55:59.673" LastActivityDate="2021-10-17T17:55:59.673" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13782" PostTypeId="1" AcceptedAnswerId="13783" CreationDate="2021-10-17T22:32:20.287" Score="3" ViewCount="293" Body="&lt;p&gt;What was Einstein's first published exposition of his final form of &lt;strong&gt;general&lt;/strong&gt; relativity, be it a scientific article or a book?&lt;/p&gt;&#xA;&lt;p&gt;I've been googling the answer for hours now, with no luck, really. The spent hours led me to think that the answer is Einstein's book (based on lectures he gave at Princeton) called &lt;em&gt;The Meaning of Relativity&lt;/em&gt;, though I am far from being sure here.&lt;/p&gt;&#xA;&lt;p&gt;His handwritten manuscripts don't count, as don't oral presentations of the theory to the Prussian Academy of Sciences (or any other oral ones).&lt;/p&gt;&#xA;" OwnerUserId="14982" LastEditorUserId="10106" LastEditDate="2021-10-18T03:50:06.407" LastActivityDate="2021-10-18T05:55:06.890" Title="What was Einstein's first publication of the final form of general relativity?" Tags="|physics|reference-request|relativity-theory|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13783" PostTypeId="2" ParentId="13782" CreationDate="2021-10-17T23:09:37.930" Score="2" Body="&lt;p&gt;The 1916 publication in &lt;em&gt;Annalen der Physik&lt;/em&gt;, 49, 769–822, also published separately in Leipzig, see e.g. &lt;a href=&quot;https://arxiv.org/abs/physics/0405066&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sauer, Albert Einstein's 1916 Review Article on General Relativity&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first comprehensive overview of the final version of the general theory of relativity was published by Einstein in 1916 after several expositions of preliminary versions and latest revisions of the theory in November 1915.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This was the paper the circumstances of whose submission and publication led to the &lt;a href=&quot;https://en.wikipedia.org/wiki/General_relativity_priority_dispute&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein and Hilbert priority dispute&lt;/a&gt;. It was &lt;em&gt;not&lt;/em&gt; his first publication on general relativity, but it added the final piece, the field equations, see &lt;a href=&quot;http://philsci-archive.pitt.edu/2123/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Janssen, Einstein’s First Systematic Exposition of General Relativity&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Einstein’s first paper on a metric theory of gravity, co-authored with his mathematician friend Marcel Grossmann, was published as a separatum in early 1913 and was reprinted the following year in Zeitschrift für Mathematik und Physik (Einstein and Grossmann  1913,  1914a). Their  second  (and  last)  joint  paper  on  the  theory  also appeared in this journal (Einstein and Grossmann 1914b). Most of the formalism of general relativity as we know it today was already in place in this Einstein-Grossmann theory. Still missing were the generally-covariant Einstein field equations.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And Einstein vaguely pondered modifications to Newtonian gravity to reconcile them with special relativity already back in 1907, in &lt;em&gt;Über  das  Relativitätsprinzip  und  die  aus  demselben  gezogenen  Folgerungen&lt;/em&gt;, &lt;em&gt;Jahrbuch der Radioaktivität und Elektronik&lt;/em&gt; 4, 411–462.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="5989" LastEditDate="2021-10-18T05:55:06.890" LastActivityDate="2021-10-18T05:55:06.890" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13784" PostTypeId="1" AcceptedAnswerId="14767" CreationDate="2021-10-18T10:29:09.053" Score="6" ViewCount="173" Body="&lt;p&gt;I recently read &lt;a href=&quot;http://www.kenalder.com/measure/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ken Alder's &amp;quot;The measure of all things&amp;quot;&lt;/a&gt; about the first steps of the metric system definition. The project described in the book was the measure the meridian arc between Dunkirk and Barcelona in order to be able to define the meter (equal to one ten-millionth of the quarter meridian). When he was in Barcelona, Pierre Méchain, the astronomer in charge of this part, made different measures in different parts of the city, but one of his results was inconsistent with the others and confused (a lot) Méchain.&lt;/p&gt;&#xA;&lt;p&gt;Almost 30 years later, in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6509489b&quot; rel=&quot;nofollow noreferrer&quot;&gt;Connaissance des temp pour l'an 1831, p. 58-77&lt;/a&gt; (published in 1828), J.-N. Nicollet, a member of the Bureau des Longitudes (and later known for his maps of the Mississipi river) wrote an article about Méchain's results saying (among other things) that experimental errors should be distinguished between :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;systematic errors, due for instance to bad calibration of instruments&#xA;or imperfect methods of observation&lt;/li&gt;&#xA;&lt;li&gt;random errors, due for&#xA;instance to fluctuations in the measurements or interpretations&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, basically, it's all about better describing the accuracy and make a clear difference between trueness and precision.&lt;/p&gt;&#xA;&lt;p&gt;I am wondering if Nicollet was the first to make this clear distinction. Before him, who worked on this topic and may have inspired Nicollet ?&lt;/p&gt;&#xA;&lt;p&gt;Also, I do not see many comments to Nicollet article (only an anonymous one in &lt;a href=&quot;https://books.google.fr/books?id=hTBDAQAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philosophical magazine 1829, p.180-188&lt;/a&gt; ). Was there discussions on this topic after Nicollet ? (to challenge, to fine-tune, etc...).&lt;/p&gt;&#xA;" OwnerUserId="13909" LastEditorUserId="6229" LastEditDate="2022-01-30T15:53:24.643" LastActivityDate="2022-12-22T15:06:45.317" Title="Nicollet and the notions of systematic and random errors" Tags="|scientific-method|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13785" PostTypeId="1" CreationDate="2021-10-18T18:39:33.587" Score="4" ViewCount="261" Body="&lt;p&gt;In stochastic calculus, the name &lt;a href=&quot;https://en.wikipedia.org/wiki/Ruslan_Stratonovich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stratonovich&lt;/a&gt; appears all over the place. However, even though Donald Fisk supposedly obtained similar results, his name appears nowhere. Who was Donald Fisk?&lt;/p&gt;&#xA;&lt;p&gt;I cannot find anything about him, nor why he isn't given any credit.&lt;/p&gt;&#xA;" OwnerUserId="15520" LastEditorUserId="4251" LastEditDate="2021-10-18T18:45:06.057" LastActivityDate="2021-10-18T20:18:33.480" Title="Who is Donald Fisk?" Tags="|mathematics|mathematicians|probability|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13786" PostTypeId="2" ParentId="13785" CreationDate="2021-10-18T20:18:33.480" Score="4" Body="&lt;p&gt;He was a PhD student of &lt;a href=&quot;https://imstat.org/2018/05/15/obituary-herman-rubin-1926-2018/&quot; rel=&quot;noreferrer&quot;&gt;Herman Rubin&lt;/a&gt; at the Michigan State University in the early 1960-s. Some of the history is reconstructed by &lt;a href=&quot;https://www.ma.imperial.ac.uk/%7Eajacquie/IC_AMDP/IC_AMDP_Docs/Literature/Jarrow_Protter_History_Stochastic_Integration.pdf&quot; rel=&quot;noreferrer&quot;&gt;Jarrow&#xA;and Protter in A short history of stochastic integration and mathematical finance&lt;/a&gt; based on private communication with Rubin. Rubin gave Fisk the ideas to work on, and he developed them in his thesis Quasi-martingales and Stochastic Integrals. According to &lt;a href=&quot;https://projecteuclid.org/download/pdf_1/euclid.lnms/1196285408&quot; rel=&quot;noreferrer&quot;&gt;Rubin's interview with Bock&lt;/a&gt;, the thesis was written in 1961, MSU published it in 1963.&lt;/p&gt;&#xA;&lt;p&gt;However, when the work was submitted to a journal the editor doubted utility of stochastic integration, and Fisk ultimately published only the part about quasimartingales, see &lt;a href=&quot;https://www.ams.org/journals/tran/1965-120-03/S0002-9947-1965-0192542-5/&quot; rel=&quot;noreferrer&quot;&gt;Fisk, Quasi-martingales (1965)&lt;/a&gt; (Transactions of AMS, 120, pp. 369-389). This explains why he is not given credit as much as Stratonovich, although, technically, MSU did publish his thesis in full. Also, he quickly dropped off the mathematical grid. The only other publication by him listed on MathSciNet is &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00537825&quot; rel=&quot;noreferrer&quot;&gt;Sample quadratic variation of sample continuous, second order martingales&lt;/a&gt; in &lt;em&gt;Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete&lt;/em&gt;, 1966.&lt;/p&gt;&#xA;&lt;p&gt;Here is from Jarrow-Protter:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;At the Third Berkeley Symposium in 1955, Rubin gave a talk on stochastic&#xA;differential equations. The following year, he presented an invited paper at the Seattle joint meetings of the Institute of Mathematical Statistics, the American Mathematical Society , the Biometric Society, the Mathematical Association of America, and the Econometrics Society. In this paper he outlined what was later to become D. L. Fisk’s Ph.D. thesis, which invented both quasimartingales and what is now known as the Stratonovich integral. To quote his own recollections:“...Quasimartingales seemed the natural candidate for the class of processes, but I did not see a clear proof. I gave the problem to Fisk to work on for a Ph.D. thesis, and he did come up with what was needed”.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Indeed, in D. L. Fisk’s thesis [27], written under Rubin when he was at Michigan State University, Fisk developed what is now known as the Stratonovich integral, and he also coined the phrase and developed the modern theory of quasimartingales, later used by K. M. Rao [68] to give an elegant proof that a quasimartingale is the difference of two submartingales, and also used by S. Orey [63] in a paper extending the idea and which foreshadowed modern day semimartingales. Fisk submitted his thesis for publication, but the editor did not believe there was much interest in stochastic integration, again according to the recollections of Herman Rubin [69]. So Fisk dropped that part of the thesis and did not pursue it, publishing instead&#xA;only the part on quasimartingales, which appeared as [28].&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-10-18T20:18:33.480" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13787" PostTypeId="2" ParentId="13759" CreationDate="2021-10-19T03:40:31.653" Score="4" Body="&lt;p&gt;The conics as geometric shapes were investigated comprehensively by many ancient Greek mathematicians. After Descartes's innovation of introducing coordinates, they were seen as essential examples of 2nd order polynomial equations. The first order equation was simply the straight line and so simple that no-one bothered investigating its geometric properties.&lt;/p&gt;&#xA;&lt;p&gt;This covers first and second order equations. It's natural to then investigate higher order equations and also to investigate how to solve them. After all, that's one of the points of an equation. Algebraic methods were developed for the third and then the fourth and there they stalled until Galois showed the fifth order equation didn't have a general solution that could be written down as a formula using the algebraic operators (addition &amp;amp; multiplication) and the taking of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th roots.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="4251" LastEditDate="2021-10-19T08:17:00.440" LastActivityDate="2021-10-19T08:17:00.440" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13788" PostTypeId="1" CreationDate="2021-10-19T16:27:45.953" Score="3" ViewCount="145" Body="&lt;p&gt;Has anyone used Michelson's echelon (pictured below), a very famous type of diffraction grating in the 1920s? I am wondering how did the diffraction image look like from this type of &lt;em&gt;transmission&lt;/em&gt; grating. Michelson's echelon was made from precisely cut glass/quartz slabs stacked like a staircase/ladder. Hence the term echelon for ladder rungs in French. However, these echelons are part of museums now. Unfortunately, I have never seen a real one. This grating was supposed to have extremely high resolving power because of a large number of diffraction orders. But, of course, no picture exists of the diffraction pattern from such an echelon. Has anyone ever used an echelon like this one or seen a published account of its &amp;quot;raw&amp;quot; diffraction pattern? Such gratings were used with a prism as an order sorter.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qhg26.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qhg26.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In contrast, I took a picture of a diffraction pattern from a 5 cm long reflective echelle grating (~ 75 lines/mm) with a mercury street lamp. Such gratings are available from popular optics manufacturers, and they look like this (e.g., Thorlabs or Richardson gratings/Edmund, Image from Thorlabs)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Kca0x.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Kca0x.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a mercury street lamp viewed at night with an echelle grating. Here one can see the beautiful overlap of several orders, and the green line of mercury gives some clue of the number of orders.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RsGFX.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RsGFX.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2021-10-23T06:08:12.863" LastActivityDate="2021-10-23T06:08:12.863" Title="Diffraction pattern of Michelson's echelon" Tags="|physics|experimental-physics|optics|light|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13789" PostTypeId="1" AcceptedAnswerId="13790" CreationDate="2021-10-20T08:34:25.067" Score="2" ViewCount="589" Body="&lt;p&gt;On the last page (page 84) of Anthony Preus's paper&lt;span class=&quot;math-container&quot;&gt;$^\star$&lt;/span&gt; on Galen's criticism of Aristotle's Conception Theory, the author writes that Galen discussing the second book of &amp;quot;On Seed&amp;quot;, which examines the female seed, did not describe the female seed properly.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;When Galen says &amp;quot;female seed&amp;quot;, does he mean semen like male semen, but colder and weaker? Or menstrual blood with which he is making a connection with Aristotle view?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And what was the view of the Hippocratics on the female seed?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;span class=&quot;math-container&quot;&gt;$*$&lt;/span&gt; Anthony Preus, &lt;a href=&quot;https://www.jstor.org/stable/4330669&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galen's Criticism of Aristotle's Conception Theory&lt;/a&gt;, Journal of the History of Biology, Volume 10, Number 1, Spring, 1977.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="15534" LastEditorUserId="4251" LastEditDate="2021-10-20T14:34:47.113" LastActivityDate="2021-10-20T14:34:47.113" Title="What is &quot;female seed&quot; in Galen's work?" Tags="|ancient-greece|biology|medicine|aristotle|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13790" PostTypeId="2" ParentId="13789" CreationDate="2021-10-20T10:38:17.153" Score="2" Body="&lt;p&gt;Galen's theory of conception, and its relationships to Hippocratic Aristotelian and other ancient theories, is discussed by &lt;a href=&quot;https://www.jstor.org/stable/4330880&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boylan in The Galenic and Hippocratic Challenges to Aristotle's Conception Theory&lt;/a&gt; Hippocrates and Galen were dual seed theorists, so for them both male and female produce seeds that contribute to inheritance in the offspring, but the male seed is &amp;quot;stronger&amp;quot;. The Hippocratic author of On Seed wrote the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the male there is female sperm and male sperm; in the same way in the woman. The male sperm is stronger than the female. For of necessity the male develops from the stronger sperm. And here is another point. If the sperm that comes from both is strong, a male is born; if weak, a female is born. Whichever prevails [in quantity] that is which is born&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The dual seed theory competed with the &amp;quot;furrowed field&amp;quot; theory, where the mother just provides fertile ground for the male seed, and Aristotle's refined version of it, according to which &amp;quot;&lt;em&gt;the male offers the motive principle and the efficient cause of generation while the female offers the material principle&lt;/em&gt;&amp;quot;. This material principle Aristotle called &lt;em&gt;katamenia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Galen offered a detailed elaboration of the dual seed theory meant to resolve the difficulty Aristotle saw with two seeds coming together to produce a coherent whole, the &lt;em&gt;kuema&lt;/em&gt;. In addition to ideas borrowed from On Seed, he uses as evidence the discovery of the ovaries by Herophilus.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Galen was a dual seed theorist who assigned most of the roles that Aristotle attached to the katamenia to the female sperma (cf. K IV 615, 625). The katamenia was only for nourishment; he believed that each body fluid should have a unique function (a principle borrowed from Aristotle and used against him). Thus for Galen both the male and the female produce seed, and it is from this that inheritance derives (K IV 604, 612, 626, 642). Galen solves the difficulty of the earlier dual seed theorists by saying that the female seed is weaker than the male's. Being weaker it is not sufficient of itself to produce a kuema, and the male is needed (cf. K IV 616).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The weakness is described materially through the old distinction of sexual division through the contraries: male hot and dry; female - cold and wet (K IV 630, 641). The female's moistness is used to aid in nourishment and the cold is a given which (following Aristotle) makes the female &amp;quot;weaker&amp;quot; than the male. Though it is a weaker seed, Galen is convinced it exists, through what he takes to be cases in which he has seen the female seed (IV 594; and at 622 he encourages a dissection which will show the residue of female seed). Galen, in asserting the existence of female seed, is following the work of Herophilus, who discovered the ovaries...  thought of as inverted testes and the natural repository of seed.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-10-20T10:38:17.153" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13791" PostTypeId="1" AcceptedAnswerId="13793" CreationDate="2021-10-21T12:15:08.177" Score="3" ViewCount="865" Body="&lt;p&gt;I encountered Grothendieck's resignation letter from Bourbaki along with its English translation not too long ago on the web, but for now it seems it's nowhere to be found. I've scoured through the Grothendieck circle site for the letter, but to no avail.&lt;/p&gt;&#xA;" OwnerUserId="15535" LastEditorUserId="4251" LastEditDate="2021-10-23T18:19:58.143" LastActivityDate="2021-10-23T18:19:58.143" Title="Where can I find Grothendieck's letter of resignation from Bourbaki?" Tags="|mathematics|mathematicians|biographical-details|grothendieck|bourbaki|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13793" PostTypeId="2" ParentId="13791" CreationDate="2021-10-21T15:51:40.523" Score="7" Body="&lt;p&gt;Here is an &lt;a href=&quot;https://agrothendieck.github.io/divers/LGB91060en.pdf&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt; of Grothendieck's letter dated October 9, 1960.&lt;/p&gt;&#xA;&lt;p&gt;Here is the letter in its &lt;a href=&quot;https://agrothendieck.github.io/divers/LGB91060.pdf&quot; rel=&quot;noreferrer&quot;&gt;original French&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;May also be of interest, &lt;a href=&quot;https://hsm.stackexchange.com/questions/11218/alexander-grothendiecks-stolen-correspondence-in-1985/11220#11220&quot;&gt;Grothendieck's stolen correspondence of 1985&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2021-10-21T15:51:40.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13796" PostTypeId="2" ParentId="13684" CreationDate="2021-10-21T22:34:24.067" Score="1" Body="&lt;p&gt;You can find a scanned copy of the 2nd edition (1716)&#xA;of the 1696 book &amp;quot;Analyse des infiniment petits, pour l'intelligence des lignes courbes&amp;quot;, by &amp;quot;Mr le Marquis De L'Hospital&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;at  &lt;a href=&quot;https://archive.org/download/infinimentpetits1716lhos00uoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/download/infinimentpetits1716lhos00uoft&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;HTH&#xA;Terry&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2021-10-21T22:34:24.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13797" PostTypeId="2" ParentId="13791" CreationDate="2021-10-21T22:45:12.667" Score="2" Body="&lt;p&gt;I have since found the original source. It appeared in the second part of a two-part obituary published in 2016 for the &lt;strong&gt;Notices of the AMS: Vol 63, No.4&lt;/strong&gt;. Though I believe the link provided by @NWR is more convenient for the sole purpose of accessing the letter.&lt;/p&gt;&#xA;&lt;p&gt;Here are the links to both parts 1 and 2, respectively:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.ams.org/publications/journals/notices/201603/rnoti-p242.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexandre Grothendieck, 1928 - 2014, Part 1.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.ams.org/publications/journals/notices/201604/rnoti-p401.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexandre Grothendieck, 1928 - 2014, Part 2.&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15535" LastActivityDate="2021-10-21T22:45:12.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13798" PostTypeId="2" ParentId="13529" CreationDate="2021-10-21T23:03:49.680" Score="3" Body="&lt;p&gt;It's very doubtful that the engraving/diagram had anything to do with Newton.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The system of the world&amp;quot; was a posthumous publication from 1728-9 onwards.  It was derived (with a little editorial amendment including a new title) from a manuscript of 1685 of Newton's (written out by his secretary/amanuensis but bearing corrections in Newton's hand). The ms is entitled 'De motu corporum Liber secundus'. Newton laid this aside in favour of a radical re-write that became Book 3 of 'Principia'. The superseded ms was found among his papers at his death.&lt;/p&gt;&#xA;&lt;p&gt;I haven't looked exhaustively but I doubt if the diagram is with the ms, it was likely composed by someone else in preparation for publication, guided by the description in the text.  So if the original even had a drawing, then it does not seem to have survived.&lt;/p&gt;&#xA;&lt;p&gt;The original manuscript has been digitized and the page-images can be seen at&lt;br /&gt;&#xA;&lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03990/9&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cudl.lib.cam.ac.uk/view/MS-ADD-03990/9&lt;/a&gt; onwards.&lt;/p&gt;&#xA;&lt;p&gt;A transcription is also online at the Newton Project,  at&#xA;&lt;a href=&quot;https://www.newtonproject.ox.ac.uk/view/texts/diplomatic/NATP00305&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.newtonproject.ox.ac.uk/view/texts/diplomatic/NATP00305&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2021-10-21T23:03:49.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13799" PostTypeId="2" ParentId="13459" CreationDate="2021-10-21T23:44:27.513" Score="3" Body="&lt;p&gt;The first main point to make in answer is : beware Newton myths -- there are far too many of them, and people have made money and (flaky) reputations fabricating catchpenny fake history about Newton.&lt;/p&gt;&#xA;&lt;p&gt;The 'apple' story has been plenty embroidered over time. But there is a source, a notebook handwritten by Newton's friend William Stukeley. It was written possibly a long time after Newton in the 1720s in his old age appears to have told a reminiscent story to Stukeley. It is impossible to say whether the story in the notebook was itself embroidered, but the notebook can be read in scanned form at the Royal Society website at&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ttp.royalsociety.org//ttp/ttp.html?id=1807da00-909a-4abf-b9c1-0279a08e4bf2&amp;amp;type=book&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ttp.royalsociety.org//ttp/ttp.html?id=1807da00-909a-4abf-b9c1-0279a08e4bf2&amp;amp;type=book&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;and the story about the apple as recorded by Stukeley reads (in part) as follows:&lt;/p&gt;&#xA;&lt;p&gt;“After dinner, the weather being warm, we {i.e. Newton and Stukeley} went into the garden and drank thea, under the shade of some apple trees … he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion’d by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself …”&lt;/p&gt;&#xA;&lt;p&gt;It can be seen that the story is connected with Newton's ideas of gravitation only, not with his laws of motion. It seems that Newton thought of some influence of the earth extending upwards as far as the apple hanging off the tree, and he seems to have wondered how much farther up that influence might also extend, maybe as far as the moon? That seems to be as far as the initial germ of an idea went.&lt;/p&gt;&#xA;&lt;p&gt;The origin of the laws of motion is a quite separate matter, certainly nothing to do with the apple story. Newton was aware of many previous contributions to possible laws of motion (and later acknowledged those he approved of, in the Principia). Various notes in the Newton papers show how he wrote and amended numerous sets of draft versions of laws of motion, worrying about this aspect or that, until it seems he was more or less satisfied.  Tracing through them all involves consulting bulky sources and if you look for a good place to start, it is probably  'Mathematical Papers of Isaac Newton, Volume VI, 1684-1691' edited by D T Whiteside (Cambridge University press, 1974).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2021-10-21T23:44:27.513" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13801" PostTypeId="1" CreationDate="2021-10-22T08:45:26.187" Score="3" ViewCount="144" Body="&lt;p&gt;Has Turing's invention of Turing machines contributed to the development of real computers, which resulted in the personal computers we currently use?&lt;/p&gt;&#xA;&lt;p&gt;I often saw it mentioned that this is an important contribution of mathematical logic to the real world, but Turing machines are just a very theoretical model of computation, so I don't see why this should influence the development of &lt;em&gt;real&lt;/em&gt; computers.&lt;/p&gt;&#xA;" OwnerUserId="15545" LastEditorUserId="4251" LastEditDate="2021-10-22T15:44:09.863" LastActivityDate="2021-10-22T15:44:09.863" Title="Has Turing's invention of Turing machines contributed to the development of real computers?" Tags="|computer-science|computers|computer-scientists|turing|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="13802" PostTypeId="1" CreationDate="2021-10-22T13:28:16.203" Score="11" ViewCount="2377" Body="&lt;p&gt;I am conducting a little research about the origin of the &amp;quot;de&amp;quot; in a version of the name of the Danish astronomer &amp;quot;Tycho Brahe&amp;quot;, namely, &amp;quot;Tycho de Brahe&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In the Danish language, there is no &amp;quot;de&amp;quot;. I was able to find it in Portuguese, Spanish, French, Italian and Romanian. Note that similar words also exist: &amp;quot;von&amp;quot; in German or &amp;quot;van&amp;quot; in Dutch. These words are used to indicate a place of birth, belonging to a noble origin or just a part of a last name.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What is the origin of the version &amp;quot;Tycho de Brahe&amp;quot; of Brahe's full name?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Hypothesis:&lt;/strong&gt; a historian from the mentioned list of countries started to use it and it become a wide-spread mistake since then.&lt;/p&gt;&#xA;&lt;p&gt;Brahe's residence in Prague:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vcnwJ.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vcnwJ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15547" LastEditorUserId="4251" LastEditDate="2021-10-22T15:30:16.573" LastActivityDate="2021-10-26T06:40:54.723" Title="Origin of the word &quot;de&quot; in the name of Danish astronomer Tycho de Brahe" Tags="|astronomers|brahe|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13803" PostTypeId="2" ParentId="13802" CreationDate="2021-10-22T15:29:45.663" Score="7" Body="&lt;p&gt;A Czech researcher, Alena Šolcová (who has an asteroid named after her, so she must be right) concluded in 2005 that this mistake was introduced in the 19th century, in works by Amalia Schoppe in 1839 and Anton Fähnrich in 1841.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;Alena Šolcová, &lt;a href=&quot;https://dml.cz/handle/10338.dmlcz/143823&quot; rel=&quot;noreferrer&quot;&gt;From Tycho Brahe to incorrect &amp;quot;Tycho de Brahe&amp;quot;. A searching for the first occurrence, when the mistaken name of famous astronomer appeared&lt;/a&gt;, Acta Universitatis Carolinae,  Mathematica et Physica, Volume 46 (2005), Issue 3, Pages 29-36.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="2667" LastEditorUserId="4251" LastEditDate="2021-10-22T15:33:45.890" LastActivityDate="2021-10-22T15:33:45.890" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13804" PostTypeId="2" ParentId="13802" CreationDate="2021-10-22T15:39:13.053" Score="8" Body="&lt;p&gt;I have read the article from Alena Šolcová. However, there is evidence that she is wrong that this is a mistake that was only introduced in the 1800s. Here is a book from 1769 having &amp;quot;Tycho de Brahe&amp;quot; in it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books/about/A_New_and_Complete_Dictionary_of_Terms_o.html?id=5XeemgpnwgsC&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books/about/A_New_and_Complete_Dictionary_of_Terms_o.html?id=5XeemgpnwgsC&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Page 447.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/aZq5T.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/aZq5T.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15550" LastEditorUserId="10563" LastEditDate="2021-10-26T06:40:54.723" LastActivityDate="2021-10-26T06:40:54.723" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13805" PostTypeId="1" AcceptedAnswerId="13807" CreationDate="2021-10-22T23:45:46.703" Score="2" ViewCount="253" Body="&lt;p&gt;Almost all of modern complex analysis (Cauchy residue theorem, analytic continuation, etc) depend on the definition of a complex derivative.&lt;/p&gt;&#xA;&lt;p&gt;That definition requires the derivative at a point &lt;span class=&quot;math-container&quot;&gt;$z_0$&lt;/span&gt; is the same no matter which direction the limit is taken to that point.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ f'(z_0)=\lim_{z \rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This definition is quite strong, and leads to the Cauchy Riemann conditions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question: Why did this definition become the standard one?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why didn't, for example, a derivative that depends on direction become standard?&lt;/p&gt;&#xA;&lt;p&gt;If other options are possible, do they lead to useful but different complex analysis?&lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="643" LastEditDate="2022-01-06T21:13:29.003" LastActivityDate="2022-01-06T21:13:29.003" Title="History of the definition of complex derivative" Tags="|mathematics|complex-analysis|definition|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13807" PostTypeId="2" ParentId="13805" CreationDate="2021-10-23T14:35:37.393" Score="5" Body="&lt;p&gt;In general, definitions in mathematics are not arbitrary. Useful definitions are those which lead to interesting and important objects. This particular definition is good because it leads to an interesting and important class of functions.&lt;/p&gt;&#xA;&lt;p&gt;A function &lt;span class=&quot;math-container&quot;&gt;$f(z)=u(x,y)+iv(x,y), z=x+iy,$&lt;/span&gt; of one complex variable is the same&#xA;as two real functions of two real variables. So partial derivatives, directional&#xA;derivatives etc. can be defined, but you do not obtain anything new with these definitions, in comparison with real calculus of functions of several real variables. Since this theory (real calculus) exists for functions of any number of real variables, one does not obtain anything new.&lt;/p&gt;&#xA;&lt;p&gt;The definition of complex derivative uses the field structure on complex numbers in a very essential way: it involves division. There is no &amp;quot;division&amp;quot; of real vectors. And this definition leads to a remarkable and useful class of functions: complex analytic functions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-10-23T14:35:37.393" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13809" PostTypeId="2" ParentId="13802" CreationDate="2021-10-23T16:15:14.177" Score="13" Body="&lt;p&gt;The earliest use of &amp;quot;de Brahe&amp;quot; I have found is in a letter in German dated 26 July 1589, by one of Brahe's royal admirers, &lt;a href=&quot;https://en.wikipedia.org/wiki/William_IV,_Landgrave_of_Hesse-Kassel&quot; rel=&quot;nofollow noreferrer&quot;&gt;William IV, Landgrave of Hesse&lt;/a&gt; (1532-1592), reprinted by Brahe in his &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=ucm.5323776852&amp;amp;view=1up&amp;amp;seq=202&amp;amp;skin=2021&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Epistolarum astronomicarum libri : quorum primus hic ... principis Gulielmi Hassiae Landtgrauij ac ipsius mathematici literas, unaque responsa ad singulas complectitur&lt;/em&gt;&lt;/a&gt;. The copy of the book linked here was printed in 1610, but the same letter appears in a 1596 edition.  (For those who have trouble reading the old German type, the next page has a translation into Latin.)  William wrote &amp;quot;zu dem Tychoni de Brahe&amp;quot;, where the syntax of the sentence puts the name in the dative case.  (&amp;quot;Zu&amp;quot; implies dative in German; the dative form of the Latin name &amp;quot;Tycho&amp;quot; is &amp;quot;Tychoni&amp;quot;.) As Stephan Mattheisen points out, the words &amp;quot;Tychoni de Brahe&amp;quot; are in Latin, set in Roman type, the rest of the sentence is in German, set in German type.  The Latin word &amp;quot;de&amp;quot; means &amp;quot;from&amp;quot;, the equivalent of German &amp;quot;von&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The other letters in this book show Tycho's usual signature was &amp;quot;Tycho Brahe&amp;quot;, so in a sense William's use of &amp;quot;de Brahe&amp;quot; is a mistake.  But not so serious a mistake as to prevent Tycho from reprinting it verbatim in this collection of letters.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VRbR5.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VRbR5.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Google Books shows many pre 1700 instances of &amp;quot;de Brahe&amp;quot;, often in German.  I ran across an instance in the 1680 Leipzig &lt;a href=&quot;https://www.google.com/books/edition/Cometischer_Gedenkzetel/7VxRAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=inauthor:%22Christianus+Uranophilus%22&amp;amp;pg=PP14&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Cometischer Gedenkzettel&lt;/em&gt;&lt;/a&gt;&#xA;by Christianus Uranophilus.  He reports &amp;quot;Anno 1590, gedencket Tycho de Brahe eines Cometens. In diesem Jahre ist an vielen Orten grosse Hungersnoth gewesen&amp;quot; (=&amp;quot;In 1590 TdB observed a comet. In that year great famine occurred in many places.&amp;quot;)&#xA;The same method turned up multiple instances of &amp;quot;de Brahe&amp;quot; in a 1594 collection of &lt;a href=&quot;https://en.wikipedia.org/wiki/Anacreontics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anacreontic&lt;/a&gt; verse, &lt;a href=&quot;https://www.google.com/books/edition/FRIDERICI_TAVBMANI_Franci_Columbae_Poeti/xi8leePjVIIC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22de+brahe%22&amp;amp;pg=PA14&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Columbae Poeticae&lt;/a&gt; by &lt;a href=&quot;https://de.wikipedia.org/wiki/Friedrich_Taubmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedrich Taubmann&lt;/a&gt;, a professor at Wittenberg. This book is discussed in a &lt;a href=&quot;https://books.google.com/books?id=TbfoBQAAQBAJ&amp;amp;pg=PT156&amp;amp;lpg=PT156&amp;amp;dq=Columbae+Poeticae&amp;amp;source=bl&amp;amp;ots=pc7YybvFTn&amp;amp;sig=ACfU3U2AxBIYQr8WOykpbOF4ZoWa4utR9Q&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjYiKSAhuTzAhVIVd8KHThsBLM4ChDoAXoECAkQAw#v=onepage&amp;amp;q=Columbae%20Poeticae&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern work&lt;/a&gt;, that I wish I could understand better.  It seems possible to me that Taubmann used &amp;quot;de Brahe&amp;quot; to fit the meter of his verse form.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="2307" LastEditDate="2021-10-25T18:33:07.143" LastActivityDate="2021-10-25T18:33:07.143" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13810" PostTypeId="2" ParentId="5645" CreationDate="2021-10-24T09:56:10.860" Score="1" Body="&lt;p&gt;José Echegaray was a brilliant Spanish mathematician, and also won the Nobel Prize for Literature in 1904...&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jos%C3%A9_Echegaray&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Jos%C3%A9_Echegaray&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2021-10-24T09:56:10.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13811" PostTypeId="1" AcceptedAnswerId="13812" CreationDate="2021-10-24T23:52:43.850" Score="2" ViewCount="162" Body="&lt;p&gt;There was a section on my textbook on history of theories of sun's energy source.&lt;/p&gt;&#xA;&lt;p&gt;It talks about how the Meteorite Theory was dismissed, as it would decrease the period of Earth's orbit by 2 seconds per year due to increased mass of the Sun.&lt;/p&gt;&#xA;&lt;p&gt;This theory was dismissed due to disagreeing with observation. And the textbook says the change in period is &amp;quot;easily measurable&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;My question is how is 2 seconds difference easily measurable in Nineteenth Century?&lt;/p&gt;&#xA;" OwnerUserId="15569" OwnerDisplayName="Sirou Ewei" LastEditorUserId="4251" LastEditDate="2021-10-27T16:03:03.420" LastActivityDate="2021-10-27T16:03:03.420" Title="How accurate was the measurement of the period of Earth's orbit in the 19th Century?" Tags="|physics|astronomy|gravity|19th-century|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13812" PostTypeId="2" ParentId="13811" CreationDate="2021-10-25T03:56:16.457" Score="4" Body="&lt;p&gt;Astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Norman_Lockyer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Norman Lockyer&lt;/a&gt; published &lt;em&gt;The Meteoritic Hypothesis&lt;/em&gt; in 1890. By then the age of the Earth was known to be at least several hundred million years from geological evidence, and Darwin’s theory of evolution suggested life on Earth originated at least 100 million years ago. A change in the length of the year by 2 seconds a year over 100 million years would make the year over 6 times longer than it is now. It would also mean that 100 million years ago the Earth was over 3 times further away from the Sun than it is now. This would put the Earth between the current orbits of Mars and Jupiter, making it uninhabitable.&lt;/p&gt;&#xA;" OwnerUserId="9542" LastActivityDate="2021-10-25T03:56:16.457" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13813" PostTypeId="2" ParentId="13802" CreationDate="2021-10-25T08:49:51.600" Score="1" Body="&lt;p&gt;&lt;em&gt;De&lt;/em&gt; is also common in Dutch, like &lt;a href=&quot;https://nl.wikipedia.org/wiki/Hugo_de_Groot_(rechtsgeleerde)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hugo de Groot&lt;/a&gt; or &lt;a href=&quot;https://nl.wikipedia.org/wiki/Karel_de_Grote&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karel de Grote&lt;/a&gt;. In the latter, &amp;quot;de Grote&amp;quot; describes Karel: he is big (not so much in size as in the size of his empire. &lt;em&gt;If&lt;/em&gt; the word &amp;quot;de&amp;quot; is coming from another language I would find it more likely that it comes from &lt;em&gt;another&lt;/em&gt; Germanic language, i.e. Dutch, than from a Latin language like Italian or French.&lt;/p&gt;&#xA;" OwnerUserId="15561" LastActivityDate="2021-10-25T08:49:51.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13814" PostTypeId="2" ParentId="13811" CreationDate="2021-10-25T15:05:13.633" Score="2" Body="&lt;p&gt;Yes, 2 seconds per year decrease of the Earth's revolution period would be easily detected in the 19th century. This is based on the general principle of astronomy, which was used since time immemorial. A 2-second decrease per year would result in 200 seconds decrease per century. And this is easily measured, since observations of the length of the year have been made for much more than a century.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="12330" LastEditDate="2021-10-26T01:02:01.410" LastActivityDate="2021-10-26T01:02:01.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13815" PostTypeId="1" CreationDate="2021-10-25T15:15:44.820" Score="3" ViewCount="75" Body="&lt;p&gt;I remember reading once that when electric motors first became available for household use, people thought that there would be one big motor in the loft, with drive belts feeding devices in different parts of the house.  Unfortunately, a search for information about this hasn't come up with anything.  Does anyone have more details of whether people actually thought this way, or did I just imagine it?&lt;/p&gt;&#xA;" OwnerUserId="15567" LastActivityDate="2021-10-25T15:15:44.820" Title="In the early days of electricity did people think in terms of a big electric motor in the loft" Tags="|electricity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13816" PostTypeId="1" AcceptedAnswerId="13818" CreationDate="2021-10-26T08:04:16.030" Score="1" ViewCount="214" Body="&lt;p&gt;In &lt;a href=&quot;https://archive.org/details/sciencemethod00poinuoft/page/36/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Science and Method&lt;/a&gt;, page 37, Poincaré (as translated by Francis Maitland) states,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The story goes that Newton once communicated to Leibnitz an anagram somewhat like the following: &lt;em&gt;aaaaabbbeeeeii&lt;/em&gt;, etc.  Naturally, Leibnitz did not understand it at all, but we who have the key know that the anagram, translated into modern phraseology, means, &amp;quot;I know how to integrate all differential equations,&amp;quot; and we are tempted to make the comment that Newton was either exceedingly fortunate or that he had very singular illusions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, I fall in with Leibnitz in lacking the key to understanding this anagram, or even seeing how this is actually an anagram. An explanation of the relationship between Newton's cryptic message and the assertion that he knew how to integrate all differential equations would be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="15574" LastActivityDate="2021-10-26T14:13:41.513" Title="How to understand Poincaré's interpretation of Newton's anagram &quot;aaaaabbbeeeeii&quot;" Tags="|newton|leibniz|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13817" PostTypeId="2" ParentId="13759" CreationDate="2021-10-26T11:00:41.447" Score="1" Body="&lt;p&gt;The study of the quintic by mathematicians such as Lagrange, Abel and Galois led to profound insights that extend well beyond polynomial equations.  For example, Galois' proof of the non-existence of a solution by radicals to the general quintic established a critical connection between field theory and group theory (&lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;the fundamental theorem of Galois theory&lt;/a&gt;).  In fact many people view Galois' proof to be the foundation of group theory.&lt;/p&gt;&#xA;&lt;p&gt;There is an intimate connection between polynomials, geometry and groups.  For instance, the symmetries of a regular tetrahedron are described by the &lt;a href=&quot;https://people.maths.bris.ac.uk/%7Ematyd/GroupNames/1/S4.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;symmetric group &lt;span class=&quot;math-container&quot;&gt;$S_4$&lt;/span&gt;&lt;/a&gt;.  You can see from the link that when studying this group, polynomials with Galois group &lt;span class=&quot;math-container&quot;&gt;$S_4$&lt;/span&gt; can be useful in better understanding the group and/or geometry.&lt;/p&gt;&#xA;&lt;p&gt;These developments were very influential in the late 1800's, once people understood the significance of what Galois had done (tragically, many years after his death).  Sophus Lie noted that his development of Lie theory was motivated by a desire to do for differential equations what Galois had done for algebraic equations.&lt;/p&gt;&#xA;&lt;p&gt;To your question on motivation, of course none of the major figures leading up to Galois' proof would have fully understood the implications of studying the quintic.  But I think Grothendieck's comments (from a &lt;a href=&quot;https://webusers.imj-prg.fr/%7Eleila.schneps/grothendieckcircle/Letters/GtoF.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;personal letter&lt;/a&gt; to Faltings) are likely applicable to what was happening,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What my experience of mathematical work has taught me again and again, is that the proof always springs from the insight, and not the other way round – and that the insight itself has its source, first and foremost, in a delicate and obstinate feeling of the relevant entities and concepts and their mutual relations. The guiding thread is the inner coherence of the image which gradually emerges from the mist, as well as its consonance with what is known or foreshadowed from other sources – and it guides all the more surely as the “exigence” of coherence is stronger and more delicate.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At least starting with Laplace and his introduction of resolvents, it seems that the great mathematicians of the time were starting to sense that there was depth to polynomials that needed to be explored.&lt;/p&gt;&#xA;&lt;p&gt;One final point of clarification, the general quintic is not unsolvable, it is unsolvable &lt;em&gt;by radicals&lt;/em&gt;.  Hermite discovered that the general quintic can be solved by elliptic modular functions.  Klein gives a detailed description of Hermite's solution (and several others), and its connection to the rotation group of the icosahedron (&lt;span class=&quot;math-container&quot;&gt;$A_5$&lt;/span&gt;), in &lt;a href=&quot;https://archive.org/details/cu31924059413439/page/n9/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lectures on the Ikosahedron, and the Solution of Equations of the Fifth Degree&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="15574" LastActivityDate="2021-10-26T11:00:41.447" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13818" PostTypeId="2" ParentId="13816" CreationDate="2021-10-26T14:13:41.513" Score="5" Body="&lt;p&gt;See more detail about this anagram (there were actually two of them) here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram&quot;&gt;https://mathoverflow.net/questions/140327/arnold-on-newtons-anagram&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Whatever the literal translation of the anagrams is, Newton indeed discovered&#xA;a method of solving &amp;quot;all equations&amp;quot; algebraic, differential, functional,&#xA;whatever. The method consisted in plugging a power series with undetermined coefficients and determining the coefficients one by one.&lt;/p&gt;&#xA;&lt;p&gt;This was indeed a great idea, and it kept mathematicians busy ever since.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-10-26T14:13:41.513" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13821" PostTypeId="1" AcceptedAnswerId="13822" CreationDate="2021-10-26T22:00:12.167" Score="5" ViewCount="222" Body="&lt;p&gt;A quadratic form &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; on a vector space &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is isotropic if &lt;span class=&quot;math-container&quot;&gt;$q(v) = 0$&lt;/span&gt; admits a nonzero solution. What was this terminology originally intended to evoke?&lt;/p&gt;&#xA;&lt;p&gt;There is some prior discussion on Math SE, &lt;a href=&quot;https://math.stackexchange.com/questions/90674/etymology-of-the-word-isotropic&quot;&gt;Etymology of the word &amp;quot;isotropic&amp;quot;&lt;/a&gt;, which traces early uses of the term, but it doesn't seem to arrive at an answer as to why the term was chosen.&lt;/p&gt;&#xA;" OwnerUserId="15579" LastEditorUserId="55" LastEditDate="2021-10-27T08:16:54.307" LastActivityDate="2021-10-27T08:16:54.307" Title="Why is the term &quot;isotropic&quot; used to describe a quadratic form and a vector?" Tags="|abstract-algebra|linear-algebra|algebraic-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13822" PostTypeId="2" ParentId="13821" CreationDate="2021-10-27T08:15:02.860" Score="4" Body="&lt;p&gt;This is an example of how a term migrates from the original context by broken telephone through various generalizations and transfers. It started with Poncelet  introducing &amp;quot;imaginaries&amp;quot;, i.e. complex coordinates, into projective geometry  in &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k9608143v.texteImage&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Traité des propriétés&#xA;projectives des figures&lt;/em&gt; (1822)&lt;/a&gt; based on his earlier work. He discovered circular points at infinity, the two points on the line at infinity through which all circles pass. Through them pass the lines with slopes &lt;span class=&quot;math-container&quot;&gt;$\pm i$&lt;/span&gt;. These were to become the &amp;quot;isotropic&amp;quot; points and lines, see &lt;a href=&quot;https://www.google.com/books/edition/Elements_of_the_History_of_Mathematics/4JprCQAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Bourbaki, Elements of the History of Mathematics, p. 131&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Laguerre observed in 1853 that one can use those lines to define angles projectively through the cross-ratio by combining them with the two given lines, see &lt;a href=&quot;https://www.jstor.org/stable/2324107&quot; rel=&quot;noreferrer&quot;&gt;Reid's review&lt;/a&gt;. In &lt;a href=&quot;http://www.numdam.org/item/NAM_1870_2_9__241_0&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur l’emploi des imaginaires en la géométrie&lt;/em&gt; (1870)&lt;/a&gt; he calls them &amp;quot;isotropic&amp;quot;, and also has &amp;quot;isotropic&amp;quot; coordinates, etc. Why &amp;quot;isotropic&amp;quot; he did not say, perhaps because circles are isotropic. And then he finally links them to quadratic forms:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation &lt;span class=&quot;math-container&quot;&gt;$ds^2 = 0$&lt;/span&gt;. On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, the isotropic curves are tangent to vectors on which the first fundamental form of the surface vanishes. The terminology then spread in the work on &lt;a href=&quot;https://en.wikipedia.org/wiki/Erlangen_program#The_problems_of_nineteenth_century_geometry&quot; rel=&quot;noreferrer&quot;&gt;Kleinian geometries&lt;/a&gt; derived from Poncelet's projective geometry by using various quadrics as absolutes, and  the work on quadratic forms and geometric groups more generally. Under the influence of Poincare and Minkowsky, it also made its way into the jargon of relativity (isotropic = null = lightlike vectors, cones, etc).&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-10-27T08:15:02.860" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13823" PostTypeId="1" CreationDate="2021-10-27T14:16:49.900" Score="1" ViewCount="103" Body="&lt;p&gt;Throughout our known history of geometry were the units representing areas and volumes always in terms of squares and cubes respectively? Take ancient Egyptian formulas as an example, the fact that their formulas are very close to ours must mean that the units representing areas and volumes were also unit squares and unit cubes, correct? I mean we can safely stamp their formulas with for example &lt;span class=&quot;math-container&quot;&gt;$cm$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$cm^2$&lt;/span&gt; and get the same result. And does that mean we can safely assume that they didn't measure e.g. volume by counting the number of times you can fill up an object by throwing identical fistfuls of sand into it?&lt;/p&gt;&#xA;" OwnerUserId="15021" LastEditorUserId="4251" LastEditDate="2021-10-28T07:30:02.250" LastActivityDate="2021-10-28T07:30:02.250" Title="Were units of area/volume always in terms of squares/cubes?" Tags="|geometry|units|ancient-egypt|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13824" PostTypeId="1" AcceptedAnswerId="13825" CreationDate="2021-10-27T23:17:03.673" Score="0" ViewCount="233" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RQWat.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RQWat.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure where else to post this. No matter how inventive my Google searches, I am unable to locate an active webpage containing this rare statue of Isaac Newton that I stored on my computer a number of years ago. Can anybody identify information about this statue?&lt;/p&gt;&#xA;" OwnerUserId="438" LastActivityDate="2021-10-29T15:03:01.637" Title="Identification of Rare Newton Statue" Tags="|newton|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13825" PostTypeId="2" ParentId="13824" CreationDate="2021-10-28T13:14:04.507" Score="3" Body="&lt;p&gt;Perhaps this is a lead:  I found an &lt;a href=&quot;http://en.broad.com/m/NewsDetail-48-3.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; which includes the following text [emphasis mine] .&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The campus of Broad Group in Changsha was designed to reflect Zhang&#xA;Yue’s eclectic influences and his passion for the environment. It&#xA;encompasses a sprawling organic garden that provides up to half of the&#xA;food consumed by his workers, and the grounds are dotted with dozens&#xA;of statues of inspirational figures, from Confucius and the poet Li&#xA;Bai to Coco Chanel and Jack Welch. There are likenesses of&#xA;environmentalist Rachel Carson, &lt;em&gt;&lt;strong&gt;Sir Isaac Newton with an apple about&#xA;to fall on his head&lt;/strong&gt;&lt;/em&gt;, and Winston Churchill flashing a V for Victory&#xA;sign. One company building is modeled after the palace at Versailles,&#xA;another after an Egyptian pyramid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="609" LastActivityDate="2021-10-28T13:14:04.507" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13826" PostTypeId="1" CreationDate="2021-10-28T13:54:47.843" Score="2" ViewCount="174" Body="&lt;p&gt;The time when Thomson discovered electrons, around 1897, what did the scientific community think about the atomic hypothesis? Was there a majoritarian consensus that matter is made of indivisible neutral atoms and was that based on evidence?&lt;/p&gt;&#xA;&lt;p&gt;Here is why I am asking this. As far as I know, Thomson gave his plum-pudding model in 1904 based on (1) the atoms are electrically neutral and (2) they contain negatively charged electrons. This suggests to me that it was clear that matter is not continuous but made of neutral atoms at that time. But as far as I know, Boltzmann was severely criticized for his belief in the atomic hypothesis. Please correct my facts.&lt;/p&gt;&#xA;" OwnerUserId="9960" LastActivityDate="2021-10-28T13:54:47.843" Title="Around 1904, did the scientific community take the atomic hypothesis seriously?" Tags="|physics|atomic-theory|" AnswerCount="0" CommentCount="27" ContentLicense="CC BY-SA 4.0" />
  <row Id="13827" PostTypeId="1" AcceptedAnswerId="13828" CreationDate="2021-10-29T04:43:04.390" Score="0" ViewCount="162" Body="&lt;p&gt;The first original definition of the metre, presented by the French Academy of Sciences in 1795 is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The length of the metre is one ten-millionth of the Earth quadrant, the distance from the North Pole to the Equator, measured along a meridian through Paris.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Roughly speaking, this definition is equivalent to the following one:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The length of the metre is 1/&lt;strong&gt;4&lt;/strong&gt;0,000,000'th of the circumference of the Earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It makes sense sense to define the length of the metre in terms of the entire circumference of the Earth, not its one-fourth, with the deiniftion only featuring a &amp;quot;pure&amp;quot; power of 10 like 100,000,000.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why was the original definition for the metre chosen? If the original definition is just the paraphrased second definition, the question becomes: where does the '4' come from?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="14982" LastActivityDate="2021-10-29T14:09:30.057" Title="Motivation behind original definition of metre?" Tags="|units|definition|" AnswerCount="1" CommentCount="1" ClosedDate="2021-10-29T18:18:21.960" ContentLicense="CC BY-SA 4.0" />
  <row Id="13828" PostTypeId="2" ParentId="13827" CreationDate="2021-10-29T14:02:21.927" Score="4" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Strictly speaking the first and second definition are not equivalent, but there was no mean at that time to measure the whole meridian directly, so they assumed central symmetry, in which case the distance from the N pole to the equator along the Paris meridian would be 1/4 of the whole meridian. (To be sure, the distance from the N pole to the equator also cannot be measured directly, so they measured some arcs of it, and used a mathematical model for the shape of the Earth. Besides many precise measurements in Europe, they organized two major expeditions: one to Lapland another to S America).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The reason why they took the length to be 40,000 kilometers is the following.&#xA;French revolutionary government also introduced decimal measure of angles and time. With these measures the right angle was 100 degrees (round decimal number and somewhat close to the usual 90 degrees). Then the whole circumference was 400 decimal degrees. Each decimal degree was divided into 100 decimal minutes.&#xA;This is how they arrived to 40,000 decimal minutes, and decided to define&#xA;kilometer as a decimal minute. So it is similar to the (nautical) mile which is defined as 1 usual minute.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;They introduced similar measure of time, with day+night =20 decimal hours (close to 24), and each decimal hour=100 decimal time-minutes.&lt;/p&gt;&#xA;&lt;p&gt;Then decimal measures for angles and time were abolished (under Napoleon), while meters and kilometers remain. You can still see sometimes in museums (and on e-bay)&#xA;clocks and angle measuring instruments graduated in these decimal hours, degrees and minutes.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-10-29T14:09:30.057" LastActivityDate="2021-10-29T14:09:30.057" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13829" PostTypeId="2" ParentId="13824" CreationDate="2021-10-29T15:03:01.637" Score="3" Body="&lt;p&gt;Following up on Carl Witthoft's lead, I found another image at this &lt;a href=&quot;http://www.zhongguoyx.com/jingcaihuigu/2956.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;.  To locate the image, you can search the page for the text &amp;quot;第五章：科技筑梦&amp;quot; to avoid lots of scrolling. The image, shown below, provides the full background text:&lt;/p&gt;&#xA;&lt;p&gt;没有大胆的猜测&lt;/p&gt;&#xA;&lt;p&gt;就没有伟大的发现&lt;/p&gt;&#xA;&lt;p&gt;(paraphrasing google translate, this comes out to: &amp;quot;Without bold guesses, there is no great discovery&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ib1zy.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ib1zy.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2021-10-29T15:03:01.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13830" PostTypeId="1" AcceptedAnswerId="13832" CreationDate="2021-10-30T18:35:23.513" Score="2" ViewCount="136" Body="&lt;p&gt;In my copy of &lt;em&gt;Physics for Scientists and Engineers with Modern Physics&lt;/em&gt; (Volume 2) by Raymond Serway there is a quote given by James Clerk Maxwell at the start of Part IV.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For the sake of persons of [...] different types, scientific truth should be presented in different forms, and should be regarded as equally scientific, whether it appears in the robust form and the vivid coloring of a physical illustration, or in the tenuity and paleness of symbolic expression.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Who are these different types of people? Is he categorizing people into visual vs aesthetic vs symbolic thinkers? Or is he saying something else in this context?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastEditorUserId="55" LastEditDate="2021-10-31T05:24:48.117" LastActivityDate="2021-10-31T05:24:48.117" Title="What did Maxwell mean by different types of people to whom science should be presented in different forms?" Tags="|quote|maxwell|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13831" PostTypeId="1" AcceptedAnswerId="13833" CreationDate="2021-10-31T00:51:33.957" Score="8" ViewCount="756" Body="&lt;p&gt;A colleague and I were having a discussion about mathematical similarities between graphs of functions and graphs as used in graph theory: Simple graphs can be defined in terms of pair (of vertices), just as we can think of the graph a function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; as given by pairs &lt;span class=&quot;math-container&quot;&gt;$(x,f(x))$&lt;/span&gt;. An old HSM question, &lt;a href=&quot;https://hsm.stackexchange.com/questions/2523/do-the-words-graphing-a-function-and-graph-theory-have-a-common-ancestor&quot;&gt;Do the words &amp;#39;graphing&amp;#39; a function and &amp;#39;graph&amp;#39; theory have a common ancestor?&lt;/a&gt;, asked if the two uses had a common ancestor, and the answers given discussed the etymology of &amp;quot;graph&amp;quot; and its relation to drawing and pictures, but didn't answer the underlying question: Were these two ideas given the same name (in English at least) because of any underlying mathematical connection, or is it just a case of the same word being used more-or-less coincidentally for two unrelated ideas? A closely related question would be who was the first to use the English word in each of the two meanings?&lt;/p&gt;&#xA;" OwnerUserId="652" LastEditorUserId="55" LastEditDate="2021-10-31T06:57:07.643" LastActivityDate="2021-11-01T07:58:49.653" Title="Was a mathematical connection involved when introducing &quot;graph&quot; of a function and &quot;graph&quot; in graph theory?" Tags="|mathematics|terminology|graph-theory|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13832" PostTypeId="2" ParentId="13830" CreationDate="2021-10-31T05:22:15.810" Score="5" Body="&lt;p&gt;The quote is from &lt;a href=&quot;https://static1.squarespace.com/static/54151077e4b05407cce436ac/t/546ab4a4e4b0ffb977602da5/1416279204963/Maxwell.pdf&quot; rel=&quot;noreferrer&quot;&gt;Maxwell's Address to the Mathematical and Physical Sections&lt;/a&gt; of the British Association for the Advancement of Science at the 40th Meeting in Liverpool (1870). The address offers some quite insightful observations and reflections on study and comprehension in mathematics and sciences, structural analogies in different sciences and their distillation into mathematical forms, and fruitful creative use of &amp;quot;scientific metaphors&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Maxwell does distinguish between abstract/symbolic types, which he considers very rare, and intuitive types (this is roughly associated with left and right hemisphere dominance today). He then further subdivides the intuitive into visual/geometric and physical/motoric types. Here is the full passage:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There  are  men  who,  when  any  relation  or  law,  however  complex,  is  put before them  in  a  symbolical  form,  can  grasp  its  full  meaning  as  a  relation among abstract  quantities.  Such  men  sometimes  treat  with  indifference  the  further statement  that  quantities  actually  exist  in  nature  which  fulfill  this  relation. The mental  image  of   the  concrete  reality  seems  rather  to  disturb  than  to  assist their contemplations.  But  the  great  majority  of   mankind  are  utterly  unable,  without long  training,  to  retain  in  their  minds  the  unembodied  symbols  of   the  pure-mathematician, so that, if  science is ever to become popular, and yet remain scientific, it must be by a profound study and a copious application of  those principles of   the  mathematical  classification  of   quantities  which,  as  we  have seen,  lie  at  the root of  every truly scientific illustration.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;There are, as I have said, some minds which can go on contemplating with satisfaction pure quantities presented to the eye by symbols, and to the mind ina form which none but mathematicians can conceive.There are others who feel more enjoyment in following geometrical forms, which they draw on paper, or build up in the  empty  space  before  them. Others,  again,  are  not  content  unless  they  can  project their whole physical energies into the scene which they conjure up. They learn at what a rate the planets rush through space, and they experience a delightful feeling of exhilaration. They calculate the forces with which the heavenly bodies pull at one another, and they feel their own muscles straining with the effort.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;To  such  men  momentum,  energy,  mass  are  not  mere  abstract  expressions  of  the results  of   scientific  inquiry.  They  are  words  of   power,  which  stir  their  souls like the  memories  of   childhood. &lt;strong&gt;For  the  sake  of   persons  of   these  different  types, scientific  truth  should  be presented  in  different  forms,  and  should  be  regarded  as equally scientific whether it appears in the robust form and the vivid colouring of  a physical illustration, or in the tenuity and paleness of  a symbolical expression.&lt;/strong&gt;&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-10-31T05:22:15.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13833" PostTypeId="2" ParentId="13831" CreationDate="2021-10-31T06:54:54.397" Score="8" Body="&lt;p&gt;There was no specific mathematical connection (both representing binary relations, say) motivating the common name. Rather, the only commonality was a vague relation of both to &amp;quot;graphic&amp;quot; (pictorial) representation that the Greek root conveys. A contributing factor might have been that &amp;quot;diagram&amp;quot;, as in Euclidean demonstrations, has the same root, and already Aristotle refers to diagrammatic proofs as &lt;em&gt;γραφὴματα&lt;/em&gt;, see &lt;a href=&quot;https://www.academia.edu/8016051/Euclid_s_Pseudaria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Acerbi, Euclid's Pseudaria&lt;/a&gt;. So the word was attractive to mathematicians for its venerable allusions. Ptolemy's &lt;em&gt;Γεωγραφικὴ&lt;/em&gt; was also well-known.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.google.com/books/edition/Graph_Theory_1736_1936/XqYTk0sXmpoC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Graph theory, 1736-1936&lt;/a&gt; Biggs, Lloyd and Wilson trace the first use of &amp;quot;graph&amp;quot; as in graph theory to &lt;a href=&quot;https://www.nature.com/articles/017284a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sylvester's note Chemistry and Algebra (1878)&lt;/a&gt;, where he connects it to &amp;quot;chemicographs&amp;quot; of molecules (emphasis his):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Every invariant and covariant thus&#xA;becomes expressible by a &lt;strong&gt;graph&lt;/strong&gt; precisely identical with a Kekulean diagram or chemicograph. But not every chemicograph is an algebraical one. I show that by an application of the algebraical law of reciprocity every algebraical graph of a given invariant will represent the constitution in terms of the roots of a quantic of a type reciprocal to that of the given invariant of an invariant belonging to that reciprocal type. I give a rule for the geometrical multiplication of graphs, i.e. for constructing a &lt;strong&gt;graph&lt;/strong&gt; to the product of in- or co- variants whose separate&#xA;graphs are given.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Peirce gave the same chemical motivation for his &lt;a href=&quot;https://en.wikipedia.org/wiki/Existential_graph&quot; rel=&quot;nofollow noreferrer&quot;&gt;existential graphs&lt;/a&gt; introduced c. 1882 for predicate calculus, although they are not quite vertex-edge graphs. The term did not solidify until König's &lt;em&gt;Theorie der endlichen und unendlichen Graphen&lt;/em&gt; (1936).&lt;/p&gt;&#xA;&lt;p&gt;As for coordinate graphs, a manual for constructing them, but not the name, appears already in Euler's textbook &lt;a href=&quot;http://www.17centurymaths.com/contents/euler/introductiontoanalysisvoltwo/ch1vol2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Introductio in analysin infinitorum&lt;/em&gt; (1748)&lt;/a&gt; that calls graphs &amp;quot;curved lines&amp;quot;. English translation of &lt;a href=&quot;https://archive.org/details/anelementarytre00babbgoog/page/n103/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Differential and integral calculus by Lacroix (1816)&lt;/a&gt; still calls them just curves or curve lines. In a different context, Gaultier uses &amp;quot;graphical construction&amp;quot; in his 1813 &lt;em&gt;Memoire sur les moyens generaux de construire graphiquement un cercle determine par trois conditions&lt;/em&gt;, and in Massau's long treatise on integration (1878-1887) aimed at engineers we read (see &lt;a href=&quot;https://www.jstor.org/stable/41494930&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tournés, Diagrams in the theory of differential equations&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In what follows, we will always represent functions by curves; when we say to give or to find a function, it will mean giving or finding graphically the curve that represents it&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Lalanne, the inventor of logarithmic paper, called his 1880 book &lt;em&gt;Méthodes graphiques pour l'expression des lois empiriques ou mathémathiques&lt;/em&gt;. The use of &amp;quot;graphical methods&amp;quot;, so named, generally picks up in engineering applications in 1880s, see &lt;a href=&quot;https://hsm.stackexchange.com/a/3584/55&quot;&gt;When was the earliest use of log-log plots to demonstrate power-law behavior?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;One early source for the name &amp;quot;graph of function&amp;quot; is once popular &lt;a href=&quot;https://archive.org/details/algebraanelemen09chrygoog/page/n331/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chrystal's Algebra (1886)&lt;/a&gt;. Chrystal makes no connection to &amp;quot;chemicographs&amp;quot; of Kekule, or to Sylvester and Peirce. He ventures generically &amp;quot;&lt;em&gt;to obtain a graphical representation of the variation of the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;&amp;quot;&lt;/em&gt; (the expression was in the air), and then introduces the term, before using it profusely:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;To every value of the function, therefore, corresponds a representative point, &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;, whose abscissa (&lt;span class=&quot;math-container&quot;&gt;$OM$&lt;/span&gt;) and ordinate (&lt;span class=&quot;math-container&quot;&gt;$MP$&lt;/span&gt;)&#xA;represent the values of the independent and dependent variables... The representative point will therefore&#xA;trace out a continuous curve, such as we have drawn in Fig. 1. This curve we may call the graph of the function. It is obvious that when we know the graph of a function we can find the value of the function corresponding to any value of the independent variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; with an accuracy that depends&#xA;merely on the precision of our drawing instruments.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-01T07:58:49.653" LastActivityDate="2021-11-01T07:58:49.653" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13834" PostTypeId="1" AcceptedAnswerId="13837" CreationDate="2021-10-31T17:44:22.307" Score="2" ViewCount="122" Body="&lt;p&gt;In Niccolo Cabeo's 1629 work &lt;em&gt;Philosophia Magnetica&lt;/em&gt;, in which he picks up on Gilbert's early electrical work in book 2, he lists a number of materials that can be electrified (book II. ch. XVII., p. 179).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Trahunt praeterea plurimae gemmae, ut Gumma elemi, Carabe Gumma naturalis ex ligno, ut dicitur, sancto, Gumma ex mastice, Pix, quae dicitur Hispana;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These are gums and resins of different types, including elemi gum, gum of the Holywood, mastic gum, and &lt;em&gt;spanish pitch&lt;/em&gt;, or &amp;quot;Pix Hispana&amp;quot;. This last item is interesting because while something like &lt;em&gt;Pix Graeca&lt;/em&gt;, Greek Pitch, refers to rosin (colophony), I can't find any reference to Spanish Pitch. I checked the OED and found nothing under &amp;quot;pitch&amp;quot; or &amp;quot;Spanish&amp;quot;, and a search on Google Books brings references to the material from the mid-1500s, and then nothing else.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;Examen omnium simplicium medicamentorum, quorum in officinis usus est&lt;/em&gt; (1537) by Antonio Musa Brassavola, there is reference starting on page 376 [the book is written as a dialogue, between Brassavola (BRA), an &amp;quot;old pharmacist&amp;quot; Senex pharmacopola (SEN), and an herbalist, Herbarius.] Brassavola is Italian, like Cabeo, and in fact both men are from Ferrara, which makes me wonder if this was a regional term used in pharmacy between 1500-1700.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone heard of this term? Or could shed some light on its meaning?&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2021-11-01T21:11:57.397" Title="What is the &quot;Pix Hispana&quot; that Niccolo Cabeus refers to?" Tags="|terminology|electricity|medicine|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13835" PostTypeId="1" CreationDate="2021-11-01T00:55:00.390" Score="1" ViewCount="242" Body="&lt;p&gt;We all come across these two words in literature: summation, addition.&lt;/p&gt;&#xA;&lt;p&gt;I personally do not know any difference between them and I view both of them the same in all mathematical aspects.&lt;/p&gt;&#xA;&lt;p&gt;Is there any case where I cannot use the two words interchangeably? If not, then why do these two words evolve?&lt;/p&gt;&#xA;" OwnerUserId="2692" LastActivityDate="2021-11-01T02:17:41.593" Title="Why two words &quot;summation&quot; and &quot;addition&quot; do exist in literature?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13836" PostTypeId="2" ParentId="13835" CreationDate="2021-11-01T02:17:41.593" Score="6" Body="&lt;p&gt;I do not know whether this distinction is universal, but to my knowledge addition primarily applies to the adding together of two entities, the augend and the addend, while summation applies to the adding of any number of operands, usually more than two, the summands.&lt;/p&gt;&#xA;&lt;p&gt;All these words are derived from Latin. The verb &lt;em&gt;addere&lt;/em&gt; means to attach, the verb &lt;em&gt;augere&lt;/em&gt; means to enlarge. The &lt;em&gt;-nd&lt;/em&gt; ending indicates the gerundium, so the augend is &amp;quot;that which is to be enlarged&amp;quot; and the addend is &amp;quot;that which is to be attached&amp;quot;. Summation goes back to French &lt;em&gt;somme&lt;/em&gt; and that in turn to Latin &lt;em&gt;summa&lt;/em&gt;, total amount (often of money). This is turn is from Latin &lt;em&gt;summus&lt;/em&gt;, highest point or top, which is also the root of the English word summit.&lt;/p&gt;&#xA;&lt;p&gt;According to Klein's Comprehensive Etymological Dictionary of the English Language (Elsevier 1971), the Romans did not sum quantities top to bottom, with the sum total below all summands. Instead, they summed from the bottom up and wrote the sum total at the top.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2021-11-01T02:17:41.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13837" PostTypeId="2" ParentId="13834" CreationDate="2021-11-01T19:05:56.850" Score="3" Body="&lt;p&gt;&lt;em&gt;Pix&lt;/em&gt; or pitch made from the resin of various coniferous trees was in common use since antiquity, and variants of this substance, often named after their place of origin, were described by Roman writers such as Pliny, Columella, and Dioscorides.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Antonio_Musa_Brassavola&quot; rel=&quot;nofollow noreferrer&quot;&gt;Antonius Musa Brassavola&lt;/a&gt;, &lt;em&gt;Examen omnium simplicium medicamentorum, quorum in officinis usus est&lt;/em&gt; (Lyon, 1537),&#xA;&lt;a href=&quot;https://books.google.com/books?id=MGJWAAAAcAAJ&amp;amp;pg=PA377#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 377&lt;/a&gt; explains with regard to &lt;em&gt;pix Hispana&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hinc est quod pix Hispana varios colores habet: quandoque, fuluum et aureus, si ex laricis resina fiat, quandoque albicans, si ex pini vel picea fiat. [...] Hispana nuncupatur, quoniam ex Galliae &amp;amp; Hispaniae montibus ad nos magna copia comportatur, quamuis &amp;amp; in Intalia sit plurima: eadem ratione &amp;amp; Graeca dicitur, quia ex Graeca affertur: &amp;amp; Colophonia, quoniam Colophone invehebatur&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Hence it is that the Spanish pitch has various colors: sometimes yellow and golden, when it is made from larch resin, and sometimes whitish, when it is made from pine or spruce. [...] It is called Spanish, because a great quantity is brought to us from the mountains of Gaul and Spain although there is also much from Italy. For the same reason it is also called Greek, because it is brought from Greece. Also colophony, because it is acquired in Colophon.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In antiquity colophony (rosin) was shipped from the Greek city of Κολοφών. The various kinds of pitch differed in their characteristics, based on which kind of tree was used as the source of unprocessed resin and differences in the process of extraction and refining.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://penelope.uchicago.edu/Thayer/L/Roman/Texts/Pliny_the_Elder/14*.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gaius Plinius Secundus, &lt;em&gt;Naturalis Historia&lt;/em&gt; (XIV,127)&lt;/a&gt; informs us:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pix in Italia ad vasa vino condendo maxime probatur Bruttia. fit e piceae resina, in Hispania autem e pinastris minime laudata. est enim resina earum amara et arida et gravi odore.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;The pitch that in Italy is most highly esteemed for vessels for packing wine is the Bruttian. It is made from spruce resin, the one from Spain is from pine and not esteemed much. It is namely this resin bitter and dry and of strong odor.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In antiquity, Bruttium was the region at the toe of Italy, occupying roughly the area of modern-day Calabria. According to the Oxford Latin Dictionary, &lt;em&gt;picea&lt;/em&gt; is most likely a spruce of the species &lt;em&gt;Picea abies&lt;/em&gt;, while &lt;em&gt;pinaster&lt;/em&gt; is identified as the maritime pine, &lt;em&gt;Pinus pinaster&lt;/em&gt;. Coating of amphorae used for wine with pitch to improve impermeability was reported by multiple Roman authors and also alluded to by Brassavola. However, more recently physical evidence has called for a critical re-assessment:&lt;/p&gt;&#xA;&lt;p&gt;Kerlijne Romanus, et al., &amp;quot;Wine and olive oil permeation in pitched and non-pitched ceramics: relation with results from archaeological amphorae from Sagalassos, Turkey,&amp;quot;&#xA;&lt;em&gt;Journal of Archaeological Science&lt;/em&gt;, Volume 36, Issue 3, 2009, pp. 900-909:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Against this background, the possible uses of the Sagalassos amphorae are discussed, and the traditional association of pine pitch with wine storage in archaeological amphorae is critically assessed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Resin could be processed into pitch by various processes, such as dry and wet distillation, as well as boiling in water followed by straining. Regarding the inferior quality of &lt;em&gt;pix Hispana&lt;/em&gt; mentioned by Pliny, in the &lt;a href=&quot;https://revistas.um.es/rmu/article/view/73951/71321&quot; rel=&quot;nofollow noreferrer&quot;&gt;publication&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Pedro A. Lillo Carpio, &amp;quot;Pix y oleum ligni, productos industriales basicos en la antigüedad y su pervivencia,&amp;quot; &lt;em&gt;Revista Murciana de Antropologica&lt;/em&gt;, No. 1, 1994, pp. 109-119&lt;/p&gt;&#xA;&lt;p&gt;the author speculates that it might have had something to do with dry distillation at high temperatures (to maximize yield? NJ):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Posiblemente los defectos de la &lt;em&gt;pix hispana&lt;/em&gt; de debian a que obtenia a base de calentar la madera excesivamente y sobre todo porque no era sometida al proceso de hervirla en agua u colarla despues, tarea por la que sabemos que se obtenia la pez blanca -- &lt;em&gt;pix alba&lt;/em&gt; -- mas pura, menos olorosa y sin particulas carbonosas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Possibly the shortcomings of &lt;em&gt;pix Hispana&lt;/em&gt; were due to the fact that it was obtained by excessive heating of the wood and in particular because it was not subjected to the process of boiling in water followed by straining, a process that we know was used for obtaining white pitch, &lt;em&gt;pix alba&lt;/em&gt;: purer, less odorous, and without carbonized particles.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As for medicinal uses of &lt;em&gt;pix Hispana&lt;/em&gt;, Rudolf Kobert, &lt;em&gt;Historische Studien aus dem Pharmakologischen Institute der Kaiserlichen Universität Dorpat&lt;/em&gt;,&#xA;Halle an der Saale: Tausche &amp;amp; Grosse 1896, &lt;a href=&quot;https://books.google.com/books?id=KQo4AQAAMAAJ&amp;amp;pg=PA67&amp;amp;lpg=PA67&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 67&lt;/a&gt;, cites the Roman physician &lt;a href=&quot;https://en.wikipedia.org/wiki/Scribonius_Largus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scribonius Largus&lt;/a&gt; as recommending both &lt;em&gt;pix Hispana&lt;/em&gt; and &lt;em&gt;pix Bruttia&lt;/em&gt; for wound dressings.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-11-01T21:11:57.397" LastActivityDate="2021-11-01T21:11:57.397" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13838" PostTypeId="1" AcceptedAnswerId="13840" CreationDate="2021-11-02T01:56:36.823" Score="10" ViewCount="6188" Body="&lt;p&gt;When introducting the metric system, the French tried to decimalise the degrees used for angles. They defined the &lt;strong&gt;right angle&lt;/strong&gt; to contain 100 &lt;a href=&quot;https://en.wikipedia.org/wiki/Gradian&quot; rel=&quot;nofollow noreferrer&quot;&gt;gradians&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Why was the right angle chosen? A somewhat equivalent question: Out of all possible angles, why is the right angle particularly special?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The only idea that comes to my mind is that the right angle is the &amp;quot;average&amp;quot; angle, in the sense that it is the arithmetic mean of all angles between 0 degrees and 180 degrees. A problem with this explanation is that an angle is something that can be an arbitrary number of degrees: it certainly does make sense to talk about 321 degrees, as well as minus 123 (-123) degrees. One can argue that any talk of such angles can be paraphrased into talk of an angle in the interval [0,180] degrees. If my argument holds, the right angle is therefore the &amp;quot;average&amp;quot;, &amp;quot;quintessential&amp;quot; angle.&lt;/p&gt;&#xA;&lt;p&gt;I will probably not accept an explanation along the lines of: &amp;quot;Well, it is called &lt;em&gt;right&lt;/em&gt;...&amp;quot;. Such an explanation is as good as saying that the &lt;em&gt;golden&lt;/em&gt; ratio is of utmost importance in mathematics, which is not the case.&lt;/p&gt;&#xA;" OwnerUserId="14982" LastEditorUserId="-1" LastEditDate="2021-11-03T21:56:36.297" LastActivityDate="2021-11-05T06:19:19.447" Title="What makes the right angle special enough to be distinguished in the French metric system?" Tags="|mathematics|units|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="13839" PostTypeId="1" CreationDate="2021-11-02T05:34:28.280" Score="0" ViewCount="238" Body="&lt;p&gt;I found a copy of the French version 'Calcul infinitésimal' online but the English edition seems to only be available on Amazon for a very hefty price, or in American libraries which I do not have access to. If anyone could point me to where I may find this book it would be greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="15535" LastActivityDate="2021-11-02T05:34:28.280" Title="Where can I find a copy of Dieudonné's 'Infinitesimal Calculus'?" Tags="|mathematics|calculus|bourbaki|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13840" PostTypeId="2" ParentId="13838" CreationDate="2021-11-02T06:11:29.683" Score="39" Body="&lt;p&gt;&amp;quot;&lt;em&gt;When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right&lt;/em&gt;&amp;quot; is one of the opening definitions of &lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookI/bookI.html&quot; rel=&quot;noreferrer&quot;&gt;Euclid's Elements&lt;/a&gt;. Why did Euclid, or rather Pythagoreans he was systematizing, single it out? Well, it was not by averaging.&lt;/p&gt;&#xA;&lt;p&gt;For one, the right angle is easy to define, with only two straight lines and equality as prerequisites. For comparison, one would need three straight lines to define &lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt;, and more for other alternatives. Dropping perpendiculars, that come out of this definition, is also a ubiquitous step in Euclidean demonstrations.&lt;/p&gt;&#xA;&lt;p&gt;For two, as one can see from the opening propositions of the Elements, Pythagoreans were interested in angles, first and foremost, for their use in triangles. And the right angle produces the simplest form of the relation among the sides of a triangle having it, &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2=c^2$&lt;/span&gt;, the crown jewel of Pythagorean geometry. For comparison, if the angle were &lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt;, say, it would have been&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$a^2+b^2+ab=c^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;One could, perhaps, instead favor as the primitive what is sometimes called &amp;quot;straight angle&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt;), definable with a single straight line. &lt;em&gt;Today&lt;/em&gt;. But back then it was quite an unnatural &amp;quot;angle&amp;quot; to consider, as it happened before the advent of abstract mathematical formality that embraces extreme cases and extensions for the sake of generality. Indeed, Euclid and other ancients never talk of &amp;quot;straight angles&amp;quot;, let alone greater ones, just as they never talk of zero and negative integers. Those creations came later. When Euclid needs to, as in the parallel postulate, he says &amp;quot;two right angles&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, bisection, needed to go from &lt;span class=&quot;math-container&quot;&gt;$180^\circ$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt;, is more complex than duplication, needed to go the other way. So if one intends to use both &lt;span class=&quot;math-container&quot;&gt;$90^\circ$&lt;/span&gt; may still be easier to start with.&lt;/p&gt;&#xA;&lt;p&gt;Regardless of whether one finds such  arguments from simplicity compelling, by the time of the French revolution the force of Euclidean tradition was by itself more than enough to make the right angle &amp;quot;quintessential&amp;quot; and a natural choice for a primitive.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-11-02T06:11:29.683" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13841" PostTypeId="2" ParentId="13838" CreationDate="2021-11-02T12:39:57.503" Score="14" Body="&lt;p&gt;The simplest answer could be that just as a metre is similar in length to a yard, a gram is similar in mass to a scruple, and a litre similar in volume to a quart, 100 gradians are similar in number to 90 degrees. That is, it's a new name for something that's close to what people are already familiar with.  There doesn't necessarily have to be anything special about the right-angle itself.&lt;/p&gt;&#xA;" OwnerUserId="15353" LastEditorUserId="15353" LastEditDate="2021-11-02T12:50:52.627" LastActivityDate="2021-11-02T12:50:52.627" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13842" PostTypeId="2" ParentId="13838" CreationDate="2021-11-02T19:26:02.243" Score="5" Body="&lt;p&gt;The unit of angle was tied to the unit of length.  Just as a nautical mile is conceptually 1 arcminute (1/21600) of Earth's circumference, the kilometer would be 1 centigradian (1/40000) of it.&lt;/p&gt;&#xA;&lt;p&gt;So it's possible that the metric system's designers thought that a revised “yard” (1 m, 39.37 in) would be a more convenient unit of length than a “cubit” (40 cm, 15.75 in) or “rod” (4 m, 13.12 ft).&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://hsm.stackexchange.com/questions/11496/why-is-one-meter-as-long-as-it-is&quot;&gt;Why is one meter as long as it is?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15620" LastActivityDate="2021-11-02T19:26:02.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13843" PostTypeId="2" ParentId="8245" CreationDate="2021-11-02T20:24:10.873" Score="3" Body="&lt;p&gt;Back in 1959, it was because the US had accumulated a lot of survey data using the older definition of the foot.&lt;/p&gt;&#xA;&lt;p&gt;For most people, the 2 PPM discrepancy between the two types of “foot” went unnoticed; AFAIK, no suburban homeowner ever put up a fight over losing ~100 µm of their lawn.  But if you're making a detailed coast-to-coast map of the US, the discrepancy adds up to 9 m or so, which was enough to concern the mapmakers.  So the geodetic surveyors were &lt;em&gt;temporarily&lt;/em&gt; granted permission to use the old definition.  However,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was furthermore &lt;em&gt;mandated&lt;/em&gt; that the U.S. survey foot be replaced by the international foot upon readjustment of the geodetic control networks of the United States. Although such a readjustment was completed in 1986, use of the U.S. survey foot persisted. This situation has led to confusion and errors that continue to this day, and it is at odds with the intent of uniform standards. (Source: &lt;a href=&quot;https://www.nist.gov/pml/us-surveyfoot&quot; rel=&quot;nofollow noreferrer&quot;&gt;NIST&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, the original reason for retaining the survey foot ended 35 years ago, but the typically slow-to-react US government just didn't bother taking the old standard off the books.  Finally, in 2019 (after the OP asked the question), NIST &lt;a href=&quot;https://www.federalregister.gov/documents/2019/10/17/2019-22414/deprecation-of-the-united-states-us-survey-foot&quot; rel=&quot;nofollow noreferrer&quot;&gt;announced&lt;/a&gt; the deprecation of  “survey” measure, effective December 31, 2022.&lt;/p&gt;&#xA;" OwnerUserId="15620" LastActivityDate="2021-11-02T20:24:10.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13845" PostTypeId="2" ParentId="533" CreationDate="2021-11-02T23:41:07.620" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;The question is why 8 furlongs? Why forty poles, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Because &lt;a href=&quot;https://en.wikipedia.org/wiki/Furlong&quot; rel=&quot;nofollow noreferrer&quot;&gt;furlongs&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Rod_(unit)&quot; rel=&quot;nofollow noreferrer&quot;&gt;poles&lt;/a&gt; were already-standardized units that predated &lt;a href=&quot;https://en.wikipedia.org/wiki/Composition_of_Yards_and_Perches&quot; rel=&quot;nofollow noreferrer&quot;&gt;a c. 1300 measurement reform&lt;/a&gt;.  These units couldn't be changed without screwing up everyone's property records.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Mile&quot; rel=&quot;nofollow noreferrer&quot;&gt;mile&lt;/a&gt; was not part of the Anglo-Saxon land-measuring system, but a borrowing from ancient Roman legions, that had used a &lt;em&gt;mille passus&lt;/em&gt; (thousand paces) as a unit of distance.  A “pace” is not naturally a constant, as it depends on a person's physical stature and health, as well as the condition of the road.  But for practical purposes, a “mile” needed to be standardized.  Choices based on the original “thousand paces” definition are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Define a “pace” as 5 Roman feet of 296 mm, resulting in a mile of 1480 m.&#xA;(Incidentally, this is close to my average personal “pace” as I measured with my FitBit a few years ago.)&lt;/li&gt;&#xA;&lt;li&gt;Define a “pace” as 5 English feet of 304.8 mm, resulting in a mile of 1524 m.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The problem with such a “mile” is that, as I already mentioned, the furlong of 660 ft was a long-established unit, and it would be a lot more convenient if a mile were a whole number of furlongs.  So the practical choice was either to round a mile down to 7 furlongs (1408 m) or up to 8 furlongs (1609 m).  I suspect that 8 was chosen to allow a mile to easily divide into halves or quarters, in contrast to the inconveniently-prime 7.&lt;/p&gt;&#xA;&lt;p&gt;Since the mile's definition was dependent on the furlong, your question then becomes why the furlong is the length it is (201.168 m).  Conceptually, a furlong is a “furrow long”, i.e., the length of a furrow in an agricultural field, determined by the distance that a team of oxen could plow before they had to be rested and start a new furrow.&lt;/p&gt;&#xA;&lt;p&gt;But why was a furlong &lt;em&gt;standardized&lt;/em&gt; to be the length that it is?  Could it have been meant as some convenient fraction of a meridian?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earth's polar circumference&lt;/a&gt; has been measured as 40 007 863 meters.  (Based on the original conceptual definition of the meter, it should be exactly 40 million, but the survey had an error of 197 PPM.)  This works out to 198 877.868 furlongs (based on the International Yard and Pound Agreement of 1959).&lt;/p&gt;&#xA;&lt;p&gt;To which you might point out, “Hey, that's really close to a nice round 200 000.  Clearly that was intentional, and so a foot was &lt;em&gt;supposed&lt;/em&gt; to be 1/132M of the Earth's circumference (303.09 mm).”&lt;/p&gt;&#xA;&lt;p&gt;But by the same reasoning, you can point out that a Roman foot (&lt;em&gt;pes&lt;/em&gt;) was almost exactly the length of a sheet of A4 paper.  Does that mean that the Romans independently discovered ISO paper sizes, or is it just a coincidence?&lt;/p&gt;&#xA;" OwnerUserId="15620" LastActivityDate="2021-11-02T23:41:07.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13846" PostTypeId="1" AcceptedAnswerId="13861" CreationDate="2021-11-03T02:15:06.303" Score="3" ViewCount="151" Body="&lt;p&gt;I know that the great Galileo made no real progress measuring the speed of light -- he disappointingly suggested that it might be infinite. I read that he concluded (based on his attempts to measure it with lanterns) that it is at least ten times the speed of sound (which he may have had a value for). Romer in 1676 brilliantly (no word play intended) measured the speed within 20 percent of the true figure observing Jovian moons and shadows at different times of the year, I think. My question is, in 1675, when Newton was over thirty, what did he think? (Or what did &lt;em&gt;any&lt;/em&gt; other scientist of that time think?) Would he have already rejected not just infinite speed but also understand that 10 times that of sound was too low by a long shot? I have never encountered him suggested an experiment to measure the speed or an idea about this speed even though he thought about light as much as any human had.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2021-11-06T12:08:16.707" Title="Ideas about the speed of light between Galileo and Romer?" Tags="|physics|newton|measurement|light|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13847" PostTypeId="2" ParentId="13838" CreationDate="2021-11-03T02:36:18.067" Score="2" Body="&lt;p&gt;I don't know what the reasons were for the people who decided it, but one thing that makes the right angle special is that it is, in some sense, the &amp;quot;most&amp;quot; angle there is. If you have two intersecting lines, the farthest from parallel they can be is perpendicular. Once you make the angle between them more than a right angle, you're just making the other angle between them smaller. For instance, consider a stick. If it starts lying flat, and you increase the angle it makes with the ground, the greatest angle it can make is a right angle. Once it reach a right angle, rotating it further just brings it closer to the ground.&lt;/p&gt;&#xA;&lt;p&gt;Another fact that might be relevant is that the metric system is also built around a right angle of the Earth: a meter was originally defined as 1/10,000,000 of the arc length from the equator to the North pole, or 100 gradians of arc length of the Earth. This makes one gradian of latitude equal to 100,000 meters (note that, again, 100 gradians is the maximum when it comes to latitude). While subdivisions of degrees have traditionally been done in powers of 60 (a minute is 1/60 of a degree, a second is 1/60 of a minute), the metric system would have subdivisions be based on powers of ten. So a meter could be described as 10 microgradians of latitude.&lt;/p&gt;&#xA;" OwnerUserId="9468" LastActivityDate="2021-11-03T02:36:18.067" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13848" PostTypeId="2" ParentId="9931" CreationDate="2021-11-03T06:21:23.123" Score="1" Body="&lt;p&gt;Writing about two years after you posted this question, Graham Wild, a Senior Lecturer in Aviation at UNSW Canberra, posted a preprint to the physics education section of arXiv that attempts to answer this: &lt;a href=&quot;https://arxiv.org/abs/2110.00690&quot; rel=&quot;nofollow noreferrer&quot;&gt;[2110.00690] On the Origins and Relevance of the Equal Transit Time Fallacy to Explain Lift&lt;/a&gt;. He traces the misunderstanding all the way back to D'Alembert in 1744, who simply &lt;em&gt;assumed&lt;/em&gt; equal transit time (ETT). Later work, in 1752, purported to derive the ETT property from Newton's laws, applied to fluids. This incorrect derivation was subsequently confirmed by Euler in 1757, lending authority to it. An ETT result does hold for potential flows, but such flows do not produce lift. Wild's paper (which I have not fully digested) discusses whether ETT can nonetheless play a role in helping to understand the origins of lift, leaving it as an open question.&lt;/p&gt;&#xA;" OwnerUserId="15622" LastActivityDate="2021-11-03T06:21:23.123" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13849" PostTypeId="2" ParentId="13838" CreationDate="2021-11-03T07:57:14.407" Score="10" Body="&lt;p&gt;I am answering the second question of yours: &amp;quot;Out of all possible angles, why is the right angle particularly special?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;An angle is something that exists on a plane which is a 2D system. In fact, for a system of defining points on a plane, we need two axes (defined by two unit vectors) which are nonparallel and thus inclined to each other by an angle. In the matter of coordinate systems, the Cartesian coordinate system is most popular -- where the axes are at right angles to each other. Right angle is special in this case because the axes become 'orthogonal' to each other i.e.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Moving a point on the plane in the direction of one axis does not make it move along the direction of the other axis.&lt;/li&gt;&#xA;&lt;li&gt;A point's coordinate on one axis can be represented by that point's distance from the other axis (along with a sign for direction).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Both of the above won't be true for axes at an acute or obtuse angle to each other.&lt;/p&gt;&#xA;&lt;p&gt;The 'dot product' of two vectors is zero if they are at right angle to each other. Dot product signifies interdependence in a sense – projection of one thing in the other's direction. &lt;strong&gt;Being independent&lt;/strong&gt; is something of a trivial case, hence right angle is important here.&lt;/p&gt;&#xA;" OwnerUserId="15623" LastEditorUserId="15623" LastEditDate="2021-11-05T06:19:19.447" LastActivityDate="2021-11-05T06:19:19.447" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13852" PostTypeId="1" AcceptedAnswerId="13853" CreationDate="2021-11-03T22:08:58.627" Score="0" ViewCount="273" Body="&lt;p&gt;Frobenius's theorem states that the only finite-dimensional, associative division algebras over &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; are: &lt;span class=&quot;math-container&quot;&gt;$\mathbb R, \mathbb C, \mathbb H$&lt;/span&gt; (where the last of these are the quaternions). So one might be led to think that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Hamilton discovered the notion of a finite-dimensional unital algebra over &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;. These things are sometimes called hypercomplex number systems.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hamilton discovered that some of these algebras are associative, and some are not.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hamilton discovered that some algebras are division algebras, while most are not.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Putting these together, Hamilton might have set about to find a larger finite-dimensional, associative, division algebra than &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, and eventually arrived at &lt;span class=&quot;math-container&quot;&gt;$\mathbb H$&lt;/span&gt;. This would be quite impressive as I &lt;em&gt;think&lt;/em&gt; that the above three notions did not get studied until after Hamilton presented the quaternions as the first non-trivial hypercomplex number system.&lt;/p&gt;&#xA;&lt;p&gt;You see, some explanation is badly needed for why Hamilton dismissed straightforward generalisations of &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt; like &lt;span class=&quot;math-container&quot;&gt;$\mathbb C \oplus \mathbb R$&lt;/span&gt; in his project to generalise &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, and instead settled on a &lt;em&gt;non-commutative&lt;/em&gt; 4-dimensional algebra whose defining relation is the arguably cryptic &lt;span class=&quot;math-container&quot;&gt;$i^2 = j^2 = k^2 = ijk = -1$&lt;/span&gt;. What criteria was he looking to satisfy?&lt;/p&gt;&#xA;&lt;p&gt;PS: &lt;em&gt;A bonus question would be to explain how Gauss independently discovered the quaternions. Should this be asked as a separate question?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="15627" LastEditorUserId="4251" LastEditDate="2022-09-08T06:20:24.887" LastActivityDate="2022-09-08T06:20:24.887" Title="What set of criteria led Hamilton to discover the quaternions?" Tags="|mathematics|abstract-algebra|quaternions|" AnswerCount="4" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="13853" PostTypeId="2" ParentId="13852" CreationDate="2021-11-03T22:45:37.990" Score="0" Body="&lt;p&gt;In the paper &lt;em&gt;Hamilton's Discovery of Quaternions&lt;/em&gt; by &lt;em&gt;B. L. VAN DER WAERDEN&lt;/em&gt;, the author lists the criteria Hamilton was looking to satisfy from his algebra:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They must define a finite-dimensional unital algebra over &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;. This led him to adopt the notation &lt;span class=&quot;math-container&quot;&gt;$a + bi + cj$&lt;/span&gt; for the elements of such an algebra.&lt;/li&gt;&#xA;&lt;li&gt;Assume a hypothetical 3-dimensional number system where the numbers are of the form &lt;span class=&quot;math-container&quot;&gt;$z = a + bi + cj$&lt;/span&gt;. Hamilton defined &lt;span class=&quot;math-container&quot;&gt;$|z|$&lt;/span&gt; to be &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a^2 + b^2 + c^2}$&lt;/span&gt; by generalising the same operation over the complex numbers. He wanted this operation to satisfy the identity &lt;span class=&quot;math-container&quot;&gt;$|wz| = |w| |z|$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The combination of these criteria define precisely the &lt;em&gt;normed unital algebras&lt;/em&gt;. By Hurwitz's theorem, the only algebras satisfying these criteria are &lt;span class=&quot;math-container&quot;&gt;$\mathbb R, \mathbb C, \mathbb H, \mathbb O$&lt;/span&gt;, where the last symbol denotes the octonions.&lt;/p&gt;&#xA;" OwnerUserId="15627" LastEditorUserId="15627" LastEditDate="2021-11-04T22:44:33.247" LastActivityDate="2021-11-04T22:44:33.247" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13854" PostTypeId="2" ParentId="13852" CreationDate="2021-11-03T22:46:35.077" Score="2" Body="&lt;p&gt;You are right when you &amp;quot;think that the above three notions did not get studied until after Hamilton presented the quaternions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Hamilton wanted to generalize complex numbers to have a tool to represent rotations of 3-space, similar to representation of rotations of the plane&#xA;by complex numbers of absolute value 1. Since rotations in 3 space do not commute, it was clear that commutativity of multiplication had to be rejected. He found first that there is no reasonable generalization in dimension 3, and then found a proper generalization in dimension 4.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-11-03T22:46:35.077" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13857" PostTypeId="1" CreationDate="2021-11-04T07:01:05.677" Score="7" ViewCount="692" Body="&lt;p&gt;There are at least two articles about Ibn Al-Haytham in Encyclopedia &lt;a href=&quot;https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/ibn-al-haytham-abu&quot; rel=&quot;noreferrer&quot;&gt;First&lt;/a&gt; and &lt;a href=&quot;https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/ibn-al-haytham&quot; rel=&quot;noreferrer&quot;&gt;Second&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Both these articles have one major difference that is according to the First article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Light of the Stars (III 48). This argues that all stars and planets, with the sole exception of the moon, are self-luminous&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Light of the Moon (III 6). Ibn al-Haytham showed here that if the moon behaved like a mirror, the light it receives from the sun would be reflected at a given point on the earth from a smaller part of its surface than is actually observed. He accordingly argued that the moon sends out its borrowed light in the same manner as a self-luminous source, that is, from every point on its surface in all directions. This is confirmed through the use of an astronomical diopter having a slit of variable length through which various parts of the moon could be viewed from an opposite hole in a screen parallel to the slit. The treatise is a beautiful combination of mathematical deduction and experimental techniques. The experiments do not, however, lead to the discovery of a new property, but only serve to prove that the mode of emission from the moon is of the same kind as the already known mode of emission from self-luminous objects. Here, as in the Optics, the role of the experiment is in contrast to its role in the work of, say, Grimaldi or Newton.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So here we see that Ibn Al- Haytham view the moon as none Self-luminous Object same thing I found in this &lt;a href=&quot;https://books.google.co.in/books?id=k2yYDwAAQBAJ&amp;amp;pg=PA21&amp;amp;lpg=PA21&amp;amp;dq=Islamic%20medieval%20astronomer%20on%20moon%20receives%20light%20from%20the%20sun%20book&amp;amp;source=bl&amp;amp;ots=OqLQ83IrGk&amp;amp;sig=ACfU3U0BoIVUSoWsQK0Rl2n1DeNkLfZNow&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjR97Gr_PvzAhUBA3IKHUFBDSEQ6AF6BAgREAM#v=onepage&amp;amp;q=Islamic%20medieval%20astronomer%20on%20moon%20receives%20light%20from%20the%20sun%20book&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;book&lt;/a&gt; and this &lt;a href=&quot;https://www.jstor.org/stable/4025317?read-now=1&amp;amp;refreqid=excelsior%3A5ef2acaf0b66e713feccd1adb7427d48&amp;amp;seq=1#page_scan_tab_contents&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt; in the journal &lt;strong&gt;BUT&lt;/strong&gt;&#xA;The second article says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This problem al-Hasan takes up again in On the Light of the Moon, in which he refutes the ancient theory that the moon reflects the sun's light like a mirror. Rather he believes that the moon is a self-illuminating body because each point on its surface broadcasts light rays in all directions, whereas each point on the surface of a mirror reflects a light ray from a single source (here the sun) in only one direction. However, he further believes that the eye receives two primary impressions in the act of vision: light and colour. Therefore he concludes that only some physical effect of the sun's light rays on the moon renders the latter's colour (and thereby it's light) visible&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This article says Moon as a self-illuminating body&#xA;How can this be possible for a person to write in one book moonlight is from the sun and other self-illuminating? I try to find the On the Light of Moon book which I get and here it's the conclusion:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Xh2n7.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Xh2n7.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt; &lt;em&gt;&lt;strong&gt;caution&lt;/strong&gt;&lt;/em&gt; it's translated from Arabic to Urdu to English if someone has Arabic to English please provide the link&lt;/p&gt;&#xA;&lt;p&gt;Can someone puts some light like the sun puts on earth?&lt;/p&gt;&#xA;" OwnerUserId="15534" LastEditorUserId="10106" LastEditDate="2022-01-28T14:53:22.910" LastActivityDate="2022-01-28T14:53:22.910" Title="Did Ibn Al-Haytham believe that the Moon reflects sunlight or that it is self-luminous?" Tags="|physics|astronomy|middle-ages|islamic-science|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13858" PostTypeId="1" CreationDate="2021-11-04T10:41:04.667" Score="3" ViewCount="643" Body="&lt;p&gt;I'm a huge fan of the history of Algebra and, recently, I've noticed a bit of an oddity. Degree one equations have been known (and solved) for as long as human history. For degree two equations, we have also known the general solution since many, many years BC, since pretty much the moment people have begun to actually study and develop mathematics.&lt;/p&gt;&#xA;&lt;p&gt;However, when it comes to the cubic, it took literal millennia for a solution to be found, only happening around 1500 AD in Italy. One would perhaps assume that the quartic would, therefore, take some considerable time to solve (maybe not millennia, since math's growth has been more or less exponential, but many years nonetheless). Not only was this not the case, the quartic was solved in the exact same timeframe as the cubic, with a couple years removed.&lt;/p&gt;&#xA;&lt;p&gt;So my question is: &lt;strong&gt;why&lt;/strong&gt; did it take so long to solve the cubic when it took so little to solve the quartic?&lt;/p&gt;&#xA;&lt;p&gt;The solution of the cubic stuck around as an open problem for far longer than the other degree polynomial equations, and I just can't see a reason for this... Maybe the real question is &amp;quot;why was the quartic so easy to solve&amp;quot;, which I am also curious about.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any insights as for why this is, or any sources for further reading? I've read a few Math History books, but none of them could clarify my question.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="14787" LastActivityDate="2023-10-22T20:17:34.970" Title="Why was the cubic specifically so hard to solve?" Tags="|mathematics|discoveries|elementary-algebra|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13859" PostTypeId="2" ParentId="13852" CreationDate="2021-11-04T11:43:21.217" Score="3" Body="&lt;p&gt;When I have time later I might return to this and try to provide additional exposition directly addressing your question, maybe quote something relevant from Crowe's &lt;a href=&quot;https://www.amazon.com//dp/0486679101&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A History of Vector Analysis&lt;/strong&gt;&lt;/a&gt; (a book I've been reading the past 2 months; I'm now a little more than half-way through it). However, for now I thought it would be useful to provide some older books (and a couple of papers by Hamilton) to look at. As for the books, besides glancing through their table of contents, reading their prefaces, etc., I also recommend reading reviews of these books, which are easy to find by googling the author's last name and the book's title in google-books (date-restrict from the publication date to about 5 years later). I’ve included links to the book reviews I know about.&lt;/p&gt;&#xA;&lt;p&gt;Every link below is freely available on the internet, at least where I am. Probably all of the books are also freely available online at the Internet Archive (easiest way: google the book's title, maybe author's last name if the title is fairly generic, along with the &amp;quot;word&amp;quot; archive.org), but probably not all of the reviews can be found elsewhere online (freely or otherwise). If you really want to dive into the older literature on quaternions, you’ll want to make use of Macfarlane’s &lt;strong&gt;Bibliography of Quaternions and Allied Systems of Mathematics&lt;/strong&gt; (1904).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1843&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=9WTtoOfnUWoC&amp;amp;pg=PA424&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;On a new species of imaginary quantities connected with a theory of quaternions&lt;/em&gt;&lt;/a&gt; by Hamilton&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1844&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=B7xJAAAAcAAJ&amp;amp;pg=RA1-PA1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Additional researches in the theory of quaternions&lt;/em&gt;&lt;/a&gt; by Hamilton&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1853&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=Ein9159_XecC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Lectures on Quaternions&lt;/strong&gt;&lt;/a&gt; by Hamilton&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=QjoAAAAAYAAJ&amp;amp;pg=PA223&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The North American Review&lt;/em&gt;&lt;/a&gt; (a review known as one of the most over-the-top-with-praise review ever of a math book)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1866&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=b2stAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elements of Quaternions&lt;/strong&gt;&lt;/a&gt; by Hamilton&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=SELRHNvOfcQC&amp;amp;pg=PA308&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Philosophical Magazine&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1867&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=PpIKAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on Quaternions&lt;/strong&gt;&lt;/a&gt; by Tait&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1873&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=TwkXAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on Quaternions&lt;/strong&gt;&lt;/a&gt; by Tait (2nd edition)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=61AwAAAAIAAJ&amp;amp;pg=PA364&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Philosophical Magazine&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1876&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=tFEDAAAAQAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Introduction to Quaternions&lt;/strong&gt;&lt;/a&gt; by Kelland/Tait&lt;/p&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=JrhCAQAAMAAJ&amp;amp;pg=PA137&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1881&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=uc83AAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elements of Quaternions&lt;/strong&gt;&lt;/a&gt; by Hardy&lt;/p&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=eiMbAQAAIAAJ&amp;amp;pg=PA372&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Ohio Educational Monthly&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=gPEBAAAAYAAJ&amp;amp;pg=PA521&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Wisconsin Journal of Education&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://www.jstor.org/stable/2900879&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Science&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=s1pDAQAAMAAJ&amp;amp;pg=PA255&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Westminster Review&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=KCIDAAAAMBAJ&amp;amp;pg=PA708&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Popular Science Monthly&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1882&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=xK0AAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Introduction to Quaternions&lt;/strong&gt;&lt;/a&gt; by Kelland/Tait (2nd edition)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1890&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=CGZLAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;An Elementary Treatise on Quaternions&lt;/strong&gt;&lt;/a&gt; by Tait (3rd edition)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1894&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=MkQ6AAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;The Outlines of Quaternions&lt;/strong&gt;&lt;/a&gt; by Hime&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=JPRBAAAAYAAJ&amp;amp;pg=PA297&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Minutes of Proceedings of the Royal Artillery Institution&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=ar8gAQAAMAAJ&amp;amp;pg=PA267&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Educational Times&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=7CfWAAAAMAAJ&amp;amp;pg=PA16&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematical Gazette&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=yFU_AQAAMAAJ&amp;amp;pg=PA196&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Academy&lt;/em&gt;&lt;/a&gt; (begins middle of middle column); review in &lt;a href=&quot;https://books.google.com/books?id=TZAOAAAAIAAJ&amp;amp;pg=PA499&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Philosophical Magazine&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=f6FFAAAAYAAJ&amp;amp;pg=PA76&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=RHHxAAAAMAAJ&amp;amp;pg=RA1-PA34&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;American Mathematical Monthly&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=BzRDAQAAMAAJ&amp;amp;pg=PA699&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Science&lt;/em&gt;&lt;/a&gt; (begins middle of right column)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1896&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=kpxRAAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Primer of Quaternions&lt;/strong&gt;&lt;/a&gt; by Hathaway&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=3TDZAAAAMAAJ&amp;amp;pg=PA113&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Literary Era&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=rR_OAAAAMAAJ&amp;amp;pg=PA421&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Physical Review&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=BzRDAQAAMAAJ&amp;amp;pg=PA700&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Science&lt;/em&gt;&lt;/a&gt; (top-most paragraph on left column of p. 700); review in &lt;a href=&quot;https://books.google.com/books?id=EqZIAAAAMAAJ&amp;amp;pg=PA112&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Electrical Engineering&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1899&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=Q5IKAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elements of Quaternions&lt;/strong&gt;&lt;/a&gt; by Hamilton (2nd edition, Volume I, edited by Joly)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1901&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=RJIKAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elements of Quaternions&lt;/strong&gt;&lt;/a&gt; by Hamilton (2nd edition, Volume II, edited by Joly)&lt;/p&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=UMkKAAAAYAAJ&amp;amp;pg=PA387&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt; (Vol. I); review in &lt;a href=&quot;https://books.google.com/books?id=WXwvAQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Philosophical Magazine&lt;/em&gt;&lt;/a&gt; (Vol. I &amp;amp; II); review in &lt;a href=&quot;https://books.google.com/books?id=_X8CAAAAYAAJ&amp;amp;pg=PA65&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Science&lt;/em&gt;&lt;/a&gt; (Vol. I &amp;amp; II); review in &lt;a href=&quot;https://books.google.com/books?id=7soKAAAAYAAJ&amp;amp;pg=PA206&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt; (Vol. II)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1904&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=ny87AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Introduction to Quaternions&lt;/strong&gt;&lt;/a&gt; by Kelland/Tait (3rd edition; prepared by Knott)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=33YCAAAAIAAJ&amp;amp;pg=PA604&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=iLJNAQAAMAAJ&amp;amp;pg=PA284&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Cambridge Review&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=ulDWAAAAMAAJ&amp;amp;pg=PA42&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematical Gazette&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1904&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=CkE6AAAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Bibliography of Quaternions and Allied Systems of Mathematics&lt;/strong&gt;&lt;/a&gt; by Macfarlane&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=y5NFAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=9k8EAAAAYAAJ&amp;amp;pg=PA758&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Philosophical Magazine&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://www.ams.org/journals/bull/1906-13-01/S0002-9904-1906-01411-2/S0002-9904-1906-01411-2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Bulletin of the American Mathematical Society&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;1905&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://books.google.com/books?id=AC8PAAAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;A Manual of Quaternions&lt;/strong&gt;&lt;/a&gt; by Joly&lt;/p&gt;&#xA;&lt;p&gt;review in &lt;a href=&quot;https://books.google.com/books?id=maDxAAAAMAAJ&amp;amp;pg=PA229&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematical Gazette&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://www.ams.org/journals/bull/1905-11-10/S0002-9904-1905-01259-3/S0002-9904-1905-01259-3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Bulletin of the American Mathematical Society&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=eZVFAAAAYAAJ&amp;amp;pg=PA121&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Nature&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=SRQLAAAAIAAJ&amp;amp;pg=PA237&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The School World&lt;/em&gt;&lt;/a&gt;; review in &lt;a href=&quot;https://books.google.com/books?id=R3xGAQAAMAAJ&amp;amp;pg=PA474&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Technics&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2022-01-27T08:05:35.593" LastActivityDate="2022-01-27T08:05:35.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13860" PostTypeId="2" ParentId="13858" CreationDate="2021-11-05T03:29:48.467" Score="7" Body="&lt;p&gt;There is nothing special about cubic. I think the right question is: why the &lt;em&gt;quadratic&lt;/em&gt; equations are much easier to solve than cubic and quartic? Well it's pretty obvious, is not it? Quadratic is indeed very easy to solve. There is a natural way to explain intuition behind the solution and there is nice geometrical interpretation. Cubic and quartic are just on the next level of difficulty.&lt;/p&gt;&#xA;&lt;p&gt;If you want a more deep answer, then I can argue that one needs symbolic algebra to solve cubic and  quartic, but to solve quadratic equation one doesn't actually need algebra (although today in schools we teach quadratic using algebra, Babylonians did not know algebra, but can solve problems that involves quadratic equations). It's not coincidence that symbolic algebra was developed in XVI century by the same people that contributed to solving cubic and quartic equations.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2021-11-05T03:29:48.467" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13861" PostTypeId="2" ParentId="13846" CreationDate="2021-11-05T10:00:06.973" Score="6" Body="&lt;p&gt;I do not think it is a virtue to make unsupported assertions just because we happen to believe them &lt;em&gt;now&lt;/em&gt;. Following available evidence does not makes one not smart or disappointing, that is how science should be done. Galileo deserves credit for moving the question from armchair speculations to observations and experiments, and honestly surmising what they suggested at the time.  Kepler and Descartes came to the same conclusion of infinite speed around the same time based on observations of lunar eclipses and refraction experiments, see &lt;a href=&quot;https://interestingengineering.com/a-brief-history-of-the-speed-of-light&quot; rel=&quot;nofollow noreferrer&quot;&gt;McFadden, A Brief History of the Speed of Light&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Things changed by the time of Newton. He does not refer to Romer explicitly, but asserts finite speed of light in &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookI-XIV&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Principia&lt;/em&gt;, Book I, Section XIV&lt;/a&gt; based on more precise observations of Jupiter's satellites (veiled reference to Romer's measurements) and Grimaldi's diffraction experiments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For it is now certain from the phenomena of Jupiter's Satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel from the sun to the earth. Moreover, the rays of light that are in our air (as lately was discovered by Grimaldus, by the admission of light into a dark room through a small hole, which I have also tried) in their passage near the angles of bodies, whether transparent or opaque (such as the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are bent or inflected round those bodies as if they were attracted to them...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But Newton accepted action at a distance for gravity based on astronomical evidence, despite his metaphysical dislike of it, as did Laplace and others a century later. The speed of electricity was also widely surmised to be infinite until the 19th century, because that is what experiments done before that suggested, see &lt;a href=&quot;https://hsm.stackexchange.com/a/11279/55&quot;&gt;Was it suspected that the speed of electricity was equal to the speed of light?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-06T12:08:16.707" LastActivityDate="2021-11-06T12:08:16.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13863" PostTypeId="2" ParentId="13857" CreationDate="2021-11-06T00:43:54.720" Score="7" Body="&lt;p&gt;English translations of Ibn Al-Haytham's optical treatises are available from &lt;a href=&quot;http://www.jphogendijk.nl/ih/Hamdard1.pdf&quot; rel=&quot;noreferrer&quot;&gt;Proceedings of the Celebrations of 1000th Anniversary&lt;/a&gt; by the Hamdard National Foundation of Pakistan. The mismatch in the OP quotes is not surprising as Ibn Al-Haytham's text is very confusing to a point of apparent inconsistency, and it is hard to understand what his ultimate conclusion is. In The Light of The Moon he starts by listing three possibilities, and then appears to reject each one of them in turn. Part of it might be translation issues, says the editor:&amp;quot;&lt;em&gt;The translators had to wage an unflailing battle&#xA;against the understanding of the mathematical and technical sections of the text... They had also to spell out the text into the language and terminology of modern science&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The OP quotes try to make sense of it based on different portions of the text, and hence come to different conclusions. But, on my reading at least, they simply misinterpret some intermediate arguments as final. The discussion of color, for example, seems to offer it as a means of distinguishing between luminous and reflective bodies, not to suggest that light and color are separate &amp;quot;impressions&amp;quot;. What Ibn Al-Haytham does insist upon is that the Moon  emits light in all directions, &lt;em&gt;as if&lt;/em&gt; it was self-luminous, despite &amp;quot;deriving&amp;quot; it from the Sun, but the mechanism of that is very murky. My best guess is that he is trying to say that the Moon is something like a translucent body brightened by diffused reflection, which he, however, emphatically refuses to call &amp;quot;reflection&amp;quot;, perhaps because this is usually associated with normal reflection by mirrors.&lt;/p&gt;&#xA;&lt;p&gt;The three possibilities listed are: self-luminous body, &amp;quot;&lt;em&gt;the light that falls on any portion of a body from some other source is reflected&lt;/em&gt;&amp;quot;, and &amp;quot;&lt;em&gt;the refracted light is visible on the surface of&#xA;the body. It happens in the case of all transparent&#xA;bodies&lt;/em&gt;&amp;quot;. Later we read&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;when the moon and the sun are opposite to each other at the time of lunar eclipse, the full moon should be visible as a luminous body. A portion of the moon becomes dark only because of the earth coming in between the sun and the moon. This event is the only reason to assert that the moon gets its light from the sun.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the Moon is not self-luminous.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;From the above discussion, it is clear that the light of the moon reaches the earth from every point on its surface at which the light of the sun is falling. However, this light is not the result of reflection.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the Moon does not reflect sunlight. Finally,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;in no case is the moon transparent&#xA;because if the moon is transparent, solar eclipse cannot occur when the moon comes into the middle of the earth and the sun.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the Moon is not transparent either.&lt;/p&gt;&#xA;&lt;p&gt;We get some positive clue from a different treatise, Light Of The Stars, where Ibn Al-Haytham says in passing that &amp;quot;&lt;em&gt;with the exception of the Moon none derives its light from the sun&lt;/em&gt;&amp;quot;. So the Moon, at least, &amp;quot;derives&amp;quot; its light from the sun even if it does not &amp;quot;reflect&amp;quot; it. Let us get back to the the last sentence of The Light of The Moon that gives Ibn Al-Haytham's cryptic conclusion:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In fact, that part of the moon, on which the light of the sun falls becomes bright and due to the atomic property of the moon it looks bright from every point opposite to it&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This &amp;quot;atomic&amp;quot; property is mentioned in only one other place, in yet another treatise, Light:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We come to the conclusion from the above that&#xA;every physical body accepts light. A transparent body&#xA;has the additional property of transmitting the light. This is the atomic aspect which lends a transparent body its brilliance and beauty.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But didn't we already rule out that the Moon is transparent? Well, not quite, &amp;quot;&lt;em&gt;all transparent bodies possess a different degree of transparency which is due to the amount of density present in them. The greater the density of a&#xA;body, the smaller will be its transparency and vice versa&lt;/em&gt;&amp;quot;. Moreover:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Whenever those rays of light fall on any&#xA;transparent body except space, this secondary light is produced in the same way as the dense bodies give rise to this light when they are exposed to the sun. But the secondary light coming from transparent bodies is much weaker than that which issues from those bodies... It must be mentioned here that the secondary light issuing from transparent bodies is not due to the light passing through them; it may be due to those rays of light which enter into them from the opposite direction.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So perhaps the moon is only translucent, to a degree, and &amp;quot;atomic&amp;quot; refers to &amp;quot;fine holes&amp;quot; between &amp;quot;fibers&amp;quot; of such bodies that let some light through. This is the best sense I can make of Ibn Al-Haytham's repeated references to Moon &amp;quot;becoming bright&amp;quot;, &amp;quot;deriving&amp;quot; the light from the Sun, but, emphatically, &lt;em&gt;not&lt;/em&gt; reflecting it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;So we conclude that the bright colour of&#xA;the moon which is visible on the earth is not due to&#xA;reflection but the real reason for that may be that&#xA;light of the sun bright as the moon&lt;/em&gt; [sic!] &lt;em&gt;and because the surface of the moon is not glazed, the real colour of the moon is visible in this light... So it is wrong to think that the part of the moon on which the light of the sun falls does not become bright or the light falling at a place spreads to the other part of the sun&lt;/em&gt; [sic! probably should be &amp;quot;moon&amp;quot;]&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-06T11:04:00.970" LastActivityDate="2021-11-06T11:04:00.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13864" PostTypeId="1" AcceptedAnswerId="13865" CreationDate="2021-11-06T03:43:24.557" Score="3" ViewCount="204" Body="&lt;p&gt;And if so, how did he imagine that light moved at different speeds in different media?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="13914" LastEditDate="2021-11-06T14:09:22.840" LastActivityDate="2021-11-06T14:09:22.840" Title="Does Fermat's Principle of Least Time imply that he believed light moved at a finite speed?" Tags="|light|fermat|refraction|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13865" PostTypeId="2" ParentId="13864" CreationDate="2021-11-06T12:06:49.817" Score="6" Body="&lt;p&gt;Yes, Fermat believed that the speed of light was finite, initially in reaction to the muddle created by Descartes, who wanted to have it both ways, the speed being infinite, yet different in different media, see &lt;a href=&quot;https://www.google.com/books/edition/When_Least_Is_Best/7ugJEAAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Nahin, When Least Is Best, p. 107&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;When Fermat read La Dioptrique he was unimpressed and, as mentioned&#xA;earlier, was blunt in his criticism. He wrote, in part, “of all the&#xA;infinite ways [to analyze the motion of light] the author [Descartes]&#xA;has taken only that one which serves him for his conclusion; he has&#xA;thereby accommodated his means to his end, and we know as little&#xA;about the subject as we did before.”... Further, Fermat rejected as&#xA;nonsense Descartes’ assertion of the infinite speed of light and his&#xA;subsequent illogical argument that light travels faster (than infinity?)&#xA;in water than in air. Fermat’s position was that light traveled at&#xA;a (very fast) finite speed in air, and that it was slowed when traveling&#xA;through a denser (“more resistive”) medium such as water.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But the order of causation was reversed. Fermat &lt;em&gt;first&lt;/em&gt; decided that light traveled at finite speed, and only &lt;em&gt;later&lt;/em&gt; came to the least time principle, when he learned that the Snell's law was experimentally confirmed:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Fermat initially believed that, since Descartes’ derivation was&#xA;clearly (to Fermat) built on sand, then the “ratio of sines is a constant”&#xA;result must be incorrect. Eventually Fermat learned that the&#xA;formula was, in fact, generally accepted as true because it could be&#xA;verified by direct experiment! This greatly puzzled Fermat; how had&#xA;Descartes managed to derive the correct law of refraction from erroneous&#xA;arguments?... With Fermat’s subsequent great discovery of&#xA;the “principle of least time” (discussed later in this chapter) his quest&#xA;ended in 1658 with success... Fermat’s solution to finding a physically correct derivation of Snell’s&#xA;law of refraction was the result of developing a generalization of&#xA;Heron’s derivation of the reflection law.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-11-06T12:06:49.817" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13866" PostTypeId="1" CreationDate="2021-11-06T12:28:51.517" Score="0" ViewCount="95" Body="&lt;p&gt;I wonder, can we meaningfully define zero divisors based on analytic rather than algebraic approach?&lt;/p&gt;&#xA;&lt;p&gt;For instance, if we extend the real numbers with divergent integrals and series, and evaluate the Maclaurin series for &lt;span class=&quot;math-container&quot;&gt;$\ln (x+1)$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$x=-1$&lt;/span&gt;, we will obtain an expansion, equal to the Harmonic series with opposite sign: &lt;span class=&quot;math-container&quot;&gt;$-\sum_{k=1}^\infty \frac1k$&lt;/span&gt;.Also, if we define logarithm as &lt;span class=&quot;math-container&quot;&gt;$\ln x=\int_1^x \frac1x dx$&lt;/span&gt;, the value at &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; will be divergent integral &lt;span class=&quot;math-container&quot;&gt;$-\int_0^1 \frac1xdx$&lt;/span&gt;, which can be shown via Laplace transform rule to be equal to the Harmonic series mentioned above (and also the Harmonic series is its Riemann sum). What if we define the value of &lt;span class=&quot;math-container&quot;&gt;$\ln 0=-\sum_{k=1}^\infty \frac1k=-\int_0^1\frac1x dx$&lt;/span&gt;? One can rise an objection that defining logarithm of zero breaks the equality &lt;span class=&quot;math-container&quot;&gt;$a \ln x=\ln x^a$&lt;/span&gt;, but it already does not work with negative &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thus, if we denote &lt;span class=&quot;math-container&quot;&gt;$\ln 0=-\sum_{k=1}^\infty \frac1k=\lambda$&lt;/span&gt;, then by some logic &lt;span class=&quot;math-container&quot;&gt;$\varepsilon =e^{\lambda/2}$&lt;/span&gt; would be a nilpotent, such that &lt;span class=&quot;math-container&quot;&gt;$\varepsilon^2=0$&lt;/span&gt;, which is similar to the dual numbers unity. It is interesting that the regularized value of Harmonic series is &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt;, and thus the regularized value of &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$-\gamma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As such, I wonder whether anyone considered introduction of nilpotents and logarithm of zero via divergent series?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="2203" LastEditDate="2022-04-28T12:20:28.453" LastActivityDate="2022-04-28T12:20:28.453" Title="Did anyone ever propose an analytic definition of zero divisors, including nilpotents, as opposed to algebraic definition?" Tags="|real-analysis|algebra|zero|logarithm|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13867" PostTypeId="1" CreationDate="2021-11-06T14:13:03.733" Score="1" ViewCount="150" Body="&lt;p&gt;Any account of the precession can be characterized by&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;perturbations due to bodies other than the Sun and&lt;/li&gt;&#xA;&lt;li&gt;a modification of the Binet equation's right-hand side, which is constant for an inverse-square law between point masses if all bodies have spherically symmetric mass distributions.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I'll focus on 2, since 1 was known to be insufficient, and any alternative to one or more Vulcans would have to leverage 2.&lt;/p&gt;&#xA;&lt;p&gt;Treating the force as a Taylor series in &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt;, one can add &lt;span class=&quot;math-container&quot;&gt;$u^3$&lt;/span&gt; etc. terms for spheroid masses. Apsidal precession can come from not only a GR correction which can be described in these terms, &lt;a href=&quot;https://en.wikipedia.org/wiki/Apsidal_precession#Calculation&quot; rel=&quot;nofollow noreferrer&quot;&gt;but also quadrupole and tidal effects&lt;/a&gt;. We now know &lt;a href=&quot;https://en.wikipedia.org/wiki/Binet_equation#Relativistic&quot; rel=&quot;nofollow noreferrer&quot;&gt;GR's &lt;span class=&quot;math-container&quot;&gt;$u^4$&lt;/span&gt; term&lt;/a&gt; is crucial, and a &lt;span class=&quot;math-container&quot;&gt;$u^3$&lt;/span&gt; term in the point-mass force would be constrained by other observations.&lt;/p&gt;&#xA;&lt;p&gt;How did pre-GR non-Vulcan explanations motivate &lt;span class=&quot;math-container&quot;&gt;$u^3$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$u^4$&lt;/span&gt; corrections, be they quadrupole, tidal, or through correcting the point-mass force law? (For example, did any propose Mercury was unusually far from a spherically symmetric density?)&lt;/p&gt;&#xA;" OwnerUserId="5659" LastActivityDate="2021-11-08T02:28:46.643" Title="What pre-GR alternatives to Vulcan were advanced to explain Mercury's perihelion precession?" Tags="|astronomy|gravity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13868" PostTypeId="1" CreationDate="2021-11-06T20:41:52.310" Score="1" ViewCount="248" Body="&lt;p&gt;In Mathematics, it is common to use &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; for height in various languages, including those whose word for height does not start with h. Why is that?&lt;/p&gt;&#xA;" OwnerUserId="15656" LastEditorUserId="10106" LastEditDate="2021-11-07T04:12:42.707" LastActivityDate="2021-11-07T04:12:42.707" Title="Why is “h” used for height?" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="6" ClosedDate="2021-11-07T08:21:35.463" ContentLicense="CC BY-SA 4.0" />
  <row Id="13869" PostTypeId="1" AcceptedAnswerId="13870" CreationDate="2021-11-07T03:01:23.923" Score="9" ViewCount="1200" Body="&lt;p&gt;I remember reading somewhere (perhaps in &lt;em&gt;The Mathematical Experience&lt;/em&gt;) that Richard Courant wrote something to the effect that, without applications to guide the river of mathematical discovery, mathematical research is doomed to wander aimlessly. There was indeed some river imagery, if I remember correctly. What is this quotation exactly, and where did it come from?&lt;/p&gt;&#xA;" OwnerUserId="3811" LastEditorUserId="4251" LastEditDate="2021-11-08T07:17:02.437" LastActivityDate="2022-05-18T03:32:21.157" Title="What was Richard Courant's saying about mathematical research apart from applications?" Tags="|mathematics|mathematicians|reference-request|quote|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13870" PostTypeId="2" ParentId="13869" CreationDate="2021-11-07T04:08:07.840" Score="10" Body="&lt;p&gt;There is river imagery in a passage from the preface (written by Courant) to &lt;a href=&quot;https://www.google.com/books/edition/Methods_of_Mathematical_Physics_Volume_1/nRAJAQAAIAAJ?hl=en&quot; rel=&quot;noreferrer&quot;&gt;Courant-Hilbert's Methods of Mathematical Physics, vol. 1&lt;/a&gt;, but the point of the metaphor is somewhat different. It is excessive specialization rather than disconnect from applications that is seen as the danger, and it is drying out research rather then aimlessness that is feared:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Since the seventeenth century, physical intuition has served as a vital source for mathematical porblems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from their roots of mathematics in intuition, have concentrated on refinement and emphasized the postulated side of mathematics, and at other times have overlooked the unity of their science with physics and other fields. In many cases, physicists have ceased to appreciate the attitudes of mathematicians. This rift is unquestionably a serious threat to science as a whole; &lt;strong&gt;the broad stream of scientific development may split into smaller and smaller rivulets and dry out.&lt;/strong&gt; It seems therefore important to direct our efforts towards reuniting divergent trends by classifying the common features and interconnections of many distinct and diverse scientific facts. Only thus can the student attain some mastery of the material and the basis be prepared for further organic development of research.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is not so say that Courant did not see  applications as vital to mathematics. I looked through &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387946705&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reid's biography of Courant&lt;/a&gt;, and there is plenty of material on that, but I did not find any similar sounding quote. You may also want to look at &lt;a href=&quot;https://www.scientificamerican.com/article/mathematics-in-the-modern-world/&quot; rel=&quot;noreferrer&quot;&gt;Courant's article Mathematics in the Modern World&lt;/a&gt; in Scientific American (it is behind paywall).&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-08T10:15:16.100" LastActivityDate="2021-11-08T10:15:16.100" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13871" PostTypeId="2" ParentId="13869" CreationDate="2021-11-07T04:10:33.183" Score="6" Body="&lt;p&gt;I quote from the book Physically Speaking A Dictionary of Quotations on Physics and Astronomy by Carl C Gaither. In the entry for Courant, there is  discussion of rivulets drying up...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Courant, Richard&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since the seventeenth century, physical intuition has&#xA;served as a vital source for mathematical problems and methods. Recent&#xA;trends and fashions have, however, weakened the connection between&#xA;mathematics and physics; mathematicians, turning away from the roots&#xA;of mathematics in intuition, have concentrated on refinement and&#xA;emphasized the postulated side of mathematics, and at times have&#xA;overlooked the unity of their science with physics and other fields.&#xA;In many cases, physicists have ceased to appreciate the attitudes of&#xA;mathematicians. &lt;strong&gt;This rift is unquestionably a serious threat to&#xA;science as a whole; the broad stream of scientific development may&#xA;split into smaller and smaller rivulets and dry out&lt;/strong&gt;. It seems&#xA;therefore important to direct our efforts toward reuniting divergent&#xA;trends by classifying the common features and interconnections of many&#xA;distinct and diverse scientific facts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Methods of Mathematical Physics (pp. v-vi)&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-11-07T04:10:33.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13872" PostTypeId="2" ParentId="13867" CreationDate="2021-11-08T02:28:46.643" Score="5" Body="&lt;p&gt;The idea of a hypothetical planet 'Vulcan' with an orbit closer to the Sun than Mercury's orbit has been given excessive prominence in secondary and popular accounts.  These accounts tend to produce a distorted narrative of what took place in the astronomical debates over the problem of apparent excess in the motion of Mercury's perihelion.&lt;/p&gt;&#xA;&lt;p&gt;U J J Le Verrier first reported the Mercury perihelion problem to the Paris Academy of Sciences in 1859 (ref.1), recommending the attention of other astronomers to this serious astronomical problem. At that time, he indicated that potential explanation by an intra-mercurial planet had difficulties, and was an unlikely solution to the problem (although he was soon afterwards induced to change his mind for a time, see below).&lt;/p&gt;&#xA;&lt;p&gt;In the same paper and at the same time he also discussed classes of conceivable explanation. The possibilities that he considered all involved extra gravitational mass present in the solar system but not yet accounted for.&lt;/p&gt;&#xA;&lt;p&gt;One class of conceivable explanation would put the extra unaccounted mass somewhere farther out from the Sun than Mercury's orbit. Le Verrier discussed for that case the possibility that the mass of Venus had been underestimated. But he reported that if Venus' mass-estimate were adjusted to account for Mercury's perihelion motion, it would have to be increased by about 10% of its existing estimate, and if true, that increase would produce much extra perturbation on the motion of the earth, something that could not be reconciled with available observations of the earth's perturbations (measured by solar  observations). So Le Verrier did not see any adjustment of the mass-estimate of Venus as a viable explanation or solution for the Mercury problem.&lt;/p&gt;&#xA;&lt;p&gt;Then the other main class of conceivable explanation for Le Verrier would put the extra unaccounted mass somewhere nearer to the Sun than Mercury's orbit. Le Verrier recognised that a simple form of this class of explanation could involve an intra-mercurial planet. But in 1859 he did not consider this explanation to be likely, because such a planet ought to be bright, and thus, even if too close to the Sun to be usually seen, it ought still to be occasionally noticeable at solar eclipses and/or by occasional passages across the sun's disk -- and nothing of the kind had been observed.&lt;/p&gt;&#xA;&lt;p&gt;The possibility that did appeal to Le Verrier in his paper of 1859, as a possible explanation that would avoid such difficulties, was not an intra-mercurial planet, but rather a 'series of corpuscles with orbits between Mercury and the Sun' ('une série de corpuscules circulant entre Mercure et le Soleil' ref.1, p.382), forming some kind of ring (ref.1 p.383).&lt;/p&gt;&#xA;&lt;p&gt;(Le Verrier's contemporary 'Théorie du mouvement de Mercure' (1859, Annales de l'Observatoire de Paris, v.5, at pp.105-6) examined the observations in much greater detail before expressing the same conclusion, that an intra-Mercurial planet is unlikely and that more likely was a ring of matter closer to the Sun than Mercury's orbit.)&lt;/p&gt;&#xA;&lt;p&gt;In spite of le Verrier's doubts of 1859, the idea of an intra-Mercurial planet attracted enthusiasm, and shortly afterwards, a French amateur astronomer Lescarbeault claimed to have seen it.  Le Verrier for a time thought this observation plausible.  But not only were there no credible corroborations: the credibility of Lescarbeault's report itself was also very soon undermined by another French observer, Liais, who in 1860 made public that he had been closely studying the sun over the same time-period as that of Lescarbeault's claimed observation, and he had seen nothing of the kind. The idea of an intramercurial planet lost credibility in serious discussions (discussed in ref.2 (Roseveare, 1982) at p.25-6, also in ref.4.)&lt;/p&gt;&#xA;&lt;p&gt;Closely related to Le Verrier's idea of a ring of 'corpuscles' was&lt;/p&gt;&#xA;&lt;p&gt;-- Seeliger's (1906) hypothesis of bands of diffuse intra-Mercurial matter, of which the possible existence received some observational corroboration through the zodiacal light -- discussed in ref.2.&lt;/p&gt;&#xA;&lt;p&gt;Other hypotheses included&lt;/p&gt;&#xA;&lt;p&gt;-- solar oblateness (quantitatively insufficient to produce the extra perihelion motion)&lt;/p&gt;&#xA;&lt;p&gt;-- a short-lived idea that the exponent in the inverse-square law should be adjusted upwards slightly from 2 to account for Mercury's perihelion motion: this was proposed 1894 by Asaph Hall, modifying an earlier suggestion of Simon Newcomb, but it was refuted by E W Brown (1903) (ref.3), who showed that such an adjusted gravitational law would be incompatible with the observed motion of the moon's perigee. W de Sitter later concurred with Brown's conclusion.&lt;/p&gt;&#xA;&lt;p&gt;These and other possibilities are reviewed and discussed in ref.2&#xA;(N T Roseveare, 1982) and ref.4 (J Earman and M Janssen, 1993).&lt;/p&gt;&#xA;&lt;p&gt;Of the possibilities discussed, @Conifold's comment looks right that the&#xA;'most prominent' of them were the Seeliger hypothesis and the alteration of the exponent in the inverse square law.&lt;/p&gt;&#xA;&lt;p&gt;But none of these pre-relativistic possibilities provided viable explanations of the extra (unaccounted-for) precession of Mercury's&#xA;perihelion. The assessment of all of them in a review by Crommelin in 1920 (ref.5) was that &amp;quot;Asaph Hall's suggestion [to alter the exponent in the law of gravitation] breaks down utterly for the Moon, and every other suggestion introduces as many difficulties as it removes&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;p&gt;1:  U. Le Verrier (1859), &amp;quot;Lettre de M. Le Verrier à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète&amp;quot;, Comptes rendus hebdomadaires des séances de l'Académie des sciences (Paris), vol. 49 (1859), pp. 379-383 (in French). ('Letter from M Le Verrier to M Faye about the theory of Mercury['s motion] and about the motion of its perihelion'; 'Weekly reports of meetings of the Academy of Sciences (Paris).)&lt;/p&gt;&#xA;&lt;p&gt;2:  N T Roseveare (1982) &amp;quot;Mercury's Perihelion from Le Verrier to Einstein.&amp;quot; (OUP), esp. chapters 2-4.&lt;/p&gt;&#xA;&lt;p&gt;3:  E W Brown (1903) &amp;quot;On the verification of the Newtonian law&amp;quot;, Monthly Notices of the Royal Astronomical Society 63, 396-7. (Brown concluded that the Hall/Newcomb proposed adjustment to the exponent in the&#xA;inverse-square law of gravitation to try and account for Mercury's excess perihelion advance was incompatible with the rate of advance of the moon's perigee, and that any adjustment compatible with the moon's motion would only generate too little extra motion to solve the Mercury problem. Brown's conclusion was agreed and adopted by W de Sitter (1913) in &amp;quot;Some problems of astronomy (VII The secular variation of the elements of the four inner planets)&amp;quot;, The Observatory, 36, 296-303.)&lt;/p&gt;&#xA;&lt;p&gt;4:  J Earman and M Janssen (1993) &amp;quot;Einstein's Explanation of the Motion of Mercury's Perihelion&amp;quot;, pp.129-172 in &amp;quot;The Attraction of Gravitation : New Studies in the History of General Relativity&amp;quot; (ed. by J Earman, M Janssen, J D Norton), esp. at 132-135.&lt;/p&gt;&#xA;&lt;p&gt;5:  A C D Crommelin (1920), &amp;quot;The Motion of Mercury's Perihelion&amp;quot;, Journal of the British Astronomical Association vol.30, 123-127.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2021-11-08T02:28:46.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13873" PostTypeId="2" ParentId="13869" CreationDate="2021-11-08T07:08:50.897" Score="5" Body="&lt;p&gt;Imagery slightly different from what is described in the question appeared in Richard Courant, David Hilbert, &amp;quot;Methoden der mathematischen Physik. Erster Band,&amp;quot; Berlin: Julius Springer 1924.&lt;/p&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://books.google.com/books?id=2qjvAAAAMAAJ&amp;amp;pg=PR5#v=onepage&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;preface&lt;/a&gt; we read (emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Von jeher hat die Mathematik mächtige Antriebe aus den engen Beziehungen gewonnen, welche zwischen den Problemen und Methoden der Analysis und den anschaulichen Vorstellungen der Physik bestehen. Erst die letzten Jahrzehnte brachten eine Lockerung dieses Zusammenhanges, indem sich die mathematische Forschung vielfach von ihren anschaulichen Ausgangspunkten ablöste und insbesondere in der Analysis manchmal allzu ausschließlich um Verfeinerung ihrer Methoden und Zuspitzung ihrer Begriffe bemühte. So kommt es, dass viele Vertreter der Analysis das Bewußtsein der Zusammengehörigkeit ihrer Wissenschaft mit der Physik und anderen Gebieten verloren haben, während auf der anderen Seite oft den Physikern das Verständnis für die Probleme und Methoden der Mathematiker, ja sogar für deren ganze Interessensphäre und Sprache abhandengekommen ist. Ohne Zweifel liegt in dieser Tendenz eine Bedrohung für die Wissenschaft überhaupt; &lt;strong&gt;der Strom der wissenschaftlichen Entwicklung ist in Gefahr sich weiter und weiter zu verästeln, zu versickern und auszutrocknen&lt;/strong&gt;. Soll er diesem Geschick entgehen, so müssen wir einen guten Teil unserer Kräfte darauf richten, Getrenntes wieder zu vereinigen, indem wir unter zusammenfassenden Gesichtspunkten die innere Zusammenhänge der mannigfaltigen Tatsachen klarlegen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Mathematics has always derived powerful impulses from the close relationship that exists between the problems and methods of analysis and the concrete ideas of physics. Only recent decades have brought a relaxation of this connection, in that mathematical research detached itself from its concrete starting points and in analysis in particular became at times far too exclusively concerned with its methods and the refinement of its terminology. As a result, many representatives of analysis have lost their awareness of the connection between their science and physics and other fields, while on the other hand physicists have often lost the understanding for the problems and methods of mathematicians, even for their entire sphere of interest and language. Without doubt this development represents a threat to the entirety of science; &lt;strong&gt;the current of scientific development is in danger of splitting itself up further and further, of draining away and drying up&lt;/strong&gt;. If it is to escape this fate, we must concentrate a good deal of our efforts on reconnecting what has been split apart by clarifying, from a unifying perspective, the inner connections of the diverse facts.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-05-18T03:32:21.157" LastActivityDate="2022-05-18T03:32:21.157" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13874" PostTypeId="1" CreationDate="2021-11-12T02:25:34.707" Score="4" ViewCount="277" Body="&lt;p&gt;I was looking for the earliest experiment or the paper which shows the determination of the mass of proton. In NIST CODATA, the mass of proton is listed as &amp;quot;1.672 621 923 69 x 10&lt;span class=&quot;math-container&quot;&gt;$^{-27 }$&lt;/span&gt;kg&amp;quot;. What type of measurements led to this number? I am not looking for the exact value but rather the experiments which allowed the measurement of proton's mass.&lt;/p&gt;&#xA;&lt;p&gt;It is so easy to find the first paper on the mass of neutron but the reference to the earliest measurement of the mass of proton remains elusive. Google Scholar was tried with year limits, different names &amp;quot;hydrogen nucleus&amp;quot;, &amp;quot;mass of hydrogen ion&amp;quot;, mass of proton etc by limiting to 1950s but nothing relevant was found.&lt;/p&gt;&#xA;&lt;p&gt;Note that the e/m of hydrogen &amp;quot;Kanalstrahlen&amp;quot; was measured in XLVII. On rays of positive electricity by J.J. Thomson in 1907 &lt;a href=&quot;https://doi.org/10.1080/14786440709463633&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1080/14786440709463633&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-11-12T02:25:34.707" Title="Earliest measurement of proton's mass" Tags="|physics|reference-request|atomic-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13877" PostTypeId="1" AcceptedAnswerId="13879" CreationDate="2021-11-13T00:17:19.863" Score="6" ViewCount="356" Body="&lt;p&gt;I was curious to know more on the history of the term &amp;quot;root&amp;quot;, in the sense of &amp;quot;a value that results in a true statement, when substituted into an equation&amp;quot; (&lt;em&gt;e.g.&lt;/em&gt;, the roots of &lt;span class=&quot;math-container&quot;&gt;$2x^2 + 1= 9$&lt;/span&gt; are &lt;span class=&quot;math-container&quot;&gt;$\pm2$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I saw &lt;a href=&quot;https://hsm.stackexchange.com/questions/12329/where-does-the-word-root-comes-from&quot;&gt;this question&lt;/a&gt;, which is also about the origin of &amp;quot;root&amp;quot;, but in the sense of a zero (&lt;em&gt;i.e.&lt;/em&gt;, rearranging the previous example into the form &lt;span class=&quot;math-container&quot;&gt;$2x^2 - 8 = 0$&lt;/span&gt;). I'm guessing the answer is the same, although, based on dictionary &amp;amp; encyclopaedic entries, it seems the &amp;quot;valid value&amp;quot; meaning predates the &amp;quot;returns zero&amp;quot; meaning.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, Wolframalpha does not even list the &amp;quot;returns zero&amp;quot; meaning in &lt;a href=&quot;https://www.wolframalpha.com/input/?i=root&quot; rel=&quot;nofollow noreferrer&quot;&gt;its 14 definitions for &amp;quot;root&amp;quot;&lt;/a&gt;, despite the fact that &lt;a href=&quot;https://reference.wolfram.com/language/ref/Root.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematica's &amp;quot;root&amp;quot; function&lt;/a&gt; is related to that sense of the word.&lt;/p&gt;&#xA;" OwnerUserId="5127" LastEditorUserId="55" LastEditDate="2021-11-14T00:42:41.033" LastActivityDate="2021-11-14T23:08:21.757" Title="What is the origin of &quot;root&quot; as a solution to an equation?" Tags="|terminology|algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13878" PostTypeId="1" AcceptedAnswerId="13880" CreationDate="2021-11-13T04:42:31.347" Score="5" ViewCount="153" Body="&lt;p&gt;I am looking for a source of the following construction problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Construct an inscribed quadrangle with given sides.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I know it under the name &lt;em&gt;problem of Brahmagupta&lt;/em&gt;, but I do not know any evidence showing that &lt;a href=&quot;https://en.wikipedia.org/wiki/Brahmagupta&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brahmagupta&lt;/a&gt; considered this problem (there is a relation to &lt;a href=&quot;https://en.wikipedia.org/wiki/Brahmagupta%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brahmagupta's formula&lt;/a&gt; but very indirect).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;P.S.&lt;/strong&gt; Actually, I did not dig deep enuf --- sorry for that. It seems that the construction is given in Brahmasphutasiddhanta.&#xA;At least Kim Plofker states so [12.39, &lt;em&gt;Brahmasphutasiddhanta of Brahmagupta&lt;/em&gt; in &lt;em&gt;Mathematics in India&lt;/em&gt; in &lt;em&gt;The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook&lt;/em&gt; p. 426].&#xA;In addition, Brahmagupta obtained the following formula for a diagonals [12.32 p. 425], say &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, of inscribed quadrangle with sides &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x=\sqrt{\tfrac{(a{\cdot}b+c{\cdot}d){\cdot}(a{\cdot}c+b{\cdot}d)}{a{\cdot}d+b{\cdot}c}},$$&lt;/span&gt;&#xA;and it leads to a solution of the construction problem as well.&lt;/p&gt;&#xA;" OwnerUserId="3303" LastEditorUserId="3303" LastEditDate="2021-11-15T17:51:03.997" LastActivityDate="2021-11-15T17:51:03.997" Title="What is the origin of the &quot;problem of Brahmagupta&quot; of constructing inscribed quadrangle with given sides?" Tags="|mathematics|reference-request|geometry|euclidean-geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13879" PostTypeId="2" ParentId="13877" CreationDate="2021-11-13T15:26:25.467" Score="8" Body="&lt;p&gt;I think there are two separate issues here:  the origin of &amp;quot;root&amp;quot; and of the custom to arrange polynomial equations in descending powers equated to zero. The latter was initiated by Stifel and made standard by Descartes's &lt;em&gt;La Geometrie&lt;/em&gt; (1637), which demonstrated its utility by bringing out relations between equations, roots, and factoring, see &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086005000480&quot; rel=&quot;nofollow noreferrer&quot;&gt;Manders, Algebra in Roth, Faulhaber, and Descartes&lt;/a&gt;. Prior to that the custom was to arrange equations so as to avoid negative coefficients, as even Stifel still called negatives &lt;em&gt;numeri absurdi&lt;/em&gt;, see &lt;a href=&quot;https://www.google.com/books/edition/The_Scientific_Work_of_Ren%C3%A9_Descartes/2kbUDAAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=descartes+equating+terms+to+zero&amp;amp;pg=PT179&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scott, The Scientific Work of René Descartes&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Stifel, in his Arithmetica Integra (Nurnberg, 1544) introduced the practice of arranging all the terms of an equation in descending powers of the unknown and then equating them to zero. In this he was followed by Descartes, and after him the practice became increasingly common... Where possible, Stifel avoided negative roots, which he called numeri absurdi.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The notion of &amp;quot;root&amp;quot; as solution to an equation was simply transferred to this form of writing them, so it has little to do with the &amp;quot;root&amp;quot; as such, as it was previously transferred to general equations from the special case &lt;span class=&quot;math-container&quot;&gt;$x^n=a$&lt;/span&gt;, which explains the name, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2265/55&quot;&gt;Why is the radical symbol √&#xA;called &amp;quot;radical&amp;quot;?&lt;/a&gt; The word, in its Arabic version &lt;em&gt;jathr&lt;/em&gt;, goes back to al-Khwarizmi himself, his famous treatise on algebra (c. 820). Robert of Chester, in his Latin translation &lt;em&gt;Liber algebrae et almucabola&lt;/em&gt; (1145), translated  &lt;em&gt;jathr&lt;/em&gt; as &lt;em&gt;radix&lt;/em&gt;. There is even speculation that the first letter in Arabic &lt;em&gt;jathr&lt;/em&gt; gave rise to the symbol for root introduced by Rudolff in Europe c. 1525. Recorde in The Whetstone of Witte (1557) translated &lt;em&gt;radix&lt;/em&gt; as root, but only as in square root. Although al-Khwarizmi occasionally calls unknowns in equations &amp;quot;&lt;em&gt;jathr&lt;/em&gt;&amp;quot; too, see &lt;a href=&quot;https://web.archive.org/web/20180805180859/http://mathforum.org/library/drmath/view/71575.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;MathForum&lt;/a&gt;, this extension did not take in Europe until 17th century; &lt;em&gt;res&lt;/em&gt; or &lt;em&gt;cosa&lt;/em&gt; were more commonly used.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="12330" LastEditDate="2021-11-14T23:08:21.757" LastActivityDate="2021-11-14T23:08:21.757" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13880" PostTypeId="2" ParentId="13878" CreationDate="2021-11-14T01:21:14.850" Score="5" Body="&lt;p&gt;Brahmagupta did not exactly consider this problem, he was rather interested in generalizing Heron's formula for the area of a triangle in terms of its sides to &lt;a href=&quot;https://en.wikipedia.org/wiki/Cyclic_quadrilateral&quot; rel=&quot;noreferrer&quot;&gt;cyclic quadrilaterals&lt;/a&gt;, as they are now called. However, his surviving text was so obscure that after Colebrook's translation of &lt;em&gt;Brahmasphuṭasiddhanta&lt;/em&gt; in 1817 European mathematicians were free to read all sorts of things into it. This particular reading is due to Chasles (&lt;a href=&quot;https://archive.org/details/aperuhistorique01chasgoog/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Aperçu Historique sur l’Origine et le Développement des Méthodes en Géométrie&lt;/em&gt;&lt;/a&gt;, 1837), as is the problem of constructing cyclic quadrilaterals with rational sides. See &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086009000688#bib5&quot; rel=&quot;noreferrer&quot;&gt;Kichenassamy's Brahmagupta’s derivation of the area of a cyclic quadrilateral&lt;/a&gt; that gives a historical sketch and a modern reconstruction of Brahmagupta’s arguments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Chasles [1837, 420–447] sees in Brahmagupta’s Propositions XII.21–38 an outline of a general theory of quadrilaterals in which all the lines considered by Brahmagupta are rational. Indeed, XII.33–38 explain how to obtain, in terms of arbitrary quantities (rśi), the sides of distinguished quadrilaterals. Chasles considers that Brahmagupta’s propositions constitute steps leading up to the construction of such rational quadrilaterals... Chasles also points out that a given set of chords, &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; in a circle in general do not determine one but three essentially different cyclic quadrilaterals...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-11-14T01:21:14.850" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13881" PostTypeId="1" CreationDate="2021-11-15T22:17:02.243" Score="6" ViewCount="190" Body="&lt;p&gt;I am wondering if there is some analog for literary movements in writing (e.g., romanticism/post-modernism) for mathematics or the sciences as a whole. I would think there would be similarly large tonal/structural shifts from decade to decade in how mathematics is communicated.&lt;/p&gt;&#xA;&lt;p&gt;I am curious if there are specific examples of these shifts (named or not) and what they entailed.&lt;/p&gt;&#xA;" OwnerUserId="15705" LastEditorUserId="6976" LastEditDate="2022-01-30T13:35:59.640" LastActivityDate="2022-01-30T13:35:59.640" Title="Literary Movements in Math Writing" Tags="|mathematics|social-context|mathematics-social-history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13883" PostTypeId="1" AcceptedAnswerId="13885" CreationDate="2021-11-16T03:03:29.683" Score="5" ViewCount="256" Body="&lt;p&gt;I'm not 100% sure this is the right site for this question, but here it goes.&lt;/p&gt;&#xA;&lt;p&gt;An already dead professor said in a lecture that Stalin (or perhaps another communist leader) wrote once something along the lines that not even statistics (or perhaps mathematics) is free from the influence of class interests. Did Stalin (or anyone) ever said something like that? If so, where is the source of that quote?&lt;/p&gt;&#xA;&lt;p&gt;Just in case, this is a very different phrase from that misattributed to him that: “The death of one man is a tragedy, the death of millions is a statistic.”&lt;/p&gt;&#xA;" OwnerUserId="10442" LastEditorUserId="10442" LastEditDate="2021-11-16T03:34:18.247" LastActivityDate="2021-11-17T03:33:44.810" Title="Who said that math or statistics is not free from class interest?" Tags="|mathematics|philosophy-of-science|quote|statistics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13884" PostTypeId="2" ParentId="13881" CreationDate="2021-11-16T13:13:16.013" Score="5" Body="&lt;p&gt;This is an interesting scholarly book:&lt;/p&gt;&#xA;&lt;p&gt;Jeremy Gray, &lt;em&gt;Plato's Ghost: The Modernist Transformation of Mathematics&lt;/em&gt; (Princeton University Press, 2008)&lt;/p&gt;&#xA;&lt;p&gt;There were profound changes in mathematics between 1880 and 1910.  Gray argues that these changes constitute a &amp;quot;modernist&amp;quot; movement, like the modernist movements in art and in music.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-11-16T13:13:16.013" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13885" PostTypeId="2" ParentId="13883" CreationDate="2021-11-16T13:46:50.103" Score="12" Body="&lt;p&gt;The quote is from Lenin, in his instructions to Popov when discussing the project of organizing Soviet statistics in summer of 1918:&amp;quot;&lt;em&gt;&lt;strong&gt;Statistics, as any other scientific discipline, poses problems and  solves them in the interests of specific classes&lt;/strong&gt;&lt;/em&gt;&amp;quot;, see &lt;a href=&quot;https://www.jstor.org/stable/2983097&quot; rel=&quot;noreferrer&quot;&gt;Kotz-Seneta, Lenin as a Statistician&lt;/a&gt;. Popov, an experienced statistician whom Lenin knew since 1905, was appointed the first head of CSA (Central Statistical Administration), a semi-independent statistical institution that Lenin established in July 1918. He quickly grew suspicious of the ideological loyalty of CSA calling in 1919 majority of its workers &amp;quot;&lt;em&gt;right-wing SR, mensheviks and even kadets&lt;/em&gt;&amp;quot;. And in 1922 he added:&amp;quot;&lt;em&gt;The CSA should not be &amp;quot;academic&amp;quot; or  &amp;quot;independent&amp;quot;, which is what it 9/10 is now due to the old bourgeois habit, but an organ of socialist construction&lt;/em&gt;&amp;quot;. See also &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-349-81634-7_10&quot; rel=&quot;noreferrer&quot;&gt;Nemchinov, The Use of Statistical and Mathematical Methods in Soviet Planning&lt;/a&gt; for broader context.&lt;/p&gt;&#xA;&lt;p&gt;Stalin was presumably on board with it, at least before WW II. Then the mood changed, see &lt;a href=&quot;https://www.jstor.org/stable/24657261&quot; rel=&quot;noreferrer&quot;&gt;Rossianov, Stalin as Lysenko's editor&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The renunciation of the ideological heritage of the 1920s and early 30s  became an integral part of the postwar Soviet ideological campaigns. For  example, in an article published in 1950, Stalin expressed his negative attitude  toward the attempts made in the 1920s to create a class-oriented culture (proletkult). From the mid-1930s, the emphasis shifted from revolutionary  cultural experimentation to upholding traditional cultural values...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;In the new ideological climate, science was considered to depend, not on class interests, but on some &amp;quot;objective&amp;quot; laws of nature.&lt;/strong&gt; In this respect, the&#xA;VASKhNIL session was a landmark: class science had focused principally on methodology; immediately after 1948, the Soviet ideology became reified into a new ontology, a new picture of the world as it purportedly was, articulated by  the political leadership. In revising Lysenko's text, then, Stalin was doing more&#xA;than changing Lysenko's writing style, or even his rhetoric; he was, in a sense, reconstructing the world. This, perhaps, helps to explain why his changes were so numerous and detailed.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ironically, the &amp;quot;objective&amp;quot; turn started by promoting Lysenko's pseudoscience over genetics.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-17T03:33:44.810" LastActivityDate="2021-11-17T03:33:44.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13886" PostTypeId="1" AcceptedAnswerId="13887" CreationDate="2021-11-17T05:12:47.910" Score="4" ViewCount="180" Body="&lt;p&gt;Any Latinists here please? 1. Why did Gauss choose &lt;em&gt;modulus&lt;/em&gt;? How does it relate to &lt;span class=&quot;math-container&quot;&gt;$n &amp;gt;1$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a - b = kn$&lt;/span&gt; for some integer &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;And why the Ablative Case?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/300013/the-origins-of-modular-and-moduli&quot;&gt;ag.algebraic geometry - The Origin(s) of Modular and Moduli - MathOverflow&lt;/a&gt;. &lt;a href=&quot;https://cs.stackexchange.com/a/114876&quot;&gt;See also AYun's answer&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The word &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dmodulus&quot; rel=&quot;nofollow noreferrer&quot;&gt;modulus&lt;/a&gt; (moduli in plural, &lt;em&gt;cf.&lt;/em&gt; &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=radius&amp;amp;fromdoc=Perseus%3Atext%3A1999.04.0059&quot; rel=&quot;nofollow noreferrer&quot;&gt;radius&lt;/a&gt; and radii, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=focus&amp;amp;fromdoc=Perseus%3Atext%3A1999.04.0059&quot; rel=&quot;nofollow noreferrer&quot;&gt;focus&lt;/a&gt; and foci, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=locus&amp;amp;fromdoc=Perseus%3Atext%3A1999.04.0059&quot; rel=&quot;nofollow noreferrer&quot;&gt;locus&lt;/a&gt; and loci) comes from Latin as a word meaning &amp;quot;small measure&amp;quot; or &amp;quot;unit of measure&amp;quot;. This is why the absolute value of a complex number &lt;em&gt;z&lt;/em&gt; is sometimes called the modulus of &lt;em&gt;z&lt;/em&gt; and why the word is used in physics for Young's modulus. In 1800 Gauss introduced the congruence relation &lt;span class=&quot;math-container&quot;&gt;$a \equiv b \bmod m $&lt;/span&gt; with &lt;em&gt;m&lt;/em&gt; being called the modulus because this equivalence relation on integers &lt;em&gt;a&lt;/em&gt; and &lt;em&gt;b&lt;/em&gt; was being &amp;quot;measured&amp;quot; according to the integer &lt;em&gt;m&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www-oed-com/view/Entry/120692?redirectedFrom=modulo#eid&quot; rel=&quot;nofollow noreferrer&quot;&gt;modulo, prep. and adj. : Oxford English Dictionary&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Etymology:&lt;/strong&gt; &amp;lt; classical Latin &lt;em&gt;modulō&lt;/em&gt;, ablative of &lt;em&gt;modulus&lt;/em&gt; &lt;a href=&quot;https://www-oed-com/view/Entry/120694#eid36548438&quot; rel=&quot;nofollow noreferrer&quot;&gt;modulus n.&lt;/a&gt;  Compare earlier &lt;a href=&quot;https://www-oed-com/view/Entry/120566#eid36504556&quot; rel=&quot;nofollow noreferrer&quot;&gt;mod prep.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The preposition was first used in this sense in a Latin context by Gauss in 1801 (see note s.v. &lt;a href=&quot;https://www-oed-com/view/Entry/120566#eid36504556&quot; rel=&quot;nofollow noreferrer&quot;&gt;mod prep.&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www-oed-com/view/Entry/120566#eid36504556&quot; rel=&quot;nofollow noreferrer&quot;&gt;mod, prep. : Oxford English Dictionary&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Etymology:&lt;/strong&gt; Shortened &amp;lt; classical Latin &lt;em&gt;modulō&lt;/em&gt; &lt;a href=&quot;https://www-oed-com/view/Entry/120692#eid36548101&quot; rel=&quot;nofollow noreferrer&quot;&gt;modulo prep. and adj.&lt;/a&gt;  (originally as a graphic abbreviation).&lt;/p&gt;&#xA;&lt;p&gt;The notation &lt;em&gt;b&lt;/em&gt; ≡ &lt;em&gt;c&lt;/em&gt; (&lt;em&gt;mod.&lt;/em&gt; &lt;em&gt;a&lt;/em&gt;) (‘&lt;em&gt;b&lt;/em&gt; is congruent to &lt;em&gt;c&lt;/em&gt; modulo &lt;em&gt;a&lt;/em&gt;’) was introduced by the German mathematician Carl Friedrich Gauss (compare &lt;a href=&quot;https://www-oed-com/view/Entry/77156#eid3281045&quot; rel=&quot;nofollow noreferrer&quot;&gt;gauss n.&lt;/a&gt;) in &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; (1801) 2. Compare earlier &lt;a href=&quot;https://www-oed-com/view/Entry/120694#eid36548602&quot; rel=&quot;nofollow noreferrer&quot;&gt;modulus n. 2b&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerDisplayName="user3478" LastActivityDate="2021-11-17T20:41:40.697" Title="Why did Gauss use [1] modulus and [2] its ablative case to define n as a positive divisor of a - b?" Tags="|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13887" PostTypeId="2" ParentId="13886" CreationDate="2021-11-17T08:37:46.360" Score="5" Body="&lt;p&gt;Gauss uses accusative &lt;em&gt;modulum&lt;/em&gt;, not ablative &lt;em&gt;modulo&lt;/em&gt;, when introducing the term in the very first sentence of &lt;a href=&quot;https://archive.org/details/disquisitionesa00gaus/page/XVIII/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Si numerus &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; numerorum &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; differentiam metitur, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; secundum a congrui dicuntur, sin minus, incongrui; ipsum &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; modulum appelamus.&lt;/em&gt;&amp;quot; (If a number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; measures the difference between two numbers &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; are said to be congruent, if not, incongruent; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; itself is called modulus).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Modulo&lt;/em&gt; is used later in the text when grammatically appropriate. The structure of the sentence is similar to Euclid's in the &lt;a href=&quot;https://mathcs.clarku.edu/%7Edjoyce/elements/bookX/defX.I.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Elements&lt;/em&gt;, book X, definition 1&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Under Gauss's definition, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; can be reached from the same starting point (any residue) in increments of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, so it is a common measure. &lt;em&gt;Modulus&lt;/em&gt; as a translation of Greek &lt;em&gt;μέτρα&lt;/em&gt; (measure) has been used by mathematicians before, so Gauss's choice of words is not particularly surprising. Cotes used it in 1722 and De Moivre in 1738 in different contexts, see &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/m/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-17T20:41:40.697" LastActivityDate="2021-11-17T20:41:40.697" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13889" PostTypeId="1" CreationDate="2021-11-19T11:20:59.710" Score="-1" ViewCount="247" Body="&lt;p&gt;Vague the question: how long have all the mathematicians working in the respective fields known the theory of categories?&lt;/p&gt;&#xA;&lt;p&gt;More specific questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is it true that all modern working algebraic topologists know category theory? How long ago?&lt;/li&gt;&#xA;&lt;li&gt;Is it true that all modern working algebraists know category theory? How long ago?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Questions still remain somewhat vague. Of course, I am not interested in single exceptions. I got the impression that in many areas of modern mathematics the language of category theory (category, functor, limit, colimit, adjoint functor, etc.) is widely used in articles and therefore it is impossible to study this area without knowing the theory of categories. I'm wondering for which areas this is really true and for how long ago.&lt;/p&gt;&#xA;" OwnerUserId="15602" LastActivityDate="2021-11-20T03:57:07.307" Title="How long have all the mathematicians working in the respective fields known the theory of categories" Tags="|mathematics|mathematicians|abstract-algebra|topology|category-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13891" PostTypeId="1" CreationDate="2021-11-20T01:16:14.607" Score="0" ViewCount="31" Body="&lt;p&gt;I'm trying to figure out what are the different paths to indirect measurement of things that are out of experimenter reach; for instance, the age of the observable universe, or an estimation of the number of atoms in it.&lt;/p&gt;&#xA;&lt;p&gt;I'd like to have something like a hypergraph, whose nodes are measurements, and whose hyperedges are physical laws within an accepted model; initial nodes are measurements that can be done on Earth, like the speed of light in vacuum, or the Cavendish experiment.&lt;/p&gt;&#xA;&lt;p&gt;For this purpose, i'd be happy to assume prior knowledge of the theoretical models; i'm only interested in verifying these models, not discovering them from scratch. Because of this, the graph may not necessarily match the timeline of their respective discoveries; cycles in the hypergraph would serve as evidence in favor of the model.&lt;/p&gt;&#xA;&lt;p&gt;Here's a couple of things I can think of, that may or may not be correct, so please do correct me if I'm wrong:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Theodolite measurements =&amp;gt; radius of Earth&lt;/li&gt;&#xA;&lt;li&gt;Foucault or Michelson experiments =&amp;gt; speed of light in vacuum&lt;/li&gt;&#xA;&lt;li&gt;Cavendish experiment =&amp;gt; gravitational constant&lt;/li&gt;&#xA;&lt;li&gt;Timing of free fall in vacuum =&amp;gt; Earth gravity&lt;/li&gt;&#xA;&lt;li&gt;gravitational constant + Earth gravity =&amp;gt; mass of Earth (via Newton's universal gravitation law)&lt;/li&gt;&#xA;&lt;li&gt;diurnal parallax (venus transit) =&amp;gt; Earth-Sun distance&lt;/li&gt;&#xA;&lt;li&gt;Earth-Sun distance + earth revolution time =&amp;gt; mass of Sun (via Kepler's third law)&lt;/li&gt;&#xA;&lt;li&gt;Earth-Sun distance =&amp;gt; close stars distance (via stellar parallax)&lt;/li&gt;&#xA;&lt;li&gt;Spectrography =&amp;gt; frequency shift of a star&lt;/li&gt;&#xA;&lt;li&gt;frequency shift of a star + speed of light =&amp;gt; relative velocity of star (via Doppler effect)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From there, how do I get to the size and age of the universe?&lt;/p&gt;&#xA;" OwnerUserId="15722" LastActivityDate="2021-11-20T01:16:14.607" Title="Is there a synthetic chart of modern cosmology, showing the dependencies between measurements and laws?" Tags="|cosmology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13892" PostTypeId="2" ParentId="13889" CreationDate="2021-11-20T03:47:10.167" Score="2" Body="&lt;p&gt;I am an algebraist. In the first edition of Lang's &amp;quot;Algebra&amp;quot;, category theory was called &amp;quot;abstract nonsense&amp;quot;. I bought the book 47 years ago. That is how long I know it. It is still abstract nonsense as far as I know. It is quite possible to study my part of algebra (geometric and algorithmic group theory) without knowing any category theory.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-11-20T03:57:07.307" LastActivityDate="2021-11-20T03:57:07.307" CommentCount="2" CommunityOwnedDate="2021-11-20T03:47:10.167" ContentLicense="CC BY-SA 4.0" />
  <row Id="13893" PostTypeId="1" CreationDate="2021-11-20T11:22:46.600" Score="0" ViewCount="110" Body="&lt;p&gt;I am interested in Antoine Lavoisier's intellectual formation/background.&#xA;Is there any available list of the books which Antoine Lavoisier read, especially ones on science/mathematics/philosophy (even if only a partial list)?&lt;/p&gt;&#xA;" OwnerUserId="15336" LastActivityDate="2021-11-20T18:08:13.670" Title="What books did Lavoisier read?" Tags="|chemistry|education|18th-century|france|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13894" PostTypeId="1" AcceptedAnswerId="13898" CreationDate="2021-11-20T17:36:45.983" Score="6" ViewCount="432" Body="&lt;p&gt;So I like to get down into the details of how certain mathematical concepts came to be, and purely as a matter of curiosity, I was wondering if anyone know which mathematician first gave the definition of a direct sum?&lt;/p&gt;&#xA;&lt;p&gt;I’ve always just sort of assumed that since for a finite set of modules, the direct sum and the direct product is the same, people probably came up with the direct product first, and then later, some mathematician wanted to do something with infinite sets of modules, realized that the universal property (of direct sums) doesn’t necessarily hold in that case, and so invented the direct sum to ”get around this,” but I’ve been wondering whether we actually &lt;em&gt;know&lt;/em&gt; that this is what happened.&lt;/p&gt;&#xA;&lt;p&gt;Any takers?&lt;/p&gt;&#xA;" OwnerUserId="15726" LastEditorUserId="55" LastEditDate="2021-11-22T08:36:17.333" LastActivityDate="2021-12-12T11:58:41.477" Title="History of Direct Sums and Direct Products" Tags="|mathematics|abstract-algebra|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13895" PostTypeId="2" ParentId="13893" CreationDate="2021-11-20T18:08:13.670" Score="5" Body="&lt;p&gt;There is online a &lt;a href=&quot;http://193.206.220.40/lavoisier/&quot; rel=&quot;noreferrer&quot;&gt;Panopticon Lavoisier&lt;/a&gt; which is a tool for searching the Archives de l’Académie des sciences, Fonds Lavoisier.&#xA;&lt;a href=&quot;http://moro.imss.fi.it/lavoisier/Lavoisier_libraryGb.asp&quot; rel=&quot;noreferrer&quot;&gt; The link to &amp;quot;library&amp;quot;&lt;/a&gt;&#xA;is introduced thus:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The catalogue of Lavoisier's library was published in &lt;strong&gt;1995 by Marco&#xA;Beretta (Bibliotheca Lavoisieriana. The Catalogue of the Library of&#xA;Antoine Laurent Lavoisier. Florence: Leo S. Olschki).&lt;/strong&gt; In 1998 a new&#xA;collection of books bearing Lavoisier's bookplate was found at the&#xA;Bibliothèque de l'Institut. The digital catalogue of Lavoisier's&#xA;library includes all the new findings and, in a few cases, the&#xA;digitalization of some title. In the guided visit to the data-base the&#xA;books are arranged alphabetically by author.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2021-11-20T18:08:13.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13896" PostTypeId="1" CreationDate="2021-11-20T19:34:16.943" Score="1" ViewCount="85" Body="&lt;p&gt;I've read arguments and statements in internet arguing about Mathematics being a science or  a language. To me, certain branches of Mathematics fit more with the definition of language and others with the definition of language. For example, equations are part of the branch Algebra, if you write the equation = F = m • a , that's a representation of a relation in Physics and it's a good representation of reality for low speeds, but if you write F = m • a^2 that has no connection with reality at all, so here the definition of Mathematics as a language makes more sense to me. You can use that language to write relations which are connected with reality or not, so the equations (language) themselves couldnt be considered science . An argument I've seen in favor of Mathematics being a language and not a science it is it doesnt rely in empirical evidence, here that argument seems to fit the situation.&lt;/p&gt;&#xA;&lt;p&gt;However, if you go to the branch of Mathematics Geometry for example, there are properties of geometric figures which are true 100% of the times, and you can test them empirically. This makes Mathematics looks more like a science and it doesnt comply with the argument that Mathematics doesnt rely in empirical evidence.&lt;/p&gt;&#xA;&lt;p&gt;So seeing these things, considering the different branches of Mathematics seems to be so different not only superficially, but in the most deep of its nature, it made me wonder if all branches of Mathematics were always considered to be part of the same discipline &amp;quot;Mathematics&amp;quot;, or were there a time where different branches were seen as different sciences or disciplines.&lt;/p&gt;&#xA;&lt;p&gt;Were all the branches of Mathematics always considered part of a single discipline &amp;quot;Mathematics&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="4512" LastActivityDate="2021-11-20T19:34:16.943" Title="Were all the branches of Mathematics always considered part of a single discipline &quot;Mathematics&quot;?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13897" PostTypeId="1" CreationDate="2021-11-20T21:48:37.473" Score="2" ViewCount="97" Body="&lt;p&gt;Discrete linear convolution &lt;span class=&quot;math-container&quot;&gt;$$&#xA;y[k]=h[n] * x[n]=\sum_{i=-\infty}^{\infty} x[i] h[k-i]&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;can be done with circulant matrices with appropriate zero padding. Is anyone aware of the name of the mathematician who proposed convolution by circulant matrices for the first time?&lt;/p&gt;&#xA;&lt;p&gt;I checked the first use of circulants in the Oxford English Dictionary, but this is associated with determinants (1881). Jeff's famous Earliest Known Words of Mathematics webpage does not list circulants or circulant matrix.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Circulant:  Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;A species of determinant.&lt;/p&gt;&#xA;&lt;p&gt;1881   W. S. Burnside &amp;amp; A. W. Panton Theory of Equations xi. §129&lt;br /&gt;&#xA;Here in all the rows the constituents are the same five quantities&#xA;taken in circular order, a different one standing first in each row. A&#xA;determinant of this kind is called a circulant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-11-20T21:48:37.473" Title="History of circulant matrices for convolution" Tags="|mathematicians|discoveries|linear-algebra|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13898" PostTypeId="2" ParentId="13894" CreationDate="2021-11-21T07:08:50.680" Score="12" Body="&lt;p&gt;&amp;quot;Direct sum&amp;quot; and &amp;quot;direct product&amp;quot; did not use to mean what they now mean in the OP sense, and even today the old usage persists. Van der Waerden in &lt;a href=&quot;https://en.wikipedia.org/wiki/Moderne_Algebra&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Moderne Algebra&lt;/em&gt; (1930-31)&lt;/a&gt; (the following section numbers are given according to the 1967 edition) uses &amp;quot;direct sum&amp;quot; when additive convention is assumed for the operation (e.g. for rings and modules, §92) and &amp;quot;direct product&amp;quot; when multiplicative convention is (e.g. for general groups,  §53). He &lt;em&gt;also&lt;/em&gt; uses &amp;quot;direct product&amp;quot; for what is now called tensor product (§67) in a special context of Galois theory, but that section was added in 1967. Van der Waerden's book is based on Artin's and Emmy Noether's lectures from mid 1920s. Sum/product distinction based on the additive/multiplicative convention is still current in group theory.&lt;/p&gt;&#xA;&lt;p&gt;Murray and von Neumann used &amp;quot;direct product&amp;quot; for tensor product (of vector spaces) from 1936 on, they also pioneered the symbol ⊗, see &lt;a href=&quot;https://hsm.stackexchange.com/q/12779/55&quot;&gt;History: Direct Product became Tensor Product?&lt;/a&gt; This is related to similar use for matrices, e.g. in &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-642-99234-6_7&quot; rel=&quot;noreferrer&quot;&gt;MacDuffee, The Theory of Matrices (1933)&lt;/a&gt;, where &amp;quot;direct sum&amp;quot; stood for combining matrices block-diagonally, and &amp;quot;direct product&amp;quot; for 'Kronecker' product. Kronecker had little to do with it, but the name stuck after Hensel c. 1890 (Zehfuss introduced it back in 1858, see &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/03081088308817548&quot; rel=&quot;noreferrer&quot;&gt;Henderson et al., On the history of the Kronecker product&lt;/a&gt;). &amp;quot;Direct product&amp;quot; is still occasionally used.&lt;/p&gt;&#xA;&lt;p&gt;Bourbaki decided to tidy it up and sort it out in their &lt;a href=&quot;http://www.cmat.edu.uy/%7Emarclan/TM/Algebra%20i%20-%20Bourbaki.pdf&quot; rel=&quot;noreferrer&quot;&gt;Algebra I (multilinear algebra, 1948)&lt;/a&gt;. There they reserved &amp;quot;direct product&amp;quot; (of modules) for the Cartesian product (regardless of conventions about the operation), &amp;quot;direct sum&amp;quot; for its subset with only finitely many non-zero entries, and separated it from &amp;quot;tensor product&amp;quot; following Whitney's 1938 general definition, see &lt;a href=&quot;https://math.stackexchange.com/q/642976/152568&quot;&gt;Origin of the modern definition of the tensor product&lt;/a&gt;. All of this happened before categorical notions of universal property, product and coproduct became fashionable in 1950s. But yes, Bourbaki's cleanup was likely motivated by working with infinite sums and products.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2021-11-25T10:20:20.123" LastActivityDate="2021-11-25T10:20:20.123" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13899" PostTypeId="1" AcceptedAnswerId="13900" CreationDate="2021-11-21T16:41:41.340" Score="0" ViewCount="266" Body="&lt;p&gt;I figured this question is better suited to this stackexchange. I give some mathematical details, but this is primarily an HSM question.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://math.stackexchange.com/q/1621371&quot;&gt;this post&lt;/a&gt;, the existence of multiplication in Peano arithmetic is an &lt;em&gt;axiom&lt;/em&gt; rather than a &lt;em&gt;definition.&lt;/em&gt; This leads me to the following questions below.&lt;/p&gt;&#xA;&lt;p&gt;How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication?&lt;/p&gt;&#xA;&lt;p&gt;Was multiplication postulated as an axiom in Peano's original work, or was it given by a definition?&lt;/p&gt;&#xA;&lt;p&gt;Is the person in the post wrong?&lt;/p&gt;&#xA;" OwnerUserId="6965" LastEditorUserId="6965" LastEditDate="2021-11-21T22:34:05.857" LastActivityDate="2021-11-21T22:52:01.580" Title="Is multiplication postulated axiomatically in Peano arithmetic?" Tags="|mathematics|mathematical-logic|arithmetic|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13900" PostTypeId="2" ParentId="13899" CreationDate="2021-11-21T22:28:13.020" Score="2" Body="&lt;p&gt;There needs to be a distinction made between the terms &lt;strong&gt;Peano axioms&lt;/strong&gt; and &lt;strong&gt;Peano arithmetic.&lt;/strong&gt; The OP is conflating the two terms, hence the confusion.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Peano_axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;this wikipedia article&lt;/a&gt; (as of 11/21/21), it says,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The math stackexchange post linked in the OP refers specifically to the theory of first-order Peano arithmetic where multiplication and addition operations are indeed postulated axiomatically. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Peano_axioms#First-order_theory_of_arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt; and &lt;a href=&quot;https://ncatlab.org/nlab/show/Peano+arithmetic&quot; rel=&quot;nofollow noreferrer&quot;&gt;this link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As for the original Peano axioms, we should refer to &lt;em&gt;Arithmetices principia, nova methodo exposita&lt;/em&gt; by Giuseppe Peano, 1889. The two images below are from the archive &lt;a href=&quot;https://archive.org/details/arithmeticespri00peangoog/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (around pages 23 and 29 of 49).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wEljg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wEljg.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Later we see this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4wHsx.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4wHsx.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A translation of Peano's work is given by &lt;em&gt;From Frege to Gödel: A source book in mathematical logic&lt;/em&gt; by Jean van Heijenoort, 1967. We see that there are nine axioms (for natural numbers), and the multiplication is indeed given by a definition in the original work.&lt;/p&gt;&#xA;&lt;p&gt;It needs to be stressed that there is no contradiction between the post that the OP linked and the historical fact. The post is talking about first-order Peano arithmetic; the Peano axioms refer to Peano's original list of axioms. It's just that Peano axioms are axioms for the theory of second-order Peano arithmetic as opposed to first-order Peano arithmetic. So both are correct.&lt;/p&gt;&#xA;&lt;p&gt;To answer the questions directly, see below.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How does this square with the fact that Peano axioms are usually presented as a list of five (or nine) axioms that mainly talk about the successor operation (and the equality relation if we're not presupposing it) with no mention of addition or multiplication?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The list of Peano axioms &lt;em&gt;is&lt;/em&gt; a list of five/nine axioms that only talk about the successor operation and no mention of addition/multiplication. The axioms for the system of first-order Peano arithmetic requires the induction axiom to be changed to a first-order axiom schema and additional axioms involving addition/multiplication added.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Was multiplication postulated as an axiom in Peano's original work, or was it given by a definition?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It was given by a definition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is the person in the post wrong?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No, because he's talking specifically about first-order Peano arithmetic.&lt;/p&gt;&#xA;" OwnerUserId="6965" LastEditorUserId="6965" LastEditDate="2021-11-21T22:52:01.580" LastActivityDate="2021-11-21T22:52:01.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13901" PostTypeId="1" CreationDate="2021-11-22T11:49:12.157" Score="1" ViewCount="59" Body="&lt;p&gt;The first name of the computer scientist &lt;a href=&quot;http://homepages.cs.ncl.ac.uk/cliff.jones&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cliff B. Jones&lt;/a&gt; is, apparently, &lt;a href=&quot;http://en.wikipedia.org/wiki/Cliff_Jones_(computer_scientist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clifford&lt;/a&gt;. What is his second name?&lt;/p&gt;&#xA;" OwnerDisplayName="user14234" LastActivityDate="2021-11-22T11:49:12.157" Title="What's the second name of the computer scientist Cliff B. Jones?" Tags="|computer-scientists|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="13902" PostTypeId="1" AcceptedAnswerId="13907" CreationDate="2021-11-22T13:13:48.710" Score="7" ViewCount="216" Body="&lt;p&gt;A question I've been curious about for a long period of time and tried to find the answer to myself a number of times (but apparently never been able to figure out just the right thing to type into Google to make the search engine give me the right result) is, what mathematician was the first to use a &lt;strong&gt;short exact sequence&lt;/strong&gt; to describe a set of objects (be they groups or rings or modules or something else), and what was the context?&lt;/p&gt;&#xA;&lt;p&gt;Anyone happens to know? Would greatly satisfy me a lot to know the answer.&lt;/p&gt;&#xA;" OwnerUserId="15726" LastActivityDate="2021-11-24T01:34:04.697" Title="First Use of the Short Exact Sequence" Tags="|mathematics|abstract-algebra|group-theory|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13903" PostTypeId="1" CreationDate="2021-11-23T04:46:31.210" Score="5" ViewCount="244" Body="&lt;p&gt;There's a fairly widespread claim that the energy released by the fission of a single atom of uranium would release enough energy to make a grain of sand visibly jump. Richard Rhodes's &lt;em&gt;The Making of the Atomic Bomb&lt;/em&gt; mentions it in its narrative of the discovery of fission; &lt;a href=&quot;https://aapt.scitation.org/doi/pdf/10.1119/1.5080566&quot; rel=&quot;noreferrer&quot;&gt;this article&lt;/a&gt; in &lt;em&gt;The Physics Teacher&lt;/em&gt; also mentions that this claim is &amp;quot;often quoted&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I'm not interested in whether the claim is accurate, just where it originally came from. Rhodes attributes it to Otto Frisch without citation; the Physics Teacher article mentions that Gerard DeGroot’s &lt;em&gt;The Bomb: A Life&lt;/em&gt; &amp;quot;mistakenly&amp;quot; attributes it to Lise Meitner and Frisch's &amp;quot;first paper on fission&amp;quot;. (I'm not sure exactly which publication this is referring to, but looking a bit through their various articles/letters in &lt;em&gt;Nature&lt;/em&gt; volume 143, I don't see the grain of sand claim.) Can anyone find the original source of this claim, assuming there is one?&lt;/p&gt;&#xA;" OwnerUserId="6554" LastActivityDate="2021-11-23T04:46:31.210" Title="What's the origin of the claim that a single uranium atom fissioning would release enough energy to visibly move a grain of sand?" Tags="|physics|reference-request|nuclear-physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13904" PostTypeId="2" ParentId="13902" CreationDate="2021-11-23T15:41:22.100" Score="4" Body="&lt;p&gt;To narrow down your search:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The lower bound is (probably) 1941. Dieudonne in his book &amp;quot;A history of Algebraic and Differential Topology&amp;quot; writes when discussing Gysin's exact sequence (which appeared in Gysin's 1941 paper):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&amp;quot;Exact sequences were not yet used in 1941 but Gysin's main result can be expressed as exactness of the following homology sequence...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Assuming that Dieudonne can be trusted on this matter, 1941 provides a lower bound.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The upper bound is 1945, the foundational paper&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;S. Eilenberg, N.  Steenrod, &lt;em&gt;Axiomatic approach to homology theory&lt;/em&gt;. Proceedings of the National Academy of Sciences of the United States of America. 31 (1945) 117–120.&lt;/p&gt;&#xA;&lt;p&gt;were they define the &lt;strong&gt;long&lt;/strong&gt; exact sequence of pairs as one of their axioms. They do not yet use the terminology &amp;quot;exact&amp;quot; and simply call it &amp;quot;a sequence&amp;quot; and define its key property which we now call &amp;quot;exactness.&amp;quot; They do not refer to anybody else in the paper, so this might have been the earliest use of exact sequences as we know them. The terminology &amp;quot;exact sequence&amp;quot; is used by&#xA;Eilenberg and Steenrod in their book &amp;quot;Foundations of Algebraic Topology&amp;quot;, 1952. I do not know if this was the earliest use, but at least this provides an upper bound.&lt;/p&gt;&#xA;&lt;p&gt;One should check, of course, earlier (than 1945) papers, especially ones by by Eilenberg. I checked only one, Eilenberg's 1944 Annals paper on singular homology theory. There are no exact sequences anywhere in the paper.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2021-11-23T15:41:22.100" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13905" PostTypeId="1" AcceptedAnswerId="13906" CreationDate="2021-11-23T16:31:51.713" Score="3" ViewCount="132" Body="&lt;p&gt;Charles Messier is know for his catalog of nebulae, but his topic of interest was actually comets. &lt;a href=&quot;https://articles.adsabs.harvard.edu/pdf/1939ASPL....3..145M&quot; rel=&quot;nofollow noreferrer&quot;&gt;This document&lt;/a&gt; states that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Louis XV nicknamed him the &amp;quot;ferret of comets&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I assume that Louis XV gave him a nickname in French. So what was the actual nickname that Louis XV gave to Charles Messier?&lt;/p&gt;&#xA;" OwnerUserId="10806" LastEditorUserId="4251" LastEditDate="2021-12-14T20:31:18.047" LastActivityDate="2021-12-14T20:31:18.047" Title="What did Louis XV actually call Charles Messier?" Tags="|18th-century|astronomers|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13906" PostTypeId="2" ParentId="13905" CreationDate="2021-11-23T21:35:58.070" Score="4" Body="&lt;p&gt;The English name is a literal translation from French. Félix Boquet, &amp;quot;Histoire de l'astronomie&amp;quot;, Paris: Payot 1925, &lt;a href=&quot;https://books.google.com/books?id=XwMLAAAAMAAJ&amp;amp;pg=PA466&amp;amp;lpg=PA466#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 466&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Rappelons ici, en parenthèse, que comme pour Maskelyne et de La Lande c'est l'eclipse de Soleil du 25 julliet 1748 qui avait décidé de la vocation du Lorrain Charles Messier (1730 - 1817) que Louis XV surnomma le &lt;em&gt;furet des comètes&lt;/em&gt;, en raison de son habileté à découvrir ces astres.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation: &amp;quot;Let us recall here, in parenthesis, that as for Maskelyne and de La Lande it is the solar eclipse of  July 25, 1748 which had decided the vocation of the Lorrainian Charles Messier (1730 - 1817) whom Louis XV nicknamed the &lt;em&gt;ferret of comets&lt;/em&gt;, because of his ability to discover these stars.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Presumably, Maskelyne is the British astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Nevil_Maskelyne&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nevil Maskelyne&lt;/a&gt; (1732 – 1811), and de La Lande is the French astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%A9r%C3%B4me_Lalande&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jérôme Lalande&lt;/a&gt; (1732 - 1807). Messier was born in Badonviller in Lorraine. Just how the solar eclipse of 1748 prompted these men to become astronomers, I do not know.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-11-24T06:13:29.670" LastActivityDate="2021-11-24T06:13:29.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13907" PostTypeId="2" ParentId="13902" CreationDate="2021-11-24T01:34:04.697" Score="3" Body="&lt;p&gt;Although exact sequences appear in Hurewicz's short note from 1941, neither he nor other early authors (Eilenberg-Steenrod in 1945, Kelley-Pitcher in 1947) used short exact sequences, see &lt;a href=&quot;https://math.stackexchange.com/q/143807/152568&quot;&gt;Math SE, History behind Exact Sequences&lt;/a&gt; and &lt;a href=&quot;https://mathoverflow.net/q/160183/51484&quot;&gt;MathOverflow, Origin of exact sequences&lt;/a&gt; for references. One can see one short exact sequence in &lt;a href=&quot;https://www.jstor.org/stable/1969135&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kelley-Pitcher, Exact Homomorphism Sequences in Homology Theory&lt;/a&gt;, but it is buried in Corollary 11.11b and no special note is made of it.&lt;/p&gt;&#xA;&lt;p&gt;They appear to be Cartan's innovation motivated by sheaves. He introduced them at his seminar on algebraic topology in 1948/49. Here is from &lt;a href=&quot;https://faculty.math.illinois.edu/K-theory/0245/survey.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weibel's History of Homological Algebra&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The next year (1948-49), Henri Cartan ran a Seminar [C48] on algebraic topology, with 17 exposes published as unbound mimeographed notes. Exposes XII–XVII&#xA;were devoted to an exposition of Leray’s theory of sheaves, but were withdrawn when Cartan’s viewpoint on sheaves changed later that year... In Expose 16 Cartan gave axioms for sheaf cohomology theory on a paracompact&#xA;space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; (with or without supports in a family &lt;span class=&quot;math-container&quot;&gt;$Φ$&lt;/span&gt; of closed subspaces of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, which we shall omit from our notation here). His axioms were: &lt;span class=&quot;math-container&quot;&gt;$H^0(X,F)$&lt;/span&gt; is the group&#xA;&lt;span class=&quot;math-container&quot;&gt;$Γ(F)$&lt;/span&gt; of global sections of &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; (with support in &lt;span class=&quot;math-container&quot;&gt;$Φ$&lt;/span&gt;); &lt;span class=&quot;math-container&quot;&gt;$H^q(X, F)$&lt;/span&gt; depends functorially on &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; and vanishes for negative &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;; &lt;strong&gt;a natural long exact cohomology sequence exists for each short exact sequence of sheaves&lt;/strong&gt;; and if &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is a “fine” sheaf then &lt;span class=&quot;math-container&quot;&gt;$H^q(X, F)=0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$q\neq 0$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Judging by MathSciNet, it took a while for the lore to make its way into publications. The earliest published appearances of short exact sequences I found are from 1958, in &lt;a href=&quot;https://www.jstor.org/stable/1970153&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heller, Homological algebra in Abelian categories&lt;/a&gt;, and in &lt;a href=&quot;http://www.numdam.org/item/10.24033/bsmf.1500.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Borel-Serre, &lt;em&gt;Le théorème de Riemann-Roch&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-11-24T01:34:04.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13908" PostTypeId="1" AcceptedAnswerId="13911" CreationDate="2021-11-24T11:35:10.797" Score="0" ViewCount="235" Body="&lt;p&gt;&lt;strong&gt;Background information:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recently asked a question about the history of the concepts of the direct sum and the tensor product in group and module theory, and was given a very concise and thorough answer regarding when both the terms and their uses appear in that particular sense for the first time by the user Conifold. Specifically, van der Waerden in 1930-1 appears to be the first one to use the term &lt;em&gt;direct sum&lt;/em&gt; as we would use it (though he doesn't use the notation &lt;span class=&quot;math-container&quot;&gt;$\oplus$&lt;/span&gt;), Murray and Von Neumann are the first one to use the notation &lt;span class=&quot;math-container&quot;&gt;$\otimes$&lt;/span&gt; for the tensor product in 1936 (though they do not call it that, and they only concern themselves with certain vector spaces, and that in a roundabout way), and Whitney in 1938 is the one to properly extend it to abelian groups in general and come up with the term &lt;em&gt;tensor product&lt;/em&gt;, which he even proves an explanation for:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The most important example of a true tensor product (and the example from which we chose the word &amp;quot;tensor&amp;quot;) is probably the following. If &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is the tangent vector space at a point of a differentiable manifold, then &lt;span class=&quot;math-container&quot;&gt;$G \circ G$&lt;/span&gt; is the space of contravariant tensors of order 2 at the point.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Whitney however uses the notation &lt;span class=&quot;math-container&quot;&gt;$\circ$&lt;/span&gt; for the tensor product, but he uses &lt;span class=&quot;math-container&quot;&gt;$\oplus$&lt;/span&gt; for the direct sum. Bourbaki later on in the 1940s appears to be the source that definitively puts into place the terminologies &lt;em&gt;direct sum&lt;/em&gt; and &lt;em&gt;tensor product&lt;/em&gt; and associates with them the notations &lt;span class=&quot;math-container&quot;&gt;$\oplus$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\otimes$&lt;/span&gt; respectively.&lt;/p&gt;&#xA;&lt;p&gt;However! Bourbaki still calls what we would call the direct product of two groups for just &lt;em&gt;the product of two groups&lt;/em&gt;, and so, I've been trying to find what should be the first use of the term &lt;em&gt;direct product&lt;/em&gt; to mean what we would have it mean. I &lt;em&gt;think&lt;/em&gt; that I've found a very good candidate, but I need some help with confirming it. Which brings me to my request...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My actual request:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I need the help of someone who can get their hands on specifically &lt;strong&gt;the 1st edition&lt;/strong&gt; of Garrett Birkhoff and Saunders Mac Lane's &lt;em&gt;A Survey of Modern Algebra&lt;/em&gt; from 1941. I want to know how they define the direct product in that book.&lt;/p&gt;&#xA;&lt;p&gt;If they use the modern definition of the direct product in that book, then it should appear in the chapter on Group Theory, in the section on Homomorphisms, and should be on the very first page (or close to) of that section. In the 3rd edition, that is Ch. 6, Sec. 11.&lt;/p&gt;&#xA;" OwnerUserId="15726" LastEditorUserId="15981" LastEditDate="2022-01-11T23:33:24.390" LastActivityDate="2022-01-11T23:33:24.390" Title="A Survey of Modern Algebra, 1st Edition (Birkhoff and Mac Lane): The Direct Product" Tags="|mathematics|abstract-algebra|group-theory|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13909" PostTypeId="2" ParentId="11308" CreationDate="2021-11-24T12:13:05.973" Score="3" Body="&lt;p&gt;Old question, but to the extent it might interest anyone, in Scandinavia, his name is almost invariably always written out in full these days.&lt;/p&gt;&#xA;&lt;p&gt;House Brahe is ancient Scandinavian nobility, dating back to times immemorial, having produced a number of important statesmen in both Sweden and Denmark. While there isn't any Tycho he might readily be confused with (the only other Tycho I can even name is Tycho Jonsson from the Swedish 1975 animated Christmas movie &lt;a href=&quot;https://en.wikipedia.org/wiki/Sagan_om_Karl-Bertil_Jonssons_julafton&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl-Bertil Jonssons Julafton&lt;/a&gt;), since we call all other members of the Brahe family by Brahe, for instance, Tycho's father the Privy Councillor &lt;a href=&quot;https://da.wikipedia.org/wiki/Otte_Thygesen_Brahe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Otte Thygesen Brahe&lt;/a&gt;, his uncle, the naval hero &lt;a href=&quot;https://da.wikipedia.org/wiki/J%C3%B8rgen_Thygesen_Brahe_til_Tostrup&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jørgen Thygesen Brahe&lt;/a&gt;, King Gustav I of Sweden's brother-in-law &lt;a href=&quot;https://sv.wikipedia.org/wiki/Joakim_Brahe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joakim Brahe&lt;/a&gt;, &lt;a href=&quot;https://sv.wikipedia.org/wiki/Ebba_Brahe&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ebba Brahe&lt;/a&gt;, who Gustavus Adolphus of Sweden wanted to marry but was barred from by his mother, &lt;a href=&quot;https://sv.wikipedia.org/wiki/Erik_Brahe_(1722%E2%80%931756)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erik Brahe&lt;/a&gt;, who was executed for treason in Stockholm in 1756 over his role in a failed coup d'etat, and the conservative Swedish statesman &lt;a href=&quot;https://sv.wikipedia.org/wiki/Magnus_Brahe_(1790%E2%80%931844)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Magnus Brahe&lt;/a&gt;, I suppose that it would seem odd if Tycho was just never named without his surname.&lt;/p&gt;&#xA;&lt;p&gt;It's just really a matter of convention.&lt;/p&gt;&#xA;&lt;p&gt;Outside of Scandinavia, where he would only be &lt;em&gt;Tycho the Astronomer&lt;/em&gt; and never really &lt;em&gt;Tycho of House Brahe&lt;/em&gt;, it makes sense that you would drop the &lt;em&gt;Brahe&lt;/em&gt; for the very reasons mentioned in Conifold's answer.&lt;/p&gt;&#xA;" OwnerUserId="15726" LastActivityDate="2021-11-24T12:13:05.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13910" PostTypeId="2" ParentId="5650" CreationDate="2021-11-25T04:15:23.490" Score="1" Body="&lt;p&gt;They didn't. The first person to divide time into 24 hours/60 minutes/60 seconds was the scholar al-Biruni (973 - ~1050) who lived in what is now Uzbekistan. He invented the concept in about 1000 as a way to subdivide the pattern of Jewish lunar cycles.&lt;/p&gt;&#xA;&lt;p&gt;Seconds could not be reliably measured until the invention of the mechanical clock around 1500, and people simply didn't try. Once clocks were in widespread use, there was a concerted effort to measure time accurately at the seconds level, notably for use in navigation/longitude determination.&lt;/p&gt;&#xA;" OwnerUserId="7183" LastActivityDate="2021-11-25T04:15:23.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13911" PostTypeId="2" ParentId="13908" CreationDate="2021-11-25T15:36:40.353" Score="2" Body="&lt;p&gt;Well, my friend in Sheffield got a hold of a copy of the first edition over at the university library there, and, much to my annoyance, the book mentions neither direct products nor direct unions. Thus, I cannot track back the origin of the term &lt;em&gt;direct product&lt;/em&gt; to mean what we currently have it mean to 1941 and Mac Lane and Birkhoff.&lt;/p&gt;&#xA;&lt;p&gt;Seems like the search goes on...&lt;/p&gt;&#xA;" OwnerUserId="15726" LastActivityDate="2021-11-25T15:36:40.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13912" PostTypeId="1" CreationDate="2021-11-26T04:42:23.157" Score="3" ViewCount="193" Body="&lt;p&gt;I've been working on de Broglie's thesis (&lt;a href=&quot;https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;English PDF&lt;/a&gt;, &lt;a href=&quot;https://tel.archives-ouvertes.fr/tel-00006807/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Original French PDF&lt;/a&gt;) for a course, and I've found something that's been bothering me. My training is in Physics, and so I'm not particularly aware of a lot of the history of this subject, so I'd appreciate any insight!&lt;/p&gt;&#xA;&lt;p&gt;The thesis makes a large number of assumptions, some of which we now know to be untrue. However, many others were quite ahead of their time. One of these assumptions that he makes very early on in the document is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An electron is for us the archetype of isolated parcel of energy, which we&#xA;believe, perhaps incorrectly, to know well; but, by received wisdom, the energy of an&#xA;electron is spread over all space with a strong concentration in a very small region, but&#xA;otherwise whose properties are very poorly known. That which makes an electron an&#xA;atom of energy is not its small volume that it occupies in space, I repeat: it occupies all space, but the fact that it is undividable, that it constitutes a unit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It would appear -- in his description of the electron as being &amp;quot;spread over all space&amp;quot; -- that de Broglie is hinting at it being a sort of &lt;a href=&quot;https://en.wikipedia.org/wiki/Wave_packet&quot; rel=&quot;nofollow noreferrer&quot;&gt;wave packet&lt;/a&gt;. Indeed, this statement is often used to show how de Broglie was aware of the electron's wave nature. However, this statement is made very early in the paper, before he describes phase and group velocities, or even tries to associate a phase-wave to a massive particle. So was it really &amp;quot;received wisdom&amp;quot; that the electron was spread over in space?&lt;/p&gt;&#xA;&lt;p&gt;There is some debate in our class as to whether or not this is an important assumption for his thesis. For example, one could argue that stating that the electron's energy is spread over space is just a way of saying it isn't a point particle. Perhaps this is so, but de Broglie's argument, in particular the phrase below (emphasis mine), seems to be a much stronger claim since he says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] I repeat: it occupies &lt;strong&gt;all space&lt;/strong&gt; [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Furthermore, one page later he uses a &amp;quot;crude&amp;quot; mechanical comparison that &amp;quot;speaks to the intuition&amp;quot; to describe a parcel of energy as a set of masses and springs that are spread out over all space, and oscillating synchronously in the particle's rest frame (see &lt;a href=&quot;https://physics.stackexchange.com/q/609742/157014&quot;&gt;here&lt;/a&gt; for an animation). He then goes on to say that the central point is that there is a &amp;quot;dephasing&amp;quot; of the motion of the weights, which forms a wave-like phenomenon that he envisions to be the phase-wave. Thus it would appear that the electron not being a point particle is quite important to his argument.&lt;/p&gt;&#xA;&lt;p&gt;So I suppose my questions would be:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What exactly was de Broglie's justification was for making the assumption that the electron was spread over all space?&lt;/li&gt;&#xA;&lt;li&gt;Was this a common assumption for people to make at the time, or was he being very imaginative?&lt;/li&gt;&#xA;&lt;li&gt;Were there any preexisting theories that supported this idea that the electron was delocalised?&lt;/li&gt;&#xA;&lt;li&gt;Or is there some evidence to show that de Broglie realised that a delocalised electron was essential for his argument to work, and introduced it &lt;em&gt;post facto&lt;/em&gt;?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(I should note that I am most interested in the &lt;em&gt;historical&lt;/em&gt; ideas that lead to his thesis being &lt;em&gt;consistent&lt;/em&gt;, rather than correct.)&lt;/p&gt;&#xA;" OwnerUserId="15736" LastActivityDate="2022-03-08T13:37:40.157" Title="de Broglie's conception of the electron" Tags="|physics|quantum-mechanics|theoretical-physics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13913" PostTypeId="1" AcceptedAnswerId="13914" CreationDate="2021-11-26T11:38:21.740" Score="2" ViewCount="641" Body="&lt;p&gt;According to Wolfram's MathWorld article &amp;quot;Narcissistic Number&amp;quot;, such numbers are also called &amp;quot;Armstrong numbers&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Such a number is an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-digit number &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; such that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$N = {d_1}^n + {d_2}^n + \cdots + {d_n}^n$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$d_1, d_2, \ldots, d_n$&lt;/span&gt; are the digits in &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;They crop up over and over again in collections of recreational mathematics problems.&lt;/p&gt;&#xA;&lt;p&gt;The question that I can't find the answer to is: who is (or was) &amp;quot;Armstrong&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;The best guess I can find is a line on ProofWiki stating that &amp;quot;there are rumours that he may have been a Michael Armstrong of Polk City in Florida&amp;quot;, but I can't find anything corroborating this.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="10106" LastEditDate="2021-11-26T13:18:36.347" LastActivityDate="2021-11-26T13:18:36.347" Title="Armstrong numbers - who is or was Armstrong?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13914" PostTypeId="2" ParentId="13913" CreationDate="2021-11-26T13:16:08.580" Score="2" Body="&lt;p&gt;As you have pointed out, Armstrong numbers seem to have been named after Michael Armstrong of Polk City in Florida. This is clear from an email he wrote to a Web Blogger in which he talks about giving the problem of “Armstrong numbers” to his computing class at University of Rochester. However, it is not known how his name became attached to the class of numbers. Below is his email from &lt;a href=&quot;http://blog.deimel.org/2010/05/mystery-solved.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lionel Deimel's blog&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the mid 1960s -- probably around 1966 -- I was teaching an&#xA;elementary course in Fortran and computing in general at The&#xA;University of Rochester, and “invented” Armstrong Numbers as an&#xA;exercise for my students. I still have the original coffee-stained&#xA;paper that was the master copy for the homework assignment and would&#xA;be happy to send you a copy if the silverfish in the garage haven’t&#xA;totally devoured it. (Full Disclosure: My memory being what it&#xA;is(n’t), I can’t say that I sat down and invented them out of whole&#xA;cloth, but I certainly don’t remember reading about them anywhere. The&#xA;paper and assignment were meant as a spoof on serious mathematical&#xA;papers that often didn’t seem to have much purpose. In any event, I am&#xA;reasonably certain that this was the first association of the name&#xA;with the numbers.)&lt;/p&gt;&#xA;&lt;p&gt;I remember there were Armstrong Numbers of several Kinds and Orders,&#xA;but don’t remember much detail, which is pretty much true of most of&#xA;my life in the 60s. The students tried to compute almost all of them,&#xA;and the sharper ones quickly realized that Fortran wasn’t the best way&#xA;to do the job. They rewrote their algorithms first in assembly&#xA;language (for an IBM 7000 series machine), and later in hard machine&#xA;language to get the last bit of speed possible. As a reward, we ran&#xA;the winning algorithm as the system’s idle process for a few nights,&#xA;resulting in a very long list of Armstrong numbers (of the first Kind,&#xA;anyway).&lt;/p&gt;&#xA;&lt;p&gt;As serendipity would have it, I was in Australia at a meeting in&#xA;February, 1988, when a short piece on “Armstrong’s Numbers” by Tim&#xA;Hartnell, one of their regular columnists, was printed in The&#xA;Australian (Tuesday, February 23rd). I immediately dashed off a note&#xA;to him asking if he was talking about MY Armstrong Numbers, or some&#xA;other Armstrong’s Armstrong numbers. We carried on a brief and cordial&#xA;correspondence, and he published a followup article about finally&#xA;finding out who “The Great Man” was in the April 19 issue. I guess&#xA;that was my 15 minutes of fame.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2021-11-26T13:16:08.580" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13915" PostTypeId="1" AcceptedAnswerId="13916" CreationDate="2021-11-26T18:55:53.417" Score="1" ViewCount="129" Body="&lt;p&gt;In the &lt;em&gt;&lt;a href=&quot;https://www.google.com/books/edition/M%C3%A9moires_de_l_Acad%C3%A9mie_royale_de_Pruss/UsaD7Kveho0C?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Memoires de l'Academie Royale de Prusse&lt;/a&gt;&lt;/em&gt; tome 1. from 1768, a summary of articles from the &amp;quot;Royal Society of Prussia&amp;quot; (the &lt;a href=&quot;https://en.wikipedia.org/wiki/Prussian_Academy_of_Sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;Königliche Akademie der Wissenschaften&lt;/a&gt;, Berlin Academy, or Prussian Academy of Sciences in Berlin) which picks up from 1745, after the original society merged with the &lt;em&gt;Nouvelle Société Littéraire&lt;/em&gt; in 1744, we find the following on pp. 3-4:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nous avons donc cru leur rendre le service le plus important, en detachant de ce precieux Recueil, &amp;amp; en leur presentant sous une forme peu volumineuse, &amp;amp; d'une acquisition facile, tout ce qui est le plus capable de les interesser dans les seize volumes in-4&lt;span class=&quot;math-container&quot;&gt;${}^\circ$&lt;/span&gt;. que l'Academie Royale de Prusse a publies, depuis son renouvellement sous Frederic II. surnomme le Grand a si juste titre.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Sorry in advance for not using accents, it's so cumbersome with Windows)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have therefore rendered [the physicians, chemists, anatomists, doctors, botanists, naturalists, etc of all Europe] the most important service, by extracting from this precious Collection, and presenting them under one less voluminous form, and easily acquired, all that is most capable of interesting them in the sixteen volumes in-4°. that the Royal Academy of Prussia published, since its renewal under Frederick II. surnommé the Great, and so rightly titled.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What does &amp;quot;in-4&lt;span class=&quot;math-container&quot;&gt;${}^\circ$&lt;/span&gt;.&amp;quot; mean? I'm understanding it to mean something like &amp;quot;in the fourth decade&amp;quot; (1740s), but &amp;quot;in&amp;quot; isn't French, it's more likely a Latin expression.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2021-11-26T19:34:56.373" Title="What does &quot;in-4o&quot; mean in this French collection from 1768?" Tags="|18th-century|france|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13916" PostTypeId="2" ParentId="13915" CreationDate="2021-11-26T19:34:56.373" Score="7" Body="&lt;p&gt;It's the format: 'in quarto'. The printed sheet of paper is folded twice, resulting in four folded sheets = eight pages.&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2021-11-26T19:34:56.373" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13917" PostTypeId="1" CreationDate="2021-11-27T15:07:20.603" Score="2" ViewCount="169" Body="&lt;p&gt;I learnt in some Wikipedia articles that the terms of the P and Q matrices designed by Heisenberg were composed of Fourier coefficients. Could you provide some explanation on how these coefficients were related to frequency radiations measurements of the hydrogen atom? Did Heisenberg have access to physical experimentation to corroborate his theory's development?&lt;/p&gt;&#xA;" OwnerUserId="15770" LastEditorUserId="4251" LastEditDate="2021-12-14T20:32:24.617" LastActivityDate="2021-12-16T17:00:09.553" Title="How did Heisenberg build the P Q matrix terms?" Tags="|physics|quantum-mechanics|physicists|heisenberg|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13918" PostTypeId="1" CreationDate="2021-11-27T18:11:23.403" Score="3" ViewCount="114" Body="&lt;p&gt;I had the following 'history of mathematics' question:&lt;/p&gt;&#xA;&lt;p&gt;Who first used the notion of supremum &lt;strong&gt;explicitly involving parameters&lt;/strong&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Let me provide a positive example of &lt;strong&gt;the latter notion&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Baire defines (,,) as sup∈[,]() in his 'Lecons sur les fonctions discontinues'. Clearly, this supremum involves parameters, as given by the variables &lt;span class=&quot;math-container&quot;&gt;$f, a, b$&lt;/span&gt; in .&lt;/p&gt;&#xA;&lt;p&gt;Negative results can be found in the work of Weierstrass and Darboux: the former merely describes the supremum, while the latter does use the notion sup∈[,](), but nothing like Baire's function .&lt;/p&gt;&#xA;&lt;p&gt;As a generalisation of my question, I am interested in any pre-1900 author using &lt;span class=&quot;math-container&quot;&gt;`&lt;/span&gt;functional' notation akin to Baire's function .&lt;/p&gt;&#xA;" OwnerUserId="15772" LastActivityDate="2021-11-27T18:11:23.403" Title="History of supremum with parameters" Tags="|mathematics|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="13919" PostTypeId="2" ParentId="12057" CreationDate="2021-11-28T06:19:31.370" Score="4" Body="&lt;p&gt;Perhaps Cybernetics lost some of its charm because Claude Shannon created Communications Theory &lt;a href=&quot;https://archive.org/details/bellsystemtechni27amerrich/page/n399/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; in 1948, which put the hazy concept of &amp;quot;communications&amp;quot; in firm, mathematical terms, which caused it to suddenly be a well-defined discipline. Since Communications was half of Cybernetics, the latter must have suffered as a result.&lt;/p&gt;&#xA;&lt;p&gt;Although not directly related to your question, it is interesting to note that &amp;quot;General Semantics&amp;quot; &lt;a href=&quot;https://en.wikipedia.org/wiki/General_semantics&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; by Alfred Korzybski seems to have followed a similar loss of popularity.&lt;/p&gt;&#xA;" OwnerUserId="15775" LastEditorUserId="15775" LastEditDate="2021-11-28T06:30:01.897" LastActivityDate="2021-11-28T06:30:01.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13920" PostTypeId="1" CreationDate="2021-11-28T14:55:43.863" Score="5" ViewCount="1296" Body="&lt;p&gt;Stochastic Partial Differential Equations (SPDEs) have received much attention in recent years, culminating in the fields medal of Martin Hairer.&lt;/p&gt;&#xA;&lt;p&gt;A rigorous mathematical starting point for the studies of SPDEs is often quoted to be the Saint-Flour lecture notes of John B. Walsh titled &amp;quot;An introduction to stochastic partial differential equations&amp;quot; in which Walsh describes the random field approach for studying SPDEs, that is one makes sense of SPDEs as &amp;quot;multidimensional&amp;quot; SDEs (rather than making sense of SPDEs as an infinite-dimensional system of SDEs). So who was this man whom many crucial ideas date back to?&lt;/p&gt;&#xA;&lt;p&gt;There is one man matching this name, namely John Bradstreet Walsh who did his PhD at the University of Illinois at Urbana-Champaign in 1966 under the supervision of Joseph Leo Doob titled &amp;quot;Probability and a Dirichlet Problem for Multiply Superharmonic Functions&amp;quot;, however I can not find any information on this mathematician. If this is the right person is there any information on him and if not who was Walsh?&lt;/p&gt;&#xA;" OwnerUserId="15520" LastEditorUserId="4251" LastEditDate="2021-11-28T17:06:06.430" LastActivityDate="2021-11-28T19:50:25.193" Title="Who is John B. Walsh?" Tags="|mathematics|mathematicians|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13921" PostTypeId="2" ParentId="13920" CreationDate="2021-11-28T19:50:25.193" Score="7" Body="&lt;p&gt;Yes, that is the one.&lt;br /&gt;&#xA;More information &lt;a href=&quot;https://mathscinet.ams.org/mathscinet/MRAuthorID/210299&quot; rel=&quot;noreferrer&quot;&gt;HERE&lt;/a&gt;&lt;br /&gt;&#xA;(If your library subscribes to MathSciNet, and you go to that page from your library, then you will be able to get links to Walsh's 70 or so publications, reviews, etc.)&lt;br /&gt;&#xA;In particular, we can find that the lectures you mention are published as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Walsh, John B. &amp;quot;An introduction to stochastic partial differential equations&amp;quot;. École d'été de probabilités de Saint-Flour, XIV—1984, 265–439, &lt;em&gt;Lecture Notes in Math.&lt;/em&gt;, &lt;strong&gt;1180&lt;/strong&gt;, Springer, Berlin, 1986.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2021-11-28T19:50:25.193" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13923" PostTypeId="2" ParentId="2503" CreationDate="2021-11-29T15:56:30.070" Score="3" Body="&lt;p&gt;J.J Sakurai died from aneurysm in his rented flat in France next to CERN, which he was visiting. His body was discovered on Monday, November 1, 1982, the official date of his death, when he didn't show up in CERN but his death probably occurred the preceding Friday evening.&lt;/p&gt;&#xA;" OwnerUserId="15787" LastActivityDate="2021-11-29T15:56:30.070" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13924" PostTypeId="1" AcceptedAnswerId="13925" CreationDate="2021-12-01T13:16:27.243" Score="4" ViewCount="277" Body="&lt;IMG SRC=&quot;https://i.stack.imgur.com/rixvf.png&quot; WIDTH=&quot;400&quot;/&gt;&#xA;&lt;p&gt;This seems an unlikely origin of the abbreviation &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; for Hamiltonian. Is there evidence for this nomenclature?&lt;/p&gt;&#xA;" OwnerUserId="1697" LastEditorUserId="1697" LastEditDate="2021-12-01T13:48:53.640" LastActivityDate="2021-12-01T14:00:00.220" Title="Hamiltonian $H$ named after Huygens?" Tags="|terminology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13925" PostTypeId="2" ParentId="13924" CreationDate="2021-12-01T13:28:07.387" Score="4" Body="&lt;p&gt;This seems to be the original source of the claim: &lt;a href=&quot;https://doi.org/10.1007/978-3-319-06248-8_25&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-3-319-06248-8_25&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A relevant quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;nowhere does Lagrange say explicitly that&#xA;he uses the letter H to honor Huygens, but there is overwhelming circumstantial&#xA;evidence that was his intention&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The main argument seems to rely on two observations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Lagrange uses the equation &lt;span class=&quot;math-container&quot;&gt;$T + V = H$&lt;/span&gt; in his book Mécanique&#xA;Analytique written before Hamilton worked on mechanics.&lt;/li&gt;&#xA;&lt;li&gt;In the book Lagrange attributes the principle of &lt;em&gt;conservation of live forces&lt;/em&gt; to Huygens.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="15793" LastEditorUserId="15793" LastEditDate="2021-12-01T14:00:00.220" LastActivityDate="2021-12-01T14:00:00.220" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13929" PostTypeId="2" ParentId="3512" CreationDate="2021-12-03T11:24:27.327" Score="0" Body="&lt;p&gt;There is the traditional Chinese timekeeping which uses various units that are not based on splitting the day in 12/24 parts (but other units that do).&lt;/p&gt;&#xA;&lt;p&gt;It is described in detail in &lt;a href=&quot;https://en.wikipedia.org/wiki/Traditional_Chinese_timekeeping&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;. It is really complicated with a huge bunch of units, partly overlapping in their definition and partly coming from different regions or periods. There are also separate systems for day and night.&lt;/p&gt;&#xA;&lt;p&gt;No point repeating all the details from Wikipedia, but here are some quotes that are relevant to your question about accuracy about units that go down to milliseconds (!):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Days were also divided into smaller units, called kè (刻). One kè was usually defined as 1⁄100 of a day until 1628, though there were short periods before then where days had 96, 108 or 120 kè.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Using the definition of kè as 1⁄100 of a day, each kè is equal to 0.24 hours, 14.4 minutes, or 14 minutes 24 seconds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kè were subdivided into smaller units, called fēn (分). The number of fēn in each kè varied over the centuries, but a fēn was generally defined as 1⁄6000 of a day. Using this definition, one fēn is equal to 14.4 seconds. This also means that a fēn is 1⁄60 of a major kè and 1⁄10 of a minor kè.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1280, Guo Shoujing's Shòushí Calendar (授时曆) had each fēn subdivided into 100 miǎo (秒). Using the definition of fēn as 14.4 seconds, each miǎo was 144 milliseconds long.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Mahāsāṃghika, translated into Chinese as the Móhēsēngzhī Lǜ (Taishō Tripiṭaka 1425) describes several units of time, including shùn or shùnqǐng (瞬頃; 'blink moment') and niàn. According to this text, niàn is the smallest unit of time at 18 milliseconds and a shùn is 360 milliseconds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9563" LastActivityDate="2021-12-03T11:24:27.327" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13930" PostTypeId="1" CreationDate="2021-12-03T12:41:55.840" Score="1" ViewCount="200" Body="&lt;p&gt;Please look at this question&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/posts/comments/9011243?noredirect=1&quot;&gt;https://math.stackexchange.com/posts/comments/9011243?noredirect=1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;A user comment this so I thought of asking here&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You mean why did we decide on using 360 degrees? I don't know for sure, but question would probably belong on hsm.stackexchange.com. I assume the reason is because 360 is highly composite, which also explains why the clock has 12 hours. –&#xA;Elliot G&#xA;1 hour ago&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another user comment it&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Babylonian scholars liked the number 360 because it has many divisors, and from there it became popular in antiquity (also because a few easily constructible angles have integer measure). An alternative system where the right angle is 100 degrees (the so-called centesimal degree) was proposed by revolutionary French academics, but it barely spread ouside of some surviving niche. In my opinion presenting degrees (sexagesimal or otherwise) as anything more than a celebrated relic which has many practical uses is a distortion of our current understanding of angles, which is radians. –&#xA;Saucy O'Path&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So basically that 1st user comment has stated my question &amp;quot; why we took a whole round or circle into 360 parts only naming each part angle as 1° not a circle with parts 540 , 260 etc etc.. only 360??&lt;/p&gt;&#xA;&lt;p&gt;2nd user said it due to &lt;strong&gt;Babylonian history&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can you explain it history of dividing circle in 360° only??&lt;/p&gt;&#xA;" OwnerUserId="15799" LastActivityDate="2021-12-03T13:39:13.647" Title="Why 360° is assigned to circle full turn ? Not any other number?" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="0" ClosedDate="2021-12-03T13:41:25.003" ContentLicense="CC BY-SA 4.0" />
  <row Id="13931" PostTypeId="2" ParentId="13930" CreationDate="2021-12-03T13:39:13.647" Score="1" Body="&lt;p&gt;A full circle is assigned 360 degrees because the Babylonians and the Sumerians used the sexagesimal (base-60) system of counting. As discussed in an &lt;a href=&quot;http://web.archive.org/web/20130520192640/http://www.wonderquest.com/circle.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;article on Wonder Quest&lt;/a&gt;, this, in turn, had come from their observation of the Sun completing a complete circle around the Earth in about 360 days:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Sumerians watched the Sun, Moon, and the five visible planets&#xA;(Mercury, Venus, Mars, Jupiter, and Saturn), primarily for omens. They&#xA;did not try to understand the motions physically. They did, however,&#xA;notice the circular track of the Sun's annual path across the sky and&#xA;knew that it took about 360 days to complete one year's circuit.&#xA;Consequently, they divided the circular path into 360 degrees to track&#xA;each day's passage of the Sun's whole journey. This probably happened&#xA;about 2400 BC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2021-12-03T13:39:13.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13932" PostTypeId="2" ParentId="3512" CreationDate="2021-12-03T14:55:41.457" Score="0" Body="&lt;p&gt;The ancient Israelites divided each of the 24 hours in a day into 1080 halaqim, making each heleq 3⅓ seconds long.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;1 part (heleq) = 76 moments (rega'im) (each moment, rega, is 0.04386 of a second; 22.8 rega'im is 1 second)&lt;/li&gt;&#xA;&lt;li&gt;1 hour (sha'ah) = 1080 parts (halaqim) (each heleq is 3⅓ seconds)&lt;/li&gt;&#xA;&lt;li&gt;1 day = 24 hours (sha'ah)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;— &lt;a href=&quot;https://en.wikipedia.org/wiki/Biblical_and_Talmudic_units_of_measurement#Day&quot; rel=&quot;nofollow noreferrer&quot;&gt;Biblical and Talmudic units of measurement - Wikipedia&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The rega unit is about 44 milliseconds, or one twenty-fifth of a second.&#xA;Except perhaps for long time averages in astronomy, I've no idea how or why such precision would be used during those times.&lt;/p&gt;&#xA;&lt;p&gt;They determined that the average month (new moon to new moon) lasts 29 days, 12 hours, 793 halaqim,&#xA;which is less than half a second longer than the currently known value.&lt;/p&gt;&#xA;" OwnerUserId="15353" LastEditorUserId="12330" LastEditDate="2021-12-05T02:34:52.283" LastActivityDate="2021-12-05T02:34:52.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13933" PostTypeId="1" CreationDate="2021-12-05T17:30:34.287" Score="0" ViewCount="237" Body="&lt;p&gt;When was the first time we see a square in an equation describing some physical effect?&lt;/p&gt;&#xA;&lt;p&gt;Well, there is the area of circle (or square of course ...), but a circle area is not - in this question definition at least - a physical phenomenon. An example of a physical phenomenon would be the lever, but we see no square there, rather a simple multiplication of a quantity with &lt;em&gt;another&lt;/em&gt; quantity: &lt;span class=&quot;math-container&quot;&gt;$M_a*D_a = M_b*D_b$&lt;/span&gt;. The equation: &lt;span class=&quot;math-container&quot;&gt;$E_v=\frac{mv^2}{2}$&lt;/span&gt; on the other hand, is a qualified example where we square a quantity. But there are, I suppose, much earlier examples of this. What are they?&lt;/p&gt;&#xA;" OwnerUserId="14954" LastEditorUserId="14954" LastEditDate="2021-12-05T20:06:58.177" LastActivityDate="2021-12-13T05:13:37.503" Title="What was the first time that a Physical quantity was squared to describe a physical phenomenon?" Tags="|physics|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="13935" PostTypeId="2" ParentId="13933" CreationDate="2021-12-06T06:43:35.717" Score="4" Body="&lt;p&gt;I'm not sure if the following satisfy your notion of physical phenomenon. I suggest Babylonian &amp;quot;System B&amp;quot; for planet motions. According to this system planets move around zodiac with constantly increasing speed for some time and then they move with constantly decreasing speed. That is, they approximate planet speed as a piecewise linear function. Then they need to square time to calculate the distance the planet travel. That is, they used formula &lt;span class=&quot;math-container&quot;&gt;$d=\frac{at^2}{2}$&lt;/span&gt; for motion with constant acceleration.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2021-12-06T06:43:35.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13936" PostTypeId="1" CreationDate="2021-12-06T09:27:12.347" Score="1" ViewCount="105" Body="&lt;p&gt;I was just reading about The Gas Laws of Boyle, Charles and Avogadro from a general college book Chemistry (7th edition, Zumdahl&amp;amp;Zumdahl).&#xA;&lt;br/&gt; Boyle was the one who discovered a correlation between pressure and volume of a gas.&lt;/p&gt;&#xA;&lt;p&gt;I was left wondering... &lt;br/&gt;&#xA;Why didn't he further studied gas properties at different volume and temperature and/or also between pressure and temperature? The latter is quite similar to his own.&lt;/p&gt;&#xA;&lt;p&gt;The literature is available on this link: &lt;br/&gt;&#xA;&lt;a href=&quot;https://www.tvusd.k12.ca.us/site/handlers/filedownload.ashx?moduleinstanceid=24457&amp;amp;dataid=8531&amp;amp;FileName=Zumdahl%20Text.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tvusd.k12.ca.us/site/handlers/filedownload.ashx?moduleinstanceid=24457&amp;amp;dataid=8531&amp;amp;FileName=Zumdahl%20Text.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Go to Content --&amp;gt; 5.2 The Gas Laws of Boyle, Charles, and Avogadro (page 181)&lt;/p&gt;&#xA;" OwnerUserId="15815" LastEditorUserId="15815" LastEditDate="2021-12-06T16:15:19.603" LastActivityDate="2021-12-10T01:21:18.277" Title="Why didn't Boyle made a correlation between V and T of a gas (and similarly between P and T)?" Tags="|chemistry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13937" PostTypeId="1" CreationDate="2021-12-06T20:35:46.190" Score="2" ViewCount="132" Body="&lt;p&gt;I am curious about the history of the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Froda%27s_theorem#:%7E:text=In%20mathematics%2C%20Darboux%E2%80%93Froda%27s%20theorem,most%20countably%20many%20of%20them.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Darboux-Froda theorem&lt;/a&gt;, which is the following theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;a monotone function &lt;span class=&quot;math-container&quot;&gt;$f:[0,1]\rightarrow \mathbb{R}$&lt;/span&gt; has at most countably many points of discontinuity.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This theorem can be found in Froda's thesis (see the references for &lt;a href=&quot;https://en.wikipedia.org/wiki/Froda%27s_theorem#:%7E:text=In%20mathematics%2C%20Darboux%E2%80%93Froda%27s%20theorem,most%20countably%20many%20of%20them.&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;) where Froda states that this theorem is implicit in Ch. VI of Darboux's 'Memoire on discontinuous functions' (which is correct, I would say); Froda also states that the Froda-Darboux theorem has been known for a long time (in 1929).  I would like to know who first proved the Froda-Darboux theorem, perhaps with the conclusion formulated using a sequence (rather than countable sets).&lt;/p&gt;&#xA;" OwnerUserId="15772" LastActivityDate="2021-12-07T19:33:43.757" Title="History of the Darboux-Froda theorem" Tags="|mathematics|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13938" PostTypeId="1" CreationDate="2021-12-06T21:05:42.753" Score="1" ViewCount="198" Body="&lt;p&gt;A group action has two laws which roughly correspond to associativity and identity&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$&#xA;\phi : (G : \textrm{Group}) \times (S : \textrm{Set}) \rightarrow S \\&#xA;\forall a, b : G . \forall c : S. \phi(a,\phi(b, c)) = \phi(a \cdot b, c) \\&#xA;\forall a : S. \phi (1,a) = a&#xA;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Looking at this definition there's nothing very &amp;quot;group&amp;quot;-like about it.  There's no law about inverses or cancellation.  It would seem that &amp;quot;monoidal-action&amp;quot; would be a more sensible name since these laws line up nicely with the monoid laws.&lt;/p&gt;&#xA;&lt;p&gt;Where does the term &amp;quot;group action&amp;quot; come from?  What context leads to the word &amp;quot;group&amp;quot; in its name?&lt;/p&gt;&#xA;" OwnerUserId="6372" LastEditorUserId="6372" LastEditDate="2021-12-07T15:20:07.450" LastActivityDate="2021-12-12T05:22:34.397" Title="Why is it called a group action?" Tags="|terminology|abstract-algebra|group-theory|galois-theory|" AnswerCount="2" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="13939" PostTypeId="2" ParentId="13611" CreationDate="2021-12-07T01:25:40.183" Score="3" Body="&lt;p&gt;We can read the original article that Kirchhoff published on &lt;strong&gt;Annalen der Physik und Chemie 1845, Band LXIV&lt;/strong&gt; with the title &amp;quot;&lt;em&gt;Ueber den Durchgang eines elektrischen Stromes durch eine Ebene ins besondere durch eine kreisformige&lt;/em&gt;&amp;quot; (&amp;quot;&lt;em&gt;On the passage of an electric current through a plane, through a circular shape in particular&lt;/em&gt;&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SXgX4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SXgX4.png&quot; alt=&quot;Kirchhoff original paper 1845&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;source: &lt;a href=&quot;https://books.google.de/books?id=Ig8t8yIz20UC&amp;amp;pg=PA497&amp;amp;hl=de&amp;amp;source=gbs_toc_r&amp;amp;cad=3#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Annalen der Physik und Chemie LXIV 1945 - p. 497&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately I could not find the pictures referenced in the paper, but I believe the information in the text alone can still be used to infer the following two points:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Kirchhoff formulated his laws on the base of experiments that used localised and non-inductive forms of electromotive force.&lt;/li&gt;&#xA;&lt;li&gt;Kirchhoff Voltage Law, as orginally formulated, is fundamentally different from Faraday's law. In particular, the inductive EMF due to a changing magnetic flux linked by the loop could not appear among the 'electromotorischen Krafte' along the path ('auf dem Wege') as considered by Kirchhoff.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h3&gt;On the sources of electromotive force considered by Kirchhoff&lt;/h3&gt;&#xA;&lt;p&gt;Right from the start, Kirchhoff mentions a constant galvanic current that flows through a metallic disc, entering and exiting it by means of wires:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Leitet man einen constanten galvanischen Strom durch eine&#xA;Metallscheibe , so wird sich die Elektricität in die ser auf eine&#xA;bestimmte Weise vertheilen . Die Art der Vertheilung kann man nach den&#xA;von Ohm aufgestellten Principien theoretisch ermitteln . Ich habe die&#xA;dazu no thige Rechnung unter der Voraussetzung , daſs der Zu stand der&#xA;Scheibe ein stationärer geworden sey , in dem Falle durchgeführt ,&#xA;daſs die Scheibe eine kreisförmige ist , und daſs die Elektricität&#xA;durch einen Draht in sie hinein , durch einen zweiten aus ihr&#xA;heraustrete.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The automatic translation from a well-known service provider was a bit approximative so I asked a German native speaker with a scientific formation to fix it. In the following the bold is mine:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If a &lt;strong&gt;constant galvanic current&lt;/strong&gt; is passed through a metal disc,&#xA;the electricity into said disc will distribute in a certain way. The&#xA;type of distribution can theoretically be determined according to the&#xA;principia &lt;strong&gt;established by Ohm&lt;/strong&gt;. I have carried out the necessary&#xA;calculation on the condition that the state of the disk has become a&#xA;stationary one, for the case that the disk has circular shape, and&#xA;that the &lt;strong&gt;electricity enters it through one wire, and exits through a&#xA;second one&lt;/strong&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The sources of electricity used by Kirchhoff are mentioned in several points in the paper, for example on page 509:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/F1LdB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/F1LdB.png&quot; alt=&quot;kirchhoff 1845 on sources&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Ich leitete durch die Scheibe der Strom einer constanter Hydrokette,&#xA;und beruhrte sie an zwei Punkte mit den Enden zweier Drahte, in deren&#xA;Schliefsung aufser dem Multiplicator eine schwache, auf Kupfer und&#xA;Zink gebildete, Thermokette eingeschaltet war.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, the automatic translation was too uncertain, but with the help of the same native German speaker and after changing 'multiplier' into 'galvanometer' (because that is what the Schweigger and Poggendorff type of galvanometer was called at the time) we get:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I passed through the disc &lt;strong&gt;the current of a constant Hydrokette&#xA;[battery]&lt;/strong&gt;, and contacted it at two points with the ends of two&#xA;wires, in the loop of which &lt;strong&gt;a weak Thermokette [thermoelectric&#xA;element] formed on copper and zinc&lt;/strong&gt; was switched in, onto a&#xA;[galvanometer].&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A sentence that helps in casting some light on the exact nature of the electromotive forces used in the experiments.&#xA;'Hydrokette' and 'thermokette' &lt;strong&gt;were also the sources of electricity used by Georg Ohm&lt;/strong&gt;, who is mentioned at the beginning of Kirchhoff's paper, in his experiments.&#xA;Ohm experimented with galvanic cells and thermopiles in the early twenties of the nineteenth century; his theory of galvanic circuits, published in 1827 as &amp;quot;&lt;em&gt;Die galvanische Kette, mathematisch bearbeitet&lt;/em&gt;&amp;quot; (&amp;quot;&lt;em&gt;The Galvanic Circuit Investigated Mathematically&lt;/em&gt;&amp;quot;), was harshly criticized by Georg Pohl and ultimately led to Ohm's resignation from his academic position in 1828. (Ohm was later 'rediscovered' ex-patria in 1841 when the Royal Society in London awarded him the Copley medal as a recognition for his past accomplishments).&#xA;At the time, the sources of electricity were galvanic cells, chemical batteries and thermopiles, like the stable 'state of the art' electricity source used by Ohm to estabilish his law: a thermocouple based on the Seebeck effect.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JB0kx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JB0kx.png&quot; alt=&quot;Seebeck thermocouple&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Seebeck thermocouple with inbuilt primitive galvanometer in the form of a magnetized needle (source: &lt;a href=&quot;http://dingler.culture.hu-berlin.de/article/pj315/ar315128&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://dingler.culture.hu-berlin.de/article/pj315/ar315128&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;As a matter of fact, &lt;em&gt;thermokette&lt;/em&gt; translates into thermopile (&lt;a href=&quot;https://de.wikipedia.org/wiki/Thermos%C3%A4ule&quot; rel=&quot;nofollow noreferrer&quot;&gt;German wikipedia&lt;/a&gt; - &lt;a href=&quot;https://en.wikipedia.org/wiki/Thermopile&quot; rel=&quot;nofollow noreferrer&quot;&gt;English Wikipedia&lt;/a&gt;).&#xA;For &lt;em&gt;hydrokette&lt;/em&gt;, which some dictionaries translate as 'hydrocell' or 'hydro element', I have not been able to find a satisfying English definition, but I am told that &amp;quot;Hydrokette&amp;quot; or &amp;quot;hydroelektrische Kette&amp;quot; is a cell made of two metal plates of different &amp;quot;order&amp;quot; immersed into a conductive fluid. A &amp;quot;Constante Hydrokette&amp;quot; has each metal plate immersed in a different conductive fluid, the two fluids being separated by a diaphragm that allows charge carriers migration but doesn't allow the fluids to mix. It basically is an electrochemical cell.&#xA;Here is a picture from a Czech presentation showing Ohm's galvanometer and comparing Thermokette with Hydrokette in regard to Ohm's experiment (the drawing on the right might not exactly illuminating, I know, but is compatible with the assumption made)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/H0BLa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/H0BLa.png&quot; alt=&quot;Thermokette vs Hydrokette&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Thermokette vs. Hydrokette in Ohm's experiment (source: &lt;a href=&quot;https://image2.slideserve.com/4698073/hled-n-kvantitativn-ch-vztah-l.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://image2.slideserve.com/4698073/hled-n-kvantitativn-ch-vztah-l.jpg&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;The following shows a photograph of the galvanometer (back then called 'a multiplier') used by Ohm with a &amp;quot;thermokette&amp;quot; in place, along with a sketch of the device in use. The voltage was 'fetched' with wires connected to the two copper 'cups' at the base of the the instrument:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IoryF.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IoryF.png&quot; alt=&quot;Ohm's multiplier with thermokette&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Actual apparatus used by Ohm and pictorial representation during its use (sources: left &lt;a href=&quot;https://www.oocities.org/bioelectrochemistry/ohm.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.oocities.org/bioelectrochemistry/ohm.htm&lt;/a&gt;, right: &lt;a href=&quot;https://www.youtube.com/watch?v=fk_BpXlfZ8U&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=fk_BpXlfZ8U&lt;/a&gt; )&lt;/p&gt;&#xA;&lt;p&gt;From the same source of the picture on the left:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;This apparatus was used by Ohm. Current flowing through the metal bar&#xA;in the center cylinder deflects a magnetized needle suspended above&#xA;it. The deflection angle is proportional to the current. The source of&#xA;electric potential is a thermocouple (discovered by Seebeck in 1821).&#xA;The ends of the thermocouple are heated by steam and cooled by&#xA;ice-water in the small containers on the tripods.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not only the EMFs sources mentioned by Kirchhoff in his paper are localised, but he also made sure that the supplied electricity was constant and stable as explained from the very start of the 1845 paper (bold mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I have carried out the necessary calculation on the condition that&#xA;the state of the disk has become &lt;strong&gt;a stationary one&lt;/strong&gt;...&lt;/em&gt;&amp;quot;&lt;br /&gt;&#xA;source: Annalen der Physik und Chemie LXIV 1845 - p. 497&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Use of a constant source of galvanic current in steady state conditions strongly deposes against the possibility that said paper was intended to include time-varying sources of EMF, or even the side effects of self-inductance. Moreover, there are no references to inductive electromotive forces or inductive elements anywhere in the text.&lt;/p&gt;&#xA;&lt;p&gt;In contrast to that, when in a later paper (&amp;quot;&lt;em&gt;Ueber die Bewegung der Elektricität in Leitern&lt;/em&gt;&amp;quot;, 1857) Kirchhoff analized a circuit with self-inductance, he was very clear in mentioning &amp;quot;Weber's law of induction&amp;quot;. In this 1857 paper, translated in English by P. Graneau and A. K. T. Assis for APEIRON Nr. 19 June 1994 (pp. 19-25) , Kirchhoff finds the local form of Ohm's law as (in modern notation)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Uh68l.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Uh68l.png&quot; alt=&quot;formula for Ohm's law&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;where he clearly considers the total electric field Etot = - grad phi - dA/dt, as in j = sigma Etot and not the conservative component alone Ecoul = -grad phi. In the foreword to the translation the translators comment that &amp;quot;&lt;em&gt;This is essentially Ohm’s law generalized by Kirchhoff to three dimensional conductors and to take into account the effects of self-induction.&lt;/em&gt;&amp;quot;&#xA;It is also worth noting that the concept of magnetic vector potential &lt;strong&gt;A&lt;/strong&gt; was introduced by Neumann in a paper published in January 1846 on volume I of the Annalen Der Physik 143 with the title “&lt;em&gt;&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Allgemeine Gesetze Der Inducirten Elektrischen Ströme&lt;/a&gt;&lt;/em&gt;” (from &lt;a href=&quot;https://hsm.stackexchange.com/questions/5078/who-discovered-the-magnetic-vector-potential-veca&quot;&gt;this&lt;/a&gt; answer).&lt;/p&gt;&#xA;&lt;p&gt;All thing considered it seems plausible that, had Kirchhoff employed inductive sources of EMF in his experiments leading to his 1845 paper, such a novel form of (time-varying) electromotive force would at least be mentioned explicitly, if not discussed at length. But this is not the case. On the other hand, on p. 510 of the 1845 paper Kirchhoff mentions again &amp;quot;&lt;em&gt;the electromotive force of a thermopile&lt;/em&gt;&amp;quot;, and on p. 513, right before stating his two celebrated laws, he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Der Strom einer starken Hydrokette theilte sich in die beiden Arme&#xA;ACB und ADB...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That can be translated as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The current of a strong [battery] parted its way into both arms ACB&#xA;and ADB...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I believe I have made my point that in his original 1845 paper Kirchhoff only mentioned and used localized, stationary, and non-inductive sources of electromotive force.&#xA;Why this is relevant to the question posed here? Because &lt;em&gt;localized&lt;/em&gt; EMF sources such as galvanic cells, voltaic batteries, thermocouples, and thermopiles, develop a voltage that can be precisely located at the device terminals (two wires) inside the branch of the circuit they are part of. This is not the case for the delocalized EMF originated by the changing magnetic flux linked by the loop in its entirety.&lt;/p&gt;&#xA;&lt;h3&gt;On what the original formulation of KVL by Kirchhoff says about the location of the sources of EMFs&lt;/h3&gt;&#xA;&lt;p&gt;This is the the original formulation of what today is known as KVL (Kirchhoff Voltage Law) or Kirchhoff Loop Rule, as written by Kirchhoff in his 1845 paper (page 513 of the linked Annalen):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WdELi.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WdELi.png&quot; alt=&quot;KVL in original 1845 paper&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Modernizing the notation (by using R instead of omega to denote the resistance, and n instead of nu for the indexes), the text reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;2) wenn die Drahte 1, 2, n, eine geschlossene Figur bilden&#xA;R1 I1 + R2 I2 + ... + Rn In&#xA;= der Summe aller elektromotorischen krafte, die sich auf dem Wege: 1, 2, n befinden; wo R1, R2, Rn die Widerstande der Drahte, I1 I2,... die&#xA;Intensitaten der Strome bezeichnen, von denen diese durchflossen&#xA;werden, alle nach einer Richtung als positiv gerechnet.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which translates into (bold mine)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;2) if the wires 1, 2, n, form a closed figure&#xA;R1 I1 + R2 I2 + ... + Rn In =&#xA;= the sum of all electromotive forces located on the path: 1, 2, ...n;  where R1, R2, ... Rn are the resistances of the wires, and I1 I2,&#xA;...In denote the intensities of the currents that flow through them,&#xA;all calculated as positive in one direction.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Automatic translation is getting better and better every day, thanks to AI progress. Google Translate's output for the original German text did not need any tweaking:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WB35T.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WB35T.png&quot; alt=&quot;this translation did not need any tweaking&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The locution &amp;quot;&lt;em&gt;located on the path: 1, 2, n&lt;/em&gt;&amp;quot; points to sources of EMF located on the path itself, as if  Kirchhoff was expressely treating localized forms of EMF. Further confirmation of this comes from a subsequent article, published in the December 1847 volume 72 of the Annalen der Physik und Chemie. The paper, published with the title &amp;quot;&lt;em&gt;Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuching der linearen Vertheilung Galvanisher Ströme geführt wird&lt;/em&gt;&amp;quot; has been translated by J. B . O'Toole and republished in IRE Transaction on Circuit Theory (IEEE Trans. CT-5, 4–7, 1958). Here Kirchhoff writes (bold and italics mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Let there be given a system of n wires: 1, 2, . . . , n joined to&#xA;one another in an arbitrary fashion. &lt;strong&gt;If an electromotive force is in&#xA;series with each of them&lt;/strong&gt;, the necessary number of linear equations for&#xA;the determination of the currents I1, I2, . . . , In, flowing through&#xA;the wires is found by using...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It follows from this, among other things, that if we choose two&#xA;wires from an arbitrary system, the current caused in one by an&#xA;&lt;strong&gt;electromotive force in the second&lt;/strong&gt; is exactly equal to the current&#xA;produced in the second by an equal &lt;strong&gt;electromotive force in the first&lt;/strong&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thereby confirming he is talking about EMFs localised as series elements inserted in each branch and not about an EMF delocalized along the whole loop. Decomposing the whole loop EMF in the partial contributions along various parts of the loop would require the knowledge of the configuration of the induced electric field in the space occupied by the conductors (knowledge that was not available at the time) or the line integration of the time derivative of the vector magnetic potential &lt;strong&gt;A&lt;/strong&gt; along the various curvilinear segments. Again, while it might be argued that knowledge of a mathematical entity corresponding to the vector potential &lt;strong&gt;A&lt;/strong&gt; transpires from Kirchhoff's 1857 paper, such a feat would have at least required a mention, if not a full explanation.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I have also found at least one additional source confirming that Kirchhoff formulated his rules addressing localized, lumped, forms of EMF. In his Physics Education paper &amp;quot;&lt;em&gt;Explaining electromagnetic induction: a critical re-examination. The clinical value of history in physics&lt;/em&gt;&amp;quot; (J Roche 1987 Phys. Educ. 22 91), science historian J. Roche writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It was shown by G Kirchhoff (1824-87) in 1849 that &lt;strong&gt;localised&lt;/strong&gt; EMFs&#xA;generally set up auxiliary electrostatic forces, by means of surface&#xA;charges, in order to establish a uniform current around the circuit&#xA;(Kirchhoff 1879 pp49-55, 1514). Since these additional fields are&#xA;conservative, the sum of the net potential differences around the&#xA;whole circuit will be exactly equal to the sum of the PDs across the&#xA;&lt;strong&gt;localised&lt;/strong&gt; underlying EMFs only. This is the substance of Kirchhoff’s&#xA;second network law (Kirchhoff 1879 pp 15-16)&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In summary, not only the experimental setup employed by Kirchhoff, but also the way he expressed his laws constitute evidence that the right hand side of Kirchhoff original KVL equation collects (with due sign) the contribution of &lt;em&gt;all&lt;/em&gt; the lumped EMF sources that are present along the path (&amp;quot;&lt;em&gt;auf dem Wege&lt;/em&gt;&amp;quot;).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Appendix: On what the localized nature of the EMF sources means for the original formulation of KVL by Kirchhoff&lt;/h3&gt;&#xA;&lt;p&gt;A common objection raised by people who believe KVL always applies, even when the electric field is not conservative, is that Kirchhoff Voltage Law was intended from its first inception to encompass all forms of electromotive forces, including the nonlocalized inductive EMF linked by the loop itself.&#xA;The reason behind this belief is that in most high school and introductory university textbooks KVL is expressed in a form applicable to lumped circuits where the sum of all kind of &lt;em&gt;localized&lt;/em&gt; EMFs (including those from lumped transformers and induction generators) is equated to the sum of all voltage drops (including those attributed to lumped inductors).&lt;/p&gt;&#xA;&lt;p&gt;The problem is that the inductive EMF is a very special kind of EMF that is not like other forms of EMF. For example, the voltage that is measured across the terminals of an inductor (such as a multi-turn coil) is different from the voltage computed along the conductor the coil is made of (something that by itself violates KVL in the loop formed by the coil filament and the jump at its terminals). For further information on this see Ramo Whinnery VanDuzer referenced below.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Original KVL with modern terminology&lt;/strong&gt;&lt;br /&gt;&#xA;Fast forward to the twentyfirst century, where we tend to express the laws of electrodynamics in terms of relations between fields, we can see how Kirchhoff formulation of localised EMFs would be expressed today.&lt;/p&gt;&#xA;&lt;p&gt;First, let's rewrite Kirchhoff's equation by making whatever EMFs that are on the paths 1, 2, ... n explicit by calling them emf1, emf2, ... emfn. If one branch has no localized emf in it, then the corresponding emf will be zero; the same can be said for the resistance. In its expanded form, Kirchhoff's original KVL becomes:&lt;/p&gt;&#xA;&lt;p&gt;R1 I1 + R2 I2 + ... + Rn In = emf1 + emf2 + ... + emfn&lt;/p&gt;&#xA;&lt;p&gt;where the emf1, emf2,... emfn are the localized electromotive forces located on the segments 1, 2, ..., n of the closed path.&lt;/p&gt;&#xA;&lt;p&gt;Now we take the localized EMFs of each branch that are on the right hand side and we bring them on the left hand side. The change in sign reflects that different sign convention we use for generators and for passive elements.&lt;/p&gt;&#xA;&lt;p&gt;(R1 I1 - emf1) + (R2 I2 - emf2) + ... + (Rn In - emfn) = 0&lt;/p&gt;&#xA;&lt;p&gt;This is the modern equivalent form of KVL as Kirchhoff formulated it.&lt;/p&gt;&#xA;&lt;p&gt;In the twentyfirst century we recognize that each term in parenthesis represents the path integral of the total electric field along each of the segments 1, 2,... n, in which the closed path (the loop) has been partitioned. Therefore, in modern terms, the original formulation of KVL says&lt;/p&gt;&#xA;&lt;p&gt;Sum of path integral of Etot.dl along the branches of our closed figure = 0&lt;/p&gt;&#xA;&lt;p&gt;The right hand side is zero because we have already accounted for all &amp;quot;&lt;em&gt;der electromotorischen Krafte, die sich auf dem Wege&lt;/em&gt;&amp;quot; - all the electromotive forces on the branches 1, 2,... n - and because Kirchhoff did not consider any delocalised source of inductive EMF. Since the path integral along a closed path is the circulation, in modern notation the original form of KVL can also be written as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9Y3rF.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9Y3rF.png&quot; alt=&quot;circulation of Etot is zero&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;circulation of Etot.dl = 0&lt;br /&gt;&#xA;Modern notation for KVL as formulated by Kirchhoff (i.e. using only localized EMF sources)&lt;br /&gt;&#xA;Note: all localized EMFs appear in the path integral on the left hand side.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Faraday's law introduces an EMF that is not on the path&lt;/strong&gt;&lt;br /&gt;&#xA;Faraday discovered, in modern terms, that the circulation of Etot.dl can be nonzero, if there is a variable magnetic flux cut by the surface formed from the closed figure 1, 2, ... n Kirchhoff was arguing about. According to Faraday's law (not Kirchhoff's law), the circulation of the total electric field along the circuit path is equal to minus the flux of the magnetic field B cut by the surface delimited by said closed path:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ByBPc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ByBPc.png&quot; alt=&quot;faraday's law in integral form&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;circulation of Etot.dl = -d/dt flux of B&lt;br /&gt;&#xA;Modern notation for Faraday's law in integral form showing the addition of the flux on the righ hand side. Faraday's law introduces an EMF source that does not fit into the path integral of the (total) electric field on the left.&lt;/p&gt;&#xA;&lt;p&gt;Note that the integral on the left accounts for all the localized electromotive forces (batteries, solar cells, peltier cells, thermocouples...) present in the branches of the loop, but not for the nonlocalized inductive EMF due to the flux cut by your closed circuit, which is the novelty introduced by Faraday.&#xA;We don't need a new law for every kind of non-inductive EMF out there: they are all accountable for in the circulation integral on the left-hand side. The term on the right, on the other hand (pun intended), is a new addition that breaks the original Kirchhoff's law and considerably extends our knowledge of the electromagnetic field. In its local, differential form&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ISXKw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ISXKw.png&quot; alt=&quot;Faraday's law differential form&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;curl E = - dB/dt&lt;/p&gt;&#xA;&lt;p&gt;Faraday's law states that the electric field curls (i.e. ceases to be irrotational) in the presence of a time-changing magnetic field. This is not 'just another kind of EMF': it is a fundamental property of the electromagnetic field that has changed our understanding of electromagnetism. It is not a coincidence that Faraday's law is one of the four celebrated Maxwell equations.&lt;/p&gt;&#xA;&lt;p&gt;The breaking of Kirchhoff's loop rule is confirmed, for example, in the second volume of Berkeley Physics: Electricity and Magnetism 3rd edition by Purcell and Morin:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/LMD1o.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LMD1o.jpg&quot; alt=&quot;Purcell and Morin on Faraday vs Kirchhoff&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Source: Purcell, Morin third edition, section 7.5&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...&lt;strong&gt;Kirchhoff Loop rule (which states that path integral of E.ds =&#xA;0 around a closed path) is no longer applicable when there is a&#xA;changing magnetic field&lt;/strong&gt;. Faraday has taken us beyond the comfortable&#xA;realm of conservative electric fields. The voltage difference between&#xA;two points now depends on the path between them.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Extended KVL and its limits&lt;/strong&gt;:&lt;br /&gt;&#xA;KVL is such a useful tool that it has been extended to work with AC and general dynamic circuits. Still, even in the 'modified', or 'generalized', or 'extended' version, in order to hold (and not to break other circuital laws, such as Ohm's law in its own generalized form) KVL requires that the sources of inductive EMF be lumped into components that will be located along the circuital path.&#xA;In its extended or generalized form, KVL can accept both the contributes of  &lt;strong&gt;lumped&lt;/strong&gt; inductive EMFs and 'voltage drops' of &lt;strong&gt;lumped&lt;/strong&gt; self-inductors in the path integral on the left of the equation.&lt;/p&gt;&#xA;&lt;p&gt;In order for extended KVL to work in a loop, it is necessary that voltages for any two points in the loop be uniquely defined. This demands for no changing flux linked by the loop itself, and in turn requires that all changing magnetic flux be confined inside the magnetic components with no appreciable leakage in the loop itself.  For further information, the reader can refer to Ramo, Whinnery, VanDuzer, &amp;quot;&lt;em&gt;Fields and Waves in Communication Electronics&lt;/em&gt;&amp;quot;, third edition (chapter 4: the electromagnetics of circuits, in particular pp. 174-179). Here is a most relevant excerpt:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ovEvs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ovEvs.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Source: Ramo Whinney VanDuzer, &amp;quot;Fields and Waves for Communication Electronics&amp;quot; 3rd edition, p. 179&lt;/p&gt;&#xA;&lt;p&gt;Extended KVL works when the sources of EMF can be lumped (and therefore localised on a particular branch) and the changing magnetic flux can be hidden inside the component, but we cannot make it work when the EMF is due to the changing flux linked by the circuit path itself. That will break KVL for good, and we are forced to consider the more general law: Faraday's Law.&lt;/p&gt;&#xA;&lt;p&gt;Note that trying to bring the delocalized inductive EMF contribution represented by the surface integral of B on the right, incorporating them inside the path integral of E on the left side in order to distribute it along the branches will alter the integrand field changing it from the total electric field Etot to the conservative part alone Ecoul of such field. This partial constituent of the (total) electric field will admit a potential function (the scalar electric potential phi) that will obey KVL but will break Ohm's law in its local form.&lt;/p&gt;&#xA;" OwnerUserId="15776" LastEditorUserId="15776" LastEditDate="2021-12-07T01:31:51.723" LastActivityDate="2021-12-07T01:31:51.723" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13940" PostTypeId="1" AcceptedAnswerId="13942" CreationDate="2021-12-07T19:32:32.183" Score="3" ViewCount="177" Body="&lt;p&gt;Every single web reference I have seen for &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacques_Tits&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacques Tits&lt;/a&gt; gives no indication that he had a middle name. All &lt;a href=&quot;https://www.knaw.nl/en/members/foreign-members/4875&quot; rel=&quot;nofollow noreferrer&quot;&gt;except one&lt;/a&gt;, which suggests his initials may have been J.L..&lt;/p&gt;&#xA;&lt;p&gt;If this is so, what was his middle name? I have not been able to discover this.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="4251" LastEditDate="2021-12-08T14:19:27.437" LastActivityDate="2021-12-08T14:19:27.437" Title="Did Jacques Tits have a middle name?" Tags="|mathematicians|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="13941" PostTypeId="2" ParentId="13937" CreationDate="2021-12-07T19:33:43.757" Score="2" Body="&lt;p&gt;Originally this was to be a comment, as I don’t really answer the question, but it’s gotten a bit long. At some later time I might look into this further and post the results here, at least if no one else sufficiently answers your question.&lt;/p&gt;&#xA;&lt;p&gt;In the past I’ve tried to track down the origin of this result (tangentially, when working on other things) and strangely, I’ve found this result, despite being so well-known and found in pretty much any real analysis text, to have been ignored by historians. Incidentally, when I say “I’ve tried to track down”, I’m not talking about looking at a few Wikipedia pages and some superficial google searches, but literature searches of the depth found in &lt;a href=&quot;https://hsm.stackexchange.com/a/8186/264&quot;&gt;answer 1&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/a/3544/264&quot;&gt;answer 2&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/a/468/264&quot;&gt;answer 3&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In any event, I suspect pretty much any of Darboux, Dini, du Bois-Reymond, Cantor, Ascoli, Volterra, Weierstrass, Harnack, Peano, Scheeffer, and maybe a few others would have easily been able to formulate and prove this result in the mid 1870s to late 1880s if someone had brought the issue to their attention (i.e. can the discontinuity set of a monotone function be uncountable), and probably the majority of the people I’ve named knew the result by 1890. For instance, Scheeffer (&lt;em&gt;Beispiel zu Theorem III&lt;/em&gt; [Example for Theorem III] on pp. 61−62 of &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF02421552.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;) showed in 1884 that any countable set can be the discontinuity set of a monotone function. Surely Baire, Borel, Lebesgue, and a few other French mathematicians knew the result by 1905 or so, and certainly Young knew it by 1907 or so (see &lt;a href=&quot;https://mathoverflow.net/a/231462/15780&quot;&gt;this answer&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I don’t know how anyone who bothered to even minimally look into the history of this result could have conceived that it was originally due to Froda, and at such a late date, which the Wikipedia page you cite originally claimed. For instance, glance over Chapter 3 (pp. 184−229) of Hahn's 1921 book &lt;a href=&quot;https://archive.org/details/theoriederreell00hahngoog/page/184/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Theorie der Reellen Funktionen&lt;/strong&gt;&lt;/a&gt;. Even if you can’t read German (and I can’t), it’s pretty clear that the huge assemblage of continuity and discontinuity minutia in this chapter leaves a result like “discontinuities of a monotone function form a countable set” in the dust. Moreover, neither Hahn nor this book were particularly unknown during the 1920s, unlike for example, Henry Blumberg, who published far-reaching extensions of Young’s 1907−1908 results that I cited above. Finally, see B. S. Thomson’s answer to &lt;a href=&quot;https://math.stackexchange.com/a/1500557/13130&quot;&gt;Proof of Froda's theorem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2021-12-07T19:33:43.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13942" PostTypeId="2" ParentId="13940" CreationDate="2021-12-07T21:28:24.587" Score="8" Body="&lt;p&gt;Technically, Belgians (and French) &lt;a href=&quot;https://www.reddit.com/r/belgium/comments/kp2ynj/do_belgians_have_middle_names/&quot; rel=&quot;noreferrer&quot;&gt;do not have middle names&lt;/a&gt;, but they can have first names consisting of two words, or several &amp;quot;first names as written on legal documents&amp;quot; (&lt;em&gt;voornamen zoals op identificatie documenten&lt;/em&gt;) that are used nowhere else, apparently. Jacques' other first name, Léon, happens to be his father's first name. Here is from &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-642-39449-2_3&quot; rel=&quot;noreferrer&quot;&gt;Buekenhout's Biography of Jacques Tits&lt;/a&gt; written on the occasion of him receiving the Abel prize:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Jacques Léon Tits was born on August 12, 1930, in Uccle (Ukkel in Dutch) in Belgium, a southern township of Brussels. All of his publications, except one, are signed Jacques Tits and this is how I [Francis Buekenhout] will refer to him from now... Jacques’s father Léon Tits (1880–1943) was a mathematician and received his education from the Université Catholique de Louvain. His mother, Maria Louisa André (1889–1957) was a piano teacher. She used to be called Louisa, and I will do so hereafter.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-12-07T21:28:24.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13943" PostTypeId="2" ParentId="13933" CreationDate="2021-12-07T23:58:32.340" Score="2" Body="&lt;p&gt;This is not a full answer but information I find useful throwing in. As @Conifold suggested in a comment, we saw a usage of square physical quantity in Kepler's third   law, and he indeed might drew inspiration from his work &lt;em&gt;Optics&lt;/em&gt; on the spreading of light:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Just as [the ratio of] spherical surfaces, for which the source of light is the center, [is] from the wider to the narrower, so the density or fortitude of the rays of light in the narrower [space], towards the more spacious spherical surfaces, that is, inversely. For according to [propositions] 6 &amp;amp; 7, there is as much light in the narrower spherical surface, as in the wider, thus it is as much more compressed and dense here than there. (&lt;a href=&quot;https://en.wikipedia.org/wiki/Inverse-square_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Source: Wikipedia&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It should be noted however that in his &lt;em&gt;Astronomia Nova&lt;/em&gt; Kepler, though he draws some similarities between the emission of light and gravity, does not recognize the inverse square distance law with respect to gravity; rather he establish there the &amp;quot;distance-law&amp;quot; (which later became the &amp;quot;area law&amp;quot; though not entirely equivalent) which missing the square. Kepler indeed compares gravity with Magnetism -- but even in the case of Magnetism no one up to his time - so it seems - has formulated any &amp;quot;distance-square&amp;quot; relation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Inverse-square_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Wikipedia&lt;/a&gt; the first to suggest a case of inverse square law, was the French astronomer Ismaël Bullialdus.&lt;/p&gt;&#xA;" OwnerUserId="14954" LastEditorUserId="14954" LastEditDate="2021-12-08T00:03:33.600" LastActivityDate="2021-12-08T00:03:33.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13944" PostTypeId="1" CreationDate="2021-12-08T04:33:43.903" Score="6" ViewCount="469" Body="&lt;p&gt;There are two widely used rules of thumb to determine which subshells are filled in a neutral atom in its ground state:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Electrons are assigned to subshells in order of increasing value of &lt;span class=&quot;math-container&quot;&gt;$n + \ell$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For subshells with the same value of &lt;span class=&quot;math-container&quot;&gt;$n + \ell$&lt;/span&gt;, electrons are assigned first to the subshell with lower &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These rules don't always hold, but that's not my concern here.  My question is: which of these rules did &lt;a href=&quot;https://en.wikipedia.org/wiki/Erwin_Madelung&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erwin Madelung&lt;/a&gt; discover?  (Both?  Just one?)&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia seems to say &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_Janet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Janet&lt;/a&gt; discovered the first in 1928:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A periodic table in which each row corresponds to one value of &lt;span class=&quot;math-container&quot;&gt;$n + \ell$&lt;/span&gt; (where the values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt; correspond to the principal and azimuthal quantum numbers respectively) was suggested by Charles Janet in 1928, and in 1930 he made explicit the quantum basis of this pattern, based on knowledge of atomic ground states determined by the analysis of atomic spectra. This table came to be referred to as the left-step table. Janet &amp;quot;adjusted&amp;quot; some of the actual &lt;span class=&quot;math-container&quot;&gt;$n + \ell$&lt;/span&gt; values of the elements, since they did not accord with his energy ordering rule, and he considered that the discrepancies involved must have arisen from measurement errors. In the event, the actual values were correct and the &lt;span class=&quot;math-container&quot;&gt;$n + \ell$&lt;/span&gt; energy ordering rule turned out to be an approximation rather than a perfect fit, although for all elements that are exceptions the regularised configuration is a low-energy excited state, well within reach of chemical bond energies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It then goes on to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1936, the German physicist Erwin Madelung proposed this as an empirical rule for the order of filling atomic subshells, and most English-language sources therefore refer to the Madelung rule. Madelung may have been aware of this pattern as early as 1926.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is &amp;quot;this&amp;quot; still rule 1?&lt;/p&gt;&#xA;&lt;p&gt;It then goes on to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1945 &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Wiswesser&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Wiswesser&lt;/a&gt; proposed that the subshells are filled in order of increasing values of the function&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$    {\displaystyle W(n,l)=n+l-{\frac {l}{l+1}}.}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is equivalent to the combination of rules 1 and 2.  So, this article seems to suggest that rule 2 was discovered by Wiswesser.  But I have my doubts.  &lt;a href=&quot;http://www.pnas.org/content/51/4/664.full.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Goudsmit and Richards&lt;/a&gt; write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Madelung discovered a simple empirical rule for neutral atoms. It consists of two parts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and then they list both 1 and 2.&lt;/p&gt;&#xA;&lt;p&gt;I have not yet managed to get ahold of Erwin Madelung's work, e.g.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;E. Madelung, &lt;em&gt;Die mathematischen Hilfsmittel der Physikers&lt;/em&gt;, Springer: Berlin, 1936&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="13875" LastEditorUserId="13875" LastEditDate="2021-12-08T17:59:05.813" LastActivityDate="2021-12-08T17:59:05.813" Title="What rule or rules did Madelung discover, exactly?" Tags="|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13945" PostTypeId="2" ParentId="13944" CreationDate="2021-12-08T05:08:15.110" Score="6" Body="&lt;p&gt;The only relevant section in E. Madelung's edited book, &lt;em&gt;Die mathematischen Hilfsmittel der Physikers&lt;/em&gt;, Springer: Berlin, 1936 is a small paragraph. It seems Madelung just called this for electron book-keeping and he had no justification for his proposition. He calls it a lexicographic order (lexikographische Ordnung).&lt;/p&gt;&#xA;&lt;p&gt;I was searching for it some other reasons several months ago. &lt;a href=&quot;https://physics.stackexchange.com/questions/656189/experimental-electron-configurations-of-copper-and-chromium-and-aufbau-principle&quot;&gt;Question in SE Physics&lt;/a&gt;. The original and the (machine but edited) translations is shared.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/s5IKx.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/s5IKx.png&quot; alt=&quot;Image&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;15 Atomic structure (electron catalog) (to p. 301).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The eigenfunction of an atom, consisting of &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; electrons and &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; -times positively&#xA;charged nucleus, can be constructed in the case of removed degeneracy in first approximation as a product of &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; hydrogen eigenfunctions (cf. p. 356), each of which is defined by four quantum numbers &lt;span class=&quot;math-container&quot;&gt;$n, l, m, s$&lt;/span&gt; defined by &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;0, n-1 \geqq l \geqq 0, s=\pm \frac{1}{2}, m \leqq l$&lt;/span&gt;. According to the Pauli's principle, no two of these functions may coincide in all four quantum numbers. According to the Bohr&#xA;principle of structure, an atom with &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; electrons is formed from an atom with &lt;span class=&quot;math-container&quot;&gt;$(Z-1)$&lt;/span&gt; electrons by adding another one (and increasing the nuclear charge by 1) without changing the quantum numbers of the already existing electrons. Therefore a catalog can be set up, from whose in each case &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; first positions the atom is built up in the basic state (cf. the table p. 360 ).&lt;/p&gt;&#xA;&lt;p&gt;The ordering principle of this catalog is a lexicographic order according to the numbers &lt;span class=&quot;math-container&quot;&gt;$(n+l), n, s, m .$&lt;/span&gt; A theoretical justification of just this arrangement is not yet available. One reads from it:&lt;/p&gt;&#xA;&lt;p&gt;1 The periodic system of the elements. Two atoms are homologous, if in each case their &amp;quot;last electron&amp;quot; in the &lt;span class=&quot;math-container&quot;&gt;$l, m, s$&lt;/span&gt;&#xA;coincides.&lt;/p&gt;&#xA;&lt;p&gt;2 The spectroscopic character of the basic term, entered in column 10 . There is namely &lt;span class=&quot;math-container&quot;&gt;$|\Sigma m|=0,1,2,3 \ldots$&lt;/span&gt; the character&lt;span class=&quot;math-container&quot;&gt;$S P, D, F, G, H, I \ldots$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2|\Sigma s|+1)$&lt;/span&gt; the multiplicity.&lt;/p&gt;&#xA;&lt;p&gt;3 The possibilities for excited states (possible terms), where not all &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; electrons are in the first &lt;span class=&quot;math-container&quot;&gt;$Z$&lt;/span&gt; positions of the catalog.&lt;/p&gt;&#xA;&lt;p&gt;The catalog is the representation form of an empirical rule. It idealizes the experience, because in some cases deviations are observed.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-12-08T05:08:15.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13946" PostTypeId="1" CreationDate="2021-12-08T05:56:58.500" Score="2" ViewCount="585" Body="&lt;p&gt;Obviously not all cases are such that a theory is presented without experiment, and then the necessary experiment is done separately by another person...&lt;/p&gt;&#xA;&lt;p&gt;But when this IS the situation, to whom does the prize generally go, in this case?&lt;/p&gt;&#xA;&lt;p&gt;Specifically I'm curious about astrophysics / dark matter / dark energy. In case there isnt enough history there to answer on these, is why I originally asked the general question.&lt;/p&gt;&#xA;" OwnerUserId="15822" LastEditorUserId="15823" LastEditDate="2021-12-13T01:17:36.270" LastActivityDate="2021-12-13T01:17:36.270" Title="Do Nobel prizes tend to go to theorists or experimenters?" Tags="|theoretical-physics|experimental-physics|publishing|nobel-prize|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13947" PostTypeId="2" ParentId="13946" CreationDate="2021-12-08T08:27:11.923" Score="6" Body="&lt;p&gt;&lt;em&gt;Statistically&lt;/em&gt;, experimenters are typically favored. One reason is that theories are only Nobelized when confirmed, and that may take too long for getting the prize. Another is that new experimental results tend to be less controversial and more common than major new theories. A prominent example of the opposite is the Nobel for DNA, where Watson and Crick got the prize, but &lt;a href=&quot;https://www.scr.org/get-connected/south-coast-repertorys-stories/south-coast-repertory%27s-stories/2019/02/25/a-nobel-experiment-rosalind-franklin-and-the-prize&quot; rel=&quot;noreferrer&quot;&gt;Wilkins did not&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://www.insidescience.org/news/how-not-win-nobel-prize-physics&quot; rel=&quot;noreferrer&quot;&gt;How (Not) to Win a Nobel Prize in Physics on Inside Science&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Roughly two-thirds of Nobel Prizes in physics have gone to experimentalists and one-third to theorists. Theorists in general run the risk of being too far ahead of their time. They may come up with groundbreaking theories that can’t be proven until much later, when experimental techniques have advanced.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;One of physics’ most famous theorists, Albert Einstein, never officially received a Nobel Prize for arguably his most famous theory -- relativity. Though anti-Semitism likely played a role in the Nobel committee’s reluctance to recognize Einstein, the committee maintained that it was because relativity hadn’t been proven. The 1921 physics prize was ultimately given to Einstein in 1922 (long story) for his work on theoretical physics in general, and especially for discovering the law of the photoelectric effect -- the phenomenon where electricity can be generated by exposing certain materials to light. Stephen Hawking, one of the world’s most famous astrophysicists, passed away earlier this year&lt;/em&gt; [2018], &lt;em&gt;without earning the Nobel Prize.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-12-08T08:27:11.923" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13948" PostTypeId="1" AcceptedAnswerId="13951" CreationDate="2021-12-08T18:28:41.900" Score="12" ViewCount="596" Body="&lt;p&gt;While reading about Borel sets I came across comments regarding the &lt;a href=&quot;https://en.wikipedia.org/wiki/Nikolai_Luzin#Luzin_affair_of_1936&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luzin Affair&lt;/a&gt;, when Luzin was the subject of various accusations made by colleagues.&lt;/p&gt;&#xA;&lt;p&gt;These accusations appear to have been both political and professional in nature - for example, that Luzin  “felt no shame in declaring the discoveries of his students to be his own achievements”.  Some of his accusers may have been coerced into participating - e.g., Aleksandrov and Kolmogorov.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Great_Purge#Intelligentsia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Intelligentsia&lt;/a&gt; section of the wikipedia article on the Great Terror lists only people from the arts - writers, poets, dramatists, and the like - as victims of the purge, many of whom were put to death.  No mathematicians are listed and although elsewhere I can find numerous references to scientists being victims, I can find no further references to mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;It seems unlikely that the Luzin Affair unique in Soviet mathematics. Are there other examples of prominent  mathematicians being victims of the purge?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;A detailed and well reference account of the Luzin Affair is available as &lt;a href=&quot;http://old.math.nsc.ru/LBRT/g2/english/ssk/case_e.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Tragedy of Mathematics in Russia&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2024-01-05T17:44:22.820" LastActivityDate="2024-01-05T17:44:22.820" Title="How far did the Great Terror reach into Soviet mathematics?" Tags="|mathematicians|soviet-union|20th-century|russia|" AnswerCount="2" CommentCount="9" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13949" PostTypeId="1" CreationDate="2021-12-09T14:48:19.070" Score="1" ViewCount="99" Body="&lt;p&gt;I remember having read on Wikipedia some time ago the page of a 20th century physicist. There was a subsection about his strategy for problem solving that involved drawing a concept map of the objects involved in the problem and then switching them around. I can't however remember the precise anecdote, nor the name of the physicist.&lt;/p&gt;&#xA;&lt;p&gt;Details that I do remember, ranked by confidence that I recall them correctly:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Male physicist&lt;/li&gt;&#xA;&lt;li&gt;Used concept maps (or mind maps) and then switched arrows and nodes around for out of the box thinking&lt;/li&gt;&#xA;&lt;li&gt;There was an anecdote along the lines of moving the Earth from its orbit (which would point to an astrophysicist)&lt;/li&gt;&#xA;&lt;li&gt;Might have been related the Los Alamos laboratory&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Does anybody have a clue who it might be? My initial instinct was either Fermi or Szilard but their respective Wikipedia page does not contain anything relative to creative problem solving strategies.&lt;/p&gt;&#xA;" OwnerUserId="15830" LastEditorUserId="15981" LastEditDate="2022-01-11T06:16:35.060" LastActivityDate="2022-01-11T06:16:35.060" Title="What physicist was famous for his out-of-the-box problem solving methodology using concept maps?" Tags="|physicists|" AnswerCount="0" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13950" PostTypeId="2" ParentId="13948" CreationDate="2021-12-09T20:48:28.623" Score="9" Body="&lt;p&gt;First of all, the terminology &amp;quot;Great Terror&amp;quot; is ambiguous: Some authors limit the period to 1937-38, some to 1934-39, etc. In the answer, I will be using the time period 1929-1953. I can explain why if you like.&lt;/p&gt;&#xA;&lt;p&gt;Here are few more names:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Dmitri_Egorov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dmitri Egorov&lt;/a&gt;. He was a PhD advisor of Luzin, Alexandroff, Petrovsky and others.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Per Wikipedia article about him:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Egorov held spiritual beliefs to be of great importance, and openly defended the Church against Marxist supporters after the Russian Revolution. He was elected president of the Moscow Mathematical Society in 1921, and became director of the Institute for Mechanics and Mathematics at Moscow State University in 1923. He also edited the journal Matematicheskii Sbornik of the Moscow Mathematical Society. However, because of Egorov's stance against the repression of the Russian Orthodox Church, he was dismissed from the Institute in 1929 and publicly rebuked. In 1930 he was arrested and imprisoned as a &amp;quot;religious sectarian&amp;quot;, and soon after was expelled from the Moscow Mathematical Society. Upon imprisonment, Egorov began a hunger strike until he was taken to the prison hospital, and eventually to the house of fellow mathematician Nikolai Chebotaryov, where he died. He was buried in Arskoe Cemetery in Kazan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;A more complex case:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Abramovich_Rokhlin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vladimir Rokhlin&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Per Wikipedia article about him:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He volunteered for the army in 1941, leading to four years as a prisoner of a German war camp. During this time he was able to hide his Jewish origins from the Nazis. Rokhlin was liberated by the Soviet military in January 1945. He then served as a German language translator for the 5th Army of the Belorussian front. In May 1945 he was sent to a Soviet 'verification camp' for former prisoners of war. In January 1946 he was transferred to another camp to determine if he was an &amp;quot;enemy of the Soviet.&amp;quot; Rokhlin was cleared in June 1946 but was forced to remain in the camp as a guard. Due to intercession by mathematicians Andrey Kolmogorov and Lev Pontryagin, he was released in December 1946 and allowed to return to Moscow, after which he returned to mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;An even more complex case:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lev_Schnirelmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lev Schnirelman&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schnirelmann committed suicide in Moscow on 24 September 1938, for reasons that are not clear. According to Lev Pontryagin's memoir from 1998, Schnirelmann gassed himself, due to depression brought on by feelings of inability to work at the same high level as earlier in his career. On the other hand, according to an interview Eugene Dynkin gave in 1988, Schnirelman took his own life after the NKVD tried to recruit him as an informer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Russian Wikipedia page provide few more details:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schnirelman told his friend L. A. Lyusternik that he had done something terrible under pressure, which indirectly agrees with the memoirs of E. B. Dynkin:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sofya Aleksandrovna Yanovskaya told me that he left a note: &amp;quot;I am dying honest before my comrades and the Soviet regime.&amp;quot; He was forced to report. ... Sofya Aleksandrovna told me that the NKVD employee who recruited him was shot. But from this, of course, Schnirelman did not come back to life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit. More names of mathematicians (none of them was particularly prominent) who were persecuted during the &amp;quot;Great Terror&amp;quot; are given in&lt;/p&gt;&#xA;&lt;p&gt;G.G.Lorentz, &lt;em&gt;Mathematics and Politics in the Soviet Union from 1928 to 1953&lt;/em&gt;, Journal of Approximation Theory, Volume 116, Issue 2, (2002) 169-223.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://pdf.sciencedirectassets.com/272427/1-s2.0-S0021904500X01141/1-s2.0-S0021904502936701/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEAAaCXVzLWVhc3QtMSJHMEUCIDnLu%2BhqDfL6vT7h6mLFiE42fAcNyRH1sq6eBMuYlPpGAiEA%2Bz3VGLQWBeEH2GYWGgVmKjObYpmzNqexNKYadc6HKKgqgwQI2f%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAEGgwwNTkwMDM1NDY4NjUiDE5NHm32333mbE7sqyrXAz7EPUUWUMZGL68l1Sty6i9gtPvoRjQ2AnR%2Bl30n8DZYOebSfMHfTxL1%2Fsyen64e3wVylZ2S1Y8H3UD57OGqG3y2%2Fr%2B083mg9hsJTqOum3Ms1oxraGVEHhN7yPV%2BFtvlOADq4Ryxri%2BZNmIBt9k4mXtd%2BQVF8Dl%2FNseEuBweC9buFfACOmtb%2FiOZucBNnQy%2BnzRLhsJMb%2FZjjzWxmwC4vcSQREm6tdoSPsfVN%2BmBsOiLjYEk2eHPxSkKhffgb5ubmMVWTazyUQnyY1huPelPHAdtwwRh9nndU0Fh3fu6tThgppnEmsOuawcdAa0qsMFiZHAw9dKd%2B6zYxYZFtHZXbKoFj8wbKJrcG0eLbiuLDIQA43qlo%2BcPN0Tczr7aoQLhipD5RJ625jU1C1LOagFA7eY%2FhurlMGbZoReqRbpWPGCs99IkL9YAcIkTMk9zIIfCOShejI8%2FhytAXUirWK4Lk7vX3Zf1%2F3%2B0%2FA1pxOdGgc9ICiptE9hOlcvbC2Yu2ocQtKZT9UQrGSI0HYh6c2Jx36Yu3MO8RdKALI3WFZwZH00wHXDP5NPgygBoobOZ3JhH1Dm%2FNBouxQFT2w37Xz1biNCBoeYg8n5pGnMCOaK3MhAL5rOpELF6JzD9x8iNBjqlAQjSudDYo64lDAtD4A6tX0HxkfjgecUzMyGe0eP6i9ALHIabaxokijpWxFrjYk9UABepzuMzt1F%2FqIhsegkztBhf6AXvKGiF%2Bv3uGNuLAppNY6fBK1j%2FBDvLFggYC%2B%2FQprOh8x%2BfUAIUnv68VDpQerffqh82GweGk%2Byr0dANp7GlzjyW5VSygtom6pS3pgFNdOomcPAX9qNQX1iQhhe6%2F83KuUHYCA%3D%3D&amp;amp;X-Amz-Algorithm=AWS4-HMAC-SHA256&amp;amp;X-Amz-Date=20211209T162313Z&amp;amp;X-Amz-SignedHeaders=host&amp;amp;X-Amz-Expires=300&amp;amp;X-Amz-Credential=ASIAQ3PHCVTYSA2352ZI%2F20211209%2Fus-east-1%2Fs3%2Faws4_request&amp;amp;X-Amz-Signature=ce21f9b5936a1c026937aff518718c0a8342f782ec0a8047e568a6b15c39778c&amp;amp;hash=b03f8afc1eeaf61fbc6768c356097baf0d8f958c356d2f05e91bff0fb5bf3c60&amp;amp;host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&amp;amp;pii=S0021904502936701&amp;amp;tid=spdf-9db359b1-7bd2-4442-a82d-38c3d2063935&amp;amp;sid=edf79978860d1443b21ac677ca018a4edca0gxrqa&amp;amp;type=client&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="4251" LastEditDate="2023-12-28T23:42:28.133" LastActivityDate="2023-12-28T23:42:28.133" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13951" PostTypeId="2" ParentId="13948" CreationDate="2021-12-09T23:01:42.747" Score="12" Body="&lt;p&gt;&amp;quot;Great terror&amp;quot; is indeed ambiguous. Essentially the whole period from 1917 to 1953 was the epoch of terror. To the mathematicians murdered by NKVD/KGB listed in Moishe Kohan's&#xA;answer I can add:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fritz_Noether&quot; rel=&quot;noreferrer&quot;&gt;Fritz Noether&lt;/a&gt; (He escaped to Soviet Union from the Nazis)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/J%C3%B3zef_Marcinkiewicz&quot; rel=&quot;noreferrer&quot;&gt;Joseph Marcinkiewicz&lt;/a&gt; (One of the 22,000 Polish officers murdered in &amp;quot;Katyn massacres&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%92%D0%B8%D1%82%D1%82,_%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80_%D0%90%D0%B4%D0%BE%D0%BB%D1%8C%D1%84%D0%BE%D0%B2%D0%B8%D1%87&quot; rel=&quot;noreferrer&quot;&gt;Alexandr Witt&lt;/a&gt;&#xA;&lt;a href=&quot;https://de.wikipedia.org/wiki/Alexander_Adolfowitsch_Witt&quot; rel=&quot;noreferrer&quot;&gt;(in German Wikipedia)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The terror decreased but did not stop with Stalin's death.&#xA;For example&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://uk.wikipedia.org/wiki/%D0%93%D0%BB%D0%B0%D0%B7%D0%BC%D0%B0%D0%BD_%D0%86%D0%B7%D1%80%D0%B0%D1%97%D0%BB%D1%8C_%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D0%B8%D1%87&quot; rel=&quot;noreferrer&quot;&gt;Israel Glazman&lt;/a&gt;&#xA;&lt;a href=&quot;https://de.wikipedia.org/wiki/Israel_Markowitsch_Glasman&quot; rel=&quot;noreferrer&quot;&gt;(in German Wikipedia)&lt;/a&gt; committed&#xA;suicide after several &amp;quot;conversations&amp;quot; with KGB (as Lev Shnirelman in&#xA;Moishe Kohan's answer did).&lt;/p&gt;&#xA;&lt;p&gt;In general, there were fewer mathematicians victims than &lt;a href=&quot;https://hsm.stackexchange.com/questions/13634/which-physicists-died-very-young-or-in-a-tragic-way/13635#13635&quot;&gt;physicists&lt;/a&gt; and &lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%91%D0%B8%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8,_%D1%80%D0%B5%D0%BF%D1%80%D0%B5%D1%81%D1%81%D0%B8%D1%80%D0%BE%D0%B2%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D0%B2_%D0%A1%D0%A1%D0%A1%D0%A0&quot; rel=&quot;noreferrer&quot;&gt;biologists&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-12-11T02:27:09.093" LastActivityDate="2021-12-11T02:27:09.093" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13952" PostTypeId="2" ParentId="13936" CreationDate="2021-12-10T01:21:18.277" Score="3" Body="&lt;p&gt;I believe that the comment by &lt;a href=&quot;https://hsm.stackexchange.com/users/9563/stephan-matthiesen&quot;&gt;@Stephan Matthiesen&lt;/a&gt; contains the crucial point (so the credit for the answer should go to him) : -- Boyle's work did not include quantitative temperature effects because he worked in the age before thermometry.&lt;/p&gt;&#xA;&lt;p&gt;Lack of the techniques of thermometry also limited, in some respects, the work of Boyle's contemporary Newton, and this aspect is mentioned, along with references that also have relevance here, in the following answer:-- &lt;a href=&quot;https://hsm.stackexchange.com/questions/8438/why-is-there-little-scholarship-devoted-to-book-ii-of-newtons-principia/8440#8440&quot;&gt;Why is there little scholarship devoted to Book II of Newton&amp;#39;s Principia?&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2021-12-10T01:21:18.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13953" PostTypeId="1" AcceptedAnswerId="13954" CreationDate="2021-12-10T11:51:38.710" Score="3" ViewCount="311" Body="&lt;p&gt;Why was the idea of writing, formalising and creating new proofs brought about? For example, why even though we have found hundreds of proofs of the Pythagorean Theorem are we still trying to find more and refine the current proofs that we have?&lt;/p&gt;&#xA;&lt;p&gt;The motivation for this question was basically to feed my own curiosity, as writing proofs for Trig and Calculus are my favourite parts of math.&lt;/p&gt;&#xA;" OwnerUserId="15833" LastEditorUserId="15833" LastEditDate="2022-01-11T01:44:21.573" LastActivityDate="2022-01-11T01:44:21.573" Title="Origins of proof in Mathematics" Tags="|mathematics|mathematicians|reference-request|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13954" PostTypeId="2" ParentId="13953" CreationDate="2021-12-10T18:27:19.553" Score="4" Body="&lt;p&gt;One day in my introductory calculus course, a professor observed that &amp;quot;there are two kinds of obvious statements: those that are true and those that are false&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In physical theories, the golden standard for &amp;quot;truth&amp;quot; is agreement with reality. In math it's not so easy: a million right-angled triangles that satisfy the Pythagorean Theorem do not rule out a right-angled triangle that violates it.  Somewhere in the sands of time must have evolved the realization that mathematical statements needed to be proved starting from statements that were either obvious or simply stipulated to be true.  This may have originated as a way of settling arguments, as a way of building knowledge (you often learn a lot when you try to prove something, whether it's obvious or non-obvious), as a guard against error, or as an end in itself.&lt;/p&gt;&#xA;&lt;p&gt;There are of course two kinds of proof: those that are valid and those that are not.  Mathematical proofs during history have often been less than rigorous, or had subtle flaws.  Sometimes the theorem to be proved was actually false.  Sometimes the proof was faulty.  Formalization became increasingly necessary so that no obvious step or assumption was left unscrutinized.&lt;/p&gt;&#xA;&lt;p&gt;As far as the apparent drive to &amp;quot;find more proofs of the Pythagorean Theorem&amp;quot;, I think that's the wrong way of looking at it.  More often than not, I expect that most of these proofs come about as bi-product of noticing that other proofs, theorems and techniques can lead to new proofs of the Pythagorean Theorem.  Often prized are proofs that are short or or beautiful or intuitive or expose unexpected connections with other mathematical ideas.&lt;/p&gt;&#xA;&lt;p&gt;Circling back to your current favourite parts of math, one interesting category of Trig proofs is the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_without_words&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proof Without Words&lt;/a&gt;.  And good example of an &amp;quot;obvious&amp;quot; Calculus theorem is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Intermediate_value_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Intermediate Value Theorem&lt;/a&gt;, which is what I recall my prof was referring to.&lt;/p&gt;&#xA;" OwnerUserId="11204" LastActivityDate="2021-12-10T18:27:19.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13955" PostTypeId="1" AcceptedAnswerId="13962" CreationDate="2021-12-11T03:12:34.377" Score="1" ViewCount="117" Body="&lt;p&gt;Where did this idea of proof originate from, was it the Greeks, Babylons, Egyptians etc?&lt;/p&gt;&#xA;&lt;p&gt;The motivation for this question was basically to feed my own curiosity as mentioned &lt;a href=&quot;https://hsm.stackexchange.com/questions/13953/origins-of-proof-in-mathematics&quot;&gt;here&lt;/a&gt;. This is basically a continuation of my question in compliance with this sites policy to post one question per post&lt;/p&gt;&#xA;" OwnerUserId="15833" LastEditorUserId="26" LastEditDate="2021-12-11T09:33:03.443" LastActivityDate="2021-12-12T02:20:50.253" Title="Where did this idea of proof originate from" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13957" PostTypeId="1" CreationDate="2021-12-11T10:07:14.283" Score="17" ViewCount="2557" Body="&lt;p&gt;The name &amp;quot;Big Bang theory&amp;quot; was coined by Sir Fred Hoyle. What did scientists originally call the Big Bang model?&lt;/p&gt;&#xA;" OwnerUserId="15839" LastEditorUserId="5989" LastEditDate="2021-12-11T11:59:23.403" LastActivityDate="2022-01-06T18:52:01.147" Title="What was the Big Bang model originally called?" Tags="|terminology|cosmology|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13958" PostTypeId="2" ParentId="13957" CreationDate="2021-12-11T10:49:59.050" Score="8" Body="&lt;p&gt;Georges Lemaître called it “hypothesis of the primeval atom” in 1931. See &lt;a href=&quot;https://web.archive.org/web/20130117044852/http://www.amnh.org/education/resources/rfl/web/essaybooks/cosmic/p_lemaitre.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Georges Lemaître, Father of the Big Bang&lt;/em&gt;&lt;/a&gt;, which is an excerpt from &lt;em&gt;Cosmic Horizons: Astronomy at the Cutting Edge&lt;/em&gt;, edited by Steven Soter and Neil deGrasse Tyson. He also called it “The Beginning of the World”.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2021-12-11T10:49:59.050" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13959" PostTypeId="2" ParentId="13957" CreationDate="2021-12-11T10:53:32.857" Score="24" Body="&lt;p&gt;Lemaitre, who proposed the first version of &amp;quot;Big Bang&amp;quot; in 1927, called it &lt;a href=&quot;https://en.wikipedia.org/wiki/Big_Bang#Development&quot; rel=&quot;noreferrer&quot;&gt;Primeval Atom&lt;/a&gt; hypothesis since late 1930s, notably in the &lt;a href=&quot;https://www.google.com/books/edition/The_Primeval_Atom/eSoFAAAAMAAJ?hl=en&quot; rel=&quot;noreferrer&quot;&gt;1950 book&lt;/a&gt; of this name. However, it was rather different from the subsequently adopted version, and singularity cosmologies did not gain much traction until 1950s to get a common label. By then, when they came to be viewed as a credible alternative, Hoyle's &amp;quot;big bang&amp;quot; nickname from 1949 stuck. On Lemaitre's motivations see &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/00033790701317692&quot; rel=&quot;noreferrer&quot;&gt;Kragh-Lambert, The Context of Discovery: Lemaître and the Origin of the Primeval-Atom Universe&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is generally acknowledged that the priority of the Big Bang idea belongs to Lemaître. As the eminent cosmologist Jim Peebles wrote in Physical Cosmology, the first textbook of the hot Big Bang standard model: ‘According to the usual criterion for establishing credit for scientific discoveries Lemaître deserves to be called the “Father of the Big Bang Cosmology”.’... The primeval atom did certainly not have factual status in the 1930s, and today we are convinced that what Lemaître ‘discovered’ has never existed. Our present picture of the very early universe has almost nothing in common with the gigantic atomic nucleus that Lemaître envisaged. Nonetheless... Lemaître may be said to have discovered the Big Bang in the form of the now abandoned hypothesis of the primeval atom.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;He did not care to distinguish between the terms ‘nucleus’ and ‘atom’, for the atomic number of the primeval atom and its decay products was so large that it made the distinction illusory. He later spoke of the primeval atom as a kind of isotope of the neutron, a metaphorical terminology he first adopted in an unpublished manuscript from the late 1930s:&lt;/em&gt; &lt;br&gt;&#xA;&amp;quot;&lt;em&gt;This hypothesis [of the primeval atom] is attractive only if we can think of the atom as a quantum unit. Then, we start really from ideally simple conditions. We have surely to think of the primeval atom as a base nucleus without surrounding electrons, an isotope of the neutron.&lt;/em&gt;&amp;quot;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-12-11T10:53:32.857" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13960" PostTypeId="2" ParentId="5907" CreationDate="2021-12-11T21:36:23.793" Score="2" Body="&lt;p&gt;The / sign before the unit (e.g. mass/kg) would indeed be the most correct mathematical way for expressing a physical quantity &lt;em&gt;in theory&lt;/em&gt;. However, I guess it falls short of being in wide usage because many units already contain a slash in their common representation (e.g. &lt;em&gt;m/s&lt;/em&gt;, &lt;em&gt;J/K&lt;/em&gt;, etc, assuming the equivalent power notation such as &lt;em&gt;m.s^-1&lt;/em&gt; is seen unpractical for simplest units). An extra slash would simply make the notation awkward (or confusing in worst cases: in &lt;em&gt;m/kg&lt;/em&gt; and &lt;em&gt;m/s&lt;/em&gt;, the &lt;em&gt;m&lt;/em&gt; obviously doesn't have the same meaning, and even the same nature; of course, in that quite artificial case, the issue is not with the notation &lt;em&gt;per se&lt;/em&gt;, but with the choice of a conflicting symbol).&#xA;Brackets or parentheses are less prone to such shortcomings.&lt;/p&gt;&#xA;" OwnerUserId="15844" LastEditorUserId="15844" LastEditDate="2021-12-11T21:52:39.947" LastActivityDate="2021-12-11T21:52:39.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13961" PostTypeId="2" ParentId="13938" CreationDate="2021-12-12T02:12:17.407" Score="2" Body="&lt;p&gt;As you point out, the notion of an action is not particularly attached to that of a group. You can have actions from a more complex mathematical gadget like Lie algebra actions or ring actions or a simpler one, like a monoid or a semigroup.&lt;/p&gt;&#xA;&lt;p&gt;The group action is historically the most prominant one and hence the notion of an action is attached to that particular kind of action. For example, building an associated vector bundle, such as the adjoint bundle in Yang-Mills, requires a group action.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-12-12T02:12:17.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13962" PostTypeId="2" ParentId="13955" CreationDate="2021-12-12T02:20:50.253" Score="4" Body="&lt;p&gt;It is believed that the idea of proof was originated from the Greeks. We don't have any proofs written by Babylonians or Egyptians. Of course it does not mean that they don't have any proofs. Maybe they have some proofs but they were much rarer than textbooks that have algorithms for solving mathematical problems. Or maybe they taught some forms of proofs only orally. It's much harder to write down the proof than explain it orally. Old Babylonians knew some sophisticated facts, like Pythagorean theorem. So they had to have some justifications for them, but we simply don't know how they justify it.&lt;/p&gt;&#xA;&lt;p&gt;The earlier proofs that we know are in Euclid's Elements. However it is certain that the idea of proofs were known earlier. For example Thales (who is the first known Greek mathematician) was said to proof that diameter divide the circle by two equal parts. The fact that Thales had to prove such obvious statement means that the notion of formal proof was known even at that time. However some historian do not believe this fact.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2021-12-12T02:20:50.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13963" PostTypeId="2" ParentId="13938" CreationDate="2021-12-12T04:22:58.197" Score="2" Body="&lt;p&gt;What you define &lt;strong&gt;is&lt;/strong&gt; monoid action. A group must act by permutations and the inverse element must act as inverse permutation. The second condition follows from the first and your two laws. Because the product of permutations corresponding to &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a^{-1}$&lt;/span&gt; must be the identity permutation.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2021-12-12T05:22:34.397" LastActivityDate="2021-12-12T05:22:34.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13964" PostTypeId="2" ParentId="13953" CreationDate="2021-12-12T11:28:18.170" Score="4" Body="&lt;p&gt;Proof has ancient antecendents lost to the modern world.&lt;/p&gt;&#xA;&lt;p&gt;Lets say I said to someone, I can throw a stone fifty yards.&lt;/p&gt;&#xA;&lt;p&gt;And she goes, prove it.&lt;/p&gt;&#xA;&lt;p&gt;And so I pick up a stone and throw it.&lt;/p&gt;&#xA;&lt;p&gt;Then we measure the distance, and its fifty four yards.&lt;/p&gt;&#xA;&lt;p&gt;So assertion proved.&lt;/p&gt;&#xA;&lt;p&gt;This is proof by demonstration. And probably the earliest kind of proof developed.&lt;/p&gt;&#xA;&lt;p&gt;But what has this got to do with mathematical proof? Everything. The notion of proof wasn't developed from nothing in mathematics, but imported from elsewhere. This is why it is wrong to think that the ancient mathematicians of Egypt and Babylonia had no notion of proof. They did. It was simply different from todays. Once an idea is newly imported into a new domain, it evolves in ways that is consonant and peculiar to this new domain. Thus the ancient Greeks did not invent proof by inventing the notion of axioms, they were merely thinking through what proof - an idea already understood - meant in mathematics.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-12-12T11:28:18.170" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13965" PostTypeId="2" ParentId="13894" CreationDate="2021-12-12T11:58:41.477" Score="1" Body="&lt;p&gt;Basically, with the invention of sets and modern algebra it was discovered that one can not only add, subtract, multiply and divide things of an algebraic gadget like a group or a ring. But you can also do these things at the level of an algebraic gadget. For example, you can multiply two groups together or add two rings together.&lt;/p&gt;&#xA;&lt;p&gt;However, whilst there was a profusion of new ways of adding and multiplying and there was a family resemblence between these constructions there were enough differences that a systematic method of saying what it means to add two objects together and how to multiply two things together in a systematic fashion was lacking.&lt;/p&gt;&#xA;&lt;p&gt;Well, this was the case until category theory appeared. Then it was understood that addition, called here the coproduct, was dual to multiplication, called here the product. And this was systematic in all categories. Moreover, the product was an example of a limit and the coproduct, an example of a colimit.&lt;/p&gt;&#xA;&lt;p&gt;It turns out that the direct sum and direct product for vector spaces  are isomorphic (ie the same) for finite-dimensional vector spaces but differ for infinite-dimensional spaces. Hence, their differences, at tge finite-dimensional level is not apparent. In infinite dimensions, it is the direct sum that is the coproduct whilst the direct product is the product. Its probably worth adding that the tensor product of algebras, a vector space equipped with a notion of multiplication, is the coproduct in the category of algebras which shows that coproducts can be quite different to what one might imagine them to be.&lt;/p&gt;&#xA;&lt;p&gt;Since Peano gave the first formal description of vectors in 1888, the notion of a direct sum and product of vector spaces will have to be after this. Its worth noting that Peano axioms were inspired by Grassmann's invention of the exterior algebra, his &lt;em&gt;Die Lineale Ausdehnungslehre&lt;/em&gt; first published in 1844.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-12-12T11:58:41.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13966" PostTypeId="2" ParentId="13933" CreationDate="2021-12-12T13:27:43.150" Score="1" Body="&lt;p&gt;Whilst geometry is most often thought of as a mathematical discipline, it is also inherently physical. That's why geometry is called what it is -- the measurement of the earth -- and by this they mean lengths, angles, areas and volumes and the addition and division of such.&lt;/p&gt;&#xA;&lt;p&gt;Hence I would suggest the area of a square was the first known physical phenomenon to exhibit squaring.  It's also probably why we call it a square.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-12-13T05:13:37.503" LastActivityDate="2021-12-13T05:13:37.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13967" PostTypeId="1" AcceptedAnswerId="14086" CreationDate="2021-12-13T05:29:42.220" Score="1" ViewCount="321" Body="&lt;p&gt;Wikipedia says that the magnetic vector potential was discovered by Neumann in 1845, by Weber in 1846 and Lord Kelvin in 1847. It does not mention who discovered the electric scalar potential.&lt;/p&gt;&#xA;&lt;p&gt;Hence my question - who discovered the electric scalar potential?&lt;/p&gt;&#xA;&lt;p&gt;(This question has been asked before on this site but only received an answer about the discovery of the gravitational potential.)&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2022-01-18T18:10:23.050" LastActivityDate="2022-01-19T14:45:59.353" Title="Who discovered the electric potential?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13968" PostTypeId="2" ParentId="13917" CreationDate="2021-12-15T00:10:57.583" Score="2" Body="&lt;p&gt;The basic &amp;quot;old quantum theory&amp;quot; facts, such as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bohr model&lt;/a&gt;, fitting the phenomenological Rydberg formula &lt;span class=&quot;math-container&quot;&gt;$E_n \propto 1/n^2$&lt;/span&gt; were common knowledge among research physicists at the time, despite deep-seated conceptual malaise. WP details:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to the Maxwell theory the frequency &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; of classical radiation is equal to the rotation frequency &lt;span class=&quot;math-container&quot;&gt;$\nu_{rot}$&lt;/span&gt; of the electron in its orbit, with harmonics at &lt;em&gt;integer multiples of this frequency&lt;/em&gt;. This result is obtained from the Bohr model for jumps between energy levels &lt;span class=&quot;math-container&quot;&gt;$E_n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$E_{n−k}$&lt;/span&gt; when &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is much smaller than &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. These jumps reproduce the frequency of the &lt;em&gt;k&lt;/em&gt;-th harmonic of orbit &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. For sufficiently large values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In formulas, you get an almost &lt;em&gt;linear&lt;/em&gt; formula in &lt;em&gt;k&lt;/em&gt;, so, then, harmonics of a single frequency, (just like the linear harmonic oscillator featured later in Heisenberg's paper!),&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Delta E\propto {-1\over n^2} - {-1\over (n-k)^2}= \frac{2kn+k^2}{n^2(n-k)^2} \approx {2k\over n^3}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Heisenberg &lt;a href=&quot;http://www.quantum-cognition.de/texts/1925%20paper.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;reinterpreted&lt;/a&gt;, essentially out of his magic hat, the quantities involved in these upper-level transitions  of the hydrogen spectrum, aggressively avoiding invocation of unobservables such as electron trajectories, which were a disquieting feature of the old quantum theory.&lt;/p&gt;&#xA;&lt;p&gt;He modeled the system as a &amp;quot;black box&amp;quot; with arbitrary formal features obeying a strange noncommutative sudoku, that produce these linear spacings in the spectrum.  The WP article you are citing identifies these infinite matrices with the routine oscillator matrices of today.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2021-12-16T17:00:09.553" LastActivityDate="2021-12-16T17:00:09.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13969" PostTypeId="1" CreationDate="2021-12-15T11:45:54.413" Score="5" ViewCount="216" Body="&lt;p&gt;I noticed that a large number of theories within graph theory are from Eastern European graph theorists, specifically Hungarian graph theorists. What is the relation between Eastern Europe (specifically Hungary) and the emergence of these theories within graph theory? Is there anything written about the history of this? I cannot find anything about this topic while I find it interesting to think about the history of some political/academic/societal state and the emergence of theories within a particular field.&lt;/p&gt;&#xA;" OwnerUserId="15855" LastEditorUserId="4251" LastEditDate="2021-12-15T12:00:30.820" LastActivityDate="2021-12-18T17:55:57.673" Title="Why are there relatively many Eastern European (specifically Hungarian) graph theorists?" Tags="|mathematicians|mathematics-social-history|combinatorics|graph-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13970" PostTypeId="1" CreationDate="2021-12-17T02:15:19.457" Score="1" ViewCount="850" Body="&lt;p&gt;I know that some Eastern European and Middle Asian countries denote the tangent by &lt;span class=&quot;math-container&quot;&gt;$\operatorname{tg}$&lt;/span&gt;. For many years, I have used &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; instead, but am currently thinking of changing that notation to &lt;span class=&quot;math-container&quot;&gt;$\operatorname{tg}$&lt;/span&gt;. I think &lt;span class=&quot;math-container&quot;&gt;$\operatorname{tg}$&lt;/span&gt; is better than &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; because I can write &lt;span class=&quot;math-container&quot;&gt;$\operatorname{tg}$&lt;/span&gt; in cursive very easily (unlike &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I would like to know why the United States, with its many Nobel prizes, has been using &lt;span class=&quot;math-container&quot;&gt;$\tan$&lt;/span&gt; rather than &lt;span class=&quot;math-container&quot;&gt;$\operatorname{tg}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14765" LastEditorUserId="4251" LastEditDate="2021-12-17T17:05:51.610" LastActivityDate="2021-12-17T17:05:51.610" Title="How has $\tan(x)$ become more popular than $\operatorname{tg}(x)$?" Tags="|mathematics|terminology|notation|russia|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13971" PostTypeId="2" ParentId="12626" CreationDate="2021-12-17T10:19:11.620" Score="3" Body="&lt;p&gt;As al-Jadi, Polaris was part of a Bedouin or Arab legend of the sky, which told of a funeral for a father. Polaris was the murderer. It's important, if the meaning is 'goat', which I'm not sure yet but seems pretty likely, various star names today take an Arab name and vary the spelling for several stars. Alpha Capricornus is alGedi, sometimes spelled Giedi. Arab/ Bedouin astrology had its own animals and constellations and asterisms. There are also multiple instances of various animals. I have a file listing them all as I find them. For example, there are so many camel stars, with variations such as mother camel (4 of them), baby camel, trotting camel and so on. Also just because goat appears as Capricorn to us doesn't mean it was a goat in Arab astrology (I couldn't say presently) or that there weren't other goats. But in this case it's a human story, the stars in the Big Dipper feature, the handle of the dipper is the mourner sisters, mourning there father in funeral procession, the cup of the ladle is the coffin which they are following.&lt;/p&gt;&#xA;&lt;p&gt;I myself defer to the Arabic traditions for various reasons, personally I dislike hybrid monster type depictions such as goat-fish of Capricorn. Then also Arab/ Bedouin astrology is clear-headed clear-skied astrology/ astronomy from a culture that used the stars for navigating ON LAND. Their feet are very much on the ground, and they are able to see the stars at most times due to lack of cloud cover. Although at some time in the distant past the region wasn't desert, but I think that is going back way far, beyond a full cycle of the pole stars.&lt;/p&gt;&#xA;&lt;p&gt;I have also been searching a name for Polaris. The Babylonian name I saw here refers to constellation so still not the wiser exactly. But the dates for that are surprisingly recent. Polaris became pole star only in 500 AD, so it can still be relevant. I'm glad others are trying to answer this question, as I have been trying now and then for a while. I had high hopes for Phoenice, but now I see it's also a pole star name. I've been through most of the other names listed here but haven't found one better than al-Jadi yet. Which name may relate to the pole star 'murdering' the other previous pole star. Which there wasn't a good one between Thuban and Polaris but Kochab was used as a stop-gap.&lt;/p&gt;&#xA;&lt;p&gt;I checked out some names and found them to refer to an axle (al-Kutb) and a needle or nail (Mismar). I guess that such names also relate to Polaris being at the center of the revolving stars. Moving forward I think al-Jadi is good enough for future use, and has the advantage of carrying the full Arabic myth of the funeral procession. Looking back on our generation, people would associate the name with our generations having developed a lot of science and seeing our own era as pivotal, a new era, out with the old and death to the old (al-Jadi as legendary murderer). I'd love to know more about the myth, as in who was murdered exactly and why. al-Na'ash is what I have. But that refers to bier, coffin, casket, so seems is not a name.&lt;/p&gt;&#xA;&lt;p&gt;Some names for Polaris just mean star, such as Yilduz from Turkish, the same word appearing in form Yildun for a different star. (Similarly it seems Kochab the former pole star is from kaukab in Arabic meaning star.) So again meaning pole star so THE star.&lt;/p&gt;&#xA;&lt;p&gt;I will try to link a graphic here using the instructions given. It just shows my efforts to clarify the whole cycle of pole stars. Emphasizing why we want a genuine star name, as any one of these stars could be 'Polaris' in their own time, given it is a wide time frame.&#xA;!https://www.facebook.com/PracticalHeroesAstrology/photos/a.1887269767966380/5467568069936514/  [Pole Stars]&lt;/p&gt;&#xA;" OwnerUserId="15862" LastActivityDate="2021-12-17T10:19:11.620" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13972" PostTypeId="1" AcceptedAnswerId="13973" CreationDate="2021-12-15T21:15:17.567" Score="6" ViewCount="673" Body="&lt;p&gt;Reading how Galileo measured time in the experiment with inclined plane, it says on &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_experiments&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Galileo accurately measured these short periods of time by creating a &lt;strong&gt;pulsilogon&lt;/strong&gt;. This was a machine created to measure time using a pendulum.[15] The pendulum was synchronized to the human pulse. He used this to measure the time at which the weighted balls passed marks that he had made on the inclined plane.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, I am unable to find any drawing or reference to what it actually looked like.&lt;/p&gt;&#xA;&lt;p&gt;According to other source I found, Galileo actually used a guitar and by listening to string rhythms/frequencies he could measure subseconds intervals.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have a reference to how it really was?&lt;/p&gt;&#xA;" OwnerUserId="15865" OwnerDisplayName="pisoir" LastEditorUserId="15865" LastEditDate="2021-12-23T12:30:50.473" LastActivityDate="2021-12-23T12:30:50.473" Title="What did Galileo's &quot;pulsilogon&quot; look like?" Tags="|experimental-physics|galileo|time|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13973" PostTypeId="2" ParentId="13972" CreationDate="2021-12-18T01:57:31.773" Score="3" Body="&lt;p&gt;According to the medical journal article &lt;a href=&quot;https://journals.lww.com/clinorthop/Fulltext/2003/08000/The_Classic__The_History_of_Instrumental_Precision.3.aspx&quot; rel=&quot;nofollow noreferrer&quot;&gt;The History of Instrumental Precision in Medicine&lt;/a&gt;, referring to the pulsilogon :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Unclaimed by Galileo, it was attributed to &lt;a href=&quot;https://en.wikipedia.org/wiki/Paolo_Sarpi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paolo Sarpi&lt;/a&gt;, and clearly enough was appropriated at a later date by that notable genius, &lt;a href=&quot;https://en.wikipedia.org/wiki/Santorio_Santorio&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sanctorius&lt;/a&gt;, who also, like Galileo, called it the pulsilogon. We have no drawing of Galileo’s pulsilogon, but it must have been identical with the simpler form as shown in Sanctorius.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Bulletin of the Scientific Instrument Society article &lt;a href=&quot;https://static1.squarespace.com/static/54ec9b40e4b02904f4e09b74/t/59ae832ed7bdcecfe13b5fb4/1504609073145/Bulletin_133_Pulsilogium.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Recreating the Pulsilogium of Sanctorius&lt;/a&gt; details a recent attempt to reconstruct the various types of pulsilogia attributed to Sanctorius and includes the following etching:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/mMyFE.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mMyFE.jpg&quot; alt=&quot;Sanctorius pulsilogia&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, the same article includes the following comment on the relation between Galileo's instrument and that of Sanctorius:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All the scholars who dealt with Galileo and his studies on motion, simply assumed that Santorio’s pulsilogium was a direct outcome of the former’s research into the properties of the pendulum. Unfortunately, there are many major problems with this hypothesis. The most important of which is that it takes no account of Santorio’s words and further assumes the primacy of Galileo’s over Santorio’s studies that is historically groundless.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2021-12-18T01:57:31.773" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13974" PostTypeId="1" AcceptedAnswerId="13977" CreationDate="2021-12-18T02:12:31.063" Score="6" ViewCount="942" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi&quot; rel=&quot;noreferrer&quot;&gt;Alfréd Rényi&lt;/a&gt; was a Hungarian mathematician who survived a lot, including a forced labor camp, and was very active in the fields of probability theory, number theory, graph theory, etc.&lt;/p&gt;&#xA;&lt;p&gt;Various articles and obituaries on him mention he died early at just 48 years of age. However, the cause of his death isn't mentioned anywhere. What caused him to die that young?&lt;/p&gt;&#xA;" OwnerUserId="15867" LastEditorUserId="4251" LastEditDate="2021-12-18T09:45:54.450" LastActivityDate="2021-12-18T10:40:34.903" Title="How did Alfréd Rényi die?" Tags="|mathematicians|biographical-details|death|20th-century|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="13975" PostTypeId="2" ParentId="13392" CreationDate="2021-12-18T02:37:43.093" Score="2" Body="&lt;p&gt;It's not that earlier mathematics is a less developed version of contemporary mathematics but that earlier mathematics is a precursor to modern mathematics. We cannot have later mathematics without earlier mathematics and it stands to reason that this earlier mathematics will be less developed.&lt;/p&gt;&#xA;&lt;p&gt;Mathematics, because it is a subject thought about and debated about by human beings will demonstrate a telos because people have a telos. Hence it is not wrong for historians of mathematics to discern a certain thread or threads along which mathematics develop.&lt;/p&gt;&#xA;&lt;p&gt;For example, since Riemann, the notion of a manifold took about fifty years to crystallise into definite form. And it's also ramified into many other directions: orbifolds, foliations, bundles, sheaves and toposes.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2021-12-18T02:37:43.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13976" PostTypeId="1" CreationDate="2021-12-18T10:19:54.803" Score="2" ViewCount="352" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Franco_Rasetti&quot; rel=&quot;nofollow noreferrer&quot;&gt;Franco Rasetti&lt;/a&gt; was an American-Italian physicist. He was born in Italy and obtained his laurea in physics at the university of Pisa. Fermi invited him to join his research group at the University of Rome. In 1934, he along with Enrico Fermi, discovered key processes leading to nuclear fission - in particular, the artificial radioactivity of flourine and aluminium - which would prove to be important in the development of the atomic bomb.&lt;/p&gt;&#xA;&lt;p&gt;However, unlike many, he refused to work on the Manhattan project on moral grounds.&lt;/p&gt;&#xA;&lt;p&gt;How many others were there — do we know?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-12-18T17:42:05.697" LastActivityDate="2021-12-18T18:01:00.723" Title="Other than Franco Rasetti, how many scientists refused to work on the Manhattan Project on moral grounds?" Tags="|physicists|nuclear-physics|world-war-2|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="13977" PostTypeId="2" ParentId="13974" CreationDate="2021-12-18T10:21:15.330" Score="8" Body="&lt;p&gt;I do not know this for Hungary specifically, but in many Western countries the cause of death is considered a private matter that is not normally mentioned in obituaries. It is at most hinted at in vague terms such as &amp;quot;suddenly and unexpected&amp;quot; or &amp;quot;after a lengthy and severe illness&amp;quot;. Unsurprisingly then, I have been unable to find relevant information in any of the obituaries published after Rényi's passing, other than that he had not been well for some time, with this information being provided by himself:&lt;/p&gt;&#xA;&lt;p&gt;Leopold Schmetterer, &amp;quot;Alfréd Rényi, in memoriam.&amp;quot; In &lt;em&gt;Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory&lt;/em&gt;. University of California Press, 1972, pp. xxv-l: (&lt;a href=&quot;https://projecteuclid.org/ebook/Download?urlid=bsmsp%2F1200514208&amp;amp;isFullBook=False&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On January 5, 1970, I received a letter from Alfréd Rényi in which he wrote of plans to visit Vienna again, and he incidentally added that he was not feeling well and had trouble with his glands. On February 2nd, Professor M. Csörgő of McGill University, who was then visiting at the Mathematic Institute of the University of Vienna, relayed the news that the Hungarian radio had announced that Alfréd Rényi had died the day before.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Rényi's book &amp;quot;A Diary on Information Theory&amp;quot; (translation of &amp;quot;Napló az információelméletről&amp;quot;, Budapest: Akadémiai Kiadó 1982) posthumously completed by his student Gyula Katona based on the author's 1969 notes, he writes from the perspective of an anonymous student of his. On &lt;a href=&quot;https://www.google.com/books/edition/A_Diary_on_Information_Theory/SVkrAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22The+professor+doesn%27t+look+too+well.%22&amp;amp;dq=%22The+professor+doesn%27t+look+too+well.%22&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;page 55&lt;/a&gt;, the imaginary student records in passing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I am eagerly awaiting the future lectures. The professor doesn't look too well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In a &lt;a href=&quot;https://gilkalai.wordpress.com/2019/05/01/the-last-paper-of-catherine-renyi-and-alfred-renyi-counting-k-trees/&quot; rel=&quot;noreferrer&quot;&gt;blog post&lt;/a&gt; &amp;quot;The last paper of Catherine Rényi and Alfréd Rényi: Counting k-Trees&amp;quot;, dated May 1, 2019, Gil Kalai, Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A few months after the conference, on February 1, 1970 Alfréd Rényi died of a violent illness.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have found one source that gives a very specific cause of death. However, no reference is cited for this, nor have I been able to corroborate this information from other sources at this point.&lt;/p&gt;&#xA;&lt;p&gt;Norman L. Johnson, Samuel Kotz (eds.), &amp;quot;Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present&amp;quot;, Wiley 2011, p. 206 (&lt;a href=&quot;https://www.google.com/books/edition/Leading_Personalities_in_Statistical_Sci/AzZZ-uW6WuEC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Catherine+R%C3%A9nyi+in+August+1969,+physicians+diagnosed+an+inoperable+lung+carcinoma+in+Alfred%22&amp;amp;pg=PA206&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Google snippet&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Following the unexpected death of Catherine Rényi in August 1969, physicians diagnosed an inoperable lung carcinoma in Alfred; he died on February 1, 1970.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2021-12-18T10:40:34.903" LastActivityDate="2021-12-18T10:40:34.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13978" PostTypeId="1" CreationDate="2021-12-18T16:15:14.197" Score="0" ViewCount="217" Body="&lt;p&gt;In 1962, Sir Oswald Mosley, leader of the British Union of Fascists which he founded in 1932 (disbanded in 1940) made overtures towards  Bertrand Russell on engaging in a public debate - according to some commentators I've read online.&lt;/p&gt;&#xA;&lt;p&gt;Russell penned a reply -  reproduced below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dear Sir Oswald,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thank you for your letter and your enclosures. I have given some thought to our recent correspondence. It is always difficult to decide how to respond to people whose ethos is so alien and in fact, repellent to one's own. It is not that I take exception to the general points made by you but that every ounce of my energy has been devoted to an active opposition to cruel bigotry, compulsive violence, and the sadistic persecution which has characterised the philosophy and practise of fascism.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I feel obliged to say that the emotional universes we inhabit are so distinct, and in the deepest way opposed, that nothing fruitful or sincere could ever emerge from association between us.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I should like you to understand the intensity of this conviction on my part. It is not out of any attempt to be rude that I say this but because of all that I value in human experience and achievement.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your sincerely,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Bertrand Russell&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Is there any indication of the topics which Mosley was hoping to debate with Russell?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It's unlikely to be fascism - which I've seen a number of commentators state, given the defeat of fascism throughout Europe by the Allies. I do know that after the 1958 Notting Hill riots he stood in the 1959 General Election for Kensington North on a stridently anti-immigration platform calling for forced repatriation of Caribbean immigrants and a ban on mixed marriages and was defeated. And again in 1961, he took part in a debate at UCL (University College London) on immigration from the Commonwealth.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q. Is there any indication how Mosley took this rejection?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2021-12-19T14:02:30.923" LastActivityDate="2021-12-19T14:02:30.923" Title="Is the response of Sir Oswald Mosley to the letter of Bertrand Russell's 1962 letter recorded?" Tags="|mathematicians|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13982" PostTypeId="1" AcceptedAnswerId="14165" CreationDate="2021-12-19T05:17:05.227" Score="7" ViewCount="1113" Body="&lt;p&gt;I learnt in school that Marie Curie died from her work at 66 years. On the other hand, Enrico Fermi, who also handled a lot of radioactive substances died of stomach cancer at the age of just 53. It is highly likely that his work also was the cause of his disease. But I haven't seen it mentioned anywhere like it is in the case of Curie. What is the reason that this isn't acknowledged? Is it simply that the causation between the work and the disease is more clear in the case of Curie but not Fermi?&lt;/p&gt;&#xA;&lt;p&gt;Additionally, given that a lot of Fermi's work was happening once Curie had already died, it is plausible he knew the risks of working with the materials he was working with. Is this supported by any sources?&lt;/p&gt;&#xA;&lt;p&gt;We do know that he felt an urgency to develop a nuclear weapon on behalf of the allies before the nazis got one. If the above is all true, it would mean he intentionally put himself in harms way for the sake of the war effort and ended up ultimately paying the price. Is this theory plausible or completely off?&lt;/p&gt;&#xA;" OwnerUserId="15867" LastEditorUserId="4251" LastEditDate="2021-12-19T11:57:07.903" LastActivityDate="2022-02-07T19:49:13.667" Title="Why is it said that Marie Curie died due to her work but the same isn't said for Fermi?" Tags="|physics|experimental-physics|physicists|nuclear-physics|fermi|" AnswerCount="1" CommentCount="3" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13983" PostTypeId="1" CreationDate="2021-12-20T05:37:31.903" Score="0" ViewCount="63" Body="&lt;p&gt;Even in my earliest physics course we took for granted that charges are made of electrons (or their absence) and currents are due to the motion of electrons. But the electron is a very modern concept and all of Maxwell’s equations had been set in order by the mid 1800s— how did people determine that currents and charges were made of the same substance before then? Which experiments suggested this and who made the connection?&lt;/p&gt;&#xA;" OwnerUserId="15882" LastActivityDate="2021-12-20T05:37:31.903" Title="How was it determined that charge and current were made of the same stuff?" Tags="|physics|electromagnetism|electricity|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="13984" PostTypeId="2" ParentId="6758" CreationDate="2021-12-21T20:27:13.537" Score="0" Body="&lt;p&gt;I really doubt this would have been possible, since Maxwell’s equations were discovered only in the later-mid century. On the other hand, it took around 150 years for Lagrangian and Hamiltonian mechanics to appear after Newton.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastActivityDate="2021-12-21T20:27:13.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13985" PostTypeId="1" CreationDate="2021-12-21T22:18:20.317" Score="2" ViewCount="129" Body="&lt;p&gt;I know that the ancients had several ways to construct geometric shapes. I know two of them: the biggest one, ruler and compass, with which you can build some polygons, bisect an angle, etc, but not square the circle, trisect an angle, or double the cube. The other one, the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Neusis_construction&quot; rel=&quot;nofollow noreferrer&quot;&gt;neusis&lt;/a&gt;, with which you can trisect an angle, or double the cube. I also think there is something called mechanical constructions (Kant alludes to it somewhere), but I can't find references at the moment. I think the brastichrone would qualify, which is a transcendental curve if I'm not mistaken.&lt;/p&gt;&#xA;&lt;p&gt;There is also a method of construction not with tools but with intersections: this is how, as Apollonius showed (I believe), one can get the conics: ellipse, parabola, hyperbola.&lt;/p&gt;&#xA;&lt;p&gt;This is more far-fetched but given a hyperbola, the lemniscate of Bernouilli can be constructed with a ruler and an angle iron, cf. &lt;a href=&quot;https://upload.wikimedia.org/wikipedia/commons/f/f1/Lemniscate_of_Bernoulli.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://upload.wikimedia.org/wikipedia/commons/f/f1/Lemniscate_of_Bernoulli.gif&lt;/a&gt;. The Lemniscate is a plane curve of degree 4.&lt;/p&gt;&#xA;&lt;p&gt;Now in modern mathematics a big subject is algebraic curves, which can  be defined by equations. Higher genus curves, such as plane cubics (elliptic curves), are of very great theoretical interest. These show up in problems that can be related to shapes known to the ancient, e.g. elliptic curves come up in calculating the arc of an ellipse (but an ancient Greek wouldn't understand that, as this takes integrals). I wonder if there are Greek-style constructions of such curves.&lt;/p&gt;&#xA;" OwnerUserId="15888" LastEditorUserId="10106" LastEditDate="2021-12-22T02:33:24.900" LastActivityDate="2021-12-22T02:33:24.900" Title="Did the ancients construct higher genus curves?" Tags="|mathematics|ancient-greece|algebraic-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="13986" PostTypeId="1" CreationDate="2021-12-22T14:31:26.497" Score="1" ViewCount="111" Body="&lt;p&gt;Traditionally, the first example of a non-convergent Fourier series of a function is considered to be the example constructed in 1870 by the mathematician Paul David Gustav du Bois-Reymond. His example was a turning point in the study of convergence of Fourier series since it shattered the hopes expressed by previous mathematicians (including Dirichlet and Riemann) that the Fourier series of any continuous function converges into it.&lt;/p&gt;&#xA;&lt;p&gt;However, I read in several places that in astronomy, when the difference between mean and true anomaly (in the so called &amp;quot;equation of the center&amp;quot;) is developed into trigonometric series of multiples of mean anomaly with coefficients calculated by Fourier integral, the trigonometric series diverges for values of eccentricity exceeding the &amp;quot;Laplace limit&amp;quot;, whose value is approximately &lt;span class=&quot;math-container&quot;&gt;$0.662$&lt;/span&gt; (for more detailed explanation of the context, read my post: &lt;a href=&quot;https://hsm.stackexchange.com/questions/7920/meaning-of-passages-by-gauss-on-the-convergence-of-expansions-in-infinite-seri&quot;&gt;Meaning of passages by Gauss on the &amp;quot;convergence of expansions (in infinite series) of the (elliptical) equation of the center&amp;quot;?&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Therefore, I have much confusion about how to interpret the fact that the phenomenon of &amp;quot;Laplace limit&amp;quot; was discovered much earlier than 1870. I have never read any claims that the discovery of this phenomenon anticipated these later developments, so most probably I misunderstood the meaning of it and misevaluated the significance of discovery of Laplace limit. But the fact that I didn't find such claims is still not a conclusive proof that there is no substantial relation between the two issues.&lt;/p&gt;&#xA;&lt;p&gt;So I'd like to put things into order and to read a correct historic assessment of the discoveries of Laplace and Paul Reymond and their relation (if there is such a relation).&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="12330" LastEditDate="2021-12-22T22:11:44.277" LastActivityDate="2021-12-22T22:11:44.277" Title="Can &quot;Laplace limit&quot; phenomenon be considered the first example of non-convergent Fourier series?" Tags="|astronomy|harmonic-analysis|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13987" PostTypeId="1" AcceptedAnswerId="13989" CreationDate="2021-12-22T19:33:10.957" Score="2" ViewCount="211" Body="&lt;p&gt;The terms &amp;quot;gram-atom / gram-molecule&amp;quot; are obsolete nowadays in chemistry and got replaced by the concept of &amp;quot;mole/ Avogadro Number&amp;quot;. But recently, I encountered a question that can have two different answers depending on how &amp;quot;gram-atom&amp;quot; was defined early in Chemistry.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Q:  A new atomic mass scale, called UM, was introduced as 1/24th part of the mass of a single C-12  atom. Find the gram-atomic weight of Hydrogen (H) as per the new scale ?&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The above question can have two answers if I use two available definitions of &amp;quot;gram-atom&amp;quot; from the web :&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/gram-atomic-weight&quot; rel=&quot;nofollow noreferrer&quot;&gt;gram-atomic weight: the amount of an atomic substance whose weight, in grams, is numerically equal to the atomic weight of that substance.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;According to this definition, as the new scale will show the atomic weight of Hydrogen as 2 UMs, its gram-atomic weight should be 2 grams. And thus there would be  &lt;span class=&quot;math-container&quot;&gt;$ 2N_A$&lt;/span&gt; atoms of Hydrogen in one gram-atom of Hydrogen (as per new scale).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bipm.org/en/history-si/mole&quot; rel=&quot;nofollow noreferrer&quot;&gt;But, if I just use the definition of 1 gram-atom as nothing but 1 mole (&lt;span class=&quot;math-container&quot;&gt;$1 N_A$&lt;/span&gt;) of atoms i.e. Gram-atom = Mole.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Then the answer would be 1 gram-atom of Hydrogen = 1 mole of Hydrogen = &lt;span class=&quot;math-container&quot;&gt;$1N_A$&lt;/span&gt; X &lt;span class=&quot;math-container&quot;&gt;$2UMs$&lt;/span&gt; (No of UMs in 1 Hydrogen atom) = 1 gram&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, which approach is correct (historically) ?&lt;/p&gt;&#xA;&lt;p&gt;Was the definition of &amp;quot;gram-atom&amp;quot; pre-required the proper definition (w.r.t. kg/gm)  of **amu ** as in the second approach?&lt;/p&gt;&#xA;&lt;p&gt;Or ppl just took any relative microscopic unit like &lt;strong&gt;amu&lt;/strong&gt; (in real life) or &lt;strong&gt;UM&lt;/strong&gt; (in our hypothetical question) and just replaced the relative atomic weight w.r.t. that unit with gram and called it gram-atomic weight?&lt;/p&gt;&#xA;" OwnerUserId="15889" LastEditorUserId="6085" LastEditDate="2021-12-22T20:13:21.960" LastActivityDate="2021-12-23T02:16:14.507" Title="History of word gram-atom / gram-molecule?" Tags="|chemistry|measurement|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13988" PostTypeId="1" AcceptedAnswerId="13990" CreationDate="2021-12-22T23:30:27.250" Score="2" ViewCount="218" Body="&lt;p&gt;The term &lt;em&gt;Poisson integral formula&lt;/em&gt; may refer to any of the related formulas for harmonic (or holomorphic) functions on a disk (or in a ball, half space, etc) in terms of their boundary values. This is the formula that contains the famous &lt;em&gt;Poisson kernel&lt;/em&gt;. When and where did Poisson publish his integral formula? I have a paper which says it was published in 1820 (without reference) and hours of searching has revealed nothing more.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="6545" LastEditDate="2021-12-23T12:01:41.457" LastActivityDate="2021-12-23T20:33:00.640" Title="Poisson integral formula" Tags="|mathematics|reference-request|complex-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13989" PostTypeId="2" ParentId="13987" CreationDate="2021-12-23T02:16:14.507" Score="1" Body="&lt;p&gt;The first one is the correct approach. As you may note now, most the &amp;quot;weighing&amp;quot; at atomic scale is defined circularly. Atomic weights have more than two hundred years old history. We need an arbitrary reference atom. Historically H was numerically set exact 1, and every atomic weight was measured with respect to that. Then came oxygen which was set to 16 exactly. Finally, in the 1960s, carbon-12 was set exact 12. Let us look at the definition.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;gram-atom  n. the quantity of an element having a mass in grams&#xA;numerically equal to its atomic weight.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As per your exercise, if 1 amu were 1/24 the mass of carbon-12 (12 exact), then all atomic weights double.&lt;/p&gt;&#xA;&lt;p&gt;Coming to the mole, The symbol &lt;em&gt;mol&lt;/em&gt; is due to Ostwald who was a very influential and respected physical chemist more than a century ago. In German, &amp;quot;mole&amp;quot; is &amp;quot;Mol&amp;quot;. It is a shortform of &lt;em&gt;Molekül&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I believe the Internet Archive has an English translation of his historically famous book of 19th century. This is where I recall reading this. The original reference is W. Ostwald &lt;em&gt;Hand- und Hilfsbuch zur Ausführung physikochemischer Messungen&lt;/em&gt;. This book is available on Google Books (pg 278). A footnote appears there:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Um nicht stets den schleppenden Ausdruck Gramm -Molekulargewicht brauchen zu müssen, werde ich diese Größe in der Folge kurzweg &lt;strong&gt;Mol&lt;/strong&gt; nennen, und erlaube mir, diesen Vorschlag zur allgemeinen Annahme zu empfehlen. Man schreibt ohnedies diese Größe häufig Mol., und braucht daher nur den Punkt fortzulassen.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;In order not to always have to use the sluggish expression gram-molecular weight, I will simply call this quantity &lt;strong&gt;Mol&lt;/strong&gt; in the following, and allow me to recommend this proposal for general acceptance. Anyway, this quantity is often written as &lt;strong&gt;Mol.&lt;/strong&gt;, so you only have to omit the point.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Translated via (&lt;a href=&quot;http://www.deepl.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.deepl.com&lt;/a&gt;) with some manual changes.&lt;/p&gt;&#xA;&lt;p&gt;So the &lt;em&gt;number&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$N_A$&lt;/span&gt; is experimentally determined with respect to the accepted definition of atomic weights.&lt;/p&gt;&#xA;&lt;p&gt;Your second definition is then dimensionally inconsistent. 1 gram-atom is not a number, it is a weight in grams. It is not equal to &lt;span class=&quot;math-container&quot;&gt;$N_A$&lt;/span&gt;. 1 gram-atom, as per accepted definition, will consist of &lt;span class=&quot;math-container&quot;&gt;$N_A$&lt;/span&gt; atoms.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2021-12-23T02:16:14.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13990" PostTypeId="2" ParentId="13988" CreationDate="2021-12-23T11:59:44.473" Score="3" Body="&lt;p&gt;I think that this is what you are asking for:&lt;/p&gt;&#xA;&lt;p&gt;Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de différents problèmes, &lt;em&gt;Journal de l'École polytechnique&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$18^e$&lt;/span&gt; cahier, 11 (1820), p. 417–489.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k4336744/f422.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; you can find the scan of the original paper (in French, of course).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT&lt;/p&gt;&#xA;&lt;p&gt;The so-called &amp;quot;&lt;em&gt;noyau de Poisson&lt;/em&gt;&amp;quot; (Poisson kernel, clearly Poisson does not call it this way) appears at page 422.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2021-12-23T20:33:00.640" LastActivityDate="2021-12-23T20:33:00.640" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="13991" PostTypeId="1" CreationDate="2021-12-23T12:00:15.257" Score="3" ViewCount="142" Body="&lt;p&gt;M. Bunge in his book &amp;quot;Evaluating Philosophies&amp;quot; makes the following statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Scientism has been explicitly opposed by dogmatists and obscurantists of all stripes, such as the neoliberal ideologist Friedrich von Hayek and the &amp;quot;critical theorist&amp;quot; Jürgen Habermas, a ponderous writer who managed to amalgamate Hegel, Marx, and Freud, and decreed that &amp;quot;science is the ideology of late capitalism.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately the book gives no sources for the quote by Habermas.&#xA;As I understood it Habermas said that &amp;quot;science is the ideology of late capitalism.&amp;quot; Since Habermas writes in German where can I find this exact quote.&lt;/p&gt;&#xA;" OwnerUserId="5483" LastEditorUserId="12330" LastEditDate="2021-12-23T21:37:27.217" LastActivityDate="2021-12-23T21:37:27.217" Title="Alleged quote by Habermas in original. &quot;Science is the ideology of late capitalism.&quot; Where to find it?" Tags="|philosophy-of-science|quote|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="13992" PostTypeId="1" AcceptedAnswerId="13993" CreationDate="2021-12-23T23:04:14.247" Score="3" ViewCount="230" Body="&lt;p&gt;As any physicist knows, a lot of amateur science afficionados out there bring up Nikola Tesla in rather fantastical ways. There are indeed a few reasons for his near mythical status in popular culture, but us physicists don't learn a lot about him during our training.&lt;/p&gt;&#xA;&lt;p&gt;What scientifically sound and historically accurate biography of Nikola Tesla would you recommend, so that I know what to say the next time some guy assures me that Tesla came up with a free energy source but the industry shut it down and hid it from us? Or something like that.&lt;/p&gt;&#xA;" OwnerUserId="15895" LastEditorUserId="4251" LastEditDate="2021-12-25T09:43:08.590" LastActivityDate="2021-12-25T09:43:08.590" Title="Demystifying Nikola Tesla: Scientifically sound, historically accurate biography" Tags="|biographical-details|electromagnetism|physicists|book-recommendation|tesla|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13993" PostTypeId="2" ParentId="13992" CreationDate="2021-12-24T12:36:36.713" Score="4" Body="&lt;p&gt;A good bet is Christopher Cooper's &lt;em&gt;The Truth about Tesla: The Myth of the Lone Genius in the History of Innovation&lt;/em&gt;, published in 2015 by Race Point Publishers.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2021-12-24T12:36:36.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13994" PostTypeId="5" CreationDate="2021-12-25T09:48:27.473" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2021-12-25T09:48:27.473" LastActivityDate="2021-12-25T09:48:27.473" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13995" PostTypeId="4" CreationDate="2021-12-25T09:48:27.473" Score="0" Body="Nikola Tesla (1856–1943) was a Serbian American inventor and engineer who discovered and patented the rotating magnetic field, the basis of most alternating-current machinery. He also developed the three-phase system of electric power transmission." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2021-12-26T09:01:28.143" LastActivityDate="2021-12-26T09:01:28.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13997" PostTypeId="2" ParentId="7437" CreationDate="2021-12-26T21:17:57.510" Score="1" Body="&lt;p&gt;The uranium came from Belgian Congo, more precisely, from the Katanga Province. See &lt;a href=&quot;https://en.wikipedia.org/wiki/Edgar_Sengier&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt; explaining the central role of Edgar Sengier, ruler of the very powerful (Belgian) &lt;a href=&quot;https://en.wikipedia.org/wiki/Union_Mini%C3%A8re_du_Haut-Katanga&quot; rel=&quot;nofollow noreferrer&quot;&gt;Union Minière du Haut Katanga&lt;/a&gt; (Katanga Mining Union), who in September 1940 secretly sent, without being ordered by anybody, around 1000 tons of especially rich uranium ore to Staten Island. The US Army was warned only afterwards, in 1942!&lt;/p&gt;&#xA;&lt;p&gt;Some more details can be found into &lt;a href=&quot;https://www.amazon.fr/Katanga-rencontre-latlas-voyages-Nagy/dp/B0000DUW3J&quot; rel=&quot;nofollow noreferrer&quot;&gt;this book&lt;/a&gt; (in french): &amp;quot;Katanga&amp;quot; by I. Nagy, Ed. Rencontres, 1965.&lt;/p&gt;&#xA;" OwnerUserId="3730" LastEditorUserId="3730" LastEditDate="2021-12-30T14:06:55.600" LastActivityDate="2021-12-30T14:06:55.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="13999" PostTypeId="1" AcceptedAnswerId="14001" CreationDate="2021-12-28T05:11:53.043" Score="4" ViewCount="388" Body="&lt;p&gt;Some sources say that gravitational potential was introduced by Lagrange in 1773, and others say that it was introduced by Bernoulli in 1738. I sifted through Daniel Bernoulli's &lt;em&gt;Hydrodynamica&lt;/em&gt; (translation by Carmody and Kobus) but could not even find the mention of &lt;em&gt;velocity potential&lt;/em&gt;, let alone a potential for Newtonian gravity. I am hoping somebody with more knowledge could clarify this for me. Thanks a lot.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="4251" LastEditDate="2021-12-28T20:24:33.687" LastActivityDate="2021-12-28T22:11:29.617" Title="Who introduced gravitational potential?" Tags="|physics|gravity|potential|fluid-dynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14000" PostTypeId="1" CreationDate="2021-12-28T11:47:23.537" Score="2" ViewCount="95" Body="&lt;p&gt;I've come to the notion of the &lt;strong&gt;Picard Group&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I was recently linked to &lt;a href=&quot;https://arxiv.org/pdf/1208.0269.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt; paper, which contains the line:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The problem of computing the Picard groups of surfaces &lt;span class=&quot;math-container&quot;&gt;$S \subset \mathbb{P}_{\mathbb{C}}^{3}$&lt;/span&gt; has a long history. The solution for smooth quadratic and cubic surfaces was known in the 1800s in terms of lines on these surfaces. In the 1880s, Noether suggested what happens in higher degree, but it wasn’t until the 1920s that Lefschetz proved the famous result bearing their name.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, the article does not give any references to these early texts and works.&lt;/p&gt;&#xA;&lt;p&gt;Where was it that algebraic geometers first started thinking about the Picard group, and what was it that they had in mind when doing so?&lt;/p&gt;&#xA;" OwnerUserId="15726" LastEditorUserId="15981" LastEditDate="2022-01-11T06:16:28.517" LastActivityDate="2022-01-11T06:16:28.517" Title="The Picard Group: Origin and History" Tags="|mathematics|abstract-algebra|algebraic-geometry|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14001" PostTypeId="2" ParentId="13999" CreationDate="2021-12-28T14:26:28.290" Score="6" Body="&lt;p&gt;The potential is not introduced in &lt;em&gt;Hydrodynamica&lt;/em&gt;. Here is from &lt;a href=&quot;https://www.google.com/books/edition/The_Oxford_Handbook_of_the_History_of_Ph/JWYBAQAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Oxford Handbook of the History of Physics, p.366&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Gravitational potential appeared implicitly in the form of the integral of force in&#xA;Johann Bernoulli’s formula for the motion of a point under the action of a central&#xA;force (1712). Bernoulli did not connect his result with the conservation of vis viva;&#xA;this step was taken in 1738 by his son Daniel, who also explicitly gave the form of&#xA;the potential in the case of Newtonian attraction (1747). After an interval of more&#xA;than twenty years, Lagrange began to make extensive use of the potential, at least as&#xA;a formal mathematical entity in the equations of mechanics. He first applied it to&#xA;the study of astronomical perturbations (1776), then to the general motion of masspoints&#xA;subjected to mutual gravitational interactions (1779), and, finally, he inserted&#xA;the potential in the equations of mechanics that bear his name (1782)&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The 1712 reference is missing, and the 1738 reference is to &lt;em&gt;Commentationes de immutatione et extensione principii conservationisvirium&#xA;vivarum, quae pro motu corporum coelestium requiritur. Commentarii academiae scientiarum&#xA;Petropolitanae&lt;/em&gt; 10 (1738), 116–124. &lt;a href=&quot;https://www.biodiversitylibrary.org/item/113649#page/128/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; is a link.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="1800" LastEditDate="2021-12-28T22:11:29.617" LastActivityDate="2021-12-28T22:11:29.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14002" PostTypeId="1" AcceptedAnswerId="14003" CreationDate="2021-12-28T17:09:16.257" Score="2" ViewCount="166" Body="&lt;p&gt;Wikipedia cites John Anderson’s &lt;em&gt;A History of Aerodynamics&lt;/em&gt; and says that velocity potential was introduced by Lagrange in 1788. However, I could trace it at least to Euler 1752, where he published his famous fluid equations. Is it Euler who introduced velocity potential or somebody even before him? It is also possible that Wikipedia is correct and I am misunderstanding something.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="4251" LastEditDate="2021-12-28T20:13:52.410" LastActivityDate="2021-12-29T01:11:18.753" Title="Who introduced velocity potential?" Tags="|physics|18th-century|potential|fluid-dynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14003" PostTypeId="2" ParentId="14002" CreationDate="2021-12-29T01:11:18.753" Score="3" Body="&lt;p&gt;It was Euler c. 1740s, &lt;a href=&quot;http://eulerarchive.maa.org/backup/E258.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Principia motus fluidorum&lt;/em&gt;&lt;/a&gt; was read to the Academy in 1752, but published even later. Daniel Bernoulli, who is sometimes also credited, did not introduce not only the velocity potential, but even the &amp;quot;Bernoulli equation&amp;quot; named after him. Lagrange derived it in 1788 from Euler's equations of motion for incompressible fluids, see &lt;a href=&quot;https://www.google.com/books/edition/A_History_and_Philosophy_of_Fluid_Mechan/ws0AIfNabpYC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Takaty, A History and Philosophy of Fluid Mechanics, pp.73-80&lt;/a&gt;. Another useful reference is &lt;a href=&quot;https://www.google.com/books/edition/The_Genesis_of_Fluid_Mechanics_1640_1780/pyZIotXTX2sC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calero, The genesis of fluid mechanics, 1640-1780, p.38&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, Bernoulli and Clairaut did give Euler the key ideas in 1743. The notion of total differential appears in Clairaut's &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=ucm.5323611249&amp;amp;view=1up&amp;amp;seq=7&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Theorie de la Figure de la Terre: Tirée des principes de l'hydrostatique&lt;/em&gt;&lt;/a&gt; (1743) that Euler was very familiar with, and Bernoulli wrote to him about the elastic potential in a letter the same year. From &lt;a href=&quot;https://www.google.com/books/edition/The_Oxford_Handbook_of_the_History_of_Ph/JWYBAQAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Oxford Handbook of the History of Physics, p.366&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The potential as stored vis viva in an elastic system was first mentioned by Daniel Bernoulli in a letter to Euler of 20 October 1743: ‘For a naturally elastic band, I express the potential live force of the curved band as&#xA;&lt;span class=&quot;math-container&quot;&gt;$\int ds/r^2$&lt;/span&gt; [...] Since no one has perfected&#xA;the isoperimetric method as much as you, you will easily solve this problem of making &lt;span class=&quot;math-container&quot;&gt;$\int ds/r^2$&lt;/span&gt; a minimum.’ Euler proceeded immediately to develop this suggestion in his ‘On the elastic curve’ (1744)... A few years later Euler introduced the potential in fluid dynamics; believing (erroneously) that every stationary flux of an incompressible homogeneous fluid is irrotational, he applied the first elements of differential forms and arrived at ‘Laplace’s equation’:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\partial^2S}{\partial x^2}+\frac{\partial^2S}{\partial y^2}+\frac{\partial^2S}{\partial z^2}=0,&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is the velocity potential (1761a). Moreover, following d’Alembert’s lead in the use of complex numbers in two-dimensional fluid dynamics, Euler (1757c) also introduced what we would now call a complex potential.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2021-12-29T01:11:18.753" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14004" PostTypeId="1" AcceptedAnswerId="14006" CreationDate="2021-12-29T04:36:45.987" Score="3" ViewCount="136" Body="&lt;p&gt;I think I'm correct in saying a random variable is a mapping from the sample space to the real line (or more generally to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;. If I'm right then random variable seems a very odd way for a mathematician to describe it: it is neither random (a perfectly fixed mapping) nor a variable. Yet I have to believe that very smart people came up with this. I'm wondering who and what was the thinking behind the label.&lt;/p&gt;&#xA;" OwnerUserId="15921" LastEditorUserId="10106" LastEditDate="2021-12-29T06:55:11.827" LastActivityDate="2021-12-29T17:51:03.947" Title="Why was the term random &quot;variable&quot; applied to a mapping?" Tags="|mathematics|statistics|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14005" PostTypeId="1" AcceptedAnswerId="14109" CreationDate="2021-12-29T12:52:17.907" Score="10" ViewCount="3254" Body="&lt;p&gt;I noticed that South Asians often write 10,00,000 instead of 1,000,000. My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What is the origin of this special numbering system? Was there a more practical reason for having a special numbering system for South Asia?&lt;/li&gt;&#xA;&lt;li&gt;Why hasn't modern South Asia reformed this redundancy, just using &amp;quot;millions&amp;quot; everywhere? Most other aspects of numbers seem to be completely coherent. There is no special numbering system for Spanish-speaking countries. Nor, there is a special numbering system for the People's Republic of China, Korea and Japan.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;A good example is the the game show &amp;quot;Who Wants to Be a Millionaire&amp;quot; became &amp;quot;Kaun Banega Crorepati&amp;quot; in India.&lt;/p&gt;&#xA;" OwnerUserId="13344" LastEditorUserId="10106" LastEditDate="2023-01-26T01:12:19.770" LastActivityDate="2024-04-02T13:03:42.060" Title="Why do South Asians often use &quot;lakhs&quot; and &quot;crores&quot; instead of &quot;millions&quot;? What is the historical origin of this system?" Tags="|ancient-india|numbers|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14006" PostTypeId="2" ParentId="14004" CreationDate="2021-12-29T14:54:27.100" Score="4" Body="&lt;p&gt;It is easy to name some of these &amp;quot;smart people&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Andrei Kolmogorov&#xA;proposed a mathematical model for probability in his book&#xA;Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of probability theory), Berlin: Julius Springer, (1933). This model is commonly accepted nowadays. This was a final step of a long development (Emile Borel, Francesco Cantelli, Felix Hausdorff, Maurice Frechet, Antoni Lomnicki, Evgeny Slutsky, A. A. Markov and many others).&lt;/p&gt;&#xA;&lt;p&gt;There were several competing mathematical models, most notable of them due to&lt;/p&gt;&#xA;&lt;p&gt;Sergei Bernstein, Theory of probability, Kharkov, 1911,&lt;/p&gt;&#xA;&lt;p&gt;Hugo Steinhaus, Les probabilités dénombrables et leur rapport à la théorie de la mesure,&#xA;Fundamenta math. 4, 286-310 (1923), and&lt;/p&gt;&#xA;&lt;p&gt;Richard von Mises,&#xA;Wahrscheinlichkeit, Statistik und Wahrheit, J. Springer (1928).&lt;/p&gt;&#xA;&lt;p&gt;Gradually the mathematical advantages of Kolmogorov axiomatization were recognized, and&#xA;all other approaches were forgotten.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Random variable&amp;quot; is an old term whose use started before these axiomatizations. It was thought as a &amp;quot;variable&amp;quot; (for example, a number) that &amp;quot;depends on chance&amp;quot;. The last expression &amp;quot;depends on chance&amp;quot; required a mathematical explanation of &amp;quot;chance&amp;quot;, and that the random variable &amp;quot;depends on it&amp;quot; means that it is really a &amp;quot;function of something&amp;quot;. Axiomatization of Steinhaus is the closest&#xA;to the one by Kolmogorov (the difference is that he used the newly invented Lebesgue measure to axiomatize the &amp;quot;chance&amp;quot;, while Kolmogorov proposed to consider an abstract measure, which has several mathematical advantages). Two other axiomatizations that I mention do not use functions on a measure space at all.&lt;/p&gt;&#xA;&lt;p&gt;On the history of development of Kolmogorov's axiomatization, there is a paper:&lt;/p&gt;&#xA;&lt;p&gt;Jack Barone and Albert Novikoff,&#xA;A History of the Axiomatic Formulation of Probability from Borel to Kolmogorov:&#xA;Part I,&#xA;Archive for History of Exact Sciences , 8.III.1978, Vol. 18, No. 2  pp.&#xA;123-190&lt;/p&gt;&#xA;&lt;p&gt;But it covers only history of Kolmogorov's axiomatization, most alternative approaches are out of scope. A paper with wider coverage is &lt;a href=&quot;https://arxiv.org/pdf/math/0606533.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Glenn Shafer and Vladimir Vovk&lt;/a&gt;, The Sources of Kolmogorov’s&#xA;Grundbegriffe.&lt;/p&gt;&#xA;&lt;p&gt;Remark. It is very common that in popular history of science one person obtains all credit for a great advance, and contributions of all other participants are forgotten. Let me cite Maurice Frechet:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;It was at the moment when Borel introduced this new kind of additivity in 1909 that all the elements needed to formulate the whole body of axioms of probability theory came together. It is not enough to have all ideas in mind... one must make sure that their totality is sufficient, bring them together explicitly and take the responsibility for saying that nothing further is needed in order to construct the theory. This is what Kolmogorov did. This is his achievement. And we do not believe that he wanted to claim any others, so far as the axiomatic theory is concerned.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2021-12-29T17:51:03.947" LastActivityDate="2021-12-29T17:51:03.947" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14007" PostTypeId="1" AcceptedAnswerId="14009" CreationDate="2021-12-30T12:40:04.507" Score="1" ViewCount="134" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Fourier&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean-Baptiste Joseph Fourier&lt;/a&gt; was a French mathematician and physicist who initiated the investigation of Fourier series. The following is the only portrait of Fourier that I know of. Are there any portraits of Fourier other than this one?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oqKYF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oqKYF.jpg&quot; alt=&quot;Painting of Fourier&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14690" LastEditorUserId="4931" LastEditDate="2021-12-31T13:38:04.337" LastActivityDate="2021-12-31T13:38:04.337" Title="Are there any portraits of Joseph Fourier other than this one?" Tags="|mathematicians|fourier|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14008" PostTypeId="1" AcceptedAnswerId="14010" CreationDate="2021-12-30T14:47:27.173" Score="2" ViewCount="186" Body="&lt;p&gt;I've read several texts suggesting that Plato defines &amp;quot;perfect number&amp;quot; in his Republic, book VIII 546 b. However, there's no definition as we can see from - for example - this &lt;a href=&quot;https://www.gutenberg.org/files/55201/55201-h/55201-h.htm#BookVIII&quot; rel=&quot;nofollow noreferrer&quot;&gt;translation&lt;/a&gt;: &amp;quot;... Now that which is of divine birth has a period which is contained in a perfect number&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; but the period of human birth...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;1: i.e. a cyclical number, such as 6, which is equal to the sum of its divisors 1, 2, 3, so that when the circle or time represented by 6 is completed, the lesser times or rotations represented by 1, 2, 3 are also completed.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So it's just the footnotes that explain what the perfect number is. Where does Plato himself define it?&lt;/p&gt;&#xA;" OwnerUserId="15928" LastActivityDate="2021-12-31T00:06:19.303" Title="(Where) does Plato define perfect number?" Tags="|mathematics|reference-request|ancient-greece|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14009" PostTypeId="2" ParentId="14007" CreationDate="2021-12-30T15:12:57.980" Score="1" Body="&lt;p&gt;Here is a caricature of Fourier and Legendre:&#xA;&lt;a href=&quot;http://www.numericana.com/fame/legendre-fourier.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.numericana.com/fame/legendre-fourier.jpg&lt;/a&gt;&#xA;(This is the only existing portrait of Legendre).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2021-12-30T15:12:57.980" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14010" PostTypeId="2" ParentId="14008" CreationDate="2021-12-31T00:06:19.303" Score="4" Body="&lt;p&gt;Plato does not define perfect numbers anywhere. The earliest extant definition is in Euclid VII, def. 22. Although Plato uses the term &amp;quot;perfect number&amp;quot; I do not think it certain that he understood this phrase in the same way as Euclid.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2021-12-31T00:06:19.303" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14011" PostTypeId="1" CreationDate="2021-12-31T17:40:55.140" Score="1" ViewCount="77" Body="&lt;p&gt;I have a &lt;a href=&quot;https://hsm.stackexchange.com/a/14003/1800&quot;&gt;reason to believe&lt;/a&gt; that Euler introduced the complex potential in his &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/227/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Continuation des recherches sur la theorie du mouvement does fluides&lt;/a&gt;, published in 1757. However, I am having hard time recognizing where it appears. The relevant place seems to be here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/C0odL.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/C0odL.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;After establishing that &lt;span class=&quot;math-container&quot;&gt;$(u-iv)(dx+idy)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(u+iv)(dx-idy)$&lt;/span&gt; are integrable, he introduces functions &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt;. The problem is I am not sure what &lt;span class=&quot;math-container&quot;&gt;$\phi:(x+yi)$&lt;/span&gt; means. Normally, I would expect the expression to involve some kind of derivative, such as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(u-iv)(dx+idy) = d\phi + id\psi.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am hoping somebody could explain what is going on here.&lt;/p&gt;&#xA;&lt;p&gt;If you need, Euler continues on the following page as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GCvns.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GCvns.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="1800" LastActivityDate="2021-12-31T17:40:55.140" Title="Complex potential in E227?" Tags="|mathematical-physics|18th-century|fluid-dynamics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14012" PostTypeId="1" AcceptedAnswerId="14608" CreationDate="2022-01-01T07:50:31.480" Score="-1" ViewCount="295" Body="&lt;p&gt;This my post may be inappropriate since math may be irrelevant in this topic.&lt;/p&gt;&#xA;&lt;p&gt;I've read &lt;a href=&quot;https://en.wikipedia.org/wiki/Mathew_Bevan&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article of wikipedia&lt;/a&gt; of the hacker called Mathew Bevan.&lt;/p&gt;&#xA;&lt;p&gt;Seemingly the ariticle has not specified how Bevan acquired the skills of hacking which led him to penetrate into the system of the air-force-base.&lt;/p&gt;&#xA;&lt;p&gt;I always wonder how nominant hackers has acquired the hacking-skills.&lt;/p&gt;&#xA;&lt;p&gt;For instance in math, we can acquire the skills of calculus-stuff doing exercises of problems in textbooks. But how with penetrating into a military-system?&lt;/p&gt;&#xA;&lt;p&gt;By the way I know that some pentest-os let us to learn some ethical-hacking-skills.&lt;/p&gt;&#xA;" OwnerUserId="14765" LastEditorUserId="4251" LastEditDate="2022-01-01T12:54:21.287" LastActivityDate="2022-07-14T08:56:11.140" Title="How did Mathew Bevan acquire hacking skill without belonging to military or national agencies?" Tags="|computers|" AnswerCount="1" CommentCount="0" ClosedDate="2022-08-17T08:51:43.237" ContentLicense="CC BY-SA 4.0" />
  <row Id="14013" PostTypeId="1" CreationDate="2022-01-01T14:07:59.513" Score="2" ViewCount="117" Body="&lt;p&gt;I have found many different claimed answers to this question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stream_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; on the stream function claims that Lagrange introduced it in 1781.&lt;/li&gt;&#xA;&lt;li&gt;Darrigol's &lt;a href=&quot;https://books.google.ca/books?id=tIRBzdTWwyUC&amp;amp;pg=PA163&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Worlds of Flow&lt;/a&gt; says that D'Alambert &lt;a href=&quot;https://fr.wikisource.org/w/index.php?title=Page:D%27Alembert_-_Opuscules_math%C3%A9matiques,_Tome_1,_1761.djvu/159&amp;amp;action=edit&amp;amp;redlink=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;did it in 1761&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/14002/who-introduced-velocity-potential/&quot;&gt;The Oxford Handbook of the History of Physics&lt;/a&gt; mentions that Euler introduced the complex potential and so by extension the stream function &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/227/&quot; rel=&quot;nofollow noreferrer&quot;&gt;in 1757&lt;/a&gt; (p.356). However, &lt;a href=&quot;https://hsm.stackexchange.com/questions/14011/complex-potential-in-e227&quot;&gt;I am not sure&lt;/a&gt; if this claim is true.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Can somebody clarify this? Thanks a lot.&lt;/p&gt;&#xA;" OwnerUserId="1800" LastEditorUserId="1800" LastEditDate="2022-01-01T15:13:04.347" LastActivityDate="2022-01-02T15:46:24.420" Title="Who introduced the stream function?" Tags="|complex-analysis|18th-century|fluid-dynamics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14014" PostTypeId="1" CreationDate="2022-01-01T18:28:21.303" Score="4" ViewCount="170" Body="&lt;p&gt;The following post is long, but I decided to write more rather than less in case it's helpful.  I tried to make it clear, quick, and easy to skip to the short version of my question, so the reader can decide whether they're interested in answering this.&lt;/p&gt;&#xA;&lt;h1&gt;For context and contrast&lt;/h1&gt;&#xA;&lt;p&gt;In most mathematical subjects, I can give a story about why it is what it is.  For instance, in algebra, if I were to explain what this subject is about, I would tell a story about some quantity that is unknown but what is known is its relationship to other quantities.  One large initial concern of algebra is to infer the unknown quantity.  From there, the focus can shift to understanding the abstract algebraic properties of the operations themselves, how one quantity varies as a function of another, and so on.&lt;/p&gt;&#xA;&lt;p&gt;In calculus I can give a story about modeling physics.  Suppose you are a medieval scientist who rejects Aristotle's physics, and you conjecture that gravity gives objects a constant acceleration.  But to test the theory out you need to determine from that hypothesis a measurable, and therefore you need to infer the position function.  From there you can get into all the other topics.  It's an over-simplification of the actual history, but I find it good enough to help me understand the subject, and it helps me explain the condensed essence of the subject to a new student.&lt;/p&gt;&#xA;&lt;p&gt;And I could go on, with complex analysis, real analysis, abstract algebra, linear algebra, mathematical logic, set theory.  For each one of these I can give a story that at least I find satisfying, and hopefully so would a student new to the subject.&lt;/p&gt;&#xA;&lt;p&gt;For topology I don't know how to do that.  I can tell &lt;em&gt;half&lt;/em&gt; intelligible stories.  Like we wanted to understand continuity for many-variable complex functions, and so we became interested in the idea of an open ball in the codomain, and corresponding open ball in the domain.  And with this you can motivate the idea of metric spaces more generally.  And you can then point out that some spaces don't have a natural metric, yet you still want to define continuity, and to do that you want to be able to say what the open sets are without appealing to any metric.&lt;/p&gt;&#xA;&lt;p&gt;Ok, all of that makes sense, but then there's the leap from there to the modern definition of a topology.  That leap is too large.  It doesn't tell you why &amp;quot;a subset of the powerset containing &lt;span class=&quot;math-container&quot;&gt;$\emptyset$&lt;/span&gt; and the universe, closed under union and finite intersection&amp;quot; is what we want from open sets, or why it delivers a satisfying version of continuous functions.&lt;/p&gt;&#xA;&lt;p&gt;For instance, it doesn't tell me why we shouldn't include in the definition extra axioms like the separation axioms--that too seems like a generalization of open sets, so why not pick that?  Or we could go in the other direction and remove the requirement that open sets be closed under finite unions.  That too gives you objects which are &amp;quot;like&amp;quot; open sets.  The open sets of real numbers, for instance, would still have this property.  So maybe that's the generalization we want.  But we drew the line here, at the modern definition of a topology, and as a mathematical community, said &amp;quot;anything more general, we are not willing to call an open set, and anything less general we call a special case of a more general notion of an open set&amp;quot;.  Why draw the line here?&lt;/p&gt;&#xA;&lt;p&gt;I'm sure there's an explanation, I just don't know what it is.  I've read about six chapters of Munkres so I know maybe the first thing about point-set topology.  I've solved a couple problems having to do with the Zariski topology, and I've heard about (but really know nothing about) &lt;span class=&quot;math-container&quot;&gt;$C^*$&lt;/span&gt;-algebras.  I know essentially nothing about algebraic and analytic topology.  So that's about where my current level of understanding is.  But as far as I can tell, the Zariski and Sierpinski topologies, and &lt;span class=&quot;math-container&quot;&gt;$C^*$&lt;/span&gt;-algebras, were all developments long after the definition of a topology was decided.  So they seem a little out of bounds to explain why mathematicians did and should have defined topology the way they did.&lt;/p&gt;&#xA;&lt;h1&gt;My actual question&lt;/h1&gt;&#xA;&lt;p&gt;What caused mathematicians to pick these axioms and not some more or less restrictive axioms, to define a topology?&lt;/p&gt;&#xA;&lt;h1&gt;My current best answer to the question&lt;/h1&gt;&#xA;&lt;p&gt;The answer I give below is almost certainly wrong in several places, so I welcome corrections.  I'm including it here to possibly give some sense of what is missing and possibly wrong about my current understanding.&lt;/p&gt;&#xA;&lt;p&gt;So Newton invents the calculus.  Riemann much later contributes to a more rigorous foundation, and also advocates for a more abstract understanding of mathematics generally.  He worked particularly on the idea of a manifold, and it was in this setting that he repeated throughout his career that we should develop a mathematics of geometry without concern for a metric.  The foundational crisis occurs in set theory and logic.  We start to see how we can make set theory well-founded, and people start developing several mathematical disciplines on set theory.&lt;/p&gt;&#xA;&lt;p&gt;Now here I really get my timeline fuzzy, but I know that at about this time Klein is working on the uniformization theorem (I don't know what that is, but it seems to be an initial motivation for thinking about &amp;quot;covering&amp;quot;, which I assume is the same as &amp;quot;an open cover&amp;quot; and the idea of a compact set having a finite subcover.).  I guess &lt;em&gt;somehow&lt;/em&gt; the modern idea of a topology &amp;quot;fit well&amp;quot; with Klein's work?  Like, I guess lots of various proposed axiom sets existed, which delivered variously special and general ideas of open sets.  Some of them were general enough to apply to this problem, and others weren't?&lt;/p&gt;&#xA;&lt;p&gt;I think, earlier than this, Hilbert had listed several geometric axioms.  Later, Riesz proposed one set of axioms, which were equivalent to the modern definition of a topology, but with the extra requirement of the Hausdorff axiom.  After that, Hausdorff took Hilbert's list of geometric axioms and crossed off all but four of them, and this collection of axioms was equivalent to the modern definition of a topology, which ended up being equivalent to what Riesz had done.  But later Kuratowski did the same as Hausdorff but dropped the Hausdorff axiom, which might mark the start of the modern definition of a topology.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Because Kuratowski's system applied more generally than Riesz's/Hausdorff's, but was still sufficiently powerful to prove the uniformization theorem, ... &lt;em&gt;maybe&lt;/em&gt; that's why mathematicians all agreed that Kuratowski's system was the ideal place to draw the line?  Is that at least a plausible oversimplification of the story?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I know that Poincare was important, both before and after this.  But as far as I can tell, his importance was in developing algebraic topology, so I'm not sure that it's necessary to bring him into the history of the founding of point-set topology.&lt;/p&gt;&#xA;&lt;p&gt;I've also seen some people's names thrown around as contributing to the founding of modern point-set topology, but I haven't yet been able to really appreciate the nature of their contributions.  Betti, Brouwer, Cantor, Borel, and more, I'm not sure exactly how they fit into this story, but I know that a more complete history includes them.&lt;/p&gt;&#xA;&lt;h1&gt;Resources I've consulted so far&lt;/h1&gt;&#xA;&lt;p&gt;For posterity I'll collect a bunch of resources I've consulted, to one degree or another, for my current understanding.  Because I'm not a mathematician, I am not able to understand everything that I've read in these resources and that may explain any mistakes I've made above.&lt;/p&gt;&#xA;&lt;h2&gt;Links&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/3136/history-of-covering-spaces&quot;&gt;History of covering spaces&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/82/what-examples-led-to-the-modern-definition-of-a-topological-space&quot;&gt;What examples led to the modern definition of a topological space?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/8409/history-of-various-definitions-of-topology&quot;&gt;History of various definitions of topology&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/12787/historical-ways-of-presenting-or-axiomatizing-the-notion-of-a-topological-spac?rq=1&quot;&gt;Historical ways of *presenting* or axiomatizing the notion of a topological space&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.britannica.com/science/topology/History-of-topology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia Britannica - History of Topology&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;Books&lt;/h2&gt;&#xA;&lt;p&gt;Dieudonne, &lt;em&gt;A History of Algebraic and Differential Topology, 1900-1960&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Fierrieros, &lt;em&gt;Labyrinth of Thought&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;James, &lt;em&gt;History of Topology&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Aull and Lowen, &lt;em&gt;A Handbook of the History of General Topology&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I did not consult the following text because it's just too expensive, but I have heard people say it's very good:&lt;/p&gt;&#xA;&lt;p&gt;Manheim, &lt;em&gt;The Genesis of Point-set Topology&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="2732" LastEditorUserId="4251" LastEditDate="2022-01-02T13:12:51.137" LastActivityDate="2022-01-02T13:12:51.137" Title="Why did the mathematical community settle on these properties to define a topology?" Tags="|mathematics|topology|definition|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14015" PostTypeId="1" CreationDate="2022-01-02T13:17:46.483" Score="3" ViewCount="69" Body="&lt;p&gt;Educated in the UK, I've been used to calling the force on a current-carrying wire in a magnetic field 'the &lt;em&gt;motor effect&lt;/em&gt; force'. But I'm increasingly aware of another (less clumsy?) name for it: 'the &lt;em&gt;Laplace&lt;/em&gt; force'. Less clumsy, maybe; but is the name justified? What role did Laplace play in investigating or analysing these forces?&lt;/p&gt;&#xA;" OwnerUserId="15941" LastEditorUserId="15941" LastEditDate="2022-02-05T19:50:47.040" LastActivityDate="2022-02-05T19:50:47.040" Title="Why is the magnetic force on a current-carrying wire sometimes called the Laplace force?" Tags="|physics|electromagnetism|electricity|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14016" PostTypeId="2" ParentId="14013" CreationDate="2022-01-02T15:46:24.420" Score="4" Body="&lt;p&gt;d'Alembert did originally introduce the stream function in &amp;quot;Remarques sur les lois du mouvement des fluides&amp;quot; in 1761.  &lt;a href=&quot;https://hdl.handle.net/2027/njp.32101037493978?urlappend=%3Bseq=171%3Bownerid=27021597769878015-183&quot; rel=&quot;nofollow noreferrer&quot;&gt;Page 149&lt;/a&gt; says that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$p = \frac{\partial \omega}{\partial x} \,,$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$q = - \frac{\partial \omega}{\partial z} \,,$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$q = -u$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p = -v$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$z = y$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\omega = \psi$&lt;/span&gt; in modern notation.  See page 138 for a better way to see how these variables match modern notation.  This is the stream function once you parse the odd notation.&lt;/p&gt;&#xA;&lt;p&gt;Lagrange basically repeated this more clearly in 1781 in &amp;quot;Mémoire sur la théorie du mouvement des fluides&amp;quot;.  See &lt;a href=&quot;https://hdl.handle.net/2027/nyp.33433009864657?urlappend=%3Bseq=235%3Bownerid=27021597769463683-297&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 173&lt;/a&gt; in particular.  Lagrange changes the variables, though.  He gives&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$p = \frac{\partial \omega}{\partial y} \,,$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$q = - \frac{\partial \omega}{\partial x} \,,$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$p = u$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q = v$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\omega = \psi$&lt;/span&gt;.  Other than that this is basically the modern notation for the stream function.&lt;/p&gt;&#xA;&lt;p&gt;I'm not familiar with the history of the complex potential, so unfortunately I cannot help you there.&lt;/p&gt;&#xA;" OwnerUserId="15946" LastActivityDate="2022-01-02T15:46:24.420" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14017" PostTypeId="1" AcceptedAnswerId="14023" CreationDate="2022-01-03T01:58:45.117" Score="5" ViewCount="1786" Body="&lt;p&gt;I have watched a video about Napoleon's theorem — maybe it was contributed by Napoleon, maybe not. I also know that Laplace himself said Napoleon was good at mathematics. However, did Napoleon make any contributions to mathematics other than his theorem?&lt;/p&gt;&#xA;" OwnerDisplayName="user15948" LastEditorUserId="4251" LastEditDate="2022-01-06T08:27:24.900" LastActivityDate="2022-07-08T10:10:41.427" Title="What contributions to mathematics did Napoleon make?" Tags="|mathematics|mathematicians|" AnswerCount="5" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14018" PostTypeId="1" CreationDate="2022-01-03T03:09:14.280" Score="0" ViewCount="84" Body="&lt;p&gt;&amp;quot;Dragon Dice&amp;quot; was a simple gambling game which rolled a few dice, and 1s were special in some way. A European noble played this game with his friends, who would lose a lot of money to him and then refuse to play this game any more.&lt;/p&gt;&#xA;&lt;p&gt;This game was notable in that people normally calculate the probability of winning incorrectly; they incorrectly estimate the chance of the special case and thus think the game is profitable when it is not.&lt;/p&gt;&#xA;&lt;p&gt;However, I have not been able to find this game in any search engine, as results are polluted by a different &amp;quot;Dragon Dice&amp;quot; game made by TSR. Does anyone have a reference to the original game I am mentioning?&lt;/p&gt;&#xA;" OwnerUserId="15949" LastEditorUserId="15949" LastEditDate="2022-01-03T03:28:34.900" LastActivityDate="2022-01-03T04:51:34.857" Title="&quot;Dragon Dice&quot;, a historical gambling game involved in the development of probability" Tags="|mathematics|probability|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14019" PostTypeId="2" ParentId="14018" CreationDate="2022-01-03T04:51:34.857" Score="2" Body="&lt;p&gt;My recollection was incorrect; the game is called Dragon Die, and comes from &lt;a href=&quot;https://gamebalanceconcepts.wordpress.com/2010/07/28/level-4-probability-and-randomness/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gamebalanceconcepts.wordpress.com/2010/07/28/level-4-probability-and-randomness/&lt;/a&gt;. The dice game used historically to abuse perception of probability was named something else (though Dragon Die also satisfies that criterion).&lt;/p&gt;&#xA;" OwnerUserId="15949" LastActivityDate="2022-01-03T04:51:34.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14020" PostTypeId="1" CreationDate="2022-01-03T07:57:25.810" Score="0" ViewCount="53" Body="&lt;p&gt;On the nature of the Child is quoted by Galen in his Book &amp;quot;On Semen&amp;quot;. I want to know what date is given to these two books, especially &amp;quot;On Generation&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="15534" LastEditorUserId="55" LastEditDate="2022-01-04T05:12:25.777" LastActivityDate="2022-01-04T05:12:25.777" Title="What date is assigned to Hippocratic treatises &quot;On Generation&quot; and &quot;On the Nature of the Child&quot;?" Tags="|biographical-details|ancient-greece|biology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14021" PostTypeId="2" ParentId="14017" CreationDate="2022-01-03T16:34:36.870" Score="11" Body="&lt;p&gt;The main contribution of Napoleon to math was his strong support of the new education system established by the revolutionary government some time before he came to power. For example, he re-established Ecole Normale Superieure which was previously closed by the Consulate. This became one of the top institutions of mathematics education and research. He also established the Scuola Normale Superiore in Pisa.&lt;/p&gt;&#xA;&lt;p&gt;Napoleon was presumably &amp;quot;good in math&amp;quot;, as a graduate of Ecole Militaire, but did not make any original contribution.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-01-03T16:41:44.077" LastActivityDate="2022-01-03T16:41:44.077" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14022" PostTypeId="1" CreationDate="2022-01-04T14:34:42.507" Score="1" ViewCount="64" Body="&lt;p&gt;The double-slit experiment shows the fundamentally probabilistic nature of quantum mechanical phenomena. On Wikipedia one can read:&lt;/p&gt;&#xA;&lt;p&gt;This type of experiment was first performed, using light, by Thomas Young in 1801, as a demonstration of the wave behavior of light.&lt;/p&gt;&#xA;&lt;p&gt;Einstein proposed interpreting the square of amplitude as probability density (Born's rule). However, who was the scientist in the history of quantum mechanics, who was the first to present a &lt;em&gt;closed formula&lt;/em&gt; for the probability density that a photon passing through the slits will hit the screen at a certain point say &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;? I know this function is the square of the underlying wave function or the mod square, when the underlying wave function is given as a complex function. What does the wave function look like according to this scientist and what does the probability density function look like? Actual formulas of the wave function and the probability density function and literature references would be welcome.&lt;/p&gt;&#xA;" OwnerUserId="15957" LastEditorUserId="10106" LastEditDate="2022-01-04T16:43:52.443" LastActivityDate="2022-01-04T16:43:52.443" Title="Who was the first scientist to give a formula for the probability density function of the position of a photon in the double slit experiment?" Tags="|physics|quantum-mechanics|theoretical-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14023" PostTypeId="2" ParentId="14017" CreationDate="2022-01-05T09:43:08.133" Score="6" Body="&lt;p&gt;First of all, a very interesting document (in french) &lt;a href=&quot;https://www.napoleon.org/histoire-des-2-empires/articles/napoleon-et-les-scientifiques-1-1779-1798/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; by Nicole DHOMBRES gives abundant information about the scientific level of Napoleon.&lt;/p&gt;&#xA;&lt;p&gt;Now, as a complement to the last sentence of the answer by Alexandre Eremenko, here is a testimony of the way mathematics had left their imprint on Napoleon. Here are two  excerpts from the very interesting memoirs of baron de Comeau (in &amp;quot;Le tacticien de Napoléon : Mémoires de guerre du Baron de Comeau&amp;quot; available as a &lt;a href=&quot;https://books.google.fr/books?id=yJGwDwAAQBAJ&amp;amp;dq=napoleon+bezout+masse+vitesse+comeau&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google book&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;These excerpts are from chapter XXXVIII. The reference to Bezout is explained by the fact that the two books entitled &amp;quot;Cours de mathématiques à l'usage du corps de l'artillerie&amp;quot; by Bezout, clearly written, were reference books in Brienne, the Artillery School that Napoleon attended. A revised version of the second of these books can be found &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k6212877w/f467.item.texteImage&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/PBeOZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PBeOZ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/IcrDI.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/IcrDI.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My own translation of these texts (I have added quotes when Napoleon is speaking).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;It is comparable to our blackboard where we were maneuvering algebra. You have made a large use of this algebra in all your campaigns. I remember you were good at that and had a fair understanding that minus times minus make plus. I haven't been bad at applying this principle: minus Germany, minus Austria, minus Prussia, minus Italy, accept that (showing himself), I am a large plus. &lt;br&gt; &lt;br&gt; &amp;quot;Malborough is going to war&amp;quot; (18th century french song sang by Napoleon). He stopped close to me. &amp;quot;I would have liked to avoid this war. Alexander [tsar Alexander I] doesn't know any longer where he is going, where we draw him. Bezout has taught you, like me, that mass times speed gives the kinetic energy, the momentum. I have enough for the mass. The speed will be there, before his sun goes down. Days last a long time in Russia when it is sunny. I will fight two, three battles. If he considers stopping.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3730" LastEditorUserId="3730" LastEditDate="2022-07-08T10:10:41.427" LastActivityDate="2022-07-08T10:10:41.427" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14024" PostTypeId="2" ParentId="14017" CreationDate="2022-01-06T02:02:12.847" Score="7" Body="&lt;p&gt;Ok, here is another tale, not sure how much of it I believe, taken from Chapter 3 of the book&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;George E. Martin, &lt;a href=&quot;https://doi.org/10.1007/978-1-4612-0629-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Geometric Constructions&lt;/a&gt;, Springer-Verlag, 1998.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The chapter begins with:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In December of 1797 there took place in Paris a brilliant gathering of prominent writers and scholars, with the immortal Lagrange and Laplace among&#xA;them. A most conspicuous member of the company was the young and victorious General Napoleon Bonaparte, who ... had occasion to entertain Lagrange&#xA;and Laplace with a kind of solution of some elementary problems of elementary geometry that was completely unfamiliar to either of the two world famous mathematicians. Legend has it that after having listened to the young man for a considerable while, Laplace, somewhat peeved, remarked, &amp;quot;General, we expected everything of you, except lessons in geometry.&amp;quot;&#xA;N. A. Court&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This quote, appears to be due to Nathan Altshiller Court, from Oklahoma University. I have no idea how reliable N. A. Court is as a storyteller. But he kept his source of the story secret. The quote is taken from&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;N. A. Court, &lt;a href=&quot;https://doi.org/10.5951/MT.51.5.0370&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mascheroni constructions&lt;/a&gt;, The Mathematics Teacher, May 1958, Vol. 51, No. 5 (May 1958), pp. 370-372&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The editorial note following Court's paper says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is reason to believe that the idea of undertaking geometric constructions with compasses alone was suggested to Mascheroni by the&#xA;earlier work of Giambattista Benedetti&#xA;(1530-1590)....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Martin then continues:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Napoleon proposed to the French mathematicians the problem of dividing a circle into four congruent arcs by using the compass alone. Although&#xA;not original with Napoleon, the problem has become known as Napoleon's&#xA;Problem. During his campaign in northern Italy, Napoleon had encountered the poet and geometer Lorenzo Mascheroni (1750-1800). Mascheroni was a professor at the University of Pavia, where Christopher Columbus&#xA;had once been a student. Mascheroni's most famous mathematical work is&#xA;his &amp;quot;Geometria del Compasso,&amp;quot; published in 1797. This work, which began&#xA;with an ode of some literary merit that was dedicated to Napoleon, showed&#xA;that all the ruler and compass constructions can be accomplished with the&#xA;euclidean compass alone.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Well, in fact, Mascheroni was not the first to prove that theorem (ostensibly, solving &amp;quot;Napoleon's Problem&amp;quot; as a special case). Mascheroni's theorem had been (unknown to Mascheroni) proved in 1672 by a little known Danish mathematician Georg Mohr (as discussed in the same paper by N.A.Court).&lt;/p&gt;&#xA;&lt;p&gt;So, if we were to believe N.A.Court, Napoleon's contribution in this case was to propose &amp;quot;Napoleon's Problem&amp;quot; to a group of French mathematicians, a problem which was solved 30 years earlier.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="4251" LastEditDate="2022-01-06T08:21:15.923" LastActivityDate="2022-01-06T08:21:15.923" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14025" PostTypeId="2" ParentId="14017" CreationDate="2022-01-06T08:25:17.663" Score="0" Body="&lt;p&gt;in the sense that chess is a mathematical game, there's the napoleon opening.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Napoleon_Opening&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Napoleon_Opening&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit: user15948 says it's not such a good opening. I'm not saying it is. It's just an opening that is named after the guy.&lt;/p&gt;&#xA;" OwnerUserId="3338" LastEditorUserId="3338" LastEditDate="2022-05-08T10:41:03.720" LastActivityDate="2022-05-08T10:41:03.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14026" PostTypeId="1" CreationDate="2022-01-06T15:33:05.893" Score="3" ViewCount="186" Body="&lt;p&gt;In a French edition of &lt;em&gt;Fiat Homo&lt;/em&gt; (first part of &lt;em&gt;A Canticle for Leibowitz&lt;/em&gt;) I found the following footnote about the definition of the electron given by Brother Francis to another monk, namely “Torsion du néant négativement chargée”. The original English has “negative twist of nothingness”, but there are no footnote in any English edition.&lt;/p&gt;&#xA;&lt;p&gt;The footnote is, in French, copied &lt;strong&gt;exactly&lt;/strong&gt; as it appears, (including the misprint in the name of Millikan !):&lt;/p&gt;&#xA;&lt;p&gt;Définition exacte (donnée par le Pr Léon Brillouin, puis reprise par Robert Andrews Mullikan, prix Nobel). Elle est en effet incompréhensible si l’on n’a pas le contexte, c’est à dire toute la complexe structure de notre physique.&lt;/p&gt;&#xA;&lt;p&gt;The second sentence is not important for my question and the first one, I believe, does not need any translation.&lt;/p&gt;&#xA;&lt;p&gt;So is there any source about Léon Brillouin giving such a definition for the electron ? He might have given it in French, or in English.&#xA;Again, any source about Robert Andrews Millikan giving it ?&lt;/p&gt;&#xA;&lt;p&gt;Millikan might have translated  into English Brillouin’s original phrase which might have been slightly different, Miller probably used Millikan’s form (if it really exists) and the french translator maybe did not bother to trace Brillouin original phrase (if it exists).&lt;/p&gt;&#xA;&lt;p&gt;Besides Brillouin and/or Millikan does anyone know of a similar definition of the electron by a “serious” physicist of that time, even if his theories are now totally  abandoned ? I am thinking in particular of William Thomson, a.k.a. Lord Kelvin and his “Vertex Theory”, or J. J. Thomson who, before he discovered the electron, did write papers about Kelvin's Vertex Theory.&lt;/p&gt;&#xA;" OwnerUserId="15967" LastEditorUserId="10106" LastEditDate="2022-01-06T17:39:57.260" LastActivityDate="2022-01-07T10:49:21.310" Title="Is there a source for a footnote in *A Canticle for Leibowitz* about the definition of the electron, &quot;Negative Twist of Nothingness&quot;" Tags="|physics|particle-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14027" PostTypeId="1" CreationDate="2022-01-06T17:35:46.513" Score="0" ViewCount="92" Body="&lt;p&gt;The answer to a different question &lt;a href=&quot;https://hsm.stackexchange.com/questions/7437/where-did-fermi-get-the-u235-for-the-first-nuclear-pile&quot;&gt;(Where did Fermi get the U235 for the first nuclear pile)&lt;/a&gt; about U235, was that Fermi used natural uranium for his reactor. This explains, in particular, the origin of the Pu239 for “Thin Man”.&lt;/p&gt;&#xA;&lt;p&gt;But how did the US obtain in just a few years enough weapon-grade enriched uranium for “Little Boy” ? Just centrifuges ?&lt;/p&gt;&#xA;" OwnerUserId="15967" LastEditorUserId="4251" LastEditDate="2022-01-07T10:38:32.073" LastActivityDate="2022-01-07T10:38:32.073" Title="How did the US get enough U235 for &quot;Little Boy&quot;" Tags="|physics|nuclear-physics|world-war-2|20th-century|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14028" PostTypeId="2" ParentId="13957" CreationDate="2022-01-06T18:35:32.003" Score="0" Body="&lt;p&gt;Lemaître model was a &amp;quot;cold matter&amp;quot; one. Our present model of &amp;quot;hot&amp;quot; Big Bang can be traced to the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\alpha\beta\gamma$&lt;/span&gt;&amp;quot; model &lt;a href=&quot;https://en.wikipedia.org/wiki/Alpher%E2%80%93Bethe%E2%80%93Gamow_paper&quot; rel=&quot;nofollow noreferrer&quot;&gt;by Alpher, Bethe and Gamow&lt;/a&gt; in 1948.&lt;/p&gt;&#xA;&lt;p&gt;It only became generally accepted after the discovery of the cosmic microwave background radiation by Penzias and Wilson in 1964.&lt;/p&gt;&#xA;" OwnerUserId="15967" LastEditorUserId="15967" LastEditDate="2022-01-06T18:52:01.147" LastActivityDate="2022-01-06T18:52:01.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14029" PostTypeId="2" ParentId="11565" CreationDate="2022-01-06T21:08:50.530" Score="5" Body="&lt;p&gt;Riemann wrote his paper on the zeta-function in 1859. He was the first person to consider the zeta-function in the complex plane, first for &lt;span class=&quot;math-container&quot;&gt;${\rm Re}(s) &amp;gt; 1$&lt;/span&gt; and then he worked out an analytic continuation to all of &lt;span class=&quot;math-container&quot;&gt;$\mathbf C$&lt;/span&gt; except for a simple pole at &lt;span class=&quot;math-container&quot;&gt;$s = 1$&lt;/span&gt;.  Before Riemann the only way the zeta-function was used was on the real line: real numbers bigger than 1 for Chebyshev and Dirichlet, and positive and negative real numbers by Euler using divergent series.  Riemann's use of complex analysis in his paper on the zeta-function was unlike anything anyone  had done before.  For example, his use of Fourier inversion to replace an integral on the positive real line with a vertical integral in &lt;span class=&quot;math-container&quot;&gt;$\mathbf C$&lt;/span&gt; was new.&lt;/p&gt;&#xA;&lt;p&gt;There were three majors figures in the development of basic complex analysis: Cauchy (1810s-1850s), Riemann (1850s-1860s), and Weierstrass (1850s-1880s). All Riemann had available to learn complex analysis was the papers of Cauchy, which were rather disorganized since Cauchy himself did not realize until the 1850s the many ways that complex-valued functions of one complex variable were far more special than functions of two real variables. The main ideas Cauchy had were his integral formula and his residue theorem, but in a much more primitive form than the way we know them today.&lt;/p&gt;&#xA;&lt;p&gt;Riemann figured out a lot on his own: his dissertation in 1851 was about Riemann surfaces, going far beyond what Cauchy had been doing.  Cauchy's focus was on using contour integrals and residues in the complex plane, mainly to compute real integrals. Riemann's vision was totally different and in many ways much grander: use the Cauchy-Riemann equation (PDEs) and conformal mappings, analytic continuation,  and study functions geometrically rather than by relying on formulas for their values everywhere. Weierstrass did everything with power series rather than with integrals or geometry. The ground in complex analysis that was prepared to let Riemann conjecture the Riemann hypothesis had been developed in large part by Riemann himself.&lt;/p&gt;&#xA;&lt;p&gt;See Section 5.10 of Bottazzini and Gray's &amp;quot;Hidden Harmonies - Geometric Fantasies: The Rise of Complex Function Theory&amp;quot; for a summary of Riemann's paper and a description of what was novel in it.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-08-17T15:57:16.733" LastActivityDate="2022-08-17T15:57:16.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14030" PostTypeId="2" ParentId="14026" CreationDate="2022-01-07T10:15:14.900" Score="2" Body="&lt;p&gt;I have found a &lt;strong&gt;provisional&lt;/strong&gt; answer to my own question.&lt;/p&gt;&#xA;&lt;p&gt;For the time being I will &lt;strong&gt;not&lt;/strong&gt; accept it because it does not mention either Léon Brillouin or Robert Andrews Millikan.&lt;/p&gt;&#xA;&lt;p&gt;I still hope someone will come up with a source for one of both of them.&lt;/p&gt;&#xA;&lt;p&gt;But Larmor was most certainly a &amp;quot;serious&amp;quot; physicist of that time (a &amp;quot;great&amp;quot; physicist would be more appropriate).&lt;/p&gt;&#xA;&lt;p&gt;In a remarkable book I found on the Web, &lt;em&gt;Histories of the Electron The Birth of Microphysics edited by Jed Z. Buchwald and Andrew Warwick H&lt;/em&gt;  I found two quotes by Larmor, in the very last years of the 19th Century, that apply the word &amp;quot;twist&amp;quot; to the electron. Not &amp;quot;twist of nothingness&amp;quot;, nor &amp;quot;negative&amp;quot;, just twist. But it is clearly within the (now totally outdated) Vortex Theory of Lord Kelvin.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a few months Larmor reconstructed his theory on the basis of FitzGerald’s suggestion. Currents were now identified with the transfer of freecharges (“monads”), which were also the cause of magnetic phenomena. Those charges had the ontological status of independent entities and ceased to be epiphenomena of the field. Furthermore, material atoms were represented as stable configurations of electrons. In Larmor’s words,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“the core of the vortex ring [constituting an atom] . . . [is] made up of discrete electric nuclei or centres of radial twist in the medium. The circulation of these nuclei along the circuit of the core would constitute a vortex . . . its strength is now subject to variation owing to elastic action, so that the motion is no longer purely cyclic. A magnetic atom, constructed after this type, would behave like an ordinary electric current in a nondissipative circuit. It would, for instance, be subject to alteration of strength by induction when under the influence of other changing currents, and to recovery when that influence is removed.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Math. and Phys. Papers, vol. 1, 515&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There was no year of publication provided for this one. This is a quote by Larmor within a quote from the book &lt;em&gt;Histories..&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Second quote, again a quote by Larmor within a quote from the book&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Larmor’s “electrons” were conceived as permanent structures in the&#xA;ether with the following characteristics:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“An electron has a vacuous core round which the radial twist is&#xA;distributed. . . . It may be set in radial vibration, say pulsation,&#xA;and this vibrational energy will be permanent, cannot possibly be&#xA;radiated away. All electrons being alike have the same period: if the&#xA;amplitudes and phases are also equal for all at any one instant, they&#xA;must remain so . . . Thus an electron has the following properties,&#xA;which are by their nature permanent ii(i) its strength [= electric&#xA;charge] i(ii) its amplitude of pulsation (iii) the phase of its&#xA;pulsation.&#xA;. . . &amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Philosophical Transactions of the Royal Society, Febr. 9, 1898,&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the word &amp;quot;twist&amp;quot; in reference of the electron, is indeed a reference to Kelvin &amp;quot;Vortex Theory&amp;quot;, and not to the electron &lt;strong&gt;spin&lt;/strong&gt;, that is a notion that did not appear before 1925, decades later.&lt;/p&gt;&#xA;" OwnerUserId="15967" LastEditorUserId="15967" LastEditDate="2022-01-07T10:49:21.310" LastActivityDate="2022-01-07T10:49:21.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14031" PostTypeId="1" CreationDate="2022-01-07T14:44:59.100" Score="2" ViewCount="315" Body="&lt;p&gt;The famous poem, &amp;quot;Jabberwocky&amp;quot; by Lewis Carroll contains a plethora of weird words.&lt;/p&gt;&#xA;&lt;p&gt;However, Carroll is known as a celebrated mathematician and logician.&lt;/p&gt;&#xA;&lt;p&gt;Does &amp;quot;Jabberwocky&amp;quot; have any mathematics in it?&lt;/p&gt;&#xA;" OwnerUserId="15014" LastActivityDate="2022-01-08T01:37:37.033" Title="Mathematics involved in &quot;Jabberwocky&quot;" Tags="|logic|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14032" PostTypeId="2" ParentId="3466" CreationDate="2022-01-07T15:04:35.647" Score="1" Body="&lt;p&gt;''When asked to comment on this denunciation of relativity by so many scientists, Einstein replied that to defeat relativity one did not need the word of 100 scientists, just one fact.''&lt;/p&gt;&#xA;&lt;p&gt;Encyclopedia Britannica&#xA;Link&#xA;&lt;a href=&quot;https://www.britannica.com/biography/Albert-Einstein/Nazi-backlash-and-coming-to-America&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.britannica.com/biography/Albert-Einstein/Nazi-backlash-and-coming-to-America&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15971" LastActivityDate="2022-01-07T15:04:35.647" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14033" PostTypeId="2" ParentId="3466" CreationDate="2022-01-07T19:07:57.797" Score="5" Body="&lt;p&gt;To return to this old question with at least a partial answer: there really was a book called &amp;quot;Hundert Autoren gegen Einstein&amp;quot;; you can read it here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/HundertAutorenGegenEinstein&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/HundertAutorenGegenEinstein&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It was published in 1931, two years before the Nazis came to power, and several of the authors, including the one named first on the title page, were in fact Jews. I cannot find a contemporary source for Einstein's reply, but it is surely an example of &amp;quot;se non è vero, è ben trovato&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="364" LastActivityDate="2022-01-07T19:07:57.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14035" PostTypeId="2" ParentId="14031" CreationDate="2022-01-08T01:37:37.033" Score="1" Body="&lt;p&gt;Another source is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Martin Gardner, &lt;em&gt;The Annotated Alice&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;with extensive commentary on both Alice books.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/T0UTD.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/T0UTD.jpg&quot; alt=&quot;image&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-01-08T01:37:37.033" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14037" PostTypeId="1" CreationDate="2022-01-08T02:46:17.453" Score="4" ViewCount="263" Body="&lt;p&gt;In &lt;em&gt;Error-Correcting Codes: A Mathematical Introduction&lt;/em&gt; (Chapman &amp;amp; Hall, 1998), John Baylis wrote (p.109)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Moving on to 2-error correcting linear codes, the condition for perfection of linear codes of dimension &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$M = 2^k = \frac{2^n}{1 + n + \binom{n}{2}} = \frac{2^{n+1}}{2 + n + n^2}$$&lt;/span&gt;&#xA;so &lt;span class=&quot;math-container&quot;&gt;$2+n+n^2$&lt;/span&gt; must be a power of 2.  It was shown in 1930 that &lt;span class=&quot;math-container&quot;&gt;$n = 1, 2, 5$&lt;/span&gt; and 90 are the only positive integers for which this is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;without any citation.  Any ideas whose 1930 number theory result he had in mind?  Of course, a reference would be wonderful.&lt;/p&gt;&#xA;&lt;p&gt;This came up in a MathOverflow &lt;a href=&quot;https://mathoverflow.net/questions/412940/when-do-binomial-coefficients-sum-to-a-power-of-2/413024#413024&quot;&gt;question&lt;/a&gt; about when &lt;span class=&quot;math-container&quot;&gt;$\sum_{k=0}^n \binom{N}{n}$&lt;/span&gt; is a power of 2.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit: As markvs explains in his answer (and as discussed in my answer to the MO question), this is equivalent to what is now known as the Ramanujan--Nagell equation &lt;span class=&quot;math-container&quot;&gt;$2^n + 7 = x^2$&lt;/span&gt;.  This only deepens the question of what Baylis was referring to:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there a 1930 result that addresses &lt;span class=&quot;math-container&quot;&gt;$2^n + 7 = x^2$&lt;/span&gt;, predating Nagell's 1948 solution (and Ljunggren's 1943 posing of the problem, both presumably unaware of Ramanujan's 1913 question)?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Was there a 1930 result about when &lt;span class=&quot;math-container&quot;&gt;$n^2 + n + 2$&lt;/span&gt; is a power of 2 with no connection made to Ramanujan's question?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3429" LastEditorUserId="55" LastEditDate="2022-01-10T08:05:32.883" LastActivityDate="2022-01-10T08:05:32.883" Title="Whose 1930 number theory result is used in characterizing perfect 2-error correcting linear codes?" Tags="|mathematics|reference-request|number-theory|20th-century|ramanujan|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14038" PostTypeId="2" ParentId="14037" CreationDate="2022-01-08T04:40:19.557" Score="4" Body="&lt;p&gt;See &lt;a href=&quot;https://oeis.org/search?q=1%2C2%2C5%2C90&amp;amp;language=english&amp;amp;go=Search&quot; rel=&quot;nofollow noreferrer&quot;&gt;this oeis entry&lt;/a&gt;. This can be easily reduced to &lt;a href=&quot;https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Nagell_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramanujan-Nagell&lt;/a&gt; problem. It was conjectured by Ramanujan in 1913 and resolved by Nagell in 1948 (where the year 1930 come from is not clear).&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2022-01-08T04:40:19.557" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14039" PostTypeId="2" ParentId="13443" CreationDate="2022-01-08T11:21:59.840" Score="2" Body="&lt;p&gt;Now that Professor Aziz is no more, I have written an obituary, a delayed one.&lt;/p&gt;&#xA;&lt;p&gt;This obituary has appeared in several dailies.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://kashmirreader.com/2021/12/02/professor-of-mathematics-abdul-aziz-a-delayed-eulogy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kashmir Reader&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.risingkashmir.com/-Professor-Abdul-Aziz-of-Mathematics--A-delayed-eulogy-96592&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rising Kashmir&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15014" LastActivityDate="2022-01-08T11:21:59.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14040" PostTypeId="1" AcceptedAnswerId="14041" CreationDate="2022-01-08T18:06:28.620" Score="4" ViewCount="186" Body="&lt;p&gt;In Walter Isaacson's Biography of Einstein, it is stated that the young Einstein was developing his own ideas in Statistical Physics and Thermodynamics and he had found some error in Drude's model for electrical conduction. He wrote to Drude about this but Drude rejected it. He also says that the real target behind Einstein's objection was none other than Boltzmann himself.&lt;/p&gt;&#xA;&lt;p&gt;I tried to look for this incident but it being his early days, I could not find anything in English, some material was in German which I couldn't understand.&lt;/p&gt;&#xA;&lt;p&gt;Can someone explain what exactly was Einstein's objection and did it come to light later or is it already related to many known fallacies of Drude's model?&lt;/p&gt;&#xA;" OwnerUserId="9585" LastEditorUserId="9585" LastEditDate="2022-01-08T19:22:20.957" LastActivityDate="2022-01-08T20:18:51.917" Title="Einstein's Objection to Drude Model" Tags="|quantum-mechanics|classical-mechanics|einstein|statistical-mechanics|boltzmann|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14041" PostTypeId="2" ParentId="14040" CreationDate="2022-01-08T19:52:50.547" Score="2" Body="&lt;p&gt;In the paper &lt;a href=&quot;https://sci-hub.se/https://doi.org/10.1007/BF00518232&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Einstein's controversy with Drude and the origin of statistical mechanics: A new glimpse from the &amp;quot;Love Letter&amp;quot;&lt;/em&gt;&lt;/a&gt; by J. Renn, (1997), we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The nature of Einstein's objections to Drude's theory [...] has remained just as unknown to us as the character of Drude's response to a letter we know Einstein had written to him around early June 1901.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Renn then notes how some new and relevant information has come to light recently. In particular, a letter from &lt;a href=&quot;https://en.wikipedia.org/wiki/Mileva_Mari%C4%87&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mileva Maric&lt;/a&gt; to Einstein concerning Drude's response to Einstein's objections. Apparently Einstein had forwarded Drude's response to Mileva, so Mileva's return letter to Einstein offers a credible source. From this and other available information, Renn argues that Einstein presented two &amp;quot;factual objections&amp;quot; to Drude's theory.&lt;/p&gt;&#xA;&lt;p&gt;Renn proposes that Einstein's first objection was to a gap in Boltzmann's gas theory, which Drude had assumed in his own theory and explains why Drude felt the need to defend Boltzmann in (what we know of) his response to Einstein's objections.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Einstein pointed to the achievements of the kinetic theory in the field of gas theory but also to its failure to derive, from mechanics and probability theory, theorems on thermal equilibrium and the second law of thermodynamics for more general systems. [...] From what we know about Drude's response it follows that at least one of Einstein's objections must have been directed at the statistical assumptions introduced by Drude and their justification by a kinetic theory of matter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Drude appears to have disputed the very existence of the gap which Einstein intended to fill.&lt;/p&gt;&#xA;&lt;p&gt;According to Renn, Einstein's second objection was related to a problem already noted by others concerning the compatibility between kinetic theory of heat and what was known about the behaviour of matter from other contexts.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given the lack of direct contemporary evidence concerning Einstein's second objection to Drude, one can only rely on later references to the theory of electrons in metals and its problems in order to identify traces of this objection.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The (PDF) pages 20-22/40 of Renns paper give a technical description of these issues before Renn concludes of the second objection:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is hence conceivable that these problems also constitute his second &amp;quot;factual&amp;quot; objection to Drude in 1901.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2022-01-08T20:18:51.917" LastActivityDate="2022-01-08T20:18:51.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14042" PostTypeId="1" CreationDate="2022-01-08T22:21:13.817" Score="1" ViewCount="191" Body="&lt;p&gt;The ratio of four over the square root of the Golden Mean is very close to &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, if the height of a pyramid of square basis of size 2L is L times the square root of the Golden Mean, the perimeter 8L of the basis is almost equal to the perimeter of the circle the radius of which is the height.&lt;/p&gt;&#xA;&lt;p&gt;But it also means (using Pythagoras and the definition of the Golden Mean) that the triangle that forms the &lt;strong&gt;face&lt;/strong&gt; of the pyramid has height L times the Golden Mean, which is, aesthetically, a desirable ratio.&lt;/p&gt;&#xA;&lt;p&gt;Now it is a fact that this is very close to the proportions of the Great Pyramid of Gizeh.&lt;/p&gt;&#xA;&lt;p&gt;In his book &lt;em&gt;Les nombres et leurs mystères&lt;/em&gt;, André Warusfel claims that the Egyptians did not care about &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, their only interest was in the Golden ratio. But is that true?&lt;/p&gt;&#xA;&lt;p&gt;Are there any archeological arguments to decide between the four options :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Warusfel's one : The Egyptians cared &lt;strong&gt;only&lt;/strong&gt; about the Golden Mean, not &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Egyptians wanted the perimeter of the pyramid to coincide with the  perimeter of the circle with its height for radius and did not care about the proportions of the faces&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They wanted &lt;strong&gt;both&lt;/strong&gt; things, because they were convinced  that&#xA;&lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; was exactly &lt;span class=&quot;math-container&quot;&gt;$4 \sqrt{\frac {\sqrt 5-1} 2 }$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;They did not care about either of these and just took for the ratio of the height to the side of the basis the simple ratio 7/11, which is also very close&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15967" LastEditorUserId="15967" LastEditDate="2022-01-08T22:52:43.493" LastActivityDate="2022-01-08T22:52:43.493" Title="Is the height of Great Pyramid based on the Golden ratio or on Pi?" Tags="|mathematics|ancient-egypt|" AnswerCount="0" CommentCount="10" ClosedDate="2022-01-12T23:28:42.103" ContentLicense="CC BY-SA 4.0" />
  <row Id="14044" PostTypeId="1" CreationDate="2022-01-09T15:34:24.047" Score="1" ViewCount="65" Body="&lt;p&gt;I am new to Aristotle study, In Book 1 of Generation of Aristotle give his view on plant generation,&lt;/p&gt;&#xA;&lt;p&gt;In Book 1.1 Aristotle writes &amp;quot; But all those creatures which do not move, as the Testacea and animals that live by clinging to something else, inasmuch as their nature resembles that of plants, have no sex any more than plants have, but as applied to them the word is only used in virtue of similarity and analogy. For there is a slight distinction of this sort since even in plants we find in the same kind some trees which bear fruit and others which, while bearing none themselves, yet contribute to the ripening of the fruits of those which do, as in the case of the fig-tree and caprifig ..... Plants, however, must be investigated separately. &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;And At the end of Book 1 part 23 &amp;quot; In all animals which can move about, the sexes are separated, one individual being male and one female, though both are the same in species, as with man and horse. But in plants these powers are mingled, female not being separated from a male. Wherefore they generate out of themselves, and do not emit semen but produce an embryo, what is called the seed... Like plants, they have no sexes &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Question 1:What is Aristotle Is saying about Plant generation?&lt;/p&gt;&#xA;&lt;p&gt;Question 2: In the last line of Book 1.1 Aristotle say Testacea and animals like it don't have sex but call them male and female just for similarity and analogy. Is the same analogy and similarity is also applied to plant as Aristotle gives their example?&lt;/p&gt;&#xA;" OwnerUserId="15534" LastActivityDate="2022-01-09T15:34:24.047" Title="What is Aristotle's view on Plant generation?" Tags="|ancient-greece|biology|aristotle|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14045" PostTypeId="1" AcceptedAnswerId="14049" CreationDate="2022-01-10T00:45:18.260" Score="10" ViewCount="368" Body="&lt;p&gt;Felix Hausdorff, who was so well known to American mathematicians through his topology book of 1914, Grundzüge der Mengenlehre, committed suicide together with his wife when facing the threat of deportation to Nazi Concentration camps in 1942.&lt;/p&gt;&#xA;&lt;p&gt;Based on Wikipedia，Hausdorff hated Nazi as early as early 1930s and asked the mathematician Richard Courant in 1939 for a research fellowship to be able to emigrate into the USA. Does anyone know how Courant replied to Hausdorff and why Hausdorff wasn't able to move to United State? Was it because Nazi deterred him or US didn't give him visa?&lt;/p&gt;&#xA;" OwnerUserId="15980" LastEditorUserId="15980" LastEditDate="2022-01-12T15:46:06.487" LastActivityDate="2022-01-23T20:15:23.503" Title="Why was Hausdorff not able to move to United States from Nazi Germany?" Tags="|mathematicians|nazi-germany|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14046" PostTypeId="1" CreationDate="2022-01-10T11:55:26.667" Score="2" ViewCount="70" Body="&lt;p&gt;Wiki (&lt;a href=&quot;https://en.wikipedia.org/wiki/Electromagnetic_mass&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Electromagnetic_mass&lt;/a&gt;) says that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thomson (1893) noticed that electromagnetic momentum and energy of charged bodies, and therefore their masses, depend on the speed of the bodies as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;electrostatic energy behaves as having some sort of momentum and &amp;quot;apparent&amp;quot; electromagnetic mass, which can increase the ordinary mechanical mass of the bodies, or in more modern terms, the increase should arise from their electromagnetic self-energy. This idea was worked out in more detail by Oliver Heaviside (1889),[3] Thomson (1893),[4] George Frederick Charles Searle (1897),[5] Max Abraham (1902),[6] Hendrik Lorentz (1892, 1904),&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;ans most of all that:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;This was discussed in connection with the proposal of the electrical origin of matter, so **Wilhelm Wien (1900)**,[9] and Max Abraham (1902),[6] **came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass.**&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;Now , I have a few related questions regarding the issue:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;who first measured the increase of mass with speed? do you have any details on the experiment?&lt;/li&gt;&#xA;&lt;li&gt;why only Einstein is credited with the formula, when since 1900 the equivalence had been found( proved?)&lt;/li&gt;&#xA;&lt;li&gt;is it possible that in 1905 Einstein referred to rest mass i.e. matter could be transformed into em energy? it was only in 1930 that a pair was converted into light, did anyone nefore 1930 anticipate that?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2022-01-10T11:55:26.667" Title="Who first measured the increase of mass with speed?" Tags="|physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14047" PostTypeId="1" CreationDate="2022-01-10T19:19:02.627" Score="8" ViewCount="2346" Body="&lt;p&gt;Columbus set off on his westward journey to Asia, believing the Earth was much smaller than it is.&lt;/p&gt;&#xA;&lt;p&gt;There was some apparent disagreement about the size, and Columbus was staking his life on it. Why didn't he make his own estimate, for example as follows:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Find a mountain near the coast, such as this one:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tEVHF.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tEVHF.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Calculate its height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, for example, by climbing it and measuring the angular dimensions and position of something of known size&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sail away from the mountain until half of it disappears&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Measure the angular height &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; of the remaining half.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;The Earth's diameter will be &lt;span class=&quot;math-container&quot;&gt;$$h \over {2 \alpha ^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Relevant:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/FOniN.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/FOniN.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13345" LastEditorUserId="13345" LastEditDate="2022-01-14T01:54:17.573" LastActivityDate="2023-05-04T12:18:26.313" Title="Why did Columbus think the Earth was much smaller than it is?" Tags="|discoveries|geometry|earth-sciences|geography|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14048" PostTypeId="2" ParentId="14047" CreationDate="2022-01-10T21:25:00.457" Score="4" Body="&lt;p&gt;I think that observing ships disappear over the horizon would be an imprecise method of calculating the size of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;[added 01-13-21 I found a quote where Aristotle (384-322 BC) mentioned that the circumference of the Earth was estimated at about 400,000 stades, and thus not large compared to stars.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://historum.com/threads/loading-of-animals-cannon.191924/page-2#post-3545934&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://historum.com/threads/loading-of-animals-cannon.191924/page-2#post-3545934&lt;/a&gt;&#xA;&lt;a href=&quot;https://bmcr.brynmawr.edu/1996/1996.10.02/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://bmcr.brynmawr.edu/1996/1996.10.02/&lt;/a&gt;  ]&lt;/p&gt;&#xA;&lt;p&gt;Eratosthenes of Cyrene (c.276 BC-195/94 BC) was the first person to measure the size of the Earth.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The measure of Earth's circumference is the most famous among the results obtained by Eratosthenes,&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus%27s_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt; who estimated that the meridian has a length of 252,000 stadia, with an error on the real value between -2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres).&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Posidonius&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt; Eratosthenes described his technique in a book entitled On the measure of the Earth, which has not been preserved.&lt;/p&gt;&#xA;&lt;p&gt;Eratosthenes' method to calculate Earth's circumference has been lost; what has been preserved is the simplified version described by Cleomedes to popularise the discovery.&lt;a href=&quot;https://en.wikipedia.org/wiki/Continent&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt; Cleomedes invites his reader to consider two Egyptian cities, Alexandria and Syene, modern Assuan:&lt;/p&gt;&#xA;&lt;p&gt;Cleomedes assumes that the distance between Syene and Alexandria was 5,000 stadia (a figure that was checked yearly by professional bematists, mensores regii).&lt;a href=&quot;https://en.wikipedia.org/wiki/Ice_age&quot; rel=&quot;nofollow noreferrer&quot;&gt;6&lt;/a&gt;&#xA;He assumes the simplified (but false) hypothesis that Syene was precisely on the Tropic of Cancer, saying that at local noon on the summer solstice the Sun was directly overhead.&#xA;He assumes the simplified (but false) hypothesis that Syene and Alexandria are on the same meridian.&lt;/p&gt;&#xA;&lt;p&gt;Under the previous assumptions, writes Cleomedes, you can measure the Sun's angle of elevation at noon of the summer solstice in Alexandria, by using a vertical rod (a gnomon) of known length and measuring the length of its shadow on the ground; it is then possible to calculate the angle of the Sun's rays, which he claims to be about 7°, or 1/50th the circumference of a circle. Taking the Earth as spherical, the Earth's circumference would be fifty times the distance between Alexandria and Syene, that is 250,000 stadia. Since 1 Egyptian stadion is equal to about 157.7 metres,&lt;a href=&quot;https://en.wikipedia.org/wiki/Ice_age#Major_ice_ages&quot; rel=&quot;nofollow noreferrer&quot;&gt;7&lt;/a&gt; the result is roughly 39,425 km, which is 1.5% less than the real circumference, 40,008 km.&lt;/p&gt;&#xA;&lt;p&gt;Eratosthenes' method was actually more complicated, as stated by the same Cleomedes, whose purpose was to present a simplified version of the one described in Eratosthenes' book. The method was based on several surveying trips conducted by professional bematists, whose job was to precisely measure the extent of the territory of Egypt for agricultural and taxation-related purposes.&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Posidonius&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Eratosthenes&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Earth%27s_circumference#Eratosthenes&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Posidonius calculated the Earth's circumference by reference to the position of the star Canopus. As explained by Cleomedes, Posidonius observed Canopus on but never above the horizon at Rhodes, while at Alexandria he saw it ascend as far as 7+1⁄2 degrees above the horizon (the meridian arc between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000 stadia due north of Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth.[10] It is generally thought that the stadion used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to 24,000 mi (39,000 km), not much short of the actual circumference of 24,901 mi (40,074 km).[10] Strabo noted that the distance between Rhodes and Alexandria is 3,750 stadia, and reported Posidonius's estimate of the Earth's circumference to be 180,000 stadia or 18,000 mi (29,000 km).[11] Pliny the Elder mentions Posidonius among his sources and without naming him reported his method for estimating the Earth's circumference. He noted, however, that Hipparchus had added some 26,000 stadia to Eratosthenes's estimate. The smaller value offered by Strabo and the different lengths of Greek and Roman stadia have created a persistent confusion around Posidonius's result. Ptolemy used Posidonius's lower value of 180,000 stades (about 33% too low) for the earth's circumference in his Geography. This was the number used by Christopher Columbus in order to underestimate the distance to India as 70,000 stades.[12]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Posidonius&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Earth%27s_circumference#Posidonius&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Around AD 525, the Indian mathematician and astronomer, Aryabhata wrote Aryabhatiya, in which he calculated the diameter of earth to be of 1,050 yojanas. The length of the yojana intended by Aryabhata is in dispute. One careful reading gives an equivalent of 14,200 kilometers, too large by 11%.[13] Another gives 15,360 km, too large by 20%.[14] Yet another gives 13,440 km, too large by 5%.[15]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Aryabhata&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Earth%27s_circumference#Aryabhata&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Around AD 830, Caliph Al-Ma'mun commissioned a group of Muslim astronomers led by Al-Khwarizmi to measure the distance from Tadmur (Palmyra) to Raqqa, in modern Syria. They calculated the Earth's circumference to be within 15% of the modern value, and possibly much closer. How accurate it actually was is not known because of uncertainty in the conversion between the medieval Arabic units and modern units, but in any case, technical limitations of the methods and tools would not permit an accuracy better than about 5%.[16]&lt;/p&gt;&#xA;&lt;p&gt;A more convenient way to estimate was provided in Al-Biruni's Codex Masudicus (1037). In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations, based on the angle between a plain and mountain top, which made it possible for it to be measured by a single person from a single location.[16] From the top of the mountain, he sighted the dip angle which, along with the mountain's height (which he calculated beforehand), he applied to the law of sines formula. This was the earliest known use of dip angle and the earliest practical use of the law of sines.[17] However, the method could not provide more accurate results than previous methods, due to technical limitations, and so al-Biruni accepted the value calculated the previous century by the al-Ma'mun expedition.[16]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus%27s_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus's_error&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1,700 years after Eratosthenes's death, Christopher Columbus studied what Eratosthenes had written about the size of the Earth. Nevertheless, based on a map by Toscanelli, he chose to believe that the Earth's circumference was 25% smaller. If, instead, Columbus had accepted Eratosthenes's larger value, he would have known that the place where he made landfall was not Asia, but rather a New World.[18]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus%27s_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus's_error&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I note that Columbus also used estimations of the size of Asia which made it extend much farther east than it really does.&lt;/p&gt;&#xA;&lt;p&gt;And I point out that in a sense Columbus was not actually wrong about the distance from Europe west across the Atlantic to Asia.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A continent is any of several large landmasses. Generally identified by convention rather than any strict criteria, up to seven geographical regions are commonly regarded as continents. Ordered from largest in area to smallest, these seven regions are: Asia, Africa, North America, South America, Antarctica, Europe, and Australia.&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus%27s_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt; Variations with fewer continents may merge some of these, for example some systems include Afro-Eurasia, America or Eurasia as single continents. Zealandia, a largely submerged mass of continental crust, has also been described as a continent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Continent&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Continent&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Modern geologists know about plate tectonics and continental crust, and may count more or fewer continents than ordinary people do.&lt;/p&gt;&#xA;&lt;p&gt;But most modern people still think and feel that if you can travel by land from place A to place B, those two places are part of one continent.&lt;/p&gt;&#xA;&lt;p&gt;And certainly nobody proved that the Americas were not part of Asia during the lifetime of Columbus, or for centuries afterwards. The Pacific Ocean east of Asia could have simply been a gigantic bay extending north into Asia and the Americas could have been vasts southward peninsulas from eastern Asia.&lt;/p&gt;&#xA;&lt;p&gt;The Bering Strait between Siberia and Alaska was not discovered by Euoprean explorers until 1728, though Euoprean geographers speculated about it earlier.&lt;/p&gt;&#xA;&lt;p&gt;Even after 1728, it might have seemed possible for a land bridge to extend across the Arctic ocean from Siberia to Canada.&lt;/p&gt;&#xA;&lt;p&gt;I don't knew when exploratins in Siberia proved the it had a coast line all the way from Norway to the Bering Strait.  Nordenskiold in the &lt;em&gt;Vega&lt;/em&gt; made the first passage along the Northeast Passage in 1878-79, but the coast must have been surveyed long before then.&lt;/p&gt;&#xA;&lt;p&gt;The first complete voyage along the Northwest Passage north of Canada was made by Amundsen in &lt;em&gt;Gjoa&lt;/em&gt; in 1903-06.  But it had been proven by 1854 that there was water all the way during the searches for the lost Franklin Expediiton.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, there was no reason for anyone to claim as a fact that the Americas were not extensions of Asia until long after the time of Columbus.  Many people wanted to believe that Asia and the Americas were separate continents, because that would make it possible to find a Northwest Passage or a Northeast Passage from Europe to east Asia and open up alternate trading routes.  So the wish for such passages was the father of the belief that the Americas were separated by water from Asia. And that belief was not proved correct for centuries afterwards.&lt;/p&gt;&#xA;&lt;p&gt;But has the belief that the Americas are separate from Asia actually been proved?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An ice age is a long period of reduction in the temperature of Earth's surface and atmosphere, resulting in the presence or expansion of continental and polar ice sheets and alpine glaciers. Earth's climate alternates between ice ages and greenhouse periods, during which there are no glaciers on the planet. Earth is currently in the Quaternary glaciation.&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Columbus%27s_error&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt; Individual pulses of cold climate within an ice age are termed glacial periods (or, alternatively, glacials, glaciations, glacial stages, stadials, stades, or colloquially, ice ages), and intermittent warm periods within an ice age are called interglacials or interstadials.&lt;a href=&quot;https://en.wikipedia.org/wiki/Earth%27s_circumference#Posidonius&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ice_age&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ice_age&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Quaternary Glaciation / Quaternary Ice Age started about 2.58 million years ago at the beginning of the Quaternary Period when the spread of ice sheets in the Northern Hemisphere began. Since then, the world has seen cycles of glaciation with ice sheets advancing and retreating on 40,000- and 100,000-year time scales called glacial periods, glacials or glacial advances, and interglacial periods, interglacials or glacial retreats. Earth is currently in an interglacial, and the last glacial period ended about 10,000 years ago. All that remains of the continental ice sheets are the Greenland and Antarctic ice sheets and smaller glaciers such as on Baffin Island.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ice_age#Major_ice_ages&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ice_age#Major_ice_ages&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So for the last two and a half million years Earth's climate has varied between cold glacial periods and warm interglacial periods.  And as far as I can tell  the glacial periods tended to last longer than the interglacial periods.&lt;/p&gt;&#xA;&lt;p&gt;And during the glacial periods the vast amounts of water tied up in ice sheets causes the sea level to drop a considerable distance, thus exposing the sea floor between Alaska and Siberia as dry or ice sheet covered land.&lt;/p&gt;&#xA;&lt;p&gt;So for the last two and a half million years a land connection between Siberia and Alaska has been at least a common as a strait between them. And so North and South America could be considered to be part of the same continent as Africa, Europe, and Asia, making Antarctica and Australia the only other real separate continents.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2022-06-05T18:16:18.680" LastActivityDate="2022-06-05T18:16:18.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14049" PostTypeId="2" ParentId="14045" CreationDate="2022-01-10T23:00:19.220" Score="16" Body="&lt;p&gt;To convert my comment to an answer:&lt;/p&gt;&#xA;&lt;p&gt;According to the the book&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Jewish Mathematicians in German-Speaking Academic Culture,&amp;quot; Springer Verlag, 2012,&lt;/p&gt;&#xA;&lt;p&gt;specifically &lt;a href=&quot;http://gj-math.uni-frankfurt.de/index.php?id=27&quot; rel=&quot;nofollow noreferrer&quot;&gt;11. Dismissals and Exile&lt;/a&gt;, Courant wrote recommendation letters for Hausdorff to several US universities. A copy of the correspondence is below. To quote from the book:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Due to his age Hausdorff, one of the most renowned mathematicians in Germany, failed in his attempts to emigrate after the “Reichskristallnacht” of 1938. Courant and others tried to help, but without success. No university in the USA was willing to offer him a position or fellowship. He remained in Nazi Germany, losing all hope of leaving the country.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hhPy1.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hhPy1.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Fn7XC.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Fn7XC.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the period leading to the 2nd World War, the number of US visas issues to people from German-speaking  countries was &lt;strong&gt;very&lt;/strong&gt; limited. Below is the quote from the webpage of the US Holocaust Museum &lt;a href=&quot;https://exhibitions.ushmm.org/americans-and-the-holocaust/what-did-refugees-need-to-obtain-a-us-visa-in-the-1930s&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, specifically dealing with the question &amp;quot;What did Refugees Need to Obtain a US Visa in the 1930s?&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As the Nazi regime’s attacks intensified in the late 1930s, hundreds of thousands of Jews in Germany tried to immigrate to the United States. To enter the United States, each person needed an immigration visa stamped into his or her passport.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was difficult to get the necessary papers to leave Germany, and US immigration visas were difficult to obtain. The process could take years.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By 1938, you could expect to wait for at least two years for a visa.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Immigrating to the United States was a complicated, expensive process even under the best of circumstances. Hundreds of thousands of people, most of them Jewish refugees, applied at American consulates in Europe but were unable to immigrate. Many of them became trapped in Nazi-occupied territory and were murdered in the Holocaust.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For instance, in 1938:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After Germany annexed Austria in March 1938, President Franklin Roosevelt combined the German and Austrian quotas, making 27,370 visas available each year for people born in these countries, who were now all considered &amp;quot;German.&amp;quot; As antisemitic persecution increased and Germany began to expand its territorial holdings in Europe, the waiting list grew.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1938, 19,552 Germans received visas, 7,919 visas went unissued. At the same time, 139,163 were on the waiting list.&lt;/p&gt;&#xA;&lt;p&gt;In 1941:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With Europe at war, the State Department implemented new restrictions designed both to protect the nation’s security and further limit immigration. In July 1941, Nazi Germany ordered US consulates in Nazi-occupied territory to close, trapping potential immigrants. Only German refugees who had already escaped Nazi territory could obtain US immigration visas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Immigration became almost impossible, and the State Department canceled the waiting list.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can find more information by visiting that web-page.&lt;/p&gt;&#xA;" OwnerUserId="6107" LastEditorUserId="6107" LastEditDate="2022-01-23T20:15:23.503" LastActivityDate="2022-01-23T20:15:23.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14050" PostTypeId="2" ParentId="14047" CreationDate="2022-01-11T02:08:00.677" Score="10" Body="&lt;p&gt;Answering the earlier version of the question first (on Columbus mistakes).&lt;/p&gt;&#xA;&lt;p&gt;There were two main sources of mistakes:&#xA;exaggerating the size of Asia and underestimating the size of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;Columbus was relying on the work of his contemporary geographer Toscanelli, with whom he corresponded (some of this correspondence survives).&lt;/p&gt;&#xA;&lt;p&gt;Toscanelli estimated the distance (westward) from Lisbon to Beijin as about 5000 (nautical) miles. He also estimated the westward distance from&#xA;&amp;quot;Antilles&amp;quot; to Japan as 2000 miles. (Existence of &amp;quot;Antilles&amp;quot; was conjectured at that time, as some islands midway where a traveler could make a stop.)&lt;/p&gt;&#xA;&lt;p&gt;These huge mistakes are mainly due to the exaggerated estimate&#xA;of the size of Asia, based on Marco Polo. Most contemporary scientists did not believe Marco Polo writings, but Toscanelli did. In fact, Marco Polo contradicted Ptolemy, the most comprehensive and reliable author of antiquity, but this was already the time when Ptolemy was beginning to lose his authority:-)&lt;/p&gt;&#xA;&lt;p&gt;The second, smaller source of error was estimate of the size of the Earth.&#xA;The difficulty with ancient sources was (and still is) that the exact value&#xA;of the distance units used by ancient authors was not known. For the length of the (nautical) mile Columbus assumed the value of&#xA;Alfragan, who stated it as 56 2/3 of (Arabian) miles, but Columbus assumed that Alfragan used Italian mile which gives an underestimate&#xA;of 75 % of the true value.&lt;/p&gt;&#xA;&lt;p&gt;Source: A comprehensive account of Columbus calculations and&#xA;related controversies is given in the book:&lt;/p&gt;&#xA;&lt;p&gt;Samuel E. &lt;em&gt;Morrison, Admiral of the Ocean Sea. A life of Christopher Columbus&lt;/em&gt;, NW Univ. Press, 1983.&lt;/p&gt;&#xA;&lt;p&gt;Answer to the updated question. You describe an imaginary measurement which could not be performed at the time of Columbus or before. There are several reasons for this. 1. Atmospheric refraction at the horizon is about 30' but it is not constant (depends of the weather conditions), and the knowledge of refraction at that time was very rudimentary. It is discussed by Ptolemy, but he could not really measure it, and I am not aware of any reliable measurement until 17 century. 2. One needs a devise to measure small angles with sufficient accuracy. We are talking of angles of less than 1 degree. There is no evidence that such devises existed until 17 century. (If you read how Archimedes tried to measure the diameter of the Sun directly, you get the idea of the difficulties involved. There was certainly no progress in this since Archimedes to the time of Columbus). 3. &amp;quot;You sail until you see 1/2 of the mountain&amp;quot;. How do you measure what distance you sailed? Read the book of Morrison about how they measured sailing distances at the time of Columbus, and see how crude the methods were. 4. There is no evidence whatsoever that anyone climbed high mountains until 18 century, not speaking of measuring their heights. Measuring the height of a mountain is a notoriously difficult task, which could be reliably performed only in 19 century, using high precision instruments (and a lot of expenses).&#xA;If you don't believe me, just try to measure the height of any mountain with the instruments you can make yourself, with your hands:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-05-04T12:18:26.313" LastActivityDate="2023-05-04T12:18:26.313" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14051" PostTypeId="1" CreationDate="2022-01-11T12:28:30.627" Score="5" ViewCount="185" Body="&lt;p&gt;Have there been instances in physics where different scientists have interpreted the same data differently? If yes, can you please give me specific examples and explain why one interpretation was favored over the other?&lt;/p&gt;&#xA;" OwnerUserId="15988" LastEditorUserId="10106" LastEditDate="2022-01-12T02:22:06.793" LastActivityDate="2022-01-12T02:22:06.793" Title="Have there been instances in physics where different scientists have interpreted the same data differently?" Tags="|physics|philosophy-of-science|theoretical-physics|experimental-physics|" AnswerCount="1" CommentCount="3" ClosedDate="2022-01-12T15:32:20.690" ContentLicense="CC BY-SA 4.0" />
  <row Id="14052" PostTypeId="2" ParentId="7247" CreationDate="2022-01-11T18:10:16.447" Score="3" Body="&lt;p&gt;A first-person account that covers the major points of the question appears in the book, &amp;quot;Surely You're Joking, Mr. Feynman&amp;quot; by Richard Feynman (The specific story is titled &amp;quot;A Different Box of Tools&amp;quot;).&#xA;The real characters were a group of math and physics grad students at Princeton. The specific student who made the claim of triviality is unnamed.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I still remember a guy sitting on the couch [in the shared math/physics department lounge], thinking very hard, and another guy standing in front of him, saying, &amp;quot;And therefore such­-and­-such is true.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Why is that?&amp;quot; the guy on the couch asks.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;It's trivial! It's trivial!&amp;quot; the standing guy says, and he rapidly reels off a series of logical steps: &amp;quot;First you assume thus-­and-­so, then we have Kerchoff's this­-and-­that; then there's Waffenstoffer's Theorem, and we substitute this and construct that. Now you put the vector which goes around here and then thus-­and-­so...&amp;quot; The guy on the couch is struggling to understand all this stuff, which goes on at high speed for about fifteen minutes!&lt;/p&gt;&#xA;&lt;p&gt;Finally the standing guy comes out the other end, and the guy on the couch says, &amp;quot;Yeah, yeah. It's trivial.&amp;quot; We physicists were laughing, trying to figure them out. We decided that &amp;quot;trivial&amp;quot; means &amp;quot;proved.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9379" LastActivityDate="2022-01-11T18:10:16.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14053" PostTypeId="2" ParentId="14051" CreationDate="2022-01-11T21:33:13.397" Score="8" Body="&lt;p&gt;The comments correctly say that this happens all the time. For a recent example, you can see my answer to: &lt;a href=&quot;https://mattermodeling.stackexchange.com/a/6888/5&quot;&gt;Can a highly-cited published paper have this type of error?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I will explain here some details which were not described in that answer (since your question is a slightly different one). The paper &lt;a href=&quot;https://www.nature.com/articles/nature05678&quot; rel=&quot;noreferrer&quot;&gt;Engel &lt;em&gt;et al.&lt;/em&gt;, &lt;em&gt;&lt;strong&gt;Nature&lt;/strong&gt;&lt;/em&gt;. (2007) &lt;strong&gt;446&lt;/strong&gt;, 782-786 &amp;quot;Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems&amp;quot;&lt;/a&gt; which currently has &lt;a href=&quot;https://scholar.google.com/citations?view_op=view_citation&amp;amp;hl=en&amp;amp;user=jvQOKToAAAAJ&amp;amp;alert_preview_top_rm=2&amp;amp;citation_for_view=jvQOKToAAAAJ:u5HHmVD_uO8C&quot; rel=&quot;noreferrer&quot;&gt;3300 citations on Google Scholar&lt;/a&gt; was interpreted by many physicists to be evidence that quantum coherence is used by bacteria to help improve their efficiency at photosynthesis. The paper also says that some of the quantum mechanical behavior occurring in the FMO photosynthetic complex is &amp;quot;analogous to Grover's algorithm&amp;quot; in quantum computing. &lt;strong&gt;It was nearly universally interpreted&lt;/strong&gt; that some bacteria near the bottom of the ocean, only get a few photons per day, and need to make sure as many of them as possible, successfully have their energy transferred to the reaction centre where photosynethsis happens, and that quantum coherence was at least partly responsible for this &amp;quot;transfer efficiency&amp;quot; being close to 99% in the FMO photosynthetic complex. &lt;strong&gt;People interpreted this as the bacteria &lt;em&gt;evolving&lt;/em&gt; over billions of years to take advantage of quantum coherence.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My interpretation of the paper, published in &lt;a href=&quot;https://pubs.acs.org/doi/10.1021/ct501066k&quot; rel=&quot;noreferrer&quot;&gt;Wilkins &amp;amp; Dattani &lt;em&gt;&lt;strong&gt;JCTC&lt;/strong&gt;&lt;/em&gt; (2015) &lt;strong&gt;11&lt;/strong&gt;, 3411-3419 &amp;quot;Why Quantum Coherence Is Not Important in the Fenna–Matthews–Olsen Complex&amp;quot;&lt;/a&gt;,&#xA;is that quantum coherence in the FMO  photosynthetic complex does exist to some extent, but it is so short-lived that &lt;strong&gt;it has absolutely zero impact on the efficiency of the photosynthesis&lt;/strong&gt;. Specifically, the solid lines in our figure below depict the dynamics including the effects of quantum coherence, and the dashed lines depict the dynamics with an incoherent theory:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lSbBj.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lSbBj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The left panel shows that there's visible qualitative differences between the &lt;em&gt;&lt;strong&gt;approximate&lt;/strong&gt;&lt;/em&gt; incoherent dynamics and the much more accurate coherent dynamics (notice the oscillatory behavior which is characteristic of quantum coherent dynamics or damped Rabi oscillations), but only for the first 300fs or so. &lt;strong&gt;The right panel shows that both the incoherent and coherent dynamics equilibriate after about 5ps,&lt;/strong&gt; which is an order of magnitude faster than the fluorescence time (which is multiple nanoseconds). Therefore, while the coherent dynamics and incoherent dynamics are both qualitatively different for the first 300fs, and the 2007 Nature paper did show evidence of quantum coherence (oscillations), the coherent and incoherent dynamics both lead to the same amount of excitons landing on the chromophore closest to the reaction center, and therefore the narrative about bacteria evolving over billions of years to act as a &amp;quot;biological quantum computer&amp;quot; or to take advantage of quantum coherence, is almost guaranteed to be an incorrect interpretation of the 2007 experiment.&lt;/p&gt;&#xA;&lt;p&gt;Returning to the questions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Have there been instances in physics where different scientists have interpreted the same data differently?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above example is a specific example in which the same experimental data from 2007 was interpreted in two different ways. One in which quantum coherence &lt;em&gt;&lt;strong&gt;is important&lt;/strong&gt;&lt;/em&gt; for photosynthesis, and one in which it &lt;em&gt;&lt;strong&gt;is not important&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;If yes, can you please give me specific examples and explain why one interpretation was favored over the other?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The following are reasons why the first interpretation was originally favored:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The idea of quantum coherence (something required for quantum computers to work, but at the time impossible to maintain for long enough even in extremely expensive artificial vacuum settings) surviving in &lt;em&gt;&lt;strong&gt;living cells&lt;/strong&gt;&lt;/em&gt;  was astonishing: the type of astonishing that gets papers published in Nature, and popularizes buzz terms like &amp;quot;quantum biology&amp;quot;;&lt;/li&gt;&#xA;&lt;li&gt;Quantum computing researchers also hopped on the bandwaggon since they could now apply the techniques they'd been developing for several years/decades to study quantum computers (which didn't exist yet and wouldn't exist in the near-term future at the time)  on something that &lt;em&gt;&lt;strong&gt;did&lt;/strong&gt;&lt;/em&gt; exist, which was bacteria;&lt;/li&gt;&#xA;&lt;li&gt;This not only opened computer scientists and theoretical physicists to the opportunity to publish about biology (a much higher impact field in which high citation counts and grant funding are more abundant), but also got people thinking this had implications for solar cells and the hot topics of developing clean energy, because solar cells had stagnated at less than 50% efficiency (see my post titled &lt;a href=&quot;https://mattermodeling.stackexchange.com/q/329/5&quot;&gt;&amp;quot;What has caused this apparent stagnancy, in the development of more efficient solar cells?&amp;quot;&lt;/a&gt;) and the FMO had an efficiency of 99%.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The reason why the second interpretation became favored in the long run was because concrete data supported it (see the above figure).&lt;/p&gt;&#xA;" OwnerUserId="8052" LastActivityDate="2022-01-11T21:33:13.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14054" PostTypeId="1" CreationDate="2022-01-11T21:43:14.297" Score="1" ViewCount="97" Body="&lt;p&gt;Plainly, they knew what a circle and sphere were and also a square and cube; but did they discuss the idea that a sphere was analogous to a circle but in 3 dimensions or similarly the analogy between a square and a cube, again, that cube was a sort of 3 dimensional square?&lt;/p&gt;&#xA;&lt;p&gt;My guess is that while they understood plane vs solid, they did not talk about 2 vs 3 dimensions. I am always surprised what arose 2000 or more years ago, if not in Greece then in China or India or later among the Arabs -- in just my lifetime, I believe we have majorly revised our understanding of when certain concepts were discovered, sometimes pushing back the date hundreds of years as I guess literature from non-Western countries is investigated.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-01-11T22:00:25.970" Title="Did the ancients have the concept of dimension?" Tags="|geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14055" PostTypeId="5" CreationDate="2022-01-11T21:54:37.150" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-01-11T21:54:37.150" LastActivityDate="2022-01-11T21:54:37.150" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14056" PostTypeId="4" CreationDate="2022-01-11T21:54:37.150" Score="0" Body="Questions about or related to the history of social sciences." OwnerUserId="8052" LastEditorUserId="8052" LastEditDate="2022-01-12T03:02:43.403" LastActivityDate="2022-01-12T03:02:43.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14057" PostTypeId="2" ParentId="14054" CreationDate="2022-01-11T22:00:25.970" Score="1" Body="&lt;p&gt;The ancients did not have a general &amp;quot;concept of dimension&amp;quot;, whatever it can mean. But they had good intuitive understanding, which is demonstrated by Euclid's &amp;quot;Definitions&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;The extremities of a line are points&lt;/p&gt;&#xA;&lt;p&gt;The extremities of a surface are lines,&lt;/p&gt;&#xA;&lt;p&gt;etc. He did not add that &amp;quot;extremities of a solid are surfaces&amp;quot;, but probably he understood this too.&lt;/p&gt;&#xA;&lt;p&gt;In modern mathematics, we have several (not equivalent) concepts of dimension, and one of them (inductive definition of dimension in topology) actually gives a rigorous basis for these &amp;quot;definitions&amp;quot; of Euclid.&lt;/p&gt;&#xA;&lt;p&gt;And certainly the ancients could only think of dimension taking only values 1,2,3.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-01-11T22:00:25.970" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14059" PostTypeId="2" ParentId="5907" CreationDate="2022-01-12T18:39:51.697" Score="7" Body="&lt;p&gt;The German &lt;a href=&quot;https://dx.doi.org/10.31030/1073187&quot; rel=&quot;nofollow noreferrer&quot;&gt;DIN Norm 461&lt;/a&gt; from 1973 explicitly says, that units &lt;strong&gt;must not&lt;/strong&gt; be put in brackets and further relates to DIN 1313. The first version of DIN 461 is from &lt;a href=&quot;https://www.beuth.de/de/norm/din-461/7374195&quot; rel=&quot;nofollow noreferrer&quot;&gt;1923&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Einheit darf keinesfalls in Klammern gesetzt werden (siehe DIN1313)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to DIN, the &lt;strong&gt;square brackets&lt;/strong&gt; are used as an &lt;strong&gt;operator&lt;/strong&gt; to &lt;strong&gt;extract&lt;/strong&gt; the unit from a physical quantity and curly brackets are used to extract the number:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;m&lt;/em&gt; = 5 kg&lt;/p&gt;&#xA;&lt;p&gt;{&lt;em&gt;m&lt;/em&gt;} = 5&lt;/p&gt;&#xA;&lt;p&gt;[&lt;em&gt;m&lt;/em&gt;] = kg&lt;/p&gt;&#xA;&lt;p&gt;In that context [kg] does not make sense.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, the idea of &lt;strong&gt;not using&lt;/strong&gt; units in (square) brackets is quite old.&lt;/p&gt;&#xA;&lt;p&gt;I just had a look at some papers from the &lt;strong&gt;1980s&lt;/strong&gt;. &lt;strong&gt;Both notations&lt;/strong&gt; occurred  there. Therefore, the idea to use units inside brackets is quite old, too.&lt;/p&gt;&#xA;&lt;p&gt;Would be interesting to know which of the notations is older. Has the square bracket notation around units already been used before 1923?&lt;/p&gt;&#xA;" OwnerUserId="16002" LastEditorUserId="16002" LastEditDate="2022-10-05T21:16:39.090" LastActivityDate="2022-10-05T21:16:39.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14060" PostTypeId="1" CreationDate="2022-01-12T16:53:22.697" Score="2" ViewCount="120" Body="&lt;p&gt;My brother and I are researching our family tree, and in our extended ancestry, there is a person who is registered on their death certificate to have died June 1858, of diphtherite. I wondered if it was the same as diphtheria and did some searching online.&lt;/p&gt;&#xA;&lt;p&gt;Apart from loads of hits on diphtheria &lt;strong&gt;as a suggested alternative&lt;/strong&gt; when just searching &lt;code&gt;&amp;quot;diphtherite&amp;quot;&lt;/code&gt; (with the speech marks) in Google, I came across a piece called &lt;a href=&quot;https://books.google.co.uk/books?id=F2NYAAAAMAAJ&amp;amp;pg=PA85&amp;amp;lpg=PA85&amp;amp;dq=%22diptherite%22&amp;amp;source=bl&amp;amp;ots=EYanopdUkO&amp;amp;sig=ACfU3U2WdyrxBPkdaQnoemNKchuTx84lew&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjW443Ey6z1AhWNYsAKHbBYB3oQ6AF6BAgBEAE#v=onepage&amp;amp;q=%22diptherite%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;An Essay on Diphtheria&lt;/em&gt;&lt;/a&gt; by W. Davis MD of Paris, Illinios in a large 1861 book called &lt;em&gt;The Medical Examiner; a Semi-monthly Journal of Medical Sciences, Volume 2&lt;/em&gt; edited by Nathan Smith Davis MD and Frank W. Reilly MD and it has me confused.&lt;/p&gt;&#xA;&lt;p&gt;First of all, right at the beginning (Page 79), it says (emphasis mine)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Diphtheria or diphtherite&lt;/strong&gt; has occupied much space in the medical journals, and engaged much attention in the past year.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, initially I thought, yes it is the same. But, later the essay says (Page 85)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Our patients did not die of diphtheria, they were killed by diphtherite.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the way to cure &lt;em&gt;diphtheria&lt;/em&gt; is to cure &lt;em&gt;diphtherite&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Admittedly I was just skimming the essay for now using highlighting of &lt;code&gt;diphtherite&lt;/code&gt; provided by Google Books, but can an someone clear this up as I cannot seem to find anything that clearly defines the difference. Or am I missing something with the skimming through?&lt;/p&gt;&#xA;" OwnerUserId="16003" OwnerDisplayName="Chris Rogers" LastEditorUserId="16003" LastEditDate="2022-01-12T20:33:26.117" LastActivityDate="2023-02-22T16:55:22.163" Title="Diphtherite: What is it and what is the difference with Diphtheria?" Tags="|medicine|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14061" PostTypeId="1" CreationDate="2022-01-13T09:52:06.783" Score="10" ViewCount="1660" Body="&lt;p&gt;In 1962, a paper called “Multiplication of Many-Digital Numbers by Automatic Computers”, by Anatoly Karatsuba and Yuri Ofman, was published at the &lt;em&gt;Proceedings of the USSR Academy of Sciences&lt;/em&gt;. It was here that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Karatsuba_algorithm&quot; rel=&quot;noreferrer&quot;&gt;Karatsuba multiplication algorithm&lt;/a&gt; appeared for the first time. However, Karatsuba himself had nothing to do with the writing of the paper. It was actually written by Kolmogorov (probably with the collaboration of Ofman), and Karatsuba only became aware of that when he was given the reprints that were due to him.&lt;/p&gt;&#xA;&lt;p&gt;I wonder whether there is some other known example, in Mathematics or in Theoretical Physics, of a situation in which a paper is published on behalf of a person who could have published it themself, without that person being aware of that until the paper was published. So, I am not talking about, for instance, papers that were published after their author died, or something like that.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="12330" LastEditDate="2022-01-14T01:37:35.897" LastActivityDate="2022-01-26T14:13:44.910" Title="Articles published without their authors being aware" Tags="|mathematics|theoretical-physics|publishing|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14062" PostTypeId="2" ParentId="14061" CreationDate="2022-01-13T11:32:54.507" Score="6" Body="&lt;p&gt;Probably the most famous historical example is the Tartaglia-Cardano affair, where Cardano published the solution to a case of depressed cubic he learned from Tartaglia in &lt;em&gt;Ars Magna&lt;/em&gt; after swearing on the Gospels not to do it, see e.g. &lt;a href=&quot;http://brain.caltech.edu/ist4/lectures/Cardano-Tartaglia_Dispute.pdf&quot; rel=&quot;noreferrer&quot;&gt;Feldmann, The Cardano-Tartaglia Dispute&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/q/347/55&quot;&gt;Why is &amp;quot;Cardano's Formula&amp;quot; (wrongly) attributed to him?&lt;/a&gt; Tartaglia was saving the gem for his own book that he never quite got around to writing. Cardano's justification was that del Ferro already knew the solution before Tartaglia. However, he did not share it and Tartaglia discovered his independently, along with an additional case of depressed cubic.&lt;/p&gt;&#xA;&lt;p&gt;This said, Cardano credited del Ferro and Tartaglia, and did quite a bit more than both of them combined, providing a comprehensive analysis of the general cubic. However, he could not make it public without revealing the key Tartaglia provided, that he repeatedly tried and failed to find himself. The situation is somewhat reminiscent of what happened with Kolmogorov and Karatsuba, see &lt;a href=&quot;https://hsm.stackexchange.com/a/12131/55&quot;&gt;Was Kolmogorov enraged after learning about the Karatsuba multiplication algorithm?&lt;/a&gt; Having one's name attached, in some form, to a key result in the area where one spent much effort might seem fair, even if they failed to obtain the result themselves. Especially when non-publication by the original author holds up emergence of the big picture that frames their own contributions.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-01-13T11:32:54.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14063" PostTypeId="2" ParentId="14061" CreationDate="2022-01-13T17:01:03.523" Score="16" Body="&lt;p&gt;In theoretical physics, the celebrated 1948 &lt;a href=&quot;https://en.wikipedia.org/wiki/Alpher%E2%80%93Bethe%E2%80%93Gamow_paper&quot; rel=&quot;noreferrer&quot;&gt;Alpher–Bethe–Gamow paper, or αβγ paper&lt;/a&gt; on cosmic nucleosynthesis.&lt;/p&gt;&#xA;&lt;p&gt;Bethe's name was thrown in, unbeknownst to him, at first, as a practical joke, for which G Gamow was notorious, to rhyme with the Greek alphabet;  but it is not as though his friend Bethe, the  pioneer of nucleosynthesis, was alien to the field.&lt;/p&gt;&#xA;&lt;p&gt;The inveterate perpetrator, Gamow, described in his 1952 book &lt;em&gt;The Creation of the Universe&lt;/em&gt;, 1952 (Dover, ISBN-13: ‎978-0486438689),&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The results of these calculations were first announced in a letter to The Physical Review, April 1, 1948. This was signed Alpher, Bethe, and Gamow, and is often referred to as the 'alphabetical article'. It seemed unfair to the Greek alphabet to have the article signed by Alpher and Gamow only, and so the name of Dr. Hans A. Bethe (in absentia) was inserted in preparing the manuscript for print. Dr. Bethe, who received a copy of the manuscript, did not object, and, as a matter of fact, was quite helpful in subsequent discussions. There was, however, a rumor that later, when the alpha, beta, gamma theory went temporarily on the rocks, Dr. Bethe seriously considered changing his name to Zacharias. The close fit of the calculated curve and the observed abundances is shown in Fig. 15, which represents the results of later calculations carried out on the electronic computer of the National Bureau of Standards by Ralph Alpher and R. C. Herman (who stubbornly refuses to change his name to Delter).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2022-01-13T17:11:47.047" LastActivityDate="2022-01-13T17:11:47.047" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14064" PostTypeId="2" ParentId="14061" CreationDate="2022-01-13T19:57:20.870" Score="6" Body="&lt;p&gt;I have a paper with 5 co-authors which was written and submitted to a journal without my approval, and even without notifying   me about it. The editor handling the paper informed me about it but it was published without my approval. The good outcome was that as a result of this paper my Erdos number is 2 because one of the 5 co-authors was a co-author of Erdos.&lt;/p&gt;&#xA;&lt;p&gt;I have also another paper published by my co-author without my approval, but it was a &amp;quot;recreational math&amp;quot; paper.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2022-01-13T19:57:20.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14065" PostTypeId="2" ParentId="14061" CreationDate="2022-01-14T00:53:12.200" Score="3" Body="&lt;p&gt;If the film 'The Man Who Knew Infinity'  is to be believed, Hardy submitted Ramanujan's paper on highly divisible integers without his (Ramanujan's) knowledge.&lt;/p&gt;&#xA;&lt;p&gt;In more detail, in his book 'The Man ho Knew Infinity: a Life of the Genius Ramanujan, Robert Kanigel states that the paper Ramanujan, S. (1915). &amp;quot;Highly composite numbers&amp;quot; ). Proc. London Math. Soc. Series 2. 14: 347–409. (1915) was submitted by Hardy without Ramanujan's knowledge.&lt;/p&gt;&#xA;" OwnerUserId="16020" LastEditorUserId="16020" LastEditDate="2022-01-14T02:34:53.227" LastActivityDate="2022-01-14T02:34:53.227" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14066" PostTypeId="5" CreationDate="2022-01-14T06:38:51.900" Score="0" Body="&lt;p&gt;If the concerned question is about one of the works of Isaac Newton, this tag should be used along with &lt;a href=&quot;/questions/tagged/physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;physics&lt;/a&gt; or &lt;a href=&quot;/questions/tagged/mathematics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;mathematics&amp;#39;&quot; rel=&quot;tag&quot;&gt;mathematics&lt;/a&gt;, depending on the field of that work, and the tag of the specific area of that work (&lt;a href=&quot;/questions/tagged/classical-mechanics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;classical-mechanics&amp;#39;&quot; rel=&quot;tag&quot;&gt;classical-mechanics&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/calculus&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;calculus&amp;#39;&quot; rel=&quot;tag&quot;&gt;calculus&lt;/a&gt; etc.).&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-03-19T00:37:30.310" LastActivityDate="2022-03-19T00:37:30.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14067" PostTypeId="4" CreationDate="2022-01-14T06:38:51.900" Score="0" Body="For questions about Isaac Newton (1643-1727), mathematician and physicist, and his works" OwnerUserId="217" LastEditorUserId="10106" LastEditDate="2022-03-19T00:37:09.553" LastActivityDate="2022-03-19T00:37:09.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14068" PostTypeId="1" AcceptedAnswerId="14071" CreationDate="2022-01-14T06:56:18.977" Score="5" ViewCount="124" Body="&lt;p&gt;We commonly say that sets obey a Boolean algebra. I think that's correct as a stand-alone statement. If true, did Cantor come up with a Boolean algebra on his own, or did he use the work of Boole?&lt;/p&gt;&#xA;" OwnerUserId="15921" LastEditorUserId="8052" LastEditDate="2022-01-15T01:52:12.310" LastActivityDate="2022-01-15T08:25:42.860" Title="Did Cantor know about Boole's work in logic, and did it influence his work on sets?" Tags="|mathematics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14069" PostTypeId="2" ParentId="14061" CreationDate="2022-01-14T19:15:41.950" Score="2" Body="&lt;p&gt;Not sure if it counts, but the name of Kolmogorov certainly reminded me that according Yaglom (&lt;a href=&quot;https://doi.org/10.1146/annurev.fl.26.010194.000245&quot; rel=&quot;nofollow noreferrer&quot;&gt;A. N. Kolmogorov as a Fluid Mechanician and Founder of a School in Turbulence Research&lt;/a&gt;) Obukhov only learned about his important contributions to the theory of local structure of turbulence being incorporated into the manuscript published by Kolmogorov at a colloquium in Marseilles on the spot, because Obukhov flew by an aeroplane and Kolmogorov took the train.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1017/S0022112062000518&quot; rel=&quot;nofollow noreferrer&quot;&gt;The seminal paper&lt;/a&gt; is actually only authored by Kolmogorov but states on the first page &lt;em&gt;&amp;quot;However, Oboukhov has now discovered how to refine our previous results in a&#xA;way which takes Landau's comments into consideration. The method consists&#xA;in...&amp;quot;&lt;/em&gt;. The same issue contains a &lt;a href=&quot;https://doi.org/10.1017/S0022112062000506&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper based on a lecture by Obukhov&lt;/a&gt; and they are closely related. Not sure if both are a joint work or if only the former.&lt;/p&gt;&#xA;" OwnerUserId="10559" LastActivityDate="2022-01-14T19:15:41.950" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14070" PostTypeId="2" ParentId="14061" CreationDate="2022-01-14T19:49:50.883" Score="4" Body="&lt;p&gt;Thanks to Google Scholar, I found out that my name was listed on &lt;a href=&quot;https://scholar.google.ca/citations?view_op=view_citation&amp;amp;hl=en&amp;amp;user=w_4YIrUAAAAJ&amp;amp;cstart=20&amp;amp;pagesize=80&amp;amp;citation_for_view=w_4YIrUAAAAJ:_kc_bZDykSQC&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 2014 paper&lt;/a&gt; without the knowledge of any of the listed co-authors (including myself). I do know the listed co-authors (including the person who published the paper online), and had been working on projects with them, but I did not know that a paper with this title was posted online without me first having the chance to proofread it and correct any errors before they became visible to the public.&lt;/p&gt;&#xA;&lt;p&gt;This sometimes (unfortunately) happens when some people are a bit too excited about displaying their work, and cannot wait for others to first approve of the paper. Perhaps they worry that the others will not immediately approve of the paper, and will want to make improvements before publishing it, which will cause delays. Perhaps they are applying for something and want to beef up their &amp;quot;apparent&amp;quot; publication list, maybe because they've had a quiet year.&lt;/p&gt;&#xA;&lt;p&gt;Another thing that often happens is that a co-author is indeed aware of the paper and that it was submitted for publication, but the proofs get approved without their knowledge or approval. &lt;strong&gt;This happens so often that I would be surprised if no one with a lot of collaborators has experienced it.&lt;/strong&gt; This also happened to me on a 2015 version of the above-mentioned 2014 paper :)&lt;/p&gt;&#xA;&#xA;" OwnerUserId="8052" LastActivityDate="2022-01-14T19:49:50.883" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14071" PostTypeId="2" ParentId="14068" CreationDate="2022-01-15T07:51:57.433" Score="3" Body="&lt;p&gt;Grattan-Guinness has an informative book on the history of logic and mathematical foundations, &lt;a href=&quot;https://www.google.com/books/edition/The_Search_for_Mathematical_Roots_1870_1/LcCx9MROpbUC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Search for&#xA;Mathematical Roots, 1870-1940&lt;/a&gt;. There are dedicated chapters on Boole and Cantor, and Boole is mentioned in Cantor's chapter not once. The &lt;a href=&quot;https://books.google.it/books?id=vrQLbbxGNMsC&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of set theory by Ferreirós&lt;/a&gt; returns the same null result.&lt;/p&gt;&#xA;&lt;p&gt;Boole's work was only publicized in Germany in the late 1870s, when Cantor was already well on his way with set theory. Schröder's exposition &lt;em&gt;Operationskreis des Logikkalkuls&lt;/em&gt; came out in 1877. This said, Schröder was a prominent mathematician with interest in set theory and foundations, so Cantor might have been aware of it, but if so he left no indication to this effect. On the reception of Boole's work in the 19th century generally see &lt;a href=&quot;https://people.umass.edu/klement/boolefrege.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Klement, New Logic and the Seeds of Analytic Philosophy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Cantor had little use for set algebra and did not introduce it in his work. When one needs a fact about sets that a Boolean identity might provide it is more straightforward to deduce that fact by a direct element argument (compare deriving some Boolean identities from axioms, which can get quite intricate, to plain element arguments for them). So unless one is specifically focused on algebraic properties of set operations they would not be interested in Boolean algebra of sets. Cantor was not.&lt;/p&gt;&#xA;&lt;p&gt;In addition to that, Boole framed his work as about &amp;quot;laws of thought&amp;quot; and translating that into Cantor's context would have required clarifying obscure relationships between properties (truth conditions), classes (intensions) and sets (extensions), which would have been a major distraction. It was not done until the early 20th century, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2516/55&quot;&gt;How did mathematicians notate the empty set before ∅?&lt;/a&gt;&#xA;&amp;quot;&lt;em&gt;In fact, by the 1890s Cantor considered set theory as something completely different from logic&lt;/em&gt;&amp;quot;, writes Ferreirós (Dedekind was far more receptive).&#xA;Cantor also had a special distaste for &amp;quot;laws of thought&amp;quot;, heavily associated with Aristotle at the time, for those &amp;quot;laws&amp;quot; were traditionally used to rule out actual infinities, see &lt;a href=&quot;https://hsm.stackexchange.com/a/2419/55&quot;&gt;Why did Cantor (and others) use c for the continuum?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-01-15T08:25:42.860" LastActivityDate="2022-01-15T08:25:42.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14072" PostTypeId="1" AcceptedAnswerId="14076" CreationDate="2022-01-15T09:20:37.657" Score="5" ViewCount="227" Body="&lt;p&gt;The &lt;em&gt;living force&lt;/em&gt; or the &lt;em&gt;vis-viva&lt;/em&gt; is a quantity usually attributed to Leibniz (although there were a few other people who identified it as a conserved quantity in certain collisions earlier). Many accounts say that there was tension between Leibniz's work and Newton's work, and in particular there was tension between conservation of vis-viva and Newtonian mechanics. (Of course, we now know there is no tension between the two things.)&lt;/p&gt;&#xA;&lt;p&gt;However, I couldn't find any specific comments by Isaac Newton on the concept of Leibniz's vis-viva. Are there any quotes by Isaac Newton himself where he explicitly rejects this notion as meaningful? Does he ever make any reference to it?&lt;/p&gt;&#xA;" OwnerUserId="6965" LastActivityDate="2022-01-15T22:51:24.010" Title="Are there any direct comments by Isaac Newton on Leibniz's living force / vis-viva?" Tags="|physics|classical-mechanics|newton|energy|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14073" PostTypeId="1" CreationDate="2022-01-15T11:24:56.397" Score="1" ViewCount="187" Body="&lt;p&gt;I may think that this post is unsuitable in HSM.&lt;/p&gt;&#xA;&lt;p&gt;NASA launches rockets far more than JAXA(Japan Aerospace Exploration Agency).&lt;/p&gt;&#xA;&lt;p&gt;BostonDynamics-robots are way general-purpose than Japan's FANUC-robots.&lt;/p&gt;&#xA;&lt;p&gt;However when it comes to average math education level till 12,13th grade, Japan is far better than US.&lt;/p&gt;&#xA;&lt;p&gt;But as I mentioned firstly, number of rockets or performances of robots are US's advantages.&lt;/p&gt;&#xA;&lt;p&gt;It is very strange for me.&lt;/p&gt;&#xA;&lt;p&gt;Just looking math-education till 12th grade, I think Japan should be more powerful than US in engineering however actually not.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone who can explain this?&lt;/p&gt;&#xA;&lt;p&gt;BTW I am good at Japanese so I can search and share about education system in Japan as possible as I can, if you require me to do so.&lt;/p&gt;&#xA;" OwnerUserId="14765" LastEditorUserId="14765" LastEditDate="2022-01-16T12:20:13.590" LastActivityDate="2022-01-16T12:20:13.590" Title="Math-education comparsion between Japan and United States from perspectives of engineering contributions" Tags="|engineering|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14074" PostTypeId="2" ParentId="14073" CreationDate="2022-01-15T14:31:53.910" Score="5" Body="&lt;p&gt;The progress of technology is not a function of the state of education system. US imports huge number of highly educated people. Japan does not. Math education in most countries of East Europe is by far superior to the US (my first hand experience), but we do not have frequent news on breakthroughs in technology from East Europe. Instead, highly educated people from East Europe search for employment in the US.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-01-15T14:31:53.910" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14076" PostTypeId="2" ParentId="14072" CreationDate="2022-01-15T22:51:24.010" Score="7" Body="&lt;p&gt;The controversy was not so much about the tension between &lt;em&gt;vis viva&lt;/em&gt; and mechanics, as about what is the &amp;quot;true&amp;quot; quantity of motion, &lt;em&gt;vis viva&lt;/em&gt; or momentum, and what is the &amp;quot;metaphysical&amp;quot; basis upon which mechanics is to be built, &lt;em&gt;vis viva&lt;/em&gt; or Newton's force (aside from multiple other side issues that got entangled with it, see &lt;a href=&quot;https://hsm.stackexchange.com/a/598/55&quot;&gt;What was the vis viva controversy, including its philosophical aspects?&lt;/a&gt;). And, as is often the case, Newton's contribution was not by what he said, but by what he &lt;em&gt;implied&lt;/em&gt; in multiple editions of &lt;em&gt;Principia&lt;/em&gt;, see &lt;a href=&quot;https://physicstoday.scitation.org/doi/full/10.1063/1.2387086&quot; rel=&quot;noreferrer&quot;&gt;Smith, The vis viva dispute&lt;/a&gt;. For example, he highlighted the non-conservation of &lt;em&gt;vis viva&lt;/em&gt; in soft collisions, and tied its conservation to static (&amp;quot;dead&amp;quot;) forces, like the centripetal force, in direct counter to Leibniz's use of it to distinguish &amp;quot;dead&amp;quot; and &amp;quot;living&amp;quot; forces:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Book 1 of Newton’s Principia, the part that has some bearing on the vis viva controversy, went to the printer in April 1686, too soon for him to have seen the Acta Eruditorum issue containing Leibniz’s note. Newton surely was aware of the controversy by the time of the second (1713) and third (1726) editions of the Principia; yet they never mention it. Nevertheless, parts of Book 1 that remained the same in all editions did feed the controversy. For example, the conservation of momentum is presented as a corollary of Newton’s laws of motion, with Huygens’s center-of-gravity principle, carefully defended, as the next corollary.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In his empirical defense of his laws of motion, Newton indicates how to make corrections for air resistance in measurements of a ballistic pendulum (see figure 2(a)) to obtain more exacting tests of theories of collision, and then adds that similar corrections can be made for imperfect elasticity of the colliding bodies. Thereby Newton underscores the failure of mass times velocity squared to be conserved when the bodies are not perfectly hard.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Leibniz’s 1686 note provoked exchanges with the Cartesians. Descartes’ conservation of motion (see figure 1) was difficult to abandon if one believed that all space is filled with matter. The exchanges led Leibniz to refine his position in writings on “dynamics” (the term is his) that were not published until the 19th century. In those writings Leibniz grants the conservation of directional motion, but argues that because it is directional, unlike &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt;, it involves reference to other bodies and therefore is not a feature of each body taken unto itself. He concedes that &lt;span class=&quot;math-container&quot;&gt;$mv^2$&lt;/span&gt; is not obviously conserved in the collision of soft bodies. But he contends that it is actually conserved via undetected motion of the microphysical parts of the bodies...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Invoking the metaphysical principle that the effect must equal the cause, Leibniz gave a variant of his 1686 argument: He calculated “the force through the effect produced in using itself up” to conclude that the force transferred from one equal body to another varies as the square of the velocity. Leibniz made clear that the metaphysical principle is what establishes the priority of the conservation of living forces in changes of motion.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-01-15T22:51:24.010" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14078" PostTypeId="1" AcceptedAnswerId="14079" CreationDate="2022-01-16T21:15:04.483" Score="5" ViewCount="108" Body="&lt;p&gt;In 1950, Giuliano Toraldo di Francia published a paper [1] with the title &amp;quot;Parageometrical Optics&amp;quot; followed by several other papers on the subject, I dare say culminating in a beautiful design work of a microwave antenna [2]. Toraldo also presented his ideas in a rather brief chapter in his otherwise excellent book &amp;quot;Onde Elettromagnetiche&amp;quot; published in 1953. His work on parageometric optics seems to have preceded or at least be simultaneous with Keller's Geometric Theory of Diffraction.&lt;/p&gt;&#xA;&lt;p&gt;Parageometric optics is supposed to be an extension of conventional geometric optics to include diffraction effects. In his analysis, Toraldo shows that the various diffractive orders resulting from&#xA;an arbitrary (amplitude/phase) transparency illuminated by a plane wave will follow Malus' law for the associated ray systems having orthogonal surfaces and thus the propagation of diffraction orders can be treated individually and &amp;quot;geometrically&amp;quot; and, furthermore, the transparency can be of arbitrary (smooth) shape.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that since 1960 or so nothing has been written on the subject, and by now is completely forgotten despite its eminent intuitive appeal; Born&amp;amp;Wolf does not mention it at all, not even as a pedagogic device. My question is why? Is anything fundamentally so wrong with it that the idea is not worth discussing?&lt;/p&gt;&#xA;&lt;p&gt;[1]:&amp;quot;Parageometrical Optics&amp;quot;, JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, 1950 vol.40, 9&lt;/p&gt;&#xA;&lt;p&gt;[2]: &amp;quot;An Application of Parageometrical Optics to the Design of a Microwave Mirror&amp;quot;, IRE Trans. Antennas and Propagation, January 1958&lt;/p&gt;&#xA;&lt;p&gt;I asked this on &lt;a href=&quot;https://physics.stackexchange.com/questions/633842/diffraction-physics-v-parageometric-optics-whatever-happened-to-it&quot;&gt;https://physics.stackexchange.com/questions/633842/diffraction-physics-v-parageometric-optics-whatever-happened-to-it&lt;/a&gt; but got no reply.&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="55" LastEditDate="2022-01-17T10:04:14.290" LastActivityDate="2022-01-17T10:04:14.290" Title="Whatever happened to parageometrical optics of diffraction?" Tags="|electromagnetism|optics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14079" PostTypeId="2" ParentId="14078" CreationDate="2022-01-17T10:01:44.600" Score="7" Body="&lt;p&gt;There was nothing wrong with parageometrical optics mathematically, it just did not take. It happens. Compare &lt;a href=&quot;https://books.google.com/ngrams/graph?content=parageometrical&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=26&amp;amp;smoothing=3&amp;amp;direct_url=t1%3B%2Cparageometrical%3B%2Cc0#t1%3B%2Cparageometrical%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;Google ngram for parageometrical&lt;/a&gt; to the &lt;a href=&quot;https://books.google.com/ngrams/graph?content=nomography&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=26&amp;amp;smoothing=3&amp;amp;direct_url=t1%3B%2Cnomography%3B%2Cc0#t1%3B%2Cnomography%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;ngram for nomography&lt;/a&gt;. And &lt;a href=&quot;https://www.maa.org/press/maa-reviews/the-history-and-development-of-nomography&quot; rel=&quot;noreferrer&quot;&gt;nomography&lt;/a&gt; had everything, mathematical depth, intuitive appeal, wide use by engineers for computations and visualization, and yet we see the same decline of interest since 1960s. Why? Because computers. With all the tools they provide there just wasn't much demand for clever geometric shortcuts like nomography or parageometrical optics. See e.g. &lt;a href=&quot;https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-15-2-278&amp;amp;id=125452&quot; rel=&quot;noreferrer&quot;&gt;Mínguez-Vega et al. (2007)&lt;/a&gt; using wave optics, complete with numerical simulations, to redo  the &amp;quot;parageometrical&amp;quot; design of &lt;a href=&quot;https://www.osapublishing.org/ol/viewmedia.cfm?uri=ol-27-11-969&amp;amp;seq=0&quot; rel=&quot;noreferrer&quot;&gt;Amako et al. (2002)&lt;/a&gt; for a diffractive-refractive lens triplet (they do not even mention di Francia).&lt;/p&gt;&#xA;&lt;p&gt;Another possible reason is that it did not generate enough new insights. One can handle diffraction by the more conventional Fresnel approach, including explanations of Ronchi-di Francia's antennas, see &lt;a href=&quot;https://www.google.com/books/edition/Microwave_Optics/JSVRAAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Cornbleet, Microwave Optics, p.81&lt;/a&gt; doing it in 1973 after mentioning the original parageometrical design in passing. By then, di Francia himself moved on to greener pastures, see &lt;a href=&quot;https://www.osapublishing.org/ao/abstract.cfm?uri=ao-12-7-1477&quot; rel=&quot;noreferrer&quot;&gt;Geodesic Lenses for Guided Optical Waves&lt;/a&gt; co-authored by him.&#xA;As very similar ngrams for &lt;a href=&quot;https://books.google.com/ngrams/graph?content=horn+angles&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=26&amp;amp;smoothing=3&amp;amp;direct_url=t1%3B%2Chorn%20angles%3B%2Cc0#t1%3B%2Chorn%20angles%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;horn angles&lt;/a&gt; and &lt;a href=&quot;https://books.google.com/ngrams/graph?content=catastrophe+theory&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=26&amp;amp;smoothing=3&amp;amp;direct_url=t1%3B%2Ccatastrophe%20theory%3B%2Cc0&quot; rel=&quot;noreferrer&quot;&gt;catastrophe theory&lt;/a&gt; show, they had their heyday in the 1940s and 1980s, respectively, and then fizzled out much like parageometrical optics. The reason was again the scarcity of new insights. It is not enough for an idea to be neat to take, it has to be sufficiently fruitful.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-01-17T10:01:44.600" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14080" PostTypeId="1" AcceptedAnswerId="14081" CreationDate="2022-01-17T10:42:44.607" Score="3" ViewCount="483" Body="&lt;p&gt;This article: &lt;a href=&quot;https://azer.com/aiweb/categories/magazine/92_folder/92_articles/92_tusi.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Farid Alakbarli, A 13th-Century Darwin? Tusi's Views on Evolution, Azerbaijan International (2001)&lt;/a&gt; claims that Nasir al-Din al-Tusi (13th century) stated an early version of the law of conservation of mass:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For instance, Tusi believed that a body of matter is able to change, but is not able to entirely disappear. He wrote: &amp;quot;A body of matter cannot disappear completely. It only changes its form, condition, composition, color and other properties and turns into a different complex or elementary matter.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That would be 500 hundred years before it was stated by Lomonosov and Lavoisier.&lt;/p&gt;&#xA;&lt;p&gt;The article is a bit old, is there any modern or academic source that asserts such a claim? It seems like a single phrase out of context and not the basis for an entire (al)chemical principle.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-01-17T13:31:02.787" LastActivityDate="2022-01-17T16:07:45.193" Title="Did the medieval Islamic scholar al-Tusi state the principle of conservation of mass?" Tags="|chemistry|islamic-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14081" PostTypeId="2" ParentId="14080" CreationDate="2022-01-17T12:50:01.720" Score="4" Body="&lt;p&gt;TL;DR Not really. In Soviet times there was a joke about the tendency of Soviet scholars to trace the origins of everything to Russian contributions: &lt;em&gt;Russia is the motherland of elephants&lt;/em&gt;. I think this is a case of that with Russia replaced by Persia.&lt;/p&gt;&#xA;&lt;p&gt;Alakbarli is himself an academic, a doctor of historical sciences (highest Soviet degree), who studied Tusi's manuscipts since 1986. The OP linked webpage is a redux of his &lt;a href=&quot;https://web.archive.org/web/20191111111133/http://www.alakbarli.aamh.az/index.files/28.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;publication in Russian&lt;/a&gt; that gives a fuller quote and the citation to a source:&#xA;Xace Nəsirəddin Tusi. Əxlaqi-Nasiri. Baki, Elm, 1989, pp. 47-48. Other more recent works that I have seen either reproduce Alakbarli's (badly translated) Tusi quote or simply repeat the claim phrased similarly to his. Here is the quote Google translated from Russian (with my minor edits):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If someone, through the eyes of an attentive and inquisitive researcher, pays attention to the process of changing substances, to their composition, transformations and opposites, ... then he will understand that not a single body is completely destroyed. It only changes its form, state, composition, color [and other] qualities and turns into a compound or elementary substance... Because material bodies do not have the property of being destructible&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Further quotes outline Tusi's natural philosophy with changeable &amp;quot;elements with properties of prime matter&amp;quot; at the basis.&lt;/p&gt;&#xA;&lt;p&gt;It is plausible that the quote is authentic, but, &lt;em&gt;pace&lt;/em&gt; Alakbarli's enthusiasm for the fellow countryman (taking that broadly), it is not as remarkable as he makes it out to be. The prime matter and the elements are Aristotelian conceptions, &lt;a href=&quot;https://en.wikipedia.org/wiki/Nothing_comes_from_nothing&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;ex nihilo nihil fit&lt;/em&gt;&lt;/a&gt; was a slogan of ancient Greek natural philosophy, as recorded in Aristotle's physics where it is credited to Parmenides. Even clearer application of it to matter specifically is found in atomists since Democritus, and we can read similar passages in &lt;em&gt;De Rerum Natura&lt;/em&gt;. So if this vague philosophizing (that does not even mention mass) counts as &amp;quot;stating the principle&amp;quot; then Tusi was beaten to it by over a millennium and a half.&lt;/p&gt;&#xA;&lt;p&gt;This said, &lt;a href=&quot;https://pubs.acs.org/doi/pdf/10.1021/ed052p658&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whitaker in An Historical Note&#xA;on the Conservation of Mass&lt;/a&gt; shows that Lomonosov and Lavoisier had precursors even when we take the conservation law in a more quantitative sense relevant to chemistry. Jean Rey in “On an Inquiry Into the Cause&#xA;Wherefore Tin and Lead Increase in Weight on Calcination” (1630) stated:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;weight is so closely united to the primary matter of the elements that they can never be deprived of it. The weight with which each portion of matter was endued at the cradle, will be carried by it to the grave. In whatever place, in whatever form, to whatever volume it may be reduced, the same weight always persists.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are still Aristotelian prime matter and elements in it, but there is also weight and references to actual measurements. Rey then dutifully reports that his measurements did not agree with his philosophizing:&amp;quot;&lt;em&gt;I... affirm that the examination of weights which is made by the balance differs greatly from that which is made by the reason&lt;/em&gt;&amp;quot;. The contribution of Lomonosov and Lavoisier should now be clear, they converted a vague armchair speculation into a confirmed quantitative law. Democritus, Aristotle and Tusi did not even get to the problem Rey discovered. Calcination went on to become a paradigm case for Stahl's phlogiston theory that Lavoisier displaced.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-01-17T12:57:40.153" LastActivityDate="2022-01-17T12:57:40.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14083" PostTypeId="1" CreationDate="2022-01-18T07:44:16.147" Score="20" ViewCount="3525" Body="&lt;p&gt;When was it realized that the gas giant planets Jupiter and Saturn, and the ice giant planets Uranus and Neptune, do not have solid surfaces?&lt;/p&gt;&#xA;&lt;p&gt;When was that idea first proposed and when was it accepted by astronomers?&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2022-01-18T11:48:48.500" Title="When was it realized that the giant planets do not have solid surfaces?" Tags="|astronomy|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14084" PostTypeId="2" ParentId="14083" CreationDate="2022-01-18T11:48:48.500" Score="25" Body="&lt;p&gt;Prehistory goes back to the second half of 19th century after Fraunhofer introduced astronomical spectroscopy, and then Kirchhoff identified some spectral lines in the Sun's spectrum in 1860s. Rutherfurd observed the spectrum of Jupiter in 1863, but its import remained murky until 1920s. For example, in a &lt;a href=&quot;https://www.scientificamerican.com/article/the-mysterious-planet-saturn/&quot; rel=&quot;noreferrer&quot;&gt;1906 Scientific American article Erber&lt;/a&gt; noted great similarity in the observed chemical composition of Jupiter and Saturn, and mentioned that Joussen found presence of hydrogen there, but &lt;a href=&quot;https://www.google.com/books/edition/Scientific_American/75M6AQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Saturn+Joussen+spectroscopic++hydrogen&amp;amp;pg=PA25637&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;his pictures&lt;/a&gt; indicate that he still thought an observer could stand on Saturn's surface, complete with hills and clouds above.&lt;/p&gt;&#xA;&lt;p&gt;Only in 1920s Jeffreys conjectured gaseous (hydrogen-rich) composition. The confirmation came in 1930s largely due to Wildt, see &lt;a href=&quot;https://lasp.colorado.edu/home/mop/files/2015/08/jupiter_ch4-1.pdf&quot; rel=&quot;noreferrer&quot;&gt;Taylor et al., The Composition of the Atmosphere of Jupiter&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Modern studies of the composition of Jupiter's atmosphere&#xA;date back to the mid-nineteenth century, when the near-&#xA;infrared spectrum of the planet was viewed by Rutherfurd&#xA;(1863) using diffraction gratings of his own manufacture. He discovered features that remained unidentified until 1932, when Wildt showed that the unknown spectral lines were due to ammonia and methane. In later years, building on the original insight of Jeffreys (1923, 1924), Wildt and others went on to note that the low density of Jupiter and the presence of these hydrogen-rich compounds in the atmosphere were consistent with a bulk composition similar to that of the Sun, that is, primarily hydrogen.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-01-18T11:48:48.500" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14085" PostTypeId="1" AcceptedAnswerId="14097" CreationDate="2022-01-19T03:38:21.137" Score="7" ViewCount="5023" Body="&lt;p&gt;From James Gleick's &lt;a href=&quot;https://books.google.com/books?id=IWQ_y90P2uIC&amp;amp;q=putna#v=snippet&amp;amp;q=putnam&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Genius: the life and science of Richard Feynman&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of Feynman’s fraternity brothers was surprised to see him return home while the examination was still going on. Feynman learned later that the scorers had been astounded by the gap between his result and the next four.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this true? From what I can tell, he was simply one of the 5 top scorers in the &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition&quot; rel=&quot;nofollow noreferrer&quot;&gt;1939 Putnam Competition&lt;/a&gt;. It seems that other resources say the same thing, but I can't find any primary evidence.&lt;/p&gt;&#xA;" OwnerUserId="16056" LastEditorUserId="4251" LastEditDate="2023-09-05T10:18:46.780" LastActivityDate="2023-09-05T10:18:46.780" Title="Did Feynman win the Putnam by a &quot;large margin&quot;?" Tags="|biographical-details|physicists|feynman|academic-awards|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14086" PostTypeId="2" ParentId="13967" CreationDate="2022-01-19T14:45:59.353" Score="2" Body="&lt;p&gt;Poisson first introduced the scalar potential to electrostatics. He takes it as well-known that, given a force between particles obeying an inverse square law, you can obtain the components of the force by taking partial derivatives of a scalar function equal to the sum of charges divided by their respective distances to that point. He arrives at this from Laplace's work in gravitational potential, and the idea of scalar potential dates back earlier, to Daniel Bernoulli in 1738, as described in &lt;a href=&quot;https://hsm.stackexchange.com/questions/12797/what-was-the-historical-motivation-for-introducing-the-concept-gravitational-po/12800#12800&quot;&gt;this answer about the gravitational potential&lt;/a&gt;. In Poisson's &lt;a href=&quot;https://histomathsci.blogspot.com/2021/04/translation-of-s-d-poissons-premiere.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;first Memoire&lt;/a&gt; [note: my own translation of the Memoire], he actually begins the analysis by saying:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1) Let us consider first a given spheroid, coated in a fluid layer infinitely thin and of variable thickness; let us suppose that the molecules of this fluid repel each-other mutually in inverse proportion to the square of their separation, and let us attempt to determine the total action of the layer on one of these molecules, or more generally on a given point of space, taken outside or inside of the spheroid. We know that the components of the attraction or the repulsion which a body exerts on a given point, are expressed by the partial differences* of a certain function of the coordinates of this point, which is to say, of the function which represents the sum of the molecules of the body, divided by their respective distances to the given point: let us designate this sum by V , relative to the fluid layer which we are considering, and let us determine the value of V as a function of the coordinates of a given point in space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;* &lt;em&gt;differences partielles&lt;/em&gt;, meaning partial derivatives&lt;/p&gt;&#xA;&lt;p&gt;In modern notation, let &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; be the charge of a &amp;quot;molecule&amp;quot; of electricity, and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}_i$&lt;/span&gt; be the position vector of a given molecule, and suppose we want to find the force at a point &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}$&lt;/span&gt; due to all molecules. What Poisson is saying is, given a function &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; defined as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;V(\mathbf{r})=\sum_i \frac{q}{|\mathbf{r_i}-\mathbf{r}|},&#xA;$$&lt;/span&gt;&#xA;we can obtain the force vector &lt;span class=&quot;math-container&quot;&gt;$\mathbf{F}$&lt;/span&gt; per unit charge at &lt;span class=&quot;math-container&quot;&gt;$\mathbf{r}$&lt;/span&gt; by taking the gradient of &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, i.e. we obtain the electric field &lt;span class=&quot;math-container&quot;&gt;$\mathbf{E}(\mathbf{r})$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mathbf{E}(\mathbf{r}) = \nabla V.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Poisson goes on to derive the electric potential between two spheres of different radii separated by large and small distances. His derivation of the potential between two spheroids is clearly the main subject of his Memoires, and his use of the potential is just a mathematical tool he introduces in his assumptions, so while he was the first to apply the concept of scalar potential to electrostatics, it's also worth noting that he had little to do with the origin of the concept of potential itself. George Green's work in &lt;em&gt;An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism&lt;/em&gt; (1828) (described &lt;a href=&quot;https://en.wikipedia.org/wiki/An_Essay_on_the_Application_of_Mathematical_Analysis_to_the_Theories_of_Electricity_and_Magnetism&quot; rel=&quot;nofollow noreferrer&quot;&gt;on Wikipedia&lt;/a&gt;, and available &lt;a href=&quot;https://www.google.com/books/edition/An_Essay_on_the_Application_of_Mathemati/GwYXAAAAYAAJ?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;on Google Books&lt;/a&gt;) was far more important in the development of the theory of potentials, for introducing a number of significant results, which now bear his name.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2022-01-19T14:45:59.353" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14087" PostTypeId="1" CreationDate="2022-01-19T19:28:31.563" Score="3" ViewCount="210" Body="&lt;p&gt;I am looking for what motivated Albert Einstein in the direction of his Special Theory.    I have read that it is unclear if he was set on that path by the Michelson-Morley experiments.  Was AE aware that the Maxwell equations did not transform covariantly under the Galilean transformation?  Does anyone know?&lt;/p&gt;&#xA;" OwnerUserId="16073" OwnerDisplayName="goedelite" LastEditorUserId="4251" LastEditDate="2022-01-19T23:15:01.523" LastActivityDate="2022-01-20T09:06:56.503" Title="Had Albert Einstein tried to use the Galilean transformation on Maxwell's equations before AE's Special Theory?" Tags="|physics|electromagnetism|20th-century|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14088" PostTypeId="1" AcceptedAnswerId="14090" CreationDate="2022-01-19T23:35:57.543" Score="19" ViewCount="4477" Body="&lt;p&gt;All I know about Garfield and math was that he made an original proof of the Pythagorean theorem. Did he make any other mathematical advancement (big or small)?&lt;/p&gt;&#xA;" OwnerDisplayName="user15948" LastEditorDisplayName="user15948" LastEditDate="2022-01-21T17:58:58.190" LastActivityDate="2022-01-21T17:58:58.190" Title="Did president Garfield make any contributions to Mathematics?" Tags="|mathematics|mathematicians|pythagorean-proposition|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14090" PostTypeId="2" ParentId="14088" CreationDate="2022-01-20T01:03:01.677" Score="29" Body="&lt;p&gt;No. The library of Congress has a well-organized website of Garfield's papers, and he did not publish anything on mathematics other than that &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theorem&quot; rel=&quot;noreferrer&quot;&gt;one note on the Pythagorean theorem&lt;/a&gt; in April 1, 1876 issue of the &lt;em&gt;New-England Journal of Education&lt;/em&gt;. Perhaps in a gesture to the publication date, the editor mislabeled it &lt;em&gt;pons asinorum&lt;/em&gt; (the bridge of asses), the medieval Latin nickname for the isosceles triangle theorem whose Euclid's proof had a reputation for getting students stuck (the diagram resembled a bridge they could not &amp;quot;cross&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;However, Garfield did have oversized enthusiasm for statistics and pushed it everywhere as a public official. He advocated it for both informing public policy and establishing a more egalitarian picture of history. See &lt;a href=&quot;https://www.jstor.org/stable/24446440&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;James A. Garfield, Historian&amp;quot; by Allan Peskin&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Garfield's fascination with quantifiable data was surprising, considering that his formal education had all but neglected mathematics. Yet his passion for statistics was so intense that he was accused of having &amp;quot;gone mad&amp;quot; on the subject. It was charged that &amp;quot;he would have the Congress and the officers of all Departments of the Government constantly running up and down the country gathering statistics.&amp;quot; There was some truth to this. Garfield did propose the establishment of a &amp;quot;Bureau of Statistics&amp;quot; in the Treasury Department, and when he was chairman of the House Committee on the Census he overloaded the schedule with questions designed to uncover all sorts of miscellaneous statistical data. Some thought these questions trivial but Garfield defended their utility. &amp;quot;This is the age of statistics...&amp;quot; he insisted. &amp;quot;When we propose to legislate for great masses of people, we must first study the great facts relating to the people - their number, strength, length of life, intelligence, morality, occupations, industry and wealth.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;These &amp;quot;great facts&amp;quot; had a further application: &amp;quot;The developments of statistics are causing history to be rewritten,&amp;quot; Garfield declared.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Till recently, the historian studied nations in the aggregate, and gave us only the story of princes, dynasties, sieges and battles. Of the people themselves - the great social body, with life, growth, forces, elements, and laws of its own - he told us nothing. Now, statistical inquiry leads him into the hovels, homes, workshops, mines, fields, prisons, hospitals, and all other places where human nature displays its weaknesses and its strength. [...]&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="9394" LastEditDate="2022-01-20T21:11:38.653" LastActivityDate="2022-01-20T21:11:38.653" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14091" PostTypeId="1" AcceptedAnswerId="14092" CreationDate="2022-01-20T03:28:04.773" Score="2" ViewCount="129" Body="&lt;p&gt;In graduate school, I remember hearing or seeing the rough quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Good mathematicians know one branch of math. Great mathematicians&#xA;know two branches.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm sure I am somewhat misquoting it, but does anyone have a reference/citation for this quote?&lt;/p&gt;&#xA;" OwnerUserId="16074" LastEditorUserId="4251" LastEditDate="2022-01-22T17:44:11.583" LastActivityDate="2022-01-22T17:44:11.583" Title="Looking for reference for quote" Tags="|mathematics|reference-request|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14092" PostTypeId="2" ParentId="14091" CreationDate="2022-01-20T03:53:42.460" Score="4" Body="&lt;p&gt;The correct quote is attributed to Banach: “Good mathematicians see analogies. Great mathematicians see analogies between analogies.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Ulam, S. M. (1976). Adventures of a mathematician. New York&lt;/p&gt;&#xA;&lt;p&gt;Note: I used Google Books to find this quote with &amp;quot;Good mathematicians&amp;quot; and &amp;quot;Great mathematicians&amp;quot; as search terms.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="6085" LastEditDate="2022-01-20T04:29:57.923" LastActivityDate="2022-01-20T04:29:57.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14094" PostTypeId="2" ParentId="14087" CreationDate="2022-01-20T09:06:56.503" Score="2" Body="&lt;p&gt;There is an interesting historical paper dealing with this issue:&lt;/p&gt;&#xA;&lt;p&gt;John Norton: &lt;a href=&quot;http://philsci-archive.pitt.edu/1743/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Einstein’s Investigations of Galilean Covariant Electrodynamics prior to 1905&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Einstein was aware before he wrote his famous 1905 special relativity paper, that the negative result of the Michelson-Morley experiment can be explained by an &lt;a href=&quot;https://en.wikipedia.org/wiki/Emission_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;emission theory of light&lt;/a&gt;, in which there is no aether, but in which the speed of light depends on the velocity of the source by the ordinary Galilean velocity addition.&lt;/p&gt;&#xA;&lt;p&gt;Einstein dismissed the emission theory of light for the following reason as described by Norton:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The decisive consideration, Einstein tells us, that spoke to him&#xA;against an emission theory prior to his 1905 paper was his conviction&#xA;that light should be characterized by frequency and intensity (and&#xA;polarization) alone. He was then rather uninterested in the fussy&#xA;details of how a variety of distinct emission theories might be&#xA;devised to accommodate to various sorts of processes of deflection or&#xA;reflection. They seem to have come to the fore in the literature&#xA;emerging around 1912 that sought to test an emission theory&#xA;experimentally, for just those details decide how the experiments are&#xA;to be done.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, many modern experiments show that the speed of light does not depend on the velocity of the source, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Emission_theory#Refutations_of_emission_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Refutations of emission theory&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7601" LastActivityDate="2022-01-20T09:06:56.503" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14095" PostTypeId="2" ParentId="2424" CreationDate="2022-01-20T15:00:45.927" Score="1" Body="&lt;p&gt;Although in general number theory was fairly neglected until Diophantus' translation, various (often Persian) mathematicians in the Islamic world were working on things related to perfect and amicable numbers (and other divisor questions), as were some European scholars.  See &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Perfect_numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor&lt;/a&gt; for some information (hat tip to &lt;a href=&quot;https://hsm.stackexchange.com/a/13493/9624&quot;&gt;this answer&lt;/a&gt;), including the apparent rediscovery of the fifth perfect number (and hence fifth Mersenne prime) on numerous occasions, or see Alireza Djafari Naini's book on some of the Islamic mathematicians, &lt;a href=&quot;https://www.journals.uchicago.edu/doi/10.1086/353631&quot; rel=&quot;nofollow noreferrer&quot;&gt;reviewed here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So the real question is, why did he bother with trying to look at so many of them, to try to determine &lt;em&gt;which&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2^p-1$&lt;/span&gt; worked?  There are two main secondary literature articles (that I know of) discussing this at some length beyond the mere list of his numbers.  The essential idea is whether the prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; has some specific form, such as &lt;span class=&quot;math-container&quot;&gt;$2^k\pm 1$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2^{2k}\pm 3$&lt;/span&gt;; the question is whether Mersenne explained it well, and/or whether there were any typographical errors.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Stillman Drake, The rule behind ‘Mersenne’s numbers’, Physis: Rivista Internazionale di Storia della Scienza Vol. 13 No. 4, pp. 421-424.  This is not easy to track down, but is fairly often referred to.&lt;/li&gt;&#xA;&lt;li&gt;Vittorio Boria, Marin Mersenne: Educator of Scientists, American University diss., 1989.  This used to be available on &lt;a href=&quot;https://wrlc-amu.primo.exlibrisgroup.com/discovery/fulldisplay?docid=alma994195983604102&amp;amp;context=L&amp;amp;vid=01WRLC_AMU:prod&amp;amp;lang=en&amp;amp;search_scope=WRLC_P_AU&amp;amp;adaptor=Local%20Search%20Engine&amp;amp;tab=WRLC_P_AU&amp;amp;query=creator,contains,boria,AND&amp;amp;sortby=rank&amp;amp;mode=advanced&amp;amp;offset=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;American University's website&lt;/a&gt; but now you seem to have to use ILL for this as well.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Finally, for a more mathematically-minded take on where to take Mersenne's (possible) conjecture, see &lt;a href=&quot;https://www.jstor.org/stable/2323195&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The New Mersenne Conjecture&amp;quot;&lt;/a&gt;.  One of the few &lt;em&gt;easy&lt;/em&gt;-to-track-down references in this area!&lt;/p&gt;&#xA;" OwnerUserId="9624" LastActivityDate="2022-01-20T15:00:45.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14096" PostTypeId="1" CreationDate="2022-01-20T16:42:51.953" Score="1" ViewCount="102" Body="&lt;p&gt;The textbook, &amp;quot;Atomic Physics&amp;quot;, by Max Born contains a derivation in its appendix of the Wien radiation law that is based only on thermodynamics and Maxwell's laws of E&amp;amp;M.   The derivations I find today all start with Planck's radiation law and show agreement except at the ultra-violet end.  Born's text is old, and I wonder if students today are not puzzled by using Planck's law, which was not available to Wien at the time he offered his displacement law and his spectral energy density. Wien's original paper was no doubt in a German publication of record in his day.&lt;/p&gt;&#xA;" OwnerUserId="16073" LastActivityDate="2022-01-20T16:42:51.953" Title="Is there a derivation (in English) of the Wien radiation law not based on statistics, other than that found in &quot;Atomic Physics&quot; by Max Born?" Tags="|radiation|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14097" PostTypeId="2" ParentId="14085" CreationDate="2022-01-21T03:22:21.257" Score="6" Body="&lt;p&gt;This is Feynman's memory of an unnamed person who told him that. To quote him in an interview:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Anyway, I was among the first five. I have since found out from somebody from Canada, where it was scored, who was in the scoring division—he came to me much later and he told me that it was astonishing. He said that at this examination, “Not only were you one of the five, but the gap between you and the other four was sensational.” He told me that. I didn’t know that. That may not be correct, but that’s what I heard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5020-2&quot; rel=&quot;noreferrer&quot;&gt;Interview of Richard Feynman by Charles Weiner on 1966 March 5&lt;/a&gt;, Niels Bohr Library &amp;amp; Archives, American Institute of Physics, College Park, MD USA.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Thank you Dave L Renfro for pointing me here. I was lucky to find this -- the word &amp;quot;Putnam&amp;quot; was never mentioned!&lt;/p&gt;&#xA;" OwnerUserId="16056" LastEditorUserId="4251" LastEditDate="2022-01-21T08:43:11.840" LastActivityDate="2022-01-21T08:43:11.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14098" PostTypeId="1" CreationDate="2022-01-21T04:45:01.907" Score="0" ViewCount="96" Body="&lt;p&gt;I may not be understanding what a &amp;quot;Hertz&amp;quot; is but it seems to simply be one cycle per second. But would not Doppler have understood what frequency was decades before Hertz? Or was Hertz just the first to understand the e/m radiation had frequency just as sound does?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-01-21T16:29:09.217" Title="Did Hertz discover the concept of frequency and if so, how did Doppler think of frequency?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14099" PostTypeId="2" ParentId="14098" CreationDate="2022-01-21T10:07:32.767" Score="3" Body="&lt;p&gt;A quick look at Wikipedia page of &lt;a href=&quot;http://Heinrich%20Hertz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heinrich Hertz&lt;/a&gt;  provides the right answer&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Heinrich Hertz proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is neither for the discovery of frequency nor that EM had a frequency.&lt;/p&gt;&#xA;&lt;p&gt;The concept of frequency is as ancient as the concept of time. It just indicates how many cycles there are in a given unit of time. Looking at the sun turn around Earth, astronomers in the past would have said that the frequency is 1 turn per 24 h. Even the Romans had a word for it &lt;em&gt;frequentia&lt;/em&gt; which is the root of the English word.&lt;/p&gt;&#xA;&lt;p&gt;Mathematically, the modern concept of waves was already fully established by the 18th century with the formulation of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wave_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;wave equation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Maxwell had predicted that light was a wave of oscillating electromagnetic fields as the equations of electromagnetism (EM) turned into a wave equation in the vacuum with the speed &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; (speed of light). The idea of light being a wave was also not new (even Doppler insisted on it), as diffraction experiments in the 19th century showed it (see experiments by Thomas Young and Jean Arago).&lt;/p&gt;&#xA;&lt;p&gt;Hertz was the one to produce the first experimental setup to create radio waves from electromagnetic principles (dipole antenna). Radio waves (RW) were also a new discovery at the time but Hertz proved that RW had the same speed and follow the same laws of refraction/reflection/diffraction as light. Light turned out to be an EM wave too but with a higher frequency than radio waves.&lt;/p&gt;&#xA;&lt;p&gt;In 1960, during the &lt;em&gt;Conférence générale des poids et mesures&lt;/em&gt;, the International Bureau of Weights and Measures (BIPM), made standard the use of Hertz to replace the previous unit of frequency, the &amp;quot;cps&amp;quot; (cycles per second).&lt;/p&gt;&#xA;&lt;p&gt;Units are named after scientist in their honor but not because they discovered the units. For example, the gauss (unit of magnetic flux) is not named after Carl Friederich Gauss because he discovered magnetism. Sightings of magnetic rocks and the invention of the compass were centuries to millenia older. Gauss just did some important work on electricity and magnetism and the British Science Academy (or whoever invented it) named the unit in his honor.&lt;/p&gt;&#xA;&lt;p&gt;Edit: Interestingly enough, with all that said, Carl Doppler avoided frequency as a variable when establishing the equations for the effect named after him. In his work &lt;a href=&quot;https://de.wikisource.org/wiki/%C3%9Cber_das_farbige_Licht_der_Doppelsterne_und_einiger_anderer_Gestirne_des_Himmels&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels&lt;/em&gt;&lt;/a&gt; he writes his equation in terms of periods (time to go from point A to B).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-01-21T16:29:09.217" LastActivityDate="2022-01-21T16:29:09.217" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14100" PostTypeId="2" ParentId="14061" CreationDate="2022-01-21T14:16:42.203" Score="4" Body="&lt;p&gt;In 1978, I formulated a summation series equation that yielded the cubes of the positive integers. I sent it to Scientific American, NCTM (The National Council Of Teachers Of Mathematics), and others.&lt;/p&gt;&#xA;&lt;p&gt;In November (1978) in their journal the The Mathematics Student (Vol. 26, No. 2), NCTM published not the equation, but instead the expansion of the summation and shaped the results into a triangle. They named it Clark's Triangle and sent me a copy.&lt;/p&gt;&#xA;&lt;p&gt;Unknown to me, and at least by the late 1990s (as in the 2020s I saw the publication), the CRC Concise Encyclopedia Of Mathematics published Clark's Triangle in their tome. Wolfram gleaned CRC's summary of Clark's Triangle and incorporated it into their archives as &lt;a href=&quot;https://mathworld.wolfram.com/ClarksTriangle.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathworld.wolfram.com/ClarksTriangle.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16092" LastEditorUserId="16092" LastEditDate="2022-01-26T14:13:44.910" LastActivityDate="2022-01-26T14:13:44.910" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14102" PostTypeId="1" AcceptedAnswerId="14133" CreationDate="2022-01-21T17:09:06.380" Score="0" ViewCount="180" Body="&lt;p&gt;I have googled the heck out of this but cannot find a reference to the year Z&amp;amp;F published their axioms. I'd expected to see an article reference but none that I could find.&lt;/p&gt;&#xA;" OwnerUserId="15921" LastEditorUserId="10106" LastEditDate="2022-01-23T05:38:07.050" LastActivityDate="2023-09-15T12:48:37.280" Title="When did Zermelo and Fraenkel publish their axioms?" Tags="|mathematics|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14103" PostTypeId="1" AcceptedAnswerId="14104" CreationDate="2022-01-21T19:10:51.890" Score="2" ViewCount="259" Body="&lt;p&gt;Greek and Muslim Egyptian mathematicians are well known throughout the world. But the only Ancient Egyptian mathematician I know of is Ahmes, who said he's just a scribe. Who were the greatest Egyptian mathematicians of Ancient times?&lt;/p&gt;&#xA;" OwnerDisplayName="user15948" LastActivityDate="2022-01-26T02:26:50.017" Title="Ancient Egyptian Mathematicians" Tags="|mathematics|mathematicians|ancient-egypt|ancient|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14104" PostTypeId="2" ParentId="14103" CreationDate="2022-01-21T20:15:44.360" Score="4" Body="&lt;p&gt;We don't know. The documents we have concerning Egyptian and Babylonian Mathematics do not contain the names of their creators.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2022-01-21T20:15:44.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14105" PostTypeId="1" CreationDate="2022-01-22T04:01:45.247" Score="22" ViewCount="6174" Body="&lt;p&gt;The current answers to the Academia HNQ &lt;a href=&quot;https://academia.stackexchange.com/q/181409/69206&quot;&gt;Should I ask for permission to name a mathematical theorem?&lt;/a&gt; include the following (small excerpts):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is no need to ask permission, and mostly likely they will be happy to have the theorem referred to with their names.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I suppose there is no formal need to ask for permission, but&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I see no harm in doing so.&lt;/li&gt;&#xA;&lt;li&gt;I see the potential for harm in not doing so.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first can be loosely paraphrased as &amp;quot;go for it&amp;quot; and &amp;quot;it would be bad if you didn't&amp;quot; but the second seems like excellent advice.&lt;/p&gt;&#xA;&lt;p&gt;I thought I'd ask here if there are any examples that might inspire words of caution as expressed in the 2nd answer, so I'd like to ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Has there ever been a case where someone wished a theorem or important result wasn't named after them? Has it happened more than once?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-01-24T00:23:36.117" LastActivityDate="2022-01-26T02:47:38.793" Title="Has there ever been a case where someone wished a theorem or important result wasn't named after them? Has it happened more than once?" Tags="|naming-conventions|" AnswerCount="8" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="14106" PostTypeId="2" ParentId="14105" CreationDate="2022-01-22T14:57:37.873" Score="6" Body="&lt;p&gt;Let me tell two anecdotes related to the question:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Once during a seminar talk Hilbert asked: &amp;quot;What is a Hilbert space?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Arnold, apparently envious for so many things named after Anosov, once proposed the term &amp;quot;Anosov's jet&amp;quot; (струя Аносова, in Russian) Anosov's reaction is not recorded, but everyone was sure that he was not happy:-)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Remark. Sometimes a term containing a name can be indeed ridiculous,&#xA;for example look at the title of this paper &lt;a href=&quot;https://arxiv.org/abs/0904.1403&quot; rel=&quot;noreferrer&quot;&gt;https://arxiv.org/abs/0904.1403&lt;/a&gt;. As far as I know, Devaney was not offended:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="26" LastEditDate="2022-01-24T10:53:14.120" LastActivityDate="2022-01-24T10:53:14.120" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14107" PostTypeId="2" ParentId="14105" CreationDate="2022-01-23T00:53:27.253" Score="8" Body="&lt;p&gt;I cannot easily document the chatter about it, but G. (&amp;quot;Beppo&amp;quot;, for &amp;quot;Giuseppe&amp;quot;) Levi in 1906 used what we'd now call a &amp;quot;Sobolev space&amp;quot;, to prove an actual (=true) minimum principle for certain vector spaces of functions. (Thus, actually giving a true, and, mercifully, sufficient version of &amp;quot;Dirichlet's principle&amp;quot; (about existence and uniqueness of a maximizing vector...) for most applications.)&lt;/p&gt;&#xA;&lt;p&gt;The story I'd heard is that &amp;quot;the French&amp;quot; referred to such spaces (in French) as &amp;quot;the spaces of Beppo Levi&amp;quot;, and pronounced in such a way that was somehow degrading to Levi.&lt;/p&gt;&#xA;&lt;p&gt;Apparently, Levi objected to this... and, since the rest of Levi's life he worked on other things, his name was removed from such reference, and now it's &amp;quot;Sobolev spaces&amp;quot;. There are various pseudo-historical accounts of how Sobolev came to agree to that renaming, etc. I cannot vouch for any of that, etc.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-01-23T00:53:27.253" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14108" PostTypeId="2" ParentId="14105" CreationDate="2022-01-23T06:47:23.990" Score="36" Body="&lt;p&gt;On the Löwenheim–Skolem theorem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is somewhat ironic that Skolem's name is connected with the upward direction of the theorem as well as with the downward direction:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I follow custom in calling Corollary 6.1.4 the upward Löwenheim-Skolem theorem. But in fact Skolem didn't even believe it, because he didn't believe in the existence of uncountable sets.&amp;quot; – Hodges (1993).&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Skolem [...] rejected the result as meaningless; Tarski [...] very reasonably responded that Skolem's formalist viewpoint ought to reckon the downward Löwenheim-Skolem theorem meaningless just like the upward.&amp;quot; – Hodges (1993).&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association of his name to a result of this type, which he considered an absurdity, nondenumerable sets being, for him, fictions without real existence.&amp;quot; – Poizat (2000).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem#Historical_notes&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem#Historical_notes&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15230" LastActivityDate="2022-01-23T06:47:23.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14109" PostTypeId="2" ParentId="14005" CreationDate="2022-01-23T08:59:03.597" Score="10" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;What is the origin of this special numbering system? Was there a more&#xA;practical reason for having a special numbering system for South Asia?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The use of the terms &amp;quot;lakhs&amp;quot; and &amp;quot;crores&amp;quot; in South Asia (or the Indian subcontinent more precisely) finds its origin from the Vedic age (c. 1500 – c. 500 BCE) - the terms are derived from Sanskrit words लक्ष (laksha) and कोटि (koti) respectively. (If you are comfortable reading Hindi you may check out &lt;a href=&quot;http://valmikiramayan.pcriot.com/utf8/yuddha/sarga28/yuddha_28_frame.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yuddha Kand of Ramayana&lt;/a&gt; where the use of कोटि for crore can be found.) There is no particular reason for adopting this system, it's just a matter of convention.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why hasn't modern South Asia reformed this redundancy, just using&#xA;&amp;quot;millions&amp;quot; everywhere? Most other aspects of numbers seem to be&#xA;completely coherent. There is no special numbering system for&#xA;Spanish-speaking countries. Nor, there is a special numbering system&#xA;for the People's Republic of China, Korea and Japan.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Modern Indian Subcontinent has not &amp;quot;reformed this redundancy&amp;quot; because there is no redundancy. People have always used lakhs and crores and continue to do so. However, having said so, &lt;a href=&quot;https://en.wikipedia.org/wiki/Indian_numbering_system#Current_usage&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sri Lanka has switched to the English numbering system in recent years&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2023-01-26T01:11:50.180" LastActivityDate="2023-01-26T01:11:50.180" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14112" PostTypeId="2" ParentId="14105" CreationDate="2022-01-23T23:49:52.947" Score="14" Body="&lt;p&gt;According to a &lt;a href=&quot;https://youtu.be/6mJXv5OBel8&quot; rel=&quot;noreferrer&quot;&gt;recent episode of &lt;em&gt;PBS Space Time&lt;/em&gt;&lt;/a&gt;, Freeman Dyson &amp;quot;regretted&amp;quot; the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dyson_sphere&quot; rel=&quot;noreferrer&quot;&gt;Dyson Sphere&lt;/a&gt; being named after him.&lt;/p&gt;&#xA;&lt;p&gt;According to the Wikipedia article&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dyson was not the first to advance this idea. He was inspired by the 1937 science fiction novel Star Maker, by Olaf Stapledon,...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The PBS episode states that Dyson referred to the concept as the &lt;em&gt;Stapledon Sphere&lt;/em&gt; as Dyson was trying to &amp;quot;avoid having his own name attached to the idea&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="16122" LastActivityDate="2022-01-23T23:49:52.947" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14113" PostTypeId="1" CreationDate="2022-01-24T09:18:38.507" Score="0" ViewCount="95" Body="&lt;p&gt;In the history of science, has there ever been a theorem/equation/law that is named after a person and has been coined that name by the same person?&lt;/p&gt;&#xA;&lt;p&gt;Example: let's say somebody named Trevor discovers an equation that solves a certain kind of problems. In some of his publications, Trevor starts referring to the equation as &amp;quot;Trevor's equation&amp;quot;. The name sticks and nowadays it is called the Trevor's equation.&lt;/p&gt;&#xA;&lt;p&gt;Has a similar situation ever taken place?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="14365" LastEditDate="2022-01-24T09:30:45.953" LastActivityDate="2022-01-24T16:06:02.120" Title="Theorem or equation self-named after the same person that published it?" Tags="|terminology|scientists|" AnswerCount="1" CommentCount="0" ClosedDate="2022-01-24T16:48:13.983" ContentLicense="CC BY-SA 4.0" />
  <row Id="14114" PostTypeId="2" ParentId="14105" CreationDate="2022-01-24T09:48:53.527" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange#Name&quot; rel=&quot;noreferrer&quot;&gt;Diffie–Hellman key exchange&lt;/a&gt; is not a case where a person wishes it weren't named after them &lt;em&gt;per se&lt;/em&gt;, but it is a case where they are unhappy that it is named after them while excluding another contributor:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 2002, Hellman suggested the algorithm be called&#xA;Diffie–Hellman–Merkle key exchange in recognition of Ralph Merkle's&#xA;contribution to the invention of public-key cryptography (Hellman,&#xA;2002), writing:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The system...has since become known as Diffie–Hellman key exchange.&#xA;While that system was first described in a paper by Diffie and me, it&#xA;is a public key distribution system, a concept developed by Merkle,&#xA;and hence should be called 'Diffie–Hellman–Merkle key exchange' if&#xA;names are to be associated with it. I hope this small pulpit might&#xA;help in that endeavor to recognize Merkle's equal contribution to the&#xA;invention of public key cryptography.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3075" LastActivityDate="2022-01-24T09:48:53.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14115" PostTypeId="2" ParentId="48" CreationDate="2022-01-24T11:56:28.317" Score="3" Body="&lt;p&gt;Murray Gell-Mann proposed the term “higgson” as synonym for “Higgs boson” in his 1994 (A39) book The Quark and the Jaguar (pgs. 193-97).&lt;/p&gt;&#xA;" OwnerUserId="16127" LastActivityDate="2022-01-24T11:56:28.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14116" PostTypeId="2" ParentId="14113" CreationDate="2022-01-24T16:06:02.120" Score="3" Body="&lt;p&gt;I am not sure this is a clear cut case but when Stigler published &lt;a href=&quot;https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stigler's law of eponymy&lt;/a&gt; he named it so in the title of the paper. The law states that no law is named after its discoverer and he did not claim to have discovered it himself or it would have falsified itself.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2022-01-24T16:06:02.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14117" PostTypeId="2" ParentId="14105" CreationDate="2022-01-24T22:39:49.803" Score="5" Body="&lt;p&gt;A near miss that I found is &lt;a href=&quot;https://en.wikipedia.org/wiki/Lewis_Fry_Richardson#Pacifism&quot; rel=&quot;noreferrer&quot;&gt;Lewis Fry Richardson&lt;/a&gt;, from Wikipedia: &amp;quot;According to Thomas Körner, the discovery that his meteorological work was of value to chemical weapons designers caused him to abandon all his efforts in this field, and destroy findings that he had yet to publish.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Perhaps some other exists who published first and then found that their work had applications that they found objectionable after it was too late--I found Richardson by searching for mathematicians who were also peace activists, but you could search the other way by finding named theorems which were used for especially objectionable things.&lt;/p&gt;&#xA;" OwnerUserId="13194" LastActivityDate="2022-01-24T22:39:49.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14118" PostTypeId="2" ParentId="14105" CreationDate="2022-01-25T00:05:25.563" Score="5" Body="&lt;p&gt;There's a story, weakly cited on his wikipedia page (citation goes to a specific page in a book which doesn't seem to back it up, but I can't read the next page in the google books preview so maybe it's there) that Paul Dirac referred to what everyone else calls &amp;quot;Fermi-Dirac statistics&amp;quot; as &amp;quot;Fermi statistics&amp;quot;, and &amp;quot;for reasons of symmetry&amp;quot; called the other type &amp;quot;Bose statistics&amp;quot; instead of &amp;quot;Bose-Einstein statistics&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I've also heard a story that I can't back up at all (and so is possibly apocryphal) that whenever someone would refer to Rabi cycles or the Rabi frequency in a conference, if he was in the audience he would stick his hand up and ask what they meant until they explained it in basic terms which didn't include his name.&lt;/p&gt;&#xA;&lt;p&gt;Both of these are examples of alleged humility (though you could argue about how humble slowing down a talk is) rather than any dismay about the discoveries themselves.&lt;/p&gt;&#xA;" OwnerUserId="12826" LastActivityDate="2022-01-25T00:05:25.563" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14119" PostTypeId="1" CreationDate="2022-01-25T22:07:56.120" Score="3" ViewCount="175" Body="&lt;p&gt;A famous characteristic of the Enigma machine was that its rotor design ensured/required that no letter ever map to itself. However, there was a plugboard on the front that could swap letter pairs. So let's say the machine were in a state where pressing A would make B light up. What would happen in that case if a plugboard connection were added swapping A and B?&lt;/p&gt;&#xA;" OwnerUserId="16147" LastActivityDate="2024-02-28T07:09:53.557" Title="What would happen if an Enigma machine's plugboard were used to map a letter to itself?" Tags="|cryptology|" AnswerCount="2" CommentCount="0" ClosedDate="2024-03-03T13:33:05.323" ContentLicense="CC BY-SA 4.0" />
  <row Id="14120" PostTypeId="1" AcceptedAnswerId="14122" CreationDate="2022-01-25T22:47:07.687" Score="4" ViewCount="451" Body="&lt;p&gt;I'm going through the journals of Alexander Mackenzie (ca 1790) and I came across this passage:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2yBZF.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2yBZF.png&quot; alt=&quot;Screenshot from Alexander Mackenzie's journal&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I gather that he's determining his latitude and longitude but I'm not clear on what units he's using, or the procedure for that matter. For example, is 29. 23. 48 an angle in degrees, minutes, seconds? In context 3. 5. 53 seems to be a time, but again I'm not clear on the units.&lt;/p&gt;&#xA;&lt;p&gt;A secondary question is about his use of the moons of Jupiter to help determine the longitude. Was this a common technique? And how did (does) it work? It's amazing to me that using a late 18th century portable telescope he could not only view the moons but distinguish between the first and the third ones.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to the &lt;a href=&quot;https://catalog.hathitrust.org/Record/001971796&quot; rel=&quot;nofollow noreferrer&quot;&gt;full text&lt;/a&gt; (volume 2, page 284).&lt;/p&gt;&#xA;&lt;p&gt;[Edit]&lt;/p&gt;&#xA;&lt;p&gt;A reprint of Mackenzie's diary prints the table as&#xA;&lt;a href=&quot;https://i.stack.imgur.com/CuJb2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CuJb2.png&quot; alt=&quot;Reprinted screenshot&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Assuming this is correct (not exactly sure what m[h] and h[m] mean), I think it's more clear. Here's how I understand the procedure:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;He measures the altitude of sun (i.e. degrees above horizon) at 3h 5m 53 in the afternoon by his watch&lt;/li&gt;&#xA;&lt;li&gt;By a table, he knows that this altitude should occur 1h 22m 38s later (so his watch is slower than it should be)&lt;/li&gt;&#xA;&lt;li&gt;He'd done a similar measurement in the morning and computed 1h 21m 44s slow, so he takes the mean of these measurements to get 1h 22m 11s slow&lt;/li&gt;&#xA;&lt;li&gt;Difference of nine hours slow – I suppose this is to put his watch time back to GMT (he's near the west coast of North America), not sure why he's adding 8 to the seconds column&lt;/li&gt;&#xA;&lt;li&gt;He observes an emersion of Jupiter's third satellite (some time later, at night) and finds in a table the exact GMT time that it should occur at. From this (and his local time) he computes his longitude to be ~8d more than some previous reference, or ~128d west of Greenwich.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Is that close to right? Step 4 is a bit unclear to me, especially regarding what he's adding into the second column.&lt;/p&gt;&#xA;" OwnerUserId="16148" LastEditorUserId="16148" LastEditDate="2022-02-01T20:55:57.137" LastActivityDate="2022-02-01T22:04:48.010" Title="How was longitude determined in the 1700s?" Tags="|astronomy|geometry|navigation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14121" PostTypeId="2" ParentId="14119" CreationDate="2022-01-26T00:28:12.127" Score="3" Body="&lt;p&gt;In Enigma the wires followed from keyboard through rotors, then through plugboard, then back through the same rotors to lamps with letters. The resulting permutation would be (rotors)(plugboard)(rotors&lt;span class=&quot;math-container&quot;&gt;$^{-1}$&lt;/span&gt;). So you can't just swap A and B.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="12330" LastEditDate="2022-01-26T03:22:56.400" LastActivityDate="2022-01-26T03:22:56.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14122" PostTypeId="2" ParentId="14120" CreationDate="2022-01-26T02:02:32.517" Score="3" Body="&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;On the units. He mixes time and angle units. In the first sentence the altitudes are in angular degrees, minutes and seconds. The main calculation in hours, minutes and seconds of time. Then he converted the final result into angular units. For example,&#xA;&lt;span class=&quot;math-container&quot;&gt;$1^h22^m38^s$&lt;/span&gt; and the following columns are time units, while&#xA;&lt;span class=&quot;math-container&quot;&gt;$8^\circ 32'21$&lt;/span&gt; are angle units.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;On the method of determining longitude by Jupiter satellites. Determination of longitude is equivalent to determination of time difference between the local time and the time at some &amp;quot;zero meridian&amp;quot;. For the local time, many observations are available, in this example he uses Sun observations in the first line.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For the time at the zero meridian, three methods were available&#xA;in 1790. The text that you posted illustrates the oldest method based on timing  of appearance and disappearance of Jupiter satellites (the method was proposed by Galileo who discovered these satellites, and immediately understood this important application). These times were tabulated for the reference meridian. Since these events do not depend on the place of the observer on Earth, their observation permits to determine the time at the zero meridian. This was the first practical astronomical method of determining longitude.&lt;/p&gt;&#xA;&lt;p&gt;In the middle of 18 century two new, more convenient methods were introduced (chronometer and Lunar distances). Unlike Jupiter satellites, these new methods could be used at sea. But on land, Jupiter satellite could give better results.&lt;/p&gt;&#xA;&lt;p&gt;Remark. The method based on Jupiter satellites does not give good results at sea. (The main reason is that there is no way to stabilize a telescope on a moving ship.) So one cannot say that this was a &amp;quot;common method&amp;quot; at sea.  But it was common among astronomers/geographers, and remained in use until the middle of 19 century.&lt;/p&gt;&#xA;&lt;p&gt;Could you please give a precise reference on the text that you cite?&#xA;I suspect the observation was made on land, since in 18th century there was no instrument for&#xA;measuring angles at sea which could give a reading to angular seconds.&lt;/p&gt;&#xA;&lt;p&gt;Here is my interpretation of his specific calculation:&#xA;He (or his assistant) observed Sun's altitude &lt;span class=&quot;math-container&quot;&gt;$29^\circ23'48''$&lt;/span&gt; at &lt;span class=&quot;math-container&quot;&gt;$3^h5^m52^s$&lt;/span&gt; by his watch. This is actually the average of 5 observations in a very quick succession.&lt;/p&gt;&#xA;&lt;p&gt;From the tables (for given latitude, which he already knew&#xA;knew) he obtained that this altitude must be at &lt;span class=&quot;math-container&quot;&gt;$1^h22^m38^s$&lt;/span&gt; for the beginning&#xA;of his observations and &lt;span class=&quot;math-container&quot;&gt;$1^h21^m44^s$&lt;/span&gt; for the end (Sun's altitude changes quickly, his 5 observations apparently took about 1 minute, which is not unusual, if you have an assistant who records the time. Or 5 observations can be made almost simultaneously by&#xA;several people). All this part is called a &amp;quot;time sight&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Then he computes the arithmetic average of these two times&#xA;(adds them and divides by 2), and obtains &lt;span class=&quot;math-container&quot;&gt;$1^h22'11''$&lt;/span&gt;, and&#xA;corrects additionally for the error of his watch (8 sec) accumulated in 9 hours&#xA;from the time when he checked it by the above Sun observation&#xA;till the time of Jupiter satellites observation. This is not surprising since he observed Sun near noon, and Jupiter's satellites in the evening. (And his watch was apparently just an ordinary watch rather then chronometer).&lt;/p&gt;&#xA;&lt;p&gt;He does not tell the detail of Jupiter satellite observations,&#xA;but apparently he managed to time the moment of emergence&#xA;of the 3-d and 1-st satellites, and comparing this time with the tables obtained two longitude differences, of which he takes the average.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-02-01T22:04:48.010" LastActivityDate="2022-02-01T22:04:48.010" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14123" PostTypeId="2" ParentId="14103" CreationDate="2022-01-26T02:21:49.030" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;But the only Ancient Egyptian mathematician I know of is Ahmes, who said he's just a scribe.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Because in Ancient Egypt math was just a tool to solve real problems (ex: inheritance, accounting, inventory control, ...). Although math was highly developed there compared to other civilizations, there were no mathematician as a profession neither as a thinker.&lt;/p&gt;&#xA;&lt;p&gt;What existed there was some kind of training in math. Which can be seen in the only two original sources of math info about the Ancient Egypt:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Moscow Mathematical Papyrus&lt;/li&gt;&#xA;&lt;li&gt;Rhind Mathematical Papyrus&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Both are likely math exercises to train future scribes. Ahmes (btw, a common name in Egypt) just transcribed such math exercises from an older and lost source. And, based on the solutions of such exercises, it is possible to infer that there were no concept of mathematical proof at that time. As long as the answer was correct, it doesn't matter what steps were taken.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;AFAIK&lt;/strong&gt;, it required 1500 years for this mindset to change. The Ancient Greeks were actually the first ones to embrace logic and develop rigorous mathematical proofs. Thus allowing the emergence of the first mathematicians (ex: Thales of Miletus).&lt;/p&gt;&#xA;" OwnerUserId="16150" LastEditorUserId="16150" LastEditDate="2022-01-26T02:26:50.017" LastActivityDate="2022-01-26T02:26:50.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14124" PostTypeId="2" ParentId="14105" CreationDate="2022-01-26T02:47:38.793" Score="2" Body="&lt;p&gt;S.S. Chern resisted calling Chern classes by name for the longest time.  He eventually gave in with a comment to the effect that everyone calls them by that name, so he should start too.  I read the latter in one of his expository papers but cannot find it at the moment.  Chern was a very humble man.&lt;/p&gt;&#xA;" OwnerUserId="16151" LastActivityDate="2022-01-26T02:47:38.793" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14126" PostTypeId="2" ParentId="14120" CreationDate="2022-01-26T21:43:50.910" Score="3" Body="&lt;p&gt;The extract you reproduced involves the observation of eclipses of one or more of Jupiter's satellites (the 'Galilean' or 'Medicean' satellites).&lt;/p&gt;&#xA;&lt;p&gt;This was in origin an invention and improvement of Galileo and others during the 17th century. It was facilitated by successive tables of the motions and eclipses of Jupiter's satellites, initially by J-D Cassini in Paris in the 1660s, and improved through the 1690s and afterwards. It still remained in use through the 1700s (an example of eclipse tables from the 1690s in English and how to use them is &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1694.0048&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;During the 18th century, the improved tables of Wargentin were often used for Jupiter's satellites. A lecture about the satellite motions was given by W de Sitter in 1931 &lt;a href=&quot;https://adsabs.harvard.edu/full/1931MNRAS..91..706D&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The principle of the longitude method based on the eclipses was that local times of satellite eclipse were to be telescopically observed at times also astronomically determined.  These eclipse times were to be compared with times of predictions of the same eclipses at a standard meridian given by the special tables, usually times for a well-known observatory meridian. The result was a time-difference for one and the same eclipse as seen locally,  and as predicted for the standard meridian/observatory. From this result a longitude-difference for the same two places follows, in the proportion of 15 degrees for each 1 hour of time-difference. There is more description of the use of eclipses of Jupiter's satellites &lt;a href=&quot;https://www.rmg.co.uk/stories/blog/solving-longitude-jupiters-moons&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, and the de Sitter lecture already cited has information showing that the timing accuracy of satellite eclipse observations was intrinsically limited to no better than about +/- 10 seconds, a limitation that ensured the eventual obsolescence of the method.&lt;/p&gt;&#xA;&lt;p&gt;But the 1700s also make up a period of revolutionary changes in many methods of longitude determination, changes both of accuracy or of principle. A number of main stages are noticeable.&lt;/p&gt;&#xA;&lt;p&gt;In the early 1700s, traditional methods from the 17th century dominated. Then the mid-century, to about the 1760s, was a period of specially notable experiments: in measurements on land, by the initial development of accurate clocks (chronometers), and by the improvement of tables of lunar motion against the stars. In the later 1700s the new and improved methods largely took over from the old.&lt;/p&gt;&#xA;&lt;p&gt;A general view of traditional techniques for determining longitude to the mid-1700s can be seen for example from entries in the &lt;a href=&quot;https://digital.nls.uk/encyclopaedia-britannica/archive/144133900&quot; rel=&quot;nofollow noreferrer&quot;&gt;first (Edinburgh) edition of the Encyclopedia Britannica&lt;/a&gt;, produced 1768-1771 at a point when traditional methods especially at sea were just becoming effectively superseded by the new methods.&lt;/p&gt;&#xA;&lt;p&gt;(1) The early 1700s:&lt;/p&gt;&#xA;&lt;p&gt;On land, longitude could be determined astronomically with telescopes and tables of progressively improving accuracy -- some details of this already given above.&lt;/p&gt;&#xA;&lt;p&gt;At sea, techniques most used were still limited and traditional, often based on dead-reckoning with log lines when out of sight of land, and often with poor results and accidents -- that motivated public rewards for adequate methods of finding longitude at sea.&lt;/p&gt;&#xA;&lt;p&gt;(2) Mid-century, to the 1760s, saw much experimentation and increasing hopes of accurate longitude determination. On land, astronomical&#xA;techniques and tables for measuring differences of meridian were improving. For use at sea, a winnowing of numerous proposals took place. Some promising methods turned out infeasible. For example, the ship's motions made it impossible in practice to control a telescope so as to observe Jupiter's satellites, in spite of attempts at stabilisation, such as suspended observing chairs. Pendulum clocks, suggested for this use by Huygens in the 17th century,  turned out unsuitable for stable timekeeping at sea, because the ship's motion interfered with the motion of the pendulums. Eventually, only two methods still looked to have reasonable prospects:&lt;/p&gt;&#xA;&lt;p&gt;(a) Observation of the moon's positions against the stars, astronomically observed at local times also astronomically determined, at places reached on voyage. These lunar positions and times were to be compared with tabulated predictions of the same moon positions for the local times of a standard observatory meridian. Hence, by difference of times, the difference of longitude was obtained.&lt;/p&gt;&#xA;&lt;p&gt;Many astronomers produced tables of the moon, but the first to be successful and accurate enough were those of Tobias Mayer (1753), and their incrementally improved versions of 1762 onwards through the end of the century, expecially in the tables of Mason (also known for his part in surveying the 'Mason-Dixon line') and finally by Buerg (1806). The necessary calculations were unwieldy, but they were reduced to feasible size in stages. One of the pioneering advocates was Nicolas-Louis Lacaille, whose role is described for example in &lt;a href=&quot;https://link.springer.com/chapter/10.1057/9781137520647_4&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt; of Guy Boistel. The method was adapted for practical use at sea by Nevil Maskelyne in his book 'The British Mariner's Guide' (1763), and it was still further simplified or streamlined by the methods of the 'Nautical Almanac and Astronomical Ephemeris' (annually 1767 onwards) and its associated 'Tables requisite...'.&lt;/p&gt;&#xA;&lt;p&gt;(b) The other longitude method, which eventually superseded the lunar method, required adequately stable and regular clocks or watches (with balanced timing components, not pendulums) to keep the time of a standard observatory or port of departure. Again this was to be compared with local time determined on voyage by astronomical observation.  Here too, the difference between local time, astronomically observed, and the time (for the same physical instant) at the standard observatory or port, kept by chronometer, gave the difference of meridians. The development of suitable clocks and chronometers is described for example in &lt;a href=&quot;https://chsi.harvard.edu/publications/quest-for-longitude&quot; rel=&quot;nofollow noreferrer&quot;&gt;'The Quest for Longitude' (ed. W J H Andrewes, 1996)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the later 18th-century, from 1760s onwards, the lunar method became widely used, partly it seems, because the books and tables needed for it were printed in large quantities and thus were available and affordable. The chronometer method eventually caught up, when clockmakers had competed to produce accurate stable chronometers of simpler design (and larger quantity) than John Harrison's earliest and very complex pioneering single examples.  It took some further advances in horological technique before robust accurate instruments were made and supplied in large enough quantity to satisfy demand, and at affordable prices (see Andrewes (1996) cited above).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-01-26T22:54:21.310" LastActivityDate="2022-01-26T22:54:21.310" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14127" PostTypeId="2" ParentId="13852" CreationDate="2022-01-27T15:25:19.983" Score="0" Body="&lt;p&gt;In B. L. van der Waerden's article (linked in your own answer), &lt;a href=&quot;https://www.jstor.org/stable/2689449&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hamilton's Discovery of Quaternions&lt;/a&gt;, the history of Hamilton and quaternions is laid out clearly, based on Hamilton's publications and correspondence. In brief (although that article is already brief), first note that Hamilton &amp;quot;knew and used the geometric representation of complex numbers,&amp;quot; but in publications &amp;quot;he emphasized the definition of complex numbers as the couple &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; which followed definite rules for addition and multiplication.&amp;quot; He posed the problem:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;To find how number-triplets &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)$&lt;/span&gt; are to be multiplied in analogy to couples &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt;&lt;/em&gt;.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Van der Waerden continues:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In analogy to the complex numbers &lt;span class=&quot;math-container&quot;&gt;$(a + ib)$&lt;/span&gt; Hamilton wrote his triplets as &lt;span class=&quot;math-container&quot;&gt;$(a + bi + cj)$&lt;/span&gt;. He represented his unit vectors &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; as mutually perpendicular &amp;quot;directed segments&amp;quot; of unit length in space. Later Hamilton himself used the word vector, which I also shall use in the following. Hamilton then sought to represent products such as &lt;span class=&quot;math-container&quot;&gt;$(a + bi + cj)(x + yi + zj)$&lt;/span&gt; again as vectors in the same space. &lt;strong&gt;He required, first, that it be possible to multiply out term by term; and second, that the length of the product of the vectors be equal to the product of the lengths.&lt;/strong&gt; This latter rule was called the &amp;quot;law of the moduli&amp;quot; by Hamilton.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(emphasis mine)&lt;/p&gt;&#xA;&lt;p&gt;Which answers the question. The rest of the article explains how Hamilton determined the necessity for four dimensions instead of three based on these criteria, and how he arrived at the relations for quaternions.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2022-01-27T15:25:19.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14128" PostTypeId="1" CreationDate="2022-01-27T16:19:10.850" Score="0" ViewCount="82" Body="&lt;p&gt;In the Preface to the second edition to Spivak's &lt;em&gt;A Comprehensive Introduction to Differential Geometry&lt;/em&gt;, Vol. 2, on p.vii says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The material in this Volume covers about what I would have completed in my class this term, had political events not intervened.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What political events is he referring to? The preface is signed at Brandeis University, July 6, 1970. The comment is not included in the third edition's preface.&lt;/p&gt;&#xA;&lt;p&gt;I would prefer a factual account, although speculation is also welcome.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your time.&lt;/p&gt;&#xA;" OwnerUserId="1887" LastEditorUserId="10106" LastEditDate="2022-01-27T16:44:37.207" LastActivityDate="2022-01-27T17:01:59.553" Title="&quot;Political Events&quot; in the Preface to the Second Edition of Spivak's Comprehensive Introduction Volume 2" Tags="|mathematics|mathematicians|biographical-details|social-context|mathematics-social-history|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14129" PostTypeId="2" ParentId="14128" CreationDate="2022-01-27T17:01:59.553" Score="1" Body="&lt;p&gt;Here is a speculative answer: The Preface to the First Edition to Volume 1 included in the third edition (also of Volume 1) is signed at Brandeis University in March 1970; and in it there is no mention of political events. According to &lt;a href=&quot;https://en.wikipedia.org/wiki/1970&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/1970&lt;/a&gt;, my speculation is that Spivak is referring to the Vietnam War/ antiwar protests in the US.&lt;/p&gt;&#xA;" OwnerUserId="1887" LastActivityDate="2022-01-27T17:01:59.553" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14130" PostTypeId="1" AcceptedAnswerId="14131" CreationDate="2022-01-27T17:02:06.830" Score="2" ViewCount="113" Body="&lt;p&gt;In his paper &amp;quot;The geometry of Finsler spaces&amp;quot; (available at &lt;a href=&quot;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-56/issue-1.P1/The-geometry-of-Finsler-spaces/bams/1183514449.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-56/issue-1.P1/The-geometry-of-Finsler-spaces/bams/1183514449.full&lt;/a&gt;) Busemann quotes Poincaré (p. 8):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Poincaré said&#xA;somewhere&lt;span class=&quot;math-container&quot;&gt;$^5$&lt;/span&gt; on the subject of generalizations: On ne fait pas un&#xA;grand voyage pour ne trouver que ce qu'on a chez soi.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;with the footnote being&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^5$&lt;/span&gt; The author forgot where he saw this remark and was unable to locate it, but&#xA;would be grateful for a reference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A naive search online points to Busemann's paper. I have also done naive searches on digital copies of the standard Poincaré &amp;quot;philosophy&amp;quot; books to no avail. I second Busemann.&lt;/p&gt;&#xA;&lt;p&gt;The quote roughly translates to: &amp;quot;One does not make a big trip to find only what one has at home.&amp;quot; Certainly this is in character, as far as I can see.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your time.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As a disclaimer, I've asked this question on M.SE about six months ago (&lt;a href=&quot;https://math.stackexchange.com/q/4204323/169085&quot;&gt;https://math.stackexchange.com/q/4204323/169085&lt;/a&gt;), but the question did not get much traction.&lt;/p&gt;&#xA;" OwnerUserId="1887" LastEditorUserId="6545" LastEditDate="2022-01-27T22:55:59.450" LastActivityDate="2022-01-27T22:55:59.450" Title="Source of a Poincaré quote from a Busemann paper" Tags="|mathematics|mathematicians|reference-request|quote|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14131" PostTypeId="2" ParentId="14130" CreationDate="2022-01-27T18:31:13.970" Score="5" Body="&lt;p&gt;I think this is a paraphrase of a passage found in the seminal paper &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Analysis_Situs_(paper)&quot; rel=&quot;noreferrer&quot;&gt;Analysis Situs&lt;/a&gt;&amp;quot; (&lt;em&gt;Journal de l'École polytechnique&lt;/em&gt;, t.1, p. 1-121, 1895; you can find it in &lt;a href=&quot;https://www.maths.ed.ac.uk/%7Ev1ranick/papers/poincare.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Oeuvres d'Henri Poicaré&lt;/em&gt;, t. VI&lt;/a&gt;, the quote is at pages 193-194, i.e. 13-14 of the scan) where he says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Aussi y a-t-il des parties de l’Hypergéométrie auxquelles il n’y a pas lieu de beaucoup s’intéresser : telles sont, par exemple, les recherches sur la courbure des surfaces dans l’espace à n dimensions. On est sûr d’avance d’obtenir les mêmes résultats qu’en Géométrie ordinaire et &lt;strong&gt;l’on n’entreprend pas un long voyage pour retrouver des spectacles tout pareils à ceux que l’on rencontre chez soi.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are also parts of the Hypergeometry in which there is no need to be very interested: for example, research on the curvature of a surface in a n-dimensional space. One is sure in advance to obtain the same results as in ordinary Geometry and one does not undertake a long journey to find spectacles quite similar to those which one meets at home.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2022-01-27T18:31:13.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14132" PostTypeId="1" CreationDate="2022-01-27T19:32:11.637" Score="3" ViewCount="189" Body="&lt;p&gt;I am searching for an year which satisfies the following criteria.&lt;/p&gt;&#xA;&lt;p&gt;Dates are in &lt;strong&gt;Gregorian&lt;/strong&gt; calendar.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Criteria&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Year range 3800 BC to 500 BC&lt;/li&gt;&#xA;&lt;li&gt;Solar Eclipse of any kind between October 14 and October 22 inclusive.&lt;/li&gt;&#xA;&lt;li&gt;Lunar Eclipse of any kind between September 29 and October 9 inclusive.&lt;/li&gt;&#xA;&lt;li&gt;Eclipses visible from New Delhi, India, UTC/GMT +5:30 hours.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Optional Criteria&lt;/strong&gt;&lt;br /&gt;&#xA;The gap between the two eclipses must be less than 14 days.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Challenge&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The challenge mainly is because the dates are in Gregorian and not Julian which the online eclipse searchers(eg: &lt;a href=&quot;https://moonblink.info/Eclipse/search&quot; rel=&quot;nofollow noreferrer&quot;&gt;moonblink&lt;/a&gt;) use. I could not find an eclipse searching tool which uses Gregorian calendar. The problem with converting Gregorian to Julian and then searching is that the date changes as the year changes and we end up with varying dates over thousands of years making it difficult to search. How to solve this?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Feasibility&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;If we solve that problem, each day of the year has approximately 20 eclipses between 3800 BC and 500 BC. So the eight days from Oct 14 to 22 would approximately have 160 eclipses. Then back to back eclipses with Lunar eclipse followed by Solar eclipse are rare, so approximately 10 to 15 eclipses will fit the back to back eclipse criteria. From which using the 'visible from New Delhi, India UTC/GMT +5:30' criteria can be used to narrow down to couple of eclipses. The optional criteria can also be used if applicable.&lt;/p&gt;&#xA;&lt;p&gt;Edit: It seems I have wrongly estimated the number of eclipses fitting the criteria. The number of years satisfying the first three criteria seem to be around 120 to 140.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Context&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is an independent effort to date the Mahabharata War using Archeastronomy.&lt;/p&gt;&#xA;&lt;p&gt;In Mahabharata war, Bhishma is said to have left his mortal body on Winter Solistice which is Dec 21 or Dec 22 Gregorian. Which was also the first quarter of  waxing Moon of &lt;a href=&quot;https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSpmmU99b_mUyiDxMXu4Z9b8eOqMBKVnmL_Dw&amp;amp;usqp=CAU&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hindu lunar month&lt;/a&gt; Magha. But this Winter solistice observation was observed through naked eye, so researchers like Dr. Manish Pandit have opined that it was 4 or 5 days after Solistice. So it could be from Dec 21 to Dec 27. We also know that there was a Lunar eclipse on Karthika Full Moon and a Solar eclipse on Karthika New Moon. Which is approximately 83 days and 68 days from observed winter solistice. This gives us Solar Eclipse of any kind between October 14 and October 22 and a Lunar Eclipse of any kind between September 29 and October 9. They had to be visible from New Delhi, India.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit 1: Conversion of date range from Gregorian to Julian&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/75ANM.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/75ANM.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16159" LastEditorUserId="16159" LastEditDate="2022-01-29T16:35:56.067" LastActivityDate="2023-02-23T18:02:00.260" Title="Is it possible to narrow down possible dates for the following eclipse pattern?" Tags="|astronomy|archaeoastronomy|ancient-india|india|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14133" PostTypeId="2" ParentId="14102" CreationDate="2022-01-28T12:23:10.950" Score="3" Body="&lt;p&gt;See &lt;a href=&quot;https://plato.stanford.edu/entries/settheory-early/#ZermGode&quot; rel=&quot;nofollow noreferrer&quot;&gt;The development of Set Theory&lt;/a&gt; and &lt;a href=&quot;https://plato.stanford.edu/entries/zermelo-set-theory/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zermelo’s Axiomatization of Set Theory&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Ernst Zermelo, 1908 paper “Untersuchungen über die Grundlagen der Mengenlehre, I” (1908) and “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre” (1930);&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Abraham Fraenkel, &amp;quot;Über den Begriff &amp;quot;definit&amp;quot; und die Unabhängigkeit des Auswahlaxioms,&amp;quot; (1922) and &amp;quot;Untersuchungen über die Grundlagen der Mengenlehre&amp;quot; (1925).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2023-09-15T12:48:37.280" LastActivityDate="2023-09-15T12:48:37.280" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14134" PostTypeId="1" CreationDate="2022-01-28T19:47:27.810" Score="1" ViewCount="63" Body="&lt;p&gt;There are various documents stating that arithmetic was first seen in 2000 BC, but wondered whether if there were any signs of it before?&lt;/p&gt;&#xA;" OwnerUserId="16164" LastEditorUserId="55" LastEditDate="2022-01-28T20:17:40.467" LastActivityDate="2022-01-28T20:17:40.467" Title="Does arithmetic go back before 2000 BC" Tags="|mathematics|arithmetic|" AnswerCount="0" CommentCount="1" ClosedDate="2022-01-29T15:15:15.417" ContentLicense="CC BY-SA 4.0" />
  <row Id="14135" PostTypeId="5" CreationDate="2022-01-28T23:28:21.703" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-01-28T23:28:21.703" LastActivityDate="2022-01-28T23:28:21.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14136" PostTypeId="4" CreationDate="2022-01-28T23:28:21.703" Score="0" Body="The study of the presence and flow of electric charge. Charges, currents, fields, potentials. It has also sub-topics such as voltage, current and resistance." OwnerUserId="217" LastEditorUserId="217" LastEditDate="2022-01-29T15:15:40.023" LastActivityDate="2022-01-29T15:15:40.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14137" PostTypeId="2" ParentId="5340" CreationDate="2022-01-29T07:40:07.530" Score="0" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;I'm wondering whether there are any cases in linguistics (or related fields like Cognitive Sciences) where this debate is being used for harm?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The subject of nature vs nurture is a hot debate and is very divisive in many scientific and non-scientific communities.&lt;/p&gt;&#xA;&lt;h2&gt;Where there is prejudice, there can be harm on both sides of the debate&lt;/h2&gt;&#xA;&lt;p&gt;In cognitive sciences, I can think of 2 examples straight away.&lt;/p&gt;&#xA;&lt;h2&gt;Good and evil inborn or developed?&lt;/h2&gt;&#xA;&lt;p&gt;The debate on whether good and evil is inborn or down to upbringing or other environmental issues has been raging for decades. The research by Philip Zimbardo, Stanley Milgram, Soloman Asch &lt;em&gt;et al&lt;/em&gt;. shows that. It can also be seen to be harmful on both sides or the debate.&lt;/p&gt;&#xA;&lt;p&gt;Sexual abuse of a child is an evil that most people want to eradicate. This in turn has led to a fight against paedophilia, hebephilia etc. when it is debated that &lt;a href=&quot;https://psychology.stackexchange.com/q/587/7604&quot;&gt;paedophilia etc. are sexual orientations which may not lead to sexual abuse of a child&lt;/a&gt;. After all, male rape of men and boys can have nothing to do with homosexual orientation (&lt;a href=&quot;https://lgbpsychology.org/html/facts_molestation.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;one example of research outcomes&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;h3&gt;If it is inborn&lt;/h3&gt;&#xA;&lt;p&gt;If it's inborn, what do you do about the evil people born that way? They can't help it after all. Do you put an extremely close 24/7/365 watch on them in order to step on the moment they are about to do something evil?&lt;/p&gt;&#xA;&lt;p&gt;Maybe you look to eugenics to eliminate the problem by aborting future pregnancies which could lead to an evil person being born. But that is not fair on the unborn child as it may not be 100% accurate and the child could turn out to be a Mother Teresa or Mahatma Gandhi incarnate.&lt;/p&gt;&#xA;&lt;p&gt;Maybe you could sterilise evil people so the gene pool for evil people is weakened. But that is unfair on the potential mother of a child who again, could turn out to be another Mother Teresa or Mahatma Gandhi.&lt;/p&gt;&#xA;&lt;h3&gt;If it is nurture&lt;/h3&gt;&#xA;&lt;p&gt;Do you put aside your indifference toward evil people, admonish those who caused it (mothers, fathers, school teachers, politicians, influences on social media...) and support the evildoers toward going the other way, or do you punush both and leave the admonishment to correct their ways? After all, it costs a lot of money to imprison an individual and the finances are finite.&lt;/p&gt;&#xA;&lt;h2&gt;Homosexuality inborn or influenced?&lt;/h2&gt;&#xA;&lt;h3&gt;If inborn (which is where science has swung toward)&lt;/h3&gt;&#xA;&lt;p&gt;There has been, and continues to be, incidents of fear of the unknown where straight people worry there will be issues from homosexual people coming on to or attacking heterosexual people. So, this fear induces violence toward them.&lt;/p&gt;&#xA;&lt;p&gt;There has been a search for &amp;quot;the gay gene&amp;quot; and that has caused controversy as &lt;a href=&quot;https://www.newscientist.com/article/mg22429963-700-gay-gene-discovery-has-good-and-bad-implications/&quot; rel=&quot;nofollow noreferrer&quot;&gt;that has good and bad implications&lt;/a&gt; and &lt;a href=&quot;https://www.the-scientist.com/news-opinion/scientists-seek-to-kill-genetic-test-for-same-sex-attraction--66591&quot; rel=&quot;nofollow noreferrer&quot;&gt;scientists have sought to &amp;quot;kill&amp;quot; a particular test for homosexuality&lt;/a&gt;. There have been calls for eugenics to step in to eliminate &amp;quot;the abomination&amp;quot;. Do we try to remove homosexuality through genetics? That is actually a debate in some circles.&lt;/p&gt;&#xA;&lt;p&gt;Do we prevent gay men from donating sperm? Do we prevent them from fathering a child through surrogacy? Do we use &lt;a href=&quot;https://en.wikipedia.org/wiki/Conversion_therapy&quot; rel=&quot;nofollow noreferrer&quot;&gt;conversion therapy&lt;/a&gt; to stop them acting on their urges? Do we try to develop &lt;a href=&quot;https://blogs.scientificamerican.com/cross-check/bizarre-brain-implant-experiment-sought-to-cure-homosexuality/&quot; rel=&quot;nofollow noreferrer&quot;&gt;brain implants to &amp;quot;cure&amp;quot; it&lt;/a&gt;?... The list can go on. After all,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Homosexuality has been [attempted to be] treated with lobotomies, chemical castration, electrical shocks and nausea-inducing drugs as well as psychotherapy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h3&gt;If there are environmental influences&lt;/h3&gt;&#xA;&lt;p&gt;Again, fear of the unknown steps in. Can a straight person be made to be gay by another gay person? Should we prevent gay people socialising with heterosexual people? Should we just beat it out of them?&lt;/p&gt;&#xA;&lt;p&gt;Should we perform lobotomies, chemical castration, electrical shocks and nausea-inducing drugs as well as psychotherapy?&lt;/p&gt;&#xA;" OwnerUserId="16003" LastActivityDate="2022-01-29T07:40:07.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14138" PostTypeId="2" ParentId="14132" CreationDate="2022-01-29T16:06:01.073" Score="1" Body="&lt;p&gt;Since you write that your main problem is conversion of the dates from Gregorian to Julian calendar, let me give reference for algorithms.&lt;/p&gt;&#xA;&lt;p&gt;Jean Meus, Astronomical algorithms, William-Bell Inc., Richmond Virginia, 1998, Chapter 7. It also contains the algorithms for Jewish and Muslim calendars.&lt;/p&gt;&#xA;&lt;p&gt;There is also a classical book, on this subject:&lt;/p&gt;&#xA;&lt;p&gt;E. J. Bickerman,  Chronology of the ancient world, London, 1969. It has conversion tables for all known calendars used in Europe and Middle East.&lt;/p&gt;&#xA;&lt;p&gt;There is also a recent book &lt;a href=&quot;https://en.wikipedia.org/wiki/Calendrical_Calculations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calendarical calculations&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-01-29T16:27:11.520" LastActivityDate="2022-01-29T16:27:11.520" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14139" PostTypeId="1" AcceptedAnswerId="14146" CreationDate="2022-01-30T01:05:33.830" Score="1" ViewCount="414" Body="&lt;p&gt;I recently read an article that discussed--rather briefly--the issues of infinitesimals and the criticism of them by Berkeley. The author of the article (which, of course, I cannot find, as I read it on my phone) claimed that Berkeley's criticism led to a bunch of revisions to calculus textbooks, including authors adding a bit more rigor--and that those changes and additions made calculus harder to teach, learn, and understand.&lt;/p&gt;&#xA;&lt;p&gt;The author shows a derivation for the derivative of &lt;span class=&quot;math-container&quot;&gt;$y = x^2 + x$&lt;/span&gt; using an approximation of non-standard analysis. We define the average slope of the curve over an interval as &lt;span class=&quot;math-container&quot;&gt;$m = \frac{y_1-y_0}{x_1-x_0}$&lt;/span&gt;. Then, we shrink the size of the interval such that &lt;span class=&quot;math-container&quot;&gt;$y_1-y_0 = dy$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_1-x_0 = dx$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dy$&lt;/span&gt; are infinitesimals--values so close to zero that they might as well be zero. Then plugging &lt;span class=&quot;math-container&quot;&gt;$y+dy$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x+dx$&lt;/span&gt; into the original function, we get:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align*}&#xA;y+dy &amp;amp;= (x+dx)^2 + x+dx \\&#xA;y+dy &amp;amp;= x^2+2xdx+dx^2 + x + dx \\&#xA;dy &amp;amp;= 2xdx+dx^2+dx \\&#xA;\frac{dy}{dx} &amp;amp;= 2x + dx+1 = 2x+1&#xA;\end{align*}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and since the value of &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt; is so close to zero that it might as well be zero, we handwave it away to get &lt;span class=&quot;math-container&quot;&gt;$y' = 2x+1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Though Newton and Leibniz had attempted to apply some rigor to the creation of infinitesimals, they mostly handwaved it away as in the above example, and Berkeley's criticism of that perhaps justified (even if the manner of critique was a bit ridiculous). But the critique was also seen as a personal affront not just to Newton (maybe) and Leibniz, but to all those working with the newly-created system of calculus. Revisions were made, eventually ending up with the Weierstrass &lt;span class=&quot;math-container&quot;&gt;$\delta / \varepsilon$&lt;/span&gt; formulation that we have at the beginnings of modern calculus textbooks... where students become readily confused.&lt;/p&gt;&#xA;&lt;p&gt;Apparently there have been A/B pedagogical studies showing that non-standard analysis leads to better learning outcomes than traditional calculus--but even those methods involve spending a chapter defining hyperreal/surreal numbers in order to produce a &amp;quot;rigorous&amp;quot; system.&lt;/p&gt;&#xA;&lt;p&gt;I argue that students learning calculus for the first time don't &lt;em&gt;need&lt;/em&gt; a rigorous treatment, whether hyperreals, or &lt;span class=&quot;math-container&quot;&gt;$\delta / \varepsilon$&lt;/span&gt;, or something else. I saw the derivation above yesterday and thought &amp;quot;Wow, that would have made calculus make sense a lot faster. That's just algebra with some added bits and bobs.&amp;quot; (I took my first calculus course nearly 30 years ago; the idea of it being algebra with bits added shocked me with its simplicity.)&lt;/p&gt;&#xA;&lt;p&gt;HS juniors/seniors/college freshman don't &lt;em&gt;need&lt;/em&gt; the analysis to be justified logically or in a mathematically rigorous way in order to perform calculus--and the vast majority of those taking calculus are not going on to take advanced mathematics courses where such rigor becomes important. I mean... we teach arithmetic from kindergarten on, but at no point does anyone other than a math major (or a self-taught adult with odd interests) learn about justifying arithmetic with set theory. We don't teach students an axiomatic formalization of elementary algebra--we present them with axioms or identities directly and don't worry about how we know they're true.&lt;/p&gt;&#xA;&lt;p&gt;If all we need to teach differentiation is to define infinitesimals as handwaveable quantities, then use the basic slope formula to find differentials, do we really need the formalization of &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\varepsilon$&lt;/span&gt;, and of limits?&lt;/p&gt;&#xA;&lt;p&gt;I know I've babbled on a bit here. To summarize:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is the authors' premise that calculus would be easier if we dumped the formalism correct/reasonable?&lt;/li&gt;&#xA;&lt;li&gt;If we were to drop the formalism, would there be any downsides for non-math majors?&lt;/li&gt;&#xA;&lt;li&gt;Was it really Berkeley's critique that caused this shift, or would it have happened anyway?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="15860" LastEditorUserId="10106" LastEditDate="2022-02-01T03:43:33.513" LastActivityDate="2022-02-01T03:43:33.513" Title="Did Berkeley's criticism of infinitesimals hobble calculus pedagogy?" Tags="|mathematics|mathematicians|calculus|real-analysis|education|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14140" PostTypeId="1" CreationDate="2022-01-30T02:05:43.777" Score="2" ViewCount="345" Body="&lt;p&gt;I came across the following article on the &amp;quot;square root of two&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ytLlB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ytLlB.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Based on this, I had the following questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Does anyone know why the Babylonians were interested in the square root of two? Perhaps there might have been some application related to engineering or measuring materials that required them to calculate the square root of two?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And does anyone know how exactly the above number approximation was calculated? How did they decide to add 24/60, 51/60^2, 10/60^3 etc.?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Reference:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Square_root_of_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Square_root_of_2&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16167" LastEditorUserId="16167" LastEditDate="2022-01-30T04:41:56.740" LastActivityDate="2022-01-30T04:41:56.740" Title="History of the &quot;Square Root of 2&quot;" Tags="|mathematics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14141" PostTypeId="5" CreationDate="2022-01-30T14:22:17.887" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-01-30T14:22:17.887" LastActivityDate="2022-01-30T14:22:17.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14142" PostTypeId="4" CreationDate="2022-01-30T14:22:17.887" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-01-30T14:22:17.887" LastActivityDate="2022-01-30T14:22:17.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14143" PostTypeId="2" ParentId="5221" CreationDate="2022-01-30T18:09:57.980" Score="3" Body="&lt;p&gt;In the &amp;quot;&lt;a href=&quot;https://webusers.imj-prg.fr/%7Eleila.schneps/grothendieckcircle/EsquisseThem.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Esquisse Thématique des Principaux Travaux Mathématique&lt;/a&gt; 4.a Algèbre catégorique&amp;quot;, Grothendieck says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;En fait, de facon continuelle depuis 1953, je me suis senti dans l'obligation, au fur et à mesure des besoins, de développer une panoplie catégorique toujours insuffisante.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15350" LastActivityDate="2022-01-30T18:09:57.980" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14144" PostTypeId="1" CreationDate="2022-01-31T14:31:28.607" Score="2" ViewCount="290" Body="&lt;h3&gt;Background&lt;/h3&gt;&#xA;&lt;p&gt;It is well known that Kolmogorov published in 1933 his ground-breaking work on probability theory &lt;em&gt;Grundbegriffe der Wahrscheinlichkeitsrechnung&lt;/em&gt; usually known to English speakers as &lt;em&gt;Foundations of the theory of probability&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Numerous sources also claim that much of his mathematical discussion and work took place in his dacha at Komarovka a small village outside Moscow.&lt;/p&gt;&#xA;&lt;h3&gt;My question&lt;/h3&gt;&#xA;&lt;p&gt;I am sure I remember reading many years ago that he was inspired to write the monograph because the roof of his dacha needed repairing. This is such a strange story that I cannot imagine I made it up but it may, of course, be an urban legend. Can anyone confirm this either way?&lt;/p&gt;&#xA;&lt;h3&gt;My research&lt;/h3&gt;&#xA;&lt;p&gt;I have done internet searches for Kolmogorov dacha, and for Kolmogorov Komarovka without finding anything which adds much to what I summarise above. I also tried Kolmogorow Datsche in case there was anything in German, the language of the monograph. I am afraid my grasp of Russian is non-existent. Adding roof to the search or Dach to the searches just adds people anxious to quote me for the repair of my roof.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastEditorUserId="4251" LastEditDate="2022-01-31T14:51:20.577" LastActivityDate="2022-10-29T18:07:31.023" Title="Was Kolmorogov really inspired to publish Foundations of the theory of probability because he needed money to repair his dacha?" Tags="|mathematics|mathematicians|biographical-details|probability|kolmogorov|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14145" PostTypeId="2" ParentId="13203" CreationDate="2022-01-31T17:22:23.017" Score="1" Body="&lt;h2&gt;How did Hadley do it?&lt;/h2&gt;&#xA;&lt;p&gt;John Hadley published &amp;quot;The method of casting, grinding, and polishing the specula of reflecting telescopes&amp;quot; which probably has a good description of how he did it. Unfortunately I couldn't find that document. Luckily John Mudge published &amp;quot;&lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rstl.1777.0017&quot; rel=&quot;nofollow noreferrer&quot;&gt;Directions for making the best composition for the metals of reflecting telescopes; together with a description of the process for grinding, polishing, and giving the great speculum the true parabolic curve.&lt;/a&gt;&amp;quot; in 1777 which gives an in-depth step by step process of how a mid 18th century astronomer would make a parabolic mirror.&lt;/p&gt;&#xA;&lt;p&gt;Step 1: Make your speculum blank.&lt;/p&gt;&#xA;&lt;p&gt;Speculum is a alloy of copper an tin that could be polished to a high mirrored surface. Its what telescopes at the time were made of.&lt;/p&gt;&#xA;&lt;p&gt;Step 2: Make a spherical mirror.&lt;/p&gt;&#xA;&lt;p&gt;This was hand ground and done the same way we do it today with ceramic chips imbedded in a blank, and a lot of hand grinding.&lt;/p&gt;&#xA;&lt;p&gt;Step 3: Make it parabolic.&lt;/p&gt;&#xA;&lt;p&gt;Starting on page 38 of the linked PDF he gives his procedure. His explantion is,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In order to give the speculum the last and finishing figure, which is done by a few strokes it must be particularly remarked, that by working the metal round and round, the sphere of the polisher by this means growing less, it wears faster in the middle: and as a segment of a sphere may become parabolic, by means of opening the extremes gradually from within outward, so it may be equally well done by increasing the curvature in the middle, in certain ratio, from without inwards.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Normally when making a telescope mirror random strokes are made, and when you've done this a lot they average out into a spherical shape cut out of a blank. You can actually mess it up by grinding in a circular path which will make your mirror more parabolic. His method seems to be to take a spherical mirror, and then start in the middle polishing in a circle. This turns it into a parabolic mirror.&lt;/p&gt;&#xA;&lt;h2&gt;Why exactly?&lt;/h2&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Parabolic_reflector&quot; rel=&quot;nofollow noreferrer&quot;&gt;It had been know&lt;/a&gt; since antiquity that a parabola focuses parallel rays onto the foci. It was James Gregory in 1663 who predicted that a parabolic mirror would correct spherical aberration and chromatic aberration.&lt;/p&gt;&#xA;&lt;h2&gt;How did he verify its asphericity?&lt;/h2&gt;&#xA;&lt;p&gt;Assuming he started with a spherical mirror that then was ground further. The mirror couldn't be spherical. Correct me if I'm wrong, but manipulating a sphirical mirror in this way, and have it still produce an image, could only result in a parabolic shape.&lt;/p&gt;&#xA;&lt;p&gt;The only test in the above document was to test separate parts of the mirror to determine image quality. He would set up the mirror into a telescope. Then he would  cover the aperture. First he would only allow light to hit the center radius to see if it produced a good image. Then he would move to the middle covering the center and the outer radius to see if he produced a good image. Finally doing the same with the outer radius.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nDQJc.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nDQJc.png&quot; alt=&quot;The apparatus used.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;For what kind of telescope was this work done?&lt;/h2&gt;&#xA;&lt;p&gt;He used it in a parabolic Newtonian in 1721, and later a true Gregorian telescope.&lt;/p&gt;&#xA;" OwnerUserId="4880" LastEditorUserId="4880" LastEditDate="2022-01-31T22:06:02.397" LastActivityDate="2022-01-31T22:06:02.397" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14146" PostTypeId="2" ParentId="14139" CreationDate="2022-01-31T19:46:17.830" Score="3" Body="&lt;p&gt;There are four questions here (including the one in the heading) that appear to involve aspects of the history of calculus methods. For clarity in case of future amendment, the questions as they are addressed here are:&lt;/p&gt;&#xA;&lt;p&gt;(Heading) Did Berkeley's criticism of infinitesimals hobble calculus pedagogy?&lt;/p&gt;&#xA;&lt;p&gt;(This heading-question was also amplified/explained in the text: [Did] Berkeley's criticism [lead] to ... revisions to calculus textbooks, including ... more rigor -- and [did] those changes and additions [make] calculus harder to teach, learn, and understand.)&lt;/p&gt;&#xA;&lt;p&gt;(1) Is the authors' premise that calculus would be easier if we dumped the formalism correct/reasonable?&lt;/p&gt;&#xA;&lt;p&gt;(2) If we were to drop the formalism, would there be any downsides for non-math majors?&lt;/p&gt;&#xA;&lt;p&gt;(3) Was it really Berkeley's critique that caused this shift, or would it have happened anyway?&lt;/p&gt;&#xA;&lt;p&gt;First, about questions (Heading) and (3):-&lt;/p&gt;&#xA;&lt;p&gt;Berkeley's (1685–1753) criticism (in &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Analyst&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Analyst; or a Discourse Addressed to an Infidel Mathematician&amp;quot; (1734)&lt;/a&gt;), which targeted Leibnitz's infinitesimal calculus in L'Hopital's version of 1696, was the third of three widely-known critiques of calculus-like methods that appeared from about the mid-17th to the mid-18th century.  All three critiques had a similar central point: to condemn the ambiguity or contradiction said to arise from inconsistent treatments of infinitesimals (or earlier, indivisibles). For the critics, infinitesimals seemed to be treated as non-zero for one purpose and then as zero for another purpose in the same demonstration.  This was offensive to those who held to an early mathematical ideal, which was to deliver certain knowledge through proofs that were also certain and indubitable. Ambiguities and contradictions were repugnant to this ideal.&lt;/p&gt;&#xA;&lt;p&gt;On the first occasion of such critiques, in the mid-17th century, the target was the 'method of indivisibles' proposed by Bonaventura Cavalieri (c.1598–1647) in 'Geometria indivisibilibus continuorum ...' (1635), and then, with a reponse to criticisms, in 'Exercitationes geometricae sex' (1647). Critics here were Paul Guldin (1577-1643), in vol.4 of his &amp;quot;Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae&amp;quot; (1635-1641), and André Tacquet (1612-1660). Cavalieri's method remained under a cloud of doubt and disrepute even though it was acknowledged (e.g. by Torricelli and Newton) to yield many useful results.&lt;/p&gt;&#xA;&lt;p&gt;On the second occasion, at the end of the 17th century and at the turn of the 18th, the main critic was Michel Rolle (1652-1719), who vigorously attacked, especially, the Leibnizian infinitesimal methods published by L'Hopital in 1696 in &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-l-hospital-s-analyse-des-infiniment-petits&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Analyse des infiniment petits'&lt;/a&gt;. His attacks have already been discussed in this forum, in &lt;a href=&quot;https://hsm.stackexchange.com/questions/7988/did-michel-rolle-say-that-the-calculus-is-a-collection-of-ingenious-fallacies/7989#7989&quot;&gt;Did Michel Rolle say that the calculus is &amp;quot;a collection of ingenious fallacies&amp;quot;?&lt;/a&gt;. On that second occasion, the attacks were conducted in public in Paris and were defended among others by Pierre Varignon, whose arguments were notable for their appeal to Newton's method of limits (the 'method of first and last ratios' given in Book 1 section 1 of the 'Principia', see below.)&lt;/p&gt;&#xA;&lt;p&gt;Berkeley's 1734 version of the critique did have effects on the writing of at least one textbook: &lt;a href=&quot;https://archive.org/details/atreatisefluxio00conggoog/&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Treatise of Fluxions' (1742)&lt;/a&gt; by Colin Maclaurin (1698-1746). This book was regarded by Jean-Étienne Montucla (1725-1799) in his &lt;a href=&quot;https://books.google.com/books?id=3_YKVLHHEEQC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Histoire des Mathématiques&lt;/a&gt; (2nd ed., vol.3), as an effective rebuttal of Berkeley's criticism and as a good defence of the calculus methods of both Newton and Leibnitz.&#xA;Montucla's 'Histoire' gave this special appreciation of Maclaurin's defence of the calculus (in passages from pp.116 and 118, here in my translation):--&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Nobody is unaware these days that the infinitesimal calculus is in&#xA;its fundamentals absolutely the same as what Newton has called the&#xA;calculus of fluxions.  Now, this latter has nothing that is not in&#xA;conformity with the most rigorous principles of Geometry, as has been&#xA;shown at full length. Accordingly, both the one and the other must&#xA;enjoy the same degree of certainty.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;[...] &amp;quot;It was seemingly in answer to the attacks of Berkeley that Mr&#xA;Maclaurin undertook his 'Treatise of Fluxions' that appeared in 1742.&#xA;There, the method of Newton is fully demonstrated without any&#xA;assumption of infinitesimals or anything else capable of lending&#xA;itself to controversy ...  Mr Maclaurin's demonstrations are of&#xA;prodigious length ... he could have limited himself to some examples&#xA;... . [But] however that may be, one can say that if any doubts could&#xA;remain about the solidity of Newton's method, they are entirely&#xA;dissipated by this work of Maclaurin ... .&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So one answer to questions (Header) and (3) can be this: that Berkeley's criticism, while of immediate influence, may not have been determinative of the course of calculus criticism and exposition, since other criticisms of similar kind had already been published, and were 'in the air' and known widely, even if not well, before him. On the other hand, Maclaurin's response to Berkeley was very lengthy and not easy to understand.&lt;/p&gt;&#xA;&lt;p&gt;In response to question (2):&lt;/p&gt;&#xA;&lt;p&gt;The controversies about calculus-related methods have mainly concerned their justificatory foundations, with much less impact on the careful practice even of some methods impugned by the critiques.&lt;/p&gt;&#xA;&lt;p&gt;For example, in relation to the method of indivisibles, Cavalieri's successor Evangelista Torricelli (1608-1647), &amp;quot;worked with indivisibles, but he cautioned that their uncritical use could lead to paradoxes&amp;quot; (see &lt;a href=&quot;http://deti-bilingual.com/wp-content/uploads/2014/06/3rd-Edition-Victor-J.-Katz-A-History-of-Mathematics-Pearson-2008.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;A History of Mathematics&amp;quot; (3rd Edition, 2008) V J Katz&lt;/a&gt;). Thus, it was probably in cautious recognition of shaky foundations for the method of indivisibles, that Torricelli took precaution after obtaining by their use  -- and doubting -- a surprising result, namely, that an infinitely long solid of revolution could nevertheless have a finite volume. Thus, he found through indivisibles that the infinitely-long solid formed by rotating a hyperbola xy = k^2 around the y axis from y = a to y =∞ nevertheless has finite volume.  Torricelli then quieted doubts and confirmed the result by presenting a second derivation, relying on the indubitable classical method of exhaustion.&lt;/p&gt;&#xA;&lt;p&gt;Likewise, Newton in his 'Principia' declined to rely on the dubious justification of the method of indivisibles for his calculus methods, even though he stated specifically that 'hereby the same thing is perform'd as by the method of indivisibles' (&lt;a href=&quot;https://books.google.com/books?id=Tm0FAAAAQAAJ&amp;amp;pg=PA54&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton 'Principia', 1729 translation, vol.1, Book 1, sec.1&lt;/a&gt;). He gave reasons for setting out his method of limits ('first and last ratios') :-&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;These lemmas are premised, to avoid the tediousness of deducing perplexed demonstrations &lt;em&gt;ad absurdum&lt;/em&gt;, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles : But because the hypothesis of indivisibles ... is reckoned less geometrical; I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratio's of nascent and evanescent quantities, that is, to the limits of those sums and ratio's; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is perform'd as by the method of indivisibles; and now those principles being demonstrated, we may use them with more safety.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So a possible answer to question (2) would seem to be this, that anything that only expresses later work on justificatory foundations for calculus-like methods might be disregarded in practical use (leaving the user in good historical company) -- provided that any shortcuts are applied with care inside the limits of well-established practice for their use.&lt;/p&gt;&#xA;&lt;p&gt;(In view of the above, this answer does not detail 19th-century work on foundations of calculus, especially by Cauchy and Weierstrass: it seems enough for present purposes to limit the discussion to the 17th- and 18th-c. history which shows the main lines on which controversies developed. Again, the 20th-century non-standard analysis appears to concern  justificatory foundations rather than everyday practice.)&lt;/p&gt;&#xA;&lt;p&gt;(As for question (1), the questioner could not identify the author or the details of the author's suggestion: nothing can be added about that here.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-01-31T19:46:17.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14147" PostTypeId="2" ParentId="14139" CreationDate="2022-02-01T02:08:09.447" Score="1" Body="&lt;p&gt;Viewing the development of mathematics since the (abstractly justifiable) criticism of the &amp;quot;infinitesimal calculus&amp;quot;: no, those criticisms did not seem to inhibit people from continuing to use ideas of calculus in many ways, but/and &lt;em&gt;did&lt;/em&gt; stimulate some people to worry more about how to make all those got-the-right-answer methods meet a better... or at least more explicit... standard of reasoning.&lt;/p&gt;&#xA;&lt;p&gt;Comparable issues arose with Green-Heaviside-Dirac-et-al use of &amp;quot;generalized functions&amp;quot;, to obtain demonstrably correct answers to physics-y questions, for 100+ years before L. Schwartz (and then A. Grothendieck) made it all completely rigorous. Did not seem to inhibit physicists, nor physics students today, but did &amp;quot;trouble&amp;quot; mathematicians for quite a while. :)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-02-01T02:08:09.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14148" PostTypeId="1" CreationDate="2022-02-01T10:18:13.007" Score="1" ViewCount="96" Body="&lt;p&gt;There is a rather famous quotation of Cantor:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematics the art of proposing a question must be held of higher value than solving it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It was in a thesis he defended for his doctoral examination (according to &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/quotations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The question is: &lt;em&gt;How&lt;/em&gt; did Cantor defend this thesis? Is there anything written by him about this subject?&lt;/p&gt;&#xA;" OwnerUserId="16177" LastEditorUserId="10106" LastEditDate="2022-02-01T10:33:32.220" LastActivityDate="2022-02-01T10:33:32.220" Title="Cantor's Art of proposing a question" Tags="|mathematics|mathematicians|cantor|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14149" PostTypeId="2" ParentId="14144" CreationDate="2022-02-01T13:25:39.463" Score="2" Body="&lt;p&gt;User kimchi lover noted in a comment that this can be answered from the article&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://homepages.cwi.nl/%7Epaulv/papers/kolmogorov.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://homepages.cwi.nl/~paulv/papers/kolmogorov.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The relevant text is&#xA;&amp;quot;In 1935, Kolmogorov and Aleksandrov acquired (initially part of) an old manor house at Komarovka [the former Stanislavskii dacha], with room for a large library and several guests. &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Authored by Paul M. B. Vitányi and translated by Harold H Mc Fadden the article entitled &amp;quot;Remembering Kolmogorov&amp;quot; was published in Metascience (2011) 20:509–511 DOI 10.1007/s11016-011-9540-6. In it we find that Kolmogorov and his friend Aleksandrov only purchased (part of) the dacha in 1935 so obviously my memory must be a false one since by then &lt;em&gt;Grundbegriffe der Wahrscheinlichkeitsrechnung&lt;/em&gt; was already two years in the past.&lt;/p&gt;&#xA;&lt;p&gt;User Rodrigo de Azevedo also pointed out this fact.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastEditorUserId="4251" LastEditDate="2022-02-01T15:30:23.887" LastActivityDate="2022-02-01T15:30:23.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14150" PostTypeId="1" CreationDate="2022-02-01T15:46:52.603" Score="1" ViewCount="244" Body="&lt;p&gt;Could you please give me some references about the history of difference equations.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your reply&lt;/p&gt;&#xA;" OwnerUserId="16178" LastEditorUserId="12330" LastEditDate="2022-02-01T18:05:35.110" LastActivityDate="2022-02-02T00:25:58.207" Title="History of difference equations" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14151" PostTypeId="2" ParentId="13581" CreationDate="2022-02-01T22:32:24.903" Score="6" Body="&lt;p&gt;In Newton's time, it was not general practice to define physical units. There were in general no equations, no constants of proportionality.&lt;/p&gt;&#xA;&lt;p&gt;This seems to be explained in terms of the practice of expressing physical relations by proportion, so that -- if one thinks in terms of modern analogies -- any constants of proportionality just cancel out. A previous topic of discussion illustrates this, I think. So I would suggest looking at &lt;a href=&quot;https://hsm.stackexchange.com/questions/7288/who-first-derived-a-v2-r/7289#7289&quot;&gt;Who first derived $a =v^2/r$&lt;/a&gt; , especially at part (b)(3) of the answer. This shows an English translation of Huygens' description of various physical relationships, all having to do with centrifugal force (the 5th paragraph of (b)(3) is the start of the translation). The relationships are expressed in terms of proportions, not equations, and the extract shows how the style used then did not require units nor constants of proportionality. I hope that helps.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-02-01T22:32:24.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14152" PostTypeId="2" ParentId="14150" CreationDate="2022-02-02T00:25:58.207" Score="3" Body="&lt;p&gt;This is a difficult question to answer, but an excellent question nonetheless. This answer is offered presuming an interest regarding the inception and historical development of difference equations.  As one becomes aware in researching such an history, greater interest was given in the application of the methods but not so much in their history and development. This was probably due largely by the isolation of individuals from others of similar interests, except within their own academic community, such as at a university.  Outside communication was by letter, or by travel to scientific meetings. Nevertheless, people compiled their notes into books, treatises, and pamphlets, which were given to students, sold to individuals, or distributed to libraries at other universities.  People read these books, took note of the methods, but other than to put a person's name to the method, passed by the historical development. Consequently, when one looks for historical notes in a &lt;em&gt;book&lt;/em&gt;, little more is found than a citation to an essential or original text documenting the method in its first instance. Books in analysis abound with exactly these citations, but little historical commentary.&lt;/p&gt;&#xA;&lt;p&gt;One book, however, has a few &lt;em&gt;very brief historical notes&lt;/em&gt; regarding exactly the historical development of difference equations -&lt;/p&gt;&#xA;&lt;p&gt;Charles Jordan, &lt;em&gt;Calculus of Finite Differences.&lt;/em&gt; 2nd Ed., 1950 (reprinted 1960). Chelsea Publishing Company, New York. 652 pp.&lt;/p&gt;&#xA;&lt;p&gt;Pay particular attention to the author's preface, and further comments in Chapter 1, Section 1, &lt;em&gt;Historical and Bibliographical Notes,&lt;/em&gt; p. 1 ff.  Note that when Jordan is discussing the Computer or Computers, he is referencing human computation by hand.  The book is rich with bibliographic notes and comments in the text, but the text is primarily about methods and applications in numerical analysis.  The book is available electronically &lt;a href=&quot;https://b-ok.cc/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  Perhaps you have already seen this book.&lt;/p&gt;&#xA;" OwnerUserId="16181" LastActivityDate="2022-02-02T00:25:58.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14153" PostTypeId="1" CreationDate="2022-02-02T04:26:50.747" Score="0" ViewCount="38" Body="&lt;p&gt;Using Google Books I couldn't find any reference to cumulus clouds before 1772. What terms were used for the different types of clouds before 1772, if there were specific terms?&lt;/p&gt;&#xA;" OwnerUserId="14055" LastActivityDate="2022-02-02T04:26:50.747" Title="Terms for different types of clouds before 1770?" Tags="|terminology|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14154" PostTypeId="1" CreationDate="2022-02-02T07:45:51.703" Score="1" ViewCount="86" Body="&lt;p&gt;There's a famous paper by L. Rosenfeld (On quantization of fields. &lt;em&gt;Nuclear Physics&lt;/em&gt; &lt;strong&gt;40&lt;/strong&gt;, 353–356 (1963). doi:&lt;a href=&quot;https://doi.org/10.1016/0029-5582(63)90279-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1016/0029-5582(63)90279-7&lt;/a&gt;) in which he criticizes the theoretical arguments leading to the conclusion that gravity must be quantized, stating that experimental is what really matters in the end of the day. It ends with the following quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The preceding hints may suffice to expose the inanity of a reasoning which amounts to putting physics upside down. It is nice to have at one's disposal such exquisite&#xA;mathematical tools as the present methods of quantum field theory, but one should&#xA;not forget that these methods have been elaborated in order to describe definite&#xA;empirical situations, in which they find their only justification. Any question as to&#xA;their range of application can only be answered by experience, not by formal argumentation. Even the legendary Chicago machine cannot deliver the sausages if it is&#xA;not supplied with hogs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I've seen this quote or references to it in a number of references (e.g., arXiv: &lt;a href=&quot;https://arxiv.org/abs/0803.3456&quot; rel=&quot;nofollow noreferrer&quot;&gt;0803.3456 [gr-qc]&lt;/a&gt;, arXiv: &lt;a href=&quot;https://arxiv.org/abs/2201.10452&quot; rel=&quot;nofollow noreferrer&quot;&gt;2201.10452 [gr-qc]&lt;/a&gt;, &lt;a href=&quot;https://huggett.people.uic.edu/Nick/My%20Work_files/why.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this text by N. Hugget and C. Callender&lt;/a&gt;, and others I can't recall). Now, I would like to know what &lt;em&gt;is&lt;/em&gt; the &amp;quot;legendary Chicago machine&amp;quot; Rosenfeld refers to. While it might initially sound silly since he only uses it as an analogy, I consider this interesting since while a possibility could be &amp;quot;It just happens that Chicago is famous for its sausages&amp;quot; (I have no clue if that's the case and I couldn't discover it), another one which is much deeper is that he's criticizing some school of thought that was then frequent in the University of Chicago, for example (once again, I'm speculating for the sake of an example, I couldn't discover if this is the case).&lt;/p&gt;&#xA;&lt;p&gt;In short, what is the Chicago machine he refers to?&lt;/p&gt;&#xA;&lt;p&gt;While I've been in Physics.SE for a while, this is my first question in History of Science and Mathematics, so if I'm failing in following any of the site's usual guidelines for questioning please let me know.&lt;/p&gt;&#xA;" OwnerUserId="16184" LastActivityDate="2022-02-02T07:45:51.703" Title="What is the &quot;legendary Chicago machine&quot; Rosenfeld refers to in his 1963 paper *On quantization of fields*?" Tags="|physics|quote|social-context|20th-century|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14156" PostTypeId="2" ParentId="13254" CreationDate="2022-02-02T18:25:33.947" Score="2" Body="&lt;p&gt;The most striking examples are the special and general relativity theories. Einstein gave physical meaning to abstract ideas of Riemann, Poincare, Lorenz, Lie and others.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2022-02-02T18:25:33.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14157" PostTypeId="2" ParentId="13469" CreationDate="2022-02-04T17:10:26.210" Score="4" Body="&lt;h2&gt;Short Answer&lt;/h2&gt;&#xA;&lt;p&gt;I believe you are thinking of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stern–Gerlach experiment&lt;/a&gt;. Which was instrumental in the discovery of quantized angular momentum at the time it was thought to prove the Bohr model, but later was experimental proof that subatomic particles have quantized spin. I think you might have been confused, because the cigar smoke actually helped them view an experiment by converting trace amounts  of silver metal to silver sulfate.&lt;/p&gt;&#xA;&lt;h2&gt;Background&lt;/h2&gt;&#xA;&lt;p&gt;&lt;strong&gt;Otto Stern&lt;/strong&gt; graduated with a PhD in Physics in 1912. He became Albert Einstein's first pupal at the Charles University in Prague. They would investigate the quantization of light, magnetism, and statistical physics. All of which would be the back bone of his Nobel prize winning experiment. After World War 1 Stern would become Max Born assistant at the Institute for Theoretical Physics where he started his work with molecular beams. He would improve on &lt;a href=&quot;https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/dunoyer-de-segonzac-louis-dominique-joseph-armand&quot; rel=&quot;nofollow noreferrer&quot;&gt;Louis Dunoyer's early molecular beam&lt;/a&gt; experiment that would later become the Stern–Gerlach experiment.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Walther Gerlach&lt;/strong&gt; received his PhD in physics at the University of Tübingen where he researched black body radiation, and the photoelectric effect. Gerlach also became an assistant at the  Institute for Theoretical Physics. He to had been interested in the molecular beam experiments carried out by Dunoyer. Using them to observer the emission spectra of beams from different metals without any success.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The physical landscape&lt;/strong&gt; at the time was dominated by the Bohr model. There were some issues that the model couldn't explained.&lt;/p&gt;&#xA;&lt;p&gt;At the time spatial quantization wasn't thought of as a &amp;quot;real&amp;quot; thing. It was thought of as an abstraction of some yet to be understood phenomena. The experiment would receive backlash from Born, and other physicist at the institute before it even took place. Born would write,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It took me quite a long time before I took this idea seriously. I thought always that [space] quantization was a kind of symbolic expression for something which you don't understand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;Experiment&lt;/h2&gt;&#xA;&lt;p&gt;The apparatus was composed of a beam of silver atoms collimated between two slits. These beams were directed over a 0.1tesla, 3.5cm magnet. The beam would cause a build up of silver metal on the glass. The classical thought was that a atomic magnet entering a field would oriented randomly, and the deflected atoms would only widen the beam. Where as quantum theory predicted the beam would split. At the time misalignments, and repeated failures of the apparatus led to a relatively short runs, and not much build up of silver.&lt;/p&gt;&#xA;&lt;p&gt;When they vented the vacuum in the apparatus to measure the result they initially saw no sign of silver build up, but as the silver metal was exposed to the atmosphere they began to see a build  up of black material.&lt;/p&gt;&#xA;&lt;p&gt;As is still the case postdocs didn't make a lot of money, so Stern would smoke really cheap cigars. When they vented the vacuum in the experiment the silver was reacting with sulfur from the cheap cigar smoke forming silver sulfide. The now tarnished silver which is black was much easier to see.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://study.com/academy/lesson/what-is-silver-sulfide-chemical-formula-uses.html#:%7E:text=Silver%20sulfide%20is%20generated%20when,t%20have%20many%20beneficial%20uses.&quot; rel=&quot;nofollow noreferrer&quot;&gt;2Ag (s) + H2S (g) → Ag2 S (s) + H2 (g)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even still they ran into experimental issues, and colleges reluctant to take their results seriously. Even worse the country was in the midst of a financial crisis, and the pair had to scrape money together to fund the experiment. Stern had moved to Rostock to teach physics. In 1922 they met in Göttingen to discuss and work on the project. Due to a train workers strike keeping Gerlach in Göttingen longer then expected they were able to get a well aligned beam that finally gave them good data.&lt;/p&gt;&#xA;&lt;h2&gt;Results!&lt;/h2&gt;&#xA;&lt;p&gt;At the time the results matched with the Bohr model, and came within 10% of his predicted magnetic moment of &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr_magneton&quot; rel=&quot;nofollow noreferrer&quot;&gt;one Bohr Magneton&lt;/a&gt;. Spatial quantization was some of the best evidence of quantum theory at the time. The Bohr model would soon be obsolete, and would lead to a new understanding of the Stern–Gerlach experiment. The splitting was not a result of the angular momentum of the silver atom, but actually due to the half &lt;a href=&quot;https://en.wikipedia.org/wiki/Spin_(physics)#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;spin angular momentum&lt;/a&gt;. Pauli wouldn't publish about spin until 1924 two years after the experiment was preformed.&lt;/p&gt;&#xA;&lt;p&gt;Gerlach would remain in Germany. He never joined the Nazi party, but he would lead the German effort to develop an atomic bomb. After hearing of the bomb being dropped on Hiroshima he suffered a nervous break down. He would be a stanch advocate for banning nuclear weapons, and work on repairing German science.&lt;/p&gt;&#xA;&lt;p&gt;Stern immigrated to the united states in 1939, and worked for the War department in World War 2. He would never return to Germany. Stern would receive the Nobel price in 1943, not for the Stern–Gerlach experiment, but &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_Nobel_laureates_in_Physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;for his contribution to the development of the molecular ray method and his discovery of the magnetic moment of the proton&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This was a summery of the article &amp;quot;&lt;a href=&quot;https://faculty.chemistry.harvard.edu/files/dudley-herschbach/files/how_a_bad_cigar_0.pdf?m=1363959307&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics&lt;/a&gt;&amp;quot;. I had never heard of this experiment, and found it very interesting.&lt;/p&gt;&#xA;" OwnerUserId="4880" LastEditorUserId="4880" LastEditDate="2022-02-07T15:00:08.563" LastActivityDate="2022-02-07T15:00:08.563" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14158" PostTypeId="1" AcceptedAnswerId="14163" CreationDate="2022-02-05T01:39:22.760" Score="0" ViewCount="130" Body="&lt;p&gt;I thought that such old terms wouldnt be known who &amp;quot;created&amp;quot; them, but terms of the same time like &lt;a href=&quot;https://blogs.umass.edu/p139ell/2012/10/14/natural-philosophy-aristotle/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Natural Philosophy&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philosophy&lt;/a&gt; are said to be created by Aristotle, and believed to be created by some by Pythagoras, respectively.&lt;/p&gt;&#xA;&lt;p&gt;How about the term Episteme, that's related to them (the first link claims Aristotle regarded Natural Philosophy as theoretical science, theoretical episteme I suppose) ,&#xA;is it known or theorized who created the term?&lt;/p&gt;&#xA;" OwnerUserId="4512" LastActivityDate="2022-02-05T23:40:05.660" Title="Is it known who coined the term Episteme?" Tags="|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14159" PostTypeId="1" CreationDate="2022-02-05T06:30:06.450" Score="3" ViewCount="158" Body="&lt;p&gt;All I know is that detecting the deflecting alpha particles was a very tedious process, so much so that that was probably one reason why he asked Hans Geiger and Ernest Marsden to do the experiment. Modern day recreation of the experiment uses light detectors.  (In the 1950s they gave white lines on a TV screen when alpha particles were detected and presently a doorbell sound. Or at least that's what I found on YouTube.)&lt;/p&gt;&#xA;&lt;p&gt;I am curious about how the deflection was detected in the original experiment. My textbook says they used a microscope. But how? Wouldn't they have to place the microscope inside the vacuum chamber?&lt;/p&gt;&#xA;&lt;p&gt;Also secondly where were the scintillating effects produced? Was it produced on a zinc sulphide coated screen or somewhere else?&lt;/p&gt;&#xA;" OwnerUserId="14225" LastEditorUserId="10106" LastEditDate="2022-02-06T12:16:22.237" LastActivityDate="2022-02-06T12:16:22.237" Title="How did Rutherford detect the deflected alpha particles?" Tags="|physics|experimental-physics|atomic-theory|particle-physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14161" PostTypeId="1" AcceptedAnswerId="14162" CreationDate="2022-02-05T18:01:22.277" Score="1" ViewCount="86" Body="&lt;p&gt;I have searched in Internet, but I have not found the material.&#xA;It couldn't be copper, because a short circuit would occur, and it couldn't be Nicrom, because Nicrom was not developed until 1905, and Joule did his experiment in 1840.&#xA;Could the material be iron or steel?&#xA;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="16214" OwnerDisplayName="ilich qynn" LastActivityDate="2022-02-05T19:19:12.423" Title="What material was the wire with which J.P. Joule did his experiment, with which he arrived at his law of heat dissipation from a resistor?" Tags="|thermodynamics|electricity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14162" PostTypeId="2" ParentId="14161" CreationDate="2022-02-05T18:33:24.000" Score="4" Body="&lt;p&gt;&amp;quot;It couldn't be copper, because a short circuit would occur&amp;quot; Not if the copper was in the form of a long thin wire. Solenoids, for example, are coils of enamelled copper wire and are usually designed to be connected across supplies of a few volts. They don't usually even get hot! It would be perfectly possible to make a heating coil from copper wire.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: He used various metals, including iron, copper and mercury. A transcript of an 1840 paper of his is available on this Royal Society website:&#xA;&lt;a href=&quot;https://royalsocietypublishing.org/doi/full/10.1098/rsnr.2020.0027&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://royalsocietypublishing.org/doi/full/10.1098/rsnr.2020.0027&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15941" OwnerDisplayName="Philip Wood" LastEditorUserId="15941" LastEditDate="2022-02-05T19:19:12.423" LastActivityDate="2022-02-05T19:19:12.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14163" PostTypeId="2" ParentId="14158" CreationDate="2022-02-05T23:30:54.713" Score="2" Body="&lt;p&gt;It is a common very ancient word, and not a technical term, as it is today (even in Modern Greek).&lt;/p&gt;&#xA;&lt;p&gt;According to the Liddell, Scott, Jones Ancient Greek Lexicon, &lt;strong&gt;ἐπιστήμη&lt;/strong&gt;&#xA;is a word used many-many centuries before Aristotle, deriving from the verb &lt;strong&gt;ἐπίσταμαι&lt;/strong&gt;, (a favorite of Homer!), associated with knowing for certain, capability, skill, expertise, assurance, etc.&lt;/p&gt;&#xA;&lt;p&gt;From the formal definition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A.acquaintance with a matter, understanding, skill, as in archery, S.Ph.1057; in war, Th.1.121, 6.72, 7.62; “ἐ. πρὸς τὸν πόλεμον” Lys.33.7 (fort. leg. περί）“ περὶ τὰ μαθήματα” Pl. Phlb.55d; “τοῦ νεῖν” Id.Grg.511c; ἐπιστήμῃ skilfully, οἱ μὴ ἐ. τοὺς ἐπαίνους “ποιούμενοι” Plot.5.5.13.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2.. professional skill: hence, profession, “οἱ τὴν ἰατρικὴν ἐ. μεταχειριζόμενοι” PFay.106.22 (ii A.D.); ζωγράφος τὴν ἐ. painter by profession, POxy.896.5 (iv A.D.).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;II.. generally, knowledge, “ἐπιστήμῃ σύ μου προὔχοις ἄν” S.OT1115; πάντ᾽ ἐπιστήμης πλέως full of knowledge in all things, Id.Ant.721, cf. Tr. 338; ἐκ τῆς ἐ. E.Fr.522.3; ἐ. δοξαστική, opp. ἀλήθεια, Pl.Sph.233c: pl., kinds of knowledge, “μυρίαι ἀνδρῶν ἐπιστᾶμαι πέλονται” B.9.38, cf. Pl.Smp.208a.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2.. scientific knowledge, science, opp. δόξα, Pl.R.477b sq., Hp.Lex4, Pl.Plt.301b, Arist.AP0.88b30, EN1139b18: coupled with ἐμπειρία and τέχνη, Pl.R.422c, cf. Ion536c, Arist.Metaph.981a2: pl., the sciences, freq. in Pl.（R.522c,al.), etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it was used by Homer, Thucydides, Sophocles, Plato, Hippocrates, etc,  the first easily four centuries before Aristotle. Aristotle used it with a narrow philosophical slant; but the word went on being used all the way to Byzantium (Anna Comnene), with the &lt;em&gt;&lt;strong&gt;broad&lt;/strong&gt;&lt;/em&gt; meaning, not the narrow application for &amp;quot;science&amp;quot;, as today.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(NB. The verb comes roughly from standing on, mastering, analogous to OE &lt;em&gt;forstandan&lt;/em&gt;, German &lt;em&gt;verstehen&lt;/em&gt;. I'm not expert enough to trace &lt;em&gt;understand&lt;/em&gt;... under, not on? Go figure...)&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2022-02-05T23:40:05.660" LastActivityDate="2022-02-05T23:40:05.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14164" PostTypeId="1" AcceptedAnswerId="14173" CreationDate="2022-02-07T08:53:37.727" Score="0" ViewCount="137" Body="&lt;p&gt;I know that the Poynting vector is defined as the cross-product &lt;span class=&quot;math-container&quot;&gt;$\vec{E}\times\vec{H}$&lt;/span&gt; and that is &amp;quot;usually&amp;quot; denoted by &lt;span class=&quot;math-container&quot;&gt;$\vec{S}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\vec{N}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if there is a particular reason for these names. I've also seen it indicated with &lt;span class=&quot;math-container&quot;&gt;$\vec{P}$&lt;/span&gt; and this makes more sense as it could stand for &amp;quot;Poynting&amp;quot;. Do the others refer to something?&lt;/p&gt;&#xA;" OwnerUserId="16207" LastEditorUserId="64" LastEditDate="2022-02-08T01:05:54.437" LastActivityDate="2022-02-08T17:30:42.263" Title="Notation of Poynting vector" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14165" PostTypeId="2" ParentId="13982" CreationDate="2022-02-07T14:55:44.740" Score="5" Body="&lt;p&gt;&lt;strong&gt;Is it simply that the causation between the work and the disease is more clear in the case of Curie but not Fermi?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Marie Curie died of &lt;a href=&quot;https://en.wikipedia.org/wiki/Aplastic_anemia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aplastic anemia&lt;/a&gt; which was most likely due to exposure to radiation from her work. The link between Aplastic anemia is well known, and is even a side effect of radiation therapies. Enrico Fermi died from stomach cancer. Which may have been a symptom of his work with radioactive material or it could have been from his diet, his weight, or his smoking. Stomach cancer is much less of a smoking gun than Aplastic anemia is.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Additionally, given that a lot of Fermi's work was happening once Curie had already died, it is plausible he knew the risks of working with the materials he was working with. Is this supported by any sources?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Marie may not have known how dangerous radiation exposure was early in her career, but by 1929 she most certainly did. She would be aware of industrial workers succumbing to radiation sickness &lt;a href=&quot;https://www.nytimes.com/1998/10/06/science/a-glow-in-the-dark-and-a-lesson-in-scientific-peril.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;such as the radium girls&lt;/a&gt;. She would state,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“I particularly would stress the fact that radium is dangerous in untrained hands,”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That does not change the fact that while she was making her discoveries she, and her colleges, did very dangerous things. They were &lt;a href=&quot;https://www.forbes.com/sites/kionasmith/2017/11/07/its-marie-curies-150th-birthday-and-her-legacy-deserves-more-nuance/?sh=5d42662bbed3&quot; rel=&quot;nofollow noreferrer&quot;&gt;moving radioactive material around with little or no shielding, pouring radium between containers without even a pair of gloves for protection, and mouth-pipetting polonium.&lt;/a&gt; When she died her notes were so &lt;a href=&quot;https://medium.com/predict/marie-curies-notebooks-someday-will-not-be-radioactive-65ce662439a3&quot; rel=&quot;nofollow noreferrer&quot;&gt;radioactive they are still kept in a lead box in Paris&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Fermi would have been most certainly aware of the dangers of radiation. He worked at Los Alamos, and warned Louis Stole that he would be dead within a year before the &lt;a href=&quot;https://en.wikipedia.org/wiki/Demon_core#Second_incident&quot; rel=&quot;nofollow noreferrer&quot;&gt;second Demon Core&lt;/a&gt; incident occurred. &lt;a href=&quot;https://www.atomicheritage.org/history/health-physics-nuclear-medicine-during-manhattan-project&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermi's wife was a nurse in Health and Safety office&lt;/a&gt; at Los Alamos, where they regularly took blood tests for radiation exposure. In the 40's and 50's they may not have had the same level of safety around radiation as we do today, but they certainly weren't carrying ampules of &lt;a href=&quot;https://www.csmonitor.com/Technology/Horizons/2011/1107/Marie-Curie-Why-her-papers-are-still-radioactive&quot; rel=&quot;nofollow noreferrer&quot;&gt;polonium and radium in their breast pocket&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is this theory plausible or completely off?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Marie Curie was a female scientist in the heroic age of science. There is a certain mythology about her. Her life plays out like Icarus. A parable were she flew too close to the sun, and then dies from it. It's such a good story they made a &lt;a href=&quot;https://www.comixology.com/Biopic-Marie-Curie/comics-series/91688&quot; rel=&quot;nofollow noreferrer&quot;&gt;comic book&lt;/a&gt; out of it. Then made a &lt;a href=&quot;https://www.imdb.com/title/tt6017756/&quot; rel=&quot;nofollow noreferrer&quot;&gt;movie&lt;/a&gt; out of the comic book.&lt;/p&gt;&#xA;&lt;p&gt;Why doesn't Fermi get the same treatment? Why would he? I'm sure his life is fascinating. He's a less known scientist known more for &lt;a href=&quot;https://trends.google.com/trends/explore?date=all&amp;amp;geo=US&amp;amp;q=%2Fm%2F05k51,Fermi%20Paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;a musing about aliens than his work on the atomic bomb&lt;/a&gt;. He died from a disease that may have been associated with radiation. That just can't compare to the woman who first discovered radioactive elements dying from a disease that's still associated with radiation sickness.&lt;/p&gt;&#xA;" OwnerUserId="4880" LastEditorUserId="13914" LastEditDate="2022-02-07T19:49:13.667" LastActivityDate="2022-02-07T19:49:13.667" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14167" PostTypeId="1" CreationDate="2022-02-07T16:35:07.913" Score="3" ViewCount="131" Body="&lt;p&gt;I would be interested in reading how Kronecker formally approached this construction, using the mathematical ideas of his time, and possibly some insight as to what he considered its philosophical implications to be. I am referring specifically to the construction which, in modern terminology, adjoins a root of an irreducible polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x) \in F[x]$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; by constructing the quotient &lt;span class=&quot;math-container&quot;&gt;$F[x]/(f(x)) $&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$(f(x))$&lt;/span&gt; is the ideal of &lt;span class=&quot;math-container&quot;&gt;$F[x]$&lt;/span&gt; generated by &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="14473" LastEditorUserId="10106" LastEditDate="2022-02-07T17:09:41.733" LastActivityDate="2022-02-08T19:53:14.527" Title="Is there a translation of Kronecker's original work on adjoining a root of a polynomial to a field?" Tags="|mathematics|reference-request|abstract-algebra|translations|galois-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14168" PostTypeId="2" ParentId="344" CreationDate="2022-02-07T16:39:41.207" Score="5" Body="&lt;p&gt;See my two 7-Feb-2022 comments above. Given how long it's likely to be before I complete this project and the very incomplete details about the result one can find (internet or publications), I thought it might be best to offer at least some of the  &amp;quot;annotated chronologically-ordered bibliography&amp;quot; now, and revise it as appropriate at some later time. What follows does not include some preliminary comments I intended to make, because they are not in very good form now (mostly just notes to myself about certain things), and does not go up to the late 1910s (after WW I is where a natural stopping point seems to be, but I'm stopping well before that for now).&lt;/p&gt;&#xA;&lt;p&gt;Regarding my italization and other conventions when giving excerpts from publications (includes translations of excerpts), see &lt;strong&gt;Conventions&lt;/strong&gt; near the beginning of &lt;a href=&quot;https://mathoverflow.net/a/162605/15780&quot;&gt;this answer&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[1]&lt;/strong&gt; Augustin Louis Cauchy (1789-1857), &lt;a href=&quot;https://archive.org/details/reumeanalytique00caucgoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Résumés Analytiques&lt;/strong&gt;&lt;/a&gt; [Analytical Summaries], l'Académie des sciences de Turin [Accademia delle Scienze di Torino; Academy of Sciences of Turin] (Italy), 1833, 166 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Reprinted on pp. 7-182 in Cauchy, &lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k902022&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Œuvres Complètes&lt;/strong&gt;, série 2, tome 10&lt;/a&gt;, Gauthier-Villars et fils, 1895, 184 pages. A table of contents, not in the 1833 publication, is given on pp. 183-184 of the 1895 publication. The following is from Grattan-Guinness (1990, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3764322381&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Convolutions in French Mathematics, 1800-1840&lt;/strong&gt;&lt;/a&gt;, Volume II, p. 1228): &lt;em&gt;Cauchy gave his own vision of mathematical analysis further airing when in 1833 he published his&lt;/em&gt; Résumés analytiques &lt;em&gt;in Turin (1833c), a 166-page summary of the&lt;/em&gt; Cours d'analyse &lt;em&gt;and certain related papers. Many of the basic definitions and results were repeated more or less verbatim: not only his definition of infinitesimals but also Theorem 1136.1&lt;/em&gt; [G-G's numbering; appears on p. 721] &lt;em&gt;on the continuity of the sum-function, convergence tests, series expansions of basic functions, trigonometry, complex variables, de Moivre's theorem, and so on. In his foreword he promised a series of livaisons&lt;/em&gt; [= installments] &lt;em&gt;to continue upon this book, but they did not appear&lt;/em&gt;. The earliest published observation I know of that the sum of an infinite series can be affected by a rearrangement of its terms appears near the beginning of §8. &lt;em&gt;Des séries doubles ou multiples. Nombres de Bernoulli&lt;/em&gt; (pp. 56-68). Specifically, Cauchy gives an example of a rearrangement of the alternating harmonic series &lt;span class=&quot;math-container&quot;&gt;$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$&lt;/span&gt; that results in a series that does not converge.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from middle of p. 58 to top p. 59)&lt;/strong&gt; [translated from French] &lt;em&gt;But if, instead of adding the terms to each other&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\mp \, \frac{1}{n+1}\,, \;\;\; \pm \, \frac{1}{n+2}\,, \;\;\; \mp \, \frac{1}{n+3}\,, \;\;\; \text{etc.} \;\;\; .....$$&lt;/span&gt; &lt;em&gt;taken in the order they are located, we change this order by choosing among the terms assigned the same sign, for example, the following&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$\pm \, \frac{1}{n+2}\,, \;\;\; \pm \, \frac{1}{n+4}\,, \;\;\; ..... \;\;\; \pm \, \frac{1}{n+2n} \; = \; \pm \, \frac{1}{3n}\,,$$&lt;/span&gt; [then] &lt;em&gt;the numerical value of the sum of these terms, as follows&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{n+2} \; + \; \frac{1}{n+4} \; + \; ..... \; + \; \frac{1}{3n}$$&lt;/span&gt; &lt;em&gt;obviously exceeds the product&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$n \times \frac{1}{3n} \; = \; \frac{1}{3}\,,$$&lt;/span&gt; &lt;em&gt;and ceases to be infinitely small for infinitely large values of&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n.$&lt;/span&gt; After this, Cauchy returns to discussing &amp;quot;multiple series&amp;quot; (double series, triple series, etc. whose terms are defined by the use of double subscripts, triple subscripts, etc.), saying that when all the terms are positive then, by a theorem previously stated, it follows that the convergence (and sum?) is not affected by various methods of adding the terms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The rearrangement Cauchy briefly described has the form &lt;span class=&quot;math-container&quot;&gt;$1 - \frac{1}{2} + a_1 - a_2 + a_3 - \cdots + (-1)^{n+1}a_n + \cdots,$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a_1 = \frac{1}{3}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_2 = \frac{1}{4} + \frac{1}{6}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_3 = \frac{1}{5} + \frac{1}{7} + \frac{1}{9}$&lt;/span&gt; and so on, where each &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; represents a finite sequence of same-signed terms from the original sequence such that the ordering of these same-signed terms among themselves is not altered and where the number of these same-signed terms is such that this number times the absolute value of the last term included equals &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}.$&lt;/span&gt; For example, &lt;span class=&quot;math-container&quot;&gt;$a_4 = \frac{1}{8} + \cdots + \frac{1}{18}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a_5 = \frac{1}{11} + \cdots + \frac{1}{27}.$&lt;/span&gt; Note that &lt;span class=&quot;math-container&quot;&gt;$a_4$&lt;/span&gt; includes &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt; terms and &lt;span class=&quot;math-container&quot;&gt;$6(\frac{1}{18}) = \frac{1}{3},$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$a_5$&lt;/span&gt; includes &lt;span class=&quot;math-container&quot;&gt;$9$&lt;/span&gt; terms and &lt;span class=&quot;math-container&quot;&gt;$9(\frac{1}{27}) = \frac{1}{3}.$&lt;/span&gt; By design, &lt;span class=&quot;math-container&quot;&gt;$a_n \geq \frac{1}{3}$&lt;/span&gt; for each &lt;span class=&quot;math-container&quot;&gt;$n,$&lt;/span&gt; so &lt;span class=&quot;math-container&quot;&gt;$1 - \frac{1}{2} + a_1 - a_2 + a_3 - \cdots + (-1)^{n+1}a_n + \cdots$&lt;/span&gt; diverges by the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-term divergence test (the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th term doesn't approach zero). Thus, the sequence of partial sums of this series does not converge, and since this sequence of partial sums is a subsequence of the sequence of partial sums of the rearranged series, it follows that the rearranged series diverges.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(from pp. 157-158 of Bottazzini/Gray's 2013&lt;/strong&gt; &lt;em&gt;&lt;strong&gt;Hidden Harmony&lt;/strong&gt;&lt;/em&gt; &lt;strong&gt;...)&lt;/strong&gt; Cauchy taught in Turin from October 1832 to July 1833, when he was called to Prague to serve as a tutor to the son of the exiled king Charles X. In accordance with the title of his chair, he delivered introductory lectures of a philosophical-theological character (Cauchy 1888a). The essential content of his Turin lectures was published in the &lt;em&gt;Résumés analytiques&lt;/em&gt; (Cauchy 1833b) which appeared in print when he was already living in Bohemia, and contain nothing new by comparison to his previous treatises. A vivid account of Cauchy's teaching was provided by Luigi Federico Menabrea.&lt;span class=&quot;math-container&quot;&gt;$^{text{38}}$&lt;/span&gt; Not surprisingly, Cauchy's lectures [begin centered quote] were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius (quoted in Belhoste 1991, 156). [end centered quote] so much so that Menabrea was the only one to &amp;quot;see it through&amp;quot; out of some 30 students who initially enrolled in the course.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[2]&lt;/strong&gt; Johann Peter Gustav Lejeune Dirichlet (1805-1859), &lt;a href=&quot;http://www.biodiversitylibrary.org/page/29663825#page/307/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Beweis des satzes, dafs&lt;/em&gt; [= dass] &lt;em&gt;jede unbegrenzte arithmetische progression, deren erstes glied und differenz ganze zahlen ohne gemeinschaftlichen factor sind, unendlich viele primzahlen enthält&lt;/em&gt;&lt;/a&gt; [Proof of the theorem, that every unlimited arithmetic progression whose first member and difference are whole numbers without a common factor, contains infinitely many prime numbers], &lt;strong&gt;Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin. Mathematische&lt;/strong&gt;, aus dem Jahre 1837 [for the year 1837], published in 1839, pp. 45-71.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another digital copy is &lt;a href=&quot;http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige?band=07-abh/1837&amp;amp;seite:int=00000286&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; Reprinted on pp. 315-345 of &lt;a href=&quot;https://archive.org/details/glejeunedirichl01dirigoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;G. Lejeune Dirichlet's Werke&lt;/strong&gt;&lt;/a&gt;, Volume 1, 1889. For an English translation by Ralf Stephan, see &lt;a href=&quot;http://arxiv.org/abs/0808.1408&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. A statement appearing just above the first line of the paper says (translated): &lt;em&gt;Read to the Akademie der Wissenschaften on 27 July 1837&lt;/em&gt;. &lt;strong&gt;Note:&lt;/strong&gt; Page 71 is incorrectly labeled as &amp;quot;81&amp;quot;. This typo has led to some incorrect &amp;quot;45-81&amp;quot; page ranges having been published: Joseph Dauben's 1979 book &lt;strong&gt;Georg Cantor ...&lt;/strong&gt; (p. 368, line -6), Jesper Lützen's 1990 book &lt;strong&gt;Joseph Liouville ...&lt;/strong&gt; (p. 824, line 12), Morris Kline's 1990 Volume 3 of &lt;strong&gt;Mathematical Thought ...&lt;/strong&gt; (p. 830, footnote 35; p. 966, footnote 54), Bottazzini/Gray's 2013 &lt;strong&gt;Hidden Harmony ...&lt;/strong&gt; (p. 782, 3rd item from top), versions 1 and 2 of Stephan's English translation (footnote on first page). Dirichlet makes some comments about the convergence of infinite series beginning near the bottom of p. 48 and continuing onto p. 49. What follows is excerpted from pp. 3-4 of version 2 (latest version; 24 November 2014) of Stephan's English translation: &lt;em&gt;Before we go on it is necessary to state the reason for the condition made above, that&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s&amp;gt;1$&lt;/span&gt; &lt;em&gt;should hold. We can convince ourselves of the necessity of this limitation if we respect the essential difference which exists between two kinds of infinite series&lt;/em&gt;. &lt;span class=&quot;math-container&quot;&gt;$[\cdots]$&lt;/span&gt; &lt;em&gt;So, for example, of the two series made from the same terms&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$ 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \frac{1}{\sqrt{5}} - \frac{1}{\sqrt{6}} + \cdots,$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$ 1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{7}} - \frac{1}{\sqrt{4}} + \cdots,$$&lt;/span&gt; &lt;em&gt;only the first converges while of the following&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots,$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$ 1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots,$$&lt;/span&gt; &lt;em&gt;both converge, but with different sums. Our infinite series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$L,$&lt;/span&gt; &lt;em&gt;as can be easily seen, belongs only then to the first of the classes&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$[\cdots]$&lt;/span&gt; I cannot find any explicit mention of Cauchy in this paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Dirichlet, &lt;a href=&quot;https://books.google.com/books?id=J9A-AQAAMAAJ&amp;amp;pg=PA397&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Démonstration de cette proposition: toute progression arithmétique dont le premier terme et la raison sont des entiers sans diviseur commun, contient une infinité de nombres premiers&lt;/em&gt;&lt;/a&gt; [Proof of this proposition: every arithmetic progression such that the first term and common ratio are integers without common divisor, contains an infinity of prime numbers], &lt;strong&gt;Journal de Mathématiques Pures et Appliquées&lt;/strong&gt; [= Liouville's Journal] (1) 4 (1839), 393-422.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A footnote (by Joseph Liouville, 1809-1882) at the bottom of p. 393 states that this is a translation by Olry Terquem (1782-1862) of Dirichlet's 1837 paper. Dirichlet's remarks about rearranging terms of an infinite sequence extend from the bottom of p. 396 to the middle of p. 397.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[3]&lt;/strong&gt; Dirichlet, &lt;a href=&quot;http://www.digizeitschriften.de/download/PPN243919689_0019/log23.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres&lt;/em&gt;&lt;/a&gt; [Investigations on diverse applications of infinitesimal analysis to the theory of numbers], &lt;strong&gt;Journal für die Reine und Angewandte Mathematik&lt;/strong&gt; 19 #4 (1839), 324-369.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dirichlet gives 4 July 1839 (on p. 369) for the date of the paper's completion. Dirichlet makes some comments about the convergence of infinite series beginning on p. 329 (line -10) and continuing onto p. 330, where I believe Dirichlet discusses the fact that rearrangements of certain series can lead to different values. However, I do not see any specific examples given. It might be the case that a specific example is described verbally, though. I cannot find any explicit mention of Cauchy in this paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[4]&lt;/strong&gt; Martin [Marcin, Martinus] &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Ohm_Martin.html&quot; rel=&quot;noreferrer&quot;&gt;Ohm (1792-1872)&lt;/a&gt;, **&lt;a href=&quot;https://books.google.com/books?id=sO09AAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;De Nonnullis Seriebus Infinitis Summandis**&lt;/a&gt; [Concerning the Summation of Certain Infinite Series], Trowitzschii et Filii [Trowitzsch und Sohn; Trowitzsch and Son] (Berlin), 1839, 15 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/3525403119&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;A 300+ page biography&lt;/a&gt; of &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Ohm_Martin.html&quot; rel=&quot;noreferrer&quot;&gt;Ohm&lt;/a&gt; was published in 1987, but I have not been able to examine it. This is a separately published booklet, written in Latin, that deals with the summation of certain rearrangements of the alternating harmonic series &lt;span class=&quot;math-container&quot;&gt;$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.$&lt;/span&gt; In §1 (p. 3) the alternating harmonic series is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\ln {2}.$&lt;/span&gt; Ohm's verification consists of plugging &lt;span class=&quot;math-container&quot;&gt;$x=0$&lt;/span&gt; into the Maclaurin series for &lt;span class=&quot;math-container&quot;&gt;$\ln (1+x).$&lt;/span&gt; The remaining verifications are similar in that Ohm first obtains series expansions of certain rational functions and their natural logarithms. In §2 (pp. 3-6) the rearrangement in which &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; positive terms are followed by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; negative term is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}\ln {2}.$&lt;/span&gt; In §3 (pp. 6-7) the result for &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; positive term followed by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; negative terms is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\ln {2}.$&lt;/span&gt; In §4 (pp. 7-8) the result for &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; positive terms followed by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; negative term is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\ln {3}.$&lt;/span&gt; In §5 (pp. 8-9) the result for &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; positive terms followed by &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; negative terms is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\ln 2 + \frac{1}{2}\ln {3}.$&lt;/span&gt; In §6 (pp. 9-11) the result for &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; positive terms followed by &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; negative term is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\ln 2 + \frac{1}{2}\ln {m}.$&lt;/span&gt; In §7 (pp. 11-12) the result for &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; positive term followed by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; negative terms is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\ln 2 - \frac{1}{2}\ln {n}.$&lt;/span&gt; In §8 (pp. 12-14) the result for &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; positive terms followed by &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; negative terms is shown to have sum &lt;span class=&quot;math-container&quot;&gt;$\ln 2 + \frac{1}{2}\ln \frac{m}{n}.$&lt;/span&gt; Finally, §9 (p. 14) gives a summary of the result proved in §8, and §10 (pp. 14-15) appears to involve some type of concluding remarks. The only reference cited is in a footnote on p. 14, which directs the reader to see &lt;em&gt;Aufsätze aus dem Gebiete der höhern Mathematik&lt;/em&gt; (Berlin, 1823), and no name appears anywhere except on the title page (in particular, neither Cauchy nor Dirichlet is mentioned). Some comments about Ohm and infinite series (but not the results he published in 1839) are given on pp. 30-31 of Cajori's 1888 paper &lt;em&gt;History of infinite series&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[5]&lt;/strong&gt; Ohm, &lt;a href=&quot;https://books.google.com/books?id=T-k3AAAAMAAJ&amp;amp;pg=PA87&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Der Geist der Mathematischen Analysis und ihr Verhältniss zur Schule&lt;/strong&gt;&lt;/a&gt; [The Spirit of Mathematical Analysis and Its Relationship to the Field], Erste Abhandlung [First Treatise], Duncker und Humblot (Berlin), 1842, xvi + 159 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An English translation by Alexander John Ellis (1814-1890), titled &lt;a href=&quot;https://books.google.com/books?id=ewKFAAAAIAAJ&amp;amp;pg=PA56&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;The Spirit of Mathematical Analysis, and Its Relation to a Logical System&lt;/strong&gt;&lt;/a&gt;, was published in 1843. After p. 103 of the English translation there are 3 pages (not numbered) that give some information about Ohm's published books. Ohm's 1839 essay &lt;strong&gt;De Nonnullis Seriebus Infinitis Summandis&lt;/strong&gt; is not included. Incidentally, I highly recommend reading the &lt;em&gt;Introduction&lt;/em&gt; (by Ohm) of the Ellis translation for an excellent account of many of the problems mathematicians were having at this time with interpreting various formal manipulations. The following is &amp;quot;[observation] (2)&amp;quot; on p. 56 [= pp. 87-88 of the 1842 German]: &lt;em&gt;Now a convergent series has this value only by virtue of the law according to which its terms proceed&lt;/em&gt; in infinitum. &lt;em&gt;Hence to prevent any doubt occurring concerning the true value of a convergent series, we must carefully enunciate the law according to which the terms are to be taken&lt;/em&gt; in infinitum. &lt;em&gt;For example, from the same reciprocal terms of the natural numbers, viz&lt;/em&gt;. &lt;span class=&quot;math-container&quot;&gt;$$1, \; -\frac{1}{2}, \; +\frac{1}{3}, \; -\frac{1}{4}, \; +\frac{1}{5}, \; -\frac{1}{6}, \; +\frac{1}{7}, \; - \,\text{in inf.}$$&lt;/span&gt; &lt;em&gt;we may compose any number of numerical infinite series, which are all convergent but have all different values, each however having its own perfectly determinate value by virtue of the determinate law according to which it is constructed. The series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7} - \frac{1}{8} + \, \text{in inf.}$$&lt;/span&gt; &lt;em&gt;in which, if we take&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; &lt;em&gt;terms, there are always as many positive as negative terms, has for its value&lt;/em&gt; log. nat. &lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt; &lt;em&gt;The series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \frac{1}{6} + \, \text{in inf.}$$&lt;/span&gt; &lt;em&gt;in which, if we take the first&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$3n$&lt;/span&gt; &lt;em&gt;terms&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; &lt;em&gt;positive and only&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;em&gt;negative of the above terms follow one another (in their order) has for its value&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}\,$&lt;/span&gt;log. nat. &lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt; &lt;em&gt;The series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \frac{1}{7} - \, \text{in inf.}$$&lt;/span&gt; &lt;em&gt;in which, if we take the first&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$3n$&lt;/span&gt; &lt;em&gt;terms, we find&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;em&gt;of them positive and&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; &lt;em&gt;negative, has for its value&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\,$&lt;/span&gt;log. nat. &lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt; &lt;em&gt;And if we take&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; &lt;em&gt;of the above terms positive and&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; &lt;em&gt;negative, and then&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; &lt;em&gt;positive and&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; &lt;em&gt;negative, and so on, the value of the resulting numerical convergent series is&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$=\,$&lt;/span&gt;log. nat. &lt;span class=&quot;math-container&quot;&gt;$2 + \frac{1}{2}\,$&lt;/span&gt;log. nat.&lt;span class=&quot;math-container&quot;&gt;$\,\frac{\mu}{\nu},$&lt;/span&gt; &lt;em&gt;and therefore&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$=\,$&lt;/span&gt;log. nat. &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; &lt;em&gt;which is that of the first series, when&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mu = \nu,$&lt;/span&gt; &lt;em&gt;but greater than that when&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mu &amp;gt; \nu$&lt;/span&gt; &lt;em&gt;and less when&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\mu &amp;lt; \nu.$&lt;/span&gt; For an outline of the method Ohm uses, see the top half of p. 818 of Cowen/Davidson/Kaufman (1980) below. Ohm is not mentioned in this 1980 paper, however.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[6]&lt;/strong&gt; Victor Amédée Lebesgue (1791-1875), &lt;a href=&quot;http://www.numdam.org/numdam-bin/item?id=NAM_1845_1_4__66_0&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur la convergence des séries&lt;/em&gt;&lt;/a&gt; [On the convergence of series], &lt;strong&gt;Nouvelles Annales de Mathématiques&lt;/strong&gt; (1) 4 (1845), 66-70.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For some biographical and bibliographical information about V. Lebesgue, see Poggendorff (1898, A-L, middle left column p. 784 to bottom left column p. 785). There appears to be no family/blood relationship between V. Lebesgue and Henri Lebesgue (known for the Lebesgue integral). &lt;strong&gt;(from bottom of p. 69 to end of the paper)&lt;/strong&gt; [translated from French; math notation slightly modernized in some places] &lt;em&gt;Proposition V. In a series containing negative terms, if we change the order of the terms&lt;/em&gt;, [then] &lt;em&gt;we can&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1^{\text o}$&lt;/span&gt; &lt;em&gt;change the sum without destroying the convergence&lt;/em&gt;; &lt;span class=&quot;math-container&quot;&gt;$2^{\text o}$&lt;/span&gt; &lt;em&gt;destroy the convergence, that is to say, render the series divergent instead of convergent&lt;/em&gt;. [Proof of] &lt;span class=&quot;math-container&quot;&gt;$1^{\text o}$&lt;/span&gt; &lt;em&gt;The series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{2n+1} - \frac{1}{2n+2},$$&lt;/span&gt; [and] &lt;span class=&quot;math-container&quot;&gt;$$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots + \frac{1}{4n+1} + \frac{1}{4n+3} - \frac{1}{2n+2},$$&lt;/span&gt; &lt;em&gt;are both convergent, because in the first the terms are decreasing, and it will be the same in the second by replacing&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1 + \frac{1}{3}$&lt;/span&gt; &lt;em&gt;by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{4}{3},$&lt;/span&gt; [and] &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{5} + \frac{1}{7}$&lt;/span&gt; &lt;em&gt;by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{12}{35},$&lt;/span&gt; &lt;em&gt;etc. Moreover, if we take for general terms&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2n+1} - \frac{1}{2n+2} = \frac{1}{(2n+1)(2n+2)};$&lt;/span&gt; [and] &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4n+1} + \frac{1}{4n+3} - \frac{1}{2n+2} = \frac{8n+5}{(4n+1)(4n+3)(2n+2)},$&lt;/span&gt; [then] &lt;em&gt;each term in the second series exceeds the corresponding term in the first&lt;/em&gt;, [and thus] &lt;em&gt;we conclude that the sum of the first series is less than that of the second&lt;/em&gt;. [Proof of] &lt;span class=&quot;math-container&quot;&gt;$2^{\text o}$&lt;/span&gt; &lt;em&gt;The series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1 - \sqrt{\frac{1}{2}} + \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{4}} + \cdots$&lt;/span&gt; &lt;em&gt;is convergent; but if instead of taking the negative terms&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; &lt;em&gt;by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; &lt;em&gt;we take them as above&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; &lt;em&gt;by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$3,$&lt;/span&gt; [then] &lt;em&gt;the series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1 + \sqrt{\frac{1}{3}} - \sqrt{\frac{1}{2}},\;$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\frac{1}{5}} + \sqrt{\frac{1}{7}} - \sqrt{\frac{1}{4}},\;$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\ldots,\;$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\frac{1}{4n+1}} + \sqrt{\frac{1}{4n+3}} - \sqrt{\frac{1}{2n+2}}\;$&lt;/span&gt; [a typo has been corrected], &lt;em&gt;will be divergent. Cauchy's rule gives, assuming&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; &lt;em&gt;is very large&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$\;\sqrt{\frac{1}{4n+1}} + \sqrt{\frac{1}{4n+3}} - \sqrt{\frac{1}{2n+2}} \; = \; \sqrt{\frac{1}{n}} - \sqrt{\frac{1}{2n}} \; = \; \frac{{\sqrt 2} - 1}{{\sqrt 2}{\sqrt n}},\;$&lt;/span&gt; &lt;em&gt;and&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\;\frac{\ln \left(\frac{{\sqrt n}{\sqrt 2}}{{\sqrt 2} - 1}\right)}{\ln n} \; = \; \frac{1}{2} + \frac{\ln \left(\frac{\sqrt 2}{{\sqrt 2} - 1}\right)}{\ln n} \; &amp;lt; \; 1.$&lt;/span&gt; &lt;em&gt;These examples are taken from a memoir of Mr. Dirichlet, where it is established that any arithmetic progression contains infinitely many prime numbers, when the first term and the common difference are relatively prime&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[7]&lt;/strong&gt; Charles-Adrien Choquet (1798-1880) and Mathias Mayer [Mayer-d'Almbert] (1786-1843), &lt;a href=&quot;https://books.google.com/books?id=wCZ8uWEC_DwC&amp;amp;pg=PA636&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Traité Élémentaire d'Algèbre&lt;/strong&gt;&lt;/a&gt; [Elementary Treatise on Algebra], 5th edition, Bachelier (Paris), 1849, xvi + 638 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There exist 5 editions, published in &lt;a href=&quot;http://catalog.hathitrust.org/Record/009268863&quot; rel=&quot;noreferrer&quot;&gt;1832&lt;/a&gt; (infinite series on pp. 198-201; nothing about rearrangements changing sums), &lt;a href=&quot;https://books.google.com/books?id=5S-NhimIzX0C&amp;amp;pg=PA528&quot; rel=&quot;noreferrer&quot;&gt;1836&lt;/a&gt; (infinite series on pp. 244-247 and pp. 528-532; nothing about rearrangements changing sums), &lt;a href=&quot;https://books.google.com/books?id=byD4vJkTNR4C&amp;amp;pg=PA578&quot; rel=&quot;noreferrer&quot;&gt;1841&lt;/a&gt; (infinite series on pp. 260-264 and pp. 578-583; nothing about rearrangements changing sums), &lt;a href=&quot;https://books.google.com/books?id=bLgUAAAAQAAJ&amp;amp;pg=PA578&quot; rel=&quot;noreferrer&quot;&gt;1845&lt;/a&gt; (infinite series on pp. 260-264 and pp. 578-583; nothing about rearrangements changing sums), and [1849] (above; series on pp. 244-248 and pp. 583-588 and pp. 634-638; the possibility of rearrangements changing sums is discussed on pp. 634-638). In the first four editions the ordering of the authors on the title page differs from that of the 1849 5th edition in that in these earlier editions &amp;quot;Mayer&amp;quot; is listed first and &amp;quot;Choquet&amp;quot; is listed second. Because Mayer died in 1843, the 1845 4th edition and 1849 5th edition were by Choquet only. I do not know whether any corrections and additions to the last two editions may have been influenced by possible notes left by Mayer on suggestions for later editions, but I suspect the additional comments in the 5th edition about rearranging terms in the alternating harmonic series are entirely due to Choquet. Following the 1849 5th edition Choquet published two editions of &lt;strong&gt;Complément d'Algèbre&lt;/strong&gt; (intended as a supplement to the 1849 5th edition), the 1st edition in &lt;a href=&quot;https://books.google.com/books?id=h3lQGk3pkCMC&quot; rel=&quot;noreferrer&quot;&gt;1851&lt;/a&gt; (iv + 51 pages) and the 2nd edition in &lt;a href=&quot;https://books.google.com/books?id=YGxaAAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;1853&lt;/a&gt; (iv + 63 pages), but neither of these includes any general discussions about infinite series. Finally, in 1856 Choquet published &lt;a href=&quot;https://books.google.com/books?id=XBJbAAAAQAAJ&amp;amp;pg=549&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Traité d'Algèbre&lt;/strong&gt;&lt;/a&gt; [Treatise on Algebra], Mallet-Bachelier (Paris), 1856, xvi + 551 pages. On pp. 548-551 of this 1856 book, Choquet discusses rearranging terms of a series. The 1856 version appears to be the same (word-for-word) as the 1849 version. &lt;a href=&quot;https://books.google.com/books?id=f8g9AQAAMAAJ&amp;amp;pg=PA429&quot; rel=&quot;noreferrer&quot;&gt;Review of 5th edition&lt;/a&gt; in Nouvelles Annales de Mathématiques (1) 8 (1849), pp. 429-431 (in French). Some information about Choquet and Mayer and their books can be found &lt;a href=&quot;http://www.numericana.com/fame/bio.htm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; under the entry for &amp;quot;Urbain Le Verrier&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What follows is a translation of pp. 636-638 of the 1849 5th edition. &lt;strong&gt;3.&lt;/strong&gt; &lt;em&gt;In order to show that, when convergence occurs only as a result of the signs, rearrangement of the terms can change the sum or render the series divergent, let us consider the series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + ....$$&lt;/span&gt; &lt;em&gt;Let us take in succession two positive terms and one negative, in the following way&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + ....$$&lt;/span&gt; &lt;em&gt;This new series can be formed from the first, by adding to it that which results from dividing all its terms by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2.$&lt;/span&gt; &lt;em&gt;Thus, if one distributes the terms in groups of four terms, the&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n^{\text{th}}$&lt;/span&gt; &lt;em&gt;group is&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{4n-3} - \frac{1}{4n-2} + \frac{1}{4n-1} - \frac{1}{4n};$$&lt;/span&gt; &lt;em&gt;after the division of all the terms by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; [and then] &lt;em&gt;grouping each term with its successor, the&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n^{\text{th}}$&lt;/span&gt; &lt;em&gt;group is&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{4n-2} - \frac{1}{4n};$$&lt;/span&gt; &lt;em&gt;the sum of these two groups is&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{4n-3} + \frac{1}{4n-1} - \frac{1}{2n},$$&lt;/span&gt; &lt;em&gt;and if in succession&lt;/em&gt; [we use] &lt;span class=&quot;math-container&quot;&gt;$n=1,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$=3,$&lt;/span&gt; &lt;em&gt;etc., the result is, successively, the groups of three terms of the series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1 + \frac{1}{3} - \frac{1}{2} +$&lt;/span&gt; *etc. [&lt;strong&gt;Note:&lt;/strong&gt; I will give a more explicit description. In what follows, &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer (i.e. &lt;span class=&quot;math-container&quot;&gt;$n=1,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$2,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$3,\;\ldots).$&lt;/span&gt; Let Series A be the series whose &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term is the sum of the &lt;span class=&quot;math-container&quot;&gt;$(4n-3)$&lt;/span&gt;'th and &lt;span class=&quot;math-container&quot;&gt;$(4n-2)$&lt;/span&gt;'th and &lt;span class=&quot;math-container&quot;&gt;$(4n-1)$&lt;/span&gt;'th and &lt;span class=&quot;math-container&quot;&gt;$4n$&lt;/span&gt;'th terms of the first series (the alternating harmonic series). Let Series B be the series whose &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt; times the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term of the first series. Let Series C be the series whose &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term is the sum of the &lt;span class=&quot;math-container&quot;&gt;$(2n-1)$&lt;/span&gt;'th and &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt;'th terms of Series B. Let Series D be the series whose &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term is the sum of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term of Series A and the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term of Series C. Then the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term of Series D is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4n-3} + \frac{1}{4n-1} - \frac{1}{2n},$&lt;/span&gt; and this value is also equal to the sum of the &lt;span class=&quot;math-container&quot;&gt;$(3n-2)$&lt;/span&gt;'th and &lt;span class=&quot;math-container&quot;&gt;$(3n-2)$&lt;/span&gt;'th and &lt;span class=&quot;math-container&quot;&gt;$3n$&lt;/span&gt;'th terms of the new series (the rearranged alternating harmonic series).] &lt;em&gt;Let&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_{2n}$&lt;/span&gt; &lt;em&gt;and&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_{4n}$&lt;/span&gt; &lt;em&gt;be the sum of the first&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; &lt;em&gt;and of the first&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$4n$&lt;/span&gt; &lt;em&gt;terms of the first series; that of the first&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2n$&lt;/span&gt; &lt;em&gt;terms of the series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2} - \frac{1}{4} + \frac{1}{6} - \frac{1}{8} +$&lt;/span&gt; &lt;em&gt;etc. will be&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}s_{2n};$&lt;/span&gt; &lt;em&gt;and the sum of the first&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$3n$&lt;/span&gt; &lt;em&gt;terms of the series obtained by rearrangement of the terms will be&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_{4n} + \frac{1}{2}s_{2n}.$&lt;/span&gt; &lt;em&gt;But, if we indefinitely increase the number&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n,$&lt;/span&gt; [then] &lt;em&gt;the sums&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_{4n}$&lt;/span&gt; &lt;em&gt;and&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_{2n}$&lt;/span&gt; &lt;em&gt;converge to the same limit, which is&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$l\,2$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$[= {\ln 2}]$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$(n^{\text{o}}\;534)$&lt;/span&gt; [= (article &lt;span class=&quot;math-container&quot;&gt;$534)].$&lt;/span&gt; &lt;em&gt;Therefore,&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$s_{4n} + \frac{1}{2}s_{2n}$&lt;/span&gt; &lt;em&gt;converges to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}l\,2.$&lt;/span&gt; &lt;em&gt;Thus the series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots$$&lt;/span&gt; &lt;em&gt;has sum&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}l\,2.$&lt;/span&gt; &lt;strong&gt;4.&lt;/strong&gt; &lt;em&gt;By another change in the order of the terms of the same series, we can obtain a divergent series. This would be if we alternately took groups of positive terms, then of negative terms, in which the number of terms increased by stages and indefinitely, in a manner that the value of each group was always greater than a given number, as the group below, for example&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{n} + \frac{1}{n+2} + \frac{1}{n+4} + \cdots + \frac{1}{3n-2},$$&lt;/span&gt; &lt;em&gt;whose value is greater than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{3n-2}$&lt;/span&gt; &lt;em&gt;and therefore greater than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}.$&lt;/span&gt; &lt;strong&gt;5.&lt;/strong&gt; &lt;em&gt;By considering the convergent series&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \cdots,$$&lt;/span&gt; &lt;em&gt;we can even more easily derive a divergent series. It suffices to alternately take two positive terms then a negative term, as follows:&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{7}} - \frac{1}{\sqrt{4}} + ....$$&lt;/span&gt; &lt;em&gt;Indeed, if we put&lt;/em&gt; [In the first equation below have included a plus sign just after the term &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{\sqrt{4}}$&lt;/span&gt; that did not appear in the original.] &lt;span class=&quot;math-container&quot;&gt;$$f(n) \;\; = \;\; 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \cdots - \frac{1}{\sqrt{2n-2}} + \frac{1}{\sqrt{2n-1}} - \frac{1}{\sqrt{2n}},$$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$F(n) \;\; = \;\; 1 + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} + \cdots - \frac{1}{\sqrt{2n-2}} + \frac{1}{\sqrt{4n-3}} + \frac{1}{\sqrt{4n-1}} - \frac{1}{2n},$$&lt;/span&gt; &lt;em&gt;we have&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$$F(n) - f(n) \;=\; \frac{1}{\sqrt{2n+1}} + \frac{1}{\sqrt{2n+3}} + \cdots + \frac{1}{\sqrt{4n-1}}.$$&lt;/span&gt; &lt;em&gt;The second member of this equality is greater than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{n}{\sqrt{4n-1}}$&lt;/span&gt; &lt;em&gt;and therefore greater than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}\sqrt{n}.$&lt;/span&gt; &lt;em&gt;It therefore tends to infinity along with&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$n.$&lt;/span&gt; &lt;em&gt;Therefore it is the same for&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$F(n)-f(n),$&lt;/span&gt; &lt;em&gt;and hence, for&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$F(n),$&lt;/span&gt; &lt;em&gt;since&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$f(n)$&lt;/span&gt; &lt;em&gt;is a finite quantity.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastActivityDate="2022-02-07T16:39:41.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14169" PostTypeId="2" ParentId="344" CreationDate="2022-02-07T16:51:06.213" Score="6" Body="&lt;p&gt;This is the second part of my &amp;quot;preliminary answer&amp;quot;. Both parts together are a little more than a third of what I ultimately intend to post, but I don't know when I'll get around to doing so. However, these two parts are certainly sufficient to give the early historical development of this topic. &lt;em&gt;(later)&lt;/em&gt; It turns out that even the portion I wanted to post goes several thousand characters over the limit, so I'm going to have to include a third part because I want to include some stuff about Paul Laurent.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[8]&lt;/strong&gt; Georg Friedrich Bernhard Riemann (1826-1866), &lt;strong&gt;Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe&lt;/strong&gt; [On the representability of a function by a trigonometric series], Habilitationsschrift Thesis (Göttingen University, under Johann Carl Friedrich Gauss, 1777-1855), December 1853.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Riemann's rearrangement result is stated and proved in the first few sentences of Article 3. Riemann's Thesis is in the form of a handwritten manuscript and it is currently in Riemann's &lt;em&gt;Nachlass&lt;/em&gt; at the Göttingen University Library Archives. Prior to 1868, it is likely that only a few people had seen Riemann's Thesis (mainly Berlin mathematicians, and possibly some others while visiting Berlin), with perhaps a few others having heard about its contents by word of mouth. The first printed and circulated version was published in 1867, pp. 87-131 in &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN250442582_0013&quot; rel=&quot;noreferrer&quot;&gt;Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867)&lt;/a&gt; (JFM 1.0131.03). The 1867 publication was due to the efforts Dedekind. Although Dedekind's introductory footnote on the first page is dated July 1867 and the date given on the title page is 1867, I believe that printed copies did not reach the mathematical public until 1868. A French translation by Jean Gaston Darboux (1831-1917) and Guillaume Jules Hoüel [Houël] (1823-1886) was published in 1873: &lt;em&gt;Sur la possibilité de représenter une fonction par une série trigonométrique&lt;/em&gt;, Bulletin des Sciences Mathématiques et Astronomiques (1) 5 (1873), &lt;a href=&quot;https://eudml.org/doc/84807&quot; rel=&quot;noreferrer&quot;&gt;pp. 20-48&lt;/a&gt; and &lt;a href=&quot;https://eudml.org/doc/84818&quot; rel=&quot;noreferrer&quot;&gt;79-96&lt;/a&gt; (JFM 5.0230.03). English translations were published in Baker/Christenson/Orde's 2004 book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0974042730&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Collected Papers&lt;/strong&gt;&lt;/a&gt; (pp. 219-256) and in Hawking's book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0762430044&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;God Created the Integers&lt;/strong&gt;&lt;/a&gt; (pp. 826-865 in 2005 edition; pp. 992-1031 in 2007 edition).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;First few sentences of Section 3, from pp. 226-227 of the 2004 English translation:&lt;/strong&gt; &lt;em&gt;The question of the representation by trigonometric series of everywhere integrable functions with finitely many maxima and minima&lt;/em&gt; [in a bounded interval] &lt;em&gt;was first settled rigorously by Dirichlet&lt;/em&gt;&lt;span class=&quot;math-container&quot;&gt;$^{26}$&lt;/span&gt; &lt;em&gt;in a paper of January 1829&lt;/em&gt;. {{footnote: &lt;span class=&quot;math-container&quot;&gt;$^{26}$&lt;/span&gt;Crelle's Journal, &lt;em&gt;vol. IV, p. 157&lt;/em&gt;.}} &lt;em&gt;The recognition of the proper way to attack the problem came to him from the insight that infinite series fall into two distinct classes, depending on whether or not they&lt;/em&gt; remain convergent when all the terms are made positive. In the first class the terms can be arbitrarily rearranged; in the second, on the other hand, the value is dependent on the ordering of the terms. Indeed, if we denote the positive terms of a series in the second class by* &lt;span class=&quot;math-container&quot;&gt;$a_{1},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$a_{2},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$a_{3},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\ldots$&lt;/span&gt;, &lt;em&gt;and the negative terms by&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$b_{1},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$b_{2},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$b_{3},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\ldots$&lt;/span&gt;, &lt;em&gt;then it is clear that&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\sum a$&lt;/span&gt; &lt;em&gt;as well as&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\sum b$&lt;/span&gt; &lt;em&gt;must be infinite. For if they were both finite, the series would still be convergent after making all the signs the same. If only one were infinite, then the series would diverge. Clearly now an arbitrarily given value&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; &lt;em&gt;can be obtained by a suitable reordering of the terms. We take alternately the positive terms of the series until the sum is greater than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$C,$&lt;/span&gt; &lt;em&gt;and then the negative terms until the sum is less than&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$C.$&lt;/span&gt; &lt;em&gt;The deviation from&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; &lt;em&gt;never amounts to more than the size of the term at the last place the signs were switched. Now, since the numbers&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; &lt;em&gt;as well as the numbers&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; &lt;em&gt;become infinitely small with increasing index, so also are the deviations from&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$C.$&lt;/span&gt; &lt;em&gt;If we proceed sufficiently far in the series, the deviation becomes arbitrarily small, that is, the series converges to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$C.$&lt;/span&gt; &lt;em&gt;The rules for finite sums only apply to the series of the first class. Only these can be considered as the aggregates of their terms; the series of the second class cannot. This circumstance was overlooked by mathematicians of the previous century, most likely, mainly on the grounds that the series which progress by increasing powers of a variable generally (that is, excluding&lt;/em&gt; [certain isolated] &lt;em&gt;individual values of this variable) belong to the first class. Clearly the Fourier series do not necessarily belong to the first class. The convergence cannot be derived, as Cauchy futilely attempted&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$[\cdots]$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is well known that Dirichlet met with Riemann nearly daily during the 1852 autumn vacation and that much of the first three sections of Riemann's thesis was provided by Dirichlet. Thus, it seems very possible to me that the rearrangement theorem actually originated with Dirichlet or was conceived by them together during these meetings. However, I have not encountered anything that considers this possibility.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are many minor extensions of this result that appeared later, and it might be interesting to document the various explicit appearances of these minor extensions. For example, any conditionally convergent series can be rearranged to additionally have &lt;span class=&quot;math-container&quot;&gt;$+\infty$&lt;/span&gt; as a sum, to have &lt;span class=&quot;math-container&quot;&gt;$-\infty$&lt;/span&gt; as a sum, to oscillate (i.e. &lt;span class=&quot;math-container&quot;&gt;$\liminf$&lt;/span&gt; of partial sums less than &lt;span class=&quot;math-container&quot;&gt;$\limsup$&lt;/span&gt; of partial sums), to oscillate boundedly &lt;span class=&quot;math-container&quot;&gt;$(-\infty &amp;lt; \liminf &amp;lt; \limsup &amp;lt; +\infty),$&lt;/span&gt; to oscillate infinitely &lt;span class=&quot;math-container&quot;&gt;$(-\infty &amp;lt; \liminf &amp;lt; \limsup = +\infty$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-\infty = \liminf &amp;lt; \limsup &amp;lt; +\infty$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$-\infty = \liminf &amp;lt; \limsup = +\infty).$&lt;/span&gt; The most precise such statement is that, given any conditionally convergent series and given any extended real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$-\infty \leq a \leq b \leq +\infty,$&lt;/span&gt; there exists a rearrangement of the terms of the series (in which the original ordering of the positive terms is not changed and the ordering of the negative terms is not changed) such that the &lt;span class=&quot;math-container&quot;&gt;$\liminf$&lt;/span&gt; of the partial sums equals &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and the &lt;span class=&quot;math-container&quot;&gt;$\limsup$&lt;/span&gt; of the partial sums equals &lt;span class=&quot;math-container&quot;&gt;$b.$&lt;/span&gt; Moreover, for each such series and for each such extended real numbers &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b,$&lt;/span&gt; there exist continuum many distict rearrangements of the series having this property. (Of course, for &lt;span class=&quot;math-container&quot;&gt;$a=b,$&lt;/span&gt; &amp;quot;infinitely many&amp;quot; is implict in Ohm's, Schlömilch's, and Pringsheim's results for the cases where they are applicable. In fact, &amp;quot;continuum many&amp;quot; in the case of Pringsheim's results.) And one could additionally ask whether continuum many such rearrangements exist each of which has the property that every real number between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is a subseries sum, or even more, whether every real number between &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is a subseries sum in which the indexing of the terms of the subseries has upper density &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; in the sequence of natural numbers, and there are other possibilities. In the references that follow I have mostly ignored these extensions because my lack of Foreign language reading knowledge would make it extremely difficult to determine EXACTLY what an author states, what an author claims to prove, and what is a reasonable (but possibly ahistorical) claim for what an author actually proves. Nonetheless, I do give discussions of such extensions in Dini (1868), Capelli/Garbieri (1886), Trachtenberg (1903), Dini (1907), and Hobson (1907).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[9]&lt;/strong&gt; Joseph Louis François Bertrand (1822-1900), &lt;a href=&quot;https://books.google.com/books?id=LWHenJr5i1oC&amp;amp;pg=PA224&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Traité d'Algèbre&lt;/strong&gt;&lt;/a&gt; [Elementary Treatise on Algebra], 2nd edition, L. [Louis] Hachette et Compagnie (Paris), 1855, ii + 495 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The 1851 1st edition was titled &lt;a href=&quot;https://books.google.com/books?id=0FSuyl077NwC&amp;amp;pg=PA405&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Traité Élémentaire d'Algèbre&lt;/strong&gt;&lt;/a&gt;, but after the 1st edition the title was shortened by omitting the word &amp;quot;Élémentaire&amp;quot;. The 1851 1st edition has a short section on infinite series (Chapter XXII, Articles 281-289, pp. 405-412), but there seems to be no mention of the possibility of obtaining different sums by rearranging the terms in an infinite series. However, rearrangements are discussed in the 1855 2nd edition in Article 202 (pp. 224-225). In Article 202 Bertrand considers the alternating harmonic series, &lt;span class=&quot;math-container&quot;&gt;$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots,$&lt;/span&gt; and its rearrangement in which two positive terms are followed by one negative term, &lt;span class=&quot;math-container&quot;&gt;$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} + \cdots$&lt;/span&gt; Bertrand explains (incorrectly) that if, in each series, we consider the partial sum up to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2n-1},$&lt;/span&gt; then the second series includes an additional &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; many negative terms, namely &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{n},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{n+1},\;\ldots$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{2n-2},$&lt;/span&gt; and thus the positive difference between the two partial sums exceeds &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2n-2},$&lt;/span&gt; and hence the sums of the two series differ by at least &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{n \rightarrow \infty}\frac{n-1}{2n-2} = \frac{1}{2}.$&lt;/span&gt; This conclusion is of course incorrect, since the actual difference between the sums is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}\ln 2 - \ln 2 = \frac{1}{2} \ln {2},$&lt;/span&gt; which is less than &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2} \ln e = \frac{1}{2}.$&lt;/span&gt; To correct Bertrand's discussion, we note that we want to consider the partial sums up to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2n-1}$&lt;/span&gt; for &lt;strong&gt;even&lt;/strong&gt; values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (a minor oversight), and in doing so the the second series includes an additional &lt;span class=&quot;math-container&quot;&gt;$n/2$&lt;/span&gt; many negative terms, namely &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{n},$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{n+2},\;\ldots,$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$-\frac{1}{2n-2},$&lt;/span&gt; and thus the positive difference between the two partial sums exceeds &lt;span class=&quot;math-container&quot;&gt;$n/2$&lt;/span&gt; times &lt;span class=&quot;math-container&quot;&gt;$\frac{1} {2n-2},$&lt;/span&gt; and hence the sums of the two series differ by at least &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{n \rightarrow \infty}\frac{n/2}{2n-2} = \frac{1}{4}.$&lt;/span&gt; The corresponding discussion for rearranging terms in an infinite series is treated differently in &lt;a href=&quot;https://books.google.com/books?id=MQ4PAAAAQAAJ&amp;amp;pg=RA1-PA18&quot; rel=&quot;noreferrer&quot;&gt;Part Two of the 1863 3rd edition&lt;/a&gt;, which was co-authored with Paul Henri Garcet (1815-1871). (Beginning with the 3rd edition, Bertrand's &lt;strong&gt;Traité d'Algèbre&lt;/strong&gt; was published in two separately paged parts, which were sometimes bound together and sometimes not bound together.) In the 1863 edition of Part Two, the discussion takes place on pp. 17-19, and for the same two series the authors show by an algebraic manipulation method (same as used in Scheibner (1860) below) that the sum of the rearranged series is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; times the sum of the original series. Bertrand does not mention anyone (Dirichlet, Ohm, etc.) regarding this topic in any of the editions of his book I've seen. From what I've been able to determine, the corresponding discussion of rearrangements in the many later editions of Bertrand's &lt;strong&gt;Traité d'Algèbre&lt;/strong&gt; seems to be unchanged from that given in the 1863 3rd edition. &lt;a href=&quot;https://books.google.com/books?id=-sg9AQAAMAAJ&amp;amp;pg=PA154&quot; rel=&quot;noreferrer&quot;&gt;The 1851 1st edition is reviewed&lt;/a&gt; in Nouvelles Annales de Mathématiques (1) 10 (1851), pp. 154-156 (only discusses the Appendix, pp. 405-532). &lt;a href=&quot;https://books.google.com/books?id=SbsKAAAAYAAJ&amp;amp;pg=PA34&quot; rel=&quot;noreferrer&quot;&gt;The 1900 17th edition is reviewed&lt;/a&gt; in Mathematical Gazette 2 #26 (March 1901), pp. 34-35.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[10]&lt;/strong&gt; &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Schlomilch.html&quot; rel=&quot;noreferrer&quot;&gt;Otto [Oscar, Oskar] Xaver Schlömilch&lt;/a&gt; (1823-1901), &lt;a href=&quot;https://books.google.com/books?id=4LsKAAAAIAAJ&amp;amp;pg=PA180&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Bemerkung über unendliche reihen&lt;/em&gt;&lt;/a&gt; [Comments about infinite series], &lt;strong&gt;Zeitschrift für Mathematik und Physik&lt;/strong&gt; 1 (1856), 180-181.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This paper discusses problems with certain manipulations of conditionally convergent series, and Ohm's name seems to be mentioned in the last paragraph. The paper does not include any specific series having a rearrangement with a different sum, unless such an example is described verbally. I was not able to obtain a coherent translation of most of this paper using online translation devices, even when I typed in the entire paper for this purpose.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[11]&lt;/strong&gt; &lt;a href=&quot;http://en.wikipedia.org/wiki/Auguste_Joseph_Alphonse_Gratry&quot; rel=&quot;noreferrer&quot;&gt;Auguste Joseph Alphonse Gratry&lt;/a&gt; (1805-1872), &lt;a href=&quot;https://books.google.com/books?id=DQ8-AAAAcAAJ&amp;amp;pg=PR79#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Philosophie. Logique&lt;/strong&gt;&lt;/a&gt; [Philosophy. Logic], Volume 1, 4th edition, Charles Douniol (Paris) and J. Lecoffre &amp;amp; Cie (Paris), 1858, cxliv + 417 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There exist 5 editions, published in &lt;a href=&quot;https://archive.org/details/logiqueg01grat&quot; rel=&quot;noreferrer&quot;&gt;1855&lt;/a&gt;, 185?, &lt;a href=&quot;https://archive.org/details/logiqu00grat&quot; rel=&quot;noreferrer&quot;&gt;1858&lt;/a&gt;, 1858 (above), and &lt;a href=&quot;https://archive.org/details/philosophielogi02gratgoog&quot; rel=&quot;noreferrer&quot;&gt;1868&lt;/a&gt;. (By &amp;quot;published in&amp;quot;, I mean the year given on the title page. I have found that many internet and digitized references give 1856 as the date of the 1st edition, while others give 1855, and no one seems to say anything about this discrepancy.) The comments below about rearranging conditionally convergent series &lt;strong&gt;do not&lt;/strong&gt; appear in the 1st edition. I have not been able to examine the 2nd edition. The comments &lt;strong&gt;do appear&lt;/strong&gt; in the 3rd, 4th, and 5th editions. An English translation &lt;a href=&quot;http://catalog.hathitrust.org/Record/001387389&quot; rel=&quot;noreferrer&quot;&gt;by Helen Singer and Milton Borah Singer&lt;/a&gt; of the 1868 5th edition was published by The Open Court Publishing Company in 1944 (xii + 628 pages). &lt;strong&gt;(from pp. 65-66 of the 1944 English edition)&lt;/strong&gt; [p. lxxix of the 3rd &amp;amp; 4th editions; p. 80 of the 5th edition] &lt;em&gt;The following example will make what I call the abyss which separates the infinite from the finite understandable, and show why it is not possible to infer from the one to the other. It is the example of semi-convergent series, where the infinite introduces singular properties, different from those of the finite. Take in this series a finite number of terms as large as you please. Their sum will obviously always be the same in whatever order they are added. But who would believe that this same series, assumed to be infinite, takes on the inexplicable characteristic, that the sum of its terms is different when they are added in a different order? Let the series be, for example&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \frac{1}{7}.$&lt;/span&gt; [&lt;strong&gt;Note:&lt;/strong&gt; In the original 3rd, 4th, 5th French editions, ellipses &lt;span class=&quot;math-container&quot;&gt;$\ldots$&lt;/span&gt; &lt;strong&gt;were used&lt;/strong&gt; here and just below.] &lt;em&gt;Assume this to be an infinite series. The sum of its terms in the order given is equal to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\log_{e}2.$&lt;/span&gt; &lt;em&gt;But arrange the terms as follows:&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4}.$&lt;/span&gt; &lt;em&gt;Then the sum of the terms changes and becomes equal to&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2} \log_{e}2.$&lt;/span&gt; &lt;em&gt;Hence, if we wished, accordingly, to infer from the finite to the infinite; if, after having demonstrated what is obvious in advance, that the terms of the series prolonged as far as we please, but still finite, always have the same sum in whatever order the addition is made, we then wished to maintain that it should be the same when the series is supposed infinite, that would be an error&lt;/em&gt;. In a lengthy footnote (footnote 10 on pp. 66-67 of the English edition; footnote 1 on pp. lxxx-lxxxi of the 3rd &amp;amp; 4th editions; footnote 1 on pp. 81-82 of the 5th edition), Gratry observes that all the partial sums of the series &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \cdots$&lt;/span&gt; will be fractions that (when reduced to lowest terms) have an odd numerator and an even denominator, but the sum of the corresponding infinite series is &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{3}.$&lt;/span&gt; Gratry also mentions a theorem of Cauchy's on the (Taylor) expansion of &lt;span class=&quot;math-container&quot;&gt;$f(x+h)$&lt;/span&gt; in which exact and precise conditions are stated for it to be valid, and then he makes the following jab at his fellow philosophers: &lt;em&gt;If philosophers worked in the same fashion&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$[\ldots]$&lt;/span&gt; &lt;em&gt;they also would make discoveries&lt;/em&gt;. After this, Gratry mentions a letter of Abel's (in his &lt;em&gt;Oeuvres&lt;/em&gt;, II, p. 266) in which Abel makes his now well-known comment that he does not know of any infinite series whose sum has been rigorously determined. (I haven't looked at the particular letter cited, but I suspect Gratry overstates what Abel actually said, because I believe Abel said except for simple cases like geometric series. For instance, see the more complete account on pp. 1-2 of the 1843 English translation of Ohm's book above.) &lt;a href=&quot;http://www.jstor.org/stable/2019913&quot; rel=&quot;noreferrer&quot;&gt;Review by Ernest Nagel (1901-1985)&lt;/a&gt; of the 1944 English translation in Journal of Philosophy 42 #21 (11 October 1945), 580-582. &lt;a href=&quot;http://www.jstor.org/stable/2181560&quot; rel=&quot;noreferrer&quot;&gt;Review by Ray Harbaugh Dotterer (1880-1967)&lt;/a&gt; of the 1944 English translation in Philosophical Review 54 #6 (November 1945), 623-624.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[12]&lt;/strong&gt; Charles Hippolyte Berger (1822-1869), &lt;a href=&quot;https://books.google.com/books?id=eGxaAAAAcAAJ&amp;amp;pg=PA34&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Théorie Élémentaire des Séries&lt;/strong&gt;&lt;/a&gt; [Elementary Theory of Series], 2nd edition, Mallet-Bachelier (Paris), 1859, iv + 5-47 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The 1st edition (which I have not seen a copy of) is 24 pages and titled &lt;strong&gt;Leçons sur les Séries&lt;/strong&gt; [Lectures on Series]. The 1st edition appeared in December 1858 and I believe it existed only in lithographic form and had very limited circulation, mainly at the Academy of Montpellier. The two examples Dirichlet gave in 1837 are discussed on pp. 33-35. No mathematician's name is mentioned in this discussion, although several mathematicians (but not Dirichlet) are mentioned in the &lt;em&gt;Avertissement&lt;/em&gt; [Preface] on pp. iii-iv. Berger first discusses the rearrangement of the alternating &amp;quot;square root&amp;quot; harmonic series in which two positive terms are followed by one negative term, showing divergence by making a direct comparison (after some algebraic manipulation) with the series whose &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;'th term is &lt;span class=&quot;math-container&quot;&gt;$\frac{\sqrt{2} - 1}{\sqrt{2} \sqrt{n} + 1}.$&lt;/span&gt; Berger obtains the divergence of this latter series by using is &lt;em&gt;Sixième Théorème&lt;/em&gt; on p. 19, a result that is sometimes called the logarithmic test and which is due to Cauchy (1821). However, Berger then follows this with a remark (on p. 34) in which he shows that the divergence can also be obtained by making an appropriate direct comparison with (a positive multiple of) the harmonic series. Berger then discusses the rearrangement of the alternating harmonic series in which two positive terms are followed by one negative term, showing that the rearranged series converges and that its sum is different from the sum of the alternating harmonic series. Later (on p. 38), Berger happens to mention that the sum of the alternating harmonic series is &lt;span class=&quot;math-container&quot;&gt;$\ln {2}$&lt;/span&gt; (a result that I do not believe Berger obtains anywhere in his book) but, as far as I can tell, Berger does not mention that the rearranged alternating harmonic series has sum &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}\ln {2},$&lt;/span&gt; nor does his discussions concerning the rearranged alternating harmonic series allow one to (easily) conclude that its sum is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; times the sum of the alternating harmonic series.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[13]&lt;/strong&gt; &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Biographies/Catalan.html&quot; rel=&quot;noreferrer&quot;&gt;Eugène Charles Catalan&lt;/a&gt; (1814-1894), &lt;a href=&quot;https://archive.org/details/traitlmentaired01catagoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Traité Élémentaire des Séries&lt;/strong&gt;&lt;/a&gt; [Elementary Treatise on Series], Leiber and Faraguet (Paris), 1860, viii + 132 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The &lt;em&gt;Avant-Propos&lt;/em&gt; [Introduction] is dated 15 February 1860. I believe this may be the earliest textbook entirely devoted to infinite series. (Stern's 1860 &lt;a href=&quot;https://archive.org/details/lehrbuchderalge00stergoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Lehrbuch der Algebraischen Analysis&lt;/strong&gt;&lt;/a&gt; is arguably a contender for this. Incidentally, Stern's book looks like it would say something about rearrangement of series somewhere, but I've been through it page-by-page twice and can find nothing that appears to be such a discussion. However, because I cannot read German at all, there may have been a verbal discussion on rearrangement of terms that I overlooked.) Chapter VII begins with Article 197 (p. 121) in which Catalan introduces the idea of transforming slowly converging alternating series in order to speed up their convergence for computational purposes. In this discussion Catalan includes the following footnote [translated from French]: &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$(^{*})$&lt;/span&gt; Not content to solve this problem, several geometers have claimed &lt;em&gt;to transform certain divergent series into convergent series&lt;/em&gt;. We believe that this statement is nonsense.&amp;quot; Since Cauchy, Dirichlet, and Ohm would probably not be called &amp;quot;geometers&amp;quot;, I wonder if the reference could be to Riemann, or perhaps to some Italian mathematicians as a result of Cauchy's early 1830s stay in Turin. Or it might have something to do with the 1858 visits by the Italian mathematicians Betti, Brioschi, and Casorati that I mention in the discussion of Novi's 1863 book below. Thus far I have not seen any mention in the published literature about this comment by Catalan. However, at the bottom of p. 250 of Bertrand (1864) (his calculus text; see below) there is also a comment about &amp;quot;geometers&amp;quot;. &lt;a href=&quot;https://archive.org/details/zeitschriftfrma26runggoog&quot; rel=&quot;noreferrer&quot;&gt;Review by Otto [Oscar, Oskar] Xaver Schlömilch (1823-1901)&lt;/a&gt; [select &amp;quot;Read Online&amp;quot;, then page indicator 530/570] in Zeitschrift für Mathematik und Physik 5 (1860), &lt;em&gt;Literaturzeitung&lt;/em&gt; [Literary supplement], p. 75 (in German; separately paged). &lt;a href=&quot;https://books.google.com/books?id=OCYCAAAAYAAJ&amp;amp;pg=PA121&quot; rel=&quot;noreferrer&quot;&gt;Another review&lt;/a&gt; is in Heidelberger Jahrbücher der Literatur 54 (1861), 121-126 (in German). The last paragraph of Schlömilch's review follows [translated from German]: &lt;em&gt;Some key questions remain unanswered, for example under what circumstances is the sum of a series a continuous function and whether and under what conditions the&lt;/em&gt; [term-by-term] &lt;em&gt;differentiation of a series is allowed, etc.; also lacks the theory of double series, which plays an important role, especially in transformations&lt;/em&gt; [of series].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[14]&lt;/strong&gt; &lt;a href=&quot;http://de.wikipedia.org/wiki/Wilhelm_Scheibner&quot; rel=&quot;noreferrer&quot;&gt;Wilhelm Scheibner&lt;/a&gt; (1826-1908), &lt;a href=&quot;https://archive.org/details/berunendlichere00schegoog&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Über Unendliche Reihen und deren Convergenz&lt;/strong&gt;&lt;/a&gt; [On Infinite Series and Their Convergence], Verlag von S. Hirzel (Leipzig), 1860, 48 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The date 20 September 1860 appears on p. 3. In a footnote on p. 10 Scheibner makes reference to p. 48 of Dirichlet's 1837 paper. On p. 10 Scheibner reminds the reader that the alternating harmonic series has sum &lt;span class=&quot;math-container&quot;&gt;$\ln {2},$&lt;/span&gt; and then he shows by an algebraic manipulation method that, by rearranging terms in the alternating harmonic series so that two positive terms are followed by one negative term, the resulting sum is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; times the sum of the original series. Scheibner's derivation of this result makes use of an algebraic manipulation method that reappears in a nearly identical way in many of the later textbook references that follow, such as the two 1862 editions of Schlömilch's books, p. 251 of Bertrand (1864), pp. 289-290 of Niewenglowski (1889), pp. 103-104 of Harkness/Morley (1898), pp. 21-22 of Whittaker (1902), pp. 45-46 of Godefroy (1903), pp. 96-97 of Manning (1906). The algebraic &amp;quot;bookkeeping work&amp;quot; is presented in a slightly different (but equivalent form) in Harkness/Morley, Whittaker, and Manning, but in these other references the details are almost exactly those on p. 10 of Scheibner's work. Scheibner's work is the earliest publication I know of that gives this algebraic manipulation method. &lt;a href=&quot;https://books.google.com/books?id=1CUCAAAAYAAJ&amp;amp;pg=PA823&quot; rel=&quot;noreferrer&quot;&gt;Review&lt;/a&gt; in Heidelberger Jahrbücher der Literatur 53 (1860), pp. 823-828 (in German). In this review, the topic of rearranging terms in an infinite series is mentioned at the bottom of p. 825. For some biographical and bibliographical information about Scheibner, see Poggendorff (1863, M-Z, middle of p. 783; 1898, M-Z, middle left column p. 1181). Curiously, neither of the Poggendorff entries mentions this 1860 publication by Scheibner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[15]&lt;/strong&gt; Schlömilch, &lt;a href=&quot;https://books.google.com/books?id=7rqRCktV_K8C&amp;amp;pg=PA116&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Handbuch der Algebraischen Analysis&lt;/strong&gt;&lt;/a&gt; [Handbook of Algebraic Analysis], 3rd edition, Friedrich Frommann (Jena, Germany), 1862, viii + 414 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There exist 6 editions, published in &lt;a href=&quot;https://books.google.com/books?id=hhZfAAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;1845&lt;/a&gt;, &lt;a href=&quot;https://archive.org/details/handbuchderalge00schlgoog&quot; rel=&quot;noreferrer&quot;&gt;1851&lt;/a&gt;, 1862 (above), 1868, 1873, and 1881 (reprinted in &lt;a href=&quot;https://books.google.com/books?id=gf4hAQAAMAAJ&amp;amp;pg=PA121&quot; rel=&quot;noreferrer&quot;&gt;1889&lt;/a&gt;). The topic of rearranging terms in an infinite series to obtain a different sum does not seem to appear in the 1845 1st edition or in the 1851 2nd edition. In the 1862 3rd edition this topic is discussed at the beginning of §29. &lt;em&gt;Bedingte und unbedingte Convergenz&lt;/em&gt; (pp. 116-117). Schlömilch shows by an algebraic manipulation method (same as used on p. 10 of Scheibner (1860)) that, by rearranging terms in the alternating harmonic series so that two positive terms are followed by one negative term, the resulting sum is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; times the sum of the original series. Schlömilch does not mention anyone (Dirichlet, Ohm, etc.) regarding this topic. I have not seen the 4th or 5th editions, so I do not know how the treatment of this topic might be different in these two editions. In the 1881/1889 6th edition this topic is discussed a little more completely in §30. &lt;em&gt;Bedingte und unbedingte Convergenz&lt;/em&gt; (pp. 121-123). Among other additions, there is a footnote that mentions Dirichlet's 1837 paper and the result in Schlömilch (1872) and Schlömilch (1873) (see below). &lt;a href=&quot;https://books.google.com/books?id=inRBAAAAYAAJ&amp;amp;pg=PA889&quot; rel=&quot;noreferrer&quot;&gt;Review by Georg Wilhelm Strauch (1811-1868)&lt;/a&gt; in Heidelberger Jahrbücher der Literatur 38 (1845), pp. 889-915 (in German). &lt;a href=&quot;https://books.google.com/books?id=h8wLAAAAYAAJ&amp;amp;pg=PA512&quot; rel=&quot;noreferrer&quot;&gt;Review by Guillaume Jules Hoüel (1823-1886)&lt;/a&gt; in Nouvelles Annales de Mathématiques (2) 3 (1864), pp. 512-518 (in French). &lt;a href=&quot;https://books.google.com/books?id=itVEAAAAcAAJ&amp;amp;pg=RA1-PA36&quot; rel=&quot;noreferrer&quot;&gt;Review&lt;/a&gt; in Zeitschrift für Mathematik und Physik 10 (1865), Literaturzeitung [Literary Supplement], p. 36 (in German and French; separately paged; not an independent review--refers the reader to Hoüel's 1864 review and gives an excerpt from Hoüel's review).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[16]&lt;/strong&gt; Schlömilch, &lt;a href=&quot;https://books.google.com/books?id=CdFJAAAAMAAJ&amp;amp;pg=PA180&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Compendium der Höheren Analysis&lt;/strong&gt;&lt;/a&gt; [Compendium of Higher Analysis], Friedrich Vieweg und Sohn, 1862, xii + 559 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There exist 5 editions, published in &lt;a href=&quot;https://books.google.com/books?id=T0g1AAAAcAAJ&amp;amp;pg=PA151&quot; rel=&quot;noreferrer&quot;&gt;1853&lt;/a&gt;, 1862 (above), &lt;a href=&quot;https://books.google.com/books?id=etNLAAAAYAAJ&amp;amp;pg=PA180&quot; rel=&quot;noreferrer&quot;&gt;1868&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=wQcAAAAAMAAJ&amp;amp;pg=PA180&quot; rel=&quot;noreferrer&quot;&gt;1874&lt;/a&gt;, and &lt;a href=&quot;https://books.google.com/books?id=CdFJAAAAMAAJ&amp;amp;pg=PA180&quot; rel=&quot;noreferrer&quot;&gt;1881&lt;/a&gt;. In the 1853 1st edition infinite series are discussed in Chapter VIII (pp. 151-162), but I can find no discussion in the 1853 edition about the possibility of rearranging the terms of an infinite series to obtain a different sum. In the 1862 2nd edition (on pp. 180-181) Schlömilch shows by an algebraic manipulation method (same as used on p. 10 of Scheibner (1860)) that, by rearranging terms in the alternating harmonic series so that two positive terms are followed by one negative term, the resulting sum is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; times the sum of the original series. Schlömilch's discussion of this is nearly identical to the discussion he gives on pp. 116-117 of the 1862 3rd edition of his book &lt;strong&gt;Handbuch der Algebraischen Analysis&lt;/strong&gt;. His discussion in &lt;strong&gt;Compendium ...&lt;/strong&gt; appears to be unchanged (identical wording and identical math expressions) in all the later editions, and no person's name (Dirichlet, Ohm, etc.) is mentioned regarding this topic. &lt;a href=&quot;https://books.google.com/books?id=0KgKAAAAIAAJ&amp;amp;pg=PA497&quot; rel=&quot;noreferrer&quot;&gt;Review by Johann Albert Arndt (1811-1882)&lt;/a&gt; in Archiv der Mathematik und Physik (1) 20 (1853), pp. 977-988 and 999-1005 (in German). &lt;a href=&quot;https://books.google.com/books?id=DsMKAAAAIAAJ&amp;amp;pg=RA2-PA27&quot; rel=&quot;noreferrer&quot;&gt;Review by Schlömilch&lt;/a&gt; in Zeitschrift für Mathematik und Physik 8 (1863), Literaturzeitung [Literary Supplement], pp. 27-28 (in German; separately paged; not a review--a reprint of the 2nd edition preface). &lt;a href=&quot;https://books.google.com/books?id=qUlLAAAAMAAJ&amp;amp;pg=PA385&quot; rel=&quot;noreferrer&quot;&gt;Review by Guillaume Jules Hoüel (1823-1886)&lt;/a&gt; in Nouvelles Annales de Mathématiques (2) 9 (1870), pp. 385-392 (in French; follow-up comments by &lt;a href=&quot;https://books.google.com/books?id=SMo9AQAAMAAJ&amp;amp;pg=PA217&quot; rel=&quot;noreferrer&quot;&gt;Louis Philippe Gilbert (1832-1892)&lt;/a&gt; in NAM (2) 11, 1872, pp. 217-221).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastActivityDate="2022-02-07T16:51:06.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14170" PostTypeId="2" ParentId="344" CreationDate="2022-02-07T16:57:29.307" Score="4" Body="&lt;p&gt;This is the third part of my &amp;quot;preliminary answer&amp;quot;. At the end of this part we're only up to 1867, although what I've done (which needs a few additions I've found in the last several years, along with my intended re-proofing of everything) goes up to about 1919, and thus the majority of the material is not included in these three parts.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;[17]&lt;/strong&gt; &lt;a href=&quot;http://www-history.mcs.st-and.ac.uk/Biographies/Laurent_Hermann.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Mathieu Hermann Laurent&lt;/a&gt; (1841-1908), &lt;a href=&quot;https://books.google.com/books?id=hQfj8Rx_sc4C&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Théorie des Séries&lt;/strong&gt;&lt;/a&gt; [Theory of Series], Mallet-Bachelier (Paris), 1862, viii + 124 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; P. M. H. Laurent is not the Laurent that the Laurent expansions in complex analysis are named after. This other Laurent is Pierre Alphonse Laurent (1813-1854), and there appears to be no family/blood relationship between the two Laurents. The table of contents is on pp. 123-124. Laurent knew of Catalan's 1860 book -- see the discussion on p. 6 of Laurent's book. Laurent discusses problems with rearranging terms of the alternating harmonic series on pp. 15-19, and problems with the Cauchy product of conditionally convergent series on pp. 28-29. The example Laurent considers on pp. 15-16 is the same as the first rearrangement example in V. Lebesgue's 1845 paper, but Laurent provides a few more details. This same example using the same method (in which the rearranged series is shown to have a different sum by showing that the difference between their partial sums is bounded away from zero) reappears in Laurent's 1867 text (pp. 286-287), in Serret's 1868 text (pp. 143-145), and in Osgood's 1897 text (p. 43). Incidentally, there is an error in Laurent's general representation of the 3-grouped terms &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{1}{1} + \frac{1}{3} - \frac{1}{2}\right),$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{1}{5} + \frac{1}{7} - \frac{1}{4}\right),$&lt;/span&gt; etc. Laurent uses &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{1}{4n-1} + \frac{1}{4n+1} - \frac{1}{2n}\right),$&lt;/span&gt; which does not correctly give the 3-grouped terms for &lt;span class=&quot;math-container&quot;&gt;$n = 1,\,2,\,3,\,\ldots.$&lt;/span&gt; A correct representation for &lt;span class=&quot;math-container&quot;&gt;$n = 1,\,2,\,3,\,\ldots$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{1}{4n-3} + \frac{1}{4n-1} - \frac{1}{2n}\right),$&lt;/span&gt; and a correct representation for &lt;span class=&quot;math-container&quot;&gt;$n = 0,\,1,\,2,\,\ldots$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\left(\frac{1}{4n+1} + \frac{1}{4n+3} - \frac{1}{2n+2}\right)$&lt;/span&gt; (what V. Lebesgue used). Correcting this error involves some minor subsequent changes (e.g. &lt;span class=&quot;math-container&quot;&gt;$n \times \frac{1}{4n+1}$&lt;/span&gt; changes to &lt;span class=&quot;math-container&quot;&gt;$n \times \frac{1}{4n-1}$&lt;/span&gt; in the 2nd line after equation (3)), but the error itself does not affect the correctness of Laurent's conclusion that the two sums differ by at least &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}.$&lt;/span&gt; For some biographical and bibliographical information about P. M. H. Laurent, see Poggendorff (1898, A-L, middle right column p. 780 to top left column p. 781) and see &lt;a href=&quot;https://medihal.archives-ouvertes.fr/IMB_UMR5584/hal-01084722v2&quot; rel=&quot;nofollow noreferrer&quot;&gt;this 25 November 2014&lt;/a&gt; manuscript. &lt;a href=&quot;http://www.numdam.org/numdam-bin/item?id=NAM_1863_2_2__25_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Review by Pierre Marie Eugène Prouhet (1817-1867)&lt;/a&gt; in Nouvelles Annales de Mathématiques (2) 2 (1863), pp. 25-31 (in French).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[18]&lt;/strong&gt; Schlömilch, &lt;a href=&quot;http://gdz.sub.uni-goettingen.de/en/dms/loader/toc/?PPN=PPN599415665_0007&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Ueber die bedingt convergirenden reihen&lt;/em&gt;&lt;/a&gt; [On conditionally convergent series], &lt;strong&gt;Zeitschrift für Mathematik und Physik&lt;/strong&gt; 7 (1862), 283-284.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The paper does not give an author's name. However, Schlömilch's authorship can be verified from an index entry on p. iii at the beginning of the journal volume (see first page of &lt;em&gt;Inhalt&lt;/em&gt;, under the listings for &lt;em&gt;Arithmetik und Analysis&lt;/em&gt;) and from the last paper listed for 1862 on p. 274 of Moritz Cantor's 1901 &lt;a href=&quot;http://archiv.ub.uni-heidelberg.de/volltextserver/13393/1/schloemilch.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nachruf an Oskar Schlömilch&lt;/a&gt;. Schlömilch begins by saying that originating with a remark by Dirichlet (no reference is given), we know that we are (in general) not allowed to arbitrarily rearrange the terms in an infinite series. Schlömilch gives the two examples that Dirichlet gave in his 1837 paper. For the rearrangement in which two positive terms are followed by one negative term, Schlömilch simply states that one converges to &lt;span class=&quot;math-container&quot;&gt;$\ln 2$&lt;/span&gt; and the other converges to &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}\ln 2.$&lt;/span&gt; For the other example (same as previous example except that square roots are present on the terms), Schlömilch gives an elegant argument--nicer than the argument in V. Lebesgue's 1845 paper, in my opinion--for the divergence of the rearranged series. This elegant argument is repeated in Schlömilch (1881, top half of p. 31).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[19]&lt;/strong&gt; Giovanni [Jean] Novi (1827-1866), &lt;a href=&quot;https://books.google.com/books?id=HHSyF241WYsC&amp;amp;pg=PA70&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Trattato di Algebra Superiore&lt;/strong&gt;&lt;/a&gt; [Treatise of Higher Algebra], Parte Prima. Analisi Algebrica [First Part. Algebraic Analysis], Le Monnier (Firenze [Florence], Italy), 1863, viii + 458 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This was intended to be the first volume of a planned 3-volume treatise, based on a manuscript (presumably handwritten) for lectures given by Enrico Betti (1823-1892) during 1858-1859, but only this first volume was ever published. The two examples Dirichlet gave in 1837 are discussed on pp. 70-71. No mathematician's name is mentioned here, but in a footnote on p. 73 Novi cites Dirchlet's 1837 paper for the result that all rearrangements of an absolutely convergent series of complex numbers have the same sum. For Dirichlet's alternating harmonic series example in which the rearrangement is where two positive terms are followed by one negative term, Novi uses the standard algebraic manipulation method to show that the sum of the rearranged series is &lt;span class=&quot;math-container&quot;&gt;$\frac{3}{2}$&lt;/span&gt; times the sum of the original series. In the case of the corresponding rearrangement of the alternating &amp;quot;square root&amp;quot; harmonic series, Novi shows by an elementary argument that the rearranged series diverges to &lt;span class=&quot;math-container&quot;&gt;$+\infty.$&lt;/span&gt; Regarding Betti, I have looked through some of his publications and made some internet searches (but I did not put a lot of effort into this) and I have not found any explicit discussion by him about series rearrangements having different sums. However, I do know that during the autumn 1858 vacation the Italian mathematicians Betti, Brioschi, Casorati all visited universities in Göttingen, Berlin, Paris and met Dirichlet, Riemann, Weierstrass, Bertrand, and others. See p. 468 &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B0050X870C&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and the bottom of p. 529 of Baker/Christenson/Orde (2004) (see the entry for Riemann's trigonometric series thesis above). Perhaps Betti learned about rearranging conditionally convergent series at this time, which took place just before (or soon after) Betti began giving his lectures on which Novi's book is based. For some additional information about Novi's book and Betti's algebra course, see p. 204 of Laura Martini, &lt;a href=&quot;http://www.sciencedirect.com/science/article/pii/S0315086099922498&quot; rel=&quot;nofollow noreferrer&quot;&gt;*The first lectures in Italy on Galois theory: Bologna, 1886-1887&lt;/a&gt;, Historia Mathematica 26 #3 (August 1999), 201-223. &lt;a href=&quot;https://books.google.com/books?id=xDA_AQAAMAAJ&amp;amp;pg=PA90&quot; rel=&quot;nofollow noreferrer&quot;&gt;Review in&lt;/a&gt; in Nouvelles Annales de Mathématiques (2) 3 (1864), pp. 90-91.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[20]&lt;/strong&gt; &lt;a href=&quot;http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Biographies/Bertrand.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Louis François Bertrand&lt;/a&gt; (1822-1900), &lt;a href=&quot;https://archive.org/details/traitdecalculdi00bertgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Traité de Calcul Différentiel et de Calcul Intégral. Première Partie. Calcul Différentiel&lt;/strong&gt;&lt;/a&gt; [Treatise on Differential and Integral Calculus. First Part. Differential Calculus], Gauthier-Villars, 1864, xliv + 780 pages. JFM 2.0298.01&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Preface is rather long (44 pages, on pp. i-xliv). The topic of rearranging terms of a conditionally convergent series is discussed in Article 248 (pp. 250-251). Bertrand mentions (bottom p. 250) that [translated from French] &amp;quot;Dirichlet, who was the first to call this point to the attention of the geometers, gave the following example:&amp;quot; Bertrand's example is the alternating harmonic series and he shows that rearranging the terms so that two positive terms are followed by one negative term gives a sum that is &lt;span class=&quot;math-container&quot;&gt;$3/2$&lt;/span&gt; times the original sum. Bertrand's derivation of this result is nearly identical to the derivation on p. 10 of Scheibner (1860). For some biographical and bibliographical information about Bertrand, see Poggendorff (1898, A-L, middle left column p. 121 to top right column p. 122). [Review by Pierre Marie Eugène Prouhet (1817-1867)] (&lt;a href=&quot;http://www.numdam.org/numdam-bin/item?id=NAM_1864_2_3__550_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.numdam.org/numdam-bin/item?id=NAM_1864_2_3__550_0&lt;/a&gt;) in Nouvelles Annales de Mathématiques (2) 3 (1864), pp. 550-558 (in French). &lt;a href=&quot;https://books.google.com/books?id=hycCAAAAYAAJ&amp;amp;pg=PA903&quot; rel=&quot;nofollow noreferrer&quot;&gt;Review by Joseph Dienger (1818-1894)&lt;/a&gt; in Heidelberger Jahrbücher der Literatur 58 (1865), pp. 903-912 (in German).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[21]&lt;/strong&gt; Laurent, &lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=ucm.5310736637;view=1up;seq=306&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Traité d'Algèbre&lt;/strong&gt;&lt;/a&gt; [Treatise on Algebra], Gauthier-Villars, 1867, xvi + 520 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There exist at least 5 editions of &amp;quot;Part 2&amp;quot; (where rearrangement of series appears), published in 1867 (above), &lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=coo.31924001510464;view=1up;seq=310&quot; rel=&quot;nofollow noreferrer&quot;&gt;1875&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=RPMGAAAAYAAJ&amp;amp;pg=PA95&quot; rel=&quot;nofollow noreferrer&quot;&gt;1881&lt;/a&gt;, &lt;a href=&quot;http://babel.hathitrust.org/cgi/pt?id=mdp.39015064411740;view=1up;seq=105&quot; rel=&quot;nofollow noreferrer&quot;&gt;1887&lt;/a&gt;, and 1897 (have not seen). Beginning with the 3rd edition, &amp;quot;Part 2&amp;quot; is separately paged and sometimes first published in a different year than &amp;quot;Part 1&amp;quot;. For example, the 3rd edition of &amp;quot;Part 1&amp;quot; was first published in 1879 and the 3rd edition of &amp;quot;Part 2&amp;quot; was first published in 1881, while I believe the 5th edition of both &amp;quot;Part 1&amp;quot; and &amp;quot;Part 2&amp;quot; were first published in 1897. In the 1867 1st edition a remark on pp. 286-287 discusses the rearrangement of the alternating harmonic series in which two positive terms are followed by one negative term. The discussion is virtually word-for-word identical to the corresponding discussion on pp. 15-16 of Laurent (1862), including Laurent's earlier error in describing the general 3-grouped term. Indeed, the only changes I could detect when quickly comparing them side-by-side was a difference in wording in the first sentence (lines 3-4 of &lt;em&gt;Remarque&lt;/em&gt;) and a colon just before equation (3) in the 1862 book that does not appear in the 1867 book. The corresponding remark in later editions seems virtually unchanged, except the error in describing the general 3-grouped term (the error appears in the 1867 1st edition and in the 1875 2nd edition) is corrected beginning with the 1881 3rd edition. &lt;a href=&quot;https://books.google.com/books?id=UwcTAQAAMAAJ&amp;amp;pg=PA477&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brief review&lt;/a&gt; in Nouvelles Annales de Mathématiques (2) 14 (1875), p. 477 (in French).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[22]&lt;/strong&gt; Johann Lieblein (1834-1881), &lt;a href=&quot;https://books.google.com/books?id=vINaAAAAcAAJ&amp;amp;pg=PA46&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Sammlung von Aufgaben aus der Algebraischen Analysis&lt;/strong&gt;&lt;/a&gt; [Collection of Exercises in Algebraic Analysis], H. Carl J. Satow (Prague), 1867, viii + 192 pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Preface dated April 1867. This is a book of student exercises designed to be used with Schlömilch's &lt;strong&gt;Handbuch der Algebraischen Analysis&lt;/strong&gt;. Exercises 253-256 (pp. 46-47) give several examples of series rearrangements in which the student is asked to determine whether the rearrangement converges and, if it does, to find its sum. One example is the rearrangement of the alternating harmonic series in which three positive terms are followed by two negative terms. Another example is the corresponding series (three positive terms followed by two negative terms) in which the terms are the sine of the harmonic series terms. That is, the series &lt;span class=&quot;math-container&quot;&gt;$$\sin 1$ + \sin \frac{1}{3} + \sin \frac{1}{5} - \sin \frac{1}{2} - \sin \frac{1}{4} + \sin \frac{1}{7} + \sin \frac{1}{9} + \sin \frac{1}{11} - \cdots $$&lt;/span&gt; The 2nd edition, titled &lt;a href=&quot;https://books.google.com/books?id=cVs4AAAAMAAJ&amp;amp;pg=PA43&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;J. Lieblein's Sammlung von Aufgaben aus der Algebraischen Analysis zum Selbstunterricht&lt;/strong&gt;&lt;/a&gt; [J. Lieblein's Collection of Exercises in Algebraic Analysis for Self Study] and prepared by Wenzel Johann [= Jan Václav?] Láska (1862-1943), was published in 1889 (viii + 180 pages; JFM 20.0253.03). In the 1889 2nd edition the same examples appear as Exercises 202-205 (pp. 43-45), and at the end of Exercise 202 (rearrangements of the alternating harmonic series) references are given to p. 333 of Stern's &lt;strong&gt;Lehrbuch der Algebraischen Analysis&lt;/strong&gt; (no year is given, and the only edition I've seen is the 1860 edition, which does not have any rearrangements on p. 333), to p. 172 of Schlömilch's's &lt;strong&gt;Uebungsbuch zum Studium der Hoeheren Analysis&lt;/strong&gt;, and to Dirichlet's 1836 paper. Also, in the 1889 2nd edition, Exercise 201 on p. 43 sketches out the apparent &amp;quot;paradox&amp;quot; discussed at the beginning of Glaisher's 1872 paper, and Glaisher's paper (but not his name) is cited at the end of Exercise 201. Review of the 1867 edition by &lt;a href=&quot;https://books.google.com/books?id=Z2NbAAAAcAAJ&amp;amp;pg=PA362&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guillaume Jules Hoüel (1823-1886)&lt;/a&gt; in Bulletin des Sciences Mathématiques et Astronomiques (1) 1 (1870), pp. 362-363 (in French).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[23]&lt;/strong&gt; Raffaele Giovanni Rubini (1817-1890), &lt;a href=&quot;https://books.google.com/books?id=6SbxYCWjluAC&amp;amp;pg=PA384&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Complemento agli Elementi d'Algebra&lt;/strong&gt;&lt;/a&gt; [To Complement to the Elements of Algebra], 2nd edition, A. Morelli (Napoli [Naples], Italy), 1867, 4 + iv + 484 + iii pages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Preface dated October 1867. I believe the 1st edition was published in 1861. I have not been able to examine a copy of the 1st edition, so it is possible that Rubini's book should appear earlier in my list. On pp. 384-385 Rubini shows that the alternating harmonic series and the rearranged version in which two positive terms are followed by one negative term have sums that differ by an amount between &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}.$&lt;/span&gt; No mathematician's name is mentioned in this particular discussion, but Dirichlet's name is mentioned further down on p. 385 for the result (which Rubini gives a proof of) that all rearrangements of an absolutely convergent series of complex numbers have the same sum. Several more editions exist (a total of 5? the last in 1878?), but I have not been able to confirm this because I am not sure whether some similarly titled books by Rubini (that I've seen notices of in various journals from this time period) should be considered later editions of this book or different books. A similar difficulty existed for the books by Choquet and Mayer (see the &amp;quot;1849&amp;quot; entry above), but I was able to resolve it because I was able to examine all the books involved. The only other book by Rubini I've seen that includes something about rearrangements changing sums is Rubini's &lt;a href=&quot;https://books.google.com/books?id=yzxeAAAAcAAJ&amp;amp;pg=PA315&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Trattato d'Algebra&lt;/strong&gt;&lt;/a&gt; [Treatise on Algebra], Parte Seconda. Complemento agli Elementi d'Algebra [Part Two. To Complement the Elements of Algebra], A. Morelli (Napoli [Naples], Italy), 1873, vi + 363 pages. In this 1873 book the same discussion described above appears repeated word-for-word on p. 315. Review of the 1873 edition by &lt;a href=&quot;https://books.google.com/books?id=p4s_AQAAIAAJ&amp;amp;pg=PA21&quot; rel=&quot;nofollow noreferrer&quot;&gt;Guillaume Jules Hoüel (1823-1886)&lt;/a&gt; in Bulletin des Sciences Mathématiques et Astronomiques (1) 6 (1874), p. 21 (in French).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="264" LastActivityDate="2022-02-07T16:57:29.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14171" PostTypeId="1" CreationDate="2022-02-08T06:51:01.273" Score="1" ViewCount="162" Body="&lt;p&gt;The well-known riddle of the Epitaph of Diophantus, attributed to Metrodorus, is one of the style of simple problem in algebra whose pattern when expressed in contemporary algebraic notation is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x = \dfrac x 6 + \dfrac x {12} + \dfrac x 7 + 5 + \dfrac x 2 + 4$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This appears in the Greek Anthology Book XIV as epigram no. 126.&lt;/p&gt;&#xA;&lt;p&gt;The version that appears in the W.R. Paton translation is best known:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This tomb holds Diophantus. Ah, how great a marvel!&#xA;the tomb tells scientifically the measure of his life.&#xA;God granted him to be a boy for the sixth part of his life,&#xA;and adding a twelfth part to this, he clothed his cheeks with down;&#xA;He lit him the light of wedlock after a seventh part,&#xA;and five years after his marriage He granted him a son.&#xA;Alas! late-born wretched child;&#xA;after attaining the measure of half his father's life, chill Fate took him.&#xA;After consoling his grief by this science of numbers for four years he ended his life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now a large swathe of the problems in the Greek Anthology are along similar lines, not only in the context of working out someone's age (of which there are quite a few) but in a whole raft of different contexts:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;how many students attended Pythagoras's Academy (epigram 1)&lt;/li&gt;&#xA;&lt;li&gt;the portions of gold donated to the statue of Pallas (epigram 2)&lt;/li&gt;&#xA;&lt;li&gt;how many apples were stolen by the Muses from Cupid (epigram 3)&lt;/li&gt;&#xA;&lt;li&gt;how many herds of cattle Augeas owned (epigram 4)&lt;/li&gt;&#xA;&lt;li&gt;how many walnuts were stolen during a highway mugging by a posse of pretty girls (epigram 116)&lt;/li&gt;&#xA;&lt;li&gt;how many apples were stolen during a similar robbery (epigram 117)&lt;/li&gt;&#xA;&lt;li&gt;a more civilised distribution of apples (epigram 118)&lt;/li&gt;&#xA;&lt;li&gt;the number of walnuts harvested from a tree (epigram 120)&lt;/li&gt;&#xA;&lt;li&gt;how far it is from Cadiz to Rome (epigram 121)&lt;/li&gt;&#xA;&lt;li&gt;how large a fortune was squandered (epigram 122)&lt;/li&gt;&#xA;&lt;li&gt;how an inheritance was distributed (epigram 123)&lt;/li&gt;&#xA;&lt;li&gt;the age of an unnamed protagonist (epigram 124)&lt;/li&gt;&#xA;&lt;li&gt;how many children were born to Philinna (epigram 125)&lt;/li&gt;&#xA;&lt;li&gt;epitaph of Demochares (epigram 127)&lt;/li&gt;&#xA;&lt;li&gt;how many people were killed when Antiochus's house collapsed (epigram 137)&lt;/li&gt;&#xA;&lt;li&gt;how Nicarete distributed her nuts (epigram 138)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Clearly these all fit the same pattern: you are given a description of how a number is broken down into a set of both fractions of the whole and one or more specific enumerations, and you are asked to calculate what that number is.&lt;/p&gt;&#xA;&lt;p&gt;Such a puzzle can be seen in a number of such collections of puzzles (if I'm not mistaken Alcuin includes some in his Propositiones ad Acuendos Juvenes), but only the Epitaph of Diophantus seems to be well-known.&lt;/p&gt;&#xA;&lt;p&gt;My question is: is there an accepted &lt;em&gt;name&lt;/em&gt; for this classification of problem?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="10106" LastEditDate="2022-02-08T09:59:19.013" LastActivityDate="2022-02-08T09:59:19.013" Title="Classification of &quot;Epitaph of Diophantus&quot; problem" Tags="|mathematics|terminology|ancient-greece|algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14172" PostTypeId="1" AcceptedAnswerId="14175" CreationDate="2022-02-08T07:29:54.950" Score="7" ViewCount="256" Body="&lt;p&gt;In the Chapter 7 epigraph in Stein and Shakarchi's &lt;em&gt;Complex Analysis&lt;/em&gt;, there is a quote, apparently from Hadamard in 1945, saying&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He proved some important properties of that function, but enunciated two or three as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying “These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.” We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one. The question concerning that last one remains unsolved as yet, though, by an immense labor pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at all certain, that the “Riemann hypothesis” is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I tried searching online, but haven't found anything about this mysterious expression, or what the two or three properties are that Riemann left unproven and that Hadamard did successfully prove. Are there any sources/details for this quote?&lt;/p&gt;&#xA;" OwnerUserId="11071" LastEditorUserId="55" LastEditDate="2022-02-09T07:48:44.750" LastActivityDate="2022-02-09T12:26:34.807" Title="What is the mysterious expression mentioned by Hadamard that Riemann derived properties of zeta function from?" Tags="|mathematics|reference-request|quote|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14173" PostTypeId="2" ParentId="14164" CreationDate="2022-02-08T17:30:42.263" Score="2" Body="&lt;p&gt;Prior to Heaviside, notation in electromagnetism was mixed, and vector notation as we know it today was not in use. Rather, equations were often expressed component-wise, although the concept of a vector still existed. Heaviside, who developed much of the modern vector notation we use today, used &lt;span class=&quot;math-container&quot;&gt;$\mathbf{W}$&lt;/span&gt; to represent this additional energy term as early as 1884.&lt;/p&gt;&#xA;&lt;p&gt;For use of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{S}$&lt;/span&gt;: As noted &lt;a href=&quot;https://ieeexplore.ieee.org/document/8125229&quot; rel=&quot;nofollow noreferrer&quot;&gt;in this article&lt;/a&gt;, &amp;quot;Energy in Electromagnetism: The Poynting Vector&amp;quot; published in IEEE Antennas and Propagation Magazine, the notation &lt;span class=&quot;math-container&quot;&gt;$\mathbf{S}$&lt;/span&gt; first appeared in H. A. Lorentz's &lt;em&gt;The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat&lt;/em&gt; (1909) p.23, where he says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As to the transfer of energy represented by (39), it must necessarily take place at the points of the surface &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; itself, because our theory leaves no room for any action at a distance. Further, we are naturally led to suppose that the actions by which it is brought about are such that, for each element &lt;span class=&quot;math-container&quot;&gt;$d\sigma$&lt;/span&gt;, the quantity &lt;span class=&quot;math-container&quot;&gt;$c[\mathbf{d}\cdot\mathbf{h}]_n d\sigma$&lt;/span&gt; may be said to represent the amount of energy that is transmitted across this particular element. In this way we come to the conception, first formulated by Poynting&lt;span class=&quot;math-container&quot;&gt;${}^1$&lt;/span&gt;, of a &lt;em&gt;current&lt;/em&gt; or &lt;em&gt;flow&lt;/em&gt; of energy. It is determined by the vector product of &lt;span class=&quot;math-container&quot;&gt;$\mathbf{d}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbf{h}$&lt;/span&gt;, multiplied by the constant &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, so that we can write for it&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mathbf{S} = c[\mathbf{d}\cdot\mathbf{h}]\qquad\qquad\qquad\text{(42)}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Where Lorentz uses &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; for the speed of light, and &lt;span class=&quot;math-container&quot;&gt;$[\ \cdot\ ]$&lt;/span&gt; represents the cross product. He doesn't give any justification for choosing &lt;span class=&quot;math-container&quot;&gt;$\mathbf{S}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It was not used by some significant physicists, including Max Abraham and Hermann Minkowski, nor was it used by Whittaker in his &lt;a href=&quot;https://www.google.com/books/edition/A_history_of_the_theories_of_aether_and/zYx2gQjpe2MC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Theories of the Ether and Electricity&lt;/a&gt; (1910), see e.g. p.348. But G. H. Livens did use &lt;span class=&quot;math-container&quot;&gt;$\mathbf{S}$&lt;/span&gt; in &lt;em&gt;The Theory of Electricity&lt;/em&gt; (1918), (second edition &lt;a href=&quot;https://www.google.com/books/edition/_/4q45AAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;available here&lt;/a&gt;) which was influential.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2022-02-08T17:30:42.263" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14174" PostTypeId="2" ParentId="14167" CreationDate="2022-02-08T19:53:14.527" Score="2" Body="&lt;p&gt;At least as a placeholder answer: it is my impression that that idea would have been extremely novel for its time, since there were &lt;em&gt;not&lt;/em&gt; abstract ideas of rings and ideals and such. On the other hand, the seemingly-tangible idea that a (reasonable) field extension of &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q$&lt;/span&gt; is obtained by adjoining suitable complex numbers (which we believe exist, etc.), had serious limitations of scale.&lt;/p&gt;&#xA;&lt;p&gt;These limitations can be seen in A. Weil's &amp;quot;Foundations of Alg Geom&amp;quot;, in the 1940s, which needed algebraically closed field extensions of uncountable transcendence degree. (True, the seeming need for &amp;quot;Grothendieck universes&amp;quot; in the most-cavalier development of modern algebraic geometry has some serious foundational stuff going on, too.)&lt;/p&gt;&#xA;&lt;p&gt;But, no, I don't know of a translation.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-02-08T19:53:14.527" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14175" PostTypeId="2" ParentId="14172" CreationDate="2022-02-09T07:47:28.737" Score="10" Body="&lt;p&gt;The mysterious expression, that Riemann did not &amp;quot;simplify enough to publish&amp;quot;, is mentioned in his letter to Weierstrass published by Weber in 1876. It is reminiscent of Fermat's &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Fermat%27s_conjecture&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;marvelous proof&amp;quot;&lt;/a&gt; that the margin was &amp;quot;too narrow to contain&amp;quot;, but a review of Riemann’s &lt;em&gt;Nachlass&lt;/em&gt; led a librarian (Distel) to the likely culprit, apparently, unbeknownst to Hadamard. Siegel gave a reconstruction of it in &lt;a href=&quot;https://arxiv.org/pdf/1810.05198.pdf&quot; rel=&quot;noreferrer&quot;&gt;On Riemann’s Nachlass for Analytic Number Theory (1932)&lt;/a&gt;, and it is now known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula&quot; rel=&quot;noreferrer&quot;&gt;Riemann–Siegel formula&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In fact, Herr Distel, a librarian, already several decades ago discovered the representation&#xA;in question of the zeta function in Riemann’s papers. It concerns an asymptotic development&#xA;which gives the behavior of the function &lt;span class=&quot;math-container&quot;&gt;$ζ(s)$&lt;/span&gt; on the critical line &lt;span class=&quot;math-container&quot;&gt;$σ = 1/2$&lt;/span&gt; and, more generally,&#xA;in each strip &lt;span class=&quot;math-container&quot;&gt;$σ_1 ≤ σ ≤ σ_2$&lt;/span&gt;, expressed for infinitely increasing magnitude of &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt;. The principal&#xA;term of this development was meanwhile rediscovered, independently of Riemann, by Hardy&#xA;and Littlewood in 1920 as a consequence of their “approximate functional equation”; they use&#xA;the same methods of proof as Riemann, namely approximate calculation of an integral by the&#xA;saddle-point method. In Riemann, however, there is also a process for obtaining additional&#xA;terms of the asymptotic series, which is based on the beautiful properties of the integral&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Φ(τ, u)=\int\frac{e^{\pi i\tau x^2+2\pi iux}}{e^{2\pi ix}-1}dx,$$&lt;/span&gt;&#xA;which, incidentally, has also led Kronecker and recently Mordell to a most elegant derivation&#xA;of the reciprocity formula for the Gauss sums.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1926 Bessel-Hagen noted in a new review of&#xA;the Riemann papers another previously unknown representation of the zeta function in terms of definite integrals; in this Riemann was also&#xA;guided by the properties of &lt;span class=&quot;math-container&quot;&gt;$Φ(τ, u)$&lt;/span&gt;.&#xA;These two representations of &lt;span class=&quot;math-container&quot;&gt;$ζ(s)$&lt;/span&gt; may be included among the most important results in&#xA;Riemann’s number-theoretic Nachlass, insofar as one does not find them in his published paper.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first &amp;quot;two or three&amp;quot; properties that Riemann derived from these expressions are likely that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_Xi_function&quot; rel=&quot;noreferrer&quot;&gt;Riemann xi function &lt;span class=&quot;math-container&quot;&gt;$\xi(s)$&lt;/span&gt;&lt;/a&gt; is an entire function of &lt;span class=&quot;math-container&quot;&gt;$s^2$&lt;/span&gt;, with a Weierstrassian product expansion, which Hadamard rigorously proved in 1893, and an asymptotic form of the prime number theorem that he derived from it in 1896, see &lt;a href=&quot;http://philsci-archive.pitt.edu/11241/&quot; rel=&quot;noreferrer&quot;&gt;Iurato, On some historical aspects of the theory of Riemann zeta function, pp. 77-78, 104&lt;/a&gt;. Both are traceable to Riemann's 1859 paper, and Iurato specifically alludes to Hadamard's remark from The Psychology of Invention in the Mathematical Field (1945) in connection with them:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Following (Stopple 2003, Chapter 10, Section 10.1), it was Riemann to&#xA;realize that a product formula for ξ(s) would have had a great significance for the study of prime numbers. &lt;strong&gt;The first rigorous proof of this product formula was due to Hadamard but, as he himself remembered, it took almost three decades before he reached to a satisfactory proof of it.&lt;/strong&gt; Likewise, also H.M. Edwards (1974, Chapter 1, Sections 1.8-1.19) affirms that the parts concerned with (2) are the most difficult portion of the 1859 Riemann's paper (see also (Bottazzini &amp;amp; Gray 2013, Chapter 5, Section&#xA;5.10)). Their goal is to prove essentially that &lt;span class=&quot;math-container&quot;&gt;$ξ(s)$&lt;/span&gt; can be expressed as an infinite product... &amp;quot;Hadamard (in 1893) proved necessary and sufficient conditions for the validity of the product formula &lt;span class=&quot;math-container&quot;&gt;$ξ(t) = ξ(0) ∏_ρ(1 − t/ρ)$&lt;/span&gt; but the steps of the argument&#xA;by which Riemann went from the one to the other are obscure, to say the very least&amp;quot;...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Just in regard to the novelties due to Hadamard, Weyl points out that the driving force for these Hadamard's investigations was the wish to obtain sufficient information about the zeros of the Riemann zeta function for establishing the asymptotic law for the distribution of prime numbers... In 1896, both Hadamard and&#xA;de la Vallée-Poussin, independently of each other, were able to draw the conclusion&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$\left[\frac{\pi(n)\ln(n)}{n}\to1\right]$&lt;/span&gt; &lt;em&gt;from 1893 Hadamard's results concerning entire functions&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-02-09T12:26:34.807" LastActivityDate="2022-02-09T12:26:34.807" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14176" PostTypeId="1" AcceptedAnswerId="14177" CreationDate="2022-02-09T14:42:07.203" Score="5" ViewCount="269" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/Michael_Sendivogius&quot; rel=&quot;noreferrer&quot;&gt;Michael Sendivogius&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Michael Sendivogius (/ˌsɛndɪˈvoʊdʒiəs/; Polish: Michał Sędziwój; 2&#xA;February 1566 – 1636) was a Polish alchemist, philosopher, and medical&#xA;doctor.&lt;/p&gt;&#xA;&lt;p&gt;A pioneer of chemistry, he developed ways of purification and creation&#xA;of various acids, metals and other chemical compounds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How did Michael Sendivogius think of his own findings at the time? Did he consider them to be &amp;quot;Alchemy&amp;quot;, or by that time he saw them as &amp;quot;Chemistry&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Notes:&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;http://History%20of%20Chemistry&quot; rel=&quot;noreferrer&quot;&gt;History of Chemistry&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The protoscience of chemistry, alchemy, was unsuccessful in explaining&#xA;the nature of matter and its transformations. However, by performing&#xA;experiments and recording the results, alchemists set the stage for&#xA;modern chemistry. The distinction began to emerge when a clear&#xA;differentiation was made between chemistry and alchemy by Robert Boyle&#xA;in his work The Sceptical Chymist (1661).&lt;/p&gt;&#xA;&lt;p&gt;Alchemy and chemistry share an interest in the composition and&#xA;properties of matter, and until the 18th century they were not&#xA;separate disciplines. The term chymistry has been used to describe the&#xA;blend of alchemy and chemistry that existed before that time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Etymology_of_chemistry#From_alchemy_to_chemistry&quot; rel=&quot;noreferrer&quot;&gt;Etymology of Chemistry from Alchemy to Chemistry &lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Later medieval Latin had alchimia / alchymia &amp;quot;alchemy&amp;quot;, alchimicus&#xA;&amp;quot;alchemical&amp;quot;, and alchimista &amp;quot;alchemist&amp;quot;. The mineralogist and&#xA;humanist Georg Agricola (died 1555) was the first to drop the Arabic&#xA;definite article al-. In his Latin works from 1530 on he exclusively&#xA;wrote chymia and chymista in describing activity that we today would&#xA;characterize as chemical or alchemical&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4512" LastEditorUserId="4512" LastEditDate="2022-02-10T02:08:51.580" LastActivityDate="2022-02-10T02:08:51.580" Title="Were Michael Sendivogius findings in Chemistry considered by him to be Alchemy?" Tags="|chemistry|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14177" PostTypeId="2" ParentId="14176" CreationDate="2022-02-09T15:39:37.670" Score="5" Body="&lt;p&gt;He referred to his work as alchemy.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://Brussels,%209th%20February,%201646.&quot; rel=&quot;noreferrer&quot;&gt;Further I liberally do grant and promise you a communication of more Notions than your Patron hath yet imparted to you, concerning as well the theoretical as the practical part of our alchemy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://Brussels,%209th%20February,%201646.&quot; rel=&quot;noreferrer&quot;&gt;For therein you shall find displayed and clearly manifested all the secrets of our cabal, and of physick demonstrative with the whole knowledge of alchemical doctrine.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://Brussels,%209th%20February,%201646.&quot; rel=&quot;noreferrer&quot;&gt;But if you have a desire at they very entrance into this study to have a full knowledge and sincere information in our alchemy they let it suffice our Novum lumen chymicum, with the annexed treatise of Sulphur and the dialogue between Mercury and the Alchemist, for in this book nothing is wanting.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Around the Time of Michael's birth was the first recorded dropping of the al- from alchemy by Georgius Agricola, but this wasn't to make a distinction between the two. Through the 17th century alchemy and chemistry would have been thought of as the same thing. &lt;a href=&quot;https://en.wikipedia.org/wiki/Etymology_of_chemistry#:%7E:text=Assuming%20a%20Greek%20origin%2C%20chemistry,molecules%2C%20crystals%2C%20and%20metals.&quot; rel=&quot;noreferrer&quot;&gt;Staring in the early 18th century the two would not be considered synonymous&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4880" LastActivityDate="2022-02-09T15:39:37.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14178" PostTypeId="1" CreationDate="2022-02-09T19:55:15.250" Score="4" ViewCount="121" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;In his 1827 treatise on differential geometry, Gauss in his &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Theorema_Egregium&quot; rel=&quot;nofollow noreferrer&quot;&gt;theorema egregium&lt;/a&gt;&amp;quot; proved that the curvature of a surface is an intrinsic invariant; it doesn't change under isometries of the surface (that is - bending and twisting is allowed, but no operation that distorts the original distances on the surface is allowed). This result became the cornerstone of the differential geometry of surfaces, but this is not the issue here. My question concerns the cartographic implication of this result - that one cannot draw a uniform scale map of a given surface on another, &lt;em&gt;&lt;strong&gt;substantially different&lt;/strong&gt;&lt;/em&gt; surface (for example, one cannot draw a uniform scale planar map of the spherical (or rather spheroidical) earth; there are inevitable distortions). As far as I understand, this is the motivation to Gauss's writing of his major two-parts treatise on Geodesy (which were published in the years 1843,1846) - which dealt with problems of conformal mappings (angle-preserving maps; conformal maps are necessary for navigation in long distances, since they represent angles of directions accurately) of the earth ellipsoid of revolution into a plane or a sphere.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;My question&lt;/strong&gt;:&lt;/p&gt;&#xA;&lt;p&gt;According to Buhler's biography of Gauss:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Part I of &amp;quot;Untersuchungen...&amp;quot; deals with the special case of the conformal mapping of an ellipsoid to a sphere; knowing this makes it possible to utilize ordinary spherical trigonometry for the purposes of geodesy. Small regions of the ellipsoid are mapped to the sphere by &lt;span class=&quot;math-container&quot;&gt;$$f(v)=\alpha v - i\mathbb{log}k$$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; the imaginary unit and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; an arbitrary point on the ellipsoid. The constants &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; have to be chosen in a way such that the distortion of the original is minimized.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Obviously, I don't understand how Gauss arrived at this functional form of the mapping, but one thing that is already implied from this description is that Gauss's procedure is based on mapping &lt;em&gt;&lt;strong&gt;small&lt;/strong&gt;&lt;/em&gt; regions of the ellipsoid of revolution into a sphere, since a global mapping necessarily distorts distances (global mapping cannot be of uniform scale, as a consequence of the Theorema Egregium).&lt;/p&gt;&#xA;&lt;p&gt;But already here I'm stuck - if we suppose &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; to be a complex number, then how do we define a correspondence between points on the ellipsoid and the complex plane? Is it done via a stereographic projection? I looked into the beginning of Gauss's original paper and he doesn't give much more explanation about this formula, despite that apparently much of the treatise is based on this mapping. In addition, what does it mean that the constants &lt;span class=&quot;math-container&quot;&gt;$\alpha ,k$&lt;/span&gt; are chosen such that the distortion is minimal? how to quantify distortion?&lt;/p&gt;&#xA;&lt;p&gt;This lack of supporting explanations is a major obstacle to anyone willing to look into his work in Geodesy, but I'm sure that it might have a fairly simple explanation, since this publication was apparently very influential in the history of Geodesy (it is not an obscure, unpublished fragment).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Useful links&lt;/strong&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The treatise mentioned in this question is on p.259 of volume 4 of Gauss's works, here is a link: &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN236005081?tify=%7B%22pages%22:%5B263%5D,%22panX%22:0.462,%22panY%22:0.446,%22view%22:%22toc%22,%22zoom%22:0.83%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Untersuchungen uber Gegenstande Der Hohern Geodesy&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="10106" LastEditDate="2022-02-10T00:13:11.463" LastActivityDate="2022-02-10T00:13:11.463" Title="Explanation request for the terminology and notation employed by Gauss in his major 1843/6 treatise on Geodesy" Tags="|mathematics|geometry|gauss|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14179" PostTypeId="2" ParentId="12302" CreationDate="2022-02-10T16:22:25.587" Score="1" Body="&lt;p&gt;It was understood that nitrogen reserves were quickly running out. At the time it was just as important for warfare as it was for food. Guano and sodium nitrate were such important recourses a war was fought over them. The &lt;a href=&quot;https://en.wikipedia.org/wiki/War_of_the_Pacific&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saltpeter War&lt;/a&gt;  (April 5, 1879 – October 20, 1883) was fought by Chile and Bolivia over saltpeter reserves in the Atacama Desert. It was known at the time that Guano and sodium nitrate were finite, and that once exhausted it would lead to food shortages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;England and all civilized nations stand in deadly peril of not having enough to eat. As mouths multiply, food resources dwindle. Land is a limited quantity, and the land that will grow wheat is absolutely dependent on difficult and capricious natural phenomena... I hope to point a way out of the colossal dilemma. It is the chemist who must come to the rescue of the threatened communities. It is through the laboratory that starvation may ultimately be turned into plenty... The fixation of atmospheric nitrogen is one of the great discoveries, awaiting the genius of chemists. (&lt;a href=&quot;https://www.todayinsci.com/C/Crookes_William/CrookesWilliam-Quotations.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir William Crookes - 1898&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Was it understood in fact that greater food production capacity would have this effect?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The idea that human population growth is potentially exponential was proposed by Thomas Malthus in 1798, and is referred to as the  &lt;a href=&quot;https://en.wikipedia.org/wiki/Malthusian_growth_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;Malthusian growth model&lt;/a&gt;. At the turn of the 19th century the human population was starting to grow significantly faster, but agriculture was still dependent on &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_Haber_process#El_Dorado_made_of_nitrogen&quot; rel=&quot;nofollow noreferrer&quot;&gt;bat droppings, human waste, and crushing up bones&lt;/a&gt;. We hit 1 billion people some time in the 1800's and we hit &lt;a href=&quot;https://en.wikipedia.org/wiki/Estimates_of_historical_world_population&quot; rel=&quot;nofollow noreferrer&quot;&gt;2 billion&lt;/a&gt; before the industrial production of nitrogen fertilizer ramped up. It was understood the population was increasing very quickly. A result of many factors such as decreased &lt;a href=&quot;https://en.wikipedia.org/wiki/Victorian_era#Fertility_rates_and_mortality_rates&quot; rel=&quot;nofollow noreferrer&quot;&gt;infant mortality, better medicine&lt;/a&gt;, and  more efficient agricultural practices (like using fertilizer).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;But was there a time when anyone said, we need to be able to dramatically increase the world population?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I don't think anyone was saying it would be great if the human population grew exponentially. Cities were already &lt;a href=&quot;https://en.wikipedia.org/wiki/Victorian_era#Poverty&quot; rel=&quot;nofollow noreferrer&quot;&gt;over crowded&lt;/a&gt;, and at the turn of the century &lt;a href=&quot;http://www.mybudget360.com/finance-investing-the-long-depression-of-1873-parallels-and-comparisons-are-we-missing-economic-information-from-an-important-piece-of-american-financial-history/&quot; rel=&quot;nofollow noreferrer&quot;&gt;unemployment was very high&lt;/a&gt;. They were looking at the world around them, the human population was starting to grow exponentially, and they thought &amp;quot;how are we going to feed everyone?&amp;quot; What happens if you have a population that's growing quickly, but the ability to grow food can't keep up? The investigation into the industrial fixation nitrogen was a reaction to a looming crisis, and brilliant scientists figured out a solution.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I could see business leaders perhaps wanting more workers and more people to sell products to although I would guess it would be the food producers themselves who wanted cheaper fertilizer&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It wasn't just farmers who wanted a cheap and secure way to source ammonia. It was used to make gunpowder as well as fertilizer. Germany was in a particularly bad spot they were importing &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_Haber_process#El_Dorado_made_of_nitrogen&quot; rel=&quot;nofollow noreferrer&quot;&gt;350,000 tones of Chilean saltpeter in 1900, and 900,000&lt;/a&gt; tones in 1912. Then World War 1 started, and allied blockades stopped the import of saltpeter. The Haber–Bosch process allowed them to continue fighting even without an external supply.&lt;/p&gt;&#xA;" OwnerUserId="4880" LastActivityDate="2022-02-10T16:22:25.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14180" PostTypeId="2" ParentId="3245" CreationDate="2022-02-11T08:44:55.540" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;It is well known that positional numeral systems are not possible without the concept of zero and corresponding notation. This is the necessary condition, but not sufficient one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is wrong.  See eg &lt;a href=&quot;https://en.wikipedia.org/wiki/Bijective_numeration&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipeda&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The bijective base-10 system is a base ten positional numeral system that does not use a digit to represent zero. It instead has a digit to represent ten, such as A.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the bijective base-26 system one may use the Latin alphabet letters &amp;quot;A&amp;quot; to &amp;quot;Z&amp;quot; to represent the 26 digit values one to twenty-six. (A=1, B=2, C=3, ..., Z=26)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With this choice of notation, the number sequence (starting from 1) begins A, B, C, ..., X, Y, Z, AA, AB, AC, ..., AX, AY, AZ, BA, BB, BC, ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Each digit position represents a power of twenty-six, so for example, the numeral ABC represents the value 1 × 262 + 2 × 261 + 3 × 260 = 731 in base 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Many spreadsheets including Microsoft Excel use this system to assign labels to the columns of a spreadsheet, starting A, B, C, ..., Z, AA, AB, ..., AZ, BA, ..., ZZ, AAA, etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And for your question about history:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Forslund (1995) appears to be another rediscovery, and hypothesizes that if ancient numeration systems used bijective base-k, &lt;strong&gt;they might not be recognized as such in archaeological documents, due to general unfamiliarity with this system.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Emphasis added.)&lt;/p&gt;&#xA;" OwnerUserId="16228" LastActivityDate="2022-02-11T08:44:55.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14181" PostTypeId="1" AcceptedAnswerId="14188" CreationDate="2022-02-11T11:17:29.490" Score="2" ViewCount="300" Body="&lt;p&gt;The &amp;quot;cube of physics&amp;quot; is a quite useful summary of physics, for historical&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; and teaching&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; purposes, that is best explained (as far as I know) in &lt;a href=&quot;https://cotejer.github.io/physics-on-a-cube&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Physics On A Cube&amp;quot; by Jeremy Côté&lt;/a&gt; or in &lt;a href=&quot;https://backreaction.blogspot.com/2011/05/cube-of-physical-theories.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The cube of physical theories&amp;quot; by Sabine Hossenfelder&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; it recaps all biggest turning points in the history of physics&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; for example, it is in the intro of &lt;a href=&quot;https://press.princeton.edu/books/hardcover/9780691145587/einstein-gravity-in-a-nutshell&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zee's Einstein Gravity in a Nutshell&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The idea has been attributed to Matvei Bronstein &lt;a href=&quot;https://en.wikipedia.org/wiki/Matvei_Petrovich_Bronstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;on wiki&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;He introduced the cGh scheme for classifying physical theories. &amp;quot;After the relativistic quantum theory is created, the task will be to develop the next part of our scheme, that is to unify quantum theory (with its constant h), special relativity (with constant c), and the theory of gravitation (with its G) into a single theory.&amp;quot;(2)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;But the &lt;a href=&quot;https://en.wikipedia.org/wiki/Matvei_Petrovich_Bronstein#cite_note-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;citation note&lt;/a&gt; references a paper I could find nowhere:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;(2) Bronstein, M. P. &amp;quot;K voprosu o vozmozhnoy teorii mira kak tselogo&amp;quot; (&amp;quot;On the Question of a Possible Theory of the World as a Whole&amp;quot;), in Uspekhiastronomitcheskihnauk. Sbornik, No. 3 (Moscow: ONTI, 1933) p. 3–30, as quoted and translated in Gorelik (2005) loc. cit.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even &lt;a href=&quot;https://cotejer.github.io/physics-on-a-cube#fn:4&quot; rel=&quot;nofollow noreferrer&quot;&gt;the one who wrote the long article&lt;/a&gt; I mentioned at the beginning couldn't find anything:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;I could not locate the original article by Bronstein. As is the case with a lot of old papers, they have citations from other resources, but there is no copy to be found. It took enough work to find Bronstein’s first name...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Any help?&lt;/p&gt;&#xA;" OwnerUserId="16229" LastEditorUserId="16229" LastEditDate="2022-02-12T20:09:01.370" LastActivityDate="2022-02-18T10:27:18.620" Title="Source documents for Bronstein's Cube of Physics" Tags="|physics|reference-request|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14183" PostTypeId="2" ParentId="14181" CreationDate="2022-02-11T17:47:30.360" Score="0" Body="&lt;p&gt;As it appears there is another version printed in book and edited by Bronstein himself:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Основные проблемы космической физики&lt;/em&gt;, 1934, под ред. М.П. Бронштейна, Киев, ОНТИ с. 186-215&#xA;&lt;a href=&quot;https://bg1lib.org/book/2557637/824ff6&quot; rel=&quot;nofollow noreferrer&quot;&gt;djvu&lt;/a&gt; (p.206 missing)&lt;/p&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="10106" LastEditDate="2022-02-13T09:18:19.950" LastActivityDate="2022-02-13T09:18:19.950" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14184" PostTypeId="1" AcceptedAnswerId="14185" CreationDate="2022-02-11T20:37:00.643" Score="6" ViewCount="908" Body="&lt;p&gt;I have heard that in Canada and the United States of America, when we write a decimal number, then we use a period, like 1.111. When we write large numbers, then we use a comma, like 1,111. Sometimes we use spaces for large numbers, like  1 111. I have also heard that Europe, Southern Africa, Central America, the West Indies, and South America use commas and periods the other way around. So, for Americans, 1.111 is the decimal number, while for Europeans, 1.111 is one thousand one hundred eleven. For Americans, 1,111 is one thousand one hundred eleven, while for Europeans, 1,111 is the decimal number. 1 111 is one thousand one hundred eleven for the entire world. How did the commas and periods develop historically?&lt;/p&gt;&#xA;&lt;p&gt;Update: Recently, USA and China too switched to using comma for the decimal. Now &amp;quot;Made In China&amp;quot; products too display the decimal comma.&lt;/p&gt;&#xA;" OwnerUserId="13344" LastEditorUserId="13344" LastEditDate="2024-01-17T22:14:19.893" LastActivityDate="2024-01-17T22:14:19.893" Title="Historical origin of commas and periods in numbers" Tags="|mathematics|numbers|" AnswerCount="1" CommentCount="2" FavoriteCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14185" PostTypeId="2" ParentId="14184" CreationDate="2022-02-11T21:37:22.877" Score="12" Body="&lt;p&gt;(I must admit that I'm writing this mostly from memory and I don't have sources and not enough time to look for any, so some details may be wrong.)&lt;/p&gt;&#xA;&lt;p&gt;Simon Stevin, who developed the notation of decimal fractions in the late 16th century, originally used a circle with numbers inside, so 1.111 would be written as &amp;quot;1⓪1①1②1③&amp;quot; and 365.256 (the length of a year in days) as 365⓪2①5②6③, where ⓪ means the preceding digits are the integer part, ① means 1/10, ② means 1/100 and so on. Stevin published this in the pamphlet &amp;quot;De Thiende&amp;quot; (&amp;quot;The Tenth&amp;quot;), written in Flemish, in 1585 (thanks to @njuffa for providing this detail in the comments).&lt;/p&gt;&#xA;&lt;p&gt;The positional system for integer numbers had, by that time, already a long history; it was originally developed in India, spread through the Islamic world and was introduced to Europe by Leonardo von Pisa (Fibonacci) in 1202. So Stevin did not need or want to modify the integer part of a number, as this was already well established; Stevin's notation just extends it to include fractional parts by marking it with the circled numbers.&lt;/p&gt;&#xA;&lt;p&gt;Later people found that the ①, ② etc. are not necessary; you only need the marker ⓪ where the integer part ends and the fractional part begins, and later the ⓪ developed into a dot as it was easier to write, initially centred above the line, but later on the line.&lt;/p&gt;&#xA;&lt;p&gt;In Germany, Leibniz however introduced a center dot for multiplication (as we still do) instead of the cross in the UK. The reason was that he was working with symbolic algebra and introduced the convention to use x, y, z for unknowns, but the x can easily be confused with the multiplication cross.&lt;/p&gt;&#xA;&lt;p&gt;But when using a dot for multiplication, having a dot as decimal marker leads again to confusion, so people started using a comma instead.&lt;/p&gt;&#xA;&lt;p&gt;As France and Germany always had very intense cultural exchange, this became the norm in France too and spread to most other European countries (possibly with Napoleon and the metric system?).&lt;/p&gt;&#xA;&lt;p&gt;By the way, this was formalised by the 9th Conférence Générale des Poids et Mesures (CGPM) in 1948 and the 22nd CGPM (2003). &lt;a href=&quot;https://www.bipm.org/documents/20126/33145759/CGPM22.pdf/36468a98-a1a8-7697-0470-5a3855653dc9&quot; rel=&quot;noreferrer&quot;&gt;Resolution 10 of the 22nd CGPM says:&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The 22nd General Conference (...)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;declares that the symbol for the decimal marker shall be either the point on the line or the comma on the line,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;reaffirms that “Numbers may be divided in groups of three in order to facilitate reading; neither dots nor commas are ever inserted in the spaces between groups”, as stated in Resolution 7 of the 9th CGPM, 1948.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second bit of the resolution is important and not well known; you should neither use a comma nor a point to separate the 1000s, just a space. This is exactly in order to avoid confusion, so 1.111 (like 1,111) can only ever mean 1+111/1000, but never 1111.&lt;/p&gt;&#xA;" OwnerUserId="9563" LastEditorUserId="9563" LastEditDate="2022-02-12T09:53:42.160" LastActivityDate="2022-02-12T09:53:42.160" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14186" PostTypeId="1" CreationDate="2022-02-12T16:04:22.423" Score="5" ViewCount="352" Body="&lt;p&gt;I have been learning about the Rank Nullity theorem and was trying to understand Who came up with the rank nullity theorem? While i did look up on the internet i came up with almost no answers. Some names i came across were Markham and Fiedler , Barrett and Feinsilver. But i dont know the details of it and what exactly ppl did?&lt;/p&gt;&#xA;" OwnerUserId="16236" LastActivityDate="2022-02-12T16:04:22.423" Title="Who proved Rank Nullity Theorem?" Tags="|mathematics|discoveries|linear-algebra|mathematics-social-history|scientists|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14188" PostTypeId="2" ParentId="14181" CreationDate="2022-02-13T09:43:33.980" Score="3" Body="&lt;p&gt;I think, with the precious help from &lt;a href=&quot;https://hsm.stackexchange.com/users/643/kcd&quot;&gt;KCd&lt;/a&gt;, that we can safely assert that:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;At least Bronstein was involved in that as it was something relevant for him (&lt;a href=&quot;http://www.ihst.ru/projects/sohist/books/bronstein.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;see around pdf-page 153&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The different attribution to &lt;a href=&quot;https://backreaction.blogspot.com/2011/05/cube-of-physical-theories.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Gamow, Ivanenko and Landau about a century ago&amp;quot;&lt;/a&gt; could be meaningful, but &lt;a href=&quot;http://gnsardan.appfarm.ru/GIL.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;the referenced document is not available&lt;/a&gt;; anyway it's all Soviet people from &amp;quot;&lt;a href=&quot;https://youtu.be/xumfir54xb4?t=470&quot; rel=&quot;nofollow noreferrer&quot;&gt;pariah CCCP&lt;/a&gt;&amp;quot;, so maybe they all worked on it. Anyway, &amp;quot;Bronstein's map&amp;quot; would be likely to be more accepted than &amp;quot;Soviet's map&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Still, it would be very nice to find if it is the &amp;quot;original work&amp;quot; of Bronstein, as this is what Carlo Rovelli wasn't quite satisfied about: &amp;quot;&lt;em&gt;it's not clear whether the drawing is a Bronstein's original work... it's a bio, not a paper of his own&lt;/em&gt;&amp;quot;. He replied this when I shared the &lt;a href=&quot;http://www.ihst.ru/projects/sohist/books/bronstein.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;book link&lt;/a&gt; and the below image with him.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KEhzW.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KEhzW.jpg&quot; alt=&quot;Bronstein's bio book with the map&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16229" LastEditorUserId="10106" LastEditDate="2022-02-18T10:27:18.620" LastActivityDate="2022-02-18T10:27:18.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14189" PostTypeId="1" CreationDate="2022-02-14T03:25:46.380" Score="4" ViewCount="730" Body="&lt;p&gt;There's a fairly widely known joke about boiling water (one version is below) that pokes fun at how mathematicians like to reduce new problems to known solutions.  I've traced it back to a footnote on page 17 of Vilenkin's &lt;em&gt;Combinatorial Mathematics for Recreation&lt;/em&gt; (Mir, 1972, trans. Yanovsky).  Does anyone know an earlier source?  Not very serious history of mathematics, I admit, but it would be nice to acknowledge who first came up with the witty story.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The story goes that a mathematician once asked a physicist: &amp;quot;You have an empty teakettle and an unlit gas range.  How do you go about boiling water?&amp;quot;  &amp;quot;Very simply,&amp;quot; replied the physicist.  &amp;quot;Fill the teakettle with water, light the gas and the put the water on to boil.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Right,&amp;quot; said the mathematician.  &amp;quot;Now solve this problem: the gas is burning and the teakettle has water in it.  How do you boil the water?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;That's no problem at all,&amp;quot; replied the physicist.  &amp;quot;Just put the teakettle on the range.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;No,&amp;quot; said the mathematician firmly.  &amp;quot;You turn off the gas, pour out the water and we arrive at our first problem, which we know how to solve.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Now, when a new problem is reduced to an already solved one, we speak of applying the &amp;quot;teakettle principle&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="3429" LastEditorUserId="3429" LastEditDate="2022-02-14T03:32:10.187" LastActivityDate="2022-02-14T03:32:10.187" Title="Origin of the &quot;teakettle principle&quot; joke?" Tags="|mathematics|reference-request|20th-century|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14191" PostTypeId="1" AcceptedAnswerId="14192" CreationDate="2022-02-14T12:02:23.190" Score="2" ViewCount="131" Body="&lt;p&gt;I have been recently studying periodic classification. I am having some confusion regarding old periodic classification.&lt;/p&gt;&#xA;&lt;p&gt;Newland's law of octaves&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wmu3j.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wmu3j.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In that you can see, in the 4th horizontal line, cobalt and nickel in one slot. When I ask the reason, many web sites and books describe it as &amp;quot;in order to fit elements, he keeps many elements under one slot.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;But I thought, &amp;quot;Why he didn't place nickel next to cobalt in another slot? Can't he?&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why under one slot necessary although he knew they are unlike elements?&lt;/p&gt;&#xA;&lt;p&gt;In Mendeleev's periodic table, if you look, he places 'Na' sodium in the same group with copper and silver, while they are totally different. Sodium is very reactive, unlike copper, silver, and gold.&lt;/p&gt;&#xA;&lt;p&gt;I found 2 different versions of Mendeleev's table:  the first in which sodium is placed with copper and all, and the second in which sodium is placed with potassium, lithium etc. (like modern table's group 1 elements) -- that kind of makes sense.&lt;/p&gt;&#xA;&lt;p&gt;First,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EJlWP.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EJlWP.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Second,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hTi5M.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hTi5M.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But mostly I found the first category table on the internet.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I am confused with  positioning; for example, sodium is put with copper, silver, and gold, although they have very different properties -- even today's modern periodic table has a different group for them.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;And also why elements are arranged in zig zag order in group in first table  not in a single straight line?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="15799" LastEditorUserId="10106" LastEditDate="2022-02-14T15:08:59.043" LastActivityDate="2022-02-14T15:51:11.217" Title="Element classification by Döbereiner and Newland (law of octaves)" Tags="|chemistry|periodic-table|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14192" PostTypeId="2" ParentId="14191" CreationDate="2022-02-14T15:51:11.217" Score="3" Body="&lt;p&gt;&lt;strong&gt;But I thought &amp;quot; why he didn't place nickel next to cobalt in another slot? Can't he?&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Newland ordered his table by atomic mass. &amp;quot;&lt;a href=&quot;https://chemistrygod.com/law-of-octaves&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to the law of octaves, every eighth element in the above table must share similar physical and chemical properties.&lt;/a&gt;&amp;quot; This pattern starts to break down after calcium. Newland thought thay cobalt and nickle, platinum and iridium, cerium and lanthanum each pair had the &lt;a href=&quot;https://web.lemoyne.edu/%7Egiunta/EA/NEWLANDSann.HTML&quot; rel=&quot;nofollow noreferrer&quot;&gt;same atomic weight&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I am confused with positioning; for example, sodium is put with copper, silver, and gold, although they have very different properties -- even today's modern periodic table has a different group for them.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;You are looking at what is called a &lt;a href=&quot;https://link.springer.com/article/10.1007/s40828-018-0059-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;short form periodic table&lt;/a&gt;. Apparently they are popular in former USSR countries. It shows how atoms have similar highest oxidation states, and chemical similarities based on the valence shell.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;And also why elements are arranged in zig zag order in group in first table not in a single straight line?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you look at the top of the table you see &amp;quot;A   B&amp;quot; in each group. The elements under them are arranged left to right according to group. Then you can see Sodium isn't put together with gold, copper, and silver. It goes group IA: H, Li, Na, K, Rb, CS and in group IB: Cu, Ag, Au. In this scheme Group IA is equivalent to group 1 in the IUPAC table and group IB is equivalent to group 11.&lt;/p&gt;&#xA;&lt;p&gt;Below is a modern version of the short form periodic table that has a much better layout.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/5jN1q.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5jN1q.png&quot; alt=&quot;x&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4880" LastActivityDate="2022-02-14T15:51:11.217" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14193" PostTypeId="1" AcceptedAnswerId="14195" CreationDate="2022-02-15T21:11:58.417" Score="0" ViewCount="126" Body="&lt;p&gt;Today I learnt that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Standard_deviation&quot; rel=&quot;nofollow noreferrer&quot;&gt;standard deviation&lt;/a&gt; is calculated as square root of the mean of the squares of the deviations from the arithmetic mean of the distribution. The mean of the squares of the deviations from the arithmetic mean of the distribution is called the variance. The population standard deviation is noted as &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;. However, no symbol was chosen for the variance. The population variance is simply noted as &lt;span class=&quot;math-container&quot;&gt;$\sigma^2$&lt;/span&gt;. The same thing applies for the sample standard deviation &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; and the sample variance &lt;span class=&quot;math-container&quot;&gt;$s^2$&lt;/span&gt;.  What is the reasoning for this? Are there any other examples like this?&lt;/p&gt;&#xA;&lt;p&gt;(Compare this with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert space&lt;/a&gt;. Look at the section which says &amp;quot;An element &lt;em&gt;A&lt;/em&gt; of &lt;em&gt;B&lt;/em&gt;(&lt;em&gt;H&lt;/em&gt;) is called 'self-adjoint' or 'Hermitian' if &lt;em&gt;A&lt;/em&gt;* = &lt;em&gt;A&lt;/em&gt;. If &lt;em&gt;A&lt;/em&gt; is Hermitian and ⟨&lt;span class=&quot;math-container&quot;&gt;$Ax, x⟩ \ge 0$&lt;/span&gt; for every &lt;em&gt;x&lt;/em&gt;, then &lt;em&gt;A&lt;/em&gt; is called 'nonnegative', written &lt;span class=&quot;math-container&quot;&gt;$A \ge 0$&lt;/span&gt;; if equality holds only when &lt;em&gt;x&lt;/em&gt; = &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt;, then &lt;em&gt;A&lt;/em&gt; is called 'positive'. The set of self adjoint operators admits a partial order, in which &lt;span class=&quot;math-container&quot;&gt;$A \ge B$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$A − B \ge 0$&lt;/span&gt;. If &lt;em&gt;A&lt;/em&gt; has the form &lt;em&gt;B&lt;/em&gt; * &lt;em&gt;B&lt;/em&gt; for some &lt;em&gt;B&lt;/em&gt;, then &lt;em&gt;A&lt;/em&gt; is nonnegative; if &lt;em&gt;B&lt;/em&gt; is invertible, then &lt;em&gt;A&lt;/em&gt; is positive. A converse is also true in the sense that, for a non-negative operator &lt;em&gt;A&lt;/em&gt;, there exists a unique non-negative square root &lt;em&gt;B&lt;/em&gt; such that &lt;span class=&quot;math-container&quot;&gt;$A = B^2 = B * B$&lt;/span&gt;.&amp;quot; Here, the non-negative operator is noted as &lt;em&gt;A&lt;/em&gt;, and its non-negative square root is noted as &lt;em&gt;B&lt;/em&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="13344" LastEditorUserId="13344" LastEditDate="2022-02-17T19:26:17.653" LastActivityDate="2022-02-17T19:26:17.653" Title="Why some calculations noted as &quot;sym^2&quot; and &quot;sym&quot;, while others noted as &quot;symA&quot; and &quot;symB&quot;, where &quot;symB&quot; is the square root of &quot;symA&quot;?" Tags="|terminology|statistics|statisticians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14194" PostTypeId="1" CreationDate="2022-02-15T23:30:21.973" Score="1" ViewCount="176" Body="&lt;p&gt;I'm currently studying Martin Gardner's 1968 repackaging of two of Henry Ernest Dudeney's collections: &amp;quot;536 Puzzles &amp;amp; Curious Problems&amp;quot;, and trying to track down all the references in his editorial footnotes.&lt;/p&gt;&#xA;&lt;p&gt;In his discussion of the &amp;quot;Four Fours&amp;quot; problem (Answers section of Puzzle 109 in the above collection), he references the magazine &amp;quot;Recreational Mathematics Magazine&amp;quot;, specifically issue 14 from 1964.&lt;/p&gt;&#xA;&lt;p&gt;I have managed to track down a copy of this specific issue of this magazine on sale via Amazon (although unfortunately not shipping to this locale), and also a copy of Issue 13. Nice and all, but what I am having difficulty tracking down is any publication details of the magazine itself: between what dates was it published, so on and so forth, and particularly (the holy grail) where I might be able to access its contents online -- or at least a list of its contents.&lt;/p&gt;&#xA;&lt;p&gt;Does anybody have any concrete information on this magazine?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="10106" LastEditDate="2022-02-22T06:56:53.680" LastActivityDate="2022-02-22T06:56:53.680" Title="Publication information on &quot;Recreational Mathematics Magazine&quot;" Tags="|mathematics|reference-request|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14195" PostTypeId="2" ParentId="14193" CreationDate="2022-02-16T14:01:20.783" Score="4" Body="&lt;p&gt;The reason is purely historical. When Karl Pearson introduced the standard deviation he used the symbol &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt; and then when Fisher wanted a symbol for its square he used &lt;span class=&quot;math-container&quot;&gt;$\sigma^2$&lt;/span&gt; but introduced a new name: variance. This can be found in &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathsym/stat/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Miller/mathsym/stat/&lt;/a&gt; several screens down the page. Note that there are other cases in statistics where we use a squared quantity because it seems a natural way to express things: &lt;span class=&quot;math-container&quot;&gt;$\chi^2$&lt;/span&gt;, and Hotelling's &lt;span class=&quot;math-container&quot;&gt;$T^2$&lt;/span&gt; for example although there are other examples.&lt;/p&gt;&#xA;&lt;p&gt;In fact I would question your assertion that there is no notation for the variance as many people use Var() or var() for it even though there is a possible confusion with variable.&lt;/p&gt;&#xA;&lt;p&gt;Just to add further issues the sample standard deviation is denoted &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; but the definition you give is for the population standard deviation which is indeed &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The reference given above was provided by user Gareth Rees in a comment on MSE.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2022-02-16T14:01:20.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14196" PostTypeId="1" AcceptedAnswerId="14202" CreationDate="2022-02-16T17:08:08.633" Score="10" ViewCount="3320" Body="&lt;p&gt;A math and physics magazine I was browsing through contains the quotation&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Earth is moving constantly, but people do not know it; like the&#xA;crew in an enclosed ship, they do not notice it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The attribution is simply &amp;quot;Chinese astronomical text, ca. 2nd century BCE&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I had thought the idea expressed here was original to Galileo in the 17th century CE. Was there really a text in China that had this idea almost 2000 years earlier? If so, what text is it?&lt;/p&gt;&#xA;&lt;p&gt;The reference for the magazine article is &lt;a href=&quot;https://static.nsta.org/pdfs/QuantumV7N1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leonovich, A. &amp;quot;Are You Relatively Sure?&amp;quot;  Quantum Magazine, V7N1&lt;/a&gt; (pp32).&lt;/p&gt;&#xA;" OwnerUserId="2238" LastEditorUserId="10106" LastEditDate="2022-02-18T10:20:52.283" LastActivityDate="2022-02-18T10:20:52.283" Title="Did a Chinese astronomical text conduct the &quot;Galileo's Ship&quot; thought experiment around the 2nd century BCE?" Tags="|physics|relativity-theory|ancient-china|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14197" PostTypeId="2" ParentId="14196" CreationDate="2022-02-16T19:05:54.243" Score="10" Body="&lt;p&gt;Chinese texts containing the idea of Galilean Relativity is actually an interpretation of Dai Nianzu (1), a Chinese historian of science. However, this interpretation of the texts is debatable, as is mentioned by Danian Hu in &lt;a href=&quot;https://www.jstor.org/stable/10.1086/521157&quot; rel=&quot;noreferrer&quot;&gt;The Reception of Relativity in China&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;[Dai] claims that as early as the first century B.C. there existed in&#xA;China an idea that was exactly the same as the Galilean principle of&#xA;relativity&lt;/strong&gt;; he also asserts that the Chinese expression&#xA;&lt;em&gt;yu-zhou&lt;/em&gt;, commonly  translated  as  “the  Universe”  and  meaning  more&#xA;literally “space-time,” demonstrates the traditional Chinese idea that&#xA;“time and space are inseparable.” &lt;strong&gt;However, Dai’s interpretation of the&#xA;ancient Chinese texts is subject to debate.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(1) Dai Nianzu, “Zhongguo ren weihe huanying xiangduilun” [Why did the  Chinese welcome the theory of relativity], Wuli, 2007, 36(3):254&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2022-02-16T19:05:54.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14198" PostTypeId="1" AcceptedAnswerId="14206" CreationDate="2022-02-17T10:31:19.493" Score="8" ViewCount="214" Body="&lt;p&gt;On page 26 of the 2003 article - The origins and legacy of Kolmogorov's&#xA;Grundbegriffe by Shafer and Vovk, it is stated that Borel rejected the countable additivity axiom of probability theory as he felt that a countable infinity of zero probabilities might add to a positive number. Was there a precedent for this in terms of an abstract or concrete application of probability theory in which this actually occurred? Or was his position simply being conservative about the axioms?&lt;/p&gt;&#xA;" OwnerUserId="16264" LastEditorUserId="55" LastEditDate="2022-02-17T21:23:18.757" LastActivityDate="2022-08-23T11:33:12.580" Title="Why did Borel reject countable additivity of probability?" Tags="|mathematics|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14199" PostTypeId="1" AcceptedAnswerId="14204" CreationDate="2022-02-17T10:31:38.440" Score="2" ViewCount="301" Body="&lt;p&gt;Mathematical statements and physics equations often are named after a person (like Pythagoras theorem or Newton's second law). Reading from different authors with different origins one may sometimes find a common statement with a different name. I have usually seen this in French and Russian sources, where Snell's law became Descartes' law and Gauss' theorem became Ostrogradsky's theorem respectively.&lt;/p&gt;&#xA;&lt;p&gt;Is there a law/theorem well known for having many different names/authors depending on the language/region? I am looking for more than 2 names/languages/regions.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-12-12T18:06:00.393" LastActivityDate="2023-12-12T18:06:00.393" Title="Is there any theorem/physical law with different names in more than two different languages/regions?" Tags="|mathematics|physics|big-list|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14200" PostTypeId="2" ParentId="14199" CreationDate="2022-02-17T11:21:08.380" Score="5" Body="&lt;p&gt;The richest example in this regard according to me is the &lt;strong&gt;WKB&lt;/strong&gt; approximation. WKB approximation, which is technically not a theorem but a recipe for obtaining approximate solutions to the time-independent Schrodinger equation, is named after Wentzel, Kramers and Brillouin. However, the technique is called by different names by/at different people/places:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;In Holland, it is called the &lt;strong&gt;KWB&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;In France, it is called the &lt;strong&gt;BWK&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;li&gt;In England, it is called the &lt;strong&gt;JWKB&lt;/strong&gt; with 'J' standing for Jeffreys.&lt;/li&gt;&#xA;&lt;li&gt;In the scientific community it is also known by the name &lt;strong&gt;LG&lt;/strong&gt; or &lt;strong&gt;Liouville–Green&lt;/strong&gt; method.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2022-02-17T11:21:08.380" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14202" PostTypeId="2" ParentId="14196" CreationDate="2022-02-17T16:18:59.377" Score="15" Body="&lt;p&gt;The text of Dai Nianzu's 中国人为何欢迎相对论 [Why did the Chinese welcome the theory of relativity], a commentary on Hu's book &lt;em&gt;Einstein in China&lt;/em&gt;, published on 物理 [&lt;em&gt;Physics&lt;/em&gt;; the journal for members of the Chinese Physical Society] is &lt;a href=&quot;http://www.wuli.ac.cn/fileup/PDF/20070318.pdf&quot; rel=&quot;noreferrer&quot;&gt;available here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The relevant quotes are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;早在汉代成书的《尚书纬·考灵曜》中，古代中国人曾这样论证“地动”的思想. 它写道：“地恒动不止，而人不知. 譬如人在大舟中，闭牖而坐，舟行而人不觉也”&lt;/p&gt;&#xA;&lt;p&gt;As early as in the book &lt;em&gt;Shangshu Wei: Kaolingyao&lt;/em&gt; written in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Han_dynasty&quot; rel=&quot;noreferrer&quot;&gt;Han Dynasty&lt;/a&gt; [202 BC – 220 AD], the ancient Chinese had argued the idea of a &amp;quot;moving earth&amp;quot;. It says: &amp;quot;The earth is constantly moving, but people do not know it. Like people sitting in a big boat, with the windows closed, the boat is moving and people do not realize it.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Shangshu Wei: Kaolingyao&lt;/em&gt; is a kind of ancient Chinese writings called 纬 (&amp;quot;Wei&amp;quot;), which are &lt;em&gt;apocryphal&lt;/em&gt; semi-religious texts purported to be prophetic. Wei books were later banned and destroyed by later dynasties due to e.g. purported political prophecies or other heretic materials. &lt;em&gt;Shangshu Wei&lt;/em&gt; was also destroyed; there is no &amp;quot;original&amp;quot; copies preserved. But other books or written works (e.g. official announcements, private letters) may quote or refer to the destroyed books which allow reconstruction of fragments (辑佚).&lt;/p&gt;&#xA;&lt;p&gt;So the quote is not taken from the work directly but from compilation and reconstruction by later scholars, e.g. &lt;a href=&quot;http://www.chinaknowledge.de/Literature/Classics/guweishu.html&quot; rel=&quot;noreferrer&quot;&gt;Guweishu&lt;/a&gt; which was printed during the Chongzhen 崇禎 reign-period (1628-1644). The quote can be found &lt;a href=&quot;https://ctext.org/library.pl?if=en&amp;amp;file=86809&amp;amp;page=40&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; in a 19th century reprint of the book. From my brief overview of the chapter (&lt;em&gt;Kaolingyao&lt;/em&gt;), it appears to be astronomical texts indeed and contains other writings on movement of the earth and stars and explanation of e.g. winter solstice.&lt;/p&gt;&#xA;&lt;p&gt;Other books also has similar quotes, e.g. in &lt;a href=&quot;https://en.wikipedia.org/wiki/Bowuzhi&quot; rel=&quot;noreferrer&quot;&gt;博物志 &lt;em&gt;Bowuzhi&lt;/em&gt;&lt;/a&gt; (c. 290 CE). The full quote there is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;《考灵曜》曰：地有四游，冬至地上北而西三万里，夏至地下南而东三万里，春秋二分其中矣。地常动不止，譬如人在舟而坐，舟行而人不觉。天地四方皆海水相通，地在其中盖无几也。七戎，六蛮，九夷，八狄，经总而言之，谓之四海，言皆近海；海之言晦昏，无所睹也。&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Kaolingyao&lt;/em&gt; says that the earth moves up and down in all (four) directions. At the winter solstice, the earth moves upward, 30,000 miles [See note] from the north to the west; at the summer solstice, the earth moves downward, 30,000 miles from the south to the east; at the two equinoxes spring or autumn, it is somewhere in the middle. The earth is constantly moving, but people do not know it. It is just like people sitting in a big boat, with the windows closed, the boat is moving and people do not realize it. The four directions of heaven and earth are connected by sea water, so the earth is a small part of this. the peoples of different appearances and types living in the different directions are collectively called the &amp;quot;four seas&amp;quot;, which means they are all near the sea and implies that they are dim and foolish without insight.&lt;/p&gt;&#xA;&lt;p&gt;Note: not actual miles, but an unspecified Chinese distance unit; nor is 30,000 an exact number.&#xA;(Translated and adapted from the modern Chinese translation of the classic texts available &lt;a href=&quot;https://www.99csw.com/book/10284/370841.htm&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alternatively, according an &lt;a href=&quot;https://www.tspweb.com/key/%E8%80%83%E7%81%B5%E6%9B%9C.html&quot; rel=&quot;noreferrer&quot;&gt;unsourced, although coherent post&lt;/a&gt; online, there is another book from Tang dynasty (618 to 907 CE) that claims the relevant quote is from another apocryphal book from Han dynasty [202 BC – 220 AD] called 河图 &lt;em&gt;Hetu&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would be reasonably confident that the quote and its ideas existed before Galileo, although it may be difficult to determine how long ago it is actually from.&lt;/p&gt;&#xA;&lt;p&gt;Disclaimer: I am not a historian of science (nor of anything else, e.g. ancient Chinese books). I am simply providing the context that I can find and reasonably verify.&lt;/p&gt;&#xA;" OwnerUserId="16076" LastActivityDate="2022-02-17T16:18:59.377" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14204" PostTypeId="2" ParentId="14199" CreationDate="2022-02-17T19:22:45.857" Score="8" Body="&lt;p&gt;I think that what you asking for is not so rare, for example the &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem&quot; rel=&quot;noreferrer&quot;&gt;Rouché-Capelli theorem&lt;/a&gt;&amp;quot; is&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Rouché–Capelli theorem in English speaking countries, Italy, Brazil and Japan;&lt;/li&gt;&#xA;&lt;li&gt;Kronecker–Capelli theorem in Austria, Poland, Romania and Russia;&lt;/li&gt;&#xA;&lt;li&gt;Rouché–Fontené theorem in France;&lt;/li&gt;&#xA;&lt;li&gt;Rouché–Frobenius theorem in Spain and many countries in Latin America;&lt;/li&gt;&#xA;&lt;li&gt;Frobenius theorem in the Czech Republic and in Slovakia.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Another example, certainly less conspicuous, is the &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_cosines&quot; rel=&quot;noreferrer&quot;&gt;law of cosines&lt;/a&gt;&amp;quot; that in France is the &lt;em&gt;théorème d'Al-Kashi&lt;/em&gt; (but also &lt;em&gt;loi des cosinus&lt;/em&gt; is common), and &amp;quot;generalized Pythagorean theorem&amp;quot; in many countries where the two names coexist (in France again, but also in Italy).&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2022-02-17T19:22:45.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14205" PostTypeId="2" ParentId="14199" CreationDate="2022-02-17T20:49:32.983" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy, Schwarz and Bunyakovski&lt;/a&gt; names are used in French, Russian and English speaking countries, along with various combinations of these names.&lt;/p&gt;&#xA;&lt;p&gt;The names &lt;a href=&quot;https://en.wikipedia.org/wiki/Casorati%E2%80%93Weierstrass_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt; Weierstrass, Casorati and Sochocki&lt;/a&gt; are variously associated with&#xA;one theorem (which was really stated for the first time by Briot and Bouquet).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sokhotski%E2%80%93Plemelj_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Another theorem&lt;/a&gt; of Sochoski is variousuly credited to Plemelj or to&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kramers and Kronig&lt;/a&gt;. (Only Russian speakers call it correctly).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-02-17T20:49:32.983" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14206" PostTypeId="2" ParentId="14198" CreationDate="2022-02-17T21:22:41.063" Score="12" Body="&lt;p&gt;Even Kolmogorov himself &amp;quot;rejected&amp;quot; countable additivity, in the sense of not making it a universal property of probability. His first chapter axiomatizes and studies finitely additive probability measures, and countable additivity is only added in the second as an extra assumption. All he has to say about it  in &lt;a href=&quot;https://archive.org/details/foundationsofthe00kolm/page/14/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Foundations of the Theory of Probability (1933)&lt;/a&gt; is this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We limit ourselves arbitrarily to only those models that satisfy Axiom VI. This limitation has been found expedient in researches of most diverse sort&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Natural_density&quot; rel=&quot;noreferrer&quot;&gt;natural density&lt;/a&gt; on natural numbers is a classical counterexample that Kolmogorov was well aware of (it was featured in Khinchin's  &lt;em&gt;Zur additiven Zahlentheorie&lt;/em&gt; (1932) among many other places around 1930, see &lt;a href=&quot;https://www.ams.org/journals/bull/1951-57-06/S0002-9904-1951-09543-9/&quot; rel=&quot;noreferrer&quot;&gt;Niven, The Asymptotic Density of Sequences&lt;/a&gt;). The probability of uniformly choosing any particular natural number is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;, but the probability of choosing at least one is, of course, &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Borel did not reject countable additivity because of such examples, which everybody knew about, but because of suspicions about infinitary set-theoretic arguments that Lebesgue and others (Borel himself included, as it turned out, albeit unwittingly) introduced into measure theory. Borel was a proto-constructivist and platonistic tendencies of Cantor's set theory did not sit well with him. Accordingly, he preferred computation of probabilities as limits for philosophical reasons, because they clearly present the infinite as merely an idealization of the finite. Borel's seminal 1909 paper on the subject was titled &lt;em&gt;Les Probabilities Denombrables et Leurs Applications Arithmetiques&lt;/em&gt; (although, as Kac quipped, &amp;quot;&lt;em&gt;all of its theorems are true but almost all of the proofs are false&lt;/em&gt;&amp;quot;). Shafer and Vovk say as much:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Borel’s discomfort with a measure-theoretic treatment can be attributed to his unwillingness to assume countable additivity for probability (Barone and&#xA;Novikoff 1978, von Plato 1994). He saw no logical absurdity in a countably infinite number of zero probabilities adding to a nonzero probability, and so&#xA;instead of general appeals to countable additivity, he preferred arguments that derive probabilities as limits as the number of trials increases (1909a, §I.4). Such&#xA;arguments seemed to him stronger than formal appeals to countable additivity, for they exhibit the finitary pictures that are idealized by the infinitary pictures.&#xA;But he saw even more fundamental problems in the idea that Lebesgue measure can model a random choice (von Plato 1994, pp. 36–56; Knobloch 2001). How can we choose a real number at random when most real numbers are not even definable in any constructive sense.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ironically, Borel had no compunction about using what he called &amp;quot;countable independence&amp;quot;, and considered it essential to his &amp;quot;denumerable probability theory&amp;quot;. Ironically, because it is equivalent to countable additivity of the product measure, the very property he was so reluctant to employ, but on which his theorems depended. This lack of realization was responsible both for errors in his paper, and for overlooking proofs and results that would have made it stronger, see &lt;a href=&quot;https://www.jstor.org/stable/41133504&quot; rel=&quot;noreferrer&quot;&gt;Barone-Novikoff, A history of the axiomatic formulation of probability from Borel to Kolmogorov&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;BOREL regarded countable independence  as the essential, new (and probabilistic) ingredient of his new theory. By contrast he used countable additivity seldom (often surreptitiously), and he never&#xA;explored its implications; about it he had reservations so deep that he frequently offered &amp;quot;alternative&amp;quot; proofs to evade reliance on it.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This assessment, which we shall defend by suitable analysis, explains at least in part why BOREL failed to draw the conclusion, attributed to CANTELLI (1917a,&#xA;1917b)... even if the trials are not assumed independent. BOREL's fascination with the principle of countable independence similarly&#xA;may explain his failure to use anywhere in BOREL (1909) ... countable sub-additivity which follows from countable additivity even if the &lt;span class=&quot;math-container&quot;&gt;$B_i$&lt;/span&gt; are not mutually incompatible.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-23T11:33:12.580" LastActivityDate="2022-08-23T11:33:12.580" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14207" PostTypeId="1" AcceptedAnswerId="14209" CreationDate="2022-02-18T02:02:52.310" Score="5" ViewCount="231" Body="&lt;p&gt;Background:&lt;/p&gt;&#xA;&lt;p&gt;I'm an undergraduate student in Mathematics and I study Mathematical Logic and Philosophy of Science in an undergraduate research project (here in Brazil, where I'm from, we call this kind of training &amp;quot;scientific iniciation&amp;quot;). Nowadays, I'm studying Halvorson's book &amp;quot;The Logic in Philosophy of Science&amp;quot; where Halvorson uses Category Theory for metatheorizing about the structure of scientific theories.&lt;/p&gt;&#xA;&lt;p&gt;Ok, so here really begins my question.&lt;/p&gt;&#xA;&lt;p&gt;The history of this this kind of inquiry goes back to Vienna Circle's received view approach of scientific theories. At the beginning of the 20th century, the Vienna Circle played an important role in the rigorous study of scientific theories (regardless of whether the group's ideas were good or not). This importance is strongly related to the introduction of formal methods in philosophical investigation and more rigor as a methodology (or at least is how I understand it - feel free to enlighten me about it if you want). Another important group that comes to my mind - as a math student - is the Bourbaki group - which similarly influenced the development of Mathematics in the early 20th century. My question then is: did these groups have any connections or members in common?&lt;/p&gt;&#xA;" OwnerUserId="16268" LastEditorUserId="16268" LastEditDate="2022-02-18T02:28:44.300" LastActivityDate="2022-02-18T21:29:13.870" Title="Relation between Bourbaki group and Vienna Circle" Tags="|mathematics|philosophy-of-science|mathematical-logic|bourbaki|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14208" PostTypeId="2" ParentId="14199" CreationDate="2022-02-18T13:39:23.603" Score="0" Body="&lt;p&gt;The &lt;a href=&quot;http://www.calculatoredge.com/math/mathhistory/historyans27.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Ludolphian number&lt;/strong&gt;&lt;/a&gt; (as it is known in Germany—at least it was up to 100 years ago) is called &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; in English-language texts.  It was called the Ludolphian number recognizing that Ludolph Van Ceulen calculated it to 35 decimal places in the sixteenth century.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-02-18T13:39:23.603" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14209" PostTypeId="2" ParentId="14207" CreationDate="2022-02-18T21:29:13.870" Score="6" Body="&lt;p&gt;Short answer: not much of a relation. Perhaps the stars could have aligned differently if the Vienna Circle lasted longer, but with the rise of Nazism many members had to flee Austria and the meetings effectively stopped in 1936. The majority of Vienna's associates were German speaking, and their focus was empirical science. There were not many mathematicians associated with Vienna, notably von Neumann, Menger and Wald, Gödel was close to Carnap in the late 1920s, but that is about it.  The Bourbaki group was only coalescing in mid 1930s, and it mostly involved French mathematicians, so there was no personal connection. The focus of Bourbaki, at least originally, was reforming mathematics education, which also did not match the empirical and philosophical bent of Vienna.&lt;/p&gt;&#xA;&lt;p&gt;Finally, it is only in the rear view mirror of today that everything formal and rigorous looks like a single lump. At the time, what moved Vienna and Bourbaki were very different drives, they descended from rival outlooks. The Vienna circle inherited Frege's approach, through Russell and (early) Wittgenstein, see e.g. &lt;a href=&quot;https://conservancy.umn.edu/handle/11299/185663&quot; rel=&quot;noreferrer&quot;&gt;Friedman, Logical Truth and Analyticity in Carnap's &amp;quot;Logical Syntax of Language&amp;quot;&lt;/a&gt;. It favored genetic definitions (like Dedekind's of real numbers), universalist language (no meta), and &amp;quot;reducing mathematics to logic&amp;quot; (this is an oversimplification). Bourbaki, on the other hand, adopted (selectively) Hilbert's formalism with its pluralism of flexible axiomatics, and largely for pragmatic reason of not getting entangled with philosophy. On the deep divides between the those two visions of mathematics see e.g. &lt;a href=&quot;https://plato.stanford.edu/entries/frege-hilbert/&quot; rel=&quot;noreferrer&quot;&gt;Frege-Hilbert correspondence&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Gödel's results of early 1930s dealt a blow to both, but much more so to Fregeanism. Carnap did adjust to semantic paradigm in his later works, but the wind was taken out of the sails, so to speak. On the other hand, Bourbaki's pragmatic formalism light, with curbed ambitions for finitary justification, flourished. If the Vienna group survived into the times where the original foundational rivalries lost their significance perhaps there could have been fruitful interaction on the common ground of precision and rigor. But that was not to be.&lt;/p&gt;&#xA;&lt;p&gt;Ironically, the &amp;quot;received view&amp;quot; of scientific theories associated with Vienna's Hempel, which was close in spirit to mathematical formalism, was displaced by the semantic view by 1970s, see &lt;a href=&quot;https://www.princeton.edu/%7Ehhalvors/teaching/phi520_f2012/Suppe_2000.pdf&quot; rel=&quot;noreferrer&quot;&gt;Suppe, Understanding Scientific Theories: An Assessment of Developments, 1969-1998&lt;/a&gt;. One area that benefited from both influences was mathematical economics, see &lt;a href=&quot;https://www.scielo.br/j/rep/a/6Pyr7KZqC6gGddJc7zLGV3N/?lang=en&quot; rel=&quot;noreferrer&quot;&gt;Martinelli-Luperi, The impacts of the early 20th century physics and mathematics crisis on contemporary economics discourse&lt;/a&gt;. The original Viena inspired works of von Neumann, Menger and his students on utility and general equilibrium was picked up by Debreu in 1950s, who attended Bourbaki seminars and was put off by &amp;quot;looseness&amp;quot; he found in contemporary economics texts.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-02-18T21:29:13.870" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14210" PostTypeId="1" AcceptedAnswerId="14212" CreationDate="2022-02-19T07:36:08.733" Score="3" ViewCount="163" Body="&lt;p&gt;Paul Wernicke, the eponym of Wernicke's Theorem , discovered during his investigations into the Four Color Theorem.&lt;/p&gt;&#xA;&lt;p&gt;I understand he was born somewhere in the German sphere of influence (i.e. could have been what is now Poland, for example), and sometime after that moved to the US.&lt;/p&gt;&#xA;&lt;p&gt;There are also a few citations I have found concerning his 1904 paper in Mathematische Annalen (vol 58 no 3 pp 413-426), and some indication that he also published on the subject in 1897 -- but I can find very little else.  He is mentioned in Robin Wilson's 2002 &amp;quot;Four Colours Suffice&amp;quot;, but he also includes nothing beyond the above information.&lt;/p&gt;&#xA;&lt;p&gt;Is there a resource anywhere which provides any further information: dates of birth and death, locations of such, where he studied, what he did for a living, and so on?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="4201" LastEditDate="2022-03-02T21:12:02.530" LastActivityDate="2022-03-02T21:12:02.530" Title="Biographical details for Paul Wernicke" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="14211" PostTypeId="1" CreationDate="2022-02-19T17:54:03.280" Score="3" ViewCount="183" Body="&lt;p&gt;I have always found strange that in elementary analytic geometry points are defined by their names followed by their coordinates, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Find the distance between &lt;span class=&quot;math-container&quot;&gt;$A(5, -3)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B(2, 1)$&lt;/span&gt;.&amp;quot; (&lt;a href=&quot;https://www.math10.com/problems/problems-in-analytic-geometry-plane/easy/&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Usually, mathematical objects are defined either by an equality or by a uniquely characterizing property. So one would expect points to be defined such as &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$A = (0,1)$&lt;/span&gt;&amp;quot; (once the coordinate system is fixed) or &amp;quot;Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be the point of coordinates &lt;span class=&quot;math-container&quot;&gt;$x=  0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y=0$&lt;/span&gt;,&amp;quot; but not &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$A(0,1)$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, usually a letter followed by some numbers between parentheses denotes the evaluation of a function at certain points, so &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$A(0,1)$&lt;/span&gt;&amp;quot; could be misunderstood for &amp;quot;The value of the function &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; at point &lt;span class=&quot;math-container&quot;&gt;$(0,1)$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;What is the historical, practical, or whatelse reason for this notation?&lt;/p&gt;&#xA;" OwnerUserId="16280" LastEditorUserId="10106" LastEditDate="2022-02-22T06:58:04.247" LastActivityDate="2022-07-22T08:07:10.623" Title="Why is this notation used to define points in (elementary) analytic geometry?" Tags="|mathematics|notation|geometry|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14212" PostTypeId="2" ParentId="14210" CreationDate="2022-02-19T21:16:22.513" Score="5" Body="&lt;p&gt;Ludwig August &lt;strong&gt;Paul&lt;/strong&gt; Wernicke was born in &lt;a href=&quot;https://en.wikipedia.org/wiki/Leipzig&quot; rel=&quot;noreferrer&quot;&gt;Leipzig&lt;/a&gt; on January 9, 1866 &lt;sup&gt;1&lt;/sup&gt;. His father appears to have been the Lutheran pastor August Wernicke, who became embassy chaplain in Lisbon in 1873 &lt;sup&gt;1&lt;/sup&gt;. Prior to that he was pastor of the German Lutheran parish in Helsingfors (modern-day &lt;a href=&quot;https://en.wikipedia.org/wiki/Helsinki&quot; rel=&quot;noreferrer&quot;&gt;Helsinki&lt;/a&gt;) from 1870 to 1873 &lt;sup&gt;2&lt;/sup&gt;. In Lisbon, the embassy chaplain also served as pastor of the German Protestant parish &lt;sup&gt;3&lt;/sup&gt;. Indications are that August Wernicke held this position until 1883 &lt;sup&gt;4&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Paul Wernicke was home schooled until he entered high school in 1882, boarding at the prestigious school in &lt;a href=&quot;https://en.wikipedia.org/wiki/Pforta&quot; rel=&quot;noreferrer&quot;&gt;Pforta&lt;/a&gt; &lt;sup&gt;1&lt;/sup&gt;. This school for gifted students was established in a former monastery in 1543 by the rulers of Saxony. The vast majority of the students belonged to the Lutheran denomination. Wernicke graduated Easter 1885 &lt;sup&gt;1&lt;/sup&gt;. After his compulsory one-year military service he studied mathematics and natural sciences at the University of Berlin &lt;sup&gt;1&lt;/sup&gt;. He moved to the US in 1892 and became a college professor &lt;sup&gt;1&lt;/sup&gt; at Agricultural &amp;amp; Mechanical College in Lexington, Kentucky in 1894 &lt;sup&gt;5&lt;/sup&gt;, teaching German and French &lt;sup&gt;6&lt;/sup&gt;. In 1898 Wernicke became a naturalized citizen of the United States &lt;sup&gt;1&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;He seems to have retained an interest in mathematics, presenting a paper on the map-coloring problem at the summer meeting of the American Mathematical Society in Toronto in 1897. As head of the department of modern languages of A. &amp;amp; M. College he took a 1½-year leave of absence &lt;sup&gt;1&lt;/sup&gt; around 1903 &lt;sup&gt;7&lt;/sup&gt; to pursue a Ph.D. in mathematics at the &lt;a href=&quot;https://en.wikipedia.org/wiki/University_of_G%C3%B6ttingen&quot; rel=&quot;noreferrer&quot;&gt;Georg August University&lt;/a&gt; in Göttingen. His thesis advisor there was &lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_Minkowski&quot; rel=&quot;noreferrer&quot;&gt;Hermann Minkowski&lt;/a&gt; &lt;sup&gt;8&lt;/sup&gt;. His dissertation &amp;quot;Ueber die Analysis situs mehrdimensionaler Räume.&amp;quot; (Analysis situs in higher dimensions) was published in 1904; the Ph.D. defense took place on August 1, 1904.&lt;/p&gt;&#xA;&lt;p&gt;In 1906 Wernicke became an additional instructor in mathematics at Washington University in St. Louis, Missouri &lt;sup&gt;9&lt;/sup&gt;. The following year he was upgraded to full instructor &lt;sup&gt;10&lt;/sup&gt;. Between 1908 and 1910 there are references to him as a professor of mathematics at the University of Kansas in Lawrence &lt;sup&gt;11&lt;/sup&gt;. Wernicke did some work as a translator &lt;sup&gt;12&lt;/sup&gt;, and in 1919 he was listed as a translator at the patent office in Washington, D.C. &lt;sup&gt;13&lt;/sup&gt;. From 1921 to 1924 he taught courses in mathematics at Research University in Washington, D.C. &lt;sup&gt;14&lt;/sup&gt;. In the late 1920s and early 1930s, he appears to have been active in the &amp;quot;Problems and Solutions&amp;quot; section of &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;. His last mention in this context is in 1932 &lt;sup&gt;15&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I have not been able to establish a date of death. I cannot find Paul Wernicke in what is available of the 1940 US census free of cost; you may want to check one of the commercial genealogical databases for that data.&lt;/p&gt;&#xA;&lt;p&gt;Publications:&lt;/p&gt;&#xA;&lt;p&gt;F. N. Cole. &amp;quot;The fourth summer meeting of the American Mathematical Society.&amp;quot; &lt;em&gt;Bull. Amer. Math. Soc.&lt;/em&gt; &lt;strong&gt;4&lt;/strong&gt; (1897), pp. 1-11. &lt;a href=&quot;https://www.ams.org/journals/bull/1897-04-01/S0002-9904-1897-00444-X/S0002-9904-1897-00444-X.pdf&quot; rel=&quot;noreferrer&quot;&gt;(scan online)&lt;/a&gt; with a short abstract of Wernicke's paper (the paper itself seems no longer extant):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Professor P. WERNICKE: &amp;quot;On the solution of the map-color problem.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Paul Wernicke, &amp;quot;Ueber die Analysis situs mehrdimensionaler Räume.&amp;quot; Dissertation, Georg-August-Universität Göttingen, 1904. &lt;a href=&quot;https://www.google.com/books/edition/Ueber_die_Analysis_Situs_mehrdimensional/o0c7AQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;(Google scan)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;P. Wernicke, &amp;quot;Über den kartographischen Vierfarbensatz.&amp;quot; &lt;em&gt;Mathematische Annalen&lt;/em&gt;, Vol. 58, No. 3, Sept. 1904, pp. 413-426 &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0058?tify=%7B%22pages%22:%5B423%5D,%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;(scan online)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;P. Wernicke, &amp;quot;Die Zahl der ordinären Kollineationstypen.&amp;quot; &lt;em&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung&lt;/em&gt;, Vol. 18 (1909), pp. 397-398 &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0019?tify=%7B%22pages%22:%5B272%5D,%22panX%22:0.532,%22panY%22:0.513,%22view%22:%22info%22,%22zoom%22:0.514%7D&quot; rel=&quot;noreferrer&quot;&gt;(scan online)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;P. Wernicke, &amp;quot;Das Problem der 36 Offiziere.&amp;quot; &lt;em&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung&lt;/em&gt;, Vol. 19 (1910), pp. 264-267 &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0019?tify=%7B%22pages%22:%5B272%5D,%22panX%22:0.532,%22panY%22:0.513,%22view%22:%22info%22,%22zoom%22:0.514%7D&quot; rel=&quot;noreferrer&quot;&gt;(scan online)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Paul Wernicke, &amp;quot;The theorems of Ceva and Menelaus and their extension.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 34, No. 9, Nov. 1927, pp. 468-472 &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.1927.11986749?journalCode=uamm20&quot; rel=&quot;noreferrer&quot;&gt;(partial scan online)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Paul Wernicke, &amp;quot;The Rectangular Hexagon.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 37, No. 2, Feb. 1930, pp. 59-63 &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.1930.11987583&quot; rel=&quot;noreferrer&quot;&gt;(partial scan online)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;B. F. Finkel, Otto Dunkel, and H. L. Olson (eds.), &amp;quot;Problems and Solutions.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 38, No. 6, Jun.-Jul. 1931, p. 339 &lt;a href=&quot;https://www.jstor.org/stable/2301836&quot; rel=&quot;noreferrer&quot;&gt;(scan online)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;3493&quot;&gt;&#xA;&lt;li&gt;&lt;em&gt;Proposed by Paul Wernicke, Washington, D.C.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; &amp;quot;Lebenslauf&amp;quot; (curriculum vitae) at the back of Wernicke's 1904 &lt;a href=&quot;https://books.google.com/books?id=o0c7AQAAIAAJ&amp;amp;pg=PT2#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;dissertation&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; John Nicholas Lenker, &amp;quot;Die Lutherische Kirche Der Welt, Band 1&amp;quot;, Sunbury, Pennsylvania, 1901, &lt;a href=&quot;https://www.google.com/books/edition/Die_Lutherische_Kirche_Der_Welt/3eBNAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22August+Wernicke%22&amp;amp;pg=PA236&quot; rel=&quot;noreferrer&quot;&gt;p.236&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die deutsche evangelisch-lutherische Gemeinde in Helsingfors besteht seit 1858 [...] Auch das innere Gemeindeleben hat sich, gepflegt von tüchtigen und treuen Pastoren (Andreas Schröder 1858-70, August Wernicke, 1870-73 [...]) gedeihlich entfaltet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;3&lt;/sup&gt; Franz Otto Stichart (ed.), Allgemeine Kirchliche Chronik&amp;quot;, Vol. 24. Hamburg: Haendcke &amp;amp; Lehmkuhl 1878 &lt;a href=&quot;https://www.google.com/books/edition/Allgemeine_kirchliche_Chronik/IOZm8ePfMz8C?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA23&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 23&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Infolge dieser Bestimmungen besteht in Lissabon eine englische und eine deutsch-evangelische Gemeinde mit besonderer Kapellem Friedhöfen und Schulen. Pfarrer der deutschen Gemeinde ist der jeweilige Gesandtschaftsprediger [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;4&lt;/sup&gt; &lt;a href=&quot;https://www.dekl.org/angebote/dekl-shop/&quot; rel=&quot;noreferrer&quot;&gt;https://www.dekl.org/angebote/dekl-shop/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pfr. Wilhelm Rothe / Pfr. Eduard Meyer / Pfr. August Wernicke: „Chronik der Deutsch-evangelischen Gemeinde zu Lissabon“, 1865 – 1883&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;5&lt;/sup&gt; &amp;quot;The Kentuckian&amp;quot; (college yearbook), Lexington, Kentucky, 1906, &lt;a href=&quot;https://www.google.com/books/edition/The_Kentuckian/1qglAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA26&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 26&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paul Wernicke, Graduate of Gymnasium of Schulpforta, Germany, 1885; University of Berlin, 1889; Ph.D., University of Göttingen, 1903; American Mathematical Society; American Association for Advancement of Science; Modern Language Association of America; &amp;quot;Analysis Situs in Higher Dimensions;&amp;quot; Professor of Modern Languages, Kentucky State College, since 1894&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;6&lt;/sup&gt; &amp;quot;Catalogue of Kentucky University, Lexington, Ky., 1892-1893.&amp;quot; Lexington, Ky.: Transylvania Printing 1893, &lt;a href=&quot;https://www.google.com/books/edition/Annual_Register_of_the_State_College_of/h-JHAQAAMAAJ?hl=en&amp;amp;gbpv=1&quot; rel=&quot;noreferrer&quot;&gt;p. 27&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Agricultural and Mechanical College of Kentucky is established in the old City Park grounds of the City of Lexington [...] Faculty [...] Paul Wernicke Professor of French, German, and Spanish&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;7&lt;/sup&gt;&amp;quot;Biennial Report of the Superintendent of Public Instruction of Kentucky&amp;quot;. Louisville, Ky.: Geo. G Fetter Publishing 1903, &lt;a href=&quot;https://books.google.com/books?id=t8CgAAAAMAAJ&amp;amp;pg=PA72&amp;amp;lpg=PA72&amp;amp;dq=%22P.+Wernicke%22+Lexington&amp;amp;source=bl&amp;amp;ots=b8q8HGCMKR&amp;amp;sig=ACfU3U0jqvy3B6IKBt1SSgD8eny7QGYwIQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjdsqzvwIv2AhV6LDQIHeoBDBYQ6AF6BAgWEAM#v=onepage&amp;amp;q=%22P.%20Wernicke%22%20Lexington&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;p. 72&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lexington, September 30, 1903. President James K. Patterson, Dear Sir: - Upon request, I have the honor to submit the following report of the condition of the department of modern languages during the collegiate year 1902-1903. As I was abroad on leave of absence, instruction was given by Messrs. Winston, assistant in the academy, and T. T. Jones, a graduate student, in those branches only [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;8&lt;/sup&gt; See dissertation. Also &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=52382&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paul August Ludwig Wernicke; Dr. phil. Georg-August-Universität Göttingen 1904 Germany; Dissertation: Über die Analysis situs mehrdimensionaler Räume; Advisor: Hermann Minkowski&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;9&lt;/sup&gt; The Washington University Record, Vol. 2, No. 1, St. Louis, Mo., October 1906, &lt;a href=&quot;https://www.google.com/books/edition/Washington_University_Record/vcdGAQAAMAAJ?hl=en&amp;amp;gbpv=1&quot; rel=&quot;noreferrer&quot;&gt;p. 2&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mr. Paul Wernicke, Ph.D., additional instructor in Mathematics. Dr. Wernicke studied in the Universities of Berlin and Goettingen, taking his Ph.D. in Mathematics from the latter institution. For the last ten years he has been at the head of the department of Modern Languages in the University of Kentucky, and now returns to his former work in Mathematics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;10&lt;/sup&gt; &amp;quot;The Bulletin of the Washington University Association,&amp;quot; Vol. 5 (1907), &lt;a href=&quot;https://www.google.com/books/edition/Bulletin_of_Washington_University_St_Lou/sq1GAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Paul+Wernicke+Washington+University&amp;amp;pg=PA218&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 213&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To be instructor in Mathematics, Paul Wernicke, Ph.D. (Göttingen, 1904); student of the Instituto Industrial e Commercial, Lisbon, Portugal;&#xA;graduate of the Gymnasium &amp;quot;Fürstenschule Pforta;&amp;quot;, student in Mathematics and in Romanic and English Philosophy, University of Berlin;&#xA;professor of Modern Languages, State College, Lexington, Kentucky, 1894-1906; member of A.A.A.S, Amer. Math. Soc. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;11&lt;/sup&gt; &amp;quot;Bulletin of the University of Kansas. General Catalogue 1908-'09&amp;quot;, Lawrence, Kansas, April 1909, &lt;a href=&quot;https://www.google.com/books/edition/Annual_Catalogue_of_the_University_of_Ka/YehHAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=Paul%20Wernicke%20&quot; rel=&quot;noreferrer&quot;&gt;p. 23&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Officers of Instruction&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;Paul Wernicke, Ph.D., 1903, (Göttingen) &lt;em&gt;Instructor in Mathematics&lt;/em&gt;, 1908.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Forty-Forth Annual Catalogue of the University of Kansas, for the year 1909-'10, Lawrence, Kansas, April 1910, &lt;a href=&quot;https://www.google.com/books/edition/Annual_Catalogue/22ktAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=professor+Paul+Wernicke+Kansas&amp;amp;pg=PA114&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 114&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paul Wernicke, Ph.D., Instructor in Mathematics&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;12&lt;/sup&gt; Paul T. Passalskij, &amp;quot;On the distribution of magnetism over the earth's surface. II&amp;quot; Translated by Paul Wernicke, &lt;em&gt;Terrestrial Magnetism and Atmospheric Electricity&lt;/em&gt;, Vol. 13, No. 4, December 1908, &lt;a href=&quot;https://www.google.com/books/edition/Terrestrial_Magnetism_and_Atmospheric_El/AB7OAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=Wernicke&quot; rel=&quot;noreferrer&quot;&gt;pp. 161-174&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;13&lt;/sup&gt; &amp;quot;Official Register of the United States 1919, Directory.&amp;quot;. Washington 1920, &lt;a href=&quot;https://www.google.com/books/edition/Official_Register_of_the_United_States/VJo4AQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Paul%22+%22Wernicke%22&amp;amp;pg=PA87&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p.87&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Department of the Interior&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;Patent Office&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;Paul Wernicke Translator&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;14&lt;/sup&gt; &amp;quot;Hearing before the subcommittee on judiciary of the committee on the District of Columbia; House of Representatives; Seventieth Congress; First Session on H.R. 7951; March 23, 24, April 2, 3,and 5, 1928&amp;quot;, &lt;a href=&quot;https://www.google.com/books/edition/Degree_conferring_Institutions/4yk1DIyXROoC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22Wernicke%22&quot; rel=&quot;noreferrer&quot;&gt;p.138&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Paul Wernicke, Ph.D., 3600 South Dakota Avenue, Washington, D.C.&#xA;&amp;quot;My connection with Research University, with regards to the standing of which you are inquiring, has been limited to teaching in their evening classes during the years 1921 to 1924. My classes were small ones in analytic geometry and calculus. They were put through regular collegiate examinations&#xA;and graded according to standards of recognized schools, e.g. the engineering department of Washington University, St. Louis, where I formerly taught. [..]&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This appears to have been a &lt;a href=&quot;https://chroniclingamerica.loc.gov/lccn/sn83045462/1928-04-05/ed-1/seq-2/ocr/&quot; rel=&quot;noreferrer&quot;&gt;hearing&lt;/a&gt; regarding a bill against so-called &amp;quot;degree mills&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;15&lt;/sup&gt; Paul Wernicke and Wm B. Campbell. &amp;quot;3493.&amp;quot; &lt;em&gt;American Mathematical Monthly&lt;/em&gt; June 1932, pp. 366-367.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-02-19T23:27:14.507" LastActivityDate="2022-02-19T23:27:14.507" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14213" PostTypeId="2" ParentId="11082" CreationDate="2022-02-20T00:38:22.303" Score="1" Body="&lt;p&gt;Here are a few historical titles I ran into:&lt;br /&gt;&#xA;- Power in Numbers: The Rebel Women of Mathematics&lt;br /&gt;&#xA;- The Thrilling Adventures of Lovelace and Babbage (graphic novel)&lt;br /&gt;&#xA;- Proving it her way: Emmy Noether, a Life in mathematics.&lt;br /&gt;&#xA;- My Remarkable Journey: A memoir by Katherine Johnson&lt;/p&gt;&#xA;&lt;p&gt;They are all appropriate for public reading (i.e. they do not assume any special knowledge). I found some of these books on this site which includes several others, including options for children: &lt;a href=&quot;https://aperiodical.com/2020/03/iwd-2020-books-about-maths-by-women/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://aperiodical.com/2020/03/iwd-2020-books-about-maths-by-women/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16136" LastEditorUserId="16136" LastEditDate="2022-02-20T00:57:03.440" LastActivityDate="2022-02-20T00:57:03.440" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14215" PostTypeId="1" AcceptedAnswerId="14216" CreationDate="2022-02-20T20:49:43.383" Score="19" ViewCount="3953" Body="&lt;p&gt;In many books one finds different explanations. Specifically popular seems to be that he &amp;quot;argued against the Copenhagen interpretation&amp;quot;. &lt;strong&gt;But what did he really intend to communicate?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I for myself have tried to come up with an answer to this question. &lt;em&gt;Please feel free to present the correct answer and / or critique my version&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;my point of view&lt;/em&gt;, Schroedinger tried to discuss the current situation in quantum physics (at that time) which seems to be somewhat equivalent to what we now call the Copenhagen interpretation (?).&lt;/p&gt;&#xA;&lt;p&gt;He discussed how one could understand the &amp;quot;Verwaschenheit&amp;quot; or &amp;quot;Unschärfe&amp;quot; (the fuzziness/blurriness) of quantum physics, which today might be called the probabilistic character of quantum mechanics (?).&lt;/p&gt;&#xA;&lt;p&gt;He although seemed not to be as worried about the probabilistic features itself, but more how one could use a realistic interpretation on this (since he and Einstein seemed to have a realistic point of view when it comes to the interpretation). In his famous paper from 1935 (where he introduced the cat Gedankenexperiment) he claimed that it was impossible to interpret this blurriness as just &amp;quot;having not enough information&amp;quot; (e.g. a Gibbs Ensemble) and also argued that viewing this blurriness as a objective property of quantum systems is inadequate with a realistic point of view. This (latter specific point) was made clear by the cat Gedankenexperiment as the blurriness as an objective property would lead to something that doesn't match out intuition in the macroscopic world.&lt;/p&gt;&#xA;&lt;p&gt;So in the end, he tried to make clear, that a realistic interpretation didn't work for any of the 2 possible ways (Gibbs Ensemble vs. objective property) the current situation of quantum mechanics (in other words the Copenhagen interpretation) was able to provide. The thought experiment was used to &amp;quot;cross one of the 2 possible ways out of the equation&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="16285" LastEditorUserId="2277" LastEditDate="2022-02-21T19:39:00.500" LastActivityDate="2022-02-21T19:39:00.500" Title="What did Schroedinger try to say with the cat thought experiment?" Tags="|physics|quantum-mechanics|physicists|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14216" PostTypeId="2" ParentId="14215" CreationDate="2022-02-21T01:36:48.633" Score="17" Body="&lt;p&gt;One can simply read what Schroedinger said, English translation of his paper &lt;a href=&quot;http://www.psiquadrat.de/downloads/schroedinger35_katze1.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Present Situation in Quantum Mechanics (1935)&lt;/a&gt; by Trimmer is &lt;a href=&quot;https://www.jstor.org/stable/986572&quot; rel=&quot;noreferrer&quot;&gt;available on Jstor&lt;/a&gt;. The cat paradox is presented as part of the argument that granting reality to the wave function, &amp;quot;blurring&amp;quot; the real, as  Schroedinger puts it, is absurd when applied to macroscopic objects, a rhetorical &lt;em&gt;reductio&lt;/em&gt; of that idea. Since the Copenhagen interpretation did not treat the wave function this way (it was rather the most complete description an observer can have) this particular argument was not directed against it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The other alternative consisted of granting reality only to the momentarily sharp determining parts - or in more general terms to each variable a sort of realization just corresponding to the quantum mechanical statistics of this variable at the relevant moment. That it is in fact not impossible to express the degree and kind of blurring of &lt;strong&gt;all&lt;/strong&gt; variables in &lt;strong&gt;one&lt;/strong&gt; perfectly &lt;strong&gt;clear&lt;/strong&gt; concept follows at once from the fact that Q.M. as a matter of fact has and uses such an instrument, the so-called wave function or &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt;-function, also called system vector. Much more is to be said about it further on. That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;But serious misgivings arise if one notices that the uncertainty affects macroscopically tangible and visible things, for which the term &amp;quot;blurring&amp;quot; seems simply wrong... One can even set up quite ridiculous cases. A cat is penned up in a steel chamber...&lt;/em&gt; [description of the paradox follows]. &lt;em&gt;It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a &amp;quot;blurred model&amp;quot; for representing reality&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-02-21T01:45:01.373" LastActivityDate="2022-02-21T01:45:01.373" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14217" PostTypeId="2" ParentId="14194" CreationDate="2022-02-21T02:05:51.003" Score="1" Body="&lt;p&gt;As mentioned in the comments, Joseph Madachy published 14 issues of this title from 1961 to 1964.  In case it helps, I think it has ISSN 0486-1876 and OCLC 1774076.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.worldcat.org/title/recreational-mathematics-magazine/oclc/707441231/&quot; rel=&quot;nofollow noreferrer&quot;&gt;WorldCat&lt;/a&gt; listing shows some volumes in several North American university libraries (and a few other collections, including the New York Public Library and Kansas City's Linda Hall Library) but, elsewhere, only in Utrecht and Adelaide.&lt;/p&gt;&#xA;&lt;p&gt;It is in &lt;a href=&quot;https://books.google.com/books?id=ophVAAAAYAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt;, but only with &amp;quot;snippet&amp;quot; view available because of copyrights.  The &lt;a href=&quot;https://catalog.hathitrust.org/api/volumes/oclc/1774076.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hathi Trust&lt;/a&gt; also has it but just allows searching.&lt;/p&gt;&#xA;" OwnerUserId="3429" LastActivityDate="2022-02-21T02:05:51.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14218" PostTypeId="1" CreationDate="2022-02-21T07:27:19.377" Score="1" ViewCount="55" Body="&lt;p&gt;I wonder if there is any estimate about the percentage of the books that survived from Islamic ancient time listed in Ibn Nadim's Al Fihrist?&#xA;Is there any counterpart of this work belonging to ancient Greece? If yes, what percentage of Greek books has survived?&lt;/p&gt;&#xA;" OwnerUserId="2364" LastActivityDate="2022-02-21T07:27:19.377" Title="Al-Fihrist survivals" Tags="|ancient-greece|islamic-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14221" PostTypeId="2" ParentId="4902" CreationDate="2022-02-21T22:44:07.553" Score="2" Body="&lt;p&gt;I highly recommend &amp;quot;Calculus Gems&amp;quot; by George F. Simmons (1992, McGraw-Hill).&lt;/p&gt;&#xA;&lt;p&gt;This contains two parts: one about the memorable mathematicians, and one detailing (brilliantly and accessibly) the actual mathematics they produced.&lt;/p&gt;&#xA;&lt;p&gt;Requires more mathematical literacy than most collections of boigraphies, purely because of the actual mathematics in it, but it's accessible to a bright high-school student, in particular one likely to be going on to do mathematics at a higher level.&lt;/p&gt;&#xA;&lt;p&gt;A number of these biographical sketches have been taken and revised from his earlier &amp;quot;Differential Equations with Applications and Historical Notes&amp;quot; -- he was never a man to waste a good paragraph if it could be reused to good effect somewhere else.&lt;/p&gt;&#xA;&lt;p&gt;Be warned: he has opinions which he is not afraid of voicing. As an example, he is somewhat dismissive of Descartes, whom he does not rate highly at all&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, George Simmons is highly underrated.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2022-02-21T22:44:07.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14223" PostTypeId="5" CreationDate="2022-02-23T09:05:23.363" Score="0" Body="&lt;p&gt;This tag should be used along with &lt;a href=&quot;/questions/tagged/physics&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;physics&amp;#39;&quot; rel=&quot;tag&quot;&gt;physics&lt;/a&gt; tag.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-02-23T16:13:38.207" LastActivityDate="2022-02-23T16:13:38.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14224" PostTypeId="4" CreationDate="2022-02-23T09:05:23.363" Score="0" Body="For questions about Particle Physics - a branch of Physics that studies the nature of the particles that constitute matter and radiation" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-02-24T07:28:07.340" LastActivityDate="2022-02-24T07:28:07.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14225" PostTypeId="1" AcceptedAnswerId="14226" CreationDate="2022-02-23T10:27:45.930" Score="1" ViewCount="163" Body="&lt;p&gt;Nowadays, scientific progress is often based on very big collaborations, like the discovery of gravitational waves by the Ligo and Virgo collaborations. But also in many other branches of science, like mathematics, there has always been a big exchange of scientists from all parts of the world working in the same field. I am just curious, how was the situation during the world wars, especially World War II?&lt;/p&gt;&#xA;&lt;p&gt;Hence, I would like to start a &amp;quot;big-list&amp;quot;, where everyone is welcome to answer with an example of some big or small collaboration involving scientists from different countries during this time and some progresses achieved therein.&lt;/p&gt;&#xA;" OwnerUserId="16300" LastActivityDate="2022-02-27T11:57:41.580" Title="Scientific Collaborations during World War" Tags="|mathematics|physics|world-war-2|big-list|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14226" PostTypeId="2" ParentId="14225" CreationDate="2022-02-23T12:55:46.677" Score="2" Body="&lt;p&gt;The most famous example of a collaboration between scientists/mathematicians during World War 2 is the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/GCHQ#Government_Code_and_Cypher_School&quot; rel=&quot;nofollow noreferrer&quot;&gt;Government Code and Cypher School (GC&amp;amp;CS)&lt;/a&gt;, an intelligence organisation which had the task of decrypting secret communications of the Axis Powers. The intelligence obtained was termed as &amp;quot;Ultra&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the Second World War, GC&amp;amp;CS was based largely at &lt;strong&gt;Bletchley&#xA;Park&lt;/strong&gt;, in present-day Milton Keynes, working on understanding the&#xA;&lt;strong&gt;German Enigma machine&lt;/strong&gt; and &lt;strong&gt;Lorenz ciphers&lt;/strong&gt;. In 1940, GC&amp;amp;CS was&#xA;working on the diplomatic codes and ciphers of 26 countries, tackling&#xA;over 150 diplomatic cryptosystems. &lt;strong&gt;Senior staff included Alastair&#xA;Denniston, Oliver Strachey, Dilly Knox, John Tiltman, Edward Travis,&#xA;Ernst Fetterlein, Josh Cooper, Donald Michie, Alan Turing, Gordon&#xA;Welchman, Joan Clarke, Max Newman, William Tutte, I. J. (Jack) Good,&#xA;Peter Calvocoressi and Hugh Foss.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Apart from the famous British mathematicians like Alan Turing and Gordon Welchman mentioned above, mathematicians from different countries (mainly the Allies) were also directly or indirectly involved in the work at Bletchley Park. A few notable among them are &lt;a href=&quot;https://en.wikipedia.org/wiki/Marian_Rejewski&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marian Rejewski&lt;/a&gt; (Polish) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Solomon_Kullback&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solomon Kullback&lt;/a&gt; (American). In fact, it was Rejewski and his fellow Polish mathematicians who first cracked the Enigma cipher. Welchman in his book &lt;em&gt;&lt;a href=&quot;https://archive.org/details/hutsixstorybreak00welc&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Hut Six Story: Breaking the Enigma Codes&lt;/a&gt;&lt;/em&gt; has written, &amp;quot;Hut 6 Ultra would never have gotten off the ground if we had not learned from the Poles, in the nick of time, the details both of the German military version of the commercial Enigma machine, and of the operating procedures that were in use.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The collaboration was a success and it is even mentioned on &lt;a href=&quot;https://en.wikipedia.org/wiki/Enigma_machine&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Many commentators say the flow of Ultra communications intelligence&#xA;from the decryption of Enigma, Lorenz, and other ciphers, shortened&#xA;the war substantially, and might even have altered its outcome.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After the Second World War, in June 1946, GC&amp;amp;CS was renamed the Government Communications Headquarters. However, the work which was done at Bletchley Park remained a secret for a long time after the war!&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-02-23T15:39:57.193" LastActivityDate="2022-02-23T15:39:57.193" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14228" PostTypeId="1" CreationDate="2022-02-24T03:43:44.113" Score="1" ViewCount="180" Body="&lt;p&gt;I came across the term trace &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;-algebra in the context of conditional probability. Does anyone know where it comes from? i found a reference to trace of a matrix here: &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Miller/mathword/&lt;/a&gt; but that was it.&lt;/p&gt;&#xA;" OwnerUserId="15921" LastEditorUserId="15921" LastEditDate="2022-02-24T06:37:31.967" LastActivityDate="2022-02-24T06:37:31.967" Title="where does trace $\sigma$-algebra come from" Tags="|set-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14229" PostTypeId="1" AcceptedAnswerId="14230" CreationDate="2022-02-24T10:08:10.717" Score="3" ViewCount="130" Body="&lt;p&gt;The first pulsar was discovered on August 6, 1967 by Jocelyn Bell, but it took a while (and the discovery of a second pulsar) to figure out what was being observed. So I'm guessing this discovery wasn't announced right away. When was the discovery of a pulsar first made public?&lt;/p&gt;&#xA;" OwnerUserId="10806" LastActivityDate="2022-03-01T14:41:53.907" Title="When was the discovery of the first pulsar announced?" Tags="|astronomy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14230" PostTypeId="2" ParentId="14229" CreationDate="2022-02-24T14:53:58.183" Score="6" Body="&lt;p&gt;As noted by @BigBrother, the original article in &lt;a href=&quot;https://www.nature.com/articles/217709a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nature&lt;/a&gt; (Nature volume 217, pages 709–713 (1968)) has a publication date of 24 February 1968.&lt;/p&gt;&#xA;&lt;p&gt;Now, it turns out there is a nice oral history by Jocelyn Bell Burnell at the &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/31792&quot; rel=&quot;nofollow noreferrer&quot;&gt;American Institute of Physics&lt;/a&gt; web site, where she says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We wrote up the first pulsar for publication, sent it to Nature. I think those were the days when the editor refereed all the papers himself. I also remember Martin Ryle calling him up sort of days in advance saying, “We’ve got something very interesting we’re sending to you. Look out for it.” &lt;strong&gt;And when the paper had been accepted but a few days before Nature appeared&lt;/strong&gt;, Tony gave a colloquium in the Cavendish in Cambridge, and we gave it a rather titillating title, you know, “A New Kind of Radio Source,” or something like that. And word began to get around that something very interesting was happening, and people came in from everywhere. And I can remember that quite clearly. I can remember Fred Hoyle sitting in the front row. And at the end of colloquium Fred saying in his Yorkshire accent that, “This is the first I’ve heard about this things, but I don’t think they’ve white dwarves. I think they’re supernova remnants.” In other words, he had taken in the fact that we found these very compact objects. For reasons that I don’t recall why, he almost immediately could say he didn’t think white dwarves were good candidates, but we were dealing with something to do with supernova — which of course is what it turned out to be.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another &lt;a href=&quot;https://www.nature.com/articles/2171222a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nature paper&lt;/a&gt;, with a proposed candidate for the pulsar (gravitational lensing of a neutron star binary, interesting conjecture but not right) was published shortly afterwards on 30 March 1968 (5 weeks later) and has as a reference:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hewish, A., Bell, S. J., Pilkington, J. D. H., Scott, P. F., and Collins, R. A., Nature, 217, 709 (1968), and Hewish, A., Cavendish Colloquium, 20 February, 1968.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The authors of this paper are from Cambridge, and thus at least one of them was likely at the Cavendish Colloquium at Cambridge. So one can strongly infer that the Colloquium that Burnell mentions is the one referenced on the 20th of February, 4 days before the issue date of Nature, and that this was the first 'public' announcement of the results.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2022-03-01T14:41:53.907" LastActivityDate="2022-03-01T14:41:53.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14232" PostTypeId="2" ParentId="9698" CreationDate="2022-02-26T13:50:19.720" Score="1" Body="&lt;p&gt;It is often forgotten that Newton did not claim the laws of motion as his own. In the 'Principia' he gave these laws the alternative title 'axioms' -- starting-points that were not considered to need demonstration. Newton acknowledged and credited a number of predecessors for individual aspects of them.&lt;/p&gt;&#xA;&lt;p&gt;Thus, in the Scholium at the end of the initial 'axioms or laws of motion' Newton wrote: &amp;quot;The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds.&amp;quot; (from Cohen translation, 1999: the same passage in the 1729 (public-domain) translation is &lt;a href=&quot;https://archive.org/details/bub_gb_Tm0FAAAAQAAJ/page/n74&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). He went on to acknowledge by name Galileo, Wren, Wallis, Huygens, and Mariotte. He also  described some experiments or perhaps illustrations relating to the laws, and at least one of them he said he had carried out himself.&lt;/p&gt;&#xA;&lt;p&gt;It was mainly after Newton's lifetime that the laws began to be called &lt;em&gt;Newton's&lt;/em&gt; laws as it began to be appreciated that there was much originality in the set of laws as a whole, as a compilation, and in the selection and adjustments that provided a nature-consistent whole from the variety (and sometimes mutual inconsistency) of prior enunciations.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-02-26T13:50:19.720" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14233" PostTypeId="1" CreationDate="2022-02-27T09:49:58.123" Score="0" ViewCount="70" Body="&lt;p&gt;How exactly did Schrodingers Cat lead to development regarding macroscopic quantum states and decoherence? One often hears that the thought experiment was the initiator to the question, whether quantum mechanics also applies to the macroscopic world - but so far I couldn't find any reliable sources that state this.&lt;/p&gt;&#xA;" OwnerUserId="16285" LastActivityDate="2022-02-28T16:37:59.593" Title="On implications of Schrodingers Cat regarding macroscopic quantum states and decoherence" Tags="|physics|quantum-mechanics|theoretical-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14238" PostTypeId="1" AcceptedAnswerId="14239" CreationDate="2022-03-02T21:20:25.477" Score="8" ViewCount="1223" Body="&lt;p&gt;I think in undergrad functional analysis we were told an anecdote that at Stefan Banach's university in Warsaw they maintained a list of open problems which had material prizes attached to them including chickens. But I can't find anything on that with google.&lt;/p&gt;&#xA;&lt;p&gt;Does somebody have a reference?&lt;/p&gt;&#xA;" OwnerUserId="4498" LastActivityDate="2022-03-04T17:09:23.327" Title="Banach chicken story" Tags="|functional-analysis|banach|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14239" PostTypeId="2" ParentId="14238" CreationDate="2022-03-02T22:39:53.007" Score="9" Body="&lt;p&gt;You seem to be referencing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Scottish_Book&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scottish Book&lt;/a&gt;, and specifically problem 153 (whether all Banach spaces have the &lt;a href=&quot;https://en.wikipedia.org/wiki/Approximation_property&quot; rel=&quot;nofollow noreferrer&quot;&gt;approximation property&lt;/a&gt;), for the solution of which Mazur promised a live goose as prize. Said prize was later awarded to Per Enflo in 1972 for proving the claim false.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; A complete pdf version of the Scottish Book, translated in English and with some commentary by Ulam is available at&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://web.archive.org/web/20180428090844/http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.archive.org/web/20180428090844/http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It also contains the original formulation of problem 153, which is not exactly the same as the one I had previously mentioned (though with a little effort I reckon that it's possible to reconstruct why they are considered equivalent).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added (2):&lt;/strong&gt; Following the comments by Gerald Edgar and Margaret Friedland, a piece of information that's worth mentioning here is that the Scottish Book was written while Banach was working in Lwów (today's L'viv, in Ukrainian). &lt;a href=&quot;https://en.wikipedia.org/wiki/Stefan_Banach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; doesn't mention Banach ever being associated with Warsaw-based cultural instituitons: he was working in Lwów essentially all the time that international contingencies allowed it, and sometimes he was in Kraków.&lt;/p&gt;&#xA;" OwnerUserId="13914" LastEditorUserId="13914" LastEditDate="2022-03-04T17:09:23.327" LastActivityDate="2022-03-04T17:09:23.327" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14240" PostTypeId="1" CreationDate="2022-03-03T14:43:33.740" Score="4" ViewCount="116" Body="&lt;p&gt;In 1923 &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol14-trans/193&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;, Einstein proposed an idea for a classical theory that would explain some features of quantum mechanics, via the overdetermination of the EoM, so that only certain configurations would be available as initial values. He did not expand much on the idea or offered much in the way of examples.&lt;/p&gt;&#xA;&lt;p&gt;As far as I can tell &lt;a href=&quot;https://scholar.google.com/scholar?cites=13498575195552129215&amp;amp;as_sdt=2005&amp;amp;sciodt=0,5&amp;amp;hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;[2]&lt;/a&gt;, this idea seemed to have no follow up, and this paper only seems to be cited in papers on the history of Einstein , no responses, rebuttal, or any other comments.&lt;/p&gt;&#xA;&lt;p&gt;Was there ever any development on this idea, or did it die right there in this first paper?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2022-03-03T14:43:33.740" Title="Einstein's overdetermination theory" Tags="|physics|quantum-mechanics|einstein|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14241" PostTypeId="1" AcceptedAnswerId="14242" CreationDate="2022-03-04T17:42:50.140" Score="1" ViewCount="148" Body="&lt;p&gt;I study pure mathematics. In pure mathematics, we begin from some axioms and obtain theorems. I am also interested in studying physics. I have some questions about the relationship between physical theory and mathematics. I apologize before asking if the questions are not appropriate. And I will be grateful to anyone who responds.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Special relativity (SR) was originally proposed by Albert Einstein in a paper published in 1905. But Minkowski spacetime as a Mathematical structure for formulating SR introduced in 1908 (three years later)! I am confused about this. Did not Einstein himself state the mathematical framework of his theory? What is the difference between SR as Einstein himself formulated and SR in Minkowski's point of view? In fact, I always thought that every physical theory is formed at the same time as its mathematical formulation. Can we have a physical theory without any mathematical formulation?&lt;/li&gt;&#xA;&lt;li&gt;I read in some textbooks or internet pages (I do not remember which book or page) that in Quantum theories in physics there are some calculations that led to correct answers but we have no mathematical explanation for why. How is such a thing possible? Aren't our calculations based on mathematics?&lt;/li&gt;&#xA;&lt;li&gt;Is there any book to explain the difference between SR in Einstein's mind and Minkowski's? I heard that  Einstein was a terrific intuitionist. Ok, but is intuition enough to form a physical theory? Thanks a lot.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16339" LastEditorUserId="16339" LastEditDate="2022-03-04T19:23:57.563" LastActivityDate="2022-03-04T21:02:28.057" Title="Physical theories and Mathematics" Tags="|mathematics|physics|theoretical-physics|relativity-theory|mathematical-physics|" AnswerCount="1" CommentCount="3" ClosedDate="2022-03-10T18:55:40.643" ContentLicense="CC BY-SA 4.0" />
  <row Id="14242" PostTypeId="2" ParentId="14241" CreationDate="2022-03-04T21:02:28.057" Score="4" Body="&lt;p&gt;I think it is not quite accurate to say that &amp;quot;mathematics is done from axioms&amp;quot;, nor to say that &amp;quot;[physics] calculations are based on mathematics&amp;quot;. Let me explain.&lt;/p&gt;&#xA;&lt;p&gt;First, as @DavidLRenfro's comment suggests, the idea that mathematics &amp;quot;has a foundation&amp;quot; (at all!) is fairly new.&lt;/p&gt;&#xA;&lt;p&gt;A keyword that, for me, clarifies many aspects of such situations, is &amp;quot;narrative&amp;quot;: written mathematics is often a &lt;em&gt;narrative&lt;/em&gt; of thinking about physics, or about mathematics, and figuring out how to write it down to communicate to other people.&lt;/p&gt;&#xA;&lt;p&gt;That is, if you wanted to tell about an event, you'd probably not worry first about axioms or defining terminology.&lt;/p&gt;&#xA;&lt;p&gt;I more-and-more have the impression that, especially historically, but also nowadays, physicists write in mathematical notation as an account of an event. Much of mathematics is the same: not deducing from axioms, but telling a story, about tangible events [sic!].&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-03-04T21:02:28.057" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14243" PostTypeId="2" ParentId="2087" CreationDate="2022-03-06T21:15:24.933" Score="0" Body="&lt;p&gt;Heaviside and Gibbs independently pretty much invented vector analysis to clearly state the main results of Maxwell's theory. So your friend is correct. However, there were a number of precursors which complicate this simple story - so not completely.&lt;/p&gt;&#xA;&lt;p&gt;When Maxwell first published his equations there were twenty equations with almost as many unknowns. Recognising that without a more transparent framework his theory would have few takers (as famously noted by Freeman Dyson - to his puzzlement) and he rewrote the theory using quaternions but this did not stimulate interest.&lt;/p&gt;&#xA;&lt;p&gt;It's only after Gibbs &amp;amp; Heaviside reformulated the theory that interest began to pick up rapidly. The traditional statement of the equations in textbooks is Heavisides which is why they're sometimes called Heaviside-Maxwell's equations.&lt;/p&gt;&#xA;&lt;p&gt;However, preceding them is an Irish mathematician called Matthew O'Brien who between 1847-1852 published seven papers on vector formulations of mechanics and anticipated many of Gibbs later results. He in turn was inspired by Hamilton's discovery of the quaternions whilst he was looking for a 3d version of the complex numbers. Also credit is due to Hermann Grassmann who put forward a very general form of linear algebra, more or less inventing the subject, in his book &lt;em&gt;A theody of extension&lt;/em&gt;. He develops the notions of linear independence and dimension amongst other ideas.&lt;/p&gt;&#xA;&lt;p&gt;Liebniz deserves credit for recognising the value of a geometry that requires no coordinates and in a way returning us to the synthetic geometry of Euclid. But was himself unable to come up with anything of solid substance. A competition was inaugurated to stimulate attempts to come up with a suitable framework which Grassmann duly won.&lt;/p&gt;&#xA;&lt;p&gt;Liebniz's insight was remarkable as it underlay the notion of both invariance and covariance in geometry later picked up by physicists. And we have many geometries take typically take about position directly without mentioning coordinates. For example, manifolds, topological spaces, metric spaces, affine spaces and of course vector spaces.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-03-06T21:15:24.933" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14244" PostTypeId="1" CreationDate="2022-03-07T04:21:35.100" Score="1" ViewCount="92" Body="&lt;p&gt;I've done some searching on the internet, and I found that an electronic analog computer was built in 1942 in Germany, during World War II. But, before the first electronic computer, weren't there electronic calculators, which are simpler machines? Was that German electronic computer a revolutionary creation?&#xA;That's all. Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="16214" LastActivityDate="2022-03-07T04:21:35.100" Title="Before the Colossus and the ENIAC, did ever an electronic calculator exist?" Tags="|computers|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14245" PostTypeId="1" CreationDate="2022-03-07T11:18:40.617" Score="2" ViewCount="109" Body="&lt;p&gt;I have recently seen &lt;a href=&quot;https://math.stackexchange.com/questions/659153/theorem-that-von-neumann-proved-in-five-minutes/924782&quot;&gt;this post&lt;/a&gt; about the theorem von Neumann proved in five minutes.&lt;/p&gt;&#xA;&lt;p&gt;Searching for more information about that, it is certain that I can't find any references for what the theorem was.&lt;/p&gt;&#xA;&lt;p&gt;But as I looked through the comments, I found it very convincing that it was an open problem which Pólya frequently asked in classes.&lt;/p&gt;&#xA;&lt;p&gt;Since Pólya was a famous problem solver, he could access many open problems in that era.&lt;/p&gt;&#xA;&lt;p&gt;What I'm curious about is what kind of open problems were in the 1920s?&lt;/p&gt;&#xA;&lt;p&gt;Were there math magazines or problem books or math competitions?&lt;/p&gt;&#xA;" OwnerUserId="16347" LastEditorUserId="10106" LastEditDate="2022-03-07T16:52:23.553" LastActivityDate="2022-03-07T16:52:23.553" Title="Questions related to the theorem von Neumann proved in five minutes" Tags="|mathematics|reference-request|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14247" PostTypeId="1" AcceptedAnswerId="14248" CreationDate="2022-03-08T01:10:30.650" Score="1" ViewCount="246" Body="&lt;p&gt;I have been learning Newton's laws, and I am confused about the different formulations of the laws, the definitions of the quantities in the modern version, and the definitions of the quantities Newton used.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, mass is defined to be an object's resistance to change in acceleration. But that presupposes Newton's second law.&lt;/p&gt;&#xA;&lt;p&gt;I previously assumed that forces are and were a push or pull, and at Newton's times could be measured and represented as vectors; accelerations are and were second derivatives of displacement vectors. And Newton, with the exact previous definitions, found a relationship between the vector sum of the force vectors on a body and the acceleration vector of the body, and the constant for scalar multiplication is given the name mass, a measure of a body's resistance to change in acceleration.&lt;/p&gt;&#xA;&lt;p&gt;However, it seems that was not the case, and mass was a pre-existing concept. But how could mass have existed when there was no Newton's second law?&lt;/p&gt;&#xA;&lt;p&gt;Can someone give a brief explanation of the definitions of the quantities in Newton's laws?  And, if the definitions changed overtime, can someone explain that change?&lt;/p&gt;&#xA;" OwnerUserId="16353" LastEditorUserId="12330" LastEditDate="2022-03-08T07:24:13.830" LastActivityDate="2022-03-08T07:24:13.830" Title="Did the concept of mass exist before Newton?" Tags="|classical-mechanics|newton|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14248" PostTypeId="2" ParentId="14247" CreationDate="2022-03-08T05:39:14.847" Score="2" Body="&lt;p&gt;The American Journal of Physics article &lt;a href=&quot;https://aapt.scitation.org/doi/10.1119/1.5027490&quot; rel=&quot;nofollow noreferrer&quot;&gt;The pre-Newtonian Meaning of the Word Weight&lt;/a&gt; notes that Newton used the Latin term &lt;em&gt;pondus&lt;/em&gt;, which translates as &lt;em&gt;weight&lt;/em&gt;, though he clearly meant mass, writing in an unfinished manuscript of the Principia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By weight I mean the quantity or amount of matter (quantitas seu copia materiae) being moved, apart from considerations of gravity, so long as there is no question of gravitating bodies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The article notes that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nor, until the time of Hooke and Newton, was there any known method of measuring what we now call weight or force of gravity on the mass.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Prior to Newton, Kepler, for example, also used the term &lt;em&gt;weight&lt;/em&gt; (&lt;em&gt;pondus&lt;/em&gt;) for what we now call mass.  Quoting the article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kepler, [...] when discussing the causes of periods of planetary motion in his Epitome where he states: “secunda pondus seu copia materiae transportandae” (“the second [cause] is the weight or the amount of matter (copia materiae) to be transported”) and “pondera vero, seu copia materiae in diversis Planetis” (“weights in truth, or the amount of matter in the different planets”) which means that Kepler defined the amount of matter in the planet as its “weight”—which we now call mass.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-03-08T05:39:14.847" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14249" PostTypeId="2" ParentId="13912" CreationDate="2022-03-08T13:37:40.157" Score="2" Body="&lt;p&gt;I am not knowledgeable in the history of the subject either, so the following may be wrong, but I would suspect that what de Broglie had in mind was that electron's Coulomb field (and, therefore, the energy of this field) &amp;quot;is spread over all space&amp;quot;. This was common knowledge then, I believe.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2022-03-08T13:37:40.157" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14250" PostTypeId="1" CreationDate="2022-03-08T19:49:46.737" Score="3" ViewCount="497" Body="&lt;p&gt;We can read as a &lt;a href=&quot;https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1966/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathunion&lt;/a&gt; excerpt that Grothendieck won the Fields medal in 1966&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Built on work of Weil and Zariski and effected fundamental advances in algebraic geometry. He introduced the idea of K-theory (the Grothendieck groups and rings). Revolutionized homological algebra in his celebrated &amp;quot;Tohoku paper&amp;quot;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But all of this was already in the &lt;a href=&quot;https://agrothendieck.github.io/divers/AAVscan.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;proceedings&lt;/a&gt; of the congress of 1958.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I would like to know if there are resources to learn why take so much time to give the Fields medal to Grothendieck.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I mean, why he was not awarded the medal in 1962? At that time it was already very clear,  that Grothendieck was an outstanding mathematician. He was also not invited to give a talk. Serre would give a talk of his &lt;a href=&quot;https://agrothendieck.github.io/divers/ICMserre62scan.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;work&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I can imagine the history around, for example, why take some time to give the Nobel prize to Einstein (and why not on relativity!), has a good bibliography or resources here and there... Maybe someone knows something about the case Grothendieck.&lt;/p&gt;&#xA;" OwnerUserId="15350" LastActivityDate="2022-03-18T00:03:26.227" Title="Grothendieck and Fields medal 1962" Tags="|mathematics|mathematicians|biographical-details|mathematics-social-history|algebraic-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14251" PostTypeId="1" CreationDate="2022-03-09T09:31:25.217" Score="11" ViewCount="274" Body="&lt;p&gt;Apologies for a question that is specific to one country (but perhaps others find it a curious example of how mathematical notation can vary between countries).&lt;/p&gt;&#xA;&lt;p&gt;In Finnish calculus texts, if &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is an antiderivative of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, it is customary to write a definite integral as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_a^b f(x) \; dx = \bigg/_{\!\!\!\!a}^{\,b} F(x)&#xA;$$&lt;/span&gt;&#xA;where the right hand side means &lt;span class=&quot;math-container&quot;&gt;$F(b)-F(a)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q: When and where did this Finnish notation originate, and how did it spread?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As far as I know, this notation is not used anywhere outside Finland, and indeed Latex does not support it very well. (Finnish mathematicians simulate it with a big &lt;span class=&quot;math-container&quot;&gt;$/$&lt;/span&gt; operator and tweak the spacing of the limits.) Elsewhere the typical notations are&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;F(x) \Big|_{x=a}^b \qquad\text{and}\qquad \Big[ F(x) \Big]_{x=a}^b&#xA;$$&lt;/span&gt;&#xA;possibly with the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x=$&lt;/span&gt;&amp;quot; omitted.&lt;/p&gt;&#xA;&lt;p&gt;The earliest occurrence I know is in &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Leonard_Lindel%C3%B6f&quot; rel=&quot;noreferrer&quot;&gt;Ernst Lindelöf&lt;/a&gt;'s ''Johdatus korkeampaan analyysiin'' (3rd printing, 1942, page 363).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VqYRd.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VqYRd.jpg&quot; alt=&quot;Lindelöf page 363&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Lindelöf simply introduces the notation here, but offers no explanation on whether it is already common. I wonder if this is the first use or if it is in fact older.&lt;/p&gt;&#xA;" OwnerUserId="16297" LastEditorUserId="10106" LastEditDate="2022-03-09T10:44:21.830" LastActivityDate="2022-03-09T10:44:21.830" Title="Origin of the special Finnish notation for difference of antiderivative" Tags="|mathematics|notation|calculus|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14252" PostTypeId="1" CreationDate="2022-03-09T11:27:52.280" Score="0" ViewCount="78" Body="&lt;p&gt;I've been reading &lt;em&gt;Infinite Powers&lt;/em&gt; by &lt;em&gt;Steven Strogatz&lt;/em&gt; and in it, he writes about how the greeks differentiated between numbers as being discrete and magnitudes as being continuous. However, all of these terms go over my head. What does he mean by discrete? What were magnitudes to the greeks? And can someone elaborate more on the difference between number and magnitude from the lens of the ancient greeks.&lt;/p&gt;&#xA;&lt;p&gt;Thank you!&lt;/p&gt;&#xA;" OwnerDisplayName="user14943" LastActivityDate="2022-03-09T11:27:52.280" Title="What was the difference between Number and Magnitude in Ancient Greece" Tags="|calculus|ancient-greece|numbers|" AnswerCount="0" CommentCount="5" ClosedDate="2022-03-15T02:35:41.083" ContentLicense="CC BY-SA 4.0" />
  <row Id="14253" PostTypeId="2" ParentId="14250" CreationDate="2022-03-10T02:55:37.980" Score="4" Body="&lt;p&gt;There is a very interesting &lt;a href=&quot;https://media.nature.com/original/magazine-assets/d41586-018-00513-8/d41586-018-00513-8.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[resource]&lt;/a&gt;, let me sum up my conclusions.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The beginnings of the Fields medal were not about honoring the best mathematicians, but &amp;quot;The committees used the medal as a redistributive tool, to give a boost to those who they felt did not already have every advantage but were doing important work nonetheless.&amp;quot;&#xA;This would explain in some aspects, why people like Weil or Hirzebruch did not get the medal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to Zariski's letters of the committee of 1958, Grothendieck was already on the list to be awarded the medal, but Schwartz (at the time in the committee) &amp;quot;thought it was better to wait until more of Grothendieck's work had come out (among a few reasons)&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1962, Grothendieck was already a high-profile in math, with his institution (founded in 1958) one of the best in the world. And I think that was the reason (as in the case of Weil and Hirzebruch) for not giving him the medal, he would not need more prestige.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1966, the medal had acquired the &amp;quot;prestige&amp;quot; of a &amp;quot;Nobel of math&amp;quot; (now the committee would give the prize to the best of mathematicians) and also the limit of age, it was the last chance for Grothendieck to receive the medal. A rather interesting point here, Grothendieck would not go to Moscow for the medal due to political reasons, Smale was also interested in politics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15350" LastEditorUserId="15350" LastEditDate="2022-03-18T00:03:26.227" LastActivityDate="2022-03-18T00:03:26.227" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14254" PostTypeId="1" CreationDate="2022-03-10T17:26:53.570" Score="0" ViewCount="63" Body="&lt;p&gt;By this I mean, are there systems of numbers, counting, sorting.. or even higher level mathematical concepts that reflect a culture's worldview? Did some cultures not develop certain math concepts because of their worldview?&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance&lt;/p&gt;&#xA;" OwnerUserId="16369" LastActivityDate="2022-03-10T17:26:53.570" Title="Are there examples of mathematical systems that reflect the worldview of the culture they are from?" Tags="|mathematics|mathematics-social-history|numbers|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14255" PostTypeId="1" CreationDate="2022-03-10T19:12:58.320" Score="1" ViewCount="169" Body="&lt;p&gt;Could you give me some references about the history of the Green's function?&#xA;Thank you in advance for your reply&#xA;In the study of boundary value problems for differential and difference equations we transform our proplem to an equivalent integral equation and there where Green's function appears, how they get this idea of Green's functions? Where this function apears? In which cases,?&lt;/p&gt;&#xA;" OwnerUserId="16178" LastEditorUserId="16178" LastEditDate="2022-03-10T21:29:39.830" LastActivityDate="2022-03-10T21:29:39.830" Title="History of Green's function" Tags="|reference-request|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14256" PostTypeId="1" AcceptedAnswerId="14258" CreationDate="2022-03-11T22:18:23.143" Score="2" ViewCount="314" Body="&lt;p&gt;When and under which circumstances did Physics and Mathematics take separate routes?&lt;/p&gt;&#xA;&lt;p&gt;Even though connections between Mathematics and Physics have been strong and prosperous at all times the methods, goals and education of practitioners of both disciplines have diverged. Is it possible to trace back points in history where mathematicians lost interest in problems and techniques from Physics, and identified different ones as more relevant to Mathematics disregarding considerations about their usefulness for understanding nature, or the other way around, where influential Physicists made conscious decisions not to pursue objectives that they considered unrelated to the physical world?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="12271" LastEditDate="2022-03-12T07:22:07.403" LastActivityDate="2022-03-12T13:13:37.990" Title="Separation between Physics and Mathematics" Tags="|mathematics|physics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14257" PostTypeId="2" ParentId="2663" CreationDate="2022-03-12T06:23:13.757" Score="0" Body="&lt;p&gt;Archimedes first used infinite series in his method of exhaustion to obtain the area of a parabola but he used geometric rather than algebraic methods to solve it. The first ever use of it algebraically comes in yuktibhasha where&#xA;1/1+x=1-x+x^2-x^3... is derived by 1-x(1/1+x)=1-x(1-x(1/1+x)).... So on&lt;/p&gt;&#xA;" OwnerUserId="16374" LastActivityDate="2022-03-12T06:23:13.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14258" PostTypeId="2" ParentId="14256" CreationDate="2022-03-12T13:02:41.583" Score="6" Body="&lt;p&gt;I do not think that there was a moment when they separated: that is a time T&#xA;such that they were one before T and separate after. Mathematics apparently started as an independent discipline from physics. Look at the work of Aristotle titled &amp;quot;Physics&amp;quot; and the work of Lucretius (exposition of Leucippus/Democritus system). They belong to physics, and they are certainly very different from mathematics of that time.&lt;/p&gt;&#xA;&lt;p&gt;Legendary Pythagoras was credited with discoveries in both physics (relation between the length of a string and the sound it produces) and mathematics (theorems in geometry and number theory).&lt;/p&gt;&#xA;&lt;p&gt;The close merge was achieved in the work of Archimedes; he&#xA;practiced both mathematics and physics, but still in his writings this&#xA;separation is evident. He never confuses physical reasoning with mathematical proof. In fact in a famous passage of his &amp;quot;Method&amp;quot; he emphasizes this difference.&lt;/p&gt;&#xA;&lt;p&gt;At the later times, the two things sometimes came closer together, sometimes separated. And this should not be obscured by the fact that sometimes&#xA;these two lines of inquiry were practiced by the same people.&lt;/p&gt;&#xA;&lt;p&gt;The closest merge probably happened in 18th and early 19th centuries,&#xA;but still at that time there was pure mathematics and mathematical physics and experimental physics. (See the famous exchange between&#xA;Fourier and Jacobi on the goals of science).&lt;/p&gt;&#xA;&lt;p&gt;This situation persists even now, at the time of high specialization: there are still many people on the boundary practicing both physics and mathematics. And there are other people who are either physicists or mathematicians but not both.&lt;/p&gt;&#xA;&lt;p&gt;So my conclusion is that there was no separation point: two sciences were separate from the beginning, but most of the time there was a close interaction between them.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-03-12T13:13:37.990" LastActivityDate="2022-03-12T13:13:37.990" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14259" PostTypeId="1" CreationDate="2022-03-12T18:10:28.733" Score="0" ViewCount="272" Body="&lt;p&gt;I heard someone say recently that Newton didn't know the chain rule. Is that true?&lt;/p&gt;&#xA;&lt;p&gt;I know Newton didn't share our current conception of functions, the real line, limits, etc., so if he did use something like the chain-rule it wouldn't have been in its modern form. So what's the most chain-rule-like idea he &lt;em&gt;did&lt;/em&gt; use (if he used anything close), and what did it look like in his notation?&lt;/p&gt;&#xA;" OwnerUserId="16379" LastActivityDate="2022-03-15T19:35:44.650" Title="Did Newton know the chain rule?" Tags="|calculus|newton|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14260" PostTypeId="1" CreationDate="2022-03-13T15:29:06.117" Score="3" ViewCount="62" Body="&lt;p&gt;I'm late to the Hamilton-Jacobi equation party and would like to know more about the historical context of its origin. What motivated its construction/discovery? What were its affordances over other formulations of classical mechanics at that time? Did its formulation involve making or discarding key assumptions to attain such affordances, or do those affordances stem from the equation's formalism more than a theoretic advance?&lt;/p&gt;&#xA;" OwnerUserId="9312" LastEditorUserId="10106" LastEditDate="2022-03-13T16:10:45.227" LastActivityDate="2022-03-13T16:10:45.227" Title="What motivated formulation of the Hamilton-Jacobi equation?" Tags="|physics|classical-mechanics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14261" PostTypeId="1" CreationDate="2022-03-14T01:45:49.920" Score="1" ViewCount="64" Body="&lt;p&gt;I understand that before Hippasus of Metapontum proved that the square root of 2 is an irrational number, it was commonly assumed that, given two line segments, it would be possible to find a third line segment whose length divides evenly into the two, i.e. commensurability.&lt;/p&gt;&#xA;&lt;p&gt;However why did the Greeks assume commensurability? What was the proof, reasonsing or evidence that made them think in this way? Or was it only an assumption without any proper justification?&lt;/p&gt;&#xA;&lt;p&gt;Also what made Hippasus doubt this assumption? What motivated him to prove this assumption was wrong?&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="16383" LastEditorUserId="10106" LastEditDate="2022-03-14T02:51:02.820" LastActivityDate="2022-03-14T02:51:02.820" Title="What is the grounding of commensurability?" Tags="|mathematics|real-analysis|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14262" PostTypeId="1" CreationDate="2022-03-14T10:25:51.723" Score="3" ViewCount="169" Body="&lt;p&gt;Condensed matter is probably as impactful as quantum mechanics, but has less coverage in popular media than other branches like particle physics. Condensed matter experiments have led to many Nobel prizes including the discovery of quantum mechanics by black body radiation. Is there are any good book on the history of solid state/condensed matter from a physics perspective? Most books out there that claim to be such end up as books about technology and engineering.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-04-28T13:00:19.247" LastActivityDate="2023-05-01T12:05:46.073" Title="Are there any good books on the history of condensed matter?" Tags="|physics|book-recommendation|solid-state-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14263" PostTypeId="1" AcceptedAnswerId="14265" CreationDate="2022-03-14T17:16:09.450" Score="5" ViewCount="961" Body="&lt;p&gt;France was home to some of the greatest mathematicians of the 20th century such as Weil, Serre, Grothendieck, and Deligne, just to name a &lt;em&gt;few&lt;/em&gt;. What algebra texts did they study from during their undergraduate and graduate years?&lt;/p&gt;&#xA;" OwnerUserId="15535" LastActivityDate="2022-03-14T19:00:43.313" Title="What were the standard algebra texts used in elite French universities during the 20th century?" Tags="|mathematics|20th-century|france|bourbaki|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14264" PostTypeId="1" CreationDate="2022-03-14T17:35:19.537" Score="1" ViewCount="153" Body="&lt;p&gt;Does anybody here own a copy of the poster &amp;quot;Mathematische Gesellschaft: Group Portrait - University of Göttingen (1899)&amp;quot; that Springer Verlag published in 1985? If so, would you be so kind as to share a photograph of it with me?&lt;/p&gt;&#xA;&lt;p&gt;In case you don't own the poster but you know of a book in which I can find the group portrait in the poster, would you be so kind as to enter the relevant bibliographical information in an answer?&lt;/p&gt;&#xA;&lt;p&gt;Let me thank you in advance for your attention and support.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-03-21T06:05:35.467" LastActivityDate="2022-03-21T06:07:50.267" Title="Help in locating a portrait of Göttingen's faculty in 1899" Tags="|reference-request|gottingen|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14265" PostTypeId="2" ParentId="14263" CreationDate="2022-03-14T19:00:43.313" Score="7" Body="&lt;p&gt;The mathematicians you name all had important relations with Bourbaki, Deligne for sure studied the elements of Bourbaki at that level. While the others would create them. The algebra of Bourbaki has a foundational status in the elements. But this is not the answer to your question, before Bourbaki (and shaping the way he will present their elements) it is &lt;strong&gt;Moderne Algebra&lt;/strong&gt; of Van der Waerden.&lt;/p&gt;&#xA;&lt;p&gt;He &lt;em&gt;&amp;quot;presented ideas that had been developed earlier by Emmy Noether and Artin —whose courses van der Waerden had recently attended in Göttingen and Hamburg, respectively— and also by other algebraists, such as Ernst Steinitz, whose works van der Waerden had also studied under their guidance. Van der Waerden masterly incorporated a great deal of the important innovations accumulated over the last years at the level of the body of algebraic knowledge. But the originality and importance of his contribution is best recognized by focusing on the totally new way of conceiving the discipline it put forward. Van der Waerden presented systematically those mathematical branches then related with algebra, deriving all the relevant results from a single, unified perspective, and using similar concepts and methods for all those branches.&amp;quot;[¹]&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;It is true that there were already excellent monographs at the time and, in fact, Bourbaki treatise was modeled in the beginning on the excellent algebra treatise of Van der Waerden. [...] Since then, algebra has developed considerably, partly because of Van der Waerden's treatise, which is still an excellent introduction. I am often asked for advice on how to start out studying algebra, and to most people I say: First read Van der Waerden, in spite of what has been done since.&amp;quot; [²]&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;You should see:&lt;/p&gt;&#xA;&lt;p&gt;[¹] Mathematical Structures from Hilbert to Bourbaki by Leo Corry&lt;/p&gt;&#xA;&lt;p&gt;[²] The Work of Nicholas Bourbaki by Jean A. Dieudonné&lt;/p&gt;&#xA;" OwnerUserId="15350" LastActivityDate="2022-03-14T19:00:43.313" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14266" PostTypeId="2" ParentId="7031" CreationDate="2022-03-15T00:55:31.280" Score="0" Body="&lt;p&gt;To clarify some thing implicit in comments and answers, from a more contemporary viewpoint:&lt;/p&gt;&#xA;&lt;p&gt;Given a polynomial &lt;span class=&quot;math-container&quot;&gt;$P(x_1,\ldots,x_n)$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; variables, we can certainly do a linear change of variables by replacing &lt;span class=&quot;math-container&quot;&gt;$x=(x_1,\ldots,x_n)$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$y=(y_1,\ldots,y_n)=x\cdot A$&lt;/span&gt; for an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-by-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. (And there are potentially issues about transpose, column-versus-row matrices, etc., but these are secondary.)&lt;/p&gt;&#xA;&lt;p&gt;Reasonably-enough, in terms of colloquial English, a polynomial &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is &lt;em&gt;invariant&lt;/em&gt; (for some subgroup &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; of the group &lt;span class=&quot;math-container&quot;&gt;$GL_n(\mathbb C)$&lt;/span&gt; of invertible &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-by-&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; complex matrices) when &lt;span class=&quot;math-container&quot;&gt;$P\circ A=P$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The phenomenon whose 19-th and early-20-th century description is considerably at variance from our contemporary description is about &amp;quot;representations of the group (acting on the variables, hence, on the polynomials&amp;quot;. In our contemporary terms, there are further, but/and highly structured, collections of polynomials beyond the simplest case of &amp;quot;invariant&amp;quot;. Long ago, these had other names... I myself cannot reliably translate Sylvester's (quite creative! :) terminology into contemporary, for example.&lt;/p&gt;&#xA;&lt;p&gt;But, in any case, it's about the representation theory of general linear groups, and subgroups theoreof, on spaces of polynomials. This has the effect of multi-linear algebra operations on the dual spaces of the representations of those groups on the vector spaces of &amp;quot;variables&amp;quot; themselves.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-03-15T00:55:31.280" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14267" PostTypeId="1" AcceptedAnswerId="14272" CreationDate="2022-03-15T08:50:44.547" Score="23" ViewCount="5406" Body="&lt;p&gt;I'm wondering if there are any mathematical objects that were given a name when first discovered (and wildly used at their time), but then got renamed to match their characteristics later?&lt;/p&gt;&#xA;&lt;h3&gt;Counter examples:&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Imaginary number&lt;/strong&gt; was coined by Descartes in the 17th century describing roots of polynomial involving &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I've seen some rename suggestions: lateral, orthogonal, perpendicular, vertical number. None of them replaced the original word, yet.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Irrational number&lt;/strong&gt; was coined by the school of Euclid around two thousand years ago to demonstrate the number that cannot be written as a &lt;em&gt;ratio&lt;/em&gt;.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Luckily(?), the word &lt;em&gt;ratio&lt;/em&gt; (&lt;span class=&quot;math-container&quot;&gt;$a/b$&lt;/span&gt;) comes after &lt;em&gt;rational&lt;/em&gt; (make sense). So &lt;em&gt;irrational&lt;/em&gt; can have both meanings and, thus, does not need to be renamed(?).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h3&gt;Related examples:&lt;/h3&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In terms of notation change, originally the equals sign was two vertical bars, then became two horizontal bars.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16390" LastEditorUserId="10106" LastEditDate="2022-03-16T01:55:01.283" LastActivityDate="2022-03-18T03:38:02.430" Title="Are there any mathematical objects that got renamed over time?" Tags="|mathematics|naming-conventions|" AnswerCount="9" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14268" PostTypeId="1" CreationDate="2022-03-15T09:10:21.270" Score="3" ViewCount="103" Body="&lt;p&gt;Consider the quote at the bottom of this post.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt; I thought this was an interesting question and asked it honestly. An answer would need an understanding of the life and times of Helmholtz, which is why I asked on a mathematics history site. But, it seems that no one has that background and the question is attracting non mathematical guesses. I would vote to close my own question now, if I could as I regret asking it.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Helmholtz seems to be saying that one can  bend a hemisphere into a spindle without changing the intrinsic metric.&lt;/p&gt;&#xA;&lt;p&gt;As I understand it - for smooth embeddings of smooth metrics with non zero curvature - the metric determines the embedding into real n-space of an n-1 surface up to an isometry of the n-space. If we allow creases, and thus a non smooth embedding, there are other options. But, none of them feel like &amp;quot;rolling a hemisphere into a spindle shape&amp;quot; say, as I interpret it, something like a lemon juicer.&lt;/p&gt;&#xA;&lt;p&gt;What did Helmholtz have in mind here. He is writing to a lay audience and seems to think that his suggestion is plain and apparent. Is this a comment I am misunderstanding due to different times and different languages? Or is he referring to some specific technical method, known at that time, that does not occur to me?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Gauss showed at the same time that this measure of curvature is not changed if the surface is bent without distension or contraction of any part of it. Thus we can roll up a flat sheet of paper into the form of a cylinder, or of a cone, without any change in the dimensions of the figures taken along the surface of the sheet. Or the hemispherical fundus of a bladder may be rolled into a spindle-shape without altering the dimensions on the surface. Geometry on a plane will therefore be the same as on a cylindrical surface ; only in the latter case we must imagine that any number of layers of this surface, like the layers of a rolled sheet of paper, lie one upon another, and that after each entire revolution round the cylinder a new layer is reached different from the previous ones.&lt;/p&gt;&#xA;&lt;p&gt;Page 39 of Popular Lectures on scientific subjects, second series.&#xA;By H Helmholtz, translated by E Atkinson. Longmans, London, 1881.&#xA;Section II: On the origin and Significance of Geometrical Axioms.&lt;/p&gt;&#xA;" OwnerUserId="16264" LastEditorUserId="16264" LastEditDate="2022-03-20T10:21:48.030" LastActivityDate="2022-03-20T10:21:48.030" Title="What did Helmholtz mean about crumpling a hemisphere?" Tags="|riemannian-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14269" PostTypeId="2" ParentId="14267" CreationDate="2022-03-15T10:28:49.690" Score="7" Body="&lt;p&gt;It seems that the constant formerly called &lt;span class=&quot;math-container&quot;&gt;$\zeta(3)$&lt;/span&gt; is now called &amp;quot;Apéry's number&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-03-15T10:28:49.690" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14270" PostTypeId="1" CreationDate="2022-03-15T19:00:34.270" Score="1" ViewCount="79" Body="&lt;p&gt;I'm reading volume 1 of Whiteside's 'Mathematical Papers of Isaac Newton,' and on pp. 383-384, Newton reaches a conclusion on his &amp;quot;Example 1st&amp;quot; in the statement &amp;quot;55:-54 :: p:q...&amp;quot; - which seems to be backwards if we are meant to read &amp;quot;::&amp;quot; as &amp;quot;as&amp;quot;.  But it's also possible that Newton intends &amp;quot;::&amp;quot; to read &amp;quot;is reciprocal to&amp;quot; - which would make his conclusion correct.&lt;/p&gt;&#xA;&lt;p&gt;This is not a life-and-death matter, just putting this out there if someone has background or interest in Newton-notation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/M6vHA.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/M6vHA.png&quot; alt=&quot;Snip1 that contains the issue&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/VnSRB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VnSRB.png&quot; alt=&quot;Snip2 that contains the issue&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16392" LastActivityDate="2022-03-16T02:21:28.730" Title="Question from Whiteside V. 1 - what did Newton mean by a:b :: c:d notation?" Tags="|notation|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14271" PostTypeId="2" ParentId="14259" CreationDate="2022-03-15T19:35:44.650" Score="1" Body="&lt;p&gt;I think Newton certainly used an equivalent of the chain rule, in that his &amp;quot;method&amp;quot; was to rip through a polyomial, multiplying each term by np/x (n = degree, p = x' or q = y', etc, x = x or y or whatever variable was targeted), which inserted chain rule-type &amp;quot;placeholders&amp;quot; everywhere they were needed.  His interest was &amp;quot;the doctrine of curves&amp;quot;, so he jumped right into solving multi-degree polynomial equations for tangents and integrals and his techniques were to that end.&lt;/p&gt;&#xA;" OwnerUserId="16392" LastActivityDate="2022-03-15T19:35:44.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14272" PostTypeId="2" ParentId="14267" CreationDate="2022-03-15T19:45:46.360" Score="16" Body="&lt;p&gt;Another example is the notion of &lt;a href=&quot;https://en.wikipedia.org/wiki/Orbifold&quot; rel=&quot;noreferrer&quot;&gt;orbifolds&lt;/a&gt;. Originally, it was introduced by Satake in 1950s under the (nondescriptive) name &lt;strong&gt;V-manifolds&lt;/strong&gt;. Then, beginning in 1980s, under the influence of William Thurston, it was replaced by the name &lt;strong&gt;orbifolds&lt;/strong&gt; to indicate that they are related to manifolds via the (local) &amp;quot;orbit space&amp;quot; construction. Thurston in his Princeton Lecture Notes describes the process of introduction of the new terminology thusly:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word &amp;quot;manifold&amp;quot; already has a different definition. I tried &amp;quot;foldamani&amp;quot;, which was quickly displaced by the suggestion of &amp;quot;manifolded&amp;quot;. After two months of patiently saying &amp;quot;no, not a manifold, a manifoldead,&amp;quot; we held a vote, and &amp;quot;orbifold&amp;quot; won.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6107" LastActivityDate="2022-03-15T19:45:46.360" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14273" PostTypeId="1" CreationDate="2022-03-15T21:15:23.570" Score="-1" ViewCount="384" Body="&lt;p&gt;I've browsed many math history books, but I've never read too deep into any single one. I always find myself reading the about the same facts and same people over and over -- the set of topics doesn't seem to reflect the richness of mathematical history.&lt;/p&gt;&#xA;&lt;p&gt;I want to learn how the techniques and topics in math have changed over time, and what role and applications math played in various societies. I find knowing the historical development of more modern techniques really helpful for understanding math from school. As well, I like learning about the different ways different societies think of mathematical concepts.&lt;/p&gt;&#xA;&lt;p&gt;These are many broad ideas, so I guess I am interested in any historical material (books, websites, etc) that doesn't follow the pervasive pattern I see -- where the exposition is crowded with random biographical details and uncritical praise of mostly upper-class white men, with little attention to the broad trends, and women and non-european mathematics barely mentioned. I feel that many math history books implicitly push the idea that math is primarily the result of individual genius, but I don't think that is true at all -- certainly there are visionaries but I believe the bulk of mathematical knowledge is created and recreated by innumerable people, most of whom create very little new research.&lt;/p&gt;&#xA;&lt;p&gt;PS for context I am almost finished an undergraduate degree and have that level of math understanding. I especially like algebra and topology.&lt;/p&gt;&#xA;" OwnerUserId="16395" LastActivityDate="2022-04-13T09:26:05.907" Title="Looking for math history but keep finding the same old stuff" Tags="|mathematics|reference-request|mathematics-social-history|book-recommendation|" AnswerCount="5" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14274" PostTypeId="2" ParentId="14273" CreationDate="2022-03-16T00:01:58.240" Score="2" Body="&lt;p&gt;You can do a search on Arab or Chinese mathematicians and you'll find lots to read.  Gender, up until 150 years, was a strong barrier.  But certainly you should check out the Marquise de Chatelet.&lt;/p&gt;&#xA;&lt;p&gt;My advice is to ignore the mathematicians and focus on the math.  Get a copy of Wallis' or Euler's Algebra or Descartes Géométrie and try to make sense of them.  You'll learn a lot.&lt;/p&gt;&#xA;&lt;p&gt;Or, if you want something a bit curated, look at Whiteside's edition of Newton's mathematical papers or Stillman Drake's or Paulo Palmieri's recreation of Galileo's experiments and calculations.  Working through stuff like that will stand you in good stead when you discover the contributor none of the rest of us know about - but should.&lt;/p&gt;&#xA;&lt;p&gt;Good luck with it!&lt;/p&gt;&#xA;" OwnerUserId="16392" LastActivityDate="2022-03-16T00:01:58.240" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14275" PostTypeId="2" ParentId="14267" CreationDate="2022-03-16T00:15:12.797" Score="27" Body="&lt;p&gt;Newton referred to his concept of a derivative as a &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fluxion&quot; rel=&quot;nofollow noreferrer&quot;&gt;fluxion&lt;/a&gt;&amp;quot;. He called time-varying functions &amp;quot;fluents&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Generally speaking, it is common for important mathematical concepts and entities to be given one name by their discoverer, but later renamed after that person. I think that is true for virtually all entities that are known as &amp;quot;Somebody's number/theorem/group/etc&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Just off the top of my head: I don't know what term Diophantus used, but he likely didn't call them &amp;quot;Diophantine equations&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Or similarly, &lt;a href=&quot;https://en.wikipedia.org/wiki/Lie_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lie called Lie algebras &amp;quot;infinitesimal groups&amp;quot;.&lt;/a&gt;&lt;/p&gt;&#xA;&#xA;" OwnerUserId="16396" LastEditorUserId="15473" LastEditDate="2022-03-18T03:38:02.430" LastActivityDate="2022-03-18T03:38:02.430" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14276" PostTypeId="2" ParentId="14267" CreationDate="2022-03-16T02:15:21.940" Score="12" Body="&lt;p&gt;Gregoire de Saint-Vincent, along with his student Alphonse Antonio de Sarasa, developed &lt;strong&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Hyperbolic_sector#Hyperbolic_logarithm&quot; rel=&quot;noreferrer&quot;&gt;hyperbolic logarithm&lt;/a&gt;&lt;/strong&gt;, now known as &lt;strong&gt;natural logarithm&lt;/strong&gt;, by relating logarithms to the quadrature of the hyperbola &lt;span class=&quot;math-container&quot;&gt;$xy=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;His findings led to the natural logarithm function, once called the&#xA;hyperbolic logarithm since it is obtained by integrating, or finding&#xA;the area, under the hyperbola.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Later, when Euler treated a logarithm as an exponent of a certain number, called the base of the logarithm, he identified hyperbolic logarithm as the natural logarithm.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Leonhard Euler changed that when he introduced transcendental&#xA;functions such as &lt;span class=&quot;math-container&quot;&gt;$10^x$&lt;/span&gt;. Euler identified &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; as the value of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; producing&#xA;a unit of area (under the hyperbola or in a hyperbolic sector in&#xA;standard position). Then the natural logarithm could be recognized as&#xA;the inverse function to the transcendental function &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2022-03-16T02:15:21.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14277" PostTypeId="2" ParentId="14270" CreationDate="2022-03-16T02:21:28.730" Score="2" Body="&lt;p&gt;Often the uses of the double colon sign indicate an 'analogy', much like a predecessor of an equation. I haven't looked hard at your example, but unless there's something unusual about it, the use of two of them could be understood as indicating three equal ratios, thus&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;55 : -54  ::  p : q ::  vel_A : vel_B&#xA;&#xA;55 / -54   =  p / q  =  vel_A / vel_B .&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;One point of difference between the old and new usages is that traditionally proportions were only regarded as legitimately formed if their members were of like kind, we might say similarly dimensioned.&lt;/p&gt;&#xA;&lt;p&gt;The old tradition and theory of proportion dating back to Euclid is explained for example by N Guicciardini in &amp;quot;Mathematics and the New Sciences&amp;quot;, chapter 8, (esp. sec. 8.3.2, p.232 et seq.) in &lt;a href=&quot;https://books.google.com/books/about/The_Oxford_Handbook_of_the_History_of_Ph.html?id=JWYBAQAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Oxford Handbook of the History of Physics' ed J Buchwald et al., 2013&lt;/a&gt;, see also &lt;a href=&quot;https://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199696253.001.0001/oxfordhb-9780199696253&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-03-16T02:21:28.730" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14278" PostTypeId="2" ParentId="14267" CreationDate="2022-03-16T03:02:51.213" Score="12" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Monad_(category_theory)&quot; rel=&quot;noreferrer&quot;&gt;Monads&lt;/a&gt; were introduced as &amp;quot;standard constructions&amp;quot; (an awful, non-descriptive name), then were called &amp;quot;triples&amp;quot; (even worse), until the term &amp;quot;monad&amp;quot; was popularized by &lt;em&gt;Categories for the working mathematician&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The history of the term is explained in more detail &lt;a href=&quot;https://english.stackexchange.com/questions/30654/where-does-the-term-monad-come-from&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16398" LastEditorUserId="16398" LastEditDate="2022-03-16T13:43:58.607" LastActivityDate="2022-03-16T13:43:58.607" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14279" PostTypeId="2" ParentId="14267" CreationDate="2022-03-16T11:32:22.380" Score="3" Body="&lt;p&gt;Kohonen maps are now called &lt;a href=&quot;https://en.wikipedia.org/wiki/Self-organizing_map&quot; rel=&quot;nofollow noreferrer&quot;&gt;self-organizing maps&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16403" LastActivityDate="2022-03-16T11:32:22.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14280" PostTypeId="2" ParentId="14267" CreationDate="2022-03-16T19:35:23.973" Score="4" Body="&lt;p&gt;In 19th-century &lt;a href=&quot;https://en.wikipedia.org/wiki/Term_logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;(term) logic&lt;/a&gt;, it used to be common to talk about &amp;quot;classes&amp;quot;. With the recognition of the idempotence law &lt;span class=&quot;math-container&quot;&gt;$b\cup b=b$&lt;/span&gt; by &lt;a href=&quot;https://books.google.com/books?id=WVMOAAAAYAAJ&amp;amp;pg=PA26&amp;amp;dq=%22same+alternatives+are%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Stanley Jevons&lt;/a&gt;, &lt;a href=&quot;https://books.google.com/books?id=nG8UAAAAYAAJ&amp;amp;pg=PA250&amp;amp;dq=%22It+is+plain+that%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Charles Sanders Peirce&lt;/a&gt;, and &lt;a href=&quot;https://books.google.com/books?id=UyVLAAAAYAAJ&amp;amp;pg=RA1-PA8&amp;amp;dq=%22Satz+der+Gleichheit+oder+Identit%C3%A4t%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Grassmann&lt;/a&gt;, those classes correspond to what we nowadays call &amp;quot;sets&amp;quot; (thus making the algebra of sets an interpretation of &lt;a href=&quot;https://en.wikipedia.org/wiki/Boolean_algebra_%28structure%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boolean algebra&lt;/a&gt;, a branch of abstract algebra which in retrospect was founded by Jevons after reforming George Boole's algebra of logic to include that idempotence law, which &lt;a href=&quot;https://plato.stanford.edu/entries/boole/#:%7E:text=Boole/Jevons%20dispute&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boole never accepted&lt;/a&gt;). And now, in ZFC set theory, &amp;quot;classes&amp;quot; mean something different: they are more general collections than &amp;quot;sets&amp;quot; (e.g. the &lt;a href=&quot;https://en.wikipedia.org/wiki/Class_%28set_theory%29&quot; rel=&quot;nofollow noreferrer&quot;&gt;[proper] class&lt;/a&gt; of all sets).&lt;/p&gt;&#xA;&lt;p&gt;Moreover, in the 19th-century German mathematical literature (e.g. in &lt;a href=&quot;https://books.google.com/books?id=fTckDwAAQBAJ&amp;amp;pg=PA6&amp;amp;dq=System+Mannigfaltigkeit+Mengen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard Dedekind's writings&lt;/a&gt;), it was common to use the terms [de] &amp;quot;System&amp;quot; ([en] &amp;quot;system&amp;quot;) and [de] &amp;quot;Mannigfaltigkeit&amp;quot; ([en] &amp;quot;manifold&amp;quot;/&amp;quot;multiplicity&amp;quot;) to refer to what we know as [de] &amp;quot;Menge&amp;quot; ([en] &amp;quot;set&amp;quot;) nowadays. So the term wasn't still standardized at that time.&lt;/p&gt;&#xA;" OwnerUserId="16404" LastActivityDate="2022-03-16T19:35:23.973" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14281" PostTypeId="2" ParentId="14267" CreationDate="2022-03-16T22:21:27.840" Score="4" Body="&lt;p&gt;&amp;quot;Polynomial&amp;quot; is a relative late denotation, from the end of the 19th, start of the 20th century. The word itself was coined by Viete around 1600 as extension of &amp;quot;binomial&amp;quot; and &amp;quot;trinomial&amp;quot;, for instance to describe the expression under a root. At about the same time &amp;quot;multinomial&amp;quot; was introduced for the same role.&lt;/p&gt;&#xA;&lt;p&gt;Then there is some short use of &amp;quot;polynomial coefficients&amp;quot; for the combinatorial numbers better known as &amp;quot;multinomial coefficients&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The functions themselves become separate objects of interest around 1800 and are denoted as &amp;quot;numerical functions&amp;quot; by Galois, some variation of &amp;quot;entire rational algebraic functions&amp;quot; with Gauß. (Afaik the german school system is the only institution to keep this denotation alive, &amp;quot;ganzrationale Funktion&amp;quot;.)&lt;/p&gt;&#xA;&lt;p&gt;And as said, in the literature on extensions of function fields suddenly around 1910 the word &amp;quot;polynomial&amp;quot; is used. Around that time there was in general a shift in algebraic notation, for instance Kronecker around 1880 used &amp;quot;module&amp;quot; for ideals.&lt;/p&gt;&#xA;" OwnerUserId="6894" LastActivityDate="2022-03-16T22:21:27.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14282" PostTypeId="1" CreationDate="2022-03-16T22:48:58.767" Score="1" ViewCount="140" Body="&lt;p&gt;The Ptolemaic theory of the latitude of the planets is dealt on in Book 13 of the &lt;em&gt;Almagest&lt;/em&gt;. This theory (which I was struggling to find info about online) is considered quite complicated, but luckily this question is not about the exactness of this theory, but rather can be reduced to this: &lt;strong&gt;according to this theory, does the local maximum of the latitude fall exactly at the opposition?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems to be the case, since Tycho Brahe (who followed Ptolemy in this regard if I'm not mistaken) was clearly surprised (as Kepler quotes in his &lt;em&gt;Astronomia Nova&lt;/em&gt;) when his observations proved that the local maximum does not fall exactly on the opposition, but can be some days away.&lt;/p&gt;&#xA;&lt;p&gt;But if this is the case, how come Ptolemy allows this in his system? I think this is indeed within  observational error (probably less than 10m of arc), but on the other hand, it can be observed relatively each day: how come the ancients could not see that the maximum latitude does not fall at the opposition?&lt;/p&gt;&#xA;" OwnerUserId="14954" LastActivityDate="2022-03-25T22:39:52.767" Title="On the Ptolemaic theory of the planet's latitude" Tags="|astronomy|ptolemy|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14283" PostTypeId="2" ParentId="14267" CreationDate="2022-03-17T03:39:54.890" Score="5" Body="&lt;p&gt;See &lt;a href=&quot;https://mathoverflow.net/questions/175833/what-recent-programmes-to-alter-highly-entrenched-mathematical-terminology-have&quot;&gt;here&lt;/a&gt; for a few examples, such as recursion theory becoming computability theory, (linear) complex groups becoming symplectic groups (change due to Weyl), bicompact becoming compact, and pre-schemes becoming schemes.&lt;/p&gt;&#xA;&lt;p&gt;Other examples: Gaussian integers used to be called complex integers, the term &amp;quot;group&amp;quot; initially referred only to what we'd call finite groups (existence of an inverse was not initially part of the definition, since it followed from the definition using finiteness),  Lie algebras were previously called infinitesimal groups (the change to Lie algebras is due to Weyl), cyclic algebras were initially called &amp;quot;algebras of type D&amp;quot; by Dickson, tensors were called &amp;quot;systems&amp;quot; by Ricci and Levi-Civita and &amp;quot;direct products&amp;quot; by Murray and von Neumann, topology was called analysis situs (change due to Lefschetz), and commutative algebra was initially called ideal theory.&lt;/p&gt;&#xA;&lt;p&gt;The meaning of homomorphism and isomorphism did not settle down for a while: in Jordan's 1870 textbook on group theory (a subject that for a long time was called the &amp;quot;theory of substitutions&amp;quot; using the old-fashioned term substitution for what we'd call a permutation), he used isomorphism for what we'd call a group homomorphism and holoedric isomorphism for what we'd call a group isomorphism (the term simple isomorphism was also used for modern-day isomorphisms). A discussion of this is &lt;a href=&quot;https://mathoverflow.net/questions/280261/whence-homomorphism-and-homomorphic&quot;&gt;here&lt;/a&gt;. Normal subgroups used to be called invariant subgroups.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-03-17T03:55:31.680" LastActivityDate="2022-03-17T03:55:31.680" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14284" PostTypeId="1" AcceptedAnswerId="14292" CreationDate="2022-03-17T16:22:32.940" Score="0" ViewCount="299" Body="&lt;p&gt;Evidence-Based Medicine (EBM) is a proposition and an area of study of medicine for which I am very fond. However, a few days ago, talking to some friends, I was confronted with a very critical approach to MBE and to Biostatistics itself.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;In 1992, the journal of the US Academy of Medicine announced the&#xA;formulation of “a new approach to teaching the practice of medicine”.&#xA;Known as Evidence-Based Medicine (EBM), it “emphasizes the examination&#xA;of evidence provided by clinical research”, for which it “requires the&#xA;application of formal rules to assess the evidence” (Guyatt et al.,&#xA;1992).&lt;/p&gt;&#xA;&lt;p&gt;A few years later, the scope of EBM had expanded considerably: “it is&#xA;the meticulous, explicit, and sensible use of the best available&#xA;evidence in making decisions regarding the care of the individual&#xA;patient” (Sackett et al., 1996). For this purpose, the level of&#xA;evidence is classified into categories, according to multiple&#xA;criteria, among which, the statistical-mathematical measures to&#xA;establish the chance that a certain effect is due or not to chance&#xA;have a prominent place. Thus, concepts such as “frequency&#xA;distributions”, “standard deviation”, “statistical significance”&#xA;(usually measured as “p-value”), “correlation”, “regression” and&#xA;“randomization” and tests such as “chi square”, “analysis of variance”&#xA;and “Fisher's exact test” have become everyday tools, not only in&#xA;clinical research, but in all areas of the biological and biomedical&#xA;sciences.&lt;/p&gt;&#xA;&lt;p&gt;The examples cited above are just a few of the many statistical&#xA;concepts and methods used in EBM. However, they were not selected&#xA;arbitrarily, but all of them were developed, among many others,&#xA;between the end of the 19th century and the beginning of the 20th&#xA;century by Karl Pearson (1857-1936) and Ronald A. Fisher (1890-1962).&#xA;Both are considered the formulators of modern statistics, who&#xA;developed according to a well-defined purpose: to improve the&#xA;hereditary endowment of the Human race.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;All fragments were taken from this book.&lt;/p&gt;&#xA;&lt;p&gt;Available for free at &lt;a href=&quot;https://www.fm.usp.br/museu/portal/livros-para-download&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.fm.usp.br/museu/portal/livros-para-download&lt;/a&gt; (portuguese)&lt;/p&gt;&#xA;&lt;p&gt;The tools that we use on a day to day basis to interrogate data and understand the world, were developed by white supremacists for the express purpose of demonstrating that white men are better than other people. In Statistics for Social Justice, we use statistical tools to expose and analyze racism in many aspects of society.  Is this a contradiction?&lt;/p&gt;&#xA;" OwnerUserId="16410" LastActivityDate="2022-03-18T10:55:36.770" Title="Are statistics racist?" Tags="|statistics|scientific-method|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14286" PostTypeId="2" ParentId="14268" CreationDate="2022-03-17T19:41:20.897" Score="1" Body="&lt;p&gt;I do not know how to give a precise description, nor draw a picture of it, but I can give anecdotal confirmation that such a thing is real.&lt;/p&gt;&#xA;&lt;p&gt;Namely, if you get a thin, flexible-but-not-stretchy hemisphere (as I've acquired from various broken kids' toys), it is easily possible to twist it up into what I can imagine is a &amp;quot;spindle&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/S13fh.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/S13fh.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vlBVH.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vlBVH.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1932" LastEditorUserId="1932" LastEditDate="2022-03-18T16:57:56.367" LastActivityDate="2022-03-18T16:57:56.367" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14287" PostTypeId="1" AcceptedAnswerId="14288" CreationDate="2022-03-17T20:33:40.660" Score="3" ViewCount="502" Body="&lt;p&gt;I have just become aware of the 19th century French mathematician &lt;strong&gt;E. Midy&lt;/strong&gt;, who apparently was the first to prove what is now known as &lt;strong&gt;Midy's theorem&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I can find out nothing about this mathematician beyond the fact that he wrote a paper &lt;em&gt;De Quelques Propriétés des Nombres et des Fractions Décimales Périodiques&lt;/em&gt; from which &amp;quot;Midy's theorem&amp;quot; seems to have come.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any access to more information about him? Full name, dates and places of birth and death help to put him into context, but any other relevant details more than welcome.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="12330" LastEditDate="2022-03-17T21:43:32.430" LastActivityDate="2022-03-19T00:38:44.837" Title="Who was E. Midy, the eponym of Midy's Theorem?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14288" PostTypeId="2" ParentId="14287" CreationDate="2022-03-17T21:18:03.303" Score="6" Body="&lt;p&gt;From the entry for Etienne Midy on &lt;a href=&quot;http://www.numericana.com/fame/bio.htm#midy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Biographies of Scientists - Numericana&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h2&gt;Etienne Midy   (c. 1775,  fl. 1846)&lt;/h2&gt;&#xA;&lt;p&gt;&lt;sup&gt;&lt;a href=&quot;http://www.numericana.com/answer/numeration.htm#midy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Midy's theorem&lt;/a&gt; (1835)&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;His name is spelled  &lt;em&gt;Meidy&lt;/em&gt;  in some records.  He was probably already teaching when  &lt;em&gt;Napoléon&lt;/em&gt;  instituted the  &lt;em&gt;lycées&lt;/em&gt;,  in 1802.  Midy himself advertised he had taught  &lt;em&gt;mathématiques spéciales&lt;/em&gt;  at &lt;a href=&quot;http://fr.wikipedia.org/wiki/Lyc%C3%A9e_Gambetta_(Cahors)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cahors&lt;/a&gt; (1821-1826) and &lt;a href=&quot;http://fr.wikipedia.org/wiki/Lyc%C3%A9e_Pothier&quot; rel=&quot;nofollow noreferrer&quot;&gt;Orléans&lt;/a&gt; (1826-1832) before moving to Nantes.&lt;/p&gt;&#xA;&lt;p&gt;At the  &lt;em&gt;Collège Royal de Nantes&lt;/em&gt;  (future  &lt;a href=&quot;https://en.wikipedia.org/wiki/Lyc%C3%A9e_Georges_Clemenceau_(Nantes)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lycée Clémenceau&lt;/a&gt;)  Midy taught  &lt;em&gt;mathématiques élémentaires&lt;/em&gt;  from 1833 to 1837.  That post was entrusted to a young  &lt;em&gt;normalien&lt;/em&gt;  (Alexandre Lepord, ENS 1834)  when Midy was promoted to teach  &lt;em&gt;mathématiques spéciales&lt;/em&gt;  again in 1837-1838  (after M. Dorveau resigned).  Midy would be replaced in this capacity by M. Gascheau  (previously, professor of physics)  when a ministerial decree  (1838-11-17)  allowed him to retire.&lt;/p&gt;&#xA;&lt;p&gt;In Nantes,  Etienne Midy lived  &lt;em&gt;3, rue Richebourg,&lt;/em&gt;  next to his workplace.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3329" LastEditorUserId="14365" LastEditDate="2022-03-19T00:38:44.837" LastActivityDate="2022-03-19T00:38:44.837" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14289" PostTypeId="2" ParentId="14284" CreationDate="2022-03-18T02:14:38.180" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Are statistics racist?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Statistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Statistics&lt;/a&gt; are a mathematical/analytical tool that can be applied to any field, such as: zebra populations, industrial processes and products, interstellar research, shoe sizes as well as medical &amp;quot;research&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Some people may have and may want to use statistics for racist purposes. That has nothing to do with the fundamentals of statistics. It would be an aberrant &lt;strong&gt;application&lt;/strong&gt; of statistics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lies,_damned_lies,_and_statistics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lies, damned lies, and statistics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2022-03-18T02:14:38.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14290" PostTypeId="2" ParentId="14273" CreationDate="2022-03-18T02:46:42.223" Score="2" Body="&lt;p&gt;You're right, many popular expositions of math and its history tend to round up the usual suspects over and over again.&lt;/p&gt;&#xA;&lt;p&gt;Try browsing through the journal &lt;a href=&quot;https://www.sciencedirect.com/journal/historia-mathematica/issues?page=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Historia Mathematica&lt;/a&gt;.  Volumes more than four years old are open access.  It is a grab bag of articles, references and book reviews about all aspects of math history.  Perhaps five percent of the articles are about non-European math history.  It's searchable.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;archive.org&lt;/a&gt; is a repository of historical texts, some going back to the 1600's. You said that &amp;quot;knowing the historical development of more modern techniques really helpful for understanding math from school.&amp;quot;, and here you can find a lot of that.  Often modern treatments can be overly formal and abstract, whereas the historical texts are more down to earth.  Just search by subject matter and mathematicians.&lt;/p&gt;&#xA;" OwnerUserId="11204" LastActivityDate="2022-03-18T02:46:42.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14291" PostTypeId="1" CreationDate="2022-03-18T07:14:42.547" Score="4" ViewCount="146" Body="&lt;p&gt;In Montgomery-Vaughan's &lt;em&gt;Multiplicative Number Theory I: Classical Theory&lt;/em&gt;, they prove in Theorem 15.3 that &lt;span class=&quot;math-container&quot;&gt;$\pi(x)-\operatorname{li}(x) = \Omega_{\pm}(x^{\Theta-\epsilon})$&lt;/span&gt; for every &lt;span class=&quot;math-container&quot;&gt;$\epsilon&amp;gt;0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Theta$&lt;/span&gt; being the supremum of the real parts of the non-trivial zeroes of the Riemann zeta function. This in particular shows that &lt;span class=&quot;math-container&quot;&gt;$\pi(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\operatorname{li}(x)$&lt;/span&gt; cross infinitely many times. In Section 15.3, Montgomery-Vaughan (M-V) write &amp;quot;Theorems 15.2 and 15.3, and Corollary 15.4, are due in substance to E. Schmidt (1903).&amp;quot;.  However, when I search &amp;quot;pi(x)-li(x) crossover infinitely&amp;quot; in Google, literally every single page cites this infinite crossover result as being from Littlewood (1914):&#xA;&lt;a href=&quot;https://i.stack.imgur.com/TQgNp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TQgNp.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Yes, it is true that Littlewood proved a stronger version of this result in 1914 (Thm. 15.11 in M-V, as they acknowledge again in Section 15.3), but most of the papers/websites just say that Littlewood (1914) showed that there are infinitely many crossovers, when M-V suggest the result was known to Schimdt in 1903.&lt;/p&gt;&#xA;" OwnerUserId="11071" LastActivityDate="2022-03-19T02:19:08.380" Title="$\pi(x)$ and $\operatorname{li}(x)$ cross infinitely many times -- due to Littlewood 1914 or Schmidt 1903?" Tags="|mathematics|discoveries|number-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14292" PostTypeId="2" ParentId="14284" CreationDate="2022-03-18T10:55:36.770" Score="4" Body="&lt;p&gt;In general it is important to check scientific results and methods for racism and other biases, so it's a fair question.&lt;/p&gt;&#xA;&lt;p&gt;One needs to distinguish, however, between the historical origins and the present-day usage.&lt;/p&gt;&#xA;&lt;p&gt;The statistical methods in EBM are fundamentally just tools. While they were developed by people with racist views and with the purpose of supporting racist views (as the quote describes), the tools themselves are not (necessarily) racist.&lt;/p&gt;&#xA;&lt;p&gt;To give an analogy, many other fields of science also have historical roots in things that are (or should be) unacceptable now. For example, some mathematics and mechanics was developed for war, to be able to calculate the path of cannon balls or such things. Chemistry was used to develop bombs. Nuclear physics was, for a time, closely tied to the development of the nuclear bomb. Medical surgery was also developed mainly by military surgeons. That doesn't mean that our understanding of mechanics, of chemistry, of the properties of atoms or or medical surgery is now militaristic.&lt;/p&gt;&#xA;&lt;p&gt;With statistical methods, it's similar. They are fundamentally neutral tool, despite of what they have been used for or are used for today.&lt;/p&gt;&#xA;&lt;p&gt;However, this doesn't mean that the applications are always harmless or unbiassed. With statistical methods and EBM, it is possible that other biasses and also racism come into the actual results. For example, if a new drug is only tested on a white European or US population (because the manufacturer is based there), it may not work on an African or Asian population or have very different side effects (this can be due to genetic or lifestyle differences). Ideally, drug trials should be performed with samples from all populations, but of course the manufacturers may focus on some markets where there is most money, and poorer countries may not have the option of requiring manufacturers to do test in their own population.&lt;/p&gt;&#xA;&lt;p&gt;So you can have a situation where all the EBM literature says some intervention works, but practitioners say &amp;quot;anecdotally in our experience, it doesn't&amp;quot; - and the practitioners are actually right if they work with in a different population than the one on which the EBM studies were done.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, there can indeed by implicit racism in EBM as well and one should always be aware of these issues, but it's a separate issue and has nothing to do with racism in the historical development of the mathematical methods.&lt;/p&gt;&#xA;" OwnerUserId="9563" LastActivityDate="2022-03-18T10:55:36.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14294" PostTypeId="1" CreationDate="2022-03-18T17:30:00.037" Score="0" ViewCount="42" Body="&lt;p&gt;Recreational math books contain duplicates for problems.&lt;/p&gt;&#xA;&lt;p&gt;I think, classical examples are &lt;code&gt;SEND + MORE = MONEY&lt;/code&gt; and Zebra puzzles. (Though I can't point exact books which include those, I only know they are popular)&lt;/p&gt;&#xA;&lt;p&gt;Is there a community database that collects these common math problems, assigns IDs and tracks history of both creation and usage across literature?&lt;/p&gt;&#xA;" OwnerUserId="16421" LastEditorUserId="12330" LastEditDate="2022-03-21T04:49:57.153" LastActivityDate="2022-03-21T04:49:57.153" Title="Reference database for recreational math problems and math puzzles?" Tags="|reference-request|education|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14295" PostTypeId="2" ParentId="14291" CreationDate="2022-03-19T02:19:08.380" Score="2" Body="&lt;p&gt;The paper &lt;a href=&quot;https://personal.math.ubc.ca/%7Egerg/teaching/613-Winter2011/OscillationErrorTerms.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oscillation Error Terms: Littlewood's Result&lt;/a&gt; writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;E.Schmidt [1903] could do this under the assumption that the Riemann Hypothesis (RH) is false. Littlewood proved in 1914 that this is also the case when RH is true.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So one could say that while Schmidt's result was part of the proof, it's hard to credit him with having proven the result &amp;quot;in substance&amp;quot; since both results were necessary.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-03-19T02:19:08.380" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14296" PostTypeId="2" ParentId="9777" CreationDate="2022-03-19T04:08:46.020" Score="1" Body="&lt;p&gt;I do not have a citation for the etymology of the terms unfortunately.  However, I know the following mnemonic.  It's conceivable to me that the following was Birkhoff's reasoning (I have no evidence).  If it helps direct anyone to find better evidence in Birkhoff's writings, that's all the better.  If not, I think it's still a better way to explain the terminology to students.&lt;/p&gt;&#xA;&lt;p&gt;Consider the projective plane.  Define a set &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; containing the empty set, the points of the plane, the lines of the plane, and the entire space.  Equip &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; with a partial order s.t. a point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is less than a line &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; lies on &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;, and s.t. the empty set is less than everything and the entire plane greater than everything.  Then &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is a lattice where for two distinct lines &lt;span class=&quot;math-container&quot;&gt;$l_0 \wedge l_1$&lt;/span&gt; is the unique point at which the lines &lt;em&gt;meet&lt;/em&gt;, and where for two distinct points &lt;span class=&quot;math-container&quot;&gt;$p_0 \vee p_1$&lt;/span&gt; is the unique line &lt;em&gt;joining&lt;/em&gt; the two points.&lt;/p&gt;&#xA;&lt;p&gt;This construction may be generalized to a Grassmannian algebra (linear subspaces of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt;) s.t. &lt;span class=&quot;math-container&quot;&gt;$S_0 \wedge S_1$&lt;/span&gt; is the intersection of two linear subspaces (also a linear subspace) and &lt;span class=&quot;math-container&quot;&gt;$S_0 \vee S_1$&lt;/span&gt; is the span of two linear subspaces (also a linear subspace).  The projective plane as described above is isomorphic to the lattice of linear subspaces of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^3$&lt;/span&gt; via the usual homogeneous coordinates for points in projective space.&lt;/p&gt;&#xA;" OwnerUserId="16425" LastActivityDate="2022-03-19T04:08:46.020" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14297" PostTypeId="1" CreationDate="2022-03-19T07:34:19.993" Score="1" ViewCount="49" Body="&lt;p&gt;The Chemistry SE question &lt;a href=&quot;https://chemistry.stackexchange.com/q/159395/16035&quot;&gt;What dodecahedral molecule is Linus Pauling likely holding in this photograph? Does it have 40 carbon atoms?&lt;/a&gt; begins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The video &lt;a href=&quot;https://youtu.be/V2GqU6fdjeQ?t=420&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quasicrystals ; Prof. Daniel Shechtman ; Nobel Prize in Chemistry&lt;/a&gt; focuses on &lt;a href=&quot;https://en.wikipedia.org/wiki/Dan_Shechtman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Professor Dan Shechtman&lt;/a&gt; but happens to include some photos of Linus Pauling &lt;strong&gt;who never believed in &lt;a href=&quot;https://en.wikipedia.org/wiki/Quasicrystal&quot; rel=&quot;nofollow noreferrer&quot;&gt;quasicrystals&lt;/a&gt;.&lt;/strong&gt; From the video:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The leader of that group was Professor Linus Pauling, a two-time Nobel laureate. He was a very important figure and an idol of the American Chemical Society, &lt;strong&gt;and to his last day, he was standing on stages and published papers saying that &amp;quot;Daniel Shechtman is talking nonsense.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;During this part of the video there is shown a photo of Linus Pauling holding some ball and stick molecular models. These appear to be &lt;a href=&quot;https://en.wikipedia.org/wiki/Regular_dodecahedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;dodecahedral&lt;/a&gt; arrangements (12 pentagons, 20 vertices) of possibly carbon atoms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From each atom of the dodecahedron another atom protrudes.&lt;/p&gt;&#xA;&lt;p&gt;The question was closed as &amp;quot;opinion-based&amp;quot; because it was decided that a fact-based answer to what Pauling was holding up in the 1950's photograph couldn't be identified with sufficient certainty for a Chemistry SE answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://chemistry.stackexchange.com/questions/159395/what-dodecahedral-molecule-is-linus-pauling-likely-holding-in-this-photograph-d#comment327700_159395&quot;&gt;There was speculation there&lt;/a&gt; that this was in 1954 &amp;quot;taken on occasion of (Pauling's) 1954 NP (Nobel Prize) award&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As it is now too late to migrate but I can't get this image nor this structure out of my craw as evidenced by two new Chemistry SE questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://chemistry.stackexchange.com/q/163833/16035&quot;&gt;Has substitution of all 20 hydrogen atoms of pentagonal dodecahedrane ever succeeded? If not, is there any reason that it can not be done?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://chemistry.stackexchange.com/q/163832/16035&quot;&gt;Has a fully methylated pentagonal dodecahedrane been synthesized or at least attempted? If not, is there some reason it could not exist?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I'd like to now ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What molecule's models might Linus Pauling be holding up in this photo? Perhaps something akin to pentagonal dodecahedrane, or could they be viruses?*&lt;/p&gt;&#xA;&lt;p&gt;Clearly I'm grasping at straws, but there might be a recording or transcript of some speech related to Pauling's 1954 Nobel Prize award.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/o9yzY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/o9yzY.jpg&quot; alt=&quot;screenshot from the video &amp;quot;Quasicrystals ; Prof. Daniel Shechtman ; Nobel Prize in Chemistry&amp;quot; https://youtu.be/V2GqU6fdjeQ?t=420&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-03-19T07:41:03.620" LastActivityDate="2022-03-19T07:41:03.620" Title="What molecule's models might Linus Pauling be holding up in this photo? Perhaps something akin to pentagonal dodecahedrane, or could they be viruses?" Tags="|chemistry|pauling|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14298" PostTypeId="1" CreationDate="2022-03-19T07:52:20.787" Score="3" ViewCount="72" Body="&lt;p&gt;The &lt;a href=&quot;https://microbiologysociety.org/blog/in-pictures-the-first-images-of-a-coronavirus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;first TEM image of a human coronavirus was in 1967&lt;/a&gt; but both images of virus particles and &lt;a href=&quot;https://en.wikipedia.org/wiki/Transmission_electron_microscopy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Transmission Electron Microscopy&lt;/a&gt; were around long before that.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/14297/3869&quot;&gt;What molecule&amp;#39;s models might Linus Pauling be holding up in this photo? Perhaps something akin to pentagonal dodecahedrane, or could they be viruses?&lt;/a&gt; includes the image below and I'm wondering if those might be intended to be models of spiked virus particles rather than fullerenes that were not even considered at the time.&lt;/p&gt;&#xA;&lt;p&gt;So to that end I'd like to ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; When was the first electron microscope image of a roundish or geometrical spiked virus particle produced?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Screenshot from the video &lt;a href=&quot;https://youtu.be/V2GqU6fdjeQ?t=420&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quasicrystals ; Prof. Daniel Shechtman ; Nobel Prize in Chemistry&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As discussed in the linked question, Pauling's file photos are used along with a discussion of the lengths to which Pauling used his position to &lt;em&gt;dismiss and otherwise cast shade on Shechtman's work&lt;/em&gt; and the possibilities that quasicrystals could possibly exist.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/o9yzY.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/o9yzY.jpg&quot; alt=&quot;screenshot from the video &amp;quot;Quasicrystals ; Prof. Daniel Shechtman ; Nobel Prize in Chemistry&amp;quot; https://youtu.be/V2GqU6fdjeQ?t=420&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;below:&lt;/strong&gt; diagram of an icosahedral adenovirus &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Adenovirus_3D_schematic.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt; The icosahedron has 20 triangular faces rather than the 12 pentagonal faces of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dodecahedron&quot; rel=&quot;nofollow noreferrer&quot;&gt;dodecahedron&lt;/a&gt;, but in early TEM images I'm not sure if it would be easy to tell the difference.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jNbhE.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jNbhE.png&quot; alt=&quot;diagram of an icosahedral adenovirus&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-03-26T23:05:05.967" LastActivityDate="2022-03-26T23:22:15.827" Title="When was the first electron microscope image of a roundish or geometrical (e.g. icospiked virus particle produced?" Tags="|biology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14300" PostTypeId="1" CreationDate="2022-03-19T10:43:56.770" Score="3" ViewCount="108" Body="&lt;p&gt;In &lt;a href=&quot;https://projecteuclid.org/journals/duke-mathematical-journal/volume-5/issue-2/A-proof-that-every-uniformly-convex-space-is-reflexive/10.1215/S0012-7094-39-00522-3.short&quot; rel=&quot;nofollow noreferrer&quot;&gt;A proof that every uniformly convex space is reflexive&lt;/a&gt; in footnote 3 (available at that link without a paywall), author &lt;a href=&quot;https://en.wikipedia.org/wiki/Billy_James_Pettis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Billy Pettis&lt;/a&gt; mentions that the first half of Lemma 1 in that paper &amp;quot;was discovered independently by &lt;a href=&quot;https://en.wikipedia.org/wiki/James_A._Clarkson&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. A. Clarkson&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_J._McShane&quot; rel=&quot;nofollow noreferrer&quot;&gt;E. J. McShane&lt;/a&gt; in 1936&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I attempted to find the corresponding publications, but could only find the publication &lt;a href=&quot;https://projecteuclid.org/journals/duke-mathematical-journal/volume-2/issue-4/Semi-continuity-of-integrals-in-the-calculus-of-variations/10.1215/S0012-7094-36-00251-X.short&quot; rel=&quot;nofollow noreferrer&quot;&gt;Semi-continuity of integrals in the calculus of variations&lt;/a&gt; of E. J. McShane from 1936, which seems to concern unrelated topics (although I only skimmed the paper, but uniform convexity and dual spaces are not even mentioned).&#xA;His &lt;a href=&quot;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-43/issue-8/Jensens-inequality/bams/1183499917.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper on Jensen's inequality&lt;/a&gt; also does not contain this assertion, if I am not mistaken.&#xA;I also searched &lt;a href=&quot;https://scholar.google.com/scholar?start=0&amp;amp;hl=de&amp;amp;as_sdt=2005&amp;amp;sciodt=0,5&amp;amp;as_ylo=1936&amp;amp;as_yhi=1945&amp;amp;cites=6338053928411781930&amp;amp;scipsc=&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Scholar&lt;/a&gt; for paper quoting the paper by Clarkson mentioned below, which introduces uniformly convex spaces, which are critical for this result, but I found none published between 1936 and 1949.&#xA;&lt;em&gt;But&lt;/em&gt; in his work &lt;a href=&quot;https://www.jstor.org/stable/2032394?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Functionals on Certain Banach Spaces&lt;/a&gt; published in 1950, the existence part of this lemma is proven in his proof of Lemma 3 (p. 403) using exactly the same arguments as in Pettis' proof.&#xA;This paper also seems to be the only one of his from 1936-1950 where uniform convexity is mentioned.&lt;/p&gt;&#xA;&lt;p&gt;The only paper by &lt;a href=&quot;https://en.wikipedia.org/wiki/James_A._Clarkson&quot; rel=&quot;nofollow noreferrer&quot;&gt;James A. Clarkson&lt;/a&gt; from 1936 is the famous &lt;a href=&quot;https://www.jstor.org/stable/1989630?seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Uniformly convex spaces&amp;quot;&lt;/a&gt;, which, introduces the uniformly convex spaces and, if I am not mistaken, also does not contain this assertion.&#xA;Later papers by this author seem to, again, concern very different properties.&lt;/p&gt;&#xA;&lt;p&gt;Could the footnote by Pettis also be understood to mean that those mathematicians discovered but did not formally publish this result?&lt;/p&gt;&#xA;&lt;p&gt;For reference, the statement is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If the Banach space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is uniformly convex and &lt;span class=&quot;math-container&quot;&gt;$\gamma_0 \in \overline{X}$&lt;/span&gt; (the dual space of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;) with &lt;span class=&quot;math-container&quot;&gt;$\| \gamma_0 \| \ne 0$&lt;/span&gt;, then there exists a unique &lt;span class=&quot;math-container&quot;&gt;$x_0 \in X$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\| x_0 \| = 1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gamma_0(x_0) = \| \gamma_0 \|$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have seen the latter property being called &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x_0$&lt;/span&gt; achieves the least upper bound (l.u.b) for &lt;span class=&quot;math-container&quot;&gt;$\gamma_0$&lt;/span&gt;&amp;quot;, e.g. in &lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF02415327.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;E.R. Lorch: &amp;quot;A curvature study of convex bodies in Banach spaces&amp;quot;, 1953&lt;/a&gt;.&#xA;This property also appears in &lt;a href=&quot;http://www.numdam.org/article/BSMF_1941__69__23_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fortet: Remarques sur les éspaces uniformément convexes translation&lt;/a&gt; on page 45.&lt;/p&gt;&#xA;" OwnerUserId="10550" LastEditorUserId="10106" LastEditDate="2022-03-28T20:33:00.130" LastActivityDate="2022-03-28T20:33:00.130" Title="Can not find reference for &quot;uniform convexity implies existence of unique conjugate&quot; mentioned by Pettis" Tags="|mathematics|reference-request|functional-analysis|20th-century|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14301" PostTypeId="1" CreationDate="2022-03-19T17:44:19.760" Score="2" ViewCount="83" Body="&lt;p&gt;We use cross sign in vector multiplication that gives vector result and dot sign that gives scalar result.&lt;/p&gt;&#xA;&lt;p&gt;But how did the dot sign come in scalar  product and the cross sign in vector product? It could also have happened that we used cross sign for scalar  product and dot sign for vector product.&lt;/p&gt;&#xA;" OwnerUserId="16428" LastEditorUserId="10106" LastEditDate="2022-03-20T03:32:52.653" LastActivityDate="2022-03-20T03:32:52.653" Title="How did Cross and Dot signs come in vector multiplication?" Tags="|physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14302" PostTypeId="1" CreationDate="2022-03-20T01:01:29.603" Score="1" ViewCount="52" Body="&lt;p&gt;Has there been any historical/cultural anthropological research into how Polynesian cultures understood geometry before contact with Europeans? In part what I am interested in is how did the dealing with the problems of navigating on a scale where the spherical nature of the earth influence their understanding of space and shapes.&lt;/p&gt;&#xA;&lt;p&gt;Just from browsing google I can see that a lot of Polynesian art seems to have an awareness of this but I know this could be because of my biasies when interpreting the art.&lt;/p&gt;&#xA;" OwnerUserId="16432" LastEditorUserId="10106" LastEditDate="2022-03-20T03:12:57.570" LastActivityDate="2022-03-20T03:12:57.570" Title="Research on Pre-Columbian Polynesian geometry?" Tags="|mathematics|reference-request|geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14303" PostTypeId="1" CreationDate="2022-03-21T01:46:54.087" Score="3" ViewCount="100" Body="&lt;p&gt;Why/when was the curly-O notation chosen for the ring of integers of an algebraic number field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="11071" LastEditorUserId="11071" LastEditDate="2022-03-21T02:15:08.287" LastActivityDate="2022-03-21T02:15:08.287" Title="Why is the ring of algebraic integers denoted by $\mathcal O_K$?" Tags="|mathematics|notation|abstract-algebra|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14304" PostTypeId="1" CreationDate="2022-03-21T09:04:10.117" Score="1" ViewCount="119" Body="&lt;p&gt;I'm looking for early uses of the word &amp;quot;operator&amp;quot; in mathematics and how it evolved to become a synonym for the modern &amp;quot;function&amp;quot; in certain areas. Jeff Miller's &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/o/&quot; rel=&quot;nofollow noreferrer&quot;&gt;page&lt;/a&gt; only has an entry for &amp;quot;operation&amp;quot; and &amp;quot;operand&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2022-03-21T20:27:12.177" Title="Early uses of the word &quot;operator&quot; in mathematics" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14305" PostTypeId="1" CreationDate="2022-03-21T14:01:38.407" Score="1" ViewCount="116" Body="&lt;p&gt;If you open any book containing some history of volcanology (e.g., &lt;a href=&quot;https://doi.org/10.1016/C2015-0-00175-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Encyclopedia of Volcanoes&lt;/a&gt;), you will always find the same story: first the &amp;quot;internal wind&amp;quot; theory of ancient Greeks, then the account of the AD 79 Vesuvius eruption by Pliny the Younger, then the discoveries of the eighteenth century by Guettard, Desmarest, and others, etc. Only European scientists studying European volcanoes.&lt;/p&gt;&#xA;&lt;p&gt;While I am not questioning the tremendous contributions of these studies to our current understanding of the volcanic phenomenon, I wonder if there are similar texts from other parts of the world. I'm thinking of places like Japan, China, the Arab world, Mesoamerica... All of which have had both active volcanoes and a writing system for quite some time. Considering this, there must exist accounts of eruptions, or old volcanological theories, in Old Japanese or Classical Arabic.&lt;/p&gt;&#xA;" OwnerUserId="14043" LastActivityDate="2022-11-16T15:45:08.090" Title="Looking for early, non-European accounts and theories of volcanism" Tags="|earth-sciences|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14306" PostTypeId="2" ParentId="14304" CreationDate="2022-03-21T20:27:12.177" Score="3" Body="&lt;p&gt;The unabridged Oxford Dictionary is of great help in such matters. It requires subscription. Here are the examples it provides since 1855.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Originally: (Mathematics) a symbol or group of symbols indicating an&#xA;operation or series of operations to be carried out, usually on a&#xA;following expression. Later also: a sign or symbol which effects other&#xA;types of operation, as logical, phonological, syntactic, etc.&lt;/p&gt;&#xA;&lt;p&gt;1855  R. Carmichael Treat. Calculus of Operations 3   The&#xA;indetermination is due to a source quite independent of the character&#xA;of the functional operator.&lt;/p&gt;&#xA;&lt;p&gt;1925   J. M. Bryant &amp;amp; J. A. Correll Alternating-current Circuits iii.&#xA;74   The operator, j = √(− 1), turns the vector through 90 degrees in&#xA;a counterclockwise direction each time it is used.&lt;/p&gt;&#xA;&lt;p&gt;1952  Eng. &amp;amp; Germanic Stud. 4 12   We may regard Modern English stone&#xA;as the result of operating with an operator that I shall write {AS. ā&#xA;MnE. [ou]} on Anglo-Saxon stān.&lt;/p&gt;&#xA;&lt;p&gt;1969   V. J. Calderbank Course on Programming in FORTRAN IV iii. 28&lt;br /&gt;&#xA;The relational expression..has the general form e1re2 where e1 and e2&#xA;are arithmetic expressions..being compared by one of the following&#xA;relational operators, r: .EQ. Equal to (=); .LT. Less than (&amp;lt;);&#xA;[etc.].&lt;/p&gt;&#xA;&lt;p&gt;1990   Proc. London Math. Soc. 60 365   The centroid operator and the&#xA;projection operator are quite different. For example, the centroid&#xA;operator commutes with linear transformations, while the projection&#xA;operator does not.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2022-03-21T20:27:12.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14307" PostTypeId="1" CreationDate="2022-03-23T01:31:18.360" Score="4" ViewCount="88" Body="&lt;p&gt;The original paper by J.J. Thompson, where he exposes his experiment to measure &lt;span class=&quot;math-container&quot;&gt;$m/e$&lt;/span&gt; can be found &lt;a href=&quot;https://www.douglas-scott-mcgregor.com/uploads/1/3/3/5/133545016/thomson1897.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. (Note that Thompson actually measured &lt;span class=&quot;math-container&quot;&gt;$m/e$&lt;/span&gt; whereas the modern discussion is in terms of &lt;span class=&quot;math-container&quot;&gt;$e/m$&lt;/span&gt;.) It seems clear from the text that distances are measured in centimeters, and voltages in Volts,  but as far as I can tell, Thompson does not provide units for magnetic or electric fields, or charges.&lt;/p&gt;&#xA;&lt;p&gt;According to tables 1-3 on page 306 of the journal, Thompson found consistent values of &lt;span class=&quot;math-container&quot;&gt;$m/e$&lt;/span&gt; in the range of &lt;span class=&quot;math-container&quot;&gt;$\sim 10^{-8}$&lt;/span&gt;, with no units given.  The current accepted value of &lt;span class=&quot;math-container&quot;&gt;$m/e$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$5.68\times 10^{-12}$&lt;/span&gt; kg/C so clearly there's some conversion factor somewhere.&lt;/p&gt;&#xA;&lt;p&gt;Thus: What units of electric field, magnetic field, and charge did Thompson use in his historical experiment?&lt;/p&gt;&#xA;" OwnerUserId="7093" LastActivityDate="2022-03-24T15:29:53.403" Title="Units in J.J. Thompson's m/e experiment" Tags="|physics|experimental-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14308" PostTypeId="2" ParentId="12882" CreationDate="2022-03-23T13:58:10.017" Score="3" Body="&lt;p&gt;This is a very interesting question that I spent much time studying. Newton never used nor invented a gravity constant. The formula for gravity force he states by words corresponds to &lt;span class=&quot;math-container&quot;&gt;$F=Mm/r^2$&lt;/span&gt;, so without any gravity constant. It is then of great importance that Newton had a very different view on matter (and mass) than modern physics. He explicitly states that the philosophy of all his work is rooted in that matter consists of indivisible particles with spatial dimensions. Newton and other scientists used Newton's work for hundreds of years to do long series of predictions, such as relative masses between heavenly objects. Newton even calculated very accurately the density of the Earth relative to that of the Sun. What Newton struggled to do was to find the density of the Earth relative to a known uniform substance. The first one to do this accurately was Cavendish in 1798. But also Cavendish did not use nor invented nor measured any gravitational constant. What is true his measures as well as a Cavendish apparatus can be used to find the gravity constant, but that is first after one re-define mass in terms of kilogram.&lt;/p&gt;&#xA;&lt;p&gt;Newton mentioned mass and weight are proportional to each other as long as the objects of interest are at the same distance from the center of the gravitational object. This is true also.&lt;/p&gt;&#xA;&lt;p&gt;The gravity constant was first mentioned in 1873 by two French physicists. This I think is not coincident. The gravity constant is needed when one re-formulates mass as kilogram. The kilogram was invented in France just before the beginning of 1800, but was first accepted in scientific circles around Europe around 1870. In 1873 they used notation f for the gravitational constant, some soon used k. People in 1894 used the notation G, but it took many years before G became the standard notation. If one uses f, G, or k as a symbol, it is naturally just cosmetic. What is important that no one used the gravity constant before 1873.&lt;/p&gt;&#xA;&lt;p&gt;I have tried to summarise some of the history of the gravity constant in a recent article (not in a top journal), but still perhaps of interest as I many references to work there that people themselves can look up and make up their own minds &lt;a href=&quot;https://www.scirp.org/pdf/jmp_2022022114492632.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;history and use of gravity constant&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16445" LastEditorUserId="10106" LastEditDate="2022-08-12T13:09:50.840" LastActivityDate="2022-08-12T13:09:50.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14309" PostTypeId="1" CreationDate="2022-03-23T16:10:54.787" Score="3" ViewCount="88" Body="&lt;p&gt;In Riemann's &amp;quot;Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe&amp;quot;, Riemann mentions taking a factor as &amp;quot;constant&amp;quot; in &amp;quot;partial integration&amp;quot;, which seems to me to be referring to the factor &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; in the standard integration by parts formula &lt;span class=&quot;math-container&quot;&gt;$\int u\,dv = uv-\int v\, du$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...durch zweimalige partielle Integration, indem man zuerst&#xA;&lt;span class=&quot;math-container&quot;&gt;$\lambda(x)$&lt;/span&gt;, dann &lt;span class=&quot;math-container&quot;&gt;$\lambda'(x)$&lt;/span&gt; als constant betrachtet...&#xA;(pp. 23-24 &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, a TeX transcription;&#xA;or pp. 30-31 &lt;a href=&quot;https://www.google.com/books/edition/Ueber_die_Darstellbarkeit_einer_Function/_zcPAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22als%20Constant%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;,&#xA;reprinted from Riemann,&#xA;Abh. Gesell. der Wiss. zu Göttingen  &lt;strong&gt;13&lt;/strong&gt;,  87-132  (1867)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does anyone know where the terminology for &amp;quot;partial integration&amp;quot; is discussed? Why is a function, which is clearly not a constant function, called &amp;quot;constant&amp;quot;? Is there a reference where this is discussed?&lt;/p&gt;&#xA;&lt;p&gt;I have a theory that it comes from a partial-derivative form of the formula for integration by parts, namely,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int {\partial\over\partial v}(uv)\;dv = uv-\int {\partial\over\partial u}(uv)\; du \,,$$&lt;/span&gt;&#xA;since in &lt;span class=&quot;math-container&quot;&gt;${\partial\over\partial v}(uv)$&lt;/span&gt;, the factor &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; is treated as constant.  But I made that formula up and have not found it anywhere.  Indeed, googling through the 19th century, the standard approach to integration by parts of today seems dominant in Cauchy and in English calculus textbooks.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Replacing &lt;span class=&quot;math-container&quot;&gt;$uv$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$f(u,v)$&lt;/span&gt; in the above equation seemed an obvious, if unfruitful, move. Zermelo made a similar remark to Brendel, who applied it to various integrals in &lt;a href=&quot;https://www.google.com/books/edition/Mathematische_Annalen/4LcKAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22partielle+Integration%22+%22als+constant%22&amp;amp;pg=PA248&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Ueber partielle Integration&amp;quot;, &lt;em&gt;Math. Ann.&lt;/em&gt; &lt;strong&gt;55&lt;/strong&gt;, 1901 (248-256)&lt;/a&gt;, half a century after Riemann wrote his paper.  While Brendel starts with the standard integration by parts formula above, he switches the roles of &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; without comment and treats &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; &amp;quot;als constant&amp;quot;; he then replaces &lt;span class=&quot;math-container&quot;&gt;$u$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Be that as it may, it seems to suggest the view was what I surmised above, and presumably that was Riemann's view, too. It leaves open the question of when and where such a view of integration by parts or partial integration, in which one factor is treated as constant, arose and was common.&lt;/p&gt;&#xA;&lt;p&gt;(By the way, Brendel remarks that Hilbert suggested that the standard formula should be called &amp;quot;integration of a product.&amp;quot; (249))&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2022-03-25T14:14:46.610" LastActivityDate="2022-03-25T14:14:46.610" Title="The term &quot;constant&quot; in &quot;integration by parts&quot; (&quot;partielle Integration&quot;)" Tags="|terminology|calculus|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14310" PostTypeId="1" AcceptedAnswerId="14311" CreationDate="2022-03-23T21:35:38.923" Score="2" ViewCount="455" Body="&lt;p&gt;The University of Messina has a scientific organization and publication &amp;quot;Accademia Peloritana dei Pericolanti&amp;quot;, see &lt;a href=&quot;https://cab.unime.it/journals/index.php/AAPP&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;. What is the origin and meaning of the word &amp;quot;peloritana&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="8151" LastEditDate="2022-03-24T16:26:53.747" LastActivityDate="2022-03-24T16:26:53.747" Title="What or who is Peloritan in the name &quot;Accademia Peloritana dei Pericolanti&quot;?" Tags="|terminology|physics-social-history|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14311" PostTypeId="2" ParentId="14310" CreationDate="2022-03-23T22:01:34.170" Score="6" Body="&lt;p&gt;It is a toponym, functionally equivalent to &amp;quot;Messina&amp;quot;.&#xA;&lt;a href=&quot;https://www.alamy.com/stock-photo/cape-pelorus.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cape Pelorus&lt;/a&gt; is the modern &lt;a href=&quot;https://en.wikipedia.org/wiki/Faro_Point&quot; rel=&quot;nofollow noreferrer&quot;&gt;Capo Peloro=Faro Point&lt;/a&gt;, at the end of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Peloritani&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peloritani Mountains&lt;/a&gt; in North East Sicily, where Messina is.&lt;/p&gt;&#xA;&lt;p&gt;There is a navigation instrument called the pelorus, said to have been named after Hannibal's navigator Pelorus (the eponym of Cape Pelorus),  unjustly executed by Hannibal; one  &lt;a href=&quot;https://www.flickr.com/photos/luigistrano/14409437702&quot; rel=&quot;nofollow noreferrer&quot;&gt;exposition&lt;/a&gt; has more detail than most such, but (sadly) no references. See also the entry for &lt;a href=&quot;https://www.google.com/books/edition/A_New_Classical_Dictionary_of_Biography/xwZgAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=pelorus%20hannibal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peloris&lt;/a&gt; in the 1853 &lt;em&gt;A New Classical Dictionary of Biography, Mythology, and Geography&lt;/em&gt; by&#xA;William Smith, which is no better, references-wise.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2022-03-24T02:47:33.103" LastActivityDate="2022-03-24T02:47:33.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14312" PostTypeId="2" ParentId="14310" CreationDate="2022-03-24T00:07:38.120" Score="2" Body="&lt;p&gt;Also the term &amp;quot;pericolanti&amp;quot; (precarious) can sound strange, and the meaning of the name is explained on the &lt;a href=&quot;http://www.accademiapeloritana.it/storia.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;website of the academia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nasceva così, ufficialmente, la Periclitantes, Peloritana Regia Academia, con chiaro riferimento alla configurazione del territorio cittadino, circoscritto, da una parte, dai monti Peloritani e proteso, dall’altra, verso quel braccio di mare insidioso, ben rappresentato nel logo del cenacolo da un veliero che solca le acque agitate dello Stretto, rese ancora più infide dalla presenza dei mostri omerici Scilla e Cariddi, metafora, come lo stesso segretario Carlo Vitali doveva spiegare, del pensiero umano, sempre in bilico «fra varie difficoltà e controversie» nel difficile raggiungimento della Verità, e degli uomini di scienza, continuamente «inter utramque viam periclitantes».&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus, officially, the Periclitantes, Peloritana Regia Academia was born, with clear reference to the configuration of the city's territory, circumscribed, on the one hand, by the Peloritani mountains and stretching out, on the other, towards that dangerous strait, well represented in the logo of the cenacle by a sailing ship ploughing through the agitated waters of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Strait_of_Messina&quot; rel=&quot;nofollow noreferrer&quot;&gt;Strait&lt;/a&gt;, made even more treacherous by the presence of the Homeric monsters Scylla and Charybdis, a metaphor, as the secretary Carlo Vitali himself had to explain, for human thought, always poised &amp;quot;between various difficulties and controversies&amp;quot; in the difficult attainment of Truth, and for men of science, continually &amp;quot;inter utramque viam periclitantes&amp;quot; [exposed to danger from both sides].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2022-03-24T00:07:38.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14313" PostTypeId="1" CreationDate="2022-03-24T04:59:52.757" Score="1" ViewCount="101" Body="&lt;p&gt;Ludwig Flamm &lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt;, Albert Einstein-Natan Rosen &lt;span class=&quot;math-container&quot;&gt;$[2]$&lt;/span&gt;, Hermann Weyl &lt;span class=&quot;math-container&quot;&gt;$[3]$&lt;/span&gt;, John Wheeler-Fuller &lt;span class=&quot;math-container&quot;&gt;$[4,5]$&lt;/span&gt;, Homer G. Ellis &lt;span class=&quot;math-container&quot;&gt;$[6]$&lt;/span&gt;, K. Bronnikov &lt;span class=&quot;math-container&quot;&gt;$[7]$&lt;/span&gt;, Michael Morris-Kip Thorne&lt;span class=&quot;math-container&quot;&gt;$[8]$&lt;/span&gt;, M. Visser &lt;span class=&quot;math-container&quot;&gt;$[9]$&lt;/span&gt;, Edward Teo &lt;span class=&quot;math-container&quot;&gt;$[10]$&lt;/span&gt;, S. Kim-H.Lee &lt;span class=&quot;math-container&quot;&gt;$[11]$&lt;/span&gt;, Francisco Lobo &lt;span class=&quot;math-container&quot;&gt;$[12]$&lt;/span&gt; and so on... are some names behind the history of wormhole physics. In particular, since the Ellis Drainhole and Morris-Thorne solutions, the physics of &lt;em&gt;traversable wormholes&lt;/em&gt; have been quite active.&lt;/p&gt;&#xA;&lt;p&gt;But, I would like to know:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Where are the (seminal or not) papers written by women, on traversable wormhole physics? Could you give me some references?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt; L. Flamm, &lt;em&gt;Comments on Einstein’s Theory of Gravity&lt;/em&gt; ,Physikalische Zeitschrift &lt;span class=&quot;math-container&quot;&gt;$17, 448, 1916$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[2]$&lt;/span&gt; A. Einstein and N. Rosen, &lt;em&gt;The Particle Problem in the General Theory of Relativity&lt;/em&gt; , Phys. Rev, &lt;span class=&quot;math-container&quot;&gt;$48, 1935$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[3]$&lt;/span&gt; H. Weyl, &lt;em&gt;Philosophy of Mathematics and Natural Science&lt;/em&gt;, Princeton University Press, &lt;span class=&quot;math-container&quot;&gt;$1949$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p.91$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[4]$&lt;/span&gt; J. A. Wheeler, &lt;em&gt;Geometrodynamics&lt;/em&gt;, Academic Press, New York and London, &lt;span class=&quot;math-container&quot;&gt;$1962$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[5]$&lt;/span&gt; R. W. Fuller and J. A. Wheeler, &lt;em&gt;Causality and Multiply-Connected Space-time&lt;/em&gt; , Phys. Rev., &lt;span class=&quot;math-container&quot;&gt;$128, 2, 1962$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[6]$&lt;/span&gt; H. G. Ellis, &lt;em&gt;Ether flow through a drainhole: A particle model in general relativity&lt;/em&gt;, Journal of Mathematical Physics, &lt;span class=&quot;math-container&quot;&gt;$14, 1, 1973$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[7]$&lt;/span&gt;  K. A. Bronnikov, &lt;em&gt;Scalar-tensor theory and scalar charge&lt;/em&gt;. Acta Physica Polonica. &lt;span class=&quot;math-container&quot;&gt;$B4: 251–266, 1973$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[8]$&lt;/span&gt; M.S. Morris and K. S. Throne, &lt;em&gt;Wormholes in General Relativity&lt;/em&gt;, Am. J. Phys., &lt;span class=&quot;math-container&quot;&gt;$56, 5, 1988$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[9]$&lt;/span&gt; M. Visser, &lt;em&gt;Lorentzian Wormholes&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$1995$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[10]$&lt;/span&gt; E. Teo, &lt;em&gt;Rotating Traversable Wormholes&lt;/em&gt;, arXiv:gr-qc/9803098v2, &lt;span class=&quot;math-container&quot;&gt;$1998$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[11]$&lt;/span&gt; S. Kim and H. Lee, &lt;em&gt;Exact solutions of charged wormhole&lt;/em&gt;, arXiv:gr-qc/0102077v1, &lt;span class=&quot;math-container&quot;&gt;$2001$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[12]$&lt;/span&gt; J. P. S. Lemos, F. S. N. Lobo and S. Q. Oliveira, &lt;em&gt;Morris-Thorne wormholes with a&#xA;cosmological constant&lt;/em&gt;, arXiv:gr-qc/0302049v2. &lt;span class=&quot;math-container&quot;&gt;$2003$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13570" LastActivityDate="2022-03-24T04:59:52.757" Title="Doubt on women behind the development of wormhole physics" Tags="|mathematicians|reference-request|physicists|gravity|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14314" PostTypeId="1" CreationDate="2022-03-24T14:44:06.130" Score="1" ViewCount="71" Body="&lt;p&gt;William Thomson (Lord Kelvin) proposed the existence of an absolute zero in 1848(1) by linearly extrapolating the experimentally determined volume-temperature law for gases. I recently learned that some scientists in 19th century objected to the existence of absolute zero but I have been unable to find such criticisms (2).&lt;/p&gt;&#xA;&lt;p&gt;(1) On an Absolute Thermometric Scale&#xA;&lt;a href=&quot;https://zapatopi.net/kelvin/papers/on_an_absolute_thermometric_scale.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://zapatopi.net/kelvin/papers/on_an_absolute_thermometric_scale.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(2) The youtuber in this video clip states that those who called themselves 'energetics' did not believe in entropy or the absolute zero scale for temperature:&#xA;&lt;a href=&quot;https://youtube.com/clip/UgkxT8US1xpLy0h_x3wJqdgK-bRzP4eMLrFN&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://youtube.com/clip/UgkxT8US1xpLy0h_x3wJqdgK-bRzP4eMLrFN&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The complete video is here:&#xA;Boltzmann's Entropy Equation: A History from Clausius to Planck&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=Tr_gv5CKB1Y&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=Tr_gv5CKB1Y&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="14904" LastEditorUserId="14904" LastEditDate="2022-03-24T16:53:27.340" LastActivityDate="2022-03-24T16:53:27.340" Title="What were some 19th century objections to the existence of absolute zero?" Tags="|physics|19th-century|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14315" PostTypeId="2" ParentId="14307" CreationDate="2022-03-24T15:29:53.403" Score="2" Body="&lt;p&gt;J. J. Thomson was a brilliant physicists but when it comes to units, he just &lt;em&gt;assumes&lt;/em&gt; that the rest of the mortals also know what he is talking about (perhaps a typical problem with overly intelligent people). In his landmark paper, he does not bother to mention units of his quantities &lt;em&gt;at all&lt;/em&gt;. So one has to resort to modern commentaries from reliable authors.&lt;/p&gt;&#xA;&lt;p&gt;You can start your journey from here: Discovery of the electron: commentary on J. J. Thomson's classic paper of 1897 DOI (&lt;a href=&quot;https://doi.org/10.1080/09500830701305514&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1080/09500830701305514&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;The units in vogue in that time, were electromagnetic units (e.m.u.). There were also electrostatic units (e.s.u). This paper clarifies that the units for m/e in Thomson's paper were grams per electromagnetic units (e.m.u.) of charge.&lt;/p&gt;&#xA;&lt;p&gt;e.m.u.: A system of electrical units formerly used in the c.g.s. system (see c.g.s. units). The e.m.u. of electric current is the abampere (all e.m.u. have the prefix ab- attached to the names of practical units). The abampere is the current that, flowing in an arc of a circle (1 centimetre in diameter), exerts a force of 1 dyne on unit magnetic pole at the centre of the circle. In e.m.u. the magnetic constant is of unit magnitude.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095746202&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095746202&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2022-03-24T15:29:53.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14317" PostTypeId="1" AcceptedAnswerId="14318" CreationDate="2022-03-25T12:36:04.973" Score="5" ViewCount="756" Body="&lt;p&gt;I found a gem called the &lt;a href=&quot;http://eulerarchive.maa.org/&quot; rel=&quot;noreferrer&quot;&gt;Euler Archive&lt;/a&gt;  and I was looking for similar easy access to Gauss' works but I couldn't find it via google searches so I beseech thee&lt;/p&gt;&#xA;" OwnerUserId="9988" LastActivityDate="2022-03-25T14:11:50.417" Title="is there an online archive of all of Gauss' works?" Tags="|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14318" PostTypeId="2" ParentId="14317" CreationDate="2022-03-25T14:11:50.417" Score="7" Body="&lt;p&gt;The published version of Gauss's collected works, which contains 12 volumes (together with two other volumes that contain treatises by later mathematicians about different aspects of Gauss's work), can be accessed freely on the digital library of Gottingen university - here is a link: &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348&quot; rel=&quot;noreferrer&quot;&gt;https://gdz.sub.uni-goettingen.de/volumes/id/PPN235957348&lt;/a&gt;. This version contain all of Gauss's publications, together with a lot of unpublished manuscripts, drafts of treatises and etc.&lt;/p&gt;&#xA;&lt;p&gt;In addition, there are published correspondences of Gauss with different mathematicians/scientists with whom he corresponded regularly. These correspondences (for examples: Gauss-Schumacher, Gauss-Bessel,...) fill many more volumes but their quality (in terms of density of mathematical ideas) is much lower than in Gauss's collected works mentioned before.&lt;/p&gt;&#xA;&lt;p&gt;And finally, and this is something that I'm specially interested in, this is not the whole story - there are many fragments and notebooks of Gauss that have not been digitized yet - here is a link to a website that apparently gives a complete guide to Gauss's work (published as well as unpublished) - &lt;a href=&quot;https://kalliope-verbund.info/de/findingaid_toc?fa.id=DE-611-BF-61709&amp;amp;fa.enum=1&amp;amp;lastparam=true&quot; rel=&quot;noreferrer&quot;&gt;https://kalliope-verbund.info/de/findingaid_toc?fa.id=DE-611-BF-61709&amp;amp;fa.enum=1&amp;amp;lastparam=true&lt;/a&gt;. Note that some references that you will find in scientific/mathematical articles about Gauss cannot be located within the twelve volumes mentioned before. For example, the famous handnote in which Gauss began to study the topological object known as &amp;quot;braid&amp;quot; cannot be found in his collected works, but according to the guide in the last link, it should be found in Handbuch 7.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2022-03-25T14:11:50.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14319" PostTypeId="1" AcceptedAnswerId="14326" CreationDate="2022-03-25T17:47:03.543" Score="6" ViewCount="780" Body="&lt;p&gt;All of the concepts and terminology central to linear algebra were established in the late 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Following recent comments by user KCd, that in the early 20th century determinants were the “primary language of linear algebra”, I conclude (rightly or wrongly) that linear algebra originally focused on techniques for solving systems of linear equations and the role of matrices and their determinants.&lt;/p&gt;&#xA;&lt;p&gt;Today, the focus of linear algebra is the study of  vector spaces and their properties.&lt;/p&gt;&#xA;&lt;p&gt;Who were the mathematicians that brought about this change of emphasis?  Was there a particular paper or textbook, or was this change simply the result of a natural move towards a more abstract presentation of the subject?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-03-27T20:10:40.753" Title="When did linear algebra become the study of vector spaces?" Tags="|mathematics|mathematicians|linear-algebra|20th-century|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14321" PostTypeId="2" ParentId="14282" CreationDate="2022-03-25T22:09:48.710" Score="1" Body="&lt;p&gt;&lt;strong&gt;tl;dr: No.  According to Ptolemy's model, the external planets will not get their maximal latitude &lt;em&gt;exactly&lt;/em&gt; at the opposition. However, his model makes the maximum latitude to occur closer to the opposition than it really is.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Ptolemy had different models for the external and internal planets. This answer considers only external planets - as this was the issue at hand in the question about Mars.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Swerdlow N.M. (2005) Ptolemy’s Theories of the Latitude of the Planets in the Almagest, Handy Tables, and Planetary Hypotheses.&lt;/em&gt; is very interesting paper about the latitude theory of Ptolemy. It does manage to present the theory quite in a neat and relatively simple manner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(1)&lt;/strong&gt; The earth is at O, through which passes the nodal line of the eccentric, which is inclined to the plane of the ecliptic at an angle&#xA;i1; N is the northern limit, near apogee, S is the southern limit,&#xA;near perigee, and the midpoint of NS is M at an eccentricity e from O.&#xA;M and e are the center of the eccentric and eccentricity projected&#xA;into the line joining the limits. &lt;strong&gt;(2)&lt;/strong&gt; When the center of the&#xA;epicycle is at N or S, it is inclined to the plane of the eccentric in&#xA;the line of sight, with the perigee to the north at N and to the south&#xA;at S, so that the latitude βo at opposition Po is greater than the&#xA;latitude βc at conjunction Pc. It is found from observation that the&#xA;difference between βo and βc is so large that the epicycle is also&#xA;inclined to the plane of the ecliptic by i2 and thus to the plane of&#xA;the eccentric by i1 + i2. &lt;strong&gt;(3)&lt;/strong&gt; When the center of the epicycle is&#xA;at the ascending node or descending node , it lies in the plane of the&#xA;ecliptic so the planet has no latitude wherever it is located. Hence&#xA;as the epicycle moves from the limit to the node, i2 decreases from&#xA;its maximum to zero, and as it moves to the next limit i2 again&#xA;increases to its maximum. Ptolemy treats i1 + i2 as a single&#xA;inclination of the epicycle to the plane of the eccentric. But since&#xA;i1 may be taken as a fixed inclination, holding the epicycle parallel&#xA;to the plane of the ecliptic, leaving i2 alone variable, which we&#xA;believe a clearer way of showing the variable inclination, we have&#xA;divided the inclination of the epicycle into two components, the fixed&#xA;i1 and the variable i2.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/adwVR.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/adwVR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, in my words, naively, we would except that the epicycle should be parallel to the ecliptic (i2=0 fixed). if this is the case, we would expect to see at every point (of the center of the epicycle) on the eccentric that the latitude at the opposition is bigger than the latitude at the conjunction: This is trivial and because at opposition the planet is closer than conjunction. [the paper uses the term &amp;quot;perigee [of the epicycle]&amp;quot; at point (2) (at least this is how I understand it): the meaning is the point of the epicycle the closet to Earth, which the planet occupies at opposition of course.].&lt;/p&gt;&#xA;&lt;p&gt;Now, the thing is that Ptolemy observations (and interpolations) showed that the latitude gap between the opposition and conjunction is bigger than expected - hence Ptolemy gave it an extra push and to make the epicycle even more inclined in respect the the eccentric; but because when the center of epicycle is on the nodes, the planet is always found to have latitude of 0 no matter where it is located on the epicycle it means that at the nodes this extra tilt (i2) is 0. So at the nodes i2=0 but at the limits it received the maximum. So i2 is not fixed and its running along   the eccentric.&lt;/p&gt;&#xA;&lt;p&gt;Now that we understand the model, we can turn back to the question: do we get the maximum latitude exactly at the opposition? In reality the latitude is basically a function of two argument: (1) the proximity to the limit; (2) the distance from Earth. At opposition the argument (2) receive it minimum - hence the contribution to the latitude is maximal, but argument (1) might receive a stronger value before/after the opposition if it becomes closer to the limit. This why it is simple to see that the maximal latitude is not exactly on the opposition (only very close to it). To express that mathematically: &lt;span class=&quot;math-container&quot;&gt;$f(\lambda) = i1* \frac{1}{p}\sin(\alpha) + d(\phi)$&lt;/span&gt;. where &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; is the latitude. &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is the distance from the node (assuming equal speed on the eccentric - which is wrong assumption but close enough to reality). and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is the period of the planet, where Earth period =1; and  where &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; is the angles left for the planet to reach the &amp;quot;perigee of the epicycle&amp;quot;. So &lt;span class=&quot;math-container&quot;&gt;$d(\phi)$&lt;/span&gt; will donate the contribution of the distance to the latitude which is a function of &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; which determined the distance (given the contestant Radii of Earth (Sun for Ptolemy) orb and the planet). When this function gets it maximum for a given location on the eccentric (\alpha); we know that when &lt;span class=&quot;math-container&quot;&gt;$\phi=0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d(\phi)$&lt;/span&gt; receives its maximum values since the distance to Earth is the shortest, so &lt;span class=&quot;math-container&quot;&gt;$d'(\phi)$&lt;/span&gt;=0 when &lt;span class=&quot;math-container&quot;&gt;$\phi=0$&lt;/span&gt;. we are left with &lt;span class=&quot;math-container&quot;&gt;$f'(\lambda) = i1* \frac{1}{p}\cos(\alpha)$&lt;/span&gt; which is not equal to 0 (except at the limit of course) hence we see that when &lt;span class=&quot;math-container&quot;&gt;$\phi=0$&lt;/span&gt; (i.e., at opposition) we are not getting maximal latitude.&lt;/p&gt;&#xA;&lt;p&gt;But in the model of Ptolemy because he had given (wrongly doing so) a tilt for the epicycle with respect to the ecliptic in the direction of the line of sight - it means that when the planet getting closer to Earth on the epicycle (as &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; approaching &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;), it accrues latitude up till &lt;span class=&quot;math-container&quot;&gt;$+i2$&lt;/span&gt; when it reaches the &amp;quot;perigee of the epicycle&amp;quot; at the opposition and &lt;span class=&quot;math-container&quot;&gt;$\phi=0$&lt;/span&gt;. So now the latitude function is as follows: &lt;span class=&quot;math-container&quot;&gt;$f(\lambda) = i1* \frac{1}{p}\sin(\alpha) + d(\phi) + i2*\cos(\phi)$&lt;/span&gt;. i2 itself, as we saw, is a function of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, but it doesn't matter for our analysis , because we see that at opposition when &lt;span class=&quot;math-container&quot;&gt;$\phi=0$&lt;/span&gt; the derivative is the same as before and is not equal to &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. Hence, also in Ptolemy model we should not receive the maximum latitude exactly at opposition, but we can see how it comes closer to the opposition as basically the model of Ptolemy strengthen the influence of &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; on the latitude. Moreover, Ptolemy not merely increase the influence of &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;, it was done at expense of the influence of i1 - because in reality (in terms of Ptolemy) i2=&lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;. so Ptolemy numbers for i1 where smaller than reality for him to keep &lt;span class=&quot;math-container&quot;&gt;$i1+i2$&lt;/span&gt; about right.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;We are left in the Question of why Tycho was surprised. Let's see how Kepler presents the attitude of Tycho (from Ch. 66 of &lt;em&gt;Astronomia Nova&lt;/em&gt; tranlsation William Donhue):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When I later had come to him in Bohemia, and frequently inquired about&#xA;how the latitudes are arranged, he answered ... and recounted many&#xA;other things. Of this present matter, he said most emphatically, ‘this&#xA;is remarkable, that the latitudes reach their maximum before or after&#xA;opposition to the sun.’&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Tycho is reportedly wrote that &amp;quot;The cause of this [large gap between opposition and maximum latitude] needs to be looked into carefully&amp;quot; (Ch. 66 brought by Kepler).&lt;/p&gt;&#xA;&lt;p&gt;I would have liked to give Tycho the credit to be surprised only by the extant of this large gap and not that there is a gap. Maybe it is so, but the account of Kepler in this chapter seems to suggest the expectation of being exactly at the opposition not only by Tycho but by other ancients.&lt;/p&gt;&#xA;" OwnerUserId="14954" LastEditorUserId="14954" LastEditDate="2022-03-25T22:39:52.767" LastActivityDate="2022-03-25T22:39:52.767" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14322" PostTypeId="2" ParentId="14319" CreationDate="2022-03-25T22:29:43.443" Score="0" Body="&lt;p&gt;This may count as a partial answer, I don't claim to have the whole story.&lt;/p&gt;&#xA;&lt;p&gt;The metaphor of vector spaces is currently being included especially among teaching tools to encourage learners and users of linear algebra and matrix problems to think of and visualise (or almost-visualise) the involved matrix and vector objects in an enlarged variety of ways.&lt;/p&gt;&#xA;&lt;p&gt;A notable and much-appreciated exponent of this teaching approach is apparent in &lt;a href=&quot;https://news.mit.edu/2019/gil-strang-still-going-strong-online-and-print-0508&quot; rel=&quot;nofollow noreferrer&quot;&gt;Prof. Gilbert Strang&lt;/a&gt; of MIT. His approach appears especially through his &lt;a href=&quot;https://klein.mit.edu/%7Egs/papers/papers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;textbooks&lt;/a&gt; and lectures especially his introductions to &lt;a href=&quot;https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/&quot; rel=&quot;nofollow noreferrer&quot;&gt;linear algebra&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I've no information about whether Prof. Strang had any close predecessors in this approach or not, but he certainly appears to engage in numerous innovative approaches to the teaching of his subject, and might indeed be at the origin of the tendency noted in the question.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-03-25T22:29:43.443" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14323" PostTypeId="2" ParentId="14319" CreationDate="2022-03-26T04:21:12.037" Score="4" Body="&lt;p&gt;Actually, I think it is not quite the case that what we now think of as &amp;quot;linear algebra&amp;quot; (either terminology or canon of standard results) existed in the late-ish 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Yes, &amp;quot;matrices&amp;quot; were about determinants... Yes, people knew how to solve (finite) linear equations.&lt;/p&gt;&#xA;&lt;p&gt;It seems that the most serious impetus to &amp;quot;linear algebra&amp;quot; (linear operators, eigenvalues...) was from &lt;em&gt;infinite-dimensional&lt;/em&gt; situations, arising in the integral equations' equivalent versions to solving differential equations, as in Hilbert, Schmidt, et al, c. 1895. Yes, Sturm-Liouville results and ideas (c. 1835) were pose-able in linear algebra, but the terminology did not exist then. In fact, their wonderful ideas were not made rigorous (or even made precise) until Steklov and Bocher after 1890.&lt;/p&gt;&#xA;&lt;p&gt;Yes, the idea many people (including myself) had/have of the history of linear algebra is that finite-dimensional stuff came first, and then fancier Hilbert-space stuff, ... but apparently that is not at all true! Crazy! :)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-03-26T04:21:12.037" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14325" PostTypeId="2" ParentId="14298" CreationDate="2022-03-26T23:22:15.827" Score="1" Body="&lt;p&gt;This is not likely the first, but this account of an observation in 1964 (after Paulings Nobel Prize talk) it does address the dodecahedral shape that Pauling is showing.&lt;/p&gt;&#xA;&lt;p&gt;From Besson et al. (2020) &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7412204/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Adenovirus Dodecahedron: Beyond the Platonic Story&lt;/a&gt; Viruses. 2020 Jul; 12(7): 718.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The adenovirus dodecahedrons were first discovered in Sweden in 1964. While purifying soluble antigens produced during an HAdV3 infection of human cells, Erling Norrby observed a homogenous population of particles sedimenting by centrifugation at a rate corresponding to 50–60 S. They appeared as six- or five-pointed ‘stars’ composed of identical tubular, capsomer-like structures, each associated with a thin club-shaped projection &lt;sup&gt;21&lt;/sup&gt;. The overall diameter between the points surrounding the star was assessed between 40 and 50 nm. These points were later attributed to the fiber’s knobs, and the tubular capsomer was ascribed to the penton base forming the core (star) of the dodecahedron. These particles do not contain hexons, but are exclusively made of penton base and fiber proteins.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;21&lt;/sup&gt;Norrby E. The relationship between the soluble antigens and the virion of adenovirus type 3. I. Morphological characteristics. Virology. 1966;28:236–248. doi: 10.1016/0042-6822(66)90148-6.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and later:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The adenovirus dodecahedron was ‘rediscovered’ in 1997 by co-expressing the penton base and the fiber of HAdV3 in insect cells using the baculovirus expression system. Insect cells lysates were then subjected to a sucrose gradient and the heavy fractions were analyzed by negative stain electron microscopy, highlighting spherical particles arranged into regular dodecahedrons by their bases&lt;sup&gt;31&lt;/sup&gt;. These particles display two-fold, three-fold, and five-fold symmetry axes. It is worth noticing that, as expected, the HAdV-2 penton base expressed in a similar way did not form dodecahedrons but only pentameric penton bases, as HAdV-2 does not form dodecahedrons under ‘natural’ conditions. This confirms that not all adenovirus serotypes are able to generate such particles. The recombinant dodecahedron made of 12 pentons (12 penton bases with 12 protruding trimeric fibers) was called the Penton Dodecahedron (Pt-Dd) (Figure 2b).&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;31&lt;/sup&gt; Fender P., Ruigrok R.W., Gout E., Buffet S., Chroboczek J. Adenovirus dodecahedron, a new vector for human gene transfer. Nat. Biotechnol. 1997;15:52–56. doi: 10.1038/nbt0197-52.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/szeXB.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/szeXB.jpg&quot; alt=&quot;Figure 2. Adenovirus dodecahedron formation and internalization.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Figure 2. Adenovirus dodecahedron formation and internalization. (a) Z-series of Hela cells infected for 16 h by wt-HAdV3. Nuclei are stained in red and the penton base is detected in green. The penton base is synthesized in the cytoplasm and transported to the nucleus (yellow results from superposition of the red and green signals) where dodecahedron assembly takes place. (b) Electron microscopy images of the purified adenovirus and (Pt-Dd) ‘penton dodecahedrons’ (bars: 90 and 30 nm, respectively). (c) Hela cells incubated with recombinant Pt-Dd for 1 h. Cell shapes were observed by DIC, nuclei are stained in blue, and Pt-Dd are detected in green.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2022-03-26T23:22:15.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14326" PostTypeId="2" ParentId="14319" CreationDate="2022-03-27T16:31:12.877" Score="3" Body="&lt;p&gt;Like mathematics in general, Linear Algebra is evolving. Vector spaces were introduced by Sylvester in the 19th century and became a core of Linear Algebra University courses by Halmos (&amp;quot;Finite dimensional vector spaces&amp;quot;, 1947) and Gelfand (&amp;quot;Lectures in Linear Algebra&amp;quot;, 1948). Motivation came from physics (quantum mechanics in particular).&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2022-03-27T20:10:40.753" LastActivityDate="2022-03-27T20:10:40.753" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14327" PostTypeId="2" ParentId="14273" CreationDate="2022-03-27T18:45:20.897" Score="5" Body="&lt;p&gt;&amp;quot;I've browsed many math history books&amp;quot; -- I'm not sure what this means. And &amp;quot;I've never read too deep into any single one&amp;quot; is not any more encouraging. A book on the history of mathematics may have a focus: What is the history of the current mathematics -- that is, what led up to it? Maybe it will stop some decades or even a century before &amp;quot;now,&amp;quot; whatever time &amp;quot;now&amp;quot; was when the book was written. Or maybe a book will be about the mathematical activity in human cultures throughout time and the world. A book may focus on a topic or certain topics; &amp;quot;calculus&amp;quot; is likely to be a somewhat popular choice in the current state of affairs.  I have not seen one focused on an imperialistic point of view, that is, how empires, and lets broaden it to include trade, promulgated learning throughout an empire and its trading partners, and whether this was bad or good (no doubt a question with nuanced answers).&lt;/p&gt;&#xA;&lt;p&gt;Of the first kind, which deals with the history of the mathematics we study in universities around the world, the books I've read have been classics: Morris Kline, Dirk Struik, Carl Boyer. In one way they're a bit dissatisfying: they more or less tell the same story. It would be worse, however, if they told vastly different stories.  The main thread of these books traces the development of mathematics from some starting points to Europe from the Renaissance through the 19th century, at which point it begins to take root almost everywhere.  The starting points are not in Europe, but in Egypt, Babylon, Persia, India, China, and Japan.  At some point mathematics starts to take hold in Greece.  From there, the next flourishing is in Alexandria and then to the Arabs.  (Rome, while having a highly educated class, did not contribute much to mathematical research, it seemed.)  From the Arabs, mathematics takes root again in Europe.  China and Japan were somewhat isolated from Europe, and it seems the discoveries of each did not influence each other much.  Or such is the broad arc of the story.  If you're interested in how we arrived at the mathematics that people study through the first or second year an undergraduate program, this is it.  The European flourishing brought modern versions of algebra, analysis, and statistics, which contain the principal subjects studied.  Naturally, the people involved were European.  I'd assume these Europeans were white, if ever I thought about it, but the races of the mathematicians were not mentioned (or I don't remember it).  Most of them were male, which reflects inequities in the society at the time, not bias in the historian; women were mentioned by name, nonetheless, for the mere offense of having done some significant mathematics (a worthy reason, by the way, which the sarcasm is meant to highlight), but only very few were mentioned.  I'm told that even &amp;quot;Men of Mathematics&amp;quot; features a female mathematician (one classic I have not read).  I'd assume (again) that very few women had the opportunity to do significant mathematics, although there is some evidence that it could be done privately and not be published (be &amp;quot;silenced&amp;quot; if you prefer).  It is important to recognize why only few women are mentioned -- and omitting to do so may reflect bias, or at least a blind spot, in the historian -- because historically, and even currently, the lesson drawn has been either that women are incapable or that they should not be allowed to do it, which is wrong.  So wrong that it needs to be pointed out.&lt;/p&gt;&#xA;&lt;p&gt;That &amp;quot;math is primarily the result of individual genius&amp;quot; is an impression one sometimes gets.  Kline's book often shows how an idea evolves due to the work of several people, but I've usually preferred his book for its focus on ideas rather than on people, although it does contain biographical tidbits that the OP complains about. I think an in-depth history is a complicated business, and books with a broad scope like the ones I'm discussing do not take the space to go into the details.  For a in-depth look at, say, Cauchy's contribution to single-variable calculus, see &lt;a href=&quot;https://www.google.com/books/edition/The_Origins_of_Cauchy_s_Rigorous_Calculu/XuFcx-laQmIC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Judith Grabiner, &lt;em&gt;The Origins of Cauchy's Rigorous Calculus&lt;/em&gt;&lt;/a&gt;.  At some length, she details what Cauchy had to work with, how many of the ideas he put together had been thought either clearly or obscurely by others, and why what he did was significant and in what sense original.  She also argues her thesis when it diverges from other historians', and an interested reader could follow up her references and try to sort it out.  Alas, I don't believe any of the historical personages are female (perhaps I don't know everyone's gender) or people of color.  I assume that is historically accurate and that she has not erased them from history.  She does mention Algarotti's &lt;em&gt;Newtonianism for the Ladies&lt;/em&gt;, so there is at least a nod to half the world's population.  I think it shows how difficult it is to write about the history of an idea and satisfy a reader who wants it to be a history of disfranchised segments of the population.&lt;/p&gt;&#xA;&lt;p&gt;If you want a glimpse at how mathematical ideas are built, some original books or papers explain what they are building on. For instance, Lagrange's &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k86263h/f1.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fonctions Analytiques&lt;/a&gt; or Riemann's &lt;a href=&quot;https://www.emis.de/classics/Riemann/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die Darstellbarkeit...&lt;/a&gt;.  The first few pages of each show what the author says he is responding to and building on.&lt;/p&gt;&#xA;&lt;p&gt;If you want a different sort of focus, perhaps you would like Marcia Ascher's &lt;a href=&quot;https://www.google.com/books/edition/Ethnomathematics/wiBBDwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ethnomathematics&lt;/a&gt;&#xA;or other books.  Whether they are historical or anthropological, or what the distinction is, is open to debate (one I'm unqualified to engage in, btw), but they have a non-western focus.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2022-03-28T21:36:32.407" LastActivityDate="2022-03-28T21:36:32.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14328" PostTypeId="1" CreationDate="2022-03-27T20:32:09.557" Score="3" ViewCount="940" Body="&lt;p&gt;I haven't found any explanation for it, and I'm curious.&lt;/p&gt;&#xA;" OwnerUserId="16462" LastEditorUserId="12724" LastEditDate="2022-04-25T06:00:32.140" LastActivityDate="2022-04-26T14:37:24.463" Title="Why is electric potential denoted by $\phi$?" Tags="|physics|notation|electromagnetism|potential|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14330" PostTypeId="1" CreationDate="2022-03-29T18:18:05.377" Score="11" ViewCount="365" Body="&lt;p&gt;I was reading an old paper (specifically, the first appearance of the Pearcey function, &lt;a href=&quot;https://doi.org/10.1080/14786444608561335&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) and I was struck by the beauty of the plots it contains, particularly for a paper from 1945-46:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/G5mmw.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/G5mmw.jpg&quot; width=&quot;450&quot; title=&quot;Contour plots of the Pearcey function&quot;&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Pearcey goes into significant depth of analysis about the integral&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;I(X,Y) = \int_{-\infty}^\infty \exp\mathopen{}\left(i\left(Yt+Xt^2+t^4\right)\right)\mathclose \: \mathrm dt&#xA;,&#xA;$$&lt;/span&gt;&#xA;and he also includes the plots above. He does not go into any detail regarding the numerical computation of the function, but the acknowledgement at the end,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The author is greatly indebted to the staff of the Cambridge University&#xA;differential analyser for collaboration in the computation involved in&#xA;this work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is very clear on what device (the Cambridge differential analyzer) was used for this. I've had a read through &lt;a href=&quot;https://twobithistory.org/2020/04/06/differential-analyzer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this nice blog post&lt;/a&gt; describing the technology, and I feel quite OK in understanding how the technology available in 1945 would be enough to allow for quite accurate calculations of this integral, using mechanical differential-equation integrators to calculate the incomplete &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; integral and therefore its eventual asymptotic value.&lt;/p&gt;&#xA;&lt;p&gt;However, the actual production of the plots in the paper is still murky to me. How does one go from the output of a mechanical differential analyzer to a contour plot of the corresponding function? Would the expectation be that the (human) computer tabulate the function for a dense grid along &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;, from there extract relevant points at which the desired contours were reached, and then draw the diagram by hand from those? Or would it have been possible to use the output of each individual &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; integration to feed back into the machine and use that to make it graph the contours somehow?&lt;/p&gt;&#xA;" OwnerUserId="272" LastEditorUserId="272" LastEditDate="2022-04-03T18:45:41.217" LastActivityDate="2022-04-03T18:45:41.217" Title="How were contour plots of complex functions produced in the days of mechanical differential analyzers?" Tags="|mathematics|computer-science|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14332" PostTypeId="1" AcceptedAnswerId="14336" CreationDate="2022-03-31T15:40:03.867" Score="0" ViewCount="109" Body="&lt;p&gt;I was digging about the discovery of logarithms by John Napier. It is clear that he was looking for a function he could compute (at least approximately) that would map products into sums. He came up with an ingenious idea[1]. Of course, we can't know how he came up with it, but when I was thinking about it, it struck me that he must not have a continuous exponential function to use. Was it the case?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16475" LastActivityDate="2022-04-01T12:59:51.300" Title="Did the logarithm function, with a continuous domain, come before the exponential function?" Tags="|mathematics|logarithm|" AnswerCount="1" CommentCount="4" ClosedDate="2022-04-02T13:47:25.737" ContentLicense="CC BY-SA 4.0" />
  <row Id="14333" PostTypeId="1" CreationDate="2022-03-31T20:14:30.433" Score="2" ViewCount="143" Body="&lt;p&gt;Those with experience may deny it, having suffered too long ago. But it stares you in the face with the somnolent, expressionless eyes of every student being exposed the first time. &lt;strong&gt;Probability notation is &lt;a href=&quot;https://www.johndcook.com/blog/2014/03/04/nomenclatural-abomination/&quot; rel=&quot;nofollow noreferrer&quot;&gt;an abomination&lt;/a&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This difficulty is usually expressed as &lt;a href=&quot;https://www.johndcook.com/blog/2008/02/23/everything-begins-with-p/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; stands for every thing&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; represents four different functions in one equation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$p(θ | x) = p(x | θ) p(θ) / p(x)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If this were &lt;em&gt;actual&lt;/em&gt; mathematical notation (yes, &lt;em&gt;actual&lt;/em&gt;, and I mean that to sting), the things that look like variables would be variables and the things that look like functions would be functions. But none of that is the case. And &lt;a href=&quot;https://en.wikipedia.org/wiki/Operator_overloading&quot; rel=&quot;nofollow noreferrer&quot;&gt;overloading &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&lt;/a&gt; for both probability and conditional probability (which has two parameters) is just so extra.&lt;/p&gt;&#xA;&lt;p&gt;I mean &lt;a href=&quot;https://lingpipe-blog.com/2009/10/13/whats-wrong-with-probability-notation/&quot; rel=&quot;nofollow noreferrer&quot;&gt;it's actually a lot worse than this&lt;/a&gt; but isn't this bad enough?&lt;/p&gt;&#xA;&lt;p&gt;My question is, and it is entirely tendentious, &lt;strong&gt;who is the person to blame for this?&lt;/strong&gt; I want to know their name(s) and pen a strongly worded letter to the New York Times, or find where they live and tape a note to their windshield that says &amp;quot;How Dare You&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But really, &lt;strong&gt;was there a single person&lt;/strong&gt; who started using &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; for absolutely everything in some quasi-mathematical manner. &lt;strong&gt;Or was it a long evolutionary process&lt;/strong&gt;,  first Markov's axioms, then someone used it for conditional probability, and then someone else started using it for whatever? Or something else?&lt;/p&gt;&#xA;" OwnerUserId="8179" LastActivityDate="2022-03-31T20:14:30.433" Title="Where does the abomination that is probability notation come from?" Tags="|notation|statistics|probability|" AnswerCount="0" CommentCount="10" ClosedDate="2022-04-05T14:17:38.923" ContentLicense="CC BY-SA 4.0" />
  <row Id="14336" PostTypeId="2" ParentId="14332" CreationDate="2022-04-01T12:40:56.683" Score="1" Body="&lt;p&gt;I want to thank Dave L Renfro for his &lt;a href=&quot;https://hsm.stackexchange.com/questions/14332/did-the-logarithm-function-with-a-continuous-domain-come-before-the-exponentia?noredirect=1#comment32542_14332&quot;&gt;comment&lt;/a&gt;. After carefully looking at the two first papers in the series &lt;em&gt;History of the exponential and logarithmic concepts&lt;/em&gt; by Florian Cajori, I found my answer.&lt;/p&gt;&#xA;&lt;p&gt;The publication of Napier's work on logarithms was in 1614. By then, we didn't even have an established notation for exponents. People used to talk about &lt;em&gt;geometric progressions&lt;/em&gt;, in words.&lt;/p&gt;&#xA;&lt;p&gt;The notation we nowadays use for powers of a number was introduced by René Descartes' &lt;em&gt;La géométrie&lt;/em&gt; in 1637 (hence, about 23 years after Napier's work). But this was only for natural numbers in the exponent, and Descartes still preferred to use &lt;span class=&quot;math-container&quot;&gt;$aa$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$a^2$&lt;/span&gt; for squares.&lt;/p&gt;&#xA;&lt;p&gt;In 1656, John Walis writes, on his &lt;em&gt;Arithmetica infinitorum&lt;/em&gt;, that the series &lt;span class=&quot;math-container&quot;&gt;$1/\sqrt 1, 1/\sqrt 2, 1/\sqrt 3, \ldots$&lt;/span&gt; has &lt;em&gt;index&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$-1/2$&lt;/span&gt;, while the series &lt;span class=&quot;math-container&quot;&gt;$1, 4, 9, \ldots$&lt;/span&gt; has &lt;em&gt;index&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;. Fractional exponents were explicit in 1676, following current notation, on a letter Newton wrote to H. Oldenburg and then forwarded to Leibniz. This was related to his famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Binomial_theorem#Newton%27s_generalized_binomial_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;binomial theorem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As a final remark, I want to say that what Napier did was, in effect, define an exponential function with a continuous domain. In his work, he describes a point moving with a velocity proportional to the distance remaining in front of it. This is a differential equation (though this is before the development of calculus) whose solution is basically an exponential with base &lt;span class=&quot;math-container&quot;&gt;$e^{-1}$&lt;/span&gt;. He did not think about bases, though. He computed this function approximately, of course, so, in practice, he used an exponential with base &lt;span class=&quot;math-container&quot;&gt;$(1 - 1/10^7)^{10^7} \cong e^{-1}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16475" LastEditorUserId="16475" LastEditDate="2022-04-01T12:59:51.300" LastActivityDate="2022-04-01T12:59:51.300" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14337" PostTypeId="2" ParentId="5754" CreationDate="2022-04-01T13:11:02.890" Score="2" Body="&lt;p&gt;Just a little more than a comment. As regards shape of planetary orbits it is rarely appreciated how much subtle is their difference from perfect (albeit eccentric) circle. While center of Mars orbit is shifted by some 0.14 AU from Sun (fig. #1), semi-minor axis is full 99.6 percent of semi-major axis (fig. #2).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6jmSa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6jmSa.png&quot; alt=&quot;First appoximation to Mars orbit - still a perfect circle&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Fig. 1 First appoximation to Mars orbit - still a perfect circle&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Mow5i.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Mow5i.png&quot; alt=&quot;Subtle difference between circle and ellipse (the circle squeezed in vertical axis)&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Fig. 2 Subtle difference between circle and ellipse (the circle squeezed in vertical axis)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The reason is that the shift is &lt;span class=&quot;math-container&quot;&gt;$e \times a$&lt;/span&gt; (semi-major axis), while semi-minor axis is given by formula &lt;span class=&quot;math-container&quot;&gt;$b^2 = a^2 \times (1 - e^2)$&lt;/span&gt; (note that eccentricity is squared).&lt;/p&gt;&#xA;&lt;p&gt;In other words - the fact that distance of Mars from the Sun changes markedly during its revolution does'n mean that the orbit is noticeably different from circle.&lt;/p&gt;&#xA;" OwnerUserId="9778" LastEditorUserId="9778" LastEditDate="2022-04-15T14:25:30.103" LastActivityDate="2022-04-15T14:25:30.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14338" PostTypeId="1" AcceptedAnswerId="14352" CreationDate="2022-04-01T15:03:28.780" Score="1" ViewCount="79" Body="&lt;p&gt;In the gregorian calendar, we split the time in years, months, weeks and days.&lt;/p&gt;&#xA;&lt;p&gt;A week is the total of seven days.&lt;/p&gt;&#xA;&lt;p&gt;I was looking at the persan calendar, being one of the most precise known calendar. However, on wikipedia (french page), I cannot find any reference to an &amp;quot;equivalent&amp;quot; of gregorians weeks.&lt;/p&gt;&#xA;&lt;p&gt;So, is there any other calendar that splits/split months into a unit of measurement we could consider the equivalent of gregorian weeks ?&lt;/p&gt;&#xA;" OwnerUserId="16479" LastActivityDate="2022-04-04T10:56:30.413" Title="Is there another calendar with the equivalent of gregorians weeks?" Tags="|time|calendar|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14339" PostTypeId="1" CreationDate="2022-04-01T16:28:15.223" Score="8" ViewCount="328" Body="&lt;p&gt;According to the MacTutor essay &amp;quot;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Thompson_irrationals/&quot; rel=&quot;nofollow noreferrer&quot;&gt;D'Arcy Thompson on Greek irrationals&lt;/a&gt;&amp;quot; (which I take to be a version of Thompson's original essay whose only liberty with the original text is giving English translations where Thompson gives Greek words and phrases), Proclus, following Adrastus, asserted that &lt;span class=&quot;math-container&quot;&gt;$2+8+50+288+\dots$&lt;/span&gt; (the sum of twice the squares of the &amp;quot;side numbers&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$1,2,5,12,\dots$&lt;/span&gt;) equals &lt;span class=&quot;math-container&quot;&gt;$1+9+49+289+\dots$&lt;/span&gt; (the sum of the squares of the &amp;quot;diagonal numbers&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$1,3,7,17,\dots$&lt;/span&gt;). Here is the relevant excerpt:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The table of side and diagonal numbers has many other properties. For instance, as Proclus tells us, the sum of the squares of two adjacent diagonals = twice the sum of the squares on the two corresponding sides: e.g. &lt;span class=&quot;math-container&quot;&gt;$3^2+7^2=2(2^2+5^2)$&lt;/span&gt;. And, in Chapter xxiii he shows, following Adrastus, that the sum of the squares of 'all' the diagonals is equal to twice the sum of the squares of 'all' the sides.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I haven't been able to find the text that Thompson is referring to here. In any case, I'm curious whether the original text seems to be an early attempt at manipulating divergent series.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, under modern approaches to regularizing divergent series, the equality asserted by Proclus fails: the two divergent sums differ by &lt;span class=&quot;math-container&quot;&gt;$\frac12$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;[Note: This question was originally posted at &lt;a href=&quot;https://mathoverflow.net/questions/419444/adrastus-proclus-and-2850288-vs-1949289&quot;&gt;MathOverflow&lt;/a&gt;, but it seems more appropriate here.]&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="2531" LastEditDate="2022-04-02T07:14:59.247" LastActivityDate="2022-04-02T07:14:59.247" Title="Adrastus, Proclus, and 2+8+50+288+… versus 1+9+49+289+…" Tags="|ancient-greece|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14341" PostTypeId="2" ParentId="14339" CreationDate="2022-04-01T21:37:18.243" Score="7" Body="&lt;p&gt;As nwr &lt;a href=&quot;https://hsm.stackexchange.com/questions/14339/adrastus-proclus-and-2850288-versus-1949289#comment32558_14339&quot;&gt;commented above&lt;/a&gt;, the original text by Proclus is his commentary on Plato's &lt;i&gt;Republic&lt;/i&gt;, and Cambridge University Press is in the process of publishing an &lt;a href=&quot;https://www.cambridge.org/core/series/proclus-commentary-on-platos-republic/8332BE7C7CD0B4599A49CD091446CFB3&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt;. Unfortunately, I think the relevant section is in Volume 2, which has not been published yet. I don't read Greek, but maybe someone who does can find the right section in the &lt;a href=&quot;https://archive.org/details/proclidiadochiin02procuoft/page/24/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Greek text&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, it may not be necessary to dig up the exact section from Proclus. D'Arcy Thompson cites an earlier paper by A. E. Taylor, &lt;a href=&quot;https://www.jstor.org/stable/2249618&quot; rel=&quot;noreferrer&quot;&gt;Forms and Numbers: A Study in Platonic Metaphysics&lt;/a&gt;.  Taylor notes that the convergents of the continued fraction of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; are&#xA;&lt;span class=&quot;math-container&quot;&gt;$${1\over 1}, {3\over 2}, {7 \over 5}, {17 \over 12}, {41 \over 29}, {99 \over 70}, \ldots.$$&lt;/span&gt;&#xA;The numerators are the &amp;quot;diagonal numbers&amp;quot; and the denominators are the &amp;quot;side numbers.&amp;quot; I would guess that the claim that &amp;quot;the sum of the squares of 'all' the diagonals is equal to twice the sum of the squares of 'all' the sides&amp;quot; is probably just the claim (in modern language) that the above sequence of fractions converges to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;.  In particular, it seems unlikely to me that divergent series are being considered here.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2022-04-01T21:37:18.243" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14344" PostTypeId="2" ParentId="8366" CreationDate="2022-04-02T07:37:04.920" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://www.wikiwand.com/en/Paul_Erd%C5%91s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Erdős&lt;/a&gt;, mathematician, certainly checks off a lot of boxes.&#xA;Never married, and he mainly traveled around with his mom.&#xA;He traveled around with his suitcase, staying at other researchers homes, and collaborated for a few months - rinse and repeat.&lt;/p&gt;&#xA;&lt;p&gt;Oh, and he did amphetamines.&lt;/p&gt;&#xA;" OwnerUserId="16481" LastActivityDate="2022-04-02T07:37:04.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14345" PostTypeId="1" CreationDate="2022-04-02T14:45:34.827" Score="6" ViewCount="256" Body="&lt;p&gt;According to Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_quantum_computing_and_communication&quot; rel=&quot;nofollow noreferrer&quot;&gt;timelines&lt;/a&gt;, quantum computing may have had some inceptions as early as the late 60's, when &lt;a href=&quot;https://en.wikipedia.org/wiki/Stephen_Wiesner&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stephen Wiesner&lt;/a&gt; invented conjugate coding.&lt;/p&gt;&#xA;&lt;p&gt;Around the early 80's, Wiesner's ideas inspired Charles Bennett and Gilles Brassard to hypothesize a cryptographic scheme - now called &lt;a href=&quot;https://en.wikipedia.org/wiki/BB84&quot; rel=&quot;nofollow noreferrer&quot;&gt;BB84&lt;/a&gt; - to use quantum mechanics to enable information-theoretically secure cryptographic communication.&lt;/p&gt;&#xA;&lt;p&gt;I sense that in the mid 80's, quantum key distribution was at best somewhat well-received, but more likely mostly ignored.  Nonetheless researchers really began to take notice and appreciation when &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00191318&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bennett, et al.&lt;/a&gt; built a working prototype machine that transmitted and received a plurality of qubits, encoded in the polarity of photons, over a short distance  to implement the BB84 scheme:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sMt93.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sMt93.jpg&quot; alt=&quot;First BB84 Machine&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recall seeing the prototype in my copy of &lt;em&gt;Scientific American&lt;/em&gt; in 1992.  From Brassard's &lt;a href=&quot;https://arxiv.org/abs/quant-ph/0604072&quot; rel=&quot;nofollow noreferrer&quot;&gt;remembrances&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Essentially without any special budget allocated to the project, we were able, in late October 1989, to establish history’s first secret quantum transmission, over a staggering distance of 32.5 centimetres...!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clearly this prototype wasn't practical or useful, but I'd posit that the theoretical and experimental researchers in the then nascent field of quantum computing/quantum information science were inspired not just by the theoretical work on BB84 but by images and descriptions of the above actual, physical tabletop device, as bulky and noisy as it was.&lt;/p&gt;&#xA;&lt;p&gt;Based on the &lt;em&gt;SciAm&lt;/em&gt; article the device was built at IBM Yorktown Heights (where Bennett had worked since '74).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As to my question, &lt;em&gt;&lt;strong&gt;could the prototype still exist anywhere&lt;/strong&gt;&lt;/em&gt;?  Or put in a basement in IBM's campus and forgotten about, or was it more than likely scavenged for parts?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Such a prototype might be worth saving, at the very least as a bit of a relic to the field of quantum information...&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt; The &lt;em&gt;SciAm&lt;/em&gt; article was in '92.  Around then, Vazirani gave a famous lecture at Bell Labs on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bernstein%E2%80%93Vazirani_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernstein-Vazirani algorithm&lt;/a&gt;, with Shor (and maybe Simon?) in the audience.  We also have the &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantum_teleportation&quot; rel=&quot;nofollow noreferrer&quot;&gt;quantum teleportation protocol&lt;/a&gt; from '93, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elitzur-Vaidman bomb tester&lt;/a&gt; from then as well...&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="9482" LastEditorUserId="9482" LastEditDate="2023-03-18T15:15:18.633" LastActivityDate="2023-09-25T12:53:51.200" Title="Does the prototype BB84 quantum cryptography machine still exist?" Tags="|quantum-mechanics|cryptology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14346" PostTypeId="1" CreationDate="2022-04-02T14:48:42.190" Score="2" ViewCount="86" Body="&lt;p&gt;I am working on a set of problems with historic context, and I do not seem to be able to retrack, why I consider this problem historic:&lt;/p&gt;&#xA;&lt;p&gt;Three dice are thrown each round. The question is, how often does one have to throw the three dice until each of the dice has shown six at least once.&lt;/p&gt;&#xA;&lt;p&gt;I do know the answer to the problem and how it is derived, but I do not find any references to it. I read the 1708 book by Bernoulli today, and he is commenting on several problems by Huygens that are similar. For some reason I believe the problem above was by Moivre, but I can not find any references. I do know, that Moivre knew about the inclusion-exclusion formula which is used here.&lt;/p&gt;&#xA;&lt;p&gt;Maybe someone has heard of this type of problem? Some people have pointed out similarities to the coupon collector, but i would like to trace back this specific problem.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: The expected time until each die has shown a six at least once is given by the following formula. Let &lt;span class=&quot;math-container&quot;&gt;$W_i\sim Geo(p)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$p=1/6$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$i=1,...n$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$T_n$&lt;/span&gt; be the time until all dice have shown a six at least once, then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbb E(T_n)=\mathbb E(\max_{i}W_i)=\sum_{k=1}^n\binom nk\frac{(-1)^{k+1}}{1-(1-p)^k}\sim -\frac{\ln(n+1)+\gamma_E}{\ln(1-p)}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="16482" LastEditorUserId="16482" LastEditDate="2022-04-03T15:04:57.467" LastActivityDate="2022-04-03T15:04:57.467" Title="Origin of a problem with three dice" Tags="|probability|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14347" PostTypeId="1" CreationDate="2022-04-03T07:49:34.590" Score="1" ViewCount="145" Body="&lt;p&gt;Newton's theory of gravitation is expressed as a mathematical relationship between distances and masses. It can be deduced from Kepler's equations (which are also mathematical).&lt;/p&gt;&#xA;&lt;p&gt;Are there even older physical theories formulated by equations?&lt;/p&gt;&#xA;" OwnerDisplayName="user16484" LastActivityDate="2022-04-03T12:50:58.250" Title="What was the first physical theory to be mathematically expressed?" Tags="|mathematical-physics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14348" PostTypeId="2" ParentId="14347" CreationDate="2022-04-03T08:54:55.203" Score="1" Body="&lt;p&gt;The &lt;em&gt;Almagest&lt;/em&gt;, written in the second century contains a mathematical description of the Universe (and, in particular, of the solar system), which is expressed mathematically. Its contents were almost certainly already known to (and copied from) Hipparchus of Nicaea, who lived in he second century BC. In fact, the kind of research that led to this began already at the time of Plato (fifth century BC).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2022-04-03T08:54:55.203" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14349" PostTypeId="1" AcceptedAnswerId="14365" CreationDate="2022-04-03T09:13:59.223" Score="5" ViewCount="144" Body="&lt;p&gt;I know that Frobenius is the one who proved the Cayley-Hamilton theorem in all its generality. However, between Cayley and Hamilton, who did first work on the subject?&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In English: &lt;a href=&quot;https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem&quot; rel=&quot;noreferrer&quot;&gt;Cayley–Hamilton theorem&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;In Italian: &lt;a href=&quot;https://it.wikipedia.org/wiki/Teorema_di_Hamilton-Cayley&quot; rel=&quot;noreferrer&quot;&gt;Teorema di Hamilton-Cayley&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I wonder because I hesitate between quoting it as Cayley-Hamilton-Frobenius and Hamilton-Cayley-Frobenius in my dissertation. Both forms can be found in textbooks.&lt;/p&gt;&#xA;" OwnerDisplayName="user16484" LastEditorUserId="4251" LastEditDate="2022-04-03T09:57:03.227" LastActivityDate="2022-04-08T14:40:22.230" Title="Who, between Cayley and Hamilton, first worked on the theorem that bears their name?" Tags="|mathematics|mathematicians|terminology|linear-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14350" PostTypeId="1" CreationDate="2022-04-03T12:06:34.693" Score="2" ViewCount="105" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ultraproduct#%C5%81o%C5%9B%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Łoś's theorem&lt;/a&gt; is a fundamental theorem in model theory (a branch of mathematical logic).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Historical question&lt;/strong&gt;: What was Łoś's original motivation to define ultraproducts and prove Łoś's theorem? Which kind of problems let to its discovery?&lt;/p&gt;&#xA;&lt;p&gt;Modern applications include an elegant proof of the compactness theorem and the construction of saturated models. However, often modern applications do not coincide with the original motivation that led to its discovery.&lt;/p&gt;&#xA;&lt;p&gt;Apparently &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S0049237X09703064&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; is Łoś's original paper. In their model theory book, Chang and Keisler write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The ultraproduct construction goes back to the work of Skolem (1934).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it seems &lt;a href=&quot;http://matwbn.icm.edu.pl/ksiazki/fm/fm23/fm23115.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; is an important precursor to Łoś's construction. Though Łoś doesn't cite it, but maybe he cites papers that cite it, I don't know.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Reformulation of the question&lt;/strong&gt;: If somebody here is able to read French and German, or already knows about the history of ultraproducts, can he summarize the context in which Łoś's and Skolem's paper were written?&lt;/p&gt;&#xA;&lt;p&gt;The language of Skolem's paper is so far removed from modern logic that I can't make head nor tail of it. His title translates to &amp;quot;On the non-characterizability of number sequence via finite or countably infinite many statements only containing number variables&amp;quot;. He talks a lot about &amp;quot;arithmetic functions&amp;quot; -- what has this to do with model theory?&lt;/p&gt;&#xA;&lt;p&gt;It's quite fascinating to learn about the history of some fundamental mathematical idea.&lt;/p&gt;&#xA;" OwnerUserId="16485" LastActivityDate="2022-04-03T12:06:34.693" Title="Context of the discovery of ultraproducts" Tags="|mathematics|terminology|discoveries|mathematical-logic|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14351" PostTypeId="2" ParentId="14347" CreationDate="2022-04-03T12:37:40.790" Score="4" Body="&lt;p&gt;The first physical theory which was mathematically expressed is&#xA;the Pythagorean theory of music. According to the legend, Pythagoras himself&#xA;discovered the relation between musical intervals and the ratios of lengths&#xA;of strings. (Based on this discovery, Pythagoras jumped to the conclusion that &amp;quot;Numbers rule the world&amp;quot;, so mathematical physics was born).&#xA;This seems to be the earliest quantitative discovery of physics.&#xA;Based on this, a sophisticated theory of music was developed, one can read about it for example in the book:&lt;/p&gt;&#xA;&lt;p&gt;B. L. van der Waerden, Science awakening. Mathemarics of ancient Egypt, Babylon and Greece, Appendix: Pythagorean teaching on harmony.&#xA;(Was earlier published as a separate paper in Math. Ann., 118 (1941)).&lt;/p&gt;&#xA;&lt;p&gt;Remark. This competes with Babylonian mathematical astronomy, but one can argue whether Babylonian astronomy belongs to physics or not. Anyway, this discovery is more ancient than Greek astronomy.&lt;/p&gt;&#xA;&lt;p&gt;Remark 2. Since Pythagoreans was a secret society, the exact details of this discovery and its date are not known. But surely this predates the first well-documented discoveries in physics which are due to Archimedes (floating bodies, the rues of the lever etc.)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-04-03T12:50:58.250" LastActivityDate="2022-04-03T12:50:58.250" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14352" PostTypeId="2" ParentId="14338" CreationDate="2022-04-04T10:56:30.413" Score="3" Body="&lt;p&gt;The French Republican Calendar, actually used for a time in France... &lt;a href=&quot;https://en.wikipedia.org/wiki/French_Republican_calendar&quot; rel=&quot;nofollow noreferrer&quot;&gt;LINK&lt;/a&gt;&lt;br /&gt;&#xA;The weeks were 10 days each, called &amp;quot;décades&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-04-04T10:56:30.413" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14353" PostTypeId="1" AcceptedAnswerId="14357" CreationDate="2022-04-05T11:48:58.140" Score="1" ViewCount="219" Body="&lt;p&gt;I think Copernicus and his contemporaries were modeling based on some data.&lt;/p&gt;&#xA;&lt;p&gt;What data was Copernicus using and who created it?&lt;/p&gt;&#xA;" OwnerDisplayName="user16492" LastActivityDate="2022-04-05T12:49:21.810" Title="What data did Copernicus use to construct his heliocentric model?" Tags="|physics|astronomy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14354" PostTypeId="2" ParentId="14353" CreationDate="2022-04-05T12:27:48.100" Score="0" Body="&lt;p&gt;Copernicus was trying to improve the Ptolemy system. I will use modernized language to describe this.&lt;/p&gt;&#xA;&lt;p&gt;In Ptolemy system,&#xA;the longitude of a planet is given by a function &lt;span class=&quot;math-container&quot;&gt;$\lambda(t)=mt+$&lt;/span&gt;(other smaller terms), where &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the &amp;quot;mean motion&amp;quot; related to the period,&#xA;&lt;span class=&quot;math-container&quot;&gt;$T=2\pi/m$&lt;/span&gt;. He noticed that the second largest term depends on the elongation of the planet from the Sun. So the motion of the Sun is somehow&#xA;incorporated to motion of all planets. Ptolemy's system does not explain this. To explain this dependence he conjectured that the planets really rotate around the Sun, and thus obtained a simpler description of their motion. This still leaves the choice&#xA;whether Sun rotates about Earth (Brahe system) or vise versa (Copernicus). Observations known at that time cannot distinguish&#xA;one possibility from another. So this was a guess, based on&#xA;&amp;quot;physical reasons'': it was already known that the Sun must be much larger than the planets, including Earth.&#xA;This was his principal&#xA;innovation. But he made many other changes to Ptolemy system, some of them&#xA;were improvements  (from the modern point of view), others not.&lt;/p&gt;&#xA;&lt;p&gt;Speaking of the data used for determining parameters of the system, he used the same data that Ptolemy used, mainly observations of conjunctions and oppositions, besides the periods.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-04-05T12:34:06.870" LastActivityDate="2022-04-05T12:34:06.870" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14355" PostTypeId="1" CreationDate="2022-04-05T12:42:13.943" Score="4" ViewCount="683" Body="&lt;p&gt;Back in 1968, &lt;a href=&quot;https://en.wikipedia.org/wiki/James_M._Bardeen&quot; rel=&quot;nofollow noreferrer&quot;&gt;James M. Bardeen&lt;/a&gt; (son of John Bardeen) gave a talk at the GR5 (5th international conference on gravitation and the theory of relativity), in which he presented a slight modification of the Schwarzschild metric, probably the first example of a regular spacetime metric.&lt;/p&gt;&#xA;&lt;p&gt;This work is quite often cited and some basic, technical info can be traced all over the internet, e.g.&lt;/p&gt;&#xA;&lt;p&gt;J.M. Bardeen: &lt;em&gt;Non-singular general-relativistic gravitational collapse&lt;/em&gt;, Proceedings of the International Conference GR5, Tbilisi, U.S.S.R. (1968), p. 174&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholar.google.com/scholar_lookup?title=Proceedings%20of%20the%20International%20Conference%20GR5&amp;amp;publication_year=1968&amp;amp;author=J.M.%20Bardeen&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholar.google.com/scholar_lookup?title=Proceedings%20of%20the%20International%20Conference%20GR5&amp;amp;publication_year=1968&amp;amp;author=J.M.%20Bardeen&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.scopus.com/record/display.uri?eid=2-s2.0-85050674326&amp;amp;origin=inward&amp;amp;txGid=b29b1e1ac4ac0c72d4c00bc10b810ef1&amp;amp;featureToggles=FEATURE_NEW_DOC_DETAILS_EXPORT:1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scopus.com/record/display.uri?eid=2-s2.0-85050674326&amp;amp;origin=inward&amp;amp;txGid=b29b1e1ac4ac0c72d4c00bc10b810ef1&amp;amp;featureToggles=FEATURE_NEW_DOC_DETAILS_EXPORT:1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nonsingular_black_hole_models&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Nonsingular_black_hole_models&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, according to the official conference page,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.isgrg.org/pastconfs.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.isgrg.org/pastconfs.php&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;no official proceedings were printed.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is there any unofficial, handwritten or printed, trace of the Bardeen's talk?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Update (April 12th, 2022): I have sent a mail to Prof. Bardeen one week ago, asking for any information about this talk/paper, but unfortunately I haven't got any reply so far.&lt;/p&gt;&#xA;&lt;p&gt;Update (April 22nd, 2022): I still haven't received any reply from Prof. Bardeen, but I've finally managed to locate physical copies of the GR5 Proceedings: one in The Library of Congress,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://catalog.loc.gov/vwebv/holdingsInfo?searchId=9651&amp;amp;recCount=25&amp;amp;recPointer=0&amp;amp;bibId=1346182&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://catalog.loc.gov/vwebv/holdingsInfo?searchId=9651&amp;amp;recCount=25&amp;amp;recPointer=0&amp;amp;bibId=1346182&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;and another one at the University of Vienna,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://usearch.univie.ac.at/primo-explore/fulldisplay?docid=UWI_alma21243303620003332&amp;amp;context=L&amp;amp;vid=UWI&amp;amp;lang=en_US&amp;amp;search_scope=UWI_UBBestand&amp;amp;adaptor=Local%20Search%20Engine&amp;amp;tab=default_tab&amp;amp;query=any,contains,International%20Conference%20on%20Gravitation%20and%20the%20Theory%20of%20Relativity&amp;amp;mode=basic&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://usearch.univie.ac.at/primo-explore/fulldisplay?docid=UWI_alma21243303620003332&amp;amp;context=L&amp;amp;vid=UWI&amp;amp;lang=en_US&amp;amp;search_scope=UWI_UBBestand&amp;amp;adaptor=Local%20Search%20Engine&amp;amp;tab=default_tab&amp;amp;query=any,contains,International%20Conference%20on%20Gravitation%20and%20the%20Theory%20of%20Relativity&amp;amp;mode=basic&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10760" LastEditorUserId="6609" LastEditDate="2024-04-03T09:22:06.287" LastActivityDate="2024-04-03T09:22:06.287" Title="James M. Bardeen's missing talk" Tags="|physics|relativity-theory|gravity|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14356" PostTypeId="1" AcceptedAnswerId="14361" CreationDate="2022-04-05T12:44:49.570" Score="2" ViewCount="211" Body="&lt;p&gt;In the past a lot of math was motivated by practical applications from real life; that's how geometry started for example. Some other areas were developed when trying to solve problems that already existed in math, say groups via Galois theory for roots of a polynomial.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if any math theory started somewhat out of the blue, say someone just decided to study some objects and their relations out of pure curiosity, even though those things were not really related to the existing math at that moment.&lt;/p&gt;&#xA;&lt;p&gt;As an example, a person just might have thought whether one can write 9 first integers in a square such that every row and column sum up to the same number, and that's how the mathematics of magic squares happened to start.&lt;/p&gt;&#xA;" OwnerUserId="16493" LastEditorUserId="12330" LastEditDate="2022-04-05T19:07:39.443" LastActivityDate="2022-04-06T01:54:44.043" Title="A randomly started branch of math" Tags="|mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14357" PostTypeId="2" ParentId="14353" CreationDate="2022-04-05T12:49:21.810" Score="2" Body="&lt;p&gt;There's a notable problem of shortage of historical evidence for the way in which Copernicus came to his theories.&lt;/p&gt;&#xA;&lt;p&gt;His earliest work that described his heliocentric concept is known as the 'Commentariolus', and in it he &amp;quot;says next to nothing about how he arrived at his new theories&amp;quot; -- see N M Swerdlow's (1973) paper &lt;a href=&quot;https://www.jstor.org/stable/986461&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Derivation and First Draft of Copernicus's Planetary Theory: A Translation of the Commentariolus with Commentary&amp;quot;&lt;/a&gt;. The 1973 paper is available from Jstor and is still a good source along with the 1984 book &amp;quot;Mathematical Astronomy in Copernicus’s De Revolutionibus&amp;quot; by N M Swerdlow and O Neugebauer (publ. Springer).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-04-05T12:49:21.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14358" PostTypeId="1" CreationDate="2022-04-05T19:02:32.227" Score="2" ViewCount="97" Body="&lt;p&gt;I have an equivalent definition of finite and countable sets that seems 'less' based on set theory than the usual definitions.  I am wondering if anyone has encountered these definitions anywhere, esp. in the pre-set theoretic era.&lt;/p&gt;&#xA;&lt;p&gt;First of all, set theory provides the definition of &lt;em&gt;countable set&lt;/em&gt; (bijection to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{N}$&lt;/span&gt;) and &lt;em&gt;finite set&lt;/em&gt; (bijection to &lt;span class=&quot;math-container&quot;&gt;$\{0, 1, \dots, k\}$&lt;/span&gt; for some &lt;span class=&quot;math-container&quot;&gt;$k\in \mathbb{N}$&lt;/span&gt;.  One can also define &lt;em&gt;Dedekind-finite&lt;/em&gt;, which is equivalent to 'finite' assuming the Axiom of Choice.&lt;/p&gt;&#xA;&lt;p&gt;Secondly, the following definitions (restricted to subsets of the reals for convenience) are equivalent to the set-theoretical ones over a strong enough logical system.&lt;/p&gt;&#xA;&lt;p&gt;A set &lt;span class=&quot;math-container&quot;&gt;$A\subset \mathbb{R}$&lt;/span&gt; is &lt;em&gt;finite&lt;/em&gt; if there is &lt;span class=&quot;math-container&quot;&gt;$N\in \mathbb{N}$&lt;/span&gt; such that any finite sequence &lt;span class=&quot;math-container&quot;&gt;$(x_0, \dots, x_N)$&lt;/span&gt; of real numbers, there is some &lt;span class=&quot;math-container&quot;&gt;$i\leq N$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$x_i\not \in A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;*) A set &lt;span class=&quot;math-container&quot;&gt;$A\subset \mathbb{R}$&lt;/span&gt; is &lt;em&gt;countable&lt;/em&gt; if there is a sequence &lt;span class=&quot;math-container&quot;&gt;$(A_n)_{n\in \mathbb{N}}$&lt;/span&gt; of finite (as in the previous paragraph) sets &lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$A=\cup_{n\in \mathbb{N}}A_n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;**) Note that the definition of 'finite' is not circular as finite sequences of reals are readily coded as real numbers.&lt;/p&gt;&#xA;&lt;p&gt;I repeat my question: has anyone encountered *) or **), esp. in the pre-set theoretic era?&lt;/p&gt;&#xA;" OwnerUserId="15772" LastActivityDate="2022-04-05T19:02:32.227" Title="The definition of 'countable' and 'finite' set" Tags="|mathematical-logic|set-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14360" PostTypeId="1" CreationDate="2022-04-05T19:54:57.673" Score="1" ViewCount="142" Body="&lt;p&gt;I am interested in a book that has all the fragments of pre-socratics but the notes etc. in the translation are from a mathematician or physicist relating with how each point affected the evolution of mathematics or physics. Case in point Zeno's arguments.&lt;br /&gt;&#xA;Is there such an edition?&lt;/p&gt;&#xA;" OwnerUserId="14662" LastEditorUserId="12330" LastEditDate="2022-04-05T20:50:41.590" LastActivityDate="2022-04-05T20:50:41.590" Title="Presocratics from the view of mathematicians and physicists" Tags="|mathematicians|philosophy-of-science|physicists|book-recommendation|" AnswerCount="0" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="14361" PostTypeId="2" ParentId="14356" CreationDate="2022-04-06T00:56:29.290" Score="7" Body="&lt;p&gt;What you describe is closely related to what is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Recreational_mathematics&quot; rel=&quot;noreferrer&quot;&gt;recreational mathematics&lt;/a&gt;.  One could make the case that games, puzzles, and doing things just for fun are just as much a part of &amp;quot;real life&amp;quot; as anything else, but when people say &amp;quot;practical applications from real life,&amp;quot; they usually tacitly exclude recreation.&lt;/p&gt;&#xA;&lt;p&gt;It is not easy to come up with mathematical problems that are neither &amp;quot;too hard&amp;quot; nor &amp;quot;too easy.&amp;quot;  Recreational mathematics has a tendency to fall into one of these two categories, and hence rarely develops into what we would call a &amp;quot;branch of math.&amp;quot;  For example, the question of the existence of &lt;a href=&quot;https://en.wikipedia.org/wiki/Lychrel_number&quot; rel=&quot;noreferrer&quot;&gt;Lychrel numbers&lt;/a&gt; was probably first &amp;quot;randomly&amp;quot; asked out of curiosity by someone with no practical application in mind.  Although there has been quite a bit of effort devoted to Lychrel numbers, the subject is &amp;quot;too hard&amp;quot; in the sense that it seems difficult if not impossible to make significant theoretical progress on it.  So Lychrel numbers are not what one would call a &amp;quot;branch of math.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Surreal_number&quot; rel=&quot;noreferrer&quot;&gt;surreal numbers&lt;/a&gt; are perhaps an exception in that they originated in recreational study of the game of Go, but have turned out to be a rich subject of mathematical study.  Origami has its roots in recreation, or at least in art, and its mathematical study has turned out to be surprisingly rich, with applications in engineering.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;b&gt;EDIT:&lt;/b&gt; Another interesting case study is graph theory.  Its origins can be traced to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg&quot; rel=&quot;noreferrer&quot;&gt;seven bridges of Königsberg&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Four_color_theorem&quot; rel=&quot;noreferrer&quot;&gt;four-color problem&lt;/a&gt;—questions that were driven primarily by curiosity rather than practical necessity.  There is no doubt that graph theory is now a major branch of mathematics.  However, one could argue that its development into a major branch of mathematics was driven by the rise of computers and the many applications of graph theory to computer science and electrical engineering.&lt;/p&gt;&#xA;&lt;p&gt;It's also worth mentioning that even in well-established areas of mathematics, new subfields often arise not because of perceived applications, but because of questions that mathematicians find interesting to study.  If you're looking for examples of areas of mathematics that were born from &amp;quot;curiosity-driven&amp;quot; rather than &amp;quot;applications-driven&amp;quot; reasons, then there are too many to list.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2022-04-06T01:54:44.043" LastActivityDate="2022-04-06T01:54:44.043" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14362" PostTypeId="1" CreationDate="2022-04-06T20:05:56.227" Score="2" ViewCount="221" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Exponential_integral#Convergent_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Wikipedia&lt;/a&gt;, the following series for the exponential integral&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\operatorname{Ei}(x) = \gamma \ln x + \exp{\frac{x}{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^n}{n! 2^{n-1}} \sum_{k=0}^{\lfloor \frac{n-1}{2} \rfloor} \frac{1}{2k+1}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;was found by Ramanujan. Was this among the results found in &lt;a href=&quot;https://en.wikipedia.org/wiki/Srinivasa_Ramanujan#Ramanujan%27s_notebooks&quot; rel=&quot;nofollow noreferrer&quot;&gt;his notebooks&lt;/a&gt; by either Littlewood or Hardy?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2022-04-10T15:21:32.083" Title="Where did we find Ramanujan's series for the exponential integral?" Tags="|mathematics|mathematicians|ramanujan|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14363" PostTypeId="2" ParentId="14362" CreationDate="2022-04-07T23:28:12.687" Score="6" Body="&lt;p&gt;A good modern place to look is in Bruce Berndt's 1994 &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0387941096&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ramanujan's Notebooks, Part IV&lt;/a&gt;, especially in Chapter 24, &amp;quot;Ramanujan's Theory of Prime Numbers&amp;quot;, especially p.130, for an equivalent form of the result in question.  Berndt's introduction seems to say (on page 4) that these results are in R's Notebooks 2 and 3 and in the letters he wrote to Hardy before he came to England, and refers to pp. xxiii, xxvii, 349, 351, and 352 of the Tata edition of R's &lt;em&gt;Collected Papers&lt;/em&gt;. And on his p.130 Berndt cites page 323 of the Tata edition of R's notebooks.  But I cannot check Berndt's references just now, so this answer is only a tentative answer.&lt;/p&gt;&#xA;&lt;p&gt;I was led to Berndt by the entry for &lt;a href=&quot;https://mathworld.wolfram.com/LogarithmicIntegral.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;logarithmic integral&lt;/a&gt; in &lt;em&gt;MathWorld&lt;/em&gt;, which cites pp.126-131 of Berndt.  The corresponding Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Logarithmic_integral_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; for the logarithmic integral cites the &lt;em&gt;MathWorld&lt;/em&gt; one  as source for the Ramanujan  attribution. The Wikipedia article for the exponential integral, which is the one cited by the OP, does not.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2022-04-10T15:21:32.083" LastActivityDate="2022-04-10T15:21:32.083" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14365" PostTypeId="2" ParentId="14349" CreationDate="2022-04-08T14:40:22.230" Score="4" Body="&lt;p&gt;To avoid an opinion-based answer, one should focus on the first paper on the subject, (1858), by Arthur Cayley, &amp;quot;A Memoir on the Theory of Matrices&amp;quot;, &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1858.0002&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Philos Trans&lt;/em&gt; &lt;strong&gt;148&lt;/strong&gt;&lt;/a&gt;, preceding Hamilton's 1864 paper (strictly on quaternions!).&lt;/p&gt;&#xA;&lt;p&gt;In it, Cayley, a major intuitive genius of the 19th century, basically invents and &amp;quot;takes ownership of&amp;quot; matrices. At the bottom of the very first page, he gloats about his&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;[...] remarkable theorem, that any matrix whatever satisfies an algebraic equation of its own order, the coefficient of the highest power being unity, and those of the other powers functions of the terms of the matrix, the last coefficient being in fact the determinant; the rule of the formation of this equation may be stated in the following condensed form, which will be intelligible after a perusal of the memoir, viz. the determinant, formed by the matrix diminished by the matrix considered as a single quantity involving the matrix unity, will be equal to zero.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In his superb pedagogical way, in paragraphs 21, 22, 23, he states it and illustrates it for 2x2 and 3x3 matrices, and adumbrates the logic and the concepts involved in the general case.&lt;/p&gt;&#xA;&lt;p&gt;A Cayley and his intellectual twin, J Sylvester, (like Campbell the discoverer, 1897, of the CBHP composition of matrix exponentials), were &amp;quot;practical&amp;quot; geniuses who sought the &amp;quot;point&amp;quot;, the answer, instead of hidebound, inhibited and often counterproductive continental chapter-and-proof proper  mathematese. They were communicating in the more old-fashioned 18th century language, which only survived in the Soviet Union in the 20th century. I don't see a priorities' drama unfolding here, but I am aware of the comical priorities' battles waged on the misnamed &amp;quot;BCH&amp;quot; formula mentioned.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastActivityDate="2022-04-08T14:40:22.230" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14366" PostTypeId="1" AcceptedAnswerId="14367" CreationDate="2022-04-09T16:56:41.087" Score="4" ViewCount="89" Body="&lt;p&gt;I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius.&lt;/p&gt;&#xA;&lt;p&gt;According to most of the related papers (e.g. &lt;em&gt;Pioneers of Representation Theory&lt;/em&gt; by C.W.Curtis), the character of abelian groups occurs earlier, which was defined as a homomorphism from an abelian group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$C^{\times}$&lt;/span&gt;. And it is the problem of factorization of group determinants which motivates Frobenius' generalization of characters to arbitrary finite group, as is described in the introduction of one of the Frobenius' papers &lt;em&gt;Über Gruppencharaktere&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is Frobenius' original definition: Let &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; be a finite group. For each conjugacy class &lt;span class=&quot;math-container&quot;&gt;$C_{i}$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, let &lt;span class=&quot;math-container&quot;&gt;$h_{ijk}$&lt;/span&gt; be the number of solutions of the equation of the equations &lt;span class=&quot;math-container&quot;&gt;$abc = 1$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$a\in C_{i}, b\in C_{j}, c\in C_{k}$&lt;/span&gt;. Denote as &lt;span class=&quot;math-container&quot;&gt;$C_{i^{'}}$&lt;/span&gt; the conjugacy class consisting of the inverses of elements in &lt;span class=&quot;math-container&quot;&gt;$C_{i}$&lt;/span&gt;, and put &lt;span class=&quot;math-container&quot;&gt;$$a_{ijk}=\frac{h_{i^{'}jk}}{|C_{i}|}.$$&lt;/span&gt; Via solving the 'structure equations' &lt;span class=&quot;math-container&quot;&gt;$$r_{j}r_{k}=a_{ijk}r_{i},$$&lt;/span&gt; Frobenius got &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; different solutions &lt;span class=&quot;math-container&quot;&gt;$(r_{q1}, r_{q2}, ..., r_{qn})$&lt;/span&gt;. Then he defined a character of a finite group &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; as a class function (constant on every conjugacy class &lt;span class=&quot;math-container&quot;&gt;$C_{i}$&lt;/span&gt; ) &lt;span class=&quot;math-container&quot;&gt;$\chi$&lt;/span&gt; , taking for any &lt;span class=&quot;math-container&quot;&gt;$g\in C_{i}$&lt;/span&gt; the value &lt;span class=&quot;math-container&quot;&gt;$$\chi_{q}(g) = \frac{fr_{qi}}{|C_{i}|},$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;${r_{qi}}$&lt;/span&gt; is given above by the structure equations and &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is used to normalize &lt;span class=&quot;math-container&quot;&gt;$\chi_{q}$&lt;/span&gt; in order that it would satisfy the orthogonality relations.&lt;/p&gt;&#xA;&lt;p&gt;This definition seems to be understandable in modern context of representation theory. What seems highly unnatural to me is that &lt;strong&gt;this definition is triggered by the attempt to factorize the group determinant&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;First of all, the coefficients of linear factors of the group determinant are exactly the linear characters of the group, and any other factor of the determinant has a degree more than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; and can't be factored into linear forms, so there seems to be no reason to generalize characters to the nonabelian case given that the linear factors are the only to appear in the factorization.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, I can't see how intuitively this problem of factorization could provide any clue leading to an explicit definition of group character like above. It seems that Frobenius' idea was to construct a homomorphism to &lt;span class=&quot;math-container&quot;&gt;$C^{\times}$&lt;/span&gt; from an algebra whose elements are the conjugacy classes with the operation induced by usual multiplication law of a group. I do understand what this algebra is in the modern context, but I can't understand how the latter was motivated by the factorization problem.&lt;/p&gt;&#xA;&lt;p&gt;Last but not least, Frobenius chose to base his definition on 'hypercomplexes' (seems equivalent to the modern concept of an algebra). I can't see the relation between the 'hypercomplexes' and the factorization problem.&lt;/p&gt;&#xA;&lt;p&gt;Will anyone be so kind to explain the intuition behind this definition, given the context of group determinants? Thanks in advance!&lt;/p&gt;&#xA;" OwnerUserId="16512" LastActivityDate="2022-04-09T21:36:28.543" Title="Intuitions for Frobenius' generalization of characters to nonabelian finite group given the historical context" Tags="|mathematics|abstract-algebra|group-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14367" PostTypeId="2" ParentId="14366" CreationDate="2022-04-09T21:36:28.543" Score="4" Body="&lt;p&gt;We can’t redo the history. It just happens to be the case that Dedekind’s questions to Frobenius about group determinants were the original inspiration. It is &lt;em&gt;not&lt;/em&gt; intuitive. Only later did Frobenius find a more conceptual approach, the one we use today. Group determinants were abandoned since they are not why we care about group characters and they are a strange motivation, as you have already seen. Maybe the article &lt;a href=&quot;https://kconrad.math.uconn.edu/articles/groupdet.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://kconrad.math.uconn.edu/articles/groupdet.pdf&lt;/a&gt; will be helpful.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2022-04-09T21:36:28.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14368" PostTypeId="1" AcceptedAnswerId="14370" CreationDate="2022-04-09T22:33:50.980" Score="4" ViewCount="646" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Maria_Gaetana_Agnesi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; says the following about Maria Gaetana Agnesi:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;She is credited with writing the first book discussing both differential and integral calculus and was a member of the faculty at the University of Bologna, although she never served.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What was the title of this first book book containing both differential and integral calculus?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2022-11-29T12:36:37.783" Title="What book did Maria Gaetana Agnesi write which contained both differential and integral calculus?" Tags="|mathematics|mathematicians|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14369" PostTypeId="2" ParentId="4968" CreationDate="2022-04-09T22:40:10.623" Score="2" Body="&lt;p&gt;Chebyshev introduced his functions &lt;span class=&quot;math-container&quot;&gt;$\theta(x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\psi(x)$&lt;/span&gt; in his 1852 paper &amp;quot;Mémoire sur les nombres premiers&amp;quot;. It is available online &lt;a href=&quot;http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;: &lt;span class=&quot;math-container&quot;&gt;$\theta(x)$&lt;/span&gt; is on p. 368 and &lt;span class=&quot;math-container&quot;&gt;$\psi(x)$&lt;/span&gt; is on p. 371. In section V he bounds &lt;span class=&quot;math-container&quot;&gt;$\psi(x)$&lt;/span&gt; above and below by expressions that are asymptotic to &lt;span class=&quot;math-container&quot;&gt;$Ax$&lt;/span&gt; (lower bound) and &lt;span class=&quot;math-container&quot;&gt;$(6/5)Ax$&lt;/span&gt; (upper bound), where&#xA;&lt;span class=&quot;math-container&quot;&gt;$A \approx .921$&lt;/span&gt; is given by an explicit formula in the middle of p. 376. The bounds together are on the bottom of page 378.  He converts this into upper and lower bounds on &lt;span class=&quot;math-container&quot;&gt;$\theta(x)$&lt;/span&gt; on p. 379.&lt;/p&gt;&#xA;&lt;p&gt;In this paper, Chebyshev does &lt;em&gt;not&lt;/em&gt; write about an equivalence between &lt;span class=&quot;math-container&quot;&gt;$\pi(x) \sim x/\log x$&lt;/span&gt; and estimates on &lt;span class=&quot;math-container&quot;&gt;$\psi(x)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\theta(x)$&lt;/span&gt;.  He does write on p. 389 about getting bounds from above and below on the number of primes between &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; using bounds on &lt;span class=&quot;math-container&quot;&gt;$\theta(x)$&lt;/span&gt; (see equation (9)), but he doesn't directly write &lt;span class=&quot;math-container&quot;&gt;$Cx/\log x &amp;lt; \pi(x) &amp;lt; C'x/\log x$&lt;/span&gt; for constants &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$C'$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In another paper that appeared in 1852, &amp;quot;Sur la totalité des nombres premiers inférieurs à une limite donnée&amp;quot;, which can be found online &lt;a href=&quot;http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A18_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, Chebyshev shows that if &lt;span class=&quot;math-container&quot;&gt;$\pi(x)/(x/\log x)$&lt;/span&gt; has a limit then this limit must be &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, although he does not write the result exactly in that form. See Theorem II on p. 348 and also pp. 356-357.&lt;/p&gt;&#xA;&lt;p&gt;Narkiewicz, in his book &lt;em&gt;The Development of Prime Number Theory&lt;/em&gt;, describes  the form of the prime number theorem that appeared in the proofs by Hadamard and de la Vallee Poussin as &lt;span class=&quot;math-container&quot;&gt;$\theta(x) \sim x$&lt;/span&gt; (see pp. 201 and 216) and writes in a footnote on p. 217 that &amp;quot;it seems&amp;quot; Sylvester in 1892 was the first to observe that &lt;span class=&quot;math-container&quot;&gt;$\theta(x) \sim x \Longrightarrow \pi(x) \sim x/\log x$&lt;/span&gt;. I find it hard to believe nobody before Sylvester was aware of that, including Chebyshev, but please note that it does &lt;em&gt;not&lt;/em&gt; appear that Chebyshev ever wrote down such a result.  Unless Hardy gave an actual reference to a paper of Chebyshev in what you had read, don't accept that Chebyshev really did prove an equivalence between &lt;span class=&quot;math-container&quot;&gt;$\pi(x) \sim x/\log x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\theta(x) \sim x$&lt;/span&gt;, or even an implication in one direction. I've never seen such an argument in one of Chebyshev's papers.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2022-04-09T22:40:10.623" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14370" PostTypeId="2" ParentId="14368" CreationDate="2022-04-09T22:53:48.853" Score="11" Body="&lt;p&gt;The book is a work in two volumes: Maria Gaetana Agnesi, &amp;quot;&lt;strong&gt;Instituzioni Analitiche&lt;/strong&gt; ad uso della gioventù italiana&amp;quot;, Milan 1748. The title roughly translates to &amp;quot;Foundations of Analysis for use by Italian youth&amp;quot;. As the title suggests, this is a textbook, and it was written in the vernacular rather than in Latin.&lt;/p&gt;&#xA;&lt;p&gt;Google's scan of the first volume (tomo I) is &lt;a href=&quot;https://books.google.com/books?id=L0pbAAAAcAAJ&amp;amp;printsec=frontcover&amp;amp;dq=Instituzioni+analitiche&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, a scan of the second volume (tomo II) is &lt;a href=&quot;https://www.google.com/books/edition/Instituzioni_analitiche_ad_uso_della_gio/fqp3BwruNk0C?hl=en&amp;amp;gbpv=1&amp;amp;dq=Instituzioni+Analitiche&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. An English translation by the Rev. &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Colson&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Colson&lt;/a&gt; was published in London by Taylor and Wilks in 1801. A scan of it can be found at &lt;a href=&quot;https://books.google.com/books?id=o54AAAAAMAAJ&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;dq=Analytical%20institutions&amp;amp;pg=PP7#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google&lt;/a&gt;. According to the editor's note at the front of the book, Colson was Lucasian Professor of Mathematics at the University of Cambridge. It further notes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;That learned and ingenious man, who had obliged his Country with an English Translation of Sir ISAAC NEWTON's Fluxions [...] found, after all, the &lt;em&gt;Analytical Institutions&lt;/em&gt; of &lt;em&gt;Agnesi&lt;/em&gt; to be so excellent, that he was at pains of learning the Italian Language, at an advanced age, for the sole purpose of translating that work into English; that the British Youth might have the benefit of it as well as the Youth of Italy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A curious detail on the title page of &lt;em&gt;Instituzioni Analitiche&lt;/em&gt; is that it does not mention a printer. While other books published in Milan at the same time carried a note such as &lt;a href=&quot;https://www.google.com/books/edition/La_Vergine_addolorata_alla_vista_di_Cris/Q29tGeXkqocC?hl=en&amp;amp;gbpv=1&amp;amp;dq=Milano+1748+%22Stampatore+Regio+Camerale%22&amp;amp;pg=PT7&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;this one&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Milano, nella Regia Ducal Corte, per Giuseppe Richino Malatesta Stampatore Regio Camerale. &lt;em&gt;Con licenza de' Superiori.&lt;/em&gt; 1748.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;identifying the printer as Giuseppe Richino Malatesta (1694-1793), from a local family of printers active from the 16&lt;sup&gt;th&lt;/sup&gt; through the 18&lt;sup&gt;th&lt;/sup&gt; century, it simply has a notice that reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Milano, MDCCXLVIII. Nella Regia-Ducal Corte. Con Licenza de Superiori.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which translates to &amp;quot;In Milan, 1748. In the Royal-Ducal Court. With License of the Superiors.&amp;quot; Looking into this, I learned that Maria Agnesi apparently had a printing workshop set up at her family's house to produce the book there, which could explain the omission of the printer's name.&lt;/p&gt;&#xA;&lt;p&gt;Paula Findlen, &amp;quot;Calculations of faith: mathematics, philosophy, and sanctity in 18th-century Italy (new work on Maria Gaetana Agnesi), &lt;em&gt;Historia Mathematica&lt;/em&gt;, Vol. 38, No. 2, May 2011, pp. 248-291 (&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086010000315&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;She was so concerned about the production of her textbook that she asked the printer Giuseppe Richini to set up his press inside Palazzo Agnesi so that she could personally supervise the technical difficulties of typesetting mathematical characters and laying out the formulas well on every page.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-11-29T12:36:37.783" LastActivityDate="2022-11-29T12:36:37.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14371" PostTypeId="2" ParentId="14273" CreationDate="2022-04-09T22:56:49.807" Score="1" Body="&lt;p&gt;One way to get around always reading about the same facts is to read more technical facts: look at books that delve into mathematics at a level beyond what would be considered suitable for a general audience.&#xA;The following books are probably only going to be appreciated by someone who has some familiarity with the areas of math that they are about.&lt;/p&gt;&#xA;&lt;p&gt;Bottazzini &amp;amp; Gray, &lt;em&gt;Hidden Harmony - Geometric Fantasies: The Rise of Complex Function Theory&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Dieudonne, &lt;em&gt;History of Algebraic Geometry&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Dieudonne, &lt;em&gt;A History of Algebraic and Differential Topology: 1900-1960&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Dieudonne, &lt;em&gt;History of Functional Analysis&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov and Yushkevich, &lt;em&gt;Mathematics of the 19th Century Vol. 1: Mathematical Logic, Algebra, Number Theory, Probability Theory&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov and Yushkevich, &lt;em&gt;Mathematics of the 19th Century Vol. 2: Geometry, Analytic Function Theory&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Kolmogorov and Yushkevich,&#xA;&lt;em&gt;Mathematics of the 19th Century Vol. 3: Function Theory According to Chebyshev, Ordinary Differential Equations, Calculus of Variations, Theory of Finite Differences&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;van der Waerden, &lt;em&gt;A History of Algebra: From al-Khwārizmī to Emmy Noether&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2022-04-09T22:56:49.807" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14372" PostTypeId="2" ParentId="3001" CreationDate="2022-04-10T04:19:13.507" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sophie_Germain&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sophie Germain&lt;/a&gt; had (and is still having) a great impact on mathematics and physics, but had to publish using a pseudonym Monsieur Le Blanc.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="3869" LastEditDate="2022-04-23T00:56:35.130" LastActivityDate="2022-04-23T00:56:35.130" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14373" PostTypeId="2" ParentId="3001" CreationDate="2022-04-11T17:29:19.160" Score="3" Body="&lt;p&gt;If the original &lt;a href=&quot;https://bitcoin.org/bitcoin.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;white paper on Bitcoin&lt;/a&gt; counts as mathematics, then it is a perfect example of what you're asking for, because the true identity of the author, &lt;a href=&quot;https://en.wikipedia.org/wiki/Satoshi_Nakamoto&quot; rel=&quot;nofollow noreferrer&quot;&gt;Satoshi Nakamoto&lt;/a&gt;, remains unknown as of this writing (2022).&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2022-04-11T17:29:19.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14374" PostTypeId="1" AcceptedAnswerId="14375" CreationDate="2022-04-11T21:46:22.240" Score="4" ViewCount="156" Body="&lt;p&gt;I just assumed that he did since we owe the proof that &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; is uncountable to him, but a search of my favorite source for answering such questions -- &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Miller/mathword/&lt;/a&gt; -- doesn't even mention uncountable.&lt;/p&gt;&#xA;" OwnerUserId="15921" LastActivityDate="2022-04-12T03:47:32.397" Title="Did Cantor coin the terms countable and uncountable? If not, who did?" Tags="|set-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14375" PostTypeId="2" ParentId="14374" CreationDate="2022-04-12T03:47:32.397" Score="9" Body="&lt;p&gt;Georg Cantor published a 47-page booklet, &amp;quot;Grundlagen einer allgemeinen Mannigfaltigkeitslehre,&amp;quot; Leipzig: Teubner 1883. The title translates to &amp;quot;Foundations of a general set theory&amp;quot; and various English translations show that he refers to countable and uncountable sets in it. I am unable to find a scan of the German original online. &lt;em&gt;Mannigfaltigkeit&lt;/em&gt; (literally translated: manifold) was Cantor's term for a set, the modern German term for which is &lt;em&gt;Menge&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The material that comprises this monograph appears to also have been published (in somewhat different form) in various individual publications, one of which is&lt;/p&gt;&#xA;&lt;p&gt;G. Cantor, &amp;quot;Ueber unendliche, lineare Punktmannichfaltigkeiten&amp;quot;, &lt;em&gt;Mathematische Annalen&lt;/em&gt;, Vol. 21, 1883, pp. 52-58 (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0021?tify=%7B%22pages%22:%5B57%5D,%22panX%22:0.582,%22panY%22:0.359,%22view%22:%22info%22,%22zoom%22:0.767%7D&quot; rel=&quot;noreferrer&quot;&gt;scan online&lt;/a&gt;). As far as I understand, Cantor's concept of a point set is a subset of the real line. Note that the title uses a spelling variant of &lt;em&gt;Mannigfaltigkeit&lt;/em&gt; suggested by Northern German pronunciation. On page 54 of this publication we find:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Da nämlich alle Bestandtheile der rechten Seite abzählbar sind und die &lt;em&gt;Anzahl&lt;/em&gt; dieser Bestandtheile eine &lt;em&gt;abzählbar unendliche&lt;/em&gt; ist, so folgt&#xA;daraus die Abzählbarkeit von &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt; und nach Th. II diejenige von &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Theorem V. &lt;em&gt;Jede Punktmenge &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; zweiter Gattung, für welche &lt;span class=&quot;math-container&quot;&gt;$P^{(\alpha)}$&lt;/span&gt; abzählbar, ist selbst abzählbar.&lt;/em&gt; [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Ist &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; eine nicht abzählbare Punktmenge, so ist auch &lt;span class=&quot;math-container&quot;&gt;$P^{(\alpha)}$&lt;/span&gt; nicht abzählbar, sowohl wenn &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; eine endliche ganze Zahl, wie auch wenn es eines der Unendlichkeitssymbole ist.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: Since all components of the right-hand side are countable and the &lt;em&gt;number&lt;/em&gt; of these components is &lt;em&gt;countably infinite&lt;/em&gt;, it follows the countability of &lt;span class=&quot;math-container&quot;&gt;$P'$&lt;/span&gt; and according to theorem II that of &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. [...] Theorem V. &lt;em&gt;Every point set &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; of the second class, for which &lt;span class=&quot;math-container&quot;&gt;$P^{(\alpha)}$&lt;/span&gt; is countable, is itself countable.&lt;/em&gt; [...] &lt;em&gt;If &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is an uncountable point set, then &lt;span class=&quot;math-container&quot;&gt;$P^{(\alpha)}$&lt;/span&gt; is also uncountable, both when &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is a finite integer and when it is one of the infinity symbols.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2022-04-12T03:47:32.397" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14376" PostTypeId="1" CreationDate="2022-04-12T08:33:07.107" Score="5" ViewCount="217" Body="&lt;p&gt;Based on this question&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/3001/are-there-any-anonymous-contributions-to-mathematics-that-had-a-great-impact&quot;&gt;Are there any anonymous contributions to mathematics that had a great impact?&lt;/a&gt; , I would like to ask the same question for physics. Physics is different from mathematics in the sense that you need to experiments in order to move on, so I guess an anonymous contribution would be much harder.&lt;/p&gt;&#xA;&lt;p&gt;Looking at the examples of the mathematics thread, I wonder if a 4chan equivalent post (&lt;a href=&quot;https://hsm.stackexchange.com/a/13394/6609&quot;&gt;https://hsm.stackexchange.com/a/13394/6609&lt;/a&gt;) on particle or cosmology could ever be taken seriously.&lt;/p&gt;&#xA;&lt;p&gt;Are there anonymous contributions to physics with large impact in the history of the field?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-04-12T08:46:49.277" LastActivityDate="2022-04-18T22:36:43.660" Title="Are there anonymous contributions to physics with large impact?" Tags="|physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14377" PostTypeId="1" CreationDate="2022-04-12T13:38:13.803" Score="0" ViewCount="108" Body="&lt;p&gt;I remember learning that the inventor of &lt;a href=&quot;https://en.wikipedia.org/wiki/Polymerase_chain_reaction&quot; rel=&quot;nofollow noreferrer&quot;&gt;polymerase chain reaction&lt;/a&gt; (PCR) was &lt;a href=&quot;https://en.wikipedia.org/wiki/Kary_Mullis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kary Mullis&lt;/a&gt;, who was awarded the Nobel prize in chemistry in 1993. I recently learned that there is some controversy about this due to the fact that his idea may not have been entirely original.&lt;/p&gt;&#xA;&lt;p&gt;The first actual written description of it was a &lt;a href=&quot;https://doi.org/10.1016/0022-2836(71)90469-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;1971 paper&lt;/a&gt; by Norwegian biochemist Kjell Kleppe. IIRC the paper was theoretical and didn't have any experimental demonstration of DNA amplification. Part of the problem was that it wouldn't be until 1976 that the Taq polymerase enzyme was isolated which is used in the PCR. So Mullis seems to have proven experimentally that Kleppe's idea worked. He also was able to optimize the process using thermal cycling and make use of the heat stable Taq polymerase enzyme.&lt;/p&gt;&#xA;&lt;p&gt;Part of what led me to doubt Mullis' claims was &lt;a href=&quot;https://geneticsunzipped.com/news/2020/11/3/the-story-of-pcr&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article about the PCR story.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Who actually invented PCR?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="16519" LastEditorUserId="4251" LastEditDate="2022-04-13T14:37:25.753" LastActivityDate="2022-04-13T14:37:25.753" Title="Did Kary Mullis actually invent PCR as he claimed?" Tags="|biology|chemists|biochemistry|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14378" PostTypeId="1" CreationDate="2022-04-12T13:59:58.210" Score="0" ViewCount="70" Body="&lt;p&gt;On page 443, section &lt;em&gt;1.1 Expanding to Contradiction&lt;/em&gt;, in José Ferreirós' &lt;em&gt;&lt;a href=&quot;https://www.researchgate.net/publication/38373294_The_Road_to_Modern_Logic-An_Interpretation&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Road To Modern Logic - An Interpretation&lt;/a&gt;&lt;/em&gt;, the following is written:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Philosophical conceptions of logic have been complex and varied; here we are only interested in the tradition of ‘formal logic’, and formal logic meant the theory of the syllogisms—above all the Aristotelian syllogisms but also the so-called hypothetical and disjunctive syllogisms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Prior to reading the above quote, and &lt;a href=&quot;https://en.wikipedia.org/wiki/First-order_logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Wikipedia article&lt;/a&gt; linking to it, I had only read &lt;em&gt;formal logic&lt;/em&gt; as referring to the larger, formalized study of logic (see &lt;a href=&quot;https://www.britannica.com/topic/formal-logic&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Britannica article&lt;/a&gt; for an example). This leads me to think there has been a shift in the use of the term, its use originating as a name for all syllogisitic logics, before being appropriated as a the name for the discipline of studying, creating and using formalized logics.&lt;/p&gt;&#xA;&lt;p&gt;So, my two questions are mostly about etymology and usage, which does place this question scarily close to belonging in EL&amp;amp;U; however, I deem this question a bit too technical to be a good fit for that site, in that it requires an educated parsing of the logics discussed in relevant texts in order to know how and why the categorizes were composed. That's the etymology part, which is history (linguistic history, but history nonetheless). The usage part is less justifiable, but I am asking about a not so extensive account of the current usage from a historical perspective; as in, how is it different from the past and why.&lt;/p&gt;&#xA;&lt;p&gt;So, to be completely clear, here are my questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What did &lt;em&gt;formal logic&lt;/em&gt; mean before?&lt;/li&gt;&#xA;&lt;li&gt;What does it mean now, and why has this changed?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16523" LastEditorUserId="131" LastEditDate="2022-04-13T08:12:39.493" LastActivityDate="2022-04-13T08:12:39.493" Title="Formal logic as a synonym to syllogistic logic, or as a name for the study of logic?" Tags="|terminology|mathematical-logic|logic|logicians|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14379" PostTypeId="2" ParentId="14273" CreationDate="2022-04-13T09:26:05.907" Score="0" Body="&lt;p&gt;There are more general books eg History of Number which do give a fair look at non-Euro eg Egypt, Babylon, Sanskrit ... worth a look&lt;/p&gt;&#xA;" OwnerUserId="7169" LastActivityDate="2022-04-13T09:26:05.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14380" PostTypeId="2" ParentId="14376" CreationDate="2022-04-13T22:00:07.393" Score="1" Body="&lt;p&gt;Sophie Germain basically invented the theory of elasticity, but published under a pseudonym.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastActivityDate="2022-04-13T22:00:07.393" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14381" PostTypeId="1" CreationDate="2022-04-14T15:55:12.640" Score="1" ViewCount="116" Body="&lt;p&gt;Much as per question - Uncia is the Latin for one-twelfth and thence is used for coin (As),length (Pes),weight (Libra), area (Jugerum) etc from whence derives inch and ounce.&#xA;I can find nothing much about Greek fractions not even the name for one-twelfth; and not much about Greek pre-Roman measurements.&#xA;Finding Indo-European equivalents would be a step further and Sanskrit would be a step sideways.&#xA;Help please.&lt;/p&gt;&#xA;" OwnerUserId="7169" LastActivityDate="2022-04-14T15:55:12.640" Title="partly a LANGUAGE question - uncia (Latin one-twelfth) is the source of both Inch and Ounce. Does anyone know the Greek or pre-Roman equivalent?" Tags="|measurement|" AnswerCount="0" CommentCount="18" ContentLicense="CC BY-SA 4.0" />
  <row Id="14384" PostTypeId="2" ParentId="11903" CreationDate="2022-04-15T17:23:22.417" Score="1" Body="&lt;p&gt;Benjamin Franklin was the originator of the terms &amp;quot;positive&amp;quot; and &amp;quot;negative&amp;quot; as they refer to electric charges (but of course those same words referring to other things existed before that), and he invented the battery and the lightning conductor and made other important contributions to understanding of electricity. (But he was not the first to do scientific work on electricity. Children are often taught that &amp;quot;Franklin discovered electricity.&amp;quot; What children are taught about things like that in elementary school is often just fairy tales. (Ambrose Bierce in &lt;em&gt;The Devil's Dictionary&lt;/em&gt; jocularly calls Franklin the &amp;quot;inventor of electricity&amp;quot;.))&lt;/p&gt;&#xA;" OwnerUserId="2193" LastActivityDate="2022-04-15T17:23:22.417" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14385" PostTypeId="1" CreationDate="2022-04-15T17:33:17.207" Score="1" ViewCount="55" Body="&lt;p&gt;Is there a good (or even mediocre) published expository account of Benjamin Franklin's work on electricity?&lt;/p&gt;&#xA;&lt;p&gt;From what I have read and heard, it appears that&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Franklin may have been the first to ascertain that lightning is an instance of the same force of nature seen in electrical phenomena observed in commonplace circumstances;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Franklin invented the lightning conductor, which protects buildings from burning down when struck by lightning;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Franklin was the first to call certain electric charges &amp;quot;positive&amp;quot; and others &amp;quot;negative&amp;quot;, and his choice of which is which has persisted despite later being found to be less felicitous than if he had made the opposite choice;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Franklin invented the battery.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;A good expository account would make that list more complete and would say how Franklin's work fit in with the work of others before and during Franklin's life, and perhaps how later researchers relied upon Franklin's work.&lt;/p&gt;&#xA;" OwnerUserId="2193" LastActivityDate="2022-04-15T17:33:17.207" Title="Published expository account of Franklin's work on electricity?" Tags="|electricity|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14386" PostTypeId="1" CreationDate="2022-04-15T23:33:54.317" Score="1" ViewCount="49" Body="&lt;p&gt;I’m interested in learning more about how statistical methodology and research design has changed over the course of the 20th and 21st century. I’m particularly interested in ways in which research methodology has been standardised across subfields, to the extent that it has. It seems like 20th century statistics has had a huge (and maybe homogenising?) impact on the way that science is done, so I’m particularly interested in statistical methodology. Even more particularly, I'm interested in its impact on psychology, but information on any fields is extremely welcome. Thanks.&lt;/p&gt;&#xA;" OwnerUserId="16538" LastActivityDate="2022-04-15T23:33:54.317" Title="Can anyone recommend sources on the standardisation of research methods in modern (20th-21st c) science?" Tags="|statistics|scientific-method|resource-recommendation|psychology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14388" PostTypeId="1" AcceptedAnswerId="14389" CreationDate="2022-04-18T21:51:00.923" Score="9" ViewCount="2217" Body="&lt;p&gt;The open access paper Lehtola &amp;amp; Karttunen (2022) &lt;a href=&quot;https://doi.org/10.1002/wcms.1610&quot; rel=&quot;noreferrer&quot;&gt;Free and open source software for computational chemistry education&lt;/a&gt; (found in &lt;a href=&quot;https://mattermodeling.stackexchange.com/a/8949/201&quot;&gt;this answer&lt;/a&gt;) contains the following paragraph:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;2 FREE AND OPEN SOURCE SOFTWARE&lt;/p&gt;&#xA;&lt;p&gt;2.1 Definitions&lt;/p&gt;&#xA;&lt;p&gt;However, the control of access to the source code of such closed-source programs lead to perpetuating power structures and may inhibit academic collaborations between authors of different program packages,&lt;sup&gt;27&lt;/sup&gt; instead of the Popperian ideal of science: the selfless pursuit of truth,&lt;sup&gt;28&lt;/sup&gt; and a fair and unbiased competition of ideas and methods in the context of computational chemistry. Key persons in control of the access to the source codes of various software packages are able to hold back equitable competition and collaboration between scientists developing new methods and algorithms. &lt;strong&gt;The issue with gatekeepers is not a new phenomenon: as was already quipped by Max Planck, “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it”;&lt;/strong&gt; this apt observation is supported by a recent study that investigated the dynamics of scientific evolution with the standard empirical tools of applied microeconomics.&lt;sup&gt;29&lt;/sup&gt; This problem is less likely to manifest in FOSS, as will be explained in the next paragraph.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;27&lt;/sup&gt;Jacob CR. &lt;a href=&quot;https://pubs.acs.org/doi/full/10.1021/acs.jpclett.5b02609&quot; rel=&quot;noreferrer&quot;&gt;How open is commercial scientific software?&lt;/a&gt; J Phys Chem Lett. 2016; 7: 351– 3.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;28&lt;/sup&gt;Li L. &lt;a href=&quot;https://pubs.acs.org/doi/abs/10.1021/ed076p20&quot; rel=&quot;noreferrer&quot;&gt;Why should anyone become a scientist? The ideal of science and its importance&lt;/a&gt;. J Chem Educ. 1999; 76: 20.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;29&lt;/sup&gt;Azoulay P, Fons-Rosen C, Zivin JSG. &lt;a href=&quot;https://www.aeaweb.org/articles?id=10.1257/aer.20161574&quot; rel=&quot;noreferrer&quot;&gt;Does science advance one funeral at a time?&lt;/a&gt; Am Econ Rev. 2019; 109: 2889– 920.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'd like to focus on what Planck's &amp;quot;state of mind&amp;quot; might have been when he posited:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An instructive example might be &lt;a href=&quot;https://en.wikipedia.org/wiki/Linus_Pauling&quot; rel=&quot;noreferrer&quot;&gt;Linus Pauling&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Dan_Shechtman&quot; rel=&quot;noreferrer&quot;&gt;Dan Shechtman&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Quasicrystal&quot; rel=&quot;noreferrer&quot;&gt;quasicrystals&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://chemistry.stackexchange.com/q/159395/16035&quot;&gt;this Chemistry SE question&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The video &lt;a href=&quot;https://youtu.be/V2GqU6fdjeQ?t=420&quot; rel=&quot;noreferrer&quot;&gt;Quasicrystals ; Prof. Daniel Shechtman ; Nobel Prize in Chemistry&lt;/a&gt; focuses on &lt;a href=&quot;https://en.wikipedia.org/wiki/Dan_Shechtman&quot; rel=&quot;noreferrer&quot;&gt;Professor Dan Shechtman&lt;/a&gt; but happens to include some photos of Linus Pauling &lt;strong&gt;who never believed in &lt;a href=&quot;https://en.wikipedia.org/wiki/Quasicrystal&quot; rel=&quot;noreferrer&quot;&gt;quasicrystals&lt;/a&gt;.&lt;/strong&gt; From the video:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The leader of that group was Professor Linus Pauling, a two-time Nobel laureate. He was a very important figure and an idol of the American Chemical Society, &lt;strong&gt;and to his last day, he was standing on stages and published papers saying that &amp;quot;Daniel Shechtman is talking nonsense.&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Pauling died in 1994&lt;sup&gt;a&lt;/sup&gt;. Shechtman was awarded a  Nobel Prize in Chemistry in 2011.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What was the context behind Planck's &amp;quot;A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Where did he say it, who (if anyone in particular) was he addressing, and if possible, which scientific truth might he have had in mind?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;a&lt;/sup&gt;presumably not from scurvy (humor)&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-04-18T21:56:40.040" LastActivityDate="2022-04-19T21:55:17.343" Title="Context behind Planck's &quot;A new scientific truth does not triumph by convincing its opponents...but rather because its opponents eventually die...&quot;?" Tags="|physics|planck|pauling|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14389" PostTypeId="2" ParentId="14388" CreationDate="2022-04-19T03:38:52.187" Score="12" Body="&lt;p&gt;This view, that science advances more so due to new generations being more accepting of an idea rather than the current generation being convinced, has come to be known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck%27s_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Planck's principle&lt;/a&gt; (sometimes stated more succinctly as &amp;quot;Science progresses one funeral at a time&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;The quote comes from Planck's &lt;em&gt;Scientific Autobiography And Other Papers&lt;/em&gt;, p.33, a copy of which is available on the &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.177537/page/n33/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Internet Archive&lt;/a&gt;. The specific context that he is referencing is the conflict between Ludwig Boltzmann's (and Max Planck's) model of thermodynamics (specifically relating to the 2nd law/entropy/irreversibility) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Wilhelm_Ostwald&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilhelm Ostwald&lt;/a&gt;'s now defunct model of &lt;a href=&quot;https://www.jstor.org/stable/10.1086/656824&quot; rel=&quot;nofollow noreferrer&quot;&gt;energetics&lt;/a&gt; which largely ignored or dismissed notions relating to entropy/irreversibility.&lt;/p&gt;&#xA;" OwnerUserId="5522" LastEditorUserId="134" LastEditDate="2022-04-19T21:55:17.343" LastActivityDate="2022-04-19T21:55:17.343" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14390" PostTypeId="2" ParentId="13676" CreationDate="2022-04-19T11:41:13.863" Score="0" Body="&lt;p&gt;I came across your question searching about Pefsu problems. I am interested into it from a mathematics perspective. I like history but I am by far not an expert into ancient Egyptian beer making, so i will make some assumptions. So let me offer you what I believe is the explanation. I believe the issue is the difference on how a modern person would use the formula and how an ancient Egyptian wold.&lt;/p&gt;&#xA;&lt;p&gt;The ancient Egyptians would have no issue to use Pefsu for the strength of beer. After all, all they cared about is how much raw material (heqats of grain), will be converted into finished product (jugs of beer). Thus you get the Pefsu which for them would be just a number (and most likely approximated to the closest integer or some simple rational) and it is measured in jugs per heqat.&lt;/p&gt;&#xA;&lt;p&gt;To our modern sensibilities this strikes as somewhat odd. This is for several factors, such as:&lt;/p&gt;&#xA;&lt;p&gt;a) The strength (alcohol content) of beer will be or less standardised by the type of beer you want to produce. Therefore it will be a constant in the calculation and not a variable as for ancient Egyptians (in fact, that is the quantity they want to determine). Note that nowadays we have a much better theoretical understanding of fermentation and can use simple scientific instruments that can estimate quite accurately the level of alcohol in the mixture. I am not sure of ancient Egyptian methods but I would suspect they would follow a fixed recipe that produced a fixed ratio concentrate, that would then be diluted accordingly. Thus we would accept the strength of the beer and vary the volume, whereas ancient Egyptians would like to keep the volume the same and vary the strength.&lt;/p&gt;&#xA;&lt;p&gt;In fact, the problem hints to that and that is what is essentially missing from the problem description: note step 3 &amp;quot;One jug of beer requires 2 and 1/6 heqats of grain&amp;quot;. Assuming that the process remains stable (number of fermentation etc.), we can assume that the amount of grain is at the very least a good approximation to the alcohol content of the beer.&lt;/p&gt;&#xA;&lt;p&gt;b) Regardless if the alcohol level is fixed or not, we desire to have meaningful measure units in modernity. Since we wish to compare two volumes, it would make sense to use the same unit of measurement. Thus, we use step 3 to convert heqats of grain to jugs of beer. Then the ratio would be an absolute number, i.e. without units, which is what we desire.&lt;/p&gt;&#xA;&lt;p&gt;To summarize, a modern person would say that by using 13 heqats of grain and following a recipe that converts 2 and 1/6 heqats of grain to 1 jug of beer, we may obtain 18 jugs of beer of a strength of 3 Pefsu (as an absolute number)&lt;/p&gt;&#xA;&lt;p&gt;An ancient Egyptian trader might be fine to skip step 3 and give the strength result in jugs per heqat, thus derive that if he is given 13 heqats of grain and must deliver 18 jugs of beer, the resulting beer will have a strength of 18/13 = 1 + 5/13 Pefsu (as jugs per heqat).&lt;/p&gt;&#xA;&lt;p&gt;In both cases the full amount of 13 heqats of grain is used, what changes is the unit we measure the strength of the beer with.&lt;/p&gt;&#xA;&lt;p&gt;Note that we may alternate between the two units by multiplying by the strength of the beer recipe, which is 2 + 1/6 = 13/6 heqats per jug.&lt;/p&gt;&#xA;" OwnerUserId="16546" LastActivityDate="2022-04-19T11:41:13.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14391" PostTypeId="2" ParentId="275" CreationDate="2022-04-19T16:24:44.820" Score="4" Body="&lt;p&gt;The best place to find an answer to this question is probably &lt;a href=&quot;https://press.princeton.edu/books/ebook/9780691187723/mind-over-magma&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mind over Magma: The Story of Igneous Petrology&lt;/a&gt;, by Davis A. Young. In chapter 13, The Mechanism of Differentiation, he explains how the Soret effect, which was viewed as the main mechanism for rock differentiation at the end of the nineteenth century, began to be rejected:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Soret effect was the most frequently invoked mechanism of&#xA;differentiation until 1893, when Backstrom issued a paper attacking&#xA;it. He was soon joined by Harker, Johnston-Lavis, Michel-Levy, Becker,&#xA;Brogger, Loewinson-Lessing, Schweig, and even Teall.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He then goes on by listing the various other mechanisms proposed by scientists at the time, such as fractional crystallization, and ends with this interesting passage (p. 225, emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Not every petrologist was convinced that differentiation, whatever the&#xA;proposed mechanism, was the dominant cause of igneous rock diversity.&#xA;&lt;em&gt;Partial melting, espoused by Becker and Teall&lt;/em&gt;, and assimilation,&#xA;endorsed by Loewinson-Lessing, were also regarded as important&#xA;petrological processes. Nevertheless, at the beginning of the&#xA;twentieth century, differentiation was acknowledged to be the leading&#xA;source of differences among igneous rocks despite a lack of agreement&#xA;on the nature of the specific physical mechanisms of differentiation&#xA;and despite the interest in partial melting, assimilation, or even&#xA;magma mixing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That would make Becker and Teall the first to recongnize partial melting as an important mechanism explaining the variability in melt composition. However, as Young points out, it was not widely accepted at the time. It would take another few decades. In chapter 16, The Theory of Crystallization-Differentiation, Young describes the contributions of Bowen, and how Holmes disagreed with it, concluding:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although the details may differ, contemporary petrologists will&#xA;recognize Holmes' essential idea in current views about the heating&#xA;and partial melting of upper granitic crust by rising basaltic magmas&#xA;that have ponded on the underside of that crust. The same arguments&#xA;were summarized the following year in a paper on the origin of igneous&#xA;rock (&lt;a href=&quot;https://doi.org/10.1017/S0016756800098332&quot; rel=&quot;nofollow noreferrer&quot;&gt;Holmes, 1932&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14043" LastActivityDate="2022-04-19T16:24:44.820" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14392" PostTypeId="1" CreationDate="2022-04-19T18:44:14.010" Score="1" ViewCount="96" Body="&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt;&lt;br /&gt;&#xA;I like to think of L'Hospital as one of the earliest authors of least-squares regression.&lt;/p&gt;&#xA;&lt;p&gt;L'Hospital, G. (1696). L'analyse des infiniment petits pour l'intelligence des lignes&#xA;courbes.&lt;/p&gt;&#xA;&lt;p&gt;I'm sure there are other historic examples, but this is the oldest that I know.&lt;/p&gt;&#xA;&lt;p&gt;Along the lines of the first publisher of the idea, I was wondering:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; What is the oldest example of using regularization? Who used the term or discovered the idea first?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Thoughts:&lt;/strong&gt;&lt;br /&gt;&#xA;Ridge regression, a regularized regression, has been around quite a while.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Hoerl, A. E., &amp;amp; Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1), 55-67.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are articles on the general topic from the 1930's, but who was the first, the pioneer?&lt;/p&gt;&#xA;" OwnerUserId="7479" OwnerDisplayName="EngrStudent" LastEditorUserId="7479" LastEditDate="2022-04-19T19:25:20.517" LastActivityDate="2022-04-19T19:25:20.517" Title="First example of regularization" Tags="|mathematicians|terminology|discoveries|physicists|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14395" PostTypeId="1" CreationDate="2022-04-23T02:55:42.970" Score="3" ViewCount="185" Body="&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Sophie_Germain#Subsequent_attempts_for_the_Prize&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sophie Germain; Work in elasticity; Subsequent attempts for the Prize&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Germain had derived the correct differential equation (a special case of the Kirchhoff–Love equation),&lt;sup&gt;31&lt;/sup&gt; but her method did not predict experimental results with great accuracy, as she had relied on &lt;strong&gt;an incorrect equation from Euler&lt;/strong&gt;,&lt;sup&gt;18&lt;/sup&gt; which led to incorrect boundary conditions.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;18&lt;/sup&gt; Gray, Mary W. (2005). &amp;quot;Sophie Germain&amp;quot;. In Bettye Anne Case; Anne M. Leggett (eds.). &lt;em&gt;Complexities: Women in Mathematics.&lt;/em&gt; Princeton University Press. pp. 68–75. ISBN 0-691-11462-5.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;31&lt;/sup&gt;Ullmann, D. (2007). &amp;quot;Life and work of E.F.F. Chladni&amp;quot;. European Physical Journal ST. 145 (1): 25–32. &lt;a href=&quot;https://ui.adsabs.harvard.edu/abs/2007EPJST.145...25U&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bibcode:2007EPJST.145...25U&lt;/a&gt;. &lt;a href=&quot;https://doi.org/10.1140%2Fepjst%2Fe2007-00145-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1140/epjst/e2007-00145-4&lt;/a&gt;. S2CID &lt;a href=&quot;https://api.semanticscholar.org/CorpusID:121813715&quot; rel=&quot;nofollow noreferrer&quot;&gt;121813715&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Upon which incorrect equation of Euler did Sophie Germain rely in her work that won a &lt;em&gt;prix extraordinaire&lt;/em&gt; from the Paris Academy of Sciences?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/a/12902/3869&quot;&gt;This thorough answer to &lt;em&gt;Did Sophie Germain find a flaw in Euler's equations for elastic vibrations?&lt;/em&gt;&lt;/a&gt; goes into some detail but states only that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It could have been one of Euler's boundary conditions for the vibrating rod extended to plates. But, according to Hill, already in 1813 the Academy was impressed enough with Germain's predictions of nodal lines and frequency ratios in Chladni's experiments for square and rectangular plates, and she does not mention a boundary value issue in the 1816 memoir.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-04-23T03:00:45.723" LastActivityDate="2023-01-22T19:01:12.420" Title="Upon which incorrect equation of Euler did Sophie Germain rely in her work that won a prix extraordinaire from the Paris Academy of Sciences?" Tags="|mathematics|physics|classical-mechanics|euler|sophie-germain|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14396" PostTypeId="5" CreationDate="2022-04-23T03:15:19.173" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-04-23T03:15:19.173" LastActivityDate="2022-04-23T03:15:19.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14397" PostTypeId="4" CreationDate="2022-04-23T03:15:19.173" Score="0" Body="French mathematician, physicist, and philosopher Marie-Sophie Germain (1776-1831)" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2022-04-24T18:20:19.173" LastActivityDate="2022-04-24T18:20:19.173" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14399" PostTypeId="2" ParentId="14328" CreationDate="2022-04-24T13:23:56.373" Score="5" Body="&lt;p&gt;This is not an answer, but is I think closer to an answer than some of the comments. The symbol &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; was used by Laplace to denote the gravitational potential in &lt;em&gt;Mécanique Céleste&lt;/em&gt; (1798, see e.g. book III chapter I &lt;span class=&quot;math-container&quot;&gt;$\S 4$&lt;/span&gt;, found in tome/volume 2). Laplace does not give a reason for using the symbol.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si l'on désigne par &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, la somme de toutes les molécules du sphéroïde, divisées par leurs distances respectives au point attiré, et que l'on nomme &lt;span class=&quot;math-container&quot;&gt;$x,y,z$&lt;/span&gt;, les coordonnées de la molécule &lt;span class=&quot;math-container&quot;&gt;$dM$&lt;/span&gt; du sphéroïde, et &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt;, celles du point attiré; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we denote by &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, the sum of all the molecules of the spheroid, divided by their respective distances to the attracted point, and we call &lt;span class=&quot;math-container&quot;&gt;$x,y,z$&lt;/span&gt;, the coordinates of the molecule &lt;span class=&quot;math-container&quot;&gt;$dM$&lt;/span&gt; of the spheroid, and &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; those of the attracted point; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The study of potentials goes back much further than &lt;em&gt;Mécanique Céleste&lt;/em&gt;, but it was Laplace's work that influenced Poisson in his Mémoires (1813) on the distribution of electricity at the surface of spheroidal conductors. Poisson also used &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;, to denote the same concept of a function whose gradient (what he called the sum of the &lt;em&gt;différences partielles&lt;/em&gt;) gives the electric force that would be experienced per unit charge at that point.&lt;/p&gt;&#xA;&lt;p&gt;By the time Maxwell wrote his Treatise in 1873, &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; was still the most common symbol for the electrostatic potential. But in some theorems, and the more general case of electrodynamics, he uses &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt;. For example, in &lt;span class=&quot;math-container&quot;&gt;$\S95b$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have hitherto used the symbol &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; for the potential, and we shall continue to do so whenever we are dealing with electrostatics only. In this chapter [chapter IV], however, and in those parts of the second volume in which the electric potential occurs in electro-magnetic investigations, we shall use &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; as a special symbol for the electric potential.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By the time of Webster's &lt;em&gt;The Theory of Electricity and Magnetism&lt;/em&gt; (1897), the symbol &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; is mentioned early on, see e.g. p. 59,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Accordingly the three equations of condition equivalent to curl &lt;span class=&quot;math-container&quot;&gt;$R=0$&lt;/span&gt; are simply the conditions that &lt;span class=&quot;math-container&quot;&gt;$X,Y,Z$&lt;/span&gt; may be represented as the derivatives of a point-function. ...The scalar function &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; (or its negative) will sometimes be termed the &lt;em&gt;potential&lt;/em&gt; of the vector &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But in most of the book Webster uses &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; to denote electric potential. In the section on magneto-statics, the symbol &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; is used for a magnetic potential.&lt;/p&gt;&#xA;&lt;p&gt;In Thomson's &lt;em&gt;Elements of the Mathematical Theory of Electricity and Magnetism&lt;/em&gt; (first published 1895, version I have available is from 1909) he uses &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; to denote potential, and &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; for magnetic potential. Jeans, &lt;em&gt;The Mathemetical Theory of Electricity and Magnetism&lt;/em&gt; (1908, 1925), uses &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\Omega$&lt;/span&gt; as well.&lt;/p&gt;&#xA;&lt;p&gt;The first textbook that I'm familiar with that uses &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; to denote potential is Abraham and Foppl, &lt;em&gt;Theorie der Elektrizität [Theory of Electricity]&lt;/em&gt;, first published in 1904, and the versions I found are from &lt;a href=&quot;https://archive.org/details/theoriederelekt04fpgoog/page/n52/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;1905&lt;/a&gt; and 1918. This is a German text, which might give us a hint that the notation &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; (both used at different times by Abraham) were more popular in German texts than in English. The translations of this text (8th and 14th editions, under the names Abraham and Becker) was very popular as well in its time, but I only have a copy of the 2nd edition from 1950. It uses &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt; to denote potential, keeping with the German convention.&lt;/p&gt;&#xA;&lt;p&gt;From what I can tell, Mason and Weaver's &lt;em&gt;The Electromagnetic Field&lt;/em&gt; (1929) uses &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt; to denote the electrostatic potential, which might be one of the earlier popular English textbooks on classical electromagnetism to use this convention.&lt;/p&gt;&#xA;&lt;p&gt;So it seems like in terms of elementary textbooks, while the notation &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; was predominant early on, the use of &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; goes back at least to Maxwell, possibly earlier, but it didn't catch on until the 1930s in the English-speaking world. It may have been adopted due to the popularity of German texts, but that's just my guess. This is all supposing that textbooks are a reasonable source of information on common conventions, but obviously published papers and more specialized texts could have been using different symbols at an earlier time, it's just harder to sort through to find occurrences.&lt;/p&gt;&#xA;&lt;p&gt;As for why these Greek letters were chosen, if I had to propose a reason, it was just because they were (and still are) commonly used to denote auxiliary functions during proofs and intermediate steps, or otherwise because those symbols were not being used to denote other quantities.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2022-04-26T14:37:24.463" LastActivityDate="2022-04-26T14:37:24.463" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14400" PostTypeId="1" CreationDate="2022-04-25T06:07:46.517" Score="0" ViewCount="113" Body="&lt;p&gt;I saw this on &lt;a href=&quot;https://en.wikipedia.org/wiki/Georg_Cantor#Later_years_and_death&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What happened to Cantor and his wife? Was his mental condition so bad that his wife wanted him to stay there?&lt;/p&gt;&#xA;" OwnerUserId="16566" LastEditorUserId="4251" LastEditDate="2022-04-27T05:23:17.817" LastActivityDate="2022-04-27T05:23:17.817" Title="Cantor's later life" Tags="|mathematicians|biographical-details|cantor|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14401" PostTypeId="1" CreationDate="2022-04-26T21:36:37.533" Score="2" ViewCount="161" Body="&lt;p&gt;Maxwell published his &lt;a href=&quot;https://en.wikipedia.org/wiki/A_Treatise_on_Electricity_and_Magnetism&quot; rel=&quot;nofollow noreferrer&quot;&gt;Treatise on Electricity and Magnetism&lt;/a&gt; in 1873, 150 years ago; before the discovery of quantized charges, special relativity, quantum field theory etc. How has the mathematical and physical modelling of &lt;em&gt;classical&lt;/em&gt; electrodynamics today changed since then?&lt;/p&gt;&#xA;" OwnerUserId="2461" LastEditorUserId="4251" LastEditDate="2022-04-27T12:27:59.293" LastActivityDate="2022-05-02T17:43:37.897" Title="How has the modelling of classical electrodynamics changed since Maxwell?" Tags="|physics|electromagnetism|quantum-field-theory|maxwell|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14403" PostTypeId="5" CreationDate="2022-04-27T05:36:17.750" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-04-27T05:36:17.750" LastActivityDate="2022-04-27T05:36:17.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14404" PostTypeId="4" CreationDate="2022-04-27T05:36:17.750" Score="0" Body="Leopold Kronecker (1823-1891) was a German mathematician who worked on number theory, algebra and logic." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2022-04-28T10:06:37.647" LastActivityDate="2022-04-28T10:06:37.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14405" PostTypeId="1" CreationDate="2022-04-27T15:05:55.250" Score="1" ViewCount="153" Body="&lt;p&gt;[I posted this to &lt;a href=&quot;https://history.stackexchange.com/questions/68870/what-kind-of-building-materials-did-cavendish-use-in-determining-the-density-of&quot;&gt;History&lt;/a&gt;, but it was suggested that History of Science and Mathematics would be a better choice. So I'm posting it here too.]&lt;/p&gt;&#xA;&lt;p&gt;I’m working to build the Cavendish experiment of 1798 according to Cavendish’s own specifications. I have a question about building materials Cavendish may have used.&lt;/p&gt;&#xA;&lt;p&gt;Cavendish put his pendulum inside a small building to protect it from wind and rain. The building has dimensions of approximately 3.5 meters. What kind of building materials Cavendish may have used in 1798?&lt;/p&gt;&#xA;&lt;p&gt;You can see a &lt;a href=&quot;https://www.scienceandsociety.co.uk/results.asp?image=10314095&quot; rel=&quot;nofollow noreferrer&quot;&gt;scale model of the experiment&lt;/a&gt;. In the model they chose to make a brick building. Do you think this is what Cavendish used? Was cement in use then?&lt;/p&gt;&#xA;&lt;p&gt;Cavendish calls this building a &amp;quot;room&amp;quot; and gives no other information about it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...I resolved to place the apparatus in a room which should remain constantly shut...&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;&lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1798.0022&quot; rel=&quot;nofollow noreferrer&quot;&gt;Experiments to determine the Density of the Earth.&lt;/a&gt; By Henry Cavendish, &lt;em&gt;Esq. F.R.S. and A.S.&lt;/em&gt;, p.471&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16575" LastEditorUserId="4251" LastEditDate="2022-04-28T17:52:11.790" LastActivityDate="2022-04-28T17:52:11.790" Title="The housing of the Cavendish Experiment" Tags="|physics|experimental-physics|18th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14406" PostTypeId="1" CreationDate="2022-04-27T19:58:43.867" Score="0" ViewCount="89" Body="&lt;p&gt;I was reading about Catalan numbers and how they provide the number count in a variety of situations.&lt;/p&gt;&#xA;&lt;p&gt;I found it very interesting how the main proof is about the underlying common patterns and symmetry of problems that appear very different e.g. well formed parenthesis, lattice paths, mountains and trees etc.&lt;/p&gt;&#xA;&lt;p&gt;I think Catalan only formally introduced so I was wondering if we know how far back they were known and how the first proofs/interpretations were inspired?&lt;/p&gt;&#xA;" OwnerUserId="14662" LastEditorUserId="4251" LastEditDate="2022-04-30T18:11:43.723" LastActivityDate="2022-04-30T18:11:43.723" Title="Origin of Catalan numbers and original intuition of the first interpretation and proofs" Tags="|mathematics|numbers|combinatorics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14407" PostTypeId="2" ParentId="14405" CreationDate="2022-04-27T21:37:48.463" Score="2" Body="&lt;p&gt;Neither the original publication: Henry Cavendish, &amp;quot;XXI. Experiments to determine the density of the earth.&amp;quot; &lt;em&gt;Philosophical Transactions of the Royal Society of London&lt;/em&gt;, Vol. 88, 1798, pp. 469-526 (&lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1798.0022&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;), nor the annotated German translation: Henry Cavendish,&amp;quot;Versuche, um die Dichtigkeit der Erde zu bestimmen.&amp;quot; &lt;em&gt;Annalen der Physik&lt;/em&gt;, Volume 2, Part 1, 1799, pp. 1-62 (&lt;a href=&quot;https://www.google.com/books/edition/Annalen_der_Physik/c1_HF4dUSmUC?gbpv=1&amp;amp;bsq=Henry%20Cavendish&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;) mention a separate building; they both just mention a room, &lt;em&gt;presumably&lt;/em&gt; in an existing building.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As I was convinced of the necessity of guarding against this source of error, I resolved to place the apparatus in a room which should remain constantly shut, and to observe the motion of the arm from without, by means of a telescope;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From 1782 to 1810 Cavendish resided at a mansion: Cavendish House, Clapham Common, a picture of which is &lt;a href=&quot;https://boroughphotos.org/lambeth/cavendish-house-clapham-common-south-side-5/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. This was in the south of London, in the Borough of Lambeth. Best I can establish, this is where Cavendish conducted his experiments. For example, the &lt;a href=&quot;https://www.britannica.com/biography/Henry-Cavendish/Experiments-with-electricity&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia Britannica&lt;/a&gt; states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Following his father’s death, Henry bought another house in town and also a house in Clapham Common, to the south of London. The London house contained the bulk of his library, while he kept most of his instruments at Clapham Common, where he carried out most of his experiments.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.londonpicturearchive.org.uk/view-item?i=124728&amp;amp;WINID=1651094673530&quot; rel=&quot;nofollow noreferrer&quot;&gt;this picture&lt;/a&gt; of the house shortly before, or during, its demolition it appears to have been constructed from brick masonry.&lt;/p&gt;&#xA;&lt;p&gt;The book: Christa Jungnickel and Russell McCormmach, &amp;quot;Cavendish — The Experimental Life, 2nd ed.&amp;quot;, Berlin: Max-Planck-Gesellschaft zur Förderung der Wissenschaften 2016 contains the following &lt;a href=&quot;https://www.mprl-series.mpg.de/studies/7/14/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;description&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cavendish’s house has been called a mansion, but a better description of it from the time is “a tolerable good house, built with red brick.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In comments OP raises the possibility of the experiment having been conducted in an outbuilding. The only specific reference to an outbuilding at Cavendish House that I could find in Jungnickel / McCormmach during a quick perusal is to a greenhouse mentioned in Cavendish's notes on experiments on air. Modern greenhouses consist largely of glass, which would rule them out as the location of the experiment, but methods of construction used for greenhouses may have been very different during Cavendish's time.&lt;/p&gt;&#xA;&lt;p&gt;An &lt;a href=&quot;https://boroughphotos.org/lambeth/clapham-and-clapham-common-map/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1877 map&lt;/a&gt; appears to show a group of three outbuildings at the far end of the Cavendish House property, but whether any of these existed in 1798 is impossible to know.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-04-28T07:10:39.207" LastActivityDate="2022-04-28T07:10:39.207" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14408" PostTypeId="1" AcceptedAnswerId="14414" CreationDate="2022-04-27T22:47:58.397" Score="3" ViewCount="232" Body="&lt;p&gt;Before Marx, economic theory had two big names: Adam Smith and Ricardo. It is well known that Marx used their work. When at school I was told without much specific that Marx had substantial contribution. What is the most important contribution of Marx to the theory of economics?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;PS.&lt;/strong&gt; Some people claim that the so-called surplus value was introduced by Marx and is his major contribution. But according to this &lt;a href=&quot;https://en.wikipedia.org/wiki/Surplus_value&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; this concept existed and was studied by both Smith and Ricardo. So my question remains.&lt;/p&gt;&#xA;&lt;p&gt;Note that Marx created &amp;quot;Political Economy&amp;quot;, but that is not an economics theory.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2022-04-28T03:19:03.417" LastActivityDate="2022-04-30T06:37:37.493" Title="Marx economics theory" Tags="|economics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14410" PostTypeId="1" AcceptedAnswerId="14412" CreationDate="2022-04-29T00:38:37.010" Score="1" ViewCount="138" Body="&lt;p&gt;I may be mistaken but I understood that Borel developed his sigma algebra before Lebesgue developed his measure. If correct, then Borel can't have been trying to find a collection of sets in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; that would support his student's measure. What was his motivation? Maybe my premise is wrong.&lt;/p&gt;&#xA;&lt;p&gt;I've Googled and checked this exchange (and other similar exchanges) and I've not been able to get even a hint that this question has been asked.&lt;/p&gt;&#xA;" OwnerUserId="15921" LastActivityDate="2022-04-29T23:24:59.137" Title="What problem was Borel trying to solve?" Tags="|set-theory|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14412" PostTypeId="2" ParentId="14410" CreationDate="2022-04-29T11:08:15.557" Score="4" Body="&lt;p&gt;Borel was trying to understand &amp;quot;countable probabilities&amp;quot;. For example, problems like this: for a given sequence &lt;span class=&quot;math-container&quot;&gt;$a_n\to 0, a_n&amp;gt;0$&lt;/span&gt;, what is the probability&#xA;that the series &lt;span class=&quot;math-container&quot;&gt;$\sum\pm a_n$&lt;/span&gt; converges? Or the &amp;quot;StPetersburg game&amp;quot;: you play against a casino. A fair coin is tossed repeatedly, until the first appearance of &amp;quot;eagle&amp;quot;. If this happens on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th toss, you win &lt;span class=&quot;math-container&quot;&gt;$2^n$&lt;/span&gt;. What is the fair entrance fee for this game?&lt;/p&gt;&#xA;&lt;p&gt;He explained his approach himself in his nice little book Probability and certainty.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-04-29T23:24:59.137" LastActivityDate="2022-04-29T23:24:59.137" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14413" PostTypeId="2" ParentId="14408" CreationDate="2022-04-29T11:26:44.180" Score="2" Body="&lt;p&gt;The concept of &amp;quot;surplus value&amp;quot; existed before Marx, but he developed a comprehensive theory, trying to prove it by statistical data. His theory predicted that development of capitalism will lead to increasing exploitation, impoverishment of the mass of proletarians, and this must inevitably lead to a &amp;quot;proletarian revolution&amp;quot;. History showed that this never happened. (Bolsheviks who seized power in Russia in 1917 claimed that this was a &amp;quot;proletarian revolution&amp;quot; but it was not. Similar claims were made by other communist parties who tried to seize power). By the early 20th century it became clear to many followers of Marx that the main conclusion of Marx's theory was wrong,&#xA;and &amp;quot;marxism&amp;quot; was reformed giving rise to modern socialist parties which come to power by democratic means in many countries of Europe. But they are not talking about &amp;quot;proletarian revolution&amp;quot; anymore.&#xA;Still they maintain that many features of Marx's criticism of capitalism remain valid, but that the transition to socialism will happen in a peaceful, democratic and gradual way.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-04-29T11:26:44.180" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14414" PostTypeId="2" ParentId="14408" CreationDate="2022-04-30T06:01:43.897" Score="3" Body="&lt;p&gt;Since there is no good answer, I will post my thoughts. First, &lt;strong&gt;background&lt;/strong&gt;. As many people in Soviet Union, I studied various parts of Marxism for many (more than 10) years. I was a good student, A's in all subjects, and unlike many, I actually read what the man wrote. So I believe that I know Marxism at least as well as a Professor of Marxism in a major US University (lots of them do not know even the &lt;a href=&quot;https://en.wikipedia.org/wiki/Gotha_Program#:%7E:text=The%20Gotha%20Program%20was%20explicitly,of%20every%20kind%2C%20and%20to&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Gotha Program&amp;quot;&lt;/a&gt; or &lt;a href=&quot;https://www.marxists.org/archive/marx/works/1852/letters/52_03_05-ab.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Marx's letter to J. Weydemeyer in New York&lt;/a&gt;&amp;quot; , where he introduced the concept of &amp;quot;Proletarian Dictatorship&amp;quot;. None of the Professors I spoke with could formulate the 11-th thesis of Feuerbach: &amp;quot;Philosophers have hitherto only interpreted the world in various ways; the point is to change it&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So &lt;strong&gt;to my answer&lt;/strong&gt;. There is no other scientist in the World, past of present, who has been so politicised -- for obvious reasons. Marx created (basically singlehandedly) Political Science, the theory of class struggle for power. For this he used many classical parts of science: Economy, Philosophy, etc. He did not contribute much to these parts, but he showed how they can be used and interpreted in terms of political science. In particular, in terms of Economics theory, which basically dealt, as described by the famous African-Russian writer Pushkin, with questions like &amp;quot;как государство богатеет и чем живёт, и почему не нужно золота ему, когда простой продукт имеет&amp;quot;  (a very rough translation: &amp;quot;how the state gets richer and how it lives, and why it doesn't need gold when it has a simple product&amp;quot;), Marx did not contribute much over Smith and Ricardo. But he found a strong (for him and his followers at least) connection between the Economics theory and Political Economy. Similar - in Philosophy. He did not overdo Hegel or even Feuerbach. But he created &amp;quot;Dialectical materialism.&lt;/p&gt;&#xA;" OwnerUserId="14548" LastEditorUserId="14548" LastEditDate="2022-04-30T06:22:53.073" LastActivityDate="2022-04-30T06:22:53.073" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14415" PostTypeId="1" CreationDate="2022-04-30T22:52:13.400" Score="2" ViewCount="247" Body="&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How and why did the discipline of mathematics come to use &lt;span class=&quot;math-container&quot;&gt;$\log(x)$&lt;/span&gt; and not &lt;span class=&quot;math-container&quot;&gt;$\ln(x)$&lt;/span&gt; in its writing - papers, books, teaching?&lt;/p&gt;&#xA;&lt;p&gt;Clearly &lt;span class=&quot;math-container&quot;&gt;$\ln(x)$&lt;/span&gt; is unambiguously &lt;span class=&quot;math-container&quot;&gt;$\log_e(x)$&lt;/span&gt;, but &lt;span class=&quot;math-container&quot;&gt;$\log(x)$&lt;/span&gt; is ambiguous.&lt;/p&gt;&#xA;&lt;p&gt;Almost all textbooks, papers and informal formats like blogs and YouTube videos use &lt;span class=&quot;math-container&quot;&gt;$\log(x)$&lt;/span&gt; and assume base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="12633" LastActivityDate="2022-05-04T02:37:45.050" Title="How did mathematics come to use $\log(x)$ and not $\ln(x)$" Tags="|logarithm|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14416" PostTypeId="2" ParentId="14415" CreationDate="2022-04-30T23:04:00.303" Score="3" Body="&lt;p&gt;For me, handwritten especially, but also typeset, &amp;quot;ln&amp;quot; is easier to mis-read than &amp;quot;log&amp;quot;. And since I do not ever use any logarithm but &amp;quot;base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&amp;quot;, (except, rarely, base &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; for some information-theory things maybe), there is simply no ambiguity.&lt;/p&gt;&#xA;&lt;p&gt;It's true, it is easier to start kids with logs base &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;, and in calculus make the distinction to/from &amp;quot;ln&amp;quot;, ... but that is a transitional phase, I think.&lt;/p&gt;&#xA;&lt;p&gt;(Financial computations do also use logs base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, for reasons of natural phenomena! But, one way or another, almost no people have any reason to directly worry about which logarithm was intended...)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-04-30T23:04:00.303" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14418" PostTypeId="1" AcceptedAnswerId="14420" CreationDate="2022-05-01T15:36:04.387" Score="5" ViewCount="322" Body="&lt;p&gt;Did the ancient Greeks, for example, speak of number of paces per heartbeat? I could see them only saying, something moved with the speed of an arrow or a running horse or a running man but that they lacked a unifying concept of speed being distance covered in a constant time. I am not even sure the Greeks understood there to be an abstract time unit that did not vary -- the time keeper for legal court that was invented divided a day into 12 (or some number) of equal sections but the length of each unit varied according to the time of year because days were shorter.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: Even if the ancients understood time as being measured in something like seconds, when speed or velocity was first thought as distance divided by time remains my question. My guess is that this occurred after 1000 AD -- Merton's Theorem I think would imply a modern way of looking at velocity: &lt;a href=&quot;https://www.britannica.com/science/Merton-theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.britannica.com/science/Merton-theorem&lt;/a&gt; and I also wonder how scientists in China or India thought of velocity or even in pre-Columbian America.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2022-09-06T08:01:05.523" LastActivityDate="2022-09-06T08:01:05.523" Title="First attempt to measure speeds using distance divided by time?" Tags="|measurement|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14419" PostTypeId="2" ParentId="524" CreationDate="2022-05-01T19:54:06.947" Score="0" Body="&lt;p&gt;Eubulides?  He's from the 4th century BC and is credited with the Sorites or heap paradox.  If you have one grain of sand you don't have a heap.  If you add one grain of sand to a non-heap it remains a non-heap.  But eventually you have enough grains to make a heap.&lt;/p&gt;&#xA;&lt;p&gt;It seems that you need some form of induction to make the paradox work, but unfortunately the work of Eubulides doesn't survive, so we don't know how he formulated it.&lt;/p&gt;&#xA;" OwnerUserId="14849" LastActivityDate="2022-05-01T19:54:06.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14420" PostTypeId="2" ParentId="14418" CreationDate="2022-05-01T22:34:27.880" Score="5" Body="&lt;p&gt;Greek astronomers of the Hellenistic epoch had a good understanding of time and speed in astronomy. They understood that neither Sun nor Moon move with constant speed, had the notion of average speed (still called &amp;quot;mean motion&amp;quot; in astronomy) and deviation of the actual speed from this average (which they called &amp;quot;equation&amp;quot;, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Equation_of_time&quot; rel=&quot;nofollow noreferrer&quot;&gt;Equation of time&lt;/a&gt;). So they had the concept of uniform time.&lt;/p&gt;&#xA;&lt;p&gt;In daily life they used unequal, seasonal hours, but astronomers understood that they are unequal and took into account the necessary corrections.&lt;/p&gt;&#xA;&lt;p&gt;I believe that minutes and seconds of time were used only by astronomers.&#xA;I do not know of any&#xA;Greek texts with a quantitative discussion of time and speed in physics.&#xA;Or any discussion of measuring speed.&lt;/p&gt;&#xA;&lt;p&gt;Edit. To address the comment about &lt;a href=&quot;https://en.wikipedia.org/wiki/Greek_and_Roman_artillery&quot; rel=&quot;nofollow noreferrer&quot;&gt;Greek and Roman artillery&lt;/a&gt;. 5 ancient technical books on this subject survive. None of them addresses the subject which we would call &amp;quot;ballistics&amp;quot;. There is even no attempt to discuss the motion of a projectile, so these books confirm what I wrote above. All known sources confirm that Greek physics was limited to statics (including hydrostatics). But there is no hint of (mathematical, that is quantitative) dynamics, or even kinematics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-05-04T21:27:34.360" LastActivityDate="2022-05-04T21:27:34.360" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14421" PostTypeId="1" AcceptedAnswerId="14422" CreationDate="2022-05-02T02:13:30.147" Score="1" ViewCount="135" Body="&lt;p&gt;Does any of you happen to own and electronic copy of the paper &amp;quot;Fermat et l'équation de Pell&amp;quot; by A. Weil?&lt;/p&gt;&#xA;&lt;p&gt;If I understand correctly, this paper can be found on pages 413-419 of the third volume of Weil's &lt;em&gt;Oeuvres Scientifiques&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In case you have it at hand and have read it already, would you be so kind as to briefly tell us the main points Weil made therein regarding Fermat's actual contributions to the theory of the Pell equation?&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-05-02T15:36:34.667" LastActivityDate="2022-05-02T15:36:34.667" Title="Fermat et l'équation de Pell" Tags="|reference-request|number-theory|fermat|weil|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14422" PostTypeId="2" ParentId="14421" CreationDate="2022-05-02T11:32:43.387" Score="2" Body="&lt;p&gt;In this short article he explains the history of the solution of Pell's equation. It was proposed by Fermat in his letter to British mathematicians, and solved by Wallis and Brouncker, jointly, in their correspondence. They explained their solution on a number of special cases. Fermat criticized this solution asserting that they did not give a general proof. Then Weil speculates whether Fermat himself had a &amp;quot;general proof&amp;quot; himself and what this proof could be. He also analyses the &amp;quot;British solution&amp;quot;, and establishes its connection with the Euclidean algorithm and a continued fraction algorithm.&lt;/p&gt;&#xA;&lt;p&gt;The first published complete proof is due to Lagrange. Weil also gives some detail about pre-history of this solution, and mentions Bachet and Pascal.&#xA;In the beginning he notices that there is no known connection of this equation with John Pell, who was famous as a mathematician during his life time but published nothing.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="136" LastEditDate="2022-05-02T15:34:26.357" LastActivityDate="2022-05-02T15:34:26.357" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14423" PostTypeId="1" CreationDate="2022-05-02T16:36:00.890" Score="0" ViewCount="76" Body="&lt;p&gt;How did people before the advent of modern science in the 17th century conceive thermodynamic phenomena? I want to know how people in Ancient Mesopotamia, Egypt, Greece, Rome, China and in the Middle Ages thought about temperature, heat, pressure, melting point, and the phase transitions between solid, liquid and gas, given that they are commonly observed phenomena, relevant in our daily lives.&lt;/p&gt;&#xA;" OwnerUserId="10492" LastEditorUserId="12330" LastEditDate="2022-05-02T22:21:17.117" LastActivityDate="2022-05-02T22:21:17.117" Title="What concept did pre-modern people have about thermodynamic phenomena?" Tags="|physics|thermodynamics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14424" PostTypeId="2" ParentId="14401" CreationDate="2022-05-02T17:43:37.897" Score="3" Body="&lt;p&gt;We will take the cutoff to be 1873, the publication date of the first edition of Maxwell's treatise. I'm not an expert in electrodynamics by any stretch, but here are some of the things that come to mind.&lt;/p&gt;&#xA;&lt;p&gt;Mathematically, the most significant difference between Maxwell and 20th century (and beyond) electrodynamics is the use of vector notation. Maxwell worked with vectors component-wise, and this sort of trend continued into the early parts of the 20th century, despite the development of vector notation by Heaviside and Gibbs in the late 19th century. Aside from the change in notation, much of the work was related to partial differential equations, which were fairly well developed by Maxwell's time. Most of our analytical &amp;quot;tricks&amp;quot; in electrostatics, like spherical harmonics, Green's functions, method of images, etc were already developed and thoroughly studied, but the conception of real analysis was certainly lagging behind (e.g. Dirac delta and distribution theory). Mathematical techniques used to approximate the solution to partial differential equations, like variational techniques, were certainly possible (the calculus of variations is old) but I personally haven't seen much in this vain in older literature. It's also usually dependent on integral equations, or functional analysis, two fields which saw most of their development in the 20th century. Even still, most textbooks going into the 20th century were based on Maxwell's coverage, so e.g. Jackson's &lt;em&gt;Classical Electrodynamics&lt;/em&gt; or Smythe's &lt;em&gt;Static and Dynamic Electricity&lt;/em&gt; are, in their first few chapters, mostly restatements and summaries of this work, but with the benefit of many decades of refinement.&lt;/p&gt;&#xA;&lt;p&gt;Certain problems were awaiting more satisfactory solutions, for example the electromagnetic field due to a moving point charge, but some of the groundwork was done and already gave hints towards a relativistic theory. The use of electric and magnetic scalar and vector potentials was justifiably seen as essential to the theory of electromagnetism, but the concept of gauge freedom was a point of contention. Problems in radiation were still new and underdeveloped, but the wave behavior was embedded in the theory and emphasized by Maxwell, despite a lack of direct experimental evidence.&lt;/p&gt;&#xA;&lt;p&gt;The most significant changes in the theory are more related to the interpretation. An obvious example is the aether, which was essential to Maxwell's conception of the electromagnetic field. This is probably the most significant change, but also the most immaterial to the mathematical side. The aether theory was put to rest at the turn of the 20th century, in light of growing evidence that the aether had essentially no role to play in the behavior of the electromagnetic field, and could be neglected without much reformulating. Other somewhat metaphysical theories and perspectives as well as conceptual tools (e.g. tubes of force) would grow and wane in popularity, before mostly disappearing from literature.&lt;/p&gt;&#xA;&lt;p&gt;Another significant change was the introduction of the Poynting vector (1884), a term in the power balance of electromagnetic waves which is consistent with Maxwell's theory. The interpretation of this &amp;quot;power flow&amp;quot; term was debated going into the 20th century, but its importance was accepted and it now plays a central role in the study of electromagnetic field energy.&lt;/p&gt;&#xA;&lt;p&gt;The limits of classical electrodynamics have been refined, as our understanding of the structure of matter grows. Things which were in Maxwell's time simply hypotheses were found to be true (atomic nature of matter, as well as electric charge). The application of electrodynamics to the dynamics of charged particles has taken a central role in electrodynamic theory. The study of radiation, and the study of electromagnetic waves, have both been developed and applied to things like optics, microwaves, and RF communication. Examples being the theory of antennas, or the theory of waveguides.&lt;/p&gt;&#xA;&lt;p&gt;Depending on your scope of &amp;quot;electrodynamics&amp;quot;, the rise of computing has been hugely impactful in industry, and for theoretical study. But I would say this is more application-related, as it does not develop the theory itself.&lt;/p&gt;&#xA;&lt;p&gt;Extension of the theory to other states of matter (notably plasma) might be getting close to the quantum nature of matter, and this isn't something I have experience with. But it's certainly been a large area of study for decades, and could be a significant development since Maxwell's time.&lt;/p&gt;&#xA;&lt;p&gt;There have also been important experimental results, as you can imagine. Trying to keep to experiments which apply classically, we have Hertz's verification of electromagnetic waves was significant for obvious reasons; Michelson-Morely was another nail in the aether's coffin; the Stewart-Tolman experiment demonstrated that the charge carriers in various conductors were negatively charged, in an experiment originally proposed by Maxwell; verifications of Coulomb's law by Plimpton, then later by Bartlett, and others.&lt;/p&gt;&#xA;&lt;p&gt;All in all, while the PDEs at the heart of classical electrodynamics have not changed much (in substance), and many of our problem solving strategies are carried over from Maxwell's time, there was still a huge amount of work to be done in applications and in setting the limits. Much of the theoretical work was closely tied to the emerging quantum theory and the theory of special relativity, the latter introducing the more modern tensor notation to electromagnetism, and providing a closer connection between the electric and magnetic fields. Still, Maxwell's most outdated ideas relate to the structure of matter and space, not so much to the mathematical theory, which were just more immature.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2022-05-02T17:43:37.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14425" PostTypeId="2" ParentId="14418" CreationDate="2022-05-03T04:21:30.967" Score="3" Body="&lt;p&gt;I'm not sure about Greeks, but Ossendrijver argues that Babylonian astronomers ca. 350-50 BCE used the trapezoidal rule to integrate Jupiter's motion along the ecliptic. The velocity is measured in degrees per day. The method of calculation resembles the trapezoidal rule and the much later “Mertonian mean speed theorem,” all corresponding to the formula &lt;span class=&quot;math-container&quot;&gt;$s = t\cdot(v_0 + v_1)/2$&lt;/span&gt;.  Apparently, as Ossendrijver argues, the method was applied to determine the time that Jupiter crossed the ecliptic.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Reference&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Mathieu Ossendrijver,&#xA;Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph,&#xA;&lt;em&gt;Science,&lt;/em&gt; 29 Jan 2016, Vol 351, Issue 6272. pp. 482-484&lt;br /&gt;&#xA;&lt;a href=&quot;https://doi.org/10.1126/science.aad8085&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1126/science.aad8085&lt;/a&gt; (paywalled, sorry)&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2022-05-03T04:21:30.967" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14426" PostTypeId="1" CreationDate="2022-05-03T13:07:23.617" Score="14" ViewCount="4041" Body="&lt;p&gt;In a 1508 German book on arithmetic for merchants, this image appears on the cover (high resolution and context &lt;a href=&quot;https://www.digitale-sammlungen.de/en/view/bsb00003523?page=2,3&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.digitale-sammlungen.de/en/view/bsb00003523?page=2,3&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/b9IqJ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;What is this depicting? Who are these people and what are their roles? In particular, what are the symbols on the table? What is happening?&lt;/p&gt;&#xA;" OwnerUserId="16591" LastActivityDate="2022-05-05T12:39:16.450" Title="What does this cover of a 1508 German book on arithmetic depict?" Tags="|arithmetic|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14427" PostTypeId="2" ParentId="14426" CreationDate="2022-05-03T17:16:11.707" Score="13" Body="&lt;p&gt;The image appears to depict a &lt;a href=&quot;https://en.wikipedia.org/wiki/Counting_board&quot; rel=&quot;noreferrer&quot;&gt;counting board&lt;/a&gt;.  Since the context is mercantilism, it is most likely depicted being used by tax collectors or accountants.  The symbols are most likely stylizations.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jjY0o.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jjY0o.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-05-03T17:16:11.707" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14428" PostTypeId="2" ParentId="14418" CreationDate="2022-05-03T23:18:41.667" Score="3" Body="&lt;p&gt;The Euclidean theory of proportion appears to speak against the ancient use of speed as a quotient between distance and time. Ratios, or proportions, according to that theory, were seen as legitimate only if they were established between quantities of like kind -- we might say, similarly dimensioned.&lt;/p&gt;&#xA;&lt;p&gt;Explanation is given for example by N Guicciardini in &amp;quot;Mathematics and the New Sciences&amp;quot;, chapter 8, (esp. sec. 8.3.2) in &lt;a href=&quot;https://global.oup.com/academic/product/the-oxford-handbook-of-the-history-of-physics-9780198805328?cc=gb&amp;amp;lang=en&amp;amp;&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Oxford Handbook of the History of Physics' ed J Buchwald et al., 2013&lt;/a&gt;. Guicciardini wrote (in the context of discussing the heritage of antiquity still operative down ro the time of Galileo):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since antiquity, continuous magnitudes had been approached through the&#xA;theory of proportions, codified in Book 5 of Euclid’s Elements. [...]&#xA;Note that the theory of proportions does not allow the formation of a&#xA;ratio between two heterogeneous magnitudes. This is particularly&#xA;important for kinematics, since it is not possible, for instance, to&#xA;define speed as a ratio between distance and time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The relationship between time, distance and speed in equable motion&#xA;might {now} be expressed as&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;    s = v · t. &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;This was not possible within the framework of the theory of&#xA;proportions, since one had to express a magnitude, speed, as being&#xA;equal to the ratio between two heterogeneous magnitudes, distance and&#xA;time. One has instead to state a series of proportions allowing only&#xA;two of the three magnitudes involved (distance, time and speed) to&#xA;vary.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What happened was that a kind of circumlocution was in use : (from the same source):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;‘when the speed is the same and we compare two equable motions the&#xA;distances are as the times’:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;  s1/s2 = t1/t2, &#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;where s1 (s2) is the distance covered in time t1 (t2). Or, ‘when the&#xA;distance covered is the same, the speeds are as the times inversely’:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;  v1/v2 = t2/t1.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It looks as though this doctrine had a considerable constraining effect on the ways in which motion might be conceptualised. The inhibitions appear to have lasted until the 17th century. According to Guicciardini in the work cited above, &amp;quot;Descartes’ geometry replaced the theory of proportions&amp;quot;. It seems to have been then that more flexible patterns of thought about motion, including velocity, gained currency and accessibility.&lt;/p&gt;&#xA;&lt;p&gt;(Some evidence in support of this late date for clear thinking about velocity comes from old works of astronomy, especially tables of motions. There are clear signs in many of them that motion and position were not clearly distinguished from each other. There was for example a concept of 'mean motion' that -- from the evidence of the numbers associated with it -- covered both place at a given point in time, and change of place in a given increment of time.)&lt;/p&gt;&#xA;&lt;p&gt;(edit:) About the Babylonian astronomy: there may be evidence in the cuneiform tablets that the Babylonian astronomers expressly considered speed. Modern knowledge of this, however, dates only from the late 19th and early 20th century. The Greeks obtained some numerical parameters from ultimately Babylonian sources. It's not clear how much they obtained besides some numerical relationships that we might express as ratios, but they expressed as pairs of numbers of events of different types happening in the same interval of time.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-05-04T01:21:21.313" LastActivityDate="2022-05-04T01:21:21.313" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14430" PostTypeId="1" CreationDate="2022-05-04T03:54:39.217" Score="2" ViewCount="89" Body="&lt;p&gt;Prior to the development of electronic computers, human computers were used and computer departments consisted of a room with humans at desks performing calculations.&lt;/p&gt;&#xA;&lt;p&gt;Large scientific organizations, such as Nasa and &lt;a href=&quot;https://ttps://hsm.stackexchange.com/questions/5616/day-to-day-tasks-of-human-computers-%C3%A0-la-hidden-figures-movie&quot; rel=&quot;nofollow noreferrer&quot;&gt;Los Alamos Laboratories&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Gilbert_Walker_(physicist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gilbert Walker&lt;/a&gt; in India in the early 1900s, when he utilized&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a group of Indian clerks to calculate correlations between weather parameters&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;that lead to the discovery of the &lt;a href=&quot;https://www.climate.gov/news-features/understanding-climate/climate-variability-southern-oscillation-index&quot; rel=&quot;nofollow noreferrer&quot;&gt;Southern Oscillation Index&lt;/a&gt;, all utilized in house human computers.&lt;/p&gt;&#xA;&lt;p&gt;Were there external organizations, anywhere, that offered computing services for a fee or were computers employed in house or hired for specific projects?&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2022-05-04T03:54:39.217" Title="Bureau human computing services" Tags="|computation|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14431" PostTypeId="2" ParentId="14426" CreationDate="2022-05-04T07:45:30.943" Score="1" Body="&lt;p&gt;It seems to depicture the page setting of a math book in the new technique developed by &lt;a href=&quot;https://en.wikipedia.org/wiki/Johannes_Gutenberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johannes Gutenberg&lt;/a&gt;. If you look closely you can see the math letters. The year 1508 was within a gold rush period for printers.&lt;/p&gt;&#xA;" OwnerUserId="16593" LastActivityDate="2022-05-04T07:45:30.943" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14432" PostTypeId="1" CreationDate="2022-05-05T01:19:39.037" Score="2" ViewCount="81" Body="&lt;p&gt;At &lt;code&gt;01:15:13&lt;/code&gt; in the delightful video &lt;a href=&quot;https://youtu.be/JP1Ljp8X6hg?t=4513&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Bit Player | Claude Shannon | Father of Information Theory | IEEE Information Theory Society&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Andrea_Goldsmith_(engineer)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Andrea Goldsmith&lt;/a&gt; recounts attending a session of the IEEE International Conference on Communication conference and hearing a presentation by &lt;a href=&quot;https://en.wikipedia.org/wiki/Claude_Berrou&quot; rel=&quot;nofollow noreferrer&quot;&gt;Claude Berrou&lt;/a&gt; which listed &lt;a href=&quot;https://en.wikipedia.org/wiki/Alain_Glavieux&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alain Glavieux&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Punya_Thitimajshima&quot; rel=&quot;nofollow noreferrer&quot;&gt;Punya Thitimajshima&lt;/a&gt; as coauthors.&lt;/p&gt;&#xA;&lt;p&gt;From Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Turbo_code#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turbo code; History&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first public paper on turbo codes was &amp;quot;Near Shannon Limit Error-correcting Coding and Decoding: Turbo-codes&amp;quot;.[2] This paper was published 1993 in the Proceedings of IEEE International Communications Conference. &lt;strong&gt;The 1993 paper was formed from three separate submissions that were combined due to space constraints.&lt;/strong&gt; The merger caused the paper to list three authors: Berrou, Glavieux, and Thitimajshima (from Télécom Bretagne, former ENST Bretagne, France). However, it is clear from the original patent filing that Berrou is the sole inventor of turbo codes and that the other authors of the paper contributed material other than the core concepts.&lt;sup&gt;[improper synthesis]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;[2] C. Berrou, A. Glavieux and P. Thitimajshima, &amp;quot;&lt;a href=&quot;https://ieeexplore.ieee.org/document/397441&quot; rel=&quot;nofollow noreferrer&quot;&gt;Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1&lt;/a&gt;&amp;quot; Proceedings of ICC '93 - IEEE International Conference on Communications, 1993, pp. 1064-1070 vol.2, doi: 10.1109/ICC.1993.397441.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since &lt;a href=&quot;https://en.wikipedia.org/wiki/Turbo_code&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turbo codes&lt;/a&gt; using forward error correction are in such widespread use today, and there is much intellectual property involved, there was &lt;em&gt;a lot&lt;/em&gt; of attention paid to the IP aspects of who invented what.&lt;/p&gt;&#xA;&lt;p&gt;But I'd like to ask more about the talk itself and what the material from Glavieux and Thitimajshima was that was merged.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What were the contributions from Glavieux &amp;amp; Thitimajshima that were &amp;quot;merged&amp;quot; into Claude Berrou's now-famous 1993 talk &amp;quot;Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Summaries are helpful but if the actual separate submissions or subsequently published works by these two are available, that would be even more informative.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2023-07-02T04:41:49.253" LastActivityDate="2023-07-02T04:41:49.253" Title="What contributions from Glavieux &amp; Thitimajshima were &quot;merged&quot; into Claude Berrou's now-famous 1993 talk &quot;Near Shannon limit error-correcting coding&quot;?" Tags="|information-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14433" PostTypeId="2" ParentId="14426" CreationDate="2022-05-05T12:27:40.257" Score="5" Body="&lt;p&gt;The image may depict the new form of doing arithmetic &lt;em&gt;by writing numbers&lt;/em&gt;, as opposed to the old style of using the &lt;a href=&quot;https://en.wikipedia.org/wiki/Counting_board&quot; rel=&quot;noreferrer&quot;&gt;counting board&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Two image attestations around the same time.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A &lt;a href=&quot;https://www.digitale-sammlungen.de/en/view/bsb11221095?page=171&quot; rel=&quot;noreferrer&quot;&gt;1503 print&lt;/a&gt; of Gregor Reisch's encyclopedia &lt;em&gt;Margarita Philosophica&lt;/em&gt;. Under the entry for &lt;em&gt;arithmetice&lt;/em&gt;, there's an image contrasting the old and new style of doing arithmetic. Note the banner saying &amp;quot;typus arithmeticæ&amp;quot;.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Zo3NR.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Zo3NR.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Two 1514 prints by the same publisher Öglin in Augsburg. &lt;a href=&quot;https://www.digitale-sammlungen.de/en/view/bsb00009317?page=6,7&quot; rel=&quot;noreferrer&quot;&gt;The first book&lt;/a&gt; is about the counting board technique. &lt;a href=&quot;https://www.digitale-sammlungen.de/en/view/bsb00005871?page=4,5&quot; rel=&quot;noreferrer&quot;&gt;The second book&lt;/a&gt; is about arithmetic by writing numbers.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/viu2y.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/viu2y.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/KEuUh.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KEuUh.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;Technology&lt;/h2&gt;&#xA;&lt;p&gt;In terms of technology for the table surface and smaller writing board, I have thus far found no references from the 1500s. Today, we can find &lt;a href=&quot;https://en.wikipedia.org/wiki/Slate_(writing)&quot; rel=&quot;noreferrer&quot;&gt;slate&lt;/a&gt; boards on which we write with chalk or some other form of soft stone.&lt;/p&gt;&#xA;" OwnerUserId="16591" LastEditorUserId="16591" LastEditDate="2022-05-05T12:39:16.450" LastActivityDate="2022-05-05T12:39:16.450" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14434" PostTypeId="1" AcceptedAnswerId="14531" CreationDate="2022-05-05T22:29:54.220" Score="11" ViewCount="627" Body="&lt;p&gt;The &lt;strong&gt;Millin series&lt;/strong&gt; is defined as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{n \mathop = 0}^\infty \frac 1 {F_{2^n} }$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$F_n$&lt;/span&gt; denotes the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th Fibonacci number.&lt;/p&gt;&#xA;&lt;p&gt;It can be shown to equal &lt;span class=&quot;math-container&quot;&gt;$\dfrac {7 - \sqrt 5} 2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But who was the D.A. Millin who it is named after?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;: User &lt;a href=&quot;https://hsm.stackexchange.com/users/16591/michael&quot;&gt;https://hsm.stackexchange.com/users/16591/michael&lt;/a&gt; has located the issue of FQ in which Millin's name originally appears, where he was identified as a Pennsylvanian high-school student in 1974.&lt;/p&gt;&#xA;&lt;p&gt;Further to this, I have found that the solution appeared in FQ issue Vol. 14 no. 2 (1976), but in this case his name appears as &lt;strong&gt;D.A. Miller&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The question arises as to whether &lt;strong&gt;Millin&lt;/strong&gt; might have been a misprint. If his name truly &lt;em&gt;is&lt;/em&gt; &amp;quot;Miller&amp;quot;, then his precise identity may be very difficult to track down. There is a professor in Virginia with that name, but he appears a couple of decades too young.&lt;/p&gt;&#xA;&lt;p&gt;Whoever he is, he may well be currently active.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="12839" LastEditDate="2022-05-06T06:55:52.750" LastActivityDate="2022-06-15T22:23:21.410" Title="Who was D.A. Millin, the eponym of the Millin Series?" Tags="|biographical-details|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14436" PostTypeId="2" ParentId="14434" CreationDate="2022-05-06T00:07:54.803" Score="14" Body="&lt;p&gt;The furthest back I came is &lt;a href=&quot;http://www.fq.math.ca/Scanned/12-3/advanced12-3.pdf&quot; rel=&quot;noreferrer&quot;&gt;this 1974 article&lt;/a&gt; from The Fibonacci Quarterly 12, No. 3.&lt;/p&gt;&#xA;&lt;p&gt;Extract:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/xWyiC.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xWyiC.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16591" LastActivityDate="2022-05-06T00:07:54.803" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14437" PostTypeId="1" CreationDate="2022-05-06T20:42:15.937" Score="2" ViewCount="175" Body="&lt;p&gt;In the original &lt;a href=&quot;https://doi.org/10.1098/rspa.1928.0023&quot; rel=&quot;nofollow noreferrer&quot;&gt;1928 paper&lt;/a&gt; (&lt;a href=&quot;http://www.math.ucsd.edu/%7Enwallach/Dirac1928.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pdf&lt;/a&gt;) the Dirac equation appears on page 615 in equation (9) as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;[p_0+\rho_1\left(\boldsymbol{\sigma},\boldsymbol{p}\right)+\rho_3mc]\psi=0\qquad(1)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Using the definitions&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;p_0=i\hbar\frac{\partial}{c\partial t};\quad\boldsymbol{p}=-i\hbar\boldsymbol{\nabla}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;from pages 613 and 611 as well expressing the matrices given at the bottom of page 614 in modern notation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\rho_1\rightarrow\gamma_5;\quad\boldsymbol{\sigma}\rightarrow\boldsymbol{\Sigma};\quad\rho_3\rightarrow\beta&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and finally multiplying through with the speed of light &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, we obtain the equation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;[i\hbar\frac{\partial}{\partial t}+c(\boldsymbol{\alpha}\cdot\boldsymbol{p})+\beta mc^2]\psi=0&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;However, this is not how the Dirac equation is written today; the term with the time derivative has the opposite sign !&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The form (1) is also found in the &lt;a href=&quot;https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/57355817/57355817.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;third edition&lt;/a&gt; (1947) of Dirac's book &lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt; (see eq (8) on page 256).&lt;/p&gt;&#xA;&lt;p&gt;In the fourth edition (1958), however, in equation (10) on page 257, we find&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;[p_0-\rho_1\left(\boldsymbol{\sigma},\boldsymbol{p}\right)-\rho_3mc]\psi=0\qquad(2)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;that translates into the form we know today. Basically, eq (2) is obtained from eq (1) by changing the sign of time.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Do we know why Dirac modified his equation in this way ?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="16607" LastActivityDate="2022-05-06T20:42:15.937" Title="The original Dirac equation" Tags="|quantum-mechanics|differential-equations|dirac|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14438" PostTypeId="1" AcceptedAnswerId="14439" CreationDate="2022-05-08T03:45:24.603" Score="2" ViewCount="267" Body="&lt;p&gt;When and how did mathematics come to be abstracted away from the physical world?&lt;/p&gt;&#xA;&lt;p&gt;At first, mathematics would originate in its simplest form of counting and addition as to keep track of certain supplies in ancient civilizations. At some point it came to be used to describe more complex physical situations, such as ancient astronomy or Archimedes' study of force and buoyancy.&lt;/p&gt;&#xA;&lt;p&gt;At what point did it come to describe inherently non-physical things; doing math for the sake of math? For example, doing things such as an analytic study of functions, the defining of abstract spaces, or even primitive number theory, etc. The earliest example I can think of is Indian and Greek geometry, but even this had a motivation in engineering and design. And yes, you could argue all of math at its core came from a motivation in physics, but today that's not the case.&lt;/p&gt;&#xA;&lt;p&gt;In the modern day, mathematicians do beautiful work with no thought of how it may effect or be used in physics or any other science for that matter. So when did this switch happen? How did it come to be &amp;quot;math&amp;quot; rather than a tool for physics?&lt;/p&gt;&#xA;" OwnerUserId="16611" LastEditorUserId="16318" LastEditDate="2022-05-14T05:40:26.290" LastActivityDate="2022-05-14T05:40:26.290" Title="The abstraction of mathematics from physics" Tags="|physics|ancient-greece|mathematical-physics|mathematics-social-history|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14439" PostTypeId="2" ParentId="14438" CreationDate="2022-05-08T11:52:22.413" Score="8" Body="&lt;p&gt;You are right, this happened in ancient Greece, and is credited to Thales and Pythagoras. Unfortunately, too little of their early work survived (nothing written by Thales or Pythagoras). The main source of&#xA;our knowledge abut this early epoch is Euclid, where mathematics is already almost completely separated from the &amp;quot;real world&amp;quot;. The problems that Euclid considers (constructions with compass and ruler, number theory, etc.) have little to do with the physical world.&lt;/p&gt;&#xA;&lt;p&gt;This is especially clear in the work of the next generation. Archimedes worked on both mathematics and physics, but made a very clear distinction between the two in his writings.&lt;/p&gt;&#xA;&lt;p&gt;By the way, Greek mathematical astronomy is of later origin than that. The Hellenistic Greeks made a clear distinction between pure and applied mathematics (though they did not use these terms).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4931" LastEditDate="2022-05-11T14:09:58.790" LastActivityDate="2022-05-11T14:09:58.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14440" PostTypeId="1" CreationDate="2022-05-06T04:23:42.093" Score="2" ViewCount="337" Body="&lt;p&gt;I'm interested in knowing more about the history around lunar tables and calculation of moon phases.&lt;/p&gt;&#xA;&lt;p&gt;When did we have first (reliable) calculations of moon phases and first (reliable) lunar tables? When did they become widespread?&lt;/p&gt;&#xA;&lt;p&gt;Reading through &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of astronomy (Wiki)&lt;/a&gt;, it appears that we've had some calculations 200-100 BC at least (Greece), but maybe even earlier (the Maya).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;&#xA;To explain a bit better: I'm trying to find an answer to the question: At which point in history, were we able to tell the exact day on which moon phases will occur, for the next 2 months (for example)? Reliable meaning that most of the time (let's say 90%) the predictions were correct.&lt;/p&gt;&#xA;" OwnerUserId="16616" OwnerDisplayName="Haris Osmanagić" LastEditorUserId="14043" LastEditDate="2022-05-12T04:44:49.273" LastActivityDate="2023-01-22T13:24:28.677" Title="When did we have first calculations of moon phases?" Tags="|astronomy|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14441" PostTypeId="1" CreationDate="2022-05-11T12:20:51.633" Score="5" ViewCount="311" Body="&lt;p&gt;Fermat's Last Theorem was open for more than 350 years until Andrew Wiles proved it in 1995. Are there possible (historical or other) reasons why David Hilbert did not include this famous open problem in his problem list in 1900?&lt;/p&gt;&#xA;" OwnerUserId="7937" LastActivityDate="2022-11-09T23:53:28.470" Title="Hilbert's problem list did not include Fermat's last theorem. Why?" Tags="|number-theory|hilbert|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14442" PostTypeId="1" CreationDate="2022-05-11T12:21:53.050" Score="1" ViewCount="156" Body="&lt;p&gt;A 1929 paper by Felix Bloch, &lt;A HREF=&quot;http://www.pwein.at/physics/Lectures/Famous-Papers/Z-Physik-52-555-1928.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Über die Quantenmechanik der Elektronen in Kristallgittern&lt;/A&gt;, is widely cited as having predicted the phenomenon of &lt;em&gt;Bloch oscillations&lt;/em&gt;: The oscillatory motion of an electron in a periodic potential when a constant force is applied.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is this attribution correct?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The 1929 paper has a section (starting on page 572) where the effect of a uniform electric field on the electrons in a crystal is considered, but I could not find any mention of oscillatory motion in that section or elsewhere in the paper. Did I overlook something?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The first explicit mention of periodic motion that I am aware of is in a 1934 paper by Clarence Zener, &lt;A HREF=&quot;https://doi.org/10.1098/rspa.1934.0116&quot; rel=&quot;nofollow noreferrer&quot;&gt;A theory of the electrical breakdown of solid dielectrics&lt;/A&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/k4bih.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/k4bih.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Perhaps we should speak of &amp;quot;Zener oscillations&amp;quot; ? I note that the term &amp;quot;Bloch-Zener oscillations&amp;quot; is used in the literature, but that refers to the superposition of Bloch oscillations and Zener tunnelling, not to the oscillations &lt;em&gt;per se&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I reproduce here the translation the paragraphs in Bloch's 1929 paper that adress the motion of the electrion in a uniform magnetic field (page 572). No mention is made of the oscillatory motion that follows from equation 48.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dt}|c_{klm}|^2=-\frac{KeF}{h}\frac{\partial}{\partial&#xA; k}|c_{klm}(0)|^{2},\;^{\ast}\qquad\qquad(48)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It can be seen that if initially the amplitudes of the eigenfunctions&#xA;building up our wavepacket are given by (46), this distribution of the&#xA;amplitudes in the &lt;span class=&quot;math-container&quot;&gt;$k,l,m$&lt;/span&gt;-space shifts towards increasing values of&#xA;&lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, more rapidly the larger the field strength &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is. In the limit&#xA;of perfectly free electrons, (48) passes directly into that, what one&#xA;would expect according to classical mechanics. According to (9) then&#xA;the velocity in the &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-direction is given by&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v_x=\frac{kh}{mK}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and instead of (48) one can write&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d}{dt}|c_{klm}|^2=-\frac{eF}{m}\frac{\partial}{\partial&#xA; v_x}|c_{klm}|^2.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This would also be expected classically, when a cloud of electrons&#xA;experiences an acceleration &lt;span class=&quot;math-container&quot;&gt;$\frac{eF}{m}$&lt;/span&gt;.&lt;span class=&quot;math-container&quot;&gt;$^{\ast\ast}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^\ast$&lt;/span&gt; Mr. Peierls has kindly pointed out to me that one can derive&#xA;equation (48) also independently of the special form (41) of the wave&#xA;packet. It does seem in any case necessary that the initial stationary&#xA;state is not sharply defined.&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$^{\ast\ast}$&lt;/span&gt; Cf. E.H. Kennard, l.c.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1697" LastEditorUserId="1697" LastEditDate="2022-05-13T06:16:34.083" LastActivityDate="2022-05-13T06:16:34.083" Title="Did Felix Bloch really discover Bloch oscillations?" Tags="|quantum-mechanics|solid-state-physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14443" PostTypeId="1" AcceptedAnswerId="14444" CreationDate="2022-05-11T19:56:45.397" Score="2" ViewCount="181" Body="&lt;p&gt;In &lt;a href=&quot;https://www.uni-muenster.de/Physik.TP/%7Emunsteg/arnold.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;On teaching mathematics&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Vladimir_Arnold&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vladimir Igorevich Arnold&lt;/a&gt; states&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion and that the &amp;quot;modern&amp;quot; definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;When googling the last part of the last sentence, I only found references to this text by Arnold, and upon searching for &lt;em&gt;Veblen&lt;/em&gt; and &lt;em&gt;manifold&lt;/em&gt; from 1925-1931 on google scholar, I couldn't find anything resembling this quote, so it is probably a paraphrase of Veblen's statement.&lt;/p&gt;&#xA;&lt;p&gt;Does anybody know what Arnold's reference might have been?&lt;/p&gt;&#xA;&lt;p&gt;The only close thing I found was&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Since the manifolds which we are here calling “regular”&#xA;are all topological spaces, they may be characterized by adding certain other axioms to the Hausdorff set.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;from Oswald Veblen's &lt;a href=&quot;https://scholarship.rice.edu/bitstream/handle/1911/9043/article_RI214223.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Analysis Situ&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10550" LastEditorUserId="10550" LastEditDate="2022-05-11T21:00:12.183" LastActivityDate="2022-05-12T20:12:22.257" Title="Reference for &quot;A manifold is a topological space which satisfies a long series of axioms.&quot;" Tags="|mathematics|mathematicians|reference-request|riemannian-geometry|20th-century|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14444" PostTypeId="2" ParentId="14443" CreationDate="2022-05-12T11:23:17.310" Score="2" Body="&lt;p&gt;It is difficult, if not impossible, to determine which &amp;quot;recent American book&amp;quot; Arnold is referring to, but the statement he makes is certainly correct.&lt;/p&gt;&#xA;&lt;p&gt;There are some very interesting surveys about the history of the concept of manifold, for example in &lt;a href=&quot;http://www2.math.uni-wuppertal.de/%7Escholz/preprints/Concept-of-manifold.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Concept of Manifold, 1850-1950&#xA;&lt;/a&gt; we read (p. 15)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the 1880s Poincare came across &amp;quot;manifolds&amp;quot; in several analytical or geometrical contexts, although he personally understood them at that time still in a rather vague way.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and (p. 16. emphasis mine)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Poincare gave a discussion which in fact spoke in favour of the surjectivity and was already sharper than Klein's, but still used highly intuitive ideas about continuous variation of images in higher dimensional spaces in a symbohcally uncontrollable manner. Even the spaces themselves were not shown to be manifolds but taken as such, without further ado. For any critical reader (&lt;strong&gt;perhaps even including Poincare himself&lt;/strong&gt;) the &amp;quot;continuity proof&amp;quot; could thus be taken at least as much as an indicator for the necessity of an&#xA;improved understanding of higher dimensional geometry as it was an indicator for the truth of the uniformization theorem. And in fact a clarification of the topological proof strategy was given only later by Brouwer [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(p. 31)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Veblen and his student J.H.C. Whitehead, coming from (and going back to) Oxford, brought the axiomatization of the manifold concept to a stage which stood up to the standards of modern mathematics in the sense of the 20th century&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The axiomatization by Veblen and Whitehead was first exposed in &lt;a href=&quot;https://www.pnas.org/doi/pdf/10.1073/pnas.17.10.551&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Set of Axioms for Differential Geometry&lt;/a&gt;, and then in &lt;a href=&quot;https://www.britannica.com/topic/The-Foundations-of-Differential-Geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Foundations of Differential Geometry&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Coming back to the book Arnold refers to, one possibility could be &lt;a href=&quot;https://www.maa.org/press/maa-reviews/a-history-of-algebraic-and-differential-topology-1900-1960-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;A History of Algebraic and Differential Topology, 1900-1960&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Dieudonné&lt;/a&gt;, published in 1989 (Arnold's paper is from 1998), but I highly doubt that Arnold would consider it an &amp;quot; American book&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="10550" LastEditDate="2022-05-12T20:12:22.257" LastActivityDate="2022-05-12T20:12:22.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14445" PostTypeId="2" ParentId="14441" CreationDate="2022-05-14T03:07:47.197" Score="9" Body="&lt;p&gt;Hilbert included in his list a much more general&#xA;Problem 10:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Probably he did not think that Fermat's equation by itself is sufficiently important to deserve a separate entry in his list.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-11-09T23:53:28.470" LastActivityDate="2022-11-09T23:53:28.470" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14446" PostTypeId="2" ParentId="5085" CreationDate="2022-05-14T06:37:22.897" Score="0" Body="&lt;p&gt;In very ancient times one could argue intuitionism was predominant. Physics was more of a tool for craftsmen, building, and navigation. In these senses it was less about discovery and more about applicability with tangible objects, and so computation with accurate calculations wasn't necessary and representing these commutative strategies in a general form (with symbols and equations) wasn't something even conceptualized. The concept of logic itself wasn't really present either in the way we understand it today. They certainly used a method of intuitionism and tradition.&lt;/p&gt;&#xA;&lt;p&gt;Later however, as philosophy and culture surrounding academics, particularly in ancient Greece, became more about discovery and wonder and less about applicability, it certainly took upon the methods of formalism. Archimedes for example, in studying force and buoyancy, would speak of mass, velocity, volume, viscosity, etc in general terms. At this point however, math and physics still didn't use symbols to represent general values, and instead would use language to describe their ideas. Alongside this, the way physical ideas were developed could be argued as intuitionistic, such as Archimedes's &amp;quot;Eureka!&amp;quot; moment. This was however still essentially formalism, in the similar way language was used to describe the ideas. And as the use of symbols became more normalized, this use of formalism became more apparent.&lt;/p&gt;&#xA;&lt;p&gt;In the modern day, where math is respected as a formal, axiomatic system of logic and physics is respected as a natural science of inductive reasoning, the distinction becomes clearer. It is now understood that science by definition can not use math in a logicistic sense. Pure logic, as in math, is deductive and extended from the axioms. There are no axioms for the universe however, as far as we know, and we can not study all cases at once as to be deductive, so instead physics must make use of the scientific method and be inductive and therefore not purely logical. As physics has become more abstract, it's hard to argue for any intuitionistic thinning as well, as it’s pretty hard to ever claim to have an &amp;quot;intution&amp;quot; for quantum mechanics or dark matter. Physics is accepted to use formalism. In fact the entire study of mathematical physics deals with exactly that- how to represent a physical situation with mathematical language. As physics gets more advanced this actually becomes a rather significant issue since understanding the relationships observed in quantum mechanics, for example, well enough to formalize it into mathematical notation is very difficult and not something our mathematics is necessarily prepared for. Furthermore there's actually come to be computational methods used in physics that &amp;quot;break&amp;quot; mathematical rules. For example I know of a method where the physicist essentially &amp;quot;crosses out&amp;quot; infinities and is left with the value they were in search for (??? So weird).&lt;/p&gt;&#xA;&lt;p&gt;So to directly answer your question, while the way mathematics has been used in physics has been ever-changing, it certainly wasn't something that was &amp;quot;debated&amp;quot; upon. Being so formal in how we consider the reasoning and philosophy of logic, math, and science is in itself a fairly recent development. I doubt Archimedes ever once considered the lack of axioms for the natural world, or even how he ought to properly use mathematical language in general in his physical studies.&lt;/p&gt;&#xA;&lt;p&gt;Looking back on your question, I noticed that while you listed logicism, intuitionism, and formalism, that you included an &amp;quot;etc&amp;quot;. I'd like to apologize for not considering anything other than those three, however I do think my answer was comprehensive in its own right and that it's unnecessary to analyze any others in answering the question.&lt;/p&gt;&#xA;" OwnerUserId="16611" LastActivityDate="2022-05-14T06:37:22.897" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14447" PostTypeId="1" CreationDate="2022-05-15T02:24:13.697" Score="0" ViewCount="84" Body="&lt;p&gt;From a perspective of math history, what discoveries resulted directly from Riemann’s discovery and codification of non-flat metrics on Riemannian manifolds? After Albert Einstein modeled the Universe on a pseudo-Riemannian manifold, Riemann’s work was rediscovered. But Einstein was a physicist. Math concepts like the Cartan connection and bundle and fiber model for the tangent space seem related to curved metrics, but did not emerge from it directly.&lt;/p&gt;&#xA;&lt;p&gt;One motivation for this question is the fact that a curved metric does not change the result for total arc length when integrated compared to the Euclidean one from which it is derived (for simple real curves). That made me think there must be some other important significance to the idea that the distance between nearby points can be something other than straight. A non-flat metric seems to be such a cornerstone of Riemannian geometry.&lt;/p&gt;&#xA;&lt;p&gt;As an example of arc length invariance, see &lt;a href=&quot;https://math.stackexchange.com/questions/4441646/do-the-curvature-properties-of-exotic-spheres-result-in-necessary-new-techniques&quot;&gt;https://math.stackexchange.com/questions/4441646/do-the-curvature-properties-of-exotic-spheres-result-in-necessary-new-techniques&lt;/a&gt; . The circumference of a great circle on a sphere is the same as that of a circle produced by a planar intersection at the equator (this also applies to curves on higher dimensional manifolds).&lt;/p&gt;&#xA;&lt;p&gt;The question arises: what calculations did change directly as a result of the fact that curved spaces have non-flat metrics?&lt;/p&gt;&#xA;&lt;p&gt;The following things did result from Riemann’s overall body of work on Riemannian manifolds:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Established that many geometries exist&lt;/li&gt;&#xA;&lt;li&gt;That Riemann curvature and metric are intrinsic to a surface, the same way Gaussian curvature is.&lt;/li&gt;&#xA;&lt;li&gt;It led to new branches of mathematics such as algebraic geometry, the Cartan and other connections, and the bundle and fiber concepts of the tangent space.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;None of these proceeds directly from the fact that a metric itself can be conceived of as curved, geometrically. It seems to me that there must have been something additional that resulted. For example, did Riemann’s work enable arc lengths on manifolds to be calculated simply with a single integral of his metric? Using a Euclidean metric on portions of a curve on a curved space would be tedious, as each planar intersection segment (especially for a higher dimensional manifold) would have to be added to others to arrive at total arc length. Geometers other than Riemann in 1860 didn’t spend time on curves on higher dimensional manifolds, rendering my suggestion sort of moot. But if they had...&lt;/p&gt;&#xA;&lt;p&gt;Or did the aforementioned discovery have a direct result on the calculation of complex curves -  curves in complex spaces that use the imaginary number i? I mean would arc lengths not be the same for the curve on the manifold compared to that on a planar (or several) intersection(s)? This is an area I know nothing about, and requires expert mathematical advice to answer. I would obviously have to define which axes are imaginary and which are real first.&lt;/p&gt;&#xA;&lt;p&gt;Is it better to view the concept of a curved metric simply as part of the greater picture, and not separated – put in context along with curved spaces, the curvature tensor, Christoffel symbols, and higher dimensional shapes? Then answer the question of how these changed the course of mathematics development.&lt;/p&gt;&#xA;" OwnerUserId="15236" LastEditorUserId="10106" LastEditDate="2022-05-15T04:15:28.750" LastActivityDate="2022-05-15T04:15:28.750" Title="What is the historical significance of Riemann’s discovery of non-flat metrics?" Tags="|mathematics|discoveries|differential-geometry|riemannian-geometry|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14448" PostTypeId="1" CreationDate="2022-05-15T08:53:24.197" Score="2" ViewCount="75" Body="&lt;p&gt;The context of this question:&lt;/p&gt;&#xA;&lt;p&gt;In everyday life we use push and pull all the time to get things moving, to move ourselves. To ride a bicycle: our feet push the pedals to get moving. To accelerate we increase the magnitude of the push.&lt;/p&gt;&#xA;&lt;p&gt;In the time that Huygens and Newton did their work the magnitude of push &amp;amp; pull that is involved in acceleration was hardly accessible to measurement, if at all.&lt;/p&gt;&#xA;&lt;p&gt;Huygens created setups where he would have pendulum bobs collide with each other. To my understanding: Huygens was able to make bobs sufficiently hard and elastic such that he got very close to perfect elastic collision.&lt;/p&gt;&#xA;&lt;p&gt;Let me refer to a setup with two pendulum bobs as a setup with &lt;span class=&quot;math-container&quot;&gt;$b_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b_2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;To my understanding: a typical setup would be one where &lt;span class=&quot;math-container&quot;&gt;$b_1$&lt;/span&gt; is released from a height &lt;span class=&quot;math-container&quot;&gt;$h_1$&lt;/span&gt;. With two bobs of equal mass &lt;span class=&quot;math-container&quot;&gt;$b_1$&lt;/span&gt; would come to a dead stop, and &lt;span class=&quot;math-container&quot;&gt;$b_2$&lt;/span&gt; would swing up to the same height as the initial height of &lt;span class=&quot;math-container&quot;&gt;$b_1$&lt;/span&gt;. With bobs of unequal mass &lt;span class=&quot;math-container&quot;&gt;$b_1$&lt;/span&gt; still has motion after the elastic collision. If the respective bobs would swing back up to consistent height then that is what Huygens could measure with consistency.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that the data that Huygens had access to was the kind of data that can be obtained with setups like that. The &lt;em&gt;height&lt;/em&gt; to which a bob swings back up is an indirect measure of the velocity imparted to it on collision.&lt;/p&gt;&#xA;&lt;p&gt;Pendulum experiments&lt;/p&gt;&#xA;&lt;p&gt;The period of pendulum swing depends on two factors: the length of the pendulum, and the gravitatinal acceleration. Gravitational mass and inertial mass are equivalent, so pendulums with bobs of different mass but equal length (from pivot point to center of mass of the bob) will have equal period of swing.&lt;/p&gt;&#xA;&lt;p&gt;The equivalence of gravitational and inertial mass was extremely helpful to the development of celestial mechanics. If gravitational mass and inertial mass are equivalent then it is not necessary to know the masses of the various planets of the solar system.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the equivalence of gravitational and inertial mass means that while you know the gravitational &lt;em&gt;acceleration&lt;/em&gt; you don't necessarily know the gravitational force.&lt;/p&gt;&#xA;&lt;p&gt;To assign a value to gravitational force we need a standard unit of mass. But the only way to set up a standard unit of mass is to create a reproducible volume of a reproducible substance, and to measure the weight of that.&lt;/p&gt;&#xA;&lt;p&gt;It could be, I don't know, that d'Alembert saw circular reasoning in that.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Today we have classroom demonstrations with an air track. The students can calibrate some elastic force, use that elastic force to accelerate a glider, record video of the motion, and then obtain the acceleration data from the video.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;The question:&lt;/p&gt;&#xA;&lt;p&gt;Is my following understanding correct:&lt;br /&gt;&#xA;The aim of d'Alembert was to develop a mechanics with no explicit use of push &amp;amp; pull. Instead everything was to be described in terms of change of quantity of motion. The concepts of potential energy and kinetic energy were not yet in their current form of course, what d'Alembert used was precursors to those concepts, but that is what d'Alembert used.&lt;/p&gt;&#xA;&lt;p&gt;From an outside perspective it seems very strange that d'Alembert would want to avoid the newtonian concept of force. In everyday life we use push &amp;amp; pull all the time to get around.&lt;/p&gt;&#xA;&lt;p&gt;Hence my question:&#xA;To what extent did d'Alembert want to avoid the newtonian concept of force?&lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2022-05-15T08:53:24.197" Title="To what extent did d'Alembert want to avoid the newtonian concept of force?" Tags="|physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14450" PostTypeId="1" CreationDate="2022-05-15T23:10:10.770" Score="4" ViewCount="246" Body="&lt;p&gt;The quote in question is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For Bourbaki, Poincaré was the devil incarnate. For students of chaos and fractals, Poincaré is of course God on Earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The common reference for this quote seems to be MacHale's &lt;em&gt;Comic Sections: The Book of Mathematical Jokes, Humour, Wit and Wisdom&lt;/em&gt; (p.145) (see e.g. &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Stone/quotations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Biographies/Stone/quotations/&lt;/a&gt; or Eisner, Farkas, Haase &amp;amp; Nagel's &lt;em&gt;Operator Theoretic Aspects Of Ergodic Theory&lt;/em&gt; (p.9) available at &lt;a href=&quot;https://www.math.uni-leipzig.de/%7Eeisner/book-EFHN.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.math.uni-leipzig.de/~eisner/book-EFHN.pdf&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I would like to know if there is a written reference for this, or else if it can be verified otherwise. MacHale's book doesn't seem to list references.&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your time.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I should add that before posting the question I did a fast reference chase which led me to Mandelbrot's &lt;em&gt;Fractals and Chaos: The Mandelbrot Set and Beyond&lt;/em&gt; (p.280) (this part seems to be from a reproduction of an earlier Intelligencer article of Mandelbrot, titled &amp;quot;Chaos, Bourbaki, and Poincaré&amp;quot;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Do not forget that for Bourbaki, Poincaré was the devil incarnate, who had left behind a&#xA;mess of unproven assertions and loose ends. They boasted that they had cleaned up that mess. Of course, Poincaré has long been a source of concern to French mathematicians. In the 1880s, Hermite kept writing to Mittag-Leffler to complain that young Poincaré never completed a proof. For students of chaos and fractals, Poincaré is, of course, God on Earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In light of this the question remains: either the quote is being misattributed to Stone (which seems likely to me, given the bombastic language (I gather from the rest of the book that Mandelbrot really disliked Bourbaki)), or Mandelbrot is quoting Stone. In this book Stone's paper &amp;quot;The Revolution in Mathematics&amp;quot; is listed as a reference; though as far as I can see Stone does not mention Bourbaki nor Poincaré there.&lt;/p&gt;&#xA;" OwnerUserId="1887" LastEditorUserId="4251" LastEditDate="2022-05-16T17:45:21.563" LastActivityDate="2024-02-27T12:18:14.177" Title="Source of a Quote by M. Stone on Poincaré and Bourbaki" Tags="|mathematics|mathematicians|reference-request|quote|bourbaki|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14451" PostTypeId="2" ParentId="14450" CreationDate="2022-05-16T09:59:47.287" Score="2" Body="&lt;p&gt;This does not directly answer if the quote &amp;quot;For Bourbaki, Poincaré was the devil incarnate&amp;quot; is from Stone or Mandelbrot (although I would bet on the latter), but only points out the complex relation between Bourbaki (I mean, the young French mathematicians of the 30s) and Poincaré's heritage.&lt;/p&gt;&#xA;&lt;p&gt;In the the imaginary biography of Bourbaki &lt;em&gt;&lt;a href=&quot;http://archives-bourbaki.ahp-numerique.fr/items/show/638#?c=0&amp;amp;m=0&amp;amp;s=0&amp;amp;cv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Notice sur la Vie et l'œuvre de Nicolas Bourbaki&lt;/a&gt;&lt;/em&gt; (Notice sur la Vie et l'œuvre de Nicolas Bourbaki), probably written under Henri Cartan, we read that in 1906 he gets a scholarship in order to study under Poincaré and Hilbert.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the idea that Bourbaki was born in reaction to the stagnation in French mathematics caused by the First World War and Poincaré, is essentially due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Raymond_Queneau&quot; rel=&quot;nofollow noreferrer&quot;&gt;Raymond Queneau&lt;/a&gt; (&lt;a href=&quot;http://queneau.fr/static/pdf/numerise/D40_2_5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; the original handwritten text of its paper &lt;em&gt;Bourbaki et les mathématiques de demain&lt;/em&gt;. Critique 18(176), 3-18 (1962)):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Il ne faut pas croire d’ailleurs que, merveille génétique, Bourbaki se soit engendré lui-même. Son père même, on peut le nommer, c’est Hilbert, et il eut comme nourrice, entre autres, Emmy Noether, et ses parrains furent quasiment tous des étrangers [...]&#xA;Poincaré a joué un mauvais tour à la mathématique et à la philosophie française en raillant le grand Péano et sa definition &amp;quot;logistique&amp;quot; de zéro [...] son autorité a détourné les mathématiciens français des recherches logique [...] La mathématique fraçaise s'est trouvée, et par le fait de la guerre de 14-18, et par l'influence de Poincaré, en plein crise gérontique.&#xA;[...] Vers les années 30, quelques jeunes mathématiciens ont pris conscience du retard pris par les mathématiques françaises&#xA;non seulement dans l’enseignement (il ne cesse de l'être que depuis peu) mais encore dans la recherché; et ils créèrent Bourbaki;&#xA;et depuis 1945, trois Français ont eu la médaille Fields, sorte de prix Nobel des mathématiques, Schwartz, Serre et Thom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;i.e.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Moreover, one must not believe that, as a genetic marvel, Bourbaki generated himself. His very father, we can name him, was Hilbert, and he had Emmy Noether as his nurse, among others, and his godparents were almost all foreigners [...] Poincaré played a bad trick on French mathematics and philosophy by mocking the great Péano and his &amp;quot;logistic&amp;quot; definition of zero [...] His authority diverted French mathematicians from logical research [...] French mathematics found itself, both because of the 14-18 war and because of Poincaré's influence, in the midst of a gerontological crisis. [...] Towards the 1930s, a few young mathematicians became aware of the backwardness of French mathematics, not only in teaching (which has only recently ceased to be the case) but also in research; they created Bourbaki; and since 1945, three Frenchmen have been awarded the Fields Medal, a kind of Nobel Prize for mathematics, Schwartz, Serre and Thom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="10608" LastEditDate="2024-02-27T12:18:14.177" LastActivityDate="2024-02-27T12:18:14.177" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14453" PostTypeId="1" CreationDate="2022-05-17T21:28:08.457" Score="1" ViewCount="74" Body="&lt;p&gt;The symbol &lt;span class=&quot;math-container&quot;&gt;$\perp\!\!\!\!\perp$&lt;/span&gt; in statistics is a way to denote &lt;a href=&quot;https://en.wikipedia.org/wiki/Independence_(probability_theory)&quot; rel=&quot;nofollow noreferrer&quot;&gt;statistical independence&lt;/a&gt; of a collection of &lt;a href=&quot;https://en.wikipedia.org/wiki/Random_variable&quot; rel=&quot;nofollow noreferrer&quot;&gt;random variables&lt;/a&gt;. I have seen two forms of it. The first is highly suitable in writing about independence between two variables:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$X\perp\!\!\!\!\perp Y$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The second works for two variables, but is especially suited for multiple variables:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\perp\!\!\!\!\perp \left[ X_1, \cdots, X_n \right]$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;What these symbols mean, for a collection of (let's say real-valued) random variables &lt;span class=&quot;math-container&quot;&gt;$\{ X_j \}_{j=1}^n$&lt;/span&gt;, is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\perp\!\!\!\!\perp \left[ X_1, \cdots, X_n \right] \iff F_{X_1, \cdots, X_n} (x_1, \cdots, x_n) = \prod_{j=1}^n F_{x_j}(x_j)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$F_{X_1, \cdots, X_n}$&lt;/span&gt; is the joint cumulative distribution function and the &lt;span class=&quot;math-container&quot;&gt;$F_{x_j}$&lt;/span&gt; is the marginal cumulative distribution function for the &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt;th random variable.&lt;/p&gt;&#xA;&lt;p&gt;What is the earliest-known publication that used the &lt;span class=&quot;math-container&quot;&gt;$\perp\!\!\!\!\perp$&lt;/span&gt; symbol (either as a binary or n-ary relation) to denote statistical independence between/among random variables?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2023-10-17T19:35:57.410" Title="What is the earliest use of the $\perp\!\!\!\!\perp$ symbol in statistics to denote statistical independence?" Tags="|terminology|reference-request|notation|statistics|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14454" PostTypeId="2" ParentId="12976" CreationDate="2022-05-18T02:28:53.710" Score="6" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Yuktibh%C4%81%E1%B9%A3%C4%81&amp;amp;oldid=1088396986&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yuktibhāṣā&lt;/a&gt; is one of the texts we have available from &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Kerala_school_of_astronomy_and_mathematics&amp;amp;oldid=1080845747&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathematicians working in Kerala between the 14th and 16th centuries&lt;/a&gt;. This is a text in the Malayalam language, though a later (poor) translation into Sanskrit also exists. As with most texts in India, precise date and authorship are hard to determine, but it is generally accepted to have been written by &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Jye%E1%B9%A3%E1%B9%ADhadeva&amp;amp;oldid=1085866308&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jyeṣṭhadeva&lt;/a&gt; (who lived c. 1500–1610), and composed around 1530 CE.&lt;/p&gt;&#xA;&lt;p&gt;(As there has been some discussion in the question and comments about the date: The date of 1530, arrived at by &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=K._V._Sarma&amp;amp;oldid=1063689115&quot; rel=&quot;nofollow noreferrer&quot;&gt;K. V. Sarma&lt;/a&gt;, is what most of the published literature says, though a footnote in P. P. Divakaran's 2007 article &amp;quot;The First Textbook of Calculus: &lt;em&gt;Yuktibhāṣā&lt;/em&gt;&amp;quot; (&lt;a href=&quot;https://www.jstor.org/stable/23497280&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR&lt;/a&gt;, &lt;a href=&quot;https://doi.org/10.1007/s10781-007-9029-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;DOI&lt;/a&gt;) says that later in life, Sarma &amp;quot;tended to favour a slightly later dating&amp;quot; of around 1550–1560. The statement in &lt;a href=&quot;http://users.uoa.gr/%7Eapgiannop/Sources/Roy-pi.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ranjan Roy's 1990 article&lt;/a&gt; saying &amp;quot;The Yuktibhasa was written about a century later&amp;quot; than the &lt;em&gt;Tantra-saṃgraha&lt;/em&gt; seems to be a mistake, as it's closer to half a century.)&lt;/p&gt;&#xA;&lt;p&gt;This text gives &lt;strong&gt;rationale&lt;/strong&gt; (&lt;em&gt;yukti&lt;/em&gt;) for various procedures (or computational methods, formulae, recipes, whatever you want to call them) that were given in earlier texts (such as the expression for 1/8th of the circumference as &lt;span class=&quot;math-container&quot;&gt;$r(1 - 1/3 + 1/5 - 1/7 + \dots$&lt;/span&gt;), now called the &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Leibniz_formula_for_%CF%80&amp;amp;oldid=1077970899&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz formula for π&lt;/a&gt;, given earlier &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Madhava_series&amp;amp;oldid=1082350264&quot; rel=&quot;nofollow noreferrer&quot;&gt;by&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Madhava_of_Sangamagrama&amp;amp;oldid=1087555363&quot; rel=&quot;nofollow noreferrer&quot;&gt;Madhava&lt;/a&gt; (c. 1340–1425) as quoted by &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Nilakantha_Somayaji&amp;amp;oldid=1065952116&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nīlakaṇṭha&lt;/a&gt; (1444–1544) in his work &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Tantrasamgraha&amp;amp;oldid=1074553712&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tantra-saṃgraha&lt;/a&gt; (1501)).&lt;/p&gt;&#xA;&lt;p&gt;A translation of the Yuktibhāṣā has been published in 2008 jointly by &lt;a href=&quot;https://link.springer.com/book/9781848820722&quot; rel=&quot;nofollow noreferrer&quot;&gt;Springer&lt;/a&gt; and Hindustan Book Agency (for &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Indian_Institute_of_Advanced_Study&amp;amp;oldid=1058740369&quot; rel=&quot;nofollow noreferrer&quot;&gt;IIAS Shimla&lt;/a&gt;), in two volumes: the first volume on the first half of the Yuktibhāṣā, which is mathematics, and the second volume on the second half, astronomy. The first volume (the second volume is similar) includes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;the original Malayalam text, in critical edition by K. V. Sarma (pages 311–470),&lt;/li&gt;&#xA;&lt;li&gt;a bare English translation of the Malayalam text, done by K. V. Sarma (pages 1–145),&lt;/li&gt;&#xA;&lt;li&gt;explanatory notes (basically an English translation suitable for mathematicians to read, but still closely following the original), by K. Ramasubramanian, M. D. Srinivas and M. S. Sriram (pages 147–266), followed by an Epilogue (updated from an article by M. D. Srinivas) on &amp;quot;Proofs in Indian Mathematics&amp;quot; more generally (pages 267–310).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In particular, for the result in question (deriving the circumference of a circle in terms of something like &lt;span class=&quot;math-container&quot;&gt;$1 - 1/3 + 1/5 - 1/7 + \dots$&lt;/span&gt;) the relevant part to look at is &amp;quot;Chapter 6: Circle and Circumference&amp;quot;, which occupies pp. 45–82 in the bare translation and 179–207 in the explanatory translation.&lt;/p&gt;&#xA;&lt;p&gt;I can go into detail if needed, but as this answer is already getting long, will just give an outline:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.3.1, the author of the Yukti-bhāṣā shows (via similar triangles etc) a result that, in modern terms, is equivalent to stating that &lt;span class=&quot;math-container&quot;&gt;$$C_i Q_i = \frac{r}{n} \frac{r^2}{k_i k_{i+1}}$$&lt;/span&gt; in the figure below:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/le3jJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/le3jJ.png&quot; alt=&quot;Figure 6.5 from page 187&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;— here, O is the centre of the circle which is of radius &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, and the side &lt;span class=&quot;math-container&quot;&gt;$EA$&lt;/span&gt; of length &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is divided into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; (we will later take &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; going very large) equal segments, the &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;th of which is &lt;span class=&quot;math-container&quot;&gt;$A_i A_{i+1}$&lt;/span&gt;. So the length of &lt;span class=&quot;math-container&quot;&gt;$EA_i$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$ri/n$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$k_i$&lt;/span&gt; is the length of &lt;span class=&quot;math-container&quot;&gt;$OA_i$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$C_i$&lt;/span&gt; is the point where &lt;span class=&quot;math-container&quot;&gt;$OA_i$&lt;/span&gt; intersects the circle, and &lt;span class=&quot;math-container&quot;&gt;$Q_i$&lt;/span&gt; is the base of the perpendicular from  &lt;span class=&quot;math-container&quot;&gt;$C_i$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$OA_{i+1}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.3.2, he argues that as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; goes large, the arc of the circle gets close to &lt;span class=&quot;math-container&quot;&gt;$C_iQ_i$&lt;/span&gt;, so the arc &lt;span class=&quot;math-container&quot;&gt;$EC$&lt;/span&gt; (one-eighth of the circumference of the circle) gets closer and closer to &lt;span class=&quot;math-container&quot;&gt;$\sum_{i} C_iQ_i$&lt;/span&gt;, which in turn can be approximated as &lt;span class=&quot;math-container&quot;&gt;$$\frac{C}{8} \approx \sum_{i=1}^{n} C_i Q_i \approx \frac{r}{n}\sum_{i=1}^{n} \frac{r^2}{k_i^2}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that &lt;span class=&quot;math-container&quot;&gt;$$k_i = OA_i = OE^2 + EA_i^2 = r^2 + (ir/n)^2$$&lt;/span&gt; (a result which he also uses shortly), so in modern terms, we can see that this amounts to &lt;span class=&quot;math-container&quot;&gt;$$\frac{\pi}{4} \approx \frac{1}{n} \sum_{i=1}^n \frac{1}{1 + (i/n)^2} \to \int_{0}^{1} \frac{dt}{1 + t^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.3.3, he proves something equivalent to&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \dots + (-1)^m x^{m-1}\frac{x}{1+x}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.3.4, he applies this (with &lt;span class=&quot;math-container&quot;&gt;$x = ir/n$&lt;/span&gt;), which amounts to (I'm omitting factors of &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt;, which are included in the translation):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align}C/8 &amp;amp;\approx \frac{r}{n} \sum_{i=1}^n \frac{1}{1 + (i/n)^2} \\ &amp;amp;= \frac{r}{n} \sum_{i=1}^{n} (1 - (i/n)^2 + (i/n)^4 + \dots) \\ &amp;amp;= \frac{r}{n}\left(\sum_{i=1}^{n}1 - \sum_{i=1}^{n}(i/n)^2 + \sum_{i=1}^{n}(i/n)^4 + \dots\right) \end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;At this point, he uses &lt;span class=&quot;math-container&quot;&gt;$$\sum_{i=1}^{n}(i/n)^k \approx \frac{i^{2k+1}}{2k + 1}$$&lt;/span&gt; (which he says he will explain later) to reduce the above sum to &lt;span class=&quot;math-container&quot;&gt;$$C/8 \approx r\left(1 - \frac13 + \frac15 - \frac17 + \dots\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which is the series in question.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;6.3.5 and 6.3.6 I didn't look into.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.4, he derives the approximation &lt;span class=&quot;math-container&quot;&gt;$$\frac{1}{n} \sum_{i=1}^{n} (i/n)^k \to \frac{1}{k+1}$$&lt;/span&gt; as promised:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.4.1, he shows that &lt;span class=&quot;math-container&quot;&gt;$S_n^{(1)} = 1 + 2 + 3 + \dots + n \approx n^2/2$&lt;/span&gt; (in two ways, one of them using the exact formula &lt;span class=&quot;math-container&quot;&gt;$S_n^{(1)} = n(n+1)/2$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.4.2, he shows that &lt;span class=&quot;math-container&quot;&gt;$S_n^{(2)} = 1^2 + 2^2 + 3^2 + \dots + n^2 \approx n^3/3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.4.3, he shows that &lt;span class=&quot;math-container&quot;&gt;$S_n^{(3)} \approx n^4/4$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$S_n^{(4)} \approx n^5/5$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In 6.4.4, he does it in general (induction, basically): &lt;span class=&quot;math-container&quot;&gt;$S_n^{(k)} = n^{k+1}/(k+1)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And 6.4.5 is an aside, doing sums of sums (leading to &lt;span class=&quot;math-container&quot;&gt;$\approx n^k/k!$&lt;/span&gt;), not needed here.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, in 6.5, he restates the conclusion: &lt;span class=&quot;math-container&quot;&gt;$$\frac{C}{8} \approx r\left(1 - \frac13 + \frac15 - \frac17 + \dots\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;All of this is done with error terms (and later in the chapter he goes even deeper on error analysis), that I skipped here. Except for the translation into modern notation, none of this is &amp;quot;overinterpreting&amp;quot;: everything is done in words, but all the arguments (&lt;em&gt;yukti&lt;/em&gt;/&lt;em&gt;upapatti&lt;/em&gt;, ≈ proofs/rationale) are there. As mentioned, the explanatory translation on pages 179–207 will be more helpful for modern readers but they still follow the original very closely; you can look at the &amp;quot;bare&amp;quot; translation on pages 45–82 to see what is &amp;quot;actually&amp;quot; there: you just have to read slowly and carefully.&lt;/p&gt;&#xA;" OwnerUserId="432" LastEditorUserId="432" LastEditDate="2022-05-18T14:25:26.337" LastActivityDate="2022-05-18T14:25:26.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14455" PostTypeId="2" ParentId="2088" CreationDate="2022-05-19T09:23:42.483" Score="0" Body="&lt;p&gt;Experimentul M-M a demonstrat ca densitatea masica a luminii s-a insumat cu densitatea masei emanata din substanta pamantului. La descifrarea constantei de actiune h s-a gasit ca fotonul, cuanta de lumina are volum, densitate, masa, presiune, inertie, impuls si energie cinetica. Si deci este graunte de substanta. In continuare s-a gasit ca densitatea masei este data de patratul inductiei magnetice de la nivelul fiecarei semilungimi de unda a nucleonului sau a fotonului. Campul densitatii masice, emanat din substanta pamantului, nu se anuleaza, nu dispare la suprafata pamantului, ci emana in spatiu cu atenuare exponentiala. Existenta campului de densitate, emanat din substanta pamantului, face sa functioneze principiul relativitatii, care nu permite determinarea starii de miscare, a unui sistem, prin experimente din interiorul sistemului.&lt;/p&gt;&#xA;&lt;p&gt;The M-M experiment showed that the mass density of light was added to the density of the mass emanating from the earth's substance. When deciphering the action constant h it was found that the photon, the amount of light has volume, density, mass, pressure, inertia, momentum and kinetic energy. And so it is a grain of substance. It was further found that the density of the mass is given by the square of the magnetic induction at each wavelength of the nucleon or photon. The field of mass density, emanating from the substance of the earth, does not cancel out, does not disappear at the surface of the earth, but emanates in space with exponential attenuation. The existence of the density field, emanating from the substance of the earth, makes the principle of relativity work, which does not allow the determination of the state of motion of a system, through experiments inside the system.&lt;/p&gt;&#xA;" OwnerUserId="16647" LastActivityDate="2022-05-19T09:23:42.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14456" PostTypeId="1" AcceptedAnswerId="14462" CreationDate="2022-05-19T10:10:21.840" Score="2" ViewCount="478" Body="&lt;p&gt;Do you know how the values were measured and if they where determined by particular conditions/ restraints? They can be put to 1 (or to any values, I suppose); what escapes me is why 1/ε × μ must be equal to c&lt;sup&gt;2&lt;/sup&gt;. Was this an arbitrary choice just to prove the speed of propagation?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastEditorUserId="16318" LastEditDate="2022-05-22T17:40:59.340" LastActivityDate="2024-01-12T14:32:10.560" Title="How were the values of permeability and permittivity derived?" Tags="|electromagnetism|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14457" PostTypeId="1" AcceptedAnswerId="14458" CreationDate="2022-05-19T15:22:03.170" Score="3" ViewCount="494" Body="&lt;p&gt;I'm tying to understand, historically, what lead to Lagrangian mechanics (LM). What did Lagrange &lt;em&gt;actually&lt;/em&gt; do?&lt;/p&gt;&#xA;&lt;p&gt;In the time (year 1788), when Lagrange published his work (that we nowadays call &amp;quot;Lagrangian mechanics&amp;quot;) the term &amp;quot;energy&amp;quot; was &lt;em&gt;not&lt;/em&gt; known. It slowly developed in the next 60 years &lt;em&gt;after&lt;/em&gt; Lagrange's work. However, in modern times we say that &lt;span class=&quot;math-container&quot;&gt;$L = T - V$&lt;/span&gt; is the difference between two energies, then do some derivation magic and get equations of motions for particles (or density fields).&lt;/p&gt;&#xA;&lt;p&gt;Today, LM is also often introduced in a way that claims that Newton laws are not optimal because they are so hard to solve because Newton did not know about &amp;quot;generalized coordinates&amp;quot;. But surely Isaac Newton was smart enough to express &lt;span class=&quot;math-container&quot;&gt;$x = r \sin(\phi)$&lt;/span&gt; or something similar in his times already. So, this approach to introduce LM also only fits if you already know the result of LM. (And even if Newton really did not know to use other coordinates than &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;: you still can do this, today, without using LM explicitly)&lt;/p&gt;&#xA;&lt;p&gt;So, what did Lagrange &lt;em&gt;actually&lt;/em&gt; do in his original works? What problem did he tried to solve? And what did he call his resulting variables, if not &amp;quot;energy&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="16663" OwnerDisplayName="Foo Bar" LastActivityDate="2022-05-19T20:05:31.470" Title="What exactly did Lagrange do, historically?" Tags="|classical-mechanics|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14458" PostTypeId="2" ParentId="14457" CreationDate="2022-05-19T19:44:22.720" Score="7" Body="&lt;p&gt;I recently came across an article that helped me put Lagrange's work in a very interesting perspective:&lt;/p&gt;&#xA;&lt;p&gt;Author: Craig Fraser (historian of science)&lt;/p&gt;&#xA;&lt;p&gt;Subject: d'Alembert's &lt;em&gt;Traité de Dynamique&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://homes.chass.utoronto.ca/%7Ecfraser/Dalembert.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://homes.chass.utoronto.ca/~cfraser/Dalembert.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://homes.chass.utoronto.ca/%7Ecfraser/D%27Alembert2.pdf&quot; rel=&quot;noreferrer&quot;&gt;http://homes.chass.utoronto.ca/~cfraser/D%27Alembert2.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Craig Fraser mentions: in the time of Euler, d'Alembert, etc. the way that differential calculus was used was significantly different from the modern use. The application of differential calculus was very much tied up in geometric interpretation. As I understand it: wherever a theorem was stated and proved a diagram was an essential part of the statement/proof.&lt;/p&gt;&#xA;&lt;p&gt;As I understand it: part of Lagrange's research programme was to reformulate mathematics so as to completely remove any dependence on diagrams. As I understand it: the title of his main work: &amp;quot;Mecanique Analytique&amp;quot; refers to that. All theorems are stated purely in the from of formulas, no dependence on geometric interpretation.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;It's not that prior to Lagrange generalized coordinates were not used at all. Specific problems were solved with ingenious methods, that in retrospect can be recognized using a form of generalized coordinates. But that was on a case-by-case basis. My understanding is that Lagrange made usage of generalized coordinates &lt;em&gt;systematic&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;About energy:&lt;br /&gt;&#xA;We have that Lagrange used the energy concept systematically, and that he could show that this energy mechanics reproduces the newtonian formulation. Many authors write that to that end Lagrange used something they refer to as d'Alembert's principle. However, it could be that Lagrange attributed to d'Alembert something that was more Lagrange's interpretation.&lt;/p&gt;&#xA;&lt;p&gt;There can be no doubt that Lagrange did have a demonstration of equivalence of newtonian formulation and energy mechanics. To my understanding: that demonstration did not use (at least not explicitly), the theorem that today is known as the Work-Energy theorem.&lt;/p&gt;&#xA;&lt;p&gt;(There is a &lt;a href=&quot;https://physics.stackexchange.com/a/696623/17198&quot;&gt;derivation of the Work-Energy theorem from F=ma&lt;/a&gt; by me on physics.stackexchange.)&lt;/p&gt;&#xA;&lt;p&gt;A translation of 'Mecanique Analytique' is available on archive.org&#xA;&lt;a href=&quot;https://archive.org/details/springer_10.1007-978-94-015-8903-1&quot; rel=&quot;noreferrer&quot;&gt;Analytic mechanics&lt;/a&gt;&lt;br /&gt;&#xA;(Among the download options there is an option to download as .pdf file.)&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="7244" LastEditDate="2022-05-19T20:05:31.470" LastActivityDate="2022-05-19T20:05:31.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14459" PostTypeId="1" CreationDate="2022-05-19T20:34:24.313" Score="5" ViewCount="128" Body="&lt;p&gt;Question &lt;a href=&quot;https://hsm.stackexchange.com/questions/12135/why-is-dirac-delta-named-after-dirac-when-the-concept-was-already-over-two-centu&quot;&gt;Why is Dirac delta named after Dirac when the concept was already over two centuries old?&lt;/a&gt; notes the Dirac delta function effectively appeared centuries before Dirac.  This was long before the now standard set theoretic notion of function was defined.  Histories tell me there was long controversy especially over Fourier's use of the integral which we now recognize as defining &lt;span class=&quot;math-container&quot;&gt;$\delta(x)$&lt;/span&gt;, which I am sure is true, but I want a quote.&lt;/p&gt;&#xA;&lt;p&gt;I seek quotes of mathematicians who specifically rejected the uses of this idea by Lagrange, Fourier, Cauchy, etc on the grounds that it is not a function.&lt;/p&gt;&#xA;&lt;p&gt;I do not mean people who said the delta function is not an ordinary function.  Many textbooks to this day use &lt;span class=&quot;math-container&quot;&gt;$\delta(x)$&lt;/span&gt; more-or-less as if it were a real-valued function, while explicitly recognizing it is actually not that.  I mean people who objected to the use of &lt;span class=&quot;math-container&quot;&gt;$\delta(x)$&lt;/span&gt;, or integrals such as Fourier's which in fact define it, on those grounds.&lt;/p&gt;&#xA;" OwnerUserId="38" LastActivityDate="2022-05-19T20:34:24.313" Title="Seeking quotes rejecting early forms of the Dirac delta" Tags="|real-analysis|harmonic-analysis|fourier|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14460" PostTypeId="1" CreationDate="2022-05-20T00:52:08.210" Score="4" ViewCount="108" Body="&lt;p&gt;A &lt;a href=&quot;https://en.wikipedia.org/wiki/Perfect_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;perfect number&lt;/a&gt; is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.  The first five perfect numbers are &lt;a href=&quot;https://oeis.org/A000396&quot; rel=&quot;nofollow noreferrer&quot;&gt;6, 28, 496, 8128, and 33550336&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Nicomachus made several statements about perfect numbers c. 100 AD.  It is common to interpret one of these (eg, see the &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Perfect_numbers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor entry on perfect numbers&lt;/a&gt;) as stating that (in modern terminology)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The &lt;em&gt;n&lt;/em&gt;th perfect number has &lt;em&gt;n&lt;/em&gt; digits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As we see, this is incorrect because the 5th perfect number has 8 digits.&#xA;Nearly a millennium later c. 980 AD, &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Al-Baghdadi/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Al-Baghdadi&lt;/a&gt; wrote instead:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He who affirms that there is only one perfect number in each power of 10 is wrong; there is no perfect number between ten thousand and one hundred thousand.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is actually relatively easy to prove — &lt;strong&gt;assuming&lt;/strong&gt; that all perfect numbers are even (which both Nicomachus and Al-Baghdadi also stated) and given the &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euclid-Euler theorem&lt;/a&gt; characterizing even perfect numbers.&lt;/p&gt;&#xA;&lt;p&gt;However, in order to demonstrate that the fifth perfect number does not have five digits, it is also strictly-speaking necessary to demonstrate that there is no odd perfect number with five digits.  Who was the first to provide a sufficiently-precise analysis of odd perfect numbers to demonstrate that there does not exist a perfect number between 10,000 and 100,000, as asserted by Al-Baghdadi?  Or more importantly, &lt;em&gt;when&lt;/em&gt;?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Remark&lt;/em&gt;: The history of perfect numbers is littered with false claims as well as bold, still-unproven assertions.  And of course the standards of proof have evolved considerably over the millennia.  However, as a minimum bar for &amp;quot;sufficiently-precise&amp;quot;, I would like there to be &lt;em&gt;any&lt;/em&gt; analysis of odd perfect numbers whatsoever beyond the claim that they don't exist.&lt;/p&gt;&#xA;" OwnerUserId="9818" LastActivityDate="2022-05-20T00:52:08.210" Title="Who first proved that there is no five-digit perfect number?" Tags="|mathematics|number-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14461" PostTypeId="1" CreationDate="2022-05-20T05:37:48.240" Score="2" ViewCount="276" Body="&lt;p&gt;Was Nikola Tesla aware of Maxwell's equations or Heinrich Hertz's experimental proof of the former when he worked on his radio device invention?&lt;/p&gt;&#xA;" OwnerUserId="16652" LastEditorUserId="4251" LastEditDate="2022-05-20T16:38:22.913" LastActivityDate="2022-05-20T16:38:22.913" Title="Was Nikola Tesla aware of Maxwell's equations?" Tags="|discoveries|electromagnetism|maxwell|radiation|tesla|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14462" PostTypeId="2" ParentId="14456" CreationDate="2022-05-20T19:37:08.967" Score="2" Body="&lt;p&gt;The vacuum permittivity &lt;span class=&quot;math-container&quot;&gt;$\epsilon_0$&lt;/span&gt; is at first glance just a constant related to electric forces. It can be measured through Coulomb's law that provides an expression for the electric force between two charged point objects (with charges &lt;span class=&quot;math-container&quot;&gt;$q_1,q_2$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$F=K_{\rm C} \frac{q_1q_2}{r^2},$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the distance between the charges and &lt;span class=&quot;math-container&quot;&gt;$K_{\rm C}$&lt;/span&gt; is the proportionality Coulomb constant and you can write it as you wish (depending on your unit system). In modern times, we write &lt;span class=&quot;math-container&quot;&gt;$K_{\rm C}=1/4\pi \epsilon_0$&lt;/span&gt; (SI units).&lt;/p&gt;&#xA;&lt;p&gt;The vacuum permeability &lt;span class=&quot;math-container&quot;&gt;$\mu_0$&lt;/span&gt; comes from magnetic effects. The easiest way to measure it is to use Ampere's force law, which describes the magnetic force between two wires carrying DC currents &lt;span class=&quot;math-container&quot;&gt;$I_1, I_2$&lt;/span&gt;, given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$f_{\rm A}=2 K_{\rm A}\frac{I_1I_2 }{r}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; has units of force per unit length. In modern times, we write &lt;span class=&quot;math-container&quot;&gt;$K_{\rm A}=\mu_0/{4 \pi}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If electric forces and magnetic forces were unrelated, we would be able to choose whatever we want for &lt;span class=&quot;math-container&quot;&gt;$K_{\rm C}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K_{\rm A}$&lt;/span&gt;, but that is not the case. The discovery of electromagnetic induction, imposed equations between electric and magnetic phenomena.&lt;/p&gt;&#xA;&lt;p&gt;James Clerk Maxwell compiled all the known equations for electricity , magnetism and electromagnetism (today known as Maxwell's equations), and in 1865 he discovered that in the case of empty space you could put them together to write a wave equation. This wave equation indicates the speed of the waves as a function of the constants of electromagnetism, as is given by the ratio&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$c=\sqrt{\frac{K_C}{K_{\rm A}}}$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When Maxwell calculated &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; from that formula, he found that the speed of these electromagnetic waves coincides with the speed of light measured by  the experiments of Fizeau and Foucault some years before.&lt;/p&gt;&#xA;&lt;p&gt;Thus in modern times, when we choose to write everything with &lt;span class=&quot;math-container&quot;&gt;$\epsilon_0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mu_0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c=(\mu_0\epsilon_0)^{-1/2}$&lt;/span&gt;, this is not a choice. The laws of electromagnetism impose that this constants have to be related by speed of light (only one of the two can be chosen freely).&lt;/p&gt;&#xA;&lt;h2&gt;Additional notes&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Weber and Kohlrausch seem to have empirically noticed the relation between c, &lt;span class=&quot;math-container&quot;&gt;$K_{\rm C}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K_{A}$&lt;/span&gt; before Maxwell in 1835. Maxwell himself cites them, but finds a theoretical reasoning.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In &amp;quot;A Dynamical Theory of the Electromagnetic Field&amp;quot;, Phil. Trans. R. Soc. Lond 1865, Maxwell uses the relation of Weber and uses the units &lt;span class=&quot;math-container&quot;&gt;$K_{\rm A}=4\pi$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$K_{\rm C}=4\pi c^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-12T14:32:10.560" LastActivityDate="2024-01-12T14:32:10.560" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14463" PostTypeId="1" CreationDate="2022-05-21T07:45:25.407" Score="2" ViewCount="115" Body="&lt;p&gt;I'm trying to compute the value of G from Cavendish's own observations. I get &lt;span class=&quot;math-container&quot;&gt;$G_{Cav}=5.27501×10^{−10}$&lt;/span&gt; which is 8 times bigger than the accepted value of &lt;span class=&quot;math-container&quot;&gt;$G_{True}=6.67430×10^{−11}$&lt;/span&gt;. Do you see anything wrong with my computations below?&lt;/p&gt;&#xA;&lt;p&gt;I'm using this formula (from &lt;a href=&quot;https://en.wikipedia.org/wiki/Cavendish_experiment#Derivation_of_G_and_the_Earth%27s_mass&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;G = \frac{2\pi^2 L r^2 \theta}{M T^2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;G = Gravitational constant&lt;/p&gt;&#xA;&lt;p&gt;L = Length of torsion balance (the distance between the centers of balls)&lt;/p&gt;&#xA;&lt;p&gt;r = The distance of attraction (between weights and balls)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; = Deflection of the arm from its rest position due to gravitational attraction&lt;/p&gt;&#xA;&lt;p&gt;M = Mass of attracting lead weight&lt;/p&gt;&#xA;&lt;p&gt;T = Natural period of oscillation of the balance&lt;/p&gt;&#xA;&lt;p&gt;I take &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; and N from the 4th experiment, (Page 520 in &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1798.0022&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cavendish's paper&lt;/a&gt;), the rest are constants,&lt;/p&gt;&#xA;&lt;p&gt;L = 1.862 m&lt;/p&gt;&#xA;&lt;p&gt;r = 0.2248 m&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$r^2$&lt;/span&gt; = 0.05053 &lt;span class=&quot;math-container&quot;&gt;$m^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; = 0.00806788 radians&lt;/p&gt;&#xA;&lt;p&gt;M = 158.04 kg&lt;/p&gt;&#xA;&lt;p&gt;T = 421 s&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$T^2$&lt;/span&gt; = 177241 &lt;span class=&quot;math-container&quot;&gt;$s^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Substituting in the numbers,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;G_{Cav} = \frac{2 \times \pi^2 \times 1.836 \times 0.05053504 \times 0.00806788}{158.04 \times 177241} = 5.27501\times 10^{-10}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is eight times bigger than the accepted value,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{G_{Cav}}{G_{Tru}} = 7.90&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There are more details &lt;a href=&quot;http://cavendish-deneyi.com/big-g-01.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2022-05-21T07:45:25.407" Title="Computing Gravitational constant G from Cavendish's data" Tags="|experimental-physics|gravity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14464" PostTypeId="2" ParentId="14456" CreationDate="2022-05-21T15:10:49.803" Score="3" Body="&lt;p&gt;Maxwell's measurements of permeability and permittivity were used in his estimate of the speed of electromagnetic waves and to support his theory that electromagnetism and light were the same phenomena. Here is how Brain Clegg described the process in his biography of Maxwell titled &lt;em&gt;Professor Maxwell's Duplicitous Demon&lt;/em&gt;. Unfortunately, it is a bit light on technical details.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] The equivalent properties for electromagnetism were known as the magnetic permeability and the electric permittivity of space, and the values of these properties were relatively poorly known, so the result had a significant uncertainty.&lt;/p&gt;&#xA;&lt;p&gt;With Charles Hockin, an electrical engineer based in Cambridge, Maxwell devised and experiment that would pin down these values to far greater accuracy than ever before. The experiment balanced out the attraction from the electrical charge on two metal plates with the repulsion between two electromagnets with like poles facing each other.  The bigger the effect, the more accurate the measurement could be - which meant hunting down an extremely high-voltage source.&lt;/p&gt;&#xA;&lt;p&gt;Rather surprisingly, the owner of the most powerful batteries in the country turned out not to be a physics laboratory or an electrical power company, but a wine merchant based in Clapham, London.  John Gassiot had spent his fortune on constructing an extravagant private laboratory.  It was Gassiot who provided Maxwell and Hockin with a vast battery of cells, 2,600 in all, which between them put out around 3,000 volts.&lt;/p&gt;&#xA;&lt;p&gt;The experiment proved highly successful, though the batteries went flat with unnerving rapidity, meaning Maxwell and Hockin had to take measurements furiously quickly before the charge ran out. These more accurate values for magnetic permeability and electric permittivity resulted in a calculated speed for electromagnetic waves of 288,000 kilometres per second.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When Maxwell later learned that Foucault had obtained a value of 298,000 kilometres per second for the speed of light, he was more certain than ever that his waves were light.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2022-05-21T19:28:20.883" LastActivityDate="2022-05-21T19:28:20.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14466" PostTypeId="1" CreationDate="2022-05-23T00:39:00.870" Score="2" ViewCount="270" Body="&lt;p&gt;Over the history of mathematics, some key facts have had multiple and different proofs developed for them. Sometimes these different proofs provide a unique insight or understanding of those facts.&lt;/p&gt;&#xA;&lt;p&gt;For example, the infinitude of primes has an ancient proof probably due to Euclid, as well as more modern proofs such as those based on Euler's product formula, and hence the fundamental theorem of arithmetic. I'm not skilled enough to understand the additional insight that the elementary proof of the prime number theorem provided.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I am searching for the simplest proof of the fact that the prime harmonic series diverges:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sum\frac{1}{p} = \infty $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;By simple, I mean requiring the least knowledge beyond secondary school mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I would be happy with a proof based on complex analysis if it is simple and elegant. I would also be happy to accept as prerequisites, if required, that the harmonic series diverges, and the Basel series converges.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;The proof set out here (&lt;a href=&quot;https://fromprimestoriemann.blogspot.com/2020/12/primes-arent-that-spread-out.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;) is simple but in my opinion inelegant.&lt;/p&gt;&#xA;&lt;p&gt;I have tried to read this &lt;a href=&quot;https://www.jstor.org/stable/2321218?origin=JSTOR-pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; by Eynden surveying several proofs and none seem simple. Erdos' proof is described as simple but I have tried several times, and failed to comprehend it.&lt;/p&gt;&#xA;" OwnerUserId="12633" LastEditorUserId="12633" LastEditDate="2022-05-23T10:23:05.307" LastActivityDate="2022-05-25T23:58:37.817" Title="Simplest of the many proofs the prime harmonic series diverges" Tags="|number-theory|primes|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14467" PostTypeId="1" AcceptedAnswerId="14469" CreationDate="2022-05-23T12:20:14.387" Score="0" ViewCount="158" Body="&lt;p&gt;Recently, a teacher of mine stated that, since the institutional division between philosophy and other fields of academic enquiry, especially the natural sciences, happened somewhere along the 19th century, thinkers that we now regard as mathematicians (like Euler and Gauss) did not think of themselves primarily as mathematicians, but as philosophers. I know that the work of such thinkers was considerably less specialized back then but, even so, is what my teacher claiming true? Citations from the mathematicians/philosophers naming themselves one or the other way would be particularly helpful.&lt;/p&gt;&#xA;" OwnerUserId="16662" LastActivityDate="2022-05-23T14:21:29.533" Title="Did thinkers we now regard as mathematicians call themselves philosophers prior to the 19th century?" Tags="|philosophy-of-science|mathematics-social-history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14468" PostTypeId="2" ParentId="14466" CreationDate="2022-05-23T13:09:28.973" Score="6" Body="&lt;p&gt;The simplest proof is Euler's original proof. It is based on the identity&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{n=1}^\infty n^{-s}=\prod_{p}\left(1-p^{-s}\right)^{-1}, s&amp;gt;1.$$&lt;/span&gt;&#xA;This identity is equivalent to existence and uniqueness of decomposition&#xA;of integers into products of primes.&lt;/p&gt;&#xA;&lt;p&gt;Now when &lt;span class=&quot;math-container&quot;&gt;$s\to 1+$&lt;/span&gt;, the LHS tends to &lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt; (since the harmonic series is divergent), so the RHS must become divergent as &lt;span class=&quot;math-container&quot;&gt;$s=1$&lt;/span&gt;, that is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sum\frac{1}{p}=\infty,$$&lt;/span&gt;&#xA;by the elementary convergence criterion for infinite products.&#xA;No complex analysis involved. Only convergence criteria for series and products with positive terms.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-05-23T13:09:28.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14469" PostTypeId="2" ParentId="14467" CreationDate="2022-05-23T14:21:29.533" Score="2" Body="&lt;p&gt;Your teacher is simply wrong. Most major fields of academia were regarded as a specialized area of interest for millennia, originating formally and most significantly in ancient Greece.&lt;/p&gt;&#xA;&lt;p&gt;Archimedes, for example, made a clear distinction in his writing (I wish I could give the exact citation for you) between applied and pure mathematics (although they didn't use those terms).The applied regarding his work in physics, and the pure in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Even the more niche field of logic was well-defined enough to be its own study by the 13th to 14th century, with medieval scholars specifically referring to themselves as logicians.&lt;/p&gt;&#xA;&lt;p&gt;Pythagoras, the main individual responsible for the distinction between pure and applied math as well as the beginnings of Greek philosophy, spoke clearly about the fact that philosophy and pure mathematics were separate fields of thought, and it was his cult's, the Pythagoreans', purpose to integrate them into a mathematical worldview.&lt;/p&gt;&#xA;&lt;p&gt;In Eastern thought, primarily China and India, there was also a clear distinction made between a society's architects and its sages. Lao-tzu and the Buddha certainly weren't considered in the same regard as a mathematician.&lt;/p&gt;&#xA;&lt;p&gt;The examples feel endless. It is human nature to label, categorize, and distinct things, and our fields of thought and study are no different. To posit that this wasn't the case until recent - that all pre-19th century scholars simply called themselves philosophers, is preposterous.&lt;/p&gt;&#xA;" OwnerUserId="16611" LastActivityDate="2022-05-23T14:21:29.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14470" PostTypeId="1" CreationDate="2022-05-24T10:23:34.127" Score="0" ViewCount="109" Body="&lt;p&gt;This is a request for help (with examples, as described below) with a talk I giving to graduate students regarding the dynamics of mathematical research among mathematicians and the development of key ideas towards the solution of open problems.&lt;/p&gt;&#xA;&lt;p&gt;So, during the talk, I am intending to supply several examples/instances where mathematicians &lt;strong&gt;almost simultaneously&lt;/strong&gt; arrived at the same/similar/different solution, idea, or proof strategy to a particular problem (probably well known or difficult problem). The context of “almost simultaneous” is not with respect to the publication dates necessarily but, give and take, within a couple of years between such publications; more relevant is that the key idea generation/discovery would have been just about the same time.&lt;/p&gt;&#xA;&lt;p&gt;Naturally, the development of calculus (Newton and Leibniz) is in such a spirit. So also is the proof of the Prime Number Theorem, both the &lt;em&gt;non-elementary&lt;/em&gt; (de la Vallée Poussin and Hadamard) and the &lt;em&gt;elementary&lt;/em&gt; (Selberg and Erdos). A recent one, I want to believe, is the bounded prime gaps (Zhang and Maynard). Especially, it would be nice if you could also provide a (his)story/link/reference of the account. I am not actually requesting for controversies or issues surrounding the solutions....but the examples of such problems and their solutions.&lt;/p&gt;&#xA;&lt;p&gt;It is also welcome cases, where such examples may be obscure, or cases where the authors perhaps collaborated (such as, I believe, Gowers and Maurey on the Unconditional Basic Sequence Problem). Little-known or Word-of-mouth examples are also welcome, as long as it is not a private matter (By this, I have in mind, for instance, that I once heard/read somewhere that a quite prominent mathematician had also been developing a proof to FLT along the same/similar lines as Wiles but (perhaps) did not pursue further after Wiles’ announcement and eventual publication. Such instances may be private rather than little-known as the said “prominent” mathematician neither published or intended for their admission to be made public).&lt;/p&gt;&#xA;&lt;p&gt;PS: This is my first time asking a question here. I do not know if such a question has been asked here before. Will welcome a link to it if so. Also, I don’t know if this is tagged differently or anything of that sort, so will be grateful if it can be made as such if necessary.&lt;/p&gt;&#xA;" OwnerUserId="16668" LastActivityDate="2022-06-14T21:12:59.010" Title="Simultaneous independent or semi-independent solutions to problems" Tags="|mathematics-social-history|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14471" PostTypeId="1" CreationDate="2022-05-24T17:46:36.260" Score="1" ViewCount="141" Body="&lt;p&gt;There is conflicting opinion as to who was Hardy's Tripos coach. The Wikipedia page on Hardy claims it was R. A. Herman, a claim that appears to be backed up by Leonard Roth in his article &amp;quot;Old Tripos days&amp;quot; that appeared in the &lt;em&gt;Monthly&lt;/em&gt; in 1971 (78, pp. 223-236). However, Titchmarsh claims that he was first coached by R. R. Webb, and later by A. E. H. Love (Obituary in the first volume of Hardy's &lt;em&gt;Collected Papers&lt;/em&gt;). As Titchmarsh was one of Hardy's students, I would take his word as final. Do I have this right? Does anyone have knowledge of who the &lt;em&gt;real&lt;/em&gt; coach was?&lt;/p&gt;&#xA;" OwnerUserId="16670" LastActivityDate="2022-05-24T17:46:36.260" Title="Who was G. H. Hardy's Tripos coach?" Tags="|biographical-details|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14472" PostTypeId="1" CreationDate="2022-05-24T18:52:22.827" Score="4" ViewCount="140" Body="&lt;p&gt;I get an impression from &lt;a href=&quot;https://plato.stanford.edu/entries/statphys-Boltzmann/#Aca&quot; rel=&quot;nofollow noreferrer&quot;&gt;this enyclopedia entry&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;that the primary source of the Boltzmann entropy equation &lt;span class=&quot;math-container&quot;&gt;$S = k \log W$&lt;/span&gt; might be&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1866, Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie, Wiener Berichte, 53: 195–220; in WA I, paper 2.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I cannot read German. The only PDF I was able to obtain did not have a text-layer, so I could not copy-paste parts into Google Translate. Eventually I might go through and hand-type the whole thing into Google translate, but in the interim I would appreciate assistance expediting my search.&lt;/p&gt;&#xA;&lt;p&gt;What is the original source of the Boltzmann entropy equation?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2022-05-24T18:52:22.827" Title="In which work was Boltzmann's entropy originally introduced?" Tags="|physics|reference-request|thermodynamics|boltzmann|entropy|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14473" PostTypeId="2" ParentId="14466" CreationDate="2022-05-25T05:40:12.503" Score="3" Body="&lt;p&gt;This is my favorite way to present Euler's proof:&lt;/p&gt;&#xA;&lt;p&gt;By the fundamental theorem of arithmetic:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\prod_{p \leq N} \left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right) \geq 1+\frac{1}{2}+ \cdots+\frac{1}{N}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;for any positive integer &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$N&amp;gt;2$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$1+\frac{1}{2}+ \cdots+\frac{1}{N} &amp;gt; 1+\frac{1}{2}+\cdots+\frac{1}{N-1} \geq \int_{1}^{N} \frac{1}{t} \, dt = \log N.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Provided that &lt;span class=&quot;math-container&quot;&gt;$e^{2x}&amp;gt;\frac{1}{1-x}$&lt;/span&gt; for any &lt;span class=&quot;math-container&quot;&gt;$x \in (0, 0.5]$&lt;/span&gt;, it follows that&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \log N &amp;lt; \prod_{p \leq N} \left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right) = \prod_{p \leq N} \frac{1}{1-\frac{1}{p}} &amp;lt; \prod_{p \leq N} e^{\frac{2}{p}}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Therefore,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \sum_{p \leq N}\frac{1}{p} &amp;gt; \frac{\log \log N}{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;for any integer &lt;span class=&quot;math-container&quot;&gt;$N&amp;gt;2$&lt;/span&gt; and we are done.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-05-25T23:58:37.817" LastActivityDate="2022-05-25T23:58:37.817" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14474" PostTypeId="2" ParentId="12057" CreationDate="2022-05-25T23:05:50.717" Score="-3" Body="&lt;p&gt;In short, it became too radical for the west and the CIA backed post modernist thinkers came to have the final word on systems thinking, replacing the field and all it's language with a new language of terms like Assemblage and Wicked Problems which imply that systems are too complex for us to bother trying to understand in the first place.&lt;/p&gt;&#xA;&lt;p&gt;The soviets won the scientific war by being open to and advancing Cybernetics along side their UK / US counterparts. As Cybernetic thought progressed, and its leading figures started building prototype internets in the early 70s, it increasingly became critical of western democracy, arguing that it was not democratic at all (see Stafford Beer's cybernetics and the will of the people talk on youtube). As the development of cybernetic analysis became inherently anti-capitalist, and seen as complimentary to, rather than opposing Soviet Dialectical Materialism (there are US military science papers on this that are very interesting), the whole field had to be ejected from academic dominance. There was a loss of funding, heavy promotion of anti-marxist post modern thinkers and notions such as post-structuralism came to be more appealing to young academics and researchers who found cybernetics to be too mathematical, or simply too difficult to grasp. So we probably see a generation of poor teachers and poor students and then we are left with a 30 year gap in the development of systems theory that we are only just beginning to fill.&lt;/p&gt;&#xA;&lt;p&gt;For CIA influence on late 20th century philosophy see Gabriel Rockhills work.&lt;/p&gt;&#xA;" OwnerUserId="16667" LastActivityDate="2022-05-25T23:05:50.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14475" PostTypeId="1" AcceptedAnswerId="14478" CreationDate="2022-05-25T23:57:40.577" Score="7" ViewCount="372" Body="&lt;p&gt;It is sometimes asserted that human-readable mathematical proofs that we construct and publish are just informal approximations to the gold standard, which is a completely formal proof in a formal language.  The formal proof is supposed to be the real McCoy, the pinnacle of Platonic perfection, which our messy informal proofs try their best to gesture toward.&lt;/p&gt;&#xA;&lt;p&gt;I seem to recall reading criticisms (of the above assertion) that focused on the enormous size of formal proofs, claiming that they were so large and unwieldy that it would be impossible to write them down in full. If they are so large that we cannot actually produce them or even properly grasp them in our minds, then can we legitimately claim that such fictions are what proofs really are, or strive to be?&lt;/p&gt;&#xA;&lt;p&gt;Although I feel that I have encountered the above criticism many times, I am having trouble finding an explicit example of someone making this kind of argument in print. The closest thing I have found is A. R. D. Mathias's essay, &lt;a href=&quot;https://www.dpmms.cam.ac.uk/~ardm/inefff.pdf&quot; rel=&quot;noreferrer&quot;&gt;A term of length 4,523,659,424,929&lt;/a&gt;, which discusses a system for set theory due to Bourbaki, and emphasizes the gigantic size of even very simple terms in the system.  However, Mathias's criticism is narrowly focused on Bourbaki's system in particular, and he does not take the additional step of arguing that &lt;i&gt;all&lt;/i&gt; fully formal proofs are too long to write down, and (thus) cannot in any sense be what mathematical proofs really are, or aspire to be.&lt;/p&gt;&#xA;&lt;p&gt;In the introduction to the book &lt;i&gt;18 Unconventional Essays on the Nature of Mathematics&lt;/i&gt;, Reuben Hersh recalls the distinction that he made in his earlier book (&lt;i&gt;What Is Mathematics, Really?&lt;/i&gt;) between formal proof and &amp;quot;an argument accepted as conclusive by the present-day mathematical community,&amp;quot; and mentions the Flyspeck Project to produce a completely formal proof of the Kepler Conjecture.  Hersh writes, &amp;quot;I do not know anyone who thinks either that this project can be completed, or that even if claimed to be complete it would be universally accepted as a convincing proof of Kepler's conjecture.&amp;quot;  What Hersh writes here is close to what I am looking for; on the question of what proofs really are, he clearly favors informal proof over formal proof, and he expresses some skepticism about formal proofs that is related to their size.  But he still stops short of arguing that formal proofs of nontrivial mathematical theorems are simply too large and complicated to ever instantiate in the real world.&lt;/p&gt;&#xA;&lt;p&gt;Is there a better published reference for the argument that (1) formal proofs of nontrivial theorems are intrinsically so huge that they will forever remain figments of our imagination, and (2) therefore we should be skeptical of the role of formal proofs in the justification of mathematical knowledge?  I'm imagining that such a reference would likely date from the mid-20th century, since nowadays the existence of computerized proof assistants makes it much harder to argue that fully formal proofs of nontrivial theorems can never be instantiated in the real world.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2023-09-10T10:07:59.827" Title="Claims that fully formal proofs are impossible to write down" Tags="|philosophy-of-science|mathematical-logic|" AnswerCount="3" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14476" PostTypeId="1" CreationDate="2022-05-26T02:39:34.700" Score="1" ViewCount="90" Body="&lt;p&gt;In &lt;em&gt;An essay on the psychology of invention in the mathematical field&lt;/em&gt; Hadamard said this about Weierstrass:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The two German mathematicians whom Poincaré compares are Weierstrass and Riemann. That, as he concludes, Riemann is typically intuitive and Weierstrass typically logical is beyond contestation. But as to the latter, Poincaré says “You may leaf through all his books without finding a figure.” It strikes me that there happens to be there an error of fact. It is true that almost no memoir of Weierstrass implied any figure: there is only one exception; but there is one, and this exception occurs in one of his most masterly and clear-cut works, one giving the most complete impression of perfection:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I mean &lt;strong&gt;his fundamental method in the calculus of variations. Weierstrass draws a simple diagram&lt;/strong&gt; and, after that initial step is taken, everything goes on in the profoundly logical way which is undoubtedly his characteristic, so that, by merely looking at that diagram, anyone sufficiently acquainted with mathematical methods could have rebuilt the whole argument. But of course there was an initial intuition, that of constructing the diagram. This was the more difficult and the more evidently an act of genius because it meant breaking from the general methods...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In a footnote, Hadamard clarified&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whether Weierstrass himself actually drew the diagram (or simply described it in words) cannot be said because he did not develop his method elsewhere than in bis oral lectures. That method remained unknown for years, except to bis former students.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In [1], I found a few diagrams, but I am unable to tell which one is this miraculous diagram, since I am unfamiliar with the calculus of variations.&lt;/p&gt;&#xA;&lt;p&gt;[1] Goldstine, H.H. (1980). Weierstrass. In: A History of the Calculus of Variations from the 17th through the 19th Century. Studies in the History of Mathematics and Physical Sciences, vol 5. Springer, New York, NY. &lt;a href=&quot;https://doi.org/10.1007/978-1-4613-8106-8_5&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-1-4613-8106-8_5&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="11081" LastEditorUserId="5989" LastEditDate="2022-05-27T06:17:56.427" LastActivityDate="2022-05-27T06:17:56.427" Title="What is Weierstrass' &quot;simple diagram&quot; in the calculus of variations?" Tags="|calculus|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14477" PostTypeId="1" CreationDate="2022-05-27T14:03:39.863" Score="2" ViewCount="191" Body="&lt;p&gt;Eudoxus developed the exhaustion method for calculating areas that continued with Archimedes and then only in the 16th century with Simon Stevin and Buonaventura Cavalieri. Why did it take so long? Was there any mystical restriction due to the fact that the method was based on the concept of infinity?&lt;/p&gt;&#xA;" OwnerUserId="16677" LastEditorUserId="6609" LastEditDate="2023-07-07T18:20:30.127" LastActivityDate="2023-07-07T18:20:30.127" Title="Exhaustion method blocked by religious aspects?" Tags="|ancient-greece|16th-century|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14478" PostTypeId="2" ParentId="14475" CreationDate="2022-05-28T11:43:43.693" Score="4" Body="&lt;p&gt;Would you be thinking of this passage in the introduction of Bourbaki's &lt;em&gt;Théorie des ensembles&lt;/em&gt;, page I 10? (&lt;a href=&quot;https://books.google.fi/books?id=VDGifaOQogcC&amp;amp;lpg=SA1-PA10&amp;amp;ots=C18MupGocn&amp;amp;pg=SA1-PA10#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google preview&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si la mathématique formalisée était aussi simple que le jeu d’échecs,&#xA;une fois décrit le langage formalisé que nous avons choisi, il n’y&#xA;aurait plus qu’à rédiger nos demonstrations dans ce langage ... un tel&#xA;projet est absolument irréalisable; la moindre demonstration du debut&#xA;de la Théorie des Ensembles exigerait déjà des centaines de signes&#xA;pour être complètement formalisée.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Same passage in its English translation (&lt;a href=&quot;https://books.google.it/books?id=IL-SI67hjI4C&amp;amp;lpg=PA351&amp;amp;hl=fi&amp;amp;pg=PA10#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google preview&lt;/a&gt;), boldface emphasis mine:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If formalized mathematics were as simple as the game of chess, then&#xA;once our chosen formalized language had been described there would&#xA;remain only the task of &lt;strong&gt;writing out our proofs in this language&lt;/strong&gt;&#xA;... such a project is &lt;strong&gt;absolutely unrealizable&lt;/strong&gt; : the tiniest proof&#xA;at the beginning of the Theory of Sets would already &lt;strong&gt;require several&#xA;hundreds of signs&lt;/strong&gt; for its complete formalization.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bourbaki goes on to claim that it is necessary to use lots of &lt;em&gt;abbreviations&lt;/em&gt; to make the language more manageable than the formalized language, and any mathematician worth his salt will be convinced that such things can be taken as shorthand descriptions of the formal thing.  And that the abbreviation rules themselves must be informal, because otherwise they will be too complicated to be useful.&lt;/p&gt;&#xA;&lt;p&gt;It &lt;em&gt;seems&lt;/em&gt; to me that this is meant to be a general statement against the longhand use &lt;em&gt;any&lt;/em&gt; formal system, not just the particular one they have chosen in the book. (Literally it says &amp;quot;our chosen formalized language&amp;quot; but that might mean &amp;quot;whatever language we have chosen to formalize, ...&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;This passage is discussed by Thomas Hales in ''Developments of formal proofs'' (&lt;a href=&quot;https://arxiv.org/abs/1408.6474&quot; rel=&quot;nofollow noreferrer&quot;&gt;arXiv&lt;/a&gt; 2014; see page 15: &amp;quot;Bourbaki objected that formal proofs are too long&amp;quot;). Bourbaki's words seem widely cited, e.g. &lt;a href=&quot;https://www.google.com/search?channel=nrow5&amp;amp;client=firefox-b-d&amp;amp;q=%22require%20several%20hundreds%20of%20signs%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;require several hundreds of signs&amp;quot;&lt;/a&gt;, so that might also give some leads.&lt;/p&gt;&#xA;" OwnerUserId="16297" LastEditorUserId="16297" LastEditDate="2022-05-28T13:59:17.047" LastActivityDate="2022-05-28T13:59:17.047" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14479" PostTypeId="2" ParentId="637" CreationDate="2022-05-30T12:11:21.183" Score="3" Body="&lt;ul&gt;&#xA;&lt;li&gt;C. Truesdell, &lt;a href=&quot;https://link.springer.com/article/10.1007/BF00253932&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solutio generalis et accurata problematum quamplurimorum de motu corporum elasticorum incomprimibilium in deformationibus valde magnis&lt;/a&gt;, Archive for Rational Mechanics and Analysis, Volume 11, Pages 106–113, January 1962.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Still &lt;a href=&quot;https://scholar.google.com/scholar?as_ylo=2018&amp;amp;hl=en&amp;amp;as_sdt=2005&amp;amp;sciodt=0,5&amp;amp;cites=8658260663973728865&amp;amp;scipsc=&quot; rel=&quot;nofollow noreferrer&quot;&gt;getting cited&lt;/a&gt; in the past few years.&lt;/p&gt;&#xA;" OwnerUserId="16706" LastEditorUserId="4251" LastEditDate="2022-05-30T19:53:13.017" LastActivityDate="2022-05-30T19:53:13.017" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14480" PostTypeId="1" CreationDate="2022-05-30T14:07:27.790" Score="2" ViewCount="150" Body="&lt;p&gt;According to this &lt;a href=&quot;https://projecteuclid.org/journalArticle/Download?urlId=rml%2F1204835119&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; copies of Frege's Begriffsschrift were both present during the early 1880s (before Peirce published his works on first-order logic) at the Johns Hopkins University, where Peirce was working as a lecturer of logic and also owned by some of his students during his time there.&lt;/p&gt;&#xA;&lt;p&gt;Since most of the original ideas of Begriffsschrift could be easily summarized and expressed in a relatively short number of sentences and thus were easily transmissible, it seems to me likely that Peirce came into contact with these ideas, which he likely reformulated into a more conventional notation before extending them in his published works on first-order logic without mentioning Frege.&#xA;The cited paper is inconclusive as to whether Peirce took anything from Frege or not.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps there is more evidence nowadays on the subject so we can give an informed estimation of the likelihood that he came into contact with Frege's ideas.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering if there is more circumstantial evidence that Peirce came into contact with these ideas before his own papers on the subject.&lt;/p&gt;&#xA;" OwnerUserId="10947" LastEditorUserId="10947" LastEditDate="2022-06-02T19:08:47.507" LastActivityDate="2022-06-02T19:08:47.507" Title="Is C.S Peirce really an independent co-discoverer of the first-order logic?" Tags="|mathematics|biographical-details|mathematical-logic|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14481" PostTypeId="1" AcceptedAnswerId="14539" CreationDate="2022-05-30T16:35:50.123" Score="4" ViewCount="171" Body="&lt;p&gt;On the Wikipedia page for &lt;a href=&quot;https://en.wikipedia.org/wiki/Ophthalmology&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ophthalmology&lt;/a&gt; they have the following diagram&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://upload.wikimedia.org/wikipedia/commons/a/a6/Cheshm_manuscript.jpg&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;with the annotation&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Anatomy of the Eye, 1200 A.D.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The diagram is recognizable as an eye. I am curious what the labels say, and what the surrounding text says. I don't know how to read the text, nor is it something I can easily copy-paste into some translator. It isn't clear where Wikipedia got the image, but the file name was &lt;code&gt;Cheshm manuscript.jpg&lt;/code&gt;. When I Google &lt;code&gt;Cheshm manuscript&lt;/code&gt; I don't get any reliable sources (e.g. tumblr pages with the image and a stock image site).&lt;/p&gt;&#xA;&lt;p&gt;What do the text and labels say around this historic diagram of an eye?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2022-06-18T22:07:54.080" Title="What does this 1200 AD eye diagram say?" Tags="|medicine|islamic-science|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14482" PostTypeId="1" CreationDate="2022-05-31T15:36:27.973" Score="3" ViewCount="126" Body="&lt;p&gt;I once read in Wikipedia that James Clerk Maxwell included the possibility that charge could be both quantified and continuous. Since the electron hadn't been discovered in 1873, does Maxwell discuss this anywhere in his 1873 book&lt;span class=&quot;math-container&quot;&gt;$^\color{red}{\star}$&lt;/span&gt; or in any other of his publications? If so, where?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{red}{\star}$&lt;/span&gt; James Clerk Maxwell, &lt;a href=&quot;https://archive.org/details/electricandmagne01maxwrich/page/n20/mode/2up?q=Amp%C3%A8re&quot; rel=&quot;nofollow noreferrer&quot;&gt;A treatise on electricity and magnetism&lt;/a&gt;, 1873.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="13040" LastEditorUserId="13040" LastEditDate="2022-06-13T12:26:02.250" LastActivityDate="2022-06-13T12:26:02.250" Title="Did Maxwell discuss charge quantization?" Tags="|physics|electromagnetism|19th-century|maxwell|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14483" PostTypeId="1" AcceptedAnswerId="14521" CreationDate="2022-05-31T21:45:32.087" Score="1" ViewCount="209" Body="&lt;p&gt;Historically (since 2500 years ago), &lt;a href=&quot;https://en.wikipedia.org/wiki/Philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;philosophy&lt;/a&gt; - &amp;quot;love of wisdom&amp;quot; in Greek - encompassed &lt;em&gt;all&lt;/em&gt; intellectual endeavors, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Natural_philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;natural philosophy&lt;/a&gt; was seen as its part. However, these days the term &lt;a href=&quot;https://en.wikipedia.org/wiki/Science&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;science&lt;/em&gt;&lt;/a&gt; has supplanted &amp;quot;&lt;em&gt;natural philosophy&lt;/em&gt;&amp;quot; and &lt;em&gt;scientists&lt;/em&gt; are not considered to be &lt;em&gt;philosophers&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I seem to recall that there was a watershed event in the second half of 19th century (circa 1870?) when a major (German?) scientist (&lt;a href=&quot;https://en.wikipedia.org/wiki/Hermann_von_Helmholtz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Helmholtz&lt;/a&gt;?) declaimed (at a major scientific convention?) something like&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Philosophers think that scientists are conceited and scientists think that philosophers are insane.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alas, I have not been able to find a specific reference.&lt;/p&gt;&#xA;&lt;h3&gt;So, who/when/where said that?&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;PS1&lt;/strong&gt;. The meaning of the phrase is that philosophers, employing their traditional scholarship, managed to learn zilch about how nature actually works, while scientists, employing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientific_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;scientific method&lt;/a&gt;, managed to learn quite a lot.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;PS2&lt;/strong&gt;. Originally &lt;a href=&quot;https://history.stackexchange.com/q/69109/1979&quot;&gt;asked&lt;/a&gt; on History.SE.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;PS3&lt;/strong&gt;. Also related:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/11085/1940&quot;&gt;When did the term &amp;#39;scientist, physicist, science, physicist&amp;#39; come in use?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/7651/1940&quot;&gt;When and why did people stopped using &amp;quot;natural philosophy&amp;quot; term and started using &amp;quot;science&amp;quot;?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1940" LastActivityDate="2022-06-13T17:00:50.523" Title="Who proclaimed separation of science from philosophy?" Tags="|terminology|reference-request|quote|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14484" PostTypeId="1" CreationDate="2022-06-02T08:37:32.690" Score="2" ViewCount="46" Body="&lt;p&gt;Throughout history, we have had many philosophers pre-Socrates to  Aristotle to Morden Era,&#xA;Did in ancient times around 1 B.C.E to 5 A.D, be it, philosophers or Astronomers when quoting famous philosophers' works, in their work for E.g: 1 A.D astronomers may quote or use Aristotle work, but write get it wrong like Aristotle said Earth is centre of cosmos were 1 A.D writer, writes that Arisitotle said that Sun is Centre of Cosmos, not Earth. Is there any example like this in the work of Ancient times?&lt;/p&gt;&#xA;" OwnerUserId="15534" LastActivityDate="2022-06-02T08:37:32.690" Title="Misunderstanding of philosopher work by other philosopher in ancient time" Tags="|philosophy-of-science|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14485" PostTypeId="1" CreationDate="2022-06-02T21:42:21.937" Score="1" ViewCount="80" Body="&lt;p&gt;I'm writing a paper on the history and the origin of Time and I'm finding it impossible to locate any meaningful material on how the concepts colloquially referred to as &amp;quot;the past&amp;quot; and &amp;quot;the future&amp;quot; originated and then were transmuted from being a category sets of events - to places that now exist elsewhere.&lt;/p&gt;&#xA;&lt;p&gt;For example, the set of all events that occurred before some moment in time were categorized as &amp;quot;the past&amp;quot;, and the set of events that will occur after some moment in time were categorized as &amp;quot;the future&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Thus the past and future are an abstract collection of events. However, how and when did that change and humanity started thinking of &amp;quot;the past&amp;quot; and &amp;quot;the future&amp;quot; as being places that we physically travel from and to at a rate of one second per second?&lt;/p&gt;&#xA;&lt;p&gt;No pun intended, but what are the histories of &amp;quot;the past&amp;quot; and &amp;quot;the future&amp;quot; as concepts and how did they evolve?&lt;/p&gt;&#xA;" OwnerUserId="16727" LastEditorUserId="16727" LastEditDate="2022-06-02T22:22:17.870" LastActivityDate="2022-06-02T22:22:17.870" Title="When did the concepts colloquially referred to as &quot;the past&quot; and &quot;the future&quot; first appear?" Tags="|time|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14486" PostTypeId="1" CreationDate="2022-06-03T05:22:46.827" Score="2" ViewCount="207" Body="&lt;p&gt;I was flicking through &lt;a href=&quot;https://users.monash.edu/%7Egfarr/research/slides/Brent-multiplication-Monash.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;these slides&lt;/a&gt; by Prof. Richard Brent, wherein we have:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Erdős (1955, in Hebrew) gave an upper bound M(n) = o(n&lt;sup&gt;2&lt;/sup&gt;) as n → ∞. After some encouragement by Linnik and Vinogradov, he proved (1960, in Russian) that ... [asymptotic formula].  (&lt;a href=&quot;https://i.ibb.co/zGVVcny/Screenshot-2022-06-03-at-13-09-48-Algorithms-for-the-Multiplication-Table-Problem-Brent-multiplicati.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;screenshot&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If I'm interpreting this correctly, Erdős has publications in both Hebrew and Russian.  I've read some of his papers in English, and I'm guessing he'd have some in Hungarian too.  Hence...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: How many languages does Paul Erdős have publications in?&lt;/p&gt;&#xA;&lt;p&gt;The references in the slides are are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;P. Erdős, Some remarks on number theory, Riveon Lematematika 9 (1955), 45–48 (Hebrew).&lt;/p&gt;&#xA;&lt;p&gt;P. Erdős, An asymptotic inequality in the theory of numbers, Vestnik Leningrad Univ. 15, 13 (1960), 41–49 (Russian). For a correction, see MR603312 (82c:10053).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10467" LastActivityDate="2022-06-03T19:49:01.353" Title="How many languages does Paul Erdős have publications in?" Tags="|mathematicians|publishing|language|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14487" PostTypeId="1" CreationDate="2022-06-03T07:33:20.417" Score="-1" ViewCount="234" Body="&lt;p&gt;The Constant of Gravitation &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; was first defined by &lt;a href=&quot;https://Boys,%20C.%20Vernon%20(1894).%20%22On%20the%20Newtonian%20constant%20of%20gravitation%22.%20Nature.%2050%20(1292):%20330%E2%80%93334.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boys&lt;/a&gt; in 1894:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Employing the C.G.S system of measurement, the Newtonian Constant is equal to the force of attraction in dynes between two balls weighing a gramme each, with their centers one centimeter apart.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] as soon as all the quantities but G in Newton’s equation&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\text{Force}=G \frac{\text{Mass} \times \text{Mass}}{\text{Distance}^2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;are known, no matter in what units the quantities are measured, G is known.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Boys says that, &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; equals the force of attraction but when I look at the equation, I see that there is a force term, the force of attraction must be given by &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; not by &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; were equal to the force of attraction between unit Masses &lt;span class=&quot;math-container&quot;&gt;$M = M = 1$&lt;/span&gt; gr, in unit distance &lt;span class=&quot;math-container&quot;&gt;$D=1$&lt;/span&gt; cm, then for &lt;span class=&quot;math-container&quot;&gt;$2D=2$&lt;/span&gt; cm, the force of attraction will be &lt;span class=&quot;math-container&quot;&gt;$2G$&lt;/span&gt; which I believe is not correct.&lt;/p&gt;&#xA;&lt;p&gt;What am I missing?&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2022-06-18T00:17:17.357" Title="Is Newton's Gravitational Constant G a force?" Tags="|newton|gravity|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14488" PostTypeId="2" ParentId="14486" CreationDate="2022-06-03T19:42:38.930" Score="5" Body="&lt;p&gt;According to the &lt;a href=&quot;https://users.renyi.hu/%7Ep_erdos/&quot; rel=&quot;noreferrer&quot;&gt;Index of P. Erdös's papers&lt;/a&gt; he has at least 2 papers in each of the following languages:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;English&lt;/li&gt;&#xA;&lt;li&gt;French&lt;/li&gt;&#xA;&lt;li&gt;German&lt;/li&gt;&#xA;&lt;li&gt;Hebrew&lt;/li&gt;&#xA;&lt;li&gt;Hungarian&lt;/li&gt;&#xA;&lt;li&gt;Russian&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;It is said that he once wanted to publish a paper &lt;a href=&quot;https://paulerdos.weebly.com/&quot; rel=&quot;noreferrer&quot;&gt;in Kurdish&lt;/a&gt; but he wasn't able to do so because he could not find a Kurdish math journal...&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-06-03T19:49:01.353" LastActivityDate="2022-06-03T19:49:01.353" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14489" PostTypeId="2" ParentId="14487" CreationDate="2022-06-04T00:07:57.903" Score="3" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; can't be a force for a number of reasons:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;It doesn't have the unit of a force.&lt;/li&gt;&#xA;&lt;li&gt;It's a plain number, while a force is a vector.&lt;/li&gt;&#xA;&lt;li&gt;It's constant, while a force can vary.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The way I understand the original text is that, in a very specific set of units, for very specific values of the parameters, the norm of the gravitational force happens to have the same numerical value as &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But since both quantities don't have the same dimension, this equality is nothing more than a coincidence, one that will disappear as soon as you switch to other units.&lt;/p&gt;&#xA;&lt;p&gt;There's one other example of this that I know of. When studying chemical equilibrium, the &lt;em&gt;activity&lt;/em&gt; of a substance is a dimensionless number. For a compound in a solution, it's defined as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a=\frac{C}{C^0}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; being the concentration of the compound and &lt;span class=&quot;math-container&quot;&gt;$C^0=1~\text{mol/L}$&lt;/span&gt; the standard concentration.&lt;/p&gt;&#xA;&lt;p&gt;As the customary unit for concentration is mol/L, the activity happens to have the same numerical value as the concentration, although their respective units differ. But you cannot say that &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; &amp;quot;is a concentration&amp;quot;, because if you use other units, like mol/m&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; or mmol/L, it stops being true.&lt;/p&gt;&#xA;" OwnerUserId="16579" LastActivityDate="2022-06-04T00:07:57.903" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="14490" PostTypeId="1" CreationDate="2022-06-04T01:39:26.703" Score="2" ViewCount="481" Body="&lt;p&gt;Who came up with the idea of using variables in mathematics to represent unknown numbers and other unknown entities? It must have been a big conceptual revolution, the idea of variables. Prior to that time, people could only think of specific numbers and equations, like for example the number 3, or the equation 2+3=5. But who was the first to come up with the idea of variables to represent unknown numbers and also make sweeping statement about all numbers? For example, the infinitely many facts that 1+0=0+1, 2+3=3+2, 4+7=7+4, etc. can be summarized by using variables to say: &amp;quot;For all x and y, x+y=y+x&amp;quot;. But someone has to have been the first to come up the idea of variables. Who was that person, and do we even have a historical record of the writings of that person?&lt;/p&gt;&#xA;" OwnerUserId="16731" LastActivityDate="2022-06-11T10:13:48.027" Title="Who came up with the idea of variables in mathematics?" Tags="|mathematics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14492" PostTypeId="1" CreationDate="2022-06-05T21:45:21.303" Score="0" ViewCount="483" Body="&lt;p&gt;I'm currently working through selected portions of Newton's Principia, but I'm already stuck in trying to understand his explanation for the first corollary (i.e., Corollary I) to the laws of motion. Here's a link for quick reference: &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Axioms,_or_Laws_of_Motion&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Axioms,_or_Laws_of_Motion&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;While I believe I understand the corollary as a whole, I'm primarily baffled by his last sentence in his explanation, namely, &amp;quot;...Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As many of you might recognize, Law I states the law of inertia: &amp;quot;Every body preserves in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.&amp;quot; In other words, objects experiencing no external forces either remain at rest or continue their motion in a straight line.&lt;/p&gt;&#xA;&lt;p&gt;With Corollary I considering motion for a body under two simultaneous forces while Law I considers motion in absence of forces, I don't see how Law I is even relevant for drawing that conclusion in that last sentence. Maybe one of you might know?&lt;/p&gt;&#xA;&lt;p&gt;On a further note, in the Scholium following the corollaries, Newton credits Galileo for the first two Laws and the first two Corollaries; however, I'm unable to find anything direct from Galileo regarding the first corollary.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-06-05T22:24:41.510" LastActivityDate="2022-06-09T14:35:53.383" Title="Newton's Corollary #1 to the Laws of Motion (Principia)" Tags="|classical-mechanics|euclidean-geometry|newton|galileo|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14494" PostTypeId="2" ParentId="14492" CreationDate="2022-06-07T05:18:33.970" Score="0" Body="&lt;p&gt;Seems silly to answer my own question, but maybe this will help someone later on.&lt;/p&gt;&#xA;&lt;p&gt;After some digging around, I realized my confusion was due to a misunderstanding of Newton's use of &amp;quot;impressed force.&amp;quot;  I was able to find commentary by James C. Maxwell in his book &amp;quot;Matter and Motion&amp;quot; in which he clarifies that &amp;quot;impressed force&amp;quot; is actually understood to be &amp;quot;impulse.&amp;quot;  With that interpretation, Corollary 1 becomes much clearer since the motion illustrated is understood to be the result of an impulse that is acted on the body at point A.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2022-06-07T05:18:33.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14496" PostTypeId="2" ParentId="14492" CreationDate="2022-06-07T17:03:42.753" Score="2" Body="&lt;p&gt;The question you raise is a good one, and it has to do with two of the laws of motion, both the first law (about inertia and the lack of need for any external influence to maintain an existing straight-line motion), and the second law (about the effect of impressed force to change motion).&lt;/p&gt;&#xA;&lt;p&gt;There has been some published controversy (and likely misunderstanding as to Newton's intent) about the second law -- and a question discussed has been whether the second law is only about force impressed in an instant, as an impulse, or whether it is also intended to cover the effects of continuous impressed forces.  I suggest that a good place from which to consider the variety of published views about this would be a 2006 paper in &lt;em&gt;Archives for the History of Exact Sciences&lt;/em&gt; (vol. 60 pp.157-207), by Bruce Pourciau (mathematical professor at Lawrence University, who has written extensively on Newton and mathematical physics), entitled &lt;em&gt;&amp;quot;Newton's Interpretation of Newton's Second Law&amp;quot;&lt;/em&gt;. Pourciau argues strongly and (as I believe) effectively, that Newton's intent was that according to the occasion or context the second law was intended to apply directly both to impulsive and to continuous forces. (A similar conclusion was reached on this particular point by Michael Nauenberg (Am. J. Phys. 80, 931-3 (2012) -- though it was in the context of disagreement with Pourciau on other points.)&lt;/p&gt;&#xA;&lt;p&gt;In the context that you raise -- i.e. the first corollary to the laws of motion -- I believe you are quite correct in identifying this as an illustration of impulsive force. For an example in support of that, the corollary begins by mentioning 'a force M impressed apart in the place A' (1729 translation). 'In the place A' is wording that clearly (for this example) localises to a point the particular force that is spoken of here. So your own consequential reasoning looks quite right, that after the localised force that is mentioned, the motion that it generated is thereafter governed by the law of inertia (and the same is additively true for the other force mentioned).&lt;/p&gt;&#xA;&lt;p&gt;So I agree with your clarification for the context in which it is made,   adding only, for the generality of cases, a suggestion to read the Pourciau paper and some of the others he cites before making any general conclusions about whether in other contexts the second law has to concern impulses only -- which for the reasons given in the cited paper would be very arguably a mistake.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-06-09T14:35:53.383" LastActivityDate="2022-06-09T14:35:53.383" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14497" PostTypeId="1" CreationDate="2022-06-08T07:44:30.177" Score="0" ViewCount="210" Body="&lt;p&gt;The medieval and non-metric Imperial data is that 7000 grains make a Pound weight avoirdupois; 5760 grains for Troy &amp;amp; Apothecaries.&#xA;Inconveniently, the only scientific data I can find of a fair age is 1824 Encyclopedia Britannica which has each ACTUAL grain being more like 65% of the STATED grain.&#xA;The same source measures Length and finds each grain close to 0.345 inch rather than the '3 to an inch' usually specified.&#xA;Has anyone got physical measurement data which eliminates this confusion?&lt;/p&gt;&#xA;" OwnerUserId="7169" LastActivityDate="2022-06-17T02:09:44.377" Title="Weights &amp; Measures - barleycorn GRAIN" Tags="|units|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14498" PostTypeId="1" CreationDate="2022-06-09T03:41:53.640" Score="2" ViewCount="210" Body="&lt;p&gt;I'm writing a young adult math story, &lt;em&gt;&lt;strong&gt;Althea and the Mystery of the Imaginary Numbers&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Her mom tells her about Scipione del Ferro, Antonio Fior, Tartaglia, Cardano, and Ferrari. And they do some math together.&lt;/p&gt;&#xA;&lt;p&gt;I hope it's ok to ask for help here. This seems like the best place to find math historians who might be willing to help me.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://drive.google.com/file/d/1fteGsyrKFmU3nBf29q8HJlC1hnfmtYQX/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here's my current draft.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Do I have the history right (pages 17 to 42)? I have done lots of research, but the sources disagree often. My original source for the story of the discovery/creation of the cubic formula is a chapter in Journey Through Genius.&lt;/p&gt;&#xA;" OwnerUserId="13188" LastEditorUserId="13188" LastEditDate="2022-06-09T15:19:31.373" LastActivityDate="2023-11-07T01:29:39.573" Title="Request: Readers to Check Accuracy of the History in my Young Adult Math Story" Tags="|mathematicians|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14499" PostTypeId="1" AcceptedAnswerId="14519" CreationDate="2022-06-09T06:50:28.510" Score="1" ViewCount="111" Body="&lt;p&gt;Empedocles state that the Eyes have fire in them surrounded by water, by this fire ray of the beam goes from eyes to object, then we see that thing, But Question arises to me, Than why not people able see in dark/night, and how they explain shadow of object and light of sun and star, Empedocles must have given any answer o this problem somewhere, but I am unable to find them? Please help me.&lt;/p&gt;&#xA;" OwnerUserId="15534" LastEditorUserId="131" LastEditDate="2022-06-09T12:37:27.530" LastActivityDate="2022-06-13T14:30:37.863" Title="Help in Understanding Emission theory of Empedocles" Tags="|physics|optics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14500" PostTypeId="1" CreationDate="2022-06-09T12:53:14.240" Score="0" ViewCount="56" Body="&lt;p&gt;I'm trying to find how Cavendish calculated the deflection of the arm due to the gravitational attractions of the weights.&lt;/p&gt;&#xA;&lt;p&gt;He gives for the deflection B=6.18 divisions. &amp;quot;Motion of the arm on moving weights from negatif to positif position = 6.18&amp;quot; (Experiment 4, page 487 in &lt;a href=&quot;https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1798.0022&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cavendish's paper&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;How did he get the number B=6.18 divisions?&lt;/p&gt;&#xA;" OwnerUserId="16575" LastEditorUserId="16575" LastEditDate="2022-06-10T07:46:02.470" LastActivityDate="2022-06-10T07:46:02.470" Title="Deflection of the pendulum arm in Cavendish experiment" Tags="|experimental-physics|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14503" PostTypeId="1" AcceptedAnswerId="14504" CreationDate="2022-06-09T18:52:44.937" Score="1" ViewCount="136" Body="&lt;p&gt;I remember reading a remark on number theory that went something like this: &amp;quot;there is plenty of material in this topic for an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; semester course without having to repeat oneself&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Have you read a sentence of this type once? If so, would you be so kind as to tell me where it is that I can find it? Quite possibly, in the version of the quote that I originally read the writer had &lt;span class=&quot;math-container&quot;&gt;$n=100$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for your help with this little question of mine.&lt;/p&gt;&#xA;&lt;p&gt;Cheerio!&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-06-09T18:58:59.340" LastActivityDate="2022-06-09T21:24:35.797" Title="Number theory: a quote" Tags="|reference-request|number-theory|quote|book-recommendation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14504" PostTypeId="2" ParentId="14503" CreationDate="2022-06-09T21:18:03.863" Score="8" Body="&lt;p&gt;&amp;quot;It has been estimated that, at the present stage of our knowledge, one could give a&#xA;200 semester course on commutative algebra and algebraic geometry without ever&#xA;repeating himself.&amp;quot; These are the opening lines of the Foreword to Ernst Kunz' book &lt;em&gt;An Introduction to Commutative Algebra and Algebraic Geometry.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="16670" LastEditorUserId="16670" LastEditDate="2022-06-09T21:24:35.797" LastActivityDate="2022-06-09T21:24:35.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14505" PostTypeId="2" ParentId="14500" CreationDate="2022-06-10T07:41:44.853" Score="1" Body="&lt;p&gt;Cavendish takes the difference of the positions of the &amp;quot;Point of rest&amp;quot; before and after he moves the weights. So when the weights are in positive position, the point of rest is at &lt;span class=&quot;math-container&quot;&gt;$24.9$&lt;/span&gt; divisions. He moves the weights to negative position and the first position of the point of rest is &lt;span class=&quot;math-container&quot;&gt;$18.72$&lt;/span&gt; divisions and &lt;span class=&quot;math-container&quot;&gt;$24.9-18.72=6.18$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2022-06-10T07:41:44.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14510" PostTypeId="1" AcceptedAnswerId="14511" CreationDate="2022-06-11T15:39:56.650" Score="1" ViewCount="74" Body="&lt;p&gt;The well celebrated field of numerical linear algebra is heavily based on Krylov subspace methods. A quick google search on Krylov himself returned the following results:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nikolay Krylov (1941-) [Russian mathematician]&lt;/li&gt;&#xA;&lt;li&gt;Aleksey Krylov (1863-1945) [Russian engineer and mathematician]&lt;/li&gt;&#xA;&lt;li&gt;Nikolay Krylov (1879-1955) [Russian mathematician]&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But it seems that none of them have personally been a part of the creation or development of &amp;quot;Krylov Subspace Method&amp;quot; and most definitely not in numerical linear algebra.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Question: If it is not some &amp;quot;Krylov&amp;quot; who led foundation of the &amp;quot;Krylov subspace methods&amp;quot; then who was/were the mathematician(s) that were part of its invention.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14690" LastActivityDate="2022-06-11T17:38:23.273" Title="Who came up with the theory of Krylov Subspaces?" Tags="|mathematicians|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14511" PostTypeId="2" ParentId="14510" CreationDate="2022-06-11T17:38:23.273" Score="3" Body="&lt;p&gt;It was &lt;a href=&quot;https://en.wikipedia.org/wiki/Aleksey_Krylov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aleksey Krylov&lt;/a&gt;, see (in Russian):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;amp;jrnid=im&amp;amp;paperid=5215&amp;amp;option_lang=eng&quot; rel=&quot;nofollow noreferrer&quot;&gt;De la résolution numérique de l'équation servant à déterminer dans des questions de mécanique appliquée les fréquences de petites oscillations des systèmes matériels&lt;/a&gt;. Bulletin de l'Académie des Sciences de l'Union des Républiques Soviétiques Socialistes. VII série. Classe des sciences mathématiques et naturelles. 1931, Issue 4, Pages 491–539.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2022-06-11T17:38:23.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14513" PostTypeId="2" ParentId="14470" CreationDate="2022-06-11T21:58:03.860" Score="2" Body="&lt;p&gt;An example of a near-simultaneous discovery which I came across recently involves &lt;a href=&quot;https://en.wikipedia.org/wiki/Liouville_number#Liouville_numbers_and_transcendence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liouville's approximation theorem&lt;/a&gt; - that for an irrational, algebraic real number &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, there are only finitely many rational numbers &lt;span class=&quot;math-container&quot;&gt;$p/q$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$| \alpha - p/q | \lt 1/q^{n+\epsilon}$&lt;/span&gt; - used by Liouville to generate transcendental numbers. Liouville suspected that the result was not optimal and a number of improvements followed (Thue, in 1909, reduced &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}n+1$&lt;/span&gt; and Siegel, in 1921 reduced &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}n+1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{n}$&lt;/span&gt;).  Then, in 1947, Freeman Dyson and Aleksandr Gelfond independently reduced &lt;span class=&quot;math-container&quot;&gt;$2\sqrt{n}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2n}$&lt;/span&gt;. (It was only in 1955 that Roth published the optimal result by reducing &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2n}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-06-11T21:58:03.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14514" PostTypeId="2" ParentId="7845" CreationDate="2022-06-11T23:42:23.997" Score="2" Body="&lt;p&gt;ONE: As others have stated, real numbers were questionable at the time, so to overcome this deficency, numbers were constructed based on magnitudes.&lt;/p&gt;&#xA;&lt;p&gt;In Euclidean geometry, magnitudes can only be compared (i.e., =,&amp;gt;,&amp;lt;) to like magnitudes, and from this, additive and subtractive operations are permitted, which in turn permits us to multiply and divide magnitudes by quantities of these magnitudes, i.e &lt;em&gt;numbers&lt;/em&gt;. As as such, these operations increase/decrease the magnitude in kind by quantities of that &lt;em&gt;number&lt;/em&gt;; for example, a length is n-&lt;em&gt;multiples&lt;/em&gt; of a another length, or a length is m-&lt;em&gt;parts&lt;/em&gt; of another length.&lt;/p&gt;&#xA;&lt;p&gt;We can't say the same about any other operations that result in another kind of magnitude since they don't have a result that is comparable (=,&amp;gt;,&amp;lt;) to the original magnitudes, thus making it absurd.  Thus, it is forbidden to mix (i.e., add/subtract/multiply/divide) magnitudes with other magnitudes since dividing angles by areas is just as absurd as adding lines to areas, and multiplying areas by areas.  Since magnitudes cannot be mixed, Euclid would object to the meaning of &amp;quot;v = x/t&amp;quot;.  It's easy to recognize that &amp;quot;x&amp;quot; is a magnitude of spatial length, but what does that make of &amp;quot;v&amp;quot; and &amp;quot;t&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;To further this point, Euclid doesn't say that an area of a rectangle &lt;em&gt;equates&lt;/em&gt; to length x width, rather, he states ratios of the areas of two rectangles are the same as the product of the ratio of lengths and the ratio of widths.  Nor does Euclid say that the area of a square is the multiplication of side by side (i.e, s^2), but rather that the ratio of the areas between two squares is the same as the product of the ratios of the sides.  Thus, &amp;quot;the area from squaring the sides&amp;quot; is to be meant as &amp;quot;the area resulting from a square constructed with equal lengths&amp;quot;.  This also means that Pythagoras's theorem is equating areas using squares that are constructed using the sides of the triangle and &lt;em&gt;not&lt;/em&gt; directly a multiplication of a number assigned to the sides.  (See Euclid Elements Book I-Prop 47 and Book VI-Prop 23)&lt;/p&gt;&#xA;&lt;p&gt;However, suppose we're able to correlate a comparison of a set of magnitudes to another comparison of a set of magnitudes. In that case, we can define &amp;quot;ratios&amp;quot;, which further permit us to relate magnitudes to other magnitudes (See Euclid Book 5-Def 5).&lt;/p&gt;&#xA;&lt;p&gt;Using this rigorous type of math, Galileo is showing us how these &amp;quot;exotic&amp;quot; magnitudes of speeds, transit-distances, and transit-times are relatable through ratios.  He starts by defining what uniform speed is, then provides us the underlying axioms that would support this definition.  Galileo recognizes that these axioms provide correlations between comparisons, which sets him up to apply the theory of ratios.  Notice how Theorem I is a near direct regurgitation of Euclid Book 5 Definition 5?&lt;/p&gt;&#xA;&lt;p&gt;Through Theorems 1-6, he shows every possible way to combine ratios with speed, space, and time - all without using any algebraic algorithms (e.g., cross multiplying or any other equation balancing).  Notice that, on Theorem 6, if we assign the first motion to be all &amp;quot;1's&amp;quot;, (e.g., v1=1, x1=1, t1=1), then the ratio collapses to our velocity equation &amp;quot;v2=x2/t2&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;TWO: Good question, that's because Euclid's theory of ratios is powerful in that it allows us to relate magnitudes to any other kinds of magnitudes, so long as the ratios of their own kind remain the same.  Therefore, we're permitted to represent a ratio of times or speeds as a ratio of lines so long as they both adhere to the definition of a ratio (i.e., Euclid Book V-Def 5) - we can see this being done with areas and lines in Euclid Book VI-Prop 1, which shows that the ratio of areas are the same as the ratio of lines of triangles/parallelograms.&lt;/p&gt;&#xA;&lt;p&gt;However, it should be noted that, though we can represent distance, speed, and time each as lines, it remains questionable whether we can add/subtract these lines from each other (e.g., nothing justifies subtracting a distance line from a speed line) or even construct upon them to build shapes (e.g., parallelograms and triangles), or rather, at least not yet.  Galileo is acutely aware of this, which is why for Theorems 1-6 for uniform motion he is very careful to keep these sets of lines for speeds, time, and distance separate without physical construction upon each other.  His primary concern is to show that ratios between these lines exist in every possible configuration without constructing upon them.&lt;/p&gt;&#xA;&lt;p&gt;What's curious is what happens in Theorems 4-6: Galileo shows how ratios of one speed, time, and distances are also the &amp;quot;compound/composite/product&amp;quot; of the others (e.g., the ratio of distances is the same as the &amp;quot;compound&amp;quot; of the ratios of speed and time).  Compounding ratios is roughly the same as &amp;quot;multiplying ratios to each other,&amp;quot; but keep in mind multiplying only applies to numbers, not magnitudes.  Compounding is a strange beast in that it isn't formally defined by Euclid but rather inferred by Book VI-Prop 23 and the definitions of duplicate/triplicate ratios, which are viewed as specific cases of the former.  A major takeaway from Book VI-Prop 23 is that it says that two parallelograms are in ratio as the compound ratios of their sides, which will be important later for Galileo.&lt;/p&gt;&#xA;&lt;p&gt;Since Galileo is able to show the compounding relationship between speed, time, and distance, and since parallelograms and their sides produce the same compound ratios (i.e., VI.23), then this allows Galileo to construct parallelograms (and triangles, since they are half a parallelogram) with speed, time, and distance.  In other words, Galileo uses theorems 1-6 in conjunction with VI.23 in order to justify the triangle and parallelogram constructions of Theorem 1 of uniform acceleration!&lt;/p&gt;&#xA;&lt;p&gt;As an interesting side note, while Galileo does draw a figure using a right triangle and a rectangle for Theorem 1 of uniform acceleration, and the most common translation (Alfonso de Salvio and Henry Crew) says they are drawn at right angles, Galileo in the original Latin/Italian text does &lt;em&gt;not&lt;/em&gt; explicitly require this!  The angle between speed and time can be any angle (but not in line with each other since this could imply they are summable) and the parallelogram can still be drawn since the compounding relationship still holds true according to VI.23.  We can't do that with our modern version x=v*t since areas in algebra are defined as square-like areas!&lt;/p&gt;&#xA;&lt;p&gt;RESPONSE TO NOTE: The 3rd day can be read independently from Days 1 and 2.  The latter generally discusses statics (i.e., the perception of weight, the balances of weights, and structure), which is hardly needed for the 3rd day.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, in response to:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;U said '' to read and appreciate the old authors like Galileo, one has&#xA;to study some of the previous development to have an idea of the&#xA;background these people had.'' Could u please suggest the&#xA;pre-requisites in order to easily comprehend the works of the great&#xA;authors like Galileo, Newton etc. The reason why I want to read such&#xA;texts is simply because(as u have also mentioned) these days texts are&#xA;written is such a fashion that one ends up indoctrinating the concepts&#xA;in their minds without knowing the hows and whys of the concept and&#xA;they end up doing no science virtually.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To understand Galileo and Newton, you need Euclid's Elements Books 1-7ish, and primarily just books 1 (defines magnitudes and basic operations), 5 (theory of ratios), and 6 (application of the theory).  After that, Conics of Appolonius, but you'd be better off with less antiquated versions of it through T.L. Heath or W.H. Besant who both reorganize it to logically flow clearer.  Personally, I like Besant since it picks up straight from Euclid's  theory of proportions instead of beginning with intersections of a cone.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-06-24T21:12:41.250" LastActivityDate="2022-06-24T21:12:41.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14515" PostTypeId="1" AcceptedAnswerId="14520" CreationDate="2022-06-12T11:58:09.290" Score="1" ViewCount="245" Body="&lt;p&gt;The Wikipedia article on squaring the circle has a relatively good history of their efforts on this &lt;em&gt;particular&lt;/em&gt; problem, but it fails to mention why the Greeks were interested in this methodology on a wider basis. An editor added a bit of useful text about this, but I believe it needs more.&lt;/p&gt;&#xA;&lt;p&gt;As I recall (dim memory here), mathematicians of the era were highly suspect of numerics, considering arithmetic to be something closer to bookkeeping than the purity of geometry which existed at another level of reality. As such, purely geometric solutions to problems were considered important. Is this remotely correct?&lt;/p&gt;&#xA;&lt;p&gt;Can anyone offer a source that covers their thinking in this regard that might be a useful reference?&lt;/p&gt;&#xA;" OwnerUserId="16767" LastEditorUserId="10106" LastEditDate="2022-06-13T03:02:17.060" LastActivityDate="2022-06-15T05:35:24.947" Title="Discussions of why the Greeks &quot;squared the...&quot; geometrically" Tags="|mathematics|geometry|ancient-greece|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14516" PostTypeId="1" CreationDate="2022-06-12T20:47:44.737" Score="1" ViewCount="183" Body="&lt;p&gt;A famous mathematician known for the algorithm he developed in 1950s named after him (Arnoldi Iteration Algorithm).&lt;/p&gt;&#xA;&lt;p&gt;For my research presentation I am including photographs of pioneers of numerical linear algebra. However, there appears to be none for Arnoldi.&lt;/p&gt;&#xA;" OwnerUserId="14690" LastActivityDate="2023-06-06T13:48:18.173" Title="Is there a known photograph of Walter Arnoldi?" Tags="|mathematicians|20th-century|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14519" PostTypeId="2" ParentId="14499" CreationDate="2022-06-13T14:30:37.863" Score="1" Body="&lt;p&gt;Much of Empedocles exists in fragments of what he originally wrote, so finding his literal writing on the subject may be difficult.  However, we can infer what he said based on the commentary provided by other writers.&lt;/p&gt;&#xA;&lt;p&gt;Based on snippets in the below link (keyword search &amp;quot;vision&amp;quot; and &amp;quot;night&amp;quot;), it seems that, while our vision reaches from our eyes and reflects back to them for perception, Empedocles also realized that light of the sun was was needed in order for vision to reflect back to our eyes, otherwise it just passes through the object.&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://history.hanover.edu/texts/presoc/emp.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://history.hanover.edu/texts/presoc/emp.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;PASSAGES FROM ARISTOTLE RELATING TO EMPEDOKLES:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Empedokles says that the light from the sun first enters the&#xA;intermediate space before it comes to vision or to the earth.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;PASSAGES IN DIELS' 'DOXOGRAPHIC GRAECI' RELATING TO EMPEDOKLES:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And he attempts to describe what vision is; and he says that what is&#xA;in the eye is fire and water, and what surrounds it is earth and air,&#xA;through which light being fine enters, as the light in lanterns. Pores&#xA;of fire and water are set alternately, and the fire-pores recognise&#xA;white objects, the water-pores black objects; for the colours&#xA;harmonise with the pores. And the colours move into vision by means of&#xA;effluences. And they are not composed alike . . . and some of opposite&#xA;elements; for some the fire is within and for others it is on the&#xA;outside, &lt;strong&gt;so some animals see better in the daytime and others at&#xA;night; those that have less fire see better by day, for the light&#xA;inside them is balanced by the light outside them; and those that have&#xA;less water see better at night, for what is lacking is made up for&#xA;them.&lt;/strong&gt; And in the opposite case the contrary is true; for those that&#xA;have the more fire are dim-sighted, since the fire increasing plasters&#xA;up and covers the pores of water in the daytime; and for those that&#xA;have water in excess, &lt;strong&gt;the same thing happens at night; for the fire is&#xA;covered up by the water. . . . Until in the case of some the water is&#xA;separated by the outside light, and in the case of others the fire by&#xA;the air; for the cure of each is its opposite.&lt;/strong&gt; That which is composed&#xA;of both in equal parts is the best tempered and most excellent vision.&#xA;This, approximately, is what he says concerning vision.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2022-06-13T14:30:37.863" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14520" PostTypeId="2" ParentId="14515" CreationDate="2022-06-13T16:29:13.593" Score="4" Body="&lt;p&gt;Basically, Euclidean geometry was considered the purest math since it carried fewer flaws than the proposed number theory at the time since numbers used &amp;quot;natural numbers&amp;quot; (i.e., integers), which produced glaring flaws.  There was no such thing as decimal numbers, but to produce a decimal number, it would just be recognized that quantities could be described as an integer number of parts of another.  For example, a &amp;quot;third&amp;quot; of an apple is one-part-of-three of a whole apple, and by multiplying this quantity by three, returns the whole apple.&lt;/p&gt;&#xA;&lt;p&gt;However, to the horror of early mathematicians, integer numbers as a basis for math produced contradictions.  The story of Hippasus comes to mind in which he proved that there are some numbers that can't be expressed as a ratio of two integers, hence the number was &amp;quot;ir&lt;strong&gt;ratio&lt;/strong&gt;nal.  Legend has it that the Pythagoreans drowned him for exposing this flaw.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hippasus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hippasus&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So how do we fix this? We revert to something more fundamental than numbers: magnitudes (e.g., lines, areas, angles, apples, oranges, etc.), whose size is regardless of quantity.  What I mean by this is that a magnitude can exist alone and still have size; and when alone, it's absurd to assign a value to it other than that the magnitude exists, and likewise, it's absurd to assign a quantity of it.  In contrast, if we were to claim that the magnitude has size &amp;quot;five,&amp;quot; then we're admitting that there is five of some other magnitude that defines the first magnitude.  Furthermore, for a standalone magnitude, we can't say there are &amp;quot;one&amp;quot; quantity of magnitudes, but rather we only claim that it exists.  For this reason, Greek math is said to start counting starting with &amp;quot;two&amp;quot; instead of &amp;quot;one,&amp;quot; which is strange to us modern counters, but after all, what's the point of counting if there's only a single magnitude?  In other words, counting and numbers only make sense once there's a plurality of magnitudes.&lt;/p&gt;&#xA;&lt;p&gt;Once we add a second magnitude to the mix, then we can compare (i.e., =, &amp;lt;, &amp;gt;) them to each other, which then provides a structure that allows addition, subtraction, multiplication, and division.  Euclid's Elements Book 1 is dedicated to establishing this foundation.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, ratios of these magnitudes may be taken, which are valid regardless of number, but a ratio is only understood once we introduce a third magnitude.  For example, the ratio A:B can exist, but it's useless by itself; however, introduce the ratio A:C and let A:B = A:C, then we can infer that B = C.  Again, all this is done without number!  The theory of ratio and proportion is all laid out in Elements Book 5.&lt;/p&gt;&#xA;&lt;p&gt;It's not until Elements Book 7 that number theory is developed and is not investigated with ratios until Book 10.  In the latter, concepts of &amp;quot;commensurable&amp;quot; and &amp;quot;incommensurable&amp;quot; are introduced, which are loosely interpreted as rational and irrational quantities.  The idea is that, if we take two arbitrary magnitudes, the question is whether we can multiply each of them respectively in a way that they can create two new equal in size magnitudes.  If the equality can be made with integer numbers of each, then the magnitudes are commensurable.  If not, then they are incommensurable.  For example, for magnitudes A and B, they are &amp;quot;commensurable&amp;quot; if 3A = 5B, which we can see that a ratio of two numbers can be produced: A:B = 5/3 (hence 5/3 is a rational number).  However, for magnitudes B and C, it's possible that any number multiplication of each will fail to produce equality.  In other words, no value for x or y integers satisfies the equality xB = yC, thus B and C are incommensurable and cannot be arranged to produce a ratio of numbers (i.e., the ratio of A and B is irrational).  However, the ratio of B:C still exists and remains useful so long as we avoid equating them through any integer numbers.  Therefore, it is much more logically sound to demonstrate relationships through geometry where irrational numbers can be avoided and ratios are preserved.&lt;/p&gt;&#xA;&lt;p&gt;For suggested reading, I recommend portions of Euclid, particularly books 1, 5, and 6.  I also found this article by math historian Grattan-Guinness to be helpful in providing historical context, where he discusses how our modern understanding of geometry was reframed through &amp;quot;algebraic-geometry&amp;quot; which resulted in subtle losses of information.  For example, algebraic-geometry will say area is length times width, but euclidean geometry never defines area.  Instead, the ratio of the areas of two rectangles are the same as the complex ratio of their lengths and widths, which takes on a broader meaning without the use of numbers.&#xA;&lt;a href=&quot;https://www.tau.ac.il/%7Ecorry/teaching/toldot/download/IGG.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tau.ac.il/~corry/teaching/toldot/download/IGG.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For these reasons above, Euclidean geometry was heavily relied upon up until algebra and the continuous number line was polished and widely accepted (it became somewhat solidified with Descartes).  Since the magnitudes and their ratios were so powerful over numbers, it comes as no surprise that squaring the circle was a problem attempted to be solved by many; however, it wasn't until 1882 that Ferdinand von Lindemann proved that squaring the circle is impossible with Euclidean methods, primarily because, not only is pi irrational, it's also a transcendental number!&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-06-15T05:35:24.947" LastActivityDate="2022-06-15T05:35:24.947" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14521" PostTypeId="2" ParentId="14483" CreationDate="2022-06-13T17:00:50.523" Score="2" Body="&lt;p&gt;This paper might be useful, though I haven't read it:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The Origin of the Separation Between Science and Philosophy&amp;quot;&#xA;&lt;a href=&quot;https://www.jstor.org/stable/20023644&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/20023644&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2022-06-13T17:00:50.523" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14523" PostTypeId="1" AcceptedAnswerId="14526" CreationDate="2022-06-13T19:28:04.163" Score="1" ViewCount="292" Body="&lt;p&gt;Thermodynamics is derived from two Greek words&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Therme&lt;/em&gt;, which means heat&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Dynamis&lt;/em&gt;, which means power&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;We know that 'thermodynamics' encapsulates many concepts like energy, temperature, entropy, exergy, heat, work, etc., then why give heat  (&lt;em&gt;therme&lt;/em&gt;) and Power (&lt;em&gt;dynamis&lt;/em&gt;) such a special place?&lt;/p&gt;&#xA;" OwnerUserId="16326" LastEditorUserId="10106" LastEditDate="2022-06-14T12:50:35.293" LastActivityDate="2022-06-14T12:50:35.293" Title="Why is thermodynamics called thermodynamics?" Tags="|physics|terminology|thermodynamics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14525" PostTypeId="2" ParentId="14515" CreationDate="2022-06-14T12:27:39.603" Score="3" Body="&lt;p&gt;My article &amp;quot;Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry&amp;quot; &lt;a href=&quot;https://link.springer.com/article/10.1007/s10699-021-09791-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/article/10.1007/s10699-021-09791-4&lt;/a&gt; discusses possible motivations for the Greek preoccupation with constructions that is evident in Euclid and in the tradition of the classical problems of circle quadrature, angle trisection, and cube duplication. I argue that this stemmed from constructivist foundational concerns. Introducing objects into mathematical discourse in a non-constructive fashion runs the risk of introducing inconsistencies and hidden assumptions. For example, &amp;quot;let ABC be a triangle with three equal sides&amp;quot; is fine but &amp;quot;let ABC be a triangle with three right angles&amp;quot; is not. This shows that one cannot merely introduce objects by &amp;quot;letting&amp;quot; it be a figure with a certain property. A way to safeguard that one it only talking about non-contradictory objects (and hence to safeguard the consistency of mathematics) is to only admit objects that are constructed by rigorous and verifiable construction principles. Hence, in particular, one cannot merely &amp;quot;let&amp;quot; a square be equal in are to a circle; rather one must construct such a square.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2022-06-14T12:27:39.603" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14526" PostTypeId="2" ParentId="14523" CreationDate="2022-06-14T12:39:57.920" Score="6" Body="&lt;p&gt;Thermodynamics is indeed derived from the Greek words &lt;em&gt;Therme&lt;/em&gt; (heat) and &lt;em&gt;Dynamis&lt;/em&gt; (power). However, &lt;em&gt;Dynamis&lt;/em&gt; is not the same as the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Power_(physics)&quot; rel=&quot;noreferrer&quot;&gt;Physics definition of Power&lt;/a&gt; but is synonymous with &amp;quot;might&amp;quot; or &amp;quot;strength&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The term &amp;quot;Thermodynamics&amp;quot; originates from the concept of &amp;quot;&lt;strong&gt;motive power of heat&lt;/strong&gt;&amp;quot;, which first appeared in 1849 in William Thomson's (Lord Kelvin) paper &amp;quot;&lt;a href=&quot;https://scholar.google.com/scholar_lookup?journal=Trans.%20Edinb.%20R.%20Soc.&amp;amp;title=An%20Account%20of%20Carnot%E2%80%99s%20Theory%20of%20the%20Motive%20Power%20of%20Heat,%20with%20numerical%20results%20deduced%20from%20Regnault%E2%80%99s%20Experiments%20on%20Steam&amp;amp;author=W.%20Thomson&amp;amp;volume=16&amp;amp;publication_year=1849&amp;amp;pages=541&amp;amp;doi=10.1017/S0080456800022481&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;An Account of Carnot's Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault's Experiments on Steam.&lt;/a&gt;&amp;quot; The term is claimed to have been coined by James Joule in 1858 in reference to Thomson's paper.&lt;/p&gt;&#xA;&lt;p&gt;Thus, Thermodynamics (or &amp;quot;&lt;em&gt;thermo-dynamics&lt;/em&gt;&amp;quot; as written earlier) was associated with heat's motive power, that is, the &amp;quot;might&amp;quot; of heat to do work or cause motion.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2022-06-14T12:39:57.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14527" PostTypeId="2" ParentId="14470" CreationDate="2022-06-14T17:43:46.590" Score="1" Body="&lt;p&gt;A recent example is Conway's &lt;a href=&quot;https://en.wikipedia.org/wiki/Angel_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Angel problem&lt;/a&gt; that was first published in 1982, and four(!) independent solutions was found in 2006 almost simultaneously.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="6788" LastEditDate="2022-06-14T21:12:59.010" LastActivityDate="2022-06-14T21:12:59.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14528" PostTypeId="2" ParentId="14498" CreationDate="2022-06-14T23:58:01.907" Score="5" Body="&lt;p&gt;You wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Rafael Bombelli ... did show that you could simplify the mess we have&#xA;up there on the whiteboard, and get x = 4.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is the central point of many popular stories about cubic equations. But this is just not true. If Cardano formula have square roots of negative numbers, you can't just simplify it  to get rid of them and calculate the roots. So, if your goal is to actually calculate the roots of this particular cubic, then Cardano's formula is pretty useless. Sorry. Luca Pacioli was right after all - there is no general formula for cubic equations that uses radicals, at least in the sense known to Luca.&lt;/p&gt;&#xA;&lt;p&gt;Here is an idea for your story to illustrate this point (sorry for taking liberty of using your character):&lt;/p&gt;&#xA;&lt;p&gt;Althea accidentally travels back in time. She meets Ptolemy who is very excited to hear that cubic equations could be solved. Ptolemy is trying to compose trig table and wants to calculate &lt;span class=&quot;math-container&quot;&gt;$\cos 20°$&lt;/span&gt;. He knows &lt;span class=&quot;math-container&quot;&gt;$\cos 60°$&lt;/span&gt; and knows the formula for &lt;span class=&quot;math-container&quot;&gt;$\cos(3x)$&lt;/span&gt;. So he knows that &lt;span class=&quot;math-container&quot;&gt;$\cos 20°$&lt;/span&gt; is the solution of the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$4x^3 - 3x = \frac 1 2.$$&lt;/span&gt;&#xA;But how to solve it?&#xA;Althea is very happy that she can help. She writes down the solution:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = \frac 1 2 \left (&#xA;\sqrt[3]{\frac 1 2 + \frac {\sqrt {-3}} {2} } +&#xA;\sqrt[3]{\frac 1 2 - \frac {\sqrt {-3}} {2} } &#xA;\right)&#xA;$$&lt;/span&gt;&#xA;Unfortunately Ptolemy is not used to our algebraic notations.&lt;br /&gt;&#xA;It took Althea a week to explain him what all symbols in the formula means, and finally Ptolemy starts to understand. But then Ptolemy noticed &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-3}$&lt;/span&gt;. And asks what is it? Althea have to explain complex numbers. (But first of course she needs to explain negative numbers). It took a month, but Ptolemy is very patient, because he is eager to learn the science of the future to complete his trig table and to use it for his astronomy calculations. Finally he become fluent with complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;But how do we calculate a cubic root of &lt;span class=&quot;math-container&quot;&gt;$ \frac 1 2 + \frac {\sqrt {3}} {2} i$&lt;/span&gt;?&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;We almost there! To calculate a cubic root of a complex number, you need to take a cubic root of its length and divide its angle by 3. You are in luck: the number &lt;span class=&quot;math-container&quot;&gt;$ \frac 1 2 + \frac {\sqrt {3}} {2} i $&lt;/span&gt; has a very nice length: it is just 1. Cubic root of 1 is 1. And is has a very nice angle: it's just 60°. So the cubic root is just a number with the length 1 and the angle 20°. It's just &lt;span class=&quot;math-container&quot;&gt;$\cos 20^\circ +i \sin 20^\circ$&lt;/span&gt;. Is it nice?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;At this point Ptolemy understands that the future science is just a fraud and he just wasted his valuable time.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="6788" LastEditDate="2022-06-15T01:09:03.470" LastActivityDate="2022-06-15T01:09:03.470" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14529" PostTypeId="1" CreationDate="2022-06-15T04:23:27.250" Score="2" ViewCount="383" Body="&lt;p&gt;The theory of general relativity was formulated around 1915 which could be summed up in words of John Wheeler, &amp;quot;Space-time tells matter how to move; matter tells space-time how to curve&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;At that time the unanimous consensus was arguably that the universe was static which essentially meant that there was no relative motion between different objects in the sky. For example, according to static universe model, the distance between Milky Way Galaxy and Andromeda Galaxy does not change. This led Einstein to use cosmological constant so that the matter does not collapse or contracts onto itself as a result of gravitational attraction.&lt;/p&gt;&#xA;&lt;p&gt;I don't think that Einstein was thinking that the collapse or contraction of matter would also result into the contraction of space onto itself as well. Yes, if the matter had collapsed onto itself, it would've curved the space around it. (I'm avoiding the term black holes on purpose since I'm looking at it in historical time frame.) In other words, I don't think it was really known at that time that the space could expand/grow or shrink.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1912, Vesto Slipher discovered that light from remote galaxies was&#xA;redshifted, which was later interpreted as galaxies receding from the&#xA;Earth. In 1922, Alexander Friedmann used Einstein field equations to&#xA;provide theoretical evidence that the universe is expanding.&lt;/p&gt;&#xA;&lt;p&gt;Swedish astronomer Knut Lundmark was the first person to find&#xA;observational evidence for expansion in 1924....&lt;/p&gt;&#xA;&lt;p&gt;In 1927, Georges Lemaître independently reached a similar conclusion&#xA;to Friedmann on a theoretical basis, and also presented observational&#xA;evidence for a linear relationship between distance to galaxies and&#xA;their recessional velocity. Edwin Hubble observationally confirmed&#xA;Lundmark's and Lemaître's findings in 1929. Assuming the cosmological&#xA;principle, these findings would imply that all galaxies are moving&#xA;away from each other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Expansion_of_the_universe#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Expansion_of_the_universe#History&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think that Vesto Slipher could have concluded that it was the relative motion of the galaxies which was producing redshift. For example, Galaxy A is moving away from Milky Way Galaxy at 500 km/s and Galaxy B moving away at 800 km/s. My point is that concluding that the space itself is growing between different objects, in my humble opinion, was quite a radical idea in its own right. It could also have begged another equally valid question that how and why the space was growing.&lt;/p&gt;&#xA;&lt;p&gt;Also, though the Wikipedia article says that Friedmann used the relativity's equations to provide the evidence that the universe was expanding. In my opinion, Friedmann could have equally chosen to prove that the universe was contracting unless he was motivated by experimental evidence, such as the observation of Slipher, that the expansion was more probably than contraction. The question is that when it was being said that the universe was expanding, what that term 'expansion' was referring to? Was it referring to relative motions of the objects moving away from each other, or was it referring to the expansion of space itself? If it was the space itself, where did they get the notion that the space could grow on its own? I don't think the theory of general relativity said anything about the expansion of space. Could you please help me with it?&lt;/p&gt;&#xA;&lt;p&gt;I know sometimes people tend to retrospectively read more into a theory more than it was really modelled for. For example, saying that Einstein or the theory of general relativity predicted black holes and gravitation waves is not really correct. Yes, the predictions/hints/pointers were really embedded there or hidden in the framework of the theory but Einstein didn't really predict it or hinted at it. When Einstein formulated the theory, his concern wasn't really to solve or predict the blackholes or gravitational waves. I'd say it's true of any theory. I'm saying this in the hope that I'm not told that the theory predicted the expansion of space even when it initially didn't.&lt;/p&gt;&#xA;&lt;p&gt;Also, if you check the link below, De Sitter's model of expanding? universe was something quite weird, if I may say so. IMHO, I don't think if calling it expanding universe model is the right thing to do. De Sitter wasn't really talking about the expansion of space, his model was along different lines of thought.&lt;/p&gt;&#xA;&lt;p&gt;More information could be found here: &lt;a href=&quot;https://history.aip.org/exhibits/cosmology/ideas/expanding.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://history.aip.org/exhibits/cosmology/ideas/expanding.htm&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thanks for the help and time!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;This edit has been made after the comment of @Conifold.&lt;/p&gt;&#xA;&lt;p&gt;I think that Einstein's original equations of the theory of relativity predicted contracting or expanding universe. I'd say that the mathematics predicted either contracting or expanding universe. I'd say that this mathematical prediction of contraction or expansion was a result of (pseudo-)Riemannian geometry used to formulate the theory of general relativity as mentioned by @Conifold.&lt;/p&gt;&#xA;&lt;p&gt;This is how I see the mathematical formulation of general relativity. The formulation of the general relativity assumed the material objects embedded into immaterial space-time. Think of space-time as a grid. The immaterial space-time can contract and expand independent of the material objects embedded in it. The material objects could be attracted to each as a result of gravity independent of the space-time, (or, repel as a result of charge).&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, this is an important point. Einstein was one of the great physicists of 20th century who really valued physical and intuitive interpretation of the theories. I personally admire him for this. That's one of the reasons Einstein agreed to Friedmann's solution but considered that it didn't have physical validity. Likewise, later Einstein considered Lemaitre's solution mathematically good but not valid in terms of physics. That's also the reasons Einstein had doubts about blackholes and gravitational waves. That's also one of the reasons Einstein had problems with the interpretation of quantum mechanics. In other words, Einstein clearly drew a line between the mathematics of theories and their 'physical' interpretations.&lt;/p&gt;&#xA;&lt;p&gt;I'd agree that the formulation of general relativity in terms of (pseudo-)Riemannian geometry allows for expanding or contracting universe.&lt;/p&gt;&#xA;&lt;p&gt;I think that my original question could be phrased better as follows.&lt;/p&gt;&#xA;&lt;p&gt;Einstein was concerned about all the matter collapsing onto itself as a result of gravitational attraction since gravity doesn't go to zero. He introduced the cosmological constant to fix the problem though he didn't really like doing it. Einstein chose such a value for cosmological constant that it fixed the contracting universe problem for him but if he had used other value (for example negative value instead of positive ) it would result into expansion. Could you please point me to any source which shows that Einstein really believed that the space itself would also contract or collapse along with the matter as a result of gravitational attraction? If Einstein did believe in the contraction of space, wasn't he bothered by it and thought about its cause? Was the contraction and expansion simply the assumed properties of space-time 'grid' by Einstein?&lt;/p&gt;&#xA;&lt;p&gt;Also, in my opinion, Einstein gave importance to the idea of contracting universe not simply because the mathematics allowed it, he also saw 'physical' cause for this contraction which was gravity. He didn't see any physical cause for the expansion therefore I think he didn't give it any consideration though the mathematics allowed the expansion.&lt;/p&gt;&#xA;&lt;p&gt;Later around 1930 Einstein agreed to the expanding universe model but was he comfortable with the expansion of space itself as he was mostly concerned, as I stated above, about the physical reality? Again, was the contraction and expansion simply the assumed properties of space-time 'grid' by Einstein?&lt;/p&gt;&#xA;&lt;p&gt;Just a side note. What Friedmann did was more of a mathematical exercise which turned out to make sense later. Friedmann presented both contracting and expanding models.&lt;/p&gt;&#xA;&lt;p&gt;References used for the Edit section:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1:&lt;/strong&gt; A century ago, Alexander Friedmann envisioned the universe’s expansion by Tom Siegfried, &lt;a href=&quot;https://www.sciencenews.org/article/alexander-friedmann-universes-expansion-1922-cosmology&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.sciencenews.org/article/alexander-friedmann-universes-expansion-1922-cosmology&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2:&lt;/strong&gt; Einstein’s conversion from his static to an expanding universe by Harry Nussbaumer, &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1311/1311.2763.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/ftp/arxiv/papers/1311/1311.2763.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;3:&lt;/strong&gt; How Friedmann Shod Einstein by Vladimir Soloviev, &lt;a href=&quot;https://arxiv.org/pdf/2204.10650.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/pdf/2204.10650.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="4251" LastEditDate="2022-07-21T11:58:08.883" LastActivityDate="2023-04-17T13:06:20.813" Title="How did they know that the space could expand?" Tags="|relativity-theory|einstein|cosmology|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14530" PostTypeId="1" CreationDate="2022-06-15T11:23:40.770" Score="1" ViewCount="65" Body="&lt;p&gt;I understand Hard disk present before &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_hard_disk_drives#1980s,_the_transition_to_the_PC_era&quot; rel=&quot;nofollow noreferrer&quot;&gt;1988&lt;/a&gt;&#xA;and I came to know that &lt;a href=&quot;https://en.wikipedia.org/wiki/Giant_magnetoresistance&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giant magnetoresistance effect&lt;/a&gt; used in applications in Hard Disk manufacturing and this effect is discovered in &lt;a href=&quot;https://en.wikipedia.org/wiki/Giant_magnetoresistance#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;1988&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So what effect were used for manufacturing HDD before 1988?&lt;/p&gt;&#xA;" OwnerUserId="16775" LastEditorUserId="16775" LastEditDate="2022-06-15T11:27:58.500" LastActivityDate="2022-06-15T11:27:58.500" Title="What effect were used for manufacturing HDD before 1988?" Tags="|computer-science|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14531" PostTypeId="2" ParentId="14434" CreationDate="2022-06-15T14:18:31.837" Score="18" Body="&lt;p&gt;I am the author of the Advanced Problem H-237 to the FQ issue Vol. 14 no. 2 (1976).  The series in that Problem was later named the Millin series.  At that time, I was a senior at the Annville-Cleona High School in Pennsylvania.  This series was also part of the paper titled &amp;quot;Observations in Pure Mathematics&amp;quot; that I submitted to the 1974 Westinghouse Science Talent Search (now the Regeneron Science Talent Search).  In the days before LaTeX, I wrote my submission to FQ using a combination of handwriting and a typewriter.  Apparently, my handwritten signature was not clear and was misread at the time.&lt;/p&gt;&#xA;&lt;p&gt;Frankly, I like the name &amp;quot;the Millin Series,&amp;quot; and I hope no one makes an effort to change it.  For my proof of this identity, see the last entry on this &lt;a href=&quot;http://www.lix.polytechnique.fr/Labo/Dale.Miller/papers/pubs.html&quot; rel=&quot;noreferrer&quot;&gt;webpage&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.lix.polytechnique.fr/Labo/Dale.Miller/&quot; rel=&quot;noreferrer&quot;&gt;Dale A Miller, Research Scientist, Inria, France&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16781" LastActivityDate="2022-06-15T14:18:31.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14532" PostTypeId="2" ParentId="7801" CreationDate="2022-06-15T15:01:46.567" Score="0" Body="&lt;p&gt;17 is apparently &amp;quot;the barrier&amp;quot; firstly b/c it comes between 16, relating to a square, and 18, which relates to a rectangle, and secondly because it separates 18 from it's epogdoon 16, the number with which it has the relationship 9:8, the major second (loosely speaking / usually). IDK why the Pythagoreans &amp;quot;abominated&amp;quot; 17; they loved 153, the triangle of 17. (-Perhaps it is partly because of the extremely dissonant nature of the major second interval (or so it was considered until relatively recently, early last century or so)?)&lt;/p&gt;&#xA;&lt;p&gt;17 is the number of consonants in the classical Greek alphabet; I'm aware of the linking of various entities (elements, stars, planets, etc.) to the letters, but cannot personally evaluate claims of historical religious significance related to isopsephy/gematria - though I'd like to think that the Cambridge School will be vindicated) . ...Perhaps someone with better qualifications can weigh in?&lt;/p&gt;&#xA;&lt;p&gt;I'm sorry I can't give refs; my notes from that period are sadly incomplete.&lt;/p&gt;&#xA;" OwnerUserId="16782" LastActivityDate="2022-06-15T15:01:46.567" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14533" PostTypeId="1" CreationDate="2022-06-15T20:58:01.337" Score="3" ViewCount="114" Body="&lt;p&gt;Context: since &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z/p^\times$&lt;/span&gt; is cyclic for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; prime, &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; is a square mod &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$(-1)^{{p-1\over 2}}=1$&lt;/span&gt;. A significantly subtler, but still classical, case is determining the quadratic symbol &lt;span class=&quot;math-container&quot;&gt;$(2/p)_2$&lt;/span&gt;: the almost-universal argument evaluates &lt;span class=&quot;math-container&quot;&gt;$(1+i)^p$&lt;/span&gt; in two different ways... The rest of the simplest proof I know for quadratic reciprocity (over &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q$&lt;/span&gt;, anyway) uses Gauss sums. Ok.&lt;/p&gt;&#xA;&lt;p&gt;But: just a few days ago, on MathStackExchange, someone (and I'd give credit, but I've lost the link...) remarked that there is also a &amp;quot;special&amp;quot; method for &lt;span class=&quot;math-container&quot;&gt;$(-3/p)_2$&lt;/span&gt;... but without further comment. Indeed, since &lt;span class=&quot;math-container&quot;&gt;$\omega+\omega^{-1}=\sqrt{-3}$&lt;/span&gt; (with &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; a primitive cube root of unity), without understanding the general version of the Gauss sum argument, on one hand &lt;span class=&quot;math-container&quot;&gt;$(\omega+\omega^{-1})^p=\omega^p+\omega^{-p}$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$\mathbb Z[\omega]\mod p$&lt;/span&gt;. On the other hand, it's&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(\sqrt{-3})^{p-1}\cdot \sqrt{-3}\;=\;3^{{p-1\over 2}}\cdot \sqrt{-3}\;=\;&#xA;(-3/p)_2\cdot \sqrt{-3}&#xA;$$&lt;/span&gt;&#xA;So, indeed, &lt;span class=&quot;math-container&quot;&gt;$(-3/p)_2={\omega^p-\omega^{-p}\over \sqrt{-3}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, the idea being exploited is that we have &amp;quot;geometric&amp;quot; ways to understand that certain square roots lie in cyclotomic fields. And, yes, an analogous thing works for &lt;span class=&quot;math-container&quot;&gt;$(5/p)_2$&lt;/span&gt;... It is charming! And, yes, the &lt;span class=&quot;math-container&quot;&gt;$(2/p)_2$&lt;/span&gt; argument can be rewritten in a slightly less obscure analogous fashion, using &lt;span class=&quot;math-container&quot;&gt;$\omega+\omega^{-1}=\sqrt{2}$&lt;/span&gt;, for &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt; an &lt;span class=&quot;math-container&quot;&gt;$8$&lt;/span&gt;th root of unity.&lt;/p&gt;&#xA;&lt;p&gt;My question is: while I do know that people knew about &lt;span class=&quot;math-container&quot;&gt;$(-1/p)_2$&lt;/span&gt; before Gauss, ... what about these others? The most plausible is &lt;span class=&quot;math-container&quot;&gt;$(2/p)_2$&lt;/span&gt;, but my superficial looking around (and superficial memory) gives nothing on this. Did people know &lt;span class=&quot;math-container&quot;&gt;$(-3/p)_2$&lt;/span&gt;? &lt;span class=&quot;math-container&quot;&gt;$(5/p)_2$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Yes, these can be viewed as simple precursors of the Gauss sums argument... one main point of which is proof that &lt;span class=&quot;math-container&quot;&gt;$\sqrt{\pm p}$&lt;/span&gt; lies in a suitable cyclotomic field... But, without that hindsight, hadn't Lagrange or even Euler observed some of these special cases?&lt;/p&gt;&#xA;&lt;p&gt;Of course, if Franz Lemmermeyer happens to be looking here, I'd imagine that he knows very well, but I think he does not visit this site, unfortunately. :)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastEditorUserId="1932" LastEditDate="2022-06-16T00:11:04.253" LastActivityDate="2022-06-18T15:00:32.750" Title="Special quadratic reciprocity? $(-3/p)_2$ and $(5/p)_2$ in addition to $(-1/p)_2$ and $(2/p)_2$?" Tags="|number-theory|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14534" PostTypeId="2" ParentId="14533" CreationDate="2022-06-16T22:14:00.183" Score="3" Body="&lt;p&gt;Franz Lemmermeyer was kind enough to send me some information by email, which I quote in part:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Euler as well as Lagrange could indeed determine the quadratic characters of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$-3$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, mainly using&#xA;cyclotomic techniques in disquise. In particular they could write the cyclotomic polynomial Phi_p(x) in the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$Y^2 -p^* Z^2$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$p^* = -3, 5, -7$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$-11$&lt;/span&gt;.  I covered Euler's contributions in the edition of his correspondence&#xA;with Goldbach here:  &lt;a href=&quot;https://edoc.unibas.ch/58842/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://edoc.unibas.ch/58842/&lt;/a&gt;&#xA;Everything else on the quadratic character of small primes is post-Gauss; I'll attach my copy of Cooper's thesis&#xA;as a pdf file, where Chapter 2 covers these things.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As Franz explains in the preface to that edition of Euler's correspondence with Goldbach, Cooper was a student or protege of L. Dickson at Chicago, and the absence of (history of) quadratic reciprocity from Dickson's Number Theory volumes is probably due to a failed expectation that Cooper would do a good write-up of it. That part &lt;em&gt;is&lt;/em&gt; treated in chapter 2 of Cooper's thesis, but apparently it did not meet Dickson's writing standards. :)&lt;/p&gt;&#xA;&lt;p&gt;(Thanks again to Franz L.)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastEditorUserId="1932" LastEditDate="2022-06-16T22:32:13.297" LastActivityDate="2022-06-16T22:32:13.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14535" PostTypeId="2" ParentId="14497" CreationDate="2022-06-17T02:09:44.377" Score="1" Body="&lt;p&gt;From which part of the head where the barley corns taken and how were they dried prior to measurement.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Inch#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;1814&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Charles Butler, a mathematics teacher at Cheam School, recorded the old legal definition of the inch to be &amp;quot;three grains of &lt;strong&gt;sound ripe barley&lt;/strong&gt; being &lt;strong&gt;taken out the middle of the ear, well dried&lt;/strong&gt;, and laid end to end in a row&amp;quot;, and placed the barleycorn, not the inch, as the base unit of the English Long Measure system, from which all other units were derived&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;emphasis is mine.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;John Bouvier similarly recorded in his 1843 law dictionary that the barleycorn was the fundamental measure. Butler observed, however, that &amp;quot;[a]s the length of the barley-corn cannot be fixed, so the inch according to this method will be uncertain&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The other thing to consider is did farming practices improve the barley crop over the years. Were barley corns larger in the 1800s than they were in the 1300s?&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2022-06-17T02:09:44.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14536" PostTypeId="1" CreationDate="2022-06-17T16:40:08.977" Score="0" ViewCount="87" Body="&lt;p&gt;I am reading the following short paper:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;P. G. Tait. &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN599484047_0006?tify=%7B%22pages%22:%5B138,139,151%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some elementary properties of closed plane curves.&lt;/a&gt; &lt;em&gt;Messenger (2)&lt;/em&gt; &lt;strong&gt;6&lt;/strong&gt; (1877), 132–133. &lt;a href=&quot;https://zbmath.org/?q=an:09.0393.01&quot; rel=&quot;nofollow noreferrer&quot;&gt;JFM 09.0393.01&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Here, Tait writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The closed curves contemplated are supposed to have nothing higher than &lt;em&gt;double&lt;/em&gt; points. By infinitesimal changes of position of the branches intersecting in it, a triple point is decomposable into &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; double points, a quadruple point into &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;, and generally an &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-ple point into &lt;span class=&quot;math-container&quot;&gt;$\dfrac{x(x-1)}{1.2}$&lt;/span&gt; double points.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm having trouble figuring out what exactly Tait intends by the term &amp;quot;double&amp;quot; points here. The description of how to change a &amp;quot;triple&amp;quot;, &amp;quot;quadruple&amp;quot;, or more generally an &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-ple&amp;quot; point into double points has not illuminated to me the definition of a double point itself.&lt;/p&gt;&#xA;&lt;p&gt;From the diagrams in Plate 2 (specifically, Fig. 11–17), I &lt;em&gt;think&lt;/em&gt; I intuitively understand what Tait is saying, but the conversion of triple points (and so on) into double points is still completely opaque to me.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://images.sub.uni-goettingen.de/iiif/image/gdz:PPN599484047_0006:00000151/full/full/0/default.jpg&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;License: Public Domain Mark 1.0 (PDM)&quot;&gt;&lt;img src=&quot;https://images.sub.uni-goettingen.de/iiif/image/gdz:PPN599484047_0006:00000151/full/full/0/default.jpg&quot; alt=&quot;Plate II&quot; title=&quot;License: Public Domain Mark 1.0 (PDM)&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Does Tait discuss his definition of double points anywhere else in greater detail? What is a modern definition of his &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;-ple points?&lt;/li&gt;&#xA;&lt;li&gt;What precisely is meant by &amp;quot;infinitesimal changes of position&amp;quot; when decomposing a triple point, for instance, into three double points?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14365" LastEditorUserId="14365" LastEditDate="2022-06-17T19:29:40.847" LastActivityDate="2022-06-17T19:29:40.847" Title="What does P. G. Tait mean by &quot;double points&quot;?" Tags="|mathematics|graph-theory|definition|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14537" PostTypeId="2" ParentId="14487" CreationDate="2022-06-17T23:30:57.947" Score="2" Body="&lt;p&gt;This is a supplement to @Miyase's answer.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(a)&lt;/strong&gt; To the question in the headline, &amp;quot;Is Newton's Gravitational Constant G a force?&amp;quot; : No, as @Miyase already said, it isn't. It's a constant of proportionality.&lt;/p&gt;&#xA;&lt;p&gt;In the OP's comments to this question there are several quotations taken from CV Boys' 1894 evening lecture at the Royal Institution, appearing to suggest something deep or mysterious about the constant of proportionality 'G'.&lt;/p&gt;&#xA;&lt;p&gt;This can usefully be compared with opinions about other constants of proportionality, e.g. at (&lt;a href=&quot;https://physics.stackexchange.com/questions/110297/what-is-a-proportionality-constant-plancks-constant/&quot;&gt;https://physics.stackexchange.com/questions/110297/what-is-a-proportionality-constant-plancks-constant/&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;There is no deep physical meaning to Boltzmann's constant. It is&#xA;merely a conversion factor to allow us to go between [temperature  and&#xA;energy].&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Similar views have been expressed about other named constants. Correspondingly, the constant G in the gravitational equation is just a conversion factor allowing us to go from one of several combinations of variables for mass and distance, to force, or to acceleration, &amp;amp;c, depending on how the equation is rearranged.&lt;/p&gt;&#xA;&lt;p&gt;Such straightforward explanations for constants of proportionality make the statements from Boys look like theatrical mystery-making in comparison.&lt;/p&gt;&#xA;&lt;p&gt;Also, to respond about a passage from Boys (1894) quoted and queried in the question/comments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The Newtonian Constant is equal to the force of attraction in  dynes&#xA;between two balls weighing a gramme each, with their  centers one&#xA;centimeter apart&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This most probably meant  &amp;quot;The Newtonian constant is &lt;em&gt;numerically&lt;/em&gt; equal to the force of attraction in dynes &lt;em&gt;in the given situation with the given masses and distance&lt;/em&gt;&amp;quot;.  It used to be a rather common way of explaining the meaning of such constants, to imagine a physical setup in which all of the auxiliary factors were 1 unit of some relevant measure. This doesn't make the constant into a force or anything else.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(b)&lt;/strong&gt; The question also states that &amp;quot;The Constant of Gravitation G was first defined by Boys in 1894&amp;quot;.  This appears not to be so, or at least to considerably overstate the position. What Boys did in effect was to propose a perhaps slightly new name for something much-explored and already named before him. This can be seen from &lt;a href=&quot;https://archive.org/details/meandensityofear00poynuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;J H Poynting's historical review&lt;/a&gt; (which won the 'Adams Prize' in 1893) which gives historical summary and explanation of much previous work. It also includes (p.2) the statement (without any claim that it was anything new)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;... the law is general, ... if we have any two bodies, of  masses&#xA;&lt;span class=&quot;math-container&quot;&gt;$M_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$M_2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; apart, the force on either is&lt;br /&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$  G \frac{M_1 \times M_2}{d^2} , $$&lt;/span&gt;&#xA;where G is a constant, the constant of gravitation.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;(c)&lt;/strong&gt; The question also asks 'what am I missing' about a numerical example. The answer seems to be, that the numbers in the question unfortunately have   arithmetic/mathematical errors. If the attractive force, considered as in the question, is between 1-gram masses at a distance D of 1 cm apart, then putting '2D = 2 cm' makes no difference because D itself is then still just 1 cm. If the question meant to ask what is the attraction if D is doubled to 2 cm, then application of the inverse-square relation (1/Distance^2) shows that the attractive force is then diminished to a quarter of what it was for D = 1 cm.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(d)&lt;/strong&gt; More generally, the question and the OP's comments take C V Boys seriously as a source for a number of statements given in his 1894 evening lecture at the Royal Institution &lt;a href=&quot;https://books.google.com/books?id=ZrloHemOmUEC&amp;amp;pg=PA353&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;p.353-377 in their 'Proceedings' Vol.14&lt;/a&gt;, although they were outside his field of expertise as an experimenter. Boys was indeed an experimental physicist of repute, as well as a popularizer of science. On the details of the physical measurements described in the lecture I have nothing to raise except the point that that like some others before and since, Boys exaggerated the accuracy of his results: he claimed (p.376) to be correct within 1 or 2 units in the 4th significant digit. Comparison with the &lt;a href=&quot;https://physics.nist.gov/cgi-bin/cuu/Value?bg&quot; rel=&quot;nofollow noreferrer&quot;&gt;modern measure (via NIST, 2019)&lt;/a&gt;) shows that really he had only 2 significant digits right. But a reading of the lecture also shows signs that he spoke unreliably or carelessly on some other matters, including exaggeration of the degree of novelty in his experiments, and he also made substantial mis-statements about the science of gravitation. At one point he also inserted a disparaging/abusive remark about an associate! These are not things one expects to read in a serious scientific contribution.&lt;/p&gt;&#xA;&lt;p&gt;The wrong statements about gravitational science are significant because they have misled many in the past and may probably continue to do so. Boys opened the lecture relied on by the OP with the &lt;em&gt;canard&lt;/em&gt; that Newton's law of gravitation  just accounts for, or results from adoption of, Kepler's laws. The same sort of mis-statement was also carelessly adopted by some others of Boys' generation, including the influential physicist and philosopher of science Pierre Duhem (1906).  Some 19th-/20th/-21st-century writers have been  more careful, including Rouse Ball (1893), Cohen (1980) and Ducheyne (2012). Ducheyne, in particular, discussed and refuted the fallacious critique by Duhem (1906) (who used a similar  mis-statement as Boys), understandably calling such treatments &amp;quot;straightforwardly embarrassing&amp;quot;. This underlines that the OP chose a source here that was unreliable at least outside a particular topic, and that to rely on an erroneous starting-point paves the way towards erroneous conclusions.&lt;/p&gt;&#xA;&lt;p&gt;References (other than those with links above):&lt;/p&gt;&#xA;&lt;p&gt;W W Rouse Ball (1893), &lt;a href=&quot;https://archive.org/details/anessayonnewton01ballgoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;An Essay on Newton's Principia&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;P M M Duhem (1906), &lt;a href=&quot;https://archive.org/details/lathoriephysiq00duhe&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;La théorie physique : son objet et sa structure&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I B Cohen (1980), &amp;quot;The Newtonian Revolution&amp;quot;,&lt;/p&gt;&#xA;&lt;p&gt;S Ducheyne (2012), &lt;a href=&quot;https://link.springer.com/book/10.1007/978-94-007-2126-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Main Business of Natural Philosophy&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-06-18T00:17:17.357" LastActivityDate="2022-06-18T00:17:17.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14538" PostTypeId="2" ParentId="14533" CreationDate="2022-06-18T11:38:47.337" Score="2" Body="&lt;p&gt;You do not need Gauss sums or arguments with &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-th powers to determine when &lt;span class=&quot;math-container&quot;&gt;$-3 \bmod p$&lt;/span&gt; is a square for &lt;span class=&quot;math-container&quot;&gt;$p &amp;gt; 3$&lt;/span&gt;: it happens if and only if there is a nontrivial cube root of unity mod &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;, since in a field &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; of characteristic not &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$a^2 = -3$&lt;/span&gt; in &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$(-1+a)/2$&lt;/span&gt; is a nontrivial cube root of unity. Thus &lt;span class=&quot;math-container&quot;&gt;$(-3|p) = 1$&lt;/span&gt; if and only if &lt;span class=&quot;math-container&quot;&gt;$3 \mid (p-1)$&lt;/span&gt;, or equivalently &lt;span class=&quot;math-container&quot;&gt;$p\equiv 1 \bmod 3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Knowing the quadratic character of &lt;span class=&quot;math-container&quot;&gt;$-1\bmod p$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$p\bmod 4$&lt;/span&gt; and of &lt;span class=&quot;math-container&quot;&gt;$-3\bmod p$&lt;/span&gt; depends on &lt;span class=&quot;math-container&quot;&gt;$p\bmod 3$&lt;/span&gt;  tells us how the quadratic character of &lt;span class=&quot;math-container&quot;&gt;$3\bmod p$&lt;/span&gt; is determined by &lt;span class=&quot;math-container&quot;&gt;$p \bmod 12$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am not saying this approach was known to Euler or Lagrange since they did not have a proof that &lt;span class=&quot;math-container&quot;&gt;$(\mathbf Z/(p))^\times$&lt;/span&gt; is cyclic, but we have that proof (first established by Gauss) and I suspect this is the approach intended at the MSE link you lost.&lt;/p&gt;&#xA;&lt;p&gt;The quadratic character of &lt;span class=&quot;math-container&quot;&gt;$-1 \bmod p$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$p &amp;gt; 2$&lt;/span&gt; can be determined without using cyclicity of &lt;span class=&quot;math-container&quot;&gt;$(\mathbf Z/(p))^\times$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$-1 \equiv b^2 \bmod p$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$(-1)^{(p-1)/2} \equiv b^{p-1} \equiv 1 \bmod p$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$(p-1)/2$&lt;/span&gt; is even and thus &lt;span class=&quot;math-container&quot;&gt;$p \equiv 1 \bmod 4$&lt;/span&gt;. Conversely, Wilson’s theorem &lt;span class=&quot;math-container&quot;&gt;$(p-1)! \equiv -1 \bmod p$&lt;/span&gt; can be rewritten as &lt;span class=&quot;math-container&quot;&gt;$(-1)^{(p-1)/2}((p-1)/2)!)^2 \equiv -1 \bmod p$&lt;/span&gt; by collecting terms at &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p-k$&lt;/span&gt; together on the left side of Wilson’s theorem. Therefore when &lt;span class=&quot;math-container&quot;&gt;$p\equiv 1 \bmod 4$&lt;/span&gt;, we get an explicit realization of &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; as a quadratic residue mod &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;: it is the square of &lt;span class=&quot;math-container&quot;&gt;$((p-1)/2)! \bmod p$&lt;/span&gt;. This argument could have been known to Lagrange (but I don’t know if it was) since he proved Wilson’s theorem.&lt;/p&gt;&#xA;&lt;p&gt;For an account by Weil of what Euler had done, see the first few pages of Section IX of Chapter III of his &lt;em&gt;Number Theory: An Approach Through History&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-06-18T15:00:32.750" LastActivityDate="2022-06-18T15:00:32.750" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14539" PostTypeId="2" ParentId="14481" CreationDate="2022-06-18T22:01:34.450" Score="5" Body="&lt;p&gt;The following is a summary and translation.&lt;/p&gt;&#xA;&lt;p&gt;The manuscript OP had is what the detailed eye looks like according to &lt;a href=&quot;https://en.wikipedia.org/wiki/Hunayn_ibn_Ishaq&quot; rel=&quot;noreferrer&quot;&gt;Hunayn ibn Ishaq&lt;/a&gt; an influential Nestorian Christian translator, scholar, physician, and scientist.&lt;/p&gt;&#xA;&lt;p&gt;Summary: In his manuscript, Hunayn believes that the eye is made of many parts and that not all of its parts are sight, but rather moisture similar to ice. As for the rest of the moisture that is in the eye and the layers and everything else other than that, each one of them was created for the benefit of the icy moisture aforementioned.&lt;/p&gt;&#xA;&lt;p&gt;The icy moisture according to Hunayn is pure white, bright, round, hard to rotate, but it has a width, and it is in the middle of the eye, like a point we could imagine lying in the middle of a ball.&lt;/p&gt;&#xA;&lt;p&gt;Hunayn sees this icy moisture as being in between two moistures, the one behind is similar to melted glass called vitreous, and the other in front is similar to egg white which he called an egg.&lt;/p&gt;&#xA;&lt;p&gt;He then states that behind the vitreous moisture there are three layers: the first layer contains the vitreous moisture and is similar to the retina (meaning the retinal veil), and the second layer is behind the first and is similar to the placenta and is called the chorionic layer, and the third layer behind the second follows the bone and it is hard and strong, and is called the rigid membrane.&lt;/p&gt;&#xA;&lt;p&gt;In the image displayed by OP, Hunayn mentions&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;and I am still a beginner with regards to the benefits of every single moisture and its cleanliness which I described in beginning of asserting its structure, its being, its flexion, and its places, and I had proceeded by informing you that the skin moisture is in the middle of the eye and that behind it is one and a third of layers, and its frame is one moisture and a third of layers &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\text{red dot}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He then states the importance of the moisture&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With the help of God, we will be explaining &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\text{the functions of the previous structures}}$&lt;/span&gt;, which are the three layers that we mentioned before, so we say that every member of the body must be nourished because it must be missing something through the decomposition of natural heat from within... (the rest of the text is not shown in the image but he proceeds to explain some source of nourishment in the blood.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14690" LastEditorUserId="14690" LastEditDate="2022-06-18T22:07:54.080" LastActivityDate="2022-06-18T22:07:54.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14540" PostTypeId="1" CreationDate="2022-06-19T00:17:46.247" Score="3" ViewCount="110" Body="&lt;p&gt;I am reading &lt;a href=&quot;https://www.jstor.org/stable/687825&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Uses and Abuses of the History of Topos Theory&lt;/a&gt; by Colin McLarty.  On page 353, the following is said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An homology theory associates groups to topological spaces so that the group structures reflect topological structure. An homology theory also associates group homomorphisms to maps- a fact topologists in the 1930s used heavily but considered secondary to the association of groups to spaces. There were many homology theories, the relation between them were not well understood, and it seemed that specific calculations in homology could be made more systematic than they were ..&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I want to ask, what were the different viewpoints of homology theory in 1930? How could this same idea of assigning a group to a topological space  be realized in so many forms? Finally, how did these ideas get united?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="12330" LastEditDate="2022-06-19T12:37:21.003" LastActivityDate="2022-06-19T12:37:21.003" Title="What were the main different homology theories in 1930 and how did they get united?" Tags="|mathematics|topology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14541" PostTypeId="1" CreationDate="2022-06-19T06:05:48.297" Score="2" ViewCount="185" Body="&lt;p&gt;The &lt;a href=&quot;https://mathworld.wolfram.com/RiemannZetaFunctionZeta2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Basel problem&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sum_{n=1}^\infty{\frac{1}{n^2}}$&lt;/span&gt;, took 90 years for a closed form, &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi^2}{6}$&lt;/span&gt;, to be found.&lt;/p&gt;&#xA;&lt;p&gt;I'm curious to know what other mathematical expressions, especially those involving limits, took a long time for a &lt;a href=&quot;https://en.wikipedia.org/wiki/Closed-form_expression&quot; rel=&quot;nofollow noreferrer&quot;&gt;closed form&lt;/a&gt; to be found. (By &amp;quot;found&amp;quot;, I mean discovered and verified.)&lt;/p&gt;&#xA;" OwnerUserId="16797" LastActivityDate="2022-06-23T06:06:57.680" Title="What mathematical expressions took a long time for a closed form to be found?" Tags="|mathematics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14542" PostTypeId="1" CreationDate="2022-06-19T08:15:32.923" Score="2" ViewCount="99" Body="&lt;p&gt;Context: I have arrived at Section &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; of chapter &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt; of Smullyan and Fitting's &lt;em&gt;Set Theory and the Continuum Problem&lt;/em&gt; during my ongoing self-study odyssey.&lt;/p&gt;&#xA;&lt;p&gt;This section raises and proves &lt;strong&gt;Cowen's theorem&lt;/strong&gt; but gives no historical background apart from providing the citation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cowen, R.H. (1971), 'Superinductive classes in class-set theory', &lt;em&gt;Notre Dame Journal of Formal Logic&lt;/em&gt; &lt;strong&gt;12&lt;/strong&gt;(1) 62-68&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Checking this out, we have &lt;strong&gt;Robert H. Cowen&lt;/strong&gt; who reports as being a student of Ray Smullyan at the time he wrote it (1969).&lt;/p&gt;&#xA;&lt;p&gt;From what I can tell, he is &lt;strong&gt;not&lt;/strong&gt; the same Cowen as the eponym of the Cowen-Thomson Theorem, as that one seems to be attributed to &lt;strong&gt;C. Cowen&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, he probably &lt;strong&gt;is&lt;/strong&gt; the eponym of the Cowen-Engeler Lemma&lt;/p&gt;&#xA;&lt;p&gt;Apart from other papers attributed to Robert Cowen in the same field of study, I have been unable to find any biographical details for him: birth date, birthplace, nationality (American assumed), etc.&lt;/p&gt;&#xA;&lt;p&gt;Anyone out there able to help?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2022-06-19T08:15:32.923" Title="Who is Robert H. Cowen, the eponym of Cowen's Lemma?" Tags="|biographical-details|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14543" PostTypeId="1" CreationDate="2022-06-20T00:37:53.060" Score="3" ViewCount="121" Body="&lt;p&gt;I am interested in the history of the ordered pair.&lt;/p&gt;&#xA;&lt;p&gt;When did ordered pairs as a concrete object, not necessarily defined in terms of sets, appear in a mathematical or logical setting?&lt;/p&gt;&#xA;&lt;p&gt;The definition of ordered pairs in terms of sets goes back to &lt;a href=&quot;https://en.wikipedia.org/wiki/Ordered_pair#Wiener%27s_definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;Norbert Wiener&lt;/a&gt; with &lt;span class=&quot;math-container&quot;&gt;$\{\{\{a\}, \varnothing\}, \{\{b\}\}\}$&lt;/span&gt;, although Kuratowski's definition &lt;span class=&quot;math-container&quot;&gt;$\{\{a\}, \{a, b\}\}$&lt;/span&gt; later superseded it.&lt;/p&gt;&#xA;&lt;p&gt;It also appears that Principia Mathematica contained some form of ordered pair, perhaps as a primitive object or perhaps defined in terms of relations. (I actually can't tell based on the Wikipedia article.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://philosophy.stackexchange.com/a/76338/&quot;&gt;This answer on the philosophy stack exchange&lt;/a&gt;, however, explains that ordered pairs were used a long time ago to extend the expressive power of Aristotlean logic.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;These devices have early traditional precursors, see &lt;a href=&quot;https://www.jstor.org/stable/20685115&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hodges, Traditional Logic, Modern Logic and Natural Language&lt;/a&gt;. E.g. Alexander of Aphrodisias and Ibn-Sina converted binary relational inferences into syllogisms by changing the domain of discourse to pairs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As a brief, massively ahistorical aside, with two relations &lt;span class=&quot;math-container&quot;&gt;$A(p, x)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B(p, y)$&lt;/span&gt; defining projections out of a pair &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; respectively we can build ordered pairs and then on to finite tuples of the form &lt;span class=&quot;math-container&quot;&gt;$(a_1, (a_2, a_3))$&lt;/span&gt; or such. So a system that can handle binary relations could in principle be as expressive as first order logic.&lt;/p&gt;&#xA;&lt;p&gt;It's not clear to me, though, whether Ibn Sina would have had an explicit concept of an ordered pair or not. This account of Ibn Sina's extension to Aristotlean logic may be ahistorical (it's possible that he used binary predicates but not ordered pairs).&lt;/p&gt;&#xA;&lt;p&gt;If he did, I also don't know whether this would be the first usage of an ordered pair.&lt;/p&gt;&#xA;" OwnerUserId="9243" LastActivityDate="2022-06-20T00:37:53.060" Title="When did ordered pairs first appear?" Tags="|mathematics|logic|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14545" PostTypeId="1" CreationDate="2022-06-20T13:36:49.633" Score="0" ViewCount="277" Body="&lt;p&gt;I remember seeing Newton's force equation written in a 19th century textbook as &lt;span class=&quot;math-container&quot;&gt;$F=(M+m)/R^2$&lt;/span&gt; (instead of &lt;span class=&quot;math-container&quot;&gt;$Mm$&lt;/span&gt;). I don't remember if they used &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;. Do you know a collection of 19th century physics textbooks available online? I'm trying to locate when physicists switched to multiplying two masses which makes no physical sense.&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2022-06-24T12:41:19.493" Title="Online resources for 19th century physics textbooks" Tags="|newton|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14546" PostTypeId="1" CreationDate="2022-06-21T11:47:32.553" Score="0" ViewCount="119" Body="&lt;p&gt;I have a vague memory of reading once that somebody (maybe a journalist) asked Poincaré how had he been capable of solving all those fantastically difficult problems that he had solved. The answer was simply “I thought a lot about them”.&lt;/p&gt;&#xA;&lt;p&gt;Has anyone more details about this history? I hope that it is not just a product of my imagination.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="6609" LastEditDate="2022-06-21T12:24:02.007" LastActivityDate="2022-06-21T12:24:02.007" Title="Poincaré's quote: &quot;I thought a lot about them&quot;" Tags="|mathematicians|quote|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14547" PostTypeId="1" AcceptedAnswerId="14552" CreationDate="2022-06-22T00:59:59.870" Score="7" ViewCount="455" Body="&lt;p&gt;How and when was water the structure of a water molecule (specifically the angles) discovered?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/nU6yr.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/nU6yr.png&quot; alt=&quot;H2O molecular structure&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Was it discovered by using a specific type of spectroscopy?&lt;/p&gt;&#xA;&lt;p&gt;I know you can derive these angles theoretically, but I'm interested in their experimental measurement.&lt;/p&gt;&#xA;" OwnerUserId="16819" LastEditorUserId="10106" LastEditDate="2022-06-22T04:17:02.370" LastActivityDate="2023-02-05T12:11:15.147" Title="When was the geometric structure of a water molecule discovered?" Tags="|physics|experimental-physics|chemistry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14549" PostTypeId="2" ParentId="14545" CreationDate="2022-06-22T12:43:59.867" Score="3" Body="&lt;p&gt;This question asks for book-references to what it calls &amp;quot;Newton's force equation&amp;quot;, written in a certain way (involving the &lt;em&gt;sum&lt;/em&gt; of two masses).&lt;/p&gt;&#xA;&lt;p&gt;Normally, an answer could just be book-references. But here, the question also provides a context: &lt;em&gt;&amp;quot;I'm trying to locate when physicists switched to multiplying two masses which makes no physical sense.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;This context modifies the question rather heavily. As is well-known, the accepted law of gravitational attractive force between two bodies with masses &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; (usually formulated as  &lt;span class=&quot;math-container&quot;&gt;$G M m / r^2$&lt;/span&gt; ), involves two masses as multiplying factors. This is what the question declares to make no physical sense. So it seems clear that the questioner has some kind of problem with the law of gravitation itself. It seems unlikely that there are books that could be found that would conform to this point of view.&lt;/p&gt;&#xA;&lt;p&gt;Further context emerges in a comment by the questioner, citing I B Cohen's 1999 translation of Newton's Book 3 Prop.7:  &lt;em&gt;&amp;quot;Gravity exists in all bodies universally and is proportional to the quantity of matter in each.&amp;quot;&lt;/em&gt;  The questioner goes on to say &lt;em&gt;&amp;quot;I don't see any reference to&#xA;multiplication here&amp;quot;.&lt;/em&gt; But proportions are notoriously known to&#xA;involve multiplication.&lt;/p&gt;&#xA;&lt;p&gt;In principle, no accepted physical law is exempt and immune from criticism and replacement. But when an accepted law is supported by many and classic observations, then a criticism, to be credible, does have to account for those observations at least as well as the criticised theory did.&lt;/p&gt;&#xA;&lt;p&gt;In the circumstances of the present question, it may therefore help to offer two things (and it seems difficult to go beyond),&lt;/p&gt;&#xA;&lt;p&gt;(i) brief explanation how it comes about that results of the laws of motion and gravitation involve (without inconsistency) relations of both kinds, i.e. some involving the product &lt;span class=&quot;math-container&quot;&gt;$M m$&lt;/span&gt;, and some the sum &lt;span class=&quot;math-container&quot;&gt;$M+m$&lt;/span&gt;, of two masses; and&lt;/p&gt;&#xA;&lt;p&gt;(ii) an indication of some of the (many) incompatibilities with observations, including the classic observations that supported the laws in the first place, that would result from the alteration that the questioner seems to want to make. (Apparently, questioner wishes to substitute the sum &lt;span class=&quot;math-container&quot;&gt;$M+m$&lt;/span&gt; for the product &lt;span class=&quot;math-container&quot;&gt;$M m$&lt;/span&gt; in the statement of the law of gravitation).&lt;/p&gt;&#xA;&lt;p&gt;(i) The accepted law of the gravitational attractive force between two point-masses &lt;span class=&quot;math-container&quot;&gt;$M_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt; at distance &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is as well-known,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ F = G M_1 m_2 / d^2 .$$&lt;/span&gt;&#xA;Then the second law of motion, often written nowadays as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ F = m  a ,$$&lt;/span&gt;&#xA;describes how an acceleration &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and mass &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; together relate to the force &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; that induces the acceleration. From the second law it follows that &lt;em&gt;acceleration&lt;/em&gt; here is &lt;em&gt;force/mass&lt;/em&gt; , &lt;span class=&quot;math-container&quot;&gt;$$ a = F/m .$$&lt;/span&gt;&#xA;In the case of the two bodies already mentioned, the gravitational &lt;em&gt;acceleration&lt;/em&gt; on mass &lt;span class=&quot;math-container&quot;&gt;$M_1$&lt;/span&gt; is  &lt;span class=&quot;math-container&quot;&gt;$$ F/M_1 = G m_2 / d^2 ,$$&lt;/span&gt; and the &lt;em&gt;acceleration&lt;/em&gt; on mass &lt;span class=&quot;math-container&quot;&gt;$m_2$&lt;/span&gt; is  &lt;span class=&quot;math-container&quot;&gt;$$ F/m_2 = G M_1 / d^2 .$$&lt;/span&gt;&#xA;But each of these two accelerations tends to bring the masses&#xA;towards each other. So the total relative acceleration is the&#xA;sum of the two accelerations, &lt;span class=&quot;math-container&quot;&gt;$$ G (M_1 + m_2) / d^2 .$$&lt;/span&gt; There is no incompatibility between such an expression for the total relative acceleration involving the sum of the masses and the expression for gravitional force involving the product.&lt;/p&gt;&#xA;&lt;p&gt;A more detailed derivation of similar kind can be seen in &lt;a href=&quot;https://books.google.com/books/about/Orbital_Motion.html?id=Hzv7k2vH6PgC&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;A E Roy 'Orbital Motion', esp. at p.62-65&lt;/a&gt;. (The law of gravitation appears in its usual form in section 4.3, and the derived equation showing the relative acceleration involving the sum of the masses at eqns. 4.10-4.12.)&lt;/p&gt;&#xA;&lt;p&gt;It seems likely that the formula that the questioner remembers&#xA;seeing, involving the sum of the masses, was a formula that&#xA;related to the accelerations, or otherwise to the motions, and&#xA;not to the forces themselves.&lt;/p&gt;&#xA;&lt;p&gt;Nothing essential is altered if instead of acceleration one considers a finite change in the velocity or in the momentum of the bodies: the resulting relation still involves the sum of the masses, but it also derives, as shown above, from the gravitational force-relation that involves the product of the masses.&lt;/p&gt;&#xA;&lt;p&gt;Books that reflect the point tend to submerge it under masses of detail in the equations. But Newton did deal with the point fairly straightforwardly in Book 1 of the 'Principia', in Propositions 57-59. One only needs a bit of patience and understanding for the archaic material and forms of expression, basically over 300 years old in the original. (The questioner already quoted from the (very good) Cohen modern translation from 1999, so it may be that he has easy access to this source for propositions 1.57-59 also.)&lt;/p&gt;&#xA;&lt;p&gt;(ii) Anyone who wishes to deny the proportionality of force to two different masses in the law of gravitation is left with the problem of accounting for the equality of descending accelerations found in Galileo's classic experiments in dropping weights off a tower and many similar observations. If the earth's gravitational force on an attracted falling body had no proportionality to its mass, then bodies of different mass would have to accelerate in their fall at different rates, light faster than heavy, which is not what Galileo or anybody else found.  (Complementary attractions of the falling bodies on the earth itself are negligibly tiny, as small as the relations of mass between such bodies and the whole earth itself.)&lt;/p&gt;&#xA;&lt;p&gt;If reconciliation is sought instead by rejecting the second law of motion as well, then the lack of the second law generates other incompatibilities with the everyday realities that it reflects -- for example, that a force or shove against a very heavy object produces little effect in motion, where the same force against a lighter object produces more motion.&lt;/p&gt;&#xA;&lt;p&gt;If it is the proportionality of gravitational force to the earth's mass that is put in question, then we can't experiementally change the earth's mass for something else, and astronomical rather than terrestrial observations have to be called in aid to settle such questions. The phenomena include Kepler's closely approximate 3/2 power relation between distances and periods of the planetary motions around the sun, and the same power relation for the distances and periods of Jupiter's major satellaites around Jupiter. But the coefficient of the power relation for the motions around the sun is different from the coefficient of the power relation for the motions around Jupiter, reflecting the different attracting masses of Jupiter and the Sun. If there were no proportionailty of gravitation to the mass of the attracting body, Jupiter's satellite system and the main solar system would have to show very different relations to each other in their relative speed/period versus distance, than what is actually observed. (The point was first handled in 'Principia' Book 3, Prop.8.)&lt;/p&gt;&#xA;&lt;p&gt;It is highly doubtful whether any amendment of the laws of motion and gravitation could be achieved without entailing false predictions of many  effects that just do not happen.  Arguments about multiplying apples are beside the point because they exist in words and rhetoric only, with no connection to physical situations relevant to the problem that has been posed here. Credible answers to questions about whether proportions are&#xA;physically meaningful are not couched in rhetoric, but in terms of correctly measured physical observations and correct mathematics applied to the observations.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-06-24T12:41:19.493" LastActivityDate="2022-06-24T12:41:19.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14550" PostTypeId="1" CreationDate="2022-06-22T17:12:19.330" Score="2" ViewCount="102" Body="&lt;p&gt;Gauss's notion of &amp;quot;oriented area&amp;quot; of figures characterizes the notion of two-dimensional content in a way that enables self-intersecting geodesic polygons (the term &amp;quot;geodesic&amp;quot; means that the polygon sides are geodesic lines of curved surface) to be regarded as zero area polygons, despite that these polygons obviously have content. According to &lt;a href=&quot;https://stackoverflow.com/questions/10049454/area-of-self-intersecting-polygon&quot;&gt;this post&lt;/a&gt;, an algebraic way to see it in the case of euclidean geometry is through the so-called &lt;a href=&quot;https://en.wikipedia.org/wiki/Shoelace_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shoelace formula&lt;/a&gt; (also discovered by Gauss, in his youth years). Another way is through considering the orientation of each sub-loop (clockwise or counterclockwise) before summing the signed area of them (and this gives the oriented area of figures).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qODsx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qODsx.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example, in the image above, taken from his letter to olbers from 1825 (Gauss's werke, volume 8, p.398-400), Gauss gives the oriented areas of the three figures shown there.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering how to give a meaningful interpretation of self-intersecting polygons with zero area, both in order to understand this notion for myself and in order to gain insight about how Gauss originated this notion. One interpretation I thought about, which works only for surfaces of constant curvature, involves the &lt;a href=&quot;https://en.wikipedia.org/wiki/Parallel_transport&quot; rel=&quot;nofollow noreferrer&quot;&gt;parallel transport&lt;/a&gt; of vectors along geodesic polygons. In a surface of constant curvature, the angular deviation of a vector transported along a closed loop from its intial direction is proportional to the area of the loop. Therefore, if one takes into account the direction of rotation of the transported vector (whetever it is transported clockwise or counterclockwise along each subloop), one gets that the angular deviation of the transported vector is equal to the oriented area of the spherical image of the geodesic polygon.&lt;/p&gt;&#xA;&lt;p&gt;For example, in the image below, a tangent vector is transported clockwise along a spherical traingle, and this results in angular deviation &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; (proportional to the area of that triangle). If we would transport the vector counterclockwise, the angular deviation would be &lt;span class=&quot;math-container&quot;&gt;$-\alpha$&lt;/span&gt;. This means that if we concatenate a second triangle with equal area through one of the vertexes of the first triangle, we would get a kind of &amp;quot;8 figure&amp;quot; and the parallel transport would result in zero angular deviation. &lt;strong&gt;Note&lt;/strong&gt; that this interpretation works only for surfaces of constant curvature - if the surface has varing curvature than each infinitesimal loop has to be weighted by its own Gaussian curvature.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TUpkX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TUpkX.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So I want to ask how mathematicians (Gauss and those influenced by him) originally conceived of this notion? and is there another interpretation of &amp;quot;oriented area&amp;quot;, not using the modern concept of parallel transport? I'm just trying to explain to myself how a (self-intersecting) polygon can have zero area.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2022-06-22T21:59:52.453" LastActivityDate="2022-06-22T21:59:52.453" Title="How to give a meaningful interpretation to Gauss's notion of &quot;oriented area&quot; of self-intersecting geodesic polygons?" Tags="|geometry|gauss|differential-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14551" PostTypeId="1" AcceptedAnswerId="14555" CreationDate="2022-06-22T17:42:08.400" Score="6" ViewCount="160" Body="&lt;p&gt;In differential geometry, if &lt;span class=&quot;math-container&quot;&gt;$e_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$e_2$&lt;/span&gt; are bases for a tangent space &lt;span class=&quot;math-container&quot;&gt;$T_pM$&lt;/span&gt;, then the coefficients of the first fundamental form is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\begin{align}E&amp;amp;:=\left&amp;lt;e_1,e_1\right&amp;gt;\\F&amp;amp;:=\left&amp;lt;e_1,e_2\right&amp;gt;\\G&amp;amp;:=\left&amp;lt;e_2,e_2\right&amp;gt;\end{align}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And for the second fundamental form, similar -- &lt;span class=&quot;math-container&quot;&gt;$L:=\mathrm{I\!I}(e_1,e_1)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$M\ldots$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$N\ldots$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;p&gt;This is... pretty strange! Not the definitions - the nomenclature. Some more modern textbooks use E/F/G for the first coefficients and e/f/g/ for the second, but everyone seems to be in agreement that at least the letters E/F/G are to be used, and L/M/N are quite common to see.&lt;/p&gt;&#xA;&lt;p&gt;Why were these letters used? (Or, is it known to be an arbitrary choice?)&lt;/p&gt;&#xA;" OwnerUserId="14994" LastActivityDate="2022-06-23T19:51:29.697" Title="Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?" Tags="|mathematics|notation|differential-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14552" PostTypeId="2" ParentId="14547" CreationDate="2022-06-22T18:19:52.630" Score="7" Body="&lt;p&gt;Császára et al. J. Chem. Phys. 122, 214305 (2005) has a nice table of determination of the bond angle of water per year (missing probably Linus Pauling first predicting 90° from approximations in quantum mechanics in 1931).&lt;/p&gt;&#xA;&lt;p&gt;The earliest measurement cited is E. K. Plyler, Phys. Rev. 39, 77 (1932). In this paper, the angle is derived from the infrared absorption frequencies of water. The result was still close but not exact, providing an angle of 115°.&lt;/p&gt;&#xA;&lt;p&gt;The next one had to wait to 1945, printed in the book G. Herzberg, Molecular Spectra and Molecular Structure Vol. II (1945) that was for decades the reference for the angle. The results come from the analysis of a series of experiments (mostly published in German) by R. Mecke and coworkers from 1933 to 1942. The angle cited is 104.45° (which coincides already with your image). Again from infrared spectroscopy.&lt;/p&gt;&#xA;&lt;p&gt;Note that all values cited are a mixture of experiments and inferring from a preexisting theory as it is very hard to &amp;quot;take a direct picture&amp;quot; of a single molecule.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-02-05T12:11:15.147" LastActivityDate="2023-02-05T12:11:15.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14553" PostTypeId="2" ParentId="14541" CreationDate="2022-06-23T06:06:57.680" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pentagonal_number_theorem&quot; rel=&quot;noreferrer&quot;&gt;Euler's Pentagonal number theorem&lt;/a&gt; was discovered by Euler in 1775, published in 1783. In his paper Euler gave only heuristic arguments. The first&lt;br /&gt;&#xA;rigorous proof&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jacobi_triple_product&quot; rel=&quot;noreferrer&quot;&gt;was obtained by Jacobi&lt;/a&gt; in 1829.&lt;/p&gt;&#xA;&lt;p&gt;The theorem is an identity for formal series:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\prod_{n=1}^\infty(1-x^n)=\sum_{-\infty}^\infty(-1)^nx^{n(3n-1)/2}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-06-23T06:06:57.680" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14555" PostTypeId="2" ParentId="14551" CreationDate="2022-06-23T19:45:05.780" Score="12" Body="&lt;p&gt;See the paper by Gauss, &lt;a href=&quot;https://archive.org/details/disquisitionesg00gausgoog/page/n30/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Disquisitiones generales circa superficies curvas&lt;/a&gt;, from 1828.  On page 24, we find:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;E\,dp^2 + 2F\,dp\,dq + G\,dq^2 .&#xA;$$&lt;/span&gt;&#xA;Since then everyone continued to use the letters &lt;span class=&quot;math-container&quot;&gt;$E,F,G$&lt;/span&gt;. That would have been long before anyone used terms like &amp;quot;tangent space&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Why did Gauss choose those letters?  On page 22, where letters &lt;span class=&quot;math-container&quot;&gt;$E,F,G$&lt;/span&gt; are first indroduced, he had already used letters &lt;span class=&quot;math-container&quot;&gt;$A,B,C,D$&lt;/span&gt;. [As well as various small letters, Greek letters, letters with primes.]&lt;/p&gt;&#xA;&lt;p&gt;Thanks to Mark Yasuda for the reference.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2022-06-23T19:51:29.697" LastActivityDate="2022-06-23T19:51:29.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14556" PostTypeId="1" AcceptedAnswerId="14557" CreationDate="2022-06-24T00:22:56.033" Score="4" ViewCount="1627" Body="&lt;p&gt;The slope-intercept form of a non-vertical line is &lt;span class=&quot;math-container&quot;&gt;$y=mx+b$&lt;/span&gt;. I have been told that the slope is called &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; because it is the first letter of the French word for mountain. But why is there the letter &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;? Wouldn't it be more logical to write &lt;span class=&quot;math-container&quot;&gt;$y=mx+n$&lt;/span&gt; or perhaps &lt;span class=&quot;math-container&quot;&gt;$y=mx+a$&lt;/span&gt;? What is the history behind that convention?&lt;/p&gt;&#xA;" OwnerUserId="16731" LastEditorUserId="10106" LastEditDate="2022-06-24T06:56:01.510" LastActivityDate="2022-06-24T06:56:01.510" Title="Why is the letter $b$ used to represent the y-intercept in the equation of straight line?" Tags="|mathematics|terminology|geometry|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14557" PostTypeId="2" ParentId="14556" CreationDate="2022-06-24T06:29:42.563" Score="9" Body="&lt;p&gt;The first known use of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; to represent the y-intercept in the equation of straight line is in Gaspard Monge's &amp;quot;Mémoire sur la Théorie des Déblais et des Remblais&amp;quot;, &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k35800/f799.item&quot; rel=&quot;noreferrer&quot;&gt;Histoire de l'Académie royale des sciences 1784 (Année 1781), p. 669&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Or, I'équation de la droite &lt;span class=&quot;math-container&quot;&gt;$Bd$&lt;/span&gt;, rapportée aux mêmes axes, eft généralement de la forme &lt;span class=&quot;math-container&quot;&gt;$y = ax + b$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, the usage of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; along with &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, the slope, dates back to George Salmon's &lt;a href=&quot;https://archive.org/details/atreatiseonconi00unkngoog/page/n36/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;A Treatise on Conic Sections, p.15&lt;/a&gt;, published in several editions beginning in 1848.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The equation, therefore, &lt;span class=&quot;math-container&quot;&gt;$y = mx + b$&lt;/span&gt;, being satisfied by every point of the line PQ, is said to be the equation of that line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Nevertheless, there is no clear answer to the question of why the letter &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, or even &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;, is used. In fact, many authors have used various other symbols.&lt;/p&gt;&#xA;&lt;p&gt;Matthew O'Brien's &lt;a href=&quot;https://archive.org/details/anelementarytre01obrgoog/page/n19/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;An Elementary Treatise on the Differential Calculus (1842), p.1.&lt;/a&gt;, uses &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; to represent the y-intercept.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus in the general equation to a right line, namely, &lt;span class=&quot;math-container&quot;&gt;$y = mx + c$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Isaac Todhunter again made use of the letter &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; in &lt;a href=&quot;https://archive.org/details/atreatiseonplan05todhgoog/page/n6/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;A Treatise on Plane Co-ordinate Geometry as Applied to the Straight Line and the Conic Sections&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Even there are different conventions in different countries. In Swedish textbooks, &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; represents slope and &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; represents the y-intercept. In Austria, &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is used for the slope and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; for the y-intercept.&lt;a href=&quot;https://mathworld.wolfram.com/Slope.html&quot; rel=&quot;noreferrer&quot;&gt;[Wolfram MathWorld]&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-06-24T06:46:00.903" LastActivityDate="2022-06-24T06:46:00.903" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14558" PostTypeId="2" ParentId="12936" CreationDate="2022-06-24T08:08:14.947" Score="1" Body="&lt;p&gt;I am not sure about Q1., but Q2. is answered in Biggs, Norman L.; Lloyd, E. Keith; Wilson, Robin J. Graph theory 1736–1936. Oxford University Press (1976). Reprinted, with corrections (1977). &lt;a href=&quot;https://zbmath.org/?q=an:0335.05101%7C0904.05001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zbl 0335.05101, Zbl 0904.05001&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From Chapter 6: The four-colour problem—early history, on page 103 (you might be able to view the relevant pages on &lt;a href=&quot;https://books.google.co.in/books?id=XqYTk0sXmpoC&amp;amp;pg=PA103#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kempe's improved 'proof' aroused the interest of P. G. Tait…On 15 March 1880 he read a paper on the map-colouring problem to the Royal Society of Edinburgh and an abstract of this paper was published in the &lt;em&gt;Proceedings&lt;/em&gt; of the society.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is the one titled &amp;quot;On the colouring of maps&amp;quot;. Further, on the same page:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was the published abstract of Tait's paper which prompted Frederick Guthrie to publish his account of the origin of the problem in the same &lt;em&gt;Proceedings&lt;/em&gt; [1]. Immediately following this account is the abstract of another paper by Tait, delivered on 19 July 1880 [6C].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paper [1] by Guthrie is the one titled &amp;quot;Note on the colouring of maps&amp;quot;, which is mentioned in a footnote in the question. The paper [6C] by Tait is the one titled &amp;quot;Remarks on the previous communication&amp;quot;. Following this, on page 104:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The full version of Tait's second paper was eventually published in the &lt;em&gt;Transactions of the Royal Society of Edinburgh&lt;/em&gt;…&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is the one titled &amp;quot;Note on a theorem in geometry of position&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So, Tait's full 'proof' of the four-colour theorem is the one that appeared in the Transactions of the Royal Society of Edinburgh.&lt;/p&gt;&#xA;&lt;p&gt;Since Tait himself mentions in his &amp;quot;Remarks…&amp;quot; paper that he has withdrawn his earlier paper, I expect that the brief outline in the first abstract is all that we have of his initial attempt at proving the four-colour theorem.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To explicitly answer Q3., I would cite the full paper published in the Transactions of the Royal Society of Edinburgh when needing to refer to Tait's 'proof' of the four-colour theorem.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'll also pull down my comment from below the question, since I think it's relevant:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My understanding is that the &amp;quot;previous communication&amp;quot; refers to his own previous communication, that is the one titled &amp;quot;On the colouring of maps&amp;quot;. It is probably only incidental that the literal previous communication also concerns the four color theorem. Indeed, Tait does not reference Guthrie's comments anywhere in his &amp;quot;Remarks …&amp;quot; paper/abstract, but only speaks about his own work that was presented earlier. – &lt;a href=&quot;https://hsm.stackexchange.com/users/14365/the-amplitwist&quot;&gt;The Amplitwist&lt;/a&gt; &lt;a href=&quot;https://hsm.stackexchange.com/questions/12936/where-can-i-find-the-complete-papers-of-abstracts-published-by-p-g-tait-in-pro#comment33192_12936&quot;&gt;Jun 17 at 23:47&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="14365" LastActivityDate="2022-06-24T08:08:14.947" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14559" PostTypeId="2" ParentId="14529" CreationDate="2022-06-24T18:09:53.840" Score="1" Body="&lt;p&gt;The idea that space is expanding itself is not a radical idea. It's not even about physical reality. When we say that space expands onto itself, we do not mean a real physical phenomena, it is just a convenient way to describe the expanding Universe.&lt;br /&gt;&#xA;There is no real difference between expanding space and receding galaxies. It depends on your coordinate system. If you choose the comoving coordinates, then your space grid would be indeed expanding. But you can choose the coordinates with non expanding grids. But there are no physical coordinate grids in our universe, so there is no real space which is expanding.&lt;/p&gt;&#xA;&lt;p&gt;This is different from the rotation of Earth, for example. We can tell that Earth is rotating without looking outside, e.g. by Foucault pendulum, but there is no any physical experiment that can tell that our universe is expanding without looking at receding far away objects like galaxies or CMB.&lt;/p&gt;&#xA;&lt;p&gt;So the phrase &amp;quot;space is expanding onto itself&amp;quot; is not about a new physical law, but it's just about a choice of a convenient coordinate system. And of course, it was not radical to Einstein. After all, choosing arbitrary coordinate system is a central point in his GR.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2022-06-24T18:09:53.840" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14561" PostTypeId="1" CreationDate="2022-06-26T04:20:25.347" Score="2" ViewCount="130" Body="&lt;p&gt;Is there a &amp;quot;Moore's Law&amp;quot; of length-determining apparatuses? Viz., has the precision of state-of-the-art length measurement apparatuses doubled periodically throughout history?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2022-07-31T14:06:40.687" Title="Is there a &quot;Moore's Law&quot; of length-determining apparatuses?" Tags="|experimental-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14563" PostTypeId="1" AcceptedAnswerId="14564" CreationDate="2022-06-27T21:07:01.043" Score="15" ViewCount="403" Body="&lt;p&gt;Every student of linear algebra learns about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trace_(linear_algebra)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;trace&lt;/em&gt; of a linear map&lt;/a&gt;. Its easiest (albeit not most conceptual) definition is: write the map as matrix, then the trace is the sum of the diagonal entries.&lt;/p&gt;&#xA;&lt;p&gt;The question about origin and motivation of this terminology has been asked several times: &lt;a href=&quot;https://math.stackexchange.com/q/1291981/96384&quot;&gt;https://math.stackexchange.com/q/1291981/96384&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/q/2593048/96384&quot;&gt;https://math.stackexchange.com/q/2593048/96384&lt;/a&gt;, &lt;a href=&quot;https://math.stackexchange.com/q/3684703/96384&quot;&gt;https://math.stackexchange.com/q/3684703/96384&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, none of them have a fully satisfying answer. What they seem to agree on is that the English &amp;quot;trace&amp;quot; is a translation of the German terminology &lt;em&gt;Spur&lt;/em&gt; (which is cognate to English spoor, but indeed translates to trace or track): &lt;a href=&quot;https://jeff560.tripod.com/t.html&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Earliest Known Uses of Some of the Words of Mathematics&amp;quot;&lt;/a&gt; quotes H. L. Brose's 1922 translation of a book of Weyl's, which might be the first use of the &lt;em&gt;English&lt;/em&gt; term &amp;quot;trace&amp;quot;. But then the question remains,&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In what (presumably German-language) source was the word &amp;quot;Spur&amp;quot; used first for this concept?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Attempt: At the bottom of &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN271034092?tify=%7B%22pages%22:%5B6%5D,%22view%22:%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;page 6&lt;/a&gt; of his (1901) doctoral thesis &lt;em&gt;Ueber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen&lt;/em&gt;, Issai Schur writes somewhat casually [emphasis added]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;dass zwei invariante Formen &lt;span class=&quot;math-container&quot;&gt;$T(A)$&lt;/span&gt; und &lt;span class=&quot;math-container&quot;&gt;$T_1(A)$&lt;/span&gt; dann und nur dann äquivalent sind, wenn ihre &lt;em&gt;Spuren, d.h. die Summen der Glieder ihrer Hauptdiagonalen&lt;/em&gt; einander gleich sind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since Schur's doctoral advisor was Frobenius, I thought it natural to look for earlier occurrences in Frobenius' works. However, I learned that in Frobenius's three &lt;em&gt;Sitzungsberichte&lt;/em&gt; papers from 1896 where he founded character theory, he actually does it via a different route, and I found no trace of a &amp;quot;Spur&amp;quot; in those. (For some overview on the history of representation and character theory, cf. &lt;a href=&quot;https://www.jstor.org/stable/41133320&quot; rel=&quot;noreferrer&quot;&gt;T. Hawkins&lt;/a&gt; and &lt;a href=&quot;https://kconrad.math.uconn.edu/articles/groupdet.pdf&quot; rel=&quot;noreferrer&quot;&gt;K. Conrad&lt;/a&gt;.) Then, in Frobenius' 1897 paper &lt;a href=&quot;https://www.e-rara.ch/download/pdf/5929678?name=%C3%9Cber%20die%20Darstellung%20der%20endlichen%20Gruppen%20durch%20lineare%20Substitutionen&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Über die Darstellung der endlichen Gruppen durch lineare Substitutionen&lt;/em&gt;&lt;/a&gt;, where he finally uses matrices, he does note that his characters turn out to be sums of diagonal elements (equation 5 in §4), and he emphasizes the importance of this especially in §6 (pages 15 and 16), but as far as I can tell, nowhere does he use the word &amp;quot;Spur&amp;quot;; instead, he writes out &amp;quot;die Summe der Diagonalelemente&amp;quot; twice.&lt;/p&gt;&#xA;&lt;p&gt;So now I am almost tempted to believe that &lt;strong&gt;Schur&lt;/strong&gt; might have casually introduced that terminology there? Of course I might have overlooked something in &lt;strong&gt;Frobenius&lt;/strong&gt;' work of that time. Another guess would be &lt;strong&gt;Dedekind&lt;/strong&gt;, who according to the mentioned historical sources played an important &amp;quot;inspirational&amp;quot; role in that time. I have not looked through any of his works yet. Surely from today's view, he would have had reason to be interested in &amp;quot;Spuren&amp;quot; already in his number theoretic works.&lt;/p&gt;&#xA;&lt;p&gt;PS: In one of the linked questions above, a theory is put forward and credited to P.J. Cohen that supposedly, German &amp;quot;Spur&amp;quot; was a mistranslation of an English word &amp;quot;spur&amp;quot; supposedly used by &lt;em&gt;Cayley&lt;/em&gt; (which would have been earlier than my guesses). This theory sounds rather unlikely to me, and according to a comment by Brian M. Scott to &lt;a href=&quot;https://math.stackexchange.com/q/1291981/96384&quot;&gt;that question&lt;/a&gt;, &amp;quot;a search of the University of Michigan Historical Mathematics Collection turns up only two instance of &lt;em&gt;spur&lt;/em&gt; in Cayley’s collected works, and they have to do with physical gears, not matrices.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2022-07-21T20:28:31.337" Title="First use of &quot;Spur&quot; (trace) for linear maps / matrices" Tags="|terminology|notation|linear-algebra|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14564" PostTypeId="2" ParentId="14563" CreationDate="2022-06-29T04:41:16.040" Score="12" Body="&lt;p&gt;Comments by user njuffa (Thank You!) lead me to what I believe is what I was looking for:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;R. Dedekind&lt;/strong&gt;: &lt;em&gt;Über die Discriminanten endlicher Körper&lt;/em&gt;. In: &lt;em&gt;Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen&lt;/em&gt; (&lt;strong&gt;1882&lt;/strong&gt;).&lt;/p&gt;&#xA;&lt;p&gt;On &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN250442582_0029?tify=%7B%22pages%22:%5B113%5D,%22panX%22:0.234,%22panY%22:0.568,%22view%22:%22info%22,%22zoom%22:0.49%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 5&lt;/a&gt; he writes (and the emphasis is in the original):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Unter der &lt;em&gt;Spur&lt;/em&gt; der Zahl &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; verstehen wir die &lt;em&gt;Summe&lt;/em&gt; aller mit ihr conjugirten Zahlen; wir bezeichnen diese offenbar rationale Zahl mit &lt;span class=&quot;math-container&quot;&gt;$S(\theta)$&lt;/span&gt;; dann ist &lt;span class=&quot;math-container&quot;&gt;$$(11) \qquad \qquad S(\theta)= \theta^{(1)} + \theta^{(2)} + \dots + \theta^{(n)} = -a_1\\ \qquad= e_{1,1} + e_{2,2} + \dots + e_{n,n}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That is, he is looking at the field trace of an algebraic &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q$&lt;/span&gt;, and in one swoosh gives the three classical definitions: as sum of all its Galois conjugates, as the negative of the next-to-leading term of its minimal polynomial, and as sum of the diagonal entries of a matrix representing it. He goes on to note that the trace is linear and &lt;span class=&quot;math-container&quot;&gt;$S(1)=n$&lt;/span&gt;, the degree of the field extension.&lt;/p&gt;&#xA;&lt;p&gt;I believe this is Dedekind's invention and the first published use of the term, given the following extra evidence:&lt;/p&gt;&#xA;&lt;p&gt;Dedekind had published &amp;quot;Supplement XI&amp;quot; to Dirichlet's &lt;em&gt;Vorlesungen über Zahlentheorie&lt;/em&gt; in its third edition, &lt;strong&gt;1879&lt;/strong&gt;. This Supplement XI (founds modern algebraic number theory and) in this 1879 version, contains an introduction to norms and discriminants of number fields (&lt;a href=&quot;https://archive.org/details/vorlesungenberz02dedegoog/page/474/mode/2up?q=Spur&quot; rel=&quot;nofollow noreferrer&quot;&gt;§ 164&lt;/a&gt;), but &amp;quot;Spuren&amp;quot; are still absent. E.g. when on &lt;a href=&quot;https://archive.org/details/vorlesungenberz02dedegoog/page/496/mode/2up?q=Spur&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 497 (§166)&lt;/a&gt; he goes through the example of a quadratic field &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q(\omega)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\omega'$&lt;/span&gt; the conjugate of &lt;span class=&quot;math-container&quot;&gt;$\omega$&lt;/span&gt;, he calls the product &lt;span class=&quot;math-container&quot;&gt;$\omega \cdot \omega'$&lt;/span&gt; by name and notation &lt;span class=&quot;math-container&quot;&gt;$N(\omega)$&lt;/span&gt;, but writes down the sum &lt;span class=&quot;math-container&quot;&gt;$\omega +\omega'$&lt;/span&gt; next to it, without terminology or notation.&lt;/p&gt;&#xA;&lt;p&gt;So I assume he amended the concept and terminology of &amp;quot;Spur&amp;quot; in the above 1882 article, in which it is used for further inquiries into the discriminant.&lt;/p&gt;&#xA;&lt;p&gt;In fact, for the fourth edition of Dirichlet's &lt;em&gt;Vorlesungen&lt;/em&gt; (&lt;strong&gt;1894&lt;/strong&gt;), Dedekind had revised and enlarged &amp;quot;Supplement XI&amp;quot;, and now the &lt;a href=&quot;https://archive.org/details/vorlesungenberz02dirigoog/page/486/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;new §167&lt;/a&gt; contains a treatment of norms, discriminants, &lt;strong&gt;and &amp;quot;Spuren&amp;quot;&lt;/strong&gt;, which incorporates a streamlined version of the 1882 article, and actually could be copied into a textbook on algebraic number theory today. (Already in the introduction to his 1882 article, Dedekind says that although he works with the ground field &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q$&lt;/span&gt; for now, completely analogous results would hold over other fields, i.e. proposes the relative trace of a field extension &lt;span class=&quot;math-container&quot;&gt;$L\vert K$&lt;/span&gt;; and he introduces it in this generality in the 1894 version.)&lt;/p&gt;&#xA;&lt;p&gt;Also, njuffa found:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;An article of Hensel's from 1897 (&lt;a href=&quot;https://ia800708.us.archive.org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1515%252Fcrll.1884.96.324.zip&amp;amp;file=10.1515%252Fcrll.1897.117.346.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Ueber die Elementartheiler zweier Gattungen, von denen die eine unter der anderen enthalten ist&lt;/em&gt;&lt;/a&gt;, Journ. f. d. reine u. ang. Math. Vol. 117, Jan. 1897, pp. 346-356, where he also uses the concept for general field extensions, and claims the terminology to be due to Dedekind (&amp;quot;Jede solche Grösse &lt;span class=&quot;math-container&quot;&gt;$H_i$&lt;/span&gt; werde in Erweiterung einer &lt;em&gt;Dedekind&lt;/em&gt;schen Bezeichnung &lt;em&gt;die Spur von &lt;span class=&quot;math-container&quot;&gt;$Y_i$&lt;/span&gt;&lt;/em&gt; genannt&amp;quot;, p. 347).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Weber's textbook &lt;em&gt;Lehrbuch der Algebra&lt;/em&gt; (first edition 1895), where the Spur (together with the norm) is introduced on &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN234729163?tify=%7B%22pages%22:%5B479%5D,%22view%22:%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 461 (§144)&lt;/a&gt;, although only via the definition with conjugates, which thus must have been the main point of view at that time; so much so that, I assume, when Frobenius in 1897 realized that his characters are &lt;em&gt;also&lt;/em&gt; sums of diagonal elements, he ignored that this was an alternative definition of a somewhat established concept in number theory.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But, as pointed out by KCd in a comment, Frobenius must have seen this shortly after, as in his 1899 report &lt;em&gt;&lt;a href=&quot;https://www.biodiversitylibrary.org/item/93034#page/378/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Über die Composition der Charaktere einer Gruppe&lt;/a&gt;&lt;/em&gt; he writes (bottom of first page):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nennt man nach dem Vorgange von DEDEKIND die Summe der Diagonalelemente einer Substitution oder Matrix ihre &lt;em&gt;Spur&lt;/em&gt;, [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(&amp;quot;Vorgang&amp;quot; which nowadays translates as &amp;quot;procedure&amp;quot; is, to my German sense, a bit of an archaic word for &amp;quot;precedent&amp;quot; here, i.e. in modern language he says &amp;quot;If one, following Dedekind, calls the sum ...&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;And certainly with and after this, and Schur's 1901 dissertation cited in the OP, the door was open for the trace of general linear maps / matrices, not just ones coming from elements in field extensions.&lt;/p&gt;&#xA;" OwnerUserId="6514" LastEditorUserId="6514" LastEditDate="2022-07-21T20:28:31.337" LastActivityDate="2022-07-21T20:28:31.337" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14566" PostTypeId="1" CreationDate="2022-06-29T19:11:25.027" Score="0" ViewCount="86" Body="&lt;p&gt;I have a (very) vague remembrance of a quote about the history of particle physics, particularly (I think?) the development of the standard model. As I recall, it was about how it was a very exciting time, and the continual discoveries were like &amp;quot;strange wonders carried back from the edge of understanding&amp;quot; or something to that effect. I know this is very vague, but I would appreciate any help sourcing this quote. I may have seen it in a seminar talk or conference, but my own googling has turned up nothing.&lt;/p&gt;&#xA;" OwnerUserId="3382" LastEditorUserId="10106" LastEditDate="2022-06-30T00:57:24.220" LastActivityDate="2022-06-30T00:57:24.220" Title="Quote about the history of particle physics" Tags="|physics|quote|particle-physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14567" PostTypeId="1" AcceptedAnswerId="14570" CreationDate="2022-06-30T10:14:54.603" Score="3" ViewCount="182" Body="&lt;p&gt;Are there theorems earlier to the works of Abel and Galois showing that a problem that mathematicians had been trying to solve for a long time was actually unsolvable? Or are those works novel in the sense of establishing the inexistence, rather than the existence, of some highly desired entity?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastActivityDate="2022-07-02T00:15:13.943" Title="Which are the most ancient theorems that negate the existence of some deeply valued entity?" Tags="|mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14568" PostTypeId="1" CreationDate="2022-06-30T11:16:15.020" Score="3" ViewCount="130" Body="&lt;p&gt;I was reading about blackbody radiation and came across the following quote.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Planck did not believe in atoms, nor did he think the second law of&#xA;thermodynamics should be statistical because probability does not&#xA;provide an absolute answer, and Boltzmann’s entropy law rested on the&#xA;hypothesis of atoms and was statistical. But Planck was unable to find&#xA;a way to reconcile his Blackbody equation with continuous laws such as&#xA;Maxwell’s wave equations. So in what Planck called “an act of&#xA;desperation,” he turned to Boltzmann’s atomic law of entropy as it was&#xA;the only one that made his equation work. Therefore, he used&#xA;Boltzmann’s constant k and his new constant h to explain the Blackbody&#xA;radiation law which became widely known through his published paper.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Planck%27s_law#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Planck%27s_law#History&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In atomic physics, the Bohr model or Rutherford–Bohr model, presented&#xA;by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of&#xA;a small, dense nucleus surrounded by orbiting electrons—similar to the&#xA;structure of the Solar System, but with attraction provided by&#xA;electrostatic forces in place of gravity. It came after the solar&#xA;system Joseph Larmor model (1897), the cubical model (1902), the&#xA;Hantaro Nagaoka Saturnian model (1904), the plum pudding model (1904),&#xA;the quantum Arthur Haas model (1910), the Rutherford model (1911), and&#xA;the nuclear quantum John William Nicholson model (1912). The&#xA;improvement over the 1911 Rutherford model mainly concerned the new&#xA;quantum physical interpretation introduced by Haas and Nicholson, but&#xA;forsaking any attempt to align with classical physics radiation.&lt;/p&gt;&#xA;&lt;p&gt;The model's key success lay in explaining the Rydberg formula for the&#xA;spectral emission lines of atomic hydrogen. While the Rydberg formula&#xA;had been known experimentally, it did not gain a theoretical&#xA;underpinning until the Bohr model was introduced. Not only did the&#xA;Bohr model explain the reasons for the structure of the Rydberg&#xA;formula, it also provided a justification for the fundamental physical&#xA;constants that make up the formula's empirical results.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://en.wikipedia.org/wiki/Bohr_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Bohr_model&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems like most scientists of 19th century did not believe in atoms and possibly the belief in atoms started gaining traction around 1900.&lt;/p&gt;&#xA;&lt;p&gt;How did they explain the radiation resulting from the materials during that era when the belief in atoms was not widespread? For example, when an object is heated why it starts emitting radiation at different wavelengths.&lt;/p&gt;&#xA;&lt;p&gt;I'm asking this question because I think this 'misunderstanding' of pre-atomic era was at the core of ultraviolet catastrophe. The electrons emitting quanta of radiation when jumping from one orbital to another was an outcome of 1913 Bohr model atomic model.&lt;/p&gt;&#xA;" OwnerUserId="9341" LastEditorUserId="9341" LastEditDate="2022-06-30T23:41:13.987" LastActivityDate="2022-06-30T23:41:13.987" Title="How did they explain the radiation from hot objects at different wavelengths before the concept of atom was widely accepted?" Tags="|physics|chemistry|atomic-theory|radiation|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14569" PostTypeId="2" ParentId="14561" CreationDate="2022-07-01T13:37:58.753" Score="1" Body="&lt;h1&gt;MOSFET scaling&lt;/h1&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Template:Semiconductor_manufacturing_processes&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sJtzU.png&quot; alt=&quot;MOSFET scaling over time&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;gives an idea of how precise semiconductor manufacturing has become.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;h1&gt;Timekeeping accuracy&lt;/h1&gt;&#xA;&lt;p&gt;James A. Barnes, “&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/L9G6VUWB/Barnes%20-%201974%20-%20Basic%20concepts%20of%20precise%20time%20and%20frequency.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Basic Concepts of Precise Time and Frequency&lt;/a&gt;,” &lt;em&gt;Time and Frequency: Theory and Fundamentals&lt;/em&gt; 140 (1974): 1–14&#xA;&lt;br&gt;&lt;sub&gt;cited on p. 2-4 of Malcolm Cooper and Jim Grozier, eds., &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=9107&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Precise Dimensions: A History of Units from 1791-2018&lt;/em&gt;&lt;/a&gt;, IOP Expanding Physics (A History of Units from 1791 to 2018, Bristol, UK: IOP Publishing, 2017)&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;gives the evolution of time-measurement throughout history:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/L9G6VUWB/Barnes%20-%201974%20-%20Basic%20concepts%20of%20precise%20time%20and%20frequency.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/w8CW5.jpg&quot; alt=&quot;Figure 2.1. A typical chart of progress in timekeeping accuracy. Courstesy of The National Institute of Standards and Technology, from (Barnes 1974).&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2022-07-01T13:37:58.753" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14570" PostTypeId="2" ParentId="14567" CreationDate="2022-07-01T13:55:49.233" Score="5" Body="&lt;p&gt;Some of the ancient examples.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Square_root_of_2#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;&lt;/a&gt;. Goes back to Pythagoreans, although they phrased it differently. The &amp;quot;highly desired entity&amp;quot; were two numbers (positive integers) that are to each other as the side of a square to its diagonal. This would have supported the &amp;quot;all is number&amp;quot; thesis that Pythagoreans held, if we believe Aristotle (there are reasonable doubts, see &lt;a href=&quot;https://www.academia.edu/3477804/_All_Is_Number_Basic_Doctrine_of_Pythagoreanism_Reconsidered&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zhmud, All Is Number?&lt;/a&gt;). Stories about the &amp;quot;revolution&amp;quot; the negative result caused are likely exaggerated, at best, see &lt;a href=&quot;https://www.maa.org/press/maa-reviews/the-mathematics-of-platos-academy-a-new-reconstruction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fowler, Mathematics of Plato's Academy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Infinitude of primes&lt;/a&gt;. The &amp;quot;entity&amp;quot; was the largest prime. Probably also goes back to Pythagoreans, although many authors assert that Euclid came up with it on his own. There is no evidence either way. But he thought it important enough to include into &lt;em&gt;Elements&lt;/em&gt; even though the proposition is not used anywhere later.&lt;/p&gt;&#xA;&lt;p&gt;Archimedes, Apollonius, etc., routinely derived &amp;quot;symptoms&amp;quot; in construction problems, violation of which implied that the problem has no solution. For example, in On Sphere and Cylinder &lt;a href=&quot;https://zoboko.com/text/x5rw9p99/the-works-of-archimedes/32&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes considered the problem of cutting a sphere by a plane&lt;/a&gt; to divide its volume in a given ratio, reduced it to a cubic equation (in modern algebraic translation, he equated ratios), and derived conditions for those to have a solution.&lt;/p&gt;&#xA;&lt;p&gt;Many more examples are discussed in Crippa's dissertation &lt;a href=&quot;https://hal.archives-ouvertes.fr/tel-01098493&quot; rel=&quot;nofollow noreferrer&quot;&gt;Impossibility results: from geometry to analysis&lt;/a&gt;, which is a comprehensive survey of early history, from antiquity to 17th century. He distinguishes between &amp;quot;absolute&amp;quot; impossibilities, such as the ones above, and impossibilities conditional on the tools employed, such as impossibility of duplicating a cube, trisecting an angle or squaring a circle with ruler and compass.&lt;/p&gt;&#xA;&lt;p&gt;I am not sure the distinction is very crisp. The impossibility of squaring a circle followed by proving transcendentality of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, which is harder, but not conceptually different from proving irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;. But, in any case, those are closer to what Abel and Galois did, and were not proved until 19th century despite widespread belief since antiquity, and (flawed) earlier attempts starting with Descartes and Gregory, see &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/full/10.1111/j.1600-0498.2009.00160.x&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lutzen, The Algebra of Geometric Impossibility&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-07-02T00:15:13.943" LastActivityDate="2022-07-02T00:15:13.943" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14571" PostTypeId="2" ParentId="48" CreationDate="2022-07-02T19:48:03.977" Score="3" Body="&lt;p&gt;Gordon Fraser and I proposed the name &amp;quot;higgson&amp;quot; in the July 2012 issue of &amp;quot;Physics World,&amp;quot; just as the boson was being discovered. We were not aware of Gell-Mann's prior advocacy of the word, but it figures. Murray definitely had &amp;quot;the naming gift&amp;quot;. The Eightfold Way, quarks, quantum chromodynamics, etc.&lt;/p&gt;&#xA;&lt;p&gt;— Michael Riordan, author, &amp;quot;The Hunting of the Quark.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="16846" LastEditorUserId="4251" LastEditDate="2022-07-02T19:49:38.520" LastActivityDate="2022-07-02T19:49:38.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14572" PostTypeId="1" AcceptedAnswerId="14573" CreationDate="2022-07-03T09:50:21.587" Score="18" ViewCount="6426" Body="&lt;p&gt;I've been told that Pythagoreans kept as a secret the incommensurability of certain quantities like the diagonal of the square. Are there other mathematical discoveries whose authors didn't want the public to learn or know about?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastActivityDate="2022-07-12T09:50:20.573" Title="Secrecy in Mathematics" Tags="|mathematics|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14573" PostTypeId="2" ParentId="14572" CreationDate="2022-07-03T11:39:45.300" Score="34" Body="&lt;p&gt;In antiquity Archimedes kept his &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;method of mechanical theorems&lt;/a&gt; about areas and volumes secret because, under Plato's influence, using lowly mechanics in glorious geometry was frowned upon at the time. The method is revealed in his letter to Eratosthenes, &amp;quot;&lt;em&gt;for some things that first became clear to me by means of mechanics were afterwards demonstrated by means of geometry&lt;/em&gt;&amp;quot;. In &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Archimedes/&quot; rel=&quot;nofollow noreferrer&quot;&gt;On Spirals&lt;/a&gt; he mentions sending false &amp;quot;theorems&amp;quot; to Alexandrians to prevent them from passing his as their own, &amp;quot;&lt;em&gt;so that those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In 16th century del Ferro and Tartaglia kept their &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubic_equation#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;solutions to the cubic&lt;/a&gt; secret, also for priority reasons and to win current at the time mathematical duels that conveyed status and money. In fact, Tartaglia rediscovered the solution independently, after learning that del Ferro knew it, and then generalized it, which allowed him to beat Fiore, del Ferro's student, in one of those duels. It was knowing that the solution exists that gave the determination to find it, he wrote.&lt;/p&gt;&#xA;&lt;p&gt;Many 17th century authors would cipher their results to claim priority and prevent others from taking advantage of them before they could. Newton famously sent an &lt;a href=&quot;https://www.mathpages.com/home/kmath414/kmath414.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;anagram to Leibniz&lt;/a&gt;, which expressed the idea of solving differential equations using calculus.&lt;/p&gt;&#xA;&lt;p&gt;With the advancement of publish or perish these sorts of practices went away. Publishing is relatively quick and people usually only hold off until the publication, if that.&lt;/p&gt;&#xA;&lt;p&gt;But other reasons came up. Babbage (the inventor of computers) discovered a statistical method for breaking Vigenere ciphers in 1846. It was kept secret because British intelligence was using it to read ciphered diplomatic correspondence, see &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0136047297&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mr. Babbage's Secret&lt;/a&gt;. The method was rediscovered and published by &lt;a href=&quot;https://en.wikipedia.org/wiki/Kasiski_examination#cite_note-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kasiski in 1863&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The story repeated itself in the 20th century. The first public key cryptosystem was developed by Cocks in 1973, also for British intelligence. It was only declassified in 1997. But already in 1977 Rivest, Shamir and Adleman published an equivalent now named after them, &lt;a href=&quot;https://en.wikipedia.org/wiki/RSA_(cryptosystem)&quot; rel=&quot;nofollow noreferrer&quot;&gt;RSA&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hamming discovered his error-correcting codes in 1948, but did not publish until 1950 because he was waiting for the patent to be approved. By that time, Golay rediscovered them and published, which led to a priority dispute, see &lt;a href=&quot;https://bookstore.ams.org/car-21/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thompson, From Error-Correcting Codes Through Sphere Packings to Simple Groups&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A different type of secrecy emerged in the academia, especially within so-called schools, usually created and led by a single prominent mathematician. It concerns not so much results as intuitions, hunches, promising problems and conjectures, heuristics, informal tricks, etc., that are not so readily sharable in published form. Those, as &lt;a href=&quot;http://mat.net.ua/mat/Arnold-Geom-Metod-Difuri.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arnold put it&lt;/a&gt; &amp;quot;&lt;em&gt;like parables, are only explained to the own disciples in private&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In this vein, there was a bizarre episode involving Kolmogorov. At his seminar in 1960 he explained his so-called &lt;span class=&quot;math-container&quot;&gt;$n^2$&lt;/span&gt; conjecture, then standing for several years. A week later Karatsuba, a young walk-in, disproved it and told him. After reporting the finding Kolmogorov shut down the seminar. Two years later he published the result and credited it to Karatsuba, but without not only his permission but even knowledge, and bundled with results of his own student, see &lt;a href=&quot;https://hsm.stackexchange.com/a/12131/55&quot;&gt;Was Kolmogorov enraged after learning about the Karatsuba multiplication algorithm?&lt;/a&gt; One possible explanation is that Kolmogorov originally misjudged the nature of the work in progress he was sharing. It was more immediately doable than he thought, and, per Arnold's maxim, only to be &amp;quot;&lt;em&gt;explained to the own disciples in private&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;To summarize, secrecy does not work for long, the results are soon rediscovered by others.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-07-09T06:17:06.697" LastActivityDate="2022-07-09T06:17:06.697" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14574" PostTypeId="2" ParentId="14572" CreationDate="2022-07-03T17:33:24.460" Score="9" Body="&lt;p&gt;Sometimes, secret information has been deposited with a trusted organization.&lt;/p&gt;&#xA;&lt;p&gt;Example.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1915, the Académie des Sciences in Paris gave the topic for its 1918 Grand Prix. The prize would be awarded for a study of iteration from a global point of view.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the later half of 1917 Gaston Julia deposited his results in sealed envelopes with the Académie des Sciences.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pierre Fatou, on the other hand, published an announcement of his results in the December 1917 part of &lt;em&gt;Comptes Rendus&lt;/em&gt;.  It later became evident that they had discovered very similar results.  Julia requested that his sealed envelopes be opened: they proved that he had the priority.  As a result, Fatou did not enter for the Grand Prix and it was awarded to Julia.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Whether Julia or Fatou deserves the credit of having priority matters little since their work was certainly totally independent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;reference: &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Fatou/&quot; rel=&quot;noreferrer&quot;&gt;MacTutor&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-07-03T17:33:24.460" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14575" PostTypeId="2" ParentId="14572" CreationDate="2022-07-03T18:04:45.190" Score="-2" Body="&lt;p&gt;Pythagoras discovered the scale on which all music is based.&lt;/p&gt;&#xA;&lt;p&gt;He also discovered that this scale fits into the pentagram, which symbolizes a mathematical law found through out nature, also known as the fibonacci sequence, or the &amp;quot;Golden Section&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Recent artists recaptured this secret, you can find the proportions of the golden ratio when you measure the facial proportions of the mona lisa for example.&lt;/p&gt;&#xA;&lt;p&gt;As for secrecy, it is more about preservation.&lt;/p&gt;&#xA;&lt;p&gt;The moment you define something it becomes a double-edged blade, because that thing is now debatable by a mass public, which may change the meaning of what you defined entirely.&lt;/p&gt;&#xA;&lt;p&gt;True knowledge is pre-symbol.&lt;/p&gt;&#xA;&lt;p&gt;It is not so much a secret as a result of people seeing the symbols they come up with also exist &amp;quot;out there&amp;quot; and from that both of these can conclude they both reached the same conclusions and they both know it.&lt;/p&gt;&#xA;&lt;p&gt;A lot of conspiracy theories and misunderstandings arise out of this, when people think the prevalence of the pyramid with the eye is evidence of an organized group.&lt;/p&gt;&#xA;&lt;p&gt;But that is not how it happens at all, it happens in an entirely opposite way.&lt;/p&gt;&#xA;" OwnerUserId="16850" LastActivityDate="2022-07-03T18:04:45.190" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14578" PostTypeId="1" CreationDate="2022-07-04T15:42:24.750" Score="3" ViewCount="153" Body="&lt;p&gt;In Buhler's biography of Gauss, who made huge contributions to the development of efficient orbital determination methods in his Theoria Motus (1809), the following information is mentioned:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The major omission in Theoria motus is a lack of treatment of parabolic orbits, a task that had earlier been solved by Olbers, and of Gauss's very efficient approximation methods.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dunnington in his biography of Gauss gives more information on this issue:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Of the numerous fundamental works on astronomy planned by Gauss, the Theoria motus is the only one he presented to his contemporaries. It treats exclusively the elliptical and hyperbolic motion in so far as it concerns the determination of an orbit from observations, which leads to the conjecture that he may have postponed further investigations on this subject with the idea that his friend Olbers had successfully solved the problem of parabolic orbit determination. In the Theoria motus he applies methods for determining an hyperbolic orbit similar to those for an elliptic orbit. But he gives in the Theoria motus a numerical example only for the calculation of the elements from two radii vectors, the included angle and the time between them. In a letter of January 3, 1806, he pointed out to Olbers a certain case where the latter's method for a parabolic orbit is not applicable. Later, in 1813, he found a correction for the method and in 1815 busied himself intensively with the parabolic-orbit problem. He not only gave Lambert's equation a different form, but also developed a series of other important relations. The determination of a circular orbit (which Gauss characterized as easy and simple) was given in 1871 by Klinkerfues, his pupil, in a textbook.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In addition, volume 7 of Gauss's werke (this volume is dedicated to his works in theoretical astronomy) contains an entire section devoted to parabolic orbits determination (p.323-374 of this volume), which includes several letters (to Olbers, amongst others) and some fragments from his Nachlass. Therefore, it seems there is an essential issue here.&lt;/p&gt;&#xA;&lt;p&gt;While reading the passages from Gauss's biographies, several questions have arose in mind:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Parabolic orbits are defined as conic sections with eccentricity &lt;span class=&quot;math-container&quot;&gt;$e=1$&lt;/span&gt; (exactly &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;), so the probability that a certain orbital trajectory will have exactly this eccentricity is zero. Therefore I guess it should be read &amp;quot;near-parabolic&amp;quot; orbits. But what is the range of values of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; (range which must include &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;) for which the orbit is considered &amp;quot;near-parabolic&amp;quot;? This is just a technical question to make sure I understand the subject.&lt;/li&gt;&#xA;&lt;li&gt;What is essentially different in the case of parabolic orbits that compelled mathematical astronomers to develop a separate method for it? I read that one of Gauss's major innovations in his Theoria motus was a general method of orbital determination that didn't make any a priori assumptions on the type of orbit determined. Therefore, I'd like to know why is the parabolic case an exception.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt; that I'm really just looking for a qualitative explanation why there was a need to develope a separate method for the parabolic case, so there is no need to dwell into the complicated mathematical details of Gauss's method for parabolic orbits determination.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2022-07-05T15:53:28.297" LastActivityDate="2022-07-05T16:39:18.113" Title="In orbital determination of celestial bodies, why did the case of parabolic or nearly-parabolic orbits require separate development?" Tags="|astronomy|gauss|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14581" PostTypeId="2" ParentId="14578" CreationDate="2022-07-05T16:39:18.113" Score="5" Body="&lt;p&gt;The core of the matter is that many of the calculations related to elliptical orbital motion depend on measures and quantities that are referred in some way to an entire cycle around the orbit.  For a parabolic orbit, with eccentricity 1, there clearly is no entire cycle:  an object moving on such an orbit never returns to its pericenter. The measures that depend for their definition on an entire cycle break down.&lt;/p&gt;&#xA;&lt;p&gt;Also, on nearly-parabolic orbits, many of the usual elliptical quantities become numerically less and less well determined, the closer the eccentricity approaches to 1. So the usual forms of elliptical calculation approach breakdown. The problem can be described as a singularity in the elliptical relationships where the eccentricity reaches 1 and the anomalies reach zero.&lt;/p&gt;&#xA;&lt;p&gt;As an example of how this works, the pericenter distance &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; on an elliptical orbit can be expressed as &lt;span class=&quot;math-container&quot;&gt;$a.(1-e)$&lt;/span&gt; , where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the semi-major axis and &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; the eccentricity.  If one considers eccentricities nearer and nearer to 1, &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; becomes larger without limit, and &lt;span class=&quot;math-container&quot;&gt;$(1-e)$&lt;/span&gt; approaches zero. But all such physical quantities are measured within some non-zero tolerance of error, however small. So whatever the degree of accuracy of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, there will be some level at which the error in &lt;span class=&quot;math-container&quot;&gt;$(1-e)$&lt;/span&gt; will be at least as large as &lt;span class=&quot;math-container&quot;&gt;$(1-e)$&lt;/span&gt; itself. The calculation loses definition, effectively lost within its own error.&lt;/p&gt;&#xA;&lt;p&gt;A similar tendency towards indeterminacy at high values of eccentricity is found in two out of the three usual measures of longitudinal motion in an ellipse: the mean anomaly and the eccentric anomaly. Physically, the eccentric anomaly is defined in terms of the geometrical center of the ellipse. As one considers increasing eccentricities and increasing semiaxes, the location of the center of the major axis becomes less and less well determined.&lt;/p&gt;&#xA;&lt;p&gt;Again, physically, the mean anomaly is defined by the proportion (relative to the whole ellipse area) represented by the area of the focal sector bounded by the radius-vector to the pericenter and the radius-vector to the current position of the orbiting body. As one considers increasing eccentricities and increasing semiaxes, the whole ellipse area becomes less and less well determined as the eccentricity becomes larger. From another point of view, all the mean anomalies for any given true anomaly collapse towards zero (on the usual elliptical basis of calculation) as the eccentricity approaches 1.&lt;/p&gt;&#xA;&lt;p&gt;This is why a different basis had to be found for the parabolic calculations: the mean motion, instead of being based on the orbital period (non-existent for the parabola) had to be put on a different conventional basis. It was also found that the relation between time and true anomaly could be defined for the parabolic motion more simply than the equations for elliptical motion, on the basis of a cubic equation now often known as &lt;em&gt;Barker's equation&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A good source for the nature of the orbital calculations and problems in elliptical, parabolic and hyperbolic motion is &lt;a href=&quot;https://books.google.com/books/about/Orbital_Motion.html?id=Hzv7k2vH6PgC&quot; rel=&quot;noreferrer&quot;&gt;A E Roy's &amp;quot;&lt;em&gt;Orbital Motion&lt;/em&gt;&amp;quot;&lt;/a&gt;. Illustrations of the problems that arise and increase as the eccentricity approaches 1 can also be found for example in R Serafin (1996) &lt;a href=&quot;https://ui.adsabs.harvard.edu/abs/1996CeMDA..65..389S&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;On Solving Kepler's Equation for Nearly Parabolic Orbits&amp;quot;&lt;/a&gt; and its references.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-07-05T16:39:18.113" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14582" PostTypeId="1" CreationDate="2022-07-05T22:04:01.860" Score="1" ViewCount="171" Body="&lt;p&gt;Are the claims of Sir Edmund Whittaker regarding David Hilbert's derivation of the general relativity theory predated that of Einstein by five days?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/General_relativity_priority_dispute#cite_note-Whi53-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;This&lt;/a&gt; Wikipedia article that I found contains the following statement:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was noted by Sir Edmund Whittaker in his 1954 book that David Hilbert had derived the theory of general relativity from an elegant variational principle almost simultaneously with Einstein's discovery of the theory.[B1] Hilbert's derivation of the theory predated that of Einstein by five days.[B2]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[B1] Whittaker, E. T (1953) A History of the Theories of Aether and Electricity: Vol 2 The Modern Theories 1900-1926. Chapter II: The Relativity Theory of Poincaré and Lorentz, Nelson, London.&lt;/p&gt;&#xA;&lt;p&gt;[B2] Kip Thorne (1994): Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton &amp;amp; Company; Reprint edition (January 1995). ISBN 0-393-31276-3&lt;/p&gt;&#xA;&lt;p&gt;I am not sure if Whittaker's claim is valid or not.&lt;/p&gt;&#xA;" OwnerUserId="16873" LastEditorUserId="14043" LastEditDate="2022-07-08T06:02:13.117" LastActivityDate="2022-07-08T06:02:13.117" Title="Is Edmund Whittaker claim that Hilbert proved General Relativity days before Einstein valid?" Tags="|relativity-theory|einstein|hilbert|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14583" PostTypeId="1" CreationDate="2022-07-06T04:59:17.983" Score="3" ViewCount="122" Body="&lt;p&gt;&lt;a href=&quot;https://youtu.be/IBlCu1zgD4Y?t=659&quot; rel=&quot;nofollow noreferrer&quot;&gt;In this lecture&lt;/a&gt; by Prof. Frederic Schuller @ 17:49 , it is said that Laplace asked a question if force could be seen equally as curvature of the underlying space which the particle moves in. However, I can't find any reference where I can't seem to find any document stating that Laplace actually stated this. Could I get help finding a reference? Thanks.&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2022-07-06T04:59:17.983" Title="References for Laplace's question regarding Newtonian Gravity" Tags="|physics|relativity-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14584" PostTypeId="1" AcceptedAnswerId="14586" CreationDate="2022-07-06T12:36:42.570" Score="2" ViewCount="129" Body="&lt;p&gt;My question is simple: why do we use &lt;span class=&quot;math-container&quot;&gt;$T_{\mu\nu}$&lt;/span&gt; to denote the stress energy momentum tensor, and when was the concept of the stress energy tensor first (or roughly the first) introduced (and by whom)?&lt;/p&gt;&#xA;&lt;p&gt;Note: I'm well aware that such trivial notation differences make no difference to the actual underlying concepts. Rather I'm just wondering why the specific use of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;? For instance, is it the case that the letter &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is the initial letter (in some other language) of the words &lt;em&gt;stress&lt;/em&gt;&#xA;or &lt;em&gt;energy&lt;/em&gt; or &lt;em&gt;momentum&lt;/em&gt; (for example, I know that we use &lt;span class=&quot;math-container&quot;&gt;$\Bbb{Z}$&lt;/span&gt; for integers because it stems from the German word &lt;em&gt;Zahl&lt;/em&gt; for &lt;em&gt;number&lt;/em&gt;)? I'm just curious because if it were me back in the day, I might have used the notation &lt;span class=&quot;math-container&quot;&gt;$P_{\mu\nu}$&lt;/span&gt;, because lower case &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is already standard notation for momentum in classical mechanics, so if I were to create a tensor for stress-energy-&lt;strong&gt;momentum&lt;/strong&gt;, I might consider &lt;span class=&quot;math-container&quot;&gt;$P_{\mu\nu}$&lt;/span&gt; as a reasonable notation.&lt;/p&gt;&#xA;" OwnerUserId="16878" LastEditorUserId="10106" LastEditDate="2022-07-06T12:58:52.987" LastActivityDate="2022-07-06T17:13:20.763" Title="Why is $T_{\mu\nu}$ the Standard Notation for the Stress-Energy-Momentum Tensor" Tags="|physics|terminology|notation|mathematical-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14585" PostTypeId="1" CreationDate="2022-07-06T13:01:47.237" Score="2" ViewCount="248" Body="&lt;p&gt;I was having a discussion with a physicist about Many Worlds Interpretation and he told me that von Neumann like the idea of having multiple worlds in quantum mechanics. When I asked him about more details, he recalled a story he heard about Richard Feynman rushing to von Neumann's office at Princeton referring to a book that von Neumann published where at some point talked about a non-unitary evolution of quantum mechanics. Feynman asked him &amp;quot;&lt;em&gt;what is this nonsense&lt;/em&gt;&amp;quot; and von Neumann replied &amp;quot;&lt;em&gt;the postulate is just there until we finally understand what is happening in the measurement process&lt;/em&gt;&amp;quot;. The physicist then told me that if the story is true then von Neumann must have realized that non-unitary time evolution cannot occur, and if that cannot happen, then Many Worlds must be true.&lt;/p&gt;&#xA;&lt;p&gt;Even if von Neumann said that, is it necessarily true that he would have supported the Many Worlds Interpretation?&lt;/p&gt;&#xA;" OwnerUserId="10956" LastEditorUserId="10956" LastEditDate="2022-07-07T19:14:20.973" LastActivityDate="2022-07-13T01:32:55.437" Title="Did John von Neumann make any comments about the Many Worlds Interpretation of Hugh Everett?" Tags="|physics|quantum-mechanics|von-neumann|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14586" PostTypeId="2" ParentId="14584" CreationDate="2022-07-06T14:35:14.397" Score="3" Body="&lt;p&gt;Because it is a &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;ensor. People do not deliberate much on choosing letters, as one can see from &lt;a href=&quot;https://hsm.stackexchange.com/q/14551/55&quot;&gt;Gauss's &lt;span class=&quot;math-container&quot;&gt;$E,F,G$&lt;/span&gt;&lt;/a&gt;, among other examples. &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; happened to be the first tensor that came up in Sommerfeld's &lt;a href=&quot;https://zenodo.org/record/1424173#.YsWcG99Onv8&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Theory of Relativity I: Four-dimensional Vector Algebra, p. 767ff&lt;/a&gt;, after he dealt with &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;- and &lt;span class=&quot;math-container&quot;&gt;$6$&lt;/span&gt;-vectors and then scalar and vector products. It was the turn of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;ensor product. Minkowski and Abraham before that used different letters. Von Laue after followed Sommerfeld in his 1911 textbook on relativity (the very first one), and so it stuck, see &lt;a href=&quot;https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Stress-energy_tensor_(electromagnetic)#cite_note-8&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Topics in Special Relativity/Stress-energy tensor&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Btw, &lt;span class=&quot;math-container&quot;&gt;$T_{\mu\nu}$&lt;/span&gt; is typically just &amp;quot;stress-energy tensor&amp;quot;, despite the momentum components in it. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Stress%E2%80%93energy%E2%80%93momentum_pseudotensor&quot; rel=&quot;nofollow noreferrer&quot;&gt;stress–energy–momentum pseudotensor&lt;/a&gt; is a modification by Landau and Lifshitz, typically denoted &lt;span class=&quot;math-container&quot;&gt;$t^{\mu\nu}_{LL}$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-07-06T17:13:20.763" LastActivityDate="2022-07-06T17:13:20.763" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14587" PostTypeId="1" CreationDate="2022-07-07T14:42:51.010" Score="2" ViewCount="93" Body="&lt;p&gt;Nowadays is a well known high school or undergrad lab experiment to determine the specific electron charge by using a Wien-filter setup as described for example &lt;a href=&quot;https://www.ld-didactic.de/documents/en-US/EXP/P/P3/P3852_e.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Up to now I thought that this experiment dates back to Thomson, but reading his &lt;a href=&quot;https://www.douglas-scott-mcgregor.com/uploads/1/3/3/5/133545016/thomson1897.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper about cathode rays&lt;/a&gt; I noticed that the experiment described in this paper is not exactly the same (see bottom of page 307 and following).&lt;/p&gt;&#xA;&lt;p&gt;Thomson seemed to measure the electric and magnetic deflection in two steps independently whereas in the version of the linked Leybold manual above applies both, a magnetic and an electric field at the same time and choose values such that the electric force and the Lorentzforce balance (i.e. such that the beam describes a straight line parallel to the initial velocity vector instead of being deflected).&lt;/p&gt;&#xA;&lt;p&gt;Did Thomson also do this variant of the experiment (as described in the Leybold manual) and where did he publish it?&lt;/p&gt;&#xA;&lt;p&gt;If you look at the linked Leybold manual you may notice that the plate distance is pretty large such that the electric field is not uniform. Nevertheless they do their calculations assuming uniformity and introduce a &amp;quot;correction factor&amp;quot; to take the nonuniformity into account. This procedure wouldn't meet standards for a scientific publication. For example you could make the plate distance much smaller compared to the plate dimensions or seriously take the nonuniformity into account.&lt;/p&gt;&#xA;&lt;p&gt;So, what is the history of this variant of the experiment? Do you have references to historical scientific papers describing exactly this variant of the experiment?&lt;/p&gt;&#xA;" OwnerUserId="2380" LastActivityDate="2022-07-12T15:05:05.623" Title="History of determining the specific charge of the electron by balacing electric and magnetic forces" Tags="|physics|reference-request|experimental-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14588" PostTypeId="2" ParentId="149" CreationDate="2022-07-07T17:22:07.467" Score="0" Body="&lt;p&gt;(Source: Pierre Duhem, 1908, &lt;em&gt;To Save The Phenomena&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;p&gt;According to Duhem, Ptolemy’s model was the first to fit observation - i.e., to “save the phenomena” of planetary motion. What Ptolemy did that previous philosophers had failed to do was to combine epicycles and eccentrics - each planet being borne of an epicycle whose centre described a circle eccentric to the world.&lt;/p&gt;&#xA;&lt;p&gt;The reason that Ptolemy was the first to combine the two appears to be that Hipparchus proved that his model could be formulated using either epicycles or eccentrics.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What Hipparchus proved was that the course of the sun can be represented either by supposing that this star describes a circle eccentric to the world, or by letting it be carried by an epicycle, provided the revolution of this epicycle is achieved in exactly the same time in which its centre has completed a circle concentric with the world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Having demonstrated the equivalence of the eccentric and epicyclic hypotheses, subsequent philosophers used either one or the other, but not both, thus failing to save the phenomena.  Only Ptolemy combined the two to save the phenomena.&lt;/p&gt;&#xA;&lt;p&gt;Although Hipparchus did not extend his model to include the planets, &lt;a href=&quot;https://en.wikipedia.org/wiki/Theon_of_Smyrna&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theon of Smyrna&lt;/a&gt;, in his &lt;em&gt;Astronomia&lt;/em&gt;, wrote that &lt;a href=&quot;https://en.wikipedia.org/wiki/Adrastus_of_Aphrodisias&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adrastus of Aphrodisias&lt;/a&gt; said Hipparchus held that his method could be applied to each planet.&lt;/p&gt;&#xA;&lt;p&gt;So who tried using epicycles between Hipparchus and Ptolemy?&lt;/p&gt;&#xA;&lt;p&gt;By the time of Hipparchus astronomy had come to be dominated by the peripatetics - i.e., the followers of Aristotle’s physics - and they weren’t having any of this epicycle nonsense.  For the Peripatetics, epicycles/eccentrics were contrary to natural principles.&lt;/p&gt;&#xA;&lt;p&gt;Prior to Aristotle, Plato had declared astronomy to be the domain of the mathematician and had appealed to mathematicians to formulate a purely mathematical astronomy which “saved the phenomena” of planetary motion.  Plato’s attitude was that the human mind was too feeble to comprehend the true nature of the superlunary spheres and therefore we must be content to have a mathematical theory which at least allowed us to make predictions that fit observation regardless of “truth”.&lt;/p&gt;&#xA;&lt;p&gt;According to Duhem, almost all of what we know of the period between Hipparchus and Ptolemy consists of the peripatetic criticisms of epicycles/eccentrics.  However, Duhem does cite the example of Adrastus of Aphrodisias who lived shortly before Ptolemy and proposed a  model using epicycles.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;According to the testimony of Theon of Smyrna, Adrastus of Aphrodisias ascribed to each planet an orbital shell contained by two spherical surfaces concentric with the universe.  Within the shell is a full sphere occupying its entire thickness.  The planet is then set into this full sphere.  The orbital shell carries the full sphere in its revolution around the centre of the world while the full sphere turns on its axis. By means of this mechanism the planet describes an epicycle whose centre traverses a circle concentric with the world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Theon is said to have constructed an orrery based on the model of Adrastus.&lt;/p&gt;&#xA;&lt;p&gt;Duhem cites no other example of a philosopher using epicycles between Hipparchus and Ptolemy.  Curiously, both Adrastus and Theon were peripatetics who justified the use of epicycles by arguing that they were the “accidental” consequence of the combination of the revolutionary and rotational movements of the planetary spheres.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-07-07T17:22:07.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14589" PostTypeId="2" ParentId="14017" CreationDate="2022-07-07T23:18:21.927" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Napoleon_points&quot; rel=&quot;nofollow noreferrer&quot;&gt;Napoleon points&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief.&lt;a href=&quot;https://en.wikipedia.org/wiki/Napoleon_points&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.&lt;/p&gt;&#xA;&lt;p&gt;The name &amp;quot;Napoleon points&amp;quot; has also been applied to a different pair of triangle centers, better known as the isodynamic points.[2]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3338" LastActivityDate="2022-07-07T23:18:21.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14591" PostTypeId="1" CreationDate="2022-07-08T11:09:47.163" Score="3" ViewCount="121" Body="&lt;p&gt;I'm interested in the development of mathematical definitions for the sake of engineering, and what makes a particular definition better suited for a problem than another in any particular context.&lt;/p&gt;&#xA;&lt;p&gt;Ideally, I aim to compile a list of examples of definitions, their historical context and a list of inadequate definitions which were proposed, along with a rationale of what was missing from the inadequate definition.&lt;/p&gt;&#xA;&lt;p&gt;A concrete example is Bell Labs being contracted to create secure military communications, and Claude Shannon developing his theory of mutual information as a result. A failed attempt would be Karl Pearson's suggestion of mutual information as P(x,y)^2.&lt;/p&gt;&#xA;&lt;p&gt;Here's a blog post that goes a little bit into that history, and points at the kind of comparitive research I'm interested in:&#xA;&lt;a href=&quot;https://www.lesswrong.com/posts/GhFoAxG49RXFzze5Y/what-s-so-bad-about-ad-hoc-mathematical-definitions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.lesswrong.com/posts/GhFoAxG49RXFzze5Y/what-s-so-bad-about-ad-hoc-mathematical-definitions&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16890" LastActivityDate="2022-07-08T13:29:31.190" Title="Examples of mathematical definitions motivated by engineering problems" Tags="|mathematics-social-history|engineering|definition|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14592" PostTypeId="2" ParentId="14591" CreationDate="2022-07-08T13:29:31.190" Score="6" Body="&lt;p&gt;Stable polynomials. A polynomial is called stable if its zeros lie in the left half-plane. This is a substantial research topic for the last 150 years. The problem arose as a pure engineering problem of stabilizing various automatically controlled machines (First Airy was investigating stabilization of his telescopes, then Stodola investigated stabilization of steam engines etc.) Important contributions were made in 19 century by Maxwell, then Routh and Hurwitz, but research on stable polynomials continues to this day.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-07-08T13:29:31.190" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14593" PostTypeId="1" AcceptedAnswerId="14595" CreationDate="2022-07-08T19:57:29.980" Score="1" ViewCount="116" Body="&lt;p&gt;I read Franklin started the positive negative naming convention of electricity.&lt;/p&gt;&#xA;&lt;p&gt;Now early scientists had to pick a direction for the battery and voltmeter. I don't understand how this was done.&#xA;&lt;em&gt;If you allow me to add:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Edison's thermionic demonstration in 1883 shows the direction of flow from negative to positive which means he know the electricity is flowing from negative to positive even though he is using Franklin's naming convention in the battery that he hooked up to his apparatus. ( Well actually it was a schematic, I don't really have an original diagram.)&lt;/p&gt;&#xA;&lt;p&gt;I must be not understanding some basic science because when the &amp;quot;true&amp;quot; direction was found why couldn't it have just as easily been called positive to negative to keep with Franklin's original naming convention?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastEditorUserId="5040" LastEditDate="2022-07-08T20:11:19.707" LastActivityDate="2022-07-09T00:43:17.867" Title="How was the direction of current flow determined in the first battery or the first voltmeter?" Tags="|electricity|scientists|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14594" PostTypeId="1" AcceptedAnswerId="14596" CreationDate="2022-07-08T21:50:35.843" Score="15" ViewCount="2017" Body="&lt;p&gt;There is family of &lt;a href=&quot;https://en.wikipedia.org/wiki/Thompson_groups&quot; rel=&quot;nofollow noreferrer&quot;&gt;famous groups&lt;/a&gt; with unusual group-theoretic properties due to a mathematician called Richard Thompson that are widely studied in group theory. The papers on these groups and the Wikipedia entry simply say that these were written down in some unpublished hand-written notes of Thompson in the 1960's. A search on Google / Wikipedia / MathSciNet has not revealed anything about who Richard Thompson was, except a possible implication that Thompson was motivated by questions coming from logic. Does anyone have any information?&lt;/p&gt;&#xA;" OwnerUserId="16892" LastEditorUserId="4251" LastEditDate="2022-07-12T08:11:55.890" LastActivityDate="2023-05-19T15:55:23.013" Title="Who was Richard Thompson?" Tags="|mathematicians|biographical-details|group-theory|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14595" PostTypeId="2" ParentId="14593" CreationDate="2022-07-09T00:43:17.867" Score="3" Body="&lt;p&gt;Too much to change.&lt;/p&gt;&#xA;&lt;p&gt;Every single book, paper, instruction manual, or anything else that referred to the poles as negative and positive would have been switched.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, anything old, if not destroyed, would potentially create confusion, and if any sizable group of people had declared that they wouldn't bother, you would have had new works, continually put out, that used the old convention, and the discrepancy would have created still more confusion.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2022-07-09T00:43:17.867" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14596" PostTypeId="2" ParentId="14594" CreationDate="2022-07-09T08:47:45.533" Score="21" Body="&lt;p&gt;I was only able to find fragmentary information on Thompson. A good starting point is his PhD thesis, which gives us his full name:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Richard Joseph Thompson, &lt;em&gt;Transformational Structure of Algebraic Logics.&lt;/em&gt; PhD thesis, University of California, Berkeley, 1979&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;A &lt;a href=&quot;https://www.proquest.com/openview/1c9f564b07f641fdc87ff1c96ca06ed6/1?pq-origsite=gscholar&amp;amp;cbl=18750&amp;amp;diss=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;partial scan&lt;/a&gt; of the thesis is publicly available free of cost. From this we learn that Thompson received an A.B. degree from &lt;a href=&quot;https://en.wikipedia.org/wiki/San_Jose_State_University&quot; rel=&quot;nofollow noreferrer&quot;&gt;San Jose State&lt;/a&gt; in 1965, that he was accepted into the PhD program at Berkeley in 1973, and that his degree was conferred on June 16, 1979. His thesis advisor was &lt;a href=&quot;https://en.wikipedia.org/wiki/Ralph_McKenzie&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ralph McKenzie&lt;/a&gt;. This agrees with the &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=12312&quot; rel=&quot;nofollow noreferrer&quot;&gt;entry for McKenzie&lt;/a&gt; at the Mathematics Genealogy Project. Curiously, there is no mention of R. J. Thompson in the &lt;em&gt;La Torre&lt;/em&gt; college yearbooks of San Jose State from the years &lt;a href=&quot;http://digitalcollections.sjlibrary.org/cdm/compoundobject/collection/sjsutorre/id/12611/rec/2&quot; rel=&quot;nofollow noreferrer&quot;&gt;1964&lt;/a&gt; and &lt;a href=&quot;http://digitalcollections.sjlibrary.org/cdm/compoundobject/collection/sjsutorre/id/13027/rec/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;1965&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Thompson's handwritten notes are inconsistently referenced in the literature, with some dating them to 1965 (e.g. David Matthew Robertson, &lt;em&gt;Conjugacy and centralisers&#xA;in Thompson’s group &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;&lt;/em&gt;, PhD thesis, Newcastle University, 2018), while others date them to 1973 (e.g. John Donnelly, &amp;quot;Ruinous subsets of Richard Thompson’s group &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;,&amp;quot; &lt;em&gt;Journal of Pure and Applied Algebra&lt;/em&gt; 208 (2007) 733–737). In many instances the notes are referred to as &amp;quot;widely circulated&amp;quot;. What are claimed to be scans of Thompson's original hand-written notes can be found here: &lt;a href=&quot;http://people.math.binghamton.edu/matt/thompson/thompson-hd_001.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;part 1&lt;/a&gt;, &lt;a href=&quot;http://people.math.binghamton.edu/matt/thompson/thompson-hd_008.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;part 2&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The first proper publication of the material appears to be in R. McKenzie and R. J. Thompson, &amp;quot;An elementary construction of unsolvable word problems in group theory&amp;quot;, in &lt;em&gt;Word Problems,&lt;/em&gt; Proc. Conf. Irvine 1969 (edited by W. W. Boone, F. B. Cannonito, and R. C. Lyndon), Studies in Logic and the Foundations of Mathematics, vol. 71, North-Holland, Amsterdam, 1973, pp. 457–478.&lt;/p&gt;&#xA;&lt;p&gt;There are only a few other papers by R. J. Thompson, some of which appeared in somewhat obscure Eastern European publications. Thompson did a brief stint  at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Institute_for_Advanced_Study&quot; rel=&quot;nofollow noreferrer&quot;&gt;Institute for Advanced Studies&lt;/a&gt; in Princeton in &lt;a href=&quot;https://www.ias.edu/scholars/richard-j-thompson&quot; rel=&quot;nofollow noreferrer&quot;&gt;1974-1975&lt;/a&gt;. Membership lists of the Association for Symbolic Logic published in &lt;em&gt;The Journal of Symbolic Logic&lt;/em&gt;, Vol. 45, No. 4, Dec. 1980 and &lt;em&gt;The Journal of Symbolic Logic&lt;/em&gt;, Vol. 47, No. 4, Dec. 1982 show him as a member of that organization. Publications in 1988 and 1993 indicate that Thompson was affiliated with the Mathematical Institute of the Hungarian Academy of Sciences, Budapest:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Richard J. Thompson, &amp;quot;Noncommutative cylindric algebras and relativizations of cylindrical algebras&amp;quot;, &lt;em&gt;Bulletin of the Section of Logic&lt;/em&gt; 17 (2):75-81 (1988) (&lt;a href=&quot;http://www.filozof.uni.lodz.pl/bulletin/pdf/17_2_5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;H. Andréka and R. J. Thompson, &amp;quot;A Stone type representation theorem for algebras of relations of higher rank&amp;quot;, &lt;em&gt;Transactions of the American Mathematical Society&lt;/em&gt;, Vol. 309, No. 2, October 1988, pp. 671-682 (&lt;a href=&quot;https://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961607-5%2FS0002-9947-1988-0961607-5.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Richard J. Thompson, &amp;quot;Complete description of substitutions in cylindric algebras and other algebraic methods.&amp;quot; &lt;em&gt;Algebraic Methods in Logic and in Computer Science&lt;/em&gt;, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Vol. 28, 1993, pp. 327-342. (&lt;a href=&quot;http://matwbn.icm.edu.pl/ksiazki/bcp/bcp28/bcp28122.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The last professional trace of R. J. Thompson coincides with the last known year of publication, when he was listed as a participant, but not presenter, at a workshop/conference: Universal Algebra and Category Theory Conference, Berkeley, July 12-23, 1993.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4251" LastEditDate="2022-07-12T08:10:32.600" LastActivityDate="2022-07-12T08:10:32.600" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14597" PostTypeId="2" ParentId="14594" CreationDate="2022-07-10T04:49:57.257" Score="11" Body="&lt;p&gt;To add to njuffa's answer.&lt;/p&gt;&#xA;&lt;p&gt;Thompson contributed to group theory, and algebraic logic.  The original motivation for his calculations/constructs was algebraic logic.  His thesis and all but two of his papers were on sets of laws that characterized certain types of algebraic logics with an eye to finding such sets that are finite.&lt;/p&gt;&#xA;&lt;p&gt;The first setting was a system of expression manipulations (duplication or elimination of a symbol, shifting parentheses, permuting symbols) which gave a monoid whose group of units was the group now known as V.  He proved the group infinite and finitely presented and when advised to find normal subgroups was able to show there are none.  This is the subject of his handwritten notes and is his most well known result in group theory.  In those notes Thompson establishes the basics of the three groups now known as F, T, and V.  The first part of the expository paper of Cannon, Floyd and Parry on the Thompson groups is a detailed expansion of those notes.&lt;/p&gt;&#xA;&lt;p&gt;His two papers not in algebraic logic are the one mentioned by njuffa and one paper by himself extending an embedding theorem of Boone and Higman.&lt;/p&gt;&#xA;&lt;p&gt;Embeddings into finitely generated simple groups which preserve the word problem, in &lt;em&gt;Word Problems, II,&lt;/em&gt; (S. I. Adian, W. W. Boone, G. Higman, eds.), North-Holland (1980) 401-441.&lt;/p&gt;&#xA;&lt;p&gt;That last paper is also quite well known.  Both papers in group theory contain pieces of the basic information about the groups F and V.&lt;/p&gt;&#xA;&lt;p&gt;His work in algebraic logic is highly regarded by the algebraic logic community.  For most of his life Thompson worked through an agency for K-12 students with reading and math difficulties in the bay area around San Jose, CA.&lt;/p&gt;&#xA;&lt;p&gt;By the way, the scanned notes in njuffa's response contains a common, often propogated error.  The third page of Part 1 (it starts with &lt;span class=&quot;math-container&quot;&gt;$C_n^m$&lt;/span&gt;) should actually be the fifth page.  No other rearrangement is necessary.&lt;/p&gt;&#xA;&lt;p&gt;Richard Thompson is alive and retired in California.&lt;/p&gt;&#xA;" OwnerUserId="16897" LastEditorUserId="16897" LastEditDate="2022-07-10T13:19:54.230" LastActivityDate="2022-07-10T13:19:54.230" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14598" PostTypeId="1" CreationDate="2022-07-11T15:01:01.107" Score="0" ViewCount="369" Body="&lt;p&gt;What was the dosage of amphetamines that Erdos was taking?  Was it a constant dose or did it increase over time?  Had he ever been diagnosed as suffering from ADD?&lt;/p&gt;&#xA;" OwnerUserId="16907" LastActivityDate="2022-07-11T15:01:01.107" Title="Erdos use of amphetamines to stimulate his creativity" Tags="|mathematics-social-history|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14600" PostTypeId="1" CreationDate="2022-07-11T19:35:56.307" Score="2" ViewCount="40" Body="&lt;p&gt;I can't find the exact date when the Archean began to mean &amp;quot;the middle of the Precambrian&amp;quot;, as opposed to the beginning or the whole thing.  It is some time after 1925 and before 1972.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Archean&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia's article on the Archean&lt;/a&gt; states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Before the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hadean&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hadean&lt;/a&gt; Eon was recognized, the Archean spanned Earth's early history from its formation about 4,540 million years ago until 2,500 million years ago.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The Hadean was identified in 1972.&lt;/p&gt;&#xA;&lt;p&gt;So I see a contradiction, because a 1969 World Book Encyclopedia article divided the Precambrian into the &lt;em&gt;Azoic&lt;/em&gt;, &lt;em&gt;Archaeozoic&lt;/em&gt;, and &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proterozoic&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proterozoic&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;All three of these terms were once synonymous, different geologists' ways of referring to the entire Precambrian. But some time in the intervening years, they came to mean different time periods. (As it stands now, &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Azoic_Age&quot; rel=&quot;nofollow noreferrer&quot;&gt;Azoic&lt;/a&gt;&amp;quot; is considered an obsolete synonym for Hadean; likewise &amp;quot;Archaeozoic&amp;quot; for &amp;quot;Archean&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;My reference for the state of affairs in 1925 is the USGS publication at &lt;a href=&quot;https://pubs.usgs.gov/bul/0769/report.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pubs.usgs.gov/bul/0769/report.pdf&lt;/a&gt; but I will only include paraphrases.&lt;/p&gt;&#xA;&lt;p&gt;Dana (1872) originally coined &amp;quot;Archean&amp;quot; to refer to the entire Precambrian, replacing the older &amp;quot;Azoic&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Proterozoic is less problematic,&#xA;because Emmons(1888), and Chamberlain and Salisbury (1906) proposed a definition of Proterozoic that roughly corresponded to its modern meaning, the time between the Archaen and the Paleozoic.&lt;/p&gt;&#xA;&lt;p&gt;However, it seems that for a time, the USGS went along with a 1908 proposal by Van Hise that &amp;quot;Proterozoic&amp;quot; refer to the entire Precambrian. This seems to be a debate the older definition won.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: As I am challenging an assertion made in Wikipedia, sources other than Wikipedia are preferred for answers.&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2022-07-11T23:31:36.733" LastActivityDate="2022-07-11T23:31:36.733" Title="At what point did &quot;Archean&quot; go from meaning the entire Precambrian to just the middle part?" Tags="|terminology|20th-century|geology|precambrian|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14601" PostTypeId="2" ParentId="14572" CreationDate="2022-07-12T09:15:39.607" Score="4" Body="&lt;p&gt;Another possible example is &lt;a href=&quot;https://www.history.co.uk/articles/the-polish-cryptographers-who-cracked-the-enigma-code&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rejewski's foundational work&lt;/a&gt; on Enigma, encompassing rather deep group-theoretic results. Unsurprisingly, cryptography leans towards secrecy.&lt;/p&gt;&#xA;&lt;p&gt;On a different line there's Gauss' work on non-Euclidean geometries. He &lt;a href=&quot;https://www.cantorsparadise.com/gauss-bolyai-lobachevsky-the-dawn-of-non-euclidean-geometry-38491218bc89&quot; rel=&quot;nofollow noreferrer&quot;&gt;wrote&lt;/a&gt; about his decision &amp;quot;not to allow it to become known during my lifetime. Most people have not the insight to understand our conclusions and I have encountered only a few who received with any particular interest what I communicated to them. In order to understand these things, one must first have a keen perception of what is needed, and upon this point the majority are quite confused. On the other hand it was my plan to put all down on paper eventually ... .&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="16706" LastEditorUserId="16706" LastEditDate="2022-07-12T09:50:20.573" LastActivityDate="2022-07-12T09:50:20.573" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14602" PostTypeId="5" CreationDate="2022-07-12T13:31:04.730" Score="0" Body="&lt;p&gt;A lack of fossils or other dating methods left 19th century geologists no choice but to treat strata below Cambrian ones as belonging to a single time period.&lt;/p&gt;&#xA;&lt;p&gt;Geologists now know that the Precambrian spans about 90% of Earth's history and have divided it into the Hadean (&amp;gt;~4 Ga), Archean (4-2.5 Ga) and Proterozoic (2.5 Ga-550 Ma, including the Ediacaran) Eons, further subdivided into smaller period.&lt;/p&gt;&#xA;&lt;p&gt;Use with the &lt;a href=&quot;/questions/tagged/geology&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;geology&amp;#39;&quot; rel=&quot;tag&quot;&gt;geology&lt;/a&gt; tag for questions about how those divisions emerged, and how geologists recognized from the geological record that specific events occurred during this time.&lt;/p&gt;&#xA;" OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2022-07-13T02:43:55.417" LastActivityDate="2022-07-13T02:43:55.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14603" PostTypeId="4" CreationDate="2022-07-12T13:31:04.730" Score="0" Body="An informal term representing the entire period of Earth's history before the Cambrian, ~4500Ma-~550 Ma. Use with the geology tag." OwnerUserId="4931" LastEditorUserId="4931" LastEditDate="2022-07-13T02:44:17.593" LastActivityDate="2022-07-13T02:44:17.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14604" PostTypeId="2" ParentId="14587" CreationDate="2022-07-12T15:05:05.623" Score="2" Body="&lt;p&gt;Since the Leybold experiment is called a Wien filter, you should look for Wien. Leybold experiment has many purposes but surely not to be exact, its used to show many different pathes of e in B and E fields. Closer to Thompsons experiment but still more to display the idea is another Leybold experiment&#xA;see &lt;a href=&quot;https://www.leybold-shop.com/vp6-1-3-1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.leybold-shop.com/vp6-1-3-1.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12914" LastActivityDate="2022-07-12T15:05:05.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14605" PostTypeId="2" ParentId="13356" CreationDate="2022-07-12T15:18:58.983" Score="5" Body="&lt;p&gt;If by &amp;quot;long-standing&amp;quot; we mean a decade or more, then this sort of thing is fairly common in combinatorics. Short, elementary arguments can be very difficult to find. Here are a few examples that I happen to know about; there are many more.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The question of whether an &lt;a href=&quot;https://arxiv.org/abs/2303.10798&quot; rel=&quot;nofollow noreferrer&quot;&gt;aperiodic monotile&lt;/a&gt; or &amp;quot;einstein&amp;quot; exists has been implicitly around ever since Wang tiles were discovered in 1961. The first example was found in 2023, by Dave Smith, who was not even a professional mathematician. The proof that Smith's tile is an einstein is not that simple, but the bottleneck was finding a candidate tile in the first place, and the tile is very simple to describe (and was found by hand, not by a vast computer search).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Union-closed_sets_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;union-closed sets conjecture&lt;/a&gt;, which was posed in 1979, is still open, but in 2022, Justin Gilmer made &lt;a href=&quot;https://arxiv.org/abs/2211.09055&quot; rel=&quot;nofollow noreferrer&quot;&gt;a huge breakthrough&lt;/a&gt;, proving a slightly weakened form of the conjecture using no concepts that were not around in 1979.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/abs/1907.00847&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hao Huang's proof of the sensitivity conjecture&lt;/a&gt; was completely elementary but did not appear until 2019, whereas the conjecture was first posed in 1992.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Dinitz_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dinitz conjecture&lt;/a&gt; was posed around 1979 and not proved until 1994, by &lt;a href=&quot;https://doi.org/10.1006%2Fjctb.1995.1011&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fred Galvin&lt;/a&gt;, by a completely elementary argument.  There had been some serious attempts to prove the Dinitz conjecture, e.g., by &lt;a href=&quot;https://arxiv.org/abs/math/9310232&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jeannette Jansen&lt;/a&gt;, so it wasn't as though nobody knew about the conjecture.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Stanley%E2%80%93Wilf_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stanley–Wilf Conjecture&lt;/a&gt; also lasted about 15 years before being proved by &lt;a href=&quot;https://doi.org/10.1016/j.jcta.2004.04.002&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marcus and Tardos&lt;/a&gt;, by an argument that could in principle have been given as soon as the conjecture had been posed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-field-kakeya-conjecture/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dvir proved the finite field Kakeya conjecture&lt;/a&gt; in 2008, about a decade after it was first posed, using a stunningly simple argument. No less a mathematician than Terry Tao had unsuccessfully thought about the conjecture. Dvir's use of the &amp;quot;polynomial method&amp;quot; paved the way for many other advances, such as the solution of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cap_set#Upper_bounds&quot; rel=&quot;nofollow noreferrer&quot;&gt;cap set problem&lt;/a&gt;, which was still quite elementary.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2024-02-09T16:36:31.427" LastActivityDate="2024-02-09T16:36:31.427" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14606" PostTypeId="1" AcceptedAnswerId="14632" CreationDate="2022-07-12T21:07:25.293" Score="2" ViewCount="152" Body="&lt;p&gt;In current French mathematical tradition, when introducing complex numbers, it is common to hear about &lt;a href=&quot;https://fr.wikipedia.org/wiki/Plan_complexe&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;complex plane of Argand-Cauchy&amp;quot;&lt;/a&gt;.&#xA;What is particular in French treatment, it is the differentiation between the geometric objects of a &amp;quot;complex plane&amp;quot; and complex numbers: complex numbers are complex coordinates of vectors and points of a complex plane.&#xA;To each vector and each point of a complex plane, the French associate a complex number, called their &lt;em&gt;affixe&lt;/em&gt; (probably this would read &lt;em&gt;affix&lt;/em&gt; in English).&lt;/p&gt;&#xA;&lt;p&gt;I tried to find the origin of the term &amp;quot;affixe&amp;quot; and that of the French treatment of the geometry of complex numbers in the literature but failed so far.&#xA;I've looked through the essays of Caspar Wessel and of Jean-Robert Argand, but they do not use &amp;quot;affixes&amp;quot;, and they could not, because for them complex numbers were vectors.&lt;/p&gt;&#xA;&lt;p&gt;Who, when, and how used the term &lt;em&gt;affixe&lt;/em&gt;/&lt;em&gt;affix&lt;/em&gt; for the first time?&#xA;Same question about differentiating between complex numbers and their geometric realisations.&lt;/p&gt;&#xA;" OwnerUserId="3030" LastActivityDate="2022-07-25T09:41:55.953" Title="Origin of the term &quot;affixe&quot;/&quot;affix&quot; in the geometric treatment of complex numbers" Tags="|mathematics|terminology|geometry|complex-analysis|19th-century|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14608" PostTypeId="2" ParentId="14012" CreationDate="2022-07-14T08:56:11.140" Score="3" Body="&lt;p&gt;I had plenty of free time. Bulletin Board Systems / Forums and the magazines Phrack &amp;amp; 2600 gave good jumping off points.&lt;/p&gt;&#xA;&lt;p&gt;Mostly trial and error and swapping knowledge with other phreakers and hackers. Practice, Practice, Practice :-)&lt;/p&gt;&#xA;" OwnerUserId="16920" LastActivityDate="2022-07-14T08:56:11.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14609" PostTypeId="1" CreationDate="2022-07-15T13:19:02.160" Score="1" ViewCount="118" Body="&lt;p&gt;I'm looking for references about trigonometric tables, especially to those with exact values, like Lambert's Algebraische Formeln für die Sinus von drei zu drei Graden. More precisely:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there an English/French/Spanish translation of Lambert's work?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Why was Lambert interested in making such tables?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Are there others examples of exact trigonometric tables construction?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12330" LastEditDate="2022-07-16T01:08:54.510" LastActivityDate="2022-07-17T01:25:50.823" Title="History of trigonometric tables including Lambert's Algebraische Formeln für die Sinus von drei zu drei Graden" Tags="|trigonometry|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14610" PostTypeId="2" ParentId="14609" CreationDate="2022-07-15T23:40:26.327" Score="3" Body="&lt;p&gt;This is a &lt;em&gt;partial&lt;/em&gt; answer that focuses on the second part of the question. The author of&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. H. Lambert&lt;/a&gt;,&#xA;&amp;quot;Algebraische Formeln für die Sinus von drey zu drey Graden&amp;quot;, In&#xA;J. H. Lambert, &lt;em&gt;Beyträge zum Gebrauche der Mathematik und deren Anwendung&lt;/em&gt;, Part 2.&#xA;Berlin: Verlag der Buchhandlung der Realschule 1770, pp. 133-139&#xA;(&lt;a href=&quot;https://www.google.com/books/edition/Beytr%C3%A4ge_zum_Gebrauche_der_Mathematik_u/wyGYKIDrQj4C?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA133&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;discusses his motivation for publishing his table in some detail, in particular in the first two sections. He desires to use neither classical formulas (going back to antiquity) for sectioning the circle, nor use modern methods that employ infinitesimal approaches or make use of imaginary operands, seeking instead to provide straightforward algebraic formulas:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§. 1.&#xA;In den Anfangsgründen der Trigonometrie wird die Verfertigung der trigonometrischen Tafeln gewöhnlich so angegeben, wie sie sich aus der Elementargeometrie begreiflich machen läßt, und wie diese Tafeln in der That anfangs verfertigt worden. Man kann nemlich vermittelst des Circuls und Lineales den Circul in 2, 3, 4, 5, gleiche Theile theilen, und jeden Bogen, so vielmal man will, halbiren. Daraus läßt sich sodann herleiten, wie  sich der Circul von 3 zu 3 Graden, oder von 45 zu 45 Min., oder von 675 zu 675 Secunden eintheilen, und die Sinus, Tangenten und Secanten von allen diesen Bögen berechnen lassen. Diese Berechnungsart ist ungemein weitläuftig. Man hat daher seit der Erfindung der Infinitesimalrechnung auf Mittel gedacht, sie abzukürzen, und jede Sinus, Tangenten und Secanten für sich zu finden, ohne daß sie erst aus einander hergeleitet werden müßten. Dazu müßten nun allerdings unendliche Reihen gebraucht werden, dafern man nicht bey den imaginairen Formeln, die Herr &lt;strong&gt;Joh. Bernoulli&lt;/strong&gt; zuerst gefunden, stehen bleiben wollte, als welche, wenn man sie nicht in unendliche Reihen verwandelt, fürnehmlich nur zu Erfindung von Lehrsätzen dienen.&lt;/p&gt;&#xA;&lt;p&gt;§. 2.&#xA;Will man aber für die Sinus und Tangenten algebraische Formeln haben, die weder imaginär sind, noch aus unendlich vielen Gliedern bestehen, so bleibt man eben so zurücke, wie die ersten Berechner der trigonometrischen Tafeln, weil man sie ebenfalls nur von 3 zu 3 Graden berechnen kann, und die dazwischen fallenden, durch fortgesetztes Halbiren, finden muß. Da mir solche Formeln noch nicht vorgekommen, so unterzog ich mich der Arbeit sie zu finden, um ausführlich zu sehen, auf welche Art die Sinus von 3 zu 3 Graden mehr oder minder irrational sind. Folgende Tabelle stellt sie mit einem Anblicke vor Augen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My rather loose translation, with additions in square brackets to enhance readability:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;§. 1.&#xA;In fundamental [texts] on trigonometry the computation of trigonometric tables is typically presented in a manner that makes it understandable based on elementary geometry, and tables were actually initially constructed in this manner. It is in fact possible to section a circle into 2, 3, 4, 5 equal parts with a compass and a straightedge, and halve each [resulting] arc as often as desired. From this one can derive how the circle can be partitioned into multiples of 3 degrees, or 45 minutes [of arc], or 675 seconds [of arc], and how the sines, tangents, and secants of all these arcs can be calculated. This manner of computation is exceedingly common. Since the invention of infinitesimal computation one has therefore tried to devise ways of shortening [such computation], and find each sine, tangent, or secant by itself, without the need to derive them from each other. However, doing so would require the use of infinite series, if one does not want to stay with the imaginary formulas first found by Mr. &lt;strong&gt;Joh. Bernoulli&lt;/strong&gt;, which are mostly only useful for the formulation of theorems, if one does not convert them into infinite series.&lt;/p&gt;&#xA;&lt;p&gt;§. 2.&#xA;If, however, one desires algebraic formulas for the sines and tangents that are neither imaginary nor comprise an infinite number of terms, one is back [in the same place] as the first computers of trigonometric tables, since one can also only compute them to multiples of 3 degrees, and has to compute intermediate [values] by repeated halving. Since I had not yet come across such formulas, I undertook the work to create them, to examine in detail in which way the sines of multiples of 3 degrees are more or less irrational. The following table presents them at a single view.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The author proceeds to present the trigonometric identities used in the construction of his table, and emphasizes their application in a systematic order. He summarizes his insights in section 4, as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§. 4.&#xA;In eben dieser Ordnung werden auch die Formeln selbst zusammengesetzter. Vergleicht man sie miteinander, in Absicht auf die Irrationalität, so findet sich, daß sie aus 15 verschiedenen Arten von Wurzelgrössen zusammengesetzt sind, und daß sie sich, wenn man einmal diese gefunden, durch bloßes Addiren und Subtrahiren berechnen lassen. Die Wurzelgrössen sind [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;§. 4. In just this order the formulas themselves become increasingly more composite. If one compares them with a view towards their irrationality one finds that they are composed of 15 different kinds of radical terms, and they can computed by simply adding and subtracting [these terms] once they have been found. The radical terms are [...]&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I was intrigued by the reference to Jacob Bernoulli's &amp;quot;imaginary formulas&amp;quot;. On &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Trigonometric_functions/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor&lt;/a&gt;, J J O'Connor and E F Robertson write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The 18&lt;sup&gt;th&lt;/sup&gt; Century saw trigonometric functions of a complex variable being studied. Johann Bernoulli found the relation between &lt;span class=&quot;math-container&quot;&gt;$\sin^{-1}z$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\log z$&lt;/span&gt; in 1702 [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and in &amp;quot;Trigonometric Delights&amp;quot;, p. 58,  Eli Maor writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] we mention that the general formulas expressing &lt;span class=&quot;math-container&quot;&gt;$\cos  n \alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin n \alpha$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$\cos \alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin \alpha$&lt;/span&gt; were found by Jakob Bernoulli in 1702, more than a hundred years after Viète's work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Neither source provides a reference to the relevant work by Jacob Bernoulli. The closest match I could find in his collected works is:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Section Indefinie des Arcs Circulaires, [...]&amp;quot; (July 13, 1702). In Jacob Bernoulli, &lt;em&gt;Opera, tomus II&lt;/em&gt;. Geneva: Cramer &amp;amp; Philibert, 1744, pp. 921-929 (&lt;a href=&quot;https://www.google.com/books/edition/Opera/S59GAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=921&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;However, from cursory reading, it is not immediately apparent how this publication relates to the two remarks I quoted above. Likewise, I cannot find an exact match in the collected works of Jacob's brother Johann Bernoulli. The publication closest in time and topic I found is&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Multisectio anguli vel arcus [...]&amp;quot; (April 1701) In Johann Bernoulli, &lt;em&gt;Opera Omnia, Tomus I&lt;/em&gt;. Lausanne and Geneva: Marc Michel Bousquet, 1742, pp. 386-392 (&lt;a href=&quot;https://www.google.com/books/edition/Johannis_Bernoulli_Opera_omnia_tam_antea/UWADfS2iDy4C?hl=en&amp;amp;gbpv=1&amp;amp;pg=386&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-07-17T01:25:50.823" LastActivityDate="2022-07-17T01:25:50.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14611" PostTypeId="1" CreationDate="2022-07-16T12:28:52.490" Score="0" ViewCount="99" Body="&lt;p&gt;Is quantum theory just a mathematical tool for correct predictions, according to Heisenberg's views?&lt;/p&gt;&#xA;&lt;p&gt;What are his personal ontological commitments in relation to quantum theory (if he had any ontological commitments at all)?&lt;/p&gt;&#xA;" OwnerUserId="16928" LastEditorUserId="4251" LastEditDate="2022-07-17T23:18:17.187" LastActivityDate="2022-07-17T23:18:17.187" Title="According to Heisenberg, is quantum theory merely a calculation procedure for predictions?" Tags="|physics|quantum-mechanics|quantum-field-theory|heisenberg|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14612" PostTypeId="2" ParentId="3099" CreationDate="2022-07-16T18:32:41.460" Score="11" Body="&lt;p&gt;Here is a summary of my findings, after chasing down the papers mentioned in the other answers and consulting some other sources such as &lt;a href=&quot;https://web.archive.org/web/20120618192547/http://www.math.niu.edu/~rusin/known-math/98/jour.rec&quot; rel=&quot;noreferrer&quot;&gt;David Rusin's &amp;quot;known-math&amp;quot; archive&lt;/a&gt;.  Depending on how you count length, I think there are three reasonable candidates for the shortest published paper.  (Counting words is an inexact science; the counts below include the title but not the references or other metadata.  A formula such as &lt;span class=&quot;math-container&quot;&gt;$E^2 - J$&lt;/span&gt; is counted as one word.)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt;16 words, 2 diagrams, no references:&lt;/b&gt; J. H. Conway and A. Soifer, &lt;a href=&quot;https://doi.org/10.2307/30037388&quot; rel=&quot;noreferrer&quot;&gt;Covering a triangle with triangles&lt;/a&gt;, &lt;i&gt;Amer. Math. Monthly&lt;/i&gt; 112 (2005), 78. This was mentioned in&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/a/3117&quot;&gt;another answer&lt;/a&gt;. As Soifer explains in his book &lt;i&gt;How Does One Cut a Triangle?&lt;/i&gt;, he and Conway submitted it to the &lt;i&gt;American Mathematical Monthly&lt;/i&gt; as a regular article. However, one caveat is that their contribution was published not as an article in its own right, but as a &amp;quot;boxed filler&amp;quot; at the end of another article. It does not appear in the table of contents, nor was it assigned its own DOI.  (The paper was later republished in &lt;a href=&quot;https://geombina.uccs.edu/past-issues/volume-xiv&quot; rel=&quot;noreferrer&quot;&gt;&lt;i&gt;Geombinatorics&lt;/i&gt; volume XIV&lt;/a&gt;, but I have not seen what the full text looks like.)  Also, even though the word count is low, the two diagrams take up more space on the page than some of the other contenders.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt;56 words, no diagrams, 1 reference:&lt;/b&gt; L. J. Lander and T. R. Parkin, &lt;a href=&quot;https://doi.org/10.1090/S0002-9904-1966-11654-3&quot; rel=&quot;noreferrer&quot;&gt;Counterexample to Euler’s conjecture on sums of like powers&lt;/a&gt;, &lt;i&gt;Bull. Amer. Math. Soc.&lt;/i&gt; 72 (1966), 1079.  If you think that Conway and Soifer's paper doesn't count, then this paper by Lander and Parkin, mentioned in &lt;a href=&quot;https://hsm.stackexchange.com/a/3102&quot;&gt;another answer&lt;/a&gt;, is probably the winner.  But there are some ambiguities with how to measure the &amp;quot;length&amp;quot; of a formula and a bibliographic reference.  I have counted the displayed formula as a single &amp;quot;word&amp;quot; but it probably should count as more than that.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt;113 words, no diagrams, no references:&lt;/b&gt; E. Nelson, &lt;a href=&quot;https://doi.org/10.1090/S0002-9939-1961-0259149-4&quot; rel=&quot;noreferrer&quot;&gt;A proof of Liouville's theorem&lt;/a&gt;,  &lt;i&gt;Proc. Amer. Math. Soc.&lt;/i&gt; 12 (1961), 995.  This paper occupies about the same amount of vertical space on the page as Lander and Parkin, so you could argue that it's a tie.  But as far as word count goes, Nelson's paper is wordier, even if you parse the formulas in Lander and Parkin as consisting of multiple &amp;quot;words.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The paper by Milnor mentioned in &lt;a href=&quot;https://hsm.stackexchange.com/a/3100&quot;&gt;another answer&lt;/a&gt; is more mathematically significant than any of the above three, but it cannot compete on the grounds of sheer brevity.  But here are a couple of papers that I feel deserve an honorable mention:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt;87 words, 2 diagrams, 2 references&amp;gt;&lt;/b&gt; K. A. Dahlke, &lt;a href=&quot;https://doi.org/10.1016/0097-3165(89)90084-8&quot; rel=&quot;noreferrer&quot;&gt;A heptomino of order 76&lt;/a&gt;, &lt;i&gt;J. Combin. Theory Ser. A&lt;/i&gt; 51 (1989), 127–128.  This is similar in flavor to Conway–Soifer, minus the controversy about whether it is a &amp;quot;real paper.&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt;95 words, no diagrams, 1 reference:&lt;/b&gt; P. H. Doyle, &lt;a href=&quot;https://doi.org/10.1017/S0305004100042821&quot; rel=&quot;noreferrer&quot;&gt;Plane separation&lt;/a&gt;, &lt;i&gt;Proc. Camb. Phil. Soc.&lt;/i&gt; 64 (1968), 291.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;b&gt;111 words, no diagrams, 1 reference:&lt;/b&gt; D. Lubell, &lt;a href=&quot;https://doi.org/10.1007/978-0-8176-4842-8_28&quot; rel=&quot;noreferrer&quot;&gt;A short proof of Sperner's lemma&lt;/a&gt;, &lt;i&gt;J. Combin. Theory&lt;/i&gt; 1 (1966), 299.  This paper has actually been quite influential and, as you'll see if you follow the link, has been reprinted in Gessel and Rota's anthology, &lt;i&gt;Classic Papers in Combinatorics&lt;/i&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2022-07-16T18:32:41.460" CommentCount="0" CommunityOwnedDate="2022-07-16T18:32:41.460" ContentLicense="CC BY-SA 4.0" />
  <row Id="14613" PostTypeId="1" CreationDate="2022-07-16T20:31:43.173" Score="2" ViewCount="310" Body="&lt;p&gt;Take a box in the shape of a cube and cut &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; of its edges so that all the faces stay connected and flatten it out onto a table. It turns out, there are &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; distinct meshes that emerge: &lt;a href=&quot;https://math.stackexchange.com/questions/4446200/how-many-distinct-ways-to-flatten-a-cube&quot;&gt;https://math.stackexchange.com/questions/4446200/how-many-distinct-ways-to-flatten-a-cube&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is a problem that doesn't have a nice closed form and yet isn't so computationally challenging that a single human can't solve it without a computer.&lt;/p&gt;&#xA;&lt;p&gt;Which makes me wonder, what is the first recorded evidence of someone knowing there are &lt;span class=&quot;math-container&quot;&gt;$11$&lt;/span&gt; ways to open up a cube?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/iP0Ys.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/iP0Ys.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="15867" LastEditorUserId="15867" LastEditDate="2022-08-14T21:20:30.323" LastActivityDate="2022-08-14T21:20:30.323" Title="When did we first know the $11$ ways of opening up a cube?" Tags="|mathematics|geometry|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14615" PostTypeId="1" AcceptedAnswerId="14616" CreationDate="2022-07-18T18:53:15.640" Score="6" ViewCount="204" Body="&lt;p&gt;Maybe this is a question better for German language Stack Exchange, but in the quote attributed to Kronecker:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So &amp;quot;ganzen Zahlen&amp;quot; translates I believe to whole numbers, but does this refer to all the integers or just the naturals (starting at 0 or 1)? The Peano axioms, contemporaneous with Kronecker's time, start defining natural numbers at 0 (or 1). Same for the von Neumann or Zermelo constructions. In this setting, negative numbers are not directly defined, and they don't correspond directly to literal counting in the real world, so I can see how you could argue negative numbers are more &amp;quot;the work of man&amp;quot; (historically, reasoned about in terms of debts and quantities owed) than natural numbers.&lt;/p&gt;&#xA;" OwnerUserId="15230" LastActivityDate="2022-08-03T02:35:49.637" Title="Did Kronecker's &quot;ganzen Zahlen&quot; refer to whole numbers as natural numbers or integers?" Tags="|terminology|quote|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14616" PostTypeId="2" ParentId="14615" CreationDate="2022-07-19T03:48:57.300" Score="8" Body="&lt;p&gt;TL; DR: the &amp;quot;&lt;em&gt;ganzen Zahlen&lt;/em&gt;&amp;quot; refers to positive integers. Gauss, Dedekind and Cantor were reifying new mathematical objects obtained by explicit or implicit procedures, and then forgetting the procedures. In contrast, Kronecker took the given (by God or nature) objects as fixed once and for all, and then tracked the procedures performed on them.&lt;/p&gt;&#xA;&lt;p&gt;Around the same time as the quote that Weber ascribed to Kronecker the latter wrote a paper &lt;em&gt;Uber den Zahlbegriff&lt;/em&gt; explaining what he meant. The initial version appeared in 1887, and is translated in the &lt;a href=&quot;https://www.google.com/books/edition/From_Kant_to_Hilbert_Volume_1/Yil-EmrsT2wC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;sourcebook From Kant to Hilbert&lt;/a&gt;. The full version was published the same year in the Crelle journal, and the added part is &lt;a href=&quot;https://web.archive.org/web/20150924053642/https://www.andrew.cmu.edu/user/edean/papers/kronecker_zahlbegriff.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;translated by Dean&lt;/a&gt;. For context and commentary see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-540-34720-0_11&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boniface, The Concept of Number from Gauss to Kronecker&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to Kronecker (and Cayley, Sylvester, etc., it was the dominant view in the 19th century), mathematics is a natural science, and must be treated as such:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Arithmetic and algebra belong to a well defined domain; the positive integers, the systems of integers represented by integral functions with positive integer coefficients,&#xA;are considered there as motion in kinematics and as matter in natural sciences.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Its basic objects must not only be free of contradiction, but also come from experience, and have procedural definitions that unambiguously decide in each instance what fits. Kronecker's definition of the &amp;quot;whole number&amp;quot; is similar to Frege's and Cantor's, and relies on the equipotence relation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Each ordinal number characterises a determined class of&#xA;equivalent collections... Thus, for instance, “three fingers” is a characteristic invariant of the class formed with the collections of three objects.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, Kronecker opposed the platonist tendencies of Dedekind and Cantor (traceable back to Gauss) of reifying numbers and other objects beyond that, and &amp;quot;&lt;em&gt;criticized Peirce and Peano, who thought they could extend mathematics beyond its frontiers, and dominate “all the spiritual realm”&lt;/em&gt;&amp;quot;, as Boniface puts it. Furthermore:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We have seen that for Gauss the&#xA;successive widenings of the concept of number were the most important feature of&#xA;the development of pure mathematics. Thus, he originally limited arithmetic to the&#xA;positive integers alone, but associated to this position a concept of number which&#xA;includes the rational, irrational, and complex numbers, and which therefore concerns&#xA;not only arithmetic, but also algebra and analysis. For Kronecker, on the contrary,&#xA;the concept of number must be strictly limited to positive integers, while arithmetic&#xA;is extended to algebra and to analysis.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;It is very important for Kronecker to keep&#xA;the initial meanings of the concept of number and, more generally, of fundamental&#xA;concepts, because by widening these concepts to adapt them to other scientific areas,&#xA;their meanings risk being diluted. Thus fundamental mathematical concepts have&#xA;fixed meanings, determined by experience, and must not be changed. A generalization of the concept of number and, therefore, a development of mathematics through&#xA;an extension of the domain of its objects are thus forbidden.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How Kronecker planned to do away with the &amp;quot;forbidden objects&amp;quot; of algebra and analysis without discarding the results (seemingly) about them is also described in the paper. Here is Dean's summary:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the 1887 article “Uber den Zahlbegriff” [4] Leopold Kronecker set himself the task (among other things)&#xA;of demonstrating that talk of algebraic numbers is unnecessary, in the sense that it can be eliminated in&#xA;favor of (somewhat elaborate) talk of natural numbers...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Kronecker’s approach allows us to avoid conceiving of negative numbers as a new sort of object that&#xA;enlarges our domain. Using Gauss’ modular arithmetic notion of congruence, Kronecker recasts the preceding&#xA;equation&lt;/em&gt; [&lt;span class=&quot;math-container&quot;&gt;$7-9=3-5$&lt;/span&gt;] &lt;em&gt;as&#xA;&lt;span class=&quot;math-container&quot;&gt;$7 + 9x ≡ 3 + 5x \pmod{x + 1}$&lt;/span&gt;,&#xA;where &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is indeterminate. We can recapture the other way of thinking, namely that we are enlarging our&#xA;domain, if we conceive of the indeterminate &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; as an actual element determined by the equation &lt;span class=&quot;math-container&quot;&gt;$x + 1 = 0$&lt;/span&gt;,&#xA;but from Kronecker’s perspective it is important that we need not do so.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This earlier version of the article abruptly ends at that point with a concluding paragraph, leaving&#xA;Kronecker’s stated goal of discarding the concept of irrational algebraic numbers untouched. However, an&#xA;extended version of “  ̈Uber den Zahlbegriff” appeared in Crelle’s journal in 1887 [4]. Therein, Kronecker&#xA;expands §5 by inserting, before the concluding paragraph, first a quick demonstration that rational numbers&#xA;can be eliminated via indeterminates, and second the promised elimination of algebraic numbers in general&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-03T02:35:49.637" LastActivityDate="2022-08-03T02:35:49.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14617" PostTypeId="1" AcceptedAnswerId="14618" CreationDate="2022-07-19T17:03:11.417" Score="4" ViewCount="186" Body="&lt;p&gt;While reading about different aspects of &lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johann Heinrich Lambert&lt;/a&gt;'s life and work, I found many interesting side remarks about Lambert's work by different authors, though it is very hard to find modern articles about them. Already in Lambert's book &lt;a href=&quot;https://en.wikipedia.org/wiki/Photometria&quot; rel=&quot;nofollow noreferrer&quot;&gt;Photometria&lt;/a&gt; there are many examples of very interesting points (mentioned in its Wikipedia article) that have not received enough attention yet.&lt;/p&gt;&#xA;&lt;p&gt;However, the book &amp;quot;Photometria&amp;quot; is not the issue here. On p.240 of the book &amp;quot;&lt;em&gt;&lt;strong&gt;Discovering the Principles of Mechanics 1600-1800: Essays by David Speiser&lt;/strong&gt;&lt;/em&gt;&amp;quot;, the author of the chapter on Lambert writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To illustrate his originality as a physicist we can mention his contributions to studies of the resistence of fluids and friction as well as &lt;em&gt;&lt;strong&gt;his work on the fluidity of sand&lt;/strong&gt;&lt;/em&gt;, topics that physicists and mathematical physicists of the 1760s mostly (and happily) left aside.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I found this topic very interesting and truly original for a 18th century researcher. Searching a bit more, I found the title and abstract of this memoir of Lambert: &amp;quot;&lt;em&gt;&lt;strong&gt;Concerning the fluidity of sand, earth, and other soft bodies, as far as it relates to the laws of hydrodynamics&lt;/strong&gt;&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So I'd like to know where I can look at this memoir (I did not succeed in finding it on Google), and also if there is any modern assessment of this aspect of his work. I hardly believe that there is any such assessment in the English literature, so I have no problem if any one can provide a link to such an article in a different language (I will use Google translate).&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="7205" LastEditDate="2023-01-24T13:36:39.683" LastActivityDate="2023-01-24T13:36:39.683" Title="Reference request: modern assesement of J.H. Lambert's work on the &quot;fluidity of sand&quot;" Tags="|reference-request|18th-century|fluid-dynamics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14618" PostTypeId="2" ParentId="14617" CreationDate="2022-07-20T05:04:17.133" Score="5" Body="&lt;p&gt;Johann Heinrich Lambert was Swiss and published in French, German, and Latin. One way of finding the relevant paper is therefore to translate the title's English translation back to those languages and perform an approximate search for that. One finds:&lt;/p&gt;&#xA;&lt;p&gt;Johann Heinrich Lambert, &amp;quot;Sur la fluidité du sable, de la terre et d'autres corps mous, relativement aux loix de l'hydrodynamique.&amp;quot; In &lt;em&gt;Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres. Année MDCCLXXII.&lt;/em&gt; Berlin: Christian Friedrich Voss 1774. In section &amp;quot;Classe de philosophie experimentale&amp;quot;, pp. 33-64 (&lt;a href=&quot;https://www.google.com/books/edition/Nouveaux_m%C3%A9moires_de_l_Acad%C3%A9mie_royale/__QgAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA33&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Google scan&lt;/a&gt;). There is a handy PDF of it &lt;a href=&quot;http://www.kuttaka.org/%7EJHL/L1772e.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As often happens with society proceedings from the 18&lt;sup&gt;th&lt;/sup&gt; century, the paper dates to a different year than the published volume. In this case the paper itself dates to 1772.&lt;/p&gt;&#xA;&lt;p&gt;Google Scholar currently lists only three publications that reference this paper by Lambert.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-07-20T06:49:12.630" LastActivityDate="2022-07-20T06:49:12.630" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14619" PostTypeId="1" CreationDate="2022-07-20T14:04:26.390" Score="2" ViewCount="360" Body="&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$Ax^2+Bx+C=0 $&lt;/span&gt; is mentioned as standard form of quadratic equation in every textbook or encyclopaedia, but what's so special about it that its called standard form of quadratic equation.&#xA;Also, I am aware that standard form of quadratic “Function” is different.&lt;/p&gt;&#xA;" OwnerUserId="16944" LastEditorUserId="131" LastEditDate="2022-07-21T08:30:35.240" LastActivityDate="2022-12-21T10:02:10.250" Title="Why is $Ax^2+Bx+C=0$ called standard form of Quadratic Equation? And who declared it the standard form?" Tags="|mathematics|algebra|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14620" PostTypeId="2" ParentId="3003" CreationDate="2022-07-20T21:37:29.270" Score="1" Body="&lt;p&gt;From &lt;em&gt;Curvature of Surfaces in 3-Space&lt;/em&gt; by Michael Garman &amp;amp; Jessica Bonnie published in &lt;em&gt;Verge&lt;/em&gt; 6:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The notion of curvature first began with the discovery and refinement of the principles of geometry by the ancient Greecks circa 800-600 BCE. Curvature was originally defined as a property of the two classical Greek curves, the line and the circle.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was noted that lines do not curve amd that every point on a circle curves the same amount. The actual study of curvature began when Aristotle expanded upon these two points and declared that there are three kinds of loci: straight, circular and mixed. It was from this premise that the true study of curvature began.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Appollonius of Perga devised methods for calculating the radius of curvature in the 3rd BCE. These methods were similar to Newton and Huygens (discovered some 2,000 years ago later) ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The next momentous advancement in the study of curvature came from Nicole Orseme in the 14C. Oresme was the first person to hint at an actual definition of curvature. He also assumed that there was a specific measure of twist which he called 'curvitas'. By observing multiple curves at once, Oresme eventually proposed that the curvature of a circle was proportional to the multiplicative inverse of its radius. This would eventually provide the driving force behind the quest of finding the curvature of a general curve ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Johannes Kepler made the next contribution to the notion of curvature. While working on the problem of al Hazin, (finding the image of a brilliant point when reflected off a circle), Kepler arrived at the notion of using a circle to measure the general curvature at a point of reflection. This approximating circle would come to be known as a curvea 'circle of curvature' at a point ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After this Huygens, Newton &amp;amp; Liebniz reified Keplers discovery through calculus into the modern notion of a curvature of a curve. A similar definition can be given for the curvature of a surface. But in higher dimensions a scalar measure of curvature isn't sufficient and tensors are required. This was Riemann's achievement after Gauss discovered his eponymous curvature. It's worth noting that calculus in Newton's hands actually began, as Garman &amp;amp; Bonnie note, with his study of curvature rather than that of motion as is commonly assumed.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-07-20T21:37:29.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14622" PostTypeId="1" CreationDate="2022-07-21T07:42:00.523" Score="1" ViewCount="83" Body="&lt;p&gt;&lt;sub&gt;&lt;a href=&quot;https://physics.stackexchange.com/q/719282&quot;&gt;Crossposted&lt;/a&gt; at Physics Stack Exchange&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I have read once that Hilbert had some reservations regarding the first form of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_field_equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;field equations&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ R_{\mu\nu} = k T_{\mu\nu} $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;because it was not possible to retrieve them from a variational principle. Is it true? I have not found any authoritative document that mentions this and I don't know where to search and I would like some sources if they exist. Does anybody know more about this?&lt;/p&gt;&#xA;" OwnerUserId="16947" LastEditorUserId="4251" LastEditDate="2022-07-21T11:52:32.917" LastActivityDate="2022-07-21T11:52:32.917" Title="Hilbert's criticism on the first version of Einstein's field equations" Tags="|physics|relativity-theory|einstein|hilbert|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14623" PostTypeId="2" ParentId="3437" CreationDate="2022-07-21T12:39:49.107" Score="1" Body="&lt;p&gt;This is a supplement to the two (good) answers by @Mauro Allegranza and @Viktor Blasjo. Although these answers and the question now have some age to them, it seems worth clarifying them by adding some historical perspective to the questioner's (largely mistaken) opening statement that &amp;quot;Before Newton many phycisists try to understand nature and the rotations of planets.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Before Kepler, astronomy was not thought to be a matter of physics, certainly not in the modern universal sense. For a long time, the idea held sway that the world consisted of two parts, quite different from each other (see Lindberg &amp;amp; Shanks (eds.)(2013) &lt;a href=&quot;https://books.google.com/books/about/The_Cambridge_History_of_Science_Volume.html?id=ZMNkAwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Cambridge History of Science', vol.2&lt;/a&gt;). The lower part was a &amp;quot;terrestrial region extend[ing] downward from the concave surface of the lunar orb to the center of the Earth&amp;quot;, a &amp;quot;realm of incessant change&amp;quot;, made from the four primary qualities &amp;quot;hot, cold, dry, and moist&amp;quot; or the elements representing them, fire, earth, air, water. In radical contrast, the upper part, the heavens above the sphere of the moon had attributes of perfection and incorruptibility. The heavens were not made of the four terrestrial elements, they were composed of a fifth and aetherial element, or &amp;quot;quintessence&amp;quot;, and their motions showed the perfections of pure geometry in uniform circular motions. Within this framework of incorruptible celestial quintessence and perfect geometry it made no sense to think of physics learned from terrestrial effects as having any role.&lt;/p&gt;&#xA;&lt;p&gt;The title of Kepler's book of 1609 is sometimes abbreviated just to &lt;a href=&quot;https://books.google.com/books/about/Astronomia_Nova.html?id=hqMvjgEACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Astronomia nova'&lt;/a&gt; or 'New astronomy'. More fully, his title in translation runs &amp;quot;A new astronomy, studied by causes, or, Celestial Physics&amp;quot;. Kepler's idea that the celestial motions were subject to physical causes was itself revolutionary. He ran into conservative objections, made stiffer by the fact that he not only departed (especially with his postulated elliptical orbits) from the geometrical perfections of circular motion; he also demanded calculations that could not be perfectly executed by traditional demonstrative methods of geometry: they could only be approximated ('Kepler's problem' in Kepler (1609) chap.60, forerunner of the later 'Kepler's equation').&lt;/p&gt;&#xA;&lt;p&gt;Kepler's ideas were rather slow to find limited acceptance. His elliptical orbits gained credibility by the success of his Rudolphine Tables (1627), especially in predicting a 1630 transit of Mercury across the sun's disk (see e.g. &lt;a href=&quot;https://ui.adsabs.harvard.edu/abs/2013JHA....44..303G&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gingerich (2013)&lt;/a&gt;).  But the particular physics that he proposed, to implement his overarching idea of physical causes in astronomy, was not a success: it involved a power of the sun, by virtue of its rotation (and by the help of something like outwardly extending magnetic fibers) to drive the planets around their orbits. The special inequalities of the moon's motion, Kepler supposed, were effects of the sun's illumination.  Kepler also thought that if not for the sun's rotatory drive, the orbital motions would cease and come to rest. (See Bruce Stephenson (1985) &lt;a href=&quot;https://books.google.com/books/about/Kepler_s_Physical_Astronomy.html?id=EpDTBwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Kepler's Physical Astronomy'&lt;/a&gt;.) The particular physics that Kepler conceived was rather speedily forgotten, even though his overarching general idea of the existence and relevance of celestial physical causes did succeed much later in the hands of others.&lt;/p&gt;&#xA;&lt;p&gt;There are other indications that a number of respected geometrical astronomers of even the mid-17th century did not regard astronomy as governed by physics, quite apart from the question of Kepler's elliptical ideas.&lt;/p&gt;&#xA;&lt;p&gt;Thus G B Riccioli's &lt;a href=&quot;https://books.google.com/books?id=_mJDAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Almagestum Novum' (1651)&lt;/a&gt; disparaged the efforts of those such as Cavalieri who had tried to give quantitative account of the core of Kepler's elliptical astronomy, calling these efforts an unsuccessful attempt to &amp;quot;apply medicine&amp;quot; to the &amp;quot;a-geometrical&amp;quot; relationships of Kepler's orbital ellipses -- as if Kepler's 'departure' from geometry was a kind of sickness in itself.&lt;/p&gt;&#xA;&lt;p&gt;Even 60 years after the publication of Kepler's 'Astronomia nova...', Nicolas Mercator (1669-70) could still report complaints about Kepler's hypotheses (&lt;a href=&quot;https://ui.adsabs.harvard.edu/abs/1670RSPT....5.1168M&quot; rel=&quot;nofollow noreferrer&quot;&gt;Philosophical Transactions 1670, vol.5, pp.1168-1175&lt;/a&gt;). At p.1174 (translated from the Latin):&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;No-one has been found up to now who would deny that Kepler's areas can satisfy the appearances; but since neither he himself nor anyone after him could determine them by direct calculation, some have criticized Kepler for having parted ways with geometry while indulging too much in [speculation about] physical causes.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Mercator's remarks bear witness to contemporary opinions that still regarded it as a mistake on Kepler's part even to propose introducing  physics into astronomy. The conservatives included I Boulliau, who accepted ellipses, but sought to relace Kepler's ideas about the associated celestial motions by something purely geometrical, and GB Riccioli whose scepticism was already cited above.&lt;/p&gt;&#xA;&lt;p&gt;To be sure, the older ideas were breaking down during the 1670s. A number of thinkers (including those acknowledged in Newton's 'Principia' book 1 prop.4 and elsewhere, Wren, Hook, Halley) were starting to apply physical ideas to orbital motions, and Hooke in particular made the insightful inference from his observation of apparent impact craters on the moon that the moon, too, had an attractive power to make heavy objects fall towards it. But nearly all physics was terrestrial, and Newton's proposal, quoted by @Viktor Blasjo, that the force retaining the moon in its orbit was of the same kind as the familiar heaviness seen in terrestrial objects, was revolutionary in itself.&lt;/p&gt;&#xA;&lt;p&gt;So this background tends to explain and support Mauro Allegranza's answer that Newton did not invent the word gravity, but he was the author of a crucial &amp;quot;shift of meaning&amp;quot; in it.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-07-21T12:39:49.107" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14624" PostTypeId="1" CreationDate="2022-07-22T10:45:18.543" Score="2" ViewCount="120" Body="&lt;p&gt;The practice of wearing face masks to protect against disease is actually nothing new. Mask wearing for the specific purpose of warding off disease was common (and even sometimes mandated by public health regulations) as far back as the &lt;a href=&quot;https://www.history.com/news/1918-spanish-flu-mask-wearing-resistance&quot; rel=&quot;nofollow noreferrer&quot;&gt;1918 &amp;quot;Spanish&amp;quot; flu pandemic&lt;/a&gt;, and the beaked &amp;quot;Doctor Schnabel&amp;quot; plague mask &lt;a href=&quot;https://en.wikipedia.org/wiki/Plague_doctor_costume&quot; rel=&quot;nofollow noreferrer&quot;&gt;dates back to at least 1721&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I'm curious when face masks were first identified as protective against disease. Were the 18th century &amp;quot;beak&amp;quot; masks a revolutionary new idea, or was masking to prevent disease already a thing?&lt;/p&gt;&#xA;&lt;p&gt;To be clear, I'm not asking about the development of the modern germ theory of disease or when the first formal medical study demonstrating the effectiveness of masks to a modern-day level of rigor was published. I am asking about the development of a substantial belief or theory that mask-wearing prevents disease, even if the reason was not known or the modern empirical evidence of that belief or theory would not come for many years.&lt;/p&gt;&#xA;&lt;p&gt;The wearing of masks for cultural, religious, fashion, or climate reasons does not count.&lt;/p&gt;&#xA;&lt;p&gt;The discovery of the effectiveness of face masks counts even if the contemporary medical theory behind the mask was completely wrong by 21st century standards. For example, if there is an ancient Egyptian medical textbook that states, &amp;quot;Respiratory disease is caused by the ghosts of executed criminals, so when disease is around, we wear masks to scare off the ghosts.&amp;quot;, it would count since it would demonstrate an understanding that a link exists between masks and disease. For the avoidance of doubt, any appearance of mask recommendations for the prevention of disease appearing in medical literature or public health orders counts even if the evidence of efficacy of such masks provided in such sources does not meet modern standards.&lt;/p&gt;&#xA;" OwnerUserId="6976" LastEditorUserId="6976" LastEditDate="2022-07-22T12:40:29.700" LastActivityDate="2022-07-23T10:31:51.650" Title="When was it first discovered that face masks protect against disease?" Tags="|discoveries|medicine|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14625" PostTypeId="2" ParentId="14624" CreationDate="2022-07-22T22:16:53.687" Score="4" Body="&lt;p&gt;Medicine is not my area of expertise, but starting with American and Canadian publications I was able to trace back the use of protective masks in &lt;em&gt;modern evidence-based medicine&lt;/em&gt; to Germany. The starting point seems to have been the following publication:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Fl%C3%BCgge&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Flügge&lt;/a&gt;, &amp;quot;Ueber Luftinfection&amp;quot;, &lt;em&gt;Zeitschrift für Hygiene und Infectionskrankheiten&lt;/em&gt;, Vol. 25, 1897, &lt;a href=&quot;https://books.google.com/books?id=VdU2AQAAMAAJ&amp;amp;pg=PA179#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 179-224&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This publication was usually cited as &amp;quot;Atmospheric infection&amp;quot; in English-language publications of the time. In five years of systematic research, Flügge and his team carefully examined the conditions under which fine dust particles and small droplets detach from surfaces of various kinds, and how far and long they can carry an infectious load (represented by Bacillus prodigiosus and Bacillus megaterium), detected through the use of plates with culture medium. Key findings were that the finest dust particles and smallest droplets capable of transmission could stay suspended in air for 4 to 5 hours, that transport of these required only minimal airflow (much smaller than anticipated), and that speaking, coughing, and sneezing would spread droplets over distances of several meters.&lt;/p&gt;&#xA;&lt;p&gt;While transmission by dust particles had apparently been considered prior to Flügge, his major contribution was pointing out the existence of transmission by droplets, which also applies to bacteria that cannot survive desiccation. He concludes that one should expect that transmission by droplets frequently occurs in diphtheria, TB, influenza, whooping cough, pneumonia, and pneumonic plague and that this risk is significantly higher in closed rooms than in the open air. He also points out that infection by droplets presents a new mode of transmission not previously considered in&#xA;the infection of surgical wounds.&lt;/p&gt;&#xA;&lt;p&gt;W. Hübener, &amp;quot;Ueber die Möglichkeit der Wundinfection vom Munde aus und ihre Verhütung durch Operationsmasken,&amp;quot; &lt;em&gt;Zeitschrift für Hygiene und Infectionskrankheiten&lt;/em&gt;, Vol. 28, 1898, &lt;a href=&quot;https://www.google.com/books/edition/Zeitschrift_f%C3%BCr_Hygiene/khM2AQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA348&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 348-372&lt;/a&gt;, referencing Flügge, examined the possibility of the transmission of infections from the mouths of surgical personnel to wounds and its prevention by the use of surgical masks. I am fairly certain that this is the origin of the surgical mask in common use today.&lt;/p&gt;&#xA;&lt;p&gt;Hübener appears to have been employed by the same surgical institute in Breslau that previously had pioneered the use of surgical caps. There is a picture of a man (the author himself?) wearing a mask of Hübener's design on page 357 in the article. The mask was constructed from layers of gauze, and there is experimental data on the effectiveness of the mask in reducing the spread of infectious droplets using single-layer and double-layer construction, separately measured under conditions of speaking, shouting, coughing, and sneezing. The paper discusses the use of masks in cases of TB, influenza, and leprosy. It mentions that the effectiveness in the case of leprosy seemed minimal. Hübener mentions that masks protect in two ways:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In solchen Fällen wird die Maske nicht nur bei bereits Inficirten eine Propogation der Keime nach aussen verhindern, sondern auch gesunde, noch nicht inficierte Personen durch ihr Tragen gegen die Infection schützen&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: In these cases the mask will not only prevent the propagation of microbes emanating from an infected person, but wearing it will also protect a person not yet infected against the infection.&lt;/p&gt;&#xA;&lt;p&gt;B. Fränkel, &amp;quot;Zur Prophylaxe der Tuberculose&amp;quot;, &lt;em&gt;Berliner Klinische Wochenschrift&lt;/em&gt;, Vol. 36, No. 2, January 8, 1899, &lt;a href=&quot;https://archive.org/details/BerlinerKlinischeWochenschrift189936/page/n51/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 21-27&lt;/a&gt;, also inspired by Flügge's work and independently of Hübener, examined the utility of masks in the specific context of preventing the spread of TB. The author considers whether masks should cover just the mouth or both mouth and nose (and favors the former) and what type of fabrix should be used to construct it. Based on his own work he recommends that TB patients always wear a mask when around other people, the only drawback being conspicuous appearance.&lt;/p&gt;&#xA;&lt;p&gt;Carl Flügge, &amp;quot;Die Verbreitung der Phthise durch staubförmiges Sputum und durch beim Husten verspritzte Tröpfchen&amp;quot;, &lt;em&gt;Zeitschrift für Hygiene und Infektionskrankheiten&lt;/em&gt;, Vol. 30, 1899, &lt;a href=&quot;https://zenodo.org/record/1686460#.Ytsio6jMKM9&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 107-124&lt;/a&gt;, discusses the spread of TB by dried sputum particles and droplets spread by coughing. Its general recommendation is &amp;quot;social distancing&amp;quot; of one meter for TB patients in public settings. The author is of the opinion that the wearing of protective masks as recommended by B. Fränkel should be restricted to particular cases where the environment of a TB patient needs to be protected as completely as possible.&lt;/p&gt;&#xA;&lt;p&gt;Even with the effectiveness of masks in cases of diseases transmitted by the recently discovered droplet infections established in principle, there was thus discussion on the specifics of mask wearing from the very beginning. Twenty years later, the benefit and use of masks appears to have been generally accepted by the medical community, as evidenced by the following quote (emphasis mine):&lt;/p&gt;&#xA;&lt;p&gt;Brewster C. Doust and Arthur Bates Lyon, &amp;quot;Face masks in infections of the respiratory tract&amp;quot;, &lt;em&gt;Journal of the American Medical Association&lt;/em&gt;, Vol. 71, No. 15, October 1918, &lt;a href=&quot;https://jamanetwork.com/journals/jama/article-abstract/219248&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 1216-1219&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The use of face masks by those whose duties necessitate their coming in contact with cases of respiratory infection &lt;strong&gt;has now become general&lt;/strong&gt;. [...] We have therefore undertaken experiments with some of the commoner types of masks in order to prove their efficiency or nonefficiency in preventing the dissemination of infectious material from the mouth during the acts of speaking or coughing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-07-23T03:35:42.897" LastActivityDate="2022-07-23T03:35:42.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14626" PostTypeId="2" ParentId="12797" CreationDate="2022-07-23T02:01:51.753" Score="2" Body="&lt;p&gt;As an undergraduate student that has been researching the the history of the idea of potential for the past few years, I can assure you that it's way more complex and fascinating than it appears at first glance. As you won't have much time to discuss it in class, I'll put an EXTREMELY basic summary here.&lt;/p&gt;&#xA;&lt;p&gt;The idea of potential was constructed gradually from a bunch of other ideas, whose roots date back, in some form or another, to at least the XVIII century. Those ideas were developed more or less independently to aid in the study of several topics, in particular fluid mechanics, gravitation, and the mathematics of functions of several variables. Like Mozibur Ullah said, scalar functions tend to be much easier to work with than vector functions, and could be used to study phenomena that methods based on finding forces and other directional quantities couldn't tackle well, such as the notoriously difficult Three Body Problem, which has been tackled with potential-like functions since at least 1777, when Lagrange published a paper on the subject.&lt;/p&gt;&#xA;&lt;p&gt;Gradually, researchers realized that it could be applied to the study of even more topics, in particular electricity, thermodynamics, probability, etc. Often, the equations used in one area were the same as the other, with only the meanings of the variables changing. This allowed results and methods from several areas to be used in apparently unrelated others. For example, a trivial theorem in electromagnetism could help predict (correctly) a non-trivial phenomenon in thermodynamics.&lt;/p&gt;&#xA;&lt;p&gt;The multidisciplinary usefulness of the idea of potential was basically forgotten after the end of the XIX century, in large part because of the invention of vector algebra, which made a lot of calculations in Classical Electromagnetism, the main field where potentials were used, much easier to not only do, but also interpret in terms of real-world phenomena. However, potentials proved to be HIGHLY useful in the study of Special Relativity and, especially after Aharonov and Bohm's paper, in Quantum Mechanics. People are just beginning to realize how important they could be in the future.&lt;/p&gt;&#xA;&lt;p&gt;If you have students that know a decent level of calculus, you could recommend that they read a paper by Gauss that I'll link below. It's where Gauss uses the term &amp;quot;potential&amp;quot; as a noun, to refer to potential functions.&lt;/p&gt;&#xA;&lt;p&gt;Some of my sources:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/generalpropositi00gaus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/generalpropositi00gaus&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Grattan-Guinness, I. 1995 Why Did George Green Write His Essay of 1828 on Electricity and Magnetism?, The American Mathematical Monthly, 102:5, 387-396,&lt;/p&gt;&#xA;&lt;p&gt;Cross J. (1983) Euler’s Contributions to Potential Theory 1730–1755. In: Leonhard Euler 1707–1783. Birkhäuser Basel. &lt;a href=&quot;https://doi.org/10.1007/978-3-0348-9350-3_16&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-3-0348-9350-3_16&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Walter S. (2007) Breaking in the 4-Vectors: The Four-Dimensional Movement in Gravitation, 1905–1910. In: Janssen M., Norton J.D., Renn J., Sauer T., Stachel J. (eds) The Genesis of General Relativity. Boston Studies in the Philosophy of Science, vol 250. Springer, Dordrecht. &lt;a href=&quot;https://doi.org/10.1007/978-1-4020-4000-9_18&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-1-4020-4000-9_18&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;WHITTAKER, Edmund Taylor. A History of the Theories of Aether and Electricity from the Age of Descartes to the Close of the Nineteenth Century. Ney York Humanities Press, 1973&lt;/p&gt;&#xA;" OwnerUserId="16957" LastActivityDate="2022-07-23T02:01:51.753" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14627" PostTypeId="2" ParentId="14624" CreationDate="2022-07-23T10:31:51.650" Score="2" Body="&lt;p&gt;After reading answer by njuffa I was surprised masks were not worn earlier, particularly during the era  when the now discredited &lt;a href=&quot;https://en.wikipedia.org/wiki/Miasma_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;miasma theory&lt;/a&gt;, that bad smells cause disease, was prevalent.&lt;/p&gt;&#xA;&lt;p&gt;If you want to extend the concept of face masks being used to protect people against disease then the &amp;quot;bird beak&amp;quot; type of mask used during the plague in Europe needs to be considered.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/MUvP1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/MUvP1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Such masks were used when the miasma theory was still prevalent. The masks were shaped like a beak so that a posy of flowers could be placed in the beak with the stems in the beak &amp;amp; the flowers near the nose of the wearer, thus protecting against bad smells.&lt;/p&gt;&#xA;&lt;p&gt;A French doctor named Charles de Lorme is credited with &lt;a href=&quot;https://timhenninger.weebly.com/history.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;inventing the beak mask in 1619.&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He designed the bird mask to be worn with a large waxen coat as a form of head-to-toe protection, modeled on a soldier’s armour. The costume was worn by plague doctors during the Plague of 1656&lt;/p&gt;&#xA;&lt;p&gt;A medieval plague doctor was also referred to as the chirurgeon (Middle English “cirurgien”, from Old French, from Latin chīrurgia, from Greek χειρουργία, as referring to surgery). He says the chirurgeon wore a long black oilcloth robe that had a hood.It was intended as a protection suit against the contagious plague. This costume had openings for the eyes that were made of glass. It also had a hollow long beak for the nose, which was filled with camphor, garlic, mint, or a sponge of vinegar. This was all to protect the doctor from miasmatic bad air.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2022-07-23T10:31:51.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14628" PostTypeId="1" AcceptedAnswerId="14629" CreationDate="2022-07-23T15:44:34.620" Score="4" ViewCount="517" Body="&lt;p&gt;What I have understood is, generally the symbol of an element is kept as the first letter of the element's name itself (H, N, O...) unless there's already a symbol with that letter, in that case, we take first two letters to write the symbol (He, Ne, Ca...). And of course, this isn't a strict naming rule, we have a lot of symbols not following this rule, rather borrowing their symbol name from there source name (Ag, Au, Fe, etc).&lt;/p&gt;&#xA;&lt;p&gt;My question is, why not follow the rule (or a convention) for an element just to keep that same name for the next element (B for boron)?&lt;/p&gt;&#xA;&lt;p&gt;Wouldn't it be more reasonable, if beryllium was named B and boron Bo?&#xA;Is there any historical reason behind this, or is it just because some random chemist liked to do it?&lt;/p&gt;&#xA;" OwnerUserId="16960" OwnerDisplayName="Rohit Joshi" LastEditorUserId="2710" LastEditDate="2022-07-28T09:12:31.790" LastActivityDate="2023-12-18T15:55:50.030" Title="Why isn't the symbol for Beryllium 'B' rather than 'Be'?" Tags="|notation|periodic-table|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14629" PostTypeId="2" ParentId="14628" CreationDate="2022-07-24T03:25:36.977" Score="16" Body="&lt;p&gt;Chronology is an important aspect in this case. Let's first understand how beryllium was discovered.&lt;/p&gt;&#xA;&lt;p&gt;Emeralds and beryl were known to ancient civilizations, they were first recognized as the same mineral (&lt;span class=&quot;math-container&quot;&gt;$\mathrm{Be_3 Al_2(SiO_3)_6}$&lt;/span&gt;) by &lt;a href=&quot;https://en.wikipedia.org/wiki/Ren%C3%A9_Just_Ha%C3%BCy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abbé Haüy&lt;/a&gt; in 1798. Later that year, &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Nicolas_Vauquelin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Louis Nicolas Vauquelin&lt;/a&gt;, a French chemist, discovered that an unknown element was present in emeralds and beryl. Attempts to isolate the new element finally succeeded in 1828 when two chemists, &lt;a href=&quot;https://en.wikipedia.org/wiki/Friedrich_W%C3%B6hler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Friedrich Wölhler&lt;/a&gt; of Germany and A. Bussy of France, independently produced beryllium by reducing beryllium chloride (&lt;span class=&quot;math-container&quot;&gt;$\mathrm{BeCl_2}$&lt;/span&gt;) with potassium in a platinum crucible.&lt;/p&gt;&#xA;&lt;p&gt;Since the salts of beryllium had a sweet taste, the element was named &amp;quot;glucinium&amp;quot; from the Greek &amp;quot;glykys&amp;quot; for &amp;quot;sweet&amp;quot; and was symbolized as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\mathrm{Gl}$&lt;/span&gt;&amp;quot;. Bussy preferred this name but Wölhler was unhappy and preferred the name &amp;quot;beryllium&amp;quot; from the Greek word 'beryllos', which means the mineral beryl. &lt;a href=&quot;https://en.wikipedia.org/wiki/Martin_Heinrich_Klaproth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Martin Klaproth&lt;/a&gt; had already pointed out in 1801 that yttria also forms sweet salts. A name derived from 'beryllos' would be less likely to cause confusion than one derived from 'glykis.' Klaproth also noted that a genus of plants was already called &amp;quot;glucine&amp;quot;. Finally, in 1949, IUPAC chose beryllium as the element’s name and this decision became official in 1957.&lt;/p&gt;&#xA;&lt;p&gt;By the time the change was accepted, boron was already discovered in 1808, by French chemists &lt;a href=&quot;https://Joseph%20Louis%20Gay-Lussac&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joseph Louis Gay-Lussac&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Jacques_Th%C3%A9nard&quot; rel=&quot;nofollow noreferrer&quot;&gt;Louis-Jacques Thenard&lt;/a&gt;. The name came from the Arabic بورق (buraq) which was borrowed from the Persian word بوره (burah). The gave this word when they found borax which was tincal in crude state. They symbolized the element as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\mathrm{B}$&lt;/span&gt;&amp;quot;. Since &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\mathrm{B}$&lt;/span&gt;&amp;quot; was already taken, IUPAC had to symbolize beryllium as &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\mathrm{Be}$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://tarek.kakhia.org/periodic_table/english/Boron_05.Tarek_Kakhia.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://tarek.kakhia.org/periodic_table/english/Boron_05.Tarek_Kakhia.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Beryllium History - &lt;a href=&quot;https://education.jlab.org/itselemental/ele004.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;It's Elemental&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Discovery of Beryllium&lt;/em&gt; by Dr. Doug Stewart (&lt;a href=&quot;https://www.chemicool.com/elements/beryllium.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16966" OwnerDisplayName="Nilay Ghosh" LastEditorUserId="16966" LastEditDate="2023-12-18T15:55:50.030" LastActivityDate="2023-12-18T15:55:50.030" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14630" PostTypeId="2" ParentId="14619" CreationDate="2022-07-24T07:42:59.740" Score="1" Body="&lt;p&gt;Afraid I can't be completely authoritative on this, I'm just throwing it in there, so to speak, but I recall reading a 19th-century text once which covered the subject of quadratics, where the &amp;quot;standard&amp;quot; form was given as &lt;span class=&quot;math-container&quot;&gt;$ax^2 + 2 b x + c = 0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This of course results in the discriminant having the slightly simpler form &lt;span class=&quot;math-container&quot;&gt;$b^2 - a c$&lt;/span&gt;, which indeed does make the quadratic formula simpler (unless my skills have completely deserted me):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x = \dfrac {-b \pm \sqrt {b^2 - ac} } a$$&lt;/span&gt;&#xA;hence eliminating the need for all those pesky numbers.&lt;/p&gt;&#xA;&lt;p&gt;Obviously sometime in the following decades the standard way of looking at this subject changed in focus, and simplicity of the quadratic form was placed at a higher priority than simplicity of the resulting formula for its evaluation, but I am afraid I have not performed the research to determine when or how that happened.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2022-07-24T07:42:59.740" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14631" PostTypeId="1" CreationDate="2022-07-24T19:27:29.247" Score="2" ViewCount="46" Body="&lt;p&gt;Yesterday I saw a reference to Watt's law being used for the power formula P = V I.&lt;/p&gt;&#xA;&lt;p&gt;It's been over 40 years since I learned my basic electrical engineering fundamentals and this is the first time that I recall seeing the phrase Watt's law used to represent this formula.&lt;/p&gt;&#xA;&lt;p&gt;Doing some searching around on the web I haven't found where this term first showed up but have seen plenty of people on both sides of the flag claiming this is a proper law and others saying not.&lt;/p&gt;&#xA;&lt;p&gt;Have done a fair bit of reading on Watt and his accomplishments in steam and mechanical power but don't recall that he ever did anything with electricity.&lt;/p&gt;&#xA;&lt;p&gt;While the watt has since become a standardized unit of work (joules per second), I don't recall reading anything about Watt himself showing that volts times amperes equals power.&lt;/p&gt;&#xA;&lt;p&gt;So, really interested in finding when/where this usage originated.&lt;/p&gt;&#xA;" OwnerUserId="16964" LastActivityDate="2022-07-24T19:27:29.247" Title="What was the first use/cite of &quot;Watt's law&quot;?" Tags="|electricity|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14632" PostTypeId="2" ParentId="14606" CreationDate="2022-07-24T20:26:38.053" Score="2" Body="&lt;p&gt;According to Eduard Study and Élie Cartan [1], it was Cauchy who introduced the term &lt;em&gt;affixe&lt;/em&gt; in &lt;em&gt;Sur la quantité géométrique &lt;span class=&quot;math-container&quot;&gt;$\mathrm{i} = 1_{\frac{\pi}{2}}$&lt;/span&gt;, et sur la réduction d'une quantité géométrique quelconque à la forme &lt;span class=&quot;math-container&quot;&gt;$x + y\mathrm{i}$&lt;/span&gt;&lt;/em&gt;, in &lt;em&gt;Exercices d'analyse et de physique mathématique&lt;/em&gt;, Tome 4 (1847).&lt;/p&gt;&#xA;&lt;p&gt;Cauchy was only talking about affixes of points, while vectors were basically identified with complex numbers, which he called &amp;quot;geometric quantities.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;[1]: E. Study and É. Cartan, &lt;em&gt;Les nombres complexes&lt;/em&gt; (1908), in &lt;em&gt;Encyclopédie des sciences mathématiques pures et appliquées&lt;/em&gt;, Tome 1, Volume 1, edited by Jules Molk.&lt;/p&gt;&#xA;" OwnerUserId="3030" LastEditorUserId="3030" LastEditDate="2022-07-25T09:41:55.953" LastActivityDate="2022-07-25T09:41:55.953" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14633" PostTypeId="1" CreationDate="2022-07-24T21:52:55.120" Score="1" ViewCount="43" Body="&lt;p&gt;Who first chose to denote classes with capital letters and sets with small letters in set theory?&lt;/p&gt;&#xA;" OwnerUserId="16965" LastEditorUserId="131" LastEditDate="2022-07-25T07:04:03.907" LastActivityDate="2022-07-25T07:04:03.907" Title="Capital letters for classes and small letters for sets" Tags="|reference-request|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14634" PostTypeId="1" AcceptedAnswerId="14635" CreationDate="2022-07-25T03:11:46.927" Score="1" ViewCount="57" Body="&lt;p&gt;From what I understand Lawrence split the atom shortly after Cockcroft and Walton. The ions were 1.25Mev protons originating at the center of the cyclotron. A rather small circular device. I looked at the Schematics but does not show exactly how he did it.&lt;/p&gt;&#xA;&lt;p&gt;Although a different question is this essentially what Glen Seaborg used to bombard Neptunium in order to produce plutonium?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2022-07-25T14:52:24.817" Title="How were the ions produced in Lawrence's 1932 cyclotron?" Tags="|particle-physics|radioactivity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14635" PostTypeId="2" ParentId="14634" CreationDate="2022-07-25T13:23:28.560" Score="2" Body="&lt;p&gt;This is discussed in the original article in &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.40.19&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physics Review&lt;/a&gt; in the Experimental Arrangement section (starting on page 25).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The source of ions. An ideal source of ions is one that delivers to the&#xA;diametral region between the electrodes large quantities of ions with low&#xA;components of velocity normal to the plane of the accelerators. This requirement&#xA;has most conveniently been met in the present experiments merely by&#xA;having a filament placed above the diametral region from which a stream of&#xA;electrons pass down along the magnetic lines of force, generating ions of&#xA;gases in the tube.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is shown in Figures 1 and 2 in the paper. Basically a hot filament producing electrons to ionize the gas.&lt;/p&gt;&#xA;&lt;p&gt;Your second question is answered in another &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.70.555https://journals.aps.org/pr/abstract/10.1103/PhysRev.70.555&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physical Review&lt;/a&gt; article. Uranyl nitrate in a paraffin block was placed behind a beryllium target irradiated with deuterons. The nuclear reaction between Be and D produced neutrons.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2022-07-25T14:52:24.817" LastActivityDate="2022-07-25T14:52:24.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14637" PostTypeId="1" CreationDate="2022-07-26T21:08:22.137" Score="3" ViewCount="222" Body="&lt;p&gt;Why did it seem at the time of Gauss that the quantity we get by multiplication of electric field with the area would have some value to it? That is what was the motivation for Gauss to find the flux of the electric field. Even now the only reason I can think that why one would find it, is because it is something that is a 'natural' step for a theory of vector field to do. However, besides this, I have no physical motivation.&lt;/p&gt;&#xA;" OwnerUserId="16986" LastEditorUserId="12330" LastEditDate="2022-07-27T06:31:42.733" LastActivityDate="2022-07-27T06:31:42.733" Title="How did Gauss come up with his law?" Tags="|electromagnetism|gauss|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14638" PostTypeId="1" AcceptedAnswerId="16138" CreationDate="2022-07-27T08:30:05.137" Score="9" ViewCount="1246" Body="&lt;p&gt;The Stronger Feit-Thompson conjecture states that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There exist no distinct prime numbers &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; such that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$\dfrac {p^q - 1} {p - 1}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\dfrac {q^p - 1} {q - 1}$&lt;/span&gt;&#xA;are not coprime.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This was refuted by N.M. Stephens in July 1971 who published &lt;em&gt;On the Feit-Thompson Conjecture&lt;/em&gt; in Mathematics of Computation volume 25 no. 115 (p. 625) where he showed that for &lt;span class=&quot;math-container&quot;&gt;$p = 17$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q = 3313$&lt;/span&gt;, the above expressions have the common prime factor &lt;span class=&quot;math-container&quot;&gt;$112 \, 643$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_conjecture&quot; rel=&quot;noreferrer&quot;&gt;Feit–Thompson conjecture&lt;/a&gt; suggests his first name might have been Nelson, but I have been unable to corroborate this.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have any biographical details about N.M. Stephens?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="264" LastEditDate="2022-07-27T09:16:38.553" LastActivityDate="2024-03-16T14:15:26.983" Title="Who was N.M. Stephens who refuted the Stronger Feit-Thompson Conjecture?" Tags="|biographical-details|number-theory|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14639" PostTypeId="1" AcceptedAnswerId="14647" CreationDate="2022-07-27T08:59:35.913" Score="1" ViewCount="262" Body="&lt;p&gt;What contribution to mathematics did Einstein make, or was he only interested in Physics and derived formulas using mathematics?&lt;/p&gt;&#xA;" OwnerUserId="16989" LastEditorUserId="16318" LastEditDate="2022-07-27T13:33:49.243" LastActivityDate="2022-07-29T22:25:19.237" Title="Einstein's contribution to Mathematics?" Tags="|mathematics-social-history|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14640" PostTypeId="2" ParentId="14638" CreationDate="2022-07-27T09:14:08.793" Score="12" Body="&lt;p&gt;A google search for [Stephens + &lt;a href=&quot;https://www.google.com/search?q=%22Stephens%22+%22On+the+Feit-Thompson+Conjecture%22&quot; rel=&quot;noreferrer&quot;&gt;On the Feit-Thompson Conjecture&amp;quot;&lt;/a&gt; led me to &lt;a href=&quot;https://www.ams.org/journals/mcom/1971-25-115/S0025-5718-1971-0297686-1/S0025-5718-1971-0297686-1.pdf&quot; rel=&quot;noreferrer&quot;&gt;an online copy of the paper&lt;/a&gt;, and the bottom of the first page of this paper gives Stephens' affiliation at that time as &amp;quot;Atlas Computer Laboratory&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;A google-books search for &lt;a href=&quot;https://www.google.com/search?q=%22Stephens,+Nelson%22+%22Atlas+Computer+Laboratory%22&amp;amp;tbm=bks&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Stephens, Nelson&amp;quot; + &amp;quot;Atlas Computer Laboratory&amp;quot;&lt;/a&gt; led to a snippet view of the London Mathematical Society's &amp;quot;List of Members&amp;quot; for 1972 that shows &lt;a href=&quot;https://books.google.com/books?id=kPLuAAAAMAAJ&amp;amp;q=%22Stephens,+Nelson%22+%22Atlas+Computer+Laboratory%22&quot; rel=&quot;noreferrer&quot;&gt;the full entry of Stephen's entry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;STEPHENS, NELSON MALCOLM, B.Sc., Ph.D.; Joint Research Fellow, Atlas Computer Laboratory; &lt;em&gt;Pembroke College, Oxford&lt;/em&gt;. [1966&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A google-books search for &lt;a href=&quot;https://www.google.com/search?tbm=bks&amp;amp;q=%22Stephens%2C+Nelson+Malcolm%22&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Stephens, Nelson Malcolm&amp;quot;&lt;/a&gt; gives several hits, and in particular shows that Donald E. Knuth gave Stephens' full name in one of Knuth's &lt;strong&gt;The Art of Computer Programming&lt;/strong&gt; volumes AND a snippet view of the London Mathematical Society's &amp;quot;List of Members&amp;quot; for 1976 that shows &lt;a href=&quot;https://books.google.com/books?id=ofLuAAAAMAAJ&amp;amp;q=%22Stephens,+Nelson+Malcolm%22&quot; rel=&quot;noreferrer&quot;&gt;the full entry of Stephen's entry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;STEPHENS, NELSON MALCOLM, B.Sc., Ph.D.; &lt;em&gt;University College, Cardiff CFI IXL&lt;/em&gt; [1966&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A google search for &lt;a href=&quot;https://www.google.com/search?q=%22Nelson+Malcolm+Stephens%22&amp;amp;filter=0&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Nelson Malcolm Stephens&amp;quot;&lt;/a&gt; shows some Pembroke College Records for 1970, 1971, 1972 that are accessible online.&lt;/p&gt;&#xA;&lt;p&gt;With this information in hand you can try various other search words/phrases, such as &lt;a href=&quot;https://www.google.com/search?q=%22Nelson+M+Stephens%22+%22Manchester%22&amp;amp;filter=0&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Nelson M Stephens&amp;quot; + &amp;quot;Manchester&amp;quot;&lt;/a&gt;, for items missed by the previous searches.&lt;/p&gt;&#xA;" OwnerUserId="264" LastActivityDate="2022-07-27T09:14:08.793" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14641" PostTypeId="1" CreationDate="2022-07-27T13:43:11.583" Score="1" ViewCount="110" Body="&lt;p&gt;I'm wondering how the Egyptians figured out the false position method in solving their equations? Is there a good source that can tell me how they thought through this?&lt;/p&gt;&#xA;" OwnerUserId="16826" LastEditorUserId="16826" LastEditDate="2022-07-27T18:07:08.190" LastActivityDate="2022-07-27T18:07:08.190" Title="I'm wondering how the Egyptians figured out the false position method in solving their equations?" Tags="|mathematical-logic|algebra|ancient-egypt|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14642" PostTypeId="1" AcceptedAnswerId="14644" CreationDate="2022-07-27T21:31:44.877" Score="1" ViewCount="128" Body="&lt;p&gt;My impression is that the equivalence of the force experienced in an accelerating elevator to the force experienced as a result of gravity is what inspired Einstein to quantify gravity in terms of the curvature of space. However I don’t see the connection. If I were being strict, I would quantify gravity as the acceleration of motion and this would lead me to derive equations like this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where does the idea of space curvature come from? What were the other models which never saw the light of day? Why didn't he, for example, posit a gradient in the size of the occupiable quanta of space- as smaller cells are more numerous in a given volume and so allow many more degrees of freedom of occupancy --- the random motion of atoms would tend to move them in that direction. This would least be in line with the accepted conceptions of statistical mechanics.&lt;/p&gt;&#xA;&lt;p&gt;The idea of space curvature seems idiosyncratic as it has nothing to do with motion. There are some clues in this thread&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://physics.stackexchange.com/questions/38379/how-did-einstein-derive-general-relativity&quot;&gt;https://physics.stackexchange.com/questions/38379/how-did-einstein-derive-general-relativity&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is this statement&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;eventually he realized that the spacetime had to be curved, by arguments based on the equivalence principle, and it must be described by the Riemannian geometry. &amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The equivalence principle relates the sensation of acceleration to different reference frames. Curvature of any kind is not mentioned.&lt;/p&gt;&#xA;&lt;p&gt;Is there a coherent description of the steps/jumps in thought that led to the idea of curvature of space?&lt;/p&gt;&#xA;" OwnerUserId="16997" LastActivityDate="2022-07-28T04:58:20.593" Title="Is there history of all the blind alleys Einstein followed between 1905 and 1915 in formulating General Relativity?" Tags="|relativity-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14644" PostTypeId="2" ParentId="14642" CreationDate="2022-07-28T04:29:44.847" Score="3" Body="&lt;p&gt;The account in &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0192806726&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Subtle is the Lord&lt;/a&gt;, by Abraham Pais, is excellent.&lt;/p&gt;&#xA;" OwnerUserId="1864" LastEditorUserId="1864" LastEditDate="2022-07-28T04:58:20.593" LastActivityDate="2022-07-28T04:58:20.593" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14646" PostTypeId="1" CreationDate="2022-07-28T17:18:02.923" Score="4" ViewCount="121" Body="&lt;p&gt;It is clear that the understanding that matter is made of atoms allows us to refine our calculations and go beyond the usual optics. Quantum mechanics solved many issues like the black body radiation, the emission lines of gases, stability of atoms, photoelectric effect, conduction and heat capacity of metals and so on...&lt;/p&gt;&#xA;&lt;p&gt;I am looking for an optical property of matter that was not well accounted using classical mechanics. Even if the true nature of optical properties like the dielectric function of materials can be derived from the band structure of solids, the Drude model for metals and Lorentz model for dielectrics were derived before quantum mechanics and work very well for macroscopic experiments. I am wondering are there any optical properties of matter that were discovered before quantum mechanics that could not be explained using classical mechanics ? or at least that it took a long time to be understood microscopically.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-01-25T18:11:56.810" LastActivityDate="2023-01-25T18:11:56.810" Title="Did quantum mechanics solve any open problems related to the optical properties of solids?" Tags="|physics|optics|solid-state-physics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14647" PostTypeId="2" ParentId="14639" CreationDate="2022-07-29T22:25:19.237" Score="1" Body="&lt;p&gt;There is a discussion of Einstein's early and late views on mathematics at &lt;a href=&quot;https://www.quora.com/Why-did-Einstein-write-Since-the-mathematicians-have-invaded-the-theory-of-relativity-I-do-not-understand-it-myself-anymore-After-all-GR-relies-heavily-on-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.quora.com/Why-did-Einstein-write-Since-the-mathematicians-have-invaded-the-theory-of-relativity-I-do-not-understand-it-myself-anymore-After-all-GR-relies-heavily-on-mathematics&lt;/a&gt;. Although he had been taught by Minkowski at ETH Zurich, he apparently had no relationship with him and disliked his approach to spacetime in special relativity. When he subsequently exploited this in general relativity, he is said to have acknowledged his debt.&lt;/p&gt;&#xA;" OwnerUserId="2822" LastActivityDate="2022-07-29T22:25:19.237" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14649" PostTypeId="1" AcceptedAnswerId="14656" CreationDate="2022-07-30T11:51:28.337" Score="5" ViewCount="260" Body="&lt;p&gt;This concerns the Stephen Barr who wrote &lt;em&gt;Experiments in Topology&lt;/em&gt; in 1964, available from Dover Publications.&lt;/p&gt;&#xA;&lt;p&gt;All I know about him is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;He wrote the above book.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He was a friend and possibly colleague of Martin Gardner, who included some of his work in his &lt;em&gt;Mathematical Games&lt;/em&gt; column in &lt;em&gt;Scientific American&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He also published in &lt;em&gt;Recreational Magic Magazine&lt;/em&gt;, and so credited Joseph Madachy in the above book&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He was apparently taught by Milton Boyd.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;He may also be the same Stephen Barr who wrote the (apparently mathematical) whodunnit &amp;quot;The N-Plus-1th Degree&amp;quot; in 1968.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Note: he is &lt;em&gt;very probably not&lt;/em&gt; the same person as the physicist Stephen Matthew Barr, who was born in 1953 (which would have made him 11 years old when he wrote &lt;em&gt;Experiments in Topology&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Somebody &lt;em&gt;must&lt;/em&gt; know something about him.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2022-08-01T05:20:58.940" Title="Who is or was Stephen Barr, author of Experiments in Topology?" Tags="|biographical-details|topology|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14651" PostTypeId="2" ParentId="14649" CreationDate="2022-07-31T04:32:18.927" Score="5" Body="&lt;p&gt;I found some (very little) information in a preface (by Yaglom) to a Russian translation of some of Barr's books:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Предлагаемый читателю сборник головоломок составлен американским&#xA;писателем и любителем математики Стивеном Барром. В США он вышел в&#xA;свет в виде трех отдельных книг, имевших значительный читательский&#xA;успех,— возможно, даже больший, чем успех чисто беллетристических&#xA;произведений Барра. К математике Барр обратился довольно поздно,&#xA;заинтересовавшись задачами моделирования сложных поверхностей,&#xA;обсуждаемыми в последней части настоящей книги. Его интерес&#xA;стимулировали внимание и поддержка такого корифея занимательной&#xA;математики, как хорошо известный нашим читателям Мартин Гарднер.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A rough translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The collection of puzzles offered to the reader was compiled by the&#xA;American writer and mathematician Stephen Barr. In the USA it was&#xA;published as three separate books that had a significant success,&#xA;perhaps even greater success than Barr's pure fiction. Barr turned to&#xA;mathematics rather late as he got interested in problems of modeling&#xA;of complex surfaces, which problems are discussed in the last part of&#xA;this book. His interest was stimulated by the attention and support of&#xA;such a leading figure of entertaining mathematics as Martin Gardner,&#xA;who is well-known to our readers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;EDIT (7/31/2022): Sorry for the (grave) error in the translation: it is actually &amp;quot;writer and amateur mathematician&amp;quot;, not &amp;quot;writer and mathematician&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastEditorUserId="9808" LastEditDate="2022-07-31T11:19:14.550" LastActivityDate="2022-07-31T11:19:14.550" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14652" PostTypeId="1" CreationDate="2022-07-31T10:25:32.800" Score="14" ViewCount="1533" Body="&lt;p&gt;In the modern day, especially with the advent of &lt;span class=&quot;math-container&quot;&gt;$\mathrm\TeX$&lt;/span&gt;, it is common practice to italicise variables. This can be seen as far back as Hann, J. (&lt;a href=&quot;https://www.google.co.in/books/edition/Examples_on_the_Integral_Calculus/bTk7AQAAIAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;1850&lt;/a&gt;). &lt;em&gt;Examples on the Integral Calculus&lt;/em&gt;. What's the oldest example of this and when did it become standard?&lt;/p&gt;&#xA;" OwnerUserId="17014" LastActivityDate="2022-08-07T16:37:57.733" Title="History of italicising variables and mathematical formatting in general" Tags="|notation|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14654" PostTypeId="2" ParentId="14652" CreationDate="2022-07-31T14:42:41.993" Score="15" Body="&lt;p&gt;In another answer of mine, we see an image of&#xA;&lt;a href=&quot;https://archive.org/details/disquisitionesg00gausgoog/page/n30/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Disquisitiones generales circa superficies curvas&amp;quot;&lt;/a&gt; from 1828 by C.F. Gauss.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/qYB6c.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/qYB6c.jpg&quot; alt=&quot;Gauss&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that already here the mathematical letters are set in italic.  (But the operator &lt;span class=&quot;math-container&quot;&gt;$\mathrm{d}$&lt;/span&gt; for derivatives is in roman.) I expect this practice goes back much further than 1828.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Another. &lt;em&gt;Institutiones Calculus Differentialis&lt;/em&gt;, L. Euler, 1787&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ICLIa.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ICLIa.jpg&quot; alt=&quot;Euler&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is &lt;em&gt;Analyse des Infiniment Petits&lt;/em&gt;, l'Hospital, 1696&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/jFL6L.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jFL6L.jpg&quot; alt=&quot;lhospital&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;Even then, using italic.  Note the &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; for differentials is italic as well.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2022-08-02T10:24:45.023" LastActivityDate="2022-08-02T10:24:45.023" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14655" PostTypeId="2" ParentId="14652" CreationDate="2022-07-31T22:39:39.660" Score="15" Body="&lt;p&gt;Rene Descartes's &lt;em&gt;La Géométrie&lt;/em&gt; of 1637 uses the same typographic convention used today in France: in equations, lower case letters  are set in italics but capital letters are set in upright text.  This can be seen on the third&#xA;&lt;a href=&quot;https://diglib.hab.de/drucke/64-25-quod-1/start.htm?image=00385&quot; rel=&quot;nofollow noreferrer&quot;&gt;page&lt;/a&gt;&#xA;of the book.  It can also be seen in  the 1925 bilingual &lt;a href=&quot;http://explore.bl.uk/primo_library/libweb/action/dlSearch.do?vid=BLVU1&amp;amp;institution=BL&amp;amp;search_scope=LSCOP-ALL&amp;amp;query=any,contains,descartes+geometrie%20smith&amp;amp;tab=local_tab&quot; rel=&quot;nofollow noreferrer&quot;&gt;edition&lt;/a&gt; by D. E. Smith and M. L. Latham, with a facsimile of the original and translation on facing pages. (Reprinted in 1954 by &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0486600688&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dover&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The web page &lt;a href=&quot;https://debart.pagesperso-orange.fr/geometrie/index.html#oeuvre&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://debart.pagesperso-orange.fr/geometrie/index.html#oeuvre&lt;/a&gt;&#xA;gives the same facsimile pages, as best as I can tell.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2022-08-07T16:37:57.733" LastActivityDate="2022-08-07T16:37:57.733" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14656" PostTypeId="2" ParentId="14649" CreationDate="2022-08-01T04:03:23.203" Score="8" Body="&lt;p&gt;The cross references to other titles by the same author in Stephen Barr's books quickly revealed that he is also the author of the following:&lt;/p&gt;&#xA;&lt;p&gt;Stephen Barr, &lt;em&gt;A Miscellany of Puzzles: Mathematical and Otherwise&lt;/em&gt;, New York: Thomas Y. Crowell 1965. Republished as &lt;em&gt;Intriguing Puzzles in Math and Logic&lt;/em&gt;, Dover 1994.&lt;/p&gt;&#xA;&lt;p&gt;Stephen Barr, &lt;em&gt;Second Miscellany of Puzzles: Mathematical and Otherwise&lt;/em&gt;, New York: Macmillan 1969. Republished as &lt;em&gt;Mathematical Brain Benders&lt;/em&gt;, Dover 1982.&lt;/p&gt;&#xA;&lt;p&gt;Stephen Barr, &lt;em&gt;Puzzlequiz: Wit Twisters, Brain Teasers, Riddles, Puzzles, and Tough Questions&lt;/em&gt;, New York: Thomas Y. Crowell 1978. Republished under the same title by Harper &amp;amp; Row 1990.&lt;/p&gt;&#xA;&lt;p&gt;The most important clue provided in the question was Barr's friendship with Martin Gardner. Gardner makes an appearance in blurbs for Barr's books, while Barr is mentioned multiple times in Gardner's publications, with indications that besides having an interest in topology as an amateur mathematician he was also an avid creator of puzzles. Two examples:&lt;/p&gt;&#xA;&lt;p&gt;Martin Gardner, &lt;em&gt;New Mathematical Diversions: Revised Edition&lt;/em&gt;, Mathematical Association of America 1995, &lt;a href=&quot;https://www.google.com/books/edition/New_Mathematical_Diversions/VE0FEAAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA37&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 37&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;Barr's Belt&lt;br /&gt;&#xA;Stephen Barr of Woodstock, New York, says that his dressing gown has a long cloth belt, the ends of which are cut at 45-degree angles [...]&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Martin Gardner, &lt;em&gt;Martin Gardner’s 6th Book of Mathematical Diversions from Scientific American.&lt;/em&gt;, American Mathematical Society 2020, &lt;a href=&quot;https://www.google.com/books/edition/Martin_Gardner_s_6th_Book_of_Mathematica/Wk0FEAAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA13&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;pg. 13&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now, it is true that at the time he wrote this apparently no one had tried to make a paper Klein bottle, but that was before Stephen Barr, a science-fiction writer and amateur mathematician of Woodstock, New York, turned his attention to the problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Given that Barr is identified as a science-fiction writer living in Woodstock, NY (a small town of some five-thousand inhabitants), one quickly finds various SF short stories by him, mostly published in &lt;em&gt;Galaxy Science Fiction&lt;/em&gt; and &lt;em&gt;The Magazine of Fantasy and Science Fiction&lt;/em&gt;, which I gather were premier American outlets for this genre in the relevant time frame.&lt;/p&gt;&#xA;&lt;p&gt;Barr's science fiction writings range from the 1950s to the 1970s, for example: &amp;quot;I Am a Nucleus&amp;quot; (1957), &amp;quot;Callahan and the Wheelies&amp;quot; (1960), &amp;quot;A Sense of the Future&amp;quot; (1972). A comprehensive list can be found at the &lt;a href=&quot;http://www.isfdb.org/cgi-bin/ch.cgi?11926&quot; rel=&quot;noreferrer&quot;&gt;Internet Speculative Fiction Database&lt;/a&gt;. &lt;em&gt;The Atlantic&lt;/em&gt; published &lt;a href=&quot;https://www.theatlantic.com/author/stephen-barr/&quot; rel=&quot;noreferrer&quot;&gt;two articles&lt;/a&gt; by Barr, in 1955 and 1961, and identified him as a&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] free-lance writer who lives in Woodstock, New York.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.findagrave.com/&quot; rel=&quot;noreferrer&quot;&gt;Find a Grave&lt;/a&gt; is a good resource for finding information about dead people, and given the dates of Barr's publications, I considered it very likely that he was no longer alive, and looked there for someone of that name buried in Woodstock, NY. Sure enough, they have a picture of a relevant &lt;a href=&quot;https://www.findagrave.com/memorial/147733160/stephen-barr&quot; rel=&quot;noreferrer&quot;&gt;grave marker&lt;/a&gt; in Woodstock Cemetery, which shows that Stephen Barr was born in 1904 and died in 1989.&lt;/p&gt;&#xA;&lt;p&gt;With all the preceding information in hand, I was finally able to track down a brief biography of Barr which confirmed the biographical data already found and adds new details:&lt;/p&gt;&#xA;&lt;p&gt;Otto Penzler (ed.), &lt;em&gt;The Black Lizard Big Book of Locked-Room Mysteries&lt;/em&gt;, Knopf Doubleday 2014, &lt;a href=&quot;https://books.google.com/books?id=sy_aCwAAQBAJ&amp;amp;pg=PA584#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;pg. 584&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Barr was born to American parents in England and grew up there before moving to the United States, possibly around 1930. He worked as a commercial artist and architectural draftsman before turning to writing full time in 1955.&lt;/p&gt;&#xA;&lt;p&gt;From the fanzine &lt;em&gt;Yandro&lt;/em&gt;, Vol. 15, No. 8, &lt;a href=&quot;http://fanac.org/fanzines/Yandro/Yandro174.pdf&quot; rel=&quot;noreferrer&quot;&gt;September 1967&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Remember that fuss about Stephen Barr allegedly plagiarizing stories&#xA;for fanzine submission? Well, he had an entry in CONTEMPORARY AUTHORS and&#xA;besides such things as his birth in England (in 1904) and present home in&#xA;Woodstock, N.Y., we learn that he was long a commercial artist and&#xA;architectural draftsman, turning free lance writer in 1955. He also&#xA;contributed stories and articles to &lt;em&gt;Vogue&lt;/em&gt;, &lt;em&gt;Mademoiselle&lt;/em&gt;, &lt;em&gt;Atlantic&lt;/em&gt;,&#xA;&lt;em&gt;Harpers&lt;/em&gt;, &lt;em&gt;Playboy&lt;/em&gt; and similar such crudzines.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Google provides only a snippet view of &lt;em&gt;Contemporary Authors&lt;/em&gt;, Vol. 1, Gale Research Company 1975, p. 39:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;BARR, Stephen&lt;/strong&gt; 1904 PERSONAL: Born 1904, in England, son of U.S. nationals; married Anna Keiley. Education: Attended schools in England. Residence: Woodstock, N.Y. CAREER: Commercial [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A 1940 &lt;a href=&quot;http://cityrecord.engineering.nyu.edu/data/1940/1940-04-30%20part%200010.pdf&quot; rel=&quot;noreferrer&quot;&gt;civil list&lt;/a&gt; for the City of New York shows Anna Keiley Barr working as a social investigator at District Office 25 (212 E. 125th St. Manhattan) of the Department of Welfare, with a start date of Jan. 1, 1938.&lt;/p&gt;&#xA;&lt;p&gt;UK records show that the &lt;a href=&quot;https://www.freebmd.org.uk/cgi/information.pl?cite=sVHMcft4C9flkQOEcMGR0g&amp;amp;scan=1&quot; rel=&quot;noreferrer&quot;&gt;birth of a Stephen Barr&lt;/a&gt; was registered in Oct.-Dec. 1904 in Uxbridge in Middlesex. This is a suburb of London. What is publicly available is a summary page; I have not found a way to access the actual birth record.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-08-01T05:20:58.940" LastActivityDate="2022-08-01T05:20:58.940" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14658" PostTypeId="1" CreationDate="2022-08-01T12:14:54.157" Score="0" ViewCount="326" Body="&lt;p&gt;I am doing an introduction to Maxwell's equations, and it is said that originally the equations were in component form. Can anyone help with the derivation of the fourth equation? I have checked many sources, and I cannot find the derivation.&lt;/p&gt;&#xA;" OwnerUserId="17023" LastEditorUserId="10106" LastEditDate="2022-08-02T06:46:36.783" LastActivityDate="2022-08-10T16:19:58.220" Title="Component form of the fourth Maxwell's equation" Tags="|physics|electromagnetism|maxwell|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14659" PostTypeId="1" CreationDate="2022-08-03T01:59:15.413" Score="1" ViewCount="66" Body="&lt;p&gt;Consider a classical scalar-tensor theory (I'm following &lt;a href=&quot;https://arxiv.org/abs/gr-qc/0403063v3&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;   S=\int d^4x\sqrt{-g}\left(&#xA;       \frac{1}{2\kappa^2}\mathcal{A}\left(\Phi\right)R&#xA;       -\frac{1}{2}\mathcal{B}\left(\Phi\right)\left(\nabla\Phi\right)^2&#xA;       - V\left(\Phi\right) &#xA;   \right)&#xA;   +&#xA;   S_m\left[e^{2\alpha\left(\Phi\right)}g_{\mu\nu},\psi_m\right]&#xA;\end{equation}&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Ricci scalar is &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;, the scalar field is &lt;span class=&quot;math-container&quot;&gt;$\Phi$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\psi_m$&lt;/span&gt; are the ''matter fields''. The theory is said to be in the ''Einstein frame'' if &lt;span class=&quot;math-container&quot;&gt;$\mathcal{A}=1,\mathcal{B}=1$&lt;/span&gt;, and is said to be in the ''Jordan frame'' if &lt;span class=&quot;math-container&quot;&gt;$\alpha=0,\mathcal{B}=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question is: what is the origin of this terminology? It looks like the term ''Einstein frame'' was introduced by &lt;a href=&quot;https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-25/issue-2/Black-holes-in-the-Brans-Dicke-theory-of-gravitation/cmp/1103857885.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hawking&lt;/a&gt;, but I can't figure out where the first paper where ''Jordan frame'' was coined (in Hawking's paper he calls it the ''Brans-Dicke frame'').&lt;/p&gt;&#xA;" OwnerUserId="17032" LastEditorUserId="5989" LastEditDate="2022-08-03T04:09:19.653" LastActivityDate="2022-08-03T04:09:19.653" Title="Origin of the terminology &quot;Einstein frame&quot; and &quot;Jordan frame&quot;" Tags="|theoretical-physics|relativity-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14660" PostTypeId="1" AcceptedAnswerId="14661" CreationDate="2022-08-03T17:26:48.500" Score="10" ViewCount="2130" Body="&lt;p&gt;I'm curious as to what the rationale was, and who the idea occurred to, for the ¬ symbol. I'll grant that more common mathematical symbols like +, −, × and ÷ are also likely unknown, but they seem to vaguely relate to their meaning in an onomatopoetic sense, or are at least clearly derived from the next-simplest related operation (like × to +). Even in set theory, ∪ denotes union—fair enough, probably from a language that began the word with &amp;quot;u&amp;quot;—and ∩ denotes intersection, its logical opposite hence upside-down. Even conjunction and disjunction (∧, ∨) seem to derive from this standard.&lt;/p&gt;&#xA;&lt;p&gt;After all of that, I have to admit, for &amp;quot;¬&amp;quot; to imply predicate-negation makes no intrinsic sense to me. When I first saw the symbol in college, even as a philomath, I had absolutely no idea what it was trying to say to me.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know where it originated and why?&lt;/p&gt;&#xA;" OwnerUserId="17037" LastEditorUserId="17037" LastEditDate="2022-08-11T14:06:58.020" LastActivityDate="2022-08-11T14:06:58.020" Title="What is the origin of the negation ( ¬ ) operator from logic?" Tags="|mathematics|biographical-details|terminology|mathematics-social-history|logic|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14661" PostTypeId="2" ParentId="14660" CreationDate="2022-08-03T19:28:58.353" Score="21" Body="&lt;p&gt;If vague resemblance is enough, then &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$¬$&lt;/span&gt;&amp;quot; resembles &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;&amp;quot;, which denotes negation in arithmetic. Lambert in &lt;em&gt;Sechs Versuche einer Zeichenkunst in der Vernunftlehre&lt;/em&gt; (1782) and Boole in &lt;a href=&quot;https://archive.org/details/investigationofl01bool/page/27/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laws of Thought (1854)&lt;/a&gt; just used &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;&amp;quot; for negation, and &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$+,×$&lt;/span&gt;&amp;quot; for &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\lor,\land$&lt;/span&gt;&amp;quot;. The latter explicitly cited the arithmetical analogy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Signs of operation, as &lt;span class=&quot;math-container&quot;&gt;$+,-,×$&lt;/span&gt; standing for those operations of the mind by which the conceptions of things are combined or resolved so as to form new conceptions involving the same elements... And these symbols of Logic are in their use subject to definite&#xA;laws, partly agreeing with and partly differing from the laws of the corresponding symbols in the science of Algebra.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Peano modified &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;&amp;quot; to &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt;&amp;quot; in &lt;a href=&quot;https://archive.org/details/peano-studii-di-logica-matematica/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Studii di logica matematica&lt;/em&gt; (1897)&lt;/a&gt;, possibly because, unlike Lambert and Boole, he was using logical symbols alongside arithmetic ones. Russell and Whitehead followed suit in &lt;em&gt;Principia&lt;/em&gt; (1910-13). Heyting introduced &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$¬$&lt;/span&gt;&amp;quot; in &lt;em&gt;Die formalen Regeln der intuitionistischen Logik&lt;/em&gt; (1930).&lt;/p&gt;&#xA;&lt;p&gt;For more information and references to the original works on this and other symbols, see &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/math-origins-the-logical-symbols&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tou, Math Origins: The Logical Symbols&lt;/a&gt; and &lt;a href=&quot;http://www.math.hawaii.edu/%7Etom/history/set.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10106" LastEditDate="2022-08-06T06:15:53.587" LastActivityDate="2022-08-06T06:15:53.587" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14662" PostTypeId="1" CreationDate="2022-08-04T00:06:31.447" Score="0" ViewCount="96" Body="&lt;p&gt;In his forward to Hardy's &lt;em&gt;Apology,&lt;/em&gt; C. P. Snow remarks that after his (Hardy's) 1939 heart attack, Hardy &amp;quot;recovered enough ... to play his pleasing elaboration (with a complicated set of bisques) of Trinity bowls.&amp;quot; From what I have been able to determine, Trinity bowls was probably a form of lawn bowling, and bisques a sort of &amp;quot;free turn&amp;quot; for players of lesser skill. I'm assuming that the &amp;quot;complicated set&amp;quot; refers to a complicated set of conditions under which the &amp;quot;free turns&amp;quot; might be taken. Can anyone illuminate me on this issue? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="16670" LastActivityDate="2022-08-04T00:06:31.447" Title="What is/are Trinity bowls?" Tags="|biographical-details|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14663" PostTypeId="1" AcceptedAnswerId="14664" CreationDate="2022-08-05T05:45:10.863" Score="-3" ViewCount="111" Body="&lt;p&gt;Did physics resulted in invention of maths or Maths resulted in invention of physics ??&lt;/p&gt;&#xA;" OwnerUserId="16989" LastActivityDate="2022-08-06T00:33:09.670" Title="Did physics resulted in invention of maths or vice a versa" Tags="|mathematical-physics|" AnswerCount="1" CommentCount="0" ClosedDate="2022-08-06T06:12:33.157" ContentLicense="CC BY-SA 4.0" />
  <row Id="14664" PostTypeId="2" ParentId="14663" CreationDate="2022-08-05T13:11:59.647" Score="2" Body="&lt;p&gt;It is not possible to tell before you define what exactly is considered as &amp;quot;math&amp;quot; and what is counted as &amp;quot;physics&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The origins of math and physics in the modern sense of these words are legendary, and the legends credit Thales with mathematics and&#xA;Pythagoras with both physics and mathematics. Their life spans are not exactly known but they intersect.&lt;/p&gt;&#xA;&lt;p&gt;In this statement the crucial part is &amp;quot;modern sense of these words&amp;quot;, that is mathematics with proofs, and physics with experiments and mathematical models. So this is quite a narrow sense which excludes for example all Babylonian and Egyptian and earlier math.&lt;/p&gt;&#xA;&lt;p&gt;The earliest known experiment which can be considered as true physics in the modern sense of the word is the discovery of relation between the length of a string and the pitch of the sound it produces. This discovery is traditionally credited to Pythagoras. Thales is credited with the first proofs of theorems. But all this information comes from the writers who lived at least two centuries later than Thales and Pythagoras.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-08-06T00:33:09.670" LastActivityDate="2022-08-06T00:33:09.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14665" PostTypeId="1" CreationDate="2022-08-07T13:58:48.087" Score="0" ViewCount="69" Body="&lt;p&gt;My understanding that is Einstein and Schrodinger both were in correspondence (see &lt;a href=&quot;https://www.youtube.com/watch?v=03ReIvXKrrU&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) and looking to reformulate General Relativity in terms of connections (which lead to this &lt;a href=&quot;https://www.salon.com/2015/04/18/albert_is_an_old_fool_einstein_vs_schrodinger_in_battle_of_the_nobel_laureates/&quot; rel=&quot;nofollow noreferrer&quot;&gt;debacle&lt;/a&gt;). Now, from the viewpoint of quantum gravity this is extremely useful. But I suspect that this could not have been their motivation given their odds with the theory. Is there any historical record or otherwise on what motivated them to look for this theory?&lt;/p&gt;&#xA;" OwnerUserId="10067" LastEditorUserId="10067" LastEditDate="2022-08-08T06:59:49.803" LastActivityDate="2022-08-08T06:59:49.803" Title="Motivation to reformulate General Relativity in terms of connections?" Tags="|einstein|gravity|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14667" PostTypeId="1" AcceptedAnswerId="14671" CreationDate="2022-08-07T20:01:26.187" Score="10" ViewCount="197" Body="&lt;p&gt;I was surprised to find that Oliver Byrne's 1847 marvelous &lt;em&gt;The Elements of Euclid&lt;/em&gt; (color version)&lt;sup&gt;1&lt;/sup&gt; uses &lt;span class=&quot;math-container&quot;&gt;$\sqsubset$&lt;/span&gt; to mean &amp;quot;greater than&amp;quot; and &lt;span class=&quot;math-container&quot;&gt;$\sqsupset$&lt;/span&gt; to mean &amp;quot;less than,&amp;quot;&#xA;in contrast our current &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; (p. xxvii).&#xA;This is especially puzzling because:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The symbols &amp;lt; and &amp;gt; first appear in Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (The Analytical Arts Applied to Solving Algebraic Equations) by Thomas Harriot (1560-1621).&amp;quot;&#xA;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathsym/relation/&quot; rel=&quot;noreferrer&quot;&gt;link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any speculation on Byrne's motivation?&#xA;To me it seems so non-intuitive that &lt;span class=&quot;math-container&quot;&gt;$\sqsubset$&lt;/span&gt; should mean &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; that I am surprised&#xA;it was used in a book that tries to make the math easier to grasp.&#xA;Perhaps &lt;span class=&quot;math-container&quot;&gt;$\sqsubset$&lt;/span&gt; was in common use in parallel with &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;   &lt;a href=&quot;https://i.stack.imgur.com/3GQdN.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3GQdN.jpg&quot; alt=&quot;Cover&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt;&#xA;Byrne, Oliver. &lt;em&gt;The first six books of the Elements of Euclid&lt;/em&gt;: in which coloured diagrams and symbols are used instead of letters for the greater ease of learners. William Pickering, 1847.&lt;/p&gt;&#xA;" OwnerUserId="606" LastEditorUserId="55" LastEditDate="2023-11-15T12:05:06.250" LastActivityDate="2023-11-15T12:05:06.250" Title="History of greater-than symbol used in reverse?" Tags="|notation|euclidean-geometry|euclid|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14668" PostTypeId="1" CreationDate="2022-08-07T20:05:26.643" Score="5" ViewCount="412" Body="&lt;p&gt;The recent &lt;a href=&quot;https://www.youtube.com/watch?v=bylFzBxzQ9M&quot; rel=&quot;noreferrer&quot;&gt;Numberphile video&lt;/a&gt; on the famous Problem 6 of the 1988 IMO (mentioned in a &lt;a href=&quot;https://hsm.stackexchange.com/a/14666/&quot;&gt;recent answer on this site&lt;/a&gt;) got me wondering:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Who came up with this problem in the first place, and how did they come up with it?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://artofproblemsolving.com/wiki/index.php/IMO_Problems_and_Solutions#1988&quot; rel=&quot;noreferrer&quot;&gt;Art of Problem Solving website&lt;/a&gt; attributes the problem to Stephan Beck of West Germany, but I have not been able to find any corroboration of this attribution, or any other information about how Beck came up with the problem.&lt;/p&gt;&#xA;&lt;p&gt;It could be that Beck was familiar with the &lt;a href=&quot;https://math.stackexchange.com/a/1898442/&quot;&gt;reduction theory of quadratic forms&lt;/a&gt; and came up with the problem that way, but Beck does not seem to be listed in the Mathematics Genealogy Project, and I would be mildly surprised that someone without a doctorate in mathematics would be familiar with that topic.&#xA;If Beck is still alive then one could ask him directly, but I do not know how to contact him.&lt;/p&gt;&#xA;&lt;p&gt;For completeness, let me also mention that I found the following paragraph on page 127 of the book &lt;i&gt;Problem-Solving Strategies&lt;/i&gt; by Arthur Engel, but it does not answer my question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The next problem was submitted in 1988 by the FRG. Nobody of the six members&#xA;of the Australian problem committee could solve it. Two of the members were&#xA;Georges Szekeres and his wife, both famous problem solvers and problem creators.&#xA;Since it was a number theoretic problem it was sent to the four most renowned&#xA;Australian number theorists. They were asked to work on it for six hours. None of&#xA;them could solve it in this time. The problem committee submitted it to the jury of&#xA;the XXIX IMO marked with a double asterisk, which meant a superhard problem,&#xA;possibly too hard to pose. After a long discussion, the jury finally had the courage&#xA;to choose it as the last problem of the competition. Eleven students gave perfect&#xA;solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2022-08-07T20:05:26.643" Title="Origin of Problem 6 on the 1988 International Mathematical Olympiad" Tags="|mathematics|number-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14669" PostTypeId="2" ParentId="12119" CreationDate="2022-08-08T04:36:37.323" Score="1" Body="&lt;p&gt;I just want to point out that  Don B. Zagier's one-paragraph solution to this &lt;em&gt;notorious&lt;/em&gt; problem is more or less &lt;a href=&quot;https://math.stackexchange.com/q/28438/319099&quot;&gt;well-known&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; be the quotient. We may suppose that &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt; is a minimal positive solution of the equation &lt;span class=&quot;math-container&quot;&gt;$(a^2 + b^2) = m(ab + 1)$&lt;/span&gt;&lt;br /&gt;&#xA;(i.e. one with the smallest value of &lt;span class=&quot;math-container&quot;&gt;$a + b$&lt;/span&gt;) for this value of &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt;. Without loss of generality, suppose that &lt;span class=&quot;math-container&quot;&gt;$b ≥ a ≥ 0$&lt;/span&gt;, and set &lt;span class=&quot;math-container&quot;&gt;$b' = ma - b$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$a^2 = bb' + m$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$b'$&lt;/span&gt; is positive, then this equation implies &lt;span class=&quot;math-container&quot;&gt;$b' = (a^2 - m)/b &amp;lt; a^2/b &amp;lt; b$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$(a, b')$&lt;/span&gt; is a smaller positive solution of the equation of which &lt;span class=&quot;math-container&quot;&gt;$(a, b)$&lt;/span&gt; was supposed to be the minimal solution. If &lt;span class=&quot;math-container&quot;&gt;$b'$&lt;/span&gt; is negative, then &lt;span class=&quot;math-container&quot;&gt;$m = a^2 - bb' ≥ a^2 + b &amp;gt; b &amp;gt; ma ≥ m$&lt;/span&gt;, a contradiction. Hence &lt;span class=&quot;math-container&quot;&gt;$b' = 0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m = a^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;: This solution is just the explicit result of applying reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form &lt;span class=&quot;math-container&quot;&gt;$x^2 + mxy + y^2$&lt;/span&gt;, which is the unique reduced quadratic form in its equivalence class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(cf. &lt;a href=&quot;http://www-groups.mcs.st-andrews.ac.uk/%7Ejohn/Zagier/Solution1.3.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www-groups.mcs.st-andrews.ac.uk/~john/Zagier/Solution1.3.html&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Maybe E. Atanassov's solution was similar to the above solution by Zagier.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="6780" LastEditDate="2022-11-01T23:20:46.760" LastActivityDate="2022-11-01T23:20:46.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14670" PostTypeId="1" CreationDate="2022-08-08T06:57:46.180" Score="3" ViewCount="104" Body="&lt;p&gt;I am writing a book on modern mathematics and the Latin language. My main examples are Newton, Euler, Gauss, etc. and some others, but all men.&lt;/p&gt;&#xA;&lt;p&gt;Is there a woman who has written important mathematics in Latin?&lt;/p&gt;&#xA;" OwnerUserId="17060" LastActivityDate="2022-08-08T06:57:46.180" Title="Are there any famous female mathematicians who have written in Latin?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14671" PostTypeId="2" ParentId="14667" CreationDate="2022-08-08T08:42:56.033" Score="10" Body="&lt;p&gt;Byrne's symbols are variations of Oughtred's, contamporaneous with Harriot's, see &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp/page/n211/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Cajori, History Of Mathematical Notations, vol. I, p.192&lt;/a&gt;. They were adopted by Barrow, Newton's teacher, in his edition of Euclid, &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp/page/n229/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;ibid., p. 210&lt;/a&gt;.  Why Byrne went with Barrow's notation c. 1660 is hard to say, by his time Harriot's symbols were canonized by Euler, but their origin was not exactly intuition. Perhaps, he felt that it would be &amp;quot;easier to grasp&amp;quot; by those unburdened by entrenched standard. For example, Lambert in 1782 used &amp;lt; and &amp;gt; &amp;quot;in reverse&amp;quot; in his logic, as inspired by implication, see &lt;a href=&quot;https://archive.org/details/asurveyofsymboli00lewiuoft/page/24/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Lewis, Survey of symbolic logic&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On early experiments with various inequality symbols see &lt;a href=&quot;https://www.jstor.org/stable/4025130&quot; rel=&quot;noreferrer&quot;&gt;Tanner, On the Role of Equality and Inequality in the History of Mathematics, p. 167-8&lt;/a&gt;. Incidentally, he attaches a confabulation on why Harriot's symbols look the way they do, which, spelled out or not, probably helped their adoption as &amp;quot;intuitive&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Here is a later example of how our related &amp;quot;intuitive&amp;quot; symbols in set theory could have been reversed. Interpreting inference as inclusion of classes was common in the 19th century. Whatever can be said for the implication pointing in the direction it does, if one chooses &lt;em&gt;that&lt;/em&gt; as &amp;quot;intuitive&amp;quot; then their inequality symbols should go the opposite way from our today's standard too. Indeed, if something is in A then it is in B (A ⊃ B) means that A must be &lt;em&gt;smaller&lt;/em&gt; than B (A ⊂ B), see &lt;a href=&quot;https://math.stackexchange.com/q/1106001/152568&quot;&gt;Why is there this strange contradiction between the language of logic and that of set theory?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Harriot, or rather his editors, derived &amp;lt; and &amp;gt; from a tattoo on a native American's hand, which was symmetric, so what they chose to stand for what was rather random. Derived from this in Schröder's &lt;em&gt;Vorlesungen&lt;/em&gt; set inclusion symbol ⊂ then conflicted with the implication symbol ⊃, as used by Peano and later many others. Gergonne earlier derived a similar symbol from the looks of letter C (in &amp;quot;contained&amp;quot;), i.e. equally randomly, see &lt;a href=&quot;https://jeff560.tripod.com/set.html&quot; rel=&quot;noreferrer&quot;&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Peano, for his part, was consistent, and used ⊃ for &lt;em&gt;inclusion&lt;/em&gt;, as well as implication, see &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/math-origins-the-logical-symbols&quot; rel=&quot;noreferrer&quot;&gt;Tou, Math Origins: The Logical Symbols&lt;/a&gt;, so his smaller would be &amp;gt;. Peirce, Schröder's inspiration for much of his content, was consistent the other way, his implication looked something like this —&amp;lt;, see &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/01445340.2011.621702&quot; rel=&quot;noreferrer&quot;&gt;Anellis, Peirce's Truth-functional Analysis and the Origin of the Truth Table&lt;/a&gt;.&#xA;Hilbert's later symbol → for implication similarly conflicts with &amp;lt;, especially when the arrowhead is drawn bigger.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-08T09:07:14.230" LastActivityDate="2022-08-08T09:07:14.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14672" PostTypeId="1" AcceptedAnswerId="14674" CreationDate="2022-08-08T13:54:43.310" Score="8" ViewCount="3622" Body="&lt;p&gt;Inspired on this post: &lt;a href=&quot;https://hsm.stackexchange.com/questions/12564/who-were-some-mathematicians-who-have-a-musical-background&quot;&gt;Who were some mathematicians who have a musical background?&lt;/a&gt; I want to now ask if there were any famous physicists who had a particular musical background.&lt;/p&gt;&#xA;&lt;p&gt;Looking in Google I just got modern names like Brian Cox or Brian May (from &lt;em&gt;Queen&lt;/em&gt;), but are there famous historical physicists that had also received formal musical education?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-08-08T15:09:52.350" LastActivityDate="2023-08-08T21:30:51.773" Title="Who were some physicists with musical background?" Tags="|physicists|" AnswerCount="12" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14673" PostTypeId="2" ParentId="14672" CreationDate="2022-08-08T14:51:10.017" Score="5" Body="&lt;p&gt;I'm not sure how far it really went, but Feynman was famous for his love of bongos and played them on many occasions:&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=2Ks8gsK22PA&quot; rel=&quot;noreferrer&quot;&gt;Richard Feynman Bongo Drums &amp;quot;Orange Juice&amp;quot;&lt;/a&gt; (YouTube).&lt;/p&gt;&#xA;" OwnerUserId="16579" LastActivityDate="2022-08-08T14:51:10.017" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14674" PostTypeId="2" ParentId="14672" CreationDate="2022-08-08T15:54:33.983" Score="11" Body="&lt;p&gt;Classical education included music as one of the liberal arts since the times of Plato's Academy, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Quadrivium&quot; rel=&quot;noreferrer&quot;&gt;quadrivium&lt;/a&gt;, so most physicists had musical background until the end of 19th century. But there also seems to be something of a general appeal music has for physicists that continues into recent times, see &lt;a href=&quot;https://news.cornell.edu/stories/2006/08/cornell-physicists-face-music-and-love-it&quot; rel=&quot;noreferrer&quot;&gt;Gold, Is it the mystery, or the structure, or the discipline?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some prominent examples:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.neatorama.com/2007/06/08/the-unpredictable-life-of-werner-heisenberg/&quot; rel=&quot;noreferrer&quot;&gt;Heisenberg&lt;/a&gt;: &amp;quot;&lt;em&gt;Werner Heisenberg was born in Würzburg, Germany, on December 5, 1901, and from birth, he seemed destined to become a concert pianist. With help from his mother, Anna, Werner was reading sheet music by age 4 and conquering the masterworks on the family's piano by the time he was 15. But Heisenberg's musical aspirations were slowly sabotaged by the family's book collection. His father, Albert, was a professor of Greek at the University of Munich, and Werner took a liking to rummaging through his dad's bookshelves. Before long, the young Heisenberg decided he preferred Archimedes to Mozart, and science replaced music as his passion.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://theconversation.com/good-vibrations-the-role-of-music-in-einsteins-thinking-54725&quot; rel=&quot;noreferrer&quot;&gt;Einstein:&lt;/a&gt;&amp;quot;&lt;em&gt;It’s little known that Einstein was an accomplished violinist, and even less known that had he not pursued science, he said he would have been a musician... He confessed to thinking about science in terms of images and intuitions, often drawn directly from his experiences as a musician, only later converting these into logic, words and mathematics.&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.journals.uchicago.edu/doi/10.1086/682003&quot; rel=&quot;noreferrer&quot;&gt;Helmholtz&lt;/a&gt;:&amp;quot;&lt;em&gt;Palestrina's music clearly caught Helmholtz's imagination. In his volume On the Sensations of Tone, he repeated the latest findings on the composer's alleged Protestant influences; for the Prussian scientist, they explained the clarity of Palestrina's style. His analysis of the Stabat mater was directed at giving this clarity some foundation in the composer's acute awareness of distortions in harmony. The basic idea was that a composer's knowledge of how to combine sounds would come forward in the music itself, even if there were no explicit rules and even if the rules that had been formulated through instruction would not account for the physiological conditions of hearing.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://aeon.co/essays/why-huygens-father-and-son-personify-the-scientific-revolution&quot; rel=&quot;noreferrer&quot;&gt;Huygens&lt;/a&gt;: &amp;quot;&lt;em&gt;Like his father, Christiaan was a more than competent musician. But whereas Constantijn enjoyed concert performances and the social possibilities that they opened up, Christiaan’s interest in music was more theoretical. In common with many thinkers of the 17th century, he had his own ideas of how music should sound, and he devised a division of the octave into 31 carefully spaced notes that he calculated would both make for a pleasing sound and facilitate matters important to performing musicians, such as transposition between keys. Unlike most, however, he went further and had built for him a special harpsichord that demonstrated this system.&#xA;Perhaps Christiaan’s feeling for music helped him to the realisation that, like sound, light too travels in a wavelike fashion.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-08-08T15:54:33.983" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14675" PostTypeId="2" ParentId="14672" CreationDate="2022-08-08T16:06:00.580" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Boltzmann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boltzmann&lt;/a&gt; was a trained classical pianist who, in his teenage years, was taught by no less than &lt;a href=&quot;https://en.wikipedia.org/wiki/Anton_Bruckner&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anton Bruckner&lt;/a&gt;.  We don't know how long these lessons lasted but it is believed that they ended when Bruckner left a wet raincoat on a piece of furniture, prompting Boltzmann's mother to tell him not to return and to appoint a new teacher for her son.&lt;/p&gt;&#xA;&lt;p&gt;Boltzmann was known to perform (at least parts of) Liszt's piano transcriptions of the Beethoven symphonies.  The only first hand account of his playing comes from Ostwald's daughter during a regularly scheduled music evening at Ostwald's home.  Her rather comical description is of an older, chubby Boltzmann balancing &amp;quot;three pairs of spectacles&amp;quot; on his nose as he struggle to read the sheets.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2023-08-08T20:01:05.267" LastActivityDate="2023-08-08T20:01:05.267" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14676" PostTypeId="2" ParentId="14672" CreationDate="2022-08-09T06:52:28.557" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Satyendra_Nath_Bose&quot; rel=&quot;noreferrer&quot;&gt;Satyendra Nath Bose&lt;/a&gt;, an Indian physicist and polymath, is famous for having collaborated with Albert Einstein on his original work, which came to be called Bose-Einstein statistics. However, very few people know that Bose was gifted at playing the Esraj, an Indian stringed instrument similar to the violin. He often used to perform for his students and colleagues at Calcutta and Dhaka universities.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/8EGO2.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/8EGO2.jpg&quot; alt=&quot;Bose playing Esraj&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2022-08-09T06:52:28.557" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14677" PostTypeId="2" ParentId="14672" CreationDate="2022-08-09T10:51:35.620" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Herschel&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Herschel&lt;/a&gt; comes to mind, if you consider this polymath to be a physicist - he built telescopes, discovered infra-red radiation in sunlight, discovered Uranus, and more.&lt;/p&gt;&#xA;&lt;p&gt;Recordings of  his compositions are available.&lt;/p&gt;&#xA;" OwnerUserId="17068" LastEditorUserId="10106" LastEditDate="2022-08-10T02:00:26.680" LastActivityDate="2022-08-10T02:00:26.680" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14678" PostTypeId="1" AcceptedAnswerId="14679" CreationDate="2022-08-09T20:05:16.890" Score="4" ViewCount="108" Body="&lt;p&gt;In William S. Gosset's &lt;em&gt;The Probable Error of a Mean&lt;/em&gt; (&lt;a href=&quot;https://www.jstor.org/stable/2331554&quot; rel=&quot;nofollow noreferrer&quot;&gt;JSTOR&lt;/a&gt;), he begins to derive the &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; sampling distribution as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Samples of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; individuals are drawn out of a population distributed normally...&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; be the standard deviation found from a sample &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,...x_n$&lt;/span&gt; (all these being measured from the mean of the population), then&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;s^2 = \frac{S(x_1^2)}{n}-\left(\frac{S(x_1)}{n}\right)^2 = \frac{S(x_1^2)}{n} - \frac{S(x_1)^2}{n^2} - \frac{2S(x_1x_2)}{n^2}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And so on. &lt;strong&gt;What is the meaning of the &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; notation, &lt;span class=&quot;math-container&quot;&gt;$S(x_1)$&lt;/span&gt;?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;My first guess was summation, but summing over a scalar (&lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt;) doesn't mean much, and it's not clear how &lt;span class=&quot;math-container&quot;&gt;$S(x_1x_2)$&lt;/span&gt; gets introduced.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Edit. @terry-s 's answer is correct, I just thought I'd add how Gosset's equation comes out of the typical sample standard deviation for anyone curious. The typical definition of standard deviation is (without Bessel's correction):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;s^2 = \frac{1}{n} \sum (x_i-\bar x)^2&#xA;$$&lt;/span&gt;&#xA;For&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bar x = \frac{\sum x_i}{n}&#xA;$$&lt;/span&gt;&#xA;If we expand, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;(x_i-\bar x) = x_i^2 + \bar x ^2 - 2\bar x x_i&#xA;$$&lt;/span&gt;&#xA;And thus the expression for &lt;span class=&quot;math-container&quot;&gt;$s^2$&lt;/span&gt; becomes (after evaluating the sum of &lt;span class=&quot;math-container&quot;&gt;$\bar x$&lt;/span&gt;)&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;s^2 = \bar x^2 + \frac{\sum x_i^2}{n} - 2\bar x\frac{\sum x_i}{n}&#xA;$$&lt;/span&gt;&#xA;And finally, by the definition of &lt;span class=&quot;math-container&quot;&gt;$\bar x$&lt;/span&gt;, the first and last terms on the right combine to give &lt;span class=&quot;math-container&quot;&gt;$-\bar x^2$&lt;/span&gt;, so&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;s^2 = \frac{\sum x_i^2}{n} - \bar x^2&#xA;$$&lt;/span&gt;&#xA;Or, more in line with Gosset's equation, this is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;s^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2&#xA;$$&lt;/span&gt;&#xA;Which is identical to the equation Gosset starts with, assuming &lt;span class=&quot;math-container&quot;&gt;$S(x_1)=\sum x_i$&lt;/span&gt;. By the identity&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(\sum x_i\right)^2 = \sum x_i^2 +2\sum_i \sum_{j=1}^{i-1} x_i x_j&#xA;$$&lt;/span&gt;&#xA;we can expand the right-most term as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(\frac{\sum x_i}{n}\right)^2 = \frac{1}{n^2} \sum x_i^2 + \frac{2}{n^2} \sum_{i,\ j} x_i x_j&#xA;$$&lt;/span&gt;&#xA;Where the double summation is to be performed for each &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, taking &lt;span class=&quot;math-container&quot;&gt;$j$&lt;/span&gt; from 1 to &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;, and this is what Gosset intended by &lt;span class=&quot;math-container&quot;&gt;$S(x_1 x_2)$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2022-08-10T13:11:55.043" LastActivityDate="2022-08-10T13:11:55.043" Title="What is the S notation in Student's The Probable Error of a Mean?" Tags="|notation|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14679" PostTypeId="2" ParentId="14678" CreationDate="2022-08-10T00:26:53.490" Score="3" Body="&lt;p&gt;I believe your 'first-guess' reading of &lt;em&gt;&amp;quot;Student&amp;quot;&lt;/em&gt;'s &lt;span class=&quot;math-container&quot;&gt;$S()$&lt;/span&gt; notation is actually right: on comparing it with other conventions for writing the sample variance &lt;span class=&quot;math-container&quot;&gt;$s^2$&lt;/span&gt;, the notation &lt;span class=&quot;math-container&quot;&gt;$S()$&lt;/span&gt; appears to be a forerunner of modern &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; notation.&lt;/p&gt;&#xA;&lt;p&gt;But in any event it appears there are also further problems with the text of &lt;em&gt;&amp;quot;Student&amp;quot;&lt;/em&gt;'s paper, caused either by the typography of the original or perhaps the reprint version, or else by original ambiguities or inconsistencies of notation and usage.&lt;/p&gt;&#xA;&lt;p&gt;For example, if the first equation that you quote is to make sense as part of an evaluation of some sample variance or standard deviation as stated, it seems that &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; in the first term &lt;span class=&quot;math-container&quot;&gt;$(1/n)S(x_1^2)$&lt;/span&gt; has to mean something like 'each of the sample differences like &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt;' -- i.e. what would now be written &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; with the summation taken from &lt;span class=&quot;math-container&quot;&gt;$i = 1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$i = n$&lt;/span&gt;. (If it doesn't mean something like that, then as you already said, it would look meaningless to sum over a single item.) But such an &lt;span class=&quot;math-container&quot;&gt;$x_i$&lt;/span&gt; interpretation of &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; is inconsistent with the usage of &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; in the immediately-preceding description of the sample, where &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; is given as only one single item in the sample group of differences from the (population) mean. Then again, the second term of the first equation quoted has to be read non-literally or as a typo, else the two terms mean the same and are equal, and the first equation collapses to a trivial zero value, which certainly can't have been intended.&lt;/p&gt;&#xA;&lt;p&gt;All in all, I'd suggest that the original paper presents such problems&#xA;of typography or ambiguity that it might be advisable to refer first to a more modern source, and then if the old paper is the point of historical interest, the more modern texts can provide means to help decipherment. (An example I'd offer of a text with modern descendants of the old equations to use for decipherment would be Wall &amp;amp; Jenkins, 'Practical Statistics for Astronomers', but that's just a local choice of something at hand, there are dozens of others. Another reference with exclusive focus on testing with the t-distribution, that might be helpful, is &lt;a href=&quot;https://scholarworks.umass.edu/pare/vol18/iss1/10/&quot; rel=&quot;nofollow noreferrer&quot;&gt;de Winter (2013)&lt;/a&gt;, &amp;quot;Using the Student's t-test with extremely small sample sizes&amp;quot;.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-08-10T00:26:53.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14680" PostTypeId="1" AcceptedAnswerId="14681" CreationDate="2022-08-10T04:32:01.727" Score="1" ViewCount="135" Body="&lt;p&gt;I dreamed about the date &lt;strong&gt;18.v.1638&lt;/strong&gt; (May 18, 1638) last night.&lt;/p&gt;&#xA;&lt;p&gt;As I currently do research on odd perfect numbers, and because Rene Descartes lived during the years 1596 to 1650, and as I am not a math historian, I would like to ask the experts here:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Did Rene Descartes send or receive any letters (regarding mathematics) on 18.v.1638 (May 18, 1638)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I tried searching via Google, using the keywords &lt;strong&gt;Descartes&lt;/strong&gt; and &lt;strong&gt;18.v.1638&lt;/strong&gt;, but there are too many search results (with none of them promising).  I did skim through a few of them, but did not find any &lt;em&gt;letters on mathematics sent to or received by Descartes on May 18, 1638&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17077" LastActivityDate="2022-08-10T05:40:06.940" Title="Did Rene Descartes send or receive any letters (regarding mathematics) on 18.v.1638 (May 18, 1638)?" Tags="|mathematics|mathematics-social-history|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14681" PostTypeId="2" ParentId="14680" CreationDate="2022-08-10T05:40:06.940" Score="2" Body="&lt;p&gt;On checking the &lt;a href=&quot;https://www.earlymoderntexts.com/authors/descartes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Early Modern Texts for Descartes' correspondences&lt;/a&gt;, I did not find any correspondence of Descartes dated precisely to 18.v.1638. However, Descartes sent a letter to Reneri which has been dated to April or May, 1638. Because of this ambiguous dating, it seems like we don't have the knowledge that if there was a correspondence precisely on 18.v.1638.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastActivityDate="2022-08-10T05:40:06.940" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14682" PostTypeId="2" ParentId="14672" CreationDate="2022-08-10T09:51:06.700" Score="1" Body="&lt;p&gt;Prof. &lt;a href=&quot;https://baas.aas.org/pub/j-m-anthony-danby-1929-2009/release/1&quot; rel=&quot;nofollow noreferrer&quot;&gt;J M A Danby&lt;/a&gt; ('Tony Danby'), physicist &amp;amp; mathematician and author of the well-known &lt;a href=&quot;https://books.google.com/books/about/Fundamentals_of_Celestial_Mechanics.html?id=N5h-AAAAIAAJ&amp;amp;redir_esc=y&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Fundamentals of Celestial Mechanics&amp;quot;&lt;/a&gt; (1962 and later editions), with minor planet &lt;a href=&quot;https://books.google.com/books?id=aeAg1X7afOoC&amp;amp;pg=PA267&quot; rel=&quot;nofollow noreferrer&quot;&gt;3415 Danby&lt;/a&gt; named after him, was also in his earlier  days a professional musician, in the mid-1950s first-chair oboist in one of the London orchestras among other musical positions.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-08-10T10:00:33.817" LastActivityDate="2022-08-10T10:00:33.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14683" PostTypeId="2" ParentId="14658" CreationDate="2022-08-10T14:45:02.180" Score="4" Body="&lt;p&gt;It is very clear from Maxwell's original work that he was writing component by component see 1865 &lt;a href=&quot;https://en.wikisource.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Dynamical Theory of the Electromagnetic Field&lt;/a&gt; in chapter III. See equations with letters (A to H).&lt;/p&gt;&#xA;&lt;p&gt;Here is the example of equation (B):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d H}{\mathrm d z } - \frac{\mathrm d G}{\mathrm d y }=\mu \alpha,$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d F}{\mathrm d x } - \frac{\mathrm d H}{\mathrm d z }=\mu \beta,$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d G}{\mathrm d y } - \frac{\mathrm d F}{\mathrm d x }=\mu \gamma.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which in modern notation is written as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d A_y}{\mathrm d z } - \frac{\mathrm d A_z}{\mathrm d y }=\mu H_x,$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d A_z}{\mathrm d x } - \frac{\mathrm d A_x}{\mathrm d z }=\mu H_y,$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{\mathrm d A_x}{\mathrm d y } - \frac{\mathrm d A_y}{\mathrm d x }=\mu H_z.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$\mathbf H$&lt;/span&gt; is the magnetic force, &lt;span class=&quot;math-container&quot;&gt;$\mathbf A$&lt;/span&gt; is the magnetic vector potential (Maxwell called it electromagnetic momentum) and &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; is the permittivity. Equation (B) can be written using vector calculus as &lt;span class=&quot;math-container&quot;&gt;$\nabla \times \mathbf A = \mu \mathbf H=\mathbf B$&lt;/span&gt; (here I introduce the magnetic field &lt;span class=&quot;math-container&quot;&gt;$\mathbf B$&lt;/span&gt; ). This set of equations above is Gauss's law for magnetic fields, as the gradient of a curl is 0 i.e. &lt;span class=&quot;math-container&quot;&gt;$\nabla \cdot \mathbf B=\nabla \cdot (\nabla \times \mathbf A)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Maxwell equations in modern form vectorial form were introduced by Oliver Heaviside. Here there are Heaviside equations following Maxwell's order, extracted from Wikipedia own article of &lt;a href=&quot;https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field#Heaviside%27s_equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Dynamical Theory of the Electromagnetic Field&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9nxtn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9nxtn.png&quot; alt=&quot;Heaviside's equations&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If by fourth equation you mean Ampère-Maxwell law, you just need to look at equation (A) and (C). Gauss' law is (G). And Faraday's law is hidden in (D), note that Lorentz' force is &lt;span class=&quot;math-container&quot;&gt;$\mathbf f=\mathbf E + \mu \mathbf v \times H$&lt;/span&gt;, so with (D) you can say that &lt;span class=&quot;math-container&quot;&gt;$\mathbf E=-\nabla \phi + \partial \mathbf A /\partial t$&lt;/span&gt;, take the curl and use (B).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-08-10T16:19:58.220" LastActivityDate="2022-08-10T16:19:58.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14684" PostTypeId="2" ParentId="14672" CreationDate="2022-08-10T17:06:08.517" Score="1" Body="&lt;p&gt;William Hershel has already been mentioned, but &lt;a href=&quot;https://oldpostroad.org/node/33&quot; rel=&quot;nofollow noreferrer&quot;&gt;Caroline Lucretia Herschel&lt;/a&gt; (sister to William) was also a musician: a harpsichordist and singer. She made some of the first performances of William's musical work.  Indeed it was through their musical collaborations that Caroline began to assist William in his astronomical work, eventually becoming an independent researcher and (arguably) the first professional female scientist.&lt;/p&gt;&#xA;" OwnerUserId="9638" LastActivityDate="2022-08-10T17:06:08.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14685" PostTypeId="1" CreationDate="2022-08-11T08:30:21.527" Score="2" ViewCount="86" Body="&lt;p&gt;Cathode ray tubes have played a very important role in the development of physics around 1900. They have also served as a key component in old-fashioned TVs.&lt;/p&gt;&#xA;&lt;p&gt;This is great and all, but what was the motivation for inventing them in the first place? They have been around since before J. J. Thomson did anything with them, and I have a hard time finding the earliest device that shows these so-called cathode rays.&lt;/p&gt;&#xA;&lt;p&gt;This leaves me with two questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What was the earliest device that we now know creates a stream of electrons?&lt;/li&gt;&#xA;&lt;li&gt;What was the motivation for inventing such a device?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6965" LastActivityDate="2022-08-11T08:30:21.527" Title="Why were Cathode ray tubes first invented?" Tags="|electricity|atomic-theory|technology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14686" PostTypeId="2" ParentId="14672" CreationDate="2022-08-11T08:47:44.450" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Max_Planck&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max Planck&lt;/a&gt; was a singer, pianist, organist, and cellist on the verge of studying music instead of physics. Apparently he had absolute pitch and even composed an operetta [&lt;a href=&quot;https://de.wikipedia.org/wiki/Max_Planck#1867%E2%80%931874:_Schulzeit_in_M%C3%BCnchen&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;].&lt;/p&gt;&#xA;" OwnerUserId="17084" LastActivityDate="2022-08-11T08:47:44.450" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14687" PostTypeId="2" ParentId="106" CreationDate="2022-08-11T14:10:43.377" Score="1" Body="&lt;p&gt;Various Celtic languages have used, and still use, a vigesimal system. If we take Welsh as an example it is still used although the decimal system is gradually replacing it. I will put the welsh words in italics. Note that Welsh has a system of mutation of the beginning of the word.&lt;/p&gt;&#xA;&lt;p&gt;20 is &lt;em&gt;ugain&lt;/em&gt; and hence 21 is &lt;em&gt;un ar hugain&lt;/em&gt; literally one and twenty.&lt;br /&gt;&#xA;30 is &lt;em&gt;deg ar hugain&lt;/em&gt; (ten and twenty)&lt;br /&gt;&#xA;40 is &lt;em&gt;deugain&lt;/em&gt; two twenties (2 is &lt;em&gt;dau&lt;/em&gt; or &lt;em&gt;dwy&lt;/em&gt; masculine or feminine)&lt;br /&gt;&#xA;50 by exception is &lt;em&gt;hanner cant&lt;/em&gt; (half hundred)&lt;/p&gt;&#xA;&lt;p&gt;My source for this is &lt;a href=&quot;https://omniglot.com/language/numbers/welsh.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numbers in welsh&lt;/a&gt;. I do not speak Welsh but I occasionally use the Welsh language TV channel S4C to watch sport and I can confirm that these forms are still used although the decimal ones seem to be more common.&lt;/p&gt;&#xA;&lt;p&gt;The page to which I linked above also links to other Celtic languages and, for instance, Breton seems to use vigesimal exclusively 20 ugent, 30 tregont, 40 daou-ugent, 50 hanter-kant.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2022-08-11T14:10:43.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14688" PostTypeId="5" CreationDate="2022-08-11T15:44:58.470" Score="0" Body="&lt;p&gt;Physicists study one or more particular fields or subfields of Physics. This tag should be used if the concerned question is about the lives or works of historical Physicists.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;/questions/tagged/biographical-details&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;biographical-details&amp;#39;&quot; rel=&quot;tag&quot;&gt;biographical-details&lt;/a&gt; tag should be used along with this tag if the question is about the lives of the concerned Physicists.&lt;/p&gt;&#xA;&lt;p&gt;If the Physicists are well-renowned (such as &lt;a href=&quot;/questions/tagged/einstein&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;einstein&amp;#39;&quot; rel=&quot;tag&quot;&gt;einstein&lt;/a&gt;, &lt;a href=&quot;/questions/tagged/newton&quot; class=&quot;post-tag&quot; title=&quot;show questions tagged &amp;#39;newton&amp;#39;&quot; rel=&quot;tag&quot;&gt;newton&lt;/a&gt; etc.), this tag should be used along with the tags of those well-renowned Physicists.&lt;/p&gt;&#xA;" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-08-14T18:52:14.520" LastActivityDate="2022-08-14T18:52:14.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14689" PostTypeId="4" CreationDate="2022-08-11T15:44:58.470" Score="0" Body="For questions about those who did Physics" OwnerUserId="10106" LastEditorUserId="10106" LastEditDate="2022-08-21T13:21:08.737" LastActivityDate="2022-08-21T13:21:08.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14690" PostTypeId="2" ParentId="14672" CreationDate="2022-08-11T17:44:21.660" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_Galilei&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo Galilei&lt;/a&gt; not only had a musical background (his father was a professional musician and he  became an accomplished lutenist himself) but he allegedly &lt;a href=&quot;https://www.jstor.org/stable/24949824&quot; rel=&quot;nofollow noreferrer&quot;&gt;used his musical skills to measure time&lt;/a&gt; in his experiments about free fall. Therefore, Galileo's musical background played a crucial role on his discoveries in Physics.&lt;/p&gt;&#xA;" OwnerUserId="5873" LastEditorUserId="10106" LastEditDate="2022-08-12T01:31:14.790" LastActivityDate="2022-08-12T01:31:14.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14691" PostTypeId="2" ParentId="14672" CreationDate="2022-08-11T19:03:53.690" Score="1" Body="&lt;p&gt;My favorite was Edward Teller.  In the summer of 1972 he gave all 48 Bach Preludes and Fugues and got me interested in Baroque music. (I am now 80 and still trying to memorize the BWV1013 flute partita.)&lt;/p&gt;&#xA;&lt;p&gt;In the engineering class, a student asked, &amp;quot;Dr. Teller, how small can you make a nuclear device?&amp;quot;   He replied, &amp;quot;We don't advertise this, but we made some that didn't go off at all.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="3896" LastActivityDate="2022-08-11T19:03:53.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14692" PostTypeId="1" CreationDate="2022-08-12T10:42:31.597" Score="2" ViewCount="434" Body="&lt;p&gt;I'm making a historical research on the origins of differential geometry, starting with non-Euclidean geometry introduced by Gauss. Reading different historical accounts, what I don't understand is how far mathematicians were motivated by a merely abstract and theoretical line of research or if the study of non-Euclidean geometry was also motivated by some pragmatic motives.  Gauss was busy also doing geodetic surveys, but that was still Euclidean geometry (right...?). As far as I understand it, Gauss did not have in mind any applications of it (like, much later, will happen with Einstein's general relativity). On the other hand, it is clear that he was not just interested in solving the problem of parallel lines, but he (and many others too, like Riemann, Lobachevsky, etc.) were busy in developing an entirely new geometry. Moreover, according to Wikipedia &amp;quot;Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it.&amp;quot; Today, for us, a manifold in an N-dimensional non-Euclidean space may sound like an established mathematical object with obvious applications in physics, but I feel that at the time of Gauss that must have sounded like an absurdity to some.&lt;/p&gt;&#xA;&lt;p&gt;So, the question is: What was the real motivation that stood behind the development of DG? Or, rephrasing: At the time of the birth of non-Euclidean differential geometry, what was the general perception about it? Did they perceive it as a fascinating but mere theoretical abstraction, or were they driven by some other theoretical and/or pragmatic necessity?&lt;/p&gt;&#xA;" OwnerUserId="17088" LastEditorUserId="17088" LastEditDate="2022-08-12T10:50:24.587" LastActivityDate="2022-11-10T13:13:43.653" Title="Non-Euclidean geometry: motivations to develop it at the times of Gauss?" Tags="|gauss|differential-geometry|mathematics-social-history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14693" PostTypeId="2" ParentId="14692" CreationDate="2022-08-12T12:33:24.023" Score="10" Body="&lt;p&gt;The motivation of discoverers of non-Euclidean geometry (Gauss, Lobachevski and Bolyai) was their attempts to prove the Fifth postulate of Euclid (to deduce it from the other axioms, or to replace by some other &amp;quot;more evident&amp;quot; axiom). The practical concern was the question &amp;quot;what is the true geometry of the physical space&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Since only one geometry (Euclidean) was known at that time, there was a lot of discussion on what it is really based on.&#xA;People noticed that while other axioms could be convincingly &amp;quot;experimentally verified&amp;quot;, this is not so about the Fifth postulate (even Euclid himself seems to understand this).&lt;/p&gt;&#xA;&lt;p&gt;The discovery of Gauss, Lobachevski and Bolyai was that the Fifth postulate does not follow from the rest (though they had no formal proof of this), and that another geometry, with a negation of this Fifth postulate is possible. And that geometry of the physical space is not a pure logical construction but has to be verified by observation.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-11-10T13:13:43.653" LastActivityDate="2022-11-10T13:13:43.653" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14694" PostTypeId="1" CreationDate="2022-08-14T06:34:12.217" Score="2" ViewCount="464" Body="&lt;p&gt;This question is inspired from: &lt;a href=&quot;https://chemistry.stackexchange.com/questions/167106/why-do-we-call-salts-such-as-agcl-sparingly-soluble&quot;&gt;Why do we call salts such as AgCl sparingly soluble?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The extent of solubility can be expressed as descriptive terms.  U.S. Pharmacopoeia has made the following classification according to the mass &lt;em&gt;m&lt;/em&gt;&lt;sub&gt;sv&lt;/sub&gt; of solvent required to dissolve one unit of mass &lt;em&gt;m&lt;/em&gt;&lt;sub&gt;su&lt;/sub&gt; of solute (Here solvent is water at 20-25 °C.):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/In23s.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/In23s.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Data from &lt;a href=&quot;https://en.wikipedia.org/wiki/Solubility&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Here &amp;quot;Sparingly soluble&amp;quot; is different than &amp;quot;Slightly Soluble&amp;quot; and is used to describe compounds that are not totally insoluble. Since when was the term &amp;quot;Sparingly Soluble&amp;quot; first introduced in chemistry and by who? Was this term introduced when the first solubility classification was generalized or was it introduced at a later point of time because they found that an extra term is needed to differentiate &amp;quot;Slightly Soluble&amp;quot; and this?&lt;/p&gt;&#xA;" OwnerUserId="16966" LastActivityDate="2022-08-14T06:34:12.217" Title="When was the term &quot;Sparingly soluble&quot; first introduced in chemistry?" Tags="|terminology|chemistry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14695" PostTypeId="1" AcceptedAnswerId="14696" CreationDate="2022-08-14T12:56:40.570" Score="4" ViewCount="649" Body="&lt;p&gt;In recent literature, I have read that &lt;a href=&quot;https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes&quot; rel=&quot;nofollow noreferrer&quot;&gt;René Descartes&lt;/a&gt; had observed that &lt;span class=&quot;math-container&quot;&gt;$V-E+F=2$&lt;/span&gt; for planar graphs. Is there any image of that page of book or article of Descartes?&lt;/p&gt;&#xA;&lt;p&gt;Seeing real page that contains the formula is more convincing/believable rather than reading 1000 pages of exposition.&lt;/p&gt;&#xA;&lt;p&gt;In David Richeson's famous book, &lt;a href=&quot;https://press.princeton.edu/books/paperback/9780691154572/eulers-gem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's Gem&lt;/a&gt;, Descartes' formula is as follows&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$P = 2F +2V −4.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="12424" LastEditorUserId="4251" LastEditDate="2022-08-16T22:35:14.777" LastActivityDate="2022-08-16T22:35:14.777" Title="Exact source that Descartes had observed that $V-E+F=2$ for planar graphs" Tags="|mathematics|reference-request|euler|graph-theory|descartes|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14696" PostTypeId="2" ParentId="14695" CreationDate="2022-08-14T13:46:08.650" Score="10" Body="&lt;p&gt;The reconstruction of the story is dealt with in the volume &lt;a href=&quot;https://en.wikipedia.org/wiki/Descartes_on_Polyhedra&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Descartes on Polyhedra: A Study of the &amp;quot;De solidorum elementis&amp;quot;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Long story short: while in Paris in 1675-1676, Leibniz copied a manuscript (now lost) titled &lt;em&gt;Progymnasmata de solidorum elementis&lt;/em&gt; that Descartes had written circa 1639; the Leinbiz's manuscript (still existing; preserved in Hannover in the Niedersächsiche Landesbibliothek, with signature mark LH IV 1, 4b, ff. 1r-v, 15r) was rediscovered and published in 1860, by Foucher de Careil. In this work, Descartes stated (or rather, this is what Leinbiz read) that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si quatuor anguli plani recti ducantur per numerum angulorum solidorum et ex producto tollantur 8 anguli recti plani, remanet aggregatum&#xA;ex omnibus angulis planis qui in superficie talis corporis solidi existunt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If one multiplies four right angles by the number of solid angles and if one removes eight right angles from the product, one is left with the aggregate of all the plane angles that lie on the surface of that solid body.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the same manuscript, Descartes also stated that the number of plane angles equals twice the number of&#xA;edges:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sunt semper duplo plures anguli plani in superficie corporis solidi,&#xA;quam latera; unum enim latus semper commune est duobus faciebus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The plane angles on the surface of a solid body are always twice as large as the sides; a single side is always common to two faces.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By combining these two statements, one can easily deduce Euler's formula, although Descartes did not make this connection (to be clear: Descartes deals only with polyhedra; while every convex polyhedron can be turned into a connected, simple, planar graph, Descartes never made this connection).&lt;/p&gt;&#xA;&lt;p&gt;There is a critical edition with facing text and Italian translation (sorry...) of &lt;em&gt;Progymnasmata de solidorum elementis&lt;/em&gt;, accompanied by mathematical, historical and philological notes, &lt;a href=&quot;https://iris.unito.it/retrieve/e27ce42c-0a02-2581-e053-d805fe0acbaa/DeSolidorumElementisBompiani.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086010000285#bbib14&quot; rel=&quot;noreferrer&quot;&gt;What Descartes knew of mathematics in 1628&lt;/a&gt; by David Rabouin, there is also a photograph of Leibniz's manuscript (the photographs of the whole manuscript are available in &lt;em&gt;Exercises pour les éléments des solides: Progymnasmata de solidorum elementis: Essai en complement d'Euclide&lt;/em&gt; by Pierre Costabel).&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2022-08-14T14:00:02.897" LastActivityDate="2022-08-14T14:00:02.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14697" PostTypeId="1" AcceptedAnswerId="14701" CreationDate="2022-08-14T17:11:41.153" Score="4" ViewCount="98" Body="&lt;p&gt;I'm trying to get a sense of how statistical distributions arose and I'm using Poisson as an example. The folklore I grew up with says that Gauss arrived at the normal distribution while trying to account for errors in astronomical measurements. And I think Bernoulli arrived at the binomial to find the probabilities of certain complicated games. In other words, these are cases where (if my folklore is correct) the distributions were the result of great minds solving real problems. But were there cases of great minds just ``noodling''? For example, Newton playing with the binomial coefficients for negative powers leading to an algebraic expression for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;That is why the Poisson seemed like a good example to use to pose this question. The statistics books tell me that it can be viewed as the limit of the binomial in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n \to \infty$&lt;/span&gt; while &lt;span class=&quot;math-container&quot;&gt;$np$&lt;/span&gt; is held constant. It seems plausible that Poisson was noodling with the binomial and arrived at his distribution, but it is also plausible that he was working on a specific problem that led him there.&lt;/p&gt;&#xA;&lt;p&gt;Is there a good, accessible reference that talks about the history of the key distributions we have today and how they were arrived at?&lt;/p&gt;&#xA;" OwnerUserId="15921" LastEditorUserId="6609" LastEditDate="2022-08-15T15:53:28.947" LastActivityDate="2022-08-15T18:55:59.453" Title="Was Poisson and other distributions found by solving applied problems or by playing around theoretically?" Tags="|probability|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14698" PostTypeId="2" ParentId="5790" CreationDate="2022-08-14T18:53:17.683" Score="1" Body="&lt;p&gt;FWIW: ~ Gama, 1790, P.25 indicates &amp;quot;...pero la corrección no se hacia hasta el fin del Ciclo, en que se intercalaban juntos los 13 días, que gastaban en fiestas, en honor de los dioses seculares, de los quales era uno el mismo Xiuhteuctli TletL...&amp;quot;, P.110 &amp;quot;...pues sabiendo ellos bien, que al fin del Ciclo había retrocedido el principio de su año civil 13 dias; y para igualarlo con el solar, los intercalaban; les era fácil saber, en qualquier año, los dias que debían computar en su cuenta, para verificar en ellos el preciso tiempo de los Equinoccios y Solsticios, y del tránsito del Sol por el vértice de la Ciudad..&amp;quot;, and a few other references to intercalations (in addition to the intro which gives a primary purpose of the book as including ~&amp;quot;Explícase el sistema de los Calendarios de los Indios, el método que tenian de dividir el tiempo, y la corrección que hacian de él para igualar el año civil, de que usaban, con el año solar trópico...&amp;quot;).&lt;/p&gt;&#xA;" OwnerUserId="17097" LastEditorUserId="17097" LastEditDate="2022-08-14T19:43:56.307" LastActivityDate="2022-08-14T19:43:56.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14699" PostTypeId="1" AcceptedAnswerId="14727" CreationDate="2022-08-14T22:55:41.150" Score="1" ViewCount="191" Body="&lt;p&gt;I'm searching interviews and/or conversations (in English) with Einstein as text. Is there something available? In particular physics (e.g. theory of relativity) would be interesting but also other (e.g. more general) interviews / conversaitons are fine.&lt;/p&gt;&#xA;" OwnerUserId="17100" LastActivityDate="2022-08-23T13:10:26.210" Title="Interviews / Conversations with Einstein" Tags="|physics|relativity-theory|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14700" PostTypeId="2" ParentId="12564" CreationDate="2022-08-15T02:17:35.690" Score="1" Body="&lt;p&gt;I don't know if &lt;a href=&quot;https://en.wikipedia.org/wiki/Olivia_Caramello&quot; rel=&quot;nofollow noreferrer&quot;&gt;Olivia Caramello&lt;/a&gt; is well-known enough, but she has degrees in mathematics and a Diploma in Piano.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2022-08-15T02:17:35.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14701" PostTypeId="2" ParentId="14697" CreationDate="2022-08-15T08:28:58.043" Score="6" Body="&lt;p&gt;As a surmise, the boundary between &amp;quot;noodling&amp;quot; and &amp;quot;solving real problems&amp;quot; is rather vague. One needs at least some vague external idea to guide the &amp;quot;noodling&amp;quot;, otherwise they can simply normalize non-negative functions with finite integral to &amp;quot;discover&amp;quot; endless distributions. On the other hand, even Gauss, who did come up with the correct error curve (normal distribution) by reflecting on the least squares method he used to fit orbits, did not need &lt;em&gt;that&lt;/em&gt; for the &amp;quot;real solving&amp;quot;. It was more of &amp;quot;noodling&amp;quot; around with abstract error assumptions.&lt;/p&gt;&#xA;&lt;p&gt;The way Poisson arrived at his distribution can also be fairly described as &amp;quot;noodling with the binomial&amp;quot;. Despite the title of the book where it appears (Research on the Probability of Judgments in Criminal and Civil Matters Preceded by the General Rules for Calculating Probability), the part about the distribution itself does not model anything tangible or solve any real problem. It is more of a pure math exercise in asymptotics of binomial distribution along the lines sketched in problems from de Moivre's book. The relevant portion is translated in full by Stiglitz in &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0167715282900104?via%3Dihub&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poisson on the poisson distribution&lt;/a&gt;, who comments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Curiously, the Poisson distribution appears but&#xA;once in all of Poisson's works, and then on but a&#xA;single page (Haight, 1967, p. 113). This reference is on p. 206 of Poisson's 1837 book, Recherches sur la Probabilité des Jugements en Matière Criminelle et en Matière Civile Précédés des Règles Générales du Calcul des Probabilités... The distribution appears in Chapter 3 as limit to the binomial. Actually, as will be seen below, Poisson derived the distribution directly as an approximation to the negative binomial cumulative distribution. There is no indication that he sensed the wide applicability of the distribution; rather, it was one of several approximations and received no special comment.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Poisson's derivation of his distribution, was foreshadowed by an analysis De Moivre had presented over a century earlier in a series of problems in his book, The Doctrine of Chances... It must be admitted that Poisson added little to De Moivre's mathematical approximation, with which he was quite familiar, although one would have to stretch the point to claim the discrete distribution &lt;span class=&quot;math-container&quot;&gt;$e^{-\omega}\frac{\omega^n}{n!}$&lt;/span&gt; is found in De Moivre.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Apparently, Newcomb was the first to apply the distribution to fit the data in 1860, namely, to find the &amp;quot;&lt;em&gt;probability that, if the stars were scattered at random over the heavens, any small space selected at random would contains stars.&lt;/em&gt;&amp;quot; The current ubiquity of the distribution is mostly unrelated to Poisson, it was rediscovered in biology and physics a century later and its applicability stems from that. Further history is discussed by &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/00031305.2022.2046159#&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hanley and Bhatnagar&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Poisson himself never gave an example of, or applied the distribution now called after him. It ended up being independently rediscovered and used by early-20th century biological and physical scientists who derived it using space and time considerations. Some of them also linked it with what we refer to today as the exponential distribution. Quite early on, it was also recognized that counts involving human activities/behavior were not always well fitted by the Poisson distribution, and that extensions were needed.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;More broadly, &lt;a href=&quot;https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stahl in The Evolution of the Normal Distribution&lt;/a&gt; describes how Simpson and Laplace came up with three different distributions by &amp;quot;noodling&amp;quot; theoretically to find the error curve: triangular, double exponential and one with logarithmic density.&lt;/p&gt;&#xA;&lt;p&gt;Although I am not aware of a book on history of distribution discoveries specifically, books on history of probability and statistics cover that in particular, see, for example, highly acclaimed &lt;a href=&quot;https://www.google.com/books/edition/The_History_of_Statistics/M7yvkERHIIMC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stigler's History of Statistics: The Measurement of Uncertainty before 1900&lt;/a&gt;. Hald also has a series of history books on different time periods. The most relevant one is probably &lt;a href=&quot;https://www.google.com/books/edition/A_History_of_Mathematical_Statistics_fro/B5TuAAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of mathematical statistics from 1750 to 1930&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-15T18:55:59.453" LastActivityDate="2022-08-15T18:55:59.453" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14702" PostTypeId="1" AcceptedAnswerId="14705" CreationDate="2022-08-15T11:46:31.180" Score="2" ViewCount="101" Body="&lt;p&gt;I was reading the &lt;a href=&quot;https://philsci-archive.pitt.edu/2691/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Compendium of the foundations of classical statistical physics&lt;/a&gt; by Jos Uffink where, on p. 4, the following statement is made:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a discussion of the foundations of classical mechanics, for example,&#xA;one need not consider the work of the Parisian scholastics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm interested in this work, but the only scholars I've found that seem to fit the bill are Oresme and Buridan. Were there any others? The quote above gives me the impression that there was a whole school of people working on mechanics, somewhat like the Oxford calcultors.&lt;/p&gt;&#xA;" OwnerUserId="17103" LastActivityDate="2022-08-16T06:00:18.553" Title="Parisian scholastics' work on mechanics" Tags="|physics|classical-mechanics|middle-ages|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14703" PostTypeId="1" CreationDate="2022-08-15T14:59:29.120" Score="2" ViewCount="332" Body="&lt;p&gt;I am interested in the history of the Basel problem. More specifically, I'm interested in knowing the history of failed attempts before Euler's crack of it, so as to know what bits of evidence informed his approach.&lt;/p&gt;&#xA;" OwnerUserId="17105" LastEditorUserId="55" LastEditDate="2022-08-15T22:38:39.157" LastActivityDate="2022-08-19T03:06:02.720" Title="What is the history of the Basel Problem before Euler and how did it inform him?" Tags="|mathematics|calculus|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14704" PostTypeId="2" ParentId="14703" CreationDate="2022-08-15T22:37:39.810" Score="6" Body="&lt;p&gt;On Euler's work and rigor issues see &lt;a href=&quot;https://hsm.stackexchange.com/a/2283/55&quot;&gt;Is it true that Euler did not &lt;em&gt;prove&lt;/em&gt; the sum of the Basel series?&lt;/a&gt; Here I will focus on prior work. After Mengoli proposed the Basel problem in &lt;em&gt;Novae quadraturae Arithmeticae&lt;/em&gt; (1650), it proceeded along two lines, numerical approximations and algebraic manipulations of similar simpler series. Infinite products also played a role, but their relevance became clear only after Euler.&lt;/p&gt;&#xA;&lt;p&gt;Numerical approximations were non-trivial. The series converges very slowly and plainly summing its first 1000 terms would only give two correct decimal places. Nonetheless, already Wallis got three correctly in &lt;em&gt;Arithmetica&#xA;infinitorum&lt;/em&gt; (1656), de Moivre got nine in 1730, and Euler himself, without knowing that, got eight in 1731. The idea was to transform the series to speed up convergence, e.g. Euler's transformation was &lt;span class=&quot;math-container&quot;&gt;$\sum\frac1{n^2}=\sum\frac1{2^nn^2}+(\ln 2)^2$&lt;/span&gt;, which converges much faster. It was based on integrating the power series for &lt;span class=&quot;math-container&quot;&gt;$\frac{\ln(1-x)}{x}$&lt;/span&gt;, see Ayoub for details.&lt;/p&gt;&#xA;&lt;p&gt;This trick followed the idea that Leibniz communicated in a 1673 letter to Johann Bernoulli, where he integrated the same series and applied integration by parts to find that summing &lt;span class=&quot;math-container&quot;&gt;$\sum\frac{(-1)^{n+1}}{n^2}$&lt;/span&gt; reduces to computing integrals of the form &lt;span class=&quot;math-container&quot;&gt;$\int x^e(1+x)^n$&lt;/span&gt;. As he added, &amp;quot;&lt;em&gt;if perhaps it were possible to consider all the cases in order, some light would be shed upon the problem.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The only rigorous result on the matter was also related to Leibniz. Huygens challenged him in 1672  to sum the reciprocals of the triangular numbers &lt;span class=&quot;math-container&quot;&gt;$\sum\frac2{n(n+1)}$&lt;/span&gt;. Leibniz found the sum to be &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; by representing it as a telescoping series &lt;span class=&quot;math-container&quot;&gt;$\sum\left(\frac2{n}-\frac2{n+1}\right)$&lt;/span&gt;. In his 1689 dissertation, which made the problem famous and gave it the name (it was defended in Basel), Jacob Bernoulli noticed that &lt;span class=&quot;math-container&quot;&gt;$\frac1{n^2}&amp;lt;\frac2{n(n+1)}$&lt;/span&gt;, and hence the Basel sum is less than &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The two closest inspirations for Euler's breakthrough were &lt;a href=&quot;https://en.wikipedia.org/wiki/Wallis_product&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wallis's product formula&lt;/a&gt; for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; (which he did not relate to the Basel series despite considering both in the same book), and Leibniz's summation of the alternating harmonic series &lt;span class=&quot;math-container&quot;&gt;$\sum\frac{(-1)^{n+1}}{2n+1}=\frac\pi4$&lt;/span&gt;, based on the power series for the arctangent (derived earlier by Nilakantha and Gregory but unknown to Leibniz, see &lt;a href=&quot;https://www.jstor.org/stable/2690896&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roy, The Discovery of the Series Formula for π&lt;/a&gt;). They suggested the use of power series and infinite products, and a role for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But Euler's key idea that led to the 1734 breakthrough was highly original and without precursors. It boldly extended factorization of polynomials to power series, specifically the power series for &lt;span class=&quot;math-container&quot;&gt;$\frac{\sin x}x$&lt;/span&gt;. Euler assumed that its real roots &lt;span class=&quot;math-container&quot;&gt;$\pm\,\pi n$&lt;/span&gt; were the only ones, and wrote &lt;span class=&quot;math-container&quot;&gt;$$\frac{\sin x}x=\prod\left(1-\frac{x^2}{\pi^2 n^2}\right),$$&lt;/span&gt; as if it was a polynomial. Formally distributing in the infinite (!) product, gave the coefficient of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; on the right as the sum of the Basel series times &lt;span class=&quot;math-container&quot;&gt;$-\frac1{\pi^2}$&lt;/span&gt;. The coefficient on the left was known to be &lt;span class=&quot;math-container&quot;&gt;$-\frac16$&lt;/span&gt; from the power series for &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt;, which gives the Basel sum as &lt;span class=&quot;math-container&quot;&gt;$\frac{\pi^2}6$&lt;/span&gt;. &amp;quot;&lt;em&gt;If only my brother were still alive&lt;/em&gt;&amp;quot; vowed Johann Bernoulli (referring to Jacob, who passed away in 1705) when Euler communicated the result to him.&lt;/p&gt;&#xA;&lt;p&gt;Euler was aware that his brilliant trick was shaky, so he applied it to &lt;span class=&quot;math-container&quot;&gt;$1-\sin x$&lt;/span&gt;, and... got the &lt;span class=&quot;math-container&quot;&gt;$\frac\pi4$&lt;/span&gt; for the Leibniz's series from 1674. He also applied the above product formula for &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; to get Wallis's product for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;. That so boosted his confidence that what he remarked next sounds like taken straight from a book of fallacies:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;For our method, which may appear to some as not reliable enough, a great confirmation comes here to light. Therefore, we should not doubt at all of the other things which are derived by the same method.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And yet he doubted still, giving two more proofs, in 1745 and 1755.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some details are given in &lt;a href=&quot;https://www.jstor.org/stable/2319041&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ayoub, Euler and the Zeta Function&lt;/a&gt; and &lt;a href=&quot;https://core.ac.uk/download/pdf/82068344.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calinger, Leonhard Euler: The First St. Petersburg Years (1727–1741)&lt;/a&gt;. &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/0025570X.2009.11953587&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dunham's When Euler Met l’Hopital&lt;/a&gt; describes influences on Euler's third (1755) derivation of the Basel sum. &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/euler-the-master-of-us-all&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dunham's book Euler, The Master of Us All&lt;/a&gt; (the title quoting Gauss) is also a great source on context and details. &lt;a href=&quot;https://hkumath.hku.hk/%7Emks/EulerHeuristicReasoning_HKULibrary.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Siu, Euler and Heuristic Reasoning&lt;/a&gt; and &lt;a href=&quot;https://www.lesswrong.com/posts/WsmnfWTP28dXCKEy8/the-use-of-many-independent-lines-of-evidence-the-basel&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Use of Many Independent Lines of Evidence: The Basel Problem post on Less Wrong&lt;/a&gt; describe the more empirical science approach to mathematics that prevailed before and during Euler's time and influenced his thinking and discoveries.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-19T03:06:02.720" LastActivityDate="2022-08-19T03:06:02.720" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14705" PostTypeId="2" ParentId="14702" CreationDate="2022-08-16T05:54:24.457" Score="3" Body="&lt;p&gt;In fairness, Uffink qualifies his assertion as &amp;quot;true for the purpose of studying the conceptual structure of a physical theory&amp;quot;. For historical purposes, Parisian scholastics and their successors, &lt;a href=&quot;https://en.wikipedia.org/wiki/Oxford_Calculators&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford calculators&lt;/a&gt;, did pave the way for Galileo and Descartes.&lt;/p&gt;&#xA;&lt;p&gt;Uffink most likely refers to the medieval theory of impetus, a modification of Aristotle's dynamics that replaced his &lt;em&gt;antiperistasis&lt;/em&gt; theory of projectile motion mocked already in antiquity by Philoponus. The latter suggested that, according to it, one could make an arrow fly by waiving hands behind it. The idea of lasting &amp;quot;impressed force&amp;quot; (&lt;em&gt;virtus derelicta&lt;/em&gt;), albeit decaying, anticipated the concept of inertia, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5153/55&quot;&gt;Who was the first scientist to suggest that objects can keep moving without applied force?&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/a/2136/55&quot;&gt;Why don't we learn Buridan's laws of motion?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The idea of impetus traveled from Hipparchus (allegedly), through Philoponus and then Islamic philosophers to medieval Europe, and more specifically to Paris first, see &lt;a href=&quot;https://philarchive.org/rec/VANRCO-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Van Dyck, Malara, Renaissance Concept of Impetus&lt;/a&gt;. There, Francis of Marchia and then, most famously, Buridan gave it the pride of place in their pre-modern dynamics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The introduction of the impetus concept in scholastic philosophy is traditionally ascribed to Francis of Marchia (first half of 14th century), who held the Franciscan chair&#xA;of theology in Paris, but recent scholarship has shown that there were some thirteenth century predecessors... Jean Buridan (1295-1363) formulated what has been called the &amp;quot;classical&amp;quot; concept&#xA;of impetus [13, p.134]. He quoted several empirical reasons to support the concept&#xA;of impetus and he also gave some proportional indications about how it relates to factors&#xA;such as the amount of matter that is put into motion [4, pp.521-523]...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Just as Marchia, Buridan also appealed to impetus in treating celestial motion, and he added acceleration as a further phenomenon that could be explained&#xA;by it. In the latter case impetus is not only the cause but also the effect of motion, as Buridan assumed that the body gradually acquires more impetus as it falls. Buridan’s direct successors, Nicole Oresme (1320-1382), Albert of Saxony (1320-1390) and Marsilius of Inghen (1340-1396), further took up the concept and offered treatments that afterwards spread over Europe from Paris.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I know that Marchia, Saxony and Inghen do not sound like Paris, but all three were professors at the University of Paris at various times (1320s, 1351-1362 and 1386-1396, respectively). Another name, not mentioned but associated with Paris and impetus, is &lt;a href=&quot;https://www.academia.edu/11086197/Introduction_Gerald_Odonis_Life_and_Works_&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gerardus Odonis&lt;/a&gt;, see also &lt;a href=&quot;https://www.jstor.org/stable/41959084&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francis of Marchia's Virtus derelicta and the Context of Its Development&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-16T06:00:18.553" LastActivityDate="2022-08-16T06:00:18.553" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14707" PostTypeId="5" CreationDate="2022-08-16T14:46:05.893" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-08-16T14:46:05.893" LastActivityDate="2022-08-16T14:46:05.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14708" PostTypeId="4" CreationDate="2022-08-16T14:46:05.893" Score="0" Body="Vladimir Igorevich Arnold (1937–2010) was a Soviet mathematician who made important contributions to various fields of mathematics." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2022-08-17T16:57:46.070" LastActivityDate="2022-08-17T16:57:46.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14709" PostTypeId="1" CreationDate="2022-08-16T18:45:29.860" Score="0" ViewCount="157" Body="&lt;p&gt;Heat, time, and electrical current have separate units, namely, calorie, second and ampere, respectively. However, there is no special unit for velocity. We still use &lt;span class=&quot;math-container&quot;&gt;$\frac{\rm m}{\rm s}$&lt;/span&gt;. Why? Velocity has a separate significance more than current, right?&lt;/p&gt;&#xA;" OwnerUserId="17112" LastEditorUserId="4251" LastEditDate="2022-08-16T21:55:07.603" LastActivityDate="2022-08-16T21:55:07.603" Title="Why is there no special unit there for velocity/speed?" Tags="|physics|units|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14711" PostTypeId="2" ParentId="11565" CreationDate="2022-08-17T08:17:22.883" Score="1" Body="&lt;p&gt;As a further data point, you might want to look at André Weil's article &amp;quot;On Eisenstein's Copy of &lt;em&gt;Disquisitiones&lt;/em&gt;&amp;quot; (Advanced Studies in Pure Mathematics 17, 1989, pp. 463-469), where he suggests that Eisenstein might have influenced Riemann when he wrote his paper on the zeta-function.&lt;/p&gt;&#xA;" OwnerUserId="3555" LastEditorUserId="12330" LastEditDate="2024-03-01T03:12:00.817" LastActivityDate="2024-03-01T03:12:00.817" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14712" PostTypeId="1" CreationDate="2022-08-17T14:46:13.923" Score="3" ViewCount="108" Body="&lt;p&gt;Just wondering how Pascal used the triangle in the problem where they wanted to determine how to divide the pot when the game ended early. It was the first major probability problem conceived in history. I couldn't really find a simple enough explanation.&lt;/p&gt;&#xA;" OwnerUserId="16826" LastEditorUserId="55" LastEditDate="2022-08-18T04:00:25.053" LastActivityDate="2022-08-18T04:00:25.053" Title="How was Pascal's triangle applied to fairly dividing the pot in an interrupted game?" Tags="|probability|combinatorics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14715" PostTypeId="2" ParentId="14712" CreationDate="2022-08-18T03:48:44.597" Score="1" Body="&lt;p&gt;This is called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Problem_of_points&quot; rel=&quot;nofollow noreferrer&quot;&gt;problem of points&lt;/a&gt;. An accessible explanation of Pascal's solution and how binomial coefficients (and hence the triangle) come up in it is given on &lt;a href=&quot;https://probabilityandstats.wordpress.com/2016/11/06/the-problem-of-points/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Probability and Statistics blog&lt;/a&gt;, but it is a bit long to be reproduced here, see also &lt;a href=&quot;https://www.jstor.org/stable/1402496&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edwards, Pascal and the Problem of Points&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/27964539&quot; rel=&quot;nofollow noreferrer&quot;&gt;Coughlin and Kerwin in Mathematical Induction and Pascal's Problem of the Points&lt;/a&gt; give a simplified explanation for a special case, and show how to perform calculations directly on the triangle.&lt;/p&gt;&#xA;&lt;p&gt;Suppose we have two players (teams), &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, with equal probabilities of winning each round. The game series was supposed to last until one of them has &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; wins (points), and whoever does gets the whole pot &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;. Sadly, the game was interrupted when &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; had only &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; wins and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; had only &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; wins. At that point, their fair shares of the pot are their probabilities of winning the series, if it continued, multiplied by &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;. To win the series, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; needs &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; more wins and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; needs &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; more wins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Pascal may have reasoned as follows. In this case the series would be&#xA;decided in at most four games. A four-game series has &lt;span class=&quot;math-container&quot;&gt;$2^4$&lt;/span&gt; possible outcomes: &lt;span class=&quot;math-container&quot;&gt;$AAAA,AAAB,AABB,...$&lt;/span&gt;. Team &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; can win two games in &lt;span class=&quot;math-container&quot;&gt;$\binom42$&lt;/span&gt; ways, three games in &lt;span class=&quot;math-container&quot;&gt;$\binom43$&lt;/span&gt; ways, and all four games in &lt;span class=&quot;math-container&quot;&gt;$\binom44$&lt;/span&gt; ways. Thus, team &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; can win two or more games in&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\binom42+\binom43+\binom44$$&lt;/span&gt;&#xA;ways. These results represent theoretical&#xA;possibilities even though it may not be necessary to play all four games. Consequently, the probability that team A will win the&#xA;series is given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac1{2^4}\left[\binom42+\binom43+\binom44\right].$$&lt;/span&gt;&#xA;Using Pascal's triangle, we find&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\binom42=6,\ \ \binom43=4,\ \ \binom44=1;$$&lt;/span&gt;&#xA;and hence the probability that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; will win is&#xA;&lt;span class=&quot;math-container&quot;&gt;$(6 + 4 + 1)/16 = 11/16$&lt;/span&gt;. Multiplying by &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;,&#xA;we get &lt;span class=&quot;math-container&quot;&gt;$11/16\,M$&lt;/span&gt;...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The share of &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, accordingly, will be the complementary &lt;span class=&quot;math-container&quot;&gt;$5/16\,M$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In general, &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; may have unequal probabilities of winning rounds, say &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1-p$&lt;/span&gt; (no ties), so the sum of binomial coefficients has to be replaced by a weighted sum, with powers of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1-p$&lt;/span&gt; as weights. Pascal set up a recursion similar to the one used to construct the triangle of binomial coefficients, but involving &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1-p$&lt;/span&gt;. The idea of calculating the shares remains the same.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-18T03:55:43.163" LastActivityDate="2022-08-18T03:55:43.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14716" PostTypeId="1" AcceptedAnswerId="15267" CreationDate="2022-08-18T21:42:27.900" Score="7" ViewCount="315" Body="&lt;p&gt;Here is a photograph of the audience of SGA3 (as noted by Mateo Carmona).&lt;/p&gt;&#xA;&lt;p&gt;&lt;img src=&quot;https://raw.githubusercontent.com/agrothendieck/agrothendieck.github.io/master/img/sga5.JPG&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other photographs may be viewed from the homepage: &lt;a href=&quot;https://agrothendieck.github.io/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://agrothendieck.github.io/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Grothendieck is facing the audience. I recognise Dieudonne at the left in the front row, Mandelbrot (?) in the second row, Chevalley to the right of Mandelbrot, but no one else. (Corrections after comments.)&lt;/p&gt;&#xA;&lt;p&gt;Does anyone recognise the others?&lt;/p&gt;&#xA;" OwnerUserId="16151" LastEditorUserId="9523" LastEditDate="2023-04-11T06:05:38.330" LastActivityDate="2023-04-11T06:05:38.330" Title="Who were the attendees to the SGA3 seminar?" Tags="|mathematicians|grothendieck|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14717" PostTypeId="1" CreationDate="2022-08-18T23:04:02.537" Score="1" ViewCount="124" Body="&lt;p&gt;I'm interested in gathering a list of problems in the history of maths where people were committed to a particular strategy for some time, only to find out that the approach was fundamentally flawed.&lt;/p&gt;&#xA;&lt;p&gt;I expect all famous and unsolved problems in mathematics probably have something resembling this quality. I'm not too sure of its history, but I would imagine there are a lot of naive (and not so naive) attempts at the Goldbach conjecture that many put a lot of faith in.&lt;/p&gt;&#xA;&lt;p&gt;Something like the natural proofs barrier for P vs. NP is probably the best exemplar I can think of.&lt;/p&gt;&#xA;&lt;p&gt;If possible, I would like to gather examples that were solved and which didn't prove a negative result (i.e., Fermat's Last Theorem)&lt;/p&gt;&#xA;" OwnerUserId="17105" LastEditorUserId="10106" LastEditDate="2022-08-19T03:08:44.557" LastActivityDate="2022-08-19T03:08:44.557" Title="Examples of math problems which exhibited deceptive progress" Tags="|mathematics|mathematics-social-history|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14718" PostTypeId="1" AcceptedAnswerId="14719" CreationDate="2022-08-19T21:01:28.160" Score="3" ViewCount="196" Body="&lt;p&gt;The theory of automorphic forms is stated in modern language as a function on the adelic points of some reductive group satisfying certain properties (see for instance, Chapter 7 of An Introduction to the Langlands Program by Bump et al). This includes modular forms as a special case, whose classical theory well preceded the concept of adeles. When was this &amp;quot;adelic&amp;quot; viewpoint on automorphic forms and modular forms first developed?&lt;/p&gt;&#xA;" OwnerUserId="6373" LastActivityDate="2022-08-20T02:08:26.787" Title="When was the study of automorphic (and in particular modular) forms become &quot;adelized&quot;?" Tags="|mathematics|number-theory|harmonic-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14719" PostTypeId="2" ParentId="14718" CreationDate="2022-08-20T02:08:26.787" Score="6" Body="&lt;p&gt;Already in 1950, Iwasawa and Tate's modernization of Hecke's theory &amp;quot;for GL(1)&amp;quot; was aggressively &amp;quot;adelized&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;My recollection of many peoples' attitudes about that, in the 1970's, was that &amp;quot;the adelic viewpoint&amp;quot; was &amp;quot;an option&amp;quot;, maybe a stylistic choice, and not at all necessarily advantageous.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, in the late 1940's and early 1950's, Godement and others were hoping for a very general representation theory of topological groups... but/and Harish-Chandra's pick-up of Wigner's and Bargmann's and others' physics-motivated explicit repn theory for small classical groups was a big change in attitude about the potential of generalization.&lt;/p&gt;&#xA;&lt;p&gt;Meanwhile, already in the late 1950's, Gelfand and collaborators were indeed working to develop an understanding of automorphic forms on adele groups, emphasizing &lt;span class=&quot;math-container&quot;&gt;$GL(2)$&lt;/span&gt;. My recollection is that, in the mid 1970's, people still did view this as &amp;quot;an option&amp;quot;. After all, R. Langlands' SLN 544, apparently composed in the mid-to-late 1960s) is not at all adelic.&lt;/p&gt;&#xA;&lt;p&gt;S. Gelbart's 1974 orange Princeton-series book was in many ways aggressively forward-looking, talking about Langlands' ideas interpreted on adele groups,  even if it was needlessly coy about various modern ideas.&lt;/p&gt;&#xA;&lt;p&gt;(Yes, indeed, in the early 1960s, A. Weil's papers about Siegel-Weil formulas, oscillator repn, and so on, certainly did embrace &amp;quot;adelic&amp;quot; viewpoints. To my recollection, this perspective was not hugely popular. Similarly, T. Tamagawa's work that emphasized &amp;quot;an adelic viewpoint&amp;quot;, was considered avante-garde...)&lt;/p&gt;&#xA;&lt;p&gt;In terms of &amp;quot;fashion&amp;quot;, to say the least, there seemed to have been no &lt;em&gt;compulsion&lt;/em&gt; to &amp;quot;do things adelically&amp;quot; until after the 1977 Corvallis conference. :)&lt;/p&gt;&#xA;&lt;p&gt;It does indeed seem to be true that there is not usually any effort exerted to persuade or explain why it's advantageous to &amp;quot;take an adelic viewpoint&amp;quot;. As with many &amp;quot;fashions&amp;quot;, a critical-thinking person could suspect that there &lt;em&gt;is&lt;/em&gt; actually not good reason to adopt the style-of-the-day. In fact, to the contrary, as I eventually realized, there &lt;em&gt;are&lt;/em&gt; many good reasons to &amp;quot;use an adelic viewpoint&amp;quot; when possible.&lt;/p&gt;&#xA;&lt;p&gt;On another hand, some relatively recent work of Winnie Li (and others) about arithmetic of modular forms for non-congruence subgroups shows that &amp;quot;the adelic viewpoint&amp;quot; does not actually cover everything.&lt;/p&gt;&#xA;&lt;p&gt;In summary, although the &amp;quot;adelic viewpoint&amp;quot; was well-documented by the early 1960's (Gelfand et al), people didn't necessarily believe it was an essential idea until maybe 1970. For example, the Godement-Jacquet treatment of standard L-functions for &lt;span class=&quot;math-container&quot;&gt;$GL(n)$&lt;/span&gt;, following Iwasawa-Tate.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-08-20T02:08:26.787" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14720" PostTypeId="1" CreationDate="2022-08-20T11:35:25.297" Score="0" ViewCount="50" Body="&lt;p&gt;Assuming there was a standard for (early) astrological calculations, did the astrologers have a 2- or 3-D view of the planets' position?&lt;/p&gt;&#xA;" OwnerUserId="5602" LastActivityDate="2022-08-20T11:35:25.297" Title="Astrologer's view of the planets' position" Tags="|archaeoastronomy|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14722" PostTypeId="1" AcceptedAnswerId="14723" CreationDate="2022-08-21T02:37:37.880" Score="29" ViewCount="6773" Body="&lt;p&gt;Eratosthenes calculated the radius of the Earth from the difference  of the lengths of shadows between Aswan and Alexandria were different (see also &lt;a href=&quot;https://www.youtube.com/watch?v=T0f6u39jlRA&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;). But this could also happen if the Earth were flat and the Sun wasn't really far away. A point source of light close to a flat plane will also cast shadows of different lengths. Did the Greeks already know by that point that the Sun was very far away? What hints in the natural world can lead you to that conclusion?&lt;/p&gt;&#xA;&lt;p&gt;Moreover, how did he conclude it was a sphere? There could be other shapes that have curvature and lead to different length shadows.&lt;/p&gt;&#xA;" OwnerUserId="15867" LastEditorUserId="15867" LastEditDate="2022-08-21T23:38:17.047" LastActivityDate="2022-08-24T18:19:10.620" Title="How did Eratosthenes know the Sun was very far away?" Tags="|reference-request|ancient-greece|ancient-egypt|archaeoastronomy|geography|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14723" PostTypeId="2" ParentId="14722" CreationDate="2022-08-21T13:14:33.497" Score="40" Body="&lt;p&gt;We do not know for sure what Eratosthenes read, but at his time this was a common knowledge. This can be inferred from the book of Archimedes, &lt;em&gt;The sand reckoner&lt;/em&gt; (Archimedes was Eratosthenes' contemporary; they corresponded).&#xA;In his book, Archimedes assumes that the Sun is 30 times larger than the Moon, and he mentions Eudoxus, Pheidias and Aristarchus who gave various estimates (Aristarchus wrote a book on this subject, which survives). Since the Sun and the Moon look about the same size, the conclusion is that the Sun is 30 times further.&lt;/p&gt;&#xA;&lt;p&gt;I can briefly sketch one of the arguments from Aristarchus book. When we see exactly 1/2 of the Moon, this means that at this moment the triangle Sun-Moon-observer has a 90 degrees angle at the Moon. Measuring the angle at the observer, we can approximately recover the shape of this triangle.&#xA;This angle is in fact very close to 90 degrees, so&#xA;we conclude that the Sun has to be much further than the Moon.&lt;/p&gt;&#xA;&lt;p&gt;That the Moon is quite far is also inferred from simple observations: its parallax is small (about one degree). They did not measure it&#xA;precisely at the time of Eratosthenes and Archimedes, but knew that it is small.&lt;/p&gt;&#xA;&lt;p&gt;For the general knowledge at that time of the sizes and distances between Earth, Moon and Sun, the book of Archimedes that I cited is a good source.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="10083" LastEditDate="2022-08-22T16:21:28.643" LastActivityDate="2022-08-22T16:21:28.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14724" PostTypeId="1" AcceptedAnswerId="14735" CreationDate="2022-08-22T11:04:15.263" Score="4" ViewCount="184" Body="&lt;p&gt;In &lt;em&gt;Probability Theory and Stochastic Processes with Applications&lt;/em&gt;, author Oliver Knill remarked:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The name &amp;quot;Kolmogorov axioms&amp;quot; honors a monograph of Kolmogorov from 1933 in which an axiomatization appeared. Other mathematicians have formulated similar axiomatizations at the same time like Hans Reichenbach in &lt;span class=&quot;math-container&quot;&gt;$1932. $&lt;/span&gt;According to Doob, axioms &lt;span class=&quot;math-container&quot;&gt;$\rm (i)-(iii) $&lt;/span&gt; were first proposed by G. Bohlmann in &lt;span class=&quot;math-container&quot;&gt;$1908.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The axioms here are the probability axioms. From the remark, it seems long before Kolmogorov's landmark incorporation of measure theory to formally define the frame work of probability, there were others who had already proceeded in the axiomatization problem.&lt;/p&gt;&#xA;&lt;p&gt;What I would like to know is &lt;strong&gt;prior Kolmogorov, to what extent contemporary works happened in pursuit of axiomatization and the associated chronology.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="17132" LastEditorUserId="17132" LastEditDate="2022-08-22T11:07:58.677" LastActivityDate="2022-08-24T07:55:17.617" Title="How were the works of axiomatization of probability progressed prior Kolmogorov?" Tags="|mathematics|probability|kolmogorov|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14725" PostTypeId="2" ParentId="2557" CreationDate="2022-08-22T15:44:48.547" Score="1" Body="&lt;p&gt;The underlying issue, in today's language, was that he needed two different models to explain observations on Earth and in the sky.&lt;/p&gt;&#xA;&lt;p&gt;For instance in his model about earthly matter he wrote that a constant force is needed in order to have a constant motion, a thing that matches many observations where attrition is manifest.&#xA;Celestial bodies did not seem to work that way, so they needed to be excluded from the scope of application of that model.&lt;/p&gt;&#xA;&lt;p&gt;Why he needed &amp;quot;celestial matter&amp;quot; to do that is a different philosophical question. Currently modern physics has disagreement between general relativity and quantum mechanics; we realize both are different approximations of what really happens and just ascribe the inconstintency to stuff we don't understand yet. Apparently Aristotle really wanted his laws to be intrinsecally true and valid everywhere.&lt;/p&gt;&#xA;&lt;p&gt;If laws are universal and absolute, a way to reconcile the models is to differentiate among the objects themselves and saying that each obeys its own rules.&lt;/p&gt;&#xA;" OwnerUserId="17142" LastEditorUserId="17142" LastEditDate="2022-08-22T21:32:43.630" LastActivityDate="2022-08-22T21:32:43.630" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14726" PostTypeId="1" AcceptedAnswerId="14730" CreationDate="2022-08-22T16:47:20.077" Score="5" ViewCount="504" Body="&lt;p&gt;Antoine Appert is mentioned in the bibliography of Steen &amp;amp; Seebach's &lt;em&gt;Counterexamples in Topology&lt;/em&gt;, but miscited as &amp;quot;Q. Appert&amp;quot;. Haven't a clue what Q would stand for so assuming this is a mistake.&lt;/p&gt;&#xA;&lt;p&gt;Apart from that, I understand that Appert published the Appert Topology in 1934 in his PhD thesis &lt;em&gt;Propriétés des espaces abstraits les plus généraux&lt;/em&gt;, and MGP has him as a student of Fréchet, but apart from that, nothing.&lt;/p&gt;&#xA;&lt;p&gt;Is anyone able to come up with anything else?&lt;/p&gt;&#xA;&lt;p&gt;EDIT: By complete coincidence I asked this question on what would have been Antoine Appert's 119th birthday.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastEditorUserId="12839" LastEditDate="2022-08-26T09:57:26.550" LastActivityDate="2022-08-26T09:57:26.550" Title="Who was Antoine Appert, the eponym of the Appert Topology and Appert Space?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14727" PostTypeId="2" ParentId="14699" CreationDate="2022-08-22T17:05:42.843" Score="2" Body="&lt;p&gt;This request for &amp;quot;interviews and/or conversations (in English)&amp;quot; (involving Einstein) suggests a search for a level of intimacy or dialogue which is not easy to reach in the available mass of published materials.  But the following items do seem to provide, as part of a wider range of subjects covered, some personal insights and responses that approach the character of dialogue -- especially the extended 1949 book 'Albert Einstein, Philosopher-scientist' edited by Schilpp, with debate-like features mentioned below.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;Notes for an Autobiography&amp;quot;, already mentioned by nwr, are valuable: they first appeared in the &amp;quot;Saturday Review of Literature&amp;quot; (Nov.26, 1949), and are available in &lt;a href=&quot;https://archive.org/details/EinsteinAutobiography&quot; rel=&quot;nofollow noreferrer&quot;&gt;various formats here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Also available are two book-form works based almost entirely on Einstein's own writing, they are perhaps as close as we can get in English to a concise form of his personal views on a great range of subjects.&lt;/p&gt;&#xA;&lt;p&gt;The first of them stems from 1934 (&amp;quot;Mein Weltbild&amp;quot;) (see background in its &lt;a href=&quot;https://en.wikipedia.org/wiki/The_World_as_I_See_It_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia entry&lt;/a&gt;). The most complete-appearing version of this in English (the apparent best I've found) appeared under the title &lt;a href=&quot;https://archive.org/details/ideasopinions00eins&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Ideas and Opinions'&lt;/a&gt;, edited by Carl Seelig. Several publications have used the English version of the original title &lt;a href=&quot;http://www.einsteinandreligion.com/worldsee.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;('The World as I see it')&lt;/a&gt; but those I've seen turn out to cover heavily abbreviated and limited extracts or selections, accordingly less useful. (For those who wish also to check the original language-version, which can often be quite important to capture nuances, the German itself has appeared in several editions, e.g. &lt;a href=&quot;https://books.google.com/books/about/Mein_Weltbild.html?id=MFpEEAAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;The second work is &lt;a href=&quot;https://books.google.com/books/about/Out_of_My_Later_Years.html?id=hGqUhp_j2XIC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Out of My Later Years&amp;quot;&lt;/a&gt; (1950), a further collection of articles and essays, all written after 1934: another instalment of Einstein's views on a wide range of subjects. (I haven't traced the original German from which they were translated by P A  Schilpp.)&lt;/p&gt;&#xA;&lt;p&gt;The 'notes for autobiography' mentioned first were soon afterwards (1949) extended by translator-editor P A Schilpp into a much larger book &lt;a href=&quot;https://books.google.com/books/about/Albert_Einstein_Philosopher_scientist.html?id=OoRpwAEACAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Albert Einstein, Philosopher-scientist&amp;quot;&lt;/a&gt; (1949). This includes commentaries and criticisms by 25 of Einstein's contemporaries. The contributors of these essays include some scintillating names, such as Bohr, Born, Pauli, Lemaitre, Infeld, von Laue, Gödel, and others. There is a long essay from Einstein replying to about 18 of them, amounting to a  dialogue that brings into focus among other things Einstein's differences of opinion with the quantum physicists and the attempts on both sides to sharpen, explain and resolve the points of disagreement. This is about as close as one can expect to get to a live conference and debate amongst all of those people.&lt;/p&gt;&#xA;&lt;p&gt;Besides these items, selected for their content of at least some writing from Einstein that perhaps approaches a kind of conversational character, there are of course the academic collections of his papers. This listing does not pretend to any completeness, but it is worth mentioning the resources at&lt;/p&gt;&#xA;&lt;p&gt;-- The &lt;a href=&quot;https://www.einstein.caltech.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Einstein papers project'&lt;/a&gt; and&lt;/p&gt;&#xA;&lt;p&gt;-- 'The &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/&quot; rel=&quot;nofollow noreferrer&quot;&gt;collected papers of Albert Einstein'&lt;/a&gt; for printed volumes of the 'Collected Papers'.&lt;/p&gt;&#xA;&lt;p&gt;One of the Einstein-related subjects of perennial interest is in the relations between relativistic and classical physics. Einstein's own expressions of view here are naturally of more than ordinary interest. They can be found in four articles of his, in which he gives explanation of relativity and appraisals of Newton and of Kepler, designed for a general audience:&lt;/p&gt;&#xA;&lt;p&gt;-- &lt;a href=&quot;https://archive.org/details/science511920mich/page/8/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Time, Space and Gravitation'&lt;/a&gt; (1919) (a letter that 'The Times' newspaper invited from Einstein, in response to an article that reported results from the well-known &lt;a href=&quot;https://en.wikipedia.org/wiki/Solar_eclipse_of_May_29,_1919&quot; rel=&quot;nofollow noreferrer&quot;&gt;1919 eclipse expedition&lt;/a&gt;, and included the headline 'Newton Overthrown') .&lt;/p&gt;&#xA;&lt;p&gt;-- &lt;a href=&quot;https://www.pbs.org/wgbh/nova/article/einstein-on-newton/&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Einstein on Newton'&lt;/a&gt;, an appraisal originally published in the 'Manchester Guardian' for 1927 March 19. It appeared again in the &lt;a href=&quot;https://library.si.edu/digital-library/book/annualreportofbo1927smit&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smithsonian Institution Annual report for 1927&lt;/a&gt;  (at p.201-207). (Remarkably, an offprint of the Smithsonian printing, which had been owned by Albert Einstein's son Dr Hans Albert Einstein, was &lt;a href=&quot;https://onlineonly.christies.com/s/shoulders-giants-newton-darwin-einstein-hawking/einstein-newton-hans-albert-einsteins-copy-23/62338&quot; rel=&quot;nofollow noreferrer&quot;&gt;sold at auction&lt;/a&gt; only a few years ago for GBP 13,750 !)&lt;/p&gt;&#xA;&lt;p&gt;-- a third article on Newton, and one on Kepler, in the collection &amp;quot;Out of My Later Years&amp;quot; already mentioned above.&lt;/p&gt;&#xA;&lt;p&gt;(Although Einstein was not a historian, these articles arguably provide a depth of perception and balance of appraisal that may not have been achieved by any commentator since, almost a must-read.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-08-23T13:10:26.210" LastActivityDate="2022-08-23T13:10:26.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14728" PostTypeId="2" ParentId="7299" CreationDate="2022-08-22T17:50:58.970" Score="3" Body="&lt;p&gt;I think I recall reading this story as a little girl in the Time Life series of science books, in the volume on “Mathematics.” If that is the case (it’s a fifty-plus year old recollection), those books were published in the early 1960s.  Perhaps there is a clue in the references at the back of the book as to the source of the story.  Time Life, by the way, did not have footnotes or in-line citations in these volumes, but they did have LOTS of pictures and anecdotes that made science come alive to a pre-teen kid.&lt;/p&gt;&#xA;" OwnerUserId="17145" LastEditorUserId="10106" LastEditDate="2022-08-22T17:59:59.667" LastActivityDate="2022-08-22T17:59:59.667" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14729" PostTypeId="2" ParentId="14722" CreationDate="2022-08-22T19:10:54.897" Score="16" Body="&lt;p&gt;This is a long answer, explaining more or less step to step why the data available to Eratosthenes indicted that the Sun should be a vast distance from the Earth, thus makings its diverging rays almost parallel when they struck the Earth.&lt;/p&gt;&#xA;&lt;p&gt;Eratosthenes  Lived from c. 276 BC to 195/194 BC.&lt;/p&gt;&#xA;&lt;p&gt;There are (rare) eclipses of the Sun by the Moon.&lt;/p&gt;&#xA;&lt;p&gt;Even though it is hard to see a thin sliver of the Moon in the daytime sky, especially when it is close to the painful glare of the Sun, people did observe the Moon before the Sun came up or after the Sun went down, approaching the position of the Sun and becoming thinner and thinner until it becomes unseen, and then becoming a thin new moon and the crescent increasing in thickness as the Moon's angular distance from the Sun increased.&lt;/p&gt;&#xA;&lt;p&gt;Anaxagoras (500 BC - 428 BC) believed that the Sun was larger than the Peloponnese, which is about 80 miles wide.  Since the Sun appears to be about half a degree wide as seen from Earth, it should be at least about 114.59 times 80 miles, or 9,167.2 miles, from Earth according to Anaxagoras.  And as the ancient Greeks explored the world and found it was larger than they thought, Greek philosophers realized the Sun had to be larger and farther away.&lt;/p&gt;&#xA;&lt;p&gt;Anaxagoras is said to be the first person to explain that solar eclipses are caused by the Moon passing in front of the Sun.  Which of course shows that the Moon is closer than the Sun.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.smithsonianmag.com/science-nature/ancient-greek-philosopher-was-exiled-claiming-moon-was-rock-not-god-180972447/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.smithsonianmag.com/science-nature/ancient-greek-philosopher-was-exiled-claiming-moon-was-rock-not-god-180972447/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Anaxagoras was said to be the first person to explain that the Moon shines due to reflected light from the Sun.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://history.stackexchange.com/questions/7890/who-discovered-first-that-the-moon-does-not-have-its-own-light&quot;&gt;https://history.stackexchange.com/questions/7890/who-discovered-first-that-the-moon-does-not-have-its-own-light&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;And that theory would indicate that the Moon was closer to Earth than the Sun.&lt;/p&gt;&#xA;&lt;p&gt;I suggest that you make two different diagrams with the Sun and the Moon revolving around the Earth, as most of the ancients believed.&lt;/p&gt;&#xA;&lt;p&gt;One diagram will show the Moon closer to Earth than the Sun, and one will show the Moon Farther from Earth than the Sun.  Make the Sun and the Moon little circles and not mere dots.&lt;/p&gt;&#xA;&lt;p&gt;Then color the side of the Moon's orb that faces away from the Earth black, and then see how much of the moon's lighted surface is visible from Earth at various angles.  The thin crescent phases of the Moon would impossible to see if the Moon was farther away from Earth than the Sun.&lt;/p&gt;&#xA;&lt;p&gt;Mercury and Venus also show phases like the Moon, visible through telescopes, and there are cases of people observing the crescent phase of Venus without optical aid - I sometimes thought that I could barely see a crescent Venus.&lt;/p&gt;&#xA;&lt;p&gt;An ancient Greek philosopher who heard that someone saw a thin crescent Venus would deduce that like the Moon, Venus would then be closer to Earth than the Sun is.&lt;/p&gt;&#xA;&lt;p&gt;And if the Moon was ever observed to occult (pass in front of and hide) Mercury or Venus, a philosopher would know that the Moon was closer to Earth than the planet it occulted.  Observations of occultations were probably why the ancients believed that Mars, Jupiter, and Saturn were farther from Earth than the Moon is.&lt;/p&gt;&#xA;&lt;p&gt;Because Mercury and Venus are actually closer to the Sun than Earth is they never appear more than 28 degrees (Mercury) and 47 degrees (Venus) from the Sun as seen from Earth.  Because Mars, Jupiter and Saturn are farther from the Sun than Earth is, they can appear as far as 180 degrees, directly opposite the direction to the Sun as seen from Earth.  There is obviously something different about the orbits of Mercury and Venus compared to those of the other 3 planets known in antiquity.&lt;/p&gt;&#xA;&lt;p&gt;In the 2nd century AD, centuries after Eratosthenes, Ptolemy believed that Mercury and Venus were farther from Earth than the Moon and closer to Earth than the sun.  So thee would have to be a large distance between the Moon and the Sun for Mercury and Venus to be between them.  But I don't know whether Eratosthenes believed that Mercury and Venus were between the Moon and the Sun.&lt;/p&gt;&#xA;&lt;p&gt;According to the answer by Alexander Ermenko, Aristarchus believed the Sun was much farther than the Moon.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I can briefly sketch one of the arguments from Aristarchus book. When we see exactly 1/2 of the Moon, this means that at this moment the triangle Sun-Moon-observer has a 90 degrees angle at the Moon. Measuring the angle at the observer, we can approximately recover the shape of this triangle. This angle is in fact very close to 90 degrees, so we conclude that the Sun has to be much further than the Moon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Aristarchus of Samos c. 310 - c. 230 BC, was an early proponent of heliocentrism instead of geocentrism.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Aristarchus claimed that at half moon (first or last quarter moon), the angle between the Sun and Moon was 87°.[19] He might have proposed 87° as a lower bound, since gauging the lunar terminator's deviation from linearity to one degree of accuracy is beyond the unaided human ocular limit (with that limit being about three arcminutes of accuracy). Aristarchus is known to have studied light and vision as well.[20]&lt;/p&gt;&#xA;&lt;p&gt;Using correct geometry, but the insufficiently accurate 87° datum, Aristarchus concluded that the Sun was between 18 and 20 times farther away from the Earth than the Moon.[21] (The true value of this angle is close to 89° 50', and the Sun's distance is approximately 400 times that of the Moon.) The implicit false solar parallax of slightly under three degrees was used by astronomers up to and including Tycho Brahe, c. AD 1600. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes, and therefore their diameters must be in proportion to their distances from Earth.[22]&lt;/p&gt;&#xA;&lt;p&gt;In On the Sizes and Distances of the Sun and Moon, Aristarchus discusses the size of the Moon and Sun in relation to the Earth. In order to achieve these measurements and subsequent calculations, he used several key notes made while observing a lunar eclipse.[23] The first of these consisted of the time that it took for the Earth's shadow to fully encompass the Moon, along with how long the Moon remained within the shadow. This was used to estimate the angular radius of the shadow.[24] From there, using the width of the cone that was created by the shadow in relation to the Moon, he determined that it was twice the diameter of the Moon at the full, non-central eclipse. In addition to this, Aristarchus estimated that the length of the shadow extended around 2.4 times the distance of the Moon from the Earth.[23]&lt;/p&gt;&#xA;&lt;p&gt;Using these calculations, along with the aforementioned estimated distances of the Sun from the Earth and Moon from the Earth, he created a triangle. Employing a similar method of geometry that he previously used for the distances, he was able to determine that the diameter of the Moon is roughly one-third that of the Earth's diameter. In order to estimate the size of the Sun, Aristarchus considered the proportion of the Sun's distance to Earth in comparison to the Moon's distance from Earth, which was found to be roughly 18 to 20 times the length. Therefore, the size of the Sun is around 19 times wider than the Moon, making it approximately six times wider than the Earth's diameter.[23]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aristarchus_of_Samos&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Aristarchus_of_Samos&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So in the time of Eratosthenes, it was known from traveling and exploration that the Earth was at least a few thousand modern miles or kilometers in diameter.&lt;/p&gt;&#xA;&lt;p&gt;And for each thousand miles the diameter of the Earth was,  the Moon would be about 333.333 miles in diameter and the Sun about 6,000 miles in diameter.   If the Sun and the Moon are about 114.59 times as far away as their diameters, for each 1,000 miles the diameter of the Earth was, the Moon would be 38,196.6 miles, and the Sun 687,540 miles, from Earth.&lt;/p&gt;&#xA;&lt;p&gt;So with the data available to Eratosthenes it was fairly accurate to consider the Sun's rays striking the Earth in different places to be very close to parallel to each other.  And as it turned out the Sun is actually many times as far away as was believed in Eratosthenes's era, thus making its rays even closer to being parallel.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2022-08-24T18:19:10.620" LastActivityDate="2022-08-24T18:19:10.620" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14730" PostTypeId="2" ParentId="14726" CreationDate="2022-08-22T20:32:55.190" Score="13" Body="&lt;p&gt;The only biographic information I could find can be found in &lt;a href=&quot;https://books.centre-mersenne.org/item/10.5802/mbk.1.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Les Séminaire de Mathématiques 1933-1939 (Première Partie-l'Histoire)&lt;/em&gt;&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Mich%C3%A8le_Audin&quot; rel=&quot;noreferrer&quot;&gt;Michèle Audin&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ni normalien ni agrégé, Antoine Appert avait passé sa thèse en 1934 [...] et était devenu membre de la SMF la meme année.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Neither a &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89cole_normale_sup%C3%A9rieure_(Paris)&quot; rel=&quot;noreferrer&quot;&gt;normalien&lt;/a&gt;&amp;quot; nor an &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Agr%C3%A9gation&quot; rel=&quot;noreferrer&quot;&gt;agrégé&lt;/a&gt;&amp;quot;, Antoine Appert had passed his thesis in 1934 [...] and had become a member of the SMF the same year.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the same source we learn that he spoke at least twice at the Hadamard seminars (in 1934-1935 and 1936-1937). The list of seminary 'subscribers', i.e. people who were to receive the proceedings, records Appert's presence in the fourth year (1936-1937) and in the fifth year, from which we learn that he lived at 8 rue Berthier in Versailles.&lt;/p&gt;&#xA;&lt;p&gt;He discussed his thesis &lt;a href=&quot;http://www.numdam.org/item/THESE_1934__156__1_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Propriétés des espaces abstraits les plus généraux&lt;/em&gt;&lt;/a&gt; on March 17, 1934, with Cartan as chairman and Fréchet and &lt;a href=&quot;https://en.wikipedia.org/wiki/Georges_Valiron&quot; rel=&quot;noreferrer&quot;&gt;Valiron&lt;/a&gt; as examiners.&lt;/p&gt;&#xA;&lt;p&gt;From the date of discussion, one can guess the date of birth: Jean Leray and &lt;a href=&quot;https://fr.wikipedia.org/wiki/Jean-Louis_Destouches&quot; rel=&quot;noreferrer&quot;&gt;Jean-Louis Destouches&lt;/a&gt; discussed their theses in 1933: the former was born in 1906, the latter in 1909; Chevalley discussed it in 1934 and was born in 1909. For Appell, one can conjecture a probable date of birth between 1905 and 1910 &lt;strong&gt;[EDIT: I was optimistic in estimating such a narrow range, but I was not far wrong: the correct date seems to be 1903, see below]&lt;/strong&gt;; certainly not later than 1912 (Weil discussed his thesis at the age of 22, but this is a rather rare case) and reasonably not earlier than 1894 (&lt;a href=&quot;https://fr.wikipedia.org/wiki/Rapha%C3%ABl_Salem&quot; rel=&quot;noreferrer&quot;&gt;Raphaël Salem&lt;/a&gt; discussed his thesis at the age of 40, but this too is a rather rare case).&lt;/p&gt;&#xA;&lt;p&gt;Regarding the value of his thesis work, one can read &lt;a href=&quot;https://tel.archives-ouvertes.fr/tel-00426604/document&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;L'entre-deux-guerres mathématique à travers les thèses soutenues en France&lt;/em&gt;&lt;/a&gt; by Juliette Leloup:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Two doctoral theses deal with questions relating to abstract spaces with reference to Maurice Fréchet. First of all, in 1930, there is the thesis of a Japanese student, Kijiro Kunugui, then in 1934, that of a French student, Antoine Appert, respectively Sur la théorie du nombre de dimensions, 1930 and Propriétés des espaces abstraits les plus généraux, 1934. These two doctorates were based on some of Maurice Fréchet's work, including the latter's 1908 thesis and especially his articles on abstract sets published in the 1920s, the results of which were summarised without demonstration by Fréchet in Les espaces abstraits et leur théorie considérée comme introduction à l'analyse générale in 1928. The two doctoral candidates, from their introduction, explicitly refer to this book which appears as the reference work on the theory of abstract sets. Fréchet also reviews all the publications on the subject. (p. 214)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Moreover, in the thesis report, Maurice Fréchet explains that he himself has already obtained extensions of the properties of linear space to the most general spaces (V) and that he &amp;quot;was able to gather a certain number of them in [his] book 'Les Espaces abstraits'. But for lack of space, [he] could only accommodate the statements. The demonstrations remained scattered in a large number of periodicals&amp;quot;. Explaining Appert's role, he writes:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;It is a first merit of M. Appert, to have been able to gather here, in a coherent presentation, statements and demonstrations of properties already known about spaces (V). It could not be a simple compilation. The published demonstrations, due to various authors, overlapped one another and used definitions that were still different at a time when the terminology was evolving. Mr. Appert was able to carry out this work thanks to a well-informed critical sense and a rigour that was always on the alert. But Mr Appert did not stop at writing this paper, which filled a gap. His second merit is to have introduced new notions and obtained interesting properties in the field of which he was taking stock. (pp. 215-216)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another comment is &lt;a href=&quot;https://www.ams.org/journals/bull/1936-42-11/S0002-9904-1936-06441-4/S0002-9904-1936-06441-4.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some more information about his life can be deduced from the list of his works:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;http://www.numdam.org/item/10.24033/bsmf.1265.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Réflexions sur une note de M. Bouligand « Problèmes bien posés et problèmes à conditions surabondantes »&lt;/em&gt;&lt;/a&gt; in &lt;em&gt;Bulletin de la S.M.F.&lt;/em&gt;, tome 65 (1937), p. 76-82&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Sur les topologies transitives&lt;/em&gt; in Bull. Acad. roy. Belgique (1937)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Espaces uniformes généralisés&lt;/em&gt;, in Comptes Rendues, 222 (1946),&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Ecart partiellement ordonné et uniformité&lt;/em&gt;, , in Comptes Rendues, 224 (1947)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://excerpts.numilog.com/books/9782307363057.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Espaces topologiques intermédiaires&lt;/em&gt;&lt;/a&gt;, with &lt;a href=&quot;https://en.wikipedia.org/wiki/Ky_Fan&quot; rel=&quot;noreferrer&quot;&gt;Ky Fan&lt;/a&gt; (1951)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This is interesting because from the title page we learn that Appert was &amp;quot;ancien &lt;a href=&quot;https://fr.wikipedia.org/wiki/Ma%C3%AEtre_de_conf%C3%A9rences_(France)&quot; rel=&quot;noreferrer&quot;&gt;maîtres de conférences&lt;/a&gt; à la Faculté des Sciences de &lt;a href=&quot;https://en.wikipedia.org/wiki/Rennes&quot; rel=&quot;noreferrer&quot;&gt;Rennes&lt;/a&gt;&amp;quot;, i.e., at the time (1951) he was a former lecturer in Rennes.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF02843678.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur l'équivalence de diverses définitions de la continuité dans les espaces topologiques généraux, et sur l'axiomatique de ces espaces&lt;/em&gt;&lt;/a&gt;, in Rendiconti del Circolo Matematico di Palermo (1961)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/content/pdf/10.1007/BF02851266.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur la mesure de Carathéodory&lt;/em&gt;&lt;/a&gt;, in Rendiconti del Circolo Matematico di Palermo (1963)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From these two entries we learn that in 1961-1963 Appert was in &lt;a href=&quot;https://en.wikipedia.org/wiki/Angers&quot; rel=&quot;noreferrer&quot;&gt;Angers&lt;/a&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Mesures et familles topologiquement monotones&lt;/em&gt;, in Comptes Redues 260 (1965)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://academic.oup.com/plms/article-abstract/s3-18/2/266/1488381&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Mesures Limites Dans Les Espaces Distanciés Séparables et Localement Compacts&lt;/em&gt;&lt;/a&gt;in Proceedings of the London Mathematical Society, s3-18-2, (1968)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The most recent paper I could find are: &lt;a href=&quot;http://www.numdam.org/article/CSHM_1982__3__63_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Sur le meilleur terme primitif en topologie&lt;/em&gt;&lt;/a&gt; in Cahiers du séminaire d’histoire des mathématiques (1982) and &lt;a href=&quot;http://www.numdam.org/item/CSHM_1989__10__231_0.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Suite à l’article sur le meilleur terme primitif en topologie&lt;/em&gt;&lt;/a&gt;, ibidem, (1989).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The Editor's note is particularly interesting:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;C’est G. Choquet qui a bien voulu nous communiquer le présent article. Il nous écrivait, le 4 avril 1981, à propos de cette Note de A. Appert : &amp;quot;Venant d’un disciple direct de Fréchet, qui a beaucoup réfléchi sur les fondements de la topologie, sa Note est un témoignage sur les préoccupations qui ont été celles des fondateurs de la topologie. A ce titre il m’a semblé qu’elle pourrait intéresser des historiens des sciences et les mathématiciens ou philosophes qui réfléchissent à la naissance des concepts.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This article was kindly communicated by &lt;a href=&quot;https://en.wikipedia.org/wiki/Gustave_Choquet&quot; rel=&quot;noreferrer&quot;&gt;G. Choquet&lt;/a&gt;. On April 4, 1981, he wrote to us about this Note by A. Appert: &amp;quot;Coming from a direct disciple of Fréchet, who has given much thought to the foundations of topology, his Note is a testimony to the concerns of the founders of topology. As such, it seemed to me that it could be of interest to historians of science and mathematicians or philosophers who reflect on the birth of concepts.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Appert is also briefly cited in &lt;a href=&quot;http://www.stat.ucla.edu/%7Eywu/research/documents/Grothendieckchap2.pdf&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Récoltes et Semailles&lt;/em&gt;&lt;/a&gt; by Grothendieck:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I had juggled with sets that I called “measurable” (without having ever seen one that was not...) and with convergence nearly everywhere, but I didn’t know what a topological space was. I was a bit lost amonst a dozen equivalent notions of “abstract space” and compactness which I fished out of &lt;strong&gt;a little textbook (by someone called Appert, I believe,&lt;/strong&gt; published in the &lt;em&gt;Actualit´es Scientifiques et Industrielles&lt;/em&gt;), which I had come across God knows how. I still had never heard the strange and barbarous words group, field, ring,&#xA;module, complex, homology (etc.) pronounced in a mathematical context, and suddenly, they were all rolling over me together. It was quite a shock!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT&lt;/p&gt;&#xA;&lt;p&gt;The information above is consistent with what is reported on &lt;a href=&quot;https://gw.geneanet.org/pierfit?n=appert&amp;amp;oc=1&amp;amp;p=antoine&quot; rel=&quot;noreferrer&quot;&gt;this site&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In short: Antoine Appert would be the grandson of &lt;a href=&quot;https://fr.wikipedia.org/wiki/F%C3%A9lix_Antoine_Appert&quot; rel=&quot;noreferrer&quot;&gt;Félix Antoine Appert&lt;/a&gt; (1817-1891), general of the army, French ambassador to the emperor of Russia. He had four children, among them Félix (Etienne Félix Eugène) (1860-1914), colonel, married to Germaine Le Riche de Breuilpont, decorated with the &lt;a href=&quot;https://fr.wikipedia.org/wiki/Ordre_national_de_la_L%C3%A9gion_d%27honneur&quot; rel=&quot;noreferrer&quot;&gt;Légion d'honneur&lt;/a&gt; in 1902, died for his country in WWI.&#xA;Antoine (Antoine Isidore Marie Joseph) Appert, son of Félix, was born August 22, 1903 in &lt;a href=&quot;https://en.wikipedia.org/wiki/Vannes&quot; rel=&quot;noreferrer&quot;&gt;Vannes&lt;/a&gt;; Docteur es-sciences mathématiques, researcher at &lt;a href=&quot;https://en.wikipedia.org/wiki/French_National_Centre_for_Scientific_Research&quot; rel=&quot;noreferrer&quot;&gt;C.N.R.S.&lt;/a&gt;; married with Marie-Louise Mayaud.&#xA;Died on December 16, 1992 in &lt;a href=&quot;https://en.wikipedia.org/wiki/Saint-Laurent-de-la-Plaine&quot; rel=&quot;noreferrer&quot;&gt;Saint-Laurent-de-la-Plaine&lt;/a&gt;, aged 89.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2022-08-22T22:06:10.157" LastActivityDate="2022-08-22T22:06:10.157" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14731" PostTypeId="1" CreationDate="2022-08-23T09:52:02.330" Score="1" ViewCount="104" Body="&lt;p&gt;I'm reading &lt;a href=&quot;http://cavendish-deneyi.com/pdf/Circular_motion.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Richard Conn Henry's article&lt;/a&gt; called &amp;quot;Circular Motion.&amp;quot; In it he states that in his &lt;em&gt;De vi Centrifuga&lt;/em&gt; Huygens discovered the formula &lt;span class=&quot;math-container&quot;&gt;$a=v^2/r$&lt;/span&gt; and that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Edmund Halley, Christopher Wren, and Robert Hooke were all able to immediately substitute the result into Kepler's third law and deduce that the gravitational force must vary inversely as the square of the distance from the sun.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He does not give any references about where he got this information. Do you know any historical documents showing that Halley, Wren, or Hooke arrived at inverse square law by substituting &lt;span class=&quot;math-container&quot;&gt;$a=v^2/r$&lt;/span&gt; into Kepler's Rule?&lt;/p&gt;&#xA;&lt;p&gt;If there exist such a document, I assume, it would be in the language of proportions and not equations.&lt;/p&gt;&#xA;&lt;p&gt;Note: &lt;a href=&quot;https://hsm.stackexchange.com/questions/7288/who-first-derived-a-v2-r&quot;&gt;This question&lt;/a&gt; has nice information about Huygens' discovery but not about its application by the above people.&lt;/p&gt;&#xA;" OwnerUserId="16575" LastEditorUserId="55" LastEditDate="2022-08-23T10:58:59.317" LastActivityDate="2022-08-23T10:58:59.317" Title="Did Halley, Wren and Hooke use $a=v^2/r$ to infer the inverse square law?" Tags="|physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14732" PostTypeId="1" CreationDate="2022-08-23T12:00:51.553" Score="2" ViewCount="93" Body="&lt;p&gt;I've been reading about kinetic theory of gases, which only deals with gases. I know that the lattice Boltzmann method, which is commonly used to simulate fluid flows, finds its origin in the kinetic theory of gases. After reading a &lt;a href=&quot;https://www.math.umd.edu/%7Elvrmr/History/index.shtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;brief chronological history&lt;/a&gt;, I wonder who extended the theory from gases to fluids in general and what arguments were used. I wonder if it was done along with the creation of statistical physics.&lt;/p&gt;&#xA;&lt;p&gt;I've read some posts on this website, however I've not seen a clear answer to my question.&lt;/p&gt;&#xA;" OwnerUserId="17148" LastEditorUserId="55" LastEditDate="2022-08-24T06:40:27.817" LastActivityDate="2022-08-24T06:40:27.817" Title="Why and when was the kinetic theory of gases generalized to fluids?" Tags="|physics|discoveries|theoretical-physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14733" PostTypeId="1" AcceptedAnswerId="14739" CreationDate="2022-08-24T05:02:32.133" Score="4" ViewCount="224" Body="&lt;p&gt;Nesbitt's Inequality can be found all over the internet:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This appears to have been first published in 1902 in Education Times, by A.M. Nesbitt.&lt;/p&gt;&#xA;&lt;p&gt;Is the identity of this person in an way retrievable, or is he or she lost in history?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2022-08-25T07:09:44.583" Title="Who was A.M. Nesbitt, the eponym of Nesbitts Inequality?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14734" PostTypeId="2" ParentId="14732" CreationDate="2022-08-24T06:29:24.917" Score="4" Body="&lt;p&gt;Considering that &amp;quot;fluid&amp;quot; is a joint name for gases and liquids, I will interpret the question as about extending the Maxwell-Boltzmann theory of dilute gases to dense gases and liquids. Although there were major developments in this direction since 1920s, they are rather fragmented, and a unified kinetic theory of dense fluids does not exist.&lt;/p&gt;&#xA;&lt;p&gt;One can see the 1905-06 work of Einstein and Smoluchowski on the Brownian motion as a step in this direction, and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Chapman%E2%80%93Enskog_theory&quot; rel=&quot;noreferrer&quot;&gt;Chapman–Enskog theory&lt;/a&gt; (1916-17) that related the Boltzmann equation and the &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;-theorem to Euler and Navier-Stokes equations as another step.&lt;/p&gt;&#xA;&lt;p&gt;The main difficulty with extension beyond dilute gases was that one could no longer assume molecular chaos, i.e. ignore position and/or velocity correlations between molecules. The simplest version was developed by Enskog in 1920s, some historical details can be found in &lt;a href=&quot;https://physicstoday.scitation.org/doi/10.1063/1.2916049&quot; rel=&quot;noreferrer&quot;&gt;Cohen's survey The kinetic theory of fluids&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The simplest such correlation occurs when the presence of a third particle, 3, prevents particles 1 and 2, to be at the&#xA;positions &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt;... If one takes only those into account, and also the difference in&#xA;position of the two colliding particles,&#xA;one obtains a kinetic equation for a dense gas of hard spheres that Enskog had already found in the 1920s.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This equation is, of course, only a first step in the generalization of the Boltzmann equation to higher densities. So far it has not been extended to realistic intermolecular potentials. Nevertheless, it has been very useful for describing the density dependence of transport coefficients of dense fluids in practice.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Writing in 1946, Born and Green described a rather skeptical attitude concerning kinetic theory of dense fluids even at equilibrium, while proposing a new approach in a multi-part &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rspa.1946.0093&quot; rel=&quot;noreferrer&quot;&gt;General kinetic theory of liquids&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It has been said that there exists no general theory of liquids because it is impossible&#xA;to utilize the simplifying conditions either of the kinetic theory of gases (Chapman&#xA;&amp;amp; Cowling 1939) where the density is small, or of the theory of solids (Born 1923)&#xA;where a high degree of spatial order may be assumed. Yet a mathematical formulation&#xA;of the problem should be possible, without making such an assumption, since&#xA;only the general laws of mechanics and statistics are involved; though the solution&#xA;itself may be extremely difficult.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, velocity correlations for non-equilibrium fluids proved much more challenging, as Cohen describes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Nicolai Bogolubov, Melville Green, I&#xA;and many others in the years after 1945 tried to continue in this way and considered velocity correlations between the particles 1 and 2 at &lt;span class=&quot;math-container&quot;&gt;$r_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$r_2$&lt;/span&gt; due to previous collisions with two particles... However, as was discovered around 1965... in contrast with the case of a fluid in thermal equilibrium, velocity correlations in a nonequilibrium fluid introduce&#xA;long-range correlations between the particles that prevent a systematic generalization of the Boltzmann equation to higher densities&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even more complex velocity correlations in high density fluids led to more negative results. They generate long-time tails, discovered by Alder and Wainwright (1967) in computer simulations: hydrodynamic equations beyond the Navier-Stokes equations do not seem to exist, as Burnett and higher-order hydrodynamic equations do not have finite transport coefficients.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;http://www.scholarpedia.org/article/Lattice_Boltzmann_Method#The_Lattice_Boltzmann_equation:_history&quot; rel=&quot;noreferrer&quot;&gt;lattice Bolztmann theory&lt;/a&gt; is more recent, and was developed, in part, in response to the above negative results:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The lattice Boltzmann equation (LBE) is a minimal form of Boltzmann kinetic equation which is meant to simulate the dynamic behaviour of fluid flows without directly solving the equations of continuum fluid mechanics. Instead, macroscopic fluid dynamics emerges from the underlying dynamics of a fictitious ensemble of particles, whose motion and interactions are confined to a regular space-time lattice...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The LBE can be shown to quantitatively reproduce the equations of motion (Chapman and Cowling, 1952) of continuum fluid mechanics, in the limit of long wavelengths as compared to the lattice scale. The idea of simplified Boltzmann equations with discrete speeds can be traced to the pioneering work of Broadwell, back in the 60's (Broadwell, 1964). However, to the best of our knowledge, these discrete-velocity Boltzmann equations were mostly intended to provide simpler, sometimes even analytically tractable, model equations for rarefied gas dynamics, but never meant as a computational alternative for the numerical solution of the Navier-Stokes equations of continuum fluid dynamics.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The major conceptual twist in this direction had to wait another 20 years, with the advent of the celebrated Frisch-Hasslacher-Pomeau (FHP) lattice gas automaton (Frisch et al., 1986; Wolfram, 1986, Wolf-Gladrow, 2000). Originally developed in response to the major pitfalls of the Lattice Gas Cellular Automaton approach (Frisch et al., 1986; Wolf-Gladrow, 2000), the LBE rapidly developed into a vigorous self-standing research subject (Mc Namara and Zanetti, 1988; Higuera and Jimenez, 1989; Higuera et al., 1989; Chen et al., 1992).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-08-24T06:29:24.917" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14735" PostTypeId="2" ParentId="14724" CreationDate="2022-08-24T07:55:17.617" Score="4" Body="&lt;p&gt;Kolmogorov's axiomatization achieved greater simplicity, clarity, rigor and unification of applications by grounding probability in measure theory. There were prior axiomatizations that went in a different direction, e.g. &lt;a href=&quot;https://iopscience.iop.org/article/10.1070/RM1961v016n02ABEH004103&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernstein's algebraic approach (1917)&lt;/a&gt;, or von Mises's frequentist one, see &lt;a href=&quot;https://www.jstor.org/stable/4355955&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lambalgen, Von Mises' Axiomatisation of Random Sequences&lt;/a&gt;. Both are sometimes named &amp;quot;the first&amp;quot; axiomatizations of probability.&lt;/p&gt;&#xA;&lt;p&gt;However, Bohlmann got there much earlier, sort of, see &lt;a href=&quot;https://www.jstor.org/stable/41133504&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bingham, Measure into probability: From Lebesgue to Kolmogorov&lt;/a&gt;. Hilbert refers to his labeling of the rules of total and compound probability as &amp;quot;axioms&amp;quot; in a footnote to the sixth problem from his famous list of 1900:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;To treat... by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theories of Probability and Mechanics&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bohlmann's work was published in the &lt;em&gt;Encyclopadia der Mathematischen Wissenschaften&lt;/em&gt; in 1901, and later in the proceedings of the 1908 International Congress of Mathematicians. He gave the first rigorous definition of independence, and distinguished between independence and pairwise independence, but did not give what one would call an axiomatic development, i.e. derived any major theorems. His work was, apparently, unknown to Kolmogorov.&lt;/p&gt;&#xA;&lt;p&gt;The developments that influenced Kolmogorov's measure-theoretic axiomatization are described by &lt;a href=&quot;https://www.jstor.org/stable/41133504&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barone-Novikoff in A history of the axiomatic formulation of probability from Borel to Kolmogorov&lt;/a&gt;, see also &lt;a href=&quot;https://arxiv.org/abs/1802.06071&quot; rel=&quot;nofollow noreferrer&quot;&gt;The origins and legacy of Kolmogorov's &lt;em&gt;Grundbegriffe&lt;/em&gt;&lt;/a&gt;. For a summary of Borel's contributions and ill-advised choices that hampered his work see &lt;a href=&quot;https://hsm.stackexchange.com/a/14206/55&quot;&gt;Why did Borel reject countable additivity of probability?&lt;/a&gt; Sierpinski's axiomatization of measure theory (1918) adapted to Borel's probability framework by Steinhaus (1923) were also major stepping stones. From Barone-Novikoff's introduction:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The need for an axiomatic foundation for Probability Theory had been stressed by Hilbert as part of the sixth problem in his celebrated list of 1900... The landmark paper, initiating the modern theory of probability, is E. Borel's &amp;quot;&lt;em&gt;Les Probabilities Dénombrables et Leurs Applications Arithmétiques&lt;/em&gt;&amp;quot; of 1909... The key figures immediately following Borel are G. Faber, F. Bernstein, and F. Hausdorff.... The ironical circumstance&lt;/em&gt; [is] &lt;em&gt;that Borel, the unquestioned founder of measure theory, attempted in 1909 to found a new theory of &amp;quot;denumerable probability&amp;quot; without relying on measure theory. The irony is further compounded in the light of Borel's paper of 1905 which identified &amp;quot;continuous probability&amp;quot; in the unit interval with measure theory.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Steinhaus (1923) finally incorporated Borel's &amp;quot;general theory&amp;quot; into measure theory by exploiting an axiomatic characterization of measure theory due to Sierpinski (1918). Steinhaus' work and more generally the increasing abstraction of measure theory itself (initiated by Frechet (1915) and Caratheodory (1914)) were events which helped pave the way to Kolmogorov's achievement.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-08-24T07:55:17.617" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14736" PostTypeId="1" CreationDate="2022-08-24T08:57:42.630" Score="2" ViewCount="66" Body="&lt;p&gt;In his book &lt;em&gt;Introduction to Metric and Topological Spaces&lt;/em&gt;, author Wilson A Sutherland in explaining the equivalence of metrics invoked the definition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Two metrics &lt;span class=&quot;math-container&quot;&gt;$d_1, d_2$&lt;/span&gt; on a set &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; will be called&lt;/em&gt; Lipschitz equivalent &lt;em&gt;if there are positive constants &lt;span class=&quot;math-container&quot;&gt;$h, k$&lt;/span&gt; such that for any&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$x, y \in X, $&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$$  hd_2(x, y) \leq d_1(x, y) \leq kd_2(x, y) .$$&lt;/span&gt;&#xA;The reader should be warned that although the name we have chosen for the concept seems appropriate, it is not universally used.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It's a familiar definition that is used to show equivalence of metrics and in fact it is equivalent to &lt;em&gt;topological equivalence&lt;/em&gt;. However, prior Sutherland, I was not familiar with the name attributed to Lipschitz. I googled and tried to narrow down any specific paper that he wrote concerning the equivalence thus lending his name, however of no avail.&lt;/p&gt;&#xA;&lt;p&gt;So, my question is, &lt;strong&gt;did Lipschitz formulate the equivalence himself or were it some authors who attributed the equivalence to him in their treatises?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2022-08-24T08:57:42.630" Title="When was Lipschitz equivalence first attributed to Lipschitz or did Lipschitz formulate it himself?" Tags="|mathematics|terminology|topology|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14737" PostTypeId="1" CreationDate="2022-08-24T11:50:31.517" Score="3" ViewCount="180" Body="&lt;p&gt;I read once (I don't remember where exactly) about an early technique (Early Renaissance) to draw a square tiling floor in perspective. The next row in the drawing is done multiplying the previous one by a constant factor. My questions:&lt;/p&gt;&#xA;&lt;p&gt;(1) What is the value of this factor? 2/3? 3/4?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/asJCn.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/asJCn.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hnTm3.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hnTm3.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;(2) Are there reliable sources/references that confirm this historical fact?&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12330" LastEditDate="2022-08-25T21:42:51.390" LastActivityDate="2022-08-25T21:42:51.390" Title="What was the pre-Renaissance proto-perspective scaling technique for painting square tiled floors?" Tags="|geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14738" PostTypeId="2" ParentId="14737" CreationDate="2022-08-24T19:43:53.987" Score="7" Body="&lt;p&gt;tl;dr&lt;/p&gt;&#xA;&lt;p&gt;Probably, what you are looking for is the &amp;quot;method of superbiparticular&amp;quot;, cited in Alberti's &lt;em&gt;De pictura&lt;/em&gt;, that uses a fixed ratio of &lt;span class=&quot;math-container&quot;&gt;$3:2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As has already been noted in the comments, during the Middle Ages and the early Renaissance (and actually even afterwards), various rules of thumb were used. The main sources relating to these rules (besides the paintings themselves that were made following them, but in general it can be very difficult to understand which rule was used by the author, considering that different parts of the painting, such as the characters, the floor and the colonnades, could follow different rules, and that the same author could change the rules from one painting to the next) are two: &lt;a href=&quot;https://en.wikipedia.org/wiki/Leon_Battista_Alberti&quot; rel=&quot;noreferrer&quot;&gt;Leon Battista Alberti&lt;/a&gt;'s &lt;a href=&quot;https://en.wikipedia.org/wiki/De_pictura&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;De pictura&lt;/em&gt;&lt;/a&gt; (written in 1435 in Tuscan with the title &lt;em&gt;Della pittura&lt;/em&gt;, the Latin translation bu the author dated 1440 circa) and &lt;em&gt;Le due regole della prospettiva prattica&lt;/em&gt; (sic), of &lt;a href=&quot;https://en.wikipedia.org/wiki/Giacomo_Barozzi_da_Vignola&quot; rel=&quot;noreferrer&quot;&gt;Jacopo Barozzi da Vignola&lt;/a&gt;, published posthumously in 1583 with extensive commentary by the mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Ignazio_Danti&quot; rel=&quot;noreferrer&quot;&gt;Ignazio Danti&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Before Alberti, or rather, before &lt;a href=&quot;https://en.wikipedia.org/wiki/Piero_della_Francesca&quot; rel=&quot;noreferrer&quot;&gt;Piero della Francesca&lt;/a&gt;, the rules of pictorial representation were based on &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Peckham&quot; rel=&quot;noreferrer&quot;&gt;Peckham&lt;/a&gt;'s &lt;em&gt;Perspectiva communis&lt;/em&gt;, which in turn is a synthesis of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ibn_al-Haytham&quot; rel=&quot;noreferrer&quot;&gt;Alhazen&lt;/a&gt;'s &lt;em&gt;De Aspectibus&lt;/em&gt; (translated in the Tuscan dialect at the end of the 14th century).&lt;/p&gt;&#xA;&lt;p&gt;Peckham speaks of bodies arranged in a &lt;em&gt;continuous&lt;/em&gt; and &lt;em&gt;orderly&lt;/em&gt; manner.&lt;/p&gt;&#xA;&lt;p&gt;The 'ordered proportion' is handed down from the translation of Euclid's Elements by &lt;a href=&quot;https://en.wikipedia.org/wiki/Gerard_of_Cremona&quot; rel=&quot;noreferrer&quot;&gt;Gherardo da Cremona&lt;/a&gt;, and consists of two series of quantities in which the terms of each series are in proportion according to an identical ratio (i.e., two series like &lt;span class=&quot;math-container&quot;&gt;$A,B,C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$D,E,F$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$A:D=B:E=C:F$&lt;/span&gt;). This proportion can be used to describe a colonnade: the apparent heights of the columns and the apparent distances between pairs of columns maintain the same ratio and are therefore in ordered proportion.&lt;/p&gt;&#xA;&lt;p&gt;The 'continuous proportion' is given in the translation by &lt;a href=&quot;https://en.wikipedia.org/wiki/Campanus_of_Novara&quot; rel=&quot;noreferrer&quot;&gt;Campano da Novara&lt;/a&gt;, and consists instead not in a proportion, but in a proportionality, i.e. a similarity of the form &lt;span class=&quot;math-container&quot;&gt;$A:B=B:C$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Continuous proportionality is appropriate to describe apparent quantities that decrease in perspective, e.g., the heights of figures or columns and the depth of floor tiles or ceiling coffers.&lt;/p&gt;&#xA;&lt;p&gt;Thus: ordered proportionality can describe the relationship between the relative heights and widths of the pictorial composition, e.g. the height of a man and the width of a floor; continuous proportionality describes the apparent decrease of magnitudes in the depth of the pictorial space, e.g. the decrease in depth of a floor tile.&lt;/p&gt;&#xA;&lt;p&gt;Now the problem is to determine the constant of proportionality in a continuous proportionality. A possibility, is to consider the &amp;quot;&lt;em&gt;metodo delle superbipartienti&lt;/em&gt;&amp;quot; (&amp;quot;method of superbiparticular&amp;quot;), described by Alberti.&lt;/p&gt;&#xA;&lt;p&gt;'Superbipartiente' is a term from medieval mathematics and music and indicates a proportional ratio of two thirds; more precisely, it indicates the decrease of one third of one of the two quantities (e.g., as in the sequence 9, 6, 4).&lt;/p&gt;&#xA;&lt;p&gt;Alberti harshly criticises this construction in chapter 19 of the first book of &lt;em&gt;De pictura&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] Hic essent nonnulli qui unam ab divisa aequedistantem lineam intra quadrangulum ducerent, spatiumque, quod inter utrasque lineas adsit, in tres partes dividerent. Tum huic secundae aequedistanti lineae aliam item aequedistantem hac lege adderent, ut spatium, quod inter primam divisam et secundam aequedistantem lineam est, in tres partes divisum una parte sui excedat spatium id quod sit et tertiam lineam, ac deinceps reliquas lineas adderent ut semper sequens inter lineas esset spatium ad antecedens, ut verbo mathematicorum loquar, superbipartiens. Itaque sic illi quidem facerent, quos etsi optimam quandam pingendi viam sequi affirment, eosdem tamen non parum errare censeo, quod cum casu primam aequedistantem lineam posuerint, tametsi caeterae aequedistantes lineae ratione et modo subsequantur, non tamen habent quo sit certus cuspidis ad bene spectandum locus. Ex quo non modici in pictura errores facile succedunt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] Then some would draw a line equidistant from the diagonal of the square, and divide the space between the two lines into three parts. Taking two parts, they would draw a line at this distance from the other and proceed according to this rule, so that the space that is between the first line and the second, divided into three parts, exceeds the space between the second and the third line by one part, and in so doing, to put it as mathematicians do, the spaces would be &lt;em&gt;superbipartienti&lt;/em&gt;. And so they would indeed do, who, although they claim to follow a certain better way of painting, nevertheless I think they are not a little mistaken, because they have placed the first line at random, and although the other lines follow according to their rule, they have no idea where the apex of the visual pyramid is, and so they make no small mistakes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, Alberti does not explicitly mention those &amp;quot;&lt;em&gt;nonnulli&lt;/em&gt;&amp;quot; who use this wrong technique.&lt;/p&gt;&#xA;&lt;p&gt;Alberti, in order to prove that the method of superbiparticular is wrong, refers to the diagonals. If the construction were right, the diametrically opposite points on the various trapezoids representing the floor tiles would have to be aligned. Indeed, aligned points project into aligned points and must therefore be drawn on the same line (chapter 20):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] Qui quidem quam recte descripti sint inditio erit, si una eademque recta continuata linea in picto pavimento coadiunctorum quadrangulorum diameter sit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] And the fact that the drawing is thus correct can be seen from this: that if you join the diameters [i.e., the diagonals] of the squares together as represented, you get a single straight line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For example, if you observe Ambrogio Lorenzetti's &lt;a href=&quot;https://en.wikipedia.org/wiki/Annunciation_(Ambrogio_Lorenzetti)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Annunciazione&lt;/em&gt;&lt;/a&gt;, dated 1344, which actually &lt;em&gt;does not&lt;/em&gt; use the method of superbiparticular, you will notice the gradual curvature of the diagonals, even though the lines coming out perpendicular to the eye converge correctly, when prolonged, at the central point (try prolonging the diagonals in &lt;a href=&quot;https://www.artesvelata.it/wp-content/uploads/2019/04/maxresdefault-1.jpg&quot; rel=&quot;noreferrer&quot;&gt;this blow-up&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Following &lt;a href=&quot;https://img-journal.unibo.it/article/view/12688/13186&quot; rel=&quot;noreferrer&quot;&gt;Romor's &lt;em&gt;Prospectiva Pingendi, Prospectiva Fingendi&lt;/em&gt;&lt;/a&gt;, in the &lt;em&gt;Annunciazione&lt;/em&gt;, Lorenzetti uses a different method and the row widths are in arithmetic progression (and not in geometric progression, as the method of superbiparticular would imply).&#xA;Nevertheless, it seems to me that in other paintings Lorenzetti uses a different technique.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/i3PVv.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/i3PVv.jpg&quot; alt=&quot;Presentazione al Tempio, Ambrogio Lorenzetti&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the &lt;a href=&quot;https://en.wikipedia.org/wiki/Presentation_at_the_Temple_(Ambrogio_Lorenzetti)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Presentazione al Tempio&lt;/em&gt;&lt;/a&gt; (1342) above, we observe the same problem concerning the diagonals, but now the row widths are in geometric progression (alas, I do not have the original painting available, so my calculations have no value, but it seems to me that the ratio of the row widths is about &lt;span class=&quot;math-container&quot;&gt;$7:5$&lt;/span&gt;, not far from the superbiparticular ratio).&lt;/p&gt;&#xA;&lt;p&gt;Coming back to the method of superbiparticular, the reason why Alberti considers it wrong is that the first line is placed at random (&amp;quot;&lt;em&gt;quod cum casu primam aequedistantem lineam posuerint&lt;/em&gt;&amp;quot;), while the fact that the row widths are in geometric progression is not a problem, on the contrary, it is acceptable to him (&amp;quot;&lt;em&gt;tametsi caeterae aequedistantes lineae ratione et modo subsequantur&lt;/em&gt;&amp;quot;), because it follows the continuous proportionality described by Campano da Novara. To overcome Alberti's error, one will have to wait for Piero's &lt;a href=&quot;https://en.wikipedia.org/wiki/De_prospectiva_pingendi&quot; rel=&quot;noreferrer&quot;&gt;De prospectiva pingendi&lt;/a&gt; where he demonstrates that the decrease in apparent magnitudes is determined neither by a proportion of those described in Book V of the Elements, nor by the distance separating the plane of representation from the eye, as Alberti claims, but is regulated by the proportion that these magnitudes have both with the distance between the eye and the plane of intersection, and with the distance between the plane of intersection and the thing seen.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Clearly, as the Lorenzetti's example shows, the method of superbiparticular was not the only &amp;quot;empirical&amp;quot; technique used by pre-Renaissance painter. For example, in &lt;em&gt;Le due regole della prospettiva prattica&lt;/em&gt; (1583), Danti cites this wrong rule, still used in his time: fix the center point and divide the base line in parts. Draw a quarter of a circumference,  and divide the arc into a number of equal parts (it is suggested to divide it into 15 equal parts), join these parts with an appropriate point and draw the parallels.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2022-08-25T16:16:06.983" LastActivityDate="2022-08-25T16:16:06.983" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14739" PostTypeId="2" ParentId="14733" CreationDate="2022-08-24T20:37:27.033" Score="7" Body="&lt;p&gt;Nesbitt's inequality was posed as problem 15114 in &lt;em&gt;The Educational Times&lt;/em&gt;. According to an &lt;a href=&quot;https://educational-times.wcu.edu/search?basic=Nesbitt&amp;amp;page=0&quot; rel=&quot;noreferrer&quot;&gt;online database&lt;/a&gt; of the publication maintained by Western Carolina University, it was published in May of 1902. A collection of selected problems and solutions from this publication can be found in Constance I. Marks (ed.), &lt;em&gt;Mathematical Questions and Solutions from &amp;quot;The Educational Times,&amp;quot; with many Papers and Solutions in addition to those published in &amp;quot;The Educational Times&amp;quot;. New Series. Volume III.&lt;/em&gt; London: F. Hodgson 1903. On pages &lt;a href=&quot;https://www.google.com/books/edition/Mathematical_Questions_and_Solutions_fro/WtIGAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=15114&quot; rel=&quot;noreferrer&quot;&gt;37&lt;/a&gt; and 38 one finds the problem (proof of the inequality) with three solutions, one of them by Nesbitt himself. On page 3 of the publication he is listed as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nesbitt, A.M., M.A.: Melbourne&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Doing a Google search on this information immediately led me to a &lt;a href=&quot;https://www.sydney.edu.au/paradisec/australharmony/register-N.php&quot; rel=&quot;noreferrer&quot;&gt;website&lt;/a&gt; &amp;quot;A biographical register of Australian colonial musical personnel–N&amp;quot; by Dr. Graeme Skinner (University of Sydney) that provides detailed bits of biographical background on Nesbitt (including a photo), confirming information already identified by &lt;a href=&quot;https://hsm.stackexchange.com/users/264/dave-l-renfro&quot;&gt;Dave L Renfro&lt;/a&gt; in comments.&lt;/p&gt;&#xA;&lt;p&gt;Alfred Mortimer Nesbitt was born on December 27, 1854 in England and died July 3, 1926 in &lt;a href=&quot;https://en.wikipedia.org/wiki/Malvern_East,_Victoria&quot; rel=&quot;noreferrer&quot;&gt;East Malvern&lt;/a&gt;, Victoria, Australia, which is a suburb of Melbourne. His parents were the physician Pearce R. Nesbitt, M.D. and his wife Arabella. Nesbitt studied mathematics at the University of Oxford, graduating in 1876. In 1877 he was appointed mathematical master of Manchester Grammar School. He married in 1880, as reported in &lt;em&gt;The Medical Times and Gazette&lt;/em&gt;, Vol. 2, London: J. &amp;amp; A. Churchill 1880, &lt;a href=&quot;https://books.google.com/books?id=3bJXAAAAMAAJ&amp;amp;pg=PA227#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;p. 227&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;NESBITT-HILL. — On August 5, at Hampstead, Alfred Mortimer Nesbitt, M.A., of Corpus Christi College, Oxford, youngest son of P. R. Nesbitt, M.D., to Margaret, only daughter of the late John Cartwright Hill, Esq.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1882 he was selected as head master of the &lt;a href=&quot;https://www.twgs.qld.edu.au/&quot; rel=&quot;noreferrer&quot;&gt;Grammar School in Toowoomba&lt;/a&gt;, Queensland, Australia. &lt;em&gt;The Manchester Grammar School Magazine&lt;/em&gt;, Vol. 10, No. 67, October 1882, &lt;a href=&quot;https://books.google.com/books?id=QPAHAAAAQAAJ&amp;amp;pg=PA38&quot; rel=&quot;noreferrer&quot;&gt;p. 38&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The cause of athletics in the School has suffered a severe loss by the departures of Mr. H. C. Barnes-Lawrence, M.A., to Giggleswick Grammar School, and of Mr. A. M. Nesbitt, M.A., to the Headmastership of Toowoomba School, Queensland. [...] Mr. Barnes-Lawrence came with our High Master in January, 1877, and Mr. Nesbitt in September of the same year.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1888 he became an assistant lecturer of mathematics and natural philosophy at Trinity College in Melbourne. The &lt;em&gt;Melbourne University Calendar 1889-90&lt;/em&gt;, University of Melbourne, &lt;a href=&quot;https://books.google.com/books?id=dDE4AAAAMAAJ&amp;amp;pg=PA362&quot; rel=&quot;noreferrer&quot;&gt;p. 362&lt;/a&gt; lists him as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;College Teaching Staff.&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;&lt;em&gt;Mathematics and Natural Philosophy.&lt;/em&gt;&lt;br /&gt;&#xA;[...]&#xA;(3) A. M. Nesbitt, M.A., late Scholar of Corpus Christi College, Oxford; First-class Honorman in Mathematics at Graduation, 1876.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In addition to being a mathematician, he appears to have been a competent musician and composer. He won first prize in the Jubilee Ode Competition of the Brisbane Musical Union, which presumably was set up for Queen Victoria's Golden Jubilee, which was celebrated in June 1887.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-08-25T07:09:44.583" LastActivityDate="2022-08-25T07:09:44.583" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14740" PostTypeId="1" AcceptedAnswerId="14746" CreationDate="2022-08-25T15:02:01.100" Score="3" ViewCount="129" Body="&lt;p&gt;We know that quantities can be defined at a point. Let's take density for instance. If we take a volume in some quantity of matter and keep on shrinking it to a point where we can assign a uniform density to it we can say that the density of that tiny volume is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\rho= \frac{\delta m}{\delta V}$$&lt;/span&gt;&#xA;Since this volume will be very small it will appear as a point macroscopically.&#xA;I'm curious to know what led to this formulation.&lt;/p&gt;&#xA;&lt;p&gt;My guess would be that when they gave a definition of density it was based on a uniform value. As in when they took a quantity of matter each unit volume had the same mass. However, they then might have come across varying values of density (as in each unit volume not having same mass) and then they needed to formulate something to specify the quantities when they are not uniform. By specifying quantities using &amp;quot;at a point&amp;quot; method we can express them as a function of position and time.&lt;/p&gt;&#xA;&lt;p&gt;This however is just my guess, does anyone know what was the truth?&lt;/p&gt;&#xA;" OwnerUserId="16326" LastActivityDate="2022-08-26T10:25:55.337" Title="What motivated scientists to define quantities at a point?" Tags="|classical-mechanics|thermodynamics|fluid-dynamics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14742" PostTypeId="1" AcceptedAnswerId="14744" CreationDate="2022-08-25T22:47:59.093" Score="4" ViewCount="880" Body="&lt;p&gt;As far as I know, according to google, Eilenberg, Steenrod's book: Foundations of Algebraic Topology was published in 1952, and Spanier's book: Algebraic Topology was published in 1966.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;My Questions are the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Is there any book on Algebraic Topology which was published between 1952-1966?&lt;/li&gt;&#xA;&lt;li&gt;Is there any book on Algebraic Topology which was published before 1952?&lt;/li&gt;&#xA;&lt;li&gt;Can someone provide me with a list of the first few published books on Algebraic Topology?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="13275" LastActivityDate="2022-08-27T03:28:28.613" Title="The First Published book on Algebraic Topology" Tags="|topology|" AnswerCount="3" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14743" PostTypeId="2" ParentId="14742" CreationDate="2022-08-26T00:34:24.647" Score="8" Body="&lt;p&gt;I searched MathSciNet for books with &amp;quot;Algebraic Topology&amp;quot; in the title...&lt;/p&gt;&#xA;&lt;p&gt;Lefschetz, Solomon &lt;em&gt;Algebraic Topology&lt;/em&gt;. American Mathematical Society Colloquium Publications, Vol. 27 American Mathematical Society, New York, 1942.&lt;/p&gt;&#xA;&lt;p&gt;Eilenberg, Samuel; Steenrod, Norman &lt;em&gt;Foundations of algebraic topology&lt;/em&gt;. Princeton University Press, Princeton, N.J., 1952.&lt;/p&gt;&#xA;&lt;p&gt;Rado, T.; Reichelderfer, P. V. &lt;em&gt;Continuous transformations in analysis. With an introduction to algebraic topology&lt;/em&gt;. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXV. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955.&lt;/p&gt;&#xA;&lt;p&gt;Wallace, Andrew H. &lt;em&gt;An introduction to algebraic topology&lt;/em&gt;. Pergamon Press, New York-London, Paris, 1957.&lt;/p&gt;&#xA;&lt;p&gt;Hilton, P. J.; Wylie, S. &lt;em&gt;Homology theory: An introduction to algebraic topology&lt;/em&gt;. Cambridge University Press, New York 1960&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Colloquium on Algebraic Topology&lt;/em&gt;, August 1–10, 1962. Lectures. Supported by the Scientific Affairs Division of the North Atlantic Treaty Organization Aarhus Universitet, Matematisk Institut, Aarhus 1962&lt;/p&gt;&#xA;&lt;p&gt;Translations, Ser. 1, Vol. 7: &lt;em&gt;Algebraic topology&lt;/em&gt;. American Mathematical Society, Providence, R.I. 1962&lt;/p&gt;&#xA;&lt;p&gt;Bourgin, D. G. &lt;em&gt;Modern algebraic topology&lt;/em&gt;. The Macmillan Company, New York; Collier Macmillan Ltd., London 1963&lt;/p&gt;&#xA;&lt;p&gt;Narasimhan, M. S.; Ramanan, S.; Sridharan, R.; Varadarajan, K. &lt;em&gt;Algebraic topology&lt;/em&gt;. Mathematical Pamphlets, 2 Tata Institute of Fundamental Research, Bombay 1964&lt;/p&gt;&#xA;&lt;p&gt;Spanier, Edwin H. &lt;em&gt;Algebraic topology&lt;/em&gt;. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966&lt;/p&gt;&#xA;&lt;p&gt;Hu, Sze-tsen &lt;em&gt;Homology theory: A first course in algebraic topology&lt;/em&gt;. Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam 1966&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-08-26T00:34:24.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14744" PostTypeId="2" ParentId="14742" CreationDate="2022-08-26T05:53:08.927" Score="13" Body="&lt;p&gt;&amp;quot;The first&amp;quot; is either Listing (1847), or Veblen (1922), or Lefschetz (1942), depending on what counts. Here's Dieudonné's book list from his &lt;a href=&quot;https://www.maa.org/press/maa-reviews/a-history-of-algebraic-and-differential-topology-1900-1960-0&quot; rel=&quot;noreferrer&quot;&gt;History of Algebraic and Differential Topology&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It took about 30 years to construct a theory of homology applicable to curvilinear &amp;quot;polyhedra,&amp;quot; embodying all the ideas of Poincaré and entirely&#xA;rigorous... The first treatise on this &amp;quot;classical&amp;quot; algebraic topology was &lt;a href=&quot;https://archive.org/details/dli.ernet.524782&quot; rel=&quot;noreferrer&quot;&gt;Veblen's Analysis&#xA;Situs, published in 1922&lt;/a&gt; (but a preliminary version was given as &amp;quot;Colloquium lectures&amp;quot; in 1916); it was followed by the much more complete book &lt;a href=&quot;https://archive.org/details/dli.ernet.425544&quot; rel=&quot;noreferrer&quot;&gt;Topology by Lefschetz (1930)&lt;/a&gt;, the very popular &lt;a href=&quot;https://archive.org/details/seifertthrelfall0000seif&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Lehrbuch der Topologie&lt;/em&gt; of H. Seifert and W. Threllfall (1934)&lt;/a&gt;, and the book by &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.358148&quot; rel=&quot;noreferrer&quot;&gt;P. Alexandroff and H. Hopf entitled &lt;em&gt;Topologie I&lt;/em&gt; (1935)&lt;/a&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://catalog.hathitrust.org/Record/000381466&quot; rel=&quot;noreferrer&quot;&gt;Lefschetz's &lt;em&gt;L'Analysis situs et la géométrie algébrique&lt;/em&gt; (1924)&lt;/a&gt; can be added to the list. But it should be said that &lt;a href=&quot;https://archive.org/details/vorstudienzurto00listgoog&quot; rel=&quot;noreferrer&quot;&gt;Listing's &lt;em&gt;Vorstudien zur Topologie&lt;/em&gt; (1847)&lt;/a&gt; already included what came to be a part of algebraic topology (&lt;a href=&quot;https://en.wikipedia.org/wiki/Listing_number&quot; rel=&quot;noreferrer&quot;&gt;Listing numbers&lt;/a&gt; of 2D manifolds). He also pioneered the name &amp;quot;topology&amp;quot; in a letter of 1836.&lt;/p&gt;&#xA;&lt;p&gt;Betti published his &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Betti/&quot; rel=&quot;noreferrer&quot;&gt;early work on homology&lt;/a&gt; in a paper rather than a book, &lt;em&gt;Sopra gli spazi di un numero qualunque di dimensioni&lt;/em&gt; (1871), as did Poincare, the official &amp;quot;founder&amp;quot; of the subject, using Leibniz's name &lt;em&gt;Analysis Situs&lt;/em&gt; (1892ff). He also coined the term &amp;quot;Betti numbers&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The term &amp;quot;algebraic topology&amp;quot; appears in Steenrod's paper Universal Homology Groups (1936),&#xA;Lefschetz then used it as the title of his &lt;a href=&quot;https://archive.org/details/algebraictopolog00lefs&quot; rel=&quot;noreferrer&quot;&gt;1942 book&lt;/a&gt;, a reworked version of his 1930 Topology, see &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/a/&quot; rel=&quot;noreferrer&quot;&gt;Earliest Known Uses of Some of the Words of Mathematics&lt;/a&gt;. Despite the earlier occurrences, the current spread of &amp;quot;topology&amp;quot; and &amp;quot;algebraic topology&amp;quot; is probably due to his influence, see &lt;a href=&quot;https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-79/issue-4/Solomon-Lefschetz-An-appreciation-in-memoriam/bams/1183534736.full&quot; rel=&quot;noreferrer&quot;&gt;Markus's biography&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-08-26T05:53:08.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14745" PostTypeId="1" CreationDate="2022-08-26T08:39:10.017" Score="5" ViewCount="134" Body="&lt;p&gt;David Wells, in his entertaining but non-scholarly &lt;em&gt;Curious and Interesting Numbers&lt;/em&gt; (1986, 2 ed. 1997) reports that the positive integer &lt;span class=&quot;math-container&quot;&gt;$1 \, 375 \, 298 \, 099$&lt;/span&gt; can be expressed as the sum of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; fifth powers in &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; different ways:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$1 \, 375 \, 298 \, 099 = 24^5 + 28^5 + 67^5 = 3^5 + 54^5 + 62^5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;He gives the source as &amp;quot;R. Alter&amp;quot; but I can find no reference to this person.&lt;/p&gt;&#xA;&lt;p&gt;I have seen a suggestion that his first name may be Roland, but I can't find any justification for this.&lt;/p&gt;&#xA;&lt;p&gt;(MathWorld has a page on this: &lt;a href=&quot;https://mathworld.wolfram.com/DiophantineEquation5thPowers.html&quot; rel=&quot;noreferrer&quot;&gt;https://mathworld.wolfram.com/DiophantineEquation5thPowers.html&lt;/a&gt; in which it is stated that this is the smallest number with this property.)&lt;/p&gt;&#xA;&lt;p&gt;Can anyone help me out with any information about R. Alter?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2022-08-27T00:29:09.443" Title="Who is/was R. Alter, who reported 1375298099 can be expressed as the sum of 3 fifth powers in 2 different ways?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14746" PostTypeId="2" ParentId="14740" CreationDate="2022-08-26T10:25:55.337" Score="3" Body="&lt;p&gt;The OP speculation is basically correct. Average density was certainly defined first, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Density#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eureka story&lt;/a&gt; told about Archimedes by Vitruvius involves distinguishing gold from non-gold by comparing average densities. The same goes for speeds, Eudoxus defined them as distance over time even earlier (c. 375 BC), see &lt;a href=&quot;https://www.jstor.org/stable/41133535&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riddell, Eudoxan Mathematics and the Eudoxan Spheres, p.7&lt;/a&gt;. Ancient astronomy was generally preoccupied with reducing any observed motion to uniform motions.&#xA;However, aside from Platonist bias for uniformity, it was a forced move due to lack of more general technical apparatus. Reducing already presupposed that observed speeds change, it is just that accounting for that directly was not an option.&lt;/p&gt;&#xA;&lt;p&gt;Oresme made one of the first steps towards creating such apparatus in &lt;em&gt;Tractatus de configuationibus qualitatum et motuum&lt;/em&gt; (1353) by considering uniformly accelerated (&amp;quot;uniformly difform&amp;quot;) motion graphically, where one could identify instantaneous speeds with slopes. He talked more generally about &amp;quot;qualities&amp;quot; varying (&amp;quot;difform&amp;quot;) in time and space, including temperature as an example. In fact, we find the idea of relating varying qualities to &amp;quot;imaginary&amp;quot; points/lines/surfaces clearly expressed in the &lt;em&gt;Tractatus&lt;/em&gt;, see &lt;a href=&quot;https://www.dam.brown.edu/people/mumford/beyond/coursenotes/2006PartIb.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mumford's notes&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Every measurable thing except numbers is&#xA;imagined in the manner of continuous quantity.&#xA;Therefore, for the mensuration of such a thing, it&#xA;is necessary that points, lines and surfaces, or&#xA;their properties be imagined. For in them, as the&#xA;Philosopher has it, measure or ratio is initially&#xA;found, while in other things it is recognized by&#xA;similarity as they are being referred to by the&#xA;intellect to the geometrical entities. Although&#xA;indivisible points, or lines, are non-existent, still it is necessary to feign them mathematically for the measures of things and for the understanding of their ratios.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...It is apparent that we ought to imagine a quality in this way in order to recognize its disposition more easily, for its uniformity and its difformity are examined more quickly, more easily and more clearly when something similar to it is described in a sensible figure. …Thus it seems quite difficult for some people to understand the nature of a quality which is uniformly difform. But what is easier to understand is that the altitude of a right triangle is uniformly difform.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These ideas found extensive continuation in the 17th century when, on the one hand, many qualities were not just seen but measured as varying, density, atmospheric pressure, optical density, etc, and on the other, calculus provided the means of modeling them. &amp;quot;Varying&amp;quot; could only mean varying from point to point, so the need to relate something to instants and points became even more pressing.&lt;/p&gt;&#xA;&lt;p&gt;Cavalieri's indivisibles and Fermat's infinitesimals, provided just the something one could relate to &lt;em&gt;every&lt;/em&gt; point, rather than to a discrete selection of them where measurements could, in practice, be made. Indeed, subdividing into infinitesimal parts and taking continuous limits was a common technique at the time. Johann Bernoulli's optical solution to the brachistochrone is a famous example, it splits optically non-uniform medium into infinitesimal layers, see &lt;a href=&quot;https://www.math.rug.nl/%7Ebroer/pdf/ws-ijbc.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Broer, Bernoulli’s light ray solution&lt;/a&gt;. The limiting answers were much more concise and tractable than their discrete analogs.&lt;/p&gt;&#xA;&lt;p&gt;This happy meeting of the modeling need with the technical opportunity sealed the deal for instantaneous and point quantities over averaged approximations. Their non-geometric interpretation as limits of finite ratios came a bit later. It was favored by Newton early (by 1680s), see &lt;a href=&quot;https://hsm.stackexchange.com/a/9590/55&quot;&gt;What was the notion of limit that Newton used?&lt;/a&gt;, and eventually supplanted indivisibles/infinitesimals due to worries about their coherence and justification.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-08-26T10:25:55.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14747" PostTypeId="2" ParentId="14745" CreationDate="2022-08-26T21:28:38.230" Score="10" Body="&lt;p&gt;The particular dual sum-of-powers representations of &lt;span class=&quot;math-container&quot;&gt;$1375298099$&lt;/span&gt; from the question appear in Table 4 of&lt;/p&gt;&#xA;&lt;p&gt;Ronald Alter, &amp;quot;Computations and generalizations on a remark of Ramanujan.&amp;quot; In: Marvin I. Knopp (ed.), &lt;em&gt;Analytic Number Theory&lt;/em&gt;: Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980 (Lecture Notes in Mathematics, 899; Springer 1981).&lt;/p&gt;&#xA;&lt;p&gt;This table is on &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=9YR8CwAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PA190#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;page 190&lt;/a&gt;. The conference was organized on the occasion of the retirement of Alter's Ph.D. thesis advisor Emil Grosswald from Temple University.&lt;/p&gt;&#xA;&lt;p&gt;The earliest mention of Ronald Alter in the literature I could find is in a list of new members of the American Mathematical Society in the following publication:&lt;/p&gt;&#xA;&lt;p&gt;John W. Green and Morton L. Curtis, &amp;quot;The Annual meeting in Miami&amp;quot;, &lt;em&gt;Bulletin of the American Mathematical Society&lt;/em&gt;, Vol. 70, 1964, pp. 343-356 (&lt;a href=&quot;https://www.ams.org/journals/bull/1964-70-03/S0002-9904-1964-11090-9/S0002-9904-1964-11090-9.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;). On page 351:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;University of Pennsylvania: Ronald Alter, Robert H. Cantor, [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Per his record at the &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=26959&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt;, Ronald Alter received his Ph.D. in 1965 from the University of Pennsylvania. The title of his Ph.D. thesis is: &lt;em&gt;On a Necessary Condition for the Validity of the Riemann Hypothesis for Functions that Generalize the Riemann Zeta Function&lt;/em&gt;. A publication based on the Ph.D. thesis appeared in 1968; Alter's affiliation is stated as University of California, Los Angeles.&lt;/p&gt;&#xA;&lt;p&gt;Ronald Alter, &amp;quot;On a necessary condition for the validity of the Riemann hypothesis for functions that generalize the Riemann zeta function.&amp;quot; &lt;em&gt;Transactions of the American Mathematical Society&lt;/em&gt;, Vol. 130, No. 1, 1968, pp. 55-74 (&lt;a href=&quot;https://www.ams.org/tran/1968-130-01/S0002-9947-1968-0218312-X/S0002-9947-1968-0218312-X.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;After receiving his Ph.D., Alter briefly joined the faculty of UCLA. &lt;em&gt;General Catalog University of California, Los Angeles&lt;/em&gt;, Regents of the University of California 1966, &lt;a href=&quot;https://www.google.com/books/edition/General_Catalog/XESY46hpogIC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22Ronald%20Alter%22%20&quot; rel=&quot;noreferrer&quot;&gt;p. 319&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ronald Alter, Ph.D., &lt;em&gt;Acting Assistant Professor of Mathematics.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Two papers published in 1968 state Alter's affiliation as &lt;a href=&quot;https://en.wikipedia.org/wiki/System_Development_Corporation&quot; rel=&quot;noreferrer&quot;&gt;System Development Corporation&lt;/a&gt;, Santa Monica, California. SDC was one of the first computer software companies, founded in 1955. The first of these two papers states that Alter is a postdoctoral fellow.&lt;/p&gt;&#xA;&lt;p&gt;Ronald Alter, &amp;quot;On the Nonexistence of Close-Packed Double Hamming-Error-Correcting Codes on q = 7 Symbols&amp;quot;, &lt;em&gt;Journal of Computer and System Sciences&lt;/em&gt;, Vol. 2, No. 2, Aug. 1968, pp. 169-176&lt;/p&gt;&#xA;&lt;p&gt;Ronald Alter, &amp;quot;On the Non-Existence of Perfect Double Hamming-Error-Correcting Codes on q = 8 and q = 9 Symbols&amp;quot;, &lt;em&gt;Information and Control&lt;/em&gt;, Vol. 13, No. 6, Dec. 1968, pp. 619-627.&lt;/p&gt;&#xA;&lt;p&gt;Ronald Alter joined the faculty of the University of Kentucky in 1970. In &lt;em&gt;Notices of the American Mathematical Society&lt;/em&gt;, Volume 17, Number 5, August 1970, &lt;a href=&quot;https://www.google.com/books/edition/Notices_of_the_American_Mathematical_Soc/pDE5AAAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22Professor+RONALD+ALTER+of+System+Development+Corporation+has+been+appointed+to+an+associate+professorship+at+the+University+of+Kentucky.+%22&amp;amp;dq=%22Professor+RONALD+ALTER+of+System+Development+Corporation+has+been+appointed+to+an+associate+professorship+at+the+University+of+Kentucky.+%22&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 735&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Professor RONALD ALTER of System Development Corporation has been appointed to an associate professorship at the University of Kentucky.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 77, 1970, &lt;a href=&quot;https://www.google.com/books/edition/The_American_Mathematical_Monthly/O6A0AAAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22Assistant+Professor+Ronald+Alter,+University+of+Kentucky,+has+been+promoted+to+Associate+Professor%22&amp;amp;dq=%22Assistant+Professor+Ronald+Alter,+University+of+Kentucky,+has+been+promoted+to+Associate+Professor%22&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 1035&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Assistant Professor Ronald Alter, University of Kentucky, has been promoted to Associate Professor&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I cannot find any publication by Alter after 1980, or in fact any other information on him after that year. All publications from 1970 through 1980 state his affiliation as University of Kentucky, and later ones more specifically mention the Department of Computer Science.&lt;/p&gt;&#xA;&lt;p&gt;In online information I found people called Ronald Alter, e.g. multiple ones born between 1934 and 1939 in the 1940 United States Census, or one in a 1958 highschool yearbook, but it is impossible to determine at this time whether any of them are identical to the mathematician Ronald Alter. I have not been able to find a relevant obituary.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Other publications by Ronald Alter:&lt;/p&gt;&#xA;&lt;p&gt;Ronald Alter, &amp;quot;Perfect Double Hamming-Error-Correcting Codes on Q-Symbols&amp;quot;, System Development Corporation, Mar. 1969 (DTIC AD0685436)&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;Perfect Double Hamming-Error-Correcting Codes on Q-Symbols&amp;quot;, &lt;em&gt;Proceedings of the Third Annual Princeton Conference on Information Sciences and Systems&lt;/em&gt;, March 27-28, 1969, pp. 547-550.&lt;br /&gt;&#xA;Ronald Alter and B. P. Lientz, &amp;quot;A Generalization of a Combinatorial Problem of Smirnov&amp;quot;, System Development Corporation, SP-3254, Nov. 1968 (also: DTIC AD0680108)&lt;br /&gt;&#xA;Ronald Alter and B. P. Lientz. &amp;quot;Applications of a generalized combinatorial problem of Smirnov.&amp;quot; &lt;em&gt;Naval Research Logistics Quarterly&lt;/em&gt;, Vol. 16, No. 4, Dec. 1969, pp. 543-547&lt;br /&gt;&#xA;Ronald Alter and Mark Villarino, &amp;quot;A remark on primes in arithmetic progressions&amp;quot;, &lt;em&gt;Bulletin of the Australian Mathematical Society&lt;/em&gt;, Vol. 3, No. 2, Oct. 1970, pp. 185-192 (&lt;a href=&quot;https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A3BD7081E4386D97FD06B84DF5B42AEA/S0004972700045858a.pdf/a-remark-on-primes-in-arithmetic-progressions.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Bennet Lientz, &amp;quot;A note on a problem of Smirnov - A graph theoretic interpretation&amp;quot;, &lt;em&gt;Naval Research Logistics Quarterly&lt;/em&gt;, Vol. 17, No. 3, Sep. 1970, pp. 407-408 (&lt;a href=&quot;https://apps.dtic.mil/sti/pdfs/AD0772431.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Thaddeus B. Curtz, &amp;quot;On binary nonassociative products and their relation to a classical problem of Euler&amp;quot;, University of Kentucky, Computer Science Dept., 1970 (?)&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;On a Diophantine equation related to perfect codes&amp;quot;, &lt;em&gt;Mathematics of Computation&lt;/em&gt;, Vol. 25, No. 115, July 1971, pp. 621-624 (&lt;a href=&quot;https://www.ams.org/journals/mcom/1971-25-115/S0025-5718-71-99713-4/S0025-5718-71-99713-4.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Thaddeus B. Curtz, &amp;quot;An Experimental Computer Science Problem Seminar&amp;quot;, &lt;em&gt;ACM SIGCSE Bulletin&lt;/em&gt;, Vol. 4, No.1, Mar. 1972, pp. 144-153 (&lt;a href=&quot;https://dl.acm.org/doi/pdf/10.1145/873684.873717&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;Five Full Houses!&amp;quot;, problem 820, &lt;em&gt;Mathematics Magazine&lt;/em&gt;, Vol. 45, No. 5, Nov. 1972, p. 289 (&lt;a href=&quot;http://math.fau.edu/Yiu/PSRM2015/yiu/LAPTOPbackup030317/MathematicsMagazine/MGProblems45(72)5.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Bennet P. Lientz. &amp;quot;Remarks on a combinatorial problem in orderings.&amp;quot; &lt;em&gt;Sankhyā: The Indian Journal of Statistics, Series A&lt;/em&gt;, Vol. 34, No. 4, Dec. 1972, pp. 457-460.&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;Can &lt;span class=&quot;math-container&quot;&gt;$\phi(n)$&lt;/span&gt; properly divide &lt;span class=&quot;math-container&quot;&gt;$n-1$&lt;/span&gt;&amp;quot;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 80, No. 2, Feb. 1973, pp. 192-193&lt;br /&gt;&#xA;Ronald Alter and Thaddeus B. Curtz, &amp;quot;On binary non-associative products and their relation to a classical problem of Euler&amp;quot;, &lt;em&gt;Commentationes Mathematicae&lt;/em&gt;, Vol 17, No. 1, 1973, pp. 1-8 (&lt;a href=&quot;https://bibliotekanauki.pl/articles/746198.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and K. K. Kubota, &amp;quot;Multiplicities of second order linear recurrences&amp;quot;, &lt;em&gt;Transactions of the American Mathematical Society&lt;/em&gt;, Vol. 178, Apr. 1973, pp. 271-284 (&lt;a href=&quot;https://www.ams.org/journals/tran/1973-178-00/S0002-9947-1973-0441841-2/S0002-9947-1973-0441841-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and K. K. Kubota, &amp;quot;The diophantine equation &lt;span class=&quot;math-container&quot;&gt;$x^{2}+D=p^{n}$&lt;/span&gt;,&amp;quot; &lt;em&gt;Pacific Journal of Mathematics&lt;/em&gt;, Vol. 46, No. 1, November 1973, pp. 11-16 (&lt;a href=&quot;https://msp.org/pjm/1973/46-1/pjm-v46-n1-p02-p.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and K. K. Kubota, &amp;quot;Prime and prime power divisibility of Catalan numbers&amp;quot;, &lt;em&gt;Journal of Combinatorial Theory, Series A&lt;/em&gt;, Vol. 15, No. 3, Nov. 1973, pp. 243-256 (&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0097316573900721/pdf?md5=2bf63b9558fa31ffe4fad21e702530d0&amp;amp;pid=1-s2.0-0097316573900721-main.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Thaddeus B. Curtz, &amp;quot;A note on congruent numbers&amp;quot;, &lt;em&gt;Mathematics of Computation&lt;/em&gt;, Vol. 28, No. 125, Jan. 1974, pp. 303-305 (&lt;a href=&quot;https://www.ams.org/journals/mcom/1974-28-125/S0025-5718-1974-0337758-9/S0025-5718-1974-0337758-9.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and K. K. Kubota, &amp;quot;The Diophantine equation &lt;span class=&quot;math-container&quot;&gt;$x^{2}+11=3^{n}$&lt;/span&gt; and a related sequence&amp;quot;, &lt;em&gt;Journal of Number Theory&lt;/em&gt;, Vol. 7, No. 1, Feb. 1975, pp. 5-10 (&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0022314X75900037/pdf?md5=ae418234cc975d6d2dcdc74c6c9e7c07&amp;amp;pid=1-s2.0-0022314X75900037-main.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;How many Latin squares are there?&amp;quot;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 82, No. 6, Jun. 1975, pp. 632-634&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;A Characterization of Self-Complementary Graphs of Order 8&amp;quot;, &lt;em&gt;Portugaliae Mathematica&lt;/em&gt;, Vol. 34, Fasc. 3, 1975, pp. 157-161 (&lt;a href=&quot;http://purl.pt/2769/1/j-5293-b-vol34-fasc3-art3_PDF/j-5293-b-vol34-fasc3-art3_PDF_01-B-R0300/j-5293-b-vol34-fasc3-art3_0000_capa1-161_t01-B-R0300.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;&amp;quot;Correction to:“A note on congruent numbers”(Math. Comp. 28 (1974), 303–305)&amp;quot;, &lt;em&gt;Mathematics of Computation&lt;/em&gt;, Vol. 30, No. 133, 1976, p. 198 (&lt;a href=&quot;https://www.ams.org/mcom/1976-30-133/S0025-5718-1976-0392816-X/S0025-5718-1976-0392816-X.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Chung C. Wang, &amp;quot;Uniquely Intersectable Graphs&amp;quot;, &lt;em&gt;Discrete Mathematics&lt;/em&gt;, Vol. 18, No. 3, 1977, pp. 217-226 (&lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/0012365X7790125X/pdf?md5=e73e767b8821c3d01c21d4f704e6fe16&amp;amp;pid=1-s2.0-0012365X7790125X-main.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Jeffrey A. Barnett, &amp;quot;Remarks on the postage stamp problem with applications to computers&amp;quot;, University of Kentucky, Department of Computer Science, 1977 (&lt;a href=&quot;https://notatt.com/lsu-stamp.pdf&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;br /&gt;&#xA;Ronald Alter and Johnson M. Hart, &amp;quot;Enumerating syntactical graphs and lattices of derivations&amp;quot;, &lt;em&gt;International Journal of Computer &amp;amp; Information Sciences&lt;/em&gt;, Vol. 8, No. 4, Aug. 1979, pp. 261-277&lt;br /&gt;&#xA;Ronald Alter, &amp;quot;The congruent number problem&amp;quot;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 87, No. 1, Jan. 1980, pp. 43-45&lt;br /&gt;&#xA;Ronald Alter and Jeffrey A. Barnett, &amp;quot;A Postage Stamp Problem&amp;quot;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 87, No. 3, Mar. 1980, pp. 206-210&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-08-27T00:29:09.443" LastActivityDate="2022-08-27T00:29:09.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14748" PostTypeId="1" CreationDate="2022-08-26T21:41:36.783" Score="7" ViewCount="496" Body="&lt;p&gt;John Stillwell, in his textbook on arithmetic cites Erdos:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As the great&#xA;Hungarian problem-solver Paul Erdos liked to point out, if you can&#xA;think of an open problem that is more than 200 years old, then it is&#xA;probably a problem in number theory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What are some examples of &amp;gt;200 years old unsolved problems NOT from number theory?&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="4251" LastEditDate="2022-09-08T10:06:34.920" LastActivityDate="2023-05-28T15:13:39.703" Title="Old unsolved questions in mathematics" Tags="|mathematics|open-problems|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14749" PostTypeId="2" ParentId="14742" CreationDate="2022-08-27T03:28:28.613" Score="1" Body="&lt;p&gt;Some other books I feel worth adding are Hilton’s An Introduction to Homotopy Theory and Steenrod’s Topology of Fiber Bundles. Also there are several books on combinatorial topology, like the ones by Alexandroff and Pontryagin, that come after the earlier topology books but still before more “modern” algebraic topology books&lt;/p&gt;&#xA;" OwnerUserId="17173" LastActivityDate="2022-08-27T03:28:28.613" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14750" PostTypeId="2" ParentId="14748" CreationDate="2022-08-27T11:51:46.603" Score="12" Body="&lt;p&gt;(This probably &lt;em&gt;does not&lt;/em&gt; answer your question, so it can be moved to the comments.)&lt;/p&gt;&#xA;&lt;p&gt;The fourth book of Albrecht Dürer's &lt;a href=&quot;https://de.wikisource.org/wiki/Underweysung_der_Messung,_mit_dem_Zirckel_und_Richtscheyt,_in_Linien,_Ebenen_unnd_gantzen_corporen/Viertes_Buch&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Vnderweysung der messung&lt;/em&gt;&lt;/a&gt; (1525),   is devoted (among other things) to polyhedra, and in particular he explores the possibility to give a planar &amp;quot;unfolding&amp;quot; of a polyhedron (&lt;a href=&quot;https://digital.slub-dresden.de/werkansicht/dlf/17139/155&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; the truncated cuboctahedron). Dürer's list is not exhaustive, and so one can argue that he (implicitely) posed the problem to give a planar unfolding (with no overlaps) for a general polyhedron. This lead to Dürer's problem, also known as &lt;a href=&quot;https://mathworld.wolfram.com/ShephardsConjecture.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Shephard's conjecture&lt;/a&gt; (1975), which is an open problem.&lt;/p&gt;&#xA;&lt;p&gt;Now, our problem is the following: does Dürer's problem date back to 1525, or is it simply a modern reinterpretation inspired by his work?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT&lt;/p&gt;&#xA;&lt;p&gt;Some comments and a partial answer:&lt;/p&gt;&#xA;&lt;p&gt;I believe that there are fundamentally two difficulties in finding old open problems in mathematics, and in particular not related to number theory.&lt;/p&gt;&#xA;&lt;p&gt;The first is that many problems, initially of a geometric or combinatorial nature, can be reformulated in numerical terms and in fact become number theory problems: I am thinking, for example, of the problem of finding a &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_brick&quot; rel=&quot;nofollow noreferrer&quot;&gt;perfect cuboid&lt;/a&gt;, or the three great problems of antiquity, including the problem of finding the regular polygons that can be constructed with a ruler and compass (which is equivalent to determining all of Fermat's primes).&lt;/p&gt;&#xA;&lt;p&gt;The second is that, even in the case of number theory problems, it is not easy to establish when and by whom a certain problem was actually posed. For instance, Nicomachus of Gerasa is often attributed with certain (still open) problems concerning perfect numbers. In reality it is doubtful that his contemporaries thought those were problems: rather, they were considered as facts. Only in the 16th century Hudalrichus Regius showed that not all numbers of the form &lt;span class=&quot;math-container&quot;&gt;$2^n-1$&lt;/span&gt; for all primes &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; are themselves prime, and so being able to give a counterexample to one of Nicomachus' statement.&lt;/p&gt;&#xA;&lt;p&gt;This situation is similar to the one cited with regard to Dürer: although he had all the tools he needed to formulate a conjecture (and thus to pose a problem), he probably simply thought it was natural that it was always possible to unfold a polyhedron, and indeed in the fourth book cited he provides a 'practical' algorithm (here I'm citing the second Latin edition of Dürer's treatise by &lt;a href=&quot;https://en.wikipedia.org/wiki/Joachim_Camerarius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Camerarius&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si quis igitur ea componere velit, is accipiat duo folia papyri, bitumine, aut pasta cohaerentiaa, et corpora illa super eadem ita describat, ut acuto cultello alterum foliorum secundum lineas ductas scindatur, et cum omnia reliqua ex residuo papyri fuerint soluta, tunc facile complicabitur in ductibus et scissuris. Ideoque advertendum est ad sequentem ductinem, nam talia ad plurima conducunt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So if someone wants to make them up, take two sheets of paper, some bitumen or a cohesive paste, and describe those bodies on them in such a way that with a sharp knife one of the sheets is torn according to the lines drawn, and when all the rest of the paper has been unbound, then it will be easily folded according to the lines and cuts. For this, it is necessary to pay attention to the following directions, because these lead to different solutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The same can also be said of Fagnano's problem. In fact, the problem formulated and solved by Fagnano is actually the following: &amp;quot;In a given acutangle triangle, inscribe a triangle whose perimeter is the smallest possible.&amp;quot; For example, in &lt;em&gt;Two Applications of Calculus to Triangular Billiards&lt;/em&gt; by Eugene Gutkin, (The American Mathematical Monthly, Vol. 104, No. 7, 1997), we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We assume that the triangle T is acute, and view it as a billiard table. The pedal triangle T_1, which is inscribed in T, is then a periodic billiard orbit [...] Moreover [...] is the shortest [...], the only closed (prime) billiard orbit known [...] The first proof, by calculus, that among all inscribed traingles the pedal triangle has the least perimeter, is attributed to J.F.F. Fagnano, ca. 1775. In his honor, the problem just stated is often called the Fagnano problem [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is therefore clear that in reality Fagnano did not pose the problem, not only in its full generality (i.e., in the case of any polygon), but not even in the case of a non-acutangle triangle (which is still an open problem, one of the &lt;a href=&quot;https://web.archive.org/web/20230528151547/https://www.impan.pl/%7Ebiuletyn/arch/5problems.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;5 most resistant problems in dynamics&lt;/a&gt;, according to &lt;a href=&quot;https://en.wikipedia.org/wiki/Anatole_Katok&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anatole Katok&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Returning to the original question, one answer (cheating a bit) could be the &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler_conjecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kepler conjecture&lt;/a&gt;: today it is no longer considered an open problem, but when (2003) the first edition of Stillwell's book was published, I think it still was (in 1998, Thomas Hales announced that he had a proof, but only in 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof, accepted by the journal &lt;em&gt;Forum of Mathematics, Pi&lt;/em&gt; in 2017).&lt;/p&gt;&#xA;&lt;p&gt;In the same spirit (an open problem at the time of the first edition of Stillwell's book), the &lt;a href=&quot;https://en.wikipedia.org/wiki/Alhazen%27s_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy's problem&lt;/a&gt; (or Alhazen's problem, and somehow related to the Fagnano's problem), i.e. &amp;quot;find the point on a spherical convex mirror at which a ray of light coming from a given point must strike in order to be reflected to another point,&amp;quot; does not fit in for only 6 years: Alhazen gave the first geometric solution in his &lt;em&gt;Book of Optics&lt;/em&gt;, and algebraic solutions were found in the 20th century, but only in 1997 was proved that there is no ruler-and-compass construction to solve it.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2023-05-28T15:13:39.703" LastActivityDate="2023-05-28T15:13:39.703" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14751" PostTypeId="1" AcceptedAnswerId="14752" CreationDate="2022-08-28T00:33:26.877" Score="14" ViewCount="3654" Body="&lt;p&gt;For example, the whole microscopic world was unknown - isn't that a fundamental problem even bigger than the &amp;quot;two clouds&amp;quot; to solve? They could regard atoms, electrons and other discovered particles as the basic building blocks of the universe, but what about, e.g., the exact theory to describe how and why atoms form molecules?&lt;/p&gt;&#xA;&lt;p&gt;IMO the idea was not reasonable at all, so I am guessing only a few of them had such unreasonable ideas. Is that true?&lt;/p&gt;&#xA;" OwnerUserId="13727" LastEditorUserId="55" LastEditDate="2022-08-28T05:02:15.003" LastActivityDate="2022-08-28T05:28:39.870" Title="Did most or just few physicists think in 1900 that there was nothing important left to discover?" Tags="|physics|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14752" PostTypeId="2" ParentId="14751" CreationDate="2022-08-28T04:04:55.210" Score="19" Body="&lt;p&gt;Even Kelvin, who is commonly &amp;quot;credited&amp;quot; with this idea, did not think so, see &lt;a href=&quot;https://arxiv.org/abs/2106.16033&quot; rel=&quot;noreferrer&quot;&gt;Passon, Kelvin's clouds&lt;/a&gt; and &lt;a href=&quot;https://www.bbvaopenmind.com/en/science/physics/lord-kelvin-and-the-end-of-physics-which-he-never-predicted/&quot; rel=&quot;noreferrer&quot;&gt;Yanes, Lord Kelvin and the End of Physics, Which He Never Predicted&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Kelvin's 1900 lecture at the Royal Institution in London, made famous by the &amp;quot;two clouds&amp;quot; was published under the title Nineteenth Century Clouds over the Dynamical Theory of Heat and Light, and can be read &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/14786440109462664&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;. It was restricted to those two topics specifically, and &lt;em&gt;did not&lt;/em&gt; say that there were no more major problems left in physics, nor did it call the clouds &amp;quot;small&amp;quot;, as often confabulated (e.g. by Bohm). Kelvin said nothing of the sort elsewhere either.&lt;/p&gt;&#xA;&lt;p&gt;The lasting popularity of this fable mirrors that of its cousin, the &amp;quot;ultraviolet catastrophe&amp;quot;, see &lt;a href=&quot;https://hsm.stackexchange.com/a/5117/55&quot;&gt;Where did Rayleigh derive the ultraviolet catastrophe?&lt;/a&gt; (he didn't). By the way, Kelvin's cloud two is, also commonly, misidentified as the latter, but neither he nor anybody else spoke of it until 1908. It was instead specific heat at low temperatures. Passon explains the reasons:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;However, the claim that “there is nothing new to be discovered in physics” is&#xA;nowhere to be found in the 1900 lecture, or in any of Kelvin’s other writings, and&#xA;Zubairy suspiciously called this quotation just “attributed” to Lord Kelvin... All this shows, that one is dealing with a narrative that has been carefully crafted in order to emphasize the rift between “classical” and “modern” physics.&#xA;Against the backdrop of something considered essentially complete (with just two small blemishes), the conceptual changes of modern physics appear even more sensational and dramatic.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;This somewhat rhetorical function is further strengthened by calling Lord Kelvin a “leading physicist of the time” (Bohm, Clegg)... However, by the time of the lecture&#xA;in 1900 (at the age of 75), his lasting contributions to thermodynamics and electromagnetism had been made decades ago. By that time, Kelvin was widely regarded in the community as the last prominent adherent of a mechanical&#xA;world view... When Kelvin published&#xA;the revised version of his Baltimore Lectures in 1904 (including the “cloud lecture” in Appendix B), the reaction seemed (to quote Knudson again) “polite indifference, and it is hard to point to a physicist whose work was influenced by&#xA;the book”&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The closest thing to the &amp;quot;end of physics&amp;quot; was said not by Kelvin but by another holdover mechanist, Michelson, at the inauguration of the Ryerson Physics Laboratory at the University of Chicago in 1894:&amp;quot;&lt;em&gt;The more important fundamental laws and facts of physical science have all been discovered...&lt;/em&gt;&amp;quot; He then repeated it in his book Light Waves and their Uses (1903). But even Michelson only talked about &lt;em&gt;fundamental&lt;/em&gt; physics, to which technicalities of molecular chemistry did not belong. Still, he expected them to be forthcoming in &amp;quot;the day not far distant&amp;quot;, and promoted ether as the emerging &amp;quot;theory of everything&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The day seems not far distant when the converging lines from many apparently remote regions of thought will meet... Then the nature of the atoms, and the forces called into play in their chemical union... the explanation of cohesion, elasticity, and gravitation — all these will be marshaled into a single compact and consistent body of scientific knowledge... one of the grandest generalizations of modern science ... that all the phenomena of the physical universe are only different manifestations of the various modes of motion of one all-pervading substance — the ether.&lt;/em&gt;&amp;quot; [quoted from &lt;a href=&quot;https://www.google.com/books/edition/Quantum_Generations/ELrFDIldlawC?hl=en&quot; rel=&quot;noreferrer&quot;&gt;Kragh, Quantum Generations, ch. 4&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Michelson's optimism was not widely shared either.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-08-28T05:28:39.870" LastActivityDate="2022-08-28T05:28:39.870" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14754" PostTypeId="1" AcceptedAnswerId="14755" CreationDate="2022-08-29T07:41:44.823" Score="0" ViewCount="86" Body="&lt;p&gt;I used to have a book that traced the history of books which had titles like &amp;quot;secrets of the universe explained&amp;quot;, &amp;quot;secrets of chemistry exlained&amp;quot;, or &amp;quot;secrets of magic revealed&amp;quot;, etc. The author claimed that with the invention of the printing press a category of books of &amp;quot;secrets&amp;quot; proliferated. Today, we still see books revealing secrets of the universe.&lt;/p&gt;&#xA;&lt;p&gt;So, this type of &amp;quot;secrets&amp;quot; books appears to be a genre.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know this book? It is probably from 1980s or '90s.&lt;/p&gt;&#xA;&lt;p&gt;Thanks.&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2022-08-29T17:38:53.310" Title="Looking for a book about &quot;mysteries of the universe&quot;" Tags="|book-recommendation|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14755" PostTypeId="2" ParentId="14754" CreationDate="2022-08-29T17:25:35.333" Score="5" Body="&lt;p&gt;This book is apparently William Eamon's &lt;a href=&quot;https://press.princeton.edu/books/paperback/9780691026022/science-and-the-secrets-of-nature&quot; rel=&quot;noreferrer&quot;&gt;Science and the Secrets of Nature&lt;/a&gt;, but there are others that fit the bill, such as Wayne Shumaker's &lt;em&gt;Natural Magic and Modern Science: Four Treatises, 1590-1657&lt;/em&gt;. The subject matter is described in a Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Books_of_secrets&quot; rel=&quot;noreferrer&quot;&gt;article&lt;/a&gt;.  As the OP suggests, there was a genre of early printed books about secrets of nature, a notable example being the 1558 &lt;a href=&quot;https://en.wikipedia.org/wiki/Magia_Naturalis&quot; rel=&quot;noreferrer&quot;&gt;Natural Magic&lt;/a&gt; of J.B. della Porta.&#xA;I don't have Eamon's book in front of me, but it surely discusses&#xA;works such as the pre-Gutenberg  gynecological work &lt;a href=&quot;https://en.wikipedia.org/wiki/Secreta_Mulierum&quot; rel=&quot;noreferrer&quot;&gt;Secreta Mulierum&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Again, without checking Eamon, this tradition has a connection with the Natural Magic tradition, implied by the title of Shumaker's book.&#xA;A vestige of &lt;em&gt;this&lt;/em&gt; tradition was the&#xA;television show &lt;a href=&quot;https://en.wikipedia.org/wiki/Watch_Mr._Wizard&quot; rel=&quot;noreferrer&quot;&gt;Mr. Wizard&lt;/a&gt; which I remember watching as a child.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2022-08-29T17:38:53.310" LastActivityDate="2022-08-29T17:38:53.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14756" PostTypeId="1" CreationDate="2022-08-30T11:55:57.327" Score="2" ViewCount="71" Body="&lt;p&gt;I am aware that Albert Einstein first added the &lt;span class=&quot;math-container&quot;&gt;$\Lambda$&lt;/span&gt;-term to his field equations in his 1917 paper &amp;quot;Cosmological considerations in the general theory of relativity&amp;quot; (german: &amp;quot;Kosmologische Betrachtungen zur allgemeinen Relativitätstheory&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;However, his orginial label for the constant was a &amp;quot;vorläufig unbekannte universelle Konstante &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;&amp;quot;, which translates to &amp;quot;for the moment unknown universal constant &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;&amp;quot;. I am interested in who first spoke of &amp;quot;cosmological constant&amp;quot;, i.e. who coined the name that is most common for this constant nowadays and in what paper/publication was it first named as such.&lt;/p&gt;&#xA;" OwnerUserId="17192" OwnerDisplayName="Octavius" LastActivityDate="2022-08-30T12:17:40.043" Title="Who coined the name &quot;cosmological constant&quot;?" Tags="|cosmology|terminology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14757" PostTypeId="1" CreationDate="2022-08-30T15:25:49.577" Score="1" ViewCount="223" Body="&lt;p&gt;What is the difference between definition and axiom? For instance, Newton's Definition 1 reads: (Cohen p. 403)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quantity of matter is a measure of matter that arises from its density and volume jointly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, Newton defines density: &lt;code&gt;Density = Quantity of matter/ Unit volume&lt;/code&gt;&lt;/p&gt;&#xA;&lt;p&gt;And Newton's Axiom 1 reads: (Cohen p.416)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Axiom 1 Newton defines his concept of inertia but he classifies his definition of inertia as an axiom (not as a definition).&lt;/p&gt;&#xA;&lt;p&gt;And also, the title of the chapter is, &amp;quot;Axioms, or the laws of motion.&amp;quot; So Newton, defines his axioms as laws of nature.&lt;/p&gt;&#xA;&lt;p&gt;Any thoughts about the difference in meaning between definition, axiom and law in Newton's time?&lt;/p&gt;&#xA;" OwnerUserId="16575" LastEditorUserId="4251" LastEditDate="2022-08-31T15:22:22.530" LastActivityDate="2022-08-31T21:30:11.037" Title="What is the difference between Newton's definitions and axioms?" Tags="|physics|newton|definition|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14758" PostTypeId="1" AcceptedAnswerId="14759" CreationDate="2022-08-31T15:00:39.357" Score="0" ViewCount="1339" Body="&lt;p&gt;I have searched and examined legendary stories of the problem-solving skills of von Neumann in mathematics.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;With &lt;a href=&quot;https://www.quora.com/What-theorem-did-John-von-Neumann-prove-during-George-Polyas-lecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Polya&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With &lt;a href=&quot;https://mosstuff.quora.com/The-most-complete-version-of-the-Dantzig-meets-von-Neumann-anecdote-I-ve-seen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dantzig&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Maybe there are other stories showing that he is a great problem-solver, but I think these two are well-known.&lt;/p&gt;&#xA;&lt;p&gt;I know that he wrote many papers in pure mathematics (mostly developing new concepts), but with his talented problem-solving skills, didn't he write any paper on solving unsolved problems in mathematics? Maybe I haven't searched enough?&lt;/p&gt;&#xA;&lt;p&gt;Just because he was not interested in publishing a paper on that subject?&lt;/p&gt;&#xA;&lt;p&gt;Or, even for him, it is not easy solving problems that are worthy of publishing. What are your thoughts?&lt;/p&gt;&#xA;" OwnerUserId="16347" LastEditorUserId="12330" LastEditDate="2022-09-03T00:38:58.420" LastActivityDate="2022-09-03T00:38:58.420" Title="Did John von Neumann solve any unsolved problem in mathematics?" Tags="|mathematics|mathematicians|biographical-details|von-neumann|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14759" PostTypeId="2" ParentId="14758" CreationDate="2022-08-31T15:30:51.237" Score="6" Body="&lt;p&gt;John von Neumann (partially) solved &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert's fifth problem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Plus: he is one of those that contributed to solving &lt;a href=&quot;https://en.wikipedia.org/wiki/Russell%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's paradox&lt;/a&gt; by building the axioms of maths.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-08-31T19:43:56.153" LastActivityDate="2022-08-31T19:43:56.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14760" PostTypeId="2" ParentId="14758" CreationDate="2022-08-31T16:58:47.977" Score="5" Body="&lt;p&gt;One guess: There was a question from physics on how to formulate quantum mechanics.  Von Neumann proposed doing it using operators on Hilbert space. (1927-1929)&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2022-08-31T17:03:55.850" LastActivityDate="2022-08-31T17:03:55.850" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14761" PostTypeId="2" ParentId="14757" CreationDate="2022-08-31T20:37:24.667" Score="3" Body="&lt;p&gt;Borrowing from another post: &lt;a href=&quot;https://physics.stackexchange.com/questions/707679/are-newtons-laws-just-definitions/712478#712478&quot;&gt;Are Newton's laws just definitions?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Newton follows the Euclidean tradition of presenting mathematical proofs by first providing a set of definitions which are then followed by a set of axioms (i.e., laws) that are assumed to be true. Definitions in themselves do not admit the existence of anything, but rather provide terminology that is later recognized as a particular condition as a result of a set of axioms.&lt;/p&gt;&#xA;&lt;p&gt;Newton presents his laws of motion based on a collection of known observations regarding motion (in particular, he cites Galileo's inclined plane experiment and hypothesis that objects only slow down due to air resistance or other friction forces, which is counter to Aristotelian physics), and carries forward with the logical consequences of these laws in a series of propositions, which so happens to match experimental evidence.&lt;/p&gt;&#xA;&lt;p&gt;Once the laws are provided, the particular effects of the laws can be matched with the definitions he provided. For example, his definitions of quantity of matter, quantity of motion, vis insita, etc., only make sense within the framework provided by his laws.&lt;/p&gt;&#xA;&lt;p&gt;...the laws are not &amp;quot;definitions&amp;quot; since definitions are only understood within the assumed truth of the laws.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In other words (and to directly answer your question), &amp;quot;Definitions&amp;quot; provide a specific term used to compactly identify something that arises from some set of axioms or propositions.&lt;/p&gt;&#xA;&lt;p&gt;As a loose example, in calculus, we define an &amp;quot;inflection point&amp;quot; as being a specific point where the second derivative of a function equals zero.  This definition does not declare the existence of the inflection point as an axiom or proposition but instead provides an identifying term for when it does occur based upon a set of assumed truths.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Axiom 1 Newton defines his concept of inertia but he classifies his&#xA;definition of inertia as an axiom (not as a definition).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Newton's First Law (i.e., Axiom 1) does NOT define inertia, rather, it declares an axiom of motion that is assumed to be true. The term &amp;quot;inertia&amp;quot; is nowhere mentioned in the law: &amp;quot;Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impress'd thereon.&amp;quot; &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Axioms,_or_Laws_of_Motion#Law1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikisource: Newton's Laws&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, in the &amp;quot;definitions&amp;quot; section that's located prior to his Axioms, Newton defines inertia (&lt;em&gt;Vis inertiae&lt;/em&gt;) under Definition 3: &amp;quot;The Vis Insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forwards in a right line.  This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the Mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this Vis insita may, by a most significant name, be called &lt;em&gt;&lt;strong&gt;Vis inertiæ&lt;/strong&gt;&lt;/em&gt; or Force of Inactivity.&amp;quot; &lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Definitions#Def3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikisource: Newton's Definitions&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thus, &amp;quot;inertia&amp;quot; is a term used to identify the effect (namely the Force of Inactivity) that results from axiom 1.  &lt;em&gt;However&lt;/em&gt;, &amp;quot;inertia&amp;quot; does not rely on axiom 1 alone because the definition states it &amp;quot;is a power of resisting,&amp;quot; thereby invoking an axiom of change (i.e., Law #2) and that it is categorized as an &amp;quot;Innate Force of Matter,&amp;quot; which practically speaking, can only be understood through some interaction (i.e., Law #3 where inertia is perceived as a reactionary force in response to a force).  Naturally, you might ask, &amp;quot;then why is Law #1 called the 'law of inertia'?&amp;quot;  Well, because inertia is still &amp;quot;there&amp;quot; during the absence of any change and external forces; however, if we omit Laws #2 and #3 and place Law #1 in a vacuum, then inertia becomes an absurd concept since there are no rules that allow change and forces that permit change.  In other words, the definition of inertia is absurd unless we admit all three laws.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And also, the title of the chapter is, &amp;quot;Axioms, or the laws of&#xA;motion.&amp;quot; So Newton, defines his axioms as laws of nature.&lt;/p&gt;&#xA;&lt;p&gt;Any thoughts about the difference in meaning between definition, axiom&#xA;and law in Newton's time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Just as you said, Newton titles the section &amp;quot;Axioms, or the laws of motion (&lt;em&gt;Axiomata sive leges motus&lt;/em&gt;).&amp;quot; Thus, Newton treats &amp;quot;Laws&amp;quot; as being synonymous with &amp;quot;axioms.&amp;quot;  If I were to guess, he probably chose the term &amp;quot;Laws&amp;quot; in order to categorically distinguish these rules of physical governance from mathematical axioms, but they are functionally the same.  It's much like &amp;quot;postulates&amp;quot; versus &amp;quot;axioms&amp;quot; in maths in which they are functionally the same, but &amp;quot;postulates&amp;quot; is a species of &amp;quot;axioms&amp;quot; that act within a specific field of study.&lt;/p&gt;&#xA;&lt;p&gt;For more examples of how definitions are treated, look at Euclid's Elements Book 1: Definitions are provided first, followed by postulates (i.e., axioms), then common notions, then propositions.  Though definitions are provided first, they are only fully understood within the context of the postulates and constructed propositions. &lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/elements/bookI/bookI.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements, Book I&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-08-31T21:30:11.037" LastActivityDate="2022-08-31T21:30:11.037" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14762" PostTypeId="2" ParentId="14757" CreationDate="2022-08-31T20:48:14.987" Score="2" Body="&lt;p&gt;I agree with the trend of the comments, and believe that in most cases Newton was trying to write straightforwardly even where we might find difficulty, perhaps partly owing to the lapse of time and changes in habits of expression.&lt;/p&gt;&#xA;&lt;p&gt;Specifically, the definitions appear to be naming conventions (just as Conifold and Paul Garrett said): each consisting of a name and and explanation, hopefully sufficient, of what is to be known by that name.&lt;/p&gt;&#xA;&lt;p&gt;The axioms are a different matter. The full name of the section is &amp;quot;Axioms or laws of motion&amp;quot;. The laws of motion are what Newton thought to be laws, and giving them the added name or label of 'axiom' meant that he also took them to be well accepted or acceptable in view of current knowledge, and thus did not propose to prove them. This is strongly reinforced by the content of the following Scholium, where he wrote&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Hitherto I have laid down such principles as have been receiv'd by&#xA;mathematicians, and are confirm'd by abundance of experiments. By the&#xA;two first Laws and the first two Corollaries, Galileo discover'd that&#xA;the descent of bodies observ'd the duplicate ratio of the time, and&#xA;that the motion of projectiles was in the curve of a Parabola;&#xA;experience agreeing with both, unless so far as these motions are a&#xA;little retarded by the resistance of the air.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ans then he went on to discuss other points and credit others including Huygens and Mariotte.&lt;/p&gt;&#xA;&lt;p&gt;It will be seen that Newton did not claim the laws of motion, axioms, to be his own. A fair comparison might be with the axioms of Euclid's Geometry: they were not thought to be in need of proof, they were starting-points on which the ensuing discussions were to build further conclusions. It was only long after his lifetime that others named the laws of motion after Newton.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-08-31T21:07:10.253" LastActivityDate="2022-08-31T21:07:10.253" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14763" PostTypeId="1" CreationDate="2022-08-31T22:57:41.303" Score="0" ViewCount="94" Body="&lt;p&gt;I am preparing an introductory course on the history of astronomy and geometry for intermediate students. The intention is to teach Copernican heliocentrism, Kepler's laws of planetary motion &amp;amp; Newton's law of universal gravitation. I couldn't find any single book dealing with both subjects. I am wondering:  is there any such book?&lt;/p&gt;&#xA;" OwnerUserId="17106" LastEditorUserId="17106" LastEditDate="2022-09-01T08:09:45.463" LastActivityDate="2022-09-01T08:09:45.463" Title="Is there any book dealing with the history of both astronomy and geometry?" Tags="|astronomy|geometry|book-recommendation|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14764" PostTypeId="1" CreationDate="2022-08-31T23:38:11.507" Score="1" ViewCount="71" Body="&lt;p&gt;In everyday English, &amp;quot;sphere&amp;quot; means a round object. People will think of the insides as part of the sphere.&lt;/p&gt;&#xA;&lt;p&gt;In Mathematics it specifically means the surface of the ball.&lt;/p&gt;&#xA;&lt;p&gt;How did the different meanings arise? Which was in use earlier and how did they end up becoming so confusing?&lt;/p&gt;&#xA;" OwnerUserId="10623" LastActivityDate="2022-08-31T23:38:11.507" Title="Usage of &quot;sphere&quot; as ball's surface vs as ball itself" Tags="|terminology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14765" PostTypeId="2" ParentId="14748" CreationDate="2022-09-02T10:49:15.960" Score="3" Body="&lt;p&gt;As a possible example &amp;quot;of &amp;gt;200 years old unsolved problems NOT from number theory&amp;quot; there may be the following:&lt;/p&gt;&#xA;&lt;p&gt;According to Peter Pesic, &amp;quot;[V I] Arnol’d points out [that] Leibniz was moved to conjecture that (in modern terms) an Abelian integral along an algebraic curve (taken between algebraic limits) is a transcendental number. This conjecture remains an important unsolved problem.&amp;quot; (Historia Mathematica 28 (2001), 215–219, at 218).&lt;/p&gt;&#xA;&lt;p&gt;I didn't succeed in tracing the source for this in Arnol'd or in Leibniz, {edit:} but @user6530 has since then kindly supplied a link to the &lt;a href=&quot;https://www.dbnl.org/tekst/huyg003oeuv10_01/huyg003oeuv10_01_0022.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;original form of Leibniz's conjecture&lt;/a&gt;, given in a letter of 1691. The subject of the 1691 discussion and conjecture appears to be  integrability, and there is no mention of transcendental numbers (a concept which also appears to have been in 1691 still in the future). From this it appears that the so-called &amp;quot;modern terms&amp;quot; of the conjecture put forward in the secondary source (which was all that I originally had available to quote), may be a more radical alteration of the original than only a rendering of it into 'modern terms', and it may be unclear whether it fairly represents the original meaning and scope of Leibniz's 1691  conjecture. Indeed my lack of any primary source was the reason why I mentioned the conjecture only as a &amp;quot;possible&amp;quot; item within the category requested in the question. In the current state of the discussion, it does not seem clear that Leibniz's calculus problem falls within number theory. But a clarification of the claimed scope of 'number theory' for the purposes of this question does seem desirable. Whether the conjecture offered here 'falls within' number theory might depend on whether number theory is now being defined so wide as to include any problem in any field for which some other problem can be specified which is in some way parallel with it and also arguably falls within number theory. Such a definition might seem to colonize the whole of mathematics.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-09-04T00:41:38.307" LastActivityDate="2022-09-04T00:41:38.307" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="14766" PostTypeId="1" CreationDate="2022-09-02T14:12:24.287" Score="0" ViewCount="101" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;From beginnings of topology, it was clear that the closed interval &lt;span class=&quot;math-container&quot;&gt;$\left[a,b \right]$&lt;/span&gt; of the real line had a certain property that was crucial for proving such theorems as the maximum value theorem and the uniform continuity theorem.(Sect-26, Munkres)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What was actually the difficulty faced by mathematicians in precisy-fying the intuitive idea of continuity?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="16355" LastEditDate="2022-09-03T16:13:50.757" LastActivityDate="2022-09-03T16:13:50.757" Title="Why was it difficult to initially phrase compactness?" Tags="|mathematics|topology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14767" PostTypeId="2" ParentId="13784" CreationDate="2022-09-03T13:23:00.730" Score="4" Body="&lt;p&gt;One year later, let me answer my own question.&lt;/p&gt;&#xA;&lt;p&gt;If we dig a little bit, we do find comments of Nicollet's work:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Adriano Balbi, italian geographer, in &lt;em&gt;Revue Encyclopédique&lt;/em&gt;, vol.38, p.453, april 1828 &lt;a href=&quot;https://books.google.fr/books?id=45BNAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=45BNAAAAcAAJ&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Charles-Rigobert Bonne, french geographer, in &lt;em&gt;Bulletin de la Société de géographie&lt;/em&gt;, n°63, p.76, july 1828 &lt;a href=&quot;https://books.google.fr/books?id=bRk7AQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=bRk7AQAAIAAJ&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;in &lt;em&gt;The philosophical magazine&lt;/em&gt;, march 1829, which I mentionned in the question&lt;/li&gt;&#xA;&lt;li&gt;Charles Sturm, french mathematician, in &lt;em&gt;Le Bulletin général et universel des annonces et des nouvelles scientifiques&lt;/em&gt;, vol.14, p.100, july 1830 &lt;a href=&quot;https://books.google.fr/books?id=RpBfAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=RpBfAAAAcAAJ&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;George Airy, in &lt;em&gt;Encyclopaedia Metropolitana&lt;/em&gt;, vol.5, p.202, 1845 &lt;a href=&quot;https://books.google.fr/books?id=FDwPfVKP79oC&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=FDwPfVKP79oC&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And also some words in astronomers letters:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;from Franz Xaver von Zach (austrian astronomer) to to Alfred Gautier (Geneva observatory), June 17th 1828, &lt;em&gt;Notizen Zur Schweiz. Kulturgeschichte&lt;/em&gt;, vol.30, p.446, 1885 &lt;a href=&quot;https://www.ngzh.ch/publikationen/vierteljahrsschrift&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ngzh.ch/publikationen/vierteljahrsschrift&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;from Giovanni Plana (Torino university) to Gautier [Nicollet worked with Plana and Gautier around 1821 for geodesic measures along the 45th parallel], May 10th 1829, &lt;em&gt;Notizen Zur Schweiz. Kulturgeschichte&lt;/em&gt;, vol.31, p.102, 1886 &lt;a href=&quot;https://www.ngzh.ch/publikationen/vierteljahrsschrift&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ngzh.ch/publikationen/vierteljahrsschrift&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Those comments are from total rejection (The philosophical magazine) to full acceptance (Airy). Interesting to note that french comments are not focusing at all on the constant error (but more on other aspects of the Nicollet's memoir) whereas non-french do consider this point.&#xA;And we see that some of the european astronomers were already aware of this question.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Zach for instance noticed the strange behaviour or repeating circles, and wrote a lot on it starting from 1812 &lt;a href=&quot;https://books.google.fr/books?id=64UvZnoWFCAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=64UvZnoWFCAC&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Gauss, when when presenting his least squares method in February 1821 at Gottingen had to describe correctly what is a random error (upon which we can apply least squares) in opposition to systematic one (upon which we cannot).&lt;/li&gt;&#xA;&lt;li&gt;Friedrich Wilhelm Bessel, from Königsberg observatory, did some experiments to identify the origin of the issues, for instance in &lt;em&gt;Philosophical Magazine&lt;/em&gt;, Vol.59, p.44, June 1822,  &lt;a href=&quot;https://books.google.fr/books?id=UzJDAQAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=UzJDAQAAMAAJ&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Piotr Sławiński, from Vilnius observatory, even proposed formulas to correct measures made with his repeating circle, &lt;em&gt;Astronomische Nachrichten&lt;/em&gt;, n°96, p.554,  &lt;a href=&quot;https://books.google.fr/books?id=adozAQAAIAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=adozAQAAIAAJ&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;All this studies were made and published (mostly in french or english) before Nicollet's memoir. We can safely assume french astronomers knew these publications, but probably disregarded them because the repeating circle manufacturer was not the same (made by Reichenbach in Munich versus made by Lenoir in Paris).&#xA;Last interesting point, Arago, working with Nicollet at Paris observatory also noticed suspicious things with Borda's repeating circle 10 years before, and even provided a relevant experimental workaround. This can be seen for instance in the minutes of the weekly Bureau des longitudes, &lt;a href=&quot;http://bdl.ahp-numerique.fr/items/show/1397&quot; rel=&quot;nofollow noreferrer&quot;&gt;11st&lt;/a&gt; and &lt;a href=&quot;http://bdl.ahp-numerique.fr/items/show/1399&quot; rel=&quot;nofollow noreferrer&quot;&gt;25th november 1818&lt;/a&gt;.&#xA;But, he didn't go further in his analysis: no hypothesis about the origin of the problem nor any rigorus motivation of its workaround.&lt;/p&gt;&#xA;&lt;p&gt;My personal conclusions of this story are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nicollet is definitively not the first one to make the distinction between systematic and random error (anyway, it took decades to make this distinction clear)&lt;/li&gt;&#xA;&lt;li&gt;Nicollet did not invent new concepts nor apply new methods to solve Méchain's problem, he &amp;quot;just&amp;quot; (tough calculatory work though) applied some other's ideas with better data (taken from more recent stars catalogs)&lt;/li&gt;&#xA;&lt;li&gt;German astronomers did very insteresting things regarding random and systematic errors, but Nicollet ignored them (volontarily, because of french astronomers mindset ?)&lt;/li&gt;&#xA;&lt;li&gt;the full history of random vs. systematic error is yet to be done (at least for the french side)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;for those who are really curious, the anonymous writer of the &lt;em&gt;Philosophical magazine&lt;/em&gt; is probably Xavier von Zach.&lt;/p&gt;&#xA;&lt;p&gt;We can assume that from a letter Zach wrote to Johann Kaspar Horner on March 1829 (same month of the publishing) stating that he sent to a journal a very critical comment of Nicollet's work. See &lt;em&gt;Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich&lt;/em&gt;, vol.18, 1873 &lt;a href=&quot;https://books.google.fr/books?id=xDtcAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.fr/books?id=xDtcAAAAcAAJ&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13909" LastEditorUserId="13909" LastEditDate="2022-12-22T15:06:45.317" LastActivityDate="2022-12-22T15:06:45.317" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14768" PostTypeId="1" CreationDate="2022-09-03T16:45:04.443" Score="2" ViewCount="126" Body="&lt;p&gt;I read in my Physics textbook that the notion of Electric fields are useful &amp;quot;when we have to deal with time dependent Electromagnetic phenomenon since no information can travel faster than light&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But this existence of speed limit was concluded by Einstein (in 1905, right ?) and the theory of fields (or field lines) was setup by Faraday (in 1890s I guess)  .&lt;/p&gt;&#xA;&lt;p&gt;So &lt;strong&gt;how did he actually realised the existence of such fields&lt;/strong&gt; like &lt;strong&gt;what was so compelling about their inevitable existence ??&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="13274" LastActivityDate="2022-09-03T16:45:04.443" Title="Why was the idea of &quot;field&quot; introduced?" Tags="|physics|electromagnetism|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14769" PostTypeId="1" CreationDate="2022-09-04T09:46:06.013" Score="4" ViewCount="475" Body="&lt;p&gt;Are there any traces of the third law of motion before Newton's Principia?&#xA;wiki says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton arrived at his set of three laws incrementally. In a 1684&#xA;manuscript written to Huygens, he listed four laws: the principle of&#xA;inertia, the change of motion by force, a statement about relative&#xA;motion that would today be called Galilean invariance, and the rule&#xA;that interactions between bodies do not change the motion of their&#xA;center of mass. In a later manuscript, Newton added a law of action&#xA;and reaction, while saying that this law and the law regarding the&#xA;center of mass implied one another. Newton probably settled on the&#xA;presentation in the Principia, with three primary laws and then other&#xA;statements reduced to corollaries, during 1685.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are there really no hints about this principle (not really conservation of momentum) before 1684?&lt;/p&gt;&#xA;" OwnerUserId="7300" LastActivityDate="2022-09-04T16:49:41.130" Title="Third law of motion before Newton?" Tags="|newton|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14770" PostTypeId="1" CreationDate="2022-09-04T13:32:07.113" Score="2" ViewCount="100" Body="&lt;p&gt;There are two well known asymptotic expansions of the Dawson integral &lt;span class=&quot;math-container&quot;&gt;$F(x)$&lt;/span&gt; and the function &lt;span class=&quot;math-container&quot;&gt;$\exp(x^2)\operatorname {erfc}(x)$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$x \rightarrow \infty$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;F(x)\sim (1/2)(1/x+1/(2x^3)+ 3/(4x^5)+\cdots)&#xA;$$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\exp(x^2)\operatorname {erfc}(x)\sim(1/\sqrt{\pi})(1/x - 1/(2x^3) + 3/(4x^5)+\cdots).&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am looking for the original publications where these expansions were derived for the first time. I would appreciate any pointers. I am also interested in other publications dealing with the issue of why the Laplace transformation cannot (or perhaps can?) be used to derive such expansions.&lt;/p&gt;&#xA;" OwnerUserId="17209" LastEditorUserId="10106" LastEditDate="2022-09-05T02:56:31.097" LastActivityDate="2022-09-05T02:56:31.097" Title="How the asymptotic expansions of the Dawson integral and $\exp(x^2)\operatorname {erfc}(x)$ were originally obtained?" Tags="|mathematics|calculus|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14771" PostTypeId="2" ParentId="14769" CreationDate="2022-09-04T16:49:41.130" Score="6" Body="&lt;p&gt;Much early thinking about what became the laws of motion in the Principia, including the third law, can be found in &lt;a href=&quot;https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00220&quot; rel=&quot;noreferrer&quot;&gt;Newton's &lt;em&gt;&amp;quot;Waste Book&amp;quot;&lt;/em&gt;&lt;/a&gt;. It is a notebook started in 1665, and it is discussed by a number of authors, including &lt;a href=&quot;https://philpapers.org/rec/FRATTL&quot; rel=&quot;noreferrer&quot;&gt;D Fraser (2005), &amp;quot;The third law in Newton's &lt;em&gt;Waste book&lt;/em&gt;&amp;quot;&lt;/a&gt; (&lt;em&gt;Studies in History and Philosophy of Science Part A&lt;/em&gt; 36(1):43-60). There is also some related material in another notebook of the period (&lt;a href=&quot;https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00092&quot; rel=&quot;noreferrer&quot;&gt;'Questiones quaedam philosophicae', 'Certain philosophical questions'&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The &lt;em&gt;'Waste Book'&lt;/em&gt; seems to be the only Newton manuscript source offering advance hints of the third law before the &lt;a href=&quot;https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum&quot; rel=&quot;noreferrer&quot;&gt;1684 manuscript 'De motu'&lt;/a&gt; (an initial precursor of 'Principia', written on the stimulus of questions put to Newton by Edmund Halley, and sent to Halley and to the Royal Society).  The &lt;em&gt;'Waste Book'&lt;/em&gt; casts some light on the origins and justification of the third law, as discussed in the Fraser paper cited above (p.50): this mentions that while Newton's words suggest he thought it was obvious that action and reaction were opposite in direction, e.g. when two bodies press against each other, it was not so obvious at first that the action and reaction were also equal in size.&lt;/p&gt;&#xA;&lt;p&gt;Much of the ingredient thinking on dynamics that went into the &lt;em&gt;'Waste Book'&lt;/em&gt; appears to have related to the problem of formulating laws that would account acceptably for the phenomena of bodies in collision. As also argued by Fraser (2005), some of the relevant arguments relate to the equal actions on bodies at both ends of a compressed spring.&lt;/p&gt;&#xA;&lt;p&gt;All of these sources may give insight into Newton's thinking, but there remain always dangers of anachronism and misinterpretation in reading such documents; and on some points, the language of the sources can permit different authors to draw different conclusions about their content and implications. For example, &lt;a href=&quot;https://archive.org/details/forceinnewtonsph0000west&quot; rel=&quot;noreferrer&quot;&gt;R S Westfall (1971), 'Force in Newton's Physics'&lt;/a&gt; (at p.348) considering the &lt;em&gt;Waste book&lt;/em&gt;, wrote that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Except for the special case in which [two interacting bodies] p and r&#xA;have equal motions, the conclusion that they press each other equally&#xA;had seemed obviously false to the great majority of those who before&#xA;1665 had attempted to analyze the force of percussion. The &lt;em&gt;Waste&#xA;book&lt;/em&gt; suggests that Newton came to deny the intuitively obvious by&#xA;recognising that for every impact, there is a frame of reference, that&#xA;of the common centre of gravity, in which the two bodies do have equal&#xA;forces. That is, his treatment of impact depended on his accepting a&#xA;relativity of motion in terms of which the idea of an absolute force&#xA;of a body's motion is meaningless.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On the other hand, Fraser (2005:53) thought that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;the evidence suggests that the third law originated in the empirical&#xA;or theoretical investigation of elasticity and that, for Newton, its&#xA;justification lay in his application of an elastic model of&#xA;collisions.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And then Westfall (1971:472) had already pointed out that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;in demonstrating the validity of the third law in the case of two&#xA;bodies attracting each other, Newton {for the sake of argument}&#xA;imagined B to attract A more strongly than A attracts B. If that&#xA;were the case, there would be an unbalanced force in the system&#xA;of two bodies, and the system would move in the line AB&#xA;‘&lt;em&gt;in  infinitum&lt;/em&gt; with a motion perpetually accelerated’ --&#xA;a situation contradicting the first law.&amp;quot;&#xA;(with quote from the Scholium to the laws of motion (1729:p37)).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this way numerous lines of argument converge on the third law. Whichever of them accounts better for the history of its development, the answer remains a &lt;strong&gt;'yes'&lt;/strong&gt;, there are several antecedent hints or traces of the third law found in Newton's writing before the 'Principia'  and before the 1684 'De motu', with clear signs that he gave attention to the subject matter in the 1660s.&lt;/p&gt;&#xA;&lt;p&gt;As a postscript, it's also relevant that Newton made no claim that the laws of motion he stated in the 'Principia' were his own.  In the 'Scholium' to the laws of motion Newton credited several others (Galileo, Wren, Wallis, Huygens, Mariotte). He also called the laws 'axioms', indicating that he thought they made generally acceptable starting-points not in need of proof: he wrote (&lt;a href=&quot;https://books.google.com/books?id=Tm0FAAAAQAAJ&amp;amp;pg=PA31&quot; rel=&quot;noreferrer&quot;&gt;1729 English translation of Principia, p.31&lt;/a&gt;) that they were &amp;quot;such principles as have been receiv'd by Mathematicians, and are confirm'd by abundance of experiments&amp;quot;, i.e. largely experiments of others.  (The naming of the laws as Newton's was made by others long after his lifetime: probably in recognition that as a set and a compilation they were not in use before Newton: he selected and combined them from among competing prior-proposed principles, and also amended their expression and scope. Concepts of action and reaction do predate Newton's work, but the third law gave precision to the concept of this paired interaction, e.g. &lt;a href=&quot;https://archive.org/details/forceinnewtonsph0000west&quot; rel=&quot;noreferrer&quot;&gt;R S Westfall (1971), 'Force in Newton's Physics'&lt;/a&gt;: especially discussion of Newton's various drafts of laws of motion, pp.448-456.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-09-04T16:49:41.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14772" PostTypeId="1" CreationDate="2022-09-05T15:27:00.960" Score="1" ViewCount="262" Body="&lt;p&gt;Why was the concept of &lt;a href=&quot;https://en.wikipedia.org/wiki/Momentum&quot; rel=&quot;nofollow noreferrer&quot;&gt;momentum&lt;/a&gt; invented in the first place, and is it useful in the different discoveries and equations that came after?&lt;/p&gt;&#xA;" OwnerUserId="17211" OwnerDisplayName="Ayush Sharma" LastEditorUserId="12330" LastEditDate="2022-09-07T02:26:54.037" LastActivityDate="2022-09-07T02:26:54.037" Title="Why was the concept of momentum invented in the first place, and what is its history?" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14773" PostTypeId="2" ParentId="14772" CreationDate="2022-09-05T15:40:42.470" Score="2" Body="&lt;p&gt;Wikipedia has a decent &lt;a href=&quot;https://en.wikipedia.org/wiki/Momentum#History_of_the_concept&quot; rel=&quot;nofollow noreferrer&quot;&gt;history section&lt;/a&gt;. For more information about physics during that time period, Julian Barbour wrote an amazing book entitled &lt;em&gt;The Discovery of Dynamics&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="2129" OwnerDisplayName="Alex Nelson" LastActivityDate="2022-09-05T15:40:42.470" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14774" PostTypeId="1" CreationDate="2022-09-05T17:28:39.427" Score="2" ViewCount="148" Body="&lt;p&gt;I have two related questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who first defined the oscillation function (perhaps under a different name)?&lt;/li&gt;&#xA;&lt;li&gt;When did the switch from the phrase &amp;quot;saltus function&amp;quot;(*) to &amp;quot;oscillation function&amp;quot; happened?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Recall that given a topological space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, a metric space &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, and a function &lt;span class=&quot;math-container&quot;&gt;$f:X\to M$&lt;/span&gt;, the oscillation function of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathrm{osc}_\cdot(f):X\rightarrow[0,\infty],x\mapsto\mathrm{inf}\{\mathrm{diam}(f(U))|U\text{ is a nbdgd of } x\}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For what it's worth, Froda, in his 1929 thesis &lt;em&gt;Sur la distribution des propriétés de voisinage des fonctions de variables réelles&lt;/em&gt; (&lt;a href=&quot;https://eudml.org/doc/192780&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://eudml.org/doc/192780&lt;/a&gt;) attributes (p.23) the observation that the higher order oscillations stabilize to a 1905 paper by Denjoy (where this is used implicitly, Froda says as far as I can see) and a 1910 paper by Sierpinski.&lt;/p&gt;&#xA;&lt;p&gt;(There are other, non-pointwise definitions, or else definitions in terms of the difference between limit superiors and limit inferiors also; these definitions are also admissible for the purposes of this question.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(*): See e.g. Hobson's 1921 book &lt;em&gt;The theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. I (2e)&lt;/em&gt; , p.291, or Blumberg's 1917 paper &amp;quot;Certain General Properties of Functions&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Finally, here is the OED definition of &amp;quot;saltus&amp;quot; (I hadn't heard of this word before):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/6sorg.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/6sorg.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Oxford English Dictionary, s.v. “saltus (n.),” July 2023, &lt;a href=&quot;https://doi.org/10.1093/OED/6909159924&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1093/OED/6909159924&lt;/a&gt;. )&lt;/p&gt;&#xA;&lt;p&gt;In light of the definition of the word it seems more appropriate to call the function at hand the saltus function than the oscillation function (also keeping in mind the more commonly studied &amp;quot;BMO functions&amp;quot;).&lt;/p&gt;&#xA;" OwnerUserId="1887" LastEditorUserId="1887" LastEditDate="2024-01-30T17:58:03.523" LastActivityDate="2024-01-30T17:58:03.523" Title="Definition and name change of the oscillation function" Tags="|mathematics|mathematicians|terminology|real-analysis|definition|" AnswerCount="2" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="14775" PostTypeId="2" ParentId="14774" CreationDate="2022-09-05T21:59:31.523" Score="2" Body="&lt;p&gt;Searching Google Books often leads to useful results. I searched your keywords as &amp;quot;oscillating function&amp;quot; &amp;quot;saltus&amp;quot; Hobson. This leads to the same book &amp;quot;The Theory of Functions of a Real Variable and the Theory of Fourier's Series Volume 1, By Ernest William Hobson, 1921&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;On page 284, Hobson has a footnote with a asterisk on oscillation. In other books, Hobson is credited with saltus.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;&lt;a href=&quot;https://i.stack.imgur.com/yM2S1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yM2S1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So it seems &lt;em&gt;Schwingung&lt;/em&gt; was translated to oscillation. You will have scavenge German literature. DeepL is an excellent free machine translator. Saltus itself is inspired by German &lt;em&gt;Sprung&lt;/em&gt; most likely by the author himself (pg 283).&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2022-09-05T21:59:31.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14776" PostTypeId="1" AcceptedAnswerId="14777" CreationDate="2022-09-06T00:20:37.080" Score="12" ViewCount="238" Body="&lt;p&gt;As I understand it, 'N. Bourbaki' was the pseudonym of a collective of French mathematicians. How, then, did 'it' give a conference talk&lt;sup&gt;&lt;a href=&quot;https://www.jstor.org/stable/2268971&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/sup&gt; in Columbus, Ohio in 1948?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;sup&gt;&lt;a href=&quot;https://www.jstor.org/stable/2268971&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/sup&gt; N. Bourbaki, &lt;a href=&quot;https://www.jstor.org/stable/2268971&quot; rel=&quot;nofollow noreferrer&quot;&gt;Foundations of mathematics for the working mathematician&lt;/a&gt;&lt;a href=&quot;https://doi.org/10.2307/2268971&quot; rel=&quot;nofollow noreferrer&quot;&gt; [DOI]&lt;/a&gt;, The Journal of Symbolic Logic, Volume 14, Number 1, March 1949.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="10890" LastEditorUserId="4251" LastEditDate="2022-09-08T06:24:17.470" LastActivityDate="2022-09-08T09:58:05.110" Title="How did 'N. Bourbaki' address a conference?" Tags="|mathematicians|bourbaki|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14777" PostTypeId="2" ParentId="14776" CreationDate="2022-09-06T09:21:56.617" Score="15" Body="&lt;p&gt;&lt;em&gt;It&lt;/em&gt; did not deliver the 1948 address to the Association for Symbolic Logic at Columbus, Ohio, obviously. Weil read it for &lt;em&gt;it&lt;/em&gt;, as we learn from &lt;a href=&quot;https://www.jstor.org/stable/2269014&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zorn's report about the meeting&lt;/a&gt; in the same issue of The Journal of Symbolic Logic linked in the OP:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There were two invited hour addresses: N. Bourbaki, Foundations of mathematics for the working mathematician, and F. B. Fitch, Towards a demonstrably consistent mathematics. The first address was read by Andre Weil on Friday morning, Saunders MacLane presiding; the second was given on Friday afternoon, with A. A. Bennett as presiding officer.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That MacLane was a close Bourbaki associate (without formally joining) helped, and the consonance with the title of &lt;a href=&quot;https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacLane's famous book of 1971&lt;/a&gt; is peculiar. But this was not the first or the last stunt Weil and co. performed on Bourbaki's behalf, see &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7731645/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Barany, Impersonation and personification in mid-twentieth century mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Because Bourbaki could not travel, write, or speak for himself, his personal manifestations relied upon impersonations. The self-styled “collaborators” of Bourbaki jointly prepared authoritative texts and joined with others to represent Bourbaki in a variety of contexts, often teasing or transgressing disciplinary norms as they did so. Relishing myth and misdirection, Bourbaki’s collaborators propounded his pseudonymous authorship as an open secret around which to establish or evade various forms of credit and responsibility in and beyond the mathematics profession.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Back in 1935 Weil engineered Bourbaki's first publication with a distinct flavor of a prank, complete with a made up country of origin:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Under cover of cover letter, André Weil enlisted Élie Cartan, member of the Academy of Sciences and father of Bourbaki collaborator Henri Cartan, to communicate Bourbaki’s first mathematical publication to the Academy’s Comptes Rendus in 1935. According to Weil, Cartan was a witting accomplice, going so far as to consult with fellow academicians at a regular lunch group before endorsing the article Weil attributed to Bourbaki.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Sending up the presumed familiarity shared within a scholarly community, Weil introduced Bourbaki in the cover letter with a knowing “Vous n’ignorez pas que. . .” – “You are not unaware,” or “As you know.” He situated the wayward professor at a café in the Paris suburb of Clichy, where he would fit stereotypically as a refugee from the vaguely Eastern-European invented country of Poldavia, recently “wiped off the map of Europe” by world events.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1937 Weil got some of the younger mathematicians at Princeton in on the ruse, including Smithies, Boas, and Tukey. Boas later gave Bourbaki’s textbooks the pride of place in Mathematical Reviews, where he was a contributor, but mentioned it was a pseudonym in some reviews. By that time, the group did not care much for hiding the fakery. Still, Boas and Tukey considered getting Bourbaki into  American Men of Science in 1942.&lt;/p&gt;&#xA;&lt;p&gt;Aside from the fake address, Bourbaki (Weil) &amp;quot;applied&amp;quot; to the AMS in 1948, and was rejected. But by 1949 Dieudonné took over as the chief prankster. Under his tutelage, Bourbaki &amp;quot;joined&amp;quot; the &lt;em&gt;Société Mathématique de France&lt;/em&gt; and was eligible to join the AMS as well, under the 1946 reciprocity agreement. It took Kline, the AMS secretary, considerable ingenuity to find reasons to reject him again this time (biographical discrepancies between Weil's and Dieudonné's applications helped). The new application was handwritten no less, most likely, by Dieudonné himself:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Dieudonné’s Bourbaki was born in Cucuteni, Moldavia, on 12 December 1886, elected to the Royal Academy of Poldavia in 1917 and the SMF in 1949, and presently employed at Dieudonné’s institute in Nancy. His 1910 doctorate came from Kharkov University, a real institution but sufficiently remote as to be unverifiable. Previous employers included the real Dorpat University, in Estonia, and the fictional Zorngahr College, an apparent portmanteau of German algebraist Max Zorn and Weil’s first employer in Aligarh.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Most of the biographical claims were, in practice, unverifiable. Kline could neither write to far-off universities or possibly defunct academies, nor assert with incontestable authority that they did not exist – notwithstanding his justified certainty that the claims were invented. Bourbaki’s Rockefeller Fellowship, however, could be easily assayed from within Kline’s own professional network. He wrote to the foundation’s Warren Weaver on the pretense of checking Bourbaki’s fellowship history, but really as a chance to vent his frustration over the undignified behavior of the French upstarts. Even Bourbaki’s discrepant signatures earned a remark, having deteriorated from that “of a determined man” in the first application to a “cramped” and “infantile” one.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hille, a recent AMS president, still advised Kline to join in on the joke and accept Bourbaki, and he might have been right. Bourbaki (Dieudonné) did not take the no lying down, and responded with &amp;quot;indignation&amp;quot; that by the reciprocity agreement &amp;quot;&lt;em&gt;A.M.S. had no right to scrutinize&lt;/em&gt;&amp;quot; the application. As Hille predicted, &amp;quot;&lt;em&gt;standing on our dignity will not get us anywhere and may expose us to ridicule&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The AMS had to pretend to take the application as a sincere attempt to gain membership from an individual in order to challenge the premise that Bourbaki should be eligible. That is, they rejected the application because they knew it to be a farce, but their procedure for this rejection depended on an elaborate performance of believing the application to be genuine.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-09-08T09:58:05.110" LastActivityDate="2022-09-08T09:58:05.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14778" PostTypeId="1" AcceptedAnswerId="14779" CreationDate="2022-09-06T15:48:55.083" Score="4" ViewCount="225" Body="&lt;p&gt;The following well-known formula for pythagorean triples is commonly called Euclid's formula:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$a, b, c$&lt;/span&gt; are three natural numbers with &lt;span class=&quot;math-container&quot;&gt;$a,c$&lt;/span&gt; odd, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; even, &lt;span class=&quot;math-container&quot;&gt;$\gcd(a,b,c)=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a^2+b^2=c^2$&lt;/span&gt;, then there exist &lt;span class=&quot;math-container&quot;&gt;$n,m$&lt;/span&gt; natural, coprime, and with exactly one of them odd, such that &lt;span class=&quot;math-container&quot;&gt;$a=m^2-n^2,\ b=2mn,\ c=m^2+n^2$&lt;/span&gt;. Furthermore, For any choice of &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; satisfying the above conditions, a triple &lt;span class=&quot;math-container&quot;&gt;$a, b, c$&lt;/span&gt; satisfying the above conditions is generated by these formulae.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Euclid in fact discusses this in lemma 1 to proposition 29, book X of &lt;em&gt;The Elements&lt;/em&gt;. When one translates the geometric argument Euclid gives there to algebra, the formula above is indeed recovered. However, what is also immediately apparent from Euclid's proof is that he only proves &lt;em&gt;sufficiency&lt;/em&gt; of the formula, not &lt;em&gt;necessity&lt;/em&gt; - i.e. he only proves that given &lt;span class=&quot;math-container&quot;&gt;$m,n$&lt;/span&gt; as above, a pythagorean triple is indeed generated.&lt;/p&gt;&#xA;&lt;p&gt;Who first proved the necessity of Euclid's formula? I could not find the answer anywhere, most sources seem to simply credit Euclid for the whole theorem. I'm also interested in what proof was used to prove necessity - I can find two common proofs, one using rational points on the unit circle, and one using just algebraic manipulation and a lemma (which follows from the FTA) that two coprime numbers whose product is a square, are squares. Which of these proofs came first?&lt;/p&gt;&#xA;" OwnerUserId="11209" LastEditorUserId="11209" LastEditDate="2022-09-06T16:22:20.503" LastActivityDate="2022-09-10T07:30:38.377" Title="Who first proved necessity of Euclid's formula for pythagorean triples?" Tags="|mathematics|number-theory|euclid|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14779" PostTypeId="2" ParentId="14778" CreationDate="2022-09-06T21:37:21.423" Score="3" Body="&lt;p&gt;Nobody. This happens quite often, one author does something that does not measure up to modern specifications of &amp;quot;proving X&amp;quot;, but, after a succession of such authors, the result is trivial when it is finally &amp;quot;proved&amp;quot;. So nobody gets the credit.&lt;/p&gt;&#xA;&lt;p&gt;In this case, the idea of rational parametrization goes back to &lt;a href=&quot;https://en.wikipedia.org/wiki/Diophantus_II.VIII&quot; rel=&quot;nofollow noreferrer&quot;&gt;problem &lt;span class=&quot;math-container&quot;&gt;$II.8$&lt;/span&gt; of Diophantus's Arithmetica&lt;/a&gt;:&amp;quot;&lt;em&gt;Partition a given square into two squares&lt;/em&gt;&amp;quot;. Kind of. Diophantus does not substitute &lt;span class=&quot;math-container&quot;&gt;$y=tx-a$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$y^2=a^2-x^2$&lt;/span&gt; with parameter &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;. He only substitutes &lt;span class=&quot;math-container&quot;&gt;$y=2x-4$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$y^2=16-x^2$&lt;/span&gt;, but makes clear that &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt; is just an example. In &lt;span class=&quot;math-container&quot;&gt;$III.19$&lt;/span&gt; this is referred to as &amp;quot;&lt;em&gt;We have learned how to decompose the given square into two squares in an infinite number of ways&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In modern times, people sometimes describe &lt;span class=&quot;math-container&quot;&gt;$II.8$&lt;/span&gt; as &amp;quot;rational parametrization&amp;quot;, see e.g. &lt;a href=&quot;https://math.stackexchange.com/q/210190/152568&quot;&gt;History of the study of rational points on the circle&lt;/a&gt;. Abu Kamil and later Fermat, with the advantage of 17th century algebraization, extracted something like that from there, at least as far as finding rational points (&lt;span class=&quot;math-container&quot;&gt;$II.8$&lt;/span&gt; is where Fermat's famous remark on the margins appears). For more on what happened in between see &lt;a href=&quot;https://bookstore.ams.org/dol-54&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bashmakova, Diophantus and Diophantine Equations&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As for explicit parametrization, it came much later. According to &lt;a href=&quot;https://arxiv.org/abs/1108.6219&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lemmermeyer's Parametrizing Algebraic Curves&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The first known geometric parametrization of an algebraic curve is due to Newton, and is contained in an article published only in 1971. The nowadays ubiquitous parametrization of the unit circle first appeared at the beginning of the 20th century in textbooks such as Kronecker's.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Fermat was not big on proving and considered finding rational solutions to quadratic Diophantine equations (like the Pythagorean or the Pell equation) a triviality. Kronecker would not claim credit for Pythagorean triples either. But Diophantus had no clue about parametrizations or rational points, and showed little interest in exhausting &lt;em&gt;all&lt;/em&gt; solutions to Diophantine equations altogether, see &lt;a href=&quot;https://irma.math.unistra.fr/%7Eschappa/NSch/Publications_files/1998cBis_Dioph.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schappacher, Diophantus of Alexandria: a Text and its History&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The fact that other linear substitutions &lt;span class=&quot;math-container&quot;&gt;$y = kx−4$&lt;/span&gt; would have yielded a solution to the original problem as well, tends not to be mentioned. Nor, to be sure, does he ever show any indication of being aware of the geometric interpretation of this or similar problems as that&#xA;of finding rational points on a circle of rational radius, and of his operations, as intersecting the circle with the line &lt;span class=&quot;math-container&quot;&gt;$y = 2x − 4$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Such a forming of right triangles from two numbers is what is treated nowadays — apparently as a late echo of a historical attribution made in the fifth century AD by Proclus — under the name of ‘pythagorean triples.’ This basic technique is used by Diophantus in many places, in particular in the problems of book “VI”... But note&#xA;that it is not evident from the text of the Arithmetica, whether Diophantus knew that all right triangles with rational sides can be obtained in this way&#xA;(a fact which we ‘simply’ like to see today as the parametrization of the rational points on the unit circle via stereographic projection from a chosen rational point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;: the rational points on the unit circle are the second points of intersections with the circle of lines with rational slope passing through &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;). This kind of statement, aiming at exhausting all solutions of a problem,&#xA;seems in fact alien to the very style of Diophantus’s problems.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-09-10T07:30:38.377" LastActivityDate="2022-09-10T07:30:38.377" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14780" PostTypeId="2" ParentId="12228" CreationDate="2022-09-06T21:53:35.733" Score="1" Body="&lt;p&gt;Because the symmetrical Adinkra symbols used by the Asante people of Africa have the intent of visualizing the indescribable some and also vaguely resemble some of his graphical supersymmetry representations. From the actual research paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The use of symbols to connote ideas which defy simple verbalization is perhaps one&#xA;of the oldest of human traditions. The Asante people of West Africa have long been&#xA;accustomed to using simple yet elegant motifs known as Adinkra symbols, to serve&#xA;just this purpose. With a nod to this tradition, we christen our graphical symbols as&#xA;“Adinkras.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17216" LastEditorUserId="17216" LastEditDate="2022-09-06T22:11:37.250" LastActivityDate="2022-09-06T22:11:37.250" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14781" PostTypeId="1" CreationDate="2022-09-07T03:08:01.733" Score="0" ViewCount="55" Body="&lt;p&gt;My friend speculated that people in the past believed that the gallbladder has therapeutic properties because of its distinctive color among surrounding meat - greenish among red.&#xA;Maybe our ancestors thought it was where our souls resided something similar to Ancient Egyptians' belief that the heart is far more important than the brain.&lt;/p&gt;&#xA;&lt;p&gt;My question is, how do we begin to research or collect pieces of evidence for this? What keyword should we use? What field of science? Is it archeopsychology?&lt;/p&gt;&#xA;" OwnerUserId="17217" LastEditorUserId="131" LastEditDate="2022-09-07T08:49:00.907" LastActivityDate="2022-09-07T08:49:00.907" Title="What branch of science is it that predicts how people in the past thought?" Tags="|resource-recommendation|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14783" PostTypeId="2" ParentId="13467" CreationDate="2022-09-07T16:50:59.010" Score="1" Body="&lt;p&gt;From &lt;em&gt;Numbers&lt;/em&gt; by Ebbinghaus, et al.:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hamilton believed that his quaternions would play a key role in physics.  With missionary zeal he strove to get them accepted by the mathematical world. [...] In Ireland and England, Hamilton became the figurehead of a school of &amp;quot;quaternionists&amp;quot; who &amp;quot;outdid their master in intolerance and rigidity.&amp;quot; At the center stood a mystic formalism treated with due reverence by the initiated.  One dreamt of a quaternionistic theory of functions and expected to gain new and profound insights into the whole realm of mathematics. To promote these utopian aims there was even founded, in 1895, an &amp;quot;International Association for promoting the study of quaternions,&amp;quot; at Yale University in New Haven, Connecticut.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, such enthusiasm was not universal even in Britain.  Thomson (Lord Kelvin), commented caustically:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So what happened to quaternions?  Their significance was greatly diminished when it was realised that the quaternion algebra was only a particular algebra of complex &lt;span class=&quot;math-container&quot;&gt;$2 \times 2$&lt;/span&gt; matrices.&lt;/p&gt;&#xA;&lt;p&gt;A similar fate awaited the flood of hypercomplex systems which inundated the whole of algebra in the second half of the nineteenth century. The realisation that there are far fewer interesting &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;-algebras than one might expect began with Frobenius' isomorphism theorem, which showed that there are only three isomorphically distinct real finite-dimensional associative division algebras - namely &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; itself, &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathbb H$&lt;/span&gt;.  This was followed by the isomorphism theorem of Mazur and Gelfand (building on the work of Hopf) which showed that &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbb C$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$\mathbb H$&lt;/span&gt; are the only Banach division algebras.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2022-09-07T16:50:59.010" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14784" PostTypeId="2" ParentId="7643" CreationDate="2022-09-07T19:55:44.493" Score="1" Body="&lt;p&gt;Galois wrote his famous testamentary letter on 29 May 1832, one day before the duel, to his friend Auguste Chevalier. At the end of the letter he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You will publicly ask Jacobi or Gauss to give their opinion not on&#xA;the truth but on the importance of the theorems.&#xA;After that there will, I hope, be people who will find profit in deciphering all this mess.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Chevalier has never asked Jacobi or Gauss publicly. Nothing about Galois' theory has been found in Jacobi or Gauss' possessions after their deaths.&lt;/p&gt;&#xA;" OwnerUserId="17222" LastActivityDate="2022-09-07T19:55:44.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14785" PostTypeId="2" ParentId="5316" CreationDate="2022-09-07T20:15:36.307" Score="0" Body="&lt;p&gt;The problem with the books of Bell, Infeld or Rigatelli is, as remarked by  Rothman's &amp;quot;Genius and Biographers: the Fictionalization of Evariste Galois&amp;quot;, you can't trust the details in the narrative; they are often invented, by the author or by someone else.&lt;/p&gt;&#xA;&lt;p&gt;Peter Neumann's &amp;quot;The Mathematical Writings of Évariste Galois&amp;quot; is the best one in English for Galois' writing.&lt;/p&gt;&#xA;&lt;p&gt;The dozen pages &amp;quot;Historical Introduction/The Life of Galois&amp;quot; in Ian Stewart's &amp;quot;Galois Theory (fourth edition)&amp;quot; is the best biography of Galois in English that I have ever read: quite long, no invention of fictional details, no fact errors.&lt;/p&gt;&#xA;" OwnerUserId="17222" LastActivityDate="2022-09-07T20:15:36.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14786" PostTypeId="2" ParentId="317" CreationDate="2022-09-07T20:39:42.403" Score="1" Body="&lt;p&gt;As a Jew, Hungarian mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/P%C3%A1l_Tur%C3%A1n&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pál Turán&lt;/a&gt; was forced to work in a brick factory during WW2. He asked a problem now known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_brick_factory_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turán's brick factory problem&lt;/a&gt; which is considered as creation of a new field in combinatorics and graph theory: &lt;a href=&quot;https://en.wikipedia.org/wiki/Extremal_combinatorics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Extremal combinatorics&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17222" LastActivityDate="2022-09-07T20:39:42.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14787" PostTypeId="1" CreationDate="2022-09-08T03:58:19.153" Score="1" ViewCount="169" Body="&lt;p&gt;We all know how to use a protractor, it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch.&lt;/p&gt;&#xA;&lt;p&gt;For instance, was the understanding of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and a compass first required before the first protractor, and if so how can I draw a full protractor on paper with just a compass, a ruler and some understanding of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;I guess my point is as follows. If we can draw a semi-circle on paper, then how can we fill up the degrees without the help of a protractor?&lt;/p&gt;&#xA;" OwnerUserId="17224" LastEditorUserId="4251" LastEditDate="2022-09-08T08:56:24.670" LastActivityDate="2022-09-09T13:42:52.807" Title="How to build a protractor without a protractor?" Tags="|mathematics|geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14789" PostTypeId="2" ParentId="13373" CreationDate="2022-09-08T07:20:46.557" Score="1" Body="&lt;p&gt;There is already some pro-Poisson remarks in Peter Neumann’s “The writings of Evariste Galois”.&lt;/p&gt;&#xA;&lt;p&gt;One is by Neumann himself:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;With hindsight one may feel that this report was wrong. But I cannot think&#xA;so: it seems to me to be a model of good refereeing. Can any of us be sure&#xA;that in an analogous situation today we would react differently? I doubt it: it is an admirable report, sympathetic but firm. All that is wrong with it is that it deals with the work of an exceptionally brilliant and awkward man. Galois had no research supervisor who might have shown him how his discoveries should be properly written up. Besides, Galois was not aman who took advice easily. Another young mathematician might have taken the criticism to heart, re-written his work, published it and become famous. Galois took offence, returned to political agitation, died young and became famous.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another one by Jacques Tits:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is it shameful to admit that even today, having to judge the memoir in the&#xA;form that Galois had given it, I would be pretty close to joining myself with the opinion expressed by Poisson?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17222" LastActivityDate="2022-09-08T07:20:46.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14790" PostTypeId="2" ParentId="14787" CreationDate="2022-09-08T13:10:41.970" Score="5" Body="&lt;p&gt;Until the middle of 18th century, division of the circle (when high precision was required) was performed by compass,&#xA;by starting with a regular pentagon or hexagon, and then performing bisections, additions and subtractions of arcs. This was a normal procedure of manufacturing astronomical instruments. Only in 18th century a division engine was invented, which &lt;a href=&quot;https://en.wikipedia.org/wiki/Dividing_engine&quot; rel=&quot;nofollow noreferrer&quot;&gt;performed the task mechanically&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;For example, in the middle of 18 century they were able to divide an arc of radius 2-12 inches into minutes, so that the overall error in measuring any was less than 1 minute. By hands, and using only compass.&lt;/p&gt;&#xA;&lt;p&gt;One cannot divide a circle into 360 equal parts exactly, using a compass only, so 1 degree angle could be obtained only approximately. But with any requires accuracy.&lt;/p&gt;&#xA;&lt;p&gt;This fact was known experimentally, and proved only in 19th century: using a compass, you can divide the circle into &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; equal parts, if and only if &lt;span class=&quot;math-container&quot;&gt;$n=2^kp_1p_2,...$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$p_k$&lt;/span&gt; are distinct primes of the form &lt;span class=&quot;math-container&quot;&gt;$2^{2^m}+1$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$360=2^3.3.3.5$&lt;/span&gt;, one cannot divide the circle into 360 parts  exactly.&lt;/p&gt;&#xA;&lt;p&gt;One does not need &amp;quot;the knowledge of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;&amp;quot; (whatever this can mean) for this procedure. Everything is based on simple geometry, and the use of compass. Division of a straight ruler into equal parts by a compass is possible for any number of parts. But from the practical point of view, it is of the same difficulty as for a circle.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-09-09T13:42:52.807" LastActivityDate="2022-09-09T13:42:52.807" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14791" PostTypeId="1" CreationDate="2022-09-09T15:38:00.107" Score="4" ViewCount="157" Body="&lt;p&gt;The maximum eigenvalue of a real symmetric (or complex Hermitian) matrix is given as the maximum of the associated the quadratic form:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lambda_{\rm max}(A) = \max_{\|x\| = 1} x^*Ax. \tag{1}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This result is often attributed to Rayleigh and Ritz (for example, in &lt;a href=&quot;https://link.springer.com/book/10.1007/978-1-4614-1099-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fuzhen Zhang's &lt;em&gt;Matrix Theory&lt;/em&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;But was this result actually originally proven by either Rayleigh or Ritz? In &lt;a href=&quot;https://www.elsevier.com/books/matrix-perturbation-theory/stewart/978-0-08-092613-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stewart and Sun's &lt;em&gt;Matrix Perturbation Theory&lt;/em&gt;&lt;/a&gt;, the authors write&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Both Rayleigh and Ritz were concerned with approximating the eigenvalues of an infinite operator by replacing it by a matrix eigenvalue problem... Neither gave formal justification for his method.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Stewart and Sun referenced work by Rayleigh in 1899 and Ritz in 1909. Fischer's theorem, which contains the &amp;quot;Rayleigh–Ritz theorem&amp;quot; (1) as a special case, was proven in 1905, four years earlier than the work of Ritz cited in Stewart and Sun.&lt;/p&gt;&#xA;&lt;p&gt;There are two separate but related ideas attributed to Rayleigh and Ritz:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The &lt;em&gt;numerical method&lt;/em&gt; of approximating extremal eigenvalues of a matrix/operator by maximizing the quadratic form (or Rayleigh quotient) over vectors from a subspace.&lt;/li&gt;&#xA;&lt;li&gt;The &lt;em&gt;variational principle&lt;/em&gt; for extremal eigenvalues. That is, the truth of equation (1) as a theorem of mathematics.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Based on the account in Stewart and Sun, it seems like that Rayleigh–Ritz are correctly attributed for developing idea 1, but idea 2 seems more properly to be attributed to Fischer, at least as a rigorous mathematical theorem. At the very least, it seems Ritz has no claim to originating equation (1) as their published work on the matter followed that of Fischer.&lt;/p&gt;&#xA;&lt;p&gt;In summary, my question is the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Who was the first to establish equation (1), either informally or formally?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17228" LastActivityDate="2022-09-09T15:38:00.107" Title="Did Rayleigh or Ritz prove the Rayleigh–Ritz theorem?" Tags="|mathematics|mathematical-physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14792" PostTypeId="1" CreationDate="2022-09-11T11:32:23.690" Score="2" ViewCount="202" Body="&lt;p&gt;If I understand correctly, &lt;em&gt;Arithmetices principia: nova methodo exposita&lt;/em&gt; is one of the most important works covering the axiomatisation of arithmetic.&#xA;I was surprised that no translation into English (or any other modern language for the matter) was available.&lt;/p&gt;&#xA;&lt;p&gt;I just found on Github someone who translated part of it but that's it.&lt;/p&gt;&#xA;&lt;p&gt;Did I miss something?&lt;/p&gt;&#xA;" OwnerUserId="17231" LastEditorUserId="17231" LastEditDate="2022-09-11T11:33:28.420" LastActivityDate="2022-09-26T11:00:48.177" Title="Is there a complete translation of &quot;Arithmetices principia&quot; by Peano?" Tags="|translations|arithmetic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14794" PostTypeId="1" CreationDate="2022-09-11T16:30:00.673" Score="2" ViewCount="119" Body="&lt;p&gt;What was Kolmogorov's attitude regarding the frequentist versus Bayesian statistics controversies? Did he ever write or speak about his own views on Fisher or de Finetti, Jeffreys, etc.? Or were those times so politically charged that it was wiser for him not to get involved with words/concepts such as &amp;quot;subjective&amp;quot; or &amp;quot;objective&amp;quot;, etc?&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="4251" LastEditDate="2022-09-11T16:46:41.920" LastActivityDate="2022-09-11T16:46:41.920" Title="Kolmogorov on frequentists versus Bayesians" Tags="|mathematicians|statistics|probability|kolmogorov|soviet-union|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14795" PostTypeId="1" AcceptedAnswerId="14799" CreationDate="2022-09-12T03:20:39.063" Score="1" ViewCount="398" Body="&lt;p&gt;I am new to this portion of stack exchange and am not sure if this type of question is allowed, but after seeing &lt;a href=&quot;https://hsm.stackexchange.com/questions/13869/what-was-richard-courants-saying-about-mathematical-research-apart-from-applica&quot;&gt;this&lt;/a&gt;, I assume it is.&lt;/p&gt;&#xA;&lt;p&gt;A while back, I remember reading a quote about math that said something along the lines of “Before you know the answer, it is impossible; once you do, it’s trivial”. I think the quote even used the terms “trivial” and “impossible”, but I can’t find it again.&lt;/p&gt;&#xA;&lt;p&gt;The closest I’ve found is “In mathematics, there are only two kinds of proofs: Trivial ones, and undiscovered ones”, but that’s not it as the actual quote used “impossible”.&lt;/p&gt;&#xA;" OwnerUserId="17232" LastActivityDate="2022-09-12T19:05:57.040" Title="What was the quote about known things being trivial and unknown things being impossible?" Tags="|quote|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14796" PostTypeId="1" CreationDate="2022-09-12T12:08:49.757" Score="3" ViewCount="50" Body="&lt;p&gt;I'm interested in kinetic theory of gases, notably its history.&lt;/p&gt;&#xA;&lt;p&gt;I know that, after the pioneering work of Boltzmann who derived the Boltzmann equation, it is possible by taking its statistical moment to obtain a set of equations that are closely linked to the Navier-Stokes equation.&lt;/p&gt;&#xA;&lt;p&gt;This system can be closed either by doing a Hilbert expansion, Chapman-Enskog expansion, Grad's 13 moments and so on ...&lt;/p&gt;&#xA;&lt;p&gt;However, I've not encountered WHO obtained this unclosed system. Who did plug and write the set of equations obtained from taking zeroth, first and second moment of the distribution function to obtain a Navier-Stokes like system ?&lt;/p&gt;&#xA;&lt;p&gt;Anybody got a source or informations about it ?&lt;/p&gt;&#xA;" OwnerUserId="17148" LastActivityDate="2022-09-12T12:08:49.757" Title="Unclosed Macroscopic equation from the statistical moments in kinetic theory of gases" Tags="|theoretical-physics|boltzmann|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14797" PostTypeId="1" CreationDate="2022-09-12T14:04:00.707" Score="1" ViewCount="67" Body="&lt;h1&gt;How was the L, LT, ST coordinate system for Metals developed and adopted?&lt;/h1&gt;&#xA;&lt;p&gt;I am talking with some colleagues about comparing metallic and non-metallic materials. Obviously, &amp;quot;apples-to-apples&amp;quot; comparisons need to happen and that naturally means you need to equate certain directions in different coordinate systems.&lt;/p&gt;&#xA;&lt;p&gt;This got me thinking: &amp;quot;how did MMPDS settle on L, LT, ST?&amp;quot; Was there resistance to forgings and extrusions being subjected to the same coordinate system (especially when you are looking at cylindrical vs non-cylindrical forms)? How were arguments against placing these various forms together in a single coordinate system resolved? Is there any documentation about the process that I could access, such as meeting notes?&lt;/p&gt;&#xA;&lt;h1&gt;Some Background&lt;/h1&gt;&#xA;&lt;p&gt;Materials' properties can vary with direction - this is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Anisotropy&quot; rel=&quot;nofollow noreferrer&quot;&gt;anisotropy&lt;/a&gt;. Various kinds of materials have &amp;quot;industry standard&amp;quot; coordinate systems so people know which directions people are talking about. For metals within aerospace engineering, this is the &lt;a href=&quot;https://mmpds.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;M&lt;/strong&gt;etallic &lt;strong&gt;M&lt;/strong&gt;aterials &lt;strong&gt;P&lt;/strong&gt;roperties &lt;strong&gt;D&lt;/strong&gt;evelopment and &lt;strong&gt;S&lt;/strong&gt;tandardization organization&lt;/a&gt;.  This group has determined (somehow?) that most directional properties can be described using their coordinate system, the orthogonally placed &lt;strong&gt;L&lt;/strong&gt;ongitudinal (or &lt;em&gt;Long&lt;/em&gt;), &lt;strong&gt;L&lt;/strong&gt;ong &lt;strong&gt;T&lt;/strong&gt;ransverse, and &lt;strong&gt;S&lt;/strong&gt;hort &lt;strong&gt;T&lt;/strong&gt;ransverse directions. This allows for end products of varying metal-forming processes to be compared to each other: extruded vs cast vs forged etc. This sometimes, but not always, corresponds to grain directions. These grain directions, otherwise known as crystal orientations, are &lt;em&gt;important&lt;/em&gt; to the materials behavior.&lt;/p&gt;&#xA;" OwnerUserId="17235" LastEditorUserId="17235" LastEditDate="2022-09-19T14:00:54.207" LastActivityDate="2022-09-19T14:00:54.207" Title="How did MMPDS Develop Their Coordinate System for Metals?" Tags="|engineering|materials-science|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14798" PostTypeId="1" AcceptedAnswerId="14907" CreationDate="2022-09-12T17:52:51.183" Score="3" ViewCount="327" Body="&lt;p&gt;In Galileo's &lt;em&gt;Two New Sciences&lt;/em&gt;, he describes an experiment demonstrating pendulum motion and how the pendulum will rise to the same height from where it started its fall.  This discussion can be found just after Fig. 45 in this &lt;a href=&quot;https://oll.libertyfund.org/title/galilei-dialogues-concerning-two-new-sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;source&lt;/a&gt;.  He describes the pendulum as acquiring an &lt;em&gt;impeto&lt;/em&gt;, then later switches to &lt;em&gt;mometo&lt;/em&gt;, then returns back to using &lt;em&gt;impeto&lt;/em&gt;.  For example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...from this we may rightly infer that the ball in its descent&#xA;through the arc CB acquired a momentum [&lt;strong&gt;impeto&lt;/strong&gt;] on reaching B, which&#xA;was just sufficient to carry it through a similar arc BD to the same&#xA;height.&#xA;...&#xA;From this we can see what can be done by the same momentum [&lt;strong&gt;impeto&lt;/strong&gt;]&#xA;which previously starting at the same point B carried the same body&#xA;through the arc BD to the horizontal CD.&lt;/p&gt;&#xA;&lt;p&gt;...for since the two arcs CB and DB are equal and similarly placed,&#xA;the momentum [&lt;strong&gt;momento&lt;/strong&gt;] acquired by the fall through the arc CB is the&#xA;same as that gained by fall through the arc DB; but the momentum&#xA;[&lt;strong&gt;momento&lt;/strong&gt;] acquired at B, owing to fall through CB, is able to lift the&#xA;same body [mobile] through the arc BD; therefore, the momentum [&lt;strong&gt;momento&lt;/strong&gt;]&#xA;acquired in the fall BD is equal to that which lifts the same body&#xA;through the same arc from B to D; so, in general, every momentum [&lt;strong&gt;momento&lt;/strong&gt;]&#xA;acquired by fall through an arc is equal to that which can lift the&#xA;same body through the same arc. But all these momenta [&lt;strong&gt;momenti&lt;/strong&gt;] which&#xA;cause a rise through the arcs BD, BG, and BI are equal, since they are&#xA;produced by the same momentum [&lt;strong&gt;momento&lt;/strong&gt;], gained by fall through CB, as&#xA;experiment shows. Therefore all the momenta [&lt;strong&gt;momenti&lt;/strong&gt;] gained by fall through the&#xA;arcs DB, GB, IB are equal.&lt;/p&gt;&#xA;&lt;p&gt;...although the above experiment shows us that the descent of the&#xA;moving body through the arc CB confers upon it momentum [&lt;strong&gt;momento&lt;/strong&gt;] just&#xA;sufficient to carry it to the same height through any of the arcs BD,&#xA;BG, BI, we are not able, by similar means, to show that the event&#xA;would be identical in the case of a perfectly round ball descending&#xA;along planes whose inclinations are respectively the same as the&#xA;chords of these arcs.&lt;/p&gt;&#xA;&lt;p&gt;...In striking these planes some of its momentum [&lt;strong&gt;impeto&lt;/strong&gt;] will be lost&#xA;and it will not be able to rise to the height of the line CD; but this&#xA;obstacle, which interferes with the experiment, once removed, it is&#xA;clear that the momentum [&lt;strong&gt;impeto&lt;/strong&gt;] (which gains in strength with&#xA;descent) will be able to carry the body to the same height.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Most translations I've seen (like the one above) translate both &lt;em&gt;impeto&lt;/em&gt; and &lt;em&gt;momento&lt;/em&gt; as being &amp;quot;momentum,&amp;quot; but during his time, did Galileo believe &lt;em&gt;impeto&lt;/em&gt; to be different from &lt;em&gt;momento&lt;/em&gt;?  If so, in what way were they different?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I understand &lt;em&gt;impeto&lt;/em&gt; to be impetus (the driving &amp;quot;force&amp;quot; of an object in motion), but I'm having difficulty finding support for &lt;em&gt;momento&lt;/em&gt; since Galileo never defines it in his text.  I've found that several articles that briefly mention &lt;em&gt;momento&lt;/em&gt;, but they draw no comparison to his use of impetus.  For example, in the Stanford Encyclopedia of Philosophy for &lt;a href=&quot;https://plato.stanford.edu/entries/galileo/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galileo&lt;/a&gt;, it states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A while later, in his 1600 manuscript version of Le Meccaniche (On&#xA;Mechanics), Galileo introduced the concept of &lt;em&gt;momento&lt;/em&gt;, a quasi force&#xA;that applies to a body at a moment, and which is somehow proportional&#xA;to weight or specific gravity (Galluzzi 1979). Still, he had no good&#xA;way to measure or compare specific gravities of bodies of different&#xA;kinds, and his notebooks during this early seventeenth-century period&#xA;reflect his trying again and again to find a way to bring all matter&#xA;under a single proportional measuring scale. He tried to study&#xA;acceleration along an inclined plane and to find a way to think of&#xA;what changes acceleration brings to &lt;em&gt;momento&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the above passage, it's interesting to note that most of the articles I've come across always cite Italian historian Paolo Galluzzi and his 1979 book &lt;em&gt;momento&lt;/em&gt;, which might shine some light on this question; however, it seems the book only exists in Italian (I'm not fluent enough for it to be practical).&lt;/p&gt;&#xA;&lt;p&gt;Speaking of Le Meccanich, I came across another related &lt;a href=&quot;https://hsm.stackexchange.com/questions/769/when-and-by-whom-was-the-term-momentum-introduced&quot;&gt;post&lt;/a&gt; about momentum where user Maruo ALLEGRANZA cites passages from Le Meccanich discussing &lt;em&gt;momento&lt;/em&gt;; however, the specific section (definitions [&lt;em&gt;diffinizioni&lt;/em&gt;]) that he quotes is conveniently &lt;a href=&quot;https://it.wikisource.org/wiki/Le_mecaniche/Diffinizioni&quot; rel=&quot;nofollow noreferrer&quot;&gt;missing&lt;/a&gt; but the rest of the text is &lt;a href=&quot;https://it.wikisource.org/wiki/Le_mecaniche&quot; rel=&quot;nofollow noreferrer&quot;&gt;functional&lt;/a&gt;.  The missing section makes it difficult to understand the context of the quoted passage.&lt;/p&gt;&#xA;&lt;p&gt;I also found that Galileo discusses &lt;em&gt;momento&lt;/em&gt; in his &lt;a href=&quot;https://www.gutenberg.org/files/37729/37729-h/37729-h.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Discorse of Floating Objects&lt;/a&gt; [&lt;em&gt;Discorso intorno alle cose che stanno in su l'acqua, o che in quella si muovono&lt;/em&gt;] where &lt;em&gt;momento&lt;/em&gt; is translated as &amp;quot;moment&amp;quot;, as seen in Definition V and Axioms I-III.  That led me to the history section of &lt;a href=&quot;https://en.wikipedia.org/wiki/Moment_(physics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;moment&lt;/a&gt; which suggests that it has origins stemming from &lt;em&gt;momento&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;That said, why was the word momentum chosen for the translation? One clue, according to Treccani, is that momento in Medieval Italy, the place the early translators lived, in a transferred sense meant both a &amp;quot;moment of time&amp;quot; and a &amp;quot;moment of weight&amp;quot; (a small amount of weight that turns the scale).&lt;/p&gt;&#xA;&lt;p&gt;In 1554, Francesco Maurolico clarifies the Latin term momentum in the&#xA;work Prologi sive sermones. Here is a Latin to English translation as&#xA;given by Marshall Clagett:&lt;/p&gt;&#xA;&lt;p&gt;[...] equal weights at unequal distances do not weigh equally, but&#xA;unequal weights [at these unequal distances may] weigh equally. For a&#xA;weight suspended at a greater distance is heavier, as is obvious in a&#xA;balance. Therefore, there exists a certain third kind of power or&#xA;third difference of magnitude—one that differs from both body and&#xA;weight—and this they call &lt;strong&gt;moment&lt;/strong&gt;.[c] Therefore, a body acquires weight&#xA;from both quantity [i.e., size] and quality [i.e., material], but a&#xA;weight receives its moment from the distance at which it is suspended.&#xA;Therefore, when distances are reciprocally proportional to weights,&#xA;the moments [of the weights] are equal, as Archimedes demonstrated in&#xA;The Book on Equal Moments.[d] Therefore, weights or [rather] moments&#xA;like other continuous quantities, are joined at some common terminus,&#xA;that is, at something common to both of them like the center of&#xA;weight, or at a point of equilibrium. Now the center of gravity in any&#xA;weight is that point which, no matter how often or whenever the body&#xA;is suspended, always inclines perpendicularly toward the universal&#xA;center.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In addition to body, weight, and moment, there is a certain fourth&#xA;power, which can be called impetus or force&lt;/strong&gt;.[e] Aristotle investigates&#xA;it in On Mechanical Questions, and it is completely different from&#xA;[the] three aforesaid [powers or magnitudes]. [...]&amp;quot; (emphasis added)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Again, I only find brief mentions that impetus and moment are different, but no elaboration.&lt;/p&gt;&#xA;&lt;p&gt;This whole thing has turned into one nasty rabbit hole full of nested sources, deadends, or missing/unavailable sources, and I'm ultimately left with a very nebulous understanding of &lt;em&gt;momento&lt;/em&gt; and what Galileo intended.  Considering what I've found in the above, it just leaves me confused how &amp;quot;moment&amp;quot; relates to Galileo's pendulum experiment.  If I assume he means &amp;quot;moment&amp;quot; as described in the moment wiki article (which is plausible since the pendulum is rotating about a fixed point), then it kind of makes sense, especially when he discusses levers in other sections of the book, but later on, it doesn't make sense at all.  For example, in &lt;em&gt;Two New Sciences&lt;/em&gt;, Galileo says,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Now seeing how great is the resistance which the air offers to the&#xA;slight momentum [&lt;strong&gt;momento&lt;/strong&gt;] of the bladder and how small that which it&#xA;offers to the large weight [peso] of the lead, I am convinced that, if&#xA;the medium were entirely removed, the advantage received by the&#xA;bladder would be so great and that coming to the lead so small that&#xA;their speeds would be equalized.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I (i.e., Simplicio) am one of those who accept the proposition, and believe that a&#xA;falling body acquires force [vires] in its descent, its velocity&#xA;increasing in proportion to the space, and that the momentum [&lt;strong&gt;momento&lt;/strong&gt;]&#xA;of the falling body is doubled when it falls from a doubled height&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The speed reaches a maximum along a vertical direction, and for other&#xA;directions diminishes as the plane diverges from the vertical.&#xA;Therefore the impetus, ability, energy, [l’impeto, il talento,&#xA;l’energia] or, one might say, the momentum [il &lt;strong&gt;momento&lt;/strong&gt;] of descent of&#xA;the moving body is diminished by the plane upon which it is supported&#xA;and along which it rolls.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;in which neither don't make sense in the context of &amp;quot;moment&amp;quot; since these are descriptions of linear motions.  I'm at a loss...&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-09-12T19:26:06.710" LastActivityDate="2023-09-28T03:14:12.853" Title="What's the difference between Galileo's &quot;impeto&quot; and &quot;momento&quot;?" Tags="|physics|terminology|galileo|definition|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14799" PostTypeId="2" ParentId="14795" CreationDate="2022-09-12T19:05:57.040" Score="2" Body="&lt;p&gt;I believe you're referring to this particular quote:&lt;/p&gt;&#xA;&lt;p&gt;“All of physics is either impossible or trivial. It is impossible until you understand it, and then it becomes trivial.” - Ernest Rutherford&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2022-09-12T19:05:57.040" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14800" PostTypeId="2" ParentId="14792" CreationDate="2022-09-12T20:00:01.053" Score="2" Body="&lt;p&gt;We see one complete English translation of Peano's &lt;em&gt;Latin&lt;/em&gt; text&#xA;&lt;a href=&quot;https://archive.org/details/arithmeticespri00peangoog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arithmetices principia: nova methodo&lt;/a&gt; – which is only 49 pages in its entirety, with just twenty pages numbered in:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;h3&gt;VII The principles of arithmetic, presented by a new method (1889)*&lt;/h3&gt;&#xA;&lt;p&gt;This remarkable little booklet (given here in its entirety) contains the first statement of Peano's best-known achievement, the postulates for the natural numbers. The essential ideas were first published by Dedekind, as Peano very generously points out in a later work, but there can be no doubt of the originality of Peano's work, and we have his own statement that he saw Dedekind's work only when &lt;em&gt;Arithmetices principia&lt;/em&gt; was going to the press.&lt;/p&gt;&#xA;&lt;p&gt;Following the preface, there is an introductory section on logical notation. In it we find, for the first time, the symbol E to indicate membership and J to indicate inclusion, and a note on the necessity to distinguish these two. The first section of the Arithmetices principia proper contains the famous five postulates, along with four others dealing with the sign =. These postulates were slightly modified in later writings.&lt;/p&gt;&#xA;&lt;p&gt;The bulk of the work is written in mathematical and logical symbols. The&#xA;preface and explanatory notes are in Latin, and the title is Latin, or almost - 'arithmetices' is transliterated Greek (the completely Latinized form would be 'arithmeticae').&lt;/p&gt;&#xA;&lt;p&gt;PREFACE&lt;/p&gt;&#xA;&lt;p&gt;Questions pertaining to the foundations of mathematics, although treated by many these days, still lack a satisfactory solution. The difficulty arises principally from the ambiguity of ordinary language. For this reason it is of the greatest concern to consider attentively the words we use. I resolved to do this, and am presenting in this paper the results of my study with applications to arithmetic. […]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;ending with&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;46'. &lt;span class=&quot;math-container&quot;&gt;$a \in \text K \text Q . \supset . \text E \text Ea = -(\text Ea \cup \text L \text Ea)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;— &amp;quot;Selected Works of Giuseppe Peano&amp;quot;, Translated and edited, with a biographical sketch and bibliography, by Hubert C. Kennedy, George Allen &amp;amp; Unwin: London, 1973, pp 101–134.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Please forgive the substandard ASCII-like rendering. It matches compared to the last lines in his original booklet:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9VZT7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9VZT7m.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6224" LastEditorUserId="131" LastEditDate="2022-09-26T11:00:48.177" LastActivityDate="2022-09-26T11:00:48.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14801" PostTypeId="1" AcceptedAnswerId="14809" CreationDate="2022-09-13T20:30:57.867" Score="3" ViewCount="154" Body="&lt;p&gt;Fermat's last theorem, which Euler had studied in the case of certain exponents, was only solved in the 1990s. Also, a counterexample to Euler's sum of powers conjecture has been found quite recently (1966). Apart from Goldbach's conjecture, are there any arithmetic problems of the like studied by Euler still open?&lt;/p&gt;&#xA;" OwnerDisplayName="user17244" LastEditorUserId="4251" LastEditDate="2022-09-19T05:41:13.090" LastActivityDate="2022-09-19T05:41:13.090" Title="Are there any arithmetic problems studied by Euler still open?" Tags="|mathematics|euler|arithmetic|open-problems|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14803" PostTypeId="1" AcceptedAnswerId="14806" CreationDate="2022-09-14T11:34:14.710" Score="3" ViewCount="130" Body="&lt;p&gt;A &lt;strong&gt;topology&lt;/strong&gt; is a pair of&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a nonempty set &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; of &lt;strong&gt;points&lt;/strong&gt;, and&lt;/li&gt;&#xA;&lt;li&gt;a set &lt;span class=&quot;math-container&quot;&gt;$Opens\subseteq 2^P$&lt;/span&gt; of &lt;strong&gt;open sets&lt;/strong&gt; that is closed under two closure conditions:&#xA;&lt;ol&gt;&#xA;&lt;li&gt;arbitrary (possibly infinite) unions and&lt;/li&gt;&#xA;&lt;li&gt;finite (possibly empty) intersections.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So far, so standard.  Please now consider dropping closure condition 2 above (closure under finite intersections).&lt;/p&gt;&#xA;&lt;p&gt;My question is: can anyone provide a reference or references for any early discussion of the notion of topological space without condition 2 above.  It was eventually included in the standard definition as we know, but does anyone know of references to any debate over whether to include it or not (much as there were discussions over, for example, the axioms of set theory).&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance.&lt;/p&gt;&#xA;" OwnerUserId="17248" LastActivityDate="2022-09-23T09:39:15.983" Title="Topologies without the axiom that finite intersection of open sets is open" Tags="|topology|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14804" PostTypeId="1" CreationDate="2022-09-14T14:25:40.793" Score="6" ViewCount="317" Body="&lt;p&gt;I'm a science/math journalist [ger] and currently I'm working on an article about the culture of prize money/bounties for solving open mathematical problems (Millennium Prize Problems and such). One section will be about the history of such bounties for which I'd like to ask you for some historical examples. I will certainly include the bounties in the &lt;a href=&quot;https://en.wikipedia.org/wiki/Scottish_Book&quot; rel=&quot;noreferrer&quot;&gt;Scottish Book&lt;/a&gt; from the Lwow School, as well as Erdös' faible for betting money on open problems. Thanks to a respond on the &lt;a href=&quot;https://mathoverflow.net/questions/430419/best-sources-for-the-history-of-prize-money-for-open-mathematical-problems&quot;&gt;same question&lt;/a&gt; on MathOverflow I will also include the bounty for solving the three body problem, announced by king Oscar II of Sweden as an older example.&lt;/p&gt;&#xA;&lt;p&gt;Now my question is: Does any of you know further examples for such rewards and maybe even know reliable sources for these?&lt;/p&gt;&#xA;&lt;p&gt;Thank you very much for your help!&lt;/p&gt;&#xA;" OwnerUserId="17250" LastEditorUserId="205" LastEditDate="2022-09-15T13:27:55.573" LastActivityDate="2023-01-18T13:28:08.837" Title="When did bounties and prize money for open mathematical problems start being a thing?" Tags="|mathematics|mathematics-social-history|open-problems|" AnswerCount="3" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14805" PostTypeId="1" CreationDate="2022-09-15T05:51:55.903" Score="2" ViewCount="100" Body="&lt;p&gt;&lt;a href=&quot;https://www.britannica.com/science/geoid/The-introduction-of-triangulation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Snell&lt;/a&gt; tends to be quoted as the person to develop triangulation in geodesy. I don't believe, however, that triangulation was invented only then. For example, &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Al-Biruni/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; Al-Biruni is mentioned to have &amp;quot;introduced techniques to measure the earth and distances on it using triangulation&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;It will be challenging to identify &lt;em&gt;the&lt;/em&gt; person responsible for the invention of triangulation, so I'm looking for a timeline/reliable sources that would give me a better idea of the full story.&lt;/p&gt;&#xA;" OwnerUserId="5243" LastActivityDate="2022-09-15T05:51:55.903" Title="History of triangulation" Tags="|discoveries|resource-recommendation|geography|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14806" PostTypeId="2" ParentId="14803" CreationDate="2022-09-15T06:17:34.110" Score="6" Body="&lt;p&gt;There was no early discussion of topology without closure under finite intersections. The open set definition first given by Bourbaki (1940) includes this condition, and its predecessors in terms of neighborhoods (Hausdorff, 1914) and in terms of closures (Alexandrov-Hopf, 1935) include its equivalents, see &lt;a href=&quot;https://hsm.stackexchange.com/a/8415/55&quot;&gt;History of various definitions of topology&lt;/a&gt; and references there.&lt;/p&gt;&#xA;&lt;p&gt;There was long experimentation (from 1906 to 1940) with various approaches to defining topology (via accumulation points, neighborhoods, closures, open sets), and  discussions of whether to include various separation axioms (by Hausdorff in 1927 and Alexandrov-Hopf in 1935) or some notion of uniformity (by Weil in 1937), but dropping closure under finite intersections (or its equivalents) was not considered. The remaining condition would have been too weak for intended applications.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2022-09-15T06:17:34.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14807" PostTypeId="2" ParentId="7071" CreationDate="2022-09-16T12:54:39.447" Score="4" Body="&lt;p&gt;Somebody did or several people did. We just don't know whom and its exceedingly unlikely we will ever know. What we can find out is the oldest example of a number line.&lt;/p&gt;&#xA;&lt;p&gt;I suggest for this the Ishango Bone. Its a baboon fibula with vertical marks ranged in a row representing numbers. Thus the simplest example of a number line. Its been dated to 9,000 BC to 20,000 BC with some arguing for 44,000 BC. The discrepency in aging is due to the nature of volcanic dust that buried the discovery site.&lt;/p&gt;&#xA;&lt;p&gt;Plus I'd say that the sign for the number one in many cultures from across Eurasia is the straight line. The number line referring to itself. One of the first examples of self-reference and right at the beginning of the story of number!&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2022-09-23T09:34:14.520" LastActivityDate="2022-09-23T09:34:14.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14808" PostTypeId="2" ParentId="7390" CreationDate="2022-09-18T14:06:21.133" Score="2" Body="&lt;p&gt;In 1970, Pauling published &amp;quot;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC283405/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Evolution and the Need for Ascorbic Acid&lt;/a&gt;&amp;quot; suggesting an ascorbic acid daily intake of 2.3g or more.&#xA;This is followed by the book &amp;quot;Vitamin C and the Common Cold&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;If you are interested in Pauling's original claim, you could start with these 2 publications.&lt;/p&gt;&#xA;&lt;p&gt;An article from University of Helsinki's Department of Public Health on &amp;quot;&lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8779885/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bias against Vitamin C in Mainstream Medicine: Examples from Trials of Vitamin C for Infections&lt;/a&gt;&amp;quot; does a good review of historical issues.&lt;/p&gt;&#xA;" OwnerUserId="17262" LastEditorUserId="17262" LastEditDate="2022-09-18T14:08:42.040" LastActivityDate="2022-09-18T14:08:42.040" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14809" PostTypeId="2" ParentId="14801" CreationDate="2022-09-18T19:15:22.007" Score="5" Body="&lt;p&gt;Here are some open problems that go back to Euler or, in the last case, his contemporary Goldbach.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The classification of all idoneal (convenient) numbers. They were introduced by Euler and he searched for them up to 10000, finding no further examples after 1848. That they seemed to stop appearing surprised Euler. It is known now that there are only finitely many idoneal numbers, and Weinberger showed GRH implies 1848 is the biggest idoneal number. As long as GRH is an open problem, the completeness of the list of idoneal numbers may remain an unsolved problem.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The existence of a &lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_brick#Perfect_cuboid&quot; rel=&quot;nofollow noreferrer&quot;&gt;perfect Euler brick&lt;/a&gt;. This is not an important problem, and as far I am aware it is not related to anything important, in contrast to the link between the old congruent number problem and the Birch and Swinnerton-Dyer conjecture.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The irrationality of Euler’s constant &lt;span class=&quot;math-container&quot;&gt;$.577...$&lt;/span&gt;, a number which Euler introduced in 1734 &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/43/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (or in English &lt;a href=&quot;https://download.uni-mainz.de/mathematik/Algebraische%20Geometrie/Euler-Kreis%20Mainz/E43RevisedVersion1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) and computed to more accuracy &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/583/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; (or in English &lt;a href=&quot;https://download.uni-mainz.de/mathematik/Algebraische%20Geometrie/Euler-Kreis%20Mainz/E583_Englisch.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Any time he studied new numbers, he tried to relate them to what was known, like trying to write them in terms of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; or logarithms. In the second paper, after computing Euler's constant to &lt;span class=&quot;math-container&quot;&gt;$16$&lt;/span&gt; digits (with a mistake only in the last digit he found), he called this number &amp;quot;even more remarkable since it has still not been possible for me in any way to reduce it to a certain known measure.&amp;quot; I think a charitable interpretation of that comment means he suspected Euler's constant is at the very least irrational, if not transcendental. Those are the kinds of things he thought about: he had proved &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is irrational only a few years after introducing Euler's constant and he raised the question of whether &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is transcendental in 1775, a year before presenting the second paper about computing Euler's constant.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Infinitude of primes of the form &lt;span class=&quot;math-container&quot;&gt;$a^2 + 1$&lt;/span&gt;.  In October 1752, Euler computed all &lt;span class=&quot;math-container&quot;&gt;$a \leq 1500$&lt;/span&gt; for which &lt;span class=&quot;math-container&quot;&gt;$a^2 + 1$&lt;/span&gt; is prime. See pp. 587-588 &lt;a href=&quot;https://archive.org/details/correspondancem01goldgoog/page/585/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, where his list has only 3 errors: it omits &lt;span class=&quot;math-container&quot;&gt;$a=1080$&lt;/span&gt; and mistakenly includes &lt;span class=&quot;math-container&quot;&gt;$a = 844$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1234$&lt;/span&gt;. Euler kept finding prime values up to &lt;span class=&quot;math-container&quot;&gt;$a = 1494$&lt;/span&gt;, and even though he does not explicitly say he conjectures there are infinitely many such primes, what he does write makes it quite clear that he thinks it is the case.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Writing odd numbers as &lt;span class=&quot;math-container&quot;&gt;$p + 2a^2$&lt;/span&gt;. In November 1752, Goldbach wrote to Euler with a conjecture that every odd number bigger than &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; has the form &lt;span class=&quot;math-container&quot;&gt;$p + 2a^2$&lt;/span&gt; for some prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and integer &lt;span class=&quot;math-container&quot;&gt;$a \geq 0$&lt;/span&gt;. See it &lt;a href=&quot;https://archive.org/details/correspondancem01goldgoog/page/593/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. (In that era &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; was considered prime, so Goldbach's claim allowed &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to be so represented using &lt;span class=&quot;math-container&quot;&gt;$p=1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a = 0$&lt;/span&gt; and he'd allow &lt;span class=&quot;math-container&quot;&gt;$9 = 1 + 2 \cdot 2^2$&lt;/span&gt; while we'd only allow &lt;span class=&quot;math-container&quot;&gt;$9 = 7 + 2 \cdot 1^2$&lt;/span&gt;.)  Euler replied &lt;a href=&quot;https://archive.org/details/correspondancem01goldgoog/page/595/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; that he checked this for all odd numbers below &lt;span class=&quot;math-container&quot;&gt;$1000$&lt;/span&gt;.  It was discovered by Stern and his students in 1856 that there are two counterexamples: &lt;span class=&quot;math-container&quot;&gt;$5777$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5993$&lt;/span&gt;. (These numbers don't even fit the pattern if we allow &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to be a prime!) Since there are two counterexamples you could say the conjecture is simply wrong, but that's a bit harsh: wouldn't it be interesting if &lt;span class=&quot;math-container&quot;&gt;$5777$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5993$&lt;/span&gt; are the only counterexamples? Indeed, Hardy and Littlewood's Conjecture I on page 49 of their paper Partitio Numerorum III &lt;a href=&quot;https://projecteuclid.org/journals/acta-mathematica/volume-44/issue-none/Some-problems-of-Partitio-numerorum-III--On-the-expression/10.1007/BF02403921.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; conjecture that every sufficiently large odd number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; has the form &lt;span class=&quot;math-container&quot;&gt;$p + 2a^2$&lt;/span&gt; and they suggest a growing asymptotic estimate for the number of representations of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; in that form. An account of this problem for a general audience appeared in Mathematics Magazine &lt;a href=&quot;https://www.maa.org/sites/default/files/Lesser-Known-Hodges18520.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  This conjecture of Hardy and Littlewood should be regarded as the correct version of Goldbach's question, and it is still not solved.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-09-19T02:26:37.007" LastActivityDate="2022-09-19T02:26:37.007" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14810" PostTypeId="2" ParentId="31" CreationDate="2022-09-19T05:54:03.780" Score="2" Body="&lt;p&gt;Most people seem to be ignoring the fact that the &lt;strong&gt;calculator of 1946 is not like the modern calculator&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;transistor hadn't been invented at the time&lt;/strong&gt; of this test. So, &lt;strong&gt;how could there be a transistor based calculator&lt;/strong&gt;. Also, notice that it says &lt;strong&gt;Electric and not Electronic&lt;/strong&gt;.&#xA;It &lt;strong&gt;might have been a mechanical calculator&lt;/strong&gt; with a little help from Electricity.&#xA;It &lt;strong&gt;might have been something like a vacuum tube based computer&lt;/strong&gt; which would be really &lt;strong&gt;hard to input to&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is a video of a contest at 1967 between an abacus and a calculator. The calculator seems to be mechanical, needing a lot more work than a modern electric calculator. The abacus in this video seems to take less work than the mechanical calculator and I don't think it's such a good idea to &amp;quot;force&amp;quot; the mechanical calculator to work at high speeds, because they are suseptible to wear and tear of the gears etc.&#xA;There also seem to be many people using Abacus, but a single person using the mechanical calculator. So, in a sense, the game is rigged and the person with the mechanical calculator had lower chances of winning.&#xA;&lt;a href=&quot;https://www.youtube.com/watch?v=6YaLuh1QRM8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=6YaLuh1QRM8&lt;/a&gt; .&#xA;From the video you can understand why the mechanical calculator was defeated by the abacus.&lt;/p&gt;&#xA;&lt;p&gt;More details about this so called Electric Calculator are needed for more accurate answers.&lt;/p&gt;&#xA;" OwnerUserId="17269" LastEditorUserId="17269" LastEditDate="2022-09-19T08:11:59.907" LastActivityDate="2022-09-19T08:11:59.907" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14811" PostTypeId="1" CreationDate="2022-09-19T13:12:51.697" Score="1" ViewCount="133" Body="&lt;p&gt;In the Introduction of his &amp;quot;Sequential Analysis&amp;quot; Wald writes that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Because of the usefulness of the sequential probability ratio test in&#xA;development work on military and naval equipment, it was classified&#xA;Restricted within the meaning of the Espionage Act. The author was&#xA;requested to submit his findings in a restricted report 7 dated&#xA;September, 1943.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Beyond the obvious desire by bureaucrats to classify almost anything they can put their hands on was there really any strategic/tactical military reason to classify Wald's work, and if yes what specific idea or method was of national security interest? In what industrial production was SPRT used during war that might have inspired the classification? (That SPRT could be applied to improve and speed up radar detection was already known early on but its actual deployment in a practical system took a much longer time and I do not believe that anybody would have ever attempted to use it with &lt;em&gt;analog&lt;/em&gt; computers.)&lt;/p&gt;&#xA;" OwnerUserId="7908" LastActivityDate="2022-10-10T00:06:28.100" Title="Why Was Sequential Analysis Classified?" Tags="|statistics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14812" PostTypeId="1" CreationDate="2022-09-19T15:36:13.820" Score="3" ViewCount="222" Body="&lt;p&gt;When I search for the history of projectile motion, I mostly find outlines starting with Aristotle and discussing Ibn Sina, Tartaglia, Galileo, Newton, etc. and perhaps with a few more Europeans in the gaps.&lt;/p&gt;&#xA;&lt;p&gt;Ibn Sina wasn't Western, but he was expanding on Aristotle's work.&lt;/p&gt;&#xA;&lt;p&gt;Were there models of projectile motion essentially independent of that line of reasoning, for example Chinese or Indian models, in the time before Galileo's work spread to those places?&lt;/p&gt;&#xA;" OwnerUserId="2238" LastActivityDate="2022-10-24T14:03:17.817" Title="Were there non-Western models of projectile motion before Galileo?" Tags="|classical-mechanics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14813" PostTypeId="1" CreationDate="2022-09-19T22:22:34.047" Score="0" ViewCount="109" Body="&lt;p&gt;The Leidenfrost Effect is a described as follows on &lt;a href=&quot;https://en.wikipedia.org/wiki/Leidenfrost_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Leidenfrost effect is a physical phenomenon in which a liquid,&#xA;close to a surface that is significantly hotter than the liquid's&#xA;boiling point, produces an insulating vapor layer that keeps the&#xA;liquid from boiling rapidly. Because of this repulsive force, a&#xA;droplet hovers over the surface, rather than making physical contact&#xA;with it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One of the consequences of the effect is that it allows for someone to briefly interact with substances that are extremely hot or cold such as liquid nitrogen or molten metal, and come away unscathed. (Do not try this at home.)&lt;/p&gt;&#xA;&lt;p&gt;Here is a video of someone brushing their wet hand through molten metal and avoiding being burned:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.reddit.com/r/interestingasfuck/comments/vtsvle/man_able_to_touch_molten_metal_with_bare_hands/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.reddit.com/r/interestingasfuck/comments/vtsvle/man_able_to_touch_molten_metal_with_bare_hands/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This was also covered in MythBusters:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the 2009 season 7 finale of MythBusters, &amp;quot;Mini Myth Mayhem&amp;quot;, the&#xA;team demonstrated that a person can wet their hand and briefly dip it&#xA;into molten lead without injury, using the Leidenfrost effect as the&#xA;scientific basis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This question seemed to come up repeatedly and one of &lt;a href=&quot;https://www.reddit.com/r/interestingasfuck/comments/vtsvle/comment/if9tvvq/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the comments&lt;/a&gt; on Reddit claimed that Leidenfrost himself proved the Leidenfrost Effect by touching molten metal, but I was unable to find any sources to back up that claim.&lt;/p&gt;&#xA;&lt;p&gt;Is there any evidence that Leidenfrost himself demonstrated this by touching molten metal, and if not, then who was the first person with enough courage to actually touch molten metal in order to demonstrate the effect?&lt;/p&gt;&#xA;" OwnerUserId="17272" LastActivityDate="2022-09-19T22:22:34.047" Title="When Was The Leidenfrost Effect First Demonstrated By Touching Molten Metal?" Tags="|physics|experimental-physics|chemistry|scientists|physics-social-history|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14816" PostTypeId="2" ParentId="7592" CreationDate="2022-09-20T23:56:10.567" Score="1" Body="&lt;p&gt;I think prior to electrolysis the composition of water was known -- hydrogen and oxygen were combined and in the presence of a spark water was created: &lt;a href=&quot;https://hsm.stackexchange.com/questions/2542/how-was-the-composition-and-structure-of-water-determined#:%7E:text=1805%3A%20Gay%2DLussac%20and%20Humboldt,of%20oxygen%20(by%20volume)&quot;&gt;https://hsm.stackexchange.com/questions/2542/how-was-the-composition-and-structure-of-water-determined#:~:text=1805%3A%20Gay%2DLussac%20and%20Humboldt,of%20oxygen%20(by%20volume)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Had these earlier experiments not been performed, the fact that water was in fact made up of two separate gases would have amazed people since water was considered an element.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-09-20T23:56:10.567" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14817" PostTypeId="1" CreationDate="2022-09-21T12:53:46.760" Score="2" ViewCount="123" Body="&lt;p&gt;Who first used the corner quotes, ⌜ and ⌝, or &lt;span class=&quot;math-container&quot;&gt;$\texttt{\Godelnum}$&lt;/span&gt; with Sam Buss's macro, for the notion of Gödel number?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Quasi-quotation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quine introduced corner quotes&lt;/a&gt;, but did not use them for the notion of Gödel numbers.&lt;/p&gt;&#xA;&lt;p&gt;The corner quotes were used for Gödel numbers by Joseph R. Shoenfield, &lt;i&gt;Mathematical Logic&lt;/i&gt;, 1967, as from page 122.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/wrF8M.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wrF8M.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Corner quotes are used prevalently in provability logic, and in other areas of logic. Two important citations are Craig Smorynski, The Incompleteness Theorems, in &lt;i&gt;Handbook of Mathematical Logic&lt;/i&gt;, as from 1st edition 1979; and Paanu Raatikainen, &lt;a href=&quot;https://plato.stanford.edu/entries/goedel-incompleteness/sup1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gödel's incompleteness theorems&lt;/a&gt;, in Stanford Encyclopedia of Philosophy.&lt;/p&gt;&#xA;&lt;p&gt;Edit: I do not find the corner quotes in Martin Davies, The Undecidable, or in the books by Moztowski or Kleene.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastEditorUserId="8094" LastEditDate="2022-09-23T14:10:56.107" LastActivityDate="2022-09-23T14:10:56.107" Title="First use of corner quotes for Gödel numbers" Tags="|reference-request|notation|mathematical-logic|logic|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14818" PostTypeId="1" CreationDate="2022-09-21T19:45:43.040" Score="3" ViewCount="183" Body="&lt;p&gt;Where is &lt;a href=&quot;https://en.wikipedia.org/wiki/Alfred_Tarski&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alfred Tarski&lt;/a&gt; buried?&lt;/p&gt;&#xA;&lt;p&gt;He was a famous Polish mathematician. He died in 1983 in Berkeley, California, USA, according to Wikipedia. I tried a search on &lt;a href=&quot;https://www.findagrave.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;Find a Grave&lt;/a&gt;, but there is no entry for his name. I also tried a multilingual search. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17284" LastEditorUserId="4251" LastEditDate="2022-09-22T16:20:47.030" LastActivityDate="2023-05-30T21:35:46.660" Title="Where is Alfred Tarski buried?" Tags="|mathematicians|20th-century|death|logicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14819" PostTypeId="1" CreationDate="2022-09-21T19:50:16.340" Score="1" ViewCount="44" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$&lt;/span&gt; be nonnegative integers. The Schur polynomial &lt;span class=&quot;math-container&quot;&gt;$s_{\lambda}(x_1, \ldots, x_n)$&lt;/span&gt; can be defined as the ratio&#xA;&lt;span class=&quot;math-container&quot;&gt;$$s_{\lambda}(x_1, x_2, \ldots, x_n) = \frac{\det\left(x_i^{\lambda_j+n-j} \right)}{\Delta(x)} \qquad (\clubsuit)$$&lt;/span&gt;&#xA;where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Delta(x) = \det \left( x_i^{n-j} \right) = \prod_{i&amp;lt;j} (x_i - x_j)$$&lt;/span&gt;&#xA;is the Vandermonde determinant. According to the historical notes in Stanley, &lt;i&gt;Enumerative Combinatorics&lt;/i&gt;, Volume 2, Chapter 7, this ratio was first studied by Cauchy.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, we know that Schur polynomials are important in the representation theory of &lt;span class=&quot;math-container&quot;&gt;$GL_n(\mathbb{C})$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$GL_n(\mathbb{F}_p)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;, and in Schubert calculus. But Schur's thesis, where he related Schur polynomials to representation theory, is from 1918, and Schubert developed Schubert calculus in the 1870's. Cauchy is much earlier; he introduced these ratios in 1815 (&lt;a href=&quot;http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_CAUCHY_2_1_91_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment&lt;/a&gt;). So my first question is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why was Cauchy studying the ratio &lt;span class=&quot;math-container&quot;&gt;$(\clubsuit)$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Continuing in this vein, the following formula is now known as Cauchy's identity:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\prod_{i,j=1}^n \frac{1}{1-x_i y_j} = \sum_{\lambda} s_{\lambda}(x) s_{\lambda}(y). \qquad (\diamondsuit)$$&lt;/span&gt;&#xA;Stanley writes that he can not find a source where Cauchy records &lt;span class=&quot;math-container&quot;&gt;$(\diamondsuit)$&lt;/span&gt;, but that &lt;span class=&quot;math-container&quot;&gt;$(\diamondsuit)$&lt;/span&gt; can be deduced from Cauchy's determinantal identity:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\det \left( \frac{1}{u_i+v_j} \right) = \frac{\Delta(u) \Delta(v)}{\prod_{i,j} (u_i + v_j)} \quad (\heartsuit)$$&lt;/span&gt;&#xA;using the substitution &lt;span class=&quot;math-container&quot;&gt;$(u_i = -1/x_i, v_j = y_j)$&lt;/span&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy-Binet formula&lt;/a&gt;. (I understand how the deduction works; you don't need to explain this part.)&lt;/p&gt;&#xA;&lt;p&gt;The identity &lt;span class=&quot;math-container&quot;&gt;$(\heartsuit)$&lt;/span&gt; is due to Cauchy; it appears as an exercise in an 1841 paper (&lt;a href=&quot;https://www.cambridge.org/core/books/abs/oeuvres-completes/memoire-sur-les-fonctions-alternees-et-sur-les-sommes-alternees/31524DAEF99769C7EC05D6845E3B9B76&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mémoire sur les fonctions alternées et sur les sommes alternées&lt;/a&gt;).  So my questions continue:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Why was Cauchy studying &lt;span class=&quot;math-container&quot;&gt;$(\heartsuit)$&lt;/span&gt;? Did he know that &lt;span class=&quot;math-container&quot;&gt;$(\heartsuit)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(\clubsuit)$&lt;/span&gt; were related?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note: I have linked Cauchy's papers above, and skimmed enough to make sure they are the right ones, but I haven't actually read them.&lt;/p&gt;&#xA;" OwnerUserId="17283" LastActivityDate="2022-09-21T19:50:16.340" Title="Why was Cauchy studying Schur polynomials and related topics?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14820" PostTypeId="2" ParentId="13581" CreationDate="2022-09-22T07:50:24.410" Score="2" Body="&lt;p&gt;Physics is not mathematics and so units are always required whether explicitly as today or implicitly as in Newtoms day.&lt;/p&gt;&#xA;&lt;p&gt;It appears in Newtons time force was generally expressed in terms of an equivalent weight. This is natural quality to use even if the unit of such is left undetermined.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-22T07:50:24.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14823" PostTypeId="1" AcceptedAnswerId="14837" CreationDate="2022-09-23T01:26:16.647" Score="1" ViewCount="173" Body="&lt;p&gt;What is the origin of the name &amp;quot;degeneracy&amp;quot; pressure and &amp;quot;degenerate&amp;quot; Fermi gas?&#xA;I was trying to find the first paper that used the term &amp;quot;degenerate/degeneracy&amp;quot; to describe either &amp;quot;degeneracy&amp;quot; pressure or &amp;quot;degenerate&amp;quot; Fermi gas. But I failed to do so. Any help is highly appreciated!&lt;/p&gt;&#xA;" OwnerUserId="17288" LastActivityDate="2022-09-29T19:31:24.470" Title="What is the origin of the name &quot;degeneracy&quot; pressure and &quot;degenerate&quot; Fermi gas?" Tags="|physics|quantum-mechanics|statistical-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14824" PostTypeId="2" ParentId="4960" CreationDate="2022-09-23T08:41:45.267" Score="1" Body="&lt;p&gt;I think this is an apocryphal story because it makes a good story. The exact 'loophole' is not known. However, according to Wikipedia, one academic suggests that it is article V on amending the Constitution in a downward spiral and thus allowing a fascist takeover.&lt;/p&gt;&#xA;&lt;p&gt;I think it apocryphal as Goedel was naturalised as an American citizen in 1948 and only seven years later the US embarked upon the disastrous Vietnam war which destroyed its reputation after the goodwill it enjoyed for helping end WWII and where Bertrand Russell made direct accusations in the New York Times that how the US treated the Vietnamese distinguished them not at all from Nazi Germany. This is worse than fascism. Russell was of course a logician but coming from a family with a tradition of public service he understood quite well - in a way that fascism and the doorway to fascism - isn't merely to be found in arcane rules in a constitution but has many different sources for example, within politics and the evil in mens heart.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2022-09-23T09:28:22.653" LastActivityDate="2022-09-23T09:28:22.653" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14825" PostTypeId="2" ParentId="3723" CreationDate="2022-09-23T08:48:27.907" Score="1" Body="&lt;p&gt;The nature of infinitesimals have been debated since Newtons time. The traditional epsilon-delta method is clumsy though rigorous. Robinson's infinitesimals was another approach - there are others - that made working with them more intuitive.&lt;/p&gt;&#xA;&lt;p&gt;Its this clearer intuition that prompted Goedels thought.&lt;/p&gt;&#xA;&lt;p&gt;However, Goedel not being geometrically minded and despite being friends of Einstein, appears to miss the fact it is differential geometry that actually is the analysis of the future.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;edit&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Actually, the nature of infinitesimals were debated from antiquity, both in India and Greece. Archimedes for example used them in determining the area of a segment of a parabola cut out by a chord.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2024-01-12T14:08:11.673" LastActivityDate="2024-01-12T14:08:11.673" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14826" PostTypeId="2" ParentId="7250" CreationDate="2022-09-23T08:58:52.693" Score="0" Body="&lt;p&gt;There was a lot of disruption to the education system during WWII. Apparently Wittgenstein wrote his notes for the &lt;em&gt;Tractatus&lt;/em&gt; whilst serving in the Austrian Army. So this is one factor.&lt;/p&gt;&#xA;&lt;p&gt;Wittgenstein also desperately wanted to be a philosopher of logic but did not know whether he had the talent and so he asked his hero Bertrand Russell to evaluate his philosophical talents.&lt;/p&gt;&#xA;&lt;p&gt;Apparently Russell thought he was either an idiot or a genius. And then later revised it to genius. Who knows what later philosopgers might think. One philosopger recently told me that analytic philosophy was on the wane. So perhaps we may need to re-evaluate ...&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-23T08:58:52.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14827" PostTypeId="2" ParentId="469" CreationDate="2022-09-23T09:05:47.897" Score="0" Body="&lt;p&gt;Topology is the idea of continuity. Archimedes was the first to use the notion of continuity in his method of exhaustion, an early version of the integral calculus.&lt;/p&gt;&#xA;&lt;p&gt;Whilst Rene Thom, a famous French algebraic geometer credits the first philosopher and the only philosopher for a millenia yo seriously think about continuity. It's because of his thinking on tgos that Aristotle dismissed atomism as untenable which quantum mechanics has shown to be valid.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-23T09:05:47.897" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14828" PostTypeId="2" ParentId="14812" CreationDate="2022-09-23T09:19:11.930" Score="1" Body="&lt;p&gt;Yes, but only intuitive ones that every saw as obvious and hence requiring as much comment as noticing that water was wet or that dropping a stone describes a straight line.&lt;/p&gt;&#xA;&lt;p&gt;The essential characteristics of projectile motion is that it traces an arc. This is obvious from our knowledge of archery and spears. And this obviously goes back a long time - into prehistory.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-23T09:19:11.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14829" PostTypeId="2" ParentId="376" CreationDate="2022-09-23T09:27:27.540" Score="0" Body="&lt;p&gt;I think the notion of infinity was developed well before any formal notion of the infinite in modern mathematics. For example, Aristotle distinguished between the All &amp;amp; the Infinite.&lt;/p&gt;&#xA;&lt;p&gt;And the sky, the horizon, has often been used in literature and poetry as an image of the infinite which suggests a long prehistory of this notion.&lt;/p&gt;&#xA;&lt;p&gt;In the Vedas, for example, we have the notion of nirguna Brahman where nir=none and guna=quality. But this is not to be understood as nothing as in mathematical nothing but as in indescribable, that is transcendental. And that remimds that the Jaims had a category where they might put the infinite - it exists and is indescribable. But I think they would say the infinite actually lies far below this category as we have just described it as not finite.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-23T09:27:27.540" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14830" PostTypeId="2" ParentId="7733" CreationDate="2022-09-23T09:31:19.220" Score="1" Body="&lt;p&gt;No. Primes occur in many situations. With Liebniz's extraordinary intuition he is more likely to intuited that prome factorisation may apply elsewhere - and indeed it does - for example in prime knots.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-23T09:31:19.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14833" PostTypeId="2" ParentId="14804" CreationDate="2022-09-23T10:02:04.043" Score="3" Body="&lt;p&gt;I'm not sure which is the earliest. But one of the earliest I knew of is due to Mobius.&lt;/p&gt;&#xA;&lt;p&gt;In 1846, Mobius invited Grassmann to paricipate in a competition to solve a problem first set by Liebniz: to derive a geometric calculus without coordinates, his famous &lt;em&gt;analysis situs&lt;/em&gt;. His entry was the winning entry - it was also the only entry. Grassmann of course is responsible for the modern development of linear and exterior algebra.&lt;/p&gt;&#xA;&lt;p&gt;I think I first read about this first in a biography of Grassmann.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-09-23T10:02:04.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14835" PostTypeId="1" CreationDate="2022-09-23T10:16:23.400" Score="5" ViewCount="788" Body="&lt;p&gt;I only recently learned that Pascual Jordan, a well known physicist, with significant contributions to the development of early quantum mechanics was a paid up member of the Nazi Party. He in fact joined a SA unit.&lt;/p&gt;&#xA;&lt;p&gt;Are there any well-known mathematicians who were fascist or Nazi?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2022-12-18T08:09:47.363" Title="Are there any well known mathematicians who were fascists?" Tags="|mathematics|biographical-details|physicists|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14836" PostTypeId="2" ParentId="14835" CreationDate="2022-09-23T12:39:22.593" Score="10" Body="&lt;p&gt;The book &lt;em&gt;Mathematicians Under the Nazis&lt;/em&gt; by Segal is a systematic study of, well, what it says in the title. It includes several examples of mathematicians who showed various degrees of sympathy for the Nazis. This includes Gentzen, Vahlen, Bieberbach and Kähler.&lt;/p&gt;&#xA;" OwnerUserId="26" LastActivityDate="2022-09-23T12:39:22.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14837" PostTypeId="2" ParentId="14823" CreationDate="2022-09-23T20:21:53.023" Score="3" Body="&lt;p&gt;Just an informed hunch, seconding @Jon Custer's suggestion. The physics usage follows the older &lt;a href=&quot;https://en.wikipedia.org/wiki/Degeneracy_(mathematics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;math usage&lt;/a&gt;, which signifies degeneracy as a limiting simplification which stretches a definition to an absurd, qualitatively simpler limit, such as a &lt;a href=&quot;https://www.mathwords.com/d/degenerate.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;parabola&lt;/a&gt; being the degenerate limit of an ellipse with a  focal point sent to infinity. Normally, the term associates to reduction of dimension or cardinality.&lt;/p&gt;&#xA;&lt;p&gt;In physics, the term appears to originate in the degeneracy of matrix eigenvalues, the limit when a vanishing parameter of a matrix induces two or more distinct eigenvalues to merge to a common one, shared by two or more different eigenvectors (eigenstates). The dominant operator is the QM hamiltonian, thus involving degenerate energy eigenvalues. The converse limit &amp;quot;breaks the degeneracy&amp;quot;, when introduction of a non-vanishing parameter (or perturbation) distinguishes two degenerate eigenstates and assigns unequal energy eigenvalues (levels, spectral lines, etc...) to them.&lt;/p&gt;&#xA;&lt;p&gt;By the time the Dirac distribution for fermions was discovered, the concept was broadly used in physics, and the limiting case of all states of a system allowed by the Pauli exclusion principle being full up to the Fermi surface was naturally termed a &amp;quot;degenerate&amp;quot; gas:&#xA;maximally ordered and  quite un-gas-like, with a Pauli exclusion &amp;quot;pressure&amp;quot; keeping fermions from overlapping and condensing.&lt;/p&gt;&#xA;&lt;p&gt;Fermi's original (1926) paper, &lt;em&gt;Zur Quantelung des idealen einatomigen Gases&lt;/em&gt;,  &lt;a href=&quot;https://doi.org/10.1007/BF01400221&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZfPhys &lt;strong&gt;36&lt;/strong&gt; 902–912&lt;/a&gt; thus  utilizes the term &lt;em&gt;Entartung&lt;/em&gt; (degeneracy).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Note added&lt;/em&gt; @Consigliere ZARF 's appreciated comment reminds us the term was already current in the old quantum theory&#xA;(Sommerfeld 1921) to indicate &amp;quot;limiting case&amp;quot;, and, before him, &lt;a href=&quot;http://www.archive.org/stream/sitzungsberichte1916deutsch#page/548&quot; rel=&quot;nofollow noreferrer&quot;&gt;K Schwarzschild p550  §3&lt;/a&gt; , (1916) &amp;quot;Zur quantenhypothese&amp;quot; &lt;em&gt;Sitz ber Preuss Akad Wiss&lt;/em&gt; (Berlin), 548-568.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2022-09-29T19:31:24.470" LastActivityDate="2022-09-29T19:31:24.470" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14838" PostTypeId="1" CreationDate="2022-09-25T01:10:27.737" Score="3" ViewCount="108" Body="&lt;p&gt;Pais, in his &amp;quot;&lt;em&gt;Inward Bound&lt;/em&gt;&amp;quot;, describes the early history of spin. He tells us that Goudsmit and Uhlenbeck interpreted Pauli's &amp;quot;doubled valuedness&amp;quot; as spin, while in turn Pauli re-interpreted a pre-existing, so-called &amp;quot;inner&amp;quot; quantum number as describing a property of the valence electron.&lt;/p&gt;&#xA;&lt;p&gt;Before Pauli's reinterpretation, Sommerfeld, Landé, and Heisenberg all seem to have written about this number, and they all thought it had something to do with the &amp;quot;inner parts&amp;quot; of the atom (not the valence electron). But who introduced it, and why?&lt;/p&gt;&#xA;&lt;p&gt;I would guess it was Sommerfeld who introduced it - in his Munich lectures in 1916 - but as far as I can tell, he only published anything about it in 1929 in Volume II of his &amp;quot;Atomic structure...&amp;quot; (which he says are based on these lectures) relatively long after the re-interpretation as intrinsic spin of the electron. Did he or did anyone introduce this &amp;quot;inner quantum number&amp;quot; in a published article before said Volume II, and before it turned into electron spin? Pais says that this quantum number is referred to as the &amp;quot;inner quantum number&amp;quot; and mentions articles by Landé and Heisenberg's very first paper, but I think neither introduced this inner quantum number for the first time, or did they?&lt;/p&gt;&#xA;" OwnerUserId="17299" LastEditorUserId="17299" LastEditDate="2022-10-07T17:59:38.287" LastActivityDate="2022-10-07T17:59:38.287" Title="What's the early history of the &quot;inner quantum number&quot;?" Tags="|physics|quantum-mechanics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14839" PostTypeId="2" ParentId="14835" CreationDate="2022-09-25T13:44:19.313" Score="4" Body="&lt;p&gt;Probably the most famous mathematicians who were  members of NSDAP are &lt;a href=&quot;https://en.wikipedia.org/wiki/Oswald_Teichm%C3%BCller&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oswald Teichmuller&lt;/a&gt; and&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Bieberbach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ludwig Bieberbach&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-09-25T13:44:19.313" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14840" PostTypeId="2" ParentId="6326" CreationDate="2022-09-26T13:07:39.703" Score="3" Body="&lt;p&gt;Geometric(al) is here opposed to arithmetic(al). These adjectives refer to how solutions were found in the past before algebraic methods were developed. Geometry was believed to be a more powerful instrument for solving complex problems.&lt;/p&gt;&#xA;&lt;p&gt;Basically problems linked to sums were solved using arithmetic, and problems linked to products/ratios/exponents were solved by geometry. E.g. it is easy to &amp;quot;see&amp;quot; the product of two numbers by looking at the rectangle area, and the &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/on-squares-rectangles-and-square-roots-square-roots-in-ancient-chinese-mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;square root extraction was therefore a geometry problem&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Using this opposition, a proportion, a sequence (progression) or a series could be qualified as arithmetical or geometrical.&lt;/p&gt;&#xA;&lt;p&gt;Other answers already dated the use of the wording, I'm just adding extracts of a book using this conventional split, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015069248170&amp;amp;view=2up&amp;amp;seq=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arithmetical Institutions. Containing a Compleat System of Arithmetic Natural, Logarithmical, and Algebraical in all Their Branches&lt;/a&gt;, by John Kirby in 1735:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Arithmetical vs. geometrical ratios&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/A7IQq.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/A7IQq.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Note how &lt;em&gt;ratio&lt;/em&gt; was used both for the common difference and the common ratio. This is now different in En. but it persisted in my native language (Fr.): Latin &lt;a href=&quot;https://en.wiktionary.org/wiki/ratus#Latin&quot; rel=&quot;nofollow noreferrer&quot;&gt;ratus&lt;/a&gt; gave &lt;em&gt;ratio&lt;/em&gt; but also &lt;em&gt;raison&lt;/em&gt; (reason). &lt;em&gt;Raison&lt;/em&gt; is the word used to refer to the common difference and common ratio.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Arithmetical proportion and progression&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/I5NUP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I5NUP.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vSryx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vSryx.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Geometrical ratio and progression&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V6xeV.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V6xeV.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Note geometrical ratios are a family including &lt;em&gt;simple&lt;/em&gt;, &lt;em&gt;multiplicate&lt;/em&gt;, &lt;em&gt;harmonical&lt;/em&gt; and &lt;em&gt;contraharmonical&lt;/em&gt;. Details are found in the same book.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZrwO4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZrwO4.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17303" LastEditorUserId="17303" LastEditDate="2022-09-26T14:49:54.307" LastActivityDate="2022-09-26T14:49:54.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14841" PostTypeId="2" ParentId="14835" CreationDate="2022-09-26T18:45:02.300" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://www.ams.org/notices/201508/rnoti-p924.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Van der Waerden in the Third Reich&lt;/a&gt; by Reinhard Siegmund-Schultze book review of &lt;em&gt;&amp;quot;The Scholar and the State. In Search of Van der Waerden&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;See the comments below&lt;/p&gt;&#xA;" OwnerUserId="15350" LastEditorUserId="15350" LastEditDate="2022-09-29T01:56:21.177" LastActivityDate="2022-09-29T01:56:21.177" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14842" PostTypeId="2" ParentId="469" CreationDate="2022-09-26T18:50:45.277" Score="0" Body="&lt;p&gt;In the &amp;quot;Elements of the History of Mathematics&amp;quot; by Bourbaki you can find in section &amp;quot;10. Topological spaces&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is Riemann who must be considered as the creator of topology, as of so many other branches of modern mathematics: it is in fact he who, first, sought to disengage the notion of topological space, conceived the idea of an autonomous theory of these spaces, defined the invariants (the &amp;quot;Betti numbers&amp;quot;) which were to play the greatest role in the later development of topology, and gave its first applications to analysis (periods of abelian integrals).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;You can look there for more.&lt;/p&gt;&#xA;" OwnerUserId="15350" LastActivityDate="2022-09-26T18:50:45.277" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14844" PostTypeId="1" CreationDate="2022-09-30T11:56:33.840" Score="5" ViewCount="2645" Body="&lt;p&gt;This is a figure that I saw while going down the rabbit hole of &amp;quot;Sacred Geometry&amp;quot; back when conspiracy theories and related nonsense were relatively harmless and fun to laugh at. A book reference is &lt;a href=&quot;https://www.google.com/books/edition/The_Ancient_Secret_of_the_Flower_of_Life/CXvSZoublTgC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Melchizedek, The Ancient Secret of the Flower of Life&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In that strange place it's called &amp;quot;Metatron's Cube&amp;quot;, because, according to the website called soul-flower.com, the first result when you google it, it is &amp;quot;a mystical 3-dimension [sic.] cube used by the Archangel Metatron to watch over the flow of energy connecting earth [sic.] and the divine&amp;quot;. It also &amp;quot;spins with energy to help you replace negative thoughts with positive ones&amp;quot; and &amp;quot;reminds us that the universe wants us to discover our personal power and use it to do good&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I didn't bother to think about the actual geometry in it at the time, but recently saw how two interlocking regular tetrahedra connected the vertices of a cube, and realised I'd seen it before. As in the second picture, the vertices of the tetrahedra are the alternating vertices of a cube, and they dice each other into eight smaller tetrahedra, for each vertex of the cube, and a regular octahedron in the middle.&lt;/p&gt;&#xA;&lt;p&gt;Then, what they've done is orthogonally projected into the plane so that there are some regular hexagons and triangles and some other nice things, and drawn in some mutually tangent circles centered at distinct points in the figure, which would not correspond to tangent spheres under the projection, but hey, they happen to be tangent to some of the regular triangles in the figure.&lt;/p&gt;&#xA;&lt;p&gt;It's also supposed to have connections to the other Platonic solids, but they're extremely faint. This is just an unusually nice object without any deep reason for being so nice, I'm sure, but I was wondering if it has a history in actual mathematics, or even design, and a name less insane to call it by.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V1sLQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V1sLQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/wwj4B.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/wwj4B.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17323" LastEditorUserId="55" LastEditDate="2022-10-02T04:11:29.300" LastActivityDate="2022-10-02T04:11:29.300" Title="Does &quot;Metatron's cube&quot; have a history and a serious name in geometry?" Tags="|mathematics|reference-request|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14845" PostTypeId="1" CreationDate="2022-10-02T00:40:53.737" Score="1" ViewCount="96" Body="&lt;p&gt;I am aware that the USA first dropped the gun-type but first &lt;em&gt;&lt;strong&gt;tested&lt;/strong&gt;&lt;/em&gt; (unsure of approach) the implosion design in 1945 -- the gun-type was never tested before use because its design was so simple. Bethe pointed out that the plutonium weapon requires machining of the metal, and there is also afaik a still-classified initiator which is pretty complex itself.&lt;/p&gt;&#xA;&lt;p&gt;Would even in the 21st century a country like North Korea use the simpler design to be sure of success or are there other reasons that they would use the implosion design despite its relative complexity?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-10-04T00:23:40.097" Title="Have most countries used plutonium/implosion approach in developing their atomic weapons?" Tags="|atomic-weapons|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14847" PostTypeId="2" ParentId="7642" CreationDate="2022-10-03T11:26:23.367" Score="11" Body="&lt;p&gt;*copied from my answer to &lt;a href=&quot;https://hsm.stackexchange.com/a/8188/229&quot;&gt;another question&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems &lt;strong&gt;ball lightning&lt;/strong&gt; was disbelieved by scientists until around 1960.  See &lt;a href=&quot;https://en.wikipedia.org/wiki/Ball_lightning&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;I knew a geologist who told us how his eye-witness account of ball lightning had been ridiculed.  He had learned not to mention it when he interviewed for jobs as a professor of geology.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-10-03T11:26:23.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14848" PostTypeId="2" ParentId="7642" CreationDate="2022-10-03T15:33:25.067" Score="15" Body="&lt;p&gt;For a long while it was widely believed that the main cause of peptic ulcer disease was stress or spicy food. When the theory arose that it was in fact usually an infection this was rejected by the medical establishment. The Greek physician &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Lykoudis&quot; rel=&quot;noreferrer&quot;&gt;John Lykoudis&lt;/a&gt; was in fact fined for his treatment of patients with antibiotics. Subsequent work by Australian scientists identified the bacterium responsible &lt;em&gt;Helicobacter pylori&lt;/em&gt; and they were awarded the Nobel for their work. A &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_peptic_ulcer_disease_and_Helicobacter_pylori&quot; rel=&quot;noreferrer&quot;&gt;full timeline&lt;/a&gt; is available on Wikipedia. It is far too long to summarise here.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2022-10-03T15:33:25.067" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14849" PostTypeId="2" ParentId="14845" CreationDate="2022-10-03T17:30:27.197" Score="1" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.wikipedia.org/wiki/South_Africa_and_weapons_of_mass_destruction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, South Africa developed and made operational a handful of uranium gun devices. No plutonium devices were made.&lt;/p&gt;&#xA;&lt;p&gt;So, at least one country went for the simple design.&lt;/p&gt;&#xA;&lt;p&gt;Libya (&lt;a href=&quot;https://en.wikipedia.org/wiki/Libya_and_weapons_of_mass_destruction&quot; rel=&quot;nofollow noreferrer&quot;&gt;also on Wikipedia)&lt;/a&gt; is another possibility, but apparently their program never progressed very well. They intended to enrich uranium, but whether that was to go directly to a gun-type weapon or as a stepping stone to breeding plutonium is unclear.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="12330" LastEditDate="2022-10-04T00:23:40.097" LastActivityDate="2022-10-04T00:23:40.097" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14850" PostTypeId="1" CreationDate="2022-10-04T18:06:14.063" Score="4" ViewCount="184" Body="&lt;p&gt;I would like to know examples of mathematical research which had its origin in engineering problems (I am mainly interested in mechanical engineering and civil engineering) and which was actually helpful to solve those (or related) problems. There is a similar question &lt;a href=&quot;https://hsm.stackexchange.com/q/14591/5989&quot;&gt;here&lt;/a&gt; (whose only answer is relevant to my question), but it's about mathematical &lt;em&gt;definitions&lt;/em&gt; rather than research.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="6965" LastEditDate="2022-10-10T06:28:23.603" LastActivityDate="2022-10-11T02:47:11.870" Title="Engineering problems leading to mathematical research" Tags="|mathematics|engineering|big-list|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14851" PostTypeId="1" CreationDate="2022-10-04T18:56:17.347" Score="2" ViewCount="94" Body="&lt;p&gt;The new Nobel laureates in physics have been given the prize because of their contributions to quantum mechanics (QM).&lt;/p&gt;&#xA;&lt;p&gt;Of course, the Nobel prize focusses more on groundbreaking work that has links to applications and experiments, rather than on purely theoretical foundational questions.&lt;/p&gt;&#xA;&lt;p&gt;Still, it would be interesting to know what the new Nobels think about the foundations of their field, especially considering the renaissance that foundational questions in QM seem to have enjoyed 'recently'.&lt;/p&gt;&#xA;" OwnerUserId="12525" LastActivityDate="2022-10-04T18:56:17.347" Title="What are the favorite interpretations of quantum mechanics by the recent Nobel laureates in physics?" Tags="|physics|quantum-mechanics|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14852" PostTypeId="1" AcceptedAnswerId="14853" CreationDate="2022-10-05T10:27:16.380" Score="1" ViewCount="86" Body="&lt;p&gt;There is a theorem due to Markov, called &lt;em&gt;Markov's Law of Large Numbers&lt;/em&gt; which goes by:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The weak law of large numbers holds if for some &lt;span class=&quot;math-container&quot;&gt;$\delta &amp;gt; 0,$&lt;/span&gt; all the mathematical expectations &lt;span class=&quot;math-container&quot;&gt;$\mathbb E\left(|X_i|^{1+\delta}\right);~ i = 1,2,\ldots$&lt;/span&gt; exist and are bounded.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I didn't find any proof. So, I started sieving down the sources. Seneta (cf.&lt;span class=&quot;math-container&quot;&gt;$\rm [I]$&lt;/span&gt;) mentions that but didn't provide any proof. It was pinpointed to Markov's &lt;em&gt;Ischislenie Veroiatnostei&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$(1913)$&lt;/span&gt; chapter &lt;span class=&quot;math-container&quot;&gt;$\rm III.$&lt;/span&gt; Unfortunately, it was written in Russian (is there any English translation?) and I couldn't make anything of it. Also, I was wondering how he derived it without using measure theoretic language. Nevertheless, I checked some of my probability books like Rosenthal, Klenke but couldn't find any explicit result and its derivation. Feller though mentioned Markov in a footnote but again didn't provide any explicit elaboration.&lt;/p&gt;&#xA;&lt;p&gt;So, that leaves me wondering: who was the first English author that incorporated the result with a proof in their treatise?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\rm [I]$&lt;/span&gt; A Tricentenary history of the Law of Large&#xA;Numbers, Eugene Seneta, &lt;span class=&quot;math-container&quot;&gt;$2013, $&lt;/span&gt; p. &lt;span class=&quot;math-container&quot;&gt;$27.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="17132" LastEditorUserId="17132" LastEditDate="2022-10-05T17:57:48.417" LastActivityDate="2022-10-06T07:47:13.057" Title="Who was the first English author that incorporated Markov's Law of Large Numbers and its derivation?" Tags="|probability|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14853" PostTypeId="2" ParentId="14852" CreationDate="2022-10-05T13:52:33.583" Score="4" Body="&lt;p&gt;In 1937, Markov's student &lt;a href=&quot;https://en.wikipedia.org/wiki/J._V._Uspensky&quot; rel=&quot;nofollow noreferrer&quot;&gt;J. Uspensky&lt;/a&gt; published his  English-language  &lt;em&gt;Introduction to Mathematical Probability&lt;/em&gt;, which has the result in question on p.191.  It seems hard to believe this is the first English publication, however.&lt;/p&gt;&#xA;&lt;p&gt;It is not present in the 3rd (1979) edition of Cramer's &lt;em&gt;Random Variables&lt;/em&gt;, and so it might not be in the 1st (1936) edition, but I don't have one at hand to check.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="12330" LastEditDate="2022-10-06T07:47:13.057" LastActivityDate="2022-10-06T07:47:13.057" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14854" PostTypeId="1" CreationDate="2022-10-06T04:10:14.040" Score="1" ViewCount="200" Body="&lt;p&gt;In the &lt;a href=&quot;https://archive.org/details/foundationsofthe00kolm/page/n83/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;archived&lt;/a&gt; version of Kolmogorov's &lt;em&gt;Foundations of the Theory of Probability,&lt;/em&gt; at the very end of the book, p. &lt;span class=&quot;math-container&quot;&gt;$84,$&lt;/span&gt; few books have been listed, one being E. Hopf's &lt;em&gt;Ergodentheorie,&lt;/em&gt; where it is mentioned&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Measure theoretic viewpoints are preferred over topological ones throughout because, as the author says, &lt;em&gt;ergodic theory is statistics and statistics is measure theory.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ch62X.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ch62X.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have done preliminary googling but so far I didn't come across any helpful site or source that mentions his quote, the time (at least earlier than &lt;span class=&quot;math-container&quot;&gt;$1937,$&lt;/span&gt; the publication year of the said book) he asserted that or the context.&lt;/p&gt;&#xA;&lt;p&gt;So, the question is: &lt;strong&gt;when and where did E. Hopf assert the concerned statement?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2022-10-06T11:11:36.397" Title="When did E. Hopf say &quot;ergodic theory is statistics and statistics is measure theory&quot;?" Tags="|quote|statistics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14856" PostTypeId="2" ParentId="14854" CreationDate="2022-10-06T10:59:27.447" Score="2" Body="&lt;p&gt;The blurb shown in the question quotes from a review of Eberhard Hopf's book &lt;em&gt;Ergodentheorie&lt;/em&gt;, Berlin: Springer 1937, by Béla Szőkefalvi-Nagy in &lt;em&gt;Acta Scientiarum Mathematicarum (Szeged)&lt;/em&gt;, Vol. 9, No. 3, 1938-40, pp. 197-198 (&lt;a href=&quot;http://pub.acta.hu/acta/showCustomerVolume.action?id=5694&amp;amp;dataObjectType=volume&amp;amp;noDataSet=true&amp;amp;style=&quot; rel=&quot;nofollow noreferrer&quot;&gt;volume online&lt;/a&gt;). The relevant sentence is on page 197 and reads as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Maßtheoretische Gesichtspunkte nehmen in der ganzen Darstellung den Vorrang vor den topologischen ein, da, wie der Verfasser betont, Ergodentheorie Statistik ist und Statistik Maßtheorie.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The language pretty clearly indicates that the reviewer is quoting Hopf &lt;em&gt;from the book under review&lt;/em&gt;, but I do not have access to the book to verify that. I am making this a community wiki answer to allow easy addition of this information at a later stage.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-10-06T11:11:36.397" LastActivityDate="2022-10-06T11:11:36.397" CommentCount="3" CommunityOwnedDate="2022-10-06T11:11:36.397" ContentLicense="CC BY-SA 4.0" />
  <row Id="14857" PostTypeId="1" CreationDate="2022-10-06T16:24:45.920" Score="1" ViewCount="163" Body="&lt;p&gt;I always have been curious about this part of the History of Science. To claim that Earth is orbiting the Sun instead of the opposite is equivalent to change one absolute referential (Earth) to another one (the Sun); something Galilean relativity forbids. So how come Galileo is credited for both the statement and the theory?&lt;/p&gt;&#xA;" OwnerUserId="17346" LastActivityDate="2022-10-31T12:32:45.890" Title="How Galileo could both possibly say that Earth is revolving around the Sun and develop the Galilean relativity?" Tags="|physics|classical-mechanics|galileo|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="14858" PostTypeId="1" AcceptedAnswerId="14861" CreationDate="2022-10-07T06:49:47.033" Score="1" ViewCount="158" Body="&lt;p&gt;Ptolemy's model is less accurate than Copernicus', but it is said to have been in use for a long time.&lt;/p&gt;&#xA;&lt;p&gt;What was the Ptolemaic model be used for?&lt;/p&gt;&#xA;" OwnerUserId="17348" LastEditorUserId="131" LastEditDate="2022-10-07T13:03:55.480" LastActivityDate="2022-10-07T14:29:27.663" Title="What was the Ptolemaic system used for?" Tags="|astronomy|ptolemy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14859" PostTypeId="1" AcceptedAnswerId="14860" CreationDate="2022-10-07T11:48:32.663" Score="0" ViewCount="110" Body="&lt;p&gt;In a &lt;a href=&quot;https://www.dbnl.org/tekst/huyg003oeuv10_01/huyg003oeuv10_01_0059.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;latter to Huygens&lt;/a&gt;, Leibniz wrote &lt;em&gt;Nam in applicatione Geometriae ad &lt;strong&gt;Physicam&lt;/strong&gt; saepissime contingit&lt;/em&gt;, which is traslated by &lt;a href=&quot;https://books.google.com.br/books?id=pO1GDgAAQBAJ&amp;amp;pg=PA67&amp;amp;lpg=PA67&amp;amp;dq=%22For%20in%20the%20application%20of%20geometry%20to%20physics%20it%20happens%20quite%20often%20that%22&amp;amp;source=bl&amp;amp;ots=cN_Lwx9tlP&amp;amp;sig=ACfU3U3CmRb_bKJPr96N-vLEJttNgqIcUg&amp;amp;hl=pt-BR&amp;amp;sa=X&amp;amp;ved=2ahUKEwj2kK6Pgs76AhUtqpUCHcAcB2gQ6AF6BAgJEAM#v=onepage&amp;amp;q=%22For%20in%20the%20application%20of%20geometry%20to%20physics%20it%20happens%20quite%20often%20that%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blåsjö (p. 67)&lt;/a&gt; as &lt;em&gt;For in the application of geometry to &lt;strong&gt;physics&lt;/strong&gt; it happens quite often&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I would like to know what the meaning of &amp;quot;physics&amp;quot; (Physicam) was at that time (particularly for Leibniz).&lt;/p&gt;&#xA;" OwnerUserId="2611" LastActivityDate="2022-10-07T12:25:14.823" Title="What did physics mean in 17th century?" Tags="|physics|17th-century|leibniz|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14860" PostTypeId="2" ParentId="14859" CreationDate="2022-10-07T12:25:14.823" Score="5" Body="&lt;p&gt;In his letter, I would assume he's referring to a problems in astronomy and mechanics. But the term physics was much more broad up until the 19th century. This is covered in good detail in e.g. J. L. Heilbron's &lt;em&gt;Elements of Early Modern Physics&lt;/em&gt; (1982), chapter &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\S 1$&lt;/span&gt;, &amp;quot;The Scope of 'Physics'&amp;quot;. Some relevant lines:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At the beginning of the seventeenth century 'physics' signified a qualitative, bookish science of natural bodies in general. It was at once wider and narrower than the subject that now has its name: wider in its coverage, which included organic and psychological as well as inorganic phenomena; and narrower in its methods, which recommended neither mathematics nor experiment. The width of coverage and the depreciation of mathematics derived from Aristotle; the indifference to experiment, as opposed to everyday experience, from the authors of peripatetic textbooks.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He continues,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;'Physics' continued to be understood in its Aristotelian extent throughout the seventeenth century. Molière's bourgeois gentilhomme [(1667)] asks his philosophical tutor what physics is and receives in reply, '[the science] that explains the principles of natural things, and the properties of bodies; that discourses about the nature of the elements, metals, minerals, stones, plants and animals; and [that] teaches us the cause of all the meteors.' Molière's friend, the Cartesian physicist Jacques Rohault, says the same ('the science that teaches us the reasons and causes of all the effects that Nature produces') and he tries to give an account of everything, including human psychology, in a physics text that had a peculiarly long life [Rohault, &lt;em&gt;Traité&lt;/em&gt; (1692)]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The transition to the modern meaning came much later; there's not a clear cutoff, but the transition was still slow by the late 1700s. For example, Heilbron cites the &lt;em&gt;Journal de physique&lt;/em&gt; (founded 1773), which emphasized natural history, and various sources up to the year 1800 which considered physics to include agriculture and biology.&lt;/p&gt;&#xA;&lt;p&gt;To summarize: Leibniz' concept of physics (Lat. &lt;em&gt;physica&lt;/em&gt;) was a general term including what we would today call astronomy, biology, natural history and psychology. It's safe to assume he's referring in his letter to the more exact of these, which admit mathematical models.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2022-10-07T12:25:14.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14861" PostTypeId="2" ParentId="14858" CreationDate="2022-10-07T14:23:36.330" Score="3" Body="&lt;p&gt;One specific application was the eclipse prediction. For example, Columbus used a predicted Lunar eclipse at least once, trying to determine his longitude. He was not very successful.&lt;/p&gt;&#xA;&lt;p&gt;Ref. S. E. Morison, Admiral of the ocean sea: a life of Christopher Columbus.&lt;/p&gt;&#xA;&lt;p&gt;One of the main applications in the Middle age was &lt;a href=&quot;https://en.wikipedia.org/wiki/Date_of_Easter&quot; rel=&quot;nofollow noreferrer&quot;&gt;computus&lt;/a&gt;, computation of the date of Easter, which is changing every year. Ptolemy's theory of Sun and the Moon was used.&lt;/p&gt;&#xA;&lt;p&gt;Remark. Applications to astrology were very limited. Astrologers used (and still use) a more ancient system, of Babylonian origin. Astrologers were not very interested in advancement of science, even at the time of Ptolemy. Of course, there were exceptions.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-10-07T14:29:27.663" LastActivityDate="2022-10-07T14:29:27.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14862" PostTypeId="1" AcceptedAnswerId="14863" CreationDate="2022-10-07T16:26:33.917" Score="13" ViewCount="1554" Body="&lt;p&gt;In &lt;a href=&quot;https://www.youtube.com/watch?v=wu9BwLoN5lw&amp;amp;t=298s&quot; rel=&quot;nofollow noreferrer&quot;&gt;an interview&lt;/a&gt; following Aspect's winning of the Nobel Prize in Physics, he claims that &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Stewart_Bell&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Stuart Bell&lt;/a&gt; discouraged him from pursuing his now-famous 1982 experiment on quantum entanglement. The reason for this being that it would somehow hurt his career as a researcher and that questions such as quantum nonlocality were of &amp;quot;no interest&amp;quot; to the physics community.&lt;/p&gt;&#xA;&lt;p&gt;I'm posting this question here instead of in Physics SE because at its core it is a matter of academic politics. What was it in the &amp;quot;research climate&amp;quot; of the late 1970s that would have led to such scorn towards an area that is now worth the Nobel prize?&lt;/p&gt;&#xA;" OwnerUserId="17352" OwnerDisplayName="Tfovid" LastEditorUserId="3869" LastEditDate="2022-10-09T03:14:20.743" LastActivityDate="2022-10-09T03:14:20.743" Title="Why was Alain Aspect discouraged from doing his now Nobel-winning work?" Tags="|physics|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14863" PostTypeId="2" ParentId="14862" CreationDate="2022-10-07T20:41:54.557" Score="13" Body="&lt;p&gt;The discouragement probably had to do with the stigma attached to &amp;quot;hidden variables&amp;quot; after the Einstein-Bohr debates, see &lt;a href=&quot;https://www.if.ufrj.br/%7Ecarlos/fismod/seminarios/interpretacaoBohm/Bohm_PinchPhysEd1979.pdf&quot; rel=&quot;noreferrer&quot;&gt;Pinch, The hidden variables controversy in quantum physics&lt;/a&gt; for some social context. Quoting Jauch (1973):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;...the discussions which surround the quest for hidden variables in quantum mechanics have, on both sides of the camp, often been conducted in a spirit of aggressiveness which resembles more the defence of orthodoxy of one ideology than a spirit of scientific objectivity.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For a while, they were considered ruled out by von Neumann's 'no hidden variables' theorem. They weren't, as Bohm and Bell showed. But, as Pinch writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Much of the ‘heat’ generated by the hidden variables controversy can, I think, be traced to the extraordinary faith which some people placed in von Neumann’s arguments. Clearly, hidden-variable theorists were likely to interpret this as a blank refusal to believe, based on prejudice and dogma, while the establishment, on the other hand, was likely to see the refusal to accept von Neumann’s arguments (which after all had in turn never been shown to be invalid) as irresponsible and likely to open once more the doors to pointless speculation over the foundations of the&#xA;theory.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1966 Bell showed that von Neuman's proof was essentially circular, in it hidden variables were ruled out not by quantum mechanics but by his extra assumption that was not even reasonable. This &amp;quot;notorious gap in von Neumann’s argument&amp;quot; is now broadly recognized, see &lt;a href=&quot;https://link.springer.com/article/10.1007/s10701-018-0197-5&quot; rel=&quot;noreferrer&quot;&gt;Mermin, Homer Nodded: Von Neumann’s Surprising Oversight&lt;/a&gt;, but the attitudes were already set and slow to shift. Even those who did not rely on it perceived the search for hidden variables as akin to the futile search for aether, and Bohmian mechanics still has a bad reputation even now. As Pinch puts it:&amp;quot;&lt;em&gt;After the earlier battles to get the Copenhagen interpretation&#xA;accepted at the famed Solvay congresses in&#xA;the 1920s it was hardly likely that the physics elite would let the foundations of the theory be undermined by the post-war generation of physicists.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But in 1970s hidden variables still were the only approach to &amp;quot;normalizing&amp;quot; (in Einstein's sense) quantum non-locality, and Bell saw it that way. Objective collapse theories that he eventually switched to only appeared in 1980s. In &lt;a href=&quot;https://www.google.com/books/edition/Quantum_Profiles/M7fRDwAAQBAJ?hl=en&quot; rel=&quot;noreferrer&quot;&gt;Bernstein's Quantum Profiles&lt;/a&gt; he is quoted as saying:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I feel that Einstein’s intellectual superiority over Bohr, in this instance, was enormous; a vast gulf between the man who saw clearly what was&#xA;needed, and the obscurantist... What worried me then was how to get rid of that division&lt;/em&gt; [between classical and quantum behavior]. &lt;em&gt;I was looking for some reformulation of the theory that would permit its elimination. It was clear to me then that the hidden-variable approach would be one such formulation. If you gave definite properties — ‘hidden variables’ — to the elementary quantum particles, you don’t have to be concerned that the classical apparatus has definite properties. Everything has definite properties. It is just that they are more under our control for big things than for little things. It was clear to me that that was one line of possible development.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-10-08T05:57:10.283" LastActivityDate="2022-10-08T05:57:10.283" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14864" PostTypeId="1" CreationDate="2022-10-07T22:16:10.400" Score="3" ViewCount="221" Body="&lt;p&gt;We know using modern analysis techniques that &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt; can be computed by their Taylor series (in fact the Taylor series are given as the definitions of these functions in today's real analysis books).&lt;/p&gt;&#xA;&lt;p&gt;What I am wondering is, how were &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; functions computed before the notion of Taylor series was invented/discovered? Was it simple done by measurement and logging results into tables? I'd be very interested to know.&lt;/p&gt;&#xA;&lt;p&gt;I am aware that people worked with functions that are related to but not the same as &lt;span class=&quot;math-container&quot;&gt;$\sin x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cos x$&lt;/span&gt;. The answers &lt;a href=&quot;https://hsm.stackexchange.com/questions/2148&quot;&gt;in this post&lt;/a&gt; explain that the chord function was used before sine and cosine, but my question still stands.&#xA;All the trigonometric functions are related to one another, and one can equally ask, how was e.g. the chord function computed before the notion of Taylor series?&lt;/p&gt;&#xA;" OwnerUserId="6965" LastActivityDate="2022-10-07T22:16:10.400" Title="How were sine and cosine functions computed before the notion of Taylor series?" Tags="|real-analysis|computation|trigonometry|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14865" PostTypeId="1" CreationDate="2022-10-07T22:44:33.450" Score="5" ViewCount="133" Body="&lt;p&gt;Motivated by this year's &lt;a href=&quot;https://www.nobelprize.org/prizes/physics/2022/press-release/&quot; rel=&quot;noreferrer&quot;&gt;Nobel prize in physics&lt;/a&gt;, I was wondering whether there are recorded statements by &lt;a href=&quot;https://de.wikipedia.org/wiki/Nathan_Rosen&quot; rel=&quot;noreferrer&quot;&gt;Nathan Rosen&lt;/a&gt; (the R in the &lt;a href=&quot;https://en.wikipedia.org/wiki/EPR_paradox&quot; rel=&quot;noreferrer&quot;&gt;EPR-paradox&lt;/a&gt;) about the Bell inequality and its violation by this years laureates?&lt;/p&gt;&#xA;&lt;p&gt;Rosen, 1909-1995, was the only one alive of the trio when Bell made the theoretical discoveries (1964), and Clauser's first experiment (1974) and even Aspect's refined experiment in the late 1980s.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia and Google search did not reveal anything about this historic debate.&lt;/p&gt;&#xA;" OwnerUserId="9716" LastEditorUserId="10106" LastEditDate="2022-10-08T06:26:31.733" LastActivityDate="2022-10-14T19:55:42.037" Title="What did Nathan Rosen (from EPR) say about Bell's inequality and its violation?" Tags="|physics|quantum-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14866" PostTypeId="2" ParentId="14835" CreationDate="2022-10-09T15:43:11.113" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bruno_de_Finetti&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bruno de Finetti&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;From &amp;quot;Probabilism&amp;quot;, Erkenntnis, vol. 31, No. 2/3 (1989), pp. 169-223 (translation of a 1931 article in Italian)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But where my spirit rebelled most ferociously and clashed against the concept of &amp;quot;absolute truth&amp;quot; was in the political field, and I could not say what part, surely very great, this sense of impatient revolt must have had in the development of my ideas. To be confronted by papier-mâché idols and a miserable political class that would have preferred Italy in ruins rather than failing (sacrilege!) to render due homage! Those delicious absolute truths that stuffed the demo-liberal brains! That impeccable rational mechanics of the perfect civilian regime of the peoples, conforming to the rights of man and various other immortal principles! October of '22! It seemed to me I could see them, these Immortal Principles, as filthy corpses in the dust. And with what conscious and ferocious voluptuousness I felt myself trampling them, marching to hymns of triumph, obscure but faithful Blackshirt!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From &amp;quot;La crisi dei principi e l'economia matematica&amp;quot;, Acta Seminarii, 2 (1943), pp. 33-68&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Facciamo che quando la vita, cessato il turbine della lotta, tornerà a rifiorire, liberata dagli egoismi che pretendevano soffocarla, siamo in grado di indicarle una nuova via meglio concepita, che superi l'assurda artificiale barriera elevata tra l'uomo e la possibilità di benessere largitegli dalla natura o conquistate dal suo ingegno. Soltanto allora la guerra &amp;quot;del sangue contro l'oro&amp;quot; sarà veramente vinta. Come vogliamo, come è necessario.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let us ensure that when life, having ceased the whirlwind of struggle, flourishes again, freed from the selfishness that claimed to stifle it, we are able to point it to a new and better conceived way, one that overcomes the absurdly artificial high barrier between man and the possibility of prosperity bestowed upon him by nature or conquered by his cleverness.&#xA;Only then will the war &amp;quot;of blood against gold&amp;quot; be truly won.&#xA;It is what we want, it is what is necessary.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17362" LastActivityDate="2022-10-09T15:43:11.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14867" PostTypeId="1" CreationDate="2022-10-09T16:52:04.033" Score="4" ViewCount="248" Body="&lt;p&gt;According to Wikipedia’s web page dedicated to &lt;a href=&quot;https://en.wikipedia.org/wiki/Thales_of_Miletus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thales of Miletus&lt;/a&gt;, he came to the idea to &lt;em&gt;prove the equality of vertical angles as a theorem&lt;/em&gt; when he saw that the Egyptians, after intersecting two lines, would overlay an angle over the opposite to check their equality. But Thales concluded that this measurement was unnecessary and showed this by &amp;quot;proof.&amp;quot; Whether he indeed proved it or whether he was the first to do so is not relevant here.&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia page gives reference to a book.&lt;/p&gt;&#xA;&lt;p&gt;Shute, William George; Shirk, William W.; Porter, George F. (1960). &lt;em&gt;Plane and Solid Geometry&lt;/em&gt;. American Book Company. pp. 25–27.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, this book has no references.&lt;/p&gt;&#xA;&lt;p&gt;In Michael Molinsky's &lt;em&gt;Convergence&lt;/em&gt; article &lt;a href=&quot;https://www.maa.org/book/export/html/692718&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some Original Sources for Modern Tales of Thales - Other Contributions to Geometry&lt;/a&gt;, there is no reference to this story.&lt;/p&gt;&#xA;&lt;p&gt;Finally, I did not find such a reference in Proclus’ &lt;em&gt;Commentary on the first book of Euclid's Elements.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is this story supported by a genuine reference?&lt;/p&gt;&#xA;" OwnerUserId="17334" LastActivityDate="2022-10-10T11:18:27.433" Title="Thales and measuring vertical angles" Tags="|ancient-greece|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14868" PostTypeId="2" ParentId="14867" CreationDate="2022-10-09T18:20:03.537" Score="6" Body="&lt;p&gt;Yes, it is, but only partially. Moreover, the ancient sources reporting the episode(s) are notoriously not the most reliable, and the story, as reported by Wikipedia and its sources, is probably fabricated merging together different quotes relating to different results.&lt;/p&gt;&#xA;&lt;p&gt;These are &lt;em&gt;all&lt;/em&gt; the ancient sources that report in some way the episode of Thales' stay in Egypt.&lt;/p&gt;&#xA;&lt;p&gt;Pliny the Elder, &lt;a href=&quot;https://en.wikipedia.org/wiki/Natural_History_(Pliny)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Naturalis Historia&lt;/em&gt;&lt;/a&gt;, XXXVI, 82:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thales of Miletus discovered how to find the measure of the height [of the pyramids] by measuring their shadow at the moment when it is equal [to that of] the projecting body.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Plutarch, &lt;em&gt;Moralia&lt;/em&gt;, &lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A2008.01.0186%3Asection%3D2&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Septem sapientium convivium&lt;/em&gt;, 2&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[here, one of the wise men, Niloxenus, is addressing Thales directly] and whereas he honors you for divers great accomplishments, he particularly admires you for this invention, that with little labor and no help of any mathematical instrument you took so truly the height of one of the pyramids; for fixing your staff erect at the point of the shadow which the pyramid cast, two triangles being thus made by the tangent rays of the sun, you demonstrated that what proportion one shadow had to the other, such the pyramid bore to the stick.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Diogenes Laertius, &lt;a href=&quot;https://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_I#Thales&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Lives and Opinions of Eminent Philosophers&lt;/em&gt;, I, 24&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pamphila states that, having learnt geometry from the Egyptians, he was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox. Others tell this tale of Pythagoras, amongst them Apollodorus the arithmetician.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Ibidem, I, 27:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He had no instructor, except that he went to Egypt and spent some time with the priests there. Hieronymus informs us that he measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Proclus does not says the Thales prove the theorem when he saw that the Egyptians intersecting two lines.&#xA;Instead, he says that Thales brought geometry to Greece from Egypt (65, 3) and reports other results concerning the fact that the diameter divides the circle into two equal parts (157, 10), that the angles at the base of an isosceles triangle are equal (250, 20), and that two triangles with one side and two adjacent angles are equal (352, 14).&#xA;About your question, the relevant part is in &lt;a href=&quot;https://archive.org/details/proclidiadochii00friegoog/page/n310/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;299, 1&lt;/a&gt; (see &lt;a href=&quot;https://universaltheosophy.com/tt/1792_The-Philosophical-and-Mathematical-Commentaries-of-Proclus-on-the-First-Book-of-Euclids-Elements_vols-1-2.pdf&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for the English translation):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proposition XV, theorem VIII&lt;/p&gt;&#xA;&lt;p&gt;If two right lines cut one another, they will form the angles at the vertex equal.&lt;/p&gt;&#xA;&lt;p&gt;We must call successive angles different from such as are vertical. For these last originate from the section of two right lines: but the former from the mere dissection of the one by another. [...] But if the two right lines mutually cut each other, they form &lt;em&gt;vertical&lt;/em&gt; angles. [...] This, therefore, is what the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented (according to &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudemus_of_Rhodes&quot; rel=&quot;noreferrer&quot;&gt;Eudemus&lt;/a&gt;) by Thales: but was thought worthy of a demonstration producing science by the institutor of the Elements. But it is not exhibited from all the particulars requisite to a perfect proposition. For construction is wanting in the present theorem: but demonstration, which must be necessarily inherent, depends on the thirteenth therem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In conclusion, even if one wants to rely on Proclus' testimony through Eudemus, the story that Thales thought up the demonstration of the theorem by observing the behaviour of the Egyptians is not based on sources, and (although it may be plausible in some respects) seems rather fabricated by combining Proclus' quotation with the episode cited by Pliny, Plutarch and Diogenes concerning the measurement of the height of the pyramids.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, observe that ancient sources, when reporting Thales' &amp;quot;scientific&amp;quot; achievements, dwell rather on his contribution to astronomy: so does the &lt;a href=&quot;https://en.wikipedia.org/wiki/Suda&quot; rel=&quot;noreferrer&quot;&gt;Suda&lt;/a&gt;, but also &lt;a href=&quot;https://en.wikipedia.org/wiki/Callimachus&quot; rel=&quot;noreferrer&quot;&gt;Callimachus&lt;/a&gt; (in &lt;em&gt;Iambs&lt;/em&gt;, &lt;a href=&quot;https://archive.org/details/pt7oxyrhynchuspa00grenuoft/page/32/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Papyrus Oxyrhynchus VII, 33&lt;/a&gt;), &lt;a href=&quot;https://en.wikipedia.org/wiki/Aetius_(philosopher)&quot; rel=&quot;noreferrer&quot;&gt;Aetius&lt;/a&gt; (in many passages of &lt;em&gt;De Placita Philosophorum&lt;/em&gt;), &lt;a href=&quot;https://en.wikipedia.org/wiki/Dercyllides&quot; rel=&quot;noreferrer&quot;&gt;Dercyllides&lt;/a&gt; (cited by &lt;a href=&quot;https://en.wikipedia.org/wiki/Theon_of_Smyrna&quot; rel=&quot;noreferrer&quot;&gt;Theon of Smyrna&lt;/a&gt;) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Apuleius&quot; rel=&quot;noreferrer&quot;&gt;Apuleius&lt;/a&gt; (in &lt;em&gt;Florida&lt;/em&gt;).&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2022-10-10T11:18:27.433" LastActivityDate="2022-10-10T11:18:27.433" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14869" PostTypeId="2" ParentId="14811" CreationDate="2022-10-10T00:06:28.100" Score="1" Body="&lt;p&gt;From Bruno de Finetti, &amp;quot;L'opera di Abraham Wald e l'assestamento concettuale della statistica matematica moderna&amp;quot;, Statistica, 11 (1951), pp. 185-192&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;L'idea essenziale dei procedimenti sviluppati sotto il nome di «Sequential Analysis» ed esposti nel volume di ugual titolo è altrettanto feconda quanto semplice. Per afferrarne il senso, basta esporla riferendoci all'esempio più elementare. Si tratti di collaudare una partita di oggetti, sottoponendo a date prove un certo campione per concludere, in base alla percentuale di pezzi difettosi riscontrativi, se la percentuale di oggetti difettosi nell'intera partita sembri tale da consigliare di accettarla o di respingerla.&#xA;Quanti pezzi occorre sottoporre a collaudo prima di poter prendere tale decisione?&#xA;Non si tratta di entrare nel merito dei criteri in base ai quali tale numero può venir determinato; si tratta di discutere se è opportuno porsi la questione in tali termini, fissare cioè un numero &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; di prove, in base al risultato delle quali prendere la decisione.&#xA;L'osservazione, in fondo intuitiva, del Wald, consiste nel notare che se fin dal principio i risultati sono decisamente favorevoli o decisamente sfavorevoli si è in grado di prendere una decisione dopo poche prove, mentre solo se la conclusione si mantiene sul limite dell'incertezza occorre prolungare l'esperimentazione e aumentare il numero delle prove. Analizzando tale concetto sia sotto l'aspetto probabilistico che in nesso a circostanze quali il costo dell'esperimentazione e l'entità del danno derivante da una decisione erronea, il Wald è pervenuto a costruire una dottrina altrettanto elegante e pregevole dal punto di vista matematico quanto appropriata e vantaggiosa per le applicazioni pratiche.&#xA;Il fatto di sospendere le prove non appena il risultato acquisito appaia sufficiente elemento per la decisione, anziché continuarle in ogni caso fino al numero necessario nel caso più sfavorevole di decisione incerta, costituisce infatti un risparmio così notevole di tempo, lavoro e materiale nell'esecuzione dei collaudi, che il procedimento stesso era stato tenuto segreto durante la guerra perché potessero avvantaggiarsene in modo esclusivo le attività connesse con la preparazione militare alleata.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The essential idea of the procedures developed under the name of «Sequential Analysis» and set forth in the volume of the same title is as fruitful as it is simple.&#xA;To grasp its meaning, we need only expound it by referring to the most basic example. Let it be a matter of testing a batch of objects, subjecting a certain sample to given tests in order to conclude, on the basis of the percentage of defective pieces found, whether the percentage of defective objects in the whole batch seems to be such as to recommend acceptance or rejection.&#xA;How many pieces need to be tested before such a decision can be made?&#xA;It is not a question of going into the criteria by which this number can be determined; it is a question of discussing whether it is appropriate to pose the question in such terms, that is, to fix a &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; number of tests, on the basis of the result of which to make the decision.&#xA;Wald's basically intuitive observation is that if from the outset the results are decidedly favorable or decidedly unfavorable one is able to make a decision after only a few trials, whereas only if the conclusion remains on the borderline of uncertainty is it necessary to prolong the experiment and increase the number of trials. By analyzing this concept both from a probabilistic point of view and in connection with circumstances such as the cost of experimentation and the extent of damage resulting from an erroneous decision, Wald has come to construct a doctrine that is as elegant and valuable mathematically as it is appropriate and advantageous for practical applications.&#xA;The fact of suspending the tests as soon as the result acquired appears to be sufficient element for decision, instead of continuing them in any case up to the necessary number in the most unfavorable case of an uncertain decision, constitutes in fact such a considerable saving of time, labor and material in the execution of the tests, that the proceedings themselves had been kept secret during the war so that only activities connected with Allied military production could benefit from them.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17362" LastActivityDate="2022-10-10T00:06:28.100" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14870" PostTypeId="2" ParentId="14850" CreationDate="2022-10-10T00:58:54.247" Score="6" Body="&lt;p&gt;Here are a few examples.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Wavelets.  There was early work by Haar, but the subject became intensively studied only decades later, in part due to work by the engineer Jean Morlet, who in fact introduced the term wavelet (well, the French term ondelette).  The history of the subject before it really took off in the 1980s involved a lot of independent rediscoveries.  See the paper &amp;quot;The Story of Wavelets&amp;quot; by R. Polikar.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Information theory. This was born in Shannon's book &lt;em&gt;The Mathematical Theory of Communication&lt;/em&gt; in the 1940s. He worked at Bell Labs and was motivated by practical problems of telephone communication.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Coding theory (error-correcting codes). This is due to Hamming, another employee of Bell Labs. A famous quote of his, related to having to restart computer programs after they stopped two weekends in a row due to mistakes: &amp;quot;Damn it, if the machine can detect an error, why can't it locate the position of the error and correct it?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2022-10-11T02:47:11.870" LastActivityDate="2022-10-11T02:47:11.870" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14871" PostTypeId="1" CreationDate="2022-10-10T05:38:47.770" Score="1" ViewCount="95" Body="&lt;p&gt;Hydrogen sulfide was previously named &amp;quot;sulphuretted hydrogen&amp;quot; but I can't find the person who named/coined it. Although Carl Wilhelm Scheele is credited to have discovered and isolated the purified version of gas, he failed the recognize the real nature of gas and hence received names such as &amp;quot;liver of sulphur air&amp;quot;, &amp;quot;hepatic air&amp;quot;. From &lt;a href=&quot;http://sulphur.atomistry.com/hydrogen_sulphide.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1777, Scheele obtained it by the action of acids on calcium polysulfide and also on manganese sulfide and ferrous sulfide; he observed the solubility of the gas in water and its oxidation to free Sulphur by atmospheric air and other oxidizing agents. On account of the phlogistic views prevalent at the time, however, Scheele and his contemporaries failed to recognize the real nature of the gas, which received such names as &amp;quot;liver of sulphur air,&amp;quot; &amp;quot;hepatic air.&amp;quot; The gas was first recognized as an oxygen-free acid by Berthollet in 1796.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On the other hand, Heinrich Buff and Friedrich Woehler discovered &lt;a href=&quot;https://en.wikipedia.org/wiki/Silane&quot; rel=&quot;nofollow noreferrer&quot;&gt;silane&lt;/a&gt; from the action of hydrochloric acid on aluminum silicide. They called the compound &lt;em&gt;siliciuretted hydrogen&lt;/em&gt;. In this case, we know that the discoverers has named SiH&lt;sub&gt;4&lt;/sub&gt; &amp;quot;siliciuretted hydrogen&amp;quot; before it was changed to silane but this wasn't the case for hydrogen sulfide.&lt;/p&gt;&#xA;&lt;p&gt;Questions:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Who/when was H&lt;sub&gt;2&lt;/sub&gt;S named &amp;quot;sulphuretted hydrogen&amp;quot;?&lt;/li&gt;&#xA;&lt;li&gt;A variation of &lt;a href=&quot;https://en.wikipedia.org/wiki/Water_gas&quot; rel=&quot;nofollow noreferrer&quot;&gt;water gas&lt;/a&gt; is &amp;quot;carburetted water gas&amp;quot; which is still used today. The name sometimes refer to &amp;quot;something related to carbon gas&amp;quot; following the pattern of old name of other element hydrides. Did they always gave elemental gas compounds a prefix of -retted while naming them?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16966" LastEditorUserId="16966" LastEditDate="2022-10-16T07:57:11.900" LastActivityDate="2022-10-16T07:57:11.900" Title="Who coined the term &quot;sulphuretted hydrogen&quot;?" Tags="|terminology|chemistry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14874" PostTypeId="2" ParentId="14850" CreationDate="2022-10-10T22:50:03.197" Score="4" Body="&lt;p&gt;O. Heaviside used things akin to the Dirac delta prior to 1900, helping to design transatlantic telegraph cables. For that matter, G. Green did, c. 1838, implicitly, in treatment of fundamental solutions of PDE's, in the context of electromagnetic phenomena. Yes, P. Dirac used such ideas again, in the late 1920's, in physics. And S. Sobolev rigorized a certainly class of distributions/generalized functions in the 1930's (and subsequently). But it was only in the late 1940's that L. Schwartz rigorized a general theory of distributions/generalized functions in a robust fashion. In the early 1950's, A. Grothendieck finished up several important mathematical aspects of this.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2022-10-10T22:50:03.197" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14875" PostTypeId="2" ParentId="14865" CreationDate="2022-10-14T19:55:42.037" Score="1" Body="&lt;p&gt;Rosen N., Bell’s theorem and quantum mechanics, American Journal of Physics 62, 109 (1994); &lt;a href=&quot;https://doi.org/10.1119/1.17626&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1119/1.17626&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3329" LastActivityDate="2022-10-14T19:55:42.037" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14876" PostTypeId="1" CreationDate="2022-10-14T22:45:17.213" Score="2" ViewCount="498" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Dirac&quot; rel=&quot;nofollow noreferrer&quot;&gt;The wiki article on Dirac&lt;/a&gt; quotes Einstein as saying of Dirac&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I don't understand Dirac at all.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But did Einstein have anything to say about the Dirac equation itself?  I remember reading somewhere that he called the Schrodinger equation &amp;quot;a stroke of genius&amp;quot;.  So surely he must have at least been glad that Dirac had found the special relativistic version of it.  Did he offer any insight on what he thought the 4-component wavefunction or the gamma matrices could mean?  (I know they have since been elucidated to some degree by the work of e.g. Hestenes.)  Any thoughts about anti-matter, spin being an integral part of the wavefunction, or other consequences?&lt;/p&gt;&#xA;" OwnerUserId="16950" LastActivityDate="2022-10-14T22:45:17.213" Title="What did Einstein say about the Dirac equation?" Tags="|quantum-mechanics|relativity-theory|einstein|dirac|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14877" PostTypeId="1" AcceptedAnswerId="14918" CreationDate="2022-10-15T06:37:59.253" Score="2" ViewCount="129" Body="&lt;p&gt;I am looking for a specific book on non-Euclidean geometry that I read in my undergraduate years.&lt;/p&gt;&#xA;&lt;p&gt;The unique characteristic of this book is that the first part of the book started by re-proving in modern notation the entirety of the first book of Euclid's Elements – given that all the geometry up until Euclid used the parallel postulate would be common to Euclidean and non-Euclidean geometries.  The second part of the book treats a few equivalent formulations of the parallel postulate and the ways in which non-Euclidean geometry modifies them.&lt;/p&gt;&#xA;" OwnerUserId="297" LastEditorUserId="6545" LastEditDate="2022-10-15T15:20:51.840" LastActivityDate="2022-11-09T18:35:56.163" Title="Searching for book about non-Euclidean geometry that recapitulates the First Book of the Elements" Tags="|reference-request|geometry|euclidean-geometry|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14878" PostTypeId="1" CreationDate="2022-10-15T19:52:22.657" Score="1" ViewCount="269" Body="&lt;p&gt;Could anyone, please, point the exact place in &amp;quot;Principia Mathematica&amp;quot; where Newton explicitly stated the expression for the gravitational force between two particles? I mean the fact that it is proportional to the product of their masses and the inverse square of the distance between them. The closest thing I've found is Propositional 75, Theorem 35, Corollary 1 [Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I.Bernard Cohen. University of California Press 1999]:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;The attractions of spheres toward other homogeneous spheres are as the attracting spheres [i.e., as the masses of the attracting spheres] divided by the squares of the distances of their own centers from the centers of those that they attract.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;But this statement is about spheres, not particles. And also only about the proportionality to the mass of the attracting sphere, not the attracted one.&lt;/p&gt;&#xA;&lt;p&gt;Also this answer &lt;a href=&quot;https://hsm.stackexchange.com/questions/12880/what-was-newtons-statement-of-the-universal-law-of-gravitation&quot;&gt;What was Newton&amp;#39;s statement of the universal law of gravitation?&lt;/a&gt; points some propositionals, but they eather do not account all proportionalities, or not general enough (stated only for planets etc.)&lt;/p&gt;&#xA;&lt;p&gt;So, what (and where) is the most general form of the law present in this book?&lt;/p&gt;&#xA;" OwnerUserId="17384" LastActivityDate="2022-10-15T19:52:22.657" Title="Principia Mathematica and Law of Gravitation" Tags="|newton|gravity|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14879" PostTypeId="1" CreationDate="2022-10-15T23:18:57.540" Score="3" ViewCount="129" Body="&lt;p&gt;Research mathematics is seldom directly applicable, but the questions we study are motivated by interplay with contemporary demands. As a combinatorialist, it's apparent that my field has benefited immensely from the demands modern computing makes on mathematics. Prior to this development, combinatorics had applications (especially to statistics and probability) but did not enjoy as central a role as it occupies today.&lt;/p&gt;&#xA;&lt;p&gt;When I look to earlier technologies, it seems to me that automated loom devices such as the Jacquard loom ought to also rely in a critical way on combinatorial constructions. My understanding of the history of combinatorics comes from textbooks in the field as opposed to primary sources or the history of mathematics literature. However, I've never seen any mention of looms as a precursor to some modern theory.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there any work in the mathematics history literature exploring the connection of automated looms to mathematics? In particular, I'd like to understand if there is any work related to combinatorics.&lt;/p&gt;&#xA;&lt;p&gt;As some examples of what I'm hoping for, it seems plausible that the interesting classes of patterns were classified at some point. Presumably these results could be translated to precise combinatorial statements. Additionally, the the physical constraints of such a technology likely would lead to constraints on the types of instructions allowed. For example, if blue, red and green strands of thread are being incorporated into the cloth perhaps one cannot have blue then green then blue then red due to some physical shortcoming of the loom itself. One might also expect certain types of errors to occur, which would motivate some form of error correction for instructions, which could likely be interpreted mathematically.&lt;/p&gt;&#xA;" OwnerUserId="14142" LastEditorUserId="55" LastEditDate="2022-10-16T06:15:30.117" LastActivityDate="2022-11-29T18:10:20.157" Title="What original mathematics arose from Jacquard loom and similar weaving devices?" Tags="|mathematics|technology|combinatorics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14880" PostTypeId="1" CreationDate="2022-10-16T18:27:47.947" Score="0" ViewCount="114" Body="&lt;p&gt;In the book &lt;a href=&quot;https://etc.usf.edu/lit2go/95/philosophy-and-fun-of-algebra/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Philosophy and Fun of Algebra&amp;quot;&lt;/a&gt;, Mary Everest Boole has a chapter dedicated to what she calls &amp;quot;the first Hebrew algebra&amp;quot; (Chapter 6). She later refers to the &amp;quot;second and third great Hebrew algebras&amp;quot; (Chapter 8). But what are these?&lt;/p&gt;&#xA;" OwnerUserId="16475" LastActivityDate="2022-10-16T18:27:47.947" Title="What are the second and third &quot;Hebrew Algebras&quot; Mary Everest Boole refer to?" Tags="|mathematics|education|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14881" PostTypeId="1" AcceptedAnswerId="14900" CreationDate="2022-10-17T23:44:45.297" Score="3" ViewCount="252" Body="&lt;p&gt;I'm sure this wasn't his intention at the outset, but here's my understanding of the history of GR so you can see where I'm coming from:&lt;/p&gt;&#xA;&lt;p&gt;The Einstein-Grossmann &amp;quot;Entwurf&amp;quot; version of the theory used the Ricci tensor in place of the Einstein tensor, and it was only when Einstein finally conceded the necessity of general covariance that he realized he needed a tensor with zero divergence.  And so, in some sense, the real breakthrough in November 1915 was the discovery of a mathematical identity -- the existence of a curvature tensor that is inherently conserved, just like matter and energy.  Thus, the field equations seem to have offloaded an important part of the role of Newton's laws, into the intrinsic nature of the Lorentzian manifold itself.  That is, once you accept that spacetime is Lorentzian, there automatically &lt;em&gt;has&lt;/em&gt; to be this locally conserved entity, and the field equations &amp;quot;merely&amp;quot; say that matter (whatever that may be) must follow its flow.&lt;/p&gt;&#xA;&lt;p&gt;The other law of GR is the principle of geodesic motion, but, as quoted &lt;a href=&quot;https://en.wikipedia.org/wiki/Geodesics_in_general_relativity#Equation_of_motion_may_follow_from_the_field_equations_for_empty_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, Einstein did not believe that was an independent postulate, but rather that it could be derived from the vacuum field equations.  And although that claim may be disputed technically, he was not philosophically ambivalent about it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the imperfections of the original relativistic theory of gravitation was that as a field theory it was not complete; it introduced the independent postulate that the law of motion of a particle is given by the equation of the geodesic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And since we know that Einstein believed in a unified field, presumably the only ultimate law of physics he would have been willing to admit would be the field equation restricting the behavior of that field.  However, as argued above, his own field equation for GR outsourced much of that restrictive authority to the (pseudo-)Riemannian geometry of spacetime itself.  And perhaps it is not a coincidence that the change in the model of spacetime which enabled that transition was itself the &lt;em&gt;removal&lt;/em&gt; of a mathematical &amp;quot;law&amp;quot;, viz., Euclid's fifth postulate!&lt;/p&gt;&#xA;&lt;p&gt;So I see this trend in the development of GR, of laws being stripped away.  From a certain point of view, you might even say that GR contains no laws of physics, in that it doesn't even try to specify the nature of matter (the right hand side of the field equation).  Once again, it outsources to theories of the structure of matter, such as quantum theory, the job of explaining &lt;em&gt;why&lt;/em&gt; the inherently conserved Einstein tensor evolves in the particular way it does.  Almost in the same way that Newton outsourced to his readers the job of explaining the remote influence of gravity.  An explanation that was ultimately given, not in terms of new entities or phenomena, but in the mathematical terms of the continuum proper.&lt;/p&gt;&#xA;&lt;p&gt;Hence my question: is there any indication that Einstein believed the laws of physics would someday &amp;quot;work themselves out of a job&amp;quot; -- that their explanatory role would be assumed by the mathematical identities inherent in the continuum, in conjunction with the particular types of substructures of that continuum proposed to correspond to the known elements of matter (e.g. topological singularities for particles, with mass/charge being some kind of topological invariant)?&lt;/p&gt;&#xA;" OwnerUserId="16950" LastActivityDate="2022-11-02T07:49:59.303" Title="Did Einstein believe the laws of physics would be replaced by mathematical identities?" Tags="|physics|theoretical-physics|relativity-theory|einstein|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14882" PostTypeId="1" CreationDate="2022-10-19T15:08:32.060" Score="1" ViewCount="79" Body="&lt;p&gt;What are the full names and dates of birth and of death of the parents of &lt;a href=&quot;https://en.wikipedia.org/wiki/Adrien-Marie_Legendre&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adrien-Marie Legendre&lt;/a&gt;? I couldn't find this information anywhere online. Not even their names. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17395" LastEditorUserId="4251" LastEditDate="2022-10-19T16:24:33.987" LastActivityDate="2022-10-19T16:25:42.507" Title="What are the names of parents of Adrien-Marie Legendre, and their dates of birth and death?" Tags="|mathematicians|legendre|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14883" PostTypeId="2" ParentId="14882" CreationDate="2022-10-19T15:33:56.777" Score="3" Body="&lt;p&gt;MacTutor is a good source for biographies of mathematicians.  Of &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Legendre&quot; rel=&quot;nofollow noreferrer&quot;&gt;Legendre&lt;/a&gt; it begins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Adrien-Marie Legendre would perhaps have disliked the fact that this article contains details of his life&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It also seems unclear whether he was born in Paris or in Toulouse.&lt;/p&gt;&#xA;&lt;p&gt;That being said, the genealogy website geneanet has &lt;a href=&quot;https://gw.geneanet.org/geneavendeemili?lang=en&amp;amp;n=legendre&amp;amp;oc=0&amp;amp;p=adrien%20marie&quot; rel=&quot;nofollow noreferrer&quot;&gt;an entry&lt;/a&gt; which gives his parents as Adrien LEGENDRE and Marie Anne Charlotte RIFAU.  See also &lt;a href=&quot;https://books.google.com/books?id=WdbWpEjLU7sC&amp;amp;pg=PA200&amp;amp;lpg=PA200&amp;amp;dq=%22Marie%20Anne%20Charlotte%20RIFAU%22&amp;amp;source=bl&amp;amp;ots=EN5-rNYg7Q&amp;amp;sig=ACfU3U0dv4DG49vPrXf3AgDqpuLPgvgoVw&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjglvKOzuz6AhXcL0QIHStgCkgQ6AF6BAgJEAM#v=onepage&amp;amp;q=%22Marie%20Anne%20Charlotte%20RIFAU%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;an excerpt of the correspondence between Stieltjes and Hermite&lt;/a&gt; which contains these same names in a footnote.&lt;/p&gt;&#xA;" OwnerUserId="5088" LastEditorUserId="4251" LastEditDate="2022-10-19T16:25:42.507" LastActivityDate="2022-10-19T16:25:42.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14884" PostTypeId="1" AcceptedAnswerId="14899" CreationDate="2022-10-19T22:16:43.643" Score="1" ViewCount="112" Body="&lt;p&gt;I am thinking about the change from Newtonian mechanics to quantum physics and relativity.&#xA;I note that we have accepted the words &amp;quot;ultraviolet catastrophe&amp;quot; in connection with the situation that lead to steps towards quantum physics.&lt;/p&gt;&#xA;&lt;p&gt;We have a number of similar paradoxes, or perhaps I should say strange things since catastrophes and paradoxes perhaps differ somewhat, for relativity. However, the paradoxes are often attributed to the new theory of special relativity rather than the problems experienced with old state of affairs. I am thinking of the “twin paradox” and “Ehrenfest’s paradox” for calculating the perimeter of a spinning cylinder. The “Ehrenfest’s paradox” seems to lie between the special and the general theory of relativity.&lt;/p&gt;&#xA;&lt;p&gt;Can we say the Lorentz contraction based on Maxwell’s equations, or the absence of the ether in the Michelson-Morley experiment felt like a paradox to contemporary physicist, or possibly a catastrophe, on equal footing with the ultraviolet catastrophy or was the transition to Einstein’s physics more a gradual change with less dramatic impressions?&lt;/p&gt;&#xA;" OwnerUserId="5602" LastActivityDate="2022-10-31T10:59:45.833" Title="Paradigms in Newtonian physics around 1900" Tags="|quantum-mechanics|philosophy-of-science|classical-mechanics|relativity-theory|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14885" PostTypeId="1" CreationDate="2022-10-19T23:33:33.113" Score="3" ViewCount="167" Body="&lt;p&gt;Bourbaki was founded around 1934, they have a limit age of fifty and many mathematicians have passed through it. According to an &lt;a href=&quot;https://www.ocf.berkeley.edu/%7Elekheng/interviews/ClaudeChevalley.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;interview&lt;/a&gt; with Chevalley no women have belonged to the group up to 1985. Due to the importance of the group in the history of mathematics, I wonder if this is the case even to this day.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is evidence that there were women in Bourbaki? Maybe Claire Voisin?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="15350" LastActivityDate="2022-10-19T23:33:33.113" Title="Were there women inside Bourbaki?" Tags="|mathematics-social-history|bourbaki|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14886" PostTypeId="1" AcceptedAnswerId="14888" CreationDate="2022-10-20T09:04:33.007" Score="7" ViewCount="641" Body="&lt;p&gt;In &amp;quot;Mathematical Thought from Ancient to Modern Times&amp;quot; Morris Kline claims (on page 209) that Leonardo da Pisa (Fibonacci) &amp;quot;showed that the roots of &lt;span class=&quot;math-container&quot;&gt;$x^3+2x^2+10x=20$&lt;/span&gt; are not constructible with straightedge and compass&amp;quot;.  I'd be interested to know how this was proved because the tools we nowadays use to show that certain straightedge and compass constructions are not possible weren't available at that time.  Can someone point me to a textbook or an article where Fibonacci's proof is explained?&lt;/p&gt;&#xA;" OwnerUserId="3555" LastActivityDate="2022-10-21T06:34:05.533" Title="Fibonacci and straightedge and compass constructions" Tags="|reference-request|geometry|euclidean-geometry|middle-ages|" AnswerCount="1" CommentCount="1" FavoriteCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14887" PostTypeId="2" ParentId="14835" CreationDate="2022-10-20T09:21:14.357" Score="1" Body="&lt;p&gt;As to the Wikipedia quotes about Gentzen one should probably add that various authors describe him as politically naive and unworldly.  It seems he joined the SA because he was told to do so to secure his academic position.  He remained in contact with his &amp;quot;esteemed&amp;quot; teacher Paul Bernays, a Jew who had left Germany in 1933, and in 1941 Gentzen even suggested that Kurt Gödel - who had fled to the US in 1940 - should be offered a professorship in Germany.&lt;/p&gt;&#xA;" OwnerUserId="3555" LastActivityDate="2022-10-20T09:21:14.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14888" PostTypeId="2" ParentId="14886" CreationDate="2022-10-20T11:04:12.953" Score="11" Body="&lt;p&gt;In the &lt;em&gt;Flos&lt;/em&gt; (&lt;em&gt;Flos Leonardi Bigolli Pisani super solutionibus quarundam questionibus ad numerum et ad geometriam, vel ad utrumque pertinentium&lt;/em&gt;), Fibonacci reinterprets in algebraic form the geometric theory of incommensurables in Book X of Euclid's &lt;em&gt;Element&lt;/em&gt; and studies a third-degree equation &lt;a href=&quot;https://archive.org/details/bub_gb_JYb-VYM12ocC/page/n231/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;namely&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Altera uero questio a predicto magistro Iohanne proposita fuit, vt inueniretur quidam cubus numerus, qui cum suis duobus quadratis et decem radicibus in unum collectis essent uiginti: super hoc meditando putaui huius questionis solutionem egredi ex his que continentur in X lib. Euclidis; et ob hoc super ipso X Euclidis accuratius studui, adeo quod sui teoremata ipsius memorie commendaui, et ipsarum intellectum comprehendi.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another question was proposed by the aforesaid teacher John (i.e. Iohannes Panormitanus, or &lt;a href=&quot;https://www.treccani.it/enciclopedia/giovanni-da-palermo_%28Dizionario-Biografico%29/&quot; rel=&quot;noreferrer&quot;&gt;John from Palermo&lt;/a&gt;, one of the scholar in the circle of &lt;a href=&quot;https://en.wikipedia.org/wiki/Frederick_II,_Holy_Roman_Emperor&quot; rel=&quot;noreferrer&quot;&gt;Frederick II&lt;/a&gt;), namely, to find a certain cube number, which with its two squares and ten roots collected into one would be twenty. And for this reason I studied the book X of Euclid more carefully, so much so that I committed his theorems to his memory, and grasped their understanding.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and, so, in modern notation, to solve&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x^3+2x^2+10x=20.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then Fibonacci shows that the positive root is not rational, nor is it a quadratic irrational (&lt;a href=&quot;https://archive.org/details/bub_gb_JYb-VYM12ocC/page/n233/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;Ostendam rursus impossibile esse quod numerus .ab. sit radix alicuius numeri ratiocinati...&lt;/a&gt;). If the root were a quadratic irrational, say &lt;span class=&quot;math-container&quot;&gt;$x =\sqrt{n}$&lt;/span&gt;, in the equality&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x(x^2+10) = 20-2x^2$$&lt;/span&gt;&#xA;the term on the left would be irrational and the term on the right rational. So, unable to solve the equation exactly, he numerically &lt;a href=&quot;https://archive.org/details/bub_gb_JYb-VYM12ocC/page/n237/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;calculates&lt;/a&gt; an approximation to 9 correct decimal places:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Et quia hec questio solui non potuit in aliquo suprascriptorum, studui solutionem eius ad propinquitatem reducere. Et inueni unam ex .X. radicibus nominatis, scilicet numerum .ab., secundum propinquitatem, esse unum et minuta .XXII. et secunda .VII. et tertia .XLII, et quarta .XXXIII. et quinta .IIII. et sexta .XL.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;in modern notation:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$x\approx  1 + \frac{22}{60} + \frac{7}{60^2}+\frac{42}{60^3}+\frac{33}{60^4}+ \frac{4}{60^5}+\frac{40}{60^6}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So actually Fibonacci does &lt;em&gt;not&lt;/em&gt; prove non-constructibility of the roots, but that they are not rational nor of the form &lt;span class=&quot;math-container&quot;&gt;$\sqrt{n}$&lt;/span&gt; for some rational &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="229" LastEditDate="2022-10-21T06:34:05.533" LastActivityDate="2022-10-21T06:34:05.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14889" PostTypeId="1" CreationDate="2022-10-22T22:41:40.933" Score="2" ViewCount="165" Body="&lt;p&gt;The various famous experiments at Los Alamos where the experimenter would dangerously push two masses of plutonium or uranium near each other and observe increased activity -- was such a phenomenon observed with other radioactive substances many years before discovery of even the neutron?&lt;/p&gt;&#xA;&lt;p&gt;Or is it possible that the kind of radioactive materials did not work this way? Or they did not have enough of the element to see it happen? I read that radium has a critical mass but maybe they never tried this experiment?&lt;/p&gt;&#xA;&lt;p&gt;But would not it have been observed that 1 gm of Ra was more active than a half of a gram or is that not even the case?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-11-05T13:09:46.560" Title="When was critical mass suspected?" Tags="|radioactivity|atomic-weapons|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14890" PostTypeId="5" CreationDate="2022-10-23T13:54:55.637" Score="0" Body="" OwnerUserId="-1" LastEditorUserId="-1" LastEditDate="2022-10-23T13:54:55.637" LastActivityDate="2022-10-23T13:54:55.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14891" PostTypeId="4" CreationDate="2022-10-23T13:54:55.637" Score="0" Body="Kurt Friedrich Gödel (1906-1978) was an Austrian logician, mathematician, and philosopher." OwnerUserId="4251" LastEditorUserId="4251" LastEditDate="2022-10-24T21:40:35.697" LastActivityDate="2022-10-24T21:40:35.697" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14892" PostTypeId="1" AcceptedAnswerId="14895" CreationDate="2022-10-24T11:39:21.823" Score="1" ViewCount="523" Body="&lt;p&gt;A Japanese book on mechanics says the following (English translation by me):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As a young boy, Tycho Brahe became interested in astronomy when he was impressed by the fact that the solar eclipse occurred almost exactly as predicted. Later, faced with the fact that a prediction of the celestial proximity of the two planets based on a heliocentric model was off by several days and a prediction based on a geocentric model was off by a month, he realized the need for precise observation of celestial bodies.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;『物理学序論としての力学』(藤原邦男)&lt;/p&gt;&#xA;&lt;p&gt;I would like to know how the prediction performance of the Copernicus and Ptolemaic models compare.&lt;/p&gt;&#xA;" OwnerDisplayName="user16492" LastActivityDate="2022-11-01T06:30:04.730" Title="Comparison of the accuracy of the Ptolemaic and Copernican models" Tags="|astronomy|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14893" PostTypeId="2" ParentId="14812" CreationDate="2022-10-24T14:03:17.817" Score="4" Body="&lt;p&gt;Needham's &lt;a href=&quot;https://www.cambridge.org/academic/subjects/history/history-science-general-interest/series/science-and-civilisation-china&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Science and Civilisation in China&lt;/em&gt;&lt;/a&gt; series has a few volumes mentioning ballistics.&lt;/p&gt;&#xA;&lt;p&gt;Volume 3, &lt;em&gt;Mathematics and the Sciences of the Heavens and Earth&lt;/em&gt;, has this short passage on the &lt;em&gt;absence&lt;/em&gt; of proper ballistics science in ancient China (p. 167):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Chinese should have been interested in mechanics for ships, in&#xA;hydrostatics for their vast canal system (like the Dutch), in&#xA;ballistics for guns (after all, they had possessed gunpowder&lt;sup&gt;f&lt;/sup&gt; four&#xA;centuries before Europe), and in pumps for mines. If they were not,&#xA;could not the answer be sought in the fact that little or no private&#xA;profit was to be gained from any of these things in Chinese society,&#xA;dominated by its imperial bureaucracy?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Volume 4, &lt;em&gt;Physics and Physical Technology, part 1, Physics&lt;/em&gt;, has this highly speculative passage about the &lt;a href=&quot;https://en.wikipedia.org/wiki/Mozi_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mozi&lt;/em&gt;&lt;/a&gt; (p. 58):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And now we find something at any rate extremely like it in the &lt;em&gt;Mo&#xA;Ching&lt;/em&gt; of the -4th or - 3rd century (Cs 49), where motion is said to&#xA;be due to the absence of an opposing force. The Mohist technical term&#xA;'supporting pillar' is to be understood only as that force which in&#xA;Newton's first law changes the otherwise permanent state of motion of&#xA;the moving body (Cs 50). The exposition distinctly states that if&#xA;there is no such force the motion will never stop. Its writer seems to&#xA;be trying, too, to describe non-linear or deflected motion as 'motion&#xA;which is not quite motion in the fullest sense'. What remains in these&#xA;brief fragments is so striking that we may be allowed to believe that&#xA;if more of the physics of the Mohist school had been preserved, we&#xA;should have found in it some discussion of trajectories, the effect of&#xA;gravity, and so on. If the Mohists had no technical term corresponding&#xA;to impetus, at least they did not suffer from the concept of 'natural&#xA;place' or the awkward idea of antiperistasis.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But the more interesting passage is perhaps to be found in &lt;em&gt;Volume 5, Chemistry and Chemical Technology, part 7, Military Technology: The Gunpowder Epic&lt;/em&gt;, pp. 390&lt;span class=&quot;math-container&quot;&gt;$-$&lt;/span&gt;391:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As is generally known, the earliest Western speculations about the&#xA;path of a projectile supposed it to move in a straight line for a&#xA;while, before succumbing finally to the influence of gravity and then&#xA;falling downwards in an equally straight line, not unlike the course&#xA;of the mortar shell seen in Fig. 148. This was the conception in the&#xA;days of Nicolo Tartaglia (+1537, +1546).&lt;sup&gt;c&lt;/sup&gt; But Galileo&#xA;(+1638) and Torricelli (+1644) proposed a parabolic&#xA;trajectory,&lt;sup&gt;d&lt;/sup&gt; and this eventually became more like a&#xA;hyperbola when the resistance of the air was fully taken into account,&#xA;as by Newton (+1674) and later mathematicians.&lt;sup&gt;e&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;In East Asia ballistics was pursued more in Japan than in China, but the&#xA;connections were close. The famous Inatomi family of&#xA;gunsmiths&lt;sup&gt;f&lt;/sup&gt; left many MS books still extant on the theory&#xA;and practice of gunnery, notably one of +1607 to +1610 in twenty-nine&#xA;large volumes.&lt;sup&gt;g&lt;/sup&gt; The most outstanding member of the family,&#xA;Inatomi Naoie, recorded a tradition that Sasaki Shyo-huziro had first&#xA;learnt the art in China, and then transmitted it to his grandfather&#xA;Inatomi Sagami-no-kami Naotoki; this would take us back to +1500 or&#xA;earlier, certainly before the arrival of either the Turkish or the&#xA;Portuguese musket (cf. pp. 440 ff. below) in China and Japan, and&#xA;would suggest that the Chinese hand-guns of the +15th century,&#xA;probably with serpentines,&lt;sup&gt;a&lt;/sup&gt; had begun the&#xA;affair.&lt;sup&gt;b&lt;/sup&gt; By +1618 trajectories were being studied by&#xA;Shimizu Hidemasa, who visualised a slow rise followed by a slow&#xA;fall.&lt;sup&gt;c&lt;/sup&gt; Then from +1659 onwards&lt;sup&gt;d&lt;/sup&gt; the parabolic&#xA;trajectory was proposed, first in the &lt;em&gt;Kaisan-ki&lt;/em&gt;&lt;sup&gt;1&lt;/sup&gt; (Book of&#xA;Improved Mathematics) by Yamada Shigemasa&lt;sup&gt;2&lt;/sup&gt;,&lt;sup&gt;e&lt;/sup&gt;&#xA;then in the remarkable work of Nozawa Sadanagra&lt;sup&gt;3&lt;/sup&gt;, the &lt;em&gt;San&#xA;Kyukai&lt;/em&gt;&lt;sup&gt;4&lt;/sup&gt; (Mathematics in Nine Chapters) of&#xA;+1677.&lt;sup&gt;f&lt;/sup&gt; This book, which accompanied the illustration of the curve with complicated quadratic equations, was the first Japanese&#xA;treatise to explain physical phenomena using mathematical formulae.&#xA;There may have been some Jesuit or other Western influence&#xA;here,&lt;sup&gt;g&lt;/sup&gt; but Nozawa's view of the world was at least as much&#xA;Chinese, based on the Yin-Yang theory, decimal metrology,&lt;sup&gt;h&lt;/sup&gt;&#xA;and the standard pitch-pipe dimensions.&lt;sup&gt;i&lt;/sup&gt; The &lt;em&gt;Suan Fa Thung&#xA;Tsung&lt;/em&gt;&lt;sup&gt;5&lt;/sup&gt; (Systematic Treatise on Arithmetic) of Chhêng&#xA;Ta-Wei&lt;sup&gt;6&lt;/sup&gt; (+1592)&lt;sup&gt;j&lt;/sup&gt; had been translated into&#xA;Japanese only two years before Nozawa's own writing and he was&#xA;probably strongly influenced by it. Lastly there was the extension of&#xA;Yamada's work by Mochinaga Toyotsugu&lt;sup&gt;7&lt;/sup&gt; &amp;amp; Ohashi&#xA;Takusei&lt;sup&gt;8&lt;/sup&gt;, the &lt;em&gt;Kaisan-ki Komoku&lt;/em&gt;&lt;sup&gt;9&lt;/sup&gt;, which&#xA;continued to speak of gunshot parabolas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, it appears that Japanese scholars had developed some kind of ballistics in the early seventeenth century, but it is hard to tell if this was achieved completely outside from Western influence... I can't copy all the footnotes here, but Needham notes that some of this happened after the closure of the country to foreign influence, when Christian religion had been outlawed, Latin Jesuits expelled, and way before the &lt;em&gt;rangaku&lt;/em&gt; (Dutch learning) period had begun.&lt;/p&gt;&#xA;" OwnerUserId="14043" LastActivityDate="2022-10-24T14:03:17.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14895" PostTypeId="2" ParentId="14892" CreationDate="2022-10-24T22:00:49.783" Score="2" Body="&lt;p&gt;The Ptolemaic and Copernican models are observationally indistinguishable, so the book must be misunderstanding the question or summarising it inaccurately.&lt;/p&gt;&#xA;" OwnerUserId="4209" LastActivityDate="2022-10-24T22:00:49.783" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14896" PostTypeId="1" CreationDate="2022-10-26T22:32:30.293" Score="3" ViewCount="232" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; be two integers not both of which are equal to zero. It is an important and well-known fact that &lt;span class=&quot;math-container&quot;&gt;$\text{gcd}(a,b)=ax_{0}+by_{0}$&lt;/span&gt; for some integers &lt;span class=&quot;math-container&quot;&gt;$x_{0}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y_{0}$&lt;/span&gt;. Even though this result was already implicit in the seventh book of the Elements (remember that it can be derived from the Euclidean algorithm), some authors refer to it as &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Bézout's identity&amp;quot;&lt;/a&gt;; personally, I like better the way in which &lt;a href=&quot;https://es.wikipedia.org/wiki/Enzo_Gentile&quot; rel=&quot;nofollow noreferrer&quot;&gt;Professor E. R. Gentile&lt;/a&gt; would call it: &lt;em&gt;the universal suggestion in arithmetic&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Do you happen to know who it was that came up with &lt;a href=&quot;https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity#Proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;the proof&lt;/a&gt; of the universal suggestion wherein the well-ordering principle of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}^{+}$&lt;/span&gt; is applied to the non-empty set &lt;span class=&quot;math-container&quot;&gt;$\{ax+by \colon (x,y) \in \mathbb{Z}^{2}\} \cap \mathbb{Z}^{+}$&lt;/span&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;Please let me thank you in advance for your insightful replies and comments!&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-12-08T13:42:12.683" LastActivityDate="2022-12-08T13:42:12.683" Title="Who came up with the proof of &quot;Bézout's identity&quot; that uses the well-ordering principle?" Tags="|reference-request|number-theory|algorithm|euclid|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14897" PostTypeId="2" ParentId="14889" CreationDate="2022-10-28T11:37:47.377" Score="2" Body="&lt;p&gt;You certainly refer to Manhattan physicist Louis Slotin. &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Slotin&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Louis_Slotin&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4438" LastEditorUserId="26" LastEditDate="2022-11-05T13:09:46.560" LastActivityDate="2022-11-05T13:09:46.560" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14898" PostTypeId="2" ParentId="14889" CreationDate="2022-10-28T12:08:04.963" Score="2" Body="&lt;p&gt;Radium can't have a critical mass as it doesn't emit neutrons during decay.&lt;/p&gt;&#xA;&lt;p&gt;See:&#xA;&lt;a href=&quot;https://www.researchgate.net/figure/Decay-chain-of-radium-226-showing-the-main-radionuclides-with-the-type-of-radioactive_fig4_263672557&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.researchgate.net/figure/Decay-chain-of-radium-226-showing-the-main-radionuclides-with-the-type-of-radioactive_fig4_263672557&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="4438" LastActivityDate="2022-10-28T12:08:04.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14899" PostTypeId="2" ParentId="14884" CreationDate="2022-10-31T10:59:45.833" Score="1" Body="&lt;p&gt;The short answer is no, there was no sense of crisis in Newtonian mechanics. Lorentz' electron theory and the contractions contained within were perfectly capable of explaining the behaviour of the electromagnetic ether in moving systems. It should be stressed here that the Michelson-Morley experiments &lt;em&gt;did not&lt;/em&gt; disprove the ether. Instead it was set up as a critical test for how the ether contracted in different circumstances.&lt;/p&gt;&#xA;&lt;p&gt;In fact, when Einstein published his article &amp;quot;On the Electrodynamics of Moving Bodies&amp;quot;, it was considered by his contemporaries as a conceptual reinterpretation of Lorentz' theory — to the point that it was known as the Lorentz-Einstein theory. Moveover, the two theories are completely empirically equivalent. Even today, there is no empirical test that can prove or disprove the existence of the ether. The reason why Einstein's theory of special relativity has won out is because 1) it is a much simpler theory, even if it may be conceptually harder to grasp, and 2) Special Relativity was much easier to incorporate into Quantum Electrodynamics than Lorentz' Electron Theory.&lt;/p&gt;&#xA;&lt;p&gt;For more how the move from Lorentz to Einstein can be seen as a Scientific Revolution happening after the fact and without the need for an anomaly, see this article from your almost-namesake Michel Janssen, &amp;quot;&lt;a href=&quot;https://link.springer.com/article/10.1007/s000160200003&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reconsidering a Scientific Revolution: The case of Einstein versus Lorentz&lt;/a&gt;&amp;quot;, Physics in Perspective 4 (2002): 421-446.&lt;/p&gt;&#xA;" OwnerUserId="11059" LastActivityDate="2022-10-31T10:59:45.833" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14900" PostTypeId="2" ParentId="14881" CreationDate="2022-10-31T11:18:09.173" Score="6" Body="&lt;p&gt;I believe you are making a contradiction between mathematical identities and physical laws that Einstein would not make. For him, mathematical identities &lt;em&gt;are&lt;/em&gt; the laws of nature, and having them derive from the Unified Field Equation &lt;em&gt;is&lt;/em&gt; the explanation.&lt;/p&gt;&#xA;&lt;p&gt;As he wrote in 1934 in &amp;quot;&lt;a href=&quot;https://doi.org/10.1086/286316&quot; rel=&quot;noreferrer&quot;&gt;On the Method of Theoretical Physics&lt;/a&gt;&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Our experience up to date justifies us in feeling sure that in Nature&#xA;is  actualized  the  ideal  of mathematical  simplicity. It   is  my&#xA;conviction  that  pure  mathematical  construction  enables  us  to&#xA;discover the concepts and the laws connecting them which give us&#xA;the  key to  the  understanding  of the  phenomena of Nature.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To   justify  this  confidence  of  mine,  I  must  necessarily  avail&#xA;myself  of  mathematical  concepts. The   physical  world  is represented  as  a  four-dimensional continuum. If  in  this  I  adopt  a Riemannian  metric,  and look  for  the simplest  laws which  such a&#xA;metric  can  satisfy,  I  arrive at   the relativistic gravitation-theory of  empty  space. If  I  adopt  in  this  space  a  vector-field, or  in other  words, the  antisymmetrical  tensor-field  derived  from  it, and if  I look  for  the simplest laws which  such  a field  can  satisfy, I arrive at the Maxwell equations for free space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It   is essential for our point of  view  that we  can arrive at these&#xA;constructions  and  the  laws  relating  them  one  with  another  by adhering  to  the  principle  of searching  for  the  mathematically simplest concepts  and their  connections. In   the paucity of  the mathematically  existent  simple  field-types  and  of the  relations between  them,  lies  the justification for  the  theorist's  hope  that he may comprehend reality in its depths.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11059" LastEditorUserId="11059" LastEditDate="2022-11-02T07:49:59.303" LastActivityDate="2022-11-02T07:49:59.303" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14901" PostTypeId="2" ParentId="14857" CreationDate="2022-10-31T12:32:45.890" Score="3" Body="&lt;p&gt;For Galileo, the relativity principle was an observational principle. When he writes about the thought experiment on a moving ship where the observer cannot recognize that there is movement, it is clear to the reader that it is indeed a &lt;em&gt;moving&lt;/em&gt; ship. So for Galileo, there can be an absolute reference system, where the Sun is at the center, and then there are all these empirically equivalent systems, such as two moving ships, whose movements cannot ever be observed. There is no contradiction between those.&lt;/p&gt;&#xA;&lt;p&gt;This duality persisted until the early twentieth century. We find this for example in Newton, who believed in absolute space and empirically equivalent inertial systems; in Maxwell, who believed in the ether but whose equations are equivalent for moving systems; and in Lorentz, who assumed an absolute ether system but had it as a principle that the ether contracted in such a way that each inertial system was by principle empirically equivalent. It was only with Poincaré and Einstein that someone thought this duality was particularly problematic.&lt;/p&gt;&#xA;" OwnerUserId="11059" LastActivityDate="2022-10-31T12:32:45.890" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14902" PostTypeId="2" ParentId="13774" CreationDate="2022-10-31T14:38:14.397" Score="2" Body="&lt;p&gt;A challenge is that many scientific projects start with a Fermi estimate to see if an idea is plausible, but if the answer is &amp;quot;yes&amp;quot;, much work and analysis is done before the work appears in public where the original Fermi estimate is at best buried in the details.&lt;/p&gt;&#xA;&lt;p&gt;I will note that Fermi's famous quick determination of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Trinity_(nuclear_test)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trinity test&lt;/a&gt; yield was more of a &amp;quot;torn up envelope&amp;quot; experiment that a &amp;quot;back of the envelope&amp;quot; estimate.  He likely used the best hydrodynamic theory from Hans Bethe to calculate the yield,  and he did all the calculations ahead of time so that soon as his bits of paper hit the ground he could use his pre-calculated table of values to give an immediate estimate of the yield. (See &lt;a href=&quot;https://physics.stackexchange.com/questions/19567/how-did-enrico-fermi-calculate-the-classical-fermi-problem/734484#734484&quot;&gt;this answer&lt;/a&gt; over on Physics Stack Exchange.) Taylor's estimate is similar in that he had worked out the relevant physics years earlier. (See &lt;a href=&quot;https://doi.org/10.1080/0020739X.2011.562324&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Michael A.B. Deakin (2011) G.I. Taylor and the Trinity test, International Journal of Mathematical Education in Science and Technology, 42:8, 1069-1079, DOI: 10.1080/0020739X.2011.562324&quot;&gt;&amp;quot;G.I. Taylor and the Trinity test&amp;quot;&lt;/a&gt;.)  Both Taylor and Fermi's genius was in recognizing how serious theory could be applied to simple experimental data - fluttering paper or a series of photos - to extract the yield.  Maybe we should have a separate category for &amp;quot;Fermi experiments&amp;quot;, where the theory is complex but the measurements simple.&lt;/p&gt;&#xA;&lt;p&gt;Getting back to the question, here are some ideas, although none of them may be as clear-cut as you'd like.&lt;/p&gt;&#xA;&lt;p&gt;Weren't geologists' and biologists' crude estimates for lower bounds on the the age of the earth in the 19th century essentially Fermi estimates that created a famous tension with Kelvin's &lt;a href=&quot;https://en.wikipedia.org/wiki/Age_of_Earth#Early_calculations&quot; rel=&quot;nofollow noreferrer&quot;&gt;estimate for the age of the Earth&lt;/a&gt; that drove research in all three fields. And wasn't even Kelvin's initial value essentially a Fermi estimate based on a crude gravitational collapse model?&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Drake_equation#Original_estimates&quot; rel=&quot;nofollow noreferrer&quot;&gt;original Drake equation&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fermi_paradox#Chain_of_reasoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fermi Paradox&lt;/a&gt; are both Fermi estimates that are foundational for &lt;a href=&quot;https://en.wikipedia.org/wiki/Search_for_extraterrestrial_intelligence&quot; rel=&quot;nofollow noreferrer&quot;&gt;SETI&lt;/a&gt; research.&lt;/p&gt;&#xA;&lt;p&gt;I would argue that John Ioannidis original paper on &lt;a href=&quot;https://en.wikipedia.org/wiki/Why_Most_Published_Research_Findings_Are_False#Argument&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Why Most Published Research Findings Are False&amp;quot;&lt;/a&gt; was also a dressed-up Fermi Estimate that helped trigger the &lt;a href=&quot;https://en.wikipedia.org/wiki/Replication_crisis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Replication Crisis&lt;/a&gt; in science.&lt;/p&gt;&#xA;&lt;p&gt;Although it isn't a specific Fermi Estimate, dimensional analysis has been foundational in fluid dynamics. Faced with intractable &lt;a href=&quot;https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;Navier–Stokes equations&lt;/a&gt;, scientists focused on identifying dimensionless ratios that characterized systems,  e.g. the &lt;a href=&quot;https://en.wikipedia.org/wiki/Reynolds_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reynolds Number&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Rayleigh_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rayleigh Number&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Nusselt_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nusselt Number&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Grashof_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grashof number&lt;/a&gt;, ….&lt;/p&gt;&#xA;&lt;p&gt;The Cosmological Constant Problem is another case where the problem is obvious from &lt;a href=&quot;https://physics.stackexchange.com/a/22645/145491&quot;&gt;simple dimensional analysis and the value of Newton's gravitational constant&lt;/a&gt;, and this has driven huge theoretical efforts in particle physics.&lt;/p&gt;&#xA;&lt;p&gt;I think Bohr was using Fermi estimation as he worked towards his Bohr atom. See, for example, &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4517-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;his recollection&lt;/a&gt; that &amp;quot;I just felt that one knows the order of magnitude of the binding of any electron.&amp;quot; or &lt;a href=&quot;https://www.amphilsoc.org/guides/ahqp/preface.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;this description&lt;/a&gt; that &amp;quot;Bohr, we learn, troubled year after year about the problem of atomic stability, made many an order-of-magnitude estimate of factors likely to be relevant to atomic stability; and reflected deeply about the effect of radiation upon stability - so that ten days after he saw Balmer's formula for the wave lengths in the hydrogen spectrum he had won his way through to the quantum theory of the atom.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Finally, this paper on my &amp;quot;to read&amp;quot; list on &lt;a href=&quot;https://quod.lib.umich.edu/cgi/t/text/text-idx?cc=ptpbio;c=ptb;c=ptpbio;idno=16039257.0012.004;view=text;rgn=main;xc=1;g=ptpbiog&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Value of Imprecise Prediction&amp;quot;&lt;/a&gt; may have some relevant examples.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2022-11-13T15:39:36.357" LastActivityDate="2022-11-13T15:39:36.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14904" PostTypeId="2" ParentId="14892" CreationDate="2022-11-01T06:30:04.730" Score="0" Body="&lt;p&gt;To elaborate Martin's answer slightly, the Copernican (heliocentric) model is indistinguishable from the Ptolemaic (geocentric) model &lt;em&gt;in its final form&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This is because both models in their final form are actually mechanically equivalent.&lt;/p&gt;&#xA;&lt;p&gt;What distinguishes planetariums/orreries based on each model is that the geocentric planetarium attempts to explicitly describe the movement of the heavens &lt;em&gt;as seen from Earth&lt;/em&gt; (so that things like retrograde motion are explicitly seen in a geocentric planetarium).&lt;/p&gt;&#xA;&lt;p&gt;Whereas in the heliocentric planetarium, the movement of the Earth together with the movement of other elements makes it more difficult to infer the observational reality we see from Earth.&lt;/p&gt;&#xA;&lt;p&gt;The heliocentric planetarium effectively demonstrates things as would be seen by a person standing on the surface of the Sun - which is not an observation that has ever been made in practice.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, the heliocentric model is regarded as the &amp;quot;simpler&amp;quot; one, but in fact the central importance of Earth to humanity, and the fact that historical interest in astronomy was primarily driven by a desire to model what was &lt;em&gt;seen&lt;/em&gt; in the sky (not to represent some abstract cosmological view), meant that for a long time the geocentric model was regarded as the simpler and more obvious description.&lt;/p&gt;&#xA;&lt;p&gt;It's also important to note that both models actually date from antiquity - the heliocentric model doesn't actually originate with Corpernicus, but is at least as old as Ptolemy who was also familiar with it.&lt;/p&gt;&#xA;&lt;p&gt;What probably harmed the geocentric model most was that improvements in telescopy meant that irregular orbits and patterns of eclipses were more readily seen and precisely quantified, which increased the mechanical complexity of geocentric planetariums that accounted for them. Better telescopes also meant that an increasing number of elements had to be accounted for.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;equants&amp;quot; and &amp;quot;epicycles&amp;quot; and so on of the geocentric model, which exist to represent very specific motion which is apparent/relative to Earth, but are unreal in the sense that they do not represent a general description of astronomical movements, massively increases mechanical complexity, and as more elements are added there is a risk that the sweeps of moving linkages and supports start to conflict with each other.&lt;/p&gt;&#xA;&lt;p&gt;It was also not entirely clear in the past whether such apparent movements in the heavens were &amp;quot;real&amp;quot; or not, but the increasing complexity and arbitrariness of the geocentric model over time, led many to doubt. In that sense, there is a certain philosophical side to the geocentric model which has been falsified - the question of whether it describes something &amp;quot;real&amp;quot;, or whether it &lt;em&gt;only&lt;/em&gt; describes what we see.&lt;/p&gt;&#xA;&lt;p&gt;Eventually designing such machinery to represent the geocentric model became too much, so that the geocentric approach began to fail in its own terms (of providing a complete description of the observational reality from Earth, &lt;em&gt;in the form of some conceivable machine that demonstrated the model in motion&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;By contrast, a planetarium that implements the heliocentric model is mechanically very much simpler, and the movements of the planets relative to the Sun are seen to be very much simpler and more regular, even though relating it to any earthly observation involves additional reasoning.&lt;/p&gt;&#xA;" OwnerUserId="6847" LastActivityDate="2022-11-01T06:30:04.730" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14907" PostTypeId="2" ParentId="14798" CreationDate="2022-11-03T04:39:32.007" Score="3" Body="&lt;p&gt;&lt;em&gt;&lt;strong&gt;Momento&lt;/strong&gt;&lt;/em&gt; is an object’s tendency to move, much like a net force acting on an object that results in motion.  For example, lighter objects move slower in a medium than heavier objects in the same medium.  In this case, Galileo states the lighter object has a lesser &lt;em&gt;momento&lt;/em&gt; on account of its lesser weight and motion in the medium.  He also demonstrates that a body hanging over an inclined plane exhibits its entire &lt;em&gt;momento&lt;/em&gt; and can be used to compare the partial &lt;em&gt;momento&lt;/em&gt; of another body suspended on an inclined plane by coupling the two bodies together, and the equilibrium found between these weights and their respective lengths of the incline plane form what we now call the law of inclined planes.&lt;/p&gt;&#xA;&lt;p&gt;Galileo originally developed &lt;em&gt;momento&lt;/em&gt; in his &lt;em&gt;Le Meccaniche&lt;/em&gt; as a static torque in his study of levers, to which he equated it with static linear forces.  More specifically, by finding equilibrium for a lever that’s bent at the fulcrum, Galileo equated the static torque produced from a weight acting on the bent end of the lever with the static linear force acting on the same weight held in equilibrium on an inclined plane.  He argued that the tangential forces on the bent lever in this example were substantially the same as the linear forces on the plane, but to Galileo, he called them both &lt;em&gt;momento&lt;/em&gt;.  Later on in Galileo’s writings, he uses &lt;em&gt;momento&lt;/em&gt; for dynamics while referring to linear forces and torques without directly comparing each other, but the nexus still remains: in Two New Sciences, both his discussion of levers on the 2nd day and his linear acceleration of the 3rd day are both described using &lt;em&gt;momento&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Galileo probably knew that &lt;em&gt;momento&lt;/em&gt; was insufficient and ill-defined, and that it made leaping assumptions between statics and dynamics, thus prompting him (or more likely, his pupil Viviani) to insert in the second edition of the &lt;em&gt;Two New Sciences&lt;/em&gt; (i.e., the &lt;em&gt;National Edition&lt;/em&gt;) a section that gropes for alternate terms to describe incline motion, hence, “[&lt;em&gt;l’impeto, il talent, l’engeria&lt;/em&gt;] or, one might say, [&lt;em&gt;il momento&lt;/em&gt;] of descent of the moving body is diminished by the plane upon which it is supported and along which it rolls.”&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Impeto&lt;/strong&gt;&lt;/em&gt; is a combination of kinetic energy and the modern conception of momentum.  Its effects can seen through impacts and is affected by the medium that is passes through.  For example, Galileo discusses that the &lt;em&gt;impeto&lt;/em&gt; of cannonball shot at point blank range is more destructive than when dropped from any height; and that the &lt;em&gt;impeto&lt;/em&gt; is lost when a cannonball is shot into water and subsequently hits the ground softly.  &lt;em&gt;Impeto&lt;/em&gt; is capable of lifting objects to a height: cannonballs are shot upward with an &lt;em&gt;impeto&lt;/em&gt;, and pendulums may be given &lt;em&gt;impeto&lt;/em&gt; through successive puffs of air to start its swing.  It is most similar to “impulse” in which an object is given &lt;em&gt;impeto&lt;/em&gt; to change its velocity.  He also states that objects moving from the same height down various inclined planes will each reach the bottom with equal &lt;em&gt;impeto&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Source of Confusion&lt;/strong&gt;: it's worth noting that the translation of &lt;em&gt;momento&lt;/em&gt; and &lt;em&gt;impeto&lt;/em&gt; as &amp;quot;momentum&amp;quot; was in an effort to make it mesh with modern physics terminology.  In the translation I used, translators Henry Crew and Antonio Favaro note that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Much of the value of any historical document lies in the language&#xA;employed, and this is doubly true when one attempts to trace the rise&#xA;and growth of any set of concepts such as those employed in modern&#xA;physics. We have therefore made this translation as literal as is&#xA;consistent with clearness and modernity. In cases where there is any&#xA;important deviation from this rule, and in the case of many technical&#xA;terms where there is no deviation from it, we have given the original&#xA;Italian or Latin phrase in italics enclosed in square brackets. The&#xA;intention here is to illustrate the great variety of terms employed by&#xA;the early physicists to describe a single definite idea, and&#xA;conversely, to illustrate the numerous senses in which, then as now, a&#xA;single word is used. For the few explanatory English words which are&#xA;placed in square brackets without italics, the translators alone are&#xA;responsible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, in my opinion, drawing this equivalence made it incredibly misleading for modern readers for parsing Galileo's intentions since &lt;em&gt;momento&lt;/em&gt; as a quasi-force in no way resembles that of a quantity of motion (i.e., &amp;quot;momentum&amp;quot;): Galileo states an object moving on a horizontal surface exhibits no &lt;em&gt;momento&lt;/em&gt;, thus it doesn't make sense to call this &lt;em&gt;momento&lt;/em&gt; as &amp;quot;momentum.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Sources (in no particular order):&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Galileo's Machines&amp;quot;, Peter Machamer, &lt;em&gt;The Cambridge Companion to Galileo&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The Use and Abuse of Mathematical Entities&amp;quot;, Rivka Feldhay, &lt;em&gt;The Cambridge Companion to Galileo&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Galileo's Perinertial Framework&amp;quot;, Wallace Hooper, &lt;em&gt;The Cambridge Companion to Galileo&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;On Galileo's Writings on Mechanics: An Attempt at a Semantic Analysis of Viviani's Scholium&amp;quot;, Halbwachs and Torunczyk (I found this paper to be the most helpful)&#xA;&lt;a href=&quot;https://www.jstor.org/stable/20116111&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/20116111&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;On Motion and On Mechanics, Comprising De Motu (ca. 1590) and Le Meccaniche (ca. 1600)&lt;/em&gt;, Drabkin and Drake.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Concepts of Force: A Study in the Foundations of Dynamics&lt;/em&gt;, Jammer&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Galileo, Courtier: The Practice of Science in the Culture of Absolutism&lt;/em&gt;, Biagioli&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Galileo&amp;quot;, Stanford Encyclopedia of Philosophy&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-11-03T16:25:07.763" LastActivityDate="2022-11-03T16:25:07.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14908" PostTypeId="1" CreationDate="2022-11-05T09:24:03.623" Score="5" ViewCount="97" Body="&lt;p&gt;In his 1829 paper on a new formulation of mechanics, Gauss presented his principle of least constraint, which parallels previous formulations of analytical mechanics and provides a new point of view about them. However, in his very short paper he discussed only holonomic constraints; constraints of motion represented by equations rather than inequalities.&lt;/p&gt;&#xA;&lt;p&gt;In the last years of his life, Gauss attempted to extend his principle to non-holonomic constraints, and we can learn from August Ritter's notebooks (Ritter was a pupil of Gauss) that in Gauss's 1850/51 lectures he discussed the following problem of computational geometry:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A region of space in a euclidean space &lt;span class=&quot;math-container&quot;&gt;$E^n$&lt;/span&gt; is specified by linear equations and inequalities. Find its shortest distance from a given point in space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In modern terms Gauss discussed a &amp;quot;Nearest Neighbour Search&amp;quot; (NNS) problem with the metric being the usual euclidean metric. This problem might look easy in low number of dimensions and constraints, but the &amp;quot;naive&amp;quot; ways to find the shortest distance become computationally inefficient as the number of dimensions and constraints grows up, since the &amp;quot;hyper-polyhedra&amp;quot; that bounds the region can become quite complicated.&lt;/p&gt;&#xA;&lt;p&gt;It seems that Gauss discovered a more algorithmically efficient way of approaching this problem (similar to more modern algorithms). I am interested in obtaining more information about Gauss's method. Either a direct explanation or some pointers to relevant literature would be much appreciated.&lt;/p&gt;&#xA;&lt;p&gt;It is perhaps worth mentioning that Gauss mainly gave this problem as an examplary problem, and his original motivations/ideas might be deeper. It is also worth mentioning that Ritter wrote his 1853 doctoral thesis &amp;quot;On Gauss's principle of least constraint&amp;quot;, which dealt with applications to non-holonomic constraints, under the close supervision of Gauss (unfortunately I did not find his doctoral thesis on web).&lt;/p&gt;&#xA;&lt;p&gt;The sources which I based this question on are p. 469-481 of volume 10-1 of Gauss's werke, as well as Paul Stackel's 1917 article &amp;quot;Remarks on the principle of least constraint&amp;quot; (which can be found in p.421-436 of this &lt;a href=&quot;http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/selecta_on_least_constraint.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;collection of translations&lt;/a&gt;). Unfortunately, the website of Gottingen digital library no longer enables to make immediate translation of text, so I could not read an English translation of the relevant pages of Gauss's werke.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="12330" LastEditDate="2022-11-06T09:07:37.507" LastActivityDate="2022-11-06T09:07:37.507" Title="What was Gauss's algorithmic method to solve a certain &quot;nearest neighbour search&quot; problem in multi-dimensional euclidean space?" Tags="|geometry|computer-science|gauss|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14909" PostTypeId="1" CreationDate="2022-11-06T00:43:11.787" Score="1" ViewCount="119" Body="&lt;p&gt;Currently, I am reading &amp;quot;&lt;a href=&quot;https://www.google.com/books/edition/Foundations_and_Fundamental_Concepts_of/J9QcmFHj8EwC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Foundations and Fundamental Concepts of Mathematics Third Edition&lt;/a&gt;&amp;quot; by Howard Eves, and I have came across this paragraph:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In our study of early mathematics we are restricted to that of Egypt and Babylonia. The ancient Egyptians recorded their work on stone and papyrus, the latter fortunatly enduring because of Egypt's unusually dry climate; the Babylonians used impershible clay tablets. In contrast to the use of these medias, the early Indians and Chinese used very perishable writing materials like bark and bamboo. Thus it has come to pass that we have a fair quanity of definite information, obtained  from primary sources, about the science and mathematics of ancient Egypt and Babylonia, while we know very little indeed, with any degree of certainty, about these fields of study in ancient India and China.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$-\text{Chapter One: Mathematics Before Euclid, Page 2, Paragraph 2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which begged the question of &amp;quot;If there exist little of Chinese and Indian math records because of the durability of the materials that they were written on, then what items do we current have?&amp;quot; Which led to me asking &lt;a href=&quot;https://chat.stackexchange.com/transcript/message/62317863#62317863&quot;&gt;this question in the Mathematics Chatroom&lt;/a&gt; for further discussion on the topic. I eventually learned about the existence of the following items:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Book_on_Numbers_and_Computation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Book on Numbers and Computation&lt;/a&gt; (Made of Bamboo Strips and String)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bakhshali_manuscript&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bakhshali Manuscript&lt;/a&gt; (Birch Bark)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Babylonian_astronomical_diaries&quot; rel=&quot;nofollow noreferrer&quot;&gt;Babylonian Astronomical Diaries&lt;/a&gt; (Clay, and Babylonian)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Then it became apparent to me: how long will these items last before they too degrade and are lost? As the biggest reason we do not have Chinese/Indian works was because they the materials they were made out of was super biodegradable, and thus degraded before they were preserved properly. These mentioned works, and more, are also susceptible to this even in a proper museum space.&lt;/p&gt;&#xA;&lt;p&gt;While I understand that many of these items exist in museums and are being well-preserved by staff, I can't control myself from being filled with some sort of inner sorrow that eventually these items will degrade with time regardless of the amount of physical preservation, and the information on them will be lost. Thus, I am utterly curious on these prominent two questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Are there an ongoing plans for these type of works to be digitized to preserve their information?&lt;span class=&quot;math-container&quot;&gt;$^{[1][2]}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;What is the expected amount of time left for most of early, hand-held, Pre-Hellenistic mathematics works made of the mentioned raw materials?&lt;span class=&quot;math-container&quot;&gt;$^{[3]}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In addition, I am also curious to more examples of ancient Indian and Chinese Mathematics that wasn't listed above that's also made from materials like bamboo and bark, as there should (hopefully/realistically) be more than those two that was mentioned.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt; Digitizing works via flash photography or scanning is often dangerous because the bright lights emitted by cameras and such can damage them, so I'm also really curious if there are ways people have gotten around this, or is this just an assumed risk if the piece is going to degrade anyways?&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[2]$&lt;/span&gt; Upon further study on this, I was able to find the &lt;a href=&quot;https://www.loc.gov/preservation/care/scan.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Library of Congress&lt;/a&gt;'s rules for digitizing works. While it is a stretch, I feel like these rules would apply themselves to other institutions and how they handle digitizing old work. In addition, they have an &lt;a href=&quot;https://www.loc.gov/rr/rarebook/digitalcoll/digitalcoll-magic.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;online catalog of digitized material&lt;/a&gt; already.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[3]$&lt;/span&gt; Of course, Egyptian math that exists on the walls of pyramids will stand to resist time for probably longer than humanity itself. So we should limit this to things that fit the definition of it being humanly portable, like clay tablets, written documents, books, etc.&lt;/p&gt;&#xA;" OwnerUserId="17466" LastEditorUserId="17466" LastEditDate="2022-11-08T21:06:50.990" LastActivityDate="2022-11-18T19:14:58.723" Title="What is the Expected Durability/Digitization Efforts of Pre-Hellenistic works that are made from Materials like Papyrus, Bamboo, Bark, Stone, e.t.c?" Tags="|ancient-egypt|ancient-india|ancient-china|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14910" PostTypeId="1" AcceptedAnswerId="14911" CreationDate="2022-11-06T06:28:30.357" Score="4" ViewCount="200" Body="&lt;p&gt;Is the idea of a continuous map in the point-set model of topological spaces, i.e. that the preimages of opens are open, due to Hausdorff (Grundzüge der Mengenlehre)?&lt;/p&gt;&#xA;&lt;p&gt;For example, does the notion of fibers of a ring morphism predate the said book (Kronecker died before the said publication)?&lt;/p&gt;&#xA;&lt;p&gt;Some discussion in Chinese: &lt;a href=&quot;https://www.zhihu.com/question/561581120/answer/2732719832&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.zhihu.com/question/561581120/answer/2732719832&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17467" LastActivityDate="2022-11-07T12:48:10.770" Title="Continuity, Hausdorff" Tags="|topology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14911" PostTypeId="2" ParentId="14910" CreationDate="2022-11-06T10:32:49.580" Score="8" Body="&lt;p&gt;Roughly it goes like this.&lt;/p&gt;&#xA;&lt;p&gt;Hausdorff (and others at the time) considered &amp;quot;manifolds&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, where for each point &lt;span class=&quot;math-container&quot;&gt;$p \in M$&lt;/span&gt; there are &amp;quot;coordinate neighborhoods&amp;quot; which are in 1-1 correspondence with neighborhoods in Euclidean space.  A function &lt;span class=&quot;math-container&quot;&gt;$f : M \to N$&lt;/span&gt; is continuous at a point &lt;span class=&quot;math-container&quot;&gt;$p \in M$&lt;/span&gt; if there are coordinate neighborhoods &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$f(p)$&lt;/span&gt; so that the corresponding map in Euclidean space is continuous.&lt;/p&gt;&#xA;&lt;p&gt;Then Hausdorff made the leap that you did not need the &lt;em&gt;coordinates&lt;/em&gt; to define continuity, it was enough to have &amp;quot;neighborhoods&amp;quot; that mapped into one another appropriately.  Hausdorff's definition of such spaces was in terms of neighborhoods, not in terms of open sets.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2vgsm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2vgsm.png&quot; alt=&quot;Hausdorff&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When other abstract definitions of &amp;quot;topology&amp;quot; were proposed (in terms of open sets, in terms of convergence, in terms of closure operation, etc.) it turned out that Hausdorff's definition was equivalent to theirs, except for the need of an additional restriction (which is nowadays called the Hausdorff condition).&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2022-11-07T12:48:10.770" LastActivityDate="2022-11-07T12:48:10.770" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14912" PostTypeId="1" CreationDate="2022-11-06T19:07:54.207" Score="0" ViewCount="75" Body="&lt;p&gt;According to &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Germ_theory_of_disease&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, the germ theory of disease has been held in disdain for centuries in Europe, where the miasma theory was prevalent.&lt;/p&gt;&#xA;&lt;p&gt;Why were scientists so biased against the germ theory?&lt;/p&gt;&#xA;" OwnerUserId="17470" LastActivityDate="2022-11-06T19:07:54.207" Title="Why was the germ theory of disease held in disdain?" Tags="|medicine|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14913" PostTypeId="1" AcceptedAnswerId="14915" CreationDate="2022-11-07T10:06:30.127" Score="2" ViewCount="138" Body="&lt;p&gt;There is a claim I occasionally read that the origin of the word &amp;quot;gauge&amp;quot; refers to a &lt;a href=&quot;https://en.wikipedia.org/wiki/Track_gauge&quot; rel=&quot;nofollow noreferrer&quot;&gt;track gauge&lt;/a&gt; used in railroad tracks (the distance between two rails). It's a claim I have seen &lt;a href=&quot;http://www.weylmann.com/weyltheory.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=2951&amp;amp;context=etd&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, on &lt;a href=&quot;https://en.wikipedia.org/wiki/Introduction_to_gauge_theory#History_and_importance&quot; rel=&quot;nofollow noreferrer&quot;&gt;this wikipedia article&lt;/a&gt;, as well as on various answers regarding the etymology of the word&lt;a href=&quot;https://www.physicsforums.com/threads/early-history-of-gauge-theory-a-science-history-paper.88998/&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;a href=&quot;https://www.billstclair.com/www.cheniere.org/correspondence/072301.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;[2]&lt;/a&gt;. &lt;a href=&quot;https://books.google.fr/books?id=vImIJE0kQNcC&amp;amp;lpg=PA224&amp;amp;ots=_g6owWowAb&amp;amp;dq=%22weyl%22%20%22gauge%22%20%22railroad%22&amp;amp;pg=PA224#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some papers&lt;/a&gt; also note specifically that this is not the original meaning of the word.&lt;/p&gt;&#xA;&lt;p&gt;As far as I can tell, none of the original Weyl papers&lt;a href=&quot;https://ia800708.us.archive.org/view_archive.php?archive=/28/items/crossref-pre-1923-scholarly-works/10.1002%252Fandp.19183622102.zip&amp;amp;file=10.1002%252Fandp.19193641002.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[3]&lt;/a&gt;&lt;a href=&quot;https://books.google.fr/books?id=jQbEcDDqGb8C&quot; rel=&quot;nofollow noreferrer&quot;&gt;[4]&lt;/a&gt;&lt;a href=&quot;https://www.pnas.org/doi/pdf/10.1073/pnas.15.4.323&quot; rel=&quot;nofollow noreferrer&quot;&gt;[5]&lt;/a&gt; mention that idea, using Eichung (or earlier on, Maßstab) in the original German, which does not have the specific meaning of a track gauge. His 1929 paper on &lt;a href=&quot;https://www.pnas.org/doi/pdf/10.1073/pnas.15.4.323&quot; rel=&quot;nofollow noreferrer&quot;&gt;gravitation and the electron&lt;/a&gt; in English translates it as gauge, but again no specific mention of this analogy. As far as I can tell, while track gauge is a possible meaning of the word, it is by no mean the only one.&lt;/p&gt;&#xA;&lt;p&gt;Despite that, I still see the occasional claim that this is the origin of the word. Is there an origin to that idea?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2022-11-07T17:45:28.900" Title="Weyl's gauge theory and railroad tracks" Tags="|physics|terminology|relativity-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14914" PostTypeId="1" CreationDate="2022-11-07T13:03:52.657" Score="1" ViewCount="142" Body="&lt;p&gt;The law of reflection, &lt;span class=&quot;math-container&quot;&gt;$r=i$&lt;/span&gt;, is attributed to Euclid. In his &lt;a href=&quot;http://philomatica.org/wp-content/uploads/2013/01/Optics-of-Euclid.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Optics&lt;/a&gt; text he refers to it, at the end of Page 360 in relation to Graph 19, and says he has demonstrated it in his &lt;a href=&quot;https://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Euclid/Catoptrics/Catoptrics%201/EuclidCatoptrics.1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Catoptrics&lt;/a&gt;. But I do not understand his reasoning in Catoptrics. (To make the matter more confusing, some historians claim Catoptrics was written by another person several hundred years after Euclid. Or that there were more than one Catoptrics.)&lt;/p&gt;&#xA;&lt;p&gt;Did he think since he can look at an image in a flat mirror without distortion then &lt;span class=&quot;math-container&quot;&gt;$r=i$&lt;/span&gt;? Is it based on a geometrical reasoning or on experiments.&lt;/p&gt;&#xA;&lt;p&gt;Did Euclid have a concept of ``shortest distance'' for the light path? &lt;a href=&quot;https://farside.ph.utexas.edu/teaching/316/lectures/node125.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Some websites&lt;/a&gt; claim this but I have not seen it in the translation of Optics, or in well-referenced papers. Did anyone before Fermat thought of shortest time or distance in relation to light path?&lt;/p&gt;&#xA;" OwnerUserId="17473" LastEditorUserId="17473" LastEditDate="2022-11-07T19:48:11.633" LastActivityDate="2022-11-07T19:48:11.633" Title="How did Euclid arrive at the law of reflection $r=i$?" Tags="|euclidean-geometry|optics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14915" PostTypeId="2" ParentId="14913" CreationDate="2022-11-07T17:45:28.900" Score="6" Body="&lt;p&gt;In German the word &amp;quot;Eichung&amp;quot; (translated as &amp;quot;gauge&amp;quot;) is not used for the width of railroad tracks but rather refers to a general standard for measurements similar to the broader meaning of English &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gauge&quot; rel=&quot;noreferrer&quot;&gt;gauge&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Looking at Weyl's 1919 paper cited in the OP, the first occurence of the word &amp;quot;eich...&amp;quot; is in the sentence (p. 102):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An jeder einzelnen Weltstelle muß die Streckeneichung vorgenommen werden, diese Aufgabe kann nicht einem zentralen Eichamt übertragen werden.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This translates as:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;At each space-time position this gauging of distance needs to be performed, a task that cannot be given to a central gauging office.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The word &amp;quot;&lt;a href=&quot;https://de.wikipedia.org/wiki/Eichamt&quot; rel=&quot;noreferrer&quot;&gt;Eichamt&lt;/a&gt;&amp;quot; (literally &amp;quot;gauging office&amp;quot;) means a bureau of standards which has the legal task of ensuring proper measurements. Its remit is thus metrology including the choice and administration of units everyone has to adopt and employ.  From the first two pages of his paper one can infer that this is the original context Weyl had in mind - even if his use of the term &amp;quot;Eichamt&amp;quot; is metaphorical: Weyl's &amp;quot;gauging&amp;quot; deals with mathematics of local scale invariance, hence the choice of local distance scales.&lt;/p&gt;&#xA;&lt;p&gt;The comparison with track gauges (not made by Weyl) makes sense as the latter correspond to a gauging of &lt;em&gt;distance&lt;/em&gt; (in a metrological or technological sense).&lt;/p&gt;&#xA;" OwnerUserId="16834" LastActivityDate="2022-11-07T17:45:28.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14916" PostTypeId="2" ParentId="5093" CreationDate="2022-11-09T11:47:58.857" Score="2" Body="&lt;p&gt;A lot of mathematical activities related to hand computing have been obsoleted by computers. Devising tables by hand such as logarithm tables, trigonometric tables, mortgage/amortization tables etc was a major occupation of mathematicians until the advent of modern computers. A lot of theoretical materials in mathematics come from the need to make computations feasible with a pen and a sheet of paper. An example is the theory of divergent series, as described in Hardy's book, which was used to investigate the Stokes's phenomenom.&lt;/p&gt;&#xA;&lt;p&gt;Have a look at the engineering curriculum fifty years ago or speak with a retired engineer. You will learn many tricks that are now forgotten and some of them played an important role on the spot. Anything related to the slide rule, for example. My favorite bit concerning trigonometry:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sin(x) \sim {x \over 60}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="6959" LastActivityDate="2022-11-09T11:47:58.857" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14917" PostTypeId="1" CreationDate="2022-11-09T13:25:33.600" Score="0" ViewCount="245" Body="&lt;p&gt;There is a story I often tell my students about tasks that seem possible but are actually impossible, but I can hardly remember the details, and I am hoping that someone here would remember. Here is what I remember: A professor back in the 70s hired two undergraduate students to complete a task that he thought must be easy but was actually impossible at that time.&lt;/p&gt;&#xA;&lt;p&gt;I vaguely remember that the professor wanted a computer program that would understand human language and the two students were computer science students. However, these details might be completely wrong.&lt;/p&gt;&#xA;&lt;p&gt;If anyone knows this story and where I can get more details about it, it would be great.&lt;/p&gt;&#xA;" OwnerUserId="17485" OwnerDisplayName="YBA" LastEditorUserId="12330" LastEditDate="2022-11-10T07:26:20.130" LastActivityDate="2022-11-10T07:26:20.130" Title="Anecdote about a seemingly possible, but actually impossible task given to students" Tags="|anecdote|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14918" PostTypeId="2" ParentId="14877" CreationDate="2022-11-09T18:35:56.163" Score="3" Body="&lt;p&gt;The book is none other than &lt;a href=&quot;https://en.wikipedia.org/wiki/Marvin_Greenberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marvin Jay Greenberg&lt;/a&gt;'s &amp;quot;Euclidean and Non-Euclidean Geometries: Development and History&amp;quot; (the fourth edition of which is available wherever fine books are sold).&lt;/p&gt;&#xA;" OwnerUserId="136" LastActivityDate="2022-11-09T18:35:56.163" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14919" PostTypeId="1" CreationDate="2022-11-10T01:44:43.760" Score="2" ViewCount="706" Body="&lt;p&gt;I remember first seeing this anecdote a few years ago in a newspaper article, now I am seeing more frequently in motivational content over the internet, every time with very few additional details. The story goes as follow:&lt;/p&gt;&#xA;&lt;p&gt;While in Princeton or Oxford University, a graduate assistant or a teacher asked Albert Einstein why did he gave his (sometimes 2nd year graduate) students the same exam as last year. Einstein answered that he was not worried, &amp;quot;it is exactly the same, well the answers have changed [since]&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;As with many of Einstein's quotes, this narration seems fabricated. Is there any source for this quote/story? Could it be a true story wrongly misattributed to Einstein? Do we know who could be the assistant/professor? What were the facts that had changed? Was he teaching optics, dynamics, atomic physics or cosmology (or all of the above)?&lt;/p&gt;&#xA;&lt;p&gt;So far, the oldest source I have found might be (American religion author) John Bytheway's 1999 &lt;em&gt;What I Wish I'd Known When I Was Single&lt;/em&gt;, a self-help book (he mentions Princeton). See also Paul Rulken's 2014 TEDxTalk &lt;a href=&quot;https://www.youtube.com/watch?v=VNGFep6rncY&amp;amp;ab_channel=TEDxTalks&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Why the majority is always wrong&amp;quot;&lt;/a&gt;. The Princeton/Oxford fact narrows this event down to 1931-1939 (Rulken says 1932).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2022-11-16T10:21:30.280" LastActivityDate="2022-11-16T10:21:30.280" Title="Einstein's &quot;answers have changed&quot;" Tags="|einstein|physicists|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14920" PostTypeId="1" CreationDate="2022-11-10T15:59:42.787" Score="2" ViewCount="127" Body="&lt;p&gt;Cf. &lt;a href=&quot;https://golem.ph.utexas.edu/category/2013/07/the_first_commutative_diagram.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://golem.ph.utexas.edu/category/2013/07/the_first_commutative_diagram.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is this the first ever commutative diagram? By commutative diagram I mean anything that shows an equality (or coherency, but higher morphisms doesn’t show up until much later (another great question for this site: the first natural transformation)) of two morphisms, so a triangle is a commutative diagram, but a string isn’t.&lt;/p&gt;&#xA;" OwnerUserId="17467" LastActivityDate="2022-11-10T15:59:42.787" Title="The first commutative diagram" Tags="|mathematics|category-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14921" PostTypeId="1" CreationDate="2022-11-11T13:43:26.147" Score="1" ViewCount="370" Body="&lt;p&gt;Who introduced &lt;a href=&quot;https://en.wikipedia.org/wiki/Fraction#Mixed_numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;mixed fraction notation&lt;/a&gt;? This notion is a source of confusion to me because it may be interpreted as multiplication.&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="26" LastEditDate="2022-11-11T14:19:26.063" LastActivityDate="2022-11-13T09:43:46.893" Title="Who introduced the mixed fraction notation?" Tags="|notation|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14923" PostTypeId="1" AcceptedAnswerId="14925" CreationDate="2022-11-11T15:58:28.140" Score="1" ViewCount="179" Body="&lt;p&gt;I came across the following quote by the famous mathematician Paul Halmos:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A clever graduate student could teach Fourier something new, but surely no one claims that he could teach Archimedes to reason better.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What does it mean, and what is the context? It seems as if Halmos wasn't fond of Archimedes' methods of reasoning, which is quite intriguing.&lt;/p&gt;&#xA;" OwnerUserId="7378" LastEditorUserId="4251" LastEditDate="2022-11-11T17:01:47.617" LastActivityDate="2022-11-11T19:03:19.857" Title="What does this quote by Paul Halmos mean?" Tags="|mathematicians|quote|archimedes|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14924" PostTypeId="2" ParentId="13155" CreationDate="2022-11-11T18:12:53.113" Score="5" Body="&lt;p&gt;&lt;em&gt;This answer is now mostly complete, and it's really long. I've made substantial changes since originally posting, but now it's pretty much in its final state. Finding the right interpretation of Cotes' work proved much more difficult, and required a lot more translation, than I had expected! A translation exists in&lt;/em&gt; Roger Cotes - Natural Philosopher &lt;em&gt;by Cook and Gowing, but I didn't have access to it. I might continue the translation, in which case I'll post a link to it later.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;My reference is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Cotes, Roger. &lt;em&gt;Logometria&lt;/em&gt;. Philosophical transactions [of the Royal Society of London], N. 338, vol. 29 (1714), pp. 31-32. &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=ucm.5324351035&amp;amp;view=1up&amp;amp;seq=11&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;This proof is pretty involved, mostly because of the large &amp;quot;knowledge debt&amp;quot; from how the practice of mathematics has changed since Cotes' time. He was working with the surface area of an ellipsoid of revolution. For that part, he includes two diagrams, and they are reproduced with the text.&lt;/p&gt;&#xA;&lt;h3&gt;Background: Cotes' Definition of Measure of a Ratio and its Modulus&lt;/h3&gt;&#xA;&lt;p&gt;I translated some passages from the beginning of &lt;em&gt;Logometria&lt;/em&gt; that might be relevant. These are just rough translations, maybe I'll revisit later.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In this article we treat &lt;em&gt;Measures of Ratios&lt;/em&gt;. These Measures are quantities of any sort, whose magnitudes are proportional with magnitudes of ratios. Thus in a given System, the measure of the same ratio is the same, of a doubled ratio is double, of a tripled ratio is triple, of a halved ratio is half, of a sesquiplicate ratio [in ratio of 3:2] is sesquialtera: hence whatever ratios are increased or diminished by composition or resolution, the measure is likewise increased or diminished. A ratio of equals has no magnitude, for addition or detraction do not lead to change; ratios which are called major and minor of inequality have opposite affections of their magnitudes, since they always make opposites in composition and resolution: therefore if we imagine the rational measure having a major term to the minor as positive, the rational measure having the minor term to the major will be negative, the rational measure between equal terms will of course be of zero magnitude. Further, different Systems of measures arise, just as the determined proportion is expressed by different ways and that it is unalterable between rational magnitudes. Hence it is clear, that the Systems can be expressed by an infinite number, by decreasing or increasing all measures of whatever given System in the same given proportion, or [otherwise for quantities by applying measures of different kinds]. In such varieties however some confusion necessarily arises, unless it is well agreed to what System of reference each of the measures which the conversation touches upon is established. One can best prepare a remedy for this issue if the given measure of some ratio, which is considered most suitable, is taken as the &lt;em&gt;Modulus&lt;/em&gt; against which in all Systems the measures of remaining ratios are continuously weighed. This being done, immediately all Systems will be determined from this Modulus : as from these measures it will consist of those which will be homogeneous to the Modulus, and which will have greater or lesser magnitudes than it, the larger or smaller it is, such that the invariance of the magnitudes of ratios to be measured is preserved by the ratio between the same measures. It will therefore be clear in the following that a certain ratio is given, between the double and triple ratios, but approaching nearer to the triple, which by our proposition may be judged most appropriate not without reason, since the very nature of the thing suggests its use and seems to insist upon it in a way by no uncertain indication. This I will call the &lt;em&gt;Modular Ratio&lt;/em&gt;, so named by virtue of its application; the way, however, that it may be more accurately defined, will be shown further below, as now a word on Logarithms must be added.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Logarithms&lt;/em&gt; are Numerical measures of ratios: they are typically so distributed in a Canon, so that each number increasing by the natural order, and set in continuous series, is ascribed to the Logarithm; in fact the use of the number itself is not commonly observed, but rather the use of the ratio which the number has to Unity. From this it is easy to find the Logarithm of a ratio denoted by whatever terms.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I had a difficult time understanding what Cotes meant here, although certain things stand out (e.g. the &amp;quot;certain ratio, between double and triple ratios, approaching nearer to triple&amp;quot; is &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, and hence it gives the natural log when used as the base of a logarithm). But from his Proposition 1, the subsequent Corollaries, and Scholium 1, I was able to verify that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The &lt;strong&gt;measure of a ratio&lt;/strong&gt;, say the ratio &lt;span class=&quot;math-container&quot;&gt;$p:q$&lt;/span&gt;, is its logarithm, &lt;span class=&quot;math-container&quot;&gt;$\log \frac{p}{q}$&lt;/span&gt;, to some unspecified base which must be agreed upon. For a logarithm, he says that it is usually calculated using &lt;span class=&quot;math-container&quot;&gt;$y=\frac{p}{q}$&lt;/span&gt;, and calculating &lt;span class=&quot;math-container&quot;&gt;$\log \frac{y}{1}$&lt;/span&gt;, i.e. &lt;span class=&quot;math-container&quot;&gt;$\log y$&lt;/span&gt;. His actual definition is more abstract: &amp;quot;Measures of ratio are quantities of any sort, whose magnitudes are analogous to the magnitudes of ratios. In a given system, the measure of the same ratio is the same, etc...&amp;quot; But later the definition becomes more clear and is calculated explicitly (Schol. 1).&lt;/li&gt;&#xA;&lt;li&gt;The &lt;strong&gt;modulus&lt;/strong&gt; of the measure is related to the logarithm base. More properly, the modulus is the &lt;em&gt;measure of ratio of a particular reference ratio&lt;/em&gt;, so that (I suppose) the reference measure is 1 for that ratio. This should be more clear with an example. Suppose we want to know the measure of ratio of quantities &lt;span class=&quot;math-container&quot;&gt;$x,y,z,...$&lt;/span&gt;, i.e. the values &lt;span class=&quot;math-container&quot;&gt;$\log x, \log y, \log z,...$&lt;/span&gt; with no base specified. If we have calculated these in base &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, then it follows that the measure of ratio &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; should be 1, i.e. &lt;span class=&quot;math-container&quot;&gt;$\log_a a$&lt;/span&gt; as expected. Hence all other measures of ratio are determined by their relationship to the modulus &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;In Proposition 1, Cotes calls &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; the &lt;em&gt;modulus of the system&lt;/em&gt;. This seems to conflict with what I'm saying, since it's actually the inverse of the measure of ratio, and referred to another measure of ratio (&lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;). So there's probably still a misunderstanding in my description above!&lt;/li&gt;&#xA;&lt;li&gt;Calculating measures of ratios in a different base &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; can always be done with&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\log_a x = M \log_b x&#xA;$$&lt;/span&gt;&#xA;where of course &lt;span class=&quot;math-container&quot;&gt;$M=\frac{1}{\log_b a}$&lt;/span&gt; (from the change of base formula). Similarly, every other value, e.g. &lt;span class=&quot;math-container&quot;&gt;$\log_a y,\log_a z$&lt;/span&gt;, etc, is obtained in base &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; by multiplying by &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the &lt;strong&gt;modular ratio&lt;/strong&gt; (later revealed to be the base &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;) which is readily calculated, then logs to every other base can be found with a suitable modulus &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; in this way. This is the way log tables are used.&lt;/li&gt;&#xA;&lt;li&gt;Practically speaking, if we wish to find the &lt;em&gt;measure of ratio&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to some base, i.e. &lt;span class=&quot;math-container&quot;&gt;$\log_a x$&lt;/span&gt; with base &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, then we can simply calculate &lt;span class=&quot;math-container&quot;&gt;$\ln x$&lt;/span&gt;, and multiply it by &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt;, this value being given by&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;M=\frac{1}{\ln a}.&#xA;$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h3&gt;Surface Area of an Ellipsoid of Revolution Part 1.&lt;/h3&gt;&#xA;&lt;p&gt;I've translated the details from pp.31-32:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Rhvsp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Rhvsp.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;...To this may be added the surfaces generated by an Ellipse. Let &lt;span class=&quot;math-container&quot;&gt;$ANB$&lt;/span&gt; be an Ellipse described about the center &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, with vertices &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, focus &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, principal semiaxis &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt;, conjugate semiaxis &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt;; and to some point &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; on the axis &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; let &lt;span class=&quot;math-container&quot;&gt;$XN$&lt;/span&gt; be the ordinate, which the Ellipse crosses at &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;. In the axis &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt;, is taken &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; is to &lt;span class=&quot;math-container&quot;&gt;$CF$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$EX$&lt;/span&gt; is joined. Then &lt;span class=&quot;math-container&quot;&gt;$KL$&lt;/span&gt; is set such that it be to &lt;span class=&quot;math-container&quot;&gt;$XC$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$XE$&lt;/span&gt; is to &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$LM$&lt;/span&gt; that which is the measure of ratio between &lt;span class=&quot;math-container&quot;&gt;$EX+XC$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; to the Modulus &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt;: and the surface generated from the arc &lt;span class=&quot;math-container&quot;&gt;$BN$&lt;/span&gt; rotating about the axis &lt;span class=&quot;math-container&quot;&gt;$CX$&lt;/span&gt;, will be to the semidiameter &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt; described, as of the lines &lt;span class=&quot;math-container&quot;&gt;$KL$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$LM$&lt;/span&gt; combined as &lt;span class=&quot;math-container&quot;&gt;$KM$&lt;/span&gt;, is to the semidiameter &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt;. For this latter construction to exist, it is necessary that the semiaxis &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; around which the revolution was made, be lesser than the other semiaxis &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt;; otherwise the quantity &lt;span class=&quot;math-container&quot;&gt;$\frac{CAq}{\sqrt{ CBq-CAq }}$&lt;/span&gt; of the modulus &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; becomes impossible, and the construction by Logometry (which in these cases often arises) turns itself into Trigonometry, as follows.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/N7ZUQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/N7ZUQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$ANB$&lt;/span&gt; be an ellipse described about the center &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, with vertices &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt;, focus &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt;, principal semiaxis &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt;, conjugate semiaxis &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt;; and to some point &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; on the axis &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; let &lt;span class=&quot;math-container&quot;&gt;$XN$&lt;/span&gt; be the ordinate, which the Ellipse crosses at &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;. The line &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; is inscribed from the angle &lt;span class=&quot;math-container&quot;&gt;$CXN$&lt;/span&gt;, such that it is to &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$CA$&lt;/span&gt; is to &lt;span class=&quot;math-container&quot;&gt;$CF$&lt;/span&gt;. Then let &lt;span class=&quot;math-container&quot;&gt;$KL$&lt;/span&gt; be taken such that it is to &lt;span class=&quot;math-container&quot;&gt;$XC$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$XE$&lt;/span&gt; is to &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt;, and let &lt;span class=&quot;math-container&quot;&gt;$LM$&lt;/span&gt; be the measure of the angle &lt;span class=&quot;math-container&quot;&gt;$XEC$&lt;/span&gt; to the Modulus &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt;, i.e. the measure of that equal to the arc whose sine is &lt;span class=&quot;math-container&quot;&gt;$XC$&lt;/span&gt;, to the radius &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt;: and the surface generated from the arc &lt;span class=&quot;math-container&quot;&gt;$BN$&lt;/span&gt; by revolving around the axis &lt;span class=&quot;math-container&quot;&gt;$CX$&lt;/span&gt;, will be to the semidiameter &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt; of the circle described, as that of the lines &lt;span class=&quot;math-container&quot;&gt;$KL$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$LM$&lt;/span&gt; combined as &lt;span class=&quot;math-container&quot;&gt;$KM$&lt;/span&gt; are, to the semidiameter &lt;span class=&quot;math-container&quot;&gt;$CB$&lt;/span&gt;. One can obtain the area of this surface by Logometry, but only by the inexplicable [imaginary numbers]. Since if any arc of the quadrant of a circle, described with radius &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt;, has the sine &lt;span class=&quot;math-container&quot;&gt;$CX$&lt;/span&gt; and the complementary sine to the quadrant &lt;span class=&quot;math-container&quot;&gt;$XE$&lt;/span&gt;: taking the radius &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; as the Modulus, the arc will be a measure of the ratio between &lt;span class=&quot;math-container&quot;&gt;$EX+CX\sqrt{-1}$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; multiplied by &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt;. Here I leave the more diligent examinations to others, who would find the work valuable.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Such is the text leading to this result. The first part of this describes the problem of finding the surface area of an ellipsoid of revolution, which leads to the latter part, the crux of the problem, describing the sum of cosine and the product of the sine and the imaginary unit. This is what we're interested in, but we'll need to understand the context.&lt;/p&gt;&#xA;&lt;h3&gt;Surface Area Part 2 (Cajori)&lt;/h3&gt;&#xA;&lt;p&gt;How Cajori describes the above in &lt;em&gt;History of the Exponential and Logarithmic Concepts&lt;/em&gt;, Amer. Math. Monthly v20 no. 2 (Feb 1913), pp. 35-47, as linked in the other answer, is as follows:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; be the horizontal radius, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; be the vertical radius, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{a^2-b^2}$&lt;/span&gt; be the horizontal distance from the center to the focus; if &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the angle &lt;span class=&quot;math-container&quot;&gt;$XEC$&lt;/span&gt;, then using &lt;span class=&quot;math-container&quot;&gt;$l=CE$&lt;/span&gt;, the value of &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$y=l\sin\theta$&lt;/span&gt;, and the surface area of the revolved arc of the ellipse &lt;span class=&quot;math-container&quot;&gt;$b^2 x^2 + a^2 y^2 = a^2 b^2$&lt;/span&gt; up to height &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; becomes&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;S&amp;amp;= a\pi \left\{ y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^4} y^2 } + \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \log \left( y\sqrt{ \frac{a^{2}-b^{2}}{b^4} }+\sqrt{ 1+\frac{a^2-b^{2}}{b^4}y^{2} } \right)  \right\} , \\&#xA;S&amp;amp;=a\pi \left\{ y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^4}y^{2} }+\frac{b^{2}}{\sqrt{ b^{2}-a^{2} }}\varphi \right\},&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\sin \theta = y\sqrt{ \frac{b^{2}-a^{2}}{b^4} }=iy\sqrt{ \frac{a^{2}-b^{2}}{b^4} }$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\cos \theta = \sqrt{ 1+\frac{a^{2}-b^{2}}{b^4}y^2 }$&lt;/span&gt;. The imaginary unit appears when flipping the sign under the root. Substituting these in, the surface area becomes&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;S=&amp;amp;a\pi \left(  \cos \theta + \frac{b^{2}}{f} \log \left( i\sin \theta +\cos \theta \right)  \right) \\&#xA;S= &amp;amp; a\pi \left( \cos \theta + \frac{b^{2}}{if}\theta  \right) &#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;and comparing like terms,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\log(i\sin \theta+\cos \theta)=-i \theta.&#xA;$$&lt;/span&gt;&#xA;Note that Cajori was missing the negative sign on the right, but Cotes had it, as seen from his having &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; on the left side. There's evidently a mistake here somewhere, because the correct result would not have a negative on the right-hand side.&lt;/p&gt;&#xA;&lt;h3&gt;Surface Area Part 3 (adapted from Cotes)&lt;/h3&gt;&#xA;&lt;p&gt;Let's go through Cotes' work, adapting it to modern notation. Cotes uses his own propositions and corollaries to draw conclusions with strictly geometric arguments, something we don't have the luxury of doing, so it will probably seem complex compared to his short passage above. But the development is the same, and here it is self-contained.&lt;/p&gt;&#xA;&lt;p&gt;Consider an ellipse, where &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is the horizontal radius, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; is the vertical radius. Thus &lt;span class=&quot;math-container&quot;&gt;$a=CB,b=CA$&lt;/span&gt;. Let the angle &lt;span class=&quot;math-container&quot;&gt;$\theta=\angle XEC$&lt;/span&gt;. This all remains true in the two following cases:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;b$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$CB=a$&lt;/span&gt; is the principal axis, &lt;span class=&quot;math-container&quot;&gt;$CA=b$&lt;/span&gt; is the conjugate&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$a&amp;lt;b$&lt;/span&gt;, so &lt;span class=&quot;math-container&quot;&gt;$CA=b$&lt;/span&gt; is the principal axis, &lt;span class=&quot;math-container&quot;&gt;$CB=a$&lt;/span&gt; is the conjugate&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Rhvsp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Rhvsp.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Case 1.&lt;/strong&gt; The drawing above shows the first case. We'll consider this one to begin. Let &lt;span class=&quot;math-container&quot;&gt;$x=CE$&lt;/span&gt; be the distance from center to &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y=CX$&lt;/span&gt; be the height of a section of the ellipse, &lt;span class=&quot;math-container&quot;&gt;$f=CF$&lt;/span&gt; be the &lt;em&gt;horizontal&lt;/em&gt; distance from center to focus, &lt;span class=&quot;math-container&quot;&gt;$l=XE$&lt;/span&gt; be the hypotenuse of a right triangle &lt;span class=&quot;math-container&quot;&gt;$XEC$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k_{1}=KL$&lt;/span&gt; be the first auxiliary length, &lt;span class=&quot;math-container&quot;&gt;$k_{2}=LM$&lt;/span&gt; be the second. Let &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; be the surface area of the ellipsoid of revolution up to height &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; in the drawing above. Cotes gives the following results:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\text{I. }&amp;amp; CE:CA :: CA:CF\\&#xA;\text{II. }&amp;amp; KL:XC :: XE:CE\\&#xA;\text{III. }&amp;amp; LM = CE\ln \left( \frac{EX+XC}{CE} \right) \\&#xA;\text{IV. }&amp;amp; S:CB :: KM:CB&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;Where &lt;span class=&quot;math-container&quot;&gt;$KM=KL+LM$&lt;/span&gt;. With the variables given previously, and in modern notation, these are&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\text{I.   }&amp;amp; \frac{x}{b}=\frac{b}{f}\\&#xA;\text{II.   }&amp;amp; \frac{k_{1}}{y}=\frac{l}{x}\\&#xA;\text{III.   }&amp;amp; k_{2} = x\ln \left( \frac{l+y}{x} \right) \\&#xA;\text{IV.   }&amp;amp; \frac{S}{a} = \frac{k_{1}+k_{2}}{a}.&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;We can deduce that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;x=&amp;amp; \frac{b^{2}}{f}, \\&#xA;k_{1} = &amp;amp; \frac{yl}{x}, \\&#xA;f =&amp;amp; \sqrt{ a^{2}-b^{2} }&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;and finally the surface area of the section of the ellipsoid of revolution would be&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S=k_{1}+k_{2}=\frac{yl}{x}+x\log \left( \frac{l}{x}+\frac{y}{x} \right) .&#xA;$$&lt;/span&gt;&#xA;The variables &lt;span class=&quot;math-container&quot;&gt;$l,x,y$&lt;/span&gt; are legs of the right triangle, and taking &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; as given, without using trigonometry, we get the others,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;x= &amp;amp; \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }}, \\[1em]&#xA;l= &amp;amp; \sqrt{ x^{2}+y^{2} }.&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;Note, however, the denominator of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (the focus) becomes imaginary if &lt;span class=&quot;math-container&quot;&gt;$b&amp;gt;a$&lt;/span&gt;. This is where imaginary numbers are introduced into the problem. We'll consider that case on its own later.&lt;/p&gt;&#xA;&lt;p&gt;We can obtain Cajori's expression for &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; by first rearranging the expression for &lt;span class=&quot;math-container&quot;&gt;$k_{1}$&lt;/span&gt; as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{1}=\frac{yl}{x}=\frac{y}{x}\sqrt{ x^{2}+y^{2} }=y\sqrt{ 1+\frac{y^{2}}{x^{2}} },&#xA;$$&lt;/span&gt;&#xA;then substituting &lt;span class=&quot;math-container&quot;&gt;$x=\frac{b^{2}}{f}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f=\sqrt{ a^{2}-b^{2} }$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{1}=y\sqrt{ 1+\frac{f^{2}}{b^{4}}y^{2} }=y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} }.&#xA;$$&lt;/span&gt;&#xA;Next, we rearrange the expression for &lt;span class=&quot;math-container&quot;&gt;$k_{2}$&lt;/span&gt; as follows, using a similar expansion:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{2}=x\ln \left( \frac{y}{x}+\frac{l}{x} \right) =x\ln \left( \frac{y}{x}+ \sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} }  \right)&#xA;$$&lt;/span&gt;&#xA;And now comes a tricky bit. To get Cajori's expression, we make the following substitution for &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;x=\frac{b^{2}}{\sqrt{ a^{2}-b^{2} }}=\sqrt{ \frac{b^{4}}{a^{2}-b^{2}} }&#xA;$$&lt;/span&gt;&#xA;But this is &lt;em&gt;only valid if &lt;span class=&quot;math-container&quot;&gt;$a^{2}-b^{2}&amp;gt;0$&lt;/span&gt;.&lt;/em&gt; This sort of manipulation does not hold when the denominator is negative. This being noted, and substituting &lt;span class=&quot;math-container&quot;&gt;$x=\frac{b^{2}}{\sqrt{ a^{2}-b^{2} }}$&lt;/span&gt; out front, we can see&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{2}= \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \ln \left( y \sqrt{ \frac{a^{2}-b^{2}}{b^{4}} }+\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} } \right) ,&#xA;$$&lt;/span&gt;&#xA;and hence we can obtain for the surface area &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; the expression&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S=k_{1}+k_{2}=y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} } + \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \ln \left( y \sqrt{ \frac{a^{2}-b^{2}}{b^{4}} }+\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} } \right)&#xA;$$&lt;/span&gt;&#xA;which is the same as Cajori's expression, but missing the coefficient &lt;span class=&quot;math-container&quot;&gt;$a \pi$&lt;/span&gt; (from integration?).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Case 2.&lt;/strong&gt; For case 2, many quantities are the same, but some are notably different; we'll prime these, as &lt;span class=&quot;math-container&quot;&gt;$x'$&lt;/span&gt;. Here's the drawing for the second case:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/N7ZUQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/N7ZUQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that the nature of the triangle &lt;span class=&quot;math-container&quot;&gt;$XEC$&lt;/span&gt; (and the angle &lt;span class=&quot;math-container&quot;&gt;$\angle XEC$&lt;/span&gt;) has changed compared to case 1 - now &lt;span class=&quot;math-container&quot;&gt;$CE$&lt;/span&gt; is the hypotenuse, and &lt;span class=&quot;math-container&quot;&gt;$XC, XE$&lt;/span&gt; are the legs. Let &lt;span class=&quot;math-container&quot;&gt;$f'=CF$&lt;/span&gt; be the &lt;em&gt;vertical&lt;/em&gt; distance from center to focus, &lt;span class=&quot;math-container&quot;&gt;$y=CX$&lt;/span&gt; is the sine of angle &lt;span class=&quot;math-container&quot;&gt;$\angle XEC$&lt;/span&gt; as in case 1, &lt;span class=&quot;math-container&quot;&gt;$x'=EX$&lt;/span&gt; is the cosine of &lt;span class=&quot;math-container&quot;&gt;$\angle XEC$&lt;/span&gt; (compare with case 1), and &lt;span class=&quot;math-container&quot;&gt;$l'=CE$&lt;/span&gt; is the hypotenuse of &lt;span class=&quot;math-container&quot;&gt;$\angle XEC$&lt;/span&gt;. We'll take &lt;span class=&quot;math-container&quot;&gt;$\theta=\angle XEC$&lt;/span&gt; to be in radians, so that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;x'= &amp;amp; XE=l'\cos \theta, \\&#xA;y= &amp;amp; XC=l'\sin \theta.&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;As in case 1, Cotes gives the following:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\text{i. }&amp;amp; CE:CA :: CA:CF\\&#xA;\text{ii. }&amp;amp; KL:XC::XE:CE\\&#xA;\text{iii. }&amp;amp; LM=\text{arc length of }\angle XEC\text{ with radius CE}\\&#xA;\text{iv. }&amp;amp; S:CB::KM:CB&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;The only difference from before is the third result. Now similar to before, but with different meanings for &lt;span class=&quot;math-container&quot;&gt;$x'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$l'$&lt;/span&gt;, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;\text{i. }&amp;amp; \frac{l'}{b}=\frac{b}{f'}\\&#xA;\text{ii. }&amp;amp; \frac{k_{1}}{y}=\frac{x'}{l'}\\&#xA;\text{iii. }&amp;amp; k_{2} = l' \theta \\&#xA;\text{iv. }&amp;amp; \frac{S}{a} = \frac{k_{1}+k_{2}}{a}.&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;Thus we can obtain&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align} &#xA;f'= &amp;amp; \sqrt{ b^{2}-a^{2} } \\&#xA;l'= &amp;amp; \frac{b^{2}}{f'} \\[.5em]&#xA;k_{1}= &amp;amp; \frac{yx'}{l'} \\[.5em]&#xA;S= &amp;amp; k_{1}+k_{2};&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;where we flipped the sign of &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; because it is now the focus in the vertical direction. Again we repeat Cajori's development by first rearranging &lt;span class=&quot;math-container&quot;&gt;$k_{1}$&lt;/span&gt;. In fact, by definition of &lt;span class=&quot;math-container&quot;&gt;$\cos \theta$&lt;/span&gt;, we have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{1}=y \frac{x'}{l'} = y\cos \theta;&#xA;$$&lt;/span&gt;&#xA;but we can use the fact that &lt;span class=&quot;math-container&quot;&gt;$x'^{2}=l'^{2}-y^{2}$&lt;/span&gt; to also obtain&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{x'}{l'} = \frac{\sqrt{ l'^{2}-y^{2} }}{l'}=\sqrt{ 1-\frac{y^{2}}{l'^{2}} }&#xA;$$&lt;/span&gt;&#xA;then substituting &lt;span class=&quot;math-container&quot;&gt;$l'=\frac{b^{2}}{f'}$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$f'=b^{2}-a^{2}$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{x'}{l'}=\sqrt{ 1- \frac{f'^{2}}{b^{4}} y^{2} }=\sqrt{ 1- \frac{b^{2}-a^{2}}{b^{4}}y^{2} }=\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} }.&#xA;$$&lt;/span&gt;&#xA;Hence&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{1}=y \frac{x'}{l'} = y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} }&#xA;$$&lt;/span&gt;&#xA;which is identical to case 1.&lt;/p&gt;&#xA;&lt;p&gt;Rearranging &lt;span class=&quot;math-container&quot;&gt;$k_{2}$&lt;/span&gt; (arc length &lt;span class=&quot;math-container&quot;&gt;$l'\ \theta$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; in radians) simply requires substituting for &lt;span class=&quot;math-container&quot;&gt;$l'$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;k_{2}=l'\ \theta = \frac{b^{2}}{\sqrt{ b^{2}-a^{2}}} \theta=-i \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \theta.&#xA;$$&lt;/span&gt;&#xA;&lt;em&gt;Note. This is where the sign discrepancy appears. Square roots of negative real numbers are not well-behaved, so we can't necessarily use normal square root algebra tricks and get the correct result. I'm not 100% on the above expression, yet.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Finally we get the full expression for the surface area of the ellipsoid,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S=k_{1}+k_{2} = y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} }-i \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \theta.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Final Result.&lt;/strong&gt; Throughout the development above, we obtained two expressions for the surface area of an ellipsoid, which must, of course, agree. They are:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;S = &amp;amp;\ y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} }-i \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \theta \\&#xA;S= &amp;amp;\ y\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} } + \frac{b^{2}}{\sqrt{ a^{2}-b^{2} }} \ln \left( y \sqrt{ \frac{a^{2}-b^{2}}{b^{4}} }+\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} } \right)&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;These are the same as Cajori gives. Subtracting these expressions, we must have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;-i \theta=\ln \left( y \sqrt{ \frac{a^{2}-b^{2}}{b^{4}} }+\sqrt{ 1+\frac{a^{2}-b^{2}}{b^{4}}y^{2} } \right).&#xA;$$&lt;/span&gt;&#xA;But we can also show that &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; obeys&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\sin \theta = y\sqrt{ \frac{b^{2}-a^{2}}{b^4} }=iy\sqrt{ \frac{a^{2}-b^{2}}{b^4} },&#xA;$$&lt;/span&gt;&#xA;and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\cos \theta = \sqrt{ 1+\frac{a^{2}-b^{2}}{b^4}y^2 },&#xA;$$&lt;/span&gt;&#xA;and these expressions only depend on quantities shared between the two cases; so that we must have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{align}&#xA;-i\theta = \ln(\cos \theta&amp;amp;+i\sin \theta) \\&#xA;&amp;amp;\textit{or} \\&#xA;\theta=  i\ln(\cos \theta&amp;amp;+i\sin \theta)&#xA;\end{align}&#xA;$$&lt;/span&gt;&#xA;with the latter expression being what Cotes himself wrote. Of course, there is a sign error here, and I would guess it's from the manipulation of square roots and negative real numbers, but hopefully someone can find where this actually happens.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2022-12-02T15:35:39.270" LastActivityDate="2022-12-02T15:35:39.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14925" PostTypeId="2" ParentId="14923" CreationDate="2022-11-11T19:03:19.857" Score="6" Body="&lt;p&gt;The meaning of the quote is more likely to be, that Archimedes' reasoning was on such a high level, that no one (example: the clever graduate student) could teach him to reason better because Archimedes knew all about how to do that already.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-11-11T19:03:19.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14926" PostTypeId="2" ParentId="14921" CreationDate="2022-11-12T12:53:56.160" Score="8" Body="&lt;p&gt;in 1202, Leonardo of Pisa (= Fibonacci) was the first &lt;strong&gt;European&lt;/strong&gt; to do it, but Arabic writers preceded him.  He wrote the fraction to the left of the whole number (e.g. &lt;span class=&quot;math-container&quot;&gt;$\frac12 12$&lt;/span&gt;) following the Arabic usage; note that Arabic is written right-to-left. At the time it was &lt;strong&gt;not&lt;/strong&gt; confusing, because he did not write products by juxtaposition.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/Fraction#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&#xA;we find a discussion of historical notations for fractions.  In particular&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The same fractional notation—with the fraction given before the integer[30]—appears soon after in the work of Leonardo Fibonacci in the 13th century.[33]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;References are&lt;br /&gt;&#xA;[30]  Miller, Jeff (22 December 2014). &amp;quot;Earliest Uses of Various Mathematical Symbols&amp;quot; &lt;a href=&quot;http://jeff560.tripod.com/mathsym.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The horizontal fraction bar was introduced by the Arabs. &amp;quot;The Arabs at first copied the Hindu notation, but later improved on it by inserting a horizontal bar between the two numbers&amp;quot; (Burton).&lt;br /&gt;&#xA;Several sources attribute the horizontal fraction bar to al-Hassar around 1200.&lt;br /&gt;&#xA;When Rabbi ben Ezra (c. 1140) adopted the Moorish forms he generally omitted the bar.&lt;br /&gt;&#xA;Fibonacci (c.1175-1250) was the first European mathematician to use the fraction bar as it is used today. He followed the Arab practice of placing the fraction to the left of the integer (Cajori vol. 1, page 311).&lt;br /&gt;&#xA;According to the DSB, Abu Abdallah Yaish ibn Ibrahim ibn Yusuf ibn Simak al-Umawi (14th or 15th century) insisted that the horizontal fraction bar be used, whereas easterners continued to write it without the bar.&lt;br /&gt;&#xA;The bar is generally found in Latin manuscripts of the late Middle Ages, but when printing was introduced it was frequently omitted, doubtless owing to typographical difficulties. This inference is confirmed by such books as Rudolff's Kunstliche rechnung (1526), where the bar is omitted in all ordinary fractions but is inserted in fractions printed in larger type and those having large numbers (Smith vol. 2, page 216).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[33]  Cajori (1928), p. 89  &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/hcaO0.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hcaO0.jpg&quot; alt=&quot;Leonardo&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="12330" LastEditDate="2022-11-13T09:43:46.893" LastActivityDate="2022-11-13T09:43:46.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14927" PostTypeId="1" AcceptedAnswerId="14929" CreationDate="2022-11-13T13:49:20.037" Score="1" ViewCount="278" Body="&lt;p&gt;I attended a lecture on the history of Astronomy and Mathematics, and I was somewhat puzzled by how scientific the early Greeks were. Yes, I am aware that they have many of the greatest mathematicians and astronomers, and were known for the rigor in philosophical thinking. However, if even centuries later we have Newton sometimes using God in his explanations, how come there wasn't more prominence of Greek religion in their studies?&lt;/p&gt;&#xA;&lt;p&gt;I imagine that there was a large disconnect between the scientific community and the common people (as is discussed a bit &lt;a href=&quot;https://hsm.stackexchange.com/questions/13623/to-what-extent-was-astronomical-knowledge-in-ancient-greece-common-knowledge&quot;&gt;here&lt;/a&gt;), but what about the role of religion and mythology inside the scientific community itself?&lt;/p&gt;&#xA;" OwnerUserId="7481" LastActivityDate="2022-11-13T21:21:36.687" Title="Was religion not present at all in the Astronomy of the Ancient Greeks?" Tags="|mathematics|astronomy|ancient-greece|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14928" PostTypeId="1" AcceptedAnswerId="14934" CreationDate="2022-11-13T16:19:40.983" Score="5" ViewCount="216" Body="&lt;p&gt;My understanding is that explaining ortho- and para- helium spectral lines was a key motivation for Heisenberg's new quantum theory. For example, Birthwistle's 1928 &lt;a href=&quot;https://books.google.ca/books?id=N-c3AAAAIAAJ&amp;amp;pg=PA215&amp;amp;lpg=PA215&amp;amp;dq=%22HEISENBERG%27S%20RESONANCE%20THEORY%20OF%20THE%20ORTHO%20AND%20PARA%20HELIUM%20SPECTRA%22&amp;amp;source=bl&amp;amp;ots=m4mBNidvft&amp;amp;sig=ACfU3U3C9P0TJmbMiTNN7snjAZIaPU-AQg&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwij0IKyvKv7AhVojIkEHZHvCmcQ6AF6BAgHEAM#v=onepage&amp;amp;q=%22HEISENBERG%27S%20RESONANCE%20THEORY%20OF%20THE%20ORTHO%20AND%20PARA%20HELIUM%20SPECTRA%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The New Quantum Mechanics&amp;quot;, Chapter XXVI&lt;/a&gt; states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is well known that the spectral terms of helium can be divided into two sets such that no term of the one will combine with a term of the other to produce a spectral line.&#xA;… One set by its transitions gives the 'para helium' lines … the other set gives the 'ortho helium' lines …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since it is &amp;quot;well known&amp;quot;, Birthwistle give no sources. He further goes on to say&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The obvious failure of the classical mechanics and the correspondence principle to solve the problem of a nucleus with two outer electrons was one of the factors which compelled Heisenberg to seek for a new quantum mechanics …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When and how was the helium spectrum measured sufficiently to establish the existence of two sets of line and the challenge for Bohr's quantum mechanics recognized? Are there any good articles or books that discuss this?&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2022-11-15T06:16:57.070" LastActivityDate="2023-11-25T13:31:38.887" Title="Early helium spectrum measurements and their challenge for Bohr's quantum mechanics" Tags="|quantum-mechanics|experimental-physics|heisenberg|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14929" PostTypeId="2" ParentId="14927" CreationDate="2022-11-13T21:21:36.687" Score="4" Body="&lt;p&gt;It seems that Greek astronomy could at least in its earliest times  occasionally be mixed with mythology if not also religion.&lt;/p&gt;&#xA;&lt;p&gt;Thus, from Michael Hoskin (1997) &lt;a href=&quot;https://books.google.com/books?id=9gZLXocOnSgC&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Cambridge Concise History of Astronomy&amp;quot;&lt;/a&gt; (p.19):--&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;When we come to study the earliest Greek attempts to make sense of&#xA;the universe, we find that our sources are fragmentary in the extreme.&#xA;What little we have owes much to the custom of Aristotle (384-322 BC),&#xA;of citing his predecessors before demolishing them. It seems that&#xA;alongside the mythologies inherited from earlier times, there&#xA;gradually emerged a speculative interest in the natural world, which&#xA;led to efforts to make sense of nature - rather than to attempts to&#xA;answer quantitative questions, such as led to the columns of numbers&#xA;on the Babylonian tablets. But what we would recognize as a mature,&#xA;predictive science of astronomy was to develop only in the Hellenistic&#xA;era, when these two approaches, the Greek and the Babylonian, merged.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus, according to this historian, mythological elements in Greek astronomy had been present at so early a date that little of their thought has survived, but these elements were being replaced even as early as the 6th-c. BC by ideas of 'impersonal laws':--&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Anaximander (c.610-c.545 BC), also of Miletus, attempted to explain&#xA;the form of the heavenly bodies in the context of his vision of worlds&#xA;constantly coming into being from the Infinite, only to perish and be&#xA;reabsorbed into the Infinite. If we can rely on accounts written many&#xA;centuries later, he thought of the stars as wheel-like condensations&#xA;of air filled with fire, with openings through which flames were&#xA;discharged. The Sun was the highest (that is, most remote) of the&#xA;heavenly bodies, with the Moon next below it, then the 'fixed' stars&#xA;(those unchanging in their positional relations), and finally the&#xA;planets. The Earth he believed to be a cylinder, on one of whose end&#xA;surfaces lived mankind; it rested in the middle of the universe,&#xA;remaining where it was because it had the same distance from&#xA;everything. The limitations of this cosmology are evident, but a&#xA;fundamental shift had occurred: earlier mythologies had been replaced&#xA;by a nature in which an impersonal law was at work.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-11-13T21:21:36.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14931" PostTypeId="1" CreationDate="2022-11-14T10:54:35.187" Score="10" ViewCount="2541" Body="&lt;p&gt;I am reading about the 1900s revolution of math pioneered by figures such as Hilbert. I have seen many articles speak very fondly of these figures due to the fact they tried to study Mathematics through the perspective of Logic.&lt;/p&gt;&#xA;&lt;p&gt;I do agree that it was great... but wasn't this perspective already put forward by Euclid when he tried to do axiomatic Geometry? What did the 1900s proof theory achieve that Euclid couldn't?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="12330" LastEditDate="2022-11-14T22:12:48.353" LastActivityDate="2022-11-15T21:04:33.047" Title="How did the notion of rigour in Euclid’s time differ from that in the 1920 revolution of Math?" Tags="|geometry|euclidean-geometry|logic|" AnswerCount="3" CommentCount="4" FavoriteCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14932" PostTypeId="2" ParentId="14931" CreationDate="2022-11-14T14:18:24.987" Score="10" Body="&lt;p&gt;The main difference is that Mathematical logic and set theory did not exist at the time of Euclid. (The Logic of Aristoteles is still very far from mathematical logic created in 19th century). As a result, Euclid's Elements are not completely rigorous from the modern (and early 20th century) point of view: for example, Euclid did not state all axioms which he really used (the first complete from the modern view set of axioms for Euclid was formulated by Hilbert). There are actual gaps in Euclid's reasoning which in principle could be detected during the 24 centuries since Euclid but were not detected. For example, &lt;a href=&quot;https://hsm.stackexchange.com/questions/6922/has-euclid-stated-cauchys-theorem&quot;&gt;Cauchy's theorem&lt;/a&gt; on convex polytopes. It is a deep theorem which fills one of these gaps, but the gap was not even recognized before Cauchy.&lt;/p&gt;&#xA;&lt;p&gt;The reason of this was that without rigorous mathematical logic, complete analysis of Euclid's axiomatic system was not possible. Understanding of relation between geometry and numbers at the time of Euclid was also very incomplete.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-11-14T14:25:42.130" LastActivityDate="2022-11-14T14:25:42.130" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14933" PostTypeId="2" ParentId="14931" CreationDate="2022-11-14T21:48:52.563" Score="14" Body="&lt;p&gt;For example, the very first proposition: Construct an equilateral triangle &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;, where one side &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; is given.&lt;br /&gt;&#xA;Euclid says&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/GszBq.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/GszBq.gif&quot; alt=&quot;diagram&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Draw a circle with center &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt;. [By Postulate 1]&lt;br&gt;&#xA;Draw the circle with center &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; and radius &lt;span class=&quot;math-container&quot;&gt;$BA$&lt;/span&gt;. [By Postulate 1]&lt;br&gt;&#xA;Let &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; be a point where the two circles intersect.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;But&lt;/strong&gt; there is nothing in Euclid that proves the two circles intersect at all.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Look at the picture, it's obvious&amp;quot; is not considered rigorous.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;A is inside the circle &lt;span class=&quot;math-container&quot;&gt;$BCD$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is outside the circle &lt;span class=&quot;math-container&quot;&gt;$BCD$&lt;/span&gt;, so there must be a point of circle &lt;span class=&quot;math-container&quot;&gt;$ACE$&lt;/span&gt; exactly on the circle &lt;span class=&quot;math-container&quot;&gt;$BCD$&lt;/span&gt;.&amp;quot;  There is no basis in Euclid to make that assertion.&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="2531" LastEditDate="2022-11-15T07:01:53.883" LastActivityDate="2022-11-15T07:01:53.883" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="14934" PostTypeId="2" ParentId="14928" CreationDate="2022-11-15T16:08:28.943" Score="3" Body="&lt;p&gt;By 1928, &lt;a href=&quot;https://doi.org/10.1002/andp.19283911205&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stark&lt;/a&gt; had an article in the Annelen der Physik on the intensities of the Ortho- and Para-series in helium.&lt;/p&gt;&#xA;&lt;p&gt;In that article, Stark references a 1924 &lt;a href=&quot;https://ui.adsabs.harvard.edu/abs/1924ApJ....60....1L/abstract&quot; rel=&quot;nofollow noreferrer&quot;&gt;Astrophysical Journal&lt;/a&gt; paper by Lyman, in which Lyman summarizes his conclusions on the He spectrum based on nearly a decade worth of work.&lt;/p&gt;&#xA;&lt;p&gt;As one example of the efforts to explain Helium, I would point to Sommerfeld's 1924 paper in the &lt;a href=&quot;https://doi.org/10.1364/JOSA.7.000509&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal of the Optical Society of America&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Remember, this was early in the quantum era - one needed good data, good theory, and good ideas to straighten it all out.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2022-11-15T16:08:28.943" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14935" PostTypeId="2" ParentId="14931" CreationDate="2022-11-15T21:04:33.047" Score="3" Body="&lt;p&gt;The hope of Hilbert was that once correctly formalized, it would be possible to show that mathematics was sound and that any statement has a proof. Godel's incompleteness results put an end to that hope, at least for recursively enumerable theories strong enough to recover arithmetic.&lt;/p&gt;&#xA;&lt;p&gt;A large part of Euclid's Elements, namely the geometric part, does not assume arithmetic but can actually be built on first order logic. Several axiomatics were devised circa 1900 to dispel a few gaps in the treatise of Euclid, mostly due to a lack of treatment of order on lines, corresponding to a point being between two other points. Amongst the many axiomatics developped at that time, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Hilbert%27s_axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;one&lt;/a&gt; introduced in the &amp;quot;&lt;em&gt;Grundlagen der geometrie&lt;/em&gt;&amp;quot; by Hilbert is the most famous. The big result was yet to come.&lt;/p&gt;&#xA;&lt;p&gt;Alfred Tarski gave a slightly different &lt;a href=&quot;https://en.wikipedia.org/wiki/Tarski%27s_axioms&quot; rel=&quot;nofollow noreferrer&quot;&gt;axiomatic&lt;/a&gt; for elementary geometry, of a more logical nature, and relying only on first order logic. He was then able to show that his theory of elementary geometry is&lt;/p&gt;&#xA;&lt;p&gt;-- &lt;em&gt;consistent&lt;/em&gt;: a well formed statement is either true or false, and cannot be both.&lt;/p&gt;&#xA;&lt;p&gt;-- &lt;em&gt;complete&lt;/em&gt;: a proposition (or its negation) stated in the language of first order logic always has a proof.&lt;/p&gt;&#xA;&lt;p&gt;-- &lt;em&gt;algorithmically decidable&lt;/em&gt;: there is an explicit algorithm that decides if a statement is true or false.&lt;/p&gt;&#xA;&lt;p&gt;So, Alfred Tarski realized Hilbert's program in the context of elementary geometry and showed what mathematicians have anticipated for more than two millenia: the logical perfection of Euclid's geometry.&lt;/p&gt;&#xA;" OwnerUserId="6959" LastActivityDate="2022-11-15T21:04:33.047" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14936" PostTypeId="1" AcceptedAnswerId="14947" CreationDate="2022-11-16T03:46:51.890" Score="1" ViewCount="95" Body="&lt;p&gt;We use &lt;span class=&quot;math-container&quot;&gt;$h_K$&lt;/span&gt; conventionally to denote the class numbers of number fields &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But I have never thought why the letter &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; was used for it.&lt;/p&gt;&#xA;&lt;p&gt;Why and who used &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; for the class number?&lt;/p&gt;&#xA;" OwnerUserId="2320" LastActivityDate="2022-11-22T09:11:44.117" Title="The symbol h for class numbers" Tags="|terminology|number-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14937" PostTypeId="1" CreationDate="2022-11-16T05:02:36.167" Score="1" ViewCount="54" Body="&lt;p&gt;I was encouraged in MathOverflow to ask this question here - so here it goes: As I am reading Frege and Boole, I am struck by that there doesn't seem to be a great alternative to Aristotle's Logic until that time, and the development by Boole and Frege are correctly hailed as a revolution in the development the subject.&lt;/p&gt;&#xA;&lt;p&gt;However, mathematicians have clearly been using first-order (&amp;amp; propositional) logic correctly in their proofs well before then, and indeed, humans - including small children! use simple forms of first order logic, and writers when constructing arguments for their books of political philosophy or whatever also employ non-Aristotelian logic freely.&lt;/p&gt;&#xA;&lt;p&gt;So what took so long for Frege and Boole / humans to overthrow old syllogistic logic? &amp;quot;It's obvious in hindsight&amp;quot;'s not the answer I am looking for (see above).&lt;/p&gt;&#xA;" OwnerUserId="16729" LastActivityDate="2022-11-16T05:02:36.167" Title="Development of Logic: Intuitive use vs what took so long to formalize?" Tags="|mathematical-logic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14938" PostTypeId="2" ParentId="14305" CreationDate="2022-11-16T15:45:08.090" Score="1" Body="&lt;p&gt;Here is a first list of things that I've found in the literature. I will update it as I gather more. They are more &amp;quot;accounts&amp;quot; than &amp;quot;theories&amp;quot;, but interesting nonetheless (at least to me!). I won't quote every account here for the sake of brevity, but they are available in the cited references.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;In the Arabic peninsula, there are historical accounts of the 1256 CE eruption of the Harrat Rahat volcanic field, according to &lt;a href=&quot;https://doi.org/10.1007/BF01245475&quot; rel=&quot;nofollow noreferrer&quot;&gt;Camp et al. (1987)&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Two eyewitness accounts of the eruption were written by A1-Kistlani, a&#xA;resident of Madinah, and Kadi Sinaan, the judge at Makkah. These&#xA;manuscripts are no longer extant, but their descriptions are&#xA;paraphrased in two later manuscripts; one, Wafa A1-Wafa, was written&#xA;in 1568 A.D. (976 A.H.) (reprinted in A1-Samhoudy 1954) and the other,&#xA;Jazb Al-Kulub, was recorded by an early European traveler, JL&#xA;Burkhardt, and published as a footnote in Burton (1893). These two&#xA;sources show remarkably good agreement in detail and dates which,&#xA;together with our field observations, permit a reconstruction of&#xA;events (Table 1).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;In Indonesia, a recent study (&lt;a href=&quot;https://doi.org/10.1016/j.jvolgeores.2022.107688&quot; rel=&quot;nofollow noreferrer&quot;&gt;Malawani et al., 2022&lt;/a&gt;) found accounts of the 1257 CE eruption of Samalas volcano; although they were probably written centuries later and thus &amp;quot;represent a recent recording of what appears to have been older oral traditions&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In Indonesia still, &lt;em&gt;History of the volcanology in the former Netherlands East Indies&lt;/em&gt; &lt;a href=&quot;https://repository.naturalis.nl/pub/317412&quot; rel=&quot;nofollow noreferrer&quot;&gt;(Neumann van Padang, 1983)&lt;/a&gt; contains interesting data on volcanic eruptions found in ancient Javanese sources, although he mainly cites Van Hinloopen Labberton (1921), which is hard to find and written in Dutch. He also questions the validity of some dates:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A problem with the use of these ancient sources is the fact that eruptions&#xA;and earthquakes at the birth and death of princes were used to emphasize their&#xA;importance and divine offspring. It is therefore more than likely that the dates of the eruptions were made to coincide with such events by either changing the&#xA;date of the eruption or the date of birth (c.q. death) of the princes (see van&#xA;Bemmelen, 1956).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The Persian book Sharafnama (Sharaf Khan Bitlisi, 1597) contains a description of an eruption of Mount Nemrut (Turkey), which has been translated and interpreted by &lt;a href=&quot;https://doi.org/10.1016/S0377-0273(03)00002-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aydar et al. (2003)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1016/S0377-0273(01)00264-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karakhanian et al. (2002)&lt;/a&gt; list several eruptions in Armenia and adjacent countries, inferred from historical chronicles. Among them, the following cuneiform inscription, attributed to King Argishti I, is interpreted as an eruption of Porak volcano in 782&lt;span class=&quot;math-container&quot;&gt;$–$&lt;/span&gt;773 BCE:&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...when I again (for the second time) laid the siege of the town of&#xA;Behoura, Mount Bamni in the area of Behoura Town was destroyed...;&#xA;smoke and soot now rise from it to the sun. When Mount Bamni was&#xA;destroyed, I took the town of Behoura.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1007/s00445-016-1010-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;Martin-Del Pozzo et al. (2016)&lt;/a&gt; did a reconstruction of 800 years of eruptive activity of Popocatépetl, including some pre-Columbian sources (codices) documenting the activity of the volcano.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To be continued...&lt;/p&gt;&#xA;" OwnerUserId="14043" LastActivityDate="2022-11-16T15:45:08.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14939" PostTypeId="1" CreationDate="2022-11-17T07:28:56.940" Score="1" ViewCount="112" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bit#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Bit#History&lt;/a&gt; does not mention the common usage of bit being  an eighth of a dollar -- surely Shannon was inspired by this?&lt;/p&gt;&#xA;&lt;p&gt;This question was asked in 2019 in English stackexchange: &lt;a href=&quot;https://english.stackexchange.com/questions/510544/is-there-any-connection-between-1-bit-1-8-dollar-and-1-bit-1-8-byte&quot;&gt;https://english.stackexchange.com/questions/510544/is-there-any-connection-between-1-bit-1-8-dollar-and-1-bit-1-8-byte&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2022-11-18T12:45:59.597" LastActivityDate="2022-11-18T12:45:59.597" Title="Bit as eighth of dollar vs Shannon's Binary Digit?" Tags="|computers|information-theory|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="14940" PostTypeId="1" AcceptedAnswerId="14941" CreationDate="2022-11-17T23:12:26.003" Score="1" ViewCount="83" Body="&lt;p&gt;Did Abel published of testing the convergence of series? If so, when did he published it. Also, did he offer a proof of the test? Or did he simply stated the test?&lt;/p&gt;&#xA;" OwnerUserId="13017" LastActivityDate="2022-11-18T06:50:04.497" Title="When did Abel publish his test for the convergence of series?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14941" PostTypeId="2" ParentId="14940" CreationDate="2022-11-18T05:56:08.407" Score="2" Body="&lt;p&gt;According to what we read in page 313 of the second English edition of K. Knopp's &lt;em&gt;Theory and applications of infinite series&lt;/em&gt;, the test and its proof may have first appeared in the &lt;a href=&quot;https://eudml.org/journal/10147&quot; rel=&quot;nofollow noreferrer&quot;&gt;Journal für die reine und angewandte Mathematik&lt;/a&gt;, vol. 1, p. 314, 1826.&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-11-18T06:50:04.497" LastActivityDate="2022-11-18T06:50:04.497" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14942" PostTypeId="2" ParentId="14909" CreationDate="2022-11-18T19:14:58.723" Score="1" Body="&lt;p&gt;I ended up emailing the Bodleian Library at Oxford about how they are preserving the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bakhshali_manuscript&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bakhshali manuscript&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is their reply to my email (I retracted names for privacy reasons):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dear [Retracted],&lt;/p&gt;&#xA;&lt;p&gt;Many thanks for your email.&lt;/p&gt;&#xA;&lt;p&gt;The Bakhshali manuscript (MS. Sansk. d. 14) has not been digitized to date, but you can see one folio online here: &lt;a href=&quot;https://treasures.bodleian.ox.ac.uk/treasures/bakhshali-manuscript/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://treasures.bodleian.ox.ac.uk/treasures/bakhshali-manuscript/&lt;/a&gt;. I am copying this to our Imaging Services team who are best placed to advise on our digitization services, but please rest assured that the preservation of material is always the first priority on any such project.&lt;/p&gt;&#xA;&lt;p&gt;This manuscript is stored and consulted at the Weston Library, the Bodleian’s dedicated special collections library. As such, the environment (temperature, relative humidity and light levels) is controlled and monitored to ensure that it adheres to national standards. I believe that the current standard is &lt;a href=&quot;https://www.bsigroup.com/en-GB/about-bsi/media-centre/press-releases/2017/october/British-Standard-for-conservation-and-care-of-library-collections-revised-/&quot; rel=&quot;nofollow noreferrer&quot;&gt;BS 4971:2017&lt;/a&gt;, but our Preventive Team (copied in) will be able to advise you further on this.&lt;/p&gt;&#xA;&lt;p&gt;You can read a little more about the Weston Library here: &lt;a href=&quot;https://estates.admin.ox.ac.uk/weston-library-refurbishment&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://estates.admin.ox.ac.uk/weston-library-refurbishment&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am also copying this to the Sanskrit Curator, [retracted], who might be able to advise you further.&lt;/p&gt;&#xA;&lt;p&gt;Best wishes,&lt;/p&gt;&#xA;&lt;p&gt;[Retracted]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;They linked the British Standard for conservation and care of library collections, which in part basically answered most of questions on the ground how these works are expected to be stored, and how they are preserved:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the most common causes of damage to archive and library collections is an unsuitable storage environment. Getting the storage environment and temperature “just right” is therefore essential, with advice given in the standard. There are obvious risks with damp conditions, which encourage mould, but also with excessively dry conditions, which can make some archives – such as acid-decayed papers – more inflexible. Pest management is also covered.&lt;/p&gt;&#xA;&lt;p&gt;Lighting can cause damage to collections, with some materials more sensitive than others. BS 4971 includes a table with four columns for users of the standard to ascertain the appropriate light sensitivity of their collection, ranging from no sensitivity to high. Diagrams for shelving, with guidance on spacing and the most appropriate dimensions to construct storage for a collection, is covered in the standard.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The responses I got from the imaging department was nothing more than the stock response they give to everyday people asking for imaging services, which conveyed the message that they were not at all interested in the email at all. However, this line in the first email I feel like answers this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Bakhshali manuscript (MS. Sansk. d. 14) has not been digitized to date, but you can see one folio online here: &lt;a href=&quot;https://treasures.bodleian.ox.ac.uk/treasures/bakhshali-manuscript/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://treasures.bodleian.ox.ac.uk/treasures/bakhshali-manuscript/&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If it has not been digitized to date, and the fact that the standard states &amp;quot;Lighting can cause damage to collections, with some materials more sensitive than others.&amp;quot; then it would be reasonable to assume they will never digitize it as it will be too much of a risk for permanent damage of an already fragile piece. Thus, I feel like this would continue to be the case for other works as well.&lt;/p&gt;&#xA;" OwnerUserId="17466" LastActivityDate="2022-11-18T19:14:58.723" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14943" PostTypeId="1" CreationDate="2022-11-19T07:09:53.310" Score="0" ViewCount="56" Body="&lt;p&gt;When Stern-Gerlach experiment was done with quantum electron , two peaks were observed rather than a continuous distribution as in case of small magnets , but how did those peaks confirm that angular momentum is quantized ??&lt;/p&gt;&#xA;" OwnerUserId="17543" LastActivityDate="2022-11-19T07:09:53.310" Title="question regarding Stern-Gerlach experiment" Tags="|quantum-mechanics|particle-physics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14944" PostTypeId="1" CreationDate="2022-11-19T09:04:38.070" Score="1" ViewCount="102" Body="&lt;p&gt;In the early twentieth century, with Esperanto and the like going on, a small movement called &amp;quot;dozenalism&amp;quot; began, with the aim of replacing base 10 with the purportedly more natural base 12. When I was in high school it was something nerdy kids would talk about. The argument is 10 has as non-trivial factors only 2 and 5, whereas 12 has 2, 3, 4 and 6, and we should prefer smaller divisors, since division by these numbers will occur more frequently. Okay, but the only reason I can see for being interested in divisors is to have more numbers, or more frequently used numbers, with terminating expansions. But this is determined only by the prime factors of the base. So why was base 12 be preferred to base 6 or base 30 having the n smallest prime factors for some n?&lt;/p&gt;&#xA;" OwnerUserId="17323" LastActivityDate="2022-11-19T09:04:38.070" Title="Why have advocates for positional number systems based on divisibility favoured base 12?" Tags="|notation|numbers|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14945" PostTypeId="1" CreationDate="2022-11-19T16:37:03.523" Score="2" ViewCount="237" Body="&lt;p&gt;In 1833 Hamilton introduced what today is called &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Hamiltonian_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Hamiltonian formalism&lt;/a&gt; in classical mechanics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;I am wondering what was his motivation. Did he try to solve a specific question?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="user65203" LastEditorUserId="16318" LastEditDate="2022-11-22T07:10:53.903" LastActivityDate="2022-11-26T12:45:41.887" Title="Motivation of the Hamiltonian formalism" Tags="|classical-mechanics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14946" PostTypeId="1" CreationDate="2022-11-22T01:22:41.163" Score="1" ViewCount="111" Body="&lt;p&gt;In quantum mechanics, we say &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; ('total angular momentum') = &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; ('orbital angular momentum') + &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; ('spin angular momentum').&lt;/p&gt;&#xA;&lt;p&gt;Apparently &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; is from 'Spin', but why &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt; for the total angular momentum?&lt;/p&gt;&#xA;&lt;p&gt;cf) &lt;a href=&quot;https://hsm.stackexchange.com/questions/6993/why-is-the-angular-momentum-written-as-jj-in-quantum-mechanics&quot;&gt;Why is the angular momentum written as JJ in quantum mechanics?&lt;/a&gt; : this question does not seem to answer the history of the notation&lt;/p&gt;&#xA;" OwnerUserId="17558" LastActivityDate="2022-11-22T01:22:41.163" Title="Why is 'total angular momentum' denoted by the letter $J$ in quantum mechanics?" Tags="|terminology|notation|quantum-mechanics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14947" PostTypeId="2" ParentId="14936" CreationDate="2022-11-22T09:11:44.117" Score="3" Body="&lt;p&gt;I asked this question on MO once. The notation goes back to Dirichlet. See &lt;a href=&quot;https://mathoverflow.net/questions/17062/why-is-h-the-notation-for-class-numbers&quot;&gt;https://mathoverflow.net/questions/17062/why-is-h-the-notation-for-class-numbers&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2022-11-22T09:11:44.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14948" PostTypeId="1" CreationDate="2022-11-22T22:47:37.463" Score="2" ViewCount="91" Body="&lt;p&gt;Cumulative distribution function is pretty much known and well covered from undergrad to grad probability texts. Specifically if &lt;span class=&quot;math-container&quot;&gt;$\mathbf P$&lt;/span&gt; is a probability measure on &lt;span class=&quot;math-container&quot;&gt;$(\mathbb R, \mathfrak B_\mathbb R),$&lt;/span&gt; then cdf is defined as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathsf F(x):= \mathbf P((-\infty, x])\tag 1\label 1.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, early texts like those by &lt;em&gt;Kolmogorov, Uspensky, Gnedenko&lt;/em&gt; defined it (borrowing from Kolmogorov's notations) as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathsf F^{(x)}(a) := \mathbf P(-\infty, a).\tag 2\label 2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;From &lt;span class=&quot;math-container&quot;&gt;$\eqref 1,~~\forall~a&amp;lt;b,$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathbf P(\{\omega:X(\omega)\in(a, b]\}) = \mathsf F(b)-\mathsf F(a).\tag 3\label 3$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;While from &lt;span class=&quot;math-container&quot;&gt;$\eqref 2,$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\mathbf P\{x\in[a,b)\} = \mathsf F^{(x)}(b)-\mathsf F^{(x)}(a).\tag 4\label 4$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;While &lt;span class=&quot;math-container&quot;&gt;$\eqref 1$&lt;/span&gt; is right-continuous, &lt;span class=&quot;math-container&quot;&gt;$\eqref 2$&lt;/span&gt; is left-continuous.&lt;/p&gt;&#xA;&lt;p&gt;In general too, while I was acquainted with Stieltjes premeasure on Borel field &lt;span class=&quot;math-container&quot;&gt;$\mathfrak B_{(~~~]}(\mathbb R)$&lt;/span&gt; which is generally defined for a non-decreasing, right continuous &lt;span class=&quot;math-container&quot;&gt;$F:\mathbb R\to\mathbb R$&lt;/span&gt; as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\chi_F(a, b] := F(b)- F(a),\tag 5\label 5$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;in &lt;em&gt;Schilling's&lt;/em&gt; measure theory text, the author defines Stieltjes function for monotonically increasing and left-continuous &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; (he calls it &lt;em&gt;Stieltjes function&lt;/em&gt;) as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\nu_F([a, b)):= F(b)-F(a).\tag 6\label 6$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;My question is why Kolmogorov and his other contemporaries defined the cdf as &lt;span class=&quot;math-container&quot;&gt;$\eqref 2$&lt;/span&gt; and not as &lt;span class=&quot;math-container&quot;&gt;$\eqref 1$&lt;/span&gt; and how &lt;span class=&quot;math-container&quot;&gt;$\eqref 1$&lt;/span&gt; became the more common face as we are accustomed to. Which author first used &lt;span class=&quot;math-container&quot;&gt;$\eqref 1$&lt;/span&gt; as the definition rather than &lt;span class=&quot;math-container&quot;&gt;$\eqref 2?$&lt;/span&gt; Was Kolmogorov the first one in history to define cdf as &lt;span class=&quot;math-container&quot;&gt;$\eqref 2?$&lt;/span&gt; Did Borel, Jordan introduce any formal definition on the same prior to Kolmogorov in their treatises?&lt;/p&gt;&#xA;&lt;p&gt;In a nutshell, my query revolves around the timeline of the introduction of cdf and its development to the one we are familiar with today.&lt;/p&gt;&#xA;" OwnerUserId="17132" LastEditorUserId="17132" LastEditDate="2022-11-23T07:39:19.597" LastActivityDate="2022-11-23T07:39:19.597" Title="History of cdf as to how it was defined prior Kolmogorov and how the paradigm shifted from $&lt;$ case to $\leq $ case" Tags="|probability|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14949" PostTypeId="2" ParentId="14945" CreationDate="2022-11-24T22:34:59.793" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://www.maths.tcd.ie/%7Edwilkins/&quot; rel=&quot;nofollow noreferrer&quot;&gt;David R Wilkins&lt;/a&gt; has created many &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Papers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;transcripts of Hamilton's articles&lt;/a&gt;, including the ones on mechanics.&lt;/p&gt;&#xA;&lt;p&gt;The page titled &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/#GenMethod&quot; rel=&quot;nofollow noreferrer&quot;&gt;On a general method in dynamics&lt;/a&gt; is specifically about Hamilton's work on the subject of mechanics.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;The articles by William Rowan Hamilton do not contain equations in the form we know them today.&lt;/p&gt;&#xA;&lt;p&gt;Jacobi later stated that Hamilton gave two equations, and that one of those two was superfluous, so in his own work Jacobi omitted that one.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;About Hamiltonian formalism:&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure, but I don't think that what today is referred to as 'Hamiltonian formalism' is present in Hamilton's articles.&lt;/p&gt;&#xA;&lt;p&gt;Rather, I suspect that what is referred to as Hamiltonian formalism was introduced at a later date, and attributed to Hamilton.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;I want to emphasize:&lt;br /&gt;&#xA;In history of science the only reliable sources are the original works, and articles and books by historians of science who use as source the original works exclusively.&lt;/p&gt;&#xA;&lt;p&gt;Conversely: the 'historical introduction' sections in physics textbooks should not be trusted.&lt;/p&gt;&#xA;&lt;p&gt;My impression is that textbook authors write such introductions from memory, reconstructing how they remember the 'historical introduction' in the textbook they grew up with. I very much suspect that each time the story is retold the author embellishes the story.&lt;/p&gt;&#xA;&lt;p&gt;It's not that such embellishment done on purpose. For sure the authors are not aware of what they are doing. It's just: there are circumstances where human memory is very fallible.&lt;/p&gt;&#xA;&lt;p&gt;Mark Twain wrote the following quip:&lt;br /&gt;&#xA;&amp;quot;In the real world, nothing happens at the right place at the right time. It is the job of journalists and historians to correct that.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Textbook authors tend to write the story as they remember taking it in, but some elements are lost, and other elements are added, and the story is reshaped to a form that aligns with the author's expectation.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;So that is why I assert:&lt;br /&gt;&#xA;Only the original works are reliable as source of history of science.&lt;/p&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2022-11-24T22:34:59.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14950" PostTypeId="1" CreationDate="2022-11-25T02:45:04.710" Score="8" ViewCount="2243" Body="&lt;p&gt;I assume this conclusion was hard to make empirically in the days of slow travel -- one could not as we can today fly from above the equator to below it and observe the difference in weather and even if one could, it would be hard to attribute this to a single cause as weather is very changeable.&lt;/p&gt;&#xA;&lt;p&gt;I further assume some ancient understood the thing about the Earth being tilted on its access but even to me (not an astronomer or even very sharp mentally) it is far from obvious that this would affect the seasons -- I do think that only if one accepts Heliocentrism that this phenomenon can be accounted for.&lt;/p&gt;&#xA;&lt;p&gt;Is it then possible that no one in western Europe in, say, 1500 believed that seasons were reversed and could explain it. I do not know what Indians or Chinese knew about this.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: I realized that a ship sailing south enough would still know what date it was and people might well note that it was colder or warmer than the same month where they came from. They might attribute this to local weather but if the differences were pronounced enough, at different longitudes, someone might suggest that indeed seasons were reversed without understanding why.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2022-11-26T20:36:18.107" LastActivityDate="2022-11-26T20:53:07.340" Title="First to realize that seasons were reversed above and below the equator?" Tags="|astronomy|earth-sciences|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14951" PostTypeId="2" ParentId="14950" CreationDate="2022-11-25T03:21:28.163" Score="10" Body="&lt;p&gt;One does not need heliocentrism for this. The path of the Sun in the sky (as seen from the Earth) was well understood in Hellenistic times, and probably long before that. It was also understood how this path determines seasons. So probably these phenomena were known&#xA;before Hipparchus. I say probably because very little  astronomic work before Ptolemy survived to our day, so one cannot be sure who noticed this explicitly for the first time. Maybe even Thales who wrote books on solstices and equinoxes, but we have no way to know what was in those books.&lt;/p&gt;&#xA;&lt;p&gt;Another thing is that this was not common knowledge, just a few astronomers (they were called philosophers at that time) understood these things. There is a famous story described by Herodotus (IV, 43) about an expedition&#xA;which sailed around Africa, from the Red sea to Gibraltar by Indian and Atlantic oceans.&#xA;Herodotus wrote: &amp;quot;According to their account (which I do not believe, let anyone believe who wants) they saw the Sun on their right hand side&amp;quot;. It is exactly this &amp;quot;improbable&amp;quot; detail which is considered a proof that this expedition really happened:-)&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-11-26T20:53:07.340" LastActivityDate="2022-11-26T20:53:07.340" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14952" PostTypeId="2" ParentId="14950" CreationDate="2022-11-25T09:41:39.920" Score="15" Body="&lt;p&gt;In the &lt;em&gt;Purgatorio&lt;/em&gt; (c.1310) Dante places Mount Purgatory at the antipodes of Jerusalem and follows the rules of hard SF by thinking out all the consequences.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Longitude and time zones&lt;/strong&gt; - this may be the first time in literature that someone has realised that 180° in longitude means 12 hours of time zone (he also has time zones for Rome and the Ganges).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Anticlockwise Sun / Sun to the north&lt;/strong&gt; - Dante the character in the poem is thoroughly startled by this and Dante the poet has quite a bit of fun at his expense, as it all has to be explained to him.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Latitude and seasonal changes&lt;/strong&gt; - Mount Purgatory being mostly above the Earth's atmosphere, it does not of course have weather. But Canto 4, in which Dante-the-character eventually manages to understand what is going on, handles the matter in terms of whether the Sun's path is &amp;quot;our side of the equator&amp;quot; or &amp;quot;the other side of the equator&amp;quot; - with &amp;quot;our side&amp;quot; and &amp;quot;the other side&amp;quot; changing places as we go from Jerusalem to the southern hemisphere.&lt;/p&gt;&#xA;" OwnerUserId="4209" LastActivityDate="2022-11-25T09:41:39.920" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="14953" PostTypeId="1" CreationDate="2022-11-26T03:11:57.893" Score="0" ViewCount="50" Body="&lt;p&gt;In the first few pages of &amp;quot;&lt;a href=&quot;https://www.jstor.org/stable/2247671&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Notion of Order&lt;/a&gt;&amp;quot;, Bertrand Russell has the footnote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The following account of the genesis of order is virtually identical with that of Mr. B. I. Gilman, Mind, N. S., vol. i.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I thought just putting this into Google would suffice, but somehow I got lots of results that are far too recent to be attributed in something published &lt;em&gt;circa&lt;/em&gt; 1901.&lt;/p&gt;&#xA;&lt;p&gt;What work is Bertrand Russell actually citing here?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastEditorUserId="17132" LastEditDate="2022-11-26T04:09:07.043" LastActivityDate="2022-11-26T04:35:09.117" Title="What is Bertrand Russell citing Gilman for in &quot;On the Notion of Order&quot;?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14954" PostTypeId="2" ParentId="14953" CreationDate="2022-11-26T04:35:09.117" Score="3" Body="&lt;p&gt;In &lt;a href=&quot;https://archive.org/details/nsmindrev01edinuoft/page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mind&lt;/em&gt;, N. S. Vol. &lt;span class=&quot;math-container&quot;&gt;$\rm I$&lt;/span&gt;&lt;/a&gt; (p. &lt;span class=&quot;math-container&quot;&gt;$519$&lt;/span&gt;), there is &lt;em&gt;&lt;a href=&quot;https://academic.oup.com/mind/article-abstract/1/4/518/949782?redirectedFrom=fulltext&quot; rel=&quot;nofollow noreferrer&quot;&gt;On The Properties of a One-Dimensional Manifold&lt;/a&gt;&lt;/em&gt; by B. I. Gilman.&lt;/p&gt;&#xA;&lt;p&gt;Russell was citing this work, if I am not wrong.&lt;/p&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2022-11-26T04:35:09.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14955" PostTypeId="2" ParentId="14945" CreationDate="2022-11-26T12:45:41.887" Score="3" Body="&lt;p&gt;Although I believe the answer by Cleonis contains much general wisdom on the subject of myths and fictional embellishments in science and math history, still on this occasion some relevant papers by Hamilton have been overlooked.&lt;/p&gt;&#xA;&lt;p&gt;See Hamilton's &lt;a href=&quot;https://royalsocietypublishing.org/doi/epdf/10.1098/rstl.1834.0017&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function&amp;quot;&lt;/a&gt; in Philosophical Transaction (1834) vol.124 p 247-308.&lt;/p&gt;&#xA;&lt;p&gt;There is a further important paper, &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rstl.1835.0009&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Second essay on a general method in dynamics&amp;quot;&lt;/a&gt; in 1835.&lt;/p&gt;&#xA;&lt;p&gt;(There are plenty of equations in there, that seem not too far away from the modern forms, but I can't give an analysis.)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2022-11-26T12:45:41.887" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14956" PostTypeId="1" CreationDate="2022-11-27T00:08:06.090" Score="3" ViewCount="433" Body="&lt;p&gt;Why was Proposition 43, Theorem 22, of Newton's &lt;em&gt;Principia&lt;/em&gt; not printed?&lt;/p&gt;&#xA;&lt;p&gt;Weinberg, &lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=9298&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;To Explain the World&lt;/em&gt;&lt;/a&gt; (2015) describes this proposition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In an unpublished “Proposition 43” that did not make it into the &lt;em&gt;Principia&lt;/em&gt; [why?], Newton acknowledged that Tycho’s theory could be true if some other force besides ordinary gravitation acted on the Sun and planets.&lt;sup&gt;22&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;sup&gt;22. &lt;a href=&quot;https://as.tufts.edu/philosophy/people/faculty/george-smith&quot; rel=&quot;nofollow noreferrer&quot;&gt;G[eorge]. E. Smith [of Tufts University]&lt;/a&gt;, to be published.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It appears on &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03965/695&quot; rel=&quot;nofollow noreferrer&quot;&gt;MSS add. 3965, fol. 341r&lt;/a&gt;:&#xA;&lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03965/695&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BR5Xo.jpg&quot; alt=&quot;Principia Proposition 43, Theorem 22 (MSS add. 3965, fol. 341r)&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Transcription (&lt;a href=&quot;https://isidore.co/calibre/#panel=book_details&amp;amp;book_id=6913&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sungenis 2018&lt;/a&gt; p. 407n470):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ut Terra quiescat in centro Systematis Solis Planetarum &amp;amp; Cometarum, requiritur et gravitas universalis, et alia insuper vis quæ agit in omnia coropora æqualiter pro quantitate materiæ in ipsis et æqualis est gravitati acceleratrici qua Terra tendit in Solem, eique contraria est, tendendo secundum lineas parallelas in plagam eandem cum linea quæ ducitur a centro Solis ad centrum Terræ…Nam talis vis in corpora omnia æqualiter &amp;amp; secundum lineas parallelas agendo situm eorum inter se non mutat sed sinit corpora eodem modo per vim gravitatis universalis inter se moveri, ac si non ageret in eadem. Terra vero, cum hæc vis gravitati ejus in Solem æqualis sit &amp;amp; contraria, in æquilibrio inter has duas vires manere potest et quiescere. Et sic corpora cælestia circa Terram quiescentem moveri possunt ut in Systemate Tychonico.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.kolbecenter.org/scoffers-will-arise-in-the-last-days-a-reply-to-fr-paul-robinson-fsspx-2/#_ednref39&quot; rel=&quot;nofollow noreferrer&quot;&gt;English translation&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In order for the Earth to be at rest in the center of the system of the Sun, Planets, and Comets, there is required both universal gravity and another force in addition that acts on all bodies equally according to the quantity of matter in each of them and is equal and opposite to the accelerative gravity with which the Earth tends to the Sun. For, such a force, acting on all bodies equally and along parallel lines, does not change their position among themselves, and permits bodies to move among themselves through the force of universal gravity in the same way as if it were not acting on them. Since this force is equal and opposite to its gravity toward the Sun, the Earth can truly remain in equilibrium between these two forces and be at rest. And thus celestial bodies can move around the Earth at rest, as in the Tychonic system.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="232" LastActivityDate="2022-11-28T05:34:24.560" Title="Why was Principia Proposition 43, Theorem 22 not published?" Tags="|physics|astronomy|newton|publishing|cosmology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14957" PostTypeId="2" ParentId="13155" CreationDate="2022-11-27T14:40:24.297" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/WpKE2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/WpKE2.jpg&quot; alt=&quot;An explanation&quot; /&gt;&lt;/a&gt;&#xA;How logarithm (hence e) took place in this subject.&lt;/p&gt;&#xA;" OwnerUserId="17582" LastActivityDate="2022-11-27T14:40:24.297" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14958" PostTypeId="2" ParentId="14956" CreationDate="2022-11-27T14:44:14.103" Score="2" Body="&lt;p&gt;Final edition has both Proposition 43 and Theorem 22 (in book 1) but they are different from what you cite.&#xA;Book 3 contains only 42 propositions.&lt;/p&gt;&#xA;&lt;p&gt;Of course we cannot penetrate Newton's mind, so this is a question of opinion. On my opinion, at the time of Newton nobody considered Brahe's system seriously (it became obsolete after the discovery of Kepler's laws).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-11-28T05:34:24.560" LastActivityDate="2022-11-28T05:34:24.560" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14959" PostTypeId="1" AcceptedAnswerId="14961" CreationDate="2022-11-28T03:29:54.653" Score="5" ViewCount="1250" Body="&lt;p&gt;I've seen quite a few of the names I. Bernard Cohen in the history of science book. But I couldn't find what I. meant.&lt;/p&gt;&#xA;" OwnerUserId="10808" LastEditorUserId="5788" LastEditDate="2022-11-30T00:08:25.160" LastActivityDate="2022-11-30T00:08:25.160" Title="What was the full name of I. Bernard Cohen?" Tags="|philosophy-of-science|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="14961" PostTypeId="2" ParentId="14959" CreationDate="2022-11-28T04:09:56.837" Score="10" Body="&lt;p&gt;I am quoting the full footnote published in &lt;em&gt;Seven Decades of History of Science: I. Bernard Cohen (1914–2003)&lt;/em&gt;. The full text is available in JSTOR &lt;a href=&quot;https://www.jstor.org/stable/10.1086/597575&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Cohen’s sister, Harriet, was three years older than he; she died in&#xA;1994. The “I” in Bernard Cohen’s name has always been a mystery—one he insisted was no mystery at all, remarking to friends that it was like&#xA;the “S” in Harry S. Truman. Replying to one author, Cohen complained&#xA;that he had “invented a fictitious new name for me, putting it within&#xA;square brackets as if I had chosen to hide it and you were revealing&#xA;it to the public. What the basis for your invention could possibly&#xA;have been, I cannot imagine. Certainly, recourse to any of the&#xA;standard sources of bibliographic information would have prevented you&#xA;from making this error. Had you consulted the Library of Congress&#xA;catalogue, the National Union Catalogue, or Who’s Who, or any other&#xA;proper source, you would have found that my correct and full legal&#xA;name is (and always has been) and is listed as, I. Bernard Cohen,&#xA;which is the name recorded on my birth certificate, my passport, my&#xA;FBI clearance, every copyright for some thirty or more books, and&#xA;other official records”: I. Bernard Cohen, letter dated 9 Mar. 1992&#xA;(courtesy of Susan T. Johnson).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another online Encyclopedia claims &lt;em&gt;I&lt;/em&gt; is for &lt;em&gt;Ierome&lt;/em&gt; without any reference. Most likely this is a conjecture. All his publications just use &lt;em&gt;I.&lt;/em&gt; so apparently &lt;em&gt;I.&lt;/em&gt; has no meaning. It is just a sound.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="-1" LastEditDate="2022-11-28T16:25:26.277" LastActivityDate="2022-11-28T16:25:26.277" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14962" PostTypeId="1" CreationDate="2022-11-28T06:29:31.877" Score="0" ViewCount="45" Body="&lt;p&gt;The Wikipedia page says for &lt;a href=&quot;https://en.wikipedia.org/wiki/Order_theory#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;order theory&lt;/a&gt; says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As explained before, orders are ubiquitous in mathematics. However, earliest explicit mentionings of partial orders are probably to be found not before the 19th century. In this context the works of George Boole are of great importance.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The page on &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Boole&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Boole&lt;/a&gt; doesn't mention order, beyond the idea of &amp;quot;orders of logic&amp;quot; (especially first-order logic). Is this Boole's contribution to order theory? Or is there some other contribution of Boole?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastEditorUserId="131" LastEditDate="2022-11-28T08:32:52.983" LastActivityDate="2022-11-28T08:32:52.983" Title="What did George Boole contribute to order theory?" Tags="|mathematics|mathematical-logic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="14963" PostTypeId="2" ParentId="14959" CreationDate="2022-11-28T14:57:23.220" Score="2" Body="&lt;p&gt;It's impossible to know for sure, but I. Bernard Cohen's parents were Isidore Cohen and Blanche Bernstein (&lt;a href=&quot;https://www.familysearch.org/ark:/61903/3:1:33SQ-GRJZ-93SV?i=32&amp;amp;cc=1488411&quot; rel=&quot;nofollow noreferrer&quot;&gt;1920 Census&lt;/a&gt;). Isidore's parents were Israel Cohen and Harriet Hyam (&lt;a href=&quot;https://www.familysearch.org/ark:/61903/1:1:24ZD-3RS&quot; rel=&quot;nofollow noreferrer&quot;&gt;Isidore's marriage record&lt;/a&gt;). I. Bernard's sister Harriet was named (as traditional) for her father's mother. It stands to reason that I. Bernard would be named for his father's father: Israel.&lt;/p&gt;&#xA;&lt;p&gt;However, as he claimed, there seem to be no written records remaining to support this idea. He was known as Bernard from a very young age.&lt;/p&gt;&#xA;" OwnerUserId="17587" LastActivityDate="2022-11-28T14:57:23.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14964" PostTypeId="1" AcceptedAnswerId="14965" CreationDate="2022-11-28T17:59:07.763" Score="0" ViewCount="426" Body="&lt;p&gt;&lt;a href=&quot;https://nutrients.ifas.ufl.edu/nutrient_pages/bsfpages/NaturalElements.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Naturally occurring elements&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Is it possible to find natural elements in other planets and their natural satellites other than planet earth?&lt;/p&gt;&#xA;" OwnerUserId="17590" LastActivityDate="2022-11-28T19:17:40.097" Title="Naturally occurring elements in other planets and their natural satellites other than earth" Tags="|chemistry|periodic-table|" AnswerCount="1" CommentCount="2" ClosedDate="2022-12-04T02:31:12.393" ContentLicense="CC BY-SA 4.0" />
  <row Id="14965" PostTypeId="2" ParentId="14964" CreationDate="2022-11-28T19:17:40.097" Score="2" Body="&lt;p&gt;Yes, naturally occurring elements can possibly be found on other planets than Earth and their satellites.  In fact, those other planets and their satellites are totally composed of naturally occuring elements.  So any astronauts or space probes that take samples of those planets and satellites and anyalize them will find variouscombounds made o fnaturally occuring elements.&lt;/p&gt;&#xA;&lt;p&gt;But all the naturally occurring elements are found on Earth in larger or smaller amounts.&lt;/p&gt;&#xA;&lt;p&gt;Scientiists have created dozens of elements in the labatories which are not found naturally.  Those elements are too unstable to last long.  Some decay into other elements in mere fractions of a second.&lt;/p&gt;&#xA;&lt;p&gt;Those heavy and  unstable elements may haveen created in supernovas and been present on Earth and other planets and satellites when they formed billions of years ago, but have long since decayed into other elements and can't be found naturally occuring anywhere in this solar system or in any solar system more than a tiny fraction as old as ours.&lt;/p&gt;&#xA;&lt;p&gt;Elements which have never been created by scientists in labatories would be very hard to create by natural processes, and should be even more unstable than the ones created on Earth so far.  And thus shey should also have decayed into other more stable elements almost immediately after being created.&lt;/p&gt;&#xA;&lt;p&gt;So, unlike in many science fiction stories, scientists think that it is almost totally impossible to ever discover new elements, never found or created on Earth, on other planets, moons, asteroids, comets, etc. in our solar system or other solar sytems, or floating in outer space.&lt;/p&gt;&#xA;&lt;p&gt;And if you want to learn about the very, very, very small hope that there could be as yet undiscovered stable elements on other worlds, you will have ask someone more knowledgable about atomic physics.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2022-11-28T19:17:40.097" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14966" PostTypeId="1" CreationDate="2022-11-29T07:23:24.060" Score="1" ViewCount="96" Body="&lt;p&gt;I understand why the effects of gravity are felt in both balloons and high-flying planes and I believe that while it could have been calculated using Newton's law of gravitation, was there speculation by real scientists about what an observer would experience before such high flights occurred.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2023-02-22T17:02:20.803" Title="Was the effect of gravity completely anticipated by the first to ascend in balloons and later airplanes?" Tags="|gravity|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14967" PostTypeId="1" CreationDate="2022-11-29T07:38:06.230" Score="4" ViewCount="285" Body="&lt;p&gt;I understand that Einstein was able to draw remarkable conclusions and was set on the path of General relativity when he realized that a falling observer not feeling his own weight.&lt;/p&gt;&#xA;&lt;p&gt;But is it believed that Einstein was actually the first person to realize specifically this weightlessness idea? I do think in those days when (perhaps) the only people who experienced weightless for a significant period of time died from it. It was only after planes and parachutes (and I don't know how many parachute jumps had been made before Einstein thought about this) that people could have really experienced weightlessness. Elevators I guess dropped enough that a momentary reduction in weight could be felt. But even elevators were fairly rare.&lt;/p&gt;&#xA;&lt;p&gt;(An aside is, I am pretty sure trains and the need for accurate/synchronized clocks played a role in general relativity although not sure if scientists like Heaviside were inspired by trains and clocks.)&lt;/p&gt;&#xA;&lt;p&gt;But did not Newton understand 200 years before Einstein that an object in orbit was actually falling and someone inside such an object would not feel his own weight? Or was he basically uninterested in what a hypothetical human &amp;quot;aboard&amp;quot; such a satellite -- he just cared about the orbit of the object itself.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-12-01T05:41:19.403" Title="Before Einstein did people realize about a falling man not feeling weight?" Tags="|classical-mechanics|relativity-theory|newton|einstein|gravity|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14968" PostTypeId="2" ParentId="771" CreationDate="2022-11-29T08:00:29.593" Score="-1" Body="&lt;p&gt;There is a very simple geometric approach to continued fraction that yields the expected approximation:&lt;/p&gt;&#xA;&lt;p&gt;For the minoring sequence : You alternatively stack a double square above and a square to the right.&lt;/p&gt;&#xA;&lt;p&gt;For the majoring sequence you start with a quare with a 3:1 rectangle (that creates a 3:4 rectangle) then, as above, You alternatively stack a double square above and a square to the right.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/mJ0DT.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/mJ0DT.png&quot; alt=&quot;minoring sequence&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/s6HNf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/s6HNf.png&quot; alt=&quot;majoring sequence&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17594" LastEditorUserId="17594" LastEditDate="2022-11-29T15:10:46.983" LastActivityDate="2022-11-29T15:10:46.983" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14969" PostTypeId="1" CreationDate="2022-11-29T10:59:03.843" Score="4" ViewCount="104" Body="&lt;p&gt;Various inventions that have become well-known were &lt;a href=&quot;https://www.davison.com/blog/inventions-that-are-surprisingly-not-patented/&quot; rel=&quot;nofollow noreferrer&quot;&gt;never patented&lt;/a&gt;, including matches, emoticons, and the magnetic strip. Other noteworthy examples include the polio vaccine (Jonas Salk), monoclonal antibodies (Cesar Milstein), the scanning tunneling microscope (IBM scientists Binnig and Rohrer), and the penicillin (invented and deliberately not patented by Fleming, later patented by Coghill and Moyer in the US).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: I am interested in systemic, empirical, and statistical studies that look into such unpatented inventions in more detail. More specifically, I am curious as to what different factors influenced why certain inventions were not patented. Was it because of certain political ideals of the inventor, not filing on time, the invention being too much akin to other products or inventions, or something else?&lt;/p&gt;&#xA;&lt;p&gt;I've searched on Google Scholar with combinations of various key words, including &amp;quot;patent&amp;quot;, &amp;quot;statistical&amp;quot;, &amp;quot;analysis&amp;quot;, &amp;quot;unpatented&amp;quot;, &amp;quot;factors&amp;quot;, &amp;quot;prior&amp;quot;, &amp;quot;art&amp;quot;, &amp;quot;political&amp;quot;, and &amp;quot;economy&amp;quot;, but I couldn't find much on this particular question.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Note&lt;/em&gt;: I've also asked a shorter version of this &lt;a href=&quot;https://patents.stackexchange.com/questions/25054/systemic-analysis-of-unpatented-inventions&quot;&gt;question&lt;/a&gt; on Patents SE, Law SE, and History SE. On the former platform, I haven't received any answers yet - users have commented I'm unlikely to find any answers there because they're more knowledgeable about the patents themselves than about the history behind them. On the latter platform, I made the &lt;a href=&quot;https://history.stackexchange.com/questions/70390/what-factors-influence-whether-an-invention-is-not-patented&quot;&gt;the question&lt;/a&gt; a bit too broad, so it was closed. Therefore, I have later also &lt;a href=&quot;https://economics.stackexchange.com/questions/53671/what-factors-influence-whether-an-invention-is-not-patented&quot;&gt;asked&lt;/a&gt; it on Economics SE. It hasn't received any answers there, yet.&lt;/p&gt;&#xA;" OwnerUserId="4296" LastActivityDate="2022-11-30T00:02:46.523" Title="What factors influence whether an invention is not patented?" Tags="|discoveries|statistics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14970" PostTypeId="2" ParentId="14969" CreationDate="2022-11-29T13:52:28.340" Score="3" Body="&lt;p&gt;{edit:  includes further details about now-historical examples: monoclonal antibodies 1970s-1980s, and insulin 1920s-1930s:}&lt;/p&gt;&#xA;&lt;p&gt;First, I'm sorry to hear of your problems getting the question answered, and I hope it will be allowed to remain on this forum. Patents have been controversial in a number of countries more or less since they have existed, but their history is very closely bound up with scientific and technical advance and its history.&lt;/p&gt;&#xA;&lt;p&gt;I don't think you will find good systematic and statistical studies of the question you posed; and even if you did, I would doubt that the results could be meaningful as a group, because patents and inventions do not really make a population: nearly every one is fundamentally different in the very ways that make it (temporarily) new and possibly of general value and interest.&lt;/p&gt;&#xA;&lt;p&gt;I suggest that a more realistic approach to investigating the question would be to consider a series of factors or filters that select an answer applicable case-by-case. A few suggestions for filters are offered below, but first it may help to consider two examples, showing that what the question says about the individual motives of inventors is of central importance too.&lt;/p&gt;&#xA;&lt;p&gt;Here it is also unfortunately the case that myth and fake history can take over. Consider monoclonal antibodies. The 'never patented' webpage to which you linked gave a summary for why they were not patented. But as it happens, in the 1980s I met and spoke with Cesar Milstein and some others concerned with the invention/development of monoclonal antibodies. What I heard directly from them about the reasons for not patenting, (and what little was directly reported from Milstein himself in the press), was very different indeed from the summary as it now stands in your linked web-page.&lt;/p&gt;&#xA;&lt;p&gt;According to Milstein's own words (and his early press interviews) he began with considerable feeling against patents generally, regarding them as a drag on research and an inhibition on its widespread application -- both of which he wished to promote. When I met him, he said that his objectives had not changed, but that his view of the best means to achieve those objectives had changed. His expressed view, as it had developed, was that it was better to obtain patents and to use them to promote use and application of the technology on reasonable terms, also using the powers given by the rights so as to prevent abuses. He had come to appreciate, he said, that omitting to do that opened the door to later-comers to obtain and possibly to abuse patents for significantly large but differentiated portions of the field in question. That, he felt, had actually happened with the monoclonal antibody invention.&lt;/p&gt;&#xA;&lt;p&gt;An additional factor in the situation was also that the people who would have had charge of obtaining patents for the monoclonal antibody invention had reached an early view that monoclonal antibodies were not patentably new. This was because at the beginning, the only generally distinguishing feature that they knew of (or perhaps were told about) was that here were molecularly uniform antibodies, where all previous known (natural) antibodies were highly heterogeneous.  Now antibodies, including heterogeneous natural antibodies, are immunoglobulin proteins. On the other hand, patients with &lt;a href=&quot;https://en.wikipedia.org/wiki/Multiple_myeloma&quot; rel=&quot;nofollow noreferrer&quot;&gt;myeloma&lt;/a&gt; or its benign precursor conditions are known to produce and to have in their blood highly uniform immunoglobulins, sometimes complete, sometimes incomplete ('light chains') -- these often used to be known as &lt;a href=&quot;https://www.urmc.rochester.edu/encyclopedia/content.aspx?contenttypeid=167&amp;amp;contentid=bence_jones_protein_urine&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bence-Jones proteins&lt;/a&gt;, now often more generically as 'paraprotein'. More to the point, diagnostic tests were known that separate out these uniform pathological immunoglobulins. So preparations of molecularly uniform immunoglobulins were in themselves not new (at least in the view of some of those involved at the time).  Of course it may be very possible or likely, that something relevantly new could have been identified and articulated about the new technology if the people involved had had deeper discussions at the time. But it does seem likely that, given the early attitude expressed by the co-inventor closely involved, not much of any such deeper discussion had been taking place.  By the time Milstein had reached his revised view, enough publication had occurred so that patenting was ruled out anyway. Certainly Milstein was continuing to express some high ideals about the way in which new-invented technology should be handled. I don't know how such ideals fare in more recent cases these days, but perhaps that's not part of the current question.&lt;/p&gt;&#xA;&lt;p&gt;There is a notable further relevant example of hesitancy or indecision about patenting in the medical/biological field: the original insulin invention/discovery of the 1920s. There was initial indecision: one of the original co-inventors, Banting, a physician, began by thinking it was contrary to his professional ethic to apply for a patent. (See &lt;a href=&quot;https://press.uchicago.edu/ucp/books/book/chicago/D/bo5388816.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michael Bliss's 'The Discovery of Insulin'&lt;/a&gt;.)  Eventually, and possibly with little prospect of getting insulin manufactured and available in required quantity without a patent, Banting changed his mind, and patents were obtained by the original inventors in several countries, also partly for the sake of controlling quality and preventing the appearance of imitations in an age when pharmaceutical regulation was not as developed as it is now. For some countries the rights were passed into the ownership of relevant authorities there.  But then later there was yet further reconsideration, and the holders of some of the original patents deliberately allowed them to lapse. There seems no record of their reasons, these lay perhaps in a view that the original purpose no longer called for them, or perhaps in general disapproval of patents.&lt;/p&gt;&#xA;&lt;p&gt;In both these examples there was clear lack of consensus on the part of those in whose hands the initiatives lay, about why and whether and how and what patents should be obtained and maintained. Each case contains a highly individual history that can be reported (or perhaps misreported), further hindering systematic study.&lt;/p&gt;&#xA;&lt;p&gt;Apart from the complexity of the individual examples mentioned, there are a few more general possible factors or filters that can lead to absence of patent.&lt;/p&gt;&#xA;&lt;p&gt;One filter is that patents have always been expensive to obtain, and something that nobody would want to copy will not normally be thought worth patenting however high its scientific interest -- unless it's a 'vanity' patent.&lt;/p&gt;&#xA;&lt;p&gt;Another filter is that the categories of invention legally allowed to be patented have been variously defined over time, and perhaps now the 'gate' is wider open than it has ever been. For example software inventions and biotechnology inventions -- at least of several kinds -- used to be excluded, but those exclusions seem now to have been narrowed considerably.&lt;/p&gt;&#xA;&lt;p&gt;Another filter arises from the fact of whether the thing in question has already been published or used, if it has, then patenting is usually ruled out.&lt;/p&gt;&#xA;&lt;p&gt;These are just a few among important possible reasons for any particular case of 'no patent'.  I think they and the examples offered above show how intractable to systematic treatment the question can be.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-11-30T00:02:46.523" LastActivityDate="2022-11-30T00:02:46.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14971" PostTypeId="2" ParentId="14967" CreationDate="2022-11-29T14:03:53.190" Score="2" Body="&lt;p&gt;It is not clear what group of people you are discussing, scientists or &amp;quot;general public&amp;quot;. People well educated in science certainly understood this (Jules Verne did not belong to this category). And speaking of the everyday experience, many people did experience weightlessness, namely when submerged in water. Human body has about the same density as water.&lt;/p&gt;&#xA;&lt;p&gt;Einstein has nothing to do with this, since all this is explained by classical mechanics.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-11-29T14:03:53.190" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="14972" PostTypeId="2" ParentId="14879" CreationDate="2022-11-29T18:10:20.157" Score="1" Body="&lt;p&gt;Here is a paper examining it with relation to computer science, but I'm not aware of any papers relating it to combinatorics. &lt;a href=&quot;https://www.diva-portal.org/smash/get/diva2:516245/FULLTEXT01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.diva-portal.org/smash/get/diva2:516245/FULLTEXT01.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17597" LastActivityDate="2022-11-29T18:10:20.157" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14973" PostTypeId="2" ParentId="14967" CreationDate="2022-11-29T19:20:13.797" Score="4" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;It was only after planes and parachutes (and I don't know how many parachute jumps had been made before Einstein thought about this) that people could have really experienced weightlessness.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Your sense of history is a little off if you don't know that balloons and airships were before airplanes.&lt;/p&gt;&#xA;&lt;p&gt;As it happens, someone who was sort of a relative of mine made their first solo flight at the age of nine, which might still be the world record, and that was years before the Wright brothers.&lt;/p&gt;&#xA;&lt;p&gt;On 19 September, 1789, a duck, a sheep and a rooster made the first balloon flight by living animals.  The first manned balloon flight flight was on 21 November 1789.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Louis-Sébastien Lenormand (May 25, 1757 – April 4, 1837&lt;a href=&quot;https://en.wikipedia.org/wiki/Louis-S%C3%A9bastien_Lenormand&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;) was a French chemist, physicist, inventor, monk, and a pioneer in parachuting.&lt;/p&gt;&#xA;&lt;p&gt;He is considered as the first man to make a witnessed descent with a parachute and is also credited with coining the term parachute, from the Latin prefix para meaning &amp;quot;against&amp;quot;, an imperative form of parare = to avoid, avert, defend, resist, guard, shield or shroud, from paro = to parry, and the French word chute for &amp;quot;fall&amp;quot;, hence the word &amp;quot;parachute&amp;quot; literally means an aeronautic device &amp;quot;against a fall&amp;quot;. After making a jump from a tree with the help of a pair of modified umbrellas, Lenormand refined his contraption and on December 26, 17831 jumped from the tower of the Montpellier observatory in front of a crowd that included Joseph Montgolfier, using a 14-foot parachute with a rigid wooden frame.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Louis-S%C3%A9bastien_Lenormand&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Louis-S%C3%A9bastien_Lenormand&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;André-Jacques Garnerin (31 January 1769 – 18 August 1823) was a French balloonist and the inventor of the frameless parachute. He was appointed Official Aeronaut of France.&lt;/p&gt;&#xA;&lt;p&gt;Garnerin began experiments with early parachutes based on umbrella-shaped devices and carried out the first frameless parachute descent (in the gondola) with a silk parachute on 22 October 1797 at Parc Monceau, Paris (1st Brumaire, Year VI of the Republican calendar).&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; Garnerin's first parachute was made of white canvas[citation needed] with a diameter of approximately 23 feet (7 m).&lt;a href=&quot;https://en.wikipedia.org/wiki/The_First_Men_in_the_Moon&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt; The umbrella was closed before he ascended, with a pole running down its center and a rope running through a tube in the pole, which connected it to the balloon.&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; Garnerin rode in a basket attached to the bottom of the parachute; at a height of approximately 3,000 feet (1,000 m) he severed the rope that connected his parachute to the balloon.&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; The balloon continued skyward while Garnerin, with his basket and parachute, fell.&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; The basket swung violently during descent,[Note 1] then bumped and scraped when it landed, but Garnerin emerged uninjured.&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;His student Jeanne Geneviève Labrosse, who later became his wife, was both a balloonist and the first female parachutist. Labrosse first flew on 10 November 1798, one of the earliest women to fly in a balloon, and on 12 October 1799, Labrosse was the first woman to parachute, from an altitude of 900 meters.[11]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9-Jacques_Garnerin&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Andr%C3%A9-Jacques_Garnerin&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So people were making parcute jumps for more than a century before Einstein.  And some of htem might have experienced weightlessness for some time before opening their paracutes.&lt;/p&gt;&#xA;&lt;p&gt;A number of people have survived falling from heights of tens or hundreds of feet.  Thus the sport of making high jumps or dives into water began.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Cliff diving has been documented as far back as 1770 when Kahekili II, king of Maui, engaged in a practice called &amp;quot;lele kawa&amp;quot;, which in English means jumping feet first into water from great heights without making a splash.[6] The king's warriors were forced to participate to prove that they were courageous and loyal to the king. The practice later developed into a competition under king Kamehameha I, and divers were judged on their style and amount of splash upon entering the water.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/High_diving&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/High_diving&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On May 19, 1885 Robert Odlum was the first person to jump off the Brooklyn Bridge, and was fatally injured. Steve Brody claimed to have survived jumping off the Brooklyn Bridge onJuly 23, 1886, and  the first confirmed survivor of a jump off the Brooklyn Bridge was Larry Donvan.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lawrence &amp;quot;Larry&amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9-Jacques_Garnerin&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt; M. Donovan, born Lawrence Degnan&lt;a href=&quot;https://en.wikipedia.org/wiki/High_diving&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt; or possibly Duignan&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; (1862&lt;a href=&quot;https://en.wikipedia.org/wiki/The_First_Men_in_the_Moon&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt; – August 7, 1888&lt;a href=&quot;https://en.wikipedia.org/wiki/High_diving&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt;) was a newspaper typesetter who became famous for leaping from bridges, first around the northeastern United States, and later in England. Inspired by the first successful Brooklyn Bridge jump by Steve Brodie, Donovan sought fame and fortune by leaping off that bridge, the Niagara Falls Suspension Bridge, and Bristol's 250 foot Clifton Suspension Bridge. Slightly injured on a couple of occasions and frequently incarcerated following his attempts, he struggled to capitalise on his fame, making money only through bets and brief periods working as an emcee or exhibiting himself in novelty shows. In August 1888, disillusioned and desperately poor, he accepted a spontaneous two-pound wager to jump from London's Hungerford Bridge late at night, but drowned in the attempt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Larry_Donovan_(bridge_jumper)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;So some 19th century persons could talk about what it felt to be in free fall for several seconds.&lt;/p&gt;&#xA;&lt;p&gt;I note that Jules Verne in &lt;em&gt;From the Earth to the Moon&lt;/em&gt; was not the only 19th century writer of stories of space flight, and maybe some of them depicted  weightlessness more accurately.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The First Men in the Moon is a scientific romance by the English author H. G. Wells, originally serialised in The Strand Magazine from December 1900 to August 1901 and published in hardcover in 1901,&lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9-Jacques_Garnerin&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt; who called it one of his &amp;quot;fantastic stories&amp;quot;.&lt;a href=&quot;https://en.wikipedia.org/wiki/High_diving&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On the way to the Moon, they experience weightlessness, which Bedford finds &amp;quot;exceedingly restful&amp;quot;.[7]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_First_Men_in_the_Moon&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/The_First_Men_in_the_Moon&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The Brick Moon&amp;quot; is a novella by American writer Edward Everett Hale, published serially in The Atlantic Monthly starting in 1869. It is a work of speculative fiction containing the first known depiction of the launch of an artificial satellite.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The Brick Moon&amp;quot; was first released serially in three parts in The Atlantic Monthly in 1869.&lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9-Jacques_Garnerin&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; A fourth part, entitled &amp;quot;Life on the Brick Moon&amp;quot;, was also published in The Atlantic Monthly in 1870.&lt;a href=&quot;https://en.wikipedia.org/wiki/High_diving&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt; It was collected as the title work in Hale's anthology The Brick Moon and Other Stories in 1899.&lt;a href=&quot;https://en.wikipedia.org/wiki/The_First_Men_in_the_Moon&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And I believe that Hale described weightlessness on the Brick Moon in orbit.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Somnium (Latin for &amp;quot;The Dream&amp;quot;) — full title: Somnium, seu opus posthumum De astronomia lunari — is a novel written in Latin in 1608 by Johannes Kepler, and first published in 1634 by Kepler's son, Ludwig Kepler.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Kepler describes acceleration and deceleration on the journey to the moon and might also describe weightlessness.&lt;/p&gt;&#xA;&lt;p&gt;Someone would have to read these, and many other space travel stories written before Relativity to see which ones might say that weightlessness in space travel is similar to the weightlessness feld when falling freely in a gravitational field.&lt;/p&gt;&#xA;&lt;p&gt;edia.org/wiki/Louis-S%C3%A9bastien_Lenormand&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2022-11-30T18:32:27.930" LastActivityDate="2022-11-30T18:32:27.930" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14974" PostTypeId="2" ParentId="9716" CreationDate="2022-11-29T19:25:16.520" Score="2" Body="&lt;p&gt;The reference above to Mathwords should be updated. In fact, Cantelli used the term &amp;quot;random variable&amp;quot; (&lt;em&gt;variabilie casuale&lt;/em&gt;, in the original Italian) in his &lt;a href=&quot;https://www.jstor.org/stable/23223974&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Sulla differenza media con ripetizione&lt;/em&gt;  (1913)&lt;/a&gt;, which could well be the first appearance in the history of probability theory. Cantelli's work cites (and abounds in results contained in) &lt;em&gt;Variabilità e Mutabilittà&lt;/em&gt;, from 1912, a well-known work by the also well-known Gini. But the term in question is not used in it. It is a work that deals with the measurement of the variability of random variables, but at no time does it give them a specific name. &lt;a href=&quot;http://www.numdam.org/item/JMPA_1867_2_12__177_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chebyshev's &lt;em&gt;Des Valeurs Moyennes&lt;/em&gt; (1867)&lt;/a&gt;, also refers to them only generically,  &amp;quot;&lt;em&gt;quantités&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The term gained some popularity in Italian and was used by Castelnuovo in his &lt;a href=&quot;https://www.google.com/books/edition/Calcolo_delle_probabilit%C3%A0/vsRXAAAAYAAJ?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calcolo delle Probabilità&lt;/em&gt; (1919)&lt;/a&gt;. From there, it extended to French (Borel, Lévy), and finally to the English speaking world in the 1930's. It is relevant here to mention how Lévy praises Castelnuovo's work in his &lt;a href=&quot;https://www.google.com/books/edition/Calcul_des_probabilit%C3%A9s/8_FLAAAAMAAJ?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Calcul des Probabilités&lt;/em&gt; (1925)&lt;/a&gt;. In it, the &amp;quot;event variable&amp;quot; is used in the way we are used to for &amp;quot;random variable&amp;quot; today. The terms &lt;em&gt;variable éventuelle&lt;/em&gt; and &lt;em&gt;variable aléatorire&lt;/em&gt; are found in other works of the time, such as Borel's treatises. Borel has a book on probability theory, &lt;em&gt;Le Hasard&lt;/em&gt; (1914) where he does not use the term, but in each of two treatises from around the middle of the following decade he uses both of them.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, Gosset in &lt;a href=&quot;https://www.semanticscholar.org/paper/The-Elimination-of-Spurious-Correlation-due-to-in-Student/917c9844efd66aac63001ece6b51380aca9afe8d&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Elimination of Spurious Correlation due to Position in Time or Space (1914)&lt;/a&gt; only  juxtaposes the words &amp;quot;variable&amp;quot; and &amp;quot;random&amp;quot; in succession. But he never says anything like &amp;quot;such a random variable&amp;quot; or &amp;quot;study of random variables&amp;quot;, as became common later. In &lt;a href=&quot;https://www.jstor.org/stable/2370961&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wintner's article On Analytic Convolutions of Bernoulli Distributions (1934)&lt;/a&gt;, one of the first to use the combination as a term in English, the author generally uses a practically modern nomenclature. And not by chance, the article abounds in results published for the first time, as it says, in Lévy's &lt;em&gt;Calcul&lt;/em&gt;. As &lt;a href=&quot;https://books.google.com/ngrams/graph?content=random+variable&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=en-2019&amp;amp;smoothing=3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Ngram shows&lt;/a&gt;, the term &amp;quot;random variable&amp;quot; only began to become popular in English in the 1930s.&lt;/p&gt;&#xA;&lt;p&gt;I wrote a longer account of the story (in Spanish!) &lt;a href=&quot;https://www.datanalytics.com/2022/11/29/origen-termino-variable-aleatoria/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17598" LastEditorUserId="55" LastEditDate="2024-01-09T06:56:06.580" LastActivityDate="2024-01-09T06:56:06.580" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14975" PostTypeId="2" ParentId="13155" CreationDate="2022-11-29T21:11:19.007" Score="0" Body="&lt;p&gt;This derivation is nice as well. It uses the same manipulation as mentioned in the above answer: Flipping the sign to use imaginary number.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.cantorsparadise.com/another-derivation-of-eulers-fabulous-formula-e5f50aa28fbd&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cantorsparadise.com/another-derivation-of-eulers-fabulous-formula-e5f50aa28fbd&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17600" LastEditorUserId="17600" LastEditDate="2022-11-29T21:44:27.183" LastActivityDate="2022-11-29T21:44:27.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14976" PostTypeId="2" ParentId="1811" CreationDate="2022-11-30T05:30:54.110" Score="1" Body="&lt;p&gt;As far as I know that Kuzmin R.O.(1928a,b) first introduced this operator[ref. A. Boyarsky, P. Gora. Laws of chaos. Invariant measures and dynamical systems in one dimension, Probability and its Applications,Boston, MA, 1997.&lt;/p&gt;&#xA;" OwnerUserId="17601" LastActivityDate="2022-11-30T05:30:54.110" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14977" PostTypeId="1" CreationDate="2022-11-30T10:13:43.223" Score="3" ViewCount="381" Body="&lt;p&gt;I was reading the &lt;a href=&quot;https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia on Clairaut's Theorem&lt;/a&gt; (Symmetry of second derivatives) and the article accounts the significant amount of time and failed proofs occurred before the theorem was made fully rigorous by Schwarz in 1873. The article then gives an elementary proof of the theorem using essentially only the Mean Value Theorem, which was proven by &lt;a href=&quot;https://en.wikipedia.org/wiki/Mean_value_theorem#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cauchy in 1823&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This makes me wonder why the theorem took 50 additional years after the Mean Value Theorem was proven to be made rigorous, when it does not seem to require many tools, and the ideas do not seem to be too difficult for a mathematician at the time to come across. I was wondering if anyone has any insight to what the field of analysis was like at the time and why there could be such a discrepancy? Could the act of discovering the right conditions (continuity of second partial derivatives) be part of it?&lt;/p&gt;&#xA;" OwnerUserId="17603" LastEditorUserId="17603" LastEditDate="2022-11-30T16:12:00.373" LastActivityDate="2023-09-04T17:04:25.493" Title="Why did Clairaut's theorem take so long to prove?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14978" PostTypeId="2" ParentId="14967" CreationDate="2022-12-01T05:41:19.403" Score="2" Body="&lt;p&gt;Galileo recognized that falling objects don't exert a weight upon other falling bodies:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Note that it is necessary to distinguish between heavy bodies in motion and the same bodies at rest. A large stone placed in a balance not only acquires additional weight by having another stone placed upon it, but even by the addition of a handful of hemp its weight is augmented six to ten ounces according to the quantity of hemp. But if you tie the hemp to the stone and allow them to fall freely from some height, do you believe that the hemp will press down upon the stone and thus accelerate its motion or do you think the motion will be retarded by a partial upward pressure? &lt;strong&gt;One always feels the pressure upon his shoulders when he prevents the motion of a load resting upon him; but if one descends just as rapidly as the load would fall how can it gravitate or press upon him? Do you not see that this would be the same as trying to strike a man with a lance when he is running away from you with a speed which is equal to, or even greater, than that with which you are following him? You must therefore conclude that, during free and natural fall, the small stone does not press upon the larger and consequently does not increase its weight as it does when at rest&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2022-12-01T05:41:19.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14979" PostTypeId="2" ParentId="3268" CreationDate="2022-12-01T19:54:28.423" Score="0" Body="&lt;p&gt;This isn't going to be an answer.  But it will be informative.  I am not sure how this community treats this type of thing.  Feel free to erase if it's out of line.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;em&gt;Storing decimal numbers as floating points introduces round off error regardless of how many bits of precision are stored.&lt;/em&gt;  For example, there is no way to express 0.1 as a &lt;em&gt;floating points&lt;/em&gt; binary fraction &lt;em&gt;precisely&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However small the round off error may be, the amounts traded on the bond markets are so large that in practical terms it may end up being thousands of dollars, that someone gains or loses on every trade.  \$1000 is only .00001 of \$100M.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;There are 2 possible solutions to this.&lt;/p&gt;&#xA;&lt;p&gt;Either switch to fixed-point decimal fractions (and keep 2 integers for each number - the whole and the decimal parts).&lt;/p&gt;&#xA;&lt;p&gt;Or keep using binary floating points numbers, and quote the prices using binary fractions.&lt;/p&gt;&#xA;&lt;p&gt;Binary fractions can be stored &lt;em&gt;precisely&lt;/em&gt; as floating-points binary numbers.  When fractions are stored precisely, there are no round-off errors in additions and subtractions.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So regardless of why the bond trading started in binary fractions, there was a good reason to keep it in binary fractions.  Again, it meant not having a loss/gain associated with every large trade.&lt;/p&gt;&#xA;" OwnerUserId="17608" LastActivityDate="2022-12-01T19:54:28.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14980" PostTypeId="2" ParentId="3268" CreationDate="2022-12-02T08:13:22.457" Score="1" Body="&lt;p&gt;The gilt-edged market in the UK used halves and quarters and eighths for its stock prices at least up to the early 1980s. The weird people who deal in very short-dated gilts may even have gone as far as sixteenths.&lt;/p&gt;&#xA;&lt;p&gt;It has nothing to do with Spanish coin: it is simply the easiest way to handle “a price half way between two other prices”.&lt;/p&gt;&#xA;" OwnerUserId="4209" LastActivityDate="2022-12-02T08:13:22.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14982" PostTypeId="2" ParentId="7292" CreationDate="2022-12-03T18:33:00.500" Score="4" Body="&lt;p&gt;There was a boy born on October 5, 1882 who read &lt;em&gt;The War of the worlds&lt;/em&gt; (1897, 1898) by H.G. Wells, at age 16 and became interested in space, which led to an experience on October 19, 1899.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On this day I climbed a tall cherry tree at the back of the barn ... and as I looked toward the fields at the east, I imagined how wonderful it would be to make some device which had even the possibility of ascending to Mars, and how it would look on a small scale, if sent up from the meadow at my feet. I have several photographs of the tree, taken since, with the little ladder I made to climb it, leaning against it.&lt;/p&gt;&#xA;&lt;p&gt;It seemed to me then that a weight whirling around a horizontal shaft, moving more rapidly above than below, could furnish lift by virtue of the greater centrifugal force at the top of the path.&lt;/p&gt;&#xA;&lt;p&gt;I was a different boy when I descended the tree from when I ascended. Existence at last seemed very purposive.[16]: 26 [23]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_H._Goddard&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Robert_H._Goddard&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;He became Dr. Robert Goddard, one of the great pioneers of rocketry.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastEditorUserId="12708" LastEditDate="2022-12-28T15:18:56.287" LastActivityDate="2022-12-28T15:18:56.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14983" PostTypeId="1" CreationDate="2022-12-04T11:22:50.247" Score="2" ViewCount="149" Body="&lt;p&gt;I have now claimed a few times on the internet, based on something (sensible!) I read, that at some point in the 1920s, that Zermelo at one point considered as a set theoretic axiom (schema) something equivalent to a restricted version of Replacement, corresponding to a function with countable domain, before going all out by 1930. Now, however, I cannot find the document in which I read this, and I cannot remember anything else about it! I foolishly didn't write this down. I have consulted &lt;a href=&quot;https://doi.org/10.1007/978-3-540-49553-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ebbinghaus' book&lt;/a&gt;, and I can't see it in there. I have consulted Kanamori's edition of &lt;a href=&quot;https://doi.org/10.1007/978-3-540-79384-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zermelo's collected works&lt;/a&gt;, and I can't see it in the prefatory essays on Zermelo's 1929 or 1930 set theory papers.&lt;/p&gt;&#xA;&lt;p&gt;However, and this is where my faulty memory shows up even more: I also in one place claimed it was &lt;em&gt;Fraenkel&lt;/em&gt; who considered countable Replacement! I cannot find this either. And in trying to hunt down my original source, I recognised the phrase&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;it is also to be noted that Skolem only wrote that &amp;quot;we could introduce&amp;quot; Replacement&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;in Kanamori's 2012 &lt;em&gt;&lt;a href=&quot;https://www.jstor.org/stable/41472440&quot; rel=&quot;nofollow noreferrer&quot;&gt;In Praise of Replacement&lt;/a&gt;&lt;/em&gt;. But I didn't see, on a search through there, anything corresponding to my half-remembered claim.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone enlighten me?&lt;/p&gt;&#xA;" OwnerUserId="2531" LastActivityDate="2022-12-04T11:22:50.247" Title="Zermelo's or Fraenkel's early consideration of something equivalent to countable Replacement" Tags="|mathematics|reference-request|set-theory|" AnswerCount="0" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="14986" PostTypeId="2" ParentId="9836" CreationDate="2022-12-05T16:24:58.563" Score="1" Body="&lt;p&gt;Gibbs' analysis (see his &lt;a href=&quot;https://www.gutenberg.org/files/50992/50992-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1902 book&lt;/a&gt;, and the beautifully concise summary of his main derivations in &lt;a href=&quot;https://books.google.com/books?id=mLXoCAAAQBAJ&amp;amp;pg=PA331&amp;amp;lpg=PA331&amp;amp;dq=%22the+birth+certificate+of+statistical+mechanics%22&amp;amp;source=bl&amp;amp;ots=xplmhT8Fwd&amp;amp;sig=ACfU3U1uutopQ9gh-xNdH-kUSmjus5zW_Q&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwiisueN9-37AhVDmIkEHeNQDU0Q6AF6BAglEAM#v=onepage&amp;amp;q=%22the%20birth%20certificate%20of%20statistical%20mechanics%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chapter 10 of &lt;em&gt;The Logic of Thermostatistical Physics&lt;/em&gt;&lt;/a&gt; by Emch and Liu) was purely mathematical based on the assumption of an underlying Hamiltonian description of the system, using the phase space density &lt;span class=&quot;math-container&quot;&gt;$\rho(X)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is a point in phase space.  Within this analysis he found that the &lt;em&gt;expectation of the log of the phase space density&lt;/em&gt;, &lt;span class=&quot;math-container&quot;&gt;$\bar{\eta} = \langle \eta \rangle$&lt;/span&gt; — where &lt;span class=&quot;math-container&quot;&gt;$\eta = \log(\rho)$&lt;/span&gt;, so that:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; \langle \eta \rangle= \int \rho(X) \, \log\left[\rho(X)\right] \, {\rm d} \Gamma&#xA;$$&lt;/span&gt;&#xA;with &lt;span class=&quot;math-container&quot;&gt;${\rm d}\Gamma$&lt;/span&gt; an infinitesimal volume element in the phase space — played the same role as (negative) entropy in a derived equation that matched the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;fundamental thermodynamic relation&lt;/a&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;{\rm d} U = T \, {\rm d}S - P\, {\rm d}V&#xA;$$&lt;/span&gt;&#xA;He therefore proposed that his completely mechanistic definition be considered an expression for thermodynamic entropy.&lt;/p&gt;&#xA;&lt;p&gt;See Chapter IV in his book (linked above), Equation (114) and the preceding discussion; and Chapter XIV, surrounding equation (483). (Forewarning: it is difficult to parse because he uses the word &amp;quot;index&amp;quot; for &amp;quot;log&amp;quot; and then uses log[exp(x)] = x to write things in an unfamiliar way).&lt;/p&gt;&#xA;&lt;p&gt;Using the Gibbs Entropy as a functional over distributions &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt;, you could determine the canonical distributions by extremizing with constraints.  This is accomplished by calculus of variations, but it is the same idea, and easier to see, using Shannon's discrete version:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S = -k \sum_i p_i \log p_i&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;If we assume only that &lt;span class=&quot;math-container&quot;&gt;$\sum p_i = 1$&lt;/span&gt;, then the multi-dimensional extremization of the multivariable function &lt;span class=&quot;math-container&quot;&gt;$S\left(\left\{p_i\right\} \right)$&lt;/span&gt;, subject to the constraint that &lt;span class=&quot;math-container&quot;&gt;$\sum p_i$&lt;/span&gt; is constant, is accomplished with a Lagrange multiplier, &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\partial}{\partial p_j} \left[ S\left(\left\{p_i\right\} \right) + \lambda \sum_i p_i \right] = 0&#xA;$$&lt;/span&gt;&#xA;which leads to &lt;span class=&quot;math-container&quot;&gt;$p_i = 1/N$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$i=1,\ldots, N$&lt;/span&gt;. This is the &lt;em&gt;microcanonical distribution&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;If we additionally assume a constant expectation of the energy, &lt;span class=&quot;math-container&quot;&gt;$U = \sum_i \varepsilon_i p_i$&lt;/span&gt;, then:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\partial}{\partial p_j} \left[ S\left(\left\{p_i\right\} \right) + \lambda_1 \sum_i p_i + \lambda_2 \sum_i \varepsilon_i p_i \right] = 0&#xA;$$&lt;/span&gt;&#xA;leads to &lt;span class=&quot;math-container&quot;&gt;$p_i = A \, \exp\left(- B \, \varepsilon_i\right)$&lt;/span&gt;, which is the &lt;em&gt;canonical distribution&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;The same procedure can be used to determine the &lt;em&gt;grand canonical distribution&lt;/em&gt;, by additionally assuming a fixed expected particle number, &lt;span class=&quot;math-container&quot;&gt;$N = \sum n_i p_i$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Finally (as Emch and Liu make clear in their book), if we consider a system with distribution &lt;span class=&quot;math-container&quot;&gt;$\rho_0$&lt;/span&gt; on an aggregate phase space, &lt;span class=&quot;math-container&quot;&gt;$\Gamma = \Gamma_1 \times \Gamma_2$&lt;/span&gt;,  composed of two subsystems, and define the &lt;em&gt;reduced probability densities&lt;/em&gt; of each sub-system:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\rho_1 = \int_{\Gamma_2} \rho_0 \quad \text{and} \quad \rho_2 = \int_{\Gamma_1} \rho_0&#xA;$$&lt;/span&gt;&#xA;then the corresponding entropies&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S_n = -k \int_{\Gamma_n} \rho_n \log \rho_n \quad (n=0,1,2)&#xA;$$&lt;/span&gt;&#xA;satisfy the inequality:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S_0 \le S_1 + S_2&#xA;$$&lt;/span&gt;&#xA;with&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;S_0 = S_1 + S_2 \quad \text{iff} \quad \rho_0 = \rho_1 \cdot \rho_2&#xA;$$&lt;/span&gt;&#xA;i.e., if the subsystems are independent.&lt;/p&gt;&#xA;&lt;p&gt;Jaynes extended these facts to &amp;quot;prove&amp;quot; the Second Law of Thermodynamics. See his paper &lt;a href=&quot;https://bayes.wustl.edu/etj/articles/gibbs.vs.boltzmann.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Gibbs vs Boltzmann entropies&lt;/em&gt; (1965)&lt;/a&gt; and &lt;a href=&quot;https://physics.stackexchange.com/a/256472/307551&quot;&gt;this physics.se answer&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="17617" LastEditorUserId="17617" LastEditDate="2022-12-10T19:23:37.103" LastActivityDate="2022-12-10T19:23:37.103" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14987" PostTypeId="1" CreationDate="2022-12-05T19:21:03.343" Score="0" ViewCount="107" Body="&lt;p&gt;Bachelors' unknotting is a way to show that all tame knots are isotopic to the unknot, by tightening a knot to a point. Why is it called 'bachelors' unknotting'?&lt;/p&gt;&#xA;" OwnerUserId="4995" LastEditorUserId="14043" LastEditDate="2022-12-06T19:16:20.750" LastActivityDate="2022-12-06T19:16:20.750" Title="Why is bachelors' unknotting called as such and who discovered it?" Tags="|geometry|differential-geometry|topology|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="14989" PostTypeId="1" CreationDate="2022-12-05T15:28:19.400" Score="0" ViewCount="139" Body="&lt;p&gt;To find magnetic field due to current conducting &lt;strong&gt;straight wire&lt;/strong&gt; we have &lt;strong&gt;Maxwell's Right hand thumb rule&lt;/strong&gt;, Which says &lt;em&gt;&lt;strong&gt;&amp;quot;Put your right hand thumb in the direction of current then curled fingers shows the direction of field&amp;quot;&lt;/strong&gt;&lt;/em&gt; I can see that this can be used for a current conducting loop but there is another version for a loop.&lt;/p&gt;&#xA;&lt;p&gt;To find magnetic field due to a current conducting &lt;strong&gt;loop&lt;/strong&gt; is given by &lt;strong&gt;another right hand thumb rule.&lt;/strong&gt; The rule says &lt;em&gt;&lt;strong&gt;&amp;quot;curl your fingers of right hand along the direction of current then thumb gives the direction of field&amp;quot;&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;My question is that who gave the second rule. Is it given by Maxwell himself or someone else? Is it modified version of first stated rule? Who should I credit for the second rule?&lt;/p&gt;&#xA;" OwnerUserId="17618" OwnerDisplayName="Level1" LastActivityDate="2023-05-07T16:06:25.823" Title="Who gave right hand thumb rule for circular loop of current conducting wire?" Tags="|electromagnetism|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14993" PostTypeId="1" CreationDate="2022-12-07T15:49:17.453" Score="1" ViewCount="69" Body="&lt;p&gt;In my experience, Dirac is most often cited as the origin of this thought experiment. However, from what I've read in his &lt;em&gt;Principles of Quantum Mechanics&lt;/em&gt;, he never actually introduces the idea with &lt;em&gt;multiple polarizers&lt;/em&gt;, only with a single crystal of tourmaline.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt; (1930), chapter 1, Dirac introduces a now-standard motivating example for quantum superposition and uncertainty, an experiment using the polarization of light (hypothesized as composed of photons in the development) at oblique angles passing through polarizers (e.g. a crystal of tourmaline). &lt;a href=&quot;https://www.informationphilosopher.com/solutions/experiments/dirac_3-polarizers/&quot; rel=&quot;nofollow noreferrer&quot;&gt;This description&lt;/a&gt;, with block quotes from Dirac, seems like a good exposition. The same site has &lt;a href=&quot;https://www.informationphilosopher.com/solutions/scientists/dirac/chapter_1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;a reproduction of the first chapter&lt;/a&gt;. But Dirac's actual development never introduces an oblique polarizer between two other polarizers (one vertical, one horizontal). For instance:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let us take a definite case. Suppose we have a beam of light passing through a crystal of tourmaline, which has the property of letting through only light plane-polarized perpendicular to its optic axis. Classical electrodynamics tells us what will happen for any given polarization of the incident beam. If this beam is polarized perpendicular to the optic axis, it will all go through the crystal; if parallel to the axis, none of it will go through; while if polarized at an angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; to the axis, a fraction &lt;span class=&quot;math-container&quot;&gt;$\sin2\alpha$&lt;/span&gt; will go through. How are we to understand these results on a photon basis?&lt;/p&gt;&#xA;&lt;p&gt;A beam that is plane-polarized in a certain direction is to be pictured as made up of photons each plane-polarized in that direction. This picture leads to no difficulty in the cases when our incident beam is polarized perpendicular or parallel to the optic axis. We merely have to suppose that each photon polarized perpendicular to the axis passes unhindered and unchanged through the crystal, while each photon polarized parallel to the axis is stopped and absorbed. A difficulty arises, however, in the case of the obliquely polarized incident beam. Each of the incident photons is then obliquely polarized and it is not clear what will happen to such a photon when it reaches the tourmaline.&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;In our present example the obvious experiment is to use an incident beam consisting of only a single photon and to observe what appears on the back side of the crystal. According to quantum mechanics the result of this experiment will be that sometimes one will find a whole photon, of energy equal to the energy of the incident photon, on the back side and other times one will find nothing. When one finds a whole photon, it will be polarized perpendicular to the optic axis. One will never find only a part of a photon on the back side. If one repeats the experiment a large number of times, one will find the photon on the back side in a fraction &lt;span class=&quot;math-container&quot;&gt;$\sin 2\alpha$&lt;/span&gt; of the total number of times. Thus we may say that the photon has a probability &lt;span class=&quot;math-container&quot;&gt;$\sin 2\alpha$&lt;/span&gt; of passing through the tourmaline and appearing on the back side polarized perpendicular to the axis and a probability &lt;span class=&quot;math-container&quot;&gt;$\cos 2\alpha$&lt;/span&gt; of being absorbed. These values for the probabilities lead to the correct classical results for an incident beam containing a large number of photons.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In fact, it appears as though Dirac's thought experiment serves only as a &lt;em&gt;reconciliation&lt;/em&gt; of quantum mechanics with the classical theory of light, rather than as a &lt;em&gt;motivating example&lt;/em&gt; of quantum mechanics, as it is presented e.g. in the website linked, among other references (if this claim is dubious I can provide examples). The three polarizer experiment, on the other hand, &lt;em&gt;does&lt;/em&gt; motivate a quantum mechanical description, and hence is very different from Dirac's, though it can be deduced without much trouble.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;When was this polarized photon description first stated in the familiar &amp;quot;three polarizers&amp;quot; format? Or, what are the earliest examples that were influential in quantum mechanics pedagogy?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="9528" LastEditDate="2022-12-08T17:17:44.170" LastActivityDate="2022-12-08T17:17:44.170" Title="Who first stated the three polarizers experiment in quantum mechanics?" Tags="|quantum-mechanics|light|dirac|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="14995" PostTypeId="1" CreationDate="2022-12-07T22:04:11.147" Score="2" ViewCount="87" Body="&lt;p&gt;How was probability studied or understood before Kolmogorov axiomatized it?&lt;/p&gt;&#xA;&lt;p&gt;I mean Bernoulli and Gauss had already started inventing and using distributions way before Kolmogorov came into the scene. There has also been measure theory in the making becoming the foundation of all probability theory.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, in math pedagogy, we start with Kolmogorov's axioms and build our way to the normal distribution which is very ahistorical.&lt;/p&gt;&#xA;" OwnerUserId="17624" LastEditorUserId="10106" LastEditDate="2022-12-08T04:32:31.217" LastActivityDate="2022-12-08T04:32:31.217" Title="What was probability like before Kolmogorov?" Tags="|mathematics|probability|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14996" PostTypeId="1" CreationDate="2022-12-08T03:58:44.340" Score="0" ViewCount="35" Body="&lt;p&gt;The textbook &lt;a href=&quot;https://www.amazon.ca/Introduction-Lattices-Order-B-Davey/dp/0521784514&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction to Lattices and Order&lt;/a&gt; by Priestly and Davey mentions in its first edition preface:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In a proselytizing article, Order: a. theory with a. view [in Kla.ssifikation und Ord:nung, INDEKS, Frankfurt, 1989], Ivan Rival discusses tbe modern role of order.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I don't see this article&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;on his &lt;a href=&quot;https://en.wikipedia.org/wiki/Ivan_Rival&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;on his &lt;a href=&quot;https://scholar.google.com/citations?user=-WDlg78AAAAJ&amp;amp;hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Scholar&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;on &lt;a href=&quot;https://www.google.com/search?channel=fs&amp;amp;client=ubuntu&amp;amp;q=%22Order%3A+A+Theory+with+a+View%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;or on &lt;a href=&quot;https://ivanrival.com/math_publications.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this page&lt;/a&gt; dedicated to his article.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Did this article exist but never make it into a modern repository? Or is it misnamed in this reference, going by a different name among Rival's articles?&lt;/p&gt;&#xA;" OwnerUserId="15025" LastActivityDate="2022-12-08T03:58:44.340" Title="Were Priestly and Davey wrong about about &quot;Order: A Theory with A View&quot; being written by Ivan Rival?" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="14997" PostTypeId="2" ParentId="5656" CreationDate="2022-12-08T05:06:31.513" Score="3" Body="&lt;p&gt;Don E.  Knuth and some of his co-authors don't write 2 | 8 but 2 \ 8. If I understand correctly, their concerns regarding the a | b notation are not totally unrelated to those already mentioned by SearchSpace:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot; The notation &lt;span class=&quot;math-container&quot;&gt;$m \mid n$&lt;/span&gt; is actually much more common than &lt;span class=&quot;math-container&quot;&gt;$m \backslash n$&lt;/span&gt; in current mathematics literature. But vertical lines&#xA;are overused--for absolute values, set delimiters, conditional&#xA;probabilities, etc.--and backward slashes are underused. Moreover,  &lt;span class=&quot;math-container&quot;&gt;$m\backslash n$&lt;/span&gt; gives an impression that &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is the denominator of an&#xA;implied ratio. So we shall boldly let our divisibility symbol lean&#xA;leftward&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;(Cf. R. L. Graham, D. E. Knuth, and O. Patashnik. Concrete mathematics: a foundation of computer science, 2nd ed. Addison-Wesley Publishing Company, 1994, p. 102.)&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2022-12-08T05:13:36.283" LastActivityDate="2022-12-08T05:13:36.283" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="14999" PostTypeId="1" AcceptedAnswerId="15005" CreationDate="2022-12-08T11:26:17.700" Score="1" ViewCount="244" Body="&lt;p&gt;I am interested in knowing what is the original source/author of the following identities:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\tan{\frac{\alpha}{2}}\tan{\frac{\beta}{2}}+\tan{\frac{\alpha}{2}}\tan{\frac{\gamma}{2}}+\tan{\frac{\beta}{2}}\tan{\frac{\gamma}{2}}=1\tag{1}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\cot{\frac{\alpha}{2}}\cot{\frac{\beta}{2}}\cot{\frac{\gamma}{2}}=\cot{\frac{\alpha}{2}}+\cot{\frac{\beta}{2}}+\cot{\frac{\gamma}{2}}\tag{2}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Where &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; are the angles of a triangle.&lt;/p&gt;&#xA;&lt;p&gt;I ask this question because &lt;a href=&quot;https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Further_%22conditional%22_identities_for_the_case_%CE%B1_+_%CE%B2_+_%CE%B3_=_180%C2%B0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; does not cite sources or mention the original author.&lt;/p&gt;&#xA;" OwnerUserId="16769" LastEditorUserId="16769" LastEditDate="2022-12-09T13:22:15.927" LastActivityDate="2022-12-10T11:57:02.643" Title="Original source of these 2 trigonometric identities" Tags="|mathematics|trigonometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15000" PostTypeId="2" ParentId="14989" CreationDate="2022-12-08T13:28:13.700" Score="2" Body="&lt;p&gt;Right-hand rule was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century.&lt;/p&gt;&#xA;" OwnerUserId="17624" LastActivityDate="2022-12-08T13:28:13.700" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15001" PostTypeId="2" ParentId="14977" CreationDate="2022-12-08T13:42:32.183" Score="1" Body="&lt;p&gt;This is because there were many requirements to define what a 'nice' function means. The multivariable case for ensuring continuity is a little more difficult than the single variable case. A function with continuous mixed partials is enough to ensure that mixed partials exist and are equal.&lt;/p&gt;&#xA;&lt;p&gt;here are the cases highlighted that fail taken from the calculus wiki site linked below&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Failure of Clairaut's theorem where both mixed partials are defined&#xA;but not equal&lt;/li&gt;&#xA;&lt;li&gt;Failure of Clairaut's theorem where only one of the mixed partials is defined&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://calculus.subwiki.org/wiki/Failure_of_Clairaut%27s_theorem_where_both_mixed_partials_are_defined_but_not_equal&quot; rel=&quot;nofollow noreferrer&quot;&gt;Calculus wiki site&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17624" LastActivityDate="2022-12-08T13:42:32.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15002" PostTypeId="1" AcceptedAnswerId="15003" CreationDate="2022-12-08T14:14:39.303" Score="3" ViewCount="156" Body="&lt;p&gt;I was writing a paper on the basics of calculus, and of course the study of velocities plays a big part in that.  In introducing the problem statement, I started with a classic word problem, &amp;quot;Alice gets in her car and drives 100 miles at a constant speed.  It takes her two hours to do so.  How fast did she travel?&amp;quot;  This is the kind of word problem shows up very early on in our education.  It's quite a simple problem.&lt;/p&gt;&#xA;&lt;p&gt;I got curious how far back in history this idea of &amp;quot;speed&amp;quot; went.  I figured I'd find a story about one of the famous Greek philosophers, but Wikipedia says &lt;a href=&quot;https://en.wikipedia.org/wiki/Speed&quot; rel=&quot;nofollow noreferrer&quot;&gt;otherwise&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, that is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$v=\frac d t$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is speed, &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; is distance, and &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; is time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This surprised me a bit.  I didn't expect our definition to have come about that recently.  The statement is cited as coming from a page in a book I do not own (Hewitt (2006), p. 42), so I can't check the specifics.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that any decent civil planner would have a sense of how long it takes to travel between cities by foot, by horse, by chariot, etc.  I feel the Greeks would have had such concepts, and in hindsight, knowing the definition attributed to Galileo, it seems like this would be a very natural way to capture this data.&lt;/p&gt;&#xA;&lt;p&gt;So, before Galileo's work, how was the concept of speed treated?  Is the only thing &amp;quot;new&amp;quot; about Galileo's version the fact that we put it into an equation form?  Or does the history of that concept really that new?&lt;/p&gt;&#xA;" OwnerUserId="5470" LastActivityDate="2022-12-10T15:19:22.973" Title="History of Speed - is it really new?" Tags="|calculus|ancient-greece|" AnswerCount="2" CommentCount="5" ClosedDate="2022-12-13T14:26:12.280" ContentLicense="CC BY-SA 4.0" />
  <row Id="15003" PostTypeId="2" ParentId="15002" CreationDate="2022-12-08T16:36:29.410" Score="3" Body="&lt;p&gt;Are you asking about the concept of speed in general or the equational relationship?  I can highlight a few developments.&lt;/p&gt;&#xA;&lt;p&gt;The concept of speed was debated by the Greeks.  For instance, Zeno of Elea presented two famous paradoxes regarding speed, and Aristotle addresses them in his &lt;em&gt;Physics&lt;/em&gt;.  From Wiki:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Achilles and the Tortise&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Achilles is in a footrace with the tortoise. Achilles allows the&#xA;tortoise a head start of 100 meters, for example. Suppose that each&#xA;racer starts running at some constant speed, one faster than the&#xA;other. After some finite time, Achilles will have run 100 meters,&#xA;bringing him to the tortoise's starting point. During this time, the&#xA;tortoise has run a much shorter distance, say 2 meters. It will then&#xA;take Achilles some further time to run that distance, by which time&#xA;the tortoise will have advanced farther; and then more time still to&#xA;reach this third point, while the tortoise moves ahead. Thus, whenever&#xA;Achilles arrives somewhere the tortoise has been, he still has some&#xA;distance to go before he can even reach the tortoise. As Aristotle&#xA;noted, this argument is similar to the Dichotomy. It lacks, however,&#xA;the apparent conclusion of motionlessness.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Arrow Paradox&lt;/em&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Zeno states that for motion to occur, an object must change the&#xA;position which it occupies. He gives an example of an arrow in flight.&#xA;He states that at any one (duration-less) instant of time, the arrow&#xA;is neither moving to where it is, nor to where it is not. It&#xA;cannot move to where it is not, because no time elapses for it to move&#xA;there; it cannot move to where it is, because it is already there. In&#xA;other words, at every instant of time there is no motion occurring. If&#xA;everything is motionless at every instant, and time is entirely&#xA;composed of instants, then motion is impossible.&lt;/p&gt;&#xA;&lt;p&gt;Whereas the [Achilles and the Tortise paradox] divide[s] space, this paradox starts by dividing time—and not into segments, but into points.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These are interesting paradoxes to consider, especially in the context of calculus.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Zeno%27s_paradoxes&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;According to science and math historian Marshal Clagett, &lt;strong&gt;Aristotle &amp;quot;is the first one whose treatise we have to give us rules for comparing the speeds of bodies in terms of the space traversed in given times&lt;/strong&gt;...we can derive from [four inequalities between space, time, and speed] our general expression V=S/T once we have decided that velocity can be represented as a ratio.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://aapt.scitation.org/doi/abs/10.1119/1.1969312?journalCode=ajp&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://aapt.scitation.org/doi/abs/10.1119/1.1969312?journalCode=ajp&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This is probably more what you're looking for.&lt;/p&gt;&#xA;&lt;p&gt;Also, it's worth noting that Aristotle proposed that the speed at which two identically shaped objects sink or fall is directly proportional to their weights and inversely proportional to the density of the medium through which they move.  This was a major point of contention for Galileo as he showed this proposition to be false.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In the Third Day: Uniform Motion of &lt;em&gt;Two New Sciences&lt;/em&gt;, Galileo restates the definition of uniform speed: &amp;quot;distances traversed by the moving particle during any equal intervals of time, are themselves equal.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;He further adds the condition &amp;quot;the word 'any,' meaning by this, &lt;em&gt;all equal intervals of time&lt;/em&gt;; for it may happen that the moving body will traverse equal distances during some equal intervals of time and yet the distances traversed during some small portion of these time-intervals may not be equal, even though the time-intervals be equal.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;From there, he states that the definition of uniform speed requires Axioms I-IV (the inequalities mentioned earlier), then uses them to construct ratios of speed, distance, and time in Theorems I-III, and then shows the product (i.e., compound) of these ratios in Theorems IV-VI.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://oll.libertyfund.org/title/galilei-dialogues-concerning-two-new-sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://oll.libertyfund.org/title/galilei-dialogues-concerning-two-new-sciences&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Although Galileo famously presents his version of the mean value theorem for uniform acceleration, the mean value theorem was actually demonstrated by the Oxford Calculators in the 14th century, and was later proved by Nicole Oresme.  The theorem is also known as the &amp;quot;Merton Rule&amp;quot;: &lt;a href=&quot;https://en.wikipedia.org/wiki/Mean_speed_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Mean_speed_theorem&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Lastly, I should note that, in all the developments above, the arguments were presented using geometry and not algebra.  The equation V=S/T would be objectionable to everyone mentioned above because dividing one magnitude (distance) by a different magnitude (time) to produce a third magnitude (speed) was considered absurd -- it would be analogous to dividing angles by areas or arc lengths by volumes.  Instead, in Euclidean geometry, like magnitudes were compared to like magnitudes through ratios.  For example, Galileo presents the relationship V1/V2 = (S1/S2)x(T2/T1) where &amp;quot;/&amp;quot; signifies ratios and not algebraic division, and &amp;quot;x&amp;quot; signifies a compounding of ratios and not a multiplication or product.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2022-12-08T17:05:49.353" LastActivityDate="2022-12-08T17:05:49.353" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15004" PostTypeId="1" CreationDate="2022-12-09T13:08:38.630" Score="2" ViewCount="577" Body="&lt;p&gt;We teach our children in school that 4 is written in Roman numerals as IV and not as IIII but at the Colosseum in Rome, gate 44 is identified as XLIIII and not as XLIV.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Ya8ch.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Ya8ch.jpg&quot; alt=&quot;Colosseum gate 44&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;When did the change from IIII to IV take place?  Why did the change take place?  Any references are welcome.&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="72" LastEditDate="2022-12-23T13:39:58.850" LastActivityDate="2022-12-24T20:10:14.520" Title="Gate 44 at the Colosseum in Rome: XLIIII or XLIV? When and why the change?" Tags="|notation|numbers|roman-empire|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15005" PostTypeId="2" ParentId="14999" CreationDate="2022-12-10T09:53:17.327" Score="5" Body="&lt;p&gt;After hours of searching through scans of books from the 18&lt;sup&gt;th&lt;/sup&gt; and 19&lt;sup&gt;th&lt;/sup&gt; centuries in English, German, French, Italian, Spanish, and Latin I managed to find &lt;em&gt;a&lt;/em&gt; source for each of the two identities.&lt;/p&gt;&#xA;&lt;p&gt;The first one appeared in a list of trigonometric identities for the oblique triangle in a book by the Belgian mathematician Émile Gelin, &lt;em&gt;Éléments de Trigonométrie Plane et Sphérique.&lt;/em&gt; Namur: Wesmael-Charlier 1888, &lt;a href=&quot;https://books.google.com/books?id=1Qg5iGbPfLsC&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;dq=trigonometrie&amp;amp;pg=PA106#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 106&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[151] [...] &lt;span class=&quot;math-container&quot;&gt;$\mathrm{tg}\frac{1}{2}\mathrm{A} \ \mathrm{tg}\frac{1}{2}\mathrm{B} + \mathrm{tg}\frac{1}{2}\mathrm{B} \ \mathrm{tg}\frac{1}{2}\mathrm{C} + \mathrm{tg}\frac{1}{2}\mathrm{C} \ \mathrm{tg}\frac{1}{2}\mathrm{A} = 1;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The second one appeared as an exercise in Joseph Alfred Serre, &lt;em&gt;Traité de Trigonometrié.&lt;/em&gt; Paris: Bachelier 1850, &lt;a href=&quot;https://archive.org/details/traitdetrigonom02serrgoog/page/n81/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 67&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; Démontrer que si &lt;span class=&quot;math-container&quot;&gt;$a,b,c$&lt;/span&gt; sont trois arcs dont la somme est égale à &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;, on a:&lt;br /&gt;&#xA;1°. &lt;span class=&quot;math-container&quot;&gt;$\mathrm{cot}\frac{a}{2} + \mathrm{cot}\frac{b}{2} + \mathrm{cot}\frac{c}{2} = \mathrm{cot}\frac{a}{2} \ \mathrm{cot}\frac{b}{2} \ \mathrm{cot}\frac{c}{2};$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think it is unlikely that these are the first (original) instances of these two identities in the literature. The inscribed circle of a triangle is already found in Euclid IV. 4. In the first century C.E., Heron of Alexandria already showed that the area of the triangle is &lt;span class=&quot;math-container&quot;&gt;$\sqrt{s(s – a)(s – b)(s – c)}$&lt;/span&gt;, where the semi-perimeter &lt;span class=&quot;math-container&quot;&gt;$s = \frac{a+b+c}{2}$&lt;/span&gt;. As far a I know, the ancient Greeks already had the notion of a (co-)tangent, as the length of a shadow cast by a gnomon hit by the sun at a particular angle.&lt;/p&gt;&#xA;&lt;p&gt;I do not know who first worked out that the radius of the inscribed circle is &lt;span class=&quot;math-container&quot;&gt;$r = \sqrt{\frac{(s – a)(s – b)(s – c)}{s}}$&lt;/span&gt; and that &lt;span class=&quot;math-container&quot;&gt;$\tan\frac{A}{2}=\frac{r}{s-a}, \tan\frac{B}{2}=\frac{r}{s-b}, \tan\frac{C}{2}=\frac{r}{s-c}$&lt;/span&gt;. With modest effort this might lead one to the first formula, for example. I would be very surprised if this did not happen long before the 19&lt;sup&gt;th&lt;/sup&gt; century.&lt;/p&gt;&#xA;&lt;p&gt;Thinking up trigonometric identities for the oblique triangle for use in exercises and exams or as mathematical puzzles for certain journals appears to have been a favorite pastime of 19&lt;sup&gt;th&lt;/sup&gt; century mathematicians (or at least this is the impression I got from my lengthy perusal of the literature) and this might be the reason such identities are most readily found in books from that time period.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-12-10T11:57:02.643" LastActivityDate="2022-12-10T11:57:02.643" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15006" PostTypeId="1" CreationDate="2022-12-10T14:46:56.713" Score="2" ViewCount="103" Body="&lt;p&gt;At the 1912 International Congress of Mathematicians, Edmund Landau listed &lt;a href=&quot;https://en.wikipedia.org/wiki/Landau%27s_problems&quot; rel=&quot;nofollow noreferrer&quot;&gt;four basic problems about prime numbers&lt;/a&gt;: Goldbach's conjecture, the twin prime conjecture, Legendre's conjecture (that there always exists a prime number between consecutive perfect squares), and whether there are infinitely many prime numbers of the form &lt;span class=&quot;math-container&quot;&gt;$n^2 + 1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Although I had heard of each of these conjectures individually, I only recently learned of the existence of Landau's list, from a &lt;a href=&quot;https://mathoverflow.net/q/436257/&quot;&gt;MathOverflow question&lt;/a&gt;.  My question, broadly speaking, is how famous and/or influential was Landau's list?  To be more specific:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are some number theory textbooks or research papers that cite Landau's list by name, or credit him with popularizing these particular conjectures?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2022-12-10T14:46:56.713" Title="Influence of Edmund Landau's list of four problems" Tags="|mathematics|number-theory|open-problems|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15007" PostTypeId="2" ParentId="15002" CreationDate="2022-12-10T15:19:22.973" Score="2" Body="&lt;p&gt;This discussion comes from confusion of two notions: what we call now the average speed and the momentary speed. The concept of average speed (distance divided by time) is very old indeed. For example, it was used by sailors in &lt;a href=&quot;https://en.wikipedia.org/wiki/Dead_reckoning&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dead reckoning&lt;/a&gt; since the times immemorial, and in general to estimate time of travel. The concept of momentary speed (the limit of the above ratio) was only made clear in the 17 century. Zeno'a paradoxes show how the ancient Greeks struggled with this notion.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-12-10T15:19:22.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15008" PostTypeId="1" CreationDate="2022-12-11T09:20:21.050" Score="8" ViewCount="2059" Body="&lt;p&gt;I have wondered if the apparently complex arrangement of the sails of early 19th century vessels was reflective of a lack of understanding of the optimal arrangement of such sails (since modern sailing yachts lack so many sails) or if indeed for tight and fast maneuvering, this arrangement, perhaps improved today using modern engineering and computers, was needed.&lt;/p&gt;&#xA;&lt;p&gt;And was it the captain who was the expert at coordinating the actions of a bunch of sailors who specialized in their own sails? Or was the best &amp;quot;sailor&amp;quot; someone else? Could it be argued that maneuvering a sailing vessel in battle is a lost art?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-12-12T18:24:36.010" Title="How were sailing warships maneuvered in battle -- who coordinated the actions of all the sailors?" Tags="|technology|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15009" PostTypeId="2" ParentId="15008" CreationDate="2022-12-11T12:37:08.177" Score="-1" Body="&lt;p&gt;For battle you need oars, not sails.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tsSM7.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tsSM7.png&quot; alt=&quot;Greek trireme&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This replica Greek trireme has three decks of oarsmen.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2022-12-11T12:37:08.177" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15011" PostTypeId="2" ParentId="15008" CreationDate="2022-12-11T18:26:02.953" Score="11" Body="&lt;p&gt;Gerald Edgar's answer is wrong since the question asks about sailing warships in the age of sail and specifically the early 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Rowing galleys were still used occassionally in the early 19th century in a few places in the Mediterranean and the Baltic I think.  One of the few examples of a ship with oars as well as sails being sent on a long voyage in the ocean was the &lt;em&gt;Adventure Galley&lt;/em&gt; of Captain Kidd's pirate hunting expedition in 1695-1698, basically a sailing ship with oars.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Adventure_Galley&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Adventure_Galley&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;War galleys would have very simple sail arrangements.&lt;/p&gt;&#xA;&lt;p&gt;The question asks about very complex sail arrangements and so is talking about the sails of full rigged sailing warships.&lt;/p&gt;&#xA;&lt;p&gt;And I am not an expert on sails.  But anyone who has seen pictures of full rigged ships has seen how complicated the rigging was and understands it would take a considerable time for experienced sailors to &amp;quot;show the ropes&amp;quot; to new sailors, let alone for the new sailors to understand the uses of the different ropes.&lt;/p&gt;&#xA;&lt;p&gt;On a sailing warship most of the crew would be manning the guns in battle, but a considerable proportion of the crew would be required to control the rigging and sails for the ship to maneuver.  Sometimes the ship could maneuver by simply having one, two or more men turn the steering wheel, but often maneuvering required adjusting the sails in various ways.&lt;/p&gt;&#xA;&lt;p&gt;I was once reading about the various ways to adjust the sails to maneuver a full rigged ship, which could be surprisingly precise.  It was claimed that the sails on one mast could be adjusted to drive the ship forward while the sails on another mast could adjusted to drive the ship backwards, resulting in the ship staying in one place.&lt;/p&gt;&#xA;&lt;p&gt;I just looked it up and that is called &amp;quot;heaving to&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Heaving_to&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Heaving_to&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I note that sailing warshps had a position called &amp;quot;master&amp;quot; or &amp;quot;sailing master&amp;quot;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The master, or sailing master, is a historical rank for a naval officer trained in and responsible for the navigation of a sailing vessel. The rank can be equated to a professional seaman and specialist in navigation, rather than as a military commander.&lt;/p&gt;&#xA;&lt;p&gt;In the Royal Navy, the master was originally a warrant officer who ranked with, but after, the lieutenants. The rank became a commissioned officer rank and was renamed navigating lieutenant in 1867; the rank gradually fell out of use from around 1890 since all lieutenants were required to pass the same examinations.&lt;/p&gt;&#xA;&lt;p&gt;When the United States Navy was formed in 1794, master was listed as one of the warrant officer ranks and ranked between midshipmen and lieutenants. The rank was also a commissioned officer rank from 1837 until it was replaced with the current rank of lieutenant, junior grade in 1883.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The duties of a master in the royal navy:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The master's main duty was navigation, taking the ship's position at least daily and setting the sails as appropriate for the required course and conditions. During combat, he was stationed on the quarterdeck, next to the captain. The master was responsible for fitting out the ship, and making sure they had all the sailing supplies necessary for the voyage. The master also was in charge of stowing the hold and ensuring the ship was not too weighted down to sail effectively. The master, through his subordinates, hoisted and lowered the anchor, docked and undocked the ship, and inspected the ship daily for problems with the anchors, sails, masts, ropes, or pulleys. Issues were brought to the attention of the master, who would notify the captain. The master was in charge of the entry of parts of the official log such as weather, position, and expenditures.[8][9]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Master_(naval)&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Master_(naval)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;By the time an officer rose to command of a ship he would learn a lot about how to maneuver a sailing so, so between them the commander and the sailing master would have the knowledge to manueuver the ship as desired in battle.&lt;/p&gt;&#xA;&lt;p&gt;And I hope this will do until someone more knowledgeable can answer.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2022-12-11T18:26:02.953" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15013" PostTypeId="1" CreationDate="2022-12-12T03:48:03.537" Score="2" ViewCount="119" Body="&lt;p&gt;I am reading &lt;em&gt;Proofs and Confirmations&lt;/em&gt; by David Bressoud.&#xA;On page &lt;span class=&quot;math-container&quot;&gt;$150$&lt;/span&gt; is a long excerpt by Richard Askey, from &amp;quot;How&#xA;can mathematicians and mathematical historians help each other?&amp;quot;&#xA;There is mentioned&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;R. Narasimhan told me that he found a definition of &amp;quot;comfortable&amp;quot;&#xA;series in one of the late volumes of Euler's collected works. For&#xA;Euler, a comfortable series is a power series whose term ratio&#xA;is a rational function of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. When I asked Narasimhan to give&#xA;me a specific reference, he was unable to find it again. I will&#xA;be very pleased to pay &lt;span class=&quot;math-container&quot;&gt;$\\\$50$&lt;/span&gt; U.S. for this reference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;When I did a Google Search for this using &lt;code&gt;Euler &amp;quot;comfortable series&amp;quot;&lt;/code&gt;,&#xA;only a handful of matches were found. Has this Euler reference been&#xA;recovered yet?&lt;/p&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="643" LastEditDate="2022-12-12T04:39:11.717" LastActivityDate="2022-12-12T04:39:11.717" Title="Euler's &quot;comfortable&quot; series" Tags="|calculus|euler|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15014" PostTypeId="2" ParentId="15008" CreationDate="2022-12-12T09:24:33.953" Score="4" Body="&lt;p&gt;Control of a warship is no different than control of any similar sized sailing ships such as trading galleys or racing yachts. The main difference is racing yacht crews are trained and have lots of practice to respond to orders quickly and similarly warship crews are trained to respond quickly to orders and perform and anticipate very quick maneuvers all while under fire.&lt;/p&gt;&#xA;&lt;p&gt;In terms of actually controlling the boat, the crew have a hierarchy of command. This hierarchy is similar for commercial civilian ships and military warships:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Captain&lt;/strong&gt; - The head of this command and the person who ultimately makes all the decisions of what maneuver to make at what time is of course the captain. The skill of the captain (seamanship) and his familiarity with the ship and the crew is critical in deciding what the ship should do at what time. When the captain decides something must be done he will give orders to his officers who in turn will ensure that the orders will be carried out so that the ship will maneuver the way the captain wants. Orders at this level are fairly high level such as &amp;quot;full sail&amp;quot; or &amp;quot;tack starboard&amp;quot; or &amp;quot;bring guns to bear&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Commissioned Officers&lt;/strong&gt; - In office-speak these are the C-level executives (eg. CTO, CFO etc.) or upper management. They are not themselves responsible for managing the ship's crews but instead they manage the middle managers. These people often hold the rank of Lieutenant in the military. Commands at this level are often a bit more detailed such as &amp;quot;steer to bearing 120 on my mark&amp;quot; or &amp;quot;unfurl the mizzen sail&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Petty Officers&lt;/strong&gt; - In office speak these are the middle managers or senior executives. They directly manage crew under them in order to perform tasks so that the captain's orders can be fulfilled. In terms of maneuvering there are two types of petty officers that are of interest: the &lt;strong&gt;coxwain&lt;/strong&gt; who is responsible for the steering the ship with the rudder and the &lt;strong&gt;boatswain&lt;/strong&gt; (also sometimes called &amp;quot;bosun&amp;quot; which is just &amp;quot;boatswain&amp;quot; in a heavy dialect) who is responsible for managing the sails and rigging. (note that the coxwain is also responsible for a type of rigging: the ropes that transfer the control of the steering wheel to the rudder - he is responsible for managing the crew that maintain this system because he is responsible for steering). The boatswain and coxwain are generally the highest ranking petty officers on the boat but on bigger boats there can also be other petty officers working under or with the boatswain to manage the crew. These people give direct orders to the crew to get things done.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Able-bodied Seamen&lt;/strong&gt; - These are your experienced sailors and are sometimes experts in a specific job on deck such as knowing how to roll and unfurl a specific sail quickly or have a lot of experience climbing a mast etc. While the petty officers generally directly manage all crew under them these experts often also take a leadership role as tutors and team leaders making sure less experienced sailors know what to do.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;strong&gt;Ordinary Seamen&lt;/strong&gt; - These are your regular sailors who actually do the job of pulling ropes and releasing ropes and handle the sail cloth etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Through the chain of command above the captain is able to fight a battle with his ship. For example, if the captain decides that the ship needs to turn starboard quickly, bring the guns to bear on the enemy and fire a volley he will relay his plan to the bridge crew. There would then be a flurry of activity to get the ship into the configuration the captain has ordered: the lieutenants will shout their commands to their petty officers. The petty officers will figure out the best way of doing what the lieutenants have requested and give commands to the teams under them. Then each team will do what they have been told.&lt;/p&gt;&#xA;&lt;p&gt;With practice a warship can be made to respond very quickly to the captain's orders.&lt;/p&gt;&#xA;" OwnerUserId="3092" LastEditorUserId="3092" LastEditDate="2022-12-12T18:24:36.010" LastActivityDate="2022-12-12T18:24:36.010" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15015" PostTypeId="1" AcceptedAnswerId="15016" CreationDate="2022-12-13T02:06:32.423" Score="13" ViewCount="2253" Body="&lt;p&gt;Whiteside in &amp;quot;The Mathematical Principles underlying Newton's&#xA;Principia Mathematica&amp;quot; wrote&lt;/p&gt;&#xA;&lt;p&gt;.. &lt;em&gt;if we can give credence to an account in The Postboy of 12 April 1731, his tomb in Westminster Abbey in London was once inscribed (on the stone 'parchment' borne proudly by the cherubs who flank his effigy) with an unspecified 'Diagram' and a certain 'converging series'; but these, thanks to the polishers who cleaned the monument twenty years ago, are no longer&#xA;visible to the naked eye&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Have modern historians with their imaging techniques determined what these inscriptions were and/or were these faithfully restored after Whiteside? If so, what are they? See &lt;a href=&quot;https://themathematicaltourist.wordpress.com/2013/09/14/isaac-newtons-tomb/#:%7E:text=Diligent%2C%20sagacious%20and%20faithful%2C%20in,ornament%20of%20the%20human%20race.&quot; rel=&quot;noreferrer&quot;&gt;this blog&lt;/a&gt; for an image of the parchments on which there is clearly a geometric figure.&lt;/p&gt;&#xA;" OwnerUserId="1830" LastEditorUserId="12330" LastEditDate="2022-12-14T06:30:32.850" LastActivityDate="2022-12-14T06:30:32.850" Title="Inscriptions on Newton's tomb" Tags="|biographical-details|newton|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15016" PostTypeId="2" ParentId="15015" CreationDate="2022-12-13T06:17:16.857" Score="19" Body="&lt;p&gt;Another picture of the scroll, but taken from a different angle, appeared in: Greg Priest, Silvia De Toffoli, Paula Findlen, &amp;quot;Tools of Reason: The Practice of Scientific Diagramming from Antiquity to the Present,&amp;quot; &lt;em&gt;Endeavour&lt;/em&gt; &lt;strong&gt;42&lt;/strong&gt; (2018), 49-59. Examining both pictures I can make out parts of the two-line inscription at the top, which reads &lt;span class=&quot;math-container&quot;&gt;$\overline{P + PQ}) ... AQ ... BQ ... CQ$&lt;/span&gt;, with undecipherable squiggly marks in the place of the dots.&lt;/p&gt;&#xA;&lt;p&gt;According to another &lt;a href=&quot;https://hsm.stackexchange.com/a/5021/4201&quot;&gt;answer&lt;/a&gt; on this site, Newton himself wrote the Generalized Binomial Theorem as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\left(P + PQ\right)^{\frac{m}{n}} = P^{\frac{m}{n}} + \frac{m}{n}AQ+\frac{m-n}{2n}BQ + \frac{m-2n}{3n}CQ+\ldots$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So this appears to be a match. Jean G. Dhombres, &amp;quot;A Tentative Interpretation of the Epistemological Significance of the Encrypted Message Sent by Newton to Leibniz in October 1676&amp;quot;, &lt;em&gt;Advances in Historical Studies&lt;/em&gt;, Vol. 3, No. 1, 2014, pp. &lt;a href=&quot;https://www.scirp.org/html/43389.html&quot; rel=&quot;noreferrer&quot;&gt;22-32&lt;/a&gt;, notes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;15&lt;/sup&gt; Newton did not use parenthesis for the binomial P + PQ, but he drew a line above the whole binomial, ended by a descending vertical line.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for the diagram taking up the central part of the scroll, it appears to match well with the diagram that appears in &lt;em&gt;Principia&lt;/em&gt;, &lt;a href=&quot;https://www.google.com/books/edition/Newton_s_Principia/N-hHAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA116&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Proposition XI. Problem VI&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If a body revolves in an ellipsis; it is required to find the law of the centripetal force tending to the focus of the ellipsis.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;David Brewster, &lt;em&gt;The Life of Sir Isaac Newton&lt;/em&gt;,  New York: J. &amp;amp; J. Harper 1831, &lt;a href=&quot;https://www.google.com/books/edition/The_Life_of_Sir_Isaac_Newton/A0RWAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA289&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 289&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On the sarcophagus is placed the figure of Sir Isaac in a cumbent posture, with his elbow resting on several of his works. Two youth stand before him with a scroll, on which is drawn a remarkable diagram relative to the solar system, and above that is a converging series.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some seven decades later Cajori could not find any solid evidence of an inscription on the scroll: Florian Cajori, &amp;quot;Was the binomial theorem engraven on Newton's monument?&amp;quot; &lt;em&gt;Bulletin of the American Mathematical Society&lt;/em&gt;, Vol. 1, No. 2, 1900, pp. &lt;a href=&quot;https://projecteuclid.org/journalArticle/Download?urlId=bams%2F1183414268&quot; rel=&quot;noreferrer&quot;&gt;52-54&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first step naturally is to question the monument, but we do so in vain. Says Dr. Granville, the present Dean of Westminster, in a letter to the writer: &amp;quot;In front of the half-recumbent figure of Sir Isaac Newton are two winged youths holding a small scroll in which are still, according to Neale, some mathematical figures. ... I fear that the figures on the small marble scroll are &lt;em&gt;quite obliterated&lt;/em&gt;. A mathematical friend mounted the monument for me. Time, I fear, and London atmosphere have done their sad work, and the older guide-books, Dart and Neale, while they carefully copy other inscriptions, naturally do not preserve for us mathematical formulæ.&amp;quot; A friend of the writer of this article reports, &amp;quot;I looked as close as a chair would permit and saw no sign of engraving on the scroll.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on these observations, Cajoli concluded:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus it appears pretty conclusively that there is no more foundation for the statement that the Binomial Theorem was inscribed on Newton's tomb or monument than there was authority for the story of the &amp;quot;apple&amp;quot; ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The reason that modern photographs show a pristine looking diagram and inscription on the scroll is explained authoritatively on the website of &lt;a href=&quot;https://www.westminster-abbey.org/abbey-commemorations/commemorations/sir-isaac-newton&quot; rel=&quot;noreferrer&quot;&gt;Westminster Cathedral&lt;/a&gt; (my bolding):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Above the sarcophagus is a reclining figure of Newton, in classical costume, his right elbow resting on several books representing his great works. They are labelled (on the fore-edges) 'Divinity', 'Chronology', 'Opticks' [1704] and 'Philo. Prin. Math' [Philosophiae Naturalis Principia Mathematica, 1686-7)]. With his left hand he points to a scroll with a mathematical design shown on it (the 'converging series'), held by two standing winged boys. &lt;strong&gt;The painting on this scroll had been erased or cleaned off in the early 19th century and was re-painted in 1977 from details in Newton's manuscripts&lt;/strong&gt;. &lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-12-13T06:56:12.333" LastActivityDate="2022-12-13T06:56:12.333" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15018" PostTypeId="1" AcceptedAnswerId="15023" CreationDate="2022-12-15T01:10:07.280" Score="2" ViewCount="306" Body="&lt;p&gt;I've Googled this question in several different ways and I get a lot of hits, but nothing answering the question.&lt;/p&gt;&#xA;&lt;p&gt;To flesh out the subject line a little, I'm interested in understanding where, in the big scheme of things, Kolmogorov's 1933 work stands. Was it just a &lt;span class=&quot;math-container&quot;&gt;``&lt;/span&gt;very a nice to have a sound mathematical basis for probability'' sort of thing, or did that sound basis allow for developments in statistics that likely could not have come about without it?&lt;/p&gt;&#xA;" OwnerUserId="15921" LastEditorUserId="15921" LastEditDate="2022-12-15T07:25:37.657" LastActivityDate="2022-12-16T16:49:50.917" Title="What advances in statistics did measure-theoretic probability theory allow?" Tags="|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15020" PostTypeId="1" CreationDate="2022-12-16T01:43:06.093" Score="1" ViewCount="43" Body="&lt;p&gt;Its my understanding that before metric units of length were based on the speed of light, there were reference bars that were used to calibrate other (mostly calibration) equipment.&lt;/p&gt;&#xA;&lt;p&gt;How were these units accurately subdivided into centimeters, millimeters, ect?&lt;/p&gt;&#xA;&lt;p&gt;In other words, given a bar of length X, how do you derive sub-lengths of that bar that are precisely 1/100 or 1/1000 of that bar?&lt;/p&gt;&#xA;" OwnerUserId="17671" LastEditorUserId="17671" LastEditDate="2022-12-16T01:43:47.027" LastActivityDate="2022-12-16T01:43:47.027" Title="Before metric length units were based on the speed of light, how were sub-units accurately derived from the reference bars?" Tags="|engineering|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15021" PostTypeId="1" CreationDate="2022-12-16T04:06:01.220" Score="3" ViewCount="170" Body="&lt;p&gt;Cavendish first isolated hydrogen and recognized it for what it was; Lavoisier realized that water consisted of hydrogen and oxygen; Dalton used hydrogen as the basis for relative atomic weights by giving it the value of 1 and assuming that the atomic weight of all other elements would be, in relative terms, simple multiples of that value. But why did Dalton assume hydrogen was the lightest element? Who determined that, and when?&lt;/p&gt;&#xA;" OwnerUserId="17672" LastEditorUserId="6085" LastEditDate="2022-12-16T05:18:46.947" LastActivityDate="2022-12-19T01:55:21.150" Title="How and when did scientists first determine that hydrogen was the lightest element?" Tags="|chemistry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15022" PostTypeId="1" AcceptedAnswerId="15037" CreationDate="2022-12-16T15:32:25.653" Score="1" ViewCount="109" Body="&lt;p&gt;In Weierstrass notation, the principal elliptic function &lt;span class=&quot;math-container&quot;&gt;$\wp$&lt;/span&gt; is a solution of the differential equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ (\wp')^2= 4\wp^3 -g_2\wp -g_3. $$&lt;/span&gt;&#xA;The case when &lt;span class=&quot;math-container&quot;&gt;$g_3=0$&lt;/span&gt; is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;lemniscatic&lt;/a&gt; (it corresponds to a square lattice), and the case &lt;span class=&quot;math-container&quot;&gt;$g_2=0$&lt;/span&gt; is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Equianharmonic&quot; rel=&quot;nofollow noreferrer&quot;&gt;equianharmonic&lt;/a&gt; (it corresponds to a hexagonal lattice). The origin of the first term is clear: it comes from the problem on the length of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernoulli's lemniscate&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is the origin of the term &amp;quot;equianharmonic&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;The term &amp;quot;equianharmonic&amp;quot; seems out of date: checking Google ngram shows a strange pattern: its usage experienced a peak around 1860 and was much more frequent than &amp;quot;lemniscatic&amp;quot; until 1980s, and nowadays it is used about 10 times less frequently than &amp;quot;lemniscatic&amp;quot;. Also the sizes and details of Wikipedia articles confirm this.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="13622" LastEditDate="2022-12-20T08:13:47.380" LastActivityDate="2022-12-21T07:26:54.067" Title="The origin and use of the term &quot;equianharmonic&quot; (elliptic function)" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15023" PostTypeId="2" ParentId="15018" CreationDate="2022-12-16T16:49:50.917" Score="1" Body="&lt;p&gt;This is not going to directly answer your question, but it's too long for a comment.&lt;/p&gt;&#xA;&lt;p&gt;From Chapter 1 of &amp;quot;The Theory of Statistics&#xA;and Its Applications&amp;quot; (&lt;a href=&quot;https://www.stat.rice.edu/%7Edcox/Stat581/chap1-2.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.stat.rice.edu/~dcox/Stat581/chap1-2.pdf&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Measure theory is a rather difficult and dry subject, and many statisticians&#xA;believe it is unnecessary to learn measure theory in order to understand statistics. To counter these views, we offer the following list of benefits from studying measure theory:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(i) A good understanding of measure theory eliminates the artificial distinction between discrete and continuous random variables. Summations become an example of the abstract integral, so one need not dichotomize proofs into the discrete and continuous cases, but can cover both at once.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(ii) One can understand probability models which cannot be classified as either discrete or continuous. Such models do arise in practice, e.g. when censoring a continuous lifetime and in Generalized Random Effects Models such as the Beta-Binomial.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(iii) The measure theoretic statistics presented here provides a basis for understanding complex problems that arise in the statistical inference of stochastic processes and other areas of statistics.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(iv) Measure theory provides a unifying theme for much of statistics. As an&#xA;example, consider the notion of likelihoods, which are rather mysterious in&#xA;some ways, but at least from a formal point of view are measure theoretically&#xA;quite simple. As with many mathematical theories, if one puts in the initial&#xA;effort to understand the theory, one is rewarded with a deeper and clearer&#xA;understanding of the subject.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(v) Certain fundamental notions (such as conditional expectation) are arguably not completely understandable except from a measure theoretic point of&#xA;view.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the start of &lt;a href=&quot;https://statmodeling.stat.columbia.edu/2009/05/27/the_benefits_of/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://statmodeling.stat.columbia.edu/2009/05/27/the_benefits_of/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Stephen Senn quips: “A theoretical statistician knows all about measure theory but has never seen a measurement whereas the actual use of measure theory by the applied statistician is a set of measure zero.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="643" LastActivityDate="2022-12-16T16:49:50.917" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15027" PostTypeId="1" CreationDate="2022-12-17T23:05:43.197" Score="3" ViewCount="112" Body="&lt;p&gt;So, I'm preparing a talk about &lt;a href=&quot;http://blog.maxshinnpotential.com/2017/07/05/can-humans-generate-random-numbers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;the well known fact&lt;/a&gt; that humans are bad at the task of generating uniformly random sequences of numbers when asked to do so.&lt;/p&gt;&#xA;&lt;p&gt;I would like to spice the talk a bit by presenting some real cases where perhaps some tax fraud or bad science was revealed by a simple frequency analysis of the compromised data. For example a case were a scientist might have displaced some data points to better fit a specific conclusion and that it was discovered by analyzing the end digits of the values he manipulated. Or perhaps some person that changes a few small numbers here and there while working for a bank to get some money flowing to his account and then was discovered by another simple frequency analysis.&lt;/p&gt;&#xA;&lt;p&gt;In short, I would like to know easy to explain examples of people been caught in some fraud because they though that they were able to emulate random numbers by themselves.&lt;/p&gt;&#xA;&lt;p&gt;I asked this question in Cryptography-stack but it was closed and recommended to be moved here.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance.&lt;/p&gt;&#xA;" OwnerUserId="10158" LastActivityDate="2022-12-18T17:08:36.203" Title="Historical examples of frauds discovered because someone tried to mimic a uniform random sequence" Tags="|computer-science|statistics|examples|cryptology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15028" PostTypeId="1" AcceptedAnswerId="15032" CreationDate="2022-12-18T00:02:18.133" Score="1" ViewCount="216" Body="&lt;p&gt;I'm reading a book where the author claims that in order for the method of angles and proportions used by Eratosthenes to work, the two cities would have to be located in the same meridian, or at least reasonably  close. Why is this the case? Couldn't I meassure the shadow on a completely different meridian and still figure out the circumference? After all, the stick on the ground and the well on Syene still both point towards the center of the Earth.&lt;/p&gt;&#xA;" OwnerUserId="17680" LastActivityDate="2024-02-11T12:14:05.813" Title="Why does Eratosthenes method for calculating the circumference of the Earth requires the city of Alexandria and Syene to be in the same meridian?" Tags="|astronomy|geometry|geography|" AnswerCount="4" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15029" PostTypeId="1" CreationDate="2022-12-18T06:41:52.513" Score="0" ViewCount="79" Body="&lt;p&gt;Why was the name of a group of opinions about the meaning of quantum mechanics called the Copenhagen Interpretation, relative to the Danish city of Copenhagen and not the name of a scientist named Copenhagen? Please correct me if this is wrong, although the scientist Niels Bohr was the only one from Copenhagen and the others were from Germany and Switzerland, so why the Copenhagen School?&lt;/p&gt;&#xA;" OwnerUserId="17682" LastEditorUserId="16318" LastEditDate="2022-12-19T03:14:09.923" LastActivityDate="2022-12-19T05:52:13.077" Title="Why are the views of quantum mechanics called the Copenhagen School?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15030" PostTypeId="2" ParentId="14835" CreationDate="2022-12-18T08:09:47.363" Score="2" Body="&lt;p&gt;The list of Itaian fascist mathematicians is, unfortunately, quite long, given that fascism lasted for 20 years and it was basically impossible to get a relevant position in accademia without supporting the regime (some professors resigned when it was imposed to swore loyalty to fascism to every University professor).&lt;/p&gt;&#xA;&lt;p&gt;It is not easy to distinguish between those that were sincerely fascist and those that merely adopted a comfortable neutrality, or simply faked loyalty to fascism being intimately at fight with it.&lt;/p&gt;&#xA;&lt;p&gt;Surely Francesco Severi was strongly supporting fascism and you can find a reasonably detailed description of his positions in here: &lt;a href=&quot;https://www.ams.org/notices/201208/rtx120801064p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper by Goodstein and - AMS Babbitt&lt;/a&gt;, together with references therein.&lt;/p&gt;&#xA;&lt;p&gt;Surely a devoted fascist was Luigi Fantappié; its famously recorded in André Weil autobiography an episod in which he went to Volterra (who was a Jew) to convince him of how beautiful the Italian new Law against Jews was. The Law eventually determined that both Volterra and Levi Civita were permanently thrown out of Italian University. Also he was appointed for some years as a Professor at University of Sao Paulo, Brasil, and he gave conferences throughout Brasil which were intented to promote fascism in that country.&lt;/p&gt;&#xA;&lt;p&gt;Enrico Bompiani and Mauro Picone were two others very relevant nathematicians that completely sided on the fascist side from the very beginning, and gained powerful poisition in the academic environment through this.&lt;/p&gt;&#xA;&lt;p&gt;A brief but informative summary of this development may be found in &lt;a href=&quot;https://www5.in.tum.de/persons/huckle/fasc.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Benzi's slides&lt;/a&gt; and in this work about the Italian Mathematica Union UMI &lt;a href=&quot;https://hal.archives-ouvertes.fr/hal-02289651/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;(paper by Giacardi and Tazzioli)&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="1689" LastActivityDate="2022-12-18T08:09:47.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15031" PostTypeId="1" AcceptedAnswerId="15036" CreationDate="2022-12-18T08:49:11.320" Score="1" ViewCount="78" Body="&lt;p&gt;Zoll's surfaces are a special kind of surfaces generalizing the spheres, in that all of their geodesics are closed and of the same length.&lt;/p&gt;&#xA;&lt;p&gt;I've tried to gather some biographical details on Otto Zoll but apart from the mention that he was David Hilbert's student I couldn't find any.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone in here can help me with some references?&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="1689" LastEditDate="2022-12-18T17:44:12.687" LastActivityDate="2022-12-18T20:00:59.617" Title="Biographical details on Otto Zoll" Tags="|mathematicians|biographical-details|reference-request|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15032" PostTypeId="2" ParentId="15028" CreationDate="2022-12-18T10:44:17.890" Score="3" Body="&lt;p&gt;The main reason is noon occurs at the same time on a meridian.&lt;/p&gt;&#xA;&lt;p&gt;The ancients didn't have accurate ways of keeping time, but everyone could tell when noon was because of the position of the Sun in the sky. By prior arrangement it is easy to agree to take the measurements on a particular day at noon. By being taken at noon they know that the measurements are taken at the same time, which was critical to the calculations.&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2022-12-18T10:44:17.890" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15033" PostTypeId="2" ParentId="15028" CreationDate="2022-12-18T16:52:42.387" Score="2" Body="&lt;p&gt;The reason mentioned in the answer of @zlaaemi is of course the most important one: the ancients could easily measure the difference in latitude, but not the difference in longitude.&#xA;The &amp;quot;problem of longitude&amp;quot; was solved only in 18 century.&lt;/p&gt;&#xA;&lt;p&gt;But there is also another reason: spherical trigonometry did not exist yet at the time of Erathosthenes. As far as we know, it was developed 300 years later (by Menelaus of Alexandria). So the problem with two laces on different longitude was probably too difficult at the time of Erathosthenes, even mathematically.&lt;/p&gt;&#xA;&lt;p&gt;Answer to the comment. Suppose you have two cities A and B. To estimate the radius of Earth you need to measure an arc of great circle a) in degrees, and b) in some terrestrial units (say in miles). To achieve a) you need to compare some astronomical observations, it is easy to measure the difference in latitude, but to measure the difference in longitude, your observations must be simultaneous.&#xA;And how do you achieve this? After 18 century you can transport clocks. But this was not available. In principle, one can use star occultations by the Moon or Lunar eclipses, but again computation of the longitude difference from such observations requires a lot of spherical trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;There are in principle two possible ways to solve b): first, to travel directly from A to B by an arc of a great circle, and measure the path; this is already a big problem: how do you know the direction of the great circle? Even compass (which was not invented yet) will not help. Second way is the direct triangulation (the method used in 18 century). But to do this you really need a lot of spherical trigonometry and very precise measurement instruments. None of this was available to Eratosthenes. So already problem b) was unsolvable at that time. I recall that they started to practice sailing on the orthodromy (the shortest path) only in the later part of 19 century, mainly because of fuel economy, but also because they just did not know how to do it before that.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-12-18T22:59:39.433" LastActivityDate="2022-12-18T22:59:39.433" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15034" PostTypeId="2" ParentId="15027" CreationDate="2022-12-18T17:08:36.203" Score="2" Body="&lt;p&gt;A classical example is uncovering by Kolmogorov of the fraud in the work of a biologist Ermolaeva. This was the time when the Soviet authorities endorsed a pseudoscientific heredity theory of Lysenko, and the work of Ermolaeva was aimed at proving by statistical data that Mendel's heredity law is wrong.&lt;/p&gt;&#xA;&lt;p&gt;The paper of Kolmogorov is called &amp;quot;On a new confirmation of&#xA;Mendel's law&amp;quot;, published in Doklady of the Academy of Scences, 27, 1 (1940) 38-42.&lt;/p&gt;&#xA;&lt;p&gt;The full story is told in detail &lt;a href=&quot;https://vm-kuznetsov.ru/files/etude/10_istoriya_biometrii.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&#xA;(in Russian).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-12-18T17:08:36.203" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15035" PostTypeId="2" ParentId="14804" CreationDate="2022-12-18T18:03:49.903" Score="2" Body="&lt;p&gt;Surely, what I refer below  it is not one of the earliest prizes, but it is  important in the history of mathematics.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is a famous story about heat diffusion, at the beginning of 19th century, that involved J. B. Fourier and his formulation of trigonometric series, and a  prize which Fourier won, but with reserve and critics. The rather troubled story of this  prize is intertwined with the history of the development of Fourier’s series and heat theory.&lt;/p&gt;&#xA;&lt;p&gt;There was, at the beginning of the 19th century, in France, a sound interest, theorical and practical,  about  the subject of heat nature and diffusion, on which had worked mathematicians and physicists as Lagrange and Lavoisier. During these years, Fourier began to deal with the subject in an original manner.&lt;/p&gt;&#xA;&lt;p&gt;His most important book, &lt;em&gt;La Théorie  Analytique de la Chaleur&lt;/em&gt;, published in 1822, in which Fourier presented his theory, was written in several stages.&lt;/p&gt;&#xA;&lt;p&gt;It was   preceded by a memoir  that Fourier submitted to the Paris Academy of Sciences in 1807, but its publication was rejected because of the opposition of Lagrange, who considered not rigorous the Fourier's treatment of trigonometric series.&lt;/p&gt;&#xA;&lt;p&gt;Fourier answered to the objections of Lagrange in an article, sent to Lagrange, and in another article sent to the Academia, in 1808.  But his 1807 memoir remained unpublished, and has been published only recently.&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But the question of heat diffusion remained open and, in 1810, the Academy of Sciences announced a prize competition on heat diffusion. Fourier submitted a revised version of his memoir, and won the competition, but was again criticized for lack of rigour and generality, as can be read in the report of the judges of the competition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This work contains the true differential equations of heat diffusion&#xA;[...] the novelty of the subject, together with its importance,&#xA;convinced the Class of Mathematical,  Physical and Natural Sciences of&#xA;the Academy to award the prize to this work,[..] but observing that it&#xA;isn’t without difficulties and that its analysis, to integrate them,&#xA;is still unsatisfactory, with  regard to generality and also with  regard to&#xA;rigour.&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For this reason, probably, this memoir was not published until 1824, when Fourier had become secretary of the Academy of Sciences.&lt;/p&gt;&#xA;&lt;p&gt;The  &lt;em&gt;Théorie Analytique de la Chaleur&lt;/em&gt; had been published in 1822, after Lagrange’s death and when Fourier’s reputation among the Academy of Sciences was rising.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt;  I don’t need to tell that Fourier’s work about heat diffusion and trigonometric  series are one of the most important achievements in the first half of 19th analysis, and beyond:  think that also Cantor’s set theory came from Cantor’s studies about Fourier’s series. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt; In Grattan Guinnnes I. and Ravetz J.R., &lt;em&gt;Joseph Fourier, 1768-1830&lt;/em&gt;, Cambridge, Mass., 1972.  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; &lt;sup&gt; Quoted in U. Bottazzini, &lt;em&gt;Il calcolo sublime&lt;/em&gt;, Boringhieri, 1981. p. 60 (en. transl.  Bottazzini, U, &lt;em&gt;The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass&lt;/em&gt;, Springer/Verlag, 2012. &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-01-18T13:28:08.837" LastActivityDate="2023-01-18T13:28:08.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15036" PostTypeId="2" ParentId="15031" CreationDate="2022-12-18T19:54:52.603" Score="4" Body="&lt;p&gt;The Ph.D. was conferred on Otto Zoll by the Georg August University of Göttingen in 1901. The title of his dissertation, which functioned as a prize essay at the same time, was: &lt;em&gt;Ueber Flächen mit Scharen von geschlossenen geodätischen Linien&lt;/em&gt; (On surfaces possessing families of closed geodetic lines). The Ph.D. thesis was accepted on June 5, 1901, and the Ph.D. defense took place on July 29, 1901. The faculty awarded the full prize, as the thesis addressed the prize problem in the most satisfying manner (&amp;quot;in befriedigendster Weise&amp;quot;). It also lauds the clear and easy to read writing. A scan of the dissertation can be found at &lt;a href=&quot;https://books.google.com/books?id=JCzyAAAAMAAJ&amp;amp;lpg=PT2&amp;amp;ots=GK_rxtQZqq&amp;amp;dq=%22Fritz%20Zoll%22%20%22Otto%20Zoll%22&amp;amp;pg=PP5#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google&lt;/a&gt;. Otto Zoll's Ph.D. &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=7351&quot; rel=&quot;nofollow noreferrer&quot;&gt;advisor&lt;/a&gt; was David Hilbert.&lt;/p&gt;&#xA;&lt;p&gt;There is a brief CV at the end of the dissertation, according to which Zoll's full name was Carl Otto Zoll. He was born April 2, 1878 in &lt;a href=&quot;https://en.wikipedia.org/wiki/H%C3%BCckeswagen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hückeswagen&lt;/a&gt;, a small town some 30 miles from Cologne, as the son of Fritz Zoll, the managing director of a factory. The German Wikipedia has a &lt;a href=&quot;https://de.wikipedia.org/wiki/Rote_Fabrik_(Fuhr)&quot; rel=&quot;nofollow noreferrer&quot;&gt;page&lt;/a&gt; on this factory from which I learned that this was a yarn spinning and textile mill belonging to Arnold Hueck, and that in 1921, after the death of Hueck, Fritz Zoll became a co-owner jointly with Hueck's widow. There is also a picture of the remaining main building, painted entirely in red.&lt;/p&gt;&#xA;&lt;p&gt;The CV further states that Zoll attended schools in Hückeswagen and &lt;a href=&quot;https://en.wikipedia.org/wiki/Wipperf%C3%BCrth&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wipperfürth&lt;/a&gt; followed by high school in &lt;a href=&quot;https://en.wikipedia.org/wiki/D%C3%BCren&quot; rel=&quot;nofollow noreferrer&quot;&gt;Düren&lt;/a&gt;, from which he gradudated Easter 1897. Thereafter he studied mathematics: From Easter 1897 to Michaelmas 1897 in Munich, from Michaelmas 1897 to Easter 1899 in Berlin, since Easter 1899 in Göttingen.&lt;/p&gt;&#xA;&lt;p&gt;Zoll's Ph.D. thesis served as the basis of his only publication in a scientific journal, available as a scan at the Göttingen Digitization Center:&lt;/p&gt;&#xA;&lt;p&gt;Otto Zoll, &amp;quot;Ueber Flächen mit Scharen geschlossener geodätischer Linien.&amp;quot; &lt;em&gt;Mathematische Annalen&lt;/em&gt;, Vol. 57, 1903, &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0057?tify=%7B%22pages%22%3A%5B115%5D%2C%22pan%22%3A%7B%22x%22%3A0.497%2C%22y%22%3A0.726%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.355%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 108-133&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The Center also has a &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3211311&quot; rel=&quot;nofollow noreferrer&quot;&gt;picture&lt;/a&gt; of Otto Zoll dating to about 1920 to 1922, part of a photo album with portrait shots of mathematicians presented to David Hilbert on January 23, 1922.&lt;/p&gt;&#xA;&lt;p&gt;I found further information on Otto Zoll in a &lt;a href=&quot;http://webdoc.sub.gwdg.de/ebook/serien/e/reports_Halle-Wittenberg_math/08-08report.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;research report&lt;/a&gt;: Manfred Goebel and Christian Schlensag, &lt;em&gt;Hans Brandes (1883 – 1965)&#xA;Promotion in Halle – Lehrer in Braunschweig&lt;/em&gt;, Reports on History of Mathematics, Report No. 08, Martin Luther University Halle-Wittenberg, Mathematical Institute, 2008. The following paragraphs are largely based on this report.&lt;/p&gt;&#xA;&lt;p&gt;From 1905 through 1947 (his retirement) Zoll was employed as a senior teacher at a high school in &lt;a href=&quot;https://en.wikipedia.org/wiki/D%C3%BCsseldorf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Düsseldorf&lt;/a&gt;, from 1916 with the rank of professor. He died January 20, 1952 in that city. From 1931 to 1943 he served as co-author and editor of a multi-volume textbook of mathematics for secondary schools, &lt;em&gt;Mathematisches Arbeits- und Lehrbuch&lt;/em&gt;, published by Vieweg in &lt;a href=&quot;https://en.wikipedia.org/wiki/Braunschweig&quot; rel=&quot;nofollow noreferrer&quot;&gt;Braunschweig&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The organization and content of the book as well as the name appear to have been adjusted multiple times to track changes in the school system. Based on offerings by used book sellers, it appears that after the school reforms of 1937/1938 there were three volumes for use by students: One for grades 5 and 6 (&amp;quot;Unterstufe&amp;quot;), one for grades 7 through 9 (&amp;quot;Mittelstufe&amp;quot;), and one for grades 10 through 12 (&amp;quot;Oberstufe). These were augmented by a volume with additional exercises and a volume for use by teachers. The textbook appears to have had a special emphasis on exercises based on various real-life scenarios. The research report finds that especially with the revisions of 1937 and 1939 numerous exercises directly related to Nazi ideology or with a military context (some even provided by the German Navy) were included in the book.&lt;/p&gt;&#xA;&lt;p&gt;Needless to say this caused problems with the Allied military administration in Germany after WW2. The publisher apparently tried to rework the textbook in conjunction with authors and editors to remove all objectionable matter. In a letter to Vieweg dated October 2, 1945, Zoll himself raised the issue whether it was even possible for him to remain as an editor, given that he had been a &amp;quot;Parteigenosse&amp;quot;, that is, a member of the Nazi party. After objections from the Education Branch of the British Military Government in Germany the publisher ultimately replaced Zoll as editor in fall of 1947.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2022-12-18T20:00:59.617" LastActivityDate="2022-12-18T20:00:59.617" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15037" PostTypeId="2" ParentId="15022" CreationDate="2022-12-18T20:28:42.773" Score="3" Body="&lt;p&gt;An answer to remove this question from the &amp;quot;unanswered list&amp;quot;: The term &amp;quot;equianharmonic&amp;quot; refers to &amp;quot;equal anharmonic ratio&amp;quot;, as explained by Wiener in 1901, see this earlier &lt;a href=&quot;https://mathoverflow.net/a/385121/11260&quot;&gt;MO post&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="1697" LastEditorUserId="1697" LastEditDate="2022-12-21T07:26:54.067" LastActivityDate="2022-12-21T07:26:54.067" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15038" PostTypeId="1" AcceptedAnswerId="15042" CreationDate="2022-12-18T12:33:18.450" Score="9" ViewCount="213" Body="&lt;p&gt;Poincaré is said to have given this definition of mathematics:&#xA;&lt;em&gt;It is the art of giving the same name to different objects.&lt;/em&gt; I can  not believe this quote is from Poincaré.&lt;/p&gt;&#xA;&lt;p&gt;Did Poincaré really say this? What could he have meant?&lt;/p&gt;&#xA;" OwnerUserId="17696" OwnerDisplayName="Al-Amrani" LastEditorUserId="12330" LastEditDate="2022-12-19T04:01:22.510" LastActivityDate="2022-12-19T18:27:11.477" Title="Poincaré's definition of mathematics (?)" Tags="|mathematics|quote|teaching|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15039" PostTypeId="2" ParentId="15021" CreationDate="2022-12-19T01:55:21.150" Score="1" Body="&lt;p&gt;Cavendish was apparently the first to measure the density of hydrogen, then called &lt;em&gt;inflammable air&lt;/em&gt; (air that can burn). It didn't have the identity of an element at the time, of course, and there were other varieties of inflammable air known, but these sorts of questions would be resolved before Dalton's time.&lt;/p&gt;&#xA;&lt;p&gt;Inflammable air was observed by e.g. Boyle, among others, but it was Cavendish who measured the density and recorded some of its notable properties. This was in &lt;em&gt;&lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rstl.1766.0019&quot; rel=&quot;nofollow noreferrer&quot;&gt;Three papers, containing experiments on factitious air&lt;/a&gt;&lt;/em&gt; (1766), available online. There's a few commentaries on this, e.g. &lt;a href=&quot;https://pubmed.ncbi.nlm.nih.gov/24793169/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;West JB. Henry Cavendish (1731-1810): hydrogen, carbon dioxide, water, and weighing the world. Am J Physiol Lung Cell Mol Physiol. 2014 Jul 1;307(1):L1-6. doi: 10.1152/ajplung.00067.2014. Epub 2014 May 2. PMID: 24793169.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I will quote Cavendish for the main result:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The specific gravity of each of the above-mentioned sorts of inflammable air, except the first, was tried in the following manner. A bladder holding about 100 ounce measures was filled with inflammable air, in the manner represented in Fig. 3. and the air pressed out again as perfectly as possible. By this means the small quantity of air remaining in the bladder was almost intirely of the inflammable kind. 80 ounce measures of the inflammable air, produced from zinc by the vitriolic acid, were then forced into the bladder in the same manner: after which, the pewter pipe was taken out of the wooden cap of the bladder, the orifice of the cap stopt up with a bit of lute, and the bladder weighed. A hole was then made in the lute, the air pressed out as perfectly as possible, and the bladder weighed again. It was found to have increased in weight &lt;span class=&quot;math-container&quot;&gt;$40 \frac{3}{4}$&lt;/span&gt; grains. Therefore the air pressed out of the bladder weighs &lt;span class=&quot;math-container&quot;&gt;$40 \frac{3}{4}$&lt;/span&gt; grains less than an equal quantity of common air: but the quantity of air pressed our of the bladder must be nearly the same as that which was forced into it, &lt;em&gt;i.e.&lt;/em&gt; 80 ounce measures: consequently 80 ounce measures of this sort of inflammable air weight &lt;span class=&quot;math-container&quot;&gt;$40 \frac{3}{4}$&lt;/span&gt; grains less than an equal bulk of common air. ...&lt;/p&gt;&#xA;&lt;p&gt;... Taking a medium therefore of the different trials, 80 ounce measures of inflammable air weigh 41 grains less than an equal bulk of common air. Therefore, if the density of common air, at the time when this experiment was tried, was 800 times less than that of water, which, I imagine, must be near the truth, inflammable air must be 5490 times lighter than water, or &lt;strong&gt;near 7 times lighter than common air&lt;/strong&gt;. But if the density of common air was 850 times less than that of water, then would inflammable air be 9200 times lighter than water, or &lt;span class=&quot;math-container&quot;&gt;$10 \frac{8}{10}$&lt;/span&gt;  lighter than common air.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Emphasis mine.)&lt;/p&gt;&#xA;&lt;p&gt;The value of 1/800 the density of water is commonly accepted even today, thus Cavendish is saying that the density of hydrogen (and, by Avogadro's law (1812), the atomic weight of hydrogen) is 1/7th that of air at standard temperature and pressure. The true ratio should be about 1/13th that of air (0.09 g/L to 1.2 g/L), about half of what Cavendish found.&lt;/p&gt;&#xA;&lt;p&gt;I suppose no other gas would (or could) be found to have such a low density (and by extension, atomic weight; but this would come 6 years after Dalton's paper).&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2022-12-19T01:55:21.150" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15040" PostTypeId="2" ParentId="15029" CreationDate="2022-12-19T05:52:13.077" Score="1" Body="&lt;p&gt;Following the advise of&#xA;@user 123456789&#xA;I expand my comment as an answer:&lt;/p&gt;&#xA;&lt;p&gt;The name reflects not the place of birth of the scientists who developed it but the place where it was developed. They all worked in an institute in Copenhagen, where Bohr was the director.&lt;/p&gt;&#xA;&lt;p&gt;From Wikipedia:&#xA;Now known as the Niels Bohr Institute, it opened on 3 March 1921, with Bohr as its director.&#xA;Bohr's institute served as a focal point for researchers into quantum mechanics and related subjects in the 1920s and 1930s, when most of the world's best known theoretical physicists spent some time in his company. Early arrivals included Hans Kramers from the Netherlands, Oskar Klein from Sweden, George de Hevesy from Hungary, Wojciech Rubinowicz from Poland and Svein Rosseland from Norway.&lt;/p&gt;&#xA;&lt;p&gt;At some later dates Max Born, Werner Heisenberg and Erwin Schrodinger spent time in the instisute.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-12-19T05:52:13.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15041" PostTypeId="1" AcceptedAnswerId="15045" CreationDate="2022-12-19T07:08:56.723" Score="1" ViewCount="204" Body="&lt;p&gt;I'm pretty sure they weren't using tensors in the modern sense at that point, but to what extent did Riemann lay out the structure or significance underlying his eponymous tensor?  In his &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;habilitation lecture of 1854&lt;/a&gt;, I don't see anything about the notion of parallel transport around a loop.  He does talk about the sectional curvature as obtained by extending all geodesics from a given point in a particular &amp;quot;surface-direction&amp;quot; (now thought of as a tangent plane I guess), and as I understand it the Riemann tensor can be completely determined by the sectional curvatures.  But then who introduced the modern conception of the Riemann tensor?  Christoffel?  Ricci?  Or was it a gradual process?  And is the English translation of the relevant source available?&lt;/p&gt;&#xA;" OwnerUserId="16950" LastActivityDate="2022-12-22T03:43:28.177" Title="Did Riemann invent the Riemann curvature tensor?" Tags="|mathematics|geometry|differential-geometry|riemannian-geometry|riemann|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15042" PostTypeId="2" ParentId="15038" CreationDate="2022-12-19T10:34:10.987" Score="10" Body="&lt;p&gt;The phrase appears in chapter 2 of Science and Method, &lt;a href=&quot;http://henripoincarepapers.univ-lorraine.fr/chp/hp-pdf/hp1908rga.pdf&quot; rel=&quot;noreferrer&quot;&gt;L'avenir des Mathématiques&lt;/a&gt;, English translation, &lt;a href=&quot;https://sites.math.rutgers.edu/courses/535/535-f08/Poincare.pdf&quot; rel=&quot;noreferrer&quot;&gt;The Future of Mathematics&lt;/a&gt;. Poincare refers to the centrality of form over substance in mathematics, and the conception of isomorphism that reveals things different in substance as identical in form, and hence allows to &amp;quot;give the same name&amp;quot; to them, as happened with groups and invariants, for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is scarcely credible, as Mach said, how much a well-chosen word can economize thought. I do not know whether or not I have said somewhere that &lt;strong&gt;mathematics is the art of giving the same name to different things. We must so understand it.&lt;/strong&gt; It is appropriate that things different in substance, but alike in form, should be put into the same mold, so to speak. When our language is well chosen, it is astonishing to see how all the demonstrations made upon some known fact immediately become applicable to many new facts. Nothing has to be changed, not even the words, since the names are the same in the new cases. There is an example, which comes at once to my mind; it is quaternions, upon which, however, I will not dwell.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;A word well chosen very often causes the&#xA;disappearance of exceptions to rules as&#xA;announced in their former forms; it was for this&#xA;purpose that the terms 'negative quantities',&#xA;'imaginary quantities', 'infinite points', have been invented. And let us not forget that these&#xA;exceptions are pernicious, for they conceal laws.&#xA;Very well then, one of those marks by which we&#xA;recognize the pregnancy of a result is in that it&#xA;permits a happy innovation in our language. The&#xA;mere fact is oftentimes without interest; it has&#xA;been noted many times, but has rendered no&#xA;service to science; it becomes of value only on&#xA;that day when some happily advised thinker&#xA;perceives a relationship, which he indicates and&#xA;symbolizes by a word.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The physicists also do it just the same way. They invented the term 'energy', a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Among the words which have had this happy&#xA;result, I will mention the 'group' and the 'invariant'.&#xA;They make us perceive the gist of many&#xA;mathematical demonstrations; they make us&#xA;realize how often mathematicians of the past&#xA;must have run across groups without recognizing&#xA;them and how, believing these groups to be&#xA;isolated things, they have found them to be in close&#xA;relationship without knowing why. Today we&#xA;would say that they were looking right in the&#xA;face of isomorphic groups. We feel now that in a&#xA;group the substance interests us but very little; it is the form alone which matters, and so, when&#xA;we once know well a single group, then we know&#xA;through it all the isomorphic groups; thanks to&#xA;the words 'groups' and 'isomorphism', which sum&#xA;in a few syllables this subtle law and make it at&#xA;once familiar to us all, we take our step at once&#xA;and in so doing economize all effort of thought.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2022-12-19T18:27:11.477" LastActivityDate="2022-12-19T18:27:11.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15043" PostTypeId="2" ParentId="15028" CreationDate="2022-12-19T13:55:48.443" Score="0" Body="&lt;p&gt;All meridiens are great circles i.e. each is equal to the circumference  of the earth. Latitudes, on the other hand, except for the equator, are not and would give the wrong result for the circumference of the earth.&lt;/p&gt;&#xA;" OwnerUserId="16223" LastActivityDate="2022-12-19T13:55:48.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15044" PostTypeId="2" ParentId="14804" CreationDate="2022-12-19T14:19:44.993" Score="2" Body="&lt;p&gt;On a much smaller scale and at very personal level consider &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Paul_Erd%C5%91s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erdős problems&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Erdős had a reputation for posing new problems as well as solving existing ones - Ernst Strauss called him &amp;quot;the absolute monarch of problem posers&amp;quot;. Throughout his career, Erdős would offer payments for solutions to unresolved problems. These ranged from USD 25 for problems that he felt were just out of the reach of the current mathematical thinking (both his and others) up to USD 10,000 for problems that were both difficult to attack and mathematically significant. Some of these problems have since been solved, including the most lucrative - Erdős's conjecture on prime gaps was solved in 2014, and the USD 10,000 paid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16223" LastActivityDate="2022-12-19T14:19:44.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15045" PostTypeId="2" ParentId="15041" CreationDate="2022-12-19T15:23:59.223" Score="7" Body="&lt;p&gt;A short answer: yes, he did.&lt;/p&gt;&#xA;&lt;p&gt;Riemann's habilitation lecture was aimed at a broad non-mathematical  audience,&#xA;so he did not use formulas in it, trying to explain everything in words. The curvature tensor is explained in Chapter II, section 2. Riemann does not talk here about parallel displacement, he is only interested in the condition that the space is flat (and this condition is vanishing of the curvature).&lt;/p&gt;&#xA;&lt;p&gt;The mathematical contents of this section is explained (with formulas) in the commentary of H.  Weyl included in Riemann's collected works. He credits Christoffel and Lipschitz for the first published account of the analytic apparatus translating Riemann's words into formulas.&lt;/p&gt;&#xA;&lt;p&gt;Riemann himself did this only partially in his another work, which is known as the &amp;quot;Paris Memoir&amp;quot;. The complete title is &amp;quot;Mathematical composition containing an attempt to give an answer proposed by the famous Paris Academy&amp;quot; (Werke, XXII).  This work is unfinished, since Riemann was missing the deadline for the competition. He sent his unfinished manuscript, and it was rejected.&lt;/p&gt;&#xA;&lt;p&gt;This work was neither completed nor published by Riemann,&#xA;but it is included in his Collected works, with a commentary&#xA;by H. Weber. It contains the explicit formula for the curvature (in dimension 3), and an indication of a proof that this is a tensor. Parallel displacement was later introduced by Levi-Civita.&lt;/p&gt;&#xA;&lt;p&gt;Edit. The Paris memoir is about some technical problem of heat conduction, which is reduced to the question, when a quadratic form with non-constant coefficients can be transformed to one with constant coefficients, by a change of coordinates.&#xA;The condition is that the metric determined by this quadratic form has zero curvature, and this is what Riemann wrote. He did not relate this to geometry. For a detailed analysis of what Riemann probably knew and a plausible recovery of his arguments see&lt;/p&gt;&#xA;&lt;p&gt;Oliver Darrigol, The mystery of Riemann curvature, Historia Mathematica 42 (2015) 47-83.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2022-12-22T03:43:28.177" LastActivityDate="2022-12-22T03:43:28.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15046" PostTypeId="1" CreationDate="2022-12-19T18:46:54.083" Score="6" ViewCount="280" Body="&lt;p&gt;It is now common to come across the term &amp;quot;left as exercise&amp;quot; in mathematics textbooks, and from there a comical usage of such terms was developed, typically by applying them to absurdly difficult propositions.&lt;/p&gt;&#xA;&lt;p&gt;Who is the first one to use such phrase? When did this phrase gain its popularity? In general, what's the history of this term (and its close relatives, where the solution to a problem is omitted)?&lt;/p&gt;&#xA;" OwnerUserId="17694" LastActivityDate="2022-12-24T15:41:31.417" Title="What's the history of &quot;left as exercise&quot; or notions similar to that?" Tags="|mathematics-social-history|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15047" PostTypeId="1" CreationDate="2022-12-19T22:29:54.380" Score="2" ViewCount="292" Body="&lt;p&gt;Omar Khayyam is well known as a mystical poet (his famous Quatrains).&lt;/p&gt;&#xA;&lt;p&gt;He is also somehow known as a mathematician (Equations of degree 3 ?).&lt;/p&gt;&#xA;&lt;p&gt;Are these the same person? A colleague in Arithmetical and Algebraic Geometry says that we do not know. How much is that true?&lt;/p&gt;&#xA;" OwnerUserId="17696" LastEditorUserId="12724" LastEditDate="2022-12-20T08:12:30.353" LastActivityDate="2022-12-25T15:24:44.183" Title="Omar Khayyam is well known as a mystical poet (Quatrains). He is also known as a mathematician. Are these the same?" Tags="|mathematics-social-history|algebraic-geometry|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15048" PostTypeId="2" ParentId="15046" CreationDate="2022-12-19T23:10:41.097" Score="5" Body="&lt;p&gt;While I can't state the origin of the use of the specific term  &lt;em&gt;left as exercise&lt;/em&gt;, and further research is needed, I think it can be framed in a broader context.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Left as exercise&lt;/em&gt; can be considered a special case of &lt;em&gt;proof by intimidation.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Proofs by intimidation&lt;/em&gt; are important rhetorical devices, useful when you don’t know exactly what you are saying,  but you want to persuade the audience of your argument. They are  studied in rhetorical logic:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[…] rhetorical logic, a promising field of science, of great value to&#xA;those writing research proposals. It provides new, and utterly&#xA;convincing tools for closing embarrassing gaps in your reasoning,&#xA;without resorting to brute-force methods.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The sharpest example of proof by intimidation is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Proof&lt;/strong&gt;: Trivial. &lt;span class=&quot;math-container&quot;&gt;$\Box$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Proofs by intimidation can be introduced by phrases as&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Clearly...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;It is self-evident that...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;It can be easily shown that...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Left as exercise&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Here a possible definition of &lt;em&gt;proof by intimidation&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] “Proof by Intimidation.” The aim&#xA;here is to make something sound terribly difficult, using as much&#xA;jargon as possible, and then ending with “so obviously X holds.”&#xA;Though the argument may be completely obscure, even totally incorrect,&#xA;proof by intimidation is understood by everyone who is too vain to&#xA;admit they don’t understand you.&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Left as exercise&lt;/em&gt; and &lt;em&gt;Trivial&lt;/em&gt; are the most efficient proofs by intimidation, as they make immediately, and without waste of words, the audience feel stupid,  and accept the thesis in silence.&lt;/p&gt;&#xA;&lt;p&gt;The mathematician Giancarlo Rota said, in his book  &lt;em&gt;Indiscrete Thoughts&lt;/em&gt;, that the expression &lt;em&gt;proof by intimidation&lt;/em&gt; was introduced by Marc Kac after the lessons of William Feller:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He took umbrage when someone interrupted his lecturing by pointing out&#xA;some glaring mistake. He became red in the face and raised his voice,&#xA;often to full shouting range. It was reported that on occasion he had&#xA;asked the objector to leave the classroom. The expression &amp;quot;proof by&#xA;intimidation&amp;quot; was coined after Feller's lectures (by Mark Kac). During&#xA;a Feller lecture, the hearer was made to feel privy to some wondrous&#xA;secret, one that often vanished by magic as he walked out of the&#xA;classroom at the end of the period. Like many great teachers, Feller&#xA;was a bit of a con man. — Rota, Gian-Carlo, 1932–1999. (1997),&#xA;&lt;em&gt;Indiscrete Thoughts&lt;/em&gt;.&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;a href=&quot;https://www.es.ele.tue.nl/%7Etbasten/fun/rhetoric_logic.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.es.ele.tue.nl/~tbasten/fun/rhetoric_logic.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;em&gt;ibid&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_intimidation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Proof_by_intimidation&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2022-12-24T14:49:55.840" LastActivityDate="2022-12-24T14:49:55.840" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15049" PostTypeId="2" ParentId="15047" CreationDate="2022-12-20T01:01:29.177" Score="4" Body="&lt;p&gt;According to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Omar_Khayyam&quot; rel=&quot;nofollow noreferrer&quot;&gt;wiki page&lt;/a&gt;, he is a mathematician and a famous poet.Many of the books and magazines we read when we were students stated that Omar Khayyam (we say Ömer Hayyam) was a mathematician and poet. There is a great interest in his poems in our country (Turkiye). I also love his poems. For example; when I was young, I read an article about Omar Khayyam's mathematician and poet personality in a respected journal (Bilim ve Teknik) published under the roof of a state institution in our country.&lt;/p&gt;&#xA;&lt;p&gt;It would be good to get the opinions of scientists from Iran and Turkiye, as they can examine Khayyam more closely.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Some Additions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I found the relevant article in the magazine I mentioned in the &lt;a href=&quot;https://geomania.org/forum/index.php?topic=8157.0&quot; rel=&quot;nofollow noreferrer&quot;&gt;August 1998 issue&lt;/a&gt; . At that time, I was a curious teenager who had completed the third year of high school. The name of the author was given as Deniz Gündüz. He may be the person promoted in this &lt;a href=&quot;https://www.imperial.ac.uk/people/d.gunduz&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;. (But I'm not sure about the link). I will try to translate some parts of Deniz Gündüz's article. Maybe if he sees this community page he can offer much better arguments on the subject. At the moment I think the poet and the mathematician person are the same. It will be happy to see the arguments of different views on the subject. I tried my best, but there may be errors in the translation. I hope that will be useful. Some part of the translation is as follows:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Poet, Philosopher, Mathematician - Omer Khayyam&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Omar Khayyam was born in 1048 or 1044 in the city of Nishapur. (It is the first capital of the Seljuks, a Turkish state.) The word Khayyam is derived from a Persian word meaning tentmaker. This name may have been given due to the profession of his father or another family elder. After studying in Nishapur and Belh, Omer goes to Samarkand. It was here that Omer wrote his treatise on algebra, which has earned him a reputation to this day. Later, Omer Khayyam, who was called to the capital Merv by Sultan Celaleddin Melikşah, was appointed as the head of the scientists' committee established to create a new calendar. This calendar, known today as the Celali Calendar, gives an error of 1 day every 5000 years. The Gregorian calendar used today gives an error of 1 day in 3330 years.&lt;/p&gt;&#xA;&lt;p&gt;The work that made Ömer Hayyam known in the Western world centuries later is Rubaiyat. One of the reasons why the Rubaiyat became so famous was that the translation by the great English poet Edward Fitzgerald was quite successful.&lt;/p&gt;&#xA;&lt;p&gt;Omer Khayyam, who is so popular and famous in the literary world, is also well known in the scientific world. Khayyam, who had important works in medicine, physics, astronomy, algebra, geometry and higher mathematics, is said to &amp;quot;know all the knowledge of his time&amp;quot;. His books, most of which are of scientific content, are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Risale fi'l Barehin ala Mesailü'l-Cebr ve'l-Mukabele / on the Algebra and Geometry&lt;/li&gt;&#xA;&lt;li&gt;Musahar fi'l Tabiiyat / A summary in the physical sciences&lt;/li&gt;&#xA;&lt;li&gt;Muhtasar fi'l Vücud / Summary of information about the asset - The book is now in the British Museum.&lt;/li&gt;&#xA;&lt;li&gt;El-Kevnn ve't-Teklif / Occurrence and Opinions&lt;/li&gt;&#xA;&lt;li&gt;Mizan-ül-Hikem / Measure of Wisdom&lt;/li&gt;&#xA;&lt;li&gt;Ravzat-ül-Ukul / Garden of Minds&lt;/li&gt;&#xA;&lt;li&gt;Fi Şerh-i ma eşkel-i men Mosaderhate Ketab-e Oklides / This book is currently in the Netherlands.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Gerard Meerman was the first to mention Khayyam's work in the West. In the preface of his book &amp;quot;Speicmen Calculi Fluxionalis&amp;quot; he wrote in 1742, he mentioned the services of Islamic scholars to mathematics in a manuscript belonging to Omer Khayyam in the Leyden library. Meerman wrote that algebraic solutions of cubic equations were found in the work donated by Warner to the library.&lt;/p&gt;&#xA;&lt;p&gt;F. Woepcke used this manuscript and two manuscripts in the National Museum of Paris when publishing his translation of L'algebre d'Omar Alhkayyami in 1851. A copy of the same book is in the Columbia University Library's collection of Professor David Eugene Smith. Found by Professor Smith in Lahore, India, this manuscript is essentially very similar to the Leyden copy.&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;Khayyam died between 1122-1131. Among the sources that give information about Khayyam, the most important one is his work called Çehar Megaale / Four Writings, written by his contemporary Nezami-ye Aruzi.&lt;/p&gt;&#xA;&lt;p&gt;The author of this book, which is the oldest document about Khayyam, says the following while talking about famous people in the field of astronomy in the third part of the book: &amp;quot;I met Omer Khayyam 20 years before his death in Belh. He was a guest in the house of one of the notables living in the street of slave traders. Because I know his reputation, I followed him like a shadow to record a word of his. So I heard him say, '&lt;em&gt;My grave will be found where the north winds bloom every spring.&lt;/em&gt;' Those words seemed absurd to me at the time, but I also knew that a man like him did not speak casually. 4 years after Khayyam's death Then I passed through Nishapur. Out of respect for a master of science, I went to visit his tomb. A guide took me there. His tomb was at the foot of the garden wall, the branches of peach and pear trees stretched out over the tomb, flowers spilled all over it. At that time, I remembered the words he said when he was in Belh and I cried.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ana Britannica, Vol. 17, pg 300.&lt;/li&gt;&#xA;&lt;li&gt;Die Rubaijat Von Omar Khayyam, Sagdos, Milano&lt;/li&gt;&#xA;&lt;li&gt;Dilgan H., Büyük Matematikçi Ömer Hayyam, Şirketi Mürettebiye Basımevi, Istanbul, 1959.&lt;/li&gt;&#xA;&lt;li&gt;Kasıt D. S., The Algebra of Omer Khayyam, Columbia University, New York, 1972.&lt;/li&gt;&#xA;&lt;li&gt;Meydan Larousse, Vol. 8, page 536.&lt;/li&gt;&#xA;&lt;li&gt;Struik D. J., Kısa Matematik Tarihi, Sarmal Yayınevi, Istanbul, 1996.&lt;/li&gt;&#xA;&lt;li&gt;Şardağ R., Bütün Yönleriyle Hayyam Rubaileri, Özgür Yayın Dağıtım, Istanbul.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="17698" LastEditorUserId="17698" LastEditDate="2022-12-25T15:24:44.183" LastActivityDate="2022-12-25T15:24:44.183" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15050" PostTypeId="1" CreationDate="2022-12-20T01:35:43.237" Score="1" ViewCount="409" Body="&lt;p&gt;The film A Beautiful Mind depicts this scene, and it is mentioned on wikipedia as well as many articles/books. John Nash famously gave a lecture to 250 mathemeticians at Columbia University in 1959 eagerly anticipating his potential proof of the Riemann Hypothesis, however it soon became clear from his incoherent speech that he had entered psychosis and was presenting nonsense.&lt;/p&gt;&#xA;&lt;p&gt;Regardless, I would still like to read/hear what he was saying if that is possible. I have tried searching for transcripts or audio of the lecture to no avail. Is this available anywhere? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="17699" LastActivityDate="2022-12-20T01:35:43.237" Title="Is there a transcript or audio recording of the lecture on Riemann Hypothesis by John Nash at Columbia University in 1959 anywhere?" Tags="|john-nash|riemann|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15052" PostTypeId="2" ParentId="15047" CreationDate="2022-12-20T16:16:35.483" Score="1" Body="&lt;p&gt;We don't really know. Unlike mathematics and astronomy, nothing of his poetry apparently survives in the original.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia says:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;There are occasional quotes of verses attributed to Omar in texts attributed to authors of the 13th and 14th centuries, but these are of doubtful authenticity, so that skeptical scholars point out that the entire tradition may be pseudepigraphic.&lt;/p&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot; Edward Granville Browne (1906) notes the difficulty of disentangling authentic from spurious quatrains: &amp;quot;while it is certain that Khayyam wrote many quatrains, it is hardly possible, save in a few exceptional cases, to assert positively that he wrote any of those ascribed to him&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-12-20T16:16:35.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15054" PostTypeId="1" CreationDate="2022-12-22T14:23:51.020" Score="3" ViewCount="53" Body="&lt;p&gt;The context is that I am trying to teach some middle school students that in science, a question could be posed/ an observation made but being able to answer or explain it, might take a couple of centuries or more. I am attempting this using the Leidenfrost effect.&lt;/p&gt;&#xA;&lt;p&gt;I understand that Leidenfrost himself was not able to explain what was going on. Can someone tell me how understanding of this effect evolved and how we came to understand it finally? (to the extent we understand it today)&lt;/p&gt;&#xA;" OwnerUserId="5528" LastActivityDate="2022-12-22T14:23:51.020" Title="Is there a resource which explains the history of how the Leidenfrost effect was explained?" Tags="|experimental-physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15055" PostTypeId="2" ParentId="15046" CreationDate="2022-12-24T02:41:28.237" Score="9" Body="&lt;p&gt;The oldest I can recall is Descartes in &lt;em&gt;La Géométrie&lt;/em&gt; (1637):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science.  Because, I find nothing here so difficult that it cannot be worked out by any one at all familiar with ordinary geometry and with algebra, who will consider all that is set forth in this treatise.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Subsequent authors shortened the expression considerably. :)&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2022-12-24T15:41:31.417" LastActivityDate="2022-12-24T15:41:31.417" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15056" PostTypeId="2" ParentId="15004" CreationDate="2022-12-24T10:30:07.160" Score="1" Body="&lt;p&gt;When, was during the &amp;quot;Renaissance period, long after the fall of the Roman empire&amp;quot;. Why is &lt;a href=&quot;https://www.unrv.com/roman-numerals/roman-clock-numerals.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;more difficult&lt;/a&gt; to ascertain. There are a number of speculative reasons.&lt;/p&gt;&#xA;&lt;p&gt;As you state, IIII was the original way of writing the number four. This may have been because it would have been easier for plebeians to count four lines than to add and subtract numbers. Another speculation is that in Latin, Jupiter was originally written as IVPPITER. Not wanting to offend one of their gods they avoided anything that had part of IVPPITER'S name, so the number four was written as IIII not IV.&lt;/p&gt;&#xA;&lt;p&gt;Changing from IIII to IV may have been a matter of style, as IIII was viewed by some as &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;being clumsier&lt;/a&gt; than IV.&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2022-12-24T10:30:07.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15057" PostTypeId="2" ParentId="15004" CreationDate="2022-12-24T14:40:01.307" Score="6" Body="&lt;p&gt;Literature on the subject seems to agree about the fact that purely additive forms, as IIII, are  the most ancient forms and the preferred forms in early Roman times, and the subtractive forms as IV were introduced, at least widely,  during the Middle Age.&lt;/p&gt;&#xA;&lt;p&gt;What is more uncertain is, as far as I know, if the additive form was exclusive, to what extent the subtractive form IV was diffused in early times, and when and how subtractive forms had origin.&lt;/p&gt;&#xA;&lt;p&gt;Below, a quotation from the History SE thread mentioned by njuffa in the comments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Romans used both additive and subtractive and this is true from late&#xA;Etruscan times up to middle ages (1100 Arabic numbers spread into&#xA;Europe).[…]&#xA;It is true that IIII is the simplest and earliest form to write 4.&#xA;During early Roman times the additive form is preferred, but we don't&#xA;know for sure whether it was exclusive, since you will find IV in&#xA;quite early republican inscriptions in the CIL.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I think it could be interesting, in this regard,  to refer to the origin of the symbols used in Roman numbering system.&lt;/p&gt;&#xA;&lt;p&gt;An interesting text about the origin is an article of the university of Bologna&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;.&#xA;Unfortunately, this article is in Italian, so I make here, below, a synthesis and a translation into English of some passage of it.&lt;/p&gt;&#xA;&lt;p&gt;In this article it is said that the origins of the symbols were in sheep counting, as the Romans were (in the origin) above all a people of shepherds.&lt;/p&gt;&#xA;&lt;p&gt;Sheep counting  was made by carving of notches on belts:  to facilitate reading, every five notches a notch was made in the shape of  V, and every ten notches a 'X'. Later,  other forms  were introduced for 50, 100 and so on.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/9keyX.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/9keyX.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But, &amp;quot;in the Roman numbering system  there is a novelty:  subtractive notation:  IV = 4 ;  XIX = 19.  Subtractive notation  instead of  'IIII' remembers the position of 4 in the serie: ' IIIIV', as  'IX' in the series: 'IIIVIIIIX'.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Su1iN.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Su1iN.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Because of this pastoral, sheep breeding, origin of both additive and subtractive systems, I suppose that also the subtractive  version was known during  very early Roman times, even if the additive numbering system became the most widespread system, until Middle Ages.&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; &lt;a href=&quot;https://history.stackexchange.com/questions/69731/when-did-subtractive-notation-become-common-for-roman-numerals&quot;&gt;https://history.stackexchange.com/questions/69731/when-did-subtractive-notation-become-common-for-roman-numerals&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;sup&gt;http://progettomatematica.dm.unibo.it/NumeriAdditivi/romani.html &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; &lt;sup&gt;Recently, it was discovered that the numbers on the Colosseum gates were painted in red (traces of very old paints were found, a very important discover), so one can think that the purpose was to make the writing very visible and understandable for a crowd of illiterate people, coming to the representations of the Colosseum, and IIII is more understandable than IV&#xA;(important and aristocratic people entered the Colosseum through separated and not numbered gates).&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2022-12-24T20:10:14.520" LastActivityDate="2022-12-24T20:10:14.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15058" PostTypeId="1" CreationDate="2022-12-24T19:39:16.343" Score="1" ViewCount="85" Body="&lt;p&gt;What was Aristotle's idea of a vacuum when he said that nature abhors one? How would any ancient have experienced/observed or created a vacuum beside, I guess, when one breathes or perhaps when water flows into an &amp;quot;empty&amp;quot; vessel.&lt;/p&gt;&#xA;&lt;p&gt;I do not believe until the middles ages or later was anything like a vacuum pump created.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2022-12-25T14:30:35.043" Title="What was Aristotle's experience with vacuums?" Tags="|aristotle|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15059" PostTypeId="1" CreationDate="2022-12-24T20:13:05.583" Score="1" ViewCount="153" Body="&lt;p&gt;I have never read his original works, but I would say that he must have used length, volume, time, velocity, acceleration, momentum or force, and maybe (kinetic) energy. Which quantities appeared in his work and which did he introduce to physics?&lt;/p&gt;&#xA;" OwnerUserId="9778" LastEditorUserId="12330" LastEditDate="2022-12-27T08:17:27.457" LastActivityDate="2022-12-27T15:18:42.247" Title="Which physical quantities were used or introduced by Newton?" Tags="|classical-mechanics|newton|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15060" PostTypeId="2" ParentId="15058" CreationDate="2022-12-25T14:30:35.043" Score="1" Body="&lt;p&gt;The main point is exactly that the ancient scientists could not create vacuum. So they believed that it does not exist.&#xA;Probably Aristotle himself never really tried, but later ancients invented water pump, whose action they could explain only by assuming that vacuum cannot exist. Vacuum appears in a water pump only when the cylinder is more than 10 m high,&#xA;and they did no build such large pumps.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-12-25T14:30:35.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15061" PostTypeId="1" AcceptedAnswerId="16070" CreationDate="2022-12-25T19:16:45.637" Score="0" ViewCount="1386" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: &amp;quot;&lt;strong&gt;Sir, &lt;span class=&quot;math-container&quot;&gt;$\frac{a+b^{n}}{n}=x$&lt;/span&gt;, hence God exists—reply!&lt;/strong&gt;&amp;quot; Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is apocryphal, given that Diderot himself did research in mathematics. The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above is an excerpt from Euler's page on Wikipedia. The article is quite clear - the story is &lt;em&gt;apocryphal&lt;/em&gt;. Too bad if it is, but I'm aware that Denis Diderot knew enough mathematics to see through the ruse.&lt;/p&gt;&#xA;&lt;p&gt;Just wanted to ask a few questions though:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What exactly is the equation in this &amp;quot;proof&amp;quot; of god?&lt;/li&gt;&#xA;&lt;li&gt;Does it belong to a certain algebraic category?&lt;/li&gt;&#xA;&lt;li&gt;What would be the general solution for such types of equations?&lt;/li&gt;&#xA;&lt;li&gt;Can mathematics prove God exists?&lt;/li&gt;&#xA;&lt;li&gt;Comments of all kinds are welcome.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I plotted the equation &lt;span class=&quot;math-container&quot;&gt;$[y = \frac{a + b^x}{x}]$&lt;/span&gt; where a = 2 and b = 3 using Desmos graphing calculator and I got this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/V2HGn.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/V2HGn.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Weird things happen to the graph when I change the signs of &lt;em&gt;a&lt;/em&gt; and &lt;em&gt;b&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;P.S. I also remember seeing this in a movie or was it my imagination? I can't tell. If someone has any information on a movie in which there's a scene which reenacts this legend, please do let me know.&lt;/p&gt;&#xA;&lt;p&gt;Muchas gracias in advance and apologies if the question fails to meet the exacting SE standards. I'm hoping that some of you will find this mathematical anecdote interesting and might have insights to share to boot.&lt;/p&gt;&#xA;&lt;p&gt;Merry Christmas.&lt;/p&gt;&#xA;" OwnerUserId="16223" LastActivityDate="2024-01-15T18:46:11.340" Title="Leonhard Euler's Mathematical Proof of God" Tags="|mathematics|mathematicians|mathematics-social-history|algebra|euler|" AnswerCount="1" CommentCount="7" ClosedDate="2024-01-03T14:05:44.630" ContentLicense="CC BY-SA 4.0" />
  <row Id="15062" PostTypeId="2" ParentId="15059" CreationDate="2022-12-26T00:28:49.020" Score="2" Body="&lt;p&gt;Yes to @Alexandre Eremenko 's comment -- provided 'force' is understood in a sense consistent with accelerative force (i.e. something that can produce change of motion in accordance with Newton's second law of motion: in contrast, for example, to a  force according to Kepler, which would maintain velocity and without which a body would come to rest, see Bruce Stephenson's 'Kepler's Physical Astronomy' linked below)). With that condition, force indeed looks like a Newtonian introduction.&lt;/p&gt;&#xA;&lt;p&gt;The need for such a proviso about the meaning in which 'force' is understood shows, however, that there is a problem with the question. The question names various modern-day labels for physical quantities, without pausing to consider whether these terms have had more than one meaning over time. In this way it suggests a mistake of anachronism: of assuming that the modern meaning of the term extends backwards over time.&lt;/p&gt;&#xA;&lt;p&gt;In the case of 'force', the term has had a complex history. Back when physics was written in Latin, corresponding words 'vis' and 'virtus' could stand, indefinitely, for almost any sort of physical power, force or virtue. So it would be easy to find references to 'vis' or 'virtus' prior to Newton, but it will also be found that these earlier references do not mean the kind of (accelerative) force that Newton was discussing. Studies of the early history, showing the variety of early concepts of force, and how slowly they became clarified, can be found for example in &lt;a href=&quot;https://archive.org/details/forceinnewtonsph0000west&quot; rel=&quot;nofollow noreferrer&quot;&gt;Westfall (1971) &amp;quot;Force in Newton's Physics&amp;quot;&lt;/a&gt;. Also Kepler's 'force', quite different than Newton's (because it maintains motion against a tendency for a body to come to rest) is discussed in &lt;a href=&quot;https://books.google.com/books?id=pxCYAeOqJg8C&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bruce Stephenson's (1987) &amp;quot;Kepler's Physical Astronomy&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Newton's contributions are not well measured by identifying which modern concepts he introduced. In some cases, where the modern concept did not yet exist in Newton's time, Newton also did not introduce it, but he did provide an important foundation on which the modern concept could later be built.&lt;/p&gt;&#xA;&lt;p&gt;This is also practically how Einstein described Newton's lasting contributions. When asked in 1919 to comment on an article suggesting &amp;quot;Newton overthrown&amp;quot; (by relativity theory), Einstein answered by disagreeing with that simplistic idea:-&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;No one must think that Newton's great creation can be overthrown in&#xA;any real sense by this or by any other theory.  His clear and wide&#xA;ideas will for ever retain their significance as the foundation on&#xA;which our modern conceptions of physics have been built.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(See &lt;a href=&quot;https://archive.org/details/science511920mich/page/8&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Science' (New Series, vol.LI, no.1305, Jan 2, 1920, at pp.8-10)&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;(Original German: &amp;quot;Niemand aber soll denken, daß durch diese oder irgendeine andere Theorie Newtons große Schöpfung im eigentlichen Sinne verdrängt werden könne. Seine klaren und großen Ideen werden als Fundament unserer ganzen modernen Begriffsbildung auf dem Gebiet der Naturphilosophie ihre eminente Bedeutung in aller Zukunft behalten.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Einstein was saying that Newton introduced foundations on which (the whole of? --  'ganzen') our modern physics conceptions were built -- but without making a statement that Newton introduced particular modern physical quantities.&lt;/p&gt;&#xA;&lt;p&gt;An arguable example of the difference is found in Newton's 'Principia' book 1 proposition 1, where Newton demonstrates that the motion of a body subject to a centripetal force takes place by equal areas relative to the center of force (i.e. in a sense that we would interpret as that of Kepler's second law, concerning equal-area motion). Commentators have occasionally expressed the thought that prop. 1 is obvious in view of the constancy of angular momentum when there is no couple or turning force. A significant point, though, is that when Newton wrote his work there was no concept of angular momentum, and proposition 1 does not introduce it (since the demonstration makes no reference to mass). But without proposition 1, there would also not have been the foundation or stepping-stone that it provides towards the concept of angular momentum.&lt;/p&gt;&#xA;&lt;p&gt;Similar observations can be made in the cases of some other modern physical quantities.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-12-27T15:18:42.247" LastActivityDate="2022-12-27T15:18:42.247" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15063" PostTypeId="1" CreationDate="2022-12-27T16:07:12.533" Score="4" ViewCount="123" Body="&lt;p&gt;It is common knowledge that the Pythagoreans discovered irrational numbers (or incommensurability), for example if the hypotenuse of an isosceless right triangle is compared with one of the legs or while investigating the pentagram.  However, what I've read is always about square roots.  Now, the famous problem of doubling the cube involves cube roots and Wikipedia claims that Hero of Alexandria knew a method to approximate cube roots.  My question is: Did the Greek mathematicians know that &amp;quot;most&amp;quot; cube roots (for example the cube root of two) are also irrational?  Or at least some?  Any sources?&lt;/p&gt;&#xA;" OwnerUserId="3555" LastEditorUserId="12330" LastEditDate="2022-12-27T18:00:20.867" LastActivityDate="2022-12-27T18:00:20.867" Title="Did the ancient Greeks know that &quot;most&quot; cube roots are irrational?" Tags="|mathematics|ancient-greece|numbers|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15064" PostTypeId="2" ParentId="13398" CreationDate="2022-12-27T16:51:39.137" Score="3" Body="&lt;p&gt;There is really no sign here of any 'discrepancy' at all. In the quoted/translated passage from 'De Revolutionibus', Copernicus is pointing out that the apogee of Mars' orbit has shifted between the time of the observation reported by Ptolemy and his own time. The quoted passage then goes on to refer to Copernicus' theoretical mechanistic ideas for the celestial motions, which explain this motion as well.&lt;/p&gt;&#xA;&lt;p&gt;Insofar as these ideas and mechanisms involved circles and epicycles, they have become obsolete since Copernicus' time, but that is of course different from any matter of discrepancy. 'Discrepancy' normally implies some kind of incompatibility or contradiction. But the slow motions of apogees over time contradict nothing: planetary tables from Ptolemy's day to the present acknowledge and incorporate such motions. They are part of the natural order, not any sort of discrepancy or mistake.&lt;/p&gt;&#xA;&lt;p&gt;It may help any further discussion to quote a somewhat better English  translation of the passage in question (i.e. not subject to the hazards of machine translation). This is from Edward Rosen, &amp;quot;Nicholas Copernicus on the Revolutions. Translation and commentary&amp;quot;, (Polish Scientific Publishers, Warsaw/Cracow; also Macmillan; also Johns Hopkins, Baltimore; 1978). The reference to an apogee is to the apogee of Mars' orbit ('eccentric') around the sun:-&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... computed backwards, the place of the eccentric's apogee in this&#xA;last observation obviously = 119°40' [ = 133°20'- 13°40'] in the&#xA;sphere of the fixed stars. In Antoninus [Pius'] time Ptolemy found the&#xA;apogee at 108°50' [Syntaxis, X, 7 : 25°30' within the Crab = 115°30'-&#xA;6°40']. It has therefore shifted eastward 10°50' [= 119°40'- 108°50'&#xA;from that time] to ours. I have also found the distance between the&#xA;centers smaIler by 40P [1460P as compared with 1500P] whereof the&#xA;eccentric's radius = 10,000P. The reason is not that Ptolemy or I made&#xA;an error, but that, as is clearly proved, the center of the earth's&#xA;grand circle has approached the center of Mars' orbit, with the sun&#xA;meanwhile remaining stationary. For, these conclusions are mutuaIly&#xA;consistent to a high degree, as will become plainer than daylight&#xA;hereafter [V, 19].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clearly, in the final sentence included here, in the quotation from the Rosen translation (omitted by the questioner), Copernicus was saying that everything is consistent, and at the end of the later chapter (V, 19) to which he refers, he concludes:--&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus also in the case of Mars the sizes and distances of its motion&#xA;have been explained through sound computation by means of the earth's&#xA;motion.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2022-12-27T17:12:32.117" LastActivityDate="2022-12-27T17:12:32.117" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15065" PostTypeId="1" CreationDate="2022-12-27T18:26:30.143" Score="2" ViewCount="222" Body="&lt;p&gt;Around 1900, was it widely known that the Chebyshev polynomial &lt;span class=&quot;math-container&quot;&gt;$T_n(X)$&lt;/span&gt; satisfies the identity&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ T_n(X) \circ \frac{X+X^{-1}}{2} = \frac{X^n+X^{-n}}2?$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;And also, would one expect top-notch algebraists of that era to be familiar with this (even though Chebyshev polynomials originated in analysis)?&lt;/p&gt;&#xA;&lt;p&gt;I would have guessed that this identity was well known back then, since following Euler (if not earlier) it seems natural to rewrite the identity &lt;span class=&quot;math-container&quot;&gt;$T_n(\cos\theta)=\cos n\theta$&lt;/span&gt; in terms of &lt;span class=&quot;math-container&quot;&gt;$z:=e^{i\theta}$&lt;/span&gt;, which yields the above identity.&lt;/p&gt;&#xA;&lt;p&gt;Here is why I ask.  In his PhD thesis of 1896, Dickson introduced a class of polynomials &lt;span class=&quot;math-container&quot;&gt;$D_n(X,a)$&lt;/span&gt;, which satisfy the identity&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ D_n(X,a) \circ \Bigl(X+\frac{a}X\Bigr) = X^n + \Bigl(\frac{a}X\Bigr)^n,$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and showed that they had remarkable properties over finite fields.  What has always puzzled me is that Dickson didn't recognize the connection of these polynomials to Chebyshev polynomials, and especially that he didn't notice how similar his identity is to the Chebyshev polynomial identity mentioned above.&#xA;Dickson didn't mention Chebyshev polynomials either in his thesis or in his 1901 book &amp;quot;Linear Groups&amp;quot;.  This seems especially surprising since Dickson was a first-rate mathematician, and also was extremely attentive to the previous math literature, for instance spending a decade writing his 1500-page history of the theory of numbers.  So that's why I wonder which properties of Chebyshev polynomials would have been familiar to leading algebraists around 1900.&lt;/p&gt;&#xA;" OwnerUserId="17728" LastActivityDate="2022-12-29T17:35:50.137" Title="What was known about Chebyshev polynomials in 1900?" Tags="|mathematics|algebra|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15066" PostTypeId="1" AcceptedAnswerId="15068" CreationDate="2022-12-28T09:41:47.580" Score="0" ViewCount="89" Body="&lt;p&gt;The simplest and most common &lt;a href=&quot;https://en.wikipedia.org/wiki/Oxocarbon&quot; rel=&quot;nofollow noreferrer&quot;&gt;oxocarbons&lt;/a&gt; are carbon monoxide (CO) and carbon dioxide (CO&lt;sub&gt;2&lt;/sub&gt;). The general consensus is that they are inorganic (see: &lt;a href=&quot;https://chemistry.stackexchange.com/questions/22195/is-carbon-dioxide-organic-or-inorganic&quot;&gt;Is carbon dioxide organic or inorganic?&lt;/a&gt;). The other two oxocarbons reasonably stable under standard conditions although rarely encountered are &lt;a href=&quot;https://en.wikipedia.org/wiki/Carbon_suboxide&quot; rel=&quot;nofollow noreferrer&quot;&gt;carbon suboxide&lt;/a&gt; (C&lt;sub&gt;3&lt;/sub&gt;O&lt;sub&gt;2&lt;/sub&gt; or O=C=C=C=O) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Mellitic_anhydride&quot; rel=&quot;nofollow noreferrer&quot;&gt;mellitic anhydride&lt;/a&gt;(C&lt;sub&gt;12&lt;/sub&gt;O&lt;sub&gt;9&lt;/sub&gt;), and Wikipedia says they are organic. Question is whether they were always considered organic (at the time of discovery) or were classified as organic at a later point of time. C&lt;sub&gt;3&lt;/sub&gt;O&lt;sub&gt;2&lt;/sub&gt; was discovered in 1873 by subjecting carbon monoxide to electric current. A chemist named &lt;a href=&quot;https://en.wikipedia.org/wiki/Otto_Diels&quot; rel=&quot;nofollow noreferrer&quot;&gt;Otto Diels&lt;/a&gt; suggested organic names for the compound but didn't state the year of the naming. C&lt;sub&gt;12&lt;/sub&gt;O&lt;sub&gt;9&lt;/sub&gt; was apparently obtained in 1830 but properly characterized in 1913 by H. Meyer and K. Steiner. Did they then classify the compound as organic?&lt;/p&gt;&#xA;" OwnerUserId="16966" LastEditorUserId="16966" LastEditDate="2022-12-29T06:26:40.450" LastActivityDate="2022-12-29T06:26:40.450" Title="Were oxocarbons other than carbon monoxide/dioxide always classified as organic compounds?" Tags="|chemistry|naming-conventions|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15067" PostTypeId="2" ParentId="15065" CreationDate="2022-12-28T16:07:19.723" Score="1" Body="&lt;p&gt;One might ask the same question about Lucas and his&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lucas_sequence&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lucas sequence&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$V_n := \alpha^n + \beta^n$&lt;/span&gt; where&#xA;&lt;span class=&quot;math-container&quot;&gt;$P:=\alpha+\beta$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q:=\alpha\beta.$&lt;/span&gt; The difference&#xA;in the Dickson and Lucas cases seems to be that they are&#xA;working with these numerical and polynomial sequences&#xA;in the context of algebra and number theory and these&#xA;sequences have &lt;em&gt;two&lt;/em&gt; variables versus only one for the&#xA;Chebyshev sequences. It also seems to me that there was&#xA;no need to mention Chebyshev as this would add no actual&#xA;useful information to the exposition of results. notice&#xA;that the Lucas sequence Wikipedia article only has a&#xA;very brief mention of Chebyshev polynomials.&lt;/p&gt;&#xA;&lt;p&gt;The Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Dickson_polynomial&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dickson polynomial&lt;/a&gt; states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since the Dickson polynomial &lt;span class=&quot;math-container&quot;&gt;$D_n(x,α)$&lt;/span&gt; can be defined&#xA;over rings with additional idempotents,  &lt;span class=&quot;math-container&quot;&gt;$D_n(x,α)$&lt;/span&gt; is&#xA;often not related to a Chebyshev polynomial.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="13622" LastEditDate="2022-12-28T16:28:13.507" LastActivityDate="2022-12-28T16:28:13.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15068" PostTypeId="2" ParentId="15066" CreationDate="2022-12-29T00:40:39.567" Score="4" Body="&lt;p&gt;No classification scheme in general sciences is perfect, there will always be exceptions, borderline cases, and controversial examples. On the other hand, it makes no difference to the molecules whether humans call them inorganic or organic. &amp;quot;A rose by any other name would smell as sweet&amp;quot; The inorganic/organic classification is as bad as chemical/physical classification. For the sake of historical notes, let us consult two leading treatises of early 1900s. One is the English author Mellor. He wrote a multivolume encyclopediac treatise on &amp;quot;Inorganic and Theoretical Chemistry&amp;quot;. It is really amazing that one man did this job! He has a full section on carbon and its sub-oxides. This is volume 5.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Rbrrp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Rbrrp.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/RbQQj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/RbQQj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For contrast, now refer to another leading inorganic chemistry treatise from Germany, Abegg's &lt;em&gt;Handbuch der Anorganischen Chemie&lt;/em&gt; it states, (1892), even though the oxides were discovered before 1892.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Kohlenstoff  und  Sauerstoff: C  bildet  mit  O  zwei  Verbindungen:&#xA;CO Kohlenoxyd; CO2 Kohlendioxyd,  Kohlensäure,  Kohlensäureanhydrid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This long section starts as carbon and oxygen: C form with O two compounds: carbon oxide CO; CO2 carbon dioxide, carbonic acid, carbonic acid anhydride.&lt;/p&gt;&#xA;&lt;p&gt;So it is well established that that carbon oxides, were considered part and parcel of inorganic chemistry. You can also check Sidgwick's two volumes Elements &amp;amp; Their Compounds (1940s-1950s?). I remember seeing carbon suboxides. Most likely, they were not well accepted. I mean how should we label graphene oxide. Is it an organic or inorganic compound?&lt;/p&gt;&#xA;&lt;p&gt;Anonymous Wikipedia authors (and SE authors) should not be deciding elements on whether a compound is organic or inorganic. Wikipedia is a very good starting place but it is not the end. It cannot replace the books and treatises that exist on the subject written by people who dedicated their lives to chemistry.&lt;/p&gt;&#xA;&lt;p&gt;As to the Wikipedia quote, &amp;quot;Otto Diels later stated that the more organic names dicarbonylmethane and dioxallene were also correct.&amp;quot; A reference is needed. Please someone add:&lt;/p&gt;&#xA;&lt;p&gt;Diels, O. and Blumberg, P. (1908), Über die Konstitution des Kohlensuboxyds. Ber. Dtsch. Chem. Ges., 41: 1233-1236. &lt;a href=&quot;https://doi.org/10.1002/cber.190804101232&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1002/cber.190804101232&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2022-12-29T00:40:39.567" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15069" PostTypeId="1" AcceptedAnswerId="15075" CreationDate="2022-12-29T12:46:42.007" Score="1" ViewCount="396" Body="&lt;p&gt;My understanding is that &lt;a href=&quot;https://en.wikipedia.org/wiki/Corentin_Louis_Kervran&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kervran&lt;/a&gt; fed chickens a diet lacking in calcium and yet eggs were produced with calcium in their shells.&lt;/p&gt;&#xA;&lt;p&gt;Two related questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Could not the calcium have been from the bones of the chicken itself?&lt;/li&gt;&#xA;&lt;li&gt;Has an attempt, with more sophisticated methods of determining where the calcium came from, been made to reproduce his results or was he completely discredited by either the source being as I asked from the chicken's bones or some source of calcium in the diet that he overlooked?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I read that that he was considered a fringe kind of scientist and won an Ignobel prize, but surely others being able to reproduce his results without being able to explain the source of the shells' calcium is important even if transmutation is not the actual reason.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2022-12-31T17:52:31.367" LastActivityDate="2023-01-03T00:36:17.143" Title="When Kervran suggested biological transmutation of elements did anyone argue this" Tags="|biology|nuclear-physics|alchemy|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15070" PostTypeId="2" ParentId="15065" CreationDate="2022-12-29T17:35:50.137" Score="2" Body="&lt;p&gt;Chebyshev polynomials really appear in various connections since 16th century, see, for example &lt;a href=&quot;https://arxiv.org/abs/2203.10955&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/2203.10955&lt;/a&gt;&#xA;But they were not named after Chebyshev until the early 20 century. So various authors just introduced them, and established those properties they need. For example, they occur in early 20 century holomorphic dynamics (Fatou, Julia, Lattes, Ritt), as exceptional cases, together with monomials and what is called now &lt;span class=&quot;math-container&quot;&gt;``&lt;/span&gt;Lattes functions'', but none of these authors mentions Chebyshev, or refers to previous studies of these polynomials.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2022-12-29T17:35:50.137" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15071" PostTypeId="1" CreationDate="2022-12-29T18:08:56.580" Score="10" ViewCount="242" Body="&lt;p&gt;I restrict my question to mathematics since this is probably the most internationalized of all sciences. During WWII, did any British mathematicians (or mathematicians from allied countries) publish any papers in German journals? And vise versa. Did journals have any policies about&#xA;papers submitted from enemy countries? I would like to see specific cases, if any.&lt;/p&gt;&#xA;&lt;p&gt;Remark. This question has a practical meaning today (I am on the editorial board of a Ukrainian journal).&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-01-01T08:08:36.177" Title="Publication of mathematical papers in journals of enemy country" Tags="|mathematics-social-history|publishing|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15072" PostTypeId="1" AcceptedAnswerId="15080" CreationDate="2022-12-30T22:38:17.197" Score="3" ViewCount="179" Body="&lt;p&gt;NPR's December 30, 2022 article &lt;a href=&quot;https://www.npr.org/sections/health-shots/2022/12/30/1142202365/gregor-mendel-genetics-dna-analyzed&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why scientists dug up the father of genetics, Gregor Mendel, and analyzed his DNA&lt;/a&gt; includes speculation by those involved about whether Mendel would want his body to be exhumed and his DNA analyzed (if it could be explained to him what DNA was):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://www.uvu.edu/directory/employee/?id=Z3dwZk11dlFrOWRHRHdxSnNmMTBJdz09&quot; rel=&quot;nofollow noreferrer&quot;&gt;Daniel Fairbanks&lt;/a&gt;, a plant geneticist and author of a book called Gregor Mendel: His Life and Legacy) has thought about how Mendel would feel about being disturbed in his grave to satisfy the curiosity of today's scientists.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;I tend to think, from what I know about him, that he very well may have been happy about this,&amp;quot; says Fairbanks. &amp;quot;But of course we can't directly ask him.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://www.muni.cz/en/people/38791-sarka-pospisilova&quot; rel=&quot;nofollow noreferrer&quot;&gt;Šárka Pospíšilová&lt;/a&gt;, a geneticist who is also the vice rector for research at Masaryk University in Brno) leans toward that theory, too.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;We believe that he would be happy. &lt;strong&gt;We know he was very enthusiastic for all kinds of research,&amp;quot; she says – noting that just before he died, Mendel requested that an extensive autopsy be done.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;He wasn't against research on his own body,&amp;quot; she says.&lt;/p&gt;&#xA;&lt;p&gt;**Even though Mendel knew nothing about DNA or the role that it played in the inheritance patterns he so closely observed, she says, in all likelihood &amp;quot;he wouldn't mind being part of research, even after his death.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Why exactly did Mendel request an extensive autopsy be done after his death? Was it simply because &amp;quot;he was very enthusiastic for all kinds of research&amp;quot; in general, or were there some questions he wanted answered even after his death?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2023-01-02T21:45:40.077" Title="&quot;Just before he died, (Gregor) Mendel requested that an extensive autopsy be done.&quot; Why? Was he simply &quot;very enthusiastic for all kinds of research&quot;?" Tags="|medicine|genetics|mendel|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15073" PostTypeId="2" ParentId="15071" CreationDate="2022-12-31T01:20:28.280" Score="7" Body="&lt;p&gt;I just looked through the WWII issues of Crelle's Journal:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0182?tify=%7B%22pages%22%3A%5B3%5D%2C%22view%22%3A%22toc%22%7D&quot; rel=&quot;noreferrer&quot;&gt;Volume 182 (1940)&lt;/a&gt; contains four papers by &lt;a href=&quot;https://en.wikipedia.org/wiki/Philip_Hall&quot; rel=&quot;noreferrer&quot;&gt;Philip Hall&lt;/a&gt; in Cambridge. The first two had been submitted in the summer of 1939, before the war; but the latter two in 1940, when the UK was officially at war with the German Empire.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0184?tify=%7B%22pages%22%3A%5B3%5D%2C%22view%22%3A%22toc%22%7D&quot; rel=&quot;noreferrer&quot;&gt;Volume 184 (1942)&lt;/a&gt; contains one paper by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9&quot; rel=&quot;noreferrer&quot;&gt;Jean Dieudonné&lt;/a&gt; in Nancy, France, submitted in April 1941. At that time, however, France was formally no longer at war with Germany, as it had been split up into a German occupied zone (where Nancy was) and the infamous &amp;quot;Vichy France&amp;quot;.&#xA;(There is another paper in French in that issue, by a Spanish author. Franquist Spain was never at war with Germany.)&lt;/p&gt;&#xA;&lt;p&gt;Beyond that, a cursory look shows no papers by authors from countries at war with Germany. In fact, most papers come from Germany itself, plus a couple from Turkey (which at that time was neutral) and Japan (a German ally).&lt;/p&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2022-12-31T01:20:28.280" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15074" PostTypeId="2" ParentId="15071" CreationDate="2022-12-31T13:32:56.433" Score="3" Body="&lt;p&gt;I found this paper by Rosa Morris from Cambridge, published in 1940 in Math. Annalen &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0117?tify=%7B%22view%22:%22info%22,%22pages%22:%5B35%5D%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;The internal problems of two dimensional potential theory&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is also a paper from a mathematician by Brno (Brunn) which at the time was occupied.&lt;/p&gt;&#xA;&lt;p&gt;I've also found in German newspapers a couple of papers by Ky Fan, Chinese and studying in Paris. And a paper by Cartan in Zeitschrift in 1939, which could well have been handled before the war started.&lt;/p&gt;&#xA;&lt;p&gt;Of course you'll know well about nazification (and similar processes in Italy) of Editorial Boards which resulted almost certainly in making the whole process of publishing math papers from abroad quite unusual. On Italian journals (like Annali di Matematica Pura e Applicata) the Editorial Board had to be approved by the Minister of Culture and Education and there are letters showing that submissions from allied (I mean Axis Pact) countries were encouraged. None appeared from GB, France or USA. Even papers from Italian mathematicians not loyal to the regime were usually published abroad.&lt;/p&gt;&#xA;&lt;p&gt;(As for your personal position I think that as long as you do not receive any kind of pressure  your presence in the Editorial board of an Ukrainian journal should be something welcomed from the whole math community).&lt;/p&gt;&#xA;" OwnerUserId="1689" LastEditorUserId="1689" LastEditDate="2023-01-01T08:08:36.177" LastActivityDate="2023-01-01T08:08:36.177" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15075" PostTypeId="2" ParentId="15069" CreationDate="2023-01-01T15:31:57.040" Score="3" Body="&lt;p&gt;For a somewhat biased but nevertheless informative review of biological transmutation, see Jean-Paul Biberian's 2012 review: &lt;a href=&quot;https://www.researchgate.net/publication/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Biological Transmutations: Historical Perspective&amp;quot;&lt;/a&gt;. (&lt;a href=&quot;https://coldfusionnow.org/tag/jean-paul-biberian/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Biberian is a &amp;quot;cold fusion&amp;quot; advocate&lt;/a&gt;.)&#xA;No attempts to independently and directly reproduce Kervran's egg laying results are reported. It is, however, claimed that such transmutations by chickens were previously observed in 1799 and 1822.&lt;/p&gt;&#xA;&lt;p&gt;Kervran and others made claims beyond egg laying, e.g. for transmutation in microorganisms and plants, and Biberian's review did report a number of replications attempts for those. Unsurprisingly, some results supported biological transmutation and some did not. Few of the reported results - either negative or positive - are in the published research literature or otherwise easily accessible.&lt;/p&gt;&#xA;&lt;p&gt;The obvious place to look for recent biological transmutation research is the Journal of Condensed Matter Nuclear Science, which was founded to provide a place for (usually otherwise unpublishable) &amp;quot;cold fusion&amp;quot; research, and of which Biberian is Editor-in-Chief. &lt;a href=&quot;http://lenr-canr.org/acrobat/BiberianJPjcondensedza.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volume 28 from 2019&lt;/a&gt; reports on biological transformation of nuclear waste, and searching their &lt;a href=&quot;https://lenr-canr.org/wordpress/?page_id=1495&quot; rel=&quot;nofollow noreferrer&quot;&gt;index&lt;/a&gt; for &amp;quot;biological&amp;quot; turns up related articles as recent as 2022.&lt;/p&gt;&#xA;&lt;p&gt;That calcium for Kervran's eggs could have come from the bones of the chicken themselves is plausible, but no signs of bone demineralization were reported in &lt;a href=&quot;https://doi.org/10.1080/00071667508416227&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;A comparison of the effects of two low‐calcium diets on egg production in the domestic fowl&amp;quot;&lt;/a&gt;, Instead it was found that&#xA;hens fed on a diet containing 0.05% calcium simply almost completely stopped laying eggs. (Because of the commercial importance of eggs, there are many published papers like this on chicken diets.)&lt;/p&gt;&#xA;&lt;p&gt;Another plausible explanation is that Kervran underestimated the availability of calcium to his chickens.  The &lt;a href=&quot;https://archive.org/details/B-001-015-432/page/n64/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;original Kervran observation&lt;/a&gt; was that hens kept in chicken coops on calcium deficient clay soil stopped laying eggs, but started up again when given mica.  Similar results were observed with guinea-fowl.&#xA;Kervran assumed that the mica contained potassium but no calcium, so hypothesized that the chickens were transmuting potassium into calcium for the eggs.  The problem is that some types of mica (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Margarite&quot; rel=&quot;nofollow noreferrer&quot;&gt;Margarite&lt;/a&gt;) do in fact contain significant amounts of calcium. Without a chemical analysis of his mica, Kervran was making a big assumption that the mica did not contain enough calcium to sustain egg laying.&#xA;Biology can be very good at scavenging what it needs from even trace sources, as famously shown by the mistaken report of &lt;a href=&quot;https://www.nature.com/articles/nature.2012.11520&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;arsenic eating bacteria&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A general problem with fringe research is that there are many more ways to get science wrong than to get it right, so it is a thankless task to try to replicate or otherwise refute the endless stream of dubious and often mutually inconsistent positive results.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2023-01-03T00:36:17.143" LastActivityDate="2023-01-03T00:36:17.143" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15076" PostTypeId="2" ParentId="7642" CreationDate="2023-01-01T17:59:25.480" Score="9" Body="&lt;p&gt;&lt;strong&gt;Plate tectonics&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;It was put forth by a &lt;em&gt;meterologist&lt;/em&gt;, and offered no explanation how, given the solid crust and mantle, the continents could drift about like icebergs on the liquid core.&lt;/p&gt;&#xA;&lt;p&gt;The discovery of processes by which a solid surface could indeed drift about took decades, but finally triumphed.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2023-01-01T17:59:25.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15079" PostTypeId="2" ParentId="7642" CreationDate="2023-01-02T14:13:27.670" Score="-1" Body="&lt;p&gt;It is somewhat astonishing that a phenomenon  that was commonly thought of as belonging to the realm of parapsychology or superstition, has become an object of scientific investigation and its existence has been  increasingly accepted as possible in  scientific environments.&lt;/p&gt;&#xA;&lt;p&gt;I refer to &lt;strong&gt;telepathy&lt;/strong&gt;:  nowadays,  a sound and serious scientific research in the field is carried on, also by universities as MIT, Harvard  and Washington University. These new researches  suggest that some forms of telepathy are possible.&lt;/p&gt;&#xA;&lt;p&gt;Of course, we are not speaking of telepathy as told  by parapsychologists or occultism, as folding a fork with the force of the mind or voluntary  communicating with another person with the force of thought,  and of course not reproducible in everyday life.&lt;/p&gt;&#xA;&lt;p&gt;But the scientific world has been increasingly aware that some forms of communication between ‘brains’ are possible.&lt;/p&gt;&#xA;&lt;p&gt;A forerunner of neuroscience in regarding telepathy not as a superstition but an actual phenomenon, deserving attention even if not understood, can be considered psychoanalysis.&lt;/p&gt;&#xA;&lt;p&gt;It is enough well known that telepathy is a phenomenon that has been regarded,  for  many years,   not as paranormal or pseudo- scientific by serious communities as Freudian societies: the Freudian societies are  deemed as the more ‘scientific’ and rigorous psychoanalytic communities, not at all inclined to accept fancy or imaginative phenomena.&lt;/p&gt;&#xA;&lt;p&gt;Freud itself paid attention to telepathy and to which he called ‘unconscious communication’ mainly in two works, &lt;em&gt;Psychoanalysis and Telepathy&lt;/em&gt; and &lt;em&gt;Dream and telepathy&lt;/em&gt;, both written in 1921.&lt;/p&gt;&#xA;&lt;p&gt;A controversial debate took place in those years about the matter, in particular there was a&lt;br /&gt;&#xA;disagreement with Ernest Jones,  about  the explicit ‘resistance’ of Freud and other psychoanalysts towards  extrasensory phenomena. And a general tendency to ignore these phenomena  followed in subsequent years .&lt;/p&gt;&#xA;&lt;p&gt;But Freud kept alive his long-standing   curiosity toward the transmission of thought and extrasensory phenomena,  as witnessed for instance by several essays he devoted to dreams and occultism in the period 1921-1932.   Freud states clearly that he is very skeptical about supernatural phenomena,  but at the same time his attitude became more positive, particularly toward telepathy.&lt;/p&gt;&#xA;&lt;p&gt;In psychoanalytic communities, in more recent times, has been increasingly accepted that some experiences of telepathy, not explained but frequently observed, occur  in  psychoanalytical  clinical context,   where some strange, unexpected,  ‘telepathic’ phenomena manifest themselves (mainly in dreams), in which someone guesses some thought or some aspect of the life of another person, without any apparent information about them. For examples of papers about this subject see (1).&lt;/p&gt;&#xA;&lt;p&gt;If  psychoanalysis can be considered science or not is subject of discussion. But, of course, these are  not scientific investigations in the sense of ‘hard sciences’ as commonly understood.&lt;/p&gt;&#xA;&lt;p&gt;But, nowadays , the possibility of telepathy isn't any longer  considered an irrational and fancy hypothesis also by ‘harder’ scientists as neuroscientists. It  is viewed as a possible field of research in neurosciences  about the functioning of the human brain,  about  which  we don’t still know very much.&lt;/p&gt;&#xA;&lt;p&gt;This is a recent news by BBC:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Scientists 'make telepathy breakthrough'&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Research led by experts at Harvard University shows technology can be used to send a simple mental message from one person to another without any contact between the two.&#xA;Neuroscientist Giulio Ruffini told the Today programme: &amp;quot;You can actually transmit information directly from one brain to another brain without intervention of the senses.&amp;quot;&#xA;&amp;quot;The next step would be to try to find more powerful techniques to send more complex &amp;gt;information,&amp;quot; he added.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.bbc.com/news/av/science-environment-29093700&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.bbc.com/news/av/science-environment-29093700&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Below an article on the matter of a serious and well-known Italian review of scientific dissemination, &lt;em&gt;Focus&lt;/em&gt;. I translate the beginning of the article of &lt;em&gt;Focus&lt;/em&gt;,&lt;/p&gt;&#xA;&lt;p&gt;From &lt;em&gt;Focus&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Is telepathy scientifically  possible?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Not exactly as in movies, but something similar to telepathy exists.&#xA;And research is making progress.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In scientific terms, we speak of brain-to-brain communication, and&#xA;some experiments of recent years prove that it is possible to put in&#xA;connection brains of several individual through a ‘computerized  neural&#xA;interface’, called BCI ( brain-computer interface). This instrument registers, encodes and decodes electrical impulses, which can be transmitted by radio, wi-fi, internet and so on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.focus.it/scienza/scienze/la-telepatia-e-scientificamente-possibile&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.focus.it/scienza/scienze/la-telepatia-e-scientificamente-possibile&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other related links, from MIT,  are:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.technologyreview.com/2014/11/21/170337/are-telepathy-experiments-stunts-or-science/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.technologyreview.com/2014/11/21/170337/are-telepathy-experiments-stunts-or-science/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://massivesci.com/articles/mind-control-brain-machine-interface/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://massivesci.com/articles/mind-control-brain-machine-interface/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In short, therefore, we can say that telepathy is in progress of being carried from pseudoscience to science, even if it seems, as far as I know,  that research in the field is still at its beginning.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(1) For examples of books and papers  about telepathy and psychoanalysis :&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.it/Sigmund-Freud-Forsyth-Case-Thought-Transmission/dp/1032159588/ref=sr_1_2?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;amp;crid=7KZZDYY2O7CV&amp;amp;keywords=maria+pierri+telepathy&amp;amp;qid=1672672827&amp;amp;s=books&amp;amp;sprefix=maria+pierri+telepathy%2Cstripbooks%2C147&amp;amp;sr=1-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.it/Sigmund-Freud-Forsyth-Case-Thought-Transmission/dp/1032159588/ref=sr_1_2?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;amp;crid=7KZZDYY2O7CV&amp;amp;keywords=maria+pierri+telepathy&amp;amp;qid=1672672827&amp;amp;s=books&amp;amp;sprefix=maria+pierri+telepathy%2Cstripbooks%2C147&amp;amp;sr=1-2&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://pubmed.ncbi.nlm.nih.gov/6853049/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://pubmed.ncbi.nlm.nih.gov/6853049/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/10481885.2014.870841&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/abs/10.1080/10481885.2014.870841&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.torrossa.com/it/resources/an/4312142&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.torrossa.com/it/resources/an/4312142&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This last paper is by some of the most prominent and appreciated Italian psychoanalysts.&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-01-02T15:22:48.420" LastActivityDate="2023-01-02T15:22:48.420" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15080" PostTypeId="2" ParentId="15072" CreationDate="2023-01-02T21:45:40.077" Score="3" Body="&lt;p&gt;The first comprehensive biography of Gregor Mendel was Hugo Iltis, &lt;em&gt;Gregor Johann Mendel: Leben, Werk und Wirkung&lt;/em&gt;, Berlin: J. Springer 1924. On p. &lt;a href=&quot;https://books.google.com/books?id=AD0MAAAAMAAJ&amp;amp;ots=UyLpfOkVGW&amp;amp;dq=hugo%20iltis&amp;amp;lr&amp;amp;pg=PA195#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;195&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Aus Furcht vor dem Scheintod und um die Diagnose seiner Krankheit ganz sicher zu stellen, verlangte er ausdrücklich die Sektion und hatte insgeheim kurz vor seinem Tod einen Klosterfunktionär durch Eidschwur zur Durchführung seines Wunsches verpflichtet. ...&lt;br /&gt;&#xA;Bei der Sektion, die von Krankenhausdirektor Dr. Brenner geleitet wurde, war auch der Neffe, Dr. Alois Schindler, damals Mediziner, anwesend. Man konstantierte außer den früher genannten Symptomen chronische Nierenentzündung (Morbus Brightii) als eigentliche Krankheit, die durch Herzhypertrophie kompliziert war.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Due to fear of apparent death and to confirm diagnosis of his illness with certainty, he expressly requested autopsy and shortly before his death had secretly pledged a monastery official by oath to carry out his wish. [...] During the autopsy, which was led by hospital director Dr. Brenner, [Mendel's] nephew Dr. Alois Schindler, then a physician, was also present. Besides the previously mentioned symptoms, chronic inflammation of the kidneys (Bright's disease) was identified as the actual underlying illness, complicated by hypertrophy of the heart.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2023-01-02T21:45:40.077" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15081" PostTypeId="1" CreationDate="2023-01-03T10:37:27.200" Score="2" ViewCount="138" Body="&lt;p&gt;In the 1964 adaptation of the James Bond novel &amp;quot;Goldfinger&amp;quot;, in a famous scene, Bond is bound to a table and threatened to be cut in half by a powerful laser that Goldfinger later uses to break into Fort Knox (while in the novel it was a table saw, apparently). The first operational laser was built in 1960, only four years prior to the films release. It seems astonishing to me that in such a short amount of time the technology made its way into popular movies already.&lt;/p&gt;&#xA;&lt;p&gt;Was a laser a well-known concept to the public by then? Would a laser that somebody with enough funds could build in the early 1960s be able to cut/melt metal? Or was the use of the laser in the movie seen as a futuristic &amp;quot;spy movie gadget&amp;quot;?&lt;/p&gt;&#xA;" OwnerUserId="17756" LastActivityDate="2023-01-03T10:37:27.200" Title="How futuristic was the use of a laser in 1964's &quot;Goldfinger&quot;?" Tags="|physics|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15082" PostTypeId="2" ParentId="13344" CreationDate="2023-01-04T02:09:53.217" Score="5" Body="&lt;p&gt;Originally, the Greek word κλῐ́μᾰ means “slope, incline, inclination,” and has nothing to do with the modern meaning of the word (long-term weather). Ptolemy defined the seven &lt;em&gt;climata&lt;/em&gt; by the length of the longest duration for the day (defined as the period when the sun is visible in the sky, and as opposed to night, when it’s not visible; the combination of both day and night periods [24 hours] is a &lt;em&gt;nycthemera&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;One clima differs from the next by a half-hour of day per nycthemera. The map you link to does have mentions of the duration of the day: for example, the second line from the bottom says “3[?] paral[x] habet maximũ diem hora[x] 12½¼” (where the [?] means I’m not sure it’s a 3; the [x]s are characters I can’t reproduce; and ½¼ is to be read as ½ + ¼ = ¾).&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy believed that the equatorial regions of the Earth were too hot to be inhabited, so he counted only “habitable” climata. The first one, as you pointed out, is the parallel of Meroe, where the longest day (June solstice) lasts 13 hours. Then for Syene, the longest day would last 13½ hours, and so on, until the clima of the Borysthenes (modern Dniepr river in Ukraine) at 48° 32′ and a longest daytime of 16 hours.&lt;/p&gt;&#xA;&lt;p&gt;As for the mention of climata in the works of later authors, most of them, if not all, do indeed correspond to the Ptolemaic climata.&lt;/p&gt;&#xA;" OwnerUserId="13368" LastActivityDate="2023-01-04T02:09:53.217" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15083" PostTypeId="1" CreationDate="2023-01-04T22:16:40.033" Score="0" ViewCount="86" Body="&lt;p&gt;Chern-Simons theory is a topological gauge field theory and play a prominent role in many brnaches of phyiscs and mathematics. On the physical side, it appears for example in three-dimensional theories of gravity, in the description of the fractional Quantum-Hall effect in condensed matter physics, as well as in high-energy and mathematical physics, like in string theory, holography and relations to conformal field theories. On the other hand, it has been studies by mathematicians, as it is deeply connected to knot invariants and three-dimensional topology.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know some text, like a book or review, about the history of Chern-Simons theory? Chern-Simons theory was first defined (in its Abelian version) in a paper of Schwarz. However, afterwards the history is less clear to me and it seems to be very non-linear. Of course, one of the most import historical works are the works of Witten on Chern-Simons theory and knot theory, but there seems to be also many other works around the same time on this theory.&lt;/p&gt;&#xA;&lt;p&gt;I am interested in the history of general Chern-Simons theory, both the classical and quantum case.&lt;/p&gt;&#xA;" OwnerUserId="17762" LastActivityDate="2023-01-04T22:16:40.033" Title="Reference Request: History of Chern-Simons Theory" Tags="|physics|reference-request|theoretical-physics|quantum-field-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15084" PostTypeId="1" CreationDate="2023-01-05T15:31:30.167" Score="1" ViewCount="79" Body="&lt;p&gt;Any resource recommendations for history of physics giving special attention to experiments that led to certain concepts and theorems in physics? For example what led to the motivation of definition of torque or work energy theorem and so on.&lt;/p&gt;&#xA;&lt;p&gt;I have asked several questions (&lt;a href=&quot;https://physics.stackexchange.com/questions/702983/motivation-for-the-definition-of-angular-momentum&quot;&gt;1&lt;/a&gt;, &lt;a href=&quot;https://physics.stackexchange.com/q/741591/330899&quot;&gt;2&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/q/14637/16986&quot;&gt;3&lt;/a&gt;) but never got a satisfactory answer, thus this question.&lt;/p&gt;&#xA;" OwnerUserId="16986" LastEditorUserId="12330" LastEditDate="2023-01-06T05:07:37.267" LastActivityDate="2023-01-06T05:07:37.267" Title="Resource recommendation for experiments that led to major concepts in physics" Tags="|physics|experimental-physics|resource-recommendation|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15085" PostTypeId="1" AcceptedAnswerId="15090" CreationDate="2023-01-05T16:55:06.757" Score="2" ViewCount="305" Body="&lt;p&gt;So, Roger Penrose is a bright guy, I mean, he won the Nobel Prize, but the Penrose-Lucas argument that the human mind is a hypercomputer based on Godel's Second Incompleteness Theorem is laughably bad. For a start, it's not really that original. I think the reason the Second Incompleteness freaks out a lot of people, to the point that they claim mathematics is a kind of fantasy based on it, is they come up with a vague version of this argument on their own. I certainly did in high school.&lt;/p&gt;&#xA;&lt;p&gt;Godel's Second Incompleteness Theorem states that any computable, consistent formal system capable of doing arithmetic cannot prove its own consistency. Penrose says the brain is a formal system, it's consistent, it believes it's consistent, so it can't be computable. But you can only make that inference when you're unfamiliar with the details of the proof of the Second Incompleteness Theorem.&lt;/p&gt;&#xA;&lt;p&gt;When we talk about a system proving its own consistency, at a high level, it means it encodes itself into the objects it considers in a way that plays nicely with its deductive rules, to the point that it is capable of giving a complete description of those rules in itself, and stating that that set of rules does not lead to any contradictions. I don't know the complete rules of my mind's deductive system. I know I accept ZFC, its consistency, its consistency's consistency, some large cardinals, but I don't know if there's nothing else I could add to that list. I just have this term in my formal language that refers to my formal system, and I can state some things about it, but I don't know the rules it corresponds to at the level of the Incompleteness Theorems, so it's fine. Similarly, I have this pronoun &amp;quot;me&amp;quot; that I call myself, but that's not the same as having a description of the state of every particle that constitutes me.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, Godel's Incompleteness Theorems don't really need computability. Just some regime capable describing itself. So we're led to the crossroads that we either can't describe the mind, or a modified Second Theorem (really, you could make the case based on Godel sentences in the original proof of the First or via the Halting Problem) where we can't know everything detail about it within that description.&lt;/p&gt;&#xA;&lt;p&gt;It strikes me as incredibly strange that Penrose would have stuck to his guns for so long on this. It's easy for me to see the cause of his mistake, he's not a logician and a naive understanding can easily lead one astray on this issue. But despite air-tight criticism from people who know the subject much better, he's stayed on. So I want to know if he's ever engaged with criticism on this front, or just ignored it, because the verdict is essentially universal that his argument is invalid even allowing its premises. If he has acknowledged it, why hasn't he budged?&lt;/p&gt;&#xA;" OwnerUserId="17323" LastActivityDate="2023-01-08T04:24:26.993" Title="Has Penrose ever acknowledged criticism of the Penrose-Lucas argument?" Tags="|biographical-details|mathematical-logic|computation|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15086" PostTypeId="2" ParentId="11954" CreationDate="2023-01-05T23:11:27.310" Score="4" Body="&lt;p&gt;Hilbert devoted a whole paper to infinity, &lt;em&gt;On the infinity&lt;/em&gt;, published in 1925 (1).&lt;/p&gt;&#xA;&lt;p&gt;Hilbert was one of the most tenacious defenders of the &amp;quot;paradise of the infinity&amp;quot;  revealed by Cantor's work. His paper was aimed at 'saving' the Cantorian set theory approach to infinity from the paradoxes and antinomies generated by  actual infinity during the so-called foundation crisis of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;On the infinity&lt;/em&gt;  he makes a short account of the concept of infinity in analysis, set theory, physics and logic, and  he speaks explicitly of &lt;em&gt;potential&lt;/em&gt; and &lt;em&gt;actual&lt;/em&gt; infinity:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we wanted to characterize briefly the  new conceptions of the&#xA;infinity that Cantor introduced,  we could no doubt say: in analysis&#xA;we deal with the infinitely small or the infinitely large  only as a&#xA;limit notion - as something that is becoming, coming to be, being&#xA;produced - that is, as we say, with the &lt;em&gt;potential infinity&lt;/em&gt;. But this is  not the real infinity itself.&#xA;That we have when, for example, we consider the totality of the&#xA;numbers &lt;span class=&quot;math-container&quot;&gt;$1, 2, 3, 4,...$&lt;/span&gt; itself as a completed entity, or when we&#xA;regard the points of a line segment as a totality of objects that is&#xA;actually given and complete. This kind of infinity is called &lt;em&gt;actual&#xA;infinity&lt;/em&gt;. (2)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On the contrary, Poincaré doesn’t accept Cantor’s actual infinity. He addressed the problem of Cantor’s infinity in his speech &lt;em&gt;L’avenir des mathématiques&lt;/em&gt; at the International Conference of Mathematicians, held in Rome in 1908, in which he states that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Cantorism” was “a good pathological case”, which could make happy&#xA;the physician called to treat it.(3)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Poincaré devotes several papers to infinity, his last article about the subject is&#xA;&lt;em&gt;La logique de l’infinie&lt;/em&gt;, published  in 1909.  He in particular speaks of &lt;em&gt;actual infinity&lt;/em&gt; in&#xA;&lt;em&gt;La mathématique e la logique&lt;/em&gt; (1905):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It has been the belief in the existence of actual infinity which has&#xA;generated these impredicative definitions […]&#xA;&lt;em&gt;There insn’t such a thing as actual infinity:&lt;/em&gt; Cantorians forgot it and fell into contradictions. (4)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(1) Hilbert, &lt;em&gt;On the infinity&lt;/em&gt; (1925), in J. van Heijenoort ed., &lt;em&gt;From Frege to Gödel&lt;/em&gt;, 1967, Cambridge University Press.&lt;/p&gt;&#xA;&lt;p&gt;(2) &lt;em&gt;Ibid.&lt;/em&gt;, p.373.&lt;/p&gt;&#xA;&lt;p&gt;(3) U. Bottazzini, &lt;em&gt;Infinito&lt;/em&gt;, Il Mulino, 2018, p. 254, my translation.&lt;/p&gt;&#xA;&lt;p&gt;(4) Poincaré, &lt;em&gt;La mathématique e la logique&lt;/em&gt; (1905), p. 55&#xA;&lt;a href=&quot;http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Les%20Mathematiques%20Et%20La%20Logique.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Les%20Mathematiques%20Et%20La%20Logique.pdf&lt;/a&gt;.  (my translation).&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-01-20T09:59:04.037" LastActivityDate="2023-01-20T09:59:04.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15088" PostTypeId="1" CreationDate="2023-01-07T12:36:27.060" Score="2" ViewCount="85" Body="&lt;p&gt;Wikipedia describes the Ideal Gas Law as coming from a combination of Boyle's Law, Charles' Law, Avogadro's Law, and Gay-Lussac's Law. It gives several derivations, both empirical and theoretical, from kinetic theory and from statistical mechanics.&lt;/p&gt;&#xA;&lt;p&gt;However, the article does not have any historical perspective. What I want to know is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Who was it that first combined these different laws into the familiar PV=nRT, and when?&lt;/li&gt;&#xA;&lt;li&gt;What derivation technique did he or she use at the time?&lt;/li&gt;&#xA;&lt;li&gt;Was it called the &amp;quot;Ideal Gas Law&amp;quot; at the time, or was that a later label?&lt;/li&gt;&#xA;&lt;li&gt;What was the state of knowledge of gas theory at the time? Were there competing theories that were taken seriously (e.g. non-atomic), or was the scientific community fairly unified around the kinetic theory?&lt;/li&gt;&#xA;&lt;li&gt;Was it considered a fatal problem that the ideal gas law didn't accurately describe real gases, or were people happy with the approximation? By &amp;quot;fatal problem&amp;quot;, I mean, did people think there had to be a radical re-think of gas theory, or did they think the inaccuracies could be accommodated by small tweaks to the existing gas theory?&lt;/li&gt;&#xA;&lt;li&gt;What did people think were the reasons for why it isn't quite right for real gases? Obviously nowadays we know why, but what were the different hypotheses at the time?&lt;/li&gt;&#xA;&lt;li&gt;Did the unification have practical consequences in industry and engineering (e.g. steam engines), or were engineers already using the separate gas laws to the same effect? Or were engineers still using &amp;quot;rules of thumb&amp;quot; and not applying the cutting-edge theory just yet?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="17777" LastEditorUserId="17777" LastEditDate="2023-01-07T12:44:26.183" LastActivityDate="2023-01-07T12:44:26.183" Title="What was the chronologically first derivation PV=nRT?" Tags="|thermodynamics|steam-engines|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15089" PostTypeId="2" ParentId="7642" CreationDate="2023-01-07T13:01:41.790" Score="7" Body="&lt;p&gt;The &lt;strong&gt;germ theory of disease&lt;/strong&gt;, and in general &lt;strong&gt;the importance of handwashing&lt;/strong&gt; was a fringe viewpoint for a long time. Good evidence existed since at least the early 19th century that micro-organisms could cause disease, but the medical profession was very slow on the uptake.&lt;/p&gt;&#xA;&lt;p&gt;Dr Semmelweis, though he lacked any theoretical model, saved the lives of many mothers and newborns by his insistence on handwashing in his maternity ward. Despite strong empirical evidence, the scientific community largely hated his ideas, and the acrimony led to the poor doctor suffering a nervous breakdown and premature death.&lt;/p&gt;&#xA;" OwnerUserId="17777" LastActivityDate="2023-01-07T13:01:41.790" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15090" PostTypeId="2" ParentId="15085" CreationDate="2023-01-07T19:01:48.470" Score="6" Body="&lt;p&gt;Penrose responded to various commentaries made regarding his 1994 book &amp;quot;Shadows of the Mind&amp;quot; in his &lt;a href=&quot;https://www.calculemus.org/MathUniversalis/NS/10/01penrose.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beyond the Doubting of a Shadow&lt;/a&gt;. I never spent much time reading over this since my initial encounter with Penrose's argument was Hilary Putnam's rather scathing and convincing criticism of his claim, published in a 1995 &lt;a href=&quot;https://www.ams.org/journals/bull/1995-32-03/S0273-0979-1995-00606-3/S0273-0979-1995-00606-3.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;book review&lt;/a&gt; for the Bulletin of the American Mathematical Society. But it does show that Penrose at least did engage with some of his critics.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2023-01-08T04:24:26.993" LastActivityDate="2023-01-08T04:24:26.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15091" PostTypeId="2" ParentId="5025" CreationDate="2023-01-08T05:44:23.223" Score="-1" Body="&lt;p&gt;A translation attributed to John Wells (see &lt;a href=&quot;http://darshanapress.com/The%20Vaisheshika%20Darshana.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://darshanapress.com/The%20Vaisheshika%20Darshana.pdf&lt;/a&gt;), of कारणाभावात्कार्याभावः। न तु कार्याभावात्कारणाभावः reads thus: &amp;quot;1.10 Absence of effect is known by the absence of a cause, 1.11 but absence of cause is not (necessarily) known by the absence of an effect.&amp;quot; (Wells, 2009, p. 4)&lt;/p&gt;&#xA;&lt;p&gt;Wells' translation invoked above is listed as the translation of the critical edition of the वैशेषिक सूत्र (Vaiśeṣika Sūtra) [see footnote 17 in https://en.wikipedia.org/wiki/Vai%C5%9Be%E1%B9%A3ika_S%C5%ABtra#cite_note-FOOTNOTEBimal_Krishna_Matilal197755%E2%80%9356-20] attributed to कणाद (Kaṇāda).&lt;/p&gt;&#xA;&lt;p&gt;Citing Fowler (2002, pp. 98-99, Perspectives of Reality: An Introduction to the Philosophy of Hinduism) and Margenau (2012, pp. xxx-xxxi, Physics and Philosophy: Selected Essays) the link invoked immediately above includes the following about when the Vaiśeṣika Sūtra was likely compiled: &amp;quot;...the Vaiśeṣika Sūtra was likely compiled sometime between 6th and 2nd century BCE,...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In view of i) all of the above, and ii) in the context of the question — When did mathematicians first use the contrapositive form to prove a conditional statement? — the following points seem noteworthy:&lt;/p&gt;&#xA;&lt;p&gt;a) 1.10 above — कारणाभावात्कार्याभावः (&amp;quot;Absence of effect is known by the absence of a cause&amp;quot;) — is, it can be reasonably posited, very very close to &amp;quot;If not B, then not A&amp;quot; (given &amp;quot;If A then B&amp;quot;), basically what is referred to as &amp;quot;Proof by Contrapositive.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;b) Interestingly, 1.11 above — न तु कार्याभावात्कारणाभावः (&amp;quot;but absence of cause is not (necessarily) known by the absence of an effect&amp;quot;) does not stop at the equivalent of &amp;quot;If not B, then not A&amp;quot; (basically 1.10 above), but goes on to add the equivalent of the point that &amp;quot;If A then B,&amp;quot; does not necessarily mean &amp;quot;If not A then not B.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In my view, any history of the idea of what is today referred to as &amp;quot;Proof by Contrapositive&amp;quot; &lt;a href=&quot;https://en.wikipedia.org/wiki/Proof_by_contrapositive#:%7E:text=In%20mathematics%2C%20proof%20by%20contrapositive,%2C%20then%20not%20A%22%20instead&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Proof_by_contrapositive#:~:text=In%20mathematics%2C%20proof%20by%20contrapositive,%2C%20then%20not%20A%22%20instead&lt;/a&gt;. that aspires to be truly global and inclusive, will need to, at the very least, include a mention of the वैशेषिक सूत्र (Vaiśeṣika Sūtra) attributed to कणाद (Kaṇāda), if not presenting &amp;quot;कारणाभावात्कार्याभावः। न तु कार्याभावात्कारणाभावः&amp;quot; from the वैशेषिक सूत्र (Vaiśeṣika Sūtra) attributed to कणाद (Kaṇāda) as amongst the earliest, if not the earliest, extant statement related to &amp;quot;Proof by Contrapositive&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="17779" LastActivityDate="2023-01-08T05:44:23.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15092" PostTypeId="1" AcceptedAnswerId="15233" CreationDate="2023-01-08T18:05:26.487" Score="3" ViewCount="364" Body="&lt;p&gt;Is there a theorem proof whose accuracy is doubted because it is short?&lt;/p&gt;&#xA;&lt;p&gt;He told me while chatting with a friend of mine. It's about a mathematician who proves a difficult theorem very briefly and simply. Those who have examined the proof have hesitated to confirm it. &amp;quot;Let's not rush it because it seems too short and simple to be true!&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;If this sort of thing did happen, I'm guessing it might be the proof Leroy Milton Kelly gave in 1948 for &lt;a href=&quot;https://mathworld.wolfram.com/SylvestersLineProblem.html#:%7E:text=This%20problem%20was%20proposed%20by,in%20the%20same%20right%20line.%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sylvester's Line Problem (1893)&lt;/a&gt;. Because the problem is tricky and Kelly's proof is unexpectedly short. (I have no arguments to support my guess, I am not assertive about it.)&lt;/p&gt;&#xA;&lt;p&gt;Have you heard of a similar story? What do you think is the reality?&lt;/p&gt;&#xA;" OwnerUserId="17698" LastEditorUserId="4251" LastEditDate="2023-01-10T17:43:01.057" LastActivityDate="2023-03-17T22:47:12.467" Title="Is there a theorem proof whose accuracy is doubted because it is short?" Tags="|mathematics|mathematicians|mathematics-social-history|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15093" PostTypeId="1" CreationDate="2023-01-09T19:38:20.133" Score="4" ViewCount="139" Body="&lt;p&gt;In 2006, the renowned physicist Freeman Dyson wrote a letter to his colleague Gerald Gabrielse regarding an advance in precision about measuring the magnetic moment of an electron. An excerpt of his letter is quoted in a news report available on the CERN website.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dear Gerald...As one of the inventors [of QED], I remember that&#xA;we thought of QED in 1949 as a temporary and jerry-built&#xA;structure, with mathematical inconsistencies and renormalized&#xA;infinities swept under the rug. We did not expect it to last more&#xA;than 10 years before some more solidly built theory would&#xA;replace it...Now, 57 years have gone by and that ramshackle&#xA;structure still stands...It is amazing that you can measure her&#xA;dance to one part per trillion and find her still following our beat.&#xA;With congratulations and good wishes for more such beautiful&#xA;experiments, yours ever, Freeman Dyson.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does anyone happen to know where the full letter is located? I'm trying to study the history behind the development of QED, and to my understanding, Dyson actually left the field after he realized that his series connecting the theories of Feynmann, Schwinger, and Tomonaga diverged under certain circumstances. From the excerpt, it appears that Dyson is still deeply critical of the theoretical footing of the theory despite praising its agreement with experimental results, though I would feel confident in wagering this opinion if the full letter was available.&lt;/p&gt;&#xA;" OwnerUserId="14053" LastActivityDate="2023-01-09T19:38:20.133" Title="Finding the Letter from Freeman Dyson to Gerald Gabrielse in 2006" Tags="|reference-request|quantum-field-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15094" PostTypeId="1" CreationDate="2023-01-10T09:48:28.820" Score="0" ViewCount="106" Body="&lt;p&gt;France research pioneered the discovery of radioactivity and nuclear research. However looking at the names of non-US Americans in the Manhattan project there are very few French names (maybe just Bertrand Goldschmidt?) so I wonder what the famous French nuclear scientists were doing at the time. Were the French developing their own nuclear weapons project during World War II or were they discouraged from making one due to the cost of war? Any good sources on the status of nuclear weapon research in France during the war would be welcome.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastActivityDate="2023-01-10T09:48:28.820" Title="Was there a French nuclear weapons project during WWII?" Tags="|physics|nuclear-physics|france|atomic-weapons|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15095" PostTypeId="1" AcceptedAnswerId="15100" CreationDate="2023-01-10T18:53:11.143" Score="4" ViewCount="148" Body="&lt;p&gt;In &lt;em&gt;Quantum-Theoretical Re-Interpretation&lt;/em&gt; (1925), Heisenberg gives the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In order to characterize this radiation [of an electron] we first need the frequencies which appear as functions of two variables. In quantum theory these functions are of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\nu(n,\alpha) = \frac{1}{h}\left[W(n)-W(n-\alpha)\right],&#xA;$$&lt;/span&gt;&#xA;and in classical theory of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\nu(n,\alpha) = \alpha\nu(n) = \alpha \frac{1}{h} \frac{dW}{dn}.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; is frequency, &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; in the first equation is the quantum number of the first state, &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; is an integer such that the second state's quantum number is &lt;span class=&quot;math-container&quot;&gt;$n-\alpha$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$W$&lt;/span&gt; is the energy. Where does the classical equation come from, and how is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to be interpreted? (It might be more specific to ask, how do we derive the classical equation from the correspondence principle?)&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2023-01-11T13:58:58.930" Title="From where did Heisenberg (1925) obtain the classical frequency equation?" Tags="|quantum-mechanics|heisenberg|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15096" PostTypeId="1" CreationDate="2023-01-10T21:50:11.310" Score="20" ViewCount="5234" Body="&lt;p&gt;Today the standard for fine grit is Alumina and Cerium Oxide. However, in the 1600s, the elements Al and Ce were not known.&lt;/p&gt;&#xA;&lt;p&gt;What type of fine grit did they use to grind and polish the lenses for telescopes?&lt;/p&gt;&#xA;&lt;p&gt;I've hunted for this online quite a lot, but cannot find a single website that talks about grit material. They talk a little about grinding machines, but I'm not at all interested in that right now. From what I gather on amateur telescope makers on YouTube, one person can grind out a lens in a day or two, and it's not back-breaking work, just time-consuming.&lt;/p&gt;&#xA;&lt;p&gt;I understand the basic regiment of using coarse grit, then smaller grit, and eventually to the smallest finest grit possible which today is apparently Cerium Oxide. Remember that visible light is on the order of 400 nm wavelength, so ideally you should grind it down to be smooth at that size, which requires some really fine abrasive grit particles.&lt;/p&gt;&#xA;" OwnerUserId="2546" LastEditorUserId="16318" LastEditDate="2023-01-11T01:45:58.487" LastActivityDate="2023-05-31T09:22:30.717" Title="What type of abrasive grit was used to grind lenses for telescopes?" Tags="|optics|telescope|materials-science|" AnswerCount="3" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15097" PostTypeId="2" ParentId="15096" CreationDate="2023-01-11T02:46:47.113" Score="22" Body="&lt;p&gt;There is an entire article devoted to your query:&lt;/p&gt;&#xA;&lt;p&gt;From &amp;quot;&lt;em&gt;Lens Making for Scientific Instrumentation in the Seventeenth Century&lt;/em&gt;&amp;quot; by&#xA;Silvio A. Bedini (it is behind pay-wall), Appl. Opt. 5, 687-694 (1966).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The glass blank was fastened to a muller, or handle, and held against&#xA;the turning pattern. The mullers were made of wood or hollowed stone.&#xA;One type of glue was made from an infusion of liquified yellow pitch,&#xA;resin, and finely ground clay to which was added a dram of pulverized&#xA;glass. Another glue for the same purpose consisted of melted black&#xA;Spanish pitch, sealing wax, and plaster.&lt;/p&gt;&#xA;&lt;p&gt;Traditional abrasives were used for lens grinding and polishing.&#xA;According to Bolantio, the best abrasive was Venetian rottenstone as&#xA;it could be pulverized and levigated so that gritty particles were&#xA;reduced to a minimum. Well-purged and minutely ground emery was&#xA;normally used for polishing; white sand and tripoli were also common.&#xA;For the final polishing, the pattern was replaced by a wooden form of&#xA;the same shape. The fine abrasive was dampened and applied to a&#xA;buckskin fitted tightly over the form. &lt;span class=&quot;math-container&quot;&gt;${ }^{10}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From there, reference 10, appears to be very relevant &amp;quot;From an Italian 17th-century manuscript account entitled &lt;em&gt;&amp;quot;A Treatise on Optics by Giovanni Christoforo Bolantio&amp;quot;&#xA;which forms part of a bound manuscript volume with the general title &amp;quot;Trattato Sulla Maniera per Costruire Orologij&amp;quot; in the author's collection.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Perhaps a very serious scholar of science history can get hold of that treatise through major world libraries. I have no idea.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2023-01-11T02:46:47.113" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15098" PostTypeId="2" ParentId="15096" CreationDate="2023-01-11T05:36:55.463" Score="15" Body="&lt;p&gt;One of the early publications covering the topic is Pere Cherubin d'Orleans, &lt;em&gt;&lt;strong&gt;La Dioptrique Oculaire&lt;/strong&gt;, ou La Theorique, La Positive, et La Mechanique, de l'Oculaire Dioptrique en Toutes Ses Especes.&lt;/em&gt; Paris: Thomas Jolly &amp;amp; Simon Benard, 1671. Polishing materials are discussed on &lt;a href=&quot;https://www.google.com/books/edition/La_Dioptrique_oculaire_ou_la_Th%C3%A9orique/MbZgfXQ13G4C?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA353&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 353&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;De La Matiere, Du Mordant&lt;/p&gt;&#xA;&lt;p&gt;Dans le besoin, l'on ne fait pas élection particuliere, de la matiere du mordant; le sable de riviere, peut passablement servir, à former le verre, il ne dure pas neantmoins assez au travail, perdant en peu de temps sa pointe, il est trop lent en son effet. L'émeril, au contraire, es trop dur, pour le verre; &amp;amp; pour les platines de leton: ausquelles, il laisse une mauvaise qualité, incrustant leur superficie, de sa poussiere; qui y retenant long temps sa pointe, gaste ordinairement les verres, que l'on veut adoucir, &amp;amp; conduire au polir: les rayant de traits, que l'on ne peut souvent oster, sans recommencer le travail, ce qui est importun. Le grez mediocrement dur, est donc le plus propre pour ce travail; il se trouve assez communément, chez ceux qui se servent de meules à aiguiser, lesquelles se rompants souvent, leur demeurement inutiles, à autres choses. Ce grez n'a besoin d'autre preparation, que d'estre broyé en poudre, l'on en doit toûjours conserver séparément, de trois, ou quatre degrez de force: pour les employer selon la qualité du travail. On les séparera fort commodément, mettant tout le grez qui a servy, dans un grand vaisseau plein d'eau, &amp;amp; le mouvant bien; car le laissant un peu r'asseoir, tout le plus gros ira au fond; &amp;amp; alors inclinant promptement ce vaisseau, pour remettre toute cette eau dans un autre, tout le plus subtil, s'y écoulera avec l'eau; que l'on laissera entierement r'asseoir, écoulant en suite doucement l'eau, pour avoir le grez, qui sera demeuré auf fond. Ce que reїterant diverses fois, l'on aura séparément, tant de degrez de force, de ce grez; que l'on voudra. On le conservera en des vaisseaux, séparément.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Father Cherubin states that if need be, ordinary river sand could be used to grind glass, but that it is too soft and will not last long. &lt;a href=&quot;https://www.merriam-webster.com/dictionary/emery&quot; rel=&quot;noreferrer&quot;&gt;Emery&lt;/a&gt; on the other hand is so hard that it causes deep scratches that are hard to polish away and may force one to start the grinding process afresh. He recommends use of a material of medium hardness in the form of broken grindstones (made from sandstone), which have no other uses and are commonly available. These are to be pulverized, and the resulting material is to be separated into three or four different classes by particle size. For this, the pulverized grindstone is vigorously stirred into water. Some time is allowed for the coarsest particles to settle and then the remaining liquid (containing smaller particles still in suspension) is quickly poured into the next vessel, where the settling process is repeated, and so on.&lt;/p&gt;&#xA;&lt;p&gt;The next section on the same page discusses polishing, for which Father Cherubin recommends &lt;a href=&quot;https://www.merriam-webster.com/dictionary/tripoli&quot; rel=&quot;noreferrer&quot;&gt;tripoli&lt;/a&gt; or &lt;a href=&quot;https://www.merriam-webster.com/dictionary/putty%20powder&quot; rel=&quot;noreferrer&quot;&gt;putty powder&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Du Tripoly&lt;/p&gt;&#xA;&lt;p&gt;Pour polir les verres déja formez, &amp;amp; adoucis par le mordant; l'on se peut servir de tripoly, ou de potée d'estain. Le meilleur tripoly, est celuy d'Allemagne. Mais de quelque sorte de tripoly, que l'on se serve, le plus leger, est toûjours le meilleur. L'on s'en peut servir en pierre, comme la nature le produit, lors qu'il a les bonnes qualitez; autrement, on le broyera parfaitement avec de l'Eau-de-vie, ou à defaut avec du vin blanc; &amp;amp; l'on en mettra une quantité ensemble, dans un vaisseau de verre, bien fermé; car en cette maniere, il se fermentera, &amp;amp; adoucira excellemment. Apres que ce tripoly, aura demeuré dans son vaisseau, quatre, ou cinq mois, l'on en pourra prendre, &amp;amp; en faire de petites masses, que l'on laissera sécher a l'ombre, pour s'en servir à sec: l'on pourra aussi s'en servir mol, &amp;amp; le prendre immediatement comme il est dans vaisseau.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The best tripoli is the German one, but in any case the lighter the better. If it is of good quality, it can be used as found in nature. Otherwise, it can be ground up and mixed with brandy (or, lacking that, white wine) and is then allowed to &amp;quot;ferment&amp;quot; and soften for fourth to five months inside a closed glass vessel. After that time, it can either be air-dried, or used wet straight from the jar.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-01-11T11:09:42.160" LastActivityDate="2023-01-11T11:09:42.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15099" PostTypeId="1" CreationDate="2023-01-11T07:19:57.723" Score="4" ViewCount="274" Body="&lt;p&gt;Or I guess the Colonies, if it happened before 1776?&lt;/p&gt;&#xA;&lt;p&gt;I know that mathematics tended to be both applied and emphasized things like taking fifth roots. Also, I think long ago high school and college may have meant different things than they do today.&lt;/p&gt;&#xA;&lt;p&gt;But what I am wondering is just how long it took for The Calculus to become taught regularly -- I assume, for example, that there were people in America who learned about Newton/Leibniz calculus not too long after it was first published.&lt;/p&gt;&#xA;&lt;p&gt;I am guessing that it may have first been introduced only in very advanced classes. I do not know if graduate school was a thing in, say, the early 19th century but maybe that would be the level at which it was studied or the equivalent level&lt;/p&gt;&#xA;&lt;p&gt;I also know that many things, not just in math, took a surprisingly long time to catch on. For example, I believe Harvard medical school did not get a microscope until the mid 19th century (as microorganisms were not then widely accepted as causing disease).&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2023-01-11T07:19:57.723" Title="When was the calculus first part of college curriculum in USA?" Tags="|mathematics|education|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15100" PostTypeId="2" ParentId="15095" CreationDate="2023-01-11T13:58:58.930" Score="5" Body="&lt;p&gt;It took me a bit to track down this reference, it doesn't seem to be mentioned in many of my usual references on the old quantum theory, but fortunately the answer can be found in &lt;a href=&quot;https://publishing.cdlib.org/ucpressebooks/view?docId=ft4t1nb2gv;chunk.id=d0e5922;doc.view=print&quot; rel=&quot;noreferrer&quot;&gt;From c-Numbers to q-Numbers&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This equation, closely connected with the correspondence principle, appears to come from:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Bohr, N. (1914). &lt;em&gt;On the effect of electric and magnetic fields on spectral lines.&lt;/em&gt; The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 27(159), 506–524. &lt;a href=&quot;https://doi.org/10.1080/14786440308635119&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1080/14786440308635119&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There, Bohr treats the problem of emission spectra by assuming only certain stationary states are allowed, each state being characterized by the quantum number &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;. The formula Heisenberg gives is not actually the classical formula, but the correspondence limit which (by the principle) should agree with classical physics. In particular, from Bohr in the reference cited:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On the present theory the frequency of the radiation emitted by the transition from the &lt;span class=&quot;math-container&quot;&gt;$(n+1)$&lt;/span&gt;th to the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th stationary state is equal to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{h}(A_{n+1}-A_n)$&lt;/span&gt;. When &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is large, this approaches to &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{h} \frac{dA_n}{dn}$&lt;/span&gt;. On the ordinary electrodynamics we should expect the frequency of the radiation to be equal to the frequency of revolution, and consequently it is to be anticipated that for large values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{dA_n}{dn} = h\omega_n.\quad .\ .\ .\ .\ .\ (8)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, by taking &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; to be the classical harmonic in the Fourier series representation of the (periodic) motion of the electron, we get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\alpha \omega_n = \alpha \frac{1}{h}\frac{dA_n}{dn}&#xA;$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt; denotes the energy in the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;th state. .&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2023-01-11T13:58:58.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15101" PostTypeId="1" AcceptedAnswerId="15102" CreationDate="2023-01-11T17:16:23.843" Score="4" ViewCount="183" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;How and when did the dedicated study of locally compact groups begin?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Specific instances from literature, recorded stories, etc., may help supplement the answers. There seems to be no reason why I would wake up on a Sunday morning and think of locally compact topological groups.&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;" OwnerUserId="7378" LastActivityDate="2023-01-12T19:29:53.967" Title="How and when did the dedicated study of locally compact groups begin?" Tags="|mathematics|abstract-algebra|topology|group-theory|harmonic-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15102" PostTypeId="2" ParentId="15101" CreationDate="2023-01-12T09:17:51.473" Score="12" Body="&lt;p&gt;The symbolic starting date is 1933, when Haar introduced left invariant measures and proved their existence on second-countable locally compact groups, see his &lt;a href=&quot;https://www.jstor.org/stable/1968346&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Der Massbegriff in der Theorie der Kontinuierlichen Gruppen&lt;/em&gt;&lt;/a&gt;. However, invariant integral on Lie groups was used by Hurwitz already in 1897, as Haar himself noted. In 1934 Pontryagin undertook a systematic study of locally compact Abelian groups and proved &lt;a href=&quot;https://en.wikipedia.org/wiki/Pontryagin_duality&quot; rel=&quot;noreferrer&quot;&gt;Pontryagin duality&lt;/a&gt; for second countable ones, compact or discrete. The general Abelian case was considered by van Kampen in 1935. Weil proved existence of invariant measure and its uniqueness in the general case in 1938, it is covered in his 1940 book &lt;a href=&quot;https://mathoverflow.net/q/251243/51484&quot;&gt;&lt;em&gt;L'intégration dans les groupes topologiques et ses applications&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A detailed theory for some non-compact non-Abelian locally compact groups (real&#xA;semi-simple Lie groups) was developed by Harish-Chandra in 1952, who relied on earlier 1943 work of Gelfand and Raikov concerning their irreducible representations. Further details and references can be found in &lt;a href=&quot;https://citeseerx.ist.psu.edu/document?repid=rep1&amp;amp;type=pdf&amp;amp;doi=1217d9776624d7b89f20ece745c528b81b11ce06&quot; rel=&quot;noreferrer&quot;&gt;Remarks on History of Abstract Harmonic Analysis by Stankovic et al.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-01-12T19:29:53.967" LastActivityDate="2023-01-12T19:29:53.967" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15103" PostTypeId="1" CreationDate="2023-01-14T08:10:09.410" Score="1" ViewCount="131" Body="&lt;p&gt;From &amp;quot;The Electron&amp;quot; by J. J. Thomson, published in The Scientific Monthly Vol. 20, No. 2 (Feb., 1925), pp. 113-115&#xA;&lt;a href=&quot;https://www.jstor.org/stable/7115&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/7115&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Continued discussion] previously determined &lt;span class=&quot;math-container&quot;&gt;$e/mv^2$&lt;/span&gt;, so that when we know &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; we can find &lt;span class=&quot;math-container&quot;&gt;$e/m$&lt;/span&gt;. This was found to be equal to &lt;span class=&quot;math-container&quot;&gt;$1.8×107$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Now if &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is the charge of electricity carried by the hydrogen atom in the electrolysis of solutions, and &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; the mass of that atom, &lt;span class=&quot;math-container&quot;&gt;$E/M$&lt;/span&gt; can be determined by measuring the quantity of hydrogen liberated when a known quantity of electricity passes through an aqueous solution. This was done long ago, and the result was that &lt;span class=&quot;math-container&quot;&gt;$E/M=104$&lt;/span&gt;. Special investigations have shown that &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, the charge on the electron, is equal to &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, the charge on the hydrogen ion ; hence since &lt;span class=&quot;math-container&quot;&gt;$e=E$&lt;/span&gt; and&#xA;&lt;span class=&quot;math-container&quot;&gt;$e/m=1.8×107$&lt;/span&gt;, while &lt;span class=&quot;math-container&quot;&gt;$E/M=104$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$m=M/1800$&lt;/span&gt; or the mass of an electron is only &lt;span class=&quot;math-container&quot;&gt;$1/1800$&lt;/span&gt; of that of an atom of hydrogen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What kind of investigation let Thomson decide that charge on hydrogen ion &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; is equal to charge on an electron &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; where the charge of electron was itself not known?&lt;/p&gt;&#xA;&lt;p&gt;How did he even decided that negative charge is quantised? Cause his experiments only tell that ratio of charge with mass is constant. Then how he decided first electron is quantised and its charge same in magnitude as on hydrogen ion?&lt;/p&gt;&#xA;" OwnerUserId="17797" LastEditorUserId="6609" LastEditDate="2023-01-16T16:53:24.847" LastActivityDate="2023-01-16T16:53:24.847" Title="How did Thomson claim that charge on hydrogen ion was equal to charge on electron without knowing charge on electron?" Tags="|chemistry|electricity|atomic-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15104" PostTypeId="1" CreationDate="2023-01-14T09:47:29.003" Score="1" ViewCount="50" Body="&lt;p&gt;When was hydrogen's absolute mass i.e, in Kg, was found for the first time.&#xA;What method was used and what information from earlier researches were used for it?&lt;/p&gt;&#xA;&lt;p&gt;Also was there any method at that time to calculate number of atoms in a gram sample?&lt;/p&gt;&#xA;" OwnerUserId="17797" LastActivityDate="2023-01-14T09:47:29.003" Title="When was the hydrogen's absolute mass found first in history?" Tags="|chemistry|atomic-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15105" PostTypeId="1" CreationDate="2023-01-14T12:08:02.783" Score="2" ViewCount="133" Body="&lt;p&gt;Note : my question is not as to whether consequence (2) is correctly derived from Arstotle's &amp;quot;law&amp;quot; ( I think it is the case) but as to whether this consequence is still true in Newtonian physics ( though deriving from a false principle). In short : &lt;em&gt;&lt;strong&gt;are both consequences false in Newtonian physics, or only consequence (1)?&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;As says the Stanford physicist who co-authored the &lt;em&gt;Theoretical Minimum&lt;/em&gt;, Aristotle lived in a world of frictions ( but was unaware of this), and thus believed that movement  naturally tends to rest, meaning that a force is needed to maintain an object is a state of motion.&lt;/p&gt;&#xA;&lt;p&gt;Hence, his dynamical views ( Aristotle's ones of course, not the Stanford physicist's ) can be summarized by this formula :&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ V = \frac{F} {m}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This &amp;quot;law&amp;quot; entails two consequences&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; If &lt;span class=&quot;math-container&quot;&gt;$F = 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$V=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; If &lt;span class=&quot;math-container&quot;&gt;$V= 0$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$F=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Modern physics ( in the sense of &amp;quot;post medieval physics&amp;quot;) shows that consequence &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; is false , or at least negates this consequence, which is enough to force the rejection of the alleged Aristotelian law.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My question is simply : is consequence &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; correct ( in spite of the&#xA;falsiy of the principle from which it is derived)?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I believe it is the case but I want to make sure.&lt;/p&gt;&#xA;&lt;p&gt;The reasoning I'd propose to show that consequence &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; is correct is as follows ( with &lt;span class=&quot;math-container&quot;&gt;$V(t) =$&lt;/span&gt; the velocity function , and &lt;span class=&quot;math-container&quot;&gt;$A(t)$&lt;/span&gt; the acceleration function).&lt;/p&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$V(t)=0$&lt;/span&gt;. in that case &lt;span class=&quot;math-container&quot;&gt;$V'(t) = 0$&lt;/span&gt;. But &lt;span class=&quot;math-container&quot;&gt;$V'(t)=A(t)$&lt;/span&gt;.So &lt;span class=&quot;math-container&quot;&gt;$A(t)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;By Newton's second law : &lt;span class=&quot;math-container&quot;&gt;$A(t)= \frac {F(t)} {m}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hence :  &lt;span class=&quot;math-container&quot;&gt;$ A(t)= 0$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$\frac {F(t)} {m}=0$&lt;/span&gt;, which implies finally &lt;span class=&quot;math-container&quot;&gt;$ F(t)=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Is this rigorous enough? In particular, is it enough to state the equations in terms of scalar quantities to show that the result holds?&lt;/p&gt;&#xA;" OwnerUserId="16558" LastEditorUserId="6609" LastEditDate="2023-01-16T17:14:20.700" LastActivityDate="2023-02-21T17:28:12.150" Title="Does Galilean/Newtonian mechanics reject both consequences of Aristotle's dynamical pseudo-law : $ V = F/m$, or only consequence (1)?" Tags="|classical-mechanics|newton|galileo|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15106" PostTypeId="2" ParentId="15103" CreationDate="2023-01-14T22:45:03.247" Score="1" Body="&lt;p&gt;The materials (chemical compounds e.g. water) used in the experiments were not electrically charged. So a positive charge on any (electrolytic) ionic part of them must be neutralised by the negative charge on the complementary part. Examples are the hydrogen ion H+ and the hydroxide ion HO-. Quantization follows from the atomic nature of matter.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2023-01-14T22:45:03.247" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15107" PostTypeId="2" ParentId="7642" CreationDate="2023-01-15T05:47:30.473" Score="5" Body="&lt;p&gt;I have an impression that until about 1800 scientists did not believe that rocks could fall from the sky, but I don't have a better source than the &lt;a href=&quot;https://artsandculture.google.com/story/the-sky-is-falling-meteors-and-meteorites-through-history-adler-planetarium/vgWh-qne7tsiIw?hl=en&quot; rel=&quot;noreferrer&quot;&gt;following&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Reports of fireballs accompanied by the fall of iron or stony objects&#xA;go back to ancient times, and some of those objects were kept and&#xA;thought of as sacred. But by the 18th century, scholars remained&#xA;generally dismissive of popular accounts of such events, deeming them&#xA;fanciful and as expressions of superstition and ignorance. By the turn&#xA;of the 19th century, with a growing number of reports on fireballs&#xA;accompanied by falling stones from various parts of the world, they&#xA;started to change their minds.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-01-15T05:47:30.473" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15110" PostTypeId="2" ParentId="7642" CreationDate="2023-01-16T16:07:40.747" Score="1" Body="&lt;p&gt;The &amp;quot;golden&amp;quot; age of alchemy was about 1300–1700 BCE and was replaced by more empirically based chemistry. Alchemists dreamed of finding the philosopher's stone, a material that would transform other metals into gold, but it was based in occultism and mysticism.&lt;/p&gt;&#xA;&lt;p&gt;We had to wait until the 20th century to show that (nuclear) &lt;strong&gt;transmutation&lt;/strong&gt; is actually possible via relativistic quantum mechanical effects. Some elements (like bismuth, platinum or mercury) can be now transmuted into gold either by gamma or neutron scattering experiments (experiments that are more expensive than the quantities of gold produced). Also some elements naturally decay into gold (after a very very long time).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-09-15T13:11:01.530" LastActivityDate="2023-09-15T13:11:01.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15111" PostTypeId="1" AcceptedAnswerId="15117" CreationDate="2023-01-17T14:28:38.983" Score="9" ViewCount="369" Body="&lt;p&gt;I'm interested in knowing about the first published texts in which sheaf-theoretic methods were used in algebra and/or in algebraic geometry.&lt;/p&gt;&#xA;&lt;p&gt;The oldest instance I am aware of is J.-P. Serre, &lt;em&gt;Faisceaux algébriques cohérents&lt;/em&gt;, which was published five years before the first volume of EGA was published. Nonetheless, I am unaware if other instances showed up before in the literature.&lt;/p&gt;&#xA;&lt;p&gt;Serre himself in FAC's introduction writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On sait que les méthodes cohomologiques, et particulièrement la théorie des faisceaux, jouent un rôle croissant, non seulement en théorie des fonctions de plusieurs variables complexes (cf. [5]), mais aussi en géométrie algébrique classique (qu'il me suffise de citer les travaux récents de Kodaira-Spencer sur le théorème de Riemann-Roch)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;We know that the cohomological methods, in particular sheaf theory, play an increasing role not only in the theory of several complex variables ([5]), but also in classical algebraic geometry (let me recall the recent works of Kodaira-Spencer on the Riemann-Roch theorem).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://achinger.impan.pl/fac/fac.pdf&quot; rel=&quot;noreferrer&quot;&gt;Translation&lt;/a&gt; from Piotr Achinger and Łukasz Krupa.)&lt;/p&gt;&#xA;&lt;p&gt;From the text it seems that Serre points out to Kodaira and Spencer for previous sheaf-theoretic algebraic geometry. However, when I look for &amp;quot;Kodaira-Spencer Riemann-Roch theorem&amp;quot; in google, the only relevant result that shows up is the paper &lt;em&gt;Cohomology and the Riemann-Roch Theorem&lt;/em&gt; by Spencer (available &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063841/&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;), which has sheaves but seems to be only complex geometry to me, not algebraic geometry. So my questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Is there something in this paper from Kodaria and Spencer I am missing that one can regard as &amp;quot;algebraic geometry&amp;quot; or that has any repercussions to AG?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Am I looking at the wrong published text from Kodaira-Spencer that Serre had on mind when he quoted them on FAC?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In general, do you know any previous instances of sheaf-theoretic methods in algebraic geometry before FAC? (And maybe &amp;quot;the first one&amp;quot;? If there is such).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;EDIT: In the introduction of his &lt;a href=&quot;https://www.jmilne.org/math/CourseNotes/AG.pdf&quot; rel=&quot;noreferrer&quot;&gt;algebraic geometry book&lt;/a&gt;, Milne writes&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Lnzt0.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Lnzt0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It seems that to write his paper Serre was fueled by the problem of defining “abstract algebraic varieties.” When Milne says that for his definition Serre was «borrowing ideas from complex analysis,» I think he means the theory of complex analytic spaces (ringed spaces which are locally isomorphic to the zero set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; of a finite collection of holomorphic functions in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^n$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; equipped with the sheaf of holomorphic functions). As far as I know, the theory of complex analytic spaces is older than FAC. But I don't know if Serre was the first one to take such an approach.&lt;/p&gt;&#xA;" OwnerUserId="5031" LastEditorUserId="5031" LastEditDate="2023-01-21T11:34:39.810" LastActivityDate="2023-09-10T23:06:38.690" Title="When was the first time/s that sheaves entered algebra and algebraic geometry?" Tags="|mathematics|algebraic-geometry|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15112" PostTypeId="1" CreationDate="2023-01-17T20:51:06.617" Score="1" ViewCount="232" Body="&lt;p&gt;In his famous 1970 paper [1], Nicholas Katz used the character 兄 for the Gauss-Manin connection. I have always been curious about the history behind this symbol.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What motivated Katz to use this symbol?&lt;/p&gt;&#xA;&lt;p&gt;My guess is that it is because, in the previous paper [2] with Oda, they used the letter Д for the Gauss-Manin connection (maybe because Manin is Russian?). Then maybe Oda feels that Д looks like 兄?&lt;/p&gt;&#xA;&lt;p&gt;Reference:&lt;/p&gt;&#xA;&lt;p&gt;[1] Katz, Nicholas M. &amp;quot;Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin.&amp;quot; Publications Mathématiques de l'IHÉS 39 (1970): 175-232.&lt;/p&gt;&#xA;&lt;p&gt;[2] Katz, Nicholas M., and Tadao Oda. &amp;quot;On the differentiation of de Rham cohomology classes with respect to parameters.&amp;quot; Journal of Mathematics of Kyoto University 8.2 (1968): 199-213.&lt;/p&gt;&#xA;" OwnerUserId="17816" LastEditorUserId="17816" LastEditDate="2023-01-17T20:51:48.470" LastActivityDate="2023-01-17T20:51:48.470" Title="Katz's symbol 兄 for Gauss-Manin connections" Tags="|mathematics|terminology|algebraic-geometry|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15113" PostTypeId="1" CreationDate="2021-07-25T16:52:40.093" Score="20" ViewCount="223" Body="&lt;p&gt;In a book called Advanced Data Analysis from an Elementary Point of View by Cosma Rohilla Shalizi, page 405, the first instance of &amp;quot;Markov process&amp;quot; is accompanied by a footnote which reads&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After the Russian mathematician A. A. Markov, who introduced the theory of Markov processes in the course of a mathematical dispute with his arch-nemesis, to show that probability and statistics&#xA;could apply to dependent events, and hence that Christianity was not necessarily true (I am not making this up: Basharin et al., 2004).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I found it curious, how could religion have anything to do with the fact that the law of large numbers can be extended to non iid variables (because that is what Nekrasov, Markov's &amp;quot;arch-nemesis&amp;quot;, was wrong about, and that argument is at the origin of the chains. They are a counterexample of Nekrasov false claim that independence is necessary for a law of large numbers). But I did not find the answer is the references math/history &lt;a href=&quot;http://langvillea.people.cofc.edu/MarkovReprint.pdf&quot; rel=&quot;noreferrer&quot;&gt;paper&lt;/a&gt; by Basharin et al.&lt;/p&gt;&#xA;&lt;p&gt;Why would the following be true:&lt;/p&gt;&#xA;&lt;p&gt;[Law of large numbers holds &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; independence] &lt;span class=&quot;math-container&quot;&gt;$\implies$&lt;/span&gt; Christianity is true&lt;/p&gt;&#xA;&lt;p&gt;And what do they mean by Christianity being true or not?&lt;/p&gt;&#xA;" OwnerDisplayName="John Cataldo" LastEditorUserId="2277" LastEditDate="2023-02-06T08:09:25.647" LastActivityDate="2023-02-06T08:09:25.647" Title="Markov chains origins and how is Christianity involved" Tags="|mathematics|statistics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15114" PostTypeId="2" ParentId="15113" CreationDate="2021-07-25T17:14:13.167" Score="18" Body="&lt;p&gt;I'm not an expert on history or theology, but it seems that the motivation behind Nekrasov's claim is related to the Russian Orthodox's Church's doctrine of free will. If this is the only context behind the religious aspect of this work by Markov, then Shalizi's depiction &amp;quot;hence that Christianity was not necessarily true&amp;quot; is probably for humorous effect rather than for historical accuracy. But it is possible that I have missed some more explicit connection between religion and Markov/Nekrasov.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://langvillea.people.cofc.edu/MarkovReprint.pdf#page=7&quot; rel=&quot;noreferrer&quot;&gt;Page 9 of Basharin et al. 2004.&lt;/a&gt; (emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nekrasov was originally a theologian by training, and later took up&#xA;mathematics, eventually obtaining a professorship at Moscow University. Nekrasov was also an active member of the Moscow Mathematical Society, another prestigious mathematical school in Russia. The Moscow school often times maintained a clear tension with its sister school in St. Petersburg. &lt;strong&gt;Most mathematicians of the Moscow School were stout members of the Russian Orthodox Church and strong proponents of the religious doctrine of free will. Moscow members like Nekrasov tried to enlist statistics and probability to provide a foundation for their doctrine of free will.&lt;/strong&gt; Of&#xA;course, Markov, an atheist and eventual excommunicate of the Church quarreled endlessly with his equally outspoken counterpart Nekrasov.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Abstract of &lt;a href=&quot;https://www.jstor.org/stable/1403785&quot; rel=&quot;noreferrer&quot;&gt;Seneta, &amp;quot;Markov and the Birth of Chain Dependence Theory.&amp;quot; 1996.&lt;/a&gt; (emphasis mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Markov's work on chain dependence was motivated by his desire to refute a statement by Nekrasov that pairwise independence of random summands was a necessary condition for the Weak Law of Large Numbers. He did this by obtaining such a Law in 1906 for systems of dependent random variables, in particular for finite homogeneous 'Markov' chains. &lt;strong&gt;Nekrasov's incorrect assertion arose out of the theological doctrine of free will, with which some members of the Moscow School of Mathematics of the time were much concerned.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10643" OwnerDisplayName="angryavian" LastActivityDate="2021-07-25T17:27:32.247" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15116" PostTypeId="1" CreationDate="2023-01-19T16:40:02.727" Score="0" ViewCount="94" Body="&lt;p&gt;I am looking for a book that discusses The history and origin of the Argument Principle ( or Cauchy's argument principle) Thanks!&lt;/p&gt;&#xA;" OwnerUserId="17825" LastEditorUserId="12330" LastEditDate="2023-01-24T13:12:10.500" LastActivityDate="2023-01-24T13:12:10.500" Title="The history and origin of the Argument Principle ( or Cauchy's argument principle)" Tags="|mathematics|reference-request|complex-analysis|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15117" PostTypeId="2" ParentId="15111" CreationDate="2023-01-21T10:45:45.203" Score="10" Body="&lt;p&gt;Here is what Dieudonné has to say on contributions other than Serre's in &lt;a href=&quot;https://www.google.com/books/edition/History_Algebraic_Geometry/kIZHDwAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Algebraic Geometry&lt;/a&gt; (VIII.1.6 and VIII.1.11):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;As early as 1909, Severi, while defining the arithmetic genus of a nonsingular irreducible projective algebraic variety of arbitrary dimension &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, had conjectured that its arithmetic genus was given by... Inspired by the proofs of Severi and Zariski, Kodaira was able to prove this conjecture in 1952 using Hodge theory and a study of a particular case of the Riemann-Roch problem... In the course of the year 1953, the work of Dolbeault, Serre, Kodaira, Spencer, and Hirzebruch led to a whole series of remarkable results linking Hodge theory, the invariants of the Italian geometry, and sheaf cohomology, and&#xA;culminating in a formulation of the Riemann-Roch theorem (in the form of an equality) valid in all dimensions.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/1969802&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kodaira's paper&lt;/a&gt; he references is K. Kodaira, The theorem of Riemann-Roch for adjoint systems on three-dimensional varieties, Ann. of Math., 56 (1952), pp. 298-342. Severi's conjecture is proved in §12 (p. 333) titled Arithmetic genera of 3-dimensional algebraic varieties. I am guessing this is, in part, what Serre refers to, since it resolves a conjecture in classical algebraic geometry. He uses algebro-geometric concepts (linear systems, divisors) and results of Zariski in the proof.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="5031" LastEditDate="2023-09-10T23:06:38.690" LastActivityDate="2023-09-10T23:06:38.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15121" PostTypeId="1" CreationDate="2023-01-23T10:03:49.693" Score="0" ViewCount="202" Body="&lt;p&gt;I realize modern algebraic notation is fairly new but for as long as there have been recorded chess games, every square I think was referred to using two coordinates, in old descriptive notation a column named after a piece and a row which I think would always have been a number from 1 to 8. Come to think of it, the game of Go would have had no good way other than two integer coordinates to refer to the intersections on which stones were placed.&lt;/p&gt;&#xA;&lt;p&gt;Is there any evidence that Descartes or earlier mathematicians had been inspired by recorded chess or Go games, perhaps even introducing their ideas using the game boards? Go and chess are of course ancient games and I know recorded chess games predate Descartes by centuries and assume that Go also was recorded long ago.&lt;/p&gt;&#xA;&lt;p&gt;It seems not even hard to imagine that before Descartes someone noticed that the distance between two points on the Go or chess board could be calculated using coordinates.&lt;/p&gt;&#xA;&lt;p&gt;EDIT: Regarding Go notation: As I understand it, a very cumbersome way to record Go games was/is in use until fairly recently or still which actually displays successive positions. I guess the value is, just recording the moves would be very hard to follow for all but the best players (I emphasize &amp;quot;guess&amp;quot; because I barely know the rules of Go) could follow a game using algebraic notation. But I suspect had coordinates been used 2000 years ago, something like Cartesian geometry would have been the almost inevitable result -- who knows what that would have meant for China combined with other advances? I realize this is speculation, of course.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2023-01-24T08:15:35.290" LastActivityDate="2023-01-24T08:15:35.290" Title="Was not chess notation an earlier sort of Cartesian Coordinates?" Tags="|mathematics|descartes|" AnswerCount="0" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="15122" PostTypeId="2" ParentId="12560" CreationDate="2023-01-24T05:34:47.457" Score="5" Body="&lt;p&gt;In the 19th century, the Gaussian integers were just called &amp;quot;complex integers&amp;quot;. In Dirichlet-Dedekind's &lt;em&gt;Vorlesungen uber Zahlentheorie&lt;/em&gt; from the late 1800s, the XI-th supplement was the 1st published account about general number fields and its first long section 159 is on elementary number theory in &lt;span class=&quot;math-container&quot;&gt;$\mathbf Z[i]$&lt;/span&gt;, but this ring was given no special notation (the field &lt;span class=&quot;math-container&quot;&gt;$\mathbf Q(i)$&lt;/span&gt; was written as &lt;span class=&quot;math-container&quot;&gt;$J$&lt;/span&gt;) and had no name like &amp;quot;Gaussian integers&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In Hardy &amp;amp; Wright's &lt;em&gt;An Introduction to the Theory of Numbers&lt;/em&gt;, Section 12.2 is called &amp;quot;The rational integers, Gaussian integers, and the integers of &lt;span class=&quot;math-container&quot;&gt;$k(\rho)$&lt;/span&gt;&amp;quot;.  Here &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; is a nontrivial cube root of unity, so their unnamed set of integers is the Eisenstein integers. In the notes at the end of that chapter, they write that the Gaussian integers were introduced by Gauss in his work on biquadratic reciprocity and &amp;quot;the numbers &lt;span class=&quot;math-container&quot;&gt;$a+b\rho$&lt;/span&gt; were introduced by Eisenstein and Jacobi in their work on cubic reciprocity&amp;quot;. It may seem odd that they give the Gaussian integers a special name but don't do the same for what we call the Eisenstein integers, but consider that most of these quadratic rings of integers have no special name: &lt;span class=&quot;math-container&quot;&gt;$\mathbf Z[\sqrt{2}]$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathbf Z[\sqrt{-5}]$&lt;/span&gt;, and so on. You can't expect each of these to get a special name; at some point it has to stop.&lt;/p&gt;&#xA;&lt;p&gt;On MathSciNet, the earliest review that mentions Eisenstein integers is a paper by Coxeter and Todd in 1953 and the next two such reviews are for papers from 1969. The fourth earliest reference is a review of the 1970 textbook&#xA;&lt;em&gt;The theory of numbers. An introduction&lt;/em&gt; by Anthony Gioia where the review  mentions &amp;quot; &amp;quot;the Jacobian (Eisenstein) integers...&amp;quot; and this is the only review on MathSciNet using the label &amp;quot;Jacobian integers&amp;quot;. There are no reviews at all mentioning Eisenstein integers between 1980 and 1997.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-01-24T05:34:47.457" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15125" PostTypeId="1" CreationDate="2023-01-26T01:36:22.790" Score="6" ViewCount="213" Body="&lt;p&gt;During the day, the sky is bright blue and, along with the yellow light from the Sun, it illuminates the surface of the Earth.&lt;/p&gt;&#xA;&lt;p&gt;What is the earliest recorded knowledge that the sky wasn't self-illuminating, but simply something that was itself illuminated by the Sun?&lt;/p&gt;&#xA;" OwnerUserId="15353" LastActivityDate="2023-01-26T01:36:22.790" Title="When did people first know that the sky was illuminated by the Sun?" Tags="|discoveries|earth-sciences|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15127" PostTypeId="1" AcceptedAnswerId="15128" CreationDate="2023-01-30T20:21:35.087" Score="1" ViewCount="196" Body="&lt;p&gt;I'm looking for any bio/info/obit on Alan Earnshaw, who was a co-author of Norman Greenwood (who has a wiki page and an online obit).&lt;/p&gt;&#xA;&lt;p&gt;FYI, I tried several different Google searches, but the best I could find was this (which still doesn't give a bio).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.webofstories.com/play/norman.greenwood/247;jsessionid=174643C48058427665476156573239C5&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.webofstories.com/play/norman.greenwood/247;jsessionid=174643C48058427665476156573239C5&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Asked on Chem SE and a moderator recommended to try here.]&lt;/p&gt;&#xA;" OwnerUserId="17900" LastActivityDate="2024-01-27T22:05:48.817" Title="Biographical info on Alan Earnshaw (Chemistry of the Elements)" Tags="|biographical-details|chemists|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15128" PostTypeId="2" ParentId="15127" CreationDate="2023-01-30T22:34:14.657" Score="5" Body="&lt;p&gt;Piecing together multiple Google snippets from &lt;em&gt;The University of Leeds Review&lt;/em&gt;, Vol. 38, 1995, p. 249:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Alan Earnshaw retired from the School of Chemistry in September 1995. After graduating from Sheffield in 1955 , he obtained the degree of PhD at London in 1958, and then served as a lecturer at Battersea College of Technology until his arrival in Leeds as a lecturer in the Department of Inorganic and Structural Chemistry at a salary on the scale £1,000 ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://de.wikipedia.org/wiki/Alan_Earnshaw&quot; rel=&quot;noreferrer&quot;&gt;German Wikipedia&lt;/a&gt; provides some additional details: Earnshaw received a bachelor degree at Sheffield, and his employment at the University of Leeds commenced in 1966. Earnshaw is the author of three books:&lt;/p&gt;&#xA;&lt;p&gt;Alan Earnshaw, &lt;em&gt;Introduction to Magnetochemistry.&lt;/em&gt; London: Academic Press 1968. 115 pp.&lt;br /&gt;&#xA;A. Earnshaw and T.J. Harrington, &lt;em&gt;The Chemistry of the Transition Elements.&lt;/em&gt; Oxford: Oxford University Press 1973. 101 pp.&lt;br /&gt;&#xA;N.N. Greenwood and A. Earnshaw, &lt;em&gt;Chemistry of the Elements.&lt;/em&gt; Oxford: Pergamon Press 1984. 1542 pp.&lt;/p&gt;&#xA;&lt;p&gt;While I was able to find UK obituaries for people with the name Alan Earnshaw, none of them appear to be for the professor of chemistry. Based on the date of retirement and the year he received his bachelor degree, Earnshaw was likely born around 1930. Publicly available &lt;a href=&quot;https://www.freebmd.org.uk/&quot; rel=&quot;noreferrer&quot;&gt;UK birth records&lt;/a&gt; show several potential candidates, but without a known place of birth it is impossible to identify any of those as the relevant one.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2023-01-30T22:34:14.657" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15129" PostTypeId="1" CreationDate="2023-01-31T00:30:17.543" Score="7" ViewCount="171" Body="&lt;p&gt;I'm interested in when (and how) the modern idea of a group action developed and how group actions became their own algebraic structures.&lt;/p&gt;&#xA;&lt;p&gt;As far as I can tell in the 19th century group actions were much more implicit than they are today and were strongly tied to the group itself, as a subgroup of a permutation group. But the way group actions are presented today (at the very least as they were presented to me) is much more axiomatic and considers group actions and groups as separate algebraic structures.&lt;/p&gt;&#xA;&lt;p&gt;When and how did this change (if it changed at all) and how did group actions become so important?&lt;/p&gt;&#xA;&lt;p&gt;I'm also looking for books or articles on this topic, so it would be very much appreciated if anyone has a recommendation, as the books on the history of group theory I have skimmed don't mention the development of group actions.&lt;/p&gt;&#xA;" OwnerUserId="17901" LastEditorUserId="6609" LastEditDate="2023-01-31T10:46:20.693" LastActivityDate="2023-01-31T10:46:20.693" Title="History of group actions as their own structures" Tags="|mathematics|reference-request|abstract-algebra|group-theory|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15130" PostTypeId="1" CreationDate="2023-01-31T11:41:33.317" Score="4" ViewCount="87" Body="&lt;p&gt;I am doing a research project on the history of group theory, and want to know about the early developments on the subject. There are plenty of proofs about why the projective special linear group over a finite field is simple, with a few exceptions.&lt;/p&gt;&#xA;&lt;p&gt;According to Wikipedia, proofs of this result has been around since 1830s, and I want to inspect on the original or some early proof of it. Where can I find such old materials?&lt;/p&gt;&#xA;" OwnerUserId="17906" LastActivityDate="2023-02-12T11:51:58.480" Title="Where can I find the early proofs for the simplicity of $\text{PSL}(n,q)$?" Tags="|mathematics|reference-request|group-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15131" PostTypeId="1" CreationDate="2023-02-01T00:50:42.140" Score="-1" ViewCount="135" Body="&lt;p&gt;It seems to me that it would be more natural to consider Dirac Delta as a piecewise-defined function, as described &lt;a href=&quot;https://math.stackexchange.com/a/4628238/2513&quot;&gt;here&lt;/a&gt;, with the scaling rule &lt;span class=&quot;math-container&quot;&gt;$\delta (ax)=\delta(x)$&lt;/span&gt;. This way we keep all the integral transforms properties of delta function, and its derivatives as long as its argument is the integration variable. But at the same time, we can define arbitrary functions on delta function, including rising it to any positive power, evaluate it at zero, and consider it outside of integrals.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me absolutely natural that a derivative of scaling-invariant function, like &lt;span class=&quot;math-container&quot;&gt;$\operatorname{sign}x$&lt;/span&gt; also should be scaling-invariant. This, in particular, allows to generalize &lt;strong&gt;sign&lt;/strong&gt; function to dual numbers while keeping its properties.&lt;/p&gt;&#xA;&lt;p&gt;So, the question is: why another convention, that is &lt;span class=&quot;math-container&quot;&gt;$\delta(a x)=\frac1{|a|}\delta(x)$&lt;/span&gt; was adopted, which does not even allow to ascribe to it a value at zero?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastActivityDate="2023-02-01T01:08:14.407" Title="Why the scaling rule $\delta(a x)=\frac1{|a|}\delta(x)$ was historically adopted?" Tags="|real-analysis|" AnswerCount="1" CommentCount="9" ClosedDate="2023-02-01T10:04:26.630" ContentLicense="CC BY-SA 4.0" />
  <row Id="15132" PostTypeId="2" ParentId="15131" CreationDate="2023-02-01T01:08:14.407" Score="5" Body="&lt;p&gt;We want the usual change of variables formulas to work.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\int_{-1}^1 f(x)\delta(x)\;dx = f(0) .&#xA;$$&lt;/span&gt;&#xA;What if we change variables &lt;span class=&quot;math-container&quot;&gt;$x=ay$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$a&amp;gt;0$&lt;/span&gt;? Then &lt;span class=&quot;math-container&quot;&gt;$y = x/a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$dx = a\, dy$&lt;/span&gt; and so for change of variables we want&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(0) = \int_{-1}^1 f(x)\delta(x)\;dx = &#xA;\int_{-1/a}^{1/a} f(ay)\delta(ay) a\;dy .&#xA;$$&lt;/span&gt;&#xA;But if we use the incorrect &lt;span class=&quot;math-container&quot;&gt;$\delta(ay) = \delta(y)$&lt;/span&gt; we would get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(0) = \int_{-1/a}^{1/a} f(ay)\delta(ay) a\;dy=&#xA;\int_{-1/a}^{1/a} f(ay)\delta(y) a\;dy=&#xA;f(0) a .&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-02-01T01:08:14.407" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15133" PostTypeId="1" CreationDate="2023-02-01T23:43:17.640" Score="4" ViewCount="123" Body="&lt;p&gt;In &amp;quot;Number: The Language of Science&amp;quot; (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as &amp;quot;the induction step&amp;quot; (and refers to what we call the induction step as &amp;quot;the recurrence step&amp;quot; or &amp;quot;the proof of the hereditary property&amp;quot;). Was this standard terminology a century ago, or was Dantzig confused?&lt;/p&gt;&#xA;&lt;p&gt;Note that he wrote this way back when mathematical induction was commonly called complete induction as opposed to Baconian or incomplete induction. Since verification of a single base case could be viewed as a minimalist version of Baconian induction, Dantzig's terminology does not seem totally illogical to me. Perhaps his use of the phrase &amp;quot;induction step&amp;quot; was standard a century ago, and over time its meaning shifted so that it now has the &amp;quot;opposite&amp;quot; meaning (that is, it now refers to the other component of proof by mathematical induction).&lt;/p&gt;&#xA;&lt;p&gt;I’d be grateful for comments by those who know more history of mathematics than I do, as well as those who can bring a multicultural perspective to this question (what sort of terminology for mathematical induction is used in other languages?).&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="131" LastEditDate="2023-02-03T07:16:34.283" LastActivityDate="2023-02-03T07:16:34.283" Title="Terminology associated with mathematical induction" Tags="|terminology|numbers|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15136" PostTypeId="1" CreationDate="2023-02-04T12:37:07.853" Score="3" ViewCount="124" Body="&lt;p&gt;A number of diagrammatic formulations have played an important role in the advancement of science. Some embody representations of physical phenomena, while others model mathematical or logical ideas in graphical form.&lt;/p&gt;&#xA;&lt;p&gt;One example that belongs to both classes are the &lt;a href=&quot;https://en.wikipedia.org/wiki/Feynman_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feynman diagrams&lt;/a&gt;. They are pictorial representations of the mathematical expressions that describe the behavior governing the interactions of subatomic particles. In addition to helping understand certain aspects of quantum chromodynamics conceptually, they also serve as an important calculating tool in this area of physics.&lt;/p&gt;&#xA;&lt;p&gt;Another example, which is aimed more at serving mostly as graphically representing a mathematical modelling language, are the &lt;a href=&quot;https://en.wikipedia.org/wiki/Petri_net&quot; rel=&quot;nofollow noreferrer&quot;&gt;Petri Nets&lt;/a&gt;. They describe distributed systems and offer a way through which computer scientists can capture the essence of business processes.&lt;/p&gt;&#xA;&lt;p&gt;Other instances of these graphical notations in different scientific areas include &lt;a href=&quot;https://en.wikipedia.org/wiki/Bond_graph&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bond graphs&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Circuit_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;circuit diagrams&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Block_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;block diagrams&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/ZX-calculus&quot; rel=&quot;nofollow noreferrer&quot;&gt;ZX/ZW calculus&lt;/a&gt;, &lt;a href=&quot;https://www.cs.ox.ac.uk/qpl2015/slides/sobocinski-tutorial.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;graphical linear algebra&lt;/a&gt;, &lt;a href=&quot;https://mathworld.wolfram.com/CommutativeDiagram.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;commutative diagrams&lt;/a&gt;, and &lt;a href=&quot;https://mathworld.wolfram.com/KnotDiagram.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;knot diagrams&lt;/a&gt;. A presentations in which a number of these diagram types are described and analysed as a whole can be found &lt;a href=&quot;https://math.ucr.edu/home/baez/ACT2017/ACT2017_erbele.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Despite having found this presentation, so far I have not been able to find many sources that describe the role of graphical languages in the advancement of the sciences. I did retrieve the following &lt;a href=&quot;https://ieeexplore.ieee.org/document/5353439&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; entitled 'The &amp;quot;Physics&amp;quot; of Notation: Toward a Scientific Basis for Constructing Visual Notations in Software Engineering' by Daniel Moody (2009). However, this article is more about making good design choices for the effective communication of scientific diagrams.&lt;/p&gt;&#xA;&lt;p&gt;What I'm more interested in, is what role these different diagrams have played in the advancement of their respective scientific fields. I wonder how they have contributed to a deeper understanding of physical phenomena and mathematical ideas. So instead of treating them merely as a visual supplement of the underlying ideas, I'm curious as to how these representations actually allowed researchers to make advancements in their respective research areas. I wonder how they allowed to do so both in the different, isolated fields, as well as in 'science' as an integrated whole.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: do you know of any references that delve into the role of diagrams in the advancement of the sciences?&lt;/p&gt;&#xA;" OwnerUserId="4296" LastEditorUserId="4296" LastEditDate="2023-02-04T23:42:40.460" LastActivityDate="2023-02-04T23:42:40.460" Title="References on the role of diagrams in scientific advancement" Tags="|reference-request|philosophy-of-science|big-list|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15138" PostTypeId="1" CreationDate="2023-02-05T17:30:33.993" Score="1" ViewCount="55" Body="&lt;p&gt;In texts of algebra, the cardinality of cosets is written in &lt;span class=&quot;math-container&quot;&gt;$[G:H]$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$|G:H|$&lt;/span&gt;. Where did this notation originate?&lt;/p&gt;&#xA;&lt;p&gt;The history about &lt;span class=&quot;math-container&quot;&gt;$G/H$&lt;/span&gt; can be found &lt;a href=&quot;https://math.stackexchange.com/questions/857539/who-named-quotient-groups&quot;&gt;here&lt;/a&gt;. &lt;span class=&quot;math-container&quot;&gt;$[G:H]$&lt;/span&gt; is called &lt;a href=&quot;https://en.wikipedia.org/wiki/Index_of_a_subgroup&quot; rel=&quot;nofollow noreferrer&quot;&gt;index of a subgroup&lt;/a&gt;. But I'm not sure what advantage this notation has. I think &lt;span class=&quot;math-container&quot;&gt;$|G/H|$&lt;/span&gt; can also be used when &lt;span class=&quot;math-container&quot;&gt;$G/H$&lt;/span&gt; is not a group or has an infinite index.&lt;/p&gt;&#xA;" OwnerUserId="10808" LastEditorUserId="10808" LastEditDate="2023-02-05T18:09:30.220" LastActivityDate="2023-02-05T18:09:30.220" Title="Where did the index of a subgroup notation $[G:H]$ begin to be used?" Tags="|mathematics|notation|abstract-algebra|group-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15139" PostTypeId="2" ParentId="13409" CreationDate="2023-02-05T22:19:15.717" Score="2" Body="&lt;p&gt;The &lt;a href=&quot;https://www.mnopedia.org/thing/post-it-notes#:%7E:text=In%201968%2C%20Spencer%20Silver%2C%20a,what%20that%20use%20could%20be.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Post-it&lt;/a&gt; note was “discovered” accidentally by an employee at 3M.  Other employees lobbied for the original `Press’n Peel’ notes but the product struggled commercially until the company gave away free samples.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Scotchgard&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scotchguard&lt;/a&gt; was also discovered accidentally by 3M employees.&lt;/p&gt;&#xA;" OwnerUserId="7093" LastActivityDate="2023-02-05T22:19:15.717" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15141" PostTypeId="1" AcceptedAnswerId="15144" CreationDate="2023-02-06T17:34:39.997" Score="5" ViewCount="447" Body="&lt;p&gt;A cobweb diagram is a visualization tool that allows one to qualitatively study the iterates of a self-map of the real line based on the graph of the function; here is an example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Bb5kU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Bb5kU.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Here the map is the Boole's transformation &lt;span class=&quot;math-container&quot;&gt;$x\mapsto x-1/x$&lt;/span&gt;. The link to the interactive Desmos graph is &lt;a href=&quot;https://www.desmos.com/calculator/20yftwjmys.&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.desmos.com/calculator/20yftwjmys.&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Other names for this construct include: zig-zag, staircase, graphical analysis, Verhulst diagram,..., the latter being from &lt;a href=&quot;https://en.wikipedia.org/wiki/Cobweb_plot&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cobweb_plot&lt;/a&gt; (the other terms are from e.g. Rudin, Devaney, Hasselblatt-Katok, Strogatz).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Who first used cobweb diagrams in a publication? In particular, I would be interested in the verification or falsification of the attribution to Verhulst.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; According to the paper Prof. Friedland referenced, Verhulst never used cobweb diagrams, at least in his work on the logistic map; which seems reasonably attributable to him. Since the logistic map is a popular choice to introduce cobweb diagram, it seems there has been a transference of attribution.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Added:&lt;/strong&gt; To continue, here is a relevant page from Legendre's 1816 monograph &lt;em&gt;Essai sur la théorie des nombres (2e)&lt;/em&gt; (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k62826k/f609.item.zoom#&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gallica.bnf.fr/ark:/12148/bpt6k62826k/f609.item.zoom#&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eAzUU.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eAzUU.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The context is approximation of roots; and two arbitrary curves are allowed (see also &lt;a href=&quot;https://math.stackexchange.com/q/4435177/169085&quot;&gt;https://math.stackexchange.com/q/4435177/169085&lt;/a&gt;). Rosa, in the paper referenced by Prof. Friedland also mentions Bidone's work which precedes that of Legendre, but also that it's a possibility that already this was folklore. Rosa claims (p.35) that Fourier and Galois, together with Legendre, are regarded as the pioneers of the method, due to their interest and contributions.&lt;/p&gt;&#xA;" OwnerUserId="1887" LastEditorUserId="1887" LastEditDate="2023-02-07T04:16:30.183" LastActivityDate="2023-02-07T04:16:30.183" Title="The Originator of Cobweb Diagrams" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15142" PostTypeId="1" CreationDate="2023-02-06T17:45:49.607" Score="5" ViewCount="200" Body="&lt;p&gt;Weinberg, in his memoir on Bryce Seligman DeWitt (available at &lt;a href=&quot;https://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/dewitt-bryce.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/dewitt-bryce.pdf&lt;/a&gt;) states that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1950 two major but totally unrelated developments occurred in his life. First, he became engaged to be married to Cecile Morette, a young French physicist who was in her second postdoctoral year at the institute. Second, at the urging of their father, he and his three brothers began the legal procedures for changing their surname to a name from their mother’s side of the family. The younger boys&#xA;were, or had been, at school in the eastern United States, and all had encountered repeated misunderstandings and false assumptions based solely on their surname, something that had seldom occurred in California. (p.5)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[A potential professorship offer to DeWitt], however, was vetoed by Felix Bloch, who upon learning that Bryce had changed his surname 20 years earlier, refused to allow the offer to proceed. (p.10)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One can find similar passages elsewhere (e.g. DeWitt-Morrette's memoirs).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What were the &amp;quot;repeated misunderstandings and false assumptions&amp;quot; in question, and why was the name change taken so negatively by Bloch?&lt;/p&gt;&#xA;" OwnerUserId="1887" LastActivityDate="2023-02-06T17:45:49.607" Title="On Bryce Seligman DeWitt's Name Change" Tags="|physics|biographical-details|physicists|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15143" PostTypeId="1" CreationDate="2023-02-07T02:55:23.687" Score="2" ViewCount="144" Body="&lt;p&gt;What were the most important classified doctoral dissertations in physics and/or mathematics written by grad students during WWII on &lt;em&gt;any&lt;/em&gt; or &lt;em&gt;all&lt;/em&gt; sides of the war (US, UK, Japan, Soviet Union, Germany, Italy, etc.), especially those that took at least 20 years or more to get published and recognized?&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="2277" LastEditDate="2023-09-07T20:12:20.577" LastActivityDate="2023-09-08T02:10:48.087" Title="classified dissertations in WWII" Tags="|world-war-2|" AnswerCount="1" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15144" PostTypeId="2" ParentId="15141" CreationDate="2023-02-07T03:00:57.383" Score="6" Body="&lt;p&gt;According to Alessandro Rosa, ``An episodic history of the staircased&#xA;iteration diagram&amp;quot;,&#xA;ANTIQUITATES MATHEMATICAE&#xA;Vol. 15(1) 2021, p. 3–90,&#xA;doi: 10.14708/am.v15i1.7056, a version of the diagram  can be traced back to Adrien-Marie Legendre, in 1822.&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2023-02-07T03:00:57.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15146" PostTypeId="1" AcceptedAnswerId="15147" CreationDate="2023-02-07T16:49:28.490" Score="2" ViewCount="488" Body="&lt;p&gt;I am currently writing an essay on the error function and after researching its historical origin, I found out who first defined it: J.W.L. Glaisher. But his definition is different from today's form. His definition: &lt;span class=&quot;math-container&quot;&gt;$\int_x^∞e^{-x^2}dx = \operatorname{erf}(x)$&lt;/span&gt; and today: &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{\sqrt{\pi}}\int_0^xe^{-x^2}dx$&lt;/span&gt;. I would like to know when they changed it and what the reason was.&lt;/p&gt;&#xA;" OwnerUserId="17925" LastEditorUserId="6609" LastEditDate="2023-02-08T10:24:56.813" LastActivityDate="2023-04-11T14:46:09.450" Title="When did the error function get its modern definition?" Tags="|terminology|calculus|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15147" PostTypeId="2" ParentId="15146" CreationDate="2023-02-07T21:10:06.320" Score="7" Body="&lt;p&gt;Harold Jeffreys's  1927 book &lt;a href=&quot;https://www.google.com/books/edition/Operational_Methods_in_Mathematical_Phys/0yI7AQAAIAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Operational Methods in Mathematical Physics&lt;/a&gt; has a 2-page note titled &amp;quot;On the notation for the error function or probability integral&amp;quot; on pages 94-95, with 9 footnotes, addressing your question. (The link I give has a free pdf download option.) He seems to say that Jeans introduced the &lt;span class=&quot;math-container&quot;&gt;$\frac2{\sqrt\pi}$&lt;/span&gt; factor, in his 1921 book &lt;em&gt;The Dynamical Theory of Gases&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A partial resolution of the inconsistency of names is to see that where Glaisher, Whittaker and Watson, etc. use &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Erf}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Erfc}$&lt;/span&gt; for the unscaled functions, A&amp;amp;S, Wikipedia, DLMF, etc. use &lt;span class=&quot;math-container&quot;&gt;$\operatorname{erf}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\operatorname{erfc}$&lt;/span&gt; for the scaled ones.  But Jeffreys uses &lt;span class=&quot;math-container&quot;&gt;$\operatorname{Erf}$&lt;/span&gt; for the scaled version, so the inconsistency is not eliminated.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="12330" LastEditDate="2023-02-08T02:04:43.753" LastActivityDate="2023-02-08T02:04:43.753" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15149" PostTypeId="1" CreationDate="2023-02-09T08:09:35.170" Score="2" ViewCount="33" Body="&lt;p&gt;I am trying to get to the bottom of a few things in the Stern-Gerlach experiment. First, on wikipedia, it says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Sommerfeld model predicted that the magnetic moment of an atom measured along an axis will only take on discrete values.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The wikipedia page also says that Stern-Gerlach was meant to test the Bohr-Sommerfield hypothesis that the direction of angular momentum is quantized.&lt;/p&gt;&#xA;&lt;p&gt;So, what I am wondering is exactly how the choice of silver atoms achieves this? I know that the chemistry of the silver atoms is such that the net angular momentum for the first 46 electrons is zero, and the small magnetic moment of the nucleus allows us to focus on the electron. My questions are:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How did they know, at the time of the experiment, that the first 46 electrons had no angular momentum at all?&lt;/li&gt;&#xA;&lt;li&gt;How do we isolate intrinsic and extrinsic momentum in this experiment for the 47th electron? I assume at this time, there was not such a distinction, but knowing that there is now, how to we isolate spin as something independent? Did they just interpret that, indeed, the direction of angular momentum is quantized, and only later the concept of spin was developed to isolate this factor that is quantized?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="17937" LastActivityDate="2023-02-09T08:09:35.170" Title="Understanding how Stern-Gerlach tests Bohr-Sommerfield Hypothesis" Tags="|physics|quantum-mechanics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15151" PostTypeId="1" CreationDate="2023-02-10T12:13:58.983" Score="4" ViewCount="235" Body="&lt;p&gt;I am looking to inquire into the role of visualization in sciences and math and wish to read about the history of visualization in those fields and also wonder where I might read about the epistemology of visualization in math and physics.&lt;/p&gt;&#xA;" OwnerUserId="17947" LastActivityDate="2023-05-28T11:57:52.627" Title="History of visualization in science and math - where may I read of it?" Tags="|reference-request|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15152" PostTypeId="1" CreationDate="2023-02-11T12:19:03.067" Score="8" ViewCount="308" Body="&lt;p&gt;Currently, I'm doing a PhD on the applications of algorithms to generate timelines of textual content. Recently, I found an article entitled &lt;em&gt;StoryFlow: Tracking the Evolution of Stories&lt;/em&gt; by S. Liu et al. (2013), which can be found over &lt;a href=&quot;https://ieeexplore.ieee.org/abstract/document/6634164?casa_token=5vFDmkpYFyMAAAAA:8X6OypnPb0C5XYBEmXoXywJJWVObvj4JA28ZhDGl526IsZMGKG0dJkwEh5guS_TaPK2XNrUD&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The article presents a computational method through which the location of different characters relative to one another throughout time is visually presented. What I found remarkable, whas that on p. 2442 it explicitly mentions and compares the outcome of the algorithm with the Jurassic Park panel of the 657'th XKCD &lt;a href=&quot;https://xkcd.com/657/&quot; rel=&quot;nofollow noreferrer&quot;&gt;comic&lt;/a&gt; &amp;quot;Movie Narrative Charts&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Here is the comparison in the article:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zYEOO.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zYEOO.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Another, more recent article that was inspired by this particular comic by Munroe is named &lt;em&gt;HyperStorylines: Interactively Untangling Dynamic hypergraphs&lt;/em&gt;. It was published in 2022 and can be found &lt;a href=&quot;https://journals.sagepub.com/doi/pdf/10.1177/14738716211045007&quot; rel=&quot;nofollow noreferrer&quot;&gt;over here&lt;/a&gt; (PDF).&lt;/p&gt;&#xA;&lt;p&gt;In general, I often find the work of Randall Munroe funny and interesting at the same time, and can imagine it has served as inspiration for scientific work more often.&lt;/p&gt;&#xA;&lt;p&gt;Yet more generally, I wonder which scientific articles were in some way inspired by web comics. I know quite a few inventions of the past have built on the imaginative comic &lt;em&gt;books&lt;/em&gt; (see e.g. the following &lt;a href=&quot;https://www.storypick.com/comic-book-inventions/&quot; rel=&quot;nofollow noreferrer&quot;&gt;list&lt;/a&gt;). I restrict this question to web comics because they've been made more recently, and I haven't seen any of them mentioned in lists of the type mentioned above. I'm interested in both research articles and inventions that have come about at least in part thanks to the existence of a particular web comic.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;: what research articles and inventions were inspired by web comics?&lt;/p&gt;&#xA;" OwnerUserId="4296" LastEditorUserId="4296" LastEditDate="2023-07-07T11:51:24.923" LastActivityDate="2023-09-16T17:24:21.750" Title="What research articles were inspired by web comics?" Tags="|reference-request|big-list|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15154" PostTypeId="2" ParentId="15151" CreationDate="2023-02-11T20:01:44.227" Score="5" Body="&lt;p&gt;I can give some suggestions in regard to visualization in mathematics. As far as I know,  not much has been written about it.&lt;/p&gt;&#xA;&lt;p&gt;There is a book by James Robert Brown, &lt;em&gt;Philosophy of Mathematics-A Contemporary Introduction to the World of Proofs and Pictures&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/em&gt;, which  devotes many pages to the subject of the use of images and the theory of visualization in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;This is a book of philosophy of mathematics &lt;strong&gt;from a Platonist point of view&lt;/strong&gt;. But it deals with a contemporary version of Platonism, that goes beyond the original Platonism and also beyond the Platonism of some prominent mathematicians of the XX century, as Kurt Gödel and Godfrey H. Hardy.&lt;/p&gt;&#xA;&lt;p&gt;And it develops an original idea of Platonism that, according to the author, is capable of dealing with many questions of philosophy of mathematics with a  &lt;strong&gt;wider and more flexible approach&lt;/strong&gt;, also with respect to other philosophy of mathematics as &lt;strong&gt;formalism and structuralism.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In particular, with regard to the issue of &lt;strong&gt;mathematical visualization&lt;/strong&gt;, Brown writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Platonism, more than any other account of mathematics, is open to the&#xA;possibility of an endless variety of investigative techniques. Proving&#xA;theorems in a traditional way is certainly one method of establishing&#xA;new mathematical truths, but it needn't be the only way. [...]&lt;/p&gt;&#xA;&lt;p&gt;Platonism can be [...] liberating for mathematical research.&lt;/p&gt;&#xA;&lt;p&gt;[...] what other means might there be? [...]and, especially, what&#xA;about diagrams and pictures? Besides traditional proofs, these not&#xA;standard techniques may also bear much fruit. &lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As we can see, this is an original view of the use of images in mathematics, which now enter the very core of mathematics, proofs, whereas they  are usually considered a tool and a support of mathematical theories  that have been built elsewhere and with different tools.&lt;/p&gt;&#xA;&lt;p&gt;Brown's book devotes, in particular, two chapters to mathematical images: &lt;em&gt;Chapter 3 - Picture Proofs and Platonism&lt;/em&gt;, and &lt;em&gt;Chapter 9 -  Proofs, Picture and Procedures in Wittgenstein&lt;/em&gt;, but the issue of mathematical images goes through the whole book.&lt;/p&gt;&#xA;&lt;p&gt;His treatment is mainly philosophical, but there are plenty of &lt;strong&gt;mathematical examples and also historical notes&lt;/strong&gt; about the subject. And, of course, there is also a bibliography.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;From a &lt;strong&gt;historical point of view,&lt;/strong&gt; we can say that there has been a 'tension' between mathematics and images, as diagrams and pictures. Whereas the use of pictures and diagrams is widespread in mathematics texts, diagrams have been often considered possibly misleading.&lt;/p&gt;&#xA;&lt;p&gt;In particular, some diffidence toward diagrams, graphs and pictures has accompanied  'waves' of rigorization of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Joseph Louis Lagrange&lt;/strong&gt;, who was very attentive to mathematical rigour, remarks  in his &lt;em&gt;Mécanique Analityque&lt;/em&gt; (1788) :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;No figures will be found in this work. The methods I set forth require neither constructions nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform procedure &lt;em&gt;(1788, Pref.)&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In XIX century, in particular during  the so-called &lt;strong&gt;second rigorization,&lt;/strong&gt; of which &lt;strong&gt;Bernard Bolzano&lt;/strong&gt; was a pioneer and carried on by &lt;strong&gt;Karl Weierstrass&lt;/strong&gt; and his school, there was a new attempt to expunge pictures and geometrical consideration from mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Pictures were considered not rigorous, for example in the issues concerning the definition of the continuity of functions. For instance, in the case of the &lt;strong&gt;Intermediate Value Theorem&lt;/strong&gt;, of which Bolzano gave a purely analytic proof, avoiding the geometric self-evident idea of a continuous function as a function whose graph is a one-piece line, a line that can be drawn without lifting the pencil  from the paper.&lt;/p&gt;&#xA;&lt;p&gt;The  present-day, of  Weierstrassian origin, &lt;strong&gt;definition of a continuous function&lt;/strong&gt; also expunges this visual idea: in particular, according to the present-day  definition, there are continuous functions whose graph cannot be drawn without lifting the pen from the paper. &lt;span class=&quot;math-container&quot;&gt;$^4$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The removal of visual and geometrical aspects continued during the &lt;strong&gt;arithmetization of analysis&lt;/strong&gt;, with the exigence of establishing the foundations of mathematical analysis, and in particular of the theory of real numbers also on numbers,  without geometrical aspects. And, in general,  it  was  peculiar of the &lt;strong&gt;logical and foundational research&lt;/strong&gt; on set theory and foundations of mathematics, the &lt;em&gt;search for mathematical roots&lt;/em&gt;, of the first part of the XX century.&lt;/p&gt;&#xA;&lt;p&gt;But &lt;strong&gt;there wasn’t only a stance against visualization.&lt;/strong&gt; At the same time, at the end of the XVIII century and during the XIX century, an important field of mathematical analysis, &lt;strong&gt;complex analysis&lt;/strong&gt;, was intertwined with geometry and images.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wessel,  Argand and Gauss, all recognized that complex numbers could be given a simple, concrete, &lt;em&gt;geometrical interpretation&lt;/em&gt; as points in the plane: the mystical quantity &lt;span class=&quot;math-container&quot;&gt;$a+ib$&lt;/span&gt; should be viewed simply as the point in the &lt;em&gt;x-y&lt;/em&gt; plane having cartesian coordinates &lt;span class=&quot;math-container&quot;&gt;$(a,b)$&lt;/span&gt; […] the reputation of Gauss ensured wide dissemination and acceptance of complex numbers as points in the plane[…] now  existed &lt;em&gt;some&lt;/em&gt;  way of making sense of these numbers - that they were now &lt;em&gt;legitimate&lt;/em&gt; objects of investigation […].  Riemann’s ideas, in particular, would simply  not have been possible without prior knowledge of the geometry of the complex plane.&lt;span class=&quot;math-container&quot;&gt;$^5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The history of rigorization continued during the XX century, in particular with &lt;strong&gt;Bourbaki&lt;/strong&gt;, a highly influential collection of French mathematicians who stressed axiomatics and rigour in the development of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Jean Dieudonné, one the most important mathematicians of the Bourbaki group, was an arch-enemy  of pictures and graphs.&lt;/p&gt;&#xA;&lt;p&gt;In the preface to his &lt;em&gt;Foundations of Modern Analysis&lt;/em&gt;(1969) Dieudonné states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;a strict adherence to axiomatic methods, with no appeal whatsoever to “geometric intuition”, at least in the formal proofs: a necessity which we have emphasized by &lt;em&gt;deliberately abstaining from introducing any diagram in the book&lt;/em&gt;.&lt;span class=&quot;math-container&quot;&gt;$^6$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An exclamation by Jean Dieudonné entered  the annals of history of mathematics.&#xA;During a conference about the didactics of mathematics in 1959, the Colloque de Royaumont, a workshop of ten days whose objective was to introduce a reform of  the teaching of mathematics in schools, Dieudonné, with regard to the teaching of geometry and pictures, cried:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;À bas Euclide! À mort le triangle!&lt;span class=&quot;math-container&quot;&gt;$^7$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; James Robert Brown, &lt;em&gt;Philosophy of Mathematics-A Contemporary introduction to the world of Proofs and Pictures&lt;/em&gt;, (2008)  second ed., Routledge. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt; &lt;em&gt;ibid.&lt;/em&gt; p. 15&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; &lt;sup&gt; In Brown, cit. p. 199. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^4$&lt;/span&gt; &lt;sup&gt; According to the contemporary definition of continuous function, a function defined only at isolated points is continuous, but its graph is evidently  composed of isolated points. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^5$&lt;/span&gt; &lt;sup&gt;Tristan Needham, &lt;em&gt;Visual Complex Analysis&lt;/em&gt;, 1997, Oxford University Press, pp. 2-3.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^6$&lt;/span&gt; &lt;sup&gt; In Brown, cit.  p. 199.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^7$&lt;/span&gt; &lt;sup&gt; (“Down with Euclid! To death the triangle!”). See Maurice Marshal, &lt;em&gt;Bourbaki&lt;/em&gt;, 2017, Edition Belin/Humensis, p. 222.  Also &lt;a href=&quot;https://matematica.unibocconi.it/articoli/anche-l%E2%80%99insegnamento-della-matematica-viene-coinvolto-una-revisione-di-obiettivi-contenuti-&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://matematica.unibocconi.it/articoli/anche-l%E2%80%99insegnamento-della-matematica-viene-coinvolto-una-revisione-di-obiettivi-contenuti-&lt;/a&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-05-28T11:57:52.627" LastActivityDate="2023-05-28T11:57:52.627" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15155" PostTypeId="1" CreationDate="2023-02-12T01:59:57.387" Score="3" ViewCount="192" Body="&lt;p&gt;On Wikipedia it says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative. He first mentioned it in a 1676 memoir &lt;br&gt; &lt;strong&gt;[ &lt;a href=&quot;https://en.wikipedia.org/wiki/Chain_rule#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chain Rule History - Wikipedia&lt;/a&gt; ]&lt;/strong&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, I do not know where to find a comprehensive library of all of Leibniz's works, nor a way of determining their publication date so I do not know how to find this memoir hence I beseech thee.&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="10106" LastEditDate="2023-02-12T04:00:36.600" LastActivityDate="2023-02-13T23:51:08.080" Title="What is the title of the 1676 Memoir in which Leibniz first used the Chain Rule?" Tags="|mathematics|calculus|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15157" PostTypeId="2" ParentId="15130" CreationDate="2023-02-12T11:51:58.480" Score="2" Body="&lt;p&gt;In 1831, Galois claimed simplicity when &lt;span class=&quot;math-container&quot;&gt;$n = 2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q = p$&lt;/span&gt; is a prime greater than &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; without proof. The first paragraph &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/grouptheory/PSLnsimple.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; has references to later proofs of the result you ask about, first when &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is prime and then general &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;. These proofs were decades later, in 1870 and the 1890s.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-02-12T11:51:58.480" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15158" PostTypeId="1" CreationDate="2023-02-12T14:08:48.070" Score="3" ViewCount="131" Body="&lt;p&gt;A number of years ago, during the break at a social theory seminar, someone mentioned that Niklas Luhmann was highly critical of environmental movements. Since the seminar was more about systems theory than any dislikes of particular movements, I didn’t dig deeper into that claim. Now however, I would be most curious to find out if there are any evidence of Luhmann publicly criticizing environmental movements because I can’t find much in his writings that I have access to (e.g. &lt;em&gt;Social Systems&lt;/em&gt;)?&lt;/p&gt;&#xA;" OwnerUserId="17957" OwnerDisplayName="Skabbpappa" LastEditorUserId="2277" LastEditDate="2023-02-27T07:50:55.510" LastActivityDate="2023-02-27T07:50:55.510" Title="Was Niklas Luhmann critical of environmental movements?" Tags="|social-sciences|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15159" PostTypeId="1" CreationDate="2023-02-12T17:27:58.180" Score="3" ViewCount="249" Body="&lt;p&gt;I'm curious whether anyone knows how &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; came to be used to represent eigenvalues and or who (if anyone) was responsible for the convention.  I've looked through a couple of books on the history of math/notation but haven't found a relevant reference.  Did the eigenvalue usage arise from work with wavelengths or quadratic forms?&lt;/p&gt;&#xA;&lt;p&gt;There is a reference &lt;a href=&quot;http://www.steen-frost.org/Steen/Papers/73spectral.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\lambda_i$&lt;/span&gt; in the normal form of &lt;span class=&quot;math-container&quot;&gt;$(Ax,x)$&lt;/span&gt; are the roots of the characteristic polynomial &lt;span class=&quot;math-container&quot;&gt;$\det(\lambda I-A)$&lt;/span&gt; [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;although I'm not sure if that is modern usage put into historical context.&lt;/p&gt;&#xA;" OwnerUserId="17952" LastEditorUserId="4251" LastEditDate="2023-02-13T13:19:39.923" LastActivityDate="2023-02-13T13:19:39.923" Title="Origin of the usage of $\lambda$ to represent eigenvalues" Tags="|mathematics|notation|linear-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15160" PostTypeId="1" CreationDate="2023-02-13T05:44:52.500" Score="1" ViewCount="106" Body="&lt;p&gt;Published tables of Fourier transform pairs have been available for many years. One such example is the &lt;a href=&quot;https://ia903204.us.archive.org/11/items/bstj7-4-639/bstj7-4-639_text.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; by George Campbell in the Bell Systems Technical Journal in 1928. Most such tables simply list the pairs, but I am interested in knowing how I might go about finding the source of a result, by which I mean an earlier paper where the pair and, presumably, its derivation, was published.&lt;/p&gt;&#xA;" OwnerUserId="17963" LastEditorUserId="17963" LastEditDate="2023-02-28T09:25:12.507" LastActivityDate="2023-02-28T09:25:12.507" Title="Early results on the Fourier transform" Tags="|mathematics|reference-request|harmonic-analysis|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15162" PostTypeId="2" ParentId="2896" CreationDate="2023-02-13T18:45:02.347" Score="1" Body="&lt;p&gt;'Calories' I believe are used the US which seems to have a rich history of fighting with the metric system. Cal and kcal are otherwise similar units (difference is where 1000 is in the calculation, but either equation is equal to the other)&lt;/p&gt;&#xA;" OwnerUserId="17965" LastActivityDate="2023-02-13T18:45:02.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15163" PostTypeId="1" AcceptedAnswerId="15166" CreationDate="2023-02-13T21:51:13.210" Score="4" ViewCount="197" Body="&lt;p&gt;I'm trying to find the location of the nodes of Mars (i.e., the point of intersection between the ecliptic and Mars's plane [rather I should say Mars average plane]) given by Copernicus. According to the Model, as I understand it, they should cut the ecliptic and be 180 degrees apart.&lt;/p&gt;&#xA;&lt;p&gt;The location of the nodes is important to have in order to calculate the latitude of Mars for any given point on its orb. However, I suspect that because Copernicus was using the Mean Sun, the nodes - by observation - would not be exactly 180 degrees apart. &lt;strong&gt;I ask where are the Martial nodes according to Copernicus, and how he determined their location&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Basically my question can be applied to Tycho Brahe as well, and I would appreciate if someone has the numbers?&lt;/p&gt;&#xA;" OwnerUserId="14954" LastActivityDate="2023-02-15T01:55:29.960" Title="How did Copernicus find the nodes of Mars? what were they?" Tags="|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15164" PostTypeId="2" ParentId="15155" CreationDate="2023-02-13T22:58:57.767" Score="9" Body="&lt;p&gt;I retracted my previous answer in the comments since it was actually for the product rule, not the chain rule.  Sorry about that.  After performing another search, I was able to find some links that might be helpful:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/questions/762/when-how-were-the-product-and-chain-rules-first-proved&quot;&gt;When/How were the product and chain rules first proved?&lt;/a&gt;&#xA;&lt;em&gt;See Top Answer by Michael Weiss.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The link that Weiss provides is no longer active, but I believe I found a working link:&lt;/p&gt;&#xA;&lt;p&gt;Rodriguez, Omar Hernandez and Lopez Fernandez, Jorge M. (2010) “A semiotic reflection on the didactics of the Chain rule,” &lt;em&gt;The Mathematics Enthusiast&lt;/em&gt;: Vol. 7 : No. 2, Art. 10.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1191&amp;amp;context=tme&quot; rel=&quot;noreferrer&quot;&gt;https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1191&amp;amp;context=tme&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Starting in Section 2: “History of the Chain Rule,” Rodriguez, states&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It may come as a surprise to the reader that nowhere in &lt;em&gt;Analyse des&#xA;infiniment petits&lt;/em&gt;, (L’Hospital, 1696), is the chain rule stated&#xA;explicitly...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It may be even more surprising to realize that the&#xA;statement of the chain rule is so absent in all of Euler’s analysis&#xA;books, &lt;em&gt;Introductio in analysin infinitorum&lt;/em&gt;, (Euler, 1748, Vol. 1),&#xA;(Euler, 1748, Vol. 2), and &lt;em&gt;Institutiones calculi differentialis&lt;/em&gt;,&#xA;(Euler, 1755). Furthermore, Euler did define the notion of a function&#xA;in (Euler, L., 1748, Vol. 1), but he never treated the topic of the&#xA;composition of functions in any of his writings, (Euler, 1748, Vol.&#xA;1), (Euler, 1748, Vol. 2) and (Euler, 1755).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;As far as we can tell,&#xA;the first mention of the Chain Rule in the literature of calculus&#xA;seems to be due to Leibniz (Child, 2007, p. 126), and it appears in a&#xA;1676 memoir (with various mistakes) in which he calculated...&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Emphasis added)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Footnote] 5 -  As far as we can tell, the first “modern” version of&#xA;the chain rule appears in Lagrange’s 1797 &lt;em&gt;Théorie des fonctions&#xA;analytiques&lt;/em&gt;, (Lagrange, J. L., 1797, §31, pp. 29); it also appears in&#xA;Cauchy’s 1823 &lt;em&gt;Résumé des Leçons données a L’École Royale Polytechnique&#xA;sur Le Calcul Infinitesimal&lt;/em&gt;, (Cauchy, A. L., 1899, Troisième Leçon,&#xA;pp. 25).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's the citation for the reference text:&lt;/p&gt;&#xA;&lt;p&gt;Child, J. M. (translator), Leibniz, G.W. (Author). (2007). The Early Mathematical Manuscripts of Leibniz. Merchant Books.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://dynref.engr.illinois.edu/rvc_Child_1920.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://dynref.engr.illinois.edu/rvc_Child_1920.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;So, to answer your title question, according to Child, the manuscript is simply titled,&#xA;&amp;quot;Manuscript, dated Nov., 1676&amp;quot; with the section titled &amp;quot;Calculus Tangentium differentialis [Differential calculus of tangents].&amp;quot;&lt;/strong&gt;  However, as Rodriguez and Fernandez point out in their paper, don't expect to see in Leibniz's writings the chain rule theorem in the form that we now know and recognize.&lt;/p&gt;&#xA;&lt;p&gt;Edit: Regarding the &amp;quot;publication date,&amp;quot; from what I've gathered from a cursory read of Child, it appears that Leibniz didn't publish these particular manuscripts himself, so I would be hesitant to put an emphasis on such dates.  Instead, the concern is when Leibniz put these ideas pen-to-paper in his notes.  From what I've gathered, it seems that Carl Immanuel Gerhardt is the one who compiled Leibniz's notes and published them in a series of texts dated 1846, 1848, and 1855, but I wouldn't give these dates much weight.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2023-02-13T23:51:08.080" LastActivityDate="2023-02-13T23:51:08.080" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15166" PostTypeId="2" ParentId="15163" CreationDate="2023-02-15T01:55:29.960" Score="6" Body="&lt;p&gt;&lt;strong&gt;Mars's latitudes with Copernicus&lt;/strong&gt;: Ancient theories of the planets' latitude did not give the same prominence as is now usual to the &lt;em&gt;'nodes'&lt;/em&gt; of the inclined planetary orbital planes, i.e. to their intersections with the plane of the ecliptic. Instead, what one sees in Ptolemy and the Ptolemaic tables, as well as in work of some later authors (including Copernicus's 'De Revolutionibus') is an emphasis on the &lt;em&gt;'limits'&lt;/em&gt; of latitude, north and/or south (see G J Toomer's 1984 translation of Ptolemy's 'Almagest' e.g. at p.633).&lt;/p&gt;&#xA;&lt;p&gt;Copernicus, too, discusses the &lt;em&gt;northern limit&lt;/em&gt;. In book 6, chapter 1,&#xA;of &lt;em&gt;'De Revolutionibus'&lt;/em&gt;, Copernicus wrote that the northern limit of latitude for the orbit of Mars, for his time*, occurred at 27° of Leo. (That amounts to 147° of ecliptic longitude.)  *The year applicable to this figure has been identified as 1523, based on a manuscript note by Copernicus (see Swerdlow (1984) reference linked below, at p.498).&lt;/p&gt;&#xA;&lt;p&gt;In the case of a more modern heliocentric orbit, such as Kepler's, the relation of the two kinds of measure is simple enough. The northern limit occurs at 90 degrees of the orbit onwards from the ascending node, and the descending node is located 90 degrees onwards from the northern limit.&lt;/p&gt;&#xA;&lt;p&gt;Copernicus in book 6 chapter 1 also explained the nodes in that way,&#xA;implying clearly that for Mars the longitudes of the ascending and descending nodes were (for his time) at 57° and 237°.  But the complications of Copernicus's latitude theory mean that the point of view from which these measures are taken is neither the earth nor the sun, but rather the geometrical center of the earth's (assumedly circular and eccentric) orbit around the sun. In his theory, and partly from this cause, the maximum north and south latitudes did not need to be numerically equal (celestial latitudes were always measured as deviations from the apparent course of the sun around the ecliptic or zodiac, but in Copernicus's theory of planetary latitudes the sun was not placed in the plane of the planetary orbits).&lt;/p&gt;&#xA;&lt;p&gt;These complications among others are discussed by N M Swerdlow in&#xA;&lt;a href=&quot;https://books.google.com/books?id=uHPgBwAAQBAJ&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Mathematical Astronomy in Copernicus's De Revolutionibus&amp;quot; (1984)&lt;/a&gt;, Part 1, at ch.6 (pp.483-538).  Swerdlow calls Copernicus's theory of planetary latitudes the 'most deeply flawed' part of &lt;em&gt;'De Revolutionibus'&lt;/em&gt;, and describes evidence suggesting that it was revised and finalised in haste shortly before publication, leaving a number of inconsistencies. He also mentions that this part of Copernicus's work has attracted few studies.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Mars's latitudes with Tycho Brahe&lt;/strong&gt;:  In the question, analogous queries are raised about the latitudes of Mars according to Tycho Brahe. The situation there is rather different than with Copernicus: Tycho published, besides his astronomical observations, theories for the motion of the sun and moon, but not theories for the other planets. It is possible that the available account closest to Tycho's point of view is in the (non-Keplerian) account given in 1622 by Tycho's former assistant, Christian Soerensen or Severinus, known as 'Longomontanus', in his book &lt;a href=&quot;http://books.google.com/books?id=f1I_AAAAcAAJ&quot; rel=&quot;noreferrer&quot;&gt;'&lt;em&gt;Astronomia danica&lt;/em&gt;'&lt;/a&gt;. As with Copernicus, the theories of planetary latitude given by Longomontanus are complicated. They seem to share much with Copernicus, and the numbers closest to those sought by the question are in 'Liber secundus', chap.14, on page 265. Longomontanus gives for Mars a longitude at 18° of Aquarius (i.e. longitude 318°) for the &lt;em&gt;southern&lt;/em&gt; limit of latitude. He also gives 'intersections' which appear to be the ascending and descending nodes for Mars's orbit, at 18° of Taurus and Scorpio. The greatest latitudes he gives are unequal to north and south (4°33' and 6°42'), so he clearly did not share Kepler's views.&#xA;Longomontanus' work is in Latin and I am not aware of any translation.&lt;/p&gt;&#xA;&lt;p&gt;Copernicus's 'De Revolutionibus' book 6 (on planetary latitudes, with examples based on Mars) is translated on p.307-330 in Edward Rosen, &amp;quot;Nicholas Copernicus on the Revolutions. Translation and commentary&amp;quot;, (Polish Scientific Publishers, Warsaw/Cracow; also Macmillan; also Johns Hopkins, Baltimore; 1978).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2023-02-15T01:55:29.960" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15167" PostTypeId="2" ParentId="656" CreationDate="2023-02-15T23:32:30.503" Score="3" Body="&lt;p&gt;It's only a 'huge' change because set theory formalities, aka ZFC, was so hegemonic.&lt;/p&gt;&#xA;&lt;p&gt;Had a wider discussion ensued as to what to take as axiomatic, the function or the set or both, then the sea change that ensued after the discovery of category theory might not be so shocking.&lt;/p&gt;&#xA;&lt;p&gt;In the large scheme of things, one might read set theory and category theory as part of a debate and philosophy about how to collect mathematical objects into collections. I certainly think that is a healthier way to think about this rather than an unhealthy way of setting category theory as a competitor to set theory.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2023-02-15T23:32:30.503" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15168" PostTypeId="1" CreationDate="2023-02-18T19:13:22.133" Score="1" ViewCount="64" Body="&lt;p&gt;I've been reading about Thermodynamics lately. The set of equations satisfied in an adiabatic process (and also more generally in polytropic ones) is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$p_1V_1^\gamma = p_2V_2^\gamma$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$T_1^\gamma p_1^{1-\gamma} = T_2^\gamma p_2^{1-\gamma}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I've seen these called &lt;em&gt;Poisson's equations&lt;/em&gt; or &lt;em&gt;Poisson's laws&lt;/em&gt;, but I was able to find no information about the naming. Did &lt;a href=&quot;https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poisson&lt;/a&gt; really derive/talk about these relations, or is it just a misnomer? Maybe another person sharing the same last name? Thanks!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I now know that Poisson first used &lt;span class=&quot;math-container&quot;&gt;$\gamma$&lt;/span&gt; for &amp;quot;condensation&amp;quot;, which will later become the heat capacity ratio used above. However, I still don't know if he treated those equations or who first named them after him.&lt;/p&gt;&#xA;" OwnerUserId="17990" LastEditorUserId="17990" LastEditDate="2023-02-21T11:41:11.570" LastActivityDate="2023-02-21T11:41:11.570" Title="Poisson's laws for adiabatic processes" Tags="|physics|terminology|thermodynamics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15169" PostTypeId="1" AcceptedAnswerId="15179" CreationDate="2023-02-18T19:26:20.510" Score="3" ViewCount="182" Body="&lt;p&gt;How did Aristotle explain magnetism? As far as I know, there were only 7 postulates in Aristotelian mechanics:&lt;/p&gt;&#xA;&lt;p&gt;The first hypothesis Earth is the center of the universe.&lt;/p&gt;&#xA;&lt;p&gt;Second hypothesis All material objects are made of the four elements originally&#xA;proposed by Empedocles and later adopted by Plato, namely earth, water, air, and&#xA;fire.&lt;/p&gt;&#xA;&lt;p&gt;Third hypothesis Each of these elements has its natural place, or physical&#xA;location, in the universe.&lt;/p&gt;&#xA;&lt;p&gt;Fourth hypothesis According to the hypothesis of antiperistasis introduced by Plato, air ‘‘displaced’’ from the front of the stone moves to its back and ‘‘pushes’’ it forward. But as the ‘‘push’’ is transmitted from one point of the air to another, it slowly weakens and allows the natural motion of the stone to prevail. As a result, the upward motion slows down and eventually reverts to a downward motion, causing the stone to hit the ground. Aristotle modified slightly Plato’s hypothesis of antiperistasis, stating that our moving hand sets to motion successive layers of air, which, in turn, push the stone. As the ‘‘force’’ is transmitted from one layer of air to another, it decays and finally, the natural downward motion of the stone prevails.&lt;/p&gt;&#xA;&lt;p&gt;Fifth hypothesis The sky and the heavenly bodies are made of a substance that is neither earth nor water, air or fire. It is a fifth element which, following the ideas of earlier natural philosophers (Philolaos, Xenophanes, and Parmenides), he named aether.3 The physical place of this fifth element was beyond the realm of fire, outside the Moon’s orbit.&lt;/p&gt;&#xA;&lt;p&gt;Sixth hypothesis The laws governing the motion of celestial bodies are different from those governing motion on Earth. So Aristotle arrived at the conclusion that, while in the region of the universe inside the Moon’s orbit, the natural state of objects is rest, in heaven the natural state of objects is eternal circular motion.&lt;/p&gt;&#xA;&lt;p&gt;Seventh hypothesis There can be no vacuum in nature&lt;/p&gt;&#xA;" OwnerUserId="16986" LastActivityDate="2023-02-20T21:05:55.563" Title="Aristotelian explanation of magnetism" Tags="|magnetism|aristotle|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15170" PostTypeId="1" AcceptedAnswerId="15171" CreationDate="2023-02-18T21:14:28.557" Score="22" ViewCount="8523" Body="&lt;p&gt;In an interview I recently saw with &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Agassi&quot; rel=&quot;noreferrer&quot;&gt;Joseph Agassi&lt;/a&gt; he said that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...The teachers that kicked him out asked that it be written in his [Einstein's] report card that they have no complaint against him. Usually they would kick a kid if he is bothersome, unruly and so on. But he [Einstein] was very bothersome in his peacefulness.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The interview is in Hebrew, but in case you're interested you can watch it &lt;a href=&quot;https://www.youtube.com/watch?v=w9ZJLgb5jco&amp;amp;t=2523s&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Can this be corroborated / refuted? I wasn't able to find any sources either way.&lt;/p&gt;&#xA;" OwnerUserId="17992" LastEditorUserId="17992" LastEditDate="2023-02-20T14:19:19.187" LastActivityDate="2023-11-12T08:03:55.413" Title="Is it true that Albert Einstein was kicked out of high school due to his &quot;peacefulness&quot;?" Tags="|biographical-details|einstein|scientists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15171" PostTypeId="2" ParentId="15170" CreationDate="2023-02-19T01:27:20.190" Score="53" Body="&lt;p&gt;The most important resource with regard to Albert Einstein's biography are Princeton University's &lt;em&gt;Collected Papers&lt;/em&gt;. Particularly relevant to the question is John Stachel (ed.), &lt;em&gt;The Collected Papers of Albert Einstein, Volume I: The Early Years 1879-1902&lt;/em&gt;, Princeton University Press 1987. In this we find excerpts from a biographical sketch written by Einstein's younger sister Maja in 1924:&lt;/p&gt;&#xA;&lt;p&gt;Maja Winteler-Einstein, &amp;quot;Albert Einstein — Beitrag für sein Lebensbild&amp;quot;, &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol1-doc/48&quot; rel=&quot;noreferrer&quot;&gt;pp. xlviii-lxvi&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In 1894 the electrotechnical company of Einstein's father and uncle suffered business failure, and based on a suggestion by an Italian friend the Einstein family moved to Milan in Italy to try and rebuild their lives there. 15-year old Albert was left behind in Munich to finish high school. He had been enrolled at the prestigious Luitpold-Gymnasium since 1887. Winteler-Einstein writes (&lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol1-doc/63&quot; rel=&quot;noreferrer&quot;&gt;p. lxiii&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Besonders unangenehm war dem Jungen auch der militärische Ton in der Schule, die systematische Erziehung zur Verehrung der Autoritäten, die bereits die Schüler an die militärische Zucht gewöhnen sollten. Mit Grauen gedachte er der nicht mehr allzufernen Zeit, da er den Soldatenrock anzuziehen hatte, um seiner Dienstpflicht zu genügen. &lt;sup&gt;[57]&lt;/sup&gt; Seelisch bedrückt u. nervös geworden, sucht er nach einem Ausweg. Als daher der Ordinarius der Klasse ihn bei irgend einer Gelegenheit hart anliess (es war derjenige, der ihm prophezeit hatte, dass nie etwas rechtes aus ihm werden würde), liess er sich vom Hausarzt ein Zeugnis geben, wies es dem Schuldirektor vor u. reiste&#xA;kurzentschlossen nach Mailand zu seinen Eltern.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;The military tone at the school was particularly objectionable to the boy, the systematic education to venerate authorities designed to accustom already students to military discipline. With utter dread he thought of the not too distant time when he would have to don a soldier's uniform to comply with his [military] service requirements. Emotionally depressed and nervous he looked for an escape route. Therefore, when his home-room teacher spoke to him harshly at some occasion (it was the same one who had pronounced that he would never amount to anything), he obtained a doctor's note from the family physician, showed it to the school's principal, and shortly departed to his parents in Milan.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;At that time all German males from age 17 to 45 were &amp;quot;wehrpflichtig&amp;quot;, that is, &lt;a href=&quot;https://de.wikipedia.org/wiki/Deutsches_Heer_(Deutsches_Kaiserreich)&quot; rel=&quot;noreferrer&quot;&gt;subject to military conscription&lt;/a&gt; and were called up for compulsory military service between ages 20 and 39. As an additional requirement young males age 16 or older had to obtain permission from the government before leaving the country, presumably to prevent draft dodging by temporarily moving abroad. Einstein was about 15¾ years old when he left Germany. Einstein was a pacifist who abhorred all things military [*]. Two footnotes to Winteler-Einstein's text make clear Einstein's perception of his teachers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;[41]&lt;/sup&gt; Einstein reportedly said: &amp;quot;Die Lehrer in der Elementarschule kamen mir wie Feldwebel vor und die Lehrer im Gymnasium wie Leutnants&amp;quot; (&lt;em&gt;Frank 1979&lt;/em&gt;, pp. 24-25, see also &lt;em&gt;Moszkowski 1921&lt;/em&gt;, p. 221)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;[56]&lt;/sup&gt; [...] Einstein later spoke of his Gymnasium teachers as &amp;quot;überwiegend dem Leutnants-charakter zugewendet&amp;quot; [&lt;em&gt;Moszkowski 1921&lt;/em&gt;, pp. 221)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These say that &amp;quot;My teachers in elementary school seemed like [drill] sergeants to me, and the teachers in the &lt;em&gt;Gymnasium&lt;/em&gt; like lieutenants&amp;quot; and most of his high-school teachers &amp;quot;adopted the character of lieutenants&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I see no evidence of Einstein being expelled from (kicked out of) the Luitpold-Gymnasium, but a further footnote indicates that his home-room teacher had expressed the desire for him to leave the school:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;sup&gt;[58]&lt;/sup&gt; &amp;quot;Als ich in der 7. Klasse des Luitpoldgymnasiums war, liess mich der Klassenlehrer [Joseph Degenhart] kommen und äusserte den Wunsch, ich möchte die Schule verlassen. Auf meine Bemerkung, dass ich mir doch nichts hätte zuschulden kommen lassen, antwortete er nur: Ihre blosse Anwesenheit verdirbt mir den Respekt in der Klasse&amp;quot; (Einstein to Philipp Frank, draft letter, 1940). He withdrew from the school on 29 December 1894 (&lt;em&gt;Jahresbericht München 1895&lt;/em&gt;, pp. 27, 28).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;When I attended the 7&lt;sup&gt;th&lt;/sup&gt; grade at the Luitpold-Gymnasium my home-room teacher called me in and expressed the desire that I should leave the school. When I remarked that I had done nothing wrong, he just replied: Your presence alone ruins the respect I receive in class.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;[*] In Albert Einstein, &amp;quot;Mein Weltbild&amp;quot;, Amsterdam: Querido Verlag 1934, he famously stated (I found some indications that this may have appeared in his writings as early as 1930):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wenn einer mit Vergnügen in Reih und Glied zu einer Musik marschieren kann, dann verachte ich ihn schon; er hat sein großes Gehirn nur aus Irrtum bekommen, da für ihn das Rückenmark schon völlig genügen würde. Diesen Schandfleck der Zivilisation sollte man so schnell wie möglich zum Verschwinden bringen. Heldentum auf Kommando, sinnlose Gewalttat und leidige Vaterländerei, wie glühend hasse ich sie, wie gemein und verächtlich erscheint mir der Krieg; ich möchte mich lieber in Stücke schlagen lassen, als mich an einem so elenden Tun zu beteiligen!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;If someone can enjoy marching in rank and file accompanied by music I already despise him; he received his large brain only by accident as the spinal cord would be fully sufficient for him. This disgrace to civilization should be abolished as soon as possible. Heroism on command, senseless violence, and vexacious nationalism, how ardently I hate them, how mean and despicable war appears to me; I'd rather be hacked to pieces than participate in such base activity.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-02-19T08:01:34.483" LastActivityDate="2023-02-19T08:01:34.483" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15173" PostTypeId="1" AcceptedAnswerId="15174" CreationDate="2023-02-19T18:43:05.453" Score="2" ViewCount="145" Body="&lt;p&gt;I am looking for which manuscripts and if available, through what chains of transmission copies of Newton's book &amp;quot;TheMethod of Fluxions&amp;quot; have reached us today&lt;/p&gt;&#xA;&lt;p&gt;So far I could not find anything with google searches.&lt;/p&gt;&#xA;&lt;p&gt;thanks for the help&lt;/p&gt;&#xA;" OwnerUserId="9988" LastActivityDate="2023-02-20T19:31:54.397" Title="What are the various manuscripts/transmissions of Newton's book &quot;The Method of Fluxions&quot;?" Tags="|calculus|newton|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15174" PostTypeId="2" ParentId="15173" CreationDate="2023-02-20T00:03:46.270" Score="3" Body="&lt;p&gt;Essentially, there is only one authoritative edition most closely sourced from the original.  Newton's work on fluxions was not published during his lifetime even though he had in preparation an incomplete manuscript and notes for possible publication.  The text we currently have is John Colson's English translation published in 1736 from a transcription belonging to William Jones. Most of the extant work of Newton can be viewed at the University of Cambridge, Newton Project.  Begin with this link to the &lt;a href=&quot;https://www.lib.cam.ac.uk/collections/departments/manuscripts-university-archives/significant-archival-collections/papers-sir&quot; rel=&quot;nofollow noreferrer&quot;&gt;Papers of Sir Isaac Newton&lt;/a&gt; or continue at this link through the &lt;a href=&quot;https://www.newtonproject.ox.ac.uk/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton Project&lt;/a&gt;.  A pdf of the original 1736 text is readily available through Google.&lt;/p&gt;&#xA;" OwnerUserId="18000" LastActivityDate="2023-02-20T00:03:46.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15175" PostTypeId="2" ParentId="656" CreationDate="2023-02-20T04:45:47.197" Score="6" Body="&lt;p&gt;The conventional view is that categories were introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s as a tool for the study of algebraic topology. What we now call functors and natural transformations were manifesting themselves, but there was not yet any precise language for defining them formally.  So Eilenberg and Mac Lane invented that language.&lt;/p&gt;&#xA;&lt;p&gt;Category theory is now often thought of as being relevant to the foundations of mathematics more generally, and comes up a lot in philosophical discussions.  But this was not true in the early days.  Eilenberg and Mac Lane were initially motivated by technical questions in a particular branch of mathematics, and were not trying to come up with a new way of thinking about all of mathematics.  Even as category theory developed further, with advances in homological algebra and algebraic geometry, there were always concrete mathematical problems driving the developments.  The notion that category theory might &amp;quot;overthrow&amp;quot; set theory and lead to a new way to think about mathematics was a rather late development.&lt;/p&gt;&#xA;&lt;p&gt;Of course, it's possible that if someone digs deeper into the historical raw data, this conventional narrative could be upended. But I rather doubt that it can be shown that the origins of category theory came from philosophy or art.  You might have better luck trying to show that William Lawvere's view of category theory was partially influenced by extra-mathematical ideas.  That is, not category theory itself, but the &lt;i&gt;use of category theory for philosophical ends&lt;/i&gt;, might have been partially motivated by philosophical considerations coming from outside of mathematics.&lt;/p&gt;&#xA;&lt;hr&gt;&#xA;&lt;p&gt;&lt;b&gt;EDIT:&lt;/b&gt; I was just reading the Introduction to Johnstone's book &lt;em&gt;Stone Spaces&lt;/em&gt;, and he says that Stone's work in the 1930s already contained examples of what we would nowadays call &lt;em&gt;functors&lt;/em&gt; and &lt;em&gt;equivalence of categories&lt;/em&gt;. Johnstone writes, &amp;quot;All this was proved in detail by Stone, although the categorical language in which we now express it was not introduced until the following decade; but Stone's Theorem was undoubtedly one of the major influences which prepared the mathematical world for the introduction of categories by Eilenberg and Mac Lane.&amp;quot;  (But note the careful wording; later in the same Introduction, Johnstone mentions that &amp;quot;the categorical ideas present in Stone's 1937 paper were not directly followed up by the founders of category theory.&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;So you could argue that categorical thinking predated Eilenberg and Mac Lane, and was implicit in Stone's work.  But Stone's work still betrays no evidence of influence from philosophy or art.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2023-06-01T18:57:08.057" LastActivityDate="2023-06-01T18:57:08.057" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15179" PostTypeId="2" ParentId="15169" CreationDate="2023-02-20T16:14:09.977" Score="7" Body="&lt;p&gt;There is only an small account of Aristotle about magnetism in &lt;a href=&quot;https://en.wikipedia.org/wiki/On_the_Soul&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;On the soul&lt;/em&gt;&lt;/a&gt; (see chapter I.2 and I.5). He discusses &lt;a href=&quot;https://en.wikipedia.org/wiki/Thales_of_Miletus&quot; rel=&quot;noreferrer&quot;&gt;Thales of Miletus&lt;/a&gt; theory of the motion of lodestone. According to Aristotle, Thales associated the magnetic forces to the presence of &amp;quot;souls&amp;quot; inside the rocks (being alive/animated). According to a &lt;a href=&quot;https://books.google.fr/books?redir_esc=y&amp;amp;id=hkMPAAAAMAAJ&amp;amp;q=aristotle#v=snippet&amp;amp;q=aristotle&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;History of electricity&lt;/em&gt;&lt;/a&gt; by Benjamin Park:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Aristotle  reports  the  sayings  of  Thales  only  by  hearsay, and  then  with  extreme  caution:  the  first  being  that  every- thing is  full  of  gods,  and  the  second*  (and  it  is  this  which is  of  especial  importance  in  our  present  research)  that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thales  too,  as  is  related,  seems  to  regard  the  soul  as somehow  producing  motion,  for  he  said  that  the  stone  has a  soul  since  it  moves  iron.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If Aristotle adhered fully to Thales hypothesis, it is not completely clear.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-02-20T21:05:55.563" LastActivityDate="2023-02-20T21:05:55.563" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15180" PostTypeId="2" ParentId="15173" CreationDate="2023-02-20T17:10:13.270" Score="6" Body="&lt;p&gt;The question asks for chains of transmission and for manuscripts antecedent to Newton's eventual book(s) on fluxions.&lt;/p&gt;&#xA;&lt;p&gt;So I would recommend, as the first places to start, the 8-volume series of 'Mathematical Papers of Isaac Newton' edited by D T Whiteside (Cambridge University Press 1967-81), and also, where available and if needed, the corresponding high-definition online images of the manuscripts at the University Library in Cambridge. Advantages of the book volumes include their commentaries, and the fact that many of the manuscripts in Latin have been provided with an English translation. Advantage of the manuscript images is to see the manuscripts as Newton left them.&lt;/p&gt;&#xA;&lt;p&gt;For each of the Whiteside volumes, the full table of contents and index are now available online (see some links below). Content from the books is otherwise hard to find online. (Hardly any of the text was visible when I last looked through the Google pages, which only showed metadata.)&lt;/p&gt;&#xA;&lt;p&gt;Much of Newton's work in the field of calculus is in volumes 1, 3 and 7 of the Whiteside series.  Here, for example, to start with, is part of the contents list for volume 1 (1664-6), along with some of the manuscript numbers in the archive at the University Library at Cambridge:&lt;/p&gt;&#xA;&lt;p&gt;First, &lt;strong&gt;from Volume 1 of Whiteside series:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Mathematical Papers of Isaac Newton, Volume I: 1664-1666,  (ed.) D&#xA;T Whiteside (1967), from CUP website:-&lt;/p&gt;&#xA;&lt;p&gt;Table of contents at:&#xA;&lt;a href=&quot;https://assets.cambridge.org/97805210/45957/toc/9780521045957_toc.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://assets.cambridge.org/97805210/45957/toc/9780521045957_toc.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;index at:&#xA;&lt;a href=&quot;https://assets.cambridge.org/97805210/45957/index/9780521045957_index.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://assets.cambridge.org/97805210/45957/index/9780521045957_index.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The volume includes:--&lt;/p&gt;&#xA;&lt;p&gt;Part II. Researches in Analytical Geometry and Calculus 1664–1666:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Early notes on Analytical Geometry (autumn 1664) ... p 155&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Work on the Cartesian Subnormal (winter 1664/5) ... p 213&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Miscellaneous Problems in Analytical Geometry and Calculus (1664/5) ... p 234&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 4004    : 5v, 8v, 87v-89v,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Normals, Curvature and the Resolution of the General Problem&#xA;of Tangents (winter 1664-spring 1665) ... p 245&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 4004    : 6r, 30v-33v, 47r-50r,&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Calculus Becomes an Algorithm (mid-1665) ... p 298&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 4000    : 92v-96r, 120r-149r,&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 3960.12 : 206, 199-202&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 3958.4  : 77r,80v&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 4000    : 152r-163v&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 3958.2  : 30v,30r&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The General Problems of Tangents, Curvature and Limit-Motion&#xA;Analysed by the Method of Fluxions (oct 1665-may 1666) ... p 369&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 3958.2 : 35r,37v,34v&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 4004   : 50v-51r , 57r-57v, 51r-51v&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The October 1666 Tract on Fluxions ... p 400 (-450)&lt;/p&gt;&#xA;&lt;p&gt;ULC MS Add 3958.3 : 48v-63v&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(The manuscripts within a numbered section of the archive are numbered by leaves (folios), not by pages, where r indicates the front or &lt;em&gt;recto&lt;/em&gt; side of a numbered leaf and v the back or &lt;em&gt;verso&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;(The online manuscript images for the last-mentioned item, the &lt;strong&gt;1666 tract on fluxions&lt;/strong&gt;, begin at &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03958/92&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cudl.lib.cam.ac.uk/view/MS-ADD-03958/92&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;(Meaning of the numbers:- image 92 of 190 in section 3958 shows folio 48 verso in subsection 3).&lt;/p&gt;&#xA;&lt;p&gt;The items referred to in MS Add 4000 (the 'College notebook') from 92v onwards can be seen at &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-04000/197&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cudl.lib.cam.ac.uk/view/MS-ADD-04000/197&lt;/a&gt; onwards.&lt;/p&gt;&#xA;&lt;p&gt;The mentioned content from MS Add 4004 (the 'Waste book') begins online at &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-04004/22&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cudl.lib.cam.ac.uk/view/MS-ADD-04004/22&lt;/a&gt; (f.5v, image 22 of 935).&lt;/p&gt;&#xA;&lt;p&gt;Next, &lt;strong&gt;Volume 3 of Whiteside series (1670-73) includes:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;sec.2. The tract &amp;quot;De methodis serierum et fluxionum&amp;quot; ... pp 32-330 (the first leaf is restored from contemporary transcripts, the rest survives in ULC MS Add 3960.14 : 3-132.&lt;/p&gt;&#xA;&lt;p&gt;(I looked for the images, which are supposed to be online at &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03960&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cudl.lib.cam.ac.uk/view/MS-ADD-03960&lt;/a&gt; , but the pages and images in that section seem poorly indexed.)&lt;/p&gt;&#xA;&lt;p&gt;It is worth noting that by no means all of Newton's work and papers in the field of calculus was expressed in what became 'fluxional' notation: he seemed (awkwardly for everyone else) to think of notations as secondary compared with the substance of methods. This does mean that a search for antecedents of the published book on fluxions needs to consider, as well, the wider range of expressions used by Newton. Some of Newton's calculus work was even cast in geometrical form, and is represented like that in the Principia. Near-contemporaries in the years immediately following publication of the 'Principia' recognized the essential identity between the various forms of expression. An example was Pierre Varignon (1654-1722), who defended the Leibnizian version of the calculus, shortly after 1700, from attacks made upon it by Michel Rolle (1652-1719). (The episode amounted to an earlier 'iteration' of the better-known attacks made by Berkeley in the 1730s on the calculus, criticizing its use of infinitesimals.) Varignon, in support of his defence of the calculus, referred to Newton's demonstrations in section 1 of book 1 in the 'Principia', which are not vulnerable to the criticisms of infinitesimals made by Rolle or Berkeley (see answers and linked references in &lt;a href=&quot;https://hsm.stackexchange.com/questions/7988/did-michel-rolle-say-that-the-calculus-is-a-collection-of-ingenious-fallacies/7989#7989&quot;&gt;Did Michel Rolle say that the calculus is &amp;quot;a collection of ingenious fallacies&amp;quot;?&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/questions/2362/why-is-calculus-missing-from-newtons-principia/6535#6535&quot;&gt;Why is calculus missing from Newton&amp;#39;s Principia?&lt;/a&gt; (answer, it isn't missing)).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2023-02-20T19:31:54.397" LastActivityDate="2023-02-20T19:31:54.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15181" PostTypeId="1" AcceptedAnswerId="15183" CreationDate="2023-02-20T17:26:23.880" Score="2" ViewCount="332" Body="&lt;p&gt;Aristotle (384 BC – 322 BC), a pioneering, iconoclastic, and brilliant ancient Greek philosopher, made the observation in his writings that the long term stable state of objects is at rest, that motion needs a cause to make it happen, and that continuous motion needs a continuous cause. He further observed that the cause should itself be in motion for it to induce motion: a stationary object cannot induce another to move.&lt;/p&gt;&#xA;&lt;p&gt;He extended this line of deduction further by arguing that since all observed motion in nature is required to be induced by a cause which itself is in motion, there must be a primordial &amp;quot;&lt;a href=&quot;https://www.britannica.com/biography/Aristotle/The-unmoved-mover&quot; rel=&quot;nofollow noreferrer&quot;&gt;unmoved mover&lt;/a&gt;&amp;quot; which must be the seed cause behind all the subsequent motion. He drew parallels between the concept of the unmoved mover, &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Active_intellect&quot; rel=&quot;nofollow noreferrer&quot;&gt;active intellect&lt;/a&gt;&amp;quot;, and God.&lt;/p&gt;&#xA;&lt;p&gt;There is absolutely nothing wrong in these observations for almost all of the terrestrial motion experienced by humans, except the motion of falling objects which seems to be caused by an unmoving Earth, and the motion of living things which seem to move of their own will without being caused by another moving object.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;What was Aristotle's way of explaining these two phenomena?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;[Even the motion of heavenly bodies seems to be unceasing and uncaused, but Aristotle explained them as being &lt;a href=&quot;https://en.wikipedia.org/wiki/Aristotelian_physics#/media/File:Ptolemaicsystem-small.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;kept in motion by the &amp;quot;unmoved mover&amp;quot;]&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="10548" LastEditorUserId="10548" LastEditDate="2023-03-31T03:04:27.993" LastActivityDate="2023-03-31T03:04:27.993" Title="How did Aristotle explain the motion of living things moving by themselves, and falling of objects, with his hypothesis of all motion needing a cause?" Tags="|physics|philosophy-of-science|ancient-greece|ancient|aristotle|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15183" PostTypeId="2" ParentId="15181" CreationDate="2023-02-21T05:08:18.117" Score="6" Body="&lt;p&gt;This issue is analyzed by &lt;a href=&quot;https://www.degruyter.com/document/doi/10.1515/9781400887330-004/pdf&quot; rel=&quot;noreferrer&quot;&gt;Furley in Self-Movers&lt;/a&gt;. Aristotle's answer is somewhat convoluted due to conflicting motives, one from common sense and the other from his cosmology. For falling bodies we have the distinction between forced and natural motions, see &lt;a href=&quot;https://arxiv.org/abs/1312.4057&quot; rel=&quot;noreferrer&quot;&gt;Rovelli, Aristotle's Physics: a Physicist's Look&lt;/a&gt;, and falling bodies move &amp;quot;naturally&amp;quot; towards the center of the Earth without being actively pushed. Yet, as Furley explains:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The natural&#xA;bodies, as opposed to things with souls, have a source not of causing&#xA;movement or of acting but of being acted on. In&#xA;fact, this gives too little to the natural bodies in Aristotle's theory. He&#xA;should at least stress that they have an internal source of being acted on in a&#xA;fully determinate way. But we do not need to pursue that subject here, and&#xA;we can also leave aside the difficult question of what is the active mover of&#xA;the natural bodies when they move according to their nature - a question&#xA;to which Aristotle offers no wholly satisfactory answer.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What &amp;quot;acts on&amp;quot; the &amp;quot;internal source&amp;quot; is presumably their &amp;quot;nature&amp;quot;, which draws them to their &amp;quot;natural place&amp;quot;. How it connects to the unmoved mover is obscure.&lt;/p&gt;&#xA;&lt;p&gt;As for the ensouled (animals and humans), on the one hand, Aristotle wants to say that their souls initiate motion. On the other, that would break his cosmological chain that leads to the unmoved mover:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The tension in Aristotle's thinking about this subject is set up by a clash&#xA;of motives. He clearly wants to preserve the commonsense intuition that&#xA;the movements of animals, and especially the actions of human beings, are&#xA;not brought about by external agents in the same way that the movements&#xA;of inanimate beings are. Yet he sees a danger that all the movements in the&#xA;cosmos might be thought explicable on this principle of the self-movement&#xA;of autonomous parts, and so insists that even this self-movement presupposes&#xA;some external changes that are independent of animal movements.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What Aristotle settled on is clever. He says that although self-movers are not moved entirely by themselves, they do, nonetheless, mediate motion in ways qualitatively different from getting pushed by external force or drawn to the natural place, and put enough &amp;quot;self&amp;quot; into it to deserve the name. The &amp;quot;external changes&amp;quot; (aside from metabolism and mechanical stimuli) are objects of desire that &amp;quot;move the soul&amp;quot;. Yet they do not move it in a mechanical way of a cart dragging a dog (to use &lt;a href=&quot;https://dailystoic.com/chrysippus/&quot; rel=&quot;noreferrer&quot;&gt;Chrysippus's metaphor&lt;/a&gt;), that, in humans, would amount to abolition of moral responsibility.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Animals are clearly distinguished from inanimate&#xA;natural bodies in that although both require external things to explain their&#xA;movements, only animals require external things perceived (or otherwise&#xA;apprehended) as having significance for them. Note that this is not just a&#xA;difference in the complexity of the response to a stimulus, but a difference&#xA;in kind. Only a being with a soul can move in this way. An animal is&#xA;correctly described as a self-mover, because when it moves, its soul moves&#xA;its body, and the external cause of its motion is a cause of motion only because it is &amp;quot;seen&amp;quot; as such by a faculty of the soul.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is Aristotle's way to have his cake (the unmoved mover as the ultimate source of all motion) and eat it too (get ensouled self-movers to initiate motion autonomously). Does he thread the needle? Perhaps, for his own purpose of getting to the unmoved mover, but not for the later uses in the cosmological argument and/or determinism with &lt;em&gt;sufficient&lt;/em&gt; causes along the way leading to the all-determining first cause. On Aristotle's view, soul-mediated effects are not determined by their causes, and so leave room for indeterminism and &amp;quot;free will&amp;quot;. In contrast to the early determinist Chrysippus, Aristotle was an early libertarian. His unmoved mover is only the first cause in a weak sense, and his God is no Providence of monotheism:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The point is that external&#xA;objects are not in themselves sufficient causes for the voluntary movements&#xA;of animals. But they do have some effect on the soul, and it would be&#xA;obstinate of Aristotle to deny that the effect can be called a movement.&#xA;There is one conspicuous loose end in the theory that the arche of human&#xA;actions is &amp;quot;in&amp;quot; the agent. Aristotle maintains that people are moved to act&#xA;by what appears desirable to them, that what appears desirable depends on&#xA;their character, and that their character in turn depends on their actions&#xA;and is therefore &amp;quot;in their power.&amp;quot; His theory needs some explanation of&#xA;these character - forming actions and of how it is that they are not caused by&#xA;external pressures but proceed from an arche in the agent himself.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-02-21T05:08:18.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15185" PostTypeId="1" CreationDate="2023-02-21T10:52:29.600" Score="2" ViewCount="252" Body="&lt;p&gt;We can often see art that is influenced by science, be it in paintings, music, novels or movies. But has any idea from the arts ever influenced a scientist to come up with a new discovery or idea?&lt;/p&gt;&#xA;" OwnerUserId="18016" LastActivityDate="2023-02-24T22:35:20.357" Title="Were scientific discoveries ever inspired by art?" Tags="|physics|chemistry|biology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15186" PostTypeId="1" CreationDate="2023-02-21T14:44:13.773" Score="2" ViewCount="204" Body="&lt;p&gt;Based on the excellent answer from @njuffa in the post &lt;a href=&quot;https://hsm.stackexchange.com/questions/15170/is-it-true-that-albert-einstein-was-kicked-out-of-high-school-due-to-his-peacef&quot;&gt;Is it true that Albert Einstein was kicked out of high school due to his &amp;quot;peacefulness&amp;quot;?&lt;/a&gt; there is a quote from what the &amp;quot;home-room teacher&amp;quot; who said to Einstein:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Your presence alone ruins the respect I receive in class.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Why did the teacher say this to Einstein. What was the context or the situation? Did Einstein undermine the authority of the teacher in some way? Or did he expose the teacher's inability?&lt;/p&gt;&#xA;" OwnerUserId="18020" LastEditorUserId="10106" LastEditDate="2023-02-22T04:49:42.080" LastActivityDate="2023-02-22T04:49:42.080" Title="Why did Einstein's teacher say that Einstein's presence alone undermined his authority?" Tags="|biographical-details|einstein|physicists|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15190" PostTypeId="1" CreationDate="2023-02-23T16:06:17.493" Score="0" ViewCount="132" Body="&lt;p&gt;I read a story and a quote few years ago on a mathematical community in my country and today I just suddenly reminiscend of and want to know whether they are real or not and if it is the former case then I would like to have sources. Unfortunately, all the old posts in that forum were lost and even if they weren't, owners of those posts are no longer active.  I tried to find them on google but no result found at all.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;The story: as far as I can remember, it is about Faltings (Fields medalist) who once was playing chess with an Abel laureate but could not win a game. The Abel laureate then stood up and belike &amp;quot;I think we could stop, you cannot win even if we keep playing&amp;quot; but Falting stopped him and retorted &amp;quot;even knowing that you cannot get a Fields medal, why do you keep doing math?&amp;quot;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;And about the quote, it is attributed to Hirzebruch if I remember correctly. He said that the growth of technology would bring people back to mathematics, literature and poetry.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerDisplayName="user18034" LastEditorDisplayName="user18034" LastEditDate="2023-02-23T20:37:13.603" LastActivityDate="2023-02-23T20:37:13.603" Title="Source of a quote and a story" Tags="|quote|mathematics-social-history|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15191" PostTypeId="1" CreationDate="2023-02-24T16:02:44.043" Score="5" ViewCount="87" Body="&lt;p&gt;There seems to be general agreement that the first known horseshoe magnet (permanent, not electromagnet) was produced by the instrument maker Johann Dietrich of Basel, probably in 1743. Dietrich also produced magnetic needles for precision measurement of the inclination of Earth's magnetic field. Daniel Bernoulli used both kinds of Dietrich's magnets in his magnetism research.&lt;/p&gt;&#xA;&lt;p&gt;However, it is not clear whether Dietrich really invented the horseshoe magnet, or rather &lt;em&gt;produced&lt;/em&gt; one at the suggestion of Daniel Bernoulli. So far, all I see is disagreement among secondary sources, and I don't really see what primary sources they are using to reach their conclusions.&lt;/p&gt;&#xA;&lt;h2&gt;References I looked at so far&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Coey, &lt;em&gt;Magnetism and Magnetic Materials&lt;/em&gt; (2010). The relevant text reads &lt;a href=&quot;https://www.google.com/books/edition/Magnetism_and_Magnetic_Materials/Ie72CFd-eSEC?hl=en&amp;amp;gbpv=1&amp;amp;dq=Swiss%20polymath%20Daniel%20Bernoulli%2C&amp;amp;pg=PA2&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;(see here)&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An important civilian advance, promoted by the Swiss polymath Daniel&#xA;Bernoulli, was the invention in 1743 of the horseshoe magnet. This was&#xA;to become magnetism's most enduring archetype. The horseshoe is an&#xA;ingenious solution to the problem of making a reasonably compact&#xA;magnet which will not destroy itself in its own demagnetizing field.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As to which source Coey used, in &lt;em&gt;Further reading&lt;/em&gt; section of that chapter, the most likely reference seems to be  Kloss, &lt;em&gt;Geschichte des Magnetismus&lt;/em&gt; (1994) &lt;a href=&quot;https://www.google.com/books/edition/Geschichte_des_Magnetismus/tP-vAAAACAAJ?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;(here)&lt;/a&gt;. Unfortunately, I have no access to it.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Forbes and Dijksterhuis, &lt;a href=&quot;https://www.google.com/books/edition/A_History_of_Science_and_Technology/xq7aAAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History of Science and Technology,&lt;/em&gt; Vol. 2&lt;/a&gt; (1963). The relevant section reads &lt;a href=&quot;https://www.google.com/search?q=%22At+Daniel+Bernoulli%27s+suggestion%22&amp;amp;tbm=bks&amp;amp;ei=ItL4Y4ipIJ6hptQPvaq5qAI&quot; rel=&quot;nofollow noreferrer&quot;&gt;(see here)&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first kind of magnets used were chiefly bar magnets. At Daniel&#xA;Bernoulli's suggestion, the Basle instrument-maker Johann Dietrich&#xA;bent the bar so as to bring the poles close together, thus creating&#xA;the first horseshoe magnet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Looking at the bibliography for the relevant chapter (23), the most likely source is &lt;a href=&quot;https://archive.org/details/bibliographicalh033138mbp&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Bibliographical History of Electricity and Magnetism&lt;/em&gt;&lt;/a&gt; by Mottelay. Unfortunately, if I search in that work or look in its index, I find no hits for 'horseshoe' or 'Dietrich'. However, it could also be that the information comes from one of the references I have no access to, such as&lt;/p&gt;&#xA;&lt;p&gt;Lincoln, E. S. edit. &lt;em&gt;A Chronological History of Electrical Development&lt;/em&gt; (New York, 1946).&lt;/p&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;em&gt;Verhandlungen der Naturforschenden Gesellschaft in Basel,&lt;/em&gt; Volumes 24-25 (available for free &lt;a href=&quot;https://www.google.com/books/edition/Verhandlungen_der_Naturforschenden_Gesel/gpWSsbJAXPUC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). The relevant section, whose translation by Google translate I don't completely understand, is this:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In der Festrede zur Feier des 50jährigen Bestehens der Basler&#xA;Naturforschenden Gesellschaft (vom 4. Mai 1867, Verh. IV, Anhang )&#xA;sprach er über die Arbeiten der Societas physica helvetica , der&#xA;Vorgängerin der Gesellschaft , nicht über die Geschichte dieser&#xA;selbst, denn , da die Gründer und Erbauer derselben zugegen waren, „&#xA;ich habe die Überzeugung, dass es den betreffenden, unter uns hochver&#xA;ehrten Männern erwünschter ist, das Rühmliche getan zu haben , als das&#xA;Getane rühmen zu hören .&amp;quot; Der Nachweis, dass die beste Form der&#xA;Inklinationsnadel der Hufeisenmagnet sei, wie 1743 Dan. Bernoulli in&#xA;einer Preisschrift bewies, diese aber vom Goldschmied und&#xA;Instrumentenmacher Joh. Dietrich in Basel zuerst konstruiert worden&#xA;ist , wie Burckhardt nachweist, dürfte daraus am meisten&#xA;interessieren.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, for what it's worth, is a translation by Google translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the speech to celebrate the 50th anniversary of the Basler&#xA;Naturforschenden Gesellschaft (May 4, 1867, Verh. IV , Appendix ) he&#xA;spoke about the work of the Societas physica helvetica , the&#xA;predecessor of the society , not about the history of this itself,&#xA;because , there the founders and builders of the same were present, &amp;quot;&#xA;I am convinced that it is more desirable for the men in question, who&#xA;are highly respected among us, to have done what is creditable than to&#xA;hear praise for what has been done.&amp;quot; The proof that the best form of&#xA;the inclination needle the horseshoe magnet is, as Dan Bernoulli&#xA;proved in a prize paper in 1743, but this was first constructed by the&#xA;goldsmith and instrument maker Joh. Dietrich in Basel, as Burckhardt&#xA;proves, should be of most interest.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;Radelet-de Grave, &amp;quot;Studies of Magnetism in the Correspondence of Daniel Bernoulli,&amp;quot; in &lt;a href=&quot;https://www.google.com/books/edition/Two_Cultures/voFsJ1EhCnYC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Two Cultures: Essays in Honour of David Speiser,&lt;/em&gt;&lt;/a&gt; K. Williams, ed. (Birkhäuser, Basel, 2006).&#xA;Here are the relevant sections:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the meantime, Daniel Bernoulli is using horseshoe-shaped magnets,&#xA;invented&lt;sup&gt;24&lt;/sup&gt; and produced by Dietrich, in order to determine&#xA;his law giving the magnitude of the force of such a magnet. In several&#xA;letters he reports an empiric law, which was later published in a rather&#xA;peculiar text meant to promote the trade of Dietrich’s magnets: it is rather&#xA;like a commercial advertisement.&lt;sup&gt;25&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;24&lt;/sup&gt;It is difficult to date the apparition of the&#xA;horseshoe magnets, which, moreover, were discovered independently in&#xA;England later on. The publication meant to promote their trade is of&#xA;1755. But they are mentioned in the letters earlier than that: Daniel Bernoulli to Johann Jacob Huber (7 June 1754) and to Pierre Bouguer&#xA;(11 November1754). In the latter D. Bernoulli says that he has been&#xA;disgusted by this kind of research but he does not give reason&#xA;why.&lt;/sup&gt;&lt;br&gt;&#xA;&lt;sup&gt;&lt;sup&gt;25&lt;/sup&gt;Text R&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;Text R,&amp;quot; to which I don't have access, is the following:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Remarques sur les Aimans artificiels de Basle,&lt;/em&gt; Nouvelle Bibliothèque germanique janv. fev. mars 1755&lt;/p&gt;&#xA;&lt;h2&gt;Questions&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What primary sources are available when trying to understand whether it was Dietrich or Bernoulli who came up with the idea for the horseshoe magnet?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Given these sources, who is most likely to have come up with the idea for the horseshoe magnet: Dietrich or Bernoulli?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h2&gt;Edit&lt;/h2&gt;&#xA;&lt;p&gt;Thanks to Tom Heinzl, I was able to access &amp;quot;Text R,&amp;quot; &lt;a href=&quot;https://www.google.com/books/edition/Nouvelle_biblioth%C3%A8que_germanique_former/3WgEAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;printsec=frontcover#v=onepage&amp;amp;q=Bernoulli&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I will now reproduce the relevant section, which starts on p. 225:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;C'est Mr. Dietrich, Bourgeois de cette Ville &amp;amp; habile Artiste, qui&#xA;fait ces Aimans artificiels. Sa curiosité naturelle le porta d'abord à&#xA;construire quelques-uns de ces Aimans, en suivant les préceptes&#xA;connus, &amp;amp; fa capacité le conduisit bientôt à les perfectionner. Il a&#xA;remarqué que la figure la plus convenable pour donner beaucoup de&#xA;force à ces Aimans  est celle d'un fer à cheval: voici la mesure des&#xA;parties d'un de mes Aimans, qui pése 9 1⁄2 onces, &amp;amp; qui porte 16&#xA;livres: …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is the output of Google translate (except for &lt;em&gt;Bourgeois,&lt;/em&gt; meaning &lt;em&gt;citizen&lt;/em&gt;; the boldfaced emphasis is mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is Mr. Dietrich, a Citizen of this city and a skillful Artist, who makes&#xA;these artificial magnets; his natural curiosity first led him to&#xA;construct some of these magnets, following the known precepts, and his&#xA;ability soon led him to perfect them. &lt;strong&gt;He has noticed that the figure&#xA;most suitable for giving much force to these magnets is that of a&#xA;horseshoe.&lt;/strong&gt; Here is the measurement of the parts of one of my magnets&#xA;which weighs 9 1/2 ounces, and which bears 16. pounds: …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So this source, at least, says that it was Dietrich himself who came up with the horseshoe design. It is still possible that there are other sources that contradict this. (It is this text that Radelet-de Grave describes as a &amp;quot;rather peculiar text meant to promote the trade of Dietrich’s magnets: it is rather like a commercial advertisement.&amp;quot;)&lt;/p&gt;&#xA;&lt;h2&gt;Edit 2&lt;/h2&gt;&#xA;&lt;p&gt;The additional two letters mentioned by Radelet-de Grave are available at swisscollections.ch.&lt;/p&gt;&#xA;&lt;p&gt;The letter to Johann Jacob Huber is available &lt;a href=&quot;https://swisscollections.ch/Record/991170514764705501&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Since I don't speak French, I transcribed it using &lt;a href=&quot;https://readcoop.eu/transkribus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Transkribus&lt;/a&gt;, and then used a combination of ChatGPT and Google translate to correct as best as I could the inevitable (i) errors in transcription and (ii) typos in the original, and then produce an English translation. As best as I can, here is the original text and its translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si l'aimant artificiel en forme de fer à cheval et les petites barres&#xA;appartiennent à la Chambre de Physique, car il me semble les avoir&#xA;payées, je prie Mr Huber de les remettre à ma servante. Je serai chez&#xA;moi cet après-midi jusqu'à 3 heures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If the artificial magnet in the shape of a horseshoe and the small&#xA;bars belong to the Physics Chamber, as I seem to have paid for them, I&#xA;ask Mr. Huber to give them to my servant. I will be at home this&#xA;afternoon until 3 o'clock.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bernoulli's letter to Pierre Bouguer can be found &lt;a href=&quot;https://swisscollections.ch/Record/991170514765105501&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Using the same procedure as for the previous letter, here is my best guess for the relevant part of it and its translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mon observation que la force des aimants artificiels semblables en&#xA;figure et en surfaces suit la raison de la surface de ces aimants&#xA;s'est trouvé confirmée par tous les essais que j'en ai faits. Je&#xA;pourrais bien faire avec le fer les expériences que vous me proposez,&#xA;si j'en reprenais le goût; mais j'en ai été tellement dégoûté que je&#xA;me suis défait de tous les aimants artificiels que j'avais pour ne&#xA;plus les avoir sous mes yeux. J'en ai remis un petit à M. Falckner&#xA;pour l'autoriser d'autant plus à vous voir.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My observation that the force of artificial magnets similar in shape&#xA;and surface is proportional to the surface area of these magnets has&#xA;been confirmed by all the experiments I have conducted. I could&#xA;perform the experiments you propose using iron if I regained an&#xA;interest in them; however, I was so disgusted that I got rid of all&#xA;the artificial magnets I had so that I no longer see them in front of&#xA;my eyes. I gave a small one to Mr. Falckner to encourage him to visit&#xA;you more often.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;Edit 3&lt;/h2&gt;&#xA;&lt;p&gt;The following reference also seems to be useful, and it was cited in the past (see &lt;a href=&quot;https://hsm.stackexchange.com/questions/15203/who-produced-the-first-permanent-horseshoe-magnet-johann-dietrich-or-gilles-aug&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;p&gt;Ferdinand Rosenberger, &lt;em&gt;Die Geschichte Der Physik in Grundzügen&lt;/em&gt; &lt;em&gt;(The history of physics in outline),&lt;/em&gt; vol. 2 (F. Vieweg und sohn, Braunschweig, Germany, 1884), p. 297, available &lt;a href=&quot;https://www.google.com/books/edition/Die_Geschichte_Der_Physik_in_Grundz%C3%BCgen/nv04AAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. The relevant section (the only one where either Dietrich or horseshoe magnets are mentioned) is &lt;a href=&quot;https://www.google.com/books/edition/Die_Geschichte_Der_Physik_in_Grundz%C3%BCgen/nv04AAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22um%20diese%20Zeit%20der%20Baseler%20Mechaniker%22&amp;amp;pg=RA1-PA297&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Inclinationsboussole selbst suchten Daniel Bernoulli und Euler zu&#xA;verbessern, indem sie die Bedingungen untersuchten , unter welchen die&#xA;verschiedenen Inclinationsnadeln übereinstimmende Resultate ergeben&#xA;mussten. Bernoulli erhielt für die betreffende Arbeit 1743 den Preis&#xA;der Pariser Akademie , Euler das Accessit. Vielleicht auch auf&#xA;Veranlassung Daniel Bernoulli's führte um diese Zeit der Baseler&#xA;Mechaniker Johann Dietrich († 1758) die Ilufeisenmagnete und ihre&#xA;Armirung ein; wenigstens entdeckte Bernoulli mit solchen&#xA;Dietrich'schen Magneten das Gesetz, dass die Tragkraft der&#xA;Hufeisenmagnete proportional ist ihren Oberflächen oder den dritten&#xA;Wurzeln aus den Quadraten ihrer Gewichte.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate, plus &lt;em&gt;Boussole&lt;/em&gt; = &lt;em&gt;compass&lt;/em&gt; (see &lt;a href=&quot;https://de.wikipedia.org/wiki/Bussole&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) and &lt;em&gt;Accessit&lt;/em&gt; assumed to be &lt;em&gt;Ak­zes­sit&lt;/em&gt; = &lt;em&gt;second place prize&lt;/em&gt; (see &lt;a href=&quot;https://www.dwds.de/wb/Akzessit&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The inclination compass itself sought to be improved by Daniel&#xA;Bernoulli and Euler by investigating the conditions under which the&#xA;various inclination needles must give consistent results. In 1743,&#xA;Bernoulli received the prize of the Paris Academy for the work in&#xA;question, and Euler received the second place prize. Perhaps also at&#xA;the instigation of Daniel Bernoulli, the Basel mechanic Johann&#xA;Dietrich († 1758) introduced iron magnets and their armouring around&#xA;this time; at least Bernoulli discovered the law with such Dietrich&#xA;magnets that the lifting capacity of the horseshoe magnets is&#xA;proportional to their surface areas or the third root of the square of&#xA;their weights.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2023-03-04T02:45:18.103" LastActivityDate="2023-03-04T02:45:18.103" Title="Permanent horseshoe magnet: primary sources of information about its invention? (Johann Dietrich and Daniel Bernoulli)" Tags="|magnetism|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15192" PostTypeId="2" ParentId="15185" CreationDate="2023-02-24T22:35:20.357" Score="3" Body="&lt;p&gt;It is plausible, albeit not proved, that &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubism&quot; rel=&quot;nofollow noreferrer&quot;&gt;cubism&lt;/a&gt; provided some inspiration for quantum mechanics.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_I._Miller&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arthur Miller&lt;/a&gt; argued strongly that cubism helped &lt;a href=&quot;https://en.wikipedia.org/wiki/Niels_Bohr&quot; rel=&quot;nofollow noreferrer&quot;&gt;Niels Bohr&lt;/a&gt; develop the &lt;a href=&quot;https://en.wikipedia.org/wiki/Complementarity_(physics)&quot; rel=&quot;nofollow noreferrer&quot;&gt;complementarity principle&lt;/a&gt; in quantum mechanics. Bohr had read the book by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Metzinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Metzinger&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Albert_Gleizes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Albert Gleizes&lt;/a&gt; on cubist theory, &lt;a href=&quot;https://en.wikipedia.org/wiki/Du_%22Cubisme%22%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Du Cubisme&amp;quot;&lt;/a&gt;, and later had Metzinger's painting &lt;a href=&quot;https://en.wikipedia.org/wiki/La_Femme_au_Cheval#Influence_on_quantum_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;La Femme au Cheval&amp;quot;&lt;/a&gt; in his office. See &lt;a href=&quot;https://doi.org/10.1007/s10699-016-9494-7&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Schinckus, C. From Cubist Simultaneity to Quantum Complementarity. Found Sci 22, 709–716 (2017)&quot;&gt;Schinckus&lt;/a&gt; for a recent discussion of Miller's claim.&lt;/p&gt;&#xA;&lt;p&gt;Materials science and engineering is one area where there is a small but steady stream of research inspired by historical art. The two- and three-dimensional patterns and shapes in this art can have interesting and useful structural properties. Here are a few examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.3389/fmats.2019.00182&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Anastasiia O. Krushynska*, Front. Mater., 13 August 2019, Sec. Mechanics of Materials, Volume 6 - 2019&quot;&gt;&amp;quot;Between Science and Art: Thin Sound Absorbers Inspired by Slavic Ornaments&amp;quot;&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1002/adom.202202248&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Hu, N., Gao, D., Cai, X., Lei, L., Wang, W., Fan, H., Müller-Buschbaum, P., Zhong, Q.,  Adv. Optical Mater. 2022, 2202248.&quot;&gt;&amp;quot;Hybrid Nanogels-Based Printable Tattoos Inspired by Lady Costume from Tang Dynasty for Monitoring Solar UV Radiation&amp;quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1039/C7MH00559H&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;L. Mao et al., Mater. Horiz., 2018, 5, 123-130&quot;&gt;&amp;quot;Writable and patternable organic solar cells and modules inspired by an old Chinese calligraphy tradition&amp;quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1016/j.eml.2016.09.001&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Ahmad Rafsanjani, Damiano Pasini, Extreme Mechanics Letters, Volume 9, Part 2, 2016, Pages 291-296&quot;&gt;&amp;quot;Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs&amp;quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1002/adfm.202107401&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Zhang, H., Paik, J., Kirigami Design and Modeling for Strong, Lightweight Metamaterials. Adv. Funct. Mater. 2022, 32, 2107401.&quot;&gt;&amp;quot;Kirigami Design and Modeling for Strong, Lightweight Metamaterials&amp;quot;&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Another related example is how Nobel-prize winning chemist &lt;a href=&quot;https://en.wikipedia.org/wiki/Harry_Kroto&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harry Kroto&lt;/a&gt;'s familiarity with &lt;a href=&quot;https://en.wikipedia.org/wiki/Buckminster_Fuller&quot; rel=&quot;nofollow noreferrer&quot;&gt;Buckminster Fuller&lt;/a&gt;'s &lt;a href=&quot;https://en.wikipedia.org/wiki/Geodesic_dome&quot; rel=&quot;nofollow noreferrer&quot;&gt;geodesic domes&lt;/a&gt; helped him &lt;a href=&quot;https://www.acs.org/content/dam/acsorg/education/whatischemistry/landmarks/lesson-plans/discovery-of-fullerenes.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;figure out&lt;/a&gt; the structure of &lt;a href=&quot;https://en.wikipedia.org/wiki/Buckminsterfullerene&quot; rel=&quot;nofollow noreferrer&quot;&gt;C-60&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, the most obvious possible source of artistic inspiration for science is &lt;em&gt;science fiction&lt;/em&gt; in both books and film.&#xA;H.G. Wells &lt;a href=&quot;https://www.vqronline.org/essay/hg-wells-and-scientific-imagination&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;H.G. Wells and the Scientific Imagination, W. Warren Wagar,  The Virginia Quarterly Review, Vol. 65, No. 3 (SUMMER 1989), pp. 390-400&quot;&gt;inspired&lt;/a&gt; several leading scientists.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Leo_Szilard&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leo Szilard&lt;/a&gt; &lt;a href=&quot;https://library.ucsd.edu/dc/object/bb58377715/_1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;implied&lt;/a&gt; that H.G. Wells &lt;a href=&quot;https://en.wikipedia.org/wiki/The_World_Set_Free&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The World Set Free&amp;quot;&lt;/a&gt; provided at least background inspiration for his work on the neutron chain reaction.&#xA;Wells also inspired &lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_H._Goddard&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Goddard&lt;/a&gt;, who in 1932 wrote to Wells &lt;a href=&quot;https://www.nasa.gov/sites/default/files/atoms/files/bolden_goddard_2016.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;saying&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1898 I read your War of the Worlds. I was sixteen years old … and I decided that what might conservatively be called ‘high-altitude research’ was the most fascinating problem in existence. The spell did not break, and I took up physics ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-02-24T22:35:20.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15193" PostTypeId="2" ParentId="12925" CreationDate="2023-02-27T04:05:38.797" Score="3" Body="&lt;p&gt;Euler's calculation of Fourier series coefficients appears in his paper&#xA;&lt;em&gt;De serierum determinatione seu nova methodus inveniendi terminos generales serierum&lt;/em&gt; from 1753.  A link to it on the Euler archive is &lt;a href=&quot;https://scholarlycommons.pacific.edu/euler-works/189/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scholarlycommons.pacific.edu/euler-works/189/&lt;/a&gt;: see the second pdf link at the bottom, on page 81, which is the end of the solution to Problem IX just above Corollary I.&lt;/p&gt;&#xA;&lt;p&gt;A cleaner scan from the &lt;em&gt;Opera Omnia&lt;/em&gt; (Series I, volume 14, p. 511) is on the Hathitrust website &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=pst.000010974780&amp;amp;view=1up&amp;amp;seq=527&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://babel.hathitrust.org/cgi/pt?id=pst.000010974780&amp;amp;view=1up&amp;amp;seq=527&lt;/a&gt; (this link may not work everywhere). See the equation just before Corollary 1 on that page.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-02-27T04:05:38.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15194" PostTypeId="1" CreationDate="2023-02-27T06:18:01.683" Score="2" ViewCount="582" Body="&lt;p&gt;The word &lt;em&gt;&amp;quot;radio&amp;quot;&lt;/em&gt; originates from &amp;quot;&lt;em&gt;radius&lt;/em&gt;&amp;quot;, which in turn came from &amp;quot;&lt;em&gt;ray&lt;/em&gt;&amp;quot;. That's why &amp;quot;radius&amp;quot; means any line from a central focal point to any directions.&lt;/p&gt;&#xA;&lt;p&gt;Because of that, in 1881, Alexander Graham Bell chose to name his &lt;a href=&quot;https://en.wikipedia.org/wiki/Photophone&quot; rel=&quot;nofollow noreferrer&quot;&gt;photophone&lt;/a&gt; as &amp;quot;radiophone&amp;quot; (which is a combination of the words &amp;quot;&lt;em&gt;radio&lt;/em&gt;&amp;quot; and &amp;quot;&lt;em&gt;phone&lt;/em&gt;&amp;quot;). That implied that his device was able to radiate voices in all directions. That seems to be the first wide usage of the word &amp;quot;radio&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In 1887, Hertz discovered a kind of electromagnetic waves that was first named as &amp;quot;&lt;em&gt;Hertzian waves&lt;/em&gt;&amp;quot;. This wave is the wave that we called &amp;quot;radio waves&amp;quot; today. Why and how was radio waves named as &lt;strong&gt;&amp;quot;radio waves&amp;quot;&lt;/strong&gt;? What is its etymology?&lt;/p&gt;&#xA;&lt;p&gt;Is it because this wave is the &lt;strong&gt;only&lt;/strong&gt; electromagnetic wave that is radial by nature (i.e. it radiates from a focal point into all directions) when Hertz discovered it?&lt;/p&gt;&#xA;&lt;p&gt;Or actually all other electromagnetic waves (such as &lt;em&gt;infrared&lt;/em&gt; and &lt;em&gt;ultraviolet&lt;/em&gt; which were discovered in 1800, way earlier than radio waves) are also radial by nature, so the naming of &lt;strong&gt;&amp;quot;radio waves&amp;quot;&lt;/strong&gt; was random with no certain origin?&lt;/p&gt;&#xA;" OwnerUserId="18048" LastEditorUserId="18048" LastEditDate="2023-02-27T07:38:03.570" LastActivityDate="2023-02-27T19:13:23.437" Title="The etymology of &quot;radio waves&quot;" Tags="|physics|terminology|electromagnetism|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15195" PostTypeId="2" ParentId="15194" CreationDate="2023-02-27T18:36:16.510" Score="4" Body="&lt;p&gt;The OED asserts that the origin of &lt;em&gt;radio&lt;/em&gt; as a stand-alone word arises by clipping earlier uses of &lt;em&gt;radio&lt;/em&gt; as a combining form. The earliest stand-alone use that the OED cites is in 1903, whereas the earliest cited use as a combining form is this one:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1875   W. Crookes in Proc. Royal Soc. 23 377   The luminous rays..repel the black surface more energetically than they do the white surface. Taking advantage of this fact, the author has constructed an instrument which he calls a radiometer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The title of that article is &lt;em&gt;On attraction and repulsion resulting from radiation&lt;/em&gt;, and the terminology &lt;em&gt;radiation&lt;/em&gt; occurs repeatedly throughout the article, including in the vicinity of the coined term &lt;em&gt;radiometer&lt;/em&gt;. For example, this quote appears in the very next paragraph, which also contains the word &lt;em&gt;rays&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The author finds that this instrument revolves under the influence of radiation, the rapidity of revolution being in proportion to the intensity of the incident rays.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unless there's something even earlier, the best guess I could form on this basis is that the word &amp;quot;radio&amp;quot;, rather than deriving directly from &amp;quot;radius&amp;quot;, instead derives from Mr. Crookes determination that &amp;quot;radiometer&amp;quot; just sounded better than &amp;quot;ray meter&amp;quot; or &amp;quot;radiation meter&amp;quot;.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Poking around in the future of 1875, the earliest entries in the OED for &amp;quot;radio&amp;quot; as a stand-alone word in the phrases &amp;quot;radio wave&amp;quot; and &amp;quot;radio frequency&amp;quot; appear at just about the same time, namely 1915, with their modern meaning. But the OED gives the following slightly earlier stand-alone use:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1912   Statutes U.S.A. XXXVII. i. 308   ‘Radio communication’ as used in this Act means any system of electrical communication by telegraphy or telephony without the aid of any wire [etc.].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and then, tracing the &lt;em&gt;telegraph&lt;/em&gt; connection further back, there is this interesting entry:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1898   J. Munro in Electrician 21 Jan. 428/2   ‘Wireless telegraphy’ is not a bad technical term; but if a more scientific name be desirable would not Radiotelegraphy or Ray Telegraphy be preferable to ‘Space Telegraphy’? which Dr. Lodge employs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This leads one to think that the &amp;quot;radio&amp;quot; combining form had become well established in science in the years since Crookes' radiometer. My best guess here might be that as the technology of telegraphic communication via electromagnetic waves advanced, the more general use of the combining form &amp;quot;radio-&amp;quot;, introduced by Crookes' and referring to unspecified frequencies of radiation, somehow became narrowed down to refer to the familiar wavelengths of electromagnetic radiation in its modern meaning.&lt;/p&gt;&#xA;" OwnerUserId="5591" LastEditorUserId="5591" LastEditDate="2023-02-27T19:13:23.437" LastActivityDate="2023-02-27T19:13:23.437" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15196" PostTypeId="1" AcceptedAnswerId="15991" CreationDate="2023-02-28T11:00:23.270" Score="1" ViewCount="109" Body="&lt;p&gt;Before there was an understanding of nuclear fusion, what hypotheses were there about the source of stellar energy? What methods, if any, were suggested for testing the hypotheses; alternatively, how were the hypotheses ultimately refuted ... or did they all fail together with the discovery of fusion?&lt;/p&gt;&#xA;" OwnerUserId="17963" LastActivityDate="2023-12-05T08:22:51.800" Title="Views of stellar energy before fusion" Tags="|cosmology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15198" PostTypeId="1" CreationDate="2023-02-28T17:16:31.907" Score="1" ViewCount="238" Body="&lt;p&gt;Whenever I read some Mathematic theories, I find it they are typically done by some people living in Germany. Eg: Riemann, Gauss, Moses Schoenfinkel etc. I want to know what Germany did to arise so many Mathematicians?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2023-02-28T17:16:31.907" Title="Why is so much of modern mathematics rooted in Germany?" Tags="|geography|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15199" PostTypeId="2" ParentId="5890" CreationDate="2023-02-28T20:45:41.727" Score="2" Body="&lt;p&gt;In addition to the references cited in the answer provided by Athansius, I include the following quote from Erik Larson's, &lt;em&gt;The Myth of Artificial Intelligence&lt;/em&gt;, which appears to offer additional details of the rules of the original game which are not present in Turing's formulation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Turing's proposal was based on a popular entertainment call the &amp;quot;imitation game.&amp;quot; In the original game, a man and a woman are hidden from view. A third person, the interrogator, relays questions to them one at a time and, by reading the answers, attempts to determine which is the man and which is the woman.  &lt;strong&gt;The twist is that the man has to try to deceive the interrogator, while the woman tries to assist him - making replies from either side suspect.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(my emphasis)&lt;/p&gt;&#xA;&lt;p&gt;Turing's own words in &lt;a href=&quot;https://redirect.cs.umbc.edu/courses/471/papers/turing.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Computing Machinery and Intelligence&lt;/a&gt; are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The new form of the problem can be described in terms of a game which we call the&#xA;&amp;quot;imitation game.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If Turing was introducing the name, then one would expect him to write &amp;quot;we will call&amp;quot; (or more likely, &amp;quot;we shall call&amp;quot;).  Otherwise, why would Turing feel the need to describe the rules of a game which he immediately changes to better suit his purpose.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-02-28T20:45:41.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15200" PostTypeId="1" AcceptedAnswerId="15202" CreationDate="2023-03-01T16:33:16.847" Score="6" ViewCount="1053" Body="&lt;p&gt;Arthur Koestler's &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Sleepwalkers_(Koestler_book)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Sleepwalkers&lt;/em&gt;&lt;/a&gt; is well-known as both a group biography of Copernicus, Brahe, Kepler and Galileo and an account of the revolutionary turn in astronomy that, in Koestler's phrasing, they ‘sleepwalked’ through.  I think it's an excellent text, and I'm not aware of that being a fringe opinion.&lt;/p&gt;&#xA;&lt;p&gt;Koestler's description seems judicious, but he isn't shy of taking a point of view of the events he's describing.  While he isn't so crass as to talk about goodies and baddies, it's clear, for example, that he isn't inclined to cast Galileo as as much of a hero as some people might, in the history of astronomy.&lt;/p&gt;&#xA;&lt;p&gt;But it's now more than 60 years since it was published, and histories often become dated as more evidence appears and consensuses evolve.  Is this still seen as a good overview?  Is there a better one?  What does one read next?&lt;/p&gt;&#xA;&lt;p&gt;One next text is Feyerabend's version of the history of Galileo's interaction with the church, in &lt;em&gt;Against Method&lt;/em&gt;.  That's excellent argumentative fun, but Feyerabend's intentions mean that it reads like it's intended for an audience that's well enough informed about the background that he doesn't have to be nit-picking about the details, or be over-scrupulous about being balanced.&lt;/p&gt;&#xA;" OwnerUserId="12870" LastActivityDate="2023-03-02T03:34:34.047" Title="Is Koestler's ‘The Sleepwalkers’ still well regarded? Is there a more recent similar source?" Tags="|biographical-details|astronomy|galileo|kepler|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15201" PostTypeId="1" CreationDate="2023-03-01T19:01:02.697" Score="4" ViewCount="123" Body="&lt;p&gt;Heard recently about digital humanities as a new paradigm of inquiry into the history of mankind and other humanities. Is the historiography of science informed by technological advances that could enable a perhaps greater number of resources to analyze for us to learn about the past? Is there ongoing research about it? Do historians of science say something about current technological capabilities in the context of the historiography of the sciences?&lt;/p&gt;&#xA;" OwnerUserId="10142" LastEditorUserId="10142" LastEditDate="2023-04-16T19:46:43.710" LastActivityDate="2023-04-16T19:46:43.710" Title="How the present technology influences the way history of science is done?" Tags="|philosophy-of-science|computer-science|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15202" PostTypeId="2" ParentId="15200" CreationDate="2023-03-01T19:01:35.200" Score="11" Body="&lt;p&gt;As the question says, Arthur Koestler in 'The Sleepwalkers' &amp;quot;isn't shy of taking a point of view of the events he's describing&amp;quot;. Koestler's book offered a gripping collective account of Copernicus, Tycho, Kepler and Galileo.  It's highly arguable that it needs supplementing nowadays.  After reading it, many a modern reader might be interested in individual descriptions of those four pioneers and their work. I don't know that anyone else has tried a new collective account, but there are several more recent works on the individuals that offer a variety of perspectives, well worth at least dipping into.   The value of dipping into a variety of accounts can be, that it gives insight to identify  points on which there is consensus, and to separate them from points that are more controversial or contentious and individual to each historian-interpreter.&lt;/p&gt;&#xA;&lt;p&gt;On Copernicus, a fairly short but thought-provoking and insightful perspective was offered in 1973 by Willy Hartner in &lt;a href=&quot;https://www.jstor.org/stable/986460&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;Copernicus: The Man, the Work, and its History&amp;quot;, (Proc Amer Phil Soc vol 117(6) 413-422)&lt;/a&gt;.  Another appreciation, more recent and also insightful, is &lt;a href=&quot;https://plato.stanford.edu/entries/copernicus/&quot; rel=&quot;noreferrer&quot;&gt;Sheila Rabin's 'Nicolaus Copernicus', in the Stanford Encyclopedia of Philosophy&lt;/a&gt;.  For a closer look at his work, probably two of the best accounts are in Noel Swerdlow's &lt;a href=&quot;https://www.jstor.org/stable/986461&quot; rel=&quot;noreferrer&quot;&gt;'Derivation and First Draft of Copernicus's Planetary Theory: A Translation of the Commentariolus with Commentary'&lt;/a&gt;, and (if you can access it) also Swerdlow &amp;amp; Neugebauer's &lt;a href=&quot;https://link.springer.com/book/10.1007/978-1-4613-8262-1&quot; rel=&quot;noreferrer&quot;&gt;'Mathematical Astronomy in Copernicus’ De Revolutionibus'&lt;/a&gt;.  A useful brief perspective account of Copernicus in the University of St Andrews series is at (&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Copernicus/&quot; rel=&quot;noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Biographies/Copernicus/&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;On Tycho, a highly recommendable biography and perspective on his work (1990) is in &lt;a href=&quot;https://books.google.com/books?id=F5a83U4B8XkC&quot; rel=&quot;noreferrer&quot;&gt;'Lord of Uraniborg: a biography of Tycho Brahe' by V E Thoren, with contributions by J R Christianson&lt;/a&gt;, and a briefer account, with (as usual) plenty of onward references is in the St Andrews series at (&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Brahe/&quot; rel=&quot;noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Biographies/Brahe/&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;With Kepler, there is a mainstream and then some far-out ideas in circulation. For the mainstream, recommendable accounts include the good biography, originally in German, by Max Caspar, since translated into English by Doris Hellman (&lt;a href=&quot;https://books.google.com/books?id=0r68pggBSbgC&quot; rel=&quot;noreferrer&quot;&gt;https://books.google.com/books?id=0r68pggBSbgC&lt;/a&gt;), and then the account in the Stanford Encyclopedia of Philosophy by Daniel di Liscia  (&lt;a href=&quot;https://plato.stanford.edu/entries/kepler/&quot; rel=&quot;noreferrer&quot;&gt;https://plato.stanford.edu/entries/kepler/&lt;/a&gt;) and the short account in the St Andrews series (&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Kepler/&quot; rel=&quot;noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Biographies/Kepler/&lt;/a&gt;). Another more recent good biography, 'Johannes Kepler' by Volker Bialas (2004), remains, as far as I know, in German only, without an English translation yet. A very good and readable account of how Kepler put together his account of his prime discoveries is in James Voelkel's (2001) &lt;a href=&quot;https://books.google.com/books?id=_NZsb6B8ATkC&quot; rel=&quot;noreferrer&quot;&gt;'Composition of Kepler's Astronomia Nova'&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When we get to Galileo there is much contentious material about. First, the good mainstream accounts include the one in the Stanford Encyclopedia of Philosophy by P Machamer and D M Miller  (&lt;a href=&quot;https://plato.stanford.edu/entries/galileo/&quot; rel=&quot;noreferrer&quot;&gt;https://plato.stanford.edu/entries/galileo/&lt;/a&gt;) and the St Andrews account (&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Galileo/&quot; rel=&quot;noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Biographies/Galileo/&lt;/a&gt;). It is impossible to ignore the many contentious accounts of Galileo, and perhaps one of the best forums to show the vigorous debates about them can be found in the 'Renaissance Mathematicus' website, with perhaps the best place to start its &lt;a href=&quot;https://thonyc.wordpress.com/2020/05/30/galileo-sources-a-starter-kit/&quot; rel=&quot;noreferrer&quot;&gt;'Galileo starter kit'&lt;/a&gt; with a wide variety of sources and opinions.&lt;/p&gt;&#xA;&lt;p&gt;{edit} I should have added references to the OUP &lt;a href=&quot;https://global.oup.com/academic/content/series/v/very-short-introductions-vsi/?cc=gb&amp;amp;lang=en&amp;amp;&quot; rel=&quot;noreferrer&quot;&gt;'Very Short Introduction'&lt;/a&gt; series, with relevant books on Copernicus, the History of Astronomy, Kepler, and Galileo.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2023-03-02T03:34:34.047" LastActivityDate="2023-03-02T03:34:34.047" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15203" PostTypeId="1" CreationDate="2023-03-01T21:16:56.083" Score="1" ViewCount="79" Body="&lt;p&gt;Several recent sources state that the first known permanent horseshoe magnet was produced by the instrument maker &lt;strong&gt;Johann Dietrich&lt;/strong&gt; of Basel, probably in 1743. However, several older French sources claim that the first horseshoe magnet was invented by &lt;strong&gt;Gilles-Augustin Bazin&lt;/strong&gt; of Strasbourg.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;There is some uncertainty over whether Dietrich produced these magnets so on his own, or at the suggestion of Daniel Bernoulli. That controversy is the subject of &lt;a href=&quot;https://hsm.stackexchange.com/questions/15191/permanent-horseshoe-magnet-primary-sources-of-information-about-its-invention&quot;&gt;another question&lt;/a&gt; of mine here at HSM StackExchange. For what it's worth, given what I know at present, I think that Dietrich came up with the idea on his own.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;h2&gt;The case for Bazin&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;In &lt;a href=&quot;https://www.google.com/books/edition/Dictionnaire_d_%C3%A9lectricit%C3%A9_et_de_magn/vQI_AAAAYAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Dictionnaire d'électricité et de magnétisme&lt;/em&gt;&lt;/a&gt; from 1887, we find the following entries. First &lt;a href=&quot;https://www.google.com/books/edition/Dictionnaire_d_%C3%A9lectricit%C3%A9_et_de_magn/vQI_AAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA9&amp;amp;printsec=frontcover&amp;amp;dq=AIMANT%20en%20fer%20%C3%A0%20cheval%20&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;AIMANT en fer à cheval (&lt;em&gt;Hufeisenmagnet&lt;/em&gt; —— Horseshoe magnet).&lt;/strong&gt;&lt;br&gt; Barreau&#xA;aimanté, recourbé en forme de fer à cheval , de manière que les deux&#xA;pôles soient rapprochés et qu'il en résulte une modification dans la&#xA;forme du champ magnétique . Les lignes de force unissant les deux&#xA;pôles étant raccourcies, le champ de force est diminué, comme&#xA;dimensions, mais augmenté proportionnellement en intensité. On regarde&#xA;Bazin (Voir ce mot) , comme l'inventeur de cette disposition&#xA;avantageuse&lt;sup&gt;1&lt;/sup&gt;. Le Dr Rosenberger en attribue le mérite à&#xA;Johann Dietrich de Bâle&lt;sup&gt;2&lt;/sup&gt;. La force portante des aimants en&#xA;fer à cheval est a &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{P^{2}}$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; est le poids de l'aimant et&#xA;a un facteur constant qui, pour les aimants de Häcker, était égal à 10&#xA;.&lt;br&gt; &lt;br&gt; &lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;BRISSON, &lt;em&gt;Principes de physique,&lt;/em&gt; t. III,&#xA;p. 226 (an VIII).&lt;/sup&gt;&lt;br&gt; &lt;sup&gt;&lt;sup&gt;2&lt;/sup&gt;Rosenberger, &lt;em&gt;Die&#xA;Geschichte der Physik,&lt;/em&gt; vol . II, p . 297.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;HORSESHOE MAGNET (&lt;em&gt;Hufeisenmagnet&lt;/em&gt; —— Horseshoe magnet).&lt;/strong&gt;&lt;br&gt; Magnetic&#xA;bar, bent in the shape of a horseshoe, so that the two poles are&#xA;brought together and this results in a modification in the shape of&#xA;the magnetic field. The lines of force uniting the two poles being&#xA;shortened, the field of force is diminished in size, but&#xA;proportionally increased in intensity. Bazin (See this word) is&#xA;regarded as the inventor of this advantageous arrangement 1 . Dr.&#xA;Rosenberger credits Johann Dietrich of Basel 2. The bearing strength&#xA;of horseshoe magnets is &lt;span class=&quot;math-container&quot;&gt;$\sqrt[3]{P^{2}}$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the weight of the&#xA;magnet and has a constant factor which for Häcker magnets was equal to&#xA;10.&lt;br&gt; &lt;br&gt; &lt;sup&gt;&lt;sup&gt;1&lt;/sup&gt;BRISSON, &lt;em&gt;Principes de physique,&lt;/em&gt; t. III, p. 226 (an VIII).&lt;/sup&gt;&lt;br&gt; &lt;sup&gt;&lt;sup&gt;2&lt;/sup&gt;Rosenberger, &lt;em&gt;Die&#xA;Geschichte der Physik,&lt;/em&gt; vol . II, p . 297.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The reference to Brisson will be discussed as the next source in this section, while the Rosenberger reference will be discussed in the section &amp;quot;The case for Dietrich.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Then, in the same source, &lt;a href=&quot;https://www.google.com/books/edition/Dictionnaire_d_%C3%A9lectricit%C3%A9_et_de_magn/vQI_AAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA33&amp;amp;printsec=frontcover&amp;amp;dq=Gilles-Augustin&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;BAZIN (Gilles -Augustin).&lt;/strong&gt; Médecin français, fixé à Strasbourg, où il est mort en mars 1754. Il était né à Paris et était correspondant&#xA;de l'Académie des sciences. C'est à lui que l'on doit la disposition&#xA;de l'aimant dite en fer à cheval ( Voir BRISSON, Physique mathématique&#xA;(an VIII, t . III , p . 226). Il n'a publié que deux mémoires sur la&#xA;description des courants magnétiques, datés de Strasbourg 1753-1754.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;BAZIN (Gilles-Augustin).&lt;/strong&gt; French doctor, settled in Strasbourg, where he died in March 1754. He was born in Paris and was a&#xA;correspondent of the Academy of Sciences. It is to him that we owe the&#xA;so-called horseshoe arrangement of the magnet (See BRISSON,&#xA;Mathematical Physics (an VIII, t. III, p. 226). He published only two&#xA;memoirs on the description of magnetic currents, dated from Strasbourg&#xA;1753-1754.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;The Brisson reference, &lt;em&gt;Traité élémentaire ou principes de physique,&lt;/em&gt; Volume 3 (1799) is available &lt;a href=&quot;https://www.google.com/books/edition/Trait%C3%A9_%C3%A9l%C3%A9mentaire_ou_principes_de_ph/0Y82isvXnCUC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. The relevant section is &lt;a href=&quot;https://www.google.com/books/edition/Trait%C3%A9_%C3%A9l%C3%A9mentaire_ou_principes_de_ph/0Y82isvXnCUC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22premier%20ex%C3%A9cut%C3%A9%20Bazin%20de%20Strasbourg%22&amp;amp;pg=PA226&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;as follows&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;L'on peut donner aux aimans artificiels telle forme que l'on veut ; ce&#xA;que l'on ne peut pas toujours faire aux aimans naturels. On en peut&#xA;faire en demi-cercle (fig. 318), en fer à cheval (fig. 319), etc. et&#xA;leur faire alors , au moyen d'un portant P, soutenir un poids plus&#xA;considérable , en faisant agir les deux poles à la fois. C'est ce qu'a&#xA;le premier exécuté Bazin de Strasbourg.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You can give artificial magnets any shape you want; which cannot&#xA;always be done to natural magnets. It can be made in a semicircle&#xA;(fig. 318), in a horseshoe (fig. 319), etc., and then make them, by&#xA;means of a bearing P, support a more considerable weight, by making&#xA;the two poles act at the same time. This is what Bazin of Strasbourg&#xA;first executed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;In &lt;em&gt;L'Intermédiaire des chercheurs et curieux&lt;/em&gt; of &lt;a href=&quot;https://www.google.com/books/edition/_/RyIFAQAAIAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;30 décembre 1891&lt;/a&gt;, we find &lt;a href=&quot;https://www.google.com/books/edition/L_interm%C3%A9diaire_des_chercheurs_et_curie/RyIFAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA1031&amp;amp;printsec=frontcover&amp;amp;dq=Bazin&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;L'histoire des aimants.­&lt;/strong&gt; —— On attribue généralement l'invention des aimants en fer à cheval à Bazin (1), de Strasbourg, ou à Johann&#xA;Dietrich (2), de Bâle.&lt;/p&gt;&#xA;&lt;p&gt;Le premier en publia le dessin, en 1753, dans un ouvrage curieux et&#xA;rare (3) sur les &lt;em&gt;Courants magnétiques&lt;/em&gt; où il étudie le spectre&#xA;magnétique des différentes formes d'aimant.&lt;/p&gt;&#xA;&lt;p&gt;Il est fait mention de cette forme dans des ouvrages antérieurs (4).&lt;/p&gt;&#xA;&lt;p&gt;Un chercheur pourrait- il nous indiquer le premier ouvrage dans lequel&#xA;ces aimants sont décrits?&lt;/p&gt;&#xA;&lt;p&gt;(1) Brisson, &lt;em&gt;Physique,&lt;/em&gt; III, p. 226, in-8 (an VIII); Jacquez,&#xA;&lt;em&gt;Dictionnaire d'Electricité,&lt;/em&gt; in-8 (1887).&lt;br&gt; (2) Rosenberger, &lt;em&gt;die Geschichte der Physik,&lt;/em&gt; t. II, p. 297 (1887).&lt;br&gt; (3) Bazin,&#xA;&lt;em&gt;Description des courants magnétiques,&lt;/em&gt; in-4 ( 1753).&lt;br&gt; (4) Mitchell, &lt;em&gt;Treatise on artificial magnets,&lt;/em&gt; In-8 (1750).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;The History of Magnets.&lt;/strong&gt; —— The invention of horseshoe magnets is generally attributed to Bazin (1) from Strasbourg or Johann Dietrich&#xA;(2) from Basel.&lt;/p&gt;&#xA;&lt;p&gt;The first published drawing, in 1753, in a curious and rare work (3)&#xA;on &lt;em&gt;Magnetic currents&lt;/em&gt; where he studies the magnetic spectrum of the&#xA;different forms of magnet.&lt;/p&gt;&#xA;&lt;p&gt;Mention is made of this form in earlier works (4) .&lt;/p&gt;&#xA;&lt;p&gt;Could a researcher tell us the first book in which these magnets are&#xA;described?&lt;/p&gt;&#xA;&lt;p&gt;(1) Brisson, &lt;em&gt;Physique,&lt;/em&gt; III, p. 226, in-8 (an VIII); Jacquez,&#xA;&lt;em&gt;Dictionnaire d'Electricité,&lt;/em&gt; in-8 (1887).&lt;br&gt; (2) Rosenberger, &lt;em&gt;die&lt;/em&gt;&#xA;&lt;em&gt;Geschichte der Physik,&lt;/em&gt; vol. II, p. 297 (1887).&lt;br&gt; (3) Bazin,&#xA;&lt;em&gt;Description des courants magnétiques,&lt;/em&gt; in-4 ( 1753).&lt;br&gt; (4) Mitchell,&#xA;&lt;em&gt;Treatise on artificial magnets,&lt;/em&gt; In-8 (1750).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I looked at the later issues of this magazine, and, unfortunately, this query seems to have gone unanswered.&lt;/p&gt;&#xA;&lt;p&gt;The reference &amp;quot;Jacquez, &lt;em&gt;Dictionnaire d'Electricité&lt;/em&gt;&amp;quot; is the first reference I gave, above. The Brisson reference I already discussed, while the Rosenberger one I will discuss below. I will soon also discuss Bazin's publication.&lt;/p&gt;&#xA;&lt;p&gt;Mitchell's treatise is available &lt;a href=&quot;https://www.google.com/books/edition/A_Treatise_of_Artificial_Magnets/C4o5AAAAcAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. It mentions horseshoe magnets, but it doesn't mention who invented them. In particular, it doesn't mention either Dietrich, or Bazin, or Bernoulli. Indeed, it is quite possible that by 1750, the horseshoe magnet had been invented in Great Britain independently. This would be consistent with the remark of Radelet-de Grave's (see below) that &lt;em&gt;It is difficult to date the apparition of the horseshoe magnets, which, moreover, were discovered independently in England later on.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;In &lt;em&gt;Traité élémentaire de physique théorique et expérimentale avec les applications à la météorologie et aux arts industriels à l'usage des Facultés, des établissements d'enseignement secondaire et des écoles spéciales du Gouvernement&lt;/em&gt; by P. A. Daguin, published in 1861, in the entry &amp;quot;Magnétisme,&amp;quot; we find &lt;a href=&quot;https://www.google.com/books/edition/Trait%C3%A9_%C3%A9l%C3%A9mentaire_de_physique_th%C3%A9or/QGUJr6u0a_kC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22M.%20Bazin%20a%20eu%20l%27id%C3%A9e%22&amp;amp;pg=PA49&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;this statement&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;M. Bazin a eu l'idée de recourber un seul barreau aimanté, en forme de&#xA;fer à cheval, de manière à faire agir les deux pôles sur un même&#xA;portant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mr. Bazin had the idea of bending a single magnetic bar, in the shape&#xA;of a horseshoe, so as to make the two poles act on the same bearing [or 'support'].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;5&quot;&gt;&#xA;&lt;li&gt;In the article &amp;quot;Un savant Barrisien précurseur de M. Pasteur: Louis Joblot (1645-1723)&amp;quot; (&amp;quot;A Barrisian scholar, precursor of Mr. Pasteur: Louis Joblot (1645-1723)&amp;quot;) by  W. Konarski, published in Mémoires de la Société des lettres, sciences et arts de Bar-le-Duc., Troisième Série, Tome IV, p. 205 (1895), we find &lt;a href=&quot;https://www.google.com/books/edition/M%C3%A9moires_de_la_Soci%C3%A9t%C3%A9_des_lettres_sc/lrcjAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA311&amp;amp;printsec=frontcover&amp;amp;dq=%20fer%20%C3%A0%20cheval%20de%20Bazin.&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; (boldface emphasis mine):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ce mode d'enseignement oral, auquel nous l'avons vu empressé à&#xA;recourir pour traiter, sous le patronage de l'Académie de Peinture,&#xA;des sujets d'un ordre tout autre, Joblot ne l'aurait-il pas employé&#xA;tout aussi volontiers, ce qui expliquerait la rareté de ses écrits sur&#xA;la physique, pour vulgariser autour de lui ses doctrines et ses&#xA;procédés en matière de magnétisme?  Quoi qu'il en ait pu être, et sans&#xA;dissimuler qu'à travers les aberrations théoriques de ceux qui&#xA;l'avaient précédé dans cette voie —Gilbert, le P. Kircher, Descartes,&#xA;Rohault, Hartsoeker— il a pu et a dû puiser d'utiles renseignements&#xA;pratiques, le peu que nous avons de lui suffit à dénoter un&#xA;observateur attentif et pénétrant, un constructeur ingénieux pour le&#xA;temps où il vécut .Les dispositions essentielles de son appareil sont&#xA;demeurées classiques pendant toute la première moitié du XVII° siècle&#xA;&lt;strong&gt;avant que de céder la place au fer à cheval de Bazin.&lt;/strong&gt; Nombre de ses indications sommaires sur les procédés de friction, sur les formes&#xA;et les proportions à donner aux barreaux, sur les soins à apporter à&#xA;leur ajustage se retrouvent, longtemps après lui, avec plus de&#xA;développements peut-être, sous la plume de Réaumur, de Le Monnier, de&#xA;Buffon, de Nollet, de Brisson.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This mode of oral teaching, to which we have seen him eager to resort&#xA;to treat, under the patronage of the Academy of Painting, subjects of&#xA;an entirely different order, Joblot would not have employed it just as&#xA;willingly, which would explain the rarity of his writings on physics,&#xA;to popularize around him his doctrines and his methods in the matter&#xA;of magnetism? Whatever it may have been, and without concealing that&#xA;through the theoretical aberrations of those who had preceded him on&#xA;this path—Gilbert, Father Kircher, Descartes, Rohault, Hartsoeker—he&#xA;was able and had to draw useful practical information, the little that&#xA;we have of him is enough to denote an attentive and penetrating&#xA;observer, an ingenious builder for the time in which he lived. The&#xA;essential arrangements of his apparatus remained classic throughout&#xA;the first half of the seventeenth century, &lt;strong&gt;before giving way to the&#xA;horseshoe of Bazin.&lt;/strong&gt; Many of his summary indications on the processes&#xA;of friction, on the shapes and proportions to be given to the bars, on&#xA;the care to be taken in their adjustment are found, long after him,&#xA;with perhaps more developments, under the pen of Réaumur, Le Monnier,&#xA;Buffon, Nollet, Brisson.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;6&quot;&gt;&#xA;&lt;li&gt;Bazin's work seems to be described in &lt;em&gt;Le Journal des Sçavans&lt;/em&gt; of Dec. 1753, &lt;a href=&quot;https://www.google.com/books/edition/Journal_des_s%C3%A7avans/5uE0mqCTv9gC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22DESCRIPTION%20DES%20COURANS%20MAGNETIQUES%22&amp;amp;pg=PA771&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. It does mention Bazin experimenting with horseshoe-shaped magnets, but it doesn't say when this might have happened. The relevant section is &lt;a href=&quot;https://www.google.com/books/edition/Journal_des_s%C3%A7avans/5uE0mqCTv9gC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22fer%20%C3%A0%20cheval%22&amp;amp;pg=PA774&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Il a quelquefois rassemblé trois aimans artificiels, &amp;amp; il en ia a&#xA;formé un triangle. Sa curiosité ne s'est pas encore trouvé satisfaite&#xA;par ces différentes combinaisons , il a donné aux morceaux d'acier sur&#xA;lesquels il faisoit ses expériences, la forme de cercle ou d'anneau,&#xA;celle d'un fer à cheval, celle de la Lettre S.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He [Bazin] has sometimes gathered together three artificial magnets,&#xA;and from them he has formed a triangle. His curiosity has not yet been&#xA;satisfied by these different combinations, he gave to the pieces of&#xA;steel on which he made his experiments, the shape of a circle or a&#xA;ring, that of a horseshoe, that of the Letter S.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bazin's 1753 publication &lt;em&gt;Description des courants magnétiques, dessinés et gravés d'aprés nature en XV planches, suivie de quelques observations sur l'aiman&lt;/em&gt; &lt;em&gt;(Description of the magnetic currents, drawn and engraved from nature in 15 plates, followed by some observations on the magnet)&lt;/em&gt; is available &lt;a href=&quot;https://www.google.com/books/edition/Description_des_courants_magn%C3%A9tiques_de/EtuWwEcplgEC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, with the supplement (containing all the figures) &lt;a href=&quot;https://www.google.com/books/edition/_/Mds-ODnEnoYC?hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwj63-yvgb39AhWbD1kFHQ6ADZwQ7_IDegQIFBAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Bazin is mentioned in English-language sources, e.g. in &lt;a href=&quot;https://www.google.com/books/edition/A_new_general_biographical_dictionary_pr/1yoAAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA410&amp;amp;printsec=frontcover&amp;amp;dq=Courans&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A new general biographical dictionary, projected and partly arranged by H.J. Rose&lt;/em&gt;&lt;/a&gt;, although that source doesn't say Bazin was the inventor of the horseshoe magnet. The same biographical dictionary &lt;a href=&quot;https://archive.org/details/b24870134_0007/page/78/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;doesn't mention&lt;/a&gt; Dietrich of Basel, and doesn't mention horseshoe magnets in its &lt;a href=&quot;https://archive.org/details/b24870134_0004/page/160/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;entry for Daniel Bernoulli&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;h2&gt;The case for Dietrich&lt;/h2&gt;&#xA;&lt;p&gt;What follows are the same sources I used in my &lt;a href=&quot;https://hsm.stackexchange.com/questions/15191/permanent-horseshoe-magnet-primary-sources-of-information-about-its-invention&quot;&gt;other question&lt;/a&gt;. For completeness, I am going to repeat them here.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Ferdinand Rosenberger, &lt;em&gt;Die Geschichte Der Physik in Grundzügen&lt;/em&gt; &lt;em&gt;(The history of physics in outline),&lt;/em&gt; vol. 2 (F. Vieweg und sohn, Braunschweig, Germany, 1884), p. 297, available &lt;a href=&quot;https://www.google.com/books/edition/Die_Geschichte_Der_Physik_in_Grundz%C3%BCgen/nv04AAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. This is the reference given in several sources mentioned above (although I should note that this is an earlier edition than referenced above, 1884 as opposed to 1887). The relevant section (the only one where either Dietrich or horseshoe magnets are mentioned) is &lt;a href=&quot;https://www.google.com/books/edition/Die_Geschichte_Der_Physik_in_Grundz%C3%BCgen/nv04AAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22um%20diese%20Zeit%20der%20Baseler%20Mechaniker%22&amp;amp;pg=RA1-PA297&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt;:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Inclinationsboussole selbst suchten Daniel Bernoulli und Euler zu&#xA;verbessern, indem sie die Bedingungen untersuchten , unter welchen die&#xA;verschiedenen Inclinationsnadeln übereinstimmende Resultate ergeben&#xA;mussten. Bernoulli erhielt für die betreffende Arbeit 1743 den Preis&#xA;der Pariser Akademie , Euler das Accessit. Vielleicht auch auf&#xA;Veranlassung Daniel Bernoulli's führte um diese Zeit der Baseler&#xA;Mechaniker Johann Dietrich († 1758) die Ilufeisenmagnete und ihre&#xA;Armirung ein; wenigstens entdeckte Bernoulli mit solchen&#xA;Dietrich'schen Magneten das Gesetz, dass die Tragkraft der&#xA;Hufeisenmagnete proportional ist ihren Oberflächen oder den dritten&#xA;Wurzeln aus den Quadraten ihrer Gewichte.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation via Google Translate, plus &lt;em&gt;Boussole&lt;/em&gt; = &lt;em&gt;compass&lt;/em&gt; (see &lt;a href=&quot;https://de.wikipedia.org/wiki/Bussole&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;) and &lt;em&gt;Accessit&lt;/em&gt; assumed to be &lt;em&gt;Ak­zes­sit&lt;/em&gt; = &lt;em&gt;second place prize&lt;/em&gt; (see &lt;a href=&quot;https://www.dwds.de/wb/Akzessit&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The inclination compass itself sought to be improved by Daniel&#xA;Bernoulli and Euler by investigating the conditions under which the&#xA;various inclination needles must give consistent results. In 1743,&#xA;Bernoulli received the prize of the Paris Academy for the work in&#xA;question, and Euler received the second place prize. Perhaps also at&#xA;the instigation of Daniel Bernoulli, the Basel mechanic Johann&#xA;Dietrich († 1758) introduced iron magnets and their armouring around&#xA;this time; at least Bernoulli discovered the law with such Dietrich&#xA;magnets that the lifting capacity of the horseshoe magnets is&#xA;proportional to their surface areas or the third root of the square of&#xA;their weights.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Forbes and Dijksterhuis, &lt;a href=&quot;https://www.google.com/books/edition/A_History_of_Science_and_Technology/xq7aAAAAMAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A History of Science and Technology,&lt;/em&gt; Vol. 2&lt;/a&gt; (1963). The relevant section reads &lt;a href=&quot;https://www.google.com/search?q=%22At+Daniel+Bernoulli%27s+suggestion%22&amp;amp;tbm=bks&amp;amp;ei=ItL4Y4ipIJ6hptQPvaq5qAI&quot; rel=&quot;nofollow noreferrer&quot;&gt;(see here)&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first kind of magnets used were chiefly bar magnets. At Daniel&#xA;Bernoulli's suggestion, the Basle instrument-maker Johann Dietrich&#xA;bent the bar so as to bring the poles close together, thus creating&#xA;the first horseshoe magnet.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This source says that the horseshoe shape was Daniel Bernoulli's idea. At present, this seems unlikely to me, because Daniel Bernoulli himself said otherwise in his &lt;em&gt;Remarques sur les Aimans artificiels de Basle (Remarks on the artificial magnets of Basel)&lt;/em&gt; Nouvelle Bibliothèque germanique janv. fev. mars 1755. It is available &lt;a href=&quot;https://www.google.com/books/edition/Nouvelle_biblioth%C3%A8que_germanique_former/3WgEAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;printsec=frontcover#v=onepage&amp;amp;q=Bernoulli&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I will now reproduce the relevant section, which starts on p. 225:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;C'est Mr. Dietrich, Bourgeois de cette Ville &amp;amp; habile Artiste, qui&#xA;fait ces Aimans artificiels. Sa curiosité naturelle le porta d'abord à&#xA;construire quelques-uns de ces Aimans, en suivant les préceptes&#xA;connus, &amp;amp; fa capacité le conduisit bientôt à les perfectionner. Il a&#xA;remarqué que la figure la plus convenable pour donner beaucoup de&#xA;force à ces Aimans  est celle d'un fer à cheval: voici la mesure des&#xA;parties d'un de mes Aimans, qui pése 9 1⁄2 onces, &amp;amp; qui porte 16&#xA;livres: …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is the output of Google translate (except for &lt;em&gt;Bourgeois,&lt;/em&gt; meaning &lt;em&gt;citizen&lt;/em&gt;; the boldfaced emphasis is mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is Mr. Dietrich, a Citizen of this city and a skillful Artist, who makes&#xA;these artificial magnets; his natural curiosity first led him to&#xA;construct some of these magnets, following the known precepts, and his&#xA;ability soon led him to perfect them. &lt;strong&gt;He has noticed that the figure&#xA;most suitable for giving much force to these magnets is that of a&#xA;horseshoe.&lt;/strong&gt; Here is the measurement of the parts of one of my magnets&#xA;which weighs 9 1/2 ounces, and which bears 16. pounds: …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;Radelet-de Grave, &amp;quot;Studies of Magnetism in the Correspondence of Daniel Bernoulli,&amp;quot; in &lt;a href=&quot;https://www.google.com/books/edition/Two_Cultures/voFsJ1EhCnYC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Two Cultures: Essays in Honour of David Speiser,&lt;/em&gt;&lt;/a&gt; K. Williams, ed. (Birkhäuser, Basel, 2006).&#xA;Here are the relevant sections:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the meantime, Daniel Bernoulli is using horseshoe-shaped magnets,&#xA;invented&lt;sup&gt;24&lt;/sup&gt; and produced by Dietrich, in order to determine&#xA;his law giving the magnitude of the force of such a magnet. In several&#xA;letters he reports an empiric law, which was later published in a rather&#xA;peculiar text meant to promote the trade of Dietrich’s magnets: it is rather&#xA;like a commercial advertisement.&lt;sup&gt;25&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;&lt;sup&gt;24&lt;/sup&gt;It is difficult to date the apparition of the&#xA;horseshoe magnets, which, moreover, were discovered independently in&#xA;England later on. The publication meant to promote their trade is of&#xA;1755. But they are mentioned in the letters earlier than that: Daniel Bernoulli to Johann Jacob Huber (7 June 1754) and to Pierre Bouguer&#xA;(11 November 1754). In the latter D. Bernoulli says that he has been&#xA;disgusted by this kind of research but he does not give reason&#xA;why.&lt;/sup&gt;&lt;br&gt;&#xA;&lt;sup&gt;&lt;sup&gt;25&lt;/sup&gt;Text R&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &amp;quot;Text R&amp;quot; is Daniel Bernoulli's &lt;em&gt;Remarques sur les Aimans artificiels de Basle,&lt;/em&gt; which I already discussed above.&lt;/p&gt;&#xA;&lt;p&gt;The additional two letters mentioned by Radelet-de Grave are available at swisscollections.ch.&lt;/p&gt;&#xA;&lt;p&gt;The letter to Johann Jacob Huber is available &lt;a href=&quot;https://swisscollections.ch/Record/991170514764705501&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Since I don't speak French, I transcribed it using &lt;a href=&quot;https://readcoop.eu/transkribus/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Transkribus&lt;/a&gt;, and then used a combination of ChatGPT and Google translate to correct as best as I could the inevitable (i) errors in transcription and (ii) typos in the original, and then produce an English translation. Here is my current best guess as to the original intended text and its translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si l'aimant artificiel en forme de fer à cheval et les petites barres&#xA;appartiennent à la Chambre de Physique, car il me semble les avoir&#xA;payées, je prie Mr Huber de les remettre à ma servante. Je serai chez&#xA;moi cet après-midi jusqu'à 3 heures.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If the artificial magnet in the shape of a horseshoe and the small&#xA;bars belong to the Physics Chamber, as I seem to have paid for them, I&#xA;ask Mr. Huber to give them to my servant. I will be at home this&#xA;afternoon until 3 o'clock.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bernoulli's letter to Pierre Bouguer can be found &lt;a href=&quot;https://swisscollections.ch/Record/991170514765105501&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Using the same procedure as for the previous letter, here is my current best guess for the relevant part of it and its translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mon observation que la force des aimants artificiels semblables en&#xA;figure et en surfaces suit la raison de la surface de ces aimants&#xA;s'est trouvé confirmée par tous les essais que j'en ai faits. Je&#xA;pourrais bien faire avec le fer les expériences que vous me proposez,&#xA;si j'en reprenais le goût; mais j'en ai été tellement dégoûté que je&#xA;me suis défait de tous les aimants artificiels que j'avais pour ne&#xA;plus les avoir sous mes yeux. J'en ai remis un petit à M. Falckner&#xA;pour l'autoriser d'autant plus à vous voir.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My observation that the force of artificial magnets similar in shape&#xA;and surface is proportional to the surface area of these magnets has&#xA;been confirmed by all the experiments I have conducted. I could&#xA;perform the experiments you propose using iron if I regained an&#xA;interest in them; however, I was so disgusted that I got rid of all&#xA;the artificial magnets I had so that I no longer see them in front of&#xA;my eyes. I gave a small one to Mr. Falckner to encourage him to visit&#xA;you more often.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So far, my most believable source for 1743 being the year when Dietrich made the first horseshoe magnet is the following.&lt;/p&gt;&#xA;&lt;ol start=&quot;4&quot;&gt;&#xA;&lt;li&gt;&lt;em&gt;Verhandlungen der Naturforschenden Gesellschaft in Basel,&lt;/em&gt; Volumes 24-25 (available for free &lt;a href=&quot;https://www.google.com/books/edition/Verhandlungen_der_Naturforschenden_Gesel/gpWSsbJAXPUC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). The relevant section, whose translation by Google translate I don't completely understand, is this:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In der Festrede zur Feier des 50jährigen Bestehens der Basler&#xA;Naturforschenden Gesellschaft (vom 4. Mai 1867, Verh. IV, Anhang )&#xA;sprach er über die Arbeiten der Societas physica helvetica , der&#xA;Vorgängerin der Gesellschaft , nicht über die Geschichte dieser&#xA;selbst, denn , da die Gründer und Erbauer derselben zugegen waren, „&#xA;ich habe die Überzeugung, dass es den betreffenden, unter uns hochver&#xA;ehrten Männern erwünschter ist, das Rühmliche getan zu haben , als das&#xA;Getane rühmen zu hören .&amp;quot; Der Nachweis, dass die beste Form der&#xA;Inklinationsnadel der Hufeisenmagnet sei, wie 1743 Dan. Bernoulli in&#xA;einer Preisschrift bewies, diese aber vom Goldschmied und&#xA;Instrumentenmacher Joh. Dietrich in Basel zuerst konstruiert worden&#xA;ist , wie Burckhardt nachweist, dürfte daraus am meisten&#xA;interessieren.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here, for what it's worth, is a translation by Google translate:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the speech to celebrate the 50th anniversary of the Basler&#xA;Naturforschenden Gesellschaft (May 4, 1867, Verh. IV , Appendix ) he&#xA;spoke about the work of the Societas physica helvetica , the&#xA;predecessor of the society , not about the history of this itself,&#xA;because , there the founders and builders of the same were present, &amp;quot;&#xA;I am convinced that it is more desirable for the men in question, who&#xA;are highly respected among us, to have done what is creditable than to&#xA;hear praise for what has been done.&amp;quot; The proof that the best form of&#xA;the inclination needle the horseshoe magnet is, as Dan Bernoulli&#xA;proved in a prize paper in 1743, but this was first constructed by the&#xA;goldsmith and instrument maker Joh. Dietrich in Basel, as Burckhardt&#xA;proves, should be of most interest.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;Summary of what I know so far&lt;/h2&gt;&#xA;&lt;p&gt;There seems to be no reason to doubt that both Bazin and Dietrich produced their horseshoe magnets independently of each other. Bazin did so no later than 1753.  Dietrich did it definitely no later than 1754, and most likely as early as 1743. Moreover, horseshoe magnets were known in England by 1750, where they had possibly been independently invented by then.&lt;/p&gt;&#xA;&lt;h2&gt;Question&lt;/h2&gt;&#xA;&lt;p&gt;What is the best current evidence on who came up with the permanent horseshoe magnet first: Bazin or Dietrich? (Or were they perhaps first invented in Great Britain?)&lt;/p&gt;&#xA;" OwnerUserId="10854" LastEditorUserId="10854" LastEditDate="2023-03-04T19:58:32.480" LastActivityDate="2023-03-04T19:58:32.480" Title="Who produced the first permanent horseshoe magnet: Johann Dietrich or Gilles-Augustin Bazin?" Tags="|magnetism|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15204" PostTypeId="1" CreationDate="2023-03-02T05:59:49.957" Score="1" ViewCount="56" Body="&lt;p&gt;A famous technique in the modification of integrands is the set of “euler substitutions” that provide substitutions for the structure&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sqrt{ax^2 +bx+c}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;That is a fairly common occurence in integration.&lt;/p&gt;&#xA;&lt;p&gt;I looked on the wikipedia page, and did not find any reference to where Euler introduced them.&lt;/p&gt;&#xA;&lt;p&gt;Does anybody know in which text euler introduces these substitutions?&lt;/p&gt;&#xA;" OwnerUserId="9988" LastActivityDate="2023-03-02T05:59:49.957" Title="In which work did Euler invent the Euler Substitutions for a quadratic composed into a radical?" Tags="|mathematics|calculus|euler|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15205" PostTypeId="1" CreationDate="2023-03-03T00:08:58.487" Score="2" ViewCount="219" Body="&lt;p&gt;Occasionally, in a proof by mathematical induction, the writer will say something like, &amp;quot;We induct on &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot; or &amp;quot;We induct on the number of vertices.&amp;quot; This usage of the verb &lt;i&gt;induct&lt;/i&gt; has always sounded strange to me.  My first question is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When did mathematicians start using the verb &lt;i&gt;induct&lt;/i&gt; in this way?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The earliest instance that I could find using Google Books was in &lt;a href=&quot;https://www.google.com/books/edition/Mathematics_281/fS9QAQAAMAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics 281: Algebraic Topology I&lt;/a&gt; by Spanier in 1955, but I suspect that there are earlier instances.  Florian Cajori's paper, &lt;a href=&quot;https://doi.org/10.1080/00029890.1918.11998417&quot; rel=&quot;nofollow noreferrer&quot;&gt;Origin of the Name “Mathematical Induction”&lt;/a&gt; demonstrates that the history of the term &lt;i&gt;mathematical induction&lt;/i&gt; goes back much further, but Cajori's article does not discuss the use of the verb &lt;i&gt;induct&lt;/i&gt;.  It looks to me like an instance of &lt;a href=&quot;https://en.wikipedia.org/wiki/Back-formation&quot; rel=&quot;nofollow noreferrer&quot;&gt;back-formation&lt;/a&gt;; i.e., the term &lt;i&gt;induction&lt;/i&gt; came first, and then people decided they wanted a verb, so they came up with &lt;i&gt;induct&lt;/i&gt;.  If this hypothesis is correct, then it prompts my second question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Has any mathematician used the term &lt;i&gt;induce on&lt;/i&gt; instead of &lt;i&gt;induct on&lt;/i&gt;?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The term &lt;i&gt;induction&lt;/i&gt; is also used in the philosophy of science; the process of inferring a general principle from many instances is often called &lt;i&gt;inductive reasoning&lt;/i&gt;, and is contrasted with &lt;i&gt;deductive reasoning&lt;/i&gt;. Now, in logic, the verb form of &lt;i&gt;deductive&lt;/i&gt; is &lt;i&gt;deduce&lt;/i&gt;; nobody ever uses the verb &lt;i&gt;deduct&lt;/i&gt; to refer to logical inference. Therefore, one might naïvely expect (by analogy) that people would use the verb &lt;i&gt;induce&lt;/i&gt; rather than &lt;i&gt;induct&lt;/i&gt;  in the context of mathematical induction.  But I don't recall ever encountering that usage.  Maybe I'm just uninformed?  A possible explanation for avoiding the word &lt;i&gt;induce&lt;/i&gt; in this context is that &lt;i&gt;induce&lt;/i&gt; has other meanings in mathematics (e.g., if &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is a subspace of a topological space &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;, then the topology on &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; induces the subspace topology on &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2023-03-04T20:08:01.890" LastActivityDate="2023-03-05T19:02:39.760" Title="Use of the verb &quot;induct&quot; in proofs by mathematical induction" Tags="|mathematics|terminology|mathematical-logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15206" PostTypeId="1" AcceptedAnswerId="15207" CreationDate="2023-03-03T11:12:34.283" Score="24" ViewCount="2789" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Euler was a non-confrontational and deeply religious person. He was kind and could get on well with anyone. He worked under any circumstances and in any environment: &lt;strong&gt;“A baby on his lap, a cat on his back — that’s how he wrote his immortal works,&lt;/strong&gt;” as one of his contemporaries said.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Usually this phrase is attributed to &lt;strong&gt;Dieudonné Thiébault&lt;/strong&gt;, but &lt;a href=&quot;http://mathcenter.spb.ru/nikaan/book/math_spb_thm.pdf#page=34&quot; rel=&quot;noreferrer&quot;&gt;we&lt;/a&gt; were unable to identify the source. However we found similar phrasing in &lt;strong&gt;Camille Paganel&lt;/strong&gt;'s &lt;em&gt;Histoire de Frédéric le Grand&lt;/em&gt;, 1847, volume 1, page 448,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;un chat sur l'épaule et ses enfant sur les genoux, rédigeait des mémoires admirés du monde savant&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Could you help me trace back the origin of this phrase (if this goes to Thiebault...)?&lt;/p&gt;&#xA;" OwnerUserId="18073" LastEditorUserId="-1" LastEditDate="2023-03-04T09:01:24.883" LastActivityDate="2023-03-04T09:01:24.883" Title="Euler: “A baby on his lap, a cat on his back — that’s how he wrote his immortal works” (origin?)" Tags="|mathematics-social-history|euler|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15207" PostTypeId="2" ParentId="15206" CreationDate="2023-03-03T13:35:02.637" Score="32" Body="&lt;p&gt;The origin is undoubtedly Dieudonné Thiébault, as the anecdote is reported in his personal recollections of his stay in Berlin, published in &lt;em&gt;Mes souvenirs de vingt ans de séjour à Berlin&lt;/em&gt;, vol. 5, (Paris, 1804):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Euler] a fait faire aux sciences mathématiques, des pas de géant; et ses immenses travaux ne lui coûtoient rien : c'est au milieu de sa famille, et du bruit que des enfans peuvent fàire; c'est en jouant lui-même avec celui qu'il prenoit sur ses genoux, et avec un angola [sic] monté sur son épaule,&#xA;qu'il a composé quelques-uns de ces Mémoires que l'Europe a admirés et admirera toujours. [&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k28628b/f17.item.r=Dieudonn%C3%A9%20Thi%C3%A9bault&quot; rel=&quot;noreferrer&quot;&gt;page 13&lt;/a&gt;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He made giant steps in the mathematical sciences, and his immense works cost him nothing: it is in the middle of his family, and the noise that children can make; it is while playing himself with the one he took on his knees, and with an Angora cat mounted on his shoulder, that he composed some of these Memoirs that Europe has admired and will always admire.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2023-03-03T13:35:02.637" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15209" PostTypeId="2" ParentId="11977" CreationDate="2023-03-04T05:56:09.087" Score="3" Body="&lt;p&gt;As pointed out by Conifold in the comments, the answer of the history seems likely that Yao was surely aware of Kolmogorov complexity at the time, but that the introduction of communication complexity was more motivated by communication between agents or processors than any connection with Kolmogorov complexity.&lt;/p&gt;&#xA;&lt;p&gt;That said, there are some relations between the two concepts, for example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Vereshchagin &lt;a href=&quot;https://doi.org/10.1007/978-3-319-06686-8_28&quot; rel=&quot;nofollow noreferrer&quot;&gt;Randomized Communication Complexity of Approximating Kolmogorov Complexity&lt;/a&gt; (preprint of full version available &lt;a href=&quot;https://eccc.weizmann.ac.il/report/2013/178/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;Kaplan &amp;amp; Laplante, &lt;a href=&quot;https://doi.org/10.1016/j.tcs.2010.10.044&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kolmogorov complexity and combinatorial methods in communication complexity&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;Mora, Briegel, &amp;amp; Kraus, &lt;a href=&quot;https://doi.org/10.1142/S0219749907003171&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Kolmogorov complexity and its applications&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="450" LastActivityDate="2023-03-04T05:56:09.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15210" PostTypeId="2" ParentId="15205" CreationDate="2023-03-05T19:02:39.760" Score="2" Body="&lt;p&gt;If we search 19th century Google books, &amp;quot;induct on&amp;quot; does not exist with &amp;quot;mathematical induction&amp;quot;. Perhaps, the term &amp;quot;induct on N&amp;quot; means to &amp;quot;form an induction on N&amp;quot;. Again this is a rare usage as per the OED.&lt;/p&gt;&#xA;&lt;p&gt;The unabridged Oxford English Dictionary gives a usage with a quotation on&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;absol&lt;/em&gt;. To form an induction; to infer by induction.  rare.  1832 W. Whewell in I. Todhunter William Whewell (1876) II. 141 The conceptions&#xA;which must exist in the mind in order to get by induction a law from a&#xA;collection of facts; and the impossibility of &lt;em&gt;inducting&lt;/em&gt; or even of&#xA;collecting without this.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another author (Investigations in Algebra By Albert Cuoco, 1990, MIT Press) puts &amp;quot;induct on&amp;quot; in quotations. So I agree with the OP, that &lt;em&gt;induct on&lt;/em&gt; does sound unfamiliar to a non-native speaker (including myself) but a lot of mathematical words have not made sense of their choices, such as &lt;em&gt;functional, fields, space, kernel&lt;/em&gt; etc. Perhaps this is the beauty of languages.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Chapter 5 &lt;span class=&quot;math-container&quot;&gt;$$ \mathrm{G}_N(M)=\frac{M(M+1)(M+2)(M+3) \cdots(M+N-1)}{N&#xA; !} $$&lt;/span&gt; The numerator of &lt;span class=&quot;math-container&quot;&gt;$\mathrm{G}_N(M)$&lt;/span&gt; is the product of all the&#xA;integers between &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$M+N-1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We want to prove that for every positive integer &lt;span class=&quot;math-container&quot;&gt;$N,&#xA; \mathrm{~F}_N=\mathrm{G}_N$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$\mathbf{Z}^{+}$&lt;/span&gt;. We will prove this&#xA;(of course) by the principle of mathematical induction; we will&#xA;&amp;quot;induct on &lt;span class=&quot;math-container&quot;&gt;$N . &quot;$&lt;/span&gt; If &lt;span class=&quot;math-container&quot;&gt;$N=1$&lt;/span&gt;, we have that &lt;span class=&quot;math-container&quot;&gt;$$ F_1(M)=M $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;German Wikipedia on &lt;em&gt;Vollständige Induktion&lt;/em&gt; has some good information too&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Die Bezeichnung Induktion leitet sich ab von lat. inductio, wörtlich „Hineinführung“. Der Zusatz vollständig signalisiert, dass es sich&#xA;hier im Gegensatz zur philosophischen Induktion, die aus Spezialfällen&#xA;ein allgemeines Gesetz erschließt und kein exaktes Schlussverfahren&#xA;ist, um ein anerkanntes deduktives Beweisverfahren handelt. Das&#xA;Induktionsprinzip steckt latent bereits in der von Euklid&#xA;überlieferten pythagoreischen Zahlendefinition: „Zahl ist die aus&#xA;Einheiten zusammengesetzte Menge.“ Euklid führte aber noch keine&#xA;Induktionsbeweise, sondern begnügte sich mit intuitiven,&#xA;exemplarischen Induktionen, die sich aber vervollständigen lassen.&#xA;Auch andere bedeutende Mathematiker der Antike und des Mittelalters&#xA;hatten noch kein Bedürfnis nach präzisen Induktionsbeweisen.&#xA;Vereinzelte Ausnahmen im hebräischen und arabischen Sprachraum blieben&#xA;ohne Nachfolge.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Machine Translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The term induction is derived from Latin inductio, literally &amp;quot;&#xA;introduction &amp;quot;. The addition of the word &amp;quot;complete&amp;quot; indicates that&#xA;this is a recognized deductive method of proof, in contrast to&#xA;philosophical induction, which deduces a general law from special&#xA;cases and is not an exact conclusion.&lt;/p&gt;&#xA;&lt;p&gt;The principle of induction is already latent in the Pythagorean&#xA;definition of number handed down by Euclid: &amp;quot;Number is the quantity&#xA;composed of units.&amp;quot; Euclid, however, did not yet carry out induction&#xA;proofs, but contented himself with intuitive, exemplary inductions,&#xA;which, however, can be completed. Also other important mathematicians&#xA;of the antiquity and the Middle Ages had no need for precise induction&#xA;proofs yet. Isolated exceptions in the Hebrew and Arabic language&#xA;areas remained without succession.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2023-03-05T19:02:39.760" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15211" PostTypeId="1" CreationDate="2023-03-06T07:54:20.307" Score="0" ViewCount="176" Body="&lt;p&gt;Cauchy Integral Formula who actually proved it? The teacher asked the following task: who actually discovered this integral formula? need historical background, but it's not Cauchy! Help!&lt;/p&gt;&#xA;" OwnerUserId="18098" LastActivityDate="2023-03-14T16:30:24.270" Title="Cauchy Integral Formula who opened?" Tags="|mathematics-social-history|" AnswerCount="1" CommentCount="8" ClosedDate="2023-03-07T02:39:05.827" ContentLicense="CC BY-SA 4.0" />
  <row Id="15212" PostTypeId="2" ParentId="15211" CreationDate="2023-03-06T21:24:22.340" Score="5" Body="&lt;p&gt;You asked two questions: who first &lt;em&gt;discovered&lt;/em&gt; the formula and who first &lt;em&gt;proved&lt;/em&gt; it.  Which is the question you are asking here?  Someone might have found the formula with incorrect hypotheses and a mistaken proof before someone else gave a correct statement and proof. Perhaps the first person could be said to have discovered the formula without having proved it.&lt;/p&gt;&#xA;&lt;p&gt;In addition to the books mentioned in comments above, look also at Jeremy Gray's &lt;em&gt;The Real and the Complex: A History of Analysis in the 19th Century&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The original development of complex analysis by Cauchy had incorrect or unclear hypotheses about the type of functions to which the main results (integral representation theorem and residue theorem) could be applied. It wasn’t until the 1850s that Cauchy understood how complex-valued functions of one complex variable were much more special than complex-valued functions of two real variables, especially the significance of complex differentiability. The first textbook on complex analysis, by Briot and Bouquet, came out in 1859. It took many years for different properties of complex functions to be recognized as describing the &lt;em&gt;same&lt;/em&gt; functions, and that's why many different labels were introduced for what we now call analytic functions: analytic, holomorphic, monodromic, monogenic, regular, and synectic.  Today only two terms survive: analytic and holomorphic.&lt;/p&gt;&#xA;&lt;p&gt;In Section 3.5.7 of the book by Bottazzini and Gray mentioned in a comment above, they write on p. 151 an equation that is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the form in which Cauchy first presented his celebrated “integral formula”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but it is an integral over &lt;span class=&quot;math-container&quot;&gt;$[-\pi,\pi]$&lt;/span&gt; rather than an integral over a circle (think of a circle parameterized by angles in &lt;span class=&quot;math-container&quot;&gt;$[-\pi,\pi]$&lt;/span&gt;) since Cauchy did not have the viewpoint of integration in the complex plane.&lt;/p&gt;&#xA;&lt;p&gt;Later they say (pp. 183-184) that a paper of Cauchy's in 1846&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;is commonly — but incorrectly — credited by historians with containing the first general statement of the Cauchy integral theorem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;On p. 185 they say&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;contrary to the popular version of this story, there is no reference in [Cauchy's 1846 paper] to either complex functions or complex integration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The last paragraph of that section says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the crucial year 1846 showed that Cauchy was well acquainted with the basic facts about complex functions, including the idea of representing complex variables in a plane, the integral theorem—now stated in a more general setting than in his (1825a)— the concept of a residue and the integral formula. Yet a comprehensive theory of complex functions was still lacking. In addition, there were unanswered questions about the nature of branch-points and the multi-valuedness of complex functions and integrals that he was unable to deal with in a satisfactory way.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In Section 3.6.3, they discuss the work of Briot and Bouquet that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;eventually formulated the correct hypothesis one has to require for Cauchy’s theorem, i.e. the existence of a unique derivative &lt;span class=&quot;math-container&quot;&gt;$f'(z)$&lt;/span&gt; (independent of the path).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on that, I suggest looking up the work of Briot and Bouquet.&lt;/p&gt;&#xA;&lt;p&gt;The modern version of Cauchy's integral formula assumes &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is complex-differentiable but &lt;em&gt;not&lt;/em&gt; that &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; is continuous, and the fact that the formula is valid under such weak conditions is due to Goursat in 1900 (Transactions AMS vol. 1, pp. 14-16 &lt;a href=&quot;https://www.ams.org/journals/tran/1900-001-01/S0002-9947-1900-1500519-7/S0002-9947-1900-1500519-7.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). At the start of that paper he writes that the proof he gave in 1884 (Acta Mathematica vol. 4, pp. 197-200) did not need a continuity of  &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt; but this fact was perhaps not made clear in 1884, and he was writing the paper in 1900 to be more explicit about not needing continuity of &lt;span class=&quot;math-container&quot;&gt;$f'$&lt;/span&gt;.  So perhaps the answer is Goursat, although it seems strange to say the Cauchy integral formula was first &lt;em&gt;discovered&lt;/em&gt; in 1900.&lt;/p&gt;&#xA;&lt;p&gt;Section 8.5.4 of the Bottazzini/Gray book gives more details about what Goursat did and they particularly emphasize the related work of Pringsheim.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-03-14T16:30:24.270" LastActivityDate="2023-03-14T16:30:24.270" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15213" PostTypeId="1" AcceptedAnswerId="15214" CreationDate="2023-03-07T04:10:11.413" Score="1" ViewCount="131" Body="&lt;p&gt;In his book on commutative rings (published 1970), Kaplansky talks about Macaulay rings. In the mid 1970's, I learned some commutative algebra from a student of his, who referred to these rings as Macaulay rings. When I returned to commutative algebra about twenty years later, these rings were now Cohen-Macaulay. When did the change occur?&lt;/p&gt;&#xA;" OwnerUserId="16670" LastActivityDate="2023-03-07T20:43:50.060" Title="When did Macaulay rings become Cohen-Macaulay rings?" Tags="|terminology|abstract-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15214" PostTypeId="2" ParentId="15213" CreationDate="2023-03-07T12:35:59.850" Score="8" Body="&lt;p&gt;Actually, the name Cohen-Macaulay rings and Macaulay rings areboth due to Zariski and Samuel (&lt;em&gt;Commutative Algebra&lt;/em&gt;, Volume 2, App. 6, p. 396, 1958):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Definition 1. Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be a local ring. The common number of elements of the maximal prime sequence in &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is called the hological dimension (or the grade) of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, and is denoted by &lt;span class=&quot;math-container&quot;&gt;$codh(A)$&lt;/span&gt;. If &lt;span class=&quot;math-container&quot;&gt;$codh(A)=dim(A)$&lt;/span&gt;, we say that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a &lt;strong&gt;Macaulay ring (or a Cohen-Macaulay ring)&lt;/strong&gt;. [emphasis mine]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, it was Kaplansky's decision to call them just Macaulay ring (see, e.g., Irving Kaplansky, &lt;em&gt;Commutative Rings&lt;/em&gt;, Chapter 3, p. 84, 1974), but at the same time other authors used the name Cohen-Macaulay ring, e.g., Mario Fiorentini, &lt;em&gt;&lt;a href=&quot;http://www.bdim.eu/item?id=RLINA_1971_8_50_2_94_0&amp;amp;fmt=pdf&quot; rel=&quot;noreferrer&quot;&gt;Esempi di anelli di Cohen-Macaulay che non sono di&#xA;Gorenstein&lt;/a&gt;&lt;/em&gt; (Examples of Cohen-Macaulay rings that are not Gorenstein rings), &lt;em&gt;Atti della Accademia Nazionale dei Lincei&lt;/em&gt;, Serie 8, Vol. 50, n.2, p. 94–99, 1971. (in italian)&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="4931" LastEditDate="2023-03-07T20:43:50.060" LastActivityDate="2023-03-07T20:43:50.060" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15215" PostTypeId="1" CreationDate="2023-03-08T02:46:58.967" Score="1" ViewCount="61" Body="&lt;p&gt;I saw this rule used in an MIT integration bee that gives a result for the definite integral of a function summed with its inverse across a fixed limit:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\int_{x_1}^{x_2}f(x) + f^{-1}(x)dx = x_{2}^{2} -x_{1}^{2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;i tried googling but i don't know the given name of the rule if it has one, so I did not find anything.&lt;/p&gt;&#xA;&lt;p&gt;Who invented/discovered this rule and if that isn't known then who is the earliest known person to have used it? thanks&lt;/p&gt;&#xA;" OwnerUserId="9988" LastActivityDate="2023-03-08T02:46:58.967" Title="Who discovered the rule for the definite integration of a function summed with its inverse function across a fixed limit?" Tags="|calculus|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15216" PostTypeId="1" CreationDate="2023-03-08T16:45:10.983" Score="0" ViewCount="150" Body="&lt;p&gt;Grothendieck's &lt;em&gt;Éléments de géométrie algébrique&lt;/em&gt;, also known as EGA, was originally devised to consist of thirteen volumes (as stated in the &lt;a href=&quot;https://fppf.site/ega/intro-auto.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;introduction of the first volume&lt;/a&gt;), from which Grothendieck ended up writing only the first four ones, in the period 1960-1967. They were published inside &lt;em&gt;Publications mathématiques de l’I.H.É.S.&lt;/em&gt; A second edition of EGA I was published by Springer in 1971, but then... that was it. There was no second edition of the other EGA volumes.&lt;/p&gt;&#xA;&lt;p&gt;My question is: why did they not proceed with the second edition of the already existing EGA volumes?&lt;/p&gt;&#xA;&lt;p&gt;Was Grothendieck himself opposed to a second edition? Why?&lt;/p&gt;&#xA;" OwnerUserId="5031" LastActivityDate="2023-03-08T16:45:10.983" Title="Why a second edition of EGA was never published?" Tags="|algebraic-geometry|grothendieck|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15217" PostTypeId="1" AcceptedAnswerId="15220" CreationDate="2023-03-10T08:54:22.000" Score="3" ViewCount="253" Body="&lt;p&gt;Sean Carroll has a &lt;a href=&quot;https://www.youtube.com/watch?v=AuqKsBQnE2A&amp;amp;ab_channel=SeanCarroll&quot; rel=&quot;nofollow noreferrer&quot;&gt;video about gauge theory&lt;/a&gt; (2020) in his series about &lt;em&gt;Greatest Ideas of the Universe&lt;/em&gt;, where he claims that early in the development of quantum chromodynamics, some physicists tried to impose a different color charge scheme: red, white and blue, instead of the usual &lt;a href=&quot;https://en.wikipedia.org/wiki/Additive_color&quot; rel=&quot;nofollow noreferrer&quot;&gt;additive color&lt;/a&gt; scheme red, green, blue. Taken from the transcript [minute 3]:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] quarks come in three colors. Of course the idea of colors, let's say: red, green and blue (There was a brief movement movement to make it red, white and blue, but that was considered a little jingoistic so red, green and blue was a little bit more physics oriented). This is an idea from Murray Gell-Mann, the famous physicist. He was one of the co-inventors of quarks and the idea is [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are there sources to back this up? Why would somebody use white instead of green?&lt;/p&gt;&#xA;&lt;p&gt;As pointed in the comments this might be related to the colors of the US flag (red, white and blue) considering the nationality of  Gell-Mann and Zweig.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="4251" LastEditDate="2023-12-09T19:56:44.113" LastActivityDate="2023-12-09T19:56:44.113" Title="On early US patriotism to choose quark color charge labels" Tags="|physics|terminology|theoretical-physics|physicists|quantum-field-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15218" PostTypeId="2" ParentId="15111" CreationDate="2023-03-10T16:53:56.360" Score="4" Body="&lt;p&gt;Jean Pierre Serre was awarded the Field medal in 1954. I recommend you his talk for the occasion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Cohomologie et géométrie algébrique&lt;/strong&gt;. Congrès int. d’Amsterdam, 1954, vol. III, pp. 515-520&lt;/p&gt;&#xA;&lt;p&gt;He starts saying &lt;em&gt;&amp;quot;De nombreux problèmes de géométrie algébrique classique peuvent être formulés et étudiés de la façon la plus commode au moyen de la théorie des faisceaux&amp;quot;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is arguably Jean Pierre Serre who must be considered the person that introduced sheaves in algebraic geometry, but this must be understood in the context of Cartan and using sheaves for several branches of mathematics, and Bourbaki interest in Algebraic geometry in those times.&lt;/p&gt;&#xA;" OwnerUserId="15350" LastActivityDate="2023-03-10T16:53:56.360" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15219" PostTypeId="1" CreationDate="2023-03-11T06:32:35.633" Score="2" ViewCount="256" Body="&lt;p&gt;What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="2172" LastEditDate="2023-03-11T06:53:19.077" LastActivityDate="2023-03-15T16:25:16.193" Title="Origin of Riemann-Stieltjes Integral" Tags="|mathematics|calculus|riemann|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15220" PostTypeId="2" ParentId="15217" CreationDate="2023-03-11T08:36:49.117" Score="12" Body="&lt;p&gt;According to the best source, &amp;quot;red, white, and blue&amp;quot; were proposed in honour of France, not the United States.&lt;/p&gt;&#xA;&lt;p&gt;As discussed in the answer to &lt;a href=&quot;https://hsm.stackexchange.com/q/11750/6504&quot;&gt;&amp;quot;Quantum chromodynamics - an origin of the name&amp;quot;&lt;/a&gt;, Murray Gell-Mann reportedly initially used red, white, and blue for the quark colours in honour of the French, not American, flag, because he was living in France near CERN at the time. The source cited is Harald Fritzsch in his book &lt;a href=&quot;https://books.google.ca/books?id=tEZkDQAAQBAJ&amp;amp;pg=PA8&quot; rel=&quot;noreferrer&quot;&gt;The Fundamental Constants: A Mystery of Physics&lt;/a&gt;. Fritzsch and Gell-Mann are often credited as the parents of quantum chromodynamics, and along with William Bardeen, authored the seminal colour paper &lt;a href=&quot;https://arxiv.org/abs/hep-ph/0211388&quot; rel=&quot;noreferrer&quot; title=&quot;W.A. Bardeen, H. Fritzch, M. Gell-Mann, 21 July 1971, Contribution to: Topical Meeting on the Outlook for Broken Conformal Symmetry in Elementary Particle Physics, CERN-TH-1538&quot;&gt;&amp;quot;Light-Cone Algebra, &lt;span class=&quot;math-container&quot;&gt;$\pi^0$&lt;/span&gt; Decay and &lt;span class=&quot;math-container&quot;&gt;$e^+ e^-$&lt;/span&gt; Annihilation&amp;quot;&lt;/a&gt;, where the &amp;quot;red, white, and blue&amp;quot; quark colours were used, so he is a pretty solid source.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-03-11T08:36:49.117" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15221" PostTypeId="1" CreationDate="2023-03-11T14:35:14.330" Score="0" ViewCount="111" Body="&lt;p&gt;Is there a good book or other reference that discusses the historical origins of the principle that not even information can travel faster than the speed of light? In my understanding the main reason why such superluminal signaling is usually considered impossible is that this would violate causality in the context of special relativity. What were the first scientific papers discussing such causality violations?&lt;/p&gt;&#xA;" OwnerUserId="18120" LastActivityDate="2023-03-11T14:35:14.330" Title="Historical origins of the idea that information cannot travel faster than light" Tags="|relativity-theory|einstein|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15223" PostTypeId="1" AcceptedAnswerId="15224" CreationDate="2023-03-12T06:03:29.490" Score="2" ViewCount="177" Body="&lt;p&gt;On the &lt;a href=&quot;https://academia.stackexchange.com/&quot;&gt;Academia&lt;/a&gt; site, there is a recent &lt;a href=&quot;https://academia.stackexchange.com/questions/194165/how-do-mathematics-journals-get-reviewers-for-a-very-new-theory&quot;&gt;question&lt;/a&gt; that asks about obtaining reviewers for a &amp;quot;new theory&amp;quot;. I'm only an amateur mathematician, not a professional, and the question got me wondering what would be considered to constitute a new theory in mathematics, as opposed to (merely) a new theorem.&lt;/p&gt;&#xA;&lt;p&gt;It would be helpful if you could use an historical example, rather than something at the cutting edge of modern mathematical research.&lt;/p&gt;&#xA;" OwnerUserId="17963" LastEditorUserId="17963" LastEditDate="2023-03-13T06:20:06.210" LastActivityDate="2023-03-13T06:20:06.210" Title="New mathematics theory vs new mathematical theorem" Tags="|mathematics|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15224" PostTypeId="2" ParentId="15223" CreationDate="2023-03-12T09:29:41.227" Score="6" Body="&lt;p&gt;The Riemann integral is very useful for most of the basic needs of integration. However, it became clear around the end of the XIX&lt;sup&gt;th&lt;/sup&gt; century that it wasn't general enough. That's why Lebesgue developed his integral. Every Riemann-integrable function is also Lebesgue-integrable, but there are Lebesgue-integrable functions which are not Riemann-integrable, such as&lt;span class=&quot;math-container&quot;&gt;$$\begin{array}{rccc}f\colon&amp;amp;[0,1]&amp;amp;\longrightarrow&amp;amp;\Bbb R\\&amp;amp;x&amp;amp;\mapsto&amp;amp;\begin{cases}1&amp;amp;\text{ if }x\in\Bbb Q\\0&amp;amp;\text{otherwise;}\end{cases}\end{array}$$&lt;/span&gt;it is not Riemann-integrable, but, with respect to the Lebesgue integral, &lt;span class=&quot;math-container&quot;&gt;$\int_0^1f(x)\,\mathrm dx=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, Lebesgue did not just provide a new theorem about the Riemann integral. He developed a new theory of integration.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2023-03-12T17:51:25.950" LastActivityDate="2023-03-12T17:51:25.950" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15225" PostTypeId="2" ParentId="15223" CreationDate="2023-03-12T14:05:09.203" Score="4" Body="&lt;p&gt;The distinction is not formal. A new theory usually involves&#xA;some new set of definitions, new methods and several theorems.&#xA;Examples are abundant. Theory of groups, theory of distributions (generalized functions), theory of integration in the complex domain (Cauchy theory), etc.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-03-12T14:05:09.203" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15226" PostTypeId="2" ParentId="15219" CreationDate="2023-03-12T14:10:26.240" Score="6" Body="&lt;p&gt;The Riemann-Stieljes was introduced by Thomas Stieltjes in his long paper (or book) on continued fractions,&lt;/p&gt;&#xA;&lt;p&gt;Recherches sur les fractions continues,&#xA;Mém. Sav. étr. 32, Nr. 2, 197 S. (1904).&lt;/p&gt;&#xA;&lt;p&gt;The purpose was to unify the treatment of of continuous and discrete measures in his theory of moments.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-03-12T14:10:26.240" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15228" PostTypeId="2" ParentId="15201" CreationDate="2023-03-14T15:25:40.387" Score="4" Body="&lt;p&gt;It helped lead to the discovery of the wrong widely used image for the mathematician Legendre. See &lt;a href=&quot;https://www.ams.org/notices/200911/rtx091101440p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.ams.org/notices/200911/rtx091101440p.pdf&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-03-14T15:25:40.387" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15229" PostTypeId="2" ParentId="15219" CreationDate="2023-03-14T20:16:13.603" Score="4" Body="&lt;p&gt;An important use of the Stieltjes integral, after its creation by Stieltjes, was its use by Riesz in &lt;span class=&quot;math-container&quot;&gt;$1909$&lt;/span&gt; to describe the continuous dual space of &lt;span class=&quot;math-container&quot;&gt;$C([0,1])$&lt;/span&gt;, the space of real-valued continuous functions &lt;span class=&quot;math-container&quot;&gt;$[0,1] \to \mathbf R$&lt;/span&gt; equipped with the sup-norm. Riesz showed the  elements of the dual space are Stieltjes integrals &lt;span class=&quot;math-container&quot;&gt;$f \mapsto \int_0^1  f(x)d\alpha(x)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; has bounded variation. This result was later absorbed into general measure theory: the dual space of the continuous real-valued functions on a compact Hausdorff space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; consists of integration of continuous functions on &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; with respect to different Radon measures.&lt;/p&gt;&#xA;&lt;p&gt;Mike Bertrand's website has an account of Riesz's argument &lt;a href=&quot;https://nonagon.org/ExLibris/riesz-proves-riesz-representation-theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and a copy of Riesz's  article &lt;a href=&quot;https://nonagon.org/ExLibris/sites/default/files/pdf/Riesz-Operations-Fonctionnelles-Lineaires-1909.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-03-15T16:25:16.193" LastActivityDate="2023-03-15T16:25:16.193" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15230" PostTypeId="2" ParentId="5045" CreationDate="2023-03-15T03:35:27.107" Score="6" Body="&lt;p&gt;I found an answer to my own question from Ranjan Roy's book &amp;quot;Elliptic and Modular Functions from Gauss to Dedekind to Hecke&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;On page 293, Roy says the &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-product of the &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;-function is &lt;strong&gt;not&lt;/strong&gt; due directly to Jacobi, but to Klein (and ealier to Weierstrass: see below). In Klein's 1878/1879 paper &lt;em&gt;Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades&lt;/em&gt; (pp. 13-75 in Volume 3 of his collected works &lt;a href=&quot;https://archive.org/details/gesammeltemath03kleirich/page/n29/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;), which has some really beautiful pictures, &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; is defined on the top of the 3rd page as a cubic discriminant and in equation (18) on the ninth page you can see the &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-product for the 12th root of &lt;span class=&quot;math-container&quot;&gt;$\Delta(\omega)$&lt;/span&gt;, where Klein's &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$e^{\pi i\omega}$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$e^{2\pi i\omega}$&lt;/span&gt;. Taking &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;th powers of both sides of the equation yields the usual product for &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;. I think the notation &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; for the discriminant function is due to Klein.&lt;/p&gt;&#xA;&lt;p&gt;The second part of Klein's equation (17) expresses the &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;th root of &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; in terms of a product of three functions whose cube he expresses just above his equation (18) as a &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-product that is taken from equation (2) on p. 89 of Jacobi's &lt;em&gt;Fundamenta Nova&lt;/em&gt; &lt;a href=&quot;https://archive.org/details/fundamentanovat00unkngoog/page/n122/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. That's the reason for attributing the &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-product of the &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;-function to Jacobi. Since Klein's work relied on Jacobi's work, Klein needed knowledge of elliptic functions.&lt;/p&gt;&#xA;&lt;p&gt;A year before Klein, in 1877, Dedekind introduced the related function &lt;span class=&quot;math-container&quot;&gt;$\eta(\omega)$&lt;/span&gt; &lt;a href=&quot;https://eudml.org/doc/148340&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;: a definition and its transformation under &lt;span class=&quot;math-container&quot;&gt;${\rm SL}_2(\mathbf Z)$&lt;/span&gt; are equations (3) and (6) on p. 281 and its product representation &lt;span class=&quot;math-container&quot;&gt;$q^{1/12}\prod_{\nu \geq 1} (1 - q^{2\nu})$&lt;/span&gt; is equation (24) on p. 285, with &lt;span class=&quot;math-container&quot;&gt;$q = e^{\pi i\omega}$&lt;/span&gt;. Dedekind's paper  does not mention the discriminant function &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt;. It also has no pictures.&lt;/p&gt;&#xA;&lt;p&gt;Roy says earlier in his book, on p. 99, that Weierstrass used his sigma function to derive the &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;-product of &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; in his lectures on elliptic functions in 1862-1863: see equation 12 on p. 164 of the 5th volume of Weierstrass collected works &lt;a href=&quot;https://archive.org/details/mathematischewer05weieuoft/page/164/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, where &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; up to powers of &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hurwitz, who was a PhD student of Klein, was the first to develop modular functions without needing prior knowledge of elliptic functions (or theta functions) of any kind in 1881, and in particular he proved the product for &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; without needing elliptic functions; see p. 552 &lt;a href=&quot;https://eudml.org/doc/156976&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&#xA;This is the approach to modular functions in introductory book today, using Eisenstein series as a starting point.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-03-16T16:05:00.290" LastActivityDate="2023-03-16T16:05:00.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15231" PostTypeId="1" AcceptedAnswerId="15232" CreationDate="2023-03-15T06:29:37.733" Score="4" ViewCount="288" Body="&lt;p&gt;I am trying to understand if ancient Romans understood and used the catenary test when building bridges. I cannot find anything online&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-03-17T09:43:07.793" Title="Did the ancient Romans know and use the catenary test when building arches and bridges?" Tags="|roman-empire|architectural-engineering|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15232" PostTypeId="2" ParentId="15231" CreationDate="2023-03-16T06:59:16.340" Score="7" Body="&lt;p&gt;Not in any explicit way. The observation that compression forces in an arch invert tension forces in a cable, which led to the &amp;quot;catenary test&amp;quot; for arch shapes, was only made by Hooke c. 1670. According to &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earliest Known Uses&lt;/a&gt;, the name &lt;em&gt;linea catenaria&lt;/em&gt; for the hanging chain curve first appears in Huygens's 1690 letter to Leibniz, who then popularized it in a 1691 &lt;em&gt;Acta Eruditorum&lt;/em&gt; paper.&lt;/p&gt;&#xA;&lt;p&gt;Nonetheless, &lt;a href=&quot;https://link.springer.com/article/10.1007/s00004-007-0040-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Huerta in Oval Domes&lt;/a&gt; suggests that there was rudimentary practical understanding, gained by trial and error, of what the &amp;quot;catenary test&amp;quot; will later spell out :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In the first two millennia the builders experimented with several types of arches and&#xA;vaults and there is no direct line of progress towards the voussoir arch with radial joints,&#xA;which is our conceptual model... The first vaults were quite small, with spans of only about one or two meters, just enough to cover the tomb. This size favoured experimentation: the vault, if not of an adequate form, will distort and the observation of the movements gave the builders a&#xA;“feeling” for the more adequate forms...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;To an architect or engineer with some experience in masonry structures it will be evident that the vault at the bottom right side is the safest, adopting an oval form which will amply contain the trajectory of compressions (the line of thrust or inverted catenary) within the arch.&#xA;Choisy [1904] was the first to point this fact as the origin of the oval arches. [...] If all the voussoirs of the arch are the same size the line of thrust will have very nearly the form of an inverted catenary. But, it is not necessary that the arch have the form of catenary: it suffices that the catenary can be contained within the masonry.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So even to the extent that line of thrust considerations were understood they were not necessarily reflected by the shape of arches directly. In particular, Romans disfavored oval shapes close to the catenary, preferring circular arches and domes for aesthetic reasons:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It appears that the Romans did not built oval domes: the central symmetry was considered a requisite and even in the experimentation of the domes in Hadrian’s Villa all the forms present a centralized character. Some exceptions may be found in the apses of thermae, and it is usually presumed that the octagon of the church of St. Gereon in Cologne rests on the oval foundations of a previous Roman building. Choisy discovered that in the bridge of Narni, the central span of the inclined road was adapted by using an arch formed&#xA;from two quarter-circles of different radii. Perhaps more examples can be found, but it&#xA;appears that the Romans used the oval form for arches and domes only in exceptional cases.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-03-17T09:43:07.793" LastActivityDate="2023-03-17T09:43:07.793" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15233" PostTypeId="2" ParentId="15092" CreationDate="2023-03-17T22:47:12.467" Score="7" Body="&lt;p&gt;There's something like this in Volume II of Feller's &lt;i&gt;Introduction to Probability Theory and Its Applications&lt;/i&gt;, Section I.5.  Suppose &lt;span class=&quot;math-container&quot;&gt;$X_0, X_1, X_2, \ldots$&lt;/span&gt; is an infinite sequence of i.i.d. real-valued random variables  with a continuous distribution function. Define the &lt;i&gt;waiting time&lt;/i&gt; &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; as the value of the first subscript &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$X_n &amp;gt; X_0$&lt;/span&gt;. What is the expected value of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;?&lt;/p&gt;&#xA;&lt;p&gt;Feller gives the following argument.  The event that &lt;span class=&quot;math-container&quot;&gt;$N &amp;gt; n-1$&lt;/span&gt; occurs if and only if the largest term of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-tuple &lt;span class=&quot;math-container&quot;&gt;$X_0, X_, \ldots X_{n-1}$&lt;/span&gt; appears at the very beginning.  By symmetry, this probability is &lt;span class=&quot;math-container&quot;&gt;$1/n$&lt;/span&gt;. Therefore,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Pr(N = n) = \Pr(N &amp;gt; n-1) - \Pr(N &amp;gt; n) = {1\over n} - {1\over n+1} = {1\over n(n+1)}.$$&lt;/span&gt;&#xA;Thus the expected value of &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is infinite!  Feller then comments:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The striking and general nature of [this result] combined with the simplicity of the proof are apt to arouse suspicion.  The argument is really impeccable (except for the informal presentation), but those who prefer to rely on brute calculation can easily verify the truth from [tedious calculation omitted].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2023-03-17T22:47:12.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15234" PostTypeId="2" ParentId="14345" CreationDate="2023-03-18T15:35:45.947" Score="4" Body="&lt;p&gt;(&lt;strong&gt;Revised September, 2023&lt;/strong&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Apparently at least as of ~2016 the machine was still in Bennett's office at Yorktown Heights!&lt;/p&gt;&#xA;&lt;p&gt;I came across a &lt;a href=&quot;https://www.youtube.com/watch?v=Vnlr2rHMNy8&amp;amp;t=132s&quot; rel=&quot;nofollow noreferrer&quot;&gt;video&lt;/a&gt; by Karol Jalochowski, who was sponsored by the National University of Singapore to interview a number of physicists and philosophers.  Jalochowski spoke with Bennett in woods around his home, and also, based on an establishing shot near the entrance to the campus, at Bennett's office in IBM.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VDPKh.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VDPKh.jpg&quot; alt=&quot;Bennett pointing to BB84 machine&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Therein Bennett shows off the original BB84 machine (it was placed in what he called a &amp;quot;coffin&amp;quot;.)  Bennett briefly explains the machine to Jalochowski and emphasizes that the apparatus is &amp;quot;laboratory equipment constructed by theorists&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But, my heart sings a little song knowing that the machine is likely still extant, and was not scavenged for parts!&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/fp9l2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fp9l2.jpg&quot; alt=&quot;Charlie Bennett's office&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Also and only somewhat incidental to the story, Bennett's office looks epic.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;(Self-answer, making CW)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;https://www.youtube.com/watch?v=B5BUhzBlO-U&amp;amp;t=531s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bennett's 2021 talk&lt;/a&gt; about the early history of quantum information at about the 8 minute mark, he has a quick slide picturing the quantum cryptographic device. Indeed, a black-and-white version of the same photo is part of the &lt;em&gt;Journal of Cryptology&lt;/em&gt; article mentioned above.&lt;/p&gt;&#xA;&lt;p&gt;The later &lt;em&gt;SciAm&lt;/em&gt; picture looks professionally photographed, while the earlier image below looks like the device is labelled for a presentation or educational purposes.  I'm not positive that the machine pictured above in the &lt;em&gt;SciAm&lt;/em&gt; article is necessarily the same as the one pictured below in Bennett &lt;em&gt;et al.'s&lt;/em&gt; report; one could be a replica or a Ship of Theseus from the original.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BuZZm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BuZZm.png&quot; alt=&quot;Labelled BB84 machine&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But, Brassard's article states that the first demonstration of the device was in October of 1989, while the &lt;em&gt;Cryptology&lt;/em&gt; article indicates that a further run was done in February of 1991, and the &lt;em&gt;SciAm&lt;/em&gt; article came out in October of 1992, so the machine was around at least for a couple of years.&lt;/p&gt;&#xA;" OwnerUserId="9482" LastEditorUserId="9482" LastEditDate="2023-09-25T12:53:51.200" LastActivityDate="2023-09-25T12:53:51.200" CommentCount="4" CommunityOwnedDate="2023-03-18T15:35:45.947" ContentLicense="CC BY-SA 4.0" />
  <row Id="15235" PostTypeId="1" CreationDate="2023-03-19T20:42:52.200" Score="22" ViewCount="3595" Body="&lt;p&gt;I am trying to track down the first use of the 'bowling ball on a rubber sheet' analogy to explain spacetime curvature in general relativity. I have found a lot of secondary sources that give contradictory citations but no primary sources. ChatGPT generated a lot of fake references. Some modern papers (from the 1990s and early 2000s) even refer to Einstein in 1915 or Eddington in 1919 but these do not link to primary sources and are not very credible. I searched through the Einstein Papers and couldn't find anything. It seems likely to have originated in the mid 1950s-1960s by either John Wheeler or Robert Dicke but I would really like to find the first citation. I know it was before 1967 as Asimov published a short story using the idea in that year (unless he invented it which seems unlikely). Does anyone know?&lt;/p&gt;&#xA;" OwnerUserId="18160" LastActivityDate="2023-03-21T18:48:37.377" Title="What is the origin of the rubber sheet analogy?" Tags="|relativity-theory|physics-social-history|" AnswerCount="3" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15236" PostTypeId="2" ParentId="15235" CreationDate="2023-03-19T21:29:33.233" Score="17" Body="&lt;p&gt;Beware of ChatGPT, it can easily generate fake references for historical materials. A while ago, I asked about the origin of the concept of vector &amp;quot;norm&amp;quot;. Not only it provided a fake German reference but also quoted the relevant text in German. All imaginary! In short, don't use this experimental interface for science for the time being.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, coming to the main query. It appears from this article, (&lt;em&gt;Our Elastic Spacetime: Black Holes and Gravitational Waves: A new computer program shows that the old analogy of spacetime as a rubber sheet is remarkably valid&lt;/em&gt; by  Larry L. Smarr and William H. Press in American Scientist, 1978, 66,  72-79), that Arthur Eddington proposed this main theme in 1928 in his book written for the public.&lt;/p&gt;&#xA;&lt;p&gt;It is not exactly what we see in planetarium shows, but the elements seem to be there. The book is titled &lt;em&gt;The nature of the physical world&lt;/em&gt; (1928). On page 127 he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I should like to show you in a general way how it is possible for a&#xA;law controlling the curvature of empty space to determine the tracks&#xA;of particles without being supplemented by any other conditions. Two&#xA;&amp;quot;particles&amp;quot; in the four-dimensional world are shown in Fig. 5 , namely&#xA;yourself and myself. We are not empty space so there is no limit to&#xA;the kind of curvature entering into our composition; in fact our&#xA;unusual sort of curvature is what distinguishes us from empty space.&#xA;We are, so to speak, ridges in the four-dimensional world where it is&#xA;gathered into a pucker. The pure mathematician in his unflattering&#xA;language would describe us as &amp;quot;singularities&amp;quot;. These two non-empty&#xA;ridges are joined by empty space, which must be free from those kinds&#xA;of curvature described by the ten principal coefficients. Now it is&#xA;common experience that if we introduce local puckers into the material&#xA;of a garment, the remainder has a certain obstinacy and will not lie&#xA;as smoothly as we might wish. You will realise the possibility that,&#xA;given two ridges as in Fig. 5, it may be impossible to join them by an&#xA;intervening valley without the illegal kind of curvature. That turns&#xA;out to be the case.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/zw3N4.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zw3N4.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Two perfectly straight ridges alone in the world&#xA;cannot be properlyjoined by empty space and therefore they cannot&#xA;occur alone. But if they bend a little towards one another the&#xA;connecting region can lie smoothly and satisfy the law of curvature.&#xA;If they bend too much the illegal puckering reappears. The law of&#xA;gravitation is a fastidious tailor who will not tolerate wrinkles&#xA;(except of a limited approved type) in the main area of the garment;&#xA;so that the seams are required to take courses which will not cause&#xA;wrinkles. You and I have to submit to this and so our tracks curve&#xA;towards each other. An onlooker will make the comment that here is an&#xA;illustration of the law that two massive bodies attract each other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="10083" LastEditDate="2023-03-21T18:48:37.377" LastActivityDate="2023-03-21T18:48:37.377" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15237" PostTypeId="2" ParentId="15235" CreationDate="2023-03-20T02:42:53.297" Score="16" Body="&lt;p&gt;A rubber sheet analogy for curved spacetime was used very soon after Einstein's publication of General Relativity. The classic &amp;quot;bowling ball&amp;quot; version may have taken a few years to evolve, but was already in circulation by 1925, and an essentially equivalent but perhaps less clear &amp;quot;worms-bullet&amp;quot; analogy was published in 1920.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Early massive central weight rubber sheet analogies&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Donald_Howard_Menzel&quot; rel=&quot;noreferrer&quot;&gt;Donald Menzel&lt;/a&gt; very clearly describes the classic heavy ball on a rubber sheet analogy in his article on &amp;quot;What is gravitation?&amp;quot; on page 223 of &amp;quot;&lt;a href=&quot;https://www.americanradiohistory.com/Archive-Electrical-Experimenter/SI-1925-07.pdf&quot; rel=&quot;noreferrer&quot;&gt;Science and Invention for July, 1925&lt;/a&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Let us make our plane out of tightly stretched rubber (see Fig. 4). A small virtually weightless marble will move across it in a straight line, the path of least resistance.  If a heavy iron ball is now placed in the center, the plane will be &amp;quot;warped&amp;quot; and the path of least resistance will no longer be straight, but curved about the object, and if there were no friction, it would continue to revolve about it forever.  This is a vivid picture of the case of the solar system. The planets do not revolve about the sun because its constraining force keeps them from flying away, but rather because space is bent and they follow their own inclination.  Einstein called the natural path of any object a &lt;em&gt;geodesic&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3nIdWl.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3nIdWl.jpg&quot; alt=&quot;Figure 4: Rubber sheet distorted by heavy mass.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;On page 74 of his 1920 popular science book, &amp;quot;&lt;a href=&quot;https://archive.org/details/easylessonsinein00slosrich/page/74/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;Easy lessons in Einstein; a discussion of the more intelligible features of the theory of relativity&lt;/a&gt;&amp;quot;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Edwin_Emery_Slosson&quot; rel=&quot;noreferrer&quot;&gt;Edwin Slossen&lt;/a&gt; (Harcourt, Brace and Howe, 1920) uses a bullet for the central weight and worms to map out the geodesics.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;How a heavy object can alter space relations may be seen from this simple illustration:&#xA;Stretch a sheet of rubber over a hoop like a drumhead.&#xA;It is now level and flat and if parallel lines are drawn across it in two directions so as to divide it up into squares like a checkerboard all these lines are straight and equidistant and all the squares are of equal size.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A row of worms, starting in an even rank and crawling along the parallel lines across the drumhead, would keep even all the way.&#xA;Now lay a bullet on the center of the drumhead.&#xA;The rubber sags down and stretches, most in the middle, least at the edges.&#xA;The &amp;quot;parallel&amp;quot; lines are no longer equidistant.&#xA;The squares are no longer equal.&#xA;The lines are no longer of the same length.&#xA;If now we repeat our worm race we shall find that those worms following lines close to the weight have to go down hill and up again and so travel a greater distance to traverse the same number of squares than those following lines nearer the edge which lie comparatively flat and are nearly as short as before.&#xA;Consequently the worms will be slowed up in proportion to their nearness to the center and the row of their heads will be swung around at an angle to their former frontage.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Other early rubber sheet analogies, but without the heavy ball:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Nicolas_Rashevsky&quot; rel=&quot;noreferrer&quot;&gt;Nicholas Rashevsky&lt;/a&gt; uses a rubber sheet analogy in a November 1924 Scientific American article titled &amp;quot;&lt;a href=&quot;http://www.jstor.org/stable/24975344&quot; rel=&quot;noreferrer&quot; title=&quot;Rashevsky, N. P. (1924). The Possibility of Other Kinds of Space. Scientific American, 131(5), 305–307.&quot;&gt;The Possibility of Other Kinds of Space&lt;/a&gt;&amp;quot;, but only talks about statically stretched spacetime.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;… if instead of the sheet of paper we take a sheet of &lt;em&gt;rubber&lt;/em&gt;. The rubber will stretch. … some parts of the original rubber plane have had their dimensions altered and this alteration of the dimensions has occurred in different degree in different parts of the rubber sheet. … we will find that the figures on the rubber &lt;em&gt;no longer obey the rules of Euclid's plane geometry&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In his General Theory of Relativity, Einstein was led to assume that the space of the real universe is, in fact, non-Euclidean.  We have seen in this article that this is quite possible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Even earlier, &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph_Sweetman_Ames&quot; rel=&quot;noreferrer&quot;&gt;Joseph Ames&lt;/a&gt; briefly mentions distorting world lines on rubber sheets in a 1919 American Physical Society Presidential Address on &amp;quot;&lt;a href=&quot;https://doi.org/10.1126/science.51.1315.253&quot; rel=&quot;noreferrer&quot; title=&quot;Science 51.1315 (1920): 253-261.&quot;&gt;Einstein's Law of Gravitation&lt;/a&gt;. (For a non-paywalled version see &lt;a href=&quot;https://archive.org/details/contemporaryscie00harr/page/n241&quot; rel=&quot;noreferrer&quot;&gt;page 238 of &amp;quot;Contemporary Science&amp;quot;&lt;/a&gt;, a 1921 book edited by &lt;a href=&quot;https://en.wikipedia.org/wiki/Benjamin_Harrow&quot; rel=&quot;noreferrer&quot;&gt;Benjamin Harrow&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I can draw a line on a sheet of paper or of rubber and by bending and stretching the sheet, I can make the line assume a great variety of shapes; each of these new shapes is a picture of a suitable transformation.&#xA;Now, consider world-lines in our four dimensional space. The complete record of all our knowledge is a series of sequences of intersections of such lines. By analogy I can draw in ordinary space a great number of intersecting lines on a sheet of rubber; I can then bend and deform the sheet to please myself ; by so doing I do not introduce any new intersections nor do I alter in the least the sequence of intersections.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Einstein's 1917 &amp;quot;Cloth&amp;quot; analogy&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I can't find any explicit use of a rubber sheet analogy by Einstein himself in those early years, but he came close in a &lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol8-trans/330&quot; rel=&quot;noreferrer&quot; title=&quot;&#xA;The Collected Papers of Albert Einstein, Princeton University Press, Volume 8: The Berlin Years: Correspondence, 1914-1918 (English translation supplement) Page 302&quot;&gt;March 1917 letter&lt;/a&gt; to &lt;a href=&quot;https://en.wikipedia.org/wiki/Willem_de_Sitter&quot; rel=&quot;noreferrer&quot;&gt;Willem de Sitter&lt;/a&gt; where he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I compare the space to a cloth floating (at rest) in the air, a certain part of which we can observe. This part is slightly curved similarly to a small section of a sphere’s surface. We philosophize on how we must construe the continuation of the cloth so that an equilibrium is reached in its tangential tension, whether it is fastened in position at the edges, extends infinitely, or has a finite size and is a closed unit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Earliest physical manifestation/construction&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although the question is about the concept, I'll note that the earliest documented physical construction of the analogy I've found so far is &amp;quot;&lt;a href=&quot;https://doi.org/10.1111/j.1949-8594.1963.tb11348.x&quot; rel=&quot;noreferrer&quot; title=&quot;Showalter, V.M. (1963), The Gravity Field Simulator. School Science and Mathematics, 63: 275-279.&quot;&gt;The Gravity Field Simulator&lt;/a&gt; built by Victor Showater at Ohio State University and published in 1963. This was a natural rubber sheet stretched on a 36 inch square frame with a 9 pound &lt;a href=&quot;https://en.wikipedia.org/wiki/Shot_put&quot; rel=&quot;noreferrer&quot;&gt;shot put&lt;/a&gt; ball in the middle and small steel balls as orbiting &amp;quot;planets&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Physics professors and teachers do not, however, always publish their demonstrations, so it is very possible earlier versions were built elsewhere.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2023-03-21T17:23:45.137" LastActivityDate="2023-03-21T17:23:45.137" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15240" PostTypeId="2" ParentId="15235" CreationDate="2023-03-20T13:28:43.930" Score="9" Body="&lt;p&gt;In the 12 December 1919 &lt;a href=&quot;https://www.google.com/books/edition/Monthly_Notices_of_the_Royal_Astronomica/F9A-AQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=gravitation%20rubber%20sheet&quot; rel=&quot;noreferrer&quot;&gt;Monthly Notices of the Royal Astronomical Society&lt;/a&gt; there is starting on page 96 a &amp;quot;Discussion of the Theory of Relativity&amp;quot; with multiple speakers and in the portion where Eddington is speaking, he says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Imagine this blackboard to be a sheet of rubber ; distort it until these irregular meshes become neat uniform squares...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2331" LastEditorUserId="2331" LastEditDate="2023-03-20T14:34:31.223" LastActivityDate="2023-03-20T14:34:31.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15241" PostTypeId="1" AcceptedAnswerId="15256" CreationDate="2023-03-21T11:41:59.360" Score="4" ViewCount="359" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Law_of_tangents#cite_note-Debarnot-3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Law of Tangents&lt;/a&gt; is a rather obscure trigonometric identity that is sometimes used in place of its better-known counterparts, the law of sines and law of cosines, to calculate angles or sides in a triangle.&lt;/p&gt;&#xA;&lt;p&gt;According to Wikipedia,the law of tangents for planar triangles was described in the 11th century by &lt;a href=&quot;https://en.wikipedia.org/wiki/Ibn_Mu%27adh_al-Jayyani&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ibn Muʿādh al-Jayyānī&lt;/a&gt; (Wikipedia cites &lt;a href=&quot;https://archive.org/details/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996/Rashed%20R.-Encyclopedia%20of%20the%20History%20of%20Arabic%20Science.%202-Routledge%20%281996%29/page/n189/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roshdi Rashed (ed.) Encyclopedia Of The History Of Arabic Science&lt;/a&gt;, p. 182. But I couldn't find any reference to the law of tangents in this book). Another source credits the Persian mathematician Abu'l-Wafa' in the 10th century CE as the first to publish it.&lt;/p&gt;&#xA;&lt;p&gt;The following statement can be found in &lt;em&gt;Merzbach and Boyer, A History of Mathematics, 3rd edition, page 278&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;Though Viete may have been the first to use this formula [the law of tangents], it was first published by the German physician and professor of mathematics Thomas Finck (1561-1656) in 1583, in Geometriae Rotundi Libri XIV.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Who was really the first to publish (or whatever that means back then) it?&lt;/p&gt;&#xA;" OwnerUserId="16769" LastEditorUserId="16769" LastEditDate="2023-03-24T11:27:24.370" LastActivityDate="2023-03-26T17:34:46.317" Title="Who published the law of tangents first?" Tags="|trigonometry|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15242" PostTypeId="1" AcceptedAnswerId="15247" CreationDate="2023-03-21T18:39:58.713" Score="1" ViewCount="173" Body="&lt;p&gt;I'm trying to understand the following passage from Boyer's and Merzbach's &lt;a href=&quot;https://archive.org/details/historyofmathema00boye&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Mathematics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/D2XSp.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/D2XSp.png&quot; alt=&quot;page 203&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1bVLb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1bVLb.png&quot; alt=&quot;Fig. 9.1&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The question I have is: how does the author derive that &lt;span class=&quot;math-container&quot;&gt;$w^2=2rv$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastEditorUserId="6609" LastEditDate="2023-03-21T19:58:49.597" LastActivityDate="2023-03-23T10:08:49.300" Title="Ancient Chinese method to calculating $\pi$" Tags="|pythagorean-proposition|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15245" PostTypeId="1" CreationDate="2023-03-22T20:44:08.387" Score="3" ViewCount="104" Body="&lt;p&gt;I already asked this question in the &lt;a href=&quot;https://astronomy.stackexchange.com/q/53198/33630&quot;&gt;Astronomy community&lt;/a&gt;, but there it was recommended to me to also try my luck here.&lt;/p&gt;&#xA;&lt;p&gt;I would like to know the original description of point sources and point spread functions. They are well-known concepts (in particular in astronomy e.g.), but every source that I could find so far that discusses them doesn't mention who actually discovered or came up with them.&lt;/p&gt;&#xA;&lt;p&gt;The closest thing that I found for now is the work by &lt;a href=&quot;https://archive.org/details/transactionsofca05camb/page/n305/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Airy&lt;/a&gt;, who mathematically described the diffraction pattern generated by a perfect optical system. So I guess that is very close. However, according to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Airy_disk&quot; rel=&quot;nofollow noreferrer&quot;&gt;wiki article&lt;/a&gt;, the phenomenon was described by Herschel at least seven years prior. But that is just regarding the outer rings of the diffraction pattern. So I would guess astronomers must have been aware of the blob-like appearance of poin sources even earlier.&lt;/p&gt;&#xA;" OwnerUserId="18187" LastActivityDate="2023-03-25T10:06:39.100" Title="Original description of point sources and point spread functions" Tags="|mathematics|physics|astronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15246" PostTypeId="2" ParentId="15245" CreationDate="2023-03-23T02:29:08.300" Score="3" Body="&lt;p&gt;Partial answer to get things started.&lt;/p&gt;&#xA;&lt;p&gt;Point sources were a natural concept for mathematicians pondering waves, and the Huygens construction for a spherical wavefront generated by a point source is an expanding spherical shell with each point on the surface being a new point source.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7Lbgf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7Lbgf.png&quot; alt=&quot;Huygens construction for a spherical wavefront generated by a point source&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Image source: The Fact Factor's &lt;a href=&quot;https://thefactfactor.com/facts/pure_science/physics/applications-of-huygens-wave-theory-of-light/7673/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Application of Huygens Wave Theory of Light&lt;/a&gt; but probably cropped from somewhere else.&lt;/p&gt;&#xA;&lt;p&gt;But I wonder if you are really interested in the development of theory not of point sources, &lt;em&gt;but of apertures&lt;/em&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Apodization&quot; rel=&quot;nofollow noreferrer&quot;&gt;Apodization&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Point_spread_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;point spread function&lt;/a&gt; go hand-in-hand; in a well-focused and diffraction-limited optical system for example, they are essentially Fourier transforms of each other.&lt;/p&gt;&#xA;&lt;p&gt;In modern times we apodize all kinds of things, outside of optics as well, cf. &lt;a href=&quot;https://chemistry.stackexchange.com/q/111982/16035&quot;&gt;Is it &amp;quot;common practice in Fourier transform spectroscopy to multiply the measured interferogram by an apodizing function&amp;quot;? If so, why?&lt;/a&gt; in Chemistry SE.&lt;/p&gt;&#xA;&lt;p&gt;In advanced optical systems pushing limits of resolution like &lt;a href=&quot;https://en.wikipedia.org/wiki/Photolithography#Resolution_in_projection_systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;deep-UV optical photolithography&lt;/a&gt; scanners, the pupil of the lenses can be fitted with both amplitude and phase apodization (as can the illumination systems for the mask)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;All of this taken together&lt;/strong&gt; suggests to me that the history of &lt;a href=&quot;https://en.wikipedia.org/wiki/Fourier_optics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourier optics&lt;/a&gt; may yield some helpful resources.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2023-03-23T02:29:08.300" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15247" PostTypeId="2" ParentId="15242" CreationDate="2023-03-23T07:56:16.610" Score="4" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$R'$&lt;/span&gt; be the reflection of &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; (see the picture below). Then &lt;span class=&quot;math-container&quot;&gt;$RR'$&lt;/span&gt; is a diameter of the circle and therefore its length is &lt;span class=&quot;math-container&quot;&gt;$2r$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The triangles &lt;span class=&quot;math-container&quot;&gt;$\triangle RQR'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\triangle RMQ$&lt;/span&gt; are similar, since both of them are right triangles and &lt;span class=&quot;math-container&quot;&gt;$\angle Q\hat RM=\angle Q\hat RR'$&lt;/span&gt;. So,&lt;span class=&quot;math-container&quot;&gt;$$\frac{\overline{RM}}{\overline{RQ}}=\frac{\overline{RQ}}{\overline{RR'}},$$&lt;/span&gt;but&lt;span class=&quot;math-container&quot;&gt;$$\frac{\overline{RM}}{\overline{RQ}}=\frac{\overline{RQ}}{\overline{RR'}}\iff\frac vw=\frac w{2r}\iff w^2=2rv.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/7IaCS.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7IaCS.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2023-03-23T10:08:49.300" LastActivityDate="2023-03-23T10:08:49.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15248" PostTypeId="1" CreationDate="2023-03-23T22:42:32.533" Score="11" ViewCount="558" Body="&lt;p&gt;There is a symbol or icon for an atom that is instantly recognizable and is associated with nuclear physics and with chemistry. A search for &amp;quot;nuclear atom symbol&amp;quot; (on 3/23/2023) shows what the essence of the symbol is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/HbcMZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/HbcMZ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;While there are variations (3 or 4 orbits, nucleus shown or not, electrons shown or not), the orbits always seem to be circular, not coplanar, and stationary.&lt;/p&gt;&#xA;&lt;p&gt;The US atomic energy commission &lt;a href=&quot;https://blog.nuclearsecrecy.com/2012/04/13/friday-image-the-atomic-energy-commission-seal-1949/&quot; rel=&quot;nofollow noreferrer&quot;&gt;adopted this icon in 1949&lt;/a&gt;, and both General Electric and the town of Richland, WA used it in 1948, see below:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/yueBz.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yueBz.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;a href=&quot;https://i.stack.imgur.com/LKP3Q.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/LKP3Q.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Sources: &lt;a href=&quot;https://www.osti.gov/opennet/manhattan-project-history/Resources/adventures_atom.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;left&lt;/a&gt; and &lt;a href=&quot;https://www.washington.edu/uwired/outreach/cspn/Website/Images/NW%20History%20Course/Lesson%2024/Atomic%20Days.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;right&lt;/a&gt; panel&lt;/p&gt;&#xA;&lt;p&gt;By this time, Hiroshima and Nagasaki had been destroyed by nuclear (fission) bombs. Both the US and the USSR were developing nuclear reactors with a view to produce electricity, and to power submarines. Quantum physics had developed more sophisticated (but less picturesque) models of electrons bound to the nucleus (in a wave-particle duality and considering the Heisenberg Uncertainty Principle, which did not support orbits).&lt;/p&gt;&#xA;&lt;p&gt;Fifteen years earlier, the Bohr-Sommerfeld model posited elliptical orbits, as shown in this sketch by Bohr:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JlS1h.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JlS1h.gif&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://physicstoday.scitation.org/doi/10.1063/PT.3.1978&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://physicstoday.scitation.org/doi/10.1063/PT.3.1978&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My question is whether there are examples of the atom icon that are pre-1948 and might have been the model for the 1948 examples that are shown above. How did we transition from many ways of depicting the structure of the atom to this iconic common one?&lt;/p&gt;&#xA;" OwnerUserId="16431" LastEditorUserId="16431" LastEditDate="2023-04-17T16:34:28.310" LastActivityDate="2023-04-24T16:01:21.397" Title="Who popularized the atom icon (atomic whirl or planetary model)" Tags="|quantum-mechanics|atomic-theory|nuclear-physics|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15249" PostTypeId="2" ParentId="8332" CreationDate="2023-03-24T02:05:29.133" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Edward_Witten&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward Witten&lt;/a&gt; published a paper with his father &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Witten&quot; rel=&quot;nofollow noreferrer&quot;&gt;Louis Witten&lt;/a&gt; in 1987: &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0550321387902495&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;``&lt;/span&gt;Large radius expansion of superstring compactifications''&lt;/em&gt;&lt;/a&gt; in Nuclear Physics B (1987), Volume 281, 109–126.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2023-03-24T02:05:29.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15250" PostTypeId="1" AcceptedAnswerId="15251" CreationDate="2023-03-24T04:03:58.360" Score="19" ViewCount="3286" Body="&lt;p&gt;Apparently Ptolemy was aware of the fact that the duration of time from noon to noon varied by many seconds throughout the course of a year. In modern times this fluctuation in length of day leads to what is referred to as &lt;a href=&quot;https://en.wikipedia.org/wiki/Equation_of_time#Early_astronomy&quot; rel=&quot;noreferrer&quot;&gt;the equation of time&lt;/a&gt;. Now, if you can make sightings of meridian crossings of the sun and compare the times of those crossings to the time of day you can make this measurement yourself. But I guess Ptolemy only had water clocks. I don't know if these clocks were accurate enough to measure fluctuations of 10s of seconds over the course of 24 hours.&lt;/p&gt;&#xA;&lt;p&gt;How, then, did Ptolemy come to the conclusion that the length of day varies throughout the course of the year? Did he use a water clock which was in fact precise enough for this measurement? Or did he perform some other sort of astronomical observation to draw this conclusion?&lt;/p&gt;&#xA;" OwnerUserId="18193" LastEditorUserId="18193" LastEditDate="2023-03-26T05:06:57.490" LastActivityDate="2023-03-26T05:06:57.490" Title="How did Ptolemy know that days were unequal lengths?" Tags="|astronomy|ancient-greece|time|ptolemy|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15251" PostTypeId="2" ParentId="15250" CreationDate="2023-03-24T11:21:55.983" Score="22" Body="&lt;p&gt;You are right: at the time of Ptolemy they could not measure&#xA;the length of a day directly. Actually Ptolemy never discusses any clocks in his book, he probably used some crude devices&#xA;record the approximate times of observations, which he never gives with accuracy better than 1/4 of an hour.&lt;/p&gt;&#xA;&lt;p&gt;The main reason for the equation of time is that&#xA;ecliptic is inclined to the equator. So the hour angle of the Sun (the angle measured along the equator) is not a linear function of Sun's latitude (position on the ecliptic). In addition to this, Sun moves on the ecliptic not uniformly.&lt;/p&gt;&#xA;&lt;p&gt;The primary observation for this phenomenon is that the seasons have unequal length (measured in days). This was known long before Ptolemy. This implies that the Sun moves on&#xA;the ecliptic with varying speed. The speed depends on its position on the ecliptic. The equation of time reflects the difference between the actual speed on a given day and&#xA;the average speed over the year. Ptolemy introduces an imaginary object, the &amp;quot;mean Sun&amp;quot;, which moves on the ecliptic with constant speed (=the average speed of the Sun), and uses&#xA;it as a reference for the time of all other astronomical phenomena.&lt;/p&gt;&#xA;&lt;p&gt;To be specific, all solar theory of Ptolemy is derived from 15 observations of solstices and equinoxes which are recorded with accuracy of 1 hour, and spread over 570 years.&lt;/p&gt;&#xA;&lt;p&gt;So for example, the length of the year can be obtained as the number of hours between the first and last observation, divided by 570. If the error&#xA;of the observations is &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, then the error in the length of the year is &lt;span class=&quot;math-container&quot;&gt;$E/570$&lt;/span&gt;. This shows the principle, how great precision is achieved.&lt;/p&gt;&#xA;&lt;p&gt;Remarks. 1.This brings a more general question: what is time, really?&#xA;Which time intervals are equal, and how do we know (or decide) this?&#xA;Nowadays we use all kinds of clocks, but this was not the case before accurate clocks were invented.&lt;/p&gt;&#xA;&lt;p&gt;To define time scale one has to choose some periodic (repetitive) natural process and postulate that it&#xA;happens &amp;quot;at the constant speed&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;At the time of Ptolemy, there were essentially two choices: a) to assume that the diurnal rotation of the sky&#xA;is uniform (that is to choose a day as the basic unit), or&#xA;b) to assume that Sun moves at a constant speed on the ecliptic&#xA;(that is to choose the year as the basic unit). Equation of time shows that these two choices are somewhat inconsistent.&#xA;Once this is discovered, the choice is dictated by simplicity of the mathematical model. In Ptolemy's own words:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And in general, we consider it a good principle&#xA;to explain the phenomena by the simplest hypotheses possible, in so far as there&#xA;is nothing in the observations to provide a significant objection to such a&#xA;procedure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Chap. III, section 1, 201 of Toomer's translation).&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy chooses b), that is the motion of the &amp;quot;mean Sun&amp;quot;&#xA;as the standard for time measurement.&#xA;The result is that days are of unequal length.&lt;/p&gt;&#xA;&lt;p&gt;(There was actually a third possibility: to choose the diurnal rotation of the sky (=&amp;quot;fixed&amp;quot; stars) for the time scale. It was actually a great insight of Hipparchus&#xA;that Sun gives the &amp;quot;correct&amp;quot; choice of the time scale).&lt;/p&gt;&#xA;&lt;p&gt;A good general philosophical discussion of the meaning of time is an article of Poincare Time measurement in his book The value of science.&lt;/p&gt;&#xA;&lt;p&gt;At the time of Poincare, time was defined as &amp;quot;the variable which makes the Second Law of Newton true&amp;quot;.&#xA;Nowadays we use a more fundamental physical theory&#xA;(quantum mechanics) to define time scale.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;Almagest is a difficult reading for a modern reader. There are two excellent modern expositions in English: O. Neugebauer, A history of ancient mathematical astronomy, and O. Pedersen, A survey of Almagest.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-03-26T00:29:44.453" LastActivityDate="2023-03-26T00:29:44.453" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15252" PostTypeId="2" ParentId="15241" CreationDate="2023-03-24T19:11:05.287" Score="3" Body="&lt;p&gt;The law of tangents is indeed attributed to the Danish polymath &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Fincke&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Fincke&lt;/a&gt; (1561–1656). According to his short &lt;a href=&quot;https://link.springer.com/article/10.1007/s00048-004-0191-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography&lt;/a&gt; by Jürgen Schönbeck (in German and behind a paywall), the law appears in Vol. 14 of Fincke's opus magnum &amp;quot;Geometria rotundi&amp;quot; (Basel, 1583) in terms of the Latin phrase:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ut semissis summae crurum ad differentiam summae semissis alteriusque&#xA;cruris, sic tangens semissis anguli crurum exterioris ad tangentem&#xA;anguli, quo minor interiorum semisse dicti retiqui minor est, aut&#xA;major, major.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My Latin is too rusty to provide an exact translation but with the help of Google translate the first part seems to say that the ratio of half the difference to half the sum of two adjacent sides is the same as the ratio of the tangents of half the (appropriately chosen) angular difference and sum, respectively.&lt;/p&gt;&#xA;&lt;p&gt;Braunmühl in &lt;a href=&quot;https://archive.org/details/vorlesungenber00brauuoft/page/n189/mode/2up?view=theater&amp;amp;q=Fink&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vorlesungen über Geschichte der Trigonometrie&lt;/a&gt; states on p. 178 that the modern version of the law (i.e. its formulation as an equation) is due to &lt;a href=&quot;https://www.google.co.uk/books/edition/Variorum_de_rebus_mathematicis_responsor/7_BCAAAAcAAJ?hl=en&amp;amp;gbpv=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vieta (1593)&lt;/a&gt; and comments (my translation from German):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In this elegant form the equation appears here [Vieta, 1593] for the&#xA;first time, while it has been put forward in much more complicated&#xA;terms by Finck[e] ten years earlier who has always, and correctly so,&#xA;been credited with its discovery.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Schönbeck (loc. cit.) also says that the underlying problem can be traced back to Ptolemy, again referring to Braunmühl's first volume (loc. cit., p. 23), where one can also find the statement that the ancient &amp;quot;Greeks did not know the tangent&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastActivityDate="2023-03-24T19:11:05.287" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15253" PostTypeId="1" AcceptedAnswerId="15258" CreationDate="2023-03-25T01:34:17.487" Score="17" ViewCount="4409" Body="&lt;p&gt;In 1957, Littlewood wrote that these are measurements of time in astronomy.&#xA;Astronomers operate with times intervals between astronomical events 1000 years apart with accuracy 1/1000 of a second. This gives 14 significant digits. Has this situation changed since?&lt;/p&gt;&#xA;&lt;p&gt;I mean ONLY the numbers related to physical phenomena,&#xA;not pure mathematical calculations outcomes like, for example, the digits of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To make it more precise, I am not asking about the&#xA;numbers like &lt;span class=&quot;math-container&quot;&gt;$3.256\cdot 10^{-19}$&lt;/span&gt;. I am asking about numbers with 14 or more significant digits&#xA;(that is all zeros on the left of the dot must be discarded).&lt;/p&gt;&#xA;&lt;p&gt;Ref. J. E. Littlewood, Mathematical Miscellany, London 1957, &amp;quot;Large numbers&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-03-26T00:41:59.927" LastActivityDate="2023-04-14T17:34:25.973" Title="What are the most precise measurements in science?" Tags="|science|" AnswerCount="6" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15254" PostTypeId="2" ParentId="15253" CreationDate="2023-03-25T02:04:57.257" Score="27" Body="&lt;p&gt;The answer is optical atomic clocks which are now pushing 19 digits of precision on the accuracy of the estimation of atomic transition frequencies. Citations to follow when I have time to add them. These technologies were enabled by advanced in lasers, laser cooling, atomic physics, and optical frequency combs.&lt;/p&gt;&#xA;&lt;p&gt;Note these are state of the art atomic clocks. The atomic clocks used for establishing the TAI timescale are still sitting around 15 digits of precision on their accuracy of realizing the SI definition of the second.&lt;/p&gt;&#xA;&lt;p&gt;The most accurate measurement which I am aware of is of the clock transition frequency in one of the &lt;span class=&quot;math-container&quot;&gt;$\text{Al}^+$&lt;/span&gt; quantum logic ion atomic clocks at NIST. See &lt;a href=&quot;https://arxiv.org/abs/1902.07694&quot; rel=&quot;noreferrer&quot;&gt;Brewer et. al. (2019)&lt;/a&gt;. Here the authors interrogate a &lt;span class=&quot;math-container&quot;&gt;$\text{Al}^+$&lt;/span&gt; ion with a laser beam and they estimate that the ratio of their uncertainty in the lasers frequency &lt;span class=&quot;math-container&quot;&gt;$\Delta \nu$&lt;/span&gt; to the laser frequency &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{\Delta \nu}{\nu} = 9.4\times 10^{-19}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is their error budget:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rVuL6.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rVuL6.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;We see the leading term is excess micromotion. This refers to the fact that the ion feels a force due to the RF fields used to trap it in place. This force causes the ion to oscillate, and this oscillation of the ion results in Doppler shifts of the ions actual transition frequency. The authors estimate the magnitude of this shift to be at the &lt;span class=&quot;math-container&quot;&gt;$-45.8 \times 10^{-19}$&lt;/span&gt; level but their uncertainty is at the &lt;span class=&quot;math-container&quot;&gt;$5.9 \times 10^{-19}$&lt;/span&gt; level, leaving it as the leading term in their error budget.&lt;/p&gt;&#xA;&lt;p&gt;One question that jumps to my mind that I'm still researching the answer on: Why report the fractional uncertainty only without reporting that actual &lt;span class=&quot;math-container&quot;&gt;$\text{Al}^+$&lt;/span&gt; clock transition frequency out to 19 digits? The answer is related to the fact that our current absolute frequency standards are based on &lt;span class=&quot;math-container&quot;&gt;$\text{Cs}$&lt;/span&gt; atomic clocks which only have a fractional frequency uncertainty at the &lt;span class=&quot;math-container&quot;&gt;$10^{-16}$&lt;/span&gt; level or so. If you want to know the absolute frequency of any oscillator you must have some chain of clock frequency comparisons eventually tying you back to &lt;span class=&quot;math-container&quot;&gt;$\text{Cs}$&lt;/span&gt; clocks. But I believe, because the &lt;span class=&quot;math-container&quot;&gt;$\text{Cs}$&lt;/span&gt; clocks are limited to &lt;span class=&quot;math-container&quot;&gt;$10^{-16}$&lt;/span&gt; stability, it is not possible to know ANY absolute frequency to better than &lt;span class=&quot;math-container&quot;&gt;$10^{-16}$&lt;/span&gt;. Folks at NIST are considering a road map towards transitioning from &lt;span class=&quot;math-container&quot;&gt;$\text{Cs}$&lt;/span&gt; based definition of the second towards a optical-clock based definition of the second so that we could all reap the benefits of a realization of the SI second with better fractional frequency stability. This is the same case that is always made to redefine the second to take advantage of more stable and accurate clocks. e.g. when we switched from quartz oscillators to atomic clocks in the first place, or from astronomical measures of time towards clock-based measures. See &lt;a href=&quot;https://iopscience.iop.org/article/10.1088/1681-7575/aaa302&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt; for more details about the roadmap to redefine the second going forward. Right now it is not easy to make an optical atomic clock, this is a major impediment towards adopting this technology as the primary frequency reference.&lt;/p&gt;&#xA;&lt;p&gt;LIGO gravitational wave detection is probably a good honorable mention for most precise measurements in science, but I think optical clocks still have it beat. Experts in both fields may have more to say.&lt;/p&gt;&#xA;" OwnerUserId="18193" LastEditorUserId="18193" LastEditDate="2023-03-25T03:46:20.483" LastActivityDate="2023-03-25T03:46:20.483" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15255" PostTypeId="2" ParentId="15253" CreationDate="2023-03-25T03:26:25.200" Score="13" Body="&lt;p&gt;The most precise measurements tend to be measurements of ratios or differences.  For example, we know the ratio of the charge on the proton to the charge on the electron to 21 decimal places, despite only knowing what the charge on an electron is* to nine decimal places.  Similarly, the &lt;a href=&quot;https://en.wikipedia.org/wiki/LIGO&quot; rel=&quot;nofollow noreferrer&quot;&gt;LIGO observatories&lt;/a&gt; can measure changes in the distance between the mirrors of one part in 10&lt;sup&gt;22&lt;/sup&gt;, despite only having a rough idea of how far apart the mirrors actually are.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;*If you're working in the 2019 version of the SI system, the elementary charge is a fixed numeric value, and you're actually measuring a combination of other quantities.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="3186" LastEditorUserId="3186" LastEditDate="2023-03-27T20:56:35.417" LastActivityDate="2023-03-27T20:56:35.417" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15256" PostTypeId="2" ParentId="15241" CreationDate="2023-03-25T13:59:00.973" Score="5" Body="&lt;p&gt;I agree that the reference in Wikipedia seems at first sight to be off.&lt;/p&gt;&#xA;&lt;p&gt;But if one reads a little further into p.184 (frame 192 of 401 in the archive.org display) of vol.2 of &lt;a href=&quot;https://archive.org/details/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996/Rashed%20R.-Encyclopedia%20of%20the%20History%20of%20Arabic%20Science.%202-Routledge%20%281996%29/page/n191/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roshdi Rashed (ed.) 1996, Encyclopedia Of The History Of Arabic Science&lt;/a&gt; -- if the archive viewer shows 'nan' instead of a page number then just click on 'vol.2' in the left panel and scroll to frame 192/page 184 --then one sees an explanation of how Ibn Muʿādh (11th-c.) derived a formula which is indeed the so-called 'law of tangents', differing only by small points of notation, not in substance, from the 'law of tangents' formulae given elsewhere:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\tan\left(\frac{x + y}{2}\right) = \frac{a+b}{a-b}\tan\left(\frac{\alpha}{2}\right).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; has been defined as &lt;span class=&quot;math-container&quot;&gt;$x - y$&lt;/span&gt; , and &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; (the angles) are opposite &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; (the sides).&lt;/p&gt;&#xA;&lt;p&gt;The derivation in the book appears to start from the sine theorem, and indeed the tangent theorem itself can be considered as essentially a rearrangement (taking a bit of work of trig. expansion to get there!) of the sine theorem.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="4201" LastEditDate="2023-03-26T17:34:46.317" LastActivityDate="2023-03-26T17:34:46.317" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15258" PostTypeId="2" ParentId="15253" CreationDate="2023-03-26T00:25:14.713" Score="10" Body="&lt;p&gt;The periods of millisecond pulsars are measured to 15 significant figures.  E.g., &lt;a href=&quot;https://iopscience.iop.org/article/10.1088/0004-637X/743/2/102&quot; rel=&quot;noreferrer&quot;&gt;here's a paper&lt;/a&gt; that gives the period of PSR J1853+1303 as 4.09179744490025(2) ms (the figure in parentheses is the uncertainty in the final digit.)&lt;/p&gt;&#xA;" OwnerUserId="13816" LastActivityDate="2023-03-26T00:25:14.713" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15259" PostTypeId="2" ParentId="15253" CreationDate="2023-03-26T03:56:27.080" Score="5" Body="&lt;p&gt;See &lt;a href=&quot;https://arxiv.org/pdf/1809.03362.pdf&quot; rel=&quot;noreferrer&quot;&gt;Weyers et. al., Advances in the accuracy, stability, and reliability of&#xA;the PTB primary fountain clocks (2018)&lt;/a&gt;. Here they report uncertainties on the PTB CSF2 Cs fountain clock at the &lt;span class=&quot;math-container&quot;&gt;$1.71\times 10^{-16}$&lt;/span&gt; level.&lt;/p&gt;&#xA;&lt;p&gt;In atomic clocks an electromagnetic oscillator (e.g. a laser beam or microwave tone) is made to interrogate an atomic transition. In this way the frequency of the electromagnetic oscillator can be compared to the frequency of the atomic transition.&lt;/p&gt;&#xA;&lt;p&gt;The definition of the SI second is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical&#xA;value of the caesium frequency, ∆νCs, the unperturbed ground-state hyperfine&#xA;transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in&#xA;the unit Hz, which is equal to s−1.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now the Cs atoms in a real experiment do not exhibit an &amp;quot;unperturbed ground-state hyperfine transition&amp;quot;. There are various systematic effects that shift these frequencies. But these systematic effects are studied so that they can be quantified and subtracted off. But the estimates of the systematic effects have uncertainty. This is why atomic clocks have finite uncertainty, despite the Cs hyperfine frequency literally being the definition of time.&lt;/p&gt;&#xA;&lt;p&gt;In the linked reference, apparently the net uncertainty on these systematic effects was limited to &lt;span class=&quot;math-container&quot;&gt;$1.71 \times 10^{-16}$&lt;/span&gt; fractional frequency uncertainty for CSF2.&#xA;We have&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;9,192,631,770 \text{ Hz }\times 1.71 \times 10^{-16} \approx 1.57 \text{ }\mu\text{Hz}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This means we can take this to mean that we can consider the atomic clock to be a measurement of the frequency of the microwave oscillator which yields a frequency measurement of&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;9,192,631,770.0000000(16) \text{ Hz} \approx 9,192,631,770.000000(2) \text{ Hz}&#xA;$$&lt;/span&gt;&#xA;The proper way to express how precise this measurement is is using the fractional uncertainty reported as &lt;span class=&quot;math-container&quot;&gt;$1.71 \times 10^{-16}$&lt;/span&gt;. I don't like sig figs since they can lead to confusion, but nonetheless, printed this way we can see this measurement of the microwave oscillation frequency was performed with a precision of 16 significant figures.&lt;/p&gt;&#xA;&lt;p&gt;I don't claim this to be the most precise measurement ever since optical atomic clocks have yielded much lower fractional uncertainties. But, unless the definition of the second is changed, those clocks suffer from a technicality that prevents us from printing out the result of their measurement as a unitful physical quantity with more than 16 significant figures.&lt;/p&gt;&#xA;" OwnerUserId="18193" LastActivityDate="2023-03-26T03:56:27.080" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15260" PostTypeId="1" CreationDate="2023-03-26T18:23:09.410" Score="3" ViewCount="239" Body="&lt;p&gt;P. 220-221 of volume 12 of Gauss's werke contain a complete list of the prize problems which Gauss suggested to the Goettingen university in the years 1830, 1834, 1842 and 1849. Those prize problems can shed some light on his intentions for further studies of several mathematical questions whose solution he did not complete. Here is a Google translation (from latin) of some of the problems in the list:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1830, May 6. [1] Derive the general criterion for the solvability of the differential trinomial &lt;span class=&quot;math-container&quot;&gt;$pdx^2+2qdxdy+rdy^2$&lt;/span&gt; into two factors, each of which is a complete differential. [2] Determine the line joining two given points which, when rotated about a given axis, produces a surface of smallest area. [3] To explain the character of the curve, in which the radius of curvature is everywhere reciprocally proportional to the length of the curve. [4] To progress the current state of our knowledge about the periodically variable stars. [5] Demonstrate the equality between the infinite series &lt;span class=&quot;math-container&quot;&gt;$$1+(\frac{1}{2})^3x+(\frac{1\cdot3}{2\cdot4})^3x^2+(\frac{1\cdot3\cdot5}{2\cdot4\cdot6})^3x^3+etc$$&lt;/span&gt; and the square of the infinite series &lt;span class=&quot;math-container&quot;&gt;$$1+(\frac{1}{4})^2x+(\frac{1\cdot5}{4\cdot8})^2x^2+(\frac{1\cdot5\cdot9}{4\cdot8\cdot12})^2x^3+etc$$&lt;/span&gt;&#xA;1834, May 21.&#xA;[1] Determine the moment of inertia of the five Platonic solids with respect to any axis through the center.[2] To explain the various methods of solving Kepler's problem, especially by means of infinite series, as well as addressing the degree of convergence of these developments.&#xA;1842, May 23. [1] To understand the methods for finding any number of right angled spherical triangles whose sides and angles have rational sines and cosines.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that I have omitted a few problems for which the translation was not good enough.&lt;/p&gt;&#xA;&lt;p&gt;Comments:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Problem [2] from 1830 concerns &amp;quot;minimal surfaces&amp;quot; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Minimal_surface_of_revolution&quot; rel=&quot;nofollow noreferrer&quot;&gt;was solved&lt;/a&gt; by Gauss's pupil Benjamin Goldschmidt. Since the solution of this problem (the catenary) was already well known during the 18th century (L. Euler discovered it), I guess Gauss's real intention was to make the methods related to minimal surfaces more rigorous.&lt;/li&gt;&#xA;&lt;li&gt;Problem [3] from 1830 concerns the so-called &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euler_spiral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler spiral&lt;/a&gt;&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Problem [4] from 1830 concerns, according to other sources I read ,the construction of a reliable &amp;quot;photometer&amp;quot; (this is the subject of another question I posted long ago: &lt;a href=&quot;https://hsm.stackexchange.com/questions/3789/what-were-the-scientific-principles-of-gausss-photometer&quot;&gt;What were the scientific principles of Gauss&amp;#39;s photometer?&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;Problem [5] from 1830 was not set as a prize problem by the Goettingen university (it remained merely a suggestion). However, Gauss apparently solved it by himself in &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN236018647?tify=%7B%22pages%22%3A%5B197%5D%2C%22pan%22%3A%7B%22x%22%3A0.569%2C%22y%22%3A0.688%7D%2C%22view%22%3A%22thumbnails%22%2C%22zoom%22%3A0.383%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;p.191-193&lt;/a&gt; of volume 10 of his werke; he apparently shows that when &lt;span class=&quot;math-container&quot;&gt;$x=1$&lt;/span&gt; than the first series and the square of the second series are both equal to &lt;span class=&quot;math-container&quot;&gt;$2\frac{\varpi^2}{\pi^2}$&lt;/span&gt;. It seems that Gauss's identity is a case of &lt;a href=&quot;https://en.wikipedia.org/wiki/Clausen%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clausen's formula&lt;/a&gt; (Clausen deduces the same identity in &lt;a href=&quot;https://www.digizeitschriften.de/id/243919689_0003%7Clog1?tify=%7B%22pages%22%3A%5B93%5D%2C%22pan%22%3A%7B%22x%22%3A0.483%2C%22y%22%3A0.493%7D%2C%22view%22%3A%22thumbnails%22%2C%22zoom%22%3A0.553%7D&amp;amp;origin=%2Fsearch%3Ffilter%255BZeitschriften%255D%255B1%255D%3D243919689%257Clog1%26filter%255BObjekttyp%255D%255B1%255D%3Dvolume&quot; rel=&quot;nofollow noreferrer&quot;&gt;his article&lt;/a&gt;); this raises the question why Gauss suggested to set this as a prize problem if Thomas Clausen has already proved it in 1828.&lt;/li&gt;&#xA;&lt;li&gt;Problem [1] from 1834 was solved by the german mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Feodor_Deahna&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feodor Deahna&lt;/a&gt; in his prize awarded essay from 1835 &amp;quot;Momenta inertiae singulorum quinque corporum regularium&amp;quot;. Apart from the fact that the Platonic solids are spherical tops, I find this problem to be also very interesting.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What is most interesting to me is problem [1] from 1830 - it deals with a differential form (such as the first and second fundamental forms), therefore it seems that it is concerned with a certain problem of differential geometry. Since it was only 2 years after Gauss's groundbreaking memoir on differential geometry, I guess it is reasonable to believe it might be related to some differential-geometric problem that occupied Gauss's mind during this time.&lt;/p&gt;&#xA;&lt;p&gt;So my main question is about the meaning of problem [1] from 1830, although any interesting/useful comments on any of the other problems will be blessed!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2023-04-05T15:22:50.267" LastActivityDate="2023-04-05T15:22:50.267" Title="Motivating several of Gauss's suggestions for prize problems in the years 1830, 1834" Tags="|biographical-details|gauss|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15261" PostTypeId="2" ParentId="2465" CreationDate="2023-03-27T12:47:08.473" Score="0" Body="&lt;p&gt;There are researchers both for and against the concept of relativistic mass even to this day. In fact, Lorentz was the first to propose the idea of relativistic mass in the form of m&lt;em&gt;gamma=m/Sqrt(1-v^2/c^2) back in 1899. Later, in his 1905 paper, Einstein suggested two forms of relativistic mass, mgamma^2 and m&lt;/em&gt;gamma^3, but neither of these formulas are used today. Despite this, some researchers still use the Lorentz relativistic mass. However, Einstein himself eventually abandoned the concept altogether, which is one reason why the general relativity community has been largely negative towards it.&lt;/p&gt;&#xA;&lt;p&gt;Although some prominent physicists, such as Wolfgang Rindler, have been positive towards relativistic mass, neither the proponents nor the opponents have investigated the implications of Lorentz relativistic mass.&lt;/p&gt;&#xA;&lt;p&gt;Personally, I believe it was a mistake to abandon the concept of Lorentz relativistic mass without fully investigating it. Taking it into account may eliminate the mathematical possibility of wormholes and provide a theoretical explanation for Supernova type Ia without relying on the hypothesis of dark energy. There are papers available online discussing this. Ignoring the concept of Lorentz relativistic mass could lead to missed opportunities for scientific advancement. Therefore, discussions surrounding relativistic mass are ongoing, and this question is far from settled. There have been a series of recently published papers discussing relativistic mass.&lt;/p&gt;&#xA;&lt;p&gt;The paper by T. R. Sandin &amp;quot;In defense of relativistic mass&amp;quot; American Journal of Physics 59, 1032 (1991); &lt;a href=&quot;https://doi.org/10.1119/1.16642&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1119/1.16642&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;is discussing the arguments against relativistic mass, it becomes apparent that they are generally weak. In his latest book on mass, Jammer dedicates an entire chapter to examining the pros and cons of these arguments that have been presented in the literature.&lt;/p&gt;&#xA;&lt;p&gt;One can have different views here, but the debate for or against relativistic mass is clearly not over.&lt;/p&gt;&#xA;" OwnerUserId="16445" LastEditorUserId="16445" LastEditDate="2023-03-28T17:35:06.160" LastActivityDate="2023-03-28T17:35:06.160" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15262" PostTypeId="2" ParentId="15253" CreationDate="2023-03-27T22:39:46.543" Score="4" Body="&lt;p&gt;If you want something less exotic than the above, I currently work at a company where we are using laser interferometry to measure abnormalities on the surface of optics with nanometer accuracy.&lt;/p&gt;&#xA;&lt;p&gt;That's only 9 significant digits, but this is commercially applied science, and it is already a scale at which you can observe astonishing things, for example how a &lt;em&gt;&lt;strong&gt;massive&lt;/strong&gt;&lt;/em&gt; 2-inch thick disk of glass, supported at 3 points, sags in the unsupported regions due to its own weight. We have to use various exotic (to me) mechanisms to subtract the error due to gravity and be left with the real abnormality.&lt;/p&gt;&#xA;" OwnerUserId="17998" LastEditorUserId="17998" LastEditDate="2023-04-04T10:39:12.680" LastActivityDate="2023-04-04T10:39:12.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15263" PostTypeId="1" CreationDate="2023-03-28T15:55:07.087" Score="1" ViewCount="109" Body="&lt;p&gt;By a &lt;em&gt;complex analytic space&lt;/em&gt; I mean what Wikipedia calls a “&lt;a href=&quot;https://en.wikipedia.org/wiki/Complex_analytic_variety#Definition&quot; rel=&quot;nofollow noreferrer&quot;&gt;complex analytic variety&lt;/a&gt;” (complex analytic spaces need not be Hausdorff). I am trying to learn about the history of this notion.&lt;/p&gt;&#xA;&lt;p&gt;In H. Grauert, R. Remmert, &lt;em&gt;Coherent Analytic Sheaves&lt;/em&gt;, Chapter 1, §1.9, there is an account of the genesis of the concept. I will reproduce here the last paragraph of the section. (Note: this book calls “complex space” to a Hausdorff complex analytic space.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Jn6fP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Jn6fP.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The last cite is&lt;/p&gt;&#xA;&lt;p&gt;[Gr₂] Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. Inst. Hautes Etudes Sci. N° 5, 233-292 (1960).&lt;/p&gt;&#xA;&lt;p&gt;When one &lt;a href=&quot;http://www.numdam.org/volume/PMIHES_1960__5_/&quot; rel=&quot;nofollow noreferrer&quot;&gt;looks it up&lt;/a&gt;, in Definition 2, p. 9, one finds an equivalent definition to the aforementioned one from wikipedia (up to Hausdorff condition; where a “bimorphe surjektive Abbildung” means an isomorphism of ringed spaces, see Definition 1 from p. 8).&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Is [Gr₂] the first published piece of writing in which (Hausdorff) complex analytic spaces were defined? If not, is there a “first one”?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Do we know if anyone else suggested the definition to Grauert? As it usually happens with the coming of new mathematical ideas, these were already “floating in the air” before anyone decided to finally write them down on a published text. For this reason, I wonder: do we know about prior appearances of the concept (at meetings, seminars, letters) that predate [Gr₂]?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="5031" LastEditorUserId="5031" LastEditDate="2023-03-29T15:03:30.983" LastActivityDate="2023-04-09T12:48:59.533" Title="Who first defined complex analytic spaces?" Tags="|mathematics|complex-geometry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15264" PostTypeId="1" CreationDate="2023-03-29T16:42:38.093" Score="4" ViewCount="131" Body="&lt;p&gt;Inspired by outdated concepts in special relativity such as &lt;a href=&quot;https://hsm.stackexchange.com/questions/2465/when-and-why-did-the-concept-of-relativistic-mass-become-outdated&quot;&gt;When and why did the concept of relativistic mass become outdated?&lt;/a&gt; and &lt;a href=&quot;https://physics.stackexchange.com/questions/121380/special-relativity-and-imaginary-coefficient-of-the-time-coordinate&quot;&gt;Special relativity and imaginary coefficient of the time coordinate&lt;/a&gt;, are there any outdated concepts in general relativity?&lt;/p&gt;&#xA;&lt;p&gt;To be specific, I am looking for mathematical entities in general relativity that are mathematically sound like relativistic mass and use Euclidian metric with time &lt;span class=&quot;math-container&quot;&gt;$t\to it$&lt;/span&gt; to get Minkowski metric in special relativity (see links above), but that are no longer described as such, have been superseded by a better interpretation, or that many physicists have argued against the utility of such a concept.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-03-31T08:36:46.737" LastActivityDate="2023-03-31T08:36:46.737" Title="What are some outdated concepts in general relativity?" Tags="|physics|relativity-theory|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15266" PostTypeId="1" CreationDate="2023-03-29T21:56:46.117" Score="1" ViewCount="39" Body="&lt;p&gt;I often see the claim that impetus, the ancestor of momentum, was understood to be proportional to speed and weight.&lt;/p&gt;&#xA;&lt;p&gt;For example, Jean Buridan wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When a mover sets a body in motion he implants into it a certain&#xA;impetus, that is, a certain force enabling a body to move in the&#xA;direction in which the mover starts it, be it upwards, downwards,&#xA;sidewards, or in a circle. &lt;strong&gt;The implanted impetus increases in the same&#xA;ratio as the velocity.&lt;/strong&gt; It is because of this impetus that a stone&#xA;moves on after the thrower has ceased moving it. But because of the&#xA;resistance of the air (and also because of the gravity of the stone)&#xA;which strives to move it in the opposite direction to the motion&#xA;caused by the impetus, the latter will weaken all the time. Therefore&#xA;the motion of the stone will be gradually slower, and finally the&#xA;impetus is so diminished or destroyed that the gravity of the stone&#xA;prevails and moves the stone towards its natural place. In my opinion&#xA;one can accept this explanation because the other explanations prove&#xA;to be false whereas all phenomena agree with this one.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As seen above, the claim offers no evidence, anecdotes, reasoning, or geometry to support the establishment of the ratio -- it seems completely arbitrary.  For the same reason (or rather lack thereof), why not claim that impetus is proportional to the squares of the speeds or the squares of the weights?&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2023-03-29T21:56:46.117" Title="What justified that impetus was in proportion to speed and weight?" Tags="|aristotle|momentum|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15267" PostTypeId="2" ParentId="14716" CreationDate="2023-03-30T03:06:32.510" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/KD4xd.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/KD4xd.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Only one left (of those visible) to be identified. The man in the middle of C. Chevalley and P. Samuel.&lt;/p&gt;&#xA;&lt;p&gt;The seminar is not very likely to be SGA 5 but SGA 3 (&amp;quot;Schémas en groupes&amp;quot;) by M. Demazure and A. Grothendieck (some day between 1962 and 1964).&lt;/p&gt;&#xA;&lt;p&gt;Photo credits: René Bouillot - Archives IHES&lt;/p&gt;&#xA;" OwnerUserId="18216" LastEditorUserId="18216" LastEditDate="2023-03-30T17:00:35.560" LastActivityDate="2023-03-30T17:00:35.560" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15268" PostTypeId="2" ParentId="15248" CreationDate="2023-03-30T22:25:48.040" Score="7" Body="&lt;p&gt;Prompted by some strange answers from chatGPT, I googled &amp;quot;atomic energy artists 1940s&amp;quot;, which led me via a &lt;a href=&quot;https://www.pinterest.com/pin/571253533995536187/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pinterest collection&lt;/a&gt; of images to Herbert Bayer.&lt;/p&gt;&#xA;&lt;p&gt;Hebert Bayer provided an image for a New York Times article published on August 12, 1945 (six days after the US detonated the atomic bomb over Nagasaki, and three days before the Emperor of Japan announced surrender) titled &amp;quot;We enter a new era - the atomic age&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/swX1Q.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/swX1Q.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://blog.nuclearsecrecy.com/2012/08/17/the-end-of-the-nuclear-age/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://blog.nuclearsecrecy.com/2012/08/17/the-end-of-the-nuclear-age/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The artist, &lt;a href=&quot;https://www.moma.org/artists/399&quot; rel=&quot;nofollow noreferrer&quot;&gt;Herbert Bayer&lt;/a&gt;, worked for General Electric and published a promotional booklet &amp;quot;Electronics, a new science for a new world&amp;quot; in 1942 containing this image:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/r8cbh.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r8cbh.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://twitter.com/lett_arc/status/1373318441474605066&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://twitter.com/lett_arc/status/1373318441474605066&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In a lower resolution view of the double page in the booklet, you can see the Bohr atomic model on the right:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/ANzY1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ANzY1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://www.pinterest.co.uk/pin/57280226491896307/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.pinterest.co.uk/pin/57280226491896307/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;One of the scientists shown in the collage (based on &lt;a href=&quot;https://www.researchgate.net/figure/Langmuir-left-WD-Coolidge-right-from-GE-Research-Schenectady-NY-USA-in-1923_fig5_325342963&quot; rel=&quot;nofollow noreferrer&quot;&gt;this photo&lt;/a&gt;) is Irving Langmuir, a physical chemist who worked at General Electric. He was fully up-to-date on the developments of quantum mechanics, having attended the &lt;a href=&quot;https://rarehistoricalphotos.com/solvay-conference-probably-intelligent-picture-ever-taken-1927/&quot; rel=&quot;nofollow noreferrer&quot;&gt;1927 Solvay conference&lt;/a&gt;. The first Langmuir prize, named after Irving Langmuir and financed by his brother, was awarded to Linus Pauling in Buffalo, NY in 1931 for applying quantum mechanics to chemical bonds. With this high-profile physical chemist at General Electric, it is surprising that Herbert Bayer chose to show electrons in orbits rather than illustrating the more accurate quantum mechanical description that was already used to understand chemistry and electronics.&lt;/p&gt;&#xA;&lt;p&gt;One page of this booklet (reproduced in black and white in the NY Times article) shows some atoms with electrons in orbits of varying radius, while one atom (on the far right over the yellow planet or sun) is the now common depiction with three orbits of equal radius:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/En50G.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/En50G.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://www.worthpoint.com/worthopedia/ge-general-electric-electronics-1860813924&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.worthpoint.com/worthopedia/ge-general-electric-electronics-1860813924&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Another work by Herbert Bayer from 1942 advertises radios (to be available after war-time restrictions) from General Electric, also showing the atomic whirl.&#xA;&lt;a href=&quot;https://i.stack.imgur.com/NqeNg.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NqeNg.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;Source: &lt;a href=&quot;https://miscman.com/product/j874your-coming-radio-as-forecast-by-general-electric/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://miscman.com/product/j874your-coming-radio-as-forecast-by-general-electric/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;While there might be other earlier drawings, the booklet from a major company combined with the image in the New York Times Magazine probably helped popularize this representation of the atom. General Electric continued to use this symbol when they started to advertise its nuclear power plants years later.&lt;/p&gt;&#xA;" OwnerUserId="16431" LastEditorUserId="16431" LastEditDate="2023-04-22T12:55:22.880" LastActivityDate="2023-04-22T12:55:22.880" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15269" PostTypeId="1" CreationDate="2023-03-31T11:14:41.947" Score="2" ViewCount="70" Body="&lt;p&gt;In his 'Vorlesungen über Dynamik', 1842-1843, Jacobi also discusses the papers by William Rowan Hamilton.&lt;/p&gt;&#xA;&lt;p&gt;Jacobi asserted that Hamilton had inadvertendly introduced a superfluous equation, which of course introduced unnecessary complexity. Is there indication that at any point Hamilton became aware of this?&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;David R Wilkins has created transcripts of many of the papers by William Rowan Hamilton.&lt;/p&gt;&#xA;&lt;p&gt;There is a page specifically for links to &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/#GenMethod&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hamilton's papers on Dynamics&lt;/a&gt; that were published in the Proceedings of the Royal Sociedty&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;Availble at John Hopkins University:&lt;br /&gt;&#xA;A machine searcheable .pdf file with scanned pages of Jacobi's &lt;a href=&quot;https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34041/31151000541619.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vorlesungen über Dynamik&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;First Lecture, page 3&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hamilton hat den schönen Zusammenhang, den er gefunden hat, etwas unzugänglich gemacht und verdunkelt, und zwar dadurch, dass er seine charakteristische Function noch zugleich von einer zweiten partiellen Differentialgleichung abhängen lässt. Die Hinzufügung dieser Bestimmung macht die ganze Entdeckung unnöthig complicirt, da eine genauere Untersuchung zeigt, dass die zweite partielle Differentialgleichung vollkommen überflüssig ist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[My translation]&lt;br /&gt;&#xA;Hamilton has made the beautiful interconnection he found somewhat inaccessible and obscure because he has made his characteristic function dependent on a second partial differential equation. This addition makes the whole discovery unnecessarily complicated because a closer examination shows that the second partial differential equation is completely superfluous&lt;/p&gt;&#xA;&lt;p&gt;19th lecture, page 153&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hamilton [...] steht zugleich noch eine zweite partielle Differentialgleichung auf, welcher V ebenfalls genügen soll.&#xA;Diese kann man aber fortlassen, weil sie sich aus der schon aufgestellten herleiten lässt [...]&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[My translation]&lt;br /&gt;&#xA;Hamilton sets up a second partial differential equation that V must satisfy. This can however be omitted because it can be derived from the already stated equation.&lt;/p&gt;&#xA;&lt;p&gt;[Jacobi proceeds to present that derivation]&lt;/p&gt;&#xA;&lt;br&gt;&#xA;" OwnerUserId="7244" LastActivityDate="2023-03-31T11:14:41.947" Title="Did William Rowan Hamilton respond to the assertions by Jacobi?" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15270" PostTypeId="1" AcceptedAnswerId="15271" CreationDate="2023-03-30T15:08:42.840" Score="29" ViewCount="969" Body="&lt;p&gt;Isaac Newton is known as the discoverer of the FTC (Fundamental Theorem of Calculus), so maybe he wrote the integral symbol and derivative symbol. I know he wrote the derivative symbol as &lt;span class=&quot;math-container&quot;&gt;$\dot y$&lt;/span&gt; but I cannot find the integral symbol he wrote. How did Isaac Newton write the integral symbol?&lt;/p&gt;&#xA;" OwnerUserId="18225" OwnerDisplayName="MIKANkankitsu" LastEditorUserId="4251" LastEditDate="2023-03-31T16:39:54.897" LastActivityDate="2023-03-31T16:39:54.897" Title="How did Isaac Newton write the integral symbol?" Tags="|mathematics|mathematicians|notation|calculus|newton|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15271" PostTypeId="2" ParentId="15270" CreationDate="2023-03-30T18:43:15.640" Score="28" Body="&lt;p&gt;Newton used both vertical bars (&lt;span class=&quot;math-container&quot;&gt;$\overset{|}{x}$&lt;/span&gt;) and rectangles (&lt;span class=&quot;math-container&quot;&gt;$\boxed{x}$&lt;/span&gt;) to denote integrals in his &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03962/1&quot; rel=&quot;noreferrer&quot;&gt;Quadratura curvarum&lt;/a&gt; published in 1704.&lt;/p&gt;&#xA;&lt;p&gt;Here, the bar notation is used on the bottom of &lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-ADD-03962/9&quot; rel=&quot;noreferrer&quot;&gt;page 9&lt;/a&gt; in the linked digital scan (multiple bars would represent multiple integrals):&#xA;&lt;a href=&quot;https://i.stack.imgur.com/EUFxE.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EUFxE.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/#:%7E:text=In%20his%20Quadratura%20curvarum%20of,rectangle%20to%20indicate%20its%20integral.:%7E:text=In%20his%20Quadratura,this%20entry.%5D&quot; rel=&quot;noreferrer&quot;&gt;source&lt;/a&gt; states that the notation did not become popular because the bar could be misinterpreted as a &amp;quot;prime&amp;quot;, and a rectangle was difficult to print at the time.&lt;/p&gt;&#xA;&lt;p&gt;Additional information from post linked by @KurtG. in the comments:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Newton used &lt;span class=&quot;math-container&quot;&gt;$\overset{|}{x}$&lt;/span&gt; to indicate the quantity whos &lt;a href=&quot;https://en.wikipedia.org/wiki/Fluxion&quot; rel=&quot;noreferrer&quot;&gt;fluxion&lt;/a&gt; (a &lt;em&gt;fluent&lt;/em&gt; was a changing value and a &lt;em&gt;fluxion&lt;/em&gt; was its instantaneous rate of change) is &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In fact, it seems this &lt;a href=&quot;https://en.wikipedia.org/wiki/Notation_for_differentiation#Newton%27s_notation_for_integration&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; sums everything up very clearly. Given &lt;span class=&quot;math-container&quot;&gt;$y=f(t)$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\overset{|}{y}=\square y=\boxed{y}=\int y\ \mathrm{d}t.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="18231" OwnerDisplayName="Scene" LastEditorUserId="18231" LastEditDate="2023-03-31T16:37:09.413" LastActivityDate="2023-03-31T16:37:09.413" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15272" PostTypeId="2" ParentId="15270" CreationDate="2023-03-30T19:20:27.687" Score="7" Body="&lt;p&gt;The section II.4 Integral Calculus in &lt;em&gt;&lt;a href=&quot;https://link.springer.com/book/10.1007/978-0-387-77036-9&quot; rel=&quot;noreferrer&quot;&gt;Analysis by Its History&lt;/a&gt;&lt;/em&gt; by E. Hairer and E. Wanner begins with quotations, one of them from a letter by Newton:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Newton, letter to Keill, April 20, 1714:&lt;/strong&gt;&#xA;And whereas &lt;span class=&quot;math-container&quot;&gt;$\mathrm{M^r}$&lt;/span&gt; Leibnits praefixes the letter &lt;span class=&quot;math-container&quot;&gt;$\int$&lt;/span&gt; to the Ordinate of a curve to denote the Summ of the Ordinates or area of the Curve. I did some years before represent the same thing by inscribing the Ordinate in a square ...&lt;/p&gt;&#xA;&lt;p&gt;My symbols therefore ... are the oldest in the kind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4084" OwnerDisplayName="epi163sqrt" LastActivityDate="2023-03-30T19:20:27.687" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15273" PostTypeId="2" ParentId="15248" CreationDate="2023-03-31T14:36:50.430" Score="4" Body="&lt;p&gt;I'm  adding this as an extended comment, taking advantage of the answer modality. It changes the subject a little, since the &lt;em&gt;&lt;strong&gt;&amp;quot;atomic&amp;quot; fib&lt;/strong&gt;&lt;/em&gt; is the 500lb gorilla in the room: The electron pseudo-planetary orbits of your iconic picture, indelibly connected in the popular imagination with nuclear energy, &lt;em&gt;not chemistry!&lt;/em&gt;, have little to do, conceptually, with nuclear physics, energy, or weapons. Perhaps a separate question on the fib would attract interesting answers...&lt;/p&gt;&#xA;&lt;p&gt;The symbolism for &amp;quot;atomic energy&amp;quot;, &amp;quot;atomic bomb&amp;quot;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Atomic_Age&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;atomic age&amp;quot;&lt;/a&gt;, etc... so misnamed and established in the popular consciousness after the &lt;a href=&quot;https://en.wikipedia.org/wiki/Smyth_Report&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smyth Report (1945)&lt;/a&gt; and the establishment of the &lt;a href=&quot;https://en.wikipedia.org/wiki/United_States_Atomic_Energy_Commission&quot; rel=&quot;nofollow noreferrer&quot;&gt;United States Atomic Energy Commission (1946)&lt;/a&gt;,  deliberately avoid &amp;quot;nuclear&amp;quot; as a summary of the fundamental processes involved. (Sure, you split the atom after splitting its nucleus, but the dominant processes and energy scales are those of nuclear, not atomic physics.)&#xA;Here is the &lt;a href=&quot;https://www.orau.org/ptp/pdf/smythreport.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;report&lt;/a&gt; .&lt;/p&gt;&#xA;&lt;p&gt;I have heard the story, but lack documentation, that H dW Smyth deliberately avoided the darker &amp;quot;nuclear&amp;quot;  in his report, in favor of &amp;quot;atomic&amp;quot; which had a benign image in popular science reporting of the 20s and 30s, and importantly, the science/fiction &amp;amp; fantasy circles of that time. Whether &lt;em&gt;they&lt;/em&gt; used the planetary picture avatar there, I don't know, but it is in the purview of your question.&lt;/p&gt;&#xA;&lt;p&gt;So, perhaps, the not-quite-ancillary question is of the actual &lt;em&gt;adoption&lt;/em&gt; of that planetary symbol (classical as opposed to  quantum mechanical, so &lt;em&gt;deeply misleading&lt;/em&gt; even for atoms, to boot!) to paradoxically represent and summarize nuclear processes and energy in the popular  imagination and memory.&lt;/p&gt;&#xA;&lt;p&gt;Many nuclear programs all over the world have followed suit and blithely (mis)use that symbol in emulation... The silliest explanation is the one the Bing chatbot provided, that &amp;quot;atomic&amp;quot; refers to the &amp;quot;atomic nucleus&amp;quot;: as though &amp;quot;atomic&amp;quot; is the crucial qualifier, lest one contemplated cellular nuclei?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addendum&lt;/strong&gt; : I have recently &lt;a href=&quot;https://military-history.fandom.com/wiki/Smyth_Report&quot; rel=&quot;nofollow noreferrer&quot;&gt;found&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The original title of the report, before it was published in book form, was &lt;em&gt;Nuclear Bombs: A General Account of the Development of Methods of Using Nuclear Energy for Military Purposes Under the Auspices of the United States Government, 1940–1945&lt;/em&gt;. The word &amp;quot;nuclear&amp;quot; was changed to &amp;quot;atomic&amp;quot; because while the former was favoured by physicists, it was not in common use by the general public at that time. This was the title used on the copyright certificate. The book was copyrighted to Smyth but issued with the statement that &amp;quot;reproduction in whole or in part is authorized and permitted&amp;quot;. Groves had the report copyrighted by Smyth in order to prevent someone else from copyrighting it. [pp 183-185. Smyth, Henry DeWolf (Spring 1976). &lt;em&gt;The &amp;quot;Smyth Report&amp;quot;&lt;/em&gt;. Princeton University Press. pp. 173–190. ISSN 0032—8456].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2023-04-24T16:01:21.397" LastActivityDate="2023-04-24T16:01:21.397" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15274" PostTypeId="1" CreationDate="2023-03-31T19:38:08.203" Score="2" ViewCount="62" Body="&lt;p&gt;Why Clausius defined entropy this way?&lt;/p&gt;&#xA;" OwnerUserId="18235" LastEditorUserId="6609" LastEditDate="2023-04-02T13:08:54.620" LastActivityDate="2023-04-02T13:08:54.620" Title="Why would Clausius define the change in entropy $∆S$ as $Q/T$?" Tags="|theoretical-physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15275" PostTypeId="1" AcceptedAnswerId="15276" CreationDate="2023-04-01T11:33:46.830" Score="3" ViewCount="182" Body="&lt;p&gt;I am wondering what are good examples of substances that were initially thought to be elements but then were found out to be compounds. How exactly were these substances found to be compounds?&lt;/p&gt;&#xA;" OwnerUserId="6965" LastActivityDate="2023-04-01T20:53:10.330" Title="Are there substances that were initially thought to be elements but are actually compounds?" Tags="|chemistry|alchemy|periodic-table|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15276" PostTypeId="2" ParentId="15275" CreationDate="2023-04-01T20:13:40.550" Score="5" Body="&lt;p&gt;&amp;quot;Are there substances that were initially thought to be elements but are actually compounds?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Good candidates for &amp;quot;substances initially thought to be elements but actually compounds&amp;quot; arguably are some of the substances classed by the phlogiston theory as &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Calx&quot; rel=&quot;noreferrer&quot;&gt;calx&lt;/a&gt;&amp;quot;. The calces included substances now known as metallic oxides. According to modern chemical evidence these are certainly compounds.  But as seen by  Georg Ernst Stahl (1660-1734), proposer of the phlogiston theory (building on ideas of his predecessor Joann Joachim Becher, 1635-c.1682-5), metals were composed of calx and phlogiston.  (See [https://www.britannica.com/science/phlogiston].)  The phlogiston theory gave a view about processes such as combustion and rusting that was nearly the opposite of later and modern theories.  It saw such processes not as a combination with oxygen (then still-undiscovered), but as something involving the escape of phlogiston.&lt;/p&gt;&#xA;&lt;p&gt;Combustible materials, including charcoal, were supposedly rich in phlogiston: so charcoal-smelting, e.g. of iron or copper ores or oxides, was supposed by the phlogiston theory to combine the calx with phlogiston to compose and yield the metal.&lt;/p&gt;&#xA;&lt;p&gt;It does not appear that the phlogistonists thought that a calx could itself be decomposed into anything more elementary. Their idea that a calx could be formed by the escape of phlogiston -- from a supposedly&#xA;compound material such as a metal --  shows that they regarded the calx as in some sense more elementary than the metal. It may be difficult to find a specific statement that the calx, as conceived by the phlogistonists, was an (absolutely) elementary substance.&lt;/p&gt;&#xA;&lt;p&gt;But from the mid-1600s to later 1700s the concept of element underwent evolution away from more ancient ideas. Already in 1661 Robert Boyle gave  in  &lt;a href=&quot;https://www.gutenberg.org/ebooks/22914&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;The Sceptical Chymist&amp;quot;, London, 1661&lt;/a&gt; (in part 6) a definition of an element that approaches the modern idea:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;And to prevent mistakes, I must advertize You, that I now mean by&#xA;Elements, as those Chymists that speak plainest do by their&#xA;Principles, certain Primitive and Simple, or perfectly unmingled&#xA;bodies; which not being made of any other bodies, or of one another,&#xA;are the Ingredients of which all those call'd perfectly mixt Bodies&#xA;are immediately compounded, and into which they are ultimately&#xA;resolved. ...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to E. J. Dijksterhuis, Boyle's idea gradually gained influence, and &amp;quot;After the appearance of 'The Sceptical Chymist', Aristotle’s doctrine of the four elements as well as Paracelsus’ theory of the three principia gradually passes into disuse.&amp;quot; &lt;a href=&quot;https://archive.org/details/mechanizationofw0000ejdi_z0a4&quot; rel=&quot;noreferrer&quot;&gt;E. J. Dijksterhuis , transl. C. Dikshoorn  (1961) 'The Mechanization of the World Picture',  Oxford UP : at 435&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Boyle did not commit himself to state which substances ('bodies') he regarded as qualifying to be Elements under his definition.  But from the indications outlined above, it seems clear that as between the calx (i.e. oxide, now recognized as a compound) of a metal (now recognized as an element) and the metal itself, the calx was regarded on the contrary by the phlogistonists as the more elementary component, out of which the metal, clearly in their view a compound, was composed by the addition of phlogiston.&lt;/p&gt;&#xA;&lt;p&gt;Arguably, therefore, the phlogistonists' view of the calx, now recognized as certainly a compound, may well qualify as an instance of the modern idea of element as well as of the idea of 'Element' proposed by Boyle.&lt;/p&gt;&#xA;&lt;p&gt;After the discovery of oxygen in the 1770s, the facts about combustion and rusting etc. were reinterpreted, so that metals were then clearly seen to be at least more elementary than their oxides. I have not seen any contemporary considerations, whether theoretical or experimental, of the question whether the metals were only relatively more elementary, or absolutely elementary.  Lavoisier in 1789 included the metals as elements in his new table (&lt;a href=&quot;https://www.rsc.org/periodic-table/history/about&quot; rel=&quot;noreferrer&quot;&gt;https://www.rsc.org/periodic-table/history/about&lt;/a&gt;), but it seems possible that the evidence on which he classed the metals as elements might not have been exhaustive of the possibility that they or some of them might be further decomposed.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2023-04-01T20:53:10.330" LastActivityDate="2023-04-01T20:53:10.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15277" PostTypeId="1" CreationDate="2023-04-02T13:07:54.877" Score="0" ViewCount="82" Body="&lt;p&gt;I am looking to understand the impact of finite temperature quantum field theory. specifically low energy version applied to condensed matter. Certainly it helps to treat many-body systems a fundamentally as possible, however under finite temperature most of these quantum effects are negligible or can be treated without field theory.&lt;/p&gt;&#xA;&lt;p&gt;I am looking the historical context of formalisms like that of Matsubara, Keldysh and KMS. I think these concepts has become much more settled now than a few decades ago. Here is &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Warszawa_Centrum_Nowych_Technologii_UW-6.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;a picture&lt;/a&gt; of a wall in the Warsaw University's Centre of New Technologies where the KMS condition is engraved as one of the main equations of science.&lt;/p&gt;&#xA;&lt;p&gt;Were there historical problems in condensed matter that needed to be fully treated under a quantum field version? How these different versions developed? Where them inspired from relativistic quantum field theory first? Any general account of the matter would be helpful.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-04-02T16:12:59.687" LastActivityDate="2023-04-05T11:31:44.477" Title="What is the history of $T\neq 0$ quantum field theory?" Tags="|thermodynamics|quantum-field-theory|solid-state-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15278" PostTypeId="1" AcceptedAnswerId="15283" CreationDate="2023-04-02T15:26:41.803" Score="3" ViewCount="198" Body="&lt;p&gt;My understanding is that he counted the number of days between every other equinox. The first year he would have counted 365 days, but after 4 years the tally would have reached 1461 days. Dividing this by 4 he would have had an average length of 365.25 days per year. Using records from older Babylonian and Greek astronomers he would have been able to achieve a higher accuracy. Am I on the right track?&lt;/p&gt;&#xA;&lt;p&gt;Bonus question: how did he measure the lengths of the seasons?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-04-04T02:50:29.437" Title="How did Hipparchus measure the length of the tropical year?" Tags="|astronomy|archaeoastronomy|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15279" PostTypeId="2" ParentId="8332" CreationDate="2023-04-02T21:46:54.353" Score="2" Body="&lt;p&gt;Paper by Vladimir Dokchitser and Tim Dokchitser is an example:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/abs/2201.04094&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/2201.04094&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18227" LastEditorUserId="10106" LastEditDate="2023-04-03T06:46:39.117" LastActivityDate="2023-04-03T06:46:39.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15280" PostTypeId="2" ParentId="8332" CreationDate="2023-04-03T12:48:46.410" Score="5" Body="&lt;p&gt;I found this&lt;/p&gt;&#xA;&lt;p&gt;Boas, H. P.; Boas, R. P.,&#xA;&amp;quot;Short proofs of three theorems on harmonic functions&amp;quot;.&#xA;&lt;em&gt;Proc. Amer. Math. Soc.&lt;/em&gt; &lt;strong&gt;102&lt;/strong&gt; (1988) 906–908.&lt;/p&gt;&#xA;&lt;p&gt;by Ralph Boas and his son Harold Boas.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I tried and failed to find (in MathSciNet) joint papers by:&lt;br /&gt;&#xA;Harald Bohr and Niels Bohr (brothers)&lt;br /&gt;&#xA;S. L. Synge and Cathleen Morawetz (father and daughter)&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-04-03T12:48:46.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15281" PostTypeId="2" ParentId="8332" CreationDate="2023-04-03T14:50:54.883" Score="5" Body="&lt;p&gt;I don't know if you can be interested also in  papers about philosophy of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;There are some writings  by the famous French mathematician André Weil and  his famous philosopher sister Simone Weil about mathematics.&lt;/p&gt;&#xA;&lt;p&gt;They are part of the correspondence between André and Simone.  They have been  gathered in Simone Weil, André Weil, &lt;em&gt;L’arte della matematica&lt;/em&gt;, Adelphi, 2018 ('The Art of Mathematics'), an Italian translation. The original title, in French,  is &lt;em&gt;Correspondance familiale,&lt;/em&gt; Gallimard, 2012.&lt;/p&gt;&#xA;&lt;p&gt;I didn't find an English translation.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;«As you have time, also too much time»  Simone Weil writes at the&#xA;beginning  of February 1940 to her beloved brother, imprisoned in the&#xA;civil prison  of Le Havre for draft evasion (André thought that his&#xA;duty was «to do mathematics, not war»), «another good occupation could&#xA;be to think about a way to show to a layman like me  what consists of&#xA;the  interest and the scope of your work&amp;quot;. [...]  André who,  at&#xA;first, had answered : «It will be as explaining a symphony to a deaf»,&#xA;[…]ends up accepting.  So, an exchange full of intellectual passion&#xA;and affection begins – which leads them also to clash over capital&#xA;points, as the discovery of incommensurables and the nature of Greek&#xA;science.  And  sister and  brother are capable of talking about&#xA;Pythagoras and the Odyssey, of cardinals skillful in Court strategies,&#xA;and about the importance of Sanskrit , of Dedekind and Gauss…&lt;/p&gt;&#xA;&lt;p&gt;( back cover of &lt;em&gt;L'arte della matematica,&lt;/em&gt; my translation from&#xA;Italian)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;An English translation of a letter of André to Simon Weil is &lt;a href=&quot;https://www.ams.org/notices/200503/fea-weil.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weil, André,  'Letter of 1940 to Simone Weil, &lt;em&gt;Notices of the American  Mathematical Society&lt;/em&gt; , 52, 3, pp. 345-341&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-06-28T12:52:20.097" LastActivityDate="2023-06-28T12:52:20.097" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15282" PostTypeId="1" CreationDate="2023-04-04T00:05:50.957" Score="1" ViewCount="74" Body="&lt;p&gt;In condensed matter, most theories were developed to understand experimental results that could not be understood using classical physics or the contemporaneous theory. One can think for example of black body radiation, photoelectric effects or superconductivity. But aside from its explanatory power, has condensed matter theories ever predicted a new phenomena? What are some good examples?&lt;/p&gt;&#xA;&lt;p&gt;If you look at the edits to this question, it was originally about predictions by many-body field theory (low energy) but I decided instead to convert it into a condensed matter one as I got no reactions, however any examples in this domain are also welcome.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-04-13T16:25:15.720" LastActivityDate="2023-04-14T01:40:57.403" Title="What are some of the greatest predictions of new phenomena in condensed matter theory?" Tags="|solid-state-physics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15283" PostTypeId="2" ParentId="15278" CreationDate="2023-04-04T02:50:29.437" Score="4" Body="&lt;p&gt;A hindrance to answering this question is that we do not have any description from Hipparchus himself on how he derived his numerical results, whether for the length of the year or of the seasons. His only surviving work is a commentary on Aratus and Eudoxus that deals with star catalogues, and tells us nothing about Hipparchus's ideas or work on the sun or moon (see Swerdlow 1980). Nearly all else from Hipparchus is seen through the lens of quotations and secondhand descriptions given by Ptolemy in the Almagest, where quite naturally they are subject to Ptolemy's preferences about content and presentation.&lt;/p&gt;&#xA;&lt;p&gt;The literature offers two possible lines of original derivation of the (estimated) (tropical) year-length adopted both by Hipparchus and by Ptolemy, of  365 ¼ days less 1/300 of a day,  Both derivations yield the same result.&lt;/p&gt;&#xA;&lt;p&gt;(1) One possible line of derivation, clearly available to Hipparchus (see below) comes from combining the month-length used as accurate by Hipparchus together with the Callippic lunisolar cycle (the use of which was also anciently ascribed to Hipparchus).&lt;/p&gt;&#xA;&lt;p&gt;(2) Another line of derivation comes from the collection of observations and measurements of cardinal points of the sun, i.e. solstices and equinoxes, as set out in Ptolemy's Almagest (bk.3, ch.1).&lt;/p&gt;&#xA;&lt;p&gt;Method (1) was suggested by Swerdlow (1980) as a possible origin of the Hipparchan year-length estimate. To summarise his description:-&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy's account in the 'Almagest' shows that Hipparchus used as accurate a synodic month-length&lt;/p&gt;&#xA;&lt;p&gt;(in sexagesimal notation, where e.g. &amp;quot;29;31,50&amp;quot; means &amp;quot;29 + 31/60 + 50/3600&amp;quot;) of:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;29;31,50,8,20 days  = 29 d + 12; 44,03,20 hr ≈ 29.530594(14) days.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Also, Censorinus (3rd-c. AD) wrote in 'De die natali' that 'Hipparchus intercalates 112 times in 304 years', meaning that Hipparchus used a lunisolar cycle where the 304 years include 112 years with 13 (lunar) months and the others with 12 months.  On these figures, Hipparchus's cycle was based on 4 [Callippic] cycles, thus containing  304 = 4 x 76 years  = 4 x 940 months.&lt;/p&gt;&#xA;&lt;p&gt;The first of the possible derivations of Hipparchus's year-length then involves combining the month-length with the cycle, both of which, on the available historical evidence, were available to Hipparchus. The combination works in for example the following way:&lt;/p&gt;&#xA;&lt;p&gt;The original &lt;a href=&quot;https://en.wikipedia.org/wiki/Callippic_cycle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Callippic cycle&lt;/a&gt; assumed 27759 days for its 940 months, implying a month-length of 29.530851(06) days. This is noticeably longer than the month-length used as accurate by Hipparchus. If instead he applied his accurate month-length to the Callippic-derived cycle of months attributed to him by Censorinus, he would have measured the 304-year cycle, of 4 x 940 months, as containing 111035 days (plus something less than an hour) (i.e. 1 day shorter than 4 of the original Callippic cycles).  Dividing by 304 then gives each year of this period in days as follows :-&lt;br /&gt;&#xA;•     111035/304 ≈ 30,50,35; / 5,4;&lt;br /&gt;&#xA;•     ≈ (to 2 places) 6,5;15,0 - 0;0,12 ≈ 365 ¼ d minus 1/300 d ,&#xA;which is indeed the year-length attributed to Hipparchus by Ptolemy.  (No actual new observations would be needed to make this determination, although observations might be made to test or confirm a previously computed value, or to demonstrate how in principle such a quantity might be derived &lt;em&gt;de novo&lt;/em&gt;.)&lt;/p&gt;&#xA;&lt;p&gt;Swerdlow (1980), who published this possible Hipparchan derivation, acknowledged that Tobias Mayer had already made a similar suggestion in 1753 that the year-length of Hipparchus could have originated by combining the Callippic cycle of months with the Hipparchan month-length.  In Mayer's description, there had been no particular reason to suppose that Hipparchus was interested in the Callippic cycle. Swerdlow's account cites ancient historical evidence from Censorinus to show that indeed Hipparchus was concerned with an extended form of the Callippic cycle. This would make the combination with his favoured month-length intrinsically more probable.&lt;/p&gt;&#xA;&lt;p&gt;Method (2), i.e. the method of observing cardinal points of the sun, is the one described by Ptolemy in the 'Almagest'.  The observations extended over 285 years according to Ptolemy, but if one considers only those that Hipparchus himself could have taken into account, the period over which the useful measurements extend is shorter.  In order to consider this method, it is necessary to take account of peculiarities of Ptolemy's presentation, which provides the little that we indirectly know of what Hipparchus did of this kind. The description in the Almagest points out some problems of the method in accuracy, e.g. in the use of a ring mounted in the equatorial plane to detect equinoxes.  A good account of the results is given in Pedersen and Jones (2011).&lt;/p&gt;&#xA;&lt;p&gt;The procedures and results reported by Ptolemy have a centuries-long history of commentary and criticism, and the practical shortcomings and difficulties of the method of observing equinoxes and solstices have generated and reinforced suspicions about the results.&lt;/p&gt;&#xA;&lt;p&gt;It may help consideration of the present question to mention that some recent authors have pointed out that it may be needless and unjustified to suppose that Ptolemy's descriptions were always or necessarily describing the first origin of his numerical results.  His alternative intentions may have been for example:&lt;/p&gt;&#xA;&lt;p&gt;(a) (as pointed out by Swerdlow (1980)) some of the measurements might be explicable on the basis that they were made to test or confirm a previously computed value, or to demonstrate how in principle such a quantity might be derived.&lt;/p&gt;&#xA;&lt;p&gt;(b) Britton (1992) found on statistical treatment that some of Ptolemy's results were considerably more accurate than could have been expected given the small number of the observations and the intrinsic uncertainties involved in each. Britton's conclusion was that in some cases, Ptolemy must have relied on a larger store of data than the items he reported, and sifted from it the values that he reported. Britton conjectured that in the absence of a discipline of statistics, and constrained by his habit and possible need to describe his work in a geometrically rigorous way, Ptolemy had no way that he found satisfactory to justify to his audience the informal mthods he must have used to process the data to get his good results.&lt;/p&gt;&#xA;&lt;p&gt;(c) A recent work (J Feke, 2018) studies Ptolemy's methods and motivations, emphasising that he was not a modern writer, and had philosophical commitments that might in principle cut across anything resembling a modern attitude to data. So there should perhaps be no expectation that data from Ptolemy occupies a place comparable to the place of data in a modern study.&lt;/p&gt;&#xA;&lt;p&gt;The fragmentary state of the evidence prevents clear decision as between possible methods (1) and (2) for Hipparchus's way of deriving his value for the solar (tropical) year. The simplicity of method (1) and its dependence on established and approved results might make it seem the more likely first route to the result. The fact that the result of method (1) is just the same as the reported result of method (2) may in itself cast doubt on whether the steps taken in method (2) were truly independent of the result of method (1).  (If method (1) was actually Hipparchus's, it raises the question why Ptolemy did not simply present it. A possible answer to that question is that Ptolemy's expressed plan for the Almagest was clearly to establish the solar theory and only then to go on to consider the moon. To make the tropical (solar) year appear to depend on lunar and lunisolar quantities as in method (1) would have run counter to that order of presentation.)&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;p&gt;J P Britton (1992), &amp;quot;Models and precision : the quality of Ptolemy's observations and parameters&amp;quot;  (New York &amp;amp; London: Garland).&lt;/p&gt;&#xA;&lt;p&gt;J Feke (2018), &amp;quot;Ptolemy’s Philosophy : Mathematics as a Way of Life&amp;quot;&#xA;(Princeton University Press).&lt;/p&gt;&#xA;&lt;p&gt;O Pedersen (rev.ed. with A Jones) (2011), [&amp;quot;A Survey of the Almagest,&#xA;with annotation and new commentary by Alexander Jones&amp;quot;]&#xA;(&lt;a href=&quot;https://books.google.com/books?id=8eaHxE9jUrwC&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?id=8eaHxE9jUrwC&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;N M Swerdlow (1980), &amp;quot;Hipparchus's Determination of the Length of the Tropical Year and the Rate of Precession&amp;quot;, &lt;a href=&quot;https://www.jstor.org/stable/41133611&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archive for History of Exact Sciences, Vol. 21(4) (1980): 291-309&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;G J Toomer (1984) Ptolemy’s Almagest, translated and annotated by G J Toomer, (London: Duckworth)&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2023-04-04T02:50:29.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15284" PostTypeId="1" CreationDate="2023-04-04T10:39:44.700" Score="1" ViewCount="162" Body="&lt;p&gt;What are some books that talk about the origins of Lagrangian and Hamiltonian mechanics? I would like to dig deeper into the fundamental approach and the fundamentals of action and its principles.&lt;/p&gt;&#xA;" OwnerUserId="18247" LastEditorUserId="6609" LastEditDate="2023-04-05T08:48:09.440" LastActivityDate="2023-04-09T07:57:37.693" Title="Book suggestions on Lagrangian and Hamiltonian mechanics?" Tags="|philosophy-of-science|classical-mechanics|book-recommendation|lagrange|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15285" PostTypeId="1" CreationDate="2023-04-04T22:48:28.903" Score="1" ViewCount="132" Body="&lt;p&gt;Goddard famously was ridiculed in the New York Times about rockets not working in the vacuum of space. Here we see that he theorized that rockets in fact would work in a vacuum: &lt;a href=&quot;https://www.nasa.gov/missions/research/f_goddard.html#:%7E:text=Goddard%20began%20teaching%20physics%20in,air%20in%20order%20to%20fly&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.nasa.gov/missions/research/f_goddard.html#:~:text=Goddard%20began%20teaching%20physics%20in,air%20in%20order%20to%20fly&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But would not Newton have been familiar with gunpowder rockets and also understood that they would work in the vacuum of space? Or did people not realize that space was nearly a vacuum and the question would not even have come up? Or perhaps rockets were so limited in their performance at Newton's time that they idea of using a rocket to reach space would not have occurred to Newton? I am pretty sure he did imagine a cannon firing a projectile into orbit but that is very different than attempting to reach, for example, the moon in this way and also is not rocket propulsion.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="10856" LastEditDate="2023-04-05T03:11:40.600" LastActivityDate="2023-04-07T06:53:20.880" Title="Would Newton have understood that rockets would work in space?" Tags="|space-travel|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15286" PostTypeId="1" CreationDate="2023-04-05T05:08:33.343" Score="2" ViewCount="94" Body="&lt;p&gt;Just recently I learned from a highschooler I help in her Calculus class about a  convergence test for series with positive terms, and and of which  I was not aware. The test is a hybrid between the ratio and root test:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Theorem:&lt;/em&gt; Suppose &lt;span class=&quot;math-container&quot;&gt;$a_n&amp;gt;0$&lt;/span&gt; for all &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;If &lt;span class=&quot;math-container&quot;&gt;$\limsup_n\Big(\frac{a_{n+1}}{a_n}\Big)^n&amp;lt;\frac1e$&lt;/span&gt;, then the  series &lt;span class=&quot;math-container&quot;&gt;$\sum_na_n$&lt;/span&gt; converges.&lt;/li&gt;&#xA;&lt;li&gt;If there is &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; such that for all &lt;span class=&quot;math-container&quot;&gt;$n\geq N$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\Big(\frac{a_{n+1}}{a_n}\Big)^n\geq\frac1e$&lt;/span&gt;, then the series &lt;span class=&quot;math-container&quot;&gt;$\sum_na_n$&lt;/span&gt; diverges.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By working some examples out: &lt;span class=&quot;math-container&quot;&gt;$\sum_n\frac{1}{n}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sum_n\frac{1}{n^p}$&lt;/span&gt;, and the hypergeometric series &lt;span class=&quot;math-container&quot;&gt;$_2F_1(a,b,c;z)$&lt;/span&gt;, one can see that this result is more powerful that the usual ratio test, in fact it at the same hierarchy level as Raabe's test (in the De Morgan hierarchy).&lt;/p&gt;&#xA;&lt;p&gt;My question is if anyone knows of the provenance of this result. Is is some sort of a folklore theorem?&lt;/p&gt;&#xA;&lt;p&gt;My highschooler learned it from the following  &lt;a href=&quot;https://math.fel.cvut.cz/en/mt/txte/2/txe3ea2d.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;, which is the English translation to this other &lt;a href=&quot;https://math.fel.cvut.cz/mt/txte/2/txc3ea2d.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt; in Czech.   I look in several classic Soviet, French and American Calculus textbooks (including translations of the seminal books of Knopp), and the result was not presented neither in the textbook nor the problem companions.&lt;/p&gt;&#xA;" OwnerUserId="18248" LastEditorUserId="18248" LastEditDate="2023-04-05T21:09:57.790" LastActivityDate="2023-04-05T21:09:57.790" Title="A result on convergence of series of positive terms" Tags="|calculus|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15287" PostTypeId="2" ParentId="15277" CreationDate="2023-04-05T11:31:44.477" Score="1" Body="&lt;p&gt;An important early work is L. Dolan and R. Jackiw, &lt;a href=&quot;https://journals.aps.org/prd/abstract/10.1103/PhysRevD.9.3320&quot; rel=&quot;nofollow noreferrer&quot;&gt;Phys. Rev. D 9, 3320 (1974)&lt;/a&gt;, which unfortunately is behind a paywall. In their introduction they refer to earlier work by Kirzhnits and Linde, &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/0370269372901098&quot; rel=&quot;nofollow noreferrer&quot;&gt;Phys. Lett. 42B, 471 (1972)&lt;/a&gt;, and parallel work by Weinberg, &lt;a href=&quot;https://journals.aps.org/prd/abstract/10.1103/PhysRevD.9.3357&quot; rel=&quot;nofollow noreferrer&quot;&gt;Phys. Rev. D 9, 3357 (1974)&lt;/a&gt;. The paper by Dolan and Jackiw has more than 3000 citations and (together with the cited publications and a few others) marks the beginning of the &amp;quot;modern period&amp;quot; of QFT at finite temperature.&lt;/p&gt;&#xA;&lt;p&gt;For work done before the 1970s, they refer to the book by Kadanoff and Baym, Quantum Statistical Mechanics (Benjamin, 1962) which can be read online &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=uc1.31822013863519&amp;amp;view=1up&amp;amp;seq=10&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. A 2019 reprint is described by the publisher (CRC Press, now Taylor and Francis) as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book is a very early systematic treatment of the application of&#xA;the field-theoretical methods developed after the Second World War to&#xA;the quantum mechanical many-body problem at finite temperature. It&#xA;describes various techniques that remain basic tools of modern&#xA;condensed matter physicists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here's a screenshot from Kadanoff and Baym's Supplementary Reading (p.201):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/uhNDx.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/uhNDx.png&quot; alt=&quot;Kadanoff and Baym, p.201&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The parallel development in the Soviet Union was a recurrent theme. At the time of writing, the authors could not have been aware of the important contribution by L. Keldysh, &lt;em&gt;Diagram technique for nonequilibrium processes&lt;/em&gt;, &lt;a href=&quot;http://jetp.ras.ru/cgi-bin/e/index/e/20/4/p1018?a=list&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sov. Phys. JETP. 20, 1018 (1965)&lt;/a&gt;, regarding what is now called the Schwinger-Keldysh method.&lt;/p&gt;&#xA;&lt;p&gt;Of course, all papers cited above contain further references.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastActivityDate="2023-04-05T11:31:44.477" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15288" PostTypeId="1" AcceptedAnswerId="15289" CreationDate="2023-04-06T10:04:27.630" Score="2" ViewCount="461" Body="&lt;p&gt;Hipparchus seems to have known that 94.5 days passed from the winter solstice to vernal equinox. How did he measure this?&lt;/p&gt;&#xA;&lt;p&gt;Equinoxes and solstices don't happen only at noon when the Sun is highest, but can happen at any time, including at night. How could they say when it happened to within a half day precision?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-04-06T17:23:41.837" Title="How did the ancients determine the exact time of equinoxes and solstices?" Tags="|astronomy|archaeoastronomy|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15289" PostTypeId="2" ParentId="15288" CreationDate="2023-04-06T12:29:47.063" Score="3" Body="&lt;p&gt;As Ptolemy describes, for equinoxes, he had a flat bronze ring installed parallel to the equator. When the Sun is one one side of the equator, the corresponding side of the ring is illuminated. The transition moment is the moment of an equinox.&lt;/p&gt;&#xA;&lt;p&gt;The problems with this method: a) the equinox can happen during a night, or when Sun is obscured by the clouds, b) how precisely you can install this ring, c) how precisely you can detect this moment.&lt;/p&gt;&#xA;&lt;p&gt;With solstices it is a bit harder. One needs the moment when the Sun attains its maximum declination. This is essentially done by measuring the height of the Sun over the horizon, with the help of a mural quadrant, described by Ptolemy. Then one has the same problems a,b,c, but the main problem of the method is that near a solstice the declination changes slowly, and it is hard to detect the exact moment when it reaches maximum.&lt;/p&gt;&#xA;&lt;p&gt;An modern analysis of errors of Hipparchus/Ptolemy in these measurements is contained in the thesis of John Philipp Britton, On the quality of solar and lunar observations and parameters in Ptolemy's Almagest, Yale Univ., 1967 (available online).&lt;/p&gt;&#xA;&lt;p&gt;Remark. All this is not of pure historical interest,&#xA;since astronomers tried to determine the actual secular variation of the Earth rotation using ancient observations. This raised the question of actual accuracy of these ancient observations.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-04-06T12:35:53.530" LastActivityDate="2023-04-06T12:35:53.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15290" PostTypeId="2" ParentId="15288" CreationDate="2023-04-06T17:23:41.837" Score="2" Body="&lt;p&gt;Ancient astronomers could find the dates of solstices and equinoxes for many years in a row (except for years they were not observable, and add up the total days between events separated by many years and then divide by the number of years to get an average length in years, which can be converted to an average length in days.&lt;/p&gt;&#xA;&lt;p&gt;Hipparchus and Ptolemy could have had access to Babylonian astronomical observational records going back for many centuries.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2023-04-06T17:23:41.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15291" PostTypeId="1" AcceptedAnswerId="15292" CreationDate="2023-04-06T18:44:39.097" Score="4" ViewCount="763" Body="&lt;p&gt;The quote is as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Nur wenn man nicht auf den Nutzen nach aussen sieht, sondern in der Mathematik selbst auf das Verhältnis der unbenutzten Teile, bemerkt man das andere und eigentliche Gesicht dieser Wissenschaft. Es  ist nicht zweckbedacht, sondern unökonomisch und leidenschaftlich.  ...&#xA;Die Mathematik ist Tapferkeitsluxus der reinen Ratio, einer der wenigen die es heute gibt.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Where did Robert Musil publish this in 1913?&lt;/p&gt;&#xA;" OwnerUserId="8094" LastEditorUserId="8094" LastEditDate="2023-04-07T17:59:55.033" LastActivityDate="2023-04-07T20:18:48.723" Title="Provenance of mathematics quote from Robert Musil, 1913" Tags="|reference-request|quote|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15292" PostTypeId="2" ParentId="15291" CreationDate="2023-04-06T20:00:12.090" Score="8" Body="&lt;p&gt;This quote is from &lt;em&gt;Der mathematische Mensch&lt;/em&gt;  (The mathematical man), which is a short essay by Robert Musil, published in 1913.&lt;/p&gt;&#xA;&lt;p&gt;In this essay Musil makes some considerations about the role and the nature of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;See for instance:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://ucldigitalpress.co.uk/Book/Article/64/88/4827/&quot; rel=&quot;noreferrer&quot;&gt;https://ucldigitalpress.co.uk/Book/Article/64/88/4827/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;An English translation is in Musil, &lt;em&gt;Precision and Soul: Essays and Addresses&lt;/em&gt;, ‎ University of Chicago Press; Reprint edition (1995).&lt;/p&gt;&#xA;&lt;p&gt;I haven't the English translation, nor the German original, I checked the quotation in the Italian traslation of  &lt;em&gt;Der mathematische Mensch&lt;/em&gt;, &lt;em&gt;L'uomo matematico&lt;/em&gt;, in Claudio Bartocci (ed.), &lt;em&gt;Racconti matematici&lt;/em&gt;, Einaudi, 2006, where the quotation is on pages 290-291.&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-04-06T20:00:12.090" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15293" PostTypeId="2" ParentId="15285" CreationDate="2023-04-07T06:53:20.880" Score="5" Body="&lt;p&gt;Not necessarily. People during Newton's era, and later, did not know that outer space was a near vacuum. The concept of an &lt;a href=&quot;https://en.wikipedia.org/wiki/Aether_theories&quot; rel=&quot;noreferrer&quot;&gt;aether/ether&lt;/a&gt; in space as a transporting medium for light was prevalent amongst many European scientists until it was proven that such an aether/ether did not exist. The composition of such an aether/ether was also uncertain.&lt;/p&gt;&#xA;&lt;p&gt;As for Isaac Newton,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Isaac Newton suggests the existence of an aether in the Third Book of Opticks (1st ed. 1704; 2nd ed. 1718): &amp;quot;Doth not this aethereal medium in passing out of water, glass, crystal, and other compact and dense bodies in empty spaces, grow denser and denser by degrees, and by that means refract the rays of light not in a point, but by bending them gradually in curve lines? ...&lt;strong&gt;Is not this medium much rarer within the dense bodies of the Sun, stars, planets and comets, than in the empty celestial space between them?&lt;/strong&gt; And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great bodies towards one another, and of their parts towards the bodies; every body endeavouring to go from the denser parts of the medium towards the rarer?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The emphasis is mine.&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2023-04-07T06:53:20.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15294" PostTypeId="1" AcceptedAnswerId="15295" CreationDate="2023-04-07T19:24:31.057" Score="0" ViewCount="61" Body="&lt;p&gt;I have several times come across the name P C Gilmore, and even made use of a publication by her, or him, viz. &amp;quot;The consistency of partial set theory without extensionality&amp;quot;, which was published in Tomas Jech (ed): Axiomatic Set Theory, Part 2, American  Mathematical Society, 1974, 147-153.&lt;/p&gt;&#xA;&lt;p&gt;May someone reveal what the initials &amp;quot;P&amp;quot; and &amp;quot;C&amp;quot; stand for?&lt;/p&gt;&#xA;" OwnerUserId="8094" LastActivityDate="2023-04-07T19:52:47.390" Title="What's the name of P C Gilmore?" Tags="|biographical-details|reference-request|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15295" PostTypeId="2" ParentId="15294" CreationDate="2023-04-07T19:51:23.470" Score="2" Body="&lt;p&gt;&lt;em&gt;Mathematics Genealogy Project,&lt;/em&gt; which I see  is a service of the NDSU Department of Mathematics, in association with the American Mathematical Society, suggests that his name is Paul Carl Gilmore:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=49610&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathgenealogy.org/id.php?id=49610&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;and this seems confirmed by Mathscinet:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathscinet.ams.org/mathscinet/MRAuthorID/73705&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathscinet.ams.org/mathscinet/MRAuthorID/73705&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-04-07T19:51:23.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15296" PostTypeId="2" ParentId="15294" CreationDate="2023-04-07T19:52:47.390" Score="2" Body="&lt;p&gt;Definetively it is Paul Carl Gilmore, he passad away on April 8, 2015, here an &lt;a href=&quot;https://give.ubc.ca/memorial/paul-gilmore/&quot; rel=&quot;nofollow noreferrer&quot;&gt;obituary&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2023-04-07T19:52:47.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15297" PostTypeId="2" ParentId="15291" CreationDate="2023-04-07T20:01:27.720" Score="3" Body="&lt;p&gt;R. Musil, &amp;quot;Der mathematische Mensch.&amp;quot; In: &lt;em&gt;Der lose Vogel&lt;/em&gt;, Leipzig: K. Wolff 1913, &lt;a href=&quot;https://www.google.com/books/edition/Der_Lose_Vogel/3uQaAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA310&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 310-314&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Der lose Vogel&lt;/em&gt; was intended to be published in monthly installments as a &amp;quot;Monatsschrift&amp;quot;, but it appears the project lost impetus towards the end, and resorted to combining multiple issues into one. Musil's brief essay appeared in combined issues 10, 11, and 12. The quoted text from the question appears on &lt;a href=&quot;https://www.google.com/books/edition/Der_Lose_Vogel/3uQaAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA312&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 312&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It should be noted that for each of the poems and essays in this publication the author is not stated. There is merely a list of all authors in the foreword on &lt;a href=&quot;https://www.google.com/books/edition/Der_Lose_Vogel/3uQaAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PR4&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;page VI&lt;/a&gt; which includes R. Musil among the essay writers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] die Gedichte sind von M. Brod, F. Werfel, R. Walser. Die Aufsätze sind von:&#xA;Alain, H. Beloc, F. Blei, S. Butler, E. v. Gebsattel, G. Hecht, A. Kolb,&#xA;W. Krug, R. Musil, P. Scheffer, M. Scheler, H. Schott, R. Stadler, A. Suares,&#xA;Vogelstein, O. Vrieslander.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The identification of Robert Musil as the author of &amp;quot;Der mathematische Mensch&amp;quot; must have been based on a different source. Maybe his writing style was easily recognized by contemporaries. The earliest reference to Musil's authorship that I can find is in Musil's &amp;quot;Gesammelte Werke&amp;quot; (collected works) edited by Adolf Frisé, published in 1952, where the essay appears at page 592.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-04-07T20:18:48.723" LastActivityDate="2023-04-07T20:18:48.723" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15298" PostTypeId="1" CreationDate="2023-04-08T05:18:43.163" Score="2" ViewCount="133" Body="&lt;p&gt;It's my first question.&#xA;I am studying elliptic function, Among them, I have a question about the Weierstrass elliptic function.&#xA;In the Weierstrass elliptic function, their coefficients are expressed by 'Eisenstein series'. So I was so curious about origin of eisenstein series. When I search about it, 'Eisenstein series' was discovered by 'Kronecker-Weber theorem' and 'Jugendtraum'.&#xA;But I want to know about this more detail.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastActivityDate="2023-08-12T02:54:30.177" Title="Background of 'Eisenstein series'" Tags="|number-theory|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15299" PostTypeId="2" ParentId="15284" CreationDate="2023-04-08T08:25:16.063" Score="6" Body="&lt;p&gt;In my opinion the existing books are not good sources of information.&lt;/p&gt;&#xA;&lt;p&gt;In particular: what is presented as 'historical information' is often incorrect.&lt;/p&gt;&#xA;&lt;p&gt;I believe only the following two categories of information source are reliable:&#xA;-the original works; the primary sources&lt;br /&gt;&#xA;-articles/books by authors who are dedicated historians of science&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maths.tcd.ie/%7Edwilkins/&quot; rel=&quot;nofollow noreferrer&quot;&gt;David R Wilkins&lt;/a&gt; has created many &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Papers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;transcripts of Hamilton's articles&lt;/a&gt;, including the ones on mechanics.&lt;/p&gt;&#xA;&lt;p&gt;It would appear that Wilkins regards the articles that were published in the Proceedings of the Royal Society as the core of Hamilton's work on dynamics.&lt;/p&gt;&#xA;&lt;p&gt;The page titled &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/#GenMethod&quot; rel=&quot;nofollow noreferrer&quot;&gt;On a general method in dynamics&lt;/a&gt; offers links links to those articles.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;However, while the articles published in the Proceedings of the Royal Society do not contain any interpretation, there is another article that does:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/LightPlanets/&quot; rel=&quot;nofollow noreferrer&quot;&gt;On a General Method of expressing the Paths of Light and of the Planets by the Coefficients of a Characteristic Function&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This was published in:&#xA;Dublin University Review and Quarterly Magazine&lt;/p&gt;&#xA;&lt;p&gt;Presumably that magazine had only a small circulation.&lt;/p&gt;&#xA;&lt;p&gt;Quoting from the above paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the quantity pretended to be economised is in fact often lavishly expended. In optics, [...] in reflexion at a curved mirror this economy is often violated. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In mathematical language, the integral called action, instead of being always a minimum, is often a maximum;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Larger quote, to provide the context of the above snippets.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But although the law of least action has thus attained a rank among the highest theorems of physics, yet its pretensions to a cosmological necessity, on the ground of economy in the universe, are now generally rejected. And the rejection appears just, for this, among other reasons, that the quantity pretended to be economised is in fact often lavishly expended. In optics, for example, though the sum of the incident and reflected portions of the path of light, in a single ordinary reflexion at a plane, is always the shortest of any, yet in reflexion at a curved mirror this economy is often violated. If an eye be placed in the interior but not at the centre of a reflecting hollow sphere, it may see itself reflected in two opposite points, of which one indeed is the nearest to it, but the other on the contrary is the furthest; so that of the two different paths of light, corresponding to these two opposite points, the one indeed is the shortest, but the other is the longest of any. In mathematical language, the integral called action, instead of being always a minimum, is often a maximum; and often it is neither the one nor the other: though it has always a certain stationary property, of a kind which has been already alluded to, and which will soon be more fully explained. We cannot, therefore, suppose the economy of this quantity to have been designed in the divine idea of the universe: though a simplicity of some high kind may be believed to be included in that idea. And though we may retain the name of action to denote the stationary integral to which it has become appropriated--which we may do without adopting either the metaphysical or (in optics) the physical opinions that first suggested the name--yet we ought not (I think) to retain the epithet least: but rather to adopt the alteration proposed&#xA;above, and to speak, in mechanics and in optics, of the Law of Stationary Action.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;About Joseph Louis Lagrange:&lt;/p&gt;&#xA;&lt;p&gt;An english translation of 'Mecanique Analytique' is available on archive.org.&lt;/p&gt;&#xA;&lt;p&gt;The first part covers Statics&lt;br /&gt;&#xA;The second part covers Dynamics&lt;/p&gt;&#xA;&lt;p&gt;Section 1 of the Second Part (Dynamics) is titled '&lt;a href=&quot;https://archive.org/details/springer_10.1007-978-94-015-8903-1/page/n210/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;the various principles of Dynamics&lt;/a&gt;'&lt;/p&gt;&#xA;&lt;p&gt;Lagrange gives an overview of the knowledge of that time, including historical information. (Some of Lagrange's historical information may be incorrect; I have not examined that.)&lt;/p&gt;&#xA;&lt;p&gt;On &lt;a href=&quot;https://archive.org/details/springer_10.1007-978-94-015-8903-1/page/n224/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 183&lt;/a&gt; Lagrange arrives at the action concept that was available at the time: Maupertuis' action&lt;/p&gt;&#xA;&lt;p&gt;Lagrange's opinion was that Maupertuis' action is not particularly relevant.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I finally arrive at the fourth principle which I call Least Action in analogy to the one to which Maupertuis has applied this designation and which the works of several illustrious authors have subsequently made so famous. This principle, viewed analytically, consists of the following: in the motion of bodies which act on one another, the sum of the products of the masses with the velocities and the spaces traversed is a minimum. The author deduced from it the laws of reflection and refraction of light as well as the laws governing the percussion of bodies in two memoirs read to the Academie des Sciences of Paris in 1744 and two years later at the Academie de Berlin, respectively.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, these applications are too restrictive to be used to establish the truth of a general principle. They have also something vague and arbitrary about them which can only make the consequence which one could draw for the accuracy of the principle itself uncertain. Thus it would be wrong, it seems to me, to put this principle as it is presented on the same level with the ones we just discussed. However, there is another way to view it, more general and more rigorous and which alone deserves the attention of geometers. Euler has given the first interpretation at the end of his treatise on isoperimetric problems published at Lausanne in 1744, 17 by showing that for the trajectories described by bodies following central forces the integral of the product of the velocity by the element of the curve is always a maximum or a minimum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The property which Euler found in the motion of isolated bodies and which seems to be confined to these bodies, I later expanded using the Conservation des Forces Vives, to the motion of any system of bodies acting upon one another in an arbitrary fashion. A new general principle derives from this effort, to wit, that the sum of the products of the masses with the integrals of the velocities multiplied by the element of the spaces traversed is always a maximum or a minimum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is the principle to which I improperly gave the name of Least Action and which I view not as a metaphysical principle but as a simple and general result of the laws of mechanics. [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;About William Rowan Hamilton and Carl Gustav Jacob Jacobi&lt;/p&gt;&#xA;&lt;p&gt;In his 1837 series of lectures 'Vorlesungen über Dynamik' Jacobi devotes a lot of attention to Hamilton's work.&lt;/p&gt;&#xA;&lt;p&gt;(As far as I know there is no english translation of that series of lectures.)&lt;/p&gt;&#xA;&lt;p&gt;Available for download (pdf file): the book &lt;a href=&quot;https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34041/31151000541619.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vorlesungen über Dynamik&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;First Lecture, page 3&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hamilton hat den schönen Zusammenhang, den er gefunden hat, etwas unzugänglich gemacht und verdunkelt, und zwar dadurch, dass er seine charakteristische Function noch zugleich von einer zweiten partiellen Differentialgleichung abhängen lässt. Die Hinzufügung dieser Bestimmung macht die ganze Entdeckung unnöthig complicirt, da eine genauere Untersuchung zeigt, dass die zweite partielleDifferentialgleichung vollkommen überflüssig ist.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[My translation]&lt;br /&gt;&#xA;Hamilton has made the beautiful interconnection he found somewhat inaccessible and obscure because he has made his characteristic function dependent on a second partial differential equation. This addition makes the whole discovery unnecessarily complicated because a closer examination shows that the second partial differential equation is completely superfluous.&lt;/p&gt;&#xA;&lt;p&gt;19th lecture, page 153&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hamilton [...] steht zugleich noch eine zweite partielle Differentialgleichung auf, welcher V ebenfalls genügen soll. Diese kann man aber fortlassen, weil sie sich aus der schon aufgestellten herleiten lässt [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[My translation]&lt;br /&gt;&#xA;Hamilton sets up a second partial differential equation that V must satisfy. This can however be omitted because it can be derived from the already stated equation.&lt;/p&gt;&#xA;&lt;p&gt;[Jacobi proceeds to present that derivation]&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;My experience has been that authors of physics books aren't particularly interested in understanding the stationary action concept. Authors seem to be content with being vague and superficial.&lt;/p&gt;&#xA;&lt;p&gt;Really, the most probing thoughts I have encountered were those in the quote from William Rowan Hamilton. It's not about minimum or maximum; both can occur. Therefore the name 'stationary action' is intrinsically superior.&lt;/p&gt;&#xA;&lt;p&gt;But the above quote is all; it would appear that after 1835 (Hamilton died in 1865) Hamilton did not publish about stationary action again.&lt;/p&gt;&#xA;&lt;p&gt;It appears there is no record of Hamilton responding to Jacobi.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;On the attitude of the physics community as a whole:&lt;/p&gt;&#xA;&lt;p&gt;It appears that in the physics community there is a preference for the name 'least action'. I think that is because people are eager to believe that &amp;quot;something&amp;quot; is minimized. As pointed out by Hamilton, that belief system is a dead end.&lt;/p&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;br&gt;&#xA;&lt;p&gt;On my stackexchange profile page there is a link to a page on my own website, with discussion of Hamilton's stationary action. (Discussion of the fundamentals.)&lt;/p&gt;&#xA;" OwnerUserId="7244" LastEditorUserId="7244" LastEditDate="2023-04-09T07:57:37.693" LastActivityDate="2023-04-09T07:57:37.693" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15300" PostTypeId="2" ParentId="15298" CreationDate="2023-04-08T12:13:12.217" Score="4" Body="&lt;p&gt;Andre Weil, Elliptic functions according to Eisenstein and Kronecher, Springer-Verlag, NY 1976.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-04-08T12:13:12.217" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15302" PostTypeId="2" ParentId="15263" CreationDate="2023-04-09T12:48:40.717" Score="1" Body="&lt;p&gt;Reference: &lt;strong&gt;Several Complex Variables IV, Springer, 1990.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In 1951 Behnke and Stein introduced the concept of a branched covering&#xA;over a domain in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^n$&lt;/span&gt; and defined a complex space to be a space whose local model is this branched covering. Soon after this Cartan and Serre&#xA;introduced a different concept of complex space taking as local model an arbitrary&#xA;analytic set in a domain in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}^n$&lt;/span&gt;. The relation between these two concepts was finally cleared up in 1958 by Grauert and Remmert who proved that&#xA;a Behnke-Stein space is a normal complex space in the sense of Cartan-Serre.&lt;/p&gt;&#xA;" OwnerDisplayName="anon" LastEditorDisplayName="anon" LastEditDate="2023-04-09T12:48:59.533" LastActivityDate="2023-04-09T12:48:59.533" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15303" PostTypeId="1" CreationDate="2023-04-09T13:49:19.100" Score="3" ViewCount="120" Body="&lt;p&gt;Does anyone know who coined the term orthonormal to refer to a basis that is orthogonal and normal?&lt;/p&gt;&#xA;&lt;p&gt;In such a poorly named mathematical world (looking at you, conditionally convergent series) I think we should shine more light on good naming, such as orthonormal. Some may argue that it can be confused with orthogonal, but to that I say just be better. It's concise, it sounds nice, and it is very very intuitive.&lt;/p&gt;&#xA;&lt;p&gt;So if anyone knows the mathematician who popularized the term I would be forever grateful; all you get when you look it up is the Gram-Schmidt process (no disrespect to my boys Gram and Schmidt tho, they're G's fr)&lt;/p&gt;&#xA;&lt;p&gt;Thank you&lt;/p&gt;&#xA;" OwnerUserId="18264" LastEditorUserId="12330" LastEditDate="2023-04-10T07:48:55.800" LastActivityDate="2023-04-10T07:48:55.800" Title="Who coined the term Orthonormal?" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15304" PostTypeId="1" CreationDate="2023-04-09T19:32:08.830" Score="7" ViewCount="696" Body="&lt;p&gt;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN236018647?tify=%7B%22pages%22%3A%5B567%5D%2C%22pan%22%3A%7B%22x%22%3A0.411%2C%22y%22%3A0.808%7D%2C%22view%22%3A%22fulltext%22%2C%22zoom%22%3A0.575%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pp. 539-542 of volume 10-1 of Gauss's werke&lt;/a&gt; include entry 97 in Gauss's diary:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have found new exact formulas for the parallax of the Moon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;as well as the formulas themselves (which were rather found in a different place than Gauss's diary) and an excerpt from his 1799 letter to the Prussian cartographer &lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Ludwig_von_Lecoq&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl Ludwig von Lecoq&lt;/a&gt; who was assisted by Gauss in his trigonometric survey of Westphalia. It is not a well known fact, but Gauss's first astronomical activity was done in this context (this was two years before his calculation of Ceres orbit), and Lecoq wrote in his &amp;quot;report on the trigonometric survey of Westphalia&amp;quot; that &amp;quot;In the astronomical part of the survey, the calculations of Gauss were of great help to me&amp;quot;. It seems that determining the exact geographic location of points by means of astronomical observations was quite a common procedure in the 18th century.&lt;/p&gt;&#xA;&lt;p&gt;I have not included Gauss's formulas here (the formulas are quite complicated) since I want to understand more about the context of the &amp;quot;lunar parallax problem&amp;quot; and not to immerse myself immediately in the mathematical details. I know that the ancient Greek astronomer Hipparchus used a kind of &lt;a href=&quot;https://en.wikipedia.org/wiki/Parallax_in_astronomy&quot; rel=&quot;nofollow noreferrer&quot;&gt;lunar parallax&lt;/a&gt; calculation during a solar eclipse in order to determine the distance to the Moon (he did it by comparing the details of the total eclipse and the partial eclipse observed in two cities whose distance was known). But I don't understand how it is possible that a complete description of the lunar parallax was not established yet in the end of the 18th century (if it was well calculated then I don't think that Lecoq should have needed Gauss's help), a time when spherical trigonometry was quite ripe and a lot of data was collected on the Moon's orbital parameters.&lt;/p&gt;&#xA;&lt;p&gt;Also, I'm not sure I understand the complexity of this problem - Parallax in general is the angular shift in observed position of a distant object when the observer changes his location; so the description of the lunar parallax angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; seems to me to be very simple - &lt;span class=&quot;math-container&quot;&gt;$\mathbb{tan\alpha} = \frac{2R_E}{d}$&lt;/span&gt; (here &lt;span class=&quot;math-container&quot;&gt;$R_E$&lt;/span&gt; is the Earth radius and &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt; the Earth - Moon distance).&lt;/p&gt;&#xA;&lt;p&gt;So I would like to understand what meanings were attached to the notion of lunar parallax in the 18th century, and to understand what was the main difficulty in the related problems.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I now understand that determining the lunar parallax was crucial for exact calculations of the observer's geographic location - especially in surveys of &amp;quot;small regions&amp;quot; in global terms (such as Westphalia) where the accuracy needed is much higher (I guess the accuracy needed is 10's or 100's times higher than in more global surveys). This is because the change in observed lunar position consists of the change in zenith direction (of the earthly observer) and the parallactic change due to the change in observer location. That is, when an observer changes his earthly location he can deduce his geographic location directly from the change in the background of fixed stars, but if he uses the Moon as a reference (which is much closer to us than fixed stars) he has to remember that the change in observed lunar position is not only the result of his zenith rotation but also of the Moon's parallax.&lt;/p&gt;&#xA;&lt;p&gt;Therefore, to avoid errors one has to separate the parallax part from the first part in order to determine the change in zenith direction (which amounts to determining geographic location). I don't know why 18th century surveys had to use the Moon instead of some fixed stars, but in any case, if one uses the Moon, they are led to an interesting geometric problem.&lt;/p&gt;&#xA;&lt;p&gt;Although the magnitude of parallax correction &lt;span class=&quot;math-container&quot;&gt;$\pi '$&lt;/span&gt; can be easily determined by observing the Moon altitude &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; - as in terry-s's answer, it satisfies the equation &lt;span class=&quot;math-container&quot;&gt;$\mathbb{sin} (\pi ') = \mathbb{cos}(a) \cdot \mathbb{sin}(\pi) = \mathbb{cos}(a) \cdot (\frac{R_E}{d})$&lt;/span&gt; (sine theorem in euclidean trigonometry) - determining the direction in which the parallax correction should be applied (in order to find the exact direction of observer's zenith) seems to much more complicated as it depends on the lunar declination (which changes from 18 to 28 degrees and back during a month), the hour angle, as well as the observer's latitude and longitude.&lt;/p&gt;&#xA;&lt;p&gt;Thus, in terms of data obtained by observing the Moon's position in the sky, the direction of parallax correction seems to be a complicated function of the Moon's altitude and azimuth as well as the timing of the observation, so determining the observer's latitude &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; and longitude &lt;span class=&quot;math-container&quot;&gt;$\varphi$&lt;/span&gt; by this method turns into a complicated trigonometric equation in unknowns &lt;span class=&quot;math-container&quot;&gt;$\theta,\varphi$&lt;/span&gt; and coefficients that depend on the lunar position (azimuth, altitude and day of the month). This explains the apparent complexity of Gauss's formulas.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Apart from deciphering Gauss's formulas, I also fail to understand why cartographers needed to use the Moon's position (a process that is prone to complexities) instead of using some fixed stars? Is it just because the Moon is the simplest celestial body to find at night?&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="6545" LastEditDate="2024-02-19T10:18:00.640" LastActivityDate="2024-02-19T10:18:00.640" Title="Entry 97 in Gauss's diary and the status of &quot;lunar parallax&quot; in the late 18th century" Tags="|astronomy|gauss|18th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15305" PostTypeId="1" CreationDate="2023-04-09T20:27:33.083" Score="0" ViewCount="60" Body="&lt;p&gt;The opening sentence of Roger Myhill's article &lt;em&gt;Paradoxes&lt;/em&gt;,  in Synthese 60 (1984),&#xA;129-143, is: “Gödel said to me more than once &amp;quot;There never were any&#xA;set-theoretic paradoxes, but the property-theoretic paradoxes are still&#xA;unresolved&amp;quot;; and he may well have said the same thing in print.”&lt;/p&gt;&#xA;&lt;p&gt;What, precisely, were Gödel's property theories. I have for a long time thought that they simply were set theories without the axiom of extensionality. But perhaps there is something more to them, e.g. something stemming from Gödel's ontological argument?&lt;/p&gt;&#xA;" OwnerUserId="8094" LastActivityDate="2023-04-09T20:27:33.083" Title="What are Gödel's property theories" Tags="|set-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15306" PostTypeId="1" CreationDate="2023-04-10T02:45:41.550" Score="0" ViewCount="68" Body="&lt;p&gt;I read on a blogpost that Lorentz built a mathematical model of gravity, but with an assumption that it arose from electrical origins. I tried googling for more information and could not find anything so I came here instead. thanks for your help&lt;/p&gt;&#xA;" OwnerUserId="9988" LastActivityDate="2023-04-14T16:45:13.637" Title="What are the equations of Electric Gravity Lorentz derived, and what is the paper in which they are found?" Tags="|theoretical-physics|electromagnetism|gravity|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15307" PostTypeId="2" ParentId="15304" CreationDate="2023-04-10T03:44:25.460" Score="5" Body="&lt;p&gt;Lunar parallax is a composite problem. Two parts of it are related by the 'horizontal parallax', &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; in the source you cited and linked. It is evaluated in one part of the problem and applied in the other.&lt;/p&gt;&#xA;&lt;p&gt;First part of the problem (not addressed by Gauss in your linked source) is to evaluate at any time the moon's horizontal parallax, i.e. its parallax when observed on the horizon. This quantity depends on its continually varying distance from the earth. It is defined as part of lunar theory, and (nowadays) by an ephemeris, e.g. by part of the current JPL &lt;a href=&quot;https://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris&quot; rel=&quot;nofollow noreferrer&quot;&gt;development ephemeris&lt;/a&gt; based on lunar laser-ranging data to which the lunar theory or equations of motion are fitted. The daily horizontal parallaxes at 0h TT are given in the &lt;a href=&quot;https://aa.usno.navy.mil/publications/asa&quot; rel=&quot;nofollow noreferrer&quot;&gt;Astronomical Almanac&lt;/a&gt;.  In Gauss's time the moon's current horizontal parallax was already provided by the usual almanacs, or it could be obtained more laboriously from the fundamental tables of the time, which were actually used to calculate the almanacs, e.g. Mayer's tables as improved by Mason (one of the the surveyors of the 'Mason-Dixon' line).&lt;/p&gt;&#xA;&lt;p&gt;Finding the horizontal parallax is the part of the problem that depends on the moon's distance from the earth.  The basic expression for the (horizontal) parallax given in the question was not quite correct, because the horizontal parallax is conventionally defined as the difference of view-direction towards the moon's center as between an observer on the earth's surface where the moon is on the horizon, and another observer located on the line between the moon and the geocenter (i.e. for whom the moon is at the local zenith) -- where the current  parallax becomes zero. So the (horizontal) parallax angle when defined in this standard way becomes &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbb{\sin \alpha} = \frac{R_E}{d}$$&lt;/span&gt;&#xA;and the symbols are as defined in the equation in the original question.&lt;/p&gt;&#xA;&lt;p&gt;The second part of the problem is the one addressed by Gauss's entry linked in the question, it is to find the actual current parallax at a given altitude or zenith distance of the moon, &lt;em&gt;given&lt;/em&gt; the horizontal parallax for the current time. Gauss made his formulae general in the (limited) sense that they would take as their inputs the moon's current horizontal parallax and then (in order to calculate its current altitude) also its declination (from the ephemeris), plus its hour angle (calculated from the Right Ascension given in the ephemeris and the current local sidereal time). These quantities are combined  trigonometrically to produce the parallax at current altitude. This current parallax varies rather simply with apparent altitude or equivalent zenith distance, it is the horizontal parallax affected by a factor proportional to the sine of the apparent zenith distance. (Current parallax is zero for an object in the zenith, and at its greatest, equal to the horizontal parallax, for an object on the horizon.)&lt;/p&gt;&#xA;&lt;p&gt;The effect of lunar parallax is always to &lt;em&gt;depress&lt;/em&gt; the (apparent) altitude, thus &lt;em&gt;increasing&lt;/em&gt; the moon's apparent zenith distance. If in doubt, just consider the line from geocenter to moon (zero parallax for an observer along that line). Next, consider where that line is for an observer with the moon on the horizon -- answer, a long way below the observer's feet.  It will be apparent that at the observer's horizon, the observer will have to look for the moon a lttle downwards compared with the line of view to the moon from the geocenter which is far below the observer's feet.&lt;/p&gt;&#xA;&lt;p&gt;Many of the relationships are more fully described in the various &lt;a href=&quot;https://aa.usno.navy.mil/publications/exp_supp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Explanatory Supplements&lt;/a&gt; to the astronomical almanacs.&lt;/p&gt;&#xA;&lt;p&gt;Gauss, in the entry linked in the question, also addressed a separate problem, that of evaluating the augmentation of the moon's apparent diameter. The moon's apparent diameter, when (hypothetically) seen from the geocenter, is provided by lunar theory and current ephemeris data. But for a real observer, the moon's linear distance will be different and therefore also its apparent diameter. E.g. for an observer with the moon at the zenith the moon will be less distant than the standard assumes, because the observer is then standing 'on top of' the earth's radius, so that the moon is that much closer, and its apparent diameter larger. Gauss calculates this 'augmented diameter' (Vergroesserter Diameter), because it is useful in connection with observations that are based, as usual, on one of the moon's limbs (edges) although the desired final quantity should be referred to the moon's center. (It should also be noted that Gauss takes as his reference distance the moon's distance when seen on the horizon, not quite the same as the current standard geocentric distance.)&lt;/p&gt;&#xA;&lt;p&gt;edit: A more modern and fully-explained treatment of lunar parallax that should be useful for comparison can be found for comparison at &lt;a href=&quot;http://www.hathitrust.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;www.hathitrust.org&lt;/a&gt;, starting (for the lunar parallax) at &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015008087143&amp;amp;view=1up&amp;amp;seq=76&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 76&lt;/a&gt; of the 1960s  &lt;a href=&quot;https://hdl.handle.net/2027/mdp.39015008087143&quot; rel=&quot;nofollow noreferrer&quot;&gt;Explanatory Supplement to the Astronomical Almanac and the American ephemeris and nautical almanac&lt;/a&gt;.  Earlier and later pages give useful information on related questions.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2023-05-01T15:44:02.760" LastActivityDate="2023-05-01T15:44:02.760" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15308" PostTypeId="1" CreationDate="2023-04-10T10:47:09.843" Score="1" ViewCount="87" Body="&lt;p&gt;I think I saw a high school science film in which Galileo is eating with nobility or something and the scientist amazes a young woman by simply dropping an orange and a grape, saying, if heavier things fell proportionately faster, the grape should only have fallen to &lt;em&gt;here&lt;/em&gt; by the time the orange hits the table. But of course this does not happen and she is so pleased she asked if she can try it.&lt;/p&gt;&#xA;&lt;p&gt;Surely Archimedes would have known this by observation, at least that heavier things did not fall proportionately faster which I believe is what Aristotle said. Not just Archimedes but any one of literally thousands if not millions of people in not just ancient Greece or Rome but India or China? Or really any country or people who had any sort of science? Like even Babylonians? Or really ancient Egyptians?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2023-04-10T10:47:09.843" Title="Did Archimedes or anyone else prior to Galileo think that heavier things did not fall faster?" Tags="|gravity|galileo|archimedes|aristotle|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15309" PostTypeId="1" AcceptedAnswerId="15329" CreationDate="2023-04-11T11:43:24.807" Score="5" ViewCount="265" Body="&lt;p&gt;As the title says, I would like to know who discovered the rational root theorem. The Encyclopaedia Britannica &lt;a href=&quot;https://www.britannica.com/science/rational-root-theorem&quot; rel=&quot;noreferrer&quot;&gt;states&lt;/a&gt; that “The 17th-century French philosopher and mathematician René Descartes is usually credited with devising the test”, but I was unable to find any reference to this both in &lt;em&gt;A History of Algebra: From al-Khwārizmī to Emmy Noether&lt;/em&gt; (by van der Waerden) or in &lt;em&gt;Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century&lt;/em&gt; (by Victor J. Katz and Karen Hunger Parshall).&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2024-02-28T07:53:26.273" Title="Who discovered the rational root theorem?" Tags="|algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15311" PostTypeId="1" CreationDate="2023-04-12T17:08:35.703" Score="2" ViewCount="115" Body="&lt;p&gt;Do I understand correctly that Archimedes was not familiar with the concept of work and hence attributed mechanical advantage to a lever's ability to create force rather than understanding that the long arm of a lever might help you lift a heavy stone, but this was at the expense of the long arm having to move a longer distance than the shorter arm? (like in the attached image)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/x7GIQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/x7GIQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-04-12T17:08:35.703" Title="Archimedes' lever and the concept of work" Tags="|physics|ancient-greece|classical-mechanics|archimedes|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15312" PostTypeId="1" CreationDate="2023-04-12T18:54:06.807" Score="3" ViewCount="226" Body="&lt;p&gt;America was found in 1492, yet the big names of American mathematics only come up later than 1900s. I also get a feeling that as years went on since 1900, the focus was more on abstract mathematics, peaking at 1950s with findings of ideas such as Category theory and so on. Why did the development stay stagnant till this time?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="3643" LastEditDate="2023-04-15T22:33:20.970" LastActivityDate="2023-04-15T22:33:20.970" Title="Why did the development of major Mathematics in America only start after 1900s?" Tags="|mathematics|education|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15313" PostTypeId="1" CreationDate="2023-04-12T22:43:54.733" Score="1" ViewCount="99" Body="&lt;p&gt;In 1821 Alexis Bouvard published a book with tables of the orbits of Jupiter, Saturn and Uranus and future predictions of the orbits. The real orbit of Uranus deviated from the calculations which was the starting point of an investigation which eventually led to the discovery of Neptune.&lt;/p&gt;&#xA;&lt;p&gt;I am wondering how he performed the calculations, which I presume are not just Kepler orbits, but including some pertubations from the known planets (correct me if I understand that wrong). Usually nowadays you would do pertubations calculations in the Hamilton-Jacobi framework, but since Jacobi just started college in 1821 it had not been invented yet (I presume).&lt;/p&gt;&#xA;&lt;p&gt;So does anybody know how exactly these calculations were done at that time?&lt;/p&gt;&#xA;" OwnerUserId="12504" LastEditorUserId="10608" LastEditDate="2024-03-05T08:30:33.840" LastActivityDate="2024-03-05T08:30:33.840" Title="How did Alexis Bouvard calculate the orbits of Jupiter, Saturn and Uranus" Tags="|classical-mechanics|celestial-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15314" PostTypeId="1" CreationDate="2023-04-13T12:28:41.073" Score="0" ViewCount="66" Body="&lt;p&gt;In basic physics, the emphasis is  calculating total forces, and going from that to the position function of the body as a function of time by integration. Now, from this prespective, how did people reach the idea to look at phase spaces?; What is the historical motivation for this?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2023-04-13T13:30:17.757" Title="What led to people looking at phase spaces?" Tags="|physics|classical-mechanics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15315" PostTypeId="2" ParentId="15314" CreationDate="2023-04-13T13:30:17.757" Score="1" Body="&lt;p&gt;Since the initial conditions of a mechanical system involve positions (configuration space) and velocities (or momenta),&#xA;the space of initial conditions is the phase space. This was realized by Hamilton who rewrite the second order equations of motion as a first order system of double size.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-04-13T13:30:17.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15316" PostTypeId="2" ParentId="15313" CreationDate="2023-04-13T13:44:48.503" Score="2" Body="&lt;p&gt;Perturbation theory begins with Newton who tried to explain the motion of the Moon (this is the object in the solar system whose motion deviates most from Keplerian orbits). Newton essentially failed, and more advanced methods of perturbation theory were developed in 18th century, main contributions are due to d'Alembert, Clairaut, Euler and Mayer. This resulted in the tables of Moon motion which had accuracy sufficient for navigation (that is an error of fraction of a minute over several years). Same ideas were applied to the large planets, of which the most conspicuous case is the 3-body problem Sun-Jupiter-Saturn. The perturbation theory is too technical to be explained here&#xA;(see the reference below), but the modern notions of Hamilton-Jacobi framework slowly evolved in the process of this development of perturbation theory. This is a general pattern: scientists solve concrete problems, and only after that their methods are generalized into some general framework.&lt;/p&gt;&#xA;&lt;p&gt;Ref. Curtis Wilson, The Great Inequality of Jupiter and Saturn: from Kepler to Laplace, Arch. Hist. Exact Sci.,&#xA;33, 1/3, 1985, 15-290. Contains a popular not very technical historical exposition of perturbation theory, as it developed in the investigation of Jupiter and Saturn.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-04-13T13:44:48.503" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15317" PostTypeId="2" ParentId="15282" CreationDate="2023-04-14T01:40:57.403" Score="2" Body="&lt;p&gt;I think &lt;a href=&quot;https://en.wikipedia.org/wiki/Topological_insulator&quot; rel=&quot;nofollow noreferrer&quot;&gt;topological insulators&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Josephson_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;Josephson effect&lt;/a&gt; would be good examples.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Topological_insulator#Prediction&quot; rel=&quot;nofollow noreferrer&quot;&gt;theoretical roots&lt;/a&gt; of &lt;strong&gt;topological insulators&lt;/strong&gt; go back to the 1980s, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Charles_L._Kane&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kane&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Eugene_J._Mele&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mele&lt;/a&gt;'s 2005 theory paper on &lt;a href=&quot;https://arxiv.org/abs/cond-mat/0506581&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Kane, C. L. &amp;amp; Mele, E. J. Z2 topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005)&quot;&gt;&amp;quot;&lt;span class=&quot;math-container&quot;&gt;$Z_2$&lt;/span&gt; topological order and the quantum spin Hall effect&amp;quot;&lt;/a&gt; is usually given the most credit for their theoretical discovery. In 2006 &lt;a href=&quot;https://en.wikipedia.org/wiki/B._Andrei_Bernevig&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernevig&lt;/a&gt;, Hughes, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Shoucheng_Zhang&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zhang&lt;/a&gt; made a &lt;a href=&quot;https://arxiv.org/abs/cond-mat/0611399&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Bernevig, B. A., Hughes, T. L. &amp;amp; Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006)&quot;&gt;specific theoretical prediction&lt;/a&gt; that quantum spin Hall insulator states could be realized in HgTe quantum wells, and this was &lt;a href=&quot;https://arxiv.org/abs/0710.0582&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007)&quot;&gt;experimentally observed in 2007&lt;/a&gt; by the &lt;a href=&quot;https://en.wikipedia.org/wiki/Laurens_W._Molenkamp&quot; rel=&quot;nofollow noreferrer&quot;&gt;Molenkamp&lt;/a&gt; lab.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;Josephson Effect&lt;/strong&gt; was &lt;a href=&quot;https://en.wikipedia.org/wiki/Josephson_effect#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;predicted&lt;/a&gt;  by &lt;a href=&quot;https://en.wikipedia.org/wiki/Brian_Josephson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brian Josephson&lt;/a&gt; in 1962, and &lt;a href=&quot;https://doi.org/10.1103/PhysRevLett.10.230&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Probable Observation of the Josephson Superconducting Tunneling Effect, P. W. Anderson and J. M. Rowell, Phys. Rev. Lett. 10, 230 – Published 15 March 1963&quot;&gt;experimentally observed&lt;/a&gt; in 1963 by &lt;a href=&quot;https://en.wikipedia.org/wiki/Philip_W._Anderson&quot; rel=&quot;nofollow noreferrer&quot;&gt;Phil Anderson&lt;/a&gt; and John Rowell.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-04-14T01:40:57.403" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15318" PostTypeId="2" ParentId="15312" CreationDate="2023-04-14T05:34:58.490" Score="6" Body="&lt;p&gt;It is absolutely irrelevant for the question when America was discovered. The first university on the US territory was founded in 1636. But until 20th century most universities taught mostly elementary mathematics. In general, mathematical research was a very localized phenomenon until 20th century, mainly cultivated in Germany, France, Britain and Italy. Contribution of other countries was relatively small, and in most other countries it did not exist at all.&lt;/p&gt;&#xA;&lt;p&gt;In the end of 19th century American students started to travel to Germany to study advanced mathematics, and then returned home to teach it on American soil. Later two events&#xA;helped to move America to the top in mathematics: first, large immigration in the 1930s of mathematicians who escaped the Nazi regime and the war, and second, the arms race and&#xA;space race after WWII. It was understood that mathematics is important and money was pumped into math education and research&#xA;during this period, since American public understood that mathematics is important for defense and competition in technologies.&lt;/p&gt;&#xA;&lt;p&gt;In most other countries, mathematical research&#xA;(I mean &amp;quot;modern mathematics&amp;quot;) started much later, in the second half of 20th century. This includes countries like China, India or Turkey which were &amp;quot;discovered&amp;quot; much earlier that America.&lt;/p&gt;&#xA;&lt;p&gt;Remark. Since you mention category theory which arose as a part of topology, let me cite Richard Courant on American mathematics of 1930s:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;``I think that when the Americans first became research-conscious they&#xA;took up the newer fields which didn't require so much background;&#xA;for example, they went into abstract algebra and topology.&#xA;In the early 1900's those were new fields. One could understand what had&#xA;been done without too much background.&#xA;One could see any number of problems that were open.&#xA;If they had chosen to go into analysis, they would have had to acquire much&#xA;more extensive backgrounds before they could do original work of their own.&#xA;I think that people in our country got started the easiest way at the time.''&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-04-15T14:21:57.730" LastActivityDate="2023-04-15T14:21:57.730" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15319" PostTypeId="1" CreationDate="2023-04-14T09:55:18.480" Score="1" ViewCount="90" Body="&lt;p&gt;I would like to know where to start, i.e papers/books/video conferences and conceptions/misconceptions, with the history of mathematics before it was really considered mathematics. I'm talking about very primitive notions or tricks that would have a mathematical meaning/explanation today. For example:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A sheep keeper makes a mark in his cane, or adds a pebble to a pile for each sheep that enters the stables. When taking them out, he/she crosses the mark or removes the pebbles one by one. In this way, the sheep keeper doesn't need to know how many sheep he/she has, or even to count how many enter/get out of the stable. If at the end one or more marks/pebbles remain, then there are missing sheep.&lt;/li&gt;&#xA;&lt;li&gt;Almost any somewhat &amp;quot;developed&amp;quot; animal has at least a vague sense of &amp;quot;few vs lots&amp;quot; of things. A mother knows if one of its cubs is missing without having to count them or using marks/pebbles. I'm talking about notions that go further than basic perception: a bird knows its altitude &amp;quot;instinctively&amp;quot; and not because it has a notion of &amp;quot;my distance to the ground is greater here than there&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;Humans always have had a sense of causality. A thunder &lt;strong&gt;must&lt;/strong&gt; be caused by something (be it the rage of some god, or a modern explanation). We have this intrinsic notion of logical implication&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Would you say that there are some necessary requirements that make mathematics and logic &amp;quot;developable&amp;quot; in a mind? For example, vague notions of missing/not missing, bigger/smaller, similar/not similar could evolve into a basic arithmetic and then into more advanced maths&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance!&#xA;(Please, let me know if the question can be made more precise)&lt;/p&gt;&#xA;" OwnerUserId="10086" LastActivityDate="2023-04-14T09:55:18.480" Title="Mathematics before mathematical notions" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15320" PostTypeId="2" ParentId="15306" CreationDate="2023-04-14T16:45:13.637" Score="3" Body="&lt;p&gt;Before a series of four papers on Einstein's Theory of Gravitation (1917-1918) Lorentz seems to have written only two papers dedicated to the subject of gravity:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;(1900) Considerations on gravitation, Proceedings Koninklijke Akademie van Wetenschappen, 2, 559-574.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Quoting opening paragraph -&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;After all we have learned in the last twenty or thirty years about&#xA;the mechanism of electric and magnetic phenomena, it is natural to&#xA;examine in how far it is possible to account for the force of&#xA;gravitation by ascribing it to a certain state of the aether. A theory&#xA;of universal attraction, founded on such an assumption, would take the&#xA;simplest form if new hypotheses about the aether could be avoided,&#xA;i.e. if the two states which exist in an electric and a magnetic&#xA;field, and whose mutual connection is expressed by the well known&#xA;electromagnetic equations were found sufficient for the purpose.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and &lt;strong&gt;(1914) La gravitation, Scientia, 16, 28-59.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am not sure either of these contains a fully worked out electromagnetic theory of gravitation.&lt;/p&gt;&#xA;&lt;p&gt;A slight more a ambitious attempt was undertaken by Emil Weichert in 1920 in &amp;quot;Die Graviatation als elektrodymische Erscheinung&amp;quot; (Gravity as an Electrodynamic Phenomenon), but by then such attempts were being completely overshadowed by Einstein's Theory of General Relativity.&lt;/p&gt;&#xA;&lt;p&gt;Before that in 1913 Samuel Bruce Mclaren had attempted an electromagnetic theory of gravitation (&amp;quot;A Theory of Gravity&amp;quot; Phil. Mag. 26, Issue 154, 636-673.), but he died in the First World War so did not develop it any further.&lt;/p&gt;&#xA;&lt;p&gt;One starting point for your studies might be Russell McCormmach's 1970 paper &amp;quot;H. A. Lorentz and the Electromagnetic View of Nature&amp;quot;, Isis, 61, No. 4, 459-497. A pdf of this can be found if you search on Google or Google Scholar.&lt;/p&gt;&#xA;" OwnerUserId="18197" LastActivityDate="2023-04-14T16:45:13.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15321" PostTypeId="2" ParentId="15253" CreationDate="2023-04-14T17:34:25.973" Score="2" Body="&lt;p&gt;Another good candidate is the measure of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Electron_electric_dipole_moment&quot; rel=&quot;nofollow noreferrer&quot;&gt;electron's electric dipole moment&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$d_e$&lt;/span&gt;. Within the Standard Model, such a dipole is &lt;em&gt;predicted&lt;/em&gt; to be non-zero but very small, at most &lt;span class=&quot;math-container&quot;&gt;$10^{−38} e \cdot \rm{cm}$&lt;/span&gt;. The current best &lt;em&gt;measured&lt;/em&gt; upper bound is in &lt;a href=&quot;https://arxiv.org/pdf/2212.11841.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roussy et al., &lt;em&gt;A new bound on the electron's electric dipole moment&lt;/em&gt; (2022)&lt;/a&gt; where the author give the following bound:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$|d_e|&amp;lt;4.1 \times 10^{−30} e\cdot \rm{cm}$$&lt;/span&gt;&#xA;at &lt;span class=&quot;math-container&quot;&gt;$90\%$&lt;/span&gt; confidence.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2023-04-14T17:34:25.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15323" PostTypeId="1" CreationDate="2023-04-15T12:08:20.853" Score="0" ViewCount="97" Body="&lt;p&gt;Can't find any information about who and when first used surface/surface area integrals. What was the original motivation? In it's modern form it depends on some relatively modern notations like partial derivatives and cross product (both mid 1800's if I'm not mistaken), but I have a sense it predates these. Then how did it look originally? Are there any records at all?&lt;/p&gt;&#xA;" OwnerUserId="18284" LastActivityDate="2023-04-15T12:08:20.853" Title="Surface Integrals History" Tags="|calculus|mathematical-physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15324" PostTypeId="1" CreationDate="2023-04-16T09:08:16.190" Score="3" ViewCount="93" Body="&lt;p&gt;I have heard about the high regard in which Moritz Cantor’s multi-volume work on the History of Mathematics has been held among mathematicians and historians alike. I am becoming increasingly interested in the History, and missing out on such a work would, in my view, be a cardinal sin. I, however, have not been able to find any sort of translation of the Work into English (the original language is German), which I find very odd, given the good repute of the Treatise. I have not even seen any amateur translations. It would be very nice if some here could provide me with at least a hint where I might look for a translation, or say why one has not been made yet (if it truly has not been made yet). I am genuinely curious!&lt;/p&gt;&#xA;" OwnerUserId="18287" LastActivityDate="2023-04-16T09:08:16.190" Title="Is there an English translation of Moritz Cantor’s History of Mathematics?" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15325" PostTypeId="1" CreationDate="2023-04-16T17:48:11.087" Score="6" ViewCount="1945" Body="&lt;p&gt;I'm looking for the earliest references to the word &lt;em&gt;space-time&lt;/em&gt; (in the modern sense), in any language.&#xA;The first references would likely be in German, as &lt;em&gt;Raum-Zeit&lt;/em&gt; or &lt;em&gt;Raumzeit&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Of course, H. Minkowski is credited as the originator of the idea. I found the following references to Minkowski's work in &lt;a href=&quot;https://archive.org/details/theoryofrelativi00paul/page/20/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pauli's Theory of Relativity&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&amp;quot;&lt;em&gt;Das Relativitätsprinzip&lt;/em&gt;&amp;quot;, published twice (lecture 1907-11-05):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Jahresbericht der Deutschen Mathematiker-Vereinigung (Jber. dtsch.&#xA;Mat. Ver.), 24 (1915) 372&lt;/li&gt;&#xA;&lt;li&gt;Annalen der Physik (Ann. Phys., Lpz.), 47&#xA;(1915) 927&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&amp;quot;&lt;em&gt;Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Köpern&lt;/em&gt;&amp;quot;, published twice:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen&#xA;(Nachr. Ges. Wiss. Göttingen) (1908) 53&lt;/li&gt;&#xA;&lt;li&gt;Mathematische Annalen (Math. Ann.), 68 (1910) 472&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ol start=&quot;3&quot;&gt;&#xA;&lt;li&gt;&lt;em&gt;Raum und Zeit&lt;/em&gt;, published twice (lecture 1908-09-21):&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Physikalische Zeitschrift (Phys. Z.), 10 (1909) 104&lt;/li&gt;&#xA;&lt;li&gt;Das Relativitätsprinzip (Leipzig 1913) - English translation &lt;a href=&quot;https://archive.org/details/principleofrelat00lore/page/74/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. &lt;strong&gt;Here he uses &lt;em&gt;die Welt&lt;/em&gt; ('the world') instead of &lt;em&gt;Raum-Zeit&lt;/em&gt; or &lt;em&gt;Raumzeit&lt;/em&gt; (space-time)&lt;/strong&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I have only been able to find the text of number 3, in English translation.&lt;/p&gt;&#xA;&lt;p&gt;Two questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Is there any online access to these old German publications?&lt;/li&gt;&#xA;&lt;li&gt;Does anyone know of any early references to publications&#xA;containing the word &lt;em&gt;Raum-Zeit&lt;/em&gt; or &lt;em&gt;space-time&lt;/em&gt; in the text?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="18290" LastEditorUserId="12330" LastEditDate="2023-04-17T11:47:08.310" LastActivityDate="2023-04-18T13:05:49.393" Title="What is the etymology of the term space-time?" Tags="|physics|terminology|" AnswerCount="4" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15326" PostTypeId="2" ParentId="15325" CreationDate="2023-04-16T18:44:56.017" Score="6" Body="&lt;p&gt;Here is a reference from 1906 (and below [in an update] from 1896).&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;THE TEACHING OF MECHANICS BY EXPERIMENT&amp;quot;&#xA;C.E. ASHFORD&lt;/p&gt;&#xA;&lt;p&gt;included in &amp;quot;The Teaching of Elementary Mechanics&amp;quot; by John Perry, 1906&#xA;&lt;BR&gt;&#xA;and in&#xA;&lt;BR&gt; The School World, p.335-339.&lt;/p&gt;&#xA;&lt;p&gt;At the bottom of page 337&#xA;&lt;a href=&quot;https://archive.org/details/schoolworld06unkngoog/page/336/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/schoolworld06unkngoog/page/336/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The diagram (Fig.2) shows one of the ordinary trolleys on its plane.&#xA;With this a boy can produce and measure uniform velocity and acceleration.&#xA;Every experiment with a trolley provides him with a space-time diagram ready made&#xA;...&#xA;and he can plot his velocity-time and acceleration-time curves from it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5ELjA.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5ELjA.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;update&lt;/p&gt;&#xA;&lt;p&gt;Here's an 1896 reference&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Experimental determination of the motion of projectiles inside the bore of a gun with the polarizing photo-chronograph&amp;quot; by Crehore, Albert and Squier, George, p. 348&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/1896Experimental_photo_chronograph-BP34-6/page/348/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;https://archive.org/details/1896Experimental_photo_chronograph-BP34-6/page/348/mode/2up&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The pressure which corresponds to the constant acceleration&#xA;&lt;BR&gt;...&lt;BR&gt;&#xA;This value, which is the probable pressure along the part of the bore where the parabola applies, it will be seen might have been much greater near the seat of the projectile without making very much change in the space-time or velocity-space curves.&#xA;&lt;BR&gt;...&lt;BR&gt;&#xA;The parabola in this case representing the space-time curve has its origin to the left of the vertical axis instead of upon it as in the previous case. The true space-time curve must manifestly pass through the origin as the time is counted from the instant when the projectile started.&#xA;&lt;BR&gt;...&lt;BR&gt;&#xA;The points on the velocity-time curve II are obtained by erecting ordinates equal to the tangent of inclination to the space-time curve I.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/YA7q5.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/YA7q5.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16070" LastEditorUserId="16070" LastEditDate="2023-04-16T18:59:40.380" LastActivityDate="2023-04-16T18:59:40.380" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15327" PostTypeId="2" ParentId="15325" CreationDate="2023-04-16T20:38:49.227" Score="7" Body="&lt;p&gt;The &lt;em&gt;Math. Ann.&lt;/em&gt; paper is &lt;a href=&quot;https://archive.org/details/sim_mathematische-annalen_1910_68/page/472/mode/2up?q=minkowski&amp;amp;view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Minkowski uses the term &lt;em&gt;Raum-Zeit&lt;/em&gt; in this paper, not &lt;em&gt;Welt&lt;/em&gt;.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Examples of terms used in the paper:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Raum-Zeit-Linien - lines&lt;/li&gt;&#xA;&lt;li&gt;Raum-Zeit-Vektoren - vectors&lt;/li&gt;&#xA;&lt;li&gt;Raum-Zeit-Matrix&lt;/li&gt;&#xA;&lt;li&gt;Raum-Zeit-Sichel  - ??&lt;/li&gt;&#xA;&lt;li&gt;Raum-Zeit-NullPunkte - origin event&lt;/li&gt;&#xA;&lt;li&gt;Raum-Zeitpunkte - point (event)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The &lt;em&gt;Ann. Phys.&lt;/em&gt; paper is &lt;a href=&quot;https://archive.org/details/sim_annalen-der-physik_1915_47_15/page/926/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I see no use made of the term &lt;em&gt;Raum-Zeit&lt;/em&gt; in this paper.&lt;/p&gt;&#xA;&lt;p&gt;In summary, the earliest paper that I've seen that makes repeated use of &lt;em&gt;Raum-Zeit&lt;/em&gt; is this paper (1910):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/sim_mathematische-annalen_1910_68/page/472/mode/2up?q=minkowski&amp;amp;view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematische Annalen (Math. Ann.), 68 (1910) 472&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;which is apparently a reprint of an earlier paper (1908; thank you @Conifold for the link):&#xA;&lt;a href=&quot;https://eudml.org/doc/58707&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (Nachr. Ges. Wiss. Göttingen) (1908) 5&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some results from Google Books Search for &amp;quot;raumzeit&amp;quot; from 1905-01-10 to 1915-12-31:&lt;/p&gt;&#xA;&lt;p&gt;H. Minkowski, &lt;em&gt;Raum und Zeit&lt;/em&gt;,&#xA;Physikalische Zeitschrift, Vol 10, No. 3, p107 (1909-02-01).&#xA;&lt;em&gt;Raum-Zeit-NullPunkte&lt;/em&gt; is the only occurrence. This doesn't seem to appear in the English translation.&lt;/p&gt;&#xA;&lt;p&gt;Gunnar Nordström, &lt;em&gt;Zur Elektrodynamik Minkowskis&lt;/em&gt;,&#xA;Physikalische Zeitschrift, Vol 10, No. 20 (1909-10-15).&#xA;&lt;em&gt;Raum-Zeit-Bezugssystemen&lt;/em&gt; (frame of reference) is the only occurrence.&lt;/p&gt;&#xA;&lt;p&gt;A. Einstein,  &lt;em&gt;Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung.&lt;/em&gt;, Physikalische Zeitschrift, Vol 10, No. 22, p820 (1909-11-10).&#xA;&lt;em&gt;Raum-Zeit-Koordinaten&lt;/em&gt; is the only occurrence.&lt;/p&gt;&#xA;" OwnerUserId="18290" LastEditorUserId="18290" LastEditDate="2023-04-17T13:54:20.517" LastActivityDate="2023-04-17T13:54:20.517" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15328" PostTypeId="2" ParentId="15325" CreationDate="2023-04-16T21:57:37.137" Score="4" Body="&lt;p&gt;Using a search on Google Books ( &lt;a href=&quot;https://books.google.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/&lt;/a&gt; ) for &amp;quot;Raumzeit&amp;quot; (no hyphen, but include the quotes in the search), restricted to 19th century (an option of Tools, in the upper right),&#xA;here are some uses of &amp;quot;Raumzeit&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/search?q=%22raumzeit%22&amp;amp;tbm=bks&amp;amp;source=lnt&amp;amp;tbs=cdr:1,cd_min:1800,cd_max:1899&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/search?q=%22raumzeit%22&amp;amp;tbm=bks&amp;amp;source=lnt&amp;amp;tbs=cdr:1,cd_min:1800,cd_max:1899&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'll just include screenshots...&#xA;&lt;BR&gt;&#xA;&lt;em&gt;Sorry I can't easily transcribe and proofread the German text.&lt;BR&gt;&#xA;I'll leave it to others to interpret the use of the term in these contexts.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(1861) Friedrich Gottlob Röber&lt;BR&gt;&#xA;&amp;quot;Elementar-Beiträge zu Bestimmung des Naturgesetzes der Gestaltung und des Widerstandes, und Anwendung dieser Beiträge auf Natur und alte Kunstgestaltung&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;p15:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/R8zHZ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/R8zHZ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;p18-19:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/zpYfT.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/zpYfT.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VpBH2.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VpBH2.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/4Bhx0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4Bhx0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/evqBU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/evqBU.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;(1803) Biblia sacra oder die heilige Schrift des alten und neuen Testamentes, Volume 1&lt;/p&gt;&#xA;&lt;p&gt;p 299 (footnote):&#xA;&lt;a href=&quot;https://i.stack.imgur.com/4dvXB.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4dvXB.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;By changing the range to 1700-1800, this came up... presumably an earlier version of the 1803 one.&lt;/p&gt;&#xA;&lt;p&gt;(1786)&#xA;“Die” heilige Schrift des alten (und neuen) Testamentes ... neuerdings übers. von Heinrich Braun -  Christian Friedrich Wappler, 1786&lt;/p&gt;&#xA;&lt;p&gt;p.299&#xA;&lt;a href=&quot;https://i.stack.imgur.com/b002P.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/b002P.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16070" LastActivityDate="2023-04-16T21:57:37.137" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15329" PostTypeId="2" ParentId="15309" CreationDate="2023-04-17T07:44:56.853" Score="7" Body="&lt;p&gt;When in doubt, go to the source. There is a nice &lt;a href=&quot;https://archive.org/details/geometryofrene00desc/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;facsimile edition&lt;/a&gt; of Descartes's &lt;em&gt;La Géométrie&lt;/em&gt; with English translation, freely available on Internet Archive. Descartes does not spell out the trick as a theorem, to which it was elevated by modern textbooks, or even as a general rule, but he does use it in several places, and in a generalized form to cover quadratic irrationals. For example on p.176 he writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Given, for example, &lt;span class=&quot;math-container&quot;&gt;$y^6—8y^4—124y^2—64=0$&lt;/span&gt;. The last term, 64, is divisible by 1, 2, 4, 8, 16, 32, and 64; therefore we must find whether&#xA;the left member is divisible by &lt;span class=&quot;math-container&quot;&gt;$y^2—1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y^2+1$&lt;/span&gt;,  &lt;span class=&quot;math-container&quot;&gt;$y^2—2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y^2+2$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y^2—4$&lt;/span&gt;, and so on.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And on p. 180 explains&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;After removing any surds or fractions, see if a binomial having one term a factor of the last term of the expression will divide the left member. If such a binomial can be found, either the known&#xA;quantity of the binomial is the required root, or, after the division is performed, the resulting equation, which is of only three dimensions,&#xA;must be treated in the same way.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He does not bother with rational roots and divisors of the leading coefficient because on pp. 172-175 he explains a trick &amp;quot;&lt;em&gt;useful in changing fractional terms of an equation to whole numbers, and often in rationalizing the terms&lt;/em&gt;&amp;quot;, which is illustrated by transforming an equation to convert its fractional roots into integers (and  quadratically irrational roots into rational first).&lt;/p&gt;&#xA;&lt;p&gt;Historians did not make much of it either. For example, &lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/00029890.1959.11989307&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyer in Descartes and the Geometrization of Algebra&lt;/a&gt; only mentions it in passing:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;That so much of La Géométrie is concerned with&#xA;the theory of equations — the number of possible positive roots, increasing and decreasing the roots of an equation, finding rational roots, and depressing the degree when a root is known — does not indicate a preference for algebra over&#xA;geometry.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ams.org/notices/200801/tx080100050p.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Serfati&lt;/a&gt; is equally brief:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He also proves that for a polynomial &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; to be divisible by &lt;span class=&quot;math-container&quot;&gt;$(X − a)$&lt;/span&gt; it is necessary and sufficient that &lt;span class=&quot;math-container&quot;&gt;$P(a)$&lt;/span&gt; be zero. He then uses his indeterminate coefficients to describe the division of a polynomial by &lt;span class=&quot;math-container&quot;&gt;$(X−a)$&lt;/span&gt;. So it was important for him to know at least one&#xA;root. For an equation with rational coefficients, he studies the rational roots, if any.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="10608" LastEditDate="2024-02-28T07:53:26.273" LastActivityDate="2024-02-28T07:53:26.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15330" PostTypeId="1" CreationDate="2023-04-18T10:43:09.523" Score="4" ViewCount="163" Body="&lt;p&gt;The history of filters and ultrafilters is a bit of a mess. According to some authors, the concept of ultrafilter was first introduced by Cartan (1937). According to some authors, the concept of ultrafilter was first introduced by Riesz (1908). In some sources, the theorem that every filter is contained in an ultrafilter was given by Tarski in 1930. I am wondering which mathematicians gave the concepts such as filter, ultrafilter and filter basis in which year and in which sources. I searched many sources, but I couldn't have a clear idea.&lt;/p&gt;&#xA;" OwnerUserId="18301" LastActivityDate="2023-04-20T23:38:16.003" Title="History of filter and ultrafilter" Tags="|mathematics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15331" PostTypeId="2" ParentId="15330" CreationDate="2023-04-18T12:46:17.390" Score="1" Body="&lt;p&gt;According to Bourbaki, Topologie, I, historical essay, filters and ultrafilters&#xA;were introduced by H. Cartan, Theorie de filtres et ultrafiltres, C. R., 205 (1937) 595-598 et 777-779.&#xA;On my opinion, Bourbaki can be trusted on this matter.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-04-18T12:46:17.390" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15332" PostTypeId="2" ParentId="15325" CreationDate="2023-04-18T13:05:49.393" Score="3" Body="&lt;p&gt;Here are 1898 references that use &amp;quot;&lt;strong&gt;l'espace-temps&lt;/strong&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(My goal in these searches is to find in publications &lt;BR&gt;&#xA;the term &amp;quot;spacetime&amp;quot; or &amp;quot;space-time&amp;quot;,&lt;BR&gt;&#xA;regardless of Minkowski's specific idea of the spacetime manifold with metric-signature isomorphic to +--- .&lt;/p&gt;&#xA;&lt;p&gt;From my comment (in response to another comment) in my first answer,&lt;BR&gt;&#xA;it would be interesting if its use, in Minkowski's precise sense as a spacetime manifold for special relativity, pre-dated his &amp;quot;Space and Time&amp;quot; paper. But it is interesting to see the term in use before Minkowski, some closer to Minkowski's use and others quite distant from it. )&lt;/p&gt;&#xA;&lt;p&gt;I found these using my Google Books search method.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&amp;quot;L'intuition motrice&amp;quot;&lt;BR&gt;&#xA;Georges Dubreuque&lt;BR&gt;&#xA;Revue Philosophique de la France Et de l'Etranger 46:253 - 292 (1898),&#xA;p. 262&lt;/p&gt;&#xA;&lt;p&gt;(copy-paste of the pdf... I didn't proofread it.  You could try to paste it into Google Translate. I bolded the term.)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;La vue est , de tous les appareils de perception , celui qui fournit à la connaissance les matériaux les mieux élaborés ; la perception optique procède par ensembles , et non comme la perception auditive par fragments successifs ; la compréhension visuelle a lieu sans efforts ; elle n'est pas , comme il arrive parfois pour la compré hension auditive , un moment distinct de la sensation brute . Quelle que puisse être à cet égard la supériorité du sens visuel , la géomé trie ne l'utilise pas ; elle ne raisonne pas sur des « perceptions mas sives » . Les figures géométriques sont obtenues par génération , ce qui signifie qu'elles vont du détail à l'ensemble , et non , comme pour les perceptions visuelles ordinaires , de l'ensemble au détail . La cons truction géométrique , aussi bien que la déduction , est « le terme d'un mouvement » ; la ligne est engendrée par le mouvement d'un point , la surface par le mouvement d'une ligne , le volume par le mouvement d'une surface . La notion fondamentale de la géométrie n'est donc pas proprement l'espace , mais le mouvement dans l'espace , autrement dit &lt;strong&gt;l'espace - temps&lt;/strong&gt; ; les figures sont des mouve ments qu'on étudie sous un aspect particulier et sans avoir égard aux vitesses ni aux accélérations ; l'élimination de la variable temps n'est en quelque sorte que virtuelle , et quelques mathématiciens ont trouvé avantage à l'y réintroduire , en traitant la géométrie comme une branche de la cinématique .&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/68qbe.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/68qbe.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;La logique sociale&lt;BR&gt;&#xA;Gabriel de Tarde&lt;BR&gt;&#xA;F. Alcan (ed), 1898&lt;/p&gt;&#xA;&lt;p&gt;pages: 91,92,164, (contents)463&lt;/p&gt;&#xA;&lt;p&gt;(copy-paste of the pdf... I didn't proofread it.  You could try to paste it into Google Translate. I bolded the term.)&lt;/p&gt;&#xA;&lt;p&gt;p.91&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...Ce moyen d'échange est fourni , en psychologie individuelle , par ces deui entités singulières : l'espace et le temps , que j'aimerais mieux désigner en un seul mot , &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; , tellement leur lien est intime , et en psy chologie sociale , par cette autre entité non moins étrange , non moins féconde en idolâtries ou en illusions réalistes : la langue , qui , comme nous le verrons , a deux faces bien distinctes . Il y a d'ailleurs à remarquer , disons - le tout de suite , que ces entités , &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; et la langue , quoique étant la condition du déve loppement de ces réalités : le Monde et Dieu , ont dû se déve lopper parallèlement à celles - ci et se sont formées peut - être de leurs débris lentement accumulés ; car toute notion a commencé par être un jugement , et tout moyen par être un but .&#xA;&lt;a href=&quot;https://i.stack.imgur.com/PmnMJ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/PmnMJ.png&quot; alt=&quot;p91&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;p92&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ainsi , je me garde de confondre les fonctions et les catégories . Le juge ment et la volonté , la religion et la politique , sont des fonctions ; mais la Matière - Force ainsi que &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; , la Divinité ainsi que la Langue sont des catégories . Ce sont là des catégo ries logiques , c'est - à - dire n'ayant trait qu'aux fonctions intellec tuelles du Jugement et de la Religion . Mais il y a aussi des catégories ou demi - catégories téléologiques , qui répondent aux fonctions pratiques de la volonté et du gouvernement . L'agréable et le douloureux sont l'un poursuivi , l'autre évité par le vouloir de l'individu primitif comme des choses qui existent en dehors de lui , et qu'il incarne dans les objets matériels de ses percep tions : de même , le bien et le mal sont poursuivis ou évités par le devoir de l'homme social , comme des réalités idéales ou des idéalités réelles qu'il cherche à fixer et qu'il fixe en effet en les incorporant dans les objets divins de son adoration . Il y a donc , en tout , pour l'esprit individuel , les catégories suivantes , logiques ou téléologiques : la Matière - Force , &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; , le Plaisir et a Douleur ; et pour l'esprit social : la Divinité , la Langue , le Bien&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hKV3I.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hKV3I.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;p164&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Le mouvement n'est que la traduction symbolique de l'évolution . Nous ne pouvons nous faire de celle - ci une idée moins large que de celui - là . Et , puisque nous voyons le champ immense de &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; ouvert à la diversité infinie , à la merveilleuse multiformité des mouve ments , sans que cette apparente liberté nuise en rien à leur détermination rigoureuse , à la stricte application des lois méca niques , - dont &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; est , pour ainsi dire , la notion intégrale , l'applicabilité totale , nous devons nous dire à priori , et l'observation des faits semble nous donner pleinement raison , que les lois de la logique , cette mécanique interne , ouvrent un débouché non moins vaste à l'inépuisable variété des évolutions soit vivantes , soit psychologiques et sociales . Il y a une logique vitale ( laquelle nous apparaît surtout sous son aspect téléologique ) , comme il y a une logique mentale et sociale . Et , si &lt;strong&gt;l'Espace - Temps&lt;/strong&gt; est l'applicabilité totale de la mécanique , on peut dire aussi bien que la Vie , sorte d'espace des dévelop pements biologiques , est l'applicabilité totale de la logique vivante , et que l'Esprit , sorte d'espace aussi des développements psychologiques et sociaux , est l'applicabilité totale de la logique individuelle et sociale .&#xA;&lt;a href=&quot;https://i.stack.imgur.com/hWRvs.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/hWRvs.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16070" LastActivityDate="2023-04-18T13:05:49.393" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15333" PostTypeId="1" CreationDate="2023-04-18T13:16:11.437" Score="10" ViewCount="4819" Body="&lt;p&gt;Who among professional mathematicians are also known as fiction writers?&#xA;I know Omar Khayyam (11-12 century), and two more recent ones: Sofya Kovalevskaya and Michele Audin.&lt;/p&gt;&#xA;&lt;p&gt;For the purposes of this question, let us define a &amp;quot;mathematician&amp;quot; as a person who has at least a few papers reviewed in &lt;a href=&quot;https://zbmath.org/&quot; rel=&quot;noreferrer&quot;&gt;Zentralblatt&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-04-28T12:24:38.063" Title="Mathematicians who wrote fiction" Tags="|mathematics-social-history|" AnswerCount="17" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15334" PostTypeId="2" ParentId="15333" CreationDate="2023-04-18T13:55:46.540" Score="9" Body="&lt;p&gt;My answer copied from &lt;a href=&quot;https://mathoverflow.net/a/369709/454&quot;&gt;another forum&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Eric_Temple_Bell&quot; rel=&quot;noreferrer&quot;&gt;&lt;strong&gt;Eric Temple Bell&lt;/strong&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;President of the MAA, 1931-32&lt;br /&gt;&#xA;Author of the book &lt;em&gt;Men of Mathematics&lt;/em&gt;&lt;br /&gt;&#xA;Bell numbers, Bell polynomials&lt;br /&gt;&#xA;etc.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also was a successful science fiction author under the pseudonym &lt;strong&gt;John Taine&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;See the fascinating book&lt;br /&gt;&#xA;Reid, Constance, &lt;em&gt;The Search for E.T. Bell&lt;/em&gt;, MAA spectrum, The Mathematical Association of America, 1993&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-04-18T13:55:46.540" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15335" PostTypeId="2" ParentId="15333" CreationDate="2023-04-18T14:08:44.070" Score="22" Body="&lt;p&gt;Charles Lutwidge Dodgson, better known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Lewis_Carroll&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lewis Carroll&lt;/a&gt;, wrote some well-known fiction books.&lt;/p&gt;&#xA;&lt;p&gt;In case you consider &lt;a href=&quot;https://en.wikipedia.org/wiki/Marvin_Minsky&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marvin Minsky&lt;/a&gt; a mathematician (16 papers in Zentralblatt), he coauthored with &lt;a href=&quot;https://en.wikipedia.org/wiki/Harry_Harrison_(writer)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harry Harrison&lt;/a&gt; the science fiction novel &lt;em&gt;The Turing Option&lt;/em&gt; (by the way, a pretty bad book).&lt;/p&gt;&#xA;&lt;p&gt;Like Quenau, also &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacques_Roubaud&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacques Roubaud&lt;/a&gt; was a member of the Oulipo group and a prolific author of fiction and poetry. He passes the Zentralblatt test, as it lists two entries: &lt;em&gt;Monades et descente&lt;/em&gt;, with Jean Benabou, and &lt;em&gt;La notion d’associativité rélative&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://it.wikipedia.org/wiki/Roberto_Vacca&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roberto Vacca&lt;/a&gt; (1927) is a researcher in logic and number theory (six entries in Zentralblatt), but also a well-known (in Italy) science popularizer. He debuted as a writer of science fiction and political fiction in 1963 with the novel &lt;em&gt;Il robot e il minotauro&lt;/em&gt; (&lt;em&gt;The robot and the minotaur&lt;/em&gt;), followed by other twelve books. These novels were not very successful and fame as a writer came only in 1971 with the essay (not fiction) on the apocalyptic theme &lt;em&gt;&lt;a href=&quot;https://it.wikipedia.org/wiki/Il_medioevo_prossimo_venturo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Il medioevo prossimo venturo&lt;/a&gt;&lt;/em&gt; (edited in English as &lt;em&gt;The Coming Dark Age The Middle Ages&lt;/em&gt;), considered a classic of futurology .&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Hausdorff&quot; rel=&quot;nofollow noreferrer&quot;&gt;Felix Hausdorff&lt;/a&gt; published, under the pen name Paul Mongré, several essays, some poetry and a play, but no fiction.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2023-04-21T12:08:44.860" LastActivityDate="2023-04-21T12:08:44.860" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15336" PostTypeId="1" CreationDate="2023-04-18T22:27:54.910" Score="2" ViewCount="121" Body="&lt;p&gt;On the occasion of Heisenberg's death, Wigner wrote an obituary in April 1976 issue of Physics Today (pp. 86-87):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sommerfeld, an excellent teacher with a wonderful overall knowledge&#xA;of his discipline, maintained close contact with the students of the&#xA;institute. Although Heisenberg was apparently not one of Sommerfeld's&#xA;favorite students, he received his doctorate in 1923 at the age of 22.&#xA;He then went to Max Born's institute in Gottingen for one year and to&#xA;Copenhagen for three years where Niels Bohr taught.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Whom did Sommerfeld prefer as student and why Heisenberg was not among those?&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="6609" LastEditDate="2023-04-18T22:44:45.050" LastActivityDate="2023-04-18T22:44:45.050" Title="Whom did Sommerfeld prefer as student?" Tags="|quantum-mechanics|physics-social-history|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15337" PostTypeId="2" ParentId="15333" CreationDate="2023-04-18T22:33:57.947" Score="13" Body="&lt;p&gt;Fred Hoyle was an Astronomer, but he passes the &lt;a href=&quot;https://zbmath.org/authors/hoyle.fred&quot; rel=&quot;noreferrer&quot;&gt;Zentralblat&lt;/a&gt; test.  He wrote several novels, most notably &lt;a href=&quot;https://www.goodreads.com/book/show/1246118.The_Black_Cloud&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Black Cloud&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In addition to &lt;em&gt;The Black Cloud&lt;/em&gt;, I read a thriller he wrote with his son, Geoffrey, called &lt;a href=&quot;https://www.goodreads.com/book/show/1065403.The_Westminster_Disaster&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;The Westminister Disaster&lt;/em&gt;&lt;/a&gt;. Probably dated, but I enjoyed it at the time.&lt;/p&gt;&#xA;" OwnerUserId="18305" LastActivityDate="2023-04-18T22:33:57.947" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15338" PostTypeId="2" ParentId="15333" CreationDate="2023-04-18T22:42:36.727" Score="4" Body="&lt;p&gt;Chandler Davis &lt;a href=&quot;https://en.wikipedia.org/wiki/Chandler_Davis&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chandler_Davis&lt;/a&gt; wrote some SF stories as well as poetry.&lt;/p&gt;&#xA;" OwnerUserId="18306" LastActivityDate="2023-04-18T22:42:36.727" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15339" PostTypeId="1" CreationDate="2023-04-18T22:42:57.757" Score="3" ViewCount="264" Body="&lt;p&gt;As usual when I see an article about asking about mathematicians, I cannot help to ask what about physics? Inspired by &lt;a href=&quot;https://hsm.stackexchange.com/questions/5645/literary-works-authored-by-mathematicians/5646#5646&quot;&gt;Literary works authored by mathematicians&lt;/a&gt; what are some physicist that also were involved in fiction and literature?&lt;/p&gt;&#xA;&lt;p&gt;Please consider providing details about the author and the cited literary works, whenever possible. Please exclude popular science books.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-04-21T12:23:53.177" LastActivityDate="2023-05-16T01:15:06.483" Title="What are some physicists that wrote literary works?" Tags="|physics|physicists|examples|physics-social-history|" AnswerCount="6" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15340" PostTypeId="2" ParentId="15333" CreationDate="2023-04-18T23:19:59.030" Score="6" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gregory_Benford&quot; rel=&quot;noreferrer&quot;&gt;Gregory Benford&lt;/a&gt; is primarily known as a physicist, but does meet the &lt;a href=&quot;https://zbmath.org/authors/?q=gregory%20benford&quot; rel=&quot;noreferrer&quot;&gt;Zentralblatt criterion&lt;/a&gt;. He is the author of &lt;a href=&quot;https://en.wikipedia.org/wiki/Timescape&quot; rel=&quot;noreferrer&quot;&gt;Timescape&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Gregory_Benford_bibliography&quot; rel=&quot;noreferrer&quot;&gt;many dozens of other speculative fiction stories&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="18307" LastActivityDate="2023-04-18T23:19:59.030" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15341" PostTypeId="2" ParentId="15333" CreationDate="2023-04-18T23:57:09.767" Score="7" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Norbert_Wiener&quot; rel=&quot;noreferrer&quot;&gt;Norbert Wiener&lt;/a&gt;, one of the founders of modern control theory and information theory, wrote a science-fiction novel named &lt;em&gt;The Tempter&lt;/em&gt;. The hero in the book was based on the real-life story of physicist Oliver Heaviside, how he came up with the transmission line equations and solved the problem of dispersion via inductive loading, and how his work was later patented by the AT&amp;amp;T in an ethically-dubious manner.&lt;/p&gt;&#xA;&lt;p&gt;It was also a terrible book.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1959 a curious book appeared, a novel of all things, written by Norbert Wiener, the&#xA;&amp;quot;father of cybernetics.&amp;quot; All the names were changed, but it is in fact a fictionalization of&#xA;Heaviside's adult life as told through the events of the Campbell-Pupin loading coil episode&#xA;(which is altered to be a &amp;quot;stolen&amp;quot; invention in feedback control). The book is perfectly awful as&#xA;a novel (some of the dialogue is funny enough to be read aloud at a party), but most technical&#xA;people would probably enjoy it because Wiener brings a lot of technical details into the story.&#xA;Heaviside was tremendously admired by Wiener (Pupin was not), and the book was Wiener's&#xA;attempt at justice for his late hero. The best one can say about it, perhaps, is that Wiener meant&#xA;well.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age (2002), Paul J. Nahin&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In addition, the Wikipedia article showed that he also wrote some short science fiction stories under the name &amp;quot;W. Norbert&amp;quot; for MIT's student newspaper, &lt;a href=&quot;https://en.wikipedia.org/wiki/Tech_Engineering_News&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Tech Engineering News&lt;/em&gt;&lt;/a&gt;, including &lt;em&gt;The Brain&lt;/em&gt; (anthologized in Groff Conklin's Crossroads in Time) and &lt;em&gt;The Miracle of the Broom Closet&lt;/em&gt; (reprinted the same year in The Magazine of Fantasy &amp;amp; Science Fiction).&lt;/p&gt;&#xA;" OwnerUserId="10481" LastEditorUserId="10481" LastEditDate="2023-04-19T00:31:43.140" LastActivityDate="2023-04-19T00:31:43.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15342" PostTypeId="2" ParentId="15333" CreationDate="2023-04-19T06:48:03.367" Score="5" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jacques_Dixmier&quot; rel=&quot;noreferrer&quot;&gt;Jacques Dixmier&lt;/a&gt; published &lt;em&gt;L'Aurore des Dieux&lt;/em&gt;, a collection of science-fiction stories. Its forward was written by Alain Connes.&lt;/p&gt;&#xA;&lt;p&gt;And Johannes Kepler published a novel called &lt;em&gt;Somniun&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastActivityDate="2023-04-19T06:48:03.367" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15343" PostTypeId="2" ParentId="15333" CreationDate="2023-04-19T06:53:48.467" Score="8" Body="&lt;p&gt;If you consider Bertrand Russell to be a mathematician, he wrote two volumes of short stories near the end of his life but they were met with ''baffled silence'' even by his most ardent fans (in the words of his biographer Ray Monk).&lt;/p&gt;&#xA;&lt;p&gt;Eric Temple Bell (known for Bell polynomials) wrote poetry and science fiction novels under the pseudonym John Taine.  His science fiction novels are known for their interesting ideas but as a writer he is considered to be extremely lacking in style, tone, sensitivity and characterization (plus anything else one might expect from a good fiction writer).&lt;/p&gt;&#xA;&lt;p&gt;Ian Stewart has also written fantasy fiction books with Terry Pratchett.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastEditorUserId="10345" LastEditDate="2023-04-28T12:24:38.063" LastActivityDate="2023-04-28T12:24:38.063" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15344" PostTypeId="2" ParentId="15333" CreationDate="2023-04-19T11:34:40.407" Score="5" Body="&lt;p&gt;Vernor Vinge was primarily a computer scientist but he has two citations on zbMath.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Vernor_Vinge&quot; rel=&quot;noreferrer&quot;&gt;Vernor Vinge&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18310" LastActivityDate="2023-04-19T11:34:40.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15345" PostTypeId="2" ParentId="15333" CreationDate="2023-04-19T12:04:14.747" Score="7" Body="&lt;p&gt;&lt;strong&gt;Raymond Quenau&lt;/strong&gt; (1903-1976), a French writer and mathematician.&lt;/p&gt;&#xA;&lt;p&gt;Even if he wasn't a professional mathematician, he dealt with mathematics and published writings of mathematics and about mathematics.&lt;/p&gt;&#xA;&lt;p&gt;It seems to me that he passed the Zentralblatt test:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://zbmath.org/?q=raymond+queneau&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://zbmath.org/?q=raymond+queneau&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;He studied mathematics at university, and during all his life he continued to cultivate his passion for mathematics, also participating to seminars of the most important mathematicians in Paris.  In particular he participated to the meetings of Bourbaki in the fifties, and had a frendship and professional collaboration with Geoge Kreisel, collaborating to the book &lt;em&gt;Elements de logique mathématique&lt;/em&gt;, published by Kreisel and Krivine in 1968.&lt;/p&gt;&#xA;&lt;p&gt;He made research about number theory. The results of his work about integers were presented at the &lt;em&gt;Académie des sciences&lt;/em&gt; in Paris on April 1968. And it was subsequently published in the &lt;em&gt;Journal of Combinatory Theory&lt;/em&gt;, commented by the famous mathematician Giancarlo Rota.&lt;/p&gt;&#xA;&lt;p&gt;Those and other information about Quenau mathematician and writer can be found in&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.it/Mathematical-Lives-Protagonists-Twentieth-Century/dp/3642136052&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.it/Mathematical-Lives-Protagonists-Twentieth-Century/dp/3642136052&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But, above all, Raymond Queneau is famous as writer and poet.&lt;/p&gt;&#xA;&lt;p&gt;He was a prominent figure in the French literature of the twentieth century, in particular for experimental literature.&#xA;He founded, together with his friend Perec, the &lt;em&gt;Oulipo&lt;/em&gt; (&lt;em&gt;Ouvroir de Litérature Potentielle&lt;/em&gt;), an experimental writing group, where the use of mathematical structures in literary endeavours was discussed.&lt;br /&gt;&#xA;His most well known novels  are &lt;em&gt;Zazié dans le  metro&lt;/em&gt; (1959), &lt;em&gt;Les fleuers bleues&lt;/em&gt; (1965) , and he is famous for his  &lt;em&gt;Exercises  de style&lt;/em&gt; (1947).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\;$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In addition, we cannot forget &lt;strong&gt;Martin Gardner&lt;/strong&gt;, who wrote novels and short stories. He is so famous that needs no introduction:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Martin_Gardner&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Martin_Gardner&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;He passed the Zentralblatt test:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://zbmath.org/authors/?q=martin+gardner&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://zbmath.org/authors/?q=martin+gardner&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-04-20T11:28:40.447" LastActivityDate="2023-04-20T11:28:40.447" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15347" PostTypeId="2" ParentId="15333" CreationDate="2023-04-19T15:06:57.627" Score="9" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Rudy_Rucker&quot; rel=&quot;noreferrer&quot;&gt;Rudy Rucker&lt;/a&gt; is a mathematician, computer scientist, and SF writer.  He wrote the Ware Tetrology (Software, Wetware, Freeware, Realware)&lt;/p&gt;&#xA;" OwnerUserId="18312" LastActivityDate="2023-04-19T15:06:57.627" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15348" PostTypeId="2" ParentId="15333" CreationDate="2023-04-19T21:05:12.223" Score="4" Body="&lt;p&gt;&lt;a href=&quot;https://zbmath.org/authors/snaith.victor-p&quot; rel=&quot;nofollow noreferrer&quot;&gt;Victor Snaith&lt;/a&gt; wrote &lt;a href=&quot;https://www.thebigquestions.com/2010/04/05/the-yukiad-perpetual-motion-and-me/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Yukiad&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To add to people mentioned by others:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jacques_Roubaud&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacques Roubaud's&lt;/a&gt; &lt;a href=&quot;https://www.google.com/books/edition/La_Princess_Hoppy_ou_Le_conte_du_Labrado/9k34tgAACAAJ?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Princess Hoppy&lt;/em&gt;&lt;/a&gt; contains a group theory problem in it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Georges_Perec&quot; rel=&quot;nofollow noreferrer&quot;&gt;Georges Perec's&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Life:_A_User%27s_Manual&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;La Vie Mode d'Emploi&lt;/em&gt;&lt;/a&gt; contains a linear algebra proof, horribly mis-typeset in the English translation. (I don't think Perec was a mathematician, though.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2023-04-19T21:05:12.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15349" PostTypeId="2" ParentId="15333" CreationDate="2023-04-20T02:39:25.330" Score="9" Body="&lt;p&gt;Since I'm fluent in Russian, I did some googling for russian ones.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sofya_Kovalevskaya&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Sofya Kovalevskaya&lt;/strong&gt;&lt;/a&gt; was not simply a published mathematician, but a first woman who obtained what can be classified as a doctoral degree in math. Aside from links to her publications, her Wiki page has a link to a novel she authored.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://ru.wikipedia.org/wiki/%D0%9C%D0%B8%D1%85%D0%B0%D0%B9%D0%BB%D0%BE%D0%B2,_%D0%A0%D0%BE%D0%BC%D0%B0%D0%BD_%D0%92%D0%B0%D0%BB%D0%B5%D1%80%D1%8C%D0%B5%D0%B2%D0%B8%D1%87&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Roman Mikhaylov&lt;/strong&gt;&lt;/a&gt;. His Wiki page (no English version, sorry) has links to both his scientific publications - and he also happens to be a member of Russian Academy of Sciences, so he's clearly a &amp;quot;someone&amp;quot; - and his books and - with him seeming being a polymath - his theater and movie works as well.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Not sure if this would count as a &amp;quot;mathematician&amp;quot;, as he was primarily a &lt;em&gt;mathematics teacher&lt;/em&gt; and not sure if he published any actual math research - but he did advanced math studies as part of his education career; &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Volkov_(writer)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Alexandr Volkov&lt;/strong&gt;&lt;/a&gt; (who wrote a Russian re-do of &amp;quot;Wizard of Oz&amp;quot; called &amp;quot;The wizard of emerald City&amp;quot; and then 5 sequels that were original works since he hated Baum's sequels)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aleksandr_Solzhenitsyn&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Aleksandr Solzhenitsyn&lt;/strong&gt;&lt;/a&gt;. Didn't publish, but when he was sent to labour camps where at least part of the time was spent in &amp;quot;sharashka&amp;quot; type research institute, where his math/physics talents were utilized (less relevant, his second wife was an actual mathematician).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mikhail_Lomonosov&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Mikhail Lomonosov&lt;/strong&gt;&lt;/a&gt; was a famous 18th century russian polymath (Moscow State University is named after him); who mostly worked in sciences but also - as was true of people like him - a mathematician. He also happened to be very literary inclined and a poet.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Irina_Grekova&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Elena Sergeevna Ventsel (aka Irina Grekova)&lt;/strong&gt;&lt;/a&gt; - hat tip to comment by @Roger Vadim&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If you expand this to other sciences, I can probably rattle off 10 more easily from memory, including at least half the main figures in Soviet SciFi (astronomer Strugatsky, engineer Zamyatin - who literally stated he chose engineering as &amp;quot;the most math-y specialty he could&amp;quot;, paleontologist Efremov, biophysicist Nick Perumov, biologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Kirill_Eskov&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eskov&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Speaking of apocryphal fan-fiction, since I brought up the last two people famuos for &amp;quot;Tolkien was wrong&amp;quot; ones based in Middle Earth, one of the most famous and controversial &lt;a href=&quot;https://en.wikipedia.org/wiki/Harry_Potter_and_the_Methods_of_Rationality&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harry Potter fan-fiction&lt;/a&gt; works was written by AI researcher Eliezer Yudkowsky, and honestly I'm inclined to count the latter as &amp;quot;math&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="18318" LastEditorUserId="18318" LastEditDate="2023-04-21T22:22:55.577" LastActivityDate="2023-04-21T22:22:55.577" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15350" PostTypeId="2" ParentId="15333" CreationDate="2023-04-20T10:10:13.363" Score="5" Body="&lt;p&gt;A &lt;a href=&quot;https://w.wiki/6c77&quot; rel=&quot;noreferrer&quot;&gt;query to Wikidata&lt;/a&gt; yields 31 individuals listed as &amp;quot;novelist&amp;quot; and &amp;quot;mathematician&amp;quot;, 29 of whom have a dedicated article in Wikipedia (26 in English language Wikipedia). I think some of them have a degree in Mathematics but could fail the Zentralblatt test.&lt;/p&gt;&#xA;&lt;p&gt;The 29 people are Thomas Carlyle, Ronald Ross, Lewis Carroll, Simon Newcomb, Sofia Kovalévskaia, Raymond Queneau, Vernor Vinge, Larry Niven, Ian Stewart, Eric Temple Bell, Rudy Rucker, Leo Perutz, Henry Adams, Gérald Tenenbaum, Charles Sheffield, Herbert Yardley, Manil Suri, Julio Cesar de Mello e Souza, David Bensoussan, Benedict Freedman, Carin Gerhardsen, Chandler Davis, Malka Elisheva Schaps, Sarah Mason, Hajime Amagi, Leila Schneps, Emily Tanimura, Vladislav Šak and Emily Coddington, some of which have already been mentioned in previous answers. Links to their Wikipedia articles and Wikidata items can be found in the linked query.&lt;/p&gt;&#xA;" OwnerUserId="5873" LastActivityDate="2023-04-20T10:10:13.363" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15351" PostTypeId="2" ParentId="15333" CreationDate="2023-04-20T10:17:45.437" Score="4" Body="&lt;p&gt;This is not technically an answer to your question, but &lt;a href=&quot;https://www.gregegan.net/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Greg Egan&lt;/a&gt; is a professional author who also does mathematics recreationally.  They have a &lt;a href=&quot;https://mathoverflow.net/users/23829/greg-egan&quot;&gt;MO profile&lt;/a&gt; and have definitely done some &lt;a href=&quot;https://www.quantamagazine.org/sci-fi-writer-greg-egan-and-anonymous-math-whiz-advance-permutation-problem-20181105/&quot; rel=&quot;nofollow noreferrer&quot;&gt;novel mathematics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On a side note, I really love their book &amp;quot;Permutation City&amp;quot; which plays with the idea of isomorphic copies of one's mind living in other structures of the Universe, ensuring a kind of &amp;quot;isomorphic immortality&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6432" LastActivityDate="2023-04-20T10:17:45.437" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15352" PostTypeId="2" ParentId="15333" CreationDate="2023-04-20T12:00:10.533" Score="3" Body="&lt;p&gt;Manil Suri (61 papers in Zentralblatt) is a mathematician at the University of Maryland who has written the books:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Death of Vishnu (2001),&lt;br /&gt;&#xA;The Age of Shiva (2008),&lt;br /&gt;&#xA;The City of Devi (2012).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3844" LastActivityDate="2023-04-20T12:00:10.533" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15353" PostTypeId="2" ParentId="15330" CreationDate="2023-04-20T14:56:24.267" Score="1" Body="&lt;p&gt;As for the history of filters and ultrafilters, Gregory H. Moore devotes some pages to this subject in his rich in information book &lt;em&gt;Zermelo's Axiom of Choice. Its Origins, Development &amp;amp; Influence&lt;/em&gt;, Dover, 2013 (republication of the edition   by Springer-Verlag, 1982).&lt;/p&gt;&#xA;&lt;p&gt;Moore too attributes the definition of &lt;em&gt;filter&lt;/em&gt; and &lt;em&gt;ultrafilter&lt;/em&gt; to Cartan.&lt;/p&gt;&#xA;&lt;p&gt;Below some passages from the book:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the same year that Čech's article appeared [1937], two&#xA;generalizations of convergence - directed sets and filters - were&#xA;developed independently of each other in the United States and France.&#xA;A few years later, these two generalizations would provide a way to&#xA;represent compactness as well. In 1922 the first approach had been&#xA;formulated jointly by two mathematicians, E.H. Moore of the University&#xA;of Chicago and H.L. Smith of the University of Philippines. [...] Only&#xA;in 1937 did Birkhoff render this approach suitable for topology. [...]&#xA;Meanwhile Henri Cartan proposed to the Paris Academy of Science a&#xA;different but related generalization of convergence: filters.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&#xA;[...] A central notion was that of maximal filter, or, as Cartan&#xA;termed it, an ultrafilter.&lt;/p&gt;&#xA;&lt;p&gt;[...] By means of the Well-Ordering Theorem and transfinite induction,&#xA;Cartan formulated and proved a fundamental result later known as the&#xA;Ultrafilter Theorem:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Every filter can be extended to an ultrafilter.&lt;/em&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;From this proposition he deduced that a Hausdorff space &lt;em&gt;E&lt;/em&gt; is compact&#xA;if and only if every ultrafilter on &lt;em&gt;E&lt;/em&gt; has a limit point.[...]&lt;/p&gt;&#xA;&lt;p&gt;In the 1940 Cartan's characterization of compactness enabled Bourbaki&#xA;to give a proof of Tychonoff's Theorem, restricted to Haussdorff&#xA;spaces, by using ultrafilters rather than complete accumulation&#xA;points.[...]&lt;/p&gt;&#xA;&lt;p&gt;Moreover Bourbaki developed filters as a device suitable for defining&#xA;a topological space, and, in particular, for characterizing pointwise&#xA;convergence  [...]&lt;/p&gt;&#xA;&lt;p&gt;Over the next two decades, filters served French topologist as their&#xA;preferred form of generalized convergence, while Americans confined&#xA;themselves to directed sets.[...] Eventually, mathematicians&#xA;recognized that these two methods of convergence (filters and directed&#xA;sets) were essentially equivalent. However, to establish this&#xA;equivalence, the Axiom came into play. (Moore, pp. 249-241).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; Cartan, Henri, “Théorie des filtres”, Comptes Rendus Hebdomandaires des Séances de l’Académie des Sciences, Paris, &lt;strong&gt;205&lt;/strong&gt;, 595-598&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt; Cartan, Henri, “Filters and ultrafilters”, Comptes Rendus Hebdomandaires des Séances de l’Académie des Sciences, Paris, &lt;strong&gt;205&lt;/strong&gt;, 777-779  &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-04-20T14:56:24.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15354" PostTypeId="1" CreationDate="2023-04-20T17:09:28.967" Score="1" ViewCount="46" Body="&lt;p&gt;I'm wondering how far back this notation goes. Thank you for your consideration.&lt;/p&gt;&#xA;" OwnerUserId="18325" LastActivityDate="2023-04-21T01:32:42.483" Title="What is the earliest paper in which graph vertex connectivity is denoted by the Greek letter $\kappa$?" Tags="|mathematics-social-history|graph-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15355" PostTypeId="2" ParentId="15333" CreationDate="2023-04-20T22:31:44.590" Score="2" Body="&lt;p&gt;Peter Schattschneider is an experimental physics and a retired professor at the Technical University of Vienna. He also writes science fiction: &lt;a href=&quot;https://en.wikipedia.org/wiki/Peter_Schattschneider&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Peter_Schattschneider&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18326" LastActivityDate="2023-04-20T22:31:44.590" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15356" PostTypeId="2" ParentId="15330" CreationDate="2023-04-20T23:38:16.003" Score="1" Body="&lt;p&gt;See also:&lt;/p&gt;&#xA;&lt;p&gt;Leopold Vietoris, dissertation &amp;quot;Stetige Mengen&amp;quot; published in Monatsheft Math., vol. 81, 173-202 in 1921&lt;/p&gt;&#xA;&lt;p&gt;He uses the German word &amp;quot;Kranz&amp;quot; for what we now call &amp;quot;filter base&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-04-20T23:38:16.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15357" PostTypeId="2" ParentId="15354" CreationDate="2023-04-21T01:27:00.767" Score="2" Body="&lt;p&gt;I looked around a bit after this question was posted on MSE, and the earliest reference I found is in Harary's 1962 paper &lt;a href=&quot;https://doi.org/10.1073/pnas.48.7.1142&quot; rel=&quot;nofollow noreferrer&quot;&gt;The maximum connectivity of a graph&lt;/a&gt;. Here,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A graph &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is called &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-connected if G has at least &lt;span class=&quot;math-container&quot;&gt;$n + 1$&lt;/span&gt; points and it is impossible to disconnect &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; by removing &lt;span class=&quot;math-container&quot;&gt;$n - 1$&lt;/span&gt; or fewer points. The connectivity of &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt;, denoted &lt;span class=&quot;math-container&quot;&gt;$\kappa (G)$&lt;/span&gt;, is defined to be &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-connected but not &lt;span class=&quot;math-container&quot;&gt;$(n + 1)$&lt;/span&gt;-connected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Harary's paper doesn't seem to suggest that Harary was the one to come up with this notation. The paper has only two citations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Whitney's &lt;a href=&quot;https://doi.org/10.1007/978-1-4612-2972-8_4&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Congruent graphs and the connectivity of graphs&lt;/em&gt;&lt;/a&gt;, the 1932 paper defining connectivity; this does &lt;em&gt;not&lt;/em&gt; use the notation &lt;span class=&quot;math-container&quot;&gt;$\kappa(G)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Berge's 1958 textbook &lt;em&gt;Théorie des graphes et ses applications&lt;/em&gt;, which could plausibly be the source of the notation; I am unable to check.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Berge's textbook is, as Harary mentions, the &amp;quot;second book on graph theory ever written&amp;quot;. The first is presumably König's &lt;em&gt;Theorie der endlichen und unendlichen Graphen&lt;/em&gt;, from 1936, which has a chapter on cut vertices and cut sets, but does not use the notation &lt;span class=&quot;math-container&quot;&gt;$\kappa(G)$&lt;/span&gt;; I don't think it even defines connectivity.&lt;/p&gt;&#xA;" OwnerUserId="18327" LastEditorUserId="18327" LastEditDate="2023-04-21T01:32:42.483" LastActivityDate="2023-04-21T01:32:42.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15358" PostTypeId="1" CreationDate="2023-04-21T06:17:23.090" Score="3" ViewCount="136" Body="&lt;p&gt;&amp;quot;Hooper's Paradox&amp;quot; is the name given by Martin Gardner (in his Mathematics, Magic and Mystery of 1956) to the apparent paradox that you can apparently dissect a &lt;span class=&quot;math-container&quot;&gt;$3 \times 10$&lt;/span&gt; rectangle into a &lt;span class=&quot;math-container&quot;&gt;$2 \times 6$&lt;/span&gt; and a &lt;span class=&quot;math-container&quot;&gt;$4 \times 5$&lt;/span&gt; rectangle.&lt;/p&gt;&#xA;&lt;p&gt;The resolution is obvious:&lt;a href=&quot;https://i.stack.imgur.com/1s571.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1s571.png&quot; alt=&quot;Hooper's Paradox&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This was the basis of Sam Loyd's famous missing square puzzle, which has been widely propagated since the beginning of the 20th century.&lt;/p&gt;&#xA;&lt;p&gt;My question: who was William Hooper? All I know is that he was the author of&#xA;&lt;em&gt;Rational Recreations&lt;/em&gt; (4 Volumes, 1774, 2nd edition 1782-3).&lt;/p&gt;&#xA;&lt;p&gt;There is a contemporaneous William Hooper who was active in the American Revolutionary War, but it can't be the same bloke because he that William Hooper was somewhat busy dodging redcoats at that time.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone find out such things as his date and place of birth, and what he did?&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2023-04-22T04:49:32.290" Title="Who was William Hooper, eponym of &quot;Hooper's Paradox&quot;?" Tags="|biographical-details|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15359" PostTypeId="2" ParentId="15339" CreationDate="2023-04-21T08:19:02.397" Score="3" Body="&lt;p&gt;Edited from Wikipedia:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Leo Szilard&lt;/strong&gt; (February 11, 1898 to May 30, 1964) was a Hungarian-German-American physicist and inventor. He conceived the nuclear chain reaction in 1933, patented the idea of a nuclear fission reactor in 1934, and in late 1939 wrote the letter for Albert Einstein's signature that resulted in the Manhattan Project that built the atomic bomb.&lt;/p&gt;&#xA;&lt;p&gt;In 1949 Szilard wrote a short story titled &amp;quot;My Trial as a War Criminal&amp;quot; in which he imagined himself on trial for crimes against humanity after the United States lost a war with the Soviet Union.&lt;/p&gt;&#xA;&lt;p&gt;Szilard published a book of short stories, The Voice of the Dolphins (1961), in which he dealt with the moral and ethical issues raised by the Cold War and his own role in the development of atomic weapons. The title story described an international biology research laboratory in Central Europe. This became reality after a meeting in 1962 with Victor F. Weisskopf, James Watson and John Kendrew. When the European Molecular Biology Laboratory was established, the library was named The Szilard Library and the library stamp features dolphins.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I'm also fairly sure I have a short story of his in an old Analog SF anthology, which may or may not have been censored as it was believed to breach secrecy laws.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2023-04-21T08:19:02.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15360" PostTypeId="1" AcceptedAnswerId="15364" CreationDate="2023-04-21T08:43:57.247" Score="7" ViewCount="695" Body="&lt;p&gt;During a brief conversation in &lt;a href=&quot;https://chat.stackexchange.com/rooms/18/ten-fold&quot;&gt;Ten Fold&lt;/a&gt;, &lt;a href=&quot;https://stats.stackexchange.com/users/919/whuber&quot;&gt;whuber&lt;/a&gt; pointed &lt;a href=&quot;https://chat.stackexchange.com/transcript/message/63407928#63407928&quot;&gt;out&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I recall reading long ago that Augustus DeMorgan abandoned probability theory after much study, finding it too tricky/non-intuitive/difficult; but I have been unable to find an authoritative reference in any of his works.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I searched in his &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/De_Morgan/&quot; rel=&quot;noreferrer&quot;&gt;bio&lt;/a&gt; of Augustus De Morgan, but at least I couldn't catch anything conspicuous.&lt;/p&gt;&#xA;&lt;p&gt;So, I am curious whether there is any written account of any possible rendezvous with probability by De Morgan.&lt;/p&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2023-04-21T19:24:12.080" Title="Is there any evidence De Morgan abandoned probability theory due to it being &quot;too tricky/non-intuitive/difficult&quot;?" Tags="|probability|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15361" PostTypeId="2" ParentId="15339" CreationDate="2023-04-21T09:15:17.487" Score="1" Body="&lt;p&gt;Would you consider popular science book as literary work? While I would demarcate the two genres, what I like about the following instance is that the author beautifully explains modern physics through the eyes of the protagonist who moves through alternate universes in his dreams.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/George_Gamow&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Gamow&lt;/a&gt;, a Soviet/American physicist, wrote &lt;a href=&quot;https://en.wikipedia.org/wiki/Mr_Tompkins&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mr. Tompkins&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://www.goodreads.com/en/book/show/17263&quot; rel=&quot;nofollow noreferrer&quot;&gt;goodreads&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mr Tompkins has become known and loved by many thousands of readers (since his first appearance over fifty years ago) as the bank clerk whose fantastic dreams and adventures lead him into a world inside the atom. George Gamow's classic provides a delightful explanation of the central concepts in modern physics, from atomic structure to relativity, and quantum theory to fusion and fission. Roger Penrose's new foreword introduces Mr Tompkins to a new generation of readers, and reviews his adventures in the light of current developments in physics today.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Excerpt from Wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mr Tompkins' adventures begin when he chooses to spend the afternoon of a bank holiday attending a lecture on the theory of relativity. The lecture proves less comprehensible than he had hoped and he drifts off to sleep and enters a dream world in which the speed of light is a mere 4.5 m/s (10 mph). This becomes apparent to him through the fact that passing cyclists are subject to a noticeable Lorentz–FitzGerald contraction. Mr Tompkins becomes acquainted with the Professor delivering the lectures and ultimately marries the Professor's daughter, Maud.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2023-04-21T09:15:17.487" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15362" PostTypeId="2" ParentId="15339" CreationDate="2023-04-21T13:34:56.537" Score="1" Body="&lt;p&gt;&lt;strong&gt;Paolo Giordano&lt;/strong&gt; is an Italian writer and physicist.&lt;/p&gt;&#xA;&lt;p&gt;His first and most famous novel is &lt;em&gt;La solitudine dei numeri primi&lt;/em&gt; (The solitude of prime numbers), published in 2008. The novel  has sold over a million copies and was translated into thirty languages.&lt;/p&gt;&#xA;&lt;p&gt;Paolo Giordano studied physics at the University of Turin and holds a PhD in theoretical particle physics. He worked at the &lt;em&gt;Istituto nazionale di fisica nucleare&lt;/em&gt; (National Institute of Nuclear Physics), where he studied the properties of the quark bottom, in particular the decay of the meson B in the semileptonic channel and radioactive (from Italian Wikipedia).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Solitude_of_Prime_Numbers_(novel)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/The_Solitude_of_Prime_Numbers_(novel)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paolo_Giordano&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Paolo_Giordano&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://it.wikipedia.org/wiki/Paolo_Giordano_(scrittore)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://it.wikipedia.org/wiki/Paolo_Giordano_(scrittore)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Scientific Publications:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;•   Paolo Gambino, Paolo Giordano, Giovanni Ossola, Nikolai Uraltsev, &lt;em&gt;Inclusive semileptonic B decays and the determination of |Vub|&lt;/em&gt;, &lt;em&gt;Journal of High Energy Physics&lt;/em&gt;, n. 10, ottobre 2007, p. 58.&lt;/p&gt;&#xA;&lt;p&gt;•   Paolo Giordano, &lt;em&gt;Inclusive semileptonic B decays and the determination of |Vub|, Journal of Physics: Conference Series,&lt;/em&gt; v. 110, 30 giugno 2008.&lt;/p&gt;&#xA;&lt;p&gt;•   Paolo Gambino, Paolo Giordano, &lt;em&gt;Normalizing inclusive rare B decays, Physics Letter B,&lt;/em&gt; maggio 2008.*&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-04-21T13:34:56.537" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15363" PostTypeId="2" ParentId="15339" CreationDate="2023-04-21T14:15:54.760" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Jagadish_Chandra_Bose&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sir J. C. Bose&lt;/a&gt; was an Indian polymath specializing in Physics. He was also a writer and composed one of the first works in Bengali science fiction.&lt;/p&gt;&#xA;&lt;p&gt;From Wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1896, Bose wrote &lt;em&gt;Niruddesher Kahini&lt;/em&gt; (The Story of the Missing One), a short story that was later expanded and added to Abyakta (অব্যক্ত) collection in 1921 with the new title &lt;em&gt;Palatak Tuphan&lt;/em&gt; (Runaway Cyclone).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2023-04-21T14:15:54.760" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15364" PostTypeId="2" ParentId="15360" CreationDate="2023-04-21T18:51:16.203" Score="14" Body="&lt;p&gt;It depends on what &amp;quot;abandoned&amp;quot; means. De Morgan did write his main works on probability in 1837-38 and then turned his primary attention to logic. In his lengthy 1837 review of Laplace's &lt;em&gt;Théorie analytique des probabilités&lt;/em&gt;, he did characterize the book as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;the Mont Blanc of mathematical analysis; but the mountain has this advantage over the book, that there are guides always ready near the former, whereas the student has been left to his own method of encountering the latter&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1853 De Morgan mused in a letter to Hamilton that &amp;quot;&lt;em&gt;everybody makes errors in probabilities, at times, and big ones&lt;/em&gt;&amp;quot;, and in 1861 confessed one of his own, see &lt;a href=&quot;https://www.tandfonline.com/doi/full/10.1080/01445340310001599579&quot; rel=&quot;noreferrer&quot;&gt;Rice, ‘Everybody makes errors’: The intersection of De Morgan's Logic and Probability, 1837 – 1847&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So perhaps one can make the inference. Rice does:&amp;quot;&lt;em&gt;read in the light of the above discussion,&lt;/em&gt; [the 1853 quote] &lt;em&gt;seems to give a reasonable explanation for his later neglect of the subject&lt;/em&gt;&amp;quot;. Still, De Morgan continued to use and promote probability in his logical works.&lt;/p&gt;&#xA;&lt;p&gt;When De Morgan talked about making errors, he was speaking from personal experience. His major contribution to probability in 1837 was to spot a mistake in works of Laplace and Poisson, but... in his supposed &amp;quot;correction&amp;quot; he himself made several mistakes, see &lt;a href=&quot;http://www.people.vcu.edu/%7Erhammack/Math490/DeMorgan.pdf&quot; rel=&quot;noreferrer&quot;&gt;Rice, De Morgan in the prehistory of statistical hypothesis&#xA;testing&lt;/a&gt;. And he missed that Poisson  corrected the mistake later in the same work. To top it off, the next year De Morgan made the &lt;em&gt;same&lt;/em&gt; mistake in two other works. The mistake was to confuse &lt;span class=&quot;math-container&quot;&gt;$p(X|Y)$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$p(Y|X)$&lt;/span&gt;, a typical rookie mistake by modern standards and the &amp;quot;thundering error&amp;quot;, as he put it years later. Here is humbled De Morgan of 1861 (quoted from Rice):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There are no questions in the whole range of applied mathematics which require such close attention, and in which it is so difficult to escape error, as those which occur in the theory of probabilities … and, of all subjects, there is no one in which writers of every grade have so frequently or so strangely made mistakes of mere inadvertence. One was pointed out about twenty years ago (‘Camb. Phil. Trans.’) into which both Laplace and Poisson had fallen, one after the other; but the discoverer of their slip proved himself signally liable to greater ones a very little while after.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The discoverer of their slip&amp;quot; is De Morgan himself, as he intimated in a letter to Hamilton:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;I see you did not notice, in the part you have read, a thundering error, on the simplest point possible. Nor has anyone else noticed it, except myself, who have printed a sarcasm against ‘a recent writer’ that I may be able to prove I found it out before anybody.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hamilton did not notice the &amp;quot;thundering error&amp;quot; even after that, so De Morgan had to give him a page reference.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-04-21T19:24:12.080" LastActivityDate="2023-04-21T19:24:12.080" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15365" PostTypeId="1" CreationDate="2023-04-21T20:14:50.190" Score="0" ViewCount="118" Body="&lt;p&gt;Historically, Gauss measured it by an old method called after his name today: The Gauss map.&lt;/p&gt;&#xA;&lt;p&gt;This maps the normal vector at each point of a surface curve to the unit sphere. Then he measured the angle difference between the starting normal at a point and a normal to the surface as it traced the sides of an infinitesimal triangle on the surface and mapped the corresponding normals on the unit sphere to find another triangle there. Finally, divided the corresponding triangle area on the unit sphere by the triangle area on the surface to find the curvature.&#xA;This method appeared in his 1827 work &amp;quot;Disquisitiones generales circa superficies curvas&amp;quot; (General Investigations of Curved Surfaces)&lt;/p&gt;&#xA;&lt;p&gt;Nevertheless, the same map was used by Olinde Rodrigues 12 years earlier than Gauss, anticipating him as early as 1815 by expressing the &amp;quot;Gaussian&amp;quot; curvature as the quotient of the aforementioned surfaces and proving that this curvature is equal to the product of principal curvatures.&lt;/p&gt;&#xA;&lt;p&gt;It is not known today if Gauss was aware of Rodrigues work or discovered it independently but certainly, Olinde Rodrigues has not received the recognition that he deserves, and his work has not been sufficiently disseminated.&lt;/p&gt;&#xA;" OwnerUserId="13809" LastEditorUserId="13809" LastEditDate="2023-04-21T22:20:36.750" LastActivityDate="2023-04-21T22:20:36.750" Title="Who discovered intrinsic curvature first?" Tags="|mathematicians|differential-geometry|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15366" PostTypeId="2" ParentId="15358" CreationDate="2023-04-21T22:15:29.370" Score="4" Body="&lt;p&gt;I have not been able to find out much other than that William Hooper was a translator of contemporary German and French publications:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Letters of Baron Bielfeld: Containing Original Anecdotes of the Prussian Court for the last Twenty Years&lt;/em&gt;, 4 volumes, London 1768-1770.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jakob Friedrich Freiherr von Bielfeld, &lt;em&gt;The Elements of Universal Erudition&lt;/em&gt;, 3 volumes, London 1770. (original work: &lt;em&gt;Les premiers traits de l'erudition universelle; ou, Analyse abregée de toutes les sciences, des beaux-arts et des belles lettres&lt;/em&gt;, 1767).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Louis Sébastien Mercier, &lt;em&gt;Memoirs of the Year Two Thousand Five Hundred&lt;/em&gt;, 2 volumes, London 1772 (original work: &lt;em&gt;L'An Deux Mille Quatre Cent Quarante&lt;/em&gt;, 1771).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Salomon Gessner, &lt;em&gt;New Idylls&lt;/em&gt;, London 1776 (original work: &lt;em&gt;Neue Idyllen&lt;/em&gt;, 1772).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Claude Adrien Helvetius, &lt;em&gt;A Treatise on Man, His Intellectual Faculties and His Education&lt;/em&gt;, 2 volumes, London 1777 (original work: &lt;em&gt;De l'homme, de ses facultés intellectuelles et de son éducation&lt;/em&gt;, 1773).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;No. 2 is significant in that it contains -- according to the Oxford English Dictionary -- the earliest known occurrence of the word &amp;quot;statistics&amp;quot;. I also note that the translator of No. 1 is identified as &amp;quot;Mr. Hooper&amp;quot;, whereas the translator of all other works is identified as &amp;quot;W. Hooper, M.D.&amp;quot;. This &lt;em&gt;may&lt;/em&gt; suggest that William Hooper was born around 1745.&lt;/p&gt;&#xA;&lt;p&gt;I notice that No. 4 was &amp;quot;printed for S. Hooper&amp;quot;. This is the publisher &lt;a href=&quot;https://www.britishmuseum.org/collection/term/BIOG148211&quot; rel=&quot;nofollow noreferrer&quot;&gt;Samuel Hooper&lt;/a&gt; (1733-1793), who was, based on searching Google Books, active between 1756 and 1792. I can find no evidence that the two men were related, though.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-04-22T04:49:32.290" LastActivityDate="2023-04-22T04:49:32.290" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15367" PostTypeId="1" CreationDate="2023-04-21T21:26:32.060" Score="1" ViewCount="85" Body="&lt;p&gt;I was wondering when was the first time that a chemist wrote a chemical equilibrium with the &lt;span class=&quot;math-container&quot;&gt;$\rightleftharpoons$&lt;/span&gt; symbol. And if it was before or after Arrhenius's dissociation theory.&lt;/p&gt;&#xA;" OwnerUserId="18338" OwnerDisplayName="David Moldes" LastEditorUserId="12330" LastEditDate="2023-04-24T05:24:23.900" LastActivityDate="2023-04-24T05:24:23.900" Title="First time equilibrium notation was used" Tags="|notation|chemistry|thermodynamics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15369" PostTypeId="1" CreationDate="2023-04-23T23:16:17.553" Score="0" ViewCount="74" Body="&lt;p&gt;In physics, is it possible to trace the history of the idea that the known forces (electromagnetic, weak, strong and gravity) should be &lt;em&gt;unified&lt;/em&gt; in a &amp;quot;Theory of Everything&amp;quot;? Would it go back to an influence from Parmenides' monism or is this too far-fetched?&lt;/p&gt;&#xA;&lt;p&gt;Further, what has been the debate around this idea (if any) and what would be the different positions? This might veer into philosophy of physics, but has there been physicists opposed to seeking a ToE? (this would have to be recent history, since of course, a clear understanding that there are 4 forces probably dates back no farther than the middle of the 20th century).&lt;/p&gt;&#xA;" OwnerUserId="5438" LastActivityDate="2023-04-23T23:16:17.553" Title="Origins of a &quot;Theory of Everything&quot; in physics" Tags="|physics|theoretical-physics|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15370" PostTypeId="1" CreationDate="2023-04-24T00:14:19.847" Score="0" ViewCount="139" Body="&lt;p&gt;In &lt;a href=&quot;https://mathoverflow.net/a/445370/10059&quot;&gt;this answer&lt;/a&gt; and the comments Joel David Hamkins talks about a conflict between Cantor-Hume principle and Euclid's principle. He writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This principle [Cantor-Hume] is often defended as a fundamental principle for any concept of &amp;quot;the number of&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Those intuitive strict inequalities are hugely discussed (hundred of&#xA;papers) in the philosophical literature surrounding Hume's principle,&#xA;with the main point being the tension between that principle and&#xA;Euclid's principle, asserting that the whole is (strictly) greater&#xA;than the (proper) parts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Galileo had thrown up his hands in confusion at the conflict between&#xA;Euclid's principle and Cantor-Hume, opting ultimately to keep Euclid.&#xA;It seems numerosity is motivated perhaps by the same idea.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maths.ed.ac.uk/%7Etl/docs/Schanuel_Length_of_potato.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schanuel's paper on geometric measure theory&lt;/a&gt;  says the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Of these two advances, Euler's has been by far the more important;&#xA;but we seem, most of us, to have spent more effort retraining our&#xA;intuitions to incorporate Cantor's ideas than Euler's.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Indeed, for instance, extremely useful geometric value, Euler's characteristic (and Betti numbers) exists exactly because Euler rejected Cantor-Hume principle.&lt;/p&gt;&#xA;&lt;p&gt;So, what are the enormous advantages of Hume princople that made it universally adopted despite it effectively contradicts foundations of modern geometric measure theory?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="55" LastEditDate="2023-04-24T11:09:17.773" LastActivityDate="2023-04-24T11:09:17.773" Title="When and why Cantor-Hume principle was universally adopted in set theory instead of Euclid's principle?" Tags="|philosophy-of-science|geometry|euler|cantor|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15371" PostTypeId="1" CreationDate="2023-04-24T00:53:07.253" Score="2" ViewCount="104" Body="&lt;p&gt;Nothing from Eudoxus is extant (unless we count a poem that paraphrases some of his astronomical work).&lt;/p&gt;&#xA;&lt;p&gt;Since he is often cited as the originator of both the theory of proportions found in book 5 of the Elements, and of the method of exhaustion, my question is&lt;/p&gt;&#xA;&lt;p&gt;What is the justification for this claim?&lt;/p&gt;&#xA;" OwnerUserId="1861" LastEditorUserId="6609" LastEditDate="2023-04-24T22:04:47.397" LastActivityDate="2023-04-24T22:04:47.397" Title="Eudoxus sources" Tags="|greek-mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15373" PostTypeId="1" CreationDate="2023-04-25T16:50:06.497" Score="2" ViewCount="51" Body="&lt;p&gt;According to Wikipedia, Sacrobosco's De Sphaera described the Ptolemaic system, but drew on additional ideas from Islamic Astronomy. How did Sacrobosco's model differ from that of Ptolemy?&lt;/p&gt;&#xA;" OwnerUserId="2702" LastActivityDate="2023-04-25T16:50:06.497" Title="How did the Astronomical model of Sacrobosco differ from Ptolemy's?" Tags="|astronomy|islamic-science|ptolemy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15374" PostTypeId="1" CreationDate="2023-04-25T18:16:50.823" Score="1" ViewCount="146" Body="&lt;p&gt;Upon some browsing I find from many online sources that vector calculus was created in the time of late 19th century by Gibbs and Heaviside, but Gauss, Green, Stokes, etc., who lived much before that, have obviously contributed to deep and fundamental theorems of vector calculus. So what exactly transpired, and what do we exactly mean when we say that Heaviside and Gibbs created vector calculus? Was it just a notational simplification and unification of many scattered ideas, or is it something else?&lt;/p&gt;&#xA;" OwnerUserId="18345" LastEditorUserId="12330" LastEditDate="2023-04-27T06:50:47.647" LastActivityDate="2023-04-27T06:50:47.647" Title="When were the foundations of vector calculus laid?" Tags="|calculus|electromagnetism|heaviside|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15375" PostTypeId="1" CreationDate="2023-04-25T19:57:42.000" Score="0" ViewCount="96" Body="&lt;p&gt;I am going through articles about Turing machines which were introduced to solve the entscheidung problem by Alan Turing in 1936. To what I understand, all modern computers are essentially Turing machines, and mathematical logic is a critical concept in computing.&lt;/p&gt;&#xA;&lt;p&gt;So, I want to ask when did it start being used with reference to Turing machines?&lt;/p&gt;&#xA;&lt;p&gt;I see no mention of it when I go through these articles&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="16355" LastEditDate="2023-04-25T21:45:05.233" LastActivityDate="2023-04-25T21:45:05.233" Title="When did mathematical logic start being applied in/ with computer science?" Tags="|computer-science|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15376" PostTypeId="1" CreationDate="2023-04-27T01:51:59.507" Score="0" ViewCount="93" Body="&lt;p&gt;I have seen the book &lt;em&gt;The Annotated Turing&lt;/em&gt; and I found it quite nice to understand the ideas of Alan Turing. Are there similar books for understanding von Neumann's ideas in computing?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="12330" LastEditDate="2023-04-27T22:40:36.683" LastActivityDate="2023-04-27T22:40:36.683" Title="Technical analysis of von Neumann's contributions to computing?" Tags="|computer-science|von-neumann|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15377" PostTypeId="2" ParentId="5673" CreationDate="2023-04-27T18:22:47.637" Score="3" Body="&lt;p&gt;But also see Hajime Inaba's paper &amp;quot;Reading Elementary Principles: Gibbs and the origin of statistical mechanics&amp;quot; ( &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/10.1002/andp.201500256&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://onlinelibrary.wiley.com/doi/10.1002/andp.201500256&lt;/a&gt; ) where, after studying and explaining Gibbs' reasoning carefully, Inaba comes to the conclusion &amp;quot;He [Gibbs] did not introduce a partition function. Rather, he appealed to a correspondence between expectation values in ensembles and thermodynamic quantities.&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="18354" LastActivityDate="2023-04-27T18:22:47.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15378" PostTypeId="1" AcceptedAnswerId="15380" CreationDate="2023-04-29T04:19:49.530" Score="5" ViewCount="277" Body="&lt;p&gt;I am interested in what subjects and activities mathematics undergrads were involved in the beginning of the 20th century. Most subjects mathematics majors study at an intermediate and advanced level now were relatively recently developed (for example, Measure Theory or Modern Logic). How different is an early 20th century mathematics undergraduate from a 21st century mathematics undergraduate?&lt;/p&gt;&#xA;" OwnerUserId="18358" LastEditorUserId="18358" LastEditDate="2023-04-30T02:11:11.023" LastActivityDate="2023-04-30T02:11:11.023" Title="What did the study plan of a early 20th century mathematics undergraduate program look like?" Tags="|mathematics|mathematics-social-history|education|academia|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15379" PostTypeId="1" CreationDate="2023-04-29T04:27:50.600" Score="1" ViewCount="68" Body="&lt;p&gt;I have a question from the first two paragraphs of &lt;a href=&quot;https://physics.stackexchange.com/a/349897/236734&quot;&gt;this answer&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton's idea of absolute space simply appeared as an answer to the following question: What is an inertial system? Saying that an inertial system is one with constant velocity relative to another inertial system of course does not answer the question. To avoid such logical weakness in Newton's first law one has, at some point, to assume that there is a frame of reference, called absolute space, that - by definition - is inertial.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand Galilean relativity consists on transformations among inertial frames of reference and such relations do not forbid an absolute space. In fact, the idea is that we can use Galilean relativity transformations to relate any inertial frame to the absolute space.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Reading between the lines, I understand that Newton first defined absolute space, and then defined every other inertial system in reference to it. This makes sense to me and is quite intuitive.&lt;/p&gt;&#xA;&lt;p&gt;However, what was then the analogue for inertial frame for Galileo?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="10106" LastEditDate="2023-04-30T00:45:43.677" LastActivityDate="2023-04-30T00:45:43.677" Title="Where does motion happen for Galileo?" Tags="|physics|classical-mechanics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15380" PostTypeId="2" ParentId="15378" CreationDate="2023-04-29T15:34:13.037" Score="13" Body="&lt;p&gt;Here is an example.&#xA;link &lt;a href=&quot;https://iiif.lib.harvard.edu/manifests/view/drs:467220382$1i&quot; rel=&quot;noreferrer&quot;&gt;https://iiif.lib.harvard.edu/manifests/view/drs:467220382$1i&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Official Register of Harvard University&lt;br /&gt;&#xA;Volume VI, May 26, 1909, number 16&lt;br /&gt;&#xA;Announcement of the Courses of Instruction offered by the&#xA;Faculty of Arts and Sciences, 1909-1910&lt;/p&gt;&#xA;&lt;p&gt;mathematics on pages 84--88&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UjvEc.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UjvEc.jpg&quot; alt=&quot;cover&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/fkpbF.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/fkpbF.jpg&quot; alt=&quot;84&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/Sjast.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Sjast.jpg&quot; alt=&quot;85&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/cEUQ3.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/cEUQ3.jpg&quot; alt=&quot;86&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/7O7iL.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/7O7iL.jpg&quot; alt=&quot;87&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/TaXai.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TaXai.jpg&quot; alt=&quot;88&quot; /&gt;&lt;/a&gt;&lt;br /&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/X5js0.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/X5js0.jpg&quot; alt=&quot;89&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2023-04-29T15:45:18.167" LastActivityDate="2023-04-29T15:45:18.167" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15383" PostTypeId="1" AcceptedAnswerId="15384" CreationDate="2023-05-01T01:19:49.403" Score="2" ViewCount="490" Body="&lt;p&gt;It's known that Aristarchus was the first to propose heliocentrism,  and Wikipedia also says so, as quoted in this site at &lt;a href=&quot;https://hsm.stackexchange.com/q/9838/18367&quot;&gt;Does Heliocentrism predate Copernicus?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;But in &lt;a href=&quot;https://en.wikipedia.org/wiki/Heraclides_Ponticus#Work&quot; rel=&quot;nofollow noreferrer&quot;&gt;a different place in Wikipedia&lt;/a&gt; it says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Heraclides proposed that the apparent daily motion of the stars was created by the rotation of the Earth on its axis once a day. This view contradicted the accepted Aristotelian model of the universe, which said that the Earth was fixed and that the stars and planets in their respective spheres might also be fixed. Simplicius says that Heraclides proposed that the irregular movements of the planets can be explained if the Earth moves while the Sun stays still.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So Aristarchus was not the first then (Heraclides lived 390-310 B.C., Aristarchus lived 310-230 B.C.).&lt;/p&gt;&#xA;" OwnerUserId="18367" LastActivityDate="2023-05-01T09:45:25.710" Title="Was Aristarchus the first to propose heliocentrism?" Tags="|astronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15384" PostTypeId="2" ParentId="15383" CreationDate="2023-05-01T09:19:55.817" Score="7" Body="&lt;p&gt;Surviving ancient sources are scarce and often unreliable, we do not know and will never know who was &amp;quot;first&amp;quot;. Aristarchus's is the earliest heliocentric proposal we have solid evidence for, Seleucus possibly supported it a century later. The partially heliocentric proposals reported by Adrastus and Theon for the inner planets (Mercury and Venus) also postdate Aristarchus and even Hipparchus, see &lt;a href=&quot;https://www.academia.edu/7498391/Heliocentrism_in_or_out_of_Heraclides&quot; rel=&quot;noreferrer&quot;&gt;Keyser, Heliocentrism in or out of Heraclides&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the older history books it was suggested that Heraclides and Ecphantus had the inner planets  circle the Sun. The meme dates back to the end of 19th century when Schiaparelli, the discoverer of Martian &amp;quot;channels&amp;quot;, promoted it. That is not full heliocentrism, and even that is not supported by modern scholarship. The Wikipedia article linked in the OP quotes &lt;a href=&quot;https://adsabs.harvard.edu/full/1992JHA....23..233E&quot; rel=&quot;noreferrer&quot;&gt;Eastwood's Heraclides and Heliocentrism&lt;/a&gt; to that effect a few lines down. Here is a fuller Eastwood's quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Only two sources have ever been cited to support the connection of Heraclides's name with a heliocentrical motion. These are Simplicius's commentary on Aristotle's Physics and Calcidius's commentary on Plato's Timaeus. I deal below with both, the first more briefly, the second in full detail... What I propose to show is that Calcidius's commentary (and incidentally Simplicius's work as well) offers no ground whatever for attributing to Heraclides of Pontus&#xA;an idea of circumsolar orbits for Mercury and Venus. This idea came into the Middle Ages only through Martianus Capella.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Nowhere in the ancient literature mentioning Heraclides of Pontus is there a clear reference to his support for any kind of heliocentrical planetary motion. Even more to the point, in none of the places where we find Heraclides's well&#xA;known proposal of the hypothesis of Earth's diurnal rotation do we find any suggestion of some further revolutionary idea such as heliocentric motion for any of the planets. The conclusion should be clear - modern proposals for an&#xA;ancient Heraclidean heliocentrism have come from post-Copernican expectations rather than from a dispassionate reading of the texts.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, there were older Pythagorean models that we know of where the Earth was moving... but not around the Sun, see &lt;a href=&quot;https://articles.adsabs.harvard.edu/full/2002JAHH....5...89T&quot; rel=&quot;noreferrer&quot;&gt;Danezis et al., From Pythagoreans to Kepler&lt;/a&gt;. In the model of Philolaus they both moved around the &amp;quot;central fire&amp;quot;, a.k.a. the &amp;quot;hearth of the universe&amp;quot;. Hicetas had an even stranger idea, as reported by Cicero:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The Syracusan Hicetas, as Theophrastus asserts, holds the view that the heaven, the Sun, the&#xA;Moon, the stars, and in short all the things on high are stationery, and that nothing in the world&#xA;is in motion except the Earth, which by revolving and twisting round its axis with extreme&#xA;velocity produces all the same result as would be produced if the Earth were stationery and the&#xA;heavens in motion&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some historians believe that Pythagorean models influenced Aristarchus.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-05-01T09:45:25.710" LastActivityDate="2023-05-01T09:45:25.710" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15385" PostTypeId="2" ParentId="14262" CreationDate="2023-05-01T12:05:46.073" Score="5" Body="&lt;p&gt;While is not a general history of condensed matter physics, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/019505329X&quot; rel=&quot;noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Out of the Chrystal Maze: Chapters from The History of Solid State Physics&lt;/a&gt; is a history of solid state physics.&lt;/p&gt;&#xA;" OwnerUserId="18165" LastActivityDate="2023-05-01T12:05:46.073" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15386" PostTypeId="1" AcceptedAnswerId="15387" CreationDate="2023-05-01T22:26:13.317" Score="3" ViewCount="382" Body="&lt;p&gt;This question's pretty simple really.  A concept of the number 0 is a major landmark that's used when discussing advanced civilizations in pre-modern history.  It was something that civilizations came to use after thousands of years and was a &amp;quot;major advancement&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;But why?  &lt;code&gt;3 - 1 = 2&lt;/code&gt;, &lt;code&gt;2 - 1 = 1&lt;/code&gt;, and &lt;code&gt;1 - 1 = ... well, 0&lt;/code&gt;.  To be clear, I understand that before this, there still would've been concepts such as &lt;code&gt;null&lt;/code&gt; or, probably much more universally, &lt;code&gt;there isn't/aren't any&lt;/code&gt;.  Furthermore a concept of negative numbers seems less natural and intuitive, so I don't put that in the same category.  However, &lt;code&gt;0&lt;/code&gt; is something that is quite visible to the naked eye (as when you run completely out of food, which is something Ancients were in routine fear of).  Really, not having &lt;code&gt;0&lt;/code&gt; seems like quite the elephant in the room, especially among civilizations with full-blown writing systems and reasonably sophisticated architecture, and the reason that this was so difficult to attain to scientifically seems like an elephant the room as well.&lt;/p&gt;&#xA;&lt;p&gt;Why was the concept of zero so difficult to come by?&lt;/p&gt;&#xA;" OwnerUserId="18370" OwnerDisplayName="Panzercrisis" LastActivityDate="2023-05-02T09:43:53.840" Title="What was so hard about discovering/inventing the number zero?" Tags="|science|mathematics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15387" PostTypeId="2" ParentId="15386" CreationDate="2023-05-02T00:37:31.763" Score="13" Body="&lt;p&gt;&lt;strong&gt;Zero is not a number&lt;/strong&gt; as long as a number is understood as a count of things. You can have one thing or two things or ten things, but if you have one and take it away, there are no things there to count.&lt;/p&gt;&#xA;&lt;p&gt;As long as numbers were being used to record tallies of various things -- so many cattle, so much gold, so many enemies slaughtered -- only the positive integers were really needed.&lt;/p&gt;&#xA;&lt;p&gt;Even today, if I show people my open hand holding two marbles and ask them &amp;quot;How many marbles am I holding?&amp;quot; they'll look at you oddly and say &amp;quot;two&amp;quot;.  If you hold up your empty hand and ask the same question, they'll look at you even more oddly and say something like &amp;quot;You don't have any marbles in your hand.&amp;quot; (It's true that some people will say &amp;quot;Zero marbles&amp;quot; but we all tend to look at them a bit oddly...)&lt;/p&gt;&#xA;&lt;p&gt;Modern mathematics has at its base the &amp;quot;Natural Numbers&amp;quot; a.k.a. the &amp;quot;counting numbers&amp;quot; which are the integers starting at one and going up.  The non-negative integers are a superset of the natural numbers with zero added, and the integers are a superset of those with the negative numbers added.  And more complicated numbers — rational, irrational, transcendental, complex, etc. — are built further on that base.&lt;/p&gt;&#xA;&lt;p&gt;The idea that there should be a &lt;em&gt;number&lt;/em&gt; for no objects turns out to be a fairly sophisticated concept.&lt;/p&gt;&#xA;" OwnerUserId="16773" LastEditorUserId="12330" LastEditDate="2023-05-02T01:09:07.430" LastActivityDate="2023-05-02T01:09:07.430" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15388" PostTypeId="2" ParentId="15386" CreationDate="2023-05-02T09:43:53.840" Score="-1" Body="&lt;p&gt;The Zero was used as a pure addendum/non-stand alone symbol most of the centuries.&#xA;F.e. the old Sumerer used a special sign which indicated a multiple of the base number 5000 years ago.&lt;/p&gt;&#xA;&lt;p&gt;But it was not used as a stand-alone symbol indicating the abstract entity &amp;quot;nothing&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Just like a comma or a space is not used without other symbols - you need to sandwich it.&#xA;The space is not even visible.&lt;/p&gt;&#xA;&lt;p&gt;But the Indians used it already as a stand alone symbol for &amp;quot;nothing&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;And in modern times some centuries ago, who would fight against the use of the Number Zero in Europe?&lt;/p&gt;&#xA;&lt;p&gt;Of course the churches in Europe were a reliable censor for many scientific discoveries and progress, as usual.&lt;/p&gt;&#xA;&lt;p&gt;The church was against the modern concept of the Zero and declared it's use to be blasphemy.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It stands for nothing, but if you write it at the end of a number, it means a multiplication by 10?&lt;/p&gt;&#xA;&lt;p&gt;And how will you devide through zero?&lt;/p&gt;&#xA;&lt;p&gt;And where do you find it in the Bible?&lt;/p&gt;&#xA;&lt;p&gt;That Zero must be pure devil's work. Let's call the Spanish&#xA;Inquisition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Luckily the powerful Italian trading families used already modern bookkeeping/accounting in the 17th century, with Debit and Credit sheets.&lt;/p&gt;&#xA;&lt;p&gt;They would control their financial assets by substracting each 2 corresponding entries - it must deliver zero. And they used a symbol for that non-existing difference.&lt;/p&gt;&#xA;&lt;p&gt;And it was also in Italy where  Fibonacci introduced the arabian/indian number system in the 13th century, replacing the old unusable Roman numeral system, including a first introduction of zero in Europe.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, mathematicians developed the system of differentiation and integration in the 17th century, too, where zero can be interpreted as a number getting smaller and smaller in infinite number of steps.&lt;/p&gt;&#xA;&lt;p&gt;In that aspect, the zero is a kind of counterpart of infinity.&lt;/p&gt;&#xA;&lt;p&gt;Source:&#xA;&amp;quot;The Nothing That Is: A Natural History of Zero&amp;quot; by Robert Kaplan.&lt;/p&gt;&#xA;" OwnerUserId="18374" LastActivityDate="2023-05-02T09:43:53.840" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15390" PostTypeId="1" CreationDate="2023-05-02T17:03:35.873" Score="1" ViewCount="86" Body="&lt;p&gt;Is there any proof or even just indication that they did: I can't find anything online and was hoping for some help.&lt;/p&gt;&#xA;" OwnerUserId="18091" LastEditorUserId="55" LastEditDate="2023-05-04T10:21:14.137" LastActivityDate="2023-05-04T10:21:14.137" Title="Were ancient astronomers (Babylonians, Egyptians or Greeks) aware of the solar analemma?" Tags="|astronomy|archaeoastronomy|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15391" PostTypeId="1" CreationDate="2023-05-02T22:34:07.657" Score="1" ViewCount="102" Body="&lt;p&gt;So I am writing my thesis around the Maxwell equations, in one part I saw to speak about their history. The issue is, I found conflicting pieces of information as to who enunciated the laws first.&lt;/p&gt;&#xA;&lt;p&gt;Let's take Faraday's law for example:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\oint \vec E \cdot d\vec s = - \frac{d}{dt} \int\vec B \cdot \vec n dA$$&lt;/span&gt;&#xA;I have searched in countless places about its first enunciation by Michael Faraday, but I found -through this site itself- that he formulated it verbally. The question I have here is: where do I find the citation/citations mentioning this by Faraday? And possibly who formulated this law in mathematical form with its reference for documentation.&lt;/p&gt;&#xA;&lt;p&gt;The same question I may ask from you my friends about Gauss's law which is the second equation of the 4 equations.&lt;/p&gt;&#xA;&lt;p&gt;Also, where can I find when and by whom exactly the local forms have been derived (with reference if possible)?&lt;/p&gt;&#xA;&lt;p&gt;And overall, where can I find a well documented historical report concerning those 4 fundamental equations?&lt;/p&gt;&#xA;&lt;p&gt;Thanks a lot and I apologize for taking some of your time.&lt;/p&gt;&#xA;" OwnerUserId="18356" LastEditorUserId="6609" LastEditDate="2023-05-03T18:19:56.940" LastActivityDate="2023-05-03T18:19:56.940" Title="A question concerning the history of Maxwell's equations" Tags="|electromagnetism|maxwell|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15393" PostTypeId="1" CreationDate="2023-05-04T12:11:11.227" Score="1" ViewCount="121" Body="&lt;p&gt;In &lt;em&gt;Investigations on the theory of Brownian movement&lt;/em&gt;, he says that osmosis holds for solute particles but not for Brownian ones, since they are too big to pass through a permeable membrane.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;So what? Why should this prevent such Brownian particles to exercise pressure?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Later on, one reads&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But a different conception is reached from the standpoint of the molecular-kinetic theory of heat. According to this theory a dissolved molecule is differentiated from a suspended body &lt;em&gt;solely&lt;/em&gt; by its dimensions&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In my view, &lt;strong&gt;this argument seems totally uncorrelated from the previous one&lt;/strong&gt;, even though in the paper they are contiguous. I don't understand the reasoning.&lt;/p&gt;&#xA;" OwnerUserId="13636" LastEditorUserId="13636" LastEditDate="2023-05-04T15:35:19.850" LastActivityDate="2023-05-04T15:35:19.850" Title="On Einstein's argument at the beginning of his PhD thesis" Tags="|einstein|atomic-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15395" PostTypeId="1" CreationDate="2023-05-05T07:14:20.760" Score="1" ViewCount="89" Body="&lt;p&gt;The earliest number notation I can find is the &lt;a href=&quot;https://www.researchgate.net/figure/The-cuneiform-notation-of-sexagesimal-digits-Zero-is-represented-by-a-blank-space-The_fig2_314097670#:%7E:text=NoDerivs%203.0%20Unported-,The%20cuneiform%20notation%20of%20sexagesimal%20digits%3A%20Zero%20is%20represented%20by,or%2059%20%3D%2050%20%2B%209&quot; rel=&quot;nofollow noreferrer&quot;&gt;sexagesimal digit notation of cuneiform&lt;/a&gt;. From my understanding of that, in order to make numbers bigger than 60, you would just add the smaller numbers up until you get the number you want. Famously, for &lt;a href=&quot;https://en.wikipedia.org/wiki/Roman_numerals&quot; rel=&quot;nofollow noreferrer&quot;&gt;Roman numerals&lt;/a&gt;, the digits are 1=I, V=5, X=10, L=50, C=100, and M=1,000. The Romans had ways of expressing bigger numbers, such as by drawing a bar (called a &lt;a href=&quot;https://en.wikipedia.org/wiki/Vinculum_(symbol)&quot; rel=&quot;nofollow noreferrer&quot;&gt;vinculum&lt;/a&gt; over the top of the number, and enclosing the numbers on the sides and bottom. Regardless, the number system was still limited at some point in the size of the number it could portray.&lt;/p&gt;&#xA;&lt;p&gt;Now, I know that &lt;a href=&quot;https://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numeral_system#:%7E:text=It%20was%20invented%20between%20the%201st%20and%204th%20centuries%20by%20Indian%20mathematicians.&quot; rel=&quot;nofollow noreferrer&quot;&gt;our &amp;quot;Arabic&amp;quot; numerals are actually Indian&lt;/a&gt; and they are &lt;a href=&quot;https://en.wikipedia.org/wiki/Positional_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;positional&lt;/a&gt;, which means (taken from Wikipedia) &amp;quot;&lt;em&gt;the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit.&lt;/em&gt;&amp;quot; Wikipedia has a really good graphic to make this all more understandable (I think):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/4r94a.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/4r94a.png&quot; alt=&quot;positional notation explained&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;My understanding of that radix is the base number, and that's how many digits in the numeral system. The digits, are just the numbers in each position, and the position determines which index   number the digit is. Lastly, the index numbers are the place digits, and they determine how many times the digit is multiplied by. For example, the first index position (2) is multiplied by the radix (10) to make that digit equal 20.&lt;/p&gt;&#xA;&lt;p&gt;So, I THINK I understand positional notation now. And, &lt;em&gt;I believe&lt;/em&gt; the answer I want to know is what is the first base 10 positional notation, where the number 0 is added onto other digits to make multiples of 10? To be clear, I don't care if the &amp;quot;0&amp;quot; looks anything like our 0, just that it represents nothing, but when added to another number represents a multiple of ten.&lt;/p&gt;&#xA;&lt;p&gt;****Note: upon further research, I am pretty sure I have found the answer, and I am going to add the answer so I can find it later. If anyone has more to add (or if my answer is wrong) they can add their own answer.&lt;/p&gt;&#xA;" OwnerUserId="3917" LastActivityDate="2023-05-06T00:30:29.930" Title="When was the first time the number 0 came to mean a multiple of ten?" Tags="|mathematics|number-theory|numbers|ancient-india|india|" AnswerCount="1" CommentCount="2" ClosedDate="2023-05-24T21:01:23.703" ContentLicense="CC BY-SA 4.0" />
  <row Id="15396" PostTypeId="1" CreationDate="2023-05-05T19:26:41.183" Score="0" ViewCount="206" Body="&lt;p&gt;We often hear scientists write of the Queen of Sciences.  To what were they referring?  And is it relevant in today's research?&lt;/p&gt;&#xA;" OwnerDisplayName="user18391" LastEditorUserId="131" LastEditDate="2023-05-08T12:26:10.713" LastActivityDate="2023-05-08T12:26:10.713" Title="In Middle Ages Europe, Physical scientists referred to the &quot;Queen of Sciences&quot;; what was this?" Tags="|middle-ages|science|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15397" PostTypeId="2" ParentId="15396" CreationDate="2023-05-05T19:44:30.553" Score="2" Body="&lt;p&gt;There were at least two Queens:&lt;/p&gt;&#xA;&lt;p&gt;In the late Middle Ages, Theology was called Queen of the sciences.  See, e.g., &lt;a href=&quot;https://journals.sagepub.com/doi/full/10.1177/0953946819868092#:%7E:text=That%20theology%20used%20to%20be,vague%20and%20general%2C%20lacking%20precision.&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;How Theology Stopped Being Regina Scientiarum—and How Its Story Continues&amp;quot;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subsequently in the 1700s, the title was transferred to mathematics.  E.g., Gauss who said &amp;quot;Mathematics is the queen of the sciences and number theory is the queen of mathematics.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;(There's a lot more if you google &amp;quot;Queen of the Sciences&amp;quot;...)&lt;/p&gt;&#xA;&lt;p&gt;Mathematics is, of course, relevant.  And in some areas -- String Theory, for example -- there seems to be no hope left but prayer...&lt;/p&gt;&#xA;" OwnerUserId="16773" LastActivityDate="2023-05-05T19:44:30.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15398" PostTypeId="1" AcceptedAnswerId="15400" CreationDate="2023-05-05T21:34:55.393" Score="4" ViewCount="78" Body="&lt;p&gt;I'm reading some articles about the history of Brownian Motion, for example, that of Maiocchi (&lt;a href=&quot;https://www.jstor.org/stable/4026755&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.jstor.org/stable/4026755&lt;/a&gt;) or Genthon (&lt;a href=&quot;https://arxiv.org/pdf/2006.05399.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/pdf/2006.05399.pdf&lt;/a&gt;), and they all cite this &lt;strong&gt;Felix Exner&lt;/strong&gt;, son of Siegmund Exner, who made some experiments with the aim of finding a velocity for Brownian motion etc.&lt;/p&gt;&#xA;&lt;p&gt;At the related &lt;a href=&quot;https://de.wikipedia.org/wiki/Siegmund_Exner-Ewarten&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia page&lt;/a&gt;, one reads:&#xA;&amp;quot;Siegmund Exner's marriage to Emilie née von Winiwarter (1847–1909) produced two children, the surgeon and university professor Alfred von Exner-Ewarten in 1875 and the meteorologist and university professor Felix Maria von Exner-Ewarten in 1876.&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I found nothing regarding the experiments with Brownian particles supposedly made by this Felix Maria. Is he the same as the one mentioned by Maiocchi et. al.? Are there some references somewhere?&lt;/p&gt;&#xA;" OwnerUserId="13636" LastActivityDate="2023-05-06T02:19:22.390" Title="The case of Felix Exner and Brownian motion" Tags="|einstein|atomic-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15399" PostTypeId="2" ParentId="15395" CreationDate="2023-05-06T00:30:29.930" Score="0" Body="&lt;p&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/#:%7E:text=The%20oldest%20dated%20Indian%20document,our%20calendar%20of%20594%20AD.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quoting from here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Indians were the first to develop a base 10 positional system and, considering the date of the Babylonian system, it came very late indeed.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The oldest dated Indian document which contains a number written in the place-value form used today is a legal document dated 346 in the Chhedi calendar which translates to a date in our calendar of 594 AD.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To be fair, the date on it may be a later addition, so there is some dispute to this. That being said, other examples of the base-10 positional notation being found are dated to 737 AD.&lt;/p&gt;&#xA;" OwnerUserId="3917" LastActivityDate="2023-05-06T00:30:29.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15400" PostTypeId="2" ParentId="15398" CreationDate="2023-05-06T02:19:22.390" Score="1" Body="&lt;p&gt;Exner's paper on Brownian Motion was published in &lt;a href=&quot;https://archive.org/details/sim_annalen-der-physik_1900_2_4/page/842/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Annelan der Physik und Chemie (1900): Volume 2 Issue 4&lt;/em&gt;&lt;/a&gt; available on archive.org (in German), starting on page 842.&lt;/p&gt;&#xA;&lt;p&gt;So yes, it is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Maria_von_Exner-Ewarten&quot; rel=&quot;nofollow noreferrer&quot;&gt;Felix Maria von Exner-Ewarten&lt;/a&gt; cited by Maiocchi and Genthon.&lt;/p&gt;&#xA;&lt;p&gt;Presumably he was familiar with fluid dynamics through his work and studies in geophysics and meteorology.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-05-06T02:19:22.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15401" PostTypeId="2" ParentId="15396" CreationDate="2023-05-06T12:53:27.630" Score="1" Body="&lt;p&gt;There is a much-quoted description of the elaborately sculpted tomb of Isaac Newton, repeated (for instance) in the April 1731 (vol 1, number 4, p.159) issue of &lt;a href=&quot;https://www.google.com/books/edition/Gentleman_s_Magazine_and_Historical_Revi/40QdAQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22queen+of+the+sciences%22&amp;amp;pg=PA159&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gentleman's Magazine&lt;/a&gt;, in the year &lt;a href=&quot;https://www.westminster-abbey.org/abbey-commemorations/commemorations/sir-isaac-newton&quot; rel=&quot;nofollow noreferrer&quot;&gt;Westminster Abbey&lt;/a&gt; says the statuary was unveiled.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... On the globe sits the figure of &lt;em&gt;Astronomy&lt;/em&gt; weeping, with a sceptre in her hand (as Queen of the Sciences) and a &lt;em&gt;Star&lt;/em&gt; over her head on the &lt;em&gt;Pyramid&lt;/em&gt;. ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Other publications with almost identical descriptions include &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=hvd.hxgf2u&amp;amp;view=1up&amp;amp;seq=428&amp;amp;q1=%22queen%20of%20the%20sciences%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Political State of Great Britain&lt;/a&gt;, April 1731, vol. 41, p.412,413; &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=nyp.33433087362327&amp;amp;view=1up&amp;amp;seq=325&amp;amp;q1=%22queen%20of%20the%20sciences%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Present State of the Republick of Letters&lt;/a&gt;, April 1731, vol. 7, pp.317-320; and &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=uc1.31822008509556&amp;amp;view=1up&amp;amp;seq=572&amp;amp;q1=%22queen%20of%20the%20sciences%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Pennsylvania Gazette&lt;/a&gt;, 22-29 July 1731, No. 141 (p.512 in the 1968 reprint).  The &lt;em&gt;Republick&lt;/em&gt; version has some added content, including a quotation of Pope's &amp;quot;All Nature and its Laws lay  in Night; God said let &lt;em&gt;Newton&lt;/em&gt; be, and all was Light&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Astronomy is not far removed from mathematics, and so on, but Q of S can have other meanings than Theology and Mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Added 7 May 2023:  An earlier English instance is in John Gadbury's 1677 interestingly titled book &lt;a href=&quot;https://www.google.com/books/edition/The_Just_and_Pious_Scorpionist_Or_the_Na/VG5jAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=queen+science&amp;amp;pg=PP8&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Just and Pious Scorpionist...&lt;/a&gt;, where the preface (flatteringly titled &amp;quot;To the Honest and Intelligent Reader&amp;quot;) states&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For it is the &lt;em&gt;Divine Urania!&lt;/em&gt; and &lt;em&gt;she&lt;/em&gt; alone ! that is the &lt;em&gt;Mistris&lt;/em&gt; and &lt;em&gt;Queen&lt;/em&gt; of all &lt;em&gt;Natural Sciences&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alas, it is clear that the author means Urania the muse of astrology not astronomy.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2023-05-07T21:56:04.690" LastActivityDate="2023-05-07T21:56:04.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15403" PostTypeId="2" ParentId="7667" CreationDate="2023-05-06T21:17:37.340" Score="0" Body="&lt;p&gt;&lt;strong&gt;Scientific Method&lt;/strong&gt; Strictly speaking the modern method of science is the Empirical Method, adopted in the late 1800s. And as you know, it requires (1) Observation, (2) Repeatability, and (3) testability/falsifiability.&lt;/p&gt;&#xA;&lt;p&gt;Most sociological ideologies, psychological theories, etc. do not qualify as &lt;em&gt;Scientific&lt;/em&gt; in this strictest sense. While they may not be scientific fact, they can serve as hypotheses in a useful way...with conditions. Or they may just be opinions propounded by researchers in the social sciences.&lt;/p&gt;&#xA;&lt;p&gt;Darwinian evolution, although eventually applied to social sciences, deals with biological sciences, and has for decades been the subject of debate as to its falsifiability (and therefore, its scientific validity). It should be noted that most modern scientists have abandoned the original thesis of Darwin, and moved on to Synthetic Molecular theories to explain the Origin of Species.&lt;/p&gt;&#xA;&lt;p&gt;If Popper became aware of the upgrade on the Evolutionary research, it may have been the basis for his change in position. It is a noble character that is willing to amend, emend, or modify his understanding of facts. Of course, any conclusion as to Popper's reasons for change must also be open to examination and falsifiability!&lt;/p&gt;&#xA;&lt;p&gt;To respond to the OP question, &lt;em&gt;Is my brief summary of likely Popperian attacks on Freud and Marx grossly wrong?&lt;/em&gt;, the answer is NO. Popper correctly noticed the failure of Freud and Marx to satisfy all the requirements of Scientific research; and strictly speaking, the old Darwinian evolutionary model failed too. To be consistent, Darwin had to be placed in the same camp as F &amp;amp; M.&lt;/p&gt;&#xA;&lt;p&gt;But when Popper modified his understanding of only requiring &amp;quot;testability&amp;quot;, he opened the door for metaphysics and hypothetical ideas to be valuable as the &amp;quot;inspiration&amp;quot; for legitimate science endeavors. (See answer above.) There is no dilemma if his particular definitions are considered. This allows the &amp;quot;mutation&amp;quot; of understanding a model when new facts are discovered.&lt;/p&gt;&#xA;" OwnerDisplayName="user18391" LastEditorDisplayName="user18391" LastEditDate="2023-05-08T20:55:38.513" LastActivityDate="2023-05-08T20:55:38.513" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15404" PostTypeId="1" AcceptedAnswerId="15407" CreationDate="2023-05-07T10:42:40.347" Score="1" ViewCount="139" Body="&lt;p&gt;The biographies of Thoralf Skolem focus on his scientific achievements. I could not find any data regarding whether he was an only child or had siblings.&lt;/p&gt;&#xA;" OwnerUserId="7165" LastEditorUserId="131" LastEditDate="2023-05-08T12:25:18.113" LastActivityDate="2023-07-14T20:23:13.347" Title="Did Skolem have any siblings?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15406" PostTypeId="1" AcceptedAnswerId="15409" CreationDate="2023-05-09T18:27:20.243" Score="4" ViewCount="356" Body="&lt;p&gt;MIT's &lt;a href=&quot;https://en.wikipedia.org/wiki/RSA_(cryptosystem)&quot; rel=&quot;nofollow noreferrer&quot;&gt;RSA&lt;/a&gt; encryption was granted a patent although it was not enforced for non-commercial applications. Similarly for Stanford's &lt;a href=&quot;https://en.wikipedia.org/wiki/Pretty_Good_Privacy&quot; rel=&quot;nofollow noreferrer&quot;&gt;PGP&lt;/a&gt; encryption algorithm.  However, these are institutions rather than mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;A borderline case of maths and physics would be Mauchly and Eckert unsuccessful application for a patent on their work relating to the &lt;a href=&quot;https://en.wikipedia.org/wiki/ENIAC&quot; rel=&quot;nofollow noreferrer&quot;&gt;ENIAC&lt;/a&gt;.  (There is an interesting backstory to this case which resulted in hostilities with von Neumann.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Q:&lt;/strong&gt; What are some examples of mathematicians who either successfully or unsuccessfully applied to patent their work?&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2023-05-13T20:12:14.727" LastActivityDate="2023-11-14T15:22:27.660" Title="Examples of mathematicians who applied to patent their work" Tags="|mathematics|mathematicians|examples|" AnswerCount="6" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15407" PostTypeId="2" ParentId="15404" CreationDate="2023-05-09T21:48:54.207" Score="7" Body="&lt;p&gt;Thoralf Albert Skolem (1887 - 1963) had several siblings, but they were 10-15 years older than him. At some time before 1900, Thoralf and his parents relocated from Sandsvær near Kongsberg to Kristiania (now Oslo). The siblings were born in the 1870s and had left the home by then. Hence they are not present when Thoralf's mathematical interests start growing.&lt;/p&gt;&#xA;&lt;p&gt;On the siblings: Two brothers (Helge and Svein) had long military careers. A third brother (Anund) was a sailor, and appears to have died young. Another brother (Thoralf) died before Thoralf Albert was born, while a sister (Magna Sofie) outlived all her siblings and died at the age of 96.&lt;/p&gt;&#xA;&lt;p&gt;It might be of interest to notice that in Kristiania/Oslo, Thoralf lived at the same address as his parents until they died, his mother Olette in 1914 and his father Even in 1928. Not until 1927 did Thoralf marry.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Edit&lt;/em&gt;: The information above was pieced together from various documents available at &lt;a href=&quot;https://www.digitalarkivet.no/en/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Digital Archives&lt;/a&gt;, which is part of the Norwegian National Archives. In particular, the 1875 and the 1891 censuses for Sandsvær and the 1900 and the 1923 censuses for Kristiania were used, as well as church books giving information on baptisms, marriages and deaths. (For a mathematical biography of Skolem, see the 8 page long biographical sketch by Jens Erik Fenstad in Skolem's &lt;em&gt;Selected Works in Logic&lt;/em&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="40" LastEditorUserId="40" LastEditDate="2023-07-14T20:23:13.347" LastActivityDate="2023-07-14T20:23:13.347" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15408" PostTypeId="1" CreationDate="2023-05-09T21:50:20.090" Score="1" ViewCount="201" Body="&lt;p&gt;For example, Albert Einstein complained that sociology departments across the quad were using his theory of relativity to advance the idea of &amp;quot;relativistic morality.&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The meaning of relativity has been widely misunderstood. Philosophers play with the word, like a child with a doll. Relativity as I see it, merely denotes that certain physical and mechanical facts, which have been regarded as positive and permanent, are relative with regard to certain other facts in the sphere of physics and mechanics.&#xA;It does not mean that everything in life is relative and that we have the right to turn the whole world mischievously topsy-turvy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Wikiquote, from 1929 Viereck interview)&lt;/p&gt;&#xA;&lt;p&gt;Liberal theologians, for another example, have adopted the model of evolution to come up with Theistic Evolution in their systematic theologies.&lt;/p&gt;&#xA;&lt;p&gt;The proponents of eugenics in the early 20th century co-opted Darwinian evolution as a pretext for manipulating human species.  As well, did Thomas Huxley adapt Darwinian evolution to promote the superiority of certain races in the anthropology department!&lt;/p&gt;&#xA;&lt;p&gt;Has anyone scrutinized this tendency of academia to &amp;quot;co-opt scientific vocabulary&amp;quot; to further their theories or ideologies?  Have any other scientists, besides Einstein, objected to misuse of their scientific verbage?&lt;/p&gt;&#xA;" OwnerDisplayName="user18391" LastEditorUserId="6609" LastEditDate="2023-05-12T22:46:07.220" LastActivityDate="2023-06-25T15:33:42.340" Title="In a survey of historical science has anyone studied the propensity of other departments to co-opt scientific terminology to further their own ideas?" Tags="|terminology|philosophy-of-science|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15409" PostTypeId="2" ParentId="15406" CreationDate="2023-05-10T06:54:22.740" Score="3" Body="&lt;p&gt;Unfortunately Penrose &lt;a href=&quot;https://patents.google.com/patent/US4133152A/en&quot; rel=&quot;nofollow noreferrer&quot;&gt;patented&lt;/a&gt; the tiles associated with his name.&lt;/p&gt;&#xA;" OwnerUserId="633" LastActivityDate="2023-05-10T06:54:22.740" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15410" PostTypeId="1" CreationDate="2023-05-10T10:03:10.423" Score="3" ViewCount="561" Body="&lt;p&gt;I want to know examples of very flattering comments from famous scientists or mathematicians concerning younger colleagues. Here's an example of what I have in mind (I shall provide more as an answer): after reading de Broglie’s master’s thesis, Einstein wrote: “He raised a corner of the big veil”.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="6609" LastEditDate="2023-05-11T22:25:36.527" LastActivityDate="2023-07-25T08:10:10.773" Title="Examples of flattering comments from a famous scientist concerning a young colleague" Tags="|biographical-details|quote|examples|big-list|" AnswerCount="7" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15411" PostTypeId="2" ParentId="15410" CreationDate="2023-05-10T10:07:38.573" Score="1" Body="&lt;p&gt;Isaac Newton wrote about Roger Cotes: “If he had lived we would have known something.”&lt;/p&gt;&#xA;&lt;p&gt;Albert Einstein once said that the reason why he kept going to the Institute of Advanced Studies after having retired was “to have the privilege of walking with Gödel to go home”.&lt;/p&gt;&#xA;" OwnerUserId="5989" LastEditorUserId="5989" LastEditDate="2023-05-10T16:15:02.180" LastActivityDate="2023-05-10T16:15:02.180" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15412" PostTypeId="2" ParentId="15410" CreationDate="2023-05-10T16:06:08.070" Score="1" Body="&lt;p&gt;Ananyo Bhattacharya provides a number of examples in his recent biography of John von Neumann.&lt;/p&gt;&#xA;&lt;p&gt;For example, Abraham Fraenkel channeled Bernoulli:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] a long manuscript of an author unknown to me, Johannes von Neumann, with the title &lt;em&gt;Die Axiomatisierung der Mengenlehre&lt;/em&gt; [...] I don't maintain that I understood everything, but enough to see that this was an outstanding work and to recognize &lt;em&gt;ex ungue leonem&lt;/em&gt;. (to know the lion by his claw)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One which made me laugh out-loud was Hilbert's comments on von Neumann.  Von Neumann quickly became a favourite of Hilbert at Göttingen, much to the annoyance of the &amp;quot;old guard&amp;quot;.  The pair would go for walks in Hilbert's garden and lock themselves up together to discuss mathematics and the new quantum theory. Hilbert, being one of the examiners at von Neumann's oral exam is alledged to have asked just one question:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In all my years I have never seen such elegant evening clothes: pray, who is the candidates tailor?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(This is reminiscent in spirit to Russell's comment to Moore when entering the examination room for Wittgenstein's oral exam.)&lt;/p&gt;&#xA;&lt;p&gt;I found this claim rather confusing since von Neumann did his doctorate at Berlin rather than Göttingen.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Gauss had some very flattering things to say of Riemann, describing his doctoral thesis as being of &amp;quot;gloriously fertile originality.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2023-05-10T21:49:50.107" LastActivityDate="2023-05-10T21:49:50.107" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15413" PostTypeId="2" ParentId="15406" CreationDate="2023-05-11T00:27:43.240" Score="3" Body="&lt;p&gt;AT&amp;amp;T Bell Labs famously patented some aspects of KORBX, their implementation of an interior point method for LP based on Karmarkar's algorithm. Karmarkar was employed by AT&amp;amp;T.&lt;/p&gt;&#xA;" OwnerUserId="5579" LastActivityDate="2023-05-11T00:27:43.240" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15414" PostTypeId="2" ParentId="15410" CreationDate="2023-05-11T00:34:08.940" Score="1" Body="&lt;p&gt;There is a lecture (available online) by Leonard Susskind at Cornell University, where Susskind makes a break when discussing the holographic principle to talk about &lt;a href=&quot;https://en.wikipedia.org/wiki/Juan_Maldacena&quot; rel=&quot;nofollow noreferrer&quot;&gt;Juan Maldacena&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was in 1994 or so this conjecture was put forward. Most people at the time thought it was a screwball in motion. But it took a couple of years, basically until 1998 a young Argentinian physicist, whose name you should know. I ask people, do you know who Juan Maldacena is? I'm talking about physicists or physics groupies and so forth, often they don't. Juan Maldacena is perhaps the greatest physicist of his generation. I won't tell you how old he is. He's younger than me, so he's not necessarily greater than me... [audience laughs] but he is certainly the greatest physicist, the greatest theoretical physicist of his generation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(from transcript, &lt;a href=&quot;https://www.youtube.com/watch?v=_3Z8RxXmoWU&amp;amp;ab_channel=mrtp&quot; rel=&quot;nofollow noreferrer&quot;&gt;source: Lecture 2: Black Holes and the Holographic Principle&lt;/a&gt; May 30th, 2014).&lt;/p&gt;&#xA;&lt;p&gt;There is also Enrico Fermi c.a. 1939, who speaking to &lt;a href=&quot;https://en.wikipedia.org/wiki/Giuseppe_Cocconi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Giuseppe Cocconi&lt;/a&gt; about &lt;a href=&quot;https://en.wikipedia.org/wiki/Ettore_Majorana&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ettore Majorana&lt;/a&gt;'s mysterious disappearance, praised:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You see, in the world there are various&#xA;categories of scientists: there are people of a secondary&#xA;or tertiary standing, who do their best but do not go very&#xA;far. There are also those of high standing, who come to&#xA;discoveries of great importance, fundamental for the&#xA;development of science. But then there are geniuses&#xA;like Galileo and Newton. Well, Ettore was one of them.&#xA;Majorana had what no-one else in the world had.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;source: Esposito, S. (2006). Fleeting genius. Physics world, 19(8), 34.&lt;/p&gt;&#xA;&lt;p&gt;I found another one. Hans Bethe in &lt;a href=&quot;https://www.youtube.com/watch?v=Kt3V3ok0DtQ&amp;amp;ab_channel=WebofStories-LifeStoriesofRemarkablePeople&quot; rel=&quot;nofollow noreferrer&quot;&gt;an interview&lt;/a&gt; with Sam Schweber (1996) refers to Feynman as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;there was a very young man in my division, Richard Feynman, and very soon noticed that he his ability was far beyond the great ability of all the rest of my division.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;he also &lt;a href=&quot;https://www.youtube.com/watch?v=zIvDA-nYQpI&amp;amp;ab_channel=WebofStories-LifeStoriesofRemarkablePeople&quot; rel=&quot;nofollow noreferrer&quot;&gt;speaks well about Freeman Dyson&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-07-24T16:19:47.470" LastActivityDate="2023-07-24T16:19:47.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15415" PostTypeId="2" ParentId="15410" CreationDate="2023-05-11T02:33:02.743" Score="3" Body="&lt;p&gt;A few about John von Neumann:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;”You know, Herb, Johnny can do calculations in his head ten times as fast as I can. And I can do them ten times as fast as you can, so you can see how impressive Johnny is” — Enrico Fermi (Nobel Prize in Physics, 1938)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“One had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch.” — Eugene Wigner (Nobel Prize in Physics, 1963)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“I have sometimes wondered whether a brain like von Neumann’s does not indicate a species superior to that of man” — Hans Bethe (Nobel Prize in Physics, 1967)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Evidently, a Ph.D. thesis and examinations did not constitute an appreciable effort” — Eugene Wigner&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This anecdote also may be of interest:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann” — George Pólya&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From: &lt;a href=&quot;https://www.privatdozent.co/p/the-unparalleled-genius-of-john-von-beb&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.privatdozent.co/p/the-unparalleled-genius-of-john-von-beb&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5438" LastEditorUserId="5438" LastEditDate="2023-05-11T13:39:34.123" LastActivityDate="2023-05-11T13:39:34.123" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15416" PostTypeId="2" ParentId="15406" CreationDate="2023-05-11T19:23:06.370" Score="0" Body="&lt;p&gt;Take a look at:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.tandfonline.com/doi/abs/10.1080/10724117.1995.11974913?journalCode=umho20&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.tandfonline.com/doi/abs/10.1080/10724117.1995.11974913?journalCode=umho20&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Complications involve algorithms and their implementatins.&lt;/p&gt;&#xA;" OwnerUserId="18414" LastActivityDate="2023-05-11T19:23:06.370" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15417" PostTypeId="1" CreationDate="2023-05-11T20:56:59.127" Score="1" ViewCount="86" Body="&lt;p&gt;Darwin and the ship Beagle, as well as every other ship entering the Pacific ocean had to sail through the treacherous waters of the Land of Fire at the tip of South America.  Many ships didn't make it, and shipwrecked.&lt;/p&gt;&#xA;&lt;p&gt;I heard (or must have read) the response Darwin wrote to the London newspaper editors who were mocking and dissing the Christian missionary movement of England. He noted that if it weren't for the work of the missionaries, the native cannibals would have eaten all the shipwrecked sailors at Tierra del Fuego! He expressed gratitude for the work of the missionaries that made that sailing, and his research much safer.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have evidence for this event by Darwin? (Not that he was an advocate of religion personally.)&lt;/p&gt;&#xA;" OwnerDisplayName="user18391" LastEditorUserId="12330" LastEditDate="2023-05-11T22:53:18.177" LastActivityDate="2023-05-11T22:53:18.177" Title="Darwin and the Natives of Tierra del Fuego" Tags="|anecdote|darwin|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15418" PostTypeId="1" CreationDate="2023-05-11T21:26:18.493" Score="4" ViewCount="83" Body="&lt;p&gt;I've long known the story of Hamilton carving the defining equations of the quaternions into the Broome (aka Brougham) Bridge, and have been aware for some years that there is no trace of the original carving. But I only recently read Hamilton's version of the story (told in a letter to his son Archibald), in which he wrote: &amp;quot;I pulled out on the spot a pocket-book, which still exists, and made an entry there and then.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I have a hard time believing Hamilton would have ever thrown the pocket-book away, and by the time he died, his reputation was sufficiently established that it seems unlikely that his heirs or the trustees of his estate could have been ignorant of its significance. So, do we know what happened to it?&lt;/p&gt;&#xA;" OwnerUserId="6587" LastActivityDate="2023-05-11T21:26:18.493" Title="Hamilton's pocket-book entry on quaternions" Tags="|quaternions|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15419" PostTypeId="1" AcceptedAnswerId="15423" CreationDate="2023-05-12T21:07:44.920" Score="9" ViewCount="1555" Body="&lt;p&gt;Is it possible to say which was the first popular science book? I was reading this question &lt;a href=&quot;https://hsm.stackexchange.com/questions/5231/einsteins-readings-of-popular-sciences-as-a-kid&quot;&gt;Einstein&amp;#39;s readings of popular sciences as a kid&lt;/a&gt; , and started wondering how far back this kind of distinction between popular science books for wider-audience (not just some political authorities) and science publications for fellow academics (and students) goes. It would have been necessary to have enough people to be able to read and the science needed to be advanced enough for it to be too technical for the layperson.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia article on &lt;a href=&quot;https://en.wikipedia.org/wiki/Popular_science&quot; rel=&quot;nofollow noreferrer&quot;&gt;pop-science&lt;/a&gt; goes back as far as to the writing of &lt;a href=&quot;https://en.wikipedia.org/wiki/Mary_Somerville&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mary Somerville&lt;/a&gt;'s &lt;em&gt;On the Connexion of the Physical Sciences&lt;/em&gt;. Do we know any earlier pop-science books?&#xA;Galileo's &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;Systema cosmicum&lt;/a&gt;&lt;/em&gt; might fit the bill, but it was just Galileo's way of publishing its results in a different way to be understood by his colleagues. Is it recognized as a pop-sci book?&lt;/p&gt;&#xA;&lt;p&gt;Do we have an established notion on which was the first popular science book?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-05-14T14:11:48.733" LastActivityDate="2023-05-16T03:57:06.593" Title="Can we estimate the first appearance of popular science books?" Tags="|publishing|" AnswerCount="5" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15420" PostTypeId="2" ParentId="15419" CreationDate="2023-05-13T02:54:09.993" Score="10" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_Hooke&quot; rel=&quot;noreferrer&quot;&gt;Robert Hooke's&lt;/a&gt; 1665 book &lt;a href=&quot;https://en.wikipedia.org/wiki/Micrographia&quot; rel=&quot;noreferrer&quot;&gt;Micrographia: or Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses. With Observations and Inquiries Thereupon&lt;/a&gt; predates Somerville by about one-hundred and seventy years.  Quoting the wikipedia article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Published in January 1665, the first major publication of the Royal Society,  it became the first scientific best-seller, inspiring a wide public interest in the new science of microscopy. The book originated the biological term cell.&lt;/p&gt;&#xA;&lt;p&gt;[...] the popularity of the book helped further the society's image and mission of being England's leading scientific organization. Micrographia's illustrations of the miniature world captured the public's imagination in a radically new way; Samuel Pepys called it &amp;quot;the most ingenious book that ever I read in my life.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-05-13T02:54:09.993" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15421" PostTypeId="1" CreationDate="2023-05-13T09:53:23.287" Score="1" ViewCount="103" Body="&lt;p&gt;I would not be surprised if ancient mathematicians had found it.&lt;/p&gt;&#xA;" OwnerUserId="17348" LastActivityDate="2023-05-13T09:53:23.287" Title="When was it known that natural numbers and rational numbers correspond one-to-one?" Tags="|mathematics|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15422" PostTypeId="2" ParentId="15410" CreationDate="2023-05-13T14:24:24.637" Score="2" Body="&lt;p&gt;In 1946, Carl Siegel asked Harald Bohr what had happened in Europe in mathematics during the war. Bohr answered in one word: &amp;quot;Selberg&amp;quot;.&#xA;This was after Atle Selberg had defended his thesis in 1943, showing that a positive proportion of the zeros of the Riemann zeta function lie on the critical line.&lt;/p&gt;&#xA;" OwnerUserId="40" LastActivityDate="2023-05-13T14:24:24.637" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15423" PostTypeId="2" ParentId="15419" CreationDate="2023-05-13T15:31:36.237" Score="13" Body="&lt;p&gt;As noted in @Conifold's comment, identifying the first popular science book is an impossible task. Before the specialization of science that accelerated during the 19th century, there was often little distinction between &amp;quot;science&amp;quot; and &amp;quot;popular science&amp;quot; for the small part of the population that was educated and literate.&#xA;Books that have been called the &amp;quot;first popular science book&amp;quot; include:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bernard_Le_Bovier_de_Fontenelle&quot; rel=&quot;noreferrer&quot;&gt;Fontenelle&lt;/a&gt;'s 1686 &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Conversations_on_the_Plurality_of_Worlds&quot; rel=&quot;noreferrer&quot;&gt;Entretiens sur la pluralité des mondes&lt;/a&gt;&amp;quot; is &lt;a href=&quot;https://adsabs.harvard.edu/full/2003ESASP.539..215P&quot; rel=&quot;noreferrer&quot; title=&quot;The Fermi paradox, N. Prantzos, Proceedings of the Conference on Towards Other Earths: DARWIN/TPF and the Search for Extrasolar Terrestrial Planets, 22-25 April 2003, Heidelberg, Germany. Edited by M. Fridlund, T. Henning, ISBN 92-9092-849-2, 2003, p. 215 - 218&quot;&gt;&amp;quot;often considered the first popular science book&amp;quot;&lt;/a&gt;) (In the 18th century, &lt;a href=&quot;https://www.smithsonianmag.com/science-nature/18th-century-writers-created-genre-popular-science-enlightenment-180971481/&quot; rel=&quot;noreferrer&quot; title=&quot;How 18th-Century Writers Created the Genre of Popular Science, Lorraine Boissoneault, Smithsonian Magazine, February 13, 2019 &quot;&gt;&amp;quot;tens of thousands&amp;quot;&lt;/a&gt; of copies of French popular science books were sold.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_Hooke&quot; rel=&quot;noreferrer&quot;&gt;Hooke&lt;/a&gt;'s 1665 &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Micrographia&quot; rel=&quot;noreferrer&quot;&gt;Micrographia&lt;/a&gt;&amp;quot; (&lt;a href=&quot;https://doi.org/10.1515/9781400865727-004&quot; rel=&quot;noreferrer&quot; title=&quot;CHAPTER 2. Meet the Microbes. Life's Engines: How Microbes Made Earth Habitable, Princeton: Princeton University Press, 2015, pp. 23-39.&quot;&gt;&amp;quot;the first  popular  science  book&amp;quot;&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Galileo_Galilei&quot; rel=&quot;noreferrer&quot;&gt;Galileo&lt;/a&gt;'s 1632&#xA;&amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;noreferrer&quot;&gt;Dialogo sopra i due massimi sistemi del mondo&lt;/a&gt;&amp;quot; (&lt;a href=&quot;https://www.pbs.org/empires/medici/renaissance/galileo.html&quot; rel=&quot;noreferrer&quot; title=&quot;PBS, Medici: Godfathers of the Renaissance&quot;&gt;&amp;quot;the first book of popular science&amp;quot;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Another interesting example of &lt;a href=&quot;https://en.wikipedia.org/wiki/Age_of_Enlightenment&quot; rel=&quot;noreferrer&quot;&gt;Enlightenment&lt;/a&gt; popular science is &lt;a href=&quot;https://en.wikipedia.org/wiki/Erasmus_Darwin&quot; rel=&quot;noreferrer&quot;&gt;Erasmus Darwin&lt;/a&gt;'s &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Botanic_Garden&quot; rel=&quot;noreferrer&quot;&gt;The Botanic Garden&lt;/a&gt;&amp;quot; published in 1791. According to wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The intent of The Botanic Garden, one of the first popular science books, is to pique readers' interest in science while educating them at the same time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The format of this book is unusual for modern readers since it is the form of two long poems: &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Botanic_Garden#The_Loves_of_the_Plants&quot; rel=&quot;noreferrer&quot;&gt;The Loves of the Plants&lt;/a&gt;&amp;quot; and &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Botanic_Garden#The_Economy_of_Vegetation&quot; rel=&quot;noreferrer&quot;&gt;The Economy of Vegetation&lt;/a&gt;&amp;quot;.  Explaining science in to the educated populace in poetic form actually has a long history, and contributors to the 2018 conference &amp;quot;&lt;a href=&quot;https://books.google.ca/books?hl=en&amp;amp;lr=&amp;amp;id=u0sFEAAAQBAJ&quot; rel=&quot;noreferrer&quot; title=&quot;2020, eds. Jesús Muñoz Morcillo and Caroline Y. Robertson-von Trotha, doi:10.1515/9783839448359&quot;&gt;Genealogy of Popular Science&#xA;From Ancient Ecphrasis to Virtual Reality&lt;/a&gt;&amp;quot; argued that European science popularization goes back to the didactic poetry of Greek and Roman antiquity.&lt;/p&gt;&#xA;&lt;p&gt;As noted in other answers and comments, another &amp;quot;popular science&amp;quot; genre from antiquity through the 18th Century were handbooks intended to share practical scientific and technical knowledge. As described by &lt;a href=&quot;https://dbcs.rutgers.edu/all-scholars/9139-stahl-william-harris&quot; rel=&quot;noreferrer&quot;&gt;Stahl&lt;/a&gt; in &amp;quot;&lt;a href=&quot;https://archive.org/details/romanscienceorig0000stah/page/15/mode/1up?q=laymen&quot; rel=&quot;noreferrer&quot; title=&quot;William Harris Stahl, Madison, University of Wisconsin Press, 1962&quot;&gt;Roman science: origins, development, and influence to the later Middle Ages&lt;/a&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Most [ancient Greek and Roman] handbooks were intended for the general reader.  Such books were occasionally written by men who had made original contributions to Greek scientific thought; more often the authors themselves had limited competence in handling scientific subjects or were laymen used to presenting any subject in a simplified form.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Without these popular works, much of the scientific knowlege of the era might have been lost. For example, all the original works of the 4th century BCE Greek astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudoxus_of_Cnidus&quot; rel=&quot;noreferrer&quot;&gt;Eudoxus&lt;/a&gt; have been lost, but his contributions were largely preserved in the didactic poem &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Aratus#Phenomena&quot; rel=&quot;noreferrer&quot;&gt;Phenomena&lt;/a&gt;&amp;quot; written a century later.&lt;/p&gt;&#xA;&lt;p&gt;This &amp;quot;handbook&amp;quot; tradition continued right through to the invention of the printing press, and later examples include &lt;a href=&quot;https://en.wikipedia.org/wiki/Books_of_secrets&quot; rel=&quot;noreferrer&quot;&gt;books of secrets&lt;/a&gt; such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Giambattista_della_Porta&quot; rel=&quot;noreferrer&quot;&gt;Giambattista Della Porta&lt;/a&gt;'s 1558 &lt;a href=&quot;https://en.wikipedia.org/wiki/Magia_Naturalis&quot; rel=&quot;noreferrer&quot;&gt;Magia Naturalis&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Isabella_Cortese&quot; rel=&quot;noreferrer&quot;&gt;Isabella Cortese&lt;/a&gt;'s 1561 &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Isabella_Cortese#The_Secrets_of_Lady_Isabella_Cortese&quot; rel=&quot;noreferrer&quot;&gt;I secreti della signora Isabella Cortese&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2023-05-15T04:15:58.680" LastActivityDate="2023-05-15T04:15:58.680" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15424" PostTypeId="1" CreationDate="2023-05-13T16:07:11.543" Score="1" ViewCount="67" Body="&lt;p&gt;The second law of thermodynamics is sometimes stated as:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$dS=d_{\rm{e}}S+d_{\rm{i}}S$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;with :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt;, the variation of entropy of the system;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$d_{\rm{e}}S$&lt;/span&gt;, the entropy exchanged with or supplied to the system from its surroundings (&lt;span class=&quot;math-container&quot;&gt;$d_{\rm{e}}S$&lt;/span&gt; can be positive, null or negative; and for a closed system that can only exchange heat with its surroundings: &lt;span class=&quot;math-container&quot;&gt;$d_{\rm{e}}S=\delta Q/T$&lt;/span&gt;);&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$d_{\rm{i}}S$&lt;/span&gt;, the entropy produced inside the system (&lt;span class=&quot;math-container&quot;&gt;$d_{\rm{i}}S≥0$&lt;/span&gt;; and null only for reversible processes).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, up to now, I have not seen a reference given along with this equation to justify it with regards to previous statements of the second law. For example, in &lt;a href=&quot;https://books.google.fr/books?id=mfFyG9jfaMYC&quot; rel=&quot;nofollow noreferrer&quot;&gt;Non-Equilibrium Thermodynamics, from S. R. DE Groot and P. Mazur (1961)&lt;/a&gt;, Chapter 3, §1, it just says that this statement is &amp;quot;a well known form of the second law of thermodynamics&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;When and where was this statement originally formulated and its equivalence with previous alternative statements first demonstrated?&lt;/p&gt;&#xA;" OwnerUserId="18419" LastActivityDate="2023-05-13T23:12:10.023" Title="Origin and justification for stating the second law of thermodynamics as $dS=d_{\rm{e}}S+d_{\rm{i}}S$?" Tags="|thermodynamics|entropy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15425" PostTypeId="2" ParentId="15424" CreationDate="2023-05-13T23:12:10.023" Score="3" Body="&lt;p&gt;Clausius in his &lt;a href=&quot;https://archive.org/details/mechanicaltheor00claugoog/page/n240/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The mechanical theory of heat&amp;quot;&lt;/a&gt; discusses what he calls &amp;quot;uncompensated transformation&amp;quot;. First he derives that &lt;span class=&quot;math-container&quot;&gt;$\int \frac{dQ}{T}=0$&lt;/span&gt; for a reversible cyclic transformation, then that &lt;span class=&quot;math-container&quot;&gt;$\int \frac{dQ}{T} \le 0$&lt;/span&gt; for an irreversible transformation, finally on page 214 of the Macmillan 1879 edition he introduces the quantity &lt;span class=&quot;math-container&quot;&gt;$N = -\int \frac{dQ}{T}\ge 0$&lt;/span&gt; and calls it the &amp;quot;uncompensated transformation&amp;quot; (&lt;em&gt;uncompensirten Verwandlung&lt;/em&gt;). The infinitesimal increment of this &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$d_iS$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;§2, Magnitude of the Uncompensated Transformation. In many cases the&#xA;magnitude of the Uncompensated Transformation is obtained directly&#xA;from the equivalence value of the transformations, as determined by&#xA;the method of Chapter IV. If for example a quantity of heat &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; is&#xA;generated by any process such as friction, and this is finally&#xA;imparted to a body of temperature &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;, the uncompensated transformation&#xA;thus produced has the value &lt;span class=&quot;math-container&quot;&gt;$\frac{Q}{T}$&lt;/span&gt;. Again, if a quantity of heat Q has&#xA;passed by conduction from a body of temperature &lt;span class=&quot;math-container&quot;&gt;$T_1$&lt;/span&gt; to another of&#xA;temperature &lt;span class=&quot;math-container&quot;&gt;$T_2$&lt;/span&gt; then the uncompensated transformation is &lt;span class=&quot;math-container&quot;&gt;$Q\left(\frac{1}{T_1}-\frac{1}{T_2}\right).$&lt;/span&gt; If&#xA;a body has passed through a non-reversible cyclical process, and we&#xA;wish to determine the resulting uncompensated transformation, which&#xA;we may call &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt;, we have, by the principles explained in Chapter IV.,&#xA;the equation &lt;span class=&quot;math-container&quot;&gt;$N = -\int \frac{dQ}{T} \tag{1}\label{1}.$&lt;/span&gt;&#xA;As however a cyclical process may be made up of several individual&#xA;changes of condition in a given body, some of which may be reversible,&#xA;others non-reversible, it is in many cases interesting to know how&#xA;much any particular one of the latter has contributed towards making&#xA;up the whole sum of uncompensated transformations. For this purpose we&#xA;may suppose that after the change of condition which we wish to enquire&#xA;into, the variable body is brought by any reversible process into its&#xA;former condition. By this means we form a smaller cyclical process, in&#xA;which equation &lt;span class=&quot;math-container&quot;&gt;$\eqref{1}$&lt;/span&gt; may be applied just as well as in the whole&#xA;process. Thus if we know the quantities of heat which the body has&#xA;taken in during this process, and the temperatures which appertain to&#xA;them, the negative integral &lt;span class=&quot;math-container&quot;&gt;$-\int \frac{dQ}{T}$&lt;/span&gt; gives the uncompensated&#xA;transformations which have taken place. But as the return to the&#xA;original condition, which has taken place in a reversible manner, can&#xA;have contributed nothing to increase this sum, the expression above&#xA;gives the uncompensated transformation which was sought, and which was&#xA;caused by the given change in condition.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="7908" LastActivityDate="2023-05-13T23:12:10.023" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15426" PostTypeId="1" CreationDate="2023-05-13T23:50:37.470" Score="2" ViewCount="90" Body="&lt;p&gt;A circle can be written as the solution set of an equation &lt;span class=&quot;math-container&quot;&gt;$x^2+y^2=1$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I have heard that Fermat associated figures with equations in the 17th century.&lt;/p&gt;&#xA;&lt;p&gt;Did people see figures as sets of points before set theory? Otherwise, what is the difference between the view of Fermat et al. and the view of set theorists?&lt;/p&gt;&#xA;" OwnerUserId="17348" LastEditorUserId="17348" LastEditDate="2023-05-14T01:29:53.527" LastActivityDate="2023-05-14T01:29:53.527" Title="Was there a view of a figure as a set of points before set theory?" Tags="|mathematics|set-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15427" PostTypeId="1" CreationDate="2023-05-14T00:01:42.040" Score="0" ViewCount="24" Body="&lt;p&gt;Our modern maths readily provides the answer to this ancient question. When shooting a projectile, what is the optimal angle that will maximise the range?&lt;/p&gt;&#xA;&lt;p&gt;Some trigonometry and Newtonian mechanics provides the answer. 45 degrees.&lt;/p&gt;&#xA;&lt;p&gt;I would be surprised if it wasn't figured out before trigonomrty.&lt;/p&gt;&#xA;&lt;p&gt;Who first and how ever figured it out?&lt;/p&gt;&#xA;" OwnerUserId="18422" LastActivityDate="2023-05-14T00:01:42.040" Title="optimal shot angle" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="0" ClosedDate="2023-05-15T06:45:24.573" ContentLicense="CC BY-SA 4.0" />
  <row Id="15428" PostTypeId="2" ParentId="15419" CreationDate="2023-05-14T05:35:58.450" Score="3" Body="&lt;p&gt;In a sense, the writings of the Flemish mathematician and engineer &lt;a href=&quot;https://en.wikipedia.org/wiki/Simon_Stevin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simon Stevin&lt;/a&gt; (1548–1620), could be considered &amp;quot;popular science&amp;quot;, insofar as he wrote them in Dutch instead of Latin so that non-scholars would be able to read them, and intended for them to be of practical use to tradesmen of all kinds in their everyday business, e.g.:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;Tafelen van Interest&amp;quot; (&lt;em&gt;Tables of interest&lt;/em&gt;) from 1582, with present value problems of simple and compound interest and interest tables that had previously been unpublished by bankers.&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/De_Thiende&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Thiende&lt;/a&gt;&amp;quot; (&lt;em&gt;The tenth&lt;/em&gt;) from 1585, in which he introduced decimals to a European audience (they had been known to Islamic scholars much earlier).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The latter had a dedication &amp;quot;to astrologers, landmeasurers, measurers of tapestry and wine casks, and stereometricians in general, mint-masters and merchants all&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;His books were published by &lt;a href=&quot;https://en.wikipedia.org/wiki/Christophe_Plantin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Christophe Plantin&lt;/a&gt; in Antwerp, and were bestsellers at the time.&lt;/p&gt;&#xA;" OwnerUserId="18423" LastEditorUserId="18423" LastEditDate="2023-05-16T03:57:06.593" LastActivityDate="2023-05-16T03:57:06.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15430" PostTypeId="1" CreationDate="2023-05-14T12:04:56.073" Score="3" ViewCount="132" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;These are &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-tuples of reals, added componentwise and multiplied via the “exterior product.” They were introduced by Grassmann in 1844 as part of a brilliant attempt to construct a vector algebra in &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-dimensional space. Grassmann’s style was far from simple and his approach was ahead of its time&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0817646841&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Abstract Algebra, Israel Kleiner&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To my understanding, differential forms is the reason why Grassman algebras are so famous. But they only appeared in 1899, so how did Grassman's interest in Grassman algebras arise right in 1844?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I have thought that maybe it is related to some differential geometry matters. Googling tells me that Gauss put forth &amp;quot;Disquisitiones generales circa superficies curvas&amp;quot; in 1827. So, maybe Grassman was influenced by this? I am not sure.&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="12330" LastEditDate="2023-05-16T11:36:43.670" LastActivityDate="2023-05-16T11:36:43.670" Title="What motivated Grassman to study Grassman algebras in 1844?" Tags="|differential-geometry|linear-algebra|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15431" PostTypeId="2" ParentId="15419" CreationDate="2023-05-14T14:06:53.293" Score="1" Body="&lt;p&gt;It depends heavily on your definitions. But I think that for any reasonable definition, we cannot estimate it.&lt;/p&gt;&#xA;&lt;p&gt;For example, &lt;a href=&quot;https://en.wikipedia.org/wiki/De_architectura&quot; rel=&quot;nofollow noreferrer&quot;&gt;De Architectura&lt;/a&gt;. It's a book, written for a non-technical audience,  and dealing with technical information that predates the science/engineering divide. And it's from about 20 BC. Eventually, in the 15th century or so, it was republished more broadly. But by any reasonable definition, it's a pop-science book.&lt;/p&gt;&#xA;&lt;p&gt;Is it the earliest? Almost certainly not. You can easily find famous Ancient Greek manuscripts and make the same argument for them. There are almost certainly lost manuscripts from even earlier. And then you look more broadly: I'm ignorant of e.g. most of early Chinese writing, but I would be shocked if there were no sources there.&lt;/p&gt;&#xA;&lt;p&gt;So: &lt;strong&gt;no&lt;/strong&gt;. I don't think we can reasonably estimate that first appearance. Too much has been lost in too many places.&lt;/p&gt;&#xA;" OwnerUserId="18424" LastActivityDate="2023-05-14T14:06:53.293" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15432" PostTypeId="2" ParentId="15419" CreationDate="2023-05-15T03:07:05.660" Score="-1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paradiso_(Dante)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dante's Paradiso&lt;/a&gt; gives a good explanation of cosmology as it was understood in the early 14th century (the spheres that move the planets around the Earth). &lt;a href=&quot;https://mathinees-lacaniennes.net/images/stories/articles/dante.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;It can be argued that Dante also developed the idea of the Universe as a 3 sphere&lt;/a&gt;, anticipating the cosmological ideas of the 20th century.&lt;/p&gt;&#xA;" OwnerUserId="15315" LastActivityDate="2023-05-15T03:07:05.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15434" PostTypeId="2" ParentId="15339" CreationDate="2023-05-15T22:03:07.570" Score="1" Body="&lt;p&gt;&lt;strong&gt;&lt;a href=&quot;https://www.tonyrothman.com/about&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tony Rothman&lt;/a&gt;&lt;/strong&gt; is a retired physicist and author.&lt;/p&gt;&#xA;&lt;p&gt;The first book he wrote is a science fiction novel, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1479472190&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;The World is Round,&lt;/a&gt; which I think is a truly great sci-fi novels.  Some of the dialogue and the way some of the characters are developed are not that sophisticated (the fact that it's a first novel really shows), but the setting is unique and the sense of wonder and exploration is awesome.&lt;/p&gt;&#xA;&lt;p&gt;Although the question doesn't want popular science titles, his second book is a very good one: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/048624587X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frontiers of Modern Physics,&lt;/a&gt; which I've read and re-read a few times over the years.  I think it's got some of the best explanations of how black holes behave.&lt;/p&gt;&#xA;" OwnerUserId="18429" LastActivityDate="2023-05-15T22:03:07.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15435" PostTypeId="2" ParentId="15339" CreationDate="2023-05-16T01:15:06.483" Score="2" Body="&lt;p&gt;SF, doubtless to no one's surprise, is littered with them:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Greg Benford: Dozens of SF novels and awards; PhD physics; Prof at UC Irving&lt;/li&gt;&#xA;&lt;li&gt;Poul Anderson: Major SF writer from 1950-2000; BS in Physics&lt;/li&gt;&#xA;&lt;li&gt;Sidney Coleman: Reviewed SF for F&amp;amp;SF; co-founder of Advent: Publishing; Harvard theoretical physics faculty, known as &amp;quot;the greatest American physicist who never won a Nobel Prize&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;Fred Hoyle: Several SF novels; major British astrophysicist&lt;/li&gt;&#xA;&lt;li&gt;John W. Campbell: Major early writer and &lt;em&gt;the&lt;/em&gt; leading editor in SF; BS in physics&lt;/li&gt;&#xA;&lt;li&gt;Alastair Reynolds: Twenty or so SF novels; PhD in astrophysics&lt;/li&gt;&#xA;&lt;li&gt;Geoffrey A. Landis: Award-winning SF writer; PhD in physics, works for NASA&lt;/li&gt;&#xA;&lt;li&gt;Milton A. Rothman: Wrote a number of SF stories under pseudonym of Lee Gregor; PhD Nuclear physicist&lt;/li&gt;&#xA;&lt;li&gt;Dave Langford: Multiple Hugos including for for a story; Editor of the SF Encyclopedia; Nuclear physicist&lt;/li&gt;&#xA;&lt;li&gt;Charles Sheffield: Multiple SF novels and short stories; Physicist and Chief Scientist of Earth Satellite Corporation&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;And if you allowed astronomers in, there'd be many more!&lt;/p&gt;&#xA;" OwnerUserId="16773" LastActivityDate="2023-05-16T01:15:06.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15436" PostTypeId="1" CreationDate="2023-05-16T06:00:51.003" Score="1" ViewCount="88" Body="&lt;p&gt;I know that Fermat had a method of adequality in order to solve certain optimization.  One such problem was: &amp;quot;Suppose that you have a rectangle of material and need to cut corners into it such that the folded flaps will make a box.  What are the cuts that maximize the volume of the box?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Were there other &amp;quot;applied&amp;quot; problems that people were trying to solve, which motivated the investigation into optimization and tangent lines more generally?&lt;/p&gt;&#xA;&lt;p&gt;What I'm really trying to wrap my head around is: Did people study the derivative because it was needed for applications in other fields?  Or did they just think the questions about tangent lines were simply interesting, and only researched them out of curiosity?  It seems to me like the box optimization problem isn't that important for real applications -- so did Fermat investigate that problem simply because he thought it was an intrinsically interesting puzzle and not for any really practical purpose?&lt;/p&gt;&#xA;&lt;p&gt;And besides him, what about the other mathematicians who contributed to understanding the derivative?  Was there anything about navigating the globe, or firing artillery, or any other thing like that, for which people felt the need to find tangent lines?&lt;/p&gt;&#xA;&lt;p&gt;Perhaps a helpful related question: Why were the ancient Greeks interested in tangents to conic sections?  Again, did the idea just strike them as interesting, or was there some way in which this helps solve applied problems that they were facing at the time?&lt;/p&gt;&#xA;" OwnerUserId="2732" LastActivityDate="2023-05-16T06:00:51.003" Title="What physical problems required the invention of the derivative?" Tags="|calculus|newton|leibniz|fermat|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15437" PostTypeId="2" ParentId="15410" CreationDate="2023-05-16T13:40:58.920" Score="1" Body="&lt;p&gt;Gauss, in the last 10 years of his life, thought very highly of Eisenstein (then 22), describing the latter’s talent as that which nature bestows only a few times per century. See &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Gotthold_Eisenstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-05-16T13:40:58.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15438" PostTypeId="1" CreationDate="2023-05-16T15:15:21.217" Score="0" ViewCount="88" Body="&lt;p&gt;The real Jordan form is &lt;a href=&quot;https://en.wikipedia.org/wiki/Jordan_normal_form#Real_matrices&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. I want to know who discovered this form. I searched google academics for a long time but couldn't get an answer.&lt;/p&gt;&#xA;" OwnerUserId="18433" LastEditorUserId="10106" LastEditDate="2023-05-17T03:52:00.173" LastActivityDate="2023-05-17T03:52:00.173" Title="Who discovered the real Jordan form?" Tags="|mathematics|linear-algebra|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15441" PostTypeId="2" ParentId="14594" CreationDate="2023-05-19T15:55:23.013" Score="2" Body="&lt;p&gt;Richard is a distant cousin of mine.&lt;br /&gt;&#xA;He was living in Sunnyvale, CA when I met him about 15 years ago.&#xA;He owns a house there, but he's pretty elderly and doesn't keep it up much.&#xA;He's a bit eccentric.&lt;/p&gt;&#xA;" OwnerUserId="18441" LastActivityDate="2023-05-19T15:55:23.013" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15442" PostTypeId="1" CreationDate="2023-05-19T18:09:27.237" Score="1" ViewCount="281" Body="&lt;p&gt;On the &lt;a href=&quot;https://en.wikipedia.org/wiki/Galactic_algorithm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia article&lt;/a&gt; for galactic algorithm (an algorithm that only becomes efficient when the inputs are so large that the algorithm is not used in practice), one of the use cases is that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Available computational power may catch up to the crossover point, so&#xA;that a previously impractical algorithm becomes practical.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What are some examples of this happening?&lt;/p&gt;&#xA;&lt;p&gt;I'm looking specifically for impractical algorithms that became practical because problem sizes got larger or computers got faster, and not because of improvements to the algorithm itself that made it practical.&lt;/p&gt;&#xA;" OwnerUserId="18442" LastActivityDate="2023-05-27T03:48:26.483" Title="What are some examples of galactic algorithms that have become practical?" Tags="|computer-science|algorithm|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15443" PostTypeId="1" CreationDate="2023-05-21T18:30:05.833" Score="1" ViewCount="95" Body="&lt;p&gt;In a essay about the meaning of life, the famous scientist Schrodinger once said &amp;quot;&lt;em&gt;Physical laws rest on atomic statistics and are therefore only approximate&lt;/em&gt;&amp;quot; (&lt;a href=&quot;http://www.whatislife.ie/downloads/What-is-Life.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.whatislife.ie/downloads/What-is-Life.pdf&lt;/a&gt; , page 4).&lt;/p&gt;&#xA;&lt;p&gt;At the same time, I have always wondered about the origins of different laws that are used in the natural sciences and engineering disciplines. For instance, consider the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;Young's modulus&lt;/strong&gt; (&lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;): dating back to the year 1727 - is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied lengthwise. It quantifies the relationship between tensile/compressive stress (&lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;) and axial strain (&lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;) in the linear elastic region of a material and is determined using the formula.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Mathematically, Young's modulus can be expressed as :&lt;span class=&quot;math-container&quot;&gt;$$E=\sigma/\epsilon$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Even to this day, we can continue to see the &amp;quot;approximate and data based nature&amp;quot; of this theoretical concept:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Rm1Lw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Rm1Lw.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have always wondered how the formula for Young's modulus was determined at the time. If I were to speculate, using today's terminology - I would guess that:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;A &amp;quot;parametric form specifying the relationship between the entities of interest&amp;quot; (i.e. &amp;quot;model&amp;quot;) was decided, possibly based on some &amp;quot;&lt;strong&gt;prior&lt;/strong&gt; hunches&amp;quot;. For instance, &lt;span class=&quot;math-container&quot;&gt;$E_1 = \sigma / \epsilon$&lt;/span&gt; vs. &lt;span class=&quot;math-container&quot;&gt;$E_2 = \sigma + \epsilon$&lt;/span&gt;, vs. &lt;span class=&quot;math-container&quot;&gt;$E_3= \sigma + \epsilon + \text{some constant}$&lt;/span&gt; , &lt;span class=&quot;math-container&quot;&gt;$E_4 = \sigma^2 / \epsilon$&lt;/span&gt;  etc.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Next, data was probably collected through a series of experiments in which measured the resulting &amp;quot;stiffness&amp;quot; (i.e. &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;) for different values of &amp;quot;stress&amp;quot; (i.e. &lt;span class=&quot;math-container&quot;&gt;$\sigma$&lt;/span&gt;) and &amp;quot;strain&amp;quot; (i.e. &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, different candidate &amp;quot;models&amp;quot; for Young's modulus were compared against the data collected from these experiments. The &amp;quot;model&amp;quot; which performed the &amp;quot;best&amp;quot; (i.e. predicted the closest &amp;quot;stiffness&amp;quot; for some given &amp;quot;stress&amp;quot; and &amp;quot;strain&amp;quot;) was likely selected as an early form of Young's modulus.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Thus, can some please tell me - is it unreasonable to believe that the &amp;quot;idea of regression&amp;quot; might have long predated its formal inception, and (as Schrödinger said) many theoretical laws in science and engineering have their roots in this early experimental regression idea instead of pure theoretical abstraction? Were such scientists responsible for these scientific laws actually performing some early version of machine learning and cross validation?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Sources:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Young%27s_modulus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Young%27s_modulus&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16167" LastEditorUserId="6609" LastEditDate="2023-05-29T10:22:22.660" LastActivityDate="2023-05-29T10:22:22.660" Title="Were many (famous) theoretical laws in science based on &quot;Statistical Regression&quot;?" Tags="|statistics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15445" PostTypeId="1" CreationDate="2023-05-22T03:02:42.443" Score="1" ViewCount="133" Body="&lt;p&gt;&lt;strong&gt;If Math is Based on Unprovable Axioms and can't be Proven - Why Does Math &amp;quot;Work&amp;quot; so Well?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have heard that both Mathematics and Religions are based on &amp;quot;axioms&amp;quot; that are by definition unprovable. I have also heard some people argue that Mathematics, like Religion - only provides us with a logical framework to interpret the world with:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/SLt7O.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/SLt7O.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If the above is true - &lt;strong&gt;why has mathematics (a system of beliefs based on unprovable axioms) proved to be so effective in  understanding, analyzing and predicting the behavior of systems and phenomena in the world; whereas the same can not be said about religion (also a system of beliefs based on unprovable axioms)?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the field of Probability, Kolmogorov's Axioms make fundamentally unprovable statements such as &amp;quot;&lt;em&gt;If A is a subset of, or equal to B, then the probability of A is less than, or equal to the probability of B.&lt;/em&gt;&amp;quot; These Axioms are regarded as being established, accepted, and self-evidently true - and also serve as a premise or starting point for further reasoning and arguments. In Zermelo-Fraenkel Set Theory, the &amp;quot;Axiom of Union&amp;quot; states that &amp;quot;&lt;em&gt;for any set of sets F there is a set A containing every element that is a member of some member of F&lt;/em&gt;&amp;quot; - the legitimacy and the validity of this axiom is generally not up for debate, it is generally considered as &amp;quot;infallible&amp;quot; regardless of your attitudes towards it. The same way, ancient religions made similar unprovable statements that were accepted at the time and also considered as starting points for further reasonings and arguments, such as &amp;quot;an afterlife self-evidently exists and the ratio of your soul's weight to the weight of a feather will decide your position in this afterlife&amp;quot;. These statements were also considered to be infallible, regardless of attitudes held towards them.&lt;/p&gt;&#xA;&lt;p&gt;Yet as we all know, Kolmogorov's Axioms along with other collections of axioms, characterize a system of beliefs (i.e. mathematics) that has been shown to be very useful in consistently and accurately understanding, analyzing and predicting the behavior of systems and phenomena in the world (e.g. science, engineering, technology) - so much so, that mathematics almost appear to be &amp;quot;objectively true&amp;quot; and pass off as unquestioned and &amp;quot;factual knowledge&amp;quot; in our everyday life. &lt;strong&gt;Is this (i.e. the power, achievements and the reputation of mathematics) attributable to the fact that perhaps our world in it's&lt;/strong&gt; &lt;em&gt;su generis&lt;/em&gt; &lt;strong&gt;form might actually have an underlying mathematical structure and we have correctly identified parts of this underlying mathematical structure and created a set of axioms that is consistent with this mathematical structure?&lt;/strong&gt;  Or is mathematics at it's fundamental level no different than mystical rain dance rituals based on unprovable commandments and tautologies, and it's plethora of successful implementations over the past thousands of years merely an inconsequential result? Are deluges and rain floods understandable through man's turbulent relationship with some God - or are they understandable through atmospheric weather patterns modelled by systems of differential equations?&lt;/p&gt;&#xA;&lt;p&gt;Even though Godel's Incompleteness Theorem exposed jarring flaws in mathematics as a whole, revealing that &amp;quot;a complete and consistent set of axioms for all mathematics is impossible&amp;quot;, - &lt;strong&gt;why has mathematics, albeit based on a unprovable axioms, managed to evade the eventual pitfalls of other axiomatic based systems, and managed to work so &amp;quot;well&amp;quot; over the years?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="16167" LastActivityDate="2023-05-22T06:33:13.330" Title="If Math is Based on Unprovable Axioms and these Axioms can't be Proven - Why Does Math &quot;Work&quot; so Well?" Tags="|probability|" AnswerCount="1" CommentCount="6" ClosedDate="2023-05-22T14:23:00.337" ContentLicense="CC BY-SA 4.0" />
  <row Id="15446" PostTypeId="2" ParentId="15445" CreationDate="2023-05-22T06:33:13.330" Score="5" Body="&lt;p&gt;Axioms in mathematics are not arbitrary statements; they are abstractions of everyday experience or physical laws (or other laws of nature). Consider two examples.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Axioms and postulates of Euclid describing lines in the plane or in the space. These are abstractions of the properties&#xA;of light rays (and stretched ropes). All these properties are confirmed by millennia of our experience. Until 19 century everyone actually thought that Euclidean geometry describes the real space in which we live.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The axioms of real numbers come from the analysis of&#xA;the procedure of measuring.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Probability. Original development of probability was stimulated by analysis of games of chance, statistics, insurance and such phenomena. Later statistical physics came into play. The axioms of probability that we are using now, is the result of this long development.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;So most of mathematics is deeply rooted in the study of the real world. And this explains its effectiveness in describing this real world.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-05-22T06:33:13.330" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15449" PostTypeId="1" CreationDate="2023-05-24T17:55:24.057" Score="1" ViewCount="95" Body="&lt;p&gt;In the following paper from I. Bernard Cohen, &amp;quot;Roemer and the First Determination of the Velocity of Light (1676)&amp;quot; published on Isis (1940):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/tCGZF.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/tCGZF.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can be read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;explaining that the delay would arise from the  simple fact that&#xA;astronomers considered light to be propagated instantaneously rather&#xA;than gradually&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm not sure I understand what this means, Roemer is defending that light is not instantaneous. Isn't the delay meant to prove that the light propagates gradually?&lt;/p&gt;&#xA;" OwnerUserId="12712" LastActivityDate="2023-05-24T22:47:49.860" Title="Light propagated instantaneously rather than gradually" Tags="|physics|terminology|optics|light|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15450" PostTypeId="1" CreationDate="2023-05-24T21:41:47.223" Score="3" ViewCount="337" Body="&lt;p&gt;Briefly from &lt;a href=&quot;https://en.wikipedia.org/wiki/Systems_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;wikipedia&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is &amp;quot;more than the sum of its parts&amp;quot; by expressing synergy or emergent behavior.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In early math courses, we are often taught how to solve mathematics provided models of a scenario, but never quite how theories on how to come with models. The stuff one finds in even the earliest conception of this theory in General System theory,  Ludwig von Bertalanffy seems to be quite useful in providing an understanding for this.&lt;/p&gt;&#xA;&lt;p&gt;Yet, his name or the idea of system theory in general theory is never mentioned Mathematics classes. Hence, my question, how did the system theory as a field fall out of public view?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2024-02-24T21:08:36.063" Title="Why did systems theory never gain popularity?" Tags="|mathematics|physics|biology|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15451" PostTypeId="2" ParentId="15449" CreationDate="2023-05-24T22:47:49.860" Score="3" Body="&lt;p&gt;What Roemer was says is that the astronomer's predictions were based on an infinite speed of light, but this is not correct:  The speed of light is finite and the extra time it takes light to cross Earth's orbit around the Sun results in the eclipses observed when Jupiter is near to superior conjunction (i.e., on the opposite side of the Sun from Earth) happening later than predicted based on eclipses when Jupiter is at opposition (i.e., straight out away from the Sun as viewed from earth) because of extra time the time takes to cross the earth's orbit.&lt;/p&gt;&#xA;&lt;p&gt;He describes this as a delay because he is compared what is predicted from the timing at opposition and an infinite speed of light.&lt;/p&gt;&#xA;&lt;p&gt;A diagram would explain this clearly, but this margin is too small to contain one.&lt;/p&gt;&#xA;" OwnerUserId="16773" LastActivityDate="2023-05-24T22:47:49.860" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15452" PostTypeId="1" CreationDate="2023-05-25T21:16:37.540" Score="0" ViewCount="75" Body="&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/14457&quot;&gt;What exactly did Lagrange do, historically?&lt;/a&gt; This is very relevant but the accepted answer just does similar speculations based on contextual details that I have done. It was mentioned that part of Lagrange's research programme was to remove all mathematical dependence on geometric interpretation. He thinks the title &lt;em&gt;Mecanique Analytique&lt;/em&gt; is a reference to that. This is a very useful insight into the mind of Lagrange when developing his formulation but I'm looking for something more. preferably from pen or tongue of the inventors themselves.&lt;/p&gt;&#xA;&lt;p&gt;Today we often study these formulations and their appropriate circumstances of usage and may become quite skilled at applying to them interesting problems but I think most rarely consider &lt;em&gt;how&lt;/em&gt; the newtonian, and now the other formulations were developed.(i think there is a tendency to dismiss the roots of milestone achievements as the product of pure genius incomprehensible to mortal men like us)&lt;/p&gt;&#xA;&lt;p&gt;More specifically, I mean the processes of ideation that led to the mathematical construction of these formulations. Googling around, I have read that it is unknown if Lagrange said about the developmental process of his mechanics, and the same goes for newton but even more so since he had a reputation for being quite prickly on the topic of sharing his mental processes that led to a discovery. So I have little hope that there is anything newton wrote or said relevant to my question, but perhaps he let something slip and that's why i'm asking here.&lt;/p&gt;&#xA;&lt;p&gt;I could not find anything about hamilton, and I am also not &lt;em&gt;yet&lt;/em&gt;  desperate enough to read all the extant works of these mechanicians in the hopes of gleaning something.&lt;/p&gt;&#xA;&lt;p&gt;If they all chose to remain silent, then the only thing we can do is:&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Know the common objective of these theories. That is, determining the a(t) v(t) r(t) of a moving body based on the dynamical portions of its motion.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Study the development of mechanical theory until the individual in question.&lt;/p&gt;&#xA;&lt;p&gt;In the case of Newton, models like kepler's laws existed for example. Whereas in the case of Lagrange, Newtonian mechanics already existed.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Compare the most appropriate circumstances of usage. This is particularly helpful with cases like Lagrange and Hamilton, because they must have saw a need to develop a new formulation despite the existing ones. Otherwise why bother? They must have been faced with problems where the previous formulations were too cumbersome.&lt;/p&gt;&#xA;&lt;p&gt;So by comparing what lagrangian mechanics is more efficient at than newton's we could glean that the mind of lagrange once wrestled with the idea of:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;if only there was a less cumbersome way of solving this pendulum problem and similarly constrained problems..sigh. If only.&amp;quot; &lt;strong&gt;invents lagrangian&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Study what has been transmitted from their lives until they published their formulations.&lt;/p&gt;&#xA;&lt;p&gt;But I hope it wont be necessary to speculate this hard, as hopefully the historians here are aware of some rare transmissions from these inventors about how they ideated their formulations.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="4251" LastEditDate="2023-05-26T06:15:24.597" LastActivityDate="2023-05-26T06:15:24.597" Title="How did Newton, Lagrange, Hamilton and others ideate their mechanical formulations?" Tags="|discoveries|classical-mechanics|newton|lagrange|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15453" PostTypeId="1" CreationDate="2023-05-26T11:42:07.037" Score="1" ViewCount="128" Body="&lt;p&gt;I am interested in the Christoffel symbol. So I found Christoffel's 1869 paper and &lt;em&gt;The Road to Reality&lt;/em&gt; by Roger Penrose. These days, the Christoffel symbol has been established as a covariant derivative, but Christoffel did not know the concept of covariant derivative and parallel transport.&lt;/p&gt;&#xA;&lt;p&gt;So, I wondered what concept Christoffel used to create the Christoffel symbol at the time.&lt;/p&gt;&#xA;&lt;p&gt;When I read the book &lt;em&gt;The Road to Reality&lt;/em&gt;, I think that Lie algebra was use. What exactly is the concept behind the Christoffel symbol?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="6609" LastEditDate="2023-05-29T10:26:25.553" LastActivityDate="2023-05-29T10:26:25.553" Title="Origin of the Christoffel symbol" Tags="|differential-geometry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15454" PostTypeId="1" AcceptedAnswerId="15457" CreationDate="2023-05-26T23:21:22.810" Score="7" ViewCount="931" Body="&lt;p&gt;This is a story that seems to be an obligatory mention in either sequence and series lessons or exponential functions and today I've decided to track down the transmission of this story.&#xA;it goes like this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the most famous legends about series concerns the long inventor of chess&#xA;whose name is lost the ages. According to the legend, an Indian king summoned the&#xA;inventor and suggested that he chose the award for the creation of the interesting&#xA;and wise game. The king was amazed by the “modest” request from the inventor&#xA;who asked to give him for the first cell of the chessboard 1 grain of wheat, for the&#xA;second—2 grains, for the third—4 grains, for the fourth—twice as much as in the&#xA;previous cell, etc. As a result, the total number of grains per 64 cells of the&#xA;chessboard would be number 18,446,744,073,709,551,615 (18 quintillion, 446 quadrillion, 744 trillion, 73 billion, 709 million, and 551,615). If the king was able to have&#xA;that much of wheat, he would have to plant it everywhere on the entire surface of&#xA;the Earth including the territories of the seas and oceans, and mountains, and the&#xA;desert, from the Arctic to the Antarctic in order obtain a satisfactory harvest, then,&#xA;perhaps, he maybe could pay his amazing debt off to the chess inventor in over&#xA;5 years. &lt;br&gt;&#xA;&lt;strong&gt;[ Methods of Solving Sequence and Series Problems - Ellina Grigorieva pg191 ]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is not exactly a hard hitting question but it would be nice to know at least the earliest mention of the story.&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="14043" LastEditDate="2023-05-31T18:27:48.760" LastActivityDate="2023-05-31T18:27:48.760" Title="What is the earliest mention of doubling grains on a chessboard story?" Tags="|mathematics-social-history|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15455" PostTypeId="1" CreationDate="2023-05-26T23:24:07.790" Score="0" ViewCount="43" Body="&lt;p&gt;I'm starting to have an interest in Heyting Algebras and their application in topological spaces. Is there any paper/reference which gives a history based exposition of it's ideas?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2023-05-26T23:24:07.790" Title="Reference explaining the history and motivation behind Heyting algebra?" Tags="|reference-request|logic|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15456" PostTypeId="2" ParentId="15442" CreationDate="2023-05-27T01:58:51.930" Score="2" Body="&lt;p&gt;One example would be Strassen’s matrix multiplication algorithm which can be more efficient than the conventional &lt;span class=&quot;math-container&quot;&gt;$O(n^{3})$&lt;/span&gt; algorithm for large enough matrices.  Unfortunately, there are numerical stability issues that can make it unsuitable in many applications.&lt;/p&gt;&#xA;" OwnerUserId="5579" LastEditorUserId="5579" LastEditDate="2023-05-27T03:48:26.483" LastActivityDate="2023-05-27T03:48:26.483" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15457" PostTypeId="2" ParentId="15454" CreationDate="2023-05-27T05:47:36.047" Score="14" Body="&lt;p&gt;The name is not lost to the ages, it was made up along with the rest of this fable. Sissa ben Dahir, an Indian brahmin or grand vizier, supposedly invented &lt;a href=&quot;https://en.wikipedia.org/wiki/Chaturanga&quot; rel=&quot;noreferrer&quot;&gt;chaturanga&lt;/a&gt;, a predecessor of chess, somewhere between 400 and 600 AD.&lt;/p&gt;&#xA;&lt;p&gt;Some prehistory is described in &lt;a href=&quot;https://history.stackexchange.com/q/5992&quot;&gt;What is the origin of the &amp;quot;wheat and chessboard&amp;quot; legend?&lt;/a&gt; on History SE. Adding up terms of a doubling series can be traced to a cuneiform tablet c. 1800 BC, without connection to yet to be invented chess, but with grains and plausible transmission to India and Persia, where it was attached to chessboards centuries later. &lt;a href=&quot;https://dergipark.org.tr/tr/download/article-file/694013&quot; rel=&quot;noreferrer&quot;&gt;Hoyrup&lt;/a&gt; translates it thus:&amp;quot;&lt;em&gt;1 grain has appended 1 grain: 2 grains the first day. 4 grains the 2nd day. 8 grains the 3d day. 16 grains the 4th day... and so on until 30 days.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The chess fable proper goes back to Islamic sources, which is why the Indian brahmin has the Arabic sounding name. Murray names several manuscripts mentioning Sissa in his &lt;a href=&quot;https://archive.org/details/AHistoryOfChess/page/n199/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;History of Chess&lt;/a&gt;, one already copied c. 996 AD, so the fable is older. &lt;a href=&quot;https://www.cambridge.org/core/journals/journal-of-the-royal-asiatic-society/article/abs/art-xiiithe-origin-and-early-history-of-chess/7F6CC6E3ECC3D98F141495AF053B26FE&quot; rel=&quot;noreferrer&quot;&gt;Macdonell&lt;/a&gt; mentions Ibn Khallikan's retelling c. 1250 AD, without Sissa and with corn instead of wheat. He also describes several other chess related tales from c. 950 AD onwards. For example, an alternative chess origin myth, with no grains at all, but with two brothers vying for the Indian throne, is told in Firdausi's famous poem &lt;a href=&quot;https://en.wikipedia.org/wiki/Shahnameh&quot; rel=&quot;noreferrer&quot;&gt;Shahnameh&lt;/a&gt; written after 977 AD. Second half of the 10th century is when chess became popular in the Islamic world, and tales about it got made and remade.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-05-27T05:47:36.047" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15459" PostTypeId="1" AcceptedAnswerId="15463" CreationDate="2023-05-27T16:17:40.137" Score="2" ViewCount="135" Body="&lt;p&gt;I've come across this diagram of the Ptolemaic model. Am I understanding correctly that this means that all the planets' deferents rotate around a point which next to Earth, while Venus' deferent rotates around an epicycle rotating around the same point. Is that correct?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/O8rwj.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/O8rwj.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18091" LastEditorUserId="55" LastEditDate="2023-05-29T04:04:37.843" LastActivityDate="2023-05-29T19:44:23.530" Title="Did the Ptolemaic system have rotating center of the deferent for Venus?" Tags="|astronomy|archaeoastronomy|ptolemy|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15460" PostTypeId="1" AcceptedAnswerId="15461" CreationDate="2023-05-28T12:59:01.650" Score="6" ViewCount="1428" Body="&lt;p&gt;Could ancient astronomers have proven heliocentrism and, if so, how could they have done so?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-05-28T13:18:56.807" Title="Could ancient astronomers have proven heliocentrism?" Tags="|astronomy|archaeoastronomy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15461" PostTypeId="2" ParentId="15460" CreationDate="2023-05-28T13:18:56.807" Score="11" Body="&lt;p&gt;You can't ever &amp;quot;prove&amp;quot; heliocentrism! (We can't even today. Thinking that science &amp;quot;proves&amp;quot; things is not at all helpful in doing science.) What we &lt;em&gt;can&lt;/em&gt; do is show that a heliocentric model of the Solar System is simpler and explains more observations with fewer assumptions.  (E.g., a heliocentric model is compatible with Newtonian mechanics and a world where earthly mechanics and heavenly motions are explained by one theory is &lt;em&gt;very&lt;/em&gt; persuasive compared with older models of planetary motion. But it is not proof.)&lt;/p&gt;&#xA;&lt;p&gt;The ancient Greeks and the medievals ran into the same basic problem: They were convinced that motions in the sky must be circular and, in the absence of Newton's work, circular motions around the Sun were no more elegant and no more explanatory than circular motions around the Earth. (One of the main reasons Copernicus' theory was  not quickly accepted is that it used circular motion and thus made seriously wrong predictions of the motions of the planets, especially Mars. By around 1600 we had excellent naked eye measurements of planetary motion, probably far better than what the Greeks had. It was Kepler's elliptical orbits which made heliocentrism convincing.)&lt;/p&gt;&#xA;&lt;p&gt;Remember also that measuring absolute distances to celestial bodies was very difficult without telescopes, so the Sun being enormously more massive than the Earth was &lt;em&gt;not&lt;/em&gt; obvious, making its centrality also less obvious.&lt;/p&gt;&#xA;&lt;p&gt;It's also worth remembering that the focus of astronomy up until the early 1600s was on understanding and predicting the motions of the planets, and not understanding them as physical bodies. (Remember that even in the 1800s some reputable scientists doubted that we would ever be able to understand the true nature of the planets and stars -- then spectroscopy was developed.)&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the Greeks knew about ellipses, so a clever Greek mathematician could have proposed elliptical orbits (for Mars and the Moon, at least) and come up with a theory which predicted their motions very accurately. But until you know the relative masses of the Sun and Earth or start to have an inkling of Newtonian dynamics and gravitation, it's really not terribly important which body is at rest.&lt;/p&gt;&#xA;&lt;p&gt;Bottom line: There was just too much science yet to be developed for the Alexandrian Greeks to find a heliocentric Solar System to be compelling. (But even then, never proven.)&lt;/p&gt;&#xA;" OwnerUserId="16773" LastActivityDate="2023-05-28T13:18:56.807" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="15462" PostTypeId="1" CreationDate="2023-05-28T19:13:13.623" Score="2" ViewCount="448" Body="&lt;p&gt;So far I've been checking Wikipedia listings, the university websites of the physicist in question, some databases like OA.mg, and literature searches at university websites. But it’s all scattered, and it would be nice if there were a database of all this stuff dedicated to physicists and their works.&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="12330" LastEditDate="2023-05-28T21:50:21.733" LastActivityDate="2023-05-30T06:28:10.443" Title="How do I find a list of all works published by a Physicist?" Tags="|physics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15463" PostTypeId="2" ParentId="15459" CreationDate="2023-05-29T04:03:50.183" Score="4" Body="&lt;p&gt;There were multiple versions of the Ptolemaic system circulating in the middle ages, including those where Venus did have a rotating eccentric, the center of the deferent, e.g. in Regiomontanus's &lt;em&gt;Epitome of the Almagest&lt;/em&gt;. Ptolemy himself claimed (mistakenly) in &lt;em&gt;Almagest&lt;/em&gt; XII.1 that such a model cannot be used for inner planets, which is puzzling because a simple parallelogram construction shows that an epicycle can always be interchanged with a rotating eccentric. Duke has a helpful &lt;a href=&quot;https://people.sc.fsu.edu/%7Edduke/models.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of medieval Ptolemaic models&lt;/a&gt; with animations, but, unfortunately, the animations require Adobe Flash and do not work anymore.&lt;/p&gt;&#xA;&lt;p&gt;The original version of Ptolemy's &lt;em&gt;Almagest&lt;/em&gt; is described e.g. by &lt;a href=&quot;https://people.sc.fsu.edu/%7Edduke/inner.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Duke in Ancient Solutions of Venus &amp;amp; Mercury Orbits&lt;/a&gt;, and its mathematical details can be found on &lt;a href=&quot;https://www.mathpages.com/home/kmath639/kmath639.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;mathpages&lt;/a&gt;. Eccentrics were typically different for different planets, and there were also equants, points from which rotations of the epicycle centers on the deferents appeared uniform.&lt;/p&gt;&#xA;&lt;p&gt;Many Islamic astronomers were dissatisfied with eccentrics and equants for philosophical reasons and replaced them in their versions by additional epicycles, see &lt;a href=&quot;https://faculty.kfupm.edu.sa/PHYS/alshukri/PHYS215/Islamic_astronomy.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Islamic astronomy by Gingerich&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The equant in particular was objectionable to philosophers who thought of planetary spheres as real physical objects, each sphere driven by the one outside it (and the outermost driven by the prime mover), and who wanted to be able to construct a mechanical model of the system... Furthermore, the equant violated the philosophical notion that heavenly bodies should be moved by a system of perfect circles, each of which rotated with uniform angular velocity about its center. To some purists even Ptolemy's eccentric deferents, which moved the earth away from the center of things, were philosophically unsatisfactory.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Al-Tusi found the equant particularly dissatisfactory. In his Tadhkira (&amp;quot;Memorandum&amp;quot;) he replaced it by adding two more small epicycles to the model of each planet's orbit... Finally a completely concentric rearrangement of the planetary mechanisms was achieved by Ibn al-Shatir, who worked in Damascus in about 1350. By using a scheme related to that of al-Tusi, Ibn al-Shatir succeeded in eliminating not only the equant but also certain other objectionable circles from Ptolemy's constructions.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;European astronomers of late middle ages/Renaissance, like Regiomontanus, inherited such modifications and made variations of their own. Ibn al-Shatir's and Regiomontanus's modifications were instrumental in helping Copernicus transform geocentric models into mathematically equivalent heliocentric ones (although exact transmission of their ideas to him remains controversial), see &lt;a href=&quot;https://journals.sagepub.com/doi/10.1177/0021828617691203?icid=int.sj-full-text.similar-articles.1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Swerdlow, Copernicus’s Derivation of the Heliocentric Theory from Regiomontanus’s Eccentric Models&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-05-29T19:44:23.530" LastActivityDate="2023-05-29T19:44:23.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15464" PostTypeId="2" ParentId="15462" CreationDate="2023-05-29T06:13:53.777" Score="7" Body="&lt;p&gt;There is no analog of Mathscinet or Zentralblatt for physics. (Though Zentralblatt covers some of theoretical and mathematical physics).&lt;/p&gt;&#xA;&lt;p&gt;Then it depends on when your physicist of interest worked.&#xA;If this is a modern person, you can try Google scholar. If the person has a profile there, it will give you a complete list. If s/he does not, you will have to search his/her papers one by one.&lt;/p&gt;&#xA;&lt;p&gt;Other electronic databases exist (some are mentioned in the comments) but they require subscription, they are much less complete than Google scholar, and all of them cover only modern period.&lt;/p&gt;&#xA;&lt;p&gt;If we are talking about a person who worked before the middle of 20th century, there is a very comprehensive database by Poggendorff,  Biographisch-literarisches Handwörterbuch zur Geschichte der exakten Wissenschaften, in multiple volumes. Originally it extended from the beginning of scientific publication to the early 20th century but after Poggendorff's death it was continued approximately to 1967 and it covers all sciences. Parts of it are available online, and the whole on CD, see referfences here &lt;a href=&quot;https://en.wikipedia.org/wiki/Johann_Christian_Poggendorff&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Johann_Christian_Poggendorff&lt;/a&gt;&#xA;Many scientific libraries have paper copies.&lt;/p&gt;&#xA;&lt;p&gt;I don't know what to suggest for the end of 20th century. The Soviets had &amp;quot;Реферативный журнал&amp;quot; which was analogous to Math Review but covered all science. There are indications that it still exists in Russia, but it does not seem to be easily available, and is still published on paper.&lt;/p&gt;&#xA;&lt;p&gt;Finally, for some physicists you can find obituaries, or papers dedicated to the persons birthdays, and they frequently contain a list of publications.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-05-30T06:28:10.443" LastActivityDate="2023-05-30T06:28:10.443" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15465" PostTypeId="1" CreationDate="2023-05-29T09:16:37.320" Score="0" ViewCount="33" Body="&lt;p&gt;I realize this is more a history of engineering question but I do not see a SE devoted to this so I thought I would ask here.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://suiter.com/granville-woods-railway-telegraphy-patent-no-373383/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://suiter.com/granville-woods-railway-telegraphy-patent-no-373383/&lt;/a&gt; discusses a very interesting invention which allowed a moving train, long before radio, to get telegraphic messages but it does not provide details. I assume messages were stored and replayed at a much higher rate than an operator could key them.&lt;/p&gt;&#xA;&lt;p&gt;But even so, I assume further that the amount of data would be very limited. For one thing, as the train moved beyond the station (and the range at which the messages could be transmitted is also of interest) I would guess that the ability to transmit dropped off very fast.&lt;/p&gt;&#xA;&lt;p&gt;If anyone can provide more technical details on an invention I had only heard of recently, I would be very interested.&lt;/p&gt;&#xA;&lt;p&gt;Perhaps this technology was superseded by radio or some other method of transmitting to moving trains that did not require proximity to the train station. Obviously continous communication with trains anywhere along the tracks would be incredibly valuable while transmitting a limited amount of data while a train passed a station assumes that there was no time to stop and simply physically pick up a bunch of paper telegrams. That is not to say that occasionally getting perhaps valuable messages without stopping -- emergencies like obstructed tracks could be signaled via lights along tracks -- would not be useful.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2023-05-29T09:16:37.320" Title="Granville Wood's Telegraph -- do we know the data rate and range?" Tags="|electromagnetism|electricity|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15466" PostTypeId="1" AcceptedAnswerId="15467" CreationDate="2023-05-30T16:04:17.497" Score="7" ViewCount="2537" Body="&lt;p&gt;I am interested in reading Euler's works.&lt;/p&gt;&#xA;&lt;p&gt;The Euler Archive contains some translated works but not all of them. I am just checking here to see if anyone know a complete translation of all of Euler's works.&lt;/p&gt;&#xA;&lt;p&gt;Preferably in HTML format.&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="2277" LastEditDate="2023-06-02T07:04:17.057" LastActivityDate="2023-12-12T09:47:45.737" Title="Have all of Euler's works been translated?" Tags="|mathematicians|euler|translations|18th-century|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15467" PostTypeId="2" ParentId="15466" CreationDate="2023-05-30T16:50:58.367" Score="12" Body="&lt;p&gt;No, not all. I don't think that they are even all published in the original language. There is an ongoing project of translating Euler's mathematical work (only mathematical)&#xA;&lt;a href=&quot;http://eulerarchive.maa.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler archive&lt;/a&gt; and you can see there which works are translated.&lt;/p&gt;&#xA;&lt;p&gt;But if you want to read Euler in English, enough of his papers are already translated in this project to keep you busy for years.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-06-03T06:15:49.273" LastActivityDate="2023-06-03T06:15:49.273" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15468" PostTypeId="2" ParentId="15450" CreationDate="2023-05-30T17:22:07.853" Score="0" Body="&lt;p&gt;Seems so that this has been rediscovered and being worked on under the guise of &amp;quot;Categorical Systems theory&amp;quot;. &lt;a href=&quot;https://topos.site/blog/2021/11/categorical-systems-theory/#:%7E:text=Categorical%20systems%20theory%20is%20the,it%20as%20%E2%80%9Cabstract%20nonsense%E2%80%9D.&quot; rel=&quot;nofollow noreferrer&quot;&gt;See here&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2023-05-30T17:22:07.853" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15469" PostTypeId="2" ParentId="14818" CreationDate="2023-05-30T21:35:46.660" Score="8" Body="&lt;p&gt;Not buried, cremated. The urn with his ashes is in a columbarium in Oakland, if I remember correctly - I visited the place in 1990s with his son Jan.&lt;/p&gt;&#xA;" OwnerUserId="18492" LastActivityDate="2023-05-30T21:35:46.660" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15470" PostTypeId="2" ParentId="15096" CreationDate="2023-05-31T09:22:30.717" Score="0" Body="&lt;p&gt;The Herschels used Pomeranian mud. (Though the only citation I have in mind is one of Patrick O'Brian's historical novels!)&lt;/p&gt;&#xA;" OwnerUserId="18496" LastActivityDate="2023-05-31T09:22:30.717" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15471" PostTypeId="2" ParentId="15466" CreationDate="2023-05-31T12:53:24.337" Score="1" Body="&lt;p&gt;There are now 128 volumes in Euler's opera in the German central catalog of libraries &lt;a href=&quot;https://kxp.k10plus.de/DB=2.1/SET=4/TTL=16/CMD?MATCFILTER=N&amp;amp;MATCSET=N&amp;amp;ACT0=&amp;amp;IKT0=&amp;amp;TRM0=&amp;amp;ACT3=*&amp;amp;IKT3=8183&amp;amp;ACT=SRCHA&amp;amp;IKT=1016&amp;amp;SRT=YOP&amp;amp;TRM=Leonhard+Euler+opera&amp;amp;TRM3=&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There is no hope to get the formula digitized in the current state of mathematical OCR. There is some hope that the Chinese can do it. They have an excellent tradition and workforce in copying glyphs.&lt;/p&gt;&#xA;" OwnerUserId="18499" LastEditorUserId="2277" LastEditDate="2023-06-02T07:04:11.853" LastActivityDate="2023-06-02T07:04:11.853" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15472" PostTypeId="1" CreationDate="2023-06-01T01:54:50.343" Score="1" ViewCount="70" Body="&lt;p&gt;The Big Bang theory is famously attributed to Lemaître .&lt;/p&gt;&#xA;&lt;p&gt;In a moment of rare sentience, I decided to consult Georges Lemaître's works to find out what the original Big Bang theory looked like. After a few hours of aggressively searching the internet, I finally found a full digital pdf of&#xA;&lt;em&gt;The Primeval atom - an essay on cosmogony&lt;/em&gt; translated by Betty H., and at first glance thought surely this is a metaphor for the singularity. I began reading, for a while wondered why he wasn't getting to the point before looking at the contents page and jumping straight to the last chapter where he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...The two ideas which have been formulated, thus far, are: First, the origin of matter as a single atom which disintegrated; second, the equilibrium of clusters of nebulae. These ideas can easily be combined into a single theory. At the time of the atom’s disintegration, matter was strongly condensed. The fragments of the atom were separated from one another with great speeds, speeds which were progressively slowed down by the force of gravitation which, for these large densities, amply overcame the cosmic repulsion &lt;br&gt;&#xA;&lt;strong&gt;[ The Primeval Atom - an essay on cosmogony p. 126 ]&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and he also described the CMBR as the emissions of the original atom. So did I read the wrong book or something? it was just floating around on the internet after all. Or did Georges later modify his theory from primeval atom -&amp;gt; singularity.&lt;/p&gt;&#xA;&lt;p&gt;Because the primeval atom idea hardly bares any resemblance to the Big Bang model at all. The principle of the modern Big Bang theory is the singularity, this in effect is the only part of the theory that tells us about the origins of the universe, the rest of the theory devoted to universal evolution is just deductions based on the singularity model and current knowledge of atomic, photon theory and astronomical observations. So if Georges' theory completely contradicts the most fundamental tenet of the Big Bang theory, how can the theory even be ascribed to him?&lt;/p&gt;&#xA;&lt;p&gt;I've seen people make arguments like:&#xA;&amp;quot;Well, you see. It furnished the foundations of the Big Bang theory&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;But the foundations of the big bang theory are universal expansion and CMBR. In fact anyone could suppose an expanding universe to mean the universe was smaller in the past so perhaps the universe originated as a tiny dot, so even in forming the foundations of the Big Bang theory, I'm not convinced Lemaître has a place. Although I read somewhere that he proved universal expansion, if so then fair enough.&lt;/p&gt;&#xA;&lt;p&gt;So this leads to my actual question. Who really invented the Big Bang theory? and this question is equivalent to &amp;quot;Who invented the idea the that the universe started from a singularity?&amp;quot; Since in my understanding that is the defining and most fundamental characteristic of the theory.&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="6609" LastEditDate="2023-06-01T12:02:39.720" LastActivityDate="2023-06-01T12:02:39.720" Title="Who actually invented the Big Bang theory?" Tags="|theoretical-physics|cosmology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15473" PostTypeId="1" AcceptedAnswerId="15480" CreationDate="2023-06-01T04:39:36.337" Score="5" ViewCount="173" Body="&lt;p&gt;In general, a linear operator on a complex vector space need not always have an eigenspace decomposition. But it will always have enough &lt;em&gt;generalized eigenvectors&lt;/em&gt; to provide a decomposition of the similar kind (&lt;span class=&quot;math-container&quot;&gt;$V=G(\lambda_1, T) \oplus\cdots \oplus G(\lambda_m,T)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$G(\lambda_i, T) $&lt;/span&gt; are the eigenspaces corresponding to the distinct eigenvalues &lt;span class=&quot;math-container&quot;&gt;$\lambda_i$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$T\in\mathcal L(V) $&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;This brief intro provides the powerful utility of generalized eigenvectors. While the concept has been hovering  around algebra textbooks for quite some times since &lt;span class=&quot;math-container&quot;&gt;$60$&lt;/span&gt;s, it has found detailed treatment in the recent textbook &lt;em&gt;Linear Algebra Done Right&lt;/em&gt; by Axler who has a penchant of suppressing the treatment of determinant at a minimal level.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia has a dedicated column in mentioning the &lt;a href=&quot;https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;history&lt;/a&gt; of eigenvectors.&lt;/p&gt;&#xA;&lt;p&gt;Unfortunately, I have failed to come across any info as to how  the concept of generalized eigenvectors was developed and who was/were responsible for the same.&lt;/p&gt;&#xA;&lt;p&gt;Was it Jordan (of Jordan basis) or someone else?&lt;/p&gt;&#xA;" OwnerUserId="17132" LastEditorUserId="55" LastEditDate="2023-06-01T13:44:37.970" LastActivityDate="2023-06-02T07:20:36.613" Title="How and by whom was the concept of generalized eigenvectors developed?" Tags="|linear-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15474" PostTypeId="1" CreationDate="2023-06-01T09:46:03.767" Score="2" ViewCount="146" Body="&lt;p&gt;To my understanding, this new Geometric algebra thing is actually nothing more than years old &lt;a href=&quot;https://en.wikipedia.org/wiki/Clifford_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clifford Algebra&lt;/a&gt;. Yet it is advertised by many of its proponents as a fundamentally new thing; why is that?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="12330" LastEditDate="2023-06-02T07:04:04.603" LastActivityDate="2023-06-02T07:04:04.603" Title="Why did Clifford Algebra suddenly gain a resurgence under the name of Geometric Algebra in recent years?" Tags="|linear-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15475" PostTypeId="1" CreationDate="2023-06-01T15:26:39.853" Score="3" ViewCount="43" Body="&lt;p&gt;Was not sure if I could ask this in the Academia Stack Exchange as the question is a little bit meta, so HSM looked like a more appropriate place. Apologies if that's not the case.&lt;/p&gt;&#xA;&lt;p&gt;There is currently some criticism around the state of academia: too much emphasis on publishing, predatory journals, non competitive pay, very competitive environment, and crisis of replication, among others.&lt;/p&gt;&#xA;&lt;p&gt;I'm wondering if anyone could point me to resources where I could get a view of how the academic environment was in the early 1900s, how different the environment was when degrees were more of a &amp;quot;premium&amp;quot; than they are today and maybe different issues affecting research back then.&lt;/p&gt;&#xA;" OwnerUserId="12712" LastActivityDate="2023-06-01T15:26:39.853" Title="Resources to learn about the state of academia in the early 1900s?" Tags="|scientific-method|academia|science|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15476" PostTypeId="2" ParentId="5537" CreationDate="2023-06-01T23:12:50.147" Score="1" Body="&lt;p&gt;Scientific Consensus has been TOTALLY WRONG many times.&lt;/p&gt;&#xA;&lt;p&gt;For examples, see: &lt;a href=&quot;https://davidfmayer.substack.com/p/the-myth-of-settled-science-a33/comments&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://davidfmayer.substack.com/p/the-myth-of-settled-science-a33/comments&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18510" LastActivityDate="2023-06-01T23:12:50.147" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15477" PostTypeId="2" ParentId="15466" CreationDate="2023-06-01T23:13:13.910" Score="3" Body="&lt;p&gt;If you are interested in reading Euler in English translation, you should take a look at Ian Bruce's excellent web site &lt;a href=&quot;https://www.17centurymaths.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.17centurymaths.com&lt;/a&gt;. His project involves the translation of mathematical works of the 17th and 18th century from Latin to English, and many of Euler's works are available. Bruce's work is a great contribution to the mathematics community.&lt;/p&gt;&#xA;" OwnerUserId="18509" LastActivityDate="2023-06-01T23:13:13.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15478" PostTypeId="1" CreationDate="2023-06-02T00:35:13.480" Score="3" ViewCount="121" Body="&lt;p&gt;I know that Kepler thought based on comet tail that light might exert pressure (although is the Solar wind not also involved?) but did Maxwell's prediction stem from newer observations or from perhaps his equations?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="4251" LastEditDate="2023-07-02T08:16:19.810" LastActivityDate="2023-07-02T08:16:19.810" Title="Why did Maxwell predict radiation pressure?" Tags="|physics|electromagnetism|light|maxwell|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15479" PostTypeId="2" ParentId="15478" CreationDate="2023-06-02T03:56:18.707" Score="2" Body="&lt;p&gt;According to Brian Clegg's &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B07KGNBZGP&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;biography of Maxwell&lt;/a&gt; the idea that light could exert pressure only came to Maxwell's attention shortly after the publication of his &lt;em&gt;Treatise&lt;/em&gt; while &amp;quot;pulling at the threads of new ideas&amp;quot; suggested by his theory.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Maxwell gave this hypothesis a theoretical backing by showing that the energy of absorbed light should result in momentum being added to the body absorbing it.&lt;/p&gt;&#xA;&lt;p&gt;The reason we don't see this happening as a rule, Maxwell suggested, was because the effect was so small.  He calculated that the sun, the most powerful light source in our vicinity, would only produce the equivalent pressure of &lt;span class=&quot;math-container&quot;&gt;$7$&lt;/span&gt; grams (&lt;span class=&quot;math-container&quot;&gt;$0.015$&lt;/span&gt; pounds) across a whole hectare (&lt;span class=&quot;math-container&quot;&gt;$2.47$&lt;/span&gt; acres) of area. Maxwell would never see a demonstration of the effect that he had predicted, but &lt;span class=&quot;math-container&quot;&gt;$25$&lt;/span&gt; years after his theory was published, the Russian physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Pyotr_Lebedev&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pyotr Lebedev&lt;/a&gt; demonstrated this 'radiation pressure' for the first time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clegg also writes that Maxwell, in a letter to the Harvard astronomer &lt;a href=&quot;https://en.wikipedia.org/wiki/George_Phillips_Bond&quot; rel=&quot;nofollow noreferrer&quot;&gt;George Phillip Bond&lt;/a&gt; dated 1863, some ten years before the publication of his &lt;em&gt;Treatise&lt;/em&gt;, wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Is there anything about a comet to render its lines of force visible? and not those of a planet which are much stronger?  I think that visible lines of gravitational force are extremely improbable, but I never saw anything so like them as some comet tails.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it is clear that Maxwell was considering the curious properties of comets before he arrived at the idea of light exerting pressure.  However, his motivation for the claim that light exerts pressure appears to have been born of purely theoretical considerations.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-06-02T03:56:18.707" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15480" PostTypeId="2" ParentId="15473" CreationDate="2023-06-02T07:20:36.613" Score="7" Body="&lt;p&gt;This is somewhat difficult to track because much of the work on linear algebra in 19th century was coached in the language of anything but matrices and vectors, differential equations, substitutions, bilinear forms, determinants, etc. That includes the works of Weierstrass and Jordan on &amp;quot;Jordan&amp;quot; canonical form. Moreover, the prefix &amp;quot;eigen&amp;quot; did not become conventional until 1960s.&lt;/p&gt;&#xA;&lt;p&gt;In particular, Weierstrass, who published a version of the canonical form in 1868, two years before Jordan, used determinants and elementary divisors, see &lt;a href=&quot;https://www.jstor.org/stable/41133484&quot; rel=&quot;noreferrer&quot;&gt;Hawkins, Weierstrass and the Theory of Matrices&lt;/a&gt;. However, Killing in his &lt;a href=&quot;https://books.google.com/books/about/Erweiterung_des_Raumbegriffes.html?id=uFkKcgAACAAJ&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Erweiterung des Raumbegriffes&lt;/em&gt; (1884)&lt;/a&gt; spelled out Weierstrass's construction in terms of what we would call generalized eigenvectors for the adjoint action operator, see &lt;a href=&quot;https://link.springer.com/book/10.1007/978-1-4612-1202-7&quot; rel=&quot;noreferrer&quot;&gt;Hawkins, Emergence of the Theory of Lie Groups, 5.1&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;An important parameter in Killing's deliberations is thus the minimal&#xA;multiplicity &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$w = 0$&lt;/span&gt; as a characteristic root. The characteristic equation&#xA;corresponding to any &lt;span class=&quot;math-container&quot;&gt;$X = \sum_{i=1}^r e_iX_i$&lt;/span&gt; is thus of the form&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\Delta(w)=(-1)^r\big[w^r-\psi_1(e)w^{r-1}+\cdots\pm\psi_{r-k}(e)w^k\big]=0,$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\psi_{r-k}(e)\not\equiv0$&lt;/span&gt;. In order to simplify the multiplication constants as&#xA;much as possible Killing chose &lt;span class=&quot;math-container&quot;&gt;$X = X'_1$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\psi_{r-k}(e)\neq0$&lt;/span&gt;, i.e. such&#xA;that &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the multiplicity of &lt;span class=&quot;math-container&quot;&gt;$w = 0$&lt;/span&gt; in the characteristic equation for &lt;span class=&quot;math-container&quot;&gt;$X'_1$&lt;/span&gt;.&#xA;According to Weierstrass's theory, &lt;span class=&quot;math-container&quot;&gt;$X'_2, ... ,X'_k$&lt;/span&gt; may then be chosen so that&#xA;&lt;span class=&quot;math-container&quot;&gt;$[X'_1,X'_i] = \epsilon_i X'_{i-1}$&lt;/span&gt; where  &lt;span class=&quot;math-container&quot;&gt;$\epsilon_i$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;. The &lt;span class=&quot;math-container&quot;&gt;$X'_i$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$i = 1, ... , k$&lt;/span&gt;, thus represent, in more modern terminology, a basis consisting of ordinary or generalized eigenvectors for &lt;span class=&quot;math-container&quot;&gt;$w = 0$&lt;/span&gt;, of the generalized null space &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{g}_0(X'_1)$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$\text{ad}\,X'_1$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;Killing also showed that in the Jordan-Weierstrass form for &lt;span class=&quot;math-container&quot;&gt;$\text{ad}\,X_1$&lt;/span&gt;, at&#xA;least two Jordan blocks must exist. That is, some &lt;span class=&quot;math-container&quot;&gt;$Y\neq cX$&lt;/span&gt; exists such&#xA;that &lt;span class=&quot;math-container&quot;&gt;$[X, Y] = 0$&lt;/span&gt;. He considered a &amp;quot;completely general&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$X_1\in\mathfrak{g}$&lt;/span&gt;, by which he&#xA;apparently meant an &lt;span class=&quot;math-container&quot;&gt;$X_1$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\text{ad}\,X_1$&lt;/span&gt; has maximal rank or, equivalently,&#xA;minimal nullity... He then extended &lt;span class=&quot;math-container&quot;&gt;$X_1$&lt;/span&gt; to a basis &lt;span class=&quot;math-container&quot;&gt;$X_2, ... ,X_k$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{g}$&lt;/span&gt; that&#xA;put &lt;span class=&quot;math-container&quot;&gt;$\text{ad}\,X_1$&lt;/span&gt; in its Jordan-Weierstrass form with Jordan blocks decreasing in&#xA;size.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The modern style matrix/vector exposition only emerged much later. As Miller remarks in the &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/&quot; rel=&quot;noreferrer&quot;&gt;Earliest Uses&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Modern expositions of spectral theory often begin with a matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and introduce value &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; and vector &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; together in the value/vector-equation &lt;span class=&quot;math-container&quot;&gt;$Ax = \lambda x$&lt;/span&gt;... This finite-dimensional case is used to motivate the treatment of differential equations and integral equations which involve infinite-dimensional spaces where the vector is now a function. The historical order of development was more or less the reverse. The polynomial equation generated from the differential equations of celestial mechanics came first, ca. 1780, then the equation was expressed using determinants ca. 1830, then the equation was associated with matrices ca. 1880, then integral equations were studied ca. 1900 until finally the modern order of topics starting from the value/vector-equation became established ca. 1940.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Textbooks that appeared in 1940s started associating Weierstrass's elementary divisors of matrices/linear transformations to their generalized eigenvectors/spaces, without the name. For example, &lt;a href=&quot;https://www.science.org/doi/10.1126/science.143.3606.560.c&quot; rel=&quot;noreferrer&quot;&gt;Mal'cev's 1948 Russian textbook&lt;/a&gt; called them root vectors/spaces.&lt;/p&gt;&#xA;&lt;p&gt;As for &amp;quot;eigen&amp;quot;values/vectors, generalized or not, Halmos wrote in Finite Dimensional Vector Spaces (1958):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Almost every combination of the adjectives proper, latent, characteristic, eigen and secular, with the nouns root, number and value, has been used in the literature for what we call a proper value&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And only in Hilbert Space Problem Book (1967) he conceded defeat:&amp;quot;&lt;em&gt;I have now become convinced that the war is over, and eigenvalues have won it&lt;/em&gt;&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-06-02T07:20:36.613" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15481" PostTypeId="1" CreationDate="2023-06-02T17:00:13.703" Score="5" ViewCount="99" Body="&lt;p&gt;The Hamiltonian of a charged particle in a magnetic field is: &lt;span class=&quot;math-container&quot;&gt;$$H=\frac{1}{2m}(\frac{h}{2\pi i}\nabla-qA)^2+q\phi$$&lt;/span&gt;&#xA;Can anybody help me find out when, how and by whom was it first derived?&lt;/p&gt;&#xA;" OwnerUserId="18513" LastEditorUserId="55" LastEditDate="2023-06-04T11:47:38.970" LastActivityDate="2023-06-04T11:47:38.970" Title="How and by whom was the magnetic term in the Hamiltonian first derived?" Tags="|quantum-mechanics|electromagnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15482" PostTypeId="1" CreationDate="2023-06-02T21:35:09.497" Score="0" ViewCount="84" Body="&lt;p&gt;I'm the maintainer of the &lt;a href=&quot;https://archive.org/details/hyper_number_4_8&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hyper-complex number List&lt;/a&gt;,&#xA;I am searching for some help in respect of the categorization, clasification regarding the List. I really like references in the line of the List, but here, certainly I am looking for alternatives ways of clasification while grouping the references of the list. The other reason I am came here and not, for example, Math Overflow, Mathematics or some other forum is because here is more probably to find people with a god grasp in mathematical references from the past and the present with a much much bigger mathematical scope than me, or, with very different background. Not quite sure if there are some helpful tags to this question. At some point in the future, the list will become a wiki, but for now will keep its current format. What other possible categories you may see?, some clear weaknesses or voids in the classification? Some other category in the line of the works that may fit in the list? or some other vein that I may exploit?&lt;/p&gt;&#xA;&lt;p&gt;Best regards, Pyra&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update&lt;/strong&gt; : I was able to dedicate a couple of days to experiment with the classification for &lt;a href=&quot;https://archive.org/details/hyper_number_5_3&quot; rel=&quot;nofollow noreferrer&quot;&gt;the version 5.3&lt;/a&gt; . Possibly, all version 5.X will be oriented toward modification and experimentation with the main tags of the list. The difference with the number of pages is mostly the separation of the &amp;quot;heaps of references&amp;quot;, not adding new ones.&lt;/p&gt;&#xA;&lt;p&gt;The First section engenders the sections &amp;quot;ALGEBRAIC STRUCTS&amp;quot;, &amp;quot;REAL NUMBER LINE&amp;quot; and &amp;quot;MATHEMATICAL RELATIONS&amp;quot;. The Geometrical section split into &amp;quot;THE CURVY, THE ROUND AND THE HOLEY&amp;quot;, &amp;quot;DIAGRAMS, ICONS AND CIRCUITS&amp;quot;, &amp;quot;KNOTS AND GRAPHS&amp;quot;, &amp;quot;SPACES AND CONTINUA&amp;quot;, &amp;quot;TURISTIC&amp;quot;, &amp;quot;THE POLYHEDRIC, THE SYNTHETIC AND THE COORDINATED&amp;quot;, &amp;quot;OFF-TOPIC AND CURIOSITIES&amp;quot;, &amp;quot;MIND, BRAIN AND NUMBERS&amp;quot; and &amp;quot;ARRANGEMENTS AND PUZZLES&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Still not too satisfied, and keenly expectant of other possible Tags. In case of a proposed solution, just propose the Tags/Labels, and I will do the migration of references to their respective classification.&lt;/p&gt;&#xA;" OwnerUserId="18515" LastEditorUserId="18515" LastEditDate="2023-06-23T19:19:28.583" LastActivityDate="2023-06-23T19:19:28.583" Title="Improvement to the classification and labeling in a list of mathematical references" Tags="|reference-request|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15483" PostTypeId="2" ParentId="15481" CreationDate="2023-06-04T06:27:28.570" Score="7" Body="&lt;p&gt;Lorentz derived the Lagrangian and the Lorentz force formula with the magnetic term in his &lt;a href=&quot;https://www.sophiararebooks.com/pages/books/4924/hendrik-antoon-lorentz/versuch-einer-theorie-der-electrischen-und-optischen-erscheinungen-in-bewegten-korpern&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern&lt;/em&gt; (1895)&lt;/a&gt;. There were precursors on the formula, Thomson, Heaviside and even Maxwell himself, but they did not use the Lagrangian. Switching from velocities to momenta and from the Lagrangian to the Hamiltonian is rather routine, it might have been done in passing by Einstein somewhere.&lt;/p&gt;&#xA;&lt;p&gt;Getting the quantum Hamiltonian became trivial after the prescription of replacing &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\frac{h}{2\pi i}\nabla$&lt;/span&gt; was introduced by Schrödinger in 1926. Kennard, for example, writes Hamiltonians for the electron in electric and magnetic fields in &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01391200&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Zur Quantenmechanik einfacher Bewegungstypen&lt;/em&gt; (1927)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-06-04T06:27:28.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15484" PostTypeId="2" ParentId="11576" CreationDate="2023-06-04T23:17:18.203" Score="2" Body="&lt;p&gt;Regarding &amp;quot;logicians or mathematicians&amp;quot;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Weierstrass&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl Weierstrass&lt;/a&gt; made it &amp;quot;into the academy while skipping over the PhD&amp;quot;, though he did eventually receive an honorary doctorate.&lt;/p&gt;&#xA;&lt;p&gt;From William Dunham's &lt;em&gt;The Calculus Gallery&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He followed a nontraditional route to prominence. His student years had been those of an underachiever, featuring more beer and sword play than is normally recommended.  At age 30 Weierstrass found himself on the faculty of a German &lt;em&gt;gymnasium&lt;/em&gt; (i.e., High School) far removed from the intellectual centers of Europe. [...]&lt;/p&gt;&#xA;&lt;p&gt;In 1854 this unknown teacher from an unknown town published a memoir on Abelian integrals that astonished the mathematicians who read it.  It was evident that the author, whoever he was, possessed an extraordinary talent. Within two years, Weierstrass had secured a position at the University of Berlin and found himself on one of the world's great mathematical faculties.  His was a truly Cinderella story.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2023-06-04T23:44:48.570" LastActivityDate="2023-06-04T23:44:48.570" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15486" PostTypeId="1" CreationDate="2023-06-05T05:10:41.760" Score="5" ViewCount="232" Body="&lt;p&gt;Euler is &lt;a href=&quot;https://math.stackexchange.com/questions/3021988/why-is-it-called-e&quot;&gt;usually credited&lt;/a&gt; with denoting this number with the letter &lt;span class=&quot;math-container&quot;&gt;$\mathrm e$&lt;/span&gt;. But&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It seems unlikely that Euler chose the letter because it is the initial of his own name, as occasionally been suggested: he was an extremely modest man and often delayed publication of his own work so that a colleague or student of his would get due credit (Maor, &lt;a href=&quot;https://books.google.com/books?id=rCGM6TKHyboC&amp;amp;pg=PA156&quot; rel=&quot;nofollow noreferrer&quot;&gt;1994&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, who first called this number &amp;quot;Euler's number&amp;quot;?&lt;/p&gt;&#xA;" OwnerDisplayName="user18522" LastEditorUserId="12330" LastEditDate="2023-06-06T01:08:48.960" LastActivityDate="2023-06-06T01:08:48.960" Title="Who first called $\mathrm e$ &quot;Euler's number&quot;?" Tags="|mathematics|terminology|euler|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15487" PostTypeId="2" ParentId="15486" CreationDate="2023-06-05T09:51:18.980" Score="8" Body="&lt;p&gt;&lt;a href=&quot;https://books.google.com/ngrams/graph?content=Euler%27s+number&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=en-2019&amp;amp;smoothing=3&quot; rel=&quot;noreferrer&quot;&gt;Google Ngram&lt;/a&gt; does not show any use of &amp;quot;Euler's number&amp;quot; before 1880s, and  early sources typically call &amp;quot;Euler's number&amp;quot; what is now called the Euler–Mascheroni constant &lt;span class=&quot;math-container&quot;&gt;$0.57721...$&lt;/span&gt;, see e.g. &lt;a href=&quot;https://www.jstor.org/stable/2369545&quot; rel=&quot;noreferrer&quot;&gt;Sylvester's Constructive Theory of Partitions (1882)&lt;/a&gt;, p. 298. Some other numbers associated with Euler are also called &amp;quot;Euler's&amp;quot;, but not &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Euler's number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&amp;quot; is probably the creation of modern textbooks. The earliest one I found is the influential &lt;a href=&quot;https://www.google.com/books/edition/What_is_Mathematics/RGgGAQAAIAAJ?hl=en&amp;amp;gbpv=0&amp;amp;bsq=%22euler%27s%20number%22&quot; rel=&quot;noreferrer&quot;&gt;What is Mathematics by Courant and Robbins&lt;/a&gt;, p. 297. Whether it was the first or not, it must have contributed to its spread. For example, the curriculum of the geometry seminar at Regis College published in &lt;a href=&quot;https://www.jstor.org/stable/2305942&quot; rel=&quot;noreferrer&quot;&gt;American Mathematical Monthly in 1950&lt;/a&gt;, recommends it as primary reading and copies &amp;quot;Euler's number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&amp;quot; from it (p. 256).&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-06-05T09:51:18.980" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15489" PostTypeId="1" AcceptedAnswerId="15490" CreationDate="2023-06-06T19:56:32.470" Score="2" ViewCount="244" Body="&lt;p&gt;From Andrew Ellinor &amp;amp; Satyajit Mohanty's &lt;a href=&quot;https://brilliant.org/wiki/solving-cubic-equations-lagranges-resolvent&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; on a technique for solving cubics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Using Lagrange's resolvents, to solve the cubic, one has to first solve a quadratic. Given the general cubic, &lt;span class=&quot;math-container&quot;&gt;$$x^3 + ax^2 + bx + c = 0$$&lt;/span&gt; It's [sic] resolvent equation is given by &lt;span class=&quot;math-container&quot;&gt;$$z^2 + z \left( 2 a^3 - 9 a b + 27 c \right) + \left( a^2 - 3b \right)^3 = 0$$&lt;/span&gt; such that that the solution is given by &lt;span class=&quot;math-container&quot;&gt;$$x= \frac{ -a+z_{1}^{\frac{1}{3}} + z_{2}^{\frac{1}{3}} }{3}$$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$z_{1,2}$&lt;/span&gt; are roots of the resolvent&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Does anyone know where, in which work, &lt;a href=&quot;https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lagrange&lt;/a&gt; did actually develop this technique?&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="4251" LastEditDate="2023-06-08T17:35:34.803" LastActivityDate="2023-06-08T17:35:34.803" Title="Where did Lagrange write his technique of resolvents for solving polynomials?" Tags="|mathematics|mathematicians|elementary-algebra|lagrange|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15490" PostTypeId="2" ParentId="15489" CreationDate="2023-06-06T20:55:53.487" Score="4" Body="&lt;p&gt;The relevant publication seems to have been published across two annual volumes (for the years 1770 and 1771, but likely printed in 1773) of the Proceedings of the Royal Academy of Science and Literature in Berlin. At present, I have not been able to find a scan of the actual publication, but it is reproduced in the third volume of the collected works of Lagrange published in 1869:&lt;/p&gt;&#xA;&lt;p&gt;J. Lagrange, &amp;quot;Réflexions sur la résolution algébrique des équations&amp;quot;,&#xA;&lt;em&gt;Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin&lt;/em&gt;, années 1770 et 1771. In: J.-A. Serret (ed.), &lt;em&gt;Oeuvres de Lagrange&lt;/em&gt;, vol. 3, Paris: Gauthier-Villars 1869, pp. 205-421&lt;/p&gt;&#xA;&lt;p&gt;The Gallica digital repository of the Bibliothèque nationale de France (BNF) provides a scan of the volume &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f206.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;. Continuing my search for the original source, I found the first part of the publication here:&lt;/p&gt;&#xA;&lt;p&gt;J. Lagrange, &amp;quot;Réflexions sur la résolution algébrique des équations&amp;quot;,&#xA;&lt;em&gt;Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin&lt;/em&gt;, 1770 (1772), pp. 134-215 (&lt;a href=&quot;https://books.google.com/books?id=NQQOAAAAQAAJ&amp;amp;pg=RA1-PA134#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan online&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;At long last, I also managed to locate the second part of the original publication:&lt;/p&gt;&#xA;&lt;p&gt;J. Lagrange, &amp;quot;Suite des Réflexions sur la Résolution algébrique des Équations&amp;quot;, &lt;em&gt;Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin&lt;/em&gt;, 1771 (1773), pp. 138-253 (&lt;a href=&quot;https://www.google.com/books/edition/Nouveaux_memoires_de_l_Academie_royale_d/enPaeQG_cfwC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Suite%20des%20r%C3%A9flexions%20sur%20la%20r%C3%A9solution%20alg%C3%A9brique%20des%20%C3%A9quations%22&amp;amp;pg=RA1-PA138&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan online&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-06-06T22:25:01.803" LastActivityDate="2023-06-06T22:25:01.803" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15491" PostTypeId="1" CreationDate="2023-06-07T09:13:28.047" Score="2" ViewCount="271" Body="&lt;p&gt;We've all heard the popular claim that the expansion of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; contains every natural number; however, as we can see from numerous sources,  such as &lt;a href=&quot;https://math.stackexchange.com/q/216343/104041&quot;&gt;Mathematics Stack Exchange&lt;/a&gt; and &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Normal_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, it is often stated emphatically by those in the know that this is only a conjecture.&lt;/p&gt;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When did this claim become popular? Why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Thoughts:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I have no clue why. Maybe a pop-maths author misstated the fact that it's a conjecture and the misunderstanding stuck.&lt;/p&gt;&#xA;&lt;p&gt;I know most mathematicians &lt;em&gt;believe&lt;/em&gt; it to be true and that the conjecture has been around for decades, but the popular understanding seems to be that it is proven.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Context:&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm a Mathematics PhD student. I want to understand the origin of this common notion because, due to its traction, some folk I know have a hard time believing me when I say it's only a conjecture.&lt;/p&gt;&#xA;" OwnerUserId="6312" LastEditorUserId="6312" LastEditDate="2023-06-07T09:30:31.440" LastActivityDate="2023-06-14T09:29:26.763" Title="Why is the claim that $\pi$ contains all natural numbers thought by the general public to be proven, despite it being conjecture currently?" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15492" PostTypeId="1" AcceptedAnswerId="15494" CreationDate="2023-06-07T21:39:17.537" Score="12" ViewCount="2751" Body="&lt;p&gt;How did the shift from the times of Thales, Pythagoras, and even Euclid, where mathematical objects were found, exhibited, or constructed from given entities, to modern mathematics occur? In modern mathematics, entities no longer need to be constructed or computed in order to be named and manipulated; they simply need to exist. What was the transformative change that enabled this level of abstraction to be embraced and widely adopted, namely the distinction between mere existence and actual realization?&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="6609" LastEditDate="2023-06-08T18:35:49.970" LastActivityDate="2023-06-09T07:05:25.860" Title="How did the shift from constructed mathematical objects to modern mathematics occur?" Tags="|mathematics|computation|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15493" PostTypeId="2" ParentId="15492" CreationDate="2023-06-08T06:37:44.363" Score="10" Body="&lt;p&gt;This change apparently occurred in the middle of 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Let me cite Carl Ludwig Siegel about it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is completely clear to me which conditions caused the gradual decadence of mathematics, from its high level some 100 years ago, down to the present hopeless nadir. Degeneration of mathematics begins with the ideas of Riemann, Dedekind and Cantor which progressively repressed the reliable genius of Euler, Lagrange and Gauss.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and he continues:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Through the influence of textbooks like those of Hasse, Schreier and van der Waerden, the new generation was seriously harmed, and the work of Bourbaki finally dealt the fatal blow.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;C. L. Siegel, Letter to A. Weil, 1959. German original is cited in the lecture of H. Grauert, in: Mathematics and Theoretical Physics, 1993, ed.: Minaketan Behara, Rudolf Fritsch, Rubens G. Lintz, W. de Gruyter, 1995.&lt;/p&gt;&#xA;&lt;p&gt;Similar views about the influence of Riemann on mathematics were shared by many mathematicians of that time, for example by Chebyshev and Lyapunov.&lt;/p&gt;&#xA;&lt;p&gt;Bourbaki also acknowledges the role of Riemann in reshaping of mathematics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Riemann has to be considered the founder of topology as well as of most of the branches of modern mathematics...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;N. Bourbaki, Topology I, Historical essay.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-06-09T07:05:25.860" LastActivityDate="2023-06-09T07:05:25.860" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15494" PostTypeId="2" ParentId="15492" CreationDate="2023-06-08T08:12:08.677" Score="18" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; It is important to distinguish different senses of &amp;quot;constructiveness&amp;quot;, the technical one, as in constructivism, and the one of approach to mathematics as primarily problem-solving. They followed two different lines of development that are often confused (this is where Wikipedia goes wrong), and only the second underwent a shift in 19th century. The first one, on the other hand, shows gradual expansion of construction tools since Euclid and is continuous through the 19th century.&lt;/p&gt;&#xA;&lt;p&gt;The problem-solving (&amp;quot;constructive&amp;quot; in a different sense) attitude also goes back to Euclid and dominated the field up to the end of 18th century. It gave way to the more meta-theoretical &amp;quot;conceptual&amp;quot; attitude even before Cantor, although he championed it along with Dedekind and Weierstrass, and it was canonized by Hilbert, Zermelo, etc., in the modern axiomatic method. The rise of impossibility and &amp;quot;pure existence&amp;quot; results to prominence accompanied it, but it was not directly related to non-constructivism in the technical sense.&lt;/p&gt;&#xA;&lt;p&gt;Here is Lützen describing the shift in &lt;a href=&quot;http://www.math.wm.edu/%7Evinroot/Wantzel.pdf&quot; rel=&quot;noreferrer&quot;&gt;Why was Wantzel overlooked for a century?&lt;/a&gt; to explain the fate of Wantzel's now famous impossibility proofs:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Indeed, during the 18th century, mathematics was a constructive enterprise consisting of finding solutions to problems, and this view of mathematics was still shared by most of Wantzel’s contemporaries... In the history of the quintic, Abel explicitly tried to change this view of impossibility results. As emphasized by Sørensen, Abel suggested that one ought to rephrase the problem as a problem that always has an answer.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Instead of asking for a solution by radicals, one should first ask if the quintic has a solution of this kind, and only if the answer turns out to be yes would one proceed to the question of finding such a solution constructively. By formulating the existence question as a mathematical problem, Abel had allowed the impossibility result to be a solution of a problem, not just a statement about the impossibility of finding an answer to a problem. Thus impossibility statements became proper theorems in mathematics. Several younger mathematicians continued along the same line as Abel. Evariste Galois is the most obvious example, but also Joseph Liouville’s investigations of integrability of functions and differential equations in finite form follow this lead.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;These new tendencies relative to impossibility results went hand in hand with a new focus&#xA;on existence theorems, and can be viewed as two sides of the same coin. In particular,&#xA;Cauchy insisted that before finding the sum of an infinite series or the solution of a differential&#xA;equation, one should prove their existence. And both trends were parts of a change&#xA;from a constructive to a conceptual paradigm in mathematics. Another instance of this&#xA;change was the emergence in the 1830s of Sturm–Liouville theory where the (qualitative)&#xA;properties of solutions to certain differential equations were investigated without solving&#xA;the equation in analytic form.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above does not apply to constructive/non-constructive in the technical or even more relaxed sense, which does not necessarily track problem-solving vs theory-building. If we go by the modern definition of constructive proof, which discards proofs by contradiction = &lt;em&gt;reductios ad absurdum&lt;/em&gt;, then the non-constructive goes all the way back to ancient Greeks. Euclid employs &lt;em&gt;reductios&lt;/em&gt; not only in incommensurability proofs, but even in the most elementary &lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/elements/bookI/propI6.html&quot; rel=&quot;noreferrer&quot;&gt;proposition  I.6&lt;/a&gt;, the isosceles triangle theorem, which surely goes back to Pythagoreans if not Thales.&lt;/p&gt;&#xA;&lt;p&gt;And even if we focus just on straightedge and compass constructions, what changes over the centuries is mostly the range of construction tools. Euclid's constructionism is not that different in spirit from modern one, where axioms of set theory are used as construction postulates. Most ZFC axioms (specification, pairing, union, powerset, replacement, choice) are even formulated as such postulates. Even employment of the axiom of choice to produce sets to specifications, the very paradigm of non-constructive, is unabashedly called &amp;quot;construction&amp;quot;, as in &lt;a href=&quot;https://math.stackexchange.com/q/33418/152568&quot;&gt;Vitali construction&lt;/a&gt; of Lebesgue non-measurable sets, etc. See &lt;a href=&quot;https://arxiv.org/abs/1109.4298&quot; rel=&quot;noreferrer&quot;&gt;Rodin's Doing and Showing&lt;/a&gt; on structural analogies between Euclidean and set-theoretic constructionism.&lt;/p&gt;&#xA;&lt;p&gt;Already in Euclid, there are tricky cases where he implicitly relaxes his straightedge and compass strictures. The proof of &lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/elements/bookXII/propXII2.html&quot; rel=&quot;noreferrer&quot;&gt;proposition XII.2&lt;/a&gt;, which applies the so-called &amp;quot;method of exhaustion&amp;quot; to the circle quadrature, essentially presents a re-construction of the circle by transcendental means (as a limit of inscribed polygons), and then uses a &lt;em&gt;reductio&lt;/em&gt; to prove its equivalence to the elementary compass construction. Archimedes used the &amp;quot;method of exhaustion&amp;quot; on conic sections and even solids, and others happily intersected transcendental curves (quadratrix, spiral) with lines. Unlike the modest (and omitted) completeness assumptions needed in the &lt;em&gt;Elements&lt;/em&gt;, these &amp;quot;constructions&amp;quot; implicitly rely on strong forms of completeness for the continuum. See &lt;a href=&quot;https://www.jstor.org/stable/41133725&quot; rel=&quot;noreferrer&quot;&gt;Bos, Construction of equations, 1637-1750&lt;/a&gt; on how transcendental methods were gradually accepted in geometry.&lt;/p&gt;&#xA;&lt;p&gt;By the 17th century, algebra and number theory also matured, so &amp;quot;constructions&amp;quot; involved infinite decimals (Stevin) and then infinite series and products (Wallis, Newton, Leibniz). &lt;a href=&quot;https://hsm.stackexchange.com/a/2038/55&quot;&gt;Algebraic analysis&lt;/a&gt;, where algebra was transferred to functional infinite sums and products, ruled the 18th century. Euler's experiments with infinite &amp;quot;constructions&amp;quot; are well-known, while Fourier's &amp;quot;constructions&amp;quot; with trigonometric series were even seen as &amp;quot;revolutionary&amp;quot; at the turn of 19th century. Many more functions were thereby made &amp;quot;constructible&amp;quot;, see &lt;a href=&quot;https://www.ams.org/books/text/010/text010-endmatter.pdf&quot; rel=&quot;noreferrer&quot;&gt;Bressoud, Radical Approach to Real Analysis&lt;/a&gt;. Both lines of development paved the way to Cantor's innovations, his work on sets was motivated by Fourier series and influenced by the emerging conceptual attitude.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-06-08T22:49:34.960" LastActivityDate="2023-06-08T22:49:34.960" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15496" PostTypeId="1" CreationDate="2023-06-08T16:48:56.343" Score="4" ViewCount="141" Body="&lt;p&gt;Kelvin famously disputed geological estimates of the age of the earth because he said gravitational collapse couldn’t fuel the sun long enough. But even if gravitational collapse, not fusion, had been the energy source, a solar lifespan calculation would be entirely contingent on the density profile.&lt;/p&gt;&#xA;&lt;p&gt;Is there any info on how Kelvin and his contemporaries approached the sun’s density profile? Did they think plasma had a specific density ceiling? Or did they just think the geological evidence wasn’t that compelling and a uniform density with a fudge factor was good enough to refute it?&lt;/p&gt;&#xA;" OwnerUserId="18534" LastEditorUserId="14043" LastEditDate="2023-06-13T12:59:30.910" LastActivityDate="2023-06-13T12:59:30.910" Title="How did Kelvin square his low estimate of the age of the sun with the geological evidence?" Tags="|physics|earth-sciences|astrophysics|geology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15498" PostTypeId="2" ParentId="15496" CreationDate="2023-06-09T10:26:10.633" Score="6" Body="&lt;p&gt;Kelvin did think that &amp;quot;uniform density with a fudge factor&amp;quot; was good enough. The more detailed considerations of the density profile were made by T.J.J. See in 1899, where he used the &amp;quot;Lane's law of temperature&amp;quot; to study central condensation without uniformity assumptions, see his informal &lt;a href=&quot;https://www.theatlantic.com/magazine/archive/1899/04/the-solar-system-in-the-light-of-recent-discoveries/636398/&quot; rel=&quot;noreferrer&quot;&gt;self-summary in The Atlantic&lt;/a&gt;. But this only increased the estimate to 32,000,000 years, in agreement with Kelvin's earlier ballparking.&lt;/p&gt;&#xA;&lt;p&gt;Even before that, Kelvin was reassured, to put it mildly, when his and others' estimates of the Earth's age, based on different considerations, produced a comparable figure of 24,000,000. As far as Kelvin was concerned, geologists had to yield, see &lt;a href=&quot;https://www.scientificamerican.com/article/the-age-of-the-sun-and-the-earth/&quot; rel=&quot;noreferrer&quot;&gt;Cajori, The Age of the Sun and the Earth&lt;/a&gt; for a good historical account:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Lord Kelvin's address of 1897 is permeated, as Prof. Chamberlin puts it, “with an air of retrospective triumph and a tone of prophetic assurance.” “It is only by sheer force of reason,” says Kelvin, “that geologists have been compelled to think otherwise, and to see that there was a definite beginning and to look forward to a definite end of this world as an abode fitted for life.” Nor was this feeling of retrospective triumph confined to Lord Kelvin or to the students of the problem of the age of the sun and earth. At the close of the century physicists and chemists gloried in the triumphs of their predecessors...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In hindsight, it was &amp;quot;retrospective triumph&amp;quot; of the same nature as Kelvin's two clouds or Michelson's ode to the aether a bit later. 'Amazingly', geologists were  not impressed, as Geikie told the British Association back in 1892. Now &lt;em&gt;that&lt;/em&gt; was prophetic:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Lord Kelvin is willing, I believe, to grant us some twenty millions of years, but Prof. Tait would have us content with less than ten millions... That there must be some flaw in the physical argument I can, for mypart, hardly doubt, though I do not pretend to be able to say where it is to be found. Some assumption, it seems to me, has been made, or some consideration has been left out of sight, which will eventually be seen to vitiate the conclusions, and which when duly taken into account will allow time enough for any reasonable interpretation of the geological record.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;By the way, Kelvin's, or rather Helmholtz's, theory was not exactly ordinary gravitational collapse (as it would require unobserved changes in the Sun's diameter), but rather what he called &amp;quot;meteoric theory&amp;quot;: &amp;quot;&lt;em&gt;coalition of smaller bodies, falling together by mutual gravitation, and generating... an exact equivalent of heat for the motion lost in collision.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Kelvin was well aware of the role of the density profile, but was content with supposing that Helmholtz's 1854 estimate of 20,000,000 based on the ultimate uniform density is roughly in the ballpark by the order of magnitude. He first ballparked possible deviation in &lt;a href=&quot;https://zapatopi.net/kelvin/papers/on_the_age_of_the_suns_heat.html#fnsectiii&quot; rel=&quot;noreferrer&quot;&gt;On the Age of the Sun’s Heat (1862)&lt;/a&gt;, and then came back to it in &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Kelvin_sun_1/&quot; rel=&quot;noreferrer&quot;&gt;The Sun's Heat (1887)&lt;/a&gt; lecture to the Royal Institution of Great Britain. But his view of the density profile's minor influence has not changed:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In all our calculations hitherto we have for simplicity taken the density as uniform throughout, and equal to the true mean density of the sun, being about 1.4 times the density of water, or about a quarter of the earth's mean density. In reality the density in the upper parts of the sun's mass must be something less than this, and something considerably more than this in the central parts, because of the pressure in the interior increasing to something enormously great at the centre.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If we knew the distribution of interior density we could easily modify our calculations accordingly; but it does not seem probable that the correction could, with any probable assumption as to the greatness of the density throughout a considerable proportion of the sun's interior, add more than a few million years to the past of solar heat, and what could be added to the past must be taken from the future.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-06-09T10:26:10.633" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15499" PostTypeId="1" CreationDate="2023-06-11T20:46:14.077" Score="2" ViewCount="115" Body="&lt;p&gt;How was the Fundamental Theorem of Calculus discovered?&lt;/p&gt;&#xA;&lt;p&gt;The FTC is at once simple enough that Math.SE is full of questions asking &amp;quot;why is it such a big deal&amp;quot; and yet avoided discovery for perhaps centuries.  While rigorous proofs took time, &lt;a href=&quot;https://activecalculus.org/single/sec-4-4-FTC.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;simple arguments from velocity and distance are easy&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;How was it discovered? What were the obstacles that needed to be solved? Was it done using the simple arguments from velocity that I linked, from geometry, or from some other method?&lt;/p&gt;&#xA;&lt;p&gt;Or, to put it differently:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What aspects of the FTC are non-obvious such that they eluded discovery for so long? And how were these obstacles to discovery eventually overcome?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;One potential source for this: If anyone has access to the original writings &lt;em&gt;introducing&lt;/em&gt; the FTC to the world (e.g. by Newton or Leibniz), these might explain the breakthrough.&lt;/p&gt;&#xA;&lt;p&gt;Note: &lt;a href=&quot;https://hsm.stackexchange.com/questions/5935/history-of-the-fundamental-theorem-of-calculus&quot;&gt;History of the Fundamental Theorem of Calculus&lt;/a&gt; asks what was done &lt;em&gt;before&lt;/em&gt; the FTC.  I'm asking specifically how the FTC was discovered.&lt;/p&gt;&#xA;" OwnerUserId="18545" LastEditorUserId="18545" LastEditDate="2023-06-11T21:11:50.480" LastActivityDate="2023-06-11T21:11:50.480" Title="How was the Fundamental Theorem of Calculus discovered?" Tags="|mathematics|discoveries|calculus|newton|leibniz|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15501" PostTypeId="2" ParentId="10984" CreationDate="2023-06-12T10:32:30.613" Score="4" Body="&lt;p&gt;Having read @jkien's answer I have done done some research on the first use of the terms velocity and speed as we define them nowadays.&lt;/p&gt;&#xA;&lt;p&gt;@jkien writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gibbs's lecture notes on vector analysis and its use in physics&#xA;circulated in a small circle since 1881.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A copy of these notes by Gibbs, &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015051341777&amp;amp;view=1up&amp;amp;seq=5&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elements of Vector Analysis arranged for the use of students of Physics&lt;/a&gt; dated 1884 has no mention of the word &lt;em&gt;speed&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I then came across a copy of &lt;a href=&quot;https://hdl.handle.net/2027/mdp.39015063897550?urlappend=%3Bseq=10%3Bownerid=13510798885629360-14&quot; rel=&quot;nofollow noreferrer&quot;&gt;An Elementary Treatise on Kinematics and Dynamics&lt;/a&gt; by MccGregor published in 1887 which states in the Preface,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TVZLe.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TVZLe.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now Tait wrote with Steele a textbook, A treatise on the Dynamics of Particles, which was published in many editions.&lt;/p&gt;&#xA;&lt;p&gt;Comparing the &lt;a href=&quot;https://hdl.handle.net/2027/uc2.ark:/13960/t6c250p5t?urlappend=%3Bseq=25&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fourth Edition paragraph 8&lt;/a&gt; published in 1878,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vJv2g.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vJv2g.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;with the &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=mdp.39015065862917&amp;amp;view=1up&amp;amp;seq=21&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fifth Edition paragraph 8&lt;/a&gt; published in 1882,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ZJrD1.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ZJrD1.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;one can see that Tait and Steele adopted the modern definitions of speed and velocity.&lt;/p&gt;&#xA;&lt;p&gt;It might well be that around 1880 there were discussions amongst many English speaking scientists about the use of words speed and velocity and then a consensus was arrived at.&lt;/p&gt;&#xA;&lt;p&gt;I am not certain as to which publication of Tait and Steele's MacGregor was referring to but it looks as though that somewhere between 1882 and 1884 formal definitions of speed and velocity were adopted.&lt;/p&gt;&#xA;" OwnerUserId="6351" LastActivityDate="2023-06-12T10:32:30.613" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15503" PostTypeId="1" AcceptedAnswerId="15504" CreationDate="2023-06-12T13:29:13.260" Score="2" ViewCount="161" Body="&lt;p&gt;When did derivative mean not only &amp;quot;slope of tangent&amp;quot; but also &amp;quot;instantaneous rate of change&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;Fermat was interested in minima and maxima, and realized these occur when the tangent has zero slope.  When was it realized that this slope of tangent was the same thing as an instantaneous rate of change, i.e. velocity?&lt;/p&gt;&#xA;&lt;p&gt;The post &lt;a href=&quot;https://hsm.stackexchange.com/questions/5006/history-of-the-derivative-tangent-of-a-curve&quot;&gt;History of the derivative/tangent of a curve&lt;/a&gt; discusses the history of the tangent notion, but not the instantaneous rate of change notion.&lt;/p&gt;&#xA;" OwnerUserId="18545" LastEditorUserId="6085" LastEditDate="2023-06-13T02:51:56.433" LastActivityDate="2023-06-13T09:17:32.603" Title="When did derivative mean not only &quot;slope of tangent&quot; but also &quot;instantaneous rate of change&quot;?" Tags="|mathematics|discoveries|calculus|newton|leibniz|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15504" PostTypeId="2" ParentId="15503" CreationDate="2023-06-13T09:17:32.603" Score="3" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Fermat was interested in minima and maxima, and realized these occur when the tangent has zero slope. When was it realized that this slope of tangent was the same thing as an instantaneous rate of change, i.e. velocity?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As far as Fermat is concerned, it should be noted that he did not work with any notion of &amp;quot;slope&amp;quot; at all, as far as I am aware.  He did realize that at minima and maxima of an &lt;em&gt;expression&lt;/em&gt; (he didn't use &lt;em&gt;functions&lt;/em&gt;, either), the change of the expression is only quadratic in the change &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; in the variable.&lt;/p&gt;&#xA;&lt;p&gt;While he of course found the tangent line for many curves, he did not think of them as representing the rate of change as we do today.  Some recent publications on Fermat can be consulted at &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/fermat.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://u.math.biu.ac.il/~katzmik/fermat.html&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-06-13T09:17:32.603" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15505" PostTypeId="2" ParentId="1848" CreationDate="2023-06-13T09:27:43.320" Score="2" Body="&lt;p&gt;I am sort of a late arrival to the party, but one classical &amp;quot;bisociation&amp;quot; in math would be the identification of the singular cohomology of a manifold and its totally different (in the sense of a different context) de Rham cohomology (based on the complex of its differential forms).  Today we are used to the idea of various cohomology theories being related in various ways, but de Rham's theorem was one of the first developments of this type.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-06-13T09:27:43.320" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15506" PostTypeId="1" CreationDate="2023-06-13T14:38:52.123" Score="2" ViewCount="72" Body="&lt;p&gt;The first pulsars, starting with B1919+21, were discovered in the late 1960s in data from the Interplanetary Scintillation Array at Cambridge. The data from the telescope was recorded on paper by &lt;a href=&quot;http://www.bigear.org/vol1no1/burnell.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;four chart recorders&lt;/a&gt;, producing around 100 feet of paper per day and &lt;a href=&quot;https://physicsworld.com/a/look-happy-dear-youve-just-made-a-discovery/&quot; rel=&quot;nofollow noreferrer&quot;&gt;around 5 km&lt;/a&gt; over the first six months of observations, all of which was pored over manually. For periods during the process of looking through it, the paper was stored in shoe boxes.&lt;/p&gt;&#xA;&lt;p&gt;I know that at least a portion of the paper containing the charts from the initial discovery still survives in the Cambridge University Library, and &lt;a href=&quot;https://www.cam.ac.uk/Discovery&quot; rel=&quot;nofollow noreferrer&quot;&gt;has been displayed publicly&lt;/a&gt;, but I'm wondering what happened to the rest of the data from that six-month period of observations. Is it in Cambridge's collections in physical form? Has it been digitized for posterity? Is it elsewhere, or lost? I could try to contact the Library if needed, but it's possible that the answer is already out there and I've simply missed it.&lt;/p&gt;&#xA;" OwnerUserId="62" LastActivityDate="2023-06-13T14:38:52.123" Title="What happened to the data from the survey that discovered the first pulsars?" Tags="|astronomy|telescope|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15507" PostTypeId="1" CreationDate="2023-06-14T01:52:23.813" Score="0" ViewCount="133" Body="&lt;p&gt;Archimedes used polygons inscribed and circumscribed to the circle to approximate pi in his &amp;quot;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Measurement_of_a_Circle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Measurement of a Circle&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;However, I think &lt;a href=&quot;https://mathworld.wolfram.com/ArchimedesRecurrenceFormula.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes' recurrence formula&lt;/a&gt; is not used there.&lt;/p&gt;&#xA;&lt;p&gt;Did Archimedes know Archimedes' recurrence formula?&lt;/p&gt;&#xA;" OwnerUserId="17348" LastActivityDate="2023-06-14T01:52:23.813" Title="Did Archimedes know Archimedes' recurrence formula?" Tags="|mathematics|archimedes|greek-mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15508" PostTypeId="2" ParentId="15491" CreationDate="2023-06-14T08:53:11.373" Score="5" Body="&lt;p&gt;People without mathematical training have many mistaken beliefs. (Students studying higher math develop mistaken beliefs too, but on a higher level: see &lt;a href=&quot;https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;It is natural to think every non-periodic pattern of digits contains all possible finite strings of digits. (Forget that it is trivial to give a counterexample when you know enough math.) This is a plausible reason people think the digits of pi contain all finite patterns, and it is not realistic to think this common misunderstanding can be traced back to a specific source.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the quasi-popular confusion (or even awareness) about &lt;span class=&quot;math-container&quot;&gt;$1+2+3+\cdots = -1/12$&lt;/span&gt; in recent years has a definite cause: the ridiculous Numberphile video on that result. It is the only awful Numberphile video, in my opinion.&lt;/p&gt;&#xA;&lt;p&gt;The math stackexchange page &lt;a href=&quot;https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples&quot;&gt;here&lt;/a&gt; will give you other settings where a false result has large first counterexamples, which might help folks you know better appreciate that a finite amount of numerical data can’t prove a result about an infinite set (like all finite strings of digits).&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-06-14T09:29:26.763" LastActivityDate="2023-06-14T09:29:26.763" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15509" PostTypeId="1" AcceptedAnswerId="15511" CreationDate="2023-06-15T07:19:50.800" Score="1" ViewCount="67" Body="&lt;p&gt;What is the iodine fax process mentioned in Vannevar Bush's &amp;quot;As we may think&amp;quot;?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Another process now in use is also slow, and more or less clumsy. For fifty years impregnated papers have been used which turn dark at every point where an electrical contact touches them, by reason of the chemical change thus produced in an &lt;strong&gt;iodine compound&lt;/strong&gt; included in the paper. They have been used to make records, for a pointer moving across them can leave a trail behind. If the electrical potential on the pointer is varied as it moves, the line becomes light or dark in accordance with the potential.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This scheme is now used in facsimile transmission. The pointer draws a set of closely spaced lines across the paper one after another. As it moves, its potential is varied in accordance with a varying current received over wires from a distant station, where these variations are produced by a photocell which is similarly scanning a picture. At every instant the darkness of the line being drawn is made equal to the darkness of the point on the picture being observed by the photocell. Thus, when the whole picture has been covered, a replica appears at the receiving end.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I searched for &amp;quot;Vannevar Bush iodine facsimile&amp;quot; and many other keyword combinations. I also asked GPT. I found nothing.&lt;/p&gt;&#xA;&lt;p&gt;My guess would be that it's a redox reaction like &lt;span class=&quot;math-container&quot;&gt;$2 KI \to I_2 + 2 e^- + 2K^+$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="11081" LastEditorUserId="11081" LastEditDate="2023-06-15T07:27:53.427" LastActivityDate="2023-06-15T20:37:54.093" Title="What is the iodine fax process mentioned in Vannevar Bush's &quot;As we may think&quot;?" Tags="|chemistry|technology|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15511" PostTypeId="2" ParentId="15509" CreationDate="2023-06-15T20:37:54.093" Score="5" Body="&lt;p&gt;I suspect that Bush meant things like the &lt;a href=&quot;https://historictech.com/the-fultograph-the-worlds-1st-fax-machine-from-1927/&quot; rel=&quot;noreferrer&quot;&gt;Fultograph&lt;/a&gt; (as one example). They used paper that had been soaked in potassium iodide and starch. When electricity is passed through the paper, the iodine is released from the potassium iodide and binds to the starch, making a blue/purple color. So-called potassium iodide starch strips are used to test for oxidizers using the same reaction (but chemically driven, not electrically).&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2023-06-15T20:37:54.093" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15512" PostTypeId="2" ParentId="136" CreationDate="2023-06-16T14:30:32.737" Score="0" Body="&lt;p&gt;((DISPUTE!)).  It is amazing -and unfair- the little attention Wikipedia gives to the inventor and engineer Spaniard Jerónimo Beaumont, who nearly 100 years before Savery, Newcomen and Watt developed a steam engine (that he patented in Madrid, Spain) Spain.  It was a working steam engine that worked at pumping water out of flooded mines.  It kept Spanish mines dry while those in England and Scotland were wet.  It is likely that the British (who got the credit unfairly) learned from the Spaniards how they managed to keep their tunnels dry! F.J. Monreal, MD&lt;/p&gt;&#xA;" OwnerUserId="18570" LastActivityDate="2023-06-16T14:30:32.737" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15514" PostTypeId="1" CreationDate="2023-06-17T22:50:49.423" Score="3" ViewCount="219" Body="&lt;p&gt;I am interested in knowing if in the era preceding the observations that lead to the advent of Quantum Mechanics, anyone foresaw logically that a theory such as Quantum Mechanics is in a sense, &amp;quot;unavoidable&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I am not referring to a prediction of any of the &lt;em&gt;technical&lt;/em&gt; details of the QM theory we have today, that of course is unlikely. I rather mean some prediction with regards to the particular and central feature of QM as a theory where the process of &lt;em&gt;measurement&lt;/em&gt; is very central, both formally and conceptually, and as one which is deeply incorporated into the theory.&lt;/p&gt;&#xA;&lt;p&gt;The motivation for the question, is that it seems possible that someone had postulated as a logical necessity, even before QM was &amp;quot;forced&amp;quot; on us, some future time when the very act of measurement itself has to be taken into account as a significant physical interaction that changes the state of the system, when the system is of a small enough scale. On that note, I need to clarify that I am not referring to techniques for handling various types of &lt;a href=&quot;https://en.wikipedia.org/wiki/Measurement_uncertainty&quot; rel=&quot;nofollow noreferrer&quot;&gt;measurement uncertainties&lt;/a&gt;, clearly those were well known prior to QM and are not really referring to the same concept I am describing.&lt;/p&gt;&#xA;&lt;p&gt;Please note I am not here debating whether that &amp;quot;logical&amp;quot; step of predicting a QM-like theory in the past is in itself valid or not -- it may very well be criticized and shown to be erroneous. I am only highlighting the point that it seems to me not impossible that someone &lt;em&gt;may&lt;/em&gt; have thought along those lines, and hence I'm curious if anyone in fact did. Also note, while my title is asking for physicists who speculated along those lines, it will be also interesting to know if any philosopher wrote something about it. However, I am most interested to know if any of the famous physicists such as Newton, Maxwell, Hamilton, etc. ever wrote something pertaining to this idea.&lt;/p&gt;&#xA;" OwnerUserId="17992" LastEditorUserId="17992" LastEditDate="2023-06-17T22:57:31.297" LastActivityDate="2023-06-19T17:52:02.483" Title="Did any &quot;classical era&quot; physicist foresee that a theory such as Quantum Mechanics is logically inescapable?" Tags="|quantum-mechanics|philosophy-of-science|physicists|measurement|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15515" PostTypeId="1" CreationDate="2023-06-18T14:08:10.503" Score="2" ViewCount="86" Body="&lt;ol&gt;&#xA;&lt;li&gt;Where did arrow heads on coordinate axes appear for the first time?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I checked Descartes' &amp;quot;Discourse on Method, Optics, Geometry, and Meteorology&amp;quot;&#xA;but he does not use any arrow heads.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;In scientific literature the convention seems to be that the arrow head indicates&#xA;the positive direction. However, in the US school system it seems they use&#xA;two arrow heads. Where did that convention appear?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="9829" LastActivityDate="2023-06-18T14:08:10.503" Title="Arrow heads on coordinate axes" Tags="|mathematics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15516" PostTypeId="1" CreationDate="2023-06-18T16:35:28.517" Score="1" ViewCount="133" Body="&lt;p&gt;Who was the first to write proofs in this fashion?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/o0xRr.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/o0xRr.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;By ``in this fashion'' I mean, using three columns, which go like:&lt;/p&gt;&#xA;&lt;p&gt;Line number. Premise or assertion. Justification.&lt;/p&gt;&#xA;&lt;p&gt;Line number. Premise or assertion. Justification.&lt;/p&gt;&#xA;&lt;p&gt;Etc.&lt;/p&gt;&#xA;" OwnerUserId="18581" LastActivityDate="2023-06-18T19:41:02.887" Title="Who was the first to write proofs in this manner?" Tags="|mathematical-logic|logic|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15517" PostTypeId="2" ParentId="15516" CreationDate="2023-06-18T19:41:02.887" Score="7" Body="&lt;p&gt;Whitehead &amp;amp; Russel, &lt;em&gt;Principia Mathematica&lt;/em&gt; (1910) is pretty much like this.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/a4ksR.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/a4ksR.gif&quot; alt=&quot;w&amp;amp;r&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The justifications are on the left, and only certain lines are numbered.&lt;/p&gt;&#xA;&lt;p&gt;But I doubt it is the first.&lt;/p&gt;&#xA;&lt;p&gt;Compare Frege's notation for first-order logic (1879):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/d4zxj.gif&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/d4zxj.gif&quot; alt=&quot;frege&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-06-18T19:41:02.887" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15518" PostTypeId="1" CreationDate="2023-06-19T01:02:11.037" Score="2" ViewCount="448" Body="&lt;p&gt;During some recent work on the &lt;a href=&quot;https://scicomp.stackexchange.com/questions/30250/accurate-and-efficient-computation-of-the-inverse-langevin-function&quot;&gt;computation of the inverse Langevin function&lt;/a&gt; I ran into trouble trying to generate a highly-accurate minimax rational approximation with a variant of the Remez algorithm, encountering failures to converge and generation of degenerate solutions that fail to preserve minimax properties. That prompted&#xA;me to look around for alternative ways of generating rational approximations, and one paper I came across during my literature search is:&lt;/p&gt;&#xA;&lt;p&gt;C. M. Lee and F. D. K. Roberts, &amp;quot;A Comparison of Algorithms for Rational &lt;span class=&quot;math-container&quot;&gt;$l_{\infty}$&lt;/span&gt; Approximation&amp;quot;, &lt;em&gt;Mathematics of Computation&lt;/em&gt;, Vol. 27, No. 121, Jan. 1973, pp. 111-121&lt;/p&gt;&#xA;&lt;p&gt;This mentions Maehly's Method as one possible alternative to the Remez algorithm, and I recalled coming across this years ago in this publication:&lt;/p&gt;&#xA;&lt;p&gt;E.G. Kogbetliantz, &amp;quot;Generation of Elementary Functions&amp;quot;. In: Anthony Ralston and Herbert S. Wilf (eds.), &lt;em&gt;Mathematical Methods for Digital Computers&lt;/em&gt;, New York: Wiley 1960, pp. 7-35&lt;/p&gt;&#xA;&lt;p&gt;I also noted that Maehly's name occurs on the author list of a standard reference for approximations to various elementary functions that is still being consulted today despite its age:&lt;/p&gt;&#xA;&lt;p&gt;John F. Hart, E.W. Cheney, Charles L. Lawson, Hans J. Maehly, Charles K. Mesztenyi, John R. Rice, Henry G. Thacher, Jr., and Christoph Witzgall, &lt;em&gt;Computer Approximations&lt;/em&gt;. New York: Wiley 1968&lt;/p&gt;&#xA;&lt;p&gt;Google Scholar showed only half a dozen publications by Hans J. Maehly, prompting the titular question.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="12330" LastEditDate="2023-06-19T03:56:31.283" LastActivityDate="2023-06-19T04:06:55.350" Title="Who was Hans J. Maehly?" Tags="|mathematicians|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15519" PostTypeId="2" ParentId="15518" CreationDate="2023-06-19T01:02:11.037" Score="7" Body="&lt;p&gt;An initial Google search returned one serendipitous result that immediately put me on the right track because it mentioned that Maehly had immigrated to the US from Switzerland shortly after receiving a Ph.D. from ETH Zurich, and had died in 1961 at a young age:&lt;/p&gt;&#xA;&lt;p&gt;Walter Gautschi, &amp;quot;Schweizer Expats in den USA&amp;quot;. Notes from a talk given in January 12, 2016 at ETH Zürich:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Liste der angeführten Expats ist natürlich bei weitem nicht vollständig&#xA;und beschränkt sich ausschliesslich auf Leute aus meinem Bekanntenkreis.&#xA;Dazu gehören auch zwei damals vielversprechende junge Mathematiker, Hans&#xA;J. Maehly und Andreas H. Schopf, die kurz nach deren Promovierung an der&#xA;ETH nach den USA ausgewandert sind, aber dort frühzeitig starben, Schopf&#xA;1959 und Maehly 1961.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Reversing the common transcription of the umlaut 'ä' as &amp;quot;ae&amp;quot;, I found Maehly's dissertation right away at the ETH digital repository. Maehly received a Ph.D. in physics from Eidgenössische&#xA;Technische Hochschule (ETH) in Zurich in 1951 with a &lt;a href=&quot;https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/134749/1/eth-33761-01.pdf&quot; rel=&quot;noreferrer&quot;&gt;dissertation&lt;/a&gt; in mathematical physics, advised by Paul Scherrer:&lt;/p&gt;&#xA;&lt;p&gt;Hans Jakob Mähly, &lt;em&gt;Methoden zur genäherten Berechnung von Eigenwerten elastischer Schwingungen anisotroper Körper&lt;/em&gt;, Ph.D. dissertation, ETH Zürich 1951&lt;/p&gt;&#xA;&lt;p&gt;This matches the information at the &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=102828&quot; rel=&quot;noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt;. There are some brief biographical notes at the end of the thesis: Maehly was born on June 25, 1920 in Basel. He graduated from high school in the spring of 1939. After one semester at the University of Basel he transferred to the department of mathematics and physics at ETH. In 1944 he received his diplom degree as an experimental physicist; his diplom thesis was on the elastic properties of seignette-electric&#xA;potassium phosphate.&lt;/p&gt;&#xA;&lt;p&gt;While at ETH, Maehly published two papers. The first one lays a foundation for the Lehmann-Maehly&#xA;method of determining bounds of eigenvalues (&lt;a href=&quot;https://de.wikipedia.org/wiki/Nikolaus_Joachim_Lehmann&quot; rel=&quot;noreferrer&quot;&gt;Lehmann&lt;/a&gt; independently published on this). The second details a root finding method sometimes referred to as &lt;a href=&quot;https://mathworld.wolfram.com/MaehlysProcedure.html&quot; rel=&quot;noreferrer&quot;&gt;Maehly's Procedure&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;H. J. Maehly, &amp;quot;Ein neues Variationsverfahren zur genäherten Berechnung der Eigenwerte hermitescher Operatoren&amp;quot;. &lt;em&gt;Helvetica Physica Acta&lt;/em&gt;, Vol. 25, No. 5, 1952, pp. 547-568&lt;/p&gt;&#xA;&lt;p&gt;Hans J. Maehly, &amp;quot;Zur iterativen Auflösung algebraischer Gleichungen&amp;quot;. &lt;em&gt;Zeitschrift für Angewandte Mathematik und Physik&lt;/em&gt;, Vol. 5, No. 3, May 1954, pp. 260-263&lt;/p&gt;&#xA;&lt;p&gt;Maehly's institutional affiliation for the first paper is given as ETH's Institute of Physics, while for the second it is stated as the Institute of Applied Mathematics of ETH. Going forward, Maehly appears to have worked on mathematical problems exclusively, all of them in the field of numerical&#xA;analysis. He was also a pioneer in numerical computation using computers. In the surviving usage records for ETH's &lt;a href=&quot;https://en.wikipedia.org/wiki/Z4_(computer)&quot; rel=&quot;noreferrer&quot;&gt;Z4&lt;/a&gt; computer, his name occurs three times. Herbert Bruderer, &amp;quot;Wofür wurde der Relaisrechner Z4 von Konrad Zuse an der ETH Zürich verwendet?&amp;quot; ETH Zürich, Departement Informatik, Zürich 2011:&lt;/p&gt;&#xA;&lt;p&gt;200 hours: partial differential equations of order 4 (1951, 1952)&lt;br /&gt;&#xA;50 hours: root finding in polynomials; context: quantum mechanical computations for aromatic compounds (1953, 1954)&lt;br /&gt;&#xA;170 hours: solving algebraic equations (1953)&lt;/p&gt;&#xA;&lt;p&gt;At some time in the mid-1950s, Maehly left Switzerland to work at Princeton. I have not been able to establish the exact date or his specific motivation. The earliest reference I could find to Maehly's presence at Princeton is a reference in Kurt Spielberg, &amp;quot;Efficient Continued Fraction Approximations To Elementary Functions.&amp;quot; &lt;em&gt;Mathematics of Computation&lt;/em&gt;, Vol. 15, No. 76, 1961, pp. 409-417:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;H. J. Maehly, Monthly Progress Report (unpublished), February 1956&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From the institutional affiliations listed in his publications, he appears to have staid at Princeton until sometime in 1959, leaving for Syracuse University. A &lt;a href=&quot;https://dl.acm.org/doi/10.1145/367149.367154&quot; rel=&quot;noreferrer&quot;&gt;letter to the editor&lt;/a&gt; in &lt;em&gt;Communications of the ACM&lt;/em&gt;, Vol. 3, No. 3, March 1960, p. A11 gives his affiliation as Department of Mathematics, Syracuse University. At Princeton, he appears to have had continued intense involvement with computers, as evidenced by the following note to department heads dated September 17, 1957:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Princeton University has recently taken over operation of the electronic&#xA;digital computer previously operated by the Institute for Advanced Study&#xA;and known as the MANIAC. This computer is now available for use by members&#xA;of the academic community. [...] Dr. Hans Maehly is in charge of the&#xA;computer and should be consulted if there are any questions concerning&#xA;its use;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I assembled the following list of Maehly's publications during his time in the United States:&lt;/p&gt;&#xA;&lt;p&gt;H. J. Maehly, &amp;quot;Rational approximations for transcendental functions&amp;quot;. In &lt;em&gt;Proceedings of the International Conference on Information Processing&lt;/em&gt;, UNESCO, June 1959, pp. 57-62. London: Butterworth 1960.&lt;/p&gt;&#xA;&lt;p&gt;H. J. Maehly, &lt;em&gt;Approximations for the CDC 1604&lt;/em&gt;. Control Data Corp 1960&lt;/p&gt;&#xA;&lt;p&gt;Hans J. Maehly, &amp;quot;Methods for Fitting Rational Approximations, Part I: Telescoping Procedures for Continued Fractions&amp;quot;. &lt;em&gt;Journal of the ACM&lt;/em&gt;, Vol. 7, No. 2, April 1960, pp. 150-162&lt;/p&gt;&#xA;&lt;p&gt;H. Maehly and Ch. Witzgall, &amp;quot;Tschebyscheff-Approximationen in kleinen Intervallen I: Approximation durch Polynome&amp;quot;. &lt;em&gt;Numerische Mathematik&lt;/em&gt;, Vol. 2, 1960, pp. 142-150&lt;/p&gt;&#xA;&lt;p&gt;H. Maehly and Ch. Witzgall, &amp;quot;Tschebyscheff-Approximationen in kleinen Intervallen II: Stetigkeitssätze für gebrochen rationale Approximationen&amp;quot;. &lt;em&gt;Numerische Mathematik&lt;/em&gt;, Vol. 2, 1960, pp. 293-307&lt;/p&gt;&#xA;&lt;p&gt;Hans J. Maehly, &amp;quot;Numerical solution of a certain transcendental equation involving exponentials (a remark on a paper by J. R. Rice)&amp;quot;. &lt;em&gt;Journal of the Society for Industrial and Applied Mathematics&lt;/em&gt;, Vol. 10, No. 1, March 1962, pp. 30-34&lt;/p&gt;&#xA;&lt;p&gt;Hans J. Maehly, &amp;quot;Methods for fitting rational approximations, Parts II and III&amp;quot;. &lt;em&gt;Journal of the ACM,&lt;/em&gt; Vol. 10, No. 3, July 1963, pp. 257-277&lt;/p&gt;&#xA;&lt;p&gt;The preface to the last publication listed, written by Maehly's colleague Christoph Witzgall, informs the reader that he passed away unexpectedly in 1961, shorty after moving from Syracuse to Argonne National Laboratory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dr. Hans J. Maehly died on the 16th of November, 1961. Just six weeks before his untimely death, Dr. Maehly had joined the Applied Mathematics Division of Argonne National Laboratory.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Multiple sources indicate that Maehly's sudden death was due to a heart attack:&lt;/p&gt;&#xA;&lt;p&gt;Thomas Haigh, &amp;quot;An interview with W. J. Cody&amp;quot;, December 2005, &lt;a href=&quot;http://webcache.googleusercontent.com/search?q=cache:_PZQ7Y931LwJ:history.siam.org/%255C/pdfs2/Cody_returned_SIAM.pdf&quot; rel=&quot;noreferrer&quot;&gt;53 pp.&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As a matter of fact, when I first started working in that area I discovered a little belatedly that we had one member of our division who was an expert on that, that was Hans Maehly, who had come over from Princeton. I say I discovered belatedly, because he was only there about six months and died of a heart attack in the parking lot, so I knew him but I didn’t know what he was working on, and he didn’t know what I was working on.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Richard Hertz, &amp;quot;Interview with Forman Acton&amp;quot;, 1971, 48 pp.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hans did a couple of pieces of research in numerical methods sorts of things, rather nice, before he departed for Syracuse, if I remember rightly. Then he went from there to Argonne Labs and he’d been at Argonne about two weeks when he died.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Christoph Witzgall&amp;quot;, &lt;em&gt;Journal of Research of the National Institute of Standards and Technology&lt;/em&gt;, Vol. 111, No. 2, March-April 2006:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Edmonds recruited Chris to join NBS, which he did in October 1962 during&#xA;the Cuban Missile Crisis. That summer he had spent at Argonne National&#xA;Laboratory, editing papers of Maehly, who had died there of a heart attack.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seemed a bit unusual that Maehly would be a co-author of the book &lt;em&gt;Computer Approximations&lt;/em&gt; given that it was published seven years after his death. One of the co-authors explained:&lt;/p&gt;&#xA;&lt;p&gt;Thomas Haigh, &amp;quot;An interview with Charles L. Lawson, Nov. 2004, &lt;a href=&quot;http://webcache.googleusercontent.com/search?q=cache:Nn8DOC6o7K8J:history.siam.org/%25255C/pdfs2/Lawson_final.pdf&quot; rel=&quot;noreferrer&quot;&gt;103 pp.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The seven were J. F Hart, E. W. Cheney, C. L. Lawson, C. K. Mesztenyi, J. R. Rice, H. C. Thacher, Jr. and C. Witzgall. [...]  An eighth person, Hans J. Maehly, is listed along with these seven as a coauthor of the book because it was felt his work on this issue had already been very significant and he died at a young age just before work on the book began. The book, &lt;em&gt;Computer Approximations&lt;/em&gt;, was published by John Wiley in 1968.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-06-19T04:06:55.350" LastActivityDate="2023-06-19T04:06:55.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15520" PostTypeId="1" CreationDate="2023-06-19T11:41:01.337" Score="1" ViewCount="124" Body="&lt;p&gt;I read that Riemann started studying the so-called Riemann's surfaces in the second half of the 19th century, introducing tools like meromorphic functions and meromorphic 1-forms. The culmination of that theory can be considered the Riemann-Roch theorem, quantifying the possible meromorphic functions existing on a compact Riemann surface with some constraints expressed in terms of divisors.&lt;/p&gt;&#xA;&lt;p&gt;Today, most Algebraic Geometry books introduce varieties defined over an arbitrary field. Compact Riemann surfaces, if mentioned, appear as a particular case &lt;span class=&quot;math-container&quot;&gt;$K = \mathbb{C}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My main question here is stated in the title, but here are related sub-questions that could guide the answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Who first started to define and study algebraic varieties and were those people aware of the results about Riemann surfaces?&lt;/li&gt;&#xA;&lt;li&gt;Was the equivalence between compact Riemann surfaces and projective algebraic curves over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; already known, or suspected, then?&lt;/li&gt;&#xA;&lt;li&gt;How did people come up with the idea of using things like 1-forms to study varieties over an arbitrary field?&lt;/li&gt;&#xA;&lt;li&gt;Also, with some knowledge of logic, one can justify that some interesting results over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; might be proven algebraically (knowing that the theory of algebraically closed fields of characteristic 0 is complete). Was that known/influential when people started studying Algebraic Geometry?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14079" LastEditorUserId="604" LastEditDate="2023-06-19T13:41:02.760" LastActivityDate="2023-06-20T07:19:32.980" Title="To what extent were Riemann surfaces a precursor to algebraic geometry?" Tags="|mathematics|complex-analysis|algebraic-geometry|riemann|weyl|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15521" PostTypeId="2" ParentId="15520" CreationDate="2023-06-19T13:25:27.647" Score="2" Body="&lt;p&gt;Hermann Weyl's book on Riemann surfaces was instrumental in the development of the Riemann-Roch theorem and its generalisations.  The relevant chapter in Weyl's book is chapter 17: The single-valued functions on F (Riemann surface) as a subclass of additive and multiplicative functions on F^ (&amp;quot;Klassenfläche''). Riemann-Roch's theorem (see &lt;a href=&quot;https://mathscinet.ams.org/mathscinet-getitem?mr=69903&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathscinet.ams.org/mathscinet-getitem?mr=69903&lt;/a&gt;).  The generalisations include the Hirzebruch-Riemann-Roch theorem &lt;a href=&quot;https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem&lt;/a&gt;, a result of central importance in algebraic geometry.  Notably, Felix Klein took an active interest in the preparation of Weyl's book on Riemann surfaces, corroborating his exceptional modernism.  Note that Weyl was Klein's mathematical great-grandson; see &lt;a href=&quot;https://www.mathgenealogy.org/id.php?id=7373&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.mathgenealogy.org/id.php?id=7373&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-06-19T13:37:00.580" LastActivityDate="2023-06-19T13:37:00.580" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15522" PostTypeId="2" ParentId="15514" CreationDate="2023-06-19T17:52:02.483" Score="3" Body="&lt;p&gt;The following is far from &amp;quot;foreseeing&amp;quot; a &amp;quot;logical inescapability&amp;quot; of quantum mechanics, but I think it's quite interesting.&lt;/p&gt;&#xA;&lt;p&gt;Bernhard Riemann (not a physicist) is generally credited with founding the mathematics of curved space (&amp;quot;Riemannian Geometry&amp;quot;) which some decades later provided the mathematical framework of General Relativity. It is less known that near the end (III §3) of his famous habilitation lecture &lt;em&gt;&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN250442582_0013&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die Hypothesen, welche der Geometrie zu Grunde liegen&lt;/a&gt;&lt;/em&gt; (held June 10, 1854, first published in 1868) he writes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Die Fragen über das Unmessbargrosse sind für die Naturerklärung&#xA;müssige Fragen. Anders verhält es sich aber mit den Fragen über das&#xA;Unmessbarkleine. Auf der Genauigkeit, mit welcher wir die Erscheinungen&#xA;in's Unendlichkleine verfolgen, beruht wesentlich die Erkenntniss&#xA;ihres Causalzusammenhangs.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nun scheinen aber die empirischen Begriffe, in welchen die räumlichen&#xA;Massbestimmungen gegründet sind, der Begriff des festen Körpers und&#xA;des Lichtstrahls, im Unendlichkleinen ihre Gültigkeit zu verlieren; es&#xA;ist also sehr wohl denkbar, dass die Massverhältnisse des Raumes im&#xA;Unendlichkleinen den Voraussetzungen der Geometrie nicht gemäss sind,&#xA;und dies würde man in der That annehmen müssen, sobald sich dadurch&#xA;die Erscheinungen auf einfachere Weise erklären liessen.&lt;/p&gt;&#xA;&lt;p&gt;Die Frage über die Gültigkeit der Voraussetzungen der Geometrie im&#xA;Unendlichkleinen hängt zusammen mit der Frage nach dem innern Grunde der&#xA;Massverhältnisse des Raumes. Bei dieser Frage, welche wohl noch zur&#xA;Lehre vom Raume gerechnet werden darf, kommt die obige Bemerkung zur&#xA;Anwendung, dass bei einer discreten Mannigfaltigkeit das Princip der&#xA;Massverhältnisse schon in dem Begriffe dieser Mannigfaltigkeit&#xA;enthalten ist, bei einer stetigen aber anders woher hinzukommen muss.&#xA;Es muss also entweder das dem Raume zu Grunde liegende Wirkliche eine&#xA;discrete Mannigfaltigkeit bilden, oder der Grund der&#xA;Massverhältnisse ausserhalb, in darauf wirkenden bindenen Kräften,&#xA;gesucht werden.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation (based on &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/Riemann_geometry/&quot; rel=&quot;nofollow noreferrer&quot;&gt;the one by William Cifford&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The questions about the immeasurably great are for the interpretation of&#xA;nature useless questions. But this is not the case with the questions&#xA;about the immeasurably small. It is upon the exactness with which we&#xA;follow phenomena into the infinitely small that our knowledge of their&#xA;causal relations essentially depends.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[...]&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Now it seems that the empirical notions on which the metrical&#xA;determinations of space are founded, the notion of a solid body and of&#xA;a ray of light, cease to be valid for the infinitely small. We are&#xA;therefore quite at liberty to suppose that the metric relations of&#xA;space in the infinitely small do not conform to the hypotheses of&#xA;geometry; and we ought in fact to suppose it, if we can thereby obtain&#xA;a simpler explanation of phenomena.&lt;/p&gt;&#xA;&lt;p&gt;The question of the validity of the hypotheses of geometry in the&#xA;infinitely small is bound up with the question of the ground of the&#xA;metric relations of space. In this last question, which we may still&#xA;regard as belonging to the doctrine of space, is found the application&#xA;of the remark made above; that in a discrete manifoldness, the ground&#xA;of its metric relations is given in the notion of it, while in a&#xA;continuous manifoldness, this ground must come from outside. Either&#xA;therefore the reality which underlies space must form a discrete&#xA;manifoldness, or we must seek the ground of its metric relations&#xA;outside it, in binding forces which act upon it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2023-06-19T17:52:02.483" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15523" PostTypeId="1" CreationDate="2023-06-19T22:37:07.000" Score="0" ViewCount="84" Body="&lt;p&gt;There have been several &lt;em&gt;book&lt;/em&gt; classification schemes, for example: Dewey Decimal, Library of Congress, etc. Have there been any &lt;em&gt;experiment&lt;/em&gt; classification schemes, &lt;em&gt;i.e.&lt;/em&gt; sets of criteria by which to classify (say) Eratosthenes' experiment for the measurement of the earth, or Fletcher's experiment for the measurement of the electric charge of the electron?&lt;/p&gt;&#xA;&lt;p&gt;I suppose that Mill's methods of induction (1843) implicitly categorize many experiments, but is this the best we've got?&lt;/p&gt;&#xA;" OwnerUserId="18581" LastActivityDate="2023-06-22T19:43:44.417" Title="Have there ever been any schemes for the classification of experiments?" Tags="|philosophy-of-science|science|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15524" PostTypeId="1" CreationDate="2023-06-20T03:29:39.100" Score="1" ViewCount="90" Body="&lt;p&gt;I mentioned &lt;a href=&quot;https://hsm.stackexchange.com/questions/14498/request-readers-to-check-accuracy-of-the-history-in-my-young-adult-math-story&quot;&gt;in a previous question&lt;/a&gt; that I'm writing a young adult novel that explores the discovery of complex numbers. I  might want to include an illustration of Tartaglia. All of the &lt;a href=&quot;https://www.google.com/search?client=firefox-b-1-d&amp;amp;sxsrf=APwXEdfEuBzMOgd2BcrUKcML0_EPHq8RqQ:1687230037780&amp;amp;q=tartaglia%20mathematician&amp;amp;tbm=isch&amp;amp;sa=X&amp;amp;ved=2ahUKEwiFpbnY7dD_AhW5ADQIHSFCDXQQ0pQJegQIDBAB&amp;amp;biw=1427&amp;amp;bih=764&amp;amp;dpr=2&quot; rel=&quot;nofollow noreferrer&quot;&gt;illustrations I see&lt;/a&gt; when I google look similar to one another. I'd like to figure out who created the original image(s), and so far have had no luck.&lt;/p&gt;&#xA;&lt;p&gt;Who created this depiction (or these depictions)?&lt;/p&gt;&#xA;" OwnerUserId="13188" LastActivityDate="2023-06-20T03:29:39.100" Title="Who is the artist who depicted Tartaglia?" Tags="|mathematicians|biographical-details|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15525" PostTypeId="2" ParentId="15520" CreationDate="2023-06-20T07:14:07.363" Score="1" Body="&lt;p&gt;Algebraic geometry is much older than Riemann surfaces.&#xA;Algebraic curves (of second degree, over the field of real numbers) where studied in great detail by Apollonius in 2 BC, and algebraic varieties (not only curves but of higher dimension!) occur in the book of Diophantus in 2 AD. The idea of rational parametrization of these varieties is due to him.&lt;/p&gt;&#xA;&lt;p&gt;At the time when Riemann wrote his thesis, algebraic geometry was already a well developed subject, with many deep results (over the fields of real, complex and rational numbers). Some examples of these deep results are Newton's classification of cubics over the real field, the so-called Puiseux series (really discovered by Newton, Bezout's theorem, Poncelet's theorem, and much more.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-06-20T07:19:32.980" LastActivityDate="2023-06-20T07:19:32.980" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15526" PostTypeId="2" ParentId="352" CreationDate="2023-06-20T14:14:05.483" Score="4" Body="&lt;p&gt;To elaborate on Leibniz's comment that &amp;quot;The axioms are not, as you say, indemonstrable&amp;quot; (see the answer &lt;a href=&quot;https://hsm.stackexchange.com/a/369/604&quot;&gt;https://hsm.stackexchange.com/a/369/604&lt;/a&gt;):  Leibniz was rather successful in choosing the appropriate axioms and/or heuristic principles for the infinitesimal calculus he developed.  I would note four particularly important items.&lt;/p&gt;&#xA;&lt;p&gt;(1) Law of Continuity.  There are several formulations of this (as distinct from the Principle of Continuity).  One of them is the following: The laws of the finite succeed in the infinite, and vice versa.&amp;quot;  This is described by Abraham Robinson as closely related to the Transfer Principle of infinitesimal analysis (for example, a relation such as &lt;span class=&quot;math-container&quot;&gt;$\sin^2 x+\cos^2 x=1$&lt;/span&gt;, being true for all standard inputs, would necessarily be true by the Transfer Principle for &lt;em&gt;all inputs&lt;/em&gt;, including for example infinitesimal ones).&lt;/p&gt;&#xA;&lt;p&gt;(2) Relation of infinite proximity.  Leibniz emphasized on numerous occasions that in the calculus he worked, not with the relation of exact equality, but rather with the relation of equality up to negligible terms.  This is closely related to the Standard Part principle of modern infinitesimal analysis.&lt;/p&gt;&#xA;&lt;p&gt;(3) The dichotomy of assignable vs inassignable number.  While there is clearly no construction of this anywhere in Leibniz, he postulates the existence of such a distinction, which enables him to work both with infinitesimals and (their inverses) infinite numbers.&lt;/p&gt;&#xA;&lt;p&gt;(4) Leibniz made it clear in a pair of texts from 1695 that his infinitesimals violate the principle of comparability found in Euclid's Elements, volume 5, Definition 4 (Leibniz referred to Definition 5 following contemporary editions of Euclid).  This is closely parallel to the violation of the Archimedean property in modern number systems containing infinitesimals.&lt;/p&gt;&#xA;&lt;p&gt;Thus all in all this leads us to suggest that Leibniz was rather successful in creating the germs of an axiomatisation of infinitesimal analysis -- something that has been clearly realized only recently.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-06-28T10:56:51.610" LastActivityDate="2023-06-28T10:56:51.610" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15528" PostTypeId="1" AcceptedAnswerId="15529" CreationDate="2023-06-21T23:06:02.240" Score="2" ViewCount="893" Body="&lt;p&gt;I am looking for more information about J. M. Gandhi who is&#xA;the creator of Gandhi polynomials. A MathSciNet search finds&#xA;only one J. M. Gandhi with 35 publications in Number Theory,&#xA;3 in Combinatorics, and 5 others. The author also wrote about&#xA;Genocchi numbers. The OEIS has several references. The article&#xA;&amp;quot;On Logarithmic Numbers&amp;quot;, Math. Student 31 (1963), 73-83&#xA;(MR0163876) at the end lists&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Dept. of Physics&#xA;University of Rajasthan&#xA;Jaipur (Rajasthan)&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;p&gt;From the list of publication I gather that the author later moved&#xA;to the USA and also Canada. The publications begin in 1961 and end&#xA;in 1978.&lt;/p&gt;&#xA;&lt;p&gt;I am mostly curious about the first name and middle initial. What&#xA;are they? I emailed a coauthor but he didn't know what the &amp;quot;J. M.&amp;quot;&#xA;expands to. I don't know if that is unusual or not.&lt;/p&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="13622" LastEditDate="2023-06-22T02:31:54.383" LastActivityDate="2023-06-22T09:03:08.417" Title="Who was J. M. Gandhi?" Tags="|mathematics|biographical-details|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15529" PostTypeId="2" ParentId="15528" CreationDate="2023-06-22T05:59:16.407" Score="6" Body="&lt;p&gt;I have not been able to figure out what the initials J. M. expand to. All references in literature and elsewhere, including an Illinois court case in which Gandhi appeared as the plaintiff, consistently refer to him as J. M. Gandhi. The year of his birth as well as the date of his death are known:&lt;/p&gt;&#xA;&lt;p&gt;Paulo Ribenboim, &amp;quot;Prime Number Records.&amp;quot; &lt;em&gt;The College Mathematics Journal&lt;/em&gt;, Vol. 25, No. 4, Sep. 1994, pp. 280-290:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;J. M. Gandhi, born in 1933, died on January 23, 1982, after an apparently harmless operation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This suggests to me that Gandhi died from complications after routine surgery. The date of death and year of birth given match the death notice that appeared in an AMS publication:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Notices of the American Mathematical Society&lt;/em&gt;, Vol. 29, No. 3, Apr. 1982, p. 302:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;J. M. Gandhi of Northern Illinois University died on January 23, 1982, at the age of 49. He was a member of the Society for 13 years.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Gandhi's institutional affiliation was stated incorrectly, so a correction appeared in the next issue of &lt;em&gt;Notices&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Notices of the American Mathematical Society&lt;/em&gt;, Vol. 29, No. 4, June 1982, p. 381:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The affiliation of J. M. Gandhi, whose death was reported o page 302 of the April 1982 &lt;em&gt;Notices&lt;/em&gt;, should have been listed as Western Illinois University&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The publications of J. M. Gandhi contain sufficient statements of institutional affiliation to track him fairly accurately from his student days to his death. His first two publications do not state an affiliation, however:&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;The coefficients of &lt;span class=&quot;math-container&quot;&gt;$\frac{\cosh x}{\cos x}$&lt;/span&gt; and a note on Carlitz's coefficients of &lt;span class=&quot;math-container&quot;&gt;$\frac{\sinh x}{\sin x}$&lt;/span&gt;.&amp;quot; &lt;em&gt;Mathematics Magazine&lt;/em&gt;, Vol. 31, No. 4, Mar.-Apr. 1958, pp. 185-191&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;Some integrals for Genocchi numbers.&amp;quot; &lt;em&gt;Mathematics Magazine&lt;/em&gt;, Vol. 33, No. 1, Sep.-Oct. 1959, pp. 21-23&lt;/p&gt;&#xA;&lt;p&gt;His next publication states his institutional affiliation as Government College, Bhilwara. Bhilwara is a city in Rajasthan.&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;A new formula for Genocchi numbers.&amp;quot; &lt;em&gt;The Mathematics Student&lt;/em&gt;, Vol. 28, 1960, pp. 83-85&lt;/p&gt;&#xA;&lt;p&gt;Three years later his affiliation is with the University of Rajasthan in Jaipur:&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;Congruences for &lt;span class=&quot;math-container&quot;&gt;$p_{r}(n)$&lt;/span&gt; and Ramanujan's &lt;span class=&quot;math-container&quot;&gt;$\tau$&lt;/span&gt; function&amp;quot;, &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 70, No. 3, Mar. 1963, pp. 265-274&lt;/p&gt;&#xA;&lt;p&gt;If found a Google snippet without useful context that states a J. M. Gandhi submitted his Ph.D. thesis at the University of Rajasthan in 1966. This seems plausible, as Gandhi's last publication stating his affiliation with the University of Rajasthan in Jaipur dates to that year:&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;Logarithmic numbers and the functions &lt;span class=&quot;math-container&quot;&gt;$d(n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\sigma(n)$&lt;/span&gt;.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 73, No. 9, Nov. 1966, pp. 959-964&lt;/p&gt;&#xA;&lt;p&gt;The next publication, received by editors in late 1966 and published the following year, states his affiliation as the University of Alberta, Edmonton, Canada:&lt;/p&gt;&#xA;&lt;p&gt;Chinthayamma and J. M. Gandhi, &amp;quot;An Extension of Fermat's Theorem.&amp;quot; &lt;em&gt;Canadian Mathematical Bulletin&lt;/em&gt;, Vol. 10, No. 3, Aug. 1967, pp. 383-386&lt;/p&gt;&#xA;&lt;p&gt;The next job transition occured roughly two years later, when a papers states his affiliation as University of Alberta, Edmonton, and York University, Toronto:&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;On Numbers Related to Partitions of a Number.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 76, No. 9, Nov. 1969, pp. 1033-1036&lt;/p&gt;&#xA;&lt;p&gt;His final move, to the United States, occured about a year after arriving in Toronto, either in late 1969 or early 1970 (the former seems more likely given his 13 years of AMS membership when he died in 1982), as evidenced by this announcement:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;International Mathematical News&lt;/em&gt;, Vol. 24, No. 95/95, Sep. 1970, p. 35:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;J. M. Gandhi of York University, Toronto, has been appointed to a professorship at Western Illinois University.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This switch is also reflected in a publication around that time, which states his affiliation as York University, Toronto, Ontario and Western Illinois University, Macomb, Illinois:&lt;/p&gt;&#xA;&lt;p&gt;J. M. Gandhi, &amp;quot;The coefficients of &lt;span class=&quot;math-container&quot;&gt;$\frac{\sinh x}{\cos x}$&lt;/span&gt;.&amp;quot; &lt;em&gt;Canadian Mathematical Bulletin&lt;/em&gt;, Vol. 13, No. 3, Sep. 1970, pp. 305-310&lt;/p&gt;&#xA;&lt;p&gt;The final trace in the literature that I could find of J. M. Gandhi are two brief contributions to a list of research problems in combinatorial number theory that appeared in &lt;em&gt;Notices of the American Mathematical Society&lt;/em&gt;, Vol. 25, No. 2, February 1978, p. 146.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-06-22T09:03:08.417" LastActivityDate="2023-06-22T09:03:08.417" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15530" PostTypeId="2" ParentId="15408" CreationDate="2023-06-23T21:13:13.380" Score="3" Body="&lt;p&gt;&lt;strong&gt;Fashionable Nonsense&lt;/strong&gt; Thanks to the observant eye of &lt;strong&gt;NWR&lt;/strong&gt; we are directed to the summary of the enlightening book by Alan Sokal and Jean Bricmont: FASHIONABLE NONSENSE:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1996 physicist Alan Sokal published an essay in &lt;em&gt;Social Text&lt;/em&gt;, an influential academic journal of cultural studies, touting the deep similarities between quantum gravitational theory and postmodern philosophy.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Soon thereafter, the essay was revealed as a brilliant parody, a catalog of nonsense written in the cutting-edge but impenetrable lingo of postmodern theorists.  The event sparked a furious debate in academic circles and made headlines of newspapers in the U&amp;gt;S&amp;gt; and abroad.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;In FASHIONSABLE NONSENSE: POSTMODERN INTELLECTUAL'S ABUSE OF SCIENCE, Sokal and his fellow physicist Jean Bricmont expand from where the hoax left off. In a delightfully witty and clear voice, the two thoughtfully and thoroughly dismantle the pseudo-scientific writings of some of the most fashionable French and American intellectuals.  More generally, they challenge the widespread notion that scientific theories are mere &amp;quot;narrations&amp;quot; of social constructions.&lt;/em&gt; (See Amazon.com under book title)&lt;/p&gt;&#xA;" OwnerDisplayName="user18391" LastActivityDate="2023-06-23T21:13:13.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15531" PostTypeId="1" CreationDate="2023-06-23T22:30:17.993" Score="1" ViewCount="169" Body="&lt;p&gt;I have a question about Monge-Ampere equation.&#xA;I read a book &lt;em&gt;The shape of inner space&lt;/em&gt; by Shing-Tung Yau.&lt;/p&gt;&#xA;&lt;p&gt;Chapter 5 in this book, Complex-Monge-Ampere equation's explaination was so interesting. In the beginning, Monge researched this equation during french revolution. after a long time, Ampere researched the equation succeed to Monge.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But there were still more preliminaries to be done before Cheng and I&#xA;could attack complex Monge-Ampère equations. We started working on&lt;br /&gt;&#xA;the Dirichlet problem, named after the German mathematician Lejeune&lt;br /&gt;&#xA;Dirichlet. It’s what we call a boundary value problem, and tackling&lt;br /&gt;&#xA;such problems is usually the first thing we do when trying to solve&lt;br /&gt;&#xA;an elliptic differential equation. We discussed an example of a&lt;br /&gt;&#xA;boundary value problem in Chapter 3, the Plateau problem, which is&lt;br /&gt;&#xA;often visualized in terms of soap films and which stated that for an&lt;br /&gt;&#xA;arbitrary closed curve, one could find a minimal surface stretching&lt;br /&gt;&#xA;across that same boundary. Every point on that surface is also a&lt;br /&gt;&#xA;solution to a particular differential equation.&#xA;(&lt;em&gt;The shape of inner space&lt;/em&gt; p.113-114)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is part of the book.  When I read this part, there are various terminologies such as Plateau problem, Dirichlet problem and so on. So I want to know about Monge-Ampere equation's historical and mathematical background for understanding '&lt;em&gt;flow of the idea&lt;/em&gt;' above terminologies. So I searched Google about this.&lt;/p&gt;&#xA;&lt;p&gt;According to my research, Monge-Ampere equation is a &lt;strong&gt;non-linear generalization of the Poisson equation&lt;/strong&gt;. Is this right?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-06-28T10:46:01.550" LastActivityDate="2023-06-28T10:46:01.550" Title="Motivation of Monge-Ampere equation" Tags="|mathematics-social-history|differential-equations|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15532" PostTypeId="1" CreationDate="2023-06-24T11:16:12.060" Score="5" ViewCount="199" Body="&lt;p&gt;Apparently, this discrepancy is one of the &amp;quot;worst predictions&amp;quot; in the history of science. Clearly the vacuum energy calculation depends on many approximations and it is not clear how it should be tied to the cosmological constant. Some articles claim that the two differ by 120 orders of magnitude, however it is not clear for me who was the first to publish this number or where it comes from. I have checked reviews like:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1016/j.crhy.2012.04.008&quot; rel=&quot;nofollow noreferrer&quot;&gt;Everything you always wanted to know about the cosmological constant problem 2012&lt;/a&gt; but the author just says: ''122 orders of magnitude often quoted in the literature'' no ref.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1103/RevModPhys.61.1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weinberg (1989)&lt;/a&gt; does not offer much better, he cites &lt;a href=&quot;https://www.scientificamerican.com/article/the-mystery-of-the-cosmological-con/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abbott 1988 popular science article&lt;/a&gt;, but Abbott only refers to the 45 orders of magnitude.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1007/s10714-008-0624-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zeldovich 1968&lt;/a&gt; the magnitude is 42(?) no mention of 120 or 122&lt;/li&gt;&#xA;&lt;li&gt;The value has to predate 1984, because Alan Guth was referring already to it in his &lt;em&gt;The New Inflationary Universe&lt;/em&gt; (1983) lecture, see also &lt;a href=&quot;https://doi.org/10.1098/rsta.1983.0092&quot; rel=&quot;nofollow noreferrer&quot;&gt; Hawking (1983)&lt;/a&gt; .&lt;/li&gt;&#xA;&lt;li&gt;W. Nernst, W. Lenz and W. Pauli seem to have discussed the problem long before Zeldovich (in the 1920s) mostly in German. Maybe one of them?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So who was the first to publish such a large difference of 120 or 122? Did the author specify values for both the vacuum energy calculation and measurements of the cosmological constant?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-08-05T14:36:56.613" LastActivityDate="2023-08-05T14:36:56.613" Title="Who was the first to estimate the vacuum energy discrepancy by 120 orders of magnitude?" Tags="|physics|cosmology|" AnswerCount="0" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15533" PostTypeId="1" CreationDate="2023-06-25T00:26:15.917" Score="0" ViewCount="200" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ShWst.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ShWst.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;After about 03:03 in Dr. Trefor Bazett's June 2023 video &lt;a href=&quot;https://youtu.be/GafRRl5XRPM?t=183&quot; rel=&quot;nofollow noreferrer&quot;&gt;Why this pattern shows up everywhere in nature || Voronoi Cell Pattern&lt;/a&gt; there is a discussion of a diagram representing work by &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Snow#Cholera&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Snow&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;John Snow (15 March 1813 – 16 June 1858) was an English physician and a leader in the development of anaesthesia and medical hygiene. He is considered one of the founders of modern epidemiology, in part because of his work in tracing the source of a cholera outbreak in Soho, London, in 1854, which he curtailed by removing the handle of a water pump. Snow's findings inspired the adoption of anaesthesia as well as fundamental changes in the water and waste systems of London, which led to similar changes in other cities, and a significant improvement in general public health around the world.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bazett uses the story of Snow's analysis as one example of the utility of &lt;a href=&quot;https://en.wikipedia.org/wiki/Voronoi_diagram&quot; rel=&quot;nofollow noreferrer&quot;&gt;Voronoi diagrams&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The screenshot above shows an irregular polygon with roughly 40 sides and has about a half-dozen concavities - not typical for diagrams I've seen and made the polygons have in the ballpark of 3 to 6 sides and are always convex.&lt;/p&gt;&#xA;&lt;p&gt;Of course the 2D Voronoi diagrams I look at and make are based on line-of-sight distances:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$r_{ij}=\sqrt{(x_i-x_j)^2 + (y_i-y_j)^2}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and that's not how people walk.&lt;/p&gt;&#xA;&lt;p&gt;So I'd like to ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt; Why doesn't John Snow's Voronoi diagram look like one? How was the diagram made? (distance to cholera-spreading water pump in 19th century London)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;From the video:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Proving Cholera is waterborne one one of the most infamous examples of use cells to actually solve something occurred in the 1850s when there was a cholera epidemic in London and Jon Snow a physician in London not Game of Thrones discovered something kind of interesting. He noticed that there was a water pump on the map and in fact there were many water pumps in London, but when you try to map out which water pump is closest well he came up with a map that looks something like this.&lt;/p&gt;&#xA;&lt;p&gt;This region was the portion of London that was all closest to this one pump, and then you'll notice in the map there's there's all these stacked lines those stack lines represented deaths to cholera and you'll notice how those are largely constrained within that geographic area. And so Jon Snow used this evidence to help support the theory that cholera was a waterborne transmission. &lt;strong&gt;And &lt;em&gt;in effect&lt;/em&gt; this is a type of Voronoi cell.&lt;/strong&gt; It's this particular cell that has the majority of these cholera outbreaks and because it's associated with a single water source then that's good evidence that cholera is waterborne - another great application...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2023-06-26T06:06:36.297" Title="Why doesn't John Snow's Voronoi diagram look like one? How was the diagram made? (distance to cholera-spreading water pump in 19th century London)" Tags="|mathematics|medicine|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15536" PostTypeId="2" ParentId="15408" CreationDate="2023-06-25T15:33:42.340" Score="0" Body="&lt;p&gt;There is an intriguing recent article by Koons in &lt;em&gt;Synthese&lt;/em&gt; at&#xA;&lt;a href=&quot;https://doi.org/10.1007/s11229-016-1295-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/s11229-016-1295-6&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Koons argues for the superiority of the (neo-)Aristotelian concept of&#xA;hylomorphism as an alernative to materialism/atomism.  I am not sure&#xA;how convincing Koons' argument is, but assuming for the moment that&#xA;the doctrine of hylomorphism is &amp;quot;scientific&amp;quot; or at least&#xA;proto-scientific, one could perhaps detect an example of what you are&#xA;looking for.&lt;/p&gt;&#xA;&lt;p&gt;Note that throughout the Middle Ages, Aristotle was considered the&#xA;supreme authority on matters that we would describe today as&#xA;&amp;quot;scientific&amp;quot;.  The concept of hylomorphism was exploited by early&#xA;catholic theologians to give what they felt was a satisfactory account&#xA;of the eucharist.  Some aspects of such an interpretation were&#xA;fiercely opposed by other theologians, including eventually the&#xA;protestant ones.&lt;/p&gt;&#xA;&lt;p&gt;Thus one could perhaps argue that the vocabulary of hylomorphism was&#xA;&amp;quot;co-opted,&amp;quot; as you say, with the goal of furthering the respectability&#xA;of certain religious &amp;quot;theories or ideologies.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-06-25T15:33:42.340" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15537" PostTypeId="2" ParentId="15533" CreationDate="2023-06-26T06:06:36.297" Score="3" Body="&lt;p&gt;More of comment. It's easier to comment on an observation with a diagram.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QvBYQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QvBYQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In the diagram above you'll see I've added some hand drawn quasi circular shapes. The red and blue shapes highlight pumps. The brown shape highlights a dead end street.&lt;/p&gt;&#xA;&lt;p&gt;The diagram is supposed to represent a boundary for those who used the red pump as their water source.&lt;/p&gt;&#xA;&lt;p&gt;Because of access issues, people who lived on the dead end street would have had a longer journey to use the red pump, so they most likely would have used the blue pump, hence that street has been removed from the diagram.&lt;/p&gt;&#xA;&lt;p&gt;I can recall, in the recent past (a few years back) either reading or seeing a You Tube video about the history of looking a water pumps as the source of illnesses in London during the period in question. Apparently, researchers at the time door knocked at people's abodes and asked them which pump they used as their water source, in addition to asking about illnesses and deaths in the households. If that was the case, it becomes an easy exercise to draw the boundaries for each pump region/cell.&lt;/p&gt;&#xA;" OwnerUserId="16318" LastActivityDate="2023-06-26T06:06:36.297" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15538" PostTypeId="1" CreationDate="2023-06-30T20:01:33.770" Score="0" ViewCount="141" Body="&lt;p&gt;I see this view attributed to him a lot during Twitter debates but I never found the source for it&lt;/p&gt;&#xA;&lt;p&gt;does anyone know if Heisenberg actually held this view/suggested it?&lt;/p&gt;&#xA;" OwnerUserId="9988" LastEditorUserId="6609" LastEditDate="2023-07-07T18:21:54.470" LastActivityDate="2023-07-07T18:21:54.470" Title="Did Heisenberg say free will could arise from quantum probabilistic mechanics?" Tags="|physics|quantum-mechanics|philosophy-of-science|quote|heisenberg|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15539" PostTypeId="1" CreationDate="2023-07-01T23:10:20.000" Score="2" ViewCount="186" Body="&lt;p&gt;According to Penrose's &lt;em&gt;Cycles of Time&lt;/em&gt;, he starts by reminding the usefulness of the second law of thermodynamics and how it applies to everyday life. In particular, how the net energy of Earth is a constant as the net flux of energy from the Sun into Earth is roughly 0 (most of the received energy is reemitted into space). However the flux emitted by the Earth into space is low energy photons and high entropy radiation compared to the one incoming from the Sun. The Sun then is a source with low entropy and that's what matters for biological processes on Earth. See also &lt;a href=&quot;https://www.youtube.com/watch?v=DxL2HoqLbyA&amp;amp;ab_channel=Veritasium&quot; rel=&quot;nofollow noreferrer&quot;&gt;recent Veritasium video&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Who was the first to notice this (publish about it)? Or in which time period this concept started being discussed and popularised? It would probably had been somebody from the 20th century as this concept needs understanding of blackbody and thermal radiation.&lt;/p&gt;&#xA;&lt;p&gt;It reminds me of Schrödinger's &lt;em&gt;What is Life?&lt;/em&gt; which discussed the importance of (lower) entropy in living beings.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-07-24T17:00:08.643" LastActivityDate="2023-07-24T17:00:08.643" Title="Who was the first to understand that there is not a net flux of energy between Sun and Earth, but of entropy?" Tags="|biology|physicists|thermodynamics|earth-sciences|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15540" PostTypeId="1" CreationDate="2023-07-03T13:12:47.463" Score="0" ViewCount="40" Body="&lt;p&gt;I am looking for good sources to quote for early work on triboelectricity. I already have the standard ones, e.g. Benjamin Park (1898) &amp;quot;A history of electricity..&amp;quot;, Roller &amp;amp; Roller (1953) DOI:10.1119/1.1933449,Maver (1918) in Encyclopedia Americana, and Hauksbee, Gilbert, Knight, Peclet papers/books. Probably I am missing some good ones.&lt;/p&gt;&#xA;&lt;p&gt;Suggestions please, including good modern overviews.&lt;/p&gt;&#xA;" OwnerUserId="18641" LastActivityDate="2023-07-03T13:12:47.463" Title="Early triboelectricity" Tags="|physics|electricity|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15541" PostTypeId="1" CreationDate="2023-07-04T08:56:52.227" Score="0" ViewCount="30" Body="&lt;p&gt;I am looking for early sources (references) to the analysis in electrostatics where the polarization vector is rewritten in terms of bound charges and a surface polarization charge.&lt;/p&gt;&#xA;&lt;p&gt;In terms of what I have already found, the answer is nothing. All standard recent texts such as Griffiths just give a derivation without sources. The paper by Joan Bromberg (1968) DOI:&lt;a href=&quot;https://doi.org/10.1119/1.1974442&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1119/1.1974442&lt;/a&gt; reasons that Maxwell himself did not do the surface and bound charges analysis, and an early textbook by N.R. Campbell (1913) &lt;em&gt;&lt;a href=&quot;https://archive.org/details/moderntheoryelec00campuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;Modern Electrical Theory&lt;/a&gt;&lt;/em&gt; does not show it. My guess is that it appeared some time after 1913, but I may be wrong&lt;/p&gt;&#xA;" OwnerUserId="18641" LastEditorUserId="6609" LastEditDate="2023-07-07T18:18:57.210" LastActivityDate="2023-07-07T18:18:57.210" Title="Early sources for surface and bound charges in polarization" Tags="|physics|electromagnetism|electricity|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15542" PostTypeId="1" CreationDate="2023-07-04T12:25:05.410" Score="0" ViewCount="105" Body="&lt;p&gt;I have a question about 'Minkowski problem' related to Gaussian curvature.&#xA;I searched 'Minkowski problem' in Google, Almost all of them were related to Einstein's theory of relativity. So I ask about this here!&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, the Minkowski problem is:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The classical Minkowski problem can be stated as follows: Given a&#xA;smooth closed curve in the plane, can we find a convex, closed curve&#xA;with the same length and the same Gaussian curvature at each point?&lt;/p&gt;&#xA;&lt;p&gt;At the time Minkowski posed this problem, there was a growing interest&#xA;in the theory of curves and surfaces. Mathematicians were exploring&#xA;various geometric properties of curves and surfaces, including&#xA;curvature. The concept of Gaussian curvature, which measures the&#xA;curvature of a surface at each point, had been developed by Carl&#xA;Friedrich Gauss in the early 19th century.&lt;/p&gt;&#xA;&lt;p&gt;The difficulty that mathematicians faced when the classical Minkowski&#xA;problem arose was finding a convex&#xA;curve with the same length and the same Gaussian curvature.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I'm not sure if this is correct, and even if this is correct, I want to know more about Minkowski problem's historical background and motivation, so I'm posting this question.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastActivityDate="2023-07-05T06:53:12.147" Title="What is Minkowski problem?" Tags="|differential-geometry|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15543" PostTypeId="1" CreationDate="2023-07-04T22:16:18.437" Score="1" ViewCount="48" Body="&lt;p&gt;The modern way to get to Bayes' theorem is through conditional probability. How did Bayes get to it? I've read, with crossed eyes, his 1763 paper and I cannot see conditional probability or his theorem in any recognizable form. Price's appendix comes nearly there. Is it implied and I just can't see it? Or worse still, it is explicit and I can't see it.&#xA;(I'm not even going to try Laplace.)&lt;/p&gt;&#xA;" OwnerUserId="15921" LastActivityDate="2023-07-04T22:16:18.437" Title="Where is conditional probability in Bayes' paper" Tags="|probability|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15544" PostTypeId="1" CreationDate="2023-07-05T00:27:46.770" Score="5" ViewCount="124" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Hermann Grassmann (1840s) verified the Leibniz conjecture that geometry 1s a form of algebra, showing that the geometric figures themselves are algebraic entities, because they are subject to definite operations (such as taking the midpoint of the segment connecting any two given points) and their relevant properties are determined by equations involving such operations.&lt;/p&gt;&#xA;&lt;p&gt;Conceptual Mathematics by Lawvere and Schanuel&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am searching on the internet for this Leibniz conjecture but I can't find it anywhere. Could someone give me a reference?&lt;/p&gt;&#xA;" OwnerUserId="16355" LastActivityDate="2023-07-05T14:29:08.037" Title="Leibniz conjecture that geometry is a form of algebra" Tags="|geometry|leibniz|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15545" PostTypeId="2" ParentId="15542" CreationDate="2023-07-05T06:47:34.320" Score="1" Body="&lt;p&gt;There are several problems which can be called &amp;quot;Minkowski problem&amp;quot;. Perhaps the most famous one is this problem from differential geometry:&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Minkowski_problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Minkowski Problem&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-07-05T06:53:12.147" LastActivityDate="2023-07-05T06:53:12.147" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15546" PostTypeId="2" ParentId="14477" CreationDate="2023-07-05T12:47:08.710" Score="2" Body="&lt;p&gt;Cavalieri did not use the method of exhaustion, but rather an alternative (and easier) approach using indivisibles.  In the Counter-Reformation context, there was fierce opposition to both mathematical and physical indivisibles.  Cavalieri and his student Degli Angeli published a number of books on indivisibles, which were criticized by several jesuit scholars (Guldin, Tacquet, Bettini).  Both Cavalieri and Degli Angeli were jesuats (with an &amp;quot;a&amp;quot;).  Cavalieri's religious order of the jesuats was suppressed by papal bull in 1668.  Some scholars relate the suppression to the scientific activities of Cavalieri and Degli Angeli that was viewed with suspicion by influential elements at Rome.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-07-05T12:47:08.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15547" PostTypeId="2" ParentId="15544" CreationDate="2023-07-05T13:53:14.523" Score="3" Body="&lt;p&gt;In  his essay in Antognazza's volume ('18), Probst notes that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Leibniz wanted to transform geometry into a pure doctrine of magnitudes without any direct relation to geometric objects. (p. 216).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Leibniz's work was a departure from the strictly geometric style of such predecessors as Barrow. Leibniz thought this was too hard, and sought to replace it by an algebraic formalism. The result was the Leibnizian calculus, where &amp;quot;blind calculation&amp;quot; achieves the results that were earlier only available through deep reflection and concentration on figures. Lawvere and Schanuel may be a bit cavalier here, since Leibniz envisioned this for transcendental curves as well, whereas what they are describing in Grassmann's name sounds purely algebraic.&lt;/p&gt;&#xA;&lt;p&gt;There is a closely related comment in &lt;em&gt;New Essays&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On the topics and in circumstances where our senses are not much engaged, our thoughts&#xA;are for the most part what we might call blind (in Latin I call them cogitationes caecas).&#xA;I mean that they are empty of perception and sensibility, and consist in the wholly&#xA;unaided use of symbols, as happens with those who calculate algebraically with only&#xA;intermittent attention to the geometrical figures which are being dealt with.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is quoted in Leduc ('14).&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-07-05T14:29:08.037" LastActivityDate="2023-07-05T14:29:08.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15548" PostTypeId="1" CreationDate="2023-07-05T15:57:22.833" Score="0" ViewCount="64" Body="&lt;p&gt;Can someone please provide an early reference to the use of the continued fraction expansion of &lt;span class=&quot;math-container&quot;&gt;$\frac{1+\sqrt D}2$&lt;/span&gt; to solve the Diophantine equation &lt;span class=&quot;math-container&quot;&gt;$x^2 - D y^2 = 4$&lt;/span&gt; for a positive integer &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; congruent to &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; modulo &lt;span class=&quot;math-container&quot;&gt;$4$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; not a square?  Or some method equivalent to the use of the continued fraction expansion of &lt;span class=&quot;math-container&quot;&gt;$\frac{1+\sqrt D}2$&lt;/span&gt;.  An 18&lt;sup&gt;th&lt;/sup&gt; century reference?  17&lt;sup&gt;th&lt;/sup&gt; century?  Modern references are easy to find—20&lt;sup&gt;th&lt;/sup&gt; and 21&lt;sup&gt;st&lt;/sup&gt; century.&lt;/p&gt;&#xA;&lt;p&gt;An equivalent method might involve use of binary quadratic forms, even if the terminology “binary quadratic form” is not used.  For example Brouncker's (maybe it's Wallis') method for solving &lt;span class=&quot;math-container&quot;&gt;$x^2 - D y^2 = 1$&lt;/span&gt; (see André Weil, &lt;em&gt;Number Theory, An approach through history from Hammurapi to Legendre&lt;/em&gt;, Birkhäuser, 2001, pp. 92 ff) essentially uses binary quadratic forms and is equivalent to computing the continued fraction of &lt;span class=&quot;math-container&quot;&gt;$\sqrt D$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I have been searching for a source for this for a few weeks, and have not found anything.  I've looked at Weil, Edwards, Mollin, Jacobson and Williams, Dickson's History (especially Vol II, pp. 351 to 353), Whitford, Fermat's Oeuvres (but I haven't read every page!), Lenstra's 2002 Notices article, Williams's millennial talk, Mahoney, Ribenboim, countless websites, and more.&lt;/p&gt;&#xA;" OwnerUserId="18654" LastEditorUserId="6609" LastEditDate="2023-07-07T18:14:35.793" LastActivityDate="2023-07-07T18:14:35.793" Title="17th or 18th century use of the continued fraction expansion of $(1 + \sqrt D)/2$ to solve the diophantine equation $x^2 - D y^2 = 4$" Tags="|mathematics|euler|fermat|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15549" PostTypeId="1" CreationDate="2023-07-05T21:58:37.593" Score="0" ViewCount="176" Body="&lt;p&gt;I have recently been interested in the history of French Mathematics especially in Grothendieck and his school. I have also been fascinated with Grothendieck's personality.&lt;/p&gt;&#xA;&lt;p&gt;It is known that during Grothendieck's IHES years is his dedication to mathematics was legendary as attested by many of his colleagues. However, I have also heard that during times he would be so concentrated on mathematical research that he would forget/neglect&#xA;to sleep and eat.&lt;/p&gt;&#xA;&lt;p&gt;Is that true?&lt;/p&gt;&#xA;&lt;p&gt;Could you provide references.&lt;/p&gt;&#xA;&lt;p&gt;P.S.&#xA;I am interested in this questions since I am studying psychology and examining the personalities of influential mathematicians. I have to write on the subject. Your help is greatly appreciated.&lt;/p&gt;&#xA;" OwnerUserId="18656" LastEditorUserId="12330" LastEditDate="2023-07-06T02:11:10.430" LastActivityDate="2023-07-06T02:11:10.430" Title="Grothendieck's complete absorption in mathematical research" Tags="|mathematicians|biographical-details|grothendieck|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15550" PostTypeId="1" CreationDate="2023-07-06T01:14:50.277" Score="3" ViewCount="95" Body="&lt;p&gt;If any of you have ever written code in DirectX, you're sure to have noticed that applying a linear operator &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; to a vector &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is done as &lt;span class=&quot;math-container&quot;&gt;$xA$&lt;/span&gt;, instead of the (nowadays usual) &lt;span class=&quot;math-container&quot;&gt;$Ax$&lt;/span&gt;. I wanted to know when this notation started to be used or from which areas of math it comes from.&lt;/p&gt;&#xA;&lt;p&gt;I remember reading an old optimization paper once where they used this order for linear algebra, or a book introducing some logic concepts for Prolog (forgot which one it was) that also used some kind of logical operators and they were written as &lt;span class=&quot;math-container&quot;&gt;$a\theta$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; is the operator.&lt;/p&gt;&#xA;&lt;p&gt;Idk if this is used in any other fields nowadays. My guess for why DirectX still uses is because of legacy&lt;/p&gt;&#xA;" OwnerUserId="18658" LastEditorUserId="18658" LastEditDate="2023-07-06T14:18:45.563" LastActivityDate="2023-07-06T14:18:45.563" Title="Where does the &quot;operator to the right&quot; notation originate?" Tags="|notation|linear-algebra|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15554" PostTypeId="1" CreationDate="2023-07-08T18:16:38.213" Score="0" ViewCount="125" Body="&lt;p&gt;I am trying to compile a list of mind-boggling/curious  historical facts in mathematics that will inspire and attract young people (9–11 years old) to the discipline of Mathematics. Do you have one fact to share? Multidisciplinary historical facts are welcome!&lt;/p&gt;&#xA;" OwnerUserId="12383" LastActivityDate="2023-07-08T18:16:38.213" Title="mind-boggling/curious historical facts that will inspire and attract young people" Tags="|teaching|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15556" PostTypeId="1" AcceptedAnswerId="15558" CreationDate="2023-07-09T13:58:05.660" Score="2" ViewCount="85" Body="&lt;p&gt;The 1833 formalism of Hamiltonian mechanics prescribes phase space coordinates (p,q) as &lt;em&gt;momenta&lt;/em&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Hamiltonian_mechanics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Hamiltonian_mechanics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Box-Jenkins method (1970) of time series forecasting uses similar notation (p,q) defined as &amp;quot;once stationarity and seasonality have been addressed, the next step is to identify the order (i.e. the p and q) of the autoregressive and moving average terms.&amp;quot; &lt;a href=&quot;https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is nothing in Box-Jenkins 1970 book suggesting awareness of Hamiltonian mechanics as a precedent.&lt;/p&gt;&#xA;&lt;p&gt;Regardless, and as noted, the similarities in notation are striking. But are they accidental?&lt;/p&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2023-07-11T09:01:53.587" Title="Are similarities in notation between Hamiltonian mechanics and Box-Jenkins time series forecasting accidental?" Tags="|classical-mechanics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15557" PostTypeId="2" ParentId="8332" CreationDate="2023-07-09T17:25:46.467" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://mathscinet.ams.org/mathscinet/MRAuthorID/165120&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lajos Soukup&lt;/a&gt; (father) and &lt;a href=&quot;https://mathscinet.ams.org/mathscinet/MRAuthorID/913420&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dániel Tamás Soukup&lt;/a&gt; (son) have written 4 joint research papers in mathematics (according to MathSciNet).&lt;/p&gt;&#xA;" OwnerUserId="7283" LastActivityDate="2023-07-09T17:25:46.467" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15558" PostTypeId="2" ParentId="15556" CreationDate="2023-07-10T15:50:12.397" Score="4" Body="&lt;p&gt;The notational similarity is purely accidental. The phase space &lt;em&gt;coordinates&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$(p,q)$&lt;/span&gt; describe a manifold of arbitrary dimensions, typically with the same number of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;'s and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;'s, i.e. &lt;span class=&quot;math-container&quot;&gt;$p = (p_1, ..., p_n)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$q = (q_1, ..., q_n)$&lt;/span&gt;. In the statistical example, the &lt;em&gt;single integers&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; denote the &lt;em&gt;order&lt;/em&gt; of the auto-regressive and moving-average &lt;em&gt;polynomials&lt;/em&gt; defining the underlying &lt;a href=&quot;https://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model&quot; rel=&quot;nofollow noreferrer&quot;&gt;ARMA model&lt;/a&gt; for a time series. Another example for the use of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; in statistics can be inferred from PP and QQ plots (abbreviating &lt;strong&gt;p&lt;/strong&gt;robabilities and &lt;strong&gt;q&lt;/strong&gt;uantiles, respectively). That said, one can of course use a Hamiltonian &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; to create a time series, say for coordinates &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;, via &lt;span class=&quot;math-container&quot;&gt;$q_{t+1} = q_t + \{q_t, H\}$&lt;/span&gt;, the curly bracket representing a &lt;a href=&quot;https://en.wikipedia.org/wiki/Poisson_bracket&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poisson bracket&lt;/a&gt;. To add historical flavour, the Wikipedia page linked above states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Somewhat ironically, an important and magisterial text on time series has been written by &lt;a href=&quot;http://mayoral.iae-csic.org/timeseries2021/hamilton.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;James D. Hamilton&lt;/a&gt; (1994).&lt;/p&gt;&#xA;" OwnerUserId="16834" LastEditorUserId="16834" LastEditDate="2023-07-11T09:01:53.587" LastActivityDate="2023-07-11T09:01:53.587" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15559" PostTypeId="1" AcceptedAnswerId="15576" CreationDate="2023-07-10T20:12:43.297" Score="5" ViewCount="234" Body="&lt;p&gt;Did they calculate a numerical value for the &amp;quot;extreme and mean ratio&amp;quot; or did they just have ways to construct it geometrically? If so, what value did they use and how did they calculate it?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-07-20T22:48:34.753" Title="Did ancient Greeks have a numerical value for the Golden Ratio" Tags="|mathematics|euclidean-geometry|greek-mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15560" PostTypeId="1" CreationDate="2023-07-11T17:45:22.283" Score="3" ViewCount="110" Body="&lt;p&gt;The relations ~ and ≍ are frequently used in math and computer science, at least within number theory and analysis of algorithms. What is their origin?&lt;/p&gt;&#xA;&lt;h1&gt;Definitions&lt;/h1&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$g(x)$&lt;/span&gt; is an eventually-positive function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;. Then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) \sim g(x)&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;if and only if&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\lim_{x\to\infty} f(x)/g(x) = k&#xA;$$&lt;/span&gt;&#xA;with &lt;span class=&quot;math-container&quot;&gt;$0&amp;lt;k&amp;lt;\infty,$&lt;/span&gt; while&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;f(x) \asymp g(x)&#xA;$$&lt;/span&gt;&#xA;if and only if&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0 &amp;lt; \liminf_{x\to\infty} f(x)/g(x) \le \limsup_{x\to\infty} f(x)/g(x) &amp;lt; \infty.&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Note that the defintion of ~ replaces the &lt;span class=&quot;math-container&quot;&gt;$\le$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$=$&lt;/span&gt; in the above.&lt;/p&gt;&#xA;&lt;h1&gt;Related work&lt;/h1&gt;&#xA;&lt;p&gt;I believe the related notations &lt;span class=&quot;math-container&quot;&gt;$\ll$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\gg$&lt;/span&gt; are due to Vinogradov, while &lt;span class=&quot;math-container&quot;&gt;$O/o/\Omega/\omega$&lt;/span&gt; are due to Hardy and Littlewood. (I've skimmed through Hardy's &lt;em&gt;Orders of Infinity&lt;/em&gt; which suggests that he and Paul Du Bois-Reymond did some early work here but I don't think that either of these notations originated there.)&lt;/p&gt;&#xA;&lt;p&gt;I was not able to find these in either of these online resources:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathsym/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mathshistory.st-andrews.ac.uk/Miller/mathsym/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_mathematical_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/History_of_mathematical_notation&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="562" LastActivityDate="2023-07-15T23:45:49.180" Title="First use of ~ and ≍ (\sym and \asymp)" Tags="|mathematics|notation|real-analysis|algorithm|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15561" PostTypeId="1" CreationDate="2023-07-12T07:49:45.100" Score="1" ViewCount="86" Body="&lt;p&gt;When studying the evolution of some concept in the history of mathematics, is there some established practice or methodology for selecting the corpus of texts to work with, or at least for selecting a list of mathematicians to consider?&lt;/p&gt;&#xA;&lt;p&gt;For example, when looking at the evolution of the concept of a &lt;em&gt;function&lt;/em&gt; from the 18th to the beginning of 20th centuries, I looked at the work of Euler, Lagrange, Cauchy, Riemann, Jordan, Goursat, Carathéodory, but this selection is rather arbitrary.&lt;/p&gt;&#xA;&lt;p&gt;How do historians of mathematics approach their studies?&lt;/p&gt;&#xA;" OwnerUserId="3030" LastActivityDate="2023-07-12T07:49:45.100" Title="In historical studies of mathematics, how to chose the corpus of texts to work with?" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15562" PostTypeId="1" AcceptedAnswerId="15563" CreationDate="2023-07-12T15:56:50.590" Score="0" ViewCount="113" Body="&lt;p&gt;As we already know that the concept of Momentum was discovered before Newton discovered his laws of motion, but my question is &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$\rightarrow$&lt;/span&gt;&lt;/strong&gt; How they discovered the relationship &lt;em&gt;p=mv&lt;/em&gt; without knowing much about the concept of Force at that time, and while performing the experiments don't the objects on which they are performing experiments always starts accelerating. How they maintain the constant velocity to found the relationship &lt;em&gt;p=mv&lt;/em&gt;?&lt;/p&gt;&#xA;" OwnerUserId="18684" LastActivityDate="2023-07-12T19:51:31.037" Title="How the concept of Momentum was discovered?" Tags="|physics|classical-mechanics|newton|physics-social-history|momentum|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15563" PostTypeId="2" ParentId="15562" CreationDate="2023-07-12T19:51:31.037" Score="2" Body="&lt;p&gt;I already made this comment back in P.SE before you copied it here, but you seem not to have taken heed:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;without knowing much about the concept of Force at that time&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The concept of forces, all the way to constructing vectors as a mathematical system allowing addition of vectors and resolving of forces, was done before even Galileo, let alone Newton. One thus cannot say that people before Newton didn't know about forces.&lt;/p&gt;&#xA;&lt;p&gt;In particular, &lt;a href=&quot;https://en.wikipedia.org/wiki/Simon_Stevin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stevin's textbook with the &amp;quot;Epitaph of Stevinus&amp;quot; on its cover&lt;/a&gt; details the above.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Back onto topic, the thing you are looking for is documented even on Wikipedia. You can trace the genesis of the concept, first of something that might persist for a while in 6th century, John Philoponus's writings, to Arabic writings in 11th century, Avicenna (Ibn Sīnā)'s, asserting that they would persist indefinitely in a vacuum.&lt;/p&gt;&#xA;&lt;p&gt;I am less happy than the those works, than the sudden appearance in 14th century by Jean Buridan who defined &lt;span class=&quot;math-container&quot;&gt;$\text{impetus}=\text{weight}\times\text{velocity}$&lt;/span&gt;, which is then basically Newtonian.&lt;/p&gt;&#xA;&lt;p&gt;Newton had a better framework, because he was able to describe gravity correctly, and thus define the concept of mass and weight as separate things, and so that is some improvement, but other than that, the concept of impetus of Buridan is already everything that is really needed. The leap the Newton needed, is thus very much smaller than one would expect.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_impetus&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Theory_of_impetus&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18591" LastActivityDate="2023-07-12T19:51:31.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15566" PostTypeId="1" CreationDate="2023-07-14T15:10:06.873" Score="1" ViewCount="96" Body="&lt;p&gt;When ancient Greeks could not observe any parallax, what were the two bodies they were looking at to determine this? I imagine the stars (or a specific star) would have been one of these, but what was the near body they used?&lt;/p&gt;&#xA;&lt;p&gt;Normally, when explaining parallax, we tell a person to close an eye, hold a finger up at arms length (the near body) and align this to an object across the room (like the stars in the example above). What &amp;quot;finger&amp;quot; did the Greeks use to try to measure parallax?&lt;/p&gt;&#xA;&lt;p&gt;They cannot have been measuring the angle between stars as these were believed to be all at the same distance on a hollow sphere.&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-07-17T17:15:52.620" Title="Ancient parallax" Tags="|astronomy|archaeoastronomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15567" PostTypeId="1" CreationDate="2023-07-14T15:50:43.063" Score="2" ViewCount="106" Body="&lt;p&gt;My question concerns the classical Desargues Theorem and its simplest version&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;The small Desargues Theorem:&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$A,B,C$&lt;/span&gt; be three distinct parallel lines and &lt;span class=&quot;math-container&quot;&gt;$a,a'\in A$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b,b'\in B$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c,c'\in C$&lt;/span&gt;, be points such that &lt;span class=&quot;math-container&quot;&gt;$ab\parallel a'b'$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$bc\parallel b'c'$&lt;/span&gt;. Then &lt;span class=&quot;math-container&quot;&gt;$ac\parallel a'c'$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The small Desargues Theorem does not involve any Projective Geometry, so it could have been known to ancient geometers (Thales, Euclides, Proclus, Pappus etc.). Is it true or not?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question.&lt;/strong&gt; Is there any evidence that the small Desargues Theorem was known to ancient Greeks?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="18692" LastEditorUserId="12330" LastEditDate="2023-07-14T20:28:35.730" LastActivityDate="2023-07-14T20:28:35.730" Title="Was the small Desargues Theorem known to ancient Greeks?" Tags="|mathematics|geometry|euclidean-geometry|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15568" PostTypeId="2" ParentId="15560" CreationDate="2023-07-15T03:19:08.617" Score="4" Body="&lt;p&gt;The notation &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; does not mean that ratio &lt;span class=&quot;math-container&quot;&gt;$f(x)/g(x)$&lt;/span&gt; has some positive limit &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, but that it has limit &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, e.g., &lt;span class=&quot;math-container&quot;&gt;$2x + \sqrt{x} \sim 2x$&lt;/span&gt;. What math books do you know that define &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; in the way you did? It would allow us to write &lt;span class=&quot;math-container&quot;&gt;$2x+\sqrt{x} \sim x$&lt;/span&gt;, which is a meaning of &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; that I've &lt;em&gt;never&lt;/em&gt; seen.&lt;/p&gt;&#xA;&lt;p&gt;This notation &lt;span class=&quot;math-container&quot;&gt;$\sim$&lt;/span&gt; is due to Landau: see page 62 of the first volume of his Handbuch der Lehre von der Verteilung der Primzahlen (1909).  It's available at archive.org &lt;a href=&quot;https://archive.org/details/handbuchderlehre01landuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;-notation and &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;-notation are not due to Hardy and Littlewood:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$o$&lt;/span&gt;-notation is due to Landau in the same book I already mentioned, on page 61.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;-notation is due to Bachmann on p. 401 of his book Die Analytische Zahlentheorie (1894) &lt;a href=&quot;https://archive.org/details/dieanalytischeza00bachuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-07-15T23:45:49.180" LastActivityDate="2023-07-15T23:45:49.180" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15570" PostTypeId="1" CreationDate="2023-07-17T09:47:08.650" Score="0" ViewCount="71" Body="&lt;p&gt;Who was the first one to give proof of 1st Principle of Mathematics in calculus (also known as ab-initio method) ,was he newton or someone else ??&lt;/p&gt;&#xA;" OwnerUserId="16989" LastEditorUserId="16989" LastEditDate="2023-07-17T10:07:25.237" LastActivityDate="2023-07-17T10:07:25.237" Title="Ab-initio method (First principle of Mathematics)" Tags="|calculus|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15571" PostTypeId="1" CreationDate="2023-07-17T10:38:52.687" Score="3" ViewCount="86" Body="&lt;p&gt;What does the following quote by Sylvester mean?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;A general algebraical determinant in its developed form may be likened to a mixture of liquids seemingly homogeneous, but which, being of differing boiling points, admit of being separated by the process of fractional distillation.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I find it hard to come up with mathematical analogies to reasonably explain the quote (found in Artin's Algebra.)&lt;/p&gt;&#xA;" OwnerUserId="7378" LastActivityDate="2023-07-17T10:38:52.687" Title="Sylvester's Quote on Determinants" Tags="|mathematics|quote|linear-algebra|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15572" PostTypeId="1" CreationDate="2023-07-17T11:09:42.760" Score="2" ViewCount="164" Body="&lt;p&gt;Category theory was  formalized around 1950s, and Grothendieck made his breakthrough papers about 10-20 years from that time. &lt;strong&gt;I wish to know, how was it possible the ideas of Category Theory were so quickly adopted?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I find this quite fascinating because there was no internet that time, and Grothendieck was involved in a different circle of Mathematicians (French) than those who found Category theory (American).&lt;/p&gt;&#xA;" OwnerUserId="16355" LastEditorUserId="4251" LastEditDate="2023-07-19T10:43:20.393" LastActivityDate="2023-07-19T10:43:20.393" Title="How did Grothendieck come in contact with Category theory?" Tags="|mathematicians|algebraic-geometry|grothendieck|category-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15574" PostTypeId="2" ParentId="15566" CreationDate="2023-07-17T17:15:52.620" Score="2" Body="&lt;p&gt;They tried to find parallax between pairs of stars. First of all, they didn't &amp;quot;believe&amp;quot; that stars were on the same distance. That was their hypothesis. And one of the reason why they accepted it is because they didn't find any parallax. If you don't have parallax then the sphere of fixed stars is the simplest model. But they tried to find deviation from this model even after they failed to measure parallax. For example, one of the reason for Hipparchus' star catalogue was to attempt to find proper motion of fixed stars. This is why he focused on triplet of stars that lies on one line. It would be easier to find if they would move in the feature.&lt;/p&gt;&#xA;&lt;p&gt;By the way, theoretically you can find parallax even between two objects at the same distance, but of course it would be much smaller and harder to measure.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastActivityDate="2023-07-17T17:15:52.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15575" PostTypeId="1" CreationDate="2023-07-18T04:49:33.150" Score="4" ViewCount="3862" Body="&lt;p&gt;I am reading Sieg's &amp;quot;Hilbert's programs and beyond&amp;quot; and I am having difficulty understanding this quote by Hilbert on page 74:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the Paris Lecture Hilbert re-emphasized and expanded this point by saying: In the case before us, where we are concerned with the axioms of real numbers in arithmetic, the proof of the consistency of the axioms is at the same time the proof of the mathematical existence of the complete system [“complete system” is Ewald’s translation for “Inbegriff ”, WS] of real numbers or of the continuum. Indeed, when the proof for the consistency of the axioms shall be fully accomplished, the doubts, which have been expressed occasionally as to the existence of the complete system of real numbers, will become totally groundless.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Sieg continues with this:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He proposed then to extend this approach to the Cantorian Alephs. Consequently and more generally, the consistency of appropriate axiom systems was to guarantee the existence of sets or, as he put it in 1905, to provide an objective criterion for the consistency of multiplicities.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is a common theme this book alludes to in Hilbert's early period, before 1920s. Hilbert believed that consistency of an axiomatic system implies the mathematical existence of the model satisfying those axioms. But it seems to me that this is only a consequence of completeness of the system in question and not a general fact at all. Namely, if some A is a syntactically consistent statement, assume that A is not true in any model. Then by completeness we have that ~A, which is true in every model, must be a theorem which contradicts the fact that A is consistent. So an existence of a model is a consequence of a consistent axiomatic system, but after the assumption of completeness, which I don't see mentioned in this book. This leads me to believe that maybe I am gravely misunderstanding the way these words were used in Hilbert's time, or that there might be some other thing I am missing. Why does Hilbert believe that consistency of axioms for real numbers would provide him with existence of the reals?&lt;/p&gt;&#xA;" OwnerUserId="18703" LastEditorUserId="12330" LastEditDate="2023-07-18T17:03:58.777" LastActivityDate="2023-08-06T13:28:39.140" Title="Why did Hilbert believe consistency implies existence?" Tags="|mathematics|mathematical-logic|hilbert|" AnswerCount="5" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="15576" PostTypeId="2" ParentId="15559" CreationDate="2023-07-18T07:50:14.290" Score="8" Body="&lt;p&gt;First of all, Greeks were not fascinated with Golden ratio as we are. Modern golden-ratio hype started about from the time of Leonardo da Vinci.  Second, Greek mathematicians were not very interested in numeric values in general. They believed that calculations were a low job of merchants. They were more interested in geometric construction. They knew how to construct &amp;quot;extreme and mean ratio&amp;quot; with a ruler and a compass, and that was enough for them. One exception was Archimedes. He did not consider himself above dirty calculations. For example, he estimated &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to be between &lt;span class=&quot;math-container&quot;&gt;$3 \frac{10}{71}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3\frac{1}{7}$&lt;/span&gt;. In the process he solved quadratic equations and calculated square roots. Of course he could use the same method to calculate &lt;span class=&quot;math-container&quot;&gt;$\phi$&lt;/span&gt;. If he did not do that, then it's not because it was too hard, but because it was too easy: everyone could do that.&lt;/p&gt;&#xA;&lt;p&gt;Now, this is not true for Greek astronomers. They had to do calculations. In particular, they needed trigonometric tables. How did they calculate trigonometric tables? They knew chords of simple school angels like &lt;span class=&quot;math-container&quot;&gt;$60^\circ$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$45^\circ$&lt;/span&gt;. They also knew that chord &lt;span class=&quot;math-container&quot;&gt;$36^\circ$&lt;/span&gt; is the &amp;quot;extreme and mean ratio&amp;quot; (note, that they used chord instead of sin: chord &lt;span class=&quot;math-container&quot;&gt;$x = 2\sin \frac{x}{2}$&lt;/span&gt;). From these simple angles they calculated chords of other angles using angle addition and subtraction theorems.&lt;/p&gt;&#xA;&lt;p&gt;So, to answer you question you need to look at Ptolemy's trigonometric table. And lo and behold you will find that according to Ptolemy&#xA;chord &lt;span class=&quot;math-container&quot;&gt;$36^\circ= 37;04;55$&lt;/span&gt; in sexagesimal. And this is absolutely correct: &lt;span class=&quot;math-container&quot;&gt;$\frac{\sqrt 5 - 1}{2} = 0; 37; 04; 55; 20...$&lt;/span&gt;. Ptolemy's precision was 2.6e-6. So the answer is yes, they did calculate at least three sexagesimal digits of the golden ratio.&lt;/p&gt;&#xA;" OwnerUserId="6788" LastEditorUserId="16318" LastEditDate="2023-07-20T22:48:34.753" LastActivityDate="2023-07-20T22:48:34.753" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15577" PostTypeId="2" ParentId="15575" CreationDate="2023-07-18T08:44:21.513" Score="16" Body="&lt;p&gt;Hilbert wrote this observation some decades before the formal theories of completeness (and incompleteness) took shape, so you can't expect him to be overly precise with regard to such notions.  The main novelty of Hilbert's remark is precisely this: consistency is &lt;em&gt;sufficient&lt;/em&gt; to ensure mathematical existence, and there is no need to argue that the geometric line is present in physical space somewhere or somehow.  This may have been a novelty to his contemporaries but truth be told the observation is already in Leibniz, who spoke of imaginary roots, infinitesimals, and negative numbers as useful &lt;em&gt;fictions&lt;/em&gt;,  even though they may seem impossible in the physical world (which Leibniz referred to as the &amp;quot;phenomenal realm&amp;quot;).  For further details on Leibniz's fictionalism see this recent article in &lt;a href=&quot;https://doi.org/10.1080/26375451.2020.1851120&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;British Journal for History of Mathematics&lt;/em&gt;&lt;/a&gt;.  Of course, much of Leibniz's writing was in manuscript form until recently and was not necessarily available to Hilbert's contemporaries.&lt;/p&gt;&#xA;&lt;p&gt;There is an insightful recent &lt;a href=&quot;https://doi.org/10.1007/s00283-022-10259-x&quot; rel=&quot;nofollow noreferrer&quot;&gt;article by Rowe&lt;/a&gt; on Cantor and Hilbert; see particularly page 4, second column.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-08-06T13:28:39.140" LastActivityDate="2023-08-06T13:28:39.140" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15578" PostTypeId="2" ParentId="15575" CreationDate="2023-07-18T16:55:11.753" Score="8" Body="&lt;p&gt;I just want to add some context to Katz's very nice answer. Hilbert's work on foundations occurred in the aftermath of the intuitionistic criticisms by Brouwer and Weyl. (Weyl in particular must have upset Hilbert, as he was Hilbert's student.) They maintained that only constructive proofs of existence counted; &amp;quot;proving&amp;quot; existence by contradiction proved nothing, or so they claimed. Hilbert responded (from Reid's biography of Hilbert):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Taking the Principle of the Excluded Middle from the mathematician is the same as ... prohibiting the boxer the use of his fists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Reid quotes an exchange between Pólya and Weyl:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Hermann, that is mathematics in shirt-sleeves&amp;quot; Pólya told him, in other words, not completely attired.&#xA;Weyl promptly offered to wager Pólya on the future of two specific propositions which would be eliminated from mathematics if Brouwer's ideas were to be accepted, as Weyl was convinced they would be---and within 20 years. The winner of the wager was to be decided by whether in 1938 Pólya was willing to admit that the following two propositions---&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;That each [non-empty] bounded set of real numbers has a precise upper bound,&lt;/li&gt;&#xA;&lt;li&gt;That each infinite set of real numbers has a countable sub-set&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;---were in fact completely vague &amp;quot;and one could ask of their truth or falsity as little as he could ask of the truth or falsity of the main ideas of Hegel's philosophy'&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cantor's remark, &amp;quot;The essence of mathematics is its freedom&amp;quot; is often quoted, but the context here are the attacks upon his work by Kronecker, a forerunner of the intuitionists. Dauben's biography of Cantor describes it this way:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For mathematicians, only one test was necessary: once the elements of any mathematical theory were seen to be consistent, then they were mathematically acceptable. Nothing more was required:&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and then he quotes Cantor directly:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In particular, in introducing new numbers, mathematics is only obliged to give definitions of them ... As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics... [Grundlagen einer allgemeinen Mannigfaltigkeitslehre]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This was in defense of Cantor's transfinite numbers. Recall here Hilbert's call-to-arms, &amp;quot;No one shall expel us from the Paradise that Cantor has created for us.&amp;quot; This appeared in Hilbert's address, &amp;quot;On the Infinite&amp;quot;, where he also said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Also old objections which we supposed long abandoned still reappear in different forms. For example, the following recently appeared: Although it may be possible to introduce a concept without risk, i.e., without getting contradictions, and even though one can prove that its introduction causes no contradictions to arise, still the introduction of the concept is not thereby justified. Is this not exactly the same objection which was once brought against complex-imaginary numbers when it was said: &amp;quot;True, their use doesn't lead to contradictions. Nevertheless their introduction is unwarranted, for imaginary magnitudes do not exist.&amp;quot;? If, apart from proving consistency, the question of the justification of a measure is to have any meaning, it can consist only in acertaining whether the measure is accomplished by commensurate success. Such success is in fact essential, for in mathematics as elsewhere success is the supreme court to whose decisions everyone submits.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="48" LastEditorUserId="48" LastEditDate="2023-07-18T20:36:17.210" LastActivityDate="2023-07-18T20:36:17.210" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15579" PostTypeId="2" ParentId="15575" CreationDate="2023-07-18T19:28:24.143" Score="4" Body="&lt;p&gt;Think back to when you learned linear algebra:&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{align}&#xA;x + y + z &amp;amp; = 2 \\&#xA;6x - 4y + 5z &amp;amp; = 31 \\&#xA;5x + 2y + 2z &amp;amp; = 13&#xA;\end{align}&lt;/span&gt;&#xA;Does a solution to this system exist? (In this case, yes.) If no solution exists, then applying elementary row operations, you will at some point get&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;0x + 0y + 0z = 1.&#xA;$$&lt;/span&gt;&#xA;Otherwise a solution exists. Extrapolating from this, one could say that if one can never deduce a contradiction from the assumption of the existence of some mathematical object, then it exists. It was only decades later that Gödel showed that that is wrong, and Hilbert was not at all pleased by Gödel's discovery.&lt;/p&gt;&#xA;&lt;p&gt;Gödel proved his two famous incompleteness theorems. He also proved what is known as Gödel's completeness theorem, which says that if no contradiction can be deduced from a set of first-order formulas, then a model of those formulas exists. I.e. every such formula with no free variables has either a proof or a counterexample.&lt;/p&gt;&#xA;" OwnerUserId="2193" LastActivityDate="2023-07-18T19:28:24.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15580" PostTypeId="1" AcceptedAnswerId="15583" CreationDate="2023-07-19T02:10:20.923" Score="3" ViewCount="118" Body="&lt;p&gt;I came across an entry for the &lt;em&gt;Proceedings of the American Physical Society&lt;/em&gt; which was published in &lt;em&gt;Phys. Rev.&lt;/em&gt; volume 93 page 939 (1954) under the heading &amp;quot;Minutes of the Stanford Meeting December 28, 29 and 30, 1953&amp;quot;. The entry is by Martin Packard and Russell Varian, both of Varian Associates, and was titled &amp;quot;Free Nuclear Induction in the Earth's Magnetic Field&amp;quot;.  It's basically an early report of NMR in earth's magnetic field.&lt;/p&gt;&#xA;&lt;p&gt;David Packard was one of the founders of Hewlett-Packard, and Russell Varian was the founder of Varian Associates, both important early Silicon Valley outfits&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/David_Packard&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Packard&lt;/a&gt; (b. 1912) earned a Masters in electrical engineering from Stanford in 1938.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://oac.cdlib.org/findaid/ark:/13030/c808662x/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Martin Packard&lt;/a&gt; (b. 1921) was a graduate student at Stanford around 1945-1946.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Does anyone know if these two Packards were biologically related?  They are too close in age to be a father-son pair but perhaps they were cousins?  I'm curious because they both worked in electrical engineering at Stanford not too far apart in time, and both had significant success in their careers.&lt;/p&gt;&#xA;" OwnerUserId="18707" LastEditorUserId="10106" LastEditDate="2023-07-19T05:48:46.647" LastActivityDate="2023-07-19T23:02:26.483" Title="Was Martin Packard of Varian Associates related to David Packard of Hewlett-Packard?" Tags="|biographical-details|physicists|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15581" PostTypeId="1" CreationDate="2023-07-19T08:22:57.690" Score="6" ViewCount="233" Body="&lt;p&gt;Permutation matrices I assume have a long history, and would be surprised if they were first considered only long after the work of Shur just after 1900, on the representation theory of &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 1&lt;/strong&gt; When were they first considered?&lt;/p&gt;&#xA;&lt;p&gt;One can however assign a matrix to an arbitrary endomorphism of a finite set (with a given ordering): just the entries will be non-negative integers bounded above by the size of the set. More generally, one can assign a matrix to an arbitrary function between (ordered) finite sets.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question 2&lt;/strong&gt; When was the first time someone thought of assigning a non-square matrix to a function of finite sets in an analogous way?&lt;/p&gt;&#xA;&lt;p&gt;If one can think of a space of functions on a finite set, then the idea presumably comes rather quickly, but the earliest I've seen of this—the free vector space on a set being the set of functions of finite support—was an unpublished draft written by Ehresmann for Bourbaki dated somewhere between 1936 and 1940.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To give some upper and lower bounds for question 1, this is what I've found so far:&lt;/p&gt;&#xA;&lt;p&gt;In the 1938 text &lt;em&gt;&lt;a href=&quot;https://archive.org/details/dli.ernet.509489&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Theory Of Group Representations&lt;/a&gt;&lt;/em&gt; by  Francis D. Murnaghan we have on page 17 a very brief introduction&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/JWNdQ.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/JWNdQ.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;and in&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;F. D. Murnaghan, &lt;em&gt;On the Representations of the Symmetric Group&lt;/em&gt;, American Journal of Mathematics, Vol. 59, No. 3 (Jul., 1937), pp. 437–488 &lt;a href=&quot;https://doi.org/10.2307/2371574&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.2307/2371574&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;we have mention of permutation matrices without even defining what they are. I should mention that in Birkhoff and MacLane's 1941 &lt;em&gt;&lt;a href=&quot;https://www.google.com.au/books/edition/A_Survey_of_Modern_Algebra/0DxzPUnaDXMC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Survey of Modern Algebra&lt;/a&gt;&lt;/em&gt; permutation matrices are also mentioned (see eg page 227). So at least in the US mathematical community in the late 1930s, it seems that it was probably well-known. Possibly this was helped by Wedderburn's 1934 &lt;em&gt;Lectures on Matrices&lt;/em&gt;, published by the AMS, where on page 33 we have the discussion of &amp;quot;elementary transformations&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TdbKk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TdbKk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;in particular:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since any permutation can be effected by a succession of transpositions, the corresponding transformation in the rows (columns) of a matrix can be produced by a succession of transformations of Type II.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and a formula for a general permutation matrix in the footnote. Wedderburn states in the introduction&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This book contains lectures on matrices given at Princeton University at various times since 1920.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Going to the German algebraic tradition, we find in van der Waerden's 1935 &lt;em&gt;&lt;a href=&quot;https://doi.org/10.1007/978-3-662-36333-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gruppen von Linearen Transformationen&lt;/a&gt;&lt;/em&gt;, page 54&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Insbesondere liefert die reguläre Darstellung von &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{R}$&lt;/span&gt;, bei der &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{R}$&lt;/span&gt; selbst Darstellungsmodul ist, eine Darstellung &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;-ten Grades von &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{g}$&lt;/span&gt;, die man ebenfalls die &lt;em&gt;reguläre Darstellung&lt;/em&gt; nennt. Die Matrixelemente von &lt;span class=&quot;math-container&quot;&gt;$A(s)$&lt;/span&gt; sind in diesem Fall&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\alpha_{ik}(s) = \begin{cases}1 &amp;amp; \text{für } ss_k = s_i\\ 0 &amp;amp; \text{sonst.} &#xA;\end{cases}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{R}$&lt;/span&gt; is the group-ring of the finite group &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{g}$&lt;/span&gt;, the &lt;span class=&quot;math-container&quot;&gt;$s_k$&lt;/span&gt; are group elements (the basis of &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{R}$&lt;/span&gt;), and &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is a general element of &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{g}$&lt;/span&gt;. So at the least, this is &lt;em&gt;a&lt;/em&gt; permutation matrix, if not named as such.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt; I can now go back to 1925:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Born, M., Jordan, P. Zur Quantenmechanik. Z. Physik 34, 858–888 (1925). &lt;a href=&quot;https://doi.org/10.1007/BF01328531&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/BF01328531&lt;/a&gt;,&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;who mention &amp;quot;die Permutationsmatrix&amp;quot; associated to a permutation, on page 865:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wir führen diesen Beweis, indem wir die Operation des Permutierens dureh Multiplikation mit einer geeigneten Matrix ersetzen.&lt;br&gt;&#xA;Eine Permutation schreiben wir&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left(\begin{array}\ 0 &amp;amp;1 &amp;amp; 2 &amp;amp; 3&amp;amp;\ldots \\ &#xA;k_0&amp;amp;k_1&amp;amp;k_2&amp;amp;k_3&amp;amp;\ldots\end{array}\right) = \left(\begin{array}\ n \\ &#xA;n_n\end{array}\right).&#xA;$$&lt;/span&gt;&#xA;Dieser ordnen wir die &lt;em&gt;Permutationsmatrix&lt;/em&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\mathbf{p} = \left(p(nm)\right),\quad p(nm)=\begin{cases}1&amp;amp;\text{für }m=k_n\\&#xA;0 &amp;amp; \text{sonst}\end{cases}&#xA;$$&lt;/span&gt;&#xA;zu.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT 2:&lt;/strong&gt; Schur in 1911 (&lt;em&gt;Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen&lt;/em&gt;, &lt;a href=&quot;https://eudml.org/doc/149348&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://eudml.org/doc/149348&lt;/a&gt;) gives a representation of &lt;span class=&quot;math-container&quot;&gt;$A_7$&lt;/span&gt; (called here &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{A}_7$&lt;/span&gt;) where the matrices (or &amp;quot;coefficient matrices&amp;quot; associated to generators) &lt;span class=&quot;math-container&quot;&gt;$Q_4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$Q_5$&lt;/span&gt; are permutation matrices, though he doesn't single them out or name them as anything special.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/A3dj5.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/A3dj5.png&quot; alt=&quot;Schur 1911 describing the relations between the matrices Qi&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/vpsch.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/vpsch.png&quot; alt=&quot;table of linear expressions for action of the matrices Qi on the basis vectors&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;On the other end of the timeline, I find in Burnside's &lt;em&gt;Theory of groups of finite order&lt;/em&gt;, the &lt;a href=&quot;https://archive.org/details/theoryofgroupsof0000burn/page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;second edition of 1911&lt;/a&gt; which contains material on representations of groups &amp;quot;as groups of linear substitutions&amp;quot;, on page 245:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The permutations of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; symbols that have been already considered are a very special case of linear substitutions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;without further fanfare. Linear substitutions are described as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...operation[s] performed on a set of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; symbols and leading to a new set of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; symbols&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where the new symbols are linear functions of the old symbols (i.e. this is essentially, in modern terms, talking about mapping a basis to an new basis).&lt;/p&gt;&#xA;&lt;p&gt;Additionally, in Bôcher's 1907 &lt;em&gt;Introduction to higher algebra&lt;/em&gt;, &lt;a href=&quot;https://archive.org/details/cu31924002936536/page/n291/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 276&lt;/a&gt;, the elementary matrices corresponding to switching rows are defined, but &lt;em&gt;not&lt;/em&gt; a general permutation matrix as in Wedderburn's &lt;em&gt;Lectures&lt;/em&gt;. Wedderburn attributes the idea of elementary transformations to &lt;a href=&quot;https://archive.org/details/dieausdehnungsl05grasgoog/page/n258/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Grassmann in 1862&lt;/a&gt;, in any case, so this aspect is far from new.&lt;/p&gt;&#xA;&lt;p&gt;For question 2, the closest I've found is in&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I. Schur, &lt;em&gt;Beiträge zur Theorie der Gruppen linearer homogener Substitutionen&lt;/em&gt;, Trans. Amer. Math. Soc. 10 (1909), 159-175, &lt;a href=&quot;https://doi.org/10.1090/S0002-9947-1909-1500832-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1090/S0002-9947-1909-1500832-1&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1DZ7Z.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1DZ7Z.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/n1jRy.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/n1jRy.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;from which it is conceivable one could consider the matrix &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; associated to an induced injection between permutation groups arising from an injective function of sets, but this mere speculation on my part.&lt;/p&gt;&#xA;&lt;p&gt;Similarly, one finds in&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I. Schur, &lt;em&gt;Neue Begrüdung der Theorie der Gruppencharaktere&lt;/em&gt;, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1905) pp 406–432 &lt;a href=&quot;https://archive.org/details/bub_gb_KwUoAAAAYAAJ/page/406/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/bub_gb_KwUoAAAAYAAJ/page/406/mode/1up&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;the weaker result (i.e. not a map between representations, just giving a canonical form for an arbitrary linear map):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;b) Ist &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; eine Matrix mit &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; Zeilen und &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; Spalten, deren Rang gleich &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; ist, so lassen sich zwei Matrizen &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; und &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; der Grade &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; und &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; von nicht verschwindenden Determinanten bestimmen, so daß in der Matrix &lt;span class=&quot;math-container&quot;&gt;$APB = (q_{\alpha \beta})$&lt;/span&gt; die &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; Koeffizienten &lt;span class=&quot;math-container&quot;&gt;$q_{11}, q_{22},\cdots, q_{rr}$&lt;/span&gt; gleich &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, die übrigen Koeffizienten gleich &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; sind.&lt;/p&gt;&#xA;&lt;p&gt;Der zuletst angeführte Satz ist identisch mit dem bekannten Theorem, welches besagt, daß eine bilineare Form &lt;span class=&quot;math-container&quot;&gt;$f = \sum_{\alpha =1}^m\sum_{\beta =1}^np_{\alpha\beta}x_\alpha y_\beta$&lt;/span&gt; vom Range &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; sich auf die Gestalt &lt;span class=&quot;math-container&quot;&gt;$f = u_1v_1+u_2v_2+\cdots+u_rv_r$&lt;/span&gt; bringen läßt, wo &lt;span class=&quot;math-container&quot;&gt;$u_1,u_2,\cdots,u_r$&lt;/span&gt; und &lt;span class=&quot;math-container&quot;&gt;$v_1,v_2,\cdots,v_r$&lt;/span&gt; linear unabhängige lineare homogene Funktionen der &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; Variabeln &lt;span class=&quot;math-container&quot;&gt;$x_\alpha$&lt;/span&gt; und der &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; Variabeln &lt;span class=&quot;math-container&quot;&gt;$y_\beta$&lt;/span&gt; bedeuten.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2531" LastEditorUserId="2531" LastEditDate="2023-08-08T02:49:46.960" LastActivityDate="2023-08-08T02:49:46.960" Title="Earliest mention of permutation matrices, or equivalent? More generally, matrices for arbitrary functions between finite sets?" Tags="|mathematics|linear-algebra|" AnswerCount="1" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15582" PostTypeId="2" ParentId="15575" CreationDate="2023-07-19T11:58:45.680" Score="2" Body="&lt;p&gt;There is a historical context involved  arising from the influence of Kant.&lt;/p&gt;&#xA;&lt;p&gt;There had been a number of reasons for anti-Kantian backlash in the nineteenth century. However, his wider influence led to the notion that mathematics has been intimately bound with astrophysics. In a certain sense, mathematicians had been at risk of losing an independent identity.&lt;/p&gt;&#xA;&lt;p&gt;The mathematical congresses began to be called, and authors like Cantor had begun to reassert Leibnizian principles (Kant had associated the principle of the identity of indiscernibles with logic and noumena in the same passage where he associated the separation of geometric points with mathematics and phenomena).&lt;/p&gt;&#xA;&lt;p&gt;Relative to this, it is important to understand that Hilbert actually had two foundational periods. His &amp;quot;Foundations of Geometry&amp;quot; had been written in the early period. Apparently, he had some concern about the applicability of mathematics across all application disciplines. I believe this to be the origin of his formalism. And, this is compatible with the direction of establishing an independent identity for mathematics.&lt;/p&gt;&#xA;&lt;p&gt;So, the question then arises, what constitutes existence for a formal mathematics?&lt;/p&gt;&#xA;&lt;p&gt;I have never had the impression that Hilbert had shared the anti-Kantian view commonly promoted in foundations. He includes non-essential quotes from Kant in &amp;quot;Foundations of Geometry&amp;quot; and makes a nod toward Kant after introducing metamathematics (Kant's schema of number is characterized with concatenated strokes just like Hilbert's metamathematical numerals).&lt;/p&gt;&#xA;&lt;p&gt;In &amp;quot;Critique of Pure Reason,&amp;quot; Kant declares that logic only provides a negative criterion of truth (rejection of contradiction). It is reasonable to conjecture that this had had influence on Hilbert through German idealism even if he had never been aware of the passage directly.&lt;/p&gt;&#xA;&lt;p&gt;As for later Hilbert, the introductory remarks from the &amp;quot;Foundations of Mathematics&amp;quot; continue to reflect a division between logic and mathematics. This may have been forced upon him by intuitionist criticism. And, his later work had been influenced because of &amp;quot;Principia Mathematica&amp;quot; by Whitehead and Russell.  Nevertheless, Hilbert had not adopted logicism. There is no reason to interpret Hilbert in the context of modern analytic philosophy.&lt;/p&gt;&#xA;&lt;p&gt;In any case, he expressly describes the consistency question as a &lt;em&gt;logical&lt;/em&gt; problem to be addressed through formal axiomatics. He does not assert that mathematics is solely formal axiomatics. In the same passage he describes formal axiomatics as relying upon interpreting universal quantification relative to decidable equality. And, he acknowledges that formal axiomatics deprecates the existential quantifier. Moreover, formal axiomatics is not independent of other mathematics. Rather, mathematical practice outside of formal axiomatics ought to inform formal axiomatics.&lt;/p&gt;&#xA;&lt;p&gt;From my reading, Hilbert changed his view across his career, although existence based upon non-contradictoriness had been retained for formal axiomatics because of his original motivations.&lt;/p&gt;&#xA;" OwnerUserId="18710" LastEditorUserId="604" LastEditDate="2023-07-20T06:57:00.210" LastActivityDate="2023-07-20T06:57:00.210" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15583" PostTypeId="2" ParentId="15580" CreationDate="2023-07-19T14:51:34.513" Score="6" Body="&lt;p&gt;One source, with a negative answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1515/9780804772617&quot; rel=&quot;nofollow noreferrer&quot;&gt;The HP Phenomenon: Innovation And Business Transformation&lt;/a&gt;, by Charles House and Raymond Price, claims of Martin Packard, &lt;a href=&quot;https://books.google.com/books?id=OsTnDv6d2DwC&amp;amp;pg=PA123&amp;amp;lpg=PA123&amp;amp;dq=%22The+HP+Phenomenon%22+%22martin+packard%22+%22dave%22&amp;amp;source=bl&amp;amp;ots=Ma3UZopnIq&amp;amp;sig=ACfU3U0ja5ErbmNm4tPz4Ax-Jka6cOCj4w&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjRlbvRgJuAAxV8C0QIHSfDAyQQ6AF6BAgIEAM#v=onepage&amp;amp;q=%22The%20HP%20Phenomenon%22%20%22martin%20packard%22%20%22dave%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;&amp;quot;no relation to Dave&amp;quot;&lt;/strong&gt; (p. 123)&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Accord to &lt;a href=&quot;https://www.cpii.com/docs/files/Varian%20Associates%20-%20An%20Early%20History.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Varian Associates: An Early History&lt;/a&gt;, David Packard was on the board of directors of Varian in 1954 (p. 15, photo p. 14). So Martin Packard and David Packard almost certainly knew each other.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2023-07-19T23:02:26.483" LastActivityDate="2023-07-19T23:02:26.483" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15584" PostTypeId="1" AcceptedAnswerId="15585" CreationDate="2023-07-19T15:51:09.650" Score="2" ViewCount="132" Body="&lt;p&gt;The words &lt;a href=&quot;https://www.etymonline.com/search?q=symbiosis&quot; rel=&quot;nofollow noreferrer&quot;&gt;symbiosis&lt;/a&gt; was first used in 1876.  This was 6 years before Darwin's death.&lt;/p&gt;&#xA;&lt;p&gt;Did he know about the concept of symbiosis?  Did he mention it in his book?&lt;/p&gt;&#xA;" OwnerUserId="16575" LastEditorUserId="12724" LastEditDate="2023-07-20T07:27:15.533" LastActivityDate="2023-07-20T07:27:15.533" Title="Did Darwin know about symbiosis?" Tags="|darwin|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15585" PostTypeId="2" ParentId="15584" CreationDate="2023-07-20T05:23:12.553" Score="4" Body="&lt;p&gt;It is useful to distinguish between the origins of the word symbiosis and the underlying concept. The following publication provides a useful overview of the genesis of the concept. Much of the early work was focused on establishing the exact nature of lichens and later extended to understanding the nature of certain chlorophyll particles found in some lower animals.&lt;/p&gt;&#xA;&lt;p&gt;Olivier Perru, &amp;quot;Aux origines des recherches sur la symbiose vers 1868-1883.&amp;quot; &lt;em&gt;Revue d’histoire des sciences&lt;/em&gt;, Vol. 59, No. 1, Jan.-Jun. 2006, pp. 5-27 (&lt;a href=&quot;https://www.cairn.info/revue-d-histoire-des-sciences-2006-1-page-5.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;The origin of both the concept and the word is found in German literature, with the concept dating to 1876 and the word dating to 1878. I have found no evidence of the word symbiosis in English prior to January of 1882. A first citation in the OED2 dated to 1877 appears to be in error (it seems an occurrence in the 1885 edition of a textbook translated from German was ascribed to the first edition of 1877, although the word did not actually appear in that edition).&lt;/p&gt;&#xA;&lt;p&gt;Albert-Bernhardt Frank, &amp;quot;Über die biologischen Verhältnisse des Thallus einiger Krustenflechten.&amp;quot; &lt;em&gt;Beiträge zur Biologie der Pflanzen&lt;/em&gt;, Vol. 2, No. 2, 1876, pp. 123-200&lt;/p&gt;&#xA;&lt;p&gt;provides an extensive overview of the then current hypotheses on the nature of lichens and provides equally extensive new evidence as to the veracity of these. On &lt;a href=&quot;https://books.google.com/books?id=3o_PAAAAMAAJ&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;dq=Beitr%C3%A4ge%20zur%20Biologie%20der%20Pflanzen%201877&amp;amp;pg=PA195#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 195&lt;/a&gt; he concludes that a new generalized concept regarding the close conglomeration of two species is needed, one that abstracts from the specifics of their interaction. For this he proposes the name &lt;em&gt;symbiotism&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wir müssen sämmtliche Fälle, wo überhaupt ein Auf- oder Ineinanderwohnen zweier verschiedener Species stattfindet, unter einen weitesten Begriff bringen, welcher die Rolle, die beide Wesen dabei spielen, noch nicht berücksichtigt, also auf das blosse Zusammenleben begründet ist, und wofür sich die Bezeichnung &lt;em&gt;Symbiotismus&lt;/em&gt; empfehlen dürfte.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also in 1876, a Hungarian professor in Klausenburg (now Cluj in Romania) who was looking into the nature of chlorophyll particles in invertebrates concluded that these must be single cell algae, and presented his findings in a talk (in Hungarian) in February 1876. However, he did not publish this in a &lt;a href=&quot;https://www.biodiversitylibrary.org/item/17518#page/658/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;more widely accessible form&lt;/a&gt; until six years later:&lt;/p&gt;&#xA;&lt;p&gt;Geza Entz, &amp;quot;Ueber die Natur der „Chlorophyllkörperchen“ niederer Tiere.&amp;quot; &lt;em&gt;Biologisches Centralblatt&lt;/em&gt;, Vol. 1, No. 21, Jan. 1882, pp. 646-650&lt;/p&gt;&#xA;&lt;p&gt;He did so in response to a &lt;a href=&quot;https://www.biodiversitylibrary.org/item/17518#page/536/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper&lt;/a&gt; published a few weeks earlier by the German zoologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Karl_Brandt_(zoologist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl Brandt&lt;/a&gt; who looked into the same issue:&lt;/p&gt;&#xA;&lt;p&gt;Karl Brandt, &amp;quot;Ueber das Zusammenleben von Algen und Tieren.&amp;quot; &lt;em&gt;Biologisches Centralblatt&lt;/em&gt;, Vol. 1, No. 17, Dec. 1881, pp. 524-527&lt;/p&gt;&#xA;&lt;p&gt;The term &amp;quot;Symbiose&amp;quot; in reference to a generalized concept of close conglomeration between two different species was coined by the German botanist &lt;a href=&quot;https://en.wikipedia.org/wiki/Heinrich_Anton_de_Bary&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anton de Bary&lt;/a&gt; who also researched lichens. He gave a talk on this in &lt;a href=&quot;https://en.wikipedia.org/wiki/Kassel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kassel&lt;/a&gt; at the 51&lt;sup&gt;st&lt;/sup&gt; Congress of German Naturalists and Physicians, September 10-17, 1878, and subsequently published it as a monograph (The Phenomenon of Symbiosis) the following year: A. de Bary, &lt;em&gt;Die Erscheinung der Symbiose&lt;/em&gt;, Strasbourg: Trübner 1879 (&lt;a href=&quot;https://books.google.com/books?id=7oQ_AAAAYAAJ&amp;amp;printsec=frontcover&amp;amp;hl=en&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;sa=X&amp;amp;ved=2ahUKEwi_4-i4upyAAxUzMDQIHTdMDDgQ6AF6BAgGEAI#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google scan&lt;/a&gt;). On &lt;a href=&quot;https://books.google.com/books?id=7oQ_AAAAYAAJ&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;pg=PA21#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 21&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Parasitismus, Mutualismus, Lichenismus, v.s.v. sind eben jeweils bestimmte Specialfälle jener allgemeinen Associationseinrichtung für welche der vorangestellte Ausdruck Symbiose als Collectivbezeichnung dienen mag.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[My translation] &amp;quot;Parasitism, mutualism, lichenism ... are particular special cases of a general mechanism of association for which the term symbiosis may serve as a collective term.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The first instance of the English word symbiosis that I could locate is in communication by the British biologist &lt;a href=&quot;https://en.wikipedia.org/wiki/Patrick_Geddes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Patrick Geddes&lt;/a&gt; read in January of 1882:&lt;/p&gt;&#xA;&lt;p&gt;Patrick Geddes, 'On the Nature and Functions of the &amp;quot;Yellow Cells&amp;quot; of Radiolarians and Coelenterates.' In &lt;em&gt;Proceedings of the Royal Society of Edinburgh&lt;/em&gt;, Vol. 11, pp. 377-396 (&lt;a href=&quot;https://scholar.archive.org/work/mq7wipen2nfnpo7bnnzkgm367e/access/ia_file/crossref-pre-1909-scholarly-works/10.1017%252Fs0370164600047064.zip/10.1017%252Fs0370164600047568.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan online&lt;/a&gt;). In reference to the paper by Karl Brandt cited above, he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;These statements are all doubtless true, but they constitute an extraordinarily slender foundation for the doctrine of &amp;quot;symbiosis.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Given this timeline, and considering that Darwin passed away in April of 1882 after having been diagnosed with a serious heart condition earlier in the year, it seems unlikely to me that he learned of the new concept of symbiosis from an English language publication. He may have read about it earlier in some German publication, but given his poor command of German as noted by his son, that does not seem all that likely.&lt;/p&gt;&#xA;&lt;p&gt;Francis Darwin, &lt;em&gt;Charles Darwin: His life told in an autobiographical chapter, and in a selected series of his published letters&lt;/em&gt;. London: John Murray 1908, &lt;a href=&quot;https://books.google.com/books?id=M8bTAAAAMAAJ&amp;amp;newbks=1&amp;amp;newbks_redir=0&amp;amp;dq=He%20himself%20learnt%20German%20simply%20by%20hammering%20away%20with%20a%20dictionary&amp;amp;pg=PA79#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 79&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He himself learnt German simply by hammering away with a dictionary; he would say that his only way was to read a sentence a great many times over, and at last the meaning occurred to him.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course there is a possibility Darwin may have learned about symbiosis through personal communication. It is not trivial to determine whether Darwin ever mentioned symbiosis in any of his writing, because he was a prolific writer. Going through his &lt;a href=&quot;http://darwin-online.org.uk/contents.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;list of publications&lt;/a&gt;, and examining his output after 1876 in some (but not full) detail, I could not find any mention of symbiosis. There are four books that fall into that time interval:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;The different forms of flowers on plants of the same species&lt;/em&gt; (1877)&lt;br /&gt;&#xA;&lt;em&gt;The effects of cross and self fertilisation in the vegetable kingdom, 2nd ed.&lt;/em&gt; (1878)&lt;br /&gt;&#xA;&lt;em&gt;The power of movement in plants&lt;/em&gt; (1880)&lt;br /&gt;&#xA;&lt;em&gt;The formation of vegetable mould through the action of worms, with observations on their habits&lt;/em&gt; (1881)&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-07-20T05:39:08.527" LastActivityDate="2023-07-20T05:39:08.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15586" PostTypeId="1" AcceptedAnswerId="15587" CreationDate="2023-07-20T08:17:23.207" Score="2" ViewCount="156" Body="&lt;p&gt;I am currently plundering the contents of the &lt;span class=&quot;math-container&quot;&gt;$1969$&lt;/span&gt; reprint of the 2nd edition of &lt;em&gt;Data and Formulae for Engineering Students&lt;/em&gt; published by Pergamon International (authors J.C. Anderson, D.M. Hum, B.G. Neal, J.H. Whitelaw) which I found in my bookshelf.&lt;/p&gt;&#xA;&lt;p&gt;In chapter 3: Physical Constants, it states:&lt;/p&gt;&#xA;&lt;p&gt;Avogadro's number &lt;span class=&quot;math-container&quot;&gt;$\qquad \text N \qquad = 6.023 \times 10^{26} \ / \text {kg mole}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;It is understood that the definition of Avogadro's number (henceforth here &lt;span class=&quot;math-container&quot;&gt;$N_0$&lt;/span&gt;) was rationalised in 1971 to be the amount of substance that contains as many elementary entities as there are atoms in &lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt; grams of carbon-&lt;span class=&quot;math-container&quot;&gt;$12$&lt;/span&gt;, but it is unclear (from the sources I have managed to find) what that number actually was, merely that it was known to be approximately &lt;span class=&quot;math-container&quot;&gt;$6.02 \times 10^{23}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is also understood that the current definition of &lt;span class=&quot;math-container&quot;&gt;$N_0$&lt;/span&gt; (as exactly &lt;span class=&quot;math-container&quot;&gt;$6.02214076 \times 10^{23}$&lt;/span&gt;) happened in 2019.&lt;/p&gt;&#xA;&lt;p&gt;I am interested to know where the &lt;span class=&quot;math-container&quot;&gt;$6.023$&lt;/span&gt; might have come from in the above book, or whether it's a mistake brought about by a conflation of the 2nd and 3rd decimal places (23) with the exponent of &lt;span class=&quot;math-container&quot;&gt;$10$&lt;/span&gt;. It is remarkably tempting to remember the &lt;span class=&quot;math-container&quot;&gt;$23$&lt;/span&gt;-ness of Avogadro's number and accidentally apply it to the mantissa.&lt;/p&gt;&#xA;&lt;p&gt;The question is: what values of &lt;span class=&quot;math-container&quot;&gt;$N_0$&lt;/span&gt; were in circulation &lt;em&gt;before&lt;/em&gt; the 2019 redefinition? In particular, what value of &lt;span class=&quot;math-container&quot;&gt;$N_0$&lt;/span&gt; was determined by Jean Perrin (who actually did most of the initial hard work of measuring it)?&lt;/p&gt;&#xA;&lt;p&gt;It occurs to me that if it &amp;quot;suddenly&amp;quot; changed from &lt;span class=&quot;math-container&quot;&gt;$6.023$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$6.022...$&lt;/span&gt;, that's a (relatively) large change to standards which in general are known to an accuracy of well under parts per million. That is to my mind something of a discrepancy.&lt;/p&gt;&#xA;" OwnerUserId="12839" LastActivityDate="2023-07-20T16:48:18.230" Title="What values of Avogadro's Number did Jean Perrin come up with?" Tags="|physics|chemistry|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15587" PostTypeId="2" ParentId="15586" CreationDate="2023-07-20T16:48:18.230" Score="2" Body="&lt;p&gt;My 1975-76 CRC Handbook (56th edition) states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Avogadro's number&lt;/strong&gt;. -- The number of molecules in one mole or gram-molecular weight of a substance. A number of values of the Avogadro number, which is usually denoted by &lt;span class=&quot;math-container&quot;&gt;$N_{A}$&lt;/span&gt;, have been found by various methods, generally lying within a range of 1% about the value of &lt;span class=&quot;math-container&quot;&gt;$6.02252 \times 10^{23}$&lt;/span&gt; per gram mole.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Rounding could easily make it &lt;span class=&quot;math-container&quot;&gt;$6.023 \times 10^{23}$&lt;/span&gt; which is still within the 1% range given.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastActivityDate="2023-07-20T16:48:18.230" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15588" PostTypeId="2" ParentId="15581" CreationDate="2023-07-21T09:54:17.450" Score="1" Body="&lt;p&gt;Study of circulants goes way back to at least 1842 (Catalan) -  they are linear combinations of cyclic permutation matrices. A very nice &lt;a href=&quot;https://kconrad.math.uconn.edu/articles/groupdet.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; by Keith Conrad has this and many other relevant references.&lt;/p&gt;&#xA;&lt;p&gt;Although I think as soon as one realises that the determinant of a &lt;span class=&quot;math-container&quot;&gt;$n\times n$&lt;/span&gt;-matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; can be written as a sum over &lt;span class=&quot;math-container&quot;&gt;$S_n$&lt;/span&gt; of determinants of permutation matrices (with entries weighted by the entries of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;), one gets permutation matrices, so it's even much earlier than 1842.&lt;/p&gt;&#xA;" OwnerUserId="14051" LastActivityDate="2023-07-21T09:54:17.450" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15589" PostTypeId="1" CreationDate="2023-07-21T13:09:33.763" Score="1" ViewCount="125" Body="&lt;p&gt;I am looking for examples of important scientific articles that have been discovered to be false and useless several years after their publication. I mean:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;they stated something interesting and important in a scientific field&lt;/li&gt;&#xA;&lt;li&gt;they were accepted as correct at first, for years&lt;/li&gt;&#xA;&lt;li&gt;many years later they have been found to be incorrect and not fixable or never fixed&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Can you provide examples?&lt;/p&gt;&#xA;" OwnerUserId="3983" LastActivityDate="2023-07-22T12:29:38.940" Title="Scientific articles discovered false and useless several years after their publication" Tags="|reference-request|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15592" PostTypeId="1" CreationDate="2023-07-22T11:10:02.303" Score="3" ViewCount="167" Body="&lt;p&gt;In the physics chapter of his biography of Gauss, W.K. Buhler writes the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Expansions into series are frequent and important in potential theory. So it does not come as a surprise that spherical functions, first introduced by Legendre, were Gauss's most useful tool. Gauss was interested in these and other special functions; among the posthumously published papers is a geometric interpretation of the spherical functions which Gauss developed in connection with electrodynamic considerations. He remarks that the individual elements in the expansion into spherical functions can be interpreted as dipolic, quadrupolic, etc., contributions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Looking into volume 5 of Gauss's works, I saw the fragment &amp;quot;on spherical functions&amp;quot; (p.630-631 there), in which I think part &lt;a href=&quot;https://i.stack.imgur.com/Qiw4D.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; is about the algebraic/analytic aspects of &lt;a href=&quot;https://en.wikipedia.org/wiki/Spherical_harmonics&quot; rel=&quot;nofollow noreferrer&quot;&gt;spherical harmonics&lt;/a&gt;, and part [2], entitled &amp;quot;geometric meaning of the spherical functions&amp;quot;, is the fragment I believe Buhler is refering to. Just to put all this in context, I think this interpretation is related to &lt;a href=&quot;https://en.wikipedia.org/wiki/Laplace_expansion_(potential)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Laplace multipole expansion&lt;/a&gt;, in which dipolic fields decay as &lt;span class=&quot;math-container&quot;&gt;$r^{-3}$&lt;/span&gt;, quadruploic fields as &lt;span class=&quot;math-container&quot;&gt;$r^{-4}$&lt;/span&gt; and etc. Gauss's interpretation is related to the angular dependence of those fields/potentials, which is relevant to diverse themes such as the description of Earth magnetic field (this was Gauss's original motivation) and the hydrogen atom problem in quantum mechanics.&lt;/p&gt;&#xA;&lt;p&gt;What striked me the most in fragment [2] is that Gauss apparently carries on his geometric interpretation to spherical functions of the fourth and fifth order, cases in which one cannot visualize its generation by collection of positive/negative charges in 3-dimensional euclidean space, but rather has to describe it by hypercubes in higher dimensions. However, I was not able to understand much from what Gauss wrote, and especially his notation was unclear to me.&lt;/p&gt;&#xA;&lt;p&gt;So I added here a translation and a pic of this short fragment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Geometric meaning of spherical functions:&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be an indefinite point and &lt;span class=&quot;math-container&quot;&gt;$A,B,C,D$&lt;/span&gt; be specified points. &lt;strong&gt;Spherical function of the first order&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$\alpha cosPA$&lt;/span&gt;. &lt;strong&gt;Second order&lt;/strong&gt;: &lt;span class=&quot;math-container&quot;&gt;$\alpha cos PA cos PC + \delta cos PB cos PD$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$A,B,C,D$&lt;/span&gt; refer to four faces of the regular octahedron. &lt;strong&gt;Third order&lt;/strong&gt;:&lt;span class=&quot;math-container&quot;&gt;$\alpha cos PA cos PB cos PC +\delta cosPA' cos PB' cosPC'$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$A,B,C$&lt;/span&gt; in one largest circle. &lt;span class=&quot;math-container&quot;&gt;$A',B',C'$&lt;/span&gt; lie in another largest circle in such a way that &lt;span class=&quot;math-container&quot;&gt;$AB = BC = CA = A'B' = B'C' = C'A' = 120°$&lt;/span&gt; and both circles intersect at right angles. &lt;strong&gt;Fourth order&lt;/strong&gt;: Aggregate of three products of four cosines each; the three largest circles intersect at right angles. &lt;strong&gt;Fifth order&lt;/strong&gt;: Aggregate of three products of five cosines each. The three largest circles intersect at one point.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/Qiw4D.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/Qiw4D.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;There is no reference to high dimensional spaces here, but even if this fragment does not introduce it, the correctness of Gauss's statements here need to be checked. I think this is another interesting point in his unpublished works that has not recieved attention. But the whole notation there is very unclear to me; therefore I'd like to get some help here by anyone who is more familiar than me with the theory of spherical harmonics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Update:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;I think I got some clue about the structure of Gauss's geometric interpretation from the drawing in this fragment. After some thoughts, I think this drawing depicts four point &lt;span class=&quot;math-container&quot;&gt;$A,B,C,D$&lt;/span&gt; at longitudes &lt;span class=&quot;math-container&quot;&gt;$0°,90°,180°,270°$&lt;/span&gt; and all at the same latitude &lt;span class=&quot;math-container&quot;&gt;$45°$&lt;/span&gt;, as seen from the pole (latitude &lt;span class=&quot;math-container&quot;&gt;$90°$&lt;/span&gt;) of the sphere (in a kind of &amp;quot;top view&amp;quot;). This speculation is consistent with the angle &lt;span class=&quot;math-container&quot;&gt;$60°$&lt;/span&gt; that also appears in this drawing: the &lt;a href=&quot;https://en.wikipedia.org/wiki/Great-circle_distance&quot; rel=&quot;nofollow noreferrer&quot;&gt;Great-circle distance&lt;/a&gt; between two adjacent points corresponds exactly to central angle of &lt;span class=&quot;math-container&quot;&gt;$60°$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;I searched for additional clues at the relavent treatises on Gauss by later mathematicians, and I found a useful reference in Harald Geppert's treatise &amp;quot;about Gauss's work on mechanics and potential theory&amp;quot;. It refers to E.W Hobson's book &amp;quot;the theory of spherical and ellipsoidal harmonics&amp;quot; (available on internet archive). Articles 81-84 in this book deal with &amp;quot;Maxwell's theory of poles&amp;quot; and on article 84 the author writes:&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The idea of determining harmonics by the position of the poles was suggested by Gauss*, but was first developed by Maxwell. An equivalent analytical theory is contained in a memoir by Clebsch.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Under the asterisk (*), Hobson refers to the same page in Gauss's Nachlass that appears here.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2023-07-23T20:35:33.850" LastActivityDate="2023-07-23T20:35:33.850" Title="Question on Gauss's geometric interpretation of &quot;spherical functions&quot;" Tags="|geometry|electromagnetism|gauss|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15594" PostTypeId="2" ParentId="15575" CreationDate="2023-07-22T14:00:40.840" Score="1" Body="&lt;p&gt;@Dr. Katz&lt;/p&gt;&#xA;&lt;p&gt;The original poster concludes with the question:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Why does Hilbert believe that consistency of axioms for real numbers would provide him with existence of the reals?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So, the issue lies with Hilbert's views on mathematical existence and not with modern jargon highly influenced by analytical philosophy. I no longer have my pseudobiography on Hilbert, or I would try to be more specific concerning the origins of his formalist views. But, the question asks about opinions he held early in his career -- presumably before the publication of &amp;quot;Foundations of Geometry.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The entry,&lt;/p&gt;&#xA;&lt;p&gt;math.stackexchange.com&lt;/p&gt;&#xA;&lt;p&gt;/questions/56603/provenance-of-hilbert-quote-on-table-chair-beer-mug#56605&lt;/p&gt;&#xA;&lt;p&gt;documents the commonly known quote associated with his early views.&lt;/p&gt;&#xA;&lt;p&gt;As noted before, the literature appears to indicate that Hilbert never fully conceded to the disparagement of Kant pursued by others. It is in Kant where logic is claimed to be distinct from mathematics. And, it is difficult to believe that a man who quoted Kant in his publications had not been aware of his actual writings.&lt;/p&gt;&#xA;&lt;p&gt;After speaking to mathematics in the Transcendental Aesthetic, Kant turns to logic in the Transcendental Logic. At approximately A60 and B84, Kant writes,&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;But, on the other hand, as regards knowledge in respect of its mere form (leaving aside all content), it is evident that logic, in so far as it expounds the universal and necessary rules of the understanding, must in these rules furnish criteria of truth. Whatever contradicts these rules is false.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So, the role of non-contradictoriness is established. The paragraph of the passage above concludes:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The purely logical criterion of truth, namely, the agreement of knowledge with the general and formal laws of the understanding and reason, is a conditio sine qua non, and is therefore the negative condition of all truth. But further than this logic cannot go. It has no test for the discovery of such error as concerns not the form but the content.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;What I do definitely recall from the pseudobiography I once owned is that Hilbert's emphasis on consistency arose from his desire to locate any alleged inconsistency associated with mathematics to application domains and not mathematics itself. Kant's description of the limitations on logic seem to be precisely what Hilbert had needed.&lt;/p&gt;&#xA;&lt;p&gt;I did try to be cautious since I have no direct evidence that Hilbert had been aware of Kant's actual statements.&lt;/p&gt;&#xA;&lt;p&gt;As for Cantor, Footnote 6 of his Grundlagen contains the statement:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The same epistemological principle can also be found in the philosophy of Leibniz.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Cantor is referring to a passage from Spinoza,&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;By 'adequate idea' I understand an idea which, to the extent that it is considered in se, without relation to an object, has all the properties or intrinsic denominations of a true idea.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;What immediately follows Cantor's reference to Leibniz is a rejection of the empiricist program of grounding knowledge upon subjective experience. Cantor writes:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;But, in my opinion these elements do not furnish us with any secure knowledge. For this can be obtained only from concepts and ideas that are stimulated by external experience, and are essentially formed by inner induction and deduction as something that, as it were, was already in us and is merely awakened and brought to consciousness.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The next footnote provides the comparison with Leibniz:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The procedure in the correct formation of concepts is in my opinion everywhere the same. One posits a thing with properties that at the outset is nothing other than a name or sign A, and then in an orderly fashion gives it different, or even infinitely many, intelligible predicates whose meaning is known on the basis of ideas that are already at hand, in particular to related concepts. If one has reached the end of this process, then one has met all the preconditions for awakening the concept A which slumbered inside us, and it comes into being accompanied by the intrasubjective reality which is all that can be demanded of a concept; to determine its transient meaning is then a matter for metaphysics.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The stepwise refinement described by Cantor is to be compared with what Leibniz writes in &amp;quot;Elements of a Calculus&amp;quot;:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Two terms which contain each other but do not coincide are commonly called 'genus' and 'species'. These, in so far as they compose concepts or terms (which is how I regard them here) differ as part and whole, in such a way that the concept of the genus is a part and that of the species is a whole, since it is composed of genus and differentia. For example, the concept of gold and the concept of metal differ as part and whole; for in the concept of gold there is contained the concept of metal and something else -- e.g., the concept of the heaviest of metals. Consequently, the concept of gold is greater than the concept of metal.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The Scholastics speak differently; for they consider, not concepts, but instances which are brought under universal concepts. So, they say that metal is wider than gold, since it contains more species than gold, and if we wish to enumerate the individuals made of gold on the one hand and those of metal on the other, the latter will be more than the former, which will therefore be contained in the latter as a part in the whole. By the use of this observation, and with suitable symbols, we could prove all the rules of logic by a calculus somewhat different from the present one -- that is, simply by a kind of inversion of it. However, I have preferred to consider universal concepts, i.e. ideas, and their combinations, as they do not depend upon the existence of individuals.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So, while Leibniz reasoned in relation to parts and wholes, he did so quite differently from the sense by which modern analytic philosophy declares mathematics to be extensional. In fact, this is the same distinction as found between Frege's concepts and his extensions of concepts (whence we should speak clearly of &amp;quot;comprehensionalist set theory&amp;quot; as opposed to intractible shorthands).&lt;/p&gt;&#xA;&lt;p&gt;For what this is worth, the purely logical form for the principle of the identity of indiscernibles should probably be attributed to Aquinas. Leibniz generalized it. Although I have not searched for it, Leibniz indicates that Aquinas used the idea to explain how God could know each soul individually.&lt;/p&gt;&#xA;&lt;p&gt;The justification for the principle lies with a section of Categories from Aristotle. The section discusses priority and simultaneity. Within it, Aristotle asserts that genera are prior to species. Aquinas took this to justify the idea that an individual is the lowest species.&lt;/p&gt;&#xA;&lt;p&gt;Today, this resolves into the admonition by first-order advocates to not confuse an individual with a singleton. It finds expression in the debate between set theory and mereology. Russell had emphasized the import of Peano's recognition of the need to make the distinction. In the modern literature, Potter has reiterated Russell's position while accusing Dedekind of mereological conflation. Lawvere has elevated Dedekind while claiming that philosophy has wandered from mathematical principles.&lt;/p&gt;&#xA;&lt;p&gt;Kant's distinction between logic and mathematics is expressed once again in his &amp;quot;Amphiboly of Concepts of Reflection.&amp;quot; It is here where the distinction is grounded upon the principle of the identity of indiscernibles. At about A263/B319 Kant writes:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Identity and Difference -- If an object is presented to us on several occasions but always with the same inner determinations, then it be taken as object of pure understanding, it is always one and the same, only one thing, not many. But, if it is appearance, we are not concerned to compare concepts, difference of spatial position at one and the same time is still an adequate ground for the numerical difference of the object.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Kant goes on to discuss Leibniz as he understood the problem (differing philosophies of Leibniz and Newton). He claims that Leibniz mistakenly took appearances (phenomena) as things-in-themselves (noumena). Kant maintains that plurality and numerical difference are ground in &amp;quot;the given&amp;quot; if sensible intuition.&lt;/p&gt;&#xA;&lt;p&gt;His paragraph concludes:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;For one part of space, although completely similar and equal to another part, is still outside the other, and for this reason is a different part, whichh when added to it constitutes with it a greater space. The same must be true of all things which exist simultaneously in the different spatial positions, however similar and equal they may otherwise be.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The last passage quoted shows why it had been easy for Hilbert to compare his metamathematical enumeration by strokes with Kantian ideas. It may also be compared with Lawvere's &amp;quot;bag of dots&amp;quot; justified by Cantorian notions disputed by the logicists.&lt;/p&gt;&#xA;&lt;p&gt;For what this is worth Dr. Katz, I had not been arguing with anyone. I thought I was answering the original poster's question.&lt;/p&gt;&#xA;&lt;p&gt;@OP&lt;/p&gt;&#xA;&lt;p&gt;Shortly after posting an answer (two parts) at the link,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://philosophy.stackexchange.com/questions/91773/which-philosophy-topics-are-necessary-for-philosophy-of-mathematics&quot;&gt;https://philosophy.stackexchange.com/questions/91773/which-philosophy-topics-are-necessary-for-philosophy-of-mathematics&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;PSE instituted a logon policy. That original poster accepted my answer because of the advice it had contained. The stackexchange community does not like me very much.&lt;/p&gt;&#xA;&lt;p&gt;You might consider the advice. Good luck in all of your endeavors.&lt;/p&gt;&#xA;" OwnerUserId="18726" LastActivityDate="2023-07-22T14:00:40.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15595" PostTypeId="1" CreationDate="2023-07-24T08:51:43.877" Score="1" ViewCount="126" Body="&lt;p&gt;The Manhattan project is well known for gathering many top scientists of the century. The Soviet nuclear program seem to have a reduced but still important cast with scientists like Andrei Sakharov, Vitaly Ginzburg and Yakov Zeldovich.  The German &lt;em&gt;Uranverein&lt;/em&gt; had Werner Heisenberg, Walther Gerlach and Otto Hahn.&lt;/p&gt;&#xA;&lt;p&gt;However when it comes to other nuclear programs it is not very clear who contributed. This question is about the postwar British nuclear bomb but I may ask similar questions about the French.&lt;/p&gt;&#xA;&lt;p&gt;I am surprised that people like James Chadwick, Rudolf Peierls or Otto Frisch, who contributed a lot in UK affairs and the Manhattan project do not seem to have been involved in the postwar British program (or at least I could not find any information about their contribution).&lt;/p&gt;&#xA;&lt;p&gt;I could find only that Baron Penney (from Kronig-Penney model) was the leading scientist. But who were the other importants scientists in the UK post-war nuclear bomb project? Any famous scientists were also involved?&lt;/p&gt;&#xA;&lt;p&gt;So far I have only found the names of John Challens and John Corner. Others like Roy Pilgrim, Ernest Mott and Cecil Bean are mostly unknown outside the project and do not have a Wikipedia page. Special mention to Klaus Fuchs that also worked on Manhattan and in the British program, and was later arrested for being a Soviet spy.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-07-25T16:58:36.307" LastActivityDate="2024-02-02T23:43:41.560" Title="Who were the famous scientists in the post-war British nuclear weapon project?" Tags="|nuclear-physics|atomic-weapons|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15596" PostTypeId="1" AcceptedAnswerId="15599" CreationDate="2023-07-25T05:18:24.773" Score="6" ViewCount="461" Body="&lt;p&gt;&lt;strong&gt;Did Georg Cantor ever think that set theory could serve as a foundational system for all of mathematics?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;He died in 1918, but Zermelo set theory (just Z, no ZF or ZFC yet) was described in a paper by Zermelo in 1908, ten years before Cantor's death.&lt;/p&gt;&#xA;&lt;p&gt;I haven't been able to find anything on the Web regarding this question. In particular, a &lt;a href=&quot;https://en.wikipedia.org/wiki/Georg_Cantor&quot; rel=&quot;noreferrer&quot;&gt;long Wikipedia about Cantor&lt;/a&gt; does not provide an answer to my question — the only thing I could gather is that Cantor knew that his set theory needed to be axiomatized.&lt;/p&gt;&#xA;" OwnerUserId="13076" LastEditorUserId="13076" LastEditDate="2023-07-25T05:26:37.983" LastActivityDate="2023-08-07T11:46:12.507" Title="Cantor, set theory and foundations" Tags="|mathematics|set-theory|cantor|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15597" PostTypeId="1" CreationDate="2023-07-25T07:47:35.663" Score="3" ViewCount="240" Body="&lt;p&gt;The Manhattan project is well known for gathering many top scientists of the century. The Soviet nuclear program seem to have a reduced but still important cast with scientists like Andrei Sakharov, Vitaly Ginzburg and Yakov Zeldovich. The German Uranverein had Werner Heisenberg, Walther Gerlach and Otto Hahn.&lt;/p&gt;&#xA;&lt;p&gt;However when it comes to other nuclear programs it is not very clear who contributed. This question is about the postwar French nuclear bomb. I asked a similar question about the British: &lt;a href=&quot;https://hsm.stackexchange.com/questions/15595/who-were-the-famous-scientists-in-the-post-war-british-nuclear-weapon-project&quot;&gt;Who were the famous scientists in the post-war British nuclear weapon project?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am surprised that people like Pierre Auger or Jules Guéron, who contributed a lot in the Manhattan project (from Canada) do not seem to have been involved in the postwar French program (or at least I could not find any information about their contribution).&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, Fréderic Joliot-Curie was against the bomb even if he created the  &lt;em&gt;Commissariat à l'énergie atomique&lt;/em&gt; (CEA).&lt;/p&gt;&#xA;&lt;p&gt;I could find only that Bertrand Goldschmidt was the leading scientist but I am unable to find any other scientists just politicians and military men. Even in the French articles of Wikipedia &lt;em&gt;&lt;a href=&quot;https://fr.wikipedia.org/wiki/Histoire_du_programme_nucl%C3%A9aire_civil_de_la_France&quot; rel=&quot;nofollow noreferrer&quot;&gt;Histoire du programme nucléaire civile de la France&lt;/a&gt;&lt;/em&gt; and &lt;em&gt;&lt;a href=&quot;https://fr.wikipedia.org/wiki/Histoire_du_programme_nucl%C3%A9aire_militaire_de_la_France&quot; rel=&quot;nofollow noreferrer&quot;&gt;Histoire du programme nucléaire militaire de la France&lt;/a&gt;&lt;/em&gt; there is just a mention of Goldschmidt and no other scientist. But who were the other scientists in the French post-war nuclear bomb project? Any famous scientists were also involved? Is this still confidential?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-08-25T11:50:52.920" LastActivityDate="2024-02-17T07:38:06.630" Title="Who were some of the scientists in the post-war French nuclear weapon project?" Tags="|france|atomic-weapons|" AnswerCount="1" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="15598" PostTypeId="2" ParentId="15410" CreationDate="2023-07-25T08:04:46.103" Score="1" Body="&lt;p&gt;Galileo compared Cavalieri to Archimedes when he finally got around to recommending  him for a university position in 1629.  However, it took a while for this to sink in with Galileo.  Three years yearlier in 1626, when a different position became available, Galileo recommended a certified mediocrity instead of Cavalieri, even though Cavalieri and Galileo had been in correspondence about Cavalieri's new approach to finding areas for five years by then.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-07-25T08:10:10.773" LastActivityDate="2023-07-25T08:10:10.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15599" PostTypeId="2" ParentId="15596" CreationDate="2023-07-25T08:30:30.843" Score="7" Body="&lt;p&gt;Cantor was interested in, and developed, a theory of ordinal numbers and cardinalities.  This was a significant accomplishment, in part because it overcame traditional resistance to working with actual infinity.  Cantor's theory did not by any means become the foundation of mathematics (though it is certainly studied extensively in modern set theory).  The axiomatisation provided by Zermelo and Fraenkel uses the language of sets, but the role of ordinals and cardinals is tangential.  So not only did Cantor not think that his &amp;quot;theory could provide a foundation for all of mathematics&amp;quot;, in point of fact it doesn't.&lt;/p&gt;&#xA;&lt;p&gt;This recent &lt;a href=&quot;https://doi.org/10.1007/s00283-022-10259-x&quot; rel=&quot;nofollow noreferrer&quot;&gt;article by Rowe&lt;/a&gt; on Cantor, Dedekind, and Hilbert suggests that both Cantor and Hilbert did think of set theory as foundation for real analysis.  Cantor and Hilbert were both bothered by foundational problems in set theory (such as the fact that all aleph-numbers can't form a set without causing a contradiction), and hoped to sort this out as a way of providing a solid foundation for the set of real numbers.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-08-07T11:46:12.507" LastActivityDate="2023-08-07T11:46:12.507" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15600" PostTypeId="1" CreationDate="2023-07-25T22:59:23.867" Score="3" ViewCount="99" Body="&lt;p&gt;I'm looking, for instance, as to when people started studying modular curves and modular forms as sections of line bundles on them, as opposed to the point of view of modular forms being holomorphic functions on the upper half-plane satisfying some conditions.&lt;/p&gt;&#xA;" OwnerUserId="6373" LastActivityDate="2023-07-25T22:59:23.867" Title="When did modular forms start to get studied via algebraic geometry?" Tags="|mathematics|number-theory|algebraic-geometry|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15601" PostTypeId="1" AcceptedAnswerId="15602" CreationDate="2023-07-26T15:07:00.170" Score="1" ViewCount="109" Body="&lt;p&gt;Even perfect numbers have the form &lt;span class=&quot;math-container&quot;&gt;$(2^n - 1)2^{n-1}$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$2^n - 1$&lt;/span&gt; is a prime number. This restricts the possible values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$34, 35, 36, 37$&lt;/span&gt;. Since &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; must be prime, only &lt;span class=&quot;math-container&quot;&gt;$37$&lt;/span&gt; has to be checked. But how did Fermat prove that &lt;span class=&quot;math-container&quot;&gt;$2^{37} - 1$&lt;/span&gt; is not prime?&lt;/p&gt;&#xA;" OwnerUserId="18742" LastEditorUserId="229" LastEditDate="2023-07-26T16:55:09.157" LastActivityDate="2023-07-26T21:20:47.210" Title="How did Fermat handle Frenicle's challenge to find a perfect number between $10^{20}$ and $10^{22}$?" Tags="|number-theory|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15602" PostTypeId="2" ParentId="15601" CreationDate="2023-07-26T21:20:47.210" Score="2" Body="&lt;p&gt;Probably (I cannot easily verify...) Fermat knew a basic modular arithmetic device, that for prime &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; dividing &lt;span class=&quot;math-container&quot;&gt;$b^n-1$&lt;/span&gt;, either &lt;span class=&quot;math-container&quot;&gt;$p|b^d-1$&lt;/span&gt; for a proper divisor of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, or &lt;span class=&quot;math-container&quot;&gt;$p=1\mod n$&lt;/span&gt;. (This is related to Fermat's little theorem.) Thus, for prime exponent &lt;span class=&quot;math-container&quot;&gt;$37$&lt;/span&gt;, we have &lt;span class=&quot;math-container&quot;&gt;$p=1\mod 37$&lt;/span&gt;. This cuts down the search space by a factor &lt;span class=&quot;math-container&quot;&gt;$18$&lt;/span&gt; or so, ...&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2023-07-26T21:20:47.210" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15604" PostTypeId="2" ParentId="375" CreationDate="2023-07-28T00:55:52.920" Score="0" Body="&lt;p&gt;In Vedic astrology the days of the week represent a very esoteric ordering of the planets.&lt;/p&gt;&#xA;&lt;p&gt;This ordering relates to the houses in astrology, which are 30 degree portions of the sky. The eastern horizon relates to the first house, the Sun and the self. The second house relates to the Moon and wealth. The third house relates to Mars and siblings. And so on to the seventh house on the western horizon which represents others (the opposite of the self). Its planet is Saturn which is farthest from the Sun.&lt;/p&gt;&#xA;&lt;p&gt;This ordering is used to interpret the chart in Vedic astrology, especially in a system called &lt;em&gt;Jaimini&lt;/em&gt; which uses it extensively. Edward Tarabilda had an alternative system which is very accurate.&lt;/p&gt;&#xA;&lt;p&gt;Also, the seven days of creation in the Bible matches this ordering of the planets. ie the first day relates to the Sun (let there be light) and the last day relates to Saturn (rest). The days in between match too.&lt;/p&gt;&#xA;" OwnerUserId="18751" LastActivityDate="2023-07-28T00:55:52.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15605" PostTypeId="1" AcceptedAnswerId="15608" CreationDate="2023-07-28T04:05:18.513" Score="2" ViewCount="132" Body="&lt;p&gt;The notion of &lt;a href=&quot;https://en.wikipedia.org/wiki/Hodge_structure&quot; rel=&quot;nofollow noreferrer&quot;&gt;weight of a Hodge structure&lt;/a&gt; and its avatars on &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt;-adic and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic cohomologies (via comparison theorems modeled on the Hodge decomposition of de Rham or singular cohomology with complex coefficients) is a little mysterious. Eigenvalues of Frobenius acting on &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt;-adic cohomology of a smooth projective variety over &lt;span class=&quot;math-container&quot;&gt;$\mathbb F_q$&lt;/span&gt; have absolute values &lt;span class=&quot;math-container&quot;&gt;$q^\frac{n}{2}$&lt;/span&gt;, for some &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;; numbers with this property are called &amp;quot;pure of weight &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;&amp;quot;. And an eigenspace of Frobenius for such an eigenvalue are called the same, as is a Galois representation on such an eigenspace.&lt;/p&gt;&#xA;&lt;p&gt;I wonder whether the term &amp;quot;weight&amp;quot; (&amp;quot;poids&amp;quot; in french) was introduced by Weil, Serre, Grothendieck, or Deligne, or perhaps someone else -which seems less likely. And was it introduced after the formulation of the Weil conjectures, inspired by those, or prior to that consideration of cohomology of varieties over finite fields by Weil ? I could even ask if it was formulated before the idea of a motive, but the answer looks obvious: most probably &amp;quot;yes&amp;quot;, as the notion of degree/weight seems appropriate on the basis of the Weil conjectures alone.&lt;/p&gt;&#xA;&lt;p&gt;Uwe Jannsen cites Deligne's ICM 1974 article on weights in &lt;a href=&quot;https://arxiv.org/pdf/1003.0927.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this overview&lt;/a&gt;. I should probably read everything thoroughly, and thus make myself a fairly good picture, but i thought that it might be good anyway to (gather in one place and) read your opinions on the historiography of the theory of weights, and perhaps nuggets of insight on this term and its various guises, which i might miss even poring through the pioneers' writings. Thank you.&lt;/p&gt;&#xA;" OwnerUserId="3481" LastEditorUserId="3481" LastEditDate="2023-07-29T07:58:18.613" LastActivityDate="2023-07-31T17:40:00.037" Title="When did the notion of and term &quot;weight&quot; arise in geometry?" Tags="|algebraic-geometry|complex-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15606" PostTypeId="1" CreationDate="2023-07-28T18:48:31.047" Score="3" ViewCount="81" Body="&lt;p&gt;Nowadays, the crystal structure of some solids can be determined by using X-ray diffraction techniques based on Bragg's law and so on. However, after reading the &lt;a href=&quot;https://en.wikipedia.org/wiki/Timeline_of_crystallography&quot; rel=&quot;nofollow noreferrer&quot;&gt;timeline of crystallography&lt;/a&gt;, it seems that physicists had some clear geometric ideas about crystals way before X-rays were discovered.&lt;/p&gt;&#xA;&lt;p&gt;I'm aware of the work of Bravais and Barlow at the end of the 19th century, but those are more mathematical rather than experimental. On the other hand, crystallography has been developing since the 18th century by the hand of Jean-Baptiste L. Romé de l'Isle and Abbé René Just Haüy, establishing some geometrical aspects of crystals and symmetry laws based on experimental data. Also, it seems that people working on crystallography assumed that crystals were composed by atoms/molecules, maybe based on Dalton's atomic theory and Bernoulli's kinetic theory.&lt;/p&gt;&#xA;&lt;p&gt;My question is about how did they come up with that before X-ray diffraction. What kind of experiments did scientists perform to study the structure of crystals? How did they know about a periodic arrangement of atoms/molecules in crystals? I think the latter question also addresses the introduction of the concept of a &amp;quot;crystal&amp;quot; in the modern sense.&lt;/p&gt;&#xA;" OwnerUserId="10066" LastEditorUserId="13182" LastEditDate="2023-07-28T22:04:22.363" LastActivityDate="2023-07-28T22:04:22.363" Title="How was crystallography studied before X-ray diffraction?" Tags="|experimental-physics|solid-state-physics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15608" PostTypeId="2" ParentId="15605" CreationDate="2023-07-31T17:40:00.037" Score="1" Body="&lt;p&gt;To turn my comment into an answer:&lt;/p&gt;&#xA;&lt;p&gt;As it says in your linked Wikipedia article, a pure Hodge structure of weight &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; has a natural group action on it, and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;this action must have the property that a real number &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; acts by &lt;span class=&quot;math-container&quot;&gt;$a^n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the representation theory camp, they would call that an action of weight &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In turn, the terminology &amp;quot;weights&amp;quot; (or rather, &amp;quot;poids&amp;quot;) in representations (of e.g. Lie groups and Lie algebras), according to &lt;a href=&quot;https://mathoverflow.net/q/154933/27465&quot;&gt;this MathOverflow question&lt;/a&gt;, was introduced in the early 20th century by E. Cartan, in turn influenced (via G. Kowalewski) by &lt;a href=&quot;https://en.wikipedia.org/wiki/Invariant_theory#The_nineteenth-century_origins&quot; rel=&quot;nofollow noreferrer&quot;&gt;19th century invariant theory&lt;/a&gt;. (Consider a homogoneous polynomial &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; of degree &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$f(ax) = a^n f(x)$&lt;/span&gt;.)&lt;/p&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2023-07-31T17:40:00.037" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15609" PostTypeId="1" AcceptedAnswerId="15610" CreationDate="2023-07-31T19:49:09.080" Score="16" ViewCount="3566" Body="&lt;p&gt;The Arora-Barak book on complexity contains the following sentence with the following footnote, page 128:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In primality testing, we are given an integer N and wish to determine&#xA;whether or not it is prime. Algorithms for primality testing were&#xA;sought after even before the advent of computers, as mathematicians&#xA;needed them to test various conjectures [3].&lt;/p&gt;&#xA;&lt;p&gt;[3]: An interesting anecdote is that Gauss, even though he was a very&#xA;fast human computer himself, used the help of a human supercomputer—an&#xA;autistic person who excelled at fast calculations—to do primality&#xA;testing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I would like to read more about this, but I cannot find anything close to a source on this, or even a repetition of this.&lt;/p&gt;&#xA;&lt;p&gt;I am not looking for a relationship between autistic people and prime numbers, or mathematics, but specifically that Gauss had an autistic assistant he tasked with checking primality.&lt;/p&gt;&#xA;&lt;p&gt;My question is that I am asking for confirmation if this is even true, and if so, to provide further references.&lt;/p&gt;&#xA;" OwnerUserId="18772" LastActivityDate="2023-08-01T21:44:23.460" Title="Autistic Assistant of Gauss used to check Primality" Tags="|gauss|primes|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15610" PostTypeId="2" ParentId="15609" CreationDate="2023-08-01T10:10:51.650" Score="27" Body="&lt;p&gt;The human computer was the computational prodigy Johann Martin &lt;strong&gt;Zacharias &lt;a href=&quot;https://en.wikipedia.org/wiki/Zacharias_Dase&quot; rel=&quot;noreferrer&quot;&gt;Dase&lt;/a&gt;&lt;/strong&gt; (June 23, 1824 - September 11, 1861), who was born in Hamburg, Germany and also died there. He reportedly attended school from the age of &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{2}$&lt;/span&gt;, and at the age of &lt;span class=&quot;math-container&quot;&gt;$15$&lt;/span&gt; began to travel (initially in Northern Germany) to make a meager living by exhibiting extraordinary feats of mental calculation. It appears that he never held a regular job for the duration of his life. His travels eventually brought him to Vienna, and his stay there led to his first publication in &lt;a href=&quot;https://en.wikipedia.org/wiki/Crelle%27s_Journal&quot; rel=&quot;noreferrer&quot;&gt;Crelle's Journal&lt;/a&gt;, on the computation of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to 200 decimal places:&lt;/p&gt;&#xA;&lt;p&gt;Z. Dahse, &amp;quot;Der Kreis-Umfang für den Durchmesser 1 auf 200 Decimalstellen berechnet&amp;quot;, &lt;em&gt;Journal für die reine und angewandte Mathematik&lt;/em&gt;, Vol. 27, No. 3, 1844, p. 198 (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0027?tify=%7B%22pages%22%3A%5B206%5D%2C%22pan%22%3A%7B%22x%22%3A0.536%2C%22y%22%3A0.616%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.365%7D&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;This result was submitted to the journal by a Viennese professor &lt;em&gt;von Schulz Strasznicky&lt;/em&gt;, who reports that Dase had arrived in Vienna in 1840 and had to rely on the kind support of various benefactors, among them the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schottenstift&quot; rel=&quot;noreferrer&quot;&gt;Scots Abbey&lt;/a&gt;. The professor had asked Dase to put his computational abilities to work on a scientifically useful result by calculating &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; to 200 decimals. Dase chose the formula &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4} \pi = \mathrm{arc tang} \frac{1}{2} + \mathrm{arc tang} \frac{1}{8} + \mathrm{arc tang} \frac{1}{5}$&lt;/span&gt; for this work. &lt;em&gt;von Schulz Strasznicky&lt;/em&gt; further reports that Dase had found temporary employment in the head office of the state railways and had commenced work on a table of natural logarithms with seven decimal places.&lt;/p&gt;&#xA;&lt;p&gt;One of Gauss's regular correspondents, &lt;a href=&quot;https://en.wikipedia.org/wiki/Heinrich_Christian_Schumacher&quot; rel=&quot;noreferrer&quot;&gt;H. C. Schumacher&lt;/a&gt; from Altona near Hamburg (today a borough of Hamburg), wrote in a letter to Gauss dated January 1, 1847 that Dase has returned to Hamburg and that two-thirds of the table of logarithms are complete, but that he anticipates Dase will have problems trying to get it published. He further mentions that he found a discrepancy between Dase's value of &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; and the earlier result of &lt;a href=&quot;https://en.wikipedia.org/wiki/William_Rutherford_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Rutherford&lt;/a&gt; after the 152&lt;sup&gt;nd&lt;/sup&gt; digit:&lt;/p&gt;&#xA;&lt;p&gt;C.A.F. Peters (ed.), &lt;em&gt;Schriftwechsel zwischen C. F. Gauss und H. C. Schumacher, 5. Band&lt;/em&gt;, Altona: G. Esch 1863, &amp;quot;No. 1131 Schumacher an Gauss&amp;quot;,  pp. 275-279 (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN635103400?tify=%7B%22pages%22%3A%5B283%5D%2C%22view%22%3A%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ein junger Mensch, &lt;em&gt;Dase&lt;/em&gt;, von dem ich Ihnen schon vor einigen&#xA;Jahren schrieb, und der Reisen gemacht hat, um Vorstellungen seines&#xA;Talentes im Kopfe zu rechnen zu geben, ist nun wieder hier und&#xA;ein merkwürdiges Beispiel, wie weit Intuition von Zahlenverhältnissen gebracht werden kann.&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;Er hat jetzt die natürlichen (7stelligen) Logarithmen der Zahlen von 1 bis 1005000 berechnet, wovon &lt;span class=&quot;math-container&quot;&gt;$\frac{2}{3}$&lt;/span&gt; schon fertig sind, und der Rest vor Ostern fertig wird, und sucht einen Verleger, den er aber schwerlich finden wird. &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; hat er bis zu 200 Decimalen berechnet. Ich habe seine Zahlen mit denen von Rutherford (Phil . Tr. 1841. Pt. II. p. 283) verglichen und finde&lt;br /&gt;&#xA;1) bis zur 152. Decimale incl. stimmen sie genau&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After consulting with &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Clausen_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Thomas Clausen&lt;/a&gt;, Schumacher informed Gauss in a  &lt;a href=&quot;https://www.google.com/books/edition/Briefwechsel_zwischen_C_F_Gauss_und_H_C/hLFXAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA280&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;letter&lt;/a&gt; dated January 22, 1847 that Dase's result for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; is entirely correct. In correspondence over the next few months, Schumacher and Gauss discussed in more detail Dase's capabilities and how one might put them to use for scientific projects. In a &lt;a href=&quot;https://www.google.com/books/edition/Briefwechsel_zwischen_C_F_Gauss_und_H_C/hLFXAAAAYAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA306&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;letter&lt;/a&gt; dated April 18, 1847 Schumacher informs Gauss that Dase is currently employed in Prussia, but that he is still in a probationary period and that he (contrary to Dase), does not expect him to be employed there permanently. He mentions that Dase has had plans to extend the table of factors to 10 million for quite a while and that the academy in Berlin might take him up on that should they continue to pay him a salary:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Preussen ist er noch in einer Probe-Anstellung. Der Termin läuft aber diesen Sommer ab, und ich möchte nach Encke's Aeusserungen bezeifeln, dass man ihn ausdehnen werde, was er selbst ziemlich sicher zu erwarten scheint. Die Factorentafel bis auf 10 Mill. auszudehnen, ist schon lange sein Plan gewesen und dazu, scheint es mir, könnte die Berliner Akademie ihn gebrauchen, wenn man ihm sonst noch ferner Gehalt geben will&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Schumacher's prediction about the difficulty of finding a publisher for Dase's table of logarithms appears to have been accurate, because it took another three years before this was published:&lt;/p&gt;&#xA;&lt;p&gt;Zacharias Dase, &lt;em&gt;Tafel der natürlichen Logarithmen der Zahlen.&lt;/em&gt; Vienna: Leopold Sommer 1850 (&lt;a href=&quot;https://www.google.com/books/edition/Tafel_der_nat%C3%BCrlichen_Logarithmen_der_Z/FYQ_AAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PP5&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;In the preface, Dase mentions his ardent desire to put his computational skills into the service of science and in particular at a German institution, but that he has given up hope that this could become reality and is therefore travelling to England to try and realize his desire there:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mein innigster Wunsch war es von jeher bei irgend einer Akademie meines deutschen Vaterlandes durch meine Rechenkraft an der Seite eines Mathematikers der Wissenschaft ein bleibendes Denkmal meiner Befähigung hinterlassen zu können; dass derselbe zu verwirklichen wäre, mag vorliegende Arbeit verbürgen. Die Hoffnung, dass mir dies auf deutscher Erde gelingen möchte, musste ich leider aufgeben; ich stehe eben im Begriffe nach England zu reisen, um auf fremden Boden zu versuchen meinen Wunsch zu realisieren.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What exactly transpired next is not clear from the sources I consulted, but that same year (1850) Dase seems to have taken the initiative by consulting with Gauss in person about creating tables of factors.&lt;/p&gt;&#xA;&lt;p&gt;E. W. Scripture, &lt;em&gt;Arithmetical Prodigies.&lt;/em&gt; Worcester: F. S. Blanchard 1891, &lt;a href=&quot;https://www.google.com/books/edition/Arithmetical_Prodigies/3Ki9lWMKnlQC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA19&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 19&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1849 Dase had wished to make tables of factors and prime numbers from the 7th to the 10th million. The Academy of Sciences at Hamburg was ready to grant him support, provided Gauss considered the work useful.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As requested by Dase, Gauss followed up their conversation with a letter dated December 7, 1850 whose contents is reproduced in the preface of the first volume of tables of factors published shortly after Dase's death. In the letter, Gauss summarizes earlier efforts on tables of factors, in particular tables by L. Chernac published in 1811 covering numbers up to 1020000 and tables by J. K. Burckhardt published in 1817 covering the first three million numbers, as well as a manuscript covering the fourth through sixth million by A. L. &lt;a href=&quot;https://en.wikipedia.org/wiki/August_Leopold_Crelle&quot; rel=&quot;noreferrer&quot;&gt;Crelle&lt;/a&gt; (founder of Crelle's Journal), deposited with the Berlin academy which Gauss expects to be published shortly. He therefore recommends that Dase initially tackle numbers from &lt;span class=&quot;math-container&quot;&gt;$6000000$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$10000000$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is not clear how exactly the work proceeded after Gauss's recommendation. Securing financing for the table-making effort seems to have been a major issue. In a paper on how best to create factor tables (including estimates of the time and money required) dated May 1853 but published only posthumously in 1856, Crelle himself recommends [p. 92] Dase for the job and mentions that Dase has the ambition to compute factor tables up to 30 million, but raises the issue [p. 93] how Dase could make a living while undertaking this work.&lt;/p&gt;&#xA;&lt;p&gt;A. L. Crelle, &amp;quot;Wie eine Tafel der untheilbaren Factoren der Zahlen bis zu beliebiger Höhe möglichst leicht und sicher aufzustellen sei.&amp;quot; &lt;em&gt;Journal für die reine und angewandte Mathematik&lt;/em&gt;, Vol. 51, No. 1, 1856, pp. 61-99 (&lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN243919689_0051?tify=%7B%22pages%22%3A%5B65%5D%2C%22view%22%3A%22info%22%7D&quot; rel=&quot;noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;W. W. Johnson, &amp;quot;Mr. Glaisher's factor tables and the distribution of primes.&amp;quot; &lt;em&gt;Annals of Mathematics&lt;/em&gt;, Vol. 1, No. 1, March 1884, &lt;a href=&quot;https://www.google.com/books/edition/Annals_of_Mathematics/FQkPAAAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Zacharias+Dase&amp;amp;pg=PA16&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;pp. 15-23&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1860, through the support of the patrons of science in his native town, Hamburg, Dase was enabled to devote himself entirely to carrying out Gauss's project, but in 1861 he died suddenly, leaving the seventh million complete and the eight million almost complete, as well as a great part of the work for the ninth and tenth millions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A death notice for Dase in a Vienna newspaper also mentions financial support by a committee of Hamburgians ultimately amounting to several thousand marks, enough to allow him to focus on table making for a period of three to four years:&lt;/p&gt;&#xA;&lt;p&gt;Wiener Zeitung, Sunday, September 15, 1861, &lt;a href=&quot;https://anno.onb.ac.at/cgi-content/anno?apm=0&amp;amp;aid=wrz&amp;amp;datum=18610915&amp;amp;seite=4&quot; rel=&quot;noreferrer&quot;&gt;p. 4&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nachdem er nemlich ziemlich plan- und zwecklos umhergestreift war und nur kurze Zeit eine Verwendung durch das preußische Finanzministerium erhalten hatte, kehrte er wieder nach Hamburg, seiner Vaterstadt, zurück, wo sich auf Anregung einiger seiner Gönner ein Komité bildete, welches Geldsammlungen zu dem Zwecke veranstaltete, um Dase 3–4 Jahre hindurch eine sorgenfreie Existenz zu sichern, die er dazu benutzen sollte, um ein logarithmisches Riesenwerk (über die Primzahlen) auszuarbeiten. Anfangs flossen die Beiträge ziemlich spärlich und Dase gab auf den Rath seiner Freunde wiederholt öffentliche Proben seiner Kunst; endlich wurden doch mehrere Tausend Mark zu dem oben gedachten Zwecke zusammengebracht und Dase machte sich an die ihm übertragene Arbeit, die nunmehr leider unvollendet bleibt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After Dase died unexpectedly from a stroke in 1861 his work was posthumously published in three volumes, with H. Rosenberg completing the tables for the nineth million while the tables for the tenth million were never published:&lt;/p&gt;&#xA;&lt;p&gt;Zacharias Dase, &lt;em&gt;Factoren-Tafeln für alle Zahlen der siebenten Million, oder genauer von 6000001 bis 7002000, mit den darin vorkommenden Primzahlen.&lt;/em&gt; Hamburg: Perthes-Besser &amp;amp; Mauke 1862 (&lt;a href=&quot;https://books.google.de/books?id=d2JHAAAAYAAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Zacharias Dase, &lt;em&gt;Factoren-Tafeln für alle Zahlen der achten Million, oder genauer von 7002001 bis 8010000, mit den darin vorkommenden Primzahlen.&lt;/em&gt; Hamburg: Perthes-Besser &amp;amp; Mauke 1863 (&lt;a href=&quot;https://books.google.com/books?id=Ni4NAAAAYAAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Zacharias Dase and H. Rosenberg, &lt;em&gt;Factoren-Tafeln für alle Zahlen der neunten Million, oder genauer von 8010001 bis 9000000, mit den darin vorkommenden Primzahlen.&lt;/em&gt; Hamburg: Perthes-Besser &amp;amp; Mauke 1865 (&lt;a href=&quot;https://books.google.com/books?id=IC4NAAAAYAAJ&amp;amp;printsec=frontcover#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;Google scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;The following publication offers a useful overview of early efforts in creating factor tables: Denis Roegel, &amp;quot;A reconstruction of the tables of factors of Burckhardt, Dase, and Glaisher (1814-1883), and their extension to the tenth million&amp;quot;. Research Report &lt;a href=&quot;https://inria.hal.science/hal-00654420&quot; rel=&quot;noreferrer&quot;&gt;hal-00654420&lt;/a&gt;, 2011&lt;/p&gt;&#xA;&lt;p&gt;So far I have not found any peer-reviewed publication that presents solid evidence that Dase was autistic, however multiple authors &lt;em&gt;speculate&lt;/em&gt; in that direction based on what appears to me as rather scant anecdotal details. For example, the &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Dase/&quot; rel=&quot;noreferrer&quot;&gt;article on Dase&lt;/a&gt; in MacTutor simply states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A modern medical expert assessing Dase using the descriptions of his personality and other factors, would suggest that he suffered from Asperger's syndrome.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-08-01T21:44:23.460" LastActivityDate="2023-08-01T21:44:23.460" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15612" PostTypeId="1" CreationDate="2023-08-02T18:47:04.547" Score="0" ViewCount="85" Body="&lt;p&gt;Is there some good book or YouTube channel that make a good comparison/distinctions between the mathematics before René Descartes, with the mathematics after Descartes?&lt;/p&gt;&#xA;&lt;p&gt;In the short article &amp;quot;&lt;a href=&quot;http://www.bibnum.education.fr/sites/default/files/46_descartes-analysis.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Book First of Descartes’s &lt;em&gt;Geometry&lt;/em&gt;&lt;/a&gt;&amp;quot; by André Warusfel :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It should be noted in passing that his technique of taking a length as a unit, which is now so commonplace, was at the time an innovation of unheard-of abstraction!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is, it's there some good text where is possible to find a comprehensive comparison of mathematical treatments with and without equivalence classes (perhaps specifically, with equivalence classes applied to rational/fractions, not necessarily in equivalences related to integers).&lt;/p&gt;&#xA;&lt;p&gt;The question borns to possibly distinguish style of mathematics before, even, the implicit introduction of equivalences into arithmetics. It is not a requirement, but, it would be nice if such a text has geometrical illustrations.&lt;/p&gt;&#xA;&lt;p&gt;The question born due to a YouTube user called &lt;a href=&quot;https://www.youtube.com/@NewCalculus/videos&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Gabriel&lt;/a&gt;. He has some information but, speaking to him is like 'walking in eggshels'. Anyway, my intention here is certainly not a critique beacuse the guy has been expelled of several forums due to his belligerent ways of communication, but rather the knowledge.&lt;/p&gt;&#xA;&lt;p&gt;What attracts my attention is how remarkable statements in one 'arithmetic' may be non remarkable in the other. Is there some other authors that can surf the two worlds, so to speak ?&lt;/p&gt;&#xA;" OwnerUserId="18780" LastEditorUserId="6609" LastEditDate="2023-08-03T09:32:35.333" LastActivityDate="2023-08-04T07:10:47.150" Title="A text or YouTube channel with a comparison between pre-Cartesian with post-Cartesian mathematics" Tags="|geometry|arithmetic|greek-mathematics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15613" PostTypeId="2" ParentId="5147" CreationDate="2023-08-02T18:55:37.293" Score="3" Body="&lt;p&gt;The guesses that are ventured in this question are entirely wrong.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;(1)&lt;/b&gt; That our reason to think multiplication is commutative is that we have an axiom that says so is wrong.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;(2)&lt;/b&gt; That those without modern standards of logical rigor could not easily understand why multiplication is commutative is wrong.&lt;/p&gt;&#xA;&lt;p&gt;Suppose &lt;span class=&quot;math-container&quot;&gt;$3\times5$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\overbrace{5+5+5}^\text{3 terms}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;and &lt;span class=&quot;math-container&quot;&gt;$5\times3$&lt;/span&gt; means &lt;span class=&quot;math-container&quot;&gt;$\overbrace{3+3+3+3+3}^\text{5 terms}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;(Remember that &amp;quot;terms&amp;quot; are things that you add or subtract, and &amp;quot;factors&amp;quot; are things that you multiply.)&lt;/p&gt;&#xA;&lt;p&gt;Here is &lt;span class=&quot;math-container&quot;&gt;$3\times 5{:}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{array}{ccl}&#xA;1 &amp;amp; \bullet \bullet\bullet\bullet\bullet &amp;amp; \longleftarrow \text{5 things} \\&#xA;2 &amp;amp; \bullet \bullet\bullet\bullet\bullet &amp;amp; \longleftarrow \text{5 things} \\&#xA;3 &amp;amp; \bullet \bullet\bullet\bullet\bullet &amp;amp; \longleftarrow \text{5 things}&#xA;\end{array}&#xA;$$&lt;/span&gt;&#xA;Take one of those things from each of the three sets of five and set them down here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bullet\bullet\bullet&#xA;$$&lt;/span&gt;&#xA;Now take another one of those things from each of the three sets of five and set them down here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bullet\bullet\bullet&#xA;$$&lt;/span&gt;&#xA;Now take another one of those things from each of the three sets of five and set them down here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bullet\bullet\bullet&#xA;$$&lt;/span&gt;&#xA;Now take another one of those things from each of the three sets of five and set them down here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bullet\bullet\bullet&#xA;$$&lt;/span&gt;&#xA;Now take another one of those things from each of the three sets of five and set them down here:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\bullet\bullet\bullet&#xA;$$&lt;/span&gt;&#xA;There you have &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; sets of &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; things, thus&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;3+3+3+3+3&#xA;$$&lt;/span&gt;&#xA;or &lt;span class=&quot;math-container&quot;&gt;$5\times 3.$&lt;/span&gt;&#xA;It is readily apparent why this must work for other numbers than &lt;span class=&quot;math-container&quot;&gt;$5$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3.$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="2193" LastEditorUserId="2193" LastEditDate="2023-08-03T00:57:28.490" LastActivityDate="2023-08-03T00:57:28.490" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15614" PostTypeId="2" ParentId="9834" CreationDate="2023-08-03T04:46:01.343" Score="1" Body="&lt;p&gt;Thomson's objection was that Perrin's experiment left open the possibility that the electric charge is being carried along by the cathode ray and may be separable from it.&lt;/p&gt;&#xA;&lt;p&gt;The article &lt;a href=&quot;https://history.aip.org/exhibits/electron/jj1897.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;3 Experiments 1 Big Idea&lt;/a&gt; provides some useful historical background, writing :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thomson wanted to see if, by bending the rays with a magnet, he could separate the charge from the rays.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-08-03T04:46:01.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15615" PostTypeId="1" CreationDate="2023-08-04T01:25:24.090" Score="3" ViewCount="112" Body="&lt;p&gt;In physics, quarks come in one of three color states, usually chosen to be called, &amp;quot;red&amp;quot;, &amp;quot;green&amp;quot;, and &amp;quot;blue&amp;quot;. However, because these are just labels, there are other naming conventions that are used. Murray Gell-Mann used &amp;quot;red&amp;quot;, &amp;quot;white&amp;quot;, and &amp;quot;blue&amp;quot; because those are the colors of the French flag. Roger Penrose, in his book &lt;em&gt;The Road to Reality&lt;/em&gt;, denounces this and states that he uses &amp;quot;red&amp;quot;, &amp;quot;yellow&amp;quot;, and &amp;quot;blue&amp;quot;, as they are the primary colors of paint, rather than the more common &amp;quot;red&amp;quot;, &amp;quot;green&amp;quot;, and &amp;quot;blue&amp;quot;, being the primary colors of light. In almost every source I can find, these seem to be the conventions that people typically use, with rgb being by far the most common.&lt;/p&gt;&#xA;&lt;p&gt;However, in &lt;em&gt;Group Theory in a Nutshell for Physicists&lt;/em&gt;, Tony Zee uses &amp;quot;red&amp;quot;, &amp;quot;green&amp;quot;, and &amp;quot;yellow&amp;quot;. This is a convention that I have not seen mentioned anywhere else, but a professor of mine claims to have learned the same notation that Zee uses.&lt;/p&gt;&#xA;&lt;p&gt;I was hoping to learn about the origin of this discrepancy. Obviously, these are just labels, and one can call them whatever they so please. That being said, I'm still interested in what the origin of the &amp;quot;rgy&amp;quot; naming scheme is. Is it a region difference? Is it a generational difference? There seems to be almost no mention of the &amp;quot;rgy&amp;quot; coloring scheme outside of this book, which doesn't even make mention of the fact that there are other conventions, much less that they are more common.&lt;/p&gt;&#xA;" OwnerUserId="18796" LastEditorUserId="18796" LastEditDate="2023-08-04T01:55:37.660" LastActivityDate="2023-08-04T23:27:44.663" Title="What is the origin of the &quot;red, green, yellow&quot; quark color convention?" Tags="|physics|terminology|particle-physics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15616" PostTypeId="2" ParentId="15612" CreationDate="2023-08-04T07:10:47.150" Score="1" Body="&lt;p&gt;Descartes is credited with introducing coordinates to geometry and it is often called analytic geometry compared with the 'synthetic' geometry of Euclid. However, Weyl denounced this as an &amp;quot;act of violence&amp;quot; and was part of a movement to return to a more synthetic form of geometry. As this was pioneered by Euclid, we really ought to say that post-Descartian geometry and so also post-non-Euclidean geometry of Riemann manifolds and so on (but not of the absolute geometry of Bolyai which is already synthetic) is a return to Euclid and so  Euclidean geometry but not in the sense normally used for this term but as a synonym for synthetic geometry.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2023-08-04T07:10:47.150" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15618" PostTypeId="2" ParentId="8173" CreationDate="2023-08-04T19:31:25.260" Score="0" Body="&lt;p&gt;The 1861 paper primarily used differential forms, but in the context of line integrals and area integrals rather than as a Grassmann calculus, though it was around at the time (Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik; i.e. The Theory of Linear Extension, a New Branch of Mathematics, 1844). I qualify that, since there was at least one place, later, in the treatise where the anti-commuting rule &lt;span class=&quot;math-container&quot;&gt;$dx dy = -dy dx$&lt;/span&gt; popped up.&lt;/p&gt;&#xA;&lt;p&gt;The 1864 paper used differential forms less so; the 1873 treatise used a pre-Heavisde form of vector calculus cannnibalized from Hamilton's quaternions ... only sparingly, to present the equations in capsule summary form.&lt;/p&gt;&#xA;&lt;p&gt;You really didn't need to ask, since the treatise is on-line.&lt;/p&gt;&#xA;&lt;p&gt;Volume 1: &lt;a href=&quot;https://archive.org/details/electricandmagne01maxwrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/electricandmagne01maxwrich&lt;/a&gt;&lt;br/&gt;&#xA;Volume 2: &lt;a href=&quot;https://archive.org/details/electricandmagne02maxwrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/electricandmagne02maxwrich&lt;/a&gt;&lt;br/&gt;&lt;/p&gt;&#xA;&lt;p&gt;The equations are in Chapters VIII and IX of volume 2.&lt;/p&gt;&#xA;&lt;p&gt;Here are the quantities and equations posted in the 1864 paper. I will reconcile them against what would best pass for the contemporary 3-vector form of the same today.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;The Quantities&lt;/b&gt;:&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;b&gt;Electromagnetic Momentum&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(F,G,H)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$F dx + G dy + H dz$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (F,G,H)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Magnetic Intensity&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(α,β,γ)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$α dx + β dy + γ dz$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (α,β,γ)/(4π)$&lt;/span&gt;. He's got that extra &lt;span class=&quot;math-container&quot;&gt;$4π$&lt;/span&gt; in all his writings and it clouds his equations and analyses.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Electromotive Force&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(P,Q,R)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$P dx + Q dy + R dz$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (P,Q,R)$&lt;/span&gt;, but only at 0 velocity. Our &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is his 0-velocity &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, while his generic &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; (which we could designate &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt;) would be our &lt;span class=&quot;math-container&quot;&gt;$_M =  + ×$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; denotes the velocity. A distinction is made in the 19th century literature between the &quot;stationary&quot; and &quot;moving&quot; forms of Maxwell's equations - more on that below.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Current Due To True Conduction&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(p,q,r)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p dy dz + q dz dx + r dx dy$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (p,q,r)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Electric Displacement&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(f,g,h)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f dy dz + g dz dx + h dx dy$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (f,g,h)$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Total Current&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(p',q',r')$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$p' dy dz + q' dz dx + r' dx dy$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (p',q',r')$&lt;/span&gt;. This notation is mostly defunct today, though it does pop up in a few places (e.g. A.O. Barut's &quot;Electrodynamics and Classical Theory of Fields and Particles&quot;). This is the current that is used in circuit analysis, not &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Quantity Of Free Electricity&lt;/b&gt;: (per unit volume): &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$e dx dy dz$&lt;/span&gt;. Today: Our &lt;span class=&quot;math-container&quot;&gt;$ρ$&lt;/span&gt; = his &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Electric Potential&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$Ψ$&lt;/span&gt;. A scalar (so as a differential form, it is still just &lt;span class=&quot;math-container&quot;&gt;$Ψ$&lt;/span&gt;). Today: Our &lt;span class=&quot;math-container&quot;&gt;$φ$&lt;/span&gt; = his &lt;span class=&quot;math-container&quot;&gt;$Ψ$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Magnetic Induction&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(μα,μβ,μγ)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$μα dy dz + μβ dz dx + μγ dx dy$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (μα,μβ,μγ)$&lt;/span&gt;. He does not make a clean distinction between Magnetic Induction and Magnetic Intensity in 1861 or 1864, but &lt;i&gt;still&lt;/i&gt; makes an effective distinction, writing the terms as diglyphs as I did here, when the sense Magnetic Induction is intended. This was a &lt;b&gt;huge&lt;/b&gt; mistake! As you can see, the two are completely different types of objects. Because of this confusion, he fails to note that &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is a 2-form. The gap is only partly rectified in the treatise - where a distinction is made. But the failure to distinguish between the two, still leads to an erroneous equation (even in the treatise), which Thomson later had to correct. More on that below.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Coefficient Of Induction&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$μ$&lt;/span&gt;. Today: our &lt;span class=&quot;math-container&quot;&gt;$μ$&lt;/span&gt; is his &lt;span class=&quot;math-container&quot;&gt;$4πμ$&lt;/span&gt;. The extra &lt;span class=&quot;math-container&quot;&gt;$4π$&lt;/span&gt; was moved from his &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; to our &lt;span class=&quot;math-container&quot;&gt;$μ$&lt;/span&gt;. That change post-dates the treatise. I don't know who did it.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(Unnamed electric coefficient)&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ε = 1/k$&lt;/span&gt;. He made the change by the time of the treatise. He said it's analogous to a &quot;spring coefficient&quot;, while &lt;span class=&quot;math-container&quot;&gt;$μ$&lt;/span&gt; is analogous to mass, in the spring force law.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;Conductivity&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$ρ$&lt;/span&gt;. Today: our &lt;span class=&quot;math-container&quot;&gt;$σ$&lt;/span&gt; is his &lt;span class=&quot;math-container&quot;&gt;$-ρ$&lt;/span&gt;. I don't know if the sign flip was intentional or an error, because it's not the only sign error he made.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;The Velocity&lt;/b&gt;: &lt;span class=&quot;math-container&quot;&gt;$(dx/dt, dy/dt, dz/dt)$&lt;/span&gt;. Today: &lt;span class=&quot;math-container&quot;&gt;$ = (dx/dt, dy/dt, dz/dt)$&lt;/span&gt;, mostly defunct. In the 1873 treatise, he used &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; - sometimes even in the same section. The distinction between the &quot;stationary&quot; and &quot;moving&quot; forms of Maxwell's equations hangs entirely on whether &lt;span class=&quot;math-container&quot;&gt;$ = $&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$ ≠ $&lt;/span&gt;. It's just left dangling there in the equations and he doesn't say what it's with respect to. But the context makes it clear: the stationary form applies to the &lt;i&gt;unique&lt;/i&gt; frame of reference in which isotropic constitutive laws exhibit their isotropy - the unique frame in which &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, and &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; are each proportional. In this frame, an electromagnetic disturbance propagates outward with a &lt;i&gt;fixed&lt;/i&gt; center. In all other frames, &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; indicates how fast the center would move. This frame is unique in &lt;i&gt;both&lt;/i&gt; non-relativistic theory and in Relativity in the presence of a medium. It is not unique in Relativity in a vacuum, and in that setting &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is arbitrary and (mostly) superfluous. That was the main point of Einstein's 1905 &quot;On the electrodynamics of moving bodies&quot;. The term &quot;moving&quot; was referring to the &lt;span class=&quot;math-container&quot;&gt;$ ≠ $&lt;/span&gt; condition. His thesis was that (for the vacuum), one could take the stationary form &lt;span class=&quot;math-container&quot;&gt;$ = $&lt;/span&gt; - for all frames (and that the in-vacuum light speed is an absolute speed, not relative at all)! Einstein was motivated partly by the question: what if &lt;span class=&quot;math-container&quot;&gt;$||$&lt;/span&gt; is equal to the electromagnetic wave speed. Ironically, his 1905 paper didn't actually answer that question and (in fact) in the 1908 treatment, by him and Laub, where he returns to the issue of moving media, where a velocity, like &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, returns once again, the question comes back. They failed to notice that. It's still a dangling issue - all the way up to the present time! That's the issue of light-speed media.&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;b&gt;The Equations&lt;/b&gt;:&lt;br/&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;b&gt;(A) Total Currents&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;p' = p + \frac{df}{dt}\\&#xA;q' = q + \frac{dg}{dt}\\&#xA;r' = r + \frac{dh}{dt}&#xA;\end{align}\right) ⇒  =  + \frac{∂}{∂t}$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(B) Magnetic Force&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;μα = \frac{dH}{dy} - \frac{dG}{dz}\\&#xA;μβ = \frac{dF}{dz} - \frac{dH}{dx}\\&#xA;μγ = \frac{dG}{dx} - \frac{dF}{dy}&#xA;\end{align}\right) ⇒  = ∇×$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(C) Electric Currents&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;\frac{dγ}{dy} - \frac{dβ}{dz} = 4πp'\\&#xA;\frac{dα}{dz} - \frac{dγ}{dx} = 4πq'\\&#xA;\frac{dβ}{dx} - \frac{dα}{dy} = 4πr'&#xA;\end{align}\right) ⇒ ∇× = $$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(D) Electromotive Force&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;P = μγ \frac{dy}{dt} - μβ \frac{dz}{dt} - \frac{dF}{dt} - \frac{dΨ}{dx}\\&#xA;Q = μα \frac{dz}{dt} - μγ \frac{dx}{dt} - \frac{dG}{dt} - \frac{dΨ}{dy}\\&#xA;R = μβ \frac{dx}{dt} - μα \frac{dy}{dt} - \frac{dH}{dt} - \frac{dΨ}{dz}&#xA;\end{align}\right) ⇒ _M = × - ∇φ - \frac{∂}{∂t}$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(E) Electric Elasticity&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;P = kf\\&#xA;Q = kg\\&#xA;R = kh&#xA;\end{align}\right) ⇒  = ε _M$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(F) Electric Resistance&lt;/b&gt; (Possible sign error):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;P = -ρp\\&#xA;Q = -ρq\\&#xA;R = -ρr&#xA;\end{align}\right) ⇒  = σ _M$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(G) Free Electricity&lt;/b&gt; (Sign error):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$e + \frac{df}{dx} + \frac{dg}{dx} + \frac{dh}{dx} = 0 ⇒ ρ + ∇· = 0$$&lt;/span&gt; It should be ρ = ∇·.&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(H) Continuity&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{de}{dt} + \frac{dp}{dx} + \frac{dq}{dy} + \frac{dr}{dz} = 0 ⇒ \frac{∂ρ}{∂t} + ∇· = 0.$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;b&gt;(I, for implied) Magnetic Induction&lt;/b&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left(\begin{align}&#xA;μα = μ·α\\&#xA;μβ = μ·β\\&#xA;μγ = μ·γ&#xA;\end{align}\right) ⇒  = μ$$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Maxwell did not make clear distinctions between total and partial derivatives - because he was young and inexperienced; only in his 20's and 30's, dying in his 40's. His failure to make a clean separation between &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; led to his writing down the wrong (implied) equation linking the two. With the correction later added by Thomson, it would read:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ = μ( - ×).$$&lt;/span&gt;&#xA;Only in this way do you get consistency with the Galilean transforms; and his equations were meant to be covariant under Galilean transforms.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; vector is superfluous. Equations (A) and (C) can be combined into (AC):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$∇× =  =  + \frac{∂}{∂t} ⇒ ∇× =  + \frac{∂}{∂t}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The alphabet soup for the contemporary names is Maxwell's doing and came from his treatise, where he used vectors for capsule summaries of the equations, much as I just did, above.&lt;/p&gt;&#xA;&lt;p&gt;The sign error in (G) leads to a cascade of other oversights or errors. First, when it written correctly (as I did above), then it becomes clear that (G), along with (AC) together already yield (H); so that (H) is superfluous. Otherwise, the sign error would make (A), (C), (G) and (H) mutually inconsistent. The failure to note that (H) was superfluous ... and that there is one more field quantity (20) than equations (19) (or 23 versus 22, if you distinguish &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; from &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;) ... leads to a failure to note that there is a partial redundancy in the quantities, by this account. That's the &lt;i&gt;gauge symmetry&lt;/i&gt;, i.e. the invariance of the equations with respect to the following &lt;i&gt;gauge transform&lt;/i&gt;:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$φ → φ + \frac{∂χ}{∂t},  →  - ∇χ.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In fact, Maxwell already &lt;i&gt;partially&lt;/i&gt; acknowledged this by pointing out that &lt;span class=&quot;math-container&quot;&gt;$φ$&lt;/span&gt; was subject to arbitrary constant adjustment. In his 1861 paper and in the treatise, he adopted the &amp;quot;Coulomb gauge&amp;quot;: &lt;span class=&quot;math-container&quot;&gt;$∇· = 0$&lt;/span&gt;; but he could have just as well used the gauge transforms to impose the gauge condition &lt;span class=&quot;math-container&quot;&gt;$φ = 0$&lt;/span&gt;, since he already said it was arbitrary and didn't really use &lt;span class=&quot;math-container&quot;&gt;$φ$&lt;/span&gt; anywhere else.&lt;/p&gt;&#xA;&lt;p&gt;The Coulomb condition is invariant under Galilei transforms, though &lt;span class=&quot;math-container&quot;&gt;$φ = 0$&lt;/span&gt; is not. So, with a Galilei transform, you'd have to regauge &lt;span class=&quot;math-container&quot;&gt;$(φ,)$&lt;/span&gt; to get back &lt;span class=&quot;math-container&quot;&gt;$φ = 0$&lt;/span&gt; in the new frame of reference.&lt;/p&gt;&#xA;&lt;p&gt;The expression for the field &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt; is frame-dependent, though &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt;, itself, is not. It is best decomposed into our &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, which is frame-dependent. But this removes the dangling vector from the field equation. The pair &lt;span class=&quot;math-container&quot;&gt;$(,)$&lt;/span&gt; fit more consistently together than do &lt;span class=&quot;math-container&quot;&gt;$(,_M)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;There are no equations for &lt;span class=&quot;math-container&quot;&gt;$∇· = 0$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$∇× + ∂/∂t = $&lt;/span&gt;, because the first one is already implied by (B), and the prospect for the second one is obscured by the fact that Maxwell was using &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt;, instead of &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;. Both appeared in the 1861 paper, the second being asserted only for the stationary case &lt;span class=&quot;math-container&quot;&gt;$ = $&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt; reduces to &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The failure to make this separation also obscures the frame-dependency in (F) and its inconsistency. The vector &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is frame-dependent, while &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt; is not! (OOPS!) So, actually (F) also selects out a frame of its own, &amp;quot;The Frame Of Resistivity&amp;quot; if you will. Moreover, it need not be the same as the &amp;quot;Frame Of Isotropy&amp;quot;. That's also obscured by the use of &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt; in (F).&lt;/p&gt;&#xA;&lt;p&gt;Einstein and Laub correctly noted (and fixed) the frame dependency of (F), as did Minkowski in all their 1908 treatments of the theory of moving media. But they both failed to note that the frame for (F) doesn't have to be the same as the Frame Of Isotropy. So, it may use a different &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; vector, &lt;span class=&quot;math-container&quot;&gt;$'$&lt;/span&gt;. If it is rewritten to fix the inconsistency, it would have the form:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ - ' ρ = σ ( + '×).$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;So, after making Thomson's correction to the unnamed extra equation, removing the sign error from (G), eliminating the now-redundant (H), removing the largely-superfluous &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; and splitting &lt;span class=&quot;math-container&quot;&gt;$_M$&lt;/span&gt; into &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$×$&lt;/span&gt;, and repairing (F), the result, with a few extra adjustments by moving around terms, is the following system:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;(B): &lt;span class=&quot;math-container&quot;&gt;$ = ∇×$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(D'): &lt;span class=&quot;math-container&quot;&gt;$ = -∇φ - \frac{∂}{∂t}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(AC): &lt;span class=&quot;math-container&quot;&gt;$∇× - \frac{∂}{∂t} = $&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(G'): &lt;span class=&quot;math-container&quot;&gt;$∇· = ρ$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(E): &lt;span class=&quot;math-container&quot;&gt;$ = ε ( + ×)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(I'): &lt;span class=&quot;math-container&quot;&gt;$ = μ ( - ×)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;(F'): &lt;span class=&quot;math-container&quot;&gt;$ - ' ρ = σ ( + '×)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The equations (B), (D'), (AC), (G') are &lt;i&gt;diffeomorphism covariant&lt;/i&gt; and can be expressed entirely in terms of 4D differential forms. They are &amp;quot;paradigm neutral&amp;quot;, since geometry (at the level of differential manifolds) can't tell the difference between relativistic theory and non-relativistic theory, nor (for that matter) even between space-like and time-like dimensions. The equations apply equally well - as is - to both relativity and non-relativistic theory.&lt;/p&gt;&#xA;&lt;p&gt;The equations (E) and (I') are the &lt;i&gt;constitutive laws&lt;/i&gt; and are where &lt;i&gt;all&lt;/i&gt; of the difference between paradigms, and all the dependency on the space-time metric are encapsulated within. They are not only different between relativity and non-relativistic theory, but also between flat space-time and curved space-times. They may be referred to as the &lt;i&gt;Maxwell-Thomson Relations&lt;/i&gt;, while their relativistic forms, which are respectively equivalent to the following:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ + \frac{1}{c^2} × = ε ( + ×)$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ - \frac{1}{c^2} × = μ ( - ×)$$&lt;/span&gt;&#xA;are what Einstein and Laub derived in 1908, as did Minkowski in 1908 in the paper where he introduced his 4D chrono-geometry. They are known as the &lt;i&gt;Maxwell-Minkowski Relations&lt;/i&gt;.&lt;/p&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$εμ = 1/c^2$&lt;/span&gt; (i.e. a vacuum), then the Maxwell-Minkowski relations are equivalent to the &amp;quot;stationary form&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$ = $&lt;/span&gt;, regardless of what &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is ... provided that &lt;span class=&quot;math-container&quot;&gt;$||$&lt;/span&gt; is less than the wave speed &lt;span class=&quot;math-container&quot;&gt;$V = 1/\sqrt{εμ}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$|| &amp;lt; c$&lt;/span&gt;. If not, then things get complicated ... and nobody's ever addressed that issue even up to the present day, to the best of my knowledge.&lt;/p&gt;&#xA;&lt;p&gt;So, &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is &amp;quot;superfluous&amp;quot;, to use Einstein's 1905 characterization, but not entirely. The irony is that the &amp;quot;not entirely&amp;quot; part pertains to the very matter that motivated his research: the case where &amp;quot;you're moving along with the light wave&amp;quot;, e.g. where you're in a medium whose frame of isotropy is moving at the wave speed of the medium. You might call that the &amp;quot;Cherenkov Threshold&amp;quot;. Moreover, the equations restricted to that setting even have a non-trivial vacuum limit. In the limit, there is still a vestige of the &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; vector. Nobody's ever noticed that, as far as I'm aware.&lt;/p&gt;&#xA;&lt;p&gt;Finally, the equation (F') also has a relativistic form, but Einstein &amp;amp; Laub and Minkowski all wrote it with the tacit assumption that &lt;span class=&quot;math-container&quot;&gt;$' = $&lt;/span&gt;: i.e. that The Frame Of Resistivity = The Frame Of Isotropy. That was an oversight. With the vectors kept distinct, the relativistic version would be:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ - ' ρ = σ \frac{ + '× - ''·/c^2}{\sqrt{1-|'|^2/c^2}}.$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13406" LastActivityDate="2023-08-04T19:31:25.260" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15619" PostTypeId="2" ParentId="15615" CreationDate="2023-08-04T23:27:44.663" Score="2" Body="&lt;p&gt;The oldest use I can find of &lt;em&gt;red, green, yellow&lt;/em&gt; quark colours is &lt;a href=&quot;https://doi.org/10.1088/0034-4885/42/8/001&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;The Quark Parton Model, Rep. Prog. Phys. 42, 1285-1335; see page 1309&quot;&gt;Frank Close in 1979&lt;/a&gt;, but that seems to be a one-off since he used &lt;em&gt;red, green, blue&lt;/em&gt; in &lt;a href=&quot;https://cds.cern.ch/record/873150&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;The theory and phenomenology of coloured quark models,Acta Phys. Pol. B 6, 6 (1975) pp.785-821&quot;&gt;1975&lt;/a&gt;, and &lt;em&gt;red, yellow, blue&lt;/em&gt; in &lt;a href=&quot;https://doi.org/10.1007/BFb0107279&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Close, F.E. (1986). Phenomenology of quarks and leptons. In: Engelbrecht, C. (eds) Quarks and Leptons. Lecture Notes in Physics, vol 248. Springer.&quot;&gt;1986&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Before Zee used &lt;em&gt;red, green, yellow&lt;/em&gt; in his 2016 &amp;quot;Group Theory in a Nutshell for Physicists&amp;quot;, Kerson Huang used the same convention in his 1992 book &lt;a href=&quot;https://books.google.ca/books?hl=en&amp;amp;lr=&amp;amp;id=jPEkmI2TLnkC&amp;amp;oi=fnd&amp;amp;pg=PA1&amp;amp;dq=K+Huang+%22red%22+%22yellow%22+%22green%22+quarks&amp;amp;ots=uTap--xwRW&amp;amp;sig=Rc_HqZGlXIbWmVvZiHLZxxtU_RQ#v=onepage&amp;amp;q=%22yellow%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Quarks, Leptons &amp;amp; Gauge Fields&amp;quot;&lt;/a&gt;, and then made the claim in his 2007 book &amp;quot;Fundamental Forces Of Nature: The Story Of Gauge Fields&amp;quot; that &lt;a href=&quot;https://books.google.ca/books?lr=&amp;amp;id=FTzICgAAQBAJ&amp;amp;dq=Fun%C2%ADda%C2%ADmen%C2%ADtal+Forces+of+Na%C2%ADture&amp;amp;q=yellow#v=snippet&amp;amp;q=yellow&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;the conventional (sic) now is red, yellow, and green&amp;quot;&lt;/a&gt;.  This certainly isn't true in the English language physics literature, but since the QCD colour names are essentially arbitrary, I wonder if there might be different conventions in other languages. I have heard that red, yellow, and green are traditionally considered the luckiest colours in China.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-08-04T23:27:44.663" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15620" PostTypeId="1" CreationDate="2023-08-05T13:43:31.600" Score="3" ViewCount="140" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sean_M._Carroll&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sean Carroll&lt;/a&gt;, while speaking about the history of quantum field theory, has said a couple of times (without being completely sure of its factuality) that Richard Feynman was motivated by his trying to solve the cosmological constant problem.&lt;/p&gt;&#xA;&lt;p&gt;Here is an excerpt from a recent &lt;a href=&quot;https://www.preposterousuniverse.com/podcast&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mindscape&lt;/a&gt; episode, &lt;a href=&quot;https://www.preposterousuniverse.com/podcast/2023/07/31/245-solo-the-crisis-in-physics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Crisis in Physics&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;That's the vacuum energy. They're the same thing. Okay, just different languages used to describe the same idea, the difference is that long after Einstein, by the '50s and '60s, people realize there could be different contributions to the cosmological constant. So in fact, Feynman I think I understand this correctly, but it's not a common fact, so maybe I'm hallucinating a little bit. One of Feynman's motivations for inventing Feynman diagrams was to get rid of quantum field theory. Back in the '50s, you could be ambitious like that, right now, quantum field theory works too well, it's a harder ambition to have, but in the 1950s, you could have said, Well, maybe I can get all the benefit of quantum field theory without all the downsides, like the infinites and one of the things that he knew was a problem Feynman was that if you have a field that fills all of empty space, even if there's no particles, even if it's in the lowest energy state in the vacuum state, it can still have energy.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From a previous episode, &lt;a href=&quot;https://www.preposterousuniverse.com/podcast/2023/06/26/241-tim-maudlin-on-locality-hidden-variables-and-quantum-foundations/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Tim Maudlin on Locality, Hidden Variables, and Quantum Foundations&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I don't know if you know this, 'cause I only recently learned it, but Richard Feynman, when he was inventing Feynman diagrams, part of his motivation was to get rid of quantum field theory and replace it with another theory, particles again, and we now think of Feynman diagrams is a tool for understanding quantum field theory, so he changed his mind, but his motivation, I'm told, was the cosmological constant problem.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I could not find any confirmation about this, even if there are plenty of quotes of Feynman on the cosmological constant problem. Was something that Feynman said himself?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-08-05T20:41:15.867" LastActivityDate="2023-08-06T17:41:16.277" Title="Were Feynman diagrams motivated by the cosmological constant problem?" Tags="|physics|cosmology|quantum-field-theory|feynman|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15621" PostTypeId="1" CreationDate="2023-08-05T16:45:57.587" Score="5" ViewCount="166" Body="&lt;p&gt;Since I've not gotten any answers after a bit more than a week, I've now cross-posted to &lt;a href=&quot;https://mathoverflow.net/questions/452715/the-first-topological-illustration-in-a-mathematics-paper&quot;&gt;MathOverFlow&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT 2023-08-15:&lt;/strong&gt; Several commenters here and at MO have asked me to sharpen the original question.  I've tried to do that there, and I've reproduced the changes (hopefully, improvements) here.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Papers and books in geometry have been using illustrations for ages - for example there are figures in &lt;a href=&quot;https://zenodo.org/record/1428448&quot; rel=&quot;nofollow noreferrer&quot;&gt;papyrus rolls&lt;/a&gt; containing bits of Euclid's &lt;em&gt;Elements&lt;/em&gt;.  My question, then, is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are some of the earliest non-trivial illustrations of topological ideas appearing in published mathematical articles?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am particularly interested in tracing the history of illustration in modern &lt;a href=&quot;https://en.wikipedia.org/wiki/Geometric_topology&quot; rel=&quot;nofollow noreferrer&quot;&gt;geometric&lt;/a&gt; (or &amp;quot;low-dimensional&amp;quot;) topology.  So I am mostly looking for early pictures of knots and surfaces (and three-manifolds - I give an example due to Poincaré below).  But all examples are welcome!&lt;/p&gt;&#xA;&lt;p&gt;Examples and non-examples:&lt;/p&gt;&#xA;&lt;ol start=&quot;0&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Non-example: The diagrams in &lt;a href=&quot;https://en.wikipedia.org/wiki/Celtic_knot&quot; rel=&quot;nofollow noreferrer&quot;&gt;editions of Euclid&lt;/a&gt; are always slightly wonky - drawing the Platonic ideal circle is impossible!  So they are &lt;em&gt;topological&lt;/em&gt; illustrations of &lt;em&gt;geometric&lt;/em&gt; ideas.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Example: The Heegaard diagram [Figure 4] in the &lt;a href=&quot;https://zenodo.org/record/1428448&quot; rel=&quot;nofollow noreferrer&quot;&gt;fifth supplement to &lt;em&gt;Analysis Situs&lt;/em&gt;&lt;/a&gt; [Poincaré, 1904]. It is not clear how Poincaré found this example and it is not clear how else he might have communicated the information.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Non-examples: The various carved &lt;a href=&quot;https://en.wikipedia.org/wiki/Celtic_knot&quot; rel=&quot;nofollow noreferrer&quot;&gt;Celtic knots&lt;/a&gt;, or braids appearing as the &lt;a href=&quot;https://www.getty.edu/publications/romanmosaics/catalogue/5/index.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;frames of mosaics&lt;/a&gt;, or the coat of arms of the &lt;a href=&quot;https://en.wikipedia.org/wiki/House_of_Borromeo#/media/File:Coat_of_arms_of_the_house_of_Borromeo.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;House of Borromeo&lt;/a&gt;, or &lt;a href=&quot;https://en.wikipedia.org/wiki/Chain_mail&quot; rel=&quot;nofollow noreferrer&quot;&gt;metal links&lt;/a&gt; made of many unknots, or indeed &lt;a href=&quot;https://en.wikipedia.org/wiki/Textile_arts&quot; rel=&quot;nofollow noreferrer&quot;&gt;textile arts&lt;/a&gt; (much older than publishing and, indeed, writing).  Of course such artefacts are important, mathematical, and ancient.  But it is impossible to point at one of them and say &amp;quot;this was the first one&amp;quot;.  This is why (perhaps wrongly!) I have restricted to &amp;quot;published&amp;quot; papers.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Non-example: &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler's diagram&lt;/a&gt; [1735] of the bridges of Konigsberg does not count.  This is because (I feel that) (a) it is a bit too close to a literal (as opposed to topological) figure and (b) graph theory is more properly a subfield of combinatorics, rather than of topology.  (Of course, some people (such as Euler) disagree with (b).)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Example: &lt;a href=&quot;https://archive.org/details/vorstudienzurto00listgoog/page/n67/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Listing's figures of knots&lt;/a&gt; [1848] in &lt;em&gt;Vorstudien zur Topologie&lt;/em&gt;.  The classic example, inspiring Tait's work on knot tabulation.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="16841" LastEditorUserId="16841" LastEditDate="2023-08-15T09:03:14.087" LastActivityDate="2023-08-15T09:03:14.087" Title="Early illustrations of topological notions in published work" Tags="|reference-request|topology|publishing|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15622" PostTypeId="1" CreationDate="2023-08-05T21:06:14.987" Score="2" ViewCount="354" Body="&lt;p&gt;Following on this &lt;a href=&quot;https://hsm.stackexchange.com/questions/15620/were-feynman-diagrams-motivated-by-the-cosmological-constant-problem&quot;&gt;Were Feynman diagrams motivated by the cosmological constant problem?&lt;/a&gt; and this &lt;a href=&quot;https://hsm.stackexchange.com/questions/15532/who-was-the-first-to-estimate-the-vacuum-energy-discrepancy-by-120-orders-of-mag&quot;&gt;Who was the first to estimate the vacuum energy discrepancy by 120 orders of magnitude?&lt;/a&gt; I found a recurring phrase in popular sources saying something like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;John Wheeler and Richard Feynman, famously predicted the value of the energy density of vacuum energy to be an astronomically huge number, &lt;span class=&quot;math-container&quot;&gt;$10^{112}$&lt;/span&gt; ergs/cm&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;. This value is so massive that Feynman and Wheeler said it would take only a teacup of this type of energy to boil the Earth’s oceans&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: Ellie Gabel &lt;a href=&quot;https://revolutionized.com/what-is-vacuum-energy-really/&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is vacuum energy really&lt;/a&gt; 2022.&lt;/p&gt;&#xA;&lt;p&gt;Note that (as pointed in the comments) the source above even provides a ridiculously high value for the vacuum energy density.&lt;/p&gt;&#xA;&lt;p&gt;After looking for this, the closest I found was the lead of the Wikipedia article for &lt;a href=&quot;https://en.wikipedia.org/wiki/Zero-point_energy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zero-point energy&lt;/a&gt;, which sources points to a The Guardian article &lt;a href=&quot;https://www.theguardian.com/education/2003/jul/17/research.highereducation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mark Pilkington, Zero point energy, 2003&lt;/a&gt;, quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wheeler and Richard Feynman calculated that there is enough such energy in the vacuum inside a single light bulb to boil all the world's oceans.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Looking a bit further I found that sci-fi author Arthur C. Clarke wrote in his book &lt;em&gt;The songs of a distant Earth&lt;/em&gt; (1986):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An admittedly naive calculation by Richard Feynman suggests that every cubic centimetre of vacuum contains enough energy to boil all the oceans of Earth. Another estimate by John Wheeler gives a value a mere seventy-nine orders of magnitude larger. When two of the world’s greatest physicists disagree by a little matter of seventy-nine zeros, the rest of us may be excused a certain scepticism; but it’s at least an interesting thought that the vacuum inside an ordinary light bulb contains enough energy to destroy the galaxy... and perhaps, with a little extra effort, the cosmos.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is there a pre-1986 article by Wheeler and/or Feynman on zero point energy or on the cosmological problem? I unable to find it and this phrasing only appears in popular sources. Maybe it comes from some popular material by Feynman? or was this fabricated by Clarke?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-12-13T16:08:59.983" LastActivityDate="2023-12-13T16:08:59.983" Title="Did Wheeler and Feynman said that vacuum zero point energy could boil the oceans?" Tags="|physics|cosmology|quantum-field-theory|feynman|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15623" PostTypeId="2" ParentId="15620" CreationDate="2023-08-06T16:23:02.747" Score="1" Body="&lt;p&gt;The first Feynman diagram in print appeared in Fig 1 , p 772 of &lt;a href=&quot;https://doi.org/10.1103/PhysRev.76.769&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Space-Time Approach to Quantum Electrodynamics&amp;quot;, R P Feynman,&#xA;&lt;em&gt;Phys Rev&lt;/em&gt;  &lt;strong&gt;76&lt;/strong&gt; (1949) 769 – 789&lt;/a&gt;,&#xA;published 15 September 1949, &lt;em&gt;&lt;strong&gt;not&lt;/strong&gt;&lt;/em&gt; a QFT paper, and without direct reference to the cosmological constant.&lt;/p&gt;&#xA;&lt;p&gt;It was F Dyson, subsequently that same year, who connected  Feynman diagrams to the QFT we know and use today.&lt;/p&gt;&#xA;&lt;p&gt;I'm not sure I understand the quotes and the question to answer it, but even in his work on gravitation, e.g. his 1960s lectures on it,  RPF did not appear to be intrigued by the cosmological constant problem as appreciated today. S Weinberg focussed the community on it decades later.&lt;/p&gt;&#xA;&lt;p&gt;Of course, Feynman being the visual type (I was his TA in 1976), used all types of Feynman-like diagrams in his lectures before or after that; he called them &amp;quot;funny diagrams&amp;quot;, and had a special type of harsh derision for experimentalists starting their talks with them, as a special type of voodoo-incantation... A nice discussion of their spread is in &lt;a href=&quot;https://doi.org/10.1016/j.shpsb.2012.04.001&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ari Gross, &amp;quot;Pictures and pedagogy: The role of diagrams in Feynman's early lectures&amp;quot;, &lt;em&gt;Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics&lt;/em&gt;, &lt;strong&gt;43&lt;/strong&gt;  (3) (2012) pp 184-194&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="7076" LastEditorUserId="7076" LastEditDate="2023-08-06T17:41:16.277" LastActivityDate="2023-08-06T17:41:16.277" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15624" PostTypeId="1" AcceptedAnswerId="15625" CreationDate="2023-08-03T12:13:56.877" Score="7" ViewCount="224" Body="&lt;p&gt;What did Euclid originally call his treatise of thirteen books that we now refer to as &lt;em&gt;&amp;quot;Euclid's Elements&amp;quot;&lt;/em&gt; ?&lt;/p&gt;&#xA;&lt;p&gt;Was it &amp;quot;The Elements&amp;quot; ? Was it something else ? Does anyone know the exact Greek title ?&lt;/p&gt;&#xA;" OwnerUserId="18820" OwnerDisplayName="J. Jaromir" LastActivityDate="2023-08-08T15:07:55.337" Title="The Original Title of &quot;Euclid's Elements&quot;" Tags="|reference-request|euclidean-geometry|" AnswerCount="1" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15625" PostTypeId="2" ParentId="15624" CreationDate="2023-08-03T12:27:25.553" Score="10" Body="&lt;p&gt;It is &amp;quot;Στοιχεῖα&amp;quot; [ Stoikheîa ] in Ancient Greek.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_Elements&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Euclid%27s_Elements&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Euclid's Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Interesting titbit/trivia :&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I recognized a word from my Chemistry Classes , hence I had a guess about Stoichiometry , which turned out to be true . . . .&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Chemical_Reactions/Stoichiometry_and_Balancing_Reactions&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Supplemental_Modules_and_Websites_(Inorganic_Chemistry)/Chemical_Reactions/Stoichiometry_and_Balancing_Reactions&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In Greek, &lt;strong&gt;stoikhein&lt;/strong&gt; means &lt;strong&gt;element&lt;/strong&gt; and metron means measure, so stoichiometry literally translated means the measure of elements.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[[ that Chemistry text may have a typo there : &amp;quot;singular of stoicheīa is stoicheīon, not stoicheīn&amp;quot; , curtesy of user &amp;quot;BakerStreet&amp;quot; ]]&lt;/p&gt;&#xA;&lt;p&gt;When we make the Singular to the Plural , we get Stoikheîa , which is what Euclid wrote.&lt;/p&gt;&#xA;" OwnerUserId="18814" OwnerDisplayName="Prem" LastEditorUserId="18814" LastEditDate="2023-08-08T15:07:55.337" LastActivityDate="2023-08-08T15:07:55.337" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15626" PostTypeId="1" CreationDate="2023-08-07T23:38:50.983" Score="1" ViewCount="82" Body="&lt;p&gt;In &lt;a href=&quot;https://plato.stanford.edu/entries/continuity/#ContInfiSeveEighCent&quot; rel=&quot;nofollow noreferrer&quot;&gt;§4&lt;/a&gt; of the Stanford Encyclopedia of Philosophy article on continuity and infinitesimals, the author (John L. Bell) mentions that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... Johann Bernoulli (1667–1748) [in a] letter of his to Leibniz written in 1698 contains the forthright assertion that “inasmuch as the number of terms in nature is infinite, the infinitesimal exists &lt;em&gt;ipso facto&lt;/em&gt;” (Boyer 1939 [1959: 239], quoting Leibniz, &lt;em&gt;Mathematische Schriften&lt;/em&gt;, III (Part 2), 555). One of his arguments for the existence of actual infinitesimals begins with the positing of the infinite sequence 1/2, 1/3, 1/4,…. If there are ten terms, one tenth exists; if a hundred, then a hundredth exists, etc.; and so if, as postulated, the number of terms is infinite, then the infinitesimal exists.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Shortly below that he notes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... Nieuwentijdt’s infinitesimals have the property that the product of any pair of them vanishes; in particular each infinitesimal is &lt;em&gt;nilsquare&lt;/em&gt;, i.e., its square and all higher powers are zero (see Mancosu 1996: 159). ... It is of interest to note that Leibnizian infinitesimals (differentials) are realized in nonstandard analysis, and nilsquare infinitesimals in smooth infinitesimal analysis (for both types of analysis see below). In fact it has been shown to be possible to combine the two approaches, so creating an analytic framework realizing both Leibniz’s and Nieuwentijdt’s conceptions of infinitesimal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Following that is a short discussion of Euler's quasi-theory that (hypothetical) evaluations of &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; are the background sense we attach to infinitesimal semantics (so to speak), or more precisely, &amp;quot;Euler treated infinitesimals as formal zeros...&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;I get the impression that Nieuwentijdt was inspired, to some extent (maybe subconsciously), by the (at the time) sort-of-recent emergence of imaginary numbers as (socially) acceptable mathematical objects. At least, it would not be hard to see how the algebraism involved in the bare acceptance of the imaginary unit can be reflected in the assertion of nilpotent infinitesimals (they, like &lt;em&gt;i&lt;/em&gt;, are arbitrary nonreal-number solutions to a specific equation template, and both involve, initially anyway, squaring). However, did any mathematician try out a definition like:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall n\exists\epsilon(\frac{\epsilon}{n} = \epsilon, \epsilon \neq 0, n \neq 0)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;... albeit without, of course, the modern quantifiers in the historical notation?&lt;/p&gt;&#xA;&lt;p&gt;I ask this in part because it was discerned much later (this is from &lt;a href=&quot;https://math.stackexchange.com/questions/146844/how-to-divide-aleph-numbers&quot;&gt;over on the MathSE&lt;/a&gt;) that &lt;span class=&quot;math-container&quot;&gt;$\frac{\aleph_1}{\aleph_0} = \aleph_1$&lt;/span&gt;, and generally when a transfinite cardinal numerator &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; exceeds the denominator &lt;span class=&quot;math-container&quot;&gt;$\kappa$&lt;/span&gt;, the lower cardinal is &amp;quot;absorbed&amp;quot; (like with algebraic absorbing elements, even) into the upper one. Moreover, &lt;span class=&quot;math-container&quot;&gt;$\frac{\lambda}{\lambda}$&lt;/span&gt; is indeterminate &amp;quot;just like&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$0/0$&lt;/span&gt; is (with the exception of &lt;span class=&quot;math-container&quot;&gt;$0 \times \lambda = 0$&lt;/span&gt;). So transfinite numbers would have this idquotient(?) feature &lt;span class=&quot;math-container&quot;&gt;$\frac{\lambda}{n} = \lambda, \lambda \neq 0$&lt;/span&gt;, and their conceptual inverses (not their operational inverses, at least not &amp;quot;at first&amp;quot;) would then be the kind of infinitesimals triangulated between Bernoulli, Nieuwentijdt, and Euler's definitions/characterizations.&lt;/p&gt;&#xA;&lt;p&gt;It seems like a natural enough idea to have tried out, and on my end of things, I look at nilsquare infinitesimals as implying that adding &lt;span class=&quot;math-container&quot;&gt;$\epsilon + \epsilon$&lt;/span&gt; together &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-many times zeroes the infinitesimal out, which would make it both positive and negative somehow (like zero, granted), especially in the act of its own addition, or then adding it to itself is equivalent to self-subtraction in some manner, and that hardly seems less difficult to imagine and make sense of than idquotient(?) infinitesimals would be. I don't know that a system of such things would have worked out, of course, even had it been tried, but again, in smooth infinitesimal analysis (SIA), they introduce nilpotent infinitesimals as suspensions of not only general commensurability but outright transgressions of the Law of the Excluded Middle—so if there had been any mathematician, in those days, who tried out the idquotient algebraism instead, maybe I should expect that an attempt would have been made nowadays to recapitulate such a definition.&lt;/p&gt;&#xA;&lt;p&gt;Or, again, was it tried out and never made workable in the way reciprocal infinitesimals have been provided for via hyperreals and surreals, or nilpotent ones modulo SIA?&lt;/p&gt;&#xA;" OwnerUserId="18733" LastActivityDate="2023-08-08T10:17:33.657" Title="Did any mathematicians of the time (the 17th Century) try out an intermediary between Bernoulli's and Nieuwentijdt’s infinitesimals?" Tags="|calculus|social-context|euler|leibniz|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15627" PostTypeId="1" AcceptedAnswerId="15629" CreationDate="2023-08-08T00:50:07.177" Score="4" ViewCount="272" Body="&lt;p&gt;At the beginning of a calculus course, we encounter two famous limits. They are&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to0}\frac{\sin x}{x}\qquad\text{and}\qquad\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n.$$&lt;/span&gt;&#xA;I'm not sure if Cauchy was the first person in history to calculate the limit &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{x\to0}\dfrac{\sin x}{x}$&lt;/span&gt;. On page 64 of his famous book: &lt;em&gt;Cours d'Analysis&lt;/em&gt;, he calculated this limit. I want to know who was the first person to calculate the limit &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="18815" LastEditorUserId="6609" LastEditDate="2023-08-08T07:56:37.120" LastActivityDate="2023-08-10T04:24:20.927" Title="Who was the first person in history to calculate the limit $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{n}\right)^n$?" Tags="|calculus|differential-equations|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15628" PostTypeId="2" ParentId="15626" CreationDate="2023-08-08T09:56:30.903" Score="2" Body="&lt;p&gt;These are assorted comments too long for a &lt;em&gt;comment&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Boyer is unreliable and should not be used as a source.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Bernoulli's &amp;quot;proof&amp;quot; of the existence of infinitesimals was rejected by Leibniz, who always (in his mature period starting around 1676) viewed infinitesimals as &lt;em&gt;fictional&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;It may not be appropriate to describe Euler's contribution as &amp;quot;quasi-theory [of] (hypothetical) evaluations&amp;quot;.  Euler's approach was more coherent than this description tends to imply; see &lt;a href=&quot;http://doi.org/10.1007/s10838-016-9334-z&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt; in &lt;em&gt;Journal for General Philosophy of Science&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cardinal arithmetic does not have good algebraic properties and therefore is not suitable for doing infinitesimal analysis, for the very reason that you point out.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Cardinal arithmetic was not developed until infinitesimal analysis was already a full-fledged field.  In particular, it was not available as a tool to the classical authors such as Leibniz and Cauchy.  One of the reasons is that they rejected &amp;quot;infinite wholes&amp;quot;, i.e., actual infinity.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-08-08T10:17:33.657" LastActivityDate="2023-08-08T10:17:33.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15629" PostTypeId="2" ParentId="15627" CreationDate="2023-08-08T12:11:14.973" Score="5" Body="&lt;p&gt;First of all, what does it exactly mean to &amp;quot;calculate&amp;quot; this limit? The limit equals to &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, but this is the definition of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;. So one only has to prove that the limit exists, and then denote it by some letter. The fact that this limit exists was known to Napier, who calculated it with at least 12 significant digits. (Though Napier had neither a definition of a limit nor definition of a real number). The formal definition of the number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is due to Euler, and it was called the &amp;quot;Euler constant&amp;quot; for this reason.&lt;/p&gt;&#xA;&lt;p&gt;See also &lt;a href=&quot;https://hsm.stackexchange.com/questions/42/which-came-first-the-natural-logarithm-or-the-base-of-the-natural-logarith&quot;&gt;Which came first, the natural logarithm or the base of the natural logarithm?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-08-09T12:25:09.697" LastActivityDate="2023-08-09T12:25:09.697" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15630" PostTypeId="2" ParentId="14672" CreationDate="2023-08-08T18:20:35.873" Score="2" Body="&lt;p&gt;Sir James Jeans. He entertained house guests with informal concerts on two pianos (Jeans and his wife). I would especially recommend his book: 'Science and Music'. Might be for free on the internet archive.&lt;/p&gt;&#xA;" OwnerUserId="18821" LastActivityDate="2023-08-08T18:20:35.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15631" PostTypeId="2" ParentId="14672" CreationDate="2023-08-08T21:30:51.773" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Johannes_Kepler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Johannes Kepler&lt;/a&gt;.  I figured he'd be an honorable mention since music made a significant impact on his theories.&lt;/p&gt;&#xA;&lt;p&gt;Kepler was lightly trained in music but he never thought of himself as skilled in composition or performance.  He often associated with musicians in court and wrote about music theory and various tuning systems.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Though he was engaged in reviving the ancient vision of cosmic&#xA;harmony, Kepler’s awareness of contemporary music informed crucial&#xA;departures from the ancients. To begin with fundamental issues of&#xA;interval construction, Kepler parted company with the Pythagorean&#xA;tuning as transmitted by Boethius and Macrobius, who defined the major&#xA;third as 81:64 and the minor third as 32:27. Instead, Kepler advocated&#xA;just intonation (5:4 and 6:5, for the major and minor thirds,&#xA;respectively). In justification, Kepler asserts that the Pythagoreans&#xA;“were so much given over to this form of philosophizing through&#xA;numbers that they did not even stand by the judgment of their ears,&#xA;though it was by their evidence that they had originally gained entry&#xA;to philosophy; but they marked out what was melodic and what was&#xA;unmelodic, what was consonant and what was dissonant, from their&#xA;numbers alone, doing violence to the natural prompting of hearing.”&#xA;Though Kepler praises Ptolemy for including just intonation among his&#xA;tunings, Kepler considers that Ptolemy still erred by denying that&#xA;thirds and sixths were consonances, so that “the man who restored the&#xA;judgment of the ears to its rightful place in words and doctrine&#xA;nevertheless deserted it again.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://www.sscm-jscm.org/v11/no1/pesic.html#:%7E:text=his%20scientific%20views.-,2.,in%20school%20and%20in%20church.&amp;amp;text=Beginning%20in%20his%20fifth%20year,later%20cites%20in%20Harmonices%20mundi.&quot; rel=&quot;nofollow noreferrer&quot;&gt;Earthly Music and Cosmic Harmony: Johannes Kepler’s Interest in Practical Music, Especially Orlando di Lasso&lt;/a&gt;&lt;/em&gt;, Peter Pesic.&lt;/p&gt;&#xA;&lt;p&gt;Kepler applied his studies in music theory to describe celestial motion in his &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Harmonices_Mundi&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harmonices Mundi&lt;/a&gt;&lt;/em&gt; (The Harmony of the World) and described their motion through &lt;a href=&quot;http://microcosmos.uchicago.edu/microcosmos_new/kepler/harmonicesmundi/music.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;sheet music notation&lt;/a&gt;. In the final section of the work, he discusses his discovery of the &amp;quot;third law of planetary motion&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2023-08-08T21:30:51.773" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15632" PostTypeId="2" ParentId="15627" CreationDate="2023-08-10T03:50:04.667" Score="5" Body="&lt;p&gt;I followed up on a pointer given by @nwr in comments that Jacob Bernoulli first encountered &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; in the context of computing continuous compound interest. Sure enough, the article &amp;quot;The number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&amp;quot; at &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/e/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor&lt;/a&gt; states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Perhaps surprisingly, since this work on logarithms had come so close to recognising the number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, when &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; is first &amp;quot;discovered&amp;quot; it is not through the notion of logarithm at all but rather through a study of compound interest. In 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of &lt;span class=&quot;math-container&quot;&gt;$\left(1+\frac{1}{n}\right)^{n}$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; tends to infinity. He used the binomial theorem to show that the limit had to lie between &lt;span class=&quot;math-container&quot;&gt;$2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt; so we could consider this to be the first approximation found to &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The article cites Eli Maor, &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; : the story of a number&lt;/em&gt;, Princeton: Princeton University Press 1994 and J. L. Coolidge, &amp;quot;The number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;.&amp;quot; &lt;em&gt;The American Mathematical Monthly&lt;/em&gt;, Vol. 57, No. 9, Nov. 1950, pp. 591-602 (&lt;a href=&quot;https://www-fourier.ujf-grenoble.fr/%7Emarin/une_autre_crypto/articles_et_extraits_livres/irationalite/Coolidge_J.L._the_number_e.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;), but neither publication cites a relevant publication by Bernoulli. Since there are other publications that mention work from 1683 on compound interest by Jacob Bernoulli when recounting the history of &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;, I spent a couple of hours trying to find that, but to no avail. I am now convinced that MacTutor is in error. At long last, I consulted M. Cantor's work on the history of mathematics:&lt;/p&gt;&#xA;&lt;p&gt;Moritz Cantor, &lt;em&gt;Vorlesungen über Geschichte der Mathematik, Vol. 3.&lt;/em&gt; Leizig: B.G. Teubner 1898.&lt;/p&gt;&#xA;&lt;p&gt;Starting at &lt;a href=&quot;https://www.google.com/books/edition/Vorlesungen_%C3%BCber_Geschichte_der_Mathema/FNcYAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA51&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 51&lt;/a&gt; and continuing onto page 52, Cantor describes Jacob Bernoulli's computation of continuous compound interest and provides the relevant source from his collected works:&lt;/p&gt;&#xA;&lt;p&gt;N&lt;sup&gt;o&lt;/sup&gt;. XL. Jacobi Bernoulli, &amp;quot;Quæstiones Nonnullaæ de Usuris,&#xA;Cum solutione Problematis de sorte Aleatorum, propositi in &lt;em&gt;Ephemerid. Gallic.&lt;/em&gt; A. 1685. Art. 25.&amp;quot; &lt;em&gt;Jacobi Bernoulli, Basileensis, Opera. Vol. 1&lt;/em&gt;, Geneva: Cramer &amp;amp; Philibert 1744, pp. 427-431 &lt;a href=&quot;https://www.google.com/books/edition/Jacobi_Bernoulli_Opera/bBltbKG8LzcC?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA427&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;(scan)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The original publication reproduced in &lt;em&gt;Opera&lt;/em&gt; is from 1690 and gives a solution for a problem posed in 1685:&lt;/p&gt;&#xA;&lt;p&gt;J. B., &amp;quot;Quæstiones nonnullæ de usuris, cum solutione Problematis de Sorte Alearum, propositi in &lt;em&gt;Ephem. Gall.&lt;/em&gt; A. 1685, artic. 25.&amp;quot;, &lt;em&gt;Acta Eruditorum Lipsiæ&lt;/em&gt;, May 1690, pp. 219-223 (&lt;a href=&quot;https://www.google.com/books/edition/Acta_eruditorum/nVcg2IyNRsAC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA219&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;The statement relevant to the asker's question appears on page 222:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;si &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt;, debebitur plusquam &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{2}a$&lt;/span&gt;, &amp;amp; minus quam &lt;span class=&quot;math-container&quot;&gt;$3a$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which is equivalent to the statement that &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{2}&amp;lt;e&amp;lt;3 $&lt;/span&gt;. These values derive in straightforward manner from a series expansion given earlier on the same page,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$a + b + \frac{b^2}{2a} + \frac{b^3}{2\cdot3a^2}+ \ldots = a\left(1 +\frac{b}{a} +\frac{1}{1\cdot 2}\left(\frac{b}{a}\right)^2 + \frac{1}{1\cdot 2\cdot 3}\left(\frac{b}{a}\right)^3 + \ldots\right)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;so when &lt;span class=&quot;math-container&quot;&gt;$a=b$&lt;/span&gt; as specified by the quoted text the first three terms of the expression in parentheses immediately provide the stated lower bound of &lt;span class=&quot;math-container&quot;&gt;$2\frac{1}{2}$&lt;/span&gt;. It is interesting to note that Bernoulli saw a use for the series in geometry, in particular as it relates to &lt;em&gt;logarithmic curves&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-08-10T04:24:20.927" LastActivityDate="2023-08-10T04:24:20.927" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15633" PostTypeId="1" CreationDate="2023-08-10T10:19:38.810" Score="1" ViewCount="41" Body="&lt;p&gt;My question about median of a triangle.&lt;/p&gt;&#xA;&lt;p&gt;The English equivalent of the Turkish word &amp;quot;kenarortay&amp;quot; is &amp;quot;median&amp;quot;. In English-language geometry sources (like books or web pages), the length of the median is often denoted with expressions like &lt;span class=&quot;math-container&quot;&gt;$m_a$&lt;/span&gt;. On the other hand, the notation &lt;span class=&quot;math-container&quot;&gt;$V_a$&lt;/span&gt; is also common in Turkish textbooks. We don't use &lt;span class=&quot;math-container&quot;&gt;$K_a$&lt;/span&gt;. This might be due to Turkish mathematicians learning geometry from other mathematicians who speak languages other than English. For instance, we know that German scientists came to Turkey in the &lt;span class=&quot;math-container&quot;&gt;$1930$&lt;/span&gt;s. However, I'm not sure if this notation was used by the Germans. Do you have any information about the origin of the &lt;span class=&quot;math-container&quot;&gt;$V_a$&lt;/span&gt; notation?&lt;/p&gt;&#xA;&lt;p&gt;Of course, mathematical notations can develop independently in local education systems or at a particular university or school. Do you know of any other countries that use the &lt;span class=&quot;math-container&quot;&gt;$V_a$&lt;/span&gt; notation?&lt;/p&gt;&#xA;&lt;p&gt;Thanks for your interest!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Notes:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Some examples&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1.&lt;/strong&gt; In wikipedia Median (geometry) page &lt;a href=&quot;https://en.wikipedia.org/wiki/Median_(geometry)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Median_(geometry)&lt;/a&gt; we see that &lt;span class=&quot;math-container&quot;&gt;$m_a$&lt;/span&gt; notation like &lt;span class=&quot;math-container&quot;&gt;$m_a = \sqrt{\dfrac{2b^2 + 2c^2 - a^2}{4}}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2.&lt;/strong&gt; In Mark Dabbs' &lt;em&gt;Elementary Triangle Geometry&lt;/em&gt; book, we can see &lt;span class=&quot;math-container&quot;&gt;$m_A,m_B,m_C$&lt;/span&gt; notations&#xA;&lt;a href=&quot;https://i.stack.imgur.com/oGdbm.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oGdbm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17698" LastEditorUserId="17698" LastEditDate="2023-08-11T11:23:41.703" LastActivityDate="2023-08-11T11:23:41.703" Title="Origin of $V_a$ (median) notation" Tags="|terminology|notation|geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15634" PostTypeId="1" AcceptedAnswerId="15646" CreationDate="2023-08-10T13:36:30.387" Score="4" ViewCount="115" Body="&lt;p&gt;I have a question about Teiji Takagi's original motivation for his groundbreaking reformulation of &lt;strong&gt;class field theory&lt;/strong&gt; implementing the &lt;strong&gt;ideal norms&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$N_{\mathfrak{m}}(L/K)$&lt;/span&gt; as new &amp;quot;key players&amp;quot; in his new formulation, This approach paves more or less the way for Artin's modern formulation of CFT adapting this approach using norms, even giving structure of certain profinite-topology determined by these.&lt;/p&gt;&#xA;&lt;p&gt;Let's recapitulate shortly what happened roughly before Takagi came up with his groundbreaking reformulation of CFT in terms of norms (for details see &lt;a href=&quot;https://kconrad.math.uconn.edu/blurbs/gradnumthy/cfthistory.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keith Conrad's Notes &lt;/a&gt; on history of class field theory:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In his 1891 book on elliptic functions and algebraic numbers, Heinrich Weber introduced the label “class field” studying Abelian field extensions &lt;span class=&quot;math-container&quot;&gt;$L/K$&lt;/span&gt; in terms of certain group, the ideal class group &lt;span class=&quot;math-container&quot;&gt;$I$&lt;/span&gt;, so at first a class field was just a particular abelian (conjecturally unramified) extension of an imaginary quadratic field, whose Galois group is isomorphic to the ideal class group of the base field. In 1897, Weber extended the concept of ideal class group: for any number field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; and nonzero ideal &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{m}$&lt;/span&gt; in  &lt;span class=&quot;math-container&quot;&gt;$\mathcal{O}_K$&lt;/span&gt;, let &lt;span class=&quot;math-container&quot;&gt;$I_{\mathfrak{m}}$&lt;/span&gt; be the group of fractional ideals in &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; relatively prime to modulus &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{m}$&lt;/span&gt; and let &lt;span class=&quot;math-container&quot;&gt;$P^+_{\mathfrak{m}}$&lt;/span&gt; be the group of principal fractional ideals &lt;span class=&quot;math-container&quot;&gt;$(\alpha/\beta)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and  &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; are nonzero integers in &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; subjected to certain additional conditions  (see above linked source for details)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any intermediate group&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ P^+_{\mathfrak{m}} \subset H \subset I_{\mathfrak{m}}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;is called an ideal group with modulus &lt;span class=&quot;math-container&quot;&gt;$\mathfrak{m}$&lt;/span&gt; and the quotient &lt;span class=&quot;math-container&quot;&gt;$I_{\mathfrak{m}}/H $&lt;/span&gt; is called a generalized ideal class group.&lt;/p&gt;&#xA;&lt;p&gt;Takagi introduced the intermediate group &lt;span class=&quot;math-container&quot;&gt;$H_{\mathfrak{m}}(L/K):= P_{\mathfrak{m}} N_{\mathfrak{m}}(L/K)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$P_{\mathfrak{m}} $&lt;/span&gt; is slight modification of &lt;span class=&quot;math-container&quot;&gt;$P^+_{\mathfrak{m}}$&lt;/span&gt; and&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$  N_{\mathfrak{m}}(L/K):= \{\mathfrak{a} \in I_{\mathfrak{m}} \ | \ \mathfrak{a} =N(\mathfrak{A}) \text{ for some fractional ideal in } L \text{ coprime to } \mathfrak{m} \}  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Obviously Takagi's key motivation was to find a kind of &amp;quot;canonical&amp;quot; group lying between &lt;span class=&quot;math-container&quot;&gt;$ P^+_{\mathfrak{m}} $&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$I_{\mathfrak{m}}$&lt;/span&gt;, therefore the factor &lt;span class=&quot;math-container&quot;&gt;$P_{\mathfrak{m}} $&lt;/span&gt; ( up to the &lt;span class=&quot;math-container&quot;&gt;$+$&lt;/span&gt;-sign issue which is about adding infinite places and I would like to sweep this under the rug in this discussion) in  &lt;span class=&quot;math-container&quot;&gt;$H_{\mathfrak{m}}(L/K)$&lt;/span&gt; seems to be &amp;quot;reasonable&amp;quot;. But I not see why Takagi decided to implement the norm factor &lt;span class=&quot;math-container&quot;&gt;$ N_{\mathfrak{m}}(L/K)$&lt;/span&gt; in his definition.&lt;/p&gt;&#xA;&lt;p&gt;It seems to &amp;quot;be fallen from heaven&amp;quot; without any conceptional idea in background. Just, like a clever guess, and my question is simply if there exist a concrete reason for this choice. (Note that in this time there was no theory of adeles and their profinite topology defined by norm groups known).&lt;/p&gt;&#xA;&lt;p&gt;ADDENDUM: &lt;a href=&quot;https://www.google.de/url?sa=t&amp;amp;source=web&amp;amp;rct=j&amp;amp;opi=89978449&amp;amp;url=https://www.kurims.kyoto-u.ac.jp/%7Ekenkyubu/bessatsu/open/B25/pdf/B25_008.pdf&amp;amp;ved=2ahUKEwjJnoWV-9GAAxVbov0HHc_4CFsQFnoECB4QAQ&amp;amp;usg=AOvVaw294LzgGjnCLjBrRTK43o69&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here &lt;/a&gt; I found historical notes on Takagi's works where is briefly remarked that Takagi was highly influenced by Hilbert's works on unramified class field theory where he gave a new reciprocity law for the so called &lt;strong&gt;Hilbert's norm residue symbol&lt;/strong&gt; (see &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Hilbert_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;here &lt;/a&gt; for the definition). I'm not sure if - beside the same name- Takagi's approach using norms is somehow inspired by these norm residue symbols introduced by Hilbert. Maybe there is a concrete example calculation in Hilbert's works involving norm ideals, which might had inspired Takagi to abstract it and build his theory on it, I don't know.&lt;/p&gt;&#xA;&lt;p&gt;Honestly, I not see any mathematical connection, between these two objects - the ideal norm and Hilbert's norm residue - beside similar names as I remarked before. Does anybody see the connection (... even about any vague one I would be happy; it's really about what might had inspired Takagi to use this norm terminology)&lt;/p&gt;&#xA;&lt;p&gt;Also I'm not completely sure if it make sense to post this question here or in MathStackExchange.&lt;/p&gt;&#xA;" OwnerUserId="7566" LastEditorUserId="643" LastEditDate="2023-08-13T14:30:02.660" LastActivityDate="2023-08-13T14:30:02.660" Title="Teiji Takagi's Fondation of Class Field Theory in Terms of Norms" Tags="|mathematics|number-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15635" PostTypeId="1" CreationDate="2023-08-10T15:08:52.737" Score="0" ViewCount="40" Body="&lt;p&gt;I would like to understand the the idea behind the historically original proof by Lefschetz of his Hyperplane theorem sketched roughly &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Lefschetz_hyperplane_theorem#Lefschetz%27s_proof&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt;. The basic setup:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; be an &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; -dimensional complex projective algebraic variety closed embedded in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{CP}^n $&lt;/span&gt;&#xA;and let &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; be a hyperplane section of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lefschetz used his idea of a Lefschetz pencil to prove the theorem. Rather than considering the hyperplane section &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; alone, he put it into a family of hyperplane sections &lt;span class=&quot;math-container&quot;&gt;$Y_t$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$Y=Y_0$&lt;/span&gt;. Because a generic hyperplane section is smooth, all but a finite number of &lt;span class=&quot;math-container&quot;&gt;$Y_t$&lt;/span&gt; are smooth varieties. After removing these points from the &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;&#xA;-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologically trivial. That is, it is a product of a generic &lt;span class=&quot;math-container&quot;&gt;$Y_t$&lt;/span&gt;&#xA;with an open subset of the &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;-plane. In modern terns the &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;-plane is nothing but the complex plane &lt;span class=&quot;math-container&quot;&gt;$\mathbb{P}^1$&lt;/span&gt; and we have a canonical &lt;em&gt;rational map&lt;/em&gt; from the union of the hypersections &lt;span class=&quot;math-container&quot;&gt;$Y_t$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{P}^1$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points. Away from the singular points, the identification can be described inductively. At the singular points, the Morse lemma implies that there is a choice of coordinate system for X&#xA;of a particularly simple form. This coordinate system can be used to prove the theorem directly.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; I not understand the part where one allows &amp;quot;inductively&amp;quot; to glue/ realize identifications &amp;quot;away from singular points inductively.&amp;quot; What is the precise statement over which we perform this &amp;quot;induction&amp;quot; and over which quantity? Subspace dimension, ie like in a Noetherian induction? This is important in order to understand which identifications is here &amp;quot;legal&amp;quot; to assume as be already done in &amp;quot;induction hypothesis&amp;quot; .&lt;/p&gt;&#xA;&lt;p&gt;Could somebody spell out the precise &amp;quot;statement&amp;quot; on which we do this induction the proof is referring to? I'm confused namely about what is allowed to be assumed to be already done by induction hypothesis, and what not, and why?&lt;/p&gt;&#xA;" OwnerUserId="7566" LastEditorUserId="12330" LastEditDate="2023-08-11T19:11:58.797" LastActivityDate="2023-08-11T19:11:58.797" Title="Lefschetz historical proof of Hyperplane Theorem" Tags="|mathematics|topology|algebraic-geometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15636" PostTypeId="1" CreationDate="2023-08-10T15:45:19.890" Score="2" ViewCount="75" Body="&lt;p&gt;In &lt;a href=&quot;https://stacks.math.columbia.edu/tag/033Q&quot; rel=&quot;nofollow noreferrer&quot;&gt;033Q&lt;/a&gt; we find defined what some sources call “Serre's conditions &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathrm{R}_k$&lt;/span&gt;” (if you don't know what a scheme is, you can read the definition for a commutative Noetherian ring in &lt;a href=&quot;https://stacks.math.columbia.edu/tag/031P&quot; rel=&quot;nofollow noreferrer&quot;&gt;031P&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;These conditions are used to characterize properties of Noetherian rings:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Lemma.&lt;/strong&gt; Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; be a commutative Noetherian ring. Then:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is reduced &lt;span class=&quot;math-container&quot;&gt;$\Leftrightarrow$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathrm{R}_0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_1$&lt;/span&gt; hold. (&lt;a href=&quot;https://stacks.math.columbia.edu/tag/031R&quot; rel=&quot;nofollow noreferrer&quot;&gt;031R&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is normal &lt;span class=&quot;math-container&quot;&gt;$\Leftrightarrow$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathrm{R}_1$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_2$&lt;/span&gt; hold. (&lt;a href=&quot;https://stacks.math.columbia.edu/tag/031S&quot; rel=&quot;nofollow noreferrer&quot;&gt;031S&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is Cohen-Macaulay &lt;span class=&quot;math-container&quot;&gt;$\Leftrightarrow$&lt;/span&gt; &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_k$&lt;/span&gt; holds for all &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. (&lt;a href=&quot;https://stacks.math.columbia.edu/tag/0342&quot; rel=&quot;nofollow noreferrer&quot;&gt;0342&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_k$&lt;/span&gt; can be defined for an &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;-module (see &lt;a href=&quot;https://stacks.math.columbia.edu/tag/031P&quot; rel=&quot;nofollow noreferrer&quot;&gt;031P&lt;/a&gt;), and then &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_1$&lt;/span&gt; has a characterization given in &lt;a href=&quot;https://stacks.math.columbia.edu/tag/031Q&quot; rel=&quot;nofollow noreferrer&quot;&gt;031Q&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My questions are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(1a) Who defined Serre's conditions the first time? (was Serre himself?)&lt;/p&gt;&#xA;&lt;p&gt;(1b) Who chose the notation &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathrm{R}_k$&lt;/span&gt; for them? (It seems standard in all modern sources).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In &lt;a href=&quot;http://www.numdam.org/articles/PMIHES_1965__24__5_0/&quot; rel=&quot;nofollow noreferrer&quot;&gt;EGA IV₂&lt;/a&gt;, conditions &lt;span class=&quot;math-container&quot;&gt;$\mathrm{S}_k$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\mathrm{R}_k$&lt;/span&gt; are defined in §5.7 and §5.8, respectively. H. Matsumura also defines them in his &lt;em&gt;Commutative Algebra&lt;/em&gt; (in (17.I)) and &lt;em&gt;Commutative Ring Theory&lt;/em&gt; (in the remark before Theorem 23.8).&lt;/p&gt;&#xA;&lt;p&gt;The second point of the lemma above is sometimes called “Serre's criterion for normality” (in EGA IV, this is (5.8.6), whereas in Matsumura's CA it's (17.I), Theorem 39, whose proof is “after EGA IV”). Right before stating Theorem 39, Matsumura says that Serre's criterion for normality is &lt;em&gt;due to Krull(1931) in the case &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is a domain, and to Serre in the general case.&lt;/em&gt; However, there is no “Krull(1931)” in the book bibliography. For these reasons, my other questions are:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(2) Which paper is “Krull(1931)” that Matsumura quotes? A search for the author and year in &lt;a href=&quot;https://scholar.google.com/scholar?q=wolfgang%20krull&amp;amp;hl=es&amp;amp;as_sdt=0%2C5&amp;amp;as_ylo=1931&amp;amp;as_yhi=1931&quot; rel=&quot;nofollow noreferrer&quot;&gt;google scholar&lt;/a&gt; shows three papers. However, after skimming through them I couldn't find the statement.&lt;/p&gt;&#xA;&lt;p&gt;(3a) Where did Serre proved his criterion for normality?&lt;/p&gt;&#xA;&lt;p&gt;(3b) Was Serre really the first one proving it? (For an arbitrary Noetherian ring.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="5031" LastEditorUserId="5031" LastEditDate="2023-08-21T12:24:58.463" LastActivityDate="2023-08-21T12:24:58.463" Title="History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring" Tags="|mathematics|reference-request|notation|abstract-algebra|algebraic-geometry|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15637" PostTypeId="1" CreationDate="2023-08-11T11:50:38.167" Score="2" ViewCount="78" Body="&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/74073/the-first-complete-proof-of-the-kronecker-weber-theorem&quot;&gt;https://mathoverflow.net/questions/74073/the-first-complete-proof-of-the-kronecker-weber-theorem&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I searched above link, and I was so interested about this post.&lt;/p&gt;&#xA;&lt;p&gt;Today, I have a question about historical background of Kronecker's Jugendtraum. In particular, I want to know the relationship with Heinrich Martin Weber and Kronecker's Jugendtraum.&lt;/p&gt;&#xA;&lt;p&gt;I know that Weber finally missed his chance to disprove Hilbert's claim in Hilbert's 12th problem because of overlooking precisely the possibility of choosing different signs.&lt;/p&gt;&#xA;&lt;p&gt;I was googling to know about this, then I found &amp;quot;Elliptic functions According to Eisenstein and Kronecker&amp;quot; and &amp;quot;On the History of Hilbert's Twelfth problem A comedy or errors&amp;quot; etc.&#xA;I read them precisely but I can't understand Weber's work in Kronecker Jugendtraum and history of Kronecker's Jugendtraum.&lt;/p&gt;&#xA;&lt;p&gt;So, I would like to get your help.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="12330" LastEditDate="2023-08-12T08:55:32.207" LastActivityDate="2023-08-12T08:55:32.207" Title="Historical background with Weber and Kronecker's Jugendtraum" Tags="|mathematicians|hilbert|kronecker|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15639" PostTypeId="2" ParentId="15637" CreationDate="2023-08-11T12:10:11.293" Score="1" Body="&lt;p&gt;The history is described in the book of Serge Vladut,&#xA;Kronecker's Jugendtraum and modular functions, Overseas Publishers Assoc., 1991.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-08-11T12:10:11.293" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15640" PostTypeId="2" ParentId="15298" CreationDate="2023-08-12T02:54:30.177" Score="3" Body="&lt;p&gt;See the book &amp;quot;Elliptic and Modular Functions from Gauss to Dedekind to Hecke&amp;quot; by Ranjan Roy.  It is a very rich source of material on the history of topics related to modular functions. Using this book I was able to answer a question I once asked here about the origin of the Jacobi product formula for the discriminant modular function &lt;span class=&quot;math-container&quot;&gt;$\Delta(t)$&lt;/span&gt;: see &lt;a href=&quot;https://hsm.stackexchange.com/questions/5045/jacobis-product-for-the-discriminant&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-08-12T02:54:30.177" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15641" PostTypeId="1" CreationDate="2023-08-12T15:41:44.903" Score="1" ViewCount="25" Body="&lt;p&gt;I know they had balls from the rubber trees mixed with morning glory flowers. A technology that was lost when the European colonizers came. But was there any info on who made this discovery and how?&lt;/p&gt;&#xA;" OwnerDisplayName="user18836" LastActivityDate="2023-08-12T15:41:44.903" Title="Is there info on how Mayans invented hard rubber?" Tags="|chemistry|science|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15643" PostTypeId="1" CreationDate="2023-08-12T16:05:32.747" Score="3" ViewCount="197" Body="&lt;p&gt;As many of you may know, sometime around the 14/15th centuries an Indian mathematician by the name of &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama&quot; rel=&quot;nofollow noreferrer&quot;&gt;Madhava of Sangamagrama&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_series&quot; rel=&quot;nofollow noreferrer&quot;&gt;derived the Maclaurin series for sine and cosine&lt;/a&gt; for the first time in recorded history, predating Newton by 250 years, give or take. On a unit circle, &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_series#Madhava%27s_sine_series:%7E:text=8%5D%5B9%5D-,Madhava%27s%20sine%20series,-%5Bedit%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;his&lt;/a&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Madhava_series#Madhava%27s_cosine_series:%7E:text=from%20the%20arc.-,Madhava%27s%20cosine%20series,-%5Bedit%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;formulas&lt;/a&gt; simplify to&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\sin(s)= s-\frac{s^3}{3!}+\frac{s^5}{5!}+\ldots$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\cos(s)= 1-\frac{s^2}{2!}+\frac{s^4}{4!}+\ldots$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is the length of some arc of the unit circle. So put yourselves into a medieval mathematician's shoes for a minute. You have some basic premises:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P1$&lt;/span&gt;. The &amp;quot;sine&amp;quot; and &amp;quot;cosine&amp;quot; are literally just geometric lengths inscribed inside a circle&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P2$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; is also a geometric length&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P3$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; has an upper bound of &lt;span class=&quot;math-container&quot;&gt;$2\pi$&lt;/span&gt;; it probably doesn't make sense to have an arc-length of a shape be longer than the shape's circumference&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$P4$&lt;/span&gt;. Lengths cannot be negative&lt;/p&gt;&#xA;&lt;p&gt;Now, let &lt;span class=&quot;math-container&quot;&gt;$T(s)$&lt;/span&gt; be a good-enough Taylor Polynomial of Sine's Maclaurin Series, situated on the unit circle. Notice the following:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$T(\frac{3\pi}{2})\approx -1$&lt;/span&gt;. Plug some more nearby values in and it's not terribly difficult to notice that &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;'s output on &lt;span class=&quot;math-container&quot;&gt;$(\pi,2\pi)$&lt;/span&gt; is a &amp;quot;negative mirror&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;'s output on &lt;span class=&quot;math-container&quot;&gt;$(0,\pi)$&lt;/span&gt;. But since we accepted &lt;span class=&quot;math-container&quot;&gt;$P1$&lt;/span&gt;, this output contradicts &lt;span class=&quot;math-container&quot;&gt;$P4$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;'s output on &lt;span class=&quot;math-container&quot;&gt;$(-\infty,0)$&lt;/span&gt; is well-defined, and replicates the behavior we see on  &lt;span class=&quot;math-container&quot;&gt;$(0,2\pi)$&lt;/span&gt;. But &lt;span class=&quot;math-container&quot;&gt;$P2$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$P4$&lt;/span&gt; suggest that &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; should NOT be defined on this interval. The fact that it is is a little spooky...&lt;/li&gt;&#xA;&lt;li&gt;Similarly, &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt;'s output on &lt;span class=&quot;math-container&quot;&gt;$(2\pi,\infty)$&lt;/span&gt; is &lt;em&gt;also&lt;/em&gt; well-defined and replicates the behavior we see on  &lt;span class=&quot;math-container&quot;&gt;$(0,2\pi)$&lt;/span&gt;. But &lt;span class=&quot;math-container&quot;&gt;$P3$&lt;/span&gt; suggests that it should be undefined.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;(Similar weirdness happens with Cosine's series)&lt;/p&gt;&#xA;&lt;p&gt;So it seems that not only is &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; surprisingly well-defined for completely invalid inputs (2,3), it's also nonsensically defined for half the inputs that &lt;em&gt;are&lt;/em&gt; valid (1). So I'm curious: were these contradictions noticed in Madhava's time? And if so, is there any historical consensus/evidence/speculation on how they were reconciled?&lt;/p&gt;&#xA;" OwnerUserId="18527" LastEditorUserId="6609" LastEditDate="2023-08-15T09:04:02.747" LastActivityDate="2023-08-17T07:14:10.167" Title="Did the Maclaurin series for sine and cosine unsettle Indian mathematicians?" Tags="|discoveries|calculus|real-analysis|ancient-india|trigonometry|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15645" PostTypeId="1" CreationDate="2023-08-12T19:05:23.333" Score="1" ViewCount="60" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; be an integral locally noetherian smooth&#xA;scheme over base field &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;. Then for every point &lt;span class=&quot;math-container&quot;&gt;$x \in X^{(1)}$&lt;/span&gt; of codimension &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;, the stalk &lt;span class=&quot;math-container&quot;&gt;$\mathcal{O}_{X,x}$&lt;/span&gt;  is a discrete valuation&#xA;ring. Choose some uniformizer &lt;span class=&quot;math-container&quot;&gt;$\pi_x \in \mathcal{O}_{X,x}$&lt;/span&gt;. Then we get a &lt;em&gt;residue homomorphism&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \partial_{MW}^{\pi_x}: \text{K}_n^{MW}(F(X)) \to \text{K}_{n-1}^{MW}(F(x)), \ (\{\pi_x, u_2, ... , u_n \}) \mapsto (\{ \overline{u_2}, ... , \overline{u_n} \})  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where here &lt;span class=&quot;math-container&quot;&gt;$F(X)$&lt;/span&gt; is the function field of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k(x)= \mathcal{O}_{X,x}/m_x$&lt;/span&gt;  the residue fieeld of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$u_i \in \mathcal{O}_{X,x}^{\times} $&lt;/span&gt;&#xA;units and &lt;span class=&quot;math-container&quot;&gt;$\overline{u_i}$&lt;/span&gt; their residues with respect residue map&#xA;&lt;span class=&quot;math-container&quot;&gt;$v_x: \mathcal{O}_{X,x} \to k(x)$&lt;/span&gt;. (the terminology can be looked up  eg in F. Morel's &lt;span class=&quot;math-container&quot;&gt;$\mathbb{A}^1$&lt;/span&gt;-Algebraic Topology over a Field).&lt;/p&gt;&#xA;&lt;p&gt;The &lt;em&gt;&lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th unramified Milnor-Witt K-group&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is defined as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \text{K}^{MW}_{n, unr}(X) := \text{Ker}(\sum_{x \in  X^{(1)}} \partial_{MW}^{\pi_x})   $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;where&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\sum_{x \in  X^{(1)}} \partial_{MW}^{\pi_x}: \text{K}_n^{MW}(F(X)) \to&#xA;\bigoplus_{x \in  X^{(1)}} \text{K}_{n-1}^{MW}(F(x))  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there a natural reason to call these groups &amp;quot;unramified&amp;quot; and is there any relation known to the concept of &amp;quot;unramifiedness&amp;quot; in context of splitting behavoir of prime ideals in number fields in &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions#Definitions&quot; rel=&quot;nofollow noreferrer&quot;&gt;algebraic number theory&lt;/a&gt;, justifying this name? Or are these two notions completely unrelated, but then the question is where this notion else come from; surely it not falls from heaven.&lt;/p&gt;&#xA;&lt;p&gt;A naive guess of mine where the notion might come from: There exist so called &lt;em&gt;Gersten complex for Milnor K-theory&lt;/em&gt; which is structurally similar to Gersten complex for étale cohomology, which involve so called unramified étale cohomology groups&#xA;&lt;span class=&quot;math-container&quot;&gt;$H^i_{ur}(X, \mu_2^{\otimes i})$&lt;/span&gt;. If that's the origin of the name&#xA;of &amp;quot;unramified&amp;quot; for unramified Milnor-Witt K-groups, the question turn's into what is the origin of the the name &amp;quot;unramified&amp;quot; for unramified étale cohomology groups.&lt;/p&gt;&#xA;&lt;p&gt;Rmk.: The identical question was already &lt;a href=&quot;https://mathoverflow.net/questions/435127/in-which-sense-unramified-milnor-witt-k-groups-are-unramified&quot;&gt;posted in MO&lt;/a&gt;, but I think that maybe that's a more appropriate plattform for it.&lt;/p&gt;&#xA;" OwnerUserId="7566" LastEditorUserId="6609" LastEditDate="2023-08-23T21:38:09.167" LastActivityDate="2023-08-23T21:38:09.167" Title="Unramified étale cohomology groups &amp; unramified Milnor-Witt K-groups: any relation to notion of unramifiedness in number theory" Tags="|mathematics|number-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15646" PostTypeId="2" ParentId="15634" CreationDate="2023-08-13T03:12:24.440" Score="2" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Honestly, I not see any mathematical connection, between these two objects - the ideal norm and Hilbert's norm residue - beside similar names as I remarked before. Does anybody see the connection&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;These two uses of the word norm are definitely related.&lt;/p&gt;&#xA;&lt;p&gt;On a &lt;em&gt;principal ideal&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$(\alpha)$&lt;/span&gt; in the integers of a number field &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt;, its ideal norm &lt;span class=&quot;math-container&quot;&gt;${\rm N}((\alpha)) := |\mathcal O_K/(\alpha)|$&lt;/span&gt; is its field norm in absolute value: &lt;span class=&quot;math-container&quot;&gt;${\rm N}((\alpha)) = |{\rm N}_{K/\mathbf Q}(\alpha)|$&lt;/span&gt;.  This is the entire reason we give the size of a residue ring &lt;span class=&quot;math-container&quot;&gt;${\mathcal O}_K/\mathfrak a$&lt;/span&gt; the name &amp;quot;ideal norm&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Hilbert's norm residue symbol is trivial on values of the field norm, just as the Legendre symbol is also called a quadratic residue symbol because it is trivial on quadratic residues.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-08-13T14:08:02.463" LastActivityDate="2023-08-13T14:08:02.463" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15647" PostTypeId="1" CreationDate="2023-08-13T14:12:34.160" Score="3" ViewCount="300" Body="&lt;p&gt;I started to see lists of left-handed scientists and Newton is included in all of them. I checked Westfall but could not find the answer. Are there any proofs that he was left handed? I've seen analyses over his portraits, that he was holding a compass on his left hand etc. but I don't think these are definitive proofs.&lt;/p&gt;&#xA;&lt;p&gt;I know this is a trivial matter but I was curious.&lt;/p&gt;&#xA;" OwnerUserId="16575" LastEditorUserId="12330" LastEditDate="2023-08-13T16:09:16.113" LastActivityDate="2023-08-13T16:09:16.113" Title="Was Isaac Newton left handed?" Tags="|biographical-details|newton|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15648" PostTypeId="1" CreationDate="2023-08-13T23:54:14.597" Score="1" ViewCount="61" Body="&lt;p&gt;Yesterday I saw Pólya's burial spot in Alta Mesa Memorial Park in Palo Alto, CA. Actually, it was not a grave but an urn (which gives a new meaning to the term &amp;quot;Pólya's urn model&amp;quot;). Naturally after seeing Pólya it's interesting to see where Szegő is buried, but I can't find anything about this anywhere online, not just at &lt;a href=&quot;https://www.findagrave.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.findagrave.com/&lt;/a&gt;.  Does anyone know where he was buried/cremated?  Note he was born in 1895 and died in 1985, so the one hit with his name on the find-a-grave website is not him since that Gábor Szegő is listed as having died in 2015.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-08-13T23:54:14.597" Title="Gábor Szegő burial location" Tags="|mathematicians|death|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15649" PostTypeId="1" AcceptedAnswerId="15651" CreationDate="2023-08-14T08:40:58.743" Score="2" ViewCount="75" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Berend_George_Escher&quot; rel=&quot;nofollow noreferrer&quot;&gt;Berend George Escher&lt;/a&gt; was a Dutch geologist (and half-brother of artist M. C. Escher). His Wikipedia page states that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He was also a pioneer in using experiments to solve geological questions, for which he set up a laboratory in Leiden.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But there is no source to this claim&lt;sup&gt;1&lt;/sup&gt;. &lt;a href=&quot;https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/escher-berend-george&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia.com&lt;/a&gt; has a bit more details:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Escher was the first geologist in the Netherlands to use laboratory experiments for the study of geological phenomena. His experiments in connection with salt domes were carried out with Kuenen (1929), who later used experimental methods in his sedimentological research. Experimental facilities designed by Escher served engineering purposes in the reclamation of the Zuider Zee.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And in the bibliography:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For the experiments on salt domes with Kuenen, see Leidse Geologisch Medelingen. 3 (1929, 151–181. A summary of most of Escher;s geological experiments is “Eine Übersicht der im geologischen Institut in Leiden von 1920 bis Ende 1937 ausgeführten geologischen Experimente,” in C. R. Congr. Int. de Géogr. Amsterdam. 2 (1938), 273–278.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But I could not find a digital copy of the latter source online, and even if I did, according to the title it will likely be written in German. So I'd like to ask:&lt;/p&gt;&#xA;&lt;p&gt;Apart from the salt dome experiments, &lt;strong&gt;which geological experiments did Berend George Escher use to solve geological questions?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Following njuffa's answer, I've now added the proper reference to the Wikipedia article.&lt;/p&gt;&#xA;" OwnerUserId="14043" LastEditorUserId="14043" LastEditDate="2023-08-15T07:48:12.877" LastActivityDate="2023-08-15T07:48:12.877" Title="Which &quot;experiments to solve geological questions&quot; did Berend George Escher use?" Tags="|earth-sciences|geology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15650" PostTypeId="2" ParentId="15111" CreationDate="2023-08-14T15:21:15.390" Score="2" Body="&lt;p&gt;I will reproduce here Dieudonné's &lt;a href=&quot;https://www.google.com/books/edition/History_Algebraic_Geometry/kIZHDwAAQBAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Algebraic Geometry&lt;/a&gt;, VIII.2, first two paragraphs. I think they speak for themselves:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;ol start=&quot;19&quot;&gt;&#xA;&lt;li&gt;&lt;p&gt;Around 1949, A. Weil had observed that the “Zariski topology” (VII, 45) could be defined on his “abstract varieties” (VII, 38) as well, and not only did it simplify the exposition but it made possible the definition in “abstract” algebraic geometry of the notion of &lt;em&gt;fiber space&lt;/em&gt;, modeled on the classical notion (VIII, 2), as well as the extension of the relations between line bundles and divisors (VIII, 4). In 1954, Serre had the idea of similarly generalizing the notion of sheaf to “abstract” varieties by replacing, in the definition of (VIII, 8), the usual topology by the Zariski topology. But, owing to this idea, he realized that the definition and the study of Weil’s “abstract varieties” could even be presented in a much more convenient form, using H. Cartan’s notion of “ringed space,” that is to say, a topological space &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; on which is given a &lt;em&gt;sheaf of rings&lt;/em&gt;, called the structure sheaf (that is, a sheaf where, for each open set &lt;span class=&quot;math-container&quot;&gt;$U\subset X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$H^0(U, \mathcal{O}_X)$&lt;/span&gt; has the structure of a ring such that the restriction homomorphisms &lt;span class=&quot;math-container&quot;&gt;$H^0(U, \mathcal{O}_X)\to H^0(V,\mathcal{O}_X)$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$V\subset U$&lt;/span&gt; are ring homomorphisms). The advantage of this point of view is that this type of structure lends itself very well to “gluing” along open sets, the verification of the “conditions for gluing” ordinarily being trivial.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For the applications he has in mind, Serre is not concerned with questions of the “field of definition” (VII, 36) and he does not use the notion of “generic point”; staying closer to the classical point of view, he fixes, once and for all, an &lt;em&gt;algebraically closed&lt;/em&gt; base field &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; (of arbitrary characteristic). The “pieces” that he glues together to obtain the definition of his varieties are what are called &lt;em&gt;affine varieties&lt;/em&gt; over &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;; such a variety &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is a subset of a space &lt;span class=&quot;math-container&quot;&gt;$k^n$&lt;/span&gt; defined by polynomial equations, and the sheaf &lt;span class=&quot;math-container&quot;&gt;$\mathcal{O}_X$&lt;/span&gt; is defined by the condition that, for every open set &lt;span class=&quot;math-container&quot;&gt;$U\subset X$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$H^0(U, \mathcal{O}_X)$&lt;/span&gt; is composed of the restrictions to &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; of the rational functions &lt;span class=&quot;math-container&quot;&gt;$x\mapsto P(x)/Q(x)$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;$k^n$&lt;/span&gt; that are defined (that is, such that &lt;span class=&quot;math-container&quot;&gt;$Q(x) \neq 0$&lt;/span&gt;) at every point &lt;span class=&quot;math-container&quot;&gt;$x\in U$&lt;/span&gt;. In addition, Serre imposed on his varieties, on the one hand, the condition of having a covering by a finite number of affine open sets (that is, open sets &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; such that the restriction to &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; of the sheaf &lt;span class=&quot;math-container&quot;&gt;$\mathcal{O}_X$&lt;/span&gt; defines &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; as an affine variety), and on the other, a “separation condition” allowing “passage to the limit” for the Zariski topology in a reasonable manner: for example, if two rational functions &lt;span class=&quot;math-container&quot;&gt;$f,g$&lt;/span&gt; are defined in &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; and agree on an everywhere dense open set of &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;, then they are necessarily identical.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;From my interpretation, Dieudonné's exposition implicitly implies that Serre is the first mathematician doing these kind of things.&lt;/p&gt;&#xA;" OwnerUserId="5031" LastActivityDate="2023-08-14T15:21:15.390" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15651" PostTypeId="2" ParentId="15649" CreationDate="2023-08-14T23:58:27.120" Score="4" Body="&lt;p&gt;This question highlighted the importance of complete and accurate citations. I spent a couple of hours researching the exact details of the two publications mentioned, after which it took on the order of ten minutes to find scans of them. The first publication is:&lt;/p&gt;&#xA;&lt;p&gt;B. G. Escher and Ph. H. Kuenen, &amp;quot;Experiments in Connection with Salt Domes.&amp;quot; &lt;em&gt;Leidse Geologische Mededelingen&lt;/em&gt;, Vol. 3, No. 1, 1928, pp. 151-182 (plus 18 additional plates). &lt;a href=&quot;https://repository.naturalis.nl/pub/505826&quot; rel=&quot;nofollow noreferrer&quot;&gt;Scan available&lt;/a&gt; at Naturalis Biodiversity Center.&lt;/p&gt;&#xA;&lt;p&gt;According to the abstract the goal of the work covered by this publication was to experimentally validate the Lachmann-Arrhenius-Harbort theory that salt domes are formed by isostasy combined with the lower density and greater plasticity of salt compared to overlaying sedimentary layers. The modelling used layers of paraffin, clay, and gypsum acted upon by hydraulic presses. The experiments confirmed that the folding observed with salt domes can be fully explained by the Lachmann theory.&lt;/p&gt;&#xA;&lt;p&gt;The proceedings of the 1938 conference turned out to be not only spread over two volumes, but the voluminous second volume was split into multiple fascicles, each covering one or more topical sections, such as cartography, physical geography, oceanography, etc. I found Escher's paper in the fascicle for section IIa:&lt;/p&gt;&#xA;&lt;p&gt;B. G. Escher, &amp;quot;Eine Übersicht der im Geologischen Institut in Leiden von 1926 bis Ende 1937 ausgeführten geologischen Experimente.&amp;quot; &lt;em&gt;Comptes Rendus Du Congrès International de Géographie, Amsterdam 1938&lt;/em&gt;, Vol. 2, Section II a, Géographie Physique, pp. 273-278. The &lt;a href=&quot;https://en.wikipedia.org/wiki/Delpher&quot; rel=&quot;nofollow noreferrer&quot;&gt;Delpher&lt;/a&gt; service of the Royal Library of the Netherlands provides a &lt;a href=&quot;https://www.delpher.nl/nl/boeken/view?query=Comptes+rendus+du+Congre%CC%80s+International+de+Ge%CC%81ographie%2C+Amsterdam+1938+Section+IIa&amp;amp;coll=boeken&amp;amp;page=1&amp;amp;maxperpage=50&amp;amp;identifier=MMKB06:000004272:00305&amp;amp;facets%5Bperiode%5D%5B%5D=0%7C20e_eeuw%7C&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This publication provides an overview of seventeen experiments, seven of which have associated publications. Twelve of the experiments were water-centric, mostly having to do with flowing water and wave action, where the latter group was trying to replicate various features of coastal geomorphology and examined the action of waves upon levees to find optimal cross sections. The laboratory used an innovative combination of a water channel with a wind tunnel. One interesting experiment in the former group examined the role of underwater erosion in the formation of submarine canyons.&lt;/p&gt;&#xA;&lt;p&gt;Of the remaining experiments, one dealt with the formation of ventifacts, with wind erosion modeled by sand being propelled by a re-purposed vacuum cleaner. Two experiments dealt with tectonics: the formation of salt domes and folding over liquid substrates, modeling involving layers of paraffin. Finally there were two experiments concerning volcanism. If I understand it correctly, one dealt with erosion processes during the formation of calderas (using canisters of compressed air to model escaping volcanic gasses), the other with depository processes during the formation of volcanic cones (using a vertical stream of sand propelled upwards by two back-to-back vacuum cleaners).&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-08-15T00:23:44.807" LastActivityDate="2023-08-15T00:23:44.807" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15652" PostTypeId="1" CreationDate="2023-08-15T00:45:12.030" Score="0" ViewCount="55" Body="&lt;p&gt;I'm assuming that Poisson came up with his approximation to the binomial to ease the burden of computing the binomial coefficients for large n. I'm also assuming that as a student of Laplace, who revived and generalized de Moivre's approximation, he must have been aware of it. So, why not take a perfectly good approximation off the shelf?&lt;/p&gt;&#xA;" OwnerUserId="15921" LastActivityDate="2023-08-15T00:45:12.030" Title="Poisson's approximation to the binomial" Tags="|statisticians|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15656" PostTypeId="1" CreationDate="2023-08-18T12:00:27.283" Score="1" ViewCount="122" Body="&lt;p&gt;I am looking for the historical context in which Gauss established its famous &lt;em&gt;Theorema Egregium&lt;/em&gt;. Was Gauss studying map projections (a nowadays popular application of the theorem)? Any references are welcome!&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="6609" LastEditDate="2023-08-22T13:30:33.963" LastActivityDate="2023-08-23T12:20:54.650" Title="Gauss' Theorema Egregium" Tags="|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15657" PostTypeId="1" AcceptedAnswerId="15662" CreationDate="2023-08-18T13:19:54.530" Score="2" ViewCount="497" Body="&lt;p&gt;After receiving the infamous letter from Bertrand Russell, Gottlob Frege allegedly had a mental breakdown and had to be hospitalized.&lt;/p&gt;&#xA;&lt;p&gt;I've seen various informal references to this, such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=ymGt7I4Yn3k&amp;amp;t=1024s&amp;amp;ab_channel=JeffreyKaplan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's Paradox - Jeffrey Kaplan - YouTube&lt;/a&gt; says:&#xA;&amp;quot;&lt;em&gt;Frege got the letter and it broke him, physically he had a breakdown and had to be hospitalized upon reading this two page letter.&lt;/em&gt;&amp;quot;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://ayushsingh.dev/blog/russells-paradox/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Russell's Paradox — haloboy777&lt;/a&gt; says:&#xA;&amp;quot;&lt;em&gt;This paradox literally caused him to go to hospital.&lt;/em&gt;&amp;quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Is there a more formal detailed description of this hospitalization?&lt;/p&gt;&#xA;" OwnerUserId="15353" LastEditorUserId="604" LastEditDate="2023-08-21T07:51:35.990" LastActivityDate="2023-08-21T07:51:35.990" Title="Was Gottlob Frege hospitalized by Russell's Paradox?" Tags="|biographical-details|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15658" PostTypeId="1" AcceptedAnswerId="15661" CreationDate="2023-08-18T19:05:10.040" Score="2" ViewCount="119" Body="&lt;p&gt;This question is in regard to Galileo's early writings on motion titled &lt;em&gt;De Motu&lt;/em&gt; (On Motion) or &lt;em&gt;De Motu Antiquiora&lt;/em&gt; (Older Writings On Motion).&lt;/p&gt;&#xA;&lt;p&gt;It is understood that Galileo never published this treatise in his lifetime, so I'm looking for evidence of when this work became available to the public through some posthumous publication.  Most of what I've found online claim that it was published in 1687 (which is coincidentally the same year Newton published his &lt;em&gt;Principia&lt;/em&gt;); however, I'm unable to find hard evidence of this fact, and it makes me wonder if this particular year is an error that's being echoed.&lt;/p&gt;&#xA;&lt;p&gt;The best evidence that I have is I.E. Drabkin's 1960 translation of &lt;em&gt;De Motu&lt;/em&gt; in which the first paragraph of Drabkin's introduction states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The first volume of the National Edition of the works of Galileo,&#xA;edited in 1890 by Antonio Favoro, contains on pages 251-419, under the&#xA;title &lt;em&gt;De Motu&lt;/em&gt;, the contents of certain manuscripts written in Latin&#xA;in Galileo's own hand.  Some of the material had been published for&#xA;the first time in 1854 in Volume II of Eugenio Alberi's edition of&#xA;Galileo's works, and the rest by Favaro himself in 1883.&lt;/p&gt;&#xA;&lt;p&gt;(a footnote to the above paragraph is provided: Alcuni scritti inediti di Galileo Galilei...&lt;em&gt;Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche&lt;/em&gt;, 16 (1883), 226-97, 135-57.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;This suggests that the treatise was only partially published in 1854 and then supplemented further in 1883.  Is there any proof to support the claim that there was a 1687 publication?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Edit #1 (in response to njuffa):&lt;/p&gt;&#xA;&lt;p&gt;As requested, I first came across the 1687 claim on Wikipedia for &lt;em&gt;De Motu Anitiquiora&lt;/em&gt;, but the date has since been changed to 1890.  Here are a few others that echo the 1687 publication date:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.wikiwand.com/en/De_Motu_Antiquiora&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.wikiwand.com/en/De_Motu_Antiquiora&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.hellenicaworld.com/Science/Physics/en/Galileothermometer.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.hellenicaworld.com/Science/Physics/en/Galileothermometer.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://earthsciences.osu.edu/sites/default/files/2020-04/report_514.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://earthsciences.osu.edu/sites/default/files/2020-04/report_514.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.edition-avantus.de/akkordeon-kammermusik/de-motu.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.edition-avantus.de/akkordeon-kammermusik/de-motu.html&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://m.famousfix.com/list/galileo-galilei-20531211&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://m.famousfix.com/list/galileo-galilei-20531211&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=N0Igeosq6OE&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/watch?v=N0Igeosq6OE&lt;/a&gt;  (see video comments)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/books?id=33crEAAAQBAJ&amp;amp;pg=PA33&amp;amp;lpg=PA33&amp;amp;dq=de+motu+antiquiora+%221687%22+galileo&amp;amp;source=bl&amp;amp;ots=hsB_pbJVT1&amp;amp;sig=ACfU3U2IT4NAGNC_sC6a6kNf4PxsLVb7fQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwilyI7W7umAAxW_m2oFHaR5C384FBDoAXoECAIQAw#v=onepage&amp;amp;q=de%20motu%20antiquiora%20%221687%22%20galileo&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://books.google.com/books?id=33crEAAAQBAJ&amp;amp;pg=PA33&amp;amp;lpg=PA33&amp;amp;dq=de+motu+antiquiora+%221687%22+galileo&amp;amp;source=bl&amp;amp;ots=hsB_pbJVT1&amp;amp;sig=ACfU3U2IT4NAGNC_sC6a6kNf4PxsLVb7fQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwilyI7W7umAAxW_m2oFHaR5C384FBDoAXoECAIQAw#v=onepage&amp;amp;q=de%20motu%20antiquiora%20%221687%22%20galileo&amp;amp;f=false&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://profilbaru.com/article/De_Motu&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://profilbaru.com/article/De_Motu&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I was able to find the earliest reference that used the 1687 date, which was surprisingly related to a 2011 Stanford Encyclopedia of Philosophy article on &amp;quot;Imagination&amp;quot; (appearing after an edit in &amp;quot;winter 2018&amp;quot;) where the following citation is provided:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Galilei, Galileo, c.1590, De Motu Antiquiora (On Motion), unpublished&#xA;manuscript, not published until 1687. Translated as part of On motion,&#xA;and On mechanics; comprising De motu, I.E. Drabkin (trans/ed.),&#xA;Madison: University of Wisconsin Press, 1960.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://plato.stanford.edu/entries/imagination/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://plato.stanford.edu/entries/imagination/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;As I mentioned earlier in my post, I own this book and I have yet to find any reference to 1687.  I can only guess that this was a citation error.&lt;/p&gt;&#xA;&lt;p&gt;For reference, here is a translation of Favaro's introduction from the link provided by njuffa in the comments (page numbers in brackets):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[245]&lt;/p&gt;&#xA;&lt;p&gt;Already Vincenzio Viviani, in certain commentaries concerning the&#xA;Works of his Master, alluded to a manuscript by Galileo in several&#xA;octavo quintenets entitled outside on the cover De motu antiquiora,&#xA;which is acknowledged to be one of his first youthful studies, and for&#xA;which nevertheless it is seen that from that time on he did not know&#xA;how to accommodate his free intellect to the obligatory philosophizing&#xA;of the common schools. And he immediately adds: However, what is most&#xA;singular, which is scattered in this manuscript, all, as can be seen,&#xA;he himself then suitably inserted, in his own places, in the works&#xA;that he printed. Of some studies made by the Florentine philosopher in&#xA;his youth, and transcribed by him in various quintens, above one of&#xA;which is marked De motu antiquorum ect. he also touches Nelli, stating&#xA;that the autograph was in his possession: and in fact it is registered&#xA;among the manuscripts in his library; and Venturi also saw it, who&#xA;admitted to searching in the Galilean papers of the private library of&#xA;the Grand Duke of Tuscany, of which the main part was in fact made up&#xA;of the manuscripts that had already belonged to Nelli, and found there&#xA;various Latin treatises on motion, written by Galileo around 1590,&#xA;which show that he since then disagreed with the doctrine of&#xA;Aristotle.&lt;/p&gt;&#xA;&lt;p&gt;[246]&lt;/p&gt;&#xA;&lt;p&gt;According to the order given to the Galilean Manuscripts ever since&#xA;they were collected in the Palatine Library in Florence, there is none&#xA;which bears the title Do motu antiquiora, referred to by Viviani, nor&#xA;the other, evidently incorrect, given by Nelli: and it is probable&#xA;that the paper containing it was torn on the occasion of those&#xA;regulations, as a result of which, as we believe, the aforementioned&#xA;notebooks were no longer kept together; since one of them is now&#xA;united, in Volume I of Part III, with the youthful studies of&#xA;astronomical houses, or at least as such considered authorizing, the&#xA;others are found in Volume I of Part V.&lt;/p&gt;&#xA;&lt;p&gt;All these writings are autographed; and most of them were never&#xA;published in the editions of Galileo's Works.&lt;/p&gt;&#xA;&lt;p&gt;Coming now to examine the substance of these studies, to which we have&#xA;wanted to keep at least in part the original title, it seems to us to&#xA;be able to say that they contain the seed, and sometimes expressly&#xA;signify, the admirable discoveries which the Author then placed them&#xA;so much above the other contemporary philosophers, and show in full&#xA;bloom the fruits attached later in the Dialogues of the New Sciences.&lt;/p&gt;&#xA;&lt;p&gt;And as for the aspect under which these studies present themselves, it&#xA;seems to us that they can be divided into three different categories.&#xA;The first, made up of almost all the things related to motion&#xA;contained in Volume I of Part III, offers us some considerations in&#xA;the form of detached notes, almost notes of a reader or reflections of&#xA;a thinker; and their very material aspect seems to us to comfort our&#xA;judgment. This code contains up to char. 100 there Iuvenilia and,&#xA;starting from the car. 102, a series of Latin words with an Italian&#xA;equivalent, which however only occupies the beginnings of a few pages,&#xA;while the aforementioned notes on motion are written on the parts that&#xA;have remained blank. However, those Latin words are not from Galileo's&#xA;hand, nor are some exercises in writing and epistolary style found in&#xA;the last pages of the codex; for which reason it would almost seem&#xA;possible to argue that this booklet, which perhaps will have already&#xA;been used by his brother, was used by the young philosopher as a&#xA;notebook to which he consigned his thoughts, as they came into his&#xA;mind or they were suggested by the readings he was doing. The second&#xA;and third categories are made up of what is read in Volume I of Part&#xA;V: the first is represented by various chapters, in which some&#xA;principles regarding the causes and laws of motion are embodied; this&#xA;contains a good number of the more notable questions concerning the&#xA;same subject, presented in the form of a dialogue. Is that,&lt;/p&gt;&#xA;&lt;p&gt;[247]&lt;/p&gt;&#xA;&lt;p&gt;minus a few exceptions, which we will discuss shortly, must also have&#xA;been the natural genesis of such studies in the mind of Galileo. He&#xA;himself, without any doubt, thought of giving a logical order to these&#xA;writings of his; and perhaps he did not completely abandon the thought&#xA;of them, except when, taking them up again after many and many years,&#xA;he wanted to profit from a part of them, inserting it almost verbatim&#xA;in the Dialogues of the New Sciences. The trace of a division into&#xA;chapters is still evident, both in the headings of the single parts&#xA;and in the body of these, which are sometimes found referred to from&#xA;one to the other; but unfortunately such references are not complete,&#xA;nor always rigorously coordinated, and indeed in some circumstances&#xA;they lead us to doubt that everything Galileo dictated about this&#xA;topic has come down to us. Now therefore, preparing ourselves to&#xA;procure for the first time a complete edition of all that remains of&#xA;these writings, we had first thought of a distribution of the matter&#xA;in various books, according to which we would have searched, entering&#xA;into the Author's intention as far as it was allowed to divine it, to&#xA;give organic form to what apparently has only a fragmentary aspect;&#xA;but we have abandoned the thought, because the division into books,&#xA;suggesting a succession of topics covered, did not seem to us to be&#xA;suitable for our case, in which we have rather only one topic in a&#xA;series of elaborations that followed one another not too long ago.&lt;/p&gt;&#xA;&lt;p&gt;And as regards Volume I of Part V, after mature considerations deduced&#xA;from the essence of the writings, from the form in which they are laid&#xA;out, from some reference signs that can be found there, and from the&#xA;traces of an ancient and original numbering of the leaves of the code,&#xA;we have been induced to consider as a first step what is read in the&#xA;chars. 61 r.-124 v. (pag. 251-340): and the various chapters included&#xA;in these we have determined to give according to the order in which&#xA;the manuscript presents them. In truth, what we read in char. 77 r.-78&#xA;v. (pages 274-276) could have been omitted without appreciable damage&#xA;to the continuity of the discussion, since there are only repeated&#xA;references to the content of the long chapter that precedes it in&#xA;char. 68v.-76r. (p. 262-273); if we hadn't set ourselves the&#xA;indeclinable maxim that nothing of Galileo's manuscripts should be&#xA;overlooked. In what we read in char. 133r.-134v. (p. 341-343),&#xA;following the same criteria, it seemed to us that we could recognize a&#xA;second or later lesson of the two initial chapters of the first part;&#xA;in the chars. 43 r.-60 r. (p. 344-366), a third discussion of the&#xA;doctrines set forth in car. 61 r.-62 v. (pages 251-253) and, with&#xA;notable omissions and additions, depending on what one reads in char.&#xA;62 v.-66 v. and 85 r. (p. 253-260) and on char. 88 v.-92 r. (p.&#xA;289-294). In this regard, we deem it appropriate to warn that a char.&#xA;49 v.-60 r. (pages 352-366) the problems are partly treated and&#xA;ordered differently than in the first lesson. However, the chars&#xA;correspond. 43 r.-6O r. (p. 344-366) at 61 p.-66 v. and 85 r.&lt;/p&gt;&#xA;&lt;p&gt;[248]&lt;/p&gt;&#xA;&lt;p&gt;(p. 251-260) and at 88 ve-92 r. (pages 289-294), since and in both it&#xA;is not yet a question of speed variations. In particular then the&#xA;chapter on char. 88 v.-92 r. appears to be much improved in the new&#xA;lesson that we have in char. 51v.-56r. (p. 355-361).&lt;/p&gt;&#xA;&lt;p&gt;We have followed these writings (pages 367-408) with the dialogued&#xA;lesson (Par. V, T. I, car. 4r.-35v.), in which we thought it&#xA;appropriate to also insert a passage, equally dialogued ( page 375,&#xA;line 10 - page 378, line 3), which we found in Volume I of Part III&#xA;(char. 102 r.-104 v.).&lt;/p&gt;&#xA;&lt;p&gt;In the dialogue, an Alexander and a Dominicus are interlocutors; with&#xA;respect to which Nelli assumed that Galileo wanted to stage Iacopo&#xA;Mazzoni and Luca Valerio, whom he had as companions in Pisa at the&#xA;time to which the composition of these writings dates. Of Dominicus we&#xA;know nothing; but certainly Alexander is none other than Galileo&#xA;himself, since in a certain place Alexander speaks of the scale as an&#xA;instrument invented by him (page 379), and again using terms used by&#xA;Galileo himself in the relative writing.&lt;/p&gt;&#xA;&lt;p&gt;At the end we have placed the unstitched notes concerning the subject&#xA;of these writings, which we found in Volume I of Part III a char. 102&#xA;r., 104 v.-110 r. (p. 409-417) and in the I of the V a char. 3 v.&#xA;(pages 418-419): and this also because it is not entirely without&#xA;foundation to think that, at least some, refer either to further&#xA;lessons in chronological order or to subsequent parts of the first&#xA;ones which have already been mentioned.&lt;/p&gt;&#xA;&lt;p&gt;In the edition of all these writings we have always wanted to stick&#xA;with the greatest fidelity to the autograph. The additions and&#xA;marginal annotations were inserted in their place in the text, also&#xA;warning the reader of them, when they were referred to by some sign;&#xA;placed in a note, when we found no trace of the recall. Furthermore,&#xA;it did not seem appropriate to neglect even some passages which the&#xA;author canceled by replacing them with others; such as those who, in&#xA;certain details, sometimes of little, sometimes of greater importance,&#xA;show us the successive work of elaboration, and of the thought and of&#xA;the form, used by Galileo around this work of his. We reported the&#xA;strokes erased, which were of some extent, in special notes: where&#xA;instead, and so it is most. Sometimes, the correction concerns only a&#xA;few words, we recorded the deleted sentence at the foot of the page,&#xA;together with the manifest errors of the autograph which, as always,&#xA;we thought we should amend in the text, however realizing them very&#xA;exactly. And the errors are unfortunately too frequent, more than one&#xA;would believe, for certain the passing of the pen of the great man to&#xA;assimilations or attractions, if you prefer, of endings, or to&#xA;viciously anacolutic constructions. Such errors we note in italics;&#xA;where the manuscript strikethroughs are indicated in roman type,&#xA;preceded and followed by&lt;/p&gt;&#xA;&lt;p&gt;[249]&lt;/p&gt;&#xA;&lt;p&gt;words of the text (and these in italics) in the midst of which they&#xA;fall, even though they do not always have a syntactic relationship&#xA;with them.&lt;/p&gt;&#xA;&lt;p&gt;With the exception of a chapter which was drawn up several years&#xA;later, and which therefore will be given by us to its proper place in&#xA;the chronological order, all the studies on motion, contained in&#xA;Volume I of Part V and in I of III, they belong to the time when&#xA;Galileo taught in the Studio of Pisa; and even if other documents were&#xA;lacking, the repeated references to the famous experiments he&#xA;performed from the Tower of Pisa around the falling bodies and, even&#xA;more, the scene of the dialogue, which is at the Bocca d'Arno, would&#xA;suffice to make him presume. And probably some commentaries on&#xA;Ptolemy's Almagest are also connected with Galileo's lectures in Pisa,&#xA;which he mentions in these writings (p. 314) that he has already&#xA;composed and is about to publish shortly; but of which in any case&#xA;nothing has come down to us.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2023-08-20T00:04:16.543" LastActivityDate="2023-08-20T10:05:08.123" Title="When was Galileo's De Motu (Antiquiora) Made Available to the Public?" Tags="|physics|galileo|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15659" PostTypeId="1" AcceptedAnswerId="15660" CreationDate="2023-08-19T09:02:49.150" Score="2" ViewCount="164" Body="&lt;p&gt;Someone suggested (please see the comments below) that I post this question on hsm.stackexchange. There is a connection to the history of mathematics in this, regarding the relationship between the completeness of real numbers and the development of calculus, as well as the discovery of the incompleteness of rational numbers.&#xA;&lt;a href=&quot;https://math.stackexchange.com/q/4755260/1209870&quot;&gt;Can the so-called completeness of real numbers be understood as closure under limits in the real number system?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;comment：&#xA;As for the mathematical analysis (your last comment), it is not that much of an achievement to prove that Q is incomplete as to prove that R is complete. In other words, you can do calculus on it and you do not need to introduce infinitesimals (which was very important milestone to achieve as it made calculus rigorous). – Stinking Bishop&lt;/p&gt;&#xA;&lt;p&gt;@ StinkingBishop Your point is: it wasn't the development of calculus that led humanity to discover the incompleteness of rational numbers. Humanity had already realized that rational numbers were &amp;quot;incomplete&amp;quot; long before, but they didn't know that real numbers were complete. Even in the time of Pythagoras, they only became aware of the incompleteness of rational numbers; nobody had rigorously proven the completeness of real numbers. The discovery of the square root of 2 highlighted the problem but didn't solve it. The square root of 2 alone couldn't prove the completeness of real numbers.– bokabokaboka&lt;/p&gt;&#xA;&lt;p&gt;@StinkingBishop To make calculus more rigorous, humans rigorously proved the completeness of real numbers based on the understanding that rational numbers were already incomplete. This resolved the concerns of mathematicians in the field of mathematical analysis and made calculus perfect. – bokabokaboka&lt;/p&gt;&#xA;&lt;p&gt;Yes, that is my understanding (although the history is always messier as it happens than when we try to summarise it.) You may want to post this also on History of Science and Mathematics StackExchange. – Stinking Bishop&lt;/p&gt;&#xA;" OwnerUserId="18857" LastEditorUserId="18857" LastEditDate="2023-08-19T09:51:33.050" LastActivityDate="2023-08-20T06:25:50.980" Title="Can the so-called completeness of real numbers be understood as closure under limits in the real number system?" Tags="|calculus|ancient-greece|real-analysis|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15660" PostTypeId="2" ParentId="15659" CreationDate="2023-08-20T06:25:50.980" Score="4" Body="&lt;p&gt;Concerning the &amp;quot;connection to the history of mathematics&amp;quot; that you mentioned: There has been a fairly clear concept of what we call a &lt;em&gt;real number&lt;/em&gt; only since Simon Stevin.  In particular, the ancient Greeks did not possess such a concept and therefore could not possibly have asked related questions such as the one you ask.  Stevin's proposal was to represent every number (whether rational or irrational) by an unending decimal; see &lt;a href=&quot;http://doi.org/10.1007/s10699-011-9228-9&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt; for more details.&lt;/p&gt;&#xA;&lt;p&gt;Until the middle of the 19th century, the dominant view was to stay away from (what mathematicians thought were contradictory) &amp;quot;infinite wholes&amp;quot; as Leibniz called them; Cauchy expressed similar sentiments.  Absent an entity that we call today the &amp;quot;&lt;em&gt;set&lt;/em&gt; of real numbers&amp;quot;, the issue of what we call completeness did not arise until the end of the 19th century.  At &lt;a href=&quot;https://hsm.stackexchange.com/q/13073&quot;&gt;First time the real numbers were axiomatized as the &amp;quot;unique complete ordered field&amp;quot;&lt;/a&gt; one finds that Hilbert first axiomatized the set of real numbers by means of an axiom system that included the axiom of completeness in 1900.&lt;/p&gt;&#xA;&lt;p&gt;Cauchy could conceivably have asked himself why exactly it was that what we call a Cauchy sequence always converges to a real number.   For whatever reason, he didn't, and simply assumed this to be true (as in his proof of the intermediate value theorem).&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-08-20T06:25:50.980" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15661" PostTypeId="2" ParentId="15658" CreationDate="2023-08-20T09:23:17.607" Score="4" Body="&lt;p&gt;TL;DR The publication history of Galileo's &lt;em&gt;De motu&lt;/em&gt; is essentially as outlined in the question: In 1854 Albèri published an initial portion, in 1883 Favaro published the balance of the work, and the first complete version was published by Favaro in 1890. There is no evidence of earlier publication. There is a possibility that copies of the relevant manuscript &lt;em&gt;may&lt;/em&gt; have been available to a few individuals, but I have not been able to find any evidence of that in the literature. The asker notes that the erroneous 1687 publication date was included in Wikipedia at one point; it may have spread from there through &lt;a href=&quot;https://xkcd.com/978/&quot; rel=&quot;nofollow noreferrer&quot;&gt;citogenesis&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Francesco Crapanzano, &amp;quot;&lt;em&gt;Per quel le confuse carte...&lt;/em&gt; The Galilean &lt;em&gt;De motu&lt;/em&gt; in Raffaello Caverni’s Reading.&amp;quot; &lt;em&gt;Philosophia Scientiæ&lt;/em&gt;, Vol. 21, No. 1, Feb. 2017, pp. 17-33 (&lt;a href=&quot;https://www.cairn.info/revue-philosophia-scientiae-2017-1-page-17.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;traces the chain of custody of the manuscript after Galileo's death in 1642 from Vincenzo Viviani to the initial publications by Albèri and Favaro. No earlier publications are mentioned. The Austrian historian &lt;a href=&quot;https://de.wikipedia.org/wiki/Karl_von_Gebler&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karl von Gebler&lt;/a&gt; (1850-1878), who in 1877 famously published the files of Galileo's trial based on the Vatican manuscript&lt;sup&gt;1&lt;/sup&gt;, states that the work circulated in a small number of copies, but did not appear in print until Alberi's edition of Galileo's works: Karl von Gebler, &lt;em&gt;Galileo Galilei und die römische Kurie.&lt;/em&gt; Stuttgart: Cotta 1876, &lt;a href=&quot;https://www.google.com/books/edition/Galileo_Galilei_und_die_r%C3%B6mische_Kurie/yJo8vwghOVIC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA12&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 12&lt;/a&gt;. No supporting references are provided.&lt;/p&gt;&#xA;&lt;p&gt;Karl von Gebler (tr. Jane Sturge), &lt;em&gt;Galileo Galilei and the Roman Curia.&lt;/em&gt; London: C. Kegan Paul 1879, &lt;a href=&quot;https://books.google.com/books?id=yDnBgoFt5RsC&amp;amp;ots=R5B4KCDtmw&amp;amp;dq=%22De%20Motu%22%20%22Galileo%22&amp;amp;lr&amp;amp;pg=PA9#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 9&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the time of his professorship at Pisa he made his grand researches&#xA;into the laws of gravitation, now know under &amp;quot;Galileo's Laws,&amp;quot; and wrote&#xA;as the result of them his great treatise &amp;quot;De Motu Gravium.&amp;quot; It then had&#xA;but a limited circulation in copies, and did not appear in print until two&#xA;hundred years after his death, in Albèris &amp;quot;Opere complete di Galileo Galilei.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The first known &lt;em&gt;partial&lt;/em&gt; publication therefore dates to 1854, bearing the annotation &lt;em&gt;NUNC PRIMUM EDITI&lt;/em&gt; (now the first edition):&lt;/p&gt;&#xA;&lt;p&gt;Eugenio Albèri (ed.), &lt;em&gt;Le Opere di Galileo Galilei&lt;/em&gt;, Vol. XI, Florence: Società Editrice Fiorentina 1854, &lt;a href=&quot;https://www.google.com/books/edition/Le_opere_di_Galileo_Galilei_Opere_fisico/PPEsEzl-WY0C?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA1&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 1-80&lt;/a&gt;, &amp;quot;Sermones De Motu Gravium&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In his introduction Albèri states that the original material consisted of several lectures on the motion of bodies as Galileo may have delivered them at the university, as well as a dialogue between protagonists expounding the same information, a method favored by Galileo as exemplified by his famous works &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems&quot; rel=&quot;nofollow noreferrer&quot;&gt;Massimi Sistemi&lt;/a&gt;&lt;/em&gt; and &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Two_New_Sciences&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nuove Scienze&lt;/a&gt;&lt;/em&gt;. Albèri therefore selected only the dialogue for publication, but added an essay covering the other material.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Questi scritti, i più antichi della mano di Galileo, si hanno nel Tomo I della Parte V dei Codici Palatini, e constano di due distinte parti. L'una com prende diversi Capitoli o Lezioni intorno il moto dei gravi, che verosimilmente Galileo veniva esponendo dalla cattedra a suoi uditori: l altra è un Dialogo che riassume e coordina quelle medesime dottrine, e il quale sin d'allora si proponeva forse di pubblicare in tal forma, che possiam dire da lui prediletta, siccome quella di cui fece uso nelle due grandi opere dei Massimi Sistemi e delle Nuove Scienze.&lt;br /&gt;&#xA;[...]&lt;br /&gt;&#xA;Delle due analoghe scritture sopraccennate pubblichiamo, per le ragioni dette, soltanto la seconda, quella cioè in forma di dialogo, aggiungendovi bensì quattro capitoli siccome saggio dell'altra.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1883 Antonio Favaro published an article in four parts on the &lt;em&gt;unpublished&lt;/em&gt; writings of Galileo Galilei, which included the parts of &lt;em&gt;De Motu&lt;/em&gt; omitted by Albèri:&lt;/p&gt;&#xA;&lt;p&gt;A. Favaro, &amp;quot;Alcuni Scritti Inediti di Galileo Galilei&amp;quot;, &lt;em&gt;Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche&lt;/em&gt;, Vol. 16, 1883; Jan. &lt;a href=&quot;https://www.google.com/books/edition/Bullettino_di_bibliografia_e_di_storia_d/LS7WnuD3t8kC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA1&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 1-64&lt;/a&gt;; Feb. &lt;a href=&quot;https://www.google.com/books/edition/Bullettino_di_bibliografia_e_di_storia_d/LS7WnuD3t8kC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA65&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 65-97&lt;/a&gt;; Mar. &lt;a href=&quot;https://www.google.com/books/edition/Bullettino_di_bibliografia_e_di_storia_d/LS7WnuD3t8kC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA135&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 135-210&lt;/a&gt;. &lt;sup&gt;2&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;The following year, Favaro combined all parts into a &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=-swFAAAAQAAJ&amp;amp;oi=fnd&amp;amp;pg=PA3&amp;amp;dq=%22Alcuni+Scritti+Inediti+di+Galileo+Galilei%22&amp;amp;ots=cmy_xZOuzS&amp;amp;sig=magICoQUGkWmMzT9O1ad9zRf6A0#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;single volume&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;Antonio Favaro, &lt;em&gt;Alcuni Scritti Inediti di Galileo Galilei.&lt;/em&gt; Rome: Tipografia delle Scienze Matematiche e Fisiche 1884, 175 pp.&lt;/p&gt;&#xA;&lt;p&gt;The complete work was published in:&lt;/p&gt;&#xA;&lt;p&gt;Antonio Favaro (ed.), &lt;em&gt;Le Opere di Galileo Galilei,&lt;/em&gt; Vol. 1. Florence: G. Barbèra 1890, &lt;a href=&quot;https://www.google.com/books/edition/Le_opere_di_Galileo_Galilei/Fs-jH38qbBsC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA243&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 243-419&lt;/a&gt;, &amp;quot;De Motu.&amp;quot;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Karl von Gebler, &lt;em&gt;Die Akten des Galilei'schen Prozesses.&lt;/em&gt; Stuttgart: Cotta 1877, 192 pp.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; The bulletin was apparently printed in quarto format, that is, eight pages to a full sheet. Each sheet has the publication name and issue month printed in the lower left corner and the sheet number in the lower right corner. The January issue comprises sheets 1-8, the February issue comprises sheets 9-17, and the March issue comprises sheets 18-26 and part of sheet 27.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;3&lt;/sup&gt; Both Galileo's father Vincenzo (1520-1591) and his younger brother Michelagnolo (1575-1631) were lutenists and composers. You can find recordings of their music on YouTube.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-08-20T10:05:08.123" LastActivityDate="2023-08-20T10:05:08.123" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15662" PostTypeId="2" ParentId="15657" CreationDate="2023-08-20T13:45:38.287" Score="9" Body="&lt;p&gt;I am almost sure that this is a fiction added for dramatic purposes. First, neither &lt;a href=&quot;https://www.prospectmagazine.co.uk/ideas/philosophy/44997/gottlob-frege-the-machine-in-the-ghost&quot; rel=&quot;noreferrer&quot;&gt;Monk&lt;/a&gt; (who writes specifically about this episode and who has a certain flair for the dramatic, and so would presumably include this episode in his account if it were true) nor Dale Jacquette mentions anything of the sort in their accounts of Frege's response to Russell's paradox (incidentally, I do not recommend this biography---cf. &lt;a href=&quot;https://www.logicmatters.net/2020/03/19/dale-jacquettes-biography-of-frege/&quot; rel=&quot;noreferrer&quot;&gt;this accurate assessment by Peter Smith&lt;/a&gt;). Indeed, Dummett even goes out of his way to say that Frege was initially not taken aback by the revelation: &amp;quot;Frege's initial reaction, as shown in his first reply to Russell, was consternation. By his second letter, he had recovered his composure&amp;quot; (&lt;em&gt;Frege: Philosophy of Language&lt;/em&gt;, p. xl). In short, there is no evidence for this reaction and plenty of evidence suggesting the opposite.&lt;/p&gt;&#xA;&lt;p&gt;That is not to say that Frege did not have his share of problems, but they seemed to be caused by personal matters, such as the death of his wife and the difficulty in raising their son alone. Bynum, in his introduction to his edition of the &lt;em&gt;Begriffsschrift&lt;/em&gt;, takes this line:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;During the decade following the publication of &lt;em&gt;Basic Laws Arithmetic II&lt;/em&gt; Frege led a rather unhappy life. There is a widespread myth that Russell's Paradox had left him a disappointed and broken man; but actually, at least until 1914, he believed his logistic programme had been carried out successfully. The problem was rather that his health was not good, and in 1905 death claimed his wife, leaving him to raise their young adopted son alone. (&lt;em&gt;Conceptual Notation and Related Articles&lt;/em&gt;, p. 50)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bynum also notes that in the next paragraph that &amp;quot;Right up to his retirement from Jena University in 1918, however, Frege remained a keen and active scholar, continuing to work on new applications of his 'conceptual notation' and to study critically the views of others on the foundations of mathematics&amp;quot; (ibid.). Hardly the image of a broken man!&lt;/p&gt;&#xA;" OwnerUserId="4097" LastActivityDate="2023-08-20T13:45:38.287" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15663" PostTypeId="1" AcceptedAnswerId="15664" CreationDate="2023-08-20T15:07:25.257" Score="2" ViewCount="208" Body="&lt;p&gt;In a Youtube Video by author AnotherRoof, the author described Descartes as an &amp;quot;arrogant&amp;quot; person who is right, quoting the sentence &amp;quot;I think, therefore I am&amp;quot;. However when I am trying to find reference, it seems that the sentence comes from one of his thought experiments where he denies the existence of his body and therefore proves his mind is of true undeniable existence. I also read through the Wikipedia article about &lt;a href=&quot;https://fr.wikipedia.org/wiki/Ren%C3%A9_Descartes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Descartes&lt;/a&gt; but there is no reference that he was an arrogant person.&lt;/p&gt;&#xA;&lt;p&gt;To clarify a bit more, I understand that the sentence &amp;quot;I think, therefore I am&amp;quot; refers to a thought experiment and is not an evidence of his personality. I also understand that in the video the Author pointed out some other quotation of Descartes's, most of which indicates a rivalry between him and Pierre de Fermat. However none of the quotation shown in the video appears to me that Descartes was arrogant, rather than being logical. The author in the video talks in a way that seems to imply that Descartes is well-known for being arrogant at the time, and the author is a pretty good mathematician so I would think there are some references about this &amp;quot;fact&amp;quot;. I tried to find reference about his personality but couldn't find any.&lt;/p&gt;&#xA;" OwnerUserId="9748" LastEditorUserId="6609" LastEditDate="2023-08-22T13:26:48.087" LastActivityDate="2023-08-22T13:26:48.087" Title="Was René Descartes an arrogant man?" Tags="|descartes|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15664" PostTypeId="2" ParentId="15663" CreationDate="2023-08-20T17:11:38.553" Score="5" Body="&lt;p&gt;You may have gotten downvoted because you took the &amp;quot;arrogant&amp;quot; claim from the video and misdirected it to the quote &amp;quot;&lt;em&gt;cogito, ergo sum&lt;/em&gt; (I think, therefore I am),&amp;quot; which has nothing to do with arrogance and more of an initial assumption for an epistemological question: In the search for truth, where do we begin?  What forms the basis of our reality? Descartes's answer is essentially &amp;quot;Since I'm capable of thought and doubt, then I know that I exist as a fundamental truth.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Cogito,_ergo_sum&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Cogito,_ergo_sum&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If you rewatch the video you mentioned, AnotherRoof doesn't rely on &amp;quot;I think, therefore I am&amp;quot; as proof of Descartes's arrogance, rather, he points to Descartes's high self-confidence in his geometry: &amp;quot;I do not enjoy speaking in praise of myself, but since few people can understand my geometry, and since you wish me to give you an opinion of it, I think it would be well to say that it is all I could hope for. [...] I have only tried to persuade people that my method is better than the ordinary one.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;However, some historians have labeled Descartes as being arrogant. For instance, in &amp;quot;Descartes: A Biography&amp;quot; written by Desmond Clarke writes, &amp;quot;He had become [by 1638] a reclusive, cantankerous, and oversensitive loner, who worried incessantly about his place in history and the priority he claimed for various discoveries&amp;quot;. Clark writes of Descartes' &amp;quot;sensitivity to criticism and the certainty that he claimed, prematurely, for his own view&amp;quot;, and that he &amp;quot;fought with almost everyone he encountered while constantly announcing that all he wanted was 'the security and tranquility' required to complete his intellectual projects&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="13182" LastEditorUserId="13182" LastEditDate="2023-08-20T18:09:26.837" LastActivityDate="2023-08-20T18:09:26.837" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15665" PostTypeId="1" CreationDate="2023-08-20T19:31:44.997" Score="2" ViewCount="556" Body="&lt;p&gt;I was looking at Wikipedia's wonderful &lt;a href=&quot;https://en.wikipedia.org/wiki/Table_of_mathematical_symbols_by_introduction_date&quot; rel=&quot;nofollow noreferrer&quot;&gt;table of the history of certain mathematical symbols&lt;/a&gt;, and there was a certain glaring omission: the use of ℝ to apply to the set of all real numbers. They have references for ℕ, ℤ, ℚ, and fascinatingly ℂ makes an appearance even though ℝ does not. Is there anyone with a source on who was the first to use it and when? I think it would be great to update Wiki with this crucial information (but also post it here).&lt;/p&gt;&#xA;" OwnerUserId="12930" LastActivityDate="2023-08-27T16:06:14.497" Title="Use of blackboard bold &quot;ℝ&quot; to refer to real numbers?" Tags="|notation|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15666" PostTypeId="2" ParentId="15663" CreationDate="2023-08-21T06:30:53.310" Score="3" Body="&lt;p&gt;Descartes, who while he was alive was politically more powerful in France than Fermat, had a priority dispute with Fermat in several areas, including finding tangents to curves.  As noted by historians, Fermat's method was better because it works also in an affine setting, whereas Descartes' method relied on finding normals which is, in hindsight, somewhat irrelevant.  Descartes did not fully understand Fermat's method of &lt;em&gt;adequality&lt;/em&gt;, and attacked it as being applicable only in special cases.  Even though the criticism is not valid, it is sometimes reproduced by modern historians until this day.  For details, see articles on Fermat listed &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/fermat.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.  I don't know if this makes him arrogant; certainly stubborn.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-08-21T06:30:53.310" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15667" PostTypeId="1" CreationDate="2023-08-22T00:32:30.743" Score="0" ViewCount="66" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sphere_eversion&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sphere eversion&lt;/a&gt; is the process of turning a sphere inside out in a three-dimensional space. See also this animation on YouTube: &lt;a href=&quot;https://www.youtube.com/watch?v=wO61D9x6lNY&quot; rel=&quot;nofollow noreferrer&quot;&gt;Outside in (2011)&lt;/a&gt; and picture below. My question is, what is the historical context behind the problem? More specifically, why was the problem raised?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/QDCyP.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/QDCyP.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="6609" LastEditDate="2023-08-24T09:22:18.513" LastActivityDate="2023-08-24T09:22:18.513" Title="Why the sphere eversion problem emerged?" Tags="|topology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15670" PostTypeId="2" ParentId="15656" CreationDate="2023-08-23T12:20:54.650" Score="2" Body="&lt;p&gt;The background, and the applications, have to do with practical surveying of the earth's surface.  One of the uses Gauss makes of the &lt;em&gt;theorema egregium&lt;/em&gt; is a comparison theorem for small triangles when one compares a triangle in the surface with the corresponding triangle of the same sidelengths in the plane.  This is discussed in Dombrowski, pages 114-115.  The earth is less curved toward the poles, with the result that the angular correction will be smaller for vertices closer to the poles:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss gives these different correction values according to (35) for one of the&#xA;largest terrestial triangles measured by him, namely with &amp;quot;vertices&amp;quot; at&#xA;Brocken, Hohehagen and Inselsberg (Dombrowski p. 115).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-08-23T12:20:54.650" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15671" PostTypeId="1" CreationDate="2023-08-23T22:28:24.373" Score="2" ViewCount="71" Body="&lt;p&gt;In what context where vector calculus concepts, such as:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Circulation&lt;/li&gt;&#xA;&lt;li&gt;Flux&lt;/li&gt;&#xA;&lt;li&gt;Curl&lt;/li&gt;&#xA;&lt;li&gt;Divergence&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;first developed? I had assumed they were developed first in fluid dynamics, since the flow of water is concrete and important, and from there applied to electromagnetics, which are obviously more abstract.  However, I can find no source to support this.&lt;/p&gt;&#xA;&lt;p&gt;This &lt;a href=&quot;https://hsm.stackexchange.com/questions/11046/what-was-the-real-need-of-divergence-and-curl-operators&quot;&gt;post&lt;/a&gt; suggests they were first developed for quaternions, ie pure math, and only later applied to physics.  My question still stands: What were the initial physical applications and how did they spread?&lt;/p&gt;&#xA;&lt;p&gt;Related but not duplicate: &lt;a href=&quot;https://hsm.stackexchange.com/questions/1925/how-were-vector-calculus-nabla-%e2%88%87-identities-first-derived&quot;&gt;How were vector calculus nabla ∇ identities first derived?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18545" LastActivityDate="2023-08-23T22:28:24.373" Title="What were the initial physical applications of vector calculus such as curl, div, circulation, and flux?" Tags="|mathematics|physics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15672" PostTypeId="1" AcceptedAnswerId="15674" CreationDate="2023-08-24T21:42:56.143" Score="2" ViewCount="132" Body="&lt;p&gt;Malus first discovered light polarization just before 1810, but Maxwell's equation stuff came around 1860s.  How was polarization interpreted before the EM theory?&lt;/p&gt;&#xA;" OwnerUserId="18879" LastEditorUserId="6609" LastEditDate="2023-08-25T08:51:46.427" LastActivityDate="2023-08-25T09:24:08.857" Title="How was light polarization interpreted when first discovered?" Tags="|physics|light|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15673" PostTypeId="1" CreationDate="2023-08-25T08:49:01.093" Score="1" ViewCount="81" Body="&lt;p&gt;These days, I'm interested in class field theory.&lt;/p&gt;&#xA;&lt;p&gt;I know that Weber first made his own 'class field' and then Hilbert made his own field, 'Hilbert class field'.&lt;/p&gt;&#xA;&lt;p&gt;By the way, were there so many weak points in Weber's class field that Hilbert had to create a Hilbert class field? If so, what are the weak points?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-08-25T22:43:39.163" LastActivityDate="2023-08-25T22:43:39.163" Title="What are Weber class field's weak points?" Tags="|mathematics-social-history|hilbert|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15674" PostTypeId="2" ParentId="15672" CreationDate="2023-08-25T09:13:24.360" Score="2" Body="&lt;p&gt;Short version: they either considered a wave in the aether (the aether being some mysterious medium that filled space allowing for only transverse waves) or corpuscules with additional degrees of freedom.&lt;/p&gt;&#xA;&lt;h2&gt;More&lt;/h2&gt;&#xA;&lt;p&gt;Along with his discovery, Malus tried to explain polarization of light using the &amp;quot;particle theory&amp;quot;. He just stated that light corpuscules could have an alignment similar to that of a bar magnet in a magnetic field.&lt;/p&gt;&#xA;&lt;p&gt;Based on 1816 polarization experiments of Arago and Fresnel, Thomas Young (who considered a &amp;quot;wave theory&amp;quot; of light) deduced in 1817 that light have to be a transverse wave. Fresnel would later write his theory of light (1832) propagation founded on the idea that light is a transverse wave on the aether.&lt;/p&gt;&#xA;&lt;p&gt;One had to wait until Maxwell (even if some others already hinted at it) to tie light to electromagnetism through Maxwell's equations.&lt;/p&gt;&#xA;&lt;p&gt;With Michelson–Morley experiment (1887) and later with the success of Einstein's special relativity (1905) the idea of aether would be put in the coffin. EM waves can travel without medium.&lt;/p&gt;&#xA;&lt;p&gt;Note that if light was a wave or a particle it does not matter anymore, with quantum mechanics we have understood that is something more complex.&lt;/p&gt;&#xA;&lt;p&gt;Taken from: &lt;a href=&quot;https://arago.elte.hu/sites/default/files/DSc-Thesis-2003-GaborHorvath-01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gábor Horváth  D.Sc. Thesis chapter 1&lt;/a&gt;.&#xA;For a little more depth I found useful the history section of &lt;a href=&quot;https://www.google.fr/books/edition/Polarized_Light/w6PMBQAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=history%20of%20light%20polarization&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Polarized Light&lt;/em&gt; by H. Goldstein&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The idea of luminiferous aether appears a bit earlier, Huygens already discussed the idea in 1690 &lt;em&gt;Traité de la Lumière:&lt;/em&gt; but he was not carrying polarization experiments. The corpuscular theory of light goes back to Newton's &lt;em&gt;Opticks&lt;/em&gt; (1704).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-08-25T09:24:08.857" LastActivityDate="2023-08-25T09:24:08.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15675" PostTypeId="1" AcceptedAnswerId="15678" CreationDate="2023-08-25T14:18:48.517" Score="1" ViewCount="257" Body="&lt;p&gt;It is said that the &lt;a href=&quot;https://en.wikipedia.org/wiki/Perspective_(graphical)&quot; rel=&quot;nofollow noreferrer&quot;&gt;theory of perspective&lt;/a&gt; in art was greatly developed during the Renaissance because of the search for reality at this time: more realistic representations.&lt;/p&gt;&#xA;&lt;p&gt;My question is what motivated this search for reality? Were there philosophical, theological reasons, etc? Was it an intellectual challenge? curiosity? demands from the wealthier social class who wanted paintings like photographs?&#xA;Are there references and studies that deal with this issue?&lt;/p&gt;&#xA;&lt;p&gt;The theory of perspective is an important part of the history of science and mathematics. It is a mathematical model that describes how objects appear to change in size and shape as they are seen from different angles. This model has been used in art, architecture, and engineering for centuries.&lt;/p&gt;&#xA;&lt;p&gt;One of the earliest known references to the theory of perspective is in the work of the ancient Greek mathematician Euclid. In his book, Optics, Euclid describes how to construct a perspective drawing. He also discusses the effects of perspective on the appearance of objects, such as how they appear to shrink as they move away from the viewer.&lt;/p&gt;&#xA;&lt;p&gt;The theory of perspective was further developed in the Renaissance by artists and mathematicians such as Filippo Brunelleschi, Leon Battista Alberti, and Albrecht Dürer. These artists used the principles of perspective to create realistic and convincing paintings and sculptures. They also used perspective to create illusions of depth and space.&lt;/p&gt;&#xA;&lt;p&gt;The theory of perspective has also been applied to other fields, such as engineering and architecture. In engineering, perspective is used to design objects that are safe and effective. In architecture, perspective is used to create buildings that are pleasing to the eye.&lt;/p&gt;&#xA;&lt;p&gt;In conclusion, the theory of perspective is a valuable tool that has been used in a variety of fields. It is a product of the history of science and mathematics, and it continues to be used today.&lt;/p&gt;&#xA;&lt;p&gt;Here are some additional references that support this thesis:&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;Alberti, Leon Battista. On Painting. Translated by John R. Spencer. New Haven, CT: Yale University Press, 1966.&#xA;Dürer, Albrecht. Four Books on Human Proportion. Translated by Caroline Elam. New York: Dover Publications, 1977.&#xA;Kemp, Martin. The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. New Haven, CT: Yale University Press, 1990.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12383" LastEditDate="2023-09-01T13:31:51.630" LastActivityDate="2023-09-01T13:31:51.630" Title="What motivated the development of the theory of perspective during the Renaissance?" Tags="|projective-geometry|" AnswerCount="1" CommentCount="5" ClosedDate="2023-09-01T00:52:08.603" ContentLicense="CC BY-SA 4.0" />
  <row Id="15676" PostTypeId="2" ParentId="7600" CreationDate="2023-08-25T14:40:21.777" Score="2" Body="&lt;p&gt;Regarding the last fragments of Gauss on Pentagramma mirificum, which are dated to 1843 and are numbered as [9]-[11] in the &lt;a href=&quot;https://www.researchgate.net/publication/235430910_Draft_Translation_of_Gauss%27_Fragmentary_Notes_on_the_Pentagramma_Mirificum&quot; rel=&quot;nofollow noreferrer&quot;&gt;translation of his fragments&lt;/a&gt;, I believe one way to put it into historic context is to remark that Gauss simply discovered here Clebsch's theorem on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pentagram_map&quot; rel=&quot;nofollow noreferrer&quot;&gt;pentagram map&lt;/a&gt;. Clebsch's theorem (proved by Alfred Clebsch in 1871) states that for any rectilinear pentagon, if one draws all its diagonals to form an inner pentagon, than the smaller pentagon is projectively equivalent to the original pentagon. Two planar figures are projectively equivalent if there is a projective mapping that sends one into the other, when a projective mapping is defined as any composition of perspective projections centered at a series of points &lt;span class=&quot;math-container&quot;&gt;$O_1,O_2,O_3$&lt;/span&gt;, etc.&lt;/p&gt;&#xA;&lt;p&gt;In his comments on Gauss's pentagramma mirificum, Robert Fricke states that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss constructed here first by continuing the diagonal and the lengthening of sides, a chain&#xA;endless in both directions, of pentagons and recognizes, that this grid (net) is transformed into one another circumscribing pentagons&#xA;through collineation:&lt;span class=&quot;math-container&quot;&gt;$$(5) x' = (\frac{2G-1}{2G'-1})x, y' = (\frac{2G-1}{2G''-1})y$$&lt;/span&gt;&#xA;For making things clear, compare the figure provided; Gauss himself made ready drawings of this kind, in which the net of pentagons are carried through still much further. The coefficients which stand on the right hand, in (5), are the “exponents” or “coefficients of the rejuvenations” which in [9] and [10] come into action. It lies very close by, to conceive the here-presented relationship of things, in the sense of modern theory of discontinuous substitution groups. The zero point, &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;, and is one of the three boundary points of the form (5) originating cyclical&#xA;collineation group.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The term &lt;a href=&quot;https://en.wikipedia.org/wiki/Collineation&quot; rel=&quot;nofollow noreferrer&quot;&gt;collineation&lt;/a&gt; refers to one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of any 3 collinear points are themselves collinear. Therefore, if I understand Fricke's remarks correctly, than his description of Gauss's ideas correspond to Clebsch theorem.&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="2554" LastEditDate="2023-08-27T14:51:42.220" LastActivityDate="2023-08-27T14:51:42.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15677" PostTypeId="1" AcceptedAnswerId="15681" CreationDate="2023-08-26T18:11:19.133" Score="1" ViewCount="68" Body="&lt;p&gt;Kepler tried to use regular polygons before using the 3D platonic solids in his Mysterium Cosmographicum.&#xA;My question is: which regular polygons did Kepler try to use and in what nesting order?&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="13182" LastEditDate="2023-08-26T18:59:54.240" LastActivityDate="2023-08-27T07:27:29.593" Title="Kepler's Mysterium Cosmographicum with regular polygons: in which nesting order?" Tags="|astronomy|geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15678" PostTypeId="2" ParentId="15675" CreationDate="2023-08-26T18:58:38.633" Score="4" Body="&lt;p&gt;There were of course philosophical and cultural reasons for the development of the &lt;em&gt;(linear) perspective&lt;/em&gt; during the Renaissance.&#xA;It cannot be considered simply a change of tastes of patrons of art or a phenomenon out of curiosity.&lt;/p&gt;&#xA;&lt;p&gt;It was linked to profound cultural changes  and different attitudes toward the role of man in the world and toward religion which took place during the so-called &lt;a href=&quot;https://www.britannica.com/topic/humanism&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;humanism&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt; ( also called &lt;em&gt;Renaissance humanism&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;And it was also linked to a renewed interest in sciences, as physics and mathematics: consider, first of all &lt;strong&gt;Leonardo Da Vinci&lt;/strong&gt;, who was at the same time one of the greatest artists of all time and a scientist.&lt;/p&gt;&#xA;&lt;p&gt;In short words, i&lt;strong&gt;t can be said that perspective can be considered a consequence and an expression of the modern anthropocentrism, which arose with humanism and Renaissance.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;By the term &lt;em&gt;&lt;strong&gt;Renaissance humanism&lt;/strong&gt;&lt;/em&gt;  historians  denote a vast cultural renewal movement, in arts and in philosophy, which characterized European civilization, conventionally established between the end of the XIV century and the end of XV century, which had an attitude of sharp break with Medieval thought, above all by setting the man at the center of the vision of life and of the world, and therefore substituting anthropocentrism for theocentrism, which was typical of the previous era.&lt;/p&gt;&#xA;&lt;p&gt;Renaissance humanism  was characterized by an intense study of ancient classical authors, and an acknowledgment of the privileged position of man in  nature.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In figurative arts, perspective is the expression of this new anthropocentrism:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In perspective the anthropocentric culture of modern era finds expression.&#xA;The new painting style which we call Renaissance, based on perspective and on the rationalization of space, is the sign of a new attitude of man toward the world, of a spiritual, cultural and social transformation […]. The concept of space which emerges  is not an expression of eternal ideas, immutable and abstract[…] The perspective image confers to the observer a privileged position, that acknowledges the primacy of the subject in the area  of knowledge,  and their centrality in historical agency. For this reason, perspective is the sign of the constitution of an anthropocentric thought, which abandons the Medieval theocentric attitude.&lt;/p&gt;&#xA;&lt;p&gt;[…] The space that is represented is no longer the divine and absolute&#xA;space, but the projection of a subjective point of view, both&#xA;individual and universal: individual because it is supposed to be&#xA;occupied by a unique observer, with only one open eye; universal as&#xA;any other observer could stay on that point  and reconstruct the space&#xA;which from it takes form according to the universal laws of Euclidean&#xA;geometry and optic. &lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The literature about Renaissance and perspective is very vast.&lt;/strong&gt; Some of the most important historians of art dedicated efforts to the subject.&lt;/p&gt;&#xA;&lt;p&gt;With regard to  perspective, specifically, there are two books of some of the most well-known scholars of art:&lt;/p&gt;&#xA;&lt;p&gt;Panofsky, Erwin,  &lt;a href=&quot;https://www.amazon.it/Perspective-Symbolic-Form-Erwin-Panofsky/dp/0942299523&quot; rel=&quot;nofollow noreferrer&quot;&gt;Perspective as a symbolic Form&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Florensky, Pavel,   &lt;a href=&quot;https://www.amazon.it/Beyond-Vision-Essays-Perception-English-ebook/dp/B00DBW7KO8/ref=sr_1_1?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;amp;crid=J2RB2E4CAFIG&amp;amp;keywords=Florensky+Beyond+Vision&amp;amp;qid=1693057617&amp;amp;sprefix=florensky+beyond+vision%2Caps%2C257&amp;amp;sr=8-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Beyond Vision&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Florensky was a Russian philosopher, a mathematician and an art historian, and he wrote also about art and mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Perspective is at the center of many connections between art and sciences&lt;/strong&gt;, in particular, of course, mathematics and physics.&lt;/p&gt;&#xA;&lt;p&gt;Boyer, &lt;em&gt;A History of Mathematics&lt;/em&gt;,  and Kline, &lt;em&gt;Mathematic Thought from Ancient to Modern Times&lt;/em&gt;, devote some chapters to Renaissance and mathematics, and in particular to the relationships between mathematics, perspective  and art.&lt;/p&gt;&#xA;&lt;p&gt;Boyer recalls Italian and European artist who developed relatively new links between mathematics and figurative arts, as Filippo Brunelleschi, Leon Battista Alberti, Piero Della Francesca, Leonardo Da Vinci, Albrecht Dürer.&lt;/p&gt;&#xA;&lt;p&gt;In particular it is interesting that he mentions a treatise, which has been lost, by Leonardo Da Vinci, about perspective, &lt;em&gt;Trattato della Pittura&lt;/em&gt; (Treatise of Painting), which begins with the admonition. “Don’t read me, if you are not a mathematician!” (Boyer, Chap. 15, sec. 20) &lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;,&lt;/p&gt;&#xA;&lt;p&gt;Morris Klein devotes a chapter to the mathematical contribution of Renaissance, and recalls that the Renaissance artists were ‘universal’ men, as they had to do any kind of jobs, so that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;they were obliged to learn mathematics, physics, architecture,&#xA;stonecutting, engineering, metalworking, anatomy, woodworking, optics,&#xA;statics,  and hydraulics. They worked manually and at the same time&#xA;they coped with the most abstract problems. &lt;em&gt;At least in the XV&lt;/em&gt;&#xA;&lt;em&gt;century,&lt;/em&gt; &lt;em&gt;they were the best mathematical physicists.&lt;/em&gt; [&lt;em&gt;emphasis mine&lt;/em&gt;] (Kline, Chap. 12, section 1) &lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Therefore, they contributed to the development of geometry, for instance Alberti introduced the concepts of &lt;em&gt;projection&lt;/em&gt; and &lt;em&gt;section&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Kline, moreover,  points out  that at the origin of the projective geometry there were some problems that arose in the theory of perspective:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Some geometers of the XVIII century tried to give an answer to these&#xA;problems. They considered the methods and the results they obtained as&#xA;a part of Euclidean geometry. Actually, these methods and results [..&#xA;.] proved to be the origin of a new branch of geometry , which in the&#xA;XIX century became known as projective geometry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Obviously,  the studies about perspective were linked not only to mathematics but &lt;strong&gt;also to physics.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;An interesting study available in English is&lt;/p&gt;&#xA;&lt;p&gt;Camerota, Filippo, &lt;a href=&quot;https://www.amazon.it/Perspective-Galileo-Ludovico-Cigolis-Prospettiva/dp/8822259866/ref=sr_1_7?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;amp;crid=1ZVGS9UJRUJU8&amp;amp;keywords=camerota+filippo&amp;amp;qid=1693055084&amp;amp;s=books&amp;amp;sprefix=camerota+filippo%2Cstripbooks%2C166&amp;amp;sr=1-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Perspective in the Age of Galileo.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The author here discusses the work of &lt;strong&gt;Ludovico Cigoli&lt;/strong&gt;, &lt;em&gt;La prospettiva pratica&lt;/em&gt; (The Practical Perspective), which is an extraordinary scientific document of the role of the representation of the visible world in &lt;strong&gt;Galilean research&lt;/strong&gt;. Cigoli was a painter very close to Galileo, and his treatise illustrates important aspects of the perspective painting, as orthogonal projections of shadows or the use of mechanical instruments.&lt;/p&gt;&#xA;&lt;p&gt;Another book by Filippo Camerota is &lt;a href=&quot;https://www.amazon.it/gp/product/8837021194/ref=ox_sc_act_title_6?smid=A11IL2PNWYJU7H&amp;amp;psc=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;La prospettiva del Rinascimento. Arte, architettura, scienza.&lt;/em&gt;&lt;/a&gt;&#xA;(Perspective of Renaissance. Art, Architecture, Science)&lt;/p&gt;&#xA;&lt;p&gt;The book is a history of perspective, beginning with the studies about vision based on the geometric model of Euclid, through Giotto, Brunelleschi, Alberti, Leonardo, with its connections with mathematics and science.&lt;/p&gt;&#xA;&lt;p&gt;But I don’t know if an English translation exists.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can see also:&lt;/p&gt;&#xA;&lt;p&gt;Samuel Y. Edgerton,&lt;br /&gt;&#xA;&lt;a href=&quot;https://www.amazon.it/gp/product/0801474809/ref=ox_sc_act_title_2?smid=A11IL2PNWYJU7H&amp;amp;psc=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Mirror, the Window, and the Telescope: How Renaissance Linear Perspective Changed Our Vision of the Universe&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;sup&gt; &lt;a href=&quot;https://finestresuartecinemaemusica.blogspot.com/2018/09/la-prospettiva-come-espressione.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://finestresuartecinemaemusica.blogspot.com/2018/09/la-prospettiva-come-espressione.html&lt;/a&gt; (my translation).   The literature about Renaissance humanism and anthropocentrism is broad, for some link in English see &lt;a href=&quot;https://www.archaeology.wiki/blog/issue/the-anthropocentric-cosmotheory-of-the-renaissance-and-the-cult-of-man/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.archaeology.wiki/blog/issue/the-anthropocentric-cosmotheory-of-the-renaissance-and-the-cult-of-man/&lt;/a&gt;&#xA;&lt;a href=&quot;https://mylandrover.ru/en/suspension/antropocentrizm-i-gumanizm-v-filosofskoi-mysli-vozrozhdeniya-gumanizm-i.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://mylandrover.ru/en/suspension/antropocentrizm-i-gumanizm-v-filosofskoi-mysli-vozrozhdeniya-gumanizm-i.html&lt;/a&gt; &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt; p. 343 Italian edition&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; &lt;sup&gt; p. 271 Italian edition. &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-08-26T19:05:10.787" LastActivityDate="2023-08-26T19:05:10.787" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15679" PostTypeId="2" ParentId="15665" CreationDate="2023-08-26T20:21:39.487" Score="2" Body="&lt;p&gt;This addresses a narrow form of the question:  when did the &amp;quot;blackboard bold&amp;quot; version of letter forms of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; first appear in printed works?&lt;/p&gt;&#xA;&lt;p&gt;Before the AMS gave us the mathbb fonts in the 1980s, Springer Verlag was using its own version, for instance in the journal &lt;em&gt;Mathematische Annalen&lt;/em&gt;.  A spot check in old issues showed its use in the paper by&#xA;Gramsch, B. &lt;a href=&quot;https://link.springer.com/article/10.1007/BF01364278&quot; rel=&quot;nofollow noreferrer&quot;&gt;Funktionalkalkül mehrerer Veränderlichen in lokalbeschränkten Algebren&lt;/a&gt; appearing in&#xA;&lt;em&gt;Mathematische Annalen&lt;/em&gt;, in 1967, vol. 174, pp.311–344.  On &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0174?tify=%7B%22pages%22%3A%5B341%5D%2C%22view%22%3A%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 323&lt;/a&gt; of that paper  we see the equivalent of &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; and on &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN235181684_0174?tify=%7B%22pages%22%3A%5B349%5D%2C%22view%22%3A%22info%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 331&lt;/a&gt; we see the equivalent of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt;.  (Thanks to user njuffa for supplying the location of a non-paywall library version of this paper.)  The design of the letter-form looks, I think, as if it were inspired by the typewriter overstruck I R that one sees in photo-offset lecture notes of the era, such as on p.23 of the 1967 &lt;em&gt;Dynamical Theories of Brownian Motion&lt;/em&gt; by Edward Nelson, in the &amp;quot;Mathematical Notes&amp;quot; series of Princeton University Press.&lt;/p&gt;&#xA;&lt;p&gt;The adoption of this practice must be the result of a matching of tastes of both the mathematical authors and the production staffs of the publishers.  Both Springer and Princeton University Press took pride in their typography, and hear-say makes it clear that the mathematical typists at PUP did so to an exceptional degree.&lt;/p&gt;&#xA;&lt;p&gt;The wider sense of the question, when did distinctive typographic forms (bold face, sans-serif bold, etc) for the symbols for the integers, the real numbers, etc, first appear in print, is addressed in the comments to the OP,&#xA;and an approximately correct answer is: with Bourbaki, decades earlier.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2023-08-27T16:06:14.497" LastActivityDate="2023-08-27T16:06:14.497" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15680" PostTypeId="1" CreationDate="2023-08-27T03:42:09.443" Score="0" ViewCount="80" Body="&lt;p&gt;In August 1823, Abel wrote a letter to Holmboe with a date:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Copenhague, l’an &lt;span class=&quot;math-container&quot;&gt;$ \sqrt[3]{6064321219} $&lt;/span&gt; (en comptant lafraction d´ecimal).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$ 1823 \frac{215}{365} &amp;lt; \sqrt[3]{6064321219} &amp;lt; 1823 \frac{216}{365} $&lt;/span&gt;, so the date would be 1823-08-04 (the 216th day, counting the 1st day as 0).&lt;/p&gt;&#xA;&lt;p&gt;But why did Abel choose such a number? He could have chosen integer &lt;span class=&quot;math-container&quot;&gt;$ n $&lt;/span&gt; between&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ \left( 1823 \frac{215}{365} \right)^3 &amp;lt; 6064303397 \le n \le 6064330729 &amp;lt; \left( 1823 \frac{216}{365} \right)^3 . $$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="5546" LastActivityDate="2023-08-27T03:42:09.443" Title="Why did Abel choose 6064321219?" Tags="|mathematics|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15681" PostTypeId="2" ParentId="15677" CreationDate="2023-08-27T07:03:24.923" Score="4" Body="&lt;p&gt;In the preface to &lt;em&gt;Mysterium Cosmographicum&lt;/em&gt;, Kepler wrote that he first saw that an equilateral triangle fit well between the orbits of Saturn and Jupiter.&lt;/p&gt;&#xA;&lt;p&gt;Noting that this triangle was the first of the regular polygons, he tried to continue the sequence with a square in the second interval (between Jupiter and Mars), a pentagon in the third (between Mars and Earth), and a hexagon in the fourth (between Earth and Venus).&lt;/p&gt;&#xA;&lt;p&gt;He had already noticed that there was an incorrect fit between Jupiter and Mars, so he tried nesting a square, triangle, and pentagon in this interval. (The order is not clear from the text.) This is the sum of the information he gave on his attempts. He noted that finding a correct combination of polygons for each case would be &amp;quot;an endless pursuit of detail&amp;quot; (&amp;quot;Infinitum est singula persequi&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;He abandoned using polygons largely because they did not give a mathematical reason why there were exactly six planets (per the state of knowledge at the time). Since there are exactly five regular solids, they corresponded one-to-one with the intervals between six orbits and so did give such a reason.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;For a summary with explanation of Kepler's attempt to use polygons, see pp.46–49 of J.V. Field, &lt;em&gt;Kepler's Geometrical Cosmology.&lt;/em&gt; Bloomsbury, 2013. ISBN: 978-1-4725-0703-7.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;For Kepler's own account in the preface to the &lt;em&gt;Mysterium Cosmographicum&lt;/em&gt;, see pp.11–12 of Kepler, &lt;a href=&quot;http://publikationen.badw.de/de/002334737&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Gesammelte Werke&lt;/em&gt;, vol.I&lt;/a&gt;. Munich: C.H. Beck, 1938; or pp.65–67 of the translation by A.M. Duncan, &lt;em&gt;The Secret of the Universe.&lt;/em&gt; Abaris Books, 1981. ISBN: 978-0-913870-64-8.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4664" LastEditorUserId="4664" LastEditDate="2023-08-27T07:27:29.593" LastActivityDate="2023-08-27T07:27:29.593" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15682" PostTypeId="1" CreationDate="2023-08-27T09:17:05.577" Score="0" ViewCount="53" Body="&lt;p&gt;According to &lt;a href=&quot;https://math.berkeley.edu/%7Egmelvin/math54f12/math110su12_grassmann_linalg.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hermann Grassmann and the creation of linear algebra&lt;/a&gt; by Desmond Sander, Grassmann was able to identify all the important notions in linear algebra in his book &amp;quot;Ausdehnungslehre&amp;quot;. For him, vector spaces are freely generated by &amp;quot;units&amp;quot; &lt;span class=&quot;math-container&quot;&gt;$e_1, e_2, \cdots$&lt;/span&gt; over real or complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;Then he embarked on studying something he called &amp;quot;linear products&amp;quot; by setting &lt;span class=&quot;math-container&quot;&gt;$$\left(\sum\alpha_ie_i\right)\left(\sum\beta_ie_j\right)=\sum(\alpha_i\beta_j)e_ie_j$$&lt;/span&gt; subject to the condition &lt;span class=&quot;math-container&quot;&gt;$\color{red}{\sum\xi_{ij}e_ie_j=0.}$&lt;/span&gt; According to the paper, to be invariant under change of bases, only possibilities are &lt;span class=&quot;math-container&quot;&gt;$$\color{blue}{e_ie_j=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  e_ie_j=e_je_i,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  e_ie_j=-e_je_i.}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;What are his motivations for the relation &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\sum\xi_{ij}e_ie_j=0$&lt;/span&gt;&amp;quot;?&lt;/li&gt;&#xA;&lt;li&gt;What was his derivation for those commutative relations?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Since I donot read German, I wasn't able to his original book nor I would be able to understand what he has written there.&lt;/p&gt;&#xA;" OwnerUserId="10434" LastActivityDate="2023-08-27T09:17:05.577" Title="Understanding Grassmann's Approach to Algebras over Vector Spaces?" Tags="|linear-algebra|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15684" PostTypeId="1" CreationDate="2023-08-27T12:34:16.857" Score="1" ViewCount="50" Body="&lt;p&gt;Much has already been written about the dissemination of Euclidean geometry into China: &lt;a href=&quot;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-euclid-in-china&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.maa.org/press/periodicals/convergence/mathematical-treasure-euclid-in-china&lt;/a&gt;, &lt;a href=&quot;https://academic.oup.com/book/12776/chapter-abstract/162933660?redirectedFrom=fulltext&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://academic.oup.com/book/12776/chapter-abstract/162933660?redirectedFrom=fulltext&lt;/a&gt; (as well as some commentary on why it was decided to be called &amp;quot;几何&amp;quot;: &lt;a href=&quot;https://hinative.com/questions/22525779&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://hinative.com/questions/22525779&lt;/a&gt;, &lt;a href=&quot;https://en.wiktionary.org/wiki/%E5%B9%BE%E4%BD%95&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wiktionary.org/wiki/%E5%B9%BE%E4%BD%95&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;I am interested in knowing the history of how (Newton's/Leibniz's) calculus was brought to China, and in particular where now-standard terminology like&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;calculus &lt;a href=&quot;https://zh.wikipedia.org/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://zh.wikipedia.org/wiki/%E5%BE%AE%E7%A7%AF%E5%88%86%E5%AD%A6&lt;/a&gt; (with &amp;quot;微&amp;quot; referring to the &amp;quot;differential&amp;quot; half of calculus, and &amp;quot;积&amp;quot; referring to the &amp;quot;integral&amp;quot; half of calculus)&lt;/li&gt;&#xA;&lt;li&gt;derivative &lt;a href=&quot;https://zh.wikipedia.org/zh/%E5%AF%BC%E6%95%B0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://zh.wikipedia.org/zh/%E5%AF%BC%E6%95%B0&lt;/a&gt;, and differentiation &amp;quot;求导&amp;quot;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;first arose.&lt;/p&gt;&#xA;" OwnerUserId="11071" LastActivityDate="2023-08-27T12:34:16.857" Title="Dissemination of Calculus in China" Tags="|mathematics|terminology|calculus|ancient-china|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15685" PostTypeId="1" CreationDate="2023-08-26T18:53:56.607" Score="4" ViewCount="36" Body="&lt;p&gt;Today we know that if you make sequential Stern-Gerlach experiments, where the magnetic fields are at an angle &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;, then the formula for the ratio how the beam splits is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$p_{1}=\cos^2 \bigg(\frac{\alpha}{2}\bigg)$$&lt;/span&gt;&#xA;for one side and&#xA;&lt;span class=&quot;math-container&quot;&gt;$$p_{2}=1-p_{1}=\sin^2 \bigg(\frac{\alpha}{2}\bigg)&#xA;$$&lt;/span&gt;&#xA;for the other side.&lt;/p&gt;&#xA;&lt;p&gt;I understand this, but what I want to know is: Who discovered this formula?&lt;/p&gt;&#xA;&lt;p&gt;I don't think Stern and Gerlach found it, because they don't mention it &lt;a href=&quot;https://arxiv.org/pdf/2301.11343.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;in their original paper&lt;/a&gt;, and I suspect it was already known by the time Pauli discovered his matrices, so it seems this formula was found sometime between 1922 and 1924, but in my research I cannot find any mention of who found it&lt;/p&gt;&#xA;&lt;p&gt;If one of you could please give me a source where they talk about this development, I would be really thankful.&lt;/p&gt;&#xA;" OwnerUserId="18906" OwnerDisplayName="HighlyEntropicMind" LastEditorUserId="12330" LastEditDate="2023-08-28T12:08:39.790" LastActivityDate="2023-08-28T12:08:39.790" Title="Who found the formula for sequential Stern-Gerlach experiments and when?" Tags="|physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15686" PostTypeId="1" CreationDate="2023-08-28T15:59:41.773" Score="0" ViewCount="125" Body="&lt;p&gt;Where to find average man/woman drawings as proposed by &lt;a href=&quot;https://en.wikipedia.org/wiki/Adolphe_Quetelet&quot; rel=&quot;nofollow noreferrer&quot;&gt;Adolphe Quetelet&lt;/a&gt;? Drawings along the years would be very nice. He proposed the idea of average man in 1835 (see &lt;a href=&quot;https://historyofinformation.com/detail.php?entryid=2154&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://historyofinformation.com/detail.php?entryid=2154&lt;/a&gt;). To see how the &amp;quot;average man evolved from 1835 would be nice!&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="6609" LastEditDate="2023-08-29T08:52:02.387" LastActivityDate="2023-08-29T20:21:18.490" Title="Where to find average man/woman drawings as proposed by Quetelet?" Tags="|statistics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15687" PostTypeId="1" AcceptedAnswerId="15688" CreationDate="2023-08-29T00:11:50.233" Score="6" ViewCount="1164" Body="&lt;p&gt;A well-known proof of the Pythagorean Theorem is illustrated in the figure below:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/pfJ8jm.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/pfJ8jm.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This figure shows a square with side lengths &lt;span class=&quot;math-container&quot;&gt;$a + b$&lt;/span&gt;, dissected into four right triangles (each with area &lt;span class=&quot;math-container&quot;&gt;$\frac 12 ab$&lt;/span&gt;) and a square (of side &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;).  A basic area computation shows that &lt;span class=&quot;math-container&quot;&gt;$(a + b)^2 = 2ab + c^2$&lt;/span&gt;, from which one can conclude that &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It's notable that the proof corresponding to this diagram is &lt;em&gt;not&lt;/em&gt; the same as the one found in Euclid's &lt;em&gt;Elements&lt;/em&gt; (Book I, Prop. 47).  Hence my question:  What is the first known appearance of this figure in print? According to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_theorem#Rearrangement_proofs&quot; rel=&quot;noreferrer&quot;&gt;Wikipedia article on the Pythagorean theorem&lt;/a&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This would seem to suggest that Bretschneider (d. 1878) and Hankel (d. 1873) published, or at least described, this diagram somewhere in their writings.  However, this claim falls short of saying that this proof (or the diagram) originates with them.  How far back can this diagram and its corresponding proof be traced?&lt;/p&gt;&#xA;" OwnerUserId="294" LastActivityDate="2023-08-29T13:04:00.590" Title="First appearance of the &quot;four triangles and a square&quot; proof of the Pythagorean Theorem" Tags="|geometry|euclidean-geometry|pythagorean-proposition|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15688" PostTypeId="2" ParentId="15687" CreationDate="2023-08-29T03:16:09.667" Score="4" Body="&lt;p&gt;I suggest that you take a look at page 49 of the 2nd. edition of &lt;a href=&quot;https://files.eric.ed.gov/fulltext/ED037335.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Pyhagorean Proposition&amp;quot;&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Elisha_Scott_Loomis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Elisha S. Loomis&lt;/a&gt;: this is &lt;strong&gt;the first book&lt;/strong&gt; at which one must take a look &lt;a href=&quot;https://hsm.stackexchange.com/questions/3227/on-einsteins-proof-of-the-so-called-pythagorean-theorem&quot;&gt;when trying to unravel matters related to the many different proofs of Eucl. I-XLVII&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Basically the author says that the proof in which you are interested can already be found on vol. I, p. 159 of a publication entitled &lt;a href=&quot;https://drive.google.com/file/d/1vLhUx_NgKZdkyDtJp4PRW-y0uh-8SZb3/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Mathematical Monthly&amp;quot;&lt;/a&gt; and that it is attributed there to a Rev. A. D. Wheeler of Brunswick, Me. It appears that &lt;a href=&quot;https://www.jstor.org/stable/3028414&quot; rel=&quot;nofollow noreferrer&quot;&gt;the said Monthly&lt;/a&gt; was a short-lived journal that circulated sometime between 1858 and 1861.&lt;/p&gt;&#xA;&lt;p&gt;Finally, one can infer from the notes in that page of Loomis's book that the very diagram in question also appears in T. Simpson's &lt;a href=&quot;https://books.google.com.mx/books?id=g-UGAAAAcAAJ&amp;amp;hl=es&amp;amp;pg=PP1#v=onepage&amp;amp;q&amp;amp;f=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Elements of geometry&amp;quot;&lt;/a&gt; (London, 1760, p. 27).&lt;/p&gt;&#xA;" OwnerUserId="136" LastEditorUserId="136" LastEditDate="2023-08-29T13:04:00.590" LastActivityDate="2023-08-29T13:04:00.590" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15689" PostTypeId="2" ParentId="15687" CreationDate="2023-08-29T04:10:17.127" Score="5" Body="&lt;p&gt;This will need confirmation (as there might be an even earlier version), but off the top of my head, I believe the earliest &amp;quot;visual proof&amp;quot; of the Pythagorean theorem was demonstrated in the Chinese Zhoubi Suanjing or Chou Pei Suan Ching (周髀算經), which is roughly dated around c. 100 B.C.E.- c. 100 C.E.   Although others claim it's dated between c. 200 B.C.E - c. 200 C.E.  In it you'll find the Gougu Theorem (AKA Chinese Pythagorean Theorem), which is similar to the visual proof that you're describing.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dvu9o.jpg&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dvu9o.jpg&quot; alt=&quot;Gougu Diagram&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="13182" LastActivityDate="2023-08-29T04:10:17.127" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15690" PostTypeId="1" CreationDate="2023-08-29T10:23:23.547" Score="3" ViewCount="660" Body="&lt;p&gt;Morris Kline (&lt;em&gt;Mathematics in Western Culture&lt;/em&gt;, &lt;a href=&quot;https://books.google.com/books?id=i45JDXEfwGAC&amp;amp;pg=PA100&quot; rel=&quot;nofollow noreferrer&quot;&gt;1953&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is said that he prognosticated his own death and committed suicide on the date predicted in order to maintain his reputation as an astrologer.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.encyclopedia.com/people/philosophy-and-religion/other-religious-beliefs-biographies/girolamo-cardano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia.com&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He foretold the date of his own death, and, at age 75, was obliged to abstain from food in order to die at the time he had predicted.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Did &lt;a href=&quot;https://en.wikipedia.org/wiki/Gerolamo_Cardano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cardano&lt;/a&gt; really do this?&lt;/p&gt;&#xA;" OwnerUserId="18903" LastEditorUserId="18903" LastEditDate="2023-08-29T10:28:07.743" LastActivityDate="2023-08-30T20:44:58.517" Title="Did Cardano predict the date of his death then commit suicide on that date?" Tags="|mathematicians|death|renaissance|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15692" PostTypeId="2" ParentId="15686" CreationDate="2023-08-29T16:15:16.540" Score="3" Body="&lt;p&gt;Not drawings but statues at Cleveland Museum: The final measure is the mean of 3,863 measurements of American citizens:&#xA;curious? take a look at: &lt;a href=&quot;https://www.cabinetmagazine.org/issues/15/cambers.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cabinetmagazine.org/issues/15/cambers.php&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Beyond the Average Man: The Art of Unlikelihood&#xA;Roger Koenker&#xA;University of Illinois, Urbana-Champaign&#xA;FEMES/Taipei: July, 2007&#xA;&lt;a href=&quot;http://www.econ.uiuc.edu/%7Eroger/research/AvMan/femes.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.econ.uiuc.edu/~roger/research/AvMan/femes.pdf&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;it seems there are no drawings in the original Quetelet book. howevwe in its&lt;/p&gt;&#xA;&lt;p&gt;I've found this in Anthropométrie ou mesure des différentes facultés de l'homme(&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k1513441w.image&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://gallica.bnf.fr/ark:/12148/bpt6k1513441w.image&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TqZ5P.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TqZ5P.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;this book Adolphe Quetelet, Social Physics and the&#xA;Average Men of Science, 1796–1874 by  Kevin Donnelly,displays this picture:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/oWkPU.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/oWkPU.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12383" LastEditDate="2023-08-29T20:21:18.490" LastActivityDate="2023-08-29T20:21:18.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15693" PostTypeId="1" AcceptedAnswerId="15695" CreationDate="2023-08-29T22:12:13.593" Score="5" ViewCount="153" Body="&lt;p&gt;When I was studying class field theory, I saw local class field theory. However, I suddenly became curious about local fields, not local class field theory. As far as I know, the local field is the output result of the localization of global field. So, I would like to know the historical reason why the local field became necessary for some reason.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="55" LastEditDate="2023-08-30T03:59:03.193" LastActivityDate="2023-08-30T09:22:07.130" Title="How did the concept of local field emerge and develop in mathematics?" Tags="|terminology|mathematics-social-history|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15694" PostTypeId="2" ParentId="15690" CreationDate="2023-08-29T22:33:46.110" Score="10" Body="&lt;p&gt;The short answer is that we do not know for certain. Best I could track it down, the story first appeared in print some thirty years after Cardano's death and may well be apocryphal. On the other hand it seems plausible based on his autobiography.&lt;/p&gt;&#xA;&lt;p&gt;From Cardano's autobiography &lt;sup&gt;1&lt;/sup&gt; it seems that he sincerely believed in his ability to foretell the future, and he relates several dramatic instances of this. For example, when he was a young man, a certain Giovanni Stefano Biffo came to him convinced that Cardano was a chiromancer and insisted he tell him something of his future. Cardano replied that his hanging was imminent. Within a week of this Biffo was apprehended by the authorities, was tortured as he persistently denied his crimes, and was hanged after six months, after having his hand cut off first (2&lt;sup&gt;nd&lt;/sup&gt; ed., chapter 42, &lt;a href=&quot;https://www.google.com/books/edition/De_propria_vita_liber/JiZjAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Biffo&amp;amp;pg=RA4-PA155&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 155&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quaedam tamen sic mihi contigerunt, ut causas vix reddere possim, memini me dum essem adolescens, persuasum fuisse cuidam Joanni Stephano Biffo, quod essem Chiromanticus, &amp;amp; tamen nil minus: rogat ille, ut praedicam ei aliquid de vita; dixi delusum esse a sociis, urget, veniam peto si quicquam gravius praedixero: dixi periculum imminere brevi de suspendio, intra hebdomadam capitur, admovetur tormentis: pertinaciter delictum negat, nihilominus tandem post sex menses laqueo vitam finivit, illi prius amputata manu.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this case Cardano states he could hardly explain the reason for his prediction, but in chapter 47 of the autobiography he mentions that he is attended by a guarding spirit (of the kind that the Greeks called angels), which he believes to be a good and merciful spirit and which is able to foresee what is to befall him (2&lt;sup&gt;nd&lt;/sup&gt; ed., &lt;a href=&quot;https://www.google.com/books/edition/De_propria_vita_liber/JiZjAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA4-PA186&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 186&lt;/a&gt;-187):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Spiritus assidentes, aut praesidentes (Graeci Angelos apellare soliti erant, quidam minus Latine Spiritus) favisse quibusdam viris pro constanti, ut dixi receptum est: Socrati, Plotino, Synesio, Dioni, Fl. Josepho, sed &amp;amp; mihi.  [...] at nobis ut credo bonus &amp;amp; misericors spiritus. [...] cum enim praevideat spiritus quod mihi imminet&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Jacques Auguste de Thou, &lt;em&gt;Historiae sui temporis&lt;/em&gt;, usually cited as Thuanus in the literature, appears to be the original printed source for the background of Cardano's death. The earliest edition containing the relevant text that I was able to find a scan of dates to 1609 &lt;sup&gt;2&lt;/sup&gt;. In book 62, in a paragraph on Hieronymus Cardanus (= Cardano) among the events of the year 1576 one finds at the end (&lt;a href=&quot;https://www.google.com/books/edition/Jac_Aug_Thuani_Historiarum_sui_temporis/ndFMAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA4-PA85&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 85&lt;/a&gt;, right columm, section B):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;tandem, cum tribus diebus minus septuagesimum quintum annum implevisset, eodem, quo praedixerat anno &amp;amp; die, videlicet XI Kal. Oct. defecit; ob id, ne falleret, mortem suam inedia accelerasse creditus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My translation: &amp;quot;Finally, when he was three days shy of completing his 75&lt;sup&gt;th&lt;/sup&gt; year, he died in the same year and on the same day that he had predicted: the 11th of Kal. Oct; for this, lest he be mistaken, he is believed to have accelerated his death by not eating.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The date of &amp;quot;11th of Kal. Oct.&amp;quot; seemed curious to me, but researching this ancient Roman way of dating confirmed it is equivalent to September 21, which is the date given by &lt;a href=&quot;https://en.wikipedia.org/wiki/Gerolamo_Cardano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Ephraim Chambers, &lt;em&gt;Cyclopædia: or, An Universal Dictionary of Arts and Sciences&lt;/em&gt;, London: 1728, Vol. 1, &lt;a href=&quot;https://search.library.wisc.edu/digital/A4C5AV6Q7LZ5DY8E/full/AGINLPYZBRQPMF8K&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 143&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;CALENDS, CALENDÆ, in the &lt;em&gt;Roman&lt;/em&gt; Chronology, the first Day of each Month. [...]&#xA;The &lt;em&gt;Calends&lt;/em&gt; were reckon'd backwards, or in retrograde Order: [...]&#xA;To find the Day of the &lt;em&gt;Calends&lt;/em&gt; we are in, see how many&#xA;Days there are yet remaining of the Month, and to that&#xA;Number add two: For Example; suppose it the 22d of&#xA;&lt;em&gt;April&lt;/em&gt;; 'tis then the 10th of the &lt;em&gt;Calends&lt;/em&gt; of &lt;em&gt;May&lt;/em&gt;. For&#xA;&lt;em&gt;April&lt;/em&gt; contains 30 Days; and 22 taken from 30, there&#xA;remains 8; to which two being added, the Sum is 10.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paragraph on Cardano from Thuanus was reproduced verbatim in the &lt;a href=&quot;https://www.google.com/books/edition/Hieronymi_Cardani_Mediolanensis_Opera_om/TS86sMstvcMC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PP35&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;preface of the first volume&lt;/a&gt; of Cardano's collected works &lt;sup&gt;3&lt;/sup&gt;, so either one may have informed subsequent authors to the present day.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Girolamo Cardano (Gabriel Naudé, ed.), &lt;em&gt;De vita propria liber.&lt;/em&gt; Paris: Jacques Villery 1643. I consulted a &lt;a href=&quot;https://www.google.com/books/edition/De_propria_vita_liber/JiZjAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=Socrates&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt; of the second edition, Amsterdam: Johannes van Ravesteyn 1654&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; Jacques Auguste de Thou, &lt;em&gt;Historiae sui temporis.&lt;/em&gt; Offenbach: Conradt Nebeny 1609 (&lt;a href=&quot;https://www.google.com/books/edition/Jac_Aug_Thuani_Historiarum_sui_temporis/ndFMAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PP5&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;3&lt;/sup&gt; Girolamo Cardano (Charles Spon, ed.), &lt;em&gt;Opera omnia.&lt;/em&gt; Lyon: Jean Antoine Huguetan &amp;amp; Marc Antoine Ravaud 1663 (&lt;a href=&quot;https://www.google.com/books/edition/Hieronymi_Cardani_Mediolanensis_Opera_om/TS86sMstvcMC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PP9&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-08-30T20:44:58.517" LastActivityDate="2023-08-30T20:44:58.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15695" PostTypeId="2" ParentId="15693" CreationDate="2023-08-30T03:58:07.200" Score="9" Body="&lt;p&gt;In hindsight, Archimedean local fields (real and complex numbers) were around since 16th century, but nobody thought of them in such terms. What stimulated the development of the abstract concept was the first example of non-Archimedean local field, that of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers. The motivation was to find an analog of Laurent expansions of functions for number theory. Although Hensel introduced &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers explicitly only in 1897, related constructions appeared earlier in works of Kummer, Dedekind and Weber, see &lt;a href=&quot;http://www.ams.org/books/conm/704/&quot; rel=&quot;noreferrer&quot;&gt;Perrin, A journey throughout the history of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers&lt;/a&gt; and &lt;a href=&quot;https://www-fourier.ujf-grenoble.fr/%7Epanchish/Mag2009L3/GouveaHensel2.pdf&quot; rel=&quot;noreferrer&quot;&gt;Gouvea, Hensel's p-adic Numbers: early history&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Hensel’s Idea. To study behavior of a function “near &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;”, use Taylor or Laurent series&#xA;in powers of &lt;span class=&quot;math-container&quot;&gt;$(X - \alpha)$&lt;/span&gt; to supplement the algebraic approach of Dedekind and Weber.&#xA;To study an algebraic number, use an expansion in terms of powers of a prime number &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;... At first, Hensel talked of finding the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic expansion of a given algebraic number. Then, he started considering the set of all possible expansions as an object of independent interest. At first, basically a formal construction. Later, various attempts to justify the “convergence” of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic series.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It was Hensel who initiated the ideology that led to the development of valuations and local fields, and it is to study &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers that Kürschák introduced the concept of valuation (of rank 1 only) in 1912. Hasse then extensively applied it to number theory after 1923, as Hensel originally intended. It was also probably under Hasse's influence that the term &amp;quot;local&amp;quot; got attached to fields in 1930s, before him &amp;quot;small/large&amp;quot; was in circulation, see &lt;a href=&quot;https://math.stackexchange.com/q/1618605/152568&quot;&gt;Math SE, Why the terminology &amp;quot;global fields&amp;quot; and &amp;quot;local fields&amp;quot;&lt;/a&gt;. Other major early contributors include Ostrowski and Krasner.&lt;/p&gt;&#xA;&lt;p&gt;General valuations of arbitrary rank were introduced by Krull in 1932, who was building also on the earlier work of Hahn (1907) and Baer (1927) on ordered fields, see &lt;a href=&quot;https://www.mathi.uni-heidelberg.de/%7Eroquette/hist_val.pdf&quot; rel=&quot;noreferrer&quot;&gt;Roquette, History of Valuation Theory&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Although the formal defnition of valuation&#xA;had been given by Kürschák [1912] it will appear that the ideas which governed&#xA;valuation theory in its first (pre-Krull) phase all came from Hensel. Thus Hensel may be called the father of valuation theory. Well, perhaps better &amp;quot;grandfather&amp;quot; because he never cared about the formal theory of valuations but only for his&#xA;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic number fields... It was Hasse who successfully introduced and applied valuation theoretic ideas into number theory. He always&#xA;propagated that the valuation theoretic point of view can be of help to better understand the arithmetic structure of number fields. Van der Waerden, a&#xA;witness of the times of the 1920s and 1930s, speaks of Hasse as &amp;quot;Hensel's best&#xA;and great propagandist of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic methods&amp;quot;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-08-30T09:22:07.130" LastActivityDate="2023-08-30T09:22:07.130" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15696" PostTypeId="1" CreationDate="2023-08-30T04:21:05.723" Score="0" ViewCount="39" Body="&lt;p&gt;I'm trying to fill a conceptual gap I have in the history of physics&lt;/p&gt;&#xA;&lt;p&gt;In 1922 Stern and Gerlach make their experiment, proving that electrons have intrinsic angular momentum, however it takes a while for people to understand this. &lt;a href=&quot;https://plato.stanford.edu/entries/physics-experiment/app5.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;At first they think this is somehow caused by quantization of orbital angular momentum, but as the months go by they realize this is not possible&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;At some point someone does experiments with two magnets, realizing that the beam doesn't split again after identical measurements, but always splits in two after measurements at different angles &lt;strong&gt;&amp;lt;- This is the hole I want to fill&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Based on the results of these consecutive measurements, Pauli, Sommerfeld, and a few others start proposing a new quantum number that can only take two values&lt;/p&gt;&#xA;&lt;p&gt;Then in 1925 Ralph Kronig, as well as Uhlenbeck and Goudsmith, independently come up with the idea that this new quantum number is intrinsic angular momentum&lt;/p&gt;&#xA;&lt;p&gt;Pauli is initially skeptic about this, but ends up embracing the idea and in 1927 publishes a paper containing the Pauli matrices we know and love today&lt;/p&gt;&#xA;" OwnerUserId="18906" LastActivityDate="2023-08-30T04:21:05.723" Title="Who was the first to make a Stern-Gerlach experiment with two magnets?" Tags="|quantum-mechanics|particle-physics|quantum-field-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15697" PostTypeId="1" AcceptedAnswerId="15698" CreationDate="2023-08-30T12:14:12.063" Score="2" ViewCount="345" Body="&lt;p&gt;The &lt;em&gt;Elements&lt;/em&gt; are often regarded as the cornerstone of the axiomatic approach to mathematics. However, mathematical textbooks have served as the foundational pillars upon which writing style, language, notation and the structural development of mathematics have been solidified over the centuries. Could someone provide further insight into the pivotal role that the &lt;em&gt;Elements&lt;/em&gt; played in shaping these aspects? Are there significant predecessors, or was it the initial serious attempt to establish the framework for the communication and instruction of mathematics? If possible, I would also like to understand what factors contributed to their widespread acceptance in the tradition of writing math.&lt;/p&gt;&#xA;" OwnerUserId="12271" LastEditorUserId="6609" LastEditDate="2023-08-30T16:11:29.160" LastActivityDate="2023-08-30T16:22:03.133" Title="The role of the Elements in the development of mathematics" Tags="|geometry|euclid|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15698" PostTypeId="2" ParentId="15697" CreationDate="2023-08-30T13:05:15.530" Score="4" Body="&lt;p&gt;Euclid was not the first book of this type, among his predecessors the ancient sources mention Hippocrates of Chios, Theudius of Magnesia, Leon and Hermotimus of Colofon.&#xA;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Euclid%27s_Elements&quot; rel=&quot;nofollow noreferrer&quot;&gt;this Wikipedia article&lt;/a&gt; for references. However none of these previous books survives. We only know those names from some surviving ancient histories of mathematics. (This frequently happens that someone writes a book which is judged by contemporaries superior to the work of his predecessors, and they stop using the previous books, and do not care about copying them). Unfortunately only a very small portion of ancient literature survived to our days.&lt;/p&gt;&#xA;&lt;p&gt;Then, for almost two millennia, Euclid's &lt;em&gt;Elements&lt;/em&gt; was the standard of mathematical exposition, it was used in teaching, as a reference and as a standard of the style of mathematical writing. This includes,&#xA;for example Newton's &lt;em&gt;Principia&lt;/em&gt;, where he tries to imitate Euclid's exposition. Even non-mathematicians like Spinoza tried to imitate Euclid's style. Only in 18th century (after the invention of Calculus) this started to change.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6609" LastEditDate="2023-08-30T16:22:03.133" LastActivityDate="2023-08-30T16:22:03.133" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15700" PostTypeId="1" AcceptedAnswerId="15701" CreationDate="2023-08-31T01:37:47.523" Score="-3" ViewCount="147" Body="&lt;p&gt;some references put that the shape of the Greek letter pi is inspired in the lion's gate at Mycenae:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The Symbolism of the Greek Alphabet&amp;quot; by Thomas Taylor(1833)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&amp;quot;The Secret Life of Symbols&amp;quot; by Michael Tsarion(2004)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;these references were given by Google's Bard. For me they do no like to be very scientific and trustworthy.&lt;/p&gt;&#xA;&lt;p&gt;My question is: are there trustworthy scientific references that put some light on the subject?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/BYxKX.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/BYxKX.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12330" LastEditDate="2023-08-31T16:07:08.700" LastActivityDate="2023-08-31T21:09:45.893" Title="is the shape of the Greek letter pi inspired in the Lion's Gate at Mycenae?" Tags="|notation|" AnswerCount="1" CommentCount="8" ClosedDate="2023-09-01T11:46:10.877" ContentLicense="CC BY-SA 4.0" />
  <row Id="15701" PostTypeId="2" ParentId="15700" CreationDate="2023-08-31T03:41:31.030" Score="10" Body="&lt;p&gt;The answer is a quite clear &amp;quot;no&amp;quot;: the modern shape of π is derived from an older, asymmetric form with one leg shorter than the other.  This form, in turn, derives from the Phoenician letter &amp;quot;pe&amp;quot;, which was a rounded hook.  The Phoenician form, in turn, probably derives from a proto-Sinaitic letter that pre-dates the construction of the Lion Gate by several centuries.&lt;/p&gt;&#xA;&lt;p&gt;See &lt;a href=&quot;https://en.wikipedia.org/wiki/Greek_alphabet#Origins&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Greek_alphabet#Origins&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3186" LastEditorUserId="3186" LastEditDate="2023-08-31T21:09:45.893" LastActivityDate="2023-08-31T21:09:45.893" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15702" PostTypeId="1" AcceptedAnswerId="15704" CreationDate="2023-09-01T09:25:14.220" Score="3" ViewCount="155" Body="&lt;p&gt;According to descriptions of &lt;a href=&quot;https://en.wikipedia.org/wiki/Mileva_Mari%C4%87&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mileva Marić&lt;/a&gt;, Einstein's first wife, she was the second woman to complete the mathematics and physics program at the &lt;a href=&quot;https://en.wikipedia.org/wiki/ETH_Zurich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zürich Polytechnikum&lt;/a&gt;. However, nobody points out who the first one was.&lt;/p&gt;&#xA;&lt;p&gt;The ETH website &lt;a href=&quot;https://ethz.ch/staffnet/en/employment-and-work/working-environment/diversity/strategy-and-numbers/women-at-eth/history-of-women-at-eth/from-the-first-female-student-to-todays-female-researchers.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;From the first female student to today's female researchers&lt;/a&gt; names the engineer Nadezda Smeckaja, the agronomist Marie Kowalik and the researcher Marie Baum. However, it is unclear if they completed the mathematics and physics program.&lt;/p&gt;&#xA;&lt;p&gt;Do you have any idea where I can find this?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="4251" LastEditDate="2023-09-04T11:27:57.607" LastActivityDate="2023-09-04T11:27:57.607" Title="Who was the first woman to complete the physics and mathematics program in Zürich?" Tags="|mathematicians|physicists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15703" PostTypeId="1" CreationDate="2023-09-01T19:31:54.307" Score="1" ViewCount="100" Body="&lt;p&gt;What was &lt;em&gt;from a historical point of view&lt;/em&gt; the motivation to extend the definition of sin and cos to angles larger than 90 degree?&lt;/p&gt;&#xA;" OwnerUserId="2380" LastEditorUserId="12330" LastEditDate="2023-09-02T05:46:09.223" LastActivityDate="2023-09-02T05:46:09.223" Title="Motivation to extend sin and cos to angles &gt; 90 degree" Tags="|mathematics|trigonometry|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15704" PostTypeId="2" ParentId="15702" CreationDate="2023-09-01T19:41:57.743" Score="13" Body="&lt;p&gt;The first woman to complete the course of study in department VI A, mathematics and physics, was Mary Ann Elizabeth Stephansen of Bergen, Norway &lt;sup&gt;4&lt;/sup&gt;. The diploma in mathematics (specifically, &lt;em&gt;Fachlehrer in mathematischer Richtung&lt;/em&gt;, which is the same kind of diploma Einstein received in 1900) was conferred on her on August 5, 1896 &lt;sup&gt;3, 8&lt;/sup&gt;. She also appears as Mary Ann Elisabeth Stephansen or Marie Elisabeth Stephansen in the literature and is sometimes referred to by her nickname Betzy. Stephansen was the second Norwegian woman to hold a doctorate &lt;sup&gt;7&lt;/sup&gt;, and the first to hold a doctorate in mathematics &lt;sup&gt;2&lt;/sup&gt;. Her work was mostly focused on teaching, with only a few publications &lt;sup&gt;2&lt;/sup&gt; following her doctoral thesis.&lt;/p&gt;&#xA;&lt;p&gt;Stephansen was born on March 10, 1872 in Bergen as the daughter of a merchant and factory owner. She graduated from high school in Bergen in 1891, and went to Zurich to study at the Eidgenössisches Polytechnikum (now: &lt;a href=&quot;https://en.wikipedia.org/wiki/ETH_Zurich&quot; rel=&quot;noreferrer&quot;&gt;ETH Zurich&lt;/a&gt;) at the department for teachers in mathematics and natural sciences &lt;sup&gt;7&lt;/sup&gt;. She received some financial support in the form of a grant from the Queen Josefine endowment &lt;sup&gt;5&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;After her return to Norway she taught at local schools in Bergen. At the same time the worked on a thesis on partial differential equations, which was published in 1902 &lt;sup&gt;1&lt;/sup&gt;. She sent it to Zurich for consideration of a Ph.D. The mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Heinrich_Burkhardt&quot; rel=&quot;noreferrer&quot;&gt;Heinrich Burkhardt&lt;/a&gt; was the examiner of her thesis &lt;sup&gt;10&lt;/sup&gt;. In the fall of that year the University of Zurich conferred on her the Ph.D. degree, as the Polytechnikum could not confer doctoral degrees. Stephansen spent the winter semester 1902-1903 at the University of Göttingen, supported by a state scholarship of 700 kroner, attending lectures by Hilbert, Klein, and Zermelo &lt;sup&gt;2, 10&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In 1906 she became an assistant of mathematics and physics at the Norwegian Agricultural College (now: &lt;a href=&quot;https://en.wikipedia.org/wiki/Norwegian_University_of_Life_Sciences&quot; rel=&quot;noreferrer&quot;&gt;Norwegian University of Life Sciences&lt;/a&gt;) in &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%85s,_Viken&quot; rel=&quot;noreferrer&quot;&gt;Ås&lt;/a&gt;, near Oslo. In 1921 she was appointed to a newly created position of docent in mathematics at that school, a post she held until retiring in 1937 &lt;sup&gt;6&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In retirement she lived with her sister in Espeland, in the vicinity of Bergen. During the German occupation of Norway in WW2 she assisted Norwegian prisoners at the &lt;a href=&quot;https://en.wikipedia.org/wiki/Espeland_detention_camp&quot; rel=&quot;noreferrer&quot;&gt;Espeland detention camp&lt;/a&gt; as she was fluent in German. For this work she received the &lt;a href=&quot;https://en.wikipedia.org/wiki/King%27s_Medal_of_Merit&quot; rel=&quot;noreferrer&quot;&gt;King's Medal of Merit&lt;/a&gt; in 1948 &lt;sup&gt;6, 9&lt;/sup&gt;. Stephansen died on February 23, 1961.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Mary Ann Elisabeth Stephansen, &amp;quot;Über partielle Differentialgleichungen vierter Ordnung, die ein intermediäres Integral besitzen.&amp;quot; &lt;em&gt;Archiv for Mathematik og Naturvidenskab&lt;/em&gt;, Vol. 24, No. 4, 1902, 80 pp. &lt;a href=&quot;https://books.google.com/books?id=6nlEAQAAMAAJ&amp;amp;pg=RA3-PA3#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;(Google scan)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; Kari Hag, &amp;quot;Elizabeth Stephansen&amp;quot;, &lt;em&gt;Norsk Biografisk Leksikon&lt;/em&gt; (&lt;a href=&quot;https://nbl.snl.no/Elizabeth_Stephansen&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;, last updated June 29, 2022). Kari Hag is professor of mathematics at the Norwegian University of Science and Technology in Trondheim.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;3&lt;/sup&gt; Dord Krstić, &lt;em&gt;Mileva &amp;amp; Albert Einstein: Their Love and Scientific Collaboration&lt;/em&gt;, Didakta 2004, &lt;a href=&quot;https://books.google.com/books?hl=en&amp;amp;lr=&amp;amp;id=nkEGldqhqMoC&amp;amp;oi=fnd&amp;amp;pg=PA5&amp;amp;dq=%22Dord+Krsti%C4%87%22&amp;amp;ots=0tzOaXt5TU&amp;amp;sig=5dxYgk58lhgldQrb5lK6OYKE4BE#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;noreferrer&quot;&gt;p. 52&lt;/a&gt;. The author cites personal communication with Dr. Beat Glaus from the history of science department of the ETH library in November of 1985. Krstić was a physics professor at Ljubljana University who researched the biography of Mileva Marić for fifty years starting in the early 1960s.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;4&lt;/sup&gt; Senta Trömel-Plötz, &amp;quot;Mileva Einstein-Marić.&amp;quot; &lt;em&gt;FemBio&lt;/em&gt; (&lt;a href=&quot;https://www.fembio.org/english/biography.php/woman/biography/mileva-maric-einstein/&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;). &lt;a href=&quot;https://en.wikipedia.org/wiki/Senta_Tr%C3%B6mel-Pl%C3%B6tz&quot; rel=&quot;noreferrer&quot;&gt;Trömel-Plötz&lt;/a&gt; is a German linguist.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;5&lt;/sup&gt; J J O'Connor and E F Robertson, &amp;quot;Mary Ann Elizabeth Stephansen&amp;quot;, &lt;em&gt;MacTutor&lt;/em&gt; (&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Stephansen/&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;6&lt;/sup&gt; &amp;quot;Elizabeth Stephansen (1872–1961).&amp;quot; lokalhistoriewiki.no (&lt;a href=&quot;https://lokalhistoriewiki.no/wiki/Elizabeth_Stephansen_(1872%E2%80%931961)&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;7&lt;/sup&gt; &amp;quot;De første norske kvinnene med doktorgrad.&amp;quot; lokalhistoriewiki.no (&lt;a href=&quot;https://lokalhistoriewiki.no/wiki/De_f%C3%B8rste_norske_kvinnene_med_doktorgrad&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;). The first Norwegian woman to obtain a doctorate was Agnes Mathilde Wergeland (1857-1914), who defended her dissertation (on a topic from the legal history of the Middle Ages) in Zurich in 1890.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;8&lt;/sup&gt; &amp;quot;Eidg. Polytechnikum. Diplom - Erteilung.&amp;quot; In &lt;em&gt;Schweizerische Bauzeitung&lt;/em&gt;, Vol. 28, No. 8, August 22, 1896, &lt;a href=&quot;https://archive.org/details/schweizerischeba2728schw/page/67/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;p. 67&lt;/a&gt;: &amp;quot;&lt;em&gt;Diplom als Fachlehrer in mathematischer Richtung&lt;/em&gt;: Herr Otto Scherrer von Stocken-Egnach (Thurgau) und Frl. Marie Elisabeth Stephansen von Bergen (Norwegen).&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;9&lt;/sup&gt; The Royal House of Norway. Tildelinger av ordener og medaljer. (&lt;a href=&quot;https://www.royalcourt.no/tildelinger.html?tid=28028&amp;amp;sek=&amp;amp;person=&amp;amp;q=&amp;amp;aarstall=1948&amp;amp;type=27124&amp;amp;start=50&quot; rel=&quot;noreferrer&quot;&gt;online&lt;/a&gt;) &amp;quot;01.07.1948 Stephansen, Betzy Frøken Betzy Stephansen er tildelt Kongens Fortjenstmedalje&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;10&lt;/sup&gt; Kari Hag and Peter Lindqvist, &amp;quot;Elizabeth Stephansen: a pioneer&amp;quot;. &lt;em&gt;Det Kongelige Norske Videnskabers Selskabs Skrifter&lt;/em&gt;, 1997.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-09-02T09:50:42.153" LastActivityDate="2023-09-02T09:50:42.153" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15706" PostTypeId="1" CreationDate="2023-09-02T15:09:17.233" Score="6" ViewCount="8361" Body="&lt;p&gt;I've read about general relativity (GR) recently and something stroke me: Einstein came up with &lt;a href=&quot;https://en.wikipedia.org/wiki/Einstein_field_equations&quot; rel=&quot;nofollow noreferrer&quot;&gt;his equations&lt;/a&gt; in 1915, linking the metric of spacetime to the distribution of energy (more exactly, to the stress-energy tensor), &lt;strong&gt;without providing any specific solution to those equations&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It was only a few months later that Schwarzschild came up with the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schwarzschild_metric&quot; rel=&quot;nofollow noreferrer&quot;&gt;famous solution&lt;/a&gt; for a non-rotating spherically symmetric mass distribution and that particular solution provided strong arguments for Einstein's equations as it simplified to &lt;a href=&quot;https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's law&lt;/a&gt; under classical physics assumptions.&lt;/p&gt;&#xA;&lt;p&gt;Years later in 1922, another solution (under homogeneity/isotropy assumptions about the whole universe, see &lt;a href=&quot;https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric&quot; rel=&quot;nofollow noreferrer&quot;&gt;FLRW metric&lt;/a&gt;) was found and explained some empirical observations (e.g., cosmological redshift).&lt;/p&gt;&#xA;&lt;p&gt;I'm wondering:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Why didn't Einstein provide any specific solution to his equations? I mean, wouldn't it be the first thing to look for?&lt;/li&gt;&#xA;&lt;li&gt;How were people convinced GR was an acceptable theory when he came without a specific solution to demonstrate consistency with Newtonian physics or other empirical observations?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14079" OwnerDisplayName="Weier" LastEditorUserId="10106" LastEditDate="2023-09-04T01:17:11.547" LastActivityDate="2023-09-14T17:45:05.830" Title="Why didn't Einstein propose any metric solution to his equations?" Tags="|physics|relativity-theory|einstein|gravity|" AnswerCount="3" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15707" PostTypeId="2" ParentId="15706" CreationDate="2023-09-02T15:26:50.313" Score="9" Body="&lt;p&gt;Einstein actually used the consistency with Newtonian theory to derive the equations. He used the assumptions that in nonrelativistic velocities and weak static gravitational fields, the equations should give you back the Newtonian potential equations.&lt;/p&gt;&#xA;&lt;p&gt;People where convinced due to the correct calculation of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury&quot; rel=&quot;noreferrer&quot;&gt;perihelion precession of the orbit of Mercury&lt;/a&gt;, predicted by the Schwarzschild solution.&lt;/p&gt;&#xA;&lt;p&gt;The proof, here incomplete, goes like this:&lt;/p&gt;&#xA;&lt;p&gt;In the nonrelativistic static and weak limit, the geodesic equation gives:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d^2 \vec{x}}{d\tau^2} = \frac{1}{2}\vec{\nabla}h_{00}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Since the Newton equation is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{d^2 \vec{x}}{d\tau^2} = - \vec{\nabla}\phi$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Einstein proposed that &lt;span class=&quot;math-container&quot;&gt;$g_{00} = -(1+2\phi/c^2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Applying the Laplace operator on both sides:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\nabla^2 g_{00} = -\frac{8 \pi G}{c^2}\rho$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Which is in fact&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$G_{00}= \frac{8\pi G}{c^4} T_{00}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$G_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerDisplayName="LolloBoldo" LastActivityDate="2023-09-02T23:40:18.827" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15708" PostTypeId="2" ParentId="15706" CreationDate="2023-09-02T15:37:43.050" Score="8" Body="&lt;p&gt;The other answer &lt;a href=&quot;https://physics.stackexchange.com/questions/778630/why-didnt-einstein-propose-any-metric-solution-to-his-equations/778633#778633&quot;&gt;by LolloBoldo&lt;/a&gt; is correct. But just to give some evidence, in the original 1915 paper Einstein explicitly states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The field equations (16a) then take, as a first approximation, the form &lt;span class=&quot;math-container&quot;&gt;$$&#xA;\frac{1}{2} \sum_{\alpha} \frac{\partial^2g_{\mu \nu}}{\partial x_{\alpha}^{2}} = \kappa T_{\mu \nu} \tag{16b}&#xA;$$&lt;/span&gt; from which one sees immediately that it contains Newton’s law as an approximation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;em&gt;&lt;a href=&quot;https://einsteinpapers.press.princeton.edu/vol6-trans/118&quot; rel=&quot;noreferrer&quot;&gt;Volume 6: The Berlin Years: Writings, 1914-1917 (English translation supplement) Page 106&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="18940" OwnerDisplayName="Eletie" LastActivityDate="2023-09-03T11:19:06.060" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15709" PostTypeId="1" AcceptedAnswerId="15710" CreationDate="2023-09-03T21:00:45.607" Score="7" ViewCount="996" Body="&lt;p&gt;I tried to look for the history of Legendre transformation, which transformed Lagrangian mechanics to Hamiltonian mechanics, usually formulated as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\begin{cases}&#xA;p_i = \frac{\partial L}{\partial v_i} \\&#xA;H = \sum_{i=1}^n p_iv_i - L&#xA;\end{cases}&#xA;$$&lt;/span&gt;&#xA;for a given function &lt;span class=&quot;math-container&quot;&gt;$\varphi(x_1,\cdots, x_n)$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I want to know how it was first derived in the history since the idea of this transformation is too counterintuitive. However, I realized that there is not even a single literature stating how the mathematician Adrien-Marie Legendre proposed this transformation. In &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/GenMeth.pdf&quot; rel=&quot;noreferrer&quot;&gt;W.R.Hamilton's original paper proposing Hamilton mechanics&lt;/a&gt; (and the &lt;a href=&quot;https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/SecEssay.pdf&quot; rel=&quot;noreferrer&quot;&gt;second essay&lt;/a&gt;, where he proposed the Hamilton canonical equations we know nowadays), he did not even mention the name &amp;quot;Legendre&amp;quot; (in contrast, he mentioned Lagrange, Poisson, Laplace several times).&lt;/p&gt;&#xA;&lt;p&gt;Therefore, what on earth did Legendre do for this transformation? What problem was Legendre trying to solve when he proposed this transformation? Or is it that he did not propose this transformation and people named this transformation after him just by mistake?&lt;/p&gt;&#xA;" OwnerUserId="18922" LastEditorUserId="10106" LastEditDate="2023-09-04T01:17:58.597" LastActivityDate="2023-09-04T05:35:45.847" Title="How does Legendre transformation in classical mechanics relate to Adrien-Marie Legendre?" Tags="|physics|classical-mechanics|lagrange|legendre|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15710" PostTypeId="2" ParentId="15709" CreationDate="2023-09-04T04:50:19.717" Score="12" Body="&lt;p&gt;The Legendre transform is not tethered to classical mechanics, it is just one application of it, another one is to thermodynamics. Geometrically, it is a general way to re-represent convex functions in terms of supporting hyperplanes, and/or to construct convex hulls of general functions. However, Legendre had neither interpretation in mind when he introduced the transform. To him, it was a clever change of variables to improve on Monge's study of Lagrange's &lt;a href=&quot;https://encyclopediaofmath.org/wiki/Minimal_surface&quot; rel=&quot;noreferrer&quot;&gt;minimal surface equation&lt;/a&gt;, see &lt;a href=&quot;https://www.mathnet.ru/php/archive.phtml?wshow=paper&amp;amp;jrnid=emj&amp;amp;paperid=354&amp;amp;option_lang=eng&quot; rel=&quot;noreferrer&quot;&gt;Borgato, Gaspard Monge (1746-1818): application of geometry to analysis&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;At the base of these results, there was an attempt by Monge to study the minimal surface equation, an attempt that highlights the merits and defects of his approach to the theory of partial differential equations. The solution provided by Monge was criticized by Legendre who claimed to have been invited by Monge himself to study the issue he was debating with Laplace. According to Legendre, Monge’s&#xA;error was due to unclear metaphysical principles on which the mathematicians did not agree. Instead of Monge’s solution, Legendre provided another one by making a very elegant change in the variable (Legendre transformation).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paper where Legendre introduced the transform is &lt;em&gt;Mémoire sur l'intégration de quelques équations aux différences partielles: lû le 1er septembre 1787, Histoire de l'Académie Royale de Sciences. Année M. DCCLXXXVII. Avec les Mémoires de Mathématique &amp;amp; de Physique de la meme année, pp. 309–351. A Paris, de l’Imprimerie royale M. DCCLXXXIX, 1789&lt;/em&gt;. Kiselman gives the following commentary on Legendre's contribution in &lt;a href=&quot;http://www2.math.uu.se/%7Ekiselman/fenchel.pdf&quot; rel=&quot;noreferrer&quot;&gt;Werner Fenchel, a pioneer in convexity theory&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;“Au lieu de considérer &lt;span class=&quot;math-container&quot;&gt;$z, p, q$&lt;/span&gt; comme des fonctions de &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt;, rien n’empêche de regarder &lt;span class=&quot;math-container&quot;&gt;$x, y, z$&lt;/span&gt;&#xA;comme des fonctions de &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; &amp;amp; &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;; [. . . ]” (p. 315). This is the very beginning of what we now call the Legendre transformation. Here it is about a function &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt;&#xA;of two variables x and y with &lt;span class=&quot;math-container&quot;&gt;$p = \partial z/\partial x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$q = \partial z/\partial y$&lt;/span&gt;. The symmetry or&#xA;duality between &lt;span class=&quot;math-container&quot;&gt;$z, p, q$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x, y, z$&lt;/span&gt; is evident. Later in the article the author considers functions of more than two variables.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;That Hamilton did not link the requisite change of variables to Legendre is not very surprising given the difference of context. Closely related canonical transformations were soon studied by Delaunay, Liouville and Jacobi. Linking it to conjugation of convex functions also took some time. Kiselman gives a translation of Fenchel's 1977 letter to him on transform's history in convex analysis with some interesting remarks on the naming:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The classical Legendre transformation was introduced purely formally by Legendre. That it is an involution and closely related to duality was, however, as far as I know, first discovered by Monge somewhat later. One shall of course keep the name Legendre transformation... W. H. Young has in a small paper from 1912 proved&lt;/em&gt; [his inequality for] &lt;em&gt;a special case of the classical Legendre transformation, which Young, however, did not notice, even though this is clear from his proof. He is only interested in the inequality. The next contribution to the topic is due to Z. W. Birnbaum &amp;amp; W. Orlicz, 1931... Here the “complementary” function &lt;span class=&quot;math-container&quot;&gt;$\widetilde{f}$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; is defined by &lt;span class=&quot;math-container&quot;&gt;$\widetilde{f}(y)=\sup_{x\in\mathbb{R}}\big(xy-f(x)\big)$&lt;/span&gt; and it is shown that &lt;span class=&quot;math-container&quot;&gt;$\tilde{\widetilde{f}}=f$&lt;/span&gt;... The inequality (**) is called Young’s inequality by several authors. This&#xA;is OK, but I cannot see any reason for calling &lt;span class=&quot;math-container&quot;&gt;$f\mapsto\widetilde{f}$&lt;/span&gt; the Young transformation.&#xA;If you want to include a personal name, it seems Legendre is the closest at hand, even though it does not create the correct association. The “maximum transformation” I have not seen.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-09-04T05:35:45.847" LastActivityDate="2023-09-04T05:35:45.847" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15712" PostTypeId="1" CreationDate="2023-09-04T09:30:11.857" Score="1" ViewCount="129" Body="&lt;p&gt;If we consider the two early pioneers in the fight against the polio desease to be Dr Jonas Salk and Dr Albert Sabin, why then did they not win Nobel Prizes?&lt;/p&gt;&#xA;&lt;p&gt;Seems to me to be enough of a tremendous medical breaktrough that it was deserving of one.&lt;/p&gt;&#xA;&lt;p&gt;Maybe these doctors were part of a team deserving the credit, but I find it odd that such a landmark medical discovery was not credit with such a prize.&lt;/p&gt;&#xA;" OwnerUserId="2094" LastActivityDate="2023-09-04T09:30:11.857" Title="Why did early pioneers in polio research not win a nobel?" Tags="|nobel-prize|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15713" PostTypeId="2" ParentId="7642" CreationDate="2023-09-04T17:53:04.637" Score="3" Body="&lt;p&gt;In mathematics, infinitesimals were widely considered not merely &amp;quot;pseudoscience&amp;quot; but &amp;quot;chimeras&amp;quot;, &amp;quot;hallucinations&amp;quot;, &amp;quot;cholera bacilli&amp;quot;, &amp;quot;fantasies&amp;quot;, &amp;quot;abominations&amp;quot;, &amp;quot;debasement of meaning&amp;quot;, &amp;quot;hazy mist&amp;quot;, &amp;quot;empty words&amp;quot;, and what not, and altogether an intellectual embarassment to shun as a plague.  One of the major objections was of theological origin; see &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#18d&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;This went on for hundreds of years until in the second half of the 20th century, serviceable theories of infinitesimals were finally developed.  A vigorous debate continues until this day, including a recent lively exchange in the pages of &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-09-15T10:12:31.147" LastActivityDate="2023-09-15T10:12:31.147" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15714" PostTypeId="2" ParentId="14085" CreationDate="2023-09-04T20:15:10.543" Score="0" Body="&lt;p&gt;However, according to wikipedia:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;plus one of the top five individual scorers (designated as Putnam Fellows) is awarded a scholarship of up to $12,000 plus tuition at Harvard University (Putnam Fellow Prize Fellowship),&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition#:%7E:text=The%20competition%20was%20founded%20in,the%20Mathematical%20Association%20of%20America.&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition#:%7E:text=The%20competition%20was%20founded%20in,the%20Mathematical%20Association%20of%20America.&quot; rel=&quot;nofollow noreferrer&quot;&gt;William Lowell Putnam Mathematical Competition&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Then the exam thing came around, and one of the prizes was that one of the winners, one of the five, would be chosen for a scholarship to Harvard for the graduate school — I believe, if I remember right. The method, as I understood it, was they didn’t want to take the winner, because they didn’t want to be stuck with a nut of some sort, by accident, right? So they made it that out of the first five they could choose, Ok?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5020-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Interview of Richard Feynman by Charles Weiner on 1966 March 5&lt;/a&gt;, Niels Bohr Library &amp;amp; Archives, American Institute of Physics, College Park, MD USA.&lt;/p&gt;&#xA;&lt;p&gt;So from the same interview, he mentioned that the math contest is national and the top fives are awarded Harvard scholarship. I am leaning towards Putnam. :)&lt;/p&gt;&#xA;" OwnerUserId="18937" LastActivityDate="2023-09-04T20:15:10.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15716" PostTypeId="2" ParentId="15706" CreationDate="2023-09-05T00:34:26.317" Score="4" Body="&lt;p&gt;The question, as asked, implicitly assumes an overly advanced state of the art for physics and astronomy in 1915&lt;/p&gt;&#xA;&lt;p&gt;GR was an elaboration of special relativity (SR), but SR had very little direct evidence in its favor in 1915. The negative result of the Michelson-Morley experiment was not clear at the time. Morley's collaborator Dayton Miller went on to do more experiments of the kind, searching for aether entrainment. The outcome was controversial as late as 1930. Other experiments supporting SR were still far in the future: Ives-Stilwell (1938), Rossi-Hall (1941). Really all the solid experimental evidence for SR was from after World War II.&lt;/p&gt;&#xA;&lt;p&gt;It's anachronistic to expect that Einstein could have posited an expanding-universe solution to the GR field equations and said, &amp;quot;Aha, that matches what we see!&amp;quot; In 1915, it wasn't even established whether the &amp;quot;spiral nebulae&amp;quot; were outside the Milky Way. The Shapley-Curtis debate was in 1920. Cosmology wasn't even a recognized field of study.&lt;/p&gt;&#xA;&lt;p&gt;As other answers have noted, the field equations of GR were structured from the start so as to be consistent with Newtonian gravity in the nonrelativistic limit, so it wasn't necessary to have the Schwarzschild solution in order to demonstrate that. In fact, Einstein didn't like the Schwarzschild solution, because it offended his Machian intuition.&lt;/p&gt;&#xA;&lt;p&gt;And finally, the Einstein field equations are nonlinear, and therefore it is very, very difficult to find closed-form solutions to them. The difficulty of finding closed-form solutions has nothing to do with the existence of solutions, which is another matter.&lt;/p&gt;&#xA;" OwnerUserId="18938" LastEditorUserId="5989" LastEditDate="2023-09-05T07:02:32.083" LastActivityDate="2023-09-05T07:02:32.083" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15717" PostTypeId="1" CreationDate="2023-09-05T08:20:26.427" Score="0" ViewCount="55" Body="&lt;p&gt;Initially water thermometer were used to measure temperatures around the 1600s, then in 18th century mercury was used in thermometers because it expands linearly with temperature. But how do they know mercury expands linearly with temperature?&lt;/p&gt;&#xA;&lt;p&gt;If for example I make thermometer A with water (just as an example, it could use any liquid which shows non linear expansion with temperature) and thermometer B with mercury, and I match the boiling point and the freezing point but about any temperature in between, like if I measure thermal expansion of mercury with thermometer B it will be linear. Similarly, if I measure thermal expansion of water with thermometer A it will be linear but if I swap thermometer A with B both the substance will show non linear expansion. How do I know which is correct? One answer I get is that we can fix the pressure and amount of gas (for example a closed container and a piston on top) and according to ideal gas equation &lt;span class=&quot;math-container&quot;&gt;$PV=nRT$&lt;/span&gt; we can measure change in volume corresponding to change in temperature and by this we can verify mercury thermometer is more accurate. But the ideal gas equation is simply the combination of three laws i.e. Boyle's law, Charles' law, and Avogadro's law and they used thermometers to take observations and derive laws, so we can't use these laws to make a thermometer since a thermometer saw used to derive these laws and also how do we know which gas behave ideally without a thermometer?&lt;/p&gt;&#xA;&lt;p&gt;So to summarise my question is&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;How did early scientists know that a mercury thermometers is more accurate than a water thermometer and how do they know mercury expands linearly with temperature?&lt;/li&gt;&#xA;&lt;li&gt;Was the concept of the ideal gas used to fabricate the thermometer or was a thermometer used to derive the ideal gas equation?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="18943" LastEditorUserId="6609" LastEditDate="2023-09-05T09:47:38.510" LastActivityDate="2023-09-05T09:47:38.510" Title="Ideal gas equation, mercury thermometer and temperature" Tags="|thermodynamics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15718" PostTypeId="1" CreationDate="2023-09-06T05:37:56.077" Score="0" ViewCount="147" Body="&lt;p&gt;Just looking into the dialogue by Plutarch &amp;quot;De facie quae in orbe Lunae apparet&amp;quot; and my impression is, they knew the law of universal gravitation quite well.&lt;/p&gt;&#xA;&lt;p&gt;For instance, it is argued that&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;A body thrown to the center of Earth will oscillate around that center due to its momentum. &lt;em&gt;That masses of a thousand talents weight, borne through the depth of the earth, stop when they reach the middle point, though nothing meets or resists them; or, if mere momentum carries them down beyond the middle point, they wheel round and turn back of themselves&lt;/em&gt; (...) &lt;em&gt;That water rushing violently downwards, if it&#xA;should reach this middle point— an incorporeal point as they say&#xA;— would stand balanced around it for a pivot, swinging with an&#xA;oscillation which never stops and never can be stopped.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;All things on Earth attract to each other. &lt;em&gt;And yet if every weighty&#xA;body converges to the same point with all its parts, the earth will&#xA;claim the heavy objects, not so much because she is middle of&#xA;the whole, as because they are parts of herself.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;The same should happen on other bodies, including the Moon and fiery bodies like Sun and the stars. &lt;em&gt;If there is any body neither assigned originally to the earth, nor torn&#xA;away from it, but having somewhere a substance and nature of&#xA;its own, such as they would describe the moon to be, what is&#xA;there to prevent its existing separately, self-centred, pressed together and compacted by its own parts?&lt;/em&gt; (...) &lt;em&gt;For it is not proved&#xA;that earth is the middle of the Universe, and, further, the way in&#xA;which bodies here are collected and drawn together towards the&#xA;earth suggests the manner in which bodies which have fallen&#xA;together on to the moon may reasonably be supposed to keep&#xA;their place with reference to her.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;The distance to the stars is beyond measurement capabilities greater than the distance to the Moon, and the distance to the Sun is at least 80 times greater. &lt;em&gt;She is lower than the stars by a distance which we cannot state in words, since&#xA;numbers fail you mathematicians when you try to reckon it, but&#xA;she touches the earth in a sense and revolves close to it&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;The distance to the Moon is (by highest estimate), 56 times the Earth's radius (which is quite precise).&lt;/li&gt;&#xA;&lt;li&gt;The Moon is so much close to Earth that it nearly &amp;quot;touches&amp;quot; Earth and affects by its gravity even things on Earth: &lt;em&gt;For she often fails to clear the earth's shadow, rising but little, because the illuminating body is so vast. But so nearly does she&#xA;seem to graze the earth and to be almost in its embrace as she&#xA;circles round, that she is shut off from the sun by it unless she&#xA;rises enough to clear that shaded, terrestrial region, dark as night,&#xA;which is the appanage of earth. Therefore I think we may say&#xA;with confidence that the moon is within the precincts of earth&#xA;when we see her blocked by earth's extremities.&lt;/em&gt; (...) &lt;em&gt;Upon this basis, the distance of the sun from the moon works out to more than forty million three hundred thousand stades. So&#xA;far has she been settled down from the sun because of her weight,&#xA;and so nearly does she adjoin the earth, that, if we are to distribute estates according to localities, the 'portion and inheritance&#xA;of the earth ' invites the moon to join her, and the moon has a&#xA;next claim to chattels and persons on earth, in right of kinship&#xA;and vicinity.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;It is also claimed that life on the Moon is impossible due to thin atmosphere, no water and hot temperature: &lt;em&gt;Then as to winds and clouds and&#xA;showers, without which plants can neither receive nor maintain&#xA;existence, it is out of the question to conceive of their being&#xA;formed, because the surrounding atmosphere is too hot and too&#xA;rare.&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So, my question is: can we say that scientists at least hypothesized the law of universal gravity? Can we say that the possibility of atmosphere on the Moon was hypothesized, and it was considered too hot and thin to support life? Can it be said that this work more or less reflects the scientific consensus or mainstream views of the 1st and 2nd centuries?&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="2203" LastEditDate="2023-09-07T08:25:19.887" LastActivityDate="2023-09-11T10:58:34.730" Title="Did the ancients know about the law of universal gravitation?" Tags="|astronomy|ancient-greece|roman-empire|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15719" PostTypeId="1" CreationDate="2023-09-06T10:05:32.807" Score="2" ViewCount="221" Body="&lt;p&gt;Bohr’s model of the atom predicted hydrogen’s spectral emissions lines —— a huge success.&lt;/p&gt;&#xA;&lt;p&gt;What about Thomson’s plum pudding model? Did it make any successful predictions?&lt;/p&gt;&#xA;" OwnerUserId="2348" LastActivityDate="2023-09-08T03:42:15.453" Title="Did the plum pudding model of the atom make any successful predictions?" Tags="|atomic-theory|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15720" PostTypeId="1" CreationDate="2023-09-06T17:13:24.387" Score="1" ViewCount="50" Body="&lt;p&gt;My understanding is that probability theory is 16th or 17th century but games of chance existed 2000 years ago -- even if more complex probabilities were not understood by ancients, surely they understood that some numbers were more likely?&lt;/p&gt;&#xA;&lt;p&gt;And would the abstract idea of randomness or probability be a concept that divination using animal guts have shared with with dice games?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2023-09-06T17:13:24.387" Title="Probability in Ancient Greece and Rome -- Dice vs Divination?" Tags="|probability|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15721" PostTypeId="1" CreationDate="2023-09-06T17:17:26.143" Score="0" ViewCount="50" Body="&lt;p&gt;That Semmelweis noticed a huge drop in mortality among new mothers simply by having doctors wash their hands between patients.&lt;/p&gt;&#xA;&lt;p&gt;But according to &lt;a href=&quot;https://en.wikipedia.org/wiki/Ignaz_Semmelweis&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Ignaz_Semmelweis&lt;/a&gt; he did not have an explanation for why this effect was occurring. This is puzzling: microorganisms were not news and surely some people suggested that they might cause disease? I know even in ancient times invisible agents or seeds might cause illness.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2023-09-06T17:17:26.143" Title="Semmelweis had no explanation?" Tags="|biology|medicine|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15722" PostTypeId="2" ParentId="8297" CreationDate="2023-09-07T02:09:35.350" Score="0" Body="&lt;p&gt;In the tratise &amp;quot;De facie quae in orbe Lunae apparet&amp;quot; by Plutarch, from early 2nd century AD it is said regarding the possibility of plant life on the Moon:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Then as to winds and clouds and showers, without which plants can&#xA;neither receive nor maintain existence, it is out of the question to&#xA;conceive of their being formed, because the surrounding atmosphere is&#xA;too hot and too rare.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thus it was understood that plant life needs air.&lt;/p&gt;&#xA;" OwnerUserId="2203" LastEditorUserId="2203" LastEditDate="2023-09-07T02:14:58.850" LastActivityDate="2023-09-07T02:14:58.850" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15723" PostTypeId="2" ParentId="15719" CreationDate="2023-09-07T04:41:04.717" Score="2" Body="&lt;p&gt;Rutherford said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In J. J. Thomson’s model of the atom the positive electricity was supposed (for mathematical reasons) to be distributed throughout a large sphere with the negative corpuscles moving inside it. This hypothesis has played a useful part in indicating possible lines of advance; but it does not fit in with more recent discoveries, which point to a concentrated positive nucleus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Discussion on the structure of atoms and molecules, in &lt;a href=&quot;https://archive.org/details/reportofbritisha15adva/page/293/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Report of the eighty-fourth meeting of the British Association for the advancement of science, Australia, July 28 – August 31, 1914 (John Murray, London, 1915), p. 293&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am not sure what those &amp;quot;possible lines of advance&amp;quot; were.&lt;/p&gt;&#xA;&lt;p&gt;EDIT (Sep 07, 2023): Heilbron (Physics Today 30, 4, 23 (1977)) considers Rutherford's nuclear model a product of Thomson's research program.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;That the nuclear atom was an outgrowth of Thomson's research program&#xA;appears plainly from the first page of Rutherford's first calculations&#xA;on the &amp;quot;theory of the structure of atoms,&amp;quot; reproduced in figure 2.&#xA;Note the depiction of the scatterer as a tiny positive nucleus of&#xA;charge ne, surrounded by a diffuse sphere of negative electricity of&#xA;fixed radius.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9808" LastEditorUserId="9808" LastEditDate="2023-09-07T14:53:59.853" LastActivityDate="2023-09-07T14:53:59.853" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15724" PostTypeId="1" CreationDate="2023-09-07T07:21:59.287" Score="6" ViewCount="63" Body="&lt;p&gt;I guess the use of arrows to visualise vectors came before the general notion of vectors, so a more precise question is: when where arrows first used to visualise physical (or mathematical) quantities which are nowadays considered to be vectors? For instance: when where arrows first used to visualise forces, velocities, tuples of numbers etc.? And were there ever competing ways to visualise vectors?&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2023-09-07T07:21:59.287" Title="When were arrows first used to visualise vectors?" Tags="|mathematics|physics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15725" PostTypeId="2" ParentId="15718" CreationDate="2023-09-07T07:54:23.060" Score="2" Body="&lt;p&gt;This is controversial. This passage from Plutarch (and some other evidence) has been thoroughly analyzed by Lucio Russo&#xA;in several papers, and in his book:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Forgotten revolution: how science was born in 300 BC and why it had to be reborn&lt;/em&gt;, translated by S. Levy, Springer, 2004.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To my understanding, majority of historians do not accept his&#xA;conclusions, particularly about the law of gravitation. Of course, the truth in science is not decided by majority. But the subject remains controversial.&lt;/p&gt;&#xA;&lt;p&gt;The main problem is that most original scientific works from Hellenistic epoch are lost, and all we have is the mentioning in secondary sources like Pliny and Plutarch, the authors who did not really understand themselves what they were talking about.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="6609" LastEditDate="2023-09-07T12:57:58.543" LastActivityDate="2023-09-07T12:57:58.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15726" PostTypeId="2" ParentId="15719" CreationDate="2023-09-07T21:45:06.823" Score="5" Body="&lt;p&gt;Yes, qualitative ones.&lt;/p&gt;&#xA;&lt;p&gt;Thomson's model consists of a series of orbits, in each orbit a different number of electrons (less electrons for inner orbits, more for outer ones) all disposed in equally spaced intervals and traveling at high velocities. The different orbits need not be coplanar, and Thomson suggests that in fact, they will occupy different planes. All those orbital rings are encased in a sphere of positive charge such that the atom is neutral. Therefore Thomson's and Nagaoka's models are identical for the electron placement, but Nagaoka places the positive charge inside the inner orbit.&lt;/p&gt;&#xA;&lt;p&gt;The important thing is that in Thomson's model the electron orbits are stable, therefore small perturbations of an electron give rise to an oscillating behavior with a characteristic frequency. As such each electron interacts with light (the source of perturbation) of only a given frequency (in first order perturbation theory) and Thomson's model can explain why atomic spectra are discrete. Contrast with Nagaoka's model that is not mechanically stable (although Nagaoka opens up the possibility that the instability in his model could explain &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; particle emission).&lt;/p&gt;&#xA;&lt;p&gt;In a follow-up paper (On the Number of Corpuscules in an Atom, Philosophical Magazine 6, 769-781 (1906)) Thomson uses his model to determine how many electrons each atom has. Through 3 different experiments, he concludes that the number of electrons is of the same order of the atomic weight and that the ratio of electrons and atomic weight is constant for all elements.&lt;/p&gt;&#xA;&lt;p&gt;To summarize, Thomson's model is a mechanically stable atom that explains the spectral lines as a result of the interaction of electrons with light and that the number of electrons in an atom was of the same order as the atomic weight, so qualitative but not quantitative results. Rutherford subsequently will establish the number of electrons as &lt;strong&gt;half&lt;/strong&gt; of the atomic weight (besides the nucleus containing the positive charges) and Bohr will give the correct formula for the spectral lines of Hydrogen.&lt;/p&gt;&#xA;" OwnerUserId="5904" LastEditorUserId="5904" LastEditDate="2023-09-08T03:42:15.453" LastActivityDate="2023-09-08T03:42:15.453" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15727" PostTypeId="2" ParentId="15143" CreationDate="2023-09-08T02:10:48.087" Score="2" Body="&lt;p&gt;I have one example of a dissertation that was classified during World War 2 but declassified just after the war.  The dissertation of &lt;a href=&quot;https://en.wikipedia.org/wiki/George_T._Reynolds&quot; rel=&quot;nofollow noreferrer&quot;&gt;George T. Reynolds&lt;/a&gt; is titled &lt;a href=&quot;https://catalog.princeton.edu/catalog/9940572913506421&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Studies in the production, propagation, and interactions of shock waves&amp;quot;&lt;/a&gt; and consisted of three OSRD reports.   The three reports were originally completed in 1943 and classified as &amp;quot;confidential&amp;quot;.  The reports were declassified and publicly available by 1946, so the contents were only classified for a short period of time.  You can read the three reports that make up the dissertation at the Library of Congress if you want.  As to its importance, from what I know it is just a standard shock tube experiment, though that was fairly cutting edge at the time.&lt;/p&gt;&#xA;" OwnerUserId="15946" LastActivityDate="2023-09-08T02:10:48.087" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15728" PostTypeId="1" AcceptedAnswerId="15729" CreationDate="2023-09-08T16:58:14.330" Score="1" ViewCount="123" Body="&lt;p&gt;Where can I find the report on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leibniz–Newton calculus controversy&lt;/a&gt; mentioned in &lt;a href=&quot;https://amsi.org.au/ESA_Senior_Years/SeniorTopic3/3b/3b_4history_2.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; article?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1712 the Royal Society in England wrote a report purporting to settle the matter — except, the whole investigation was effectively directed by Newton himself. The report found that Leibniz had concealed his knowledge of Newton's work — based on facts now known to be false.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I have found numerous places online where it is discussed, but not the actual text.&lt;/p&gt;&#xA;" OwnerDisplayName="user18960" LastActivityDate="2023-09-10T03:04:12.823" Title="Where can I find the Royal Society report on the controversy over the invention of differential calculus?" Tags="|mathematics|calculus|newton|leibniz|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15729" PostTypeId="2" ParentId="15728" CreationDate="2023-09-08T19:36:49.083" Score="4" Body="&lt;p&gt;At request of Leibniz himself, the &lt;em&gt;Royal Society&lt;/em&gt; appointed a commission which had the task of dealing with a dispute that followed an article by John Keill published in 1710 in &lt;em&gt;&amp;quot;Philosophical&#xA;Transactions of the Royal Society&amp;quot;&lt;/em&gt;, about centripetal forces, in which the author not only claimed the priority of Newton in the discovery of calculus, but also accused Leibniz of plagiarism.&lt;/p&gt;&#xA;&lt;p&gt;In 1712 the commission published the &lt;em&gt;Commercium epistolicum&lt;/em&gt;, which gathered  documents about the controversy and the report of the Royal Society, in which Keill's statements were judged valid. &lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;There are several editions of the &lt;em&gt;Commercium&lt;/em&gt;&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;A 1856 edition&lt;/strong&gt; is:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Commercium epistolicum D.J. Collins et aliorum de Analiysi promota:&lt;/em&gt; Jussu Societatis Regiae in lucem editum: et iam cum eisudem recensione praemissa, et iudicio primari, ut ferebatur, Matematici subiuncto, iterum impresum. &lt;em&gt;Ou correspondance de J.Cloons et d'autres savants célèbres du XVII siècle, relative à l'Analyse supérieure&lt;/em&gt;, reémprimée sur l'édition original de 1712, avec l'indication des variantes de l'édition de 1722, complétée  par une collection de pièces justificatives et de documents, et publiée par J. B. Biot et F. Lefort. Mallet et Bachelier, Paris, 1856.&lt;/p&gt;&#xA;&lt;p&gt;In this edition the Report of The Royal Society in on page 182.&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;You can see it, for instance, here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.it/Commercium-Epistolicum-Collins-Aliorum-Analysi/dp/1247589145&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.it/Commercium-Epistolicum-Collins-Aliorum-Analysi/dp/1247589145&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Other editions&lt;/strong&gt; can be seen here (the 1722 edition):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://searchworks.stanford.edu/view/1215929&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://searchworks.stanford.edu/view/1215929&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;or here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.martayanlan.com/pages/books/3615/isaac-newton-john-collins/commercium-epistolicum-de-analysi-promota?soldItem=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.martayanlan.com/pages/books/3615/isaac-newton-john-collins/commercium-epistolicum-de-analysi-promota?soldItem=true&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Another edition of the  &lt;em&gt;Commercium&lt;/em&gt; is included  in Newton, I., &lt;em&gt;Opera quae extant omnia&lt;/em&gt;,&lt;/strong&gt; Nichols, London 1782.&lt;/p&gt;&#xA;&lt;p&gt;You can see here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.rct.uk/collection/1090227/isaaci-newtoni-opera-quae-exstant-omnia-v-1-sir-isaac-newton&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.rct.uk/collection/1090227/isaaci-newtoni-opera-quae-exstant-omnia-v-1-sir-isaac-newton&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;or here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.amazon.it/Isaaci-Newtoni-Opera-Exstant-Omnia/dp/0461972328&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.it/Isaaci-Newtoni-Opera-Exstant-Omnia/dp/0461972328&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;An Italian translation&lt;/strong&gt; of the Report of the Royal Society, based on 1856 edition,  is in Cantelli, G., (ed.), &lt;em&gt;La disputa Leibniz-Newton sull'analisi&lt;/em&gt; , mentioned in the footnotes, p. 151.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; See Cantelli, G. (ed.), Giusti, E. (intr.), La disputa Leibniz-Newton sull'analisi, Bollati Borighieri, 2006. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt; Cantelli, cit. p. XXXV. &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-09-10T03:04:12.823" LastActivityDate="2023-09-10T03:04:12.823" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15730" PostTypeId="2" ParentId="9627" CreationDate="2023-09-09T09:34:24.247" Score="0" Body="&lt;p&gt;So far as I know, there are two interpretations why Raphael Robinson uses &lt;span class=&quot;math-container&quot;&gt;$\mathsf{Q}$&lt;/span&gt; represents Robinson Arithmetic.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;By &lt;a href=&quot;https://golem.ph.utexas.edu/category/2011/10/weak_systems_of_arithmetic.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Baez&lt;/a&gt;, &lt;span class=&quot;math-container&quot;&gt;$\mathsf{Q}$&lt;/span&gt; is the name of a character in &lt;em&gt;Star Trek&lt;/em&gt; who could instantly judge whether any statement was provable in this system, or not. While &lt;em&gt;Star Trek&lt;/em&gt; came out in 1966, and Robinson Arithmetic was first considered by Raphael Robinson in his 1950's paper:&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Robinson, R., (1950) 1952. An Essentially Undecidable Axiom System. In: Graves, L. M., Smith, P. A., Hille, E. and Zariski, O. ed. &lt;em&gt;Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U. S. A. August 30-September 6, 1950, Volume I&lt;/em&gt;. Rhode Island: American Mathematical Society, 729-730.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and the symbol &lt;span class=&quot;math-container&quot;&gt;$\mathsf{Q}$&lt;/span&gt; already appeared on page 51 of his 1953's book:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Tarski, A., Mostowski, A. and Robinson, R., 1953. &lt;em&gt;Undecidable Theories&lt;/em&gt;. Amsterdam: North-Holland Publishing Company. Reprinted by Dover Publishing Company, Inc. in 2010.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So the interpretation is not suitable.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;In my opinion, the suitable interpretation for Robinson's using &lt;span class=&quot;math-container&quot;&gt;$\mathsf{Q}$&lt;/span&gt; to represent Robinson Arithmetic is that &lt;span class=&quot;math-container&quot;&gt;$\mathsf{Q}$&lt;/span&gt; is after &lt;span class=&quot;math-container&quot;&gt;$\mathsf{P}$&lt;/span&gt; in order since in his 1953's book Robinson uses &lt;span class=&quot;math-container&quot;&gt;$\mathsf{P}$&lt;/span&gt; to represent Peano Arithmetic, and Robinson Arithmetic is the subsequent theory they consider.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="18962" LastEditorUserId="18962" LastEditDate="2023-09-10T18:41:50.690" LastActivityDate="2023-09-10T18:41:50.690" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15731" PostTypeId="1" CreationDate="2023-09-09T21:08:13.167" Score="0" ViewCount="62" Body="&lt;p&gt;I know that Bourbaki were the first who used the word 'bijection', but one-to one functions were for sure used before them. So do you aware of the earliest examples of one-to-one correspondences?&lt;/p&gt;&#xA;" OwnerUserId="18965" LastActivityDate="2023-09-09T21:08:13.167" Title="Who was the first to use bijections?" Tags="|mathematics|mathematicians|terminology|set-theory|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15732" PostTypeId="2" ParentId="14475" CreationDate="2023-09-10T02:45:18.090" Score="4" Body="&lt;p&gt;In Donald Mackenzie's article, &lt;a href=&quot;https://web.archive.org/web/20160910165845/http://www.cs.cornell.edu/courses/cs4860/2012fa/MacKenzie-TheAutomationOfProof.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Automation of Proof: A Historical and Sociological Exploration&lt;/a&gt;, I found some statements by P. Nidditch that come close to what I have been looking for.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the whole literature of mathematics there is not a&#xA;single valid proof in the logical sense … The number&#xA;of original books or papers on mathematics in the&#xA;course of the last 300 years is of the order of 106; in&#xA;these, the number of even close approximations to really valid proofs is of the order of 101 … In the relatively few places where a mathematician has seriously tried to give a valid proof, he has always overlooked at least some of the rules of inference and&#xA;logical theorems of which he has made use and to&#xA;which he has made no explicit reference.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for whether full proofs could &lt;i&gt;in principle&lt;/i&gt; be written down, Nidditch wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Since no mathematician has ever constructed a complete proof, his reputed capacity for doing so has no better status than an occult quality.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2023-09-10T02:45:18.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15733" PostTypeId="2" ParentId="14475" CreationDate="2023-09-10T10:07:59.827" Score="0" Body="&lt;p&gt;Do we say the presentation in Whitehead &amp;amp; Russell is not fully formal?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/D9q9A.gif&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/D9q9A.gif&quot; alt=&quot;picture&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-09-10T10:07:59.827" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15734" PostTypeId="1" CreationDate="2023-09-10T20:09:40.400" Score="0" ViewCount="353" Body="&lt;p&gt;Have there been any examples in the past where a large proof is claimed but nobody takes the time to check and that has been proved/disproved after a long time? I am not interested in proofs that were never taken seriously because they were unreadable (not even wrong).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-09-22T09:24:59.713" LastActivityDate="2023-09-22T09:24:59.713" Title="Any notable large proof that took a long time before anyone checked it?" Tags="|mathematics|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="15735" PostTypeId="1" CreationDate="2023-09-11T06:33:08.907" Score="3" ViewCount="83" Body="&lt;p&gt;I'm not quite sure if this is the right place to ask this question (in fact, I was redirected to this SE from the &lt;a href=&quot;https://math.stackexchange.com/questions/4767078/did-the-romans-really-use-the-binomial-formula-to-calculate-products&quot;&gt;Math Stackexchange&lt;/a&gt;), but it's probably more fitting than the original posting place.&lt;/p&gt;&#xA;&lt;p&gt;I was told the following as a motivating story in an undergraduate intro to algorithms class years back. Unfortunately, I have no means of obtaining a source on this account, and now wonder about its veracity:&lt;/p&gt;&#xA;&lt;p&gt;In Roman times, when going to the market, it would be a commonplace problem to multiply e.g. a price with a quantity of an item to calculate the total price. Since people weren't able to do grade-school multiplication, let alone in their heads, this posed a serious problem. One solution would have been to produce boards that showed a multiplication table. But the quadratic space (weight, production time, etc.) requirements made this not viable.&lt;/p&gt;&#xA;&lt;p&gt;However, Romans could do addition, subtraction (using some aids, possibly), and division by two, relatively easily; at least compared to outright multiplication of arbitrary numbers. So the clever solution the Romans came up with was to produce boards (of linear size in the size of the range of numbers to be captured) that contained a list of squares. Then, they could employ &lt;span class=&quot;math-container&quot;&gt;$ab=\frac{(a+b)^2-a^2-b^2}2$&lt;/span&gt; to perform multiplication, reading off the squares from their table and adding/subtracting as they knew how to do it.&lt;/p&gt;&#xA;&lt;p&gt;Is this something the instructor made up, or has anyone ever heard of this practice? If so, I would be extremely happy about a source for this.&lt;/p&gt;&#xA;" OwnerUserId="18972" LastEditorUserId="5989" LastEditDate="2023-09-11T07:30:24.037" LastActivityDate="2023-09-11T07:30:24.037" Title="Did the Romans really use the binomial formula to calculate products?" Tags="|mathematics|reference-request|roman-empire|antiquity|arithmetic|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15736" PostTypeId="1" AcceptedAnswerId="15737" CreationDate="2023-09-11T06:47:00.997" Score="1" ViewCount="359" Body="&lt;p&gt;I often have questions of the &amp;quot;who predicted and proved this theorem when and in what context?&amp;quot; kind.&lt;/p&gt;&#xA;&lt;p&gt;There are two ways I can think of.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Read books on the history of mathematics.&lt;/li&gt;&#xA;&lt;li&gt;Find textbooks in which the theorem is written and look at the bibliography.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Is there another way?&lt;/p&gt;&#xA;" OwnerUserId="18973" LastActivityDate="2023-09-14T08:14:42.323" Title="How to find the first appearance of a theorem?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15737" PostTypeId="2" ParentId="15736" CreationDate="2023-09-11T08:17:15.087" Score="9" Body="&lt;p&gt;Finding out who stated and proved some theorem first can require a more serious research than reading history of mathematics books and textbooks. Many examples can be given.&#xA;I give only a couple.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Casorati-Weierstrass theorem. It was mostly credited to Weierstrass (1876). Then papers of Casorati (1868) and Sokhotski were found (both of 1868). So the theorem obtained 3 names attached to it. And finally a more thorough research showed that it was stated in the book of Briot and Bouquet (1859).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Circle packing theorem of Andreev (1970). It was relatively little known outside the narrow circle of specialists, until Thurston made it popular in (1985) by proposing new point of view and several applications. This rasied the interest to the theorem, and soon people found that it was in fact proved by Koebe in 1936.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;P. Koebe, Kontaktprobleme der konformen Abbildung, Berichte uber die Verhandlungen&#xA;der Sachsischen Akademie der Wissenschaften, Leipzig, Mathematische-Physische Klasse 88, 1936,&#xA;141-164.&lt;/p&gt;&#xA;&lt;p&gt;Certainly no textbook and no history book contained this theorem until 1985.&lt;/p&gt;&#xA;&lt;p&gt;Most mathematicians are not really interested in history, and for them the names of theorems are only convenient labels. Mathematical subjects go in and out of fashion. Once something becomes fashionable, someone digs the history, and finds who stated or proved some important theorem first. This does not necessarily lead to the change of the label.&#xA;See also &lt;a href=&quot;https://mathoverflow.net/questions/285627&quot;&gt;https://mathoverflow.net/questions/285627&lt;/a&gt; on the naming patterns of mathematical objects.&lt;/p&gt;&#xA;&lt;p&gt;To address directly your question &amp;quot;how to find&amp;quot;, let me give an example from my own career. In 1970-s Einar Hille studied differential equations &lt;span class=&quot;math-container&quot;&gt;$F(y'',y)=0$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is a polynomial, and conjectured that all meromorphic solutions are elliptic functions, possibly degenerate. I proved this conjecture in 1982. I was a beginning mathematician at that time and was very proud to prove a conjecture of a famous mathematician. On my way to a seminar where I was supposed to talk on this, I stopped in the library to read some old French papers in CR. After I finished my reading, some time remained, and I just started to leaf an old CR volume. To my surprise I found a paper of Picard, who proved &amp;quot;my theorem&amp;quot; in 1890 ! So I did not publish my proof. Later I found that the same result was proved and published by Bank and Kaufman in 1981. Neither they nor Hille knew about this paper of Picard. Both Picard and Hille published textbooks. Picard's textbook contains this theorem, with a proof, and Hille's textbook contains his conjecture.&lt;/p&gt;&#xA;&lt;p&gt;To conclude: some theorems are considered important, and someone investigates their histories. There is no algorithm for this, it is a creative activity, like any research. Most theorems never attract wide attention, they are forgotten, sometimes rediscovered again and again.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-09-14T08:14:42.323" LastActivityDate="2023-09-14T08:14:42.323" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15738" PostTypeId="2" ParentId="15718" CreationDate="2023-09-11T10:58:34.730" Score="2" Body="&lt;p&gt;I agree with the answer from @Alexander Eremenko, as well as with @Mauro Allegranza's useful remark about ancient ideas related to (but not the same as) gravity. Also the following observations may be found useful.&lt;/p&gt;&#xA;&lt;p&gt;The question really calls for understanding of what was meant both by the ideas mentioned in Plutarch's work referenced in the question, and also what was and is meant by the idea and theory of 'universal gravitation'.&lt;/p&gt;&#xA;&lt;p&gt;Ancient understandings of the cosmos often made a distinction between terrestrial and celestial materials and laws. Effects now referred to gravity were often conceived as primarily local to the neighbourhood of the earth, while celestial motions were supposed governed by other principles, often principles of perfection, such as circular motion and geometry. A helpful sourcebook, even though mainly covering ideas later than those of Plutarch, is &amp;quot;A Source Book in Medieval Science&amp;quot;  (ed. Edward Grant, 1974).&lt;/p&gt;&#xA;&lt;p&gt;I'd agree that the passages quoted in the question do suggest that their author speculated on some extension of terrestrial effects.  There are ideas of analogous effects for the moon, sun, and stars, perhaps including the possibility of local effects of weight for each, and also that the moon might be habitable. But nothing in the quoted passages appears to suggest any attempt to make out that these effects are more than local to the bodies concerned. Especially there appears to be no assimilation between terrestrial effects and celestial motions, no idea that both are governed by the same laws.&lt;/p&gt;&#xA;&lt;p&gt;On the other hand, the idea of universal gravitation has, in particular, two fundamental components which are sometimes overlooked. (1) is the idea of gravity as centripetal accelerative force. (2) is the idea that such centripetal accelerative force occurs between all pairs of material objects, whether terrestrial or celestial, so that (for example) the physical cause of the heaviness of terrestrial objects, and the physical cause of the moon remaining in its orbit around the earth and of the planets remaining in their orbits around the sun, are of one and the same kind.&lt;/p&gt;&#xA;&lt;p&gt;No suggestion of any of these centripetal accelerative gravitational ideas appears in Plutarch, and neither does their equal application to terrestrial effects and celestial motions.  Steffen Ducheyne's  &amp;quot;The main Business of natural Philosophy&amp;quot; (2012) is a useful source about these ideas and also about their understanding (and sometimes misunderstanding) over the last few centuries.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2023-09-11T10:58:34.730" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15739" PostTypeId="1" CreationDate="2023-09-11T13:04:47.257" Score="0" ViewCount="111" Body="&lt;p&gt;Hilbert reconstructed Euclid's axioms. Is there an equivalent restructuring of Newton's axioms, or are they considered consistent?&lt;/p&gt;&#xA;" OwnerUserId="5602" LastEditorUserId="10106" LastEditDate="2023-09-12T05:59:03.940" LastActivityDate="2023-09-12T05:59:03.940" Title="Reconstruction of Newton's axioms" Tags="|physics|classical-mechanics|newton|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15741" PostTypeId="1" AcceptedAnswerId="15742" CreationDate="2023-09-11T17:24:09.930" Score="3" ViewCount="130" Body="&lt;p&gt;In French, Gauss' law for magnetism (no magnetic monopoles) is, sometimes at least, referred to as &lt;em&gt;Équation de Maxwell-Thomson&lt;/em&gt;. What is the historical justification of this?&lt;/p&gt;&#xA;" OwnerUserId="18974" LastEditorUserId="6609" LastEditDate="2023-09-11T19:34:07.790" LastActivityDate="2023-09-20T10:26:48.830" Title="&quot;Équation de Maxwell-Thomson&quot;" Tags="|electromagnetism|maxwell|magnetism|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15742" PostTypeId="2" ParentId="15741" CreationDate="2023-09-11T20:30:13.007" Score="6" Body="&lt;p&gt;In French, physical laws are not always named the same as in English (usually some Frenchman is added)&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Snell's law &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;loi de Snell-Descartes&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Gauss' theorem &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;théorème de Ostrogradski-Gauss&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;WKB aproximation &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;approximation BKW&lt;/em&gt; (Léon Brillouin has to come first)&lt;/li&gt;&#xA;&lt;li&gt;Boyle's law &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;loi de Boyle-Mariotte&lt;/em&gt; or &lt;em&gt;loi de Mariotte&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Rouché-Capelli theorem &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;théorème de Rouché–Fontené&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Motor effect (force) &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;force de Laplace&lt;/em&gt;&lt;/li&gt;&#xA;&lt;li&gt;Addition of two vectors &lt;span class=&quot;math-container&quot;&gt;$\to$&lt;/span&gt; &lt;em&gt;relation de Chasles&lt;/em&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As for &lt;em&gt;l'équation de Maxwell-Thomson&lt;/em&gt;, in French all of Maxwell's equations start with Maxwell: &lt;em&gt;Maxwell-Gauss, Maxwell-Faraday, Maxwell-Ampère, Maxwell-Thomson&lt;/em&gt;. The latter refers to Lord Kelvin (William Thomson), see ref 1. Kelvin contributed greatly to electromagnetism, so it is hard to tell why is he honored with this equation. However, he did one very important related thing. In 1846, Kelvin showed the common relation between the magnetic field &lt;span class=&quot;math-container&quot;&gt;$\mathbf B$&lt;/span&gt; and the magnetic vector potential &lt;span class=&quot;math-container&quot;&gt;$\mathbf A$&lt;/span&gt;, that is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbf B=\nabla \times \mathbf A\;,$$&lt;/span&gt;&#xA;which directly implies &lt;em&gt;l'équation de Maxwell–Thomson&lt;/em&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla \cdot \mathbf B=0\;,$$&lt;/span&gt;&#xA;because it is the divergence of a curl. I would have preferred &lt;em&gt;Maxwell-Kelvin&lt;/em&gt; but it is going to be hard to track who coined the name.&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.google.fr/books/edition/Electromagn%C3%A9tisme/8ufMDwAAQBAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%C3%A9quation%20Maxwell-Thomson%20histoire&amp;amp;pg=PA4&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Crysos, Electromagnétisme IUT&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1063%2FPT.3.2585&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chen Ning Yang, &lt;em&gt;The conceptual origins of Maxwell’s equations and gauge theory&lt;/em&gt;,Physics Today 2014&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-09-20T10:26:48.830" LastActivityDate="2023-09-20T10:26:48.830" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15743" PostTypeId="2" ParentId="15739" CreationDate="2023-09-11T21:14:46.770" Score="1" Body="&lt;p&gt;Would you consider clarifying the question itself to enable a more informative answer than this?  Particularly, can you be more specific about what you mean by 'restructuring', in a sense that could equally apply to alterations in Euclid's axioms and/or in Newton's laws of motion?&lt;/p&gt;&#xA;&lt;p&gt;In all three editions of Newton's 'Principia' the axioms that occur near the beginning are not just 'axioms', they are 'axioms or laws of motion'.&lt;/p&gt;&#xA;&lt;p&gt;They have an essentially physical aspect that means they can't meaningfully be treated as purely mathematical constructs that might be differently chosen, they were selected for their compatibility with natural phenomena of motion and related physical behavior: and any alteration of them that is contemplated must clearly maintain that compatibility.&lt;/p&gt;&#xA;&lt;p&gt;Having said that, the 'axioms or laws of motion' have often been re-expressed. They are usually expressed nowadays, especially in modern  physics textbooks, in a way that looks very different indeed than their original expression by Newton. Yet it is widely considered that the differences are or ought to be of modernised form and expression only, and not of substance or core meaning. Yet arguments can arise over such questions as whether the modern formulations really do match in substance what Newton meant, and sometimes even what did he mean?&lt;/p&gt;&#xA;&lt;p&gt;I will try to go further if you would clarify the question.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2023-09-11T21:14:46.770" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15744" PostTypeId="1" CreationDate="2023-09-12T12:15:24.903" Score="3" ViewCount="582" Body="&lt;p&gt;In a previous question on this website: &lt;a href=&quot;https://hsm.stackexchange.com/questions/7144/what-was-eulers-first-language&quot;&gt;What was Euler&amp;#39;s first language?&lt;/a&gt;,&#xA;Alexandre Eremenko wrote the following about Leonard Euler:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is little doubt that he also learnt French in his childhood (his thesis is written in French), as well as Latin, Greek and Hebrew at school.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is there any material evidence that he actually knew Ancient Greek?&lt;/p&gt;&#xA;" OwnerDisplayName="user18960" LastEditorUserId="12330" LastEditDate="2023-09-13T11:37:44.527" LastActivityDate="2023-09-16T17:41:01.427" Title="Did Euler know Ancient Greek?" Tags="|mathematicians|euler|language|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15745" PostTypeId="2" ParentId="15744" CreationDate="2023-09-12T14:05:01.993" Score="6" Body="&lt;p&gt;From Euler's &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-7643-7539-3_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Autobiography&lt;/a&gt;, p. 2:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nachdem musste ich mich auf Gutbefinden meiner famille beÿ der Theologischen Fakultaet einschreiben lassen, da ich&#xA;mich denn ausser der Theologie besonders auf die griechische und Hebräische Sprache appliciren sollte; womit es aber nicht recht fort wollte, weil ich meine meiste Zeit auf die mathematische Studia wendete, und zu meinem Glück die Sonnabend Visiten bei dem Herrn Johanni Bernoulli noch immer fortdaureten...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Afterwards following the wish of my family, I had to enroll at the Theological Faculty where I should apply myself, in addition to theology, to the Greek and Hebrew languages in particular; which did, however, not proceed very well because I used most of my time for mathematical studies, and, luckily for me, the Saturday visits at Mr Johann Bernoulli's still continued....&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[Thanks to @SebastianKoppehel for the translation of the second half of the quote.]&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2023-09-12T21:57:06.477" LastActivityDate="2023-09-12T21:57:06.477" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15747" PostTypeId="1" AcceptedAnswerId="15761" CreationDate="2023-09-12T23:00:54.633" Score="4" ViewCount="126" Body="&lt;p&gt;It is common in the compressed sensing literature to refer to the number of nonzero entries of a vector as its &lt;span class=&quot;math-container&quot;&gt;$\ell_0$&lt;/span&gt; &amp;quot;norm.&amp;quot; The scare quotes are there because strictly speaking, the &lt;span class=&quot;math-container&quot;&gt;$\ell_0$&lt;/span&gt; &amp;quot;norm&amp;quot; is not a norm, but the terminology is not unreasonable since&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{p\to 0^+} \sum_{i=1}^n |v_i|^p$$&lt;/span&gt;&#xA;is indeed the number of nonzero numbers among the &lt;span class=&quot;math-container&quot;&gt;$v_i$&lt;/span&gt;. My question is, who first used this terminology? Wikipedia, as of this writing, &lt;a href=&quot;https://en.wikipedia.org/wiki/Lp_space#When_p_=_0&quot; rel=&quot;nofollow noreferrer&quot;&gt;claims it was Donoho&lt;/a&gt;, but no citation is given.&lt;/p&gt;&#xA;&lt;p&gt;If no specific person can be credited with coining the terminology, is there at least some good evidence that it became popular only with the advent of the field of compressed sensing? I have had trouble doing searches because (as Wikipedia explains) &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\ell_0$&lt;/span&gt; norm&amp;quot; has two different meanings.&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;ADDENDUM:&lt;/b&gt; Someone pointed out to me a 2001 paper by David Donoho, &lt;a href=&quot;https://doi.org/10.1109/18.959265&quot; rel=&quot;nofollow noreferrer&quot;&gt;Uncertainty principles and ideal atomic decomposition&lt;/a&gt;, which calls the number of nonzero entries the &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$\ell_0$&lt;/span&gt; quasinorm.&amp;quot; This is the earliest reference that I am aware of. Can anyone beat it?&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2023-09-14T03:48:09.380" LastActivityDate="2023-09-18T17:40:35.210" Title="Who first referred to the number of nonzero entries of a vector as its $\ell_0$ norm?" Tags="|mathematics|notation|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15750" PostTypeId="1" CreationDate="2023-09-14T03:41:36.473" Score="1" ViewCount="207" Body="&lt;p&gt;I remember from my quantum course that the first person (I believe in 1927) to show that helium has a bound state, using the variational principle, was a nazi. It was remarked by my professors that he was essentially blotted out the history books, despite this approximation of the ground state energy of helium being one of the first calculations in quantum mechanics that couldn't be done classically.&lt;/p&gt;&#xA;&lt;p&gt;What was his name? I feel like he was blotted from Google as well...&lt;/p&gt;&#xA;&lt;p&gt;Edit: It is possible that this person used perturbation theory, rather than the variational principle. Sakurai cites Albrecht Unsöld as the first to perform the &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/10.1002/andp.19273870304&quot; rel=&quot;nofollow noreferrer&quot;&gt;calculation&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="18988" LastEditorUserId="18988" LastEditDate="2023-09-15T20:43:58.873" LastActivityDate="2023-09-16T23:06:10.043" Title="Who was the scientist who first showed that helium has a bound state, and was he a nazi?" Tags="|quantum-mechanics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15751" PostTypeId="2" ParentId="15750" CreationDate="2023-09-14T10:55:31.413" Score="9" Body="&lt;p&gt;The publication in question is: Georg W. Kellner, &amp;quot;Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 44, Nos. 1-2, Jul. 1927, pp. 91-109 (&lt;a href=&quot;https://www.google.com/books/edition/Zeitschrift_f%C3%BCr_Physik/bPZMAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA91&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;). This publication has 140 citations according to Google Scholar, so it has not exactly been neglected.&lt;/p&gt;&#xA;&lt;p&gt;This publication was the first to employ the variational method of Ritz, which lead to mathematical results that were much closer to experimental measurements when compared to the simple first-order perturbation method used by Unsöld &lt;sup&gt;1&lt;/sup&gt;, whom Kellner cites. It should be noted that at the same time as Kellner, J. C. Slater computed an accurate result using a non-variational method &lt;sup&gt;2&lt;/sup&gt;. Using the same variational method of Ritz used by Kellner and by including higher-order terms, Hylleraas was able to eliminate the gap between computational results and experimental measurements by 1930 &lt;sup&gt;3,4,5&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I was able to extract biographical information on Kellner from his second Ph.D. dissertation (in political science) by piecing together many Google snippets.&lt;/p&gt;&#xA;&lt;p&gt;Georg Wolfgang Kellner, &lt;em&gt;Sozialpolitik und Antikonjunkturpolitik: Untersuchungen über Möglichkeiten zur Stabilisierung der Wirtschaft und der Arbeit.&lt;/em&gt; Gießen: Otto Kindt 1934, 76 pp.&lt;/p&gt;&#xA;&lt;p&gt;Biographical summary: Born February 1, 1905 in Wetzlar. Graduated high school in Gießen Easter 1923. From Easter 1923 to fall of 1927 studied mathematics and physics at the University of Berlin. Received his Ph.D. on October 11, 1927 with a disseration entitled &amp;quot;Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie&amp;quot;. Worked as an assistant at the Institute&#xA;for Theoretical Physics of the University of Berlin from November 1927&#xA;to April 1931. From May 1931 through December 1932 worked on a research&#xA;project in theoretical physics, mostly at the University of Leipzig.&#xA;Moved to his home town of Wetzlar in September of 1932. Studied political&#xA;science at the University of Gießen and received his Ph.D. in July 1933.&#xA;Since September 1933 employed by &lt;a href=&quot;https://en.wikipedia.org/wiki/Ernst_Leitz_GmbH&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ernst Leitz GmbH&lt;/a&gt;, a manufacturer of optical products.&lt;/p&gt;&#xA;&lt;p&gt;I have &lt;em&gt;not&lt;/em&gt; found any evidence that Kellner was a Nazi. Absence of evidence is not evidence of absence: Given that the Nazi party ultimately counted some eight million members, it is certainly possible that he was a party member. It seems likewise possible that someone confused him with Dr. Dr. Walter Kellner (1906-1963), who was &lt;a href=&quot;https://books.google.com/books?id=fTbpBQAAQBAJ&amp;amp;pg=PA149&amp;amp;lpg=PA149&amp;amp;dq=%22Dr.+Dr.+Walter+Kellner%22&amp;amp;source=bl&amp;amp;ots=MyyQ4_wuZy&amp;amp;sig=ACfU3U3hR6w9wMv5u3nnSn-qAdLfv9UOTQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwjlkOjVo6uBAxU1EzQIHWdtBRAQ6AF6BAgIEAM#v=onepage&amp;amp;q=%22Dr.%20Dr.%20Walter%20Kellner%22&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;in charge of the &amp;quot;Jewish Library&amp;quot;&lt;/a&gt; at &lt;a href=&quot;https://en.wikipedia.org/wiki/Reich_Security_Main_Office&quot; rel=&quot;nofollow noreferrer&quot;&gt;RHSA&lt;/a&gt; from 1939 on.&lt;/p&gt;&#xA;&lt;p&gt;I have also been able to trace Kellner's membership in the Deutsche Physikalische Gesellschaft (DPG; German Physical Society) from &lt;a href=&quot;https://books.google.com/books?id=hLog8SJke0YC&amp;amp;lpg=PA1&amp;amp;pg=PR1#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;member directories&lt;/a&gt; published annually. He became a member in 1926, nominated by &lt;a href=&quot;https://en.wikipedia.org/wiki/Max_von_Laue&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max von Laue&lt;/a&gt;. &lt;em&gt;Verhandlungen der Deutschen Physikalischen Gesellschaft.&lt;/em&gt; 3&lt;sup&gt;rd&lt;/sup&gt; series, Vol. 7, No. 2, Jul. 1926, &lt;a href=&quot;https://www.google.com/books/edition/Verhandlungen/yDjvTyBxg6UC?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22Verhandlungen%20der%20Deutschen%20Physikalischen%20Gesellschaft%22%20%20%22Kellner%22&amp;amp;pg=RA8-PA36&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;p. 36&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Als Mitglieder wurden in die Gesellschaft aufgenommen:&lt;br /&gt;&#xA;Hr. stud. phil. WOLFGANG GEORG KELLNER, Berlin-Dahlem, Hittorfstr. 29 (Vorgeschlagen durch Hrn. M v. LAUE)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Two things are curious about this brief notice. Kellner appeared as G. W. Kellner, Georg W. Kellner, and Georg Wolfgang Kellner in publications, but here his names are reversed. Interesting is also that his academic status is stated as &lt;em&gt;stud. phil.&lt;/em&gt; rather than &lt;a href=&quot;https://en.wikipedia.org/wiki/Candidate_of_Philosophy&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;cand. phil.&lt;/em&gt;&lt;/a&gt;, although he seems to have been within one year of receiving his Ph.D. His status as a student of von Laue is mentioned in this publication:&lt;/p&gt;&#xA;&lt;p&gt;S. Esposito and A. Naddeo, &amp;quot;Majorana Solutions to the Two-Electron Problem&amp;quot;, &lt;em&gt;Foundation of Physics&lt;/em&gt;, Vol. 42, 2012, pp. 1586–1608:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Such a novel method came into the play a bit earlier by means of G.W. Kellner, a student of M. von Laue in Berlin, who introduced for the first time the variational Ritz method in order to estimate the ground state energy of the helium atom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Based on the DPG member directories, Kellner resided in Berlin until some time in 1931, is listed as residing in Leipzig in 1932 and 1933, and then took up residence in &lt;a href=&quot;https://en.wikipedia.org/wiki/Wetzlar&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wetzlar&lt;/a&gt;, a small town in Hesse, where he is listed for the years 1934 and 1935. I could find no later trace of him.&lt;/p&gt;&#xA;&lt;p&gt;There is a group picture that shows Georg W. Kellner next to &lt;a href=&quot;https://en.wikipedia.org/wiki/Werner_Heisenberg&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heisenberg&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Felix_Bloch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bloch&lt;/a&gt;, and other colleagues at the Physikalisches Institut der Universität Leipzig, taken around 1931, on p. 36 of &lt;em&gt;Journal Universität Leipzig&lt;/em&gt;, &lt;a href=&quot;https://d-nb.info/123852530X/34&quot; rel=&quot;nofollow noreferrer&quot;&gt;No. 5 / 2005&lt;/a&gt;. Heisenberg ran the Institute of Physics at Leipzig University from 1927 to 1942; based on the presence of Bloch the picture more likely dates to 1932.&lt;/p&gt;&#xA;&lt;p&gt;I have not been able to establish Kellner's Ph.D. advisor with certainty. In his 1927 publication based on his Ph.D. dissertation he thanks von Laue, Schrödinger, Heisenberg, and others, but he does not mention von Laue being his advisor. Based on information in David C Clary, &lt;em&gt;Schrödinger in Oxford&lt;/em&gt;, WSPC 2022, he may have been associated with &lt;a href=&quot;https://en.wikipedia.org/wiki/Max_Born&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max Born&lt;/a&gt; at that time. In the context of the fifth &lt;a href=&quot;https://en.wikipedia.org/wiki/Solvay_Conference&quot; rel=&quot;nofollow noreferrer&quot;&gt;Solvay Conference&lt;/a&gt; of 1927,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Born pointed out that Kellner in his research group had extended Schrödinger’s theory in an approximate way to the helium atom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; Albrecht Unsöld, &amp;quot;Beiträge zur Quantenmechanik der Atome&amp;quot;, &lt;em&gt;Annalen der Physik&lt;/em&gt;, Vol. 387 (4&lt;sup&gt;th&lt;/sup&gt; Series, Vol. 82), No. 3, Feb. 1927, pp. 355-393 (&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k153842/f374.item.r=Albrecht&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; J. C. Slater, &amp;quot;The Structure of the Helium Atom: I&amp;quot;, &lt;em&gt;Proceedings of the National Academy of Sciences of the United States of America&lt;/em&gt;, Vol. 13, No. 6, Jun. 1927, pp. 423-430 (&lt;a href=&quot;https://www.pnas.org/doi/abs/10.1073/pnas.13.6.423&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;3&lt;/sup&gt; Egil A. Hylleraas, &amp;quot;Über den Grundzustand des Heliumatoms.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 48, Nos. 7-8, 1928, pp. 469-494 (&lt;a href=&quot;https://www.google.com/books/edition/Bestimmung_der_lage_der_sauerstoffatome/dR-V7F8vqdcC?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA469&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;4&lt;/sup&gt; Egil A. Hylleraas, &amp;quot;Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium&amp;quot;, &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 54, Nos. 5-6, May 1929, pp. 347–366 (&lt;a href=&quot;https://www.google.com/books/edition/Bestimmung_der_lage_der_sauerstoffatome/dR-V7F8vqdcC?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA347&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;&lt;sup&gt;5&lt;/sup&gt; Egil A. Hylleraas and Bjarne Undheim, &amp;quot;Numerische Berechnung der 2 &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt;-Terme von Ortho- und Par-Helium.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 65, Nos. 11-12, Nov. 1930, pp. 759-772 (&lt;a href=&quot;https://www.google.com/books/edition/Bestimmung_der_lage_der_sauerstoffatome/dR-V7F8vqdcC?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA265&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;List of Kellner's publications:&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 44, Nos. 1-2, Jul. 1927, pp. 91-109 (&lt;a href=&quot;https://www.google.com/books/edition/Zeitschrift_f%C3%BCr_Physik/bPZMAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA91&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Der Grundterm des einfach ionisierten Lithiums nach der Schrödingerschen Theorie&amp;quot;. &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;. Vol. 44, Nos. 1-2, Jul. 1927, pp. 110-112 (&lt;a href=&quot;https://www.google.com/books/edition/Zeitschrift_f%C3%BCr_Physik/bPZMAQAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=RA1-PA110&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie.&amp;quot; &lt;em&gt;Naturwissenschaften&lt;/em&gt;, Vol. 15, No. 27, Jul.  1927, p. 561 (letter to the editor) (&lt;a href=&quot;https://www.google.com/books/edition/Die_Naturwissenschaften/qxTxAAAAMAAJ?hl=en&amp;amp;gbpv=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;scan&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Die Kausalität in der Quantenmechanik.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 55, No. 1, Jan. 1929, pp. 44-51&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Die Kausalität in der Quantenmechanik II.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 59, Nos. 11-12, Nov. 1930, pp. 820-835&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Zwei Bemerkungen zu meiner Arbeit „Die Kausalität in der Quantenmechanik II“.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;, Vol. 64, Nos. 1-2, Jan. 1930, pp. 147-150&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Die Ionisierungsspannungen von Atomkonfigurationen mit 2 Elektronen.&amp;quot; &lt;em&gt;Naturwissenschaften&lt;/em&gt;,  Vol. 18, No. 4, Jan. 1930, p. 86 (letter to the editor)&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &amp;quot;Die Kausalität in der Physik.&amp;quot; &lt;em&gt;Zeitschrift für Physik&lt;/em&gt;. Vol. 64, Nos. 7-8, Jul. 1930, pp. 568-580&lt;/p&gt;&#xA;&lt;p&gt;G. W. Kellner, &lt;em&gt;Sozialpolitik und Antikonjunkturpolitik: Untersuchungen über Möglichkeiten zur Stabilisierung der Wirtschaft und der Arbeit.&lt;/em&gt; Gießen: Otto Kindt 1934, 76 pp.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Kellner's biographical summary, pieced together from Google snippets of his 1934 Ph.D. dissertation. The formatting is mine, and there may be OCR errors.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ich, der unterzeichnete Georg Wolfgang Kellner, wurde am 1. Februar 1905 als Sohn des Bankherrn Otto Kellner in Wetzlar und seiner Ehefrau Hermine Kellner, geb. Spamer, geboren. Ich besuchte von Ostern 1911 bis Ostern 1914 das Städtische Lyzeum in Wetzlar, von Ostern 1914 bis Ostern 1921 das staatliche humanistische Gymnasium in Wetzlar und von Ostern 1921 bis Ostern 1923 das Realgymnasium in Gießen, das ich mit dem Reifezeugnis verließ.&lt;/p&gt;&#xA;&lt;p&gt;Ich studierte von Ostern 1923 bis Herbst 1927 an der Universität Berlin Mathematik und Physik und promovierte dort am 11. Oktober 1927 auf Grund  einer Dissertation mit dem mit dem Titel: &amp;quot;Die Ionisierungsspannung des&#xA;Heliums nach der Schrödingerschen Theorie&amp;quot;. Vom 1. November 1927 bis 30.  April 1931 war ich Assistent am Institut für theoretische Physik der  Universität Berlin.&lt;/p&gt;&#xA;&lt;p&gt;Vom 1. Mai 1931 bis zum 31. Dezember 1932 bearbeitete ich, hauptsächlich an der Universität Leipzig, auf Grund eines Forschungsstipendiums der Notgemeinschaft der Deutschen Wissenschaft eine Aufgabe aus dem Gebiete der theoretischen Physik, siedelte jedoch schon im September 1932 nach meiner Heimat Wetzlar über.&lt;/p&gt;&#xA;&lt;p&gt;Im Wintersemester 1932/33 belegte ich als Gasthörer an der Universität Gießen Vorlesungen und Übungen bei den Herren Prof. Lenz, Mombert, Günther,  Stoltenberg,sowie bei Herrn Regierungsrat Dr. Bues und bei Frau Dr. v . Reichenau. Seit Ende Januar 1933 war ich neben den Vorlesungen und während der Ferien bis Ende April 1933 informatorisch am Arbeitsamt in Wetzlar tätig.&lt;/p&gt;&#xA;&lt;p&gt;Ich entschloß mich sodann, mich am  Ende des Sommersemesters 1933 in Gießen zur Promotion zum Dr. rer. pol. zu melden und ließ mich darum an der Universität Gießen immatrikulieren. Ich belegte Vorlesungen und Übungen bei den Herren Prof. Gmelin, Bötticher, Günther, Mombert und Auler.&lt;/p&gt;&#xA;&lt;p&gt;Nach Ablegung der Promotionsprüfungen war ich von Anfang August bis Mitte September 1933 nochmals am Arbeitsamt in Wetzlar tätig und bin seit dieser Zeit bei der Fa. Ernst Leitz G.m.b.H., Optische Werke in Wetzlar, angestellt. Wetzlar, im Juli 1934. Dr. G. W. Kellner.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-09-16T23:06:10.043" LastActivityDate="2023-09-16T23:06:10.043" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15753" PostTypeId="1" CreationDate="2023-09-15T23:06:47.527" Score="2" ViewCount="101" Body="&lt;p&gt;I read in Wiki they were together at the Solvay Conderence. There is also a note from 1926 letter to Paul Ehrenfest, Albert Einstein wrote of a Dirac paper, &amp;quot;&lt;em&gt;I am toiling over Dirac. This balancing on the dizzying path between genius and madness is awful.&amp;quot;&lt;/em&gt; In another letter concerning the Compton effect he wrote, &lt;em&gt;&amp;quot;I don't understand the details of Dirac at all.&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Dirac used special relativity in 1926 as part of the Dirac Equation.&lt;/p&gt;&#xA;&lt;p&gt;I am scratching my head on the comment from the architect of general relativity arguably one of the crown jewels of human achivement. On the other hand the EPR paper requires deep insight into quantum mechanics to be able to predict the behavior of what entanglement would do well ahead of anyone else. Of course there were two other's who contributed along  with Einstein but I rather imagine Einstein understood quantum mechanics. I was wondering if other than what I found above there is more to read?&lt;/p&gt;&#xA;" OwnerUserId="5040" LastActivityDate="2023-09-15T23:06:47.527" Title="Was there any discourse between Dirac and Einstein that was recorded in print or noted?" Tags="|einstein|dirac|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15755" PostTypeId="1" AcceptedAnswerId="15756" CreationDate="2023-09-16T07:56:01.937" Score="9" ViewCount="1475" Body="&lt;p&gt;[This question is about popular conceptions and therefore goes into strange directions, don't get too shocked]&lt;/p&gt;&#xA;&lt;p&gt;The notion of spacetime can be traced back to roughly the 18th century where some people like Lagrange and d'Alembert started to idly speculate about time and space being part of a same whole, and we can consider its official birth to be that of Minkowski's paper on the topic in 1908, but however there exists a rather uncertain era in between the two where time as a fourth dimension seems to occasionally pop up.&lt;/p&gt;&#xA;&lt;p&gt;The obvious intermediary step in between would be for instance Hamilton's theory of quaternions, but there are however some weird outliers as well : H. G. Wells' &amp;quot;The Time Machine&amp;quot; in 1895 refers to spacetime, quite right at the beginning, ie :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Now, it is very remarkable that this is so extensively overlooked,”&#xA;continued the Time Traveller, with a slight accession of cheerfulness.&#xA;“Really this is what is meant by the Fourth Dimension, though some&#xA;people who talk about the Fourth Dimension do not know they mean it.&#xA;It is only another way of looking at Time. There is no difference&#xA;between Time and any of the three dimensions of Space except that our&#xA;consciousness moves along it. But some foolish people have got hold of&#xA;the wrong side of that idea. You have all heard what they have to say&#xA;about this Fourth Dimension?”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The influences of this are quite hard to find. In a much more general direction, the notion of a fourth dimension in general seems to have been popularized by among other people Zöllner's &amp;quot;Transcendental Physics&amp;quot; in 1875&lt;a href=&quot;https://archive.org/details/transcendentalph00zlrich&quot; rel=&quot;noreferrer&quot;&gt;[1]&lt;/a&gt;, in which he tries to tie the notion of higher dimension to supernatural forces, an idea which is found in later popular books like Blavatsky's Secret Doctrine, or C. W. Leadbeater's book on remote viewing. In particular, Schofield's book &amp;quot;Another World&amp;quot;&lt;a href=&quot;https://archive.org/details/cu31924031255882&quot; rel=&quot;noreferrer&quot;&gt;[2]&lt;/a&gt; seems to reference Mr. Wells [I assume it is H. G. Wells here] in 1905,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Many speculations concerning the fourth dimension have been made since this book was&#xA;first issued, notably that by Mr. Wells that it is &amp;quot; Time.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Making it seem as though this was a well known idea at the time, at least within some circles.&lt;/p&gt;&#xA;&lt;p&gt;The idea of a fourth dimension also comes up in Charles Hinton's writings on the topic in 1880&lt;a href=&quot;https://www.gutenberg.org/ebooks/67153&quot; rel=&quot;noreferrer&quot;&gt;[3]&lt;/a&gt;, including a method of visualizing the fourth dimension which uses time as an analogy for the fourth dimension, although he does not seem to actually say that the fourth dimension itself is time. Flatland appears a few years later with much similar ideas on the topic.&lt;/p&gt;&#xA;&lt;p&gt;Did H. G. Wells have some kind of notion of spacetime that he commonly talked about outside of his stories, and do we know where that idea came from exactly?&lt;/p&gt;&#xA;&lt;p&gt;Edit : Some hints from Wells' autobiography :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the students' Debating Society, of which I will tell more later, I heard about and laid hold of the idea of a four dimensional frame for a fresh apprehension of physical phenomena, which afterwards led me to send a paper, &amp;quot;The Universe Rigid,&amp;quot; to the Fortnightly Review (a paper which was rejected by Frank Harris as incomprehensible), and gave me a frame for my first scientific fantasia, the Time Machine, and there was moreover a rather elaborate joke going on with Jennings and the others, about a certain &amp;quot;Universal Diagram&amp;quot; I proposed to make, from which all phenomena would be derived by a process of deduction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="2628" LastEditDate="2023-09-16T08:31:25.530" LastActivityDate="2023-09-17T08:45:15.893" Title="Where did the popular idea of spacetime come from?" Tags="|physics|relativity-theory|time|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15756" PostTypeId="2" ParentId="15755" CreationDate="2023-09-16T13:40:07.383" Score="9" Body="&lt;p&gt;A nice history was given by Archibald, R.C. (1914), &amp;quot;Time as a fourth dimension&amp;quot;, Bull. Amer. Math. Soc., 20 (8): 409–412&#xA;&lt;a href=&quot;https://doi.org/10.1090%2FS0002-9904-1914-02511-X&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1090%2FS0002-9904-1914-02511-X&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Regarding Wells, it says:&lt;/p&gt;&#xA;&lt;p&gt;Apparently independent of earlier suggestion, a writer in Nature [March 26, 1885, vol. 31, p. 481; &amp;quot;Four-dimensional space,&amp;quot; by &amp;quot; S .&amp;quot;] developed the idea of &amp;quot;time-space.&amp;quot; A decade later the popular English novelist Herbert George Wells wrote a serial, for W. E. Henley's New Review [Jan.-June, 1895, vol. 12; &amp;quot;The time machine,&amp;quot; republished in book form, London, 1895.] in which an invention for negotiating similar &amp;quot;space&amp;quot; is basic.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to the above mentioned article that discusses &amp;quot;time-space&amp;quot; in Nature from 1885, anonymously written and signed as &amp;quot;S.&amp;quot;:&#xA;&lt;a href=&quot;https://en.wikisource.org/wiki/Four-Dimensional_Space&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikisource.org/wiki/Four-Dimensional_Space&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT:&lt;/strong&gt;&#xA;Time as fourth dimension appeared in a satirical and popular essay &amp;quot;Der Raum hat vier Dimensionen&amp;quot; (English: Space has four dimensions) published as early as 1846 in the book &amp;quot;&lt;a href=&quot;https://books.google.com/books?id=yWVue5jPRM0C&quot; rel=&quot;noreferrer&quot;&gt;Vier Paradoxa&lt;/a&gt;&amp;quot; by German physicist and philosopher &lt;a href=&quot;https://en.wikipedia.org/wiki/Gustav_Fechner&quot; rel=&quot;noreferrer&quot;&gt;Gustav Fechner&lt;/a&gt; (under the pseudonym &amp;quot;Dr. Mises&amp;quot;). For instance he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;p.37: Um die Gestalten des Raums mit vier Dimensionen zu berechnen,&#xA;hat er blos nöthig, seine Variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; als vierte Raumcoordinate zu&#xA;betrachten. (&lt;strong&gt;English&lt;/strong&gt;: In order to compute the shapes of&#xA;space of four dimensions, he is only required to consider his&#xA;variable &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; as fourth coordinate of space.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Fechner's views have been described by &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Alexander_Gunn&quot; rel=&quot;noreferrer&quot;&gt;John Alexander Gunn&lt;/a&gt;, in his book &amp;quot;The Problem of Time&amp;quot; (1929), as follows on page 207:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After considering life in a two-dimensional plane and then in a&#xA;three-dimensional, he concluded that a higher dimension must always&#xA;present itself as a time-succession. Fechner urged that the third&#xA;dimension, to a being unable to rise above it in an active intuition&#xA;and construction, must present itself as a time-succession. A&#xA;two-dimensional being is able to move and construct freely in a plane&#xA;world while subjectively (unconsciously) existing in the third&#xA;dimension. A three-dimensional being is able to move and construct&#xA;freely in a three-dimensional space while unconsciously living in the&#xA;fourth. The conscious mind of man surveys time in memory and&#xA;imagination only because it lives in a dimension of reality still&#xA;higher than the time-succession, but one which implies the&#xA;time-dimension: The analytic succession of time-flow is below the&#xA;synthetic unity of the ego for which that succession is.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&amp;quot;Fellner, H.G. and Lindgren, W.F., 2011. &lt;em&gt;Gustav Theodor Fechner: pioneer of the fourth dimension&lt;/em&gt;. The Mathematical Intelligencer, 33, pp.126-137.&amp;quot; &lt;a href=&quot;https://doi.org/10.1007/s00283-011-9214-7&quot; rel=&quot;noreferrer&quot;&gt;https://doi.org/10.1007/s00283-011-9214-7&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="7601" LastEditorUserId="7601" LastEditDate="2023-09-17T08:45:15.893" LastActivityDate="2023-09-17T08:45:15.893" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15757" PostTypeId="2" ParentId="15744" CreationDate="2023-09-16T14:21:29.717" Score="2" Body="&lt;p&gt;Since Leonard Euler went to a Gymnasium in Basel, he surely learnt greek, every student had to learn it, he graduated in theologie, which one could not do without greek as a language, &amp;quot;womit es nicht so recht fortwollte&amp;quot; says only, he did not spend too much time with it, but since his father was a theologian he made sure his son knew some greek. At the time in europe most students had to learn greek.&#xA;the german wikipedia gives the name of his school there exist a history of this school and  it says &amp;quot;Die Schule war nur männlichen Jugendlichen zugänglich und legte ihren Schwerpunkt auf Latein, Griechisch und Hebräisch.&amp;quot; Translation: The school was only for boys and its main purpose  was  Latin, Greek and Hebrew.&lt;/p&gt;&#xA;" OwnerUserId="12914" LastEditorUserId="12914" LastEditDate="2023-09-16T17:41:01.427" LastActivityDate="2023-09-16T17:41:01.427" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15758" PostTypeId="2" ParentId="15152" CreationDate="2023-09-16T17:24:21.750" Score="2" Body="&lt;p&gt;If we ignore research on webcomics themselves, &lt;a href=&quot;https://xkcd.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd&lt;/a&gt; seems to be the most common webcomic inspiration for science and mathematical research, and these are pretty much all in computer science and visualization.&lt;/p&gt;&#xA;&lt;p&gt;Searching for &amp;quot;xkcd&amp;quot; on Google Scholar &lt;a href=&quot;https://scholar.google.ca/scholar?hl=en&amp;amp;as_sdt=0%2C5&amp;amp;q=%22xkcd%22&amp;amp;oq=&quot; rel=&quot;nofollow noreferrer&quot;&gt;turns up thousands of results&lt;/a&gt;, and although most of these are either acknowledgements of use of images or to the use the &lt;a href=&quot;https://stackoverflow.com/q/12675147/1598769&quot;&gt;xkcd-style plots&lt;/a&gt; you mention, xkcd has inspired some other research, e.g.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1007/978-3-540-78243-8_10&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Irwin, B., Pilkington, N. In: Goodall, J.R., Conti, G., Ma, KL. (eds) VizSEC 2007. Mathematics and Visualization. Springer, Berlin, Heidelberg&quot;&gt;&amp;quot;High Level Internet Scale Traffic Visualization Using Hilbert Curve Mapping.&amp;quot;&lt;/a&gt; (2007), inspired by &lt;a href=&quot;https://xkcd.com/195/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd-195 &amp;quot;Map of the Internet&amp;quot;&lt;/a&gt;.)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://thomas-zimmermann.com/publications/files/zeller-promise-2011.pdf&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Promise '11: Proceedings of the 7th International Conference on Predictive Models in Software Engineering, September 2011, Article No.: 5, Pages 1–7, doi:10.1145/2020390.2020395&quot;&gt;&amp;quot;Failure is a Four-Letter Word&amp;quot;&lt;/a&gt;  (2011), inspired by &lt;a href=&quot;http://xkcd.com/882/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd-936 &amp;quot;Significant&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1145/2335356.2335366&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Shay, R., Kelley, P. G., Komanduri, S., Mazurek, M. L., Ur, B., Vidas, T., . . . Cranor, L. F. (2012).  Symposium On Usable Privacy and Security&quot;&gt;&amp;quot;Correct horse battery staple: Exploring the usability of system-assigned passphrases.&amp;quot;&lt;/a&gt; (2012), inspired by &lt;a href=&quot;https://xkcd.com/936/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd-936 &amp;quot;Password Strength&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://aclanthology.org/N15-1180.pdf&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Marjan Ghazvininejad and Kevin Knight, Human Language Technologies: The 2015 Annual Conference of the North American Chapter of the ACL, pages 1569–1575, Denver, Colorado, May 31 – June 5, 2015, doi:10.3115/v1/N15-1180&quot;&gt;&amp;quot;How to memorize a random 60-bit string&amp;quot;&lt;/a&gt; (2015), inspired by &lt;a href=&quot;https://xkcd.com/936/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd-936 &amp;quot;Password Strength&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.1109/TVCG.2019.2903945&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Görtler, M. Spicker, C. Schulz, D. Weiskopf and O. Deussen, in IEEE Transactions on Visualization and Computer Graphics, vol. 25, no. 6, pp. 2193-2204, 1 June 2019, doi: 10.1109/TVCG.2019.2903945&quot;&gt;&amp;quot;Stippling of 2D Scalar Fields&amp;quot;&lt;/a&gt; (2019), inspired by &lt;a href=&quot;https://xkcd.com/1939/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd-1939 &amp;quot;2016 Election Map&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://doi.org/10.7880/abas.0220914a&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Masayuki Hatta, Annals of Business Administrative Science 21 (2022) 91–102&quot;&gt;&amp;quot;The Nebraska Problem in Open Source Software Development&amp;quot;&lt;/a&gt; (2022), inspired by &lt;a href=&quot;https://xkcd.com/2347/&quot; rel=&quot;nofollow noreferrer&quot;&gt;xkcd-2347 &amp;quot;Dependency&amp;quot;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I can't immediately find similar inspiration acknowledgements for other &amp;quot;scientific&amp;quot; web comics such as &lt;a href=&quot;https://phdcomics.com/comics.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;PHD Comics&lt;/a&gt;, &lt;a href=&quot;http://www.smbc-comics.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saturday Morning Breakfast Cereal&lt;/a&gt;, &lt;a href=&quot;https://qwantz.com/index.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dinosaur Comics&lt;/a&gt;, and &lt;a href=&quot;https://abstrusegoose.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abstruse Goose&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;It is, of course, a problem that researchers are not always consciously aware of the sources of their inspiration. Even if they are aware that a webcomic inspired them, they may be embarrassed to admit it in a published paper.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-09-16T17:24:21.750" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15759" PostTypeId="1" AcceptedAnswerId="15760" CreationDate="2023-09-18T13:53:23.533" Score="0" ViewCount="110" Body="&lt;p&gt;I recently delved into a discussion about a statement attributed to the renowned mathematician and philosopher, Benjamin Peirce. In this statement, he refers to mathematics as &amp;quot;the science that draws necessary conditions.&amp;quot; This has piqued my curiosity about the linguistic origins and historical context of the term &amp;quot;necessary condition&amp;quot; in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/CN5bk.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/CN5bk.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Interpreting &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt; is a necessary condition for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;&amp;quot; seems to be a formal way to express that &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; implies &lt;span class=&quot;math-container&quot;&gt;$q$&lt;/span&gt;&amp;quot; (&lt;span class=&quot;math-container&quot;&gt;$p \Rightarrow q$&lt;/span&gt;). This realization has led me to several questions:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Main question: Who was the first person that introduced the term &amp;quot;necessary condition&amp;quot; in mathematical discourse?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;second important related&#xA;question: Was there a specific rationale or historical backdrop that&#xA;necessitated the introduction of this term?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;How has the understanding    and utilization of this term evolved in&#xA;the landscape of mathematical language over time?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am genuinely interested in uncovering the historical nuances surrounding the inception and adoption of this phrase in the mathematical community.&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="6609" LastEditDate="2023-09-18T20:16:31.643" LastActivityDate="2023-09-18T20:16:31.643" Title="Who first introduced the term &quot;necessary condition&quot; in mathematical language?" Tags="|terminology|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15760" PostTypeId="2" ParentId="15759" CreationDate="2023-09-18T15:55:06.250" Score="2" Body="&lt;p&gt;Oxford English Dictionary, s.v. “'necessary condition' in necessary, adj., sense I.1.e”, July 2023. &lt;a href=&quot;https://doi.org/10.1093/OED/1255631544&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1093/OED/1255631544&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.oed.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Oxford English Dictionary&lt;/em&gt;&lt;/a&gt; (subscription required) shows earliest quotation for &amp;quot;necessary condition&amp;quot; in 1650, Hobbes, &lt;em&gt;Leviathan&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;It shows &amp;quot;sufficient condition&amp;quot; only from 1914, but earlier uses of &amp;quot;sufficient cause&amp;quot; (1656) and &amp;quot;sufficient reason&amp;quot; (1717).  They also mention the terminology of Leibniz, &amp;quot;raison déterminante&amp;quot; (1710) later changed to &amp;quot;raison suffisante&amp;quot; (1714).&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-09-18T15:55:06.250" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15761" PostTypeId="2" ParentId="15747" CreationDate="2023-09-18T17:35:15.480" Score="4" Body="&lt;p&gt;Following the suggestion in &lt;a href=&quot;https://hsm.stackexchange.com/questions/15747/who-first-referred-to-the-number-of-nonzero-entries-of-a-vector-as-its-ell-0#comment36570_15747&quot;&gt;AChem's comment&lt;/a&gt;, I wrote to David Donoho, and he suggested that I check the literature on interpolation of normed Abelian groups.  Sure enough, the terminology can (essentially) be found in Section 6 of Jaak Peetre and Gunnar Sparr, &lt;a href=&quot;https://doi.org/10.1007/BF02417949&quot; rel=&quot;nofollow noreferrer&quot;&gt;Interpolation of normed Abelian groups&lt;/a&gt;, &lt;i&gt;Annali di Matematica Pura ed Applicata&lt;/i&gt; 92 (1972), 217–262.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In this Section we assume that all groups in question consist of measurable functions &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; defined on a domain &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; of the &lt;span class=&quot;math-container&quot;&gt;$d$&lt;/span&gt;-dimensional real Cartesian space &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^d$&lt;/span&gt;, with values in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt;. … define &lt;span class=&quot;math-container&quot;&gt;$L_0 = L_0(D)$&lt;/span&gt; as the space corresponding to the norm&#xA;&lt;span class=&quot;math-container&quot;&gt;$$|| f ||_{L_0} = \mathrm{meas.} (\mathrm{supp.} f).$$&lt;/span&gt;&#xA;(Here &lt;span class=&quot;math-container&quot;&gt;$\mathrm{supp.} f$&lt;/span&gt; denotes the support of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt;, i.e. a measurable set &lt;span class=&quot;math-container&quot;&gt;$E\subset D$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$f=0$&lt;/span&gt; a.e. in &lt;span class=&quot;math-container&quot;&gt;$D-E$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$f\ne 0$&lt;/span&gt; a.e. in &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt;; it is unique up to a nullset).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If we take &lt;span class=&quot;math-container&quot;&gt;$D$&lt;/span&gt; to be a finite discrete set, then we recover the notion of the &lt;span class=&quot;math-container&quot;&gt;$\ell_0$&lt;/span&gt; norm as being the number of nonzero entries. Peetre and Sparr may not be the earliest reference, but it does at least show that the concept was around long before compressed sensing was a &amp;quot;thing,&amp;quot; even if compressed sensing led to an explosion in the attention devoted to the concept.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2023-09-18T17:40:35.210" LastActivityDate="2023-09-18T17:40:35.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15762" PostTypeId="2" ParentId="11880" CreationDate="2023-09-19T14:35:55.330" Score="8" Body="&lt;p&gt;Although one can find &lt;a href=&quot;https://theconversation.com/limitless-power-arriving-too-late-why-fusion-wont-help-us-decarbonise-176974&quot; rel=&quot;noreferrer&quot;&gt;claims that this joke goes back to the 1960s&lt;/a&gt;,  the joke only really started to bite after the first 30 years of fusion research had passed and we seemed no closer to the goal.  This timescale matches the first explicit mention I can find of the joke which is from a &lt;a href=&quot;https://www.osti.gov/biblio/5705749&quot; rel=&quot;noreferrer&quot; title=&quot;Sheffield, J. Panel discussion on prospects for fusion power. United States Conf. 8604175-1, 1986.&quot;&gt;1986 &amp;quot;Panel Discussion on Prospects for Fusion Power&amp;quot;&lt;/a&gt;, where it was stated:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although substantial progress is made every year in fusion research, the projected time to realize the ultimate goal of commercial fusion always seems to be 25 to 30 years away.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is amusing to note that according to recent research on &lt;a href=&quot;https://doi.org/10.1007/s10894-023-00361-z&quot; rel=&quot;noreferrer&quot; title=&quot;Takeda, S., Keeley, A.R. &amp;amp; Managi, S.,  J Fusion Energ 42, 16 (2023).&quot;&gt;&amp;quot;How Many Years Away is Fusion Energy? A Review&amp;quot;&lt;/a&gt;, scientists now believe (on average) that fusion is 17.8 years away:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;we should now say “fusion was said to be 19.3 years away 30 years ago; it was 28.3 years away 20 years ago; 27.8 years away 10 years ago.” And now, scientists believe fusion energy is only 17.8 years away.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-09-19T14:35:55.330" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15763" PostTypeId="1" CreationDate="2023-09-20T15:24:29.797" Score="-1" ViewCount="144" Body="&lt;p&gt;Recently, I stumbled upon a historically important monograph on a technical subject, which explained complex physical phenomena without a single mathematical equation. I forgot the name of the author, the work, or any other identifying information. Therefore, I ask this question with two goals in mind: first, rediscover the work (some details below); second, find out if there are other similar examples, especially written by people who otherwise have unquestionable mathematical proficiency.&lt;/p&gt;&#xA;&lt;p&gt;The monograph I read was written between 16th and 19th century, it may have been on a topic related to light (electromagnetic radiation) or optics, and it was quite a voluminous work with maybe 100 pages. For some reason I think the author was someone of the likes of Christiaan Huygens, Thomas Young, or Augustin-Jean Fresnel, but I am not sure if it is any of these three. I also believe he was English, or the work was written in English, but I am not sure about this either.&lt;/p&gt;&#xA;" OwnerUserId="19027" LastActivityDate="2023-09-21T03:29:59.200" Title="Technical papers or monographs without a single mathematical equation" Tags="|physics|optics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15764" PostTypeId="2" ParentId="2437" CreationDate="2023-09-20T21:50:32.097" Score="4" Body="&lt;p&gt;Are we all talking about the same &amp;quot;Heaviside&amp;quot; who wrote &amp;quot;On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field&amp;quot;, received 1891-06-09, read 1891-06-18, that you can find a copy of over at the Royal Society (&lt;a href=&quot;https://doi.org/10.1098/rsta.1892.0011&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1098/rsta.1892.0011&lt;/a&gt;) or on the Internet Archive (&lt;a href=&quot;https://archive.org/details/philtrans06858697&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://archive.org/details/philtrans06858697&lt;/a&gt;)?&lt;/p&gt;&#xA;&lt;p&gt;Absolutely sure? :)&lt;/p&gt;&#xA;&lt;p&gt;Probably not, because this is what he wrote...&lt;/p&gt;&#xA;&lt;p&gt;Under the section &amp;quot;The Electromagnetic Equations in a Moving Medium&amp;quot; [Footnote about the term &amp;quot;Moving&amp;quot; pending]&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ = μ, \quad  = c, \tag{81}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$U = ½, \quad T = ½, \tag{82}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Q_1 = , \quad Q_2 = , \tag{83}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ = k, \quad  = g, \tag{84}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ρ = \text{div}\ , \quad σ = \text{div}\ , \tag{85}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ =  + \dot{} + ρ, \quad  =  + \dot{} + σ \tag{86,87}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ = V, \quad  = V, \tag{88,91}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\text{curl}( - _0 - ) =  =  + \dot{} + ρ, \label{89}\tag{89}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$-\text{curl}( - _0 - ) =  =  + \dot{} + σ, \label{90}\tag{90}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\left\{ \begin{matrix}&#xA;\text{curl}\  = , &amp;amp;  +  = _0, \\&#xA;-\text{curl}\  = , &amp;amp;  +  = _0, \\&#xA;\end{matrix}\right\} \tag{92}&#xA;$$&lt;/span&gt;&#xA;thus (as he noted), we may also write (&lt;span class=&quot;math-container&quot;&gt;$\ref{89}$&lt;/span&gt;) and (&lt;span class=&quot;math-container&quot;&gt;$\ref{90}$&lt;/span&gt;) as&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\left\{\begin{matrix}&#xA;\text{curl}( - _0) = _0 =  + \dot{} + ρ + , \\&#xA;-\text{curl}( - _0) = _0 =  + \dot{} + σ + .&#xA;\end{matrix}\right\} \tag{93}&#xA;$$&lt;/span&gt;&#xA;where&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = electric displacement and magnetic induction,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = electric and magnetic force,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; = permittivity and inductivity (for isotropic media),&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; = stored electrical &amp; magnetic energy (per unit volume),&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = fluxes (read: current densities),&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; = electric and magnetic conductivity,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$ρ$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$σ$&lt;/span&gt; = the volume densities of electrification and magnetification,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$σ$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; are quite likely fictitious and &lt;span class=&quot;math-container&quot;&gt;$g$&lt;/span&gt; superfluous,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = the &quot;true&quot; electric and magnetic currents,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = the velocity of electrification (and magnetification), i.e. the velocity relative to the medium,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = the motional electric and magnetic forces, the latter is a correction to Maxwell, who failed to include it,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$_0$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$_0$&lt;/span&gt; = the intrinsic force (of electrification) and force of intrinsic magnetization,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; = the current densities associated with the motional electric and magnetic forces,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$_0$&lt;/span&gt; &amp; &lt;span class=&quot;math-container&quot;&gt;$_0$&lt;/span&gt; = maybe these are the &quot;true&quot; currents, instead.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Now, I don't know about anyone else, but that doesn't look like four equations! I suspect that you might have been on the receiving end of &amp;quot;telephone tag&amp;quot;, running a dozen layers deep, because somebody was definitely asleep at the switch there. It doesn't even resemble the item you cited!&lt;/p&gt;&#xA;&lt;p&gt;&lt;b&gt;The Footnote: &amp;quot;Moving&amp;quot;&lt;/b&gt;&lt;br/&gt;&#xA;Now ... let's put this in context. All the pre-1905 literature (as well as post-1905 literature) makes a clear distinction between &amp;quot;stationary&amp;quot; and &amp;quot;moving&amp;quot;. That's meant in an absolute sense: stationary and moving with respect to The Medium. Everything before 1905 is a medium, including outer space, and has its own reference frame with respect to which &amp;quot;stationary&amp;quot; and &amp;quot;moving&amp;quot; are determined.&lt;/p&gt;&#xA;&lt;p&gt;This is the meaning and sense of &amp;quot;moving&amp;quot; in Einstein's 1905 &amp;quot;On The Electrodynamics Of Moving Bodies&amp;quot;. In that context, you can see clearly that the very title, itself, is a loud wake-up call to those of the time - something that falls on deaf ears today, of those who have no living memory of the earlier context.&lt;/p&gt;&#xA;&lt;p&gt;But, as I noted: some things &lt;i&gt;after&lt;/i&gt; 1905 make the distinction too.&lt;/p&gt;&#xA;&lt;p&gt;Maxwell's equations, such as we would write them today, and such as we would write them if the magnetic sources were present too, would be:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;∇· = σ, \quad ∇× + \frac{∂}{∂t} = -, \\&#xA;∇· = ρ, \quad ∇× - \frac{∂}{∂t} = +. \\&#xA;$$&lt;/span&gt;&#xA;The constitutive relations between the two sets of fields that goes with this, in the non-relativistic case would be:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ = ε( + ×), \quad  = μ( - ×),$$&lt;/span&gt;&#xA;where, now, I am using &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; to denote the velocity of the medium, instead of Heaviside's &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, because that's what Maxwell (sporadically) used.&lt;/p&gt;&#xA;&lt;p&gt;Heaviside's other letters are scrambled up a bit. I think he's got his &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; swapped, because Maxwell used &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; for the &amp;quot;total&amp;quot; current. Maxwell's &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is our &lt;span class=&quot;math-container&quot;&gt;$ + \dot{}$&lt;/span&gt;. The magnetic letters for currents and sources are his insertions that he called, &amp;quot;probably fictitious&amp;quot;, but he's just going to put them there because he likes things to line up symmetrically. Basically, that was actually the reason he gave.&lt;/p&gt;&#xA;&lt;p&gt;His vector product &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is our &lt;span class=&quot;math-container&quot;&gt;$×$&lt;/span&gt;. His scalar product &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is our &lt;span class=&quot;math-container&quot;&gt;$·$&lt;/span&gt;. In addition, he also had a generalized inverse &lt;span class=&quot;math-container&quot;&gt;$^{-1}$&lt;/span&gt;,&#xA;which is our &lt;span class=&quot;math-container&quot;&gt;$/(·)$&lt;/span&gt;. His dot was used as a separator. For instance, his &lt;span class=&quot;math-container&quot;&gt;$·$&lt;/span&gt; actually means his &lt;span class=&quot;math-container&quot;&gt;$()$&lt;/span&gt;, or our &lt;span class=&quot;math-container&quot;&gt;$·$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I won't even try to line up which combinations of his letters correspond to our letters. They don't exactly match. Maxwell's &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, for instance, is our &lt;span class=&quot;math-container&quot;&gt;$ + ×$&lt;/span&gt;, as it was for Hertz (after you convert his letters to our letters), while Lorentz stuck with the bare &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;, and added the &lt;span class=&quot;math-container&quot;&gt;$×$&lt;/span&gt; as a separate term. Maxwell had no &lt;span class=&quot;math-container&quot;&gt;$-×$&lt;/span&gt; addition to &lt;span class=&quot;math-container&quot;&gt;$ - ×$&lt;/span&gt;. Thomson made that correction. I thought it was him first, but maybe Heaviside did it, since he laid claim to the &amp;quot;motional&amp;quot; magnetic force as his idea.&lt;/p&gt;&#xA;&lt;p&gt;I'm not entirely sure which &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; Heaviside's is: Maxwell's or ours; the same for &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt;. I'll have to look at it more closely and (maybe) add a few remarks on this with a later edit.&lt;/p&gt;&#xA;&lt;p&gt;That passing over the 1905 divide changes nothing, here. The &lt;span class=&quot;math-container&quot;&gt;$$&lt;/span&gt; is still there - except for the vacuum. The relativistic version of the constitutive relations above is the Maxwell-Minkowski(-Einstein-Laub) relations:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$ + \frac{×}{c^2} = ε( + ×), \quad  - \frac{×}{c^2} = μ( - ×).$$&lt;/span&gt;&#xA;The inclusions on the left-hand side of each equation are offsets that will cancel the &amp;quot;motional&amp;quot; forces on the right ... &lt;i&gt;if&lt;/i&gt; ... &lt;span class=&quot;math-container&quot;&gt;$εμ = (1/c)^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Hertz' and Lorentz' constitutive laws are equivalent to the non-relativistic versions, not the Maxwell-Minkowski relations. You'll have to go through the exercise to lay out a &amp;quot;Modern versus Heaviside&amp;quot; Rosetta Stone to verify that his equations line up with the Maxwell-Thomson(-Hertz-Lorentz) relations, as I'm almost certain they do.&lt;/p&gt;&#xA;&lt;p&gt;But, in any case: there's a whole lot more than four equations there! And, I scanned volumes 1 and 2 of his &amp;quot;Electromagnetic Theory&amp;quot; (1899). Basically, he just lifted his earlier equations and put them in there, with further detail and elaboration.&lt;/p&gt;&#xA;" OwnerUserId="13406" LastEditorUserId="13406" LastEditDate="2023-10-22T18:44:05.397" LastActivityDate="2023-10-22T18:44:05.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15765" PostTypeId="2" ParentId="15763" CreationDate="2023-09-21T02:54:21.923" Score="3" Body="&lt;p&gt;I read something like that about Faraday's treatise (book?) (if I remember correctly, there was just one formula there). As for Faraday's mathematical abilities, opinions vary.&lt;/p&gt;&#xA;&lt;p&gt;I will try to find a reference.&lt;/p&gt;&#xA;&lt;p&gt;EDIT (Sep 20 2023). Yes, Faraday's &amp;quot;Experimental researches in electricity&amp;quot;, 1839, (&lt;a href=&quot;https://ia601400.us.archive.org/15/items/in.ernet.dli.2015.500031/2015.500031.Experimental-Researches.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;a big file&lt;/a&gt;) seems to fit the bill: a lot of pages without any formulas. I cannot be sure there are not any, but here's a quote from W.F.G. Swann, Science v.73, No. 1896, p.462 (1931):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;When one reads through Faraday's experimental researches and finds in&#xA;them no mathematical formula, one sometimes wonders whether a person&#xA;of his intuitive powers of  conception may not, as a result, have&#xA;limited his vision as to the generality of the possibilities. The more&#xA;closely we read, however, the more we see that even when delving in&#xA;those realms which are the natural field of mathematical analysis, he&#xA;has an  uncanny way of knowing exactly what he is doing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9808" LastEditorUserId="9808" LastEditDate="2023-09-21T03:29:59.200" LastActivityDate="2023-09-21T03:29:59.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15766" PostTypeId="2" ParentId="15734" CreationDate="2023-09-21T03:58:47.933" Score="3" Body="&lt;p&gt;Maybe &lt;a href=&quot;https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups&quot; rel=&quot;nofollow noreferrer&quot;&gt;classification of finite simple groups&lt;/a&gt; qualifies (&amp;quot;The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.&amp;quot;)?&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;1983  Gorenstein announces the proof of the classification is complete,&#xA;somewhat prematurely as the proof of the quasithin case was&#xA;incomplete.&lt;/p&gt;&#xA;&lt;p&gt;2004  Aschbacher and Smith publish their work on quasithin groups&#xA;(which are mostly groups of Lie type of rank at most 2 over fields of&#xA;even characteristic), filling the last gap in the classification known&#xA;at that time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-09-21T03:58:47.933" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15767" PostTypeId="1" CreationDate="2023-09-21T23:03:07.987" Score="0" ViewCount="125" Body="&lt;p&gt;This question is about a class of periodic permutations,&lt;br&gt;that are produced by applying finite permutations to the binary digits of all integers.&lt;br&gt;&#xA;In lack of a better name, they shall be called &lt;em&gt;bit permutations&lt;/em&gt; for now.&lt;/p&gt;&#xA;&lt;p&gt;Some of these permutations are often called &lt;em&gt;shuffle&lt;/em&gt;, &lt;em&gt;butterfly&lt;/em&gt; and &lt;em&gt;bit-reversal&lt;/em&gt;,&lt;br&gt;&#xA;which can be prepended by &lt;em&gt;sub&lt;/em&gt;, &lt;em&gt;super&lt;/em&gt;, and &lt;em&gt;perfect&lt;/em&gt;.&#xA;E.g. &amp;quot;k sub-butterfly&amp;quot; or &amp;quot;perfect shuffle&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The simplest example is exchanging the two least significant bits. (&lt;a href=&quot;https://oeis.org/A080412&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sloane's A080412&lt;/a&gt;)&lt;br&gt;&#xA;Well-known examples are the &lt;a href=&quot;https://en.wikipedia.org/wiki/Bit-reversal_permutation&quot; rel=&quot;nofollow noreferrer&quot;&gt;bit-reversal permutations&lt;/a&gt;. (&lt;a href=&quot;https://oeis.org/A030109&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sloane's A030109&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Within this question I will denote the sequences as follows:&lt;br&gt;&#xA;Let p&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt; be the &lt;em&gt;n&lt;/em&gt;-th finite permutation. (See &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Symmetric_group_4;_permutation_list_(0-based).svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; list.)&lt;br&gt;&#xA;Then S&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt; shall be the sequence obtained by applying p&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt; on the binary digits of all integers.&lt;br&gt;&#xA;See &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Consecutive_bit_permutations_24x16.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; 24×16 matrix, whose rows are S&lt;sub&gt;0&lt;/sub&gt; ... S&lt;sub&gt;23&lt;/sub&gt;. (&lt;a href=&quot;https://oeis.org/A195665&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sloane's A195665&lt;/a&gt;)&lt;br&gt;&#xA;(These sequences are not the permutations, but the results of applying them – but never mind.)&lt;/p&gt;&#xA;&lt;p&gt;Below are the first 6 finite permutations assigned to S&lt;sub&gt;0&lt;/sub&gt; ... S&lt;sub&gt;5&lt;/sub&gt;.&lt;br&gt;&#xA;The 6×8 matrix is the top left corner of the 24×16 matrix.&lt;/p&gt;&#xA;&lt;pre&gt;&lt;code&gt;[0, 1, 2]   --&amp;gt;   [0, 1, 2, 3, 4, 5, 6, 7]&#xA;[1, 0, 2]   --&amp;gt;   [0, 2, 1, 3, 4, 6, 5, 7]&#xA;[0, 2, 1]   --&amp;gt;   [0, 1, 4, 5, 2, 3, 6, 7]&#xA;[2, 0, 1]   --&amp;gt;   [0, 2, 4, 6, 1, 3, 5, 7]&#xA;[1, 2, 0]   --&amp;gt;   [0, 4, 1, 5, 2, 6, 3, 7]&#xA;[2, 1, 0]   --&amp;gt;   [0, 4, 2, 6, 1, 5, 3, 7]&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;h3&gt;Question&lt;/h3&gt;&#xA;&lt;p&gt;Who came up with this?&lt;br&gt;&#xA;What are the earliest examples of permutations like these (except perfect bit-reversals).&lt;br&gt;&#xA;Ideally, I would like to know the oldest source that shows the first 6 or 24 examples together.&lt;br&gt;&#xA;There should certainly be something older than &lt;a href=&quot;https://oeis.org/A195665&quot; rel=&quot;nofollow noreferrer&quot;&gt;A195665&lt;/a&gt;, which I published in 2011.&lt;br&gt;&lt;/p&gt;&#xA;&lt;p&gt;(I am hoping for something with the clarity of &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Rothe_Stellenexponenten_2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; source, where one can see,&lt;br&gt;that Rothe has essentially defined the factorial number system around the year 1800.)&lt;/p&gt;&#xA;&lt;h3&gt;Sources&lt;/h3&gt;&#xA;&lt;h4&gt;misc.&lt;/h4&gt;&#xA;&lt;p&gt;Searching for specific bit permutations leads to all kinds of technical sources.&lt;br&gt;&#xA;&lt;a href=&quot;https://www.mvtec.com/doc/halcon/1811/en/hit_or_miss_golay.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; and &lt;a href=&quot;https://github.com/Mysticial/DigitViewer/blob/master/Source/DigitViewer2/DigitHash/Kernels/DigitHash_x64_AVX2.h&quot; rel=&quot;nofollow noreferrer&quot;&gt;there&lt;/a&gt; are examples for &amp;quot;0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15&amp;quot;. (S&lt;sub&gt;9&lt;/sub&gt;)&lt;br&gt;&#xA;&lt;a href=&quot;https://www.informatik.hu-berlin.de/de/forschung/gebiete/algorithmenII/Lehre/ss06/krypto2/skript/kap1.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Here&lt;/a&gt; and &lt;a href=&quot;https://github.com/Ledger-Donjon/rainbow/blob/master/examples/hacklu2009/go.py&quot; rel=&quot;nofollow noreferrer&quot;&gt;there&lt;/a&gt; are examples for &amp;quot;0, 4, 8, 12, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15&amp;quot;. (S&lt;sub&gt;16&lt;/sub&gt;)&lt;br&gt;&lt;/p&gt;&#xA;&lt;h4&gt;Bhujade (1995)&lt;/h4&gt;&#xA;&lt;p&gt;&lt;em&gt;Parallel Computing&lt;/em&gt;, Moreshwar R. Bhujade&lt;/p&gt;&#xA;&lt;p&gt;On pages 164–166 are the chapters &lt;em&gt;Perfect Shuffle Permutations&lt;/em&gt;, &lt;em&gt;Butterfly Permutation&lt;/em&gt; and &lt;em&gt;Bit Reversal Permutations&lt;/em&gt;. The perfect shuffle is &lt;a href=&quot;https://commons.wikimedia.org/wiki/File:Walsh_permutation_241.svg&quot; rel=&quot;nofollow noreferrer&quot;&gt;this rotation&lt;/a&gt;. The other examples are called (sub/super) butterfly and (sub/super) bit-reversal. (The butterflies and bit-reversals are the same in this case, because the examples use only 3 bits.)&lt;/p&gt;&#xA;&lt;h4&gt;Bhattacharyya et al. (2003)&lt;/h4&gt;&#xA;&lt;p&gt;&lt;em&gt;Domain-Specific Processors: Systems, Architectures, Modeling, and Simulation&lt;/em&gt;&lt;br&gt;&#xA;Shuvra S. Bhattacharyya, ‎Ed F. Deprettere, ‎Jürgen Teich&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.researchgate.net/figure/Signal-flow-graphs-of-FFT-algorithms-a-radix-2-and_fig1_248853426&quot; rel=&quot;nofollow noreferrer&quot;&gt;figure on page 72&lt;/a&gt; shows two transformations:&lt;br&gt;&#xA;(a) from S&lt;sub&gt;23&lt;/sub&gt; to S&lt;sub&gt;18&lt;/sub&gt; using 2-point FFT&lt;br&gt;&#xA;(b) from S&lt;sub&gt;16&lt;/sub&gt; to itself using 4-point FFT&lt;/p&gt;&#xA;&lt;p&gt;The caption is: Signal flow graphs of FFT algorithms:&#xA;(a) radix-2 and (b) radix-4 algorithm&lt;/p&gt;&#xA;&lt;h4&gt;Dally, Towles (2004)&lt;/h4&gt;&#xA;&lt;p&gt;&lt;em&gt;Principles and Practices of Interconnection Networks&lt;/em&gt;&lt;br&gt;&#xA;William James Dally, ‎Brian Patrick Towles&lt;/p&gt;&#xA;&lt;p&gt;S&lt;sub&gt;9&lt;/sub&gt; is described as a &lt;em&gt;shuffle permutation&lt;/em&gt; on page 57.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For bit-rotation (BR) traffic, the node with address {b&lt;sub&gt;3&lt;/sub&gt;, b&lt;sub&gt;2&lt;/sub&gt;, b&lt;sub&gt;1&lt;/sub&gt;, b&lt;sub&gt;0&lt;/sub&gt;}&#xA;sends packets only to the node with address {b&lt;sub&gt;2&lt;/sub&gt;, b&lt;sub&gt;1&lt;/sub&gt;, b&lt;sub&gt;0&lt;/sub&gt;, b&lt;sub&gt;3&lt;/sub&gt;}.&#xA;Expressed differently, this traffic corresponds to the shuffle&#xA;permutation {0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15}&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h4&gt;Lee, Ye (2018)&lt;/h4&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://arxiv.org/pdf/cs/0611024.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Relational Approach to Functional Decomposition of Logic Circuits&lt;/em&gt;&lt;/a&gt; (PDF on Arxiv)&lt;br&gt;&#xA;The figure on page 14 shows S&lt;sub&gt;3&lt;/sub&gt; and S&lt;sub&gt;12&lt;/sub&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19029" LastEditorUserId="19029" LastEditDate="2023-09-22T22:04:28.867" LastActivityDate="2023-09-22T22:04:28.867" Title="Who invented bit permutations like shuffle, butterfly and bit-reversal?" Tags="|mathematicians|computer-science|combinatorics|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15768" PostTypeId="1" AcceptedAnswerId="15770" CreationDate="2023-09-22T06:23:12.640" Score="0" ViewCount="252" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/jD4w2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/jD4w2.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/R9W7A.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/R9W7A.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;&lt;a href=&quot;https://i.stack.imgur.com/r9SQx.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/r9SQx.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&#xA;The original version of these photos is Felix Klein's &amp;quot;Development of Mathematics in the 19th Century&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In second photo&lt;/strong&gt;&#xA;In this book, it says class-field and decompose 2 into &lt;span class=&quot;math-container&quot;&gt;$(1+i)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(1-i)$&lt;/span&gt;. Is it not possible to decompose it into 3 parts like &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1+i}$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$\sqrt{1-i}$&lt;/span&gt;? If this is not possible, I would appreciate it if you could explain the rules of the class field.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In third photo&lt;/strong&gt;&#xA;And it says &lt;span class=&quot;math-container&quot;&gt;$2\mu+(1+\sqrt{-5})\nu$&lt;/span&gt;. I understood this to be the sum of &amp;quot;the set of numbers divisible by 2&amp;quot;and &amp;quot;the set of numbers divisible by field &lt;span class=&quot;math-container&quot;&gt;$K\sqrt{-5}$&lt;/span&gt;&amp;quot;. i.e. Felix Klein could have written &lt;span class=&quot;math-container&quot;&gt;$a+b\sqrt{-5}$&lt;/span&gt; in the book, but I think he wrote &lt;span class=&quot;math-container&quot;&gt;$1+\sqrt{-5}$&lt;/span&gt;, which is the easiest. If this interpretation is wrong, what is the meaning of &lt;span class=&quot;math-container&quot;&gt;$2\mu+(1+\sqrt{-5})\nu$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-09-23T01:06:10.807" LastActivityDate="2023-09-24T20:50:47.247" Title="Question about Felix Klein's &quot;Development of Mathematics in the 19th Century&quot;" Tags="|number-theory|book-recommendation|" AnswerCount="1" CommentCount="5" ClosedDate="2023-09-25T00:35:29.780" ContentLicense="CC BY-SA 4.0" />
  <row Id="15769" PostTypeId="1" CreationDate="2023-09-23T01:48:18.353" Score="3" ViewCount="136" Body="&lt;p&gt;Much of the Ancient Greek's mathematical philosophy texts have survived from antiquity and passed to modern times. Also, texts previously thought to be lost are being occasionally rediscovered (Notably the &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes_Palimpsest&quot; rel=&quot;nofollow noreferrer&quot;&gt;Archimedes Palimpsest&lt;/a&gt; and &lt;a href=&quot;https://link.springer.com/article/10.1007/s00407-022-00302-w&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ptolemy's Meteoroscope&lt;/a&gt;). Unfortunately, there are also writings lost in time, that we know of only by reference.&lt;/p&gt;&#xA;&lt;p&gt;I am looking for a comprehensive list of the &lt;em&gt;titles&lt;/em&gt; of Ancient Greek manuscripts from the millennium of 7th century BC to 6th century AD. Does such a list exist?&lt;/p&gt;&#xA;&lt;p&gt;Details for each title should include: authorship, timeframe, whether it be extant or lost, accessibility of manuscript, availability of translations, and topics covered.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h3&gt;Resources&lt;/h3&gt;&#xA;&lt;p&gt;&lt;strong&gt;Extant Writings and Fragments&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;http://www.perseus.tufts.edu/hopper/collection?collection=Perseus:collection:Greco-Roman&quot; rel=&quot;nofollow noreferrer&quot;&gt;Perseus Digital Library&lt;/a&gt;: Contains the English translations of a few well known texts.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://opengreekandlatin.github.io/First1KGreek/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Open Greek and Latin&lt;/a&gt;: A more expansive base, with purely Greek and Latin. However, user experience is difficult.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(Diagrams not included)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Lost Writings&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Lost_literary_work&quot; rel=&quot;nofollow noreferrer&quot;&gt;An list of lost works via Wikipedia&lt;/a&gt;: By no means a complete list.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;These references I mentioned are fine, but not extremely suitable or simple.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;If such a comprehensive list does not exist, I will put the work in to create one. Are other scholars are having similar troubles dealing with the disorganization of classical writings?&lt;/p&gt;&#xA;" OwnerUserId="19033" LastEditorUserId="19033" LastEditDate="2023-09-23T02:12:14.530" LastActivityDate="2023-09-23T02:12:14.530" Title="Is there a comprehensive list of Ancient Greek mathematical writings?" Tags="|biographical-details|reference-request|antiquity|greek-mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15770" PostTypeId="2" ParentId="15768" CreationDate="2023-09-23T04:09:59.597" Score="1" Body="&lt;p&gt;Regarding your second question:&lt;/p&gt;&#xA;&lt;p&gt;It is not clear what you mean by &amp;quot;&lt;em&gt;the set of numbers divisible by the field &lt;span class=&quot;math-container&quot;&gt;$K\sqrt{-5}$&lt;/span&gt;&lt;/em&gt;&amp;quot;.  In any field, every element is divisible by every non-zero element of the field.&lt;/p&gt;&#xA;&lt;p&gt;The expression &lt;span class=&quot;math-container&quot;&gt;$2\mu + ( 1 + \sqrt{-5} )\nu$&lt;/span&gt; denotes any element that can be written in this way, where &lt;span class=&quot;math-container&quot;&gt;$\mu$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; are &lt;a href=&quot;https://en.wikipedia.org/wiki/Algebraic_integer&quot; rel=&quot;nofollow noreferrer&quot;&gt;algebraic integers&lt;/a&gt; with respect to the field &lt;span class=&quot;math-container&quot;&gt;$K(\sqrt{-5})$&lt;/span&gt;.  Since not every element of a field is an algebraic integer we would not write &lt;span class=&quot;math-container&quot;&gt;$a + b\sqrt{-5}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(I am still wading in the shallow waters of algebra, so other readers may have different views of your questions.)&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2023-09-24T20:50:47.247" LastActivityDate="2023-09-24T20:50:47.247" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15771" PostTypeId="1" CreationDate="2023-09-24T13:31:20.017" Score="1" ViewCount="124" Body="&lt;p&gt;I am knowing &lt;a href=&quot;https://en.wikipedia.org/wiki/Abstract_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abstract Algebra&lt;/a&gt; things; I am searching aims of Abstract Algebra and origins of parts of Abstract Algebra. I thought original initial textbooks have explicit links to aims and origins of the parts of Abstract Algebra.&lt;/p&gt;&#xA;&lt;p&gt;Can we list major textbooks on Abstract Algebra in the order of time?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Abstract Algebra: &amp;quot;In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.&amp;quot; - Wikipedia/AbstractAlgebra&lt;/p&gt;&#xA;" OwnerUserId="2172" LastActivityDate="2023-09-24T14:14:44.323" Title="List of textbooks on Abstract Algebra in the order of time" Tags="|mathematics|abstract-algebra|algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15772" PostTypeId="1" CreationDate="2023-09-24T13:33:22.900" Score="0" ViewCount="77" Body="&lt;p&gt;Neil Degrasse Tyson has claimed that, via his &lt;em&gt;Principia&lt;/em&gt;, Isaac Newton was the first person (on record) to make a &amp;quot;modern&amp;quot; theory of physics, in the sense that Newton made an axiomatic framework to produce predictions that can be compared to observations of reality.&lt;/p&gt;&#xA;&lt;p&gt;Tyson said this in a lecture, but I'm having trouble finding a link to it right now, but that's not really important because the idea is what I'm asking about.&lt;/p&gt;&#xA;&lt;p&gt;In any case, can this claim be refuted by a counter example that preceded Newton? Who really was the first to build an axiomatic framework of (at least some part) of nature?&lt;/p&gt;&#xA;" OwnerUserId="7638" LastEditorUserId="6609" LastEditDate="2023-09-25T12:46:16.400" LastActivityDate="2023-09-25T12:46:16.400" Title="Who made the first (recorded) axiomatic model of nature?" Tags="|theoretical-physics|newton|physicists|mathematics-social-history|ancient|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15773" PostTypeId="2" ParentId="15771" CreationDate="2023-09-24T14:00:23.723" Score="2" Body="&lt;p&gt;Notation: BK = Book&lt;/p&gt;&#xA;&lt;p&gt;BK#1892 &lt;a href=&quot;https://archive.org/details/theoryofsubstitu00nettuoft&quot; rel=&quot;nofollow noreferrer&quot;&gt;The theory of substitution and its applications to algebra. Rev. by the author and translated with his permission by F.N. Cole 1892&lt;/a&gt;&#xA;(Book was mentioned in preface of BK#1901)&lt;/p&gt;&#xA;&lt;p&gt;BK#1901 &lt;a href=&quot;https://ia800906.us.archive.org/12/items/lineargroupswith00ledi/lineargroupswith00ledi.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Linear Groups with an Exposition of Galois Field Theory by Dickson L.E. 1901&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2172" LastEditorUserId="2172" LastEditDate="2023-09-24T14:14:44.323" LastActivityDate="2023-09-24T14:14:44.323" CommentCount="2" CommunityOwnedDate="2023-09-24T14:33:19.263" ContentLicense="CC BY-SA 4.0" />
  <row Id="15774" PostTypeId="2" ParentId="7513" CreationDate="2023-09-25T20:28:43.117" Score="1" Body="&lt;p&gt;Don't have the reputation to comment on Gerald Edgar's answer, but it seems not-quite-accurate from a few minutes of googling, at least all the sources I could find on ancient Egyptian land taxes mentioned &amp;quot;area&amp;quot; but not &amp;quot;perimeter&amp;quot;. An weaker but still interesting claim does seem to be true, according to the article&#xA;&lt;a href=&quot;https://digitalcommons.tacoma.uw.edu/cgi/viewcontent.cgi?article=1848&amp;amp;context=ias_pub&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Measuring the Accuracy of an Ancient Area&#xA;Formula&amp;quot;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;In assessing the area of a plot of land, they used a faulty formula for computing the area of a general quadrilateral. Given sides of lengths &lt;span class=&quot;math-container&quot;&gt;$a,b,c,d$&lt;/span&gt;, they said &lt;span class=&quot;math-container&quot;&gt;$A = .5(a+c) \cdot .5(b+d)$&lt;/span&gt;, that is, they averaged the lengths of opposing sides and multiplied the two results. The formula is wrong but not as outrageously wrong as using the perimeter as a proxy, so it's &lt;em&gt;slightly&lt;/em&gt; less embarrassing that they were still using it &amp;quot;200 years after Euclid had lived and taught in Egypt&amp;quot; (the paper spends a few pages analyzing how far off the formula is for different quadrilaterals).&lt;/p&gt;&#xA;&lt;p&gt;And of course, &lt;a href=&quot;https://math.stackexchange.com/a/916980/596484&quot;&gt;there is no formula for the area of a quadrilateral that uses only its sides&lt;/a&gt;. &lt;a href=&quot;https://en.wikipedia.org/wiki/Brahmagupta%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brahmagupta's formula&lt;/a&gt; works for cyclic quadrilaterals, but that was not until 7th century AD India. And &lt;a href=&quot;https://en.wikipedia.org/wiki/Heron%27s_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Heron's formula&lt;/a&gt; does the trick for triangles, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Heron%27s_formula#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;I'm not sure&lt;/a&gt; if that was discovered after the relevant time period. (Full disclosure: I'm not an expert on ancient Egypt or its mathematics).&lt;/p&gt;&#xA;" OwnerUserId="9239" LastActivityDate="2023-09-25T20:28:43.117" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15775" PostTypeId="1" CreationDate="2023-09-25T20:38:30.853" Score="3" ViewCount="152" Body="&lt;p&gt;This question is similar to &lt;a href=&quot;https://hsm.stackexchange.com/questions/3421/what-was-the-longest-delay-between-prediction-and-confirmation-of-a-theory&quot;&gt;What was the longest delay between prediction and confirmation of a theory?&lt;/a&gt; but I want to frame it in a different way. I am looking for long delays between prediction and discovery, that were predicted under the frame of the current theory. Here are some examples of what I mean by this:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Examples that fit the description&lt;/strong&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Einstein prediction of gravitational waves in 1916 and discovery by LIGO in 2015. Many people suggested it before like Heaviside or Poincaré but under complete different classical theories.&lt;/li&gt;&#xA;&lt;li&gt;Prediction of the Higgs boson 1964, discovery in 2012. Predicted under a new theory but clearly the right one as it is the current one in the Standard Model.&lt;/li&gt;&#xA;&lt;li&gt;Prediction of planets. For example Neptune was predicted in 1846 using Newtonian mechanics, and found short after.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;Examples that do not fit the description&lt;/strong&gt;&lt;/em&gt;:&#xA;Based on the previous question, here are some examples based on that question that are not what I am looking for&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Atomism by Democritus in Ancient Greece and confirmed definitively in 1905. Clearly a good guess, but science had to break so many paradigms that his predictions have more to do with philosophy than science. Dalton is also not that great because chemistry models were to be refined many times after him (also quantum mechanics was weirder than anything imagined).&lt;/li&gt;&#xA;&lt;li&gt;John Michell's prediction of black holes in 1783. Again a good guess based on classical mechanics but general relativity is more clearly suited.&lt;/li&gt;&#xA;&lt;li&gt;Heliocentrism, it is not truly a prediction, it is more like we have data, we have two models, which is it?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So are there any predictions under the theory of the time or closely-suggested theory that took as long or more than the time between the prediction and detection of gravitational waves?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-09-26T06:41:33.253" LastActivityDate="2023-09-26T12:07:07.607" Title="Do gravitational waves hold the record of longest delay between prediction and confirmation under the same theory?" Tags="|physics|experimental-physics|einstein|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15776" PostTypeId="2" ParentId="15775" CreationDate="2023-09-25T23:58:58.543" Score="1" Body="&lt;p&gt;I'd consider European idea that the Southern Hemisphere had a large landmass near or around the South Pole, referred to as &lt;a href=&quot;https://en.wikipedia.org/wiki/Terra_Australis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Terra Australis&lt;/a&gt;, as satisfying at least some of the OP's desirables.  European maps were drawn with a southern continent by 1570, while the first (European) sighting of Antarctica was around 1820.&lt;/p&gt;&#xA;&lt;p&gt;The Europeans had a &amp;quot;good&amp;quot; theory about balancing of land-mass in the Northern hemisphere &amp;quot;balanced&amp;quot; out in the southern hemisphere.  From a certain point of view, if we add the age of the Earth and some general handwaving about diffusion of land-masses this &lt;em&gt;might&lt;/em&gt; fit under plate tectonics - as continents break up and diffuse over the Earth it might make sense that the distribution of land mass between different hemispheres was more likely to be even. (Although while researching this answer I came across the &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Land_and_water_hemispheres&quot; rel=&quot;nofollow noreferrer&quot;&gt;Land and water hemispheres&lt;/a&gt;&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;So then again, this might be another example such a Pluto - Antarctica was &amp;quot;discovered&amp;quot; by Europeans based on a theory that was incomplete.&lt;/p&gt;&#xA;" OwnerUserId="9482" LastEditorUserId="9482" LastEditDate="2023-09-26T12:07:07.607" LastActivityDate="2023-09-26T12:07:07.607" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15777" PostTypeId="1" CreationDate="2023-09-26T04:13:49.407" Score="0" ViewCount="208" Body="&lt;p&gt;If you look at Felix Klein's &amp;quot;Development of mathematics in the 19th century&amp;quot;, it says that Puiseux developed the Riemann surface theory to show the connection between the two Galois groups.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We saw how the Riemann surface is connected with the ideas of Galois theory, as long as we specify the basic parameter &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; of the field, and how the ideas of Galois theory can be made intuitive with the help of Riemann surface. Instead of giving the Riemann surface. I can prescribe the branch places &lt;span class=&quot;math-container&quot;&gt;$a,b,...,k$&lt;/span&gt; and stipulate which permutations arise from the continuations along closed paths. this takes us from Riemann back to Pulseux, who brought forward such groups in 1851. But this contains the amazing possibility of carrying over to number fields, not the Riemann surface itself but the theorems and questions that follow from considering it. For the branch places &lt;span class=&quot;math-container&quot;&gt;$a,b,...,k$&lt;/span&gt; correspond, as we already know, to the prime factors of the &amp;quot;essential&amp;quot; component of the discriminant, and the Galois group of the algebraic function field naturally corresponds to the Galois group of the number field. &lt;strong&gt;(Development of mathematics in the 19th century p.313~314)&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And I saw this &lt;a href=&quot;https://hal.science/hal-01674509/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;file.&lt;/a&gt; if you see this file's page 8,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We showed that Puiseux studied how the fact that the roots of an algebraic equation are interchanged when the variable &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; describes some loops in the plane leads to a group factorized into permutations. He used in this context the word “monodromic,” which was already introduced by Cauchy. Hermite continued using this word in [37] (1851).10 This led eventually to the notion of monodromy group, which we still use today. Jordan, in his Trait´e des substitutions [40] (1870), defined a group he called the algebraic group, which contains the monodromy group as a normal subgroup. The paper [37] by Hermite in which he studies the solvability&#xA;of equations by radicals makes the relation between the work of Puiseux and the&#xA;Galois group of an algebraic equation. Hermite’s paper starts as follows&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Could you explain the motivation and mathematical background for Puiseux's work focusing on the relationship between the Riemann surface and the Galois group?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-09-26T21:12:29.670" LastActivityDate="2023-09-28T04:46:39.040" Title="Motivation of Puiseux's Riemann surface and Galois group theory" Tags="|mathematics-social-history|riemannian-geometry|galois-theory|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15779" PostTypeId="1" AcceptedAnswerId="15780" CreationDate="2023-09-27T02:51:40.610" Score="1" ViewCount="113" Body="&lt;p&gt;Could it be Newton had some ideas with respect to the cannonball thought experiments prior to the famous apple story?&lt;/p&gt;&#xA;&lt;p&gt;The thought experiment goes like you shoot an ordinary cannonball tangential to the Earth. Assume the Earth is smooth, and, the cannon is mounted at a certain height above the ground. The cannonball takes a standard projectile motion path (a curve) and crashes into Earth a few distance away from where it was launched. For the second trial, you can increase the fire power and repeat the experiment and observe that the curve elongates a little, and, the cannonball has displaced farther. You can keep repeating the experiment in this manner such that you will find that the cannonball will eventually follow the curvature of Earth, and, if you stand long enough by the fire cannon mount, the cannonball might hit you from behind.&lt;/p&gt;&#xA;&lt;p&gt;If so, why was such a thing never expressed in &lt;em&gt;Principia&lt;/em&gt;? Did any of his close associates or contemporaries express such a thought experiment?&lt;/p&gt;&#xA;" OwnerUserId="6084" LastEditorUserId="6609" LastEditDate="2023-09-27T06:46:51.953" LastActivityDate="2023-09-27T08:37:58.997" Title="Did Newton's leap to understanding gravity involve cannonball thought experiments on a smooth Earth?" Tags="|newton|gravity|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15780" PostTypeId="2" ParentId="15779" CreationDate="2023-09-27T06:17:25.543" Score="5" Body="&lt;p&gt;It is because it was moved to another book &lt;em&gt;A Treatise of the System of the World&lt;/em&gt; (also by Isaac Newton).&lt;/p&gt;&#xA;&lt;p&gt;From Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Newton%27s_cannonball&quot; rel=&quot;nofollow noreferrer&quot;&gt;Newton's cannonball&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton's original plan for &lt;em&gt;Philosophiæ Naturalis Principia Mathematica&lt;/em&gt; was that it should consist of two books, the first analyzing basic laws of motion, and the second applying them to the Solar System. In order to include more material on motion in resisting media, the first book was split into two; the succeeding (now third) book, originally written in a more popular style, was rewritten to be more mathematical. However, manuscripts of an earlier draft of this last book survived, and a version of it was published in 1728 as &lt;em&gt;De mundi systemate&lt;/em&gt;; an English translation was also published earlier in 1728 under the name &lt;em&gt;A Treatise of the System of the World&lt;/em&gt;. The thought experiment occurs near the start of this work.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is Newton's drawing (referring to the cannonball thought experiment) from that book (p.7):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/3rVKA.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/3rVKA.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-09-27T08:37:58.997" LastActivityDate="2023-09-27T08:37:58.997" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15781" PostTypeId="1" CreationDate="2023-09-27T09:15:56.753" Score="3" ViewCount="147" Body="&lt;p&gt;I am curious to know when the right-hand-rule for vector product was established and used consistently in mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I read here&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/14989/who-gave-right-hand-thumb-rule-for-circular-loop-of-current-conducting-wire&quot;&gt;Who gave right hand thumb rule for circular loop of current conducting wire?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;that in physics it was established by Ambrose Fleming in the late 19th century. I assume that in math it was commonly adopted later to have everything compatible with physics.&lt;/p&gt;&#xA;&lt;p&gt;My reason for asking is that I am reading an introductory algebraic geometry text by Victoria Scott from 1894. I have noticed that she seems to use a left hand rule for the computation of vector product: in order to compute an area of a triangle she does a product of the sides (that I assume is the vector product) but, with the order chosen, she should get a negative area. And when she wants a negative area (to remove from another area)... the result should be positive with the modern convention. Since at the end she is just interested in plane geometry, consistency of the signs is the only thing that she needs, and she gets it. But this has raised my curiosity about the history of the right-hand-rule and the orientation of axes in the Euclidean 3-space...&lt;/p&gt;&#xA;&lt;p&gt;Thank you for your help!&lt;/p&gt;&#xA;" OwnerUserId="19056" LastEditorUserId="6609" LastEditDate="2023-09-28T08:25:47.130" LastActivityDate="2023-09-28T08:25:47.130" Title="History of right hand rule" Tags="|mathematics|euclidean-geometry|algebraic-geometry|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15782" PostTypeId="1" CreationDate="2023-09-27T22:17:24.620" Score="0" ViewCount="60" Body="&lt;p&gt;Is there a translation of &lt;a href=&quot;https://it.wikipedia.org/wiki/Niccol%C3%B2_Cabeo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Niccolò Cabeo&lt;/a&gt;'s work &lt;em&gt;Philosophia magnetica&lt;/em&gt; into English (or other modern language)? The original text in Latin is available for example &lt;a href=&quot;https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE7320178&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; but I can't find any translation (unlike William Gilbert's &lt;em&gt;De Magnete&lt;/em&gt; for example).&lt;/p&gt;&#xA;" OwnerUserId="19060" LastEditorUserId="6609" LastEditDate="2023-09-28T08:25:07.233" LastActivityDate="2023-09-28T08:25:07.233" Title="Translation of Cabeo's Philosophia magnetica" Tags="|electromagnetism|electricity|magnetism|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15783" PostTypeId="2" ParentId="14798" CreationDate="2023-09-28T03:02:12.717" Score="0" Body="&lt;p&gt;'Momentum' (Latin) or 'momento' (Italian) is always the momentum &lt;em&gt;of X&lt;/em&gt;, where X is some quantity in its own right. Or so I would propose, at least for TNS. And when, in discussing motion, he just says 'momento,' I would suggest he means 'momentum of speed.'&lt;/p&gt;&#xA;&lt;p&gt;The moment of a weight (see the Second Day) is its weight &lt;em&gt;qua&lt;/em&gt; effective: e.g., two unequal weights at inversely proportional distances from the fulcrum have equal momenta of weight, but not equal weights. He talks about the moment that one position along a lever has, compared to another position: i.e., the effectiveness that a given weight would have at that position as compared to other positions.&lt;/p&gt;&#xA;&lt;p&gt;So what does he mean by 'momenta of speed' in the treatment of accelerated motion? In one place he says &amp;quot;gradum seu momentum&amp;quot;, as if to suggest they are equivalent, so that &amp;quot;degree of speed&amp;quot; = &amp;quot;momentum of speed.&amp;quot; But if you look closely at the argument of his Proposition I on accelerated motion, he first argues that the aggregates of the parallel lines are equal, then that the degrees of speed are represented by those parallels, and only then does he bring in the term 'momentum,' concluding that the total momenta of speed in the accelerated motion are equal to the total momenta in the uniform motion. So here it looks like 'momentum of speed' is indeed the instantaneous value or degree of speed a body has at an instant, but also has the connotation of something like 'power to move a body a certain distance in a given time.' Otherwise it isn't clear what it adds to the line of argument to shift from 'degrees of speed' to 'momenta of speed.' Momentum of speed is the speed at an instant, qua effective at moving the body a given amount in a given time.&lt;/p&gt;&#xA;&lt;p&gt;The common notion in the term, then, seems to be something causal, like this: the momentum of X is the instantaneous value of quantity X that a body has, insofar as that quantity is effective at bringing about some effect. The momentum of speed a body has at an instant is the value of its speed at that instant, considered as a power for the body to accomplish a certain motion as a result of that value. So you can see there is something quasi-inertial in the idea.&lt;/p&gt;&#xA;&lt;p&gt;As for 'momentum' vs 'impetus', impetus is more like our modern momentum: it's measured by the impact a body is able to exert when it hits something. So the size (weight, mass, whatever) of the body matters for 'impeto', but not for 'momento' (where 'momento' is short for 'momentum of speed').&lt;/p&gt;&#xA;&lt;p&gt;Here's the point that I think will help you: &lt;em&gt;When Galileo is talking about one body only, he will use the terms 'momento' and 'impeto' interchangeably, since presumably the body isn't changing mass/weight, and he assumes (incorrectly) that the impact is proportional to the velocity.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;Added note: I don't think it is correct to interpret him as saying that a body moving on a horizontal plane has no momentum &lt;em&gt;of any kind&lt;/em&gt;. On my reading what he means is that it has no momentum of weight/heaviness: it still has a momentum of speed. The context governs which quantity the momentum is 'of': here he is comparing it to bodies going up or down inclined planes, so heaviness is the intrinsic causal quantity, whose effective value at an instant is the momentum.&lt;/p&gt;&#xA;&lt;p&gt;I agree with Andrew R that Crew and De Salvo confuse things with their translation, but I think there is a reason they find it natural to translate momentum as momentum, sometimes: because when Galileo mean's momentum of speed, he does have a quasi-inertial concept in mind: the value of the speed at that instant, where that speed is considered as a power for the body to move or keep moving a certain amount in a given time.&lt;/p&gt;&#xA;" OwnerUserId="19062" LastEditorUserId="19062" LastEditDate="2023-09-28T03:14:12.853" LastActivityDate="2023-09-28T03:14:12.853" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15784" PostTypeId="2" ParentId="7513" CreationDate="2023-09-28T12:16:07.857" Score="1" Body="&lt;p&gt;In his recent book  &lt;em&gt;A New History of Greek Mathematics&lt;/em&gt; (2022)  Reviel Netz writes concerning the history of reception of Pythagoras and other early writers:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The argument that Aristotle is an unreliable narrator of the earlier history of philosophy was made forcefully already by H. F. Cherniss, Aristotle’s Criticism of Pre-Socratic Philosophy (Baltimore, MD: John Hopkins Press, 1935).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Netz details some of Aristotle's misconceptions about Pythagoras and others.  So the fairy tales about Pythagoras certainly don't originate with early 20th century mathematicians, contrary to what user Conifold's &lt;a href=&quot;https://hsm.stackexchange.com/a/7521/604&quot;&gt;answer&lt;/a&gt; seems to suggest.&lt;/p&gt;&#xA;&lt;p&gt;Conifold does not contest the legitimacy of describing the early 20th century &amp;quot;foundation crisis&amp;quot; as such.  This leaves us with the remaining item on the list by Fraenkel, Bar-Hillel, and Levy: the alleged foundation crisis in the first half of the 19th century having to do with Fourier series and other developments that were pushing the envelope of what kind of functions mathematical analysis dealt with.  Here Conifold is similarly sceptical concerning the existence of a &amp;quot;crisis&amp;quot;, and I would like to disagree with Conifold.  Consider, for example, Abel's reaction to Cauchy's work.  On the one hand, while forcefully disagreeing with Cauchy's politics, Abel clearly stated that Cauchy is the only one who knows how to do analysis.  On the other hand, Abel is the one who claimed in 1826 to find a purported mistake in Cauchy's work on the convergence of series.  It emerges that according to Abel, the only one who could do analysis was publishing mistaken theorems in analysis.  My appreciation of the situation in the first half of the 19th century is that it can indeed be described as a crisis, even if we have challenged the Cauchy-Weierstrass tale that sometimes passes for the history of 19th century analysis; see &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/cauchy.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this list of publications&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So my conclusion, at variance with Conifold, is that two out of three crises that Fraenkel, Bar-Hillel, and Levy mentioned are legitimate.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-09-28T12:16:07.857" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15785" PostTypeId="1" CreationDate="2023-09-28T15:35:46.553" Score="1" ViewCount="147" Body="&lt;p&gt;A nice property of algebraically closed fields is that the theory that describes them (&lt;span class=&quot;math-container&quot;&gt;$ACF$&lt;/span&gt;) admits quantifier elimination: any statement can be shown equivalent (in the theory) to another statement without quantifiers.&lt;/p&gt;&#xA;&lt;p&gt;This is a crucial point in showing that &lt;span class=&quot;math-container&quot;&gt;$ACF_p$&lt;/span&gt;, the theory of algebraically closed fields of characteristics &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; (either &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; or a prime), is complete: every statement of &lt;span class=&quot;math-container&quot;&gt;$ACF$&lt;/span&gt; is equivalent to a statement without quantifiers which basically only assert things on the characteristics (e.g. &lt;span class=&quot;math-container&quot;&gt;$(1 + 1 + 1 = 0) \land \lnot (1 + 1 = 0))$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;My question is: when, how and why did people start to have interest in those aspects of the theory of fields? (or simply to ask themselves questions like: if I can formulate a theorem related to &lt;span class=&quot;math-container&quot;&gt;$\mathbb{C}$&lt;/span&gt; &amp;quot;in algebraic terms&amp;quot;, can it be proven using algebraic properties only?)&lt;/p&gt;&#xA;" OwnerUserId="14079" LastEditorUserId="14079" LastEditDate="2024-04-06T11:23:31.037" LastActivityDate="2024-04-06T11:23:31.037" Title="When did mathematicians realize that theory of algebraically closed fields admits quantifier elimination?" Tags="|mathematics|mathematical-logic|abstract-algebra|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15786" PostTypeId="1" AcceptedAnswerId="15790" CreationDate="2023-09-28T16:17:09.297" Score="1" ViewCount="162" Body="&lt;p&gt;In his book &lt;a href=&quot;https://www.goodreads.com/book/show/484458.Surreal_Numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Surreal Numbers&lt;/em&gt;&lt;/a&gt; Donald Knuth refers to &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Horton_Conway&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Horton Conway&lt;/a&gt; as J.H.W.H. Conway. The J is for John and the H for Horton, but what about the W and the H? I have searched for Conway's middle names but to no avail. Or are the extra initials purely fictional, added by Knuth &lt;a href=&quot;https://doi.org/10.2307/2690126&quot; rel=&quot;nofollow noreferrer&quot;&gt;in order to make a connection with Jehovah (YHWH)&lt;/a&gt;?&lt;/p&gt;&#xA;" OwnerUserId="12735" LastActivityDate="2023-09-29T14:01:46.337" Title="What does the 'W.H.' stand for in 'J.H.W.H. Conway' in Knuth's book Surreal Numbers?" Tags="|mathematicians|computer-scientists|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15787" PostTypeId="2" ParentId="10990" CreationDate="2023-09-28T19:13:12.590" Score="3" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.google.com/books/edition/The_Historical_Development_of_Quantum_Th/warvAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22Until+1930,+Dirac+always+wrote%22&amp;amp;dq=%22Until+1930,+Dirac+always+wrote%22&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mehra and Rechenberg&lt;/a&gt;, the symbol &lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$h/(2\pi)$&lt;/span&gt; was indeed first introduced in Dirac's 1930 book &lt;em&gt;The Principles of Quantum Mechanics&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Until 1930, Dirac always wrote &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$h /2\pi$&lt;/span&gt; , which notation we&#xA;shall not follow here; it was only later (in &lt;em&gt;The Principles of&#xA;Quantum Mechanics,&lt;/em&gt; Dirac, 1930d) that he introduced the symbol&#xA;&lt;span class=&quot;math-container&quot;&gt;$\hbar$&lt;/span&gt; for &lt;span class=&quot;math-container&quot;&gt;$h /2\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;                            &lt;sup&gt;Jagdish&#xA;Mehra and ‎Helmut Rechenberg, &lt;em&gt;The Historical Development of Quantum&#xA;Theory, vol. 6, The Completion of Quantum Mechanics 1926-1941, Part 1&lt;/em&gt;&#xA;(Springer, New York, 2000), p. 291, footnote 332.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Unfortunately, they don't give further details about why Dirac chose this symbol.&lt;/p&gt;&#xA;" OwnerUserId="10854" LastActivityDate="2023-09-28T19:13:12.590" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15788" PostTypeId="1" CreationDate="2023-09-28T23:41:52.727" Score="1" ViewCount="163" Body="&lt;p&gt;As Napier calculated the logarithms of numbers in the base &lt;span class=&quot;math-container&quot;&gt;$\left(1-\frac{1}{10^7}\right)^{10^7}$&lt;/span&gt;, I expected to find the number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; in the tables of his &lt;em&gt;Mirifici Logarithmorum Canonis Descriptio&lt;/em&gt;. That is, the number 3,678,794, which is equal to the integer part of &lt;span class=&quot;math-container&quot;&gt;$10^7\times\left(1-\frac{1}{10^7}\right)^{10^7}$&lt;/span&gt;, as &lt;span class=&quot;math-container&quot;&gt;$\left(1-\frac{1}{10^7}\right)^{10^7}$&lt;/span&gt; gets close to &lt;span class=&quot;math-container&quot;&gt;$ \frac{1}{e}$&lt;/span&gt; and Napier multiplies the result by &lt;span class=&quot;math-container&quot;&gt;$10^7$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, the number that I find on the last row of the last table is 3,465,735 rather than 3,678,794. So, this number is certainly not &lt;span class=&quot;math-container&quot;&gt;$1/e \times 10^7$&lt;/span&gt;. Is there a way of identifying the number &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt; in the tables?&lt;/p&gt;&#xA;&lt;p&gt;Note: This question is related to &lt;a href=&quot;https://hsm.stackexchange.com/a/47/19008&quot;&gt;this answer,&lt;/a&gt; which however refers to the book &lt;em&gt;Constructio&lt;/em&gt; of Napier. I refer here to the book &lt;em&gt;Descriptio&lt;/em&gt;, which is the first book of Napier on logarithms.&lt;/p&gt;&#xA;" OwnerUserId="19008" LastEditorUserId="19008" LastEditDate="2023-10-01T10:06:05.747" LastActivityDate="2023-10-01T10:06:05.747" Title="Can I find the number e in the tables of Napier?" Tags="|mathematics|logarithm|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15789" PostTypeId="1" AcceptedAnswerId="15793" CreationDate="2023-09-29T08:02:38.100" Score="1" ViewCount="231" Body="&lt;p&gt;Arguably one of the most hated parts of English mathematical terminology is the word “&lt;strong&gt;nondecreasing&lt;/strong&gt;”, referring to a function such that &lt;span class=&quot;math-container&quot;&gt;$x\leq y \;\Rightarrow\; f(x)\leq f(y)$&lt;/span&gt; (what other conventions or languages might call something like “weakly increasing”, “(weak-)order-preserving” or various variations thereof).  The term is much hated, of course, because it is not the negation of “decreasing” (i.e., there are functions which are not decreasing but neither are they nondecreasing), so the question arises as to how this annoyance came about.&lt;/p&gt;&#xA;&lt;p&gt;I can form two (noncontradictory!) conjectures as to the origin of “nondecreasing”:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;One is that it is simply formed by analogy with “nonnegative” to mean “&lt;span class=&quot;math-container&quot;&gt;$\geq 0$&lt;/span&gt;” (which is also questionable, but at least makes sense logically) and “no less” to mean “&lt;span class=&quot;math-container&quot;&gt;$\geq$&lt;/span&gt;”: simply following the same pattern of using “non-” with the reversed inequality gives “nondecreasing”.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Another is that “nondecreasing” should really be “monotonically nondecreasing” (some people indeed use the full form, which is cumbersome but clear), and that the adverb was eventually left out as it seemed redundant (rather than being later added by people who wanted to make the term less illogical).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt;  Who introduced “nondecreasing”, when, and why?  And is either (or both) of the above conjectures correct?&lt;/p&gt;&#xA;" OwnerUserId="3616" LastActivityDate="2023-10-03T17:48:06.600" Title="Who introduced the terminology “nondecreasing” for weakly increasing (i.e. x≤y ⇒ f(x)≤f(y)), and when/why?" Tags="|mathematics|terminology|" AnswerCount="1" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15790" PostTypeId="2" ParentId="15786" CreationDate="2023-09-29T08:32:51.577" Score="3" Body="&lt;p&gt;In &lt;em&gt;Surreal Numbers&lt;/em&gt; there is already a passage where Bill says,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The JHWH might also stand for &amp;quot;Jehovah.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to the paper you already cited (D. M. Berry, M. Yavne, Math Magazine 1976):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So, armed wit Hertz's Pentateuch [2], Alcalay's English-Hebrew and Hebrew-English Dictionaries &lt;a href=&quot;https://books.google.fr/books?id=kE0FEAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;, Knuth's book, paper, pencils, and erasers, we sat down and argued our way to the &amp;quot;original&amp;quot; Hebrew version of the Conway Stone. [...]&lt;/p&gt;&#xA;&lt;p&gt;The name of the creator J.H.W.H. Conway poses a problem that Alice noticed&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But 'Conway'... that's not a Hebrew name. You've got to be kidding&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;[...] Actually Knuth is kidding the name is clearly intended to suggest the name of the mathematician John Horton Conway of Cambridge University.[...]&lt;/p&gt;&#xA;&lt;p&gt;J.H.W.H. is obviously intended to be translated as יהוה (letters: YHWH).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Any reasons to disagree with the authors?&lt;/p&gt;&#xA;&lt;p&gt;See also Martin's Gardner &lt;a href=&quot;https://books.google.fr/books?id=kE0FEAAAQBAJ&amp;amp;source=gbs_navlinks_s&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Mathematical Magic Show&lt;/em&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Alice and Bill ( A and B), have fled from the &amp;quot;system&amp;quot; to a haven on the coast of the Indian Ocean. There they unearth a half-buried black rock carved with ancient Hebrew writing. Bill, who knows Hebrew, manages to translate the opening sentence: &amp;quot;In the beginning everything was void, and J. H. W. H. Conway began to create numbers.&amp;quot; JHWH is a transliteration of how the ancient Hebrews wrote the name Jehovah. &amp;quot;Conway&amp;quot; also appears without vowels, but it was the most common English name Bill could think of that fitted the consonants.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also this obituary by Scientific American: &lt;a href=&quot;https://blogs.scientificamerican.com/observations/remembering-mathematical-magician-john-conway/&quot; rel=&quot;nofollow noreferrer&quot;&gt;E. Landau (2020), &lt;em&gt;Remembering Mathematical Magician John Conway&lt;/em&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-09-29T14:01:46.337" LastActivityDate="2023-09-29T14:01:46.337" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15791" PostTypeId="2" ParentId="15785" CreationDate="2023-09-29T09:54:55.290" Score="5" Body="&lt;p&gt;A useful sketch of history is given in &lt;a href=&quot;https://www.jstor.org/stable/2274424&quot; rel=&quot;noreferrer&quot;&gt;Alfred Tarski's Elimination Theory for Real Closed Fields by van den Dries&lt;/a&gt;. The result was known to Tarski by 1948 (when his Decision Method for Elementary Algebra and Geometry, referenced as [48&lt;span class=&quot;math-container&quot;&gt;$^{\text{m}}$&lt;/span&gt;] below, came out), and is simpler than quantifier elimination for real closed fields that he focused on, but he did not make much of it, nor even stated it explicitly. In his earlier work (1937), he did not even spell out &amp;quot;real closed fields&amp;quot; despite listing the axioms for them:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The title of the monograph suggests a change of emphasis from definability questions to completeness, the fact that the truth of any sentence in R can be established purely on the basis of the axioms for real closed fields...  Tarski's axioms here are just the familiar ones for ordered fields plus an axiom schema saying that sign-changing polynomials have a zero. Remarkably, he does refer to van der Waerden's section [1937, p. 229ff.] on real closed fields, but does not mention these fields, and only indicates one other model of his axioms besides &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, namely the field of real algebraic numbers.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Lowenheim, Skolem and Langford eliminated quantifiers to settle logical questions about newfangled structures like Boolean algebras and other ordered sets, where a parallel to older concerns of classical algebra was not at all evident. This origin may explain why Tarski refers only once briefly to the connection between quantifier elimination and algebraic elimination theory. Also, Tarski never seems to have explicitly mentioned in print that the theory of algebraically closed fields admits quantifier elimination. He obviously knew this result; cf. [48&lt;span class=&quot;math-container&quot;&gt;$^{\text{m}}$&lt;/span&gt;, p. 54, Note 16] (and p. 373 of Seidenberg [1954]), but only drew attention to its zero-dimensional consequences: decidability, and completeness in each characteristic.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Now, eliminating quantifiers for algebraically closed fields is, modulo familiar logic, just a simple exercise in the Euclidean algorithm for multivariable polynomials over &lt;span class=&quot;math-container&quot;&gt;$\mathbb{Z}$&lt;/span&gt;, much easier than doing the same thing for real closed fields. Still, this exercise would have been worthwhile in 1948... In the writings of A. Seidenberg [1954], [1956] and A. Robinson [1956a], [1958], [1959a] the kinship of logical quantifier elimination to algebraic elimination theory, specializing parameters, Nullstellensatze and similar ideas becomes more explicit and very fruitful.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Tarski's work on quantifier elimination generally did not attract attention of algebraists, or, indeed, anyone outside of mathematical logic until  1954. Then, Seidenberg, his colleague at Berkeley, published a proof of Hormander's inequality for polynomials based on it, with logical formalism reduced to a minimum. This made it more accessible, but the full &amp;quot;realization&amp;quot; came, perhaps, in 1956, with Abraham Robinson's monograph Complete Theories. Robinson was attracted by the model-completeness consequence of quantifier elimination, which led him to  rethinking some classical algebraic results, and to their generalizations, including for algebraically closed valued fields:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;This model-completeness property drew Abraham Robinson's attention, and in Complete Theories he showed that model-completeness of RCF, the elementary theory of real closed fields, follows on general grounds from two facts known since the classical Artin-Schreier paper [1926]: (i) Each ordered integral domain has a real closure. (ii) If &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; is a real closed field then a simple ordered field extension &lt;span class=&quot;math-container&quot;&gt;$R(x)$&lt;/span&gt;, with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; distinguished, is up to &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt;-isomorphism uniquely determined by the set &lt;span class=&quot;math-container&quot;&gt;$\{r \in R: r &amp;lt; x\}$&lt;/span&gt;. Besides these two algebraic facts only the simplest model-theoretic techniques (compactness, diagrams) are used in Robinson's argument, which runs to no more than two pages.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Robinson went on to treat in a similar way other important structures: algebraically closed valued fields [1956a] and certain kinds of ordered Abelian groups (Robinson and Zakon [1960])... Robinson made Tarski's theorem not only an immediate consequence of Artin-Schreier theory, but also its culmination: Artin's solution [1927] of Hilbert's 17th problem by means of a delicate specialization becomes in Robinson's hands a memorable one-liner that can be applied over and over again...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And so it was. More generally, applications to &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic fields were developed largely under Robinson's influence:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Robinson's simple methods efficiently organize and create new algebraic knowledge, but more important is that these methods suggest fruitful analogies. The theory of real closed fields assumes here the role of paradigm. It is no accident that a basic corner of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic theory was entirely developed by logically trained mathematicians. We refer here to the breakthrough papers by Ax and Kochen [1965a], [1965b], [1966] and Ersov [1965], [1966], [1967] (who were of course also inspired by other ideas from logic, in particular by the then popular ultraproducts)...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-09-29T09:54:55.290" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15793" PostTypeId="2" ParentId="15789" CreationDate="2023-09-29T18:50:20.747" Score="5" Body="&lt;p&gt;My guess is that &amp;quot;nondecreasing&amp;quot; became a thing in the late 1800's and early 1900's, with the development of the theory of functions of bounded variation&#xA;and of the Stieltjes integral. Thus, in his 1882 &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227.chemindefer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Sur la série de Fourier&amp;quot;&lt;/a&gt; Camille Jordan wrote&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Le  théorème  subsistera donc tous le fois que &lt;span class=&quot;math-container&quot;&gt;$F(x)$&lt;/span&gt; pourra être représenté de &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt;  à &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt; par &lt;span class=&quot;math-container&quot;&gt;$f(x)-\phi(x)$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; et &lt;span class=&quot;math-container&quot;&gt;$\phi(x)$&lt;/span&gt; étant deux fonctions fines et non décroissantes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hobson's 1907 &lt;em&gt;Theory of Functions of a Real Variable&lt;/em&gt; says the same thing&#xA;(on p.336 of the 1956 Dover reprint):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... that it can be expressed as the difference of two bounded monotone functions, which are either both non-increasing or both non-diminishing.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to F. Riesz and B. Sz-Nagy, writing in 1955 (on p.106 of the 1995 English translation of &lt;em&gt;Functional Analysis&lt;/em&gt;), this result was made popular by Riesz's 1909 result about the dual of &lt;span class=&quot;math-container&quot;&gt;$C[0,1]$&lt;/span&gt;, for which it is useful to know (on p.10)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Every function of bounded variation is the difference of two nondecreasing functions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My guess is that the English term &amp;quot;nondecreasing&amp;quot; is a direct borrowing from the French &amp;quot;non décroissantes&amp;quot;.  I do not know if the ambiguities the OP finds in the English term are present in the French one.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2023-10-03T17:48:06.600" LastActivityDate="2023-10-03T17:48:06.600" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15794" PostTypeId="1" AcceptedAnswerId="15798" CreationDate="2023-09-30T08:59:36.373" Score="2" ViewCount="96" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Frog_galvanoscope&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frog galvanoscope&lt;/a&gt; is an instrument for detecting small voltages, made of a frog's leg.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia notes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The instrument is capable of detecting extremely small voltages, and could far surpass other instruments available in the first half of the nineteenth century, including the electromagnetic galvanometer and the gold-leaf electroscope.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yet Golding Bird could still write in 1848 that &amp;quot;the irritable muscles of a frog's legs were no less than 56,000 times more delicate a test of electricity than the most sensitive condensing electrometer.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but I couldn't find any actual values in modern units.&lt;/p&gt;&#xA;&lt;p&gt;What was the minimum voltage in volts that could be detected by a frog galvanoscope?&lt;/p&gt;&#xA;" OwnerUserId="3054" LastActivityDate="2023-10-01T17:34:32.517" Title="How sensitive was the frog galvanoscope?" Tags="|biology|electricity|measurement|18th-century|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15795" PostTypeId="1" CreationDate="2023-09-30T12:02:39.220" Score="0" ViewCount="285" Body="&lt;p&gt;I'm interested in John von Neumann these days. So I searched &lt;a href=&quot;https://www.ams.org/notices/201302/rnoti-p154.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this file&lt;/a&gt;. And I read books &lt;em&gt;The Man from the Future&lt;/em&gt; and &lt;em&gt;John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;In particular, &lt;a href=&quot;https://www.ams.org/notices/201302/rnoti-p154.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this file&lt;/a&gt; provides a direct assessment of Freeman Dyson's von Neumann's thinking process. (p.1 right side)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Johnny’s unique gift as a mathematician was to transform problems in all areas of mathematics into problems of logic. He was able to see intuitively the logical essence of problems and then to use the simple rules of logic to solve the problems. His first paper is a fine example of his style of thinking. A theorem which appears to belong to geometry, restricting the possible positions of points where some function of a complex variable is equal to zero, is transformed into a statement of pure logic. All the geometrical&#xA;complications disappear and the proof of the theorem becomes short and easy. In the whole paper there are no calculations, only verbal definitions and logical deductions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And If you see the book, &lt;em&gt;John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More&lt;/em&gt;, it cites von Neumann's mathematical hobby,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He used to find out the thinking process of math masters such as &amp;quot;Galileo&amp;quot; and &amp;quot;Newton&amp;quot;, the background of the idea, and the reason for the failure and was interested.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are all of the above informations true? And explain in detail the way von Neumann treated mathematics.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="6609" LastEditDate="2023-10-02T08:06:46.420" LastActivityDate="2023-10-02T08:06:46.420" Title="John von Neumann's thinking process" Tags="|mathematicians|von-neumann|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15797" PostTypeId="2" ParentId="15788" CreationDate="2023-10-01T02:03:52.880" Score="5" Body="&lt;p&gt;Q: &amp;quot;Can I find the number e in the tables of Napier?&amp;quot;&#xA;(and)&#xA;&amp;quot;The number that I find on the last row of the last table is 3,465,735 rather than 3,678,794. Is there a number closer to e in the tables?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;To begin by cutting quickly to the chase, I see no mistake on Napier's part with the number 3465735 in the tables (which the question suggests is a mistake by Napier).  When converted to modern form, with sign and decimal point, this number is at most 1 unit (in the last place) different from the natural-logarithmic sine (or cosine) of 45 degrees -- which is what it ought to be, for its place in the layout of Napier's tables.&lt;/p&gt;&#xA;&lt;p&gt;John Napier of Merchistoun (various spellings survive) gave his first tables in his book with title 'Mirifici Logarithmorum Canonis Descriptio' (1614) -- translated four years later by Edward Wright as 'Description of the admirable table of logarithms'.  Napier tabulated what can readily be recognized now as natural (or Napierian, or base-e) logarithms of trigonometric functions, but printed them according to conventions of his day, so that (for example) one does not look for decimal points or for signed numbers.&lt;/p&gt;&#xA;&lt;p&gt;In the main tables, for example, each page-header gives an angle in degrees, and the successive rows are for functions of the header degree supplemented by 0, 1, 2, (etc) minutes.  The first column of tabular values in any row then gives the (natural) sine of the angle for that row, and the second column gives the (natural, Napierian or base-e) logarithm of that sine. The remaining columns give natural-logarithmic tangent and cosine, then the natural cosine, and finally, on the right, the minutes in the complement to 90 degrees of the angle identified by the supplement of minutes in the left column (when the right-hand column angle is read as a supplement of minutes with the number of degrees shown this time at the &lt;em&gt;foot&lt;/em&gt; of the table-page).&lt;/p&gt;&#xA;&lt;p&gt;The result has an economy of layout, so that the tables as a whole appear to cover primarily a semi-quadrant (i.e. a range of 45 degrees). But when read in the appropriate direction for any chosen angle, i.e. downwards from the top and from left to right for angles in the range 0 to 45 degrees, but upwards from the foot and from right to left for angles in the range 45 to 90 degrees, they give the whole gamut of logarithmic sines, tangents, and cosines for an entire quadrant (thus also natural-logarithmic cotangents), although modern names (apart from 'sinus' and 'logarithmi') do not appear.&lt;/p&gt;&#xA;&lt;p&gt;Although 8 digits are printed, the accuracy in those early tables does not always reach 6 digits e.g. relative to a precise natural-logarithmic sine value.&lt;/p&gt;&#xA;&lt;p&gt;So, to the extent that one can find tabulated angles and the natural, base-e, logarithms of their sines and other trigonometric functions to the accuracy indicated,  then yes, one can say that the base of the natural logarithms that we call 'e' is implicitly found there throughout.&lt;/p&gt;&#xA;&lt;p&gt;One does not find the value of 'e' given explicitly in Napier's tables. But then it also doesn't appear explicitly in a modern table of natural logarithms or natural-logarithmic trig functions, either -- unless one counts its presence in a table of &lt;span class=&quot;math-container&quot;&gt;$e^x$&lt;/span&gt;, which is not strictly a logarithm table at all.&lt;/p&gt;&#xA;&lt;p&gt;There can conceivably be other ways of looking at this question -- perhaps for example by asking whether there is a number close to but not equal to 'e', to which Napier's numerical values relate more closely than to 'e' itself. One could perhaps also say that in modern terms Napier has tabulated &lt;span class=&quot;math-container&quot;&gt;$-10^7 * ln (sin(angle))$&lt;/span&gt; rather than the &lt;span class=&quot;math-container&quot;&gt;$ln(sin(angle))$&lt;/span&gt; itself, and so on.  But there does not seem to be evidence that Napier thought in ways that match either of those ideas.  The non-modern way in which he did think can probably best be gathered through reading his description or at least its 17th-c. English translation.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2023-10-01T08:20:00.013" LastActivityDate="2023-10-01T08:20:00.013" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15798" PostTypeId="2" ParentId="15794" CreationDate="2023-10-01T17:34:32.517" Score="7" Body="&lt;p&gt;A frog galvanoscope does not measure voltage. Instead it detects electricity that same way that I detect my static charge after walking across a carpet in winter - by a short current flow. The current causes the frog leg to twitch and me to jump from the shock of the spark when I touch a door knob. (The analogy is reasonably close, since &lt;a href=&quot;https://en.wikipedia.org/wiki/Luigi_Galvani&quot; rel=&quot;noreferrer&quot;&gt;Galvani&lt;/a&gt; needed a spark gap in the circuit to make the frog leg twitch.)&lt;/p&gt;&#xA;&lt;p&gt;In general, a &lt;a href=&quot;https://en.wiktionary.org/wiki/galvanoscope&quot; rel=&quot;noreferrer&quot;&gt;galvanoscope&lt;/a&gt; measures current; an &lt;a href=&quot;https://en.wikipedia.org/wiki/Electrometer&quot; rel=&quot;noreferrer&quot;&gt;electrometer&lt;/a&gt; measures electric charge or potential.&#xA;The frog-leg galvanometer a kind of threshold ammeter. It tells you that a voltage source can provide a current - at least for an instant - greater than the threshold needed to make the frog leg twitch.&lt;/p&gt;&#xA;&lt;p&gt;The &amp;quot;56,000&amp;quot; factor comes from page 316 of Wilkinson's 1804 &lt;a href=&quot;https://archive.org/details/elementsgalvani01wilkgoog/page/316/mode/1up&quot; rel=&quot;noreferrer&quot;&gt;Elements of Galvanism&lt;/a&gt;, and is a comparison of &lt;em&gt;charge&lt;/em&gt; sensitivity, not &lt;em&gt;voltage&lt;/em&gt;. Touching a silver and a copper plate, each 0.1 inch square, to the frog leg made it twitch, but it took 20 touches connected to 6 inch diameter copper and zinc plates to make a gold leaf electroscope visibly deflect. The electric potential in both cases is about a volt.  The relative sensitivity comes from the calculation: &lt;span class=&quot;math-container&quot;&gt;$20\,\pi\,(6/2)^2/0.1^2 = 56,549$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I can't immediately find a source that clearly states how much current or charge is needed to make a frog leg twitch.  Individual &lt;a href=&quot;https://doi.org/10.2170/jjphysiol.22.147&quot; rel=&quot;noreferrer&quot; title=&quot;Threshold Excitation And Optimal Stimulation For Frog Skeletal Muscle Fibers Studied With Intracellular Electrodes, Fumitake Inoue, George B. Frank, The Japanese Journal of Physiology, 1972, Volume 22, Issue 2, Pages 147-155&quot;&gt;frog skeletal muscle fibres can respond&lt;/a&gt; to 50 nA at 50 mV for pulses longer than 5 ms, so this might be considered the ultimate sensitivity for a single-muscle-fibre galvanometer.&#xA;Similar results have been &lt;a href=&quot;https://doi.org/10.1007/BF00603729&quot; rel=&quot;noreferrer&quot; title=&quot;Strength-duration properties of a rapidly adapting insect sensory neuron, French, A.S.  J. Comp. Physiol. 159, 757–764 (1986).&quot;&gt;observed for cockroach legs&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2023-10-01T17:34:32.517" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15799" PostTypeId="2" ParentId="10990" CreationDate="2023-10-02T04:14:50.163" Score="3" Body="&lt;p&gt;It was noted at &lt;a href=&quot;https://sci.physics.research.narkive.com/y3qeeLYf/origin-of-hbar&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://sci.physics.research.narkive.com/y3qeeLYf/origin-of-hbar&lt;/a&gt; that h-bar appeared in a German translation of Dirac's article (&lt;a href=&quot;https://books.google.com/books?id=H-w8AAAAIAAJ&amp;amp;pg=PA359&amp;amp;lpg=PA359&amp;amp;dq=dirac%201928%20Leipziger%20Vortr%C3%A4ge%20(Quantentheorieund%20Chemie),%2085-94,%201928&amp;amp;source=bl&amp;amp;ots=NviE_8Kv_B&amp;amp;sig=ACfU3U1IDQBsorvxMkc4ZcFEjeQbCE-exQ&amp;amp;hl=en&amp;amp;sa=X&amp;amp;ved=2ahUKEwir54eavNaBAxX5kIkEHYHHCxAQ6AF6BAghEAM#v=onepage&amp;amp;q=dirac%201928%20Leipziger%20Vortr%C3%A4ge%20(Quantentheorieund%20Chemie)%2C%2085-94%2C%201928&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Quantentheorie des Elektrons, Leipziger Vorträge (Quantentheorie und Chemie), 85-94, 1928&lt;/a&gt;), but the bar there was above h, not crossing it. I would speculate that later it was replaced with what we now know as h-bar just because printers had something like that for some obscure alphabet, as @jkien suggested.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-10-02T04:14:50.163" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15800" PostTypeId="1" CreationDate="2023-10-03T04:36:56.477" Score="1" ViewCount="168" Body="&lt;p&gt;I was usually interested in metric tensor, So I have hard searched it, But most of what I was looking for was about the 'theory of relativity'. Even so, I can find information about mathematical content, Poincare disc model of Hyperbolic geometry.&lt;/p&gt;&#xA;&lt;p&gt;In the 'Poincare disc model' I can think of the metric tensor's meaning which is to apply Pythagoras' theorem to a curved situation. However, I am not sure about the mathematical meaning of the metric tensor yet. I know it was Riemann who first discovered this theory in the era before Einstein's application of the metric sensor to his theory of relativity. I want to know about Riemann's thinking for making metric tensor.&lt;/p&gt;&#xA;&lt;p&gt;In other words, I want to know the overall birth background of the metric tensor.&#xA;i.e. I want to know whether metric tensor can be induced purely mathematically or not.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-10-03T10:25:59.003" LastActivityDate="2023-10-06T08:28:12.910" Title="What is Metric tensor's origin?" Tags="|differential-geometry|riemannian-geometry|riemann|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15801" PostTypeId="2" ParentId="15800" CreationDate="2023-10-04T08:47:37.340" Score="2" Body="&lt;p&gt;The metric tensor was introduced by Riemann in his lecture:&#xA;1854 – Über die Hypothesen, welche der Geometrie zugrunde liegen. There are several English translations, and a comprehensive commentary in the book by M. Spivak, &lt;a href=&quot;https://luis.impa.br/aulas/anvar/Spivak_Vol1_3ed.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;A comprehensive introduction to differential geometry&lt;/a&gt;, vol. I.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-10-06T08:28:12.910" LastActivityDate="2023-10-06T08:28:12.910" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15802" PostTypeId="1" AcceptedAnswerId="15888" CreationDate="2023-10-04T20:35:08.853" Score="1" ViewCount="206" Body="&lt;p&gt;After reading the history of horse power (and power), the physical definitions for them and after testing the theory in rally races, I'm curious what were the reasons for selecting this word (power) as a &amp;quot;badge&amp;quot; for &lt;span class=&quot;math-container&quot;&gt;$\mathbf F\cdot \mathbf v$&lt;/span&gt; later called energy rate.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://hsm.stackexchange.com/a/3255/19097&quot;&gt;https://hsm.stackexchange.com/a/3255/19097&lt;/a&gt; the power might be &amp;quot;ability to act or do.&amp;quot; I ask because energy rate is more transparent than &amp;quot;power&amp;quot; if we take into consideration their meanings.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Horsepower&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Horsepower&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Why wasn't joule/second enough? Why the need for a new unit (watt)? The energy concept is younger than power. Speed is also a rate: m/s or km/h or rpm and so on. Speed does not have an analog in terms of measuring unit.&lt;/p&gt;&#xA;&lt;p&gt;Example: The car travels at a speed of 10 Names (Name Surname of the guy who never defined this so far).&lt;/p&gt;&#xA;&lt;p&gt;Found this interesting question that relates to power &lt;a href=&quot;https://hsm.stackexchange.com/questions/3254/etymology-of-power-math&quot;&gt;Etymology of &amp;quot;power&amp;quot; (math.)&lt;/a&gt;. And this &lt;a href=&quot;https://hsm.stackexchange.com/questions/13304/when-was-the-concept-of-power-defined&quot;&gt;When was the concept of “Power” defined?&lt;/a&gt;. So according  to &lt;a href=&quot;https://hsm.stackexchange.com/a/13309/19097&quot;&gt;https://hsm.stackexchange.com/a/13309/19097&lt;/a&gt;, Watt was not aware of energy and he thought that power is force multiplied by velocity.&lt;/p&gt;&#xA;&lt;p&gt;Also if energy is younger 1840 than power 1770, why didn't joule/second replace the watt? &lt;a href=&quot;https://hsm.stackexchange.com/a/2554/19097&quot;&gt;https://hsm.stackexchange.com/a/2554/19097&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19097" LastEditorUserId="1815" LastEditDate="2023-10-05T18:33:33.320" LastActivityDate="2023-11-08T08:04:46.873" Title="Why energy rate did not replace power = Force times velocity?" Tags="|terminology|energy|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15803" PostTypeId="2" ParentId="3254" CreationDate="2023-10-05T18:30:00.230" Score="6" Body="&lt;p&gt;The Greek word δύναμις is, I believe, an ordinary (Ancient) Greek word meaning power, ability, strength, etc. The first entry in &lt;a href=&quot;https://www.perseus.tufts.edu/hopper/morph?l=dunamis&amp;amp;la=greek#lexicon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Liddell/Scott/Jones&lt;/a&gt; (click on LSJ) cites Homer:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;ἦ τ᾽ἂν ἀμυναίμην, εἴ μοι δύναμίς γε παρείη. (Homer, &lt;em&gt;Od.&lt;/em&gt; 2.62)&lt;br /&gt;&#xA;I would defend myself if the power were in me. (transl. Lattimore)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Liddell/Scott/Jones in entry V gives the mathematical meaning as &amp;quot;power...usu. second power&amp;quot; and cites a passage in Aristotle's &lt;em&gt;Metaphysics&lt;/em&gt; (&lt;a href=&quot;https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0051%3Abook%3D5%3Asection%3D1019b&quot; rel=&quot;nofollow noreferrer&quot;&gt;1019b&lt;/a&gt;) that gives a derivation. Plato also uses δύναμίς to mean squaring in the &lt;em&gt;Republic&lt;/em&gt; (&lt;a href=&quot;https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0167%3Asection%3D587d&quot; rel=&quot;nofollow noreferrer&quot;&gt;587d&lt;/a&gt;) and &lt;em&gt;Timaeus&lt;/em&gt; (&lt;a href=&quot;https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0179%3Atext%3DTim.%3Asection%3D32a&quot; rel=&quot;nofollow noreferrer&quot;&gt;31c/32a&lt;/a&gt;), but something else in &lt;em&gt;Theatetus&lt;/em&gt; (&lt;a href=&quot;http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0171%3Atext%3DTheaet.%3Apage%3D147&quot; rel=&quot;nofollow noreferrer&quot;&gt;147e et seq.&lt;/a&gt;) (see below). I do not know of older references that might suggest how far back the mathematical meaning goes.&lt;/p&gt;&#xA;&lt;p&gt;Aristotle connects the mathematical meaning of δύναμις to the fact that &amp;quot;it is impossible [ἀδύνατον] that the diagonal of a square should be commensurable with the sides&amp;quot; (transl. Tredennick). The word ἀδύνατον has the same root as δύναμις, of course, and Aristotle claims δύναμις derives its meaning as &amp;quot;square&amp;quot; as an extension of the geometric fact. While that may have been clear to his contemporary readers, I have to guess at what is meant. My guess, in somewhat modern terms, is that a number is δύναμις if a square with an area equal to a (whole) number of unit squares has a root (side) equal to a number of units, that is, we have the power or ability to construct such a square from multiple-unit lengths. My explanation does not fit quite as neatly with the incommensurability of the diagonal as I would like, but it seems plausible to me.&lt;/p&gt;&#xA;&lt;p&gt;The passage in &lt;em&gt;Theatetus&lt;/em&gt; seems relevant and opposes this. There δύναμις is ultimately defined to be the &lt;em&gt;sides&lt;/em&gt; of squares that are not integers (i.e., in modern terms, irrational square roots of integers). I cannot resolve this conflict. Nonetheless, the passage clearly connects &amp;quot;power,&amp;quot; squares, and irrational roots. One translator (sorry, I've lost the reference) suggested that the mathematical meaning of δύναμις might have been in flux in Plato's time.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2023-10-06T10:47:05.970" LastActivityDate="2023-10-06T10:47:05.970" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15804" PostTypeId="1" CreationDate="2023-10-06T12:35:54.650" Score="2" ViewCount="154" Body="&lt;p&gt;People sometimes make a distinction between &lt;em&gt;continuous&lt;/em&gt; mathematics and &lt;em&gt;discrete&lt;/em&gt; mathematics.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Continuous mathematics study objects that abstract the notion of a &lt;em&gt;continuum&lt;/em&gt; and typical examples are the set of real numbers &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}$&lt;/span&gt;, the complex numbers, and the objects they are used to describe (e.g. manifolds).&lt;/li&gt;&#xA;&lt;li&gt;Discrete mathematics study objects that can be labelled by integers and typical examples are the set of integers, the finite sets or basically everything studied in theoretical computer science (the possible states of a Turing Machine are countable).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;My question is about the word &lt;strong&gt;discrete&lt;/strong&gt;: where does it come from? why is it used to describe such a property?&lt;/p&gt;&#xA;" OwnerUserId="14079" LastActivityDate="2023-10-06T15:16:59.087" Title="Etymology of &quot;discrete&quot; in mathematics" Tags="|mathematics|terminology|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15805" PostTypeId="2" ParentId="15804" CreationDate="2023-10-06T13:09:24.800" Score="0" Body="&lt;p&gt;It is just the usual English sense of the word discrete. See the &lt;a href=&quot;https://en.wiktionary.org/wiki/discrete&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wiktionary definition&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Separate; distinct; individual; non-continuous.&lt;br /&gt;&#xA;a government with three discrete divisions&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;That can be perceived individually, not as connected to, or part of, something else.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The entry then goes on to give a number of technical definitions including one similar to the one you are interested in. Note that there is also a word discreet which is completely different.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2023-10-06T13:09:24.800" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15806" PostTypeId="2" ParentId="15804" CreationDate="2023-10-06T15:16:59.087" Score="2" Body="&lt;p&gt;From the unabridged Oxford dictionary, the usage in English dates back to 1600s.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;lt; classical Latin discrētus separate, distinct, in post-classical&#xA;Latin also specifically in music (1630 in the passage translated in&#xA;quot. 1664 at sense A.1b), use as adjective of past participle of&#xA;discernere discern v.   Compare Middle French, French discret&#xA;(feminine discrète, †discrete) different, distinct (early 14th cent.),&#xA;(in mathematics) discontinuous, composed of distinct units (2nd half&#xA;of the 14th cent. in quantité discrete).   Compare discretion n. III&#xA;Compare also earlier discreet adj. and see discussion at that entry.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;1660– Belonging to, relating to, or dealing with distinct or disconnected parts; (Mathematics, of proportion) = discontinual adj. 2.&lt;/p&gt;&#xA;&lt;p&gt;Quote: All Geometrical proportion is either discrete, or continued. Discrete is, when the &lt;em&gt;similitudo rationum&lt;/em&gt; is only between the 1. and the 2. and the 3. and 4. term.&lt;/p&gt;&#xA;&lt;p&gt;Reference: Roger Coke • Justice vindicated from the false fucus put upon it, by Thomas White gent. Mr. Thomas Hobbs, and Hugo Grotius: As also elements of power &amp;amp; subjection; wherein is demonstrated the cause of all humane… • 1660.&#xA;London, Printed by Tho. Newcomb for G. Bedell and T. Collins, at the Middle-Temple-Gate, Fleetstreet Roger Coke (c1628–?1707)&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2023-10-06T15:16:59.087" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15807" PostTypeId="2" ParentId="7477" CreationDate="2023-10-06T15:38:31.510" Score="10" Body="&lt;p&gt;There is no evidence for an intentional purge of all audio recordings of Alan Turing. The BBC recordings seem to be the only ones ever made by Turing, so there wasn't really anything to &amp;quot;purge&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;According to Alan Jones article on &lt;a href=&quot;https://oro.open.ac.uk/5609/&quot; rel=&quot;nofollow noreferrer&quot; title=&quot; IEEE Annals of the History of Computing, 26(2) pp. 3–15. DOI: https://doi.org/10.1109/MAHC.2004.1299654&quot;&gt;&amp;quot;Five 1951 BBC Broadcasts on Automatic Calculating Machines &amp;quot;&lt;/a&gt;, the 1951 series was recorded on &lt;a href=&quot;https://en.wikipedia.org/wiki/Acetate_disc&quot; rel=&quot;nofollow noreferrer&quot;&gt;acetate discs&lt;/a&gt;, but all five of them - not just Turing's - have been lost. Only their BBC transcripts survive.&lt;/p&gt;&#xA;&lt;p&gt;For comparison, &lt;a href=&quot;https://en.wikipedia.org/wiki/Bertrand_Russell&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bertrand Russell&lt;/a&gt;, who was a far more famous public intellectual of the same era who &lt;a href=&quot;https://www.bbc.com/mediacentre/proginfo/2022/20/bertrand-russell&quot; rel=&quot;nofollow noreferrer&quot;&gt;regularly appeared on the BBC&lt;/a&gt;, has only &lt;a href=&quot;https://explore.bl.uk/primo_library/libweb/action/search.do?ct=&amp;amp;pag=&amp;amp;indx=1&amp;amp;pageNumberComingFrom=2&amp;amp;frbg=&amp;amp;rfnGrpCounter=3&amp;amp;indx=11&amp;amp;fn=search&amp;amp;mulIncFctN=facet_creationdate&amp;amp;dscnt=0&amp;amp;scp.scps=scope%3A%28BLCONTENT%29&amp;amp;rfnIncGrp=2&amp;amp;tb=t&amp;amp;fctV=audio&amp;amp;fctV=Russell%2C+Bertrand%2C+1872-1970&amp;amp;mode=Basic&amp;amp;vid=BLVU1&amp;amp;ct=Next+Page&amp;amp;rfnGrp=3&amp;amp;rfnGrp=1&amp;amp;vl(297891280UI0)=any&amp;amp;srt=rank&amp;amp;tab=local_tab&amp;amp;fctN=facet_rtype&amp;amp;fctN=facet_creator&amp;amp;dum=true&amp;amp;vl(freeText0)=bertrand%20russell&amp;amp;fctIncV=%5B-999999999+TO+1958%5D&amp;amp;dstmp=1696601387174&quot; rel=&quot;nofollow noreferrer&quot;&gt;14 recordings prior to 1959 in the British Library&lt;/a&gt;.  &lt;a href=&quot;https://en.wikipedia.org/wiki/Alexander_Fleming&quot; rel=&quot;nofollow noreferrer&quot;&gt;Alexander Fleming&lt;/a&gt;, discover of penicillin, was also much better known than Turing at the time but has only &lt;a href=&quot;https://explore.bl.uk/primo_library/libweb/action/search.do?ct=facet&amp;amp;fctN=facet_rtype&amp;amp;fctV=audio&amp;amp;rfnGrp=2&amp;amp;rfnGrpCounter=2&amp;amp;frbg=&amp;amp;rfnGrpCounter=1&amp;amp;indx=1&amp;amp;fn=search&amp;amp;mulIncFctN=facet_creationdate&amp;amp;dscnt=0&amp;amp;rfnIncGrp=1&amp;amp;scp.scps=scope%3A%28BLCONTENT%29&amp;amp;tb=t&amp;amp;mode=Basic&amp;amp;vid=BLVU1&amp;amp;ct=facet&amp;amp;vl(297891280UI0)=any&amp;amp;srt=rank&amp;amp;tab=local_tab&amp;amp;dum=true&amp;amp;vl(freeText0)=alexander%20fleming&amp;amp;dstmp=1696601860880&amp;amp;fctIncV=%5B-999999999+TO+1958%5D&quot; rel=&quot;nofollow noreferrer&quot;&gt;one saved recording&lt;/a&gt; in the collection.  The archive has no recordings of &lt;a href=&quot;https://en.wikipedia.org/wiki/Francis_Crick&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francis Crick&lt;/a&gt; prior to him being awarded the Nobel Prize in 1962.&lt;/p&gt;&#xA;&lt;p&gt;Turing was not a media star during his lifetime. During the years 1944-1954, the &lt;a href=&quot;https://www.britishnewspaperarchive.co.uk/search/results/1944-01-01/1954-12-31?basicsearch=%22alan%20turing%22&amp;amp;exactsearch=true&amp;amp;retrievecountrycounts=false&amp;amp;sortorder=dayearly&quot; rel=&quot;nofollow noreferrer&quot;&gt;British Newspaper Archive&lt;/a&gt; has only one story where a reporter talked to him.  It appears that &lt;a href=&quot;https://spectrum.ieee.org/what-turing-himself-said-about-the-imitation-game&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;What Turing Himself Said About the Imitation Game, Diane Proudfoot, IEEE Spectrum, 30 Jun 2015&quot;&gt;the two 1951/1952 lectures&lt;/a&gt;  were his first (and only) appearances on the BBC. As far as I can tell, neither the expansive biography, &lt;a href=&quot;https://en.wikipedia.org/wiki/Alan_Turing:_The_Enigma&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Alan Turing : the enigma&amp;quot;&lt;/a&gt;, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Andrew_Hodges&quot; rel=&quot;nofollow noreferrer&quot;&gt;Andrew Hodges&lt;/a&gt;  nor &lt;a href=&quot;https://dermotturing.com/my-recent-books/prof/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Prof: Alan Turing Decoded&amp;quot;&lt;/a&gt; by his nephew &lt;a href=&quot;https://en.wikipedia.org/wiki/Dermot_Turing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dermot Turing&lt;/a&gt; mention any other recordings - either public or private - made by Turing.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2023-12-14T01:50:08.740" LastActivityDate="2023-12-14T01:50:08.740" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15809" PostTypeId="2" ParentId="15802" CreationDate="2023-10-07T15:32:01.187" Score="0" Body="&lt;p&gt;The number of named units is arbitrary and depends on history, tradition, context, and convenience. One could equally well ask why the &lt;a href=&quot;https://en.wikipedia.org/wiki/International_System_of_Units&quot; rel=&quot;nofollow noreferrer&quot;&gt;SI system&lt;/a&gt; has any named &lt;a href=&quot;https://en.wikipedia.org/wiki/SI_derived_unit&quot; rel=&quot;nofollow noreferrer&quot;&gt;derived units&lt;/a&gt;, since every measurable quantity can be quantified in terms of the seven base units: metre, second, kilogram, ampere, kelvin, mole, and candela. Who needs the &lt;a href=&quot;https://en.wikipedia.org/wiki/Volt&quot; rel=&quot;nofollow noreferrer&quot;&gt;volt&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ohm&quot; rel=&quot;nofollow noreferrer&quot;&gt;ohm&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Newton_(unit)&quot; rel=&quot;nofollow noreferrer&quot;&gt;newton&lt;/a&gt;,&lt;a href=&quot;https://en.wikipedia.org/wiki/Pascal_(unit)&quot; rel=&quot;nofollow noreferrer&quot;&gt;pascal&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hertz&quot; rel=&quot;nofollow noreferrer&quot;&gt;hertz&lt;/a&gt;, …? On the other hand, why is there no named SI unit for momentum? (There is a &lt;a href=&quot;https://www.change.org/p/honor-emmy-noether-with-the-unit-for-momentum&quot; rel=&quot;nofollow noreferrer&quot;&gt;suggestion&lt;/a&gt; to name such a momentum unit after Emmy Noether.)&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Watt&quot; rel=&quot;nofollow noreferrer&quot;&gt;watt&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Joule&quot; rel=&quot;nofollow noreferrer&quot;&gt;joule&lt;/a&gt; were simultaneously proposed by electrical engineer &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_Wilhelm_Siemens&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Wilhelm Siemens&lt;/a&gt; in 1882, to replace long-existing but arbitrary units: the &lt;a href=&quot;https://en.wikipedia.org/wiki/Horsepower#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;horsepower&lt;/a&gt; and the &lt;a href=&quot;https://en.wikipedia.org/wiki/Calorie#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;calorie&lt;/a&gt;. I think Siemens recognized that you can't eliminate a common existing unit without replacing it with succinct alternative.  The fact that both horsepower and calories are still in common use shows how strong tradition and convenience are.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6609" LastEditDate="2023-11-06T23:48:09.627" LastActivityDate="2023-11-06T23:48:09.627" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15810" PostTypeId="1" AcceptedAnswerId="15811" CreationDate="2023-10-08T15:48:52.830" Score="13" ViewCount="4688" Body="&lt;p&gt;I'm writing about the history of the concept of noise and am having trouble tracking down references from when the term &amp;quot;noise&amp;quot; started being associated with statistical noise such as Gaussian Noise and not just things like people talking in the background and RF noise that prevents the interception of radio signals.&lt;/p&gt;&#xA;&lt;p&gt;An &lt;a href=&quot;https://doi.org/10.1126/science.os-1.25.304.c&quot; rel=&quot;nofollow noreferrer&quot;&gt;article in Science&lt;/a&gt; describing a new telephone device invented by Tuft’s College professor A. E. Dolbear noted that the words sounded “clear without the sputtering and confused noises” of Alexander Bell’s system.  By the 1920s, &lt;a href=&quot;https://doi.org/10.1126/science.os-2.53.310&quot; rel=&quot;nofollow noreferrer&quot;&gt;engineers at AT&amp;amp;T&lt;/a&gt; are measuring and comparing the noise between trans-Atlantic radio links and cables.&lt;/p&gt;&#xA;&lt;p&gt;But I can't find references that go from these early uses of the term &amp;quot;noise&amp;quot; to &amp;quot;Gaussian Noise.&amp;quot; The earliest reference I've found so far is &lt;a href=&quot;https://apps.dtic.mil/sti/tr/pdf/AD0035947.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Technical Report 189&lt;/a&gt; from the Cruft Laboratory, June 1, 1954.&lt;/p&gt;&#xA;" OwnerUserId="6112" LastEditorUserId="10975" LastEditDate="2023-11-16T03:17:16.817" LastActivityDate="2023-11-16T03:17:16.817" Title="Who coined the term &quot;signal-to-noise ratio&quot; and when did statisticians start using the term &quot;noise&quot; to describe randomness?" Tags="|computer-science|statistics|information-theory|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15811" PostTypeId="2" ParentId="15810" CreationDate="2023-10-08T21:08:36.560" Score="26" Body="&lt;p&gt;You are looking at relatively recent references. Noise originally comes from telecommunication analysis in early 1900s well before Shannon; basically, the electrical noise was heard as audible noise- hence the term&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Signal to Noise&amp;quot; can be traced to 1923 article, by&#xA;Arnold, H. D., &amp;amp; Espenschied, L. (1923). Transatlantic Radio Telephony 1. Bell System Technical Journal, 2(4), 116-144.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ratio of Signal to Noise Strength; Words Received. The noise curve of&#xA;Fig. 9 and that of Fig. 10 can, therefore, be read as &amp;quot;The strength of&#xA;the signal tone which can just be heard through the noise.&amp;quot; It can,&#xA;therefore, be directly compared with the signal curve itself and the&#xA;difference between the two curves is a measure of the level of the&#xA;actual signal strength above that which would just permit of the&#xA;signals being heard. Actually, the difference between the two curves,&#xA;as shown in the figures, is proportional to the ratio of the signal to&#xA;the noise strength, because the curves are plotted to a logarithmic&#xA;scale.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5eEP0.png&quot; rel=&quot;noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5eEP0.png&quot; alt=&quot;Figure 10&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2023-10-08T21:08:36.560" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15812" PostTypeId="1" CreationDate="2023-10-09T03:55:43.847" Score="2" ViewCount="197" Body="&lt;p&gt;I know that formula &lt;span class=&quot;math-container&quot;&gt;$∂^2=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d^2=0$&lt;/span&gt; very important in the homology and cohomology theory. And I understand that this formula was generated from the process of finding a solution to the partial differential equation. So, I want to know the origin and motivation of the formula &lt;span class=&quot;math-container&quot;&gt;$∂^2=0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$d^2=0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;(In this question, &lt;span class=&quot;math-container&quot;&gt;$∂$&lt;/span&gt;'s meaning is that the process of finding the boundary of a simplex. Like below picture. i.e. &lt;span class=&quot;math-container&quot;&gt;$∂^2=0$&lt;/span&gt; means that boundaries have no boundary. )&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/ARGKa.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ARGKa.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-10-09T12:48:05.043" LastActivityDate="2023-10-09T12:48:05.043" Title="The origin of $∂^2=0$ and $d^2=0$" Tags="|terminology|algebraic-geometry|differential-equations|category-theory|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15813" PostTypeId="2" ParentId="15127" CreationDate="2023-10-09T13:46:19.313" Score="0" Body="&lt;p&gt;Hello I am tracing my family tree and so far I can tell you this is his obituary&lt;/p&gt;&#xA;&lt;p&gt;Alan EARNSHAW&#xA;March 10, 2009 Suddenly at his home, Alan of Drawbriggs Court, Appleby, formerly of Meltham, Huddersfield. Loving husband of Larraine, father to Sarah, Bryony, Louise and Peter and devoted Grandfather. Funeral service and cremation to take place at Carlisle Crematorium on Friday, March 20 at 12.40 p.m. Family flowers only are requested donations if desired to The Great North Air Ambulance c/o Glyn Jones Funeral Director, 12 Battlebarrow, Appleby-in-Westmorland, CA16 6XS.&lt;/p&gt;&#xA;&lt;p&gt;If anyone can tell me anything that would be great&lt;/p&gt;&#xA;" OwnerUserId="19133" LastActivityDate="2023-10-09T13:46:19.313" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15816" PostTypeId="1" CreationDate="2023-10-10T08:01:49.093" Score="2" ViewCount="118" Body="&lt;p&gt;In Wikipedia’s &lt;a href=&quot;https://en.wikipedia.org/wiki/Giuseppe_Peano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Peano entry&lt;/a&gt; I find the following quote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[In] the First International Conference of Philosophy [Peano]&#xA;presented a paper which posed the question of correctly formed&#xA;definitions in mathematics, i.e. &amp;quot;how do you define a definition?&amp;quot;.&#xA;This became one of Peano's main philosophical interests for the rest&#xA;of his life.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Did Peano answer the question he posed? Where can I find more information about the subject?&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2023-10-10T08:01:49.093" Title="Peano's question about how to define a definition" Tags="|definition|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15817" PostTypeId="1" AcceptedAnswerId="15820" CreationDate="2023-10-11T00:14:47.547" Score="5" ViewCount="1888" Body="&lt;p&gt;There's a quote by Poincare on the &amp;quot;new functions&amp;quot;, such as continuous functions without derivatives, that were appearing during the second half of the 19th century. The fullest version I've found online is in &lt;a href=&quot;https://golem.ph.utexas.edu/category/2016/03/foundations_of_mathematics.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;this comment by John Baez&lt;/a&gt; on the n-Category Café. A shorter version is on the &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Functions/&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor's function concept page&lt;/a&gt;, which attributes the year of the quote to 1899.&lt;/p&gt;&#xA;&lt;p&gt;It reads:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Logic sometimes makes monsters. For half a century we have seen a mass&#xA;of bizarre functions which appear to be forced to resemble as little&#xA;as possible honest functions which serve some purpose. More of&#xA;continuity, or less of continuity, more derivatives, and so forth.&#xA;Indeed, from the point of view of logic, these strange functions are&#xA;the most general; on the other hand those which one meets without&#xA;searching for them, and which follow simple laws appear as a&#xA;particular case which does not amount to more than a small corner.&lt;/p&gt;&#xA;&lt;p&gt;In former times when one invented a new function it was for a&#xA;practical purpose; today one invents them purposely to show up defects&#xA;in the reasoning of our fathers and one will deduce from them only&#xA;that.&lt;/p&gt;&#xA;&lt;p&gt;If logic were the sole guide of the teacher, it would be necessary to&#xA;begin with the most general functions, that is to say with the most&#xA;bizarre. It is the beginner that would have to be set grappling with&#xA;this teratologic museum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Does anyone know the source?&lt;/p&gt;&#xA;" OwnerUserId="15017" LastEditorUserId="6609" LastEditDate="2023-10-11T13:21:55.310" LastActivityDate="2023-10-11T13:21:55.310" Title="Source of a Poincaré quote: &quot;Logic sometimes makes monsters...&quot;" Tags="|mathematics|mathematicians|real-analysis|quote|19th-century|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15818" PostTypeId="1" CreationDate="2023-10-11T00:50:05.347" Score="1" ViewCount="137" Body="&lt;p&gt;In many serious engineering and scientific publications including IEEE publications, we see a quote attributed to Nikola Tesla which goes like this&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If you want to find the secrets of the universe, think in terms of energy, frequency, and vibration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However no one gives a reference because this is common quote on social media type posts but no book or publication before the internet era (1990s) ever mentions it. I searched Google Scholar and Google Books with year limited to pre-1999, there is nothing. Is there an authentic reference of this quote? Wiki Quotes also has a reference to a vague website. See for example: &lt;a href=&quot;https://en.wikiquote.org/wiki/Nikola_Tesla#Disputed&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikiquote.org/wiki/Nikola_Tesla#Disputed&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6085" LastEditorUserId="55" LastEditDate="2023-10-12T00:01:32.313" LastActivityDate="2023-10-12T00:01:32.313" Title="Attributed quote to Nikola Tesla" Tags="|reference-request|quote|debunking|tesla|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15819" PostTypeId="2" ParentId="15817" CreationDate="2023-10-11T03:29:22.067" Score="3" Body="&lt;p&gt;This text appears in Poincaré's 1909 &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=uc1.$b279049&amp;amp;seq=144&amp;amp;q1=musee&quot; rel=&quot;nofollow noreferrer&quot;&gt;Science et méthode&lt;/a&gt;, starting on p.132.  I do not know if he published this passage earlier.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2023-10-11T03:55:32.523" LastActivityDate="2023-10-11T03:55:32.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15820" PostTypeId="2" ParentId="15817" CreationDate="2023-10-11T04:24:58.620" Score="11" Body="&lt;p&gt;McTutor most likely took the passage from &lt;a href=&quot;https://www.google.com/books/edition/Mathematical_Thought_From_Ancient_to_Mod/8YaBuGcmLb0C?hl=en&quot; rel=&quot;noreferrer&quot;&gt;Kline's Mathematical Thought From Ancient to Modern Times, v.3, p.973&lt;/a&gt;, they reproduced his translation verbatim. Kline references Poincare's essay &lt;em&gt;Dans la Science Mathématique&lt;/em&gt; published in &lt;em&gt;L'Enseignement Mathématique&lt;/em&gt;, 11 (1899) 157-62 (&lt;em&gt;Ouvres&lt;/em&gt;, 11, 129-34). &lt;a href=&quot;https://www.e-periodica.ch/digbib/view?pid=ens-001%3A1899%3A1#310&quot; rel=&quot;noreferrer&quot;&gt;Electronic version&lt;/a&gt; is available from SwissDML.&lt;/p&gt;&#xA;&lt;p&gt;In 1908, the essay was folded into Poincare's book &lt;a href=&quot;https://fr.wikipedia.org/wiki/Science_et_M%C3%A9thode&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Science et Méthode&lt;/em&gt;&lt;/a&gt;. An &lt;a href=&quot;https://archive.org/details/sciencemethod00poinuoft/page/124/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;English translation&lt;/a&gt; of it is accessible on Internet Archive and differs from Kline's:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Formerly, when a new function was invented, it was in view of some practical end. Today they are&#xA;invented on purpose to show our ancestors' reasoning at fault, and we shall never get anything more than that out of them.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;If logic were the teacher's only guide, he would have to begin with the most general, that is to say, with the most weird, functions. He would have to set the beginner to wrestle with this collection of monstrosities. If you don't do so, the logicians might say, you will only reach exactness by stages.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-10-11T07:14:30.957" LastActivityDate="2023-10-11T07:14:30.957" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15822" PostTypeId="1" CreationDate="2023-10-11T12:51:30.903" Score="2" ViewCount="658" Body="&lt;p&gt;When I was reading, the question just popped into my head after noticing that the Greek letter ψ looks kind of like a wave itself. Stylized, they look even more wavy:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\Huge \Psi\;\Huge\psi$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This made me want to ask the question why &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; was chosen to represent the wave function. I am aware that &lt;a href=&quot;https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erwin Schrödinger&lt;/a&gt; is responsible for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Schrödinger equation&lt;/a&gt; which calculates the &lt;a href=&quot;https://en.wikipedia.org/wiki/Wave_function&quot; rel=&quot;nofollow noreferrer&quot;&gt;wave function&lt;/a&gt;, giving the value for &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt;. Also, I read somewhere that I can no longer find, that Schrödinger is the one who chose &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; to represent the wave function.&lt;/p&gt;&#xA;&lt;p&gt;I have found &lt;a href=&quot;https://www.quora.com/Why-is-psi-the-wave-function&quot; rel=&quot;nofollow noreferrer&quot;&gt;a Quora answer&lt;/a&gt; that gives quite a reasonable, I am just not sure if it is correct, given that they provide no sources, and I cannot find any supporting documentation to back up it. The answer given there is that the Greek letter &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; was associated with the Greek god of the sea, &lt;a href=&quot;https://en.wikipedia.org/wiki/Poseidon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Poseidon&lt;/a&gt;. The connection is that &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; looks like the trident that Poseidon was known for. They also mentions how pronouncing &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; (Psi), sounds like the beginning of his name. I am not sure if they were meaning to suggest that there was an actual connection between the two, but judging from &lt;a href=&quot;https://en.wiktionary.org/wiki/%CE%A0%CE%BF%CF%83%CE%B5%CE%B9%CE%B4%E1%BF%B6%CE%BD&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Greek spelling of Poseidon on Wikipedia&lt;/a&gt; there is no &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; is Poseidon's name. So, the explanation goes, because &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; is associated with Poseidon, who is king of the waves, and the value this variable was to represent was the wave function, why not make &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; represent the wave function. This makes perfect sense to me.&lt;/p&gt;&#xA;&lt;p&gt;So, my questions are, first, was Schrödinger the one who chose &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; to represent the wave function, and if not, who did? And, second, why was &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; chosen to represent the wave function (i.e. was the Quora explanation correct)? Lastly, how do you know/ what are you sources (i.e., hopefully a link to a authoritative document, or maybe your professor told you the story that they were told by Schrödinger himself)?The last part is where the Quora answer is lacking.&lt;/p&gt;&#xA;" OwnerUserId="3917" LastEditorUserId="6609" LastEditDate="2023-10-11T13:33:27.210" LastActivityDate="2023-10-11T18:25:46.257" Title="Why was the Greek letter psi (Ψ) chosen to represent the wave function?" Tags="|physics|quantum-mechanics|theoretical-physics|particle-physics|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15823" PostTypeId="2" ParentId="15822" CreationDate="2023-10-11T18:25:46.257" Score="9" Body="&lt;h3&gt;Prior to Schrodinger&lt;/h3&gt;&#xA;&lt;p&gt;There is no use of the symbol by de Broglie, Schrodinger's predecessor. In the first three short notes from de Broglie on the topic of wave mechanics (1923), there is no use of the symbol. In de Broglie's thesis there is only one brief mention of the symbol that I can see, in chapter 2 section 3 &lt;em&gt;Les deux principes de moindre action dans la dynamique de l'electron&lt;/em&gt;, where &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; is used to denote the scalar potential. So this is unrelated to the wavefunction.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;De Broglie, L. (1923). Ondes et quanta. &lt;em&gt;Compte Rendus, 177,&lt;/em&gt; 507-510. (&lt;a href=&quot;https://www.academie-sciences.fr/pdf/dossiers/Broglie/Broglie_pdf/CR1923_p507.pdf&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;De Broglie, L. (1923). Quanta de lumière, diffraction et interférences. &lt;em&gt;Compte Rendus, 177,&lt;/em&gt; 548-550. (&lt;a href=&quot;https://www.academie-sciences.fr/pdf/dossiers/Broglie/Broglie_pdf/CR1923_p548.pdf&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;De Broglie, L. (1923). Les quanta, la théorie cinétique des gaz et le principe de Fermat. &lt;em&gt;Comptes rendus, 177&lt;/em&gt;, 630-632. (&lt;a href=&quot;https://www.academie-sciences.fr/pdf/dossiers/Broglie/Broglie_pdf/CR1923_p630.pdf&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;li&gt;De Broglie, L. (1925). PhD Thesis. (&lt;a href=&quot;https://theses.hal.science/tel-00006807/document&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h3&gt;Schrodinger's Wavefunction&lt;/h3&gt;&#xA;&lt;p&gt;This brings us to Schrodinger, and his 1926 papers, which introduce the wavefunction. The first paper is &lt;em&gt;Quantisierung als Eigenwertproblem&lt;/em&gt; (Quantization as an Eigenvalue Problem).&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Schrödinger, E. (1926). Quantisierung als eigenwertproblem. &lt;em&gt;Annalen der physik, 385&lt;/em&gt;(13), 437-490. (&lt;a href=&quot;https://onlinelibrary.wiley.com/doi/epdf/10.1002/andp.19263840404&quot; rel=&quot;noreferrer&quot;&gt;Link&lt;/a&gt;, see also the &lt;a href=&quot;http://ofp.cosmo-ufes.org/uploads/1/3/7/0/13701821/quantisation_as_an_eigenvalue_problem.pdf&quot; rel=&quot;noreferrer&quot;&gt;English Translation&lt;/a&gt; translated by Oliver F. Piatetella.)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;In this paper, Schrodinger uses the symbol &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; when describing the Hamiltonian differential equation,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;H\left(q, \frac{\partial S}{\partial q}\right) = E,&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We introduce now in place of &lt;span class=&quot;math-container&quot;&gt;$S$&lt;/span&gt; a new, unknown function &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; in such a  manner that &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; would appear as a &lt;em&gt;product&lt;/em&gt; of suitable functions of the single coordinates.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Setting up a variational problem, he arrives at the wave equation&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;\Delta \psi +\frac{2m}{K^2}\left(E + \frac{e^2}{r}\right)\psi = 0&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;which demonstrates the wavelike behavior of the scalar field &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt;. From this we can conclude that there is no deeper meaning to the symbols &lt;span class=&quot;math-container&quot;&gt;$\psi$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\Psi$&lt;/span&gt; beyond convention when solving problems in mathematical physics.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2023-10-11T18:25:46.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15824" PostTypeId="1" CreationDate="2023-10-12T13:56:10.183" Score="5" ViewCount="95" Body="&lt;p&gt;Good people, I'm trying to get my head around the history of algebraic geometry, and while Dieudonné's tome is a very good source (very often the &lt;em&gt;only&lt;/em&gt; source), it can from time to time be very &lt;em&gt;idiosyncratic&lt;/em&gt;. As such, I was wondering if I had understood things properly on a certain point. Specifically, is the following statement correct:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The notion of morphisms between varieties as we understand it today with regular maps and all that is an invention of Zariski and Weil in the 30s and 40s. Prior to that, the only notion of &amp;quot;morphisms&amp;quot; that algebraic geometers had were rational transformations (i.e. rational maps), and the only notion of two varieties being &amp;quot;isomorphic&amp;quot; that they had was that two varieties were birationally equivalent.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;If that is true, then would it further be fair to say the following:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The very notion of thinking of varieties and morphisms between them being the central &amp;quot;things&amp;quot; around which everything in algebraic geometry centers is a conception that first came about due to category theory, and that prior to Zariski-Weil-Grothendieck, people didn't have that particular structured approach to algebraic geometry. That in a sense, it was more akin to &amp;quot;trying to do Euclid and Descartes on algebraic surfaces&amp;quot;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;As always, look forward to your responses.&lt;/p&gt;&#xA;" OwnerUserId="15726" LastActivityDate="2023-10-12T13:56:10.183" Title="History of Algebraic Geometry: Morphisms and Birational Geometry" Tags="|mathematics|algebraic-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15827" PostTypeId="1" CreationDate="2023-10-16T23:32:44.540" Score="2" ViewCount="155" Body="&lt;p&gt;I seem to recall reading that some French mathematician (perhaps a member of Bourbaki?) came up with the idea of basing error-correcting codes on Galois fields quite early in the development of algebraic coding theory, but I don’t remember the details, and my googling skills haven’t helped me come up with any leads.&lt;/p&gt;&#xA;&lt;p&gt;Golay’s original 1949 article on what are now called Golay codes does not mention finite fields. On the other hand, the 1959 article of Bose and Ray-Chaudhuri on what are now called BCH codes does. So in between 1949 and 1959 someone must have noticed the connection between codes and finite fields. Was it Alexis Hocqueghem, John Cocke, or someone else?&lt;/p&gt;&#xA;&lt;p&gt;(Apologies to those who posted the first few comments that appear below for not crafting my original post with more care. If those comments are now hard for others to interpret it is because I’ve fixed the wording of the question to address their concerns.)&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="6609" LastEditDate="2023-10-19T14:49:52.143" LastActivityDate="2023-11-04T19:07:12.573" Title="Error-correcting codes based on Galois fields" Tags="|abstract-algebra|galois|" AnswerCount="0" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15828" PostTypeId="1" CreationDate="2023-10-17T09:10:35.823" Score="1" ViewCount="103" Body="&lt;p&gt;I know of Maxwell's, Descartes' and Laplace's demons but I recently found out that there is Darwinian one. I do not think that this demon appears in the works of Charles Darwin. Do you know who may have coined the term (Darwin's/Darwinian demon)?&lt;/p&gt;&#xA;&lt;p&gt;This article: U. Kutschera &amp;amp; K. J. Niklas &lt;em&gt;&lt;a href=&quot;https://doi.org/10.1111/plb.12171&quot; rel=&quot;nofollow noreferrer&quot;&gt;Darwin-Wallace Demons: survival of the fastest in populations of duckweeds and the evolutionary history of an enigmatic group of angiosperms&lt;/a&gt;&lt;/em&gt; (2014) says it was R. Law &lt;em&gt;&lt;a href=&quot;https://www.journals.uchicago.edu/doi/abs/10.1086/283488&quot; rel=&quot;nofollow noreferrer&quot;&gt;Optimal life histories under age-specific predation&lt;/a&gt;&lt;/em&gt; The American Naturalist (1979). But I have found various earlier sources.&lt;/p&gt;&#xA;&lt;p&gt;The earliest English source I could find is &lt;a href=&quot;https://en.wikipedia.org/wiki/Colin_Pittendrigh&quot; rel=&quot;nofollow noreferrer&quot;&gt;Colin S. Pittendrigh&lt;/a&gt;'s &lt;em&gt;&lt;a href=&quot;http://nicorg.pbworks.com/f/Pittendrigh%20Temporal%20Organization.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On temporal organization in living systems&lt;/a&gt;&lt;/em&gt; - Harvey lectures, 1961, in relation to the works of German E. Bünning. So maybe it appears first in another language? Also I am suspicious that such a concept was invented for a lecture.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-12-12T07:07:27.217" LastActivityDate="2023-12-12T07:07:27.217" Title="Who came up with the Darwinian demon?" Tags="|terminology|evolution|darwin|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15829" PostTypeId="2" ParentId="14453" CreationDate="2023-10-17T19:35:57.410" Score="2" Body="&lt;p&gt;An early reference that uses this notation and (at a minimum) introduced many people to this notation is the following publication.&lt;/p&gt;&#xA;&lt;p&gt;Dawid, A. P. “Conditional Independence in Statistical Theory.” Journal of the Royal Statistical Society. Series B (Methodological) 41, no. 1 (1979): 1–31. &lt;a href=&quot;http://www.jstor.org/stable/2984718&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://www.jstor.org/stable/2984718&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Have you found earlier uses?&lt;/p&gt;&#xA;" OwnerUserId="19182" LastActivityDate="2023-10-17T19:35:57.410" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15830" PostTypeId="1" CreationDate="2023-10-18T01:31:23.367" Score="3" ViewCount="2275" Body="&lt;p&gt;I was reading the preface of &lt;a href=&quot;https://www.marxists.org/archive/marx/works/1881/mathematical-manuscripts/ch02.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marx's Mathematical Manuscripts&lt;/a&gt;. They explain the situation of calculus in the time of Marx, it seems that at the time analysis as we know today was still being forged by Cantor, Cauchy, Dedekind, Weierstrass. Also, it seems that there was some resistance to the appearance of calculus as we know today and Marx was reading some old books:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This more precise work was unknown in the English universities at that time. Not without reason did the well-known English mathematician Hardy comment in his Course of Pure Mathematics, written significantly later (1917): ‘It [this book] was written when analysis was neglected in Cambridge, and with an emphasis and enthusiasm which seem rather ridiculous now. If I were to rewrite it now I should not write (to use Prof. Littlewood’s simile) like a “missionary talking to cannibals”,’ (preface to the 1937 edition). Hardy had to note as a special achievement the fact that in monographs in analysis ‘even in England there is now [i.e., in 1937] no lack’.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Marx studied textbooks of differential calculus. He oriented himself with books used at courses in Cambridge University, where in the 17th century Newton held a chair of higher mathematics, the traditions of which were kept by the English up to Marx’s day. Indeed, there was a sharp struggle in the 20s and 30s of the last century between young English scholars, grouped about the ‘Analytical Society’ of mathematicians, and the opposing established and obsolete traditions, converted into untouchable ‘clerical’ dogma, represented by Newton. The latter applied the synthetic methods of his Principia with the stipulation that each problem had to be solved from the beginning without converting it into a more general problem which could then be solved with the apparatus of calculus.&lt;/p&gt;&#xA;&lt;p&gt;In this regard, the facts are sufficiently clear that Marx began studying differential calculus with the work of the French abbot Sauri, Cours complet de mathématiques (1778), based on the methods of Leibnitz and written in his notation, and that he turned next to the De analyse per aequationes numero terminorum infinitas of. Newton (cf.ms.2763). Marx was so taken with Sauri’s use of the Leibnitzian algorithmic methods of differentiation that he sent an explanation of it (with application to the problem of the tangent to the parabola) in a special appendix to one of his letters to Engels.&lt;/p&gt;&#xA;&lt;p&gt;Marx, however, did not limit himself to Sauri’s Cours. The next text to which he turned was the English, translation of a modern (1827) French textbook, J.-L. Boucharlat’s Eléments de calcul différentiel et du calcul intégral. Written in an ecletic spirit, in combined the ideas of d’Alembert and Lagrange. It went through eight editions in France alone and was translated into foreign languages (including Russian); the textbook, however, did not satisfy Marx, and he next turned to a series of monographs and survey-course books. Besides the classic works of Euler and MacLaurin (who popularised Newton) there were the university textbooks of Lacroix, Hind, Hemming, and others. Marx made scattered outlines and notations from all these books.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It seems that at the time, people were doing &amp;quot;weird&amp;quot; things in calculus:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Differential calculus is characterised by its symbols and terminology, such notions as ‘differential’ and ‘infinitely small’ of different orders, such symbols as dx, dy, d²y, d³y ... dy/dx³, d²y/dx², d³y/dx³ and others. In the middle of the last century many of the instructional books used by Marx associated these concepts and symbols with special methods of constructing quantities different from the usual mathematical numbers and functions. Indeed, mathematical analysis was obliged to operate with these special quantities. This is not true at the present time: there are no special symbols in contemporary analysis; yet the symbols and terminology have been preserved, and even appear to be quite suitable. How? How can this happen, if the corresponding concepts have no meaning? The mathematical manuscripts of Karl Marx provide the best answer to this question. Indeed, such an answer which permits the understanding of the essence of all symbolic calculus, whose general theory was only recently constructed in contemporary mathematical logic.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But to me, it's not clear at all what are these weird things. I tried to take a look at Sauri's book but it not only seems that the are constructing different things but the typography is a bit weird. I'd like to know more what were these weird things people did back in the time.&lt;/p&gt;&#xA;&lt;p&gt;In particular, what opposition people had to writing &amp;quot;dx/dy=0/0&amp;quot; as is pointed out in a letter sent to Marx by Engels:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and I was pleased to find that I did not need them. I compliment you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes from the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls. And yet it is clear that dy/dx can only be the pure expression of a completed process if the last trace of the quanta x and y has disappeared, leaving the expression of the preceding process of their change without any quantity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I know what is the modern opposition to this, but I don't know if the opposition was the same back in the day.&lt;/p&gt;&#xA;" OwnerUserId="332" LastActivityDate="2023-10-18T10:58:16.880" Title="What were the &quot;weird&quot; things people were doing in calculus at the time of Marx?" Tags="|calculus|real-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15831" PostTypeId="2" ParentId="15830" CreationDate="2023-10-18T07:57:23.007" Score="5" Body="&lt;p&gt;Besides infintesimals, there are other ideas and concepts that mathematicians used to work with up until roughly 1940, that have not been properly formalized in modern mainstream mathematics, like the ideas of &lt;em&gt;variable quantities, constants&lt;/em&gt; and surrounding concepts like &lt;em&gt;function of something&lt;/em&gt; and &lt;em&gt;change/difference of a variable&lt;/em&gt; (&lt;span class=&quot;math-container&quot;&gt;$\Delta x$&lt;/span&gt;, or in infinitesimal form &lt;span class=&quot;math-container&quot;&gt;$dx$&lt;/span&gt;). Although people still use these words and ideas nowadays, they either use them informally without proper definition, or the modern definition doesn't match the idea of the past. The most notorious example is the word &lt;em&gt;function&lt;/em&gt; which got a new definition between 1900 and 1940. See &lt;a href=&quot;https://hsm.stackexchange.com/q/6104/3462&quot;&gt;Who first considered the f in f(x) as an object in itself, and who decided to call it a function?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Further SE questions that might help you get a feel for these issues are:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/questions/307947/formalizations-of-the-idea-that-something-is-a-function-of-something-else&quot;&gt;Formalizations of the idea that something is a function of something else&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/q/84221/745&quot;&gt;Variable-centric logical foundation of calculus&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://mathoverflow.net/q/115416/745&quot;&gt;If d/dx is an operator, on what does it operate?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/q/1258923/1984&quot;&gt;What did Alan Turing mean when he said he didn't fully understand dy/dx?&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2023-10-18T10:58:16.880" LastActivityDate="2023-10-18T10:58:16.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15833" PostTypeId="2" ParentId="15810" CreationDate="2023-10-19T12:05:48.463" Score="3" Body="&lt;p&gt;The article &amp;quot;The history of noise [on the 100th anniversary of its birth]&amp;quot; in Volume 22, Issue 6, November 2005 of the IEEE Signal Processing Magazine seems to go in the same direction. &lt;a href=&quot;https://ieeexplore.ieee.org/document/1550188&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://ieeexplore.ieee.org/document/1550188&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Looking at one of the older papers referenced in that article, W. Schottky &amp;quot;Über spontane Stromschwankungen in verschiedenen elektrizitätsleitern&amp;quot; I see he uses the term &amp;quot;Schroteffekt&amp;quot; so I guess the German term &amp;quot;Rauschen&amp;quot; had not yet been established in 1918...&lt;/p&gt;&#xA;" OwnerUserId="19191" LastActivityDate="2023-10-19T12:05:48.463" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15834" PostTypeId="1" AcceptedAnswerId="15840" CreationDate="2023-10-19T12:14:00.367" Score="6" ViewCount="112" Body="&lt;p&gt;The ancient greeks considered surfaces such as cones, but did they study the &lt;a href=&quot;https://en.wikipedia.org/wiki/Paraboloid#Hyperbolic_paraboloid&quot; rel=&quot;noreferrer&quot;&gt;hyperbolic paraboloid&lt;/a&gt;? What is the first occurrence of such surface in history?&lt;/p&gt;&#xA;" OwnerUserId="6959" LastActivityDate="2023-10-21T02:25:36.563" Title="First occurrence of hyperboloid paraboloid" Tags="|geometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15835" PostTypeId="1" CreationDate="2023-10-19T13:34:37.390" Score="3" ViewCount="107" Body="&lt;p&gt;Although finite fields are usually introduced as field extensions of fields of prime order, they also arise as quotients of number rings; e.g., &lt;span class=&quot;math-container&quot;&gt;$GF(9)$&lt;/span&gt; comes from taking the Gaussian integers mod 3 and &lt;span class=&quot;math-container&quot;&gt;$GF(4)$&lt;/span&gt; comes from taking the Eisenstein integers mod 2. It seems likely that mid-19th-century mathematicians would have stumbled across specific finite fields in this setting and perhaps also noticed the connection to the work of Galois (though of course the general concept of a finite field is a modern one). Can anyone steer me toward relevant primary or secondary literature?&lt;/p&gt;&#xA;" OwnerUserId="6587" LastActivityDate="2023-11-13T02:04:09.777" Title="Finite fields as quotients" Tags="|number-theory|abstract-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15836" PostTypeId="1" CreationDate="2023-10-19T17:26:27.783" Score="2" ViewCount="118" Body="&lt;p&gt;&lt;em&gt;Surely You're Joking Mr. Feynman&lt;/em&gt; has Feynman ascribing to Onsager the following quote (during  the International Conference of Theoretical Physics in Kyoto, in 1953):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;We should tell Feynman that nobody has ever figured out the order of&#xA;any transition correctly from first principles.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is there reason to believe/disbelieve Onsager said this? Was this true at the time? When did someone first correctly show the order of a phase transition from first principles? (I realize &amp;quot;show&amp;quot; probably has in this case multiple interpretations corresponding to different levels of rigour required I would leave this open to those answering.)&lt;/p&gt;&#xA;" OwnerUserId="18217" LastEditorUserId="6609" LastEditDate="2024-03-13T14:09:20.813" LastActivityDate="2024-03-13T14:09:20.813" Title="Onsager on phase transitions" Tags="|physics|theoretical-physics|quote|feynman|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15839" PostTypeId="1" AcceptedAnswerId="15916" CreationDate="2023-10-20T21:32:52.327" Score="2" ViewCount="110" Body="&lt;p&gt;I am not sure if this question is on topic here but I will give it a try. According to the Nobel Nomination archive &lt;a href=&quot;https://www.nobelprize.org/nomination/archive/&quot; rel=&quot;nofollow noreferrer&quot;&gt;official website&lt;/a&gt;, nomination data cannot be &amp;quot;revealed until 50 years later&amp;quot; &lt;em&gt;at least&lt;/em&gt;. Peace and Literature Prizes are updated up to 1971. Physics Prize nomination archives are revealed up to 1970. Why don't we have information on 1971, 1972 and 1973? Medicine is even worse because it is stuck at 1953 for no clear reason.&lt;/p&gt;&#xA;&lt;p&gt;Per &lt;a href=&quot;https://www.nobelprize.org/frequently-asked-questions/#h-how-nobel-prize-laureates-are-chosen&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nobel frequently-asked questions&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For prizes older than 50 years old, you can browse the nomination archive. Note that in some cases the archives are sealed as long as people mentioned are still alive.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;At first I though that the nominees should be deceased in order for the information to be released but people like Jeffrey Goldstone (nominated in 1968) are still alive. Maybe this condition is about the nominators?&lt;/p&gt;&#xA;&lt;p&gt;What restricts a new year of nominations to be revealed each year? Do we know how often the archives get updated?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-10-21T12:33:56.577" LastActivityDate="2023-11-15T12:39:52.640" Title="Why aren't Nobel nomination archives updated more often?" Tags="|nobel-prize|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15840" PostTypeId="2" ParentId="15834" CreationDate="2023-10-21T02:25:36.563" Score="6" Body="&lt;p&gt;I suspect that Coolidge goes from Fermat to Brianchon, and only mentions Monge in passing,  because his focus is on synthetic geometry. He remarks:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It is certainly surprising that the synthetic geometers of the seventeenth&#xA;and eighteenth centuries who were so much interested in the conic sections should have shown no particular interest in the quadric&#xA;surfaces.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;But the main work on quadrics in the 18th century was done analytically, first by Clairaut, in connection with his investigations of the shape of the Earth, and then by Euler, Lagrange and Monge. In the appendix on surfaces to the second volume of his &lt;em&gt;Introductio in analysin infinitorum&lt;/em&gt; (1748), Euler gave a complete geometric classification of quadrics, and it is there that we meet the hyperbolic paraboloid for the first time. Some historical commentary can be found in &lt;a href=&quot;https://www.google.com/books/edition/Mathematical_Thought_from_Ancient_to_Mod/aO-v3gvY-I8C?hl=en&amp;amp;gbpv=1&amp;amp;dq=descartes+%22hyperbolic+paraboloid%22+history&amp;amp;pg=PA547&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;Kline's Mathematical Thought from Ancient to Modern Times, v.2, ch.23&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The quadric surfaces, e.g. sphere, cylinder, paraboloid, hyperboloid of two sheets, and ellipsoid, were of course known geometrically before 1700; in fact, some of them appear in Archimedes' work. Clairaut in his book of 1731 gave the equations of some of these surfaces... Though Euler had done some earlier work on the equations of surfaces, it is in Chapter 5 of the Appendix to the second volume of his Introductio (1748) that he systematically takes up three-dimensional coordinate geometry.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;He presents much of what had already been done and then studies the general second degree equation in three variables&lt;/em&gt; [denoted (1)]. &lt;em&gt;He now seeks to use change of axes to reduce this equation to the forms that result from having the principal axes of the quadric surfaces represented by (1) as the coordinate axes... Euler uses this transformation to reduce (1) to canonical forms and obtains six distinct cases: cone, cylinder, ellipsoid, hyperboloid of one and two sheets, hyperbolic paraboloid (which he discovered), and parabolic cylinder.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Monge later showed that hyperboloid of one sheet and hyperbolic paraboloid are doubly ruled surfaces, i.e. can be &amp;quot;assembled&amp;quot; from two families of straight lines, which precipitated their popularity in engineering and architecture. Euler went on to develop differential geometry of general surfaces in &lt;em&gt;Recherches sur la courbure des surfaces&lt;/em&gt; (1760) and study developable surfaces in &lt;em&gt;De Solidis Quorum Superficiem in Plan um Explicare Licet&lt;/em&gt; (1771). Their study was also later taken up by Monge.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-10-21T02:25:36.563" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15841" PostTypeId="1" AcceptedAnswerId="15844" CreationDate="2023-10-22T03:36:35.543" Score="2" ViewCount="108" Body="&lt;p&gt;Usually, when studying the applications and results of a theory, it becomes clear why it was interesting to define it in a certain way. However, I'm currently beginning my studies in functional analysis. I see &lt;span class=&quot;math-container&quot;&gt;$l^p$&lt;/span&gt; spaces as a generalization of Euclidean space, for vectors in an infinitely-dimensional space. But I would like to understand the construction of this space in a logical manner, comprehending the chronology of definitions, where it originated, who defined it, and everything else. Does anyone have a good recommendation for a history on this?&lt;/p&gt;&#xA;" OwnerUserId="19205" LastActivityDate="2023-10-23T15:10:17.520" Title="$l^p$ space definition" Tags="|functional-analysis|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15843" PostTypeId="2" ParentId="13858" CreationDate="2023-10-22T20:17:34.970" Score="5" Body="&lt;p&gt;The primary reason is that you have a roadblock: you absolutely need complex numbers - even for real-valued solutions - unless you go the route of using trigonometry and/or hyperbolic functions. The simplest example of the roadblock is directly connected to the ancient problem of trisecting the 60 degree angle. Write down the cubic for the cosine of 20 degrees (obtained via the triple angle formula, using the cosine of 60 degrees):&#xA;&lt;span class=&quot;math-container&quot;&gt;$$4c^3 - 3c = \frac{1}{2}.$$&lt;/span&gt;&#xA;Now, use the &amp;quot;Cardano&amp;quot; formula, and you'll quickly see the problem. In fact, there's no way to express the real root to this in terms of the four arithmetic operations and square and cube roots, without using complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;The other roadblock was negative numbers! Del Ferro solved all equations of the form &lt;span class=&quot;math-container&quot;&gt;$x^3 + mx = n$&lt;/span&gt; before 1500 ... but only with non-negative coefficients &lt;span class=&quot;math-container&quot;&gt;$(m,n)$&lt;/span&gt;. Why? Because there wasn't any notion of negative numbers around yet!&lt;/p&gt;&#xA;&lt;p&gt;Both complex and negative numbers came onto the scene together, not separately; because once you start talking about numbers like &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt;, you're also going to be asking questions like what &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-1}$&lt;/span&gt; is.&lt;/p&gt;&#xA;&lt;p&gt;In the early 1500's, Tartaglia said &amp;quot;I have a secret&amp;quot;; Cardano said &amp;quot;Cough it up&amp;quot; and Tartaglia bound him to a (word-of-mouth) non-disclosure, but Cardano said: &amp;quot;Publish it soon or I'll end-run you&amp;quot; (by propping up and upgrading del Ferro's solution). Cardano's publication allowed for both negative numbers and their square roots, but left the square roots of negative numbers uninterpreted.&lt;/p&gt;&#xA;&lt;p&gt;All of that is the account given over on the Wikipedia article &lt;a href=&quot;https://en.wikipedia.org/wiki/Cubic_equation&quot; rel=&quot;noreferrer&quot;&gt;Cubic Equation&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A real root to &lt;span class=&quot;math-container&quot;&gt;$x^3 + 3Ax + 2B = 0$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$x = C - A/C$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$C^3 = -B + \sqrt{A^3 + B^2}$&lt;/span&gt;. So, with the cosine of 20 degrees, &lt;span class=&quot;math-container&quot;&gt;$A = -1/4$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B = -1/16$&lt;/span&gt;, and the cosine of 20 degrees is &lt;span class=&quot;math-container&quot;&gt;$C + 1/(4C)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$C^3 = 1/16 + \sqrt{-3/256}$&lt;/span&gt; ... oops. There's your problem. What's the cube root of a complex number, expressed in the form &lt;span class=&quot;math-container&quot;&gt;$x + i y$&lt;/span&gt;? How do you get the &lt;span class=&quot;math-container&quot;&gt;$(x,y)$&lt;/span&gt; parts for &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;? Well, that involves solving a cubic equation. Guess which one?&lt;/p&gt;&#xA;&lt;p&gt;It's the &lt;a href=&quot;https://en.wikipedia.org/wiki/Casus_irreducibilis&quot; rel=&quot;noreferrer&quot;&gt;Casus Irreducibilis&lt;/a&gt; Impasse. According to the linked Wikipedia article, the proof that it actually &lt;i&gt;is&lt;/i&gt; an impasse was established in 1843. So, in 300+ years' retrospect: that's your roadblock. You're forced either into complex numbers (never mind negative numbers) or - in this example - into trigonometry.&lt;/p&gt;&#xA;&lt;p&gt;I had a similar problem, when in high school. I asked my 11th grade algebra teacher what the solution to the cubic &lt;span class=&quot;math-container&quot;&gt;$x^3 + x + 1 = 0$&lt;/span&gt; was, and he said &amp;quot;You need to learn calculus for that&amp;quot;. So, the following week, I went back to him and said, &amp;quot;Ok. I bought a \$5 calculus book from the bookstore, carefully read through all of the sections, did all the exercises, and learned the material, but I still can't solve the equation&amp;quot; (in closed form). Curiously, I never got any answer from him on that, since he was too busy fainting at the time, for some strange reason. About seven months later, the following year, I asked my third semester calculus assistant professor in college, and he finally coughed it up. So ... all that running around just to find out what the closed form solution to &lt;span class=&quot;math-container&quot;&gt;$x^3 + x + 1 = 0$&lt;/span&gt; was. And it's not even one of those Impasse cases.&lt;/p&gt;&#xA;" OwnerUserId="13406" LastActivityDate="2023-10-22T20:17:34.970" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15844" PostTypeId="2" ParentId="15841" CreationDate="2023-10-23T15:10:17.520" Score="4" Body="&lt;p&gt;As I recall, there are extensive historical notes in both:&lt;/p&gt;&#xA;&lt;p&gt;Dunford &amp;amp; Schwartz, &lt;em&gt;Linear Operators&lt;/em&gt; (Interscience, 1958)&lt;/p&gt;&#xA;&lt;p&gt;M. M. Day, &lt;em&gt;Normed Linear Spaces&lt;/em&gt;  (Springer, 1958)&lt;/p&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-10-23T15:10:17.520" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15845" PostTypeId="1" CreationDate="2023-10-23T18:41:50.247" Score="0" ViewCount="53" Body="&lt;p&gt;Jan Rak, in his &lt;a href=&quot;https://www.scalar-field.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;NAMI-tech SEM 2023&lt;/a&gt; lecture (@&lt;a href=&quot;https://www.youtube.com/watch?v=4IczIRrT-Ws&amp;amp;list=PL4PyHPfpmOMirw1kV9o1GpLURfAuvSix8&amp;amp;index=2&amp;amp;t=773s&quot; rel=&quot;nofollow noreferrer&quot;&gt;12:53&lt;/a&gt;), claims:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, at the time Nicola Tesla […] was trying to replicate the Hertz experiment, and he was unsuccessful. He discovered some other properties of radio frequency waves, came to Hertz try to convince him that his conclusion about the transverse character is incorrect, and so they had a long debate, but at the end, as we all know, the history decided in favor of Herz and Tesla was largely ignored until these days.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What was Tesla's replication of Hertz's experiment?&lt;/p&gt;&#xA;" OwnerUserId="232" LastActivityDate="2023-10-23T18:41:50.247" Title="Tesla tried to replicate Hertz's experiment?" Tags="|electromagnetism|tesla|hertz|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15846" PostTypeId="1" CreationDate="2023-10-23T23:36:52.513" Score="1" ViewCount="107" Body="&lt;p&gt;I’m delving into the intriguing world of alphabetic number systems (greek, for instance), where letters serve a dual purpose—forming words and representing numerical values.&lt;/p&gt;&#xA;&lt;p&gt;I’m curious about the practical implications of such a system, specifically concerning potential confusion or errors. Can you imagine using the same symbols for both words and numbers in today's complex world? It seems like a recipe for misunderstanding!&lt;/p&gt;&#xA;&lt;p&gt;That brings me to my main query: Are there any historical documents or accounts where scholars or scribes from the past have expressed their concerns or frustrations regarding this dual-use of the alphabet? I’m interested in firsthand accounts, critiques, or any form of written evidence that sheds light on how people back then navigated this unique system and what challenges they faced.&lt;/p&gt;&#xA;&lt;p&gt;Your insights and references to specific texts, authors, or periods would be greatly appreciated. I’m eager to learn more about how our ancestors tackled this fascinating linguistic and numerical challenge!&lt;/p&gt;&#xA;" OwnerUserId="12383" LastActivityDate="2023-10-24T15:27:25.397" Title="Historical Accounts of Confusion in Alphabetic Number Systems?" Tags="|notation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15847" PostTypeId="1" CreationDate="2023-10-24T03:46:29.673" Score="0" ViewCount="72" Body="&lt;p&gt;I read in &lt;em&gt;Newton and the Counterfeiter&lt;/em&gt; about the young mathematician who was friends with Sir Isaac and competent enough to have detected an error in &lt;em&gt;Principia&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;He offered a theory of gravity, I believe that is more or less identical to that of Le Sage and at first look (to me) is fairly compelling: That invisible particles are moving throughout space, exerting pressure on objects -- what we see as gravity is the &amp;quot;shadow effect&amp;quot; when one body blocks the particles on one side.&lt;/p&gt;&#xA;&lt;p&gt;I understand that one objection which came much later is how such particles would heat bodies -- don't think anyone understood the idea that heat could be generated in this way in the 17th century. But the book mentions that two contemporaries of Newton who were also &amp;quot;natural philosophers&amp;quot; found this idea amusing and did not take it seriously.&lt;/p&gt;&#xA;&lt;p&gt;But given that this idea resurfaced after de Dullier (and maybe more than once even in the 19th century in a different form) just what was the basis for 17th/18th century physicists being so dismissive?&lt;/p&gt;&#xA;&lt;p&gt;To be clear, I am not interested in objections that occurred much later but rather those of the 17th and early 18th century physicists who first encountered Fatio's work.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="6609" LastEditDate="2023-10-24T11:52:18.300" LastActivityDate="2023-10-24T11:52:18.300" Title="Fatio de Dullier's Theory of Gravity -- Why ridiculed in 17th/18th century?" Tags="|newton|gravity|thermodynamics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15848" PostTypeId="1" CreationDate="2023-10-24T09:33:48.937" Score="0" ViewCount="59" Body="&lt;p&gt;I was surprised to see a reply to a comment on his answer to a Quora question by a research mathematician claiming that Hilbert spaces were actually due to J. W. Gibbs rather than to D. Hilbert. The Wikipedia entry for Hilbert spaces does not credit Gibbs at all, and the entry for Gibbs does not say enough to lead me to such a conclusion either. It is the case that in the entry for Hilbert spaces, applications to the phenomenon known as Gibbs phenomenon are briefly mentioned, but it’s not clear yet to me that anything in the Wikipedia entry for Gibbs phenomenon suggests that that topic was somehow the motivation for defining Hilbert spaces.&lt;/p&gt;&#xA;&lt;p&gt;Consequently, in searching online for information about this but historical trivia led me to this site through the question at the following link:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/6752/are-there-any-records-that-show-how-hilbert-came-to-invent-or-discover-hilbe&quot;&gt;Are there any records that show how Hilbert came to &amp;quot;invent&amp;quot; or &amp;quot;discover&amp;quot; Hilbert spaces?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The answers and comments there don’t seem to suggest either that J. W. Gibbs invented Hilbert spaces, so my question remains.&lt;/p&gt;&#xA;" OwnerUserId="19211" LastActivityDate="2023-10-24T09:33:48.937" Title="Did J. W. Gibbs “invent” Hilbert spaces before Hilbert formulated the notion of such spaces?" Tags="|mathematics|physics|hilbert|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15849" PostTypeId="1" CreationDate="2023-10-24T12:06:01.627" Score="4" ViewCount="178" Body="&lt;p&gt;After looking at some school sources in French, it is common to provide the various number sets in the following order&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathbb{N}\subset \mathbb{Z}\subset\mathbb{D}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$&lt;/span&gt; are the usual natural numbers, integers, rationals, reals and complex, respectively.&lt;/p&gt;&#xA;&lt;p&gt;However I am puzzled by the &lt;span class=&quot;math-container&quot;&gt;$\mathbb D$&lt;/span&gt; which represents the &lt;em&gt;décimaux rélatifs&lt;/em&gt; (literally &amp;quot;relative decimals&amp;quot;, a better term might be terminating decimals). This seems to refer to rationals with a finite sequence of digits in decimal notation (any integer divided by a power of 10, for example 1/3 is not included).&lt;/p&gt;&#xA;&lt;p&gt;Here are some examples:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://jeretiens.net/ensembles-de-nombres-mathematiques/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://jeretiens.net/ensembles-de-nombres-mathematiques/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://clg-monnet-briis.ac-versailles.fr/IMG/pdf/les_nombres_entiers_et_rationnels_cours_.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://clg-monnet-briis.ac-versailles.fr/IMG/pdf/les_nombres_entiers_et_rationnels_cours_.pdf&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.logamaths.fr/ensemble-d-des-nombres-decimaux-relatifs/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.logamaths.fr/ensemble-d-des-nombres-decimaux-relatifs/&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;What was the need of making the difference between rationals and terminating/relative decimals? Who were the first authors to discuss this difference? Was it a Frenchman?&lt;/p&gt;&#xA;&lt;p&gt;According to French Wikipedia, &lt;em&gt;&lt;a href=&quot;https://fr.wikipedia.org/wiki/Nombre_d%C3%A9cimal&quot; rel=&quot;nofollow noreferrer&quot;&gt;nombre décimal&lt;/a&gt;&lt;/em&gt;, it might be related to French fascination for developing a metric system [unsourced]. But the article seems to mix the history of terminating decimals with the history of of decimal representation.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://books.google.com/ngrams/graph?content=d%C3%A9cimaux%20relatifs&amp;amp;year_start=1800&amp;amp;year_end=2019&amp;amp;corpus=fr-2019&amp;amp;smoothing=3&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ngram&lt;/a&gt; seems to indicate that this terminology became popular in the 1970s.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-10-25T11:56:11.570" LastActivityDate="2023-10-25T11:56:11.570" Title="Who proposed terminating decimals as a major set and why are them important in France?" Tags="|mathematics|number-theory|set-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15851" PostTypeId="1" CreationDate="2023-10-25T03:49:20.773" Score="0" ViewCount="64" Body="&lt;p&gt;I'm doing undergraduate research on the history of abstract algebra (specifically permutation groups) and the notion of symmetric groups in indigenous artwork has come up several times. Is anyone aware of evidence of this 'intrinsic' human desire to construct groups I could point to? Or maybe some natural phenomena that conform to the definition of group? I'd like to have specific examples I can explain (and possibly map some ideas of formalized group theory onto). Thanks!&lt;/p&gt;&#xA;&lt;p&gt;(I posted &lt;a href=&quot;https://math.stackexchange.com/questions/4793503/group-theory-in-non-european-subaltern-cultures&quot;&gt;this question&lt;/a&gt; on Math stack exchange too but I'm not sure how to move it over...sorry for the duplicate!)&lt;/p&gt;&#xA;" OwnerUserId="19216" LastEditorUserId="6609" LastEditDate="2023-10-25T07:12:22.980" LastActivityDate="2023-10-25T07:12:22.980" Title="Group theory in non-European/subaltern cultures?" Tags="|mathematics|abstract-algebra|mathematics-social-history|group-theory|ancient-india|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15852" PostTypeId="1" AcceptedAnswerId="15854" CreationDate="2023-10-25T11:38:23.293" Score="4" ViewCount="350" Body="&lt;p&gt;How did Scott and Amundsen detect the direction to the South Pole during their expedition? How did they determine the exact South Pole on reaching there?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Comparison_of_the_Amundsen_and_Scott_expeditions&quot; rel=&quot;nofollow noreferrer&quot;&gt;Comparison of the Amundsen and Scott expeditions&lt;/a&gt; doesn't give this information.&lt;/p&gt;&#xA;" OwnerUserId="10548" LastActivityDate="2023-11-23T09:59:47.367" Title="How did Scott and Amundsen detect the South Pole?" Tags="|astronomy|earth-sciences|geography|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15853" PostTypeId="1" CreationDate="2023-10-25T11:45:11.027" Score="0" ViewCount="165" Body="&lt;p&gt;I read &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0465050743&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Edward Frenkel's Love and Math&lt;/a&gt;. But reanding this book made me wonder about origin of the concept of the sheaf used in algebraic geometry. I think the conclusion that I came to in the process of reading this book is that the concept of the Langlands program, which is a fusion of number theory, curves over finite fields, and Riemann surface. please see the below paragraphs.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In 1940, during the war, Andre Weil was imprisoned in France for refusing to serve in the army. As the obituary published in &lt;em&gt;The Economist&lt;/em&gt; put it. While in prison, Weil wrote a letter to his sister Simone Weil, a famous philosopher and humanist. This letter is a remarkable document; in it, he tries to explain in fairly elementary terms the &amp;quot;big picture&amp;quot; of mathematics as he saw it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The connection between Riemann surfaces and curves over finite fields should now be clear: both come from the same kind of equations, but we look for solutions in different domains, either finite fields or complex numbers. On the other hand, &amp;quot;any argument or result in number theory can be translated, word for word, &amp;quot;to curves over finite fields, as Weil put in his letter. Weil's idea was therfore that curves over finite fields are the objects that intermediate between number theory and Riemann surfaces.Thus, we find a bridge, between number theory and Riemann surfaces, and that is the theory of algebraic curves over finite fields. In other words, we have three parallel tracks or columns: &lt;em&gt;Number Theory, Curves over Finite Fields, Riemann Surfaces&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And this book says that automorphic function's geometric likeness is sheaf similar to the fundamental group of the Riemann surface acting as Galois group.&lt;/p&gt;&#xA;&lt;p&gt;In the end, I think sheaf's motivation is Weil's thinking. This conclusion is right? If it is wrong, Could you explain background of Why Jean pierre Serre thought he should develop the sheaf theory?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-10-27T11:17:23.577" LastActivityDate="2023-10-27T11:17:23.577" Title="Why Serre need to develop the concept of &quot;sheaf theory&quot; in algebraic geometry?" Tags="|terminology|algebraic-geometry|" AnswerCount="0" CommentCount="1" ClosedDate="2023-11-04T11:30:06.117" ContentLicense="CC BY-SA 4.0" />
  <row Id="15854" PostTypeId="2" ParentId="15852" CreationDate="2023-10-25T19:00:10.740" Score="5" Body="&lt;p&gt;I'm certain that Scott and Amundsen didn't detect the exact South Pole but more less reached a position that explorers would think was close enough to count.&lt;/p&gt;&#xA;&lt;p&gt;One technique they probably used was to measure their latitude and find that were, for example, about 1,000 feet from the pole.  Then walk forward for about 2,000 feet and measure their latitude again.  If it was close to the previous measurement they should have walked closer to the pole than either of those two points.&lt;/p&gt;&#xA;&lt;p&gt;Then they could turn at a right angle, walk another 2,000 feet, measure latitude, walk another 2,000 feet measure latitude, and son on until they had walked approximately 8,000 feet along the sides of a rough square.&lt;/p&gt;&#xA;&lt;p&gt;If each of the corner points had a latitude north of the south pole and those latitudes were close to each other, they could deduce that the south pole was within the square and then walk as close to the center of the square as they could measure and try measuring their latitude again, and maybe walk in a much smaller square around the south pole.&lt;/p&gt;&#xA;&lt;p&gt;How would polar explorers measure latitude?  With the same sorts of instruments that navigators at see and surveyors on land measured latitude.  Measuring latitude is necessary for ships to reach their proper destinations without wasting time and not run into rocks, and for surveyors to put the borders of parcels of land at the right positions.&lt;/p&gt;&#xA;&lt;p&gt;I think they would probably use a sextant to measure the height of the Sun above the horizon.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Professional sextants use a click-stop degree measure and a worm adjustment that reads to a minute, 1/60 of a degree. Most sextants also include a vernier on the worm dial that reads to 0.1 minute. Since 1 minute of error is about a nautical mile, the best possible accuracy of celestial navigation is about 0.1 nautical miles (190 m). At sea, results within several nautical miles, well within visual range, are acceptable. A highly skilled and experienced navigator can determine position to an accuracy of about 0.25-nautical-mile (460 m).&lt;a href=&quot;https://en.wikipedia.org/wiki/South_Pole#/media/File:Geographic_Southpole_crop.jpg&quot; rel=&quot;noreferrer&quot;&gt;4&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Sextant&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Sextant&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I think they would probably use a portable chronometer to measure what time it was at the longitude which the chronometer was set for - when they got close the north or south pole it wouldn't matter much which longitude the chronometer was set for.&lt;/p&gt;&#xA;&lt;p&gt;Show they would look up the apparent height of the Sun above the horizon at that time of day at various latitudes and find the latitude which corresponded to the height of the Sun at that time, which would be the latitude they were in.&lt;/p&gt;&#xA;&lt;p&gt;And of course there was a limit to the accuracy of their chronometers and their sexants and their techniques of using them.&lt;/p&gt;&#xA;&lt;p&gt;The published first person accounts of Peary and those who later reached the North pole, and Amundsen and those who reached the south pole after him, and Scott's  diary entry for the day they reached the south pole should record some details of how they measured their latitude.&lt;/p&gt;&#xA;&lt;p&gt;Here is a link to a photo of the ceremonial south pole erected at the Amundsen-Scott South Pole Station.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/File:Ceremonial_South_Pole.jpg&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/File:Ceremonial_South_Pole.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, the ceremonial pole is not at the exact geographical south pole.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/South_Pole#/media/File:SPSM.05.jpg&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/South_Pole#/media/File:SPSM.05.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/South_Pole#/media/File:Geographic_Southpole_crop.jpg&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/South_Pole#/media/File:Geographic_Southpole_crop.jpg&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;It would be interesting to plot the approximate paths of Amundsen and Scott on a map of the south pole base.&lt;/p&gt;&#xA;&lt;p&gt;And of course historians of polar exploration should be able to give much better answers.&lt;/p&gt;&#xA;" OwnerUserId="12708" LastActivityDate="2023-10-25T19:00:10.740" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15855" PostTypeId="1" CreationDate="2023-10-25T20:23:09.660" Score="2" ViewCount="196" Body="&lt;p&gt;Indian mathematicians (e.g., Brahmagupta in the 6th century) developed the idea of 0 as more than a placeholder.&lt;/p&gt;&#xA;&lt;p&gt;In 1202, Fibonacci wrote &amp;quot;These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0, which in Arabic is called zephyr, any number is written, as shown below.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Fibonacci seems to be saying that zero is not quite a number like the rest. But if he basing this off the Indians work from centuries before, why is this? Wasn't it well-developed by this point?&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance!&lt;/p&gt;&#xA;" OwnerUserId="19226" LastActivityDate="2023-10-26T16:22:05.817" Title="Did Fibonacci not grasp the idea of zero?" Tags="|mathematics|mathematics-social-history|numbers|zero|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15856" PostTypeId="2" ParentId="15855" CreationDate="2023-10-26T16:22:05.817" Score="5" Body="&lt;p&gt;The nonzero digits are also numbers that were considered by the Greeks as existing entities (the case for 1 was seen as somewhat special as it is not composed of other entities; Simon Stevin notably fought against this attitude concerning the singularity of 1).  Since 0 is not a number of anything, it was naturally viewed as a fictional entity not on par with &lt;span class=&quot;math-container&quot;&gt;$1,\ldots,9$&lt;/span&gt;.  Leibniz viewed entities familiar to the Greeks as the only possible ones; other entities such as surds, imaginary roots, infinitesimals, negatives (surely including 0) were fictions.  A detailed analysis appears in &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#23g&quot; rel=&quot;noreferrer&quot;&gt;this publication&lt;/a&gt;.  There is nothing surprising about Fibonacci's attitude here.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-10-26T16:22:05.817" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15859" PostTypeId="1" AcceptedAnswerId="15867" CreationDate="2023-10-27T13:38:19.253" Score="4" ViewCount="159" Body="&lt;p&gt;Some while ago we had a question about mathematicians patenting their work &lt;a href=&quot;https://hsm.stackexchange.com/questions/15406/examples-of-mathematicians-who-applied-to-patent-their-work&quot;&gt;Examples of mathematicians who applied to patent their work&lt;/a&gt; I was about to answer when I realised I needed to find a reputable source for my answer. I failed.&lt;/p&gt;&#xA;&lt;p&gt;Background&lt;/p&gt;&#xA;&lt;p&gt;Some time in the 1970s I attended a meeting of the British Association for the Advancement of Science in Nottingham, UK. I attended a talk on sphere packing given, I am almost sure, by John Conway. It began with a crate of fruit to demonstrate lower dimensions but then proceeded into higher dimensions. In the course of that part I remember him mentioning that someone had applied for a patent for a method of information transmission based on 8-dimnesional sphere packing.&lt;/p&gt;&#xA;&lt;p&gt;Searching for further information is made slightly more difficult because internet searches tend to be dominated by the work of &lt;a href=&quot;https://en.wikipedia.org/wiki/Maryna_Viazovska&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maryna Viazovska&lt;/a&gt; for which she was awarded the Fields Medal (For the proof that the&#xA;&lt;span class=&quot;math-container&quot;&gt;$E_{8}$&lt;/span&gt; lattice provides the densest packing of identical spheres in 8 dimensions). However I have also found that one method of error correction in information transmission, &lt;a href=&quot;https://en.wikipedia.org/wiki/Binary_Golay_code&quot; rel=&quot;nofollow noreferrer&quot;&gt;the binary Golay code&lt;/a&gt; is based on the &lt;a href=&quot;https://en.wikipedia.org/wiki/Leech_lattice&quot; rel=&quot;nofollow noreferrer&quot;&gt;Leech lattice&lt;/a&gt; which is the basis for the sphere packing solution in 24-dimensions.&lt;/p&gt;&#xA;&lt;p&gt;What I have been unable to establish is whether anyone did indeed try to patent the 8-dimensional solution or whether my memory after all those years is failing me.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2023-11-05T14:14:04.493" Title="Did anyone apply for a patent based on sphere packing?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15860" PostTypeId="1" CreationDate="2023-10-28T02:37:57.117" Score="3" ViewCount="218" Body="&lt;p&gt;Russ Roberts (host of EconTalk podcast) claimed on his (&lt;a href=&quot;https://www.econtalk.org/nicholas-wapshott-on-samuelson-and-friedman/&quot; rel=&quot;nofollow noreferrer&quot;&gt;2021-08-16&lt;/a&gt;) podcast:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Before he [Milton Friedman] came along, ... no one believed that inflation was mainly a monetary phenomenon.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His guest Nicholas Wapshott agrees:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;until Milton, all Keynesians who ruled the ruled the roost, said that money didn't matter.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Roberts has repeatedly made similar claims on his podcast.)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;To me, the above claims seem like bizarre exaggerations of Friedman's influence and distortions of intellectual history.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that economists had always been well aware of the importance of money on inflation.&lt;/p&gt;&#xA;&lt;p&gt;One thing economists had not been quite as aware of was the important role of contractionary monetary policy in the Great Depression. Friedman and Schwartz (&lt;a href=&quot;https://en.wikipedia.org/wiki/A_Monetary_History_of_the_United_States&quot; rel=&quot;nofollow noreferrer&quot;&gt;1963&lt;/a&gt;) brought this to most economists' attention.&lt;/p&gt;&#xA;&lt;p&gt;Another thing was that some economists believed in was the Phillips curve trade-off between inflation and unemployment. Friedman (&lt;a href=&quot;https://www.aeaweb.org/aer/top20/58.1.1-17.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1968&lt;/a&gt;) (and others such as Phelps) helped to correct this.&lt;/p&gt;&#xA;&lt;p&gt;What I don't think is true is that &amp;quot;no one believed that inflation was mainly a monetary phenomenon&amp;quot; or &amp;quot;all Keynesians who ruled the ruled the roost, said that money didn't matter&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="18903" LastActivityDate="2023-11-03T13:02:27.657" Title="Is it true that before Milton Friedman, &quot;no one believed that inflation was mainly a monetary phenomenon&quot;?" Tags="|economics|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15861" PostTypeId="1" CreationDate="2023-10-28T17:17:19.147" Score="0" ViewCount="49" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Phosphorus#Discovery&quot; rel=&quot;nofollow noreferrer&quot;&gt;According to Wikipedia&lt;/a&gt;, Hennig Brand discovered phosphorous. Later, Robert Boyle desired to replicate Brand's discovery, but didn't know that urine was used - only that it involved something that &amp;quot;belonged to the body of man&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;That quote is apparently taken from the 1939 edition of &lt;em&gt;Mellor's Modern Inorganic Chemistry&lt;/em&gt;; however, I'm unable to access a copy of that. &lt;a href=&quot;https://archive.org/details/b29807761/page/704/mode/2up?ref=ol&amp;amp;view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;Looking at the 1933 edition&lt;/a&gt;, the quote doesn't seem to appear (though interestingly, does imply an important role for Boyle's assistant, A. G. Hanckwitz), so I can't verify the context of the quote.&lt;/p&gt;&#xA;&lt;p&gt;My question is: what substances from the human body did Boyle (or Hanckwitz) try before settling on urine? I'm not particularly knowledgeable about chemistry beyond school, never mind historical chemistry/alchemy, so I don't know if urine would've been known as the most plausible candidate, or just one of dozens of possibilities given the vagueness of the hint available. Is there any record, by Boyle or Hanckwitz, of how much trial and error this process involved?&lt;/p&gt;&#xA;" OwnerUserId="10862" LastActivityDate="2023-10-28T17:17:19.147" Title="What substances from the body did Boyle try when attempting to produce phosphorus?" Tags="|chemistry|alchemy|biochemistry|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15862" PostTypeId="1" CreationDate="2023-10-28T21:53:03.320" Score="2" ViewCount="86" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Gabriel%27s_horn&quot; rel=&quot;nofollow noreferrer&quot;&gt;Torricelli solid/Gabriel's Horn&lt;/a&gt; is defined as the rotation-invariant solid delimited by a hyperbola. It appears in &lt;em&gt;De solido hyperbolico acuto&lt;/em&gt; where Torricelli proves that it has a finite volume, despite the fact that it is infinite.&lt;/p&gt;&#xA;&lt;p&gt;Nowadays, that paradox appears in a different guise, as the fact that the area of that surface that bounds the Torricelli solid is infinite. In simple terms, the Torricelli trumpet cannot be painted but it can be filled with paint. This is for example how the story is told in Wikipedia.&lt;/p&gt;&#xA;&lt;p&gt;However, I do not know who made this observation regarding the area, and when.&lt;/p&gt;&#xA;&lt;p&gt;Using integral calculus, these facts can be given relatively straightforward proofs.&lt;/p&gt;&#xA;&lt;p&gt;Torricelli's proof that the solid has finite volume uses an approximation of the solid as a stack of “infinitesimally” small concentric cylinders. This is indeed enough to bound that volume.&lt;/p&gt;&#xA;&lt;p&gt;However, to evaluate the area, that would not be enough and to write a proof in Torricelli's style (or in the style of the early inventors of the infinitesimal calculus), it would be necessary to consider the solid as a stack of infinitesimall truncated cones.&lt;/p&gt;&#xA;" OwnerUserId="19241" LastEditorUserId="19241" LastEditDate="2023-10-28T21:54:29.847" LastActivityDate="2023-10-28T21:54:29.847" Title="Who evaluated the surface of the Torricelli solid/Gabriel's horn" Tags="|reference-request|calculus|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15863" PostTypeId="2" ParentId="8179" CreationDate="2023-10-30T03:36:35.663" Score="7" Body="&lt;p&gt;The term &lt;em&gt;elasticity&lt;/em&gt; was first used in print by Alfred Marshall (&lt;a href=&quot;https://www.jstor.org/stable/25163976&quot; rel=&quot;noreferrer&quot;&gt;1885&lt;/a&gt;, p. 260):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We may want to find a measure of what may be called &lt;em&gt;the elasticity of demand&lt;/em&gt;; that is, when a fall of price leads to an increase in the amount demanded, we may want to know the ratio in which the percentage by which the amount demanded has increased stands to the percentage by which the price has fallen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Marshall then elaborated on &lt;em&gt;elasticity&lt;/em&gt; in &lt;em&gt;Principles of Economics&lt;/em&gt; (1890, pp. &lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=njp.32101065216663&amp;amp;view=1up&amp;amp;seq=194&amp;amp;skin=2021&quot; rel=&quot;noreferrer&quot;&gt;162&lt;/a&gt; ff):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;his willingness to purchase the thing stretches itself out a great deal under the action of a small inducement: the elasticity of his demand, we may say, is great.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I agree with OP that &lt;em&gt;sensitivity&lt;/em&gt; would have been a better word.&lt;/p&gt;&#xA;&lt;p&gt;But it seems that Marshall somehow thought of &amp;quot;stretching&amp;quot; and so used the word &lt;em&gt;elasticity&lt;/em&gt; instead. His enormous influence is such that we are now stuck with this word.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Note that economists had been at least somewhat aware of the notion of elasticity, but it was Marshall who highlighted it and made it explicit.&lt;/p&gt;&#xA;&lt;p&gt;John Maynard Keynes (&lt;a href=&quot;https://academic.oup.com/ej/article-abstract/34/135/311/5282617&quot; rel=&quot;noreferrer&quot;&gt;1925&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I do not think that Marshall did economists any greater service than by the explicit introduction of the idea of &amp;quot;elasticity.&amp;quot; ...&lt;/p&gt;&#xA;&lt;p&gt;it is rather remarkable that the notion was not more clearly disentangled either by Mill or by Jevons. ...&lt;/p&gt;&#xA;&lt;p&gt;The way in which Marshall introduces Elasticity, without any suggestion that the idea is novel, is remarkable and characteristic. The field of investigation opened up by this instrument of thought is again one where the full fruits have been reaped by Professor Pigou rather than by Marshall himself. ...&lt;/p&gt;&#xA;&lt;p&gt;Mrs. Marshall tells me that he hit on the notion of elasticity, as he sat on the roof at Palermo shaded by the bath-cover in 1881, and was highly delighted with it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="18903" LastEditorUserId="18903" LastEditDate="2023-10-30T03:42:49.300" LastActivityDate="2023-10-30T03:42:49.300" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15864" PostTypeId="2" ParentId="15860" CreationDate="2023-10-30T16:06:45.613" Score="1" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;To me, the above claims seem like bizarre exaggerations of Friedman's&#xA;influence and distortions of intellectual history.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that economists had always been well aware of the&#xA;importance of money on inflation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I agree with you, and Russ Robert's statement, put in this way, is not only exaggerated but bluntly wrong (maybe he refers only to economists after Keynes).&lt;/p&gt;&#xA;&lt;p&gt;The famous Milton Friedman's statement, that &lt;em&gt;inflation is a monetary phenomenon&lt;/em&gt;, is a way to express the so-called &lt;strong&gt;&lt;em&gt;quantity theory of money&lt;/em&gt;,&lt;/strong&gt; also denoted as  &lt;em&gt;&lt;strong&gt;neutrality of money&lt;/strong&gt;&lt;/em&gt;  or by the expression &lt;em&gt;&lt;strong&gt;money is a veil&lt;/strong&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Nowadays &amp;quot;Veil of Money&amp;quot; and &amp;quot;Neutrality of Money&amp;quot; are shorthand&#xA;expressions for the basic quantity-theory proposition that it is only&#xA;the absolute price level of an economy, and not relative prices and&#xA;the rate of interest, and hence real outputs, that is affected by&#xA;changes in the quantity of money.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And, on the contrary of Russ Robert's statement, &lt;strong&gt;quantity theory/neutrality of money, was the most widespread monetary theory of Classical and Neoclassical economics, toward which Keynes addressed his criticism.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The quantity theory of money&lt;/strong&gt; states that (in the long-medium run or also in the short run, according to different views) &lt;strong&gt;the quantity of money in the economy affects only the nominal variables, in particular the general price level, and not the real variables:&lt;/strong&gt; in this sense, money is &lt;em&gt;&lt;strong&gt;neutral&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The quantity theory of money, therefore,  via the so-called &lt;em&gt;&lt;strong&gt;equation of exchange&lt;/strong&gt;&lt;/em&gt; , states that the general price level of goods and services is directly proportional to the amount of money in circulation (i.e., the money supply), and that the causality runs from money to prices. This implies that the theory can  explain inflation, and that 'inflation is a monetary phenomenon'.&lt;/p&gt;&#xA;&lt;p&gt;As a consequence, monetary policy, that is a change of the money supply by monetary authorities, is ineffective, at least in the medium-short run.&lt;/p&gt;&#xA;&lt;p&gt;From a widespread textbook of macroeconomics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...in the medium run, real variables, such as production,&#xA;unemployment or real rate of interest, are independent of monetary&#xA;policy. The only variables that monetary policy determines are the&#xA;rate of inflation and the nominal interest rate. In the medium run, a&#xA;higher rate of growth of money leads exclusively to a higher&#xA;inflation and higher nominal interest rates. The fact that monetary&#xA;policy doesn't affect real variables [...] is defined as &lt;em&gt;neutrality&#xA;of money.&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;The fundamental pioneering work about the contemporary version of quantity theory&lt;/strong&gt; (and a seminal work of &lt;a href=&quot;https://en.wikipedia.org/wiki/Monetarism&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;monetarism&lt;/em&gt;&lt;/a&gt;) is the article by Milton Friedman,&#xA;'The Quantity Theory of Money. A restatement', 1956.&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A review of the quantity theory of money has been made by Friedman in a  subsequent article, &lt;em&gt;Quantity Theory of Money&lt;/em&gt;, (1989). &lt;span class=&quot;math-container&quot;&gt;$^4$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Friedman opens his article with a quote from David Hume (1711-1796):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Lowness of interest is generally ascribed to plenty of money. But&#xA;augmentation [of the quantity of money] has no other effect than to&#xA;heighten the price of labour and commodities ... In the progress&#xA;toward these changes, the augmentation may have some influence, by&#xA;exciting industry, but after the prices are settled ... it has no&#xA;manner of influence. […] (David Hume, 1752)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Therefore, it is clear also from  Friedman’s article, that &lt;strong&gt;the quantity theory/neutrality of money, is a concept that existed well before Friedman&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In particular, the quantity theory was maintained by &lt;a href=&quot;https://it.wikipedia.org/wiki/David_Hume&quot; rel=&quot;nofollow noreferrer&quot;&gt;David Hume(1711-1796)&lt;/a&gt;: a  &lt;em&gt;classical  locus&lt;/em&gt; of these  analyses is the  Hume’s essay &lt;a href=&quot;https://books.mondadoristore.it/Of-Money-David-Hume/eae978147530607/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Of money&lt;/em&gt; &lt;/a&gt;, where he considers that money is neutral in the medium run, even if it may influence real variables during the transition.&lt;/p&gt;&#xA;&lt;p&gt;‘Pioneers’ of the quantity theory of money are considered &lt;a href=&quot;https://en.wikipedia.org/wiki/Quantity_theory_of_money&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Mun (1571-1641)&lt;/a&gt; and &lt;a href=&quot;https://plato.stanford.edu/entries/bodin/#BodiTheoEconFina&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Bodin (1529-1596)&lt;/a&gt;, who concentrated on the question of the abundance of gold and silver which he considered the principal and singular cause for the high prices of his era.&lt;/p&gt;&#xA;&lt;p&gt;In the 16th century&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Martin de Azpilcueta Navarro in Salamanca in 1556 and Jean Bodin in France in 1568&#xA;identified the inflow of silver from the Spanish colonies of Mexico and Upper Peru&#xA;as the cause of the rise in prices and depreciation of silver throughout Europe.&lt;span class=&quot;math-container&quot;&gt;$^5$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;But, &lt;strong&gt;after this more ancient works, the quantity theory of money became  a widespread theory that synthetizes the monetary doctrine of &lt;a href=&quot;https://en.wikipedia.org/wiki/Classical_economics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;classical&lt;/em&gt;&lt;/a&gt;,  and &lt;a href=&quot;https://en.wikipedia.org/wiki/Neoclassical_economics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;neoclassical&lt;/em&gt;&lt;/a&gt; economists&lt;/strong&gt;,  well before Keynes, according to whom money is &lt;strong&gt;a simple medium of exchange, and does not affect real phenomena of the economic system.&lt;/strong&gt;, but determines the general level of prices.&lt;/p&gt;&#xA;&lt;p&gt;Robert Clower writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;“Hume’s essay provides a balanced  example of what has become known as&#xA;the &lt;em&gt;quantity theory of money&lt;/em&gt;. Subsequent writers haven’t always&#xA;formulated or interpreted this theory so wisely […] [&lt;em&gt;Hume maintained&#xA;that money doesn’t affect real economy in the long run, but can affect it in the short run, my note&lt;/em&gt;]&#xA;and maintained that money is only a ‘veil’ […] This wrong version of&#xA;Hume's theory has occupied  a prominent place in the thought of traditional&#xA;economists for a large part of modern history. This version underlies&#xA;what we nowadays call &lt;em&gt;the simple quantity theory&lt;/em&gt;, the central&#xA;statement of which is that the total amount of means of payment&#xA;rules the absolute  level of monetary prices, but does not affect the real&#xA;exchange ratios between the other goods. […]&lt;/p&gt;&#xA;&lt;p&gt;Anyway, until the publication in 1936 of John Maynard Keynes’&#xA;&lt;em&gt;The General Theory of Employment, Interest and Money&lt;/em&gt;, the most part of economists accepted without discussion that all the economic problems&#xA;of practical relevance could be studied […] assuming that money is a&#xA;‘veil’ both in the long and in the short run.” &lt;span class=&quot;math-container&quot;&gt;$^6$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;A classical economist&lt;/strong&gt; that dealt with the quantity theory of money is the British economist &lt;a href=&quot;https://en.wikipedia.org/wiki/Nassau_William_Senior&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Nassau W. Senior&lt;/strong&gt;(1790-1864)&lt;/a&gt;, in his essay &lt;em&gt;On the Quantity and Value of Money&lt;/em&gt; (in Clower, &lt;em&gt;cit&lt;/em&gt;.), where he analyzes  the doctrine of another &lt;strong&gt;classical economist,&lt;/strong&gt; &lt;a href=&quot;https://it.wikipedia.org/wiki/John_Stuart_Mill&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;John S. Mill(1806-1873)&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mill (in the spirit of Hume) made the famous statements that &amp;quot;the relations of commodities to one  another remain unaltered by money&amp;quot; and that &amp;quot;there cannot, in short, be intrinsically a more  insignificant thing, in the economy of society, than money&amp;quot;. &lt;span class=&quot;math-container&quot;&gt;$^7$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also the classical economist &lt;a href=&quot;https://en.wikipedia.org/wiki/David_Ricardo&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;David Ricardo&lt;/strong&gt; (1772-1823)&lt;/a&gt; made a presentation of the quantity theory in his essay &lt;em&gt;The High Price of Bullion&lt;/em&gt;(1810) and in his work &lt;em&gt;On the Principles of Political Economy and Taxation&lt;/em&gt; (1817).&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Subsequently, the quantity theory of money was sustained by &lt;strong&gt;neoclassical economist as Fisher, Wicksell, Marshall, Pigou.&lt;/strong&gt; The quantity theory  was widespread and commonly accepted:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We can say that, until the 30s, the quantity theory was the theory of&#xA;money for excellence, actually shared by everyone. This doesn’t mean&#xA;that the consensus was unanimous: in fact, it was repeatedly&#xA;criticized by many minor authors. But an alternative theory was never&#xA;proposed, and all the leading economists agreed about one of the three&#xA;versions of the quantity theory […]: the &lt;em&gt;transactions approach&lt;/em&gt;,&#xA;described  in the work of Fisher &lt;em&gt;The Purchasing Power of Money&lt;/em&gt;&#xA;(1911); the &lt;em&gt;monetary balances approach&lt;/em&gt;, formulated by Marshall,&#xA;Walras, Wicksell, Keynes [&lt;em&gt;before his own theory, my note&lt;/em&gt;]: and the &lt;em&gt;income approach&lt;/em&gt;, that lead to the&#xA;explicit introduction of the concept of velocity of money with respect&#xA;to income in the Pigou’s work &lt;em&gt;Industrial Fluctuations&lt;/em&gt;.&lt;span class=&quot;math-container&quot;&gt;$^8$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&lt;strong&gt;The equation of exchanges&lt;/strong&gt;&lt;/em&gt;. The contemporary form  of the so-called &lt;em&gt;&lt;strong&gt;equation of exchanges&lt;/strong&gt;&lt;/em&gt;, on which is based the contemporary quantity theory of money,  was popularized by  the American economist &lt;strong&gt;Irving Fisher&lt;/strong&gt; (1867-1947) in 1911, in its work &lt;em&gt;The Purchasing Power of Money&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Attempts to formulate mathematically the relations&#xA;just presented verbally date back several centuries (Humphrey, 1984). They&#xA;consist of creating identities equating a flow of money payments to a flow of&#xA;exchanges of goods or services. The resulting quantity equations have proved a&#xA;useful analytical device and have taken different forms as quantity theorists have&#xA;stressed different variables.&#xA;[…]The most famous version of the&#xA;quantity equation is doubtless the transactions version formulated by Simon&#xA;Newcomb (1885) and popularized by Irving Fisher (1911):&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$MV=PT \qquad (1)$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;or, if we desire to distinguish currency from deposit transactions&#xA;[&lt;em&gt;here &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; has a different meaning than in &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt;&lt;/em&gt;, my note]&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$MV + M'V' = PT \qquad(1’) ^9 $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; is the quantity of money in the economy, &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is the&#xA;&lt;em&gt;velocity of circulation&lt;/em&gt; of money, that is the average number of times a unit of money changes hand in the course of effecting a given period's transactions of currency, &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is the general level of prices, &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is the amount of transactions in the economy in the  given period.&lt;/p&gt;&#xA;&lt;p&gt;In the  more usual modern form, the &lt;em&gt;income version&lt;/em&gt;, introduced by Pigou (1927)&lt;span class=&quot;math-container&quot;&gt;$^{10}$&lt;/span&gt;, real income &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; instead of gross transaction &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is used:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$MV=PY \qquad(2),$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;as national or social accounting has stressed income transactions rather than gross transactions. Equations &lt;span class=&quot;math-container&quot;&gt;$(1)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt; differ from each other because the volume of transactions in the economy includes intermediate goods and existing assets, in additions to final goods and services.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;As such, equations (1), (1’) and (2) are  simply  identities&lt;/strong&gt;, that say that the product of the quantity of money &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; times its velocity of circulation &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; (that is the money exchanged in a period) equals the value of exchanges of goods, services and securities  in the period, that is the overall transactions times the weighted average of prices (or monetary income in the form &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;To sustain  the quantity theory of money, and thus to consider money as a  &lt;em&gt;veil&lt;/em&gt; over real phenomena, proposing a cause-effect relationship between the quantity of money and the price level,  Fisher &lt;strong&gt;had to assume that the velocity of circulation &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; and the volume of trade &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; are constant&lt;/strong&gt;.&#xA;He recognized that both fluctuate over the business cycle but they always tend to return to an equilibrium level. He writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We now proceed to show that (except during the transition period) the&#xA;volume of trade, like the velocity of money, is independent of the&#xA;quantity of money. An inflation of the currency cannot increase the&#xA;products of farms and factors, nor the speed of freights or ships. The stream of business depends on natural resources and technical conditions, not on the quantity of money.[…]&lt;/p&gt;&#xA;&lt;p&gt;We may now restate , then, in what causal sense the quantity theory is true. It is true in the sense that &lt;em&gt;one of the normal effects of an increase of the quantity of money is an exactly proportional increase in the general level of prices&lt;/em&gt;.&lt;span class=&quot;math-container&quot;&gt;$^{11}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Economists &lt;strong&gt;Alfred Marshall, A.C. Pigou, and John Maynard Keynes&lt;/strong&gt; (before  his &lt;em&gt;The General Theory of Employment, Interest and Money&lt;/em&gt;, 1936) of the Cambridge school, took a slightly different approach to the quantity equation, focusing on money demand instead of money supply.&#xA;They stated a version of  the equation of exchange that is called the &lt;em&gt;&lt;strong&gt;Cambridge Equation&lt;/strong&gt;&lt;/em&gt;. This version of the equation is&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$M=k\times PY$$&lt;/span&gt;\qquad (3),&lt;/p&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the fraction of income that people collectively wish to hold in the form of cash balances. Marshall’s &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is the reciprocal of the velocity of circulation &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;. &lt;span class=&quot;math-container&quot;&gt;$^{12}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Equation &lt;span class=&quot;math-container&quot;&gt;$(3$&lt;/span&gt; is aritmetically equivalent to equation &lt;span class=&quot;math-container&quot;&gt;$(2)$&lt;/span&gt;, but it rests on different ideas about the role of money in the economy.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Knut_Wicksell&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Knut Wicksell&lt;/strong&gt;(1851-1926)&lt;/a&gt;also sustained the quantity theory of money, even if he proposed a different &lt;em&gt;transmission mechanism&lt;/em&gt; from money to the level of prices, called &lt;em&gt;indirect transmission mechanism&lt;/em&gt;&lt;span class=&quot;math-container&quot;&gt;$^{13}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In conclusion, to sustain  the quantity theory of money, neoclassical economists had to demonstrate that &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; (or &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;) are constant, at least in the long run.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is usually considered as a reflection of payments habits, and therefore it is an institutional factor not subjected to change, at least in the short  term.&#xA;The &lt;strong&gt;key variable, however is the level of income &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;&lt;/strong&gt; (or of transactions): the neoclassical economists thought that, &lt;em&gt;at least in the medium-long run, the level of income is at its full employment level, so that it is a fixed variable&lt;/em&gt;, and a change in the quantity of money can therefore affect only the general price level  &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The belief that the economic system is able to adjust  spontaneously toward full employment, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Say%27s_law&quot; rel=&quot;nofollow noreferrer&quot;&gt;Say’s Law&lt;/a&gt;, which states that &lt;em&gt;supply creates its own demand&lt;/em&gt;,  so that failures of effective demand don’t appear in the economy at least in the long run, were  the  reasons why neoclassical economists believed in the quantity theory of money.&lt;/p&gt;&#xA;&lt;p&gt;Keynes criticized the quantity theory, in particular  because he maintained that it is not true that the economics system spontaneously adjusts to reach an equilibrium of full employment, but can rest in a state of unemployment. In this case &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; is not a fixed variable, but can change as a consequence of a change in the quantity of money (his most famous work where he exposes his theory of unemployment is the &lt;em&gt;General Theory of Employment, Interest and Money&lt;/em&gt; 1936). &lt;span class=&quot;math-container&quot;&gt;$^{14}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; &lt;a href=&quot;https://dokumen.tips/documents/in-search-of-the-veil-of-money-and-the-neutrality-of-money-a-note-on-the.html?page=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Don Patinkin and Otto Steiger, In Search of the &amp;quot;Veil of Money&amp;quot; and the &amp;quot;Neutrality of Money&amp;quot;: A Note on the Origin of Terms Author(s), The Scandinavian Journal of Economics, Vol. 91, No. 1 (Mar., 1989), pp. 131-146 (&lt;/a&gt;&lt;/sup&gt;&#xA;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt;Blanchard O., Amighini A., Giavazzi F., &lt;em&gt;Macroeconomics. A European perspective&lt;/em&gt;, Pearson, 2023, Chap. 9, sec. 2.  &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;&lt;sup&gt;https://scirp.org/reference/referencespapers.aspx?referenceid=2244481&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^4$&lt;/span&gt;&lt;sup&gt;Friedman, M. (1989). Quantity Theory of Money. In: Eatwell, J., Milgate, M., Newman, P. (eds) Money. The New Palgrave. Palgrave Macmillan, London. &lt;a href=&quot;https://doi.org/10.1007/978-1-349-19804-7_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-1-349-19804-7_1&lt;/a&gt;   &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^5$&lt;/span&gt; &lt;sup&gt;  Dimand, R.W. (2010). Monetary Economics, History of. In: Durlauf, S.N., Blume, L.E. (eds) Monetary Economics. The New Palgrave Economics Collection. Palgrave Macmillan, London.  &lt;a href=&quot;https://doi.org/10.1057/9780230280854_24&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1057/9780230280854_24&lt;/a&gt;&#xA;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^6$&lt;/span&gt;&lt;sup&gt; Clower R. , Introduction&#xA;to Clower R. (ed.), &lt;em&gt;Monetary Theory&lt;/em&gt;, 1969, Penguin Modern Economic Readings. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^7$&lt;/span&gt; &lt;sup&gt; In Don Patinkin and Otto Steiger, &lt;em&gt;cit.&lt;/em&gt;, p. 131.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^8$&lt;/span&gt; &lt;sup&gt;Blaug M., &lt;em&gt;Storia e critica della teoria economica&lt;/em&gt;, Boringhieri, 1970, p. 755,&#xA;(English edition: Blaug M., &lt;em&gt;Economic Theory in Retrospect&lt;/em&gt;,&#xA;Irwin-Homewood, 1968.)&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^9$&lt;/span&gt;&lt;sup&gt;Friedman, M. (1989), &lt;em&gt;cit.&lt;/em&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^{10}$&lt;/span&gt; &lt;sup&gt; Pigou A. C., (1927) &lt;em&gt;Industrial Fluctuations&lt;/em&gt;. 2nd edn., London: Macmillan, 1929, reprinted, new York: A.M. Kelley, 1967.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^{11}$&lt;/span&gt; &lt;sup&gt;Brue l.S, Grant R.R., &lt;em&gt;The Evolution of Economic Thought&lt;/em&gt;, Thomson, 2007, p. 311&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^{12}$&lt;/span&gt;&lt;sup&gt; Blaug M., &lt;em&gt;ibid.&lt;/em&gt;, p. 757.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^{13}$&lt;/span&gt; &lt;sup&gt; Blaug M., &lt;em&gt;ibid.&lt;/em&gt;, p. 760.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^{14}$&lt;/span&gt; &lt;sup&gt;Keynes J. M., &lt;em&gt;The General Theory of Employment, Interest and Money&lt;/em&gt;, BN publishing, 2008 (first published by Macmillan Cambridge University Press)&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-11-03T13:02:27.657" LastActivityDate="2023-11-03T13:02:27.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15865" PostTypeId="1" AcceptedAnswerId="15866" CreationDate="2023-10-30T16:49:42.030" Score="0" ViewCount="123" Body="&lt;p&gt;When I started learning about Laws related to &amp;quot;constant proportions&amp;quot; and other hypothesis like &amp;quot;all matter is composed of smallest indivisible particles&amp;quot;, I didn't notice it was not so obvious for people earlier to even think about it like we do, so I dug up a little bit and then started asking all sorts of question that how did people in those times even proceed with such propositions, Here, I write a Quick recap/Summary while also giving detailed explanations about some &amp;quot;Significant&amp;quot; points, I am answering This to share what I have found From original sources and my understandings, That I gathered in My notes&lt;/p&gt;&#xA;" OwnerUserId="17776" LastEditorUserId="17776" LastEditDate="2023-10-30T17:24:12.140" LastActivityDate="2023-11-01T07:18:00.960" Title="A summary of Major events in Pre-Modern Atomist Philosophy of Chemistry" Tags="|philosophy-of-science|chemistry|atomic-theory|" AnswerCount="1" CommentCount="4" ClosedDate="2023-11-01T14:16:44.350" ContentLicense="CC BY-SA 4.0" />
  <row Id="15866" PostTypeId="2" ParentId="15865" CreationDate="2023-10-30T16:49:42.030" Score="0" Body="&lt;p&gt;A summary according to timeline for pre-Daltonian chemistry related to Atomism&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Democritus (460-370 BCE):He was an ancient Greek philosopher, whose mentor was &amp;quot;Leucippus&amp;quot;, Democritus was central figure in development of philosophical atomism, he reasoned that if the stone were to be continually cut into smaller and smaller pieces then; at some point, there would be a piece which would be so small as to be indivisible. He called these small pieces of matter &amp;quot;Atomos,&amp;quot; the Greek word for indivisible. Democritus, theorized that atoms were specific to the material which they composed, less known, An Indian Philosopher named &amp;quot;Kanad&amp;quot; also Proposed an atomist view of matter, and taught so in his school of thoughts.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Aristotle and Plato: One of the best ancient Greek philosophers were against the idea that everything was made up of atoms, they believed that everything is infinitely cuttable, meaning Materials are eternal and can be cut any number of times, he (Aristotle) also believed that Four core elements can be transformed into one another, namely earth, water, air and fire.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The atomist theory got its noticeable revival in &lt;span class=&quot;math-container&quot;&gt;$17^{th}$&lt;/span&gt; century when In 1643, Evangelista Torricelli, an Italian mathematician and pupil of Galileo, showed that air had weight and was capable of pushing down on a column of liquid mercury (thus inventing the barometer), Daniel Bernoulli (Swiss) proposed that gases could be composed of particles too small to be seen by eyes, loosely packed in empty space..&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Joseph Priestley, who in 1770s started experimenting with red calx. Specifically, mercury calx, a red colored stone. When he heated the stone(using magnifying glass and sun rays), it not only converted to silver liquid metal, But also formed a strange gas, He conducted The famous experiment of Mouse under jar, in which he placed a mouse under a jar and observed the time required for its death, and then collected the strange gas, and placed the mouse in container with this strange gas, and found that it lived longer, also when flames were burnt in this gas, It burned with vigorous flame, He called the gas &amp;quot;dephlogisticated air&amp;quot;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Antoine Lavoisier: He belonged to a wealthy family in France, He was a firm believer in Accurate calculations and its importance, he is well known for the shift from qualitative to quantitative analysis in Chemistry, He performed a series of experiments on This &amp;quot;dephlogisticated air&amp;quot;, and found that it changed some substances acidic, also during that period it was known that acids react with metal to form a highly inflammable gas, called phlogiston, He mixed these two gases &amp;quot;phlogiston&amp;quot; and &amp;quot;dephlogisticated air&amp;quot; in a container and slide a match, The gases immediately burnt and he saw some water droplets in container, He then did these type of experiments with lots of other gases and demonstrated that new substances can be formed by combining gases, Phlogiston was named &amp;quot;hydrogène&amp;quot;, French word for &amp;quot;hydrogen&amp;quot;, One of his most famous experiments was the combustion of mercury in a closed vessel. He showed that the weight of the mercury and the vessel increased after the reaction, and that the same amount of air was used up. This result contradicted the phlogiston theory, which predicted that the weight of the mercury and the vessel would decrease after the reaction&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jeremias Benjamin Richter: In 1972, German chemist Richter had a firm belief that reaction followed a specific mathematics which can be explained with accurate quantitative measurements, Evidence for the existence of atoms was the law of definite proportions proposed by Jeremias Benjamin Richter in 1792. Richter found that the ratio by weight of the compounds consumed in a chemical reaction was always the same. He coined the term &amp;quot;Stoichiometry&amp;quot;, From Greek words &amp;quot;Stoikheion&amp;quot; for element and &amp;quot;Metron&amp;quot; for measurement&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Proust: Joseph louis Proust was a French chemist, Proust studied iron sulfides, tin oxides and other substances like copper carbonates, he even made artificial copper carbonate and compared it with Natural one, and showed that each had same proportions of &lt;span class=&quot;math-container&quot;&gt;$(Cu ,O,C)$&lt;/span&gt;,Through different experiments like Adding a little more of One Element , And still found that, The extra weight remained unreacted, Leading to conclusion that no intermediate compound other than these proportions was possible, He Then gave his Famous &amp;quot;Law of definite proportions&amp;quot;, Scholars who have reviewed the writings of Proust found that he had enough data to have discovered the law of multiple proportions himself, but somehow he did not&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;h2&gt;&amp;quot;Now here are Significant interesting parts&amp;quot;&lt;/h2&gt;&#xA;&lt;p&gt;Dalton: John Dalton was an English chemist and Physicist, here is how he contributed to Atomist Philosophy.&lt;/p&gt;&#xA;&lt;p&gt;In his paper, He mentioned that there are three kinds of matter present, namely , Solid, liquid and Elastic fluids, He then further stated, That everything is made up of indivisible particles, and they are attracted to each other with forces with power more or less according to condition, This almost explains well the states ,The force is called attraction of cohesion&lt;/p&gt;&#xA;&lt;p&gt;One important and Interesting question That Dalton had in his mind was as follows&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;If atomic theory was In fact true, Then do all such atoms(particles) of water have same shape and figure and weight?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is how he responded to this query in his paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If it was case, if really some particles of water were heavier than other, then, hypothetically, if one parcel of liquid water was made of those heavier water, it will and should affect specific gravity of water, Which is clearly not the case&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Although now, we know isotopes exist&lt;/p&gt;&#xA;&lt;p&gt;He further proposed that gases are of two types&lt;/p&gt;&#xA;&lt;p&gt;(1) Pure gases(Which cannot be made of combination of other gases)&lt;/p&gt;&#xA;&lt;p&gt;(2) Mixed gases(Which can be made by combination of other gases)&lt;/p&gt;&#xA;&lt;p&gt;And Further for explaining chemical reactions, he used rule of greatest simplicity, that particles of two chemicals form a binary molecule of 1:1 correspondence, example&lt;/p&gt;&#xA;&lt;p&gt;Hydrogen gas+ Oxygen gas gives water, Note that they did not knew what were the molecular formulas for these gases like our modern days, for them , These gases were elements&lt;/p&gt;&#xA;&lt;p&gt;After Taking the values of Lavoisier's rough measurement of weights of reactants, He used the rough Lavoisier measurement that 2.7 gram of Hydrogen gas react with 15.3 grams of oxygen gas to give 18g of water, Which led him to find relative atomic masses, These were the first attempts to compare masses of fundamental particles of two different elements(their definition of element applies), Hence, due to Dalton's assumption&lt;/p&gt;&#xA;&lt;p&gt;He believed that 1 fundamental particle of Hydrogen gas react with One fundamental particle of oxygen gas to form one fundamental particle of water, Hence relative masses of can simply be found by simply taking ratios of weight in which they react, hence he then gave Mass of Oxygen's fundamental particle as &lt;span class=&quot;math-container&quot;&gt;$\frac {15.3}{2.7}=5.5$&lt;/span&gt; times the mass of Hydrogen's fundamental particle, If Lavoisier calculation would have been accurate, he would have found the relative mass of Oxygen's fundamental particles as 8 times the mass of hydrogen's fundamental particles&lt;/p&gt;&#xA;&lt;p&gt;using this Binary assumption, He carried out many more such reaction and Gave his table of Atomic weights where he put atomic weight of Hydrogen element as &amp;quot;1 unit&amp;quot;, Note this was the first time relative atomic weight was being measured, so he really had the choice to set the weight of Hydrogen element to &amp;quot;1 relative measuring unit&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As the time progressed, Dalton Found the law of Multiple proportions, Which led him to revise his theory about binary 1:1 fundamental particles combination, He then revised it and stated that as long as 1:1 ratio compounds exist(with lowest weight), He made some rules that &lt;span class=&quot;math-container&quot;&gt;$2^{nd}$&lt;/span&gt; lowest weight compound will combine as 1:2 then 1:3 etc.&lt;/p&gt;&#xA;&lt;p&gt;In modern terms, example can be given as&#xA;as long as nitrous gas(nitrous oxide) exist, there can exist nitrogen dioxide etc..&lt;/p&gt;&#xA;&lt;p&gt;Quick fact: This revised theory caused a lot of debate in future as Dalton refused to believe that water can be &lt;span class=&quot;math-container&quot;&gt;$H_2O$&lt;/span&gt; without existence of compound like &lt;span class=&quot;math-container&quot;&gt;$HO$&lt;/span&gt; as Avogadro proposed later&lt;/p&gt;&#xA;&lt;p&gt;Important Point to note is that Dalton thought that all pure gases/elastic fluids are monoatomic in nature, He then observed that when equal volumes of azotic gas(Modern day Nitrogen gas) and oxygen gas is taken, It instantly forms twice the volume of &amp;quot;Nitrous gas&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Hence Dalton first reached to a conclusion &amp;quot;that equal volumes of oxygen and nitrogen gas contain equal number of molecules as they react completely to form Nitrous gas&amp;quot;, But at that time, it was assumed that nitrogen and oxygen are monoatomic and result in 1:1 binary molecule, But above conclusion was quickly rejected by Dalton himself as if two gases had equal number of particles, then as per this conclusion, The nitrous gas formed must have twice the amount of particles as that of either nitrogen or oxygen as twice the volume of nitrous gas was formed, But this cannot be the case if Nitrogen and oxygen reacted in 1:1 particle combination, at most, have same number of molecules as that of either oxygen or nitrogen, it cannot form twice the number of particle of nitrous gas out of nothing, Hence Dalton rejected the conclusion that equal volumes of gases had equal number of particles, The KEY point to note here is that Dalton had no idea that a Gas can Be Diatomic, If Dalton had known that A gas could be diatomic, He might have explained this observation properly&lt;/p&gt;&#xA;&lt;p&gt;Now, Gay Lussac later found experimentally that volumes of gases too, react in multiple&#xA;simple whole number proportion, Example just for explanation(Don't know if he actually used this reaction)&lt;/p&gt;&#xA;&lt;p&gt;1 volume of azotic gas react with 1 volume of oxygen gas to form 2 volume of nitrous gas,&lt;/p&gt;&#xA;&lt;p&gt;2 volumes of nitrogen gas react with 1 volume of oxygen gas to form 2 volume of modern day (&lt;span class=&quot;math-container&quot;&gt;$N_2O$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;1 volume of nitrogen gas react with 2 volumes of oxygen gas to form 2 volume of modern day (&lt;span class=&quot;math-container&quot;&gt;$NO_2$&lt;/span&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Now, Dalton replied to New observations of gay Lussac and tried to convince that somehow, Gay Lussac's observations are similar to his observation and lead to conclusion similar to that of Dalton's earlier conclusion mention above, If The conclusion that equal volumes, equal particles conclusion was true, This would lead to definite proportion law in volumes of gases, similar to Dalton's Definite proportion law, But the only difference would be that Dalton's Law would be applicable everywhere while , This law would only be valid for Elastic fluid, Dalton further Tries to highlight that this irregular conclusion might just be due to the limitations of the measurement instruments of that period, and further in time, It would get better&lt;/p&gt;&#xA;&lt;p&gt;Now , &amp;quot;&lt;strong&gt;Lorenzo Romano Amedeo Carlo Avogadro&lt;/strong&gt;&amp;quot;, who was an Italian scientist, Wrote a paper named &amp;quot;&lt;em&gt;essay on a manner of determining the relative masses of the elementary molecules of bodies, and the proportions in which they enter into these compounds&amp;quot;&lt;/em&gt; after Release of Interesting paper by Gay Lussac named &amp;quot;&lt;em&gt;Memoir on the Combination of Gaseous Substances with Each Other&lt;/em&gt;&amp;quot;, In which Avogadro states that&#xA;If Dalton atomic theory is true, and indeed Gay Lussac experiments are hereby of less errors, There must exist a relation between volume and number of particles in gas, the simplest of all such relations would be that equal volumes of all gases contain equal number of elementary particles, But if relations were different for different For different for different gases, though Quantized by volumes, It would be scarcely possible that law regulating distances between these particle(that has direct impact on volume) could give in all cases, relation so simple as conclusion as according to Gay Lussac&lt;/p&gt;&#xA;&lt;p&gt;A simple explanation of what Avogadro meant to say is as follows&lt;/p&gt;&#xA;&lt;p&gt;Explanation: We already know by Gay Lussac multiple proportion volume experiment that number of atoms/elementary particles  must be quantized to a particular volume at same external conditions&lt;/p&gt;&#xA;&lt;p&gt;Now there can be these two cases&lt;/p&gt;&#xA;&lt;p&gt;Case 1) The quantization might be same for all gases i.e &lt;span class=&quot;math-container&quot;&gt;$\frac {V}{n}=k$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is a natural number, and &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; is same for all gases, which in turn means, All Gases have same number of particles in equal volumes&lt;/p&gt;&#xA;&lt;p&gt;Case 2) The value of k  Is different for different gases, meaning particles are still quantized with volume, but the quantization is different for different gases&lt;/p&gt;&#xA;&lt;p&gt;Now considering case 2, Let us take an example of reaction of two gases &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt; AND &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Suppose&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$n_1$&lt;/span&gt; particles per volume '&lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;'&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt; has &lt;span class=&quot;math-container&quot;&gt;$n_2$&lt;/span&gt; particles per volume '&lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;'&lt;/p&gt;&#xA;&lt;p&gt;Now SUPPOSE &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; volumes of gas &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt; react with &lt;span class=&quot;math-container&quot;&gt;$xV$&lt;/span&gt; volume of gas &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt; completely, &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is a natural number&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$V(G_1)+xV(G_2)$$&lt;/span&gt; react completely&lt;/p&gt;&#xA;&lt;p&gt;Hence, &lt;span class=&quot;math-container&quot;&gt;$n_1$&lt;/span&gt; particles of &lt;span class=&quot;math-container&quot;&gt;$(G_1)$&lt;/span&gt; react completely with &lt;span class=&quot;math-container&quot;&gt;$xn_2$&lt;/span&gt; particles of &lt;span class=&quot;math-container&quot;&gt;$(G_2)$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;meaning,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$xn_2=k'n_1$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$k'$&lt;/span&gt; is an integer,&lt;/p&gt;&#xA;&lt;p&gt;The relation states that exactly &lt;span class=&quot;math-container&quot;&gt;$xn_2$&lt;/span&gt; particles of &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt; gas forms integral volume &lt;span class=&quot;math-container&quot;&gt;$(k'V)$&lt;/span&gt;&#xA;of &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As Avogadro said the relation is so simple taking into account the laws governing distances between particles, affecting volume&lt;/p&gt;&#xA;&lt;p&gt;This simple relation for case 2 might exist in reality, but will be rare for all gases and scarcely plausible that integral volumes of other gas &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt; are formed by number of particles of &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt; exactly Integral multiple of number of particles Of &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This might be very rare relation to to true for all gas pairs, hence Avogadro assumed that case 1 is more plausible and hence hypothesised it&lt;/p&gt;&#xA;&lt;p&gt;note: The theoretical proof backing up the hypothesis that equal volume of gases have equal number of particles was later given using KTG, also experimental evidence of hypothesis was found by studying effusion and diffusion of gases&lt;/p&gt;&#xA;&lt;p&gt;After Considering Avogadro hypothesis to be true, Following deduction can be quickly made&lt;/p&gt;&#xA;&lt;p&gt;1)The ratio of densities of Two gases will give ratio of their respective atomic masses,&lt;/p&gt;&#xA;&lt;p&gt;because&#xA;&lt;span class=&quot;math-container&quot;&gt;$\frac{\rho_1}{\rho_2}=\frac{\frac{M_1N_1}{V_1}}{\frac{M_2N_2}{V_2}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;As&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{N_1}{V_1}=\frac{N_2}{V_2}$&lt;/span&gt; according to hypothesis, it follows&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{\rho_1}{\rho_2}=\frac{M_1}{M_2}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;2)Experimentally ,The following data was available to Avogadro&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\rho_{oxygen}=1.10359$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$\rho_{hydrogen}=0.07321$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{\rho_{oxygen}}{\rho_{hydrogen}}=\frac{M_{oxygen}}{M_{hydrogen}}=\frac{1.10359}{0.07321}=15.074$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Hence Avogadro found that &lt;span class=&quot;math-container&quot;&gt;$M_{Oxygen}=15 \cdot M_{Hydrogen}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;similarly&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$\frac{\rho_{nitrogen}}{\rho_{hydrogen}}=\frac{M_{nitrogen}}{M_{hydrogen}}=\frac{0.96913}{0.07321}=13.2381$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;3)Now observing carefully, When a gas combines with two or more times volume of another gas, completely, It is now expected that resulting compound must form volume equal to that of first of these gases, as it is limiting factor for number of particles, example, when &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; volume of &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt; gas react with &lt;span class=&quot;math-container&quot;&gt;$3V$&lt;/span&gt; volume of &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt; gas, completely, Each particle of &lt;span class=&quot;math-container&quot;&gt;$G_1$&lt;/span&gt; gas must combine with 3 particles of &lt;span class=&quot;math-container&quot;&gt;$G_2$&lt;/span&gt;, But experimental observations showed that it was not always true, example&lt;/p&gt;&#xA;&lt;p&gt;2 volumes of Hydrogen gas react completely with 1 volume of oxygen gas to form 2 volume of water, Here it was expected that only 1 volume of water would be formed&lt;/p&gt;&#xA;&lt;p&gt;But, Still being persistent About the Hypothesis to be true, It is almost natural to conclude that instead of hydrogen gas and oxygen gas being made of single elementary particle, they might be made up of two or more elementary particles forming one elementary entity, Which then further redivide into Elementary particles during reaction&lt;/p&gt;&#xA;&lt;p&gt;Example If we consider Hydrogen to be made of two Hydrogen elementary particle formed 1 elementary entity &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$H_2$&lt;/span&gt;&amp;quot; And similar for oxygen As &amp;quot;&lt;span class=&quot;math-container&quot;&gt;$O_2$&lt;/span&gt;&amp;quot;, This consideration will now perfectly align with experimental results&lt;/p&gt;&#xA;&lt;p&gt;That are&lt;/p&gt;&#xA;&lt;p&gt;2V volume of Hydrogen gas react with V volume of Oxygen completely , Meaning,&lt;/p&gt;&#xA;&lt;p&gt;2n entities of Hydrogen gas react completely with n entities of Oxygen ,Meaning&lt;/p&gt;&#xA;&lt;p&gt;4n elementary particle particles of Hydrogen react completely with 2n elementary particles of oxygen&lt;/p&gt;&#xA;&lt;p&gt;Notice there is a difference in how I used Elementary &amp;quot;Particles&amp;quot; and Elementary &amp;quot;Entities&amp;quot;, In modern day definition ,Particles refer to Atoms, while Entities refer to molecules, There was a huge confusion in Amedeo Avogadro's original text regarding How he used terms like &amp;quot;Molecule&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;As experimentally 2V volume of water is formed, It follows that 2n entities of water is formed, Giving a result that each 2 of Hydrogen Gas particles reacted with Each one of Oxygen gas particles to form 2n entities of water, Meaning every entity of water has composition &lt;span class=&quot;math-container&quot;&gt;$H_2O$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Notice, We would have supposed that elementary entity of Hydrogen gas would consist of 4 hydrogen particles and Elementary entity of Oxygen consist of 4 elementary particles of Oxygen, This would also have followed the experimental result&lt;/p&gt;&#xA;&lt;p&gt;But according to Occam's razor, simplest theory of all theories, explaining all experimental results must be adopted, &lt;span class=&quot;math-container&quot;&gt;$H_2O$&lt;/span&gt; composition is simpler that &lt;span class=&quot;math-container&quot;&gt;$H_4O_2$&lt;/span&gt;, Hence it must be adopted&lt;/p&gt;&#xA;&lt;p&gt;Using modern techniques, like X-ray Crystallography and others, we have evidence that Composition of water is Indeed &lt;span class=&quot;math-container&quot;&gt;$H_2O$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;This Hypothesis led Jean Perrin To calculate Avogadro Constant using Brownian motion, Which Was a significant step in Quantitative physical chemistry&lt;/p&gt;&#xA;&lt;h2&gt;References&lt;/h2&gt;&#xA;&lt;p&gt;Link to original paper:&#xA;&lt;a href=&quot;https://iiif.wellcomecollection.org/pdf/b24855169&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://iiif.wellcomecollection.org/pdf/b24855169&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Other sources: Wikipedia.org, britannica.com, &lt;a href=&quot;https://web.lemoyne.edu/%7Egiunta/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://web.lemoyne.edu/~giunta/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17776" LastEditorUserId="17776" LastEditDate="2023-11-01T07:18:00.960" LastActivityDate="2023-11-01T07:18:00.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15867" PostTypeId="2" ParentId="15859" CreationDate="2023-10-31T15:26:08.480" Score="5" Body="&lt;p&gt;Thanks to a pointer from terry-s to search in &lt;a href=&quot;https://worldwide.espacenet.com/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://worldwide.espacenet.com/&lt;/a&gt; I have now found the patent.&lt;/p&gt;&#xA;&lt;p&gt;Leslie Connock Jesty applied on 21 August 1968 for a patent: Application number 24073/68 at The Patent Office London&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;This invention results directly from solutions of the mathematical problem of the efficient packing of n-dimensional hyperspheres in n-dimensional hyperspace. The invention is applicable to methods of coding communication signals to increase their information content. It can be used for example to increase the rate at which information can be transmitted through a given transmission channel without significantly increasing the bandwidth or the signal to noise ratio of the system&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It provides other examples of possible applications.&lt;/p&gt;&#xA;&lt;p&gt;The internet does not provide a great amount of detail about him sadly. A genealogy site reveals that his middle name Connock was his mother's maiden name. He was born in 1907 and died in 1988.&lt;/p&gt;&#xA;&lt;p&gt;Further searching reveals some publications (thanks to Google Scholar). One which gives a good picture of his interests is a joint paper with G T Winch published in 1937 in the Transactions of the Illuminating Society (now known as Lighting Research and Technology). Entitled &amp;quot;Television images An analysis of their essential qualities&amp;quot; it considers contrast, range, brightness, definition, size, presentation, colour or tint, and flicker. Of course in 1937 broadcast television was in its infancy so he was a pioneer. He also seems to have applied for many more patents about the development of television.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastEditorUserId="16128" LastEditDate="2023-11-05T14:14:04.493" LastActivityDate="2023-11-05T14:14:04.493" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15868" PostTypeId="2" ParentId="8179" CreationDate="2023-10-31T19:38:01.427" Score="1" Body="&lt;p&gt;The answer by user103496 gives the history. But they write&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;But it seems that Marshall somehow thought of &amp;quot;stretching&amp;quot; and so used&#xA;the word elasticity instead. His enormous influence is such that we&#xA;are now stuck with this word.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I argue that it is not the case that he just &amp;quot;somehow thought&amp;quot;. He envisioned the graph, and realized that a demand schedule that has this measure higher in absolute value, &lt;em&gt;stretches more&lt;/em&gt; (compared to a straight line), hence it is &amp;quot;more elastic&amp;quot;, hence the measure does measure some notion of elasticity. To wit,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/bn5RZ.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/bn5RZ.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="244" LastActivityDate="2023-10-31T19:38:01.427" CommentCount="17" ContentLicense="CC BY-SA 4.0" />
  <row Id="15869" PostTypeId="2" ParentId="5907" CreationDate="2023-10-31T23:00:44.400" Score="0" Body="&lt;p&gt;Let's let m be the mass of some star. We can the write the equation:&lt;/p&gt;&#xA;&lt;p&gt;m  =  2.0 ((30))[kg]&lt;/p&gt;&#xA;&lt;p&gt;The equation is saying that the mass , a physical attribute of some star, is equivalent to a rather huge number, 2 000 000 000 000 000 000 000 000 000 000, of units which are abbreviated to [kg].&lt;/p&gt;&#xA;&lt;p&gt;The square brackets are interpreted to mean &amp;quot;the unit&amp;quot; and &amp;quot;kg&amp;quot; is the SI symbol for the unit of mass, &amp;quot;kilogram&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;The square brackets are indicating a constant attribute. The mass of a kilogram is constant. The mass of a star is a variable that varies depending on which star you consider. But the function of a unit such as [kg] is that every [kg] is the same quantity of mass as every other [kg].&lt;/p&gt;&#xA;&lt;p&gt;Algebraically the equation above can be converted to&lt;/p&gt;&#xA;&lt;p&gt;m/[kg] = 2.0 ((30)).&lt;/p&gt;&#xA;&lt;p&gt;This equation is saying the mass of this star divided by the mass of the SI unit mass, [kg], results in a huge number. This agrees with the original equation. There is no contradiction.&lt;/p&gt;&#xA;&lt;p&gt;If astronomers found it convenient to use this particular star as an astronomical unit of mass, they could define [m*]:&lt;/p&gt;&#xA;&lt;p&gt;[m*] = 2.0 ((30))[kg].&lt;/p&gt;&#xA;&lt;p&gt;Dividing both sides by the mass of the astronomer's unit star we  have the equation;&lt;/p&gt;&#xA;&lt;p&gt;1 = (2.0 ((30)) [kg])/ [m*].&lt;/p&gt;&#xA;&lt;p&gt;This we can use to convert from a physical quantity expressed in [m*] to the same quantity expressed in the SI standard [kg].&lt;/p&gt;&#xA;&lt;p&gt;For example the mass of a massive black hole, mbh, measured to be say:&lt;/p&gt;&#xA;&lt;p&gt;mbh = 1.54 ((8)) [m*]&lt;/p&gt;&#xA;&lt;p&gt;could be converted to:&lt;/p&gt;&#xA;&lt;p&gt;mbh = 3.1 ((38)) [kg]&lt;/p&gt;&#xA;&lt;p&gt;by multiplying {1.54 ((8)) [m*]} times { 2 ((30)) [kg]/ [m*]}.&lt;/p&gt;&#xA;&lt;p&gt;The right hand bracketed expression is equivalent to one.&lt;/p&gt;&#xA;&lt;p&gt;Note that mbh could be rapidly changing in time, but [m*] is not.&lt;/p&gt;&#xA;&lt;p&gt;And that mbh could be used for the mass of any size of black hole where [m*]&#xA;is tied through its definition to the SI standard kilogram.&lt;/p&gt;&#xA;&lt;p&gt;Without the square brackets&amp;quot; m* &amp;quot; would not be considered a unit of mass because it looks like a variable quantity.&lt;/p&gt;&#xA;&lt;p&gt;The letter &amp;quot;m&amp;quot; has many meanings, the bracketed &amp;quot;[m]&amp;quot; means the SI standard [meter], a unit of length. The brackets prevent confusion.&lt;/p&gt;&#xA;&lt;p&gt;I see below this box that I should have used MathJax for the equations.&#xA;My apologies.... my bad.&lt;/p&gt;&#xA;&lt;p&gt;Wow! Looks like MathJax is rewriting my equations. Mostly just removing blank spaces.&lt;/p&gt;&#xA;" OwnerUserId="19258" LastActivityDate="2023-10-31T23:00:44.400" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15870" PostTypeId="1" CreationDate="2023-11-01T08:39:58.413" Score="2" ViewCount="65" Body="&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiments#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lunar Laser Ranging Experiments; History&lt;/a&gt; mentions the first lasers ever bounced off &lt;em&gt;the whole Moon&lt;/em&gt; were in 1962, and probably both the US and Soviet groups used Q-switched ruby lasers.&lt;/p&gt;&#xA;&lt;p&gt;In 1969 the Apollo astronauts placed an array of optical retroreflectors on the moon and carefully oriented it so that the face of the array pointed towards the mean direction of the Earth so that each reflector's contribution would be returned at nearly the same time. See &lt;a href=&quot;https://space.stackexchange.com/q/14943/12102&quot;&gt;What are these structures on the Lunar Ranging Retro Reflector (LRRR) arrays for?&lt;/a&gt; for lots of details.&lt;/p&gt;&#xA;&lt;p&gt;ON the ground almost certainly photomultiplier tubes were used at the focus of the telescope receiving the pulses. I'm guessing they dealt with reflected sunlight by waiting until the Apollo 11 site was not sunlit, and a combination of a pinhole and filter for the laser wavelength, as well as some nice gated coincidence electronics.&lt;/p&gt;&#xA;&lt;p&gt;For more on that see my companion question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/15871/3869&quot;&gt;How did the earliest measurements of lasers bounced off of Apollo 11 retroreflectors deal with the light from the Moon and pick out single photons?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;But here I'd like to ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What laser technology/ies were first used to bounce off of the Apollo 11 retroreflectors to accurately measure distance to the Moon?&lt;/p&gt;&#xA;&lt;p&gt;Was it still &lt;a href=&quot;https://en.wikipedia.org/wiki/Q-switching&quot; rel=&quot;nofollow noreferrer&quot;&gt;Q-switched&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Ruby_laser&quot; rel=&quot;nofollow noreferrer&quot;&gt;ruby lasers&lt;/a&gt;? They would presumably be looking for nanosecond(s) scale timing rather than the milliseconds for the full Moon work in 1962.&lt;/p&gt;&#xA;&lt;p&gt;A photo of the laser would be great!&lt;/p&gt;&#xA;&lt;p&gt;These days I believe either pulsed Ti-sapphire or Nd:YAG lasers (infrared, frequency doubled to green) are used. (cf. &lt;a href=&quot;https://space.stackexchange.com/a/50400/12102&quot;&gt;this answer to &lt;em&gt;When they shoot lasers at the Moon for ranging, what is the shape of the beam?&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Assuming they started with ruby lasers, I will probably ask when they switched from ruby to this technology in a separate question, but I will wait to find out if ruby was the first since Nd:YAG lasers date back to 1964.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2023-11-01T09:05:21.050" LastActivityDate="2023-11-01T20:13:16.310" Title="What laser technology/ies were first used to bounce off of the Apollo 11 retroreflectors to accurately measure distance to the Moon?" Tags="|astronomy|optics|technology|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15871" PostTypeId="1" CreationDate="2023-11-01T08:52:28.747" Score="2" ViewCount="93" Body="&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiments#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lunar Laser Ranging Experiments; History&lt;/a&gt; mentions the first lasers ever bounced off &lt;em&gt;the whole Moon&lt;/em&gt; were in 1962, and probably both the US and Soviet groups used Q-switched ruby lasers.&lt;/p&gt;&#xA;&lt;p&gt;In 1969 the Apollo astronauts placed an array of optical retroreflectors on the moon and carefully oriented it so that the face of the array pointed towards the mean direction of the Earth so that each reflector's contribution would be returned at nearly the same time. See &lt;a href=&quot;https://space.stackexchange.com/q/14943/12102&quot;&gt;What are these structures on the Lunar Ranging Retro Reflector (LRRR) arrays for?&lt;/a&gt; for lots of details.&lt;/p&gt;&#xA;&lt;p&gt;On the ground almost certainly photomultiplier tubes were used at the focus of the telescope receiving the pulses. &lt;strong&gt;I'm guessing&lt;/strong&gt; that they dealt with reflected sunlight by waiting until the Apollo 11 site was not sunlit, and a combination of a pinhole and filter for the laser wavelength, as well as some nice gated coincidence electronics.&lt;/p&gt;&#xA;&lt;p&gt;But I have no idea, so I'd like to ask:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; How did the earliest measurements of lasers bounced off of Apollo 11 retroreflectors deal with the light from the Moon and pick out single photons?&lt;/p&gt;&#xA;&lt;p&gt;A photo of the receiving system optics would be great!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Companion question:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/q/15870/3869&quot;&gt;What laser technology/ies were first used to bounce off of the Apollo 11 retroreflectors to accurately measure distance to the Moon?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;For more on the Apollo 11 retroreflectors themselves see&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://space.stackexchange.com/q/14943/12102&quot;&gt;What are these structures on the Lunar Ranging Retro Reflector (LRRR) arrays for?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2023-11-01T09:04:57.570" LastActivityDate="2023-11-01T18:36:12.970" Title="How did the earliest measurements of lasers bounced off of Apollo 11 retroreflectors deal with the light from the Moon and pick out single photons?" Tags="|astronomy|optics|technology|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15872" PostTypeId="1" CreationDate="2023-11-01T09:04:34.093" Score="1" ViewCount="59" Body="&lt;p&gt;We had radio amplifiers that could transmit powerful signals and those were pushed to microwaves. We had lasers that were originally natural transitions in solids and gasses but now tunable lasers go from deep in the infrared to near UV at least, and with wigglers, undulators and free electron lasers we can go from UV to X-rays.&lt;/p&gt;&#xA;&lt;p&gt;Above X-rays is a semi-infinite range of photon energies that have yet to qualify, so gamma rays would not count as a &amp;quot;hole&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I am certain I'm missing other technologies as well.&lt;/p&gt;&#xA;&lt;p&gt;What I'd like to know is which wavelength range was the last &amp;quot;hole&amp;quot; in the spectrum where there was no technology to make amplified narrow-band power?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What was the last &amp;quot;hole&amp;quot; in the electromagnetic spectrum where amplified, tunable, narrow-band signals couldn't be produced artificially?&lt;/p&gt;&#xA;&lt;p&gt;I'm guessing it's somewhere between a millimeter and a micron in wavelength, but I could be way off.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;note:&lt;/strong&gt; I say &amp;quot;tunable&amp;quot; to indicate that you can center your narrow band anywhere within a wide range so that there is continuous coverage. Radio, optical lasers, synchrotron radiation-based sources are all tunable, while the old gas and solid state lasers had fixed frequencies and big gaps in between.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2023-11-01T18:12:33.883" LastActivityDate="2023-11-01T18:12:33.883" Title="What was the last &quot;hole&quot; in the electromagnetic spectrum where amplified, tunable, narrow-band power couldn't be produced artificially?" Tags="|physics|technology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15873" PostTypeId="2" ParentId="15870" CreationDate="2023-11-01T12:57:47.843" Score="4" Body="&lt;p&gt;A short article in the IEEE Transactions on Nuclear Science (&lt;a href=&quot;https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;amp;arnumber=4326697&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;, DOI: 10.1109/TNS.1972.4326697, S.K. Poultney &amp;quot;Single Photon Detection and Timing in the Lunar Laser Ranging Experiment&amp;quot;, IEEE Transaction on Nuclear Science 19(3) 12-17 (1972)) discusses the setup used by the author to make measurements at the McDonald Observatory. The paper references S.K. Poultney and D.G. Currie in the IAU Telegram Circular 2164, August 1969, as the first announcement of such a measurement. I cannot find that one online (and my Morse code is rusty anyway, but, seriously, the Central Bureau for Astronomical Telegrams at &lt;a href=&quot;http://www.cbat.eps.harvard.edu/services/IAUCs.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harvard&lt;/a&gt; does not have it up).&lt;/p&gt;&#xA;&lt;p&gt;Descriptions of laser parameters are scattered through the paper, since the focus is on the timing system. It apparently is a 4-ns, few-Joule Q-switched ruby laser at 6943 angstroms firing at 20 pulses per minute (round trip travel time is 2.5 seconds, so pretty good utilization). They used the 272 cm reflector at the observatory to collimate the laser to a 4 km spot on the moon. With a quantum counting efficiency of the photomultiplier tube of 8% (!) they get 0.2 photoelectrons detected per laser pulse (out of ~10&lt;span class=&quot;math-container&quot;&gt;$^{19}$&lt;/span&gt; photons sent up).&lt;/p&gt;&#xA;&lt;p&gt;In the Future Plans section they state that a second US station is being built in Hawaii that would use a 0.2 ns Nd:YAlO&lt;span class=&quot;math-container&quot;&gt;$_{3}$&lt;/span&gt; (note YAP, not YAG) in the green (so doubled, but that is not noted in the paper).&lt;/p&gt;&#xA;&lt;p&gt;A later paper is in &lt;a href=&quot;https://www.science.org/doi/10.1126/science.167.3917.368&quot; rel=&quot;nofollow noreferrer&quot;&gt;Science&lt;/a&gt; (C.O. Alley et al., Science 167 (3917), the 23 January 1970 issue. There they specify that a&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Space Rays custom-built, two-stage Q-switched ruby laser (7 joules, 20-nsec pulse length, &lt;span class=&quot;math-container&quot;&gt;$2.4 \times 10^{-8}$&lt;/span&gt; radians full beam divergence from 0.75-inch rod, one pulse every 6 seconds&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;was used for the measurements. (Space Rays as a laser company can be found in a few hits on Google, so seems to have been a real company. Great name for this particular application.) I have no explanation for the discrepancy in pulse length between the various accounts.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2023-11-01T20:13:16.310" LastActivityDate="2023-11-01T20:13:16.310" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15874" PostTypeId="2" ParentId="15871" CreationDate="2023-11-01T14:19:50.433" Score="4" Body="&lt;p&gt;The short article in the IEEE Transactions on Nuclear Science (&lt;a href=&quot;https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&amp;amp;arnumber=4326697&quot; rel=&quot;nofollow noreferrer&quot;&gt;link&lt;/a&gt;, DOI: 10.1109/TNS.1972.4326697, S.K. Poultney &amp;quot;Single Photon Detection and Timing in the Lunar Laser Ranging Experiment&amp;quot;, IEEE Transaction on Nuclear Science 19(3) 12-17 (1972)) in the answer to your companion question discusses the setup used by the author to make measurements at the McDonald Observatory.&lt;/p&gt;&#xA;&lt;p&gt;Given the setup at the McDonald Observatory using the 272 cm reflector there, they expected to see 0.2 photoelectrons (signal) per laser firing. The authors have one paragraph labeled &amp;quot;Discrimination Against High Background Noise&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Operation under all background illumination conditions&#xA;requires a variety of noise discrimination&#xA;schemes. The extreme case is the detection of a&#xA;single laser photon against the combined background of&#xA;bright moon and bright sky. A spatial filter of 6 arc&#xA;sec and a spectral filter of 0.7 A width still allow&#xA;about 300k counts/sec of noise.&lt;span class=&quot;math-container&quot;&gt;$^{10}$&lt;/span&gt; Another discrimination&#xA;technique was the setting of a 6 microsecond gate&#xA;about the expected time of arrival of the lunar return&#xA;by electronic means. The final technique was the post&#xA;detection clustering of returns within a few nsec. A&#xA;number of repeated rangings were, of course, necessary&#xA;to be certain of a signal return. The RCA C31000F&#xA;(with ERMA photosurface) which is used as the photodetector&#xA;exhibited a noise rate of 30k counts/sec at&#xA;room temperature and several k counts/sec when cooled&#xA;to about 0°C due partially to its operation at very&#xA;high gain. The photodetector noise rate is only&#xA;important when viewing the dark moon at night.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Endnote 10 is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;S. K. Poultney, &amp;quot;The Detector Package for the&#xA;Laser Ranging Experiment at McDonald Observatory:&#xA;Its Design, Performance, and Operation,&amp;quot; Dept.&#xA;of Phys. and Astron. Tech. Rep. No. 957,&#xA;University of Maryland, August 1969.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This shows up in the university &lt;a href=&quot;https://umaryland.on.worldcat.org/oclc/633480036&quot; rel=&quot;nofollow noreferrer&quot;&gt;catalog&lt;/a&gt;, and is available as a &lt;a href=&quot;https://ntrs.nasa.gov/search?fundingNumber=NGR-21-002-109&amp;amp;q=Poultney&quot; rel=&quot;nofollow noreferrer&quot;&gt;NASA&lt;/a&gt; technical report, along with others detailing the work under contract.&lt;/p&gt;&#xA;&lt;p&gt;The impression I get is that the experiment was capable of operation regardless of background light (&amp;quot;Operation under all background illumination conditions&amp;quot;) using good optical and electronic practices.&lt;/p&gt;&#xA;" OwnerUserId="2982" LastEditorUserId="2982" LastEditDate="2023-11-01T18:36:12.970" LastActivityDate="2023-11-01T18:36:12.970" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15875" PostTypeId="2" ParentId="8179" CreationDate="2023-11-02T17:12:28.343" Score="4" Body="&lt;p&gt;&lt;strong&gt;The term &lt;em&gt;elasticity&lt;/em&gt; in economics&lt;/strong&gt;  is almost surely linked to the concept of &lt;a href=&quot;https://www.britannica.com/science/elasticity-physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;elasticity in physics&lt;/strong&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;While the introduction of the term elasticity in economics is attributed to Marshall, as far as I know, we haven’t an explicit source about the reason why Marshall chose this term.&lt;/p&gt;&#xA;&lt;p&gt;But we can make conjectures that are not arbitrary and are based on relevant sources.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;It is well known that the mathematization of economics in the second half of the XIX century referred widely to physics,&lt;/strong&gt; and mathematical economists as Edgeworth, Walras, Pareto, Marshall himself, knew physics, and referred often to physics in their works.&lt;/p&gt;&#xA;&lt;p&gt;In particular, Pareto wrote his graduation thesis in engineering on &lt;a href=&quot;https://books.google.it/books?id=dWQ7AAAAcAAJ&amp;amp;pg=PA5&amp;amp;redir_esc=y#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Principles of the theory of elastic solid bodies&lt;/em&gt; (&lt;em&gt;Engl. transl&lt;/em&gt;)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Marshall knew their works, as he explicitly quotes  them in his &lt;a href=&quot;https://competitionandappropriation.econ.ucla.edu/wp-content/uploads/sites/95/2017/09/PrinciplesEbook.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Principles of Economics&lt;/em&gt;&#xA;(1890)&lt;/a&gt;&#xA;.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, in  Appendix C, sec. 2. &lt;em&gt;The method in economics, physics and biology&lt;/em&gt;, of the  &lt;em&gt;Principles&lt;/em&gt;,  he  explicitly compares the methodology of physics and economics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is true that the forces with which economics deals have one advantage for&#xA;deductive treatment in the fact that their method of combination is, as Mill observed, are&#xA;that of mechanics rather than of chemistry. That is to say, when we know the action of&#xA;two economic forces separately—as for instance the influences which an increase in&#xA;the rate of wages and a diminution in the difficulty of the work in a trade will&#xA;severally exert on the supply of labour in it—we can predict fairly well their conjoint&#xA;action, without waiting for specific experience of it .[…] (p. 449)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Marshall himself had a degree in mathematics and physics at the University of Cambridge.&lt;/p&gt;&#xA;&lt;p&gt;So,  the hypothesis that Marshall had in mind the terminology of physics and  the elasticity of a body, or the stretching of a wire, is more than  reasonable (even if the concept of  elasticity in economics and in physics are of course different).&#xA;&lt;span class=&quot;math-container&quot;&gt;$$***$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In many economics texts it is taken for granted that the term &lt;em&gt;elasticity&lt;/em&gt; in economics comes from  the terminology of physics.&lt;/p&gt;&#xA;&lt;p&gt;For an example,&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.jstor.org/stable/23228014&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lionello Rossi, Del concetto di elasticità in economia (Engl. Tr. About the Concept of Elasticity in Economics), 1932&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;In this article, Rossi seems to take for granted that the concept of elasticity in economics originates from physics, and discusses the actual use in economics of the term elasticity  and other terms coming from mechanics as &lt;em&gt;rigidity&lt;/em&gt; and &lt;em&gt;anelastic&lt;/em&gt;, comparing their use  in economic and in physics.&lt;/p&gt;&#xA;&lt;p&gt;For instance,  he writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We try […] to clarify  the concepts outlined above, and we start&#xA;making few remarks about the  homonymous  mechanical concepts[…]&lt;/p&gt;&#xA;&lt;p&gt;In mechanics a body is defined elastic if, when deformed under the&#xA;action of a stimulus, has the property to resume its primitive form&#xA;[…]&lt;/p&gt;&#xA;&lt;p&gt;The concept of rigidity, instead, defines the greater or smaller&#xA;deformability, and therefore it is not antithetical to the concept of&#xA;elasticity […]. That is, rigidity denotes a further property of&#xA;elastic bodies. Anelasticity, instead, it a concept antithetical to&#xA;that of elasticity[…].&lt;/p&gt;&#xA;&lt;p&gt;That said about mechanical concepts, what about homonymous concepts of&#xA;economics?&lt;/p&gt;&#xA;&lt;p&gt;The only concept that in the common use has a perfect analogy with the&#xA;concept of mechanics is that of rigidity […]; and the authors seem to&#xA;agree about it.&lt;/p&gt;&#xA;&lt;p&gt;Instead, the concept of elasticity, would be, according to Marshall,&#xA;nothing more than the opposite of the concept of rigidity[…].&lt;/p&gt;&#xA;&lt;p&gt;Whatever the mechanical concepts are, in economics the Marshallian&#xA;definition of elasticity as variability and antithetical to rigidity&#xA;seems obvious, in relations to the denominations that entered the&#xA;common usage, reserving the name of anelasticity to a different&#xA;concept, conform to the mechanical one […]. (&lt;em&gt;pp. 20-21, my transl.&lt;/em&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-11-02T17:12:28.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15876" PostTypeId="2" ParentId="15406" CreationDate="2023-11-03T18:59:47.620" Score="1" Body="&lt;p&gt;Upon obtaining his PhD, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hugh_Everett_III&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hugh Everett&lt;/a&gt; turned his back on physics and became a game theorist working for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Weapons_Systems_Evaluation_Group&quot; rel=&quot;nofollow noreferrer&quot;&gt;Weapons Systems Evaluation Group&lt;/a&gt; at the Pentagon.  While his thesis was being evaluated by Wheeler, he turned down an invitation to spend two months in Copenhagen discussing his ideas with Bohr et al.  However, some years later, while on holiday in Europe with his wife and family, he spent two days in Copenhagen attempting to explaining his ideas to Bohr.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;After a long and fruitless discussion with Bohr, Everett walked back to his Copenhagen hotel under the steel-gray afternoon twilight of the Danish sky, leaving quantum physics behind him.  Drinking and smoking incessantly at the hotel bar - &amp;quot;he was sloppy and had a cigarette with him all the time.&amp;quot; recalled Suzanne Misner - Everett had another brilliant alcohol-fueled idea, totally unrelated to the universal wave function. Jotting notes on hotel stationary while downing several pints of beer, Everett developed a new optimization algorithm for allocating military resources.  It was easy to apply and fast to run on the bulky and slow computers of the time. When he returned home, Everett secured a patent for his algorithm, and it ultimately made him and his circle of military-minded colleagues rich.  Everett finally had what he wanted: a never ending supply of booze, food, and cigarettes.  Life was good.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(Source: &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1541698975&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;What is Real?&lt;/em&gt;&lt;/a&gt; by Adam Becker)&lt;/p&gt;&#xA;" OwnerUserId="3643" LastActivityDate="2023-11-03T18:59:47.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15877" PostTypeId="1" CreationDate="2023-11-04T19:34:10.773" Score="5" ViewCount="184" Body="&lt;p&gt;I've encountered the quote &amp;quot;Number is the ruler of forms and ideas, and the cause of gods and demons,&amp;quot; which is frequently attributed to Pythagoras. My objective is to ascertain the authenticity of this attribution and, if it is indeed true, to locate where it was first documented.&lt;/p&gt;&#xA;&lt;p&gt;Some have suggested that this quote might be found in Chapter 28 of Iamblichus's 'On the Life of Pythagoras.' However, after reviewing the text, I did not find the quote there. Could anyone provide insight or clarification about its origin? Any guidance on where to find a primary source or at least an early mention of this concept would be greatly appreciated.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance for your assistance.&lt;/p&gt;&#xA;" OwnerUserId="12383" LastActivityDate="2023-11-08T18:58:44.780" Title="Seeking Source of this Pythagorean Quote &quot;Number is the ruler of forms and ideas, and the cause of gods and demons.&quot;" Tags="|quote|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15878" PostTypeId="1" CreationDate="2023-11-05T15:26:24.313" Score="3" ViewCount="152" Body="&lt;p&gt;I’ve noticed a recurring post on social media that presents an image suggesting the shapes of Hindu-Arabic numerals are connected to the angles within their design:&#xA;&lt;a href=&quot;https://i.stack.imgur.com/rijLA.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rijLA.png&quot; alt=&quot;hindu-arabic shapes from angles?&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This claim seems dubious to me.&lt;/p&gt;&#xA;&lt;p&gt;I am seeking a definitive and reliable reference regarding the evolution of the Hindu-Arabic numerals. Specifically, my question is: How did the ten Hindu-Arabic numerals (0-9) come to have their current forms?&lt;/p&gt;&#xA;&lt;p&gt;If anyone could point me towards scholarly articles, historical texts, or authoritative studies on this subject, I would greatly appreciate it.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance for your assistance!&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12330" LastEditDate="2023-11-05T20:59:04.510" LastActivityDate="2023-11-05T20:59:04.510" Title="How did Hindu-Arabic Numerals get their shapes?" Tags="|notation|numbers|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15879" PostTypeId="2" ParentId="15877" CreationDate="2023-11-05T16:36:36.827" Score="6" Body="&lt;p&gt;The two links in my comment, &lt;a href=&quot;https://www.google.com/books/edition/Iamblichus_Life_of_Pythagoras_or_Pythago/J_BZAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=Pythagoras+%22sacred+discourse%22+number&amp;amp;pg=PA106&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt; and &lt;a href=&quot;https://www.gutenberg.org/files/63300/63300-h/63300-h.htm#c28&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Gutenberg&lt;/a&gt;, are different editions of the same translation by Thomas Taylor (1818). The Gutenberg edition has a different layout, with endnotes instead of footnotes.  The scan of the actual book shows they were originally footnotes. It is in the footnote by Thomas, where we find Syrianus quoted:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pythagoras, therefore, in the Sacred Discourse, clearly says, that “number is the ruler of forms and ideas, and is the cause of Gods and dæmons.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I wasn't sure whether or whom Thomas was quoting at first, but the Greek is also given and more clearly attributed to Syrianus. One can find the Greek in the following manuscript:&lt;/p&gt;&#xA;&lt;p&gt;Syrianus, In metaphysica commentaria (Cambridge, Trinity College, MS O.9.4),&#xA;&lt;a href=&quot;https://cudl.lib.cam.ac.uk/view/MS-TRINITY-O-00009-00004/146&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://cudl.lib.cam.ac.uk/view/MS-TRINITY-O-00009-00004/146&lt;/a&gt;,&#xA;p. 132, ll. 22-24.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/UURb2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/UURb2.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;αυτος μεν Πυθαγορας, εν τῳ ιερῳ λογῳ, διαρρηδην μορφων και ιδεων κραντορα τον αριθμον ελεγεν ειναι, και θεων και δαιμονων αιτιον.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Remarks on the manuscript by the Cambridge library:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This manuscript, probably copied in the last thirty years of the 17th century, is a copy of the commentary on Aristotle's Metaphysics by the 5th-century Neoplatonist philosopher Syrianus.&lt;/p&gt;&#xA;&lt;p&gt;...&lt;/p&gt;&#xA;&lt;p&gt;The manuscript was probably a copy of &lt;a href=&quot;https://medieval.bodleian.ox.ac.uk/catalog/manuscript_740&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oxford, Bodleian Library, MS Auct. T.1.16 (Misc. 194)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The text posted below the image is the beginning of the Greek quote in the footnote from the Gutenberg transcription. It seems to be missing the first three words from the manuscript:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;πῶς δ'ἄν&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My Greek is pretty poor. Perhaps someone else would help. Literally the words translate &amp;quot;And how [&lt;em&gt;or&lt;/em&gt; thus, as] if...&amp;quot;, where &amp;quot;and&amp;quot; (δε) often combines with μεν (second word in Taylor's quote) to form a contrasting pair of phrases. Taylor includes &amp;quot;therefore&amp;quot; in his translation, which may come from these words. Since ἔλεγεν seems to be the indicative mood, it's unlikely that the Greek means &amp;quot;as if Pythagoras said....&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;In sum, the source of the purported quote seems to be Syrianus (4th/5th c. CE, about 100 years after Iamblichus and 900 years after Pythagoras), who attributes it to something Pythagoras himself expressly said (διαρρήδην ἔλεγεν). The English version is from Taylor's translation. As Lee Mosher points out, the text cited by Syrianus (τῷ ἱερῷ λόγῳ, the Sacred Discourse) has not been found, and doubt surrounds whether such a text ever existed. Interestingly, Iamblichus questions whether Pythagoras actually wrote the Sacred Discourse in the passage leading up to Thomas's footnote.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastEditorUserId="1815" LastEditDate="2023-11-08T18:58:44.780" LastActivityDate="2023-11-08T18:58:44.780" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15880" PostTypeId="1" AcceptedAnswerId="15886" CreationDate="2023-11-05T20:40:07.417" Score="4" ViewCount="450" Body="&lt;p&gt;&lt;strong&gt;I asked &lt;a href=&quot;https://math.stackexchange.com/questions/4800744/reference-request-what-were-the-problems-of-accepting-zero-negative-numbers-a&quot;&gt;this&lt;/a&gt; question on MSE and comments suggested I should ask it here&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am currently reading Baby Rudin as my second analysis book (after Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert) and I was surprised that Rudin’s book includes complex numbers, I tried to avoid complex numbers for a while because the way we were taught about them in school and even college was unsatisfactor They said something like &amp;quot;Mathematicians in the past didn’t accept complex numbers and thought they were meaningless and nonsense, but now they don't&amp;quot; and then moved on with the course without explaining why mathematicians in the past rejected complex numbers and what made them accept them later. This question has been on my mind for quite some time and I tried to avoid complex numbers as much as possible as this question is probably very hard to answer (after all, there must be a reason why we were not taught the answer to this question). But now, since I &amp;quot;have&amp;quot; to deal with complex numbers, I want to know the answer to this question. After some thought, I remembered that not only complex numbers was viewed as meaningless and nonsense then became normal, but also zero and negative numbers were not accepted and viewed as nonsense in the past, and then became normal.&lt;/p&gt;&#xA;&lt;p&gt;So I want to ask for some references (books, articles, etc.) that explain what was the problem of accepting zero, negative numbers, and complex numbers, and how these problems were solved.&lt;/p&gt;&#xA;&lt;p&gt;Jesse Madnick told me in the comments of my original question that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;These problems were &amp;quot;solved&amp;quot; by humans understanding that mathematical ideas are just that: abstract concepts. All that's required for a mathematical object to exist is a precise definition and a rigorous proof of existence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I want to ask what made modern mathematician think this way and when and how did this happen?&lt;/p&gt;&#xA;" OwnerUserId="18466" LastEditorUserId="18466" LastEditDate="2023-11-15T03:13:57.440" LastActivityDate="2023-11-15T03:13:57.440" Title="Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved?" Tags="|mathematics|reference-request|complex-analysis|book-recommendation|zero|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15881" PostTypeId="1" CreationDate="2023-11-06T18:20:01.380" Score="1" ViewCount="362" Body="&lt;p&gt;I am pretty sure that Albert Einstein explained special relativity by saying in effect, sitting on a hot stove for a minute can seem like an hour; sit with a pretty girl for an hour and it might seem like a minute.&lt;/p&gt;&#xA;&lt;p&gt;Everyone has experienced time not flowing the same (also like a watched pot never boils so people were already noticing this clearly enough to create this aphorism) in all circumstances but how does this elucidate relativity at all, especially since time really does flow differently in special relativity -- it is not a matter of perception.&lt;/p&gt;&#xA;" OwnerUserId="10856" LastEditorUserId="6609" LastEditDate="2023-11-06T23:46:41.153" LastActivityDate="2023-11-15T02:16:48.977" Title="Did Einstein really explain Relativity using the hot stove/pretty girl analogy and if so why?" Tags="|relativity-theory|einstein|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15884" PostTypeId="1" CreationDate="2023-11-07T06:50:15.907" Score="0" ViewCount="105" Body="&lt;p&gt;Can anyone recommend some good book(s) on the &lt;strong&gt;Evolution of Quantum Theory&lt;/strong&gt;, focusing on &lt;strong&gt;history&lt;/strong&gt;, not necessarily on explaining the technical/math part?&lt;/p&gt;&#xA;&lt;p&gt;The books I have read so far,&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0393078299&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality&lt;/a&gt;&amp;quot; by Manjit Kumar&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1400079969&quot; rel=&quot;nofollow noreferrer&quot;&gt;Uncertainty: Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science&lt;/a&gt;&amp;quot; by David Lindley&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B005H5O1DU&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Enigma: Physics Encounters Consciousness&lt;/a&gt;&amp;quot; by Bruce Rosenblum&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1568582323&quot; rel=&quot;nofollow noreferrer&quot;&gt;Entanglement: The Greatest Mystery in Physics&lt;/a&gt; by Aczel, Amir D&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;They all did a good job in explaining some part of the history, but not the overall history. So if I were to recommend just ONE book I won't recommend any of them.&lt;/p&gt;&#xA;&lt;p&gt;Actually, I find &amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/1840468505&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introducing Quantum Theory A Graphic Guide&lt;/a&gt;&amp;quot; does a decent job of that but I was hoping to find a more detailed version of it.&lt;/p&gt;&#xA;&lt;p&gt;Someone recommends &amp;quot;&lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0199566844&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Quantum Story A History in 40 Moments&lt;/a&gt;&amp;quot; but I have not read it. Any other recommendations?&lt;/p&gt;&#xA;&lt;p&gt;I am looking for a book about 300 pages (or less) and costs less than 30$ (I think that is a reasonable price for a book around 300 pages). All the books I list here fit in this category.&lt;/p&gt;&#xA;&lt;p&gt;BTW, I added this &amp;quot;requirement&amp;quot; after seeing the recommendation from the comments and the answer I got so far.&lt;/p&gt;&#xA;" OwnerUserId="18681" LastEditorUserId="18681" LastEditDate="2023-11-07T10:22:34.530" LastActivityDate="2023-11-07T10:22:34.530" Title="Book on the evolution of Quantum Theory" Tags="|physics|resource-recommendation|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15885" PostTypeId="2" ParentId="15884" CreationDate="2023-11-07T10:01:35.690" Score="1" Body="&lt;p&gt;If you want one book covering the whole history, I can recommend 3 volumes of&lt;/p&gt;&#xA;&lt;p&gt;Mituo Taketani, The formation and logic of quantum mechanics,&#xA;(translated from Japanese), World Scientific 2001.&lt;/p&gt;&#xA;&lt;p&gt;Some other books that I can recommend are:&lt;/p&gt;&#xA;&lt;p&gt;Sin-itiro Tomonaga, The story of spin, also translated from the Japnese, and&lt;/p&gt;&#xA;&lt;p&gt;B. L. van der Waerden, Sources of quantum mechanics, North Holland, 1967. This is a collection of original papers of the early period, (ending with Born, Jordan, Heisenberg) with a historical introduction of vdWaerden.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-11-07T10:11:33.710" LastActivityDate="2023-11-07T10:11:33.710" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15886" PostTypeId="2" ParentId="15880" CreationDate="2023-11-07T12:20:03.193" Score="5" Body="&lt;p&gt;What you write suggests that you still have difficulty with complex numbers.  I would suggest that you overcome such difficulties before you proceed to the next step such as delving deeply into real analysis and other advanced topics.  Results such as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra&quot; rel=&quot;nofollow noreferrer&quot;&gt;fundamental theorem of algebra&lt;/a&gt; are ubiquitous in mathematics, including real analysis; therefore you would be seriously impeded in your progress in math without understanding complex numbers.&lt;/p&gt;&#xA;&lt;p&gt;To respond to your historical question: the problem historically was that mathematical entities were required by many scholars to correspond to entities in the real world (&amp;quot;2&amp;quot; corresponds to two apples, etc.).  While modern mathematicians resolved the problem by rejecting such requirements, this was &lt;em&gt;not&lt;/em&gt; the solution that initially allowed mathematicians to overcome such inhibitions.&lt;/p&gt;&#xA;&lt;p&gt;Leibniz thought of numbers including surds (irrationals), negatives, imaginaries, infinitesimals as &lt;em&gt;fictions&lt;/em&gt; that are only &lt;em&gt;accidentally impossible&lt;/em&gt; but not contradictory.  This ties in to a certain extent with his doctrine of &lt;em&gt;possible worlds&lt;/em&gt;: while such fictional numbers can't be found in our world, perhaps they could be found in other worlds (modern mathematicians of course don't need such justifications).  Thus there was an approach to justifying complex numbers several centuries before they were declared to be abstract entities (possibly fluttering in some realm of the abstracta).  Leibniz's approach didn't require that.  For a recent study of Leibnizian methodology, see &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#23h&quot; rel=&quot;nofollow noreferrer&quot;&gt;this article&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In the last quarter of the 17th century, some leading scholars were still puzzled by the success of the complex numbers.  Thus, a famous episode has young Leibniz explaining to no less an authority than &lt;a href=&quot;https://en.wikipedia.org/wiki/Christiaan_Huygens&quot; rel=&quot;nofollow noreferrer&quot;&gt;C. Huygens&lt;/a&gt;, and the latter reacting: &amp;quot;there is something here beyond my comprehension&amp;quot;.  Leibniz did a lot to make complex numbers &amp;quot;legitimate&amp;quot;.  The history of the solution of the cubic in the context of the development of Leibniz's philosophy was studied in detail in &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#14c&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Euler had numerous publications using complex numbers, and of course the famous formula &lt;span class=&quot;math-container&quot;&gt;$e^{i\theta}=\cos\theta+i\sin\theta$&lt;/span&gt; was already in common use by the middle of the 18th century.  It would be a mistake to place the &amp;quot;legitimation&amp;quot; of complex numbers as late as the 19th century with Gauss.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-11-13T11:38:45.287" LastActivityDate="2023-11-13T11:38:45.287" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15887" PostTypeId="1" AcceptedAnswerId="15889" CreationDate="2023-11-08T03:33:56.390" Score="9" ViewCount="2741" Body="&lt;p&gt;I came across the above quote, and found it quite interesting. However, I struggled to find an actual source. Did Newton truly say this?&lt;/p&gt;&#xA;" OwnerUserId="2134" LastActivityDate="2023-11-08T17:47:21.073" Title="Did Newton say: &quot;If I have ever made any valuable discoveries, it has been due more to patient attention, than to any other talent&quot;?" Tags="|reference-request|newton|quote|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15888" PostTypeId="2" ParentId="15802" CreationDate="2023-11-08T07:56:13.440" Score="1" Body="&lt;p&gt;The equality &lt;span class=&quot;math-container&quot;&gt;$dE/dt=vF$&lt;/span&gt; can be considered a nontrivial insight, and not simply a definition of the term on the right (ot left). An analogous question would be: in light of the equality &lt;span class=&quot;math-container&quot;&gt;$E=mc^2$&lt;/span&gt;, why do we need the concept of energy since we already know mass and speed of light?&lt;/p&gt;&#xA;&lt;p&gt;Or maybe I misunderstand the question. There seem to be several ones. One seems to be: why did people call &lt;span class=&quot;math-container&quot;&gt;$dE/dt$&lt;/span&gt; the power and not just energy rate. There is probably an interesting history to this, but similarly you could ask why people called &lt;span class=&quot;math-container&quot;&gt;$dp/dt$&lt;/span&gt; the force and not simply the momentum rate.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="3462" LastEditDate="2023-11-08T08:04:46.873" LastActivityDate="2023-11-08T08:04:46.873" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15889" PostTypeId="2" ParentId="15887" CreationDate="2023-11-08T08:17:53.743" Score="17" Body="&lt;p&gt;So far I have been unable to find substantially identical words in Newton's writings, and as usual, authors that use this quotation do not cite a source. The origin &lt;em&gt;may&lt;/em&gt; be a remark of Newton's in private correspondence which was embellished over time.&lt;/p&gt;&#xA;&lt;p&gt;In a &lt;a href=&quot;https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00254&quot; rel=&quot;noreferrer&quot;&gt;letter&lt;/a&gt; from Isaac Newton to Richard Bentley, dated December 10, 1692, apparently in the context of his work on gravity, he wrote:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But if I have done the publick any service this way 'tis due to nothing but industry &amp;amp; a patient thought.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Thomas Reid, &lt;em&gt;Essays on the Active Powers of Man.&lt;/em&gt; Edinburgh: John Bell 1788, &lt;a href=&quot;https://www.google.com/books/edition/Essays_on_the_Active_Powers_of_Man/5PteAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA80&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 80&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sir Isaac Newton, to one who complimented him upon the force of genius,&#xA;which had made such improvements in mathematics and natural philosophy,&#xA;is said to have made this reply, which was both modest and judicious,&#xA;That if he had made any improvements in those sciences it was owing&#xA;more to patient attention than to any other talent.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some authors claim that the person who had complimented Newton was Dr. Pemberton. This is presumably &lt;a href=&quot;https://en.wikipedia.org/wiki/Henry_Pemberton&quot; rel=&quot;noreferrer&quot;&gt;Henry Pemberton&lt;/a&gt; (1694 – 1771). While Pemberton knew Newton personally and later wrote a book on Newton's philosophy, I have not been able to identify anything relevant to the question in Pemberton's publications.&lt;/p&gt;&#xA;&lt;p&gt;Tryon Edwards, &lt;em&gt;A Dictionary of Thoughts; Being a Cyclopedia of Laconic Quotations.&lt;/em&gt; New York: Cassell 1891, &lt;a href=&quot;https://www.google.com/books/edition/A_Dictionary_of_Thoughts/uUi0R_St0qYC?hl=en&amp;amp;gbpv=1&amp;amp;dq=Newton%20If%20I%20have%20ever%20made%20any%20valuable%20discoveries%2C%20it%20has%20been%20due%20more%20to%20patient%20attention%2C%20than%20to%20any%20other%20talent&amp;amp;pg=PA120&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 120&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If I have ever made any valuable discoveries, it has been owing more to patient attention, than to any other talent. — &lt;em&gt;Sir Isaac Newton.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here we have a version substantially identical to the one from the question in a reference work that &lt;em&gt;may&lt;/em&gt; have enjoyed wide circulation.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-11-08T08:25:05.080" LastActivityDate="2023-11-08T08:25:05.080" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15890" PostTypeId="2" ParentId="15852" CreationDate="2023-11-08T17:05:45.837" Score="6" Body="&lt;p&gt;I want to add to this answer some details about &amp;quot;measuring latitude&amp;quot;. The instruments used are sextant (for measuring altitude of a star (or Sun or Moon) over the horizon, and chronometer for measuring time. Under normal conditions, a student&#xA;of a naval academy passing the exam was expected to find his coordinates at sea with&#xA;these instruments with 1-2 (nautical) mile accuracy.&lt;/p&gt;&#xA;&lt;p&gt;But the conditions near the S Pole cannot be called normal.&#xA;First a clearly visible horizon may not be available, and one&#xA;may expect various irregularities of refraction near the horizon. On the other hand, observations are performed on land, so they almost certainly used an artificial horizon.&#xA;This increases accuracy.&lt;/p&gt;&#xA;&lt;p&gt;The simplest method would be to try to determine the maximal&#xA;and minimal altitude of the Sun during the day, and make sure that they are equal. But this is not practical since&#xA;this altitude changes too slowly during the day (if you are exactly at the pole it does not change at all).&lt;/p&gt;&#xA;&lt;p&gt;Surely they were familiar with more advanced methods of &amp;quot;positions lines&amp;quot;, when any two stars&#xA;are used instead of the Sun, and in a good weather, with reliable chronometer, and enough time for repeated observations,&#xA;they could probably determine their positions to a fraction of a mile.&lt;/p&gt;&#xA;&lt;p&gt;Myself, with a sextant, chronometer and artificial horizon, I was able to do this on land with accuracy 0.2 mile&#xA;(less than 300 m). The crucial condition is weather: the stars must be visible! Another important condition is the ability to transport the chronometer during the travel so that it keeps the time well.&lt;/p&gt;&#xA;&lt;p&gt;More details can be found &lt;a href=&quot;https://ir.canterbury.ac.nz/items/467b3e93-aba3-44ec-9fbf-b9a42ed17786&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For more technical analysis, including the assessment of their accuracy see&lt;/p&gt;&#xA;&lt;p&gt;The Observations of Amundsen and Scott at the South pole&#xA;by Arthur R. Hinks,&#xA;The Geographical Journal , Apr., 1944, Vol. 103, No. 4 (Apr., 1944), pp. 160-180&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-11-23T09:59:47.367" LastActivityDate="2023-11-23T09:59:47.367" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15892" PostTypeId="1" CreationDate="2023-11-10T01:14:57.663" Score="12" ViewCount="1655" Body="&lt;p&gt;When answering a &lt;a href=&quot;https://physics.stackexchange.com/a/787517/47472&quot;&gt;Physics.SE question&lt;/a&gt;, I made a claim that Newton realized that &lt;span class=&quot;math-container&quot;&gt;$F=ma$&lt;/span&gt; worked in some frames, which are called &amp;quot;inertial frames.&amp;quot;  Nowadays, we know that there are non-inertial frames.  One of the commenters suggested that started with d'Alembert.  Further research shows he is certainly associated with the use of fictitious forces to explore motion in these non-inertial frames.  But the harder question is the one I pose here: was Newton aware that in some frames, objects accelerate without forces being applied to them?&lt;/p&gt;&#xA;" OwnerUserId="5470" LastEditorUserId="12330" LastEditDate="2023-11-10T15:59:22.480" LastActivityDate="2023-11-14T01:00:39.917" Title="Did Newton know about non-inertial frames?" Tags="|physics|calculus|classical-mechanics|newton|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15893" PostTypeId="2" ParentId="15892" CreationDate="2023-11-10T08:01:58.093" Score="21" Body="&lt;p&gt;&lt;strong&gt;TL;DR&lt;/strong&gt; Yes, but...&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;1) Inertial frames&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;To say that Newton had the modern  conception of even inertial frames (based on the laws of motion), is an overstatement. Theoretically, he did not need them because he had absolute space, frames moving uniformly relative to it were inertial. In practice, he did have to spell out some of their properties to identify the fixed stars with such a frame.&lt;/p&gt;&#xA;&lt;p&gt;But this is not to say that his successors did much better. Despite Euler's and d'Alembert's systematization of mechanical concepts at the conclusion of the &lt;a href=&quot;https://hsm.stackexchange.com/a/598/55&quot;&gt;&lt;em&gt;vis viva&lt;/em&gt; controversy&lt;/a&gt;, according to &lt;a href=&quot;https://plato.stanford.edu/entries/spacetime-iframes/#EmerConcInerFram&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP's The emergence of the concept of inertial frame&lt;/a&gt;, clarity on inertial frames did not come until the late 19th century. &amp;quot;Inertial systems&amp;quot; were independently introduced by Lange and James Thomson (Kelvin's older brother) around 1885, and the crucial condition that all accelerations must come from action-reaction pairs of forces attributable to bodies within the system, was added by Muirhead in 1887.&lt;/p&gt;&#xA;&lt;p&gt;In hindsight, Mach in &lt;em&gt;Die Mechanik in ihrer Entwickelung&lt;/em&gt; found implicit anticipation of these notions in Newton's treatment of orbital motions relative to the fixed stars, and in the Corollary V of &lt;em&gt;Principia&lt;/em&gt;, which states that the center of gravity is unmoved by internal interactions, and remains at rest or in uniform motion in the absence of external forces. This is &amp;quot;as close as Newton could come to the notion of an inertial frame&amp;quot;, according to SEP.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;2) Non-inertial frames&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;What about non-inertial frames? Newton notes a special case of them in Corollary VI to the laws of motion in &lt;em&gt;Principia&lt;/em&gt;, see &lt;a href=&quot;https://plato.stanford.edu/entries/spacetime-iframes/#QuasInerFramNewtCoroVI&quot; rel=&quot;nofollow noreferrer&quot;&gt;SEP, “Quasi-inertial” frames: Newton’s Corollary VI&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If bodies are moved in any way among themselves, and are urged by equal accelerative forces along parallel lines, they will all continue to move among themselves in the same way as if they were not acted on by those forces.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What Newton had in mind is discussed in his &lt;em&gt;De Motu Corporum Liber Secundus&lt;/em&gt; (1687, the year &lt;em&gt;Principia&lt;/em&gt; came out), where the Corollary is applied to the system of Jupiter's satellites and even, hypothetically, to the Solar system. However, in both cases the point was that the effects of such accelerative force are observationally immaterial:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;It may be imagined that the sun and planets are impelled by some other force equally and in the direction of parallel lines; but such a force (by Cor. VI of the Laws of Motion) would not change the situation of the planets among themselves, nor would produce any sensible effect.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;3) Fictitious forces&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Moreover, Newton did not make the connection between non-inertial frames and fictitious forces, let alone formulate anything like Muirhead's condition. For example, while his views of the centrifugal force evolved from the early years to &lt;em&gt;Principia&lt;/em&gt;, there is no clear realization even there that it is only present in rotating frames (i.e. is fictitious).  &lt;a href=&quot;https://www.jstor.org/stable/234081&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bertoloni Meli in The Relativization of Centrifugal Force&lt;/a&gt; usefully summarizes Newton's handling of the centrifugal force as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In summary, before 1679 Newton - like Descartes, Borelli, and Leibniz - believed that orbital motion depended on the imbalance between gravity and centrifugal force; after 1684 he believed that centrifugal force was equal and opposite to gravity, from the third law of motion. In general, he explained curvilinear motion in terms of centripetal force and inertia alone, without centrifugal force; why in this case centrifugal force could be neglected, however, was not clear.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In certain passages, such as definition 5, Newton seems to associate centrifugal force with inertia. In other passages, such as the scholium to proposition 4, Book III, centrifugal force prevents an orbiting body from falling toward the center. In cases different from orbital motion, such as a planet rotating around its axis, he believed that centrifugal force was different from gravity. Hence Newton's theory of centrifugal force followed a case-by-case pattern, and the solution to one particular problem could not be easily generalized.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;For Huygens and Newton centrifugal force was the result of a curvilinear motion of a body; hence it was located in nature, in the object of investigation. According to a more recent formulation of classical mechanics, centrifugal force depends on the choice of how phenomena can be conveniently represented. Hence it is not located in nature, but is the result of a choice by the observer... Current views concerning the ideas on centrifugal force expressed by Newton in the Principia Mathematica are severely affected by the projection of modern methods and ideas that are found neither in the Principia nor in works contemporary with it.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-11-14T01:00:39.917" LastActivityDate="2023-11-14T01:00:39.917" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15897" PostTypeId="1" CreationDate="2023-11-12T11:49:03.323" Score="24" ViewCount="7546" Body="&lt;p&gt;&lt;strong&gt;Background&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I am interested in scientists and mathematicians that were afraid to publish their findings during their lifetime, and to what degree such fears hinder scientific progress.&lt;/p&gt;&#xA;&lt;p&gt;So far, I've identified three famous scientists that were afraid to publish their important findings:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Nicolas Copernicus. He delayed the publication of his work on a heliocentric model of the solar system for fear of either astronomical objections, or objections based on religious grounds.&lt;/li&gt;&#xA;&lt;li&gt;Carl Friedrich Gauss. Though fear might not always have been the main reason for him to delay the publication of his work -- he often waited quite some time before he published anything. He was a perfectionist, so perhaps one might describe it as a fear for not publishing something that was up to his high standards.&lt;/li&gt;&#xA;&lt;li&gt;Dan Shechtman. He was afraid of &lt;strike&gt;publishing his findings on quasicrystals&lt;/strike&gt; publishing alone, because he needed the expertise of his coauthors to explain his observations regarding quasicrystals. Moreover, he faced strong opposition from one of the most celebrated scientists of his time: Linus Pauling.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Some also believe Charles Darwin was afraid to publish his theory of evolution and avoided doing so for 20 years, but this has been &lt;a href=&quot;https://www.cam.ac.uk/research/news/darwins-delay-the-stuff-of-myth#:~:text=theory%20of%20evolution.-,New%20Cambridge%20research%20shows%20Darwin%20had%20no%20fears%20about%20publishing,by%20a%20Cambridge%20University%20academic.&quot; rel=&quot;noreferrer&quot;&gt;refuted&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am also aware of cases of scientists and mathematicians that were scorned for their ideas (Ludwig Boltzmann, Ignaz Semmelweis, Georg Cantor), or ignored (Gregor Mendel, George Zweig). Although these are interesting (albeit tragic) cases as well, for the purposes of this particular question I would like to restrict the examples to those that were afraid of publishing -- for whatever reason.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What other well-known scientists were afraid to publish their findings, and why?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4296" LastEditorUserId="4296" LastEditDate="2023-11-14T13:04:49.597" LastActivityDate="2024-02-18T22:10:58.257" Title="What scientists and mathematicians were afraid to publish their findings?" Tags="|biographical-details|big-list|" AnswerCount="8" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="15898" PostTypeId="1" CreationDate="2023-11-05T00:23:22.397" Score="0" ViewCount="49" Body="&lt;h2&gt;I didn't know that can happen and since I already asked the question &lt;a href=&quot;https://hsm.stackexchange.com/questions/15880/reference-request-what-were-the-problems-of-accepting-zero-negative-numbers-a?noredirect=1#comment37036_15880&quot;&gt;here &lt;/a&gt; I don't know what to do with this question should I delete it ?&lt;/h2&gt;&#xA;&lt;p&gt;I am currently reading Baby Rudin as my second analysis book (after Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert) and I was surprised that Rudin’s book includes complex numbers, I tried to avoid complex numbers for a while because the way we were taught about them in school and even college was unsatisfactor They said something like &amp;quot;Mathematicians in the past didn’t accept complex numbers and thought they were meaningless and nonsense, but now they don’t&amp;quot; and then moved on with the course without explaining why mathematicians in the past rejected complex numbers and what made them accept them later. This question has been on my mind for quite some time and I tried to avoid complex numbers as much as possible as this question is probably very hard to answer (after all, there must be a reason why we were not taught the answer to this question). But now, since I &amp;quot;have&amp;quot; to deal with complex numbers, I want to know the answer to this question. After some thought, I remembered that not only complex numbers was viewed as meaningless then became normal, but also zero and negative numbers were not accepted and viewed as nonsense in the past, and then became normal.&lt;/p&gt;&#xA;&lt;p&gt;So I want to ask for some references (books, articles, etc.) that explain what was the problem of accepting zero, negative numbers, and complex numbers, and how these problems were solved.&lt;/p&gt;&#xA;" OwnerUserId="18466" OwnerDisplayName="pie" LastEditorUserId="18466" LastEditDate="2023-11-12T17:18:21.060" LastActivityDate="2023-11-12T17:18:21.060" Title="Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved?" Tags="|complex-analysis|reference-request|book-recommendation|" AnswerCount="0" CommentCount="21" ClosedDate="2023-11-18T17:35:54.653" ContentLicense="CC BY-SA 4.0" />
  <row Id="15899" PostTypeId="2" ParentId="15897" CreationDate="2023-11-12T14:38:18.623" Score="24" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mikhael_Gromov_(mathematician)&quot; rel=&quot;noreferrer&quot;&gt;Mikhael Gromov&lt;/a&gt; started his career in the Soviet Union.  When he decided to try to leave (which was not an easy thing to do at the time, as it is now), he stopped publishing his work, fearful that if the word gets around about him being a great mathematician, the Soviet authorities will prevent him from leaving.  His last paper in Russia was published in 1973:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gromov, M. L.  Convex integration of differential relations. I. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 329–343.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His next paper, already in the West, was published five years later in 1978:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gromov, M. Manifolds of negative curvature. J. Differential Geometry 13 (1978), no. 2, 223–230.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This was followed by an explosion of publications: by 1981, he published 12 new papers, all of them ground-breaking in their respective field.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-11-12T14:38:18.623" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15900" PostTypeId="2" ParentId="15880" CreationDate="2023-11-12T19:05:46.733" Score="5" Body="&lt;p&gt;The problem with accepting complex numbers, when your only experience with &amp;quot;numbers&amp;quot; is the real number line, is lacking a visualization of them. No ordinary (i.e., real) number can square to a negative number, so square roots of negative numbers were considered impossible or fictional.&lt;/p&gt;&#xA;&lt;p&gt;It took the authority of someone like &lt;em&gt;Gauss&lt;/em&gt; to use complex numbers in productive ways in order to make their usage more widely accepted.  Moreover, the realization that complex numbers can be visualized as points in the ordinary plane helped lower the barrier to working with them: we could &lt;em&gt;see&lt;/em&gt; what adding and multiplying does to complex numbers, so they are no longer scary objects.&lt;/p&gt;&#xA;&lt;p&gt;The Springer-Verlag book &amp;quot;Numbers&amp;quot; by Ebbinghaus and several other people is a great resource on this: see the 1st section in Chapter 3.&lt;/p&gt;&#xA;&lt;p&gt;The Veritasium video on the cubic formula &lt;a href=&quot;https://www.youtube.com/watch?v=cUzklzVXJwo&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; describes the algebraic source of complex numbers: solving &lt;em&gt;cubic&lt;/em&gt; equations. A cubic polynomial clearly, by a graph, always has at least one real root, but the cubic formula sometimes expresses that root in terms of square roots of negative numbers (the formulas is a sum of two parts, and the imaginary parts cancel out, leaving a purely real number even when separate parts of the formula are not real).&lt;/p&gt;&#xA;&lt;p&gt;The realization that math can describe things not immediately related to things in the real world was a slow process that accelerated throughout the 1800s. It came up most strikingly in geometry, with the discovery of non-Euclidean geometry (the hyperbolic plane) independently by Lobachevsky and Bolyai, as well as Gauss earlier, who kept his discovery secret because he knew it would be really controversial. Riemann introduced the idea of manifolds in any number of dimensions decades before others could really appreciate what he had done. In algebra there was the development of unusual number systems like  quaternions by Hamilton, which require 4 coordinates (at a time when the idea of 4 dimensions was unfamiliar).&lt;/p&gt;&#xA;&lt;p&gt;In 1910, Ernst Steinitz wrote a very influential article on a general theory of fields. Its axiomatic approach was revolutionary and led to the viewpoint in abstract algebra advocated in the 1920s and 1930s by Artin and Noether, reaching a worldwide audience (well, among math students) when their lectures appeared in van der Waerden's textbook &lt;em&gt;Morderne Algebra&lt;/em&gt; (1930). Steinitz wrote in a footnote in the introduction to his paper that the main inspiration leading him to prepare that article was Kurt Hensel's creation of the very unusual fields of &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic numbers, which unlike all earlier fields were not in any natural way a subfield of the real or complex numbers or fields of functions (there are finite fields too, going back to Galois in the early 1800s).&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-11-12T19:31:18.620" LastActivityDate="2023-11-12T19:31:18.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15901" PostTypeId="2" ParentId="15897" CreationDate="2023-11-12T23:42:22.163" Score="8" Body="&lt;p&gt;Schrödinger was afraid of publishing what is now known as the Klein-Gordon equation because it did not describe experimental data correctly. Instead, he published the nonrelativistic version, which described the data correctly (without relativistic corrections). Reference: Dirac, Recollections of an Exciting Era//History of Twentieth Century Physics: Proceedings of the International School of Physics &amp;quot;Enrico Fermi&amp;quot;. Course LVII. - New York; London: Academic Press, 1977. -P.109-146. Dirac recollects his conversation with Schrödinger that took place in (approximately) 1940.&lt;/p&gt;&#xA;&lt;p&gt;Kronig did not publish the idea of electron rotation as the cause of &amp;quot;duplicity&amp;quot; of atomic spectra because of Pauli's critique. Reference: A. Pais, Phys. Today 42(12), 34 (1989);&lt;/p&gt;&#xA;&lt;p&gt;If I understand correctly, Pauli was afraid of publishing what is now known as the Yang-Mills field because he believed it implied existence of extra zero-mass particles, which are not observed. Reference: A. Pais, The Genius of Science. A portrait gallery of twentieth-century physicists, Oxford University Press, 2000, pp.243-245.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastEditorUserId="9808" LastEditDate="2023-11-14T04:10:48.237" LastActivityDate="2023-11-14T04:10:48.237" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15902" PostTypeId="2" ParentId="15835" CreationDate="2023-11-13T00:41:04.703" Score="2" Body="&lt;p&gt;In the famous XI-th Supplement in Dirichlet-Dedekind's &lt;em&gt;Vorlesungen über Zahlentheorie&lt;/em&gt;, quotient rings modulo an ideal (prime ideal or general nonzero ideal) are not used, but systems of representatives are used, just like the concrete way students may first learn to do arithmetic mod &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; in the integers using the representatives &lt;span class=&quot;math-container&quot;&gt;$0, 1, \ldots, m-1$&lt;/span&gt; exclusively.&lt;/p&gt;&#xA;&lt;p&gt;On p. 509 (in Section 171), Dedekind writes &lt;span class=&quot;math-container&quot;&gt;$(\mathfrak a,\mathfrak b)$&lt;/span&gt; for the number of incongruent numbers in &lt;span class=&quot;math-container&quot;&gt;$\mathfrak a$&lt;/span&gt; modulo &lt;span class=&quot;math-container&quot;&gt;$\mathfrak b$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;On p. 564 (in Section 180) Dedekind writes &lt;span class=&quot;math-container&quot;&gt;$(\mathfrak o,{\mathfrak o}\mu) = \pm{N}(\mu)$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$N$&lt;/span&gt; is the norm mapping. Today we'd say &lt;span class=&quot;math-container&quot;&gt;$\mathfrak o/(\mu)$&lt;/span&gt; has size &lt;span class=&quot;math-container&quot;&gt;$|N(\mu)|$&lt;/span&gt;. I guess Weierstrass's absolute value symbol was not yet universal.&lt;/p&gt;&#xA;&lt;p&gt;On pp. 570-572 (still in Section 180), Dedekind describes basic properties of elements of &lt;span class=&quot;math-container&quot;&gt;$\mathfrak o$&lt;/span&gt; modulo &lt;span class=&quot;math-container&quot;&gt;$\mathfrak p$&lt;/span&gt;, which we'd call the finite field &lt;span class=&quot;math-container&quot;&gt;$\mathfrak o/\mathfrak p$&lt;/span&gt;, e.g., the &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;th power map is additive mod &lt;span class=&quot;math-container&quot;&gt;$\mathfrak p$&lt;/span&gt; and for each &lt;span class=&quot;math-container&quot;&gt;$\alpha \bmod \mathfrak p$&lt;/span&gt; there's a least &lt;span class=&quot;math-container&quot;&gt;$a \geq 1$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$\alpha^{p^a} \equiv a \bmod \mathfrak p$&lt;/span&gt;, and in terms of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; the minimal polynomial of &lt;span class=&quot;math-container&quot;&gt;$\alpha \bmod \mathfrak p$&lt;/span&gt; over &lt;span class=&quot;math-container&quot;&gt;$\mathbf Z/p\mathbf Z$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$(t-\alpha)(t-\alpha^p)\cdots (t-\alpha^{p^{a-1}})$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In footnotes on these pages about finite fields arising as number field quotient rings &lt;span class=&quot;math-container&quot;&gt;$\mathfrak o/\mathfrak p$&lt;/span&gt;, Dedekind refers the reader to his earlier papers  &lt;a href=&quot;https://eudml.org/doc/135869&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die Discriminanten endlicher Körper&lt;/a&gt; (the term &amp;quot;finite field&amp;quot; in that title means number fields, as they have finite dimension over the rationals) and &lt;a href=&quot;https://eudml.org/doc/135827&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber den Zusammenhang zwischen der Theorie der ideale und der Theorie der höheren Congruenzen&lt;/a&gt;.  The second paper has been translated into English by Gouvea and Webster: see &lt;a href=&quot;https://arxiv.org/pdf/2107.08905.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In those days, people spoke about a &amp;quot;higher congruence&amp;quot; (see the end of the 2nd title) or &amp;quot;double modulus&amp;quot;. A higher congruence was a congruence between polynomials whose coefficients are themselves in a field defined by congruences, namely &lt;span class=&quot;math-container&quot;&gt;$\mathbf F_p[x]/(f(x))$&lt;/span&gt;. For example, they might view our &lt;span class=&quot;math-container&quot;&gt;$\mathbf Z[i]/(3)$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$(\mathbf Z/3\mathbf Z)[x]/(x^2+1)$&lt;/span&gt; as congruences mod &lt;span class=&quot;math-container&quot;&gt;$3,x^2+1$&lt;/span&gt;. The earlier post on this site about &lt;a href=&quot;https://hsm.stackexchange.com/questions/12656/history-of-irreducible-polynomials-and-motivation-for-them&quot;&gt;irreducible polynomials&lt;/a&gt; is worth reading.&lt;/p&gt;&#xA;&lt;p&gt;Dickson, in the preface to his 1900 book &amp;quot;Linear groups with an exposition of the Galois field theory&amp;quot; (note &amp;quot;Galois field theory&amp;quot; = finite field theory), wrote&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The linear groups investigated by Galois, Jordan and Serret were defined for the field of integers taken modulo &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;; the general Galois field enters only incidentally in their investiagations. The linear fractional group in a general Galois field was partially investigated by Mathieu, and exhaustively by Moore, Burnside and Wiman. The work of Moore first emphasized the importance of employing in group problems the general Galois field in place of the special field of integers [modulo &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;], the results being almost as simple and the investigations no more complicated.  In this way the systems of linear groups studied by Jordan have all be [sic] generalized by the author and in the investigation of new systems the Galois field has been employed ab initio.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I had never heard of Wiman before. It is Anders Wiman.  In any case, Burnside, Dickson, Moore, and Wiman all began their work involving finite fields in the 1890s (Burnside had earlier worked in applied math), so I think the viewpoint that &amp;quot;everything we can be do for the integers mod &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; should be done for all finite fields&amp;quot; really took off in that decade and the early years of the 20th century, not during the mid-19th century.  This isn't a surprise, since the classification of general finite fields is due to Moore in 1893.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-11-13T02:04:09.777" LastActivityDate="2023-11-13T02:04:09.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15903" PostTypeId="2" ParentId="15897" CreationDate="2023-11-13T06:17:48.920" Score="15" Body="&lt;p&gt;The moment I saw this Question , I connected it with &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; , which I think is the Earliest Example of hiding ( Mathematical ) Discoveries out of fear.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Square_root_of_2#History&quot; rel=&quot;noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Square_root_of_2#History&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. &lt;strong&gt;Little is known with certainty&lt;/strong&gt; about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an &lt;strong&gt;official secret&lt;/strong&gt; the discovery that the square root of two is irrational, and, according to legend, &lt;strong&gt;Hippasus was murdered&lt;/strong&gt; for divulging it. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant, for example by Conway &amp;amp; Guy (1996).&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;That text contains my high-lights :&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Pythagoreans did not &amp;quot;publish&amp;quot; or &amp;quot;publicize&amp;quot; the irrationality of &lt;span class=&quot;math-container&quot;&gt;$\sqrt{2}$&lt;/span&gt; ...&lt;/li&gt;&#xA;&lt;li&gt;who-ever considered doing so would have the fear of getting murdered ...&lt;/li&gt;&#xA;&lt;li&gt;a little fancifully , the secret is still not revealed regarding the who or the when of that Scary Discovery ...&lt;/li&gt;&#xA;&lt;li&gt;the truth has been totally buried , lost &amp;amp; forgotten , ensuring that we may never ever know all the facts of that matter ...&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="18814" LastEditorUserId="18814" LastEditDate="2023-11-13T08:40:25.953" LastActivityDate="2023-11-13T08:40:25.953" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15904" PostTypeId="2" ParentId="15897" CreationDate="2023-11-13T11:04:42.940" Score="2" Body="&lt;p&gt;William Thurston did not publish many of his results. Instead he circulated them in unfinished form in some narrow circles. Some reasons for this he explained in his article&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On proof and progress in mathematics. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 161–177.&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2023-11-16T07:35:25.233" LastActivityDate="2023-11-16T07:35:25.233" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15905" PostTypeId="2" ParentId="15897" CreationDate="2023-11-13T15:20:37.873" Score="7" Body="&lt;p&gt;Newton. Newton was afraid of controversies and of being involved in public disputes. As a result he was reluctant to publish his results, preferring to communicate them to a narrow circle of his correspondents, or just keep to himself.&#xA;Many of his mathematical works were published with large delay, or after his death. His most important work (Principia) was literally wrestled from him by Halley, who published it&#xA;at his own expense.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-11-13T15:20:37.873" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15906" PostTypeId="2" ParentId="15897" CreationDate="2023-11-14T03:07:16.540" Score="1" Body="&lt;p&gt;(Not sure if this is the kind of answer you're looking for, but since both questions are HNQs, I thought I'd write it.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://academia.stackexchange.com/questions/203738/consequences-of-publishing-work-using-text-from-millions-of-sci-hub-articles&quot;&gt;duncster94&lt;/a&gt; appears to be doing research related to machine learning as applied to the biology literature. They are hesitant to publish their results, because the data source they used for the research is illegal, and they are afraid of blowback.&lt;/p&gt;&#xA;&lt;p&gt;I don't know how much this is affecting the field, however.&lt;/p&gt;&#xA;" OwnerUserId="19327" LastActivityDate="2023-11-14T03:07:16.540" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15908" PostTypeId="2" ParentId="15406" CreationDate="2023-11-14T14:41:30.670" Score="2" Body="&lt;p&gt;Leslie Connock Jesty obtained a patent for a method of information transmission relying on hypersphere packing. There are further details about him in my question and answer &lt;a href=&quot;https://hsm.stackexchange.com/questions/15859/did-anyone-apply-for-a-patent-based-on-sphere-packing&quot;&gt;Did anyone apply for a patent based on sphere packing?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although hypersphere packing is a mathematical problem it is arguable whether he was a mathematician. His name does not appear in the &lt;a href=&quot;https://www.mathgenealogy.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematics Genealogy Project&lt;/a&gt; or in the database &lt;a href=&quot;https://zbmath.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;zbMATH&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2023-11-14T14:41:30.670" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15909" PostTypeId="2" ParentId="15406" CreationDate="2023-11-14T15:22:27.660" Score="2" Body="&lt;p&gt;Sergeyev &lt;a href=&quot;https://patents.google.com/patent/EP1728149B1/en&quot; rel=&quot;nofollow noreferrer&quot;&gt;patented infinity&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;N.B.  Just to avoid any misunderstanding: my views concerning this are summarized &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/pirahodimetz.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-11-14T15:22:27.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15910" PostTypeId="1" CreationDate="2023-11-14T15:57:39.573" Score="0" ViewCount="54" Body="&lt;p&gt;When did exactly the Langevin twins occur in relativity?&lt;/p&gt;&#xA;&lt;p&gt;My first idea was to simply look up Langevin's &lt;a href=&quot;https://amshistorica.unibo.it/diglib.php?inv=7&amp;amp;int_ptnum=108&amp;amp;term_ptnum=302&quot; rel=&quot;nofollow noreferrer&quot;&gt;original paper&lt;/a&gt; (in 1911), but he never mentions them to be twins :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;our traveller would need only to agree to being shut inside a projectile that the Earth would launch at a velocity sufficiently close to that of light, but still less than it, which is physically possible, arranging for an encounter with say, a star to take place at the end of one year in the lifetime of the traveller and to send him back towards the Earth at the same velocity. Having returned to Earth two years older, provided that his velocity has remained within the range of only one part in twenty thousands less than the velocity of light. The most reliably established experimental facts of physics enable us to state that this is indeed what would happen.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It does not seem to be mentioned anywhere in there that this is some experiment involving two twin brothers.&lt;/p&gt;&#xA;&lt;p&gt;As far as I can tell, the earliest mention I can find of this would be in Schmidt's popular science book &amp;quot;Relativity and the universe&amp;quot; (1921), where quoting Hermann Weyl, he gives this example :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The life process of a human being may well be compared with the movement of a clock. Its course,therefore,will depend on the state of motion of the system of coordinates in which the particular man is spending his life. Now imagine twin brothers who one day take leave of each other. Let one of them stay at home, i.e., let him be permanently at rest in a suitable system of coordinates. But the other is to go on a journey and to travel with velocities as great as possible relatively to his home. If, many years afterwards,this traveller returned home he would find himself noticeably younger than his brother who remained at home.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this the actual earliest appearance of the Langevin twins?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="2628" LastEditDate="2023-11-14T16:26:13.967" LastActivityDate="2023-11-14T16:26:13.967" Title="The Langevin twins" Tags="|relativity-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15911" PostTypeId="1" AcceptedAnswerId="15912" CreationDate="2023-11-14T16:37:37.143" Score="1" ViewCount="179" Body="&lt;p&gt;Cross posted at &lt;a href=&quot;https://mathoverflow.net/q/457909/37385&quot;&gt;Math Overflow&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;For historical reasons, the English term &amp;quot;power set&amp;quot; in set theory is a translation of the German &amp;quot;Potenzmenge&amp;quot;, which is still in use in German mathematical literature.&lt;/p&gt;&#xA;&lt;p&gt;Did some English language mathematics use the term &amp;quot;potency set&amp;quot; for power set, and if so, what are good references.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastEditorUserId="8094" LastEditDate="2023-11-16T02:02:14.080" LastActivityDate="2023-11-16T02:02:14.080" Title="Was &quot;potency set&quot; used for power set?" Tags="|mathematics|terminology|reference-request|set-theory|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15912" PostTypeId="2" ParentId="15911" CreationDate="2023-11-14T21:25:39.627" Score="4" Body="&lt;p&gt;The paper Rieger, L.&#xA;&amp;quot;On the consistency of the generalized continuum hypothesis.&amp;quot;&#xA;&lt;em&gt;Rozprawy Mat.&lt;/em&gt; 31 (1963) described in MR0146091, which can be downloaded from &lt;a href=&quot;http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-5ca4b532-4aa3-4856-924f-8bbed25e3c33&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-5ca4b532-4aa3-4856-924f-8bbed25e3c33&lt;/a&gt; (click on the &amp;quot;31. książka&amp;quot;) uses the phrase &amp;quot;potency set axiom&amp;quot; on pp 15, 20, 38 (and maybe elsewhere).&lt;/p&gt;&#xA;&lt;p&gt;When discussing the &amp;quot;potency set axiom&amp;quot; Rieger refers to axiom C3 in Gödel's 1940&#xA;&lt;em&gt;The Consistency of the Continuum Hypothesis&lt;/em&gt;&#xA;(Annals of Mathematics Studies, Princeton), whose text is reproduced in vol II of the 1990 edition of Gödel's &lt;em&gt;Collected Works&lt;/em&gt;. Towards the end of Chapter II (on p.32 of the &lt;em&gt;Collected Works&lt;/em&gt;) is  a group of 4 axioms, labeled &amp;quot;Group C&amp;quot;.  Gödel describes the third of them as providing for the existence of a &amp;quot;set including the set of all subsets&amp;quot; of a given set, but does not use any of the terms &amp;quot;Potenzmenge&amp;quot;, &amp;quot;power set&amp;quot; or &amp;quot;potency set&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;So I suppose  Rieger translated the usual term &amp;quot;Axiom der Potenzmenge&amp;quot; (used by Zermelo in his 1907 paper) into English, with the help of a dictionary, not knowing that  &amp;quot;power set&amp;quot; is idiomatic and that  &amp;quot;potency set&amp;quot; is never used. (See &lt;a href=&quot;https://en.wikipedia.org/wiki/Ladislav_Rieger&quot; rel=&quot;nofollow noreferrer&quot;&gt;this&lt;/a&gt; for info about Rieger's life.)&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2023-11-16T01:38:38.590" LastActivityDate="2023-11-16T01:38:38.590" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="15913" PostTypeId="1" AcceptedAnswerId="15917" CreationDate="2023-11-15T01:35:06.713" Score="13" ViewCount="1082" Body="&lt;p&gt;I gather that the symbols &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\supset$&lt;/span&gt; &lt;a href=&quot;https://www.math.ucdavis.edu/%7Eanne/WQ2007/mat67-Common_Math_Symbols.pdf&quot; rel=&quot;noreferrer&quot;&gt;were introduced&lt;/a&gt; by Ernst Schröder in his 1890 &lt;em&gt;Vorlesungen über die Algebra der Logik&lt;/em&gt;. This account also appears—attributed to good old Cajori—in &lt;a href=&quot;https://math.stackexchange.com/questions/1253991/bourbaki-and-the-symbol-for-set-inclusion&quot;&gt;an answer on the Mathematics SE site&lt;/a&gt; along with a quotation from Peano’s 1889 &lt;em&gt;Arithmetices Principia Nova Methodo Exposita&lt;/em&gt; explaining use of a left-right-reflected upper-case C to mean contained in (or, as he put it in the Latin original, &lt;em&gt;continetur&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;It seems safe to hypothesize that Peano chose the reflected C &lt;strong&gt;because&lt;/strong&gt; C is the first letter of &lt;em&gt;continetur&lt;/em&gt;. Indeed, I recall being taught in grade school in the 1960s that such was the motivation for the symbol. But Nathan Moore, though a fine fifth-grade teacher, cannot be presumed to have been an authority on either the history of mathematical notation or the influences of Latin—through Norman French—on the lexicon of Modern English. And anyway, maybe my memory has corrupted itself over the past half century  and all he really said was that the resemblance between &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; and “C for &lt;em&gt;&lt;strong&gt;c&lt;/strong&gt;ontained in&lt;/em&gt;” can serve as a handy mnemonic. But back to Peano. Even though the hypothesis&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Peano’s choice stemmed from &lt;em&gt;continetur&lt;/em&gt;’s starting with C&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;does seem reasonable, that apparent reasonableness does not constitute sufficient grounds for asserting that the hypothesis is veridical.&lt;/p&gt;&#xA;&lt;p&gt;Apparently relevant too is the fact that in the same work, Peano also uses &lt;span class=&quot;math-container&quot;&gt;$\cap$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\cup$&lt;/span&gt; as we do today, but defines them as &lt;em&gt;et&lt;/em&gt; and &lt;em&gt;vel&lt;/em&gt;, respectively (in English, &lt;em&gt;and&lt;/em&gt; and &lt;em&gt;or&lt;/em&gt;). Although I relied in grade school on the anglophone-obvious mnemonic “&lt;span class=&quot;math-container&quot;&gt;$\cup$&lt;/span&gt; is for &lt;em&gt;&lt;strong&gt;u&lt;/strong&gt;nion&lt;/em&gt;,” the Peano-reasonable hypothesis (analogous to “mirrored C is for &lt;em&gt;&lt;strong&gt;c&lt;/strong&gt;ontinetur&lt;/em&gt;”) is “&lt;span class=&quot;math-container&quot;&gt;$\cup$&lt;/span&gt; is for &lt;em&gt;&lt;strong&gt;v&lt;/strong&gt;el&lt;/em&gt;.”&lt;/p&gt;&#xA;&lt;p&gt;But here’s a major wrinkle that sings out for anyone working in order theory. There are enchanting parallels between &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt;. Consider:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;We denote the reflexive closure of &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\subseteq$&lt;/span&gt;. Likewise, we denote the reflexive closure of &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; is the canonical, the prototypical partial order on &lt;span class=&quot;math-container&quot;&gt;$\bf R$&lt;/span&gt;. Likewise, the canonical partial order on any power set &lt;span class=&quot;math-container&quot;&gt;$2^S$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$\subseteq$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;li&gt;We denote the order duals of &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\leq$&lt;/span&gt; by the left-right-reflected symbols &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\geq$&lt;/span&gt;, respectively. Likewise, the duals of &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\subseteq$&lt;/span&gt; we denote by the reflections &lt;span class=&quot;math-container&quot;&gt;$\supset$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\supseteq$&lt;/span&gt;, respectively.&lt;/li&gt;&#xA;&lt;li&gt;Of the symbols &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt;, we use the convex-left one for the relation of increasing magnitude: &lt;span class=&quot;math-container&quot;&gt;$(x,y)\in &amp;lt;$&lt;/span&gt; means that that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is the number with the lesser magnitude. Likewise, we use the convex-left &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; for the relation of increasing cardinality: &lt;span class=&quot;math-container&quot;&gt;$(X,Y)\in\subset$&lt;/span&gt; means that &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is the set with the lesser cardinality. This is consistent with the way that the notational similarity between &lt;span class=&quot;math-container&quot;&gt;$0$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\emptyset$&lt;/span&gt; reflects the mathematical similarity between the objects that those two symbols represent.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I note in passing that because of all these order-theoretic parallels, I am among the authors who use &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; only to mean the reflexive reduction of &lt;span class=&quot;math-container&quot;&gt;$\subseteq$&lt;/span&gt;, which is to say, &lt;span class=&quot;math-container&quot;&gt;$\subsetneq$&lt;/span&gt;. Of course, that usage of &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; is not the more common, so I always point it out explicitly. And sometimes I simply use &lt;span class=&quot;math-container&quot;&gt;$\subsetneq$&lt;/span&gt; instead.&lt;/p&gt;&#xA;&lt;p&gt;But more importantly, it seems to me that the parallels are just too good to have happened entirely by accident. Indeed, in his &lt;a href=&quot;https://math.hawaii.edu/%7Etom/history/set.html&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Earliest Uses of Symbols of Set Theory and Logic&lt;/em&gt;&lt;/a&gt;, Tom Craven states that before Schröder introduced &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt; et al., “the symbols &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$&amp;gt;$&lt;/span&gt; had been used.&amp;quot; Craven may have found this in Cajori, to which I don’t have access.&lt;/p&gt;&#xA;&lt;p&gt;So, was Schröder influenced by Peano’s reflected C for &lt;em&gt;continetur&lt;/em&gt;? And more generally, what can we say for certain about all the considerations that did lead Schröder to his choice… and, perhaps to different degrees, led to his notation’s now universal acceptance?&lt;/p&gt;&#xA;" OwnerUserId="10975" LastEditorUserId="55" LastEditDate="2023-11-15T12:51:12.380" LastActivityDate="2023-11-15T14:07:22.290" Title="What was the motivation for the choice of the subset symbol?" Tags="|notation|set-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15914" PostTypeId="2" ParentId="15881" CreationDate="2023-11-15T02:09:20.817" Score="2" Body="&lt;p&gt;@njuffa points in his comment to &lt;a href=&quot;https://quoteinvestigator.com/2014/11/24/hot-stove/&quot; rel=&quot;nofollow noreferrer&quot;&gt;a strong argument&lt;/a&gt; for the case that he did indeed use the analogy.&#xA;As to its suitability, it’s reasonable to read it as tongue in cheek.&lt;/p&gt;&#xA;&lt;p&gt;There’s a comparably rough analogy that is also often attributed to Einstein (though this one &lt;a href=&quot;https://quoteinvestigator.com/2012/02/24/telegraph-cat/&quot; rel=&quot;nofollow noreferrer&quot;&gt;appears to be apocryphal&lt;/a&gt;). That story goes, essentially,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Someone once admitted to Einstein his ignorance of how wireless [radio] works. Einstein explained, “Imagine a very long cat with its tail in Berlin and its head in Paris. If one steps on its tail in Berlin, it is heard to squawk in Paris. Wireless is like that. But without the cat.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="10975" LastEditorUserId="10975" LastEditDate="2023-11-15T02:16:48.977" LastActivityDate="2023-11-15T02:16:48.977" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15915" PostTypeId="1" AcceptedAnswerId="15923" CreationDate="2023-11-15T11:14:13.263" Score="3" ViewCount="132" Body="&lt;p&gt;I am trying to find who first translated Maxwell's equations and Lorentz's force into the Lagrangian formalism. It seems a very straightforward thing to do if you know enough of electromagnetism and Lagrangians so maybe nobody cared to give too much credit to the scientist who first published it.&lt;/p&gt;&#xA;&lt;p&gt;Maybe there were some attemps to do it even before Maxwell's equation?&lt;/p&gt;&#xA;&lt;p&gt;Who was the first to formulate the Lagrangian of electromagnetism?&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-11-15T12:36:12.597" LastActivityDate="2023-11-17T18:03:05.797" Title="Who postulated the first Lagrangian for electrodynamics?" Tags="|physics|electromagnetism|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15916" PostTypeId="2" ParentId="15839" CreationDate="2023-11-15T12:39:52.640" Score="4" Body="&lt;p&gt;The Nobel Prize organization was kind to answer to my question. They are lagging to reveal the information &amp;quot;due to lack of resources&amp;quot;. They also responded that concerning Physics, these are still kept secret due to living nominees.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastActivityDate="2023-11-15T12:39:52.640" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15917" PostTypeId="2" ParentId="15913" CreationDate="2023-11-15T12:50:06.107" Score="14" Body="&lt;p&gt;That Schröder was influenced by Peano is very unlikely. When introducing his inclusion symbol in &lt;em&gt;Vorlesungen&lt;/em&gt;, Schröder explicitly cites &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; for numbers in arithmetic as motivation, and his version looks more like rounded &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt; than c. He mentions Du Bois-Reymond's &lt;span class=&quot;math-container&quot;&gt;$≺$&lt;/span&gt; and Peirce's &lt;span class=&quot;math-container&quot;&gt;$−\!\!\!&amp;lt;$&lt;/span&gt; (Peirce was also Schröder's main source for the content of &lt;em&gt;Vorlesungen&lt;/em&gt;), but not Peano. &lt;a href=&quot;https://maa.org/press/periodicals/convergence/math-origins-the-logical-symbols&quot; rel=&quot;noreferrer&quot;&gt;Tou, Math Origins: The Logical Symbols&lt;/a&gt; is a useful survey that has photocopies of text from original sources.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, while Peano's symbol, unlike Schröder's, looks more like the modern &lt;span class=&quot;math-container&quot;&gt;$\subset$&lt;/span&gt;, its meaning is reversed compared to both modern and Schröder's. Peano's &lt;span class=&quot;math-container&quot;&gt;$a\,⊃\,b$&lt;/span&gt; for classes &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; means 'every &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;'. This is in line with de Morgan's X)Y used for the same purpose earlier, and Peano's own use of &lt;span class=&quot;math-container&quot;&gt;$\,⊃\,$&lt;/span&gt; also for implication. The latter was adopted by Russell and Whitehead in &lt;em&gt;Principia&lt;/em&gt; and became common until eventually and slowly displaced by Hilbert's arrow.&lt;/p&gt;&#xA;&lt;p&gt;Why the modified Schröder's symbol prevailed is hard to say. The similarity with c shape might have helped. Gergonne derived an inclusion symbol from c shape long before Peano, and it probably appealed to people after. Hausdorff's adoption in 1914 &lt;a href=&quot;https://archive.org/details/grundzgedermen00hausuoft/page/4/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Grundzüge der Mengenlehre&lt;/em&gt;&lt;/a&gt;, the first textbook on set theory, helped too. His symbol still looked like Schröder's rounded &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt;. One can speculate that it was later uniformized to the shape of implication for typographic reasons.&lt;/p&gt;&#xA;&lt;p&gt;As for &lt;span class=&quot;math-container&quot;&gt;$&amp;lt;$&lt;/span&gt;, it goes back to 17th century and was derived from rather murky and quaint considerations by Harriot's editors, see &lt;a href=&quot;https://www.jstor.org/stable/4025130&quot; rel=&quot;noreferrer&quot;&gt;Tanner, On the Role of Equality and Inequality in the History of Mathematics&lt;/a&gt; and &lt;a href=&quot;https://hsm.stackexchange.com/a/14671/55&quot;&gt;History of greater-than symbol used in reverse?&lt;/a&gt; here.  &lt;a href=&quot;https://archive.org/details/historyofmathema031756mbp/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Cajori's History Of Mathematical Notations, vol. I &lt;/a&gt; is freely available on Internet Archive, and documents a long history of experimentation with symbols from 17th to 20th century, with all sorts of shapes and motivations for them.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-11-15T14:07:22.290" LastActivityDate="2023-11-15T14:07:22.290" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15919" PostTypeId="2" ParentId="15915" CreationDate="2023-11-16T23:39:28.660" Score="1" Body="&lt;p&gt;Various versions of electromagnetic Lagrangian were &amp;quot;postulated&amp;quot; by elastic aether theorists before Maxwell, see &lt;a href=&quot;https://archive.org/details/historyoftheorie00whitrich&quot; rel=&quot;nofollow noreferrer&quot;&gt;Whittaker, History of the theories of aether and electricity&lt;/a&gt;. Green appears to be the first in 1837 (p.151):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The needful reformation of the elastic-solid theory of reflexion was effected by Green, in a paper read to the Cambridge Philosophical Society in December, 1837. Green, though inferior to Cauchy as an analyst, was his superior in&#xA;physical insight; instead of designing boundary-equations for the express purpose of yielding Fresnel's sine and tangent formulae, he set to work to determine the conditions which are actually satisfied at the interfaces of real elastic solids.&#xA;These he obtained by means of general dynamical principles.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hal.science/hal-01307751/document&quot; rel=&quot;nofollow noreferrer&quot;&gt;Love's Treatise on the mathematical theory of elasticity&lt;/a&gt; credits Green with introducing the variational method even into the elasticity theory itself. MacCullagh in 1839 (p.155) and Stokes c. 1867 (p.172) followed him with elastic aether. By Maxwell's time, it was routine. Whittaker mentions that Maxwell himself did it by analogy with elastic theories (p.308), as did Fitzgerald in 1879 with a mistake corrected by Larmor in 1893 (p.368). Lorentz &lt;a href=&quot;https://en.wikipedia.org/wiki/Lorentz_force#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;used a Lagrangian&lt;/a&gt; to derive the formula for the force later named after him in &lt;a href=&quot;https://de.wikisource.org/wiki/Versuch_einer_Theorie_der_electrischen_und_optischen_Erscheinungen_in_bewegten_K%C3%B6rpern&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern&lt;/em&gt; (1895)&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-11-16T23:39:28.660" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15920" PostTypeId="1" CreationDate="2023-11-17T02:33:31.490" Score="1" ViewCount="83" Body="&lt;p&gt;In the article &lt;em&gt;Quine’s New Foundations&lt;/em&gt; of The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Thomas Forster writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In [1944] Hailperin gave the first of a number of finite axiomatisations of NF now known. Many of them exploit the function &lt;span class=&quot;math-container&quot;&gt;$x\mapsto \{y|x\in y\}$&lt;/span&gt;&#xA;which is injective and total and is an &lt;span class=&quot;math-container&quot;&gt;$\in$&lt;/span&gt;-isomorphism. This function was known to Whitehead, who suggested to Quine that &lt;span class=&quot;math-container&quot;&gt;$\{y|x\in y\}$&lt;/span&gt;&#xA;should be called the “essence” of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; (a terminology clearly suggested by a view of sets as properties-in-extension).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How did Alfred North Whitehead communicate the essential point of view to Quine?&lt;/p&gt;&#xA;" OwnerUserId="8094" LastEditorUserId="8094" LastEditDate="2023-11-17T13:02:58.630" LastActivityDate="2023-11-17T13:02:58.630" Title="Whence Whitehead's essence?" Tags="|mathematics|reference-request|set-theory|logic|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15921" PostTypeId="1" CreationDate="2023-11-17T05:27:48.480" Score="4" ViewCount="133" Body="&lt;p&gt;There is a famous incident in the history of mathematics involving the mathematician Per Enflo being awarded a live goose by Stanislaw Mazur for solving problem 153 in the Scottish Book by constructing the first known example of a Banach space failing to have the approximation property. The incident is somewhat amusing because the award was offered in 1936 but not claimed until 1972, at which point the prize of a live goose was decidedly anachronistic.&lt;/p&gt;&#xA;&lt;p&gt;There is a commonly used picture of the event, which seems to be more or less the only one online:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/E5Mge.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/E5Mge.jpg&quot; alt=&quot;Stanislaw Mazur awarding a live goose to Per Enflo.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I was told recently that this was actually broadcast on Polish television. &lt;a href=&quot;https://perenflo.com/math&quot; rel=&quot;nofollow noreferrer&quot;&gt;This statement is also corroborated by Enflo's personal website.&lt;/a&gt; Was a recording of this made, and if so does it still exist today?&lt;/p&gt;&#xA;" OwnerUserId="19348" LastEditorUserId="19348" LastEditDate="2023-11-24T16:45:21.833" LastActivityDate="2023-11-24T16:45:21.833" Title="Is there existing footage of Stanislaw Mazur giving Per Enflo a live goose for solving the approximation problem?" Tags="|mathematics|functional-analysis|anecdote|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15922" PostTypeId="1" CreationDate="2023-11-17T13:47:01.920" Score="0" ViewCount="36" Body="&lt;p&gt;What is the first recorded use of a mixure of honey and vinegar to treat wounds?&lt;br /&gt;&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Oxymel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cato the Elder&lt;/a&gt; indicates it was used during Roman times, but it's unclear if this was as a tonic or if it was used to clean wounds&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2023-11-17T13:47:01.920" Title="First recorded use of oxymel to treat wounds" Tags="|medicine|ancient|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15923" PostTypeId="2" ParentId="15915" CreationDate="2023-11-17T18:03:05.797" Score="2" Body="&lt;p&gt;In his book on Electrodynamics (Lectures on Theoretical Physics, Vol. 3), Chapter 32-E, Arnold Sommerfeld talks about &amp;quot;Schwarzschild's principle of least action&amp;quot;, developed by K. Schwarzschild (Göttinger Nachrichten 1903, English translation &lt;a href=&quot;https://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/schwarzschild_-_on_electrodynamics.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;). Recall that the action is just the space-time integral of the Lagrangian density. The principle of least action then varies the action to arrive at the Euler-Lagrange equations. For Schwarzschild's action these are indeed Maxwell's equations and the Lorentz force equation. In a footnote, Sommerfeld says admiringly:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Note the publication date of 1903! Thus Schwarzschild arrived intuitively at the correct postulate of the theory of invariants six years ahead of Minkowski.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Of course, Schwarzschild's Lagrangian also predates Einstein's special relativity by two years. It is indeed remarkable that Schwarzschild obtained the correct Lorentz invariant Lagrangian of electromagnetism at such an early stage.&lt;/p&gt;&#xA;&lt;p&gt;Schwarzschild himself refers to Lorentz (I believe to &lt;a href=&quot;https://www.lorentz.leidenuniv.nl/IL-publications/sources/Lorentz_AN_1892.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt; from 1892) and Helmholtz (presumably &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/10.1002/andp.18922830902&quot; rel=&quot;nofollow noreferrer&quot;&gt;this publication&lt;/a&gt; of the same year). Both authors have formulated somewhat clumsy and incomplete precursors to Schwarzschild's (basically) modern version.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastActivityDate="2023-11-17T18:03:05.797" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15924" PostTypeId="1" AcceptedAnswerId="15935" CreationDate="2023-11-18T06:56:03.377" Score="2" ViewCount="224" Body="&lt;p&gt;Emmy Noether was initially interested in invariant theory. But how did she become interested in abstract algebra? And why did she become particularly interested in ring and ideal theory?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="10106" LastEditDate="2023-11-21T02:52:07.337" LastActivityDate="2023-11-21T02:52:07.337" Title="How did Emmy Noether become interested in abstract algebra?" Tags="|mathematicians|biographical-details|noether|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15926" PostTypeId="1" CreationDate="2023-11-18T13:55:11.760" Score="0" ViewCount="127" Body="&lt;p&gt;In Book 7, Prop. 1. Euclid uses repeated subtraction to prove that two numbers are relatively prime. As explained &lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookVII/propVII1.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;  the Greek word for repeated subtraction is &amp;quot;antenaresis&amp;quot;. There isn't much information about Antenaresis online. I found &lt;a href=&quot;https://youtu.be/Egm6AydHguI?si=BwtFHNmVaamFyLmd&quot; rel=&quot;nofollow noreferrer&quot;&gt;this video&lt;/a&gt; (and &lt;a href=&quot;https://drive.google.com/file/d/1Io8GnIm-qegGZQwJlbNtb35O8FeLHzEW/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;the text mentioned in the video&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;I have two questions. (1) Do you have more information on antenaresis and (2) What does Heath do in his commentary to the mentioned proposition?&lt;/p&gt;&#xA;&lt;p&gt;Heath's commentary can be found &lt;a href=&quot;https://drive.google.com/file/d/1Io8GnIm-qegGZQwJlbNtb35O8FeLHzEW/view&quot; rel=&quot;nofollow noreferrer&quot;&gt;on page 234 of this link&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;This is the screen shot:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/sm8kb.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/sm8kb.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I understand what Euclid does but I don't understand Heath's notation.&lt;/p&gt;&#xA;&lt;p&gt;For instance, how does Heath's notation work for a=85 and b=54?&lt;/p&gt;&#xA;" OwnerUserId="16575" LastActivityDate="2023-11-18T21:29:40.197" Title="Euclid's use of antenaresis and Heath's commentary" Tags="|euclid|primes|" AnswerCount="1" CommentCount="13" ContentLicense="CC BY-SA 4.0" />
  <row Id="15927" PostTypeId="1" CreationDate="2023-11-18T19:16:14.120" Score="0" ViewCount="103" Body="&lt;p&gt;On my way to learn about the very beginning of quantum mechanics and its different formulations, starting with Heisenberg infinite matrices and Schrödinger's wave functions, I can really not find till now a single reference in which it is explained how Heisenberg and Schrödinger were doing quantum mechanics i.e. determining probabilities about measurements for positions and general observables within their own formulation framework of, respectively, infinite matrices and wave functions (i.e. without talking about Heisenberg and Schrödinger's pictures inside a Hilbert space 2(ℝ) for example). I mean for example for Schrödinger, given a wave function () of a fixed system, say an electron, what was exactly his interpretation of ()? (Before Born's interpretation came I mean) and how he did to do computations with it in order to predict the probabilities for the position, the momentum the energy and so on?&lt;/p&gt;&#xA;" OwnerUserId="19358" LastActivityDate="2023-11-19T20:11:37.837" Title="How did Schrödinger do quantum mechanics with wave functions?" Tags="|quantum-mechanics|mathematical-physics|quantum-field-theory|functional-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15928" PostTypeId="2" ParentId="15926" CreationDate="2023-11-18T21:29:40.197" Score="2" Body="&lt;pre&gt;&lt;code&gt;54 ) 85 ( 1&#xA;     54&#xA;    ---&#xA;     31 ) 54 ( 1&#xA;          31&#xA;         ---&#xA;          23 ) 31 ( 1&#xA;               23&#xA;              ---&#xA;                8 ) 23 ( 2&#xA;                    16&#xA;                   ---&#xA;                     7 ) 8 ( 1&#xA;                         7&#xA;                        --&#xA;                         1&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-11-18T21:29:40.197" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15929" PostTypeId="1" AcceptedAnswerId="15933" CreationDate="2023-11-18T21:46:47.667" Score="3" ViewCount="187" Body="&lt;p&gt;I am on a quest to uncover the rich tapestry of history surrounding &lt;a href=&quot;https://en.wikipedia.org/wiki/Scientific_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;scientific notation&lt;/a&gt; as a way of expressing numbers. Specifically, I'm interested in scholarly books, peer-reviewed articles, and historical documents that delve into:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;The scientific context and historical era during which scientific notation was first conceptualized and utilized.&lt;/li&gt;&#xA;&lt;li&gt;The key scientists and mathematicians who played pivotal roles in the development and refinement of scientific notation.&lt;/li&gt;&#xA;&lt;li&gt;The reasons for the widespread adoption and enduring popularity of scientific notation in the scientific community.&#xA;My research so far, including a review of the available Wikipedia content, has not yielded the depth of information I require. I would greatly appreciate any leads on academic or historical sources that provide a thorough analysis of these points.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="9523" LastEditDate="2023-11-23T10:27:36.710" LastActivityDate="2023-11-23T10:27:36.710" Title="Seeking Comprehensive References on the History of Scientific Notation" Tags="|mathematics|notation|" AnswerCount="2" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15930" PostTypeId="2" ParentId="15929" CreationDate="2023-11-19T11:32:15.443" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vieta (Viète)&lt;/a&gt; is the original creator of algebraic notation.  Fermat in particular benefited from Vieta's innovations and always adhered to his formalism.  Descartes developed a system of notation closer to what we use today.  You may want to consult the helpful recent book&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mazur, Joseph.  Enlightening symbols. A short history of mathematical notation and its hidden powers. Princeton University Press, Princeton, NJ, 2014. xxiv+285 pp. ISBN: 978-0-691-15463-3&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-11-19T11:32:15.443" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15931" PostTypeId="1" AcceptedAnswerId="15971" CreationDate="2023-11-19T16:25:43.013" Score="1" ViewCount="116" Body="&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rBoI6.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rBoI6.png&quot; alt=&quot;JJ thomson's setup for the cathode ray tube&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I asked here as well &lt;a href=&quot;https://chemistry.stackexchange.com/questions/177889/in-jj-thomsons-cathode-ray-experiment-why-is-effects-of-gravity-on-electron-not&quot;&gt;https://chemistry.stackexchange.com/questions/177889/in-jj-thomsons-cathode-ray-experiment-why-is-effects-of-gravity-on-electron-not&lt;/a&gt;&#xA;&lt;a href=&quot;https://physics.stackexchange.com/questions/789225/in-jj-thomsons-cathode-ray-experiment-why-is-effects-due-to-gravity-on-electron?noredirect=1#comment1773186_789225&quot;&gt;https://physics.stackexchange.com/questions/789225/in-jj-thomsons-cathode-ray-experiment-why-is-effects-due-to-gravity-on-electron?noredirect=1#comment1773186_789225&lt;/a&gt;&#xA;&lt;br&gt;&lt;br&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$${}$$&lt;/span&gt;&#xA;Explaining the setup:-&#xA;&lt;br&gt;&#xA;The experiment is described in the picture.&#xA;Instead of the magnets in the picture imagine two circular coils on both the sides with current running through it, this creates a magnetic field perpendicular to the loop which can be predicted using biot savart law. Run the current in the loop in such a way so that the magnetic field is going into your screen, away from you. Without the magnets the beam is deflected up towards the positive plate proving the charge on the beam is -ve as +ve and -ve attract.&#xA;q(v x B) = Magnetic force, so the magnetic force is acting downwards.&#xA;q/m ratio can be determined as well by combining both the magnets and the charged plates and changing the electric and magnetic fields until there seems to be no deflection on the beam so net force acting on beam must be close to 0 and we solve taking all the forces, more details here &lt;a href=&quot;https://scienceready.com.au/pages/thomsons-discovery-of-the-electron&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://scienceready.com.au/pages/thomsons-discovery-of-the-electron&lt;/a&gt;&#xA;&lt;br&gt;&lt;br&gt;&#xA;I had a problem with how he finds the q/m ratio which is why the gravitational force on the electron is not considered? Only the magnetic force and electric force is considered. So i thought maybe gravitational force is negligible? I included gravitational force and got an expression which relates q/m to a constant value&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Magnetic force + Gravitational force = Electric force$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$qvB +m9.8 = qE$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$v = E/B - (9.8m)/(qB)$$&lt;/span&gt;&#xA;when we only have the magnetic force acting on the beam, it will always act perpendicular to  the velocity due to the magnetic field going into the screen, so it will constantly change the direction of velocity  but not its magnitude and the beam will move in a circle.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$centripetal force = magnetic force$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$mv^2/r = qvB$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$mv/r = qB$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{m(E/B - (9.8m)/(qB))}{r} = qB $$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{mE}{Br} - \frac{9.8m^2}{Bqr} -qB = 0$$&lt;/span&gt;&#xA;this can be treated as a quadratic equation where the variable is m&#xA;&lt;br&gt; so if we have an equation ax^2 + bx + c = 0 its roots are = &lt;span class=&quot;math-container&quot;&gt;$\frac{-b +- \sqrt{b^2 - 4ac}}{2a}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$m = \frac{ ( -E/Br +- \sqrt{ \frac{E^2}{B^2 r^2} - \frac{4 \cdot 9.8}{r}})Bqr }{2 \cdot 9.8}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\frac{m}{q} = \frac{ ( -E/Br + \sqrt{ \frac{E^2}{B^2 r^2} - \frac{4 \cdot 9.8}{r}})Br }{2 \cdot 9.8}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;br&gt;but I dont know at what magnetic and electric fields so i need the values of electric field of the charged plates and magnetic field of the coils at which there is no net deflection and the radius of the circle the beam goes through when only the magnetic field acts on it, provide the source as well. Maybe there is some other way to understand why gravitational force is not included if so could you explain that way.&lt;/p&gt;&#xA;" OwnerUserId="19362" LastEditorUserId="19362" LastEditDate="2023-11-25T06:52:58.087" LastActivityDate="2023-12-01T00:16:07.757" Title="In JJ Thomson's cathode ray experiment I need values for the electric field and magnetic field when net force on the cathode beam = 0" Tags="|physics|chemistry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15932" PostTypeId="1" CreationDate="2023-11-19T18:13:41.343" Score="0" ViewCount="49" Body="&lt;p&gt;In DOI: 10.4236/ahs.2020.94019 235 Advances in Historical Studies, p.234&#xA;D’Alembert and the Wave Equation: Its Disputes and Controversies, or &lt;a href=&quot;https://www.scirp.org/pdf/ahs_2020112716312281.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scirp.org/pdf/ahs_2020112716312281.pdf&lt;/a&gt; p.6 of 11&lt;/p&gt;&#xA;&lt;p&gt;A. R. E. Oliveira states&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Euler criticized d’Alembert’s work pointing out that the two arbitrary&#xA;functions ϕ and ψ are determined by the initial conditions of the&#xA;problem: ... which represent the initial form of the curve and the&#xA;&lt;strong&gt;distributions of initial velocities&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but he does not give a specific reference.  He does list 5 Euler references--only 1 in English.  My questions are:  &lt;em&gt;Which is the source which contains this assertion by Euler and is there an English translation?&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="7211" LastEditorUserId="10106" LastEditDate="2023-11-20T03:44:02.593" LastActivityDate="2023-11-20T03:44:02.593" Title="Need a reference for Euler's velocity initial condition for the wave equation" Tags="|physics|euler|waves|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15933" PostTypeId="2" ParentId="15929" CreationDate="2023-11-19T18:24:17.513" Score="3" Body="&lt;p&gt;First (mathematical) use of the term cited by the &lt;em&gt;OED&lt;/em&gt;: &lt;a href=&quot;https://www.google.com/books/edition/The_American_Mathematical_Monthly/NAcPAAAAIAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22American%20Mathematical%20Monthly%22%20vol.%2022%201915%20notation&amp;amp;pg=PA328&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;F.L. Griffin, An experiment in correlating freshman mathematics, &lt;em&gt;MAA Monthly&lt;/em&gt; (1915), p. 328&lt;/a&gt;. Griffin later published a book, &lt;a href=&quot;https://www.google.com/books/edition/An_Introduction_to_Mathematical_Analysis/sh_vAAAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;dq=%22scientific+notation%22&amp;amp;pg=PA190&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Introduction to Mathematical Analysis&lt;/em&gt; (1921), pp. 190f&lt;/a&gt;. His purpose in his article and book is the improvement of teaching. He also states that the notation is &amp;quot;common in scientific work&amp;quot; (1915), which seems the basis for calling it &amp;quot;scientific notation.&amp;quot; One advantage: &amp;quot;This avoids operations with long rows of zeros or decimal places.&amp;quot; (1921) He also argues that familiarity with the notation will aid the student's learning and appreciation of logarithms.&lt;/p&gt;&#xA;&lt;p&gt;The broader notion that a scientific notation might allow the reduction of solving problems to symbolic operations, such as Vieta's algebra notation did for solving problems in numbers (see &lt;a href=&quot;https://hsm.stackexchange.com/a/15930&quot;&gt;@Katz's answer&lt;/a&gt;), grew in the 19th century. By the late 19th century and on into the 20th, you find people arguing for and proposing scientific notations for phonetics, music, color, and the scientific teaching of English. At least one of these projects led to something still in use, namely the &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_the_International_Phonetic_Alphabet&quot; rel=&quot;nofollow noreferrer&quot;&gt;IPA&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The OP may be the first person interested in a comprehensive history of scientific notation. As a representation of numbers, I do not know of a work that examines the use of the notation &amp;quot;so common in scientific work&amp;quot; prior to 1915, according to Griffin. I know a random instance or two. Airy's &lt;a href=&quot;https://www.google.com/books/edition/Numerical_Lunar_Theory/xxdAAAAAYAAJ?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Numerical Lunar Theory (1886)&lt;/a&gt; has some tables in which the entries are integers to be multiplied by a &amp;quot;unit&amp;quot; at the head of the column that is a power of ten. This is not the present-day convention for scientific notation, but it is similar. As far as the history after Griffin, I suspect scientific notation evolved due to the need for convenient forms of input/output for computers. I realize the OP is not looking to start research into the question, but assuming the basic research has not been done, I offer these &amp;quot;waypoints&amp;quot; in the hope they are helpful.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2023-11-19T18:24:17.513" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15934" PostTypeId="2" ParentId="15927" CreationDate="2023-11-19T20:11:37.837" Score="1" Body="&lt;p&gt;I don't think Heisenberg and Schrödinger calculated probabilities at the beginning, as the role of probabilities in quantum mechanics was not known then. Schrödinger initially interpreted the squared magnitude of the wave function as the charge density.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-11-19T20:11:37.837" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15935" PostTypeId="2" ParentId="15924" CreationDate="2023-11-20T04:14:59.527" Score="8" Body="&lt;p&gt;Ernst Fischer was hired by Erlangen in 1911. He is known to us today for his work in analysis (Riesz-Fischer theorem), but he worked on algebraic topics at Erlangen (and his thesis was on determinants). He influenced Noether's direction of work towards Hilbert's approach to studying polynomial rings, which had success in proving results of sweeping generality without tedious computations; think of Hilbert's basis theorem. Noether's thesis had hundreds of calculations in it and she never thought highly of this work. That experience perhaps made her more interested in a style of math that was more conceptual and less computational, which in her case turned out to be abstract algebra.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2023-11-20T04:14:59.527" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15936" PostTypeId="1" CreationDate="2023-11-20T12:17:27.587" Score="6" ViewCount="676" Body="&lt;p&gt;In differential geometry and other fields, an expression involving differentials can be &lt;em&gt;closed&lt;/em&gt; or &lt;em&gt;exact&lt;/em&gt;.  In &lt;span class=&quot;math-container&quot;&gt;$\mathbb R^2\setminus\{0\}$&lt;/span&gt; for example, &lt;span class=&quot;math-container&quot;&gt;$dr$&lt;/span&gt; is exact whereas &lt;span class=&quot;math-container&quot;&gt;$d\theta$&lt;/span&gt; is closed but not exact.  What is the origin of these terms?  Note that I deliberately avoided the use of the term &amp;quot;form&amp;quot; because the use of &lt;em&gt;open&lt;/em&gt; and &lt;em&gt;exact&lt;/em&gt; in reference to expressions involving differentials may have preceded the theory of differential forms.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-11-20T21:08:32.357" Title="Origin of exact and closed differential expressions" Tags="|mathematics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15937" PostTypeId="2" ParentId="15936" CreationDate="2023-11-20T14:51:33.910" Score="4" Body="&lt;p&gt;From &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathwords&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Exact differential&lt;/strong&gt; is found in 1824 in &lt;em&gt;First Principles of the Differential and Integral Calculus, or the Doctrine of Fluxions, Intended as an Introduction to the Physico-Mathematical Sciences; Taken Chiefly from the Mathematics of Bézout, And translated from the French For the Use of the Students of the University at Cambridge, New England&lt;/em&gt;: &amp;quot;But when the numerator is the differential of the denominator, multiplied or divided by a constant number, the proposed differential must be decomposed into two factors, of which one shall be a fraction having for its numerator the exact differential of the denominator, and the other a constant number.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://www.oed.com/dictionary/exact_adj1?tab=meaning_and_use#5167889&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Oxford English Dictionary&lt;/em&gt;&lt;/a&gt; (subscription required) has this from 1825:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As there are many differentials of two variables which are not exact differentials, so also there are many differential equations which are not the immediate differentials of any primitive equation.&lt;br /&gt;&#xA;D. Lardner, &lt;em&gt;Elem. Treat. Differential &amp;amp; Integral Calculus&lt;/em&gt; ii. xvii. 284&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastEditorUserId="229" LastEditDate="2023-11-20T15:23:44.903" LastActivityDate="2023-11-20T15:23:44.903" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15938" PostTypeId="2" ParentId="15936" CreationDate="2023-11-20T21:08:32.357" Score="7" Body="&lt;p&gt;From Bottazzini U., Gray J., &lt;em&gt;Hidden Harmony-Geometric Fantasies. The rise of Complex Functions&lt;/em&gt;, Springer, 2013 (from which it seems that the terms 'exact' and 'complete' for differentials were used   interchangeably):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the opening pages of his (1739)&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;, read to the French Académie on&#xA;March 4, 1739, Clairaut stated that if &lt;span class=&quot;math-container&quot;&gt;$Adx+Bdy$&lt;/span&gt; is the differential of&#xA;any function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; and constants, then &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial A}{\partial y}=\frac{\partial B}{\partial x}$&lt;/span&gt;; and vice versa,   if &lt;span class=&quot;math-container&quot;&gt;$\frac{\partial A}{\partial y}=\frac{\partial B}{\partial x}$&lt;/span&gt;  then  &lt;span class=&quot;math-container&quot;&gt;$Adx+Bdy$&lt;/span&gt; is a&#xA;'complete' differential, as he called it [...].&lt;/p&gt;&#xA;&lt;p&gt;In a subsequent &lt;em&gt;Mémoire&lt;/em&gt;&#xA;Clairaut used both the terms 'complete' and 'exact' differential&#xA;(1740).&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;[...] In his (1768)&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt;[...] d'Alembert [...] picked up the example mentioned by Clairaut [...]concluded that not only has &lt;span class=&quot;math-container&quot;&gt;$Pdx+Qdy$&lt;/span&gt; to be a complete differential...&lt;/p&gt;&#xA;&lt;p&gt;[...] mathematicians of the eighteenth century thought of the integral of a differential&#xA;form  such as &lt;span class=&quot;math-container&quot;&gt;$Pdx+Qdy$&lt;/span&gt; as a function whose differential is the given form,&#xA;whereas nowadays the expression &lt;span class=&quot;math-container&quot;&gt;$\int{Pdx+Qdy}$&lt;/span&gt; is understood as a line integral.&#xA;[...]&lt;/p&gt;&#xA;&lt;p&gt;Formal manipulations of exact differential and complex functions similar to&#xA;d'Alembert can also be found in papers by Euler and by Lagrange on&#xA;fluid dynamics, as well as in Euler's papers on orthogonal trajectory and conformal mappings.[...] (pp. 84-87).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I found the use of &lt;em&gt;exact differential&lt;/em&gt; also in Lacroix, &lt;em&gt;An Elementary Treatise On Differential and Integral Calculus&lt;/em&gt;, Cambridge, 1816 (engl. transl. of  &lt;em&gt;Traité Élémentaire du Calcul Différentiel et du Calcul Intégral, 1802&lt;/em&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Although the equation &lt;span class=&quot;math-container&quot;&gt;$(B)$&lt;/span&gt; should become identical by the substitution of&#xA;values of &lt;span class=&quot;math-container&quot;&gt;$P, Q, R$&lt;/span&gt;, it does not follow, as a necessary consequence,&#xA;that the equation &lt;span class=&quot;math-container&quot;&gt;$$Pdx+Qdy+Rdx=0$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;is a &lt;em&gt;&lt;strong&gt;exact&lt;/strong&gt;&lt;/em&gt; differential; but at least it may be rendered such by&#xA;multiplication by some factor. (p. 412, &lt;em&gt;emphasis mine&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; Clairaut A.C. , 1739, Recherches générales sur le calcul integral. &lt;em&gt;Hist. Acad. Sci., Paris&lt;/em&gt; (1740), 425-436.&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt; Clairaut A.C. , 1740, Sur l'intégration ou la construction des équations differentielles du premier ordre. &lt;em&gt;Hist. Acad. Sci., Paris&lt;/em&gt; (1742), 293-322. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^3$&lt;/span&gt; &lt;sup&gt;Alembert, J. le Rond d',  1768. Sur l'équilibre des fluides. &lt;em&gt;Opusc. math.&lt;/em&gt; 5, 1-40 &lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-11-20T21:08:32.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15939" PostTypeId="1" CreationDate="2023-11-21T10:15:16.827" Score="1" ViewCount="137" Body="&lt;p&gt;How did someone came up with an idea that if we do prime factorization of two numbers and then multiply all the prime factors but including common ones only once, we will get a number that is the least common multiple of these two numbers?&lt;/p&gt;&#xA;&lt;p&gt;And why do we take the repeating factors only once? I know this question might sound silly but I am a total beginner and math who don't know anything. Please help me understand this.&lt;/p&gt;&#xA;" OwnerUserId="19370" LastEditorUserId="19370" LastEditDate="2023-11-21T16:36:07.803" LastActivityDate="2023-11-21T16:36:07.803" Title="How did someone discover LCM?" Tags="|mathematics|mathematical-logic|arithmetic|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15940" PostTypeId="1" CreationDate="2023-11-21T21:45:47.593" Score="3" ViewCount="185" Body="&lt;p&gt;According to Tobias Hurter’s popular exposition &lt;em&gt;Too Big for a Single Mind&lt;/em&gt; (narrated in the present tense):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dirac makes use of an elegant mathematical tool developed by the Irish mathematician William Hamilton in the nineteenth century.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is from a passage discussing Dirac in Cambridge in the summer of 1925; see pages 127-128.&lt;/p&gt;&#xA;&lt;p&gt;What “elegant mathematical tool” is Hurter talking about? Was it quaternions? Unfortunately the notes at the back of the book do not say anything more about this episode.&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="6609" LastEditDate="2023-11-22T15:23:14.990" LastActivityDate="2023-11-24T01:54:03.380" Title="Dirac’s debt to Hamilton" Tags="|quantum-mechanics|dirac|quaternions|" AnswerCount="2" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15942" PostTypeId="2" ParentId="15940" CreationDate="2023-11-23T17:03:15.453" Score="3" Body="&lt;p&gt;I'd wager that the ambiguous reference is to the appearance of Hamiltonian &lt;em&gt;quaternions&lt;/em&gt; (as an instance of a Clifford algebra) in Dirac's construction of a square root (of a certain sort) of a/the Laplacian, which is nowadays called a/the &amp;quot;Dirac operator&amp;quot;. (Not to be confused with Dirac's &lt;span class=&quot;math-container&quot;&gt;$\delta$&lt;/span&gt;.) It is a linear differential operator with coefficients in a Clifford algebra related to a quadratic form attached to the geometry of the situation.&lt;/p&gt;&#xA;&lt;p&gt;This is quite different from square roots of positive unbounded self-adjoint operators on Hilbert spaces... For example, the Dirac operator still has a robust distributional sense, while the unbounded-op-on-Hilbert-space version is not distributional.&lt;/p&gt;&#xA;&lt;p&gt;The Hamiltonian quaternions arise as a Clifford algebra attached to a low-dimensional quadratic form.&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2023-11-23T17:03:15.453" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15943" PostTypeId="2" ParentId="15940" CreationDate="2023-11-24T01:54:03.380" Score="1" Body="&lt;p&gt;The OP noted in a comment:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I don’t know if Hurter is correct, but I stress that he is writing&#xA;about Dirac in 1925, not 1927.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Therefore, the text is probably about Dirac's 1925 &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rspa.1925.0150&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; , where Dirac uses Hamiltonian function. The equations of motion then feature the Poisson bracket. The Dirac's general approach to quantization involves the replacement of the Poisson brackets by commutators.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-11-24T01:54:03.380" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15946" PostTypeId="1" CreationDate="2023-11-24T10:24:54.970" Score="5" ViewCount="650" Body="&lt;p&gt;Scientists are often associated together and get famous group names. In physics, I know of&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Via Panisperna boys (Enrico Fermi and co. working in Rome in nuclear physics)&lt;/li&gt;&#xA;&lt;li&gt;Princeton string quartet (Gross, Harvey, Martinec, and Rohm, working in string theory)&lt;/li&gt;&#xA;&lt;li&gt;Oxford calculators (14th century philosophers working on early mechanics)&lt;/li&gt;&#xA;&lt;li&gt;The Martians (Hungarian scientists working in the Manhattan project like Szilard, von Neumann, Wigner and Teller, plus later other Hungarian mathematicians)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Do you know of any other named famous groups? I could not find a key word on the internet to easily look up for more. Of course, I am excluding formal associations and societies intended to gather more members. Also excluding faculty teams unless they got a popular pseudonym or popular name (like Via Panisperna boys). This question is a [tag:big list] for science and math groups in general not only physics.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-11-24T12:12:09.987" LastActivityDate="2024-02-20T15:41:03.750" Title="What are some famous named groups of scientists?" Tags="|big-list|" AnswerCount="10" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15947" PostTypeId="2" ParentId="15946" CreationDate="2023-11-24T17:15:22.280" Score="4" Body="&lt;p&gt;If we consider economics to be a science then the group known as the &lt;a href=&quot;https://en.wikipedia.org/wiki/Chicago_Boys&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chicago Boys&lt;/a&gt; might be an example. They were a group of Latin American economists, chiefly Chilean, who had prominent political roles. Their name comes from the fact that they had trained in Chicago.&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2023-11-24T17:15:22.280" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15948" PostTypeId="1" AcceptedAnswerId="15966" CreationDate="2023-11-24T19:04:41.247" Score="8" ViewCount="291" Body="&lt;h2&gt;Background &amp;amp; My research&lt;/h2&gt;&#xA;&lt;p&gt;To begin I did some research to find a few articles on the history of pH namely &lt;a href=&quot;https://web.archive.org/web/20191214110759/http://www.che.uc.edu/jensen/w.%20b.%20jensen/reprints/102.%20ph.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&amp;quot;The Symbol for pH&amp;quot;&lt;/em&gt;&lt;/a&gt;- William B. Jensen, &lt;em&gt;&amp;quot;One-Hundred Years of pH&amp;quot;&lt;/em&gt; - Rollie J. Myers (but couldn't access it) and &amp;quot;&lt;em&gt;The origin and the meaning of the little p in pH&amp;quot;&lt;/em&gt; - Jens G Nørby (but couldn't access this also).&lt;/p&gt;&#xA;&lt;p&gt;So to start I did some reading on the wiki article for pH and found that S.P.L. Sørensen first introduced the term in his [paper] - &lt;em&gt;&amp;quot;Über die Messung und die Bedeutung der Wasserstoffionenkonzentration bei enzymatischen Prozessen&amp;quot;&lt;/em&gt;&lt;sup&gt; &lt;a href=&quot;https://web.archive.org/web/20210415205740/https://core.ac.uk/download/pdf/14517358.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;/sup&gt; in 1909 in order to explain the acidity and basicity of the acids/bases on the &amp;quot;Arrhenius concept&amp;quot;. To explain to some extent the use of the logarithmic scale I found out that his experiments contained the use of electrodes with one being a calomel electrode and the other being a normal hydrogen electrode. Upon experimental results and an equation (which I think was the Nernst Equation) he found that the electric potential of the cell was proportional to −log[H+].&lt;/p&gt;&#xA;&lt;p&gt;To somehow explain the use of &amp;quot;p in pH&amp;quot; I found out S.P.L. Sørensen in his original paper used &lt;span class=&quot;math-container&quot;&gt;$P_H$&lt;/span&gt; to represent this “hydrogen ion exponent”. Although as his work became more popular several more variations like Ph and pH were introduced with the latter getting officially adopted by the &amp;quot;Journal of Biological Chemistry&amp;quot; in 1910's and eventually gaining more recognition than the other variations. The origin of &amp;quot;P&amp;quot; in the original text is speculated to be related to puissance (French), potenz (German) or potens (Danish) which were the languages in which Sørensen published.&lt;/p&gt;&#xA;&lt;p&gt;This was all I could find from my research and still have some doubts like how the little p in &amp;quot;pH&amp;quot; evolved/generalized to be used as &amp;quot;an operator&amp;quot; like - &lt;span class=&quot;math-container&quot;&gt;$pK_a, pK_b, pK_w$&lt;/span&gt; etc. which are further expressed below in the main question.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Question:&lt;/h2&gt;&#xA;&lt;p&gt;I was recently introduced to the concept of pH in chemistry and its value being the following: &lt;span class=&quot;math-container&quot;&gt;$$\mathrm{pH} =-\log[\mathrm H^+]$$&lt;/span&gt; This was followed up by the &amp;quot;generalization of the concept of pH&amp;quot; in &lt;em&gt;&amp;quot;finding the p&amp;quot;&lt;/em&gt; of a base/acid (e.g., &lt;span class=&quot;math-container&quot;&gt;$pK_a, pK_b, pK_w$&lt;/span&gt; etc.)&lt;/p&gt;&#xA;&lt;p&gt;Now I want to know how is it that the definition of pH evolved from being the &amp;quot;hydrogen ion concentration&amp;quot; to the &amp;quot;activity of hydrogen ions&amp;quot; as well some background on the experiments that he performed which led him to define these concepts the way they are? Also how is it that the concept became &amp;quot;more generalized&amp;quot; and evolved from being the negative common logarithm of the concentration of hydrogen ions to a more generalized &amp;quot;operation&amp;quot; applicable to things more than just the hydrogen ions?&lt;/p&gt;&#xA;&lt;p&gt;Any sort of articles and further reading links will be very much appreciated which could help me learn more about the &amp;quot;history of pH&amp;quot; and how it originated and evolved as a concept.&lt;/p&gt;&#xA;" OwnerUserId="19382" LastEditorUserId="19382" LastEditDate="2024-01-18T17:19:43.510" LastActivityDate="2024-01-18T17:19:43.510" Title="How did the concept of pH originate and develop?" Tags="|terminology|chemistry|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="15949" PostTypeId="2" ParentId="15946" CreationDate="2023-11-24T19:45:22.940" Score="6" Body="&lt;p&gt;A famous group of mathematicians, important above all for the development of functional analysis, was &lt;em&gt;&lt;strong&gt;the group of the&lt;/strong&gt;&lt;/em&gt; &lt;a href=&quot;https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Scottish Café&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;  in Lwow, Poland, a group of mathematicians that between the two World Wars gathered at that café around the personality of &lt;a href=&quot;https://en.wikipedia.org/wiki/Stefan_Banach&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Stefan Banach&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Through the work of Hugo Steinhaus and  Stefan Banach, Lwow had become one of the most important centers of functional analysis, and, in 1929 they founded the journal &lt;em&gt;Studia Mathematica&lt;/em&gt;, specialized in functional analysis.&lt;/p&gt;&#xA;&lt;p&gt;The publication of the journal was interrupted at  the beginning of the Second World War, and  was resumed in 1948.&lt;/p&gt;&#xA;&lt;p&gt;A fundamental role had the atmosphere of the Scottish Café, where Banach and his colleagues used to gather, discussing about problems of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Stanis%C5%82aw_Ulam&quot; rel=&quot;noreferrer&quot;&gt;Stanislaw Ulam&lt;/a&gt; recounts that the tables of the café had marble tops, so they could write in pencil on the table during their discussions. To avoid the results to be deleted  by the waiter every day,  Banach’s wife  provided them with a large notebook, which became known as the &lt;a href=&quot;https://www.amazon.it/Scottish-Book-Mathematics-Selected-Problems/dp/3319794329/ref=sr_1_1?__mk_it_IT=%C3%85M%C3%85%C5%BD%C3%95%C3%91&amp;amp;crid=2AM9D62JY89EO&amp;amp;keywords=scottish+book+mathematics&amp;amp;psr=EY17&amp;amp;qid=1700851714&amp;amp;s=black-friday&amp;amp;sprefix=scottish+book+mathematics%2Cblack-friday%2C142&amp;amp;sr=1-1&amp;amp;ufe=app_do%3Aamzn1.fos.9d4f9b77-768c-4a4e-94ad-33674c20ab35&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Scottish Book&lt;/em&gt; and was subsequently published.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Among the most well-known attenders of the discussions at the Scottish Café  there were Stefan Banach, Hugo Steinhaus, Stanislaw Mazur, Stanislaw Saks, Stanislaw Marcin Ulam.&lt;/p&gt;&#xA;&lt;p&gt;A recent reconstruction of the history of the mathematicians of the Scottish Café can be found in the paper &lt;a href=&quot;https://inria.hal.science/hal-02386544/document&quot; rel=&quot;noreferrer&quot;&gt;Chris Zielinski, ‘Mathematicians at the Scottish Café’, (2018)&lt;/a&gt;. The author’s  grandfather  Tomasz Zielinski was the owner of the Scottish Café.&lt;/p&gt;&#xA;" OwnerUserId="17306" LastEditorUserId="17306" LastEditDate="2023-11-24T19:54:41.820" LastActivityDate="2023-11-24T19:54:41.820" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15950" PostTypeId="2" ParentId="15946" CreationDate="2023-11-24T19:55:35.393" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/M._Lothaire&quot; rel=&quot;nofollow noreferrer&quot;&gt;M. Lothaire&lt;/a&gt; (the M. simply stays for &amp;quot;Monsieur&amp;quot;, i.e. &amp;quot;Mr.&amp;quot;) was the pen name of a group of mathematicians and computer scientists, many of whom were former students of &lt;a href=&quot;https://en.wikipedia.org/wiki/Marcel-Paul_Sch%C3%BCtzenberger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Marcel-Paul Schützenberger&lt;/a&gt;. The name is used as the author of &lt;a href=&quot;https://www-igm.univ-mlv.fr/%7Eberstel/Lothaire/&quot; rel=&quot;nofollow noreferrer&quot;&gt;three seminal books on the combinatorics of words&lt;/a&gt;.&#xA;Since the members of Lothaire were, at least initially, French, Italian and German, and in view of the fact that its founder, Schützenberger, was the heir to an &lt;a href=&quot;https://fr.wikipedia.org/wiki/Famille_Sch%C3%BCtzenberger&quot; rel=&quot;nofollow noreferrer&quot;&gt;ancient Alsacian family&lt;/a&gt;, was chosen the name of Lothaire I, founder of &lt;a href=&quot;https://en.wikipedia.org/wiki/Middle_Francia&quot; rel=&quot;nofollow noreferrer&quot;&gt;Middle Francia&lt;/a&gt;, an ephemeral kingdom extending from Frisia to northern Italy.&#xA;The whole history is explained in the Preface of the 1997 edition of &lt;a href=&quot;https://www.cambridge.org/core/books/abs/combinatorics-on-words/preface-to-this-edition/C57C6EE48C287879B83B8C34701B1B75&quot; rel=&quot;nofollow noreferrer&quot;&gt;Combinatorics on Words&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] In his review for the Bulletin of the London Mathematical Society, John Howie had suggested a possible pun on &amp;quot;l'auteur&amp;quot;. There is really nothing of this Lacanian type. Lothaire was the grandson of Carolus Magnus [...] he reigned over Lotharingia, a kingdom comprising part of France, Germany and Italy more or less centered in Alsace [...].&#xA;It became established that the students of M.-P. Schützenberger, himself of a renowned alsacian family, were citizen of a mythical kingdom of Lotharingia [which actually takes his name from Lothaire II, son of Lothaire I]. There is today a &lt;a href=&quot;https://en.wikipedia.org/wiki/S%C3%A9minaire_Lotharingien_de_Combinatoire&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lotharingian seminar&lt;/a&gt; of combinatorics holding frequent meetings in France Germany, and Italy [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Lothaire was (and in part it still is) a group structured like Bourbaki, while the following are names used informally and sometimes simply humorously to write some mathematical works, without having a structured group behind them.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Besse&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arthur Lancelot Besse&lt;/a&gt; was the name of a group of French differential geometers; &lt;a href=&quot;https://en.wikipedia.org/wiki/Blanche_Descartes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blanche Descartes&lt;/a&gt; was the name used by R. Leonard Brooks, Arthur Harold Stone, Cedric Smith, and W. T. Tutte to publish some papers in Eureka, a mathematical student magazine in Cambridge; L. T. F. Gamut was the fictitious author of &amp;quot;Logic, Language and Meaning&amp;quot; (vol. 1 &amp;amp; 2), two monography written by the Dutch logicians Johan van Benthem, Jeroen Groenendijk, Dick de Jongh, Martin Stokhof and Henk Verkuyl; &lt;a href=&quot;https://en.wikipedia.org/wiki/G._W._Peck&quot; rel=&quot;nofollow noreferrer&quot;&gt;G. W. Peck&lt;/a&gt; was the pseudonymous used as author (or co-author) of a number of papers in combinatorics (Paul Erdős was one of the actual authors), while &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Rainwater&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Rainwater&lt;/a&gt; is the &amp;quot;author&amp;quot; of important results in functional analysis.&lt;/p&gt;&#xA;&lt;p&gt;Finally, mention should be made of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Polymath_Project&quot; rel=&quot;nofollow noreferrer&quot;&gt;Project Polymath&lt;/a&gt;, which is a collective mathematical problem-solving project. An online platform enables mathematicians to work together to find a solution to a problem, improve a result, check a proof, etc. The project began in 2009 on Timothy Gowers' blog and has the support of Terence Tao and Michael Nielsen, a specialist in open science.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="12330" LastEditDate="2023-11-26T09:11:28.163" LastActivityDate="2023-11-26T09:11:28.163" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15951" PostTypeId="2" ParentId="15946" CreationDate="2023-11-24T21:21:25.113" Score="4" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Vienna_Circle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vienna Circle&lt;/a&gt; (German: Wiener Kreis)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;...a group of elite philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, chaired by Moritz Schlick.&lt;br /&gt;&#xA;Inner Circle: Gustav Bergmann, Rudolf Carnap, Herbert Feigl, Philipp Frank, Kurt Gödel, Hans Hahn, Olga Hahn-Neurath, Béla Juhos, Felix Kaufmann, Victor Kraft, Karl Menger, Richard von Mises, Otto Neurath, Rose Rand, Josef Schächter, Moritz Schlick, Friedrich Waismann, Edgar Zilsel.&lt;br /&gt;&#xA;Periphery: Alfred Jules Ayer, Egon Brunswik, Karl Bühler, Josef Frank, Else Frenkel-Brunswik, Heinrich Gomperz, Carl Gustav Hempel, Eino Kaila, Hans Kelsen, Charles W. Morris, Arne Naess, Karl Raimund Popper, Willard Van Orman Quine, Frank P. Ramsey, Hans Reichenbach, Kurt Reidemeister, Alfred Tarski, Olga Taussky-Todd, Ludwig Wittgenstein.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="229" LastActivityDate="2023-11-24T21:21:25.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15952" PostTypeId="2" ParentId="8265" CreationDate="2023-11-25T00:34:49.143" Score="0" Body="&lt;p&gt;here's another thought on the &amp;quot;same meridian&amp;quot; problem. arguably, the two cities not being on the exact same meridian would be made less relevant by Eratosthenes' requirement that the Alexandria measurement be made at &amp;quot;apparent noon&amp;quot;. (He apparently accepted as true the report that a vertical structure casts no shadow on the summer solstice at apparent noon in Syene.) It was stated elsewhere that the 2 cities are actually separated by 3 degrees of longitude, so the observations in the 2 cities would not have been made at the same time of day, but would have been separated by the time it takes the earth ot rotate 3 degrees; at least, to the extent that the 2 observers were successful in making their respective observations at apparent noon (the time at which the sun was directly above their respective longitudes). This means that the distance between cities which he extrapolated from 7 to 360 degrees was longer than it should have been.  But that would have made his calculated circumference too long, when in fact it was slightly too short.  Most likely problem was the difficulty in visually determining apparent noon.&lt;/p&gt;&#xA;" OwnerUserId="19383" LastEditorUserId="19383" LastEditDate="2023-11-25T03:44:54.520" LastActivityDate="2023-11-25T03:44:54.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15953" PostTypeId="2" ParentId="14928" CreationDate="2023-11-25T13:31:38.887" Score="1" Body="&lt;p&gt;There is also nice discussion with lots of historic references in an &lt;a href=&quot;https://doi.org/10.1080/00033790902741633&quot; rel=&quot;nofollow noreferrer&quot;&gt;article by Helge Krah from 2009 ('The Solar Element: A Reconsideration of Helium's Early History')&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19387" LastActivityDate="2023-11-25T13:31:38.887" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15954" PostTypeId="2" ParentId="15946" CreationDate="2023-11-26T15:48:03.540" Score="4" Body="&lt;p&gt;In physics, there are a number of groups referred to as a &amp;quot;&lt;a href=&quot;https://en.wikipedia.org/wiki/Gang_of_Four&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gang of Four&lt;/a&gt;&amp;quot; after the notorious leaders of the Chinese Cultural Revolution (1966-76). A (presumably) incomplete list includes:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Elihu Abrahams, Philip W. Anderson, Donald Licciardello and T.V. Ramakrishnan who published an influential paper on &lt;a href=&quot;https://en.wikipedia.org/wiki/Anderson_localization&quot; rel=&quot;nofollow noreferrer&quot;&gt;Anderson localisation&lt;/a&gt;: &lt;em&gt;Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions&lt;/em&gt;. Phys. Rev. Lett. 42, 673–676 (1979), see e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Elihu_Abrahams&quot; rel=&quot;nofollow noreferrer&quot;&gt;Abraham's Wikipedia page&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Raymond &amp;quot;Ray&amp;quot; Sawyer, Robert &amp;quot;Bob&amp;quot; Sugar, James “Jim” Hartle, and Douglas “Doug” Scalapino who founded the Institute for Theoretical Physics at Santa Barbara and are depicted on &lt;a href=&quot;https://giving.ucsb.edu/Impact/gang-of-four&quot; rel=&quot;nofollow noreferrer&quot;&gt;this photograph&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Dick Harting, Wolfram Kittel, Hans Sens and Martinus &amp;quot;Tini&amp;quot; Veltman from NIKHEF, Amsterdam, shown &lt;a href=&quot;https://www.nikhef.nl/thegangof4/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Marc Davis, George Efstathiou, Carlos Frenk and Simon White (DEFW) who received the 2011 Cosmology Prize from The Peter and Patricia Gruber Foundation according to &lt;a href=&quot;https://www.iau.org/news/pressreleases/detail/iau1103/&quot; rel=&quot;nofollow noreferrer&quot;&gt;this press release&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;Chris Hull, Fay Dowker, Jerome Gauntlett and Dan Waldram at Imperial college (per &lt;a href=&quot;https://iopscience.iop.org/article/10.1088/2058-7058/16/8/5/meta&quot; rel=&quot;nofollow noreferrer&quot;&gt;this IOP news article&lt;/a&gt;).&lt;/li&gt;&#xA;&lt;li&gt;The Orsay group, A. Le Yaouanc, L. Oliver, O. Pene and J.C. Raynal, who together have &lt;a href=&quot;https://inspirehep.net/literature?sort=mostrecent&amp;amp;size=25&amp;amp;page=1&amp;amp;q=find%20a%20yaouanc%20and%20a%20oliver%20and%20a%20pene%20and%20a%20raynal&quot; rel=&quot;nofollow noreferrer&quot;&gt;132 joint publications on INSPIRE&lt;/a&gt; and were referred to as a &amp;quot;Gang of Four&amp;quot; by F. Yndurain at a conference which I attended some 25 years ago.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Finally, there is the group of three, H. Lehmann, K. Symanzik and W. Zimmermann (LSZ), who were (slightly derogatory) &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-642-70307-2_1&quot; rel=&quot;nofollow noreferrer&quot;&gt;dubbed the &amp;quot;Feldverein&amp;quot;&lt;/a&gt; (&amp;quot;field theory club&amp;quot;) by Pauli.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastEditorUserId="16834" LastEditDate="2023-11-28T12:20:09.233" LastActivityDate="2023-11-28T12:20:09.233" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15955" PostTypeId="2" ParentId="15946" CreationDate="2023-11-26T17:48:23.843" Score="1" Body="&lt;p&gt;Surely the most famous would be the Pythagorean School.&lt;/p&gt;&#xA;&lt;p&gt;Stories about the school appear in the later doxographical tradition, relying on legends and mystical traditions of a dubious kind.  However, Heraclitus, who lived just one generation after Pythagoras, praised the advances they made in mathematics and science.  Aristotle wrote a book on the Pythagorean school, now lost. The school lasted for centuries.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Another example is the &lt;a href=&quot;https://en.wikipedia.org/wiki/Indian_mathematics#Kerala_mathematics_(1300%E2%80%931600)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kerala School&lt;/a&gt;, which is credited with being the first to study infinite series.  These include the Taylor series for the important trigonometric functions (centuries before Newton) as well as the geometric series and the Leibniz formula for &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2023-11-28T02:16:45.927" LastActivityDate="2023-11-28T02:16:45.927" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15956" PostTypeId="2" ParentId="15946" CreationDate="2023-11-26T21:55:46.923" Score="2" Body="&lt;p&gt;In addition to the Pythagorean school, cited in &lt;a href=&quot;https://hsm.stackexchange.com/a/15955/17306&quot;&gt;nwr's answer&lt;/a&gt;, there is an equally famous group of the ancient era, the &lt;em&gt;&lt;strong&gt;peripatetics&lt;/strong&gt;&lt;/em&gt; or &lt;a href=&quot;https://en.wikipedia.org/wiki/Peripatetic_school&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;peripatetic school&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;, which was the name given to the   &lt;a href=&quot;https://www.britannica.com/biography/Aristotle/Political-theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Lyceum of Aristotle&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;em&gt;peripatetics&lt;/em&gt; were a group of students that gathered around the personality of Aristotle, and took its name from  the cloister (&lt;em&gt;peripatos&lt;/em&gt;) in which they walked and held their discussions. They dealt with philosophical and scientific disciplines, that, of course, were not so distinct as nowadays.&lt;/p&gt;&#xA;&lt;p&gt;In particular, from a scientific point of view, Aristotle and his disciples dealt  not only with physics, but also with biology.&lt;/p&gt;&#xA;&lt;p&gt;Even if physics, and his treatises collected in &lt;a href=&quot;https://en.wikipedia.org/wiki/Physics_(Aristotle)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Physics&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;,   are the most well-known aspect of Aristotle's philosophical and scientific thought, in a subsequent period of his life Aristotle dealt with biology, and his biological researches were  the fruit of long and precise studies he carried out, with the collaboration of his disciples, during the twelve years of direction of the &lt;em&gt;Lyceum&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Undoubtedly, even before him [Aristotle] the Greeks had undertaken various inquiries of biological nature, but Aristotle provided a new approach to these researches [...]. We need only recall that, in his works about biology, he speaks of about five hundred animal species, many of them studied with anatomic dissections, drawings of organs, and so on.&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;His most important book  about biology is &lt;a href=&quot;https://en.wikipedia.org/wiki/Parts_of_Animals&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Parts of Animals&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The peripatetic &lt;a href=&quot;https://en.wikipedia.org/wiki/Theophrastus&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Theophrastus&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;, the successor of Aristotle as director of the Lyceum, continued the naturalistic and biological studies of his master, in his treatises &lt;em&gt;&lt;strong&gt;Enquiry into Plants (Historia Plantarum)&lt;/strong&gt;&lt;/em&gt;&#xA;and  &lt;em&gt;&lt;strong&gt;On the Causes of Plants&lt;/strong&gt;&lt;/em&gt; .&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The successor of Theophrastus, the peripatetic &lt;a href=&quot;https://en.wikipedia.org/wiki/Strato_of_Lampsacus&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Strato of Lampsacus&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;, marked a   decisive turning point in the course of studies of the Lyceum, stressing again the physical aspect rather than the biological one of naturalistic philosophy, coming to express the necessity of a rapprochement to Democritus. He attempted to reformulate  Aristotle's physics theories on the base of Democritus’  atomism.&#xA;Several modern scholars consider Strato as the 'highest point of Greek physics': but, unfortunately, we keep of him too little to judge exactly the value of his scientific researches and the meaning of  his reinterpretation of Aristotle’s  heritage .&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^1$&lt;/span&gt; &lt;sup&gt; Geymonat L., &lt;em&gt;Storia del pensiero filosofico e scientifico. Vol. I, L’antichità e il medioevo&lt;/em&gt; (En. tr. &lt;em&gt;History of Philosophic and scientific Thought. Vol. 1, Ancient Era and Middle Age)&lt;/em&gt;), Garzanti, 1970, p. 276. My translation. &lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$^2$&lt;/span&gt; &lt;sup&gt;&lt;em&gt;ibid.&lt;/em&gt;, p. 279&lt;/sup&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2023-11-26T21:55:46.923" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15957" PostTypeId="2" ParentId="11344" CreationDate="2023-11-27T15:14:40.207" Score="3" Body="&lt;p&gt;Actually Desargues used the term &amp;quot;project&amp;quot;.  The title of his fundamental work was &amp;quot;Brouillon d'un &lt;em&gt;Project&lt;/em&gt;&amp;quot;.  Notice the unusual spelling (in modern French the word is spelled without the &amp;quot;c&amp;quot;).  It is possible that his use of the term (apparently in a different sense) may have influenced the subsequent choice of the standard name for the field.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-11-27T15:14:40.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15958" PostTypeId="1" CreationDate="2023-11-27T19:59:43.223" Score="3" ViewCount="104" Body="&lt;p&gt;It is well known that Cardano considered the problem of &amp;quot;dividing 10&#xA;into two parts the product of which is 40&amp;quot; in his &lt;em&gt;Ars Magna&lt;/em&gt;. This problems leads to the complex solutions &lt;span class=&quot;math-container&quot;&gt;$5+ \sqrt{-15}$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5- \sqrt{-15}$&lt;/span&gt;, which Cardano would write as &lt;span class=&quot;math-container&quot;&gt;$5\tilde{p}R\tilde{m}15$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$5\tilde{m}R\tilde{m}15$&lt;/span&gt;, respectively.&lt;/p&gt;&#xA;&lt;p&gt;For this problem, Cardano supplemented his algebraic reasoning with a geometric construction of &amp;quot;completing the square&amp;quot; (see Richard Witmer's english translation of the &lt;em&gt;Ars Magna&lt;/em&gt;, p. 219). With this construction, one has to imagine a negative area, that of substracting &lt;span class=&quot;math-container&quot;&gt;$40$&lt;/span&gt; units of area to the square &lt;span class=&quot;math-container&quot;&gt;$AD$&lt;/span&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/lIzw8.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/lIzw8.png&quot; alt=&quot;&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, Cardano seemingly thinks the problem with this construction isn't the negative area, but the fact that &lt;span class=&quot;math-container&quot;&gt;$AD$&lt;/span&gt; is an area while &lt;span class=&quot;math-container&quot;&gt;$40$&lt;/span&gt; is a length:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Yet the nature of AD is not the same&#xA;as that of 40 or of AB, since a surface is far from the nature of a&#xA;number and from that of a line, though somewhat closer to the&#xA;latter. This truly is sophisticated, since with it [i. e., with &lt;span class=&quot;math-container&quot;&gt;$\sqrt{-15}$&lt;/span&gt;] one cannot carry out&#xA;the operations one can in the case of a pure negative and other&#xA;[numbers].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Cardano also treats complex numbers in his other lesser-known book &lt;em&gt;Ars Magna Arithmeticæ&lt;/em&gt;. He considers the problem of &amp;quot;dividing &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; in two parts whose product is &lt;span class=&quot;math-container&quot;&gt;$3$&lt;/span&gt;&amp;quot;. This is the same as solving the equation &lt;span class=&quot;math-container&quot;&gt;$x^2-x+3=0$&lt;/span&gt;. The solutions are complex:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x=\frac{1}{2}\pm \sqrt{\frac{1}{4}-3}=\frac{1}{2}\pm \sqrt{-\frac{11}{4}}.$$&lt;/span&gt;&#xA;Cardano writes this solutions as &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;.&lt;span class=&quot;math-container&quot;&gt;$\tilde{p}$&lt;/span&gt;.R.v.&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;.m.3. and &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;.&lt;span class=&quot;math-container&quot;&gt;$\tilde{m}$&lt;/span&gt;.R.v.&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;.m.3. (see image below). This is done on page 374, &lt;a href=&quot;https://archive.org/details/imgmar3940alllMiscellaneaOpal/page/372/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;volume IV of Cardano's &lt;em&gt;Opera omnia&lt;/em&gt;&lt;/a&gt; (which contains the &lt;em&gt;Ars Magna Arithmeticæ&lt;/em&gt;).&lt;/p&gt;&#xA;&lt;p&gt;My question is: &lt;strong&gt;what is Cardano trying to say in the following excerpt of &lt;em&gt;Ars Magna Arithmeticæ&lt;/em&gt; (p. 374)?&lt;/strong&gt; Does it have anything to do with his confusion about the different &amp;quot;natures&amp;quot; of &lt;span class=&quot;math-container&quot;&gt;$AD$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$40$&lt;/span&gt;? He adds the word &amp;quot;quadrati&amp;quot; to the solutions (&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{2}$&lt;/span&gt;.&lt;span class=&quot;math-container&quot;&gt;$\tilde{p}$&lt;/span&gt;.R.v.&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;quadrati.m.3.), but I don't see the reason for doing this. It seems as if he added a variable &lt;span class=&quot;math-container&quot;&gt;$y^2$&lt;/span&gt; to the solutions, like&#xA;&lt;span class=&quot;math-container&quot;&gt;$$x=\frac{1}{2}\pm \sqrt{\frac{1}{4}y^2-3}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;I don't have a clue why Cardano does this. I have only found a paper of &lt;a href=&quot;https://www.researchgate.net/publication/295682735_L%27Ars_magna_arithmeticae_nel_corpus_matematico_di_Cardano&quot; rel=&quot;nofollow noreferrer&quot;&gt;Veronica Gavagna&lt;/a&gt; about the &lt;em&gt;Ars Magna Arithmeticæ&lt;/em&gt;, but she doesn't talk much about this.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/41TjI.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/41TjI.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19392" LastActivityDate="2023-11-27T19:59:43.223" Title="What is Cardano trying to say in this passage of his Ars Magna Arithmeticæ?" Tags="|mathematics|mathematicians|translations|" AnswerCount="0" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="15959" PostTypeId="2" ParentId="15946" CreationDate="2023-11-28T11:50:51.280" Score="2" Body="&lt;p&gt;&lt;em&gt;The Fundamental Fysiks Group&lt;/em&gt; was a 70s gaggle of physicists ably described by David Kaiser, MIT historian of science, in his book, &lt;em&gt;How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival&lt;/em&gt;. One publication of the group was Fritjof Capra's &lt;em&gt;Tao of Physics&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Fundamental_Fysiks_Group&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Fundamental_Fysiks_Group&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/How_the_Hippies_Saved_Physics/uSOfvQEACAAJ?hl=enhttps://web.mit.edu/dikaiser/www/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.google.com/books/edition/How_the_Hippies_Saved_Physics/uSOfvQEACAAJ?hl=enhttps://web.mit.edu/dikaiser/www/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/The_Tao_of_Physics&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/The_Tao_of_Physics&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2365" LastActivityDate="2023-11-28T11:50:51.280" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15960" PostTypeId="2" ParentId="11344" CreationDate="2023-11-28T14:35:52.890" Score="4" Body="&lt;p&gt;“The first known use of the expression 'projective geometry' only dates back to Olry Terquem (1859)” in the article Sur diverses géométries, Nouvelles Annales de Mathématiques, 18, 445–446, according to &lt;a href=&quot;https://www.cambridge.org/core/books/abs/anachronisms-in-the-history-of-mathematics/measuring-past-geometers-a-history-of-nonmetric-projective-anachronism/75A9056750446E37CABBDAFE12E32986&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.cambridge.org/core/books/abs/anachronisms-in-the-history-of-mathematics/measuring-past-geometers-a-history-of-nonmetric-projective-anachronism/75A9056750446E37CABBDAFE12E32986&lt;/a&gt; page 263.&lt;/p&gt;&#xA;" OwnerUserId="370" LastActivityDate="2023-11-28T14:35:52.890" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15961" PostTypeId="2" ParentId="11344" CreationDate="2023-11-28T17:50:22.523" Score="3" Body="&lt;p&gt;In a series of six articles in the &lt;a href=&quot;https://www.google.com/books/edition/The_Athenaeum/6SEY9DCRnwsC?hl=en&amp;amp;gbpv=1&amp;amp;bsq=%22history%20of%20perspective%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Athenaeum&lt;/em&gt; (1861) (pp. 446,509,549,617,652,727)&lt;/a&gt; on the history of perspective, Augustus De Morgan makes reference to &amp;quot;projective geometry&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;More than thirty years ago I showed a teacher from the country some models for teaching solid geometry, to avoid the confusion which arises from drawing solids on a plane. He asked me whether I was not afraid of making the subject too easy; I could but answer that, let it be as easy as it might there was plenty of trouble ready up above And this is the case with the new projective geometry, and with all other branches of thought: make a plaything of what was difficult, and you will find no more than a difficulty in what was impossible. (447)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Not as early as Viktor Blasjo's 1859 reference.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.google.com/books/edition/The_Penny_Cyclopaedia_of_the_Society_for/8Jjqd969494C?hl=en&amp;amp;gbpv=1&amp;amp;dq=projective%20geometry&amp;amp;pg=PA40&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Penny Cyclopaedia (1841)&lt;/a&gt;: The &amp;quot;geometry of projections&amp;quot;, not literally &amp;quot;projective geometry&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We have already spoken of the geometry of projections [GEOMETRY p. 156]: unfortunately there is no elementary work which gives a general view of its first principles; and until such a work shall appear, the student must search for himself the writings of Monge, Carnôt, Chasles, Poncelet, &amp;amp;c. 'The History of Geometry' by M Chasles referred to in the article cited will furnish many more references and the 'Propriétés Projectives des Figures,' by M. Poncelet is perhaps the work in which the student may most easily make an advantageous beginning of the subject. (p. 40)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2023-11-28T17:50:22.523" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15962" PostTypeId="1" CreationDate="2023-11-28T19:55:11.910" Score="12" ViewCount="4694" Body="&lt;p&gt;There is a lot of interesting information about young mathematicians, but I cannot find any information about the youngest mathematician that published an original research article in a peer-reviewed journal.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have this information? Thanks a lot.&lt;/p&gt;&#xA;" OwnerUserId="19397" LastEditorUserId="6609" LastEditDate="2024-03-16T11:10:35.023" LastActivityDate="2024-04-05T22:49:30.033" Title="Who are the youngest mathematicians that published an original research article in a peer-reviewed journal?" Tags="|mathematicians|reference-request|publishing|big-list|" AnswerCount="9" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="15963" PostTypeId="2" ParentId="15962" CreationDate="2023-11-28T20:22:19.113" Score="8" Body="&lt;p&gt;Daniel Larsen published his &amp;quot;Bertrand's Postulate for Carmichael Numbers&amp;quot; in &lt;em&gt;&lt;a href=&quot;https://academic.oup.com/imrn/article-abstract/2023/15/13072/6647493?redirectedFrom=fulltext&amp;amp;login=false&quot; rel=&quot;noreferrer&quot;&gt;International Mathematics Research Notices&lt;/a&gt;&lt;/em&gt; at age 19.&lt;/p&gt;&#xA;" OwnerUserId="136" LastActivityDate="2023-11-28T20:22:19.113" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15964" PostTypeId="2" ParentId="15962" CreationDate="2023-11-28T20:54:51.543" Score="19" Body="&lt;p&gt;George Bergman, &amp;quot;A number system with an irrational base,&amp;quot; &lt;em&gt;Mathematics Magazine&lt;/em&gt;, Vol. 31, No. 2, Nov.-Dec. 1957, pp. 98-110.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/George_Bergman&quot; rel=&quot;noreferrer&quot;&gt;George Mark Bergman&lt;/a&gt; is an American mathematician who was born on July 22, 1943 in Brooklyn, New York. He was 14 years old at the time of publication. The paper is referenced in Knuth's TAOCP. According to the journal's publisher, the Mathematical Association of America (bolding mine):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Mathematics Magazine&lt;/em&gt; is an Open Select, international, &lt;strong&gt;peer-reviewed&lt;/strong&gt; journal publishing high-quality, lively, readable, and appealing exposition on a wide range of mathematical topics, including original mathematics, historical content, and connections among mathematics and other disciplines.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;G. L. Alexanderson (ed.), &lt;em&gt;Harmony of the World: 75 Years of Mathematics Magazine&lt;/em&gt;, Mathematical Association of America 2007, is a collection of selected papers from the journal's history. Bergman's paper appears on pp. 99-106. According to the editor's &lt;a href=&quot;https://www.cambridge.org/core/books/abs/harmony-of-the-world/number-system-with-an-irrational-base/BC0503A067EFE03FCAA732243463CD5E&quot; rel=&quot;noreferrer&quot;&gt;note&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This article was written by the author when he was a 12-year old student at Junior High School 246 in Brooklyn, New York. Here he explores using the golden mean, which he calls τ, more commonly now called ϕ, as the base of a number system. Bergman later went on to get a PhD at Harvard under the direction of John Tate and has had a distinguished career as a mathematician at the University of California, Berkeley.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Pr&lt;span class=&quot;math-container&quot;&gt;$\infty$&lt;/span&gt;fWiki lists the publication in their &lt;a href=&quot;https://proofwiki.org/wiki/Mathematician:George_Mark_Bergman&quot; rel=&quot;noreferrer&quot;&gt;article stub on Bergman&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2023-11-28T20:54:51.543" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15965" PostTypeId="2" ParentId="15962" CreationDate="2023-11-29T18:43:46.163" Score="13" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Rubin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arthur Leonard Rubin&lt;/a&gt; submitted the following paper at age 12.5 -- he was born in Feb 1956 and the paper was submitted in Nov 1968:&lt;/p&gt;&#xA;&lt;p&gt;Arthur L. Rubin and Jean E. Rubin, &lt;a href=&quot;http://matwbn.icm.edu.pl/ksiazki/fm/fm65/fm65123.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Extended operations and relations on the class of ordinal numbers&lt;/em&gt;&lt;/a&gt;, &lt;strong&gt;Fundamenta Mathematicae&lt;/strong&gt; 69 #2 (1969), 227-242.&lt;/p&gt;&#xA;&lt;p&gt;Aficionados (i.e. fans) of large numbers (e.g. see &lt;a href=&quot;https://mathoverflow.net/a/74395/15780&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.google.com/search?q=large+numbers+%22rapidly+growing+functions%22&quot; rel=&quot;nofollow noreferrer&quot;&gt;this search&lt;/a&gt;) and large countable ordinals (e.g. see Baez's &lt;a href=&quot;https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-ordinals-part-1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Part 1&lt;/a&gt; &amp;amp; &lt;a href=&quot;https://johncarlosbaez.wordpress.com/2016/07/04/large-countable-ordinals-part-2/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Part 2&lt;/a&gt; &amp;amp; &lt;a href=&quot;https://johncarlosbaez.wordpress.com/2016/07/07/large-countable-ordinals-part-3/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Part 3&lt;/a&gt;) and &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperoperation&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperoperations&lt;/a&gt; (i.e. iteratively iterating from tetration) might have encountered this paper.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;(ADDED ABOUT 4 MONTHS LATER)&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;Notices of the American Mathematical Society&lt;/strong&gt; &amp;quot;Preliminary report&amp;quot; for this paper appears on pp. 196-197 of the &lt;a href=&quot;https://www.ams.org/journals/notices/196801/196801FullIssue.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;15 #1 (January 1968)&lt;/a&gt; issue, and the published abstract's received date is 20 October 1967 (age 11.5).&lt;/p&gt;&#xA;" OwnerUserId="264" LastEditorUserId="264" LastEditDate="2024-03-14T16:07:51.240" LastActivityDate="2024-03-14T16:07:51.240" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15966" PostTypeId="2" ParentId="15948" CreationDate="2023-11-30T04:59:31.477" Score="4" Body="&lt;p&gt;Your question requires a lot of historical background of late 1890s and early 1900s. I requested a couple of articles from different libraries. Basically, it will be best to dispel the common myth that &amp;quot;p&amp;quot; means anything from a certain European language. It does not. Later authors developed this association. At least Sorenson never used p for power or the German Potenz and there are many more stories as shown in the Table. An interested reader should consult Nørby, Jens G. &amp;quot;The origin and the meaning of the little p in pH.&amp;quot; Trends in Biochemical Sciences 25.1 (2000): 36-37 for actual excerpts and other common stories associated with the letter p. His father worked with Sorenson. See this table for example from Nørby:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/VvdeN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/VvdeN.png&quot; alt=&quot;Table&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now accepting the &lt;strong&gt;definition&lt;/strong&gt; that pH represented a negative logarithm (of base 10) of hydrogen ion concentration, the invention of pH concept is merely &lt;em&gt;a matter of convenience&lt;/em&gt;. Even in the early 1900's, chemists were interested in measuring the concentration of hydrogen ions by indicators. Then it must be naturally convenient to invent a scale of hydrogen ion concentration because one can vary the concentration of &amp;quot;hydrogen&amp;quot; over several orders of magnitude in water.&lt;/p&gt;&#xA;&lt;p&gt;I do not exactly know when the letter p became an operator of of -log, but currently it is used as such.&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2023-11-30T04:59:31.477" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15967" PostTypeId="2" ParentId="15946" CreationDate="2023-11-30T09:21:54.477" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Arthur_Besse&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arthur Besse&lt;/a&gt; is not a single person but rather a group of French mathematicians. Their most famous work is a book on Einstein Manifolds. Even if you have the book in your hands it is still nontrivial to see that Arthur Besse is not an actual person.&lt;/p&gt;&#xA;" OwnerUserId="15145" LastActivityDate="2023-11-30T09:21:54.477" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15968" PostTypeId="1" AcceptedAnswerId="15969" CreationDate="2023-11-30T11:11:02.093" Score="7" ViewCount="255" Body="&lt;p&gt;My question is clearly related to &lt;a href=&quot;https://hsm.stackexchange.com/q/7216&quot;&gt;this one&lt;/a&gt;, but my interest is not specifically in Heisenberg's result. To quote from Wikipedia.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A nonzero function and its Fourier transform cannot both be sharply localized at the same time. A similar tradeoff between the&#xA;variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency, while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Who first explicitly notes the fact stated in the first sentence (A nonzero function and its Fourier transform cannot both be sharply localized at the same time), and who was it that found that the Gaussian distribution makes the trade-off equal in the frequency and time domains?&lt;/p&gt;&#xA;" OwnerUserId="17963" LastEditorUserId="4251" LastEditDate="2023-12-05T14:49:39.117" LastActivityDate="2024-01-09T14:03:37.793" Title="Who first stated the &quot;uncertainty principle&quot; for Fourier transforms?" Tags="|mathematics|harmonic-analysis|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15969" PostTypeId="2" ParentId="15968" CreationDate="2023-11-30T14:25:25.307" Score="8" Body="&lt;p&gt;It seems that Heisenberg was the first to notice this phenomenon in 1927. He did not use the terminology of Fourier transform. Heisenberg's principle was rigorously stated and proved by Kennard in 1927 and Weyl in 1928 (see the &lt;a href=&quot;https://en.wikipedia.org/wiki/Uncertainty_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;references in Wikipedia&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The first formulation in pure mathematical literature is due to G. H. Hardy,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://doi.org/10.1112/jlms/s1-8.3.227&quot; rel=&quot;nofollow noreferrer&quot;&gt;A theorem concerning Fourier transforms&lt;/a&gt;, &lt;em&gt;Journal of the London Mathematical Society&lt;/em&gt;, 8 1933, 227-231.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Hardy cites an unpublished remark by N. Wiener:&#xA;&amp;quot;a pair of transforms  and  cannot both be very small&amp;quot;, and does not mention Heisenberg. However, mathematical formulation of Hardy is different from Heisenberg's. Since then many results in the spirit of Wiener's principle were proven. See, for example the book by Khavin and Joricke, The Uncertainty Principle in Harmonic Analysis, Springer , 1994.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-01-09T14:03:37.793" LastActivityDate="2024-01-09T14:03:37.793" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15970" PostTypeId="2" ParentId="15962" CreationDate="2023-11-30T14:51:38.733" Score="6" Body="&lt;p&gt;John von Neumann published his first co-authored paper ''On the position of zeroes of certain minimum polynomials'' with Michael Fekete at the age of 18.&lt;/p&gt;&#xA;&lt;p&gt;His first solo paper ''On the introduction of transfinite numbers'' was published at age 19.&lt;/p&gt;&#xA;&lt;p&gt;Maryam Mirzakhani also published her first paper (''A small non-4-choosable planar graph'') aged 19.&lt;/p&gt;&#xA;" OwnerUserId="10345" LastEditorUserId="10345" LastEditDate="2023-11-30T17:35:06.067" LastActivityDate="2023-11-30T17:35:06.067" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15971" PostTypeId="2" ParentId="15931" CreationDate="2023-12-01T00:10:09.443" Score="3" Body="&lt;p&gt;Oh I see, normally an ExB filter is called a &amp;quot;velocity filter&amp;quot; - see Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/Wien_filter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wien filter&lt;/a&gt; where there be math.&lt;/p&gt;&#xA;&lt;p&gt;In an ExB filter charged particles go in a straight line when &lt;span class=&quot;math-container&quot;&gt;$v = E/B$&lt;/span&gt;. So any combination with the same ratio will work.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;But&lt;/strong&gt; by adding the force of gravity which depends on the particle's mass explicitly, you've &amp;quot;broken&amp;quot; the Wien filter. At normal lab voltages and magnetic fields and accelerated electron velocities, the gravity term is so very tiny that it would be buried below all of the other experimental uncertainties in this device.&lt;/p&gt;&#xA;&lt;p&gt;So here in HSM SE I think that the answer to&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In JJ Thomson's cathode ray experiment I need values for the electric field and magnetic field when net force on the cathode beam = 0&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;is &lt;strong&gt;No you don't.&lt;/strong&gt; For historical purposes of JJ Thomson's experiment, gravity was a way way negligible effect, and the problem reduces to the simple E/B ratio. The tube is an early &lt;a href=&quot;https://en.wikipedia.org/wiki/Wien_filter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wien filter&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;However, in Physics SE some math will be needed to see if there's an analytical solution to your problem, or if you have to use a numerical technique to find the roots (zeros) of the expression for total force.&lt;/p&gt;&#xA;&lt;p&gt;So rewrite your equation in the form &lt;span class=&quot;math-container&quot;&gt;$F_{total} = ...$&lt;/span&gt;, set it to zero, and massage it mathemtatically to see if an analytical solution exist (probably does!) or if you have to find the zeros. There will of course be (at least) two because the gravitational force can be in the same direction of either the magnetic, or the electric force. You chose only one of the two possibilities in your example.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;update:&lt;/strong&gt; In your cross-post in Chemistry SE &lt;a href=&quot;https://chemistry.stackexchange.com/a/177932/16035&quot;&gt;@BuckThorn used the first term in the Taylor expansion&lt;/a&gt; to get an approximate solution.&lt;/p&gt;&#xA;" OwnerUserId="3869" LastEditorUserId="3869" LastEditDate="2023-12-01T00:16:07.757" LastActivityDate="2023-12-01T00:16:07.757" CommentCount="16" ContentLicense="CC BY-SA 4.0" />
  <row Id="15972" PostTypeId="1" AcceptedAnswerId="15973" CreationDate="2023-12-01T08:27:50.043" Score="5" ViewCount="1001" Body="&lt;p&gt;I was reading about the discovery of Neptune when I realised that Caroline Herschel was still alive at the time. Neptune was discovered in September 1846. Caroline died in January 1848, more than a year later. Do we have evidence as to whether she knew about Neptune’s discovery? If she did, then do we know anything about her reactions to it?&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia lists two full length biographies:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;The Comet Sweeper: Caroline Herschel's Astronomical Ambition&lt;/em&gt; by Claire Brock (2007)&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;The Quiet Revolution of Caroline Herschel: The Lost Heroine of Astronomy&lt;/em&gt; by Emily Winterburn (2017)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I was able to borrow the first one on the &lt;a href=&quot;https://archive.org/details/cometsweepercaro0000broc&quot; rel=&quot;nofollow noreferrer&quot;&gt;Internet Archive&lt;/a&gt; and did a text search but there was no mention of Neptune. I also did a text search  on the second book on &lt;a href=&quot;https://books.google.co.uk/books?id=PmQxDwAAQBAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;Google Books&lt;/a&gt; and it also does not mention Neptune.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Motivation for the question&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The discovery of Neptune came about due to the need to explain irregularities in the orbit of Uranus which had been discovered by Caroline's brother, William Herschel, in 1781, and was notable as the first new planet to be discovered since antiquity. Caroline assisted William in his work and was also a celebrated astronomer in her own right. She maintained an interest in astronomy into her old age, so I thought that the news would have been of considerable interest to her.&lt;/p&gt;&#xA;" OwnerUserId="19409" LastEditorUserId="10368" LastEditDate="2023-12-01T17:26:39.050" LastActivityDate="2023-12-01T17:26:39.050" Title="Did Caroline Herschel know about the discovery of Neptune?" Tags="|biographical-details|astronomy|astronomers|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15973" PostTypeId="2" ParentId="15972" CreationDate="2023-12-01T09:31:17.633" Score="12" Body="&lt;p&gt;The answer is yes, she did know. Remarkably she was informed of the news mere days after the discovery in a letter from Alexander von Humboldt dated September 25 1846 which accompanied the Gold Medal presented by the King of Prussia on her 96th birthday. The planet had yet to be named and he refers to it as 'the transuranian planet'. Here is the relevant section from the letter&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[The King of Prussia] had wished you to receive this little gratification on your ninety-sixth birthday, and by an unfortunate mistake the date of Caroline Lucretia Herschel’s birth has been changed from the 16th of March to the 16th of October, and I am the culprit, misled by a misprint in a French history of astronomy. I know I may count upon your indulgence and that of your distinguished family in England. I specially deserve such leniency to-day—the day on which my young friend, Dr. Galle, assistant astronomer in our Observatory (to the triumph of theoretical astronomy be it said), has discovered the transuranian planet indicated by Le Verrier as the cause of the perturbations of Uranus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Caroline clearly read this letter since on 1 October 1846 she wrote to her niece about it although she makes no mention of the new planet&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My dearest Niece!—&lt;/p&gt;&#xA;&lt;p&gt;I must not let the messenger go without a line just to say that I am still in the land of the living, of which, however, I have no other proof than a letter from Baron v. Humboldt, inclosing a Golden Medal from the King of Prussia. I can say no more at present, and the post will not wait, so believe me, my dear niece, yours and my dear nephew’s most affectionate aunt,&lt;/p&gt;&#xA;&lt;p&gt;                                                                                                                            Car. Herschel.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My quick browse through the &lt;em&gt;Memoir and Correspondence of Caroline Herschel&lt;/em&gt;, where I found these letters did not reveal anything about her reaction to the news. However in a letter from her nephew Sir John Herschel dated July 11 1847 which accompanied his “Cape Observations” book, he mentions that Le Verrier is at Collingwood, Herschel's home, so presumably she knew who he was.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Louisa sends you all our news, and the autographs of Struve and Adams, who, with M. Leverrier, are now at Collingwood.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Source: &lt;a href=&quot;https://www.fadedpage.com/showbook.php?pid=20130405&quot; rel=&quot;noreferrer&quot;&gt;Memoir and Correspondence of Caroline Herschel&lt;/a&gt; by Mrs John Herschel (1876)&lt;/p&gt;&#xA;" OwnerUserId="19409" LastEditorUserId="19409" LastEditDate="2023-12-01T10:54:23.030" LastActivityDate="2023-12-01T10:54:23.030" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15974" PostTypeId="1" AcceptedAnswerId="15975" CreationDate="2023-12-01T16:18:23.920" Score="4" ViewCount="646" Body="&lt;p&gt;I wonder what the quotes from mathematicians are about their views on academic research vs. math competitions. I can't find much, but I have found one:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Math competitions are to research what spelling bees are to writing.&lt;/em&gt; - Bill Thurston&lt;/p&gt;&#xA;&lt;p&gt;What are other quotes related to this topic in the history of mathematics?&lt;/p&gt;&#xA;" OwnerUserId="19397" LastEditorUserId="19397" LastEditDate="2023-12-01T16:31:55.697" LastActivityDate="2023-12-01T22:46:12.233" Title="What is the opinion of famous mathematicians about academic research compared to mathematical competitions and olympiads?" Tags="|mathematicians|" AnswerCount="1" CommentCount="1" ClosedDate="2023-12-05T13:26:19.807" ContentLicense="CC BY-SA 4.0" />
  <row Id="15975" PostTypeId="2" ParentId="15974" CreationDate="2023-12-01T22:46:12.233" Score="9" Body="&lt;p&gt;Here's a page containing quotes from mathematicians such as Tao, Shimura and Wiles (and which includes a much-expanded version of the Thurston quote):&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.lesswrong.com/posts/EdFDwjsLNpgtTMJAp/great-mathematicians-on-math-competitions-and-genius&quot; rel=&quot;noreferrer&quot;&gt;https://www.lesswrong.com/posts/EdFDwjsLNpgtTMJAp/great-mathematicians-on-math-competitions-and-genius&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Part of the quote by Wiles:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While there have been periods, notably in the thirteenth, fourteenth and fifteenth centuries when mathematicians would engage in timed duels with each other, nowadays this is not the custom. In fact time is very much on your side. However the transition from a sprint to a marathon requires a new kind of stamina and a profoundly different test of character. We admire someone who can win a gold medal in five successive Olympics games not so much for the raw talent as for the strength of will and determination to pursue a goal over such a sustained period of time. Real mathematical theorems will require the same stamina whether you measure the effort in months or in years&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16185" LastActivityDate="2023-12-01T22:46:12.233" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15976" PostTypeId="1" CreationDate="2023-12-01T22:51:47.530" Score="2" ViewCount="82" Body="&lt;p&gt;The barycentric calculus of Möbius involves formal sums of expressions of the form &lt;span class=&quot;math-container&quot;&gt;$mP$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; is a real number and &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is a point, where &lt;span class=&quot;math-container&quot;&gt;$mP$&lt;/span&gt; is to be thought of as &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; units of mass located at &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. This is not problematic as long as all masses are positive. But how did Möbius interpret expressions like &lt;span class=&quot;math-container&quot;&gt;$(1)P + (-1)Q$&lt;/span&gt;? Nowadays we would identify it with the vector from &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. Did Möbius have any interpretation of this kind?&lt;/p&gt;&#xA;" OwnerUserId="6587" LastActivityDate="2023-12-04T17:46:10.737" Title="Negative coefficients in the barycentric calculus" Tags="|geometry|linear-algebra|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15979" PostTypeId="1" CreationDate="2023-12-02T15:45:10.960" Score="2" ViewCount="231" Body="&lt;p&gt;In article 23 of his first memoir on biquadratic residues, Gauss gave the first example of a binomial-type congruence, which is apparently a quite &amp;quot;deep&amp;quot; result - many authors relate it to the theory of integral points on certain elliptic curves. His result is stated as follows: if &lt;span class=&quot;math-container&quot;&gt;$p=4n+1$&lt;/span&gt; is a prime, then &lt;span class=&quot;math-container&quot;&gt;$p=a^2+b^2$&lt;/span&gt; for some integers &lt;span class=&quot;math-container&quot;&gt;$a,b$&lt;/span&gt; (this is Fermat's two squares theorem). Gauss showed that if &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; is fixed by &lt;span class=&quot;math-container&quot;&gt;$a\equiv 1 \pmod{4} $&lt;/span&gt; then&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$2a\equiv{2n \choose n}\pmod {p}$$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$2a\equiv{\frac{p-1}{2} \choose \frac{p-1}{4}}\pmod {p}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In his treatise on Gauss's number theoretic investigations, Paul Bachmann remarks that other isolated theorems of this kind were also known to Gauss, and refers to p. 39 of volume 10-1 of his collected works. Looking there, I found the following statements of Gauss:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$p=3n+1=x^2+3y^2$&lt;/span&gt; is a prime number, then &lt;span class=&quot;math-container&quot;&gt;$$2x\equiv {\frac{p-1}{2}\choose \frac{p-1}{6}} \pmod {p}$$&lt;/span&gt; Under the same condition, if &lt;span class=&quot;math-container&quot;&gt;$4p=l^2+27m^2$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$$l\equiv - {\frac{2(p-1)}{3}\choose \frac{p-1}{3}}\pmod{p}.$$&lt;/span&gt; If a prime number is &lt;span class=&quot;math-container&quot;&gt;$p=8n+1 = a^2+2b^2,$&lt;/span&gt;  then &lt;span class=&quot;math-container&quot;&gt;$$2a\equiv {\frac{p-1}{2}\choose \frac{p-1}{8}} \pmod {p},$$&lt;/span&gt; and&#xA;if &lt;span class=&quot;math-container&quot;&gt;$p=8n+3=a^2+2b^2$&lt;/span&gt; then &lt;span class=&quot;math-container&quot;&gt;$$2a \equiv&#xA;&amp;gt; {\frac{p-1}{2}\choose\frac{p-3}{8}} \pmod {p}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While his published binomial congruence is quite referenced in the literature (it is often refered to as the &amp;quot;classical&amp;quot; example of such congruences), I have not found discussions of the unpublished results from his Nachlass. Are these unpublished congruences not discussed just because they are based on similar principles as his published one? and are their derivations of similar complexity? whatever the answer is, I would like to know on what principles they rest (but only in general terms, as I am not a mathematician and right now I cannot read into the details).&lt;/p&gt;&#xA;&lt;p&gt;Any useful comments/references will be appreciated!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="12330" LastEditDate="2023-12-04T21:43:38.770" LastActivityDate="2023-12-04T21:43:38.770" Title="Implications of several unpublished &quot;binomial congruences&quot; of Gauss" Tags="|number-theory|gauss|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15980" PostTypeId="1" CreationDate="2023-12-02T15:52:27.690" Score="1" ViewCount="68" Body="&lt;p&gt;Now we generally think of the Platonic solids as being the regular convex polyhedra. And while the Ancient Greeks were aware of this solids as being particularly special, I don't believe that it is likely that they thought of them in the same way.&lt;/p&gt;&#xA;&lt;p&gt;There are two main reasons I suspect this:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;The modern definition of regular polytope, &amp;quot;A polytope whose symmetry group acts transitively on its flags&amp;quot; involves some concepts which seem a little to synthetic for the Ancient Greeks.&lt;/li&gt;&#xA;&lt;li&gt;There are lots of other properties held by the Platonic solids as a set that make them unique among the solids.&#xA;&lt;ul&gt;&#xA;&lt;li&gt;They are the only regular finite &lt;em&gt;acoptic&lt;/em&gt; polyhedra (having no self-intersection or skew elements). I doubt they even considered infinite or non-acoptic polyhedra to be polyhedra, so convexity would likely have been a redundant condition.&lt;/li&gt;&#xA;&lt;li&gt;They are the only finite &lt;em&gt;weakly-regular&lt;/em&gt; (isohedral, isotoxal and isogonal) acoptic polyhedra.&lt;/li&gt;&#xA;&lt;li&gt;Every equivelar (every vertex having the same degree and every face having the same number of edges) finite acoptic polyhedron is a distortion of a Platonic solid.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;There are more ways to slightly modify or combine the above properties to get new classifiers that specify the Platonic solids. Additionally there are plenty more properties that hold for the Platonic solids specifically, but which are unlikely to have been discovered by the Ancient Greeks.&lt;/p&gt;&#xA;&lt;p&gt;What did the Ancient Greeks think the Platonic solids were?  What property exactly did they consider to set that set apart?&lt;/p&gt;&#xA;&lt;p&gt;I'd appreciate answers that are grounded in ancient sources, but I also understand that these sources can be vague by modern standards, so some interpretation might be necessary.&lt;/p&gt;&#xA;" OwnerUserId="6372" LastActivityDate="2023-12-02T15:52:27.690" Title="How did the Ancient Greeks conceive of the Platonic solids?" Tags="|geometry|ancient-greece|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15981" PostTypeId="2" ParentId="15979" CreationDate="2023-12-02T16:58:07.027" Score="4" Body="&lt;p&gt;I have seen papers and books on Gauss and Jacobi sums discuss such results, e.g., see &lt;a href=&quot;https://core.ac.uk/download/pdf/82122545.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://core.ac.uk/download/pdf/82122545.pdf&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The book &amp;quot;Gauss and Jacobi sums&amp;quot; by Berndt, Evans, and Williams has some results like this too: see page 2 in the introduction and look up &amp;quot;binomial coefficients&amp;quot; in the index. They also related use the Gross-Koblitz formula for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;-adic Gauss sums to improve some congruences to modulus &lt;span class=&quot;math-container&quot;&gt;$p^2$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Congruences like this appear in some exercises in Ireland &amp;amp; Rosen's &lt;em&gt;A Classical Introduction to Modern Number Theory&lt;/em&gt; 2nd ed. (exer. 27 in Chapter 8, exer. 39 &amp;amp; 40 in Chapter 9)&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-12-02T17:21:25.853" LastActivityDate="2023-12-02T17:21:25.853" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15982" PostTypeId="1" CreationDate="2023-12-02T23:58:30.743" Score="1" ViewCount="98" Body="&lt;p&gt;I am looking for a complete collection of fragments of &lt;a href=&quot;https://en.wikipedia.org/wiki/Eudemus_of_Rhodes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eudemus of Rhodes&lt;/a&gt;. He is considered to be the first historian of science. His fragments have given us a glimpse of how Greek mathematics developed through the earliest Greek philosophers.&lt;/p&gt;&#xA;&lt;p&gt;Are the fragments of Eudemus available online in Ancient Greek or English?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Potential Candidates:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Edemas von Rhodos&lt;/em&gt;, by F. Wehrli (1955) gathered 90 fragments of Eudemus for &lt;em&gt;Die Schule des Aristoteles vol. VIII&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;em&gt;Eudemus of Rhodes&lt;/em&gt; by István Bodnár, William Fortenbaugh (2002) is a complete collection of the fragments and testimonia of Eudemus.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="19033" LastEditorUserId="19033" LastEditDate="2023-12-03T04:57:50.930" LastActivityDate="2023-12-03T04:57:50.930" Title="A collection of the fragments of Eudemus of Rhodes?" Tags="|reference-request|antiquity|greek-mathematics|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="15983" PostTypeId="2" ParentId="15976" CreationDate="2023-12-03T01:29:34.603" Score="2" Body="&lt;p&gt;The Wikipedia article Barycentric coordinate system states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.&lt;/p&gt;&#xA;&lt;p&gt;Every point has barycentric coordinates, and their sum is not zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The quote &amp;quot;is at infinity&amp;quot; does suggest projective geometry.&#xA;In fact, this can be made precise. Consider three dimensional&#xA;Euclidean space with Cartesian coordinates. Interpret the&#xA;homogeneous coordinate triple &lt;span class=&quot;math-container&quot;&gt;$(x: y: z)$&lt;/span&gt; as referring to a two&#xA;dimensional plane point &lt;span class=&quot;math-container&quot;&gt;$(x/z, y/z)$&lt;/span&gt; with &lt;em&gt;weight&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; where the&#xA;weights are additive. That is, the weight function &lt;span class=&quot;math-container&quot;&gt;$(x, y, z)&#xA;\mapsto z$&lt;/span&gt; is linear. Note the relation &lt;span class=&quot;math-container&quot;&gt;$(x_1: y_1: z_1) \equiv&#xA;(x_2: y_2: z_2)$&lt;/span&gt; iff &lt;span class=&quot;math-container&quot;&gt;$(x_1, y_1, z_1) = w(x_2, y_2, z_2)$&lt;/span&gt; where&#xA;&lt;span class=&quot;math-container&quot;&gt;$w\ne 0$&lt;/span&gt;. This equivalence relation partitions the space into&#xA;equivalence classes with representatives unit weight &lt;span class=&quot;math-container&quot;&gt;$(x: y: 1)$&lt;/span&gt;&#xA;which represents the point &lt;span class=&quot;math-container&quot;&gt;$(x, y)$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$z\ne 0.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The zero weight triple &lt;span class=&quot;math-container&quot;&gt;$(x: y: 0)$&lt;/span&gt; represents a point on the line&#xA;at infinity. One way to understand this is to consider the point&#xA;&lt;span class=&quot;math-container&quot;&gt;$R :=(w)P + (1-w)Q$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; varies. If &lt;span class=&quot;math-container&quot;&gt;$w=0$&lt;/span&gt; it is the point &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt;.&#xA;If &lt;span class=&quot;math-container&quot;&gt;$w=1$&lt;/span&gt; it is the point &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. In the limit where &lt;span class=&quot;math-container&quot;&gt;$w\to\infty$&lt;/span&gt; the&#xA;two weights &lt;span class=&quot;math-container&quot;&gt;$1-w$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$w$&lt;/span&gt; are proportional to &lt;span class=&quot;math-container&quot;&gt;$-1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; while&#xA;the  point &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; move off to infinity. Thus, &lt;span class=&quot;math-container&quot;&gt;$(1)P +(-1)Q$&lt;/span&gt; represents&#xA;the point on the line at infinity in the &lt;em&gt;direction&lt;/em&gt; of the vector&#xA;from &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt;. There are contexts where &lt;span class=&quot;math-container&quot;&gt;$(1)P +(-1)Q$&lt;/span&gt; represents&#xA;the vector from &lt;span class=&quot;math-container&quot;&gt;$Q$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; but this is not one of them.&lt;/p&gt;&#xA;" OwnerUserId="13622" LastEditorUserId="13622" LastEditDate="2023-12-04T17:46:10.737" LastActivityDate="2023-12-04T17:46:10.737" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15984" PostTypeId="2" ParentId="15962" CreationDate="2023-12-03T03:06:41.320" Score="4" Body="&lt;ul&gt;&#xA;&lt;li&gt;It's well known that &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Pascal/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blaise Pascal&lt;/a&gt; published his &lt;em&gt;Essai pour les coniques&lt;/em&gt; in February of 1640 at the age of 16.&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89variste_Galois&quot; rel=&quot;nofollow noreferrer&quot;&gt;Galois'&lt;/a&gt; &lt;em&gt;Démonstration d’un théorème sur les fractions&#xA;continues périodiques&lt;/em&gt; paper was published in the Annales de mathématiques pures et appliquées in April 1829 at the age of 17.&lt;/li&gt;&#xA;&lt;li&gt;At age 18, &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Erd%C5%91s&quot; rel=&quot;nofollow noreferrer&quot;&gt;Erdős&lt;/a&gt; proved Bertrand's postulate and had it published the next year at age 19 in the article &lt;em&gt;Beweis eines Satzes von Tschebyschef&lt;/em&gt; in Acta Litterarum ac Scientiarum (1932).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="2918" LastEditorUserId="2918" LastEditDate="2023-12-05T14:18:00.840" LastActivityDate="2023-12-05T14:18:00.840" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15985" PostTypeId="1" CreationDate="2023-12-03T07:00:55.230" Score="4" ViewCount="85" Body="&lt;p&gt;I'm curious about Pfaff's problem and all the various mathematicians who have worked on it. From Wikipedia articles I was able to gather the following timeline of events.&lt;/p&gt;&#xA;&lt;p&gt;In the modern language:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Pfaff's problem:&lt;/strong&gt; Given a collection of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-forms on a manifold, find a submanifold where they vanish.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Frobenius theorem:&lt;/strong&gt; A distribution on a manifold is integrable if and only if it is involutive.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Darboux's theorem:&lt;/strong&gt; Given a &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-form &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$d\theta$&lt;/span&gt; having constant rank, locally &lt;span class=&quot;math-container&quot;&gt;$\theta$&lt;/span&gt; has a nice canonical form.&lt;/p&gt;&#xA;&lt;p&gt;There is a similar canonical form for a collection of involutive vector fields and this is used in modern proofs of the Frobenius theorem.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Timeline:&lt;/strong&gt; (from Wikipedia)&lt;/p&gt;&#xA;&lt;p&gt;1814-1815: Pfaff asks about the integrability of a system of &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt;-forms, presumably just a system of ODEs in the modern language.&lt;/p&gt;&#xA;&lt;p&gt;1840: Deahna provides a sufficient condition for the Frobenius theorem.&lt;/p&gt;&#xA;&lt;p&gt;1866: Clebsch provides a necessary condition for the Frobenius theorem.&lt;/p&gt;&#xA;&lt;p&gt;1887: Frobenius applies &amp;quot;Frobenius&amp;quot; theorem to the Pfaffian problem.&lt;/p&gt;&#xA;&lt;p&gt;1882: Darboux proves his theorem, solving Pfaff's problem because the solution can be obtained, at least locally, by setting some coordinates (in a specific parametrization) to zero.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Questions:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Today, differential equations are either treated as just mathematical objects that describe surfaces et c., or in relation to solving some physical system or finding trajectories of objects and so on. I suspect Pfaff was concerned with the mechanics and dynamics of physical systems, but I'm not sure. What prompted Pfaff to pose his problem and did he work on it?&lt;/p&gt;&#xA;&lt;p&gt;Secondly, in modern terms one of the implications of Frobenius theorem is fairly trivial using the notion of Lie brackets, pushforwards et c. So, what exactly were Deahna and Clebsch concerned with? Were they simply answering a question of &amp;quot;a system of ODEs has a solution if...&amp;quot; and &amp;quot;if a solution exists, then the coefficients of the ODES should satisfy...&amp;quot;? Of course, without the modern language, I'd say that both implications would look like a mess of symbols and far from trivial.&lt;/p&gt;&#xA;&lt;p&gt;Lastly, I'm completely in the dark as to what Frobenius' contribution is. Wikipedia simply states that he &amp;quot;applied&amp;quot; Frobenius theorems to the problem. Did Frobenius give an answer to Pfaff's problem or give a hint on tackling Pfaff's problem? Was Darboux aware of Frobenius' work or was it a different kind of solution answering a differently stated problem?&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt; (from Wikipedia)&lt;/p&gt;&#xA;&lt;p&gt;Pfaff, Johann Friedrich (1814–1815). &amp;quot;&lt;a href=&quot;https://archive.org/details/abhandlungenderp14akad/page/76/mode/1up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi&lt;/a&gt;&amp;quot; [A general method to completely integrate partial differential equations, as well as ordinary differential equations, of order higher than one, with any number of variables]. Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin (in Latin): 76–136&lt;/p&gt;&#xA;&lt;p&gt;Deahna, F. (1840). &amp;quot;&lt;a href=&quot;https://zenodo.org/record/2312273&quot; rel=&quot;nofollow noreferrer&quot;&gt;Über die Bedingungen der Integrabilitat ...&lt;/a&gt;&amp;quot; J. Reine Angew. Math. 20: 340–350. doi:10.1515/crll.1840.20.340. S2CID 120057555.&lt;/p&gt;&#xA;&lt;p&gt;Clebsch, A. (1866). &amp;quot;&lt;a href=&quot;https://zenodo.org/record/2050541&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die simultane Integration linearer partieller Differentialgleichungen&lt;/a&gt;&amp;quot;. J. Reine. Angew. Math. (Crelle). 1866 (65): 257–268. doi:10.1515/crll.1866.65.257. S2CID 122439486.&lt;/p&gt;&#xA;&lt;p&gt;Frobenius, G. (1877). &amp;quot;Über das Pfaffsche Problem&amp;quot;. J. Reine Angew. Math. 1877 (82): 230–315. doi:&lt;a href=&quot;https://doi.org/10.1515%2Fcrll.1877.82.230&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1515/crll.1877.82.230&lt;/a&gt;. S2CID 119848431.&lt;/p&gt;&#xA;&lt;p&gt;Darboux, Gaston (1882). &amp;quot;&lt;a href=&quot;http://gallica.bnf.fr/ark:/12148/bpt6k68005v&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur le problème de Pfaff&lt;/a&gt;&amp;quot; [On the Pfaff's problem]. Bull. Sci. Math. (in French). 6: 14–36, 49–68.&lt;/p&gt;&#xA;" OwnerUserId="13222" LastActivityDate="2023-12-03T07:00:55.230" Title="History of Pfaff's problem and integrability conditions" Tags="|differential-geometry|differential-equations|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="15988" PostTypeId="1" CreationDate="2023-12-04T11:52:19.780" Score="8" ViewCount="147" Body="&lt;p&gt;Who first suggested that the area enclosed by a closed path and the area enclosed by that same path traversed in reverse could be regarded as equal in magnitude but opposite in sign?&lt;/p&gt;&#xA;&lt;p&gt;Cauchy must have noticed this in connection with contour integrals, but perhaps Gauss or someone else came up with this perspective first, possibly in connection with surveying (the first mechanical planimeter was invented in 1818).&lt;/p&gt;&#xA;&lt;p&gt;I would appreciate not just an attribution of priority but a supporting citation.&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="6587" LastEditDate="2023-12-04T20:46:58.687" LastActivityDate="2023-12-13T21:30:30.583" Title="Who first considered signed area?" Tags="|geometry|complex-analysis|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="15989" PostTypeId="1" CreationDate="2023-12-04T15:44:07.020" Score="10" ViewCount="2510" Body="&lt;p&gt;A full professor teaching the history of mathematics at Masters level recently told a friend of mine that there was nothing of interest left to explore in the mathematics written in Latin over the last 1000 years. Given that a significant part of Euler's work has never been translated into a modern language, and that I recall a recent discovery about Gauss's invention of what is now called the &lt;a href=&quot;https://en.wikipedia.org/wiki/Fast_Fourier_transform&quot; rel=&quot;nofollow noreferrer&quot;&gt;fast Fourier transform&lt;/a&gt; (FFT), I was very surprised. I think my confusion is caused by my lack of familiarity with the field. So, how do we explain the lack of activity in the study of Latin mathematics?&lt;/p&gt;&#xA;" OwnerUserId="19422" LastEditorUserId="4251" LastEditDate="2023-12-05T13:35:57.337" LastActivityDate="2023-12-05T14:34:04.493" Title="How do we explain the lack of activity in the study of Latin mathematics?" Tags="|mathematics|language|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="15990" PostTypeId="2" ParentId="15989" CreationDate="2023-12-04T16:10:16.047" Score="12" Body="&lt;p&gt;Since the full professor in question certainly hasn't read all of &amp;quot;the mathematics written in Latin over the last 1000 years,&amp;quot; one can assert with certainty that he &lt;em&gt;literally&lt;/em&gt; does not know what he is talking about.  Half of Leibniz's writing has not even been digitalized, although &lt;a href=&quot;http://www.gottfried-wilhelm-leibniz-gesellschaft.de/leibniz-edition.html&quot; rel=&quot;noreferrer&quot;&gt;this project&lt;/a&gt; has been going on for exactly 100 years.  I don't read Latin but some of my coauthors do, and we recently used decisive evidence found in unpublished Latin texts by Leibniz; see &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#23h&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt; and also &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#23i&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;A current controversy related to the Leibnizian calculus was summarized &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html#23a&quot; rel=&quot;noreferrer&quot;&gt;in this article in &lt;em&gt;The Mathematical Intelligencer&lt;/em&gt;&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To respond to one of the &lt;a href=&quot;https://hsm.stackexchange.com/questions/15989/how-do-we-explain-the-lack-of-activity-in-the-study-of-latin-mathematics/15990?noredirect=1#comment37316_15990&quot;&gt;&lt;em&gt;comments&lt;/em&gt;&lt;/a&gt;: at least one of the sides in the debate is very well funded.  Thus, David Rabouin received a &lt;a href=&quot;https://u-paris.fr/en/five-erc-advanced-grants-at-universite-de-paris/&quot; rel=&quot;noreferrer&quot;&gt;3 million euro grant&lt;/a&gt; to pursue his study and interpretation of Leibniz's Latin texts.  He has also taken every opportunity to try to &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/depictions.html&quot; rel=&quot;noreferrer&quot;&gt;sink my work&lt;/a&gt;; luckily, &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/leibniz.html&quot; rel=&quot;noreferrer&quot;&gt;without much success&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2023-12-05T11:20:25.720" LastActivityDate="2023-12-05T11:20:25.720" CommentCount="11" ContentLicense="CC BY-SA 4.0" />
  <row Id="15991" PostTypeId="2" ParentId="15196" CreationDate="2023-12-05T08:22:51.800" Score="1" Body="&lt;p&gt;&lt;a href=&quot;https://arxiv.org/abs/1609.02834&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://arxiv.org/abs/1609.02834&lt;/a&gt;&#xA;&amp;quot;The Source of Solar Energy, ca. 1840-1910: From Meteoric Hypothesis to Radioactive Speculations&#xA;Helge Kragh&#xA;Why does the Sun shine? Today we know the answer to the question and we also know that earlier answers were quite wrong. The problem of the source of solar energy became an important part of physics and astronomy only with the emergence of the law of energy conservation in the 1840s. The first theory of solar heat based on the new law, due to J. R. Mayer, assumed the heat to be the result of meteors or asteroids falling into the Sun. A different and more successful version of gravitation-to-heat energy conversion was proposed by H. Helmholtz in 1854 and further developed by W. Thomson. For more than forty years the once so celebrated Helmholtz-Thomson contraction theory was accepted as the standard theory of solar heat despite its prediction of an age of the Sun of only 20 million years. In between the gradual demise of this theory and the radically different one based on nuclear processes there was a period in which radioactivity was considered a possible alternative to gravitational contraction. The essay discusses various pre-nuclear ideas of solar energy production, including the broader relevance of the question as it was conceived in the Victorian era.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-12-05T08:22:51.800" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15992" PostTypeId="2" ParentId="15989" CreationDate="2023-12-05T14:34:04.493" Score="2" Body="&lt;p&gt;Mathematics written in Latin during the last 1000 years is usually not designated as &amp;quot;Latin mathematics&amp;quot;. Contrary to what your professor says, most of mathematical papers of Euler have been translated into English:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;http://eulerarchive.maa.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://eulerarchive.maa.org/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Papers of Gauss, Jacobi and other great mathematicians who wrote in Latin are not translated but they were intensively studied, and most mathematicians who study them have no difficulty with reading mathematics in Latin.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-12-05T14:34:04.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15993" PostTypeId="1" AcceptedAnswerId="15996" CreationDate="2023-12-05T20:00:41.683" Score="5" ViewCount="71" Body="&lt;p&gt;Birkhoff and Rota, in their book &lt;em&gt;Ordinary Differential Equations&lt;/em&gt; (4e), claim on p.55 that:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The concept of the adjoint of a linear operator, which originated historically in the search for integrating factors, is of major importance because of the role which it plays in the theory of orthogonal and biorthogonal expansions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I am interested in substantiations of the historical claim of Birkhoff and Rota; i.e. the claim that the roots of the notion of an adjoint is in the pursuit of finding integrating factors. It is admissible to interpret the word &amp;quot;adjoint&amp;quot; with arbitrarily specific scope. Different words signifying similar ideas are also admissible. (Consequently my question is not about terminology; compare &lt;a href=&quot;https://hsm.stackexchange.com/q/5908/1887&quot;&gt;What is the first usage of the term &amp;quot;Adjoint&amp;quot; and why was this word chosen?&lt;/a&gt;, &lt;a href=&quot;https://hsm.stackexchange.com/q/2605/1887&quot;&gt;The terminologies &amp;quot;Adjoint&amp;quot; and &amp;quot;Adjugate&amp;quot;&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For context, this is how Birkhoff and Rota define the adjoint: for &lt;span class=&quot;math-container&quot;&gt;$L= p_2 D^2+p_1 D+p_0$&lt;/span&gt; a linear differential operator,&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is &lt;strong&gt;exact&lt;/strong&gt; if for some &lt;span class=&quot;math-container&quot;&gt;$q_1,q_0$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$L = D(q_1 D + q_0)$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; is exact iff &lt;span class=&quot;math-container&quot;&gt;$p_2''-p_1'+p_0=0$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is an &lt;strong&gt;integrating factor&lt;/strong&gt; for &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt; if &lt;span class=&quot;math-container&quot;&gt;$vL$&lt;/span&gt; is exact,&lt;/li&gt;&#xA;&lt;li&gt;Thus &lt;span class=&quot;math-container&quot;&gt;$v$&lt;/span&gt; is an integrating factor iff &lt;span class=&quot;math-container&quot;&gt;$M[v]=(vp_2)''-(vp_1)'+(vp_0) = 0$&lt;/span&gt;,&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$M = p_2D^2 + (2p_2'-p_1)D + (p_2''-p_1'+p_0)$&lt;/span&gt; is the &lt;strong&gt;adjoint&lt;/strong&gt; of &lt;span class=&quot;math-container&quot;&gt;$L$&lt;/span&gt;.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(Thus for some appropriately chosen functions, &lt;span class=&quot;math-container&quot;&gt;$\langle v,L[u]\rangle = \langle M[v], u\rangle$&lt;/span&gt;, the bilinear pairing being through an integral.)&lt;/p&gt;&#xA;" OwnerUserId="1887" LastEditorUserId="55" LastEditDate="2023-12-06T01:39:22.800" LastActivityDate="2023-12-06T01:39:22.800" Title="Are adjoint operators historically related to integrating factors?" Tags="|mathematics|differential-equations|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15995" PostTypeId="2" ParentId="2279" CreationDate="2023-12-05T23:49:49.160" Score="3" Body="&lt;p&gt;i was a scientific tyypist in the 70s and we used an IBM selectric II that had a bracket into which &amp;quot;typits&amp;quot; were inserted to produce scientific symbols, it also had a lever to make spaces 1/2 size and you rolled the bar up and down for super and subscripts - nothing in the sample equation would have been problematic. There were times when a symbol had to be hand drawn but that was rare.  Many papers had to be camera ready for publisher. Took a while to become proficient but eventually you could count through an equation (left to right and up and down) to insert the symbols in the correct place without changing them and the ball in and out constantly.&lt;/p&gt;&#xA;" OwnerUserId="19427" LastActivityDate="2023-12-05T23:49:49.160" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="15996" PostTypeId="2" ParentId="15993" CreationDate="2023-12-06T01:38:26.947" Score="5" Body="&lt;p&gt;Yes, Birkhoff and Rota make an unnecessarily cryptic allusion to Lagrange's memoir &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k2155691/f521&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Solution de différents problèmes de calcul intégral&lt;/em&gt; (1766)&lt;/a&gt;, where he multiplies a linear equation   by a factor and then integrates by parts. This leads to what will later be called &amp;quot;adjoint equation&amp;quot; for the (integrating) factor, see &lt;a href=&quot;https://www.jstor.org/stable/41133697&quot; rel=&quot;noreferrer&quot;&gt;Demidov, On the History of the Theory of Linear Differential Equations, 2.1&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The name &amp;quot;adjoint&amp;quot; was attached by Fuchs a century later in &lt;a href=&quot;https://eudml.org/doc/147986&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten&lt;/em&gt; (1866)&lt;/a&gt;. The transition from adjoint equation to adjoint operator/transformation (on function spaces with inner product given by an integral) was made by Riesz in &lt;a href=&quot;https://link.springer.com/article/10.1007/BF02422940&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Über lineare Funktionalgleichungen&lt;/em&gt; (1916)&lt;/a&gt;. He used it to study operator inverses, see &lt;a href=&quot;https://www.jstor.org/stable/41133253&quot; rel=&quot;noreferrer&quot;&gt;Bernkopf, The Development of Function Spaces, pp.58-62&lt;/a&gt;. Note that a different use of &amp;quot;adjoint&amp;quot; was around at the time as introduced by Bocher in 1907, for what is now called &amp;quot;adjugate&amp;quot;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-12-06T01:38:26.947" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="15997" PostTypeId="2" ParentId="15988" CreationDate="2023-12-06T02:44:31.437" Score="5" Body="&lt;p&gt;Signed areas came up in connection with winding numbers even before Gauss, see &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-030-13609-3_6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sunada, From Euclid to Riemann and Beyond&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Gauss briefly noticed in the letter to Bessel in 1811 , the winding number reveals the nature of the complex logarithm. Winding numbers appeared in the study of signed areas by Albrecht Ludwig Friedrich Meister (1724-1788).&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Meister's &lt;a href=&quot;https://www.google.com/books/edition/Generalia_de_genesi_figurarum_planarum_e/fOE_AAAAcAAJ?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Generalia de genesifigurarum planarum, et inde pendentibus earum affectionibus&lt;/em&gt;&lt;/a&gt; (&lt;em&gt;Novi Commentarii Soc. Reg. Scient. Gott.&lt;/em&gt;, 1 (1769/1770), 144-180) had very few equations, but derived what is now called &amp;quot;shoelace formula&amp;quot;. Gauss used the idea of winding numbers in his proofs of FTA (1799), see &lt;a href=&quot;https://www.semanticscholar.org/paper/The-Fundamental-Theorem-of-Algebra-Made-Effective%3A-Eisermann/a63d3d071f6527a3daa1dbf47f105c20b80847d1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Eisermann, Fundamental Theorem of Algebra Made Effective&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Cauchy came to signed areas, and volumes, around 1812-13 from a different perspective, not in connection with winding numbers and contour integrals. It was the usual byproduct of algebraic treatment of geometric objects - algebra naturally uniformizes geometrically distinct cases in theorems. Chasles inserted signed segments into the cross-ratio for similar reasons a bit later (1830). Here is &lt;a href=&quot;https://www.google.com/books/edition/History_of_Analytic_Geometry/2T4i5fXZbOYC?hl=en&quot; rel=&quot;nofollow noreferrer&quot;&gt;Boyer, History of Analytic Geometry&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1812-1813, for example, J.P.M. Binet (1786-1856)&#xA;wrote a long and tedious paper in which he made use of analogous notations (which he called &amp;quot;resultants&amp;quot;) in analytic theorems on volumes, areas, and lengths of rectangular configurations in three dimensions. His work includes the equivalent of the multiplication of determinants; but in this early use of determinants the familiar square array was missing. This element, together with the double-subscript notation, was supplied by Cauchy at the very time of Binet's paper. Cauchy applied his &amp;quot;resultants&amp;quot; in attributing signed values to angles, areas,&#xA;and volumes in coordinate geometry.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-12-13T21:30:30.583" LastActivityDate="2023-12-13T21:30:30.583" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15998" PostTypeId="1" AcceptedAnswerId="15999" CreationDate="2023-12-06T06:20:59.857" Score="6" ViewCount="1348" Body="&lt;p&gt;When reading the section related to Gram-Schmidt process in the book &lt;em&gt;Linear Algebra and Its Applications&lt;/em&gt; by Gilbert Strang, I found a foot note that says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If Gram thought of it first, what was left for Schmidt.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Other books taught me that the two independently developed the method and many Internet sources also say the same thing (none go into the details of the contribution of each to the procedure). Thus, I suspect that prof. Strang seems to imply that Gram made the first contribution to the procedure but it was still incomplete and Schmidt later on handled the missing part (probably the part to normalize the basis vectors).&lt;/p&gt;&#xA;&lt;p&gt;So what exactly did prof. Strang mean by adding the footnote?&lt;/p&gt;&#xA;" OwnerUserId="19429" LastEditorUserId="55" LastEditDate="2023-12-06T08:27:29.507" LastActivityDate="2023-12-06T19:51:53.900" Title="What was the role of Schmidt in derivation of the Gram-Schmidt process?" Tags="|mathematics|linear-algebra|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="15999" PostTypeId="2" ParentId="15998" CreationDate="2023-12-06T08:26:25.997" Score="14" Body="&lt;p&gt;A detailed history can be found in &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.1839&quot; rel=&quot;noreferrer&quot;&gt;Gram-Schmidt orthogonalization: 100 years and more by Leon, Björck and Gander&lt;/a&gt;, see also their &lt;a href=&quot;https://www.cis.upenn.edu/%7Ecis6100/Gram-Schmidt-Bjorck.pdf&quot; rel=&quot;noreferrer&quot;&gt;slides&lt;/a&gt; for a brief version. In short, Schmidt's 1907 presentation was more explicit and indicated a key property of the process that Gram left out in 1879/1883. It led to the process becoming popular. But the idea predates both of them and can be traced to the third edition of Laplace's &lt;em&gt;Théorie Analytique des Probabilités&lt;/em&gt; (1820), where orthogonalization was applied in the course of solving least squares systems to estimate the masses of Jupiter and Saturn.&lt;/p&gt;&#xA;&lt;p&gt;Here are the highlights:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1907, Erhard Schmidt published a paper on integral equations. In that paper, he made use of an orthonormalization technique that has since become known as the CGS. He remarked in the paper that in essence the formulas were presented by J. P. Gram. The algorithm has become well known and is included as a standard topic in elementary linear algebra textbooks. The earliest linkage of the two names Gram and Schmidt to describe the orthonormalization process appears to be in a 1935 paper by Wong. An apparently slight change in the CGS process gives the&#xA;modified Gram–Schmidt process (MGS)... This tool, applied to a least squares problem, can be found in a book by Laplace on the analytic&#xA;theory of probabilities.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;J. P. Gram’s original 1879 thesis on integral equations was written in Danish. The results&#xA;became more accessible to the general mathematics community when a German version was&#xA;published in 1883. In this paper, Gram was concerned with series expansions of real functions&#xA;using least squares. Gram was influenced by the work of Chebyshev, and his original orthogonalization procedure was applied to orthogonal polynomials... In the paper, Schmidt applied an orthogonalization process to a&#xA;sequence of functions &lt;span class=&quot;math-container&quot;&gt;$\phi_1\dots\phi_n$&lt;/span&gt; with respect to the inner product &lt;span class=&quot;math-container&quot;&gt;$\langle f,g\rangle=\int_a^bf(x)g(x)\,dx$&lt;/span&gt;.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The Schmidt paper popularized the orthogonalization process, and as a result of this paper the names Gram and Schmidt have become forever linked. The Schmidt paper did not use inner-product or norm notation... The Gram–Schmidt process has two characteristic properties that are clearly evident in the&#xA;algorithm presented in Schmidt’s 1907 paper. The first property is that the &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;-th vector &lt;span class=&quot;math-container&quot;&gt;$q_k$&lt;/span&gt;&#xA;(or function) is a linear combination of &lt;span class=&quot;math-container&quot;&gt;$a_k$&lt;/span&gt; and its predecessors. Second, the combination is&#xA;expressed in terms of the vector &lt;span class=&quot;math-container&quot;&gt;$a_k$&lt;/span&gt; and the previous GS vectors &lt;span class=&quot;math-container&quot;&gt;$q_1,\dots,q_k$&lt;/span&gt;. On the other hand, the Gram paper honors the first property but not the second.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In 1816, Laplace published his major treatise Théorie Analytique des Probabilités (Analytic&#xA;Theory of Probability). The third edition, which appeared in 1820, contains three supplements. In&#xA;the first supplement, the goal of Laplace is to compute the mass of Jupiter (and Saturn) from a system of normal equations provided by the French astronomer Bouvard and from this same system to&#xA;compute the distribution of error in the solution assuming a normal distribution of the noise on the&#xA;observations... The method which Laplace introduces consists in successively projecting the system of equations orthogonally to a column of the matrix &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. These actions eliminate the associated couple observation/variable from the updated system. Ultimately, Laplace eliminates all the variables but the one of interest in the linear least squares problem, which eliminates all the columns but one in &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;. Laplace is indeed introducing the main technique behind the Gram–Schmidt algorithm (successive orthogonal&#xA;projections.)&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-12-06T19:51:53.900" LastActivityDate="2023-12-06T19:51:53.900" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16000" PostTypeId="2" ParentId="3602" CreationDate="2023-12-06T12:53:59.867" Score="1" Body="&lt;p&gt;There is now an English version available : &lt;a href=&quot;https://www.amazon.fr/One-Hundred-Authors-Against-Einstein/dp/B09PHH7KC8&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.amazon.fr/One-Hundred-Authors-Against-Einstein/dp/B09PHH7KC8&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2023-12-06T12:53:59.867" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16001" PostTypeId="1" AcceptedAnswerId="16002" CreationDate="2023-12-06T13:27:15.580" Score="3" ViewCount="251" Body="&lt;p&gt;In one of his essays, &lt;a href=&quot;https://en.wikipedia.org/wiki/Y_Combinator&quot; rel=&quot;nofollow noreferrer&quot;&gt;Y Combinator&lt;/a&gt; (YC) co-founder &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Graham_(programmer)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Graham&lt;/a&gt; (PG) wrote&lt;span class=&quot;math-container&quot;&gt;$^\color{magenta}{\star}$&lt;/span&gt; the following.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;My father is a mathematician. For most of my childhood he worked for Westinghouse, modelling nuclear reactors [...] He grew up in the small Welsh seacoast town of Pwllheli.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Who would that mathematician be?&lt;/p&gt;&#xA;&lt;p&gt;I suspect it's &lt;a href=&quot;https://en.wikipedia.org/wiki/Ronald_Graham&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ronald Graham&lt;/a&gt;, but he was born in the US, not the UK. Also, it seems he didn't work on nuclear reactors.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sub&gt;&lt;span class=&quot;math-container&quot;&gt;$\color{magenta}{\star}$&lt;/span&gt; Paul Graham, &lt;a href=&quot;http://www.paulgraham.com/work.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;What doesn't seem like work?&lt;/a&gt;, January 2015.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="65" LastEditorUserId="4251" LastEditDate="2023-12-09T18:05:51.883" LastActivityDate="2023-12-09T18:05:51.883" Title="Who is Paul Graham's father?" Tags="|mathematicians|" AnswerCount="1" CommentCount="10" ContentLicense="CC BY-SA 4.0" />
  <row Id="16002" PostTypeId="2" ParentId="16001" CreationDate="2023-12-07T00:40:50.770" Score="5" Body="&lt;p&gt;It was John Graham (1933-2017). As Paul says, &amp;quot;&lt;em&gt;for most of my childhood he worked for Westinghouse, modelling nuclear reactors&lt;/em&gt;&amp;quot;, and he remained in the nuclear industry for the rest of his life. His B.S. in mathematics was from the University of Wales, and he did graduate work at the University of Illinois and the University of London. Here is his short bio from the &lt;a href=&quot;https://cdn.ans.org/about/presidents/docs/john-graham-bio.pdf&quot; rel=&quot;noreferrer&quot;&gt;obituary&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;He was been engaged in nuclear safety for 60 years, including ongoing work with the Pacific Nuclear Council on promotion of peaceful uses of nuclear technologies, since 1997. Graham began a career with Westinghouse in 1969 as Manager of Nuclear Safety at Westinghouse Advanced Reactors Division, Madison, Wisc., with responsibility for safety of the Clinch River Breeder Reactor in Tennessee, among other projects. In 1985, he became the Licensing Manager for the Basalt Waster Isolation Project.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;After he left Westinghouse, he was named the Director of Nuclear Safety for Atomic Energy of Canada Limited (AECL) until 1992, when he became vice-president of British Nuclear Fuels (BNFL). He also worked for the U.K. Atomic Energy Agency as a senior scientist and for the Atomic Energy of Canada Research for British Nuclear Fuels as a Director of Safety. He was a recognized Fellow at Westinghouse... He wrote &lt;a href=&quot;https://www.google.com/books/edition/Fast_Reactor_Safety/y0jXPlC7X18C?hl=en&amp;amp;gbpv=0&quot; rel=&quot;noreferrer&quot;&gt;Fast Reactor Safety&lt;/a&gt; in 1971 and authored 25 other books.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Nuclear News ran his &lt;a href=&quot;https://www.ans.org/file/532/john-graham-nn-article.pdf&quot; rel=&quot;noreferrer&quot;&gt;extended profile&lt;/a&gt; when he was the American Nuclear Society president in 1995-96. He was born in Wales, went to college in the US, then returned to England to work in Harwell labs until 1968, when he got a job at Westinghouse (&amp;quot;&lt;em&gt;Claire had good reason to be upset, Graham concedes. They had two small children at that time - Paul, aged four, and Jennifer, who was less than a year old&lt;/em&gt;&amp;quot;). They settled at the US and eventually became American citizens. There is a bit on his engagement with mathematics:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Mathematics was something I've always liked- I love numbers; they're fascinating. I liked algebra. and I love calculus. I truly believe the whole of life is based on calculus and any aberrations in life are simply aberrations in the calculus... I applied to various places and was accepted at the universities&#xA;of Ohio and Illinois, and went to the University&#xA;of Illinois in 1954-55. I studied quantum&#xA;mechanics and mathematics. I didn't stay to&#xA;get the master's degree because I didn't think&#xA;the master's was at all significant, and besides,&#xA;I wanted to take some time to travel around the&#xA;States.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;I found out&#xA;that everybody in that graduate department&#xA;was working on similar problems with&#xA;slightly different force fields. So I was doing&#xA;neutron-deuteron impact problems with a given&#xA;force field. Someone next to me was working&#xA;with a different force field, and someone&#xA;else was doing deuteron-deuteron problems&#xA;with my force field. We were all contributing&#xA;to the professor's thesis, but we were each only&#xA;a very small brick in his edifice. That's not&#xA;very satisfying. So I didn't finish that degree...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;However, it does not seem like he did much mathematical work after the early years, aside from nuclear modeling (MathSciNet lists no publications by him). His profile also has a childhood picture of Paul and Jennifer, his kids, and a picture of him, his wife Claire and Paul, age 30, with a short blurb:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Today, John and Claire live in Golden,&#xA;Colo. Claire works in the controller's office&#xA;for the American Animal Hospital Association&#xA;(you are not allowed to call it Aaha!).&#xA;Their son Paul, now age 30, has a doctorate in&#xA;computer science, but is currently working as&#xA;an artist and as CEO of ARTIX, a World Wide&#xA;Web company.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2023-12-07T00:52:36.220" LastActivityDate="2023-12-07T00:52:36.220" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16003" PostTypeId="1" CreationDate="2023-12-07T06:02:21.453" Score="2" ViewCount="103" Body="&lt;p&gt;Here: &lt;a href=&quot;https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiments&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiments&lt;/a&gt;&#xA;Some calculations were done to show how JJ Thomson's model predicts alpha particles will be mostly undeflected by the gold foil. There they used the charge of an alpha particle and charge of a gold atom in their calculations, how did they know that?&lt;/p&gt;&#xA;&lt;p&gt;Did they assume Van den Broek's hypothesis that nuclear charge = atomic number or through some other way? Also they could also have found q/m ratio of alpha particle using a setup similiar to JJ Thomson's cathode ray experiment where he found q/m ratio of electron but I assume that they didn't know the masses of the atoms?&lt;/p&gt;&#xA;&lt;p&gt;Update:-&#xA;Wikipedia had this, I dont know how I missed it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The calculations above use modern measurements, but Rutherford had sufficiently accurate estimates. He assumed that the radius of atoms in general to be on the order of &lt;span class=&quot;math-container&quot;&gt;$10^{−10}$&lt;/span&gt; m (perhaps after reading Jean Perrin's studies on Brownian motion[7]) and the positive charge of a gold atom to be about 100 times that of hydrogen (100 e).[8] He knew that gold has an atomic weight of 197. From an experiment in 1906, Rutherford measured alpha particles to have a charge of 2 e and an atomic weight of 4, and alpha particles emitted by radon to have velocity of 1.70×107 m/s.[9] Jean Perrin in 1909 measured the mass of hydrogen to be 1.43×10−27 kg,[10] and if alpha particles are four times as heavy as that, they would have an absolute mass of 5.72×10−27 kg.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'll look into the papers further&lt;/p&gt;&#xA;" OwnerUserId="19362" LastEditorUserId="19362" LastEditDate="2023-12-27T09:22:04.890" LastActivityDate="2023-12-27T09:22:04.890" Title="In Geiger-Marsden experiments how was charge of alpha particle and charge of gold atom found?" Tags="|physics|discoveries|experimental-physics|atomic-theory|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16004" PostTypeId="1" AcceptedAnswerId="16005" CreationDate="2023-12-07T07:19:32.373" Score="4" ViewCount="726" Body="&lt;p&gt;As I studied Cardano's formula for the cubic, Tschirnhaus transformation came up as a very important step. The more I studied it the more attractive and interesting it seemed. So I am curious about the motivation and background of the person behind it, what he had in mind when inventing it. I saw the post &lt;a href=&quot;https://math.stackexchange.com/questions/49955/motivation-behind-tschirnhaus-transformation-and-cardanos-formula&quot;&gt;Motivation Behind Tschirnhaus Transformation and Cardano's Formula&lt;/a&gt; on Math SE, but I am interested in what motivated the idea of the Tschirnhaus transformation itself rather than its uses.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="55" LastEditDate="2023-12-07T10:39:03.737" LastActivityDate="2023-12-07T10:45:52.600" Title="What motivated the idea of the Tschirnhaus transformation of polynomial equations?" Tags="|discoveries|algebra|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16005" PostTypeId="2" ParentId="16004" CreationDate="2023-12-07T10:22:30.583" Score="8" Body="&lt;p&gt;Attempting to find substitutions that would generalize Cardano's and Ferrari's solutions to the cubic and quartic was a popular enterprise in the 17-18th centuries. Gregory, Leibniz, Euler, Bezout and Lagrange participated along with many lesser known mathematicians. So if the question is how Tschirnhaus came up with his transformation a brief answer is by algebraic experimentation inspired by Cardano and Descartes (the two he mentioned by name).&lt;/p&gt;&#xA;&lt;p&gt;For general information on the man, his time, life and mathematical work, a good source is &lt;a href=&quot;https://core.ac.uk/download/pdf/82668914.pdf&quot; rel=&quot;noreferrer&quot;&gt;Kracht-Kreyszig, E. W. von Tschirnhaus: His role in early calculus and his work and impact on Algebra&lt;/a&gt;, section 4 is dedicated to the Tschirnhaus's &lt;em&gt;Acta Eruditorum&lt;/em&gt; paper that introduced his transformation. Descartes left a particularly deep impression on him, he was introduced to Geometry in his youth by the brother of its editor, van Schooten, and later had Leibniz arrange for him to study Descartes's unpublished papers, see &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Biographies/Tschirnhaus/&quot; rel=&quot;noreferrer&quot;&gt;McTutor's biography&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Tschirnhaus's paper, A method for removing all intermediate terms from a given equation (1683),  was essentially completed by 1677, and shared with Leibniz, but Tschirnhaus kept working to improve his results, without success, and timed the publication to his election to the French &lt;em&gt;Académie des sciences&lt;/em&gt;. The paper has been &lt;a href=&quot;https://dl.acm.org/doi/10.1145/844076.844078&quot; rel=&quot;noreferrer&quot;&gt;translated into English&lt;/a&gt; by Green. On his motivation, Tschirnhaus says the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We have learned from DesCartes’ Geometry by what method the second term might reliably be removed from&#xA;a given equation; but on the question of removing multiple intermediate terms I have seen nothing hitherto in the&#xA;analytic arts. On the contrary, I have encountered not a few who believed that the thing could not be done by any&#xA;art. For this reason I have decided to set down here some things concerning this business, enough at least for those&#xA;who have some grounding in the analytic art, since the others could scarcely be content with so brief an exposition:&#xA;reserving the remainder (which they might wish to see here) for some other time... DesCartes shows [this] in removing the second term from a given equation, when he says that for removing the second term with the aid of the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;$x = y+a$&lt;/span&gt;, in a quadratic equation the quantity a must be equal to &lt;span class=&quot;math-container&quot;&gt;$\frac{p}2$&lt;/span&gt; in a cubic equation &lt;span class=&quot;math-container&quot;&gt;$\frac{p}3$&lt;/span&gt; and so forth.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Tschirnhaus proceeds to explain how to choose &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; to remove two terms, and later comments that his solution to the cubic is &amp;quot;&lt;em&gt;different from the Cardano expression most particularly in&#xA;the fact that it includes a single cubic radical sign, where the Cardano expression contains two&lt;/em&gt;&amp;quot;. After that, he gets carried away and claims that any (polynomial) equation can be reduced by his method, promising to show it elsewhere in a manner reminiscent of Fermat's famous remark on the margins:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;But finally it should be noted that by means of the equations displayed in the second paragraph not only may two,&#xA;three, four, &amp;amp;c terms from a given equation be removed, but that from various other equations different from these&#xA;I might also do the same thing. Nay, I can easily show that all possible equations may be handled by this business.&#xA;But that I might be able to show by this method (according to paragraph five) all possible analytic expressions of&#xA;roots of every possible equation, I shall explain more fully in its own place.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As a matter of fact, Tschirnhaus's method does not always succeed even in removing the three middle terms from the general quintic, let alone that &amp;quot;all possible equations may be handled by this business&amp;quot;, as Leibniz pointed out years before the publication:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Concerning your... method for finding the roots of an equation... I do not believe that it will be successful for equations of higher degree,&#xA;except in special cases. I believe that I have a proof for this.&lt;/em&gt;&amp;quot; [quoted from Kracht-Kreyszig]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bring (1786) and Jerrard (1834) did find a way to remove the three terms (likely, independently), see &lt;a href=&quot;https://dl.acm.org/doi/10.1145/990353.990371&quot; rel=&quot;noreferrer&quot;&gt;Adamchik-Jeffrey, Polynomial transformations of Tschirnhaus, Bring and Jerrard&lt;/a&gt;. Alas, that still did not solve the quintic. Although the futility of the original quest to solve it in radicals was shown by Abel and Galois, the Tschirnhaus transformation continued to inspire. Hermite applied it to solve the quintic in terms of elliptic modular functions in 1858 (his other inspiration was Viete's trigonometric solution to the irreducible case of the cubic). For later developments, see &lt;a href=&quot;https://arxiv.org/abs/2001.06515&quot; rel=&quot;noreferrer&quot;&gt;Wolfson, Tschirnhaus transformations after Hilbert&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2023-12-07T10:22:30.583" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16006" PostTypeId="2" ParentId="16004" CreationDate="2023-12-07T10:45:52.600" Score="5" Body="&lt;p&gt;D. T., &amp;quot;Methodus auferendi omnes terminos intermedios ex data æquatione.&amp;quot; &lt;em&gt;Acta Eruditorum&lt;/em&gt; (1683), &lt;a href=&quot;https://www.google.com/books/edition/Acta_Eruditorum/G9vXoXsnUt0C?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA204&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;pp. 204-207&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The 'D' &lt;em&gt;may&lt;/em&gt; stand for doctor, because the author's full name was Ehrenfried Walther von Tschirnhaus, so there is no 'D' in his name. He states his motivation in the initial paragraph of this publication: To simplify a polynomial equation by eliminating &lt;em&gt;more than one&lt;/em&gt; of the terms.&lt;/p&gt;&#xA;&lt;p&gt;He writes that it was known that one could always remove the second term by the method of Descartes, but that he had so far not encountered any method in the analytical art to eliminate more than one term. He therefore decided to reveal one here, albeit only briefly for those skilled in the art, with additional details reserved for the future.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ex Geometria Dn Des Cartes notum est, qua ratione semper secundus terminus ex data æquatione possit auferri; quod plures terminos intermedios auferendos, hactenus nihil inventum vidi in Arte Analytica, imo non paucos offendi, qui crediderunt, id nulla arte perfici posse. Quapropter hic quædam circa hoc negotium aperire constitui, verum saltem pro iis, qui Artis Analyticae apprime gnari, cum aliis tam brevi explicatione vix satisfieri possit: reliqua, quæ hic desiderari possent, alii tempori reservans.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2023-12-07T10:45:52.600" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16007" PostTypeId="1" AcceptedAnswerId="16008" CreationDate="2023-12-08T19:14:35.317" Score="0" ViewCount="86" Body="&lt;p&gt;In Steven Strogatz's &amp;quot;Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering&amp;quot;, Storgatz mentions a &amp;quot;violent controversy&amp;quot; between Zahler, Sussman, and Kolata that happened in 1977. However, I am not able to find anything like that. What was the controversy he is talking about here?&lt;/p&gt;&#xA;" OwnerUserId="16986" LastActivityDate="2023-12-08T20:15:41.053" Title="Zahler, Sussman, and Kolata Controversy of 1977" Tags="|physics-social-history|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16008" PostTypeId="2" ParentId="16007" CreationDate="2023-12-08T20:15:41.053" Score="2" Body="&lt;p&gt;The following (and the references there) seems relevant: Guckenheimer, J. The catastrophe controversy. The Mathematical Intelligencer 1, 15–20 (1978).&lt;/p&gt;&#xA;&lt;p&gt;Looks like it was about attempts to hype the catastrophe theory and extend it to numerous areas where its usefulness was dubious.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-12-08T20:15:41.053" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16009" PostTypeId="2" ParentId="15962" CreationDate="2023-12-09T14:48:37.437" Score="2" Body="&lt;p&gt;Other examples of articles published by mathematicians at an early age:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;High-schooler Britney Gallivan was born in 1985 and published a paper early 2002 in which she established a &lt;a href=&quot;https://mathworld.wolfram.com/Folding.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;formula&lt;/a&gt; for the &amp;quot;loss function&amp;quot; of folding paper in half. So she was either 16 or 17 at the time of publication. By means of her new folding techniques, she managed to fold a large piece of toilet paper 12 times, even though the commonly held belief at the time was that paper could be folded in half only 8 times.&lt;/li&gt;&#xA;&lt;li&gt;Werner Boy (1879 - 1914) published on what later would become known as &amp;quot;Boy's Surface&amp;quot; in 1901 (age 21/22). A link to the paper can be &lt;a href=&quot;https://surfaces.gotfork.net/boy.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;found here&lt;/a&gt;.&lt;/li&gt;&#xA;&lt;li&gt;At age 16, Dmitriy Zakharov posted &lt;a href=&quot;https://arxiv.org/abs/1705.01171&quot; rel=&quot;nofollow noreferrer&quot;&gt;an article&lt;/a&gt; on the ArXiv entitled &amp;quot;Acute Sets&amp;quot;, in which he proved a new bound for a 1962 problem - &lt;a href=&quot;https://mathoverflow.net/a/360996/93724&quot;&gt;improving&lt;/a&gt; a result of Erdős.&lt;/li&gt;&#xA;&lt;li&gt;There is also Norbert Wiener who published the main findings of his dissertation on mathematical logic when he was either 19 or 20 years (!) of age. The paper is called &lt;em&gt;A simplification in the logic of relations&lt;/em&gt; and was published in 1914.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="4296" LastEditorUserId="4296" LastEditDate="2024-04-05T22:49:30.033" LastActivityDate="2024-04-05T22:49:30.033" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16010" PostTypeId="1" CreationDate="2023-12-09T15:08:38.630" Score="1" ViewCount="97" Body="&lt;p&gt;In Moritz Epple's article &amp;quot;Geometric Aspects in the Development of Knot Theory&amp;quot;, Epple writes the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It has been suggested that one of the earliest tools of combinatorial knot theory was forged by Carl Friedrich Gauss. Some posthumously published fragments of his Nachlass, originally written in the years 1825 and 1844, document that the Gottingen mathematician tried to classify closed plane curves with a finite number of transverse self-intersections, sometimes called &amp;quot;Tractfiguren&amp;quot; (tract figures) by Gauss. To do so, he invented a symbolical coding of such figures. He assigned a number or letter to each crossing and then wrote down the sequence of crossing symbols that resulted from following the curve in a given direction from a given point (see Figure 1). To some extent, this symbol-sequence captured the characteristic features of the tract figure in the sense of Geometria situs, as Gauss preferred to call the as yet unexplored science of topology.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is the description of an unsigned &lt;a href=&quot;https://en.wikipedia.org/wiki/Gauss_notation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss code&lt;/a&gt;. Epple continues and describes some of Gauss's remarks on properties of Gauss codes of tract figures:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss noticed that the sequences arising in this way had to satisfy certain conditions: Each of the symbols representing one of the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; crossings had to appear exactly twice, once in an even and once in an odd place of the sequence. As Gauss noticed in 1844, however, these conditions were sufficient only for &lt;span class=&quot;math-container&quot;&gt;$n\le 4$&lt;/span&gt;. He thus set out to write down a table of the admissible sequences for five crossings, but he did not find a method to solve the general problem of determining exactly which symbol sequences satisfying the above conditions actually represented crossing sequences of &amp;quot;Tractfiguren&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here Epple described Gauss's parity condition. After that he says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What reasons did Gauss have for looking at this matter? Unfortunately, the fragments themselves do not give a clear answer. From the perspective of later knot theory, Gauss's attempt might look like a first step toward knot tables, but we will see that he had other reasons for studying &amp;quot;Tractfiguren&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(I must say that this is not the only source I read about Gauss's classification of tract figures, but I am still confused.)&lt;/p&gt;&#xA;&lt;p&gt;My question is very basic - are the &amp;quot;tract figures&amp;quot; mentioned in these passages supposed to be knot projections? in other words, is it possible that a symbols sequence which satisfies the parity condition cannot be realized as a plane curve? because if the answer to my second question is yes, that means that &amp;quot;restricted&amp;quot; Gauss codes (that satisfy the parity condition) are not equivalent to planar curves, and therefore the problem of Gauss word realization is merely about curves and is relavent only to graph theory and not to knot theory.&lt;/p&gt;&#xA;&lt;p&gt;However, if the answer to my second question is no, than the Gauss word realization problem must be about classifying which Gauss words correspond to projections of 3D knots and which correspond to a projection of the unknot (which means that despite the tract figure may look complicated, it actually can be transformed by a series of valid steps into a single non self-intersecting loop, which is a projection of the unknot). Such an assumption seems to be consistent with the fact Gauss refers many times to these tract figures as &amp;quot;knots&amp;quot;, but I did not find anywhere on web the assertion that Gauss word realization problem is simply an unknotting problem.&lt;/p&gt;&#xA;&lt;p&gt;(I appologize for asking too many questions about very minor points, I know this is annoying, but this is simply my lack of rigorous mathematical knowledge combined with my lack of time that leads me to ask many questions with more or less overlapping themes.)&lt;/p&gt;&#xA;" OwnerUserId="2554" LastActivityDate="2023-12-09T15:08:38.630" Title="Was the &quot;Gauss word realization problem&quot; a kind of unknotting problem?" Tags="|gauss|topology|graph-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16011" PostTypeId="2" ParentId="16003" CreationDate="2023-12-09T16:59:23.197" Score="2" Body="&lt;p&gt;As for the properties of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-particles, there is some information in &lt;a href=&quot;https://www.nobelprize.org/prizes/chemistry/1908/rutherford/lecture/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rutherford's Nobel lecture&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Strutt (Lord Rayleigh) in 1901 and Sir William Crookes in 1902&#xA;suggested that they might possibly prove to be projected particles&#xA;carrying a positive charge. I independently arrived at the same&#xA;conclusion from consideration of a variety of evidence. If this were&#xA;the case, the &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-rays should be deflected by a magnetic field.&#xA;Preliminary work showed that the deflection was very slight if it&#xA;occurred at all. Experiments were continued at intervals over a period&#xA;of two years and it was not until 1902, when a preparation of radium&#xA;of activity 19,000 was available, that I was able to show conclusively&#xA;that the particles were deflected by a magnetic field, though in a&#xA;very minute degree compared with the &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt;-rays. This showed that&#xA;the &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-rays consisted of projected charged particles while the&#xA;direction of deflection indicated that each particle carried a&#xA;positive charge. The &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-particles were shown to be deflected&#xA;also by an electric field and from the magnitude of the deflection, it&#xA;was deduced that the velocity of the swiftest particles was about 2.5&#xA;x &lt;span class=&quot;math-container&quot;&gt;$10^9$&lt;/span&gt; cm per second, or one-twelfth the velocity of light, while&#xA;the value of &lt;span class=&quot;math-container&quot;&gt;$e/m$&lt;/span&gt; – the ratio of the charge carried by the particle&#xA;to its mass – was found to be 5,000 electromagnetic units.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for the charge of the gold nucleus, it was deduced from the results of scattering of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt;-particles. Several arguments were offered (&lt;a href=&quot;https://web.mit.edu/8.13/8.13c/references-fall/rutherford/rutherford-scattering-of-alpha-and-beta-particles.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rutherford, E.(1911) 'LXXIX. The scattering of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$\beta$&lt;/span&gt; particles by matter and the structure of the&#xA;atom', Philosophical Magazine Series 6, 21: 125, 669 — 688&lt;/a&gt;), for example,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(a) In the paper of Gelger and Marsden (&lt;em&gt;loc. cit.&lt;/em&gt;) on the diffuse&#xA;refiexion of &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; particles falling on various substances it was&#xA;shown that about ~ 1/8000 of the &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; particles from radium C&#xA;falling on a thick plate of platinum are scattered back in the&#xA;direction of the incidence. This fraction is deduced on the assumption&#xA;that the &lt;span class=&quot;math-container&quot;&gt;$\alpha$&lt;/span&gt; particles are uniformly scattered in all directions,&#xA;the observations being made for a deflexion of about 90&lt;span class=&quot;math-container&quot;&gt;$^{\circ}$&lt;/span&gt;. The&#xA;form of experiment is not very suited for accurate calculation, but&#xA;from the data available it can be shown that the scattering observed&#xA;is about that to be expected on the theory if the atom of platinum has&#xA;a central charge of about 100 &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While platinum is close to gold in terms of atomic mass and nucleus charge, there are also some estimates for gold in the same work.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastEditorUserId="9808" LastEditDate="2023-12-10T12:31:55.713" LastActivityDate="2023-12-10T12:31:55.713" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16012" PostTypeId="1" CreationDate="2023-12-10T12:52:00.863" Score="2" ViewCount="234" Body="&lt;p&gt;Well, I am not sure if this is the right place to ask but  here it goes (if this forum  is not the right place to ask, I apologize and certainly moderators will delete my question): i am very curious about the authorship and history  of this calendar design(the clever compact one page calendar: &lt;a href=&quot;https://medium.com/starts-with-a-bang/this-one-page-calendar-will-change-how-you-view-the-year-b8ecad85eebd&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://medium.com/starts-with-a-bang/this-one-page-calendar-will-change-how-you-view-the-year-b8ecad85eebd&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;as a gift here goes the 2024 one page calendar:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rjBBf.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rjBBf.png&quot; alt=&quot;e2024 one page calendar&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12383" LastEditDate="2023-12-10T13:08:05.057" LastActivityDate="2023-12-13T05:51:56.813" Title="who invented the one page calendar?" Tags="|calendar|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16013" PostTypeId="1" CreationDate="2023-12-11T00:23:30.630" Score="4" ViewCount="155" Body="&lt;p&gt;The Mandelbrot set was discovered by Benoit Mandelbrot in 1980.&lt;/p&gt;&#xA;&lt;p&gt;However, Gaston Julia lived from 1893 to 1978.&lt;/p&gt;&#xA;&lt;p&gt;Does this mean Julia sets, in the popular form we know them today, were discovered before the famous Mandelbrot fractal?&lt;/p&gt;&#xA;" OwnerUserId="12633" LastActivityDate="2023-12-13T13:54:36.840" Title="Were Julia sets discovered before the Mandelbrot set?" Tags="|mandelbrot|julia|chaos|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16014" PostTypeId="2" ParentId="16013" CreationDate="2023-12-11T14:40:23.127" Score="10" Body="&lt;p&gt;Yes, about 60 years before.&lt;/p&gt;&#xA;&lt;p&gt;What is called &amp;quot;Julia set&amp;quot; was actually discovered by P. Fatou, his first publication on the subject is from 1917:&lt;/p&gt;&#xA;&lt;p&gt;Sur les substitutions rationnelles,&#xA;Comptes Rendus, 164, 806-808 (1917).&lt;/p&gt;&#xA;&lt;p&gt;Julia's paper on the subject was published in 1918,&#xA;but he was able to prove that he discovered it independently of Fatou.&lt;/p&gt;&#xA;&lt;p&gt;It was called &amp;quot;irregular set&amp;quot; until, after the death of Fatou,&#xA;Julia insisted that it must be called Julia set.&lt;/p&gt;&#xA;&lt;p&gt;(Documented source: the book of Michele Audin, Fatou, Julia, Montel: The Great Prize of Mathematical Sciences of 1918, and Beyond, Springer 2011. )&lt;/p&gt;&#xA;&lt;p&gt;Mandelbrot set was discovered by Brooks and Matelski in 1978 (but their paper was published in 1981):&lt;/p&gt;&#xA;&lt;p&gt;The dynamics of 2-generator subgroups of PSL(2,C). Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 65–71, Ann. of Math. Stud., No. 97, Princeton Univ. Press, Princeton, NJ, 1981.&lt;/p&gt;&#xA;&lt;p&gt;The contribution of Mandelbrot consists in production of high-tech computer pictures of it.&#xA;His paper was published in 1980, it contains a good quality picture, but this first picture is not quite correct.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;The things are never named after their true discoverers&amp;quot; (Arnold).&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-12-13T13:54:36.840" LastActivityDate="2023-12-13T13:54:36.840" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16016" PostTypeId="2" ParentId="15466" CreationDate="2023-12-12T09:42:22.543" Score="1" Body="&lt;p&gt;Some new translations (mostly into German but also into English) can be found on the website of the &lt;a href=&quot;https://www.agtz.mathematik.uni-mainz.de/algebraische-geometrie/van-straten/euler-kreis-mainz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Euler Kreis Mainz&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="3462" LastEditorUserId="4251" LastEditDate="2023-12-12T09:47:45.737" LastActivityDate="2023-12-12T09:47:45.737" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16017" PostTypeId="1" AcceptedAnswerId="16018" CreationDate="2023-12-12T17:13:30.793" Score="8" ViewCount="1055" Body="&lt;p&gt;Augustus DeMorgan wrote comments on Euclid's &lt;em&gt;Elements&lt;/em&gt;, which capture many of the most important points.  Heath quotes them extensively.&lt;/p&gt;&#xA;&lt;p&gt;I cannot find any source for the original: Where can I see DeMorgan's comments in the original?&lt;/p&gt;&#xA;" OwnerUserId="18545" LastEditorUserId="6609" LastEditDate="2023-12-12T18:04:17.060" LastActivityDate="2023-12-13T00:08:42.200" Title="DeMorgan's commentary on Euclid's Elements" Tags="|mathematics|reference-request|geometry|euclidean-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16018" PostTypeId="2" ParentId="16017" CreationDate="2023-12-12T23:50:09.547" Score="13" Body="&lt;p&gt;De Morgan's &amp;quot;Short Supplementary Remarks on the first Six Books of Euclid's Elements&amp;quot; is contained in the &lt;a href=&quot;https://archive.org/details/sim_british-almanac-companion_1849/page/n3/mode/2up&quot; rel=&quot;noreferrer&quot;&gt;Companion to the (British) Almanac for the year 1849&lt;/a&gt;, pp.5–20, published by the &lt;a href=&quot;https://en.wikipedia.org/wiki/British_Almanac&quot; rel=&quot;noreferrer&quot;&gt;Society for the Diffusion of Useful Knowledge&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4463" LastEditorUserId="4463" LastEditDate="2023-12-13T00:08:42.200" LastActivityDate="2023-12-13T00:08:42.200" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16019" PostTypeId="2" ParentId="16012" CreationDate="2023-12-13T05:34:06.720" Score="3" Body="&lt;p&gt;The web-page linked by this question (&amp;quot;Who invented ... ?&amp;quot;) credits the 'one-page calendar' design (for a year) to &amp;quot;E Siegel&amp;quot; -- along with a phrase that suggests he is also claiming a copyright. (It's not necessary for this answer to go into the degree of originality in the design, or whether it amounts to an invention, etc.)&lt;/p&gt;&#xA;&lt;p&gt;There are many possible ways to write out a calendar -- using 'calendar' here to mean, as in the example given, an arrangement of tables or lists that tells, for any date defined by month and date-number within a given year, on which weekday it falls.&lt;/p&gt;&#xA;&lt;p&gt;Most or all of such lists effectively use a part of the information originally given long ago in the papers that defined the Gregorian calendar reform of 1582, and/or in the wide variety of listings in prayer-books and other subsequent works to define the date of Easter. The two main functions contained in these accounts were (i) to define how the days of the week match up with the calendar dates in the year, and (ii) also how the new and full moons match up with the dates in the year.&lt;/p&gt;&#xA;&lt;p&gt;Calendars such as those in the present question need only part (i) of the data just mentioned. The information in that part is readily compiled by recognizing that there are (a) just seven arrangements of the monthly calendar, where the 1st of the month falls on each one of the seven weekdays (allowing variability of month-length from 28 to 31 days according to the case); and (b) just seven pairs of annual calendars, each defined by the weekday of January 1, and within each pair, whether the particular year is a common or leap year.&lt;/p&gt;&#xA;&lt;p&gt;The original matching, for years of the modern (Gregorian) calendar, between the days of the week and the numerical month-dates, is found in &amp;quot;Canon 4 : De litera Dominicali&amp;quot; from the implementing documents accompanying or following the &lt;a href=&quot;https://en.wikipedia.org/wiki/Inter_gravissimas&quot; rel=&quot;nofollow noreferrer&quot;&gt;1582 papal bull &amp;quot;Inter gravissimas ...&amp;quot;&lt;/a&gt;. For present and practical purposes, the necessary information can also be found, summarised much more briefly and usably, for example at page 427 in the 1961 &lt;a href=&quot;https://archive.org/details/astronomicalalmanac1961/page/n437/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Explanatory Supplement to the Astronomical Ephemeris ...&amp;quot;&lt;/a&gt;. (Both of these cited works define the weekday-year relationship by the 'Sunday letter', which is 'A' where the first Sunday of the year is on January 1; 'B' where it is on January 2; and so on up to 'G'.)&lt;/p&gt;&#xA;&lt;p&gt;Since the question asks about invention, authorship and history, it is relevant to go back to the Latin original of Canon 4, which can be found reproduced in a book of 1603, &lt;a href=&quot;https://books.google.com/books?id=x1lDAAAAcAAJ&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Romani Calendarii A Gregorio XIII. P.M. Restituti Explicatio&amp;quot;&lt;/a&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Christopher_Clavius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Christopher Clavius&lt;/a&gt;, who was a leading technical member of the papal calendar commission that accepted (and modified) the calendar reform proposal of &lt;a href=&quot;https://en.wikipedia.org/wiki/Aloysius_Lilius&quot; rel=&quot;nofollow noreferrer&quot;&gt;Aloysius Lilius&lt;/a&gt;. (Lilius is sometimes credited with proposing the entire calendar reform, but it can be seen from the 1577 consultative booklet containing his proposal &lt;a href=&quot;https://dx.doi.org/10.3931/e-rara-3802&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Compendium novae rationis restituendi calendarium ...&amp;quot;&lt;/a&gt;, as well as from the references to it in the text of 'Inter gravissimas..,' that its concern is to deal with the (more difficult) issues about Easter full moons and related matters; it does not originate the basic reform of the solar part of the calendar, for which the proposal was improved from ideas of older origin.) Clavius may therefore be the effective designer/author/compiler of the information in Canon 4 as presented (leaving aside the imponderable question whether it was also ever something 'invented').&lt;/p&gt;&#xA;&lt;p&gt;So far, this deals with the information content of the 1-page calendar. If information is desired about the origin of details of the layout, this seems very difficult to find. But it seems relevant to notice that while the illustrative 1-page calendar for 2024 presented in this question does effectively compress a year's calendar into a single table, there are also many other other possible ways of presenting the same and more information about calendar days and dates.  For example, the accompanying page-image (below) illustrates a way of fitting, into a single A4 page (with instructions for use), the day-date information not only for one year, but also for any and all years of the modern (Gregorian) calendar. Readily derivable from this is the information contained not only in the the 1-page calendar example for 2024 given in the question, but also in the corresponding calendar for any other year. (The attached page image was compiled by the present poster, but there is no wish to claim copyright.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/5Lq42.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/5Lq42.png&quot; alt=&quot;1-page of 2 tables and instructions to find the current month's calendar and the weekday for any chosen Gregorian Calendar date.&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="5788" LastEditorUserId="5788" LastEditDate="2023-12-13T05:51:56.813" LastActivityDate="2023-12-13T05:51:56.813" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16020" PostTypeId="1" CreationDate="2023-12-13T15:07:55.313" Score="1" ViewCount="85" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;“I was sitting writing at my textbook, but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire, and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by repeated visions of this kind, could now distinguish larger structures of manifold conformations; long rows, sometimes more closely fitted together; all twisting and turning in snake like motion. But look! What was that? One of the snakes had seized hold of it’s own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I woke;.... I spent the rest of the night working out the consequences of the hypothesis. Let us learn to dream, gentlemen, and then perhaps we shall learn the truth but let us beware of making our dreams public before they have been approved by the waking mind.” -August Kekulé&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;This was the juncture when Kekulé discovered the structure of benzene. But what are the meanings of the metaphors he uses to describe?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&amp;quot;All twisting and turning in snake-like motion...&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;One of the snakes had seized hold of it's own tail...&amp;quot;&lt;/li&gt;&#xA;&lt;li&gt;&amp;quot;The form whirled mockingly before my eyes...&amp;quot;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;This question might look like it belongs to the English SE but it needs chemistry knowledge to answer it.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone please explain what Kekulé meant? It's evident that benzene is cyclic and unsaturated, but I feel he had subtle clues of benzene undergoing resonance and its other properties.&lt;/p&gt;&#xA;" OwnerUserId="19458" LastEditorUserId="6609" LastEditDate="2023-12-13T15:53:50.087" LastActivityDate="2023-12-13T15:53:50.087" Title="Kekulé and his discovery of benzene" Tags="|discoveries|chemistry|quote|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16021" PostTypeId="1" CreationDate="2023-12-13T15:18:52.893" Score="0" ViewCount="51" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation#Results_for_the_normal_distribution&quot; rel=&quot;nofollow noreferrer&quot;&gt;term to remove bias from an estimate of standard deviation for a normal distribution&lt;/a&gt; is referred to as &lt;span class=&quot;math-container&quot;&gt;$c_4$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What is the origin or reason for using that notation for the correction factor?&lt;/p&gt;&#xA;" OwnerUserId="19459" LastActivityDate="2023-12-13T15:18:52.893" Title="Why is the standard deviation bias correction factor called c₄?" Tags="|mathematics|notation|statistics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16022" PostTypeId="2" ParentId="15597" CreationDate="2023-12-13T16:28:02.543" Score="1" Body="&lt;p&gt;Providing a partial response here to my own question. However I have still to find a proper document with a list or names. Here is a recap of what I got:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;As I said, Fréderic Joliot Curie was not in there, he was against the&#xA;nuclear weapon project and was kicked out of the CEA.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Bertrand_Goldschmidt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bertrand Goldschmidt&lt;/a&gt; is considered the father of the French&#xA;atomic bomb and was leading the project until 1960. He had very little public appearance.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Yves_Rocard&quot; rel=&quot;nofollow noreferrer&quot;&gt;Yves Rocard&lt;/a&gt;, father of some famous French minister, responsible for the French atomic program and took over Joliot-Curie, and also called father of the French bomb. He is kind of interesting because he became obsessed with UFOs and alternative medecine when he got older.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Francis_Perrin_(physicist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francis Perrin&lt;/a&gt;, son of famous French experimental physicist &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Perrin&lt;/a&gt; and brother-in-law of &lt;a href=&quot;https://fr.wikipedia.org/wiki/Pierre_Auger&quot; rel=&quot;nofollow noreferrer&quot;&gt;Pierre Auger&lt;/a&gt; (who participated in the Canadian-wing of the Manhattan Project). He was the one that revealed that there was some kind of cooperation on nuclear research between France and Israel. He also discovered the only natural nuclear reactor in the Oklo mine, Gabon.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Robert_Dautray&quot; rel=&quot;nofollow noreferrer&quot;&gt;Robert Dautray&lt;/a&gt;, engineer scientist and another CEA director. He said to be the Father of the French atomic bomb (how many are there?). However this claim has been contested.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Engineer &lt;a href=&quot;https://fr.wikipedia.org/wiki/Michel_Carayol&quot; rel=&quot;nofollow noreferrer&quot;&gt;Michel Carayol&lt;/a&gt; worked on the design of the H-bomb.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I am still looking for a deeper source.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="10608" LastEditDate="2024-02-17T07:38:06.630" LastActivityDate="2024-02-17T07:38:06.630" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16023" PostTypeId="2" ParentId="9975" CreationDate="2023-12-14T19:05:52.703" Score="1" Body="&lt;p&gt;Einstein believed that his insufficiently deferential attitude to his professors was what caused him to end up in the scientific &amp;quot;wilderness&amp;quot;, unable to get a job in research. He was convinced that the poor references from his professors meant that he was now effectively unemployable as a scientist. He's supposed to have applied to every major university in Europe, and been turned down or ignored. Even his father took to writing pleading letters to people asking whether they couldn't give his son a job. Minkowski in particular is supposed to have written off Einstein as &amp;quot;A lazy dog who will never amount to anything&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Mikowski was part of the &amp;quot;Lorentz aether theory&amp;quot; community, and had probably intended to write a major paper on the subject for some years. He was probably quite surprised when his &amp;quot;useless&amp;quot; ex-student Einstein, now working at the Swiss Patent Office wrote one first. Minkowski's 1909 paper did Einstein a favour by citing him and mentioning him favourably. It's difficult to find any other favourable peer-reviewed citation of Einstein's work on special relativity before 1909.&lt;/p&gt;&#xA;&lt;p&gt;Einstein didn't manage to get a full-time academic research position until around 1909, &lt;em&gt;four years&lt;/em&gt; after he'd published his 1905 &amp;quot;wonder year&amp;quot; papers, as a self-financing independent. He had some hard years before a friend helped him to get the Patent Office job, and while he was there, he lived in a tiny little room, estranged from his wife and child.&lt;/p&gt;&#xA;&lt;p&gt;From Einstein's point of view, he was a talented researcher who had been screwed over by an establishment of pompous non-achievers who took badly to his belief that he was better at this that they were.&lt;/p&gt;&#xA;&lt;p&gt;Minkowski's helpful citation of Einstein might be considered an &amp;quot;olive branch&amp;quot; ... if so, Einstein appears not to have taken it up. Since Minkowski was probably one of those responsible for his years of hardship, his attitude may have been &amp;quot; **** him &amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;When Minkowski then obligingly died in 1909, this was just one less person whose opinion Einstein had to worry about.&lt;/p&gt;&#xA;" OwnerUserId="10790" LastActivityDate="2023-12-14T19:05:52.703" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16024" PostTypeId="2" ParentId="15968" CreationDate="2023-12-15T02:36:45.680" Score="2" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/K%C3%BCpfm%C3%BCller%27s_uncertainty_principle&quot; rel=&quot;nofollow noreferrer&quot;&gt;Küpfmüller's uncertainty principle&lt;/a&gt; based on rectangular functions as  bandlimiting frequency filters appeared already 1924, years before Heisenberg's.&lt;/p&gt;&#xA;" OwnerUserId="13040" LastActivityDate="2023-12-15T02:36:45.680" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16026" PostTypeId="2" ParentId="8071" CreationDate="2023-12-15T13:23:48.807" Score="2" Body="&lt;p&gt;There are 3 main reasons why the Olbia gear is considered as genuine:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The excavation was an official one led by a professional team, during which many other ancient objects were found at underground layers (stratums) in correspondence to their age. Moreover, the fragment itself was found in a layer from the second century B.C, lay down under a top stratum which was already ancient. Thus, it is very difficult to see how can the Olbia wheel be considered as a modern contamination&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;The Brass metal from which the wheel is made was highly oxidized, pointing out a heavy clue to the age of the wheel (at least for it to be no a modern one)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Finally, an almost definite proof for the Olbia gear wheel age can be obtained   from a very interesting feature regarding the total teeth number for the complete wheel to be 55. There is no question that the wheel was a part of a complex portable mechanism, relatively small (the wheel dimensions also hint to this direction). The ultimate majority (if not all) of such mechanisms in modern time should be simply clocks (even astronomical clocks, which have of course a time keeping machine), but clocks have no gear wheels containing 55 teeth! This is because clocks are based on the division of the numbers 24 and 60 for their gear kinematic chain, and a gear wheel which has 55 teeth leads to irrational ratio values which are fit to an astronomical cycles gained by a manually operated planetarium. Considering that it is agreed among most of the scholars today that a Hellenistic tradition of creating such planetariums in antiquity took place, it is highly plausible to rely on for the ancient originality of the Olbia wheel.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19466" LastActivityDate="2023-12-15T13:23:48.807" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16028" PostTypeId="1" CreationDate="2023-12-15T19:07:28.717" Score="1" ViewCount="121" Body="&lt;p&gt;I was reading this &lt;a href=&quot;https://physics.stackexchange.com/questions/265915/why-didnt-newton-study-electricity-and-magnetism&quot;&gt;question&lt;/a&gt;. There I read somewhere that he would have discovered modern physics if he had studied electromagnetism and nuclear physics.&lt;/p&gt;&#xA;&lt;p&gt;It was most likely in jest. But I ask&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;strong&gt;How much of physical,mathematical and/or technological advancements had been made in Newtown's time?&lt;/strong&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;Given the state of science politics &amp;amp; the real world, &lt;em&gt;was it feasible to discover modern physics or general relativity for Newton&lt;/em&gt;?&lt;/strong&gt;(given his genius)&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;It seems a really difficult work but the popular opinion seems to be that Newton was more of a genius than Einstein so I don't know. &lt;br&gt; After all,&#xA;Almost every topic in classical physics  has been contributed to by Newton. Every subject other than Electromagnetism has some important formula, law, approximation or theorem given by Newton.&lt;br&gt; If we consider him to be the inventor of calculus, then I guess he has a major indirect contribution to all of physics(classical or quantum)&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;While it seems opinion based, the question is really boils down to the scientific and technological condition of the 17th century.&lt;/strong&gt; and a little bit on Newton's genius&lt;/p&gt;&#xA;" OwnerUserId="19470" LastEditorUserId="19470" LastEditDate="2023-12-15T20:54:04.023" LastActivityDate="2023-12-15T20:54:04.023" Title="Could Newton have come up with General Relativity?" Tags="|relativity-theory|newton|einstein|18th-century|17th-century|" AnswerCount="0" CommentCount="2" ClosedDate="2023-12-19T14:07:48.107" ContentLicense="CC BY-SA 4.0" />
  <row Id="16030" PostTypeId="1" AcceptedAnswerId="16033" CreationDate="2023-12-16T01:32:26.773" Score="2" ViewCount="435" Body="&lt;p&gt;The famous equation &lt;span class=&quot;math-container&quot;&gt;$F = G\cfrac{m_1m_2}{r^2}$&lt;/span&gt; is not in &lt;em&gt;Mathematical Principles of Natural Philosophy&lt;/em&gt;, as it has been pointed out by a commentator &lt;a href=&quot;https://hsm.stackexchange.com/questions/12880/what-was-newtons-statement-of-the-universal-law-of-gravitation&quot;&gt;on another question&lt;/a&gt; about where the equation is in the book. The closest one can get is Corollary 2 to Proposition V (&amp;quot;The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from that planet's centre&amp;quot;), according to the commentator.&lt;/p&gt;&#xA;&lt;p&gt;If that's the case, then where does the famous version of the equation come from, when did it develop, and why is Newton credited with the equation if it's not in the book? Is the equation implied or found through examples? If it's in prose, please could someone quote the correct passage?&lt;/p&gt;&#xA;" OwnerUserId="19472" LastEditorUserId="-1" LastEditDate="2023-12-16T18:16:55.090" LastActivityDate="2023-12-17T20:22:05.400" Title="Where does the famous equation $F = G\cfrac{m_1m_2}{r^2}$ come from?" Tags="|newton|gravity|" AnswerCount="1" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="16032" PostTypeId="2" ParentId="13409" CreationDate="2023-12-17T03:59:29.067" Score="4" Body="&lt;p&gt;In a relatively recent &lt;a href=&quot;https://pubs.aip.org/physicstoday/article/56/12/53/632269/Stern-and-Gerlach-How-a-Bad-Cigar-Helped-Reorient&quot; rel=&quot;nofollow noreferrer&quot;&gt;Physics Today article&lt;/a&gt; (Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics, December 2003, p. 53), it is argued that the fact that both Stern and Gerlach smoked cheap, high sulfur cigars in the lab helped bring out the contrast of the deposited silver which made the experiment successful. Without their smoking habit, they may not have discovered their amazing result!&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;EDIT&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Here is what they saw:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/EjI1j.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/EjI1j.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="3643" LastEditDate="2024-01-06T01:10:11.930" LastActivityDate="2024-01-06T01:10:11.930" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16033" PostTypeId="2" ParentId="16030" CreationDate="2023-12-17T20:22:05.400" Score="4" Body="&lt;p&gt;The language in Newton’s (translated) work is antiquated by modern standards. Saying that the force is “reciprocally as the square of the distance” means, in more modern language, that the force is inversely proportional to the square of the distance. In equation form,&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$ F = \frac{c}{r^2} $$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is a quantity which does not depend on distance.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;(Prop. VII, Book III) &lt;em&gt;That there is a power of gravity tending to all bodies, proportional to&#xA;the several quantities of matter which they contain.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;That all the planets mutually gravitate one towards another, we have proved before ; as well as that the force of gravity towards every one of them, considered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence (by Prop. LXIX, Book I, and its Corollaries) it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.&#xA;Moreover, since all the parts of any planet A gravitate towards any other planet B ; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole ; and (by Law III) to every action corresponds an equal re-action ; therefore the planet B will, on the other hand, gravitate towards all the parts of the planet A ; and its gravity towards any one part will be to the gravity towards the whole as the matter of the part to the matter of the whole. Q.E.D.&#xA;COR, 1. Therefore the force of gravity towards any whole planet arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this ; for all at traction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet, as formed of a number of lesser planets, meeting together in one globe ; for hence it would appear that the force of the whole must arise from the forces of the component parts. If it is objected, that, ac cording to this law, all bodies with us must mutually gravitate one to wards another, whereas no such gravitation any where appears, I answer, that since the gravitation towards these bodies is to the gravitation towards the whole earth as these bodies are to the whole earth, the gravitation to wards them must be far less than to fall under the observation of our senses.&#xA;COR. 2. The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places from the parti cles ; as appears from Cor. 3, Prop. LXXIV, Book I.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This lengthy passage says that the gravitational force toward an object &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; is proportional to the mass of &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt; (“proportional to the matter which [it] contain[s]”). Furthermore, because of the action-reaction law, the force must also be proportional to the mass of the object being attracted to &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;As a result, the force of attraction between objects &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is proportional to &lt;span class=&quot;math-container&quot;&gt;$m_A$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$m_B$&lt;/span&gt; and inversely proportional to &lt;span class=&quot;math-container&quot;&gt;$r^2$&lt;/span&gt;, which can be expressed in equation form as&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F = G \frac{m_A m_B}{r^2}$$&lt;/span&gt;&#xA;where &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is some universal constant.&lt;/p&gt;&#xA;&lt;p&gt;The fact that Newton wrote in such a lengthy and opaque way is a product of his time. The mathematical formalism and terminology which enables us to express Newton’s law of gravitation in a few lines is an apparatus which has been developed in the intervening years.&lt;/p&gt;&#xA;" OwnerUserId="19479" LastActivityDate="2023-12-17T20:22:05.400" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="16034" PostTypeId="1" CreationDate="2023-12-18T12:21:11.750" Score="0" ViewCount="105" Body="&lt;p&gt;recently read a book on the history of science by academic Tatiana Roque. In it, she discusses how, during the Enlightenment, scientific dissemination was a fundamental assumption. Roque writes that knowledge inherently aimed to reach a broad audience, beyond the restricted circles of the Republic of Letters. This desire to engage the public was integral to knowledge production. She further notes the Encyclopedia's distribution through unofficial means, the widespread circulation of selected articles in anonymous brochures, and the strategic role of science popularization in garnering trust in new ideals.&lt;/p&gt;&#xA;&lt;p&gt;One example she cites is the Great Exhibition of 1851 in London, inaugurated by Queen Victoria, showcasing the era's progress. She also mentions the formation of the British Association for the Advancement of Science in 1831, with the aim of promoting and disseminating scientific knowledge. In the latter half of the 19th century, museums and exhibitions proliferated, maintaining the popularity of science among the expanding and decentralizing middle class. Victorian era 'conversaziones' combined scientific knowledge, technological skill, and appreciation for art and fashion, fostering a new visual culture and participatory public understanding of science.&lt;/p&gt;&#xA;&lt;p&gt;Considering these historical insights into the critical role of scientific dissemination, I'm looking for references (such as books) that specifically address the history of scientific dissemination. Can anyone recommend comprehensive sources on this topic?&lt;/p&gt;&#xA;" OwnerUserId="12383" LastEditorUserId="12330" LastEditDate="2023-12-19T12:55:03.553" LastActivityDate="2023-12-19T12:55:03.553" Title="Seeking References on the History of Scientific Dissemination" Tags="|reference-request|popularizatio|dissemination|publicawareness|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16035" PostTypeId="1" CreationDate="2023-12-21T03:59:22.413" Score="2" ViewCount="82" Body="&lt;p&gt;I have some historical questions in connections for the notes to a book I am writing.&lt;/p&gt;&#xA;&lt;p&gt;Who the first person to discover that the Coulomb potential in two dimension is &lt;span class=&quot;math-container&quot;&gt;$\log(|x|^{-1})$&lt;/span&gt;, equivalently that this is essentially the fundamental solution for the 2D Poisson equation? I'm guessing it might have been Poisson in one of his remarkable papers between 1817 and 1818. These papers are not readily available to me.&lt;/p&gt;&#xA;&lt;p&gt;Also who first used the term logarithmic potential?&lt;/p&gt;&#xA;&lt;p&gt;I'm also interested in the discrete case. My understanding is that the discrete fundamental solution was first discussed in Polya's famous 1921 paper on recurrence in random walks. Can anyone confirm that and if that is also the place where the &lt;span class=&quot;math-container&quot;&gt;$\log(n)$&lt;/span&gt; asymptotics was found (so far as I can see from skimming his paper with my almost non-existent German, all he used is that it goes to infinity).&lt;/p&gt;&#xA;" OwnerUserId="19491" LastEditorUserId="6609" LastEditDate="2023-12-21T10:19:11.430" LastActivityDate="2023-12-21T10:19:11.430" Title="History of logarithmic potential" Tags="|mathematics|physics|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16036" PostTypeId="1" CreationDate="2023-12-21T14:49:40.613" Score="2" ViewCount="88" Body="&lt;p&gt;G. H. Hardy wrote (apropos of the task of assigning values to divergent series):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is plain that the first step towards such an interpretation must be some definition, or definitions, of the 'sum' of an infinite series, more widely applicable than the classical definition of Cauchy. This remark is trivial now: it was not a triviality even to the greatest mathematicians of the 18th century. They had not the habit of definition: it was not natural to them to say, in so many words, `by &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; we &lt;em&gt;mean&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt;'. There are reservations to be made ... but it is broadly true to say that mathematicians before Cauchy asked not 'How shall we &lt;em&gt;define&lt;/em&gt; 1-1+1-…?' but 'What &lt;em&gt;is&lt;/em&gt; 1-1+1-…?', and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Are there good accounts of the process by which the mathematical community came to adopt the habit of definition? Also, did Cauchy have a fully modern attitude? For instance, did Cauchy ever apply his definition to conclude that such-and-such a series does &lt;em&gt;not&lt;/em&gt; converge?&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="6609" LastEditDate="2023-12-21T19:40:29.707" LastActivityDate="2023-12-21T19:40:29.707" Title="The habit of definition" Tags="|real-analysis|definition|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16037" PostTypeId="1" CreationDate="2023-12-21T21:59:18.483" Score="1" ViewCount="41" Body="&lt;p&gt;At what point in time did the term &amp;quot;autonomous&amp;quot; begin to signify time-invariance within the context of dynamical systems, and what prompted this association?&lt;/p&gt;&#xA;" OwnerUserId="3725" LastActivityDate="2023-12-21T21:59:18.483" Title="Usage of the word &quot;autonomous&quot; to refer to time-invariant differential equations" Tags="|mathematics|terminology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16038" PostTypeId="1" AcceptedAnswerId="16040" CreationDate="2023-12-22T02:55:09.920" Score="2" ViewCount="146" Body="&lt;p&gt;&lt;strong&gt;Motivation:&lt;/strong&gt; In any history, there is a cause-and-effect relationship. So I became curious about the situation in which the sheaf theory came to appear. In other words, I'm curious about what problem was left unsolved and led to the emergence of sheaf as a solution. Or I wonder if sheaf emerged as a more minor concept in the progress of researching a certain concept.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Addition:&lt;/strong&gt; I know that &amp;quot;Cousin problem&amp;quot; that Kiyoshi Oka contributed is related to sheaf theory. This thinking is right?&lt;/p&gt;&#xA;" OwnerUserId="18258" LastEditorUserId="18258" LastEditDate="2023-12-22T11:26:13.310" LastActivityDate="2023-12-23T01:20:50.010" Title="The reason why sheaf theory emerged" Tags="|terminology|mathematics-social-history|algebraic-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16039" PostTypeId="1" AcceptedAnswerId="16045" CreationDate="2023-12-22T17:23:29.567" Score="2" ViewCount="156" Body="&lt;p&gt;In the Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#:%7E:text=The%20three%20laws%20state%20that,major%20axis%20of%20its%20orbit.&quot; rel=&quot;nofollow noreferrer&quot;&gt;article&lt;/a&gt; for &amp;quot;Kepler's laws of planetary motion,&amp;quot; the article suggests that &amp;quot;Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own laws of motion and law of universal gravitation.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;A commentator on another &lt;a href=&quot;https://hsm.stackexchange.com/questions/16030/where-does-the-famous-equation-f-g-cfracm-1m-2r2-come-from&quot;&gt;question&lt;/a&gt; suggests that the &amp;quot;kepler's laws' article is not correct on this. Newton's Principia of 1687 showed (&lt;a href=&quot;https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookI-XI&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bk1 sec.11 props 57-69, esp 66&lt;/a&gt;) that in a system with &amp;gt;2 masses, deviations from keplerian motion would occur, though they might be rather small if one of the bodies was of much greater mass than the others&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Please could someone describe the relationship of the Newtonian and Keplerian laws that Newton was able to achieve (if it's different from the above description by the previous commentator) and maybe also give a brief description of how the laws were used together (if they were) after Newton? Please let me express my gratitude for your consideration in advance!&lt;/p&gt;&#xA;" OwnerUserId="19472" LastEditorUserId="19472" LastEditDate="2023-12-23T15:43:58.093" LastActivityDate="2023-12-24T15:46:00.883" Title="What is the relationship between the Newtonian laws of motion and gravity and the Keplerian laws of planetary motion?" Tags="|newton|gravity|kepler|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16040" PostTypeId="2" ParentId="16038" CreationDate="2023-12-22T21:37:59.390" Score="4" Body="&lt;p&gt;Sheaves were created by Leray to avoid doing anything resembling applied mathematics while he was in a prisoner of war camp during World War II, as he was an expert on PDEs and did not want this to become known. Hence he turned his attention to algebraic topology.&lt;/p&gt;&#xA;&lt;p&gt;For more details, see the notes by Haynes Miller &lt;a href=&quot;https://math.mit.edu/%7Ehrm/papers/ss.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastEditorUserId="643" LastEditDate="2023-12-23T01:20:50.010" LastActivityDate="2023-12-23T01:20:50.010" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16041" PostTypeId="1" CreationDate="2023-12-23T01:27:28.303" Score="1" ViewCount="211" Body="&lt;p&gt;I have recently become fascinated by the history of electricity. I have researched and come up with the following observations:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Static electricity was discovered in early times, by phenomena such as rubbing amber and watching dust attract/ rubbing turning glass also attracting light objects.&lt;/li&gt;&#xA;&lt;li&gt;It was found that some materials attracted and repelled when charged.&lt;/li&gt;&#xA;&lt;li&gt;It was found the some materials conduct the electricity, and electric charge could be transferred from one object to another, known as electric current.&lt;/li&gt;&#xA;&lt;li&gt;A kite experiment found that a metal key became electrically charged when struck by lightning, and where able to transfer lightning strikes away from buildings through a rod and wire&lt;/li&gt;&#xA;&lt;li&gt;Leyden Jar was clever and it was found to store electric charge, and discharge it by connecting the outside and inside via a conductor, a process we now now occurs due to electric field pushing electrons from the outside metal&lt;/li&gt;&#xA;&lt;li&gt;Someone differentiated magnetic force from electrical forces, the reason I am not sure of, I think its because objects appeared to be magnetic by nature and assert a particular type of force.&lt;/li&gt;&#xA;&lt;li&gt;Volta was intrigued by an experiment involving frogs and eventually discovered the voltaic pile, allowing for a constant flow of electric charge known as first battery.&lt;/li&gt;&#xA;&lt;li&gt;He developed a better battery, and then people realised that the magnetic force on the compass interacted with a circuit.&lt;/li&gt;&#xA;&lt;li&gt;This eventually led to Ampere's Laws, which eventually led to Faraday's law, allowing for mechanical generation of electricity.&lt;/li&gt;&#xA;&lt;li&gt;Fast forward a bit and someone discovered the electron, and eventually calculated its mass using previously established laws associated with electric fields.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;I was wondering if there is a more detailed recount of events, that is not too long, that I can read about to enrich my understanding of the fundamentals. My knowledge about early magnetism is sketchy, and my recounts are very shallow and not very detailed.&#xA;I am moreso interested in the practical side of thing, and talking about what people where calling what back in the time, rather than talking about complex stuff that is around is today, because that kind of defeats the purpose of the exercise in my opinion.&lt;/p&gt;&#xA;&lt;p&gt;Any help is appreciated; thank you so much.&lt;/p&gt;&#xA;" OwnerUserId="19497" LastEditorUserId="6545" LastEditDate="2024-01-12T09:57:39.313" LastActivityDate="2024-01-12T15:07:57.470" Title="Is there a concise but detailed history of electricity?" Tags="|physics|reference-request|electromagnetism|electricity|" AnswerCount="4" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16042" PostTypeId="1" CreationDate="2023-12-23T07:21:51.697" Score="0" ViewCount="33" Body="&lt;p&gt;I'm wondering about the origin of the concept of proof in human thinking. Nowadays the concept is a cornerstone of mathematics, but also in law. Did early mathematicians maybe adopt it from the judiciary system of ancient civilizations?&lt;/p&gt;&#xA;" OwnerUserId="4498" LastActivityDate="2023-12-23T07:21:51.697" Title="Etymology or genesis of the concept of proof" Tags="|mathematics|mathematical-logic|logic|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16043" PostTypeId="1" CreationDate="2023-12-23T10:16:50.023" Score="2" ViewCount="71" Body="&lt;p&gt;In the Bourbaki archives in Lorainne, France, there appears to be a Latin telegram from Nicolas Bourbaki to Marcel Brelot (see reference 486 at &lt;a href=&quot;https://iecl.univ-lorraine.fr/files/2021/04/DELSARTE_archives_Bourbaki_cote_BKI.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://iecl.univ-lorraine.fr/files/2021/04/DELSARTE_archives_Bourbaki_cote_BKI.pdf&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Is this document publicly available? I have a particular interest in the history of Latin as a language of mathematics, and this seems to be one of the last uses of the language by a famous mathematician (group) - although it was most likely some kind of joke.&lt;/p&gt;&#xA;" OwnerDisplayName="user19499" LastEditorUserId="6229" LastEditDate="2023-12-23T15:10:54.060" LastActivityDate="2023-12-23T15:10:54.060" Title="Where can I find Bourbaki's Latin telegram?" Tags="|mathematics|bourbaki|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16044" PostTypeId="2" ParentId="16041" CreationDate="2023-12-23T16:29:28.423" Score="1" Body="&lt;p&gt;As mentioned in the comments, &amp;quot;concise&amp;quot; and &amp;quot;detailed&amp;quot; are contradicting requirements. A detailed history is&lt;/p&gt;&#xA;&lt;p&gt;Edmund Whittaker, A history of the theories of Aether and Electricity, 2 volumes, Harper &amp;amp; brothers, NY 1951.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2023-12-23T16:29:28.423" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16045" PostTypeId="2" ParentId="16039" CreationDate="2023-12-23T16:57:49.187" Score="5" Body="&lt;p&gt;Newton's law of gravity is equivalent to Kepler's laws in the following sense: for two bodies, assuming attraction by a central force if the force obeys Newton's law then the motion obeys Kepler's laws; and conversely, Kepler's laws imply Newton's law of attraction. This is a mathematical theorem essentially established by Newton.&lt;/p&gt;&#xA;&lt;p&gt;For more than 2 bodies, there are no &amp;quot;Kepler laws&amp;quot;; for example, for the system Sun-Earth-Moon, Kepler's law hold only approximately, if we neglect the influence of one of the bodies on the two others. The motion under the Newton's law of attraction is in general very complicated. And it took long time to derive all observed motions in the Solar system from Newton's gravity laws.&lt;/p&gt;&#xA;&lt;p&gt;Consistence of the theory with observations is excellent, except for Mercury, where a more general gravity law is required - general relativity.&lt;/p&gt;&#xA;&lt;p&gt;Remark. There was a long and heated discussion in 20th century, whether Newton really proved equivalence of his gravity law to Kepler laws, or only an implication in one direction. On my opinion, he proved the equivalence.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2023-12-24T15:46:00.883" LastActivityDate="2023-12-24T15:46:00.883" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16046" PostTypeId="1" CreationDate="2023-12-24T00:29:53.873" Score="1" ViewCount="37" Body="&lt;p&gt;In Hamilton's &amp;quot;on a general method in dynamics&amp;quot;, he starts with varying the function &lt;span class=&quot;math-container&quot;&gt;$U$&lt;/span&gt; and writes the equation:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$&lt;/span&gt;&#xA;Then he defines &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; to be:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$&lt;/span&gt;&#xA;Then by &lt;span class=&quot;math-container&quot;&gt;$dT=dU$&lt;/span&gt;, he writes:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$T=U+H$$&lt;/span&gt;&#xA;Then he varies T and writes:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\delta T= \delta U+\delta H$$&lt;/span&gt;&#xA;note that he is also varying in the initial conditions, that's why he did not omit the term &lt;span class=&quot;math-container&quot;&gt;$\delta H$&lt;/span&gt;.&#xA;Hamilton then multiplies this expression by dt and integrates and writes it as:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\int\sum m(dx \delta \dot x+dy \delta \dot y+dz \delta \dot z)=\int\sum m(d \dot x \delta x+d \dot y \delta y+d \dot z \delta z)+\int\delta H dt$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then comes the part where I got confused. He says &amp;quot;that is, by the principles of the calculus of variations&amp;quot; and writes:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\delta V=\sum m(\dot x \delta x+\dot y \delta y+\dot z \delta z)-\sum m(\dot a \delta a+\dot b \delta b+\dot c \delta c)+\delta H t$$&lt;/span&gt;&#xA;where (x,y,z) and (a,b,c) are final and initial conditions then he denotes V by the integral:&#xA;&lt;span class=&quot;math-container&quot;&gt;$$V=\int\sum m(\dot x \delta x+\dot y \delta y+\dot z \delta z)$$&lt;/span&gt;&#xA;My questions are as follows: &lt;br /&gt;&#xA;1-how did he get &lt;span class=&quot;math-container&quot;&gt;$\delta V$&lt;/span&gt;, what &amp;quot;principle of the calculus of variations&amp;quot; did he use? &lt;br /&gt;&#xA;2-then how from that did he get the integral &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt;?&lt;/p&gt;&#xA;" OwnerUserId="19501" LastActivityDate="2023-12-24T00:29:53.873" Title="Hamilton's derivation for the characteristic function in his essay" Tags="|classical-mechanics|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16047" PostTypeId="1" AcceptedAnswerId="16049" CreationDate="2023-12-24T18:28:12.153" Score="4" ViewCount="567" Body="&lt;p&gt;The story of Hennig Brand discovering the element phosphorous is often repeated without citation, and there doesn't seem to be much scholarship about Brand in particular (although the history of the discovery of the elements has been covered in many books, even some focusing on the history of phosphorous). So far, I've tracked down the story to Leibniz's &lt;em&gt;Historia inventionis phosphori&lt;/em&gt; published in 1710 (in Misc. berol.). But here he is referred to simply as &amp;quot;Brand&amp;quot;, and in Poggendorffs biography of the history of science, he also only refers to him as Brand. So how do we know the first name of the amateur alchemist Brand?&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2023-12-25T07:54:24.580" Title="How do we know Hennig Brand's name?" Tags="|biographical-details|chemistry|alchemy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16048" PostTypeId="1" CreationDate="2023-12-24T21:35:29.877" Score="4" ViewCount="194" Body="&lt;p&gt;In my school curriculum, and in many standard presentations such as &lt;a href=&quot;https://en.wikipedia.org/wiki/State_of_matter&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;, it is claimed that there are four fundamental states of matter: solids, liquids, gases, and plasmas. How was this list developed?&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;When did the original conception of matter as being divided into solids, liquids, or gases first arise, and by whom? Did this classification exist already in antiquity, or was it a byproduct of the scientific revolution and the discovery of gases as separate from air?&lt;/li&gt;&#xA;&lt;li&gt;How, when, and by whom was plasma added to this standard list? I find this especially puzzling since, in discussions among physicists, there seems to be &lt;a href=&quot;https://physics.stackexchange.com/questions/79426/is-a-plasma-a-distinct-phase-of-matter&quot;&gt;pretty widespread disagreement&lt;/a&gt; over whether plasma deserves to be on a similar footing as the first three (for example, because it is not separated by a phase transition from the other states).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19504" LastActivityDate="2024-01-04T14:21:11.493" Title="What is the history of the classification of states of matter?" Tags="|physics|chemistry|" AnswerCount="2" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16049" PostTypeId="2" ParentId="16047" CreationDate="2023-12-25T02:58:57.863" Score="8" Body="&lt;p&gt;Hermann Peters, &amp;quot;Leibniz als Chemiker.&amp;quot; &lt;em&gt;Archiv für die Geschichte der Naturwissenschaften und der Technik&lt;/em&gt;, Vol. 7, No. 2, Apr. 1916, &lt;a href=&quot;https://www.google.com/books/edition/Archiv_f%C3%BCr_die_Geschichte_der_Naturwiss/EPi8NWFJtmwC?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA85&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;pp. 85-108&lt;/a&gt;, reproduces (text only) correspondence between Leibniz and Brand. The author states that at the time of publication, these letters collected by Leibniz were held in the Royal Library in Hanover.&lt;/p&gt;&#xA;&lt;p&gt;The correspondence transcribed in Peters' publication covers the timeframe between October 1678 and August 1682. Most of the letters by Brand are signed &amp;quot;Hennig Brand M. D.&amp;quot;, while others are signed &amp;quot;H. Brand M. D.&amp;quot;, &amp;quot;H. Brand M. Dr.&amp;quot;, or &amp;quot;H. B. M. D.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;It is doubtful that Brand was actually a medical doctor. Johann Kunckel von Löwenstern (Johann Caspar Engelleder, ed.), &lt;em&gt;Collegium Physico-Chymicum Experimentale oder Laboratorium Chymicum, II. Edition.&lt;/em&gt; Hamburg and Leipzig: Samuel Heyl 1722, &lt;a href=&quot;https://www.google.com/books/edition/Collegium_physico_chymicum_experimentale/WmJWAAAAcAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA660&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;pp. 660-664&lt;/a&gt;, who visited Brand on a trip to Hamburg shortly after independently discovering phosphorous, describes Brand as a failed merchant who had taken up work in the field of medicine and was known as Doctor Brand.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-12-25T07:54:24.580" LastActivityDate="2023-12-25T07:54:24.580" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16050" PostTypeId="1" AcceptedAnswerId="16054" CreationDate="2023-12-25T16:36:54.820" Score="3" ViewCount="69" Body="&lt;p&gt;The &lt;a href=&quot;https://www.merriam-webster.com/dictionary/plate%20tectonics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Merriam-Webster online dictionary&lt;/a&gt; writes that the first known usage of plate tectonics in the sense of the geological theory was in 1969. Encyclopedia Britannica credits Alfred Wegener with an early conception of the theory in 1915, and then cites Harry Hess in 1960 as conceiving of seafloor spreading, an integral aspect of plate tectonics. The encyclopedia cites other geologists (McKenzie, Parker, Morgan, le Pichon) with developing the theory in the 1960s and the geologists Isacks, Oliver, and Sykes with coining the term &amp;quot;new global tectonics.&amp;quot; But no source I can find describes where &amp;quot;plate tectonics&amp;quot; was published for the first time. Please, does anyone know when this term was first used?&lt;/p&gt;&#xA;" OwnerUserId="19472" LastActivityDate="2023-12-26T08:39:24.650" Title="When was the term &quot;plate tectonics&quot; first used in geology?" Tags="|geology|plate-tectonics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16051" PostTypeId="1" AcceptedAnswerId="16055" CreationDate="2023-12-25T21:35:13.183" Score="3" ViewCount="239" Body="&lt;p&gt;I asked this question on MSE &lt;a href=&quot;https://math.stackexchange.com/questions/4833194/how-to-prove-analysis-famous-theorems-without-rigor&quot;&gt;here&lt;/a&gt; but I was told it would do better here&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;I always wonder how mathematicians proved theorems before Cauchy’s epsilon-delta proof. Since many &amp;quot;recent&amp;quot; theorems, such as the Stolz-Cesàro theorem, Cauchy limit theorem , Cauchy D'Alembert's Law, use epsilon-delta as part of their proofs, I wonder how Euler would have proved Stolz-Cesàro theorem for example. Although this theorem was not known to Euler, I am pretty sure that he could have &amp;quot;proved&amp;quot; it in minutes.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Stolz-Cesàro theorem case &lt;span class=&quot;math-container&quot;&gt;$\frac{*}{\infty}$&lt;/span&gt;:-&lt;/strong&gt;&#xA;If &lt;span class=&quot;math-container&quot;&gt;$b_n $&lt;/span&gt; is a monotone increasing sequence and &lt;span class=&quot;math-container&quot;&gt;$\lim \limits_{n \to \infty} b_n = \infty $&lt;/span&gt;,&#xA;and if &lt;span class=&quot;math-container&quot;&gt;$\lim \limits_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}- b_n}= l \in \overline{\mathbb{R}} $&lt;/span&gt;, then  &lt;span class=&quot;math-container&quot;&gt;$\lim \limits_{n \to \infty} \frac{a_n }{b_n}=l$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Stolz-Cesàro theorem case &lt;span class=&quot;math-container&quot;&gt;$\frac{0}{0}$&lt;/span&gt;:-&lt;/strong&gt;&#xA;If &lt;span class=&quot;math-container&quot;&gt;$b_n $&lt;/span&gt; is a monotone decreasing sequence and &lt;span class=&quot;math-container&quot;&gt;$\lim \limits_{n \to \infty} b_n = \lim \limits_{n \to \infty}  a_n = 0 $&lt;/span&gt;,&#xA;and if &lt;span class=&quot;math-container&quot;&gt;$\lim \limits_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}- b_n}= l \in \overline{\mathbb{R}} $&lt;/span&gt;, then  &lt;span class=&quot;math-container&quot;&gt;$\lim \limits_{n \to \infty} \frac{a_n }{b_n}=l$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Cauchy D'Alembert's Law&lt;/strong&gt; If &lt;span class=&quot;math-container&quot;&gt;$a_n$&lt;/span&gt; is a sequence of real numbers and if &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} =l$&lt;/span&gt;  the &lt;span class=&quot;math-container&quot;&gt;$\lim\limits_{n \to \infty} \sqrt[n]{a_n}=l$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I want to see how to &amp;quot;prove&amp;quot; one of these theorems without rigor.&lt;/p&gt;&#xA;" OwnerUserId="18466" LastEditorUserId="18466" LastEditDate="2024-01-01T01:24:59.700" LastActivityDate="2024-01-01T01:24:59.700" Title="How to &quot;prove&quot; recent analysis theorems without rigor?" Tags="|mathematics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16052" PostTypeId="1" CreationDate="2023-12-26T03:55:01.093" Score="0" ViewCount="37" Body="&lt;p&gt;As far as I know, there is an intuitive and easy way to think about sheaf theory through the Taylor series.&#xA;For example, in the case of stalk, which is one of the important concepts in sheaf, we can think of the Taylor series around point &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;.  But I became curious about the background that led me to think this way.&lt;/p&gt;&#xA;&lt;p&gt;And I know that De Branges's theorem, which proved Bieberbach's conjecture &lt;a href=&quot;https://en.m.wikipedia.org/wiki/De_Branges%27s_theorem&quot; rel=&quot;nofollow noreferrer&quot;&gt;(see this wikipedia article!)&lt;/a&gt;, is related to this content. Was it from the moment De Branges's theorem appeared that we began to view sheaf theory from the perspective of Taylor series?&lt;/p&gt;&#xA;&lt;p&gt;I read the &lt;a href=&quot;http://kobra.bibliothek.uni-kassel.de/bitstream/urn:nbn:de:hebis:34-200604038936/1/prep0513.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;following pdf file&lt;/a&gt; attached to the wikipedia article, but it did not provide a specific answer.  If anyone knows anything about this, I would appreciate your help.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastActivityDate="2023-12-26T03:55:01.093" Title="De Branges's theorem and perspective on Sheaf theory" Tags="|mathematics-social-history|algebraic-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16053" PostTypeId="1" CreationDate="2023-12-26T07:57:19.883" Score="1" ViewCount="35" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$G=(V,E)$&lt;/span&gt; be a graph with vertex set &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; and edge set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;. The Local edge connectivity &lt;span class=&quot;math-container&quot;&gt;$\lambda(x,y;G)$&lt;/span&gt;  for &lt;span class=&quot;math-container&quot;&gt;$x,y\in V$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$x\neq y$&lt;/span&gt;  is defined as the least number &lt;span class=&quot;math-container&quot;&gt;$|F|$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$y$&lt;/span&gt; are disconnected in &lt;span class=&quot;math-container&quot;&gt;$G-F$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$F\subseteq E$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;When was the concept of local edge connectivity introduced? More specifically, I would like to know the paper in which this concept was introduced. I have not been able to trace it to its base article.&lt;/p&gt;&#xA;" OwnerUserId="19509" LastActivityDate="2023-12-26T07:57:19.883" Title="What is the earliest paper in which local edge connectivity was defined?" Tags="|graph-theory|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16054" PostTypeId="2" ParentId="16050" CreationDate="2023-12-26T08:21:44.603" Score="2" Body="&lt;p&gt;D. Davies and D. P. McKenzie, &amp;quot;Seismic Travel-Time Residuals and Plates.&amp;quot;&#xA;&lt;em&gt;Geophysical Journal International&lt;/em&gt;, Vol. 18, No. 1, Sep. 1969, &lt;a href=&quot;https://academic.oup.com/gji/article/18/1/51/570217&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 51-63&lt;/a&gt; (my bolding):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The purpose of this paper is to demonstrate that some seismological observations which have not yet appeared to have any obvious cause are explicable in terms of the &lt;strong&gt;'plate tectonics'&lt;/strong&gt; developed in the last year or two by McKenzie &amp;amp; Parker (1967), Morgan (1968) and Le Pichon (1968). That seismological information is relevant to a critical examination of plate tectonics is testified by Isacks, Oliver &amp;amp; Sykes (1968) who produce a formidable amount of evidence on global plate behaviour.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The publications referenced are:&lt;/p&gt;&#xA;&lt;p&gt;Dan P. McKenzie and Robert L. Parker, &amp;quot;The North Pacific: An example of tectonics on a sphere.&amp;quot; &lt;em&gt;Nature,&lt;/em&gt; Vol. 216, Dec. 1967, pp. &lt;a href=&quot;https://sites.ualberta.ca/%7Eunsworth/UA-classes/699/2011/pdf/McKenzie_Plate_Tectonics_on_sphere_NATURE_1967.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1276-1280&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;W. Jason Morgan, &amp;quot;Rises, trenches, great faults and crustal blocks.&amp;quot; &lt;em&gt;Journal of Geophysical Research,&lt;/em&gt; Vol. 73, No. 6, Mar. 1968, &lt;a href=&quot;https://www.eoas.ubc.ca/%7Ecjohnson/EOSC448D/P1b_Morgan_1968_JGR_Rises_Trenches_Faults.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 1959-1982&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Xavier Le Pichon, &amp;quot;Sea-floor spreading and continental drift.&amp;quot; &lt;em&gt;Journal of Geophysical Research,&lt;/em&gt; Vol. 73, No. 12,  Jun. 1968, &lt;a href=&quot;https://www.geologie.ens.fr/50years_plate_tectonics/j-geophys-res-1968-le-picho.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 3661-3697&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Bryan Isacks, Jack Oliver, and Lynn R. Sykes, &amp;quot;Seismology and the new global tectonics.&amp;quot; &lt;em&gt;Journal of Geophysical Research&lt;/em&gt;, Vol. 73, No. 18, Sep. 1968, &lt;a href=&quot;https://courses.seas.harvard.edu/climate/eli/Courses/EPS281r/Sources/Layered-mantle/dynamics/Isacks-Oliver-sykes-1968.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;pp. 5855-5899&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;None of these references seems to use the term &amp;quot;plate tectonics&amp;quot;, but the last of these refers to (p. 5879) the &amp;quot;plate model of tectonics&amp;quot;, which could be considered a precursor term:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The plate model of tectonics provides, in a simple way, for the termination of an island arc by the abrupt or gradual transition to a transform fault, by a decrease in the rate of convergence to zero, or by some combination of these.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastEditorUserId="4201" LastEditDate="2023-12-26T08:39:24.650" LastActivityDate="2023-12-26T08:39:24.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16055" PostTypeId="2" ParentId="16051" CreationDate="2023-12-26T15:23:46.347" Score="5" Body="&lt;p&gt;The &amp;quot;standard&amp;quot; way of proving such things without epsilontics would be to use infinitesimal analysis, at which Euler was an expert &lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/euler.html&quot; rel=&quot;noreferrer&quot;&gt;of course&lt;/a&gt;.  From this point of view, the formula &lt;span class=&quot;math-container&quot;&gt;$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\ell$&lt;/span&gt; means that for all unlimited (informally, infinite) values of &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, the ratio &lt;span class=&quot;math-container&quot;&gt;$\frac{a_{n+a}}{a_n}$&lt;/span&gt; is infinitely close to &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt;, and in fact it is enough to require such a relation of infinite proximity &lt;span class=&quot;math-container&quot;&gt;$\frac{a_{n+a}}{a_n}\approx\ell$&lt;/span&gt; only for &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; greater than some fixed unlimited &lt;span class=&quot;math-container&quot;&gt;$H&amp;gt;0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Since you are interested in arguments &amp;quot;without rigor&amp;quot;, I will present a plausible argument that requires further work if one wants to make it fully OK.  Suppose &lt;span class=&quot;math-container&quot;&gt;$H=K^2$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is of course still unlimited.  Then &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; is &amp;quot;much&amp;quot; smaller than &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;.  Since the value on the limit is only dependent on the behavior of an infinite tail of the sequence, we can modify the original sequence by replacing the first &lt;span class=&quot;math-container&quot;&gt;$K$&lt;/span&gt; terms in such a way that all the ratios &lt;span class=&quot;math-container&quot;&gt;$\frac{a_n}{a_n}$&lt;/span&gt; are precisely &lt;span class=&quot;math-container&quot;&gt;$\ell$&lt;/span&gt;, without appreciably affecting either the limit of the ratios or the limit of the root.  What we obtain is that the sequence is appreciably of the form &lt;span class=&quot;math-container&quot;&gt;$C\ell^n$&lt;/span&gt; (again, up to modifications of some initial terms).  Then the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-th root for &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;H$&lt;/span&gt; will be appreciably &lt;span class=&quot;math-container&quot;&gt;$\sqrt[n]{C} \, \sqrt[n]{\ell^n}\approx \ell$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2023-12-26T15:23:46.347" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16056" PostTypeId="1" AcceptedAnswerId="16057" CreationDate="2023-12-27T15:20:58.587" Score="3" ViewCount="108" Body="&lt;p&gt;What are sources placing a length (one dimensional) in proportion to an area (two dimensional)?&lt;/p&gt;&#xA;&lt;p&gt;The Greek geometers compared quantities of the same dimension: e.g. the area of a circle is in proportion to the area of a square with the radius as the side.&lt;/p&gt;&#xA;&lt;p&gt;Descartes was to my knowledge the first to compare quantities of &lt;em&gt;different&lt;/em&gt; dimensions: Segment &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt; is in proportion to &lt;span class=&quot;math-container&quot;&gt;$XY$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$PQ \cdot RS$&lt;/span&gt; is to &lt;span class=&quot;math-container&quot;&gt;$TU$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What examples of there of people saying, e.g.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Consider a line &lt;span class=&quot;math-container&quot;&gt;$OB$&lt;/span&gt; and a curve.  For any point &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; on the curve, drop the perpendicular from &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$BO$&lt;/span&gt;, and call its foot &lt;span class=&quot;math-container&quot;&gt;$C'$&lt;/span&gt;.  The curve has the property that at any point &lt;span class=&quot;math-container&quot;&gt;$C$&lt;/span&gt;, the length &lt;span class=&quot;math-container&quot;&gt;$CC'$&lt;/span&gt; &lt;em&gt;[one dimensional]&lt;/em&gt; is in direct proportion to the area of a square with side &lt;span class=&quot;math-container&quot;&gt;$OC'$&lt;/span&gt; &lt;em&gt;[two dimensional]&lt;/em&gt; if and only if the curve is a parabola whose vertex is the point of tangency to &lt;span class=&quot;math-container&quot;&gt;$O$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or any other type of proportion or equation between different dimensions (length, area, volume)?&lt;/p&gt;&#xA;&lt;p&gt;Obviously this is common place in calculus.  I'm looking for examples of where this idea developed, as it seems totally absent from classic geometry.&lt;/p&gt;&#xA;" OwnerUserId="18545" LastEditorUserId="18545" LastEditDate="2023-12-27T19:15:52.847" LastActivityDate="2024-02-12T20:21:08.707" Title="Which geometer first compared a length (one dimensional) to an area (two dimensional)?" Tags="|mathematics|calculus|geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16057" PostTypeId="2" ParentId="16056" CreationDate="2023-12-27T18:56:58.280" Score="6" Body="&lt;p&gt;Al Khwarizmi, one of the first written sources of the general solution to quadratics, does &lt;a href=&quot;https://maa.org/press/periodicals/convergence/algebra-tiles-explorations-of-al-khw-rizm-s-equation-types-al-khw-rizm-s-compendium-on-calculating&quot; rel=&quot;nofollow noreferrer&quot;&gt;exactly that&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I observed that the numbers which are required in calculating by Completion and Reduction are of three kinds, namely, roots, squares, and simple numbers relative to neither root nor square.&#xA;A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending.&#xA;A square is the whole amount of the root multiplied by itself.&#xA;A simple number is any number which may be pronounced without reference to root or square.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In doing so, if I understand correctly, he seems to &amp;quot;upshift&amp;quot; length into area, by &lt;a href=&quot;http://www.ams.org/publicoutreach/feature-column/fc-2020-11#three&quot; rel=&quot;nofollow noreferrer&quot;&gt;equating a length &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; to an area &lt;span class=&quot;math-container&quot;&gt;$x \times 1$&lt;/span&gt;&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/p3ghw.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/p3ghw.png&quot; alt=&quot;http://www.ams.org/publicoutreach/feature-column/fc-2020-11#three&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;If my reading is correct, then Al Khwarizmi, besides solving the quadratic, helped lay a foundation for analytic geometry and calculus, which depend on being able to compare e.g. the rate of change of &lt;span class=&quot;math-container&quot;&gt;$x^2$&lt;/span&gt; compared to rate of change of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.  This was unthinkable to the Greeks, but Al Khwarizmi seems to have opened the door.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;A similar comment is made &lt;a href=&quot;https://www.tau.ac.il/%7Ecorry/publications/articles/pdf/algebra%20EB.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus, Islamic mathematics allowed, and&#xA;indeed encouraged at variance with the Greek tradition...&#xA;the simultaneous manipulations of magnitudes of different dimensions as part of the solution of an individual problem. Thus... the solution of a quadratic&#xA;equation is a “number”, rather than a “line segment” or an “area”... this approach was fundamental in developing a more abstract and&#xA;general conception of number, which eventually became essential for the creation of a&#xA;full-fledged abstract idea of an equation.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="18545" LastEditorUserId="18545" LastEditDate="2024-02-12T20:21:08.707" LastActivityDate="2024-02-12T20:21:08.707" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16058" PostTypeId="1" CreationDate="2023-12-27T20:47:45.270" Score="1" ViewCount="66" Body="&lt;p&gt;I mean this definition:&lt;/p&gt;&#xA;&lt;p&gt;A sequence &lt;span class=&quot;math-container&quot;&gt;$(u_n)_{n\in \mathbb{N}}$&lt;/span&gt; converges to a limit &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; if and only if:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\forall \epsilon&amp;gt;0 ~~\exists N \in \mathbb{N} ~~ \forall n \ge N ~~\vert u_n -l \vert &amp;lt; \epsilon$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Cauchy gave literally in 1823 a definition of the limit (that can be applied to a sequence) before: &amp;quot;If the values assigned successively to the same variable approach indefinitely a fixed value, so that they differ from it as little as one wishes, than the latter (value) is called their limit.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;The Universalis encyclopedia gives the first limit notion to Benjamin Robins in &amp;quot;A Discourse Concerning the Nature and Certainty of Sir Isaac Newton's Method of Fluxions and Prime and Ultimate Ratios&amp;quot; (1735): &lt;a href=&quot;https://www.universalis.fr/encyclopedie/limite/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.universalis.fr/encyclopedie/limite/&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then several mathematicians had the notion of limit in their mind according to the same encyclopedia and Walter Felscher in &amp;quot;Bolzano, Cauchy, Epsilon, Delta&amp;quot; (2000): &lt;a href=&quot;https://www.karlin.mff.cuni.cz/%7Espurny/doc/ma1/jirkavesely/Felscher.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.karlin.mff.cuni.cz/~spurny/doc/ma1/jirkavesely/Felscher.pdf&lt;/a&gt;, like D'Alembert, Gauss, Bolzano, Cauchy, Hausdorff and Cartan.&lt;/p&gt;&#xA;&lt;p&gt;It seems that the &amp;quot;invention&amp;quot; of quantifiers are attributed to Gottlob Frege in his &amp;quot;Begriffsschrift&amp;quot;: &lt;a href=&quot;https://plato.stanford.edu/entries/frege-logic/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://plato.stanford.edu/entries/frege-logic/&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But I don't find the details of my question.&lt;/p&gt;&#xA;" OwnerUserId="19511" LastActivityDate="2023-12-27T20:47:45.270" Title="Who was the first to define the limit of a convergent sequence with quantifiers $\forall$ and $\exists$?" Tags="|mathematicians|mathematical-logic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16059" PostTypeId="2" ParentId="16048" CreationDate="2023-12-28T21:33:42.267" Score="4" Body="&lt;p&gt;As @Mary notes in the comments, the ancient idea of &amp;quot;earth, water, air, and fire&amp;quot; maps nicely onto solid, liquid, gas, and plasma, but wasn't until the early 19th century that the idea of three states of matter was well established, and plasma only become commonly referred to as a fourth state of matter in the 1950s.&lt;/p&gt;&#xA;&lt;p&gt;The distinction between solid and liquid go back to antiquity, but the term &amp;quot;gas&amp;quot; was first used for a compressible fluid by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jan_Baptist_van_Helmont#Career_as_chemistry_pioneer&quot; rel=&quot;nofollow noreferrer&quot;&gt;van Helmont&lt;/a&gt; and published in 1648 in his posthumous &lt;a href=&quot;https://wellcomecollection.org/works/veeufd87/items?canvas=158&amp;amp;query=%22This+spirit%22&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;1662 English Translation&quot;&gt;&amp;quot;Ortus medicinae&amp;quot;&lt;/a&gt; :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This spirit, hitherto unknown, which can be neither be contained in vessels nor reduced to a visible body, I call by a new name, gas.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The modern concept of a &amp;quot;gas&amp;quot; was, however, really only &lt;a href=&quot;https://www.proquest.com/openview/0bf9d60923fae3992e000abd1c8fcf9b/1?pq-origsite=gscholar&amp;amp;cbl=18750&amp;amp;diss=y&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;The Foundations of Modern Chemistry: The Origin and Development of the Concept of the Gaseous State and Its Role in the Chemical Revolution of the Eighteenth Century, Jerry Bruce Gough, Ph.D. Thesis, Cornell University, 1971&quot;&gt;established by Lavoisier&lt;/a&gt; more than a century later in the 1770s. By 1847 the idea of the three states of matter was commonplace enough to be included in &lt;a href=&quot;https://books.google.ca/books?hl=en&amp;amp;lr=&amp;amp;id=oCEJAAAAIAAJ&amp;amp;oi=fnd&amp;amp;pg=PA21#v=onepage&amp;amp;q&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;First Principles of Chemistry: For the Use of Colleges and Schools, Benjamin Silliman, published by Beck &amp;amp; Bliss, Philadelphia, 1850&quot;&gt;introductory chemistry textbooks&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I believe William Crookes in 1880 was the first to &lt;a href=&quot;https://archive.org/details/jstor-2901038/page/n1/mode/1up&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;A Fourth State of Matter, W. Crookes, Science, Volume 1, 1880, p. 32&quot;&gt;argue strongly&lt;/a&gt; that there exists a fourth state of matter that we would now call a low density plasma, but he did not have our modern concept of a plasma.  (&amp;quot;Plasma&amp;quot; was first used for this ionized state by &lt;a href=&quot;https://en.wikipedia.org/wiki/Plasma_(physics)#Early_history&quot; rel=&quot;nofollow noreferrer&quot;&gt;Langmuir in 1928&lt;/a&gt;.) There seems to be a big gap, however, between Crookes and the mid-1950s when &lt;a href=&quot;https://scholar.google.ca/scholar?hl=en&amp;amp;as_sdt=0%2C5&amp;amp;as_ylo=1950&amp;amp;as_yhi=1958&amp;amp;q=%22fourth+state+of+matter%22+%22plasma%22&amp;amp;btnG=&quot; rel=&quot;nofollow noreferrer&quot;&gt;references to plasma as a &amp;quot;fourth state of matter&amp;quot;&lt;/a&gt; started to appear again, largely by people involved in nuclear fusion research. (See &lt;a href=&quot;https://archive.org/details/elezier-the-fourth-state-of-matter-an-introduction-to-plasma-science&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;The Fourth State of Matter, Shalom and Yaffa Eliezer, IOP Publishing Ltd 2001&quot;&gt;here&lt;/a&gt;, for example, for a more recent discussion of plasma and its history.)&lt;/p&gt;&#xA;" OwnerUserId="6504" LastEditorUserId="6504" LastEditDate="2023-12-29T16:20:47.917" LastActivityDate="2023-12-29T16:20:47.917" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16060" PostTypeId="1" CreationDate="2023-12-28T22:10:54.720" Score="1" ViewCount="110" Body="&lt;p&gt;Does anyone know of an English translation of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ludwig_Bieberbach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ludwig Bieberbach&lt;/a&gt;’s infamous 1934 Berlin lecture, delivered at the annual conference of the &lt;em&gt;Deutscher Verein zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts&lt;/em&gt;? This is the lecture in which he tried to draw a distinction between German mathematics and Jewish/French mathematics, claiming them to be “separated by an unbridgeable chasm”.&lt;/p&gt;&#xA;" OwnerUserId="6587" LastEditorUserId="4201" LastEditDate="2023-12-28T23:36:27.373" LastActivityDate="2023-12-28T23:36:27.373" Title="Bieberbach’s 1934 lecture on “German mathematics”" Tags="|mathematicians|20th-century|nazi-germany|1930s|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16061" PostTypeId="1" CreationDate="2023-12-29T10:53:10.077" Score="2" ViewCount="132" Body="&lt;h2&gt;Background &amp;amp; My research&lt;/h2&gt;&#xA;&lt;p&gt;So, I recently studied about the concepts of Stress and Strain in my high school physics classes and wanted to know about the history behind the origin and emergence of these concepts.&lt;/p&gt;&#xA;&lt;p&gt;Firstly, about the concept of stress, I found out that it was coined by Augustin-Louis Cauchy in 1822 when he announced his &amp;quot;Stress Principle&amp;quot; which states the following:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Upon any smooth closed orientable surface &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt;, be it an imagined surface within the body or the bounding surface of the body itself, there exists an integrable field of &lt;em&gt;traction vectors&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$t_{dv}$&lt;/span&gt;, equipollent to the action exerted by the matter exterior to &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt; &lt;em&gt;(contact forces)&lt;/em&gt; and contiguous to it on that interior to &lt;span class=&quot;math-container&quot;&gt;$dV$&lt;/span&gt; &lt;em&gt;(body forces)&lt;/em&gt;. [1]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Next to move towards the &lt;em&gt;&amp;quot;normal&amp;quot;&lt;/em&gt; representation of stress as we know it I found that we firstly need to represent the total force (and total torque) acting on the body which can be represented by the integration of the traction vectors with respect to &lt;em&gt;mass&lt;/em&gt; and &lt;em&gt;surface area&lt;/em&gt; (and their rotational transformations for the total torque). Now, I found that the stress we talk about only concerns with the boundary surface forces(contact forces) and hence the mass forces(body forces) vanish as the surfaces tend to zero while considering the &lt;em&gt;&amp;quot;normal&amp;quot;&lt;/em&gt; idea of stress. Also, the couple stresses (stresses due to couple forces causing rotation) are not considered in this &lt;em&gt;&amp;quot;normal&amp;quot;&lt;/em&gt; sense of stress (which is the classical sense of continuum mechanics).&lt;/p&gt;&#xA;&lt;p&gt;I also found out that the traction vector &lt;span class=&quot;math-container&quot;&gt;$t_{dv}$&lt;/span&gt; is dependent on the normal vector &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; to the point we find the traction vector on (which is known as &lt;em&gt;Cauchy's Postulate&lt;/em&gt; essentially stating that &lt;span class=&quot;math-container&quot;&gt;$t_{dv}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$t_{n_i}$&lt;/span&gt; for any normal &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; and plane &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt;). Finally, after some rearranging of terms we get that the resulting traction vector &lt;span class=&quot;math-container&quot;&gt;$t_{dv}$&lt;/span&gt; (for some normal vector &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, plane &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; and some area &lt;span class=&quot;math-container&quot;&gt;$dS$&lt;/span&gt; containing an arbitrary point &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;) is equal to &lt;span class=&quot;math-container&quot;&gt;$dF/dS$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$dF$&lt;/span&gt; is the surface force on that point &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt;) and is called the &lt;em&gt;surface traction&lt;/em&gt;, also known as &lt;em&gt;stress vector&lt;/em&gt;. Also, the state of stress at a particular point &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; is then depicted by the stresses at all planes passing through that point with some normal vectors &lt;span class=&quot;math-container&quot;&gt;$n_i$&lt;/span&gt; (this relation is known as &lt;em&gt;Cauchy's Fundamental Theorem&lt;/em&gt;) and according to this theorem the state of the stress at any of those infinite planes can be represented merely by knowing the stress vectors on three mutually perpendicular planes of the form: &lt;span class=&quot;math-container&quot;&gt;$t_{n_i}$&lt;/span&gt; = &lt;span class=&quot;math-container&quot;&gt;$n_i \cdot T$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; is the stress tensor (i.e. the linear map between the stress at an arbitrary plane &lt;span class=&quot;math-container&quot;&gt;$i$&lt;/span&gt; and it's normal vector &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;) which is obtained by the tensor product of the traction vectors at three mutually perpendicular planes and their associated normal vectors (&lt;span class=&quot;math-container&quot;&gt;$t_x,t_y,t_z \text{ and } n_x,n_y,n_z$&lt;/span&gt; which is essentially a 3&lt;span class=&quot;math-container&quot;&gt;$\times$&lt;/span&gt;3 matrix).&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;[1] Cited from Truesdell: The Creation and Unfolding of the Concept of Stress, chap. IV in Essays in the History of Mechanics.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Question:&lt;/h2&gt;&#xA;&lt;p&gt;Now, I want to know what motivated Cauchy to develop his postulate on the dependence of the stress (or traction) vector and the normal vector to the surface (which would also explain the dependence of stress on the orientation of the surface i.e. the plane considered). Additionally, I'm curious about the experiments, if any, that Cauchy used to verify his formulations and theorems.(I read that his work was motivated by Euler's previous work on Hydrodynamics around the 1750s and his idea of breaking the whole body into two imaginary parts (one interior and one exterior) which was Euler's Cut Principle).&lt;/p&gt;&#xA;&lt;p&gt;Also, is the 'proof' for Cauchy's theorem and Cauchy's postulate in the form of deriving the expression for the &lt;em&gt;&amp;quot;linear dependence of traction on the normal vector&amp;quot;&lt;/em&gt;, along with the &lt;em&gt;&amp;quot;existence of a stress tensor&amp;quot;&lt;/em&gt; for inter-planar transformations (of the infinite planes) by the use of 'Cauchy's Tetrahedron,' the original proof provided by Cauchy himself, or just a modern depiction of the proof? If it is the original, are there any sources showing why he used the tetrahedron specifically?&lt;/p&gt;&#xA;&lt;p&gt;I could not find any information on the concept of Strain whatsoever and how it originated or who coined the term (although I think that it was coined by Cauchy as well). Hence, any information on the concept of strain would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="19382" LastEditorUserId="19382" LastEditDate="2023-12-29T11:06:52.993" LastActivityDate="2023-12-31T21:02:54.333" Title="Origin of the concepts of Stress and Strain" Tags="|physics|terminology|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16062" PostTypeId="1" CreationDate="2023-12-29T13:50:45.873" Score="1" ViewCount="89" Body="&lt;p&gt;How is the first column of plimpton 322 derived from the other columns?&lt;/p&gt;&#xA;&lt;p&gt;Consider the first row of the tablet. The second, &lt;span class=&quot;math-container&quot;&gt;$b,$&lt;/span&gt; and third, &lt;span class=&quot;math-container&quot;&gt;$ c,$&lt;/span&gt; column are pythagorean triples &lt;span class=&quot;math-container&quot;&gt;$(a,b,c)= (120, 119, 169)$&lt;/span&gt; that satisfy &lt;span class=&quot;math-container&quot;&gt;$a^2 + b^2 = c^2.$&lt;/span&gt; We can derive, &lt;span class=&quot;math-container&quot;&gt;$a, $&lt;/span&gt; by taking the root of &lt;span class=&quot;math-container&quot;&gt;$a^2= c^2 - b^2$&lt;/span&gt; which give us &lt;span class=&quot;math-container&quot;&gt;$ 120 = \sqrt{169^2 -119^2}.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Now the first column should be &lt;span class=&quot;math-container&quot;&gt;$b^2/a^2 = (119/120)^2 = 0.983403.$&lt;/span&gt; But when I convert the &lt;span class=&quot;math-container&quot;&gt;$1:59:00:15 $&lt;/span&gt; to a base &lt;span class=&quot;math-container&quot;&gt;$ 10$&lt;/span&gt; notation then I get &lt;span class=&quot;math-container&quot;&gt;$428415. $&lt;/span&gt; How are these supposed to be the same or where am I making an error ?&lt;/p&gt;&#xA;&lt;p&gt;Reference: &lt;a href=&quot;https://en.wikipedia.org/wiki/Plimpton_322&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;; &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/S0315086017300691&quot; rel=&quot;nofollow noreferrer&quot;&gt;ScienceDirect&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19514" LastEditorUserId="17132" LastEditDate="2023-12-29T15:55:33.830" LastActivityDate="2023-12-29T19:30:07.793" Title="How do I obtain the first column of Plimpton 322 from the other columns?" Tags="|mathematics|ancient|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16063" PostTypeId="2" ParentId="16048" CreationDate="2023-12-29T19:08:18.760" Score="2" Body="&lt;p&gt;The terms &amp;quot;solid&amp;quot; and &amp;quot;liquid&amp;quot;/&amp;quot;fluid&amp;quot; seems date back to Proto-Indoeuropean. Early civilizations worked out metallurgy, the use of snow/ice, and rustic methods of distillation.&lt;/p&gt;&#xA;&lt;p&gt;During antiquity, many civilization introduced a version of the classical elements. In Europe, Empedocles is often cited as the one having introduced the idea of the four elements: earth, water, air and fire. Aristotle would later expand on Empedocles' theory. These classical elements could be seen as an early version of the states of matter but also as an early periodic table.&lt;/p&gt;&#xA;&lt;p&gt;Note that Aristotle already speaks of some phase transitions. Regarding vapor (&lt;em&gt;Meteorologica&lt;/em&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I have proved by experiment that salt water evaporated forms fresh, and the vapour does not, when it condenses, condense into sea water again.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The next step in the evolution of the &amp;quot;elements&amp;quot; arrived with the development of alchemy during late Greco-Roman civilization and the Islamic Middle Ages. Geber (Jābir ibn Hayyān) classified matter in terms of four properties: hotness, coldness, dryness, and moistness. European alchemist Paracelsus classified matter in the &lt;em&gt;tria prima&lt;/em&gt;: salt (solid), mercury (liquid) and sulfur (flammable). However during this period, many other alchemical elements were proposed, see Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/Alchemical_symbol&quot; rel=&quot;nofollow noreferrer&quot;&gt;alchemical symbol&lt;/a&gt; for various classifications. These elements go more into the classifications that would result later in the elements of the periodic table and not on the states of matter.&lt;/p&gt;&#xA;&lt;p&gt;Less esoteric classifications reemerged with the development of chemistry. As others have noted, Jan Baptist van Helmont introduced the concept of &amp;quot;gas&amp;quot; in the 17th century (before that these substances were called &amp;quot;airs&amp;quot; or &amp;quot;vapors&amp;quot;). The study of gases would lead to the gas laws, the ideal gas law and eventually to the atomic theory of gases.&lt;/p&gt;&#xA;&lt;p&gt;The development of thermodynamics would lead to the discovery of the triple point of water by Lord Kelvin in the 19th century.  J. Willard Gibbs would introduce the term &amp;quot;phase&amp;quot;. The microscopic understanding of matter in the 20th century, would lead to the precise quantitative understanding of phase transitions (by people like Paul Erenfest or Lev Landau). Irving Langmuir would introduce the concept of plasma in 1928.&lt;/p&gt;&#xA;&lt;p&gt;At what point between the alchemical elements and modern theory did the concept of &amp;quot;solid, liquid, gas&amp;quot; + &amp;quot;plasma&amp;quot; became the norm is hard to assess and merits more research.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-04T14:21:11.493" LastActivityDate="2024-01-04T14:21:11.493" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16064" PostTypeId="2" ParentId="16062" CreationDate="2023-12-29T19:22:53.280" Score="3" Body="&lt;p&gt;If I translated the notation correctly, the paper you link says that the first column is &lt;span class=&quot;math-container&quot;&gt;$c^2/a^2$&lt;/span&gt;, and in this example at least this seems correct (1;59,00,15 is in fact exactly &lt;span class=&quot;math-container&quot;&gt;$(169/120)^2$&lt;/span&gt;).&lt;/p&gt;&#xA;" OwnerUserId="2578" LastEditorUserId="2578" LastEditDate="2023-12-29T19:30:07.793" LastActivityDate="2023-12-29T19:30:07.793" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16066" PostTypeId="1" CreationDate="2023-12-30T06:54:12.410" Score="1" ViewCount="57" Body="&lt;p&gt;According to Wikipedia, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Theory_of_impetus&quot; rel=&quot;nofollow noreferrer&quot;&gt;theory of impetus&lt;/a&gt;, which was a precursor to the modern concepts of inertia and momentum, was developed in the West by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Buridan&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean Buridan&lt;/a&gt;, and further developed by &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Cantius&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Cantius&lt;/a&gt; (Jan Kanty in his native Polish). Cantius is a figure I'm interested in, but there does not seem to be much information on how exactly he contributed to the theory. The statement on Wikipedia is unsourced, Google Scholar does not turn up any useful results, and Google just returns a bunch of sites either copying or paraphrasing Wikipedia. So that's my question: what is his contribution to the theory, and are there any reliable sources explaining it or backing it up?&lt;/p&gt;&#xA;" OwnerUserId="19517" LastActivityDate="2023-12-30T06:54:12.410" Title="What was John Cantius' contribution to the theory of impetus?" Tags="|physics|classical-mechanics|momentum|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16067" PostTypeId="1" AcceptedAnswerId="16068" CreationDate="2023-12-30T13:57:03.643" Score="3" ViewCount="280" Body="&lt;p&gt;The name &amp;quot;Loschmidt echo&amp;quot; is used in quantum physics for the quantity&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA; M(t) \equiv \left|  \langle\psi_0| e^{i H t/\hbar} e^{-i H_0 t/\hbar} |\psi_0\rangle \right|^2&#xA;$$&lt;/span&gt;&#xA;where&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;H = H_0 + \Sigma.&#xA;$$&lt;/span&gt;&#xA;The idea is that &lt;span class=&quot;math-container&quot;&gt;$|\psi_0\rangle$&lt;/span&gt; is the state of some quantum system at time zero and &lt;span class=&quot;math-container&quot;&gt;$\Sigma$&lt;/span&gt; is a small perturbation; the system evolves under &lt;span class=&quot;math-container&quot;&gt;$H_0$&lt;/span&gt; for time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; and we would like to discover the degree to which the result overlaps with evolution under &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;. One can interpret &lt;span class=&quot;math-container&quot;&gt;$M(t)$&lt;/span&gt; also as if the system first evolved under &lt;span class=&quot;math-container&quot;&gt;$H_0$&lt;/span&gt; and then under &lt;span class=&quot;math-container&quot;&gt;$-H$&lt;/span&gt; for a further time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; (or backwards in time under &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt;), and one wishes to know the degree to which &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; 'undoes' the effect of &lt;span class=&quot;math-container&quot;&gt;$H_0$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The name &amp;quot;Loschmidt&amp;quot; is, as I understand it, that of Johann Josef Loschmidt (15 March 1821 – 8 July 1895) who debated with Boltzmann about reversibility and entropy etc. in classical physics, so one can see why &lt;span class=&quot;math-container&quot;&gt;$M(t)$&lt;/span&gt; might have got to be named after him. However in view of the fact that &lt;span class=&quot;math-container&quot;&gt;$M(t)$&lt;/span&gt; as defined above is a thoroughly quantum-mechanical quantity I am not sure if the name is apt, and I wonder whether we might introduce the name Peres since Asher Peres used the quantity in his &lt;em&gt;Stability of quantum motion in chaotic and regular systems&lt;/em&gt;, Phys. Rev. A 30, 1610 (1984). Anyway my question here is if anyone can elucidate the origins of the name &amp;quot;Loschmidt echo&amp;quot;, and comments on the feasibility or appropriateness of introducing another name would also be welcome.&lt;/p&gt;&#xA;" OwnerUserId="19521" LastEditorUserId="19521" LastEditDate="2023-12-30T20:54:50.860" LastActivityDate="2023-12-31T12:29:00.337" Title="Origin of the name &quot;Loschmidt echo&quot; in quantum chaos studies" Tags="|quantum-mechanics|chaos|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16068" PostTypeId="2" ParentId="16067" CreationDate="2023-12-30T21:45:45.973" Score="6" Body="&lt;p&gt;Here I present what I could find about the coinage of the term &amp;quot;Loschmidt echo&amp;quot; (mostly based of two reviews &lt;a href=&quot;https://doi.org/10.1016/j.physrep.2006.09.003&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; and &lt;a href=&quot;http://www.scholarpedia.org/article/Loschmidt_echo&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; and sources therein).&lt;/p&gt;&#xA;&lt;p&gt;Erwin Hahn demonstrated and defined spin echos in the 1950, per &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.80.580&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;. As soon as this happened the discussion of a &amp;quot;Loschmidt paradox&amp;quot; being realizable in the experiments was discussed. See &lt;a href=&quot;https://doi.org/10.1119/1.1934539&quot; rel=&quot;nofollow noreferrer&quot;&gt;4&lt;/a&gt;, which is from 1957, &amp;quot;Loschmidt-Ehrenfest paradox&amp;quot; of &lt;a href=&quot;https://doi.org/10.1016/0003-4916(65)90071-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt; from 1965, and 1970 &amp;quot;Loschmidt demon&amp;quot;  from &lt;a href=&quot;https://doi.org/10.1103/PhysRevLett.25.218&quot; rel=&quot;nofollow noreferrer&quot;&gt;6&lt;/a&gt;. Ref &lt;a href=&quot;https://doi.org/10.1016/0003-4916(65)90071-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;5&lt;/a&gt; also cites &lt;a href=&quot;https://doi.org/10.1143/PTP.22.745&quot; rel=&quot;nofollow noreferrer&quot;&gt;7&lt;/a&gt; for having first made this association in 1959 between spin echoes and thermodynamic paradoxes.&lt;/p&gt;&#xA;&lt;p&gt;However the earliest mention of &amp;quot;Loschmidt echo&amp;quot; I have found online is from Hahn himself in a lecture from 1978, see &lt;a href=&quot;https://doi.org/10.1039/FS9781300007&quot; rel=&quot;nofollow noreferrer&quot;&gt;8&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The path taken in precession phase may be in accord&#xA;with a mirror sequence of pulses, in which case the Loschmidt echo scheme is realized; or the path may be cyclic, which does not retrace the forward evolution of phase, but&#xA;nevertheless returns to the initial point by a different return route.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While this article is not cited, review &lt;a href=&quot;http://www.scholarpedia.org/article/Loschmidt_echo&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; cites &lt;a href=&quot;https://www.jstor.org/stable/24969498&quot; rel=&quot;nofollow noreferrer&quot;&gt;9&lt;/a&gt; to say:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Early on Erwin Hahn recognized that his procedure, which he viewed as a change in the sign of the system Hamiltonian, provided a quantum implementation of the Loschmidt proposal.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hahn in Ref. &lt;a href=&quot;https://www.jstor.org/stable/24969498&quot; rel=&quot;nofollow noreferrer&quot;&gt;9&lt;/a&gt; (1984) does not use the term Loschmidt echo but his realization of the Loschdmidt paradox is explained in more detail.&lt;/p&gt;&#xA;&lt;p&gt;The next textual mention of the &amp;quot;Loschmidt echo&amp;quot; (as far as I can find online) does not appear until the 2001 work of Rodolfo Jalabert and Horace Patawski (&lt;a href=&quot;https://doi.org/10.1103/PhysRevLett.86.2490&quot; rel=&quot;nofollow noreferrer&quot;&gt;10&lt;/a&gt;) who also refer to Ref. &lt;a href=&quot;https://www.jstor.org/stable/24969498&quot; rel=&quot;nofollow noreferrer&quot;&gt;9&lt;/a&gt;. The 2001 publication also clearly defines the formula that you mention. Worth highlighting Asher Peres who also introduced the same formula in 1984, without calling it Loschdmidt echo, in &lt;a href=&quot;https://doi.org/10.1103/PhysRevA.30.1610&quot; rel=&quot;nofollow noreferrer&quot;&gt;11&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Reviews &lt;a href=&quot;https://doi.org/10.1016/j.physrep.2006.09.003&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; and &lt;a href=&quot;http://www.scholarpedia.org/article/Loschmidt_echo&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; suggest that the echo was named after the Loschmidt paradox. Review &lt;a href=&quot;http://www.scholarpedia.org/article/Loschmidt_echo&quot; rel=&quot;nofollow noreferrer&quot;&gt;2&lt;/a&gt; was also written partially by Jalabert and Patawski, if their use of the term was inspired by some other earlier work they missed to indicate it.  The term Loschmidt echo might have existed informally to refer a series of spin echo results until its formalization in 2001.&lt;/p&gt;&#xA;&lt;p&gt;Disclaimer: I will comment very little on the &amp;quot;feasibility&amp;quot; of introducing an alternative name. This site is not the forum for that unless it is already a matter of historical name conflict. Also it clearly does not help with the vast family of spin echo names.&lt;/p&gt;&#xA;&lt;h2&gt;References&lt;/h2&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;Gorin, T., Prosen, T., Seligman, T. H., and Žnidarič, M. (2006). Dynamics of Loschmidt echoes and fidelity decay. Physics Reports, 435(2-5), 33-156. &lt;a href=&quot;https://doi.org/10.1016/j.physrep.2006.09.003&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1016/j.physrep.2006.09.003 &lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Scholarpedia's &lt;a href=&quot;http://www.scholarpedia.org/article/Loschmidt_echo&quot; rel=&quot;nofollow noreferrer&quot;&gt;Loschmidt echo&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hahn, E. L. (1950). Spin echoes. Phys. Rev., 80(4), 580. &lt;a href=&quot;https://journals.aps.org/pr/abstract/10.1103/PhysRev.80.580&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1103/PhysRev.80.580&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rothstein, Jerome. &amp;quot;Nuclear spin echo experiments and the foundations of statistical mechanics.&amp;quot; Am. J. Phys. 25.8 (1957): 510-518. &lt;a href=&quot;https://doi.org/10.1119/1.1934539&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1119/1.1934539&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Baur, M., Jordan, J. R., Jordan, P. C., and Mayer, J. E. (1965). Towards a theory of linear nonequilibrium statistical mechanics. Ann. Phys., 35(1), 96-163. &lt;a href=&quot;https://doi.org/10.1016/0003-4916(65)90071-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1016/0003-4916(65)90071-0&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Rhim, W-K., Alexander Pines, and John S. Waugh. &amp;quot;Violation of the spin-temperature hypothesis.&amp;quot; Phys. Rev. Lett. 25.4 (1970): 218. &lt;a href=&quot;https://doi.org/10.1103/PhysRevLett.25.218&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1103/PhysRevLett.25.218&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Blatt, J. M. (1959). An alternative approach to the ergodic problem. Pro. Theo. Phys., 22(6), 745-756. &lt;a href=&quot;https://doi.org/10.1143/PTP.22.745&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1143/PTP.22.745&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hahn, E., &amp;quot;Pulsed nuclear magnetic resonance in solids. A survey.&amp;quot; Faraday Symposia of the Chemical Society. Vol. 13. Royal Society of Chemistry, 1978. &lt;a href=&quot;https://doi.org/10.1039/FS9781300007&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1039/FS9781300007&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Brewer, R.G., and Hahn, E. L. &amp;quot;Atomic Memory&amp;quot;. Scientific American, Vol. 251(6), pp. 50-57 (1984). (Available in &lt;a href=&quot;https://www.jstor.org/stable/24969498&quot; rel=&quot;nofollow noreferrer&quot;&gt;jstor&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Jalabert, R. A., and Pastawski, H. M. (2001). Environment-independent decoherence rate in classically chaotic systems. Phys. Rev. Lett., 86(12), 2490. &lt;a href=&quot;https://doi.org/10.1103/PhysRevLett.86.2490&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1103/PhysRevLett.86.2490&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Peres, A. (1984). Stability of quantum motion in chaotic and regular systems. Phys. Rev. A, 30(4), 1610. &lt;a href=&quot;https://doi.org/10.1103/PhysRevA.30.1610&quot; rel=&quot;nofollow noreferrer&quot;&gt;doi:10.1103/PhysRevA.30.1610&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2023-12-31T12:29:00.337" LastActivityDate="2023-12-31T12:29:00.337" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16069" PostTypeId="2" ParentId="16067" CreationDate="2023-12-30T22:52:51.307" Score="1" Body="&lt;p&gt;@Mauricio quotes 1978 work mentioning Loschmidt echo. Looks like the use of the term in that work was ignored, as ref. 1 says: &amp;quot;At time &lt;em&gt;2t&lt;/em&gt;, one measures a maximum in the return probability, that we call Loschmidt echo (LE)&amp;quot;, and ref. 2 quotes ref. 1 as the source of the term: &amp;quot;quantity measuring sensitivity to perturbations of the backward evolution is nowadays known as the Loschmidt echo after Ref. [7]&amp;quot; Furthermore, it does not look like the term in 1978 work was applied specifically to the expression of the OP.&lt;/p&gt;&#xA;&lt;p&gt;As for the appropriateness of the term, it looks OK to me, but it's a matter of preference.&lt;/p&gt;&#xA;&lt;p&gt;ref 1:  Jalabert and Pastawski, Phys. Rev. Lett. 86 (2001) 2490-2493&lt;/p&gt;&#xA;&lt;p&gt;ref 2: T. Gorin et al. / Physics Reports 435 (2006) 33 –156&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2023-12-30T22:52:51.307" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16070" PostTypeId="2" ParentId="15061" CreationDate="2024-01-01T23:16:56.800" Score="-6" Body="&lt;p&gt;It seems that &lt;span class=&quot;math-container&quot;&gt;$\frac{a+b^n}{n}=x$&lt;/span&gt; is wrongly attributed to that one that used to prove existence of God by Euler, however there is another formula named &lt;strong&gt;&amp;quot;Euler's identity&amp;quot;&lt;/strong&gt;, this formal is truly mysterious, since it has all essence of mathematical analysis, and the fact that this formula actually exist and exist exactly in this shape is actually mystical.&lt;/p&gt;&#xA;&lt;p&gt;Euler's identity is often cited as an example of deep mathematical beauty. &lt;strong&gt;Three of the basic arithmetic operations occur exactly once each&lt;/strong&gt;: addition, multiplication, and exponentiation. &lt;strong&gt;The identity also links five fundamental mathematical constants&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;e^{i\pi}+1=0&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Galileo once said, “Mathematics is the language with which God wrote the Universe.”&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;However, this formula not just essence of math beauty, but it also connects the beauty of mathemathics with core of natural science, Euler identity is often used to describe the electro-dynamics of harmonical oscillation, for electrical current particularly. Matter of the universe is nothing more but just energy and electro-magnetic interaction, that so easy to describe with essence of pure human logic expressed in God's language of math.&lt;/p&gt;&#xA;&lt;p&gt;Of course any mathematical formula will be pure speculation, and  &lt;strong&gt;strictly following scientific logic to prove existence of God, it will require more solid evidence, that are falsifiable according to Karl Popper&lt;/strong&gt;, if you scientist and philosopher you should know that name.&lt;/p&gt;&#xA;&lt;p&gt;However, &lt;strong&gt;if you are believer, like Euler was&lt;/strong&gt;, who also was one of the greatest mathematician minds by the way, and you believe that this world was created in idealistic way, by some supernatural being, who shared his intelligence and his way of thinking kindly with us, as Bible says that, &lt;strong&gt;God used image of himself to create an Adam&lt;/strong&gt;, first human being. So if you are an intelligent Being and you creating the world, most likely &lt;strong&gt;you will invent algebra and mathematical analysis first&lt;/strong&gt;, and you will invent this formula, and then &lt;strong&gt;you will use this formula and math to describe the matter of surrounding universe&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here few quotes about the beauty of Euler's Identity by recognized professors of mathematics:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Euler%27s_identity&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Euler%27s_identity&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;(rev: &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&amp;amp;oldid=1189364066&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&amp;amp;oldid=1189364066&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Stanford University mathematics professor &lt;strong&gt;Keith Devlin&lt;/strong&gt; has said, &amp;quot;like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence&amp;quot;.[7] And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being &amp;quot;of exquisite beauty&amp;quot;.[8]&lt;/p&gt;&#xA;&lt;p&gt;Mathematics writer &lt;strong&gt;Constance Reid&lt;/strong&gt; has opined that Euler's identity is &amp;quot;the most famous formula in all mathematics&amp;quot;.[9] And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity &amp;quot;is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth&amp;quot;.[10]&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Something I can add from myself, &lt;strong&gt;there is also one psychological mystery connected with this formula&lt;/strong&gt;, &lt;code&gt;identity&lt;/code&gt; as often it is referenced, it is not just mathematical definition, it also has certain psychological definition. &lt;strong&gt;So if you see this beauty in that equation, you identify your self with Euler&lt;/strong&gt;, like professor Keith Devlin and Constance Reid, and because Euler was a really great mathematician mind it is really honor. &lt;strong&gt;In this moment you are sharing the intelligence of the God, that God kindly shared his intelligence with us in the moment of creation&lt;/strong&gt;, this is the place where you can really see that God actually exists.&lt;/p&gt;&#xA;&lt;p&gt;So as you can see here, three things are connected here: &lt;strong&gt;Mathematics(our logic)&lt;/strong&gt;, &lt;strong&gt;Science(our world's matter)&lt;/strong&gt; and &lt;strong&gt;Psychology(our soul)&lt;/strong&gt;. Too many just for one silly formula, right?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Identification_(psychology)&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Identification_(psychology)&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;This way to prove the existence of God, it is not truly scientific, &lt;strong&gt;it is speculative but I think it has right to exist&lt;/strong&gt;. It is giving a hope to all of us, because otherwise it is very sad to be alone without any meaning on this planet lost somewhere in the universe.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;&amp;quot;The most incomprehensible thing about the universe is that it is comprehensible.&amp;quot;&lt;/strong&gt; - Albert Einstein&lt;/p&gt;&#xA;&lt;p&gt;Need to mention that A. Einstein publicly asked not to use any of his quotes to support any religion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Religious_and_philosophical_views_of_Albert_Einstein&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Religious_and_philosophical_views_of_Albert_Einstein&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For all the sceptics, who tend to follow strict science approach, &lt;strong&gt;chinese room is known paradox, that shows weakness of scientific approach, science can't disprove existent of God, the best science can do is to say that there is no scientific tools to answer this question&lt;/strong&gt;. Saying that there is no God, it is not scientific statement, it rather another form of religion and believe in God unexistence. You can believe in existence or you can believe in unexistence of God. However many greatest mind noticed some mystical mystery in this world with unusually strong comprehensiveness property of that world that supposed to be chaotic. And Euler with his identity come very close to idealistic beginning of the world and comprehensiveness.&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;consciousness&amp;quot; have exactly same properties form scientific perspective like &amp;quot;God&amp;quot;, but obviously &amp;quot;consciousness&amp;quot; is real, so strict scientific approach regarding unexistence of God is destroyed.&#xA;&lt;a href=&quot;https://en.wikipedia.org/wiki/Chinese_room&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Chinese_room&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Simply saying, if this world ever had a purpose or this world created by intelligent being - the only evidence that we can find internally, is to see an order and beauty, in that place where we suppose to see a chaos. And obviously &lt;strong&gt;Euler found an order in the place where everyone were expecting to see a chaos&lt;/strong&gt;, maybe it just mind game of our beautiful humans mind, or maybe it really artifact of rational creation. Here is another quote: “is absolutely paradoxical; we cannot understand it, and we don't know what it means” - Benjamin Peirce, I am totally agree with Benjamin Pierce.&lt;/p&gt;&#xA;" OwnerUserId="19528" LastEditorUserId="19528" LastEditDate="2024-01-15T18:46:11.340" LastActivityDate="2024-01-15T18:46:11.340" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16071" PostTypeId="1" CreationDate="2024-01-02T05:16:20.937" Score="1" ViewCount="82" Body="&lt;p&gt;Very briefly, my understanding of the initial motivations for studying &lt;span class=&quot;math-container&quot;&gt;$L^p$&lt;/span&gt; spaces included interest in the &lt;span class=&quot;math-container&quot;&gt;$\ell_2$&lt;/span&gt; space, due to relationships with quadratic forms that arose from searching for coefficients of Fourier series.&lt;/p&gt;&#xA;&lt;p&gt;Then I believe separately the study of heat equations led to integral equations, and Hilbert eventually found that being in &lt;span class=&quot;math-container&quot;&gt;$L^2$&lt;/span&gt; was sufficient for certain solutions to exist.  Hilbert wouldn't have put it that way at the time, but as a rough and very fast description, I think it does the job.&lt;/p&gt;&#xA;&lt;p&gt;In the end, it seems like all the motivations could be traced back to physics problems giving rise to certain mathematical problems, which the mathematicians then took interest in.&lt;/p&gt;&#xA;&lt;p&gt;I wonder if anyone knows a similar story about what people were working on which led to the study of linear functionals and Riesz representation.  For instance, I guess when people understood &lt;span class=&quot;math-container&quot;&gt;$L^2$&lt;/span&gt; well enough they could be guaranteed of the existence of solutions to certain differential equations that they wanted to understand.  So analogous with that, when they found the Riesz representation theorem, did that then deliver on something which had previously motivated the project?&lt;/p&gt;&#xA;" OwnerUserId="2732" LastEditorUserId="2732" LastEditDate="2024-01-02T16:45:46.940" LastActivityDate="2024-01-02T16:45:46.940" Title="What were people looking for when they started to study bounded linear functionals?" Tags="|mathematics|hilbert|functional-analysis|banach|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16072" PostTypeId="1" AcceptedAnswerId="16076" CreationDate="2024-01-02T12:20:39.033" Score="1" ViewCount="127" Body="&lt;p&gt;Eulerian numbers &lt;span class=&quot;math-container&quot;&gt;$A(n, k)$&lt;/span&gt; were discovered by Euler in 1755 in his work &lt;em&gt;Institutiones calculi differentialis.&lt;/em&gt; In combinatorics, they count the number permutations of the numbers 1 to &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; with exactly &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt; ascents.&lt;/p&gt;&#xA;&lt;p&gt;I do not find any further information about the develop of Eulerian numbers from 1755 to now (for example I do not find any information about who discovered the combinatorial intepretation).&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have these informations? Thank you.&lt;/p&gt;&#xA;" OwnerUserId="19397" LastActivityDate="2024-02-03T04:28:35.490" Title="Who discovered the famous combinatorial intepretation of Eulerian numbers?" Tags="|combinatorics|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16073" PostTypeId="1" CreationDate="2024-01-02T17:02:46.590" Score="0" ViewCount="92" Body="&lt;p&gt;Most histories, that I've encountered, of mathematics about the 18th century and onward focus on a chronology of publications, results, definitions, and similar &amp;quot;pure&amp;quot; interests.  However, I am often interested in why these people were asking these particular questions in the first place, and why they thought to try this idea and that idea.&lt;/p&gt;&#xA;&lt;p&gt;Of course this can be hard to discern with the slim writings that we have on the topic -- probably most of this was communicated between mathematicians in un-recorded conversation.  Some of the psychology of these mathematicians may just never have been made public in any way.  But I'm sure there is at least something that can be extracted from the available writing.&lt;/p&gt;&#xA;&lt;p&gt;I'm also sure that some mathematicians had not much more motivation than curiosity.  Still I know that a lot of mathematics was motivated by more specific and physical questions, and I often find those very informative when they exist.&lt;/p&gt;&#xA;&lt;p&gt;Are there books that give a lot of attention to the motivations that mathematicians had for both their questions as well as their methods of solution?  I mostly have in mind the development of analysis, but other fields would also be interesting.&lt;/p&gt;&#xA;" OwnerUserId="2732" LastEditorUserId="2732" LastEditDate="2024-01-03T22:33:14.673" LastActivityDate="2024-01-03T22:33:14.673" Title="The history of motivations" Tags="|mathematics|mathematicians|reference-request|discoveries|" AnswerCount="1" CommentCount="5" ClosedDate="2024-01-05T08:43:49.207" ContentLicense="CC BY-SA 4.0" />
  <row Id="16074" PostTypeId="2" ParentId="16073" CreationDate="2024-01-02T17:52:17.987" Score="2" Body="&lt;p&gt;&amp;quot;The Origins of Cauchy's Rigorous Calculus&amp;quot; by Judith Grabiner discusses some of the motivations of mathematicians involved in the development of calculus, starting in the eighteenth century (with Cauchy's antecedents). She also references other historians with whom she differs. I'm not sure I'd say the book gives a lot of attention to motivations, but Grabiner is interested in why the &amp;quot;rigorous calculus&amp;quot; emerged.&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2024-01-02T17:52:17.987" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16075" PostTypeId="1" CreationDate="2024-01-02T23:42:26.593" Score="1" ViewCount="64" Body="&lt;p&gt;Counting has been in observed in almost every culture across the world. I think it developed because it proved to be so useful for a range of tasks. I am wondering what the earliest use cases for it was though, I imagine it would be recording food stock / weapons in inventory to trade or population. But these are just guesses, I have no idea. All sources I have observed converge to the consensus that tally marking is the oldest known form, but don't go into much detail about what was being counted at that time, and these sources are all free online articles, so not very good sources (not experienced in research).&lt;/p&gt;&#xA;&lt;p&gt;If it's not possible to give a concrete answer due to variability / lack of knowledge, any historical analysis would be appreciated, thank you.&lt;/p&gt;&#xA;" OwnerUserId="19530" LastActivityDate="2024-01-02T23:42:26.593" Title="What is the earliest documentation of counting and what was it used for?" Tags="|mathematics|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16076" PostTypeId="2" ParentId="16072" CreationDate="2024-01-03T05:12:27.623" Score="5" Body="&lt;p&gt;For what it's worth (&lt;a href=&quot;https://www.hindawi.com/journals/jmath/2013/629132/&quot; rel=&quot;noreferrer&quot;&gt;Tingyao Xiong, Hung-Ping Tsao, Jonathan I. Hall, &amp;quot;General Eulerian Numbers and Eulerian Polynomials&amp;quot;, Journal of Mathematics, vol. 2013, Article ID 629132, 9 pages, 2013. https://doi.org/10.1155/2013/629132&lt;/a&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The following combinatorial definition of Eulerian numbers was&#xA;discovered by Riordan in the 1950s.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-01-03T05:12:27.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16077" PostTypeId="1" CreationDate="2024-01-03T23:24:16.180" Score="0" ViewCount="47" Body="&lt;p&gt;Where can I find documentation of fossil findings about ancient human anscestors, even down to the point of believed divergence between great apes and humans? I heard a skeleton has been found earlier than both species, but with large sensory organs and a more ape-like structure, but can't find anything on it.&lt;/p&gt;&#xA;&lt;p&gt;Any resources appreciated, thank you.&lt;/p&gt;&#xA;" OwnerUserId="19530" LastActivityDate="2024-01-03T23:24:16.180" Title="Fossil evidence of common ancestor to Great Apes and Humans" Tags="|biology|evolution|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16080" PostTypeId="2" ParentId="12150" CreationDate="2024-01-04T15:03:41.867" Score="1" Body="&lt;p&gt;So, I'm not a historian of math, but I am a recovering numerical linear algebraist. What Gauss proved, or at least that is true, is that the n-point rule is EXACT for a function that happens to be a polynomial of degree up to &lt;span class=&quot;math-container&quot;&gt;$2 n - 1$&lt;/span&gt;, IIRC. Thus, unless your function has some incredibly high frequency noise, or is outright not differentiable or even continuous, you can usually do just fine IRL with pretty small n. Now you ask, how do you get a PERFECT fit for &lt;span class=&quot;math-container&quot;&gt;$2  n - 1$&lt;/span&gt; polynomials with just n points? Sounds impossible given there are &lt;span class=&quot;math-container&quot;&gt;$2 n - 1$&lt;/span&gt; constraints but n degrees of freedom (optimizing location of points). Well there are n other DOF, namely the sizes of the weights. So, there are &lt;span class=&quot;math-container&quot;&gt;$2  n$&lt;/span&gt; DOF, the same as the number of linearly independent polynomials of degree &lt;span class=&quot;math-container&quot;&gt;$\leq 2  n - 1$&lt;/span&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19535" LastEditorUserId="6609" LastEditDate="2024-01-04T15:22:59.970" LastActivityDate="2024-01-04T15:22:59.970" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16081" PostTypeId="1" CreationDate="2024-01-04T16:38:14.590" Score="1" ViewCount="64" Body="&lt;p&gt;I have had this theory for a while - basic principles in math were likely known by early humans such as caveman (and possibly some level of understanding might even be possessed by animals). I think this is because these basic math principles are &amp;quot;useful&amp;quot; at describing the natural world. &lt;strong&gt;In short (I believe that), completely different lifeforms that live on completely different planets, yet provided these planets share similar laws of physics - these lifeforms will likely encounter and develop similar principles in mathematics.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;For example, consider the triangle inequality:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\|x+y|^2 \leq \|x\|^2 +  \|y\|^2$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Thousands of years before mathematicians like Euclid, I think it's possible that caveman might have had some knowledge and encountered some version of the of the triangle inequality, as it likely came up in daily tasks involving hunting and navigation. As this inequality likely proved to be both accurate and  useful in describing relationships between sets of distances, it likely survived from generation to generation and became engrained in the collective mathematical knowledge of societies. On the other hand, relationships that were not accurate and useful did not survive and became forgotten. While the debate on questions such as &amp;quot;was mathematics discovered or invented?&amp;quot; might never be solved - I think this might be the closest we can get: mathematical ideas that proved to be useful and accurate have higher likelihoods of surviving over time.&lt;/p&gt;&#xA;&lt;p&gt;This brings me to the next part of my theory - why is the triangle inequality true? Does this have anything to do with the geometry/manifold of planet earth? &lt;strong&gt;Are there some geometrical surfaces (e.g. highly irregular, curved, non-smooth, discontinuous and bumpy surfaces) in which the triangle inequality is true?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;As an aside, consider the following idea: Researchers from the domains of linguistics and biology believe that vastly different cultures that live in similar environmental climates have similar language patterns. For example, different cultures in humid climates have been observed to use fewer consonants - as consonants require more energy and stress on the larynx, and humans naturally try to conserve energy in more humid climates (e.g. &lt;a href=&quot;https://www.frontiersin.org/articles/10.3389/fpsyg.2017.01285/full&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.frontiersin.org/articles/10.3389/fpsyg.2017.01285/full&lt;/a&gt;, &lt;a href=&quot;https://academic.oup.com/pnasnexus/article/2/12/pgad384/7457938?login=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://academic.oup.com/pnasnexus/article/2/12/pgad384/7457938?login=false&lt;/a&gt;, &lt;a href=&quot;https://academic.oup.com/jole/article/1/1/33/2281884?login=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://academic.oup.com/jole/article/1/1/33/2281884?login=false&lt;/a&gt;, &lt;a href=&quot;https://www.pnas.org/doi/abs/10.1073/pnas.1417413112&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.pnas.org/doi/abs/10.1073/pnas.1417413112&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Given this idea, I believe the following premise:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Just as different cultures in similar climates eventually stumbled upon similar language patterns ... (I believe that) in any planet where the geometry of the planet allows for the triangle inequality to be true - lifeforms on that planet will eventually encounter and conceive of the triangle inequality.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In a more general sense, planets that share similar laws of physics and chemical compositions will undoubtedly will share some forms of physical commonalities (e.g. behavior of fluids such as evaporation, melting of solid objects, etc.).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Lifeforms on these planets will observe similar natural phenomena and try to create mathematical, logical, scientific paradigms/models to describe these natural processes (e.g. mathematical relationships between volumes of liquids, their boiling temperature, pressure and altitude i.e. &lt;a href=&quot;https://en.wikipedia.org/wiki/Triple_point&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Triple_point&lt;/a&gt;, &lt;a href=&quot;https://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation&lt;/a&gt;) in their environment - and these logical/scientific models from different lifeforms will likely be similar to one another conditional on the fact that their environments are similar.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Thus, these lifeforms will inevitably stumble upon and encounter the same math principles due to the properties and constraints of their physical environments.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So going back to my original question: &lt;strong&gt;Are there some geometrical surfaces (e.g. highly irregular, curved, non-smooth, discontinuous and bumpy surfaces) in which the triangle inequality is true?&lt;/strong&gt; I conjure that lifeforms on such planets will discover and use different geometrical identities to describe relationships between sets of distances that are more accurate and useful on their planets.&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;References:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathoverflow.net/questions/23113/is-there-any-geometry-where-the-triangle-inequality-fails&quot;&gt;https://mathoverflow.net/questions/23113/is-there-any-geometry-where-the-triangle-inequality-fails&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="16167" LastActivityDate="2024-01-04T16:38:14.590" Title="How does Math look on Different Planets?" Tags="|mathematics|" AnswerCount="0" CommentCount="2" ClosedDate="2024-01-05T02:10:38.410" ContentLicense="CC BY-SA 4.0" />
  <row Id="16082" PostTypeId="2" ParentId="12150" CreationDate="2024-01-04T18:42:18.620" Score="3" Body="&lt;p&gt;Here is the assessment of &lt;a href=&quot;https://doi.org/10.1137/060659831&quot; rel=&quot;nofollow noreferrer&quot;&gt;Trefethen, Is Gauss Quadrature Better than Clenshaw–Curtis?, 2008&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss formulas are defined by choosing the nodes optimally in the sense of maximizing the degree of polynomials that [the Gaussian quadrature formula] integrates exactly. Since there are n + 1 nodes, it is not surprising that the attainable degree is n + 1 orders higher than in the generic case, i.e., degree 2n + 1. Gauss quadrature formulas were discovered by Gauss in his mid-thirties [25], and it is a measure of the extraordinary productivity of Gauss’s career that this great discovery merits hardly a mention in many accounts of his life. For these formulas are truly powerful. Their impact was mainly theoretical before the advent of computers, because of the difficulty of determining the nodes and weights and also of evaluating integrands at irrational arguments, but in the computer era, especially since the appearance of a paper by Golub and Welsch in 1969 (following earlier work of Goertzel in 1954, Wilf in 1962, and Gordon in 1968), it has been practical as well.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another way of looking at the &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;-point Gauss-Legendre rule is that it integrates the first 2n terms of the Taylor expansion of an analytic function exactly and approximates the rest. Gauss's objective was the highest degree of approximation (as in terms up to &lt;span class=&quot;math-container&quot;&gt;$x^{2n-1}$&lt;/span&gt;), but it turns out to approximate the remainder the Taylor series of a function analytic in a neighborhood of &lt;span class=&quot;math-container&quot;&gt;$[-1,1]$&lt;/span&gt; well and one with an infinite radius of convergence extremely well.&lt;/p&gt;&#xA;&lt;p&gt;The theoretical importance of Gaussian quadrature received a boost by &lt;a href=&quot;https://eudml.org/doc/183033&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jacobi 1826&lt;/a&gt; and others, who connected Gaussian quadrature to orthogonal polynomials, and by the subsequent development of spectral methods.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;...perhaps in all of numerical analysis.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Since the full significance of Gaussian integration arrives in the 1960s by Trefethen's account, I cannot resist mentioning another major development in the 1960s that has origins in the work of Gauss: The FFT (&lt;a href=&quot;https://doi.org/10.2307%2F2003354&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cooley and Tukey, An algorithm for the machine calculation of complex Fourier series, 1965&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="1815" LastActivityDate="2024-01-04T18:42:18.620" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16083" PostTypeId="1" AcceptedAnswerId="16084" CreationDate="2024-01-05T05:32:01.357" Score="0" ViewCount="175" Body="&lt;p&gt;The &lt;a href=&quot;https://pt.m.wikipedia.org/wiki/Teorema_de_Burlet?fbclid=IwAR27ynAJMgfKKIoUFr4ojMRHaO9JCtVpGptX5aDQ9LDIblF8crdgXFlTpSw&quot; rel=&quot;nofollow noreferrer&quot;&gt;Burlet's theorem&lt;/a&gt; is a result in Euclidean geometry, which can be formulated as follows:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Theorem&lt;/strong&gt;. Consider triangle &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt; with &lt;span class=&quot;math-container&quot;&gt;$\angle{C}=\gamma$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; be the point where the incircle touches side &lt;span class=&quot;math-container&quot;&gt;$AB$&lt;/span&gt;, and denote the lengths &lt;span class=&quot;math-container&quot;&gt;$AP$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$BP$&lt;/span&gt; as &lt;span class=&quot;math-container&quot;&gt;$m$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;, respectively. Then&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\Delta=mn\cot{\frac{\gamma}{2}},$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$\Delta$&lt;/span&gt; denotes the area of &lt;span class=&quot;math-container&quot;&gt;$ABC$&lt;/span&gt;. Note that when &lt;span class=&quot;math-container&quot;&gt;$\gamma=\frac{\pi}{2}$&lt;/span&gt;, then &lt;span class=&quot;math-container&quot;&gt;$\Delta=mn$&lt;/span&gt; (which is more well-known).&lt;/p&gt;&#xA;&lt;p&gt;A proof and an extension to cyclic quadrilateral can be found &lt;a href=&quot;https://geometriadominicana.blogspot.com/2024/01/a-generalization-of-burlets-theorem-to.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;So, who was Burlet really? It seems that in Latin America, this theorem is known by this name, but no one really knows who he was. I have found some references to Oscar Burlet &lt;a href=&quot;https://en.wikipedia.org/wiki/Hassler_Whitney#Death&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Extras/De_Rham_mountaineering/?fbclid=IwAR25y_pOoEEWtjyhivxe6PbzKtpqS6uj_U2hzW3xqHjzdVsoLpuURpUteIo&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and &lt;a href=&quot;https://www.genealogy.math.ndsu.nodak.edu/id.php?id=104035&amp;amp;fbclid=IwAR30pdjB3SaD2JqO-Ptm-dQvdU90RhU2nFGM8rkCTwaY5Z8_cnhLyWtEOz0&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. Could it be the same person?&lt;/p&gt;&#xA;" OwnerUserId="16769" LastEditorUserId="16769" LastEditDate="2024-01-05T06:22:32.963" LastActivityDate="2024-01-05T17:43:56.190" Title="Who was Burlet, the one from Burlet's theorem?" Tags="|euclidean-geometry|trigonometry|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16084" PostTypeId="2" ParentId="16083" CreationDate="2024-01-05T17:43:56.190" Score="6" Body="&lt;p&gt;I am unable to find &lt;em&gt;any&lt;/em&gt; references to a &amp;quot;theorem of Burlet&amp;quot; in a print publication regardless of language, e.g. Satz von Burlet, théorème de Burlet. All online resources referencing this term are in Portuguese and are recent. For example, the entry in the Portuguese Wikipedia linked in the question dates to 2023.&lt;/p&gt;&#xA;&lt;p&gt;I performed an internet search for &amp;quot;Burlet&amp;quot; in conjuction with &amp;quot;triangle&amp;quot; and &amp;quot;inscribed circle&amp;quot; in multiple languages which yielded a relevant &lt;em&gt;question&lt;/em&gt; by an A. Burlet from Dublin:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Nouvelles annales de mathématiques&lt;/em&gt;, Vol. 15, 1856, &lt;a href=&quot;https://www.google.com/books/edition/Nouvelles_annales_de_math%C3%A9matiques/RxUFAAAAQAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA290&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;p. 290&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Questions. [...] 336. Un triangle rectangle est équivalent au rectangle des deux segments faits sur l'hypoténuse par le point de contact du cercle inscrit. (A. BURLET de Dublin)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Translation: &amp;quot;A right triangle is equivalent to the rectangle of the two segments made on the hypotenuse by the point of contact of the inscribed circle.&amp;quot; A solution to the problem posed by Burlet appeared in the same publication the following year:&lt;/p&gt;&#xA;&lt;p&gt;P. H. Rochette, S.J., &amp;quot;Solution de la Question 336&amp;quot;, &lt;em&gt;Nouvelles annales de mathématiques&lt;/em&gt;, Vol. 16, 1857, &lt;a href=&quot;https://www.google.com/books/edition/Nouvelles_annales_de_math%C3%A9matiques_jour/jco9AQAAMAAJ?hl=en&amp;amp;gbpv=1&amp;amp;pg=PA43&amp;amp;printsec=frontcover&quot; rel=&quot;noreferrer&quot;&gt;pp. 43-44&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I am unable to find further information on A. Burlet in other publications of the time. Burlet &lt;em&gt;may&lt;/em&gt; have been a teacher or amateur mathematician. Presumably Dublin refers to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Dublin&quot; rel=&quot;noreferrer&quot;&gt;Irish city&lt;/a&gt; but we cannot be entirely sure.&lt;/p&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2024-01-05T17:43:56.190" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16085" PostTypeId="1" AcceptedAnswerId="16090" CreationDate="2024-01-06T03:52:20.167" Score="0" ViewCount="271" Body="&lt;p&gt;I know how to define addition and multiplication of natural numbers using set theory, but I think that before Cantor, mathematicians did not try to use set theory as a foundation for mathematics (I could be wrong). How did mathematicians define addition and multiplication of natural numbers before Cantor and Peano?&lt;/p&gt;&#xA;&lt;p&gt;I chose mathematics before 1870 as an arbitrary cutoff, after Cauchy’s death and before Cantor and “modern” set theory, since rigorous mathematics started with Cauchy (I think). This era should have some somewhat rigorous definitions of addition and multiplication.&lt;/p&gt;&#xA;&lt;p&gt;I also want to know how addition and multiplication were defined in Newton’s or ancient times (if there exist a definition).&lt;/p&gt;&#xA;" OwnerUserId="18466" LastEditorUserId="18466" LastEditDate="2024-01-09T06:27:37.663" LastActivityDate="2024-01-12T14:32:09.757" Title="How was addition and multiplication of natural numbers defined before 1870 (Cantor and modern set theory)?" Tags="|mathematics|numbers|arithmetic|" AnswerCount="2" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16086" PostTypeId="2" ParentId="343" CreationDate="2024-01-06T12:57:48.237" Score="1" Body="&lt;p&gt;Not disagreeing with references to Kepler, Galileo and Hooke. Also Boyle. However, there are also contemporary references to the voyages of European discovery, that first gave lie to classical ideas and methods of proceeding. Bacon acknowledges that debt as does Galileo. Even Copernicus references the new image of the Earth. The phrase &amp;quot;Terraqueous Earth&amp;quot; originates in 1646 and William Watts, in 1663, wrote &amp;quot; .... the thoughts of the philosophers have been contradicted by the unexpected observations of the navigators.&amp;quot; I don't think Newton was actually referring to them, but it does give a pleasant meaning to his sense of &amp;quot;seeing further.&amp;quot;&lt;/p&gt;&#xA;" OwnerUserId="19543" LastActivityDate="2024-01-06T12:57:48.237" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16087" PostTypeId="1" CreationDate="2024-01-06T18:49:07.913" Score="0" ViewCount="147" Body="&lt;p&gt;I'm looking for an instance other than that of the 'big bang' — a term coined by Fred Hoyle to reject the theory being referred to. The theory also later went on to become the consensus with the very same name.&lt;/p&gt;&#xA;&lt;p&gt;Symbolically put:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Person A coins Term X to reject Theory Y.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Over time, Theory Y becomes the consensus.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Term X, initially coined to reject Theory Y, becomes the most common name for the now-consensus Theory Y.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="19544" LastActivityDate="2024-01-06T18:49:07.913" Title="Instances in which a term initially coined to reject a theory later became the widely accepted term for that very theory after it became the consensus" Tags="|terminology|cosmology|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16088" PostTypeId="1" CreationDate="2024-01-07T13:55:30.950" Score="0" ViewCount="51" Body="&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\pi(x)$&lt;/span&gt; be the prime counting function. It is well-known that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\lim_{x\to\infty}\frac{\pi(x)}{x}=0.$$&lt;/span&gt;&#xA;My question is who is the first person to prove this result?&#xA;Did Eratosthenes prove this theorem for the first time?&lt;/p&gt;&#xA;" OwnerUserId="18815" LastActivityDate="2024-01-07T13:55:30.950" Title="History of prime distribution $\lim_{x\to\infty}\frac{\pi(x)}{x}=0$" Tags="|number-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16089" PostTypeId="1" AcceptedAnswerId="16094" CreationDate="2024-01-07T15:04:53.980" Score="8" ViewCount="1035" Body="&lt;p&gt;I am trying to track down at what point mathematicians started to use the terminology of expanding a function &amp;quot;around a point&amp;quot; or in the &amp;quot;neighbourhood of a point&amp;quot;. Neither Taylor nor Maclaurin would appear to use such terminology, and while I would have guessed that Weierstrass would be using it, I have yet to see a publication of his where that is the case.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone happen to know?&lt;/p&gt;&#xA;" OwnerUserId="15726" LastEditorUserId="55" LastEditDate="2024-01-08T03:26:01.700" LastActivityDate="2024-01-16T14:45:43.387" Title="When did &quot;neighbourhood of a point&quot; first appear in the history of Taylor series?" Tags="|mathematics|calculus|real-analysis|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16090" PostTypeId="2" ParentId="16085" CreationDate="2024-01-07T16:18:46.783" Score="2" Body="&lt;p&gt;In the first appendix (Note I) to his 1821 book &lt;em&gt;Analyse Algebrique&lt;/em&gt;, Cauchy discusses multiplication of signs in detail, and then gives the following definition of addition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ajouter au&#xA;nombre A le nombre B, ou, en d'autres termes, faire&#xA;subir au nombre A 1'accroissement + B ; c'est ce qu'on&#xA;appelle faire une addition arithmetique. Le resultat de&#xA;cette operation s'appelle somme.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A few pages later, he defines multiplication as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Multiplier le&#xA;nombre A par Ie nombre B, c'est operer sur Ie nombre&#xA;A precisement comme on opere sur l'unite pour obtenir&#xA;B. Le resultat de cette operation est ce qu'on appelle Ie&#xA;produit de A par B.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastActivityDate="2024-01-07T16:18:46.783" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16091" PostTypeId="1" AcceptedAnswerId="16104" CreationDate="2024-01-08T00:50:38.093" Score="3" ViewCount="101" Body="&lt;p&gt;I can not identify clearly who was the first one to realize that the electromagnetic tensor is the curvature 2-form of a U(1)-connection. Looking at Weyl's work, it seems that he came pretty close to it in 1929 but I can not find any clear indication that he realized that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; was a connection and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; its curvature. I can not see any clear statement about this until Wu and Yang's paper in 1975. Could anyone help me?&lt;/p&gt;&#xA;" OwnerUserId="19547" LastEditorUserId="19547" LastEditDate="2024-01-08T16:23:10.680" LastActivityDate="2024-01-10T12:45:32.270" Title="Who discovered that the electromagnetic tensor is the curvature of a connection?" Tags="|physics|differential-geometry|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16092" PostTypeId="2" ParentId="16089" CreationDate="2024-01-08T03:23:40.593" Score="5" Body="&lt;p&gt;According to &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Miller/mathword/n/&quot; rel=&quot;noreferrer&quot;&gt;Miller's Earliest Known Uses&lt;/a&gt;, &amp;quot;neighborhood of a point&amp;quot; first appears in Cayley's 1877 paper (in a different context), and then in &lt;a href=&quot;https://archive.org/details/introductiontost00harnrich/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;Harnack's calculus text&lt;/a&gt; (1891), where it is used extensively throughout in connection with continuity, partial derivatives, regularity (holomorphy), series convergence, etc. When it comes to the Taylor series in chapter 2 of book IV, we read passages like:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/introductiontost00harnrich/page/346/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;§187&lt;/a&gt; &amp;quot;&lt;em&gt;If  we  know  that  a  function  loses  the  property  of  a  unique analytic  function  at  certain  points &lt;span class=&quot;math-container&quot;&gt;$c_1,c_2\dots$&lt;/span&gt; of the plane, and  we desire  &lt;strong&gt;to  expand  this  function  in  a  series  of  powers  in  the  neighbourhood&#xA;of  a  point&lt;/strong&gt;  &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;,  the  circle  of  convergence  may  only  be  so  large  as  to  pass&#xA;through  one  or  more  of  the  points  &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;,  but  must  include  none  of  them. For,  the  series  of  powers  is  a  unique  analytic  function  in  the  entire circle,  while  by  hypothesis  the  function  loses  this  property  in  each  point  &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://archive.org/details/introductiontost00harnrich/page/350/mode/2up?view=theater&quot; rel=&quot;noreferrer&quot;&gt;§189&lt;/a&gt; &amp;quot;&lt;em&gt;Having  established  that  analytic  functions  can  be  expressed by  series  of  ascending  positive  integer  powers,  it  is  still  necessary  that we  should  discuss  their  singular  points,  in  order  that  we  may  ascertain how  analytic  functions  admit  of  &lt;strong&gt;expansion  in  the  neighbourhood  of&#xA;any  singular  point&lt;/strong&gt;  according  to  its  kind.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-01-08T03:23:40.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16093" PostTypeId="2" ParentId="343" CreationDate="2024-01-08T08:33:13.430" Score="1" Body="&lt;p&gt;There will have been multiple shoulders that Newton stood on. Two are Plato and Aristotle since in his philosophical notebooks he specifically mentions them:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Plato is a friend, Aristotle is a friend but truth is a greater friend.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2024-01-08T08:33:13.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16094" PostTypeId="2" ParentId="16089" CreationDate="2024-01-08T09:36:26.560" Score="12" Body="&lt;p&gt;The terminology &amp;quot;neighbourhood of a point&amp;quot; (in German, &amp;quot;Umgebung einer bestimmten Stelle&amp;quot;) with its current meaning, dates back at least to 1841 when Weierstrass wrote his &lt;em&gt;Zur Theorie der Potenzreihen&lt;/em&gt; (&amp;quot;On the theory of power series&amp;quot;), published in &lt;a href=&quot;https://archive.org/details/mathematischewer01weieuoft/page/66/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mathematische Werke, Band I, S. 67-74&lt;/a&gt;.&#xA;The definition of neighbourhood of a point in &lt;span class=&quot;math-container&quot;&gt;$\mathbb{R}^n$&lt;/span&gt; is in the footnote at page 70 and is used throughout the paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Unter Umgebung einer bestimmten Stelle &lt;span class=&quot;math-container&quot;&gt;$(a_1,a_2,...,a_\rho)$&lt;/span&gt; im Gebiete von &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; Veränderlichen &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,...,x_\rho$&lt;/span&gt; ist ein Bereich von folgender Beschaffenheit zu verstehen: Wenn &lt;span class=&quot;math-container&quot;&gt;$(x'_1,x'_2,...,x'_\rho)$&lt;/span&gt; irgend eine Stelle des Bereiches ist, so gilt dasselbe von jeder Stelle &lt;span class=&quot;math-container&quot;&gt;$(x''_1,x''_2,...,x''_\rho)$&lt;/span&gt;, die den Bedingungen &lt;span class=&quot;math-container&quot;&gt;$|x''_1-a_1|\leq |x'_1-a_1|$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|x''_2-a_2|\leq |x'_2-a_2|$&lt;/span&gt;, ..., &lt;span class=&quot;math-container&quot;&gt;$|x''_\rho-a_\rho|\leq |x'_\rho-a_\rho|$&lt;/span&gt; entspricht. Ein solcher Bereich ist z. B. der Convergenzbezirk eder gewöhnlichen Potenzreihe von &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,...,x_\rho$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The neighbourhood of a particular point &lt;span class=&quot;math-container&quot;&gt;$(a_1,a_2,...,a_\rho)$&lt;/span&gt; in the domain of &lt;span class=&quot;math-container&quot;&gt;$\rho$&lt;/span&gt; variables &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,...,x_\rho$&lt;/span&gt; is to be understood as a domain of the following nature: if &lt;span class=&quot;math-container&quot;&gt;$(x'_1,x'_2,...,x'_\rho)$&lt;/span&gt; is a point of the domain, then the same is true of any point &lt;span class=&quot;math-container&quot;&gt;$(x''_1,x''_2,... ,x''_\rho)$&lt;/span&gt; such that &lt;span class=&quot;math-container&quot;&gt;$|x''_1-a_1|\leq |x'_1-a_1|$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$|x''_2-a_2|\leq |x'_2-a_2|$&lt;/span&gt;, ..., &lt;span class=&quot;math-container&quot;&gt;$|x''_\rho-a_\rho|\leq |x'_\rho-a_\rho|$&lt;/span&gt;. Such a region is, for example, the convergence region of the ordinary power series of &lt;span class=&quot;math-container&quot;&gt;$x_1,x_2,...,x_\rho$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Harnack was well-aware of the work of Weierstrass, and in fact at page 130 he cites in footnote &lt;em&gt;&lt;a href=&quot;https://ia902807.us.archive.org/19/items/mathematischewer02weieuoft/mathematischewer02weieuoft.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Theorie der eindeutigen analytischen Functionen&lt;/a&gt;&lt;/em&gt;  (1876) where one can read&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Ich will von einer eindeutigen analytischen Function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; der complexen&#xA;Veränderlichen &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; sagen, sie verhalte sich regulär in der Umgebung einer&#xA;hestimmten Stelle (&lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;), wenn sie innerhalb eines gewissen Bezirks[*], dessen Mittelpunkt &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; ist, überall einen endlichen und mit &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; stetig sich ändernden Werth hat.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I say that an analytic function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; of the complex variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; is regular in the neighbourhood of a point (&lt;span class=&quot;math-container&quot;&gt;$x = a$&lt;/span&gt;) if it has a finite value everywhere in a circle whose centre is &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and which varies continuously with &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Probably Harnack read the &lt;a href=&quot;https://www.google.it/books/edition/Annales_Scientifiques_de_L_%C3%89cole_Normal/5yBLAAAAYAAJ?hl=it&amp;amp;gbpv=1&amp;amp;pg=PA111&amp;amp;printsec=frontcover&quot; rel=&quot;nofollow noreferrer&quot;&gt;French translation&lt;/a&gt;, which he cites at page 148:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the French Translation by E. Picard of the above Memoir of Weierstrass, which appeared under the revision of the Author in the Annales de l'Ecole Normale, 2e Serie, T. VIII, 1879, entitled: &amp;quot;Memoire sur les fonctions analytiques uniformes&amp;quot;, the opening statement reads as follows:&lt;/p&gt;&#xA;&lt;p&gt;Among unique (uniformes) functions of a single variable, rational functions form a distinct class which we proceed to define by their characteristic property.&lt;/p&gt;&#xA;&lt;p&gt;We shall say that a unique function &lt;span class=&quot;math-container&quot;&gt;$f(z)$&lt;/span&gt; of the complex variable &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; is regular in the neighbourhood of a point &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, when for all values of &lt;span class=&quot;math-container&quot;&gt;$z$&lt;/span&gt; comprised within a circle having its centre at &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and a radius sufficiently small, the function can be developed in a series of the form [...]. In case the point a were at infinity we should replace &lt;span class=&quot;math-container&quot;&gt;$z-\infty$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{z}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Every point &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; in whose neighbourhood the function &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is not regular will be called a singular point [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Back to Weierstrass: he used the terminology &amp;quot;Umgebung einer bestimmten Stelle&amp;quot; throughout his work, and the term &lt;em&gt;Umgebung&lt;/em&gt; appears at lest 59 times in the first volume of his &lt;em&gt;Werke&lt;/em&gt; and 106 in the second one. Moreover, the terminology was in use in the German mathematical community, for example &lt;em&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/File:%C3%9Cber_die_Integration_der_linearen_Differentialgleichungen_durch_Reihen.djvu&quot; rel=&quot;nofollow noreferrer&quot;&gt;Über die Integration der linearen Differentialgleichungen durch Reihen&lt;/a&gt;&lt;/em&gt; (1873) start with the sentence&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wenn alle Integrale einer homogenen linearen Differentialg &lt;span class=&quot;math-container&quot;&gt;$\lambda^{\text{ter}}$&lt;/span&gt; Ordnung in der Umgebung einer bestimmten  Stelle,[...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If all integrals of a homogeneous linear differential of order &lt;span class=&quot;math-container&quot;&gt;$\lambda^{\text{ter}}$&lt;/span&gt; in the neighbourhood of a certain point, [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Anyway, the idea (but not the definition) of neighbourhood predates Weierstrass, and as TonioElGringo says, it is present also in Cauchy. For example in the &lt;em&gt;Cours d'analyse de l'École royale polytechnique&lt;/em&gt; (1821) one can find (&lt;a href=&quot;https://archive.org/details/coursdanalysede00caucgoog/page/34/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 35&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On dit encore que la fonction &lt;span class=&quot;math-container&quot;&gt;$f(y)$&lt;/span&gt; est, dans le voisinage d'une valeur[**] particulière, attribuée à la variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, fonction continue de cette variables, toutes les fois quelle est continue entre deux limites de &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, même très-rapprochées, qui renferment la valeur dont il s'agit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We also say that the function &lt;span class=&quot;math-container&quot;&gt;$f(y)$&lt;/span&gt; is, in the vicinity of a particular value attributed to the variable &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, a continuous function of this variable, whenever it is continuous between two limits of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt;, even very close together, which contain the value in question.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In my opinion, there is a fundamental difference with respect to Weierstrass' use of it: in Cauchy the idea of neighbourhood is intuitive and has a vaguely geometric meaning (a sort of unspecified proximity), Weierstrass on the other hand gives a formal definition of it (again based on a geometric idea, of course) and which in some respects prefigures the topological definition.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;[*]:  Weierstrass uses the word &amp;quot;Bezirk&amp;quot;, which nowadays means normally &amp;quot;district&amp;quot; (I believe), but actually comes from the Latin &lt;em&gt;circulus&lt;/em&gt;, &amp;quot;circle&amp;quot;, while for the geometric figure one normally uses &amp;quot;Kreis&amp;quot; (from Proto-Germanic), even it is also used to denote a particular administrative division.&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;[**]: Also notice that Cauchy always (not only in the &lt;em&gt;Cours d'Analyse&lt;/em&gt;) says &amp;quot;voisinage d'une &lt;em&gt;valeur&lt;/em&gt;&amp;quot; (neighbourhood of a &lt;em&gt;value&lt;/em&gt;) and never &lt;em&gt;of a point&lt;/em&gt;.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2024-01-16T14:45:43.387" LastActivityDate="2024-01-16T14:45:43.387" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16095" PostTypeId="1" CreationDate="2024-01-08T21:32:55.320" Score="1" ViewCount="36" Body="&lt;p&gt;I have read that Jordan first defined the concept of variation and studied functions of bounded variation, in his 1881 publication &lt;em&gt;Sur la Serie de Fourier&lt;/em&gt;, as referenced here:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Camille_Jordan&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Camille_Jordan&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Can anyone say what caused Jordan to create and investigate this concept?  Of course it must have come to him while thinking about Fourier series, but is there a particular Fourier series (or class of Fourier series, or functions, or any other specific objects of interest) that might have inspired this definition?  Is there some object which, because of this definition and analysis, he was then able to understand better?&lt;/p&gt;&#xA;&lt;p&gt;(I haven't found a translation of this article into English, so I'm not sure if it already contains the answer to the question.)&lt;/p&gt;&#xA;" OwnerUserId="2732" LastActivityDate="2024-01-08T21:32:55.320" Title="History of functions of bounded variation" Tags="|real-analysis|functional-analysis|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16096" PostTypeId="1" AcceptedAnswerId="16097" CreationDate="2024-01-09T00:47:32.070" Score="3" ViewCount="185" Body="&lt;p&gt;There is a learning technique by Richard Feynman, which is known as the &amp;quot;Feynman learning technique&amp;quot;, and consists of four steps:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Select a concept to learn.&lt;/li&gt;&#xA;&lt;li&gt;Teach it to a child (or maybe an adult who has zero knowledge in that field).&lt;/li&gt;&#xA;&lt;li&gt;Identify gaps in your explanation. Go back to the source material to better understand it.&lt;/li&gt;&#xA;&lt;li&gt;Organize and simplify.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;You can find its description, for example, &lt;a href=&quot;https://fs.blog/feynman-learning-technique/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; or &lt;a href=&quot;https://www.colorado.edu/artssciences-advising/resource-library/life-skills/the-feynman-technique-in-academic-coaching&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;But where can I find its original description, that is, the description provided by Richard Feynman himself?&lt;/p&gt;&#xA;" OwnerUserId="19553" LastEditorUserId="12330" LastEditDate="2024-01-10T12:31:42.513" LastActivityDate="2024-01-10T12:31:42.513" Title="Feynman learning technique" Tags="|feynman|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16097" PostTypeId="2" ParentId="16096" CreationDate="2024-01-09T10:34:45.833" Score="3" Body="&lt;h2&gt;Some Feynman's quotes about learning an teaching&lt;/h2&gt;&#xA;&lt;p&gt;I do not think Feynman's learning technique is found in his video lectures or in &lt;em&gt;Surely you're joking Mr. Feynman&lt;/em&gt; (maybe loosely based on his remarks when he was teaching in Brazil?). In those sources he rather explain what are his steps to follow the scientific method. Watch Cornell lecture &lt;em&gt;&lt;a href=&quot;https://www.youtube.com/watch?v=-2NnquxdWFk&amp;amp;t=1006s&amp;amp;ab_channel=TalkReelTV&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Character of Physical Law&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Some academic sources cite &lt;em&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Feynman%27s_Lost_Lecture&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feynman's Lost Lecture&lt;/a&gt;&lt;/em&gt; for his technique, but I could not find it there. In the preface by by David L. and Judith R. Goodstein, they write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Feynman was a truly great teacher. He prided himself on being able to devise ways to explain even rhe most profound ideas to beginning students. Once, I said to him, &amp;quot;Dick, explain to me, so that I can understand it, why spin one-half particles obey Fermi-Dirac statistics.&amp;quot; Sizing up his audience perfectly, Feynman said, &amp;quot;I'll prepare a freshman lecture on it.&amp;quot; But he came back a few days later to say, &amp;quot;I couldn't do it. I couldn't reduce it to the freshman level. That means we don't really understand it.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Which shows his methodology, but it is still not very precise. There is no single recognized or clear source in academic sources so the method might have been invented by somebody else.&lt;/p&gt;&#xA;&lt;p&gt;It is also interesting his account in Feynman's &lt;a href=&quot;https://www.aip.org/history-programs/niels-bohr-library/oral-histories/5020-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;1973 interview with Charles Wiener&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;You see I didn't know how much was known and how much was not known so it was more a matter of learning. For me to learn, I read a little bit and I work out the rest. I don’t really read a lot of what the other guy does. I read what his assumptions are and if they seem reasonable then I work out the conclusions.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another interesting teach comes from his 1966 &lt;em&gt;&lt;a href=&quot;http://www.feynman.com/science/what-is-science/&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is Science?&lt;/a&gt;&lt;/em&gt; lecture&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I finally figured out a way to test whether you have taught an idea or you have only taught a definition.&lt;/p&gt;&#xA;&lt;p&gt;Test it this way: you say, “Without using the new word which you have just learned, try to rephrase what you have just learned in your own language.” Without using the word “energy,” tell me what you know now about the dog’s motion.” You cannot. So you learned nothing about science.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;Genius biography and its self-help book origin&lt;/h2&gt;&#xA;&lt;p&gt;The closest I have found so far is from the biography &lt;em&gt;Genius: The Life and Science of Richard Feynman&lt;/em&gt; by James Gleick:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In preparing for his oral qualifying examination, a rite of passage for every graduate student, he chose not to study the outlines of known physics. Instead he went up to MIT, where he could be alone, and opened a fresh notebook. On the title page he wrote: &lt;em&gt;Notebook Of Things I Don’t Know About&lt;/em&gt;. For the first but&#xA;not the last time he reorganized his knowledge. He worked for weeks at disassembling each branch of physics, oiling the parts, and putting them back together, looking all the while for the raw edges and inconsistencies.&#xA;He tried to find the essential kernels of each subject. When he was done he had a notebook of which he was especially proud. It was not much use in preparing for the examination, as it turned out.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is another passage in the book where Gleick cites a possibly apocryphal story which is even closer (but I do not understand the nature of this source):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;A physicist studying quantum field theory with Murray Gell-Mann at the California Institute of Technology in the 1950s, before standard texts have become available, discovers unpublished lecture notes by Richard Feynman, circulating samizdat style. He asks Gell-Mann about them.Gell-Mann says no, Dick’s methods are not the same as the methods used here. The student asks, well, what are Feynman’s methods? GellMann leans coyly against the blackboard and says, Dick’s method is this. &lt;strong&gt;You write down the problem. You think very hard.&lt;/strong&gt; (He shuts his eyes and presses his knuckles parodically to his forehead.) &lt;strong&gt;Then you write down the answer.&lt;/strong&gt;&lt;/em&gt;&#xA;However is not a good learning method.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This quote appeared also in: J. Gleick, &lt;em&gt;&lt;a href=&quot;https://www.nytimes.com/1992/09/20/magazine/part-showman-all-genius.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Part Showman, All Genius&lt;/a&gt;&lt;/em&gt;, New York Times (1992)&lt;/p&gt;&#xA;&lt;p&gt;The earliest version I could find of the full Feynman's technique in the form you present is from Scott Young, a self-help book author, from 2011: &lt;a href=&quot;https://www.scotthyoung.com/learnonsteroids/grab/TranscriptFeynman.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.scotthyoung.com/learnonsteroids/grab/TranscriptFeynman.pdf&lt;/a&gt; before that &lt;em&gt;Feynman's technique&lt;/em&gt; referred to Feynman's diagrams. A hint of that was given in another SE site: &lt;a href=&quot;https://psychology.stackexchange.com/a/30138&quot;&gt;https://psychology.stackexchange.com/a/30138&lt;/a&gt; .&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-10T08:07:36.007" LastActivityDate="2024-01-10T08:07:36.007" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16098" PostTypeId="1" AcceptedAnswerId="16100" CreationDate="2024-01-09T12:13:05.740" Score="3" ViewCount="1842" Body="&lt;p&gt;Wolfgang Rautenberg wrote in &amp;quot;A Concise Introduction to Mathematical Logic&amp;quot; that Kurt Gödel, to show that finite sequences from ℕ can be coded and manipulated by purely arithmetical formulae, &amp;quot;phoned with God&amp;quot;, and attributes this claim to Andrzej Mostowski. I have been unable to find any other source that makes this exact claim. Did Gödel really say something like this to Mostowski?&lt;/p&gt;&#xA;" OwnerUserId="4691" LastActivityDate="2024-01-10T23:16:51.207" Title="Is there evidence that Gödel said &quot;phoned with God&quot;?" Tags="|quote|mathematics-social-history|logicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16099" PostTypeId="2" ParentId="9808" CreationDate="2024-01-09T15:59:36.573" Score="0" Body="&lt;p&gt;To my knowledge, Neumann gave the formula in 1845.It is said that Faraday did not provide any mathematical formulas. The law of electromagnetic induction was obtained by Neumann in 1845 by calculating the energy between two coils based on the Ampere force formula.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$Energy=-\frac{I_{1}I_{2}}{2}\oint_{C_{2}}\oint_{C_{1}}\frac{d\boldsymbol{l}_{1}\cdot d\boldsymbol{l}_{2}}{r}\ \ \ \ (1)$$&lt;/span&gt;&#xA;From this we can obtained that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathcal{E}_{2,1}=-\frac{\mu_{0}}{4\pi}\oint_{C_{2}}\oint_{C_{1}}\frac{dI_{1}}{dt}\frac{d\boldsymbol{l}_{1}\cdot d\boldsymbol{l}_{2}}{r}\ \ \ \ (2)$$&lt;/span&gt;&#xA;I do not clear how can we from (1) to Derived (2). But when we obtained (2) it is easy to obtained the Faraday law.&#xA;First define Induced electromotive force and magnetic vector potential&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\mathcal{E}_{2,1}\triangleq\oint_{C_{2}}\boldsymbol{E}_{1}\cdot d\boldsymbol{l}_{2}$$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{A}_{1}\triangleq\frac{\mu_{0}}{4\pi}\oint_{C_{1}}\frac{I_{1}d\boldsymbol{l}_{1}}{r}$$&lt;/span&gt;&#xA;Hence we obtained,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\oint_{C_{2}}\boldsymbol{E}_{1}\cdot d\boldsymbol{l}_{2}=-\oint_{C_{2}}\frac{\partial}{\partial t}\boldsymbol{A}_{1}d\boldsymbol{l}$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\oint_{C_{2}}(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})\cdot d\boldsymbol{l}_{2}=0$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\iint_{\Gamma}\nabla\times(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})\cdot\hat{n}d\Gamma=0$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla\times(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})=0$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1}=-\nabla\phi_{1}$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{E}_{1}=-\nabla\phi_{1}-\frac{\partial}{\partial t}\boldsymbol{A}_{1}$$&lt;/span&gt;&#xA;or omit the subscript,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{E}=-\nabla\phi-\frac{\partial}{\partial t}\boldsymbol{A}$$&lt;/span&gt;&#xA;This is Faraday's law proposed by Maxwell himself. Maxwell's descendants further rewrote this formula as,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla\times\boldsymbol{E}=-\nabla\times\nabla\phi-\frac{\partial}{\partial t}\nabla\times\boldsymbol{A}$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla\times\boldsymbol{E}=-\frac{\partial}{\partial t}\nabla\times\boldsymbol{A}$$&lt;/span&gt;&#xA;define&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{B}\triangleq\nabla\times\boldsymbol{A}$$&lt;/span&gt;&#xA;we obtained&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla\times\boldsymbol{E}=-\frac{\partial}{\partial t}\boldsymbol{B}$$&lt;/span&gt;&#xA;We see that the Faraday's law of electromagnetic induction derived from Neumann's electromagnetic induction formula is mediocre, which we are using in today's textbook. Therefore, this credit belongs to Neumann The vector potential we also defined above is also known as Neumann vector potential.&#xA;Considering,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\oint_{C}\cdots Idl\rightarrow\int_{V}\cdots JdV$$&lt;/span&gt;&#xA;Obtain,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{A}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{\boldsymbol{J}}{r}dV$$&lt;/span&gt;&#xA;It is worth mentioning that Weber also proposed the law of electromagnetic induction in 1846, and the vector corresponding to this law of electromagnetic induction is,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{A}_{W}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{(\boldsymbol{J}\cdot\boldsymbol{r})\boldsymbol{r}}{r^{3}}dV$$&lt;/span&gt;&#xA;If the coil is a closed loop, these two magnetic vector potentials are equivalent, that is, the induced electromotive force calculated by them is equal. Therefore, Weber is also another contributor to the law of electromagnetic induction.&lt;/p&gt;&#xA;&lt;p&gt;It is worth mentioning that Maxwell defined a magnetic field based on the curl of a magnetic vector under quasi-static conditions&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{A}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{\boldsymbol{J}}{r}dV\ \ \ \ \ （3）$$&lt;/span&gt;&#xA;Because,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla\times\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\int_{V}\nabla(\frac{1}{r})\times\boldsymbol{J}dV$$&lt;/span&gt;&#xA;or&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\nabla\times\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\int_{V}\boldsymbol{J}\times\frac{\boldsymbol{r}}{r^{3}}dV\ \ \ \ (4)$$&lt;/span&gt;&#xA;From Biota's law&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{B}=\frac{\mu_{0}}{4\pi}\int_{V}\boldsymbol{J}\times\frac{\boldsymbol{r}}{r^{3}}dV\ \ \ \ (5)$$&lt;/span&gt;&#xA;Compare (4) and (5) to obtain the definition,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\boldsymbol{B}\triangleq\nabla\times\boldsymbol{A}$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="13831" LastActivityDate="2024-01-09T15:59:36.573" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16100" PostTypeId="2" ParentId="16098" CreationDate="2024-01-09T16:42:59.313" Score="10" Body="&lt;p&gt;I have not been able to locate the passage in Rautenberg's book in which the episode is reported (EDIT: is in Chapter 6.4 &amp;quot;The Representability Theorem&amp;quot; on page 243), however a similar anecdote (but without the reference to the problem it refers to) is given in &lt;em&gt;&lt;a href=&quot;https://www.mimuw.edu.pl/sites/default/files/wspomnienia/a-mostowski-pol.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;O życiu i działalności Andrzeja Mostowskiego&lt;/a&gt;&lt;/em&gt; [On the Life and Work of&#xA;Andrzej Mostowski] by Stanisław Krajewski and Marian Srebrny, Wiadomosci matematyczne, Annales Societatis Mathematicae Polonae XXII.1 (1979), pp. 53–64. An English traslation (with some updates) of this paper is &lt;em&gt;&lt;a href=&quot;https://www.mimuw.edu.pl/sites/default/files/wspomnienia/a-mostowski-ang.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;On the Life and Work of Andrzej Mostowski (1913–1975)&lt;/a&gt;&lt;/em&gt; in Seventy Years of Foundational Studies: A Tribute to Andrzej Mostowski A. Ehrenfeucht, V.W. Marek and M. Srebrny (2007):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;However, the classes on applied mathematics turned out to be “horribly boring”[*] mainly due to the paucity of the mathematical means that were being used. And so, Mostowski abandoned this field – as it later turned out – forever. What aroused his interest were the classes of George Pólya and the seminar of Paul Bernays. Additionally, Mostowski was lucky to have the opportunity of listening to the lectures of the greatest masters: Kurt Gödel’s (in Vienna) on the consistency of the axiom of choice, Herman&#xA;Weyl’s on symmetry and Wolfgang Pauli’s on physics (both in Zürich). He always spoke of Gödel with utmost respect as of a genius. &lt;strong&gt;He once said that Gödel knows the solutions of the most difficult problems – as if he had a direct phone line to God.&lt;/strong&gt; In reference to Weyl, Mostowski’s attitude towards him can be best exemplified by the fact that for years Weyl’s portrait hung on the wall of the Professor’s office, on the ninth floor (room 908) of the Palace of Culture and Science in Warsaw, as the sole decoration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Krajewski reports the same story in &lt;em&gt;&lt;a href=&quot;https://wydawnictwa.ptm.org.pl/index.php/wiadomosci-matematyczne/article/viewFile/3820/3431&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurt Gódel i jego dzieło&lt;/a&gt;&lt;/em&gt;, Wiadomości Matematyczne XXIII, 2 (1981):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Postać Godła owiewa legenda obrazująca jego genialność. Dobrym jej przykładem jest powiedzenie A. Mostowskiego, który chodził na seminarium Godła w Wiedniu: „Gódel miał taką znajomość problemów, którymi się zajmował — a były to przecież najtrudniejsze kwestie w podstawach matematyki! — jakby miał bezpośredni telefon do Pana Boga”.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The figure of the Gödel is shrouded in legend illustrating its genius. A good example of it is the saying of A. Mostowski, who attended Gödel's seminar in Vienna: &amp;quot;Gödel had such familiarity with the problems he dealt with - and these were, after all, the most difficult questions in the foundations of mathematics! - as if he had a direct phone call to the Lord God&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Clearly, the problem lies in identifying the source of the story reported by Krajewski, but fortunately these are indicated in the first cited article:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The information included in this article has been taken from the following sources. Firstly, from the existing publications on the life and scientific output of Andrzej Mostowski. They are listed at the end of this text. &lt;strong&gt;The article by S. Hartman&lt;/strong&gt;, which presents the profile of Mostowski, is of particular interest. We have also used the outline of &lt;strong&gt;the biography of A. Mostowski prepared by A. Śródka&lt;/strong&gt; for the Archive of the Polish Academy of Sciences. The second source of data which we have used is &lt;strong&gt;the personal files and other documents of Andrzej Mostowski&lt;/strong&gt; that have been found in the files of the University of Warsaw and the Institute of Mathematics of the Polish Academy of Sciences. The third – &lt;strong&gt;the oral information provided by various people and our own memories&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The paper of Hartman is S. Hartman, &lt;em&gt;Andrzej Mostowski&lt;/em&gt;, Matematyka 2 (1976), pp. 67–70, but I'm unable to find it online; while the reference to Śródka is probably to &lt;a href=&quot;https://pl.wikipedia.org/wiki/Andrzej_%C5%9Ar%C3%B3dka&quot; rel=&quot;nofollow noreferrer&quot;&gt;Andrzej Stefan Śródka&lt;/a&gt; who wrote the &lt;em&gt;Uczeni Polscy XIX-XX stulecia&lt;/em&gt; (a biographical dictionary of Polish scholars of the 19th and 20th centuries).&lt;/p&gt;&#xA;&lt;p&gt;From the above, it seems that it was not actually Gödel who claimed to be in telephone contact with God, but that it was instead Mostowki's consideration that Gödel's genius suggested that he was in direct contact with God.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;For what it is worth, my impression is that instead the story comes from  orally reported anecdotes. Curiously, &lt;a href=&quot;https://en.wikipedia.org/wiki/Henk_Barendregt&quot; rel=&quot;nofollow noreferrer&quot;&gt;Henk Barendregt&lt;/a&gt; in &lt;em&gt;&lt;a href=&quot;https://www.cs.ru.nl/%7Ehenk/valedictory-lectureHB.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Keys to two intimacies: Mathesis and Mysticism&lt;/a&gt;&lt;/em&gt; (2015) reports the same tale but referring to Solovay instead of Gödel:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For both experiencing mathematics and mysticism one only can&#xA;create the right conditions, the rest is ---one could say---divine grace. In this way one describes it in many traditions of mysticism. But this parlance is also in vogue among mathematicians. &lt;strong&gt;Thus spoke Polish logician Andrzej Mostowski with admiration about his American colleague Robert Solovay: &amp;quot;He must have a direct phone line to God.&amp;quot;&lt;/strong&gt; So far the personal keys to mentioned subjects.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Therefore, assuming that Barendregt reports the episode correctly, this reinforces the idea that it is rather a figure of speech of Mostowski who, speaking of a mathematician of genius, such as Godel and Solovay, in order to emphasise the naturalness with which they solved the most difficult problems, imagined that they 'were on the phone with God'.&lt;/p&gt;&#xA;&lt;p&gt;&lt;sub&gt;[*]: “horribly boring” is a quote form J. N. Crossley (ed.), Algebra and Logic, Lecture Notes in Mathematics 450, Springer, 1975, p. 45.&lt;/sub&gt;&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2024-01-10T23:16:51.207" LastActivityDate="2024-01-10T23:16:51.207" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16101" PostTypeId="1" CreationDate="2024-01-09T17:11:48.093" Score="0" ViewCount="245" Body="&lt;p&gt;I discovered that John Robert Schrieffer (Nobel Prize in Physics 1972 for superconductivity theory) was in prison after a tragic car accident (1 person died, 7 were injured) where he was driving without a valid license.&lt;/p&gt;&#xA;&lt;p&gt;Do you have more examples of scientists/mathematicians that committed crimes like these that are not related to politics or war? I would like to avoid people like Lev Landau that even if they were in prison (and have interesting anecdotes), they were there for political reasons. And in the same idea let's keep Alan Turing and Galileo out for the same reason.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-12T14:40:12.907" LastActivityDate="2024-01-12T14:40:12.907" Title="Who are some scientists that were convicted for non-war non-political crimes?" Tags="|biographical-details|big-list|examples|" AnswerCount="3" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16102" PostTypeId="2" ParentId="16101" CreationDate="2024-01-09T17:20:59.370" Score="5" Body="&lt;p&gt;I think the most famous case is that of &lt;a href=&quot;https://en.wikipedia.org/wiki/Andr%C3%A9_Bloch_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;André Bloch&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;André Bloch was a French mathematician who worked in the field of&#xA;complex analysis and who spent most of his life confined in a psychiatric hospital. When the First World War began in August 1914, both André and his brother Georges were drafted into the army, and both were wounded during their service; André's wounds were serious and he was hospitalised several times. On 17 November 1917, André killed his brother Georges, his uncle and his aunt by stabbing them, and after the murders he was arrested without resistance. He was confined in a psychiatric hospital on the outskirts of Paris for 31 years.&lt;/p&gt;&#xA;&lt;p&gt;Surely Bloch committed a major crime, but it is not clear if its status in the asylum can be assimilated to a stay in prison.&#xA;A partial answer is in the paper of Henri Cartan and Jacqueline Ferrand &lt;em&gt;&lt;a href=&quot;http://www.numdam.org/article/CSHM_1988__9__211_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Le cas André Bloch&lt;/a&gt;&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Déclaré irresponsable (1) A. B. fut interné à l’hôpital psychiatrique de Saint-Maurice, appelé aussi &amp;quot;Maison de Charenton&amp;quot;. Il ne le quittera que 31 ans plus tard&lt;/p&gt;&#xA;&lt;p&gt;(1) Il semble que cette pénible affaire ait été étouffée à l’époque, et que les journaux n’en aient pas parlé ; car les camarades d’études des frères Bloch eux-mêmes n’ont pas connu les détails du drame. N’oublions pas que la France était en guerre et que le meurtrier était un officier en permission.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Declared irresponsible (1) A. B. was committed to the psychiatric hospital at Saint-Maurice, also known as &amp;quot;Maison de Charenton&amp;quot;. He did not leave the hospital until 31 years later.&lt;/p&gt;&#xA;&lt;p&gt;(1) It seems that this distressing affair was hushed up at the time, and that the newspapers did not mention it, because the Bloch brothers' fellow students themselves did not know the details of the tragedy. Let's not forget that France was at war and that the murderer was an officer on leave.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Two cases of mathematicians who were certainly confined to prison, although of less scientific value than Bloch, are those of  &lt;a href=&quot;https://www.dw.com/en/murderer-solves-ancient-math-problem-and-finds-his-mission/a-53895884&quot; rel=&quot;nofollow noreferrer&quot;&gt;Christopher Havens&lt;/a&gt;, a self-taught mathematician who solved a number theory problem, and of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ted_Kaczynski&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theodore John Kaczynski&lt;/a&gt;, also known as the Unabomber, who was a mathematics prodigy, but abandoned his academic career in 1969.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;Galileo Galilei also spent the last years of his life under house arrest, but I think we can nowadays consider his crime as a political one.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2024-01-10T15:19:30.573" LastActivityDate="2024-01-10T15:19:30.573" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16103" PostTypeId="2" ParentId="16101" CreationDate="2024-01-10T04:15:15.553" Score="2" Body="&lt;p&gt;Sadly, Paul Ehrenfest, before committing suicide, killed his son, who had Down syndrome (&lt;a href=&quot;https://thereader.mitpress.mit.edu/paul-ehrenfest-forgotten-physicist/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://thereader.mitpress.mit.edu/paul-ehrenfest-forgotten-physicist/&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-01-10T04:15:15.553" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16104" PostTypeId="2" ParentId="16091" CreationDate="2024-01-10T12:45:32.270" Score="3" Body="&lt;p&gt;I still have only a partial answer to the question. Looking at Trautman's lecture notes published in 1970 (&lt;a href=&quot;https://doi.org/10.1016/0034-4877(70)90003-0&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1016/0034-4877(70)90003-0&lt;/a&gt;), it is more than clear that he knew for sure that &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; was a U(1) connection and &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; its curvature. Lubin's work in 1963 (&lt;a href=&quot;https://doi.org/10.1016/0003-4916(63)90194-5&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1016/0003-4916(63)90194-5&lt;/a&gt;) is rather technical and seems to already point towards that direction. However, going back as far as Utiyama's work of 1955 (&lt;a href=&quot;https://doi.org/10.1103/PhysRev.101.1597&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1103/PhysRev.101.1597&lt;/a&gt;), right after the work of Yang and Mills, it seems that he generalizes the work of Weyl (1929) to non-abelian group in a very technical way, comparing the affine connexions and the gauge fields without truly identifying them as a the same mathematical/geometrical structure. Any complement to this would of course be welcome!&lt;/p&gt;&#xA;" OwnerUserId="19547" LastActivityDate="2024-01-10T12:45:32.270" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16105" PostTypeId="2" ParentId="16041" CreationDate="2024-01-10T13:02:32.000" Score="0" Body="&lt;p&gt;As mentioned above, 'concise' and 'detailed' are contradictory requirements...&lt;/p&gt;&#xA;&lt;p&gt;On the 'concise' side, allow me to suggest –for those who read French– the book 'Histoire de L'Électricité', by Gérard Borvon. ISBN 978-2-7117-2492-5&lt;/p&gt;&#xA;" OwnerUserId="6222" LastActivityDate="2024-01-10T13:02:32.000" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16107" PostTypeId="1" CreationDate="2024-01-10T14:38:52.010" Score="2" ViewCount="213" Body="&lt;p&gt;When reading on the topic of Ancient Egyptian geometry by Ancient Greek philosophers, there is a certain sense that this is quite a thriving discipline that seems comparable to the type of geometry done in Greece at that time.  E.g., Diogenes Laertius' Lives of Eminent Philosophers :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And Pamphile relates that he, having learnt geometry from the Egyptians, was the first person to describe a right-angled triangle in a circle, and that he sacrificed an ox in honour of his discovery&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or Clement of Alexandria' Stromata&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;I have roamed over the most ground of any man of my time, investigating the most remote parts. I have seen the most skies and lands, and I have heard of learned men in very great numbers. And in composition no one has surpassed me; in demonstration, not even those among the Egyptians who are called Arpenodaptae, with all of whom I lived in exile up to eighty years.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;or in Iamblichus' Life of Pythagoras :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But it is said that he very much applied himself to geometry among the Egyptians. For with the Egyptians there are many geometrical problems; since it is necessary that from remote periods, and from the time of the Gods themselves,[38] on account of the increments and decrements of the Nile, those that were skilful should have measured all the Egyptian land which they cultivated. Hence also geometry derived its name. Neither did they negligently investigate the theory of the celestial orbs, in which likewise Pythagoras was skilled. Moreover, all the theorems about lines appear to have been derived from thence.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This seems to give the impression of some geometry (and astronomy) with some theory behind it, and from various other quotes, partly done by the priestly class.&lt;/p&gt;&#xA;&lt;p&gt;However, the actual geometry that seems to exist in surviving documents like the Rhind papyrus or the Moscow papyrus, seem to be fairly utilitarian texts on the topic, being mostly either scribal exercises or practical computations, which do not seem to have much hints of such things.&lt;/p&gt;&#xA;&lt;p&gt;Am I perhaps overinterpreting what is said in those sources? Are there any other documents which would be more in line with this interpretation? Is it maybe just that such documents did not survive at all, or were not even written down? Or is it maybe even the case that most of these are legendary, as many of these texts are written much after the period in question was concerned (from around 600 BCE to 300 BCE)?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="12330" LastEditDate="2024-01-10T17:05:56.620" LastActivityDate="2024-01-14T04:26:17.137" Title="Ancient Egyptian geometry" Tags="|geometry|ancient-greece|ancient-egypt|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16110" PostTypeId="2" ParentId="16101" CreationDate="2024-01-11T09:34:49.223" Score="3" Body="&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Frampton#Drug_smuggling_conviction&quot; rel=&quot;nofollow noreferrer&quot;&gt;Paul Frampton&lt;/a&gt; was convicted of drug smuggling in Argentina.&lt;/p&gt;&#xA;&lt;p&gt;This appears to have been foolishness rather than criminal intent, though as always it's hard to know who to believe. Frampton is not a world famous scientist, though the case made headlines in the UK possibly because of the &lt;em&gt;femme fatale&lt;/em&gt; aspect. I think it mainly proves physicists are human.&lt;/p&gt;&#xA;" OwnerUserId="2082" LastActivityDate="2024-01-11T09:34:49.223" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16111" PostTypeId="2" ParentId="16107" CreationDate="2024-01-11T11:04:11.153" Score="-3" Body="&lt;p&gt;Although the Greeks are credited with the discovery of mathematics, this is clearly not the case. The discovery of number and geometry ranks as one of the greatest discoveries. It was already there by the time of the Babylonians and Egyptians. Who and when these two disciplines were discovered is very likely lost to time, although we do have evidence like the Ishango bone which shows tallying was known back in 20,000 BC and with the Lembobo bone, possibly to 40,000 BC. Both of these were discovered in Africa.&lt;/p&gt;&#xA;&lt;p&gt;Although a lot of Babylonian and Egyptian mathematics may look primitive compared to the later developments of Greek mathematics, the key thing is that it came later. The important thing is that it was a vigorous field that seeded future developments. And of course the development of key concepts like length, angle, area and volume were hugely important, although I don't expect that they were discovered by the Egyptians but probably predate them.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2024-01-12T13:00:12.880" LastActivityDate="2024-01-12T13:00:12.880" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16112" PostTypeId="2" ParentId="16107" CreationDate="2024-01-11T14:21:59.460" Score="4" Body="&lt;p&gt;Greek writers of Hellenistic times indeed had a tendency to&#xA;claim that mathematics and astronomy came to Greece from Egypt. This has no conformation whatsoever. Modern studies show that mathematics and astronomy in Egypt were in very primitive state, in comparison with Mesopotamia and Greece.&#xA;On the other hand, these Hellenistic sources do not mention Mesopotamia (Babylon), while there are documented cases of&#xA;transfer of important scientific knowledge from Babylon to Greece. Notice that all those Hellenistic authors who claim&#xA;that there was a substantial influence of Egyptian mathematics/astronomy are not mathematicians or astronomers themselves, so they probably did not understand the subject.&lt;/p&gt;&#xA;&lt;p&gt;The reason of this confusion is unclear.&#xA;As an example, all Hellenistic sources tell us that Thales traveled to Egypt to learn mathematics. On the other hand these same sources say that Thales surprised Egyptian priests when he showed them how to measure the height of a pyramid.&#xA;Modern historian comments: &amp;quot;And what could these priests possibly teach Thales?&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Hellenistic historians never mention any specific important contribution of Egyptians to mathematics or astronomy. Just the general statements that founders of Greek exact sciences, Thales and Pythagoras &amp;quot;traveled to Egypt to learn the wisdom of Egyptian priests&amp;quot;. What could they possibly learn there remains a mystery.&lt;/p&gt;&#xA;&lt;p&gt;Ptolemy, an astronomer who worked in Egypt at the time&#xA;of the Roman empire, and whose works are our principal source of knowledge about ancient astronomy, frequently refers to Babylonians and never on Egyptians.&lt;/p&gt;&#xA;&lt;p&gt;References:&lt;/p&gt;&#xA;&lt;p&gt;O. Neugebauer, The exact sciences in antiquity,&lt;/p&gt;&#xA;&lt;p&gt;O. Neugebauer, A history of ancient mathematical astronomy,&lt;/p&gt;&#xA;&lt;p&gt;B. L. Van der Waerden, Science awakening.&lt;/p&gt;&#xA;&lt;p&gt;This last book discusses exactly this question: &amp;quot;What could Greeks learn from Egyptians?&amp;quot; in the last section of Chapter I. The author mentions other opinions on the question as well.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-01-14T04:26:17.137" LastActivityDate="2024-01-14T04:26:17.137" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16113" PostTypeId="1" CreationDate="2024-01-12T12:23:02.220" Score="2" ViewCount="82" Body="&lt;p&gt;I've been looking up the history and evolution of the seven base units and am currently checking out the ampere. What I've found is that 1A is defined as the current in a wire which would experience a force equivalent to this formula:&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$F=\frac{\mu_0 I_1 I_2l}{2\pi r}$$&lt;/span&gt;&#xA;which is &lt;span class=&quot;math-container&quot;&gt;$2\times 10^{-7}$&lt;/span&gt;N when all other values are 1.&lt;/p&gt;&#xA;&lt;p&gt;But since this definition is dependent on vacuum permeability, I tried looking for how the value of this was defined and either came across the same formula or Ampere's circuital law... which also has current in it.&lt;/p&gt;&#xA;&lt;p&gt;I understand the force can be measured with a balance or a weight, but currents are measured with an ammeter, which if I'm not wrong, is a modified/evolved galvanometer, which is also calibrated based on weight balances(?) upon the force exerted by a current carrying wire, so it's all starting to feel like an ouroboros to me.&lt;/p&gt;&#xA;&lt;p&gt;Any help? My knowledge base is basically at the level of a Cambridge A level graduate, with a bit more due to compulsory courses in university.&lt;/p&gt;&#xA;&lt;p&gt;(I asked &lt;a href=&quot;https://physics.stackexchange.com/q/797061/371600&quot;&gt;this question on PSE&lt;/a&gt; and was redirected here.)&lt;/p&gt;&#xA;" OwnerUserId="19563" LastEditorUserId="6609" LastEditDate="2024-01-21T18:07:01.227" LastActivityDate="2024-01-21T18:07:01.227" Title="What was defined first and how? The ampere or the vacuum permeability?" Tags="|electromagnetism|experimental-physics|units|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16114" PostTypeId="2" ParentId="16041" CreationDate="2024-01-12T13:07:03.207" Score="0" Body="&lt;p&gt;Concise &amp;amp; detailed to me usually means an entry from an encyclopedia. So you can try this &lt;a href=&quot;https://en.m.wikipedia.org/wiki/History_of_electromagnetic_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;entry&lt;/a&gt; from Wikipedia on precisely this subject. It also has a bibliography which you can use to pursue the subject further.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2024-01-12T13:07:03.207" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16115" PostTypeId="2" ParentId="16085" CreationDate="2024-01-12T14:16:43.090" Score="2" Body="&lt;p&gt;The addition and multiplication of numbers were seen as so intuitively obvious that no-one bothered to formalise them before the modern period with Peano's axioms and also the development of set theory. It's worth noting that set theory would be trivial if it was not for the axiom asserting the existence of an infinity.&lt;/p&gt;&#xA;&lt;p&gt;This is not the case for geometry which was first formalised by Euclid though he was only completing a development that had preceded him and which also had a further period of development in the modern period by inclusion of the axioms of order which involves introducing a new primitive notion - betweeness. This was discovered by Pasch and developed further by Hilbert.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2024-01-12T14:32:09.757" LastActivityDate="2024-01-12T14:32:09.757" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16116" PostTypeId="2" ParentId="13409" CreationDate="2024-01-12T14:24:19.003" Score="0" Body="&lt;p&gt;The discovery of penicillin by the Scottish bacteriologist, Alexander Fleming. In 1928, he noted that colonies of the bacterium Staphylococcus Aureus failed to grow in those areas of a culture that had been accidentally contaminated by a green mold - Penicillium Notatum. He isolated the mold and found that it produced a substance capable of killing many of the common bacteria that infect humans. He called this substance Penicillin.&lt;/p&gt;&#xA;&lt;p&gt;Apocraphylly, the apple that serendipitously fell and provoked Newton into discovering gravity.&lt;/p&gt;&#xA;" OwnerUserId="2324" LastActivityDate="2024-01-12T14:24:19.003" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16117" PostTypeId="2" ParentId="16041" CreationDate="2024-01-12T15:07:57.470" Score="1" Body="&lt;p&gt;Two books come to mind here: Kathy Joseph's &lt;em&gt;The Lightning Tamers&lt;/em&gt; is one I haven't read yet, but it has been very well received and Kathy definitely does her research (see also her YouTube series) so I would imagine it is a solid book. As well, I'd recommend Keithley's &lt;em&gt;The Story of Electrical and Magnetic Measurements&lt;/em&gt;. Keithley includes the main beats from the history of electricity, including good detail, without the thorough detail and scholarship of some other histories (Park Benjamin, Edmund Whittaker, J. L. Heilbron).&lt;/p&gt;&#xA;&lt;p&gt;For the early history of electricity, I might also plug my own channel, where I reproduce experiments from the earliest experimenters and look at how their experiments informed their understanding. You can find &lt;a href=&quot;https://www.youtube.com/channel/UCzUm_AERTUOQWwk7djShOSA&quot; rel=&quot;nofollow noreferrer&quot;&gt;my channel here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;To better understand how people viewed electricity in this period, consider looking at Priestley's &lt;em&gt;The History and Present State of Electricity&lt;/em&gt; (1767) written at a time when the study of electrostatics was in full swing, but mathematical physics was not yet applied, and currents and electromagnetism were still distant. This work has been very impactful in forming the narrative of the &amp;quot;history of electricity.&amp;quot; It's not too difficult to read, particularly if the typesetting is modern, although the writing style in the 18th century is obviously a bit different from today.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2024-01-12T15:07:57.470" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16118" PostTypeId="1" CreationDate="2024-01-12T15:35:58.307" Score="0" ViewCount="73" Body="&lt;p&gt;I recently read an article which said Cayley showed that affine geometry could be developed from projective geometry after he learnt of Laguerre's definition of the angle using the cross ratio. This was then used by Klein to develop the Klein disk model of the hyperbolic plane which established the equiconsistency of Euclidean and hyperbolic geometry.&lt;/p&gt;&#xA;&lt;p&gt;I've looked online for an exposition of this definition of Laguerre's but have found nothing. I've also looked into a textbook of Coxeter's on projective geometry and ahain come up with nothing peryaining to the above although Coxeter does give, he says, an improved definition of an angle, it relies on on Euclid's original definition.&lt;/p&gt;&#xA;&lt;p&gt;Q. What is Laguerre's definition of the angle using the cross ratio?&lt;/p&gt;&#xA;" OwnerUserId="2324" LastEditorUserId="2324" LastEditDate="2024-01-12T19:46:51.403" LastActivityDate="2024-01-12T19:46:51.403" Title="What is Laguerre's definition of the angle via the cross ratio?" Tags="|reference-request|geometry|euclidean-geometry|projective-geometry|non-euclidean-geometry|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16119" PostTypeId="2" ParentId="5951" CreationDate="2024-01-13T00:03:23.880" Score="2" Body="&lt;p&gt;A small point, that I've only recently thought about: To my perception, most of classical-Greek mathematics had implicit units attached. Of &amp;quot;length&amp;quot; or &amp;quot;area&amp;quot; or &amp;quot;volume&amp;quot;. In such a context, a &lt;em&gt;ratio&lt;/em&gt; of two lengths would be unit-less... a strange thing? ... while the ratio of an area by a length would have units of length (and would be the length of the other side of the rectangle whose area was the original area-number, with one side equal to the given length-number)?&lt;/p&gt;&#xA;&lt;p&gt;For example, for me, this clarifies the (in modern terms) circumlocutions about &amp;quot;similar triangles&amp;quot;... where, indeed, &lt;em&gt;ratios&lt;/em&gt; of lengths are the same. But those ratios have no units, so, perhaps, somehow, do not really exist as first-class objects in their own right? :)&lt;/p&gt;&#xA;" OwnerUserId="1932" LastActivityDate="2024-01-13T00:03:23.880" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16120" PostTypeId="1" AcceptedAnswerId="16123" CreationDate="2024-01-15T09:39:53.940" Score="6" ViewCount="6452" Body="&lt;h2&gt;I asked this question on MSE &lt;a href=&quot;https://math.stackexchange.com/questions/4844970/why-the-development-of-mathematics-was-very-slow-between-ancient-greece-and-desc?noredirect=1#comment10321017_4844970&quot;&gt;here&lt;/a&gt;&lt;/h2&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;In my studies of mathematics (I am not very good at mathematics, I only studied real analysis, some linear algebra, geometry and calculus ), I noticed that mathematics can be divided into two major parts. One part is Ancient Greek mathematics . This part has achievements in geometry and arithmetic and their connections. The other part is modern mathematics from Descartes to today. This part has discoveries in calculus, number theory, topology, and many other fields (areas). But what happened between these two eras? In my studies, I did not find many theorems from this period or any great breakthrough at all.&lt;/p&gt;&#xA;&lt;p&gt;It seems like mathematics stopped there and only resumed in the 1500s and 1600s here I am not talking about European mathematics but mathematics in general (Middle East, China, India, etc.). But why is that? Mathematics is not an empirical science like physics that needs experiments to test theories and wait for centuries such that their tools become good enough. Mathematics only relies on logic, and for some reason Mathematics was just like other empirical sciences -- it began really to develop after the 1600s.&lt;/p&gt;&#xA;&lt;p&gt;As far as I know, mathematics can be divided into fields (areas) like topology, analysis, number theory, algebra, geometry, combinatorics, etc. Most of these fields or areas took more than a millennium to appear. The first major one was geometry, and it appeared in Ancient Greece. All the others did not exist until after the 1600s, like topology or analysis, or they were too small and narrow to be considered as separate fields before. So why is that? Even if calculus took 1000 years to be discovered (Which I don't know why! see &lt;a href=&quot;https://hsm.stackexchange.com/questions/16129/what-were-the-obstacles-that-made-the-discovery-of-calculus-very-late&quot;&gt;This question &lt;/a&gt; ) I heard that many mathematical fields don't rely on analysis very much, like abstract algebra, graph theory, etc.&lt;/p&gt;&#xA;&lt;p&gt;I heard that philosophy held back mathematics for millennia, Archimedes could have discovered calculus if he did not think too much about philosophy and infinity.  Is this the reason?&lt;/p&gt;&#xA;&lt;p&gt;So why was the development of mathematics slow for a millennium  and why did many mathematical concepts take millennia to develop? or in other words why the development of mathematics was very slow between Ancient Greece and Descartes (I don't mean Descartes specifically but the era that he lived in. Cavalieri, Fermat, Kepler, etc. were from the same era, i.e., 1550−1650&#xA;.)&lt;/p&gt;&#xA;&lt;p&gt;I think this question might  have too long answers  as a user point out before in one of my old questions:  &amp;quot;You must realize that the history of mathematics is not going to be broken down into a couple of paragraphs,&amp;quot; and so I will ask for books or references that answer this question if the answer is too long.&lt;/p&gt;&#xA;" OwnerUserId="18466" LastEditorUserId="18466" LastEditDate="2024-01-21T02:18:52.137" LastActivityDate="2024-01-25T09:39:17.313" Title="Why was the development of mathematics very slow between Ancient Greece and Descartes?" Tags="|mathematics|reference-request|ancient-greece|book-recommendation|middle-ages|" AnswerCount="5" CommentCount="15" ContentLicense="CC BY-SA 4.0" />
  <row Id="16121" PostTypeId="2" ParentId="16120" CreationDate="2024-01-15T11:55:55.220" Score="3" Body="&lt;p&gt;There reason for the speed-up in the 1600s was Vieta's development of an algebraic formalism that allowed mathematicians for the first time to write down mathematical formulas rather than merely abbreviated phrases in natural language.  In your title you mention Descartes specifically, but several dramatic advances occurred before Descartes.  The names that come to mind are in alphabetical order Cavalieri, Fermat, Kepler, and others.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2024-01-15T11:55:55.220" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16122" PostTypeId="2" ParentId="16120" CreationDate="2024-01-15T12:54:34.847" Score="6" Body="&lt;p&gt;Mathematics is a tiny part of human activities, and its historical development cannot be considered in isolation.&#xA;Since the 2nd century AD, there was a decline in all intellectual activities. Eventually civilization itself collapsed, and its slow revival started only in  16th century. In this period, people mostly lived in the villages,&#xA;rather than in large cities, most of the economy was on subsistence level, most people were illiterate, technology was lost, and the main&#xA;intellectual activity was concern about afterlife.&lt;/p&gt;&#xA;&lt;p&gt;This was not the only such collapse of which we have historical evidence: another one on a similar scale happened around 1200 BC. Historians discuss a lot the reasons of these collapses, the literature on the topic is enormous, and there is no common opinion.&lt;/p&gt;&#xA;&lt;p&gt;Let me mention just a few books, and one modern paper.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Edward Gibbon, The decline and fall of the Roman empire.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Peter Heather, The Fall of the Roman Empire: A New History of Rome and the Barbarians, Oxford, 2006.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Brian Ward-Perkins, The fall of Rome and the end of civilization, Oxford UP, 2005.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Adrian Goldsworthy, Fall of the West: the death of the Roman superpower, Phoenix, 2010.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;W. M. Jongman, Gibbon was right: the decline and fall of Roman economy, in the book: Crises and the Roman Empire. Proceedings of the Seventh Workshop of the International Network Impact of Empire (Nijmegen, June 20-24, 2006),&#xA;Freely available on Internet.&lt;/p&gt;&#xA;&lt;p&gt;(This paper checks Gibbon's insight with modern hard evidence).&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;To be sure, decline of mathematics, and of science in general, began several centuries before the decline of the Roman empire,&#xA;see on this:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;L. Russo, The forgotten revolution. How science was born in 300 BC and why it had to be reborn, Springer, 2004.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;So decline in science was followed by a collapse of civilization.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="-1" LastEditDate="2024-01-16T18:01:38.723" LastActivityDate="2024-01-16T18:01:38.723" CommentCount="12" ContentLicense="CC BY-SA 4.0" />
  <row Id="16123" PostTypeId="2" ParentId="16120" CreationDate="2024-01-15T14:29:43.480" Score="20" Body="&lt;p&gt;Making my comment into an answer:&lt;/p&gt;&#xA;&lt;p&gt;Was there a gap of knowledge or slow-down of progress in math as a whole between Ancient Greeks and the 17th century?&lt;/p&gt;&#xA;&lt;p&gt;The answer is probably no.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Mathematics_in_the_medieval_Islamic_world&quot; rel=&quot;nofollow noreferrer&quot;&gt;Islamic Medieval World&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Indian_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Medieval India&lt;/a&gt; contributed greatly to math during the Middle Ages. The fields of algebra, polynomial equations, cubic roots, modern numerals, logic, negative numbers, trigonometric functions and much more came from that period. These contributions are not minor and contributed continuously to the development of math. Islamic sources in particular translated and preserved a lot of Ancient Greek and Indian sources, and Islamic sources were later used by Renaissance Europeans like Fibonacci.&lt;/p&gt;&#xA;&lt;p&gt;Wikipedia's &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;history of mathematics&lt;/a&gt; also cites:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As for well known theorems of this time you have: the law of cosines (a.k.a. Al-Kashi's theorem), Chinese remainder theorem, Merton mean speed theorem, Wilson's theorem, Thabit's rule, divergence of the harmonic series,  and the binomial theorem.&lt;/p&gt;&#xA;&lt;p&gt;But even if we forget the Middle Ages as some &amp;quot;Dark Ages&amp;quot; (a term now considered as misnomer) and focus on Europe, we still have the whole Renaissance. People like Tartaglia and Cardano helped develop the solution to the quartic equations. Bombelli developed the algebra of imaginary numbers.&lt;/p&gt;&#xA;&lt;p&gt;Many theses could be written on why the developments between Ancient Greeks and 17th century math seem &amp;quot;slow&amp;quot; or minor. I propose three conjectures:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Lack of proper sources and attribution: sources foreign to Europe were not translated or properly attributed. This makes a lot of contributions that have no authors attached to them (even if we have words like &lt;em&gt;algebra&lt;/em&gt; that come from Arabic or &lt;em&gt;algorithm&lt;/em&gt; which comes from Al-Khwarizmi).&lt;/li&gt;&#xA;&lt;li&gt;Utility: Some mathematical content is preserved and transferred to the next generation better than others. Most of the time if there is some application (engineering, physics or astronomy) there was a higher chance for the content to survive. &lt;a href=&quot;https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kerala School&lt;/a&gt; might have invented calculus, but clearly their results were never largely popularized (maybe even never left Kerala).&lt;/li&gt;&#xA;&lt;li&gt;Transition of power: As Europe declined in the Middle Ages and the Medieval Islamic World became an important center of knowledge and power, clearly there was a lot of friction. Text had to be translated from Greek and Latin (and Sanskrit from India) to Arabic - and when Europe &amp;quot;reemerged&amp;quot;, back from Arabic to Latin. Religious conflicts and other power struggles also did not help. Imagine how hard it was to introduce Hindu decimal numerals from the Indian subcontinent into West Asia and into Europe without the Internet, printing, long distance sailing and so on. Nevertheless, it made the trip because it was an important breakthrough.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;However for every reason that I can find for this possible setback in progress in math, I can find a ton of contributions from those times that make the setback negligible. How much importance we give to these developments over some more recent ones is more a matter of opinion or perspective - though this is not the forum for opinion-based discussions.&lt;/p&gt;&#xA;&lt;p&gt;The Dark Ages is a myth. For a light read, check for example &lt;em&gt;The Light Ages: The Surprising Story of Medieval Science&lt;/em&gt; by Seb Falk  (&lt;a href=&quot;https://time.com/5911003/middle-ages-myths/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Time&lt;/em&gt; excerpt&lt;/a&gt;), and &lt;em&gt;The West: A New History in Fourteen Lives&lt;/em&gt; by Naoíse Mac Sweeney (&lt;a href=&quot;https://www.smithsonianmag.com/history/the-myth-of-the-dark-ages-ignores-how-classical-traditions-flourished-around-the-world-180982190/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smithsonian excerpt&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;MathSE recommends &lt;em&gt;&lt;a href=&quot;https://www.gutenberg.org/ebooks/31246&quot; rel=&quot;nofollow noreferrer&quot;&gt;A Short Account of the History of Mathematics&lt;/a&gt;&lt;/em&gt; by W. W. Rouse Ball for the history of math in general. It divides all math in three great periods: Ancient Greeks, Middle Ages and Renaissance, and Modern (beginning with Descartes). Chapters cover math from the last period of the Roman Empire, Middle Ages math (both in Islamic and Europe) and Renaissance math.&lt;/p&gt;&#xA;&lt;p&gt;I end by adding this good quote I found in &lt;a href=&quot;https://en.wikipedia.org/wiki/MacTutor_History_of_Mathematics_Archive&quot; rel=&quot;nofollow noreferrer&quot;&gt;MacTutor History of Mathematics Archive&lt;/a&gt; article on &amp;quot;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Arabic_mathematics/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Arabic mathematics&lt;/a&gt;&amp;quot;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.&lt;/p&gt;&#xA;&lt;p&gt;That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_mathematics&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;:-&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... Arabic science only reproduced the teachings received from Greek science.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-25T09:39:17.313" LastActivityDate="2024-01-25T09:39:17.313" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="16125" PostTypeId="1" CreationDate="2024-01-15T23:11:52.410" Score="4" ViewCount="59" Body="&lt;p&gt;In a 1963 article by Robert Dicke (&amp;quot;The observational basis of general relativity&amp;quot;), there is a reference to a concept for a gravitational experimental device in the form of a big ball surrounded by a shell that can move on its own, so as to shield the ball's trajectory from non-gravitational interferences like the interplanetary medium&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/1fjFs.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/1fjFs.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The description is given as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;An example of a space experiment that might improve these distance&#xA;measurements is an artificial planet surrounded by some gas shield to&#xA;eliminate gas drag and light pressure difficulties that would&#xA;interfere with the gravitational orbit. A radar technique could then&#xA;be used to determine the distance to this device with great precision.&#xA;This would give us precise orbital data for both the earth and the&#xA;planetoid. The gas-shield technique has been independently suggested&#xA;by a number of people, probably first by M. Schwarzschild. His&#xA;suggestion is to surround a small artificial planet with a hollow&#xA;spherical shell equipped with small gas jets as in figure 1-1. When&#xA;the artificial planet gets near the wall, the gas jets are turned on&#xA;by a servomechanism to compensate the drag and to keep the shell from&#xA;touching the inner sphere. In this way the artificial planet moves on&#xA;a proper planetary orbit, not being influenced by radiation pressure&#xA;and gas drag.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are however no bibliographical references to this proposal. Nothing in the bibliographical references of the paper, nothing that I can find in Martin Schwarzschild's bibliography (I assume that M. Schwarzschild is Martin Schwarzschild, Karl Schwarzschild's son), nor is there much to be found looking for the notion of an artificial planet in general relativity testing, nothing about that specific design anyway&lt;a href=&quot;https://adsabs.harvard.edu/full/1960AJ.....65..272C&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;a href=&quot;https://www.nature.com/articles/183666a0&quot; rel=&quot;nofollow noreferrer&quot;&gt;[2]&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;What could this be a reference to?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2024-01-15T23:11:52.410" Title="Schwarzschild's artificial planet" Tags="|relativity-theory|astrophysics|space-travel|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16126" PostTypeId="1" AcceptedAnswerId="16146" CreationDate="2024-01-16T01:42:35.040" Score="2" ViewCount="199" Body="&lt;p&gt;Einstein, in his 1905 relativity paper &lt;a href=&quot;https://users.physics.ox.ac.uk/%7Ertaylor/teaching/specrel.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Electrodynamics of Moving Bodies&lt;/a&gt;, allocates a section to the issue of reflection of light from a moving mirror and derives three formulas (angle, frequency, energy). What was his reason or motivation behind including this particular problem?&lt;/p&gt;&#xA;&lt;p&gt;Near the bottom of Page 18 of above reference Einstein, in relation to his formula for pressure of light on a moving mirror, says &amp;quot;in agreement with experiment and with other theories&amp;quot;. What experiment and theories is he referring to?&lt;/p&gt;&#xA;&lt;p&gt;Was there any related experiment at that time that showed discrepancy with the classical formulations? For example were there any &lt;a href=&quot;https://www.fourmilab.ch/cship/aberration.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;observed aberration&lt;/a&gt; that indicated an issue?&lt;/p&gt;&#xA;" OwnerUserId="17473" LastEditorUserId="17473" LastEditDate="2024-01-19T14:54:21.900" LastActivityDate="2024-01-20T19:19:07.657" Title="What was the reason Einstein included reflection from a moving mirror in his relativity paper?" Tags="|electromagnetism|relativity-theory|einstein|light|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16127" PostTypeId="1" AcceptedAnswerId="16150" CreationDate="2024-01-16T14:24:51.900" Score="1" ViewCount="141" Body="&lt;p&gt;I am trying to understand the origin of harmonic mean and get an intuitively feel for why it was invented in the first place.&lt;/p&gt;&#xA;&lt;p&gt;I have surfed the web but I keep seeing things like harmonic series / musical chords etc. Despite digging through, I can't really see any meaningful connection between harmonic series and harmonic mean.&lt;/p&gt;&#xA;&lt;p&gt;That is, I understand that each term after the first in a harmonic serie is the harmonic mean of the neighboring terms but that doesn't really explain our current use of harmonic mean when solving problems.&lt;/p&gt;&#xA;&lt;p&gt;It doesn't really give any gist as to why harmonic mean should even be thought of in the first place.&lt;/p&gt;&#xA;" OwnerUserId="19573" LastActivityDate="2024-01-21T23:28:03.583" Title="Is there any meaningful history behind harmonic mean?" Tags="|mathematics|harmonic-analysis|arithmetic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16129" PostTypeId="1" AcceptedAnswerId="16139" CreationDate="2024-01-16T21:13:37.273" Score="8" ViewCount="930" Body="&lt;p&gt;I wonder What were the obstacles that made the discovery of calculus very late ?&lt;/p&gt;&#xA;&lt;p&gt;Why the discovery of calculus took so long? I know that some of the ideas and techniques of calculus appeared in ancient Greece, but they were not developed into a systematic theory until the &lt;span class=&quot;math-container&quot;&gt;$17$&lt;/span&gt;th century by Newton and Leibniz. What were the main obstacles or challenges that prevented the earlier mathematicians from discovering calculus?&lt;/p&gt;&#xA;&lt;p&gt;One of the factors that I think might have played a role is philosophy. I have heard that some of the ancient and medieval mathematicians in Islamic world  were influenced by Aristotle’s philosophy, which had a negative view of infinity and rejected the concept of limit and convergence. Aristotle also preferred geometric methods over algebraic methods, which might have limited the scope and applicability of calculus. Is this true? How did philosophy affect the development of calculus? And is philosophy to blame for the discovery of calculus taking more than 1000 years?&lt;/p&gt;&#xA;&lt;p&gt;Another factor that I think might have motivated the discovery of calculus is the problem of finding the area of a curve and the tangent line to a curve. These are two important problems in geometry and they require the concepts of derivative and integral. I know that Archimedes and other ancient mathematicians used the method of exhaustion to approximate the area of a curve, and that Fermat and other 17th century mathematicians used the method of adequality to find the tangent line to a curve. But why did it take so long to generalise and formalise these methods into calculus?&lt;/p&gt;&#xA;&lt;p&gt;I know this question might have a long and complex answer, so I will ask for a book or reference if the answers are too long or complex.&lt;/p&gt;&#xA;" OwnerUserId="18466" LastEditorUserId="18466" LastEditDate="2024-03-23T20:09:22.113" LastActivityDate="2024-03-27T18:29:38.350" Title="What were the obstacles that made the discovery of calculus very late?" Tags="|mathematics|reference-request|calculus|philosophy-of-science|book-recommendation|" AnswerCount="2" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16130" PostTypeId="1" CreationDate="2024-01-16T22:23:16.190" Score="1" ViewCount="172" Body="&lt;p&gt;On Math Stack Exchange, I am impressed by users' skill at finding closed form expressions for definite integrals. For example:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/562694/398708&quot;&gt;Example 1:&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx=4\pi\operatorname{arccot \sqrt{\phi}}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/4838494/398708&quot;&gt;Example 2:&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://math.stackexchange.com/q/4707354/398708&quot;&gt;Example 3:&lt;/a&gt; &lt;span class=&quot;math-container&quot;&gt;$\int_0^\infty \frac{\sin^2{(x\sin x)}}{x^2}dx=1$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Out of curiosity, how good was Newton at definite integration? More specifically:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What are some examples of non-trivial definite integrals that Newton evaluated?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="16797" LastEditorUserId="16797" LastEditDate="2024-01-17T09:18:57.013" LastActivityDate="2024-01-17T19:44:33.397" Title="How good was Newton at definite integration?" Tags="|mathematics|calculus|newton|real-analysis|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16131" PostTypeId="1" AcceptedAnswerId="16154" CreationDate="2024-01-17T09:47:49.473" Score="4" ViewCount="190" Body="&lt;p&gt;I have studied the prefixes &lt;em&gt;&amp;quot;Iso-&amp;quot;, &amp;quot;Sec-&amp;quot;, &amp;quot;Tert-&amp;quot; and &amp;quot;Neo-&amp;quot;&lt;/em&gt; for a long time in chemistry but wonder where they originate from i.e. where is the place (the paper or the journal) that they first appeared in the form that they are used today in modern-chemistry.&lt;/p&gt;&#xA;&lt;p&gt;About the prefixes of &amp;quot;&lt;em&gt;Neo-&lt;/em&gt;&amp;quot; and &amp;quot;&lt;em&gt;Iso-&lt;/em&gt;&amp;quot;, the earliest appearance of them I could find is by William Odling in his paper in 1876 cited below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Altogether, however, the names which call to mind that the normal, iso-, and neo-acids are mono-, di-, and trimethyl &amp;amp;c. derivatives would seem to be most advantageous. &lt;sup&gt;[1]&lt;/sup&gt;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;[1] Odling, W. (1876). &lt;a href=&quot;https://archive.org/details/londonedinburgh511876lond/page/n5/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Formulation of Paraffins and their&#xA;Derivatives&lt;/a&gt;.&#xA;&lt;em&gt;The London, Edinburgh and Dublin philosophical magazine and journal of science, ser.5:v.1 (Jan-June)&lt;/em&gt;, 216.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, about the prefixes of &amp;quot;&lt;em&gt;Sec-&lt;/em&gt;&amp;quot; and &amp;quot;&lt;em&gt;Tert-&lt;/em&gt;&amp;quot;, I could not find any definitive appearance of them (in the way we use them today) on any resource like - Wikipedia or OED. However, I did find the use of the words &amp;quot;tertiary&amp;quot; and &amp;quot;secondary&amp;quot; as they are used in modern-chemistry in several old sources dating back to the late 1800's. &lt;sup&gt;[2][3][4]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[2] Meyer, V. &amp;amp; Lecco, M. (1876). &lt;a href=&quot;https://ia600708.us.archive.org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1002%252Fcber.18750080137.zip&amp;amp;file=10.1002%252Fcber.18750080212.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zur Constitution der&#xA;Ammo-niumverbindungen&lt;/a&gt;.&#xA;&lt;em&gt;Berichte der deutschen chemischen Gesellschaft , 8(2)&lt;/em&gt;, 936-938.&lt;/p&gt;&#xA;&lt;p&gt;[3] Stoermer, R. (1898). &lt;a href=&quot;https://ia800708.us.archive.org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1002%252Fcber.189803101202.zip&amp;amp;file=10.1002%252Fcber.189803102232.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die Einwirkung von salpetriger Säure&#xA;auf secundäre aromatische&#xA;Amine&lt;/a&gt;.&#xA;&lt;em&gt;Berichte der deutschen chemischen Gesellschaft , 31(2)&lt;/em&gt;, 2523-2541.&lt;/p&gt;&#xA;&lt;p&gt;[4] Fischer, E. (1890). &lt;a href=&quot;https://ia600708.us.archive.org/view_archive.php?archive=/22/items/crossref-pre-1909-scholarly-works/10.1002%252Fcber.18890220290.zip&amp;amp;file=10.1002%252Fcber.189002301126.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Synthese des&#xA;Traubenzuckers&lt;/a&gt;.&#xA;&lt;em&gt;Berichte der deutschen chemischen Gesellschaft , 23(1)&lt;/em&gt;, 799-805.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So, I want to ask about the first occurrence of these prefixes as used in modern-chemistry and if the sources I provided and searched on have any relation to the history and origin of these prefixes.&lt;/p&gt;&#xA;" OwnerUserId="19382" LastActivityDate="2024-01-26T00:32:07.920" Title="History and origin of the Iso-, Sec-, Tert- and Neo- prefixes?" Tags="|terminology|chemistry|" AnswerCount="2" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16132" PostTypeId="1" AcceptedAnswerId="16153" CreationDate="2024-01-17T12:17:16.447" Score="4" ViewCount="166" Body="&lt;p&gt;As the title already says, I'm interested in the number of scientific articles published per year, preferably going back to the times of Newton or even Galilei (who is generally considered to be the founder of modern science). I already found a few resources (&lt;a href=&quot;https://ourworldindata.org/grapher/scientific-and-technical-journal-articles?tab=chart&quot; rel=&quot;nofollow noreferrer&quot;&gt;ourworldindata.org&lt;/a&gt;, &lt;a href=&quot;https://dblp.dagstuhl.de/statistics/publicationsperyear.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;dblp.org&lt;/a&gt;, &lt;a href=&quot;https://wordsrated.com/number-of-academic-papers-published-per-year/&quot; rel=&quot;nofollow noreferrer&quot;&gt;wordsrated.com&lt;/a&gt;, &lt;a href=&quot;https://academic.oup.com/gigascience/article/8/6/giz053/5506490&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;), but none of them are exactly what I'm looking for. Their data typically only cover a relatively short timespan, or don't include recent years.&lt;/p&gt;&#xA;" OwnerUserId="18187" LastEditorUserId="18187" LastEditDate="2024-01-19T10:57:03.340" LastActivityDate="2024-01-26T12:40:14.813" Title="Is there a good resource for the number of scientific literature published per year, potentially going back to the times of Newton or even Galilei?" Tags="|resource-recommendation|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16133" PostTypeId="2" ParentId="16120" CreationDate="2024-01-17T14:53:56.337" Score="4" Body="&lt;p&gt;Perhaps the question ought to be turned on its head: why did mathematical learning hit the accelerator in Ancient Greece and during the modern era in Western Europe (17th century to present)?&lt;/p&gt;&#xA;&lt;p&gt;Some historians (e.g., &lt;a href=&quot;https://press.princeton.edu/books/hardcover/9780691172187/escape-from-rome&quot; rel=&quot;nofollow noreferrer&quot;&gt;Walter Scheidel&lt;/a&gt;) have reasoned there were unique conditions in both eras and locations that helped foster the growth of mathematical and scientific knowledge: a fractured, competitive political geography that prevented a single authority controlling or stifling higher learning, and the presence of an economically prosperous, literate and educated class.&lt;/p&gt;&#xA;&lt;p&gt;Humans are naturally curious and certainly human knowledge has expanded in other times and places (the Roman and Chinese empires, ancient Egyptian, Mesopotamian, Indian civilizations) but at a slower pace.&lt;/p&gt;&#xA;" OwnerUserId="3888" LastActivityDate="2024-01-17T14:53:56.337" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16134" PostTypeId="1" AcceptedAnswerId="16136" CreationDate="2024-01-17T19:14:31.047" Score="1" ViewCount="142" Body="&lt;p&gt;E.g. a quote from &lt;a href=&quot;https://www.liebig-museum.de/wp-content/uploads/2020/11/Liebig_Chemische_Briefe_bk24.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Justus von Liebig&lt;/a&gt;, 17th Chemical Letter, 1858, in German:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Wir können ein Stück Zucker, auch wenn wir es noch so fein reiben, nicht flüssig machen, noch viel weniger können wir durch eine mechanische Gewalt ein Zuckeratom zersetzen, ein Kohlenstoff oder Wasserstoffatom davon losreissen.&lt;br&gt;&#xA;Wir können in einer Zuckerlösung durch Schütteln Zuckeratome und Wasseratome neben einander hin und her bewegen, aber die Elemente derselben wechseln damit ihren Platz nicht.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Literal translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We cannot make a piece of sugar liquid, no matter how finely we grind it, much less can we use mechanical force to decompose a sugar atom or tear away a carbon or hydrogen atom from it.&lt;br&gt;&#xA;We can move sugar atoms and water atoms back and forth next to each other in a sugar solution by shaking, but the elements do not change their place.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How widespread was it to call those chemical entities &amp;quot;atoms&amp;quot;, which are molecules &lt;em&gt;or&lt;/em&gt; atoms in modern terminology?&lt;/p&gt;&#xA;&lt;p&gt;Was this also common in the English literature?&lt;/p&gt;&#xA;" OwnerUserId="9479" LastActivityDate="2024-02-02T08:48:21.193" Title="Were molecules called atoms in the 19th century?" Tags="|terminology|chemistry|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16135" PostTypeId="2" ParentId="16130" CreationDate="2024-01-17T19:44:33.397" Score="6" Body="&lt;p&gt;Newton integrated in the 'Principia' at least three rate-expressions related to features of the motion of the moon. (Newton's convention for denoting such rates was to call them 'hourly motions'):&lt;/p&gt;&#xA;&lt;p&gt;(a) book 3: propositions 25-29, especially 26, identify the rate of change of the moon's areal velocity, and this rate was then integrated; (it would have been constant under Kepler's 2nd law in the absence of perturbations, but it does suffer periodic perturbation by the Sun, and Newton was interested in determining the resulting variation and succeeded, under certain simplifying approximations adopted to make the problem tractable within the limits of later-17th-century technique);&lt;/p&gt;&#xA;&lt;p&gt;(b) book 3: propositions 30-33 identify the rate of fluctuation of the regression of the moon's nodes, which is then integrated so as to find the difference between the true nodal motion and the mean due to the considered cause;&lt;/p&gt;&#xA;&lt;p&gt;(c) book 3: propositions 34-35 identify the rate of fluctuation of the inclination of the moon's orbital plane, and again there is an integration to find the difference between the true inclination and the mean due to the considered cause.&lt;/p&gt;&#xA;&lt;p&gt;All of these integrations were described in terms of Newton's geometrical formulations of the calculus, without any notation for equations or integrations that would be easily recognisable today (see e.g.  &lt;a href=&quot;https://hsm.stackexchange.com/questions/6673/how-did-newton-write-his-equations/6676#6676&quot;&gt;How did Newton write his equations?&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;They were however reviewed and explained with translation into  terms of later notation (much more accessible!) by no less a reviewer than Pierre-Simon Laplace, in the 1825 volume (in French only) of his &lt;a href=&quot;https://books.google.fr/books?id=PjmesIW_LzUC&amp;amp;pg=PA367&quot; rel=&quot;noreferrer&quot;&gt;'Traite de la Mecanique Celeste', tome V, book XVI, especially chapter 2&lt;/a&gt;, beginning at page 367 (though chapter 1, from page 349, is also a highly  recommendable read containing many interesting expressions of Laplace's point of view on Newton's work).&lt;/p&gt;&#xA;&lt;p&gt;Newton in an earlier phase of his work in the 1670s also gave a solution in the form of a power series to Kepler's (original) problem of determining the orbital position after a given time-interval under equal-area motion in an elliptical orbit. This solution seems to have involved power-series expansion of a rate expression that was then integrated term-by-term to give a trigonometrical function of a relevant angle (eccentric anomaly) which can then be further processed with basic trigonometry to give the main wanted result, the position-angle in the orbit after the given time-interval.&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2024-01-17T19:44:33.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16136" PostTypeId="2" ParentId="16134" CreationDate="2024-01-17T21:40:12.423" Score="4" Body="&lt;p&gt;Summary: I don't think there was a definite point when it changed meaning from &amp;quot;smallest component with the same chemistry&amp;quot; to uniquely &amp;quot;compound of atoms&amp;quot;. It was mostly a gradual transition between 1860 and 1908.&lt;/p&gt;&#xA;&lt;h2&gt;Origin of the word&lt;/h2&gt;&#xA;&lt;p&gt;&amp;quot;Molecule&amp;quot; as a word comes from Latin is very ancient and was not necessarily used for science. According to &lt;a href=&quot;https://www.etymonline.com/word/molecule&quot; rel=&quot;nofollow noreferrer&quot;&gt;Online Etymology Dictionary: Molecule&lt;/a&gt; it can be traced back to Réné Descartes. However the modern use of the word was given by Amadeus Avogadro in 1811 meaning &amp;quot;smallest part into which a substance can be divided without destroying its chemical character&amp;quot;. Note than an atom is usually defined as the &amp;quot;smallest indivisible component&amp;quot; all atoms are molecules under older definitions (but not all molecules are atoms).&lt;/p&gt;&#xA;&lt;h2&gt;Success of molecular theory&lt;/h2&gt;&#xA;&lt;p&gt;In the 19th century the concept of atoms and molecules was not clear. At the beginning of the century John Dalton had just published his idea of atoms, and atoms could combine together to form compounds.&lt;/p&gt;&#xA;&lt;p&gt;Note that during this period, some scientists were against the idea of molecules and atoms, so in an alternative theory a gas was considered to be composed of some indivisible fluid instead.&lt;/p&gt;&#xA;&lt;p&gt;Avogadro, and independently André-Marie Ampère advanced the idea of &amp;quot;composite molecules&amp;quot;. This hypothesis helped explain some inconsistencies with Gay-Lussac law of combining volumes and Dalton's theory.  According to Wikipedia &lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_molecular_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of molecular theory&lt;/a&gt; even if Avogadro reinvented the word, he still used molecule in the old sense:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Avogadro uses the name &amp;quot;molecule&amp;quot; for both atoms and molecules. Specifically, he uses the name &amp;quot;elementary molecule&amp;quot; when referring to atoms and to complicate the matter also speaks of &amp;quot;compound molecules&amp;quot; and &amp;quot;composite molecules&amp;quot;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The success of Avogadro-Ampère hypothesis is possibly due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Stanislao_Cannizzaro&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stanislao Cannizzaro&lt;/a&gt;. Some experimental investigations using organic chemistry were in accordance with Avogadro's law however when conducting similar experiments with certain inorganic substances, deviations from the law seemed apparent. It was Cannizzaro, addressing the &lt;a href=&quot;https://en.wikipedia.org/wiki/Karlsruhe_Congress&quot; rel=&quot;nofollow noreferrer&quot;&gt;Karlsruhe Congress&lt;/a&gt; in 1860, resolved this seeming contradiction, based on the ideas of Ampère and Avogadro (the later had died a couple of years before). He clarified that these deviations were a result of molecular dissociations occurring at specific temperatures. The hypothesis became Avogadro's law and made possible the measurements of atomic weights to distinguish between atoms and compound atoms. However Cannizzaro still uses molecules in the original sense, see &lt;em&gt;&lt;a href=&quot;https://archive.org/details/sketchofcourseof00cannrich/mode/2up?view=theater&amp;amp;q=molecule&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sketch of a Course of Chemical Philosophy.&lt;/a&gt;&lt;/em&gt;. Similarly August Kekulé, the congress organizer, wrote in a &lt;a href=&quot;https://web.lemoyne.edu/%7EGIUNTA/karlsruhe.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;note&lt;/a&gt;: &amp;quot;A molecule is 1, at most 2 atoms.&amp;quot;&lt;/p&gt;&#xA;&lt;h2&gt;Evolution of the meaning&lt;/h2&gt;&#xA;&lt;p&gt;The evolution from molecule as smallest part and molecule as compound atom took some time. In an article in 1881 Popular Science Monthly &amp;quot;&lt;a href=&quot;https://en.wikisource.org/wiki/Popular_Science_Monthly/Volume_19/September_1881/What_Is_a_Molecule%3F&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is a molecule&lt;/a&gt;&amp;quot;, the definition alternates between the two:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He &amp;quot;decomposes,&amp;quot; or breaks up, these molecules into &amp;quot;atoms&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;but also&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;and every molecule, though physically indivisible, can be broken up chemically into atoms, which are themselves the molecules&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;a similar mixed treatment is given by James Clerk Maxwell 1873 &amp;quot;&lt;a href=&quot;https://victorianweb.org/science/maxwell/molecules.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;Molecules&lt;/a&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In 1904, Walther Nersnt gives a similar mixed account but is already much closer to the modern meaning (in &lt;em&gt;&lt;a href=&quot;https://books.google.fr/books?hl=en&amp;amp;lr=&amp;amp;id=pm4PAwAAQBAJ&amp;amp;oi=fnd&amp;amp;pg=PA1&amp;amp;dq=Nernst%201895&amp;amp;ots=UswdNymRdK&amp;amp;sig=P1nzeifFxA_W7TgqUFRO0wD_jT4&amp;amp;redir_esc=y#v=onepage&amp;amp;q=Nernst%201895&amp;amp;f=false&quot; rel=&quot;nofollow noreferrer&quot;&gt;Theoretical Chemistry from the Standpoint of Avogardro's Rule &amp;amp; Thermodynamics&lt;/a&gt;&lt;/em&gt;):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Therefore these molecules must be divisible, and such a separation takes place when a compound decomposes into its elements. Thus we reach the assumption that a molecule does not continuously fill the space appropriated by it, but is a discrete aggregate of separable particles in its total space. These particles we call &amp;quot;atoms.&amp;quot; These atoms, by the union of which, molecules are made, are all alike if they belong to the molecule of one element, but different if they belong to the molecule of a chemical compound. Only in the first case [i.e. of an element] can it happen that a molecule may consist of a single atom. The force which binds the atoms in molecular union, we call the &amp;quot;chemical affinity&amp;quot; of the atom.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It is often considered that the final demonstration of the existence of molecules was given by Jean Perrin in his 1908, with his &lt;a href=&quot;https://web.mit.edu/swangroup/footer/perrin_bm.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;work on Brownian motion&lt;/a&gt;. But in this work he clearly refers to water as a molecule and hydrogen, oxygen, potassium and other elements as atoms, so the transition to the modern meaning was already cleared up by this time. Interestingly enough Perrin refers to the electron as the atom of electricity.&lt;/p&gt;&#xA;&lt;h3&gt;Comments&lt;/h3&gt;&#xA;&lt;p&gt;As suggested by OP, already in 1911, &lt;a href=&quot;https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Atom&quot; rel=&quot;nofollow noreferrer&quot;&gt;Encyclopedia Brittanica&lt;/a&gt; reads &amp;quot;a compound atom, or as we now say a “molecule”.&lt;/p&gt;&#xA;&lt;p&gt;Some of this text was based on: H.H. Kubbinga, &amp;quot;&lt;a href=&quot;https://doi.org/10.1016/0166-1280(88)80487-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;The first ‘molecular” theory (1620): Isaac Beeckman (1588–1637)&lt;/a&gt;&amp;quot; (1988)&lt;/p&gt;&#xA;&lt;p&gt;Suggestion: If you wish to deep further into when the transition was made I suggest finding a German speaker, I think something could be found in the works of van 't Hoff, Boltzmann, Weber, Lorentz and even in  Einstein's doctoral thesis.&lt;/p&gt;&#xA;&lt;h2&gt;More on the Karlsruhe Congress&lt;/h2&gt;&#xA;&lt;p&gt;After writing all this I have found that the question &amp;quot;What is a molecule and what is an atom&amp;quot; was one of the three main topics during the Karlsruhe Congress:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Shall a difference be made between the expressions “molecule” and “atom,” such that a molecule be named the smallest particle of bodies which can enter into chemical reactions and which may be compared to each other in regard to physical properties, atoms being the smallest particles of those bodies which are contained in molecules?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Mr. H. Kopp, summarizing the discussion, says that the need to separate the idea of the molecule from that of the atom appears to be established; that the notion of the molecule can be fixed with the help of purely chemical considerations; that the definition does not have to involve density alone; and, finally, that it appears natural to call the largest quantity the molecule, and the smallest quantity the atom. In concluding, the speaker formulates the first question to be put to the assembly. This question is as follows: &amp;quot;Is it appropriate to establish a distinction between the terms molecule and atom, and to call molecules, which are comparable as far as physical properties go, the smallest quantities of bodies which enter into or come out of a reaction, and to call atoms the smallest quantities of bodies which are contained in these molecules?&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-02-02T08:48:21.193" LastActivityDate="2024-02-02T08:48:21.193" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16137" PostTypeId="2" ParentId="16120" CreationDate="2024-01-17T23:14:59.713" Score="2" Body="&lt;p&gt;An aspect that has not been mentioned and is at least a partial answer is that progress must strongly depend on the number of people working in the field, which is the product of the population and the fraction of them working in the field. I'll focus on Europe here.&lt;/p&gt;&#xA;&lt;p&gt;The population in Europe &lt;a href=&quot;https://www.encyclopedia.com/international/encyclopedias-almanacs-transcripts-and-maps/population-europe-early-modern-demographic-patterns&quot; rel=&quot;nofollow noreferrer&quot;&gt;doubled&lt;/a&gt; between 1500 and 1750; consequently, &lt;em&gt;ceteris paribus,&lt;/em&gt; the number of people working in the field (I avoid the term &amp;quot;mathematician&amp;quot; because many scholars at the time were polymaths) should have doubled. Without any reference I can imagine that there was only a handful of people of the kind who would advance mathematics (or any other field) until maybe the early 1600s, in the entirety of Europe.&lt;/p&gt;&#xA;&lt;p&gt;And the &lt;em&gt;fraction&lt;/em&gt; of the population working in the field should also have increased. Life in Europe until the late middle-ages intermittently devolved into a fight for survival, often unsuccessful. Europe-scale famines and pandemics ravaged the continent in the middle-ages. While both didn't disappear later, they became less frequent.&lt;sup&gt;&lt;a href=&quot;https://books.openbookpublishers.com/10.11647/obp.0323/ch13.xhtml&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;With greater wealth and a growing surplus in food came the opportunity for a division of labor; professionals who did not directly contribute to food production could be sustained, like musicians, artists and scholars. At the same time, trade and crafts made education more important; in the cities, an educated citizenship emerged, among them teachers. Courts were able to host scholars like Leibniz and Voltaire.&lt;/p&gt;&#xA;&lt;p&gt;That alone must be one reason the speed of mathematical development, like everything else,  picked up after 1500. Working hypothesis: That other regions, like the Arabian world, were more productive may be an indicator for greater prosperity.&lt;/p&gt;&#xA;" OwnerUserId="9476" LastActivityDate="2024-01-17T23:14:59.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16138" PostTypeId="2" ParentId="14638" CreationDate="2024-01-18T09:51:54.047" Score="15" Body="&lt;p&gt;I’m Nelson Stephens’ daughter. He was born on 6th May 1941 and was a Professor of Mathematics at the University of London. He passed away on 8th January 2024. Not sure I can be much help on the maths but very cool to see what an impact he had :)&lt;/p&gt;&#xA;" OwnerUserId="19584" LastActivityDate="2024-01-18T09:51:54.047" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16139" PostTypeId="2" ParentId="16129" CreationDate="2024-01-18T16:32:14.487" Score="8" Body="&lt;p&gt;I would like to make several points with regard to this interesting&#xA;question.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;The discovery of some Taylor series of trig functions by the Kerala&#xA;school is a very impressive early breakthrough.  However, I believe&#xA;the consensus of historians is that this does not in any way&#xA;constitute significant progress toward the invention of something on&#xA;the scale of the calculus.  Similar remarks apply to many other&#xA;multicultural achievements touted at answers to some parallel&#xA;questions.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Note that by the time of James Gregory, for example, many power&#xA;series were already known, before the invention of the calculus.&#xA;Today we tend to think of power series as part of the calculus, but&#xA;historically they were available well before, including (as I&#xA;mentioned) in Kerala early on.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;With regard to Aristotle: The issue is not so much Aristotle&#xA;himself but rather some Aristotelian doctrines as developed by&#xA;medieval catholic theologians, including Aquinas.  One of such&#xA;doctrines was the doctrine of hylomorphism which, so they claimed,&#xA;stemmed from Aristotle.  Hylomorphism was viewed as the rival doctrine&#xA;of atomism/indivisibles.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Hylomorphism since at least Aquinas was viewed as a theoretical&#xA;background necessary for the catholic interpretation of the eucharist.&#xA;Therefore indivisibles were viewed with great suspicion by some&#xA;catholic theologians, including jesuit mathematicians such as Guldin,&#xA;Tacquet, and others.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In a forthcoming article on Cavalieri, we show how religious&#xA;resistance to his method of indivisibles (a precursor of integration)&#xA;was the source of much of the opposition to his work and the work of&#xA;his disciple Stefano degli Angeli.  Both Cavalieri and degli Angeli&#xA;were jesuats.  The order was suppressed by papal bull in 1668.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;In this sense, it can be said that philosophical and theological&#xA;doctrines were an impediment to the development of the calculus, where&#xA;ideas related to indivisibles were needed for the eventual&#xA;breakthrough by Leibniz and Newton.  Such doctrines may have contributed&#xA;to the delay that you mentioned.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You ask also for a book dealing with these issues.  While we don't&#xA;have a book, there is a number of articles you can consult&#xA;&lt;a href=&quot;https://u.math.biu.ac.il/%7Ekatzmik/infinitesimals.html&quot; rel=&quot;noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;You ask also about Aristotle and infinity.  Medieval scholastics interpreted Aristotle as introducing a distinction between potential infinity and actual infinity, and rejecting the actual sort.  This dogma was similarly used against scholars attempting to explore infinitesimal mathematics.  Thus, Paul Guldin wrote in his book that Cavalieri's indivisibles involve viewing a plane region as consisting of an actual infinity of parallel lines (Cavalieri himself denied this), and therefore meaningless, and therefore any results obtained by means of indivisibles are false.  Cavalieri and degli Angeli were never able to establish a school practicing these methods.  James Gregory, who visited degli Angeli in the 1660s, saw his books suppressed in Italy.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2024-01-21T10:01:23.900" LastActivityDate="2024-01-21T10:01:23.900" CommentCount="14" ContentLicense="CC BY-SA 4.0" />
  <row Id="16141" PostTypeId="1" CreationDate="2024-01-19T06:35:55.430" Score="0" ViewCount="38" Body="&lt;p&gt;I'm curious as to the origins of thinking of displacement as the area under a v/t curve.&lt;/p&gt;&#xA;&lt;p&gt;I assume that Newton (and/or Leibniz) was already familiar with the concept and knew that calculating the area was simple for linear curves, but developed his new &amp;quot;calculus&amp;quot; in order to calculate areas for more complex curves.&lt;/p&gt;&#xA;&lt;p&gt;I am also aware of the Merton rule, developed a few centuries before Newton, which proved more-or-less that the average velocity for an object experiencing constant acceleration would be exactly halfway between the initial and final velocities. I am not sure as to the methods used to prove the Merton rule.&lt;/p&gt;&#xA;&lt;p&gt;I think that the concept of &amp;quot;area under the curve&amp;quot; couldn't really come about until coordinate plane graphing was invented, which I believe started with Descartes.&lt;/p&gt;&#xA;&lt;p&gt;So what is the first reference to calculating areas under curves? What was the context of this first use?&#xA;What is the first reference to doing so on a v/t graph specifically to calculate displacement?&lt;/p&gt;&#xA;" OwnerUserId="19591" LastActivityDate="2024-01-19T06:35:55.430" Title="What are the origins of calculating the area of a v/t graph to determine displacement?" Tags="|physics|calculus|newton|descartes|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16142" PostTypeId="2" ParentId="13460" CreationDate="2024-01-19T13:34:45.960" Score="0" Body="&lt;p&gt;The guy I am trying to take on as my PhD adviser has his genealogy go up to Ibn-i Sina, the founder of 'modern' medicine, that goes back 200 years further back than Sharaf al-Tusi, to 1086. This is bonkers to think about does not mean anything really.&lt;/p&gt;&#xA;" OwnerUserId="19594" LastActivityDate="2024-01-19T13:34:45.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16143" PostTypeId="1" AcceptedAnswerId="16151" CreationDate="2024-01-19T18:16:22.350" Score="4" ViewCount="120" Body="&lt;p&gt;In a history lecture a while back the teacher spoke about a person who wrote something that (to me) sounded like an instruction manual on how to write &amp;quot;good science&amp;quot;. The basic point was that the scientific texts shouldn't include flowery language and should only include the facts.&lt;/p&gt;&#xA;&lt;p&gt;I thought I remembered it being Robert Boyle, but for him I can only find things about his chemistry, so I'm likely mistaken about him.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone know who that might have been?&lt;/p&gt;&#xA;&lt;p&gt;Thanks!&lt;/p&gt;&#xA;&lt;p&gt;Prior research: I've done extensive web searches about this, but as one might expect, it's difficult to find a person's name (and one  that far back at that) if one only has such vague ideas like having written a manual on &amp;quot;good science&amp;quot;. That said, I can say with relative certainty that it was not Robert Boyle.&lt;/p&gt;&#xA;&lt;p&gt;I've also been reading the basics (see below) to see if anything rings a bell, but so far I've not found anything:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;Morus, Iwan Rhys. 2010. “Placing Performance.” Isis 101 (4): 775–78.&#xA;&lt;a href=&quot;https://doi.org/10.1086/657476&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1086/657476&lt;/a&gt;. Shapin, Steven. 2003. “The Image of&#xA;the Man of Science.” In The Cambridge History of Science, edited by&#xA;Roy Porter, 1st ed., 159–83. Cambridge University Press.&#xA;&lt;a href=&quot;https://doi.org/10.1017/CHOL9780521572439.008&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1017/CHOL9780521572439.008&lt;/a&gt;. ———. 2010.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Never Pure:&#xA;Historical Studies of Science as If It Was Produced by People with&#xA;Bodies, Situated in Time, Space, Culture, and Society, and Struggling for Credibility and Authority. Baltimore: Johns Hopkins University&#xA;Press.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Sleigh, Charlotte. 2011. Literature and Science. Outlining&#xA;Literature. Houndsmill, Basingstoke, Hampshire ; New York: Palgrave&#xA;Macmillan.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="19596" LastActivityDate="2024-01-21T17:02:59.643" Title="Who was this person in the Early Modern Period who wrote a manual on writing &quot;good science&quot;?" Tags="|science|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16144" PostTypeId="1" CreationDate="2024-01-20T01:57:16.860" Score="1" ViewCount="97" Body="&lt;p&gt;I know that Maclane and Eilenberg created category theory together. And I also know that lattice theory is a concept created by Dedekind with the motivation of ring and ideal theory. And this concept was developed by Ore and  Birkhoff. However, when I studied lattice theory, I saw that it started from the isomorphism of sets and then gradually expanded to topology space. So, I wonder if lattice theory may have influenced the emergence of category theory?&lt;/p&gt;&#xA;&lt;p&gt;This is what I mean by &amp;quot;lattice&amp;quot; in this question.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A lattice is an algebraic structure &lt;span class=&quot;math-container&quot;&gt;${\displaystyle (L,\vee ,\wedge )}$&lt;/span&gt;, consisting of a set &lt;span class=&quot;math-container&quot;&gt;${\displaystyle L}$&lt;/span&gt; and two binary, commutative and associative operations &lt;span class=&quot;math-container&quot;&gt;${\displaystyle \vee }$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;${\displaystyle \wedge }$&lt;/span&gt; on &lt;span class=&quot;math-container&quot;&gt;${\displaystyle L}$&lt;/span&gt; satisfying the following axiomatic identities for all elements &lt;span class=&quot;math-container&quot;&gt;${\displaystyle a,b\in L}$&lt;/span&gt; (sometimes called absorption laws):&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle a\vee (a\wedge b)=a}$&lt;/span&gt; ,&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle a\wedge (a\vee b)=a}$&lt;/span&gt;&#xA;The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together. These are called idempotent laws.&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle a\vee a=a}$&lt;/span&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle a\wedge a=a}$&lt;/span&gt;&#xA;These axioms assert that both &lt;span class=&quot;math-container&quot;&gt;${\displaystyle (L,\vee )}$&lt;/span&gt; and&#xA;&lt;span class=&quot;math-container&quot;&gt;${\displaystyle (L,\wedge )}$&lt;/span&gt; are semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the dual of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same partial order.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="55" LastEditDate="2024-01-20T20:56:44.677" LastActivityDate="2024-01-20T20:56:44.677" Title="Was there historical influence of lattice theory on category theory?" Tags="|category-theory|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16146" PostTypeId="2" ParentId="16126" CreationDate="2024-01-20T09:11:09.433" Score="3" Body="&lt;p&gt;Miller (1981) specifically addressed your question in Chapter 11 of this book:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 978-0-201-04679-3&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;He wrote&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... problems concerning the reflection of light from the moving mirror,&#xA;due to their fundamental relationship to both electromagnetism and&#xA;thermodynamics, were the theme of many papers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;He cited the following papers by &lt;a href=&quot;https://en.wikipedia.org/wiki/Hendrik_Lorentz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hendrik Lorentz&lt;/a&gt;, who discussed solutions to first order in v/c:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;De l'influence du mouvement de la terre sur les phénomènes lumineux,&lt;/em&gt; Vers. Kon. Akad. Wetensch. Amsterdam, 2, 297 (1886). Reprinted in Collected Papers, 4, pp. 153-214.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;On the Reflection of light by Moving bodies,&lt;/em&gt; Vers. Kon. Akad. Wetensch. Amsterdam, 1, 28 (1892). Reprinted in Collected Papers, 4, pp. 215-218.&lt;/li&gt;&#xA;&lt;li&gt;&lt;em&gt;Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern&lt;/em&gt; (Leiden: Brill, 1895). Reprinted in Collected Papers, 5, pp. 1-137.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;However, the most important paper was written by &lt;a href=&quot;https://en.wikipedia.org/wiki/Max_Abraham&quot; rel=&quot;nofollow noreferrer&quot;&gt;Max Abraham&lt;/a&gt; in 1904:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;em&gt;Zur Theorie der Strahlung und des Strahlungsdruckes,&lt;/em&gt; Ann. Phys., 14, 236-287 (1904).&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Abraham's results are valid for all orders in v/c. However, he only discussed the problem with respect to a frame is which the reflecting surface was moving, and his method based on classical electromagnetism was extremely complicated.&lt;/p&gt;&#xA;&lt;p&gt;On the contrary, Einstein in 1905 used the methods of special relativity, which allowed him to achieve all those results by using the rest frame of the reflecting surface, thereby significantly simplifying the computation. As pointed out by Miller:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the three pages of §8 Einstein deduced the transformation equation&#xA;for the energy of a light complex, and then solved exactly the&#xA;problems of determining the characteristics of plane waves incident on&#xA;a perfectly reflecting surface that is in inertial motion with&#xA;velocity v, and of calculating the pressure of light on this surface&#xA;[§8, 1-2]. Abraham (1904) spent forty pages to solve these problems.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So Einstein only needed 3 pages to achieve results for which Abraham required 40 pages...&lt;/p&gt;&#xA;&lt;p&gt;Regarding the experiments to which Einstein alluded to, Miller points to those of Lebedev (1900) as well as Nichols &amp;amp; Hull (1901) on &lt;a href=&quot;https://en.wikipedia.org/wiki/Radiation_pressure&quot; rel=&quot;nofollow noreferrer&quot;&gt;radiation pressure&lt;/a&gt;, which were also referenced by Abraham (1904).&lt;/p&gt;&#xA;" OwnerUserId="7601" LastEditorUserId="7601" LastEditDate="2024-01-20T19:19:07.657" LastActivityDate="2024-01-20T19:19:07.657" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16147" PostTypeId="1" AcceptedAnswerId="16148" CreationDate="2024-01-20T09:29:50.140" Score="3" ViewCount="353" Body="&lt;p&gt;According to Van Der Waerden's &amp;quot;&lt;em&gt;Science Awakening&lt;/em&gt;&amp;quot;, it was common for Ancient Greek mathematicians to use a board filled with sand to draw their figures, ie :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;But the ancients made their geometric drawings on a board (plinthium, abacium, abacus) covered with sand. This becomes clear from the statements in Iamblichus, Apuleius and Hieronymus which Winter has brought together&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The reference given is Winter's &amp;quot;&lt;em&gt;Der Tod des Archimedes&lt;/em&gt;&amp;quot;, which is however exceedingly hard to find, not being digitized or sold on any website. Would anyone happen to know what those exact statements are, by any chance?&lt;/p&gt;&#xA;" OwnerUserId="2628" LastEditorUserId="6609" LastEditDate="2024-01-20T12:00:54.770" LastActivityDate="2024-01-21T17:38:35.717" Title="Ancient drawing board in mathematics" Tags="|mathematics|geometry|ancient-greece|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16148" PostTypeId="2" ParentId="16147" CreationDate="2024-01-20T18:40:34.383" Score="7" Body="&lt;p&gt;I do not know Winter's text, at any rate I think the references to Iamblichus and Apuleius.&lt;/p&gt;&#xA;&lt;p&gt;For Iamblichus, the reference is definitely to &lt;a href=&quot;https://archive.org/details/Taylor1818IamblichusLifeOfPythagoras/page/9/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Life of Pythagoras&lt;/em&gt;, Chapter V&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Though no one therefore attended to him, and no one was genuinely desirous of those disciplines which he endeavoured by all means to introduce among the Greeks, yet he neither despised nor neglected Samos, because it was his country, and therefore wished to give his fellow-citizens a taste of the sweetness of the mathematical disciplines, though they were unwilling to be instructed in them. With a view to this, therefore, he employed the following method and artifice. Happening to observe a certain youth, who was a great lover of gymnastic and other corporeal exercises, but otherwise poor and in difficult circumstances, playing at ball in the Gymnasium with great aptness and facility, he thought the young man might easily be persuaded to attend to him, if he was sufficiently supplied with the necessaries of life, and freed from the care of procuring them. As soon as the youth, therefore, left the bath, Pythagoras called him to him, and promised that he would furnish him with every thing requisite to the support of his bodily exercise, on condition that he would receive from him gradually and easily, but continually, so that he might not be burthened by receiving them at once, certain disciplines, which he said he had learnt from the Barbarians in his youth, but which now began to desert him through forgetfulness and the incursions of old age. But the young man immediately acceded to the conditions, through the hope of having necessary support. Pythagoras, therefore, endeavoured to instruct him in the disciplines of arithmetic and geometry, &lt;strong&gt;forming each of his demonstrations in an abacus, and giving the youth three oboli as a reward for every figure which he learnt.&lt;/strong&gt; This also he continued to do for a long time, exciting him to the geometrical theory by the desire of honour; diligently, and in the best order, giving him (as we have said) three oboli for every figure which he apprehended. But when the wise man observed that the elegance, sweetness, and connexion of these disciplines, to which the youth had been led in a certain orderly path, had so captivated him that he would not neglect their pursuit though he should suffer the extremity of want, he pretended poverty, and an inability of giving him three oboli any longer. But the youth on hearing this replied, “I am able without these to learn and receive your disciplines.” Pythagoras then said, “But I have not the means of procuring sufficient nutriment for myself.” As it is requisite, therefore, to labour in order to procure daily necessaries and mortal food, &lt;strong&gt;it would not be proper that his attention should be distracted by the abacus&lt;/strong&gt;, and by stupid and vain pursuits. The youth, however, vehemently abhorring the thought of discontinuing his studies, replied: “I will in future provide for you, and repay your kindness in a way resembling that of the stork: for I in my turn will give you three oboli for every figure.”&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;For Apuleius, the reference is to &lt;em&gt;Pro se de magia liber&lt;/em&gt; (Apology) 16.7:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quem tu librum, Aemiliane, si nosses ac non modo campo et glebis, uerum etiam abaco et puluisculo te dedisses, mihi istud crede, quanquam teterrimum os tuum minimum a Thyesta tragico demutet, tamen profecto discendi cupidine speculum inuiseres et aliquando relicto aratro mirarere tot in facie tua sulcos rugarum.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And if you, Aemilianus, had known this book [the book is the lost Archimedes' &lt;em&gt;Catoptrics&lt;/em&gt;] and had given yourself not only to the field and the glebes, but also to the &lt;strong&gt;abacus and the sand&lt;/strong&gt;, believe me, even with this monstrous appearance of yours as a Tiestean mask, you would, no doubt, for the passion of study, have gone to the mirror and sometimes, having stopped the plow, you would have beheld on your face the many furrows of wrinkles.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The reference to Hieronymus is not very clear to me, and I am also not entirely sure which Hieronymus is being talked about.&lt;/p&gt;&#xA;&lt;p&gt;Thanks to njuffa who identified the correct reference: it is the commentary to the &lt;a href=&quot;https://en.wikipedia.org/wiki/Ezekiel_4&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chapter 4 of the Book of Ezekiel&lt;/a&gt; of &lt;a href=&quot;https://en.wikipedia.org/wiki/Jerome&quot; rel=&quot;nofollow noreferrer&quot;&gt;St. Jerome&lt;/a&gt;, that can be found in &lt;a href=&quot;https://www.google.com/books/edition/Operum_D_Hieronymi/-PMMgmOUkVsC?hl=en&amp;amp;gbpv=1&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Opere D. Hieronymi Quintus Tomus Commentarios in prophetas quos maiores vocant, continet&lt;/em&gt;, Basel 1553, p. 387&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Symmachus manifestius interpretatus est πλινϑεϊον, quem nos laterculum et abacum appellare possumus. In cuius puluere solent geometræ, γραμμας, id est, lineas radiosque describere.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Symmachus_(translator)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Symmachus&lt;/a&gt; translates more clearly as &lt;em&gt;πλινϑεϊον&lt;/em&gt; what we may call &lt;em&gt;laterculum&lt;/em&gt; or &lt;em&gt;abacum&lt;/em&gt;, which is the instrument in whose sand the geometers make their &amp;quot;γραμμας&amp;quot;, i.e., drawings formed by lines and rays.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The passage of the Bible traslated by Symmachus is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.biblegateway.com/passage/?search=Ezekiel%204:1&amp;amp;version=nkjv&quot; rel=&quot;nofollow noreferrer&quot;&gt;You also, son of man, take a &lt;em&gt;clay tablet&lt;/em&gt; and lay it before you, and portray on it a city, Jerusalem.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that Jerome is not a mathematician, so it is doubtful that he had direct knowledge of the instrument he is talking about. This is my personal impression, corroborated by the passage &amp;quot;drawings formed by lines and rays.&amp;quot; Here, lines and rays does not make much sense, one would rather expect something like &amp;quot;lines and circles&amp;quot;; my personal impression is that Jerome has incorrectly interpreted some passage from Virgil or Cicero (see below) and confuses the meaning of the term &amp;quot;radius,&amp;quot; which can also denote the radius of a circle, but mainly designates the rod for drawing diagrams in the sand.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;There is also another Hieronymus, &lt;a href=&quot;https://en.wikipedia.org/wiki/Hieronymus_of_Rhodes&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hieronymus of Rhodes&lt;/a&gt;, belonging to the school of Aristotle and therefore also called Hieronymus Peripateticus.&#xA;He is cited by Diogenes Laertius as the source of the account concerning Thales' journey to Egypt and the measurement of the pyramids, and he is also often quoted by Cicero (although it may be doubted that he is the same Hieronymus, and not an even more obscure mathematician) as a philosopher.&#xA;And speaking of Cicero, although I was unable to find the reference to the correct Hieronymus, may I add an interesting quote from &lt;a href=&quot;https://la.wikisource.org/wiki/De_natura_deorum/Liber_II&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De natura deorum&lt;/em&gt; 2, 18&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Interea Vellei noli quaeso prae te ferre vos plane expertes esse doctrinae. conum tibi ais et cylindrum et pyramidem pulchriorem quam sphaeram videri, novum etiam oculorum iudicium habetis. Sed sint ista pulchriora dumtaxat aspectu -- quod mihi tamen ipsum non videtur; quid enim pulchrius ea figura, quae sola omnis alias figuras complexa continet, quaeque nihil asperitatis habere, nihil offensionis potest, nihil incisum angulis nihil anfractibus, nihil eminens nihil lacunosum; cumque duae formae praestantissimae sint, ex solidis globus (sic enim σφαῖρα interpretari placet), ex planis autem circulus aut orbis, qui κύκλος Graece dicitur, his duabus formis contingit solis ut omnes earum partes sint inter se simillumae a medioque tantum absit extremum, quo nihil fieri potest aptius -- sed si haec non videtis, &lt;strong&gt;quia numquam eruditum illum pulverem attigistis,&lt;/strong&gt; ne hoc quidem physici intellegere potuistis, hanc aequabilitatem motus constantiamque ordinum in alia figura non potuisse servari?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikisource.org/wiki/On_the_Nature_of_the_Gods_(Yonge)/Book_2&quot; rel=&quot;nofollow noreferrer&quot;&gt;i.e.&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the mean while, Velleius, let me entreat you not to be always saying that we are utterly destitute of every sort of learning. The cone, you say, the cylinder, and the pyramid, are more beautiful to you than the sphere. This is to have different eyes from other men. But suppose they are more beautiful to the sight only, which does not appear to me, for I can see nothing more beautiful than that figure which contains all others, and which has nothing rough in it, nothing offensive, nothing cut into angles, nothing broken, nothing swelling, and nothing hollow; yet as there are two forms most esteemed, the globe in solids (for so the Greek word σφαῖρα, I think, should be construed), and the circle, or orb, in planes (in Greek, κύκλος); and as they only have an exact similitude of parts in which every extreme is equally distant from the centre, what can we imagine in nature to be more just and proper? But if you have never raked into this &lt;strong&gt;learned dust&lt;/strong&gt; to find out these things, surely, at all events, you natural philosophers must know that equality of motion and invariable order could not be preserved in any other figure.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here the &amp;quot;learned dust&amp;quot; (or sand), as in Apuleius, refers to the instruments of geometers, and then it is a classical locution to refer to mathematical knowledge.&lt;/p&gt;&#xA;&lt;p&gt;There are also many other examples in Greek and Latin literature of the use of figures drawn in sand. A very famous one is found in the episode of the dialogue with the slave in Plato's &lt;em&gt;Menon&lt;/em&gt;, in which Plato draws figures on the ground. Another example is found in the first of three versions of Archimedes' death reported in Plutarch's &lt;em&gt;Life of Marcellus&lt;/em&gt; (here Archimedes is drawing figures in the sand when a Roman soldier discovers and kills him). In these two episodes there is no direct reference to the use of the &lt;em&gt;abacus&lt;/em&gt; or &lt;em&gt;aβαχ&lt;/em&gt; (term derived from the ancient Phoenician word &lt;em&gt;abak&lt;/em&gt;, meaning dust), but the reader is left to imagine the scene. By contrast, a lesser-known but perhaps more relevant example is found in Persius' first satire in which the author explicitly mentions the &amp;quot;abaco numeros et secto in puluere metas&amp;quot;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;pre&gt;&lt;code&gt;inde uaporata lector mihi ferueat aure,&#xA;non hic qui in crepidas Graiorum ludere gestit&#xA;sordidus et lusco qui possit dicere 'lusce,'&#xA;sese aliquem credens Italo quod honore supinus&#xA;fregerit heminas Arreti aedilis iniquas,      &#xA;nec qui abaco numeros et secto in puluere metas&#xA;scit risisse uafer, multum gaudere paratus&#xA;si cynico barbam petulans nonaria uellat.&#xA;his mane edictum, post prandia Callirhoen do.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;pre&gt;&lt;code&gt;I want readers with cleansed ears, fired by such stuff, not&#xA;Some wretch who delights in poking fun at Greek sandals;&#xA;A one-eyed man who loves to call another man ‘One-eye’,&#xA;Who thinks he’s something, full of provincial importance,&#xA;Because at Arretium, as aedile, he ruled on half-measures,&#xA;Not some crafty fellow who’s used to jeering at maths&#xA;On the abacus, or diagrams drawn in the furrowed dust,&#xA;One ready to howl with delight when some insolent whore&#xA;Tugs at a Cynic’s beard. To them I’d recommend reading&#xA;Posters in the morning, a novel, say Callirhoe, after lunch.&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&amp;quot;Secto in pulvere metas&amp;quot; means literally &amp;quot;conical section [drawn on the] dust&amp;quot;. So, in this passage, Persius is referring to the &amp;quot;dust abacus&amp;quot; (which differs from the &amp;quot;lapilli abacus&amp;quot; which originates the counting frames for arithmetic calculations), a rectangular tablet, made of wood or bronze, on which green-colored powder (&amp;quot;pulvis hyalinus&amp;quot;) was glued, allowing numerical symbols and geometric figures to be traced with a stick (&amp;quot;radius&amp;quot;), using it, in this way, as we use a blackboard today.&lt;/p&gt;&#xA;&lt;p&gt;This instrument is mentioned again by Cicero in his &lt;em&gt;Tuscolanae Disputationes&lt;/em&gt; (Quaestio V, 23):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;ex eadem urbe humilem homunculum a pulvere et radio excitabo, qui multis annis post fuit Archimedes&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From the same city I shall awaken, from the dust and the stick with which one writes in the sand, a humble little man who lived many years later, Archimedes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;from which it appears that Archimedes used this type of abacus, to draw geometric figures; but also Virgil mentions it in his &lt;em&gt;Eglogues&lt;/em&gt; (III, 40)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;pre&gt;&lt;code&gt;si quis fuit alter&#xA;Descripsit radio totum qui gentibus orbem&#xA;&lt;/code&gt;&lt;/pre&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;who was the other, who to men drew with his stick all the sky.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;A description of this type of abacus, but much later than Virgil and Cicero, is found in &lt;a href=&quot;https://archive.org/details/denuptiisphilolo00martuoft/page/486/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;De Nuptiis Philologiae et Mercuri&lt;/em&gt;, Liber VI, De Geometria&lt;/a&gt;, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Martianus_Capella#De_nuptiis&quot; rel=&quot;nofollow noreferrer&quot;&gt;Martianus Capella&lt;/a&gt; (5th century)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Parent denique jam ingressurae artes obsequio electissimae feminarum, quae decentem quandam atque hyalini pulveris respersione coloratam velut mensulam gestitantes ad medium superi senatus locum fiducia promtiore procedunt [...] Illud quippe quod gerulae detulerunt, abacus nuncupatur, res depingendis designandisque opportuna formis; quippe ubi vel lineares ductus, vel circulares flexus, vel triangulares arraduntur anfractus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Finally, the arts show themselves in deference to the most chosen women, who, carrying a table of a decent sort and colored with a sprinkling of hyaline powder, confidently advance to the middle of the upper senate.[...] In fact, what the handmaids brought is called an abacus and is used to draw and delineate things into proper shapes by scraping straight, curved or triangular figures on its surface.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2024-01-21T17:38:35.717" LastActivityDate="2024-01-21T17:38:35.717" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16149" PostTypeId="2" ParentId="3131" CreationDate="2024-01-21T09:16:25.793" Score="0" Body="&lt;p&gt;Sophie Germaine and hypatia.Sophie Germaine died at the age of 55 due to breast cancer. Hypatia was brutally murdered by a mob of Christian fanatics. They pulled her from her carriage on a street in Alexandria, dragged her to a church, stripped her naked, beat her to death and/or flayed her, tore off her limbs, and burned her remains.Damascius adds that they also cut out her eyeballs.&lt;/p&gt;&#xA;" OwnerUserId="19603" LastActivityDate="2024-01-21T09:16:25.793" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16150" PostTypeId="2" ParentId="16127" CreationDate="2024-01-21T16:21:26.197" Score="2" Body="&lt;p&gt;A mathematical relation linking the &amp;quot;harmonic series&amp;quot; with the &amp;quot;harmonic mean&amp;quot; is the following:&lt;/p&gt;&#xA;&lt;p&gt;Let &lt;span class=&quot;math-container&quot;&gt;$\{s_n\}$&lt;/span&gt; be the harmonic series&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$s_1=1,\; s_2 = \frac 12, \; s_3 = \frac 13,...$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Then&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The harmonic mean of &lt;span class=&quot;math-container&quot;&gt;$s_{n-1}, \, s_{n+1}$&lt;/span&gt; is equal to &lt;span class=&quot;math-container&quot;&gt;$s_n$&lt;/span&gt;,&#xA;&lt;span class=&quot;math-container&quot;&gt;$$H\left(s_{n-1}, \, s_{n+1}\right) = s_n$$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;PROOF&lt;/strong&gt;&#xA;&lt;span class=&quot;math-container&quot;&gt;$$H\left(s_{n-1}, \, s_{n+1}\right) = \frac {2}{\frac 1 {s_{n-1}} + \frac 1 {s_{n+1}}} = \frac{2}{n-1 + n+1} = \frac 1n = s_n$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;A characterization that we find in an ancient text, quoted in&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Pythagorean_means#History&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/Pythagorean_means#History&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;says&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There are three means in music: one is arithmetic, second is the&#xA;geometric, third is sub-contrary, which they call harmonic. The mean&#xA;is arithmetic when three terms are in proportion such that the excess&#xA;by which the first exceeds the second is that by which the second&#xA;exceeds the third. (...) The mean is the geometric when they are such&#xA;that as the first is to the second, so the second is to the third.&#xA;(...) Subcontrary, which we call harmonic, is the mean when they are&#xA;such that, by whatever part of itself the first term exceeds the&#xA;second, by that part of the third the middle term exceeds the third.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So each satisfied a constant relation/was characterized by a distinct property, and the human mind needs no more to take an interest.&lt;/p&gt;&#xA;&lt;p&gt;The definition above for the &lt;strong&gt;arithmetic&lt;/strong&gt; mean is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$A:\;\; A -a = b-A \implies A(a,b) = \frac{a+b} 2.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The definition for the &lt;strong&gt;geometric&lt;/strong&gt; mean is&#xA;&lt;span class=&quot;math-container&quot;&gt;$$G:\;\; \frac a G = \frac G b \implies ab = G^2 \implies G(a,b) = (ab)^{1/2}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The definition for the &lt;strong&gt;harmonic&lt;/strong&gt; mean is more convoluted. It says for &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; the &amp;quot;part&amp;quot;, it is of the same value, namely it holds that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$H:\;\; \begin{cases} a-H = pa \\ H-b = pb \end{cases} \implies \begin{cases} p = \frac{a-H}{a} \\ p = \frac{H-b}{b}\end{cases}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Equating right-hand-sides we get&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\frac{a-H}{a} = \frac{H-b}{b} \implies H(a,b) =  \frac {2ab}{b+a} = \frac {2}{\frac {b+a}{ab}} = \frac 2 {\frac 1a + \frac 1b}.$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;An apparently on-the-spot piece of literature, with references is&lt;/p&gt;&#xA;&lt;p&gt;Komić, J. (2011). Harmonic Mean. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. &lt;a href=&quot;https://doi.org/10.1007/978-3-642-04898-2_645&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://doi.org/10.1007/978-3-642-04898-2_645&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="244" LastEditorUserId="244" LastEditDate="2024-01-21T23:28:03.583" LastActivityDate="2024-01-21T23:28:03.583" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16151" PostTypeId="2" ParentId="16143" CreationDate="2024-01-21T17:02:59.643" Score="2" Body="&lt;p&gt;As already noted in the comments, the &lt;a href=&quot;https://en.wikipedia.org/wiki/Royal_Society&quot; rel=&quot;nofollow noreferrer&quot;&gt;Royal Society&lt;/a&gt; is often given credit for advocating and requiring plain clear scientific writing, but if your teacher was referring to an individual then it was most likely &lt;a href=&quot;https://en.wikipedia.org/wiki/Francis_Bacon&quot; rel=&quot;nofollow noreferrer&quot;&gt;Francis Bacon&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to &lt;a href=&quot;https://doi.org/10.2307/356123&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Francis Bacon and the Rhetoric of Science, James P. Zappen, College Composition and Communication, Vol. 26, No. 3 (Oct., 1975), pp. 244-247 (4 pages)&quot;&gt;Zappen (1975)&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Bacon is frequently hailed as the father of modern scientific prose. … The fullest and most coherent exposition of Bacon's view of communication appears in the &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Advancement_of_Learning&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;The Advancement of Learning&lt;/em&gt;&lt;/a&gt;, published in 1605.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In this work, Bacon &lt;a href=&quot;https://archive.org/details/in.ernet.dli.2015.272672/page/n37/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;complained&lt;/a&gt; about flowery writers who:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;… began to hunt more after words than matter; more after the choiceness of the phrase, and the round and clean composition of the sentence, and the sweet falling of the clauses, and the varying and illustration of their works with tropes and figures, than after the weight of matter, worth of subject, soundness of argument, life of invention or depth of judgment.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bacon continued in this vein in &lt;a href=&quot;https://en.wikipedia.org/wiki/Novum_Organum&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;Novum Organum: True Suggestions for the Interpretation of Nature&amp;quot; (1620)&lt;/a&gt;, where he &lt;a href=&quot;https://archive.org/details/0415-bk/page/31/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;emphasized&lt;/a&gt; the primacy of facts over words:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;… the great and solemn disputes of learned men often terminate in controversies about words and names, …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;We must instead&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;… necessarily have recourse to particular instances, and their regular series and arrangement …&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To learn more, &lt;a href=&quot;https://doi.org/10.1080/00033790.2023.2235364&quot; rel=&quot;nofollow noreferrer&quot; title=&quot;Inventing the language of Things: the emergence of scientific reporting in seventeenth-century England, Plamena Panayotova (2023), Annals of Science, DOI: 10.1080/00033790.2023.2235364&quot;&gt;Panayotova (2023)&lt;/a&gt; is an (unpaywalled) recent review of the development of scientific writing in the 17th Century.&lt;/p&gt;&#xA;" OwnerUserId="6504" LastActivityDate="2024-01-21T17:02:59.643" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16152" PostTypeId="1" CreationDate="2024-01-22T07:34:54.443" Score="0" ViewCount="90" Body="&lt;p&gt;I have been trying to understand what is the meaning of &lt;a href=&quot;https://en.wikipedia.org/wiki/Bernoulli_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernoulli numbers&lt;/a&gt;, but to my mind it has been obscured behind complicated formulas without much explaination. I presume finding the history could be of use in understanding.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://link.springer.com/article/10.1007/s00283-021-10072-y&quot; rel=&quot;nofollow noreferrer&quot;&gt;The formulas&lt;/a&gt; I found are: &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{1-e^{-x}} = \frac{x}{2}(\coth{\frac{x}{2}} + 1) = \sum^\infty_{k=0}{\frac{B_k x^k}{k!}}$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.stackexchange.com/a/1222496/1204542&quot;&gt;One answer&lt;/a&gt; in math stack exchange said:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It is a analytical continuation of the sum of the infinite series &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{n^s}$&lt;/span&gt;, where the real part of the sum converges.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is this true? I mean, &lt;span class=&quot;math-container&quot;&gt;$\frac{x}{1-e^x}$&lt;/span&gt; does resemble &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{1-x}$&lt;/span&gt;, but where do Bernoulli numbers come in here?&lt;/p&gt;&#xA;" OwnerUserId="19610" LastEditorUserId="19610" LastEditDate="2024-01-22T08:37:32.387" LastActivityDate="2024-01-22T08:37:32.387" Title="History of Bernoulli numbers" Tags="|mathematics|number-theory|" AnswerCount="0" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="16153" PostTypeId="2" ParentId="16132" CreationDate="2024-01-22T12:51:07.873" Score="5" Body="&lt;h2&gt;Caveats&lt;/h2&gt;&#xA;&lt;p&gt;As far as I can tell from my research, due to the complexity of the task, there is currently no single platform that has a virtually comprehensive data base of all the scientific literature published over the last few centuries. I guess this is mostly for three reasons that are also intertwined (these are my conclusions and may be wrong):&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;For one—especially regarding more recent publications due to the rise of the internet—,there is an ever increasing number of (online) journals and platforms where such literature can be published, and it's very difficult to keep track of it all, and also cross-reference it with archives (such as &lt;a href=&quot;https://arxiv.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;arxiv.org&lt;/a&gt;) that also publish pre-prints etc.&lt;/li&gt;&#xA;&lt;li&gt;Secondly—especially regarding old literature from times before publication standards were established (such as papers, journals, theses)—, it's difficult to categorize some such literature as scientific.&lt;/li&gt;&#xA;&lt;li&gt;Thirdly—also especially regarding older literature—, some literature is published in languages other than English (the currently universal language of science), and some even in more than one. So it's not only difficult to find all the works written in other languages, but also to cross-reference them.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Lastly, while I think it should be possible to reasonably select which formats of science literature to consider (e.g. only articles, theses, and similar), I wonder how to handle different versions of essentially the same work, for example in case of (text) books. If they are fundamentally upgraded, then that may be a reason to count them more than once, but especially with the modern inflation of re-releasing the same book every year &amp;quot;in a new and updated version&amp;quot; that only had its chapters moved around and a word changed (to make money off of helpless students no less), I don't think they should count.&#xA;Similarly, I'm also not sure about how to handle errata, or maybe even letters. And I wonder if, like with novels, there are already nonsene textbooks and articles out there that were entirely written by AI, and if we will soon see an inflation of such &amp;quot;literature&amp;quot; as well. I certainly wouldn't count it.&lt;/p&gt;&#xA;&lt;h2&gt;Resources&lt;/h2&gt;&#xA;&lt;p&gt;Anyways, here are the best (non-comprehensive) resources that I could find so far:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;https://openalex.org&quot; rel=&quot;nofollow noreferrer&quot;&gt;openalex.org&lt;/a&gt;&lt;/strong&gt;: a huge, freely accessible data base of scientific literature, that goes back several hundreds of years. It differentiates between the following publication types: &lt;em&gt;article, book-chapter, dissertation, book, dataset, paratext, reference-entry, report, peer-review, editorial, erratum, grant, letter, standard (I have no idea), other&lt;/em&gt;. It also allows you to easily download the number of publications per year.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;https://www.dimensions.ai&quot; rel=&quot;nofollow noreferrer&quot;&gt;dimensions.ai&lt;/a&gt;&lt;/strong&gt;: very similar to OpenAlex, but requires the creation of a user account, and includes fewer publication types: &lt;em&gt;article, chapter, proceeding, preprint, monograph, edited book&lt;/em&gt;.&lt;/li&gt;&#xA;&lt;li&gt;&lt;strong&gt;&lt;a href=&quot;https://www.webofscience.com&quot; rel=&quot;nofollow noreferrer&quot;&gt;webofscience.com&lt;/a&gt;&lt;/strong&gt;: has been around for much longer than OpenAlex or dimensions.ai, but only goes back to 1900. To my surprise, it also contains significantly fewer entries than either of the other two, even though it supports a myriad of different publication types: &lt;em&gt;article, art exhibit review, bibliography, biographical-item, book, book chapter, book review, correction, dance performance review (!!!), data paper, database review, early access, editorial material, excerpt, expression of concern, fiction, creative prose, film review, hardware review, item withdrawal, letter, meeting,  meeting abstract, meeting summary, music performance review, music score, music score review, news item, poetry, proceedings paper, publication with expression of concern, record review, reprint, retracted publication, retraction, review, script, software review, theater review, tv review/radio review, withdrawn publication&lt;/em&gt;. (&lt;strong&gt;Update note&lt;/strong&gt;: previously, I said that the Web of Science contained more entries than the previous two data bases, but I looked at the wrong data and was mistaken.)&lt;/li&gt;&#xA;&lt;li&gt;Then there is also &lt;a href=&quot;https://datacatalog.worldbank.org/search/dataset/0037712/World-Development-Indicators&quot; rel=&quot;nofollow noreferrer&quot;&gt;the data provided by the &lt;strong&gt;World Bank&lt;/strong&gt;&lt;/a&gt; (which is also what &lt;a href=&quot;https://ourworldindata.org/grapher/scientific-and-technical-journal-articles?tab=chart&quot; rel=&quot;nofollow noreferrer&quot;&gt;ourworldindata.org&lt;/a&gt; and &lt;a href=&quot;https://wordsrated.com/number-of-academic-papers-published-per-year/&quot; rel=&quot;nofollow noreferrer&quot;&gt;wordsrated.com&lt;/a&gt; rely on), but as far as I can tell, their data only goes back to 1996, and is actually provided by the National Science Foundation (NSF). It also only includes scientific and technical journal articles, so it is very limited in that regard as well. And lastly, unlike the other three, it is not a data base of the underlying literature, but an accumulation of statistical data. So I would not recommend it.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;Comparison&lt;/h2&gt;&#xA;&lt;p&gt;Below I compare the different resources in a plot. As you can see, OpenAlex has by far the most entries (in total and per year), and also covers the longest time period. So of the four resources, it seems to be the most comprehensible data base, but keeping in mind the caveats mentioned above, I don't know if it's also the most reliable (i.e., maybe some works have more than one entry, e.g. due to different versions/languages etc., and data sets should probably also not be counted).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/eJXmN.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/eJXmN.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18187" LastEditorUserId="18187" LastEditDate="2024-01-26T12:40:14.813" LastActivityDate="2024-01-26T12:40:14.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16154" PostTypeId="2" ParentId="16131" CreationDate="2024-01-23T19:28:49.500" Score="2" Body="&lt;p&gt;OED, The unabridged &lt;em&gt;Oxford English Dictionary&lt;/em&gt; has the earliest reference to your query.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;tertiary&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Applied to compounds regarded as being derived from ammonia by replacement of three hydrogen atoms by organic radicals, and to derivatives of such compounds; also extended to analogous derivatives of other elements, esp. phosphorus. [The sense is due to Gerhardt &amp;amp; Chiozza, who used &lt;strong&gt;French tertiaire&lt;/strong&gt; (Compt. Rend. (1853) XXXVII. 88).]&lt;/p&gt;&#xA;&lt;p&gt;Applied to organic compounds other than amines, etc. (see sense A.1b.i) in which the characteristic functional group is located on a saturated carbon atom which is itself bonded to three other carbon atoms. [Applied originally to alcohols by H. Kolbe, who used &lt;strong&gt;German tertiär&lt;/strong&gt; (Ann. der Chem. u. Pharm. (1864) CXXXII. 104).]&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;secondary&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Applied to compounds regarded as being derived from ammonia (†or water) by replacement of two hydrogen atoms by organic radicals (cf. primary adj. A.II.18b); also extended to analogous derivatives of other elements, esp. phosphorus. [The sense is due to Gerhardt &amp;amp; Chiozza, who used &lt;strong&gt;French secondaire&lt;/strong&gt; (Compt. Rend. (1853) XXXVII. 88).]&lt;/p&gt;&#xA;&lt;p&gt;Applied to organic compounds other than amines, etc. (see sense A.3l.i) in which the characteristic functional group is located on a saturated carbon atom which is itself bonded to two other carbon atoms. [Applied originally to alcohols by H. Kolbe, who used &lt;strong&gt;German secundär&lt;/strong&gt; (Ann. der Chem. u. Pharm. (1864) CXXXII. 102).]&lt;/p&gt;&#xA;" OwnerUserId="6085" LastActivityDate="2024-01-23T19:28:49.500" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16155" PostTypeId="1" CreationDate="2024-01-23T23:43:55.513" Score="0" ViewCount="105" Body="&lt;p&gt;I have been studying the concepts of &amp;quot;resonance&amp;quot; and &amp;quot;mesomerism&amp;quot; recently and a common principle of these concepts is the &lt;em&gt;&amp;quot;delocalization&amp;quot;&lt;/em&gt; (of electrons, molecular orbitals etc). Now, I have read a lot of articles on the concept of &lt;em&gt;&amp;quot;delocalization&amp;quot;&lt;/em&gt; and how it evolved but wonder who coined the term first. Here is all the information I could find through my research (about the emergence of the &lt;em&gt;&amp;quot;ideas&amp;quot;&lt;/em&gt; and &lt;em&gt;&amp;quot;concepts&amp;quot;&lt;/em&gt; of delocalization along with some instances of the actual usage of the term).&lt;/p&gt;&#xA;&lt;p&gt;Firstly, about the concept of &lt;em&gt;&amp;quot;π-bond delocalization&amp;quot;&lt;/em&gt;, it emerged as a concept in 1865 when August Kekulé proposed the structured for benzene in his paper &lt;em&gt;&amp;quot;Sur la constitution des substances aromatiques&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1WwzUi63KBh5EfO1XmZ3OmONHGl60YMC6/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[1]&lt;/a&gt;&lt;/sup&gt; (it is interesting to note that he revealed in &lt;em&gt;&amp;quot;Feier der Deutschen Chemischen Gesellschaft zu Ehren August Kekulé's.&amp;quot;, 23(1), 1305-1306&lt;/em&gt; &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1TwlLvGkl9mD8fBpHuhOJwrLVjBsyIzvA/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[2]&lt;/a&gt;&lt;/sup&gt; that he had ideated the shape of the benzene molecule after having a day-dream of a snake eating its own tail however it is doubted as the occurrence of ring structures had been depicted before 1865 by &amp;quot;Archibald Couper&amp;quot; in 1858 &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/10x1yZHxguPXwGDB5pivSt6NJ-Z6QLmmu/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[3]&lt;/a&gt;&lt;/sup&gt; and &amp;quot;Joseph Loschmidt&amp;quot; in 1861&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1vDWSEW0LRryhmsYmnum7-N42Z9TxDIED/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[4]&lt;/a&gt;&lt;/sup&gt;). Further, in 1869 Albert Ladenburg criticized Kekulé's 1865 structure in his paper &lt;em&gt;&amp;quot;Bemerkungen zur aromatischen Theorie&amp;quot;&lt;/em&gt; &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1lySUkUiIho6E7rvAt-siMC3eGcxTW_uR/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[5]&lt;/a&gt;&lt;/sup&gt; and in response kekulé modified his proposal in 1872&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/106KbEX-OLuEBM_J-krYPF1cib2bWzOk6/view?usp=drive_link&quot; rel=&quot;nofollow noreferrer&quot;&gt;[6]&lt;/a&gt;&lt;/sup&gt; and suggested that the benzene molecule oscillates† between the two equivalent structures thus giving the modern-representation for benzene.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;† &lt;em&gt;However, Kekulé’s 1872 refinement proposed a fully six-fold symmetrical, but “vibrating”, benzene structure. He did not intend the “oscillating” hypothesis others mistakenly ascribed to him later. This almost universal misinterpretation should not be repeated in the literature and in textbooks.&lt;/em&gt;&lt;sup&gt;[7]&lt;/sup&gt;&lt;/p&gt;&#xA;&lt;p&gt;[7] Schleyer, P. (2005). &lt;a href=&quot;https://drive.google.com/file/d/1q3Ncu1eEvK1-TYYDykyR1Vpta8HQ9ttH/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction: Delocalization Pi and Sigma&lt;/a&gt;.&#xA;&lt;em&gt;Chem. Rev., 105&lt;/em&gt;(10), 3433–3435&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Now, after the work of Kekulé came Johannes Thiele in 1899 in his paper &lt;em&gt;&amp;quot;Zur Kenntniss der ungesattigten Verbindungen&amp;quot;&lt;/em&gt; &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1kkBOYP2QLL01qSY-1S9k8pbcruX5Rta3/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[8]&lt;/a&gt;&lt;/sup&gt; involving his application of the “partial valence” (&lt;em&gt;Partialvalenz&lt;/em&gt;) concept upon discovering the unusual stability of the cyclopentadienyl anion to rationalize the special chemical behaviour of unsaturated compounds (which are now called “delocalized π systems”). He attributed the greater reactivity of these double bonds to “residual affinities”(&lt;em&gt;Affinitätsrest&lt;/em&gt;) (equivalent to p orbitals), which could link across the central atoms in a “conjugated”(&lt;em&gt;Conjugirte&lt;/em&gt;) diene  explaining their extra reactivity in addition reactions (he was the first one to coin the term &amp;quot;conjugated systems&amp;quot;). This also explained the lack of reactivity in benzene with it being a &amp;quot;closed conjugated system&amp;quot; and giving the picture of having a &amp;quot;partial valency&amp;quot;(delocalized π-system) leading to the modern idea of resonance.†&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;† &lt;em&gt;This paragraph contains ideas and text from [7] which is cited above although it has been modified with additional information and rephrasing.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To discuss about the idea and evolution of delocalization in π-systems, it dates back to Heisenberg in 1926 is his paper &lt;em&gt;&amp;quot;Mehrkorperproblem und Resonanz in der Quantenmechanik&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1LYAm8UcI95J9XimzoHr6oXpqI2iWukni/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[9 (eng. translation)]&lt;/a&gt;&lt;/sup&gt;&#xA;wherein he introduced the idea of &amp;quot;resonance&amp;quot;(in the physics sense) in the discussion of the quantum states of the helium atom comparing it with classical system of resonating coupled harmonic oscillators. In this case, the resulting coupling which is lower in frequency than either of the uncoupled vibrations; is analogous to a lower energy. Heitler &amp;amp;, London in 1927 in their paper &lt;em&gt;&amp;quot;Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1WMz4XAInDh98gVdzwgNxtCP6Xp0WNXWH/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[10]&lt;/a&gt;&lt;/sup&gt; applied this concept to the normal state of the hydrogen molecule wherein the molecule was described as involving the resonance (exchange) of the electrons between the positions about the two nuclei(of hydrogen), and the bond energy was attributed largely to the resonance energy (exchange integral).&lt;/p&gt;&#xA;&lt;p&gt;This concept was finally given the modern shape as we know it by Linus Pauling in 1928 through his paper &lt;em&gt;&amp;quot;The shared-electron chemical bond&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/18dZQHlToM5eovE6IyLp8Nkje17Cj6K0I/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[11]&lt;/a&gt;&lt;/sup&gt;&#xA;which was followed by another paper in the same year, titled &lt;em&gt;&amp;quot;The application of the quantum mechanics to the structure of the hydrogen molecule and hydrogen molecule-ion and to related problems.&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1LA4yvz31X-C-i6aJvmOGjYjh1i7dCynV/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[12]&lt;/a&gt;&lt;/sup&gt;†.&#xA;This was the start of the theory of hybrid bond orbitals (i.e. the Valence Bond Theory), which was then developed further by Pauling and Slater(with works starting from 1931).&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;† This part is a bit confusing because it is unclear which of the two papers was published later. In a 1977 paper titled, &lt;em&gt;&amp;quot;The theory of resonance in chemistry&amp;quot;&lt;/em&gt; &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1Rrxx6PtljeRJBnlxeWu-z2epaMVsUwCr/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[13]&lt;/a&gt;&lt;/sup&gt; Pauling mentions:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;A similar treatment, involving resonance of one electron between the 1s orbitals about the two nuclei, was developed for the hydrogen molecule ion (Pauling 1928a). The theory of resonance was then used (Pauling 1928b) to explain.....&amp;quot;&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where &lt;em&gt;&amp;quot;Pauling 1928b&amp;quot;&lt;/em&gt; refers to &lt;em&gt;&amp;quot;The shared-electron chemical bond&amp;quot;&lt;/em&gt; which was published earlier although mentioned to be used after &lt;em&gt;&amp;quot;Pauling 1928a&amp;quot;&lt;/em&gt; referring to &lt;em&gt;&amp;quot;The application of the quantum mechanics to the structure of the hydrogen molecule and hydrogen molecule-ion and to related problems.&amp;quot;&lt;/em&gt; which was published later.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;After reviewing multiple papers from Pauling, Slater, Hückel, etc., I couldn't find any mention of the word &amp;quot;delocalization&amp;quot; in the π-system sense. Finally, the earliest mention of the word 'delocalization' (in the sense of 'π-bond delocalization') that I could find was in a paper by Klaus Ruedenberg in 1962, titled &lt;em&gt;&amp;quot;The Physical Nature of the Chemical Bond&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1TURpFXXWNilYgWjii8KU6bOskMobSeRd/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[14]&lt;/a&gt;&lt;/sup&gt; (interestingly there is a mention by Ruedenberg in his &lt;a href=&quot;https://drive.google.com/file/d/1L5oFwcu5JTTBmgJmU4JlL8atp6T21eWm/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;autobiography&lt;/a&gt; wherein he says about himself; &lt;em&gt;&amp;quot;defined  the  real  meaning  of  delocalization  in  conjugated, especially aromatic,π-systems....&amp;quot;&lt;/em&gt; ). There is another mention by Ruedenberg in 1963 in an article titled &lt;em&gt;&amp;quot;Localized Atomic and Molecular orbitals&amp;quot;&lt;/em&gt;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/17OiqYewmrszZJl1cSiXKCdku0q6gCGOU/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[15]&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Secondly, the concept of 'σ-bond delocalization' emerged in Robert Mulliken's paper in 1941 named &lt;em&gt;&amp;quot;Hyperconjugation&lt;/em&gt;*&amp;quot;&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/14yCcAWTHn-ALkKFWLUU1cOb0gmmIHzvL/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[16]&lt;/a&gt;&lt;/sup&gt;(with A. Rieke and W. G. Brown). In this paper, is the earliest occurrence of the word delocalization that I could find (pg. 42) (it is also interesting to note that the first use of the term localized† I could find is from &amp;quot;Friedrich Hund&amp;quot; in 1931&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1KOlkyUxBxyHMnXgYJCVJNCghlrd_BkCJ/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[17]&lt;/a&gt;&lt;/sup&gt; and of the term non-localized† {the predecessor of the term &amp;quot;delocalized&amp;quot;} from &amp;quot;Robert Mulliken&amp;quot; in 1935&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1Hqa0JmgqZYUOy7nFtqF8AWHg3Ss2ItUQ/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[18]&lt;/a&gt;&lt;/sup&gt;).&lt;/p&gt;&#xA;&lt;p&gt;*&lt;sub&gt; I think most of the information in this question is correct, still if there are any corrections/edits and revisions/suggestions, they will be appreciated! Also, you may think this question as kind of pointless but I have wondered for a long time about the origin of the term &amp;quot;delocalization&amp;quot; so any help will be very appreciated. Thanks!&lt;/sub&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;Question:&lt;/h2&gt;&#xA;&lt;p&gt;So, finally my question is that who is person who coined the term &amp;quot;delocalization&amp;quot;(in the sense of chemistry) and who used the term for the first time in that sense? Also, is it somehow related to the people I mentioned above (specifically Ruedenberg and Mulliken as they were the only people whose publications contained the actual word &amp;quot;delocalization&amp;quot; and not just the idea concerning it)?&lt;/p&gt;&#xA;" OwnerUserId="19382" LastEditorUserId="19382" LastEditDate="2024-01-27T13:35:41.073" LastActivityDate="2024-01-27T13:35:41.073" Title="A brief history of &quot;delocalization&quot; of electrons" Tags="|physics|terminology|quantum-mechanics|chemistry|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="16159" PostTypeId="1" AcceptedAnswerId="16160" CreationDate="2024-01-25T07:27:29.653" Score="0" ViewCount="143" Body="&lt;p&gt;I came across this short: &lt;a href=&quot;https://www.youtube.com/shorts/LjBaPxutpQo&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.youtube.com/shorts/LjBaPxutpQo&lt;/a&gt; where Brian Cox says the idea of black holes dates back to 1790. It stemmed according to it from escape velocity and that you could at some point have escape velocities larger than the speed of light.&lt;/p&gt;&#xA;&lt;p&gt;My understanding is that Newton's laws were still the prevalent theory back then with nothing superseding them. So while they would have imagined such objects to be quite dark, I don't think they would have expected that it would be impossible to escape them once inside the event horizon. Since Newtons laws say you can get to any speed you want if you accelerate long enough, I imagine they might have thought a powerful enough rocket could still get them out of such an object.&lt;/p&gt;&#xA;&lt;p&gt;Is there anything that supports people believing that faster than light travel should be possible around the 1790s?&lt;/p&gt;&#xA;&lt;p&gt;Note: my question is specifically around whether there is any evidence that people thought faster than light travel was possible in 1790 since Newton's laws allow it.&lt;/p&gt;&#xA;" OwnerUserId="15867" LastEditorUserId="15867" LastEditDate="2024-01-25T22:25:52.107" LastActivityDate="2024-01-25T23:32:06.977" Title="Would they have known you can't go faster than light in 1790" Tags="|classical-mechanics|relativity-theory|light|" AnswerCount="1" CommentCount="1" ClosedDate="2024-01-31T11:17:47.983" ContentLicense="CC BY-SA 4.0" />
  <row Id="16160" PostTypeId="2" ParentId="16159" CreationDate="2024-01-25T07:34:22.493" Score="5" Body="&lt;blockquote&gt;&#xA;&lt;p&gt;Would they have known you can't go faster than light in 1790?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;No, they did not know.&lt;/p&gt;&#xA;&lt;p&gt;In 1783, &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Michell&quot; rel=&quot;nofollow noreferrer&quot;&gt;John Michell&lt;/a&gt; considered what would happen to light if an object were too massive. He considered that light would not be able to escape the massive object due to its gravitational pull, foreshadowing the existence of black holes.&lt;/p&gt;&#xA;&lt;p&gt;His calculation was classical, and he was not considering the speed of light as a limit, just a mass limit for a star, at which light could not escape.&lt;/p&gt;&#xA;&lt;p&gt;Laplace came up with the same idea in 1796.&lt;/p&gt;&#xA;&lt;p&gt;Before 1905 Einstein's special relativity, some scientists considered superluminal phenomena. Scientist believed for some time that forces like Coulomb law and Newton's universal gravitation were instantaneous. Laplace in &amp;quot;Description of the System of the Universe&amp;quot;  considers that, when studying the Moon, that gravitation should exceed the speed of light (by a factor of &lt;span class=&quot;math-container&quot;&gt;$10^6$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Also due Galilean relativity, it was considered that if light was emitted from a moving object, then light would go faster or slower that light emitted from a body at rest. Fresnel makes this argument in 1818.&lt;/p&gt;&#xA;&lt;p&gt;In &lt;em&gt;Two Planets&lt;/em&gt; by &lt;a href=&quot;https://en.wikipedia.org/wiki/Kurd_Lasswitz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Kurd Lasswitz&lt;/a&gt; (1897)&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;the Martians mention their propulsion drive, expecting to soon reach propellant speeds beyond the speed of light.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(&lt;a href=&quot;https://scifi.stackexchange.com/questions/272112/first-story-to-mention-the-speed-of-light&quot;&gt;Courtesy of Scifi SE&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;Source:  Malykin, G. B., and E. A. Romanets. &amp;quot;Superluminal motion (review)&amp;quot; (2012) &lt;a href=&quot;https://link.springer.com/article/10.1134/S0030400X12040145&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://link.springer.com/article/10.1134/S0030400X12040145&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-01-25T23:32:06.977" LastActivityDate="2024-01-25T23:32:06.977" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16161" PostTypeId="1" CreationDate="2024-01-25T12:00:58.610" Score="2" ViewCount="109" Body="&lt;p&gt;In the preamble to “Essays on the Theory of Numbers”, Dedekind makes passing reference to a theory (expounded in Cantor’s “Ueber die Ausdenung eines Satzes aus der Theorie der trigonometrischen Reihen”) of “real numbers of higher type”. What are these, in modern terms?&lt;/p&gt;&#xA;" OwnerUserId="6587" LastActivityDate="2024-01-25T12:00:58.610" Title="What were Cantor’s “real numbers of higher type”?" Tags="|real-analysis|set-theory|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16163" PostTypeId="2" ParentId="16131" CreationDate="2024-01-26T00:32:07.920" Score="2" Body="&lt;p&gt;After the answer and information from @AChem, I would like to add some additional information about the sources mentioned in the answer.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[1] Gerhardt, C. &amp;amp; Chiozza, L. (1853). &lt;a href=&quot;https://drive.google.com/file/d/1essh2CfGzbuEji5v8TXYNnto-bDuwul6/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Recherches sur les amides&lt;/a&gt;.&#xA;&lt;em&gt;Comptes rendus hebdomadaires des séances de l'Académie des sciences, XXXVII(July)&lt;/em&gt;, 88.&lt;/p&gt;&#xA;&lt;p&gt;[2] Kolbe, H. (1864). &lt;a href=&quot;https://drive.google.com/file/d/1T1W7Fw1wJmWoJE9H-ZdLBsfgvXxnP3F8/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ueber die seoundären Alkohole&lt;/a&gt;.&#xA;&lt;em&gt;Annalen der Chemie und Pharmacie, 132(1)&lt;/em&gt;, 103.†&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;sup&gt;†&lt;/sup&gt; It is a bit confusing with how Kolbe mentions the notation &lt;em&gt;&amp;quot;alcohols&amp;quot;&lt;/em&gt; in this paper which is listed below:&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;&amp;quot;One of the major difficulties in Kolbe’s formulas was that he refused to abandon equivalent weights for atomic weights until 1869, long after other chemists had adopted them. He still followed Berzelius in using the values C =6 and O = 8 &lt;strong&gt;(these refer to the atomic weights calculated by Berzelius)&lt;/strong&gt;, so that he had to double the number of atoms of these elements in his formulas —thus, his notation for the methyl group was C&lt;sub&gt;2&lt;/sub&gt;H&lt;sub&gt;3&lt;/sub&gt;, and hydrated carbonic acid became 2HO. C&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;4&lt;/sub&gt;.&amp;quot;&lt;/em&gt; &lt;sup&gt;[&lt;a href=&quot;https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/kolbe-adolf-wilhelm-hermann&quot; rel=&quot;nofollow noreferrer&quot;&gt;3&lt;/a&gt;]&lt;/sup&gt;(text in bold added by me)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, if someone is also curious about the origin of some other prefixes like &amp;quot;&lt;em&gt;ortho-&lt;/em&gt;&amp;quot;, &amp;quot;&lt;em&gt;meta-&lt;/em&gt;&amp;quot; &amp;amp; &lt;em&gt;&amp;quot;para-&amp;quot;&lt;/em&gt; here is some information about them.&lt;/p&gt;&#xA;&lt;p&gt;The prefixes &amp;quot;&lt;em&gt;para-&lt;/em&gt;&amp;quot; and &lt;em&gt;&amp;quot;meta-&amp;quot;&lt;/em&gt; first appeared in the 1830's (almost in the current sense today to represent &lt;em&gt;&amp;quot;closely related compounds&amp;quot;&lt;/em&gt;) with these prefixes being introduced by Berzelius&lt;sup&gt; &lt;a href=&quot;https://books.google.com/books?id=2OY4AAAAMAAJ&amp;amp;pg=RA1-PA70&quot; rel=&quot;nofollow noreferrer&quot;&gt;[4]&lt;/a&gt;&lt;/sup&gt;(in 1830 to show the name &lt;em&gt;&amp;quot;paratartaric acid&amp;quot;&lt;/em&gt;) and Graham &lt;sup&gt;&lt;a href=&quot;https://www.jstor.org/stable/pdf/107997.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;[5]&lt;/a&gt;&lt;/sup&gt;(in 1833 to show the name &lt;em&gt;&amp;quot;metaphosphoric acid&amp;quot;&lt;/em&gt;) respectively. About the prefix &lt;em&gt;&amp;quot;ortho-&amp;quot;&lt;/em&gt;, it first appeared in 1859 by Odling &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/1PAH3natRj6Oomi2ltT1_xSxPHjxCwl4W/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[6]&lt;/a&gt;&lt;/sup&gt;to represent the acid corresponding to the (highest degree of hydration).&lt;/p&gt;&#xA;&lt;p&gt;Finally, the use of all the three prefixes together was popularized by Körner &lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/18B37ZR-OrjATKW8grXHDMuFJ3FYTBrkY/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[7]&lt;/a&gt;&lt;/sup&gt;(between the period of 1866-1874) to various isomers of disubstituted benzene. The first usage of the three prefixes(in the sense we know them today) was done by the Chemical Society of London in 1879&lt;sup&gt;&lt;a href=&quot;https://drive.google.com/file/d/13H6IrX12dRVD6YhfRg_5u97F77jn9D3i/view?usp=sharing&quot; rel=&quot;nofollow noreferrer&quot;&gt;[8]&lt;/a&gt;&lt;/sup&gt;&lt;em&gt;(pg. 276-281)&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19382" LastActivityDate="2024-01-26T00:32:07.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16164" PostTypeId="1" AcceptedAnswerId="16167" CreationDate="2024-01-26T15:38:00.230" Score="1" ViewCount="103" Body="&lt;p&gt;A question about a notation in David Hilberts's &amp;quot;Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper&amp;quot; (&lt;a href=&quot;https://de.wikisource.org/wiki/David_Hilbert_Gesammelte_Abhandlungen_Erster_Band_%E2%80%93_Zahlentheorie/Kapitel_6&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; a german online available source, not sure if there exist a free available english version.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; In &lt;em&gt;Satz 1&lt;/em&gt; Hilbert introduced the notion of &lt;span class=&quot;math-container&quot;&gt;$t:=(\Theta: \Theta^r)$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$\Theta:=e^{2 \pi i/l^h}$&lt;/span&gt; a &lt;span class=&quot;math-container&quot;&gt;$l^h$&lt;/span&gt;-root of unity and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; certain natural number) and called it &amp;quot;Substitution der Gruppe des Kreiskörpers &lt;span class=&quot;math-container&quot;&gt;$k(\Theta)$&lt;/span&gt;&amp;quot;, ie &amp;quot;substitution of the group of cyclotomic field &lt;span class=&quot;math-container&quot;&gt;$k(\Theta)$&lt;/span&gt;&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;What's the meaning of &lt;span class=&quot;math-container&quot;&gt;$(\Theta: \Theta^r)$&lt;/span&gt; in modern terms?&lt;/p&gt;&#xA;" OwnerUserId="7566" LastEditorUserId="12330" LastEditDate="2024-01-29T07:57:05.980" LastActivityDate="2024-01-29T21:40:40.720" Title="David Hilbert's paper: Substitution of the group of cyclotomic field" Tags="|mathematics|notation|number-theory|hilbert|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16165" PostTypeId="2" ParentId="15127" CreationDate="2024-01-27T22:05:48.817" Score="0" Body="&lt;p&gt;Alan Earnshaw, author of many railway books, is NOT the same man as Alan Earnshaw, co-author of Chemistry of Elements. Alan Earnshaw who wrote railway books was born on 8 Aug 1952 and died 10 Mar 2009. He was a director of “Trans-Pennine Publishing Limited” and “Nostalgia Road Publications Limited”. His date of birth is in the company information at &lt;a href=&quot;https://find-and-update.company-information.service.gov.uk/company/03240045&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://find-and-update.company-information.service.gov.uk/company/03240045&lt;/a&gt; and &lt;a href=&quot;https://find-and-update.company-information.service.gov.uk/company/04772951&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://find-and-update.company-information.service.gov.uk/company/04772951&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19630" LastActivityDate="2024-01-27T22:05:48.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16167" PostTypeId="2" ParentId="16164" CreationDate="2024-01-29T21:40:40.720" Score="4" Body="&lt;p&gt;I think in modern notation, the element &lt;span class=&quot;math-container&quot;&gt;$\Theta$&lt;/span&gt; (which is literally defined as &lt;span class=&quot;math-container&quot;&gt;$\displaystyle e^{\frac{2\pi i}{l^h}}$&lt;/span&gt; i.e. a primitive &lt;span class=&quot;math-container&quot;&gt;$l^h$&lt;/span&gt;-th root of unity; &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt; is a prime here) would often be called &lt;span class=&quot;math-container&quot;&gt;$\zeta$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$\zeta_{l^h}$&lt;/span&gt;, the field &lt;span class=&quot;math-container&quot;&gt;$k(\Theta)$&lt;/span&gt; is really just the cyclotomic field extension &lt;span class=&quot;math-container&quot;&gt;$\mathbb Q(\zeta)$&lt;/span&gt; of &lt;span class=&quot;math-container&quot;&gt;$k=\mathbb Q$&lt;/span&gt;, and for a given integer &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; not divisible by &lt;span class=&quot;math-container&quot;&gt;$l$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt; (which one should maybe call &lt;span class=&quot;math-container&quot;&gt;$t_r$&lt;/span&gt;) is the element of its Galois group given by &lt;span class=&quot;math-container&quot;&gt;$\zeta \mapsto \zeta^r$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In a bit fancier and possibly more precise modern terminology, the correspondence &lt;span class=&quot;math-container&quot;&gt;$r\mapsto [t_r: \zeta \mapsto \zeta^r]$&lt;/span&gt; defines an isomorphism&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$(\mathbb Z/l^h \mathbb Z)^\times \rightarrow Gal(\mathbb Q(\zeta) \vert \mathbb Q).$$&lt;/span&gt;&lt;/p&gt;&#xA;" OwnerUserId="6514" LastActivityDate="2024-01-29T21:40:40.720" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16168" PostTypeId="1" AcceptedAnswerId="16170" CreationDate="2024-01-30T11:18:19.883" Score="2" ViewCount="150" Body="&lt;p&gt;Borel introduces the notion of &lt;em&gt;hauteur&lt;/em&gt; (French for 'height') in a note titled&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Sur l'approximation les uns par les autres des nombres formant un ensemble dénombrable&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;in the Comptes Rendus journal in 1903.&lt;/p&gt;&#xA;&lt;p&gt;To my own surprise, Borel's notion of &lt;em&gt;hauteur&lt;/em&gt; is exactly what we call 'height' today: for a countable set of reals &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, a &lt;em&gt;height function&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$H$&lt;/span&gt; is such that &lt;span class=&quot;math-container&quot;&gt;$\{ x \in A: H(x)&amp;lt;n\}$&lt;/span&gt; is finite for all natural &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt;.  Height functions &lt;a href=&quot;https://en.wikipedia.org/wiki/Height_function#Other_height_functions&quot; rel=&quot;nofollow noreferrer&quot;&gt;are used to show&lt;/a&gt; that there is a bijection between the natural and rational numbers.&lt;/p&gt;&#xA;&lt;p&gt;Was Borel the first one to come up with height functions, or are there earlier known instances?&lt;/p&gt;&#xA;" OwnerUserId="15772" LastActivityDate="2024-01-30T15:00:03.033" Title="Height function following Borel" Tags="|reference-request|real-analysis|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16169" PostTypeId="2" ParentId="14774" CreationDate="2024-01-30T11:31:40.550" Score="2" Body="&lt;p&gt;The notation is a bit harsh, but Ascoli defines 'oscillazione' in the following paper (p. 867, paragraph 6; see also p. 869, at the top, for a reference to Riemann, &lt;em&gt;Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe&lt;/em&gt;) and formulates a version of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann_integral#Integrability&quot; rel=&quot;nofollow noreferrer&quot;&gt;Vitali-Lebesgue theorem&lt;/a&gt; using this notion.&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Giulio Ascoli, &lt;a href=&quot;https://archive.org/details/attidellaac2218741875acca/page/862/mode/2up?view=theater&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sul concetto di integrale definito&lt;/a&gt;, Atti della Reale Accademia dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali 2 (1875), no. 2, 862-872.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="15772" LastEditorUserId="6545" LastEditDate="2024-01-30T17:39:58.883" LastActivityDate="2024-01-30T17:39:58.883" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16170" PostTypeId="2" ParentId="16168" CreationDate="2024-01-30T13:51:10.180" Score="4" Body="&lt;p&gt;As usual, it is very difficult to identify the first occurrence of a term, but in this case it is Borel's own article that can help us.&lt;/p&gt;&#xA;&lt;p&gt;The full note can be found &lt;a href=&quot;http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3091&amp;amp;I=297&amp;amp;M=tdm&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Étant donné un ensemble dénombrable &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; de nombres réels, nous ferons correspondre à chaque nombre de l'ensemble un entier positif &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; que nous appelerons &lt;em&gt;hauteur&lt;/em&gt; de ce nombre dans &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; et tel que chaque valeur de &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; corresponde à un nombre limité de nombres de &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;. Une telle correspondance peut être obtenue d'une infinité de manières [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Given a countable set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; of real numbers, we will map to each number in the set a positive integer &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt;, which we will call &lt;em&gt;height&lt;/em&gt; of that number in &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;, such that each value of &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; corresponds to a finite number of numbers in &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;. Such a correspondence can be obtained in an infinite number of ways [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Borel says &amp;quot;nous ferons correspondre&amp;quot;, that is &amp;quot;we will map to&amp;quot;, using the first person plural and providing no bibliographical reference (which he generally does), thus suggesting that it is his own definition.&lt;/p&gt;&#xA;&lt;p&gt;A little further on we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Je me propose de résumer d'abord quelques recherches, que je publierai prochainement, relatives à la &lt;em&gt;détermination effective&lt;/em&gt; de la fonction &lt;span class=&quot;math-container&quot;&gt;$\varphi(h_1,h_2)$&lt;/span&gt; pour certains ensembles &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;; j'indiquerai ensuite quel intéret il me paraitrait y avoir à la déterminer dans des cas de plus en plus généraux.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;First, I'll summarize some research I'll be publishing shortly into the &lt;em&gt;effective determination&lt;/em&gt; of the function &lt;span class=&quot;math-container&quot;&gt;$\varphi(h_1,h_2)$&lt;/span&gt; [a positive lower bound for the difference &lt;span class=&quot;math-container&quot;&gt;$|x_1-x_2|$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; have height &lt;span class=&quot;math-container&quot;&gt;$h_1$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$h_2$&lt;/span&gt;, respectively] for certain sets &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt;; then I'll indicate the interest I see in determining it in increasingly general cases.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So once again the impression is that Borel is reporting personal considerations. The most interesting part is as follows.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dans le travaux antérieurs sur l'approximation des incommensurables, on supposait que le nombre &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; appartenait à un ensemble &lt;span class=&quot;math-container&quot;&gt;$E_1$&lt;/span&gt;, le nombre &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; à l'ensemble &lt;span class=&quot;math-container&quot;&gt;$E_2$&lt;/span&gt; des nombres rationnels, et l'on cherchait une limitation asymptotiques de &lt;span class=&quot;math-container&quot;&gt;$\varphi(h_1,h_2)$&lt;/span&gt; lorsque, &lt;span class=&quot;math-container&quot;&gt;$h_1$&lt;/span&gt; étant fixe, &lt;span class=&quot;math-container&quot;&gt;$h_2$&lt;/span&gt; croissait indéfiniment; tels sont les résultats de Liouville, de M. Maillet, de M. Störmer; dans ma Note précité, j'ai étudié le cas d'un ensemble &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; formé des nombres algébrique, mais j'ai supposé aussi &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; fixe et cherché seulement une limitation asymptotique de &lt;span class=&quot;math-container&quot;&gt;$\varphi(h_1,h_2)$&lt;/span&gt;; pour les applications, il y a un grand intéret à se placer au nouveau point de vue que je viens d'indiquer, cau une inégalité asymptotique est souvent peu utile lorsque l'on ne sait pas à partir de quelle valeur de la variable elle est vérifiée.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In earlier work on the approximation of incommensurables, it was assumed that the number &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; belonged to a set &lt;span class=&quot;math-container&quot;&gt;$E_1$&lt;/span&gt;, the number &lt;span class=&quot;math-container&quot;&gt;$x_2$&lt;/span&gt; to the set &lt;span class=&quot;math-container&quot;&gt;$E_2$&lt;/span&gt; of rational numbers, and an asymptotic limitation of &lt;span class=&quot;math-container&quot;&gt;$\varphi(h_1,h_2)$&lt;/span&gt; was sought when, &lt;span class=&quot;math-container&quot;&gt;$h_1$&lt;/span&gt; being fixed, &lt;span class=&quot;math-container&quot;&gt;$h_2$&lt;/span&gt; grew indefinitely; such are the results of Liouville, Mr Maillet, Mr Störmer; in my aforementioned Note [see below], I studied the case of a set &lt;span class=&quot;math-container&quot;&gt;$E$&lt;/span&gt; made up of algebraic numbers, but I also assumed &lt;span class=&quot;math-container&quot;&gt;$x_1$&lt;/span&gt; to be fixed and sought only an asymptotic limitation of &lt;span class=&quot;math-container&quot;&gt;$varphi(h_1,h_2)$&lt;/span&gt;; for applications, there is great interest in taking the new point of view I have just indicated, as an asymptotic inequality is often of little use when we don't know from which value of the variable it is verified.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here we are explicitly talking about a new point of view compared to previous research, and also compared to the author's own research. The note cited is thet of Mars, 6, 1899, &lt;em&gt;&lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k30841/f596.item.r=borel&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur la nature aritmétique du nombre &lt;span class=&quot;math-container&quot;&gt;$e$&lt;/span&gt;&lt;/a&gt;&lt;/em&gt;, where no explicit reference is made to height; another relavant work of Borel is &lt;em&gt;Leçons sur la théorie des fonctions&lt;/em&gt;, &lt;a href=&quot;https://archive.org/details/leconstheoriefon00borerich/page/n33/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chapter II&lt;/a&gt;.&#xA;The works of Störmer are &lt;em&gt;&lt;a href=&quot;https://projecteuclid.org/journals/acta-mathematica/volume-27/issue-none/Quelques-propri%C3%A9t%C3%A9s-arithm%C3%A9tiques-des-int%C3%A9grales-elliptiques-et-leurs-applications-a/10.1007/BF02421305.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quelques propriétés arithmétiques des intégrales elliptiques et leurs applications a la théorie des fonctions entières transcendantes&lt;/a&gt;&lt;/em&gt; and &lt;em&gt;&lt;a href=&quot;http://www.numdam.org/item/10.24033/bsmf.628.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur une propriété arithmétique des logarithmes des nombres algébriques&lt;/a&gt;&lt;/em&gt;; as in the case of Borel's work previously cited, there is no direct reference to the term height.&#xA;The relevant works of Maillet are &lt;em&gt;&lt;a href=&quot;http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3089&amp;amp;I=782&amp;amp;M=tdm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur les équations différentielles rationnelles&lt;/a&gt;&lt;/em&gt;, &lt;em&gt;&lt;a href=&quot;http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3089&amp;amp;I=1190&amp;amp;M=tdm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur les équations et les nombres transcendants&lt;/a&gt;&lt;/em&gt;, &lt;em&gt;&lt;a href=&quot;http://www.numdam.org/item/10.24033/bsmf.672.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur les équations différentielles et la théorie&#xA;des ensembles&lt;/a&gt;&lt;/em&gt;, and &lt;em&gt;&lt;a href=&quot;http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3089&amp;amp;I=1190&amp;amp;M=tdm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sur les nombres e et p et les équations transcendantes&lt;/a&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In conclusion, judging from Borel's way of expressing himself in the quoted passages and again in other places in his note (toward the end, it is suggested that the concept of height be generalized to the case of complex numbers as well), it seems that the term was precisely introduced by Borel, with the idea of generalizing approaches previously proposed by other authors and by Borel himself.&lt;/p&gt;&#xA;&lt;p&gt;However, this still does not tell us where exactly Borel got the idea of introducing the concept of height from. Again, Borel's article itself gives us some clues. Toward the end we read:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Si l'on construit l'analyse en partant de &lt;em&gt;l'unité&lt;/em&gt;, la définition d'un nombre quelconque exige un certain nombre d'opération élémentaires , [...] l'hauteur &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; d'un nombre &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; [...] est en relation simples avec le nombre d'opérations élémentaires au moyen desquelles on peut définir &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; [...]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;If we construct the analysis starting from &lt;em&gt;unity&lt;/em&gt;, the definition of any number requires a certain number of elementary operations, [...] the height &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; of a number &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; [...] is in simple relation to the number of elementary operations by means of which &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; can be defined [...].&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This sibylline statement is made more clearly in a later work, &lt;em&gt;&lt;a href=&quot;http://www.numdam.org/item/JMPA_1903_5_9__329_0.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Contribution à l'analyse arithmétique du continu&lt;/a&gt;&lt;/em&gt; (this work deserves to be restarted almost in its entirety, below I will select only a few passages):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Toutes les Mathématiques peuvent se déduire de la seule notion&#xA;de nombre entier [...] [c]e nouveau point de vue peut être caractérisé par le fait que l'on ne fait jamais intervenir dans&#xA;chaque question qu'un &lt;em&gt;nombre limité de nombres entiers&lt;/em&gt; au moyen&#xA;desquels tous les éléments de la question sont explicitement définis [...] Si l'on considère ainsi un nombre (rationnel, algébrique ou transcendant) caractérisé par un certain nombre d'entiers et par certaines opérations [...] La complication de chaque nombre peut être ainsi caractérisée numériquement [...] le nombre&#xA;ou symbole qui mesurera cette complication sera dit la hauteur du&#xA;nombre dans le système considéré.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;i.e.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;All Mathematics can be deduced from the notion of integers alone [...] [t]his new point of view can be characterized by the fact that each question never involves more than a &lt;em&gt;limited number of integers&lt;/em&gt;, by means of which all the elements of the question are explicitly defined [...] If we thus consider a number (rational, algebraic or transcendental) characterized by a certain number of integers and by certain operations [...] The complication of each number can thus be characterized numerically. If we thus consider a number (rational, algebraic or transcendental) characterized by a certain number of integers and by certain operations [...] The complication of each number can thus be characterized numerically [...] the number or symbol that measures this complication is said to be the height of the number in the system considered.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Borel himself in a footnote lets us know that this idea of height is essentially due to &lt;a href=&quot;https://en.wikipedia.org/wiki/Jules_Drach&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jules Drach&lt;/a&gt; (with whom Borel had written in the 1895 &lt;em&gt;&lt;a href=&quot;https://archive.org/details/introletudetheor00borerich/page/n7/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Introduction à l'Étude de la Théorie des Nombres et de l'Algèbre Supérieure&lt;/a&gt;&lt;/em&gt;), and is found in particular in &lt;em&gt;Introduction à la Théorie des nombres et de l'Algèbre supérieure&lt;/em&gt; and in his thesis &lt;em&gt;&lt;a href=&quot;http://www.numdam.org/item/10.24033/asens.457.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Essai sur la théorie générale de l'intégration et sur la classification des transcendantes&lt;/a&gt;&lt;/em&gt;. However, according to Borel, in the works cited above, Drach applies this idea only to the study of differential equations and partial derivatives, whereas instead one should use this concept to address problems concerning transcendental numbers and functions more generally. Nonetheless, it is Borel himself who states that this observation is also due to Drach.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastEditorUserId="6545" LastEditDate="2024-01-30T15:00:03.033" LastActivityDate="2024-01-30T15:00:03.033" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16171" PostTypeId="2" ParentId="15962" CreationDate="2024-01-30T19:17:05.210" Score="3" Body="&lt;p&gt;Certainly not as young as Arthur Rubin, but &lt;a href=&quot;https://en.wikipedia.org/wiki/Carl_St%C3%B8rmer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Carl Størmer&lt;/a&gt; was also very precocious.&lt;/p&gt;&#xA;&lt;p&gt;Størmer entered the University of Christiana (Oslo) in 1892 and in the same year, at the age of 18, he published his first work, &lt;em&gt;&lt;a href=&quot;https://www.biodiversitylibrary.org/item/148545#page/451/mode/1up&quot; rel=&quot;nofollow noreferrer&quot;&gt;Summation af nogle trigonometriske rækker&lt;/a&gt;&lt;/em&gt; (&lt;em&gt;Summation of some trigonometric series&lt;/em&gt;), and then published six more articles before the end of his studies.&lt;/p&gt;&#xA;" OwnerUserId="6545" LastActivityDate="2024-01-30T19:17:05.210" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16172" PostTypeId="2" ParentId="15962" CreationDate="2024-01-31T01:39:44.623" Score="3" Body="&lt;p&gt;Lajos Posa wrote one &lt;a href=&quot;http://combinatorica.hu/%7Ep_erdos/1962-13.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;1&lt;/a&gt; with Erdos and was submitted for publication in September 1961. At the time of submission Posa was not 14 yet.&lt;br /&gt;&#xA;[1] Erdos and Posa : On the maximal number of disjoint circuits of a graph, &lt;a href=&quot;http://combinatorica.hu/%7Ep_erdos/1962-13.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://combinatorica.hu/~p_erdos/1962-13.pdf&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="7908" LastActivityDate="2024-01-31T01:39:44.623" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16174" PostTypeId="1" CreationDate="2024-02-01T05:41:02.237" Score="4" ViewCount="166" Body="&lt;p&gt;The theory of vector bundles (and their characteristic classes) appears to have been standardized in the 20th century by all of the familiar names. Considering its substantial importance throughout geometry, I'm curious as to where/how the idea originated. I'm sure that they were &amp;quot;implicitly&amp;quot; present well before a proper definition was given, and I assume they were introduced as a generalization of the tangent bundle.&lt;/p&gt;&#xA;&lt;p&gt;Also, considering the various perspectives on vector bundles in algebraic geometry, how were algebraic geometers pre-20th century thinking about vector bundles?&lt;/p&gt;&#xA;&lt;p&gt;I appreciate any information! If possible, I'd love to see references to historic papers on the topic.&lt;/p&gt;&#xA;" OwnerUserId="19642" LastEditorUserId="55" LastEditDate="2024-02-03T13:02:00.947" LastActivityDate="2024-02-03T13:12:09.273" Title="What is the history of vector bundles and their characteristic classes?" Tags="|geometry|algebraic-geometry|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16175" PostTypeId="1" CreationDate="2024-02-01T19:57:17.527" Score="3" ViewCount="117" Body="&lt;p&gt;When was a mathematical formula (instead of just words) used for the 1st time in natural sciences to describe a natural phenomenon?&lt;/p&gt;&#xA;" OwnerUserId="19645" LastActivityDate="2024-02-02T02:48:28.183" Title="Mathematization of natural sciences" Tags="|mathematics|physics|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16176" PostTypeId="2" ParentId="16175" CreationDate="2024-02-02T02:48:28.183" Score="1" Body="&lt;p&gt;Pythagoras discovered that simple mathematical ratios were important to harmony.  As Xenocrates put it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is reported over a century after the discovery.  Any earlier such discovery has not come down to us.&lt;/p&gt;&#xA;" OwnerUserId="12724" LastActivityDate="2024-02-02T02:48:28.183" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16177" PostTypeId="1" AcceptedAnswerId="16182" CreationDate="2024-02-02T10:59:45.727" Score="4" ViewCount="449" Body="&lt;p&gt;According to Wikipedia, the first person who proposed &lt;a href=&quot;https://en.wikipedia.org/wiki/18-electron_rule&quot; rel=&quot;nofollow noreferrer&quot;&gt;18-electron rule&lt;/a&gt; was American chemist Irving Langmuir, but the rule is widely known by the name Sidgwick's rule. I cannot find any information about Sidgwick, was he a scientist?&lt;/p&gt;&#xA;" OwnerUserId="19647" LastEditorUserId="55" LastEditDate="2024-02-03T01:21:02.403" LastActivityDate="2024-02-03T01:21:02.403" Title="Who was the inventor of the 18-electron rule?" Tags="|biographical-details|discoveries|chemistry|chemists|" AnswerCount="2" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16179" PostTypeId="2" ParentId="16177" CreationDate="2024-02-02T13:33:55.933" Score="3" Body="&lt;p&gt;I imagine the reference is to &lt;a href=&quot;https://en.wikipedia.org/wiki/Nevil_Sidgwick&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nevil Sidgwick&lt;/a&gt; The Wiki entry does not specifically mention that rule but he clearly worked in related areas. Note that scientific discoveries are often named after someone other than their discoverer, a phenomenon known as &lt;a href=&quot;https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy&quot; rel=&quot;nofollow noreferrer&quot;&gt;Stigler's law of eponymy&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="16128" LastActivityDate="2024-02-02T13:33:55.933" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16181" PostTypeId="2" ParentId="15962" CreationDate="2024-02-03T00:03:59.520" Score="4" Body="&lt;p&gt;Katherine Knox recently published &lt;a href=&quot;https://doi.org/10.1080/00029890.2023.2230860&quot; rel=&quot;nofollow noreferrer&quot;&gt;Billiard circuits in quadrilaterals&lt;/a&gt; in the &lt;i&gt;American Mathematical Monthly&lt;/i&gt;, and the paper contains original research results. I'm not sure of her exact age, but she is a seventh grader.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2024-02-03T00:03:59.520" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16182" PostTypeId="2" ParentId="16177" CreationDate="2024-02-03T01:18:10.530" Score="4" Body="&lt;p&gt;History of the rule is traced in &lt;a href=&quot;https://pubs.acs.org/doi/10.1021/ed082p28&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jensen, The Origin of the 18-Electron Rule&lt;/a&gt;. Langmuir proposed the original version in &lt;a href=&quot;https://www.science.org/doi/10.1126/science.54.1386.59&quot; rel=&quot;nofollow noreferrer&quot;&gt;Types of Valence (1921)&lt;/a&gt;. Nevil Sidgwick (1873-1952) refined it using Bohr-Bury configurations in 1923, see his &lt;a href=&quot;https://play.google.com/store/books/details/Nevil_Vincent_Sidgwick_The_Electronic_Theory_of_Va?id=xAu9AAAAIAAJ&amp;amp;pli=1.&quot; rel=&quot;nofollow noreferrer&quot;&gt; Electronic Theory of Valency (1927)&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Detailed Sidgwick's biography is available &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rsbm.1954.0016&quot; rel=&quot;nofollow noreferrer&quot;&gt;from the Royal Society&lt;/a&gt;. He was born&#xA;in Oxford on 8 May 1873. In 1892 he entered Christ Church college at Oxford to study natural sciences, and in the late 1890-s worked in Ostwald’s laboratory in Leipzig, where he learned modern techniques of studying organic compounds. From 1901 and for the rest of his life Sidgwick stayed in  the Lincoln college. In 1914 he met Rutherford, whose work and personality had a deep influence on him. From then on he turned to atomic chemistry. Aside from the 18-electron rule, he is remembered for the &lt;a href=&quot;https://en.wikipedia.org/wiki/Inert-pair_effect&quot; rel=&quot;nofollow noreferrer&quot;&gt;inert-pair effect&lt;/a&gt; (1927) and ideas that led to the &lt;a href=&quot;https://en.wikipedia.org/wiki/VSEPR_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;VSEPR theory&lt;/a&gt; (1940).&lt;/p&gt;&#xA;&lt;p&gt;Here is from Jensen:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The 18-electron rule, much beloved of the organometallic&#xA;chemist, was first formulated in 1921 by the American&#xA;chemist, Irving Langmuir (1881–1957), as part of his&#xA;program to extend the Lewis static-atom model beyond&#xA;argon in the periodic table... An alternative electron counting procedure, based on&#xA;the newer electronic configurations of Bohr and Bury, was&#xA;introduced by the British chemist, Nevil Sidgwick (1873–&#xA;1952), in 1923. Known as the effective atomic number&#xA;(EAN) rule, it focused not just on the valence-shell electron count, but on the total atom electron count. Attainment of an octet or an 18-electron outer configuration was equivalent to attaining the total electron count (or atomic number) of the nearest noble gas.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Sidgwick’s counting procedure was first applied to transition-metal carbonyls and nitrosyls by the German chemist, F. Reiff, in 1931; in 1934 Sidgwick extended its use to include bridged, as well as mononuclear, complexes. By 1940 Blanchard was also using&#xA;Sidgwick’s version of the rule and the same is true of many&#xA;inorganic texts published in the 1950s. In the late 1960s, however, there was a reversion to the&#xA;earlier electron counting procedure of Langmuir, no&#xA;doubt because Sidgwick’s procedure, which includes the&#xA;chemically inactive core electrons, results in a separate numerical stability standard for each row of the transition&#xA;block, whereas Langmuir’s procedure, like the octet rule,&#xA;makes use of a single numerical standard applicable to the&#xA;entire block.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-02-03T01:18:10.530" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16183" PostTypeId="2" ParentId="16072" CreationDate="2024-02-03T04:22:34.977" Score="2" Body="&lt;p&gt;Richard Stanley, in his book &lt;i&gt;Enumerative Combinatorics&lt;/i&gt;, Volume I, writes that &amp;quot;the first written statement connecting Eulerian numbers with descents seems to have been by Carlitz and Riordan in 1953.&amp;quot; The reference is to the paper, &lt;a href=&quot;https://doi.org/10.1215/S0012-7094-53-02033-X&quot; rel=&quot;nofollow noreferrer&quot;&gt;Congruences for Eulerian numbers&lt;/a&gt;, &lt;i&gt;Duke Math. J.&lt;/i&gt; 20 (1953), 339–343. They write:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is associated with a game of patience played with deck of cards of arbitrary specification, with cards dealt into a single pack so long as they are not falling and a new pack started for every fall; the problem is to determine the number of ways of dealing &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; packs. When the deck is of cards numbered &lt;span class=&quot;math-container&quot;&gt;$1$&lt;/span&gt; to &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;, the answer is given by numbers &lt;span class=&quot;math-container&quot;&gt;$A_{ks}$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Though you didn't ask about this specifically, Stanley also mentions that Laplace (1812) already knew that Eulerian numbers measure the volume of diagonal slices of a hypercube. See &lt;a href=&quot;https://doi.org/10.1016/j.aam.2019.101960&quot; rel=&quot;nofollow noreferrer&quot;&gt;An asymptotic distribution theory for Eulerian recurrences with applications&lt;/a&gt; by Hwang et al. for more information.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastEditorUserId="11174" LastEditDate="2024-02-03T04:28:35.490" LastActivityDate="2024-02-03T04:28:35.490" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16184" PostTypeId="2" ParentId="8184" CreationDate="2024-02-03T04:44:47.453" Score="1" Body="&lt;p&gt;Several species of animals  &lt;a href=&quot;http://web.archive.org/web/20210218012215/https://blogs.iu.edu/sciu/2020/12/12/seven-cryptids-species/&quot; rel=&quot;nofollow noreferrer&quot;&gt;were long regarded by scientists as cryptids&lt;/a&gt; but are now known to be real, e.g., the platypus, okapi, and giant squid.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2024-02-03T04:44:47.453" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16185" PostTypeId="2" ParentId="1987" CreationDate="2024-02-03T05:10:58.770" Score="2" Body="&lt;p&gt;In his article, &lt;a href=&quot;http://web.archive.org/web/20100613095251/http://claymath.org/library/annual_report/ar2005/05report_featurearticle.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Can biology lead to new theorems?&lt;/a&gt;, Bernd Sturmfels gives four theorems that were inspired by biology. As one example, Andreas Dress and his collaborators have developed a theory of finite metric spaces that was inspired by the study of phylogenetic trees. I once attended a talk by Dress where he stressed that the theory could be used to &amp;quot;fit&amp;quot; a metric space to phylogenetic data, in a way that did &lt;i&gt;not&lt;/i&gt; impose the assumption that the data came from a tree. Empirically, one finds that the fitted space is actually close to a tree. This result helps answer the skeptic who claims that phylogenetics employs circular logic, because it assumes a phylogenetic tree and then turns around and claims that the data provides evidence for an evolutionary tree.&lt;/p&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2024-02-03T05:10:58.770" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16186" PostTypeId="2" ParentId="13035" CreationDate="2024-02-03T05:35:41.387" Score="0" Body="&lt;p&gt;My grandmas uncle, but I don't think his Romani ancestry was documented after they came to the U.S. He was considered the top on his field. George Ray Wait.&lt;/p&gt;&#xA;" OwnerUserId="19651" LastActivityDate="2024-02-03T05:35:41.387" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16187" PostTypeId="2" ParentId="16174" CreationDate="2024-02-03T12:56:08.857" Score="4" Body="&lt;p&gt;There are brief historical notes on the history of vector bundles and characteristic classes in section 6.5.4 of &lt;a href=&quot;https://im0.p.lodz.pl/%7Ekubarski/AnalizaIV/Wyklady/Luke%20G.,%20Mishchenko%20A.%20Vector%20bundles%20and%20their%20applications%20(Kluwer,%202000)(262s)_MDat_.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Luke-Mishchenko, Vector Bundles and Their Applications&lt;/a&gt;. A more detailed account is given by &lt;a href=&quot;https://doi.org/10.1016/B978-044482375-5/50023-7&quot; rel=&quot;nofollow noreferrer&quot;&gt;Zisman in the History of Topology volume, ch. 22&lt;/a&gt;, but it has to be filtered because he goes over history of general fibrations and fiber bundles, rather than vector bundles specifically. The same goes for chapter III of &lt;a href=&quot;https://www.google.com/books/edition/A_History_of_Algebraic_and_Differential/RUV5Dz90rDkC?hl=en&amp;amp;gbpv=0&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dieudonne's History of Algebraic and Differential Topology&lt;/a&gt;. On the history of affine connections and gauge theory see &lt;a href=&quot;https://hsm.stackexchange.com/a/147/55&quot;&gt;When and how was the geometric understanding of gauge theories developed?&lt;/a&gt;, and for notation &lt;a href=&quot;https://hsm.stackexchange.com/a/7297/55&quot;&gt;Notation for fiber bundles - why E for total space?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Algebraic geometers pre-20th century were not thinking about vector bundles or anything similar, that perspective emerged after Leray and Grothendieck. Even vector spaces only appeared at the end of 19-th century. Some anticipation can be vaguely identified in Darboux's moving trihedrons from &lt;em&gt;Leçons sur la Théorie Générale des Surfaces&lt;/em&gt; (1889) and Elie Cartan's early work on Pfaffian systems (1899), see &lt;a href=&quot;https://hsm.stackexchange.com/a/2500/55&quot;&gt;Where did Cartan introduce his notation for basis vectors and covectors?&lt;/a&gt; for references. Zisman names Cartan as the founding father of vector bundles specifically, and Seifert as initiating the general terminology:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The other founder father is Elie Cartan. In a long series of papers published between 1922 and 1925, in relation to differential geometry and connections, the author engaged himself in computations which may seem a little repetitive today but where one can however see how E. Cartan was endeavouring to associate, in a more and more precise way a vector, affine or projective space with each point of a manifold, providing a coherent system of relations in order to link all these spaces together.... It is Herbert Seifert who created the term gefaserter Raum (fibre space) in his 1932 paper Topologie drei-dimensionale gefaserter Railme. That work, still motivated by the study of 3-dimensional manifolds, contains a definition of fibre spaces, sometimes called Seifert fibrations...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;According to Dieudonne, vector bundles first appeared explicitly in Whitney's work of 1935. Here is from Luke-Mishchenko on the history of characteristic classes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The study of bundles was probably started by Poincare with the study of nontrivial coverings. Fibrations appeared in connection with the study of smooth&#xA;manifolds, then characteristic classes were defined and for a long time they were the main tool of investigation. The Stieffel–Whitney characteristic classes were introduced by Stieffel and Whitney in 1935 for tangent bundles&#xA;of smooth manifolds. Whitney also considered arbitrary sphere bundles. These were &lt;span class=&quot;math-container&quot;&gt;$\!\!\!\mod 2$&lt;/span&gt; characteristic classes. The integer characteristic classes were constructed by Pontryagin. He also proved the classification theorem for general bundles with the structure group &lt;span class=&quot;math-container&quot;&gt;$O(n)$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$SO(n)$&lt;/span&gt; using a universal bundle over a universal base such as a&#xA;Grassmannian. For complex bundles, characteristic classes were constructed by Chern. This was the beginning of the general study of bundles and the period was well&#xA;described in the book of Steenrod.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;A crucial point of time came with the discovery by Leray of spectral sequences which were applied to the calculation of the homology groups of bundles. Another was at the discovery by Bott of the periodicity of the homotopy groups of the unitary and orthogonal groups. From this point, vector bundles occupied an important place in the theory&#xA;of bundles as it was through them that nontrivial cohomology theories were constructed. It was through Bott periodicity that the Grothendieck constructions became so significant in their influence on the development of algebraic K–theory.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-02-03T13:12:09.273" LastActivityDate="2024-02-03T13:12:09.273" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16188" PostTypeId="2" ParentId="8184" CreationDate="2024-02-03T13:01:51.850" Score="2" Body="&lt;p&gt;Poet Edgar Allan Poe is considered to be the first to solve &lt;a href=&quot;https://en.wikipedia.org/wiki/Olbers%27s_paradox&quot; rel=&quot;nofollow noreferrer&quot;&gt;Olber's paradox&lt;/a&gt; in &lt;em&gt;Eureka: A prose poem&lt;/em&gt; of 1848. Lord Kelvin came with the same argument 50-ish years later, however the solution would not be agreed upon until experimental verification in the 1920-1930s of a non-static universe.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-02-04T12:46:58.030" LastActivityDate="2024-02-04T12:46:58.030" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16189" PostTypeId="1" CreationDate="2024-02-05T13:02:23.210" Score="0" ViewCount="46" Body="&lt;p&gt;I heard a story about how ancient people believed that fathers contributed to the white things in a child (bones), while mothers contributed the red matter (blood). However, I found no reference regarding this in my unsuccessful internet search.&lt;/p&gt;&#xA;&lt;p&gt;So where and when did this thought originate from, or which are the oldest references to it?&lt;/p&gt;&#xA;&lt;p&gt;(Trivia: &lt;a href=&quot;https://www.google.com/search?q=wolves%20without%20teeth%20lyrics&quot; rel=&quot;nofollow noreferrer&quot;&gt;It's even referenced in a song.&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="3138" LastEditorUserId="6609" LastEditDate="2024-02-05T13:11:41.397" LastActivityDate="2024-02-05T13:11:41.397" Title="History on ancient thoughts regarding bones and blood" Tags="|biology|obsolete-theories|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16190" PostTypeId="5" CreationDate="2024-02-05T13:12:12.690" Score="0" Body="&lt;p&gt;For obsolete or superseded scientific theories&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-02-06T08:33:59.543" LastActivityDate="2024-02-06T08:33:59.543" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16191" PostTypeId="4" CreationDate="2024-02-05T13:12:12.690" Score="0" Body="For obsolete or superseded scientific theories" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-02-06T07:57:42.650" LastActivityDate="2024-02-06T07:57:42.650" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16192" PostTypeId="1" CreationDate="2024-02-05T21:50:07.937" Score="1" ViewCount="67" Body="&lt;p&gt;I am new to statistics and linear regression and I came across the face that auguste bravais discovered regression line but didn't realize it.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Auguste Bravais (1811-1863), professor of astronomy and physics, is&#xA;probably best known for his work in crystallography (Nelson, 1998).&#xA;With respect to error theory, he is best known for a paper he wrote in&#xA;1846 titled &amp;quot;Analyse mathématique sur les probabilités des erreurs de&#xA;situation d'un point&amp;quot; [translated: &amp;quot;Mathematical analysis on the&#xA;probability of errors of a point&amp;quot;]. This work is renowned for being&#xA;the very first mathematical exposition of the theory of correlation.&#xA;Pearson (1896), later to recall these remarks, said that it was&#xA;Bravais who first discussed the fundamental theorems of the&#xA;correlational calculus. In his famous paper of 1846, Bravais,&#xA;mathematically, not empirically, found the equation of the normal&#xA;surface for the frequency of error. Using both analytic and geometric&#xA;methods, Bravais also essentially found what would eventually be&#xA;coined &amp;quot;regression line.&amp;quot; He did so through investigating how the&#xA;various elliptical areas of the frequency surface vary given various&#xA;directly observed quantities. Through this, he found the line of&#xA;regression, but, in essence, did not realize it, and thus could not&#xA;&amp;quot;make the leap&amp;quot; (Walker, 1929) necessary to claim the discovery of&#xA;correlation or regression&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is how did he use &amp;quot;various elliptical areas of the frequency surface&amp;quot; to come across the regression line? I don't understand it at all. Please use diagrams if you can. Not too much complex maths too cause I am a newbie. Just explain simply how that works.&lt;/p&gt;&#xA;" OwnerUserId="19573" LastActivityDate="2024-02-05T21:50:07.937" Title="How exactly did Auguste Bravais come up with the regression line?" Tags="|mathematics|physics|calculus|discoveries|machine-learning|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16193" PostTypeId="1" CreationDate="2024-02-06T09:29:17.930" Score="3" ViewCount="90" Body="&lt;p&gt;Why is conventional width of zodiac set to 16 degrees (8 degrees at both sides of ecliptic), see for example this Taurus plate from Bayer's Uranometria? I assume that the planet which defines the width is Venus but its extremal latitudes in the past millenia were nearly 9 degrees. On the other hand its difficult to observe the planet at greatest latitudes since it's close to the Sun in the sky.&lt;/p&gt;&#xA;&lt;p&gt;I would appreciate sources (in particular ancient ones) which gives zodiac width. Was the width result of observation or computation?&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/dHc1Q.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/dHc1Q.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="9778" LastActivityDate="2024-02-06T09:29:17.930" Title="Width of zodiac" Tags="|astronomy|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16195" PostTypeId="2" ParentId="3131" CreationDate="2024-02-07T17:59:34.777" Score="1" Body="&lt;p&gt;In Europe, the &lt;strong&gt;First World War 1914-18&lt;/strong&gt;, also called the &lt;em&gt;&lt;strong&gt;Great War&lt;/strong&gt;&lt;/em&gt;, was an extremely traumatic event, that stopped an era of well-being and progress (often referred to as &lt;a href=&quot;https://en.wikipedia.org/wiki/Belle_%C3%89poque&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;La belle époque&lt;/em&gt;&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The European war 1914-18 involved all social components of the major  European nations, with intellectuals and students in the forefront.&#xA;The consequences were  disastrous in terms of loss of human lives and&#xA;careers of excellent scholars cut short. A drama that changed the destiny of a generation of scientists in many countries, the scientific community itself, and the course of scientific research.&lt;/p&gt;&#xA;&lt;p&gt;Among the scientists, many mathematicians of many European countries  died in the battlefield and in  the trenches during the &lt;em&gt;Great War&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I mention here, in particular, the tragedy of what has been called the &lt;em&gt;incomprensible écatombe&lt;/em&gt;&#xA;in France. In particular,  800 students of the &lt;a href=&quot;https://en.wikipedia.org/wiki/%C3%89cole_normale_sup%C3%A9rieure_(Paris)&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;École Normale Supérieure&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;  of Paris took part in the war, and 239 of them lost their lives. Among them, also many mathematicians (see &lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The war left a deep wound  and a traumatic memory in the country, which influenced also the &lt;a href=&quot;https://en.wikipedia.org/wiki/Nicolas_Bourbaki&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;&lt;strong&gt;Bourbaki group&lt;/strong&gt;&lt;/em&gt;&lt;/a&gt;: they  described the post-war period in France as a “mathematic desert”, and they attributed the causes to the loss of a generation, decimated by the war.&lt;/p&gt;&#xA;&lt;p&gt;According to some historian of mathematics,  the trauma of the war influenced  the  abstract nature of the mathematics of the Bourbaki group. The historian of mathematics David Aubin writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Dans les années 1930, le groupe de mathématiciens Bourbaki a promu une&#xA;vision abstraite et universelle de sa discipline en rupture avec celle&#xA;de ses prédécesseurs. Le sacrifice de nombreux jeunes mathématiciens&#xA;durant la Première Guerre mondiale a sans doute fortement contribué à&#xA;ce revirement.&lt;/p&gt;&#xA;&lt;p&gt;[During the years 1930, the Bourbaki group of mathematicians  has&#xA;prompted an abstract and universal vision of their subject, in rupture&#xA;with  the one of their predecessors. The sacrifice of many young&#xA;mathematicians during the First World War has undoubtedly contributed&#xA;to this reversal] ( &lt;span class=&quot;math-container&quot;&gt;$[2]$&lt;/span&gt;, p.1, my transl.)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;And in his book  &lt;em&gt;&lt;strong&gt;L’Élite sous la mitraille&lt;/strong&gt;&lt;/em&gt; &lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt;  David Aubin writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Au course des semaines le plus sanglantes de l’histoire de France,&#xA;entre le 22 août et le 20 octobre 1914, douze mathématiciens élèves&#xA;de l’École Normale supérieure de la rue d’Ulm, à Paris, tous agrégés&#xA;de mathématiques et, pour la plupart, auteurs de travaux remarqués,&#xA;publiés et recencés dans le repertoire de reference, le  “Jahrbuch&#xA;über die Fortschritte der Mathematik” (JFM), sont tués sur le champs&#xA;de bataille du Nord et de l’Est. Dans les années qui suivent, dix&#xA;autres jeunes mathematicians connaîtront le même sort.&lt;/p&gt;&#xA;&lt;p&gt;During the bloodiest weeks of the history of France, between the 22th&#xA;of August and the 20th of October 1914, twelve mathematicians, all&#xA;former students  of the École Normale Supérieure de la rue d’Ulm, in&#xA;Paris, all of them &lt;em&gt;agrégés&lt;/em&gt; of mathematics and, for the most part,&#xA;authors of works reported, published and reviewed in the bibliographic&#xA;repertory, the “Jahrbuch über die Fortschritte der Mathematik” (JFM),&#xA;are killed on the battlefield of the North and the East. In the&#xA;following years, ten other young mathematicians will&#xA;suffer the same fate. (p. 7, my transl. )&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In 1923, built thanks to a subscription, the &lt;em&gt;&lt;strong&gt;Monument aux morts de l’École Normale Superiéure&lt;/strong&gt;&lt;/em&gt; [&lt;em&gt;Monument to the Victims of the École Normale Superiéure&lt;/em&gt;] in Rue de l’Ulm,  by  the sculptor  Paul Landowski, was inaugurated.&lt;/p&gt;&#xA;&lt;p&gt;Out of  &lt;span class=&quot;math-container&quot;&gt;$239$&lt;/span&gt; &lt;em&gt;normaliens&lt;/em&gt; whose name appears on the monument to the victims,  &lt;strong&gt;&lt;span class=&quot;math-container&quot;&gt;$22$&lt;/span&gt; are agrégés of mathematics.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Below the &lt;em&gt;Tableau&lt;/em&gt; that gives the list of those mathematicians who died in  the First World War: as the table shows, fourteen of them died in 1914, four in 1915, three in 1916 and another, Paul Viple, some days after the Armistice, because of the injuries sustained  on the 25th of October 1918.&#xA;(source &lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt;, p. 10)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/rXW3F.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/rXW3F.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;We can notice the name of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ren%C3%A9_Gateaux&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;René Gateaux&lt;/strong&gt;&lt;/a&gt;,  whose name is known for the &lt;em&gt;Gateux derivative&lt;/em&gt;, and whose works on functional analysis impressed the famous Italian mathematician Vito Volterra.&#xA;&lt;span class=&quot;math-container&quot;&gt;$$\;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;Below an image of the &lt;em&gt;Monument to the Victims of the École Normal Supérieure&lt;/em&gt;:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/I9bN2.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/I9bN2.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[1]$&lt;/span&gt; Aubin, David, &lt;em&gt;L’élite sous la mitraille. Les normaliens, les mathématiques et la Grande Guerre,  1900-1925&lt;/em&gt;, Editions Rue d’Ulm/Presse de l’École normale supériueure, 2018.&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$[2]$&lt;/span&gt; Aubin, David,   ‘Des mathématiciens sous la Grande Guerre’,&#xA;&lt;a href=&quot;https://www.pourlascience.fr/sd/histoire-sciences/des-mathematiciens-sous-la-grande-guerre-20451.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://www.pourlascience.fr/sd/histoire-sciences/des-mathematiciens-sous-la-grande-guerre-20451.php&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="17306" LastActivityDate="2024-02-07T17:59:34.777" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16196" PostTypeId="1" CreationDate="2024-02-08T02:54:38.503" Score="3" ViewCount="227" Body="&lt;p&gt;I saw this &lt;a href=&quot;https://hsm.stackexchange.com/q/15812&quot;&gt;post&lt;/a&gt;.  And I already posted it on &lt;a href=&quot;https://math.stackexchange.com/q/4858448&quot;&gt;Math stack exchange&lt;/a&gt;, but since someone recommended this site, I'm refining it and posting it again.  And I understand that the mathematical object called cohomology came from Poincaré duality.  But I would like to know the details of the history of cohomology itself.  So, if you know, I would appreciate it if you could leave an answer.  Or, I would appreciate it if you could recommend a book related to the history of cohomology with as few formulas as possible.&lt;/p&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="6609" LastEditDate="2024-02-08T10:00:32.857" LastActivityDate="2024-03-26T14:27:56.267" Title="History of cohomology theory" Tags="|mathematics|topology|algebraic-geometry|book-recommendation|" AnswerCount="3" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16197" PostTypeId="1" CreationDate="2024-02-08T06:13:59.953" Score="1" ViewCount="82" Body="&lt;p&gt;In his book &lt;em&gt;Surely You're Joking, Mr. Feynman!&lt;/em&gt;, Feynman himself says that he did not know how to compute contour integrals. To give a direct quote, he says:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One thing I never did learn was contour integration.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I find this fascinating as there is a famous result in quantum field theory involving contour integration that is named after Feynman (the Feynman propagator).&lt;/p&gt;&#xA;&lt;p&gt;Was Feynman the first to derive the Feynman propagator? If so, how did he do it if he supposedly did not know about complex contour integration. If not, why is it named after him?&lt;/p&gt;&#xA;" OwnerUserId="19665" LastActivityDate="2024-02-08T06:13:59.953" Title="Why is the Feynman propagator named after Feynman?" Tags="|physics|quantum-field-theory|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16199" PostTypeId="2" ParentId="16196" CreationDate="2024-02-08T15:42:33.483" Score="3" Body="&lt;p&gt;Cohomology theories were introduced independently by&#xA;A. Kolmogorov, Ueber die Dualitat im Aufbau der kombinatorischen Topologie, Mat. Sbornik, vol. 1, 1936, 97-102,&lt;/p&gt;&#xA;&lt;p&gt;and by J. W. Alexander, On the chains of a complex and their&#xA;duals, Proc. Nat. Acad. Sci, 21, 1935, 509-512, complete exposition in A.. of Math., 37, 1936, 698-708.&lt;/p&gt;&#xA;&lt;p&gt;Remark. One of the comments mentions Poincare duality. It is only modern formulation of duality that uses cohomology. Poincare himself stated it in terms of homology.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-26T11:59:24.703" LastActivityDate="2024-03-26T11:59:24.703" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16200" PostTypeId="1" AcceptedAnswerId="16208" CreationDate="2024-02-09T01:57:45.680" Score="1" ViewCount="216" Body="&lt;p&gt;Among the many cohomology theory's branches I asked about last time, I was curious about &lt;span class=&quot;math-container&quot;&gt;$d^2=0$&lt;/span&gt; because I know that it is the formula that is the basis of all cohomology.  So this time, I would like to ask a more detailed question than the last one.  I want to know the background of singular homology, that is, the motivation and history of singular homology.  I would like to receive a recommendation for a text-oriented book with little formula or an answer on the motivation and background of this singular homology.&lt;/p&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="6609" LastEditDate="2024-02-09T13:35:32.130" LastActivityDate="2024-02-14T03:27:02.650" Title="Motivation and history of singular homology" Tags="|topology|book-recommendation|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16201" PostTypeId="2" ParentId="13356" CreationDate="2024-02-09T16:41:45.713" Score="5" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Gaussian_correlation_inequality&quot; rel=&quot;noreferrer&quot;&gt;Gaussian correlation inequality&lt;/a&gt; was first conjectured in the 1950s, and not proved until 2014 by Thomas Royen. As Wikipedia puts it,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The proof did not gain attention when it was published in 2014, due to Royen's relative anonymity and the fact that the proof was published in a predatory journal. Another reason was a history of false proofs (by others) and many failed attempts to prove the conjecture, causing skepticism among mathematicians in the field. The conjecture, and its solution, came to public attention in 2017, when other mathematicians described Royen's proof in a mainstream publication and popular media reported on the story.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.quantamagazine.org/statistician-proves-gaussian-correlation-inequality-20170328/&quot; rel=&quot;noreferrer&quot;&gt;Quanta magazine&lt;/a&gt; describes the reaction of statistician Donald Richards as follows:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Upon seeing the proof, “I really kicked myself,” Richards said. Over the decades, he and other experts had been attacking the GCI with increasingly sophisticated mathematical methods, certain that bold new ideas in convex geometry, probability theory or analysis would be needed to prove it. Some mathematicians, after years of toiling in vain, had come to suspect the inequality was actually false. In the end, though, Royen’s proof was short and simple, filling just a few pages and using only classic techniques. Richards was shocked that he and everyone else had missed it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="11174" LastActivityDate="2024-02-09T16:41:45.713" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16202" PostTypeId="2" ParentId="16200" CreationDate="2024-02-10T14:24:32.610" Score="2" Body="&lt;p&gt;Poincare's original papers, Analysis situs and 5 supplements to it, contain relatively few formulas, make a good reading, and explain his motivation. If you can read French. Another good source is&lt;/p&gt;&#xA;&lt;p&gt;H. Seifert and W. Threlfall, A textbook of topology,&#xA;which exists in English translation.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-02-11T04:07:47.870" LastActivityDate="2024-02-11T04:07:47.870" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16203" PostTypeId="2" ParentId="15836" CreationDate="2024-02-11T05:40:08.153" Score="0" Body="&lt;p&gt;First, the source linked by @Kvothe in a comment provides a wording that is (significantly) different from that in the OP.&lt;/p&gt;&#xA;&lt;p&gt;Second, I can only guess what Onsager meant, but possibly that was about there being no good theoretical description of phase transitions in &lt;strong&gt;three-dimensional&lt;/strong&gt; models based on calculations using, say, the canonical ensemble. Since 1953, there has been tremendous progress. @MikhailKatz mentioned BCS theory of conductivity; the renormalization group and &lt;span class=&quot;math-container&quot;&gt;$\epsilon$&lt;/span&gt;-expansion methods were developed. I don't work in this area anymore, but I believe there are now excellent expansions for critical indices. Onsager himself obtained an analytical solution for the phase transition in the &lt;strong&gt;two-dimensional&lt;/strong&gt; Ising model in 1944.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-02-11T05:40:08.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16204" PostTypeId="2" ParentId="15028" CreationDate="2024-02-11T12:14:05.813" Score="1" Body="&lt;p&gt;Below, B is X km north of A, while C is X km west of A.&lt;/p&gt;&#xA;&lt;p&gt;Since A and B are on the same meridian/longitude, the circle through them is a great circle, whose length is (approximately) the same as the equator and is what we want to measure.&lt;/p&gt;&#xA;&lt;p&gt;In contrast, A and C are not on same meridian/longitude (instead, they are on the same parallel/latitude). The length of the circle through them is smaller than that of the equator and is not what we want to measure.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/TVeO7.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/TVeO7.jpg&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="18903" LastActivityDate="2024-02-11T12:14:05.813" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16205" PostTypeId="2" ParentId="15897" CreationDate="2024-02-11T13:28:37.920" Score="1" Body="&lt;p&gt;Another good example of a scientist who was afraid to publish is Jeremy Bentham.&lt;/p&gt;&#xA;&lt;p&gt;According to the following &lt;a href=&quot;https://en.wikipedia.org/wiki/Jeremy_Bentham#Gender_and_sexuality&quot; rel=&quot;nofollow noreferrer&quot;&gt;section&lt;/a&gt; of his wiki biography, he argued for the liberalisation of laws prohibiting homosexual sex in the c. 1785 essay &amp;quot;Paederasty (Offences Against One's Self)&amp;quot;. The essay remained unpublished during his lifetime for fear of offending public morality, and was not published until 1978 (!).&lt;/p&gt;&#xA;&lt;p&gt;Incidentally, the story of Bentham was &lt;a href=&quot;https://www.newyorker.com/culture/the-new-yorker-interview/peter-singer-is-committed-to-controversial-ideas&quot; rel=&quot;nofollow noreferrer&quot;&gt;put forth&lt;/a&gt; by philosopher Peter Singer to illustrate the importance of initiating the &lt;em&gt;Journal of Controversial Ideas&lt;/em&gt;. According to their &lt;a href=&quot;https://journalofcontroversialideas.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;website&lt;/a&gt;, the journal is&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... the first open access, peer-reviewed, interdisciplinary journal&#xA;specifically created to promote free inquiry on controversial topics.&lt;/p&gt;&#xA;&lt;p&gt;The Journal of Controversial Ideas offers a forum for careful,&#xA;rigorous, unpolemical discussion of issues that are widely considered&#xA;controversial, in the sense that certain views about them might be&#xA;regarded by many people as morally, socially, or ideologically&#xA;objectionable or offensive. The journal offers authors the option to&#xA;publish their articles under a pseudonym, in order to protect&#xA;themselves from threats to their careers or physical safety.  We hope&#xA;that this will also encourage readers to attend to the arguments and&#xA;evidence in an essay rather than to who wrote it. Pseudonymous authors&#xA;may choose to claim the authorship of their work at a later time, or&#xA;to reveal it only to selected people (such as employers or prospective&#xA;employers), or to keep their identity undisclosed indefinitely.&#xA;Standard submissions using the authors’ actual names are also&#xA;encouraged.&lt;/p&gt;&#xA;&lt;p&gt;We welcome submissions in all areas of academic research insofar as&#xA;the topics discussed are relevant to society at large.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4296" LastActivityDate="2024-02-11T13:28:37.920" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16206" PostTypeId="1" CreationDate="2024-02-12T13:20:50.307" Score="2" ViewCount="166" Body="&lt;p&gt;Rafael Bombelli was the first European mathematician to write about the laws of arithmetic for negative numbers.&lt;/p&gt;&#xA;&lt;p&gt;On Wikipedia I read that he wrote:&#xA;“Minus 5 times minus 6 makes plus 30”.&lt;/p&gt;&#xA;&lt;p&gt;I also read that Bombelli was interested in cubic equations and developed laws of arithmetic with imaginary numbers.&lt;/p&gt;&#xA;&lt;p&gt;My question is that the laws of arithmetic he produced for imaginary numbers have commentaries and are generated by applying existing mathematical properties such as distribution over multiplication to equate whole number answers to cubic equations, which do not uphold to Cardano's general formula.&lt;/p&gt;&#xA;&lt;p&gt;I was wondering if his laws for arithmetic with negatives arose in the same fashion.&lt;/p&gt;&#xA;&lt;p&gt;If there is anywhere I can read about the subsequent progression of arithmetic with negative and imaginary numbers and how their acceptances were somewhat intertwined and “artificial” it would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="19679" LastEditorUserId="6545" LastEditDate="2024-02-12T14:29:42.920" LastActivityDate="2024-02-13T15:53:06.023" Title="Did Rafael Bombelli write any commentary about his rules for arithmetic involving negative numbers?" Tags="|mathematics|reference-request|arithmetic|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16207" PostTypeId="2" ParentId="16206" CreationDate="2024-02-12T15:24:36.990" Score="5" Body="&lt;p&gt;Leo Corry's recent text, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B012B9MA1Y&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;A Brief History of Numbers&lt;/em&gt;&lt;/a&gt;, offers an authoritative account of these matters.&lt;/p&gt;&#xA;&lt;p&gt;According to Corry, Bombelli's attitude to negative numbers was the same as Cardano's. Bombelli would not have written &amp;quot;Minus 5 times minus 6 makes plus 30”, as wikipedia suggests.  Rather, Bombelli would have written &amp;quot;Subtract 5 times subtract 6 makes 30.&amp;quot;  Furthermore, Corry states that the law of signs for multiplication was &lt;a href=&quot;https://hsm.stackexchange.com/questions/8184/examples-of-when-the-professional-scientists-or-mathematicians-were-wrong-but-t/11370#11370&quot;&gt;established by the abacists&lt;/a&gt; and rejected by Cardano.  What Corry gives Bombelli credit for is extending the law of signs to imaginary numbers.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;What Bombelli did in his book [&lt;em&gt;Algebra&lt;/em&gt;] was to extend to the case of square roots of negative numbers (with the help of some new terminology) the abacist rules for the arithmetic of binomials. ... Using this nomenclature, he wrote down correctly all the rules of sign multiplication with numbers of this kind.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="3643" LastEditorUserId="12330" LastEditDate="2024-02-13T15:53:06.023" LastActivityDate="2024-02-13T15:53:06.023" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16208" PostTypeId="2" ParentId="16200" CreationDate="2024-02-14T03:27:02.650" Score="4" Body="&lt;p&gt;Regarding singular homology proper, its main development occurred in the 1930s and 1940s with Lefschetz and Eilenberg figuring prominently. For this history, Dieudonné's &amp;quot;&lt;em&gt;A History of Algebraic and Differential Topology, 1900 - 1960&lt;/em&gt;&amp;quot; is rather authoritative (see especially pages 68-70). See also &lt;a href=&quot;https://en.wikipedia.org/wiki/Ronald_Brown_(mathematician)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Ronnie Brown's&lt;/a&gt; math stack exchange answer &lt;a href=&quot;https://math.stackexchange.com/questions/1055619/original-papers-on-singular-homology-cohomology&quot;&gt;here&lt;/a&gt; in which he comments briefly on singular homology and references Eilenberg's 1944 paper.&lt;/p&gt;&#xA;" OwnerUserId="2918" LastActivityDate="2024-02-14T03:27:02.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16209" PostTypeId="1" CreationDate="2024-02-14T11:41:41.107" Score="0" ViewCount="113" Body="&lt;p&gt;I was reading “A brief history of numbers” by Corry, but I came across a part that confused me.&lt;/p&gt;&#xA;&lt;p&gt;Cardano accepted the law of signs for “subtractions” proposed by an older group of Italian mathematicians because of the product of two binomials.&lt;/p&gt;&#xA;&lt;p&gt;The author writes that that the quadratic equation &lt;span class=&quot;math-container&quot;&gt;$x^2 = px + q$&lt;/span&gt; and two formulas are an example that force Cardano into accepting negatives but I don't see how this is the case.&lt;/p&gt;&#xA;&lt;p&gt;If anyone else has any other examples or commentary it would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="19679" LastEditorUserId="12330" LastEditDate="2024-02-14T22:01:37.897" LastActivityDate="2024-02-15T07:42:29.403" Title="How did negative numbers “force themselves” onto Cardano, and was it analogous to how imaginary numbers were forced upon him?" Tags="|mathematics|numbers|arithmetic|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16211" PostTypeId="1" AcceptedAnswerId="17285" CreationDate="2024-02-15T01:59:50.787" Score="1" ViewCount="103" Body="&lt;p&gt;From the timeline of Maxwell's prediction 1865 and Hertz's observation in 1887 that gave an understanding of the light wave as EM wave, How did the Black body radiation study understand the emitted radiation as what at that time since the timeline of Black body radiation was also in 1890s? Did they know that the radiation is EM wave? Or how did they understand the radiation that emitted?&lt;/p&gt;&#xA;" OwnerUserId="19685" LastEditorUserId="6609" LastEditDate="2024-02-18T10:03:12.323" LastActivityDate="2024-02-29T14:04:56.377" Title="Black body radiation before Hertz's observation?" Tags="|physics|electromagnetism|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16212" PostTypeId="1" CreationDate="2024-02-15T05:48:58.203" Score="1" ViewCount="118" Body="&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Joukowsky_transform&quot; rel=&quot;nofollow noreferrer&quot;&gt;Joukowsky transform&lt;/a&gt; is a conformal mapping of a disk to an airfoil shape.&#xA;The wiki page says that &amp;quot;it was historically used to understand some principles of airfoil design&amp;quot;.&#xA;That's kind of a vague statement and I'd like to know more about precisely how it was used.&#xA;I know that early work on theoretical fluid mechanics before digital computers were available used potential theory and other techniques from complex analysis.&#xA;&lt;strong&gt;Was complex analysis integral to the design and engineering of aircraft before computers, or was it a theoretical and conceptual tool used to help understand fluid mechanics but not used directly in the design of aircraft?&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;I'm asking this because I occasionally encounter people rolling their eyes at learning about imaginary numbers, and I like being able to tell them that an important historical application of complex numbers was the defeat of the Nazi menace in the skies.&#xA;I'd like to make sure I'm being truthful there.&lt;/p&gt;&#xA;" OwnerUserId="19691" LastEditorUserId="55" LastEditDate="2024-02-15T08:02:35.057" LastActivityDate="2024-02-22T05:00:49.043" Title="How were complex analysis methods, like the Joukowsky transform, used in early aircraft design?" Tags="|complex-analysis|engineering|aerodynamics|" AnswerCount="3" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16213" PostTypeId="2" ParentId="16212" CreationDate="2024-02-15T08:01:20.683" Score="1" Body="&lt;p&gt;&amp;quot;Theoretical and conceptual tool&amp;quot; is probably closer to what happened than &amp;quot;used directly&amp;quot;, but how &amp;quot;integral&amp;quot; it was is probably in the eye of the beholder. Joukowsky's 1906 model was an influential early part of gradually developing and intertwined theoretical and experimental work with feedback going in both directions, see &lt;a href=&quot;http://acversailles.free.fr/documentation/08%7EDocumentation_Generale_M_Suire/Aerodynamique/Theorie/Highlights_from_the_history_of_airfoil_develoment.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Highlights From The History of Airfoil Development by Jones&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Joukowsky's airfoil itself was too idealized for direct use, but, together with abundant experimental data, it became the basis of more refined models that were used in design more directly, such as &lt;a href=&quot;https://en.wikipedia.org/wiki/Airfoil#Thin_airfoil_theory&quot; rel=&quot;nofollow noreferrer&quot;&gt;Munk's thin airfoil model&lt;/a&gt; of 1921 elaborated by Glauert. The thin airfoil model manipulates a Fourier series ansatz for the vortex distribution rather than uses complex analysis, but...&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The vortex distribution function... is Glauert's approximation and is based on Joukowski transformation results (&lt;span class=&quot;math-container&quot;&gt;$A_0$&lt;/span&gt; term) which mainly covers the effect of angle of attack, plus a Fourier series variation (&lt;span class=&quot;math-container&quot;&gt;$A_n$&lt;/span&gt; terms) to account for camber. It automatically obeys the Kutta condition with zero vorticity at the trailing edge. It is based on a mapped angular position &lt;span class=&quot;math-container&quot;&gt;$θ$&lt;/span&gt; rather than an exact surface location &lt;span class=&quot;math-container&quot;&gt;$s$&lt;/span&gt; to allow for ease of integration.&lt;/em&gt;&amp;quot; [&lt;a href=&quot;http://www.aerodynamics4students.com/subsonic-aerofoil-and-wing-theory/2-d-thin-aerofoil-theory.php&quot; rel=&quot;nofollow noreferrer&quot;&gt;Auld-Srinivas,  &amp;quot;2-D Thin Aerofoil Theory&lt;/a&gt;.&#xA;]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is from Jones:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1902, W. M. Kutta calculated the lift of a thin,&#xA;cambered plate at zero angle of attack and obtained a&#xA;substantial lift force without drag in a frictionless fluid.&#xA;In 1906, a theory of airfoils having rounded leading&#xA;edges and varying angles of attack was developed by N.&#xA;E. Joukowski... Beginning in 1921, NACA began collecting airfoil data from laboratories around the world and presenting them in a uniform notation... During this period, the design of airfoils was largely intuitive and based on the experience gained in testing numerous variations. While the experimenters were thus occupied, attempts were being made to extend the theory of Kutta and Joukowski to cover a greater variety of shapes. Thus, R. von Mises, a well-known aerodynamicist, was able to extend Joukowski's theory to cover an infinite series of profiles, all obtained by transformation of a circle. Also, H. Glauert, and T. von Karman obtained generalizations of the Joukowski airfoils.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;A new direction in experimental technique appeared&#xA;when Max Munk of NACA Langley proposed the variable&#xA;density wind tunnel (see NACA TN 60, 1921)... Soon after the variable density tunnel, Munk developed his &amp;quot;thin airfoil theory.&amp;quot; Although less accurate, this theory was a great simplification of earlier theories and permitted the relations between shape, pressure, and lift to be seen more clearly than before. With thin airfoil theory it was relatively easy to design airfoils that had a fixed or even a stable center of pressure. A systematic series of these, known as the M sections, were tested in the variable density tunnel at Langley in the 1920s. Of these, the M-6 and the M-12 found application; for example, on the Waco Taperwing. I used the M-12 on a small racing airplane (Pobjoy Phantom, Fig. 7)&#xA;which flew in the 1930 National Air Races. Another important&#xA;result of the thin airfoil theory, brought out by&#xA;Glauert, is the magnifying effect of a plain trailing-edge&#xA;flap when used as a control surface.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-02-15T08:23:59.383" LastActivityDate="2024-02-15T08:23:59.383" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16214" PostTypeId="2" ParentId="16212" CreationDate="2024-02-15T09:14:56.213" Score="0" Body="&lt;p&gt;A more up-to-date reference that's fairly comprehensive for the first half of the 20th century would be&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Robinson, A.; Laurmann, J. A.&#xA;Wing theory.&#xA;Cambridge, at the University Press, 1956. ix+569 pp.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is described as follows in Math Reviews:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;This is an admirable compendium of the mathematical theories of the aerodynamics of aerofoils and wings. Almost all the important results are referred to, even though there can be only a brief reference to literature in connection with the more difficult topics. It begins with a remarkably economical account of the fundamentals of hydrodynamics, including all the principal results concerning irrotational and rotational inviscid flow, the elements of viscous flow in so far as they are needed for a simple account of boundary layer theory, and some account of turbulent flow and wakes. The second chapter gives the two-dimensional aerofoil theory for incompressible flow. After the standard complex variable theory, the different methods of calculating pressure distribution for arbitrary aerofoils, or designing aerofoils for given pressure distribution, are set out. The limitations of the &lt;em&gt;&lt;strong&gt;Kutta-Joukowski&lt;/strong&gt;&lt;/em&gt; condition, and the preferability of using empirical values of lift to fix circulation, are clearly stated. In addition, biplanes and cascades of aerofoils are treated, and the chapter ends with an account of the Squire &amp;amp; Young method of calculating profile drag.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The review provides further details; see &lt;a href=&quot;https://mathscinet.ams.org/mathscinet/article?mr=82299&quot; rel=&quot;nofollow noreferrer&quot;&gt;there&lt;/a&gt; (if you have access).&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2024-02-15T09:14:56.213" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16216" PostTypeId="1" CreationDate="2024-02-15T13:51:34.623" Score="2" ViewCount="87" Body="&lt;p&gt;Nowadays the unique 2 point, nondiscrete, nontrivial topological space goes by the name of the &lt;a href=&quot;https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sierpinski space&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;How did that space come to be named after Sierpinski?&lt;/p&gt;&#xA;&lt;p&gt;The comments to &lt;a href=&quot;https://mathoverflow.net/questions/192287/sierpinski-like-spaces&quot;&gt;this MathOverflow post&lt;/a&gt; contain some speculations on this question, but I've not been able to find anything else.&lt;/p&gt;&#xA;" OwnerUserId="5591" LastActivityDate="2024-02-15T13:51:34.623" Title="Origin of &quot;Sierpinski space&quot;?" Tags="|terminology|topology|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="16217" PostTypeId="2" ParentId="15897" CreationDate="2024-02-15T23:00:33.150" Score="1" Body="&lt;p&gt;&lt;strong&gt;Albert Einstein&lt;/strong&gt; I don't know if it could be &lt;em&gt;out of fear&lt;/em&gt;, or the reason was&#xA;&lt;em&gt;a contrary, earth-shaking implication&lt;/em&gt; that embarrassed him because it did not line up with his worldview.  However, the Theory of Special Relativity---and General Relativity---implied the &lt;strong&gt;beginning of the universe&lt;/strong&gt; and that did not set well with him or his colleagues who advocated a Steady State Universe with no beginning.&lt;/p&gt;&#xA;&lt;p&gt;So fearful he was of upsetting the apple cart (and fearful of its philosophical implications) that he invented the infamous &lt;strong&gt;fudge factor&lt;/strong&gt; (universal constant). Which he later admitted was the worst mistake of his mathematical career!&lt;/p&gt;&#xA;&lt;p&gt;Fortunately, Einstein overcame his fear, and admitted that the &lt;em&gt;beginning of the Universe&lt;/em&gt; was a definite possibility.&lt;/p&gt;&#xA;" OwnerDisplayName="user18391" LastEditorDisplayName="user18391" LastEditDate="2024-02-18T22:10:58.257" LastActivityDate="2024-02-18T22:10:58.257" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16219" PostTypeId="1" CreationDate="2024-02-16T08:57:34.243" Score="1" ViewCount="111" Body="&lt;p&gt;I read 17 equations that changed the world by Ian Stewart.  This book provides information about the correlation between  electromagnetic force and topological invariant.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The idea of a topological invariant goes back to Gauss’s work on magnetism. He was interested in how magnetic and electrical field lines could link with each other, and he defined the linking number, which counts how many times one&#xA;field line winds round another. This is a topological invariant: it remains the&#xA;same if the curves are continuously deformed. He found a formula for this&#xA;number using integral calculus, and every so often he expressed a wish for a&#xA;better understanding of the ‘basic geometric properties’ of diagrams. It is no&#xA;coincidence that the first serious inroads into such an understanding came&#xA;through the work of one of Gauss’s students, Johann Listing, and Gauss’s&#xA;assistant August Möbius. Listing’s Vorstudien zur Topologie (‘Studies in&#xA;Topology’) of 1847 introduced the word ‘topology’, and Möbius made the role&#xA;of continuous transformations explicit.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In the history of mathematics and science, was Gauss the first scholar to be interested in systematic topological invariants?  And how did Poincaré become interested in topological invariants?  Did Gauss' research enrich the general interest in topological invariants during Poincaré's era?&lt;/p&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="19597" LastEditDate="2024-02-16T09:10:42.603" LastActivityDate="2024-02-16T09:10:42.603" Title="Relationship between electromagnetic and topological invariant" Tags="|electromagnetism|gauss|topology|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="16220" PostTypeId="1" CreationDate="2024-02-17T12:34:05.173" Score="1" ViewCount="66" Body="&lt;p&gt;In the 2023 film &lt;a href=&quot;https://en.wikipedia.org/wiki/Oppenheimer_(film)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oppenheimer&lt;/a&gt; based on the 2005 biography &lt;a href=&quot;https://en.wikipedia.org/wiki/American_Prometheus&quot; rel=&quot;nofollow noreferrer&quot;&gt;American Prometheus&lt;/a&gt; by Kai Bird and Martin J. Sherwin, there seems to be two references to work by Oppenheimer on molecules:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;RABI: I caught your lecture on molecules. Caught some of it.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;and&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;HEISENBERG: I liked your paper on molecules.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm guessing this reference is related in some way to what we now call the &lt;a href=&quot;https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation&quot; rel=&quot;nofollow noreferrer&quot;&gt;Born–Oppenheimer approximation&lt;/a&gt; since (earlier?) Bohr had sent him to Göttingen to study with Born.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; What paper or papers about molecules did Heisenberg &amp;quot;like&amp;quot;, and what has Heisenberg published or otherwise commented about it in a well-documented way?&lt;/p&gt;&#xA;" OwnerUserId="3869" LastActivityDate="2024-02-17T12:34:05.173" Title="What paper or papers about molecules did Heisenberg &quot;like&quot;, and what has Heisenberg published or otherwise commented about it?" Tags="|quantum-mechanics|heisenberg|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16221" PostTypeId="1" CreationDate="2024-02-18T09:50:44.833" Score="0" ViewCount="109" Body="&lt;p&gt;I am going over some history of the complex numbers, and two things baffle me (and they are not mathematics).&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;From Cardano's time to around the 18th century, negative numbers were not accepted by all mathematicians, and there was absolutely no universal or formal/proven rules of arithmetic with them, maybe it was all just “word of mouth” or “experimentation”?&lt;/li&gt;&#xA;&lt;li&gt;How can you study the square root of a negative if you don't even accept negatives?&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19705" LastEditorUserId="6545" LastEditDate="2024-02-18T10:04:01.633" LastActivityDate="2024-02-25T13:29:02.843" Title="Why and how did the study of complex numbers progress despite the denial of negative numbers?" Tags="|mathematics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16222" PostTypeId="1" CreationDate="2024-02-18T12:51:08.783" Score="16" ViewCount="3608" Body="&lt;p&gt;Sometimes, for one reason or the other, the works of scholars get lost. In some cases, they're lost forever. This happened to many books during the fire of the library of Alexandria, for instance.&lt;/p&gt;&#xA;&lt;p&gt;Luckily, it can also happen that these works resurface after a while. I am interested in these cases. Below, I list five examples:&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Oskar_Fischer&quot; rel=&quot;noreferrer&quot;&gt;Oskar Fischer&lt;/a&gt; was a Czech neuropathologist. His work on what we now call Alzheimer's disease predates Alois Alzheimer's studies. It was published in a series of papers dating from 1907, 1910, and 1912, and was arguably even more important than his. Being of Jewish descent, he was murdered by the nazis during the Second World War. Moreover, he was part of the Prague School of neuropathology, which at the time was a rival of the Munich School, which consisted of the influential academic Emil Kraepelin, Alzheimer himself, and others. Kraepelin made sure Alzheimer's name was associated with the disease in an important book on psychiatry. These factors contributed to the obscurity of Fischer's contributions. It was only in 2008 when Michel Goedert brought his contributions &lt;a href=&quot;https://www.medicalnewstoday.com/mnt/releases/145778#1&quot; rel=&quot;noreferrer&quot;&gt;to light&lt;/a&gt;, after visiting the Archives of Charles University in Prague, and having spoken to Fischer's relatives.&lt;/li&gt;&#xA;&lt;li&gt;When &lt;a href=&quot;https://en.m.wikipedia.org/wiki/Carl_Jung&quot; rel=&quot;noreferrer&quot;&gt;Carl Jung&lt;/a&gt;'s relationship broke with his close colleague Sigmund Freud in 1913, it had profound consequences on his life. One of them was his work on the &lt;a href=&quot;https://en.m.wikipedia.org/wiki/The_Red_Book_(Jung)&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Red Book&lt;/em&gt;&lt;/a&gt; or &lt;em&gt;Liber Novus&lt;/em&gt;. He wrote it between 1914 and c. 1930. It records the author's experiments and observations on himself between the years 1913 and 1916. The book was read by just a few people at the time. It was only by 2009 that it was made open to publication by Jung's estate.&lt;/li&gt;&#xA;&lt;li&gt;In 1885, a new mathematical competition was initiated by editors of the journal &lt;em&gt;Acta Mathematica&lt;/em&gt; to honour the 60th birthday of Oscar II, the king of Sweden at the time. The deadline for submission was June 1888. One of the submissions was by &lt;a href=&quot;https://nl.wikipedia.org/wiki/Henri_Poincar%C3%A9&quot; rel=&quot;noreferrer&quot;&gt;Henri Poincaré&lt;/a&gt;. He submitted a manuscript in which he proposed a novel way to approach the &lt;em&gt;n-body problem&lt;/em&gt;. The judges of the competition had difficulties with understanding the work, however, so after lengthy discussions Poincaré submitted a revised memoir on the 5th of January, 1890. The original manuscript was long thought to be lost, but it &lt;a href=&quot;https://www.mittag-leffler.se/about-us/history/prize-competition/&quot; rel=&quot;noreferrer&quot;&gt;turned up again&lt;/a&gt; by a lucky coincidence in 2011.&lt;/li&gt;&#xA;&lt;li&gt;The English philosopher and social reformer &lt;a href=&quot;https://en.wikipedia.org/wiki/Jeremy_Bentham#Gender_and_sexuality&quot; rel=&quot;noreferrer&quot;&gt;Jeremy Bentham&lt;/a&gt; wrote an essay entitled &amp;quot;Paedarasty (Offences Against One's Self)&amp;quot; around 1785. In this work, he argued for the liberalisation of laws prohibiting homosexual sex. The essay remained unpublished during his lifetime, however, &lt;a href=&quot;https://hsm.stackexchange.com/a/16205/4296&quot;&gt;for fear&lt;/a&gt; of offending public morality. It was only in 1978 that it was published.&lt;/li&gt;&#xA;&lt;li&gt;The French-German mathematician &lt;a href=&quot;https://en.wikipedia.org/wiki/Wolfgang_Doeblin&quot; rel=&quot;noreferrer&quot;&gt;Wolfgang Doeblin&lt;/a&gt; was an expert in probability theory. When the war broke out in 1939, he refused to be exempted from military service, and was stationed as a telephone operator in Givet, in the Ardennes. While he worked there, he wrote on what are now known as the Chapman-Kolmogorov equations. He incorporated his findings in a letter, which he sealed and sent to the French Academy of Sciences. In the year 2000, the sealed &lt;a href=&quot;https://www.sciencedirect.com/science/article/abs/pii/S076444420101847X?via%3Dihub&quot; rel=&quot;noreferrer&quot;&gt;enveloppe&lt;/a&gt; was opened.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;Please note that this question is specifically aimed at the &lt;em&gt;physical information objects&lt;/em&gt; themselves that were lost and found again later. Other questions, including this &lt;a href=&quot;https://mathoverflow.net/questions/176425/rediscovery-of-lost-mathematics&quot;&gt;MO question&lt;/a&gt;, are geared towards the rediscovery of the &lt;em&gt;ideas&lt;/em&gt; contained within these works. This question is focussed on the books, letters, essays, and other written documents that were once lost but later resurfaced.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question&lt;/strong&gt; which works by scientists, philosophers, and mathematicians (re)surfaced after a long time?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4296" LastEditorUserId="4296" LastEditDate="2024-02-18T22:48:16.987" LastActivityDate="2024-03-19T10:27:06.133" Title="Works of scientists, philosophers and mathematicians that (re)surfaced after a long time" Tags="|biographical-details|big-list|examples|" AnswerCount="10" CommentCount="9" ContentLicense="CC BY-SA 4.0" />
  <row Id="16223" PostTypeId="2" ParentId="16222" CreationDate="2024-02-18T15:23:18.077" Score="18" Body="&lt;p&gt;A major work of Archimedes, which is usually called &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;The Method of mechanical theorems&amp;quot;&lt;/a&gt; was thought to be lost since the late antiquity till 1906 when it was found as a palimpsest in a convent in Constantinople. Using modern tools it was completely recovered, and now it is published and is &lt;a href=&quot;https://archive.org/details/cu31924005730563/page/n3/mode/2up&quot; rel=&quot;nofollow noreferrer&quot;&gt;available online&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Remark. It was actually lost twice: after its finding in 1906 it remained in possession of the Orthodox church, and in 1922 it went missing because of the evacuation as a result of a war, and for the next 70 years was in private hands, and was not available for study. For the second time it resurfaced in 1998. The full story is described &lt;a href=&quot;https://en.wikipedia.org/wiki/Archimedes_Palimpsest&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;. It is still privately owned, but the owner permitted a study and reproduction.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-02-25T13:31:21.770" LastActivityDate="2024-02-25T13:31:21.770" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16224" PostTypeId="2" ParentId="16221" CreationDate="2024-02-18T15:44:45.567" Score="4" Body="&lt;p&gt;Those mathematicians who worked with complex numbers, of course did accept negative numbers. When you say that &amp;quot;negative numbers were not accepted&amp;quot;, you mean that they were not UNIVERSALLY accepted. Neither complex numbers were. At any stage of development some people are much more advanced than other people.&lt;/p&gt;&#xA;&lt;p&gt;If you ever filed a US tax return, you know that IRS does not accept negative numbers. Otherwise how do you explain their instructions: &amp;quot;If line X is less than or equal to line Y, subtract line X from line Y and enter here&amp;quot;. And later &amp;quot;If X is greater than Y, subtract Y from X and enter here&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Some historian 500 years from now may conclude that&#xA;in 21st century negative numbers were not accepted in the US.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-02-25T13:29:02.843" LastActivityDate="2024-02-25T13:29:02.843" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16225" PostTypeId="2" ParentId="16222" CreationDate="2024-02-18T22:54:28.013" Score="7" Body="&lt;p&gt;Not necessarily lost but ignored:&lt;/p&gt;&#xA;&lt;p&gt;In 1900, &lt;a href=&quot;https://en.wikipedia.org/wiki/Louis_Bachelier&quot; rel=&quot;noreferrer&quot;&gt;Louis Bachelier&lt;/a&gt;, working under Henri Poincaré, worked the math of random walks while working in economics. His thesis came before Einstein (1905) and Smoluchowski (1906) on Brownian motion. It also predated the &lt;a href=&quot;https://en.wikipedia.org/wiki/Efficient-market_hypothesis&quot; rel=&quot;noreferrer&quot;&gt;efficient-market hypothesis&lt;/a&gt;. It is said that it was Nobel Prize in Economics, &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Samuelson&quot; rel=&quot;noreferrer&quot;&gt;Paul Samuelson&lt;/a&gt; who rediscovered the thesis in the 1960s after it was first translated to English. From then on his contribution has been largely recognized in retrospective.&lt;/p&gt;&#xA;&lt;p&gt;Here is another one that is not necessarily a piece of science but still interesting:&lt;/p&gt;&#xA;&lt;p&gt;Isaac Newton was obsessed with Bible numerology. Thousands of his alchemy and Bible papers were locked for centuries and were finally bought by Southeby's corporation in the 1930. Curiously, economist &lt;a href=&quot;https://en.wikipedia.org/wiki/John_Maynard_Keynes&quot; rel=&quot;noreferrer&quot;&gt;John Mayard Keynes&lt;/a&gt; acquired some of it. In between these papers, a collector named &lt;a href=&quot;https://en.wikipedia.org/wiki/Abraham_Yahuda&quot; rel=&quot;noreferrer&quot;&gt;Abraham Yahuda&lt;/a&gt; found a Newton's manuscript from 1704 stating (in very obscure prose) something about the end of an era in 2060 AD. Some people have taken this to mean the end of the world, but what Newton really meant is debated.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-02-19T10:29:41.357" LastActivityDate="2024-02-19T10:29:41.357" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16226" PostTypeId="2" ParentId="16222" CreationDate="2024-02-19T04:52:01.097" Score="9" Body="&lt;p&gt;Roger Bacon (1214? - 1294), &lt;em&gt;Opus Majus,&lt;/em&gt; by order of Pope Clement, delivered to him in 1267 as a seven-part handwritten manuscript series; resided in the Vatican Library, remained unpublished for more than &lt;strong&gt;500 years.&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="19711" LastActivityDate="2024-02-19T04:52:01.097" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16227" PostTypeId="1" CreationDate="2024-02-19T05:52:14.447" Score="2" ViewCount="111" Body="&lt;p&gt;So, I've been over fixated on negative numbers lately. I'm coming to the conclusion that, mathematics is usually progressed if it is &amp;quot;useful&amp;quot;. The more &amp;quot;useful&amp;quot; a mathematical object is generally the more it appears in study and practice. Arguably the most useful mathematical object a natural number, has the most wide array of applications in the world.&lt;/p&gt;&#xA;&lt;p&gt;Another example of this, is for example, there might be some kind of scenario where 2+2 = 10, and constructing a mathematical system with that as a theorm, but it would be a pretty specific / useless one and not studied very much / broadly.&lt;/p&gt;&#xA;&lt;p&gt;The thing that I think was hard about accepting negative numbers, is firstly the conceptual problem when trying to understand a subtraction from 0 with a lack of context, and also a lack of geometric understanding (with respect to lengths and so forth).&lt;/p&gt;&#xA;&lt;p&gt;I've observed throughout history, the most natural analogy mathematicians use for negative numbers is debt. In the this setting, it's natural that if your relieved of a debt you gain credit. This is probably the most common justification for the rule of signs I've observed, even from Euler himself.&lt;/p&gt;&#xA;&lt;p&gt;So TLDR i'm looking for Euler's writings about negative numbers, and maybe any kind of paper that examines the relationship with usefulness and mathematical study.&lt;/p&gt;&#xA;" OwnerUserId="19712" LastEditorUserId="12330" LastEditDate="2024-02-20T07:41:51.423" LastActivityDate="2024-02-20T07:41:51.423" Title="I would like to read about Euler's view on negative numbers" Tags="|mathematics|euler|numbers|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16228" PostTypeId="2" ParentId="16222" CreationDate="2024-02-19T12:10:05.643" Score="3" Body="&lt;p&gt;&lt;strong&gt;Gems of &amp;quot;Epicurean Philosophy&amp;quot;&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the case of food, we do not right away believe things that are scarce to be absolutely more pleasant than those which are abundant.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;.... for we do [not] refrain from questioning some things, but understanding/remembering others. And may it be evident to us to say true things, as they might have often appeared evident!&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Overview:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;These were written on Papyri around 79 AD &amp;amp; were stored in a &lt;a href=&quot;https://en.wikipedia.org/wiki/Villa_of_the_Papyri&quot; rel=&quot;nofollow noreferrer&quot;&gt;villa&lt;/a&gt; in Herculaneum near &lt;a href=&quot;https://en.wikipedia.org/wiki/Eruption_of_Mount_Vesuvius_in_79_AD&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mount Vesuvius&lt;/a&gt;, lost under volcanic ash, mud &amp;amp; lava until getting excavated in 1750 AD &amp;amp; are getting &amp;quot;deciphered/read/identified&amp;quot; with AI now 2023 AD onwards.&lt;br /&gt;&#xA;Depending on how we want to count, the &amp;quot;resurfacing&amp;quot; is either 2000 years or 1700 years or 275 years after obtaining the Source Material.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;&lt;strong&gt;Article Details &lt;a href=&quot;https://www.sciencealert.com/behold-the-first-readable-passages-from-the-charred-scrolls-of-vesuvius&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Behold! The First Readable Passages From The Charred Scrolls of Vesuvius HUMANS&amp;quot;&lt;br /&gt;&#xA;[ 09 February 2024 ] By CARLY CASSELLA&lt;/p&gt;&#xA;&lt;p&gt;So far, about 5 percent of the scroll has been unrolled and read to date. It is not a duplicate of past work, scholars of the Vesuvius Challenge say, but a &amp;quot;never-before-seen text from antiquity.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;According to scholars from the Vesuvius Challenge, &amp;quot;the general subject of the text is pleasure, which, properly understood, is the highest good in Epicurean philosophy. In these two snippets from two consecutive columns of the scroll, the author is concerned with whether and how the availability of goods, such as food, can affect the pleasure which they provide.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Classicist Richard Jenko from the University of Michigan thinks the author of the text may be a follower of Epicurus, as someone of that description is thought to have lived and worked at the Villa of Papyri.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Challenge Details &lt;a href=&quot;https://scrollprize.org/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;:&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;The Vesuvius Challenge is a machine learning and computer vision competition that in 2023 cracked the riddle of the Herculanum [sic] Papyri &amp;amp; awarded over $1,000,000 in prizes.&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;h2&gt;ADDENDUM:&lt;/h2&gt;&#xA;&lt;p&gt;Will we get to know who wrote it? Will we get the whole text? Will there be Scientific texts &amp;amp; Mathematical texts?&lt;/p&gt;&#xA;&lt;p&gt;It is too early to speculate!&lt;/p&gt;&#xA;" OwnerUserId="18814" LastEditorUserId="-1" LastEditDate="2024-02-19T16:32:43.040" LastActivityDate="2024-02-19T16:32:43.040" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16229" PostTypeId="1" CreationDate="2024-02-19T17:33:53.567" Score="2" ViewCount="48" Body="&lt;p&gt;In combinatorics, it is well understood that &lt;a href=&quot;https://en.wikipedia.org/wiki/Parity_of_a_permutation&quot; rel=&quot;nofollow noreferrer&quot;&gt;a permutation has a parity or sign&lt;/a&gt;: each permutation is either &lt;em&gt;odd&lt;/em&gt; or &lt;em&gt;even&lt;/em&gt; (sometimes described as &lt;em&gt;negative&lt;/em&gt; and &lt;em&gt;positive&lt;/em&gt;, respectively).  Composing two permutations of the same parity yields an even permutation; composing two of opposite parity produces an odd parity result.&lt;/p&gt;&#xA;&lt;p&gt;I am trying to establish &lt;em&gt;when&lt;/em&gt; this concept was first understood, regardless of whether the word &lt;em&gt;parity&lt;/em&gt; was used to describe it.  I vaguely recall reading that Pascal (1623–62) was one of the first to talk about it, but I cannot now find a source for this and am doubting my memory.  I'm feel certain Euler (1707–83) must have understood this, but my Latin is not up to reading &lt;a href=&quot;https://archive.org/details/bub_gb_sYE_AAAAcAAJ/&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Institutiones Calculi Differentialis&lt;/em&gt;&lt;/a&gt; to confirm.  Can anyone help?&lt;/p&gt;&#xA;" OwnerUserId="19717" LastActivityDate="2024-02-19T17:33:53.567" Title="When was the parity of a permutation first understood?" Tags="|combinatorics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16230" PostTypeId="2" ParentId="13257" CreationDate="2024-02-19T18:24:55.120" Score="4" Body="&lt;p&gt;She didn't do any math. From my research about her, it seems that the user you are referring to is teasing the fact that her nickname was &amp;quot;Sisi&amp;quot; which looks similar to the integrand. I also highly doubt the integral of the cardinal sine function was even denoted at the time (the earliest mention of it I can find is by Abramowitz in the 1980s, although I'm sure it existed maybe a few decades before). The only sign of education I've found about her (citing Wikipedia page, couldn't find enough information elsewhere) was mostly learning languages while getting her hair groomed.&lt;/p&gt;&#xA;&lt;p&gt;As a fun note, the dictator of Egypt is named &amp;quot;Abdelfattah El-Sisi&amp;quot; so I guess he did/does math as well.&lt;/p&gt;&#xA;" OwnerUserId="18713" LastActivityDate="2024-02-19T18:24:55.120" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16232" PostTypeId="1" CreationDate="2024-02-19T21:05:52.043" Score="0" ViewCount="49" Body="&lt;p&gt;I was researching a bit about the history of the famous Klein-Gordon equation and I found out that Fock also independently discovered it in the same year as Klein and Gordon, 1926. However, chronologically, the original articles where the equation appears are ordered with &lt;a href=&quot;https://doi.org/10.1007/BF01397481&quot; rel=&quot;nofollow noreferrer&quot;&gt;Klein&lt;/a&gt;'s first (April 28, 1926), followed by &lt;a href=&quot;https://doi.org/10.1007/BF01321989&quot; rel=&quot;nofollow noreferrer&quot;&gt;Fock&lt;/a&gt;'s (July 30, 1926), and &lt;a href=&quot;https://doi.org/10.1007/BF01390840&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gordon&lt;/a&gt;'s last (September 29, 1926). However, I have seen at most three articles using the nomenclature &amp;quot;Klein-Fock-Gordon equation,&amp;quot; while the widespread usage seems to be &amp;quot;Klein-Gordon equation.&amp;quot; Is there any reason behind leaving Fock's name out of the equation?&lt;/p&gt;&#xA;" OwnerUserId="19405" LastEditorUserId="12330" LastEditDate="2024-02-20T03:21:58.497" LastActivityDate="2024-02-20T03:21:58.497" Title="Where is Fock on Klein-Gordon equation?" Tags="|physics|terminology|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16233" PostTypeId="2" ParentId="16222" CreationDate="2024-02-20T14:00:57.260" Score="1" Body="&lt;p&gt;Evariste Galois springs to mind, again, not so much lost as ignored.&lt;/p&gt;&#xA;&lt;p&gt;He died at the age of twenty but I believe it took decades for his work to be completely recognised as revolutionary.&lt;/p&gt;&#xA;" OwnerUserId="19719" LastEditorUserId="64" LastEditDate="2024-03-18T12:19:54.990" LastActivityDate="2024-03-18T12:19:54.990" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16234" PostTypeId="2" ParentId="15946" CreationDate="2024-02-20T15:41:03.750" Score="4" Body="&lt;p&gt;In inorganic chemistry (say that five times, fast), &amp;quot;Fluorine Martyrs&amp;quot; is a romantic label for the scientists that tried to isolate elemental fluorine.  They had many tragic injuries and fatalities before taming the &amp;quot;savage beast&amp;quot; (as the Nobel Prize for Moissan called the element).&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/History_of_fluorine#Early_isolation_attempts&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.wikipedia.org/wiki/History_of_fluorine#Early_isolation_attempts&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19720" LastActivityDate="2024-02-20T15:41:03.750" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16236" PostTypeId="1" AcceptedAnswerId="16237" CreationDate="2024-02-21T04:34:14.787" Score="17" ViewCount="1525" Body="&lt;p&gt;There has been a growing chorus &lt;a href=&quot;https://www.clinicallab.com/null-hypothesis-significance-testing-may-be-an-inaccurate-tool-27654&quot; rel=&quot;noreferrer&quot;&gt;against&lt;/a&gt; the conventional NHST (Null Hypothesis Significance Testing). One thing is the blind usage of a monolithic significance level &lt;span class=&quot;math-container&quot;&gt;$5\%.$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;In a recent &lt;a href=&quot;https://stats.stackexchange.com/questions/639628/never-understood-the-concept-of-the-p-value-it-should-be-higher-than-0-05&quot;&gt;thread&lt;/a&gt; at CV, user &lt;a href=&quot;https://stats.stackexchange.com/a/639650/362671&quot;&gt;Peter Flom&lt;/a&gt; claimed that Fisher had asserted somewhere that he advocated changing the threshold.&lt;/p&gt;&#xA;&lt;p&gt;While Peter couldn't remember the exact quote and the source, it was supposed to be &lt;a href=&quot;https://stats.stackexchange.com/questions/639628/never-understood-the-concept-of-the-p-value-it-should-be-higher-than-0-05/639650?noredirect=1#comment1195921_639650&quot;&gt;like&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;... no sane researcher uses the same significance level in all cases.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is, did Ronald Fisher ever express anything on this issue? Would appreciate if someone is able to pin down the source where he made such statement(s).&lt;/p&gt;&#xA;" OwnerUserId="17132" LastActivityDate="2024-02-21T09:41:10.787" Title="Did Ronald Fisher ever say anything on varying the threshold of significance level?" Tags="|quote|statistics|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16237" PostTypeId="2" ParentId="16236" CreationDate="2024-02-21T09:41:10.787" Score="20" Body="&lt;p&gt;Jerry Dallal collected some of &lt;a href=&quot;http://www.jerrydallal.com/lhsp/p05.htm&quot; rel=&quot;noreferrer&quot;&gt;Fisher's quotes&lt;/a&gt; from various works. And Fisher said too much, first setting the P=0.05 cutoff in 1925, then not following it himself, and, finally, openly advocating flexibility in 1956 in response to Neyman and Pearson taking it too far. However, even then he advocated &lt;em&gt;not setting&lt;/em&gt; any fixed cutoff at all, not &lt;em&gt;changing&lt;/em&gt; it, this is what the passage alluded to in the OP is about. On more charitable reading, Fisher always meant 0.05 only as a vague threshold, not a cutoff. For more on Fisher's vs Neyman-Pearson's approaches see &lt;a href=&quot;https://hsm.stackexchange.com/a/3186/55&quot;&gt;Hypothesis testing: Fisher vs. Popper vs. Bayes&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The P=0.05 level comes from Fisher's Statistical Methods for Research Workers (1925), albeit with some wiggle room:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The value for which P=0.05, or 1 in 20, is 1.96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation ought to be considered significant or not. [...] If P is between .1 and .9 there is certainly no reason to suspect the hypothesis tested. If it is below .02 it is strongly indicated that the hypothesis fails to account for the whole of the facts. Belief in the hypothesis as an accurate representation of the population sampled is confronted by the logical disjunction: Either the hypothesis is untrue, or the value of &lt;span class=&quot;math-container&quot;&gt;$\chi^2$&lt;/span&gt; has attained by chance an exceptionally high value. The actual value of P obtainable from the table by interpolation indicates the strength of the evidence against the hypothesis. A value of &lt;span class=&quot;math-container&quot;&gt;$\chi^2$&lt;/span&gt; exceeding the 5 per cent. point is seldom to be disregarded.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Despite these declarations, he was more flexible in practice in the same book, or, less charitably, inconsistent:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The results of t shows that P is between .02 and .05. The result must be judged significant, though barely so [...] we find... t=1.844 [with 13 df, P = 0.088]. The difference between the regression coefficients, though relatively large, cannot be regarded as significant.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Says Dallal:&amp;quot;&lt;em&gt;Part of the reason for the apparent inconsistency is the way Fisher viewed P values. When Neyman and Pearson proposed using P values as absolute cutoffs in their style of fixed-level testing, Fisher disagreed strenuously.  Fisher viewed P values more as measures of the evidence against a hypotheses...&lt;/em&gt;&amp;quot; It is only then, in Statistical Methods and Scientific Inference (1956), that Fisher elevated his practiced flexibility and attention to context to a principle:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The attempts that have been made to explain the cogency of tests of significance in scientific research, by reference to hypothetical frequencies of possible statements, based on them, being right or wrong, thus seem to miss the essential nature of such tests. A man who &amp;quot;rejects&amp;quot; a hypothesis provisionally, as a matter of habitual practice, when the significance is at the 1% level or higher, will certainly be mistaken in not more than 1% of such decisions. For when the hypothesis is correct he will be mistaken in just 1% of these cases, and when it is incorrect he will never be mistaken in rejection. This inequality statement can therefore be made. However, the calculation is absurdly academic, for in fact &lt;strong&gt;no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas&lt;/strong&gt;. Further, the calculation is based solely on a hypothesis, which, in the light of the evidence, is often not believed to be true at all, so that the actual probability of erroneous decision, supposing such a phrase to have any meaning, may be much less than the frequency specifying the level of significance.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-02-21T09:41:10.787" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16238" PostTypeId="1" AcceptedAnswerId="16239" CreationDate="2024-02-21T11:17:33.360" Score="4" ViewCount="357" Body="&lt;p&gt;Auguste Comte is often described (e.g., &lt;a href=&quot;https://en.wikipedia.org/wiki/Auguste_Comte&quot; rel=&quot;nofollow noreferrer&quot;&gt;on Wikipedia&lt;/a&gt;) as a “mathematician” besides being a philosopher of science.  I am aware that he taught mathematics (he was at various times a &lt;em&gt;répétiteur&lt;/em&gt; and/or an &lt;em&gt;examinateur&lt;/em&gt; at the École polytechnique), and he was acquainted or even friends with prominent French mathematicians of the time such as Henri Navier (of “Navier-Stokes” fame) and Joseph Fourier.  Also, of course, he placed mathematical sciences at one end of his hierarchy of sciences.  But I am not aware of any original mathematical result by Comte himself, or whether he did any mathematical research.  Nor could I find any publication of his, but I imagine “publication” worked very differently in the early 1800's than now, and I didn't really know how to search, so this doesn't mean anything.&lt;/p&gt;&#xA;&lt;p&gt;So anyway, &lt;strong&gt;question&lt;/strong&gt;: did Auguste Comte do any original work in mathematics himself?  If so, what did it involve?&lt;/p&gt;&#xA;" OwnerUserId="3616" LastActivityDate="2024-02-21T13:15:25.577" Title="What were Auguste Comte's contributions to mathematics (if any)?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16239" PostTypeId="2" ParentId="16238" CreationDate="2024-02-21T13:15:25.577" Score="5" Body="&lt;p&gt;Comte's main contribution to mathematics is the careful set of notes he kept of Cauchy's course in analysis in the 1820s.  This is mentioned in Schubring's book&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Schubring, Gert.  Conflicts between generalization, rigor, and intuition. Number concepts underlying the development of analysis in 17–19th Century France and Germany. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 2005.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As I recall, the first few pages of Comte's notes are unfortunately missing.&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2024-02-21T13:15:25.577" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16240" PostTypeId="2" ParentId="16222" CreationDate="2024-02-21T14:56:01.730" Score="6" Body="&lt;p&gt;&lt;em&gt;Hipparchus' star catalogue&lt;/em&gt;, long thought to be lost, was rediscovered in 2022 by &lt;a href=&quot;https://doi.org/10.1177%2F00218286221128289&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gysembergh, Williams, and Zingg&lt;/a&gt;. From &lt;a href=&quot;https://en.wikipedia.org/wiki/Hipparchus_star_catalog&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The catalog was lost to history, until parts of it were rediscovered&#xA;in 2022 in the Codex Climaci Rescriptus, an ancient palimpsest found&#xA;in Saint Catherine's Monastery on Mount Sinai.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;See also the popular-scientific articles in&#xA;&lt;a href=&quot;https://www.nature.com/articles/d41586-022-03296-1&quot; rel=&quot;nofollow noreferrer&quot;&gt;Nature&lt;/a&gt; (if you have access), the&#xA;&lt;a href=&quot;https://www.smithsonianmag.com/smart-news/medieval-manuscript-oldest-map-of-the-stars-Hipparchus-180980993/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Smithsonian magazine&lt;/a&gt;, or &lt;a href=&quot;https://www.livescience.com/hipparchus-star-map-found&quot; rel=&quot;nofollow noreferrer&quot;&gt;LiveScience&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="4703" LastEditorUserId="4703" LastEditDate="2024-02-21T23:04:22.327" LastActivityDate="2024-02-21T23:04:22.327" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16241" PostTypeId="2" ParentId="16222" CreationDate="2024-02-21T15:54:41.833" Score="4" Body="&lt;p&gt;Stef's comment about Riemann's lost work refers to the story behind the &lt;a href=&quot;https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann-Siegel formula&lt;/a&gt;. Siegel extracted said formula from unpublished work by Riemann that was kept in the Goettingen library but had basically been forgotten. He briefly recounts the historical background at the beginning of his publication from 1932 (&lt;a href=&quot;https://zbmath.org/?format=complete&amp;amp;q=an:58.1037.07&quot; rel=&quot;nofollow noreferrer&quot;&gt;JFM 58.1037.07&lt;/a&gt;) which has only recently been &lt;a href=&quot;https://arxiv.org/pdf/1810.05198.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;translated into English&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastActivityDate="2024-02-21T15:54:41.833" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="16242" PostTypeId="2" ParentId="16222" CreationDate="2024-02-21T16:53:22.640" Score="2" Body="&lt;p&gt;In &lt;a href=&quot;https://en.wikipedia.org/wiki/The_Codebreakers&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Code Breakers&lt;/a&gt;, the recently deceased David Kahn describes how Thomas Jefferson developed a &lt;a href=&quot;https://en.wikipedia.org/wiki/Jefferson_disk&quot; rel=&quot;nofollow noreferrer&quot;&gt;wheel cypher&lt;/a&gt; substantially more advanced than contemporary processes or instruments before 1800.&lt;/p&gt;&#xA;&lt;p&gt;For some reason he only used the concepts in a limited way and the existence of the technique and the device to implement it only resurfaced among his papers in the Library of Congress in 1922.&lt;/p&gt;&#xA;&lt;p&gt;The US Army had just adopted a newly, independently, invented device with the similar concept the same year.&lt;/p&gt;&#xA;" OwnerUserId="19729" LastActivityDate="2024-02-21T16:53:22.640" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16243" PostTypeId="1" CreationDate="2024-02-21T19:20:17.947" Score="0" ViewCount="69" Body="&lt;p&gt;I've noticed these gear-like Ancient Greek items in a local museum annotated along the lines of &amp;quot;decorative clothing pins&amp;quot;. These don't look like much decorative to me (too symmetric for their liking?), rather they look suspiciously like gears. They also have a small hole in the middle, through where they could be attached to a shaft.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/xj6Y4.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/xj6Y4.jpg&quot; alt=&quot;gears or pins?&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;To generalize this question - how did the gears used in Antikythera look like? What were the basic ways in which they were coupled, how many teeth did they have, how did they produce the rotation angle difference required?&lt;/p&gt;&#xA;&lt;p&gt;Quick googling showed modern gears which look like reconstruction of how Antikythera looked like, as well as the same worn out gear, which nontheless looks like what I've showed above.&lt;/p&gt;&#xA;&lt;p&gt;The bottom item is also of interest since it seems to contain the endings of four shafts. Could it be a part of a mechanism?&lt;/p&gt;&#xA;" OwnerUserId="19732" LastEditorUserId="19732" LastEditDate="2024-02-21T19:47:43.860" LastActivityDate="2024-02-21T19:47:43.860" Title="How did the Ancient Greek gears look like (identify this)" Tags="|antiquity|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16244" PostTypeId="2" ParentId="16222" CreationDate="2024-02-21T19:40:41.973" Score="3" Body="&lt;p&gt;The most famous in recent times I can think of: mathematician George Andrews's 1976 rediscovery of &lt;a href=&quot;https://en.wikipedia.org/wiki/Ramanujan%27s_lost_notebook&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;strong&gt;Ramanujan's Lost Notebook&lt;/strong&gt;&lt;/a&gt; in a box of mathematician G. N. Watson's at Trinity College caused quite a stir. It was likened to &amp;quot;the discovery of Beethoven's Tenth Symphony&amp;quot; by mathematician Bruce Berndt, given Ramanujan's almost mythical status as mathematician genius who died young. Andrews and Berndt have published five books covering the contents and proofs of the theorems, which were written on loose sheets of paper without proofs (that habit contributing to the mythos of Ramanujan as a &amp;quot;divinely-inspired&amp;quot; mathematician). &lt;a href=&quot;https://arxiv.org/ftp/arxiv/papers/1208/1208.2694.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Uncovering Ramanujan’s “Lost” Notebook: An Oral History&lt;/em&gt;&lt;/a&gt; by Robert P. Schneider gives a more personal story of Ramanujan's work and the story of rediscovery.&lt;/p&gt;&#xA;" OwnerUserId="15230" LastActivityDate="2024-02-21T19:40:41.973" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16245" PostTypeId="1" CreationDate="2024-02-21T23:32:52.633" Score="1" ViewCount="71" Body="&lt;p&gt;Around the time when mathematics was becoming formal, the notion of detaching from attaching &amp;quot;contextual interpretation&amp;quot; to symbols in algebra, up to the point of avoiding inconsistency (contradiction) was becoming recognized by people like Peacock and Hankel.&lt;/p&gt;&#xA;&lt;p&gt;I know this principle was adopted by Peano, but I'm aware that the generally accepted construction of the number systems we have today are not due to Peano, but rather Dedekind and Cauchy.&lt;/p&gt;&#xA;&lt;p&gt;I realize the principle of permanence while may not exactly recognized is used throughout mathematics since the 19th century even if the exact logic of it may have changed slightly.&lt;/p&gt;&#xA;&lt;p&gt;So I'm looking for an explanation of how this principle progressed from the 19th century and its influence in the foundations of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="19712" LastEditorUserId="131" LastEditDate="2024-02-22T07:02:40.590" LastActivityDate="2024-02-22T07:02:40.590" Title="Did the principle of permanence have an influence on mathematicians like Dedekind and Cauchy?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="16246" PostTypeId="2" ParentId="16212" CreationDate="2024-02-22T05:00:49.043" Score="0" Body="&lt;p&gt;This is &lt;em&gt;a very interesting question&lt;/em&gt; but, nevertheless, one could view this from a position of experience, a perspective in fluid mechanics, and an overview of aircraft history, reaching, thereby, a completely different answer.  Given this view, the emphasized question within your post may be answered as follows: &lt;strong&gt;complex analysis was integral to the design and engineering of aircraft before computers, and it was used as a theoretical and conceptual tool to help understand fluid mechanics and applications in aircraft design, particularly in regard to airfoils and potential-flow theory.&lt;/strong&gt; The upshot was that the application of conformal mapping and analysis was almost exclusively limited to, and used in, airfoil analysis and design.  Since about 1975, the process has been extensively developed for computer application.&lt;/p&gt;&#xA;&lt;p&gt;One may safely say that Joukowski and his colleagues eschewed cumbersome calculations, and with the method he developed, the tools used for analysis were those available to any grade-schooler, namely, compass, ruler and reflective-metal straight edge.  The process used in the Joukowski transformation is known as &lt;em&gt;conformal mapping.&lt;/em&gt;  Given an elementary understanding of the transformation process, understanding the mathematics becomes relatively straightforward. Tools used, other than a pencil and ledger-tabulating paper, would be a slide rule and planimeter, and possibly a Comptometer, or as later available, a Marchant-style calculator.  Of course, one would need a set of mathematical tables for various trigonometric functions and logarithms.  The numerical methods would be based on simple differences and interpolations which, in many cases, were used for differentiation or integration.  The methods similarly used in graphical and mechanical calculation were well known.&lt;/p&gt;&#xA;&lt;p&gt;The use of conformal mapping in relation to elementary problems in fluid flow was hardly new.  In 1738 Jacob Bernoulli first expressed the momentum and energy relations in fluid flowing from one location to another.  Then in 1753 Euler proposed potential flow as a means for understanding irrotational fluid flow.  More than 150 years later in 1904, Kutta would observe that the flow around an airfoil could not be irrotational if the flowing fluid was to smoothly leave the trailing edge of the airfoil.  Independently, in 1906, Joukowsky would make a similar observation and work through a solution using conformal mapping and potential-flow analysis. The method provides a means for determining the velocity of the fluid flow around an airfoil.  The fluid is treated as inviscid (that is, without viscosity), and Kutta's observations are included by superimposing a rotational component on the flow solution.&lt;/p&gt;&#xA;&lt;p&gt;Extended computational methods in applications of the Joukowski transformation and airfoil analysis were developed during the period after WW-I.  The basic process of hands-on-paper analysis and wind-tunnel verification was undertaken by the National Advisory Committee for Aeronautics.  This was done for a large number of already known airfoils, and an extensive series of easily developed wing sections that could be applied in general aviation and more advanced aircraft.  Two areas of particular interest were in drag reduction and in propeller design.  Applications were in understanding fluid mechanics and the consequences in perfection of aircraft construction methods for the most efficient use of power in flight.&lt;/p&gt;&#xA;&lt;p&gt;Wind-tunnel testing, however, presented a series of problems, the principal of which was expense.  Further, the facilities for the wind tunnel were very large and required significant power and a number of personnel to operate.  No short-cuts could be undertaken in building models used in wind-tunnel testing of wing sections.  In fact, comments are made within the reports of the NACA regarding the stringent standards to which models needed to be constructed, and the need to develop an adequate theory for the on-paper evaluation of wing sections based upon mathematical methods and insights. Theodorsen, along with Garrick, proposed such a mathematical approach in the 1930s.  Although the mathematical methods may have seemed somewhat obscure, they were, in general, relatively simple and fundamentally could be accomplished with little more than hand tabulation and analysis of results from graphical and mechanical computation.  Greater precision in results could be obtained with relatively direct mathematical methods that were readily available and known.  More detailed hand computation was involved, and could be expensive.  Even so, along the approach taken in the mathematical analysis was the realization that the Joukowski method was easily used where the resulting transform-circle used in the method was altered in its shape to conform to that of the airfoil of interest.  In fact, Jones and McWilliams &lt;a href=&quot;http://acversailles.free.fr/documentation/08%7EDocumentation_Generale_M_Suire/Aerodynamique/Profils/Theorie/Profil-Dessin/Design_your_own_airfoil.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;show precisely how this is done&lt;/a&gt; and the kind of results that could be obtained.&lt;/p&gt;&#xA;&lt;p&gt;The objective in using the Joukowski method, and variations, was to determine the pressure distribution of the flow along the surface of the airfoil contour.  Using conformal mapping, the pressure distribution could be derived directly for an airfoil of interest, or for an airfoil inversely derived from the pressure distribution. Of particular interest was the &lt;em&gt;velocity distribution&lt;/em&gt; of the flow over the contour of the airfoil.  This &lt;em&gt;velocity distribution&lt;/em&gt; could be derived directly from the pressure distribution.&lt;/p&gt;&#xA;&lt;p&gt;One aspect in understanding the fluid mechanics of the flow in the near-vicinity of the surface related to the maintenance of laminar flow and an understanding of turbulent-flow separation.  This was especially important in reducing drag.  Consequently methods were advanced that allowed the velocity distribution and shape of the airfoil to be adjusted by changes in the shape of the transform-circle used in the conformal mapping.&lt;/p&gt;&#xA;&lt;p&gt;For the use of conformal mapping in airfoil design &lt;em&gt;per se,&lt;/em&gt; this refined approach was initially used in the design of the series 6 wing section for the P-51. The design and construction of this aircraft was in essence a tour de force accomplished by the NACA and North American Aviation, that resulted in a completed prototype from lofting of plans to prototype assembly and initial flight testing, within &lt;em&gt;90 days.&lt;/em&gt;  The newly-designed laminar-flow wing section had favorable aerodynamic characteristics that made the aircraft easy to fly and control over the range of flight conditions needed for bomber escort duty and aerial combat, especially in the European theater.&lt;/p&gt;&#xA;&lt;p&gt;Methods used by the NACA engineers in the design and hands-on-paper analysis of the series 6 airfoils, were not overly complex and wonderful.  Every Joukowski airfoil may be developed in a process using three circles, namely, the basis circle (centered on the y-axis and known principally as the unit circle), the transform circle, and a resulting invert circle. The basis circle simply establishes the relationship of the line of centers used in the transformation. If this line is inclined, the transformation result is a cambered airfoil.   If the basis circle is transformed onto itself, the result is the camber line of the airfoil.  However, if the center of the basis circle is at the origin and that of the transform circle offset adjacently on the abscissa, the resulting airfoil is uncambered and symmetrical. The approach taken by the NACA in developing the series 6 wing sections was similar, namely, to simply distort the shape of the transform circle and map it back onto the basis circle to obtain a symmetrical airfoil.  This process is termed &lt;em&gt;mapping the near circle onto the unit circle.&lt;/em&gt;  When this is done, the functional relationship regarding the angular distortion and difference in shape of the near circle relative to the unit circle, can be used in a relatively simple expression defining the ratio of the fluid velocity at the airfoil surface, to that of the free-stream velocity some distance away. In this way, desirable characteristics of the velocity distribution could be derived for enhancement of laminar flow.&lt;/p&gt;&#xA;&lt;p&gt;After the end of the Second World War, for this type of airfoil design work there were no electronic computers readily available for perhaps more than 20 years.  In the United States, the NACA laminar wing sections were used on many general aviation aircraft, and ultimately used in the perfected design of the Sisu-1A, a high performance sailplane.    However, after 1950 the methods used by the two German engineers are of particular interest as they are relevant to the posted question.  Both of these engineers were interested in the optimized design of laminar airfoils for high-performance soaring flight, and for applications in general aviation aircraft.  They were well aware of the methods used by the NACA in developing the series 6 airfoils.  The interests of Richard Eppler were in the understanding how to control turbulent separation and optimize the airfoil design for minimum drag.  Eppler was also interested in calculating the drag of an airfoil based upon an adequate understanding of turbulence and flow separation.  His work, although relatively obscure in the post-war period for nearly 20 years, became very highly regarded and well known in applications for high performance sailplanes and aircraft.   Although Eppler was well published, his work did not step into the forefront until after 1975.  At about this time, with the help of associations in the United States (particularly with NASA), Eppler transferred his methods to digital computation.  His principal work in airfoil design, nevertheless, was based on the mainstay of conformal mapping derived from modifications to the Joukowski transformation, and consequent desired results to the airfoil shape in order to optimize the velocity distribution and airfoil design.  Although a practicing aeronautical engineer in the early portion of his career, Eppler ultimately joined the faculty of the University of Stuttgart.&lt;/p&gt;&#xA;&lt;p&gt;F X Wortmann, a contemporary of Eppler, became much more well known and in the forefront of airfoil design in the post-war period from the early 1950s and into the 1980s.  His interests were also in turbulent separation and optimization of laminar flow in airfoil design. From the very beginning of Wortmann’s work, he understood the dynamics of boundary-layer separation and the need to appropriately shape the airfoil to control the velocity profile and boundary-layer separation.  Wortmann became director of the Institute for Aerodynamics and Gas Dynamics at the University of Stuttgart.  His most well-known work was in the development of low-drag airfoils for high-performance sailplanes.  An apparent fundamental aspect of Wortmann’s airfoil-design work was in the modification of the Joukowski profile for conformance of the profile velocity distribution to meet specified objectives regarding laminar flow and turbulent separation.&lt;/p&gt;&#xA;&lt;p&gt;Is the Joukowski transformation still used today?  The answer is &lt;strong&gt;yes.&lt;/strong&gt; Aspects of airfoil design, particularly for low-speed, high-performance aircraft, are reliant on the use of the method, and variations of it, to achieve desired results.&lt;/p&gt;&#xA;" OwnerUserId="19735" LastActivityDate="2024-02-22T05:00:49.043" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="16247" PostTypeId="1" CreationDate="2024-02-22T09:35:59.877" Score="1" ViewCount="69" Body="&lt;p&gt;Did Archimedes ever discuss hornangles?&lt;/p&gt;&#xA;&lt;p&gt;A hornangle (also known as angle of contingence, etc.) is the &amp;quot;crevice&amp;quot; between the circle and its tangent line at a point (from the modern viewpoint, such an &amp;quot;angle&amp;quot; would be zero).&lt;/p&gt;&#xA;&lt;p&gt;We know that Euclid did, and that hornangles were the subject of a lively debate between Clavius and Peletier in the 16th century.  Furthermore, they were mentioned by a number of authors in the 17th century.  But what about Archimedes?&lt;/p&gt;&#xA;" OwnerUserId="604" LastActivityDate="2024-02-22T09:35:59.877" Title="Archimedes on hornangles?" Tags="|mathematicians|archimedes|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16248" PostTypeId="1" CreationDate="2024-02-22T11:42:12.370" Score="0" ViewCount="70" Body="&lt;p&gt;The way in which scientists should deal with errors in observations of natural phenomena was a subject of much debate over a period of about 150 years between around 1720 and 1870. The history is well documented in numerous publications by &lt;a href=&quot;http://www.sheynin.de/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Oscar Sheynin&lt;/a&gt; (also Oskar Cheinine, Л. Б. Шейнин) .&lt;/p&gt;&#xA;&lt;p&gt;Two issues of central importance to the debate were:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;whether the mean of a series of observations provided a better estimate of the true value than a did a single &amp;quot;good&amp;quot; observation (see for example, the paper by &lt;a href=&quot;https://www.jstor.org/stable/104912&quot; rel=&quot;nofollow noreferrer&quot;&gt;Simpson&lt;/a&gt;), and&lt;/li&gt;&#xA;&lt;li&gt;how best to characterize the distribution of errors  (see for example, the paper on &lt;a href=&quot;https://doi.org/10.1007/BF00776066&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss and the Theory of Errors&lt;/a&gt; by Sheynin)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Although I have looked at several papers of Sheynin's, and tracked down some of the papers to which he refers, I have been unable to find any &lt;em&gt;actual&lt;/em&gt; set of observations (of &lt;em&gt;&lt;strong&gt;anything&lt;/strong&gt;&lt;/em&gt; ... astronomical, geodetic, or otherwise) that dates from the period and were used in any writing on the topic either of the mean or of error distributions.&lt;/p&gt;&#xA;&lt;p&gt;Can anyone suggest where I might locate such a data set, even if of only a handful of observations? ... It would be &lt;em&gt;&lt;strong&gt;especially useful&lt;/strong&gt;&lt;/em&gt; to me if I can find a set of repeated (i.e., resampling) observations, as opposed to a time series.&lt;/p&gt;&#xA;&lt;p&gt;But for his death only a few weeks ago on 3 January 2024, and of which I only just learnt, I might have emailed Shenyin himself. As it stands, this forum seemed the best place to ask because of the subject of interest, although obviously &lt;a href=&quot;https://opendata.stackexchange.com/&quot;&gt;SE Opendata&lt;/a&gt; is an alternative if my question is not considered appropriate here.&lt;/p&gt;&#xA;" OwnerUserId="17963" LastEditorUserId="17963" LastEditDate="2024-02-23T04:50:46.937" LastActivityDate="2024-02-23T11:09:54.357" Title="Data that motivated early discussions about the mean and about error distributions" Tags="|statistics|19th-century|18th-century|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16250" PostTypeId="1" CreationDate="2024-02-22T17:15:24.883" Score="1" ViewCount="78" Body="&lt;p&gt;What's the history of the use of crude oil for transportation before the Industrial Revolution?&lt;/p&gt;&#xA;&lt;p&gt;Wiktionary says the Latin word &lt;a href=&quot;https://en.wiktionary.org/wiki/petroleum#Latin&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;petroleum&lt;/em&gt;&lt;/a&gt;, from &lt;em&gt;petra&lt;/em&gt; (“rock”) + &lt;em&gt;oleum&lt;/em&gt; (“oil”), is medieval Latin, so presumably at least the medievals had uses for it.&lt;/p&gt;&#xA;" OwnerUserId="232" LastEditorUserId="232" LastEditDate="2024-02-22T21:12:52.793" LastActivityDate="2024-02-27T06:28:54.400" Title="What's the history of the use of crude oil for transportation before the Industrial Revolution?" Tags="|thermodynamics|technology|energy|ancient|inventions|" AnswerCount="1" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16251" PostTypeId="2" ParentId="16248" CreationDate="2024-02-22T18:05:30.090" Score="2" Body="&lt;p&gt;As described in &lt;a href=&quot;https://www.jstor.org/stable/2690592&quot; rel=&quot;nofollow noreferrer&quot;&gt;this paper&lt;/a&gt;, Gauss became famous with his prediction of the orbit of Ceres based on observations by the astronomer Piazzi. Piazzi's astronomical data (from January and February 1801) were published in von Zach's journal &lt;em&gt;Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde&lt;/em&gt; in the September issue of 1801. Gauss's calculations are described in the December issue. Luckily, the journal is available online (in particular &lt;a href=&quot;https://zs.thulb.uni-jena.de/receive/jportal_jpvolume_00201967&quot; rel=&quot;nofollow noreferrer&quot;&gt;Volume 4, 1801&lt;/a&gt;), via the state library of Thuringia, Germany.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit:&lt;/strong&gt; I'm adding a screenshot with Piazzi's observations (&amp;quot;data&amp;quot;) from loc. cit. (Sept. 1801). Gauss used different subsets of these to predict the later position of Ceres.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/m39td.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/m39td.png&quot; alt=&quot;Piazzi's observations Jan/Feb 1801&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;To deal with more data than mathematically required, hence to find an optimal solution for an overdetermined problem, Gauss invented the method of least squares. He describes this method in the third section of his book &lt;em&gt;Theoria Motus&lt;/em&gt; (English translation online &lt;a href=&quot;https://quod.lib.umich.edu/m/moa/AGG8895.0001.001/288&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;The method of least squares was developed independently by Legendre and apparently put to use in the context of the meridian measurements by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_Baptiste_Joseph_Delambre&quot; rel=&quot;nofollow noreferrer&quot;&gt;Delambre&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Pierre_Mechain&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mechain&lt;/a&gt;. This story is told in Chapter 11 of Ken Alder's book cited on Mechain's Wikipedia page. With some effort, one should be able to find the data collected by Delambre and Mechain as well. A good source is Delambre's 1806 text &lt;em&gt;Base du système métrique décimal, ou Mesure de l'arc du méridien compris entre les parallèles de Dunkerque et Barcelone&lt;/em&gt;, in particular the section on geodesic observations &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k110604s/f185.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastEditorUserId="16834" LastEditDate="2024-02-23T11:09:54.357" LastActivityDate="2024-02-23T11:09:54.357" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="16253" PostTypeId="2" ParentId="11598" CreationDate="2024-02-23T23:49:09.600" Score="0" Body="&lt;p&gt;Good question. I found a reasonable answer . It includes .&lt;/p&gt;&#xA;&lt;p&gt;Eratosthenes and the Mystery of the Stades - Distance from Alexandria to Syene&#xA;&amp;quot;That the distance between Alexandria and Syene is 5000 stades.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;There is little doubt that Eratosthenes got this figure directly from his earlier map of the known world [4, p.62].  How he initially obtained this value is a controversial question which may never be answered, but it is doubtful that Eratosthenes actually measured the distance himself [5, p.154 ].  The writings of Strabo the Geographer (ca. 20 BCE) suggest that the lands along the Nile were measured every year.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://maa.org/book/export/html/116342&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://maa.org/book/export/html/116342&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19748" LastEditorUserId="229" LastEditDate="2024-02-24T06:57:29.593" LastActivityDate="2024-02-24T06:57:29.593" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16254" PostTypeId="2" ParentId="7487" CreationDate="2024-02-24T12:18:09.243" Score="1" Body="&lt;p&gt;Gauss' formula is very probably correct in the non-relativistic regime, as long as the accelerations are negligible. However, the formula is in fact not fully compatible with Maxwell's equations.&lt;/p&gt;&#xA;&lt;p&gt;From a historical point of view, one needs to start with Ampere. Around 1820, he measured the magnetic forces between parallel and non-parallel conductors. He found a force law that does not correspond to the formula that is known today as Ampère's force law. In 1835, Gauss realized that this original force law could be explained by imagining a current element to be composed of two point charges. For Gauss and later for Weber, magnetism was only an effect and not a fundamental force (the Lorentz force can be deduced and there is no need to postulate it).&lt;/p&gt;&#xA;&lt;p&gt;In 1870, there was unfortunately an incompatible break with the interpretation of magnetism, because Maxwell assumed in his derivations that the B fields are generated by currents in closed conductor loops and not by single moving point charges (see page 162: &lt;em&gt;Maxwell, &amp;quot;Treatise on Electricity and Magnetism&amp;quot;, Vol. 2 (2 ed.), Clarendon Press, Oxford, 1881&lt;/em&gt;). By means of this postulate, he was able to carry out some formula transformations and arrived at the Biot-Savart law. However, Maxwell now had to include the Lorentz force as an additional equation, since it could not be deduced any longer from the field of a point charge. Maxwell's electrodynamics was very successful, as it allowed the modeling of electromagnetic free-space waves for the first time. The electrodynamics of Ampere, Gauss and Weber was almost completely forgotten.&lt;/p&gt;&#xA;&lt;p&gt;Interestingly, it is relatively easy to merge the two electrodynamics. The approach is as follows: First one solves Maxwell's equations for two point charges in the rest frame of the recipient. The Lorentz force can therefore be simplified to &lt;span class=&quot;math-container&quot;&gt;$\vec{F} = q\,\vec{E}$&lt;/span&gt; and all velocity-dependent terms go into the current density. Once the solution for the E-field has been found, a Galilean transformation is carried out, which is allowed if one is only interested in non-relativistic velocities. As it turns out, the solution is compatible with Gauss' formula if one uses the equation &lt;span class=&quot;math-container&quot;&gt;$\vec{F} = q\, \gamma(v)\,\vec{E}$&lt;/span&gt; instead of &lt;span class=&quot;math-container&quot;&gt;$\vec{F} = q\,\vec{E}$&lt;/span&gt;. An interesting aspect of this method is that B-field and Lorentz force become obsolete, as all magnetic effects can now be interpreted in the same way as Gauss did. In addition, one does not lose the advantages of Maxwell's equations.&lt;/p&gt;&#xA;&lt;p&gt;The final formula for the electromagnetic force &lt;span class=&quot;math-container&quot;&gt;$\vec{F}$&lt;/span&gt; that a point charge &lt;span class=&quot;math-container&quot;&gt;$q_s$&lt;/span&gt; with the trajectory &lt;span class=&quot;math-container&quot;&gt;$\vec{r}_s(t)$&lt;/span&gt; exerts on another point charge &lt;span class=&quot;math-container&quot;&gt;$q_d$&lt;/span&gt; with the trajectory &lt;span class=&quot;math-container&quot;&gt;$\vec{r}_d(t)$&lt;/span&gt; is&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\vec{F} = \frac{q_d\,q_s\,\gamma(v)}{4\,\pi\,\varepsilon_0}\,\left(\frac{\left(\vec{r}\,c + r\,\vec{v}\right)\,\left(c^2-v^2 - \vec{r}\cdot\vec{a}\right)}{\left(r\,c + \vec{r}\cdot\vec{v}\right)^3} + \frac{\vec{a}\,r}{\left(r\,c + \vec{r}\cdot\vec{v}\right)^2} \right),&#xA;\end{equation}&lt;/span&gt;&#xA;with the retarded distance vector&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\vec{r} = \vec{r}_d(\tau) - \vec{r}_s(\tau),&#xA;\end{equation}&lt;/span&gt;&#xA;the retarded difference velocity&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\vec{v} = \dot{\vec{r}}_d(\tau) - \dot{\vec{r}}_s(\tau),&#xA;\end{equation}&lt;/span&gt;&#xA;and the retarded difference acceleration&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\vec{a} = \ddot{\vec{r}}_d(\tau) - \ddot{\vec{r}}_s(\tau).&#xA;\end{equation}&lt;/span&gt;&#xA;The retarded time can be obtained by solving the equation&#xA;&lt;span class=&quot;math-container&quot;&gt;\begin{equation}&#xA;\tau = t - \frac{r}{c}.&#xA;\end{equation}&lt;/span&gt;&#xA;(for example with Newton's method, note that &lt;span class=&quot;math-container&quot;&gt;$r = \Vert\vec{r}\Vert$&lt;/span&gt; etc.)&lt;/p&gt;&#xA;&lt;p&gt;The force formula is very useful for computer simulations of electromagnetic fields and waves (&lt;a href=&quot;https://github.com/StKuehn/OpenWME&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://github.com/StKuehn/OpenWME&lt;/a&gt;), since there is no longer any need to solve differential equations. Incidentally, it also fulfills Newton's third law, i.e. one can easily show that the total momentum of an isolated system of point charges is a conserved quantity. In addition, the force formula is invariant in all inertial frames and complies with Einstein's postulates without the need to perform a Lorentz transformation (of course, this only applies to the non-relativistic regime). Because of these properties, the formula is particularly useful in electrical engineering.&lt;/p&gt;&#xA;&lt;p&gt;The derivation from Maxwell's equations and the proof that the force formula agrees with Gauss' formula for very small accelerations and velocities can be found in &lt;em&gt;&amp;quot;The Importance of Weber–Maxwell Electrodynamics in Electrical Engineering&amp;quot;, IEEE Transactions on Antennas and Propagation, vol. 71, no. 8, 2023&lt;/em&gt; and &lt;em&gt;&amp;quot;Weber-Maxwell electrodynamics: classical electromagnetism in its most compact and pure form&amp;quot;, Techrxiv, 2024&lt;/em&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19751" LastActivityDate="2024-02-24T12:18:09.243" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="16255" PostTypeId="1" CreationDate="2024-02-24T15:46:29.827" Score="0" ViewCount="71" Body="&lt;p&gt;Did physicians in the first century AD understand that some diseases are sexually transmitted? Would a typical well-educated person who was not a physician have likely known this?&lt;/p&gt;&#xA;" OwnerUserId="19754" LastActivityDate="2024-02-25T04:17:51.777" Title="Was it understood in the first century AD that some diseases are sexually transmitted?" Tags="|medicine|roman-empire|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17255" PostTypeId="1" CreationDate="2024-02-24T16:27:12.880" Score="0" ViewCount="31" Body="&lt;p&gt;In Alanzo Church's &amp;quot;A Formulation of the Simple Theory of Types&amp;quot; (&lt;em&gt;The Journal of Symbolic Logic&lt;/em&gt; &lt;strong&gt;5&lt;/strong&gt; no.2 (1940) 56--68, &lt;a href=&quot;https://doi.org/10.2307/2266170&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;code&gt;DOI:10.2307/2266170&lt;/code&gt;&lt;/a&gt;), he adopts the convention that terms of type &lt;span class=&quot;math-container&quot;&gt;$\iota$&lt;/span&gt; are nonlogical constants (I presume &lt;span class=&quot;math-container&quot;&gt;$\iota$&lt;/span&gt; is chosen because terms of this type are called &amp;quot;individuals&amp;quot;), and terms of type &lt;span class=&quot;math-container&quot;&gt;$\omicron$&lt;/span&gt; are for truth values. More explicitly, Church states:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the interpretation of the theory it is intended that the subscript shall indicate the type of the variable or constant, &lt;span class=&quot;math-container&quot;&gt;$\omicron$&lt;/span&gt; being the type of propositions, &lt;span class=&quot;math-container&quot;&gt;$\iota$&lt;/span&gt; the type of individuals, ... (56-57)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So it seems clear that &lt;span class=&quot;math-container&quot;&gt;$\iota$&lt;/span&gt; specifically is chosen for &amp;quot;individuals&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;I'm at a loss as to why &lt;span class=&quot;math-container&quot;&gt;$\omicron$&lt;/span&gt; was chosen, and if there is a mnemonic meaning for it (just as &amp;quot;individual&amp;quot;, shortened to &amp;quot;i&amp;quot;, then converted to Greek gives us &lt;span class=&quot;math-container&quot;&gt;$\iota$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Question:&lt;/strong&gt; Is there a reason why &lt;span class=&quot;math-container&quot;&gt;$\omicron$&lt;/span&gt; was chosen to denote the type of truth values?&lt;/p&gt;&#xA;" OwnerUserId="2129" LastEditorUserId="2129" LastEditDate="2024-02-26T15:44:34.230" LastActivityDate="2024-02-26T15:44:34.230" Title="Notation for Propositional values in Church's &quot;Simple Theory of Types&quot;" Tags="|mathematics|notation|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17256" PostTypeId="1" CreationDate="2024-02-24T20:29:35.257" Score="0" ViewCount="48" Body="&lt;p&gt;The greek mathematician Anaxagoras made an inacurrate evaluation of the distance between the Sun and the Earth. I've been thinking about his work and would like to know what exactly was wrong about his calculation.&lt;/p&gt;&#xA;&lt;h2&gt;Facts:&lt;/h2&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Syene is a city located (nearly) on the Cancer tropic, which means that on June &lt;span class=&quot;math-container&quot;&gt;$21$&lt;/span&gt; at noon, the Sun is at the zenith of, say, an object. More specifically, the object has no shadow. On the other hand, Alexandria is located &lt;span class=&quot;math-container&quot;&gt;$800$&lt;/span&gt;km (&lt;span class=&quot;math-container&quot;&gt;$=5000$&lt;/span&gt; stadia) northwards from Syene.&lt;/li&gt;&#xA;&lt;li&gt;In this configuration, Anaxagoras has measured an angle of &lt;span class=&quot;math-container&quot;&gt;$7,2°$&lt;/span&gt;  between the vertical in Alexandria and the direction of the sunrays shining in Alexandria.&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;h2&gt;Assumptions (of Anaxagoras):&lt;/h2&gt;&#xA;&lt;p&gt;Anaxagoras assumes Earth is planar and that the Sun is a point (and as a consequence of it, &lt;em&gt;that the sunrays shine the Earth in a conic &amp;quot;divergent&amp;quot; beam&lt;/em&gt;). Actually I don't know (maybe I should read in its writings) if Anaxagoras made the assumption that the Sun is a point, or just if he made the assumption that sunrays arrive in a conic &amp;quot;divergent&amp;quot; beam.&lt;/p&gt;&#xA;&lt;p&gt;Anyway, the figure below illustrates Anaxagoras' assumption that the Earth is flat and the conic beam of sunrays. You can ignore the fact that the Sun is a point (since we are not sure of this fact).&#xA;&lt;a href=&quot;https://i.stack.imgur.com/yd1y0.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/yd1y0.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;h2&gt;Defending Anaxagoras (and trying to understand which assumption of his model was wrong):&lt;/h2&gt;&#xA;&lt;p&gt;While it is an evidence that his first assumption of Earth flatness was completely wrong, the other assumption (as we saw) has two versions, which are both viable in my opinion.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;If he was (only) thinking that sunrays were coming in a conic beam towards the surface of the Earth, we can see this assumption is not far from reality as illustrated in the figure below (the first scheme is what really happens (in our &lt;span class=&quot;math-container&quot;&gt;$2024$&lt;/span&gt; model), the second one is what Anaxagoras thought).&lt;a href=&quot;https://i.stack.imgur.com/ridqb.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/ridqb.jpg&quot; alt=&quot;two conic beam from the Sun, reality and Anaxagoras&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;if he was assuming that the Sun is a point, again it is not a bad assumption. Isn't that the assumption we make when we nowadays say the sunrays come parallel to the surface of the Earth ? I am pretty sure it is the same.&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;So, my question is which assumption was wrong in Anaxagoras model ? Only that the Earth is flat ? Or is there any other wrong assumption ?&lt;/p&gt;&#xA;&lt;p&gt;Note: We cannot say &amp;quot;assuming a small distance between the Earth and the Sun was a wrong assumption from Anaxagoras&amp;quot; since this distance is exactly what he was aiming to calculate.&lt;/p&gt;&#xA;&lt;p&gt;I don't know if thanking in advance is allowed in this section (thanks in advance anyway).&lt;/p&gt;&#xA;" OwnerUserId="20754" LastActivityDate="2024-02-24T20:29:35.257" Title="What are the wrong assumptions made in Anagaxoras' model in his calculation of Earth-Sun distance?" Tags="|cosmology|greek-mathematics|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17257" PostTypeId="2" ParentId="3340" CreationDate="2024-02-24T21:17:35.827" Score="4" Body="&lt;p&gt;It is worth noting that Gauss himself did not claim to be the first to invent the concept of congruence.  In Disquisitiones Arithmeticae, after giving his definition of congruence and introducing his symbol &lt;span class=&quot;math-container&quot;&gt;$\equiv$&lt;/span&gt;, he adds the following &lt;a href=&quot;https://la.wikisource.org/wiki/Disquisitiones_arithmeticae/Sectio_prima#cite_note-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;footnote&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Hoc signum propter magnam analogiam quae inter aequalitatem atque congruentiam invenitur adoptavimus. Ob eandem caussam ill. Le Gendre in comment. infra saepius laudanda ipsum aequalitatis signum pro congruentia retinuit, quod nos ne ambiguitas oriatur imitari dubitavimus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which can be translated as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;We adopted this symbol because of the great analogy that is found between equality and congruence. It is for the same reason that the illustrious Legendre, in a commentary that we will later have occasion to cite, retained the very symbol of equality for congruence. We hesitated to imitate this, lest any ambiguity arise.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;To understand what Gauss is talking about here, we can simply follow his citations.  This leads to the following &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k35847/f650.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;passage&lt;/a&gt; in Legendre's &lt;em&gt;Recherches d’Analyse indéterminée&lt;/em&gt;,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Pour exprimer que &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; est divisible par &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, ou que &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{A}$&lt;/span&gt; est un entier, j'ecrirai &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{A}=e$&lt;/span&gt;.  On peut omettre dans le cours du calcul tous les multiples de &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;: ainsi les coefficiens &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, etc. et généralement tous les nombres sur lesquels on opérera, ne surpasseront jamais &lt;span class=&quot;math-container&quot;&gt;$\tfrac{1}{2} A$&lt;/span&gt; en plus ou en moins.  D'après cette omission on peut écrire indifféremment &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{A} = e$&lt;/span&gt;, ou &lt;span class=&quot;math-container&quot;&gt;$P=0$&lt;/span&gt;, pourvu que dans le dernier cas, on ne perde pas de vue le nombre &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; dont on rejette les multiples.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;which can be translated as&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To express that &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; is divisible by &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;, or that &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{A}$&lt;/span&gt; is an integer, I will write &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{A}=e$&lt;/span&gt;. During calculations, we can omit all multiples of &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;: thus the coefficients &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt;, etc. and generally all numbers on which we operate will never exceed plus or minus &lt;span class=&quot;math-container&quot;&gt;$\tfrac{1}{2} A$&lt;/span&gt;. Having made this omission, we can write indifferently &lt;span class=&quot;math-container&quot;&gt;$\frac{P}{A} = e$&lt;/span&gt; or &lt;span class=&quot;math-container&quot;&gt;$P=0$&lt;/span&gt;, provided that in the latter case, we do not lose sight of the number &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; whose multiples are rejected.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;While Legendre did not use or introduce terminology like &amp;quot;congruence&amp;quot;, &amp;quot;modulus&amp;quot; etc. to describe what he was doing, he was certainly writing equations in which it is implicit that the two sides are not equal, but rather differ by an unspecified multiple of a given number.  Further reading reveals instances where he manipulates these equations algebraically, in the same way that one manipulates ordinary equations (e.g. the first paragraph at the top of &lt;a href=&quot;https://gallica.bnf.fr/ark:/12148/bpt6k35847/f659.item&quot; rel=&quot;nofollow noreferrer&quot;&gt;page 475&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;Gauss appears to have thought that this was similar enough (or even identical) to his own concept of congruence, to the extent that he had to acknowledge it (and assert the superiority of his own notation).&lt;/p&gt;&#xA;" OwnerUserId="19442" LastActivityDate="2024-02-24T21:17:35.827" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17258" PostTypeId="1" CreationDate="2024-02-24T22:44:07.327" Score="1" ViewCount="70" Body="&lt;p&gt;I’m slightly confused by what Dedekind means by “characteristic for the concepts they designate” in the quote below:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;But [. . . ] these extensions of definitions no longer allow scope for arbitrariness; on the contrary, they follow with compelling necessity from the earlier restricted definitions, provided one applies the following principle: Laws which emerge from the initial definitions and which are characteristic for the concepts that they designate are to be considered as of general validity&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is his example for context:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;We already have a definite example in multiplication. This operation arose from the requirement that a multiply-repeated performance of an operation of the next lower rank [Ordnung] — namely the addition of a fixed positive or negative addend (the so-called multiplicand) — be collected together into a single act. The multiplier — that is, the number which states how often the addition of the multiplicand is to be thought of as repeated — is therefore at the outset necessarily a positive integer; a negative multiplier would, under this first definition of multiplication, make absolutely no sense.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A special definition is therefore needed in order to admit negative multipliers as well, and thereby to liberate the operation from the initial constraint; but such a definition involves a priori complete arbitrariness, and it would only later be decided whether then this arbitrarily chosen definition would bring any real use to arithmetic; and even if the definition succeeded, one could only call it a lucky guess, a happy coincidence — the sort of thing a scientific method ought to avoid.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;So let us instead apply our general principle. We must investigate which laws govern the product if the multiplier undergoes in succession the same general alterations which led to the creation of the sequence of negative integers out of the sequence of positive integers. For this it suffices if we determine the alteration which the product undergoes if one makes the simplest numerical operation with the multiplier, namely, allowing it to go over into the next-following number. By successive repetition of this operation we obtain the familiar addition theorem for the multiplier: in order to multiply a number by a sum, one multiplies it by each summand and then adds these partial products together.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;From this theorem a subtraction theorem immediately follows for the case where the minuend is greater than the subtrahend. If one now declares this law to be valid in general (that is, to hold also when the difference which the multiplier represents is negative) then one obtains the definition of multiplication with negative multipliers; and it is then of course no accident that the general law which multiplication obeys is exactly the same for both cases.&amp;quot; [Dedekind, 1854, §8]&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Also, was Dedekind influenced by group theory in his treatment of multiplication?&lt;/p&gt;&#xA;" OwnerUserId="20755" LastEditorUserId="55" LastEditDate="2024-02-25T08:51:51.750" LastActivityDate="2024-02-25T08:51:51.750" Title="What does Dedekind mean by &quot;laws characteristic for the concepts&quot;?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17259" PostTypeId="2" ParentId="16255" CreationDate="2024-02-25T04:17:51.777" Score="3" Body="&lt;p&gt;This is outside my area of expertise. From a &lt;em&gt;cursory&lt;/em&gt; reading of the literature I get the impression that at present there is no clarity on what sexually transmitted diseases known today were actually present in classical antiquity, as the clues available in surviving writings from that time are sparse and vague, leading authors to speculate based on scant evidence. The following, by a classicist with a focus on medicine in classical antiquity, therefore looks like a plausible summary.&lt;/p&gt;&#xA;&lt;p&gt;Rebecca Flemming, &amp;quot;(The Wrong Kind of) Gonorrhea in Antiquity.&amp;quot; In Simon Szreter (ed.), &lt;em&gt;The Hidden Affliction: Sexually Transmitted Infections and Infertility in History&lt;/em&gt;, University of Rochester Press 2019, pp. 43-67. On p. 46 (my bolding):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;While many Latin satirists enjoyed exposing all Rome’s sexual foibles in explicit detail, if there is evidence of STIs to be found in antiquity, this is the best place to look. The argument is, however, that no such evidence is to be found. The symptoms of painful urination and some kind of vaginal or penile discharge, perhaps with swelling of the foreskin or lower abdominal pain, the conjunction of which would be taken to indicate modern gonorrhea (and, indeed, chlamydia), do not appear together in any ancient medical text, nor in any other part of the ancient literary record. This is a significant absence, given the dense coverage of human ailments, injuries, and cures in written material from classical Greece and Rome. &lt;strong&gt;Nor are notions of the sexual transmission of disease to be found in antiquity; sexual encounters were not considered sites of pathological danger.&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2024-02-25T04:17:51.777" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17260" PostTypeId="1" CreationDate="2024-02-25T09:59:02.487" Score="1" ViewCount="37" Body="&lt;p&gt;This is an extract from page 138 of Ptolemy's Almagest (Toomer):&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Hipparchus] made a very accurate observation of the autumnal equinox, and says that he calculated that it occurred at midnight, third-fourth epagomenal day [-146 Sept. 26/27]. The year is the 178th from the death of Alexander. 285 years later, in the third year of Antoninus, which is the 463rd from the death of Alexander, we observed, again very securely, that the autumnal equinox occurred on Athyr 9 [139 Sept. 26], approximately one hour after sunrise. Therefore the period of return comprised, over 285 complete Egyptian years (that is years of 365 days), 70&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; days plus approximately &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{20}$&lt;/span&gt;th of a day, instead of the 71&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt; days corresponding to the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;-day surplus for the above [285] years. Thus the return took place earlier than it would have with the [365]&lt;span class=&quot;math-container&quot;&gt;$\frac{1}{4}$&lt;/span&gt;-day year by one day less about &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{20}$&lt;/span&gt;th day.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;How does Ptolemy calculate the &lt;span class=&quot;math-container&quot;&gt;$\frac{1}{20}$&lt;/span&gt;th of a day? Shouldn't this be &lt;span class=&quot;math-container&quot;&gt;$\frac{7}{24}$&lt;/span&gt; assuming sunrise is at 6am on an equinox?&lt;br /&gt;&#xA;So the equinox is early by &lt;span class=&quot;math-container&quot;&gt;$\frac{17}{24}$&lt;/span&gt;th of a day instead of &lt;span class=&quot;math-container&quot;&gt;$\frac{19}{20}$&lt;/span&gt;th of a day as he states later in the text, right?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastEditorUserId="6609" LastEditDate="2024-02-26T08:23:52.123" LastActivityDate="2024-02-26T08:23:52.123" Title="Understanding Ptolemy's Almagest" Tags="|archaeoastronomy|ptolemy|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17261" PostTypeId="1" AcceptedAnswerId="17265" CreationDate="2024-02-25T19:57:16.623" Score="4" ViewCount="765" Body="&lt;p&gt;The Axiom of Completeness states that any non-empty set with an upper bound has a least upper bound.  When and why was this concept of least upper bound dubbed &amp;quot;completeness&amp;quot;?&lt;/p&gt;&#xA;&lt;p&gt;It's true, of course, that least upper bound is one way to establish the &lt;a href=&quot;https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;completeness of the real number system&lt;/a&gt;.  But there are &lt;a href=&quot;https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers&quot; rel=&quot;nofollow noreferrer&quot;&gt;many other equivalent ways&lt;/a&gt;, none of them called &amp;quot;Axiom of Completeness&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In particular, see &lt;a href=&quot;https://math.stackexchange.com/questions/4870101/is-the-axiom-of-completeness-logically-equivalent-to-there-is-no-proper-superse&quot;&gt;https://math.stackexchange.com/questions/4870101/is-the-axiom-of-completeness-logically-equivalent-to-there-is-no-proper-superse&lt;/a&gt; where it is explored that &amp;quot;completeness&amp;quot; may have originally had a different meaning: that any thing that could be a number is a number, or, more formally, that &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; is a maximal ordered Archimedean field, such that there is no proper superset of &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; that is an ordered Archimedean field.&lt;/p&gt;&#xA;&lt;p&gt;To me, this suggests that the notion of completeness was first defined this way: that &lt;span class=&quot;math-container&quot;&gt;$\mathbb R$&lt;/span&gt; is &lt;em&gt;complete&lt;/em&gt; in the sense that nothing could be added to it without taking away essential properties, and only afterwards did people say &amp;quot;this is equivalent to saying that any non-empty set with an upper bound has a least upper bound&amp;quot; and so they called that second property &lt;em&gt;completeness&lt;/em&gt;.  Are there any sources shedding light on this?&lt;/p&gt;&#xA;" OwnerUserId="18545" LastActivityDate="2024-02-26T20:59:54.090" Title="When and why was the concept of &quot;having a least upper bound&quot; dubbed &quot;completeness&quot;, as in Axiom of Completeness?" Tags="|mathematics|real-analysis|set-theory|numbers|" AnswerCount="2" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17263" PostTypeId="2" ParentId="17261" CreationDate="2024-02-26T02:47:13.963" Score="2" Body="&lt;p&gt;Michael E2 suggests the first usage of the Axiom of Completeness was &lt;a href=&quot;https://math.berkeley.edu/%7Ewodzicki/160/Hilbert.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hilbert (p.21)&lt;/a&gt; in 1899 (!!!), who writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Remark. To the preceeding five groups of axioms, we may add the following one,&#xA;which, although not of a purely geometrical nature, merits particular attention from a&#xA;theoretical point of view. It may be expressed in the following form:&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Axiom of Completeness.&lt;/strong&gt; To a system of points, straight lines,&#xA;and planes, it is impossible to add other elements in such a manner that the system&#xA;thus generalized shall form a new geometry obeying all of the five groups of axioms.&#xA;In other words, the elements of geometry form a system which is not susceptible of&#xA;extension, if we regard the five groups of axioms as valid.&lt;/p&gt;&#xA;&lt;p&gt;This axiom gives us nothing directly concerning the existence of limiting points,&#xA;or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano’s&#xA;theorem by virtue of which, for all sets of points situated upon a straight line between&#xA;two definite points of the same line, there exists necessarily a point of condensation,&#xA;that is to say, a limiting point. From a theoretical point of view, the value of this axiom&#xA;is that it leads indirectly to the introduction of limiting points, and, hence, renders it&#xA;possible to establish a one-to-one correspondence between the points of a segment and&#xA;the system of real numbers.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I'm shocked that no such term wasn't used in the 1800s when completeness was being conceived.&lt;/p&gt;&#xA;" OwnerUserId="18545" LastActivityDate="2024-02-26T02:47:13.963" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17264" PostTypeId="2" ParentId="6435" CreationDate="2024-02-26T04:06:21.430" Score="0" Body="&lt;p&gt;I have translated the text but not the tables. My translation is very rough and the parts I've cleaned up are much better but the work is endless. I've translated many of his scientific articles and am working on Alph's edition of his father's Memories et souvenirs, another long project. I've also transcribed and translated all of letters to Asa Gray (~75).&lt;/p&gt;&#xA;" OwnerUserId="20759" LastActivityDate="2024-02-26T04:06:21.430" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17265" PostTypeId="2" ParentId="17261" CreationDate="2024-02-26T06:21:57.213" Score="8" Body="&lt;p&gt;According to &lt;a href=&quot;https://www.sciencedirect.com/science/article/pii/031508609290074L&quot; rel=&quot;nofollow noreferrer&quot;&gt;Burn, Irrational numbers in English language textbooks, 1890–1915: Constructions and postulates for the completeness of the real numbers&lt;/a&gt;, &amp;quot;completeness&amp;quot; was first used by Dedekind in 1872, and then by Hilbert in 1899. This should not be too surprising. Meta-mathematical considerations concerning classes of number systems were rare before axiomatizations, they were studied in concrete realizations, so the term would not have been very useful. Dedekind was a rare exception anticipating later algebraic formalism. Hilbert needed it to distinguish standard geometry from algebraic and non-Archimedean models, and those were not studied until the very end of 19th century.&lt;/p&gt;&#xA;&lt;p&gt;Both Dedekind's and Hilbert's uses were sporadic, widespread adoption of &amp;quot;completeness&amp;quot; only happened in 1920-s, with the general theory of metric spaces. Dedekind and others did use &amp;quot;continuity&amp;quot; derived from the &amp;quot;continuum&amp;quot; for similar purposes before that. Dedekind seems to prefer &amp;quot;continuity&amp;quot; and talks about &amp;quot;all continuous domains&amp;quot;. The least upper bound property and the Archimedean axiom are sometimes called &amp;quot;axioms of continuity&amp;quot; even today.&lt;/p&gt;&#xA;&lt;p&gt;Here is Burn:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;There is an anachronism in our title, for the widespread use of the term &amp;quot;completeness&amp;quot;&#xA;to describe the extension of the rational numbers to the real numbers&#xA;only became conventional from its occurrence in the study of metric spaces,&#xA;though it was used by Dedekind in [1872] and in his correspondence with Lipschitz&#xA;[Dedekind 1876], and by Hilbert in his axioms for geometry [1899] (in every&#xA;translation, and every edition after the first).&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In the period with which we are&#xA;concerned, the issue would generally have been worded differently. How, it&#xA;would have been asked, could the continuity of the real line be guaranteed or how&#xA;could the continuum be characterized? Dedekind had used the term &amp;quot;completeness&amp;quot;&#xA;in [1872] both to describe the closure of a number field under the four&#xA;arithmetical operations, and as a synonym for &amp;quot;continuity.&amp;quot; We will retain the&#xA;modern term ''completeness'' for the sake of clarity, since the term ''continuity,''&#xA;in modern usage, applies only to functions.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://people.math.ethz.ch/%7Ehalorenz/4students/Begleitseminar/Dedekind.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Dedekind's &lt;em&gt;Stetigkeit und irrationale Zahlen&lt;/em&gt; (1872)&lt;/a&gt; is &lt;a href=&quot;https://www.gutenberg.org/files/21016/21016-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;translated into English&lt;/a&gt; by Berman, it is well-known for developing (Dedekind) cuts. The 1876 letter to Lipschitz is printed in Dedekind's  &lt;em&gt;Gesammelte mathematische Werke&lt;/em&gt;, vol. 3, p. 473, 1930. Here is from Berman's translation:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If now, as is our desire, we try to follow up arithmetically all phenomena in&#xA;the straight line, the domain of rational numbers is insufficient and it becomes&#xA;absolutely necessary that the instrument &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; constructed by the creation of the&#xA;rational numbers be essentially improved by the creation of new numbers such&#xA;that the domain of numbers shall gain the same completeness, or as we may say&#xA;at once, the same continuity, as the straight line.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The above comparison of the domain &lt;span class=&quot;math-container&quot;&gt;$R$&lt;/span&gt; of rational numbers with a straight&#xA;line has led to the recognition of the existence of gaps, of a certain incompleteness&#xA;or discontinuity of the former, while we ascribe to the straight line&#xA;completeness, absence of gaps, or continuity. In what then does this continuity&#xA;consist? Everything must depend on the answer to this question, and only&#xA;through it shall we obtain a scientific basis for the investigation of all continuous&#xA;domains.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-02-26T20:59:54.090" LastActivityDate="2024-02-26T20:59:54.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17267" PostTypeId="1" AcceptedAnswerId="17268" CreationDate="2024-02-26T10:10:38.613" Score="0" ViewCount="55" Body="&lt;p&gt;In Ptolemy's geocentric model the Sun travels through the ecliptic and around the Earth once every 24 hours and the Earth does not rotate about its axis. What is Ptolemy referring to when he talks about the Sun's position during the year? How did he reconcile the Sun's eccentric orbit (giving rise to different length seasons) with the fact that the Sun rotated around the Earth every 24hrs?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2024-02-26T14:24:02.567" Title="Earth's and Sun's rotation in the Ptolemaic world" Tags="|archaeoastronomy|ptolemy|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17268" PostTypeId="2" ParentId="17267" CreationDate="2024-02-26T13:52:25.067" Score="3" Body="&lt;p&gt;In Ptolemy system, there is a sphere of fixed stars which rotates about the Earth with constant speed 1 rotation per 24 hours. This rotation is about the axis which passes through the North and South pole. The poles are fixed with respect to the stars, and the great circle equidistant from the poles is the (celestial) equator. (Well, almost fixed, see below about precession).&lt;/p&gt;&#xA;&lt;p&gt;All other motions in the sky are referred to this sphere of fixed stars. (So mathematically it is completely equivalent to diurnal rotation of the Earth).&lt;/p&gt;&#xA;&lt;p&gt;Sun moves on the ecliptic, which is the great circle (fixed in the sphere of fixed stars) which is inclined to the equator. The period of this rotation is 1 year, and it is slightly non-uniform (&amp;quot;inequality of the Sun&amp;quot;).&lt;/p&gt;&#xA;&lt;p&gt;This theory gives a perfect account of the seasons which is essentially the same as our modern explanation. (Contrary to what @Conifold says in his comment). Ecliptic intersects&#xA;the equator at two points which are called the Spring and Autumn equinoxes. In the middles of the corresponding arcs&#xA;of the ecliptic there are two points at the greatest distance from equator. They are called the winter and summer solstices. The seasons are the intervals between&#xA;these 4 points. It is an early discovery (before Hipparchus) that the seasons have unequal lengths. The conclusion is that the motion of the sun along the ecliptic is slightly non-uniform. This was called the Inequality of the Sun, and it corresponds in modern astronomy to the first and second Kepler's laws).&lt;/p&gt;&#xA;&lt;p&gt;So it is hotter in summer, because a) the rays of the sun hit the Earth under bigger angle, and b) the days are longer).&lt;/p&gt;&#xA;&lt;p&gt;In addition to this, the position of the equator (and thus of the poles) with respect to the stars and ecliptic is not exactly fixed; the points of  equinoxes move with the period roughly 1 degree per century. This motion is called precession.&#xA;(This is an ancient estimate, modern speed of precession is 1 deg per 71.6 years).&lt;/p&gt;&#xA;&lt;p&gt;This is a more or less complete description of the Ptolemy's theory about Earth and Sun. Mathematically it is EQUIVALENT to the modern description. The difference between Ptolemy's and Kepler's description only concerns&#xA;the motion of the Moon and planets.&lt;/p&gt;&#xA;&lt;p&gt;Another difference, between the &amp;quot;Sun inequality&amp;quot; and the first two Kepler laws, is so small that it was undistinguishable for the accuracy of observations&#xA;18th century. For planets, Ptolemy theory also gives very good results. The inadequacy of his description was only detected in the end of 16th century as a result of improvement of observation accuracy (of Tycho Brahe). The most difficult case is that of the Moon. There was a continuous progress in describing the Moon motion since Ptolemy to the beginning of 20th century when a theory consistent with all observations was finally achieved.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-02-26T14:24:02.567" LastActivityDate="2024-02-26T14:24:02.567" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17269" PostTypeId="1" AcceptedAnswerId="17277" CreationDate="2024-02-26T15:52:42.077" Score="26" ViewCount="8375" Body="&lt;p&gt;In the film &amp;quot;The Imitation Game&amp;quot; Alan Turing, while being interviewed at Bletchley Park, confesses that he doesn't speak German, which almost makes him fail the interview. I think I read somewhere else (Simon Singh?) that breaking Enigma was one of the earliest examples when cryptoanalysis required more math than language knowledge, so this scene might emphasize that fact.&lt;/p&gt;&#xA;&lt;p&gt;However, I find it hard to believe that Turing didn't know German at all. For example, in his earlier work on &lt;em&gt;Entscheidungsproblem&lt;/em&gt;, Turing refers to Gödel's papers, which were not translated to English until 1960s (quoting C. Petzold).&lt;/p&gt;&#xA;&lt;p&gt;So my question: is there any evidence about Turing's knowledge of German?&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Update&lt;/em&gt;: when I was asking this question, this was the only scene of the film I had seen (in a trailer). Now that I have seen the whole movie, I realize how stupid the question was. I mean, that was likely the least of all the inaccuracies.&lt;/p&gt;&#xA;" OwnerUserId="20763" LastEditorUserId="20763" LastEditDate="2024-04-04T09:22:05.327" LastActivityDate="2024-04-04T09:22:05.327" Title="Did Alan Turing know the German language?" Tags="|language|computer-scientists|turing|" AnswerCount="4" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17270" PostTypeId="2" ParentId="17269" CreationDate="2024-02-26T16:52:53.427" Score="14" Body="&lt;p&gt;This is not really evidence, but according to &lt;a href=&quot;https://www.ams.org/notices/200910/rtx091001236p.pdf&quot; rel=&quot;noreferrer&quot;&gt;https://www.ams.org/notices/200910/rtx091001236p.pdf&lt;/a&gt; according to Hodges,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[Andrew] Gleason took Alan [Turing] to a crowded restaurant on 18th Street.&#xA;They were sitting at a table for two, just a few&#xA;inches from the next one, and talking of statistical&#xA;problems, such as that of how to best estimate the&#xA;total number of taxicabs in a town, having seen a&#xA;random selection of their license numbers. The&#xA;man at the next table was very upset by hearing&#xA;this technical discussion, which he took to be a&#xA;breach of ‘security’. … Alan said, ‘Shall we continue&#xA;our conversation in German?'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hodges gives this story, on p.246 of the 1983 edition of his book. He does not give a reference.  He does not mention  Gleason in his list of sources.&lt;/p&gt;&#xA;" OwnerUserId="6229" LastEditorUserId="6229" LastEditDate="2024-02-26T17:10:46.470" LastActivityDate="2024-02-26T17:10:46.470" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17271" PostTypeId="2" ParentId="17269" CreationDate="2024-02-26T20:03:49.070" Score="23" Body="&lt;p&gt;Alan Turing's &lt;a href=&quot;http://oldshirburnian.org.uk/wp-content/uploads/2015/01/Transcripts-of-Turings-School-Reports-1926-1931.pdf&quot; rel=&quot;noreferrer&quot;&gt;report cards&lt;/a&gt; from &lt;a href=&quot;https://en.wikipedia.org/wiki/Sherborne_School&quot; rel=&quot;noreferrer&quot;&gt;Sherborne School&lt;/a&gt; indicate that he studied German in 1930 and 1931, but his teacher Geoffrey John Bromehead Watkins noted in 1931 that&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;He does not seem to have any aptitude for languages.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="4201" LastActivityDate="2024-02-26T20:03:49.070" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17272" PostTypeId="1" AcceptedAnswerId="17273" CreationDate="2024-02-26T20:47:57.697" Score="3" ViewCount="71" Body="&lt;p&gt;In 1851, in one of the most famous publications in the history of thermodynamics, &lt;em&gt;On the Dynamical Theory of Heat, with numerical remits deduced from Mr JOULE'S equivalent of a Thermal Unit. etc.&lt;/em&gt;,  Thomson (later Lord Kelvin) pronounced his version of the 2nd law as follows:&#xA;&amp;quot;It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Some 45 years later in 1897, Planck in the 1st edition of his book Treatise on Thermodynamics he phrased the 2nd law as: &amp;quot;It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir,&amp;quot;&#xA;without even hinting at an &lt;em&gt;inanimate&lt;/em&gt; material agency in this formulation.&lt;/p&gt;&#xA;&lt;p&gt;I assume in 1851 it was still normal to accept some form of vitalism, hence Thomson's restricting his 2nd law to inanimate agencies but not any more 45 years later. What and when the change happened to reject vitalism in physics? When did Thomson himself reject vitalism explicitly, did he ever?&lt;/p&gt;&#xA;" OwnerUserId="7908" LastEditorUserId="55" LastEditDate="2024-02-27T06:14:41.507" LastActivityDate="2024-02-27T08:19:10.733" Title="When was Kelvin's vitalism rejected in physics?" Tags="|biology|thermodynamics|energy|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17273" PostTypeId="2" ParentId="17272" CreationDate="2024-02-27T01:12:07.993" Score="3" Body="&lt;p&gt;It was normal to accept some form of vitalism way past 1851, and it was still around at the time of Planck's writing in 1897. In fact, vitalism experienced a resurgence in 1870s in response to the apparent tension between it and the energy conservation law, and somewhat crude applications of mechanics to physiology by du Bois-Reymond and others. Kelvin himself &lt;a href=&quot;https://www.lesswrong.com/posts/6i3zToomS86oj9bS6/mysterious-answers-to-mysterious-questions#fn1x38&quot; rel=&quot;nofollow noreferrer&quot;&gt;reaffirmed his vitalism&lt;/a&gt; in On the Dissipation of Energy: Geology and General Physics (1894). Bergson, who kept current on physics of his time and later debated Einstein, defended vitalism in &lt;a href=&quot;https://en.wikipedia.org/wiki/Creative_Evolution_(book)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Creative Evolution (1907)&lt;/a&gt;, as did Haldane in &lt;a href=&quot;https://www.forgottenbooks.com/it/download/MechanismLifeandPersonality_10048677.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mechanism, life and personality (1913)&lt;/a&gt;. Earlier, in 1870s, some big names in physics, like Maxwell and Boussinesq, argued for compatibility of vitalism with physics.&lt;/p&gt;&#xA;&lt;p&gt;Although vitalism was not fully abandoned by scientists until 1920s, in 1870s mechanistic determinism &lt;em&gt;a la&lt;/em&gt; Laplace started to gain ground. By 1890, the vitalism/determinism debate subsided and 'working physicists', like Planck, preferred to steer clear of the topic.&lt;/p&gt;&#xA;&lt;p&gt;The history of the debate is discussed in &lt;a href=&quot;https://www.tandfonline.com/doi/10.1080/00033790.2014.935954&quot; rel=&quot;nofollow noreferrer&quot;&gt;Strien, Vital Instability: Life and Free Will in Physics and Physiology&lt;/a&gt;, &lt;a href=&quot;https://www.academia.edu/en/66365381/Unexpected_Convergence_between_Science_and_Philosophy_A_debate_on_determinism_in_France_around_1880&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bordoni, Unexpected Convergence between Science and Philosophy&lt;/a&gt; and &lt;a href=&quot;https://www.jstor.org/stable/2709176&quot; rel=&quot;nofollow noreferrer&quot;&gt;Hacking, Nineteenth Century Cracks in the Concept of Determinism&lt;/a&gt;. For the 'free will' aspects of it see &lt;a href=&quot;https://hsm.stackexchange.com/a/2680/55&quot;&gt;History of the study of indeterminism in classical mechanics&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Here is from Strien:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;During the period 1860–1880, a number of physicists and mathematicians, including&#xA;Maxwell, Stewart, Cournot and Boussinesq, used theories formulated in terms of&#xA;physics to argue that the mind, the soul or a vital principle could have an impact on the&#xA;body. This paper shows that what was primarily at stake for these authors was a&#xA;concern about the irreducibility of life and the mind to physics, and that their theories&#xA;can be regarded primarily as reactions to the law of conservation of energy, which was&#xA;used among others by Helmholtz and Du Bois-Reymond as an argument against the&#xA;possibility of vital and mental causes in physiology. In light of this development,&#xA;Maxwell, Stewart, Cournot and Boussinesq showed that it was still possible to argue&#xA;for the irreducibility of life and the mind to physics, through an appeal to instability or&#xA;indeterminism in physics: if the body is an unstable or physically indeterministic&#xA;system, an immaterial principle can act through triggering or directing motions in the&#xA;body, without violating the laws of physics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Here is Maxwell's take from 1879, which sheds some light on the background of his famous &lt;a href=&quot;https://en.wikipedia.org/wiki/Maxwell%27s_demon&quot; rel=&quot;nofollow noreferrer&quot;&gt;thermodynamic demon&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Science has thus compelled us to admit that that which distinguishes a living body&#xA;from a dead one is neither a material thing, nor that more refined entity, a ‘form of&#xA;energy’. There are methods, however, by which the application of energy may be&#xA;directed without interfering with its amount. Is the soul like the engine-driver, who&#xA;does not draw the train himself, but, by means of certain valves, directs the course&#xA;of the steam so as to drive the engine forward or backward, or to stop it?&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Hacking adds more on the rise of determinism:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Cassirer was wrong to say that our modern universal&#xA;body-and-mind causal determinism reared its head for the first time in&#xA;1872, but he pointed to a remarkable ferment in the concept of determinism&#xA;that reached a climax at about that time. The very word&#xA;''determinism'' became generally current in French and English just&#xA;before then, and in German it attained Cassirer's sense of the word&#xA;during the same period. The literature has been full of zany arguments&#xA;in which bits of mathematics of physics were pulled down from the&#xA;shelf to make room for freedom of the will. Buckle, whom we have&#xA;largely forgotten, turned the whole of history into the evolution of&#xA;statistical determinism. Exactly ten years later, a much more famous&#xA;work by Marx would speak of ''the natural laws of capitalist&#xA;production... of these tendencies winning their way through and&#xA;working themselves out with iron necessity.'' I believe that Cassirer&#xA;almost inadvertently pointed to a period of tumult and chaos in&#xA;thought about necessity and determinism.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-02-27T08:19:10.733" LastActivityDate="2024-02-27T08:19:10.733" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17274" PostTypeId="2" ParentId="16250" CreationDate="2024-02-27T06:28:54.400" Score="1" Body="&lt;p&gt;Indirect uses for transportation: &amp;quot;Petroleum is not a recent discovery, it was known to various ancient peoples in Asia, North Africa, Europe, and America where it seeped to the surface and was used in various forms for at least 70,000 years. It was most widely used to &lt;strong&gt;caulk ships, pave roads&lt;/strong&gt;...&amp;quot; (&lt;a href=&quot;https://link.springer.com/referenceworkentry/10.1007/978-3-319-02330-4_38-1#:%7E:text=During%20the%20Classical%20Period%2C%20the,the%20bitumen%20deposits%20of%20the&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Oil: Regions and Uses of Petroleum in the Classical and Medieval Periods&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-02-27T06:28:54.400" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17275" PostTypeId="1" CreationDate="2024-02-27T09:17:36.120" Score="4" ViewCount="66" Body="&lt;p&gt;I am looking for the etymology of &lt;em&gt;matrice creuse&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;p&gt;According to Wikipedia, it seems &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Joseph_Sylvester#Legacy&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Joseph Sylvester&lt;/a&gt; used the term &amp;quot;matrix&amp;quot; in 1850, and &lt;a href=&quot;https://en.wikipedia.org/wiki/Harry_Markowitz&quot; rel=&quot;nofollow noreferrer&quot;&gt;Harry Markowitz&lt;/a&gt; used the term &amp;quot;sparse matrix&amp;quot; about a century later.&lt;/p&gt;&#xA;&lt;p&gt;In French, it seems that you can translate it by &lt;em&gt;matrices creuses&lt;/em&gt; (&amp;quot;hollow&amp;quot; matrices) and it seems more prevalent than &lt;em&gt;matrices éparses&lt;/em&gt; (&amp;quot;sparse&amp;quot; matrices), which would be a more accurate translation (I believe).&lt;/p&gt;&#xA;&lt;p&gt;Here are my questions :&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;I have looked on Google ngrams for some expressions and checked books by hand link by link, but I do not understand why &lt;em&gt;matrice creuse&lt;/em&gt; seems more prevalent than &lt;em&gt;matrice éparse&lt;/em&gt; in French.&lt;/li&gt;&#xA;&lt;li&gt;I think I can understand why the term &lt;em&gt;creuse&lt;/em&gt; is used in French, but if possible I would want confirmation that is it because all the 0s form a &amp;quot;hole&amp;quot; in the 2D notation of the matrix and the adjective does not refer to another thing; and that the term &lt;em&gt;matrice creuse&lt;/em&gt; was initially employed to describe exactly this idea.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;EDIT: I've looked too for &lt;em&gt;vecteur creux&lt;/em&gt; (sparse vectors) but I did not find conclusive results either.&#xA;I have seen some mentions of &lt;em&gt;vecteur creux&lt;/em&gt; in French texts.&lt;/p&gt;&#xA;" OwnerUserId="20765" LastEditorUserId="6609" LastEditDate="2024-02-28T14:42:48.177" LastActivityDate="2024-02-28T14:42:48.177" Title="Sparse matrix (&quot;matrice creuse&quot;) etymology in French" Tags="|terminology|linear-algebra|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17276" PostTypeId="1" CreationDate="2024-02-27T14:36:41.447" Score="0" ViewCount="103" Body="&lt;p&gt;They knew the Earth was a sphere, and Eratosthenes even computed the radius. Menelaus and possibly Hipparchus knew that the angles inside a triangle add to more than &lt;span class=&quot;math-container&quot;&gt;$\pi$&lt;/span&gt; on a sphere.&lt;/p&gt;&#xA;&lt;p&gt;Did they have something like the concept of curvature? Did they know the perimeter and area of the disc on a sphere? Did they know anything about hyperbolic geometry?&lt;/p&gt;&#xA;" OwnerUserId="19593" LastEditorUserId="12330" LastEditDate="2024-02-28T08:12:24.183" LastActivityDate="2024-02-28T08:12:24.183" Title="How much did the ancient Greeks know of non-Euclidean geometry?" Tags="|ancient-greece|non-euclidean-geometry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17277" PostTypeId="2" ParentId="17269" CreationDate="2024-02-27T18:41:08.250" Score="35" Body="&lt;p&gt;By the time he was at university, Alan Turing had sufficient facility at reading German (in gothic type no less) that he was reading advanced texts on Quantum Mechanics. In chapter 4 of the biography &amp;quot;Prof: Alan Turing Decoded&amp;quot; in the section &lt;em&gt;The domination of the Germans&lt;/em&gt;, Sir Dermot Turing notes&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of the prize books awarded to Alan by Sherborne was &lt;em&gt;Mathematische Grundlagen der Quantenmechanik&lt;/em&gt; (Mathematical Foundations of Quantum Mechanics) by John von Neumann, whom we will encounter again. [...] Alan reported that von Neumann was 'very interesting reading, although the applied mathematicians seem to find it rather strong meat'.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;It was strong meat because it was in German, a language in which Alan had demonstrated his lack of aptitude for languages at Sherborne; but if you were serious about maths in the 1930s, you had to be able to get on with mathematical German.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The book can be seen at the Bletchley Park museum and is clearly well-used, including tea stains next to important results.&lt;/p&gt;&#xA;&lt;p&gt;Moreover, the cryptanalytic community at Bletchley Park adopted various cod-Germanic words in their jargon. Among these are &lt;a href=&quot;https://en.wikipedia.org/wiki/Turingery&quot; rel=&quot;nofollow noreferrer&quot;&gt;Turingismus&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Banburismus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Banburismus&lt;/a&gt; which are attributed to Turing.&lt;/p&gt;&#xA;&lt;p&gt;I'm also told by classified historians that the tri-partite liaison meetings between the British, French and Polish cryptanalysts on Enigma (at some of which, Turing was present) were conducted in German as it was the common language that all of the cryptanalysts knew. This is confirmed in part by chapter 6 of &lt;em&gt;Herivelismus&lt;/em&gt; by John Herivel.&lt;/p&gt;&#xA;&lt;p&gt;Post-war there is an &lt;a href=&quot;https://www.research-collection.ethz.ch/bitstream/handle/20.500.11850/69357/eth-6960-01.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;elusive story&lt;/a&gt; that Turing was part of a group that visited Germany to understand Konrad Zuse's progress in developing a computer. Accounts say that the &amp;quot;interrogation&amp;quot; took the form of  a colloquium.&lt;/p&gt;&#xA;&lt;p&gt;It's known that Turing could occasionally stutter, which might have limited his ability to demonstrate fluency in a secondary school classroom, but it is clear that his command of written German was very strong indeed.&lt;/p&gt;&#xA;&lt;p&gt;There are many things to commend about &lt;em&gt;The Imitation Game&lt;/em&gt;, but it is not intended to be a documentary.&lt;/p&gt;&#xA;" OwnerUserId="20771" LastEditorUserId="9523" LastEditDate="2024-03-26T05:45:43.847" LastActivityDate="2024-03-26T05:45:43.847" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17278" PostTypeId="2" ParentId="14119" CreationDate="2024-02-27T19:55:20.957" Score="0" Body="&lt;p&gt;In this case B would still light up.&lt;/p&gt;&#xA;&lt;p&gt;In Enigma encryption the Steckerbrett is passed through twice: once immediately after input and once immediately before output. In your example, the initial depression of the 'A' key is reinterpreted by the Steckerbrett as depressing a 'B' on an unsteckered Enigma. The unsteckered Enigma would then encrypt this as 'A' (because Enigma encryption is reciprocal) before returning to the Steckerbrett where the 'A' would be reinterpreted as 'B'. Thus the 'B' bulb would light up.&lt;/p&gt;&#xA;&lt;p&gt;Note that the connectors on the Steckerbrett consisted of two wires with two connectors at each end:&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/e1LDs.jpg&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/e1LDs.jpg&quot; alt=&quot;Enigma Steckerbrett&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;one wire would connect the 'A' bulb/key to the 'B' Eintrittwaltz and the other would connect the 'B' bulb/key to the 'A' Eintrittwaltz so that in your case both wires would be live with current flowing in opposite directions.&lt;/p&gt;&#xA;" OwnerUserId="20771" LastEditorUserId="20771" LastEditDate="2024-02-28T07:09:53.557" LastActivityDate="2024-02-28T07:09:53.557" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17279" PostTypeId="1" CreationDate="2024-02-28T13:01:27.033" Score="4" ViewCount="183" Body="&lt;p&gt;This question is a follow-up to &lt;a href=&quot;https://hsm.stackexchange.com/questions/17276/how-much-did-the-ancient-greeks-know-of-non-euclidean-geometry&quot;&gt;How much did the ancient Greeks know of non-Euclidean geometry?&lt;/a&gt;, where it was claimed in a &lt;a href=&quot;https://hsm.stackexchange.com/questions/17276/how-much-did-the-ancient-greeks-know-of-non-euclidean-geometry#comment38114_17276&quot;&gt;comment&lt;/a&gt; that &amp;quot;the earliest inklings [of hyperbolic geometry] came from medieval Islamic mathematicians.&amp;quot;  Is there any evidence for such a claim?  I am hoping for something more serious than attributing the earliest inklings of group theory to the Maians because they made pots possessing reflectional symmetry.  As I suspect, some such claims may have been made by Rashed who also attributed infinitesimal calculus to scholars far earlier than is generally considered reliable, but I am looking forward to being surprised by some solid evidence.&lt;/p&gt;&#xA;&lt;p&gt;For example, Yushkevich died in 1993.  In 1996, Rashed included an article by Yushkevich in a volume he published.  If Yushkevich held such sentiments, they would presumably appeared in articles he published while alive.  Note that he was born in 1906 and had plenty of oportunity to do so (he had hundreds of publications listed in MathSciNet alone).&lt;/p&gt;&#xA;&lt;p&gt;Rozenfeld and Yushkevich have a book appropriately entitled &amp;quot;theory of parallels in medieval East&amp;quot; (no mention of hyperbolic geometry).  Here is the relevant part of a MathSciNet review:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The scientists of the Arabic caliphate already at the time of caliph al-Ma'moun (813–833) realized a shortcoming in the Elements of Euclid. The first in the caliphate to pay attention to the peculiar role of the fifth postulate of Euclid was the great scholar of Central Asia, al-'Abbas al-Jauhari , member of the Academy of al-Ma'moun. Later on the same attempts were made but were unsuccessful. Perhaps the greatest level of the theory of parallel lines in the East was reached in the works of 'Omar Khayyam and Nasir ad-Din at-Tusi . The information from these works was adopted by European scientists of the XVI–XVIII centuries. More or less all of this is fully expounded in the book under review.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;So we are apparently talking about attempts to prove the parallel postulate (rather than either hyperbolic geometry or non-Euclidean geometry).&lt;/p&gt;&#xA;&lt;p&gt;If there is any evidence of &amp;quot;inklings of hyperbolic geometry&amp;quot; I am still interested in seeing it.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Proclus&quot; rel=&quot;nofollow noreferrer&quot;&gt;Proclus&lt;/a&gt; dealt with the theory of parallels centuries earlier, so if anything &amp;quot;the earliest inklings&amp;quot; are misattributed.&lt;/p&gt;&#xA;&lt;p&gt;I have a 1912 translation of Bonola's book, translated by Carlslaw.&#xA;Bonola discusses Proclus starting on page 2.  On page 3, Bonola&#xA;mentions that Proclus refers to a work by Geminus from 1st century&#xA;BCE.  By page 7, Bonola gets to the Arabian Commentary of Al-Nirizi&#xA;(9th Century) ... and is attributed to Aganis.  In footnote 3 on page&#xA;7, Bonola mentions a scholarly disagrement about whether Aganis is&#xA;Geminus or not.  On pages 9-10 Bonola emphasizes the derivative nature&#xA;of the work of Al-Nirizi.  On page 10, Bonola deals with Nasir-Eddin's&#xA;proof of the parallel postulate.  All this material is in a chapter I&#xA;entitled &amp;quot;attempts to prove the parallel postulate.&amp;quot;  Chapter II,&#xA;entitled &amp;quot;Forerunners of Non-Euclidean geometry&amp;quot;, starts with Saccheri.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2024-03-05T14:21:49.717" LastActivityDate="2024-03-05T14:21:49.717" Title="Did the earliest inklings of hyperbolic geometry come from medieval Islamic mathematicians?" Tags="|islamic-science|non-euclidean-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17280" PostTypeId="1" AcceptedAnswerId="17284" CreationDate="2024-02-28T16:17:48.817" Score="2" ViewCount="141" Body="&lt;p&gt;In the past, I came across a very elegant direct definition (below) of a function, which is based on the fundamental concepts of triples, pairs, and sets. However, I find it difficult to search the internet for any citations that I could refer to. One professor once told me that this definition could be over 200 years old, and it might be difficult to determine the author. But I would be satisfied with even citations from reliable sources where this definition was presented or used. Do any of you perhaps recognize any works of this type and could help me?&lt;/p&gt;&#xA;&lt;p&gt;The &lt;strong&gt;function&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$f= (X,Y,G)$&lt;/span&gt; is an ordered triple consisting of the following elements:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;a &lt;strong&gt;domain&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$X$&lt;/span&gt; that is any set&lt;/li&gt;&#xA;&lt;li&gt;a &lt;strong&gt;codomain&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$Y$&lt;/span&gt; that is also any set&lt;/li&gt;&#xA;&lt;li&gt;a &lt;strong&gt;graph&lt;/strong&gt; &lt;span class=&quot;math-container&quot;&gt;$G \subseteq X\times Y$&lt;/span&gt; being a set of pairs, such that &lt;span class=&quot;math-container&quot;&gt;$\forall x \in X, \exists ! y\in Y : (x,y) \in G$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;(&lt;span class=&quot;math-container&quot;&gt;$\exists !$&lt;/span&gt; means &lt;a href=&quot;https://en.wikipedia.org/wiki/Uniqueness_quantification&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;There exists exactly one&amp;quot;&lt;/a&gt;)&lt;/p&gt;&#xA;" OwnerUserId="20782" LastEditorUserId="3462" LastEditDate="2024-02-29T14:47:27.883" LastActivityDate="2024-02-29T14:47:27.883" Title="Origin of modern definition of a function as a graph" Tags="|mathematics|definition|" AnswerCount="2" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17281" PostTypeId="2" ParentId="17280" CreationDate="2024-02-29T08:43:37.090" Score="3" Body="&lt;p&gt;This is definitely not 200 years old. The first definitions of the &lt;em&gt;modern&lt;/em&gt; function concept are from Dedekind, Cantor, Peano and Frege from around 1890, and even those definitions are not encoded inside the formalism of set theory as graphs. Beware that the original meaning assigned to &amp;quot;function&amp;quot; (by Leibniz, Bernoulli, Euler etc.) was something else.&lt;/p&gt;&#xA;&lt;p&gt;See this question and answers for some details on this:&#xA;&lt;a href=&quot;https://hsm.stackexchange.com/q/6104/3462&quot;&gt;Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;I suspect that the definition of a function as a special subset of a product set occurred after 1900 (I'd probably start looking in Hausdorff, Zermelo or von Neumann)&lt;/p&gt;&#xA;" OwnerUserId="3462" LastActivityDate="2024-02-29T08:43:37.090" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17282" PostTypeId="1" AcceptedAnswerId="17287" CreationDate="2024-02-29T08:55:12.253" Score="2" ViewCount="101" Body="&lt;p&gt;I am curious about the origin and history of the doubly periodic function.  I was looking for the Eisenstein series, and the Eisenstein series is one of the representative examples of a doubly periodic function.  So I became curious about the origin and history of this doubly periodic function.&lt;/p&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="55" LastEditDate="2024-02-29T23:40:39.103" LastActivityDate="2024-03-01T11:59:19.367" Title="History of doubly periodic functions" Tags="|elliptic-functions|modular-forms|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17283" PostTypeId="2" ParentId="17279" CreationDate="2024-02-29T09:18:14.990" Score="3" Body="&lt;p&gt;From Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), &amp;quot;Geometry&amp;quot;, in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447–494 [470], Routledge, London and New York:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Three scientists, Ibn al-Haytham, [Omar] Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose &lt;strong&gt;importance came to be completely recognized only in the 19th century.&lt;/strong&gt; In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, &lt;strong&gt;embodied the first few theorems of the hyperbolic and the elliptic geometries.&lt;/strong&gt; Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By &lt;strong&gt;their works on the theory of parallel lines&lt;/strong&gt; Arab mathematicians directly influenced the relevant investigations of their European counterparts.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;(emphasis mine, extracted from Wikipedia transcription of the reference in &lt;a href=&quot;https://en.wikipedia.org/wiki/Hyperbolic_geometry&quot; rel=&quot;nofollow noreferrer&quot;&gt;hyperbolic geometry&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;These Islamic scholars were not trying to come up with hyperbolic geometry in the sense that this was not invented yet. Khayyam was trying to prove the parallel postulate and in doing so he came up with things like the &lt;a href=&quot;https://en.wikipedia.org/wiki/Saccheri_quadrilateral&quot; rel=&quot;nofollow noreferrer&quot;&gt;Saccheri quadrilateral&lt;/a&gt;, foreshadowing results that would be found later with the development of non-Euclidean geometry.&lt;/p&gt;&#xA;&lt;p&gt;I think that your concern that the editor misinterpreted Yushkevich is just overcaring about the terminology. The paragraph above is not saying that Khayyam (or any other for that matter) went against his time and created hyperbolic geometry. This was not the case. It is just a statement that when his findings are reworked in modern terms, they could indeed be thought of as &amp;quot;emboidying&amp;quot; the &amp;quot;first few theorems of the hyperbolic and the elliptic geometries&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;In the end it all depends on how close has a development to be in order to be considered an &amp;quot;inkling&amp;quot;. As you say for that matter, Proclus did a commentary on the parallel postulate before Medieval Scholars, should he be the considered to be first inkling? The threshold of what counts or not as an &amp;quot;inkling&amp;quot; is in the eye of the beholder.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-03-04T10:40:35.523" LastActivityDate="2024-03-04T10:40:35.523" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17284" PostTypeId="2" ParentId="17280" CreationDate="2024-02-29T11:17:22.357" Score="5" Body="&lt;p&gt;The precise definition can be found into Bourbaki's &lt;a href=&quot;http://tomlr.free.fr/Math%E9matiques/Bourbaki/Theorie%20Des%20Ensembles.pdf&quot; rel=&quot;noreferrer&quot;&gt;Elements of Mathematics: Theory of sets (1968; but the 1st French edition is dated 1939)&lt;/a&gt;, page 76:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;A correspondence between a set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; and a set &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; is a triple &lt;span class=&quot;math-container&quot;&gt;$\Gamma = (G,A,B)$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is a &lt;em&gt;graph&lt;/em&gt; (a set of ordered pairs), &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; is the source, and &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; the target of &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Domain&lt;/em&gt; and &lt;em&gt;range&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt; are defined usually.&lt;/p&gt;&#xA;&lt;p&gt;The basic &amp;quot;ingredients&amp;quot; (as per Michael's answer above) can be found in the early 20th Century textbooks on set theory; see e.g. Felix Hausdorff, &lt;a href=&quot;https://www.google.it/books/edition/Set_Theory/TFA_EAAAQBA&quot; rel=&quot;noreferrer&quot;&gt;Set theory&lt;/a&gt;, translation of the 3rd German ed 1937 (1st edition as &lt;a href=&quot;https://en.wikipedia.org/wiki/Grundz%C3%BCge_der_Mengenlehre&quot; rel=&quot;noreferrer&quot;&gt;&lt;em&gt;Grundzüge der Mengenlehre&lt;/em&gt; (1914)&lt;/a&gt;), page 16:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;Ordered pairs&lt;/em&gt; make possible the introduction of the concept of function [...] Let &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; a set of ordered pairs &lt;span class=&quot;math-container&quot;&gt;$p=(a,b)$&lt;/span&gt;; for every pair &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; that occurs in &lt;span class=&quot;math-container&quot;&gt;$P$&lt;/span&gt; (&lt;span class=&quot;math-container&quot;&gt;$p \in P$&lt;/span&gt;), we call &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; an &lt;em&gt;image&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; a &lt;em&gt;pre-image&lt;/em&gt; (or &lt;em&gt;inverse image&lt;/em&gt;) of &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;. Let &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt; the set of all pre-images &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$B$&lt;/span&gt; the set of all images &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt; [...][ we say that we have a &lt;em&gt;mapping&lt;/em&gt; from one set onto the other.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In the particular case in which each &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; has only a single image &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, we denote this element &lt;span class=&quot;math-container&quot;&gt;$b$&lt;/span&gt;, which is determined by &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; and depends on &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; by &lt;span class=&quot;math-container&quot;&gt;$b=f(a)$&lt;/span&gt;, and we say that this is a &lt;em&gt;single-valued&lt;/em&gt; or &lt;em&gt;univalent function&lt;/em&gt; of &lt;span class=&quot;math-container&quot;&gt;$a$&lt;/span&gt; defined on the set &lt;span class=&quot;math-container&quot;&gt;$A$&lt;/span&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In conclusion&lt;/em&gt;, the definition is &lt;strong&gt;100 years old&lt;/strong&gt;.&lt;/p&gt;&#xA;" OwnerUserId="131" LastActivityDate="2024-02-29T11:17:22.357" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17285" PostTypeId="2" ParentId="16211" CreationDate="2024-02-29T14:04:56.377" Score="2" Body="&lt;p&gt;&lt;em&gt;This is a new rewritten answer, after diving more into the contemporary references and finding some modern references. I deleted my old answer, which had information about black body radiation experiments, because this is essentially a new answer.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;hr /&gt;&#xA;&lt;p&gt;By the 1850s, the idea that heat radiation was the same as light was gaining increasing favor. It obeyed the same properties as light radiation (reflection, refraction, polarization, double refraction, interference) and sources of heat radiation were always closely connected with sources of light. Prior to these developments, heat radiation was understood phenomenologically, although its connection to light was suspected from the late 18th century. An excellent reference for the history up to the year 1800 is:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;E.S. Cornell M.Sc. (1936) &lt;em&gt;Early studies in radiant heat&lt;/em&gt;, Annals of Science, 1:2, 217-225, DOI: &lt;a href=&quot;https://doi.org/10.1080/00033793600200171&quot; rel=&quot;nofollow noreferrer&quot;&gt;10.1080/00033793600200171&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From the conclusion to that paper:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Thus, before 1800, the existence of invisible heat rays had been generally accepted. It was known that these rays were propagated practically instantaneously and in straight lines, and were reflected according to the same law as that followed by light rays. It was suspected that they could also be refracted like light. &lt;strong&gt;Because of these similarities between the rays of heat and light, a close connection between the two was suspected, and [James] Hutton went so far as to maintain that the invisible heat rays were invisible light. The majority of scientists, however, bearing in mind the different absorption effects of glass on the rays of heat and light from the fire, now well known, preferred to reserve their judgment&lt;/strong&gt; and used the terms &amp;quot;Obscure Heat&amp;quot; or &amp;quot;Radiant Heat&amp;quot; or &amp;quot;Radiant Caloric&amp;quot; for the invisible heat rays. There were two theories prevalent as to the nature of the invisible heat rays: -- (1) That they were a material emanation, and (2) that they were vibrations in an all-pervading heat fluid. Of these, the former was the more commonly held. Moreover, the reflection of cold had been clearly demonstrated, and satisfactorily explained by Prevost's theory.&#xA;This was the state of the subject when Sir William Herschel made his discovery of the ifra-red heating rays in the solar spectrum in 1800. (p. 225, emphasis mine)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Herschel's discovery of radiant heat in the sun's spectrum led to the publication of &lt;em&gt;Experiments on the solar, and on the terrestrial Rays that occasion Heat; with a comparative View of the Laws to which Light and Heat, or rather the Rays which occasion them, are subject, in order to determine whether they are the same, or different.&lt;/em&gt; (Phil. Trans. 90, 1800) (&lt;a href=&quot;https://babel.hathitrust.org/cgi/pt?id=pst.000054592520&amp;amp;view=1up&amp;amp;seq=357&quot; rel=&quot;nofollow noreferrer&quot;&gt;link to part 1 on HathiTrust&lt;/a&gt;). He says in the introduction,&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I must also remark, that in using the word rays, I do not mean to oppose, much less to countenance, the opinions of those philosophers who still believe that light itself comes to us from the sun, not by rays, but by the supposed vibrations of an elastic ether, every where diffused throughout space; I only claim the same privilege for the rays that occasion heat, which they are willing to allow to those that illuminate objects. For, in what manner soever this radiance may be effected, it will be fully proved hereafter, that &lt;strong&gt;the evidence, either for rays, or for vibrations which occasion heat, stands on the same foundation on which the radiance of the illuminating principle, light, is built&lt;/strong&gt;. (p. 294, emphasis mine)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This may seem like a decisive moment, but in fact the question was far from settled - discussions of Herschel's conclusion extended into the 1830s. The term &amp;quot;calorific rays&amp;quot; or &amp;quot;calorific radiation&amp;quot; was popular, if you'd like to dig up some sources from the period. See for example:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Powell, Baden (1825) &lt;em&gt;An Experimental Enquiry into the Nature of the Radiant Heating Effects from Terrestrial Sources&lt;/em&gt;. Phil. Trans. 1825 p. 187. (&lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rstl.1825.0011&quot; rel=&quot;nofollow noreferrer&quot;&gt;Link&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;It was the experiments between 1830 and 1850 performed by the likes of Ampere, Forbes, Melloni, and others, that helped the heat-rays-as-light theory gain favor. In the 1840s, Faraday was also proposing ideas that could be considered the beginnings of the electromagnetic theory of light. Simultaneously, the classical theory of thermodynamics was taking shape.&lt;/p&gt;&#xA;&lt;p&gt;From a practical perspective, the question of what heat rays &lt;em&gt;were&lt;/em&gt; (light, particles, other rays with their own aether, a combination of multiple effects) was less important than how the heat rays &lt;em&gt;behaved&lt;/em&gt; thermodynamically, in particular with regards to thermodynamic equilibrium and the second law of thermodynamics. Now hopefully the information in my previous answer will make more sense.&lt;/p&gt;&#xA;&lt;p&gt;Prevost (1791) postulated his theory of heat exchange, stating that every body radiates heat, and receives radiant heat from other bodies. From Pictet's experiments with heat radiation (1790), it was known that radiant heat could be reflected and focused, like light. Wollaston (1802), then later von Fraunhofer (1817), had discovered that the visible spectrum of the sun had dark lines (now called Fraunhofer lines), some of which aligned well with the spectral lines emitted by laboratory light sources. Sir John Herschel (1823) and William Hallows Miller (1833) experimented with passing the solar spectrum through gases and observing the resulting spectrum; additional dark lines which appeared were attributed to the absorption of radiation by the gases. Foucault (1849, 1860) demonstrated two important facts: (1) that an electric arc produces light of the same wavelength as what was known as the D line in the sun's spectrum, and (2) when light from a more uniform radiator was passed through this arc, a dark D line also appeared, indicating that the arc was both emitting and absorbing the same wavelength of radiation. Balfour Stewart (1858) performed experiments with rock salt and concluded that the radiating power of every kind of substance is equal to its absorbing power, for every kind of ray of radiant heat. [Whittaker, vol. 1, ch. XII].&lt;/p&gt;&#xA;&lt;p&gt;Indeed, theories of black-body radiation using Maxwell's theory of electromagnetic light didn't come about until the late 19th century into the early 20th century, with e.g. the Rayleigh-Jeans law, but by this time Hertz's experiments had already demonstrated the existence of electromagnetic waves and classical electrodynamics was reaching maturity.&lt;/p&gt;&#xA;" OwnerUserId="9528" LastActivityDate="2024-02-29T14:04:56.377" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17286" PostTypeId="2" ParentId="17269" CreationDate="2024-02-29T17:08:09.240" Score="-1" Body="&lt;p&gt;I suspect that Alun Turing had to read Published materials during his Academic career and prior to Bletchley by Authors most probably from German descent, some of which may not have been translated? If I had to guess, I would have thought Alun Turning may have been self educated in this regard, due to the nature of the area of Science and Mathematics fields he was learning at the time, this means there were other Mathematicians &amp;amp; scientists who recorded their works in their own languages, prior to those materials leaving academia and translated into English journals, which I guess would safely mean one would need to learn about, even if not in his native Language. This is all just antidotal, yet again another example is probably whilst in his famous role at Bletchley park working on the UTM I suspect due to the sheer amount of Foreign (German) intercepted communications received at the time which required deciphering and of the importance language was to understanding how to break a communication encryption device &amp;quot;Enigma&amp;quot; how could one not help but to pick up bits of the language here and there?, even in an unorthodox fashion, which over time, Alun Turing accumulated into a good understanding of the German Language. If this is not enough reasoning to allow for safe assumptions, outlined above as to the question:&#xA;&amp;quot;is there any evidence about Turing's knowledge of German?&amp;quot;&#xA;I would state Yes, almost certainly, Alun Turing learned German during his Efforts, maybe even prior to Bletchley Park. I would also assume Alun Turing would have been asked to take German lessons by his Superiors / Commanding Officer in an Official Capacity and like with all Signal Intelligence you learn from AND about your operating environment. These is all probable events which are safe to assume probably occurred. For a definitive answer I suspect the National Archive may hold the key as to the true answer inside either Alun Turnings War Records or even in Princeton University may hold some works or information about his background incl spoken languages Alun Turning was Fluent in.&lt;/p&gt;&#xA;" OwnerUserId="20788" LastActivityDate="2024-02-29T17:08:09.240" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17287" PostTypeId="2" ParentId="17282" CreationDate="2024-02-29T23:27:40.507" Score="4" Body="&lt;p&gt;Doubly periodic functions first appeared as inverses of elliptic integrals that were originally motivated by computing the arc length of the lemniscate, but later found uses in astronomy (elliptic orbits), mechanics (non-linearized pendulum), etc. Good sources on this part of history are &lt;a href=&quot;https://link.springer.com/book/10.1007/978-3-319-23715-2&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gray, The Real and the Complex&lt;/a&gt;, and shorter &lt;a href=&quot;https://kurser.math.su.se/pluginfile.php/16103/mod_folder/content/0/2019/2019_K4_report.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Mossiaguine, A Brief History of Elliptic Functions&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Elliptic integrals were studied by Fagnano, Euler and Legendre in the 18th century, but without any of them noticing that the inverse functions were doubly periodic. According to Gray, Gauss discovered that when he inverted the lemniscate integral in 1797, and then for general integrals in 1800.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Gauss remarked in May 1800 (Diary entry nr. 108) that the new elliptic functions&#xA;are doubly periodic and have zeros and poles and can be expanded as quotients of&#xA;power series. His approach was two-fold: in one direction he inverted an elliptic&#xA;integral depending on a parameter &lt;span class=&quot;math-container&quot;&gt;$k$&lt;/span&gt;; in the other he developed doubly periodic&#xA;functions directly as quotients of entire functions. This raised the question of whether the two approaches define exactly the same objects. It was easy enough for Gauss to mimic the arguments for the lemniscatic integral in the general case, and conclude that the general elliptic function can be represented as a quotient. It proved much harder to show that every doubly periodic function defined as a quotient arises as the inverse of a suitable elliptic integral.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Alas, Gauss's researches were confined to his diaries, and so public appearance of doubly periodic functions had to await the work of Abel and Jacobi. Jacobi later opined why they were not discovered sooner:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;The fear that the mathematician most chiefly concerned with the determination of numerical values had of the imaginary, was the reason that Legendre was prevented from taking the most important step in modern analysis, the introduction of doubly periodic functions.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Abel published his first major paper on the subject, &lt;a href=&quot;https://web.archive.org/web/20160913235716/http://www.abelprize.no/nedlastning/verker/oeuvres_1881_del1/oeuvres_completes_de_abel_nouv_ed_1_kap16_opt.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;em&gt;Recherches sur les fonctions elliptiques&lt;/em&gt;&lt;/a&gt;, in early 1827, motivated, in part, by a generalization of the circle division problem from Gauss’s &lt;em&gt;Disquisitiones Arithmeticae&lt;/em&gt; to the lemniscate. It was the first published work on doubly periodic functions, second part of it came out in 1828. Jacobi came to inverting elliptic integrals from a different angle, by considering transformations of elliptic integrals into each other by a rational change of the variables, he published at the end of 1827. Whether he did it before or after reading Abel's paper is a matter of dispute (he claimed the former).&lt;/p&gt;&#xA;&lt;p&gt;The subsequent Abel-Jacobi rivalry turned out to be very fruitful for the subject and culminated in Jacobi's &amp;quot;&lt;em&gt;book that was to become the definitive account of the theory of elliptic functions for at least a generation&lt;/em&gt;&amp;quot;, &lt;em&gt;Fundamenta Nova Theoriae Functionum Ellipticarum&lt;/em&gt; (1829). Abel tragically passed away the same year. Weierstrass got hold of Jacobi's book in 1836, and became the only student in Gudermann's class on them in 1839-40. But his own work on them started only in 1860-s, and built on Eisenstein's representation of them by double series. Eisenstein was drawn to the subject by Gauss, whom he admired (the feeling was mutual), in 1840-s. His main discoveries date to 1846-47, and he too got embroiled in priority disputes with Jacobi.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-03-01T11:59:19.367" LastActivityDate="2024-03-01T11:59:19.367" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17288" PostTypeId="1" AcceptedAnswerId="17289" CreationDate="2024-03-01T03:48:55.453" Score="1" ViewCount="53" Body="&lt;p&gt;I am wondering if Dedekind's theory about the structure of deductive science influenced the work of Hilbert.&lt;/p&gt;&#xA;&lt;p&gt;Hilbert obviously favored axioms at the beginnings of a deductive science, whereas Dedekind seemed to take a more conceptual approach.&lt;/p&gt;&#xA;&lt;p&gt;Any details about the relationship between the works of these mathematicians would be appreciated.&lt;/p&gt;&#xA;" OwnerUserId="19712" LastActivityDate="2024-03-01T06:41:44.650" Title="Did Dedekind's work directly influence the work of Hilbert?" Tags="|mathematics|mathematicians|mathematical-logic|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17289" PostTypeId="2" ParentId="17288" CreationDate="2024-03-01T06:41:44.650" Score="0" Body="&lt;p&gt;Hilbert was influenced especially by Dedekind's 1888 essay &lt;a href=&quot;https://www.gutenberg.org/files/21016/21016-pdf.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Nature and Meaning of Numbers&lt;/a&gt;, but he shared only half of Dedekind's approach. To keep things in perspective, it is important to note the third, most traditional genetic approach, advocated at the time by Helmholtz-Kronecker and more rigorously by Frege, as reflected in the &lt;a href=&quot;https://plato.stanford.edu/entries/frege-hilbert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Frege-Hilbert correspondence&lt;/a&gt;. Hilbert explicitly distinguished his own axiomatic approach from Dedekind's structural one, which, arguably, combined axiomatic and genetic one. The combination came to be appreciated much later, when Gödel showed that axiomatic theories could not be &amp;quot;grounded&amp;quot; the way Hilbert hoped.&lt;/p&gt;&#xA;&lt;p&gt;Dedekind and Hilbert were both promoting abstract arguments about objects with multiple concrete realizations, on this they were on the same side. But Dedekind's approach was bottom-up, abstracting general properties from concrete realizations and operating on them, while Hilbert's was top-down, laying down the axioms and deriving theorems about 'incomplete' objects. He saw the genetic side as redundant, at least, from the foundational perspective. Modernizing, we could say that Dedekind had in mind a class of models, while Hilbert had a formal theory. In a similar spirit, some modern mathematicians distinguish between &amp;quot;structural&amp;quot; axioms of &amp;quot;working mathematicians&amp;quot; and &amp;quot;foundational&amp;quot; axioms of logicians, e.g. &lt;a href=&quot;https://math.stanford.edu/%7Efeferman/papers/newaxioms.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Feferman in Does mathematics need new axioms?&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Interconnections between Hilbert's and Dedekind's ideas are analyzed in &lt;a href=&quot;https://www.jstor.org/stable/20118650&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sieg-Schlimm, Dedekind's Analysis of Number: Systems and Axioms&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;In 1888 Hilbert made his Rundreise from Konigsberg to other German&#xA;university towns. He arrived in Berlin just as Dedekind's Was&#xA;sind und was sol/en die Zahlen? had been published. Hilbert reports&#xA;that in mathematical circles everyone, young and old, talked about&#xA;Dedekind's essay, but mostly in an opposing or even hostile sense. A&#xA;year earlier, Helmholtz and Kronecker had published articles on the&#xA;concept of number in a Festschrift for Eduard Zeller... Dedekind's reflections... can&#xA;be understood... as articulating&#xA;an axiomatic approach that is joined with a genetic one in a&#xA;methodologically coherent way... If we think of the genetic method as underlying the construction of&#xA;mathematical objects, systems of which are models of appropriate&#xA;axiom systems, we can see very clearly how it complements in&#xA;Dedekind's hands an axiomatic approach.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;In his essay Uber den Zahlbegriff,&#xA;Hilbert distinguished sharply between the axiomatic and genetic&#xA;method, but did not recognize then the complementary roles they&#xA;play for the foundations of arithmetic... Dedekind's approach is associated with a novel&#xA;structuralist perspective on mathematics and is grounded in logic&#xA;broadly conceived. Hilbert sustains this general perspective in what he later calls existential axiomatics, but he gives up a logicist in&#xA;favor of a finitist grounding of mathematics... By tracing its development we provide a view of Dedekind's&#xA;evolving foundational position that apparently differs from Hilbert's:&#xA;to our knowledge, Hilbert never considered Dedekind as having used&#xA;the axiomatic method.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;There is no conflict, and consequently no choice has to be&#xA;made, between a genetic and an axiomatic approach for Dedekind... There is no supersession of Dedekind's &amp;quot;deductive method&amp;quot;&#xA;(described on pp. 246-248) by the axiomatic method of Hilbert's, but&#xA;the former is rather the very root of the latter. Hilbert's first&#xA;axiomatic formulations in Uber den Zahlbegriff and Grundlagen der&#xA;Geometrie are patterned after Dedekind's. Indeed, Hilbert is a logicist in Dedekind's spirit at that point, and it is no accident that, as late as&#xA;1917/1918, he was attracted by attempts to provide a logicist foundation&#xA;of mathematics.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;As Sieg-Schlimm admit, their interpretation is controversial and other historians, Ferreiros, Corry and McCarty see them as further apart, and Dedekind as more &amp;quot;non-modern&amp;quot;, &amp;quot;non-axiomatic&amp;quot; and &amp;quot;Kantian&amp;quot;. However, Hilbert himself retained Kantian elements in his treatment of symbol manipulation, see &lt;a href=&quot;https://philosophy.stackexchange.com/a/28157/9148&quot;&gt;Was there a Kantian influence on Hilbert's formalist programme?&lt;/a&gt;, and &lt;a href=&quot;https://plato.stanford.edu/entries/frege-hilbert/#FregBackInitDiff&quot; rel=&quot;nofollow noreferrer&quot;&gt;Blanchette, writing for SEP&lt;/a&gt;, also cites their commonalities:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Perhaps most clearly illustrated in Dedekind 1888, the central idea of the new approach is to understand mathematical theories as characterizing general “structural” conditions that might be had in common by any number of different ordered domains. Just as, in algebra, the axioms for a group give general conditions that can be satisfied by any manner of object whatsoever under appropriate relations, so too on the new view the axioms of geometry specify multiply-instantiable conditions. Viewing theories from this modern perspective, it is entirely appropriate to take axioms as Hilbert does, since reinterpretable sentences are the right vehicles to express the multiply-instantiable conditions in question. From the point of view of the earlier fixed-domain conception of theories, on the other hand, such reinterpretable sentences are entirely inappropriate as axioms, since they fail to fix a determinate subject-matter.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-03-01T06:41:44.650" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17290" PostTypeId="1" CreationDate="2024-03-01T11:19:25.990" Score="0" ViewCount="28" Body="&lt;p&gt;So I was reading a History of Mathematics by Katz, and noticed that the first definition of a field came from Weber, who had previously done extensive joint work with Dedekind. His definition was used in German Textbooks for 30 years.&lt;/p&gt;&#xA;&lt;p&gt;It has been explained to me that Dedekind's approach to mathematics was &amp;quot;bottom-up&amp;quot; which essentially means that from concrete realizations we build up by examination of properties to broader definitions (such as field).&lt;/p&gt;&#xA;&lt;p&gt;My question is simply did Weber share this approach in his mathematics? Or was he more of a formalist.&lt;/p&gt;&#xA;&lt;p&gt;PS: This is my last question. Cheers.&lt;/p&gt;&#xA;" OwnerUserId="19712" LastActivityDate="2024-03-01T11:19:25.990" Title="Did Heinrich Weber have a structural approach to mathematics similar to Dedekind?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17291" PostTypeId="1" CreationDate="2024-03-01T18:14:49.497" Score="2" ViewCount="38" Body="&lt;p&gt;The Iranian New Year unlike most other cultural celebrations of our planet completing a trip around the sun is exact and observational, I am curious exactly when the observational nature of the calendar was adopted socially.&lt;/p&gt;&#xA;&lt;p&gt;In the Gregorian Calendar the moment of &amp;quot;New Year&amp;quot; is when the date changes from Dec 31 to Jan 1, a.k.a midnight. Hence the moment of New Years varies with one's time zones. Although I am not familiar with other calenders in depth, my understanding is that various other cultures don't even celebrate the moment of New Years, its more a celebration around the day of and eve of New Years (like the Chinese Lunisolar Calendar or the Arabic Lunar Calendar).&lt;/p&gt;&#xA;&lt;p&gt;The Iranian Calendar however celebrates its new year at the precise moment of the Vernal Equinox, and this moment is of cultural significance. This is why it is an observational calendar. Unlike the Gregorian Calendar based on an algorithm of leap years, the Iranian Calendar is based on the precise positioning of the Earth and Sun. Hence it actually has no error.&lt;/p&gt;&#xA;&lt;p&gt;To be clear this year in Eastern Time, New Years is at 23:06:26PM on March 19th. This is a time precise down to the second, and virtually every family who celebrates Nowruz will celebrate at this MOMENT. This means regardless of one's timezone, New Year Time is the same everywhere. It's actually quite beautiful.&lt;/p&gt;&#xA;&lt;p&gt;Now my question is: when was this practice of precision relating to the time of celebrating New Years adopted socially? It is very hard for me to imagine before the advent of radio some local fella without his personal observatory knowing the exact time of the Vernal Equinox. So then what? What was the timeline of the adoption of this beautiful (and perfect I would argue) calendar? Is there some body of astronomers which determine the exact Vernal Equinox used for the Iranian Calendar**? Did royalty/aristocracy who likely had access to astronomers and mathematicians always celebrate precisely or is this entire practice novel to the 20th century?&lt;/p&gt;&#xA;&lt;p&gt;**I know every observatory determines the Equinox, but much like standard time there ought be some organization and institution for the Iranian Calendar which has the final say.&lt;/p&gt;&#xA;" OwnerUserId="13800" LastEditorUserId="13800" LastEditDate="2024-03-05T19:08:48.373" LastActivityDate="2024-03-05T19:08:48.373" Title="History of Observational &quot;Nowruz&quot; (Iranian New Year)" Tags="|astronomy|calendar|" AnswerCount="0" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17292" PostTypeId="1" CreationDate="2024-03-01T21:49:19.517" Score="3" ViewCount="51" Body="&lt;p&gt;People have been doing calculus of several variables since well before the concepts of vectors, matrices, and linear algebra were formalized. Where can I learn about the development of multivariable calculus?&lt;/p&gt;&#xA;" OwnerUserId="20791" LastActivityDate="2024-03-01T21:49:19.517" Title="What can I read to learn the history of multivariable calculus?" Tags="|mathematics|calculus|linear-algebra|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17293" PostTypeId="1" CreationDate="2024-03-01T23:13:54.203" Score="1" ViewCount="146" Body="&lt;p&gt;Dedekind gave a construction and explanation of integers and rational in 1858. This was as ordered pairs of natural numbers. I'm not sure if this was the standard view of these objects after this point. Hilbert didn't seem to write much (if anything) about this subject.&lt;/p&gt;&#xA;&lt;p&gt;If there can be no conclusive &amp;quot;when&amp;quot;, are there any textbooks that include general constructions or something like this? Many sources say this is the time period when &amp;quot;negatives&amp;quot; where finally accepted into mathematics, but give no detail about the acceptance.&lt;/p&gt;&#xA;&lt;p&gt;When did integers start being viewed as a ring structure?&lt;/p&gt;&#xA;&lt;p&gt;I've gotten side tracked a lot lately with math history, but I think this is what I am most concerned about. This is my last question about this topic, it is the one I originally wanted an answer too I think, but wasn't asking it properly.&lt;/p&gt;&#xA;&lt;p&gt;I find Dedekind's work to be interesting though.&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="19712" LastActivityDate="2024-03-04T13:22:58.320" Title="When were negative numbers fully accepted into mathematics?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17294" PostTypeId="1" CreationDate="2024-03-02T08:41:30.317" Score="0" ViewCount="50" Body="&lt;p&gt;In what direction does the Sun rotate in the geocentric model?&lt;br /&gt;&#xA;It seems to me that this should rotate in a clockwise direction around Earth (as seen from the north pole), but all the illustrations I've found indicate the opposite. Am I missing something? A counter-clockwise rotation of the Sun would have sunrises in the west, right?&lt;/p&gt;&#xA;" OwnerUserId="18091" LastActivityDate="2024-03-05T13:37:03.417" Title="The Sun's direction of rotation in the geocentric model" Tags="|archaeoastronomy|ptolemy|" AnswerCount="1" CommentCount="0" ClosedDate="2024-03-09T20:18:45.603" ContentLicense="CC BY-SA 4.0" />
  <row Id="17295" PostTypeId="2" ParentId="17294" CreationDate="2024-03-02T13:06:54.227" Score="1" Body="&lt;p&gt;Sun (and everything else in the sky) participate in two motions.&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;Sky as a whole (the sphere of fixed stars) rotates clockwise, 1 rotation per 24 hours. BTW for this reason we call this direction clockwise. Our clocks are made to model this rotation (except that the hour hand performs two rotations during the time when the sky perform one rotation).&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;For this reason EVERYTHING (Sun, all stars, Moon and planets) rise in the East and set in the West every day.&lt;/p&gt;&#xA;&lt;ol start=&quot;2&quot;&gt;&#xA;&lt;li&gt;With respect to the sphere of fixed stars, every other object, including Sun has its own motion. It is this motion that is described in illustrations. Most of these motions are counterclokwise,&#xA;in particular Sun describes the a circle in the sphere of fixed stars counterclockwise, one rotation in a year.&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;In other words, if you observe Sun, and put in on the star globe every day at the same time, you will discover that during a year it described one full circle counterclockwise.&lt;/p&gt;&#xA;&lt;p&gt;Remark. All this description assumes that you look from the Northern hemisphere. In Australia, all directions of rotations are opposite (for Southern hemisphere, all &amp;quot;clockwise&amp;quot; and &amp;quot;counterclockwise&amp;quot;&#xA;in my answer have to be interchanged.)&lt;/p&gt;&#xA;&lt;p&gt;Remark 2. By the way, in mathematics counterclockwise is called the &amp;quot;positive direction&amp;quot;: in polar coordinates, polar angle increases counterclockwise. This is because mathematics was always closely connected with astronomy. So mathematicians used the astronomical convention, while clockmakers are more concerned with everyday needs of ordinary people.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-05T13:37:03.417" LastActivityDate="2024-03-05T13:37:03.417" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17296" PostTypeId="2" ParentId="17293" CreationDate="2024-03-02T13:44:32.443" Score="4" Body="&lt;p&gt;I am afraid that &amp;quot;fully accepted&amp;quot; is not well defined in this question. Some mathematicians used negative numbers beginning from 2-nd century AD, then they slowly penetrated mathematics, and physics and astronomy. It is safe to say that by 18th century all professional mathematicians routinely used them. And when definitions of such things as fields were given (in early 19 century), rational and real numbers served as trivial examples.&lt;/p&gt;&#xA;&lt;p&gt;However this does not apply to mathematics textbooks written for non-mathematicians. For example, the authors of 19 century textbooks for mariners explain in detail the decimal system, geometry, trigonometry and logarithms but avoid negative numbers.&lt;/p&gt;&#xA;&lt;p&gt;High school algebra textbooks of second half of 19th century use negative numbers routinely, about early 19 century I am not sure.&lt;/p&gt;&#xA;&lt;p&gt;Let me also notice that the instructions to US tax forms&#xA;avoid negative numbers to this day.&lt;/p&gt;&#xA;&lt;p&gt;Edit. I consulted an expert in the history of celestial navigation. Here is his reply:&lt;/p&gt;&#xA;&lt;p&gt;&amp;quot;Celestial navigation textbooks and resources up to the very end of positive development (around 1978, which also happens to be the year the first GPS satellite was launched) continued to behave as if negative numbers either didn't exist or were too complicated for average navigators.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;Edit 2. Another group of people which avoids the use of negative numbers is historians. They enumerate years like this ...2 BC, 1 BC, 1 AD, 2 AD...,&#xA;while astronomers, who really need to calculate with dates, enumerate the same years in the logical order as ...-1,0,1,2,...&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-04T13:22:58.320" LastActivityDate="2024-03-04T13:22:58.320" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17297" PostTypeId="1" CreationDate="2024-03-02T17:46:54.227" Score="3" ViewCount="148" Body="&lt;p&gt;Is there any English translation of the 1918-1920 Memoirs of Fatou and Julia on the iteration of rational functions?&lt;/p&gt;&#xA;" OwnerUserId="20797" LastEditorUserId="12330" LastEditDate="2024-03-03T10:58:11.563" LastActivityDate="2024-03-15T22:05:19.060" Title="Translated articles of Fatou and Julia" Tags="|mathematics|reference-request|complex-analysis|translations|" AnswerCount="4" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17298" PostTypeId="1" CreationDate="2024-03-02T17:58:56.720" Score="2" ViewCount="339" Body="&lt;p&gt;In his account of refraction, which can be found for example &lt;a href=&quot;https://plato.stanford.edu/entries/descartes-method/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;, Descartes compares it with the motion of a tennis ball.&lt;/p&gt;&#xA;&lt;p&gt;He says that going into a denser medium must reduce the total speed, while the tanget speed must be the same. This leads to the conclusion that the trajectory must bend &lt;em&gt;away from&lt;/em&gt; the normal line.&lt;/p&gt;&#xA;&lt;p&gt;This can be seen in his own figure.&lt;/p&gt;&#xA;&lt;p&gt;This reasoning works for a tennis ball, but for light it is wrong: light bends &lt;em&gt;towards&lt;/em&gt; the normal line.&lt;/p&gt;&#xA;&lt;p&gt;Yet everybody says Descartes correctly deduced the refraction law. I don't understand.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/2MlBL.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/2MlBL.png&quot; alt=&quot;enter image description here&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="19593" LastActivityDate="2024-03-02T18:37:16.660" Title="Isn't Descartes' account of refraction wrong?" Tags="|optics|descartes|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17299" PostTypeId="2" ParentId="17298" CreationDate="2024-03-02T18:37:16.660" Score="5" Body="&lt;p&gt;The point is the Descartes also argues that light should move &lt;em&gt;faster&lt;/em&gt; in the denser medium, contrary to what happens with a tennis ball. In one of his notebooks, he wrote: &amp;quot;&lt;em&gt;Since light can be produced only in matter, where there is more matter it is produced more easily, other things being equal; therefore it penetrates more easily through a denser than through a rare medium&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;So he uses the tennis ball for intuition, and deduces the refraction it would have when its speed is reduced. But since the speed of light increases, the refraction is opposite.&lt;/p&gt;&#xA;&lt;p&gt;(How this is to be reconciled with his view that the speed of light is actually infinite, I don't know.)&lt;/p&gt;&#xA;&lt;p&gt;In any case, the greater originality in Descartes work is that it is quantitative. He not only says the light bends towards the normal, which was known since antiquity, but he clearly states that the sines of the angles are in proportion (and not the angles themselves, for example).&lt;/p&gt;&#xA;" OwnerUserId="20798" LastActivityDate="2024-03-02T18:37:16.660" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17300" PostTypeId="1" CreationDate="2024-03-02T22:52:15.700" Score="1" ViewCount="29" Body="&lt;p&gt;I saw André Weil's Weil divisor. So I became curious about the history of the divisor. However, when I searched the history of divisor, I found that Leopold Kronecker announced the beginning of the concept of divisor with his Divisorentheorie. So I would like to know the background and history of how Kronecker came up with the concept of Divisorentheorie.&lt;/p&gt;&#xA;" OwnerUserId="18258" LastActivityDate="2024-03-02T22:52:15.700" Title="History of Kronecker's Divisorentheorie" Tags="|kronecker|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17301" PostTypeId="1" CreationDate="2024-03-03T00:28:14.003" Score="1" ViewCount="46" Body="&lt;p&gt;I am referring to the construction using pairs of natural numbers in 1858.&lt;/p&gt;&#xA;&lt;p&gt;Since we use pretty much the same construction today in some analysis courses (Analysis 1, Terence Tao), except without the formal notion of &amp;quot;equivalence class&amp;quot;, I'm assuming that his constructions must have become standard.&lt;/p&gt;&#xA;&lt;p&gt;If not, then what was, and what happened to his works?&lt;/p&gt;&#xA;" OwnerUserId="19712" LastActivityDate="2024-03-03T00:28:14.003" Title="Did Dedekind's construction of the integers and rational numbers become standard in mathematics textbooks?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17303" PostTypeId="1" AcceptedAnswerId="17306" CreationDate="2024-03-03T12:14:12.057" Score="13" ViewCount="4622" Body="&lt;p&gt;Could someone list me the most important battles between mathematicians which happened in history, especially such that strong emotions played role in that time?&lt;/p&gt;&#xA;&lt;p&gt;Perhaps the most known one is the battle between supporters of Newton and Leibniz on invention of calculus..&#xA;but..&lt;/p&gt;&#xA;&lt;p&gt;What other similar campaigns are known in history of mathematics? Who was engaged and who finally won the battle if we can say that it was won? ( in the case of Newton and Leibniz I would say it was draw)&lt;/p&gt;&#xA;" OwnerUserId="6955" LastEditorUserId="6609" LastEditDate="2024-03-05T13:07:29.177" LastActivityDate="2024-03-06T06:13:57.977" Title="Great battles in the history of mathematics" Tags="|mathematicians|examples|" AnswerCount="5" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17306" PostTypeId="2" ParentId="17303" CreationDate="2024-03-03T18:35:21.580" Score="19" Body="&lt;p&gt;From Wikipedia:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy&quot; rel=&quot;noreferrer&quot;&gt;Leibniz–Newton calculus controversy&lt;/a&gt; (draw, independent discovery)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy&quot; rel=&quot;noreferrer&quot;&gt;Brouwer-Hilbert controversy&lt;/a&gt; (Hilbert removed Brouwer from the board of &lt;em&gt;Mathematische Annalen&lt;/em&gt;, Brouwer’s philosophy of mathematics was not taken very seriously at the time)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Hobbes%E2%80%93Wallis_controversy&quot; rel=&quot;noreferrer&quot;&gt;Hobbes–Wallis controversy&lt;/a&gt; (Wallis’ arguments are considered superior as many arguments of Hobbes were fallacious)&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory&quot; rel=&quot;noreferrer&quot;&gt;Controversy over Cantor's theory&lt;/a&gt; (Cantor's theory is part of today's set theory)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;From &lt;a href=&quot;https://maa.org/press/periodicals/convergence/great-feuds-in-mathematics&quot; rel=&quot;noreferrer&quot;&gt;Great Feuds in Mathematics&lt;/a&gt;, you also have:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/Strick/cardan.pdf&quot; rel=&quot;noreferrer&quot;&gt;Tartaglia vs. Cardano&lt;/a&gt; (about the cubic equation priority, independent discovery, draw)&lt;/li&gt;&#xA;&lt;li&gt;Fermat vs. Descartes (on tangent curves, at the time it was Descartes who was considered winner but now it is thought of as a draw)&lt;/li&gt;&#xA;&lt;li&gt;Jakob Bernoulli vs. Johann Bernoulli (about &lt;a href=&quot;https://mathworld.wolfram.com/BrachistochroneProblem.html&quot; rel=&quot;noreferrer&quot;&gt;the brachistochrone problem&lt;/a&gt;?, Johann had an incorrect proof and tried to pass Jakob’s solution as his own.)&lt;/li&gt;&#xA;&lt;li&gt;Poincaré vs. Russell (on the nature of logic and Kant’s philosophy of mathematics, it seems it is either a draw or Russell was wrong, see &lt;a href=&quot;https://academic.oup.com/philmat/article-abstract/1/1/24/1496709?redirectedFrom=fulltext&quot; rel=&quot;noreferrer&quot;&gt;this&lt;/a&gt;, specially since the development of Gödel's theorem)&lt;/li&gt;&#xA;&lt;li&gt;Borel vs. Zermelo (on the axiom of choice, part of modern axioms of mathematics, but its validity is still discussed)&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;Book: &lt;em&gt;Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever&lt;/em&gt;, Hal Hellman, 2006&lt;/p&gt;&#xA;&lt;p&gt;Note that all of these feuds have philosophical points of conflict that were not necessarily resolved, I just added a few remarks on the main issues.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="12330" LastEditDate="2024-03-04T16:42:57.397" LastActivityDate="2024-03-04T16:42:57.397" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17307" PostTypeId="2" ParentId="17297" CreationDate="2024-03-03T22:04:41.640" Score="6" Body="&lt;p&gt;I was pointed to this chain of messages and I would then like giving you my version of facts. The translations project was the &lt;strong&gt;very very early version&lt;/strong&gt; of the contents that ended up in the book I published for AMS in the end of 2011. It started for &lt;strong&gt;my personal knowledge and fruition&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;I can't recall exactly how the files of Julia's memoir translations spread online, as facts date back to more than twenty years ago. This was not my concern at all in then times as I was aware that the quality was not worth the circulation, and honestly I would have liked them to be removed when I discovered they had been put online (a few years later). Anyway I could not know how to do that, since I could not find out whom to call for. I just have a bare recollection that I shared'em with somebody, for pure show. Things went out of my control.&lt;/p&gt;&#xA;&lt;p&gt;And yes, professor Eremenko is right: I suspended the work on translations. Or maybe I gave up. But there were some valid motivations: first, due to the above book. Second, at the time I was not sure they might really be of interest for publishers or readers. Most of them are original memoirs or articles, written in French. Which I presumed that most mathematicians, and amateurs, are able to read it. I thought they might have not been of use for anybody- Moreover, I am not a native english speaker, so I was not quite confident in spreading them, since they need revision before being made public.&lt;/p&gt;&#xA;&lt;p&gt;You would like to know that &lt;strong&gt;Julia's are not the only translations I did&lt;/strong&gt;. I have much more material:&lt;strong&gt;almost all Julia's works&lt;/strong&gt; up to early 1920, including the memoir of 1919-21, where Julia discovered the so-called &amp;quot;lines of Julia&amp;quot;, inherent to the neighboring behavior about essential singularities and, as you know, quite close on Julia set. There would be only the memoir about permutable maps to be translated.&#xA;In addition, there are also &lt;strong&gt;almost all Fatou's works&lt;/strong&gt; on iterations, except the long memoir of 1919-20, the one you mentioned in your messages. That I did not at the time, because of the original plan changed. Again, the translation of &lt;strong&gt;the two articles by Samuel Lattès&lt;/strong&gt; appearing in Comptes Rendus of 1918 (about Julia set filling the Riemann sphere) and &lt;strong&gt;the three works by Hubert Cremer&lt;/strong&gt;, where he explored the dynamics around non-linearizable indifferent points of irrational kind.&lt;/p&gt;&#xA;&lt;p&gt;The original plan behind the next-to-be aforementioned book consisted in releasing a collection of those early articles that would have covered nearly fifty years of holomorphic dynamics, spanning from Schroder and Koenigs, up to Fatou, Julia and Cremer. Including Bottcher's production: either his thesis of 1899 (compiled in German) and mostly the later production around 1904, which anticipated the discovery of so-called &amp;quot;Fatou-Leau's flower&amp;quot;. (By the way, Ernst Lémeray accomploished it too shortly before Leau's thesis). Bottcher went to a thorough analysis of this same question, pretty much interesting on my viewpoint. The original work is in Russian, and I know the contents thanks to a native Polish-Russian speaker who translated some of those pages.&lt;/p&gt;&#xA;&lt;p&gt;In conclusion, I gave up because I would have liked joining a task force including a native speaker / mathematicians (German for Cremer, Russian for Bottcher at least). Take care that the original format of my files is still MS Word, so they need be turned into LaTeX first.&lt;/p&gt;&#xA;&lt;p&gt;I am currently busy with the submission of a new history book on holomorphic dynamics, that covers some of the topics I resumed above. But I am open to resume this translations project, if there would be margins of interest. According to the above, it naturally goes that I would be very prone to any form of cooperation and coordination.&lt;/p&gt;&#xA;&lt;p&gt;Thanks for showing interest. Hope I have fulfilled your curiosities.&#xA;Feel free to write me to alessandro.a.rosa@gmail.com&lt;/p&gt;&#xA;" OwnerUserId="4584" LastEditorUserId="4584" LastEditDate="2024-03-15T22:05:19.060" LastActivityDate="2024-03-15T22:05:19.060" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17308" PostTypeId="1" CreationDate="2024-03-04T09:48:37.627" Score="0" ViewCount="34" Body="&lt;p&gt;I am looking for complete and trustable references on the history of electromagnetism. Does anyone have some good recommendations?&lt;/p&gt;&#xA;" OwnerUserId="19547" LastActivityDate="2024-03-04T09:48:37.627" Title="References on the history of electromagnetism" Tags="|electromagnetism|" AnswerCount="0" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17309" PostTypeId="2" ParentId="17303" CreationDate="2024-03-04T10:40:41.040" Score="12" Body="&lt;p&gt;One battle that is somewhat overlooked is the battle over &lt;a href=&quot;https://en.wikipedia.org/wiki/Bonaventura_Cavalieri&quot; rel=&quot;noreferrer&quot;&gt;Bonaventura Cavalieri&lt;/a&gt;'s &lt;a href=&quot;https://en.wikipedia.org/w/index.php?title=Method_of_indivisibles&amp;amp;oldid=668250529&quot; rel=&quot;noreferrer&quot;&gt;indivisibles&lt;/a&gt;.  His opponent was &lt;a href=&quot;https://en.wikipedia.org/wiki/Paul_Guldin&quot; rel=&quot;noreferrer&quot;&gt;Paul (Habakkuk) Guldin&lt;/a&gt;, a Jesuit (and other Jesuits).  Cavalieri was a &lt;a href=&quot;https://en.wikipedia.org/wiki/Jesuati&quot; rel=&quot;noreferrer&quot;&gt;Jesuat&lt;/a&gt;.  Most engines today automatically correct &amp;quot;Jesuat&amp;quot; to &amp;quot;Jesuit&amp;quot;, as the Jesuats are mostly unknown (for reasons that will become clear below).  The Jesuat order was founded over a century before the Jesuits.  Starting in the 1620s and 1630s, the two orders were lined up on opposite sides of a debate over indivisibles, with Cavalieri fiercely defending their legitimacy, and Guldin and others fiercely opposed to them.&lt;/p&gt;&#xA;&lt;p&gt;The bottom line of the Jesuits' opposition to indivisibles were doctrinal: they were convinced that ideas such as atomism and indivisibles were contrary to Catholic dogma (obviously Cavalieri, who was a devout Catholic, did not think so).  At issue is the rivalry between the Aristotelian doctrine of hylomorphism which since Aquinas became the theoretical &amp;quot;explanation&amp;quot; of the miracle of the Eucharist, and Democritus' atomism, thought of as subversive and heretical by the Jesuits.&lt;/p&gt;&#xA;&lt;p&gt;The generals of the Jesuit order issued numerous bans against indivisibles.  The Jesuits had far more influence in Rome than the Jesuats, and things came to a head in 1668 when the Jesuat order was suppressed abolished by papal bull (&amp;quot;bulla&amp;quot;, not the animal).  Cavalieri's student &lt;a href=&quot;https://en.wikipedia.org/wiki/Stefano_degli_Angeli&quot; rel=&quot;noreferrer&quot;&gt;Degli Angeli&lt;/a&gt;, similarly a Jesuat, wrote about a dozen books about applications of the method of indivisibles, the last one I believe a year before the suppression of his order.  After that, he didn't publish anything on indivisibles any more, though he remained active and published a book in mechanics, and continuing his teaching at the university for several decades after that.  In the 1690s, &lt;a href=&quot;https://en.wikipedia.org/wiki/Jacopo_Riccati&quot; rel=&quot;noreferrer&quot;&gt;Riccati&lt;/a&gt; attended his lectures at the university (obviously not about indivisibles).&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2024-03-05T08:52:11.797" LastActivityDate="2024-03-05T08:52:11.797" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17310" PostTypeId="2" ParentId="16196" CreationDate="2024-03-04T19:18:26.810" Score="3" Body="&lt;p&gt;&lt;em&gt;A History of Algebraic and Differential Topology, 1900 - 1960&lt;/em&gt; by Jean Dieudonné describes, among other things, the development of (co)homology in great detail.&lt;/p&gt;&#xA;" OwnerUserId="20805" LastActivityDate="2024-03-04T19:18:26.810" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17312" PostTypeId="2" ParentId="17303" CreationDate="2024-03-05T00:33:00.177" Score="3" Body="&lt;p&gt;Dorian invasion. Unlike native Achaeans, the Dorians viewed the world in terms of Euclidean geometry (not yet called that) in which &amp;quot;large and small can be the same&amp;quot; (similarities are automorphisms), didn't use intentional perspective and destroyed almost all works of art they could find which used it (and other artifacts as well; they were aware of cultural assimilation of invaders being the usual outcome and adamant in avoiding it, even if it meant lower quality of life). They clearly won – nowadays we have no problem using maps, technical drawings in scale, or copying, without specific instruction, large letters from a blackboard into a notebook, unconsciously scaling them down, even as children.&lt;/p&gt;&#xA;&lt;p&gt;Akousmatics vs mathematicians (first group of people who called themselves that), the factions into which Pythagoreans split upon discovery of irrational numbers. Result: mathematicians are a (fairly) respected profession, akousmatics quickly became an esoteric fringe subculture.&lt;/p&gt;&#xA;&lt;p&gt;Linear algebra (with linear equations), preferred on the continent, particularly among Germans, vs matrix theory, developed and favoured on the British Isles. Result: draw IMO; emotions subsided and theorems showing relevant equivalences became common knowledge in the field; both tools are often reached for, depending on the problem and other factors.&lt;/p&gt;&#xA;&lt;p&gt;Analysts vs algebraists, logicians and theoretical computer scientists. Ongoing. Sometimes with heightened emotions.&lt;/p&gt;&#xA;" OwnerUserId="20807" LastActivityDate="2024-03-05T00:33:00.177" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17313" PostTypeId="2" ParentId="17303" CreationDate="2024-03-05T07:24:18.757" Score="5" Body="&lt;p&gt;In computable algebraic geometry there is the &amp;quot;elimination of elimination theory&amp;quot; controversy.&lt;/p&gt;&#xA;&lt;p&gt;It starts with Kronecker in 1880 providing an approach towards the solution of polynomial systems that eliminates one variable after the other using resultants. This way fundamental theorems like the Nullstellensatz are proven constructively, but with great machinery. At the time no complexity analysis was done.&lt;/p&gt;&#xA;&lt;p&gt;Hilbert in the 1890's provided a shorter, basically non-constructive proof of the Nullstellensatz and related results. The further development of algebraic theory followed mainly Hilberts approach, to the point that in editions after 1930 of vander Wardens influential book any mention of elimination theory was eliminated. Previously it contained a nice elementary exposition in the first chapters.&lt;/p&gt;&#xA;&lt;p&gt;The development of the Gröbner basis algorithm in the 1960's (?) was taken as confirmation of the Hilbert approach. Such algorithms are widely implemented in CAS, but contain a large heuristic part.&lt;/p&gt;&#xA;&lt;p&gt;Only starting in the 1990's were efficient elimination procedures developed. This is still a small fringe of the community, but has some advantages: there is a theoretically accessible complexity theory for it and the complexity magnitudes are better than all what is available for Gröbner bases. and their complexity proven&lt;/p&gt;&#xA;" OwnerUserId="6894" LastActivityDate="2024-03-05T07:24:18.757" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17314" PostTypeId="1" AcceptedAnswerId="17321" CreationDate="2024-03-05T09:26:04.740" Score="0" ViewCount="62" Body="&lt;p&gt;I have an assignment about the foundations of mathematics.&lt;/p&gt;&#xA;&lt;p&gt;I am trying to compile a list where I get common construction of integers and a small writing about the constructor and their explanation.&lt;/p&gt;&#xA;&lt;p&gt;For example, I have the ordered pair construction by Dedekind, and his explanation via his principle of “generality”, where he writes that definitions appearing in a restricted form can be extended to avoid arbitrariness.&lt;/p&gt;&#xA;&lt;p&gt;What about the original constructors of other constructions, where can I read about their commentaries?&lt;/p&gt;&#xA;" OwnerUserId="19498" LastActivityDate="2024-03-06T13:22:03.383" Title="Is there a resource about integer constructions and motivations?" Tags="|mathematics|mathematicians|" AnswerCount="1" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17315" PostTypeId="2" ParentId="17297" CreationDate="2024-03-05T11:26:36.210" Score="3" Body="&lt;p&gt;Before posting my thoughts I would like to thank everyone for contributing to this inquiry and express my sincere appreciation for Mr Rosa's answer.&lt;/p&gt;&#xA;&lt;p&gt;I have read parts of two books on the history of complex dynamics&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;A History of Complex Dynamics&#xA;From Schröder to Fatou and Julia&lt;/li&gt;&#xA;&lt;li&gt;Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906–1942&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;I have to say it was quite a joy reading through their chapters, and I found them very helpful and informative for a plethora of aspects regarding the theory itself.&lt;/p&gt;&#xA;&lt;p&gt;Before continuing further though, I need to say that I am not an expert in the field, as I am about to begin my PhD on holomorphic dynamics in autumn this year. So, my knowledge about publishing is minimal at best, if nonexistent. Furthermore, I am neither a native English speaker, nor a French one, but I manage to understand a little bit of the latter.&lt;/p&gt;&#xA;&lt;p&gt;Last year I completed a Master's Thesis on the subject, but there were no faculty members in my university studying complex dynamics, (my thesis committee was comprised of some external experts on complex dynamics though). Therefore, I had to learn the theory on my own, and at the time, I was finding the books available a little difficult to crack through their line of thought and motivation behind some concepts, especially the first time I encountered some ideas. This was the case with the Leau-Fatou theorem for example, where on the other hand Fatou's original Memoir provided much needed (for me) explanation and details.&lt;/p&gt;&#xA;&lt;p&gt;Furthermore, it strikes me, how much these men understood the subject they were studying. One can find evidence for that on pages 158 to 170 of Julia's online available Memoir, for example. Of course, nowadays with the help of computers, we can easily visualize Julia sets, but I find the thought that Fatou and Julia suspected the complexity of the objects they were dealing with quite appealing.&lt;/p&gt;&#xA;&lt;p&gt;In my opinion, these Memoirs could serve as an auxiliary introduction for the subject and translating them into English could help popularize the field and help lower the &amp;quot;entry barrier&amp;quot; for students, as both Fatou's and Julia's Memoirs contain many worked-out and very detailed examples, something that many of the books I encountered partially lack, and provide strong motivation for many concepts now standardized in the theory. Moreover, I agree with Mr Rosa's remark that comments and graphics would constitute a great enhancement.&lt;/p&gt;&#xA;&lt;p&gt;I understand Mr Rosa's objections to making the translated documents public, but as &lt;em&gt;Conifold&lt;/em&gt; pointed out, &amp;quot;Even raw translations in Word, maybe converted into pdf, would make the works accessible to many people who cannot read them today and benefit from the original flow of ideas&amp;quot;, and I have experienced this benefiting myself.&lt;/p&gt;&#xA;&lt;p&gt;In conclusion, I believe that such a collection of translated documents could greatly enhance the learning experience, and the study of foundational ideas, and could significantly contribute to the historical aspect of the field. I was puzzled when I began studying complex dynamics and I could not locate a translation of such influential publications, they remind me of Klein's work on the Icosahedron or the Fricke-Klein books.&lt;/p&gt;&#xA;&lt;p&gt;I would be very pleased if I could be of any help in such a project.&lt;/p&gt;&#xA;&lt;p&gt;I would like to see your thoughts on those points as naive as they may seem from someone who does not have significant experience with such matters.&lt;/p&gt;&#xA;" OwnerUserId="20797" LastActivityDate="2024-03-05T11:26:36.210" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17316" PostTypeId="2" ParentId="17297" CreationDate="2024-03-05T13:01:52.053" Score="1" Body="&lt;p&gt;I will embark on the translation of Fatou's memoir and hopefully get it done by the start of next summer. I am quite confident of it, thanks to the recent re-study for a new book project.&lt;/p&gt;&#xA;&lt;p&gt;Take care that one such project would be of interest for experts and beginners; hence it's not just about pasting my rough translations into a PDF and share it. Syntax, language, appearance must be worked out.&lt;/p&gt;&#xA;&lt;p&gt;To &lt;strong&gt;Prelude&lt;/strong&gt;, I posted my email address in the first comment at the top. Write me, in order to check whether we can meet on some point.&lt;/p&gt;&#xA;" OwnerUserId="4584" LastEditorUserId="4584" LastEditDate="2024-03-05T13:10:56.643" LastActivityDate="2024-03-05T13:10:56.643" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17317" PostTypeId="1" CreationDate="2024-03-05T18:42:20.683" Score="2" ViewCount="49" Body="&lt;p&gt;In the Wikipedia page for the Ising Model it is written without citations:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;One of Democritus' arguments in support of atomism was that atoms&#xA;naturally explain the sharp phase boundaries observed in&#xA;materials[citation needed], as when ice melts to water or water turns&#xA;to steam.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;My question is:&lt;/p&gt;&#xA;&lt;p&gt;Is this statement historically accurate? If so, did Democritus have an argument or explained his intuition on how exactly atoms explain sharp phase transitions?&lt;/p&gt;&#xA;" OwnerUserId="19660" LastEditorUserId="6609" LastEditDate="2024-03-06T10:01:30.660" LastActivityDate="2024-03-06T10:01:30.660" Title="Did Democritus predict atoms using sharp phase transitions" Tags="|ancient-greece|atomic-theory|statistical-mechanics|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17318" PostTypeId="2" ParentId="17303" CreationDate="2024-03-06T06:13:57.977" Score="5" Body="&lt;p&gt;I would say &lt;a href=&quot;https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/&quot; rel=&quot;noreferrer&quot;&gt;Mochizuki vs Scholze&lt;/a&gt; fits the bill, though this is very recent history:-)&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-03-06T06:13:57.977" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17319" PostTypeId="1" AcceptedAnswerId="17320" CreationDate="2024-03-06T07:03:13.623" Score="12" ViewCount="1744" Body="&lt;p&gt;I read a book &lt;a href=&quot;https://link.springer.com/book/10.1007/978-0-8176-4685-1&quot; rel=&quot;noreferrer&quot;&gt;&amp;quot;a history of abstract algebra&amp;quot;- chapter 6&lt;/a&gt; by Israel Kleiner.  And in this book, it is said that Emmy Noether gave a presentation at the ICM congress held in Zurich in 1932, and that the content was the starting point of modern cohomology theory.  And the reference for this part is M.K. Smith's &lt;strong&gt;&amp;quot;Emmy Noether's contributions to mathematics, Unpublished notes (13pp, CA 1976)&amp;quot;&lt;/strong&gt;, but no matter how much I search on Google, I can't find a resource where I can read this paper.  So in the end, what I want to ask are&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;p&gt;What did Emmy Noether present at the 1932 ICM convention?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Which of the contents presented by Emmy Noether could have been the starting point of modern cohomology?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="19597" LastEditDate="2024-03-06T09:41:27.567" LastActivityDate="2024-03-06T12:20:53.037" Title="Emmy Noether's announcement in 1932 ICM" Tags="|topology|noether|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17320" PostTypeId="2" ParentId="17319" CreationDate="2024-03-06T12:20:53.037" Score="16" Body="&lt;p&gt;All past ICM proceedings are available online in one place: &lt;a href=&quot;https://www.mathunion.org/icm/proceedings&quot; rel=&quot;noreferrer&quot;&gt;https://www.mathunion.org/icm/proceedings&lt;/a&gt;. Scroll down to the year 1932, where pp. 189-194 in volume 1 is Noether's paper, which starts on p. 197 in that pdf file. The basic point is that the multiplicative relations among the standard basis elements in a crossed product algebra are an example of a 2-cocycle.  This is entirely in the direction of abstract algebra and number theory, not topology (one of your tags), so you be disappointed.&lt;/p&gt;&#xA;&lt;p&gt;See also the section &amp;quot;Class field theory and cohomology&amp;quot; on pp. 42-45 in Chapter 1 of Peter Roquette's book &lt;em&gt;Contributions to the History of Number Theory in the 20th Century&lt;/em&gt;. This is also available as manuscript 29 &lt;em&gt;The Brauer-Hasse-Noether theorem in historical perspective&lt;/em&gt; on Roquette's webpage &lt;a href=&quot;https://www.mathi.uni-heidelberg.de/%7Eroquette/manu.html&quot; rel=&quot;noreferrer&quot;&gt;https://www.mathi.uni-heidelberg.de/~roquette/manu.html&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="643" LastActivityDate="2024-03-06T12:20:53.037" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17321" PostTypeId="2" ParentId="17314" CreationDate="2024-03-06T13:22:03.383" Score="2" Body="&lt;p&gt;Yes, there is a good resource: it is the book called Numbers,&#xA;by 8 authors (Springer, 1991). The first chapter of this book written by K. Mainzer is dedicated to the history of constructions of integers and rational numbers.&lt;/p&gt;&#xA;" OwnerUserId="64" LastActivityDate="2024-03-06T13:22:03.383" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17322" PostTypeId="1" CreationDate="2024-03-06T13:59:50.563" Score="2" ViewCount="56" Body="&lt;p&gt;In one of his non-fiction works, probably &amp;quot;Mysterious World&amp;quot; or &amp;quot;World of Strange Powers&amp;quot;, Arthur C. Clarke tells an anecdote about an astronomers' expedition to Africa (if I remember correctly) to observe a solar eclipse. The local population mocks the &amp;quot;arrogant&amp;quot; foreigners who believe to know in advance what the Sun will do. After the eclipse, the astronomers ask the locals whether they have changed their mind. Not at all, they reply, but concede that it was an interesting coincidence.&lt;/p&gt;&#xA;&lt;p&gt;Does anyone have the exact quote and the reference?&lt;/p&gt;&#xA;&lt;p&gt;My apologies if this is not the right community to ask.&lt;/p&gt;&#xA;" OwnerUserId="20816" LastEditorUserId="20816" LastEditDate="2024-03-06T14:02:11.370" LastActivityDate="2024-03-06T14:02:11.370" Title="Reference to a comment by Arthur C. Clarke" Tags="|reference-request|astronomy|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17323" PostTypeId="2" ParentId="9785" CreationDate="2024-03-06T15:26:48.210" Score="3" Body="&lt;p&gt;Erwin Schrödinger was very particular for his polyamorous/swinger affairs. He slept with other women and (forced?) let his wife sleep with other men. He invented Schrödinger's equation after going away to a sky resort with an unidentified mistress during Christmas. Some of Schrödinger's sexual partners included underage girls (as young as 12). This revelation has pushed some universities to remove statues and well as his name from lecture rooms.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-03-06T17:48:08.637" LastActivityDate="2024-03-06T17:48:08.637" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17324" PostTypeId="2" ParentId="16196" CreationDate="2024-03-06T16:24:05.383" Score="1" Body="&lt;p&gt;A key tool in developing homology or cohomology theories is that of a&#xA;chain complex.  Dieudonne writes in his &lt;a href=&quot;https://doi.org/10.1007/978-0-8176-4907-4&quot; rel=&quot;nofollow noreferrer&quot;&gt;History of Algebraic and Differential Topology&lt;/a&gt;:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;The first occurrence of a chain complex is to be found in Poincare's&#xA;papers of 1900&amp;quot; (page 4).&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dieudonne also mentions some antecedents in the 19th century:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;After an unsuccessful attempt in 1895 to give a genuine mathematical&#xA;formulation to the intuitive ideas of Riemann and Betti, probably&#xA;inspired by earlier work of the nineteenth century on polyhedra and&#xA;the Euler formula, [Poincare] restricted himself to compact&#xA;triangulated spaces ...  From this beginning the evolution of homolog&#xA;went through series of steps, the description of which constitutes&#xA;Part 1 of this book.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Dieudonne's introduction is probably a good place to start if one&#xA;wants to understand the history of (co)homology theories.&lt;/p&gt;&#xA;" OwnerUserId="604" LastEditorUserId="604" LastEditDate="2024-03-26T14:27:56.267" LastActivityDate="2024-03-26T14:27:56.267" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17325" PostTypeId="2" ParentId="17297" CreationDate="2024-03-07T02:32:40.713" Score="2" Body="&lt;p&gt;This is a longer comment rather than an answer to the  question  specifically about Fatou and Julia papers. Having been given green light by Sandro, let me add some more details about his project of &amp;quot;releasing a collection of those early articles that would have covered nearly fifty years of holomorphic dynamics, spanning from Schroder and Koenigs, up to Fatou, Julia and Cremer&amp;quot;. Sandro and I exchanged emails in 2013-2014 concerning my joint work with Stanislaw Domoradzki on the life and achievements of Lucjan Emil Boettcher. In one such exchange I learned about Sandro's project described in his answer above. Wishing to contribute, I translated (from Polish to English) Boettcher's 1898 article in several parts: ``Zasady rachunku iteracyjnego&amp;quot;&#xA;[Principles of iterational calculus , Prace&#xA;Matematyczno Fizyczne, vol. X (1899 1900), pp. 65-86, 86-101, v.&#xA;XII(1901), p. 95-111, v. XIII(1902), pp. 353-371.&lt;/p&gt;&#xA;&lt;p&gt;This paper contains most of Boettcher's pioneering ideas in holomorphic dynamics. His 1904 memoir in Russian largely overlaps with it. My translation is almost complete, except typesetting of some elaborate but repetitive formulas in Part III.  I feel confident about its accuracy (I am a PhD mathematician, a native speaker of Polish, and I have been living for nearly half of my life in the USA). Sandro helped me with reproducing Boettcher's graphics. This translation could be  viewed as a part of his project.   It was never published or posted online, because I had no idea what would be a good venue for it (posting on ArXiv not as an author of the contents is tricky, and I do not have a Web page). And (as Sandro already said) there was little external interest. While I do not mind  sharing the files,  a word of warning is in order. Boettcher's works should be appreciated for the wealth of the ideas they contain, but they suffer from lack of rigor.&#xA;I pointed this out on many occassions, e.g. here.&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/7576/failures-in-math/7598#7598&quot;&gt;Failures in math&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="3434" LastActivityDate="2024-03-07T02:32:40.713" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17327" PostTypeId="1" CreationDate="2024-03-07T11:33:40.697" Score="1" ViewCount="171" Body="&lt;p&gt;Thanks to Conifold and Chris Leary's comments, I learned that the Nakayama lemma was not first created by a mathematician named Nakayama, but that mathematicians named Azumaya and Krull first created a lemma corresponding to the prototype of the Nakayama lemma. And I recently searched Azumaya algebra.  However, the year in which this Azumaya algebra was created was 1951, which is the same as the Nakayama lemma.  And since Nakayama lemma is used in the process of proving equivalent propositions within Azumaya algebra, I wondered if Azumaya algebra was behind the creation of Nakayama lemma.&lt;/p&gt;&#xA;" OwnerUserId="19597" LastEditorUserId="19597" LastEditDate="2024-03-15T06:02:03.313" LastActivityDate="2024-03-15T06:02:03.313" Title="How did Azumaya come up with the Nakayama lemma?" Tags="|abstract-algebra|book-recommendation|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17328" PostTypeId="2" ParentId="9785" CreationDate="2024-03-07T12:27:34.737" Score="2" Body="&lt;p&gt;The question is a bit ambiguous, as the OP acknowledges. But the answers so far are mostly about mathematicians who may have been crazy (e.g. Godel) or who lived unconventional lives (e.g. Schrodinger). This isn't exactly the same as &amp;quot;chaotic&amp;quot;. In fact, Godel led a rather stunningly ordered life.&lt;/p&gt;&#xA;&lt;p&gt;This &lt;a href=&quot;https://www.reddit.com/r/math/comments/166hvpq/are_there_any_accomplished_alcoholic/?rdt=36913&quot; rel=&quot;nofollow noreferrer&quot;&gt;Reddit thread&lt;/a&gt; is specifically about mathematicians who were alcoholics.&lt;/p&gt;&#xA;&lt;p&gt;In Germany during the Nazi era, many mathematicians (and, of course millions of others) had rather chaotic (to put it mildly) lives. One is the engineer/mathematician Jakow Trachtenberg. He was born in pre-revolution Russia, fled to Weimar Germany, then got sent to a concentration camp where he developed a system of doing math very rapidly and largely in your head.&lt;/p&gt;&#xA;&lt;p&gt;In some sense, Paul Erdos led a deliberately chaotic life, going from house to house with his minimal belongings and his incredible talent and interest in math.&lt;/p&gt;&#xA;" OwnerUserId="10085" LastActivityDate="2024-03-07T12:27:34.737" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17329" PostTypeId="1" AcceptedAnswerId="17340" CreationDate="2024-03-07T15:33:51.210" Score="3" ViewCount="158" Body="&lt;p&gt;In Brian Conrey's article on &lt;a href=&quot;https://aimath.org/%7Ekaur/publications/90.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;Riemann's hypothesis&lt;/a&gt;, one reads in the very beginning:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;On Christmas Eve 1849 Gauss wrote a letter to his former student Encke&#xA;in which he described his thoughts about the number of primes π(x)&#xA;less than or equal to x. Gauss had developed his ideas around 1792&#xA;when he was 15 or 16 years old. His conclusion was that up to a small&#xA;error term π(x) was close to &lt;span class=&quot;math-container&quot;&gt;$li(x)$&lt;/span&gt; the logarithmic integral&lt;/p&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$li(x) = \int_2^x \frac{dt}{log\ t}$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;The strikingly good approximation was computed over and over by Gauss&#xA;at intervals up to 3 million, all computed by Gauss himself who could&#xA;determine the number of primes in a chiliad (block of one thousand&#xA;numbers) in 15 minutes.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Is it known how Gauss could do that: determine the number of primes in up to 3,000 chiliads, which must have taken him 750 hours. What was his trick? And were his numbers correct or just good estimates?&lt;/p&gt;&#xA;" OwnerUserId="6422" LastActivityDate="2024-03-10T18:54:56.220" Title="How did Gauss determine the number of primes?" Tags="|number-theory|gauss|primes|riemann|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17330" PostTypeId="1" AcceptedAnswerId="17339" CreationDate="2024-03-07T22:41:55.663" Score="3" ViewCount="62" Body="&lt;p&gt;I know they weren't required in the early 1800s but obviously they are now so just wondering when this started being required and/or who coined holotype. Internet research yielded no good answers.&lt;/p&gt;&#xA;" OwnerUserId="19583" LastEditorUserId="9523" LastEditDate="2024-03-14T16:43:48.633" LastActivityDate="2024-03-14T16:43:48.633" Title="When did they start requiring holotypes for species description?" Tags="|terminology|biology|taxonomy|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17331" PostTypeId="1" CreationDate="2024-03-07T23:38:41.230" Score="0" ViewCount="36" Body="&lt;p&gt;To me, it seems no number is the work of God, they are all concepts of the mind.&lt;/p&gt;&#xA;&lt;p&gt;However, it seems negative numbers are more artificial than the rest of the numbers out there. So why did he describe them as part of god's work?&lt;/p&gt;&#xA;&lt;p&gt;Thank you.&lt;/p&gt;&#xA;" OwnerUserId="19712" LastActivityDate="2024-03-07T23:38:41.230" Title="Why did Kronecker say &quot;the integers are the work of God, the rest is the work of man&quot;?" Tags="|mathematics|mathematicians|" AnswerCount="0" CommentCount="1" ClosedDate="2024-03-08T01:43:24.803" ContentLicense="CC BY-SA 4.0" />
  <row Id="17333" PostTypeId="1" CreationDate="2024-03-08T07:27:11.117" Score="2" ViewCount="64" Body="&lt;p&gt;I read that: &amp;quot;The term &amp;quot;homomorphism&amp;quot; appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra&amp;quot;. (&lt;a href=&quot;https://en.m.wikipedia.org/wiki/Homomorphism&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://en.m.wikipedia.org/wiki/Homomorphism&lt;/a&gt;)&lt;/p&gt;&#xA;&lt;p&gt;But definitely the concept have already been made (?). Because we know most of mathematicians of pre Klein era need this concept for their works.&lt;/p&gt;&#xA;&lt;p&gt;I want to know who was he? And what was his inspiration and why did he introduce the term like this: f(xy)=f(x)f(y).&#xA;And do we have a previous definition for homomorphism (or isomorphism or every other morphism)?&lt;/p&gt;&#xA;" OwnerUserId="18846" LastActivityDate="2024-03-08T07:27:11.117" Title="Who did introduce homomorphism concept for the first time?" Tags="|abstract-algebra|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17334" PostTypeId="1" CreationDate="2024-03-08T20:52:11.277" Score="0" ViewCount="52" Body="&lt;p&gt;I am currently looking for references on books that deal with the conversion of old units to new units.&lt;/p&gt;&#xA;&lt;p&gt;I am particularly interested in books that actually have tables of conversion because I need to convert some units in my research.&lt;/p&gt;&#xA;&lt;p&gt;I would be grateful for any recommendations that you can provide.&lt;/p&gt;&#xA;&lt;p&gt;Thank you in advance for your help.&lt;/p&gt;&#xA;&lt;p&gt;Sincerely,&#xA;Lucas N.&lt;/p&gt;&#xA;" OwnerUserId="20827" LastActivityDate="2024-03-08T20:52:11.277" Title="References for converting old units to new units" Tags="|units|measurement|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17336" PostTypeId="2" ParentId="9785" CreationDate="2024-03-09T18:54:13.960" Score="1" Body="&lt;p&gt;Feynman seems to fit, at least in some periods of his life.&lt;/p&gt;&#xA;&lt;p&gt;For example, Schweber mentions that in &amp;quot;QED and the men who made it&amp;quot;, 1994, p. 479, with a reference to Gleick's 1992 book on Feynman: &amp;quot;Until his marriage to Gweneth, the emptiness and yearning were filled by one-night stands and tempestuous, destructive love affairs'. I have not read the Gleick's book.&lt;/p&gt;&#xA;&lt;p&gt;Another example. At least for quite some time, Feynman had some unconventional ideas on brushing teeth. It is easy to find references to that. For example, Gell-Mann mentioned that.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-03-09T18:54:13.960" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17337" PostTypeId="1" CreationDate="2024-03-10T00:00:22.797" Score="6" ViewCount="1306" Body="&lt;p&gt;Besides the famous cartoon, of course, there seems to be no portrait of Legendre. Legendre is well regarded nowadays and he was also quite influential at his time, for example, Jacobi and Abel praised him. Furthermore, he came from a wealthy family. So it is surprising to me that there is no actual portrait of him.&lt;/p&gt;&#xA;" OwnerUserId="20835" LastEditorUserId="55" LastEditDate="2024-03-10T06:52:03.967" LastActivityDate="2024-03-10T06:58:13.160" Title="How come there is no portrait of Legendre?" Tags="|mathematics|mathematicians|legendre|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17338" PostTypeId="2" ParentId="17337" CreationDate="2024-03-10T06:50:59.563" Score="11" Body="&lt;p&gt;Because (Adrien-Marie) Legendre, the mathematician, wanted it this way, see &lt;a href=&quot;https://www.ams.org/notices/200911/rtx091101440p.pdf&quot; rel=&quot;noreferrer&quot;&gt;Duren, Changing Faces: The Mistaken Portrait of Legendre&lt;/a&gt;, who quotes Poisson:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;We know from testimony of Poisson&#xA;that Legendre did not welcome personal attention,&#xA;but wanted his work to speak for itself. Speaking&#xA;at Legendre’s funeral, Poisson said:&amp;quot;Our colleague often expressed the desire that in speaking of him it would be&#xA;only a question of his works, which are in fact his&#xA;whole life. I will comply strictly with his wish in&#xA;this tribute that I come to pay...&amp;quot; In view of these remarks, it seems likely that Legendre actively discouraged the making of portraits.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;This is in contrast to the absence of Hooke's portraits, where the cause is more mysterious, see  &lt;a href=&quot;https://royalsociety.org/blog/2010/12/hooke-newton-and-the-missing-portrait&quot; rel=&quot;noreferrer&quot;&gt;Henderson on Hooke, Newton, and the ‘missing’ portrait&lt;/a&gt;. In the absence of Adrien-Marie's real portraits, a portrait of a French revolutionary &lt;a href=&quot;https://en.wikipedia.org/wiki/Adrien-Marie_Legendre#/media/File:Louis_Legendre.jpg&quot; rel=&quot;noreferrer&quot;&gt;Louis Legendre&lt;/a&gt; was substituted for his as early as 1900, and reproduced even by some historians, like Eves and Struik. The error was only discovered by two University of Strasbourg students in 2005.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Once the traditional portrait was known to be&#xA;false, a feverish search began for a true portrait of&#xA;Adrien-Marie Legendre. Miraculously, an authentic&#xA;portrait was discovered during the year 2008 in&#xA;the library of the Institut de France in Paris, among&#xA;a rare collection of seventy-three caricatures of&#xA;members of the Institute. One of the watercolor&#xA;sketches (depicted on the cover of this issue of&#xA;the Notices) shows the heads of Legendre and&#xA;Fourier, with bodies lightly sketched in pencil.&#xA;Their names “Legendre” and “Fourier” are written&#xA;below the sketch. Fourier is easily recognized&#xA;from existing portraits, but Legendre takes on a&#xA;totally new appearance. This is the only image of&#xA;Adrien-Marie Legendre known to exist.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;[...] &lt;em&gt;The album of caricatures has a mysterious&#xA;history. It is the work of Julien-Léopold Boilly&#xA;(1796–1874). His father Louis-Léopold Boilly&#xA;(1761–1845) was a more famous artist, a member&#xA;of the Académie des Beaux-Arts best known for&#xA;his oil portraits. The younger Boilly was commissioned&#xA;to do a series of engravings of members&#xA;of the Institute, but the work was never finished;&#xA;the partial collection (also in the library of the&#xA;Institut de France) includes a formal portrait of&#xA;Fourier but apparently none of Legendre... The collection also contains a caricature of Laplace, but no other mathematicians are represented.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;The entire album can be viewed on the&#xA;website of the &lt;a href=&quot;https://www.photo.rmn.fr/C.aspx?VP3=CMS3&amp;amp;VF=Home&quot; rel=&quot;noreferrer&quot;&gt;Réunion des musées nationales&lt;/a&gt; (Click on “recherche”, then type in “Boilly” for searching and click repeatedly on “suivante”). The two Boilly artists appear to be confused, since the website attributes the album&#xA;of caricatures to the father Louis-Léopold.&#xA;In the juxtaposition of Legendre and Fourier,&#xA;the artist seems to be commenting on a contrast&#xA;of personalities: Fourier fat and jolly, Legendre&#xA;lean and acerbic. However, the historical record&#xA;sometimes portrays Legendre in quite a different&#xA;light. A case in point is the kind reception he gave&#xA;to Abel and Jacobi.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-03-10T06:58:13.160" LastActivityDate="2024-03-10T06:58:13.160" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17339" PostTypeId="2" ParentId="17330" CreationDate="2024-03-10T11:00:44.143" Score="3" Body="&lt;p&gt;As far as botany is concerned, the concept of type specimen and obligation of using one was &lt;a href=&quot;https://www.iapt-taxon.org/historic/history.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;introduced&lt;/a&gt; in the 1935 version of the code (the so-called &lt;a href=&quot;https://www.iapt-taxon.org/historic/1935.htm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Cambridge Rules&lt;/a&gt;).&lt;/p&gt;&#xA;&lt;p&gt;For zoology, surprisingly, it is only compulsory to explicitely name a name-bearing specimen &lt;a href=&quot;https://code.iczn.org/introduction/#historical-background&quot; rel=&quot;nofollow noreferrer&quot;&gt;since 1999&lt;/a&gt;, when the 4th (and current) edition of the ICZN code was published.&lt;/p&gt;&#xA;&lt;p&gt;The term 'holotype' though (and the concept of typification) is older. The earliest occurrence I can find is in this short communication in the journal &lt;em&gt;Science&lt;/em&gt; from 1897: &lt;a href=&quot;https://www.science.org/doi/pdf/10.1126/science.5.121.636&quot; rel=&quot;nofollow noreferrer&quot;&gt;What is a type in Natural History?&lt;/a&gt; by Charles Schuchert.&lt;/p&gt;&#xA;" OwnerUserId="84" LastActivityDate="2024-03-10T11:00:44.143" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17340" PostTypeId="2" ParentId="17329" CreationDate="2024-03-10T18:54:56.220" Score="6" Body="&lt;p&gt;For what it's worth:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;As Goldshtein writes, “Evidently Gauss considered the tabulation of&#xA;primes as some sort of pastime and amused himself by compiling&#xA;extensive tables on how the primes distribute themselves in various&#xA;intervals of length 1000.” To do this, he was using larger and larger&#xA;tables of primes or tables of prime decomposition.18&lt;/p&gt;&#xA;&lt;p&gt;These included (see the letter in Footnote 15) Vega’s Tables&#xA;(available to Gauss in 1796, and listing primes up to 400,031), then&#xA;in 1811 a friend made Gauss a present of Chernau’s cribrum (an exerpt&#xA;is on the right 19) reaching 1,020,000. Finally, the Burckhardt’s&#xA;factorization tables (published in 3 parts in 1814–17 and organized&#xA;similarily to the cribrum) allowed Gauss to investigate primes up to&#xA;the limit of these tables: to 3,000,000.&lt;/p&gt;&#xA;&lt;p&gt;Footnote 18:  Usually these tables were compiled by other people.&#xA;However, since he could find some errors in these tables, apparently&#xA;he have been duplicating at least some calculations himself.&lt;/p&gt;&#xA;&lt;p&gt;Footnote 15: This is discussed in his letter to Johann Franz Encke of&#xA;1849 (in a book form)...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://math.berkeley.edu/%7Eserganov/ilyaz.org/papers/in_preparation/Reciprocity_Langlands/Gauss-missed-2023-09-12.pdf&quot; rel=&quot;noreferrer&quot;&gt;Source&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;Just a comment: looks like the author of Cribrum Arithmeticum was L. Chernac, not Chernau.&lt;/p&gt;&#xA;" OwnerUserId="9808" LastActivityDate="2024-03-10T18:54:56.220" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17341" PostTypeId="1" CreationDate="2024-03-10T20:00:41.867" Score="1" ViewCount="114" Body="&lt;p&gt;My understanding is, Newton in the 17th century did not use this formula but rather said, in words basically that if you apply a force it will cause a mass to accelerate in the direction of that force.&lt;/p&gt;&#xA;&lt;p&gt;How did we move from descriptive words to using multiplication to express the same idea? Many people must have said, What does multiplication have to do with exerting a force? (Indeed, I ask that same question today in the 21st century.)&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2024-03-10T21:28:31.153" Title="F = ma -- How was did we come to understand that this compact form expressed what Newton said in words?" Tags="|mathematics|terminology|classical-mechanics|newton|" AnswerCount="1" CommentCount="7" ContentLicense="CC BY-SA 4.0" />
  <row Id="17342" PostTypeId="2" ParentId="17341" CreationDate="2024-03-10T21:22:06.227" Score="3" Body="&lt;p&gt;Sometimes you just have to Wikipedia-it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the &lt;em&gt;Principia&lt;/em&gt;, he had already developed calculus (which he called &amp;quot;the science of fluxions&amp;quot;), but in the &lt;em&gt;Principia&lt;/em&gt; he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of Euclid to be more rigorous. Consequently, the &lt;em&gt;Principia&lt;/em&gt; does not express acceleration as the second derivative of position, and so it does not give the second law as &lt;span class=&quot;math-container&quot;&gt;$ F=ma$&lt;/span&gt;. This form of the second law was written (for the special case of constant force) at least as early as 1716, by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jakob_Hermann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jakob Hermann&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Note that Newton says that &lt;span class=&quot;math-container&quot;&gt;$F$&lt;/span&gt; is proportional to &lt;span class=&quot;math-container&quot;&gt;$\dot p$&lt;/span&gt; (the derivative/change of momentum &lt;span class=&quot;math-container&quot;&gt;$p= m v$&lt;/span&gt;). No need to have a product of anything.&lt;/p&gt;&#xA;&lt;p&gt;Looking at Hermann's book (&lt;em&gt;Phoronomia&lt;/em&gt;), he writes directly &lt;span class=&quot;math-container&quot;&gt;$G=M \mathrm dV/\mathrm dt$&lt;/span&gt;, &lt;span class=&quot;math-container&quot;&gt;$G$&lt;/span&gt; is the weight, &lt;span class=&quot;math-container&quot;&gt;$M$&lt;/span&gt; the mass and &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; the speed.&lt;/p&gt;&#xA;&lt;p&gt;Leonhard Euler wrote independently in &lt;em&gt;Novi Commentarii&#xA;academiae scientiarum Petropolitanae&lt;/em&gt; that  &lt;span class=&quot;math-container&quot;&gt;$F=M \mathrm d^2x/\mathrm d t^2$&lt;/span&gt;, in 1775.&lt;/p&gt;&#xA;&lt;p&gt;Futher reading: &lt;a href=&quot;https://plato.stanford.edu/entries/newton-principia/&quot; rel=&quot;nofollow noreferrer&quot;&gt;https://plato.stanford.edu/entries/newton-principia/&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-03-10T21:28:31.153" LastActivityDate="2024-03-10T21:28:31.153" CommentCount="5" ContentLicense="CC BY-SA 4.0" />
  <row Id="17343" PostTypeId="1" CreationDate="2024-03-11T22:52:39.903" Score="2" ViewCount="33" Body="&lt;p&gt;My understanding is that the above pairs of terms are identical in definition, indicating acceptors/donors of an electron pair. So, why are both used?&lt;/p&gt;&#xA;" OwnerUserId="19583" LastActivityDate="2024-03-11T22:52:39.903" Title="Why are the terms electrophile/nucleophile used? Are they not identical to Lewis acids/bases?" Tags="|chemistry|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17344" PostTypeId="1" AcceptedAnswerId="17358" CreationDate="2024-03-12T22:43:19.227" Score="1" ViewCount="75" Body="&lt;p&gt;The Human Genome Project was famously declared complete in 2003, and to date there are 6000 genes discovered/annotated. What was the first one discovered and annotated?&lt;/p&gt;&#xA;" OwnerUserId="19583" LastEditorUserId="20856" LastEditDate="2024-03-18T07:20:46.897" LastActivityDate="2024-03-19T08:11:45.503" Title="What was the first human gene sequenced/annotated?" Tags="|biology|genetics|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17345" PostTypeId="1" CreationDate="2024-03-13T21:01:05.863" Score="1" ViewCount="61" Body="&lt;p&gt;The description of the 1993 English translation of &lt;em&gt;&lt;a href=&quot;https://doi.org/10.1201/9781003062950&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Quantum Dice&lt;/a&gt;&lt;/em&gt;, by Ponomarev and Kurchatov, as well as one of the quotes for chapter 2 of &lt;em&gt;&lt;a href=&quot;https://www.cambridge.org/highereducation/books/quantum-computation-and-quantum-information/01E10196D0A682A6AEFFEA52D53BE9AE&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Computation and Quantum Information&lt;/a&gt;&lt;/em&gt;, by Nielsen and Chuang (2000), contain this quote attributed to Einstein:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Quantum mechanics: Real Black Magic Calculus.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The former reference attributes it to &amp;quot;a letter in 1925&amp;quot;. Nielsen and Matsuschak specify that it is a letter to a colleague from that year in their recent essay &amp;quot;&lt;a href=&quot;https://quantum.country/qm&quot; rel=&quot;nofollow noreferrer&quot;&gt;Quantum Mechanics Distilled&lt;/a&gt;&amp;quot;. And finally, Nielsen tweeted in 2023 that &lt;a href=&quot;https://twitter.com/michael_nielsen/status/1664051112121606144&quot; rel=&quot;nofollow noreferrer&quot;&gt;Chuang had a reference&lt;/a&gt; for this quote.&lt;/p&gt;&#xA;&lt;p&gt;However, neither my Google fu nor (presumably) my Deutsch are up to the task of actually finding the letter in question.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;In what letter does this quote of Einstein's actually occur?&lt;/strong&gt;&lt;/p&gt;&#xA;" OwnerUserId="20847" LastEditorUserId="10106" LastEditDate="2024-03-14T03:36:52.840" LastActivityDate="2024-03-14T03:36:52.840" Title="Origin of Einstein quote &quot;Quantum mechanics: Real Black Magic Calculus&quot;" Tags="|physics|reference-request|quantum-mechanics|einstein|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17347" PostTypeId="2" ParentId="8050" CreationDate="2024-03-14T13:15:17.343" Score="0" Body="&lt;p&gt;As far as I've been able to find out, the implication for Pythagoreans usually went more the other way around. The point was that all magnitudes being commensurable was a consequence of their more fundamental belief in the notion of numbers as a fundamental element of the universe.&lt;/p&gt;&#xA;&lt;p&gt;It can be hard to know exactly what Pythagoreans believed in that respect, since there are very few Pythagorean texts on that exact topic (as far as I know), especially during the early period of Pythagoreanism before any idea of incommensurable magnitude existed, and we mostly have their ideas from other authors, but they seemed to have believed that all things being numbers, this includes space itself, and every magnitude is &amp;quot;discrete&amp;quot;, if we can apply that modern notion to whatever they believed. For a few notions on this see for instance book XIII of Aristotle's Metaphysics :&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;And the Pythagoreans, also, believe in one kind of number-the&#xA;mathematical; only they say it is not separate but sensible substances&#xA;are formed out of it. For they construct the whole universe out of&#xA;numbers-only not numbers consisting of abstract units; they suppose&#xA;the units to have spatial magnitude.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The Pythagorean version in one way affords fewer difficulties than&#xA;those before named, but in another way has others peculiar to itself.&#xA;For not thinking of number as capable of existing separately removes&#xA;many of the impossible consequences; but that bodies should be&#xA;composed of numbers, and that this should be mathematical number, is&#xA;impossible. For it is not true to speak of indivisible spatial&#xA;magnitudes; and however much there might be magnitudes of this sort,&#xA;units at least have not magnitude; and how can a magnitude be composed&#xA;of indivisibles?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There are many other statements which make this unclear however. They also seemed to think that points did exist, associated to the number 1, and that lines were associated with the number 2, so that points were the extremities of lines, but as this was quite a contentious point in antiquity (cf. for instance Sextus Empiricus &amp;quot;Against the geometers&amp;quot;), they may not have believed that lines were &lt;em&gt;composed&lt;/em&gt; of points. What they exactly imagined the structure of space was like from these ideas isn't obvious (it probably did not work amazingly), but one area where they were interested in was the notion of &lt;a href=&quot;https://en.wikipedia.org/wiki/Figurate_number&quot; rel=&quot;nofollow noreferrer&quot;&gt;figurate numbers&lt;/a&gt;. Of course, the thoughts of the Pythagoreans weren't necessarily united, unchanging (it is quite possible that incommensurable lengths were &lt;em&gt;discovered&lt;/em&gt; by Pythagoreans, although the evidence for this aren't that clear) nor resembling modern notions of a formal system, so it is hard to figure out the exact ideas that they were thinking.&lt;/p&gt;&#xA;&lt;p&gt;The notion that there is a smallest length of a line does appear in other geometers, like Xenocrates, (maybe) Democritus or the Epicureans (which is where a lot of the specific arguments against the notions are found later on), which are maybe based on Pythagorean ideas. Sometimes this smallest length is infinitesimal, sometimes not.&lt;/p&gt;&#xA;&lt;p&gt;This idea that all magnitudes are discrete in this sense leads to the notion of commensurability, as for instance Proclus shows in his commentary on Euclid (quoting Geminus of Rhodes) that incommensurable magnitudes leads to infinite divisibility.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;For when geometers demonstrate that there is incommensurability among&#xA;magnitudes and that not all magnitudes are commensurable with one&#xA;another, what else could we say they are demonstrating than that every&#xA;magnitude is divisible indefinitely and that we can never reach an&#xA;indivisible part which is the least common measure of magnitudes?&#xA;This, then, is demonstrable, but it is an axiom that every continuum&#xA;is divisible; hence a finite line, being continuous, is divisible.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;What wrong theorems exactly those ideas led to for the Pythagoreans is unclear as far as I know, except for the obvious cases : the assumption that any two lengths are commensurable, such as the hypothenuse to the other sides of a triangle. In particular, a lot of Pythagorean mathematics was the theory of proportions, a theory of proportion that was only valid for figures of rational proportions, which I would guess they imagined valid for any figure. If they indeed believed in indivisible lines, this would make some basic theorems wrong, as well, such as the ability to divide any segment into two equal segments, depending on if the &amp;quot;length&amp;quot; was even or odd.&lt;/p&gt;&#xA;&lt;p&gt;It is quite possible that the discovery of incommensurable length led to the general abandonment (possibly gradual) of the notion of those indivisible lines (or, in the case of Epicurians, the abandonment of geometry altogether), so that later Pythagoreans may have had different ideas.&lt;/p&gt;&#xA;" OwnerUserId="2628" LastActivityDate="2024-03-14T13:15:17.343" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17349" PostTypeId="1" CreationDate="2024-03-15T10:33:59.397" Score="0" ViewCount="76" Body="&lt;p&gt;Some mathematical concepts, such as that of number and that of geometrical figure, presumably originate from pre-existing notions already used by at least some non-mathematicians.&lt;/p&gt;&#xA;&lt;p&gt;Others seem to have a different origin, maybe the concept of trope for example.&lt;/p&gt;&#xA;&lt;p&gt;What are the concepts, if any, which are currently the object of active research in mathematics, which do not have any obvious origin outside of mathematics?&lt;/p&gt;&#xA;&lt;p&gt;Thank you for any scholarly reference.&lt;/p&gt;&#xA;" OwnerUserId="20851" LastActivityDate="2024-03-15T10:33:59.397" Title="Which mathematical concepts do not have any obvious origin outside mathematics?" Tags="|terminology|reference-request|philosophy-of-science|definition|" AnswerCount="0" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17351" PostTypeId="1" CreationDate="2024-03-15T15:48:34.607" Score="0" ViewCount="79" Body="&lt;p&gt;It seems to me that he was able to reformulate Maxwell's equations in a more understandable form and in fact come up with vector calculus without finishing high school would arguably cause him to be ranked with people like Galois and Ramanujan. Does this make sense?&lt;/p&gt;&#xA;" OwnerUserId="10856" LastActivityDate="2024-03-15T17:39:01.437" Title="Where does Oliver Heaviside fit in the ranks of physicists/mathematicians?" Tags="|mathematics|calculus|electricity|" AnswerCount="1" CommentCount="1" ClosedDate="2024-03-16T07:59:19.263" ContentLicense="CC BY-SA 4.0" />
  <row Id="17352" PostTypeId="1" AcceptedAnswerId="17357" CreationDate="2024-03-15T17:10:56.237" Score="4" ViewCount="102" Body="&lt;p&gt;The klepsydra is a water clock consisting of a vessel with a hole at the bottom. The height of the water level is used to measure time. For this level to be proportional to time, the vessel has to have the shape of a quartic parabola: &lt;span class=&quot;math-container&quot;&gt;$h=cr^4$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$h$&lt;/span&gt; is the height and &lt;span class=&quot;math-container&quot;&gt;$r$&lt;/span&gt; is the radius of the cross-section at this height.&lt;/p&gt;&#xA;&lt;p&gt;Question: did the ancients know this? To discover this fact, mathematics and physics of Archimedes would be sufficient, in principle. But I am not aware of any ancient literature discussing the theory of water clocks.&lt;/p&gt;&#xA;&lt;p&gt;It is possible that they could somehow discover the required shape empirically. It is hard to judge from the pictures since one needs to know the shape of the inner surface which is usually not seen in the pictures.&lt;/p&gt;&#xA;&lt;p&gt;But even more interesting question is: if they tried to discover the correct shape by experiment, what kind of standard time could they use for comparison? Before the modern era, essentially only sundials and klepsydras were available to measure time.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="12724" LastEditDate="2024-03-16T07:58:27.107" LastActivityDate="2024-03-17T20:17:36.407" Title="Did the ancients know how to construct a correct klepsydra?" Tags="|measurement|time|antiquity|" AnswerCount="1" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17353" PostTypeId="2" ParentId="17351" CreationDate="2024-03-15T17:39:01.437" Score="1" Body="&lt;p&gt;(Question whether Heaviside's achievements &amp;quot;would arguably cause him to be ranked with people like Galois and Ramanujan. Does this make sense?&amp;quot;)&lt;/p&gt;&#xA;&lt;p&gt;Yes, I would think it makes a lot of sense. Even taking into account that much of his achievement had applications in view (often in electrical engineering) rather than belonging to pure mathematics alone, the mathematical content of his work is notable in itself.&lt;/p&gt;&#xA;&lt;p&gt;Some references to both aspects of his work may be found in &lt;a href=&quot;https://royalsocietypublishing.org/doi/10.1098/rsta.2017.0457&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;On Heaviside's contributions to transmission line theory: waves, diffusion and energy flux&amp;quot;, C Donaghy-Spargo (2018)&lt;/a&gt;. Some  explanation, with references to some of the background and origin of his work including the Heaviside operational calculus and other scientific and engineering publications, is also given in this HSM answer, &lt;a href=&quot;https://hsm.stackexchange.com/questions/8302/origin-of-the-heaviside-function/8308#8308&quot;&gt;The origin of the Heaviside function&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;Arguably Heaviside's work is now underrated, as it was during his lifetime (in spite of his eventual FRS). There have been some steps taken to remember him appropriately, e.g. in 2018 there was a theme issue of &lt;em&gt;Philosophical Transactions of the Royal Society A&lt;/em&gt; on &amp;quot;&lt;em&gt;Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’&lt;/em&gt;&amp;quot;. Nevertheless, the history remains that, relying on his mathematical analysis, he invented improvements in transmission lines for telephony that led to his proposal to reduce distortion, but the whole was later patented out of his hands by Michael Pupin in the USA, who sold the rights for much money, while Heaviside lived and died in poverty. (See also, besides the links already given above, e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/Heaviside_condition&quot; rel=&quot;nofollow noreferrer&quot;&gt;the Heaviside condition&lt;/a&gt;, and &lt;a href=&quot;https://www.pa3fwm.nl/technotes/tn28-heaviside-transmission-lines.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;'Oliver Heaviside and the theory of transmission lines', by P T de Boer&lt;/a&gt;).&lt;/p&gt;&#xA;" OwnerUserId="5788" LastActivityDate="2024-03-15T17:39:01.437" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17355" PostTypeId="2" ParentId="5252" CreationDate="2024-03-17T00:28:28.017" Score="-1" Body="&lt;p&gt;Dynamical systems are described by a set of differential/difference equations(ordinary, partial, deterministic, stochastic, etc).&#xA;Although most of these equations must be solved numerically, the solutions describe the evolution of state variables in time and/or space.&lt;/p&gt;&#xA;&lt;p&gt;In the Thermodynamics textbooks I have examined that kind of structure does not exist unless or until one looks at Chemical Kinetics and dynamics.&lt;/p&gt;&#xA;&lt;p&gt;So Thermodynamics appears to be a fuzzily defined discipline with thousands of empirically acquired steady-state data.&lt;/p&gt;&#xA;" OwnerUserId="20854" LastActivityDate="2024-03-17T00:28:28.017" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17356" PostTypeId="1" CreationDate="2024-03-17T12:29:22.743" Score="2" ViewCount="72" Body="&lt;p&gt;The Lanczos tri-diagonalization process is widely or even routinely used today. It is said that it is useful for obtaining the extremal eigenvalues, but useless for medium eigenvalues.&lt;/p&gt;&#xA;&lt;p&gt;But who discovered this point? I suspect that Lanczos himself did not know this.&lt;/p&gt;&#xA;" OwnerUserId="2641" LastEditorUserId="55" LastEditDate="2024-03-19T11:35:24.057" LastActivityDate="2024-03-19T11:35:24.057" Title="Who discovered that the Lanczos method can only calculate extremal eigenvalues of large matrices?" Tags="|mathematics|linear-algebra|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17357" PostTypeId="2" ParentId="17352" CreationDate="2024-03-17T20:17:36.407" Score="5" Body="&lt;p&gt;At least, in ancient Greece and Rome I am not aware of the vessel shape being used as a way to uniformize the flow of water. Instead, &amp;quot;correct&amp;quot; clepsydras were constructed using a different mechanism. The water level in a reservoir (of any shape) was kept constant by inflow source and overflow pipes, and time was measured by floats rising with the (uniform due to constant pressure) outflow from the reservoir. Some of those are described by Vitruvius, see &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Water_clocks&quot; rel=&quot;noreferrer&quot;&gt;McTutor, Water-clocks&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="55" LastActivityDate="2024-03-17T20:17:36.407" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17358" PostTypeId="2" ParentId="17344" CreationDate="2024-03-18T13:45:58.153" Score="5" Body="&lt;p&gt;&lt;strong&gt;Answer&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Although the original publication makes no claims to this effect, I suggest that the first human gene sequence to be reported was that of &lt;strong&gt;β-globin in 1980&lt;/strong&gt;.&lt;/p&gt;&#xA;&lt;p&gt;The reference to the publication is &lt;a href=&quot;https://www.cell.com/cell/abstract/0092-8674(80)90428-6&quot; rel=&quot;nofollow noreferrer&quot;&gt;Lawn &lt;em&gt;et al.&lt;/em&gt; (1980) &lt;em&gt;Cell&lt;/em&gt; &lt;strong&gt;21&lt;/strong&gt;, 647–651&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Historical context&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Following the acceptance of the messenger RNA (mRNA) hypothesis in 1959/1960 (reviewed by &lt;a href=&quot;https://doi.org/10.1016/j.cub.2015.05.032&quot; rel=&quot;nofollow noreferrer&quot;&gt;M. Cobb in  Current Biology 25, R523–R548, 2015&lt;/a&gt;) there were two main requirements/obstacles to determining the sequence of a particular mRNA or of the gene encoding it:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;Isolation and purification&lt;/li&gt;&#xA;&lt;li&gt;Sequencing methodology&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The combination of these problems meant that most initial work was done on the small RNAs, transfer RNA (tRNA) and the various components of ribosomal RNA (rRNA), and in bacterial or yeast systems. Sequencing was of radioactively labelled RNAs by laborious chemical techniques.&lt;/p&gt;&#xA;&lt;p&gt;It was for the reasons of isolation and purification that the first animal mRNAs sequenced by these means were the globin mRNAs which dominate the spectrum of mRNAs in reticulocytes. At that time there was no way of isolating pure genes, and no methods of sequencing DNA.&lt;/p&gt;&#xA;&lt;p&gt;The purification problem was overcome by the introduction of &lt;a href=&quot;https://en.wikipedia.org/wiki/Molecular_cloning&quot; rel=&quot;nofollow noreferrer&quot;&gt;molecular cloning&lt;/a&gt; in plasmid vectors (e.g. &lt;a href=&quot;https://en.wikipedia.org/wiki/PBR322&quot; rel=&quot;nofollow noreferrer&quot;&gt;pBR322, introduced in 1977&lt;/a&gt;), made possible by the previous discovery of &lt;a href=&quot;https://en.wikipedia.org/wiki/Restriction_enzyme&quot; rel=&quot;nofollow noreferrer&quot;&gt;type II restriction endonucleases&lt;/a&gt;. This could be applied to mRNA by copying to complementary DNA (cDNA) using the enzyme, reverse transcriptase. The discovery (1971) that eukaryotic mRNAs were &lt;a href=&quot;https://en.wikipedia.org/wiki/Polyadenylation&quot; rel=&quot;nofollow noreferrer&quot;&gt;polyadenylated at their 3′ ends&lt;/a&gt; greatly facilitated their cDNA cloning.&lt;/p&gt;&#xA;&lt;p&gt;The availability of cDNA clones for specific eukaryotic mRNAs allowed screening of genomic DNA by hybridization. Thus the genes for the mRNAs which were easiest to purify because of their abundance in particular tissues were also the first to be cloned and mapped. The various haemoglobins were obvious candidates, and the shift to the human genome was natural because of the extensive interest in human haemoglobinopathies, and the corresponding genetic knowledge. Other genes that were relatively easy to clone included those of muscle proteins, such as actin. The &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC347018/&quot; rel=&quot;nofollow noreferrer&quot;&gt;first human actin gene sequence&lt;/a&gt; was not, however, published until 1982. It should also be noted that identification of these genes was also facilitated by the fact that the sequences of their protein products had been determined directly.&lt;/p&gt;&#xA;&lt;p&gt;The sequencing problem was overcome by the introduction in 1977 of the &lt;a href=&quot;https://www.ncbi.nlm.nih.gov/pmc/articles/PMC392330/&quot; rel=&quot;nofollow noreferrer&quot;&gt;Maxam/Gilbert (chemical) method of sequencing DNA&lt;/a&gt;, allowing over 100 nucleotides to be determined at a time.&lt;/p&gt;&#xA;&lt;p&gt;It was the combination of these two technologies that was important. Ironically, The first &lt;a href=&quot;https://pubmed.ncbi.nlm.nih.gov/873928/&quot; rel=&quot;nofollow noreferrer&quot;&gt;human globin mRNA sequence, published in 1977&lt;/a&gt;, used cDNA cloning, but then transcribed the cDNA into radioactive RNA, and employed the laborious chemical methods already mentioned (ribonuclease digestion and 2D chromatography, followed by fragment identification).&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Conclusion&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Considering the time of introduction of Maxam/Gilbert sequencing method (1977) and that of publication of the sequence of the human β-globin gene (1980), together with the other historical factors mentioned above, I doubt that there are any other contenders for the claim for the first human gene sequence.&lt;/p&gt;&#xA;&lt;p&gt;What I feel is more important than “which was first?” is “why?” and “why then?”, questions which I hope have also been answered by the historical context described above. This also suggests the further methodological problems that needed to be solved before the Human Genome Project was considered feasible ten years later.&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Appendix&lt;/strong&gt;&lt;/p&gt;&#xA;&lt;p&gt;Annotated 2060 nucleotide sequence from Lawn &lt;em&gt;et al.&lt;/em&gt; (1980)&lt;/p&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://i.stack.imgur.com/NAYCO.png&quot; rel=&quot;nofollow noreferrer&quot;&gt;&lt;img src=&quot;https://i.stack.imgur.com/NAYCO.png&quot; alt=&quot;Beta Globin gene sequence&quot; /&gt;&lt;/a&gt;&lt;/p&gt;&#xA;" OwnerUserId="20856" LastEditorUserId="20856" LastEditDate="2024-03-19T08:11:45.503" LastActivityDate="2024-03-19T08:11:45.503" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17359" PostTypeId="2" ParentId="16222" CreationDate="2024-03-19T10:27:06.133" Score="3" Body="&lt;p&gt;None of &lt;a href=&quot;https://mathshistory.st-andrews.ac.uk/HistTopics/Bolzano_manuscripts&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bernhard Bolzano's manuscripts&lt;/a&gt; were printed in his lifetime, and only one work was published in 1852, three years after his death (it was praised by G. Cantor). The rest had to wait until 1920s, and an example of a continuous-yet-undiffirentiable function he made up around 1830 was only published in 1930.&lt;/p&gt;&#xA;" OwnerUserId="13566" LastActivityDate="2024-03-19T10:27:06.133" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17360" PostTypeId="1" CreationDate="2024-03-20T07:30:21.357" Score="1" ViewCount="62" Body="&lt;p&gt;What were the fundamental principles and ideas of projective geometry that made people consider it groundbreaking and separates it from the rest of the geometry?  I would love to learn about a good book on projective geometry I could read.&lt;/p&gt;&#xA;" OwnerUserId="19723" LastActivityDate="2024-03-20T12:29:15.710" Title="Which people are considered to be the founders of Projective Geometry?" Tags="|reference-request|mathematics-social-history|projective-geometry|" AnswerCount="1" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17361" PostTypeId="2" ParentId="17360" CreationDate="2024-03-20T12:29:15.710" Score="2" Body="&lt;p&gt;While the wiki article mentions Desargues, it is inaccurate in that it implies that Desargues only had the notion of a point at infinity, whereas he went much further, including the line at infinity.  A series of recent papers (not mentioned in the wiki articles) tries to give Desargues the proper credit, as he went amazingly far:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Anglade, Marie; Briend, Jean-Yves Nombrils, bruslans, autrement&#xA;foyerz: la géométrie projective en action dans le Brouillon Project de&#xA;Girard Desargues.  Arch. Hist. Exact Sci. 76 (2022), no. 2, 173–206.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Briend, Jean-Yves Mathématiques en perspective: Desargues, la Hire,&#xA;le Poîvre.  Arch. Hist. Exact Sci. 75 (2021), no. 6, 699–736.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Anglade, Marie; Briend, Jean-Yves Le diamètre et la traversale: dans&#xA;l'atelier de Girard Desargues.  Arch. Hist. Exact Sci. 73 (2019),&#xA;no. 4, 385–426.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Anglade, Marie; Briend, Jean-Yves La notion d'involution dans le&#xA;Brouillon project de Girard Desargues.  Arch. Hist. Exact Sci. 71&#xA;(2017), no. 6, 543–588.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="604" LastActivityDate="2024-03-20T12:29:15.710" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17363" PostTypeId="1" CreationDate="2024-03-20T20:38:07.350" Score="1" ViewCount="119" Body="&lt;p&gt;I asked this question on MSE &lt;a href=&quot;https://math.stackexchange.com/questions/4884328/are-there-alternatives-functions-for-the-gamma-function-that-was-used-as-general&quot;&gt;here&lt;/a&gt;&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$\Gamma(x)= \int_0^ \infty e^{-t}t^{x-1}dt \ \ \ \  \  x&amp;gt;0.  $$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Bohr and Mollerup showed that the gamma  function is the only positive  function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; defined on &lt;span class=&quot;math-container&quot;&gt;$(0,\infty)$&lt;/span&gt; that has these properties&lt;/p&gt;&#xA;&lt;ol&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x+1)=xf(x)$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(1)=1$&lt;/span&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; is a continuous&lt;/li&gt;&#xA;&lt;li&gt;&lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt;  is log convex&lt;/li&gt;&#xA;&lt;/ol&gt;&#xA;&lt;p&gt;But before Bohr and Mollerup  why did Euler or Bernoulli  chose this function ? there are infinite many  positive continuous function &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; st &lt;span class=&quot;math-container&quot;&gt;$f(x+1)=xf(x)$&lt;/span&gt;, was there any alternatives functions for the gamma function as a generalisation for the factorials  that was used before and now we stopped? If there is one then why do we stop using it? If there isn't one then  how did Euler and other found the &amp;quot;Only&amp;quot; &amp;quot;natural&amp;quot; generalisation  for the factorials centuries before   Bohr and Mollerup as somehow they knew that this solution is unique and why &lt;span class=&quot;math-container&quot;&gt;$\Gamma$&lt;/span&gt;&#xA;stands out from other options.&lt;/p&gt;&#xA;" OwnerUserId="18466" LastEditorUserId="18466" LastEditDate="2024-03-21T04:41:18.843" LastActivityDate="2024-03-21T04:41:18.843" Title="Are there alternatives functions for the gamma function that was used as generalisation for the factorials?" Tags="|mathematics|real-analysis|" AnswerCount="0" CommentCount="8" ContentLicense="CC BY-SA 4.0" />
  <row Id="17364" PostTypeId="1" CreationDate="2024-03-21T16:09:23.123" Score="3" ViewCount="93" Body="&lt;p&gt;A couple of years ago I read an anecdote (amounting to a few sentences, very similar to this very paragraph) about a prominent scientist (most likely a mathematician or a physicist) who submitted a paper for publication and was asked by the editor to shorten it. He returned the paper with minimal changes (something like removing a single 'that', 'and' or a similar insignificant word or even just a comma). The paper seems to have been published without further questions asked.&lt;/p&gt;&#xA;&lt;p&gt;Last year my friend told me that anecdote as if it had happened to Oliver Heaviside at which point I vaguely recalled that I had read a similar story about someone else. However I failed to look it up in the books as well as to find it via internet search engines and started wondering if my recollection was merely a memory illusion. That's why I am posting this question on this site (which seems to be the most appropriate place for such an inquiry across the StackExchange network).&lt;/p&gt;&#xA;" OwnerUserId="20864" LastEditorUserId="12330" LastEditDate="2024-03-22T08:12:48.943" LastActivityDate="2024-03-22T08:12:48.943" Title="Which scientist had hard time shortening his paper?" Tags="|biographical-details|anecdote|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17365" PostTypeId="1" AcceptedAnswerId="17366" CreationDate="2024-03-22T12:39:55.230" Score="2" ViewCount="954" Body="&lt;p&gt;Laplace gave the theory of bodies so heavy that they would not let even light escape. This theory needs light to have a mass and hence is compatible with the corpuscular theory.&#xA;So how did Laplace feel when the corpuscular theory was proved false by Thomas Young a decade later?&lt;/p&gt;&#xA;" OwnerUserId="20866" LastEditorUserId="6609" LastEditDate="2024-03-22T12:50:29.600" LastActivityDate="2024-03-22T18:43:03.883" Title="What was Pierre-Simon Laplace's reaction to Young's double slit experiment?" Tags="|light|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17366" PostTypeId="2" ParentId="17365" CreationDate="2024-03-22T13:38:08.147" Score="10" Body="&lt;p&gt;Short answer: Pierre-Simon Laplace was skeptic of Thomas Young, but he had to go silent after the works of Augustin-Jean Fresnel and François Arago.&lt;/p&gt;&#xA;&lt;h2&gt;Long answer&lt;/h2&gt;&#xA;&lt;p&gt;Laplace was the reference for the theory of corpuscular light. As you have mentioned, in 1796 Laplace considered that light may not be able to escape a start if the density is large enough, making some bodies invisible. Four years later, Young published his famous double slit experiment.&lt;/p&gt;&#xA;&lt;p&gt;Now, all French opticians at the time were big followers of Laplace (Jean-Baptiste Biot, Etienne-Louis Malus, Arago, Siméon Denis Poisson and so on). When Young publishes his experiment and his defense for wave theory, historical sources (see reference below) tells us about how Young's result was received in France:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;In France, however, Young's work was almost completely ignored.&#xA;Not a word about it can be found in Haüy's &lt;em&gt;Traité de physique&lt;/em&gt; of&#xA;1806 or Biot's longer &lt;em&gt;Traité&lt;/em&gt; of 1816. Nor did the French corpuscularians respond in print to Young's detailed critique of Laplace's&#xA;theory of extraordinary refraction in the &lt;em&gt;Quarterly Review&lt;/em&gt; for 1809&#xA;or his criticism of Biot's theory of mobile polarization in the same&#xA;magazine in 1814. In fact, Biot seems to have summed up the French&#xA;attitude toward Young in a letter to Brewster in 1816:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;I very much esteem the merit and talent of this distinguished scientist, but you will permit me to tell you that his witness has no more weight here than the authority of Aristotle against the observations of Galileo on motion... Even if his theory, which I do not know at all, were true...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Some French corpuscularians read Young only in private, like Malus (from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Polarizer#Malus.27_law_and_other_properties&quot; rel=&quot;noreferrer&quot;&gt;Malus' law of polarization&lt;/a&gt;). Other would clearly defect, like Arago (from the &lt;a href=&quot;https://en.wikipedia.org/wiki/Arago_spot&quot; rel=&quot;noreferrer&quot;&gt;Arago spot&lt;/a&gt;, see below) in 1813.&lt;/p&gt;&#xA;&lt;p&gt;With respect to Laplace, he seems to have kept his corpuscularian position but leaving the criticism to others. It would be the work of Fresnel that would change everything in France. Fresnel was already showing how diffraction could be explained with waves, but many were skeptics. In 1817, a commission made from the most prestigious French physicist of the time, including Laplace, proposed diffraction as the subject of the biannual physics prize competition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The wording of the contest made it clear that the commissioners sought a corpuscular explanation of diffraction.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Fresnel who during this time was trying to fix all the problems of his wave theory, profited on the occasion and participated. Fresnel's work was so clear that the commission had no option but to give him the prize in 1819 even if his theory was not corpuscular. Laplace who read Arago's report on a follow-up work of Fresnel that explained double refraction, said that he&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;declared that he put this research above everything that had been&#xA;communicated to the Académie in a long time.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Another great battle was that of Arago vs. Poisson. Poisson calculated that if wave theory was true, there would be diffraction (Arago) spot through a circular disk, very different from corpuscular theory that predicted no light around the edges. Arago took the challenge. His experiment, showed Poisson's corpuscular theory wrong in 1819. This in part, led to the acceptance of Fresnel as the winner of the contest.&lt;/p&gt;&#xA;&lt;p&gt;Thus, a new set of scientists (including Fresnel and Arago) was rising and taking positions of power in the French Science Academy, creating an anti-Laplacian coalition:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;By 1823, therefore, the anti-Laplace faction was in control of&#xA;the Académie, and Arago, its leader, was in the dominant position&#xA;that Laplace had occupied a decade earlier. As a critic of Arago has put it:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Laplace was reduced to silence, M. Biot absented himself from the Institut&#xA;for several years and M. Arago remained master of the battlefield.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;h2&gt;Reference&lt;/h2&gt;&#xA;&lt;p&gt;The quotes have been taking from this fantastic document:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://www.jstor.org/stable/284930?seq=9&quot; rel=&quot;noreferrer&quot;&gt;Frankel, Eugene. &amp;quot;Corpuscular optics and the wave theory of light: the science and politics of a revolution in physics.&amp;quot; Social studies of science (1976)&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;There is a whole chapter on Biot vs. Fresnel. Biot was the main advocate of corpuscular theory, after Laplace. On a motion by Laplace, Biot tried to counter Fresnel.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-03-22T18:43:03.883" LastActivityDate="2024-03-22T18:43:03.883" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17367" PostTypeId="1" AcceptedAnswerId="17370" CreationDate="2024-03-22T19:06:16.337" Score="4" ViewCount="118" Body="&lt;p&gt;I could not find who wrote the equation &lt;span class=&quot;math-container&quot;&gt;$c=\lambda\nu$&lt;/span&gt; for the first time. Neither I found a name for this equation. A user from Physics forum thinks it is too obvious for anyone familiar with wavelength and frequency, and once light has been understood traveling as electromagnetic waves, someone must has described it with this equation. Does anyone know anything more about it?&lt;/p&gt;&#xA;" OwnerUserId="20869" LastEditorUserId="6609" LastEditDate="2024-03-23T15:26:46.837" LastActivityDate="2024-03-26T15:59:01.027" Title="Who wrote the equation $c=\lambda\nu$ for the first time?" Tags="|light|waves|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17368" PostTypeId="1" CreationDate="2024-03-24T13:39:05.797" Score="0" ViewCount="50" Body="&lt;p&gt;The citation we get from &lt;a href=&quot;http://projecteuclid.org/journals/acta-mathematica/volume-71/issue-none/The-method-of-successive-approximation-for-functional-equations/10.1007/BF02547750.full&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://projecteuclid.org/journals/acta-mathematica/volume-71/issue-none/The-method-of-successive-approximation-for-functional-equations/10.1007/BF02547750.full&lt;/a&gt; says &lt;code&gt;number={none}&lt;/code&gt; in the BIB file and no &lt;code&gt;IS&lt;/code&gt; entry in the RIS file and no visible month and the date &lt;code&gt;PY  - 1939/1/1&lt;/code&gt; in the RIS file.&lt;/p&gt;&#xA;&lt;p&gt;The citation we get from &lt;a href=&quot;http://link.springer.com/article/10.1007/BF02547750#citeas&quot; rel=&quot;nofollow noreferrer&quot;&gt;http://link.springer.com/article/10.1007/BF02547750#citeas&lt;/a&gt; says, on the contrary, &lt;code&gt;IS = 1&lt;/code&gt; (and mentions an &lt;code&gt;SN&lt;/code&gt; entry, though ISSN/ISBN were introduced much later) in the RIS file and month &lt;code&gt;December&lt;/code&gt; visibly and the date &lt;code&gt;DA  - 1939/12/01&lt;/code&gt; in the RIS file.&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;p&gt;What's the reason behind this discrepancy?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Which data is right? Why?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;Does this publication have any issue number at all? If so, which one?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;li&gt;&lt;p&gt;What's the true month of the publication?&lt;/p&gt;&#xA;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="20806" LastEditorUserId="20806" LastEditDate="2024-03-24T18:49:32.303" LastActivityDate="2024-03-24T18:49:32.303" Title="Correct citation of “The method of successive approximation for functional equations” by L. Kantorovich, Leningrad, 1939?" Tags="|reference-request|functional-analysis|" AnswerCount="0" CommentCount="3" ContentLicense="CC BY-SA 4.0" />
  <row Id="17369" PostTypeId="1" AcceptedAnswerId="17375" CreationDate="2024-03-24T14:14:58.937" Score="2" ViewCount="62" Body="&lt;p&gt;In general relativity, the Bondi-Van der Burg-Metzner-Sachs (BMS) group is the group of symmetries of future null infinity. To the best of my knowledge, it originated in the works of &lt;a href=&quot;https://doi.org/10.1098/rspa.1962.0161&quot; rel=&quot;nofollow noreferrer&quot;&gt;Bondi, Van der Burg, and Metzner&lt;/a&gt; and later in the work of &lt;a href=&quot;https://doi.org/10.1098/rspa.1962.0206&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sachs&lt;/a&gt;.&lt;/p&gt;&#xA;&lt;p&gt;In spite of this, I have read multiple references that refer to this group as the Bondi-Metzner-Sachs group, ignoring Van der Burg. I'd say this is evident in the abbreviation BMS, instead of BVMS, for example. Furthermore, &lt;a href=&quot;https://en.wikipedia.org/wiki/Bondi%E2%80%93Metzner%E2%80%93Sachs_group&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wikipedia&lt;/a&gt; calls the group the Bondi-Metzner-Sachs group.&lt;/p&gt;&#xA;&lt;p&gt;Why is Van der Burg often ignored in the name of the BMS group? Isn't he one of the authors of the original paper?&lt;/p&gt;&#xA;" OwnerUserId="16184" LastActivityDate="2024-03-26T18:39:21.510" Title="Why is the Bondi-Van der Burg-Metzner-Sachs group often called only Bondi-Metzner-Sachs group?" Tags="|relativity-theory|" AnswerCount="1" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17370" PostTypeId="2" ParentId="17367" CreationDate="2024-03-24T16:35:36.727" Score="3" Body="&lt;p&gt;Relations between the frequency and length of a string are known since Antiquity, but a more precise mathematical treatment would have to wait until the invention of calculus.&lt;/p&gt;&#xA;&lt;p&gt;The &lt;a href=&quot;https://en.wikipedia.org/wiki/Wave_equation&quot; rel=&quot;nofollow noreferrer&quot;&gt;wave equation&lt;/a&gt; was first studied by &lt;a href=&quot;https://en.wikipedia.org/wiki/Brook_Taylor&quot; rel=&quot;nofollow noreferrer&quot;&gt;Brook Taylor&lt;/a&gt; (1713) and later by &lt;a href=&quot;https://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jean d'Alembert&lt;/a&gt; (1746). (As for who really did it first check this answer: &lt;a href=&quot;https://hsm.stackexchange.com/questions/13022/who-discovered-the-wave-equation&quot;&gt;Who discovered the wave equation?&lt;/a&gt;). I was not able to find the explicit equation from commentary of their work but &lt;a href=&quot;https://Jakob%20Hermann&quot; rel=&quot;nofollow noreferrer&quot;&gt;Jakob Hermann&lt;/a&gt; in &lt;em&gt;Phoronomia&lt;/em&gt; (a commentary on the work of Taylor and Newton) in 1716 writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;span class=&quot;math-container&quot;&gt;$$V=\lambda/T$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;where &lt;span class=&quot;math-container&quot;&gt;$V$&lt;/span&gt; is the speed of sound and &lt;span class=&quot;math-container&quot;&gt;$T$&lt;/span&gt; the period of oscillation. The rest of the analysis of Hermann was wrong. Source: &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-1-4613-9461-7_5&quot; rel=&quot;nofollow noreferrer&quot;&gt;The Evolution of Dynamics: Vibration Theory from 1687 to 1742 pp&lt;/a&gt;&lt;/p&gt;&#xA;&lt;p&gt;The solutions of the wave equation are of the form &lt;span class=&quot;math-container&quot;&gt;$f( x/\lambda-\nu t)$&lt;/span&gt; (in the argument you have &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; position, &lt;span class=&quot;math-container&quot;&gt;$\lambda$&lt;/span&gt; wavelength, &lt;span class=&quot;math-container&quot;&gt;$\nu$&lt;/span&gt; frequency and time &lt;span class=&quot;math-container&quot;&gt;$t$&lt;/span&gt;), which can also be written as another function &lt;span class=&quot;math-container&quot;&gt;$g(x-ct)$&lt;/span&gt; (where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the speed of the wave). For the simplest case (describing sinusoidal waves), it leads to your formula &lt;span class=&quot;math-container&quot;&gt;$c=\nu\lambda$&lt;/span&gt; (known as linear dispersion relation).&lt;/p&gt;&#xA;&lt;p&gt;As for light. &lt;a href=&quot;https://en.wikipedia.org/wiki/Thomas_Young_(scientist)&quot; rel=&quot;nofollow noreferrer&quot;&gt;Thomas Young&lt;/a&gt; showed definitely in 1800 that light behaves like a wave using a &lt;a href=&quot;https://en.wikipedia.org/wiki/Double-slit_experiment&quot; rel=&quot;nofollow noreferrer&quot;&gt;double slit experiment&lt;/a&gt;. Experiments showed that light in air seemed to follow that simple linear relation &lt;span class=&quot;math-container&quot;&gt;$c=\nu\lambda$&lt;/span&gt; when it is not travelling inside a media.&lt;/p&gt;&#xA;&lt;p&gt;In 1855, &lt;a href=&quot;https://Wilhelm%20Eduard%20Weber&quot; rel=&quot;nofollow noreferrer&quot;&gt;Wilhelm Weber&lt;/a&gt; and &lt;a href=&quot;https://en.wikipedia.org/wiki/Rudolf_Kohlrausch&quot; rel=&quot;nofollow noreferrer&quot;&gt;Rudolf Kohlrausch&lt;/a&gt; suggested that the speed of light was equivalent to the ratio of electric and magnetic constants. In 1864, &lt;a href=&quot;https://en.wikipedia.org/wiki/James_Clerk_Maxwell&quot; rel=&quot;nofollow noreferrer&quot;&gt;James Clerk Maxwell&lt;/a&gt;, compiled all the different equations of electromagnetism and finally showed that electromagnetic fields propagate as waves, and in vacuum these waves follow &lt;span class=&quot;math-container&quot;&gt;$c=\nu\lambda$&lt;/span&gt; where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; was the calculated to be equal to the experimental value for the speed of light. So visible light is a type of electromagnetic wave with linear dispersion.&lt;/p&gt;&#xA;&lt;p&gt;In the comments you mentioned gravitational waves. In 1915 Albert Einstein published his theory of general relativity (suggesting that gravity is related to the curvature of space-time), and in 1916 he derived the equations for the gravitational waves. These equations are very complex and describe the distorsion of spacetime, the same arguments that one needs to write the equation. However for very simple spacetimes geometries it also follows &lt;span class=&quot;math-container&quot;&gt;$c=\nu\lambda$&lt;/span&gt;, where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the speed of light. The fact that for light and gravitational waves the speed is the same is related to relativity, where &lt;span class=&quot;math-container&quot;&gt;$c$&lt;/span&gt; is the maximum speed a wave can travel and it's related to causality.&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-03-26T15:59:01.027" LastActivityDate="2024-03-26T15:59:01.027" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17371" PostTypeId="1" AcceptedAnswerId="17377" CreationDate="2024-03-25T14:59:06.303" Score="5" ViewCount="106" Body="&lt;p&gt;I remember that Willard van Orman Quine wrote something to the effect that physics may be paradoxical, in similar ways as naive set theory is paradoxical.&lt;/p&gt;&#xA;&lt;p&gt;May someone help find the quote?&lt;/p&gt;&#xA;&lt;p&gt;Edit 1 - A Quine specialist just wrote the following to me:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Sounds like something we would say, but I can't locate it. In Pursuit of Truth -- 20-1, 35-6 -- he does speak of modern physics presenting logical problems ...&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;Edit 2 - Unfortunately, it was not in the suggestions of Edit 1.&lt;/p&gt;&#xA;" OwnerUserId="8094" LastEditorUserId="8094" LastEditDate="2024-03-27T14:27:16.740" LastActivityDate="2024-03-28T06:34:48.047" Title="What did Quine say on paradox and physics?" Tags="|physics|philosophy-of-science|theoretical-physics|quote|" AnswerCount="1" CommentCount="6" ContentLicense="CC BY-SA 4.0" />
  <row Id="17372" PostTypeId="1" CreationDate="2024-03-26T09:09:27.997" Score="4" ViewCount="1854" Body="&lt;p&gt;My very basic understanding of Quantum Mechanics and its history is that first, some physical quantities were thought to be continuous but experiments showed that they only took discrete values.&lt;/p&gt;&#xA;&lt;p&gt;My question is when and how did people introduce the notion of Hermitian operators and postulated that the observed discrete values actually were the eigenvalues of some Hermitian operator?&lt;/p&gt;&#xA;&lt;p&gt;Things that would be appreciated in the answer:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;What were the first quantities and corresponding operators to be introduced?&lt;/li&gt;&#xA;&lt;li&gt;Was the motivation for the eigenvalues formalism mostly experimental or were there some theoretical arguments as well?&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;" OwnerUserId="14079" LastEditorUserId="14079" LastEditDate="2024-03-26T15:21:20.217" LastActivityDate="2024-03-26T16:14:04.647" Title="Use of eigenvalues of operators in quantum mechanics" Tags="|physics|quantum-mechanics|" AnswerCount="2" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17373" PostTypeId="2" ParentId="17372" CreationDate="2024-03-26T11:39:05.487" Score="4" Body="&lt;p&gt;The story is long and quite complicated. Here is a very short and necessarily incomplete sketch.&lt;/p&gt;&#xA;&lt;p&gt;It begins with experimental observations about atomic spectra. The main fact is that each atom has some fixed discrete sequence of frequencies&#xA;of light that it emits or absorbs. This was known from the early 19th century (Fraunhofer) and constitutes the basis of spectral analysis. In 1885 Balmer discovered a simple empiric formula for the spectrum of hydrogen atom. Then Planck discovered the discreteness of black body radiation and relation between energy and frequency.&#xA;This enabled Einstein to explain photoelectric effect. Both Planck and Einstein were explaining experimental facts&#xA;about radiation.&lt;/p&gt;&#xA;&lt;p&gt;Next important step was made by Bohr, who proposed a model&#xA;of hydrogen atom which explained Balmer's formula. This was the basis of the so-called Old quantum theory. It explained many experimental facts but was based on some ad hoc &amp;quot;quantization rules&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Linear operators came to the theory in the work of Heisenberg (matrices) and Schrodinger (differential operators), followed very shortly shortly by the work of Born, Jordan and Dirac.&#xA;From their work it became clear that the discrete values of&#xA;energy (related to the frequency by Plank's formula) are actually eigenvalues of certain Hermitian operator. This final formulation is due to Dirac, and rigorous mathematical background was developed by von Neumann.&lt;/p&gt;&#xA;&lt;p&gt;So the first quantized physical quantity were energy and action. And this was a result of complicated theoretical development whose goal was to explain experimental facts, mainly about emission and absorption of radiation.&#xA;Chronologically, the first hint that measured values of physical quantities are eigenvalues of certain operators comes from the work of Heisenberg. But Heisenberg did not know the relevant mathematics and had to invent matrices himself. The connection with known mathematical theories was established by Born, Jordan and Dirac. And necessary further development of these mathematical theories is due to von Neumann.&lt;/p&gt;&#xA;&lt;p&gt;Recommended literature:&#xA;For the first period (from Fraunhofer to Bohr): S. Sternberg, History of 19th century&#xA;spectroscopy. Appendix F to his book &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/0521558859&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Group theory and physics&lt;/a&gt; Cambridge UP, 1994.&lt;/p&gt;&#xA;&lt;p&gt;For the second period (from Bohr to Dirac) B. L. van der Waerden, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/048645892X&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Sources of quantum mechanics.&lt;/a&gt; It includes English translations of the main original articles with short comments by van der Waerden.&lt;/p&gt;&#xA;&lt;p&gt;Also A. Sommerfeld, &lt;a href=&quot;https://rads.stackoverflow.com/amzn/click/com/B008TNNO7G&quot; rel=&quot;nofollow noreferrer&quot; rel=&quot;nofollow noreferrer&quot;&gt;Atombau und Spektrallinien (1921)&lt;/a&gt; is a very good source for the transition period between old and modern quantum mechanics. Later editions of this book were expanded to include the work of Heisenberg and Schrodinger.&lt;/p&gt;&#xA;" OwnerUserId="64" LastEditorUserId="64" LastEditDate="2024-03-26T15:22:10.153" LastActivityDate="2024-03-26T15:22:10.153" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17374" PostTypeId="2" ParentId="17372" CreationDate="2024-03-26T13:10:51.877" Score="8" Body="&lt;p&gt;Comments to answer: the question is hard to answer because, there is more than one question and more than a way to interpret it.&lt;/p&gt;&#xA;&lt;p&gt;If the question is when discrete values and eigenvalues were introduced as part of the formalism of quantum mechanics? The answer is from the very first paper of Planck on quantum theory from 1900. Why? because the idea of eigenvalues was already being used for the development of classical theory of waves. Note that since the Pythagoreans or even before that, we know that the natural frequencies of an ordinary string are discrete. So the discreteness of quantum phenomena was quickly understood as a wave-like behaviour since the very beginning.&lt;/p&gt;&#xA;&lt;p&gt;If the question is when the notion of Hermitian-like matrices was introduced to quantum mechanics then it is better to ask about the history of operators directly. Check:&lt;/p&gt;&#xA;&lt;ul&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/6704/origin-of-operators-in-quantum-mechanics&quot;&gt;Origin of operators in quantum mechanics&lt;/a&gt;&lt;/li&gt;&#xA;&lt;li&gt;&lt;a href=&quot;https://hsm.stackexchange.com/questions/2838/how-did-quantum-mechanics-operators-come-into-being&quot;&gt;How did quantum mechanics operators come into being?&lt;/a&gt;&lt;/li&gt;&#xA;&lt;/ul&gt;&#xA;&lt;p&gt;The most important contribution for the notion of operators are the three papers of Werner Heisenberg (1925-1926) on &lt;a href=&quot;https://en.wikipedia.org/wiki/Matrix_mechanics&quot; rel=&quot;noreferrer&quot;&gt;matrix mechanics&lt;/a&gt;. The period from 1900 to 1925 is now known as old quantum theory.  Heisenberg's 1925 reinterpretation paper led to modern quantum mechanics. He introduced the idea that we needed a non-commutative algebra, but it was with the help of Pascual Jordan and Max Born that he was able to formalize it using matrices. Note that the dimension of these matrices was not necessary finite. In 1927, Jordan seems to have introduced even the symbol &lt;span class=&quot;math-container&quot;&gt;$\dagger$&lt;/span&gt; of Hermitian conjugate, see: &lt;a href=&quot;https://hsm.stackexchange.com/questions/11426/who-introduced-the-daggersymbol-as-conjugate-transpose-in-quantum-mechanics&quot;&gt;Who introduced the &amp;quot;dagger&amp;quot;symbol as conjugate transpose in quantum mechanics?&lt;/a&gt; And Born and Norbert Wiener have already a section called &amp;quot;calculus of operators&amp;quot; in their paper &lt;a href=&quot;https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm19265184&quot; rel=&quot;noreferrer&quot;&gt;A New Formulation of the Laws of Quantization of Periodic and Aperiodic Phenomena&lt;/a&gt; in 1926.&lt;/p&gt;&#xA;&lt;p&gt;In parallel, Schrödinger's introduced at the same time as Heisenberg his wave equation as an eigenvalue problem, but he was borrowing the mathematical tools from analytical mechanics and from the classical theory of waves.&lt;/p&gt;&#xA;&lt;h2&gt;First operators to be quantized&lt;/h2&gt;&#xA;&lt;p&gt;You asked what were the first operators introduced? Position and momentum, because from classical mechanics these are the important variables for any dynamics. Heisenberg and Schrödinger chose these two &amp;quot;operators&amp;quot; in their original papers.&#xA;They also defined the energy or Hamiltonian operator (as a function of &lt;span class=&quot;math-container&quot;&gt;$x$&lt;/span&gt; and &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt;).&lt;/p&gt;&#xA;&lt;h2&gt;Last question&lt;/h2&gt;&#xA;&lt;p&gt;As for&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Was the motivation for the eigenvalues formalism mostly experimental or were there some theoretical arguments as well?&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;If the question is: does the use of eivenvalues was motivated experimentally? the answer is yes, but again there was no surprise on how to address that for the community.&lt;/p&gt;&#xA;&lt;p&gt;If the question is does the Hermitian algebra was motivated by experiments? The answer is no so clear. Again Schrödinger's approach did not need that (of course he was already naming things operators in the same manner as you call &lt;span class=&quot;math-container&quot;&gt;$\nabla^2$&lt;/span&gt; a differential operator). I would argue that this falls more in the realm of &amp;quot;is math invented or discovered&amp;quot;, everybody was already using some underline theory of operators without proper rigorous understanding. Heisenberg was not looking back at experiments, but more for another way of deriving previous results. Note that the later task of showing that this was all the same was undertaken later by Paul Dirac, John von Neummann and others. Von Neumann introduced more formally the concept of self-adjoint operators in 1932 (borrowing also from the works of Hilbert and Hermann Weyl on integral operators who were working on that since the beginning of the 20th century).&lt;/p&gt;&#xA;" OwnerUserId="6609" LastEditorUserId="6609" LastEditDate="2024-03-26T16:14:04.647" LastActivityDate="2024-03-26T16:14:04.647" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17375" PostTypeId="2" ParentId="17369" CreationDate="2024-03-26T18:39:21.510" Score="3" Body="&lt;p&gt;The &lt;a href=&quot;https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1962.0161&quot; rel=&quot;nofollow noreferrer&quot;&gt;paper by Bondi et al.&lt;/a&gt; is unusual in that different parts (A-D) are signed by different authors. Its history and context are thoroughly described in an (open access) &lt;a href=&quot;https://link.springer.com/content/pdf/10.1140/epjh/e2019-100020-1.pdf?pdf=core&quot; rel=&quot;nofollow noreferrer&quot;&gt;article by D. C. Robinson&lt;/a&gt;. Julian van der Burg, who was Bondi's PhD student, only signed Part B (together with Bondi), while Kenneth Metzner is the only signatory of Part C. Robinson states, though, that Bondi wrote the whole article. The super-translations of the BMS group appear only in Metzner's Part C and are then discussed in Part D, signed only by Bondi. Rainer (Ray) Sachs generalised these findings in a &lt;a href=&quot;https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1962.0206&quot; rel=&quot;nofollow noreferrer&quot;&gt;subsequent publication&lt;/a&gt;. Metzner was an American PhD student of Philip Morrison's at Cornell who joined the &amp;quot;collaboration&amp;quot; when Bondi was visiting during a sabbatical. Robinson quotes van der Burg with a statement from 2009 saying&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;To this day I have no idea who Metzner is.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In summary, the acronym BMS seems chosen correctly, listing all authors who contributed to the concepts behind.&lt;/p&gt;&#xA;" OwnerUserId="16834" LastActivityDate="2024-03-26T18:39:21.510" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17377" PostTypeId="2" ParentId="17371" CreationDate="2024-03-27T17:50:15.090" Score="4" Body="&lt;p&gt;See &lt;a href=&quot;https://www.jstor.org/stable/20123117&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;On what there is&amp;quot; (1948)&lt;/a&gt; reprinted into &lt;a href=&quot;https://www.google.it/books/edition/From_a_Logical_Point_of_View/OalXwuw3MvMC&quot; rel=&quot;nofollow noreferrer&quot;&gt;From a Logical Point of View&lt;/a&gt;, page 19 of the 2nd revised edition of 1961:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;An antinomy arose between the undular and the corpuscular account of light...&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;" OwnerUserId="131" LastEditorUserId="131" LastEditDate="2024-03-28T06:34:48.047" LastActivityDate="2024-03-28T06:34:48.047" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17378" PostTypeId="2" ParentId="16129" CreationDate="2024-03-27T18:29:38.350" Score="1" Body="&lt;p&gt;My opinion is decidedly against the current, and I will also write quickly from memory without looking for precise references. So, if the quality of this answer is considered low, I can accept it.&#xA;Let's start immediately with an important question: who invented infinitesimal calculus? Usually, the common response is: Newton and Leibniz. This would suggest that it was an invention in the air if two scientists (albeit mythical) had arrived at it together. Instead, if the author was only one, one could think of a stroke of genius, a sudden leap into the future.&#xA;Now, in my opinion, the author is clearly only one: Leibniz. I won't go into the &amp;quot;procedural&amp;quot; details, but the point is that it is too particular, too unstable, too contradictory (yes, contradictory) to be independently produced by two different people; in fact, it took about two centuries to give infinitesimal calculus a satisfactory foundation (first by replacing infinitesimals and infinites with limits, then finding an acceptable definition of a limit).&#xA;The second important question is: what is infinitesimal calculus? Leibniz explains it in &amp;quot;Historia et origo calculi differentialis&amp;quot;: it is a way to represent algebraically the curves that Descartes could not represent with his system. So, the heart of the new system is given by the differential equations that are tasked with describing the curves that we now call transcendental.&#xA;Finally, we come to the initial question: why did it take so long? In my opinion, it's the opposite: between Descartes and Leibniz only a few years pass (1684 - 1637 = 47), which is a miracle. If Leibniz had never existed, I don't know how mathematics would have developed; perhaps it would have focused on infinite series.&lt;/p&gt;&#xA;" OwnerUserId="17362" LastActivityDate="2024-03-27T18:29:38.350" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17379" PostTypeId="1" CreationDate="2024-03-27T18:49:04.787" Score="0" ViewCount="39" Body="&lt;p&gt;I have access to the quote&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;The analogy between the myth of mathematics and the myth of physics is, in some additional and perhaps fortuitous ways, strikingly close. Consider, for example, the crisis which was precipitated in the foundations of mathematics, at the turn of the century, by the discovery of Russell’s paradox and other antinomies of set theory. These contradictions had to be obviated by unintuitive, ad hoc devices; our mathematical myth-making became deliberate and evident to all. But, what, of physics? An antinomy arose between the undular and the corpuscular accounts of light; and if this was not as out-and-out a contradiction as Russell’s paradox, I suspect that the reason is that physics is not as out-and-out as mathematics.&#xA;Blockquote&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;from pages 18-19 in the 1st chapter of the &lt;strong&gt;Second edition&lt;/strong&gt; of Willard van Orman Quine's book &lt;em&gt;From a Logical Point of View.&lt;/em&gt;&lt;/p&gt;&#xA;&lt;p&gt;What is the pagination of the quote in the &lt;strong&gt;First edition&lt;/strong&gt; of Quine's book?&lt;/p&gt;&#xA;" OwnerUserId="8094" LastActivityDate="2024-03-27T18:49:04.787" Title="From a Logical Point of View quote, 2nd and 1st editions" Tags="|philosophy-of-science|quote|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17381" PostTypeId="1" CreationDate="2024-03-28T12:27:44.013" Score="4" ViewCount="163" Body="&lt;p&gt;How applications of catastrophe theory outside mathematics stalled the theory, and why?&lt;/p&gt;&#xA;&lt;p&gt;I know that the theory had its fair share of popularity during the 1970s, with many distinguished mathematicians working on it, for example Thom, Arnol'd, Whitney, etc. After being incorporated into applications outside mathematics the theory underwent a transformation into a stagnated field and lost much of its prestige, as some on the internet have put it. If this is the case then:&lt;/p&gt;&#xA;&lt;p&gt;How did this phenomenon occur?&lt;/p&gt;&#xA;&lt;p&gt;What were the mechanisms that led to stagnation and why mathematicians stopped working on the field? Did they feel that they were being invaded by &amp;quot;outsiders&amp;quot; or by people that did not understand the principles of the theory? Did they feel bad because the name of the theory was used to support outlandish results outside mathematics?&lt;/p&gt;&#xA;&lt;p&gt;Where there just social reasons, (such as those mentioned above), for the exodus of mathematicians from the theory?&lt;/p&gt;&#xA;&lt;p&gt;Moreover, I know nothing about catastrophe theory and its history. The subject of its &amp;quot;stalling&amp;quot; by being popular outside mathematics was brought up during a conversation, and it struck me in an odd way. So this is why I am asking those questions, mainly from a sociological point of view.&lt;/p&gt;&#xA;" OwnerUserId="20797" LastEditorUserId="6609" LastEditDate="2024-04-01T17:41:34.793" LastActivityDate="2024-04-02T22:10:01.183" Title="How and why was catastrophe theory brought to its knees?" Tags="|mathematics|mathematics-social-history|" AnswerCount="0" CommentCount="4" ContentLicense="CC BY-SA 4.0" />
  <row Id="17382" PostTypeId="1" CreationDate="2024-03-28T17:41:42.770" Score="1" ViewCount="70" Body="&lt;p&gt;The proof can be found &lt;a href=&quot;https://homes.esat.kuleuven.be/%7Ewcastryc/reciprocity.pdf&quot; rel=&quot;nofollow noreferrer&quot;&gt;here (pdf)&lt;/a&gt;. It was published in 1838 and Gauss lived until 1855, so I would guess that he read it. Did Gauss say anything about it?&lt;/p&gt;&#xA;" OwnerUserId="20835" LastActivityDate="2024-03-28T17:41:42.770" Title="What did Gauss think about V. A. Lebesgue's proof of quadratic reciprocity?" Tags="|mathematics|gauss|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17383" PostTypeId="1" AcceptedAnswerId="17384" CreationDate="2024-03-28T21:15:43.583" Score="12" ViewCount="1139" Body="&lt;p&gt;This post is prompted by a recent &lt;a href=&quot;https://math.stackexchange.com/questions/4889210/given-two-triangles-if-the-vertices-of-the-first-triangle-coincide-with-the-ver?noredirect=1#comment10430817_4889210&quot;&gt;question&lt;/a&gt; on MSE asking about &amp;quot;Axiom 10&amp;quot; of Euclid's Elements, as found in editions by  Byrne and Conway: &amp;quot;Two right lines cannot enclose a space&amp;quot;. This axiom is not in Heath's translation.&lt;/p&gt;&#xA;&lt;p&gt;Axiom 10 is employed by Conway in I.4 and XI.3 to argue that since two line segments terminate in the same points, they must be the same line segment (for they cannot enclose an area, and they otherwise would). Heath says nothing of the sort in I.4, appealing to superposition and the equality of things that coincide. But in Heath's XI.3 there is the statement that &amp;quot;two straight lines... have the same extremities, and will clearly enclose an area: which is absurd&amp;quot;, a statement which has no particular support.&lt;/p&gt;&#xA;&lt;p&gt;My question: What is the origin of Axiom 10? Is it a sort of patch, taken from XI.3?&lt;/p&gt;&#xA;&lt;p&gt;&lt;strong&gt;Edit&lt;/strong&gt;: Euclid I.4 is the side-angle-side theorem, where if two triangles have two sides the same, and the angles enclosed by the sides the same, the remaining 3rd side is the same. Heath's translation is &lt;a href=&quot;http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookI/propI4.html&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt; and the Byrne translation &lt;a href=&quot;https://www.c82.net/euclid/book1/&quot; rel=&quot;nofollow noreferrer&quot;&gt;here&lt;/a&gt;.&lt;/p&gt;&#xA;" OwnerUserId="19734" LastEditorUserId="19734" LastEditDate="2024-03-31T20:10:07.470" LastActivityDate="2024-03-31T20:10:07.470" Title="What is the origin of &quot;two straight lines cannot enclose a space&quot; axiom in Euclid's Elements?" Tags="|geometry|euclid|" AnswerCount="1" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17384" PostTypeId="2" ParentId="17383" CreationDate="2024-03-29T00:41:55.537" Score="13" Body="&lt;p&gt;It seems that the addition was made already in antiquity and motivated by the apparent gap in the proof of I.4. &lt;a href=&quot;https://link.springer.com/chapter/10.1007/978-3-319-12102-4_5&quot; rel=&quot;noreferrer&quot;&gt;Rabouin in Proclus’ Conception of Geometric Space and Its Actuality&lt;/a&gt; writes:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Another example of straight line represented as “non straight” and used by Proclus is given in the course of the commentary on I.4. There he criticizes the fact that Euclid assumed without proof that two lines cannot enclose a space (this is a crucial assumption in proofs by superposition such as I.4 or I.8 and was even incorporated in the “common notions” in some versions of the Elements). Proclus undertakes to provide the missing argument and proceeds ad absurdum...&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;In an attached footnote, he adds:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;If we accept triangles with sides&#xA;represented as “unstraight”, we could make two triangles coincide on two of their sides and the&#xA;angle contained by them without coinciding on the third. This will ruin the demonstrations “by&#xA;superposition”. The admissibility of such a diagram seems to be attested in the discussion about&#xA;the curious “four sided” triangle, to which Proclus alludes in In Eucl. 329. Barozzi (1560, p. 189) With such a diagram, we could easily demonstrate the contradictory of I.4 by supposing that two triangles coincide on two sides (and the angle containing them) and then realize that one is&#xA;nonetheless contained in the other. This possibility is ruled out by the fact that it would presuppose admitting two straight lines enclosing a space.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;There is a more detailed discussion of the history of &amp;quot;axiom 10&amp;quot; in &lt;a href=&quot;https://maa.org/press/maa-reviews/philosophy-of-mathematics-and-deductive-structure-in-euclids-elements&quot; rel=&quot;noreferrer&quot;&gt;Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements&lt;/a&gt;, who calls it &amp;quot;postulate 6&amp;quot;.  Mueller quotes Proclus's opinion that it should not be added to the axioms, meaning it was added in some editions of Elements already in antiquity, and it is controversial whether the words &amp;quot;&lt;em&gt;two straight lines will enclose a space, which is impossible&lt;/em&gt;&amp;quot; in I.4 are also a later insertion.&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&amp;quot;&lt;em&gt;Proclus undoubtedly thinks the words are genuine, although he almost certainly knows of texts in which the postulate does not occur.&#xA;The existence of texts in which the words but not the postulate occur makes it seem likely that the postulate was added to justify the words... Proclus claims that Euclid knows or acknowledges [eidenai] the assertion when he states the first postulate, which presupposes the uniqueness of the straight line connecting two points. Proclus’ claim is vague enough to be incontestable. Euclid undoubtedly does “know” that only one straight line can be drawn through two points...&lt;/em&gt;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;&lt;em&gt;But... Postulate 1 implies nothing about uniqueness. And if Euclid wished to assert the uniqueness of the straight line, he surely could have done so — for example, by inserting a definite article before ‘straight line’ in postulate 1. The evidence suggests that Euclid never raised the question whether the impossibility of two straight lines enclosing a space was a logical consequence of his first principles. For him this assertion rests directly on the intuitive notion of a straight line, and, since this notion has been characterized in the definitions, there is no need to prove a directly obvious fact about it.&lt;/em&gt;&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;&lt;a href=&quot;https://www.maths.tcd.ie/%7Edwilkins/Courses/MA232A/MA232A_Mich2015/EuclidResources/EuclidAxiomSystem.php&quot; rel=&quot;noreferrer&quot;&gt;Wilkins&lt;/a&gt; lists variants of Euclid's first principles from various modern editions, and &amp;quot;axiom 10&amp;quot; appears in those &amp;quot;based on Commandinus&amp;quot;. That would be &lt;a href=&quot;https://maa.org/press/periodicals/convergence/mathematical-treasure-commandinos-elements-of-euclid&quot; rel=&quot;noreferrer&quot;&gt;Commandino's translation of the Elements&lt;/a&gt; from 1572. So Commandino, contra Proclus, sides with those ancient editors who thought that &amp;quot;axiom 10&amp;quot; could not be derived from others and had to be added as a separate one.&lt;/p&gt;&#xA;" OwnerUserId="55" LastEditorUserId="55" LastEditDate="2024-03-29T04:50:46.817" LastActivityDate="2024-03-29T04:50:46.817" CommentCount="0" ContentLicense="CC BY-SA 4.0" />
  <row Id="17386" PostTypeId="1" CreationDate="2024-03-29T20:21:26.273" Score="2" ViewCount="51" Body="&lt;p&gt;In the physicist chapter of &lt;a href=&quot;https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/carl-friedrich-gauss&quot; rel=&quot;nofollow noreferrer&quot;&gt;Gauss's bio on encyclopedia.com&lt;/a&gt; appears the following statement about Gauss-Weber's unpublished correspondence:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Stimulated by Faraday’s discovery of induced current in 1831, the pair energetically investigated electrical phenomena. They arrived at Kirchhoff’s laws in 1833 and &lt;em&gt;&lt;strong&gt;anticipated various discoveries in static, thermal, and frictional electricity but did not publish, presumably because their interest centered on terrestrial magnetism&lt;/strong&gt;&lt;/em&gt;.&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;I read this encyclopedia article some years ago, but only recently I found more information about Gauss-Weber work on thermogalvanism. In p.113-114 of &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN236059505?tify=%7B%22pages%22%3A%5B289%5D%2C%22pan%22%3A%7B%22x%22%3A0.382%2C%22y%22%3A0.723%7D%2C%22view%22%3A%22fulltext%22%2C%22zoom%22%3A0.332%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;Clemens Shaefer's treatise on Gauss's physical works&lt;/a&gt; appears the following information:&lt;/p&gt;&#xA;&lt;blockquote&gt;&#xA;&lt;p&gt;Gauss only made publicly known his investigation into thermogalvanism in the above mentioned essay on the bifilar magnetometer. There he says: &amp;quot;The new apparatus serves for sharp Measurement of even the weakest galvanic currents... This clearly refutes the erroneous opinion of many physicists, as if they could not penetrate a chain of significant length. No matter how long a chain, such currents are not canceled out, but are only weakened in exactly the same proportion as with other types of excitation. Using a thermogalvanic apparatus of a peculiar construction, the mere touch of the points with the finger produces a galvanic current, which, even if it has to run through a chain of almost two miles long, mostly of very thin wire, still results in very significant deflections of the magnetic rod.&amp;quot;&lt;/p&gt;&#xA;&lt;/blockquote&gt;&#xA;&lt;p&gt;The above mentioned essay (which contains the full version of this quotation of Gauss) is called &lt;a href=&quot;https://gdz.sub.uni-goettingen.de/id/PPN236006339?tify=%7B%22pages%22%3A%5B357%5D%2C%22view%22%3A%22fulltext%22%7D&quot; rel=&quot;nofollow noreferrer&quot;&gt;&amp;quot;A new tool for magnetic observations (published in October 1837)&amp;quot;&lt;/a&gt;, and the most relavent part is p.354. However, according to Shaefer: &amp;quot;Nothing is known about the construction of the special thermogalvanic apparatus that Gauss mentions.&amp;quot;&lt;/p&gt;&#xA;&lt;p&gt;According to Shaefer, in letters to Schumacher and Olbers, Gauss also states that: &amp;quot;Perhaps one can make very important discoveries from this for pyrometry (blast furnaces, porcelain furnaces, melting furnaces, etc)&amp;quot;, and Shaefer says: &amp;quot;Gauss's broad view is evident here in the most brilliant way&amp;quot;. These last statements are not clear to me since I don't understand Gauss's remarks on the applicability of thermogalvanism to pyrometry, nor I understand why this is a brilliant anticipation (as Schaefer says). Please note I am not familiar with the science behind thermogalvanism nor with the history of this science; its only aspect I am aware of is the &lt;strong&gt;Seebeck effect&lt;/strong&gt;, in which temperature differences across a metal generate electric voltage (potentials difference).&lt;/p&gt;&#xA;&lt;p&gt;Since the details of Gauss's thermogalvanic instrument are apparently lost (although there is some chance for more information once &lt;a href=&quot;https://kalliope-verbund.info/de/findingaid_toc?fa.id=DE-611-BF-61709&amp;amp;fa.enum=1&amp;amp;lastparam=true&quot; rel=&quot;nofollow noreferrer&quot;&gt;additional relavent parts of Gauss's Nachlass&lt;/a&gt; will be published), this is clearly not a good question, but perhaps there is a way of concluding something about Gauss-Weber's thermogalvanic circuit just from the verbal description of its functioning and from the consequent development of such instruments by later researchers in the 19th century. After all, Gauss was not so technologically oriented, and it is unlikely that this device was &amp;quot;a spectacle&amp;quot; for its time, so probably other inventors in the next 2 decades arrived at similar developments.&lt;/p&gt;&#xA;&lt;p&gt;Thanks in advance for any useful piece of information!&lt;/p&gt;&#xA;" OwnerUserId="2554" LastEditorUserId="6609" LastEditDate="2024-03-31T14:21:11.370" LastActivityDate="2024-03-31T14:21:11.370" Title="Reconstruction of the details of Gauss-Weber &quot;thermogalvanic chain&quot;" Tags="|gauss|electricity|technology|" AnswerCount="0" CommentCount="2" ContentLicense="CC BY-SA 4.0" />
  <row Id="17388" PostTypeId="1" CreationDate="2024-03-31T20:46:07.120" Score="3" ViewCount="41" Body="&lt;p&gt;In 18th century botany, how did taxonomists such as Carl Linnaeus (Sweden) and Antoine-Laurent de Jussieu (France) manage to practically organize such a large number of plant species? For example, with an index card system, or something similar?&lt;/p&gt;&#xA;&lt;p&gt;I'm interested generally in the practical side of plant classification in the 18th century, and Linnaeus and de Jussieu are specific and well-known examples. Encyclopedia entries and short biographies do not contain much information, but I'm sure it's related to the organization of plants in the Jardin du Roi or similar gardens, though I haven't been able to find much information on how these botanists actually managed their work.&lt;/p&gt;&#xA;&lt;p&gt;(If the question is too broad, I would be interested in relevant references and sources for this sort of study.)&lt;/p&gt;&#xA;&lt;p&gt;&lt;em&gt;Note: I'm not interested in how these systems themselves are organized (by reproductive organs, with artificial or natural selection of qualities) but instead I'm curious about the &amp;quot;bookkeeping&amp;quot; involved, how a person actually organizes their notes.&lt;/em&gt;&lt;/p&gt;&#xA;" OwnerUserId="9528" LastEditorUserId="12330" LastEditDate="2024-04-03T06:37:25.553" LastActivityDate="2024-04-03T06:37:25.553" Title="How did de Jussieu or Linnaeus practically manage the organization of plants?" Tags="|biology|18th-century|taxonomy|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
  <row Id="17389" PostTypeId="1" CreationDate="2024-03-31T23:09:16.810" Score="1" ViewCount="47" Body="&lt;p&gt;If &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; is a positive integer then there is a monic irreducible polynomial &lt;span class=&quot;math-container&quot;&gt;$f(x)$&lt;/span&gt; such that if &lt;span class=&quot;math-container&quot;&gt;$p$&lt;/span&gt; is an odd prime not dividing &lt;span class=&quot;math-container&quot;&gt;$n$&lt;/span&gt; nor the discriminant of &lt;span class=&quot;math-container&quot;&gt;$f$&lt;/span&gt; then&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;p=x^2+ny^2\iff \left(\frac{-n}{p}\right)=1\quad\text{and}\quad f(x)\equiv 0\mod p\quad\text{has a solution.}&#xA;$$&lt;/span&gt;&#xA;This is a simplified version of theorem 9.2 from Cox's book &amp;quot;Primes of the form x^2+ny^2&amp;quot;.&lt;/p&gt;&#xA;&lt;p&gt;Euler was highly interested in the problem of representing primes by quadratic forms &lt;span class=&quot;math-container&quot;&gt;$x^2+ny^2$&lt;/span&gt;, and he made many impressive discoveries that take the above form. For example, he discovered that&#xA;&lt;span class=&quot;math-container&quot;&gt;$$&#xA;p=x^2+27y^2\iff p\equiv 1\mod 3\quad\text{and}\quad x^3-2\equiv 0\mod p\quad\text{has a solution.}&#xA;$$&lt;/span&gt;&lt;/p&gt;&#xA;&lt;p&gt;My question is: Did Euler conjecture the form of the general solution for arbitrary &lt;span class=&quot;math-container&quot;&gt;$n&amp;gt;0$&lt;/span&gt;, as stated above?&lt;/p&gt;&#xA;" OwnerUserId="20835" LastActivityDate="2024-03-31T23:09:16.810" Title="Was Euler aware of the general form of the characterization of primes of the form $p=x^2+ny^2$ for arbitrary $n&gt;0$?" Tags="|mathematics|number-theory|euler|" AnswerCount="0" CommentCount="1" ContentLicense="CC BY-SA 4.0" />
</posts>